\nAir Shower Simulation and Hadronic Interactions\nJe ff Allen 1 , Antonella Castellina 2 , Ralph Engel 3 , Katsuaki Kasahara 4 , Stanislav Knurenko 5 , Tanguy Pierog 3 , Artem Sabourov 5 , Benjamin T. Stokes 6 , Ralf Ulrich 3 ,\nfor the Pierre Auger, Telescope Array, and Yakutsk Collaborations, and\nSergey Ostapchenko 7 and Takashi Sako 8\n\n1 New York University, 4 Washington Place, New York, NY, USA\n2 Osservatorio Astrofisico di Torino (INAF), Universit'a di Torino and Sezione INFN, Torino, Italy\n3 Karlsruhe Institute of Technology, Institut fur Kernphysik, Karlsruhe, Germany\n4 RISE, Waseda University, Tokyo, Japan\n5 Yu.G. Shafer Institute of Cosmophysical Research and Aeronomy, Yakutsk, Russia\n6 High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, UT, USA\n7 NTNU, Institutt for fysikk, 7491 Trondheim, Norway\n8 Solar-Terrestrial Environment Laboratory, Nagoya University, Japan\n\nAbstract. The aim of this report of the Working Group on Hadronic Interactions and Air Shower Simulation is to give an overview of the status of the field, emphasizing open questions and a comparison of relevant results of the di ff erent experiments. It is shown that an approximate overall understanding of extensive air showers and the corresponding hadronic interactions has been reached. The simulations provide a qualitative description of the bulk of the air shower observables. Discrepancies are however found when the correlation between measurements of the longitudinal shower profile are compared to that of the lateral particle distributions at ground. The report concludes with a list of important problems that should be addressed to make progress in understanding hadronic interactions and, hence, improve the reliability of air shower simulations.\n1 Introduction\nAfter the first interaction of a cosmic ray particle of very high energy in the atmosphere a multitude of subsequent interactions, leading to particle multiplication, and decay processes give rise to a cascade of secondary particles called extensive air shower (EAS) [1]. With the electromagnetic and weak interactions being well described by perturbative calculations within the Standard Model of Particle Physics, the limited understanding of strong interactions becomes the dominant source of uncertainties of shower predictions. Even though Quantum Chromodynamics (QCD) is the well-established and experimentally confirmed theory of strong interactions, only processes with very large momentum transfer can be predicted from first principles until now. It is still not possible to calculate the bulk properties of multiparticle production as needed for air shower simulation. Additional, simplifying assumptions as well as phenomenological and empirical parametrizations are needed to develop models for hadronic interactions describing various particle production processes. These additional assumptions need to be verified, parametrizations constrained, and parameters tuned by comparisons to accelerator data.\nThe simulation of extensive air showers forms one of the pillars on which the data analysis of modern experiments for ultra-high energy cosmic rays (UHECRs) rests. Improving the understanding and modeling of hadronic particle production is one of the important prerequisites for a reliable interpreta-\nEPJ Web of Conferences\ntion of UHECR data. Even though calorimetric techniques based on the measurement of fluorescence and Cherenkov light have been developed for a nearly model-independent energy determination of extensive air showers, determining the type of the primary particles and, in particular, estimating the mass of the primary particle can only be done with the help of sets of simulated reference showers. 1\nMuch e ff ort has been devoted to improving both the simulation of extensive air showers in the atmosphere and the understanding of the corresponding, and typically very complex detector response function. In the following we will concentrate on discussing the simulation of extensive air showers as this aspect of the overall data simulation chain is the same in all UHECR experiments. We will assume that uncertainties arising from simulating the detector response due to the shower particles are much smaller and can be disregarded here. This is certainly not guaranteed per se and it is the task of each collaboration to verify the quality of the detector simulation before attributing possible discrepancies to, for example, shortcomings in the understanding of air showers or hadronic multiparticle production.\nAfter recalling some basic features of air showers (Sec. 2) we will give an overview of the most frequently used code packages for air shower simulation in Sec. 3. The LHC data allow us to test the hadronic interaction models employed in these code packages, for the first time, at equivalent energies beyond that of the knee in the cosmic ray spectrum. In Sec. 4 some representative model predictions are compared to LHC data and implications are discussed. A good overall bracketing of the LHC data by model predictions is found, even though each of the models will have to be improved to obtain a satisfactory description of the LHC data. Therefore it is not surprising that the distributions of most of the shower observables are well reproduced by shower simulations (Sec. 5).\nTaking advantage of the hybrid detection setups of the latest generation of UHECR detectors, longitudinal and lateral particle distributions can be measured shower-by-shower. A comparison of the particle densities at ground with the ones expected according to the shower longitudinal profile reveals possible shortcomings of the shower predictions that are not yet understood. The overall data set of such comparisons is discussed in Sec. 6 and the results from the Auger, TA, and Yakutsk Collaborations are compared. There are strong indications that a significant part of the shortcomings of the shower predictions are related to the muonic shower component. Therefore we summarize recent developments aiming at better understanding muon production in air showers in Sec. 7.\nGiven the increased awareness of apparent deficits of current air shower predictions and related theory developments, the wealth of new data from the LHC, and the high-quality measurements of the current generation of air shower observatories, significant progress can be expected in the reliability of air shower predictions in the next few years. Open questions that need to be addressed to optimally benefit from these developments to improve the interpretation of EAS data and to solve some outstanding problems will be listed in the concluding section of this report.\n2 Physics of Air Showers\nReviews of the physics of extensive air showers and hadronic interactions can be found in, for example, [3-6] and a very instructive extension of Heitler's cascade model to hadronic showers is given in [7]. Here we give only a qualitative introduction to some aspects of shower physics that will be needed in the discussions later on.\nWith pions being the most abundant secondary particles of hadronic interactions at high energy, the di ff erence in lifetime and decay products of neutral and charged pions lead to fundamentally di ff erent ranges of interaction energies that give rise to the electromagnetic and muonic shower components. In air showers, neutral pions decay almost always before interacting again (except E π 0 > 10 19 eV) and provide high-energy photons that feed the electromagnetic shower component. In contrast, the energy of charged pions has to be degraded down to some 30 -100GeV before the probability to decay will exceed that of another interaction. In first approximation, the depth profile of charged shower particles, being dominated by e + e -of the electromagnetic component, is linked to the secondary particles of the first few interactions in the hadronic core of the shower. In contrast, the muons at ground stem from\nConference Title, to be filled\n\n\n\n\n\n
Fig. 1. Longitudinal shower profiles of electrons and muons. Shown are also the sub-showers produced by the secondary particles of the first 100 highest-energy interactions and the cumulative profile of these sub-showers [8]. The simulation was done with a modified version of CONEX [9] for a proton shower of 10 19 eV.
\n\na chain of 8 -10 successive hadronic interactions with the energy of the last interaction producing pions or kaons that lead to observed muons being in the range of 20 -200GeV [10,11]. This is illustrated in Fig. 1 showing the sub-showers produced by the secondary particles of the first 100 highest-energy interactions in an air shower [8]. While more than 50% of the electromagnetic shower particles are produced by decaying neutral pions of the first 100 interactions in an air shower, only a negligible fraction of the muonic component stems from decaying charged pions and kaons of the same interactions.\nThe longitudinal profile of showers as well as the energy and angular distributions of charged particles in a shower exhibit universality features [12-17] if considered as a function of shower age, here approximated by\ns = 3 X X + 2 X max , (1)\nwith X being the slant depth along the shower axis and X max the depth of shower maximum. Universality features are routinely used to estimate the Cherenkov light signal of showers, see, for example, [15, 18]. Not only the electromagnetic particles but also the production depth as well as the energy and transverse momentum distributions of muons can be described by functions [19,20] that depend only weakly on the assumed hadronic interaction model. Breaking down the overall shower signal at ground into di ff erent muonic and electromagnetic components, the description of showers can be improved considerably [21-23].\nEPJ Web of Conferences\nWhile the bulk of charged particles is considered for longitudinal profiles of the electromagnetic and muonic shower components, lateral particle distributions at ground are only measured at large distances from the shower axis. Particle densities at, for example, 600 or 1000 m from the shower core are only indirectly related to the overall particle multiplicity of a shower at ground. This is reflected in the fact that, for example, the last interaction for producing a muon observable at ground has a median energy of ∼ 100GeV for the KASCADE array (typical primary energy 3 × 10 15 eV; lateral distance 40 -200m, see [11]) and only 20 -30GeV for the Auger array (typical primary energy 10 19 eV; lateral distance 1000 m, see [24]). Still universality arguments can also be applied to particle densities far away from the core [21] but the model-related di ff erences between the predictions are larger. In particular, low-energy interactions can lead to a model-dependent violation of universality profiles that is reflected in both the muonic and electromagnetic shower content at large lateral distance [16, 23].\n3 Shower Simulation Packages and Hadronic Interaction Models\nThe most frequently used code packages for the simulation of air showers for the Auger, TA and Yakutsk experiments are AIRES [25], CORSIKA [26], and COSMOS [27]. In these packages, the Monte Carlo method is applied to simulate the evolution of air showers with all secondary particles above a user-specified energy threshold. The large number of secondary particles that would have to be followed in the simulation of ultra-high energy showers makes the simulations so time consuming that thinning techniques are applied. Only a representative set of particles is included in the simulation below a pre-defined energy threshold. The discarded particles are accounted for by increasing the statistical weight of the particles remaining in the simulation [28,29]. The main drawbacks of this method are artificial fluctuations related to particles with large statistical weight [30,31] and the need for treating weighted particles in the detector simulations. The Auger and TA Collaborations have developed di ff erent methods for de-thinning simulated showers, see [32] and [33], respectively. Also a limited number of fully simulated showers of the highest energies are available in each collaboration (see, for example, [34]) but the statistics is very limited and there is a lack of methods to work with such large amounts of data in an e ffi cient way.\nThe classic Monte Carlo codes are complemented by hybrid simulation programs that combine the Monte Carlo technique with either numerical solutions of cascade equations (CONEX [9] and SENECA [35]) or libraries of pre-simulated air showers (COSMOS [27]). The advantage of the hybrid codes is the much reduced CPU time needed for simulating showers at very high energy and the possibility to simulate showers either without thinning or with very low statistical weights. But there is always a remaining risk that rare shower fluctuations or local correlations might not be correctly described.\nDi ff erent codes are applied for the simulation of the electromagnetic shower component. The code used in AIRES is based on the one developed by Hillas for MOCCA [28,36]. CONEX, CORSIKA, and SENECA are interfaced to modified versions of EGS4 [37]. Similar to AIRES, COSMOS comes with a custom-developed code for electromagnetic interactions called EPICS [27]. The suppression of high-energy interactions of photons and electrons due to the Landau-Pomeranchuk-Migdal (LPM) e ff ect (for example, see [38]) is taken into account in all code packages.\nDue to the di ff erent underlying approaches for modeling hadronic interactions at low ( E < 80 -200GeV) and high energies typically always two hadronic interaction models are employed in air shower simulations. The low-energy models GHEISHA [39], FLUKA [40], and UrQMD [41] are available in CORSIKA while AIRES and COSMOS employ custom-made codes. Typical high-energy interaction models are DPMJET II [42] and III [43,44], EPOS [45-47], QGSJET 01 [48,49] and II [50-52], and SIBYLL [53-55]. A compilation of simulation results obtained with many of these low- and high-energy models can be found in [56-58].\nSeveral comparisons of the predictions of shower simulation packages for the same interaction models are available in the literature, see [4] and [61,62,59,60,63]. In the ideal case, the predictions for air showers of a given energy and primary particle depend only on the user-selected energy thresholds and on the choice of hadronic interaction models selected for the simulation. While it is possible to\nConference Title, to be filled\n\n\n\n\n\n
Fig. 2. Longitudinal shower profiles of electrons and muons. Predictions calculated with CORSIKA and COSMOS are compared for vertical proton showers of 10 19 eV [59,60]. High-energy interactions were simulated with QGSJET II.03 for energies above 80 GeV.
\n\nchose a high-energy interaction model that is supported in all shower simulation codes, less flexibility is available for the low-energy interaction models, which are often di ff erent in each of the simulation packages. Therefore the results of these comparisons typically show a good agreement of the longitudinal shower profiles, which are mainly related to hadronic interactions at high energy. Di ff erences are found for observables that are sensitive to low-energy interactions, which are, for example, the total number of muons and particle densities at large lateral distances. The di ff erences between the predictions of di ff erent simulation packages are typically of the order of 5%, increasing in some phase space regions up to ∼ 10%. For example, longitudinal shower profiles from a recent comparison between CORSIKA and COSMOS [59,60] are shown in Fig.2.\n4 Performance of Hadronic Interaction Models\nMost of the interaction models used in air shower simulations are not commonly applied in high energy physics (HEP) simulations and, conversely, HEP models are not used for air shower simulations. This is related to the fact that HEP models are typically limited to a set of primary particles that are available in accelerator experiments and optimized only for collider energies. Similarly, cosmic ray interaction models are not routinely used by HEP collaborations for acceptance correction calculations including\n\n\n
Fig. 3. Pseudorapidity distribution of charged particles measured at LHC. CMS data [64] are shown together with model predictions. Similar data sets are available from the ATLAS and ALICE Collaborations [65-67] (not shown). Below 2 TeV center-of-mass energy the models were tuned to previously available collider data from Tevatron and SPS.
\n\nthe comparison of raw, uncorrected data as this can be done with the HEP event generators as well. Furthermore dedicated HEP event generators describe highp ⊥ and electroweak physics processes in much more detail and contain many more parameters for improving the description of particular distributions than any of the cosmic ray or general purpose models. In contrast, interaction models for cosmic ray physics are tuned to describe data of accelerator measurements over a wide range of energies, often sacrificing a perfect description of some distributions in favour of a good overall reproduction of the whole data set.\nHence comparisons of model predictions to data from fixed-target and collider experiments are typically made by the authors of the models and using only fully acceptance corrected data. However, over the last years much progress has been made regarding the interaction between the cosmic ray and high energy physics communities. A number of workshops and meetings (including this one) and the direct involvement of cosmic ray physicists in HEP collaborations have lead to a much higher level of awareness and intensified communication of both communities.\nComparisons of high-energy interaction models used for air shower simulations with LHC data can be found in, for example, [69-71]. Here only some representative examples can be included for illustration. The pseudorapidity distribution of charged particles is shown together with model predictions in Fig. 3. Not only most central particle distributions are reasonably well bracketed by the model predictions, also the predicted energy flow at larger pseudorapidity is in good agreement with the LHC data. This can be seen in Fig. 4 where the energy flow measured by CMS [68] is compared to model predictions. Note that all these model results are true predictions as the models were tuned years before LHC data became available. Nevertheless, in many cases, the minimum bias data are better described by interaction models developed primarily for cosmic ray physics than the various tunes of standard HEP models [68,69].\nOn the other hand, there are also some important distributions measured at LHC that are not well described or bracketed by these pre-LHC interaction models. Examples of large deviations from measurements are multiplicity distributions of charged particles and particle production spectra at large pseudorapidity and Feynmanx . The possible impact of these deviations on air shower predictions is subject to ongoing research and not yet fully understood. In Fig. 5 the Feynmanx distribution of\n\n\n
Fig. 4. Energy flow as function of pseudorapidity in forward direction. CMS data [68] are shown together with model predictions calculated by the CMS Collaboration. The left (right) panel shows the measurements for 900 GeV (7 TeV) c.m.s. energy.
\n\n\n\n
Fig. 5. Photon spectra in very forward direction as measured by the LHCf Collaboration [72]. Superimposed are the predictions of hadronic interaction models used for air shower simulations. The lower panels show the relative di ff erence between data and model predictions.
\n\nphotons produced in forward direction is shown [72]. The data have been obtained within the LHCf experiment that is specifically built for measuring particles in the very forward direction.\nEPJ Web of Conferences\nIn general, the predictions of the di ff erent cosmic ray interaction models bracket the LHC data reasonably well [69]. This observation supports the hope that simulating air showers with di ff erent interaction models should also bracket the correct, but unknown predictions for air showers. On the other hand, this success does not mean that there is no need to improve the interaction models. There is not a single interaction model that reproduces most of new LHC data well and, hence, would be much better than the others. Each of the models needs to be re-tuned or underlying ideas be extended to obtain a better description of the data.\nManynewdata sets measured in LHC experiments and also the fixed-target experiments NA49 [73] and NA61 [74,75] became available within the last 2 -3 years and the process of tuning and modifying the interaction models to obtain an improved description is ongoing. At the time of writing these proceedings the new model versions QGSJET II.04 [70] and EPOS LHC [71] have been released as the first versions of interaction models tuned to LHC data. In addition, work is in progress to develop improved versions of SIBYLL and DPMJET that are also tuned to the new LHC data.\nFinally it should be mentioned that, in addition to accelerator data, also air shower measurements provide important information on hadronic interactions. For example, cross section measurements made with air shower observables [76] can be used to estimate not only the rise of the cross section but also to verify the model approximations used to calculate the proton-air cross section from the data on proton-proton cross sections. Even though the systematic uncertainties of air shower measurements of this kind are much higher than that of accelerator data, such measurements are the only way to study particle production at interaction energies well beyond the range of colliders [77-80].\n5 Overall Description of Shower Characteristics\nSimulating air showers with a realistic energy spectrum, primary mass composition, and arrival direction distribution, the predicted and observed distributions of the various observables of reconstructed showers can be compared. Such comparisons are very important as they are end-to-end tests of the simulation chain for the experiments. Only if good agreement is found one can cross-check the impact of quality cuts applied in the process of data analysis and avoid unexpected biases. The results of such end-to-end simulations depend, of course, on the assumed mass composition of the primary particles and the employed hadronic interaction models.\nExamples of such comparisons for fluorescence detectors can be found in [81-84]. A number of unpublished comparisons of typical observables measured with the Auger and TA surface detector arrays (zenith and azimuth angles, number of stations per event, signal size distribution) have been presented at this meeting. They all show very good agreement between the measured distributions and the corresponding Monte Carlo predictions. While the TA simulations were done for proton primaries, the Auger simulations were based on a 50 / 50 proton-iron mixture. Both collaborations used QGSJET II.03 as high-energy interaction model. The TA data are better described by a primary composition of only protons than using 100% iron. In case of the Auger simulations, which were also made assuming only proton or iron primaries, it was found that most of the surface detector observables exhibit only a very limited sensitivity to the primary mass composition.\nIt should be considered as an important success of hadronic interaction models and modern air shower simulation packages that such a good overall description of the general features of the observed showers is reached. However, di ff erent combinations of primary mass composition and shower energy can lead to very similar surface detector signals (for example, the signal at 600 or 1000 m in units of that of vertical muons). Therefore it is very important to use additional information to determine, for example, the primary energy in a composition-independent way. This is done with fluorescence telescopes in the case of the Auger Observatory [85] and Telescope Array [86], and with non-imaging Cherenkov light measurements in the Yakutsk setup [87].\n6 Detailed Comparison with Shower Data\nDetecting air showers simultaneously with fluorescence telescopes and an array of surface detectors at ground is often referred to as hybrid measurements. One of the first setups of this type was the\nConference Title, to be filled\nprototype instrument of the High Resolution Fly's Eye (HiRes) [88] that was operated in coincidence with the CASA-MIA array [89] in Utah. Already in this prototype experiment strong indications for a discrepancy between the measured density of muons at 600 m from the shower core and the one expected from simulations were found, if the composition inferred from the measurements of the longitudinal shower profiles was used [90]. While the data on the depth of shower maximum suggested the conclusion that a transition from a heavy to a light, almost proton dominated composition is seen, the density of muons stayed above or at the level of the predictions for iron primaries.\n\n\n
Fig. 6. Longitudinal and lateral profiles of one high-energy shower observed simultaneously with the fluorescence and the surface detectors of the Pierre Auger Observatory [91]. The data are compared with simulated showers for proton and iron primaries that predict the same longitudinal profiles. The simulations were made with SENECA and QGSJET II.03 / FLUKA as high-/ low-energy interaction models.
\n\nCovering energies higher than those of the early HiRes-MIA measurements, the data of the Auger Collaboration also indicate a discrepancy between the observed and predicted number of muons in air showers [91]. Di ff erent methods have been applied to obtain either the expected surface detector signal S (1000) or the muon density at 1000m from the shower core for a given primary energy or even an observed longitudinal shower profile (see [92] for a review given at this meeting). This discrepancy is most directly displayed in a shower-by-shower comparison of simulated profiles with those obtained from hybrid measurements, see Fig. 6. Compared to proton showers as reference, the measured values of S (1000) are about 1.5 to 2 times higher that the simulated ones. Studying inclined showers of about 10 19 eV with zenith angles larger than 60 · , whose particle content at ground is completely dominated by muons, gives a ratio of [91]\nN µ, data N µ, MC ∣ ∣ ∣ ∣ ∣ ∣ QGSJET , p = 2 . 13 ± 0 . 04(stat) ± 0 . 11(sys) , (2)\nwhere the systematic uncertainty due to the 22% energy scale uncertainty is not included. This ratio is about 1 . 2 for iron showers simulated with EPOS 1.99. The data on inclined showers imply that the observed discrepancy is closely related to, if not dominated by a deficiency in simulating the muon component of air showers.\nThe detectors of the Auger, TA and Yakutsk air shower arrays are di ff ering in their sensitivities to muons and electromagnetic particles, and the Yakutsk array even contains a number of shielded muon detectors. For example, the contribution of muons to the scintillator signal of the TA surface detectors will never exceed 30% for showers with zenith angles less than 45 · . In comparison, a signal fraction of 30 -80%is expected due to muons for the Auger water Cherenkov tanks in the zenith angle range from 0 -60 · . Therefore similar studies of the TA and Yakutsk Collaborations can provide a very important, independent information on a possible muon deficit of air shower simulations.\nEPJ Web of Conferences\nThe Yakutsk Collaboration compared the predicted and measured charged particle and muon lateral distributions for a number of high energy showers. Within the uncertainties implied by the unknown primary composition, no significant discrepancies between the predictions obtained with both QGSJET II / FLUKA and EPOS / UrQMD and the data are found [93].\nThe TA Collaboration studied the di ff erence between the energy one would assign to a shower based on the calorimetric fluorescence light measurement and that derived from the comparison with simulations of the surface detector signal. It was found that the fluorescence-assigned energy is about 27% lower than the energy estimated from the comparison with simulated showers [94]. Given the limited sensitivity of the TA surface detector stations, no direct study of the muon contribution to the overall signal has been done so far.\nThe observations of all three collaborations can be brought into qualitative agreement if the di ff erent energy scales of the experiments are taken into account. The factor with which the energy scales need to be re-scaled are listed in the report of the Spectrum Working Group [95]. For example, for the same shower, the Yakutsk simulations are done at an energy about two times higher than that of the Auger Observatory. Hence, also about a factor two more muons are predicted in the Yakutsk simulations in comparison to Auger simulations. Given that the Yakutsk simulations agree with their measurements, this implies two times more muons in the Yakutsk data than predicted by simulations if the Auger energy scale is used. A similar estimate can be done for the TA-Auger comparison, where one has to keep in mind that only ∼ 20%of the scintillator signal stems from muons. If the discrepancy observed by the Auger Collaboration would be attributed to the muon component only, the expected energy rescaling would be somewhat smaller than the ∼ 27% reported by the TA Collaboration [94].\nIt remains to be shown how much of the apparent discrepancy between the surface detector signals and the predictions based on calorimetric shower energy determination can be cured by adopting di ff erent energy scales in the experiments. A reduction of the systematic uncertainties of the energy assignment to air showers will be needed to quantify more reliably possible discrepancies between shower data and predictions.\n7 Hadronic Interactions and Muon Production in Air Showers\nThe observation of a possible muon deficit in simulated showers in comparison to Auger measurements, first reported already in 2007 [96], has triggered a number of theoretical studies searching for modifications of hadronic interactions that could result in an enhancement of the muonic shower component. Changes of the inelastic cross section, inelasticity of interactions, secondary particle multiplicity at high energy, and many other parameters lead only to moderate changes of the predicted number of muons (see, for example, [97,98]). Also scenarios with new physics processes such as string percolation [99] or the drastic change of interaction properties due to, for example, chiral symmetry restoration [100] have been discussed.\nSo far all proposed changes that lead to a significant increase of the number of muons in air showers are directly or indirectly based on either one or both of the following e ff ects:\n(i) An increase of the production rate of particles that do not decay, for example baryon-antibaryon pairs, leads to higher muon multiplicities of showers since these particles will stay being part of the hadronic shower component and loose their energy only by producing further hadronic particles. This e ff ect has been discussed first in [101] and is one of the di ff erences of the EPOS model with respect to the other models [102].\n(ii) A change of the type of the leading particle produced in inelastic interactions can also be very e ffi cient in reducing the energy that is transferred to the electromagnetic shower component [103]. The majority of sub-showers in an extensive air shower are initiated by charged pions. The chance probability of producing a charged pion or a neutral pion as leading particle in charged pion interactions is about 2 : 1. Replacing all leading π 0 by ρ 0 mesons, which have the same valence quarks but are spin 1 particles, leads to a drastic enhancement of muon production since neutral ρ mesons decay immediately into two charged pions [104]. Indeed, fixed-target data [105-107] indicate that, in contrast to conventional model predictions, the production of ρ 0 dominates that of π 0 for Feynmanx larger than 0 . 5.\nConference Title, to be filled\nBoth e ff ects are the reason for the higher muon multiplicities predicted with EPOS. Leading rho mesons are explicitly generated for pion interactions in the new version II.04 of QGSJET, boosting the muon multiplicity in air showers significantly. Depending on the relative importance of e ff ect (i) and energy distribution of the baryon pairs, the energy spectrum of the additionally produced muons is very soft and might be in contradiction to shower attenuation data.\nFinally it should be mentioned that also an increase of the production of kaons could lead to a higher muon multiplicity and also a harder muon energy spectrum [103,99].\n8 Conclusions and Outlook\nOver the last two decades the quality and predictive power of air shower simulations has improved significantly and a very good overall description of most of the shower features observed in experiments has been reached.\nThe predictions of independently developed shower simulation packages agree reasonably well with each other if the same hadronic interaction models are used for the simulations. While there is very good agreement for the electromagnetic shower component, the di ff erences between the predictions for muon multiplicities and lateral distributions can be as large as ∼ 5 -10%. These di ff erences are most likely related to the use of di ff erent low-energy interaction models. A comprehensive comparison of the di ff erent shower simulation packages, similar to the recent study of CORSIKA and COSMOS predictions [60], should be made to quantify the systematic uncertainties of the predictions and possibly also to identify and address shortcomings of the simulation packages.\nThe limited theoretical understanding of modeling hadronic multiparticle production together with the limitations of accelerator measurements in energy, covered phase space, and projectile-target combinations are the dominating source of systematic uncertainties of air shower predictions. Because of this, the systematic uncertainties of the model predictions cannot be estimated reliably. More work is needed to improve QCD calculations in the lowp ⊥ domain, in particular to understand screening and saturation e ff ects, and to develop alternative models to describe particle production.\nThe new LHC data provide extremely useful input for tuning hadronic interaction models. Even though the first LHC data on multiplicities and cross sections were well bracketed by the predictions of interaction models used for air shower simulations, the comparison to data revealed the need for further model developments and tuning. Improved and re-tuned versions of EPOS and QGSJET are already available and similar versions of DPMJET and SIBYLL are in preparation.\nThe interaction between the cosmic ray and high energy physics communities has intensified and the direct engagement of cosmic ray physicists in LHC and fixed-target experiments has lead to a much better understanding of the needs of the cosmic ray community. There is also very large interest from the side of LHC communities to have cosmic ray physicists being involved in the analysis of accelerator data. One example of the achieved progress is the use of cosmic ray interaction models by LHC collaborations to compare data with predictions in publications.\nBoth the Auger and TA Collaborations have found indications for a discrepancy between the expected and observed surface detector signals for showers with fluorescence energy measurement. Accounting for the di ff erent energy scales of the Auger, TA, and Yakutsk experiments the observed discrepancies are consistent with each other. Most likely, the dominating sources of the discrepancies are shortcomings of the simulation of muon production in air showers. The systematic uncertainties of the overall energy scales of the experiments of 20 -30% will have to be reduced significantly to be able to combine data from di ff erent experiments for a stronger test of the shower predictions.\nWhile the production of electromagnetic particles is clearly dominated by the electromagnetic cascade induced by photons of very high energy due to neutral pion decay early on in the shower evolution, the muonic component receives contributions from all high-energy interactions above ∼ 20GeV lab. energy. Di ff erent modifications of the simulation of particle production in low- and high-energy interactions have been considered to increase the number of muons in air showers. Progress has been made by understanding that both production of stable or long-lived hadrons such as baryon pairs and modifications to the particle types generated as leading particles are e ffi cient mechanisms to increase the muon multiplicity without changing significantly the longitudinal shower profile. It remains to\nEPJ Web of Conferences\nbe shown whether the discrepancies between the observed and predicted surface detector signals for a given shower energy can be resolved by improving low- and high-energy interaction models using conventional physics assumptions or whether these observations indeed indicate a change of the characteristics of hadronic interactions at energies beyond that of LHC.\nReferences\n\n1. P. Auger, P. Ehrenfest, R. Maze, Robley, and A. Fr'eon, Rev. Mod. Phys. 11 (1939) 288-291.\n2. D. B. Kieda, S. P. Swordy, and S. P. Wakely, Astropart. Phys. 15 (2001) 287-303.\n3. T. K. Gaisser, R. J. Protheroe, K. E. Turver, and T. J. L. McComb, Rev. Mod. 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C9 (1981) 93-104.\n106. M. Aguilar-Benitez et al. (LEBC-EHS Collab.), Z. Phys. C44 (1989) 531.\n107. N. Agababyan et al. (EHS / NA22 Collab.), Z. Phys. C46 (1990) 387-395.\n"},"sections":{"kind":"list like","value":[{"title":"Air Shower Simulation and Hadronic Interactions","content":"Je ff Allen 1 , Antonella Castellina 2 , Ralph Engel 3 , Katsuaki Kasahara 4 , Stanislav Knurenko 5 , Tanguy Pierog 3 , Artem Sabourov 5 , Benjamin T. Stokes 6 , Ralf Ulrich 3 , for the Pierre Auger, Telescope Array, and Yakutsk Collaborations, and Sergey Ostapchenko 7 and Takashi Sako 8 Abstract. The aim of this report of the Working Group on Hadronic Interactions and Air Shower Simulation is to give an overview of the status of the field, emphasizing open questions and a comparison of relevant results of the di ff erent experiments. It is shown that an approximate overall understanding of extensive air showers and the corresponding hadronic interactions has been reached. The simulations provide a qualitative description of the bulk of the air shower observables. Discrepancies are however found when the correlation between measurements of the longitudinal shower profile are compared to that of the lateral particle distributions at ground. The report concludes with a list of important problems that should be addressed to make progress in understanding hadronic interactions and, hence, improve the reliability of air shower simulations.","pages":[1]},{"title":"1 Introduction","content":"After the first interaction of a cosmic ray particle of very high energy in the atmosphere a multitude of subsequent interactions, leading to particle multiplication, and decay processes give rise to a cascade of secondary particles called extensive air shower (EAS) [1]. With the electromagnetic and weak interactions being well described by perturbative calculations within the Standard Model of Particle Physics, the limited understanding of strong interactions becomes the dominant source of uncertainties of shower predictions. Even though Quantum Chromodynamics (QCD) is the well-established and experimentally confirmed theory of strong interactions, only processes with very large momentum transfer can be predicted from first principles until now. It is still not possible to calculate the bulk properties of multiparticle production as needed for air shower simulation. Additional, simplifying assumptions as well as phenomenological and empirical parametrizations are needed to develop models for hadronic interactions describing various particle production processes. These additional assumptions need to be verified, parametrizations constrained, and parameters tuned by comparisons to accelerator data. The simulation of extensive air showers forms one of the pillars on which the data analysis of modern experiments for ultra-high energy cosmic rays (UHECRs) rests. Improving the understanding and modeling of hadronic particle production is one of the important prerequisites for a reliable interpreta-","pages":[1]},{"title":"EPJ Web of Conferences","content":"be shown whether the discrepancies between the observed and predicted surface detector signals for a given shower energy can be resolved by improving low- and high-energy interaction models using conventional physics assumptions or whether these observations indeed indicate a change of the characteristics of hadronic interactions at energies beyond that of LHC.","pages":[12]},{"title":"2 Physics of Air Showers","content":"Reviews of the physics of extensive air showers and hadronic interactions can be found in, for example, [3-6] and a very instructive extension of Heitler's cascade model to hadronic showers is given in [7]. Here we give only a qualitative introduction to some aspects of shower physics that will be needed in the discussions later on. With pions being the most abundant secondary particles of hadronic interactions at high energy, the di ff erence in lifetime and decay products of neutral and charged pions lead to fundamentally di ff erent ranges of interaction energies that give rise to the electromagnetic and muonic shower components. In air showers, neutral pions decay almost always before interacting again (except E π 0 > 10 19 eV) and provide high-energy photons that feed the electromagnetic shower component. In contrast, the energy of charged pions has to be degraded down to some 30 -100GeV before the probability to decay will exceed that of another interaction. In first approximation, the depth profile of charged shower particles, being dominated by e + e -of the electromagnetic component, is linked to the secondary particles of the first few interactions in the hadronic core of the shower. In contrast, the muons at ground stem from","pages":[2]},{"title":"Conference Title, to be filled","content":"Both e ff ects are the reason for the higher muon multiplicities predicted with EPOS. Leading rho mesons are explicitly generated for pion interactions in the new version II.04 of QGSJET, boosting the muon multiplicity in air showers significantly. Depending on the relative importance of e ff ect (i) and energy distribution of the baryon pairs, the energy spectrum of the additionally produced muons is very soft and might be in contradiction to shower attenuation data. Finally it should be mentioned that also an increase of the production of kaons could lead to a higher muon multiplicity and also a harder muon energy spectrum [103,99].","pages":[11]},{"title":"3 Shower Simulation Packages and Hadronic Interaction Models","content":"The most frequently used code packages for the simulation of air showers for the Auger, TA and Yakutsk experiments are AIRES [25], CORSIKA [26], and COSMOS [27]. In these packages, the Monte Carlo method is applied to simulate the evolution of air showers with all secondary particles above a user-specified energy threshold. The large number of secondary particles that would have to be followed in the simulation of ultra-high energy showers makes the simulations so time consuming that thinning techniques are applied. Only a representative set of particles is included in the simulation below a pre-defined energy threshold. The discarded particles are accounted for by increasing the statistical weight of the particles remaining in the simulation [28,29]. The main drawbacks of this method are artificial fluctuations related to particles with large statistical weight [30,31] and the need for treating weighted particles in the detector simulations. The Auger and TA Collaborations have developed di ff erent methods for de-thinning simulated showers, see [32] and [33], respectively. Also a limited number of fully simulated showers of the highest energies are available in each collaboration (see, for example, [34]) but the statistics is very limited and there is a lack of methods to work with such large amounts of data in an e ffi cient way. The classic Monte Carlo codes are complemented by hybrid simulation programs that combine the Monte Carlo technique with either numerical solutions of cascade equations (CONEX [9] and SENECA [35]) or libraries of pre-simulated air showers (COSMOS [27]). The advantage of the hybrid codes is the much reduced CPU time needed for simulating showers at very high energy and the possibility to simulate showers either without thinning or with very low statistical weights. But there is always a remaining risk that rare shower fluctuations or local correlations might not be correctly described. Di ff erent codes are applied for the simulation of the electromagnetic shower component. The code used in AIRES is based on the one developed by Hillas for MOCCA [28,36]. CONEX, CORSIKA, and SENECA are interfaced to modified versions of EGS4 [37]. Similar to AIRES, COSMOS comes with a custom-developed code for electromagnetic interactions called EPICS [27]. The suppression of high-energy interactions of photons and electrons due to the Landau-Pomeranchuk-Migdal (LPM) e ff ect (for example, see [38]) is taken into account in all code packages. Due to the di ff erent underlying approaches for modeling hadronic interactions at low ( E < 80 -200GeV) and high energies typically always two hadronic interaction models are employed in air shower simulations. The low-energy models GHEISHA [39], FLUKA [40], and UrQMD [41] are available in CORSIKA while AIRES and COSMOS employ custom-made codes. Typical high-energy interaction models are DPMJET II [42] and III [43,44], EPOS [45-47], QGSJET 01 [48,49] and II [50-52], and SIBYLL [53-55]. A compilation of simulation results obtained with many of these low- and high-energy models can be found in [56-58]. Several comparisons of the predictions of shower simulation packages for the same interaction models are available in the literature, see [4] and [61,62,59,60,63]. In the ideal case, the predictions for air showers of a given energy and primary particle depend only on the user-selected energy thresholds and on the choice of hadronic interaction models selected for the simulation. While it is possible to","pages":[4]},{"title":"4 Performance of Hadronic Interaction Models","content":"Most of the interaction models used in air shower simulations are not commonly applied in high energy physics (HEP) simulations and, conversely, HEP models are not used for air shower simulations. This is related to the fact that HEP models are typically limited to a set of primary particles that are available in accelerator experiments and optimized only for collider energies. Similarly, cosmic ray interaction models are not routinely used by HEP collaborations for acceptance correction calculations including the comparison of raw, uncorrected data as this can be done with the HEP event generators as well. Furthermore dedicated HEP event generators describe highp ⊥ and electroweak physics processes in much more detail and contain many more parameters for improving the description of particular distributions than any of the cosmic ray or general purpose models. In contrast, interaction models for cosmic ray physics are tuned to describe data of accelerator measurements over a wide range of energies, often sacrificing a perfect description of some distributions in favour of a good overall reproduction of the whole data set. Hence comparisons of model predictions to data from fixed-target and collider experiments are typically made by the authors of the models and using only fully acceptance corrected data. However, over the last years much progress has been made regarding the interaction between the cosmic ray and high energy physics communities. A number of workshops and meetings (including this one) and the direct involvement of cosmic ray physicists in HEP collaborations have lead to a much higher level of awareness and intensified communication of both communities. Comparisons of high-energy interaction models used for air shower simulations with LHC data can be found in, for example, [69-71]. Here only some representative examples can be included for illustration. The pseudorapidity distribution of charged particles is shown together with model predictions in Fig. 3. Not only most central particle distributions are reasonably well bracketed by the model predictions, also the predicted energy flow at larger pseudorapidity is in good agreement with the LHC data. This can be seen in Fig. 4 where the energy flow measured by CMS [68] is compared to model predictions. Note that all these model results are true predictions as the models were tuned years before LHC data became available. Nevertheless, in many cases, the minimum bias data are better described by interaction models developed primarily for cosmic ray physics than the various tunes of standard HEP models [68,69]. On the other hand, there are also some important distributions measured at LHC that are not well described or bracketed by these pre-LHC interaction models. Examples of large deviations from measurements are multiplicity distributions of charged particles and particle production spectra at large pseudorapidity and Feynmanx . The possible impact of these deviations on air shower predictions is subject to ongoing research and not yet fully understood. In Fig. 5 the Feynmanx distribution of photons produced in forward direction is shown [72]. The data have been obtained within the LHCf experiment that is specifically built for measuring particles in the very forward direction.","pages":[5,6,7]},{"title":"5 Overall Description of Shower Characteristics","content":"Simulating air showers with a realistic energy spectrum, primary mass composition, and arrival direction distribution, the predicted and observed distributions of the various observables of reconstructed showers can be compared. Such comparisons are very important as they are end-to-end tests of the simulation chain for the experiments. Only if good agreement is found one can cross-check the impact of quality cuts applied in the process of data analysis and avoid unexpected biases. The results of such end-to-end simulations depend, of course, on the assumed mass composition of the primary particles and the employed hadronic interaction models. Examples of such comparisons for fluorescence detectors can be found in [81-84]. A number of unpublished comparisons of typical observables measured with the Auger and TA surface detector arrays (zenith and azimuth angles, number of stations per event, signal size distribution) have been presented at this meeting. They all show very good agreement between the measured distributions and the corresponding Monte Carlo predictions. While the TA simulations were done for proton primaries, the Auger simulations were based on a 50 / 50 proton-iron mixture. Both collaborations used QGSJET II.03 as high-energy interaction model. The TA data are better described by a primary composition of only protons than using 100% iron. In case of the Auger simulations, which were also made assuming only proton or iron primaries, it was found that most of the surface detector observables exhibit only a very limited sensitivity to the primary mass composition. It should be considered as an important success of hadronic interaction models and modern air shower simulation packages that such a good overall description of the general features of the observed showers is reached. However, di ff erent combinations of primary mass composition and shower energy can lead to very similar surface detector signals (for example, the signal at 600 or 1000 m in units of that of vertical muons). Therefore it is very important to use additional information to determine, for example, the primary energy in a composition-independent way. This is done with fluorescence telescopes in the case of the Auger Observatory [85] and Telescope Array [86], and with non-imaging Cherenkov light measurements in the Yakutsk setup [87].","pages":[8]},{"title":"6 Detailed Comparison with Shower Data","content":"Detecting air showers simultaneously with fluorescence telescopes and an array of surface detectors at ground is often referred to as hybrid measurements. One of the first setups of this type was the","pages":[8]},{"title":"7 Hadronic Interactions and Muon Production in Air Showers","content":"The observation of a possible muon deficit in simulated showers in comparison to Auger measurements, first reported already in 2007 [96], has triggered a number of theoretical studies searching for modifications of hadronic interactions that could result in an enhancement of the muonic shower component. Changes of the inelastic cross section, inelasticity of interactions, secondary particle multiplicity at high energy, and many other parameters lead only to moderate changes of the predicted number of muons (see, for example, [97,98]). Also scenarios with new physics processes such as string percolation [99] or the drastic change of interaction properties due to, for example, chiral symmetry restoration [100] have been discussed. So far all proposed changes that lead to a significant increase of the number of muons in air showers are directly or indirectly based on either one or both of the following e ff ects: (i) An increase of the production rate of particles that do not decay, for example baryon-antibaryon pairs, leads to higher muon multiplicities of showers since these particles will stay being part of the hadronic shower component and loose their energy only by producing further hadronic particles. This e ff ect has been discussed first in [101] and is one of the di ff erences of the EPOS model with respect to the other models [102]. (ii) A change of the type of the leading particle produced in inelastic interactions can also be very e ffi cient in reducing the energy that is transferred to the electromagnetic shower component [103]. The majority of sub-showers in an extensive air shower are initiated by charged pions. The chance probability of producing a charged pion or a neutral pion as leading particle in charged pion interactions is about 2 : 1. Replacing all leading π 0 by ρ 0 mesons, which have the same valence quarks but are spin 1 particles, leads to a drastic enhancement of muon production since neutral ρ mesons decay immediately into two charged pions [104]. Indeed, fixed-target data [105-107] indicate that, in contrast to conventional model predictions, the production of ρ 0 dominates that of π 0 for Feynmanx larger than 0 . 5.","pages":[10]},{"title":"8 Conclusions and Outlook","content":"Over the last two decades the quality and predictive power of air shower simulations has improved significantly and a very good overall description of most of the shower features observed in experiments has been reached. The predictions of independently developed shower simulation packages agree reasonably well with each other if the same hadronic interaction models are used for the simulations. While there is very good agreement for the electromagnetic shower component, the di ff erences between the predictions for muon multiplicities and lateral distributions can be as large as ∼ 5 -10%. These di ff erences are most likely related to the use of di ff erent low-energy interaction models. A comprehensive comparison of the di ff erent shower simulation packages, similar to the recent study of CORSIKA and COSMOS predictions [60], should be made to quantify the systematic uncertainties of the predictions and possibly also to identify and address shortcomings of the simulation packages. The limited theoretical understanding of modeling hadronic multiparticle production together with the limitations of accelerator measurements in energy, covered phase space, and projectile-target combinations are the dominating source of systematic uncertainties of air shower predictions. Because of this, the systematic uncertainties of the model predictions cannot be estimated reliably. More work is needed to improve QCD calculations in the lowp ⊥ domain, in particular to understand screening and saturation e ff ects, and to develop alternative models to describe particle production. The new LHC data provide extremely useful input for tuning hadronic interaction models. Even though the first LHC data on multiplicities and cross sections were well bracketed by the predictions of interaction models used for air shower simulations, the comparison to data revealed the need for further model developments and tuning. Improved and re-tuned versions of EPOS and QGSJET are already available and similar versions of DPMJET and SIBYLL are in preparation. The interaction between the cosmic ray and high energy physics communities has intensified and the direct engagement of cosmic ray physicists in LHC and fixed-target experiments has lead to a much better understanding of the needs of the cosmic ray community. There is also very large interest from the side of LHC communities to have cosmic ray physicists being involved in the analysis of accelerator data. One example of the achieved progress is the use of cosmic ray interaction models by LHC collaborations to compare data with predictions in publications. Both the Auger and TA Collaborations have found indications for a discrepancy between the expected and observed surface detector signals for showers with fluorescence energy measurement. Accounting for the di ff erent energy scales of the Auger, TA, and Yakutsk experiments the observed discrepancies are consistent with each other. Most likely, the dominating sources of the discrepancies are shortcomings of the simulation of muon production in air showers. The systematic uncertainties of the overall energy scales of the experiments of 20 -30% will have to be reduced significantly to be able to combine data from di ff erent experiments for a stronger test of the shower predictions. While the production of electromagnetic particles is clearly dominated by the electromagnetic cascade induced by photons of very high energy due to neutral pion decay early on in the shower evolution, the muonic component receives contributions from all high-energy interactions above ∼ 20GeV lab. energy. Di ff erent modifications of the simulation of particle production in low- and high-energy interactions have been considered to increase the number of muons in air showers. Progress has been made by understanding that both production of stable or long-lived hadrons such as baryon pairs and modifications to the particle types generated as leading particles are e ffi cient mechanisms to increase the muon multiplicity without changing significantly the longitudinal shower profile. It remains to","pages":[11]}],"string":"[\n {\n \"title\": \"Air Shower Simulation and Hadronic Interactions\",\n \"content\": \"Je ff Allen 1 , Antonella Castellina 2 , Ralph Engel 3 , Katsuaki Kasahara 4 , Stanislav Knurenko 5 , Tanguy Pierog 3 , Artem Sabourov 5 , Benjamin T. Stokes 6 , Ralf Ulrich 3 , for the Pierre Auger, Telescope Array, and Yakutsk Collaborations, and Sergey Ostapchenko 7 and Takashi Sako 8 Abstract. The aim of this report of the Working Group on Hadronic Interactions and Air Shower Simulation is to give an overview of the status of the field, emphasizing open questions and a comparison of relevant results of the di ff erent experiments. It is shown that an approximate overall understanding of extensive air showers and the corresponding hadronic interactions has been reached. The simulations provide a qualitative description of the bulk of the air shower observables. Discrepancies are however found when the correlation between measurements of the longitudinal shower profile are compared to that of the lateral particle distributions at ground. The report concludes with a list of important problems that should be addressed to make progress in understanding hadronic interactions and, hence, improve the reliability of air shower simulations.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"After the first interaction of a cosmic ray particle of very high energy in the atmosphere a multitude of subsequent interactions, leading to particle multiplication, and decay processes give rise to a cascade of secondary particles called extensive air shower (EAS) [1]. With the electromagnetic and weak interactions being well described by perturbative calculations within the Standard Model of Particle Physics, the limited understanding of strong interactions becomes the dominant source of uncertainties of shower predictions. Even though Quantum Chromodynamics (QCD) is the well-established and experimentally confirmed theory of strong interactions, only processes with very large momentum transfer can be predicted from first principles until now. It is still not possible to calculate the bulk properties of multiparticle production as needed for air shower simulation. Additional, simplifying assumptions as well as phenomenological and empirical parametrizations are needed to develop models for hadronic interactions describing various particle production processes. These additional assumptions need to be verified, parametrizations constrained, and parameters tuned by comparisons to accelerator data. The simulation of extensive air showers forms one of the pillars on which the data analysis of modern experiments for ultra-high energy cosmic rays (UHECRs) rests. Improving the understanding and modeling of hadronic particle production is one of the important prerequisites for a reliable interpreta-\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"EPJ Web of Conferences\",\n \"content\": \"be shown whether the discrepancies between the observed and predicted surface detector signals for a given shower energy can be resolved by improving low- and high-energy interaction models using conventional physics assumptions or whether these observations indeed indicate a change of the characteristics of hadronic interactions at energies beyond that of LHC.\",\n \"pages\": [\n 12\n ]\n },\n {\n \"title\": \"2 Physics of Air Showers\",\n \"content\": \"Reviews of the physics of extensive air showers and hadronic interactions can be found in, for example, [3-6] and a very instructive extension of Heitler's cascade model to hadronic showers is given in [7]. Here we give only a qualitative introduction to some aspects of shower physics that will be needed in the discussions later on. With pions being the most abundant secondary particles of hadronic interactions at high energy, the di ff erence in lifetime and decay products of neutral and charged pions lead to fundamentally di ff erent ranges of interaction energies that give rise to the electromagnetic and muonic shower components. In air showers, neutral pions decay almost always before interacting again (except E π 0 > 10 19 eV) and provide high-energy photons that feed the electromagnetic shower component. In contrast, the energy of charged pions has to be degraded down to some 30 -100GeV before the probability to decay will exceed that of another interaction. In first approximation, the depth profile of charged shower particles, being dominated by e + e -of the electromagnetic component, is linked to the secondary particles of the first few interactions in the hadronic core of the shower. In contrast, the muons at ground stem from\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"Conference Title, to be filled\",\n \"content\": \"Both e ff ects are the reason for the higher muon multiplicities predicted with EPOS. Leading rho mesons are explicitly generated for pion interactions in the new version II.04 of QGSJET, boosting the muon multiplicity in air showers significantly. Depending on the relative importance of e ff ect (i) and energy distribution of the baryon pairs, the energy spectrum of the additionally produced muons is very soft and might be in contradiction to shower attenuation data. Finally it should be mentioned that also an increase of the production of kaons could lead to a higher muon multiplicity and also a harder muon energy spectrum [103,99].\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"3 Shower Simulation Packages and Hadronic Interaction Models\",\n \"content\": \"The most frequently used code packages for the simulation of air showers for the Auger, TA and Yakutsk experiments are AIRES [25], CORSIKA [26], and COSMOS [27]. In these packages, the Monte Carlo method is applied to simulate the evolution of air showers with all secondary particles above a user-specified energy threshold. The large number of secondary particles that would have to be followed in the simulation of ultra-high energy showers makes the simulations so time consuming that thinning techniques are applied. Only a representative set of particles is included in the simulation below a pre-defined energy threshold. The discarded particles are accounted for by increasing the statistical weight of the particles remaining in the simulation [28,29]. The main drawbacks of this method are artificial fluctuations related to particles with large statistical weight [30,31] and the need for treating weighted particles in the detector simulations. The Auger and TA Collaborations have developed di ff erent methods for de-thinning simulated showers, see [32] and [33], respectively. Also a limited number of fully simulated showers of the highest energies are available in each collaboration (see, for example, [34]) but the statistics is very limited and there is a lack of methods to work with such large amounts of data in an e ffi cient way. The classic Monte Carlo codes are complemented by hybrid simulation programs that combine the Monte Carlo technique with either numerical solutions of cascade equations (CONEX [9] and SENECA [35]) or libraries of pre-simulated air showers (COSMOS [27]). The advantage of the hybrid codes is the much reduced CPU time needed for simulating showers at very high energy and the possibility to simulate showers either without thinning or with very low statistical weights. But there is always a remaining risk that rare shower fluctuations or local correlations might not be correctly described. Di ff erent codes are applied for the simulation of the electromagnetic shower component. The code used in AIRES is based on the one developed by Hillas for MOCCA [28,36]. CONEX, CORSIKA, and SENECA are interfaced to modified versions of EGS4 [37]. Similar to AIRES, COSMOS comes with a custom-developed code for electromagnetic interactions called EPICS [27]. The suppression of high-energy interactions of photons and electrons due to the Landau-Pomeranchuk-Migdal (LPM) e ff ect (for example, see [38]) is taken into account in all code packages. Due to the di ff erent underlying approaches for modeling hadronic interactions at low ( E < 80 -200GeV) and high energies typically always two hadronic interaction models are employed in air shower simulations. The low-energy models GHEISHA [39], FLUKA [40], and UrQMD [41] are available in CORSIKA while AIRES and COSMOS employ custom-made codes. Typical high-energy interaction models are DPMJET II [42] and III [43,44], EPOS [45-47], QGSJET 01 [48,49] and II [50-52], and SIBYLL [53-55]. A compilation of simulation results obtained with many of these low- and high-energy models can be found in [56-58]. Several comparisons of the predictions of shower simulation packages for the same interaction models are available in the literature, see [4] and [61,62,59,60,63]. In the ideal case, the predictions for air showers of a given energy and primary particle depend only on the user-selected energy thresholds and on the choice of hadronic interaction models selected for the simulation. While it is possible to\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4 Performance of Hadronic Interaction Models\",\n \"content\": \"Most of the interaction models used in air shower simulations are not commonly applied in high energy physics (HEP) simulations and, conversely, HEP models are not used for air shower simulations. This is related to the fact that HEP models are typically limited to a set of primary particles that are available in accelerator experiments and optimized only for collider energies. Similarly, cosmic ray interaction models are not routinely used by HEP collaborations for acceptance correction calculations including the comparison of raw, uncorrected data as this can be done with the HEP event generators as well. Furthermore dedicated HEP event generators describe highp ⊥ and electroweak physics processes in much more detail and contain many more parameters for improving the description of particular distributions than any of the cosmic ray or general purpose models. In contrast, interaction models for cosmic ray physics are tuned to describe data of accelerator measurements over a wide range of energies, often sacrificing a perfect description of some distributions in favour of a good overall reproduction of the whole data set. Hence comparisons of model predictions to data from fixed-target and collider experiments are typically made by the authors of the models and using only fully acceptance corrected data. However, over the last years much progress has been made regarding the interaction between the cosmic ray and high energy physics communities. A number of workshops and meetings (including this one) and the direct involvement of cosmic ray physicists in HEP collaborations have lead to a much higher level of awareness and intensified communication of both communities. Comparisons of high-energy interaction models used for air shower simulations with LHC data can be found in, for example, [69-71]. Here only some representative examples can be included for illustration. The pseudorapidity distribution of charged particles is shown together with model predictions in Fig. 3. Not only most central particle distributions are reasonably well bracketed by the model predictions, also the predicted energy flow at larger pseudorapidity is in good agreement with the LHC data. This can be seen in Fig. 4 where the energy flow measured by CMS [68] is compared to model predictions. Note that all these model results are true predictions as the models were tuned years before LHC data became available. Nevertheless, in many cases, the minimum bias data are better described by interaction models developed primarily for cosmic ray physics than the various tunes of standard HEP models [68,69]. On the other hand, there are also some important distributions measured at LHC that are not well described or bracketed by these pre-LHC interaction models. Examples of large deviations from measurements are multiplicity distributions of charged particles and particle production spectra at large pseudorapidity and Feynmanx . The possible impact of these deviations on air shower predictions is subject to ongoing research and not yet fully understood. In Fig. 5 the Feynmanx distribution of photons produced in forward direction is shown [72]. The data have been obtained within the LHCf experiment that is specifically built for measuring particles in the very forward direction.\",\n \"pages\": [\n 5,\n 6,\n 7\n ]\n },\n {\n \"title\": \"5 Overall Description of Shower Characteristics\",\n \"content\": \"Simulating air showers with a realistic energy spectrum, primary mass composition, and arrival direction distribution, the predicted and observed distributions of the various observables of reconstructed showers can be compared. Such comparisons are very important as they are end-to-end tests of the simulation chain for the experiments. Only if good agreement is found one can cross-check the impact of quality cuts applied in the process of data analysis and avoid unexpected biases. The results of such end-to-end simulations depend, of course, on the assumed mass composition of the primary particles and the employed hadronic interaction models. Examples of such comparisons for fluorescence detectors can be found in [81-84]. A number of unpublished comparisons of typical observables measured with the Auger and TA surface detector arrays (zenith and azimuth angles, number of stations per event, signal size distribution) have been presented at this meeting. They all show very good agreement between the measured distributions and the corresponding Monte Carlo predictions. While the TA simulations were done for proton primaries, the Auger simulations were based on a 50 / 50 proton-iron mixture. Both collaborations used QGSJET II.03 as high-energy interaction model. The TA data are better described by a primary composition of only protons than using 100% iron. In case of the Auger simulations, which were also made assuming only proton or iron primaries, it was found that most of the surface detector observables exhibit only a very limited sensitivity to the primary mass composition. It should be considered as an important success of hadronic interaction models and modern air shower simulation packages that such a good overall description of the general features of the observed showers is reached. However, di ff erent combinations of primary mass composition and shower energy can lead to very similar surface detector signals (for example, the signal at 600 or 1000 m in units of that of vertical muons). Therefore it is very important to use additional information to determine, for example, the primary energy in a composition-independent way. This is done with fluorescence telescopes in the case of the Auger Observatory [85] and Telescope Array [86], and with non-imaging Cherenkov light measurements in the Yakutsk setup [87].\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"6 Detailed Comparison with Shower Data\",\n \"content\": \"Detecting air showers simultaneously with fluorescence telescopes and an array of surface detectors at ground is often referred to as hybrid measurements. One of the first setups of this type was the\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"7 Hadronic Interactions and Muon Production in Air Showers\",\n \"content\": \"The observation of a possible muon deficit in simulated showers in comparison to Auger measurements, first reported already in 2007 [96], has triggered a number of theoretical studies searching for modifications of hadronic interactions that could result in an enhancement of the muonic shower component. Changes of the inelastic cross section, inelasticity of interactions, secondary particle multiplicity at high energy, and many other parameters lead only to moderate changes of the predicted number of muons (see, for example, [97,98]). Also scenarios with new physics processes such as string percolation [99] or the drastic change of interaction properties due to, for example, chiral symmetry restoration [100] have been discussed. So far all proposed changes that lead to a significant increase of the number of muons in air showers are directly or indirectly based on either one or both of the following e ff ects: (i) An increase of the production rate of particles that do not decay, for example baryon-antibaryon pairs, leads to higher muon multiplicities of showers since these particles will stay being part of the hadronic shower component and loose their energy only by producing further hadronic particles. This e ff ect has been discussed first in [101] and is one of the di ff erences of the EPOS model with respect to the other models [102]. (ii) A change of the type of the leading particle produced in inelastic interactions can also be very e ffi cient in reducing the energy that is transferred to the electromagnetic shower component [103]. The majority of sub-showers in an extensive air shower are initiated by charged pions. The chance probability of producing a charged pion or a neutral pion as leading particle in charged pion interactions is about 2 : 1. Replacing all leading π 0 by ρ 0 mesons, which have the same valence quarks but are spin 1 particles, leads to a drastic enhancement of muon production since neutral ρ mesons decay immediately into two charged pions [104]. Indeed, fixed-target data [105-107] indicate that, in contrast to conventional model predictions, the production of ρ 0 dominates that of π 0 for Feynmanx larger than 0 . 5.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"8 Conclusions and Outlook\",\n \"content\": \"Over the last two decades the quality and predictive power of air shower simulations has improved significantly and a very good overall description of most of the shower features observed in experiments has been reached. The predictions of independently developed shower simulation packages agree reasonably well with each other if the same hadronic interaction models are used for the simulations. While there is very good agreement for the electromagnetic shower component, the di ff erences between the predictions for muon multiplicities and lateral distributions can be as large as ∼ 5 -10%. These di ff erences are most likely related to the use of di ff erent low-energy interaction models. A comprehensive comparison of the di ff erent shower simulation packages, similar to the recent study of CORSIKA and COSMOS predictions [60], should be made to quantify the systematic uncertainties of the predictions and possibly also to identify and address shortcomings of the simulation packages. The limited theoretical understanding of modeling hadronic multiparticle production together with the limitations of accelerator measurements in energy, covered phase space, and projectile-target combinations are the dominating source of systematic uncertainties of air shower predictions. Because of this, the systematic uncertainties of the model predictions cannot be estimated reliably. More work is needed to improve QCD calculations in the lowp ⊥ domain, in particular to understand screening and saturation e ff ects, and to develop alternative models to describe particle production. The new LHC data provide extremely useful input for tuning hadronic interaction models. Even though the first LHC data on multiplicities and cross sections were well bracketed by the predictions of interaction models used for air shower simulations, the comparison to data revealed the need for further model developments and tuning. Improved and re-tuned versions of EPOS and QGSJET are already available and similar versions of DPMJET and SIBYLL are in preparation. The interaction between the cosmic ray and high energy physics communities has intensified and the direct engagement of cosmic ray physicists in LHC and fixed-target experiments has lead to a much better understanding of the needs of the cosmic ray community. There is also very large interest from the side of LHC communities to have cosmic ray physicists being involved in the analysis of accelerator data. One example of the achieved progress is the use of cosmic ray interaction models by LHC collaborations to compare data with predictions in publications. Both the Auger and TA Collaborations have found indications for a discrepancy between the expected and observed surface detector signals for showers with fluorescence energy measurement. Accounting for the di ff erent energy scales of the Auger, TA, and Yakutsk experiments the observed discrepancies are consistent with each other. Most likely, the dominating sources of the discrepancies are shortcomings of the simulation of muon production in air showers. The systematic uncertainties of the overall energy scales of the experiments of 20 -30% will have to be reduced significantly to be able to combine data from di ff erent experiments for a stronger test of the shower predictions. While the production of electromagnetic particles is clearly dominated by the electromagnetic cascade induced by photons of very high energy due to neutral pion decay early on in the shower evolution, the muonic component receives contributions from all high-energy interactions above ∼ 20GeV lab. energy. Di ff erent modifications of the simulation of particle production in low- and high-energy interactions have been considered to increase the number of muons in air showers. Progress has been made by understanding that both production of stable or long-lived hadrons such as baryon pairs and modifications to the particle types generated as leading particles are e ffi cient mechanisms to increase the muon multiplicity without changing significantly the longitudinal shower profile. It remains to\",\n \"pages\": [\n 11\n ]\n }\n]"}}},{"rowIdx":260,"cells":{"bibcode":{"kind":"string","value":"2013EPJWC..5308013S"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1210.1427.pdf"},"content":{"kind":"string","value":"\nRadar reflection off extensive air showers\nJ. Stasielak 1 , a , S. Baur 2 , M. Bertaina 3 , J. Blumer 2 , A. Chiavassa 3 , R. Engel 2 , A. Haungs 2 , T. Huege 2 , K.-H. Kampert 4 , H. Klages 2 , M. Kleifges 2 , O. Kromer 2 , M. Ludwig 2 , S. Mathys 4 , P. Neunteufel 2 , J. Pekala 1 , J. Rautenberg 4 , M. Riegel 2 , M. Roth 2 , F. Salamida 5 , H. Schieler 2 , R. ˇ Sm'ıda 2 , M. Unger 2 , M. Weber 2 , F. Werner 2 , H. Wilczy'nski 1 , and J. Wochele 2\n\n1 Institute of Nuclear Physics PAN, Krakow, Poland\n2 Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany\n3 Universit'a di Torino and Sezione INFN, Torino, Italy\n4 Bergische Universitat Wuppertal, Wuppertal, Germany\n5 Universit'a dell'Aquila and INFN, L'Aquila, Italy\n\nAbstract. We investigate the possibility of detecting extensive air showers by the radar technique. Considering a bistatic radar system and di ff erent shower geometries, we simulate reflection of radio waves o ff the static plasma produced by the shower in the air. Using the Thomson cross-section for radio wave reflection, we obtain the time evolution of the signal received by the antennas. The frequency upshift of the radar echo and the power received are studied to verify the feasibility of the radar detection technique.\n1 Introduction\nTraditional techniques of extensive air shower (EAS) detection include counting the shower particles at the ground level and measuring fluorescence light from the excited nitrogen in the atmosphere. Detecting radio emission from the shower particles is a promising new technique that is currently in development. An alternative method is the radar technique, where a ground-based radio transmitter illuminates the ionization trail left behind the shower front and another ground-based antenna receives the scattered radio signal with its upshifted frequency.\nThe concept of implementing a radar for cosmic ray detection was first introduced about 70 years ago by Blackett and Lovell [1]. However due to the negative results of dedicated experiments, this concept was forgotten for several decades. In recent years renewed attention has been attracted to this topic, see e.g. [2,3,4,5]. Experimental e ff orts to detect EAS using the radar technique were made by several groups, e.g. [6], the LAAS group [7], [8], the MARIACHI experiment [9], or the TARA project [10], however no detection was confirmed so far.\nIn this paper we investigate the possibility of detecting EAS by the bistatic radar system. Especially, we are interested in verifying the feasibility of supplementing the CROME (Cosmic-Ray Observation via Microwave Emission) detector [11,12] with a radio transmitter and using its existing antennas as receivers for the radar echo. The CROME antenna setup consists of several microwave receivers for the frequency ranges 1 . 2 -1 . 7 (L band), 3 . 4 -4 . 2 (extended C band), and 10 . 7 -11 . 7 GHz (low Ku band). Since we expect large frequency upshifts of the radar echo and the CROME antennas are fine tuned to the GHz frequency band, we can narrow down our interest to the frequency range of 1 -100 MHz of the emitted radio waves.\nThe concept of detecting EAS using radar technique is based on the principle of scattering radio waves o ff the static plasma produced in the atmosphere by the energetic particles of the shower. The locally produced plasma decays exponentially with the lifetime τ . According to Vidmar [13], for the\nEPJ Web of Conferences\n\n\n
Fig. 1. A schematic diagram representing a bistatic radar system and reflection from the static plasma disk produced by the EAS in the atmosphere. See the text for a detailed explanation.
\n\nR\nplasma densities relevant for EAS and at low altitudes, the three-body attachment to oxygen dominates the deionization process as it depends quadratically on the oxygen density. This leads to the plasma lifetime of 10 ns at sea level and about 100 ns at altitude of 10 km. Since the received power of the radar echo is strongly diminished by the geometrical factor, the strongest signal will be obtained from the altitudes close to the ground level. Thus we can safely assume τ = 10 ns.\nThe ionization trail that results from meteor or lightning is traditionally divided into the underdense and overdense regions, depending on the local plasma frequency ν p . If the electron density is high enough that the plasma frequency exceeds the radar frequency, then the region is overdense and the radio wave is reflected from its interface. In contrast, if the local plasma frequency is lower than the frequency of the incoming radio wave, then the region is underdense and the radio wave can penetrate the ionized region. In such a case the reflections are caused by the Thomson scattering of the radio wave on individual free electrons.\nIf we consider the radar frequency in the range 1 -100 MHz and shower energies up to 10 20 eV, then the maximum size of the overdense plasma region will be several meters. Now, if we calculate the attenuation length of the radio wave in the plasma with frequency ν p = 100 MHz and ν p = 1 MHz one obtains ∼ 300 m and ∼ 3000 km, respectively. Those lengths are much larger than the size of the overdense plasma, thus the radio wave can easily penetrate the whole volume of the plasma disk produced by the EAS.\nUsing the Thomson cross-section for radio wave reflection seems to be justified from the above arguments. An exact calculation of the total power received requires the integration of the contributions from each individual electron in the shower. The final result will then depend on the individual phase factors of the scattering electrons.\n2 Radar reflection\nA schematic diagram representing a bistatic radar system is shown in Figure 1. A ground-based radio transmitter (T) irradiates a disk-like shower. The front edge of the disk is an ionization front moving towards the ground with the speed of light and leaving created static plasma behind it. The radio signal is scattered by the free electrons in the ionization trail and subsequently received by the groundbased antenna (R). The geometry of the bistatic radar system is set up by the polar coordinates of the\n\n\n
Fig. 2. The time evolution of the region from which the scattered radio waves arrive simultaneously to the detector. The arrival time t = 0 corresponds to the moment at which the shower hits the ground. The considered shower has energy of 10 18 eV and is heading towards the transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude.
\n\ntransmitter and the receiver, i.e. by the distances from the shower core to the transmitter ( dT ) and to the receiver ( dR ) together with the angles ϕ t and ϕ r .\nLet us consider the radar reflection from the disk element with coordinates ( rL , ϕ ), altitude h, and electron density ne . Its contribution to the radar echo at the receiver at time t can be expressed by\nUrc v ( t , s , rL , ϕ ) = UT √ GTe i ω t + ϕ 0 e -i ∫ r n k · d r e -i ∫ rsc n ksc · d rsc | r || rsc | sin α √ d σ T d Ω ne , (1)\nwhere GT is the transmitter antenna gain, UT [V] is the magnitude of the transmitted field, k and ksc are the wave vectors of incoming and scattered radio wave, ϕ 0 is the initial phase of the emitted signal, d σ T / d Ω is the Thomson cross-section, s is the projection of the distance between the shower core and the considered disk element on the shower axis, α is the inclination angle of the reflected radio wave, and n is the refractive index of the air. By analogy to the setup of the CROME detector, we assume that the receiver is oriented vertically upwards.\nThe signal received by the antenna at a given time t is sum of the signals scattered at di ff erent times and from di ff erent parts of the plasma disk. These individual contributions interfere with each other and only an integral over the whole volume v ol ( t ) from which they arrive simultaneously gives us the correct value. The total electric field strength of the radio wave at the receiver is given by\nUrc v ( t ) = ∫ v ol ( t ) Urc v ( t , s , rL , ϕ ) rLdrLd ϕ ds . (2)\nNote that time t is defined in such a way that t = 0 coincides with the moment at which the shower hits the ground.\nEPJ Web of Conferences\n\n\n
Fig. 3. Upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d .
\n\nFigure 2 shows the time evolution of the region of integration v ol ( t ) (see equation (2)) for a shower with an energy of 10 18 eV heading towards transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude.\nAs we can see, the volume over which we integrate the signal spans usually over the wide range of altitudes. This implies a high variation of the air density. Therefore, the refractive index and the Cherenkov angle can vary significantly over the considered region. This makes the whole analysis very complex.\n3 Frequency upshift\nDespite the fact that the radio wave is scattered on the static plasma, the region in which the plasma is created moves with relativistic velocity. Therefore, we can observe the Doppler shift of the received radar echo. Figure 3 shows the upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d .\nThe frequency upshift depends on the wave direction and the refractive index of air. It has the highest value for the case in which the viewing angle coincides with the Cherenkov cone. The typical fr is high enough to upshift the MHz signal into the GHz range. Note that fr only weakly depends on the distance from the shower core to the transmitter dT .\n4 Details of the simulation\nThe main assumptions and approximations are the following:\n\n-The shower longitudinal profile is taken to be a fit to the Gaisser-Hillas function for proton showers.\n-The lateral distribution of the shower is given by the Gora function [14].\n-Static plasma produced by the shower decays exponentially with the characteristic time τ = 10 ns.\n-We consider radar reflections only from the part of the ionization trail which is up to 40 ns behind the shower front. In other words, the length of the plasma disk produced by the extensive air shower is constrained to 4 τ . This leads to less than 2% of the total plasma electrons not being accounted for.\n-We use the US Standard Atmosphere model.\n-We assume that the initial phase of the emitted signal is ϕ 0 = 0. Choosing di ff erent values of ϕ 0 can lead to slightly di ff erent values of the received power.\n\n\n\n
Fig. 4. The waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases a vertical shower with energy 10 18 eV heading towards the transmitter is considered.
\n\n\n-The e ff ective area of the receiver antenna is AR = 1 m 2 and the transmitter emits signal over the whole upper hemisphere ( GT = 2).\n-The receiver points vertically and its sensitivity is independent of the direction and frequency of the radar echo. The detector e ffi ciency is 100%.\n\n5 Power received by the detector antenna\nThe ratio of the 'instantaneous' power PR ( t ) received by the detector antenna to the power emitted by the transmitter PT is given by\nPR ( t ) / PT = R 2 ( t ) (3)\nwhere\nR ( t ) = 1 √ 4 π ReUrcv(t) UT √ AR (4)\nThe term R ( t ) is proportional to the electric field strength detected by the receiver. It can be used in the Fourier analysis to obtain the power spectrum of the recorded signal. The 'real' power received by the detector PR can be obtained by averaging R 2 ( t ) (according to the time resolution of the detector). The ratio PR / PT ∼ GTAR .\n6 Results\nFigure 4 shows the waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases the receiver is outside the Cherenkov cone (which is the most common case) and vertical showers heading towards the transmitter are considered. The time t = 0 coincides with the moment at which the shower hits the ground.\nAs we can see, the frequency of the received signal is higher than that of the emitted one. Moreover, the amplitude grows with time, which is the geometrical e ff ect of the reflections from the lower parts of the atmosphere.\nFor showers, where the receiver is inside the Cherenkov cone, the time sequence is reversed and the lower part of the shower is seen first. Therefore, the amplitude decreases with time. However, this is the case only for very inclined showers with cores up to several meters away from the receiver.\nThe ratios of the power received by the detector to the emitted one ( PR / PT ) for vertical showers are given in table 1. It is evident that the strength of the signal decreases with the frequency ν . It is simply because the size of the region from which we get the coherent signal decreases with the wavelength and the negative interference cancels out the signal.\nEPJ Web of Conferences\n
\n\n
Table 1. The ratios of the power received by the detector to the emitted one ( PR / PT ) in dB for di ff erent frequencies ν of the emitted radio wave, di ff erent transmitter-receiver distances dR and di ff erent shower energy E . In all cases a vertical shower heading towards the transmitter is considered.
\n
\n7 Conclusions\nThe typical ratio of the received to the emitted power is in the range of 10 -10 - 10 -13 , which makes the detection of the radar reflection plausible. The C band ( ∼ 3 GHz) is particularly well-suited for this purpose because of very low noise ( < 10 K) in this frequency range. The CROME antennas supplemented with commercial high power MHz transmitter would create a radar system dedicated to EAS detection. A note should be added, that the shown time traces are for an infinite bandwidth detector -a realistic detector would only be able to detect the signal in narrow frequency range. Moreover, it will only see the shower for a certain fraction of its development.\nThe performed analysis will help to choose the frequency of the emitted wave to optimize the detection of the radar echo. The work is in progress.\n8 Acknowledgments\nIt is our pleasure to acknowledge the interaction and collaboration with many colleagues from the KASCADE-Grande and Pierre Auger Collaboration. This work has been supported in part by the KIT start-up grant 2066995641,the ASPERA project BMBF 05A11VKA, the Helmholtz-University Young Investigators Group VH-NG-413 and the National Centre for Research and Development (NCBiR).\nReferences\n\n1. P. M. S. Blackett and A. C. B. Lovell, Proc. Royal. Soc. A177 , 183 (1940)\n2. P. W. Gorham, Astroparticle Physics 15 , 177 (2001)\n3. P. W. Gorham, AIP Conference proceedings 579 , 253 (2001)\n4. M. I. Bakunov et al., Astroparticle Physics 33 , 335 (2010)\n5. H. Takai et al., Proceedings of the 32th ICRC, Beijing (2011)\n6. T. Matano et al., Canadian Journal of Physics 46 , S255 (1968)\n7. A. Lyono et al., Proceedings of the 28th ICRC, Tsukuba (2003)\n8. T. Terasawa et al., Proceedings of the 31st ICRC, Lodz (2009)\n9. M. F. Bugallo et al., Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 2661 (2008)\n10. J. Belz et al., Proceedings of the International Symposium on Future Directions in UHECR Physics, CERN (2012) (to be published)\n11. R. ˇ Sm'ıda et al., Proceedings of the 32th ICRC, Beijing (2011), arXiv:1108.0588\n12. R. ˇ Sm'ıda et al., Proceedings of the International Symposium on Future Directions in UHECR Physics, CERN (2012) (to be published)\n13. R. J. Vidmar, IEEE Transactions On Plasma Science 18 , 733 (1990)\n14. D. Gora, et al., Astroparticle Physics 24 , 484 (2006)\n"},"sections":{"kind":"list like","value":[{"title":"Radar reflection off extensive air showers","content":"J. Stasielak 1 , a , S. Baur 2 , M. Bertaina 3 , J. Blumer 2 , A. Chiavassa 3 , R. Engel 2 , A. Haungs 2 , T. Huege 2 , K.-H. Kampert 4 , H. Klages 2 , M. Kleifges 2 , O. Kromer 2 , M. Ludwig 2 , S. Mathys 4 , P. Neunteufel 2 , J. Pekala 1 , J. Rautenberg 4 , M. Riegel 2 , M. Roth 2 , F. Salamida 5 , H. Schieler 2 , R. ˇ Sm'ıda 2 , M. Unger 2 , M. Weber 2 , F. Werner 2 , H. Wilczy'nski 1 , and J. Wochele 2 Abstract. We investigate the possibility of detecting extensive air showers by the radar technique. Considering a bistatic radar system and di ff erent shower geometries, we simulate reflection of radio waves o ff the static plasma produced by the shower in the air. Using the Thomson cross-section for radio wave reflection, we obtain the time evolution of the signal received by the antennas. The frequency upshift of the radar echo and the power received are studied to verify the feasibility of the radar detection technique.","pages":[1]},{"title":"1 Introduction","content":"Traditional techniques of extensive air shower (EAS) detection include counting the shower particles at the ground level and measuring fluorescence light from the excited nitrogen in the atmosphere. Detecting radio emission from the shower particles is a promising new technique that is currently in development. An alternative method is the radar technique, where a ground-based radio transmitter illuminates the ionization trail left behind the shower front and another ground-based antenna receives the scattered radio signal with its upshifted frequency. The concept of implementing a radar for cosmic ray detection was first introduced about 70 years ago by Blackett and Lovell [1]. However due to the negative results of dedicated experiments, this concept was forgotten for several decades. In recent years renewed attention has been attracted to this topic, see e.g. [2,3,4,5]. Experimental e ff orts to detect EAS using the radar technique were made by several groups, e.g. [6], the LAAS group [7], [8], the MARIACHI experiment [9], or the TARA project [10], however no detection was confirmed so far. In this paper we investigate the possibility of detecting EAS by the bistatic radar system. Especially, we are interested in verifying the feasibility of supplementing the CROME (Cosmic-Ray Observation via Microwave Emission) detector [11,12] with a radio transmitter and using its existing antennas as receivers for the radar echo. The CROME antenna setup consists of several microwave receivers for the frequency ranges 1 . 2 -1 . 7 (L band), 3 . 4 -4 . 2 (extended C band), and 10 . 7 -11 . 7 GHz (low Ku band). Since we expect large frequency upshifts of the radar echo and the CROME antennas are fine tuned to the GHz frequency band, we can narrow down our interest to the frequency range of 1 -100 MHz of the emitted radio waves. The concept of detecting EAS using radar technique is based on the principle of scattering radio waves o ff the static plasma produced in the atmosphere by the energetic particles of the shower. The locally produced plasma decays exponentially with the lifetime τ . According to Vidmar [13], for the","pages":[1]},{"title":"EPJ Web of Conferences","content":"Figure 2 shows the time evolution of the region of integration v ol ( t ) (see equation (2)) for a shower with an energy of 10 18 eV heading towards transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude. As we can see, the volume over which we integrate the signal spans usually over the wide range of altitudes. This implies a high variation of the air density. Therefore, the refractive index and the Cherenkov angle can vary significantly over the considered region. This makes the whole analysis very complex.","pages":[4]},{"title":"2 Radar reflection","content":"A schematic diagram representing a bistatic radar system is shown in Figure 1. A ground-based radio transmitter (T) irradiates a disk-like shower. The front edge of the disk is an ionization front moving towards the ground with the speed of light and leaving created static plasma behind it. The radio signal is scattered by the free electrons in the ionization trail and subsequently received by the groundbased antenna (R). The geometry of the bistatic radar system is set up by the polar coordinates of the transmitter and the receiver, i.e. by the distances from the shower core to the transmitter ( dT ) and to the receiver ( dR ) together with the angles ϕ t and ϕ r . Let us consider the radar reflection from the disk element with coordinates ( rL , ϕ ), altitude h, and electron density ne . Its contribution to the radar echo at the receiver at time t can be expressed by where GT is the transmitter antenna gain, UT [V] is the magnitude of the transmitted field, k and ksc are the wave vectors of incoming and scattered radio wave, ϕ 0 is the initial phase of the emitted signal, d σ T / d Ω is the Thomson cross-section, s is the projection of the distance between the shower core and the considered disk element on the shower axis, α is the inclination angle of the reflected radio wave, and n is the refractive index of the air. By analogy to the setup of the CROME detector, we assume that the receiver is oriented vertically upwards. The signal received by the antenna at a given time t is sum of the signals scattered at di ff erent times and from di ff erent parts of the plasma disk. These individual contributions interfere with each other and only an integral over the whole volume v ol ( t ) from which they arrive simultaneously gives us the correct value. The total electric field strength of the radio wave at the receiver is given by Note that time t is defined in such a way that t = 0 coincides with the moment at which the shower hits the ground.","pages":[2,3]},{"title":"3 Frequency upshift","content":"Despite the fact that the radio wave is scattered on the static plasma, the region in which the plasma is created moves with relativistic velocity. Therefore, we can observe the Doppler shift of the received radar echo. Figure 3 shows the upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d . The frequency upshift depends on the wave direction and the refractive index of air. It has the highest value for the case in which the viewing angle coincides with the Cherenkov cone. The typical fr is high enough to upshift the MHz signal into the GHz range. Note that fr only weakly depends on the distance from the shower core to the transmitter dT .","pages":[4]},{"title":"4 Details of the simulation","content":"The main assumptions and approximations are the following:","pages":[4]},{"title":"5 Power received by the detector antenna","content":"The ratio of the 'instantaneous' power PR ( t ) received by the detector antenna to the power emitted by the transmitter PT is given by where The term R ( t ) is proportional to the electric field strength detected by the receiver. It can be used in the Fourier analysis to obtain the power spectrum of the recorded signal. The 'real' power received by the detector PR can be obtained by averaging R 2 ( t ) (according to the time resolution of the detector). The ratio PR / PT ∼ GTAR .","pages":[5]},{"title":"6 Results","content":"Figure 4 shows the waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases the receiver is outside the Cherenkov cone (which is the most common case) and vertical showers heading towards the transmitter are considered. The time t = 0 coincides with the moment at which the shower hits the ground. As we can see, the frequency of the received signal is higher than that of the emitted one. Moreover, the amplitude grows with time, which is the geometrical e ff ect of the reflections from the lower parts of the atmosphere. For showers, where the receiver is inside the Cherenkov cone, the time sequence is reversed and the lower part of the shower is seen first. Therefore, the amplitude decreases with time. However, this is the case only for very inclined showers with cores up to several meters away from the receiver. The ratios of the power received by the detector to the emitted one ( PR / PT ) for vertical showers are given in table 1. It is evident that the strength of the signal decreases with the frequency ν . It is simply because the size of the region from which we get the coherent signal decreases with the wavelength and the negative interference cancels out the signal.","pages":[5]},{"title":"7 Conclusions","content":"The typical ratio of the received to the emitted power is in the range of 10 -10 - 10 -13 , which makes the detection of the radar reflection plausible. The C band ( ∼ 3 GHz) is particularly well-suited for this purpose because of very low noise ( < 10 K) in this frequency range. The CROME antennas supplemented with commercial high power MHz transmitter would create a radar system dedicated to EAS detection. A note should be added, that the shown time traces are for an infinite bandwidth detector -a realistic detector would only be able to detect the signal in narrow frequency range. Moreover, it will only see the shower for a certain fraction of its development. The performed analysis will help to choose the frequency of the emitted wave to optimize the detection of the radar echo. The work is in progress.","pages":[6]},{"title":"8 Acknowledgments","content":"It is our pleasure to acknowledge the interaction and collaboration with many colleagues from the KASCADE-Grande and Pierre Auger Collaboration. This work has been supported in part by the KIT start-up grant 2066995641,the ASPERA project BMBF 05A11VKA, the Helmholtz-University Young Investigators Group VH-NG-413 and the National Centre for Research and Development (NCBiR).","pages":[6]}],"string":"[\n {\n \"title\": \"Radar reflection off extensive air showers\",\n \"content\": \"J. Stasielak 1 , a , S. Baur 2 , M. Bertaina 3 , J. Blumer 2 , A. Chiavassa 3 , R. Engel 2 , A. Haungs 2 , T. Huege 2 , K.-H. Kampert 4 , H. Klages 2 , M. Kleifges 2 , O. Kromer 2 , M. Ludwig 2 , S. Mathys 4 , P. Neunteufel 2 , J. Pekala 1 , J. Rautenberg 4 , M. Riegel 2 , M. Roth 2 , F. Salamida 5 , H. Schieler 2 , R. ˇ Sm'ıda 2 , M. Unger 2 , M. Weber 2 , F. Werner 2 , H. Wilczy'nski 1 , and J. Wochele 2 Abstract. We investigate the possibility of detecting extensive air showers by the radar technique. Considering a bistatic radar system and di ff erent shower geometries, we simulate reflection of radio waves o ff the static plasma produced by the shower in the air. Using the Thomson cross-section for radio wave reflection, we obtain the time evolution of the signal received by the antennas. The frequency upshift of the radar echo and the power received are studied to verify the feasibility of the radar detection technique.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"Traditional techniques of extensive air shower (EAS) detection include counting the shower particles at the ground level and measuring fluorescence light from the excited nitrogen in the atmosphere. Detecting radio emission from the shower particles is a promising new technique that is currently in development. An alternative method is the radar technique, where a ground-based radio transmitter illuminates the ionization trail left behind the shower front and another ground-based antenna receives the scattered radio signal with its upshifted frequency. The concept of implementing a radar for cosmic ray detection was first introduced about 70 years ago by Blackett and Lovell [1]. However due to the negative results of dedicated experiments, this concept was forgotten for several decades. In recent years renewed attention has been attracted to this topic, see e.g. [2,3,4,5]. Experimental e ff orts to detect EAS using the radar technique were made by several groups, e.g. [6], the LAAS group [7], [8], the MARIACHI experiment [9], or the TARA project [10], however no detection was confirmed so far. In this paper we investigate the possibility of detecting EAS by the bistatic radar system. Especially, we are interested in verifying the feasibility of supplementing the CROME (Cosmic-Ray Observation via Microwave Emission) detector [11,12] with a radio transmitter and using its existing antennas as receivers for the radar echo. The CROME antenna setup consists of several microwave receivers for the frequency ranges 1 . 2 -1 . 7 (L band), 3 . 4 -4 . 2 (extended C band), and 10 . 7 -11 . 7 GHz (low Ku band). Since we expect large frequency upshifts of the radar echo and the CROME antennas are fine tuned to the GHz frequency band, we can narrow down our interest to the frequency range of 1 -100 MHz of the emitted radio waves. The concept of detecting EAS using radar technique is based on the principle of scattering radio waves o ff the static plasma produced in the atmosphere by the energetic particles of the shower. The locally produced plasma decays exponentially with the lifetime τ . According to Vidmar [13], for the\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"EPJ Web of Conferences\",\n \"content\": \"Figure 2 shows the time evolution of the region of integration v ol ( t ) (see equation (2)) for a shower with an energy of 10 18 eV heading towards transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude. As we can see, the volume over which we integrate the signal spans usually over the wide range of altitudes. This implies a high variation of the air density. Therefore, the refractive index and the Cherenkov angle can vary significantly over the considered region. This makes the whole analysis very complex.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"2 Radar reflection\",\n \"content\": \"A schematic diagram representing a bistatic radar system is shown in Figure 1. A ground-based radio transmitter (T) irradiates a disk-like shower. The front edge of the disk is an ionization front moving towards the ground with the speed of light and leaving created static plasma behind it. The radio signal is scattered by the free electrons in the ionization trail and subsequently received by the groundbased antenna (R). The geometry of the bistatic radar system is set up by the polar coordinates of the transmitter and the receiver, i.e. by the distances from the shower core to the transmitter ( dT ) and to the receiver ( dR ) together with the angles ϕ t and ϕ r . Let us consider the radar reflection from the disk element with coordinates ( rL , ϕ ), altitude h, and electron density ne . Its contribution to the radar echo at the receiver at time t can be expressed by where GT is the transmitter antenna gain, UT [V] is the magnitude of the transmitted field, k and ksc are the wave vectors of incoming and scattered radio wave, ϕ 0 is the initial phase of the emitted signal, d σ T / d Ω is the Thomson cross-section, s is the projection of the distance between the shower core and the considered disk element on the shower axis, α is the inclination angle of the reflected radio wave, and n is the refractive index of the air. By analogy to the setup of the CROME detector, we assume that the receiver is oriented vertically upwards. The signal received by the antenna at a given time t is sum of the signals scattered at di ff erent times and from di ff erent parts of the plasma disk. These individual contributions interfere with each other and only an integral over the whole volume v ol ( t ) from which they arrive simultaneously gives us the correct value. The total electric field strength of the radio wave at the receiver is given by Note that time t is defined in such a way that t = 0 coincides with the moment at which the shower hits the ground.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3 Frequency upshift\",\n \"content\": \"Despite the fact that the radio wave is scattered on the static plasma, the region in which the plasma is created moves with relativistic velocity. Therefore, we can observe the Doppler shift of the received radar echo. Figure 3 shows the upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d . The frequency upshift depends on the wave direction and the refractive index of air. It has the highest value for the case in which the viewing angle coincides with the Cherenkov cone. The typical fr is high enough to upshift the MHz signal into the GHz range. Note that fr only weakly depends on the distance from the shower core to the transmitter dT .\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4 Details of the simulation\",\n \"content\": \"The main assumptions and approximations are the following:\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"5 Power received by the detector antenna\",\n \"content\": \"The ratio of the 'instantaneous' power PR ( t ) received by the detector antenna to the power emitted by the transmitter PT is given by where The term R ( t ) is proportional to the electric field strength detected by the receiver. It can be used in the Fourier analysis to obtain the power spectrum of the recorded signal. The 'real' power received by the detector PR can be obtained by averaging R 2 ( t ) (according to the time resolution of the detector). The ratio PR / PT ∼ GTAR .\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"6 Results\",\n \"content\": \"Figure 4 shows the waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases the receiver is outside the Cherenkov cone (which is the most common case) and vertical showers heading towards the transmitter are considered. The time t = 0 coincides with the moment at which the shower hits the ground. As we can see, the frequency of the received signal is higher than that of the emitted one. Moreover, the amplitude grows with time, which is the geometrical e ff ect of the reflections from the lower parts of the atmosphere. For showers, where the receiver is inside the Cherenkov cone, the time sequence is reversed and the lower part of the shower is seen first. Therefore, the amplitude decreases with time. However, this is the case only for very inclined showers with cores up to several meters away from the receiver. The ratios of the power received by the detector to the emitted one ( PR / PT ) for vertical showers are given in table 1. It is evident that the strength of the signal decreases with the frequency ν . It is simply because the size of the region from which we get the coherent signal decreases with the wavelength and the negative interference cancels out the signal.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"7 Conclusions\",\n \"content\": \"The typical ratio of the received to the emitted power is in the range of 10 -10 - 10 -13 , which makes the detection of the radar reflection plausible. The C band ( ∼ 3 GHz) is particularly well-suited for this purpose because of very low noise ( < 10 K) in this frequency range. The CROME antennas supplemented with commercial high power MHz transmitter would create a radar system dedicated to EAS detection. A note should be added, that the shown time traces are for an infinite bandwidth detector -a realistic detector would only be able to detect the signal in narrow frequency range. Moreover, it will only see the shower for a certain fraction of its development. The performed analysis will help to choose the frequency of the emitted wave to optimize the detection of the radar echo. The work is in progress.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"8 Acknowledgments\",\n \"content\": \"It is our pleasure to acknowledge the interaction and collaboration with many colleagues from the KASCADE-Grande and Pierre Auger Collaboration. This work has been supported in part by the KIT start-up grant 2066995641,the ASPERA project BMBF 05A11VKA, the Helmholtz-University Young Investigators Group VH-NG-413 and the National Centre for Research and Development (NCBiR).\",\n \"pages\": [\n 6\n ]\n }\n]"}}},{"rowIdx":261,"cells":{"bibcode":{"kind":"string","value":"2013EPSC....8..507W"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1302.6425.pdf"},"content":{"kind":"string","value":"\nPhotometric stability analysis of the Exoplanet Characterisation Observatory\nI. P. Waldmann, 1 glyph[star] E. Pascale, 2 B. Swinyard, 1 , 3 G. Tinetti, 1 A. Amaral-Rogers 2\n7\n\nL. Spencer, 2 M. Tessenyi, 1 M. Ollivier 5 and V. Coud'e du Foresto 6 ,\n1 Dept. Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK\n2 School of Physics & Astronomy, Cardiff University, Cardiff, CF24 3AA, UK\n3 STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, OX11 0QX, UK\n4 Dept. Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK\n5 Institut d'Astrophysique Spatiale, Btiment 121, Universit'e de Paris-Sud, 91405 ORSAY Cedex, France\n6 Observatoire de Paris (LESIA), 5 place Jules Janssen, F-92190 Meudon, France\n7 Center for Space and Habitability, University of Bern, Sidlerstrasse 5, CH-3012, Bern, Switzerland\n\nABSTRACT\nPhotometric stability is a key requirement for time-resolved spectroscopic observations of transiting extrasolar planets. In the context of the Exoplanet Characterisation Observatory ( EChO ) mission design, we here present and investigate means of translating space-craft pointing instabilities as well as temperature fluctuation of its optical chain into an overall error budget of the exoplanetary spectrum to be retrieved. Given the instrument specifications as of date, we investigate the magnitudes of these photometric instabilities in the context of simulated observations of the exoplanet HD189733b secondary eclipse.\nKey words: space vehicles: instruments - instrumentation: spectrographs - techniques: spectroscopic - stars:planetary systems\n1 INTRODUCTION\nThe last decade has seen a surge in exoplanetary discoveries, with ∼ 850 planets confirmed (Schneider et al. 2011) and over two thousand Kepler candidates (Borucki et al. 2011; Batalha et al. 2013) waiting for confirmation. With the precise measurements of their masses and radii we have gained a staggering wealth of information for a plethora of targets and planetary types. In this large and growing consensus of foreign worlds, some afford us the opportunity of further characterisation. By studying the exoplanets' atmospheres, we can not only infer their chemical make-up but also constrain their climates, thermodynamical processes and formation histories. The sum total of this knowledge will allow us to understand planetary science and our own solar system in the context of a much larger picture.\nUsing Hubble , Spitzer as well as ground based facilities, the success of exoplanetary spectroscopy of transiting (e.g. Beaulieu et al. 2010, 2011; Berta et al. 2012; Charbonneau et al. 2002, 2008; Deming et al. 2005, 2007; Grillmair et al. 2007, 2008; Richardson et al. 2007; Snellen et al. 2008, 2010a,b; Bean et al. 2011; Stevenson et al. 2010; Swain et al. 2008, 2009a,b; Tinetti et al. 2007a,b, 2010a; Thatte et al. 2010; Mandell et al. 2011; Sing et al. 2009, 2011; Pont\net al. 2008; Burke et al. 2007, 2010; D'esert et al. 2011; Redfield et al. 2008; Waldmann et al. 2012, 2013; Crouzet et al. 2012; Brogi et al. 2012) as well as non-transiting planets (e.g. Janson et al. 2010; Currie et al. 2011) has been remarkable in recent years.\nFollowing these recent successes and in the frame of ESA's Cosmic Vision programme, the Exoplanet Characterisation Observatory ( EChO ) has been considered as mediumsized M3 mission candidate for launch in the 2022 - 2024 timeframe (Tinetti et al. 2012b). The current 'Phase-A study' space-mission concept is a 1.2 metre class telescope, passively cooled to ∼ 50 K and orbiting around the second Lagrangian Point (L2). The current baseline for the payload consists of four integrated spectrographs providing continuous spectral coverage from 0.5 - 16 µ m at resolution ranging from R ∼ 300 to 30. For a detailed description of the telescope and current payload design studied by our instrument consortium, we refer the reader to the literature (Puig et al. 2012; Tinetti et al. 2012a,b; Swinyard et al. 2012; Eccleston et al. 2012; Reess et al. 2012; Adriani et al. 2012; Zapata et al. 2012; Pascale et al. 2012; Focardi et al. 2012; Tessenyi et al. 2012). In order to observe the spectrum of the extrasolar planet, the EChO mission uses the time-resolved spectrophotometry method to observe transiting exoplanets. The spectrum of the planet is either seen in transmission when the planet transits in front of the star\nalong our line of sight or in emission when the planet's thermal contribution is lost during secondary eclipse when the planet disappears behind its host.\nBy taking consecutive short observations we follow the transit or secondary eclipse event of an extrasolar planet. This approach results in an exoplanetary lightcurve for each individual wavelength channel. The spectrum of the planet is then derived by model fitting said lightcurves and recording the eclipse depths at each wavelength bin. The amplitudes of these modulations on the mean of the exoplanetary lightcurve depths are unsurprisingly small. Taking the case of the secondary eclipse emission spectrum we typically find the planetary contrast to be of the order of 10 -3 for hotJuptiers in an orbit around a typical K0 star, and a much less favourable 10 -5 contrast for so called 'temperate SuperEarths' orbiting M-dwarf stars (Tessenyi et al. 2012). A mission such as EChO hence needs to provide a sensitivity that either matches or exceeds these contrasts over the duration of a transit/eclipse observation lasting one (e.g. GJ1214b, Charbonneau et al. 2009) to tens of hours (e.g. HD80606b, Fossey et al. 2009). The required instrument stability over said time spans can be achieved by providing a high photometric stability. There are three main factors that may introduce photometric variability over time and may limit the photometric stability. These are:\n(i) Pointing stability of the telescope:\nThe 1 σ pointing jitter of the satellite is currently baselined to be of the order of 10 milli-arcsec from 90s to 10h of continuous observation 1 . Effectively this defines the maximum Performance Reproducibility Error (PRE) over ten hours, specifying the reproducibility of the experiment. These pointing drifts manifest themselves in the observed data product via two mechanisms: 1) the drifting of the spectrum along the spectral axis of the detector, from here on referred to as 'spectral jitter'; 2) the drift of the spectrum along the spatial direction (or 'spatial jitter'). The effect of pointing jitter on the observed time series manifests as nonGaussian noise correlated among all detectors in all focal planes of the payload and is characterised by the powerspectrum of the telescope pointing. The amplitude of the resultant photometric scatter depends on the pointing jitter power spectrum, the PSF of the instruments, the detector intra-pixel response and the amplitude of the inter- pixel variations.\n(ii) Thermal stability of the optical-bench and mirrors:\nThermal emission of the instrument, baselined at 45K, is a source of photon noise and does hence not directly contribute to the photometric stability budge beyond an achievable signal to noise ratio (SNR) of the observation. However fluctuations in thermal emissions constitute a source of correlated noise in the observations and need to be maintained at amplitudes small compared to the science signal observed. Given a ∼ 45K black-body peaks in the far-IR, we are dominated by the Wien tail of the black body distribution, resulting in steep temperature gradients and stringent requirements on thermal stability in the long wavelength instruments of the EChO payload. Additionally to variable photon noise contribution, thermal variations may impact\ndark currents and responsivities of the detector which must be taken into account.\n(iii) Stellar noise and other temporal noise sources:\nWhilst beyond the control of the instrument design, stellar noise is an important source of temporal instability in exoplanetary time series measurements (Ballerini et al. 2011). This is particularly true for M dwarf host stars as well as many non-main sequence stars. Correction mechanisms of said fluctuations must and will be an integral part of the EChO science study but goes beyond the instrument photometric stability discussion presented here.\nAlong with the achievable SNR, the photometric stability of the instrument is the deciding factor for the success of missions or facilities aiming at transit spectroscopy.\nIn this paper we study the effects of the spatial/spectral jitter and thermal variability on the photometric stability budget of the instrument and telescope.\n2 ECHO AND ECHOSIM\nGiven EChO is currently in its Phase-A study phase, we concede that a noise budget and stability analysis, such as this one, can only be preliminary. Here and in Pascale et al. (in prep.), we present methodology used for the testing and optimisation of the current instrument design. The study presented in this paper draws on the end-to-end mission simulator EChOSim , which is discussed in detail in Pascale et al. (in prep.).\nIn the simulations of this paper we assume EChO to have a 1.2 metre diameter primary mirror off-axis telescope. The light beam is simultaneously fed into five spectrographs via dichroic beam splitters: Visible (Vis), short wavelength IR (SWIR), mid-wavelength IR (MWIR1 and MWIR2) and long-wavelength IR (LWIR) instruments covering the spectral range from 0.5 - 16 µ m.\nIn order to provide an end-to-end simulation of the observation, EChOSim incorporates a full simulation of the science payload including the telescope, as well as the astrophysical scene including Zodi emissions. EChOSim uses realistic mirror reflectivities and estimates the instrument transmission function for each channel as a function of wavelength. This includes transmission, optical throughput and spatial modulation transfer function. Using tabulated emissivity values as a function of wavelength, EChOSim also estimates the thermal emission spectrum of the several optical elements of the telescope. The transmission through the dichroic chain is simulated, and the incoming radiation is split among the 5 instruments assuming realistic transmittance and reflectivity data (Pascale et al., in prep.). Dispersion and detection by the focal plane array are simulated. The dispersion of the light over the focal plane is modelled by a linear dispersion law, which is related to the sampled spectral resolving power R .\nThe full focal plane illumination due to the backgrounds and the science signal's point spread function (PSF) is calculated and convolved with realistic intra-pixel variations (see section 5). Photon noise, read noise and pointing jitter noise are calculated on a pixel by pixel basis and time series of the observation are generated. These time series are then analysed and model fitted using Mandel & Agol (2002) analytical solutions and an adaptive Metropolis-Hastings Markov\n\n\n
Figure 1. Secondary eclipse lightcurve of a hot-Jupiter type exoplanet with eclipse duration of 720min (Mandel & Agol 2002). Noise at 10 -4 level was added.
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Figure 2. Time series of 6 orbits of a hot-Jupiter (akin to HD189733b, Torres et al. 2008). The deep troughs are limbdarkened transits (Mandel & Agol 2002; Claret 2000), smaller troughs are secondary eclipses and sinusoidal variations are due to the planetary phase curve as the planetary day-side rotates in and out of view. White noise of the level of 10 -4 was added.
\n\nChain Monte Carlo (MCMC) algorithm (Haario et al. 2001, 2006; Hastings 1970). For a more detailed list of the instrument parameters assumed in these simulations see table 1 and Pascale et al. (in prep.).\n3 FREQUENCY BANDS OF INTEREST\nFigure 1 shows an example of a secondary eclipse of a typical hot-Jupiter planet. Figure 2 shows the signal observed\n\n\n
Figure 3. Power spectra of time series shown in figure 2 for different orbital periods. Blue: Period = 120 days, Green = 2.21 days (akin to HD189733b), Red = 0.4 days. The sensitive frequency range extends from 1.9x10 -4 - 1.7x10 -3 Hz.
\n\nby EChO over the duration of 6 planetary orbits of a hotJupiter. From these figures it can easily be seen that timecorrelated noise has the greatest impact on the retrieved science at temporal variation frequencies comparable to those of the transit/eclipse event, or multiples thereof. Figure 3 shows the frequency domain representation of Figure 2 given a variety of orbital periods. Here the desired signal is contained in discrete frequencies and their respective overtones. Frequency ranges beyond these shown in Figure 3 can safely be filtered out using pass-band filters, or normalisation using low order polynomials in the time domain, without impairing the transit morphology. Given the range of transit periods observed and the goal of accurate ingress and egress mapping, we find the 'crucial frequency band' to be from 1.9x10 -4 to 1.7x10 -3 Hz, outside of which slow moving trends and high-frequency noise can effectively be filtered. This approach of slow moving trend removal is well tested for Kepler data (e.g. Gilliland et al. 2010). The overall critical frequency band for EChO is determined by the longest observation expected and the need to Nyquist sample the highest expected frequencies. We hence limit our simulations of pointing jitter or temperature fluctuations to this frequency range of interest.\n4 SPECTRAL JITTER\nAs described in the introduction, the telescope pointing jitter has the effect of translating the spectrum on the detector along both, the spatial and spectral directions. Whilst the real translation is a combination of these orthogonal components, we will consider the effect of both these translations independently of each other. With the observing approach of taking consecutive spectra over a given length of time, we obtain a flux time series for each pixel or resolution element ∆ λ . Should, during the course of the observation, the stellar spectra be shifted along the detector the same pixel will not observe the same stellar flux but that corresponding to the shifted stellar spectrum. If we are to construct a time series for each resolution element of the detector, we consequently need to re-sample the stellar spectra to a common grid. The ability to re-sample the individual stellar spectra to a uniform wavelength grid depends on how well we can determine these relative shifts.\nIn this section we will simulate a series of consecutively observed stellar spectra and shift each one of them according to a pointing jitter distribution derived from the Herschel Space Telescope. We will fit stellar absorption lines of these spectra and use the derived centroids to determine the accuracy with which we can determine the spectral drift. This is translated into a final post-correction flux error. It is worth noting that inter and intra pixel variations are less critical in this case as the the Nyquist sampling adopted at instrument level spreads each resolution element over at least two adjacent pixels.\n4.1 Synthesising pointing jitter from Herschel observations\nIn order to simulate the pointing jitter of the EChO mission as realistically as possible, we synthesised the pointing patterns of the Herschel Space Telescope for which real pointing\n
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Table 1. Simulation parameters for EChOSim
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Figure 4. Herschel pointing over a three hour continuous observation.
\n\ninformation exists (Swinyard private communication). Figure 4 shows Herschel pointings recorded for a three hour time period with a sampling frequency of ∼ 1 second. The pointing jitter distance for Herschel, ∆ Her , is given by the Pythagorean argument relative to their means:\n∆ Her = √ Ra 2 + Dec 2 . (1)\nAssuming the pointing jitter in Ra and Dec are normally (Gaussian) distributed, we can state that the probability distribution function (pdf) of ∆ Her , P (∆ Her ), is given by a Rayleigh distribution. The Rayleigh distribution is the pythagorean argument of two orthogonal Gaussian distributions. However, for a real system such as Herschel, the pointing jitter is described by a Gaussian and a non-Gaussian component. These non-Gaussian components propagate to P (∆ Her ) in the form of an increased skew and broader wings than those of a pure Rayleigh distribution. We can approximate P (∆ Her ) using the more general Weibull distribution of which the Rayleigh distribution is a special case. The Weibull distribution interpolates between a Rayleigh and an Exponential distribution and is hence ideally suited to de-\n\n\n
Figure 5. Probability distribution function of the pointing jitter length of Herschel
\n\nhe broader wings and skew of the Herschel pointing jitter pdf. The Weibull distribution is given by\nP wbl ( x ; τ, κ ) = { τ κ ( x κ ) κ -1 e -( x/τ ) κ x glyph[greaterorequalslant] 0 , 0 x < 0 , (2)\nwhere τ is known as the amplitude coefficient and κ as the distribution shape coefficient. Equation 2 reduces to an Exponential and a Rayleigh distribution for κ = 1 and κ = 2 respectively. The mean and the variance of the Weibull distribution are given by:\nµ wbl = τ Γ(1 + 1 /τ ) (3)\nσ 2 wbl = τ 2 Γ(1 + 2 /τ ) -µ 2 (4)\nwhere Γ is the gamma function (Riley et al. 2002).\n4.1.1 Scaling the Herschel pointing distribution\nFigure 5 shows P (∆ Her ) (blue) obtained from the pointing information in figure 4 and the best fitting Weibull distribution, P wbl ( · ), with τ = 247 . 71 and κ = 1 . 6547. The mean of the distribution is µ = 221 . 45 mas. In order to obtain the scaled distribution to a jitter amplitude of 10 mas predicted for EChO, we maintain the shape parameter at κ = 1 . 6547 and re-derive the scaling parameter, τ , for µ = 10 mas. This yields the pointing jitter distribution P (∆ Echo ) shown in figure 6.\nWe now randomly sample from P (∆ Echo ) to obtain\n\n\n
Figure 6. Synthesised probability distribution for EChO pointing jitter.
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Figure 7. Stellar spectrum between 1.0 and 2.5 µ m. Red: down sampled spectrum at R = 300; Blue: interpolated spectrum (see step 4); Inset: spectral line used to fit for the wavelength jitter; G1: flux gradient of stellar 'black-body' at 1.29 - 1.40 µ m; G2: flux gradient at 1.46 - 1.64 µ m.
\n\n∆ Echo for M number of observations in our simulated observing run. Assuming a resolving power of R = 300 for the NIR channel (1.0-2.5 µ m), we can calculate the spectral resolving power to be 0.0159nm/mas. Hence, we can express ∆ Echo as function of spectral wavelength drift and from hence forth denote the EChO jitter as ∆ m ( λ ), where m is the m'th spectrum observed.\n4.2 Applying pointing jitter to stellar spectra\nUsing the Phoenix 2 code we generated a stellar spectrum of a sun analogue and proceeded with the following steps:\n\n(i) The input spectrum was trimmed to a wavelength range of 1.0 - 2.5 µ m and re-sampled to a resolution of R = 300 at a central wavelength λ c = 1 . 75 µ m. This yields a wavelength coverage of δλ = 5 . 8 nm per pixel. We denote the spectrum by F star ( λ ), where λ is the wavelength.\n\n(ii) A single stellar absorption line was now selected (wavelength range: 1.19 - 1.21 µ m), and the spectrum consequently trimmed. We denote the trimmed spectrum as F line ( λ ) encompassing N data points in λ .\n\n(iii) Normally distributed random noise was added to the spectrum with the signal-to-noise (SNR) chosen to be the\n\nfraction of the maximal line amplitude over the standard deviation of the Gaussian noise component\nSNR = max | ( F line ( λ )) | σ gauss (5)\nThe resulting spectrum, F final ( λ ), is hence F final ( λ ) = F line ( λ )+ N ( σ, µ ), where N ( σ, µ ) is the Gaussian noise with σ variance and µ = 0 mean. We chose a SNR of 500 for this study.\n\n(iv) The spectrum F final ( λ ) was now interpolated by a factor S using a 'piecewise cubic Hermite interpolation' (Press et al. 2007) which assures the best fit to the original time series. Here we chose S = 100. This step is required to numerically implement sub-pixel drifts and does not impair or bias the results.\n(v) Steps iii & iv were repeated M times to create the M × N dimensional matrix X , where N is the number of points in λ and M was taken to be 100. One can think of M being the time axis containing spectra F final (1 , λ ) , F final (2 , λ ) . . . F final ( M,λ ), and N being the spectral axis of the matrix X .\n(vi) Each spectrum in X was now shifted by ∆ m ( λ ) along the wavelength axis randomly towards either the blue or red part of the spectrum.\n\n4.3 Fitting 'jittered' spectra\n\n(i) Now, each spectrum F final ( m,λ ) was fitted with a Voigt profile using a Levenberg-Marquardt minimisation algorithm (Press et al. 2007) and the resulting centroids, C m ( λ ), were recorded.\n(ii) The recorded centroids were subtracted from the motion jitter to give the fitting residual\nR m ( λ ) = √ (∆ m ( λ ) -C m ( λ )) 2 and the standard deviation of the residual σ R , which was recorded.\n(iii) Steps i to ii were repeated 100 times to obtain the sampled probability distribution of P( σ R ).\n\n4.4 Translating fitting residuals to total flux error\nWe can now use the fitting residual R m ( λ ) to calculate the observed flux, F obs ( m,λ ), per wavelength range δλ and time step m :\nF obs ( m,λ ) = λ + δλ + R m ( λ ) ∫ λ + R m ( λ ) F final ( m,λ )d λ (6)\nabove we can see that R m ( λ ) constitutes a change in the integration interval of each pixel with respect to the stellar spectrum. For the case where δλ → 0, we can calculate F obs ( m,λ ) using the linear approximation\nF obs ( m,λ ) = G ( R m ( λ ) -¯ R m ( λ )) + F final ( m,λ ) (7)\nG = d F star ( λ ) d λ (8)\n\n\n
Figure 8. Snapshot of EChOSim output for the MWIR detector. Top: Focal plane illumination including the science spectrum, instrumental and astrophysical backgrounds. Bottom: Plot of the total and individually contributing fluxes.
\n\nwhere ¯ R m ( λ ) is the mean of the fitting residual and was subtracted to account for equal amounts of positive and negative drifts, and G is the local gradient of the stellar spectrum. We can now express the resulting flux error due to residual drifts along the spectral direction as\nF err ( m,λ ) = G ( R m ( λ ) -¯ R m ( λ )) (9)\nIn section 8 we show the residual flux error F err for the gradients G1 and G2 shown in figure 7.\n5 SPATIAL JITTER\nWe have investigated the effect of spatial jitter as well as thermal variability. Whereas inter/intra pixel variations can largely be ignored in the case of spectral jitter, the detector pixel responses and variations can play a significant role in the computation of the spatial jitter which is caused by the movement of the spectral PSF along the spatial direction of the detector. EChOSim fully simulates the effect of inter and intra pixel variations on a spatially resolved detector grid. We here outline the spatial jitter calculations but refer the reader to Pascale et al. (in prep.) for a more detailed discussion. The dispersed signals are sampled by each detector assuming a wavelength-depended and instrument specific PSF which is convolved with the intra-pixel dependent response of the detector. Figure 8 is a snapshot of a standard diagnostic diagram returned by EChOSim . The top panel shows the focal plane illumination of the detector array. The bottom panel shows the individual flux contributions: astrophysical ( Q point ), zodiacal light ( Q zodi ), instrument thermal emissions ( Q optics ), and their combined total ( Q tot ).\nWe can describe the focal plane illumination of a detector with a monochromatic point source using the diffraction PSF pattern (Marc Ferlet priv. com.; Pascale et al. in prep.)\np ( x, y, λ ) = 1 2 πσ x σ y e -y 2 2 σ 2 y e -[ x -x 0 ( λ )] 2 2 σ 2 x (10)\nσ x = F # λ π √ 2 /K x σ y = F # λ π √ 2 /K y (11)\nwhere x and y are detector coordinates along the spectral and spatial axes respectively, K x and K y are parameters accounting for spatial aberrations (we assume no aberrations K x = K y ) and F # is the ratio between the effective focal length and the effective telescope diameter (see table 1).\nThe one dimensional PSF for the spatial direction can now be written as\np y ( y, λ ) = 1 σ y √ 2 π e -y 2 2 σy . (12)\nBarron et al. (2007) studied the intra-pixel response of IR detectors, and their best-fit model is analytically implemented in EChOSim . The cross section of the response along the spatial axis of the detector array is hence given by:\nF ( y ) = arctan { tanh [ 1 2 l d ( y + ∆ pix 2 )]} (13) -arctan { tanh [ 1 2 l d ( y -∆ pix 2 )]}\nwhere l d is the diffusion length. The sampled PSF along the spatial axis is then given by the convolution\np s ( y, λ ) = p y ( y, λ ) × F ( y ) (14)\nFinally the signal is sampled by the detector. The detector response is given by the convolution of the point source flux of the star-planet system, Q point ( λ, t ), with the spectrally dispersed PSFs\nQ point ( i, j, t ) = QE ( λ ) Q point ( λ, t ) × p s ( y, λ ) (15)\nwhere i and j are the detector pixel indices. Note that optical efficiencies don't explicitly appear in this equation as Q point ( λ, t ) already accounts the throughput budget, with the exception of the quantum efficiency, QE ( λ ) (Pascale et al., in prep.).\nσ p ( i, j, t ) glyph[similarequal] σ p f eff p s ∂p s ∂y Q point (16)\nwhere f eff is the plate scale (in µ m rad -1 ) and σ p is the pointing jitter (in radians per second) randomly sampled from the pointing distribution P (∆ Her ). Variations in photometric errors are estimated by consecutive runs for a range of pointing jitter amplitudes from zero to 200 milli-arcsec over a ten hour observing window. The observed spectrum is reconstructed using the EChOSim pipeline and the uncertainties on the reconstructed spectra are shown in figures 12 & 13 for two cases where PSFs of different sizes are used.\nGiven the current uncertainty over the exact nature of inter and intra-pixel variations we have not attempted to decorrelate the pointing jitter in post processing (e.g. Swain et al. 2008; Burke et al. 2010; Crouzet et al. 2012) which\nwould reduce the uncertainties reported in figures 12 & 13. We must hence consider this analysis as a conservative worst case scenario.\n6 THERMAL STABILITY\nWe have also investigated the impact of thermal emissions and fluctuations of the optics on the photometric stability of the reconstructed spectrum. Whilst thermal emissions are not directly a problem to photometric stability (source of white noise), they pose constraints on the temperature fluctuations allowed over the time span of an exoplanetary transit/eclipse in the reddest wavelengths. Here varying thermal emissions may produce detector counts that are equivalent or exceed the signal amplitude expected from a planetary eclipse. We calculate the thermal contribution using\nQ thermal ( i, j, t ) = π 4 ∆ 2 p f 2 # λ i + 1 2 L ∆ p /LD ∫ λ i + 1 2 L ∆ p /LD BB λ,T d λ (17)\nwhere L is the image size of the spectrometer slit in number of pixels, LD the lateral dispersion in mm, ∆ p is the pixel size, f # is the f-number and BB is the Planck function for a given temperature T and wavelength λ . EChOSim calculates the thermal emission of the instrument optics as well as the primary, secondary and tertiary mirrors separately and coherently propagates the resulting emission to detector counts. We have explored a temperature regime ranging from 40 - 60K with a grid size of 0.2K for both, the optical bench and the mirror temperatures. This provides us with an absolute scale of the thermal contributions, see results in section 8. We have now calculated the expected signal strength of a secondary eclipse of a HD189733b like hot-Jupiter, F planet , and for a given base temperature, T 0 , calculated the temperature variation, ∆ T , required to produce a detector signal of the same strength as F planet for a given wavelength λ :\n∆ F planet ( λ ) = f λ ( T 0 +∆ T ) -f λ ( T 0 ) (18)\nwhere f λ ( T ) is the functional form of the thermal emission/temperature relation for a given wavelength seen in figures 14 & 15. This can either be numerically approximated by interpolation to a fine enough grid or by fitting a high-order polynomial function to the data.\n7 TESTCASE\nWe have now calculated the spectral and spatial jitter contributions for the hot-Jupiter HD189733b (Torres et al. 2008). We currently have insufficient information on the expected thermal stability in the longest wavelength range so we do not include thermal stability in our test-case calculations. This said, as seen in section 8, the overall thermal stability is not a decisive factor for wavelengths shorter than ∼ 14 µ m.\nFor our simulations we use realistic Phoenix stellar models for the host stars. The planetary emission model is taken from Tessenyi et al. (2012). EChOSim fully solves\n\n\n
Figure 9. Residual centroid fitting error
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Figure 10. Probability distribution of fitting residual standard deviation σ R
\n\nthe dynamical star-planet system and computes the time resolved spectra by model fitting the generated time series.\nWe have now computed the emission spectra of HD189733b with the full noise contribution (shot noise, read noise, spatial/spectral jitter) as well as for the spatial jitter and spectral jitter only. This allows for the direct comparability of the noise contributions.\n8 RESULTS\nAs described in section 4 the 'jittered' stellar absorption line was fitted using a Voigt profile and the centroids recorded. The residual between the real and fitted spectral shifts are\n\n\n
Figure 11. Fraction of residual fitting flux F err over the total flux of the star F star for both spectral flux gradients shown in figure 7. G1 = -760 and G2 = 24 Wm -2 nm -1 .
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Figure 12. Simulation using EChOSim showing spatial jitter noise as fraction of total stellar flux, studied as a function of spectral wavelength and pointing jitter amplitude ranging from 0 200 milli-arcseconds. A PSF FWHM is assumed of 0.7 and 0.5 of the detector-pixel size. Figure 13 shows the same simulation with a PSF twice this size.
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Figure 13. Simulation using EChOSim showing spatial jitter noise as fraction of total stellar flux, studied as a function of spectral wavelength and pointing jitter amplitude ranging from 0 200 milli-arcseconds. A PSF FWHM is assumed of 1.4 and 1.0 of the detector-pixel size and twice the detector pixel size as stated in table 1.
\n\nshown in figure 9. The Monte-Carlo analysis of the standard deviation of the fitting residual is shown in figure 10. As expected, the probability distribution on the parameter σ R is largely Gaussian as we randomly sample from the pointing jitter distribution P (∆ Echo ). This also shows that the centroid fitting does not introduce biases in the pointing jitter correction. The fitting residual can now be translated to a total flux error using equation 9. Taking the ratio F err /F star we can derive the relative error due to residual jitter, figure 11. The flux error is dependent on the local stellar flux gradient. From figure 7, we derived two gradients: G1 = -\n\n\n
Figure 14. Detector counts (in e-/s) due to the telescope emission, as a function of temperature and wavelength.
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Figure 15. Detector counts (in e-/s) due to the optical bench emission, as a function of temperature and wavelength.
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Figure 16. The temperature fluctuation required (∆ T mirror ) for a given mirror temperature to mimic the variation of counts observed by a secondary eclipse of a HD189733b like hot-Jupiter
\n\n760 Wm -2 nm -1 and G2 = +24 Wm -2 nm -1 . For these two gradients, figure 11 shows that the relative flux error lies between 10 -6 ∼ 2 × 10 -5 . These values are of course larger for the Wien tail of the stellar black body with spectral jitter errors of ∼ 10 -4 at places (see figures 18 & 19).\nFigures 12 & 13 show the simulated spatial jitter contributions for a pointing jitter amplitude range or 0 - 200 milliarcseconds. As the photometric error resulting from spatial jitter is largely dependent on the PSF of the instrument and the pixel dimensions, we have calculated the photometric error for a PSF FWHM of 0.7 and 0.5 of the detector-pixel size (figure 12) and that for double these values (figure 13). As seen in the figures the spatial jitter is of the order of ∼ 10 -5 -10 -4 but significantly higher for the spectral ranges of the NIR instrument (2 - 5 µ m). This is to be expected as the SWIR instrument features a smaller pixel size.\nFigures 14 & 15 show the thermal contribution of the\n\n\n
Figure 17. The temperature fluctuation required (∆ T optics ) for a given optical bench temperature to mimic the variation of counts observed by a secondary eclipse of a HD189733b like hot-Jupiter
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Figure 18. Observations of a single secondary eclipse observation of HD189733b binned to 0.1 µ m bins. Inset: native resolution of the instrument.
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Figure 19. Error budget per wavelength in terms of planetary contrast ratio of HD189733b for spectrum in figure 18. Red: total flux error; blue: spectral jitter, green: spatial jitter.
\n\ntelescope (mirrors and optics) and the optical bench temperatures respectively. This can largely be regarded as negligible for wavelengths shortward of ∼ 14 µ m but significant at longer wavelengths and temperatures above 50K. As described in section 6, this thermal emission was translated into a temperature tolerance, ∆ T optics , showing the temperature change necessary to mimic the transit depth of a secondary eclipse feature of hot-Jupiter HD189733b at a given wavelength. These tolerances are shown in figures 16 & 17 for the telescope and optical bench temperatures respectively. We find that when the telescope and optical bench are at 45K, temperature fluctuations need to be below 6K for the telescope and 500mK for the bench in order to observe a hot Jupiter. More restrictive constraints would be required when fainter signals are involved.\nHaving investigated the pointing jitter contributions for given wavelengths and pointing jitter amplitudes, we show the simulated 'observations' of a single eclipse event of HD189733b in figure 18. The blue spectra's error-bars contain the full noise contribution. Figure 19 shows a breakdown of the individual error contributions. Here red stands for the total noise contribution (including read and shot noise), blue for the spectral jitter as calculated in section 4 and green for the spatial jitter contribution as calculated in section 5.\nIt can easily be seen that spectral jitter is important at the Wien tail of the stellar black body distribution and less so at the Rayleigh-Jeans tail. The spatial jitter noise depends on the individual detectors with the effect being strongest for the SWIR detector.\n9 DISCUSSION\nOne way to reduce spatial jitter noise is to use pixels that are small compared to the PSF. In this case the effect of the spatial jitter is only governed by the intra-pixel response, and using many pixels to sample the PSF will washout both the inter and intra response variations. A higher sampling of the PSF using more pixels will also positively impact the spectral jitter as tracking of the stellar lines becomes easier. However, this will only be realistic if the noise from the detectors is sufficiently low but will allow a significant amount of de-correlation as the PSF centroid can be tracked across the spatial dimension of the array. The simulations take into account realistic intra-pixel responses in the spectral band from MWIR and LWIR. At shorter wavelength, the simulations currently have to be considered worst case scenarios because the system is not diffraction limited at these wavelength.\nFigures 14 & 15 show the detected radiation in electrons/second from the instrument and telescope (separately) for a temperature range of 40 - 60K. Further studies will include additional instrumental effects such as mechanical vibrations, thermo-mechanical distortions, variable detector dark currents (assumed fixed with temperature in this study), detector responsitivity drifts and the effect of cosmic ray impacts. These studies will look at the extent the effect can be de-correlated from the timelines when additional information is available for data processing, such as temperature sensors monitoring the optics and telescope temperatures. We also plan to provide off-axis detectors to monitor the non stellar backgrounds and therefore provide a means of directly removing the background signals. Such background removal is particularly important for fainter sources. Whereas HD189733b is photon limited, observations of faint super-earths will likely be background limited and thermal and zodiacal light emissions need to be carefully accounted for.\n10 CONCLUSION\nIn this paper we present the methodology used for a photometric stability analysis of the EChO mission and asses the photometric stability given its current 'Phase-A' design specifications. We describe how spectral and spatial jitter due to space-craft pointing uncertainties are propagated to an uncertainty on the exoplanetary spectrum measured by EChO . We furthermore investigate tolerances on the thermal stability of the space-craft's optical path. The photometric stability error budget was estimated for a simulated secondary eclipse observations of the hot-Jupiter HD189733b. As the instrument parameters are not set in stone as of date, we have throughout considered the 'worst-case' assumptions only and photometric stability errors may significantly decrease as the instrument definition phase proceeds.\nACKNOWLEDGMENTS\nThis work is supported by STFC, NERC, UKSA, UCL and the Royal Society.\nREFERENCES\n
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\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Photometric stability is a key requirement for time-resolved spectroscopic observations of transiting extrasolar planets. In the context of the Exoplanet Characterisation Observatory ( EChO ) mission design, we here present and investigate means of translating space-craft pointing instabilities as well as temperature fluctuation of its optical chain into an overall error budget of the exoplanetary spectrum to be retrieved. Given the instrument specifications as of date, we investigate the magnitudes of these photometric instabilities in the context of simulated observations of the exoplanet HD189733b secondary eclipse. Key words: space vehicles: instruments - instrumentation: spectrographs - techniques: spectroscopic - stars:planetary systems","pages":[1]},{"title":"Photometric stability analysis of the Exoplanet Characterisation Observatory","content":"I. P. Waldmann, 1 glyph[star] E. Pascale, 2 B. Swinyard, 1 , 3 G. Tinetti, 1 A. Amaral-Rogers 2 7","pages":[1]},{"title":"1 INTRODUCTION","content":"The last decade has seen a surge in exoplanetary discoveries, with ∼ 850 planets confirmed (Schneider et al. 2011) and over two thousand Kepler candidates (Borucki et al. 2011; Batalha et al. 2013) waiting for confirmation. With the precise measurements of their masses and radii we have gained a staggering wealth of information for a plethora of targets and planetary types. In this large and growing consensus of foreign worlds, some afford us the opportunity of further characterisation. By studying the exoplanets' atmospheres, we can not only infer their chemical make-up but also constrain their climates, thermodynamical processes and formation histories. The sum total of this knowledge will allow us to understand planetary science and our own solar system in the context of a much larger picture. Using Hubble , Spitzer as well as ground based facilities, the success of exoplanetary spectroscopy of transiting (e.g. Beaulieu et al. 2010, 2011; Berta et al. 2012; Charbonneau et al. 2002, 2008; Deming et al. 2005, 2007; Grillmair et al. 2007, 2008; Richardson et al. 2007; Snellen et al. 2008, 2010a,b; Bean et al. 2011; Stevenson et al. 2010; Swain et al. 2008, 2009a,b; Tinetti et al. 2007a,b, 2010a; Thatte et al. 2010; Mandell et al. 2011; Sing et al. 2009, 2011; Pont et al. 2008; Burke et al. 2007, 2010; D'esert et al. 2011; Redfield et al. 2008; Waldmann et al. 2012, 2013; Crouzet et al. 2012; Brogi et al. 2012) as well as non-transiting planets (e.g. Janson et al. 2010; Currie et al. 2011) has been remarkable in recent years. Following these recent successes and in the frame of ESA's Cosmic Vision programme, the Exoplanet Characterisation Observatory ( EChO ) has been considered as mediumsized M3 mission candidate for launch in the 2022 - 2024 timeframe (Tinetti et al. 2012b). The current 'Phase-A study' space-mission concept is a 1.2 metre class telescope, passively cooled to ∼ 50 K and orbiting around the second Lagrangian Point (L2). The current baseline for the payload consists of four integrated spectrographs providing continuous spectral coverage from 0.5 - 16 µ m at resolution ranging from R ∼ 300 to 30. For a detailed description of the telescope and current payload design studied by our instrument consortium, we refer the reader to the literature (Puig et al. 2012; Tinetti et al. 2012a,b; Swinyard et al. 2012; Eccleston et al. 2012; Reess et al. 2012; Adriani et al. 2012; Zapata et al. 2012; Pascale et al. 2012; Focardi et al. 2012; Tessenyi et al. 2012). In order to observe the spectrum of the extrasolar planet, the EChO mission uses the time-resolved spectrophotometry method to observe transiting exoplanets. The spectrum of the planet is either seen in transmission when the planet transits in front of the star along our line of sight or in emission when the planet's thermal contribution is lost during secondary eclipse when the planet disappears behind its host. By taking consecutive short observations we follow the transit or secondary eclipse event of an extrasolar planet. This approach results in an exoplanetary lightcurve for each individual wavelength channel. The spectrum of the planet is then derived by model fitting said lightcurves and recording the eclipse depths at each wavelength bin. The amplitudes of these modulations on the mean of the exoplanetary lightcurve depths are unsurprisingly small. Taking the case of the secondary eclipse emission spectrum we typically find the planetary contrast to be of the order of 10 -3 for hotJuptiers in an orbit around a typical K0 star, and a much less favourable 10 -5 contrast for so called 'temperate SuperEarths' orbiting M-dwarf stars (Tessenyi et al. 2012). A mission such as EChO hence needs to provide a sensitivity that either matches or exceeds these contrasts over the duration of a transit/eclipse observation lasting one (e.g. GJ1214b, Charbonneau et al. 2009) to tens of hours (e.g. HD80606b, Fossey et al. 2009). The required instrument stability over said time spans can be achieved by providing a high photometric stability. There are three main factors that may introduce photometric variability over time and may limit the photometric stability. These are:","pages":[1,2]},{"title":"(i) Pointing stability of the telescope:","content":"The 1 σ pointing jitter of the satellite is currently baselined to be of the order of 10 milli-arcsec from 90s to 10h of continuous observation 1 . Effectively this defines the maximum Performance Reproducibility Error (PRE) over ten hours, specifying the reproducibility of the experiment. These pointing drifts manifest themselves in the observed data product via two mechanisms: 1) the drifting of the spectrum along the spectral axis of the detector, from here on referred to as 'spectral jitter'; 2) the drift of the spectrum along the spatial direction (or 'spatial jitter'). The effect of pointing jitter on the observed time series manifests as nonGaussian noise correlated among all detectors in all focal planes of the payload and is characterised by the powerspectrum of the telescope pointing. The amplitude of the resultant photometric scatter depends on the pointing jitter power spectrum, the PSF of the instruments, the detector intra-pixel response and the amplitude of the inter- pixel variations. (ii) Thermal stability of the optical-bench and mirrors: Thermal emission of the instrument, baselined at 45K, is a source of photon noise and does hence not directly contribute to the photometric stability budge beyond an achievable signal to noise ratio (SNR) of the observation. However fluctuations in thermal emissions constitute a source of correlated noise in the observations and need to be maintained at amplitudes small compared to the science signal observed. Given a ∼ 45K black-body peaks in the far-IR, we are dominated by the Wien tail of the black body distribution, resulting in steep temperature gradients and stringent requirements on thermal stability in the long wavelength instruments of the EChO payload. Additionally to variable photon noise contribution, thermal variations may impact dark currents and responsivities of the detector which must be taken into account. (iii) Stellar noise and other temporal noise sources: Whilst beyond the control of the instrument design, stellar noise is an important source of temporal instability in exoplanetary time series measurements (Ballerini et al. 2011). This is particularly true for M dwarf host stars as well as many non-main sequence stars. Correction mechanisms of said fluctuations must and will be an integral part of the EChO science study but goes beyond the instrument photometric stability discussion presented here. Along with the achievable SNR, the photometric stability of the instrument is the deciding factor for the success of missions or facilities aiming at transit spectroscopy. In this paper we study the effects of the spatial/spectral jitter and thermal variability on the photometric stability budget of the instrument and telescope.","pages":[2]},{"title":"2 ECHO AND ECHOSIM","content":"Given EChO is currently in its Phase-A study phase, we concede that a noise budget and stability analysis, such as this one, can only be preliminary. Here and in Pascale et al. (in prep.), we present methodology used for the testing and optimisation of the current instrument design. The study presented in this paper draws on the end-to-end mission simulator EChOSim , which is discussed in detail in Pascale et al. (in prep.). In the simulations of this paper we assume EChO to have a 1.2 metre diameter primary mirror off-axis telescope. The light beam is simultaneously fed into five spectrographs via dichroic beam splitters: Visible (Vis), short wavelength IR (SWIR), mid-wavelength IR (MWIR1 and MWIR2) and long-wavelength IR (LWIR) instruments covering the spectral range from 0.5 - 16 µ m. In order to provide an end-to-end simulation of the observation, EChOSim incorporates a full simulation of the science payload including the telescope, as well as the astrophysical scene including Zodi emissions. EChOSim uses realistic mirror reflectivities and estimates the instrument transmission function for each channel as a function of wavelength. This includes transmission, optical throughput and spatial modulation transfer function. Using tabulated emissivity values as a function of wavelength, EChOSim also estimates the thermal emission spectrum of the several optical elements of the telescope. The transmission through the dichroic chain is simulated, and the incoming radiation is split among the 5 instruments assuming realistic transmittance and reflectivity data (Pascale et al., in prep.). Dispersion and detection by the focal plane array are simulated. The dispersion of the light over the focal plane is modelled by a linear dispersion law, which is related to the sampled spectral resolving power R . The full focal plane illumination due to the backgrounds and the science signal's point spread function (PSF) is calculated and convolved with realistic intra-pixel variations (see section 5). Photon noise, read noise and pointing jitter noise are calculated on a pixel by pixel basis and time series of the observation are generated. These time series are then analysed and model fitted using Mandel & Agol (2002) analytical solutions and an adaptive Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm (Haario et al. 2001, 2006; Hastings 1970). For a more detailed list of the instrument parameters assumed in these simulations see table 1 and Pascale et al. (in prep.).","pages":[2,3]},{"title":"3 FREQUENCY BANDS OF INTEREST","content":"Figure 1 shows an example of a secondary eclipse of a typical hot-Jupiter planet. Figure 2 shows the signal observed by EChO over the duration of 6 planetary orbits of a hotJupiter. From these figures it can easily be seen that timecorrelated noise has the greatest impact on the retrieved science at temporal variation frequencies comparable to those of the transit/eclipse event, or multiples thereof. Figure 3 shows the frequency domain representation of Figure 2 given a variety of orbital periods. Here the desired signal is contained in discrete frequencies and their respective overtones. Frequency ranges beyond these shown in Figure 3 can safely be filtered out using pass-band filters, or normalisation using low order polynomials in the time domain, without impairing the transit morphology. Given the range of transit periods observed and the goal of accurate ingress and egress mapping, we find the 'crucial frequency band' to be from 1.9x10 -4 to 1.7x10 -3 Hz, outside of which slow moving trends and high-frequency noise can effectively be filtered. This approach of slow moving trend removal is well tested for Kepler data (e.g. Gilliland et al. 2010). The overall critical frequency band for EChO is determined by the longest observation expected and the need to Nyquist sample the highest expected frequencies. We hence limit our simulations of pointing jitter or temperature fluctuations to this frequency range of interest.","pages":[3]},{"title":"4 SPECTRAL JITTER","content":"As described in the introduction, the telescope pointing jitter has the effect of translating the spectrum on the detector along both, the spatial and spectral directions. Whilst the real translation is a combination of these orthogonal components, we will consider the effect of both these translations independently of each other. With the observing approach of taking consecutive spectra over a given length of time, we obtain a flux time series for each pixel or resolution element ∆ λ . Should, during the course of the observation, the stellar spectra be shifted along the detector the same pixel will not observe the same stellar flux but that corresponding to the shifted stellar spectrum. If we are to construct a time series for each resolution element of the detector, we consequently need to re-sample the stellar spectra to a common grid. The ability to re-sample the individual stellar spectra to a uniform wavelength grid depends on how well we can determine these relative shifts. In this section we will simulate a series of consecutively observed stellar spectra and shift each one of them according to a pointing jitter distribution derived from the Herschel Space Telescope. We will fit stellar absorption lines of these spectra and use the derived centroids to determine the accuracy with which we can determine the spectral drift. This is translated into a final post-correction flux error. It is worth noting that inter and intra pixel variations are less critical in this case as the the Nyquist sampling adopted at instrument level spreads each resolution element over at least two adjacent pixels.","pages":[3]},{"title":"4.1 Synthesising pointing jitter from Herschel observations","content":"In order to simulate the pointing jitter of the EChO mission as realistically as possible, we synthesised the pointing patterns of the Herschel Space Telescope for which real pointing information exists (Swinyard private communication). Figure 4 shows Herschel pointings recorded for a three hour time period with a sampling frequency of ∼ 1 second. The pointing jitter distance for Herschel, ∆ Her , is given by the Pythagorean argument relative to their means: Assuming the pointing jitter in Ra and Dec are normally (Gaussian) distributed, we can state that the probability distribution function (pdf) of ∆ Her , P (∆ Her ), is given by a Rayleigh distribution. The Rayleigh distribution is the pythagorean argument of two orthogonal Gaussian distributions. However, for a real system such as Herschel, the pointing jitter is described by a Gaussian and a non-Gaussian component. These non-Gaussian components propagate to P (∆ Her ) in the form of an increased skew and broader wings than those of a pure Rayleigh distribution. We can approximate P (∆ Her ) using the more general Weibull distribution of which the Rayleigh distribution is a special case. The Weibull distribution interpolates between a Rayleigh and an Exponential distribution and is hence ideally suited to de- he broader wings and skew of the Herschel pointing jitter pdf. The Weibull distribution is given by where τ is known as the amplitude coefficient and κ as the distribution shape coefficient. Equation 2 reduces to an Exponential and a Rayleigh distribution for κ = 1 and κ = 2 respectively. The mean and the variance of the Weibull distribution are given by: where Γ is the gamma function (Riley et al. 2002).","pages":[3,4]},{"title":"4.1.1 Scaling the Herschel pointing distribution","content":"Figure 5 shows P (∆ Her ) (blue) obtained from the pointing information in figure 4 and the best fitting Weibull distribution, P wbl ( · ), with τ = 247 . 71 and κ = 1 . 6547. The mean of the distribution is µ = 221 . 45 mas. In order to obtain the scaled distribution to a jitter amplitude of 10 mas predicted for EChO, we maintain the shape parameter at κ = 1 . 6547 and re-derive the scaling parameter, τ , for µ = 10 mas. This yields the pointing jitter distribution P (∆ Echo ) shown in figure 6. We now randomly sample from P (∆ Echo ) to obtain ∆ Echo for M number of observations in our simulated observing run. Assuming a resolving power of R = 300 for the NIR channel (1.0-2.5 µ m), we can calculate the spectral resolving power to be 0.0159nm/mas. Hence, we can express ∆ Echo as function of spectral wavelength drift and from hence forth denote the EChO jitter as ∆ m ( λ ), where m is the m'th spectrum observed.","pages":[4,5]},{"title":"4.2 Applying pointing jitter to stellar spectra","content":"Using the Phoenix 2 code we generated a stellar spectrum of a sun analogue and proceeded with the following steps: (ii) A single stellar absorption line was now selected (wavelength range: 1.19 - 1.21 µ m), and the spectrum consequently trimmed. We denote the trimmed spectrum as F line ( λ ) encompassing N data points in λ . fraction of the maximal line amplitude over the standard deviation of the Gaussian noise component The resulting spectrum, F final ( λ ), is hence F final ( λ ) = F line ( λ )+ N ( σ, µ ), where N ( σ, µ ) is the Gaussian noise with σ variance and µ = 0 mean. We chose a SNR of 500 for this study.","pages":[5]},{"title":"4.4 Translating fitting residuals to total flux error","content":"We can now use the fitting residual R m ( λ ) to calculate the observed flux, F obs ( m,λ ), per wavelength range δλ and time step m : above we can see that R m ( λ ) constitutes a change in the integration interval of each pixel with respect to the stellar spectrum. For the case where δλ → 0, we can calculate F obs ( m,λ ) using the linear approximation where ¯ R m ( λ ) is the mean of the fitting residual and was subtracted to account for equal amounts of positive and negative drifts, and G is the local gradient of the stellar spectrum. We can now express the resulting flux error due to residual drifts along the spectral direction as In section 8 we show the residual flux error F err for the gradients G1 and G2 shown in figure 7.","pages":[5,6]},{"title":"5 SPATIAL JITTER","content":"We have investigated the effect of spatial jitter as well as thermal variability. Whereas inter/intra pixel variations can largely be ignored in the case of spectral jitter, the detector pixel responses and variations can play a significant role in the computation of the spatial jitter which is caused by the movement of the spectral PSF along the spatial direction of the detector. EChOSim fully simulates the effect of inter and intra pixel variations on a spatially resolved detector grid. We here outline the spatial jitter calculations but refer the reader to Pascale et al. (in prep.) for a more detailed discussion. The dispersed signals are sampled by each detector assuming a wavelength-depended and instrument specific PSF which is convolved with the intra-pixel dependent response of the detector. Figure 8 is a snapshot of a standard diagnostic diagram returned by EChOSim . The top panel shows the focal plane illumination of the detector array. The bottom panel shows the individual flux contributions: astrophysical ( Q point ), zodiacal light ( Q zodi ), instrument thermal emissions ( Q optics ), and their combined total ( Q tot ). We can describe the focal plane illumination of a detector with a monochromatic point source using the diffraction PSF pattern (Marc Ferlet priv. com.; Pascale et al. in prep.) where x and y are detector coordinates along the spectral and spatial axes respectively, K x and K y are parameters accounting for spatial aberrations (we assume no aberrations K x = K y ) and F # is the ratio between the effective focal length and the effective telescope diameter (see table 1). The one dimensional PSF for the spatial direction can now be written as Barron et al. (2007) studied the intra-pixel response of IR detectors, and their best-fit model is analytically implemented in EChOSim . The cross section of the response along the spatial axis of the detector array is hence given by: where l d is the diffusion length. The sampled PSF along the spatial axis is then given by the convolution Finally the signal is sampled by the detector. The detector response is given by the convolution of the point source flux of the star-planet system, Q point ( λ, t ), with the spectrally dispersed PSFs where i and j are the detector pixel indices. Note that optical efficiencies don't explicitly appear in this equation as Q point ( λ, t ) already accounts the throughput budget, with the exception of the quantum efficiency, QE ( λ ) (Pascale et al., in prep.). where f eff is the plate scale (in µ m rad -1 ) and σ p is the pointing jitter (in radians per second) randomly sampled from the pointing distribution P (∆ Her ). Variations in photometric errors are estimated by consecutive runs for a range of pointing jitter amplitudes from zero to 200 milli-arcsec over a ten hour observing window. The observed spectrum is reconstructed using the EChOSim pipeline and the uncertainties on the reconstructed spectra are shown in figures 12 & 13 for two cases where PSFs of different sizes are used. Given the current uncertainty over the exact nature of inter and intra-pixel variations we have not attempted to decorrelate the pointing jitter in post processing (e.g. Swain et al. 2008; Burke et al. 2010; Crouzet et al. 2012) which would reduce the uncertainties reported in figures 12 & 13. We must hence consider this analysis as a conservative worst case scenario.","pages":[6,7]},{"title":"6 THERMAL STABILITY","content":"We have also investigated the impact of thermal emissions and fluctuations of the optics on the photometric stability of the reconstructed spectrum. Whilst thermal emissions are not directly a problem to photometric stability (source of white noise), they pose constraints on the temperature fluctuations allowed over the time span of an exoplanetary transit/eclipse in the reddest wavelengths. Here varying thermal emissions may produce detector counts that are equivalent or exceed the signal amplitude expected from a planetary eclipse. We calculate the thermal contribution using where L is the image size of the spectrometer slit in number of pixels, LD the lateral dispersion in mm, ∆ p is the pixel size, f # is the f-number and BB is the Planck function for a given temperature T and wavelength λ . EChOSim calculates the thermal emission of the instrument optics as well as the primary, secondary and tertiary mirrors separately and coherently propagates the resulting emission to detector counts. We have explored a temperature regime ranging from 40 - 60K with a grid size of 0.2K for both, the optical bench and the mirror temperatures. This provides us with an absolute scale of the thermal contributions, see results in section 8. We have now calculated the expected signal strength of a secondary eclipse of a HD189733b like hot-Jupiter, F planet , and for a given base temperature, T 0 , calculated the temperature variation, ∆ T , required to produce a detector signal of the same strength as F planet for a given wavelength λ : where f λ ( T ) is the functional form of the thermal emission/temperature relation for a given wavelength seen in figures 14 & 15. This can either be numerically approximated by interpolation to a fine enough grid or by fitting a high-order polynomial function to the data.","pages":[7]},{"title":"7 TESTCASE","content":"We have now calculated the spectral and spatial jitter contributions for the hot-Jupiter HD189733b (Torres et al. 2008). We currently have insufficient information on the expected thermal stability in the longest wavelength range so we do not include thermal stability in our test-case calculations. This said, as seen in section 8, the overall thermal stability is not a decisive factor for wavelengths shorter than ∼ 14 µ m. For our simulations we use realistic Phoenix stellar models for the host stars. The planetary emission model is taken from Tessenyi et al. (2012). EChOSim fully solves the dynamical star-planet system and computes the time resolved spectra by model fitting the generated time series. We have now computed the emission spectra of HD189733b with the full noise contribution (shot noise, read noise, spatial/spectral jitter) as well as for the spatial jitter and spectral jitter only. This allows for the direct comparability of the noise contributions.","pages":[7]},{"title":"8 RESULTS","content":"As described in section 4 the 'jittered' stellar absorption line was fitted using a Voigt profile and the centroids recorded. The residual between the real and fitted spectral shifts are shown in figure 9. The Monte-Carlo analysis of the standard deviation of the fitting residual is shown in figure 10. As expected, the probability distribution on the parameter σ R is largely Gaussian as we randomly sample from the pointing jitter distribution P (∆ Echo ). This also shows that the centroid fitting does not introduce biases in the pointing jitter correction. The fitting residual can now be translated to a total flux error using equation 9. Taking the ratio F err /F star we can derive the relative error due to residual jitter, figure 11. The flux error is dependent on the local stellar flux gradient. From figure 7, we derived two gradients: G1 = - 760 Wm -2 nm -1 and G2 = +24 Wm -2 nm -1 . For these two gradients, figure 11 shows that the relative flux error lies between 10 -6 ∼ 2 × 10 -5 . These values are of course larger for the Wien tail of the stellar black body with spectral jitter errors of ∼ 10 -4 at places (see figures 18 & 19). Figures 12 & 13 show the simulated spatial jitter contributions for a pointing jitter amplitude range or 0 - 200 milliarcseconds. As the photometric error resulting from spatial jitter is largely dependent on the PSF of the instrument and the pixel dimensions, we have calculated the photometric error for a PSF FWHM of 0.7 and 0.5 of the detector-pixel size (figure 12) and that for double these values (figure 13). As seen in the figures the spatial jitter is of the order of ∼ 10 -5 -10 -4 but significantly higher for the spectral ranges of the NIR instrument (2 - 5 µ m). This is to be expected as the SWIR instrument features a smaller pixel size. Figures 14 & 15 show the thermal contribution of the telescope (mirrors and optics) and the optical bench temperatures respectively. This can largely be regarded as negligible for wavelengths shortward of ∼ 14 µ m but significant at longer wavelengths and temperatures above 50K. As described in section 6, this thermal emission was translated into a temperature tolerance, ∆ T optics , showing the temperature change necessary to mimic the transit depth of a secondary eclipse feature of hot-Jupiter HD189733b at a given wavelength. These tolerances are shown in figures 16 & 17 for the telescope and optical bench temperatures respectively. We find that when the telescope and optical bench are at 45K, temperature fluctuations need to be below 6K for the telescope and 500mK for the bench in order to observe a hot Jupiter. More restrictive constraints would be required when fainter signals are involved. Having investigated the pointing jitter contributions for given wavelengths and pointing jitter amplitudes, we show the simulated 'observations' of a single eclipse event of HD189733b in figure 18. The blue spectra's error-bars contain the full noise contribution. Figure 19 shows a breakdown of the individual error contributions. Here red stands for the total noise contribution (including read and shot noise), blue for the spectral jitter as calculated in section 4 and green for the spatial jitter contribution as calculated in section 5. It can easily be seen that spectral jitter is important at the Wien tail of the stellar black body distribution and less so at the Rayleigh-Jeans tail. The spatial jitter noise depends on the individual detectors with the effect being strongest for the SWIR detector.","pages":[7,8,9]},{"title":"9 DISCUSSION","content":"One way to reduce spatial jitter noise is to use pixels that are small compared to the PSF. In this case the effect of the spatial jitter is only governed by the intra-pixel response, and using many pixels to sample the PSF will washout both the inter and intra response variations. A higher sampling of the PSF using more pixels will also positively impact the spectral jitter as tracking of the stellar lines becomes easier. However, this will only be realistic if the noise from the detectors is sufficiently low but will allow a significant amount of de-correlation as the PSF centroid can be tracked across the spatial dimension of the array. The simulations take into account realistic intra-pixel responses in the spectral band from MWIR and LWIR. At shorter wavelength, the simulations currently have to be considered worst case scenarios because the system is not diffraction limited at these wavelength. Figures 14 & 15 show the detected radiation in electrons/second from the instrument and telescope (separately) for a temperature range of 40 - 60K. Further studies will include additional instrumental effects such as mechanical vibrations, thermo-mechanical distortions, variable detector dark currents (assumed fixed with temperature in this study), detector responsitivity drifts and the effect of cosmic ray impacts. These studies will look at the extent the effect can be de-correlated from the timelines when additional information is available for data processing, such as temperature sensors monitoring the optics and telescope temperatures. We also plan to provide off-axis detectors to monitor the non stellar backgrounds and therefore provide a means of directly removing the background signals. Such background removal is particularly important for fainter sources. Whereas HD189733b is photon limited, observations of faint super-earths will likely be background limited and thermal and zodiacal light emissions need to be carefully accounted for.","pages":[9]},{"title":"10 CONCLUSION","content":"In this paper we present the methodology used for a photometric stability analysis of the EChO mission and asses the photometric stability given its current 'Phase-A' design specifications. We describe how spectral and spatial jitter due to space-craft pointing uncertainties are propagated to an uncertainty on the exoplanetary spectrum measured by EChO . We furthermore investigate tolerances on the thermal stability of the space-craft's optical path. The photometric stability error budget was estimated for a simulated secondary eclipse observations of the hot-Jupiter HD189733b. As the instrument parameters are not set in stone as of date, we have throughout considered the 'worst-case' assumptions only and photometric stability errors may significantly decrease as the instrument definition phase proceeds.","pages":[9]},{"title":"ACKNOWLEDGMENTS","content":"This work is supported by STFC, NERC, UKSA, UCL and the Royal Society.","pages":[10]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Photometric stability is a key requirement for time-resolved spectroscopic observations of transiting extrasolar planets. In the context of the Exoplanet Characterisation Observatory ( EChO ) mission design, we here present and investigate means of translating space-craft pointing instabilities as well as temperature fluctuation of its optical chain into an overall error budget of the exoplanetary spectrum to be retrieved. Given the instrument specifications as of date, we investigate the magnitudes of these photometric instabilities in the context of simulated observations of the exoplanet HD189733b secondary eclipse. Key words: space vehicles: instruments - instrumentation: spectrographs - techniques: spectroscopic - stars:planetary systems\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Photometric stability analysis of the Exoplanet Characterisation Observatory\",\n \"content\": \"I. P. Waldmann, 1 glyph[star] E. Pascale, 2 B. Swinyard, 1 , 3 G. Tinetti, 1 A. Amaral-Rogers 2 7\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 INTRODUCTION\",\n \"content\": \"The last decade has seen a surge in exoplanetary discoveries, with ∼ 850 planets confirmed (Schneider et al. 2011) and over two thousand Kepler candidates (Borucki et al. 2011; Batalha et al. 2013) waiting for confirmation. With the precise measurements of their masses and radii we have gained a staggering wealth of information for a plethora of targets and planetary types. In this large and growing consensus of foreign worlds, some afford us the opportunity of further characterisation. By studying the exoplanets' atmospheres, we can not only infer their chemical make-up but also constrain their climates, thermodynamical processes and formation histories. The sum total of this knowledge will allow us to understand planetary science and our own solar system in the context of a much larger picture. Using Hubble , Spitzer as well as ground based facilities, the success of exoplanetary spectroscopy of transiting (e.g. Beaulieu et al. 2010, 2011; Berta et al. 2012; Charbonneau et al. 2002, 2008; Deming et al. 2005, 2007; Grillmair et al. 2007, 2008; Richardson et al. 2007; Snellen et al. 2008, 2010a,b; Bean et al. 2011; Stevenson et al. 2010; Swain et al. 2008, 2009a,b; Tinetti et al. 2007a,b, 2010a; Thatte et al. 2010; Mandell et al. 2011; Sing et al. 2009, 2011; Pont et al. 2008; Burke et al. 2007, 2010; D'esert et al. 2011; Redfield et al. 2008; Waldmann et al. 2012, 2013; Crouzet et al. 2012; Brogi et al. 2012) as well as non-transiting planets (e.g. Janson et al. 2010; Currie et al. 2011) has been remarkable in recent years. Following these recent successes and in the frame of ESA's Cosmic Vision programme, the Exoplanet Characterisation Observatory ( EChO ) has been considered as mediumsized M3 mission candidate for launch in the 2022 - 2024 timeframe (Tinetti et al. 2012b). The current 'Phase-A study' space-mission concept is a 1.2 metre class telescope, passively cooled to ∼ 50 K and orbiting around the second Lagrangian Point (L2). The current baseline for the payload consists of four integrated spectrographs providing continuous spectral coverage from 0.5 - 16 µ m at resolution ranging from R ∼ 300 to 30. For a detailed description of the telescope and current payload design studied by our instrument consortium, we refer the reader to the literature (Puig et al. 2012; Tinetti et al. 2012a,b; Swinyard et al. 2012; Eccleston et al. 2012; Reess et al. 2012; Adriani et al. 2012; Zapata et al. 2012; Pascale et al. 2012; Focardi et al. 2012; Tessenyi et al. 2012). In order to observe the spectrum of the extrasolar planet, the EChO mission uses the time-resolved spectrophotometry method to observe transiting exoplanets. The spectrum of the planet is either seen in transmission when the planet transits in front of the star along our line of sight or in emission when the planet's thermal contribution is lost during secondary eclipse when the planet disappears behind its host. By taking consecutive short observations we follow the transit or secondary eclipse event of an extrasolar planet. This approach results in an exoplanetary lightcurve for each individual wavelength channel. The spectrum of the planet is then derived by model fitting said lightcurves and recording the eclipse depths at each wavelength bin. The amplitudes of these modulations on the mean of the exoplanetary lightcurve depths are unsurprisingly small. Taking the case of the secondary eclipse emission spectrum we typically find the planetary contrast to be of the order of 10 -3 for hotJuptiers in an orbit around a typical K0 star, and a much less favourable 10 -5 contrast for so called 'temperate SuperEarths' orbiting M-dwarf stars (Tessenyi et al. 2012). A mission such as EChO hence needs to provide a sensitivity that either matches or exceeds these contrasts over the duration of a transit/eclipse observation lasting one (e.g. GJ1214b, Charbonneau et al. 2009) to tens of hours (e.g. HD80606b, Fossey et al. 2009). The required instrument stability over said time spans can be achieved by providing a high photometric stability. There are three main factors that may introduce photometric variability over time and may limit the photometric stability. These are:\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"(i) Pointing stability of the telescope:\",\n \"content\": \"The 1 σ pointing jitter of the satellite is currently baselined to be of the order of 10 milli-arcsec from 90s to 10h of continuous observation 1 . Effectively this defines the maximum Performance Reproducibility Error (PRE) over ten hours, specifying the reproducibility of the experiment. These pointing drifts manifest themselves in the observed data product via two mechanisms: 1) the drifting of the spectrum along the spectral axis of the detector, from here on referred to as 'spectral jitter'; 2) the drift of the spectrum along the spatial direction (or 'spatial jitter'). The effect of pointing jitter on the observed time series manifests as nonGaussian noise correlated among all detectors in all focal planes of the payload and is characterised by the powerspectrum of the telescope pointing. The amplitude of the resultant photometric scatter depends on the pointing jitter power spectrum, the PSF of the instruments, the detector intra-pixel response and the amplitude of the inter- pixel variations. (ii) Thermal stability of the optical-bench and mirrors: Thermal emission of the instrument, baselined at 45K, is a source of photon noise and does hence not directly contribute to the photometric stability budge beyond an achievable signal to noise ratio (SNR) of the observation. However fluctuations in thermal emissions constitute a source of correlated noise in the observations and need to be maintained at amplitudes small compared to the science signal observed. Given a ∼ 45K black-body peaks in the far-IR, we are dominated by the Wien tail of the black body distribution, resulting in steep temperature gradients and stringent requirements on thermal stability in the long wavelength instruments of the EChO payload. Additionally to variable photon noise contribution, thermal variations may impact dark currents and responsivities of the detector which must be taken into account. (iii) Stellar noise and other temporal noise sources: Whilst beyond the control of the instrument design, stellar noise is an important source of temporal instability in exoplanetary time series measurements (Ballerini et al. 2011). This is particularly true for M dwarf host stars as well as many non-main sequence stars. Correction mechanisms of said fluctuations must and will be an integral part of the EChO science study but goes beyond the instrument photometric stability discussion presented here. Along with the achievable SNR, the photometric stability of the instrument is the deciding factor for the success of missions or facilities aiming at transit spectroscopy. In this paper we study the effects of the spatial/spectral jitter and thermal variability on the photometric stability budget of the instrument and telescope.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2 ECHO AND ECHOSIM\",\n \"content\": \"Given EChO is currently in its Phase-A study phase, we concede that a noise budget and stability analysis, such as this one, can only be preliminary. Here and in Pascale et al. (in prep.), we present methodology used for the testing and optimisation of the current instrument design. The study presented in this paper draws on the end-to-end mission simulator EChOSim , which is discussed in detail in Pascale et al. (in prep.). In the simulations of this paper we assume EChO to have a 1.2 metre diameter primary mirror off-axis telescope. The light beam is simultaneously fed into five spectrographs via dichroic beam splitters: Visible (Vis), short wavelength IR (SWIR), mid-wavelength IR (MWIR1 and MWIR2) and long-wavelength IR (LWIR) instruments covering the spectral range from 0.5 - 16 µ m. In order to provide an end-to-end simulation of the observation, EChOSim incorporates a full simulation of the science payload including the telescope, as well as the astrophysical scene including Zodi emissions. EChOSim uses realistic mirror reflectivities and estimates the instrument transmission function for each channel as a function of wavelength. This includes transmission, optical throughput and spatial modulation transfer function. Using tabulated emissivity values as a function of wavelength, EChOSim also estimates the thermal emission spectrum of the several optical elements of the telescope. The transmission through the dichroic chain is simulated, and the incoming radiation is split among the 5 instruments assuming realistic transmittance and reflectivity data (Pascale et al., in prep.). Dispersion and detection by the focal plane array are simulated. The dispersion of the light over the focal plane is modelled by a linear dispersion law, which is related to the sampled spectral resolving power R . The full focal plane illumination due to the backgrounds and the science signal's point spread function (PSF) is calculated and convolved with realistic intra-pixel variations (see section 5). Photon noise, read noise and pointing jitter noise are calculated on a pixel by pixel basis and time series of the observation are generated. These time series are then analysed and model fitted using Mandel & Agol (2002) analytical solutions and an adaptive Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm (Haario et al. 2001, 2006; Hastings 1970). For a more detailed list of the instrument parameters assumed in these simulations see table 1 and Pascale et al. (in prep.).\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3 FREQUENCY BANDS OF INTEREST\",\n \"content\": \"Figure 1 shows an example of a secondary eclipse of a typical hot-Jupiter planet. Figure 2 shows the signal observed by EChO over the duration of 6 planetary orbits of a hotJupiter. From these figures it can easily be seen that timecorrelated noise has the greatest impact on the retrieved science at temporal variation frequencies comparable to those of the transit/eclipse event, or multiples thereof. Figure 3 shows the frequency domain representation of Figure 2 given a variety of orbital periods. Here the desired signal is contained in discrete frequencies and their respective overtones. Frequency ranges beyond these shown in Figure 3 can safely be filtered out using pass-band filters, or normalisation using low order polynomials in the time domain, without impairing the transit morphology. Given the range of transit periods observed and the goal of accurate ingress and egress mapping, we find the 'crucial frequency band' to be from 1.9x10 -4 to 1.7x10 -3 Hz, outside of which slow moving trends and high-frequency noise can effectively be filtered. This approach of slow moving trend removal is well tested for Kepler data (e.g. Gilliland et al. 2010). The overall critical frequency band for EChO is determined by the longest observation expected and the need to Nyquist sample the highest expected frequencies. We hence limit our simulations of pointing jitter or temperature fluctuations to this frequency range of interest.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4 SPECTRAL JITTER\",\n \"content\": \"As described in the introduction, the telescope pointing jitter has the effect of translating the spectrum on the detector along both, the spatial and spectral directions. Whilst the real translation is a combination of these orthogonal components, we will consider the effect of both these translations independently of each other. With the observing approach of taking consecutive spectra over a given length of time, we obtain a flux time series for each pixel or resolution element ∆ λ . Should, during the course of the observation, the stellar spectra be shifted along the detector the same pixel will not observe the same stellar flux but that corresponding to the shifted stellar spectrum. If we are to construct a time series for each resolution element of the detector, we consequently need to re-sample the stellar spectra to a common grid. The ability to re-sample the individual stellar spectra to a uniform wavelength grid depends on how well we can determine these relative shifts. In this section we will simulate a series of consecutively observed stellar spectra and shift each one of them according to a pointing jitter distribution derived from the Herschel Space Telescope. We will fit stellar absorption lines of these spectra and use the derived centroids to determine the accuracy with which we can determine the spectral drift. This is translated into a final post-correction flux error. It is worth noting that inter and intra pixel variations are less critical in this case as the the Nyquist sampling adopted at instrument level spreads each resolution element over at least two adjacent pixels.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4.1 Synthesising pointing jitter from Herschel observations\",\n \"content\": \"In order to simulate the pointing jitter of the EChO mission as realistically as possible, we synthesised the pointing patterns of the Herschel Space Telescope for which real pointing information exists (Swinyard private communication). Figure 4 shows Herschel pointings recorded for a three hour time period with a sampling frequency of ∼ 1 second. The pointing jitter distance for Herschel, ∆ Her , is given by the Pythagorean argument relative to their means: Assuming the pointing jitter in Ra and Dec are normally (Gaussian) distributed, we can state that the probability distribution function (pdf) of ∆ Her , P (∆ Her ), is given by a Rayleigh distribution. The Rayleigh distribution is the pythagorean argument of two orthogonal Gaussian distributions. However, for a real system such as Herschel, the pointing jitter is described by a Gaussian and a non-Gaussian component. These non-Gaussian components propagate to P (∆ Her ) in the form of an increased skew and broader wings than those of a pure Rayleigh distribution. We can approximate P (∆ Her ) using the more general Weibull distribution of which the Rayleigh distribution is a special case. The Weibull distribution interpolates between a Rayleigh and an Exponential distribution and is hence ideally suited to de- he broader wings and skew of the Herschel pointing jitter pdf. The Weibull distribution is given by where τ is known as the amplitude coefficient and κ as the distribution shape coefficient. Equation 2 reduces to an Exponential and a Rayleigh distribution for κ = 1 and κ = 2 respectively. The mean and the variance of the Weibull distribution are given by: where Γ is the gamma function (Riley et al. 2002).\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"4.1.1 Scaling the Herschel pointing distribution\",\n \"content\": \"Figure 5 shows P (∆ Her ) (blue) obtained from the pointing information in figure 4 and the best fitting Weibull distribution, P wbl ( · ), with τ = 247 . 71 and κ = 1 . 6547. The mean of the distribution is µ = 221 . 45 mas. In order to obtain the scaled distribution to a jitter amplitude of 10 mas predicted for EChO, we maintain the shape parameter at κ = 1 . 6547 and re-derive the scaling parameter, τ , for µ = 10 mas. This yields the pointing jitter distribution P (∆ Echo ) shown in figure 6. We now randomly sample from P (∆ Echo ) to obtain ∆ Echo for M number of observations in our simulated observing run. Assuming a resolving power of R = 300 for the NIR channel (1.0-2.5 µ m), we can calculate the spectral resolving power to be 0.0159nm/mas. Hence, we can express ∆ Echo as function of spectral wavelength drift and from hence forth denote the EChO jitter as ∆ m ( λ ), where m is the m'th spectrum observed.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"4.2 Applying pointing jitter to stellar spectra\",\n \"content\": \"Using the Phoenix 2 code we generated a stellar spectrum of a sun analogue and proceeded with the following steps: (ii) A single stellar absorption line was now selected (wavelength range: 1.19 - 1.21 µ m), and the spectrum consequently trimmed. We denote the trimmed spectrum as F line ( λ ) encompassing N data points in λ . fraction of the maximal line amplitude over the standard deviation of the Gaussian noise component The resulting spectrum, F final ( λ ), is hence F final ( λ ) = F line ( λ )+ N ( σ, µ ), where N ( σ, µ ) is the Gaussian noise with σ variance and µ = 0 mean. We chose a SNR of 500 for this study.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"4.4 Translating fitting residuals to total flux error\",\n \"content\": \"We can now use the fitting residual R m ( λ ) to calculate the observed flux, F obs ( m,λ ), per wavelength range δλ and time step m : above we can see that R m ( λ ) constitutes a change in the integration interval of each pixel with respect to the stellar spectrum. For the case where δλ → 0, we can calculate F obs ( m,λ ) using the linear approximation where ¯ R m ( λ ) is the mean of the fitting residual and was subtracted to account for equal amounts of positive and negative drifts, and G is the local gradient of the stellar spectrum. We can now express the resulting flux error due to residual drifts along the spectral direction as In section 8 we show the residual flux error F err for the gradients G1 and G2 shown in figure 7.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"5 SPATIAL JITTER\",\n \"content\": \"We have investigated the effect of spatial jitter as well as thermal variability. Whereas inter/intra pixel variations can largely be ignored in the case of spectral jitter, the detector pixel responses and variations can play a significant role in the computation of the spatial jitter which is caused by the movement of the spectral PSF along the spatial direction of the detector. EChOSim fully simulates the effect of inter and intra pixel variations on a spatially resolved detector grid. We here outline the spatial jitter calculations but refer the reader to Pascale et al. (in prep.) for a more detailed discussion. The dispersed signals are sampled by each detector assuming a wavelength-depended and instrument specific PSF which is convolved with the intra-pixel dependent response of the detector. Figure 8 is a snapshot of a standard diagnostic diagram returned by EChOSim . The top panel shows the focal plane illumination of the detector array. The bottom panel shows the individual flux contributions: astrophysical ( Q point ), zodiacal light ( Q zodi ), instrument thermal emissions ( Q optics ), and their combined total ( Q tot ). We can describe the focal plane illumination of a detector with a monochromatic point source using the diffraction PSF pattern (Marc Ferlet priv. com.; Pascale et al. in prep.) where x and y are detector coordinates along the spectral and spatial axes respectively, K x and K y are parameters accounting for spatial aberrations (we assume no aberrations K x = K y ) and F # is the ratio between the effective focal length and the effective telescope diameter (see table 1). The one dimensional PSF for the spatial direction can now be written as Barron et al. (2007) studied the intra-pixel response of IR detectors, and their best-fit model is analytically implemented in EChOSim . The cross section of the response along the spatial axis of the detector array is hence given by: where l d is the diffusion length. The sampled PSF along the spatial axis is then given by the convolution Finally the signal is sampled by the detector. The detector response is given by the convolution of the point source flux of the star-planet system, Q point ( λ, t ), with the spectrally dispersed PSFs where i and j are the detector pixel indices. Note that optical efficiencies don't explicitly appear in this equation as Q point ( λ, t ) already accounts the throughput budget, with the exception of the quantum efficiency, QE ( λ ) (Pascale et al., in prep.). where f eff is the plate scale (in µ m rad -1 ) and σ p is the pointing jitter (in radians per second) randomly sampled from the pointing distribution P (∆ Her ). Variations in photometric errors are estimated by consecutive runs for a range of pointing jitter amplitudes from zero to 200 milli-arcsec over a ten hour observing window. The observed spectrum is reconstructed using the EChOSim pipeline and the uncertainties on the reconstructed spectra are shown in figures 12 & 13 for two cases where PSFs of different sizes are used. Given the current uncertainty over the exact nature of inter and intra-pixel variations we have not attempted to decorrelate the pointing jitter in post processing (e.g. Swain et al. 2008; Burke et al. 2010; Crouzet et al. 2012) which would reduce the uncertainties reported in figures 12 & 13. We must hence consider this analysis as a conservative worst case scenario.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"6 THERMAL STABILITY\",\n \"content\": \"We have also investigated the impact of thermal emissions and fluctuations of the optics on the photometric stability of the reconstructed spectrum. Whilst thermal emissions are not directly a problem to photometric stability (source of white noise), they pose constraints on the temperature fluctuations allowed over the time span of an exoplanetary transit/eclipse in the reddest wavelengths. Here varying thermal emissions may produce detector counts that are equivalent or exceed the signal amplitude expected from a planetary eclipse. We calculate the thermal contribution using where L is the image size of the spectrometer slit in number of pixels, LD the lateral dispersion in mm, ∆ p is the pixel size, f # is the f-number and BB is the Planck function for a given temperature T and wavelength λ . EChOSim calculates the thermal emission of the instrument optics as well as the primary, secondary and tertiary mirrors separately and coherently propagates the resulting emission to detector counts. We have explored a temperature regime ranging from 40 - 60K with a grid size of 0.2K for both, the optical bench and the mirror temperatures. This provides us with an absolute scale of the thermal contributions, see results in section 8. We have now calculated the expected signal strength of a secondary eclipse of a HD189733b like hot-Jupiter, F planet , and for a given base temperature, T 0 , calculated the temperature variation, ∆ T , required to produce a detector signal of the same strength as F planet for a given wavelength λ : where f λ ( T ) is the functional form of the thermal emission/temperature relation for a given wavelength seen in figures 14 & 15. This can either be numerically approximated by interpolation to a fine enough grid or by fitting a high-order polynomial function to the data.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"7 TESTCASE\",\n \"content\": \"We have now calculated the spectral and spatial jitter contributions for the hot-Jupiter HD189733b (Torres et al. 2008). We currently have insufficient information on the expected thermal stability in the longest wavelength range so we do not include thermal stability in our test-case calculations. This said, as seen in section 8, the overall thermal stability is not a decisive factor for wavelengths shorter than ∼ 14 µ m. For our simulations we use realistic Phoenix stellar models for the host stars. The planetary emission model is taken from Tessenyi et al. (2012). EChOSim fully solves the dynamical star-planet system and computes the time resolved spectra by model fitting the generated time series. We have now computed the emission spectra of HD189733b with the full noise contribution (shot noise, read noise, spatial/spectral jitter) as well as for the spatial jitter and spectral jitter only. This allows for the direct comparability of the noise contributions.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"8 RESULTS\",\n \"content\": \"As described in section 4 the 'jittered' stellar absorption line was fitted using a Voigt profile and the centroids recorded. The residual between the real and fitted spectral shifts are shown in figure 9. The Monte-Carlo analysis of the standard deviation of the fitting residual is shown in figure 10. As expected, the probability distribution on the parameter σ R is largely Gaussian as we randomly sample from the pointing jitter distribution P (∆ Echo ). This also shows that the centroid fitting does not introduce biases in the pointing jitter correction. The fitting residual can now be translated to a total flux error using equation 9. Taking the ratio F err /F star we can derive the relative error due to residual jitter, figure 11. The flux error is dependent on the local stellar flux gradient. From figure 7, we derived two gradients: G1 = - 760 Wm -2 nm -1 and G2 = +24 Wm -2 nm -1 . For these two gradients, figure 11 shows that the relative flux error lies between 10 -6 ∼ 2 × 10 -5 . These values are of course larger for the Wien tail of the stellar black body with spectral jitter errors of ∼ 10 -4 at places (see figures 18 & 19). Figures 12 & 13 show the simulated spatial jitter contributions for a pointing jitter amplitude range or 0 - 200 milliarcseconds. As the photometric error resulting from spatial jitter is largely dependent on the PSF of the instrument and the pixel dimensions, we have calculated the photometric error for a PSF FWHM of 0.7 and 0.5 of the detector-pixel size (figure 12) and that for double these values (figure 13). As seen in the figures the spatial jitter is of the order of ∼ 10 -5 -10 -4 but significantly higher for the spectral ranges of the NIR instrument (2 - 5 µ m). This is to be expected as the SWIR instrument features a smaller pixel size. Figures 14 & 15 show the thermal contribution of the telescope (mirrors and optics) and the optical bench temperatures respectively. This can largely be regarded as negligible for wavelengths shortward of ∼ 14 µ m but significant at longer wavelengths and temperatures above 50K. As described in section 6, this thermal emission was translated into a temperature tolerance, ∆ T optics , showing the temperature change necessary to mimic the transit depth of a secondary eclipse feature of hot-Jupiter HD189733b at a given wavelength. These tolerances are shown in figures 16 & 17 for the telescope and optical bench temperatures respectively. We find that when the telescope and optical bench are at 45K, temperature fluctuations need to be below 6K for the telescope and 500mK for the bench in order to observe a hot Jupiter. More restrictive constraints would be required when fainter signals are involved. Having investigated the pointing jitter contributions for given wavelengths and pointing jitter amplitudes, we show the simulated 'observations' of a single eclipse event of HD189733b in figure 18. The blue spectra's error-bars contain the full noise contribution. Figure 19 shows a breakdown of the individual error contributions. Here red stands for the total noise contribution (including read and shot noise), blue for the spectral jitter as calculated in section 4 and green for the spatial jitter contribution as calculated in section 5. It can easily be seen that spectral jitter is important at the Wien tail of the stellar black body distribution and less so at the Rayleigh-Jeans tail. The spatial jitter noise depends on the individual detectors with the effect being strongest for the SWIR detector.\",\n \"pages\": [\n 7,\n 8,\n 9\n ]\n },\n {\n \"title\": \"9 DISCUSSION\",\n \"content\": \"One way to reduce spatial jitter noise is to use pixels that are small compared to the PSF. In this case the effect of the spatial jitter is only governed by the intra-pixel response, and using many pixels to sample the PSF will washout both the inter and intra response variations. A higher sampling of the PSF using more pixels will also positively impact the spectral jitter as tracking of the stellar lines becomes easier. However, this will only be realistic if the noise from the detectors is sufficiently low but will allow a significant amount of de-correlation as the PSF centroid can be tracked across the spatial dimension of the array. The simulations take into account realistic intra-pixel responses in the spectral band from MWIR and LWIR. At shorter wavelength, the simulations currently have to be considered worst case scenarios because the system is not diffraction limited at these wavelength. Figures 14 & 15 show the detected radiation in electrons/second from the instrument and telescope (separately) for a temperature range of 40 - 60K. Further studies will include additional instrumental effects such as mechanical vibrations, thermo-mechanical distortions, variable detector dark currents (assumed fixed with temperature in this study), detector responsitivity drifts and the effect of cosmic ray impacts. These studies will look at the extent the effect can be de-correlated from the timelines when additional information is available for data processing, such as temperature sensors monitoring the optics and telescope temperatures. We also plan to provide off-axis detectors to monitor the non stellar backgrounds and therefore provide a means of directly removing the background signals. Such background removal is particularly important for fainter sources. Whereas HD189733b is photon limited, observations of faint super-earths will likely be background limited and thermal and zodiacal light emissions need to be carefully accounted for.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"10 CONCLUSION\",\n \"content\": \"In this paper we present the methodology used for a photometric stability analysis of the EChO mission and asses the photometric stability given its current 'Phase-A' design specifications. We describe how spectral and spatial jitter due to space-craft pointing uncertainties are propagated to an uncertainty on the exoplanetary spectrum measured by EChO . We furthermore investigate tolerances on the thermal stability of the space-craft's optical path. The photometric stability error budget was estimated for a simulated secondary eclipse observations of the hot-Jupiter HD189733b. As the instrument parameters are not set in stone as of date, we have throughout considered the 'worst-case' assumptions only and photometric stability errors may significantly decrease as the instrument definition phase proceeds.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"This work is supported by STFC, NERC, UKSA, UCL and the Royal Society.\",\n \"pages\": [\n 10\n ]\n }\n]"}}},{"rowIdx":262,"cells":{"bibcode":{"kind":"string","value":"2013ExA....35....1B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1208.0447.pdf"},"content":{"kind":"string","value":"\nThe Astro-WISE datacentric information system\nK. Begeman, A.N. Belikov, D.R. Boxhoorn, E.A. Valentijn\nReceived: date / Accepted: date\nAbstract In this paper we present the various concepts behind the Astro-WISE Information System. The concepts form a blueprint for general scientific information systems (WISE) which can satisfy a wide and challenging range of requirements for the data dissemination, storage and processing for various fields in science. We review the main features of the information system and its practical implementation.\nKeywords Data Grid · Grid Computing · Information System · Middleware\n1 Introduction\nDigital astronomical catalogues have been built from the very first moment information technology enabled this, e.g. the first Abell catalogue of clusters of galaxies was digitally prepared and printed in the days minus signs were not available in print [1]. As soon as digital scanning devices became available photographic material was scanned and first large image surveys were digitally published, such as ESO-LV [2] and the digitized sky surveys like DPOSS [3], succeeded by the Palomar-Quest Survey [4], in the 90's followed by CCD-based surveys such as the 2MASS survey [5], the Sloan Digital Sky Survey SDSS [6].\nThe rapid accumulation of astronomical digital data and its public dissemination was, compared to other disciplines, achieved at an early stage, thanks to an open and collaborative astronomical world community who adopted the FITS image data format as early as 1979 [7].\nThe CDS, initially Centre de Donn'ees Stellaires and later renamed into Centre de Donn'ees astronomiques de Strasbourg took the lead in Europe to collect and disseminate the ever growing data sets of a zoo of astronomical observatories and projects. VizieR webservices nowadays provide access to over 9000 catalogues. Numerous astronomical data centers followed, developing specialized services, for\nexample, the Infrared Processing and Analysis Center and the bibliographical SAO/NASA Astrophysics Data System.\nIn the early 2000's it was realized that the ever growing data volumes require new approaches: the community becomes the data provider and the International Virtual Observatory and its European branch, the European Virtual Observatory, developed standards and interfaces to allow individual data centers to publish their catalogs and images in a common framework providing worldwide access to users who can query, cross-match and visualize multiple databases via the internet. A highlight formed the overplotting of data from different experiments, like Xray satellites and optical ground-based observatories with some keystrokes on the Aladin interactive software sky atlas.\nWhile the Virtual Observatory focused on the dissemination of published data it was also realized in the early 2000's that the upcoming data deluge required a new approach to the handling of the data stream from the telescope to the science-ready result. Modern experiments not only involve more data, but also require increasing precision on the various calibrations. The dependency of the final result (sometimes only a few numbers, like cosmological parameter values) on time-variable calibrations of very large data sets, involving the evaluation by large research teams distributed in smaller groups over various sites sets the basic requirements to the system handling the data. While in classical systems data is delivered in various releases to the public, often re-processing the whole set with higher versions of the code, the high data rates of modern experiments require an alternative approach, where the up-to-date result is derived on demand by the user.\nThus, we set out to design and implement an integrated datacentric system Astro-WISE in which the processing, storage and administration is integrated in a single environment, providing a living system to both the data producers and the customers. Early reports of this development and implementation have been given in [8] and [9]. To reach this goal, traceability of each individual data item handled by pipelines or any piece of code is carefully maintained, every data item beyond pixel value is kept as Metadata and made persistent and distributed in a relational database with an object-oriented view, mapping all the dependencies.\n2 Datacentric Approach to Data Processing\nThe datacentric approach facilitates scientists to cooperate in data analysis and mining by means of the internet, nowadays referred to as e-science. The AstroWISE information system connects to an own developed distributed grid processing system (based on Distributed Processing Unit, DPU), but has been also connected to the EGEE/EGI/BiGGrid-Grid [10], particularly for the operations of the Lofar radio telescope. Nowadays, this aspect of the system is referred to as cloud computing, but the Astro-WISE datacentric approach at the same time facilitates the sharing and web hosting of all the data handled by the system.\nToday, we are just taking the first small steps towards the development of approaches and systems who could handle the upcoming data deluge. The way of managing and administration the data will be a key, and will require ever improving, refining and self-organizing approaches. The key notion is that a fruitful approach towards handling the data deluge focuses on the administration, modeling,\nstandardizing and tracking of the data rather than on the processing of the data by CPUs. This datacentric approach, in which the design of operational systems is driven by data management and data standard issues rather than data processing/cpu/pipelining issues. Processing systems turn into information systems. First attempts to combine archiving and processing in the 1980s include ESO's MIDAS table file system [11], integrating common tables both accessible in source code and by user interfaces like the MIDAS monitor/prompt. The MIDAS table file system allows both users and programmers to change its content and also registering the modifications in table keywords leading to archives ready for further review and (trend) analysis. Its development was essential for the production and photometric calibration of the images of the ESO-LV galaxies. However, the 600,000 fore- and background objects visible on the images, stars and galaxies, could only be handled with great difficulties inside the system, response times running into hours per operations, demonstrating the importance of built-in scalability.\nThis MIDAS Table file system marked a stepping stone to current datacentric approaches which are enabled by the introduction of modern object-oriented programming languages, data modeling tools like UML and XML and relational databases.\nWehave combined these views, systems and requirements in order to design and build integrated information systems which have the potential for self-organizing and enrichment with new data. Both self-organizing and enrichment are processes in time. In fact, the datacentric approach implies the detailed modeling and awareness of how things change in time. Important changes range from\n\nI. new data entering the system (ingest) to\nII. new or modified source code handling these data. In turn, the new data entering the system might be\nIII. subject to physical changes (e.g. the gain of a sensor), while source code is often modified on the basis of\nIV. advancements of human understanding of the physical changes - our model of the world also changes (e.g. the cause and modeling of the gain variations of the sensor).\n\nThe ideal information system would seamlessly cope with all these changes in time, thus creating a living long term digital preservation environment. Astro-WISE is one of the first systems attempting to reach this goal.\nFor several years many authors predicted the upcoming data avalanche due to the advancements in digital sensor technology. Obviously, this has now started, both scientific experiments such as LHC and Lofar are running into the tens of Petabytes data acquisition regimes, while text, imaging, genome and internet gathered data collections like e-Bay are exceeding similar volumes. This triggered the development of Grid infrastructure that is able to handle Petabytes of data ([12] and [13]).\nAll this has happened in an approximate 20 year time span when the first large digital imaging collection was published, ESO-LV [2], containing scans of 32,000 galaxy images, 4 Gbyte of data filling a room full with tapes and populating the very first version of optical media.\nBy now, the data avalanche is multidisciplinary and worldwide and involves all aspects of our society, from science to commerce, public services and health care. Digital data is gathered, processed, distributed and accessed at a continuously\ngrowing rate. And this is only the beginning: in, say, two decades, the same room full with 4 Gbyte of tapes 20 years ago, and 40 Petabyte of tapes today could possibly contain 40 Zetabytes (40 million Petabytes) of data. The growth of the data volume produced by various types of sensor networks met with the growth of the disk capacity and search for new methods in the increasing of the memory storage density (see [14]). In order to be useful all this data has to be organized, administrated, managed and distributed.\nIt took a billion years for cells to develop into extremely complex information systems around the DNA; it has resulted into systems and approaches with an incredible complexity with memory mapping, copying, filtering and distribution mechanisms of data including feedback. Compared to this, our present data organizing mechanisms in IT are still very simple and rudimentary and we are just making the first steps to living self-organizing systems and archives required to deal with and optimally use the ocean of digital data to come.\nHowever, also in the cell stochastic processes are important, and absence of organization 'from above' are countered by survival of the fittest type of mechanisms, which appear as self-organizing. Also in human endeavored IT, standards seldom come from above, an exception being the ASCII character standards, mandated in 1968 by U.S. President Lyndon B. Johnson to be used in all computers purchased by the United States federal government.\nI have also approved recommendations of the Secretary of Commerce regarding standards for recording the Standard Code for Information Interchange on magnetic tapes and paper tapes when they are used in computer operations. All computers and related equipment configurations brought into the Federal Government inventory on and after July 1, 1969, must have the capability to use the Standard Code for Information.\nInterchange and the formats prescribed by the magnetic tape and paper tape standards when these media are used.In practice, standards are often invented by companies or scientific communities which are discriminated in a complex survival of the fittest battle, which involves that many factors that it appears stochastic.\nThe view, or better requirement, for future self-organizing information systems and its archives is that it organizes itself for all these changes in time. It could be seen as a step towards the extremely complex and self-organizing information systems in a cell.\nThe core problem for the definition of an information system is the definition of information itself. The information can be an input data flow from sensor detectors, an archive of transactions, trends deduced from the analysis of some data and even results of the data modeling or simulations. As result, information systems will host a variety of subsystems defined by its purpose. Examples are a Decision Support System gathering and analyzing trends for a specific business use case (see [15]) or Geographical Information Systems with focus on geospatial data analysis and a visualization of the data.\nNevertheless, it is possible to define primary components of any Information System following simple requirements on storage, processing and sharing data. Such a system should include a data model, storage and data processing facilities together with user interfaces. In Astro-WISE we have decided to separate and store all data beyond pixel/measurement data in a database and all pixel/measurement data in files on independent storage media.\nThis paper does not touch details of the implementation of Astro-WISE information system which are described in other papers but describes the general\noutline of the Astro-WISE information system and specify features which make Astro-WISE unique. For the more in-depth review of particular aspects of AstroWISE we recommend to read corresponding papers on the optical pipeline implemented in Astro-WISE [16], quality control realised in Astro-WISE [17], the system of user interfaces and services [18], and the hardware solution for Astro-WISE and information systems created on the basis of Astro-WISE [19].\n3 Building a Scientific Information System\nThe development of the Astro-WISE information system, the first one which implemented WISE technology, started from the very practical challenge: enable a community of researchers distributed over the world to process the data of astronomical imaging survey of the OmegaCAM 256 Megapixel camera on ESO's VST telescope at Cerro Paranal (Chile). These scientists should be able to evaluate the quality of the data, apply a number of calibrations, share the data within the team and employ distributed resources of a Petabytes scale of data storage and Teraflops capacity in data processing.\nFrom this our basic requirements on the system are derived:\n\n* Scalability of the system: any part of the system, i.e., data storage, data processing, metadata management, should be scalable with the increase of incoming data and a number of users involved in the data processing. The system should be scalable with respect to the data processing algorithms and pipelines, allowing the implementation of new pipelines and derive improved results from the same raw or intermediate data with new algorithms. The scalability of data mining should be possible, i.e. the system should satisfy all possible kinds of requests; from the retrieval of a single data item by identifier to a complicated archive study involving multiple complex queries.\n* Distributed system: any derivation of a result or search for a result should be possible by different users at different sites where the system is implemented. This makes it possible to optimally use shared resources.\n* Traceability: all activity in the system should leave a clear footprint so that it will be possible to trace the origin of any changes in the data and find an algorithm, program and user who created a data item. This allows expert knowledge to be shared amongst all users of the system.\n* Adaptability. The system should be possible to adopt for a number of different scientific use-cases, providing resources, pipelines and expertise to perform a data processing according to user's interests on the same data set.\n\nThe most general requirements listed above results in a set of more detailed and specific requirements to different components of the information system. First, the ability to share data among a number of users becomes more valuable when all these users are working with the same set of standards for the data. Second, the requirement to trace any changes in the data processed by different users with different programs and pipelines implies that the system has a common approach to treat the data items on different stages of the data processing and keeps the record of changes in the data. As result the basic requirements to the system dictate the use of a common data model and the common standards for the data\nstorage. Below we will describe the common data model features defined from the requirements to the system.\nRequirements to the Data Model:\n\n-Petabyte systems can not work when anarchy is allowed at the data acquisition. Users should operate with the same standards for the raw measurements data acquired at the experiment. Moreover, this implies a careful definition of the data taking scenarios, observing templates in astronomy, scan scenarios on sensor systems both for the scientific measurements data and for the calibrations. The raw data ingested in the system should be fully and completely described in the system so that users do not need to involve detailed knowledge about the acquired data and can share it with other users of the system. The data model must provide complete data provenance , i.e., both external data ingested in the system and the data derived inside the system should be described with sufficient level of detail to be able to rederive them in the environment external to their original one.\n-The data model should describe all the data products, on all stages of the data processing For the user it should be possible to understand the origin of each data item and to trace the data item back to the raw data. The data model must provide full data lineage , i.e., each data product and literally each bit of information must be traced back to the origin and creator of the information.\n-The model should match all scientific use-cases and should allow the flexibility to modify source code without changing the model as long as this makes sense. Obviously, here boundaries will have to be determined by means of inside, thus human, knowledge. This is the most tedious part in the design of the information system, but in practice it turns out that the various processes are well defined and dictated by physics. The data model must be flexible to the changes in the data processing .\n-Requirements to trace changes in the data triggered by different versions of the processing software and the necessity to repeat the data processing due to the changes in the pipeline implies that all parameters that affect the derived result must be preserved in the system. The user should be able to reproduce the data item and to find a trend in the data due to the changes in the processing parameters or algorithm. At the same time the user should be able to reproduce already existing item to ensure the preservation of the data and software used previously. The data model must allow both data reprocessing and data reproduction , and the reprocessing should be done with the preservation of the previously produced versions of the data.\n\nWhen properly defined the data model allows to code, instantiate and trace the data processing for all scientific use-cases.\nAlso, the data model should allow to accommodate external data, as well as provide necessary comparison and checking with these external data sets.\nWhen the data model is carefully defined our approach is to convert this into a pipeline model, in turn translated into a hierarchy of Classes. These Classes are then mapped into a database. We originally did this with object-oriented databases like Objectivity, but currently we do this in a relational database and build an object oriented view in order to maintain transparency in the class model, and its dependencies.\n\n\n
Fig. 1 Top-level definition of Astro-WISE information system with the specification for each layer.
\n\nRequirements to Infrastructure. The data model should be shared among geographically remote sites of the system, as research groups at various locations are collaboratively adding and extracting data from the distributed system. The basic requirement is that in fact any component (database, file system, code base, CPUs) should possibly be present at different sites, for a number of reasons:\n\n-hardware resources between the sites can be shared,\n-avoiding single centralized components prevent an overload of resources and duties at that node,\n-redundancy of the system to single-site malfunctions,\n-sharing of resources between partners allows to reduce the overall cost of the system.\n\nFor example, the data ingestion to the system can be done in Garching, Germany, the data processing in Groningen, The Netherlands and the image analysis in Napoli, Italy.\nRequirements to the User Interfaces. The user should be able to trace the data processing, to retrieve data items and identify all its dependencies on other items (mostly calibrations) in the system, to initiate data processing of particular data items and to find a quality estimation for the data.\nAs we can see, all these requirements put together are more challenging than requirements on common business information systems. The key difference is that we allow everything, including the code base to change in time.\nIn the following chapter we outline how our requirements formed a layout for a scientific information system capable to share a huge data volume, to process these data and fulfill scientific use-cases for a number of users.\n4 WISE Concept\nFirst of all we have to clearly distinguish between two classes of data storage systems in Astronomy: proper archives containing fixed published data and infor-\nmation systems. The main difference can be found in an ability to work with the data and to change it during the storage. For an archive stored data is a stable entity which is provided to the user as it is, meanwhile the information system allows to the user to modify their own data creating user-specific version of an archive and share this version with other users.\nThe Astro-WISE approach, and its general multi-discipline WISE approach implements a scientific information system.\nThe usual approach to building an information system is to perceive three components: the data layer, business rules and the interfaces (see Fig. 1). In our case, we separate the data layer in two parts: the pure measurement data layer, hereafter data layer, and everything beyond that, hereafter Metadata layer, ranging from file sizes, to statistics of pixel values and detected events in the measurements (galaxy, particle, word). The Metadata layer allows to implement a data model, the data layer allows to store the data as files in a standard format, while business rules implemented in Python classes bind the metadata and the data layers. Interfaces provide user access to both the business rules and the data. In the case of AstroWISE business rules present in the Metadata layer (Data Definition Language used to create a data model), data layer (the on-the-fly compression of the data on data storage grid) and-the most apparent to the user-in a number of pipelines and programs which the user defines to process the data. This last part of business rules components we will call data processing layer.\nTo implement Astro-WISE we use abstractions of storage, processing and database capabilities as a basis for the infrastructure for each of the layers of the system. The metadata layer is realized in a relational DBMS through an abstraction of the required database functionality, the data layer is put on the Astro-WISE data storage grid through an abstraction of storage and the processing grid is used to connect the user with the data and metadata layers by a number of interfaces for the data processing layer. Separation of these three infrastructures plays a key role in the flexibility of the system, as we will see below.\nThe metadata layer implements a list of necessary functionalities:\n\n1. Inheritance of data objects . Using object-oriented programming, all objects within the system can inherit key properties of the parent object. All these properties are made persistent.\n2. Full lineage . The linking (associations or references, or joins) between object instances in the database is maintained completely. Each data item in the system can be traced back to its origin. The tracing of the data object can be both forward and backward. For example, it is possible to find which raw frames were used to find magnitudes, shapes and position for this particular source and, at the same time, which sources were extracted from that particular raw frame.\n3. Consistency . At each processing step, all processing parameters and the inputs which are used are kept within the system. Astro-WISE keeps the old versions of all data items along with all parameters used to produce them and all dependencies between objects.\n4. Embarrassingly parallel and distributed processing, the administration of asynchronous processing is recorded in the metadata layer in a natural way.\n\nOur requirements on distribution and multiple users propagate as key principles of the realization of metadata and data layers and business rules which form the core of the WISE approach:\n\n1. Component based software engineering (CBSE) . This is a modular approach to software development, each module can be developed independently and wrapped in the base language of the system (Python) to form a pipeline or workflow.\n2. An object-oriented common data model used throughout the system . This means that each module, application and pipeline will deal with the unified data model for the whole cycle of data processing from the raw data to the final data product.\n3. Persistence of all the data model objects . Each data product in the data processing chain is described as an object of a certain class and saved in the archive of the specific project along with the parameters used for data processing.\n\nThe Astro-WISE system is realized in the Python programming language. It allows to wrap any program into a Python module, library or class. The use of Python also allows to combine the principles of modular programming with objectoriented programming, so that each package in the system can be built and run independently with an object-oriented data model serving as glue between modules. At the same time, the logic behind pipelines and workflows in Astro-WISE allows the execution of part of the processing chain independently from the other parts. We will describe this approach in more detail in the example of optical image processing in Section 4.1.\nThe conceptual difference between Astro-WISE and other existing systems is that Astro-WISE moves from the usual for astronomy processing-centric approach to a data-centric approach. The data processing itself becomes an integral part of the archive.\nThe typical solution for the data processing and storage for Astronomy handles the data processing and the final data product delivered to the user as two completely separated entities. The data product usually is a result of the processing for the whole survey with fixed processing parameters used for the whole set of images. The user has access to reduced images and the catalog, both data products are stable within a 'release', which usually refers to the sky coverage performed by the survey.\nIn this way the survey data center provides the user with a specific version of the data product that will not change over time, which covers most of the science use-cases.\nNevertheless there are a number of use-cases which can not be satisfied by the 'standard' version. For example, in the search for objects like brown dwarfs or quasars it is important to lower the detection threshold which implies the reprocessing of the data. The user himself has to care about such reprocessing thereby reinventing the whole data processing system for the survey and involving his own resources. The Astro-WISE system allows to work on any use-case using the standard pipelines and performing programming on a minimal level - if this is necessary at all.\nFigure 1 shows principles of WISE concept in the binding together all layers of the information system. Data processing pipelines define use-cases and data model\n\n\n
Fig. 2 Implementation of WISE concept in the data storage. Each persistent Python class inherits interfaces to the metadata and data layers.
\n\nfor the system, which is implemented in the Python classes which wrap pipeline and define the common persistent data model objects. The data layer uses this data model, and interfaces are providing an access to the system for users. The common data model effectively brings together all layers of the information system.\nThe common data model can be modified by a mutial agreement of all users of the system. Any user can propose changes in the data model, and, if changes were accepted, the system administrator implements these changes in corresponding Python classes and in the database scheme.\nIn the next sections we will review in detail the infrastructural layers of AstroWISE and their implementation.\n4.1 Metadata Layer\nThe bulk of the data stored in Astro-WISE is stored in files of some format (FITS). Each file is registered in the Astro-WISE metadata database with the unique filename. Apart from just registering each file in the system the metadata database implements the important part of WISE approach - the common data model.\nFigure 2 shows the general outline for all data models implemented in informations systems based on WISE concept. two core classes DBObject and DataObject are parent classes for the class which is storing metadata persistently and the class which is storing the data in the file. These classes include interfaces to the metadata database and the data storage, the physical implementation of the metadata database and the data (files) storage can be different for different systems.\nLet us suppose that we wish to use Astro-WISE for the data processing of optical images. Figure 3 shows typical classes of data items used by the optical data processing pipeline to reduce the data from raw images to the final science ready\n\n\n
Fig. 3 A set of Astro-WISE classes for an optical data processing pipeline. Each class is a persistent one stored in the metadata database and linked to the file stored in Astro-WISE system.
\n\n\n\n
Fig. 4 Input, intermediate and final data products of the typical processing pipeline.
\n\ncatalogues. This data model, deduced from the pipeline and enhanced keeping in mind possible scientific use-cases, is a central part of the metadata layer.\nThe data model is implemented both in the relational database (currently Oracle 11g RAC is used) and in the hierarchy of Python classes. All core classes are made persistent, i.e., any change in the object is mirrored in the corresponding tables of the database. The method used for the implementing of the data lineage is the Persistent Object Hierarchy (Fig. 2). According to this method objects of Astro-WISE are made persistent recursively, all operations and attributes of the object are saved in the metadata database.\nAs an example let us explore the data reduction task - we will create a reduced image from the raw image on one CCD chip. First of all, we will select an image we wish to process:\nawe> raw = (RawScienceFrame.filename == 'WFI.2000-01-01T08:57:15.410_3.fits')[0]\nThe image is selected by browsing the metadata database for an object of class RawScienceFrame with filename WFI.2000-01-01T08:57:15.410_3.fits . The returned object raw was retrieved from the metadata database with all attributes. In the next step we will retrieve from the database all necessary calibration objects. The logic in the function select_for_raw allows to take the calibration files for the same night of observation ( 2000-01-01T08:57:15.410 ) or closest to this night.\nawe> hot = HotPixelMap.select_for_raw(raw)\nawe> cold = ColdPixelMap.select_for_raw(raw)\nawe> flat = MasterFlatFrame.select_for_raw(raw)\nawe> bias = BiasFrame.select_for_raw(raw)\nNow we can instantiate a new object - our target - a reduced image which will be based on all the images above:\nawe> reduced = ReducedScienceFrame() awe> reduced.raw = raw awe> reduced.hot = hot awe> reduced.cold = cold awe> reduced.bias = bias awe> reduced.flat = flat\nPlease, note that all images which will be used to create a new reduced image reduced are referenced through attributes of the new image. This new image should get a unique ID, which is a name of the file where the image will be stored.\nawe> reduced.set_filename()\nNow the image can be created invoking the make() method of the ReducedScienceFrame class.\nawe> reduced.make()\nThe make() produced a file with a new image which will be stored on one of the dataservers of Astro-WISE (see Section 4.2):\nawe> reduced.store()\nFinally, the metadata database will be updated and the new image will become a part of the metadata layer.\nawe> reduced.commit()\nThe sequence described above is a general way to create a new data entity in Astro-WISE: to combine a set of references to entities which will be used to create a new one, to invoke a pipeline or a part of pipeline which will generate a new entity, save the created data in the data layer, commit a new data entity to the metadata layer. The processing parameters, for example, the method used for the overscan of the image, will be saved as a persistent attribute of the new data entity as well.\n\n\n
Fig. 5 Data storage network of Astro-WISE. Dataservers are grouped geographically with one dataserver dedicated to the external data exchange. The user requests a file from a local dataserver and, if the file is not found in the local group, the local dataserver will request the file from the other dataserver groups. As soon as file is found it is copied to the dataserver that the user contacted and provided to the user.
\n\nThe important feature of keeping the full data lineage in the system is an ability to avoid unnecessary reprocessing. The execution of the make() method of reduced object actually will start with the search in the metadata database for an object with the same attributes, and if such an object exists (and the user is allowed to retrieve it according to the user's permissions) the user will be redirected to an already existing object and the processing part of the method will be skipped. The data lineage allows to avoid unnecessary reprocessing as well as to use forward and backward chaining in the dependencies of the data items in the system.\nObjects can be deleted from the database under the following restrictions: every user can delete the data he created at the privileges level 1 (see 4.5). For higher privileges levels only the project manager is allowed to delete. To delete a data object myobject from the database Context class is used (see [16] for the full description):\nawe> Context().delete(myobject)\nObjects can be deleted if they are not referenced by other data objects. If they are, it might be desirable to invalidate the object, so that the object will stay in the system but will not be used for the data processing (unless the user specifically ask for invalidated objects). The user-friendly service for the validation of data objects is described in [17].\n4.2 Data Layer\nEach data entity in Astro-WISE has two parts: a data part which is stored in a file and a metadata which is a part of the metadata database and which keeps all\ndependencies of the data entity. The data part of the entity is stored as a file on one of the Astro-WISE dataservers. The predefined file format can be changed, but for most files the format is FITS. Nevertheless the dataserver is not limited to a single file format and can store files of any type. The dataserver is a specific solution which in its functionality is closests to the storage element of the Grid.\nThe main requirement to the dataservers as storage space is, apart from the safety, scalable size. The size should be scalable at the Terabyte level allowing to increase the storage volume from a few Terabytes to almost a Petabyte (this is the typical scale for the KIDS data archive from a few raw images at the beginning of the survey to hundreds of Terabytes with all intermediate data products at the end of observations).\nDataservers are organized in geographically close clusters, within each cluster all dataservers know about the existence of all other dataservers in the cluster. A dataserver in one cluster can contact other clusters via a dedicated dataserver in another cluster (see Fig. 5). All communications are done using a standard HTTP protocol. Two types of requests are available: to retrieve a file and to store a new file.\nThe unique name of the file is used as a unique identifier and this unique name is stored in the database as part of the metadata.\nThe user of the system does not know the actual location of the file on the underlying file system and operates with the URL of the file only. The URL has the form http:/// . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers.\nOn the request from the user to retrieve the file the dataserver can then either return the requested file or redirect the client to the dataserver that has the file. Each dataserver has a permanent data storage and a cache, the last one is used to temporarily store a file retrieved from the other dataserver. By user request, for example, http://ds.astro.rug.astro-wise.org:8000/WFI.2000-09-28T02:22:37.466_8.fits the dataserver ds.astro.rug.astro-wise.org will check cache space for the file WFI.2000-09-28T02:22:37.466_8.fits . If the file is not found, the dataserver will check it's own permanent storage space, and if there is no such file, all dataservers in a 'local' cluster ( astro.rug.astro-wise.org ) will be requested. If there is still no file found, all other clusters will be requested, and as soon as the file is located it will be copied to the cache of the ds.astro.rug.astro-wise.org and returned to the user. In addition, a file can be compressed or decompressed on-the-fly during retrieval, and a slice of the data can be retrieved by specifying the slice as part of the URL.\nThe dataserver is written in Python and can be installed on any operating system and underlying filesystem that can support long case-sensitive filenames. For the Astro-WISE Linux filesystems, XFS and GPFS are currently in use.\n4.3 Processing Layer\nThe example in the Section 4.1 involves the processing facilities of Astro-WISE, which are distributed and combined from processing facilities of all Astro-WISE\npartners. The user has the choice to send the job to one of the processing elements of Astro-WISE (including Grid computing elements) or to use the processing power at the user's disposal, for example, PC or even notebook.\nIn the core of the processing layer of Astro-WISE is a distributed processing unit (DPU). This is a three-component middleware which consists of the DPU server, DPU client and the so called DPU runner. The DPU server is a front-end to any processing system and interacts with the processing system's native queuing system, allowing the user to submit jobs to this particular queuing system, inspect or cancel them. The DPU client includes all functions and methods the user can call to interact with DPU server. The DPU runner is a program which is run on the remote processing facility and checks availability of all necessary software to run an Astro-WISE job, installing required packages if necessary. The system runs on openpbs or under its own queue management software. The DPU itself allows synchronizations of jobs as well and can also transfer parts of a sequence of jobs to other DPU's.\nIn the case of Grid computing element the DPU server will check a user's identity and will use the user's credentials (Grid certificate) to mediate with the Virtual Organization Management System. Currently OmegaCEN is using omegac Virtual Organization with an ability to submit jobs to Grid computing elements in Amsterdam and Groningen.\n4.4 Interfaces\nThe description of Astro-WISE system would be incomplete without the description of a number of interfaces provided for the user. The user can write his own applications in the Python language calling Astro-WISE libraries or can involve Astro-WISE services. The first case requires the use of the Astro-WISE Command Line Interface called AWE (Astro-WISE Environment), which can be installed on any site, PC or notebook\nThe Command Line Interface - CLI - supposes that the user writes his own programs using Python, but to browse the data or even to process observations there is no need to use the CLI. All operations required to perform this activity are possible with a set of standard web services of Astro-WISE. Of course, the use of the CLI gives to the user much more freedom in data processing. The CLI is more useful for experienced users working on a particular use-case, meanwhile web services are developed for routine operations during the data processing of the surveys.\nThe web interfaces are divided into two types: data browsing/exploration and data processing/qualification. The first group includes:\n\n* dbviewer 1 - the metadata database interface which allows browsing and querying of all attributes of all persistent classes stored in the system,\n* quick data search 2 - allows querying on a limited subset of attributes of the data model (coordinate range and object name), and provides results of all projects in the database,\n\n\n* image cut out service 3 and color image maker 4 - these two services are for the astronomical image data type and allow to create a cut out of the image or to create a pseudo-color RGB image from three different images of the same part of the sky,\n* GMap 5 - exploration tool of the Astro-WISE system using the GoogleSky interface.\n\nData processing / qualification interfaces are:\n\n* target processing 6 - the main web tool to process the data in Astro-WISE. This web interface allows users to go through pre-defined processing chains, submitting jobs on the Astro-WISE computing resources with the ability to select the computing node of Astro-WISE,\n* quality service 7 - allows to estimate the quality of the data processing and set a flag highlighting the quality of the data,\n* CalTS 8 - web interface for identifying and qualifying calibration data.\n\nAll web services are built using a set of Python classes developed for AstroWISE as a basis and uses a modular principle, which allows to create a new web service using components of older ones.\n4.5 Authorization and Authentication System\nAstro-WISE is a multi-user system which must accommodate sharing data between scientists and at the same time protect the private data of each user. Each user in the Astro-WISE system has an identity protected by a password, and, optionally, if the user wants to submit job to Grid resources he has to get a Grid certificate.\nThe authorization and authentication system is implemented on the level of the metadata database. As soon as a user logged in with his username/password, the user's privileges are checked in the database, allowing the user to browse the data according to the user's privileges while obeying the privileges of other users. The data in the metadata database are grouped by projects . A Project is a collection of resources which is associated with a group of users who can access these resources, usually one of these users has the role of Project Manager who has privileges to include new users, remove users and publish the data to the wider community.\nThe system of access to the data is based on three attributes which any data entity in Astro-WISE has: user , project and privileges . The first one identifies the user that created the data entity. The second one defines to which project the data entity belongs, the third one defines who is able to use this data entity. All these attributes are initialized the first time the data entity is made persistent in the Astro-WISE system (including the case that the entity is created by one of the Astro-WISE pipelines) and they are both persistent attributes, i.e., stored in the metadata database.\n
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Table 1 Privileges system of Astro-WISE
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\nTable 1 shows the range of values for the privileges attribute. In the case of privileges=1 only the creator of the data item can see it, the creator can raise privileges to privileges=2 , in this case all the users of the project will be able to browse this data entity. Raising privileges to 3 the user makes the data item accessible for all Astro-WISE users in all projects, and with priveleges=4 the anonymous user (a user with minimal read-only privileges in the system) can see it as well. Finally, with priveleges=5 the data item is accessible via Astro-WISE Virtual Observatory interfaces.\nThe special Python class Context was created to handle authorization and authentication. The Context allows to change privileges of all user's data items, during the change Context is checking for dependencies to prevent inconsistency in the dependencies due to the different privileges of data items. For example, if the raw image was published by the user to the project scope and some other user has created a reduced image from this raw image, the original user will not be able to downgrade privileges on the raw image to the private data scope.\n4.6 WISE architecture\nWe described above three infrastructural components of Astro-WISE: the relational DBMS (the metadata layer), dataserver (the data layer) and DPU (the processing layer). Combined with user interfaces these components build an AstroWISE node - a detached Astro-WISE site which can operate independently from other sites. Multiple nodes can be combined to form a distributed system. Each node is independent from others in the sense that it is administrated independently and can handle both the data distributed over other nodes and the data restricted to this node only and shielded from other nodes.\nFigure 6 shows the elements of the node:\n\n* Data storage servers (dataservers).\n* A metadata database to store a full description of each data file with links and references to other objects.\n* Astro-WISE programming environment (CLI) along to web services which give the user an ability to access to the stored data and launch a data processing.\n* Computing nodes for the data processing with DPU servers on top of their quing system.\n* A version control system for developers to include new modules, classes and libraries into the system.\n\nAll these elements are optional, an Astro-WISE node can be installed without dataservers (dataservers from other nodes are used), metadata database or processing facilities. In fact an Astro-WISE node can be installed on a notebook as\n\n\n
Fig. 6 Present-day Astro-WISE nodes with a typical composition of a node in a full deployment node.
\n\nAstro-WISE environment only giving the user access to the system - if the user is not satisfied with web services on the remote nodes.\nPresently Astro-WISE includes sites at Groningen University, Leiden University and Nijmegen University (The Netherlands), Argelander-Institut fur Astronomie, Bonn and Universitats-Sternwarte Munchen (Germany) and Osservatorio Astronomico di Capodimonte, Napoli (Italy).\n5 Astro-WISE: migration to other systems\nThe WISE concept for an information system is a set of principles described in Section 4, Astro-WISE is the first system which realizes this concept and serves as a basis for the further development of information systems. In this section we describe the adaptivity of the parent system to new tasks and challenges on the example of LOFAR Long Term Archive and Molgenis system.\nThe importance of the LOFAR Long Term Archive (LTA) for the development of WISE concept is a necessity to use an external infrastructure which should be included in the architecture to form a complete information system. We preserved all principles of Astro-WISE adding storage and processing which is not controlled by the system itself (BiGGrid 9 ). Additionally we integrated three different systems of Authorization and Authentication to make it possible for the LOFAR observa-\ntory to create new users and control resources in the system. The LOFAR LTA design is described in [20] and the architecture and infrastructure in [21].\nAnother significant development was achieved with the adaptation of Molgenis 10 . Molgenis is a framework written in Java to build user interfaces and databases from definitions that are written in XML. From these definitions the database tables and webserver are generated. Its origins lie in biomedical applications. Historically Molgenis focused more on interaction with the user to streamline the 'protocols' (recipes for analysis) of researchers to keep track of analyses of results. On the other hand, the WISE technology focused more on results from a processing perspective. This lead to the idea to combine the strengths of both frameworks and use some models that already existed in XML for Molgenis to generate a datamodel in Python for Astro-Wise. Then Molgenis could be extended to use the WISE infrastructure for distributed storage and processing.\nThe existing Molgenis XML model is organized in a common fashion through 'module', 'entity' and 'field' where each field has a type of int , float , str or can be a reference type such as mref or xref . Most of these have a counterparts in the WISE framework which made it possible to write a conversion tool.\nThe webserver that is generated by Molgenis has to communicate to a database that it usually creates itself. However, in this case the database had to be generated by Astro-Wise because the frameworks did not have a backend for a common database (Oracle vs. Non-Oracle). Instead of writing a new database backend for either framework it was decided to write an xmlrpc interface to encapsulate database queries and return their results. Since client software for all database flavours does not have to be present this functionality can be extended, e.g., since this is xmlrpc, a programming language independent implementation could also use the database in a way that the WISE framework dictates.\nThe approach developed and tested in the case of Molgenis allows to create a new information system based on Astro-WISE with all services starting from the data model coded in XML and to do this automatically. This approach allowed to decrease the time and resources spent for the developing and implementation of a new system significantly.\nThe next system which will be created with the approach tested on Molgenis is a data processing system for Multi Unit Spectroscopic Explorer 11 (MUSE).\n6 Conclusion and Future Work\nThe Astro-WISE information system, the first information system in Astronomy, proved to be a reliable and flexible tool for the data processing. Originally developed to process the data of KIlo Degree Survey (KIDS 12 ) it triggered development of the unique approach to the architecture of scientific information systems (WISE approach).\nBoth Astro-WISE and the WISE approach are living systems which are open to improvements. For the last 2 years further development of the WISE approach is hosted by Target Holding 13 . Target Holding is an expertise center in the Northern\nNetherlands which is building a cluster of sensor network information systems and provides cooperation between a number of scientific projects and business partners like IBM and Oracle. Target creates and supports a hardware infrastructure for hosting tens of Petabytes of data for projects in astronomy, medicine, artificial intellegence and biology.\nIn this development the WISE approach was used to create new information systems extending the original Astro-WISE on new data models, new data storage and processing capacities and new fields.\nAcknowledgements Astro-WISE is an on-going project which started from a FP5 RTD programme funded by the EC Action 'Enhancing Access to Research Infrastructures'. This work was performed as part of the Target project. Target project is supported by Samenwerkingsverband Noord Nederland. It operates under the auspices of Sensor Universe. It is also financially supported by the European fund for Regional Development and the Dutch Ministry of Economic Affairs, Pieken in de Delta, the Province of Groningen and the Province of Drenthe.\nReferences\n\n1. Abell G.O., The Distribution of Rich Clusters of Galaxies, ApJS 3, 211 (1958)\n2. Lauberts A., Valentijn E. A., 1989, The surface photometry catalogue of the ESO-Uppsala galaxies. Garching: European Southern Observatory, (1989)\n3. Djorgovski S. G., Gal R. R., Odewahn S. C., de Carvalho R. R., Brunner R., Longo G., Scaramella R., The Palomar Digital Sky Survey (DPOSS), in 'Wide Field Surveys in Cosmology', eds. S. Colombi, Y. Mellier and B. Raban, Gif sur Yvette: Editions Frontieres, 89 (1998)\n4. Djorgovski S. G., Baltay C., Mahabal A. A., Drake A. J., Williams R., Rabinowitz D., Graham M. J., Donalek C., Glikman E., Bauer A., Scalzo R., Ellman N., The PalomarQuest digital synoptic sky survey, Astron. Nach. 329, 263 (2008)\n5. Skrutskie M. F., Schneider S. E., Stiening R., Strom S. E., Weinberg M. D., Beichman C., Chester T., Cutri R., Lonsdale C., Elias J., Elston R., Capps R., Carpenter J., Huchra J., Liebert J., Monet D., Price S., Seitzer P., The Two Micron All Sky Survey (2MASS): Overview and Status, in 'The Impact of Large Scale Near-IR Sky Surveys', eds. F. Garzon et al., Dordrecht: Kluwer Academic Publishing Company, 25 (1997)\n6. Stoughton C. et al., Sloan Digital Sky Survey: Early Data Release, AJ 123, 485 (2002)\n7. Wells D.C., Greisen E.W., FITS - a Flexible Image Transport System, in 'International Workshop on Image Processing in Astronomy', Proc. of the 5th. Colloquium on Astrophysics, eds. G. Sedmak, M. Capaccioli, R.J. Allen., Osservatorio Astronomico di Trieste, Trieste, Italy, 445 (1979)\n8. Valentijn E.A., Deul E., Kuijken K.H., Observing and Data Mining with OmegaCAM, ASPC 232, 392 (2001)\n9. Valentijn E.A, Kuijken K., ASTRO-WISE - An Astronomical Wide-Field Imaging System for Europe, in 'Toward an International Virtual Observatory', Proc. of the ESO/ESA/NASA/NSF Conference, eds. P.J. Quinn and K.M. G'orski, Springer-Verlag, Berlin/Heidelberg, 19 (2004)\n10. Begeman K., Belikov A.N., Boxhoorn D., Dijkstra F., Valentijn E.A., Vriend W.-J., Zhao Z., Merging Grid Technologies: Astro-WISE and EGEE, Journal of Grid Computing 8, 199 (2010)\n11. Grosbol P., Ponz D., The Midas Table File System, Mem, Soc. Astr. It. 56, 429 (1985)\n12. Jones B., An Overview of the EGEE Project, in: Book Series Lecture Notes in Computer Science, Vol. 3664, 1 (2005)\n13. Stewart G.A., Cameron D., Cowan G.A. & McCance G., Storage and Data Management in EGEE, In Proc. Fifth Australasian Symposium on Grid Computing and e-Research (AusGrid 2007), Ballarat, Australia. CRPIT, 68 (2007)\n14. Hick J., Shalf J., Storage Technology, in: 'Scientific Data Management. Challenges, Technology and Deployment', Chapman & Hall/CRC Computational Science Series, 3 (2010)\n\n\n15. Berner E.S., Clinical Decision Support Systems: Theory and Practice , Health Informatics Serie, Springer (2007)\n16. McFarland J.P., Verdoes-Kleijn G., Sikkema G., Helmich E.M., Boxhoorn D.R., Valentijn E.A., The Astro-WISE Optical Image Pipeline: Development and Implementation, Experimental Astronomy, in press, arXiv:1110.2509v2\n17. McFarland J.P., Helmich E.M., Valentijn E.A., The Astro-WISE Approach to Quality Control for Astronomical Data, Experimental Astronomy, in press\n18. Belikov, A.N., Vriend W.-J., Sikkema G., User interfaces in Astro-WISE, Experimental Astronomy, in press\n19. Belikov A.N., Dijkstra F., Gankema J.A., van Hoof J.B.A.N., Koopman R., Information Systems Playground - The Target Infrastructure, Scaling Astro-WISE into the Petabyte range, Experimental Astronomy, in press, arXiv:1110.0937v1\n20. Valentijn E., Belikov A., LOFAR Information System Design, Mem. Soc. Astr. It. 80, 509 (2009)\n21. Begeman K., Belikov A.N., Boxhoorn D.R., Dijkstra F., Holties H., Meyer-Zhao Z., Renting G.A., Valentijn E.A., Vriend W.-J., LOFAR Information System, Future Generation Computing Systems 27, 319 (2011)\n"},"sections":{"kind":"list like","value":[{"title":"The Astro-WISE datacentric information system","content":"K. Begeman, A.N. Belikov, D.R. Boxhoorn, E.A. Valentijn Received: date / Accepted: date Abstract In this paper we present the various concepts behind the Astro-WISE Information System. The concepts form a blueprint for general scientific information systems (WISE) which can satisfy a wide and challenging range of requirements for the data dissemination, storage and processing for various fields in science. We review the main features of the information system and its practical implementation. Keywords Data Grid · Grid Computing · Information System · Middleware","pages":[1]},{"title":"1 Introduction","content":"Digital astronomical catalogues have been built from the very first moment information technology enabled this, e.g. the first Abell catalogue of clusters of galaxies was digitally prepared and printed in the days minus signs were not available in print [1]. As soon as digital scanning devices became available photographic material was scanned and first large image surveys were digitally published, such as ESO-LV [2] and the digitized sky surveys like DPOSS [3], succeeded by the Palomar-Quest Survey [4], in the 90's followed by CCD-based surveys such as the 2MASS survey [5], the Sloan Digital Sky Survey SDSS [6]. The rapid accumulation of astronomical digital data and its public dissemination was, compared to other disciplines, achieved at an early stage, thanks to an open and collaborative astronomical world community who adopted the FITS image data format as early as 1979 [7]. The CDS, initially Centre de Donn'ees Stellaires and later renamed into Centre de Donn'ees astronomiques de Strasbourg took the lead in Europe to collect and disseminate the ever growing data sets of a zoo of astronomical observatories and projects. VizieR webservices nowadays provide access to over 9000 catalogues. Numerous astronomical data centers followed, developing specialized services, for example, the Infrared Processing and Analysis Center and the bibliographical SAO/NASA Astrophysics Data System. In the early 2000's it was realized that the ever growing data volumes require new approaches: the community becomes the data provider and the International Virtual Observatory and its European branch, the European Virtual Observatory, developed standards and interfaces to allow individual data centers to publish their catalogs and images in a common framework providing worldwide access to users who can query, cross-match and visualize multiple databases via the internet. A highlight formed the overplotting of data from different experiments, like Xray satellites and optical ground-based observatories with some keystrokes on the Aladin interactive software sky atlas. While the Virtual Observatory focused on the dissemination of published data it was also realized in the early 2000's that the upcoming data deluge required a new approach to the handling of the data stream from the telescope to the science-ready result. Modern experiments not only involve more data, but also require increasing precision on the various calibrations. The dependency of the final result (sometimes only a few numbers, like cosmological parameter values) on time-variable calibrations of very large data sets, involving the evaluation by large research teams distributed in smaller groups over various sites sets the basic requirements to the system handling the data. While in classical systems data is delivered in various releases to the public, often re-processing the whole set with higher versions of the code, the high data rates of modern experiments require an alternative approach, where the up-to-date result is derived on demand by the user. Thus, we set out to design and implement an integrated datacentric system Astro-WISE in which the processing, storage and administration is integrated in a single environment, providing a living system to both the data producers and the customers. Early reports of this development and implementation have been given in [8] and [9]. To reach this goal, traceability of each individual data item handled by pipelines or any piece of code is carefully maintained, every data item beyond pixel value is kept as Metadata and made persistent and distributed in a relational database with an object-oriented view, mapping all the dependencies.","pages":[1,2]},{"title":"2 Datacentric Approach to Data Processing","content":"The datacentric approach facilitates scientists to cooperate in data analysis and mining by means of the internet, nowadays referred to as e-science. The AstroWISE information system connects to an own developed distributed grid processing system (based on Distributed Processing Unit, DPU), but has been also connected to the EGEE/EGI/BiGGrid-Grid [10], particularly for the operations of the Lofar radio telescope. Nowadays, this aspect of the system is referred to as cloud computing, but the Astro-WISE datacentric approach at the same time facilitates the sharing and web hosting of all the data handled by the system. Today, we are just taking the first small steps towards the development of approaches and systems who could handle the upcoming data deluge. The way of managing and administration the data will be a key, and will require ever improving, refining and self-organizing approaches. The key notion is that a fruitful approach towards handling the data deluge focuses on the administration, modeling, standardizing and tracking of the data rather than on the processing of the data by CPUs. This datacentric approach, in which the design of operational systems is driven by data management and data standard issues rather than data processing/cpu/pipelining issues. Processing systems turn into information systems. First attempts to combine archiving and processing in the 1980s include ESO's MIDAS table file system [11], integrating common tables both accessible in source code and by user interfaces like the MIDAS monitor/prompt. The MIDAS table file system allows both users and programmers to change its content and also registering the modifications in table keywords leading to archives ready for further review and (trend) analysis. Its development was essential for the production and photometric calibration of the images of the ESO-LV galaxies. However, the 600,000 fore- and background objects visible on the images, stars and galaxies, could only be handled with great difficulties inside the system, response times running into hours per operations, demonstrating the importance of built-in scalability. This MIDAS Table file system marked a stepping stone to current datacentric approaches which are enabled by the introduction of modern object-oriented programming languages, data modeling tools like UML and XML and relational databases. Wehave combined these views, systems and requirements in order to design and build integrated information systems which have the potential for self-organizing and enrichment with new data. Both self-organizing and enrichment are processes in time. In fact, the datacentric approach implies the detailed modeling and awareness of how things change in time. Important changes range from The ideal information system would seamlessly cope with all these changes in time, thus creating a living long term digital preservation environment. Astro-WISE is one of the first systems attempting to reach this goal. For several years many authors predicted the upcoming data avalanche due to the advancements in digital sensor technology. Obviously, this has now started, both scientific experiments such as LHC and Lofar are running into the tens of Petabytes data acquisition regimes, while text, imaging, genome and internet gathered data collections like e-Bay are exceeding similar volumes. This triggered the development of Grid infrastructure that is able to handle Petabytes of data ([12] and [13]). All this has happened in an approximate 20 year time span when the first large digital imaging collection was published, ESO-LV [2], containing scans of 32,000 galaxy images, 4 Gbyte of data filling a room full with tapes and populating the very first version of optical media. By now, the data avalanche is multidisciplinary and worldwide and involves all aspects of our society, from science to commerce, public services and health care. Digital data is gathered, processed, distributed and accessed at a continuously growing rate. And this is only the beginning: in, say, two decades, the same room full with 4 Gbyte of tapes 20 years ago, and 40 Petabyte of tapes today could possibly contain 40 Zetabytes (40 million Petabytes) of data. The growth of the data volume produced by various types of sensor networks met with the growth of the disk capacity and search for new methods in the increasing of the memory storage density (see [14]). In order to be useful all this data has to be organized, administrated, managed and distributed. It took a billion years for cells to develop into extremely complex information systems around the DNA; it has resulted into systems and approaches with an incredible complexity with memory mapping, copying, filtering and distribution mechanisms of data including feedback. Compared to this, our present data organizing mechanisms in IT are still very simple and rudimentary and we are just making the first steps to living self-organizing systems and archives required to deal with and optimally use the ocean of digital data to come. However, also in the cell stochastic processes are important, and absence of organization 'from above' are countered by survival of the fittest type of mechanisms, which appear as self-organizing. Also in human endeavored IT, standards seldom come from above, an exception being the ASCII character standards, mandated in 1968 by U.S. President Lyndon B. Johnson to be used in all computers purchased by the United States federal government. I have also approved recommendations of the Secretary of Commerce regarding standards for recording the Standard Code for Information Interchange on magnetic tapes and paper tapes when they are used in computer operations. All computers and related equipment configurations brought into the Federal Government inventory on and after July 1, 1969, must have the capability to use the Standard Code for Information. Interchange and the formats prescribed by the magnetic tape and paper tape standards when these media are used.In practice, standards are often invented by companies or scientific communities which are discriminated in a complex survival of the fittest battle, which involves that many factors that it appears stochastic. The view, or better requirement, for future self-organizing information systems and its archives is that it organizes itself for all these changes in time. It could be seen as a step towards the extremely complex and self-organizing information systems in a cell. The core problem for the definition of an information system is the definition of information itself. The information can be an input data flow from sensor detectors, an archive of transactions, trends deduced from the analysis of some data and even results of the data modeling or simulations. As result, information systems will host a variety of subsystems defined by its purpose. Examples are a Decision Support System gathering and analyzing trends for a specific business use case (see [15]) or Geographical Information Systems with focus on geospatial data analysis and a visualization of the data. Nevertheless, it is possible to define primary components of any Information System following simple requirements on storage, processing and sharing data. Such a system should include a data model, storage and data processing facilities together with user interfaces. In Astro-WISE we have decided to separate and store all data beyond pixel/measurement data in a database and all pixel/measurement data in files on independent storage media. This paper does not touch details of the implementation of Astro-WISE information system which are described in other papers but describes the general outline of the Astro-WISE information system and specify features which make Astro-WISE unique. For the more in-depth review of particular aspects of AstroWISE we recommend to read corresponding papers on the optical pipeline implemented in Astro-WISE [16], quality control realised in Astro-WISE [17], the system of user interfaces and services [18], and the hardware solution for Astro-WISE and information systems created on the basis of Astro-WISE [19].","pages":[2,3,4,5]},{"title":"3 Building a Scientific Information System","content":"The development of the Astro-WISE information system, the first one which implemented WISE technology, started from the very practical challenge: enable a community of researchers distributed over the world to process the data of astronomical imaging survey of the OmegaCAM 256 Megapixel camera on ESO's VST telescope at Cerro Paranal (Chile). These scientists should be able to evaluate the quality of the data, apply a number of calibrations, share the data within the team and employ distributed resources of a Petabytes scale of data storage and Teraflops capacity in data processing. From this our basic requirements on the system are derived: The most general requirements listed above results in a set of more detailed and specific requirements to different components of the information system. First, the ability to share data among a number of users becomes more valuable when all these users are working with the same set of standards for the data. Second, the requirement to trace any changes in the data processed by different users with different programs and pipelines implies that the system has a common approach to treat the data items on different stages of the data processing and keeps the record of changes in the data. As result the basic requirements to the system dictate the use of a common data model and the common standards for the data storage. Below we will describe the common data model features defined from the requirements to the system.","pages":[5,6]},{"title":"Requirements to the Data Model:","content":"When properly defined the data model allows to code, instantiate and trace the data processing for all scientific use-cases. Also, the data model should allow to accommodate external data, as well as provide necessary comparison and checking with these external data sets. When the data model is carefully defined our approach is to convert this into a pipeline model, in turn translated into a hierarchy of Classes. These Classes are then mapped into a database. We originally did this with object-oriented databases like Objectivity, but currently we do this in a relational database and build an object oriented view in order to maintain transparency in the class model, and its dependencies. Requirements to Infrastructure. The data model should be shared among geographically remote sites of the system, as research groups at various locations are collaboratively adding and extracting data from the distributed system. The basic requirement is that in fact any component (database, file system, code base, CPUs) should possibly be present at different sites, for a number of reasons: For example, the data ingestion to the system can be done in Garching, Germany, the data processing in Groningen, The Netherlands and the image analysis in Napoli, Italy. Requirements to the User Interfaces. The user should be able to trace the data processing, to retrieve data items and identify all its dependencies on other items (mostly calibrations) in the system, to initiate data processing of particular data items and to find a quality estimation for the data. As we can see, all these requirements put together are more challenging than requirements on common business information systems. The key difference is that we allow everything, including the code base to change in time. In the following chapter we outline how our requirements formed a layout for a scientific information system capable to share a huge data volume, to process these data and fulfill scientific use-cases for a number of users.","pages":[6,7]},{"title":"4 WISE Concept","content":"First of all we have to clearly distinguish between two classes of data storage systems in Astronomy: proper archives containing fixed published data and infor- mation systems. The main difference can be found in an ability to work with the data and to change it during the storage. For an archive stored data is a stable entity which is provided to the user as it is, meanwhile the information system allows to the user to modify their own data creating user-specific version of an archive and share this version with other users. The Astro-WISE approach, and its general multi-discipline WISE approach implements a scientific information system. The usual approach to building an information system is to perceive three components: the data layer, business rules and the interfaces (see Fig. 1). In our case, we separate the data layer in two parts: the pure measurement data layer, hereafter data layer, and everything beyond that, hereafter Metadata layer, ranging from file sizes, to statistics of pixel values and detected events in the measurements (galaxy, particle, word). The Metadata layer allows to implement a data model, the data layer allows to store the data as files in a standard format, while business rules implemented in Python classes bind the metadata and the data layers. Interfaces provide user access to both the business rules and the data. In the case of AstroWISE business rules present in the Metadata layer (Data Definition Language used to create a data model), data layer (the on-the-fly compression of the data on data storage grid) and-the most apparent to the user-in a number of pipelines and programs which the user defines to process the data. This last part of business rules components we will call data processing layer. To implement Astro-WISE we use abstractions of storage, processing and database capabilities as a basis for the infrastructure for each of the layers of the system. The metadata layer is realized in a relational DBMS through an abstraction of the required database functionality, the data layer is put on the Astro-WISE data storage grid through an abstraction of storage and the processing grid is used to connect the user with the data and metadata layers by a number of interfaces for the data processing layer. Separation of these three infrastructures plays a key role in the flexibility of the system, as we will see below. The metadata layer implements a list of necessary functionalities: Our requirements on distribution and multiple users propagate as key principles of the realization of metadata and data layers and business rules which form the core of the WISE approach: The Astro-WISE system is realized in the Python programming language. It allows to wrap any program into a Python module, library or class. The use of Python also allows to combine the principles of modular programming with objectoriented programming, so that each package in the system can be built and run independently with an object-oriented data model serving as glue between modules. At the same time, the logic behind pipelines and workflows in Astro-WISE allows the execution of part of the processing chain independently from the other parts. We will describe this approach in more detail in the example of optical image processing in Section 4.1. The conceptual difference between Astro-WISE and other existing systems is that Astro-WISE moves from the usual for astronomy processing-centric approach to a data-centric approach. The data processing itself becomes an integral part of the archive. The typical solution for the data processing and storage for Astronomy handles the data processing and the final data product delivered to the user as two completely separated entities. The data product usually is a result of the processing for the whole survey with fixed processing parameters used for the whole set of images. The user has access to reduced images and the catalog, both data products are stable within a 'release', which usually refers to the sky coverage performed by the survey. In this way the survey data center provides the user with a specific version of the data product that will not change over time, which covers most of the science use-cases. Nevertheless there are a number of use-cases which can not be satisfied by the 'standard' version. For example, in the search for objects like brown dwarfs or quasars it is important to lower the detection threshold which implies the reprocessing of the data. The user himself has to care about such reprocessing thereby reinventing the whole data processing system for the survey and involving his own resources. The Astro-WISE system allows to work on any use-case using the standard pipelines and performing programming on a minimal level - if this is necessary at all. Figure 1 shows principles of WISE concept in the binding together all layers of the information system. Data processing pipelines define use-cases and data model for the system, which is implemented in the Python classes which wrap pipeline and define the common persistent data model objects. The data layer uses this data model, and interfaces are providing an access to the system for users. The common data model effectively brings together all layers of the information system. The common data model can be modified by a mutial agreement of all users of the system. Any user can propose changes in the data model, and, if changes were accepted, the system administrator implements these changes in corresponding Python classes and in the database scheme. In the next sections we will review in detail the infrastructural layers of AstroWISE and their implementation.","pages":[7,8,9,10]},{"title":"4.1 Metadata Layer","content":"The bulk of the data stored in Astro-WISE is stored in files of some format (FITS). Each file is registered in the Astro-WISE metadata database with the unique filename. Apart from just registering each file in the system the metadata database implements the important part of WISE approach - the common data model. Figure 2 shows the general outline for all data models implemented in informations systems based on WISE concept. two core classes DBObject and DataObject are parent classes for the class which is storing metadata persistently and the class which is storing the data in the file. These classes include interfaces to the metadata database and the data storage, the physical implementation of the metadata database and the data (files) storage can be different for different systems. Let us suppose that we wish to use Astro-WISE for the data processing of optical images. Figure 3 shows typical classes of data items used by the optical data processing pipeline to reduce the data from raw images to the final science ready catalogues. This data model, deduced from the pipeline and enhanced keeping in mind possible scientific use-cases, is a central part of the metadata layer. The data model is implemented both in the relational database (currently Oracle 11g RAC is used) and in the hierarchy of Python classes. All core classes are made persistent, i.e., any change in the object is mirrored in the corresponding tables of the database. The method used for the implementing of the data lineage is the Persistent Object Hierarchy (Fig. 2). According to this method objects of Astro-WISE are made persistent recursively, all operations and attributes of the object are saved in the metadata database. As an example let us explore the data reduction task - we will create a reduced image from the raw image on one CCD chip. First of all, we will select an image we wish to process: awe> raw = (RawScienceFrame.filename == 'WFI.2000-01-01T08:57:15.410_3.fits')[0] The image is selected by browsing the metadata database for an object of class RawScienceFrame with filename WFI.2000-01-01T08:57:15.410_3.fits . The returned object raw was retrieved from the metadata database with all attributes. In the next step we will retrieve from the database all necessary calibration objects. The logic in the function select_for_raw allows to take the calibration files for the same night of observation ( 2000-01-01T08:57:15.410 ) or closest to this night. awe> hot = HotPixelMap.select_for_raw(raw) awe> cold = ColdPixelMap.select_for_raw(raw) awe> flat = MasterFlatFrame.select_for_raw(raw) awe> bias = BiasFrame.select_for_raw(raw) Now we can instantiate a new object - our target - a reduced image which will be based on all the images above: Please, note that all images which will be used to create a new reduced image reduced are referenced through attributes of the new image. This new image should get a unique ID, which is a name of the file where the image will be stored. awe> reduced.set_filename() Now the image can be created invoking the make() method of the ReducedScienceFrame class. awe> reduced.make() The make() produced a file with a new image which will be stored on one of the dataservers of Astro-WISE (see Section 4.2): awe> reduced.store() Finally, the metadata database will be updated and the new image will become a part of the metadata layer. awe> reduced.commit() The sequence described above is a general way to create a new data entity in Astro-WISE: to combine a set of references to entities which will be used to create a new one, to invoke a pipeline or a part of pipeline which will generate a new entity, save the created data in the data layer, commit a new data entity to the metadata layer. The processing parameters, for example, the method used for the overscan of the image, will be saved as a persistent attribute of the new data entity as well. The important feature of keeping the full data lineage in the system is an ability to avoid unnecessary reprocessing. The execution of the make() method of reduced object actually will start with the search in the metadata database for an object with the same attributes, and if such an object exists (and the user is allowed to retrieve it according to the user's permissions) the user will be redirected to an already existing object and the processing part of the method will be skipped. The data lineage allows to avoid unnecessary reprocessing as well as to use forward and backward chaining in the dependencies of the data items in the system. Objects can be deleted from the database under the following restrictions: every user can delete the data he created at the privileges level 1 (see 4.5). For higher privileges levels only the project manager is allowed to delete. To delete a data object myobject from the database Context class is used (see [16] for the full description):","pages":[10,11,12,13]},{"title":"awe> Context().delete(myobject)","content":"Objects can be deleted if they are not referenced by other data objects. If they are, it might be desirable to invalidate the object, so that the object will stay in the system but will not be used for the data processing (unless the user specifically ask for invalidated objects). The user-friendly service for the validation of data objects is described in [17].","pages":[13]},{"title":"4.2 Data Layer","content":"Each data entity in Astro-WISE has two parts: a data part which is stored in a file and a metadata which is a part of the metadata database and which keeps all dependencies of the data entity. The data part of the entity is stored as a file on one of the Astro-WISE dataservers. The predefined file format can be changed, but for most files the format is FITS. Nevertheless the dataserver is not limited to a single file format and can store files of any type. The dataserver is a specific solution which in its functionality is closests to the storage element of the Grid. The main requirement to the dataservers as storage space is, apart from the safety, scalable size. The size should be scalable at the Terabyte level allowing to increase the storage volume from a few Terabytes to almost a Petabyte (this is the typical scale for the KIDS data archive from a few raw images at the beginning of the survey to hundreds of Terabytes with all intermediate data products at the end of observations). Dataservers are organized in geographically close clusters, within each cluster all dataservers know about the existence of all other dataservers in the cluster. A dataserver in one cluster can contact other clusters via a dedicated dataserver in another cluster (see Fig. 5). All communications are done using a standard HTTP protocol. Two types of requests are available: to retrieve a file and to store a new file. The unique name of the file is used as a unique identifier and this unique name is stored in the database as part of the metadata. The user of the system does not know the actual location of the file on the underlying file system and operates with the URL of the file only. The URL has the form http:// / . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers. . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers. On the request from the user to retrieve the file the dataserver can then either return the requested file or redirect the client to the dataserver that has the file. Each dataserver has a permanent data storage and a cache, the last one is used to temporarily store a file retrieved from the other dataserver. By user request, for example, http://ds.astro.rug.astro-wise.org:8000/WFI.2000-09-28T02:22:37.466_8.fits the dataserver ds.astro.rug.astro-wise.org will check cache space for the file WFI.2000-09-28T02:22:37.466_8.fits . If the file is not found, the dataserver will check it's own permanent storage space, and if there is no such file, all dataservers in a 'local' cluster ( astro.rug.astro-wise.org ) will be requested. If there is still no file found, all other clusters will be requested, and as soon as the file is located it will be copied to the cache of the ds.astro.rug.astro-wise.org and returned to the user. In addition, a file can be compressed or decompressed on-the-fly during retrieval, and a slice of the data can be retrieved by specifying the slice as part of the URL. The dataserver is written in Python and can be installed on any operating system and underlying filesystem that can support long case-sensitive filenames. For the Astro-WISE Linux filesystems, XFS and GPFS are currently in use.","pages":[13,14]},{"title":"4.3 Processing Layer","content":"The example in the Section 4.1 involves the processing facilities of Astro-WISE, which are distributed and combined from processing facilities of all Astro-WISE partners. The user has the choice to send the job to one of the processing elements of Astro-WISE (including Grid computing elements) or to use the processing power at the user's disposal, for example, PC or even notebook. In the core of the processing layer of Astro-WISE is a distributed processing unit (DPU). This is a three-component middleware which consists of the DPU server, DPU client and the so called DPU runner. The DPU server is a front-end to any processing system and interacts with the processing system's native queuing system, allowing the user to submit jobs to this particular queuing system, inspect or cancel them. The DPU client includes all functions and methods the user can call to interact with DPU server. The DPU runner is a program which is run on the remote processing facility and checks availability of all necessary software to run an Astro-WISE job, installing required packages if necessary. The system runs on openpbs or under its own queue management software. The DPU itself allows synchronizations of jobs as well and can also transfer parts of a sequence of jobs to other DPU's. In the case of Grid computing element the DPU server will check a user's identity and will use the user's credentials (Grid certificate) to mediate with the Virtual Organization Management System. Currently OmegaCEN is using omegac Virtual Organization with an ability to submit jobs to Grid computing elements in Amsterdam and Groningen.","pages":[14,15]},{"title":"4.4 Interfaces","content":"The description of Astro-WISE system would be incomplete without the description of a number of interfaces provided for the user. The user can write his own applications in the Python language calling Astro-WISE libraries or can involve Astro-WISE services. The first case requires the use of the Astro-WISE Command Line Interface called AWE (Astro-WISE Environment), which can be installed on any site, PC or notebook The Command Line Interface - CLI - supposes that the user writes his own programs using Python, but to browse the data or even to process observations there is no need to use the CLI. All operations required to perform this activity are possible with a set of standard web services of Astro-WISE. Of course, the use of the CLI gives to the user much more freedom in data processing. The CLI is more useful for experienced users working on a particular use-case, meanwhile web services are developed for routine operations during the data processing of the surveys. The web interfaces are divided into two types: data browsing/exploration and data processing/qualification. The first group includes:","pages":[15]},{"title":"Data processing / qualification interfaces are:","content":"All web services are built using a set of Python classes developed for AstroWISE as a basis and uses a modular principle, which allows to create a new web service using components of older ones.","pages":[16]},{"title":"4.5 Authorization and Authentication System","content":"Astro-WISE is a multi-user system which must accommodate sharing data between scientists and at the same time protect the private data of each user. Each user in the Astro-WISE system has an identity protected by a password, and, optionally, if the user wants to submit job to Grid resources he has to get a Grid certificate. The authorization and authentication system is implemented on the level of the metadata database. As soon as a user logged in with his username/password, the user's privileges are checked in the database, allowing the user to browse the data according to the user's privileges while obeying the privileges of other users. The data in the metadata database are grouped by projects . A Project is a collection of resources which is associated with a group of users who can access these resources, usually one of these users has the role of Project Manager who has privileges to include new users, remove users and publish the data to the wider community. The system of access to the data is based on three attributes which any data entity in Astro-WISE has: user , project and privileges . The first one identifies the user that created the data entity. The second one defines to which project the data entity belongs, the third one defines who is able to use this data entity. All these attributes are initialized the first time the data entity is made persistent in the Astro-WISE system (including the case that the entity is created by one of the Astro-WISE pipelines) and they are both persistent attributes, i.e., stored in the metadata database. Table 1 shows the range of values for the privileges attribute. In the case of privileges=1 only the creator of the data item can see it, the creator can raise privileges to privileges=2 , in this case all the users of the project will be able to browse this data entity. Raising privileges to 3 the user makes the data item accessible for all Astro-WISE users in all projects, and with priveleges=4 the anonymous user (a user with minimal read-only privileges in the system) can see it as well. Finally, with priveleges=5 the data item is accessible via Astro-WISE Virtual Observatory interfaces. The special Python class Context was created to handle authorization and authentication. The Context allows to change privileges of all user's data items, during the change Context is checking for dependencies to prevent inconsistency in the dependencies due to the different privileges of data items. For example, if the raw image was published by the user to the project scope and some other user has created a reduced image from this raw image, the original user will not be able to downgrade privileges on the raw image to the private data scope.","pages":[16,17]},{"title":"4.6 WISE architecture","content":"We described above three infrastructural components of Astro-WISE: the relational DBMS (the metadata layer), dataserver (the data layer) and DPU (the processing layer). Combined with user interfaces these components build an AstroWISE node - a detached Astro-WISE site which can operate independently from other sites. Multiple nodes can be combined to form a distributed system. Each node is independent from others in the sense that it is administrated independently and can handle both the data distributed over other nodes and the data restricted to this node only and shielded from other nodes. Figure 6 shows the elements of the node: All these elements are optional, an Astro-WISE node can be installed without dataservers (dataservers from other nodes are used), metadata database or processing facilities. In fact an Astro-WISE node can be installed on a notebook as Astro-WISE environment only giving the user access to the system - if the user is not satisfied with web services on the remote nodes. Presently Astro-WISE includes sites at Groningen University, Leiden University and Nijmegen University (The Netherlands), Argelander-Institut fur Astronomie, Bonn and Universitats-Sternwarte Munchen (Germany) and Osservatorio Astronomico di Capodimonte, Napoli (Italy).","pages":[17,18]},{"title":"5 Astro-WISE: migration to other systems","content":"The WISE concept for an information system is a set of principles described in Section 4, Astro-WISE is the first system which realizes this concept and serves as a basis for the further development of information systems. In this section we describe the adaptivity of the parent system to new tasks and challenges on the example of LOFAR Long Term Archive and Molgenis system. The importance of the LOFAR Long Term Archive (LTA) for the development of WISE concept is a necessity to use an external infrastructure which should be included in the architecture to form a complete information system. We preserved all principles of Astro-WISE adding storage and processing which is not controlled by the system itself (BiGGrid 9 ). Additionally we integrated three different systems of Authorization and Authentication to make it possible for the LOFAR observa- tory to create new users and control resources in the system. The LOFAR LTA design is described in [20] and the architecture and infrastructure in [21]. Another significant development was achieved with the adaptation of Molgenis 10 . Molgenis is a framework written in Java to build user interfaces and databases from definitions that are written in XML. From these definitions the database tables and webserver are generated. Its origins lie in biomedical applications. Historically Molgenis focused more on interaction with the user to streamline the 'protocols' (recipes for analysis) of researchers to keep track of analyses of results. On the other hand, the WISE technology focused more on results from a processing perspective. This lead to the idea to combine the strengths of both frameworks and use some models that already existed in XML for Molgenis to generate a datamodel in Python for Astro-Wise. Then Molgenis could be extended to use the WISE infrastructure for distributed storage and processing. The existing Molgenis XML model is organized in a common fashion through 'module', 'entity' and 'field' where each field has a type of int , float , str or can be a reference type such as mref or xref . Most of these have a counterparts in the WISE framework which made it possible to write a conversion tool. The webserver that is generated by Molgenis has to communicate to a database that it usually creates itself. However, in this case the database had to be generated by Astro-Wise because the frameworks did not have a backend for a common database (Oracle vs. Non-Oracle). Instead of writing a new database backend for either framework it was decided to write an xmlrpc interface to encapsulate database queries and return their results. Since client software for all database flavours does not have to be present this functionality can be extended, e.g., since this is xmlrpc, a programming language independent implementation could also use the database in a way that the WISE framework dictates. The approach developed and tested in the case of Molgenis allows to create a new information system based on Astro-WISE with all services starting from the data model coded in XML and to do this automatically. This approach allowed to decrease the time and resources spent for the developing and implementation of a new system significantly. The next system which will be created with the approach tested on Molgenis is a data processing system for Multi Unit Spectroscopic Explorer 11 (MUSE).","pages":[18,19]},{"title":"6 Conclusion and Future Work","content":"The Astro-WISE information system, the first information system in Astronomy, proved to be a reliable and flexible tool for the data processing. Originally developed to process the data of KIlo Degree Survey (KIDS 12 ) it triggered development of the unique approach to the architecture of scientific information systems (WISE approach). Both Astro-WISE and the WISE approach are living systems which are open to improvements. For the last 2 years further development of the WISE approach is hosted by Target Holding 13 . Target Holding is an expertise center in the Northern Netherlands which is building a cluster of sensor network information systems and provides cooperation between a number of scientific projects and business partners like IBM and Oracle. Target creates and supports a hardware infrastructure for hosting tens of Petabytes of data for projects in astronomy, medicine, artificial intellegence and biology. In this development the WISE approach was used to create new information systems extending the original Astro-WISE on new data models, new data storage and processing capacities and new fields. Acknowledgements Astro-WISE is an on-going project which started from a FP5 RTD programme funded by the EC Action 'Enhancing Access to Research Infrastructures'. This work was performed as part of the Target project. Target project is supported by Samenwerkingsverband Noord Nederland. It operates under the auspices of Sensor Universe. It is also financially supported by the European fund for Regional Development and the Dutch Ministry of Economic Affairs, Pieken in de Delta, the Province of Groningen and the Province of Drenthe.","pages":[19,20]}],"string":"[\n {\n \"title\": \"The Astro-WISE datacentric information system\",\n \"content\": \"K. Begeman, A.N. Belikov, D.R. Boxhoorn, E.A. Valentijn Received: date / Accepted: date Abstract In this paper we present the various concepts behind the Astro-WISE Information System. The concepts form a blueprint for general scientific information systems (WISE) which can satisfy a wide and challenging range of requirements for the data dissemination, storage and processing for various fields in science. We review the main features of the information system and its practical implementation. Keywords Data Grid · Grid Computing · Information System · Middleware\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"Digital astronomical catalogues have been built from the very first moment information technology enabled this, e.g. the first Abell catalogue of clusters of galaxies was digitally prepared and printed in the days minus signs were not available in print [1]. As soon as digital scanning devices became available photographic material was scanned and first large image surveys were digitally published, such as ESO-LV [2] and the digitized sky surveys like DPOSS [3], succeeded by the Palomar-Quest Survey [4], in the 90's followed by CCD-based surveys such as the 2MASS survey [5], the Sloan Digital Sky Survey SDSS [6]. The rapid accumulation of astronomical digital data and its public dissemination was, compared to other disciplines, achieved at an early stage, thanks to an open and collaborative astronomical world community who adopted the FITS image data format as early as 1979 [7]. The CDS, initially Centre de Donn'ees Stellaires and later renamed into Centre de Donn'ees astronomiques de Strasbourg took the lead in Europe to collect and disseminate the ever growing data sets of a zoo of astronomical observatories and projects. VizieR webservices nowadays provide access to over 9000 catalogues. Numerous astronomical data centers followed, developing specialized services, for example, the Infrared Processing and Analysis Center and the bibliographical SAO/NASA Astrophysics Data System. In the early 2000's it was realized that the ever growing data volumes require new approaches: the community becomes the data provider and the International Virtual Observatory and its European branch, the European Virtual Observatory, developed standards and interfaces to allow individual data centers to publish their catalogs and images in a common framework providing worldwide access to users who can query, cross-match and visualize multiple databases via the internet. A highlight formed the overplotting of data from different experiments, like Xray satellites and optical ground-based observatories with some keystrokes on the Aladin interactive software sky atlas. While the Virtual Observatory focused on the dissemination of published data it was also realized in the early 2000's that the upcoming data deluge required a new approach to the handling of the data stream from the telescope to the science-ready result. Modern experiments not only involve more data, but also require increasing precision on the various calibrations. The dependency of the final result (sometimes only a few numbers, like cosmological parameter values) on time-variable calibrations of very large data sets, involving the evaluation by large research teams distributed in smaller groups over various sites sets the basic requirements to the system handling the data. While in classical systems data is delivered in various releases to the public, often re-processing the whole set with higher versions of the code, the high data rates of modern experiments require an alternative approach, where the up-to-date result is derived on demand by the user. Thus, we set out to design and implement an integrated datacentric system Astro-WISE in which the processing, storage and administration is integrated in a single environment, providing a living system to both the data producers and the customers. Early reports of this development and implementation have been given in [8] and [9]. To reach this goal, traceability of each individual data item handled by pipelines or any piece of code is carefully maintained, every data item beyond pixel value is kept as Metadata and made persistent and distributed in a relational database with an object-oriented view, mapping all the dependencies.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 Datacentric Approach to Data Processing\",\n \"content\": \"The datacentric approach facilitates scientists to cooperate in data analysis and mining by means of the internet, nowadays referred to as e-science. The AstroWISE information system connects to an own developed distributed grid processing system (based on Distributed Processing Unit, DPU), but has been also connected to the EGEE/EGI/BiGGrid-Grid [10], particularly for the operations of the Lofar radio telescope. Nowadays, this aspect of the system is referred to as cloud computing, but the Astro-WISE datacentric approach at the same time facilitates the sharing and web hosting of all the data handled by the system. Today, we are just taking the first small steps towards the development of approaches and systems who could handle the upcoming data deluge. The way of managing and administration the data will be a key, and will require ever improving, refining and self-organizing approaches. The key notion is that a fruitful approach towards handling the data deluge focuses on the administration, modeling, standardizing and tracking of the data rather than on the processing of the data by CPUs. This datacentric approach, in which the design of operational systems is driven by data management and data standard issues rather than data processing/cpu/pipelining issues. Processing systems turn into information systems. First attempts to combine archiving and processing in the 1980s include ESO's MIDAS table file system [11], integrating common tables both accessible in source code and by user interfaces like the MIDAS monitor/prompt. The MIDAS table file system allows both users and programmers to change its content and also registering the modifications in table keywords leading to archives ready for further review and (trend) analysis. Its development was essential for the production and photometric calibration of the images of the ESO-LV galaxies. However, the 600,000 fore- and background objects visible on the images, stars and galaxies, could only be handled with great difficulties inside the system, response times running into hours per operations, demonstrating the importance of built-in scalability. This MIDAS Table file system marked a stepping stone to current datacentric approaches which are enabled by the introduction of modern object-oriented programming languages, data modeling tools like UML and XML and relational databases. Wehave combined these views, systems and requirements in order to design and build integrated information systems which have the potential for self-organizing and enrichment with new data. Both self-organizing and enrichment are processes in time. In fact, the datacentric approach implies the detailed modeling and awareness of how things change in time. Important changes range from The ideal information system would seamlessly cope with all these changes in time, thus creating a living long term digital preservation environment. Astro-WISE is one of the first systems attempting to reach this goal. For several years many authors predicted the upcoming data avalanche due to the advancements in digital sensor technology. Obviously, this has now started, both scientific experiments such as LHC and Lofar are running into the tens of Petabytes data acquisition regimes, while text, imaging, genome and internet gathered data collections like e-Bay are exceeding similar volumes. This triggered the development of Grid infrastructure that is able to handle Petabytes of data ([12] and [13]). All this has happened in an approximate 20 year time span when the first large digital imaging collection was published, ESO-LV [2], containing scans of 32,000 galaxy images, 4 Gbyte of data filling a room full with tapes and populating the very first version of optical media. By now, the data avalanche is multidisciplinary and worldwide and involves all aspects of our society, from science to commerce, public services and health care. Digital data is gathered, processed, distributed and accessed at a continuously growing rate. And this is only the beginning: in, say, two decades, the same room full with 4 Gbyte of tapes 20 years ago, and 40 Petabyte of tapes today could possibly contain 40 Zetabytes (40 million Petabytes) of data. The growth of the data volume produced by various types of sensor networks met with the growth of the disk capacity and search for new methods in the increasing of the memory storage density (see [14]). In order to be useful all this data has to be organized, administrated, managed and distributed. It took a billion years for cells to develop into extremely complex information systems around the DNA; it has resulted into systems and approaches with an incredible complexity with memory mapping, copying, filtering and distribution mechanisms of data including feedback. Compared to this, our present data organizing mechanisms in IT are still very simple and rudimentary and we are just making the first steps to living self-organizing systems and archives required to deal with and optimally use the ocean of digital data to come. However, also in the cell stochastic processes are important, and absence of organization 'from above' are countered by survival of the fittest type of mechanisms, which appear as self-organizing. Also in human endeavored IT, standards seldom come from above, an exception being the ASCII character standards, mandated in 1968 by U.S. President Lyndon B. Johnson to be used in all computers purchased by the United States federal government. I have also approved recommendations of the Secretary of Commerce regarding standards for recording the Standard Code for Information Interchange on magnetic tapes and paper tapes when they are used in computer operations. All computers and related equipment configurations brought into the Federal Government inventory on and after July 1, 1969, must have the capability to use the Standard Code for Information. Interchange and the formats prescribed by the magnetic tape and paper tape standards when these media are used.In practice, standards are often invented by companies or scientific communities which are discriminated in a complex survival of the fittest battle, which involves that many factors that it appears stochastic. The view, or better requirement, for future self-organizing information systems and its archives is that it organizes itself for all these changes in time. It could be seen as a step towards the extremely complex and self-organizing information systems in a cell. The core problem for the definition of an information system is the definition of information itself. The information can be an input data flow from sensor detectors, an archive of transactions, trends deduced from the analysis of some data and even results of the data modeling or simulations. As result, information systems will host a variety of subsystems defined by its purpose. Examples are a Decision Support System gathering and analyzing trends for a specific business use case (see [15]) or Geographical Information Systems with focus on geospatial data analysis and a visualization of the data. Nevertheless, it is possible to define primary components of any Information System following simple requirements on storage, processing and sharing data. Such a system should include a data model, storage and data processing facilities together with user interfaces. In Astro-WISE we have decided to separate and store all data beyond pixel/measurement data in a database and all pixel/measurement data in files on independent storage media. This paper does not touch details of the implementation of Astro-WISE information system which are described in other papers but describes the general outline of the Astro-WISE information system and specify features which make Astro-WISE unique. For the more in-depth review of particular aspects of AstroWISE we recommend to read corresponding papers on the optical pipeline implemented in Astro-WISE [16], quality control realised in Astro-WISE [17], the system of user interfaces and services [18], and the hardware solution for Astro-WISE and information systems created on the basis of Astro-WISE [19].\",\n \"pages\": [\n 2,\n 3,\n 4,\n 5\n ]\n },\n {\n \"title\": \"3 Building a Scientific Information System\",\n \"content\": \"The development of the Astro-WISE information system, the first one which implemented WISE technology, started from the very practical challenge: enable a community of researchers distributed over the world to process the data of astronomical imaging survey of the OmegaCAM 256 Megapixel camera on ESO's VST telescope at Cerro Paranal (Chile). These scientists should be able to evaluate the quality of the data, apply a number of calibrations, share the data within the team and employ distributed resources of a Petabytes scale of data storage and Teraflops capacity in data processing. From this our basic requirements on the system are derived: The most general requirements listed above results in a set of more detailed and specific requirements to different components of the information system. First, the ability to share data among a number of users becomes more valuable when all these users are working with the same set of standards for the data. Second, the requirement to trace any changes in the data processed by different users with different programs and pipelines implies that the system has a common approach to treat the data items on different stages of the data processing and keeps the record of changes in the data. As result the basic requirements to the system dictate the use of a common data model and the common standards for the data storage. Below we will describe the common data model features defined from the requirements to the system.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"Requirements to the Data Model:\",\n \"content\": \"When properly defined the data model allows to code, instantiate and trace the data processing for all scientific use-cases. Also, the data model should allow to accommodate external data, as well as provide necessary comparison and checking with these external data sets. When the data model is carefully defined our approach is to convert this into a pipeline model, in turn translated into a hierarchy of Classes. These Classes are then mapped into a database. We originally did this with object-oriented databases like Objectivity, but currently we do this in a relational database and build an object oriented view in order to maintain transparency in the class model, and its dependencies. Requirements to Infrastructure. The data model should be shared among geographically remote sites of the system, as research groups at various locations are collaboratively adding and extracting data from the distributed system. The basic requirement is that in fact any component (database, file system, code base, CPUs) should possibly be present at different sites, for a number of reasons: For example, the data ingestion to the system can be done in Garching, Germany, the data processing in Groningen, The Netherlands and the image analysis in Napoli, Italy. Requirements to the User Interfaces. The user should be able to trace the data processing, to retrieve data items and identify all its dependencies on other items (mostly calibrations) in the system, to initiate data processing of particular data items and to find a quality estimation for the data. As we can see, all these requirements put together are more challenging than requirements on common business information systems. The key difference is that we allow everything, including the code base to change in time. In the following chapter we outline how our requirements formed a layout for a scientific information system capable to share a huge data volume, to process these data and fulfill scientific use-cases for a number of users.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"4 WISE Concept\",\n \"content\": \"First of all we have to clearly distinguish between two classes of data storage systems in Astronomy: proper archives containing fixed published data and infor- mation systems. The main difference can be found in an ability to work with the data and to change it during the storage. For an archive stored data is a stable entity which is provided to the user as it is, meanwhile the information system allows to the user to modify their own data creating user-specific version of an archive and share this version with other users. The Astro-WISE approach, and its general multi-discipline WISE approach implements a scientific information system. The usual approach to building an information system is to perceive three components: the data layer, business rules and the interfaces (see Fig. 1). In our case, we separate the data layer in two parts: the pure measurement data layer, hereafter data layer, and everything beyond that, hereafter Metadata layer, ranging from file sizes, to statistics of pixel values and detected events in the measurements (galaxy, particle, word). The Metadata layer allows to implement a data model, the data layer allows to store the data as files in a standard format, while business rules implemented in Python classes bind the metadata and the data layers. Interfaces provide user access to both the business rules and the data. In the case of AstroWISE business rules present in the Metadata layer (Data Definition Language used to create a data model), data layer (the on-the-fly compression of the data on data storage grid) and-the most apparent to the user-in a number of pipelines and programs which the user defines to process the data. This last part of business rules components we will call data processing layer. To implement Astro-WISE we use abstractions of storage, processing and database capabilities as a basis for the infrastructure for each of the layers of the system. The metadata layer is realized in a relational DBMS through an abstraction of the required database functionality, the data layer is put on the Astro-WISE data storage grid through an abstraction of storage and the processing grid is used to connect the user with the data and metadata layers by a number of interfaces for the data processing layer. Separation of these three infrastructures plays a key role in the flexibility of the system, as we will see below. The metadata layer implements a list of necessary functionalities: Our requirements on distribution and multiple users propagate as key principles of the realization of metadata and data layers and business rules which form the core of the WISE approach: The Astro-WISE system is realized in the Python programming language. It allows to wrap any program into a Python module, library or class. The use of Python also allows to combine the principles of modular programming with objectoriented programming, so that each package in the system can be built and run independently with an object-oriented data model serving as glue between modules. At the same time, the logic behind pipelines and workflows in Astro-WISE allows the execution of part of the processing chain independently from the other parts. We will describe this approach in more detail in the example of optical image processing in Section 4.1. The conceptual difference between Astro-WISE and other existing systems is that Astro-WISE moves from the usual for astronomy processing-centric approach to a data-centric approach. The data processing itself becomes an integral part of the archive. The typical solution for the data processing and storage for Astronomy handles the data processing and the final data product delivered to the user as two completely separated entities. The data product usually is a result of the processing for the whole survey with fixed processing parameters used for the whole set of images. The user has access to reduced images and the catalog, both data products are stable within a 'release', which usually refers to the sky coverage performed by the survey. In this way the survey data center provides the user with a specific version of the data product that will not change over time, which covers most of the science use-cases. Nevertheless there are a number of use-cases which can not be satisfied by the 'standard' version. For example, in the search for objects like brown dwarfs or quasars it is important to lower the detection threshold which implies the reprocessing of the data. The user himself has to care about such reprocessing thereby reinventing the whole data processing system for the survey and involving his own resources. The Astro-WISE system allows to work on any use-case using the standard pipelines and performing programming on a minimal level - if this is necessary at all. Figure 1 shows principles of WISE concept in the binding together all layers of the information system. Data processing pipelines define use-cases and data model for the system, which is implemented in the Python classes which wrap pipeline and define the common persistent data model objects. The data layer uses this data model, and interfaces are providing an access to the system for users. The common data model effectively brings together all layers of the information system. The common data model can be modified by a mutial agreement of all users of the system. Any user can propose changes in the data model, and, if changes were accepted, the system administrator implements these changes in corresponding Python classes and in the database scheme. In the next sections we will review in detail the infrastructural layers of AstroWISE and their implementation.\",\n \"pages\": [\n 7,\n 8,\n 9,\n 10\n ]\n },\n {\n \"title\": \"4.1 Metadata Layer\",\n \"content\": \"The bulk of the data stored in Astro-WISE is stored in files of some format (FITS). Each file is registered in the Astro-WISE metadata database with the unique filename. Apart from just registering each file in the system the metadata database implements the important part of WISE approach - the common data model. Figure 2 shows the general outline for all data models implemented in informations systems based on WISE concept. two core classes DBObject and DataObject are parent classes for the class which is storing metadata persistently and the class which is storing the data in the file. These classes include interfaces to the metadata database and the data storage, the physical implementation of the metadata database and the data (files) storage can be different for different systems. Let us suppose that we wish to use Astro-WISE for the data processing of optical images. Figure 3 shows typical classes of data items used by the optical data processing pipeline to reduce the data from raw images to the final science ready catalogues. This data model, deduced from the pipeline and enhanced keeping in mind possible scientific use-cases, is a central part of the metadata layer. The data model is implemented both in the relational database (currently Oracle 11g RAC is used) and in the hierarchy of Python classes. All core classes are made persistent, i.e., any change in the object is mirrored in the corresponding tables of the database. The method used for the implementing of the data lineage is the Persistent Object Hierarchy (Fig. 2). According to this method objects of Astro-WISE are made persistent recursively, all operations and attributes of the object are saved in the metadata database. As an example let us explore the data reduction task - we will create a reduced image from the raw image on one CCD chip. First of all, we will select an image we wish to process: awe> raw = (RawScienceFrame.filename == 'WFI.2000-01-01T08:57:15.410_3.fits')[0] The image is selected by browsing the metadata database for an object of class RawScienceFrame with filename WFI.2000-01-01T08:57:15.410_3.fits . The returned object raw was retrieved from the metadata database with all attributes. In the next step we will retrieve from the database all necessary calibration objects. The logic in the function select_for_raw allows to take the calibration files for the same night of observation ( 2000-01-01T08:57:15.410 ) or closest to this night. awe> hot = HotPixelMap.select_for_raw(raw) awe> cold = ColdPixelMap.select_for_raw(raw) awe> flat = MasterFlatFrame.select_for_raw(raw) awe> bias = BiasFrame.select_for_raw(raw) Now we can instantiate a new object - our target - a reduced image which will be based on all the images above: Please, note that all images which will be used to create a new reduced image reduced are referenced through attributes of the new image. This new image should get a unique ID, which is a name of the file where the image will be stored. awe> reduced.set_filename() Now the image can be created invoking the make() method of the ReducedScienceFrame class. awe> reduced.make() The make() produced a file with a new image which will be stored on one of the dataservers of Astro-WISE (see Section 4.2): awe> reduced.store() Finally, the metadata database will be updated and the new image will become a part of the metadata layer. awe> reduced.commit() The sequence described above is a general way to create a new data entity in Astro-WISE: to combine a set of references to entities which will be used to create a new one, to invoke a pipeline or a part of pipeline which will generate a new entity, save the created data in the data layer, commit a new data entity to the metadata layer. The processing parameters, for example, the method used for the overscan of the image, will be saved as a persistent attribute of the new data entity as well. The important feature of keeping the full data lineage in the system is an ability to avoid unnecessary reprocessing. The execution of the make() method of reduced object actually will start with the search in the metadata database for an object with the same attributes, and if such an object exists (and the user is allowed to retrieve it according to the user's permissions) the user will be redirected to an already existing object and the processing part of the method will be skipped. The data lineage allows to avoid unnecessary reprocessing as well as to use forward and backward chaining in the dependencies of the data items in the system. Objects can be deleted from the database under the following restrictions: every user can delete the data he created at the privileges level 1 (see 4.5). For higher privileges levels only the project manager is allowed to delete. To delete a data object myobject from the database Context class is used (see [16] for the full description):\",\n \"pages\": [\n 10,\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"awe> Context().delete(myobject)\",\n \"content\": \"Objects can be deleted if they are not referenced by other data objects. If they are, it might be desirable to invalidate the object, so that the object will stay in the system but will not be used for the data processing (unless the user specifically ask for invalidated objects). The user-friendly service for the validation of data objects is described in [17].\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"4.2 Data Layer\",\n \"content\": \"Each data entity in Astro-WISE has two parts: a data part which is stored in a file and a metadata which is a part of the metadata database and which keeps all dependencies of the data entity. The data part of the entity is stored as a file on one of the Astro-WISE dataservers. The predefined file format can be changed, but for most files the format is FITS. Nevertheless the dataserver is not limited to a single file format and can store files of any type. The dataserver is a specific solution which in its functionality is closests to the storage element of the Grid. The main requirement to the dataservers as storage space is, apart from the safety, scalable size. The size should be scalable at the Terabyte level allowing to increase the storage volume from a few Terabytes to almost a Petabyte (this is the typical scale for the KIDS data archive from a few raw images at the beginning of the survey to hundreds of Terabytes with all intermediate data products at the end of observations). Dataservers are organized in geographically close clusters, within each cluster all dataservers know about the existence of all other dataservers in the cluster. A dataserver in one cluster can contact other clusters via a dedicated dataserver in another cluster (see Fig. 5). All communications are done using a standard HTTP protocol. Two types of requests are available: to retrieve a file and to store a new file. The unique name of the file is used as a unique identifier and this unique name is stored in the database as part of the metadata. The user of the system does not know the actual location of the file on the underlying file system and operates with the URL of the file only. The URL has the form http:// / . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers. . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers. On the request from the user to retrieve the file the dataserver can then either return the requested file or redirect the client to the dataserver that has the file. Each dataserver has a permanent data storage and a cache, the last one is used to temporarily store a file retrieved from the other dataserver. By user request, for example, http://ds.astro.rug.astro-wise.org:8000/WFI.2000-09-28T02:22:37.466_8.fits the dataserver ds.astro.rug.astro-wise.org will check cache space for the file WFI.2000-09-28T02:22:37.466_8.fits . If the file is not found, the dataserver will check it's own permanent storage space, and if there is no such file, all dataservers in a 'local' cluster ( astro.rug.astro-wise.org ) will be requested. If there is still no file found, all other clusters will be requested, and as soon as the file is located it will be copied to the cache of the ds.astro.rug.astro-wise.org and returned to the user. In addition, a file can be compressed or decompressed on-the-fly during retrieval, and a slice of the data can be retrieved by specifying the slice as part of the URL. The dataserver is written in Python and can be installed on any operating system and underlying filesystem that can support long case-sensitive filenames. For the Astro-WISE Linux filesystems, XFS and GPFS are currently in use.\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"4.3 Processing Layer\",\n \"content\": \"The example in the Section 4.1 involves the processing facilities of Astro-WISE, which are distributed and combined from processing facilities of all Astro-WISE partners. The user has the choice to send the job to one of the processing elements of Astro-WISE (including Grid computing elements) or to use the processing power at the user's disposal, for example, PC or even notebook. In the core of the processing layer of Astro-WISE is a distributed processing unit (DPU). This is a three-component middleware which consists of the DPU server, DPU client and the so called DPU runner. The DPU server is a front-end to any processing system and interacts with the processing system's native queuing system, allowing the user to submit jobs to this particular queuing system, inspect or cancel them. The DPU client includes all functions and methods the user can call to interact with DPU server. The DPU runner is a program which is run on the remote processing facility and checks availability of all necessary software to run an Astro-WISE job, installing required packages if necessary. The system runs on openpbs or under its own queue management software. The DPU itself allows synchronizations of jobs as well and can also transfer parts of a sequence of jobs to other DPU's. In the case of Grid computing element the DPU server will check a user's identity and will use the user's credentials (Grid certificate) to mediate with the Virtual Organization Management System. Currently OmegaCEN is using omegac Virtual Organization with an ability to submit jobs to Grid computing elements in Amsterdam and Groningen.\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"4.4 Interfaces\",\n \"content\": \"The description of Astro-WISE system would be incomplete without the description of a number of interfaces provided for the user. The user can write his own applications in the Python language calling Astro-WISE libraries or can involve Astro-WISE services. The first case requires the use of the Astro-WISE Command Line Interface called AWE (Astro-WISE Environment), which can be installed on any site, PC or notebook The Command Line Interface - CLI - supposes that the user writes his own programs using Python, but to browse the data or even to process observations there is no need to use the CLI. All operations required to perform this activity are possible with a set of standard web services of Astro-WISE. Of course, the use of the CLI gives to the user much more freedom in data processing. The CLI is more useful for experienced users working on a particular use-case, meanwhile web services are developed for routine operations during the data processing of the surveys. The web interfaces are divided into two types: data browsing/exploration and data processing/qualification. The first group includes:\",\n \"pages\": [\n 15\n ]\n },\n {\n \"title\": \"Data processing / qualification interfaces are:\",\n \"content\": \"All web services are built using a set of Python classes developed for AstroWISE as a basis and uses a modular principle, which allows to create a new web service using components of older ones.\",\n \"pages\": [\n 16\n ]\n },\n {\n \"title\": \"4.5 Authorization and Authentication System\",\n \"content\": \"Astro-WISE is a multi-user system which must accommodate sharing data between scientists and at the same time protect the private data of each user. Each user in the Astro-WISE system has an identity protected by a password, and, optionally, if the user wants to submit job to Grid resources he has to get a Grid certificate. The authorization and authentication system is implemented on the level of the metadata database. As soon as a user logged in with his username/password, the user's privileges are checked in the database, allowing the user to browse the data according to the user's privileges while obeying the privileges of other users. The data in the metadata database are grouped by projects . A Project is a collection of resources which is associated with a group of users who can access these resources, usually one of these users has the role of Project Manager who has privileges to include new users, remove users and publish the data to the wider community. The system of access to the data is based on three attributes which any data entity in Astro-WISE has: user , project and privileges . The first one identifies the user that created the data entity. The second one defines to which project the data entity belongs, the third one defines who is able to use this data entity. All these attributes are initialized the first time the data entity is made persistent in the Astro-WISE system (including the case that the entity is created by one of the Astro-WISE pipelines) and they are both persistent attributes, i.e., stored in the metadata database. Table 1 shows the range of values for the privileges attribute. In the case of privileges=1 only the creator of the data item can see it, the creator can raise privileges to privileges=2 , in this case all the users of the project will be able to browse this data entity. Raising privileges to 3 the user makes the data item accessible for all Astro-WISE users in all projects, and with priveleges=4 the anonymous user (a user with minimal read-only privileges in the system) can see it as well. Finally, with priveleges=5 the data item is accessible via Astro-WISE Virtual Observatory interfaces. The special Python class Context was created to handle authorization and authentication. The Context allows to change privileges of all user's data items, during the change Context is checking for dependencies to prevent inconsistency in the dependencies due to the different privileges of data items. For example, if the raw image was published by the user to the project scope and some other user has created a reduced image from this raw image, the original user will not be able to downgrade privileges on the raw image to the private data scope.\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"4.6 WISE architecture\",\n \"content\": \"We described above three infrastructural components of Astro-WISE: the relational DBMS (the metadata layer), dataserver (the data layer) and DPU (the processing layer). Combined with user interfaces these components build an AstroWISE node - a detached Astro-WISE site which can operate independently from other sites. Multiple nodes can be combined to form a distributed system. Each node is independent from others in the sense that it is administrated independently and can handle both the data distributed over other nodes and the data restricted to this node only and shielded from other nodes. Figure 6 shows the elements of the node: All these elements are optional, an Astro-WISE node can be installed without dataservers (dataservers from other nodes are used), metadata database or processing facilities. In fact an Astro-WISE node can be installed on a notebook as Astro-WISE environment only giving the user access to the system - if the user is not satisfied with web services on the remote nodes. Presently Astro-WISE includes sites at Groningen University, Leiden University and Nijmegen University (The Netherlands), Argelander-Institut fur Astronomie, Bonn and Universitats-Sternwarte Munchen (Germany) and Osservatorio Astronomico di Capodimonte, Napoli (Italy).\",\n \"pages\": [\n 17,\n 18\n ]\n },\n {\n \"title\": \"5 Astro-WISE: migration to other systems\",\n \"content\": \"The WISE concept for an information system is a set of principles described in Section 4, Astro-WISE is the first system which realizes this concept and serves as a basis for the further development of information systems. In this section we describe the adaptivity of the parent system to new tasks and challenges on the example of LOFAR Long Term Archive and Molgenis system. The importance of the LOFAR Long Term Archive (LTA) for the development of WISE concept is a necessity to use an external infrastructure which should be included in the architecture to form a complete information system. We preserved all principles of Astro-WISE adding storage and processing which is not controlled by the system itself (BiGGrid 9 ). Additionally we integrated three different systems of Authorization and Authentication to make it possible for the LOFAR observa- tory to create new users and control resources in the system. The LOFAR LTA design is described in [20] and the architecture and infrastructure in [21]. Another significant development was achieved with the adaptation of Molgenis 10 . Molgenis is a framework written in Java to build user interfaces and databases from definitions that are written in XML. From these definitions the database tables and webserver are generated. Its origins lie in biomedical applications. Historically Molgenis focused more on interaction with the user to streamline the 'protocols' (recipes for analysis) of researchers to keep track of analyses of results. On the other hand, the WISE technology focused more on results from a processing perspective. This lead to the idea to combine the strengths of both frameworks and use some models that already existed in XML for Molgenis to generate a datamodel in Python for Astro-Wise. Then Molgenis could be extended to use the WISE infrastructure for distributed storage and processing. The existing Molgenis XML model is organized in a common fashion through 'module', 'entity' and 'field' where each field has a type of int , float , str or can be a reference type such as mref or xref . Most of these have a counterparts in the WISE framework which made it possible to write a conversion tool. The webserver that is generated by Molgenis has to communicate to a database that it usually creates itself. However, in this case the database had to be generated by Astro-Wise because the frameworks did not have a backend for a common database (Oracle vs. Non-Oracle). Instead of writing a new database backend for either framework it was decided to write an xmlrpc interface to encapsulate database queries and return their results. Since client software for all database flavours does not have to be present this functionality can be extended, e.g., since this is xmlrpc, a programming language independent implementation could also use the database in a way that the WISE framework dictates. The approach developed and tested in the case of Molgenis allows to create a new information system based on Astro-WISE with all services starting from the data model coded in XML and to do this automatically. This approach allowed to decrease the time and resources spent for the developing and implementation of a new system significantly. The next system which will be created with the approach tested on Molgenis is a data processing system for Multi Unit Spectroscopic Explorer 11 (MUSE).\",\n \"pages\": [\n 18,\n 19\n ]\n },\n {\n \"title\": \"6 Conclusion and Future Work\",\n \"content\": \"The Astro-WISE information system, the first information system in Astronomy, proved to be a reliable and flexible tool for the data processing. Originally developed to process the data of KIlo Degree Survey (KIDS 12 ) it triggered development of the unique approach to the architecture of scientific information systems (WISE approach). Both Astro-WISE and the WISE approach are living systems which are open to improvements. For the last 2 years further development of the WISE approach is hosted by Target Holding 13 . Target Holding is an expertise center in the Northern Netherlands which is building a cluster of sensor network information systems and provides cooperation between a number of scientific projects and business partners like IBM and Oracle. Target creates and supports a hardware infrastructure for hosting tens of Petabytes of data for projects in astronomy, medicine, artificial intellegence and biology. In this development the WISE approach was used to create new information systems extending the original Astro-WISE on new data models, new data storage and processing capacities and new fields. Acknowledgements Astro-WISE is an on-going project which started from a FP5 RTD programme funded by the EC Action 'Enhancing Access to Research Infrastructures'. This work was performed as part of the Target project. Target project is supported by Samenwerkingsverband Noord Nederland. It operates under the auspices of Sensor Universe. It is also financially supported by the European fund for Regional Development and the Dutch Ministry of Economic Affairs, Pieken in de Delta, the Province of Groningen and the Province of Drenthe.\",\n \"pages\": [\n 19,\n 20\n ]\n }\n]"}}},{"rowIdx":263,"cells":{"bibcode":{"kind":"string","value":"2013ExA....36..451C"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1305.3789.pdf"},"content":{"kind":"string","value":"\nExperimental Astronomy manuscript No.\n(will be inserted by the editor)\nBackground simulations for the Large Area Detector onboard LOFT\nRiccardo Campana · Marco Feroci ·\nEttore Del Monte\n·\nTeresa Mineo ·\nNiels Lund · George W. Fraser\nReceived: date / Accepted: date\nAbstract The Large Observatory For X-ray Timing (LOFT), currently in an assessment phase in the framework the ESA M3 Cosmic Vision programme, is an innovative medium-class mission specifically designed to answer fundamental questions about the behaviour of matter, in the very strong gravitational and magnetic fields around compact objects and in supranuclear density conditions. Having an effective area of ∼ 10 m 2 at 8 keV, LOFT will be able to measure with high sensitivity very fast variability in the X-ray fluxes and spectra. A good knowledge of the in-orbit background environment is essential to assess the scientific performance of the mission and optimize the design of its main instrument, the Large Area Detector (LAD). In this paper the results of an extensive Geant-4 simulation of the instrument will be discussed, showing the main contributions to the background and the design solutions for its reduction and control. Our results show that the current LOFT/LAD design is expected to meet its scientific requirement of a background rate equivalent to 10 mCrab in 2-30 keV, achieving about 5 mCrab in the most important 2-10 keV energy band. Moreover, simulations show an anticipated modulation of the background rate as small as 10% over the orbital timescale. The intrinsic photonic origin of the largest background component also allows for\nOn behalf of the LOFT collaboration.\nR. Campana, M. Feroci, E. Del Monte\nINAF/IAPS, Via Fosso del Cavaliere 100, I-00133, Roma, Italy\nand INFN/Sezione di Roma 2, viale della Ricerca Scientifica 1, I-00133, Roma, Italy.\nE-mail: riccardo.campana@inaf.it\nPresent address of R. Campana:\nINAF/IASF-Bologna, via Gobetti 101, I-40129, Bologna, Italy.\nT. Mineo\nINAF/IASF-Palermo, Via Ugo La Malfa 153, I-90146, Palermo, Italy.\nN. Lund\nDTU-Space, Elektrovej Bld. 327, DK-2800, Kgs. Lyngby, Denmark.\nG. Fraser\nSpace Research Centre, Dept. of Physics and Astronomy, University of Leicester, LE17RH, Leicester, UK.\nan efficient modelling, supported by an in-flight active monitoring, allowing to predict systematic residuals significantly better than the requirement of 1%, and actually meeting the 0.25% science goal.\nKeywords X-ray astronomy · Instrumental background · Montecarlo simulations\nPACS 07.85.Fv · 07.87.+v · 95.40.+s\n1 Introduction\nAstrophysical observations in the X-ray domain are often dominated by the background. In order to assess and maximize the scientific performance of a satellite-borne instrument during its design phase, is essential to have an early estimation of the background level and to identify its origin. This will allow to drive the optimization of the experiment design itself. Moreover, a reliable estimation of the background level is important to assess the scientific objectives and performance of the mission. The most convenient and consolidated way to compute such a background is by means of a Monte Carlo simulation, using e.g. the Geant-4 toolkit [1], in which the interactions of the space environment with a somewhat simplified mass model of the spacecraft and of the instruments are sampled and studied. A correct evaluation of the radiation environment surrounding the experiment is of course the most important ingredient in the background simulation.\nIn this paper we will show Monte Carlo simulations of the instrumental background for the Large Area Detector (LAD) instrument onboard the proposed LOFT (Large Observatory for X-ray Timing) satellite [2,3], that is a candidate for the third slot of medium-class missions (M3) of the Cosmic Vision 2015-2025 programme of the European Space Agency, with a possible launch in the 2022-24 timeframe.\nLOFT (Figure 1) has been specially designed to answer the 'Fundamental Question' #3.3 of the Cosmic Vision programme, Matter under extreme conditions . In the aim of investigating the behaviour of the matter in the most extreme physical conditions that can be found in the gravitational and magnetic fields around neutron stars and black holes, LOFT will study the rapid variability of the X-ray emission from these objects, both in the spectral and temporal domains. Moreover, LOFT will be an observatory for virtually all classes of bright X-ray sources: among them, X-ray bursters, high mass X-ray binaries, cataclysmic variables, magnetars and active galactic nuclei.\nThe main experiment onboard LOFT is the LAD [3] (Sect. 2), a collimated instrument with a collecting area 20 times bigger than its immediate predecessor (PCA, the Proportional Counter Array onboard the Rossi X-ray Timing Explorer, RXTE), and based on large-area Silicon Drift Detectors (SDDs) coupled to lead-glass Micro Channel Plate (MCP) collimators. In order to survey a large fraction of the sky simultaneously, and to trigger follow-up observations\nwith the main instrument, LOFT will host also a coded-mask Wide Field Monitor (WFM, [4]). This latter instrument, sensitive in the same energy range of LAD, will use basically the same SDD detectors, with some design differences to optimize the detector response for imaging [5,6,7].\nThe background level required to satisfy the LOFT scientific objectives, in particular regarding the relatively faint (1-10 mCrab) sources like most Active Galactic Nuclei (AGNs) and some accretion-powered X-ray pulsars, is ≤ 10 mCrab for the 2-30 keV LAD standard band. Being a collimated instrument, the LAD will not measure simultaneously and independently the source flux and the background. While some collimated instruments (e.g. BeppoSAX/PDS and RXTE/HEXTE, both at hard X-rays) employ a 'rocking' strategy, where the same instrument unit alternates observations of the source and of the background, others (e.g. RXTE/PCA, at soft X-rays, similar to LAD) rely on a background model obtained by means of dedicated blank-field pointings and by using onboard ratemeters [8]. As we will discuss in the following, LAD will adopt an approach similar to PCA. To ensure the fulfillment of the scientific objectives, in particular for low flux sources, an accurate estimation and modelling of the background level for this instrument is of particular importance. The background variations that have a significant impact on the scientific performance should be minimized, and controlled to a high degree of accuracy. The LOFT science case resulted in a requirement on background knowledge at 1%, with a goal at 0.25%. As a comparison, RXTE/PCA reached a background residual systematic level of ∼ 0.5%-1% [8,9], while BeppoSAX/PDS of ∼ 1% [10].\nThis paper is structured as follows. In Section 2 we describe the LAD instrument. In Section 3 we introduce the space environment for LOFT and discuss the main contributions to the instrumental background, while in Section 4 the main features of the LAD Monte Carlo mass model and simulator are described. The simulation results are shown in Section 5, and in Section 6 we draw our conclusions.\n2 The Large Area Detector\nThe Large Area Detector is an array of 2016 individual large-area Silicon Drift Detectors, sensitive to the 2-30 keV emission (with a extended 30-80 keV band) collimated in a ∼ 1 · field of view by means of a lead-glass microcapillary collimator plate. The instrument design is modular: an array of 4 × 4 SDDs and their front-end electronics (FEE) are placed in a module . Each module has its independent back-end electronics (MBEE) that powers and configure the FEE, manages the event readout and monitors the detector health. A panel is composed of an array of 21 modules, placed on a support grid (Figure 2).\nThe current LAD design by the LOFT consortium envisages 6 panels (each about 350 cm × 90 cm), that stems from the satellite bus and are deployed once the spacecraft reaches its orbit (Figure 1). The collimators, made of lead glass, have the same footprint as the SDDs (about 11 cm × 7 cm), a\n\n\n
Fig. 1 A sketch of the LOFT satellite and its instruments.
\n\n\n\n
Fig. 2 Left : a LAD module, showing the mounting of the collimator, SDD and the FEE. Center : Back-side view of the module, showing the radiative surface and the back-end electronics. Right : a LAD Detector Panel.
\n\nthickness of 6 mm, and 100 µ m wide square holes separated with 20 µ m thick walls (and thus having an open area ratio , f OAR , of ∼ 70%). An aluminium frame couples the collimators with the detectors frame and ensures the tight alignment constraints.\nThe working principle of the SDDs is the following: a photon is absorbed in the depleted silicon bulk, and produces a cloud of electrons whose number is proportional to the incident energy. The cloud drifts towards an array of read-out anodes, by means of a high-voltage field applied between the middle section and the two detector edges, hosting two independent rows of anodes. During the drift, the cloud size increases due to diffusion. The number of anodes that will read-out the charge cloud thus depends on the absorption point distance and anode pitch [5]. For the LAD detectors, that have a pitch of 970 µ m, the charge cloud originating from the absorption of a X-ray photon in the 2-80 keV band is at most 0.5 mm (FWHM) thus read-out by 1 or 2 anodes, depending on the impact point and the energy. At 6 keV, for example,\n∼ 40% of the events are collected by one anode and ∼ 60% by two anodes. The event multiplicity is read-out and transmitted to the ground, enabling a selection of 'single' and 'double' events, with different energy resolutions.\n3 Sources of background\nIn this section we discuss the main sources of the LOFT background. We describe the environment encountered in the baseline low-Earth orbit, with its hadronic, leptonic and photon components. Moreover, we describe the internal source of background, i.e. the natural radioactivity of the collimator material.\n3.1 The LOFT orbit\nThe LOFT orbit requirement is at an altitude of h = 600 km with an inclination of 5 · . Likewise, lower-altitude orbits (down to 550 km) and/or inclinations (down to 0 · ) are considered, with small improvements for what regards the conclusions about the background rate and properties. The lower altitude and inclinations, however, impacts on the long-term radiation damage on the detectors. For these orbits, the geomagnetic latitude range is usually θ M < 0 . 3 radians.\n\n\n
Fig. 3 Values of the trapped proton intensity (AP8MIN model) for the LOFT baseline orbit. Data obtained from the SPENVIS software ( http://www.spenvis.oma.be ). The South Atlantic Anomaly is well apparent above South America.
\n\nIn the current work we neglected the possible background component due to the activation of the instrument and satellite materials. Based on the arguments below, we estimated it to be a second-order effect. The South Atlantic Anomaly, a region of trapped high-energy protons and electrons, is grazed by the LOFT baseline orbit only in its outermost regions (Figure 3), with a shallower passage for lower altitudes and inclinations. Therefore, the effect on background due to the activation of materials by this intense radiation is expected to be negligible with respect to other sources, as confirmed by preliminary evaluations and heritage from past missions in similar orbits (e.g., BeppoSAX and AGILE). It should also be noted that in contrast to previous experiments, the LAD instrument has a very light structure per unit volume and the relatively small spacecraft assembly is seen at a small viewing angle, due to the tower supporting the LAD panels (see Figure 1). In addition to that, as for example observed by Swift/XRT [11], Silicon detectors show marginal activation effects in LEO, even when operated as focal plane instruments (more local mass) and on more inclined orbits (up to ∼ 20 · ). However, given our goal of controlling the LAD background to the highest possible accuracy, we plan to study this aspect in great detail in a future work.\n3.2 Primary cosmic rays\nFollowing the approach of Mizuno et al. [12], we assume that the primary cosmic ray spectrum is expressed as F U ( R ), a function of the particle magnetic rigidity, R = pc/Ze , where Z is the atomic number and p the particle momentum.\nThe full spectrum at a given location in the magnetosphere and a given phase of the solar cycle will be given by:\nF ( E ) = F U ( E + Zeφ ) × F M ( E,M,Z,φ ) × C ( R,h,θ M ) (1)\nwhere M and Ze are the mass and charge of the particle, E its kinetic energy, φ is a solar modulation factor, h is the orbit height and θ M is the geomagnetic latitude.\nThe second term, that includes the effects of the solar modulation on the cosmic ray particles, is given by [13]:\nF M ( E,M,Z,φ ) = ( E + Mc 2 ) 2 -( Mc 2 ) 2 ( E + Zeφ + Mc 2 ) 2 -( Mc 2 ) 2 (2)\nwhere the solar modulation potential φ is given by:\nφ = 0 . 55 GV at solar minimum (3)\nφ = 1 . 10 GV at solar maximum (4)\nThe geomagnetic cutoff function is given, for vertically incident particles, by [12]:\nC ( R,h,θ M ) = 1 1 + ( R/R cut ) -r (5)\nin which the cutoff rigidity is obtained from the Stormer equation [14]:\nR cut = 14 . 5 × ( 1 + h R E ) -2 cos 4 θ M GV (6)\nwhere R E = 6371 km is the Earth radius. Moreover, the r exponent in Eq. 5 is:\nr = 12 for protons (7)\nr = 6 for electrons and positrons (8)\nFor the 600 km, 5 · inclination LOFT orbit, the cutoff rigidity therefore is found to be in the range R cut ∼ 10.1-12.1 GV.\nSince we are interested mostly in the average background fluxes, we discard at the present stage the east-west effect for which particles coming from different directions have different cutoff rigidities. We therefore assume that the general primary spectrum, now expressed as:\nF ( R ) = F U ( E + Zeφ ) × ( E + Mc 2 ) 2 -( Mc 2 ) 2 ( E + Zeφ + Mc 2 ) 2 -( Mc 2 ) 2 × 1 1+( R/R cut ) -r (9)\nhas an uniform angular distribution with respect to the zenith angle, up to the Earth horizon, i.e. for zenith angles θ from 0 · to θ cut , where the latter is given by:\nθ cut = arcsin ( 1 h R E +1 ) (10)\nFor h = 600 km, θ cut ∼ 114 · . The solid angle subtended by the Earth at an altitude h is given by:\nΩ E = 2 π ( 1 -1 R E /h +1 √ 1 + 2 R E h ) (11)\nand therefore the accessibile sky subtends a solid angle Ω = 4 π -Ω E (Figure 4). For h = 600 km, the Earth blocks about 30% of the sky.\n3.2.1 Primary protons\nThe unmodulated value of the cosmic ray electron spectrum is given by the BESS [15] and AMS measurements [16]:\nF U ( E ) = 23 . 9 × [ R ( E ) GV ] -2 . 83 particles m -2 s -1 sr -1 MeV -1 (12)\nIn Figure 5 the primary proton spectra for different values of the solar modulation and for the LOFT orbit is reported. The variation in flux between solar minimum and maximum, at the peak flux, is ∼ 15%.\n\n\n
Fig. 4 The fraction of the sky blocked by the Earth, for a given orbit altitude h .
\n\n\n\n
Fig. 5 Model spectrum of the primary protons for the LOFT orbit. The red data points are the AMS-01 measurements of the proton spectrum, primary and secondary, for the Shuttle orbit (380 km) and low magnetic latitude ( θ M < 0 . 2), from [16].
\n\n3.2.2 Primary electrons and positrons\nThe unmodulated value of the cosmic ray electron spectrum is given by [17, 18,12]:\nF U ( E ) = 0 . 65 × [ R ( E ) GV ] -3 . 3 particles m -2 s -1 sr -1 MeV -1 (13)\nThe fraction of positrons to electrons, usually given by the ratio e + / ( e + + e -), is found to be rather independent of the energy [19], i.e. the spectrum of primary positrons has the same slope of the electron one, but a different normalization:\nF U ( E ) = 0 . 051 × [ R ( E ) GV ] -3 . 3 particles m -2 s -1 sr -1 MeV -1 (14)\nIn Figure 6 the primary e -and e + spectra for different values of the solar modulation and for the LOFT orbit are reported. The difference in flux between solar minumum and maximum is about 20% at the peak.\n\n\n
Fig. 6 Model spectrum of the primary electrons and positrons for the LOFT orbit.
\n\n3.2.3 Primary alpha particles\nWe consider only the primary helium nuclei, because the contribution of heavier nuclei is negligible with respect to the other primary particles flux [20].\nThe unmodulated flux, as given by Mizuno et al. [12] on the basis of AMS and BESS data is:\nF U ( E ) = 1 . 5 × [ R ( E ) GV ] -2 . 77 particles m -2 s -1 sr -1 MeV -1 (15)\nand is shown in Figure 7.\n5\n\n\n
Fig. 7 Model spectrum of the primary helium nuclei for the LOFT orbit.
\n\n3.3 Secondary particles\n3.3.1 Protons\nFor the low altitude equatorial Earth orbits considered, the impinging proton spectrum outside the trapped particle belt, i.e. the South Atlantic Anomaly, consists of the primary component discussed in Sect. 3.2 and of a secondary, quasi-trapped component, originating from and impacting to the Earth atmosphere (sometimes in the literature they are referred to as the 'splash' and 'reentrant' albedo components). AMS measurements [16] showed that this secondary component is composed of a short-lived and a long-lived particle population, both originating from the regions near the geomagnetic equator. Mizuno et al. [12] model the secondary equatorial proton spectrum as a cutoff\npower-law:\nF ( E ) = 0 . 123 × ( E 1 GeV ) -0 . 155 e -( E/ 0 . 51) 0 . 845 particles m -2 s -1 sr -1 MeV -1\n(16)\nand a power law for energies below 100 MeV:\nF ( E ) = 0 . 136 × ( E 100 MeV ) -1 particles m -2 s -1 sr -1 MeV -1 (17)\nThis spectrum is shown in Figure 8 together with the AMS measurements. According to the measurements performed with the NINA and NINA-II instruments [21], the extrapolation below 100 MeV is likely an overestimate.\nWe do not consider here the soft (10 keV-1 MeV), highly directional equatorial proton population [22,23], since these particles are most efficiently stopped by all the spacecraft structures surrounding the detectors. Even the small fraction that impinges on the detector through the capillary holes, albeit significant when considering the long-term radiation damage on the SDDs, leaves a negligible background signal.\n\n\n
Fig. 8 Secondary proton spectrum. The red data points are the AMS-01 measurements of the proton spectrum, both primary and secondary, for the Shuttle orbit (380 km) and low magnetic latitude ( θ M < 0 . 2, from [16]), while the black line is the Mizuno et al. [12] analytic form.
\n\n3.3.2 Electrons and positrons\nThe equatorial secondary electron spectrum can be approximated with a broken power law [12]:\nF ( E ) = 0 . 3 × ( E 100 MeV ) -2 . 2 for 100 MeV < E < 3 GeV (18)\nF ( E ) = 0 . 3 × ( 3 GeV 100 MeV ) -2 . 2 ( E 3 GeV ) -4 for E ≥ 3 GeV (19)\nand a power law for energies below 100 MeV:\nF ( E ) = 0 . 3 × ( E 100 MeV ) -1 particles m -2 s -1 sr -1 MeV -1 (20)\nAt variance with respect to the primary particles, in the geomagnetic equatorial region the positrons are predominant with respect to the electrons. The spectrum is the same, but the ratio e + /e -is about 3.3 (Figure 9).\n\n\n
Fig. 9 Secondary electron and positron spectrum. Analytic form of Mizuno et al. [12].
\n\n3.4 Photon background\nAs we will see in the following Sections, the main contribution to the overall background for the LAD instrument is due to the high energy X-ray emission transmitted and scattered in the lead-glass collimators. The main contribution comes from the CXB and the Earth albedo emission.\n3.4.1 Cosmic photon background\nFor the cosmic X-ray and γ -ray diffuse background we assume the Gruber et al. [24] analytic form, derived from HEAO-1 A4 and COMPTEL measurements, and fitted in the energy range between 3 keV and 100 GeV. This spectrum is usually used as a standard reference [25,26], and is plotted in Figure 10.\nThe first branch is valid below 60 keV:\nF ( E ) = 7 . 877 × ( E 1 keV ) -1 . 29 e -E/ 41 . 13 photons cm -2 s -1 sr -1 keV -1 (21)\nwhile for energies above 60 keV:\nF ( E ) = 0 . 0004317 × ( E 60 keV ) -6 . 5 +0 . 0084 × ( E 60 keV ) -2 . 58 (22) +0 . 00048 × ( E 60 keV ) -2 . 05 photons cm -2 s -1 sr -1 keV -1\nOther measurements of the CXB have been performed by, e.g., BeppoSAX [25], INTEGRAL [27,28] and BAT [26]. These measurements are consistent with the HEAO-1 results, albeit some of these seems to indicate a slightly higher normalization ( ∼ 8%) for the > 10 keV spectrum.\n3.4.2 Albedo γ -ray background\nThe secondary photon background is due to the interactions of cosmic rays (proton and leptonic components) with the atmosphere, and to the reflection of CXB emission. As such, it has a strong zenith dependence [29]. This albedo component has a higher flux, for unit of solid angle, than the CXB for energies above ∼ 70 keV. We assume the albedo spectrum as measured by BAT [26], that agrees in the range above 50 keV, after some corrections, with previous measurements [30,31]. This spectrum can be parameterized by the function [32,26]:\nF ( E ) = 0 . 0148 ( E 33 . 7 keV ) -5 + ( E 33 . 7 keV ) 1 . 72 photons cm -2 s -1 sr -1 keV -1 (23)\nand is plotted in Figure 11.\n\n\n
Fig. 10 The cosmic X-ray diffuse background as derived from HEAO-1 A4 measurements [24] (solid line). CXB observations performed by PDS [25] and BAT [26] are also shown.
\n\n\n\n
Fig. 11 Albedo γ -ray observations performed by BAT [26] and its parameterisation (Eq. 23).
\n\n3.5 Earth neutron albedo\nAs reported by the official ESA ECSS documents 1 , there is presently no model for atmospheric albedo neutron fluxes considered mature enough to be used as a standard. To account for the flux of neutrons produced by cosmic-ray interactions in the Earth atmosphere, we however used the results of the QinetiQ Atmospheric Radiation Model (QARM, [33,34]), that uses a response function approach based on Monte Carlo radiation transport codes to generate directional fluxes of atmospheric secondary radiation. The resulting spectrum, shown in Figure 12, is also consistent with the Monte Carlo simulations of Armstrong & Colborn [35,36].\n3\n\n\n
Fig. 12 Model spectrum of the Earth albedo neutrons for the LOFT orbit.
\n\n3.6 Natural radioactivity\nThe lead-glass used in the microcapillary plate collimators contains potassium, in a fraction of approximately 7.2% by weight. The activity due to the naturally occurring long-lived radioactive isotope 40 K (with an abundance of 0.0117% with respect to natural potassium) is to be taken into account. This isotope decays with the emission of β rays having a continuum of energies up to 1.31 MeV ( ∼ 89% branching ratio) and of 1.46 MeV photons ( ∼ 11% b.r.) due to\nelectronic capture. Let L , ρ , f OAR be the thickness, density and open fraction of the collimator. Therefore, the collimator activity for unit area is:\nA coll = A K · M K (24)\nwhere A K ∼ 30 Bq/g is the specific activity of natural potassium due to 40 K, and M K is the total potassium mass per unit area in the collimator, i.e.\nM K = L (1 -f OAR ) f K (25)\nwhere f K is the fraction by weight of potassium in the collimator glass. For the MCP lead glass, we have f K ∼ 7 . 2%. Assuming for the MCP collimator L = 6 mm, ρ = 3 . 3 g/cm 3 , f OAR = 69 . 44% we obtain M K = 436 g/m 2 . Therefore, we have A coll = 13100 Bq/m 2 . For a total geometrical LAD collimator area of ∼ 15 m 2 we have thus ∼ 196500 decays/s. Considering the branching ratio, we have therefore that every second in the collimator bulk are uniformly and isotropically generated ∼ 175000 electrons with a continuum of energies and ∼ 21000 monochromatic photons with a 1.46 MeV energy.\n4 The GEANT-4 LOFT/LAD simulator\n4.1 The mass model\nSimulations were performed using the Geant-4 Monte Carlo toolkit [1] (version 4.9.4). Geant-4 allows to describe the geometry and the materials of the instruments and of the satellite bus. Moreover, the code enables to follow the various physical interactions along the path of a primary event through the various components of the geometry, evaluating the secondary particles and energy deposits generated.\nFor the description of the electromagnetic interactions, the Low Energy Livermore library 2 is used, that trace the interactions of electrons and photons with matter down to about 250 eV, using interpolated data tables. Physical processes like photoelectric effect, Compton and Rayleigh scattering, pair production, bremsstrahlung, multiple scattering and annihilation are simulated, optionally taking into account also the effect of photon polarization. Fluorescence X-rays and Auger electrons from the various chemical elements are included. The hadronic physics, instead, is described using different models in different energy ranges (e.g. Bertini Cascade model, Quark-Gluon String model, etc. 3 ). Both elastic and inelastic processes are treated, the latter describing e.g. nuclear capture, fragmentation, de-excitation and scattering of the relevant stable and long-lived nucleons and mesons.\nIn Figure 13 is shown the mass model geometry used in the simulations for the LAD instrument. A single LAD panel is simulated, with the actual dimensions ( ∼ 350 cm × 90 cm) but with a simplified design that consists of a\nstacked-layers geometry (sketched in Figure 14). A simplified geometry for the satellite bus, the structural tower and the other five panels is assumed, using aluminium as the material with an 'effective' density that takes into account the total mass and volume.\n\n\n
Fig. 13 The Geant-4 LOFT/LAD mass model.
\n\nThe SDD array is represented as a 450 µ m thick slab of silicon, subdivided in 0.97 mm × 35 mm pixels. On its surface are placed various passive layers (Al cathode implants, SiO 2 passivation layer, undepleted Silicon bulk), the MCP collimator and the optical/thermal filter (1 µ m kapton coupled with a 80 nm Al layer). The MCP collimator has an 'effective', reduced density, to take into account the holes without resorting to a more accurate but much more timeconsuming geometrical description. Simulations on a smaller geometrical size have been performed to compare the 'effective' collimator with a complete geometrical capillary plate description (see Sect. 5.2), giving consistent results. Underneath the detectors, the FEE board is simulated as a 2 mm thick FR4 slab with an interposed 100 µ mCu conductive layer and a 100 µ mAl shielding layer, a 1 mm thick Al radiator and a 500 µ m Pb backshield. An aluminium frame encloses the module sides, while the panel structure consists of a carbonfiber reinforced plastic grid frame.\n\n\n
Fig. 14 Sketch of the simulated Geant-4 LOFT/LAD module geometry.
\n\n4.2 Primary event generation\nFor the generation of primary events, the General Particle Source 4 (GPS) approach is followed, that allows to produce arbitrary spectra in rather complex geometries. The initial events (particles or photons) are generated on the surface of a sphere of radius R , with a cosine-biased emission angle to ensure an isotropic flux. The emission angle is further restricted between 0 (normal to the spherical surface) to θ max : the emission cone then subtends a smaller sphere of radius r that surrounds the experiment. We have:\nθ max = arctan ( r R ) . (26)\nLet Φ be the energy-integrated flux, between E min and E max , expressed in particles cm -2 s -1 sr -1 . The total rate is therefore:\nN r = Φ 4 π 2 R 2 sin 2 θ max (27)\nThis derives from the integration over the 2 π emission angle for a point on the spherical surface, biased with the cosine law:\n2 π 0 dφ π/ 2 0 dθ cos θ sin θ = π (28)\n∫ ∫\nand integrated over the source sphere surface S = 4 πR 2 . The restriction on the emission angle introduces the further factor sin 2 θ max .\nTherefore the simulation time that corresponds to N events is:\ngenerated primary\nτ = N Φ 4 π 2 R 2 sin 2 θ max (29)\nIf we detect C i counts in the energy bin i , the corresponding measured flux (in counts cm -2 s -1 ), i.e. convolved with the detector response, would be:\nF i = C i τA det (30)\nwhere A det is the detector sensitive area.\n4.3 Anode multiplicity rejection filter\nFor the LAD, an anode multiplicity rejection algorithm is implemented to filter out ionization streaks from charged particles, that leave an energy deposit on more than 2 adjacent anodes. More in detail, for each of the two independent halves of a SDD tile, events are rejected if they trigger more than two adjacent anodes. Likewise, events are rejected if there is a group of non-adjacent triggering anodes within a 'distance' of 32 channels in the same half-SDD (Figure 15). Energy deposits at a further distance are treated as independent.\nSince the average energy loss of minimum ionizing particles (MIPs) in silicon is about 3.7 MeV/cm, their total energy deposit is usually above ∼ 150 keV. The anode multiplicity rejection filter, combined together with an upper threshold on the reconstructed event signal, is very effective in the suppression of the particle-induced background: a threshold of 80 keV allow to filter out 94%96% of the particle background, depending on the incoming particle type and spectrum. Of course, since this event elaboration is performed in the frontend electronics, dead-time calculations should be performed using the total incoming background rate. The acceptance efficiency of this filter, defined as the fraction of photons from a 'true', on-axis X-ray source that survive the multiplicity rejection, is very high. For a Crab-like spectrum, the efficiency is 99.98%.\n5 Simulation results\nThe simulations have been performed by generating an isotropic flux of primary events (photons or particles, the latter evaluated at the Solar minimum, the worst case condition) on a sphere surrounding the experiment, recording the energy deposits in the detector pixels and then applying the proper filter for the event multiplicity (Sect. 4.3). The resulting counts have been then properly normalized, taking into account the energy spectra and solid angles of the simulated background contributions.\n\n\n
Fig. 15 The anode multiplicity rejection filter. Four SDDs are conceptually shown, each subdivided into two independent halves. Darker rectangles represent triggering anodes. Events a and b are single-anode energy deposits, while c and d are double-anode events: all of these are accepted by the filter. On the contrary, triple-anode events ( e ) or disconnected multiple-anode events ( f, g, h ) are rejected.
\n\nBefore discussing the general results for the LAD background and a preliminary assessment of its stability, in the next two subsections we deal with some subleties that required a more detailed geometrical description and event processing.\n5.1 Capillary reflections\nA standalone, more detailed raytracing simulator has been built for the MCP to evaluate the effects of the grazing incidence angle reflectivity from the pore inner walls for the CXB photons collected in the ∼ 1 · × 1 · field of view. The code tracks photons from their incoming direction up to the detector plane modeling their interaction with the optical/thermal filter and the collimator structure. The simulated collimator has a full geometrical description, 6 mm thick filled with square pores 100 µ m wide and 20 µ m thick walls. The absorption and transmission probability of the optical/thermal filter elements are computed from tables derived from the database of the National Insti-\ntandards and Technology 5 ; the lead glass attenuation coefficient is derived from tabulated values (G. Fraser, priv. comm.). Reflectivity from the pore walls, as derived from a set of laboratory measurements (G. Fraser, priv. comm.) is well modeled by data derived from CXRO database 6 corresponding to lead glass with a 11.8 nm surface micro-roughness value. Results from this code, when reflection is not taken into account, are well in agreement with those obtained by the Geant-4 simulator (Figure 16). The effect of the wall reflectivity produces an increase of the aperture CXB (Mineo et al., in preparation), going from 40% at energies lower than 3 keV, down to ∼ 10% between 5 and 10 keV and below 5% above 10 keV.\n\n\n
Fig. 16 Comparison of results between the Geant-4 code and a stand-alone raytracing simulator developed to evaluate the effects of the reflection of grazing-incident photons on the inner walls of the collimator.
\n\n5.2 Internal 40 K activity\nNot all the potassium decay products (see Sect. 3.6) will leave a signal in the detector: the emission is isotropic, and the photon (or the electron) has to\ncross different thicknesses of lead glass, at various pitch angles, before reaching the SDD. Moreover, the efficiency of the silicon detector should be taken into account; eventually, only a fraction of the detected events will have a pulse height amplitude corresponding to the standard LAD energy band (2-30 keV).\nPrimary particles (1.46 MeV monochromatic photons and electrons having energies between 1 keV and 1311 keV) have been generated randomly in the lead-glass walls of a ∼ 1 . 2 × 1 . 2 cm complete model of a MCP geometry and mass (6 mm thickness, 100 µ m square holes, 20 µ m wall thickness, ∼ 70% OAR) coupled to a SDD detector. It is found that the decay of one 40 K atom generating a 1.46 MeV photon has a very small probability to generate a signal with equivalent amplitude between 2 and 30 keV, obtaining we obtain a background rate of only ∼ 5 × 10 -5 counts cm -2 s -1 . Therefore, for all practical purposes, we can discard the effect of the 40 K electronic capture decay channel. Electrons produced by the β decay, on the contrary, have a higher chance to cause a signal, producing a background rate of ∼ 2 × 10 -3 counts cm -2 s -1 .\n5.3 The LAD background\nThe total resulting LAD background is shown in Figure 17, while the breakdown of the count rate in its various components is shown in Table 1. No detector energy resolution smoothing of the spectra was included here.\nThe main background contribution is due to the high-energy photons from the diffuse cosmic X-ray background and the Earth albedo that leaks from and are scattered in the collimators. These two components alone accounts for about 70% of the background count rate in the 2-30 keV band.\nThe diffuse emission collected in the field of view (even including the effect of capillary reflectivity), the particle-induced and internal backgrounds are a smaller contribution to the total count rate. While the CXB emission collected in the aperture field of view has a significant contribution only below ∼ 10 keV, cosmic-ray particles and neutrons produce almost flat spectra, similarly to the internal activity background, and these components become dominant only above ∼ 20 keV. The dips in the neutron-induced spectrum are due to inelastic scattering resonances.\nFluorescence emission from the Pb contained in the collimator glass ( L -shell lines at 10.55 and 12.61 keV) is well apparent, artificially emphasized by the non-inclusion of the energy resolution in the plot. Analysis are ongoing to evaluate whether these lines are to be shielded or used as in-flight calibration lines. In the plot is also shown, as a reference to the LAD science requirement, the spectrum of a 10 mCrab point-like source (dashed line). The total background count rate in the 2-30 keV band corresponds to a flux of ∼ 9 mCrab, thus ensuring the fulfillment of the scientific requirements, and it is below 5 mCrab in the most important 2-10 keV band.\nWe can estimate a margin of error on the background rate, taking into account that the CXB emission and the Earth albedo fluxes are affected by\na maximum error of ∼ 10%-20% on their normalization [26,28]. The particle components are more uncertain [12,16] (see also Sect. 3.2). However, these have been conservatively assumed at the Solar minimum, while LOFT, if selected, will fly around the next Solar maximum ( ∼ 2024), when the particle flux is expected to be approximately 20% less intense. We consider a ∼ 50% error on these fluxes. Weighting the above uncertainties on the LAD background components, we thus estimate an overall conservative maximum margin of error on the total background rate of ∼ 20%, given the present geometrical model. Future developments in the mission design will allow for a more accurate mass model and consequently to refine these results. Therefore, we can conclude that the LOFT scientific requirements are expected to be met.\n\n\n
Fig. 17 The LAD total background and its various components discussed in the text. The spectrum corresponding to a 10 mCrab source is shown as a dashed line. Note that the latter spectrum and the aperture CXB one have been normalized relatively to the other curves according to the reduced effective area that includes the collimator OAR (i.e. ∼ 10 m 2 at 8 keV).
\n\n5.4 Background variability\nThe main components of the LAD background are shown in Figure 17. The high energy photons from CXB emission and the Earth γ -ray albedo leaking through the collimator represent the dominant background component.\n
\n\n
Table 1 The LAD background contributions. The last line shows the LAD requirement for the total background level in the 2-30 keV band.
\n
\nAlthough they are intrinsically steady and predictable, due to their different intensity and spectra a varying relative orientation of the LAD in their radiation environment will cause a small and smooth modulation of the detected background, on the orbital timescales ( ∼ 90 minutes). This is due to the varying viewing geometry along the orbit and for different attitudes. This effect has been studied through simulations, finding that the maximum expected modulation of the background is less than 10%. This value has to be compared to a factor of a few for instruments dominated by particle-induced background. For example, RXTE/PCA had up to a ∼ 250% variation on orbital timescales [8]. In the LOFT case, the effect of the other potentially varying sources, i.e. particle induced background, is greatly reduced by the very stable environment offered by the low Earth equatorial orbit and by the fact that this component accounts for less than 6% of the overall background.\nIn Figure 18 the background rate is shown as a function of the angle between the LAD pointing direction and the center of the Earth. θ E = 0 · represents the Earth center aligned with the field of view, while θ E = 180 · stands for the Earth at the instrument nadir. In practice, this corresponds to the orbital modulation for a low-declination source. The curve representing the total background (black symbols) shows a maximum modulation of ∼ 8%. The modulation of the background rate is entirely due to geometrical effect and it can be predicted and modeled, thanks to the intrinsic steadiness of the relevant sources. Each background component is well modeled as a function of the Earth location with respect to the pointing direction by a sum of two Gaussian distributions, centered at about θ E = 0 · and 180 · for the Earthoriginated components and at about θ E = 120 · and 240 · for the diffuse sky components. These distributions, also reported in Figure 18, arise from the convolution of the directional 'transparency' of the LAD instrument with the Earth-occulted field of view. Of course, when the detector points towards the zenith ( θ E = 180 · ) the contribution from the leaking CXB emission is maximum. The overall convolution of these out-of-phase components is to give a very small fluctuation of the total background. Moreover, for an actual pointing towards an astrophysical source, the range of possible Earth angles θ E is restricted, from a θ E = 90 · for a source at the orbital pole (that is nearly coin-\npole) to the full 0 · -180 · range for equatorial sources. This lowers the background modulation further below the maximum during the observation of a realistic scientific target. The geometrical model, properly calibrated using in-orbit flat fields, is anticipated to allow for a background prediction at a level significantly better than 1% in the 2-10 keV band, which is the LAD science requirement. Such a level of systematic uncertainty was indeed already reached by the past experiment RXTE/PCA, which, in the presence of a much more variable background level (250% vs 8%) and less predictable background sources, with an appropriate modeling reached the level of ∼ 1% [8,9].\nHowever, as some of the LOFT science cases (in particular the extragalactic science) will benefit from reaching a background knowledge significantly better than the requirement, an active background monitoring was designed for the LAD, to further improve the modeling. Due to the slow and smooth background variation, there is no need for a high-statistics, instantaneous monitoring of the rate. Rather, a continuous benchmark of the slow modulation will allow for a real-time verification of the background model. This active background monitoring is achieved by the introduction of a 'blocked collimator' (a collimator with the same stopping power but no holes) for an area corresponding to one Module of the LAD. This will enable the continuous monitor of all components of the LAD background, with the exception of the aperture background, accounting for ∼ 90% of the total background, and also evaluating the long-term variations (e.g. linked to the Solar cycle). Preliminary simulations for different targets (i.e., attitude configurations) show that the accuracy in the background modeling during a typical observation can be improved down to ∼ 0.1%-0.3% by using these additional data.\nThe subject of time-dependent modeling of the background is being further studied in depth to definitely assess the ultimate instrument capability for weak sources. Here we just reported a few preliminary results allowing to identify the range of interest. An exhaustive report and discussion is deferred to a dedicated paper, currently in preparation.\n5.5 Absolute background level: cosmic variance and source contamination\nIn the previous section we preliminarily addressed the issue of background stability and modeling during a single observation. This is the key parameter for most of the science driver of observations of weak sources, where relative variations of source features will be studied (e.g., the Fe line in AGNs). Anyway, a few science cases may require an estimate of the absolute value of the background. This faces an intrinsic physical property of the CXB, known as cosmic variance [37]: the flux of the CXB is not perfectly isotropic but varies on different angular scales. The variation was measured [37,38,39] as ∼ 7% (rms) on the angular scale of 1 · and ∼ 2% on 20 · -40 · . In the case of the LAD, this affects only the aperture background, as the CXB/albedo contamination is seen from all directions. The expected variation in the LAD background\n\n\n
Fig. 18 Non-aperture background modulation due to the varying position of the Earth with respect to the pointing direction. θ E is the polar angle between the LAD pointing axis and the center of the Earth. Blue points are the CXB-induced contribution, orange points the Earth γ -ray albedo, cyan points the albedo neutrons and red points the particles. The black triangles represent the total background, while the black continuous curve is the background model. All quantities have been scaled to the average isotropic background count rate in the 2-30 keV band.
\n\ndue to the cosmic variance is given by the fraction of the total rate due to the aperture CXB (13%) weighted by the variance over the field of view ( ∼ 7%), which is about 0.9%. Given its intrinsic astrophysical nature, the only way of measuring this component is through a local blank field, just as for any other imaging or non-imaging experiment. This is indeed the plan for the few science cases requiring the absolute knowledge of the background rate rather than its stability. It is worth stressing that the cosmic variance does not affect at all the background stability in time for a given target observation.\nA further small contribution to the 'local background' (i.e., for a given target/attitude) is given by the possible contamination due to bright and hard sources outside the field of view. Similar to the leaking of diffuse CXB/albedo photons through the collimator, the same effect can occur to photons from point-like sources. We investigated this effect by simulations, and the maximum contribution to the overall background is of the order of 1-3% (Crab-like sources, in 2-30 keV) down to 0.1% (softer sources, e.g. Sco X-1). This additional minor contribution will be monitored by the active background monitoring system, as well as by the Wide Field Monitor. The quantitative assessment of the small effect on the background stability when the contaminating sources\nare variable on the time scale of interest will be extensively addressed in the paper in preparation mentioned earlier.\n6 Conclusions\nLOFT will be an innovative mission that will observe compact Galactic and extragalactic objects in both the spectral and the temporal domains. The unprecedented sensitive area of the LAD instrument will open new windows in the study of the fundamental physics allowed by these natural laboratories. The scientific objectives of the mission require an accurate knowledge and minimisation of the detector background. To this end, an extensive massmodel for the LOFT/LAD experiment has been developed, using the standard Geant-4 toolkit (Sect. 4), and all the main components (photonic, leptonic, hadronic and internal, Sect. 3) of the orbital background environment have been simulated.\nThe main contribution to the overall background is found to be due to the diffuse X-rays that 'leaks' from the lead-glass collimators of the instrument. This emission originates from the diffuse cosmic X-ray background and from the Earth albedo. The particle-induced background is minimised mainly thanks to the LOFT low-inclination, low-altitude orbit and small mass for unit area of the LAD experiment, becoming dominant only at high energies, above 30 keV. A further suppression of the particle background is enabled by the particular signature that these events leave on the Silicon Drift Detectors (Sect. 4.3). The simulations show the feasibility for the current LAD instrument design to fulfill the required background level of 10 mCrab in the 2-30 keV band.\nBackground variations on orbital timescales are mostly induced by the varying geometry between the position of the Earth and the pointing direction, As such, they can be modelled and accounted for. The use of special detector units ('blocked' module) and a carefully planned blank-field pointing strategy, are foreseen to monitor these background variations enabling to reach a residual systematics level better than 0.3%. The residual contribution of strong (and variable) off-axis sources, above 30-50 keV, if needed, can be further modelled using observations performed with the other instrument onboard LOFT, the WFM.\nAn increase in the accuracy of the background determination for the LAD will be provided by further improvements in the geometrical description of the instrument itself and of the spacecraft bus structures. Furthermore, results will be refined by taking into account a complete determination of the residual activation of the detector materials following the grazing passages in the highbackground environment of the South Atlantic Anomaly, where however no scientific observations are conducted.\nAcknowledgements We are grateful to the anonymous referee whose helpful comments greatly improved the text, and to Alessandra De Rosa for useful discussions. We acknowl-\nedge financial support from ASI/INAF contract I/021/12/0. RC furthermore acknowledge support from INAF Tecno-PRIN 2009 grant.\nReferences\n\n1. S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce, M. Asai, D. Axen, S. Banerjee, G. Barrand, F. Behner, L. Bellagamba, J. Boudreau, L. Broglia, A. Brunengo, H. Burkhardt, S. Chauvie, J. Chuma, R. Chytracek, G. Cooperman, G. Cosmo, P. Degtyarenko, A. dell'Acqua, G. DePaola, D. Dietrich, R. Enami, A. Feliciello, C. Ferguson, H. Fesefeldt, G. Folger, F. Foppiano, A. 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Fraser Received: date / Accepted: date Abstract The Large Observatory For X-ray Timing (LOFT), currently in an assessment phase in the framework the ESA M3 Cosmic Vision programme, is an innovative medium-class mission specifically designed to answer fundamental questions about the behaviour of matter, in the very strong gravitational and magnetic fields around compact objects and in supranuclear density conditions. Having an effective area of ∼ 10 m 2 at 8 keV, LOFT will be able to measure with high sensitivity very fast variability in the X-ray fluxes and spectra. A good knowledge of the in-orbit background environment is essential to assess the scientific performance of the mission and optimize the design of its main instrument, the Large Area Detector (LAD). In this paper the results of an extensive Geant-4 simulation of the instrument will be discussed, showing the main contributions to the background and the design solutions for its reduction and control. Our results show that the current LOFT/LAD design is expected to meet its scientific requirement of a background rate equivalent to 10 mCrab in 2-30 keV, achieving about 5 mCrab in the most important 2-10 keV energy band. Moreover, simulations show an anticipated modulation of the background rate as small as 10% over the orbital timescale. The intrinsic photonic origin of the largest background component also allows for On behalf of the LOFT collaboration. R. Campana, M. Feroci, E. Del Monte INAF/IAPS, Via Fosso del Cavaliere 100, I-00133, Roma, Italy and INFN/Sezione di Roma 2, viale della Ricerca Scientifica 1, I-00133, Roma, Italy. E-mail: riccardo.campana@inaf.it Present address of R. Campana: INAF/IASF-Bologna, via Gobetti 101, I-40129, Bologna, Italy. T. Mineo INAF/IASF-Palermo, Via Ugo La Malfa 153, I-90146, Palermo, Italy. N. Lund DTU-Space, Elektrovej Bld. 327, DK-2800, Kgs. Lyngby, Denmark. G. Fraser Space Research Centre, Dept. of Physics and Astronomy, University of Leicester, LE17RH, Leicester, UK. an efficient modelling, supported by an in-flight active monitoring, allowing to predict systematic residuals significantly better than the requirement of 1%, and actually meeting the 0.25% science goal. Keywords X-ray astronomy · Instrumental background · Montecarlo simulations PACS 07.85.Fv · 07.87.+v · 95.40.+s","pages":[1,2]},{"title":"1 Introduction","content":"Astrophysical observations in the X-ray domain are often dominated by the background. In order to assess and maximize the scientific performance of a satellite-borne instrument during its design phase, is essential to have an early estimation of the background level and to identify its origin. This will allow to drive the optimization of the experiment design itself. Moreover, a reliable estimation of the background level is important to assess the scientific objectives and performance of the mission. The most convenient and consolidated way to compute such a background is by means of a Monte Carlo simulation, using e.g. the Geant-4 toolkit [1], in which the interactions of the space environment with a somewhat simplified mass model of the spacecraft and of the instruments are sampled and studied. A correct evaluation of the radiation environment surrounding the experiment is of course the most important ingredient in the background simulation. In this paper we will show Monte Carlo simulations of the instrumental background for the Large Area Detector (LAD) instrument onboard the proposed LOFT (Large Observatory for X-ray Timing) satellite [2,3], that is a candidate for the third slot of medium-class missions (M3) of the Cosmic Vision 2015-2025 programme of the European Space Agency, with a possible launch in the 2022-24 timeframe. LOFT (Figure 1) has been specially designed to answer the 'Fundamental Question' #3.3 of the Cosmic Vision programme, Matter under extreme conditions . In the aim of investigating the behaviour of the matter in the most extreme physical conditions that can be found in the gravitational and magnetic fields around neutron stars and black holes, LOFT will study the rapid variability of the X-ray emission from these objects, both in the spectral and temporal domains. Moreover, LOFT will be an observatory for virtually all classes of bright X-ray sources: among them, X-ray bursters, high mass X-ray binaries, cataclysmic variables, magnetars and active galactic nuclei. The main experiment onboard LOFT is the LAD [3] (Sect. 2), a collimated instrument with a collecting area 20 times bigger than its immediate predecessor (PCA, the Proportional Counter Array onboard the Rossi X-ray Timing Explorer, RXTE), and based on large-area Silicon Drift Detectors (SDDs) coupled to lead-glass Micro Channel Plate (MCP) collimators. In order to survey a large fraction of the sky simultaneously, and to trigger follow-up observations with the main instrument, LOFT will host also a coded-mask Wide Field Monitor (WFM, [4]). This latter instrument, sensitive in the same energy range of LAD, will use basically the same SDD detectors, with some design differences to optimize the detector response for imaging [5,6,7]. The background level required to satisfy the LOFT scientific objectives, in particular regarding the relatively faint (1-10 mCrab) sources like most Active Galactic Nuclei (AGNs) and some accretion-powered X-ray pulsars, is ≤ 10 mCrab for the 2-30 keV LAD standard band. Being a collimated instrument, the LAD will not measure simultaneously and independently the source flux and the background. While some collimated instruments (e.g. BeppoSAX/PDS and RXTE/HEXTE, both at hard X-rays) employ a 'rocking' strategy, where the same instrument unit alternates observations of the source and of the background, others (e.g. RXTE/PCA, at soft X-rays, similar to LAD) rely on a background model obtained by means of dedicated blank-field pointings and by using onboard ratemeters [8]. As we will discuss in the following, LAD will adopt an approach similar to PCA. To ensure the fulfillment of the scientific objectives, in particular for low flux sources, an accurate estimation and modelling of the background level for this instrument is of particular importance. The background variations that have a significant impact on the scientific performance should be minimized, and controlled to a high degree of accuracy. The LOFT science case resulted in a requirement on background knowledge at 1%, with a goal at 0.25%. As a comparison, RXTE/PCA reached a background residual systematic level of ∼ 0.5%-1% [8,9], while BeppoSAX/PDS of ∼ 1% [10]. This paper is structured as follows. In Section 2 we describe the LAD instrument. In Section 3 we introduce the space environment for LOFT and discuss the main contributions to the instrumental background, while in Section 4 the main features of the LAD Monte Carlo mass model and simulator are described. The simulation results are shown in Section 5, and in Section 6 we draw our conclusions.","pages":[2,3]},{"title":"2 The Large Area Detector","content":"The Large Area Detector is an array of 2016 individual large-area Silicon Drift Detectors, sensitive to the 2-30 keV emission (with a extended 30-80 keV band) collimated in a ∼ 1 · field of view by means of a lead-glass microcapillary collimator plate. The instrument design is modular: an array of 4 × 4 SDDs and their front-end electronics (FEE) are placed in a module . Each module has its independent back-end electronics (MBEE) that powers and configure the FEE, manages the event readout and monitors the detector health. A panel is composed of an array of 21 modules, placed on a support grid (Figure 2). The current LAD design by the LOFT consortium envisages 6 panels (each about 350 cm × 90 cm), that stems from the satellite bus and are deployed once the spacecraft reaches its orbit (Figure 1). The collimators, made of lead glass, have the same footprint as the SDDs (about 11 cm × 7 cm), a thickness of 6 mm, and 100 µ m wide square holes separated with 20 µ m thick walls (and thus having an open area ratio , f OAR , of ∼ 70%). An aluminium frame couples the collimators with the detectors frame and ensures the tight alignment constraints. The working principle of the SDDs is the following: a photon is absorbed in the depleted silicon bulk, and produces a cloud of electrons whose number is proportional to the incident energy. The cloud drifts towards an array of read-out anodes, by means of a high-voltage field applied between the middle section and the two detector edges, hosting two independent rows of anodes. During the drift, the cloud size increases due to diffusion. The number of anodes that will read-out the charge cloud thus depends on the absorption point distance and anode pitch [5]. For the LAD detectors, that have a pitch of 970 µ m, the charge cloud originating from the absorption of a X-ray photon in the 2-80 keV band is at most 0.5 mm (FWHM) thus read-out by 1 or 2 anodes, depending on the impact point and the energy. At 6 keV, for example, ∼ 40% of the events are collected by one anode and ∼ 60% by two anodes. The event multiplicity is read-out and transmitted to the ground, enabling a selection of 'single' and 'double' events, with different energy resolutions.","pages":[3,4,5]},{"title":"3 Sources of background","content":"In this section we discuss the main sources of the LOFT background. We describe the environment encountered in the baseline low-Earth orbit, with its hadronic, leptonic and photon components. Moreover, we describe the internal source of background, i.e. the natural radioactivity of the collimator material.","pages":[5]},{"title":"3.1 The LOFT orbit","content":"The LOFT orbit requirement is at an altitude of h = 600 km with an inclination of 5 · . Likewise, lower-altitude orbits (down to 550 km) and/or inclinations (down to 0 · ) are considered, with small improvements for what regards the conclusions about the background rate and properties. The lower altitude and inclinations, however, impacts on the long-term radiation damage on the detectors. For these orbits, the geomagnetic latitude range is usually θ M < 0 . 3 radians. In the current work we neglected the possible background component due to the activation of the instrument and satellite materials. Based on the arguments below, we estimated it to be a second-order effect. The South Atlantic Anomaly, a region of trapped high-energy protons and electrons, is grazed by the LOFT baseline orbit only in its outermost regions (Figure 3), with a shallower passage for lower altitudes and inclinations. Therefore, the effect on background due to the activation of materials by this intense radiation is expected to be negligible with respect to other sources, as confirmed by preliminary evaluations and heritage from past missions in similar orbits (e.g., BeppoSAX and AGILE). It should also be noted that in contrast to previous experiments, the LAD instrument has a very light structure per unit volume and the relatively small spacecraft assembly is seen at a small viewing angle, due to the tower supporting the LAD panels (see Figure 1). In addition to that, as for example observed by Swift/XRT [11], Silicon detectors show marginal activation effects in LEO, even when operated as focal plane instruments (more local mass) and on more inclined orbits (up to ∼ 20 · ). However, given our goal of controlling the LAD background to the highest possible accuracy, we plan to study this aspect in great detail in a future work.","pages":[5,6]},{"title":"3.2 Primary cosmic rays","content":"Following the approach of Mizuno et al. [12], we assume that the primary cosmic ray spectrum is expressed as F U ( R ), a function of the particle magnetic rigidity, R = pc/Ze , where Z is the atomic number and p the particle momentum. The full spectrum at a given location in the magnetosphere and a given phase of the solar cycle will be given by: where M and Ze are the mass and charge of the particle, E its kinetic energy, φ is a solar modulation factor, h is the orbit height and θ M is the geomagnetic latitude. The second term, that includes the effects of the solar modulation on the cosmic ray particles, is given by [13]: where the solar modulation potential φ is given by: The geomagnetic cutoff function is given, for vertically incident particles, by [12]: in which the cutoff rigidity is obtained from the Stormer equation [14]: where R E = 6371 km is the Earth radius. Moreover, the r exponent in Eq. 5 is: For the 600 km, 5 · inclination LOFT orbit, the cutoff rigidity therefore is found to be in the range R cut ∼ 10.1-12.1 GV. Since we are interested mostly in the average background fluxes, we discard at the present stage the east-west effect for which particles coming from different directions have different cutoff rigidities. We therefore assume that the general primary spectrum, now expressed as: has an uniform angular distribution with respect to the zenith angle, up to the Earth horizon, i.e. for zenith angles θ from 0 · to θ cut , where the latter is given by: For h = 600 km, θ cut ∼ 114 · . The solid angle subtended by the Earth at an altitude h is given by: and therefore the accessibile sky subtends a solid angle Ω = 4 π -Ω E (Figure 4). For h = 600 km, the Earth blocks about 30% of the sky.","pages":[6,7]},{"title":"3.2.1 Primary protons","content":"The unmodulated value of the cosmic ray electron spectrum is given by the BESS [15] and AMS measurements [16]: In Figure 5 the primary proton spectra for different values of the solar modulation and for the LOFT orbit is reported. The variation in flux between solar minimum and maximum, at the peak flux, is ∼ 15%.","pages":[7]},{"title":"3.2.2 Primary electrons and positrons","content":"The unmodulated value of the cosmic ray electron spectrum is given by [17, 18,12]: The fraction of positrons to electrons, usually given by the ratio e + / ( e + + e -), is found to be rather independent of the energy [19], i.e. the spectrum of primary positrons has the same slope of the electron one, but a different normalization: In Figure 6 the primary e -and e + spectra for different values of the solar modulation and for the LOFT orbit are reported. The difference in flux between solar minumum and maximum is about 20% at the peak.","pages":[9]},{"title":"3.2.3 Primary alpha particles","content":"We consider only the primary helium nuclei, because the contribution of heavier nuclei is negligible with respect to the other primary particles flux [20]. The unmodulated flux, as given by Mizuno et al. [12] on the basis of AMS and BESS data is: and is shown in Figure 7. 5","pages":[9,10]},{"title":"3.3.1 Protons","content":"For the low altitude equatorial Earth orbits considered, the impinging proton spectrum outside the trapped particle belt, i.e. the South Atlantic Anomaly, consists of the primary component discussed in Sect. 3.2 and of a secondary, quasi-trapped component, originating from and impacting to the Earth atmosphere (sometimes in the literature they are referred to as the 'splash' and 'reentrant' albedo components). AMS measurements [16] showed that this secondary component is composed of a short-lived and a long-lived particle population, both originating from the regions near the geomagnetic equator. Mizuno et al. [12] model the secondary equatorial proton spectrum as a cutoff power-law: (16) and a power law for energies below 100 MeV: This spectrum is shown in Figure 8 together with the AMS measurements. According to the measurements performed with the NINA and NINA-II instruments [21], the extrapolation below 100 MeV is likely an overestimate. We do not consider here the soft (10 keV-1 MeV), highly directional equatorial proton population [22,23], since these particles are most efficiently stopped by all the spacecraft structures surrounding the detectors. Even the small fraction that impinges on the detector through the capillary holes, albeit significant when considering the long-term radiation damage on the SDDs, leaves a negligible background signal.","pages":[10,11]},{"title":"3.3.2 Electrons and positrons","content":"The equatorial secondary electron spectrum can be approximated with a broken power law [12]: and a power law for energies below 100 MeV: At variance with respect to the primary particles, in the geomagnetic equatorial region the positrons are predominant with respect to the electrons. The spectrum is the same, but the ratio e + /e -is about 3.3 (Figure 9).","pages":[12]},{"title":"3.4 Photon background","content":"As we will see in the following Sections, the main contribution to the overall background for the LAD instrument is due to the high energy X-ray emission transmitted and scattered in the lead-glass collimators. The main contribution comes from the CXB and the Earth albedo emission.","pages":[13]},{"title":"3.4.1 Cosmic photon background","content":"For the cosmic X-ray and γ -ray diffuse background we assume the Gruber et al. [24] analytic form, derived from HEAO-1 A4 and COMPTEL measurements, and fitted in the energy range between 3 keV and 100 GeV. This spectrum is usually used as a standard reference [25,26], and is plotted in Figure 10. The first branch is valid below 60 keV: while for energies above 60 keV: Other measurements of the CXB have been performed by, e.g., BeppoSAX [25], INTEGRAL [27,28] and BAT [26]. These measurements are consistent with the HEAO-1 results, albeit some of these seems to indicate a slightly higher normalization ( ∼ 8%) for the > 10 keV spectrum.","pages":[13]},{"title":"3.4.2 Albedo γ -ray background","content":"The secondary photon background is due to the interactions of cosmic rays (proton and leptonic components) with the atmosphere, and to the reflection of CXB emission. As such, it has a strong zenith dependence [29]. This albedo component has a higher flux, for unit of solid angle, than the CXB for energies above ∼ 70 keV. We assume the albedo spectrum as measured by BAT [26], that agrees in the range above 50 keV, after some corrections, with previous measurements [30,31]. This spectrum can be parameterized by the function [32,26]: and is plotted in Figure 11.","pages":[13]},{"title":"3.5 Earth neutron albedo","content":"As reported by the official ESA ECSS documents 1 , there is presently no model for atmospheric albedo neutron fluxes considered mature enough to be used as a standard. To account for the flux of neutrons produced by cosmic-ray interactions in the Earth atmosphere, we however used the results of the QinetiQ Atmospheric Radiation Model (QARM, [33,34]), that uses a response function approach based on Monte Carlo radiation transport codes to generate directional fluxes of atmospheric secondary radiation. The resulting spectrum, shown in Figure 12, is also consistent with the Monte Carlo simulations of Armstrong & Colborn [35,36]. 3","pages":[15]},{"title":"3.6 Natural radioactivity","content":"The lead-glass used in the microcapillary plate collimators contains potassium, in a fraction of approximately 7.2% by weight. The activity due to the naturally occurring long-lived radioactive isotope 40 K (with an abundance of 0.0117% with respect to natural potassium) is to be taken into account. This isotope decays with the emission of β rays having a continuum of energies up to 1.31 MeV ( ∼ 89% branching ratio) and of 1.46 MeV photons ( ∼ 11% b.r.) due to electronic capture. Let L , ρ , f OAR be the thickness, density and open fraction of the collimator. Therefore, the collimator activity for unit area is: where A K ∼ 30 Bq/g is the specific activity of natural potassium due to 40 K, and M K is the total potassium mass per unit area in the collimator, i.e. where f K is the fraction by weight of potassium in the collimator glass. For the MCP lead glass, we have f K ∼ 7 . 2%. Assuming for the MCP collimator L = 6 mm, ρ = 3 . 3 g/cm 3 , f OAR = 69 . 44% we obtain M K = 436 g/m 2 . Therefore, we have A coll = 13100 Bq/m 2 . For a total geometrical LAD collimator area of ∼ 15 m 2 we have thus ∼ 196500 decays/s. Considering the branching ratio, we have therefore that every second in the collimator bulk are uniformly and isotropically generated ∼ 175000 electrons with a continuum of energies and ∼ 21000 monochromatic photons with a 1.46 MeV energy.","pages":[15,16]},{"title":"4.1 The mass model","content":"Simulations were performed using the Geant-4 Monte Carlo toolkit [1] (version 4.9.4). Geant-4 allows to describe the geometry and the materials of the instruments and of the satellite bus. Moreover, the code enables to follow the various physical interactions along the path of a primary event through the various components of the geometry, evaluating the secondary particles and energy deposits generated. For the description of the electromagnetic interactions, the Low Energy Livermore library 2 is used, that trace the interactions of electrons and photons with matter down to about 250 eV, using interpolated data tables. Physical processes like photoelectric effect, Compton and Rayleigh scattering, pair production, bremsstrahlung, multiple scattering and annihilation are simulated, optionally taking into account also the effect of photon polarization. Fluorescence X-rays and Auger electrons from the various chemical elements are included. The hadronic physics, instead, is described using different models in different energy ranges (e.g. Bertini Cascade model, Quark-Gluon String model, etc. 3 ). Both elastic and inelastic processes are treated, the latter describing e.g. nuclear capture, fragmentation, de-excitation and scattering of the relevant stable and long-lived nucleons and mesons. In Figure 13 is shown the mass model geometry used in the simulations for the LAD instrument. A single LAD panel is simulated, with the actual dimensions ( ∼ 350 cm × 90 cm) but with a simplified design that consists of a stacked-layers geometry (sketched in Figure 14). A simplified geometry for the satellite bus, the structural tower and the other five panels is assumed, using aluminium as the material with an 'effective' density that takes into account the total mass and volume. The SDD array is represented as a 450 µ m thick slab of silicon, subdivided in 0.97 mm × 35 mm pixels. On its surface are placed various passive layers (Al cathode implants, SiO 2 passivation layer, undepleted Silicon bulk), the MCP collimator and the optical/thermal filter (1 µ m kapton coupled with a 80 nm Al layer). The MCP collimator has an 'effective', reduced density, to take into account the holes without resorting to a more accurate but much more timeconsuming geometrical description. Simulations on a smaller geometrical size have been performed to compare the 'effective' collimator with a complete geometrical capillary plate description (see Sect. 5.2), giving consistent results. Underneath the detectors, the FEE board is simulated as a 2 mm thick FR4 slab with an interposed 100 µ mCu conductive layer and a 100 µ mAl shielding layer, a 1 mm thick Al radiator and a 500 µ m Pb backshield. An aluminium frame encloses the module sides, while the panel structure consists of a carbonfiber reinforced plastic grid frame.","pages":[16,17]},{"title":"4.2 Primary event generation","content":"For the generation of primary events, the General Particle Source 4 (GPS) approach is followed, that allows to produce arbitrary spectra in rather complex geometries. The initial events (particles or photons) are generated on the surface of a sphere of radius R , with a cosine-biased emission angle to ensure an isotropic flux. The emission angle is further restricted between 0 (normal to the spherical surface) to θ max : the emission cone then subtends a smaller sphere of radius r that surrounds the experiment. We have: Let Φ be the energy-integrated flux, between E min and E max , expressed in particles cm -2 s -1 sr -1 . The total rate is therefore: This derives from the integration over the 2 π emission angle for a point on the spherical surface, biased with the cosine law: and integrated over the source sphere surface S = 4 πR 2 . The restriction on the emission angle introduces the further factor sin 2 θ max . Therefore the simulation time that corresponds to N events is: generated primary If we detect C i counts in the energy bin i , the corresponding measured flux (in counts cm -2 s -1 ), i.e. convolved with the detector response, would be: where A det is the detector sensitive area.","pages":[18,19]},{"title":"4.3 Anode multiplicity rejection filter","content":"For the LAD, an anode multiplicity rejection algorithm is implemented to filter out ionization streaks from charged particles, that leave an energy deposit on more than 2 adjacent anodes. More in detail, for each of the two independent halves of a SDD tile, events are rejected if they trigger more than two adjacent anodes. Likewise, events are rejected if there is a group of non-adjacent triggering anodes within a 'distance' of 32 channels in the same half-SDD (Figure 15). Energy deposits at a further distance are treated as independent. Since the average energy loss of minimum ionizing particles (MIPs) in silicon is about 3.7 MeV/cm, their total energy deposit is usually above ∼ 150 keV. The anode multiplicity rejection filter, combined together with an upper threshold on the reconstructed event signal, is very effective in the suppression of the particle-induced background: a threshold of 80 keV allow to filter out 94%96% of the particle background, depending on the incoming particle type and spectrum. Of course, since this event elaboration is performed in the frontend electronics, dead-time calculations should be performed using the total incoming background rate. The acceptance efficiency of this filter, defined as the fraction of photons from a 'true', on-axis X-ray source that survive the multiplicity rejection, is very high. For a Crab-like spectrum, the efficiency is 99.98%.","pages":[19]},{"title":"5 Simulation results","content":"The simulations have been performed by generating an isotropic flux of primary events (photons or particles, the latter evaluated at the Solar minimum, the worst case condition) on a sphere surrounding the experiment, recording the energy deposits in the detector pixels and then applying the proper filter for the event multiplicity (Sect. 4.3). The resulting counts have been then properly normalized, taking into account the energy spectra and solid angles of the simulated background contributions. Before discussing the general results for the LAD background and a preliminary assessment of its stability, in the next two subsections we deal with some subleties that required a more detailed geometrical description and event processing.","pages":[19,20]},{"title":"5.1 Capillary reflections","content":"A standalone, more detailed raytracing simulator has been built for the MCP to evaluate the effects of the grazing incidence angle reflectivity from the pore inner walls for the CXB photons collected in the ∼ 1 · × 1 · field of view. The code tracks photons from their incoming direction up to the detector plane modeling their interaction with the optical/thermal filter and the collimator structure. The simulated collimator has a full geometrical description, 6 mm thick filled with square pores 100 µ m wide and 20 µ m thick walls. The absorption and transmission probability of the optical/thermal filter elements are computed from tables derived from the database of the National Insti- tandards and Technology 5 ; the lead glass attenuation coefficient is derived from tabulated values (G. Fraser, priv. comm.). Reflectivity from the pore walls, as derived from a set of laboratory measurements (G. Fraser, priv. comm.) is well modeled by data derived from CXRO database 6 corresponding to lead glass with a 11.8 nm surface micro-roughness value. Results from this code, when reflection is not taken into account, are well in agreement with those obtained by the Geant-4 simulator (Figure 16). The effect of the wall reflectivity produces an increase of the aperture CXB (Mineo et al., in preparation), going from 40% at energies lower than 3 keV, down to ∼ 10% between 5 and 10 keV and below 5% above 10 keV.","pages":[20,21]},{"title":"5.2 Internal 40 K activity","content":"Not all the potassium decay products (see Sect. 3.6) will leave a signal in the detector: the emission is isotropic, and the photon (or the electron) has to cross different thicknesses of lead glass, at various pitch angles, before reaching the SDD. Moreover, the efficiency of the silicon detector should be taken into account; eventually, only a fraction of the detected events will have a pulse height amplitude corresponding to the standard LAD energy band (2-30 keV). Primary particles (1.46 MeV monochromatic photons and electrons having energies between 1 keV and 1311 keV) have been generated randomly in the lead-glass walls of a ∼ 1 . 2 × 1 . 2 cm complete model of a MCP geometry and mass (6 mm thickness, 100 µ m square holes, 20 µ m wall thickness, ∼ 70% OAR) coupled to a SDD detector. It is found that the decay of one 40 K atom generating a 1.46 MeV photon has a very small probability to generate a signal with equivalent amplitude between 2 and 30 keV, obtaining we obtain a background rate of only ∼ 5 × 10 -5 counts cm -2 s -1 . Therefore, for all practical purposes, we can discard the effect of the 40 K electronic capture decay channel. Electrons produced by the β decay, on the contrary, have a higher chance to cause a signal, producing a background rate of ∼ 2 × 10 -3 counts cm -2 s -1 .","pages":[21,22]},{"title":"5.3 The LAD background","content":"The total resulting LAD background is shown in Figure 17, while the breakdown of the count rate in its various components is shown in Table 1. No detector energy resolution smoothing of the spectra was included here. The main background contribution is due to the high-energy photons from the diffuse cosmic X-ray background and the Earth albedo that leaks from and are scattered in the collimators. These two components alone accounts for about 70% of the background count rate in the 2-30 keV band. The diffuse emission collected in the field of view (even including the effect of capillary reflectivity), the particle-induced and internal backgrounds are a smaller contribution to the total count rate. While the CXB emission collected in the aperture field of view has a significant contribution only below ∼ 10 keV, cosmic-ray particles and neutrons produce almost flat spectra, similarly to the internal activity background, and these components become dominant only above ∼ 20 keV. The dips in the neutron-induced spectrum are due to inelastic scattering resonances. Fluorescence emission from the Pb contained in the collimator glass ( L -shell lines at 10.55 and 12.61 keV) is well apparent, artificially emphasized by the non-inclusion of the energy resolution in the plot. Analysis are ongoing to evaluate whether these lines are to be shielded or used as in-flight calibration lines. In the plot is also shown, as a reference to the LAD science requirement, the spectrum of a 10 mCrab point-like source (dashed line). The total background count rate in the 2-30 keV band corresponds to a flux of ∼ 9 mCrab, thus ensuring the fulfillment of the scientific requirements, and it is below 5 mCrab in the most important 2-10 keV band. We can estimate a margin of error on the background rate, taking into account that the CXB emission and the Earth albedo fluxes are affected by a maximum error of ∼ 10%-20% on their normalization [26,28]. The particle components are more uncertain [12,16] (see also Sect. 3.2). However, these have been conservatively assumed at the Solar minimum, while LOFT, if selected, will fly around the next Solar maximum ( ∼ 2024), when the particle flux is expected to be approximately 20% less intense. We consider a ∼ 50% error on these fluxes. Weighting the above uncertainties on the LAD background components, we thus estimate an overall conservative maximum margin of error on the total background rate of ∼ 20%, given the present geometrical model. Future developments in the mission design will allow for a more accurate mass model and consequently to refine these results. Therefore, we can conclude that the LOFT scientific requirements are expected to be met.","pages":[22,23]},{"title":"5.4 Background variability","content":"The main components of the LAD background are shown in Figure 17. The high energy photons from CXB emission and the Earth γ -ray albedo leaking through the collimator represent the dominant background component. Although they are intrinsically steady and predictable, due to their different intensity and spectra a varying relative orientation of the LAD in their radiation environment will cause a small and smooth modulation of the detected background, on the orbital timescales ( ∼ 90 minutes). This is due to the varying viewing geometry along the orbit and for different attitudes. This effect has been studied through simulations, finding that the maximum expected modulation of the background is less than 10%. This value has to be compared to a factor of a few for instruments dominated by particle-induced background. For example, RXTE/PCA had up to a ∼ 250% variation on orbital timescales [8]. In the LOFT case, the effect of the other potentially varying sources, i.e. particle induced background, is greatly reduced by the very stable environment offered by the low Earth equatorial orbit and by the fact that this component accounts for less than 6% of the overall background. In Figure 18 the background rate is shown as a function of the angle between the LAD pointing direction and the center of the Earth. θ E = 0 · represents the Earth center aligned with the field of view, while θ E = 180 · stands for the Earth at the instrument nadir. In practice, this corresponds to the orbital modulation for a low-declination source. The curve representing the total background (black symbols) shows a maximum modulation of ∼ 8%. The modulation of the background rate is entirely due to geometrical effect and it can be predicted and modeled, thanks to the intrinsic steadiness of the relevant sources. Each background component is well modeled as a function of the Earth location with respect to the pointing direction by a sum of two Gaussian distributions, centered at about θ E = 0 · and 180 · for the Earthoriginated components and at about θ E = 120 · and 240 · for the diffuse sky components. These distributions, also reported in Figure 18, arise from the convolution of the directional 'transparency' of the LAD instrument with the Earth-occulted field of view. Of course, when the detector points towards the zenith ( θ E = 180 · ) the contribution from the leaking CXB emission is maximum. The overall convolution of these out-of-phase components is to give a very small fluctuation of the total background. Moreover, for an actual pointing towards an astrophysical source, the range of possible Earth angles θ E is restricted, from a θ E = 90 · for a source at the orbital pole (that is nearly coin- pole) to the full 0 · -180 · range for equatorial sources. This lowers the background modulation further below the maximum during the observation of a realistic scientific target. The geometrical model, properly calibrated using in-orbit flat fields, is anticipated to allow for a background prediction at a level significantly better than 1% in the 2-10 keV band, which is the LAD science requirement. Such a level of systematic uncertainty was indeed already reached by the past experiment RXTE/PCA, which, in the presence of a much more variable background level (250% vs 8%) and less predictable background sources, with an appropriate modeling reached the level of ∼ 1% [8,9]. However, as some of the LOFT science cases (in particular the extragalactic science) will benefit from reaching a background knowledge significantly better than the requirement, an active background monitoring was designed for the LAD, to further improve the modeling. Due to the slow and smooth background variation, there is no need for a high-statistics, instantaneous monitoring of the rate. Rather, a continuous benchmark of the slow modulation will allow for a real-time verification of the background model. This active background monitoring is achieved by the introduction of a 'blocked collimator' (a collimator with the same stopping power but no holes) for an area corresponding to one Module of the LAD. This will enable the continuous monitor of all components of the LAD background, with the exception of the aperture background, accounting for ∼ 90% of the total background, and also evaluating the long-term variations (e.g. linked to the Solar cycle). Preliminary simulations for different targets (i.e., attitude configurations) show that the accuracy in the background modeling during a typical observation can be improved down to ∼ 0.1%-0.3% by using these additional data. The subject of time-dependent modeling of the background is being further studied in depth to definitely assess the ultimate instrument capability for weak sources. Here we just reported a few preliminary results allowing to identify the range of interest. An exhaustive report and discussion is deferred to a dedicated paper, currently in preparation.","pages":[23,24,25]},{"title":"5.5 Absolute background level: cosmic variance and source contamination","content":"In the previous section we preliminarily addressed the issue of background stability and modeling during a single observation. This is the key parameter for most of the science driver of observations of weak sources, where relative variations of source features will be studied (e.g., the Fe line in AGNs). Anyway, a few science cases may require an estimate of the absolute value of the background. This faces an intrinsic physical property of the CXB, known as cosmic variance [37]: the flux of the CXB is not perfectly isotropic but varies on different angular scales. The variation was measured [37,38,39] as ∼ 7% (rms) on the angular scale of 1 · and ∼ 2% on 20 · -40 · . In the case of the LAD, this affects only the aperture background, as the CXB/albedo contamination is seen from all directions. The expected variation in the LAD background due to the cosmic variance is given by the fraction of the total rate due to the aperture CXB (13%) weighted by the variance over the field of view ( ∼ 7%), which is about 0.9%. Given its intrinsic astrophysical nature, the only way of measuring this component is through a local blank field, just as for any other imaging or non-imaging experiment. This is indeed the plan for the few science cases requiring the absolute knowledge of the background rate rather than its stability. It is worth stressing that the cosmic variance does not affect at all the background stability in time for a given target observation. A further small contribution to the 'local background' (i.e., for a given target/attitude) is given by the possible contamination due to bright and hard sources outside the field of view. Similar to the leaking of diffuse CXB/albedo photons through the collimator, the same effect can occur to photons from point-like sources. We investigated this effect by simulations, and the maximum contribution to the overall background is of the order of 1-3% (Crab-like sources, in 2-30 keV) down to 0.1% (softer sources, e.g. Sco X-1). This additional minor contribution will be monitored by the active background monitoring system, as well as by the Wide Field Monitor. The quantitative assessment of the small effect on the background stability when the contaminating sources are variable on the time scale of interest will be extensively addressed in the paper in preparation mentioned earlier.","pages":[25,26,27]},{"title":"6 Conclusions","content":"LOFT will be an innovative mission that will observe compact Galactic and extragalactic objects in both the spectral and the temporal domains. The unprecedented sensitive area of the LAD instrument will open new windows in the study of the fundamental physics allowed by these natural laboratories. The scientific objectives of the mission require an accurate knowledge and minimisation of the detector background. To this end, an extensive massmodel for the LOFT/LAD experiment has been developed, using the standard Geant-4 toolkit (Sect. 4), and all the main components (photonic, leptonic, hadronic and internal, Sect. 3) of the orbital background environment have been simulated. The main contribution to the overall background is found to be due to the diffuse X-rays that 'leaks' from the lead-glass collimators of the instrument. This emission originates from the diffuse cosmic X-ray background and from the Earth albedo. The particle-induced background is minimised mainly thanks to the LOFT low-inclination, low-altitude orbit and small mass for unit area of the LAD experiment, becoming dominant only at high energies, above 30 keV. A further suppression of the particle background is enabled by the particular signature that these events leave on the Silicon Drift Detectors (Sect. 4.3). The simulations show the feasibility for the current LAD instrument design to fulfill the required background level of 10 mCrab in the 2-30 keV band. Background variations on orbital timescales are mostly induced by the varying geometry between the position of the Earth and the pointing direction, As such, they can be modelled and accounted for. The use of special detector units ('blocked' module) and a carefully planned blank-field pointing strategy, are foreseen to monitor these background variations enabling to reach a residual systematics level better than 0.3%. The residual contribution of strong (and variable) off-axis sources, above 30-50 keV, if needed, can be further modelled using observations performed with the other instrument onboard LOFT, the WFM. An increase in the accuracy of the background determination for the LAD will be provided by further improvements in the geometrical description of the instrument itself and of the spacecraft bus structures. Furthermore, results will be refined by taking into account a complete determination of the residual activation of the detector materials following the grazing passages in the highbackground environment of the South Atlantic Anomaly, where however no scientific observations are conducted. Acknowledgements We are grateful to the anonymous referee whose helpful comments greatly improved the text, and to Alessandra De Rosa for useful discussions. We acknowl- edge financial support from ASI/INAF contract I/021/12/0. RC furthermore acknowledge support from INAF Tecno-PRIN 2009 grant.","pages":[27,28]},{"title":"References","content":"2. B. Bertucci, A. Biland, S. Bizzaglia, S. Blasko, G. Boella, M. Boschini, M. Bourquin, L. Brocco, G. Bruni, M. Buenerd, J.D. Burger, W.J. Burger, X.D. Cai, C. Camps, P. Cannarsa, M. Capell, D. Casadei, J. Casaus, G. Castellini, C. Cecchi, Y.H. Chang, H.F. Chen, H.S. Chen, Z.G. Chen, N.A. Chernoplekov, T.H. Chiueh, Y.L. Chuang, F. Cindolo, V. Commichau, A. Contin, P. Crespo, M. Cristinziani, J.P. da Cunha, T.S. Dai, J.D. Deus, N. Dinu, L. Djambazov, I. D'Antone, Z.R. Dong, P. Emonet, J. Engelberg, F.J. Eppling, T. Eronen, G. Esposito, P. Extermann, J. Favier, E. Fiandrini, P.H. Fisher, G. Fluegge, N. Fouque, Y. Galaktionov, M. Gervasi, P. Giusti, D. Grandi, O. Grimm, W.Q. Gu, K. Hangarter, A. Hasan, V. Hermel, H. Hofer, M.A. Huang, W. Hungerford, M. Ionica, R. Ionica, M. Jongmanns, K. Karlamaa, W. Karpinski, G. Kenney, J. Kenny, W. Kim, A. Klimentov, R. Kossakowski, V. Koutsenko, M. Kraeber, G. Laborie, T. Laitinen, G. Lamanna, G. Laurenti, A. Lebedev, S.C. Lee, G. Levi, P. Levtchenko, C.L. Liu, H.T. Liu, I. Lopes, G. Lu, Y.S. Lu, K. Lubelsmeyer, D. Luckey, W. Lustermann, C. Ma˜na, A. Margotti, F. Mayet, R.R. McNeil, B. Meillon, M. Menichelli, A. Mihul, A. Mourao, A. Mujunen, F. Palmonari, A. Papi, I.H. Park, M. Pauluzzi, F. Pauss, E. Perrin, A. Pesci, A. Pevsner, M. Pimenta, V. Plyaskin, V. Pojidaev, M. Pohl, V. Postolache, N. Produit, P.G. Rancoita, D. Rapin, F. Raupach, D. Ren, Z. Ren, M. Ribordy, J.P. Richeux, E. Riihonen, J. Ritakari, U. Roeser, C. Roissin, R. Sagdeev, G. Sartorelli, A. Schultz von Dratzig, G. Schwering, G. Scolieri, E.S. Seo, V. Shoutko, E. Shoumilov, R. Siedling, D. Son, T. Song, M. Steuer, G.S. Sun, H. Suter, X.W. Tang, S.C.C. Ting, S.M. Ting, M. Tornikoski, J. Torsti, J. Trumper, J. Ulbricht, S. Urpo, I. Usoskin, E. Valtonen, J. Vandenhirtz, F. Velcea, E. Velikhov, B. Verlaat, I. Vetlitsky, F. Vezzu, J.P. Vialle, G. Viertel, D. Vit'e, H. Von Gunten, S. Waldmeier Wicki, W. Wallraff, B.C. Wang, J.Z. Wang, Y.H. Wang, K. Wiik, C. Williams, S.X. Wu, P.C. Xia, J.L. Yan, L.G. Yan, C.G. Yang, M. Yang, S.W. Ye, P. Yeh, Z.Z. Xu, H.Y. Zhang, Z.P. Zhang, D.X. Zhao, G.Y. Zhu, W.Z. Zhu, H.L. Zhuang, A. Zichichi, B. Zimmermann, Physics Letters B 490 , 27 (2000)","pages":[28,30]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Experimental Astronomy manuscript No. (will be inserted by the editor)\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Background simulations for the Large Area Detector onboard LOFT\",\n \"content\": \"Riccardo Campana · Marco Feroci · Ettore Del Monte · Teresa Mineo · Niels Lund · George W. Fraser Received: date / Accepted: date Abstract The Large Observatory For X-ray Timing (LOFT), currently in an assessment phase in the framework the ESA M3 Cosmic Vision programme, is an innovative medium-class mission specifically designed to answer fundamental questions about the behaviour of matter, in the very strong gravitational and magnetic fields around compact objects and in supranuclear density conditions. Having an effective area of ∼ 10 m 2 at 8 keV, LOFT will be able to measure with high sensitivity very fast variability in the X-ray fluxes and spectra. A good knowledge of the in-orbit background environment is essential to assess the scientific performance of the mission and optimize the design of its main instrument, the Large Area Detector (LAD). In this paper the results of an extensive Geant-4 simulation of the instrument will be discussed, showing the main contributions to the background and the design solutions for its reduction and control. Our results show that the current LOFT/LAD design is expected to meet its scientific requirement of a background rate equivalent to 10 mCrab in 2-30 keV, achieving about 5 mCrab in the most important 2-10 keV energy band. Moreover, simulations show an anticipated modulation of the background rate as small as 10% over the orbital timescale. The intrinsic photonic origin of the largest background component also allows for On behalf of the LOFT collaboration. R. Campana, M. Feroci, E. Del Monte INAF/IAPS, Via Fosso del Cavaliere 100, I-00133, Roma, Italy and INFN/Sezione di Roma 2, viale della Ricerca Scientifica 1, I-00133, Roma, Italy. E-mail: riccardo.campana@inaf.it Present address of R. Campana: INAF/IASF-Bologna, via Gobetti 101, I-40129, Bologna, Italy. T. Mineo INAF/IASF-Palermo, Via Ugo La Malfa 153, I-90146, Palermo, Italy. N. Lund DTU-Space, Elektrovej Bld. 327, DK-2800, Kgs. Lyngby, Denmark. G. Fraser Space Research Centre, Dept. of Physics and Astronomy, University of Leicester, LE17RH, Leicester, UK. an efficient modelling, supported by an in-flight active monitoring, allowing to predict systematic residuals significantly better than the requirement of 1%, and actually meeting the 0.25% science goal. Keywords X-ray astronomy · Instrumental background · Montecarlo simulations PACS 07.85.Fv · 07.87.+v · 95.40.+s\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"Astrophysical observations in the X-ray domain are often dominated by the background. In order to assess and maximize the scientific performance of a satellite-borne instrument during its design phase, is essential to have an early estimation of the background level and to identify its origin. This will allow to drive the optimization of the experiment design itself. Moreover, a reliable estimation of the background level is important to assess the scientific objectives and performance of the mission. The most convenient and consolidated way to compute such a background is by means of a Monte Carlo simulation, using e.g. the Geant-4 toolkit [1], in which the interactions of the space environment with a somewhat simplified mass model of the spacecraft and of the instruments are sampled and studied. A correct evaluation of the radiation environment surrounding the experiment is of course the most important ingredient in the background simulation. In this paper we will show Monte Carlo simulations of the instrumental background for the Large Area Detector (LAD) instrument onboard the proposed LOFT (Large Observatory for X-ray Timing) satellite [2,3], that is a candidate for the third slot of medium-class missions (M3) of the Cosmic Vision 2015-2025 programme of the European Space Agency, with a possible launch in the 2022-24 timeframe. LOFT (Figure 1) has been specially designed to answer the 'Fundamental Question' #3.3 of the Cosmic Vision programme, Matter under extreme conditions . In the aim of investigating the behaviour of the matter in the most extreme physical conditions that can be found in the gravitational and magnetic fields around neutron stars and black holes, LOFT will study the rapid variability of the X-ray emission from these objects, both in the spectral and temporal domains. Moreover, LOFT will be an observatory for virtually all classes of bright X-ray sources: among them, X-ray bursters, high mass X-ray binaries, cataclysmic variables, magnetars and active galactic nuclei. The main experiment onboard LOFT is the LAD [3] (Sect. 2), a collimated instrument with a collecting area 20 times bigger than its immediate predecessor (PCA, the Proportional Counter Array onboard the Rossi X-ray Timing Explorer, RXTE), and based on large-area Silicon Drift Detectors (SDDs) coupled to lead-glass Micro Channel Plate (MCP) collimators. In order to survey a large fraction of the sky simultaneously, and to trigger follow-up observations with the main instrument, LOFT will host also a coded-mask Wide Field Monitor (WFM, [4]). This latter instrument, sensitive in the same energy range of LAD, will use basically the same SDD detectors, with some design differences to optimize the detector response for imaging [5,6,7]. The background level required to satisfy the LOFT scientific objectives, in particular regarding the relatively faint (1-10 mCrab) sources like most Active Galactic Nuclei (AGNs) and some accretion-powered X-ray pulsars, is ≤ 10 mCrab for the 2-30 keV LAD standard band. Being a collimated instrument, the LAD will not measure simultaneously and independently the source flux and the background. While some collimated instruments (e.g. BeppoSAX/PDS and RXTE/HEXTE, both at hard X-rays) employ a 'rocking' strategy, where the same instrument unit alternates observations of the source and of the background, others (e.g. RXTE/PCA, at soft X-rays, similar to LAD) rely on a background model obtained by means of dedicated blank-field pointings and by using onboard ratemeters [8]. As we will discuss in the following, LAD will adopt an approach similar to PCA. To ensure the fulfillment of the scientific objectives, in particular for low flux sources, an accurate estimation and modelling of the background level for this instrument is of particular importance. The background variations that have a significant impact on the scientific performance should be minimized, and controlled to a high degree of accuracy. The LOFT science case resulted in a requirement on background knowledge at 1%, with a goal at 0.25%. As a comparison, RXTE/PCA reached a background residual systematic level of ∼ 0.5%-1% [8,9], while BeppoSAX/PDS of ∼ 1% [10]. This paper is structured as follows. In Section 2 we describe the LAD instrument. In Section 3 we introduce the space environment for LOFT and discuss the main contributions to the instrumental background, while in Section 4 the main features of the LAD Monte Carlo mass model and simulator are described. The simulation results are shown in Section 5, and in Section 6 we draw our conclusions.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2 The Large Area Detector\",\n \"content\": \"The Large Area Detector is an array of 2016 individual large-area Silicon Drift Detectors, sensitive to the 2-30 keV emission (with a extended 30-80 keV band) collimated in a ∼ 1 · field of view by means of a lead-glass microcapillary collimator plate. The instrument design is modular: an array of 4 × 4 SDDs and their front-end electronics (FEE) are placed in a module . Each module has its independent back-end electronics (MBEE) that powers and configure the FEE, manages the event readout and monitors the detector health. A panel is composed of an array of 21 modules, placed on a support grid (Figure 2). The current LAD design by the LOFT consortium envisages 6 panels (each about 350 cm × 90 cm), that stems from the satellite bus and are deployed once the spacecraft reaches its orbit (Figure 1). The collimators, made of lead glass, have the same footprint as the SDDs (about 11 cm × 7 cm), a thickness of 6 mm, and 100 µ m wide square holes separated with 20 µ m thick walls (and thus having an open area ratio , f OAR , of ∼ 70%). An aluminium frame couples the collimators with the detectors frame and ensures the tight alignment constraints. The working principle of the SDDs is the following: a photon is absorbed in the depleted silicon bulk, and produces a cloud of electrons whose number is proportional to the incident energy. The cloud drifts towards an array of read-out anodes, by means of a high-voltage field applied between the middle section and the two detector edges, hosting two independent rows of anodes. During the drift, the cloud size increases due to diffusion. The number of anodes that will read-out the charge cloud thus depends on the absorption point distance and anode pitch [5]. For the LAD detectors, that have a pitch of 970 µ m, the charge cloud originating from the absorption of a X-ray photon in the 2-80 keV band is at most 0.5 mm (FWHM) thus read-out by 1 or 2 anodes, depending on the impact point and the energy. At 6 keV, for example, ∼ 40% of the events are collected by one anode and ∼ 60% by two anodes. The event multiplicity is read-out and transmitted to the ground, enabling a selection of 'single' and 'double' events, with different energy resolutions.\",\n \"pages\": [\n 3,\n 4,\n 5\n ]\n },\n {\n \"title\": \"3 Sources of background\",\n \"content\": \"In this section we discuss the main sources of the LOFT background. We describe the environment encountered in the baseline low-Earth orbit, with its hadronic, leptonic and photon components. Moreover, we describe the internal source of background, i.e. the natural radioactivity of the collimator material.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3.1 The LOFT orbit\",\n \"content\": \"The LOFT orbit requirement is at an altitude of h = 600 km with an inclination of 5 · . Likewise, lower-altitude orbits (down to 550 km) and/or inclinations (down to 0 · ) are considered, with small improvements for what regards the conclusions about the background rate and properties. The lower altitude and inclinations, however, impacts on the long-term radiation damage on the detectors. For these orbits, the geomagnetic latitude range is usually θ M < 0 . 3 radians. In the current work we neglected the possible background component due to the activation of the instrument and satellite materials. Based on the arguments below, we estimated it to be a second-order effect. The South Atlantic Anomaly, a region of trapped high-energy protons and electrons, is grazed by the LOFT baseline orbit only in its outermost regions (Figure 3), with a shallower passage for lower altitudes and inclinations. Therefore, the effect on background due to the activation of materials by this intense radiation is expected to be negligible with respect to other sources, as confirmed by preliminary evaluations and heritage from past missions in similar orbits (e.g., BeppoSAX and AGILE). It should also be noted that in contrast to previous experiments, the LAD instrument has a very light structure per unit volume and the relatively small spacecraft assembly is seen at a small viewing angle, due to the tower supporting the LAD panels (see Figure 1). In addition to that, as for example observed by Swift/XRT [11], Silicon detectors show marginal activation effects in LEO, even when operated as focal plane instruments (more local mass) and on more inclined orbits (up to ∼ 20 · ). However, given our goal of controlling the LAD background to the highest possible accuracy, we plan to study this aspect in great detail in a future work.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"3.2 Primary cosmic rays\",\n \"content\": \"Following the approach of Mizuno et al. [12], we assume that the primary cosmic ray spectrum is expressed as F U ( R ), a function of the particle magnetic rigidity, R = pc/Ze , where Z is the atomic number and p the particle momentum. The full spectrum at a given location in the magnetosphere and a given phase of the solar cycle will be given by: where M and Ze are the mass and charge of the particle, E its kinetic energy, φ is a solar modulation factor, h is the orbit height and θ M is the geomagnetic latitude. The second term, that includes the effects of the solar modulation on the cosmic ray particles, is given by [13]: where the solar modulation potential φ is given by: The geomagnetic cutoff function is given, for vertically incident particles, by [12]: in which the cutoff rigidity is obtained from the Stormer equation [14]: where R E = 6371 km is the Earth radius. Moreover, the r exponent in Eq. 5 is: For the 600 km, 5 · inclination LOFT orbit, the cutoff rigidity therefore is found to be in the range R cut ∼ 10.1-12.1 GV. Since we are interested mostly in the average background fluxes, we discard at the present stage the east-west effect for which particles coming from different directions have different cutoff rigidities. We therefore assume that the general primary spectrum, now expressed as: has an uniform angular distribution with respect to the zenith angle, up to the Earth horizon, i.e. for zenith angles θ from 0 · to θ cut , where the latter is given by: For h = 600 km, θ cut ∼ 114 · . The solid angle subtended by the Earth at an altitude h is given by: and therefore the accessibile sky subtends a solid angle Ω = 4 π -Ω E (Figure 4). For h = 600 km, the Earth blocks about 30% of the sky.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"3.2.1 Primary protons\",\n \"content\": \"The unmodulated value of the cosmic ray electron spectrum is given by the BESS [15] and AMS measurements [16]: In Figure 5 the primary proton spectra for different values of the solar modulation and for the LOFT orbit is reported. The variation in flux between solar minimum and maximum, at the peak flux, is ∼ 15%.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"3.2.2 Primary electrons and positrons\",\n \"content\": \"The unmodulated value of the cosmic ray electron spectrum is given by [17, 18,12]: The fraction of positrons to electrons, usually given by the ratio e + / ( e + + e -), is found to be rather independent of the energy [19], i.e. the spectrum of primary positrons has the same slope of the electron one, but a different normalization: In Figure 6 the primary e -and e + spectra for different values of the solar modulation and for the LOFT orbit are reported. The difference in flux between solar minumum and maximum is about 20% at the peak.\",\n \"pages\": [\n 9\n ]\n },\n {\n \"title\": \"3.2.3 Primary alpha particles\",\n \"content\": \"We consider only the primary helium nuclei, because the contribution of heavier nuclei is negligible with respect to the other primary particles flux [20]. The unmodulated flux, as given by Mizuno et al. [12] on the basis of AMS and BESS data is: and is shown in Figure 7. 5\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"3.3.1 Protons\",\n \"content\": \"For the low altitude equatorial Earth orbits considered, the impinging proton spectrum outside the trapped particle belt, i.e. the South Atlantic Anomaly, consists of the primary component discussed in Sect. 3.2 and of a secondary, quasi-trapped component, originating from and impacting to the Earth atmosphere (sometimes in the literature they are referred to as the 'splash' and 'reentrant' albedo components). AMS measurements [16] showed that this secondary component is composed of a short-lived and a long-lived particle population, both originating from the regions near the geomagnetic equator. Mizuno et al. [12] model the secondary equatorial proton spectrum as a cutoff power-law: (16) and a power law for energies below 100 MeV: This spectrum is shown in Figure 8 together with the AMS measurements. According to the measurements performed with the NINA and NINA-II instruments [21], the extrapolation below 100 MeV is likely an overestimate. We do not consider here the soft (10 keV-1 MeV), highly directional equatorial proton population [22,23], since these particles are most efficiently stopped by all the spacecraft structures surrounding the detectors. Even the small fraction that impinges on the detector through the capillary holes, albeit significant when considering the long-term radiation damage on the SDDs, leaves a negligible background signal.\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"3.3.2 Electrons and positrons\",\n \"content\": \"The equatorial secondary electron spectrum can be approximated with a broken power law [12]: and a power law for energies below 100 MeV: At variance with respect to the primary particles, in the geomagnetic equatorial region the positrons are predominant with respect to the electrons. The spectrum is the same, but the ratio e + /e -is about 3.3 (Figure 9).\",\n \"pages\": [\n 12\n ]\n },\n {\n \"title\": \"3.4 Photon background\",\n \"content\": \"As we will see in the following Sections, the main contribution to the overall background for the LAD instrument is due to the high energy X-ray emission transmitted and scattered in the lead-glass collimators. The main contribution comes from the CXB and the Earth albedo emission.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"3.4.1 Cosmic photon background\",\n \"content\": \"For the cosmic X-ray and γ -ray diffuse background we assume the Gruber et al. [24] analytic form, derived from HEAO-1 A4 and COMPTEL measurements, and fitted in the energy range between 3 keV and 100 GeV. This spectrum is usually used as a standard reference [25,26], and is plotted in Figure 10. The first branch is valid below 60 keV: while for energies above 60 keV: Other measurements of the CXB have been performed by, e.g., BeppoSAX [25], INTEGRAL [27,28] and BAT [26]. These measurements are consistent with the HEAO-1 results, albeit some of these seems to indicate a slightly higher normalization ( ∼ 8%) for the > 10 keV spectrum.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"3.4.2 Albedo γ -ray background\",\n \"content\": \"The secondary photon background is due to the interactions of cosmic rays (proton and leptonic components) with the atmosphere, and to the reflection of CXB emission. As such, it has a strong zenith dependence [29]. This albedo component has a higher flux, for unit of solid angle, than the CXB for energies above ∼ 70 keV. We assume the albedo spectrum as measured by BAT [26], that agrees in the range above 50 keV, after some corrections, with previous measurements [30,31]. This spectrum can be parameterized by the function [32,26]: and is plotted in Figure 11.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"3.5 Earth neutron albedo\",\n \"content\": \"As reported by the official ESA ECSS documents 1 , there is presently no model for atmospheric albedo neutron fluxes considered mature enough to be used as a standard. To account for the flux of neutrons produced by cosmic-ray interactions in the Earth atmosphere, we however used the results of the QinetiQ Atmospheric Radiation Model (QARM, [33,34]), that uses a response function approach based on Monte Carlo radiation transport codes to generate directional fluxes of atmospheric secondary radiation. The resulting spectrum, shown in Figure 12, is also consistent with the Monte Carlo simulations of Armstrong & Colborn [35,36]. 3\",\n \"pages\": [\n 15\n ]\n },\n {\n \"title\": \"3.6 Natural radioactivity\",\n \"content\": \"The lead-glass used in the microcapillary plate collimators contains potassium, in a fraction of approximately 7.2% by weight. The activity due to the naturally occurring long-lived radioactive isotope 40 K (with an abundance of 0.0117% with respect to natural potassium) is to be taken into account. This isotope decays with the emission of β rays having a continuum of energies up to 1.31 MeV ( ∼ 89% branching ratio) and of 1.46 MeV photons ( ∼ 11% b.r.) due to electronic capture. Let L , ρ , f OAR be the thickness, density and open fraction of the collimator. Therefore, the collimator activity for unit area is: where A K ∼ 30 Bq/g is the specific activity of natural potassium due to 40 K, and M K is the total potassium mass per unit area in the collimator, i.e. where f K is the fraction by weight of potassium in the collimator glass. For the MCP lead glass, we have f K ∼ 7 . 2%. Assuming for the MCP collimator L = 6 mm, ρ = 3 . 3 g/cm 3 , f OAR = 69 . 44% we obtain M K = 436 g/m 2 . Therefore, we have A coll = 13100 Bq/m 2 . For a total geometrical LAD collimator area of ∼ 15 m 2 we have thus ∼ 196500 decays/s. Considering the branching ratio, we have therefore that every second in the collimator bulk are uniformly and isotropically generated ∼ 175000 electrons with a continuum of energies and ∼ 21000 monochromatic photons with a 1.46 MeV energy.\",\n \"pages\": [\n 15,\n 16\n ]\n },\n {\n \"title\": \"4.1 The mass model\",\n \"content\": \"Simulations were performed using the Geant-4 Monte Carlo toolkit [1] (version 4.9.4). Geant-4 allows to describe the geometry and the materials of the instruments and of the satellite bus. Moreover, the code enables to follow the various physical interactions along the path of a primary event through the various components of the geometry, evaluating the secondary particles and energy deposits generated. For the description of the electromagnetic interactions, the Low Energy Livermore library 2 is used, that trace the interactions of electrons and photons with matter down to about 250 eV, using interpolated data tables. Physical processes like photoelectric effect, Compton and Rayleigh scattering, pair production, bremsstrahlung, multiple scattering and annihilation are simulated, optionally taking into account also the effect of photon polarization. Fluorescence X-rays and Auger electrons from the various chemical elements are included. The hadronic physics, instead, is described using different models in different energy ranges (e.g. Bertini Cascade model, Quark-Gluon String model, etc. 3 ). Both elastic and inelastic processes are treated, the latter describing e.g. nuclear capture, fragmentation, de-excitation and scattering of the relevant stable and long-lived nucleons and mesons. In Figure 13 is shown the mass model geometry used in the simulations for the LAD instrument. A single LAD panel is simulated, with the actual dimensions ( ∼ 350 cm × 90 cm) but with a simplified design that consists of a stacked-layers geometry (sketched in Figure 14). A simplified geometry for the satellite bus, the structural tower and the other five panels is assumed, using aluminium as the material with an 'effective' density that takes into account the total mass and volume. The SDD array is represented as a 450 µ m thick slab of silicon, subdivided in 0.97 mm × 35 mm pixels. On its surface are placed various passive layers (Al cathode implants, SiO 2 passivation layer, undepleted Silicon bulk), the MCP collimator and the optical/thermal filter (1 µ m kapton coupled with a 80 nm Al layer). The MCP collimator has an 'effective', reduced density, to take into account the holes without resorting to a more accurate but much more timeconsuming geometrical description. Simulations on a smaller geometrical size have been performed to compare the 'effective' collimator with a complete geometrical capillary plate description (see Sect. 5.2), giving consistent results. Underneath the detectors, the FEE board is simulated as a 2 mm thick FR4 slab with an interposed 100 µ mCu conductive layer and a 100 µ mAl shielding layer, a 1 mm thick Al radiator and a 500 µ m Pb backshield. An aluminium frame encloses the module sides, while the panel structure consists of a carbonfiber reinforced plastic grid frame.\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"4.2 Primary event generation\",\n \"content\": \"For the generation of primary events, the General Particle Source 4 (GPS) approach is followed, that allows to produce arbitrary spectra in rather complex geometries. The initial events (particles or photons) are generated on the surface of a sphere of radius R , with a cosine-biased emission angle to ensure an isotropic flux. The emission angle is further restricted between 0 (normal to the spherical surface) to θ max : the emission cone then subtends a smaller sphere of radius r that surrounds the experiment. We have: Let Φ be the energy-integrated flux, between E min and E max , expressed in particles cm -2 s -1 sr -1 . The total rate is therefore: This derives from the integration over the 2 π emission angle for a point on the spherical surface, biased with the cosine law: and integrated over the source sphere surface S = 4 πR 2 . The restriction on the emission angle introduces the further factor sin 2 θ max . Therefore the simulation time that corresponds to N events is: generated primary If we detect C i counts in the energy bin i , the corresponding measured flux (in counts cm -2 s -1 ), i.e. convolved with the detector response, would be: where A det is the detector sensitive area.\",\n \"pages\": [\n 18,\n 19\n ]\n },\n {\n \"title\": \"4.3 Anode multiplicity rejection filter\",\n \"content\": \"For the LAD, an anode multiplicity rejection algorithm is implemented to filter out ionization streaks from charged particles, that leave an energy deposit on more than 2 adjacent anodes. More in detail, for each of the two independent halves of a SDD tile, events are rejected if they trigger more than two adjacent anodes. Likewise, events are rejected if there is a group of non-adjacent triggering anodes within a 'distance' of 32 channels in the same half-SDD (Figure 15). Energy deposits at a further distance are treated as independent. Since the average energy loss of minimum ionizing particles (MIPs) in silicon is about 3.7 MeV/cm, their total energy deposit is usually above ∼ 150 keV. The anode multiplicity rejection filter, combined together with an upper threshold on the reconstructed event signal, is very effective in the suppression of the particle-induced background: a threshold of 80 keV allow to filter out 94%96% of the particle background, depending on the incoming particle type and spectrum. Of course, since this event elaboration is performed in the frontend electronics, dead-time calculations should be performed using the total incoming background rate. The acceptance efficiency of this filter, defined as the fraction of photons from a 'true', on-axis X-ray source that survive the multiplicity rejection, is very high. For a Crab-like spectrum, the efficiency is 99.98%.\",\n \"pages\": [\n 19\n ]\n },\n {\n \"title\": \"5 Simulation results\",\n \"content\": \"The simulations have been performed by generating an isotropic flux of primary events (photons or particles, the latter evaluated at the Solar minimum, the worst case condition) on a sphere surrounding the experiment, recording the energy deposits in the detector pixels and then applying the proper filter for the event multiplicity (Sect. 4.3). The resulting counts have been then properly normalized, taking into account the energy spectra and solid angles of the simulated background contributions. Before discussing the general results for the LAD background and a preliminary assessment of its stability, in the next two subsections we deal with some subleties that required a more detailed geometrical description and event processing.\",\n \"pages\": [\n 19,\n 20\n ]\n },\n {\n \"title\": \"5.1 Capillary reflections\",\n \"content\": \"A standalone, more detailed raytracing simulator has been built for the MCP to evaluate the effects of the grazing incidence angle reflectivity from the pore inner walls for the CXB photons collected in the ∼ 1 · × 1 · field of view. The code tracks photons from their incoming direction up to the detector plane modeling their interaction with the optical/thermal filter and the collimator structure. The simulated collimator has a full geometrical description, 6 mm thick filled with square pores 100 µ m wide and 20 µ m thick walls. The absorption and transmission probability of the optical/thermal filter elements are computed from tables derived from the database of the National Insti- tandards and Technology 5 ; the lead glass attenuation coefficient is derived from tabulated values (G. Fraser, priv. comm.). Reflectivity from the pore walls, as derived from a set of laboratory measurements (G. Fraser, priv. comm.) is well modeled by data derived from CXRO database 6 corresponding to lead glass with a 11.8 nm surface micro-roughness value. Results from this code, when reflection is not taken into account, are well in agreement with those obtained by the Geant-4 simulator (Figure 16). The effect of the wall reflectivity produces an increase of the aperture CXB (Mineo et al., in preparation), going from 40% at energies lower than 3 keV, down to ∼ 10% between 5 and 10 keV and below 5% above 10 keV.\",\n \"pages\": [\n 20,\n 21\n ]\n },\n {\n \"title\": \"5.2 Internal 40 K activity\",\n \"content\": \"Not all the potassium decay products (see Sect. 3.6) will leave a signal in the detector: the emission is isotropic, and the photon (or the electron) has to cross different thicknesses of lead glass, at various pitch angles, before reaching the SDD. Moreover, the efficiency of the silicon detector should be taken into account; eventually, only a fraction of the detected events will have a pulse height amplitude corresponding to the standard LAD energy band (2-30 keV). Primary particles (1.46 MeV monochromatic photons and electrons having energies between 1 keV and 1311 keV) have been generated randomly in the lead-glass walls of a ∼ 1 . 2 × 1 . 2 cm complete model of a MCP geometry and mass (6 mm thickness, 100 µ m square holes, 20 µ m wall thickness, ∼ 70% OAR) coupled to a SDD detector. It is found that the decay of one 40 K atom generating a 1.46 MeV photon has a very small probability to generate a signal with equivalent amplitude between 2 and 30 keV, obtaining we obtain a background rate of only ∼ 5 × 10 -5 counts cm -2 s -1 . Therefore, for all practical purposes, we can discard the effect of the 40 K electronic capture decay channel. Electrons produced by the β decay, on the contrary, have a higher chance to cause a signal, producing a background rate of ∼ 2 × 10 -3 counts cm -2 s -1 .\",\n \"pages\": [\n 21,\n 22\n ]\n },\n {\n \"title\": \"5.3 The LAD background\",\n \"content\": \"The total resulting LAD background is shown in Figure 17, while the breakdown of the count rate in its various components is shown in Table 1. No detector energy resolution smoothing of the spectra was included here. The main background contribution is due to the high-energy photons from the diffuse cosmic X-ray background and the Earth albedo that leaks from and are scattered in the collimators. These two components alone accounts for about 70% of the background count rate in the 2-30 keV band. The diffuse emission collected in the field of view (even including the effect of capillary reflectivity), the particle-induced and internal backgrounds are a smaller contribution to the total count rate. While the CXB emission collected in the aperture field of view has a significant contribution only below ∼ 10 keV, cosmic-ray particles and neutrons produce almost flat spectra, similarly to the internal activity background, and these components become dominant only above ∼ 20 keV. The dips in the neutron-induced spectrum are due to inelastic scattering resonances. Fluorescence emission from the Pb contained in the collimator glass ( L -shell lines at 10.55 and 12.61 keV) is well apparent, artificially emphasized by the non-inclusion of the energy resolution in the plot. Analysis are ongoing to evaluate whether these lines are to be shielded or used as in-flight calibration lines. In the plot is also shown, as a reference to the LAD science requirement, the spectrum of a 10 mCrab point-like source (dashed line). The total background count rate in the 2-30 keV band corresponds to a flux of ∼ 9 mCrab, thus ensuring the fulfillment of the scientific requirements, and it is below 5 mCrab in the most important 2-10 keV band. We can estimate a margin of error on the background rate, taking into account that the CXB emission and the Earth albedo fluxes are affected by a maximum error of ∼ 10%-20% on their normalization [26,28]. The particle components are more uncertain [12,16] (see also Sect. 3.2). However, these have been conservatively assumed at the Solar minimum, while LOFT, if selected, will fly around the next Solar maximum ( ∼ 2024), when the particle flux is expected to be approximately 20% less intense. We consider a ∼ 50% error on these fluxes. Weighting the above uncertainties on the LAD background components, we thus estimate an overall conservative maximum margin of error on the total background rate of ∼ 20%, given the present geometrical model. Future developments in the mission design will allow for a more accurate mass model and consequently to refine these results. Therefore, we can conclude that the LOFT scientific requirements are expected to be met.\",\n \"pages\": [\n 22,\n 23\n ]\n },\n {\n \"title\": \"5.4 Background variability\",\n \"content\": \"The main components of the LAD background are shown in Figure 17. The high energy photons from CXB emission and the Earth γ -ray albedo leaking through the collimator represent the dominant background component. Although they are intrinsically steady and predictable, due to their different intensity and spectra a varying relative orientation of the LAD in their radiation environment will cause a small and smooth modulation of the detected background, on the orbital timescales ( ∼ 90 minutes). This is due to the varying viewing geometry along the orbit and for different attitudes. This effect has been studied through simulations, finding that the maximum expected modulation of the background is less than 10%. This value has to be compared to a factor of a few for instruments dominated by particle-induced background. For example, RXTE/PCA had up to a ∼ 250% variation on orbital timescales [8]. In the LOFT case, the effect of the other potentially varying sources, i.e. particle induced background, is greatly reduced by the very stable environment offered by the low Earth equatorial orbit and by the fact that this component accounts for less than 6% of the overall background. In Figure 18 the background rate is shown as a function of the angle between the LAD pointing direction and the center of the Earth. θ E = 0 · represents the Earth center aligned with the field of view, while θ E = 180 · stands for the Earth at the instrument nadir. In practice, this corresponds to the orbital modulation for a low-declination source. The curve representing the total background (black symbols) shows a maximum modulation of ∼ 8%. The modulation of the background rate is entirely due to geometrical effect and it can be predicted and modeled, thanks to the intrinsic steadiness of the relevant sources. Each background component is well modeled as a function of the Earth location with respect to the pointing direction by a sum of two Gaussian distributions, centered at about θ E = 0 · and 180 · for the Earthoriginated components and at about θ E = 120 · and 240 · for the diffuse sky components. These distributions, also reported in Figure 18, arise from the convolution of the directional 'transparency' of the LAD instrument with the Earth-occulted field of view. Of course, when the detector points towards the zenith ( θ E = 180 · ) the contribution from the leaking CXB emission is maximum. The overall convolution of these out-of-phase components is to give a very small fluctuation of the total background. Moreover, for an actual pointing towards an astrophysical source, the range of possible Earth angles θ E is restricted, from a θ E = 90 · for a source at the orbital pole (that is nearly coin- pole) to the full 0 · -180 · range for equatorial sources. This lowers the background modulation further below the maximum during the observation of a realistic scientific target. The geometrical model, properly calibrated using in-orbit flat fields, is anticipated to allow for a background prediction at a level significantly better than 1% in the 2-10 keV band, which is the LAD science requirement. Such a level of systematic uncertainty was indeed already reached by the past experiment RXTE/PCA, which, in the presence of a much more variable background level (250% vs 8%) and less predictable background sources, with an appropriate modeling reached the level of ∼ 1% [8,9]. However, as some of the LOFT science cases (in particular the extragalactic science) will benefit from reaching a background knowledge significantly better than the requirement, an active background monitoring was designed for the LAD, to further improve the modeling. Due to the slow and smooth background variation, there is no need for a high-statistics, instantaneous monitoring of the rate. Rather, a continuous benchmark of the slow modulation will allow for a real-time verification of the background model. This active background monitoring is achieved by the introduction of a 'blocked collimator' (a collimator with the same stopping power but no holes) for an area corresponding to one Module of the LAD. This will enable the continuous monitor of all components of the LAD background, with the exception of the aperture background, accounting for ∼ 90% of the total background, and also evaluating the long-term variations (e.g. linked to the Solar cycle). Preliminary simulations for different targets (i.e., attitude configurations) show that the accuracy in the background modeling during a typical observation can be improved down to ∼ 0.1%-0.3% by using these additional data. The subject of time-dependent modeling of the background is being further studied in depth to definitely assess the ultimate instrument capability for weak sources. Here we just reported a few preliminary results allowing to identify the range of interest. An exhaustive report and discussion is deferred to a dedicated paper, currently in preparation.\",\n \"pages\": [\n 23,\n 24,\n 25\n ]\n },\n {\n \"title\": \"5.5 Absolute background level: cosmic variance and source contamination\",\n \"content\": \"In the previous section we preliminarily addressed the issue of background stability and modeling during a single observation. This is the key parameter for most of the science driver of observations of weak sources, where relative variations of source features will be studied (e.g., the Fe line in AGNs). Anyway, a few science cases may require an estimate of the absolute value of the background. This faces an intrinsic physical property of the CXB, known as cosmic variance [37]: the flux of the CXB is not perfectly isotropic but varies on different angular scales. The variation was measured [37,38,39] as ∼ 7% (rms) on the angular scale of 1 · and ∼ 2% on 20 · -40 · . In the case of the LAD, this affects only the aperture background, as the CXB/albedo contamination is seen from all directions. The expected variation in the LAD background due to the cosmic variance is given by the fraction of the total rate due to the aperture CXB (13%) weighted by the variance over the field of view ( ∼ 7%), which is about 0.9%. Given its intrinsic astrophysical nature, the only way of measuring this component is through a local blank field, just as for any other imaging or non-imaging experiment. This is indeed the plan for the few science cases requiring the absolute knowledge of the background rate rather than its stability. It is worth stressing that the cosmic variance does not affect at all the background stability in time for a given target observation. A further small contribution to the 'local background' (i.e., for a given target/attitude) is given by the possible contamination due to bright and hard sources outside the field of view. Similar to the leaking of diffuse CXB/albedo photons through the collimator, the same effect can occur to photons from point-like sources. We investigated this effect by simulations, and the maximum contribution to the overall background is of the order of 1-3% (Crab-like sources, in 2-30 keV) down to 0.1% (softer sources, e.g. Sco X-1). This additional minor contribution will be monitored by the active background monitoring system, as well as by the Wide Field Monitor. The quantitative assessment of the small effect on the background stability when the contaminating sources are variable on the time scale of interest will be extensively addressed in the paper in preparation mentioned earlier.\",\n \"pages\": [\n 25,\n 26,\n 27\n ]\n },\n {\n \"title\": \"6 Conclusions\",\n \"content\": \"LOFT will be an innovative mission that will observe compact Galactic and extragalactic objects in both the spectral and the temporal domains. The unprecedented sensitive area of the LAD instrument will open new windows in the study of the fundamental physics allowed by these natural laboratories. The scientific objectives of the mission require an accurate knowledge and minimisation of the detector background. To this end, an extensive massmodel for the LOFT/LAD experiment has been developed, using the standard Geant-4 toolkit (Sect. 4), and all the main components (photonic, leptonic, hadronic and internal, Sect. 3) of the orbital background environment have been simulated. The main contribution to the overall background is found to be due to the diffuse X-rays that 'leaks' from the lead-glass collimators of the instrument. This emission originates from the diffuse cosmic X-ray background and from the Earth albedo. The particle-induced background is minimised mainly thanks to the LOFT low-inclination, low-altitude orbit and small mass for unit area of the LAD experiment, becoming dominant only at high energies, above 30 keV. A further suppression of the particle background is enabled by the particular signature that these events leave on the Silicon Drift Detectors (Sect. 4.3). The simulations show the feasibility for the current LAD instrument design to fulfill the required background level of 10 mCrab in the 2-30 keV band. Background variations on orbital timescales are mostly induced by the varying geometry between the position of the Earth and the pointing direction, As such, they can be modelled and accounted for. The use of special detector units ('blocked' module) and a carefully planned blank-field pointing strategy, are foreseen to monitor these background variations enabling to reach a residual systematics level better than 0.3%. The residual contribution of strong (and variable) off-axis sources, above 30-50 keV, if needed, can be further modelled using observations performed with the other instrument onboard LOFT, the WFM. An increase in the accuracy of the background determination for the LAD will be provided by further improvements in the geometrical description of the instrument itself and of the spacecraft bus structures. Furthermore, results will be refined by taking into account a complete determination of the residual activation of the detector materials following the grazing passages in the highbackground environment of the South Atlantic Anomaly, where however no scientific observations are conducted. Acknowledgements We are grateful to the anonymous referee whose helpful comments greatly improved the text, and to Alessandra De Rosa for useful discussions. We acknowl- edge financial support from ASI/INAF contract I/021/12/0. RC furthermore acknowledge support from INAF Tecno-PRIN 2009 grant.\",\n \"pages\": [\n 27,\n 28\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"2. B. Bertucci, A. Biland, S. Bizzaglia, S. Blasko, G. Boella, M. Boschini, M. Bourquin, L. Brocco, G. Bruni, M. Buenerd, J.D. Burger, W.J. Burger, X.D. Cai, C. Camps, P. Cannarsa, M. Capell, D. Casadei, J. Casaus, G. Castellini, C. Cecchi, Y.H. Chang, H.F. Chen, H.S. Chen, Z.G. Chen, N.A. Chernoplekov, T.H. Chiueh, Y.L. Chuang, F. Cindolo, V. Commichau, A. Contin, P. Crespo, M. Cristinziani, J.P. da Cunha, T.S. Dai, J.D. Deus, N. Dinu, L. Djambazov, I. D'Antone, Z.R. Dong, P. Emonet, J. Engelberg, F.J. Eppling, T. Eronen, G. Esposito, P. Extermann, J. Favier, E. Fiandrini, P.H. Fisher, G. Fluegge, N. Fouque, Y. Galaktionov, M. Gervasi, P. Giusti, D. Grandi, O. Grimm, W.Q. Gu, K. Hangarter, A. Hasan, V. Hermel, H. Hofer, M.A. Huang, W. Hungerford, M. Ionica, R. Ionica, M. Jongmanns, K. Karlamaa, W. Karpinski, G. Kenney, J. Kenny, W. Kim, A. Klimentov, R. Kossakowski, V. Koutsenko, M. Kraeber, G. Laborie, T. Laitinen, G. Lamanna, G. Laurenti, A. Lebedev, S.C. Lee, G. Levi, P. Levtchenko, C.L. Liu, H.T. Liu, I. Lopes, G. Lu, Y.S. Lu, K. Lubelsmeyer, D. Luckey, W. Lustermann, C. Ma˜na, A. Margotti, F. Mayet, R.R. McNeil, B. Meillon, M. Menichelli, A. Mihul, A. Mourao, A. Mujunen, F. Palmonari, A. Papi, I.H. Park, M. Pauluzzi, F. Pauss, E. Perrin, A. Pesci, A. Pevsner, M. Pimenta, V. Plyaskin, V. 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Zimmermann, Physics Letters B 490 , 27 (2000)\",\n \"pages\": [\n 28,\n 30\n ]\n }\n]"}}},{"rowIdx":264,"cells":{"bibcode":{"kind":"string","value":"2013FoPh...43.1478B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1301.5471.pdf"},"content":{"kind":"string","value":"\nA new approach to modifying theories of gravity\nChristian G. Bohmer ∗ and Nicola Tamanini † Department of Mathematics, University College London Gower Street, London, WC1E 6BT, UK\nOctober 3, 2018\nAbstract\nWe propose a new point of view for interpreting Newton's and Einstein's theories of gravity. By taking inspiration from Continuum Mechanics and its treatment of anisotropies, we formulate new gravitational actions for modified theories of gravity. These models are simple and natural generalisations with many interesting properties. Above all, their precise form can, in principle, be determined experimentally.\n1 Introduction\nSince Newton's formulation of the law of universal gravitation in the Principia 1687, the theory has been unchanged. Very few attempts have been made to modify Newtonian gravity, modified Newtonian dynamics being probably the one exception [1]. It took almost 330 years until a generalisation of Newton's theory was successfully constructed. Einstein's theory of gravity was radically different to any previous physical theory. It abandoned the absolute notion of time and replaced the concept of force by the curvature of a four dimensional space. However, it took only a few years when the first modifications and extensions of Einstein's theory appeared in the literature, see [2] for an excellent historical review. Ever since, modified theories of gravity have enjoyed a prominent role in theoretical physics. It is the aim of the current work to propose modifications of Newton's and Einstein's theories of gravity by applying the same idea to both of them.\nLet us start with the actions of Newtonian gravity and its relativistic analogue, the Einstein-Hilbert action\nS Newton = ∫ [ -ρϕ + 1 8 πG δ ij ∂ i ϕ∂ j ϕ ] d 3 x, (1)\nS EH = ∫ [ L matter + c 4 16 πG g µν R µν ] √ -g d 4 x. (2)\nVariation of the Newtonian action with respect to the gravitation potential ϕ yields the well-known Poisson equation ∆ ϕ = 4 πGρ where ρ is a given matter\ndistribution. Likewise, variations of the Einstein-Hilbert action with respect to the metric (which contains the gravitational potentials) gives the famous Einstein field equations G αβ = 8 πG/c 4 T αβ .\nWhen comparing the Einstein-Hilbert action (2) with other models in physics, it appears to be somewhat unnatural as one generally considers potential energies quadratic in the field strength, Hooke's law probably being the best known. However, when looking at the Einstein-Hilbert action, there are various very good reason for its form. Some of these can be motivated mathematically while others are simply observational. The field equations derived from it are close to Newton's theory of gravity and where the solutions deviate they are doing so in precisely the way to be in agreement with observations. However, there are observational facts which strongly indicate that our understanding of the gravitational force is far from complete. Dark matter and dark energy are two unknown forms of matter which are required to make the Universe work.\nLooking at (1) and (2) we note that both actions contain a contraction with respect to the metric. In the Newtonian model we simply have the flat metric for Cartesian 3-space. We are now taking some inspiration from Continuum Mechanics and in particular constitutive equations which specify the various different models studied in this field. In Maxwell's electromagnetism the constitutive equations for instance define the form of the dielectric tensor D = ε E where ε is a rank 2 tensor in general and we should really write D i = ε i j E j . When working with the Faraday tensor and its corresponding excitation, one writes H ij = 1 / 4 /epsilon1 ijmn χ mnkl F kl where for Maxwell's theory in vacuum χ mnkl = √ -g ( g mk g nl -g nk g ml ), see [3]. In many simple applications ε i j is taken to be proportional to the Kronecker delta which corresponds to a simple isotropic medium with dielectric constant /epsilon1 , namely ε i j = /epsilon1δ j i . Sometimes /epsilon1 is allowed to vary throughout the medium. In other words, a rank 2 isotropic tensor is proportional to the metric tensor. We are using this observation to argue that the Newton and Einstein-Hilbert actions are based on the assumption of an isotropic medium. Since Nature has a strong tendency to be anisotropic, this appears to be rather unnatural. By doing so, we are changing the interpretation of the terms δ ij and g µν in (1) and (2), respectively. We now view them much like the material tensors specified by constitutive equations which define the properties of the material. It is an interesting historical fact that F. Klein noted in a letter to Hilbert, as early as 1917, that the Ricci scalar in the Einstein-Hilbert action can be written as χ mnkl R mnkl , see [4]. For a recent paper inspired by similar thoughts, see [5].\nOur approach follows the principal ideas of Brans and Dicke [6]. They suggested a model where the gravitational coupling was allowed to vary in space and time and it was viewed as an additional dynamical degree of freedom in the theory. The original Brans-Dicke model can be viewed as the isotropic limit of out model, we simply choose C µν = φ ( x α ) g µν with the main difference that we do not treat the new degrees of freedom as dynamical.\nWe might want to, for sake of concreteness, speak of the properties of the vacuum here. However, we would like to be very careful and point out that this change of viewpoint has a variety of philosophical implications when interpreting the modified theories. We would like to keep these issues aside for now and proceed with the formulation of the theories. The assumption of an isotropic vacuum is certainly well supported by a host of experimental observations. However, there is clearly room for some improvement, in particular\nin general relativity. Many of the well-known modifications are is some sense rather severe. Field equations become higher than second order, new fields are introduced, locality is broken, local Lorentz invariance is broken etc. We will show that our theories are completely harmless and retain all desired properties.\n2 Modified gravity using continuum mechanics\nOne of the most conservative modifications of Newton's and Einstein's theories we can think of is therefore the following\nS Newton = ∫ [ -ρϕ + 1 8 πG C ij ∂ i ϕ∂ j ϕ ] d 3 x, (3)\nS EH = ∫ [ L matter + c 4 16 πG C µν R µν ] √ -g d 4 x, (4)\nwhere C ij and C µν are symmetric rank 2 tensors which contain information about the underlying structure of the theory. In Continuum Mechanics such objects are often referred to as material tensors or elastic coefficients [7]. We will stick with this well established notation and should clearly distinguish it from the energy-momentum tensor T αβ . We should emphasise that in this approach to gravity we are able to assume any symmetry for the metric, for instance spherical symmetry or homogeneity and isotropy. However, the symmetry of the metric is independent of the symmetry of the elastic coefficients.\nLet us firstly consider variations of the modified Newtonian theory (3) with respect to the potential φ . We find\n∂ i ( C ij ∂ j ϕ ) = 4 πGρ, (5)\nwhich reduces to the Poisson equation if we choose C ij = δ ij . As C ij can in principle be an arbitrary tensor, solutions to this equation may be quite different. As a simple example, let us consider the case when C ij = diag( c 1 , c 2 , c 3 ) with the c i being constants. In continuum mechanics or solid state physics one would speak of a crystal with three principal propagation directions. In this case, the field equation (5) can be reduced to the Poisson equation by rescaling the coordinates. Looking for a radially symmetric solution gives the interesting result\nϕ ∝ 1 √ c 1 x 2 + c 2 y 2 + c 3 z 2 . (6)\nIn this case the gravitational field of a massive body is ellipsoidal instead of being spherical. However, the strength of this effect depends on the values of the constants. If for instance, the numerical values of the three constants c 1 , c 2 , c 3 are very close to each other, then the gravitational field will look spherical unless very large distances are taking into account.\nIt should also be noted that the components of C ij do not have to be constants. They can be functions of the coordinates. Thus, the form of the gravitational field may be different in different regions of space, however, the gravitational law (5) would still be universal. If we consider C ij = χ ( t, x, y, z ) δ ij , the field equations are\n∆ ϕ = 4 πG χ ρ -∂ i ϕ∂ i χ χ . (7)\nIf the function χ is slowly varying in space | ∂ i χ | /lessmuch 1, we would have a theory which would be in very good agreement with Newton's theory on smaller scales, like the solar system. Note also that assuming a time varying χ would correspond to a time dependent gravitational constant, compare with Dirac's large number hypothesis [8].\nLet us now consider the action (4). We assume that the tensor C µν may depend on the metric, this is important so that the limit of general relativity can be recovered. Note that the variation of the Ricci tensor does not give a surface term anymore because C µν is not the metric. One has to integrate by parts twice (the Ricci tensor contains second derivatives of the metric) and move the derivatives action on the metric variations to act on the matter tensor. Denoting\nΣ µναβ = -δC µν δg αβ , (8)\nthe whole calculation yields the field equation\nΣ µναβ R µν -1 2 g αβ C µν R µν + 1 2 /square C αβ + 1 2 g αβ ∇ µ ∇ ν C µν -∇ µ ∇ ( α C β ) µ = 8 πG c 4 T αβ . (9)\nwhere we have also included the energy-momentum tensor of matter.\nRecall that in General Relativity the field equations imply the conservation equations by virtue of the twice contracted Bianchi identities. This does no longer hold due to the presence of the material tensor C µν in the field equations. Nonetheless, we can take the covariant derivative of the field equation (9) with respect to ∇ α and find the following conservation equation\n∇ α Σ µναβ R µν +Σ µναβ ∇ α R µν -1 2 g αβ R µν ∇ α C µν -C ασ ∇ α R β σ -R β σ ∇ α C ασ = 8 πG c 4 ∇ α T αβ . (10)\nThis implies that the conservation equations for T αβ is no longer a direct consequence of the gravitational field equations alone. However, if we assume that the energy-momentum tensor is derived from a diffeomorphism invariant matter action, then the energy-momentum tensor is covariantly conserved in view of Noether's theorem\n∇ α T αβ = 0 . (11)\nWe will assume this henceforth. Therefore, the conservation equation (11) imposes constraints on the components of C µν via (10) and thus its components cannot be chosen completely arbitrarily. For concreteness, we define\nJ β = ∇ α Σ µναβ R µν +Σ µναβ ∇ α R µν -1 2 g αβ R µν ∇ α C µν -C ασ ∇ α R β σ -R β σ ∇ α C ασ , (12)\nwhich gives the additional consistency equation\nJ β = 0 . (13)\nIt should also be emphasised that one has to be careful when choosing the tensor C µν and the derived quantity Σ µναβ . Namely, there is a conceptual difference between prescribing the tensor C µ ν from prescribing C µν . In the former case, we find that C µν = g µσ C ν σ and thus C µν has an explicit dependence on the metric which in turn will yield a non-trivial form for Σ µναβ . On the other, if we specify C µν a priori , then this tensor does not depend on the metric and hence the resulting variational derivative would be zero. One has to carefully distinguish both cases when analysing the field equations.\nIt is very difficult to analyse the field equations (9) given general Σ µναβ and C µν . It is also not clear what would constitute a good choice for those quantities. As a first attempt to understand the theory, we assume C µν to be conformally related to the inverse metric\nC µν = φ ( x α ) g µν , (14)\nwhere φ ( x α ) is a scalar function depending on the coordinates. This choice simplifies the field equation considerably and they are now given by\nφG αβ + g αβ /square φ -∇ α ∇ β φ = 8 πG c 4 T αβ . (15)\nUpon division by the field φ and solving for the Einstein tensor, the resulting equation\nG αβ = φ -1 8 πG c 4 T αβ + φ -1 ( ∇ α ∇ β φ -g αβ /square φ ) . (16)\nshows similarities with non-minimally coupled scalar field theories [6, see Eq. (11)]. The main difference between the current theory and most other approaches is that φ is not a dynamical degree of freedom because C µν is a prescribed tensor and has not corresponding equations of motion. The conformal model appears to agree with [6] in the limit when the Brans-Dicke parameter ω → 0. However, as variations with respect to φ are not considered, there is no propagation equation for the scalar field. By substituting (14) back into the action (4), we note that this simply corresponds to assuming the gravitational constant to be varying in time and space, see for instance [8]. Varying constants models are generally based on non-minimally coupled scalar field with kinetic term similar to Brans-Dicke theory, see [9].\n3 A Schwarzschild like solution\nLet us start by considering a static and spherically symmetric vacuum spacetime described by the metric\nds 2 = -e ν ( r ) dt 2 + e µ ( r ) dr 2 + r 2 d Ω 2 , (17)\nwhere d Ω 2 is the usual line element of the two-sphere. We also assume φ = e ξ ( r ) . When the analogue situation is analysed in General Relativity where ξ ≡ 0, one finds two independent equations which determine the two unknown functions ν ( r ) and µ ( r ). In this model, there is the additional degree of freedom ξ and\nfortunately, there are now three independent equations. These are given by\n-e µ r 2 + 1 r 2 -1 2 µ ' ξ ' -µ ' r + ξ '' +( ξ ' ) 2 +2 ξ ' r = 0 , (18)\n-e µ r 2 + 1 r 2 + 1 2 ν ' ξ ' + ν ' r +2 ξ ' r = 0 , (19)\n-1 4 µ ' ν ' -1 2 µ ' ξ ' -µ ' 2 r + ν '' 2 + 1 2 ν ' ξ ' + 1 4 ( ν ' ) 2 + ν ' 2 r + ξ '' +( ξ ' ) 2 + ξ ' r = 0 . (20)\nEqn. (19) can be solved for the function µ which can then be substituted into the other two equations. Combining those linearly gives the condition ν ' ∝ ξ ' which then allows us to reduce this problem to a single differential equation. While separation of variable and subsequent integration is possible, the resulting equation cannot be solved analytically for the unknown functions due to its high nonlinearity. However, by assuming ξ ' = -C 2 ν ' with C /lessmuch 1 we can find an approximate solution to the field equations which is given by\ne ν = 1 -C r + O ( C 4 ) , (21)\ne -µ = 1 -C r + 2 C 3 r + O ( C 4 ) , (22)\ne ξ = 1 + C 3 r + O ( C 4 ) . (23)\nOne easily verifies that this solution satisfies the field equations (18)-(20) up to O ( C 4 ). By choosing C = 2 GM we arrive at a Schwarzschild like solution with only a small difference. Clearly, this difference of the order of C 3 and therefore it would be very difficult to distinguish between this metric and the Schwarzschild metric using solar system tests. This is a promising result which indicates that this theory can pass solar system tests without great difficulty.\n4 Cosmology\n4.1 Conformal model\nNext, we want to study the cosmological implications of field equations (9). Similar to the above we assume C µν = φ ( t ) g µν and consider a FLRW universe\nds 2 = -dt 2 + a 2 ( t ) dx 2 + dy 2 + dz 2 ( 1 + k 4 r 2 ) 2 , (24)\nwhere a ( t ) is the scale factor and r 2 = x 2 + y 2 + z 2 . This yields the following cosmological field equations\n3 H 2 φ +3 H ˙ φ +3 kφ a 2 = 8 πG c 4 ρ, (25)\n-¨ φ -2 φ a a -2 H ˙ φ -H 2 φ -kφ a 2 = 8 πG c 4 p , (26)\nwhere an overdot denotes differentiation with respect to time derivative and H is the Hubble parameter H = ˙ a/a . Moreover, the conservation equation (11) gives\n˙ ρ +3 H ( ρ + p ) = 0 , (27)\nwhile the consistency equation (13) for J 0 becomes\n3 ˙ φ ( ˙ H +2 H 2 + k a 2 ) = 0 . (28)\nThe other three components vanish identically J 1 = J 2 = J 3 = 0.\nIrrespective of the choice of matter in this model, the consistency equation (13) either implies that ˙ φ = 0 which is equivalent to General Relativity, or\n˙ H +2 H 2 + k a 2 = 0 . (29)\nThe solutions to this differential equation are given by\na ( t ) = a 0 √ t -t 0 k = 0 , (30)\na ( t ) = √ a 2 0 -k ( t -t 0 ) 2 k = ± 1 , (31)\nand correspond to a radiation dominated universe. Next, we have to check whether these solutions are consistent with the remaining equations (25), (26) and (27). Firstly, we start looking for vacuum solutions, ρ = p = 0. The conservation equation (27) is trivially satisfied and one verifies that\nφ ( t ) = φ 0 √ ( t -t 0 ) k = 0 , (32)\nφ ( t ) = φ 0 ( t -t 0 ) √ a 2 0 -k ( t -t 0 ) 2 k = ± 1 , (33)\nis a solution to the remaining field equations.\nUnder the assumption that C µν = φ ( t ) g µν and ρ = p = 0 we cannot find an accelerating solution to the field equations. We could, of course, add regular matter to the field equations and seek other solutions. Let us briefly consider the situation where we include an incompressible perfect fluid or dust, for simplicity we consider k = 0 only. The scale factor is unchanged and given by (30). The conservation equation (27) implies the standard relation ρ ∝ 1 /a 3 and it turns out that\nρ ( t ) = ρ 0 ( t -t 0 ) 3 / 2 , φ ( t ) = 16 πρ 0 t +3 φ 0 3 √ t -t 0 , (34)\nis a solution to the field equations.\n4.2 Fluid like model\nHowever, we are particularly interested in the vacuum equations without ordinary matter. The reason for this is simply that we want to show that an\nadditional non-dynamical structure in the theory suffices to get a dynamical universe. One can easily find such solutions by introducing an extra degree of freedom in the material tensor. Let us choose the elastic coefficients to be\nC µ ν = diag( -/rho1, σ, σ, σ ) , (35)\nwhich means that C µν = 1 2 ( g µσ C ν σ + g νσ C µ σ ) and thus we have an explicit dependence on the metric, implying a non-zero Σ µναβ given by\nΣ µναβ = 1 4 ( g µα C νβ + g να C µβ + g µβ C να + g νβ C µα ) . (36)\nOne can interpret the quantities /rho1 and σ as the energy density and pressure of the vacuum, thereby specifying its internal structure.\nThe cosmological field equations, the conservation equation and the consistency equation of this model are given by\n-H 2 /rho1 -1 2 H ( ˙ /rho1 -˙ σ ) + kσ a 2 = 8 πG 3 c 4 ρ, (37)\n3 H 2 /rho1 +2( /rho1H )˙ + 1 2 (¨ /rho1 -¨ σ ) -kσ a 2 = 8 πG c 4 p , (38)\nH 2 ( ˙ /rho1 -3 ˙ σ ) + ˙ H ( ˙ /rho1 -˙ σ ) -2 k ˙ σ a 2 = 0 , (39)\n˙ ρ +3 H ( ρ + p ) = 0 , (40)\nwhere the conservation equation can be derived from the first three equations. In the analogue situation in General Relativity we would have two independent equations for three unknown functions while for this model we have one additional equation and two additional degrees of freedom.\nTo begin with, we consider the vacuum case ρ = p = 0 which eliminates one equation and two degrees of freedom. Thus, we are left with two independent equations and three unknown functions. In order to close this system, we choose a linear equation of state σ = w/rho1 for the spacetime structure.\nNow, we can solve Eq. (37) for ˙ /rho1 and substitute this result into Eq. (38) to arrive at a single differential equation in a ( t ). One can also check that substitution of ˙ /rho1 into Eq. (39) leads to the same differential equation, confirming that these equations are indeed not independent. This differential equation is given by\n(˙ a 2 -kw ) ( ( w -1) a a +2 w ( k + ˙ a 2 ) ) = 0 , (41)\n/negationslash\n/negationslash\n/negationslash\na ( t ) = √ kw ( t -t 0 ) , (42)\nwhere in the derivation we assumed /rho1 = 0, w = 1 and ˙ a = 0. This differential equation is the product of two equations and thus can be solved by finding a solution to either of the two equations. The first one is easily solved by\nand is valid only if kw > 0.\nThe second differential equation in (41) cannot be solved analytically for arbitrary k and w due to the non-linear nature of the equation. For w = 1 / 3 for instance we can find the two solutions\na ( t ) = 1 2 ( 1 α 2 e α ( t -t 0 ) + ke -α ( t -t 0 ) ) , (43)\na ( t ) = 1 2 ( ke α ( t -t 0 ) + 1 α 2 e -α ( t -t 0 ) ) , (44)\nwhich are valid for all k . For w = 0 one only finds one solution a ( t ) = a 0 t . For k = 0 one can solve the differential equation for all w and its solution is given by\na ( t ) = a 0 ( t -t 0 ) w -1 3 w -1 , (45)\n/negationslash\nwhich is well defined provided that w = 1 / 3 and corresponds to a power-law solution. For 0 < w < 1 / 3 this would correspond to an accelerated solution. The case w = 1 / 3 needs to be treated separately. As the exponent becomes very large, one would expect this to correspond to exponential functions, and indeed in this case\na = a 0 e λt . (46)\nThus, we were able to find solutions of the field equations modelling a universe which can undergo periods of accelerated expansion, without the need to introduce any forms of matter. All we have done is to add an additional non-dynamical structure to the theory on a very fundamental level.\n4.3 Kasner type model\nWe are now considering a Kasner type metric given by\nds 2 = -dt 2 + t 2 p 1 dx 2 + t 2 p 2 dy 2 + t 2 p 3 dz 2 , (47)\nand assume the material tensor C µ ν to be of the form\nC µ ν = diag( -1 , c 1 ( t ) , c 2 ( t ) , c 3 ( t )) . (48)\nThe field equations of this system are quite complicated. However, one notes that all field equations contain terms of the form tc ' i ( t ) and t 2 c '' i ( t ) which indicate that one can arrive at algebraic equations by choosing\nc i ( t ) = 2 γ i log( t ) , (49)\nwhere the γ i are constants. With this additional assumption (49), and considering a vacuum ρ = p = 0, the field equations are given by\np 1 γ 1 + p 2 γ 2 + p 3 γ 3 -p 1 p 2 -p 1 p 3 -p 2 p 3 = 0 , (50)\n( p 1 + p 2 + p 3 -1)( γ 1 -γ 3 + p 1 -p 3 ) = 0 , (52)\n( p 1 + p 2 + p 3 -1)( γ 1 -γ 2 + p 1 -p 2 ) = 0 , (51)\n( p 1 + p 2 + p 3 -1)( γ 2 -γ 3 + p 2 -p 3 ) = 0 , (53)\n( p 1 + p 2 + p 3 -1)( p 1 γ 1 + p 2 γ 2 + p 3 γ 3 ) = 0 . (54)\nThe structure of these equations is quite interesting as 4 of the 5 equations can be solved immediately by assuming the Kasner condition p 1 + p 2 + p 3 = 1. Note that in General Relativity this condition is necessary to solve the field equations. The condition p 1 + p 2 + p 3 = 1 and the remaining equation (50) give two algebraic relation for the 6 free parameters.\nHowever, we can also find solutions to Eqs. (50)-(54) without the Kasner condition. We start by assuming that p 1 + p 2 + p 3 = 1 which allows us to\n/negationslash\ndivide Eqs. (51)-(54) by the factor ( p 1 + p 2 + p 3 -1). We note that the three Eqs. (51)-(53) are not independent as (51) -(52) + (53) = 0 . Hence, we are left with four independent equations.\nOne easily verifies that a solution can be written in the form\nγ 1 = p 2 2 p 1 + p 2 , γ 2 = p 2 1 p 1 + p 2 , γ 3 = p 1 + p 2 , (55)\np 3 = -p 1 p 2 p 1 + p 2 . (56)\nWe should remark that Eq. (56) does imply that one of the three p i has to be negative. While the additional structure due to the material tensor C µ ν changes the underlying conceptual physics substantially, the solution shows many similarities with General Relativity.\n5 Conclusions\nIt is tempting to argue that we introduced a form of matter through the back door by choosing our material tensor, the elastic coefficients. However, it is far from clear whether this is indeed the case. Note that C µν is not a dynamical variable and thus it cannot be interpreted simply as matter. In Continuum Mechanics and when working with crystal symmetries, the tensor C µν is said to encode the symmetry properties of the material, in our case the vacuum. We simply say that the vacuum as we know it may have an internal structure which is specified by C µν . We are breaking away from the assumption that the vacuum is isotropic and structureless.\nThe form of the elastic coefficients can in principle be determined observationally. In the context of cosmology, one could start with the specific C µν given by (35) and assume it to be close to the metric g µν . When one considers models where deviations from General Relativity will vary with cosmological time, it would be most interesting to see how observational data would determine the form of C µν which provides the best fit to the data. Using our approach to gravity, we will be able to use observations directly to specify the model instead of guessing new theories and deriving their implications on a case by case basis. Ultimately, experiments and observations will determine the correct theory.\nAcknowledgements\nWe thank Friedrich Hehl and Dmitri Vassiliev for useful discussions. We would also like to thank the anonymous referees who have given us very valuable feedback.\nReferences\n\n[1] M. Milgrom, Astrophys. J. 270 , 365 (1983); Astrophys. J. 270 , 371 (1983).\n[2] H. F. M. Goenner, Living Rev. Rel. 7 , 2 (2004).\n[3] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics , Birkhauser, Boston (2003).\n\n\n[4] F. Klein, Nachrichten der Kgl. Gesellschaft der Wissenschaften zu Gottingen. Mathematisch-physikalische Klasse. (1917.) Vorgelegt in der Sitzung vom 25. Januar 1918. In Felix Klein Gesammelte Mathematische Abhandlungen, Erster Band , Section XXXI, R. Fricke and A. Ostrowski (Eds.), Springer, Berlin (1921).\n[5] R. A. Battye and J. A. Pearson, 'Massive gravity, the elasticity of spacetime and perturbations in the dark sector,' arXiv:1301.5042 [astro-ph.CO].\n[6] C. Brans and R. H. Dicke, Phys. Rev. 124 , 925 (1961).\n[7] A. E. Green and W. Zerna, Theoretical Elasticity , Oxford University Press, London (1963).\n[8] P. A. M. Dirac, Proc. Roy. Soc. Lond. A 165 , 199 (1938).\n[9] J. D. Barrow and P. Parsons, Phys. Rev. D 55 , 1906 (1997) [gr-qc/9607072]; J. -P. Uzan, Living Rev. Rel. 14 , 2 (2011) [arXiv:1009.5514 [astro-ph.CO]].\n\n"},"sections":{"kind":"list like","value":[{"title":"A new approach to modifying theories of gravity","content":"Christian G. Bohmer ∗ and Nicola Tamanini † Department of Mathematics, University College London Gower Street, London, WC1E 6BT, UK October 3, 2018","pages":[1]},{"title":"Abstract","content":"We propose a new point of view for interpreting Newton's and Einstein's theories of gravity. By taking inspiration from Continuum Mechanics and its treatment of anisotropies, we formulate new gravitational actions for modified theories of gravity. These models are simple and natural generalisations with many interesting properties. Above all, their precise form can, in principle, be determined experimentally.","pages":[1]},{"title":"1 Introduction","content":"Since Newton's formulation of the law of universal gravitation in the Principia 1687, the theory has been unchanged. Very few attempts have been made to modify Newtonian gravity, modified Newtonian dynamics being probably the one exception [1]. It took almost 330 years until a generalisation of Newton's theory was successfully constructed. Einstein's theory of gravity was radically different to any previous physical theory. It abandoned the absolute notion of time and replaced the concept of force by the curvature of a four dimensional space. However, it took only a few years when the first modifications and extensions of Einstein's theory appeared in the literature, see [2] for an excellent historical review. Ever since, modified theories of gravity have enjoyed a prominent role in theoretical physics. It is the aim of the current work to propose modifications of Newton's and Einstein's theories of gravity by applying the same idea to both of them. Let us start with the actions of Newtonian gravity and its relativistic analogue, the Einstein-Hilbert action Variation of the Newtonian action with respect to the gravitation potential ϕ yields the well-known Poisson equation ∆ ϕ = 4 πGρ where ρ is a given matter distribution. Likewise, variations of the Einstein-Hilbert action with respect to the metric (which contains the gravitational potentials) gives the famous Einstein field equations G αβ = 8 πG/c 4 T αβ . When comparing the Einstein-Hilbert action (2) with other models in physics, it appears to be somewhat unnatural as one generally considers potential energies quadratic in the field strength, Hooke's law probably being the best known. However, when looking at the Einstein-Hilbert action, there are various very good reason for its form. Some of these can be motivated mathematically while others are simply observational. The field equations derived from it are close to Newton's theory of gravity and where the solutions deviate they are doing so in precisely the way to be in agreement with observations. However, there are observational facts which strongly indicate that our understanding of the gravitational force is far from complete. Dark matter and dark energy are two unknown forms of matter which are required to make the Universe work. Looking at (1) and (2) we note that both actions contain a contraction with respect to the metric. In the Newtonian model we simply have the flat metric for Cartesian 3-space. We are now taking some inspiration from Continuum Mechanics and in particular constitutive equations which specify the various different models studied in this field. In Maxwell's electromagnetism the constitutive equations for instance define the form of the dielectric tensor D = ε E where ε is a rank 2 tensor in general and we should really write D i = ε i j E j . When working with the Faraday tensor and its corresponding excitation, one writes H ij = 1 / 4 /epsilon1 ijmn χ mnkl F kl where for Maxwell's theory in vacuum χ mnkl = √ -g ( g mk g nl -g nk g ml ), see [3]. In many simple applications ε i j is taken to be proportional to the Kronecker delta which corresponds to a simple isotropic medium with dielectric constant /epsilon1 , namely ε i j = /epsilon1δ j i . Sometimes /epsilon1 is allowed to vary throughout the medium. In other words, a rank 2 isotropic tensor is proportional to the metric tensor. We are using this observation to argue that the Newton and Einstein-Hilbert actions are based on the assumption of an isotropic medium. Since Nature has a strong tendency to be anisotropic, this appears to be rather unnatural. By doing so, we are changing the interpretation of the terms δ ij and g µν in (1) and (2), respectively. We now view them much like the material tensors specified by constitutive equations which define the properties of the material. It is an interesting historical fact that F. Klein noted in a letter to Hilbert, as early as 1917, that the Ricci scalar in the Einstein-Hilbert action can be written as χ mnkl R mnkl , see [4]. For a recent paper inspired by similar thoughts, see [5]. Our approach follows the principal ideas of Brans and Dicke [6]. They suggested a model where the gravitational coupling was allowed to vary in space and time and it was viewed as an additional dynamical degree of freedom in the theory. The original Brans-Dicke model can be viewed as the isotropic limit of out model, we simply choose C µν = φ ( x α ) g µν with the main difference that we do not treat the new degrees of freedom as dynamical. We might want to, for sake of concreteness, speak of the properties of the vacuum here. However, we would like to be very careful and point out that this change of viewpoint has a variety of philosophical implications when interpreting the modified theories. We would like to keep these issues aside for now and proceed with the formulation of the theories. The assumption of an isotropic vacuum is certainly well supported by a host of experimental observations. However, there is clearly room for some improvement, in particular in general relativity. Many of the well-known modifications are is some sense rather severe. Field equations become higher than second order, new fields are introduced, locality is broken, local Lorentz invariance is broken etc. We will show that our theories are completely harmless and retain all desired properties.","pages":[1,2,3]},{"title":"2 Modified gravity using continuum mechanics","content":"One of the most conservative modifications of Newton's and Einstein's theories we can think of is therefore the following where C ij and C µν are symmetric rank 2 tensors which contain information about the underlying structure of the theory. In Continuum Mechanics such objects are often referred to as material tensors or elastic coefficients [7]. We will stick with this well established notation and should clearly distinguish it from the energy-momentum tensor T αβ . We should emphasise that in this approach to gravity we are able to assume any symmetry for the metric, for instance spherical symmetry or homogeneity and isotropy. However, the symmetry of the metric is independent of the symmetry of the elastic coefficients. Let us firstly consider variations of the modified Newtonian theory (3) with respect to the potential φ . We find which reduces to the Poisson equation if we choose C ij = δ ij . As C ij can in principle be an arbitrary tensor, solutions to this equation may be quite different. As a simple example, let us consider the case when C ij = diag( c 1 , c 2 , c 3 ) with the c i being constants. In continuum mechanics or solid state physics one would speak of a crystal with three principal propagation directions. In this case, the field equation (5) can be reduced to the Poisson equation by rescaling the coordinates. Looking for a radially symmetric solution gives the interesting result In this case the gravitational field of a massive body is ellipsoidal instead of being spherical. However, the strength of this effect depends on the values of the constants. If for instance, the numerical values of the three constants c 1 , c 2 , c 3 are very close to each other, then the gravitational field will look spherical unless very large distances are taking into account. It should also be noted that the components of C ij do not have to be constants. They can be functions of the coordinates. Thus, the form of the gravitational field may be different in different regions of space, however, the gravitational law (5) would still be universal. If we consider C ij = χ ( t, x, y, z ) δ ij , the field equations are If the function χ is slowly varying in space | ∂ i χ | /lessmuch 1, we would have a theory which would be in very good agreement with Newton's theory on smaller scales, like the solar system. Note also that assuming a time varying χ would correspond to a time dependent gravitational constant, compare with Dirac's large number hypothesis [8]. Let us now consider the action (4). We assume that the tensor C µν may depend on the metric, this is important so that the limit of general relativity can be recovered. Note that the variation of the Ricci tensor does not give a surface term anymore because C µν is not the metric. One has to integrate by parts twice (the Ricci tensor contains second derivatives of the metric) and move the derivatives action on the metric variations to act on the matter tensor. Denoting the whole calculation yields the field equation where we have also included the energy-momentum tensor of matter. Recall that in General Relativity the field equations imply the conservation equations by virtue of the twice contracted Bianchi identities. This does no longer hold due to the presence of the material tensor C µν in the field equations. Nonetheless, we can take the covariant derivative of the field equation (9) with respect to ∇ α and find the following conservation equation This implies that the conservation equations for T αβ is no longer a direct consequence of the gravitational field equations alone. However, if we assume that the energy-momentum tensor is derived from a diffeomorphism invariant matter action, then the energy-momentum tensor is covariantly conserved in view of Noether's theorem We will assume this henceforth. Therefore, the conservation equation (11) imposes constraints on the components of C µν via (10) and thus its components cannot be chosen completely arbitrarily. For concreteness, we define which gives the additional consistency equation It should also be emphasised that one has to be careful when choosing the tensor C µν and the derived quantity Σ µναβ . Namely, there is a conceptual difference between prescribing the tensor C µ ν from prescribing C µν . In the former case, we find that C µν = g µσ C ν σ and thus C µν has an explicit dependence on the metric which in turn will yield a non-trivial form for Σ µναβ . On the other, if we specify C µν a priori , then this tensor does not depend on the metric and hence the resulting variational derivative would be zero. One has to carefully distinguish both cases when analysing the field equations. It is very difficult to analyse the field equations (9) given general Σ µναβ and C µν . It is also not clear what would constitute a good choice for those quantities. As a first attempt to understand the theory, we assume C µν to be conformally related to the inverse metric where φ ( x α ) is a scalar function depending on the coordinates. This choice simplifies the field equation considerably and they are now given by Upon division by the field φ and solving for the Einstein tensor, the resulting equation shows similarities with non-minimally coupled scalar field theories [6, see Eq. (11)]. The main difference between the current theory and most other approaches is that φ is not a dynamical degree of freedom because C µν is a prescribed tensor and has not corresponding equations of motion. The conformal model appears to agree with [6] in the limit when the Brans-Dicke parameter ω → 0. However, as variations with respect to φ are not considered, there is no propagation equation for the scalar field. By substituting (14) back into the action (4), we note that this simply corresponds to assuming the gravitational constant to be varying in time and space, see for instance [8]. Varying constants models are generally based on non-minimally coupled scalar field with kinetic term similar to Brans-Dicke theory, see [9].","pages":[3,4,5]},{"title":"3 A Schwarzschild like solution","content":"Let us start by considering a static and spherically symmetric vacuum spacetime described by the metric where d Ω 2 is the usual line element of the two-sphere. We also assume φ = e ξ ( r ) . When the analogue situation is analysed in General Relativity where ξ ≡ 0, one finds two independent equations which determine the two unknown functions ν ( r ) and µ ( r ). In this model, there is the additional degree of freedom ξ and fortunately, there are now three independent equations. These are given by Eqn. (19) can be solved for the function µ which can then be substituted into the other two equations. Combining those linearly gives the condition ν ' ∝ ξ ' which then allows us to reduce this problem to a single differential equation. While separation of variable and subsequent integration is possible, the resulting equation cannot be solved analytically for the unknown functions due to its high nonlinearity. However, by assuming ξ ' = -C 2 ν ' with C /lessmuch 1 we can find an approximate solution to the field equations which is given by One easily verifies that this solution satisfies the field equations (18)-(20) up to O ( C 4 ). By choosing C = 2 GM we arrive at a Schwarzschild like solution with only a small difference. Clearly, this difference of the order of C 3 and therefore it would be very difficult to distinguish between this metric and the Schwarzschild metric using solar system tests. This is a promising result which indicates that this theory can pass solar system tests without great difficulty.","pages":[5,6]},{"title":"4.1 Conformal model","content":"Next, we want to study the cosmological implications of field equations (9). Similar to the above we assume C µν = φ ( t ) g µν and consider a FLRW universe where a ( t ) is the scale factor and r 2 = x 2 + y 2 + z 2 . This yields the following cosmological field equations where an overdot denotes differentiation with respect to time derivative and H is the Hubble parameter H = ˙ a/a . Moreover, the conservation equation (11) gives while the consistency equation (13) for J 0 becomes The other three components vanish identically J 1 = J 2 = J 3 = 0. Irrespective of the choice of matter in this model, the consistency equation (13) either implies that ˙ φ = 0 which is equivalent to General Relativity, or The solutions to this differential equation are given by and correspond to a radiation dominated universe. Next, we have to check whether these solutions are consistent with the remaining equations (25), (26) and (27). Firstly, we start looking for vacuum solutions, ρ = p = 0. The conservation equation (27) is trivially satisfied and one verifies that is a solution to the remaining field equations. Under the assumption that C µν = φ ( t ) g µν and ρ = p = 0 we cannot find an accelerating solution to the field equations. We could, of course, add regular matter to the field equations and seek other solutions. Let us briefly consider the situation where we include an incompressible perfect fluid or dust, for simplicity we consider k = 0 only. The scale factor is unchanged and given by (30). The conservation equation (27) implies the standard relation ρ ∝ 1 /a 3 and it turns out that is a solution to the field equations.","pages":[6,7]},{"title":"4.2 Fluid like model","content":"However, we are particularly interested in the vacuum equations without ordinary matter. The reason for this is simply that we want to show that an additional non-dynamical structure in the theory suffices to get a dynamical universe. One can easily find such solutions by introducing an extra degree of freedom in the material tensor. Let us choose the elastic coefficients to be which means that C µν = 1 2 ( g µσ C ν σ + g νσ C µ σ ) and thus we have an explicit dependence on the metric, implying a non-zero Σ µναβ given by One can interpret the quantities /rho1 and σ as the energy density and pressure of the vacuum, thereby specifying its internal structure. The cosmological field equations, the conservation equation and the consistency equation of this model are given by where the conservation equation can be derived from the first three equations. In the analogue situation in General Relativity we would have two independent equations for three unknown functions while for this model we have one additional equation and two additional degrees of freedom. To begin with, we consider the vacuum case ρ = p = 0 which eliminates one equation and two degrees of freedom. Thus, we are left with two independent equations and three unknown functions. In order to close this system, we choose a linear equation of state σ = w/rho1 for the spacetime structure. Now, we can solve Eq. (37) for ˙ /rho1 and substitute this result into Eq. (38) to arrive at a single differential equation in a ( t ). One can also check that substitution of ˙ /rho1 into Eq. (39) leads to the same differential equation, confirming that these equations are indeed not independent. This differential equation is given by /negationslash /negationslash /negationslash where in the derivation we assumed /rho1 = 0, w = 1 and ˙ a = 0. This differential equation is the product of two equations and thus can be solved by finding a solution to either of the two equations. The first one is easily solved by and is valid only if kw > 0. The second differential equation in (41) cannot be solved analytically for arbitrary k and w due to the non-linear nature of the equation. For w = 1 / 3 for instance we can find the two solutions which are valid for all k . For w = 0 one only finds one solution a ( t ) = a 0 t . For k = 0 one can solve the differential equation for all w and its solution is given by /negationslash which is well defined provided that w = 1 / 3 and corresponds to a power-law solution. For 0 < w < 1 / 3 this would correspond to an accelerated solution. The case w = 1 / 3 needs to be treated separately. As the exponent becomes very large, one would expect this to correspond to exponential functions, and indeed in this case Thus, we were able to find solutions of the field equations modelling a universe which can undergo periods of accelerated expansion, without the need to introduce any forms of matter. All we have done is to add an additional non-dynamical structure to the theory on a very fundamental level.","pages":[7,8,9]},{"title":"4.3 Kasner type model","content":"We are now considering a Kasner type metric given by and assume the material tensor C µ ν to be of the form The field equations of this system are quite complicated. However, one notes that all field equations contain terms of the form tc ' i ( t ) and t 2 c '' i ( t ) which indicate that one can arrive at algebraic equations by choosing where the γ i are constants. With this additional assumption (49), and considering a vacuum ρ = p = 0, the field equations are given by The structure of these equations is quite interesting as 4 of the 5 equations can be solved immediately by assuming the Kasner condition p 1 + p 2 + p 3 = 1. Note that in General Relativity this condition is necessary to solve the field equations. The condition p 1 + p 2 + p 3 = 1 and the remaining equation (50) give two algebraic relation for the 6 free parameters. However, we can also find solutions to Eqs. (50)-(54) without the Kasner condition. We start by assuming that p 1 + p 2 + p 3 = 1 which allows us to /negationslash divide Eqs. (51)-(54) by the factor ( p 1 + p 2 + p 3 -1). We note that the three Eqs. (51)-(53) are not independent as (51) -(52) + (53) = 0 . Hence, we are left with four independent equations. One easily verifies that a solution can be written in the form We should remark that Eq. (56) does imply that one of the three p i has to be negative. While the additional structure due to the material tensor C µ ν changes the underlying conceptual physics substantially, the solution shows many similarities with General Relativity.","pages":[9,10]},{"title":"5 Conclusions","content":"It is tempting to argue that we introduced a form of matter through the back door by choosing our material tensor, the elastic coefficients. However, it is far from clear whether this is indeed the case. Note that C µν is not a dynamical variable and thus it cannot be interpreted simply as matter. In Continuum Mechanics and when working with crystal symmetries, the tensor C µν is said to encode the symmetry properties of the material, in our case the vacuum. We simply say that the vacuum as we know it may have an internal structure which is specified by C µν . We are breaking away from the assumption that the vacuum is isotropic and structureless. The form of the elastic coefficients can in principle be determined observationally. In the context of cosmology, one could start with the specific C µν given by (35) and assume it to be close to the metric g µν . When one considers models where deviations from General Relativity will vary with cosmological time, it would be most interesting to see how observational data would determine the form of C µν which provides the best fit to the data. Using our approach to gravity, we will be able to use observations directly to specify the model instead of guessing new theories and deriving their implications on a case by case basis. Ultimately, experiments and observations will determine the correct theory.","pages":[10]},{"title":"Acknowledgements","content":"We thank Friedrich Hehl and Dmitri Vassiliev for useful discussions. We would also like to thank the anonymous referees who have given us very valuable feedback.","pages":[10]}],"string":"[\n {\n \"title\": \"A new approach to modifying theories of gravity\",\n \"content\": \"Christian G. Bohmer ∗ and Nicola Tamanini † Department of Mathematics, University College London Gower Street, London, WC1E 6BT, UK October 3, 2018\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"We propose a new point of view for interpreting Newton's and Einstein's theories of gravity. By taking inspiration from Continuum Mechanics and its treatment of anisotropies, we formulate new gravitational actions for modified theories of gravity. These models are simple and natural generalisations with many interesting properties. Above all, their precise form can, in principle, be determined experimentally.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"Since Newton's formulation of the law of universal gravitation in the Principia 1687, the theory has been unchanged. Very few attempts have been made to modify Newtonian gravity, modified Newtonian dynamics being probably the one exception [1]. It took almost 330 years until a generalisation of Newton's theory was successfully constructed. Einstein's theory of gravity was radically different to any previous physical theory. It abandoned the absolute notion of time and replaced the concept of force by the curvature of a four dimensional space. However, it took only a few years when the first modifications and extensions of Einstein's theory appeared in the literature, see [2] for an excellent historical review. Ever since, modified theories of gravity have enjoyed a prominent role in theoretical physics. It is the aim of the current work to propose modifications of Newton's and Einstein's theories of gravity by applying the same idea to both of them. Let us start with the actions of Newtonian gravity and its relativistic analogue, the Einstein-Hilbert action Variation of the Newtonian action with respect to the gravitation potential ϕ yields the well-known Poisson equation ∆ ϕ = 4 πGρ where ρ is a given matter distribution. Likewise, variations of the Einstein-Hilbert action with respect to the metric (which contains the gravitational potentials) gives the famous Einstein field equations G αβ = 8 πG/c 4 T αβ . When comparing the Einstein-Hilbert action (2) with other models in physics, it appears to be somewhat unnatural as one generally considers potential energies quadratic in the field strength, Hooke's law probably being the best known. However, when looking at the Einstein-Hilbert action, there are various very good reason for its form. Some of these can be motivated mathematically while others are simply observational. The field equations derived from it are close to Newton's theory of gravity and where the solutions deviate they are doing so in precisely the way to be in agreement with observations. However, there are observational facts which strongly indicate that our understanding of the gravitational force is far from complete. Dark matter and dark energy are two unknown forms of matter which are required to make the Universe work. Looking at (1) and (2) we note that both actions contain a contraction with respect to the metric. In the Newtonian model we simply have the flat metric for Cartesian 3-space. We are now taking some inspiration from Continuum Mechanics and in particular constitutive equations which specify the various different models studied in this field. In Maxwell's electromagnetism the constitutive equations for instance define the form of the dielectric tensor D = ε E where ε is a rank 2 tensor in general and we should really write D i = ε i j E j . When working with the Faraday tensor and its corresponding excitation, one writes H ij = 1 / 4 /epsilon1 ijmn χ mnkl F kl where for Maxwell's theory in vacuum χ mnkl = √ -g ( g mk g nl -g nk g ml ), see [3]. In many simple applications ε i j is taken to be proportional to the Kronecker delta which corresponds to a simple isotropic medium with dielectric constant /epsilon1 , namely ε i j = /epsilon1δ j i . Sometimes /epsilon1 is allowed to vary throughout the medium. In other words, a rank 2 isotropic tensor is proportional to the metric tensor. We are using this observation to argue that the Newton and Einstein-Hilbert actions are based on the assumption of an isotropic medium. Since Nature has a strong tendency to be anisotropic, this appears to be rather unnatural. By doing so, we are changing the interpretation of the terms δ ij and g µν in (1) and (2), respectively. We now view them much like the material tensors specified by constitutive equations which define the properties of the material. It is an interesting historical fact that F. Klein noted in a letter to Hilbert, as early as 1917, that the Ricci scalar in the Einstein-Hilbert action can be written as χ mnkl R mnkl , see [4]. For a recent paper inspired by similar thoughts, see [5]. Our approach follows the principal ideas of Brans and Dicke [6]. They suggested a model where the gravitational coupling was allowed to vary in space and time and it was viewed as an additional dynamical degree of freedom in the theory. The original Brans-Dicke model can be viewed as the isotropic limit of out model, we simply choose C µν = φ ( x α ) g µν with the main difference that we do not treat the new degrees of freedom as dynamical. We might want to, for sake of concreteness, speak of the properties of the vacuum here. However, we would like to be very careful and point out that this change of viewpoint has a variety of philosophical implications when interpreting the modified theories. We would like to keep these issues aside for now and proceed with the formulation of the theories. The assumption of an isotropic vacuum is certainly well supported by a host of experimental observations. However, there is clearly room for some improvement, in particular in general relativity. Many of the well-known modifications are is some sense rather severe. Field equations become higher than second order, new fields are introduced, locality is broken, local Lorentz invariance is broken etc. We will show that our theories are completely harmless and retain all desired properties.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"2 Modified gravity using continuum mechanics\",\n \"content\": \"One of the most conservative modifications of Newton's and Einstein's theories we can think of is therefore the following where C ij and C µν are symmetric rank 2 tensors which contain information about the underlying structure of the theory. In Continuum Mechanics such objects are often referred to as material tensors or elastic coefficients [7]. We will stick with this well established notation and should clearly distinguish it from the energy-momentum tensor T αβ . We should emphasise that in this approach to gravity we are able to assume any symmetry for the metric, for instance spherical symmetry or homogeneity and isotropy. However, the symmetry of the metric is independent of the symmetry of the elastic coefficients. Let us firstly consider variations of the modified Newtonian theory (3) with respect to the potential φ . We find which reduces to the Poisson equation if we choose C ij = δ ij . As C ij can in principle be an arbitrary tensor, solutions to this equation may be quite different. As a simple example, let us consider the case when C ij = diag( c 1 , c 2 , c 3 ) with the c i being constants. In continuum mechanics or solid state physics one would speak of a crystal with three principal propagation directions. In this case, the field equation (5) can be reduced to the Poisson equation by rescaling the coordinates. Looking for a radially symmetric solution gives the interesting result In this case the gravitational field of a massive body is ellipsoidal instead of being spherical. However, the strength of this effect depends on the values of the constants. If for instance, the numerical values of the three constants c 1 , c 2 , c 3 are very close to each other, then the gravitational field will look spherical unless very large distances are taking into account. It should also be noted that the components of C ij do not have to be constants. They can be functions of the coordinates. Thus, the form of the gravitational field may be different in different regions of space, however, the gravitational law (5) would still be universal. If we consider C ij = χ ( t, x, y, z ) δ ij , the field equations are If the function χ is slowly varying in space | ∂ i χ | /lessmuch 1, we would have a theory which would be in very good agreement with Newton's theory on smaller scales, like the solar system. Note also that assuming a time varying χ would correspond to a time dependent gravitational constant, compare with Dirac's large number hypothesis [8]. Let us now consider the action (4). We assume that the tensor C µν may depend on the metric, this is important so that the limit of general relativity can be recovered. Note that the variation of the Ricci tensor does not give a surface term anymore because C µν is not the metric. One has to integrate by parts twice (the Ricci tensor contains second derivatives of the metric) and move the derivatives action on the metric variations to act on the matter tensor. Denoting the whole calculation yields the field equation where we have also included the energy-momentum tensor of matter. Recall that in General Relativity the field equations imply the conservation equations by virtue of the twice contracted Bianchi identities. This does no longer hold due to the presence of the material tensor C µν in the field equations. Nonetheless, we can take the covariant derivative of the field equation (9) with respect to ∇ α and find the following conservation equation This implies that the conservation equations for T αβ is no longer a direct consequence of the gravitational field equations alone. However, if we assume that the energy-momentum tensor is derived from a diffeomorphism invariant matter action, then the energy-momentum tensor is covariantly conserved in view of Noether's theorem We will assume this henceforth. Therefore, the conservation equation (11) imposes constraints on the components of C µν via (10) and thus its components cannot be chosen completely arbitrarily. For concreteness, we define which gives the additional consistency equation It should also be emphasised that one has to be careful when choosing the tensor C µν and the derived quantity Σ µναβ . Namely, there is a conceptual difference between prescribing the tensor C µ ν from prescribing C µν . In the former case, we find that C µν = g µσ C ν σ and thus C µν has an explicit dependence on the metric which in turn will yield a non-trivial form for Σ µναβ . On the other, if we specify C µν a priori , then this tensor does not depend on the metric and hence the resulting variational derivative would be zero. One has to carefully distinguish both cases when analysing the field equations. It is very difficult to analyse the field equations (9) given general Σ µναβ and C µν . It is also not clear what would constitute a good choice for those quantities. As a first attempt to understand the theory, we assume C µν to be conformally related to the inverse metric where φ ( x α ) is a scalar function depending on the coordinates. This choice simplifies the field equation considerably and they are now given by Upon division by the field φ and solving for the Einstein tensor, the resulting equation shows similarities with non-minimally coupled scalar field theories [6, see Eq. (11)]. The main difference between the current theory and most other approaches is that φ is not a dynamical degree of freedom because C µν is a prescribed tensor and has not corresponding equations of motion. The conformal model appears to agree with [6] in the limit when the Brans-Dicke parameter ω → 0. However, as variations with respect to φ are not considered, there is no propagation equation for the scalar field. By substituting (14) back into the action (4), we note that this simply corresponds to assuming the gravitational constant to be varying in time and space, see for instance [8]. Varying constants models are generally based on non-minimally coupled scalar field with kinetic term similar to Brans-Dicke theory, see [9].\",\n \"pages\": [\n 3,\n 4,\n 5\n ]\n },\n {\n \"title\": \"3 A Schwarzschild like solution\",\n \"content\": \"Let us start by considering a static and spherically symmetric vacuum spacetime described by the metric where d Ω 2 is the usual line element of the two-sphere. We also assume φ = e ξ ( r ) . When the analogue situation is analysed in General Relativity where ξ ≡ 0, one finds two independent equations which determine the two unknown functions ν ( r ) and µ ( r ). In this model, there is the additional degree of freedom ξ and fortunately, there are now three independent equations. These are given by Eqn. (19) can be solved for the function µ which can then be substituted into the other two equations. Combining those linearly gives the condition ν ' ∝ ξ ' which then allows us to reduce this problem to a single differential equation. While separation of variable and subsequent integration is possible, the resulting equation cannot be solved analytically for the unknown functions due to its high nonlinearity. However, by assuming ξ ' = -C 2 ν ' with C /lessmuch 1 we can find an approximate solution to the field equations which is given by One easily verifies that this solution satisfies the field equations (18)-(20) up to O ( C 4 ). By choosing C = 2 GM we arrive at a Schwarzschild like solution with only a small difference. Clearly, this difference of the order of C 3 and therefore it would be very difficult to distinguish between this metric and the Schwarzschild metric using solar system tests. This is a promising result which indicates that this theory can pass solar system tests without great difficulty.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"4.1 Conformal model\",\n \"content\": \"Next, we want to study the cosmological implications of field equations (9). Similar to the above we assume C µν = φ ( t ) g µν and consider a FLRW universe where a ( t ) is the scale factor and r 2 = x 2 + y 2 + z 2 . This yields the following cosmological field equations where an overdot denotes differentiation with respect to time derivative and H is the Hubble parameter H = ˙ a/a . Moreover, the conservation equation (11) gives while the consistency equation (13) for J 0 becomes The other three components vanish identically J 1 = J 2 = J 3 = 0. Irrespective of the choice of matter in this model, the consistency equation (13) either implies that ˙ φ = 0 which is equivalent to General Relativity, or The solutions to this differential equation are given by and correspond to a radiation dominated universe. Next, we have to check whether these solutions are consistent with the remaining equations (25), (26) and (27). Firstly, we start looking for vacuum solutions, ρ = p = 0. The conservation equation (27) is trivially satisfied and one verifies that is a solution to the remaining field equations. Under the assumption that C µν = φ ( t ) g µν and ρ = p = 0 we cannot find an accelerating solution to the field equations. We could, of course, add regular matter to the field equations and seek other solutions. Let us briefly consider the situation where we include an incompressible perfect fluid or dust, for simplicity we consider k = 0 only. The scale factor is unchanged and given by (30). The conservation equation (27) implies the standard relation ρ ∝ 1 /a 3 and it turns out that is a solution to the field equations.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"4.2 Fluid like model\",\n \"content\": \"However, we are particularly interested in the vacuum equations without ordinary matter. The reason for this is simply that we want to show that an additional non-dynamical structure in the theory suffices to get a dynamical universe. One can easily find such solutions by introducing an extra degree of freedom in the material tensor. Let us choose the elastic coefficients to be which means that C µν = 1 2 ( g µσ C ν σ + g νσ C µ σ ) and thus we have an explicit dependence on the metric, implying a non-zero Σ µναβ given by One can interpret the quantities /rho1 and σ as the energy density and pressure of the vacuum, thereby specifying its internal structure. The cosmological field equations, the conservation equation and the consistency equation of this model are given by where the conservation equation can be derived from the first three equations. In the analogue situation in General Relativity we would have two independent equations for three unknown functions while for this model we have one additional equation and two additional degrees of freedom. To begin with, we consider the vacuum case ρ = p = 0 which eliminates one equation and two degrees of freedom. Thus, we are left with two independent equations and three unknown functions. In order to close this system, we choose a linear equation of state σ = w/rho1 for the spacetime structure. Now, we can solve Eq. (37) for ˙ /rho1 and substitute this result into Eq. (38) to arrive at a single differential equation in a ( t ). One can also check that substitution of ˙ /rho1 into Eq. (39) leads to the same differential equation, confirming that these equations are indeed not independent. This differential equation is given by /negationslash /negationslash /negationslash where in the derivation we assumed /rho1 = 0, w = 1 and ˙ a = 0. This differential equation is the product of two equations and thus can be solved by finding a solution to either of the two equations. The first one is easily solved by and is valid only if kw > 0. The second differential equation in (41) cannot be solved analytically for arbitrary k and w due to the non-linear nature of the equation. For w = 1 / 3 for instance we can find the two solutions which are valid for all k . For w = 0 one only finds one solution a ( t ) = a 0 t . For k = 0 one can solve the differential equation for all w and its solution is given by /negationslash which is well defined provided that w = 1 / 3 and corresponds to a power-law solution. For 0 < w < 1 / 3 this would correspond to an accelerated solution. The case w = 1 / 3 needs to be treated separately. As the exponent becomes very large, one would expect this to correspond to exponential functions, and indeed in this case Thus, we were able to find solutions of the field equations modelling a universe which can undergo periods of accelerated expansion, without the need to introduce any forms of matter. All we have done is to add an additional non-dynamical structure to the theory on a very fundamental level.\",\n \"pages\": [\n 7,\n 8,\n 9\n ]\n },\n {\n \"title\": \"4.3 Kasner type model\",\n \"content\": \"We are now considering a Kasner type metric given by and assume the material tensor C µ ν to be of the form The field equations of this system are quite complicated. However, one notes that all field equations contain terms of the form tc ' i ( t ) and t 2 c '' i ( t ) which indicate that one can arrive at algebraic equations by choosing where the γ i are constants. With this additional assumption (49), and considering a vacuum ρ = p = 0, the field equations are given by The structure of these equations is quite interesting as 4 of the 5 equations can be solved immediately by assuming the Kasner condition p 1 + p 2 + p 3 = 1. Note that in General Relativity this condition is necessary to solve the field equations. The condition p 1 + p 2 + p 3 = 1 and the remaining equation (50) give two algebraic relation for the 6 free parameters. However, we can also find solutions to Eqs. (50)-(54) without the Kasner condition. We start by assuming that p 1 + p 2 + p 3 = 1 which allows us to /negationslash divide Eqs. (51)-(54) by the factor ( p 1 + p 2 + p 3 -1). We note that the three Eqs. (51)-(53) are not independent as (51) -(52) + (53) = 0 . Hence, we are left with four independent equations. One easily verifies that a solution can be written in the form We should remark that Eq. (56) does imply that one of the three p i has to be negative. While the additional structure due to the material tensor C µ ν changes the underlying conceptual physics substantially, the solution shows many similarities with General Relativity.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"5 Conclusions\",\n \"content\": \"It is tempting to argue that we introduced a form of matter through the back door by choosing our material tensor, the elastic coefficients. However, it is far from clear whether this is indeed the case. Note that C µν is not a dynamical variable and thus it cannot be interpreted simply as matter. In Continuum Mechanics and when working with crystal symmetries, the tensor C µν is said to encode the symmetry properties of the material, in our case the vacuum. We simply say that the vacuum as we know it may have an internal structure which is specified by C µν . We are breaking away from the assumption that the vacuum is isotropic and structureless. The form of the elastic coefficients can in principle be determined observationally. In the context of cosmology, one could start with the specific C µν given by (35) and assume it to be close to the metric g µν . When one considers models where deviations from General Relativity will vary with cosmological time, it would be most interesting to see how observational data would determine the form of C µν which provides the best fit to the data. Using our approach to gravity, we will be able to use observations directly to specify the model instead of guessing new theories and deriving their implications on a case by case basis. Ultimately, experiments and observations will determine the correct theory.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"Acknowledgements\",\n \"content\": \"We thank Friedrich Hehl and Dmitri Vassiliev for useful discussions. We would also like to thank the anonymous referees who have given us very valuable feedback.\",\n \"pages\": [\n 10\n ]\n }\n]"}}},{"rowIdx":265,"cells":{"bibcode":{"kind":"string","value":"2013FrPhy...8..630Z"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1302.5485.pdf"},"content":{"kind":"string","value":"\nBlack Hole Binaries and Microquasars\nShuang-Nan Zhang 1 , 2 ∗\n1 Laboratory for Particle Astrophysics, Institute of High Energy Physics,Beijing 100049, China. 2 National Astronomical Observatories, Chinese Academy Of Sciences, Beijing 100012, China.\n(Dated: October 31, 2018)\nThis is a general review on the observations and physics of black hole X-ray binaries and microquasars, with the emphasize on recent developments in the high energy regime. The focus is put on understanding the accretion flows and measuring the parameters of black holes in them. It includes mainly two parts: (1) Brief review of several recent review article on this subject; (2) Further development on several topics, including black hole spin measurements, hot accretion flows, corona formation, state transitions and thermal stability of standard think disk. This is thus not a regular bottom-up approach, which I feel not necessary at this stage. Major effort is made in making and incorporating from many sources useful plots and illustrations, in order to make this article more comprehensible to non-expert readers. In the end I attempt to make a unification scheme on the accretion-outflow (wind/jet) connections of all types of accreting BHs of all accretion rates and all BH mass scales, and finally provide a brief outlook.\nPACS numbers:\nCONTENTS\n
\n\n
\nVIII. Further developments on thermal stability of SSD\n23\n\nIX. Unification and Outlook 25\nReferences 27\n\nI. SYNOPSIS\nI will start by defining what I mean by black hole binaries (BHBs) and microquasars in this article. I decide to restrict myself to only a subclass of BHBs, namely, BH X-ray binaries (BHXBs), since these are the only class of BHBs known observationally. I will then simply refer microquasars as BHXBs for reasons discussed in Section II.\nSince many excellent, comprehensive and quite up-todate review articles on BHXBs are readily available in literature, I feel it is not necessary to write another bottomup and comprehensive review article on the same subjects at this stage. I will thus take quite an unusual approach in this article. I will first give some concise guides on several representative review articles [1-4], with some necessary updates. I will then focus on the further developments on several topics I feel deserve more discussions, i.e., BH spin measurements (Section IV), hot accretion flows (Section V), corona formation (Section VI), state transitions (Section VII) and thermal stability of SSD (Section VIII). The emphasis is thus put on understanding the accretion flows and measuring the parameters of black holes in them. Some rather general issues on BH astrophysics, such as what astrophysical BHs are and how to identify them observationally, are not discussed here but can be found from my recent book chapter entitled 'Astrophysical Black Holes in the Physical Universe' [5].\nThe usual practice of writing a review article is to end\n\n\n
FIG. 1. Illustration of a BHXB and microquasar. X-ray emission is produced from the central hot accretion disk. A jet is normally observed in the radio band. The companion star may produce optical emission lines. The disk wind may be observed with UV/X-ray absorption lines.
\n\nby listing some outstanding issues and major unsolved problems, and then to propose some possible approaches to them. I initially did not do this in the first draft. The history of astronomy tells us that major progress is almost always made by unexpected discoveries and research results; unpredictability is an essential nature of astronomy. This article is not intended to be read by funding agencies or proposal reviewers, so I thought I did not have to do it. In astronomy, knowing what has happened, but looking and doing it differently are far more important and effective than following other people's advises. However, the editors of this book suggested me to write a brief 'outlook' in the end. I thus did it nevertheless.\nII. ACRONYMS AND TERMINOLOGY\nIn Table I, I list all acronyms used in this article; most of these are quite commonly used in this community.\nA BH binary (BHB) is a gravitationally bound binary system in which one of the objects is a stellar mass BH with mass from several to tens of solar masses ( M glyph[circledot] ); the other object, i.e. its companion, can be either a normal star, a white dwarf, or a neutron star (NS). In case a binary system consists of two BHs, it is is referred to as a binary BH system, which is not covered in this article. When the companion in a BHB is a normal star, the gas from the star may be accreted to the BH and X-rays are produced, as a consequence of the heating by converting the gravitational potential energy into the kinetic energy of the gas, and a BH X-ray binary (BHXB) is referred to as such a binary system, as shown in Figure 1. The possible existence of BHXBs was first suggested by Zel'dovich & Novikov [6, 7]. The first BHXB found is Cygnus X-1 [8], now a well-studied system among many others found subsequently in the Milky Way and nearby galaxies.\nThe terminology of microquasar has some twisting in it. Historically it was first referred to the BHXB 1E1740.7 -2942 in the Galactic center region, because a double-sided jet was detected from it, mimicking some\n
\n\n
TABLE I. List of Acronyms.
\n
\nquasars with similar radio lobes, which have much larger scales [9]. Soon after, superluminal jets are observed from a BHXB GRS 1915+105 [10], which is also referred to as a microquasar. Nevertheless both BHXBs are quite unusual compared to many others, thus microquasars were considered quite unusual. However it became clear that microquasars may be quite common among BHXBs, since the discovery of a normal BHXB GRO J1655 -40 [11], whose superluminal jets were observed [12] and showed some correlations with its X-ray\nemission [13]. Subsequently several more BHXBs have been observed with superluminal jets. At this stage microquasars began to be referred to as BHXBs with relativistic jets (with bulk motion of about or larger than 90% of the speed of light), to distinguish them from NS X-ray binaries (NSXBs) that only have mildly relativistic jets (with bulk motion of about or smaller than 50% of the speed of light) [14]. However the discovery of relativistic jets from a NSXB Circinus X-1 made the situation complicated: relativistic jets are no longer uniquely linked to BHs [15].\nNow in retrospect, a microquasar can be literally and easily understood as the micro version of a quasar; however a quasar may or may not be observed with collimated jets. A quasar has been already understood as a special galaxy centered by an actively accreting supermassive BH with a mass from millions to billions of M glyph[circledot] , and thus its total light output is dominated by the BH's accretion process, in a similar way as in BHXBs. The production or lack of relativistic jets may have similar or even the same underlying physical mechanisms in BHXBs and AGNs, though their surrounding environments may modify their observed morphologies [16]. It is therefore more natural to simply refer microquasars as BHXBs; in the rest of this article, microquasar and BHXB are used interchangeably.\nWe therefore will focus on BHXBs and thus will not discuss binary systems producing gamma-rays and sometimes radio jets, which are most likely high-mass NSXBs and in which jets or pulsars' winds interact with the wind of its high-mass companion to produce the observed gamma-rays [17-23]. Such systems show very different observational characteristics, e.g. the long (1667 days) super-orbital modulation with phase offset (about 280 days) between its X-ray and radio light curves found in LS I+61 · 303 [24].\nIII. REVIEW OF REVIEWS\nHere I attempt to review the four recent review articles [1-4] I consider most useful to readers. Additional information and updates are provided when necessary. Some overlaps exists between these review articles, as expected and inevitable in bottom-up review articles. To avoid repetitions as much as possible in this article, I thus put different emphasizes on different articles, with of course my personal tastes and perceptions.\nA. The most recent Science Collection\nThe recent collection of perspectives and reviews in the Science magazine provides excellent introductions to and concise summaries of the current state of our understanding of BH physics and astrophysics [1, 25-27]. To the subjects of this article, the most relevant article in this collection is the one by Fender and Belloni enti-\n\n\n
FIG. 2. A typical Hardness-Intensity-Diagram (HID) of spectral evolution of a BHXB, following the A → B → C → D → E → F cycle (top). Steady or transient jets are present during the A → B or B → C → D stage. No jets are observed, but hot disk winds are ubiquitous during the D → E stage. (Figure 2 in Ref.[1].)
\n\ntled 'Stellar-Mass Black Holes and Ultraluminous X-ray Sources', which is focused on the observational characteristics of BHXBs [1]. In particular, the article provides an excellent description of the general picture of spectral evolutions of BHXBs with the Hardness-IntensityDiagram (HID), which is found to be well correlated with the observed jets from BHXBs, as shown in Figure 2. Actually this cycle is also well tracked by their flux variability, represented by the measured root-mean-squares (rms) above its average flux, as shown by the RMSIntensity-Diagram (RID) [28] of the BHXB GX 339-4 in Figure 3, which also shows additional horizontal tracks at intermediate intensities. Sometimes, a full HID cycle does not go into the soft state at all (Figure 4), perhaps due to a failed outburst [29].\nThe basic scenario is as follows. During the initial stage of an X-ray outburst of a BHXB (A → B), which is triggered by a sudden increase of accretion rate onto the BH, its spectrum is normally hard and steady jets are always observed. After reaching its peak luminosity, its spectrum begins to soften in a chaotic way and transient jets are normally observed (B → C → D). After this transition, the system calms down with a soft spectrum and no jets are present (D → E). Finally the system returns to its quiescent state with a hard spectrum accompanied with the reappearance of jets (E → F). Throughout this cycle, the presence of hot accretion disk winds appears to be anti-correlated with its spectral hardness and jet ejection. This empirical pattern appears to be quite universal for all BHXBs with very few exceptions, though the underlying physics is still not well understood yet.\n\n\n
FIG. 3. The Hardness-Intensity-Diagram (HID) and RMSIntensity-Diagram (RID) of the BHXB GX 339-4. The main difference from Figure 2 is that there are additional horizontal tracks at intermediate intensities, a phenomenon known as hysteresis of state transitions, which will be discussed in Section VII. (Adapted from Figure 1 in Ref.[28].)
\n\n\n\n
FIG. 4. The Hardness-Intensity-Diagram (HID) of H1743322. The main difference from Figure 3 is that there is a complete track at low intensity from the 2008 outburst, which did not go into the soft state at all. (Adapted from Figure 7 in Ref.[29].)
\n\nNevertheless putting together the above scenario is a very significant progress in this field over the last ten years. Due to the conciseness of this article, some other important subjects on BHXBs are not fully discussed and many original references are also missing.\nOther articles in this collection [25-27] are less relevant to the subjects of this article, but are still quite interesting to read, within the context of BH astrophysics and physics. The only other type of astrophysical BHs known to exist in the physical universe are supermassive BHs in the center of almost each galaxy. Volonteri [27] concisely summarized our current understanding on how they are formed and grow over the cosmic time; merging of two BHs is a key process here. Thorne [25] focused on what happens when BHs merge together to produce gravitational waves, which might be used as a new laboratory for studying gravitational physics and a new window for\nexploring the universe. Witten [26] then explained the quantum properties of BHs, in particular the basic ideas behind Hawking radiation, which might not be important at all for astrophysical BHs. Nevertheless the understanding gained through studying the quantum mechanics of BHs actually plays very important roles in developing other theories of physics, such as that in heavy ion collisions and high-temperature superconductors, as vividly described by Witten [26].\nB. The most recent ARA&A article on BHXBs, with some updates\nThe most recent ARA&A article on BHXBs by Remillard and McClintock entitled 'X-Ray Properties of Black-Hole Binaries' [2] (referred to as RM06 hereafter) provides the most complete, comprehensive and accurate review of BHXBs, which actually covers subjects much beyond just the X-ray properties of BHXBs. The Introduction of RM06 highlights the initial theoretical and observational studies of BHXBs, followed by brief comments on several main review articles on BHXBs preceding this one and the basic properties of BHs within the context of general relativity (GR). All main properties of BHXBs known at the time are summarized in the first table and figure there. In Table II, I compile the most updated data on all BHXBs currently known, including three BHXBs outside the Galaxy and not in the Large Magellanic Cloud (LMC); the currently available spin measurements for these BHs are also included for completeness. In Table II,\nf ( M ) ≡ P orb K 3 C / 2 πG = M BH sin 3 i/ (1 + q ) 2 , (1)\nwhere P orb is the orbital period, K C is the semiamplitude of the velocity curve of the companion star, M BH is BH mass, i is the the orbital inclination angle, and q ≡ M C /M BH is the mass ratio. In Figure 5, I show the updated graphical representation of most of the BHXBs listed in Table II.\nThe X-ray properties of BHXBs are characterized by their X-ray light curves, timing and spectra. Essentially all known BHXBs were discovered initially as bright Xray sources, and the majority of them were detected as transient X-ray sources with X-ray all-sky monitors. The transient properties of some of BHXBs can be interpreted by the disk instability model (DIM) (see Ref.[49] and references therein), which assumes a constant mass transfer rate from the mass donor to the accretion disk. However the accretion rate from the disk to the compact object, i.e. a white dwarf (WD), a neutron star (NS), or a BH, is normally lower than the mass supply rate in the disk, so mass is accumulated in the disk. When the accumulated mass exceeds a certain critical value, a sudden increase of accretion rate results in a nova-like outburst. DIM is most successful in explaining an outburst with fast rise and exponential decay (FRED). However as shown in RM06, many observed X-ray light curves of BHXBs\n
\n\n
TABLE II. Twenty four confirmed BHXBs and their BH masses and spins. Except those marked with references, all other data are taken from Remillard & McClintock (2006) [2] (references therein).
\n
\na : Reported in the first paper on systematic BH spin measurements [47]; b : Reported in the most recent literature; c: Postulated to be extreme retrograde Kerr BH, due to the lack of the thermal disk component above 2 keV (GRO J1719-24 also belongs to this class) [47]; d : Postulated to be non-spinning BH, due to the observed low disk temperature; e : BH mass of 7 M glyph[circledot] assumed [47]; f : Based on the inner disk radius decrease by a factor of two from the hard state to the soft state transition [47].\nare far more complicated than just FRED and their recurrent time scales are also not compatible with DIM. Perhaps disk truncation (a subject to be discussed extensively more later) and mass transfer instability are additional ingredients [50].\nX-ray emission from a BHXB is variable at all time scales, from the free-fall or Keplerian orbital time scale of milliseconds near the BH, to the various oscillations (some are even related to GR) in the disk with time scales from milliseconds to minutes, to the viscous time scales of minutes, and to the various instabilities of different time scales. Therefore timing studies of BHXBs can probe the geometry and dynamics in BHXBs. However, lack of coherent signals, such as that observed from pulsars, makes it difficult to unambiguously identify the underlying mechanisms from the detected X-ray variabilities. Nevertheless power density spectra (PDS) and rms still allow us to make progress in understanding the general characteristics of a BHXB, in particular when combined with its spectral behaviors, as already discussed briefly above.\nThe most successful understanding of BHXBs so far is the description of their thermal X-ray spectral component by the classical Shakura-Sunyaev Disk (SSD) model\n[51]. After applying GR to SSD, the temperature distribution in the disk can be obtained [52]. Applying this model to BHXBs, one can even measure the spin of their BHs [47], by assuming that the inner accretion disk boundary is the inner-most stable circular orbit (ISCO) of the BH (a subject to be discussed more in Section IV). However a power-law (PL) component is almost ubiquitous in the spectra of BHXBs. The interplay between these two components results in various spectral states, which are also found to be well correlated with their timing properties.\nHistorically these spectral states have been named in many different ways, reflecting mostly how they were identified with the observations available at the times. In RM06, three states are defined, which show distinctively different spectral and timing behaviors, as shown in Figure 6 based on the RXTE data on the BHXB GRO J1655-40. The thermal state has its X-ray spectrum dominated by the thermal disk component and very little variability. The hard state has its X-ray spectrum dominated by the a PL component and strong variability. The steep power-law (SPL) state is almost a combination of the above two states, but the PL is steeper. I will keep using these definitions throughout this article for consis-\n\n\n
FIG. 5. Schematic diagram of the dynamically confirmed BHXBs, maintained by Dr. Orosz (http://mintaka.sdsu.edu/faculty/orosz/web/). The color scale for the 17 objects with low mass companions (i.e. stars with masses less than about 3 M glyph[circledot] ) represents the temperature of the star. However, the high mass companions in Cyg X-1, LMC X-1, LMC X-3, and M33 X-7 are considerably hotter, and are thus not well represented in color scale. (Adapted from the plot maintained by Dr. Orosz (http://mintaka.sdsu.edu/faculty/orosz/web/ and Figure 1 in Ref.[48])
\n\ntency. Please refer to Table 2 in RM06 for quantitative descriptions of these three states. When a BHXB is in quiescence, i.e. not in an outburst, its spectral shape is similar to the hard state spectrum.\nReferring back to Figure 2, RM06 found that the HID can be reasonably well understood in terms of transitions between the above three states; I summarize this in Table III. After examining the extensive RXTE data on spectral evolutions of six BHXBs, the above conclusion seems to be valid generally. Thus a coherent picture seems to emerge: jets are produced only when the PL component in the spectrum is strong and winds are present only when the PL component disappears; jets and winds appear to be mutually exclusive. This suggests that transient winds should appear in the SPL state, anti-phased with transient jets, if winds quench jets as suggested recently [53, 54].\nThe robust link between the PL component and jets provides an important clue to the jet production mechanism. The PL component is believed to be produced in an optically thin and geometrically thick corona, in\n\n\n
FIG. 6. Characteristic spectra and PDS of the BHXB GRO J1655-40 in three different states, namely, steep powerlaw (SPL), thermal and hard states. Please refer to Figure 2 in RM06 for details. (Figure 2 in Ref.[2].)
\n\n
\n\n
TABLE III. Relations between HID, spectral states, jets and wind.
\n
\nwhich hot electrons up-scatter the thermal photons from the disk. As discussed in RM06, the inner region of the disk appears to be truncated quite far away from the BH in the hard state; this should be the primary reason for the low luminosity of this state, as evidence for the existence of BHs in BHXBs. However it is still not understood how the corona is formed; I will discuss this later in Section VI.\nIn the final part, RM06 discussed the exciting possibilities of using BHXBs as probes of strong gravity, including confirming these BHXBs contain true BHs, measuring the spins of BHs, relating BH spin to the Penrose process and other phenomena, and finally carrying out tests of the Kerr Metric. They also discussed in length how to measure the spins of BHs and commented the various methods of doing so. In Section IV I will re-visit the BH spin measurements in details.\nLacking of a solid surface, a BHXB is not expected to produce and actually never observed to have coherent pulses. However quasi-periodic oscillations (QPOs) are frequently detected in their X-ray light curves; a QPO is defined as a 'bump' feature in the PDS, if Q = ν/ ∆ ν > 2,\nwhere ν and ∆ ν are the peak frequency and FWHM of the bump, respectively. The observed QPOs are further divided into low frequency (LF) (0.1-30 Hz) and high frequency (HF) ( > 30 Hz) QPOs. Typical LFQPOs can be found as the peak around several Hz in the top and bottom PDS of Figure 6. In literature, LFQPOs are further divided into several subclasses and their underlying mechanisms are far from clear at this stage, though many models have been proposed and briefly discussed in RM06. HFQPOs have been detected from several sources at 40-450 Hz. Of particular interests are their stable nature when detected, which may be linked to either the mass or spin or both of the BH in a BHXB.\nIn some cases a 3:2 frequency ratio is found. Although the 3:2 HFQPO pairs could be interpreted in some epicyclic resonance models [55, 56], there remain serious uncertainties as to whether epicyclic resonance could overcome the severe damping forces and emit Xrays with sufficient amplitude and coherence to produce the HFQPOs. A revised model applies epicyclic resonances to the magnetic coupling (MC) of a BH's accretion disk to interpret the HFQPOs [57]. This model naturally explains the association of the 3:2 HFQPO pairs with the steep power-law states and finds that the severe damping can be overcome by transferring energy and angular momentum from a spinning BH to the inner disk in the MC process.\nC. Two comprehensive and long articles on modeling accretion flows in BHXBs\nPage limitations to the above review articles did not allow in depth discussions on the detailed processes and models of accretion flows in BHXBs, which are responsible for the above described energy spectra, PDS, state transitions, jets, and wind. Here I introduce two comprehensive and long articles just on this, by Done [3] and Done, Gierli'nski & Kubota [4].\n1. A beginner's guide\nThe first is intended to readers who just start research in this field [3]. It started with the basic tools in plotting spectra and variability, and then described the basic ideas and methods used to infer the inner accretion disk radius when a BHXB is in a thermal state, in order to measure the BH spin [47]. A useful introduction is made on how to make various corrections to account for various effects, including color correction, special and general relativistic effects, starting from the original work [47]. Several commonly use fitting models, i.e. DISKBB , BHSPEC and KERRBB , in the XSPEC package are also briefly introduced. An example is given to demonstrate that the expected relation L disk ∝ T 4 in [47] agrees with the data from the BHXB GX 339-4 [58], where L disk and T in are the disk's total luminosity and temperature at the inner\ndisk boundary, respectively.\nThe hard state is then briefly touched upon, using the Advection Dominated Accretion Flow (ADAF) model [59]. ADAF naturally explains the hot corona required by the observed PL component, especially at very low accretion rate. Evaporation at low accretion rate in the inner disk region has been proposed as the mechanism producing a geometry with a radially truncated disk and a hot inner flow; the latter might be the ADAF [60]. This also means that when a significant PL component is present in the spectrum, BH spin measurement cannot be done with the inferred inner disk boundary (see, however, counter evidence discussed in Sections IV and VI). This geometry is considered a paradigm that can account for many of the observed diverse phenomena from spectral evolution to timing properties.\nA brief, yet interesting discussion is given on scaling up the above models to Active Galactic Nuclei (AGNs), which host actively accreting supermassive BHs in the centers of galaxies. Applying the insights learnt from BHXBs on their spectral evolution, changing disktemperature with accretion rate and BH mass, and diskjet connections, one might be able to understand many phenomena beyond the simple AGN unification scheme, in which the different observational appearance is all attributed to an viewing angle difference.\nThe continuum emissions of both the disk thermal and PL components are modified by absorptions and added by additional spectral features along the line of sight (LOS). Absorptions in neutral media produce various photoelectric absorption edges, but in ionized media result in both absorption edges and lines. In addition to recovering the original X-ray emission of a BHXB, modeling the absorption features is important in learning the physical properties of the absorption media, such as column density, ionization and velocity along LOS. Winds from BHXBs discussed above are always detected this way. Several XSPEC fitting models for various kinds of absorption edges and lines are also introduced here, i.e. TBABS , ZTHABS , TBVARABS , ZVPHABS , TBNEW , ABSORI and WARMABS . Figure 7 shows the calculated absorption structures with ABSORI and WARMABS ; note that ABSORI does not include absorption lines, but WARMABS does.\nX-rays interacting with the surrounding medium can ionize it and heat the free electrons up by Compton scattering; hot electrons can also loose energy by interacting with lower energy photons. A Compton temperature of the plasma is reached when the above two processes reaches an equilibrium. The temperature is determined by only the spectral shape of the continuum, so the heated plasma can escape as winds from the disk if the velocity of the ions exceed that of the escape velocity at that radius, which is usually very far away from the inner disk boundary. As the continuum luminosity approaches the Eddington limit, the radiation pressure reduces the escape velocity substantially in the inner disk region such that continuum driven winds can be launched\n\n\n
FIG. 7. Calculated absorption structures of a column density N H = 10 23 cm -2 for different ionization parameter ξ = L/nr 2 , where L is the illuminating luminosity, n is the density of the medium, and r is the distance from the radiation source. Top panel : with the ABSORI model that does not include absorption lines. Bottom panel : for ξ = 100 with both ABSORI and WARMABS, the latter is based on the XSTAR photo-ionization package thus absorption lines are taken into account properly.(Adapted from Figure 17 in Ref.[3].)
\n\nalmost everywhere in the disk of a BHXB, forming a radiation driven wind, which is also called thermal wind; alternatively winds may also be driven magnetically in the inner disk region, but this is much less understood yet. In contrast, in an AGN the peak continuum emission is in the UV band, which has a much larger opacity than Compton scattering in neutral or weakly ionized medium during both photoelectric and line absorptions. This means the effective Eddington luminosity is reduced by large factors and UV line driven wind is easily produced at high velocity. This explains the relatively lower velocity (hundreds km/s) and highly ionized winds from BHXBs, but much higher velocity (thousands km/s to 0.2 c ) and weakly ionized winds from AGNs.\nMaterial illuminated by X-rays produce both fluorescence lines and reflection features, which depends on both the continuum spectral shape and ionization state of the material, as shown in Figure 8 calculated with\n\n\n
FIG. 8. Ionized reflections from a constant density slab, calculated with the XSPEC ATABLE model, REFLIONX.MOD , which includes the self-consistent line and recombination continuum emission, for different values of ξ . The inset shows a detailed view of the iron line region. For neutral material, around 1/3 of the line photons are scattered in the cool upper layers of the disk before escaping, forming a Compton down-scattered shoulder to the line. For highly ionized reflection, the upper layers of the disk are heated to the Compton temperature so that the spectral features are broadened. (Adapted from Figure 22 in Ref.[3].)
\n\nthe XSPEC ATABLE model REFLIONX.MOD, which includes the self-consistent line and recombination continuum emission. Note that for highly ionized reflection, the Compton heated upper layers of the disk broadens the spectral features; these effects are not included in the simpler XSPEC model PEXRIV . Replacing the stationary slab by a disk around a BH, both the special and general relativistic effects will further smear (broaden) the spectral features, as shown in Figure 9 for the iron line region with different values of ξ , inner disk radius r in , and viewing angle i ; these are calculated with the REFLIONX.MOD and then convolved with KDBLUR in the XSPEC package. Other similar XSPEC fitting models are DISKLINE , LAOR , and KY . The inner disk radius can in principle be determined by modeling the observed broad iron line features, that in turn can be used to measure BH spins, as will be discussed briefly in Section IV. Figure 10 shows all components of a broad band spectrum of a BHXBs, including interstellar absorption.\nWe thus have two ways discussed so far to determine r in of BHXBs by either fitting its thermal continuum or the fluorescent iron line feature. However results are not always consistent, obtained with different methods, even using the same data by different authors. The author believes that r in increases at low accrete rate, to make room for the corona occupying this space. Nevertheless\n\n\n
FIG. 9. Relativistic smearing of the iron line region. The sharp peak shown in solid grey lines are the original features. Other colored solid lines have different r in marked in units of r g = GM/c 2 for i = 30 · (solid lines); similarly those for i = 60 · are shown in dotted lines. (Adapted from Figure 25 in Ref.[3].)
\n\n\n\n
FIG. 10. Illustration of all components of the broad band energy spectrum of a BHXBs; interstellar absorption is normally prominent at low energies. (Adapted from Figure 3b in Ref.[61].)
\n\na fairly good summary on the current conflicts and confrontations is made on this issue, which I will discuss further in Section IV.\n2. An expert's handbook\nThe second one can be considered as a handbook on accretion flows on BHXBs, and really lives up to its subtitle 'Everything you always wanted to know about accretion but were afraid to ask' [4] (DGK07 hereafter). Besides its much longer length of 66 pages, the main difference from the above review articles is that it is focused on confrontations between theories and observations and intends to depict a coherence picture of the accretion physics in BHXBs. In the following I will summarize briefly the main points and conclusions reached in DGK07; those I have reviewed above and will discuss more later will be skipped for brevity.\nThe underlying physics of DIM for triggering the outbursts in BHXBs discussed above is the hydrogenionization instability, which produces the so-called 'S'curve, as shown in Figure 11; irradiation by the inner hot disk to keep the outer disk hot is required to produce the slow flux decays, e.g. the exponential decays, frequently observed in them. The outer disk radius is obviously another key parameter, which is determined by the tidal instability in a binary system. This mechanism can explain why BHXBs with high mass companions are all persistently bright, since their outer disks are always in the upper branch due to the combination of their higher average mass transfer rate and inner disk irradiation. Similarly it also explains the differences and similarities between the light curve properties of neutron star X-ray binaries (NSXBs) and BHXBs, since a NS has a lower mass. It should be noted that the additional surface emission from the NS may also help to maintain the outer disk hot and stay in the upper stable branch [62].\nThe SSD prescription assumes that the stress is proportional to pressure. Because the gas pressure is P gas ∝ T but radiation pressure is P rad ∝ T 4 , so a small temperature increase causes a large pressure increase when P rad ≥ P gas , and thus large stress increase, which in turn heats the disk even more. The opacity cannot decrease effectively to cool it down, so the disk becomes unstable when L ≥ 0 . 06 L Edd , where L Edd is the Eddington luminosity of a BHXB. However in most BHXBs their disk emissions appear to be stable up to around 0 . 5 L Edd . One way out is to assume that the stress is proportional to √ P rad P gas ∝ T 5 / 2 , so the stress increases slower in the radiation pressure dominated regime. Beyond this the disk should become unstable, as evidenced by the sometimes 'heart-beat' bursts of the super-Eddington BHXB GRS 1915+105; other BHXBs with L max > L Edd (e.g. V404 Cyg and V4641) were not observed with such instabilities, perhaps due to the combination of insufficient observational coverage and sensitivity. However similar 'heart-beat' bursts were also observed re-\n\n\n
FIG. 11. 'S'-curve due to the Hydrogen-ionization instability. At a radius R , the SSD solution gives kT ∝ ˙ m 1 / 4 , where T is the local disk temperature and ˙ m is the local mass accretion rate. A positive slope means cooling can balance heating, so T changes slowly and the disk is stable. In the unstable middle branch, thermally emitted photons are absorbed to ionize the hydrogen atoms and are thus trapped in the disk, causing a large opacity ( σ ). If the mass supply rate ˙ m in happens in the middle range, then the disk experiences a limit-cycle instability from (1) to (4): (1) T < 10 4 K, neutral hydrogen has low σ , ˙ m in > ˙ m , so the disk is built up and T increases slowly; (2) T ∼ 10 4 K, hydrogen atoms begin to be ionized and so has high σ , T increases rapidly until hydrogen is fully ionized; (3) T > 10 5 K, ionized hydrogen has modest σ , ˙ m in < ˙ m , so the disk is eaten out and T decreases slowly; (4) T ∼ 10 5 K, protons and electrons begin to recombine, T decreases rapidly until reaching the lower stable branch again. (Adapted from Figure 1 in Ref.[4].)
\n\nIGR J17091-3624, which is likely substantially sub-Eddington unless it is located much beyond 20 kpc and/or its BH mass is quite small [63].\nSuper-Eddington accretion flow (SEAF) can become stable again, if the radiation instability is overcome by an optically thick ADAF, i.e. the slim disk model, in which the trapped photons in the flow is advected inwards, thus balancing the heating generated by viscosity. Strong radiation driven winds can be easily produced; this can happen in BHXBs and NSXBs (e.g. Z-sources). Evidence exists that truncated inner disk (TID) is quite common in SEAF and outflow even dominates over inflow in SEAF [64].\nThe observed PL component in BHXBs cannot be explained by the SSD-like models and thus requires a hot accretion flow (HAF). At low accretion luminosity (e.g. the quiescent state), the HAF may be the advection dominated accretion flow (ADAF), the convection dominated accretion flow (CDAF), or the advection dominated accretion inflow/outflow solution (ADIOS). At higher luminosity (e.g. the hard state), the original hot and optically thin disk solution (i.e. the SLE solution) is unstable, because the electron heating efficiency by the Coulomb\ncoupling between protons and electrons is too low. The luminous HAF (LHAF), however, has the advection as a heating source to electrons, so the heating efficiency increases and thus electrons can cool the flow more effectively. Outflows can be produced in ADAF/ADIOS; collimated jets can also be produced if magnetic fields are involved, so an accretion flow may even be jet dominated (i.e. JDAF).\nThe interplays between the SSD and HAF may be responsible for the observed different states in BHXBs discussed above. If the PL component is produced by Compton up-scattering, then the combination of the optical depth τ and L h / L s , where L h is the heating power in electrons and L s is the cooling power in seed photons, can describe the observed variety of spectra. For example, the hard state has L h / L s glyph[greatermuch] 1, but the thermal and SPL state have L h / L s ≤ 1. The location of r in is proposed to be closely related to L h / L s , as illustrated in Figure 12. As discussed above, the steeper PL in the thermal and SPL states is mostly non-thermal in nature, so non-thermal Comptonization is required. Detailed comparisons with data suggest thermal Comptonization also cannot be ignored even in the thermal and SPL states.\nIn Figure 12, the hot inner flow (HIF) and patchy corona are responsible for thermal and non-thermal Comptonization, respectively. When a source transits from hard to SPL state, r in decreases, HIF is reduced but the patchy corona becomes dominant. The transition from SPL to soft state is then marked by the disappearance of HIF and significant reduction of the patchy corona. The above scenario obviously depends on two fundamental assumptions: (1) slab-HIF produces the non-thermal PL component; (2) slab-HIF is mostly located between the TID and the BH. Both assumptions are examined exhaustively based on the existing observations and their spectral modeling. Slab-HIF is found to be consistent with essentially all data. An alternative to slab-HIF is that the PL is produced from the jet base and beamed away from the disk; however observations suggest the PL emission is quite isotropic, thus in conflict with this. TID is found also consistent with data when the PL component becomes important, since the observed L disk ∼ T 4 in deviates significantly from a linear relation, suggesting r in is larger when the PL component becomes important. However some studies showed that r in is unchanged in the initial hard state, if the Compton up-scattering process is treated properly in Monte-Carlo simulations to recover the lost disk photons [65]; this issue will be further discussed in Section IV.\nThe above developed HIF/TID model based on the spectral evolutions of BHXBs can also be applied to explain the majority of time variability PDS of BHXBs consistently. The TID acts as a low pass filter as it cannot response effectively to variations in the HIF, so r in controls the low frequency break of the PDS: ν LFB ∼ 0 . 2( r/ 6) -3 / 2 ( m/ 10) -1 Hz, where r = r in /r g and m = M BH /M glyph[circledot] ; this predicts that ν LFB changes from 0.03 to 0.2 Hz as observed during transitions from the\n\n\n
FIG. 12. Left panel: a selection of spectra in different states taken from the 2005 outburst of GRO J1655-40; state names in parentheses are defined in RM06. Right panel: proposed accretion flow structures in these different spectra; note the SSD is truncated away from ISCO in the bottom two states. (Adapted from Figure 9 in Ref.[4].)
\n\nhard to SPL and soft states if r decreases from 20 to 6. The HIF/TID model can also explain the observed LFQPO variations with luminosity, if the LFQPOs are some kinds of characteristic frequencies related to r in . To some extend, the shape of the PDS can also be explained by this model. However the observed variability rms ∼ flux linear correlation and log-normal distribution of fluctuations may need additional ingredient, such as the proposed propagation fluctuation model. However I point out that such correlations and distributions are also observed in the light curves of gamma-ray bursts and solar flares [66, 67], which can be produced by the generic self-organized criticality mechanism [66]. Similar rms ∼ flux correlation has also been found for blazars, in agreement with the minijets-in-a-jet statistical model [68].\nThe spectral and timing properties of weakly magnetized NSXBs are known to have many similarities and differences from BHXBs; this is particularly true for the atoll sources that have similar ranges of L/L Edd to BHXBs. The essential distinction is that a NS has a solid surface, but a BH does not. Observationally the continuum spectra of atoll sources can be modeled as composed of SSD, PL emission, and blackbody emission from the NS surface (or the boundary layer between the TID and NS surface); the former two components are quite similar to BHXBs. The spectral evolutions of these NSXBs are thus driven similarly by the combinations of τ and L h / L s . However the blackbody emission is an additional source of L s , so the PL is not as hard as that in BHXBs and r in variations are less effective in changing L h / L s ; the latter means it is more difficult to find evidence of TID from modeling only the spectral evolutions in NSXBs. On the other hand, the TID/HIF model in NSXBs can produce timing behaviors in the same way as in BHXBs discussed above, consistent with observations; additional timing behaviors, such as coherent X-ray pulsations and\nkilo-Hz QPOs observed in these NSXBs, are caused by the rapid spins of the hard surfaces of the NSs.\nIV. FURTHER DEVELOPMENTS ON BH SPIN MEASUREMENTS\nA BH predicted in GR can only possess three parameters, namely, mass, spin and electric charge, known as the so-called BH no hair theorem. Even if a BH was born with net electric charge, its electric charge can be rapidly neutralized by attracting the opposite charge around it in any astrophysical setting, because the strength of electromagnetic interaction is many orders of magnitude stronger than that of gravitational interaction. Therefore an astrophysical BH may only have two measurable properties, namely, mass and spin, making BHs the simplest macroscopic objects in the universe. Practically, only Newtonian gravity is needed in measuring the BH mass in a binary system. However, GR is needed in measuring the BH spin.\nThe mass and spin of a BH has different astrophysical meanings. Its mass can be used to address the question of ' How much matter (and energy) has plunged into the BH?'. However its spin can be used to address the question of ' How did the matter (and energy) plunge into the BH?'. This is because matter and energy plunged into a BH can carry angular momentum, which is a vector with respect to the spin axis of the BH. In order to increase the total gravitating mass-energy from M i with zero spin to M f , the added rest-mass must be [69, 70]\n∆ M = 3 M i [sin -1 ( M f / 3 M i ) -sin -1 (1 / 3)] , (2)\nand its final spin becomes\na ∗ ≡ cJ/GM 2 f = ( 2 3 ) 1 / 2 M i M f [ 4 -( 18 M 2 i M 2 f -2 ) 1 / 2 ] , (3)\nwhere J is the BH's angular momentum. Clearly we have a ∗ = 1 when M f /M i = 6 1 / 2 ; further accretion simply maintains this state [70]. Therefore the required additional rest-mass to spin a BH from zero to maximum spin is ∆ M glyph[similarequal] 1 . 85 M i = 0 . 75 M f ; this is a lower limit to the accreted mass [71]. Figure 13 shows a ∗ as a function of ∆ M . Ignoring Hawking radiation of a macroscopic BH, the only way to extract the energy of a BH and thus reducing its gravitating mass is by extracting its spin energy. Recently, evidence of BH spin energy extraction to power relativistic jets has been found, from the observed correlation between the maximum radio luminosity and its BH spin of a microquasar [72, 73]; however the average jet power is not correlated with BH spin [74]. This indicates that jets may be produced by both BlandfordPayne (BP) [75] and Blandford-Znajek (BZ) [76] mechanisms; but the BZ mechanism is more powerful and responsible for producing the peak radio luminosity. This provides another possible way to estimate a BH's spin [73], similar to a recent proposal of using the peak luminosity of the disk emission to estimate a BH's spin [77]. However there is so far no independent demonstration of validness of either of the above two new methods, which are thus not discussed further in this section.\nFigure 14 shows the mass reduction, ∆ m = ( M max -M f ) /M max ( M max and M f are the BH's maximum mass and final mass, respectively), as a function of extracting efficiency glyph[epsilon1] of an extreme Kerr BH. For a 10 M glyph[circledot] BH with glyph[epsilon1] = 1 (∆ m ∼ 0 . 3), the total extracted energy is ∆ mM f c 2 glyph[similarequal] 10 54 erg and its total gravitating mass is reduced to M max / √ 2 [71]; this energy could be sufficient to power gamma-ray bursts (GRBs). It is thus plausible that supercritical accretion onto a newly born BH may spin it up and extract its spin energy to power ultrarelativistic jets; multiple spin-up and spin-down cycles may also happen during one GRB, if the collapsing material is clumpy.\nAs shown in Figure 15, the radius of the ISCO of the BH, R ISCO , is a monotonic function of the BH spin [78], beyond which radius a test particle will plunge into the BH under any perturbation; however, in Newtonian gravity a stable circular orbit can be found at any radius. It is thus reasonable to assume that the accretion disk around a BH terminates at this radius, i.e., r in = R ISCO . Therefore, a ∗ can be inferred if one can measure R ISCO in units of its gravitational radius r g = GM/c 2 . Currently three methods have been proposed to measure the BH spin in BHXBs, and all these methods rely essentially on measuring R ISCO . In case the radiative efficiency ( η ≡ L/ ˙ Mc 2 ) can be measured, a ∗ can also be determined this way, as shown in Figure 16. Actually η is a very simple function of R ISCO , i.e, η ∼ 1 /R ISCO , as shown in Figure 17.\n\n\n
FIG. 13. BH spin a ∗ vs. accreted rest-mass ∆ M , in units the final mass M f ( a ∗ = 1) (solid line, bottom axis), and in units of the initial mass M i ( a ∗ = 0) (dashed, top axis). (Adapted slightly from Figure 3 Ref.[71].
\n\n\n\n
FIG. 14. Extracted fractional energy ∆ m = ( M max -M f ) /M max as a function of efficiency glyph[epsilon1] of extraction of a BH's rotational energy. (Figure 4 in Ref.[71].
\n\nA. The first black hole spin measurement with X-ray spectral continuum fitting\nIn 1997, I and my colleagues proposed a method of measuring the BH spin in BHXBs. It started when we tried to measure the mass of the BH in GRO J1655-40 [11] by measuring r in from its X-ray continuum spectral fitting and assuming the BH is not spinning; we found the mass of the BH is around 4 M glyph[circledot] [79]. Coincidentally just when this paper was going to press, Orosz and Bailyn [80] announced their accurate measurement of the BH mass of GRO J1655-40, which is significantly larger than what we found. A co-author of our paper, Rashid\n\n\n
FIG. 17. The solid line shows numerical curve of the radiative efficiency η as a function of R ISCO . The dashed line shows η ∼ 1 /R ISCO , which approximates the numerical curve.
\n\n\n\n
FIG. 15. The radius of the innermost stable circular orbit ( R ISCO ) of a BH as a function of the spin parameter ( a ∗ ) of the BH, i.e., the dimensionless angular momentum; a negative value of a ∗ represents the case that the angular momentum of the disk is opposite to that of the BH, i.e., the disk is in a retrograde mode. Therefore a ∗ can be measured by determining the inner accretion disk radius, if the inner boundary of the disk is the ISCO of the BH.
\n\n\n\n
FIG. 16. Radiative efficiency η as a function of BH spin parameter a ∗ .
\n\nSunyaev urged me to resolve this apparent discrepancy. I immediately realized that a spinning BH of 7 M glyph[circledot] in GRO J1655-40 would be consistent with the inferred with Xray data, and added a note in proof in the paper suggesting this possibility [79]. This turns out to be the first BH spin measurement on record. I then invited two of my close friends and collaborators, Wei Cui and Wan Chen, to join me to apply this method systematically to other BHXBs; a new main conclusion in this work was that the first microquasar in the Milky Way, GRS 1915+105, also contains a spinning BH [47].\nAt the time DISKBB was the only available fitting model in the XSPEC package for determining r in from an observed X-ray continuum spectra, if the disk contin-\nuum is described by the SSD model: L disk = 4 πσr 2 in T 2 in , where T in in DISKBB is the disk temperature at r in (i.e. the peak disk temperature in SSD) and L disk can be calculated from the disk flux (after the correction to absorption), the distance to the source and disk inclination (an issue to be discussed later). To determine the physical inner disk radius from the DISKBB parameter r in , several effects must be considered: (1) electron scattering in the disk modifies the observed X-ray spectrum; (2) the temperature distribution in the disk is not accurately described by the Newtonian gravity as assumed in SSD; (3) the observed temperature distribution is different from the locally emitted one; (4) the observed flux is different from the locally emitted one. The latter three are all due to GR effects [52]. For each of the above effects, we introduced a correction factor, using the best available knowledge at the time. Since then, several improvements have been made to correct for these effects and this continuum fitting (CF) method is now quite mature in making accurate BH spin measurements, given sufficiently high quality X-ray continuum spectral measurements and accurate system parameters of the observed BHXB.\nB. Further developments and applications of the continuum fitting method\nThis CF method of measuring BH spin has since been applied widely to essentially every BHXB with a well measured X-ray continuum spectrum showing a prominent thermal accretion disk component. In particular, this method has been improved and incorporated into the widely used X-ray spectral fitting package XSPEC, e.g., KERRBB [81], BHSPEC [82], and KERRBB2 [35, 37]. Both the KERRBB and BHSPEC are relativistic models, but they have their own drawbacks and advantages (See Ref.[35] for a detailed comparison). KERRBB includes all the relativistic effects, but it requires to fix the\nspectral hardening factor. In contrast, BHSPEC could calculate the spectral hardening factor on its own; however, it does not include the returning radiation effect, which turns out to be an important factor in BH spin determination in BHXBs. KERRBB2 combines both models by generating the spectral hardening factor table from BHSPEC and using the table as the input for KERRBB . The research group led by Ramesh Narayan of Harvard University, Jeffrey McClintock at Smithsonian Astronomical Observatory (SAO), and Ronald Remillard of Massachusetts Institute of Technology (MIT) [83] has since applied this method and contributed to most of the BH spin measurements available in the community, as shown in Table II.\nThe CF method relies on two fundamental assumptions: (1) The measured r in is uniquely related to R ISCO of the BH. (2) There is no or negligible X-ray radiation from the plunging matter onto the BH beyond R ISCO . The latter has been studied with numerical simulations that include the full physics of the magnetized flow, which predict that a small fraction of the disks total luminosity emanates from the plunging region [84]. However, in the context of BH spin estimation, it has been found that the neglected inner light in the CF method only has a modest effect, i.e., this bias is less than typical observational systematic errors [85, 86].\nThe first assumption above requires that the measured r in remains stable as a BHXB changes its spectral state and luminosity. However it was noticed that r in measured is usually much smaller, sometimes even smaller than R ISCO of a extreme Kerr BH in a prograde orbit, when the X-ray spectrum contains a significant hard PL component, which is believed to be produced by inverse Compton scattering of the thermal disk photons in a hot corona. We realized that the inferred smaller r in could be due to the lost thermal disk photons in the scattering process. We then investigated this problem and confirmed that the inferred r in can be made consistent with r in inferred from the thermal disk component dominated spectrum, if the scattered photons are recovered properly by doing detailed radiative transfer in the corona [65]; the same conclusion was also reached by the Harvard/SAO/MIT group independently without knowing our much earlier results [87]. Therefore the method of BH spin measurement by X-ray continuum fitting can also be applied to some SPL state with strong PL component. The stable nature of the measured r in is proven with the textbook case of LMC X-3, when its X-ray luminosity varied over more than one order of magnitude observed in nearly two decades with many different X-ray instruments, as shown in Figure 18 [88].\nHowever, when ordered by the observed disk luminosity l D = L disk /L Edd , the measured r in shows a clear increasing trend when l D > 0 . 3, as shown in Figure 19 [88]. The similar trend has also been found in another BHXB GRS 1915+105 [35] and two NSXBs [64, 89]. It was found that r = r in /r g indeed increases physically when l D > 0 . 3, by comparing the evolution of r as a function\n\n\n
FIG. 18. The measured r = r in /r g and disk luminosity l D = L disk /L Edd (assuming M BH = 10 M glyph[circledot] ) as functions of observation time, for the BHXB LMC X-3. (Slightly adapted from Figure 1 in Ref.[88])
\n\nof l D over a large range for several BHXBs and NSXBs. Using the blackbody surface emissions of the NSs in these NSXBs is critical in evaluating any possible disk thickening due to high luminosity that would block at least part of the NS surface emission, as well as determining the actual NS mass accretion rate, which turns out to be much less than the disk mass accretion rate; this suggests that the increased radiation pressure is responsible for the increase of r and significant outflow when l D > 0 . 3 [64]. The same trend is much more pronounced in the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2, as shown in Figure 20 together with the data from other BHXBs and NSXBs [90]. However the exact value of r obtained this way should be taken with caution, since the non-negligible energy advection at high accretion rate can modify the disk structure in non-trivial ways, thus making the SSD prescription inaccurate in this case [91].\nFigure 20 also shows that as l D decreases, r again starts to increases. However r increases at higher l D and with a different slope for a NSXB than for a BHXB, which can be naturally explained as due to the 'propeller' effect of the interaction between the NS's magnetosphere and its accretion disk [92] and the 'no-hair' of the BH [90]. Figure 21 shows the radiative efficiencies of various systems; BHXBs may have either higher or lower efficiencies than NSXBs, because a BH has neither solid surface nor magnetic field [5].\nIt is interesting to compare the BH spin results in our first paper [47] and the most recent literature for the same BHXBs as listed in Table II: (1) For GRS 1915+105, A0620-003 and LMC X-3, both results are fully consistent; (2) For GRO J1655-40, the original result points to an extreme Kerr BH ( a ∗ ∼ 0 . 93) [47], somewhat different from the most recent result of mildly spinning BH ( a ∗ = 0 . 65 -0 . 75) [32]. However, the original result was obtained using the BH mass of 7 M glyph[circledot] , about 10% larger\n\n\n
FIG. 19. The measured r = r in /r g as a function of disk luminosity l D = L disk /L Edd (assuming M BH = 10 M glyph[circledot] ), for the BHXB LMC X-3. (Slightly adapted from Figure 2 in Ref.[88])
\n\n\n\n
FIG. 20. The measured r = r in /r g as a function of disk luminosity l D = L disk /L Edd , for several NSXBs and BHXBs, and the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2. All data points are taken from Ref.[64], except for that on XTE J1118+480 [93] and NGC1313 X-2 [90].
\n\nthan the currently best estimate that was used to obtain an updated BH spin in the most recent literature [32]. From Figure 15, it can be seen that a ∗ would be decreased from 0.93 to 0.87, if the BH mass is decreased by about 10%; actually a ∗ = 0 . 85 is also allowed in the new estimate [32]; (3) For Cygnus X-1, our original conclusion that a ∗ = 0 . 75 in the high/soft state and a ∗ = -0 . 75 in the low/hard state was based on the assumption that r in decreased by a factor of two when the source made a transition from its normal low/hard state to the unusual high/soft state [94]. However the more realistic constraint is that r in changed by more than a factor of 1.8-3.2 during the state transition [94], which implies that a ∗ switched from | a ∗ | > 0 . 85, fully compatible with the\n\n\n
FIG. 21. The diagonal line shows a 1 /r scaling, calibrated to take the value of 0.057 when r = 6. The thick black line is for a BH accreting systems. The range of r = 1 -9 corresponds to the ISCO of a BH with different a ∗ ; the radiative efficiency ranges between a few to several tens of percents, far exceeding the p-p fusion radiative efficiency taking place in the Sun. The case for r > 9 corresponds to TID, whose radiative efficiency can be extremely low, because energy is lost into the event horizon of the BH. The thin solid black horizontal line is for the 10% efficiency when matter hits the surface of a neutron star where all gravitational energy is released as radiation. The thin solid black diagonal line above the point marked for 'Kerr BH' is for a speculated 'naked' compact object [5], whose surface radius is extremely small and thus the radiative efficiency can be extremely high. (Figure 10.7 in Ref.[5]
\n\nlatest result of a ∗ > 0 . 95 in the high/soft state [37].\nThe above would suggest that a very low temperature disk component exists in its hard state spectrum, if Cygnus X-1 indeed harbors a Kerr BH and the accretion disk switches from a retrograde mode to a prograde mode when it makes the hard-to-soft state transition. For a supermassive BH in the center of a galaxy, it is well understood that its mass is mostly gained through accretion in its AGN phase [95]. Therefore random accretion (between prograde and retrograde modes) tends to make its final BH spin close to zero, regardless its initial BH spin. For Cygnus X-1, its current BH mass of around 10-20 M glyph[circledot] cannot be gained through post-formation accretion, since its observed average mass accretion rate ˙ M ∼ 2 × 10 -9 M glyph[circledot] /yr and the age of its companion is much less that 10 8 yr; actually the age of the companion is estimated to range from 4.6 and 7.8 million years [96]. This means that its post-formation BH mass growth is much less than a fraction of its current mass and thus its current BH spin must be quite close to that at birth. Even if accreting at Eddington rate, to grow its BH spin from 0 and the final mass to be the current observed value, the timescale is around 3 . 1 × 10 7 yr and the accreted mass is roughly 7.3 M glyph[circledot] [37]. If its accretions al-\ntes between prograde and retrograde modes, then its current BH spin should be even closer to its initial spin than that in low mass BHXBs that only stay in one accretion mode due to roche-lobe overflows. Therefore the current high BH spin in Cyg X-1 must be natal; this conclusion is also true for the other highly spinning BHs in other BHXBs [48].\nIn summary, BH spin in BHXBs can now be measured reliably with the CF method, when the luminosity of a BHXB is between ∼ 0 . 02 and ∼ 0 . 3 in Eddington unit and their system parameters are well-known a prior . The observed thermal disk spectrum can be modeled directly to obtain the BH spin with the available KERRBB2 model in XSPEC when the X-ray spectrum in dominated by this component, i.e., the source is in the thermal state. When a significant power-law component is present, the inverse Compton scattering process must be taken into account to recover the disk photons scattered into this PL component, with for example the SIMPL/SIMPLR model [37, 97] in XSPEC. Recently another way to measure BH spin using the outburst properties of BHXBs has been proposed [77], which is mentioned briefly in Section VIII; however the effectiveness of this method needs to be tested.\nC. Uncertainties of the continuum fitting method\nIn spite of the tremendous progress made so far on BH spin measurements in BHXBs, most of these BH spin measurements suffer from considerable uncertainties. Actually the major source of these uncertainties comes primarily from the uncertainties in their BH masses, accretion disk inclination angles, and distances. The accurate BH mass measurement is required because R ISCO must be in units of r g = GM/c 2 in Figure 15. The disk inclination angle and distance are also required because the total luminosity of the disk emission L disk is needed, in order to estimate the absolute value of a ∗ .\nSo far, all BH masses in X-ray binaries (XRBs) have been estimated using the Kepler's 3rd law of stellar motion, expressed in the so-called the mass function given in Equation (1). Since the only direct observables are P orb and K C , both M C and i have to be determined indirectly in order to obtain the BH mass estimate reliably. The companion's mass M C can be determined relatively reliably by the observed spectral type of the companion star and i can be estimated by modeling the observed ellipsoidal modulation of the companion's optical or infrared light curve. The observed ellipsoidal modulation is a consequence of exposing different parts of the pearshaped companion star to the observer at different orbital phases (see Figs. 1 and 5); the pear-shape is caused by the tidal force of the compact star, which also heats the side of the companion star facing it. For details of BH mass estimates using this method, please refer to [2].\nHowever model dependence and other uncertainties (such as accretion disk contamination) cannot be circum-\n\n\n
FIG. 22. Phased VIH ellipsoidal light curves corresponding to the three states, namely 'passive', 'loop' and 'active' states; the data in each state and each wave band are plotted twice for clarity. Typical differential photometric error bars are shown in the upper right corner of the passive-state light curve for each band. To produce purer ellipsoidal light curves, variability on timescales greater than 10 days has been removed from loop- and active-state data. (Figure 2 in Ref.[98].)
\n\nvented completely and thus systematic error may exist in determining their system parameters. For example, three optical states, namely 'passive', 'loop' and 'active' states, have been identified in the normally called 'quiescent' state of A0620-003 when its X-ray luminosity is very low; only during the passive state its optical light curve modulation is purely ellipsoidal, i.e., accretion disk contamination is completely negligible, as shown in Figure 22 [98]. This means that considerable systematic errors in determining its inclination angle may occur unless only the 'passive' state data are used. Unfortunately previous observations of BHXBs used to determine their inclinations did not always occur during the passive state, thus systematic errors may be common in previous results [99]. Even for GRO J1655-40, of which all previous observations were made during its passive state [99], its BH mass measured with different observations, or the same data analyzed by different groups are not exactly the same, and even not completely consistent between them, as shown in Figure 23, which show a scatter of about 20-30% to the estimated BH mass, much larger than its statistical error of a few percents.\nIt should be pointed out that the inclination i in Equation (1) refers to that of the orbital plane of the binary system. Because the accreting material initially carries the angular momentum inherited from the companion star, the formed disk should be co-planar with the orbital plane of the binary system. However, the BH spin in a BHXB cannot be changed significantly by accretion [71] (and see discussion above on Cygnus X-1). Therefore its spin axis may not coincide with the normal direc-\n\n\n
FIG. 23. The BH mass and system inclination of GRO J165540 reported at different times, ordered by their publication dates [80, 100-105]. Note that the third and fourth reports [101, 102] adopted the inclination in the second report [100], and the last (seventh) report [105] adopted the inclination in the fifth report [103].
\n\ntion of the orbital plane of the binary system. In this case the Bardeen-Petterson effect [106], due to frame-dragging by the spinning BH, can rapidly align the normal axis of the inner disk region with the spin axis of the BH, making the inner disk and binary system mis-aligned. Circumstantial evidence already exists for mis-alignment between the two axes, because the orbital inclination of GRO J1655-40 shown in the upper panel of Figure 23 is significantly different from ∼ 80 · inferred from its relativistic jets [107], if the jets are powered by extracting the spin energy of the BH via the Blandford-Znajek mechanism [76]. Nevertheless, the orbital inclination is normally used in place of the disk inclination, which is needed in BH spin measurement using the CF method, in order to infer the total disk luminosity and calculate GR correction factors. However, the inner disk inclination is currently not available essentially for any of the known BHXBs. For example, a mis-alignment of 10 · from i = 70 · can cause nearly 50% error to the total disk luminosity, which will translate into nearly 30% error in r in . A Schwarzschild BH may be estimated to have a ∗ falling any where between [ -0 . 5 , 0 . 5], if r in is uncertain within about 30%, according to Fig.(15).\nAccurate determinations of distances of astrophysical objects in the Milky Way are difficult, e.g., for BHXBs that are not standard candles. Normally some absorption features in their spectra, in conjunction with their posi-\ntions in the galactic coordinates, are used to infer their distances. For example, the distance of GRO J1655-40 is commonly taken as 3 . 2 ± 0 . 2 kpc, based primarily on observed absorption lines and somewhat on the dynamics of the observed jets. Critical examinations of all available data related to its distance, however, are in favor of a much closer distance of less than 2 kpc and more likely just 1 kpc [108, 109]. Similar conclusion is also reached to the distance of A0620-003, revising its distance from the commonly accepted 1 . 2 ± 0 . 4 kpc to ∼ 0 . 4 kpc, making its possibly the closest BHXB known so far [109]; however, a distance of 1 . 06 ± 0 . 12 pc was preferred in a more recent study [30]. Similarly the currently adopted distances of many other BHXBs may also have considerable systematic errors. If true, this would change significantly the current estimates on their masses and spins.\nTherefore future improvements of the continuum fitting method depend upon the improved measurements on their BH masses, accretion disk inclination, and distances.\nD. Future improvements of the continuum fitting method\nThe mass ratio q can be determined directly according to the law of momentum conservation, i.e.,\nM C /M BH = K BH /K C , (4)\nif the semi-amplitude of the velocity curve of the BH K BH can be observed directly, as illustrated in Figure 1. The orbital inclination i can be then calculated using Equation (1), avoiding any systematics related to the ellipsoidal light curve modeling.\nSince a BH is not directly observable, we can only hope to observe any emission or absorption line feature comoving with it. The accretion disk certainly moves with the accreting BH. However any line feature of the inner accretion disk suffers from the broadening of disk's orbital motion and distortions by relativistic effects around the BH, thus making it practically impossible, or difficult at least, for detecting the binary orbital motion of the BH. Orbital motion of double-peaked disk emission lines were observed for NSXB Sco X-1 [110], the BHXBs A0620-003 [111, 112]) and GRS 1124-68 [112]. Unfortunately a significant phase offset of velocity modulation was found from that expected based on the observed orbital motion of the companion, though the velocity semi-amplitude is consistent with the expected mass ratio [112]. Soria et al. (1998) observed the orbital motion of the double-peaked disk emission line He II λ 4686 from GRO J1655-40, and found its velocity modulation phase and semi-amplitude in agreement with the kinematic and dynamical parameters of the system [113]. However one major problem in accurately measuring the orbital motion of the primary from the observed double-peaked emission lines is how to determine reliably the line cen-\n\n\n
FIG. 24. Velocity curve of the 39 observed X-ray absorption lines with Chandra from GRO J1655-40, after subtracting the line of sight intrinsic velocity of each line at each orbital phase. The upper panel marks the velocity of each line with its 1σ error bar, slightly shifted horizontally for visual clarity. The bottom panel shows the weighted average velocity of all lines in the upper panel at each phase; the solid curve is the fitted velocity curve with its orbital period and phase fixed at the values observed previously. (Slightly adapted from Figures 2 in Ref.[114])
\n\ner, because the lines are typically asymmetric and also variable.\nWe have recently proposed to observe the Doppler shifts of the absorption lines of the accretion disk winds co-rotating with the BH around its companion star [114], since in many XRBs accretion disk winds are ubiquitous and appear to be rather stable when observed (e.g., in Ref.[115]). We verified this method using Chandra and HST high resolution spectroscopic observations of GRO J1655-40 (shown in Figure 24) and LMC X-3. Unfortunately the currently available data only covered small portions of their orbital phases and thus do not allow better constraints to their system parameters. Future more observations of these two sources and other sources with detectable absorption lines from their accretion disk winds will allow reliable and precise measurements of the BH masses and orbital inclination angles in accreting BHXBs.\nSince the accretion disk has very high density and is ionized near the BH, where the majority of the observed disk emission is produced due to the very deep gravitational potential near the BH, scattering of the primary disk emission is inevitable. The scattered light is polarized and its polarization fraction and position angle depend on the viewing direction (inclination ), scattering optical depth and the radius where the scattering occurs [116-119]. Ignoring many details, it can be shown that the polarization fraction, P ( i ), of the observed disk photons (initial disk emission plus the scattered emission) is given by\n1 P ( i ) = 1 + A cos i 1 -cos 2 i , (5)\nwhere A is a constant depending upon the scattering op-\n\n\n
FIG. 25. Polarization fraction of observed accretion disk emission as a function of its inclination.
\n\ntical depth; detailed calculations made by Chandrasekhar [120] gives P (75 · ) = 0 . 04. Note that here the disk photons are from the Raleigh-Jeans part of the multicolor blackbody spectrum with a characteristic shape of f ( ν ) ∝ ν 1 / 3 , i.e., no GR effect is included. We can therefore find the disk inclination angle by measuring the polarization fraction of this part of the disk emission, as shown in Figure 25. At energies above the Raleigh-Jeans part of the multi-color blackbody spectrum, the polarization is strongly effected by both the inclination and BH spin, as shown in Figure 26. The continuum spectra are clearly degenerated for the different combinations of inclination and BH spin, but the polarization fraction and angle as functions of energy can clearly distinguish between them [118]. Therefore X-ray spectra-polarimetry observations of BHXBs will certainly make important progresses in measuring BH spin.\nBesides using polarization measurements to obtain inner disk inclination (and BH spin), the broad iron Kalpha measurements can also be used to do so [121-123], because iron K-alpha emission is believed to come from the fluorescent emission of the disk as a result of illumination by a hard X-ray source above the disk. Naturally this emission is sensitive to the disk inclination (and BH spin). However, compared to the polarization measurement method, this method is less straight forward and may suffer from systematic uncertainties in deriving the disk inclination (and BH spin), because complicated modeling of the hard X-ray component and line emissivity from the disk is required. Therefore it is essential to use all methods discussed above to measure both inclination angles (orbital plane and inner disk) and BH spin. Studying the relationship between the results obtained with different methods is also important in its own rights, in order to understand accretion disk physics, its interaction with the BH in its center, production of relativistic jets, the origin of the BH spin, and ultimately the formation mechanisms of BHs and BHXB systems.\nThe recent dispute on the distances of GRO J1655-40\n\n\n
FIG. 26. Disk continuum flux (a), polarization fraction (b) and polarization angle (c) as functions of photon energy for different model parameters. i , inclination angle of the inner disk in degrees; ˙ m , mass accretion rate in units of Eddington rate by assuming a 10% conversion efficiency from rest mass to radiation in all cases. Other parameters: mass of the BH is 10 M glyph[circledot] ; distance to the BH is 10 kpc; spectral hardening factor is 1 . 6. (Adapted from Figures 4-6 and Table 1 in Ref.[118])
\n\nand A0620-003 exemplifies the difficulty of determining the distances of BHXBs using mostly absorption lines [108, 109]. We have recently suggested a method of using the delay time between the X-ray fluxes of an XRB and its X-ray scattering halo by interstellar dust to infer its distance [124, 125]. However this method may suffer from our incomplete knowledge of the distribution of interstellar medium. Ideally precise astrometry can determine their distances model-independently, by measuring their trigonometric parallaxes. Recently the distance to Cygnus X-1 was determined reliably and accurately this way (1 . 86 +0 . 12 -0 . 11 kpc; [126]); which is key to the consequent measurement of its BH mass and spin [36, 37]. Currently it remains challenging to measure the trigonometric parallaxes of objects at distances beyond several kpc where most BHXBs are located. Future high precision astrometry missions are expected to improve the distance estimates to these BHXBs significantly.\nTherefore we expect that the improved measurements discussed above on their BH masses, inner disk inclination, and distances will allow future improvements of BH spin measurement with the CF method.\nE. Possible application of the continuum fitting method to AGNs\nMore recently, the CF method is also applied to constrain the BH spin in an active galactic nucleus (AGN), which is powered by matter accretion onto the central supermassive BH in its center [127]. However the BH spin inferred this way for an supermassive BH is quite uncertain, because: (1) The peak temperature of the accretion disk inversely increases with the mass of a BH. Therefore, for a supermassive, its temperature would be in the ultraviolet energy range which will be strongly absorbed and hard to observe; (2) The system parameters of a supermassive BH (e.g., the mass of the BH and the inclination angle of the accretion disk) have larger uncertainties; (3) The uncertain mechanism for some components (e.g., the soft X-ray excess) in AGN spectra also increases the difficulty; (4) Some emission and absorption lines may distort the continuum spectrum substantially; and (5) In some cases the contribution of its host galaxy to the observed total continuum spectrum cannot be removed satisfactorily.\nThe polarized continuum of an AGN should be a pure accretion disk continuum, at least in the optical to near infrared band [128]. One possible way to measure the BH spin in an AGN with the CF method is to combine the observed polarized optical to near infrared continuum spectrum with the observed total UV continuum spectrum to get a broad band continuum spectrum of an AGN [129]. In principle the broadened Balmer edge features and the total UV spectrum can be used to constrain the disk inclination angle and fraction of host galaxy contamination, respectively [129]. However the quality of the currently available data is still insufficient to allow\naccurate determination of BH spin in AGNs with this method.\nNevertheless the principal method of measuring BH spin in AGNs should be using the reflected broad iron line and continuum components [121-123]. The main reason is that i can also be determined simultaneously and r in obtained this way is already in units of r g , thus avoiding naturally the uncertainties caused by the BH mass and inclination in the CF method. However cross calibration can be done between the CF and reflection fitting methods if both can be applied to the same supermassive BH.\nV. FURTHER DEVELOPMENTS ON HOT ACCRETION FLOWS\nIn this section I briefly summarize some further developments on hot accretion flows, which are believed to be responsible for the PL component of the spectra in BHXBs. The multi-waveband spectra of the hard state of the BHXB XTE J1118+480 was modeled with the TID with hot accretion flow (ADAF) geometry (panel (a) in Figure 27) and TID (with ADAF) plus jet model (panel (b) in Figure 27), with ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D [93]. The steep UV spectrum provides clear evidence for a large truncation radius for the SSD ( r = 600), and the radio to infrared spectrum dominates the jet emission, which also contribute to the hard PL component [93]. Such low accretion rate gives very low SSD luminosity l D ∼ 5 × 10 -4 (since the disk radiation efficiency η ∼ 1 /r ; see Figure 17), which is far below the turn-over luminosity of l D ∼ 10 -2 shown in Figure 20. Therefore the inferred truncation radius of XTE J1118+480 agrees with the extrapolation of data points of XTE J1817-330 down to very low disk luminosity. Large truncation radii are also reported from several other sources in the hard state (e.g., in Ref.[130]). However, the exact values of these truncation radii may have large uncertainties, since no direct detection of the inner disk peak emission was available, unlike the strong case of XTE J1118+480 [93]. For this reason I did not include these reported values in Figure 20.\nPanel (b) in Figure 27 also shows that the radiation from both the hot accretion flow and the jet contribute to the X-ray emission. However the former is roughly proportional to ˙ m 2 , whereas the latter to ˙ m . Thus with the decrease of ˙ m , the contribution from the jet becomes more and more important, thus the X-ray radiation will be dominated by the jet (when l glyph[lessorsimilar] 10 -5 -10 -6 ), as shown in Figure 28 [93]. The observational data of very low-luminosity AGNs clearly show a correlation between radio and X-ray with a correlation index of ∼ 1 . 2 [131], in excellent agreement with the prediction shown in Figure 28 [93].\nIt is well known that the highest luminosity an ADAF can produce is only about 3% L Edd . However the observed highest luminosity a hard state can reach can\n\n\n
FIG. 27. Spectral modeling results for XTE J1118+480. (a) Outer cool disk plus inner hot accretion flow model. (b) An additional jet component is included. ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D . (Adapted from Figures 1-2 in Ref.[93])
\n\nbe 10% L Edd or even higher, which can be described by LHAF shown in Figure 29, where several previously known solutions of accretion flows are all unified in a single scheme [133]. The hard state spectrum of XTE J1550-564 with L ∼ 6% L Edd is explained by LHAF very well, including the X-ray spectral slope and the value of the cutoff energy [134].\nVI. FURTHER DEVELOPMENTS ON CORONA FORMATION\nWhile the formation of the SSD in a BHXB is reasonably well understood, the formation of the corona, which is the hot accretion flow discussed in Section V, remains less understood. Over the last more than 10 years, Liu and her collaborators have developed a model to explain the formation and evolution of the corona in a BHXB or AGN, in which the mass accretion rate ˙ m in units of the Eddington ratio drives the variations of the complex accretion flows by the interaction between the cold SSD and the hot corona [60, 135-139]. Specifically, the coupling between the hot corona and the cold disk leads to mass exchange between them. The gas in the thin\n\n\n
FIG. 28. Radio (8.6 GHz)CX-ray (2C11 keV) correlation for BHXBs. The observed correlation is shown by the segment AB. Segments BCD show the predicted correlation at lower luminosities, which approaches that of a pure-jet model, as shown by the segment DE. Note that below the point C ( ∼ 10 -6 l D ), the X-ray emission is dominated by the jet and the correlation steepens. (Figures 1 in Ref.[132])
\n\n\n\n
FIG. 29. The thermal equilibrium curve of various accretion solutions. The accretion rate is in units of ˙ M Edd ≡ 10 L Edd /c 2 and the units of Σ is g cm -2 . The parameters are M BH = 10 M glyph[circledot] , α = 0 . 1, and r = 10. (Figures 1 in Ref.[133])
\n\ndisk is heated up and evaporates into the corona as a consequence of thermal conduction from the hot corona, or the corona gas condenses into the disk as a result of overcooling by, for example, external inverse Compton scattering. If ˙ m is low, evaporation occurs and can completely remove the thin disk, leaving only the hot corona in the inner region and a truncated thin disk in outer region; this provides a mechanism for ADAF at low ˙ m . If ˙ m is high, the gas in the corona partially condenses to the disk due to strong Compton cooling, resulting in disk dominant accretion. The model naturally explains the different structures of accretion flow in different spectral states as shown in Figure 30 [60, 135, 137-144]. The hysteresis observed in spectral state transitions can also be explained by different irradiations from different evolution history under the same scenario [145-147].\nFigure 30 is significantly different from the illustration\n\n\n
FIG. 30. A schematic description of the accretion flow structures in different spectral states as consequences of diskcorona interaction, which is primarily driven by mass accretion rate dotm . SPL: steep power-law state; VHS: very high state; soft: soft state; H/S: high/soft state; hard: hard state; L/H: low/hard state; Q: quiescent state. (Adapted from a similar figure provided by Prof. Bifang Liu.)
\n\nof accretion flow structures in different spectral state in Figure 12 on three aspects: (1) At intermediate ˙ m , the SSD is broken by ADAF into two parts, an outer disk and an inner disk; (2) The inner disk boundary here is always located very close to the BH, except in the very low ˙ m hard or quiescent state, which is very different from the TID scenario depicted in Figure 12; (3) The corona here covers essentially the whole accretion disk, especially the inner disk region, whereas in Figure 12 the corona is mostly located inward from the inner disk boundary. The observed soft X-ray component in the low/hard state can be explained by the existence of the inner disk [139, 140, 142, 144]. As I have discussed above, the inner disk boundary radius inferred with the CF method in the presence of a strong PL component is actually consistent with that in the soft state when the PL component is weak, after taking into account the Compton scattering in the corona [65, 87]. Actually the essential assumption behind the broad iron line/reflection fitting method of determining BH spin is that the inner disk boundary is at the ISCO when the PL component is strong. All these tend to support the existence of the inner disk at intermediate ˙ m . However it remains to be demonstrated that if the whole SSD is indeed broken into the two parts at intermediate ˙ m , as illustrated in Figure 30.\nRecently it has been suggested that the ADAF and\n\n\n
FIG. 31. Solid lines: accretion disk models at different accretion rates. Clumpy structures in either ADAF or corona (shown as insets) may be developed at intermediate accretion rates, corresponding to different spectral states. (Adapted from Figures 1 and 2 in Ref.[148])
\n\ncorona shown in Figure 30 may have clumpy structures, as shown in Figure 31 [148]. The 'clumpy' model (Figure 31) has mainly two different consequences from the 'uniform' model (Figure 30): (1) The inner disk is transient in the 'clumpy' model; (2) The 'clumpy' model can explain the variabilities observed in X-ray binaries (such as the state transitions discussed in Section VII) and radio-loud AGNs (such as BL Lac objects).\nDetermining the structure of the corona in an BHXB observationally remains difficult; the fundamental issue on whether the corona covers mostly the accretion disk or the central compact object still remains unclear so far. Since the observed similar spectral and state transitions between some NS and BH XRBs are quite similar [149], it is reasonable to assume that they have similar coronae. Recently we have used type I X-ray bursts from low-mass NSXBs to show that X-ray bursts experience negligible Comptonization and that the corona cools rapidly during the rising phase of X-ray bursts and is then heated up rapidly during the rising phase of X-ray bursts, as shown in the upper panel of Figure 32 for IGR J1747721 [150]. These results suggest that the corona cannot cover the central compact object completely (lower panel of Figure 32) and that the destruction and formation time scales of the corona are as short as seconds; such short time scales are quite difficult to understand in the above discussed evaporation model, in which the time scales are related to the viscous time scales of the accretion disk. However, this short time scale is consistent with a corona produced by magnetic reconnections in the accretion disk, in a similar way to the solar corona heating [151]; this conclusion was based on the inferred accretion flow structure of a BHXB shown in panel (A) of Figure 33, in comparison with the atmospheric structure of the Sun shown in panel (B) of Figure 33.\nIt is worth noticing that the purported coronae in both XRBs and AGNs have strikingly similar properties, e.g., they all have electron temperatures of the order of hun-\n\n\n
FIG. 32. Upper panel: Anti-correlation between the observed type I X-ray bursts from surface of the NS in IGR J1747-721 (Adapted from Figures 3 and 2 in Ref.[150]).
\n\n\n\n
FIG. 33. Schematic diagrams of the solar atmosphere (A) and accretion disk structure (B). (Figure 2 in Ref.[151]).
\n\ndreds keV, in spite that the temperatures, inner disk radii and variability time scales of their cold accretion disks all scale with their BH masses (and accretion rates) as predicted in the SSD model. This means that their coronae are scale independent. It is perhaps not coincidental that the electrons' velocities in a corona are approximately the same as the Keplerian orbital velocities of the inner disk, which are also roughly the same as the launching velocities of jets. Of course these velocities are also the varialized velocities of the central BHs. It is plausible that turbulent small scale magnetic fields lifts\n\n\n
FIG. 34. The clumps in the clumpy corona/ADAF of a BHXB are channeled into the rotating and wound-up magnetic field lines and then ejected from the system. The launched jets should be episodic and have knotted structures.
\n\nthe plasma in the 'warm layer' shown in Figure 33 to form the corona, which is then launched into the jets by the rotating large scale magnetic fields through either the BP or BZ mechanism; during these processes the magnetic fields are mostly responsible for changing the directions of motion of plasmas by the Lorentz force. If the corona is clumped as discussed in Section VI [148], then the plasma channeled into the jets should be clumpy and thus the ejected jets should be episodic and have knotted structures, in agreement with observations; this scenario is illustrated in Figure 34. Alternatively episodic ejections may also occur in the disk, similar to the coronal mass ejection on the Sun [152].\nInterestingly, the soft X-ray excess (SXE) frequently observed in AGNs can be interpreted as the warm layer found in XRBs, since the inferred plasmas parameters, with electron temperature of 0.1 to 0.3 keV and Compton scattering optical depth of around 10, are similar to that of XRBs and universal among AGNs with different BHmasses, but seem to be related only with ˙ m [153-155]. Therefore their warm layers are also scale independent. Indeed, magnetic flares, perhaps caused by magnetic reconnections in the cold disk, can produce the warm layers in both XRBs and AGNs [156, 157].\nVII. FURTHER DEVELOPMENTS ON STATE TRANSITIONS\nTo further understand the mechanism of the spectral state transitions, it is important to study those spectral state transitions in individual BHXBs during different outbursts. The advantages are obvious - uncertainties in our estimates of black hole mass, the binary properties such as the orbital period, or the source distance will not play any role in producing the observed diverse transi-\ntion properties in individual BHXBs, which can only be driven by the accretion process under different initial conditions. The sources which firstly allowed such a study were the BH transients GX 339-4 and XTE J1550-564, the NS transient Aquila X-1, and the flaring NS low-mass XRB (LMXB) 4U 1705-44, in which a remarkable correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the following soft state was found [158-160]. More recently, a comprehensive study of the hard-to-soft spectral state transitions detected in all the bright XRBs in a period of about five years with simultaneous X-ray monitoring observations with the RXTE/ASM and the Swift/BAT confirmed the correlation between the hard-to-soft transition and the peak luminosity of the following soft state [161, 162], as shown in the upper panel of Figure 35 [161]. More important was the discovery of the correlation between the transition luminosity and the rate-of-change of the luminosity during the rising phase of an outburst of transients or a flare of persistent sources, as shown in the lower panel of Figure 35 [161].\nThe above correlation implies that in most cases it is the rate-of-increase of the mass accretion rate, rather than the mass accretion rate itself, determines the hardto-soft spectral transition; this is depicted in Figure 36 [161]. In addition, the discovery of the relation between the hard X-ray peak flux and the waiting time of transient outburst in the BH transient GX 339-4 (as shown in Figure 37) supports that the total mass in the disk determines the peak soft state luminosity of the following outburst [159]. Therefore spectral states should be understood in the non-stationary accretion regime, which would be described by both ˙ m and m , as well as the total mass in the disk before an outburst.\nVIII. FURTHER DEVELOPMENTS ON THERMAL STABILITY OF SSD\nThe SSD model predicts that when the accretion rate is over a small fraction of the Eddington rate, which corresponds to L glyph[greaterorsimilar] 0 . 06 L Edd , the inner region of the disk is radiation-pressure-dominated and then both secularly [163] thermally unstable [164, 165]. However, observations of the high/soft state of black hole X-ray binaries with luminosity well within this regime (0 . 01 L Edd glyph[lessorsimilar] L glyph[lessorsimilar] 0 . 5 L Edd ) indicate that the disk has very little variability, i.e., quite stable [166]. It has been well established that the accretion flow in this state is described by the SSD model [4, 167]. Radiation magnetohydrodynamic simulations of a vertically stratified shearing box have confirmed the absence of the thermal instability [168]. Recently, the thermal stability is revisited by linear analysis, by taking into account the role of magnetic field in the accretion flow [169]. By assuming that the field responds negatively to a positive temperature perturbation, it was found that the threshold of accretion rate above which the disk becomes thermally unstable increases signifi-\n\n\n
FIG. 35. Upper panel: correlation between the transition luminosity (15-50 keV) and the peak luminosity of the following soft state (2-12 keV) in Eddington units. Lower panel: correlation between the peak luminosity of the soft state and the maximum rate-of-increase of the X-ray luminosity around the hard-to-soft transition. (Adapted from Figures 24 and 27 in Ref.[161])
\n\ncantly, compared with the case of not considering the role of magnetic field. This accounts for the stability of the observed sources with high luminosities, as shown in Figure 38. If the magnetic pressure is less than about 24% of the total pressure, then this model can explain the 'heart-beat' limit-cycle instability observed in GRS 1915+105 at its highest luminosity; this peculiar source holds the highest accretion rate (or luminosity) among BHXBs.\nObservations of GRS 1915+105 showed that t high (the duration of the outburst phase) is comparable to t low (duration of the quiescent phase) and L high is 3 to 20 times larger than L low [170]. However, numerical calculations\n\n\n
FIG. 36. Correlation between the peak fluxes of the initial low/hard states in the outbursts of GX 339-4 and the time since the latest low/hard state peak in the previous outburst. The dashed line passes the origin and the data point of maximal peak flux, showing an example of a linear relation. (Adapted from Figure 3 in Ref.[159])
\n\n\n\n
FIG. 37. Aschematic picture of the regimes of the hard state. Two assumed transient outbursts of different peak luminosities are shown. When a source is under stationary accretion, spectral transitions between the hard state and the soft state occurs at a nearly a constant luminosity L 0 . When a source is undergoing an outburst or flare, the hard-to-soft transition occurs at a luminosity above L 0 . The additional luminosity roughly proportional to ∆ L ∆ T . The soft-to-hard transitions are expected to occur around L 0 . (Figures 28 in Ref.[161])
\n\nshowed that t high is less than 5 percent of t low , and L high is around two orders of magnitude larger than L low . Some efforts have been made to improve the theory in order to explain observations either by some artificial viscosity prescription [171] or by additional assumption of the energy exchange between the disk and corona [172]. Taking into account the stress evolution process, it was found that the growth rate of thermally unstable modes can decrease significantly owing to the stress delay, which may help to understand the 'heart-beat' limit-cycle variability of GRS 1915+105 [173]. The limit-cycle properties are found to be dominated by the mass-supply rate (accre-\n·\n·\n\n\n
FIG. 38. The thermal equilibrium curves of a thin disk at 10 r g for different Φ 1 , which is defined as Φ 1 ≡ B ϕ H and is in unit of Gs · cm. Other model parameters are BH mass M BH = 10 M glyph[circledot] and viscosity parameter α = 0 . 1 . (Figure 2 in Ref.[169])
\n\ntion rate at the outer boundary) and the value of the α -viscosity parameter in the SSD model that assumes that the viscous torque is proportional to the total pressure [77]. It was also found that only the maximal outburst luminosity (in Eddington units) is positively correlated with the spin of a BH, providing another way to probe BH spin [77]; this is mainly due to the smaller inner disk radius and thus higher radiative efficiency for larger a ∗ as shown in Figures 15, 16 and 17.\nIX. UNIFICATION AND OUTLOOK\nDespite of the many progresses made over the last decades on the study of BHXBs and microquasars, there are still many outstanding and unresolved issues, which can be pierced by two big pictures. One big picture is related to the astrophysics of BHXBs and microquasars, which is centered on understanding the accretion-outflow (wind or jet) connections at different accretion rates for different types of astrophysical systems involving BHs. The other big picture is related to the fundamental physics of BHXBs and microquasars, which is centered on identifying astrophysical BHs and testing theories of strong gravity. These two pictures are actually also entangled together, because a strong gravity theory, such as GR, is needed to describe the astrophysical aspects of these systems, and we need to understand the astrophysics of BHXBs and microquasars before we can start to test any strong gravity theory[174].\nFigure 39 shows the relation between the collimationcorrected gamma-ray luminosity and the kinetic power for AGNs and gamma-ray bursts (GRBs)[175]. For at least some of GRBs, a super-Eddington accreting compact stellar object, probably a BH with around 10 M glyph[circledot] , is believed to be responsible for powering the highly relativistic jets, which produces intense gamma-rays through violent collisions of blobs in the jets[176]. The two types of AGNs shown here, flat-spectrum radio quasars (FS-\n\n\n
FIG. 39. The relation between the collimation-corrected gamma-ray luminosity and the kinetic power for AGNs and GRBs. The shaded regions display the 2 σ confidence band of the fits. The blazar and GRB best-fit models (dashed and dotted lines, respectively) follow correlations which are consistent, within the uncertainties, with the best-fit model obtained from the joint data set (solid line).(Figure 3 in Ref.[175])
\n\nRQs) and BL Lacs, have all been observed to have mildly relativistic jets. Their central engines are supermassive BHs with around 10 7 -9 M glyph[circledot] with accretion rates just below (for FSRQs) or far below (for BL Lacs) Eddington rate[177]. The good correlation over about 10 orders of magnitudes between these very different systems with very different accretion rates suggests that there must be some common mechanisms responsible for the accretionoutflow (wind/jet) connections for all accreting BHs[178].\nPutting together all related phenomenologies and some theoretical modeling of BHXBs and microquasars discussed in this article, an unification scheme is illustrated in Figure 40 with the major elements in the accretionoutflow connections in different types of astrophysical systems harboring both stellar mass BHs and supermassive BHs with accretion rates over several orders of magnitudes. The types of accreting stellar mass BHs include BHXBs and microquasars in different spectral states, as well as ultra-luminous X-ray sources (ULXs) (some of which are most likely ultra-Eddington accreting BHs[180]), and GRBs. The types of accreting supermassive mass BHs include low-luminosity AGNs (LLAGNs), BL Lac objects, normal radio quiet quasars (RQQs), FSRQs and broad-line-less luminous quasars (BL3Qs) [179] (I create the acronym 'BL3Q' here just for fun).\nThis scheme is not a theoretical model, or even toy model yet. It, however, can be used as one possible chain to pierce many observed phenomenologies together, providing a possible frame work for further theoretical developments on accretion-outflow connections. Future observations will scrutinize this unification scheme and\n\n\n
FIG. 40. Unification scheme of accretion-outflow connections of accreting BHs for different ranges of accretion rates, corresponding to different spectral states or different types of astrophysical systems. From very low to very high accretion rates in units of the Eddington rate ˙ m , these states are: hard/quiescent state, hard/low state, thermal-dominated soft state, steep power-law state, slim disk, and NDAF disk; different types of AGNs with similar accretion-outflow structures are also labeled for comparison. The ubiquitous dusty tori and BLRs are absent in all BHXBs and microquasars. The lack of dusty tori can be easily understood since there is no dust supply in them. It is then plausible that the lack of BLRs is the consequence of no dusty tori, suggesting that a BLR may be formed out of the evaporated inner dusty torus by the anisotropic radiation of the accretion disk in an AGN[179].
\n\nSome necessary elaborations and points:\n\n· ˙ m glyph[greatermuch] 10: the two-phase clumpy thick disk is in the form of neutrino dominated accretion flow (NDAF), which can extend to ISCO due to the extremely low opacity of neutrinos. The large scale magnetic field lines are wound-up by the spinning BH and rotating disk. The discontinuous jet is produced by the BZ mechanism. Most likely no such ultra-Eddington AGN exists.\n· ˙ m ∼ 1: the thin disk is truncated by radiation pressure, which drives near-spherical winds out. BL3Q is predicted as the early phase of a quasar, when the radiation pressure drives all gas in the radiation cone out so that no broadline region (BLR) can be formed[179]. Actually all gas mixed with dust in the radiation cone can be blown out at substantially below Eddington rate, since the dust has a very high opacity to emissions from visible to UV. There BL3Q should be generic in the early phase of a quasar's active cycle, if the quasar activity is triggered by enhanced mass inflow consisting of a mixture of gas and dust.\n· ˙ m ∼ 0 . 3: the thin disk extends to around ISCO and the two-phase corona is clumpy, so the jet is produced mostly via the BZ mechanism and is discontinuous. The dusty torus and BLR for in a FSRQ are not shown.\n· ˙ m ∼ 0 . 1: the thin disk extends to around ISCO and nearequatorial wind is thermally driven out; small scale and turbulent magnetic fields may be responsible for launching the plasma out of the disk via magnetic reconnections. The dusty torus and BLR in a RQQ are not shown.\n· ˙ m ∼ 0 . 03: the thin disk is made of an inner and outer part; the inner disk extends to around ISCO. The twophase ADAF is clumpy, so the jet produced via a mixture of the BP and BZ mechanisms is discontinuous. The dusty torus and BLR in a BL Lac are not shown.\n· ˙ m< 0 . 01: 'Q' refers to the very low luminosity quiescent state that normally displays a hard power-law spectrum. The large scale magnetic fields rotating with the truncated thin disk and thick ADAF channel the continuous plasma to form collimated and continuous jets via the BP mechanism. There is neither dusty torus nor BLR in LLAGNs [179].\n\nrevise it inevitably. Of particular importance is the reliable determinations of the mass and spin of accreting BHs. The BH mass allows us to determine the accretion rate in units of Eddington rate, i.e., ˙ m , which is a key parameter to the unification scheme of accretion-outflow connections. The BH spin is of course another key parameter, because it can determine the radiative efficiency of the disk and jet power in the BZ mechanism.\nOnce we have well determined BH parameters and a consistent and predictable theory describing the observed accretion-outflow connections, we are then ready to study the fundamental physics of BHs with BHXBs and microquasars, such as the properties of event horizon, spacetime around Schwarzschild and Kerr BHs, BH spin energy (and mass) extraction. For example, both the broad iron line and CF fitting methods can be used to measure\na BH's spin. However the two methods are equivalent only if the metric is accurately described with Kerr metric. Different spin measurements for a BH with the two different methods would invalidate Kerr metric, provided that we understand the accretion disk physics thoroughly. On the other hand, a correct metric is of course required to describe accurately the accretion flow around BHs. 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It includes mainly two parts: (1) Brief review of several recent review article on this subject; (2) Further development on several topics, including black hole spin measurements, hot accretion flows, corona formation, state transitions and thermal stability of standard think disk. This is thus not a regular bottom-up approach, which I feel not necessary at this stage. Major effort is made in making and incorporating from many sources useful plots and illustrations, in order to make this article more comprehensible to non-expert readers. In the end I attempt to make a unification scheme on the accretion-outflow (wind/jet) connections of all types of accreting BHs of all accretion rates and all BH mass scales, and finally provide a brief outlook.","pages":[1]},{"title":"CONTENTS","content":"VIII. Further developments on thermal stability of SSD 23","pages":[1]},{"title":"I. SYNOPSIS","content":"I will start by defining what I mean by black hole binaries (BHBs) and microquasars in this article. I decide to restrict myself to only a subclass of BHBs, namely, BH X-ray binaries (BHXBs), since these are the only class of BHBs known observationally. I will then simply refer microquasars as BHXBs for reasons discussed in Section II. Since many excellent, comprehensive and quite up-todate review articles on BHXBs are readily available in literature, I feel it is not necessary to write another bottomup and comprehensive review article on the same subjects at this stage. I will thus take quite an unusual approach in this article. I will first give some concise guides on several representative review articles [1-4], with some necessary updates. I will then focus on the further developments on several topics I feel deserve more discussions, i.e., BH spin measurements (Section IV), hot accretion flows (Section V), corona formation (Section VI), state transitions (Section VII) and thermal stability of SSD (Section VIII). The emphasis is thus put on understanding the accretion flows and measuring the parameters of black holes in them. Some rather general issues on BH astrophysics, such as what astrophysical BHs are and how to identify them observationally, are not discussed here but can be found from my recent book chapter entitled 'Astrophysical Black Holes in the Physical Universe' [5]. The usual practice of writing a review article is to end by listing some outstanding issues and major unsolved problems, and then to propose some possible approaches to them. I initially did not do this in the first draft. The history of astronomy tells us that major progress is almost always made by unexpected discoveries and research results; unpredictability is an essential nature of astronomy. This article is not intended to be read by funding agencies or proposal reviewers, so I thought I did not have to do it. In astronomy, knowing what has happened, but looking and doing it differently are far more important and effective than following other people's advises. However, the editors of this book suggested me to write a brief 'outlook' in the end. I thus did it nevertheless.","pages":[1,2]},{"title":"II. ACRONYMS AND TERMINOLOGY","content":"In Table I, I list all acronyms used in this article; most of these are quite commonly used in this community. A BH binary (BHB) is a gravitationally bound binary system in which one of the objects is a stellar mass BH with mass from several to tens of solar masses ( M glyph[circledot] ); the other object, i.e. its companion, can be either a normal star, a white dwarf, or a neutron star (NS). In case a binary system consists of two BHs, it is is referred to as a binary BH system, which is not covered in this article. When the companion in a BHB is a normal star, the gas from the star may be accreted to the BH and X-rays are produced, as a consequence of the heating by converting the gravitational potential energy into the kinetic energy of the gas, and a BH X-ray binary (BHXB) is referred to as such a binary system, as shown in Figure 1. The possible existence of BHXBs was first suggested by Zel'dovich & Novikov [6, 7]. The first BHXB found is Cygnus X-1 [8], now a well-studied system among many others found subsequently in the Milky Way and nearby galaxies. The terminology of microquasar has some twisting in it. Historically it was first referred to the BHXB 1E1740.7 -2942 in the Galactic center region, because a double-sided jet was detected from it, mimicking some quasars with similar radio lobes, which have much larger scales [9]. Soon after, superluminal jets are observed from a BHXB GRS 1915+105 [10], which is also referred to as a microquasar. Nevertheless both BHXBs are quite unusual compared to many others, thus microquasars were considered quite unusual. However it became clear that microquasars may be quite common among BHXBs, since the discovery of a normal BHXB GRO J1655 -40 [11], whose superluminal jets were observed [12] and showed some correlations with its X-ray emission [13]. Subsequently several more BHXBs have been observed with superluminal jets. At this stage microquasars began to be referred to as BHXBs with relativistic jets (with bulk motion of about or larger than 90% of the speed of light), to distinguish them from NS X-ray binaries (NSXBs) that only have mildly relativistic jets (with bulk motion of about or smaller than 50% of the speed of light) [14]. However the discovery of relativistic jets from a NSXB Circinus X-1 made the situation complicated: relativistic jets are no longer uniquely linked to BHs [15]. Now in retrospect, a microquasar can be literally and easily understood as the micro version of a quasar; however a quasar may or may not be observed with collimated jets. A quasar has been already understood as a special galaxy centered by an actively accreting supermassive BH with a mass from millions to billions of M glyph[circledot] , and thus its total light output is dominated by the BH's accretion process, in a similar way as in BHXBs. The production or lack of relativistic jets may have similar or even the same underlying physical mechanisms in BHXBs and AGNs, though their surrounding environments may modify their observed morphologies [16]. It is therefore more natural to simply refer microquasars as BHXBs; in the rest of this article, microquasar and BHXB are used interchangeably. We therefore will focus on BHXBs and thus will not discuss binary systems producing gamma-rays and sometimes radio jets, which are most likely high-mass NSXBs and in which jets or pulsars' winds interact with the wind of its high-mass companion to produce the observed gamma-rays [17-23]. Such systems show very different observational characteristics, e.g. the long (1667 days) super-orbital modulation with phase offset (about 280 days) between its X-ray and radio light curves found in LS I+61 · 303 [24].","pages":[2,3]},{"title":"III. REVIEW OF REVIEWS","content":"Here I attempt to review the four recent review articles [1-4] I consider most useful to readers. Additional information and updates are provided when necessary. Some overlaps exists between these review articles, as expected and inevitable in bottom-up review articles. To avoid repetitions as much as possible in this article, I thus put different emphasizes on different articles, with of course my personal tastes and perceptions.","pages":[3]},{"title":"A. The most recent Science Collection","content":"The recent collection of perspectives and reviews in the Science magazine provides excellent introductions to and concise summaries of the current state of our understanding of BH physics and astrophysics [1, 25-27]. To the subjects of this article, the most relevant article in this collection is the one by Fender and Belloni enti- tled 'Stellar-Mass Black Holes and Ultraluminous X-ray Sources', which is focused on the observational characteristics of BHXBs [1]. In particular, the article provides an excellent description of the general picture of spectral evolutions of BHXBs with the Hardness-IntensityDiagram (HID), which is found to be well correlated with the observed jets from BHXBs, as shown in Figure 2. Actually this cycle is also well tracked by their flux variability, represented by the measured root-mean-squares (rms) above its average flux, as shown by the RMSIntensity-Diagram (RID) [28] of the BHXB GX 339-4 in Figure 3, which also shows additional horizontal tracks at intermediate intensities. Sometimes, a full HID cycle does not go into the soft state at all (Figure 4), perhaps due to a failed outburst [29]. The basic scenario is as follows. During the initial stage of an X-ray outburst of a BHXB (A → B), which is triggered by a sudden increase of accretion rate onto the BH, its spectrum is normally hard and steady jets are always observed. After reaching its peak luminosity, its spectrum begins to soften in a chaotic way and transient jets are normally observed (B → C → D). After this transition, the system calms down with a soft spectrum and no jets are present (D → E). Finally the system returns to its quiescent state with a hard spectrum accompanied with the reappearance of jets (E → F). Throughout this cycle, the presence of hot accretion disk winds appears to be anti-correlated with its spectral hardness and jet ejection. This empirical pattern appears to be quite universal for all BHXBs with very few exceptions, though the underlying physics is still not well understood yet. Nevertheless putting together the above scenario is a very significant progress in this field over the last ten years. Due to the conciseness of this article, some other important subjects on BHXBs are not fully discussed and many original references are also missing. Other articles in this collection [25-27] are less relevant to the subjects of this article, but are still quite interesting to read, within the context of BH astrophysics and physics. The only other type of astrophysical BHs known to exist in the physical universe are supermassive BHs in the center of almost each galaxy. Volonteri [27] concisely summarized our current understanding on how they are formed and grow over the cosmic time; merging of two BHs is a key process here. Thorne [25] focused on what happens when BHs merge together to produce gravitational waves, which might be used as a new laboratory for studying gravitational physics and a new window for exploring the universe. Witten [26] then explained the quantum properties of BHs, in particular the basic ideas behind Hawking radiation, which might not be important at all for astrophysical BHs. Nevertheless the understanding gained through studying the quantum mechanics of BHs actually plays very important roles in developing other theories of physics, such as that in heavy ion collisions and high-temperature superconductors, as vividly described by Witten [26].","pages":[3,4]},{"title":"B. The most recent ARA&A article on BHXBs, with some updates","content":"The most recent ARA&A article on BHXBs by Remillard and McClintock entitled 'X-Ray Properties of Black-Hole Binaries' [2] (referred to as RM06 hereafter) provides the most complete, comprehensive and accurate review of BHXBs, which actually covers subjects much beyond just the X-ray properties of BHXBs. The Introduction of RM06 highlights the initial theoretical and observational studies of BHXBs, followed by brief comments on several main review articles on BHXBs preceding this one and the basic properties of BHs within the context of general relativity (GR). All main properties of BHXBs known at the time are summarized in the first table and figure there. In Table II, I compile the most updated data on all BHXBs currently known, including three BHXBs outside the Galaxy and not in the Large Magellanic Cloud (LMC); the currently available spin measurements for these BHs are also included for completeness. In Table II, where P orb is the orbital period, K C is the semiamplitude of the velocity curve of the companion star, M BH is BH mass, i is the the orbital inclination angle, and q ≡ M C /M BH is the mass ratio. In Figure 5, I show the updated graphical representation of most of the BHXBs listed in Table II. The X-ray properties of BHXBs are characterized by their X-ray light curves, timing and spectra. Essentially all known BHXBs were discovered initially as bright Xray sources, and the majority of them were detected as transient X-ray sources with X-ray all-sky monitors. The transient properties of some of BHXBs can be interpreted by the disk instability model (DIM) (see Ref.[49] and references therein), which assumes a constant mass transfer rate from the mass donor to the accretion disk. However the accretion rate from the disk to the compact object, i.e. a white dwarf (WD), a neutron star (NS), or a BH, is normally lower than the mass supply rate in the disk, so mass is accumulated in the disk. When the accumulated mass exceeds a certain critical value, a sudden increase of accretion rate results in a nova-like outburst. DIM is most successful in explaining an outburst with fast rise and exponential decay (FRED). However as shown in RM06, many observed X-ray light curves of BHXBs a : Reported in the first paper on systematic BH spin measurements [47]; b : Reported in the most recent literature; c: Postulated to be extreme retrograde Kerr BH, due to the lack of the thermal disk component above 2 keV (GRO J1719-24 also belongs to this class) [47]; d : Postulated to be non-spinning BH, due to the observed low disk temperature; e : BH mass of 7 M glyph[circledot] assumed [47]; f : Based on the inner disk radius decrease by a factor of two from the hard state to the soft state transition [47]. are far more complicated than just FRED and their recurrent time scales are also not compatible with DIM. Perhaps disk truncation (a subject to be discussed extensively more later) and mass transfer instability are additional ingredients [50]. X-ray emission from a BHXB is variable at all time scales, from the free-fall or Keplerian orbital time scale of milliseconds near the BH, to the various oscillations (some are even related to GR) in the disk with time scales from milliseconds to minutes, to the viscous time scales of minutes, and to the various instabilities of different time scales. Therefore timing studies of BHXBs can probe the geometry and dynamics in BHXBs. However, lack of coherent signals, such as that observed from pulsars, makes it difficult to unambiguously identify the underlying mechanisms from the detected X-ray variabilities. Nevertheless power density spectra (PDS) and rms still allow us to make progress in understanding the general characteristics of a BHXB, in particular when combined with its spectral behaviors, as already discussed briefly above. The most successful understanding of BHXBs so far is the description of their thermal X-ray spectral component by the classical Shakura-Sunyaev Disk (SSD) model [51]. After applying GR to SSD, the temperature distribution in the disk can be obtained [52]. Applying this model to BHXBs, one can even measure the spin of their BHs [47], by assuming that the inner accretion disk boundary is the inner-most stable circular orbit (ISCO) of the BH (a subject to be discussed more in Section IV). However a power-law (PL) component is almost ubiquitous in the spectra of BHXBs. The interplay between these two components results in various spectral states, which are also found to be well correlated with their timing properties. Historically these spectral states have been named in many different ways, reflecting mostly how they were identified with the observations available at the times. In RM06, three states are defined, which show distinctively different spectral and timing behaviors, as shown in Figure 6 based on the RXTE data on the BHXB GRO J1655-40. The thermal state has its X-ray spectrum dominated by the thermal disk component and very little variability. The hard state has its X-ray spectrum dominated by the a PL component and strong variability. The steep power-law (SPL) state is almost a combination of the above two states, but the PL is steeper. I will keep using these definitions throughout this article for consis- tency. Please refer to Table 2 in RM06 for quantitative descriptions of these three states. When a BHXB is in quiescence, i.e. not in an outburst, its spectral shape is similar to the hard state spectrum. Referring back to Figure 2, RM06 found that the HID can be reasonably well understood in terms of transitions between the above three states; I summarize this in Table III. After examining the extensive RXTE data on spectral evolutions of six BHXBs, the above conclusion seems to be valid generally. Thus a coherent picture seems to emerge: jets are produced only when the PL component in the spectrum is strong and winds are present only when the PL component disappears; jets and winds appear to be mutually exclusive. This suggests that transient winds should appear in the SPL state, anti-phased with transient jets, if winds quench jets as suggested recently [53, 54]. The robust link between the PL component and jets provides an important clue to the jet production mechanism. The PL component is believed to be produced in an optically thin and geometrically thick corona, in which hot electrons up-scatter the thermal photons from the disk. As discussed in RM06, the inner region of the disk appears to be truncated quite far away from the BH in the hard state; this should be the primary reason for the low luminosity of this state, as evidence for the existence of BHs in BHXBs. However it is still not understood how the corona is formed; I will discuss this later in Section VI. In the final part, RM06 discussed the exciting possibilities of using BHXBs as probes of strong gravity, including confirming these BHXBs contain true BHs, measuring the spins of BHs, relating BH spin to the Penrose process and other phenomena, and finally carrying out tests of the Kerr Metric. They also discussed in length how to measure the spins of BHs and commented the various methods of doing so. In Section IV I will re-visit the BH spin measurements in details. Lacking of a solid surface, a BHXB is not expected to produce and actually never observed to have coherent pulses. However quasi-periodic oscillations (QPOs) are frequently detected in their X-ray light curves; a QPO is defined as a 'bump' feature in the PDS, if Q = ν/ ∆ ν > 2, where ν and ∆ ν are the peak frequency and FWHM of the bump, respectively. The observed QPOs are further divided into low frequency (LF) (0.1-30 Hz) and high frequency (HF) ( > 30 Hz) QPOs. Typical LFQPOs can be found as the peak around several Hz in the top and bottom PDS of Figure 6. In literature, LFQPOs are further divided into several subclasses and their underlying mechanisms are far from clear at this stage, though many models have been proposed and briefly discussed in RM06. HFQPOs have been detected from several sources at 40-450 Hz. Of particular interests are their stable nature when detected, which may be linked to either the mass or spin or both of the BH in a BHXB. In some cases a 3:2 frequency ratio is found. Although the 3:2 HFQPO pairs could be interpreted in some epicyclic resonance models [55, 56], there remain serious uncertainties as to whether epicyclic resonance could overcome the severe damping forces and emit Xrays with sufficient amplitude and coherence to produce the HFQPOs. A revised model applies epicyclic resonances to the magnetic coupling (MC) of a BH's accretion disk to interpret the HFQPOs [57]. This model naturally explains the association of the 3:2 HFQPO pairs with the steep power-law states and finds that the severe damping can be overcome by transferring energy and angular momentum from a spinning BH to the inner disk in the MC process.","pages":[4,5,6,7]},{"title":"C. Two comprehensive and long articles on modeling accretion flows in BHXBs","content":"Page limitations to the above review articles did not allow in depth discussions on the detailed processes and models of accretion flows in BHXBs, which are responsible for the above described energy spectra, PDS, state transitions, jets, and wind. Here I introduce two comprehensive and long articles just on this, by Done [3] and Done, Gierli'nski & Kubota [4].","pages":[7]},{"title":"1. A beginner's guide","content":"The first is intended to readers who just start research in this field [3]. It started with the basic tools in plotting spectra and variability, and then described the basic ideas and methods used to infer the inner accretion disk radius when a BHXB is in a thermal state, in order to measure the BH spin [47]. A useful introduction is made on how to make various corrections to account for various effects, including color correction, special and general relativistic effects, starting from the original work [47]. Several commonly use fitting models, i.e. DISKBB , BHSPEC and KERRBB , in the XSPEC package are also briefly introduced. An example is given to demonstrate that the expected relation L disk ∝ T 4 in [47] agrees with the data from the BHXB GX 339-4 [58], where L disk and T in are the disk's total luminosity and temperature at the inner disk boundary, respectively. The hard state is then briefly touched upon, using the Advection Dominated Accretion Flow (ADAF) model [59]. ADAF naturally explains the hot corona required by the observed PL component, especially at very low accretion rate. Evaporation at low accretion rate in the inner disk region has been proposed as the mechanism producing a geometry with a radially truncated disk and a hot inner flow; the latter might be the ADAF [60]. This also means that when a significant PL component is present in the spectrum, BH spin measurement cannot be done with the inferred inner disk boundary (see, however, counter evidence discussed in Sections IV and VI). This geometry is considered a paradigm that can account for many of the observed diverse phenomena from spectral evolution to timing properties. A brief, yet interesting discussion is given on scaling up the above models to Active Galactic Nuclei (AGNs), which host actively accreting supermassive BHs in the centers of galaxies. Applying the insights learnt from BHXBs on their spectral evolution, changing disktemperature with accretion rate and BH mass, and diskjet connections, one might be able to understand many phenomena beyond the simple AGN unification scheme, in which the different observational appearance is all attributed to an viewing angle difference. The continuum emissions of both the disk thermal and PL components are modified by absorptions and added by additional spectral features along the line of sight (LOS). Absorptions in neutral media produce various photoelectric absorption edges, but in ionized media result in both absorption edges and lines. In addition to recovering the original X-ray emission of a BHXB, modeling the absorption features is important in learning the physical properties of the absorption media, such as column density, ionization and velocity along LOS. Winds from BHXBs discussed above are always detected this way. Several XSPEC fitting models for various kinds of absorption edges and lines are also introduced here, i.e. TBABS , ZTHABS , TBVARABS , ZVPHABS , TBNEW , ABSORI and WARMABS . Figure 7 shows the calculated absorption structures with ABSORI and WARMABS ; note that ABSORI does not include absorption lines, but WARMABS does. X-rays interacting with the surrounding medium can ionize it and heat the free electrons up by Compton scattering; hot electrons can also loose energy by interacting with lower energy photons. A Compton temperature of the plasma is reached when the above two processes reaches an equilibrium. The temperature is determined by only the spectral shape of the continuum, so the heated plasma can escape as winds from the disk if the velocity of the ions exceed that of the escape velocity at that radius, which is usually very far away from the inner disk boundary. As the continuum luminosity approaches the Eddington limit, the radiation pressure reduces the escape velocity substantially in the inner disk region such that continuum driven winds can be launched almost everywhere in the disk of a BHXB, forming a radiation driven wind, which is also called thermal wind; alternatively winds may also be driven magnetically in the inner disk region, but this is much less understood yet. In contrast, in an AGN the peak continuum emission is in the UV band, which has a much larger opacity than Compton scattering in neutral or weakly ionized medium during both photoelectric and line absorptions. This means the effective Eddington luminosity is reduced by large factors and UV line driven wind is easily produced at high velocity. This explains the relatively lower velocity (hundreds km/s) and highly ionized winds from BHXBs, but much higher velocity (thousands km/s to 0.2 c ) and weakly ionized winds from AGNs. Material illuminated by X-rays produce both fluorescence lines and reflection features, which depends on both the continuum spectral shape and ionization state of the material, as shown in Figure 8 calculated with the XSPEC ATABLE model REFLIONX.MOD, which includes the self-consistent line and recombination continuum emission. Note that for highly ionized reflection, the Compton heated upper layers of the disk broadens the spectral features; these effects are not included in the simpler XSPEC model PEXRIV . Replacing the stationary slab by a disk around a BH, both the special and general relativistic effects will further smear (broaden) the spectral features, as shown in Figure 9 for the iron line region with different values of ξ , inner disk radius r in , and viewing angle i ; these are calculated with the REFLIONX.MOD and then convolved with KDBLUR in the XSPEC package. Other similar XSPEC fitting models are DISKLINE , LAOR , and KY . The inner disk radius can in principle be determined by modeling the observed broad iron line features, that in turn can be used to measure BH spins, as will be discussed briefly in Section IV. Figure 10 shows all components of a broad band spectrum of a BHXBs, including interstellar absorption. We thus have two ways discussed so far to determine r in of BHXBs by either fitting its thermal continuum or the fluorescent iron line feature. However results are not always consistent, obtained with different methods, even using the same data by different authors. The author believes that r in increases at low accrete rate, to make room for the corona occupying this space. Nevertheless a fairly good summary on the current conflicts and confrontations is made on this issue, which I will discuss further in Section IV.","pages":[7,8,9]},{"title":"2. An expert's handbook","content":"The second one can be considered as a handbook on accretion flows on BHXBs, and really lives up to its subtitle 'Everything you always wanted to know about accretion but were afraid to ask' [4] (DGK07 hereafter). Besides its much longer length of 66 pages, the main difference from the above review articles is that it is focused on confrontations between theories and observations and intends to depict a coherence picture of the accretion physics in BHXBs. In the following I will summarize briefly the main points and conclusions reached in DGK07; those I have reviewed above and will discuss more later will be skipped for brevity. The underlying physics of DIM for triggering the outbursts in BHXBs discussed above is the hydrogenionization instability, which produces the so-called 'S'curve, as shown in Figure 11; irradiation by the inner hot disk to keep the outer disk hot is required to produce the slow flux decays, e.g. the exponential decays, frequently observed in them. The outer disk radius is obviously another key parameter, which is determined by the tidal instability in a binary system. This mechanism can explain why BHXBs with high mass companions are all persistently bright, since their outer disks are always in the upper branch due to the combination of their higher average mass transfer rate and inner disk irradiation. Similarly it also explains the differences and similarities between the light curve properties of neutron star X-ray binaries (NSXBs) and BHXBs, since a NS has a lower mass. It should be noted that the additional surface emission from the NS may also help to maintain the outer disk hot and stay in the upper stable branch [62]. The SSD prescription assumes that the stress is proportional to pressure. Because the gas pressure is P gas ∝ T but radiation pressure is P rad ∝ T 4 , so a small temperature increase causes a large pressure increase when P rad ≥ P gas , and thus large stress increase, which in turn heats the disk even more. The opacity cannot decrease effectively to cool it down, so the disk becomes unstable when L ≥ 0 . 06 L Edd , where L Edd is the Eddington luminosity of a BHXB. However in most BHXBs their disk emissions appear to be stable up to around 0 . 5 L Edd . One way out is to assume that the stress is proportional to √ P rad P gas ∝ T 5 / 2 , so the stress increases slower in the radiation pressure dominated regime. Beyond this the disk should become unstable, as evidenced by the sometimes 'heart-beat' bursts of the super-Eddington BHXB GRS 1915+105; other BHXBs with L max > L Edd (e.g. V404 Cyg and V4641) were not observed with such instabilities, perhaps due to the combination of insufficient observational coverage and sensitivity. However similar 'heart-beat' bursts were also observed re- IGR J17091-3624, which is likely substantially sub-Eddington unless it is located much beyond 20 kpc and/or its BH mass is quite small [63]. Super-Eddington accretion flow (SEAF) can become stable again, if the radiation instability is overcome by an optically thick ADAF, i.e. the slim disk model, in which the trapped photons in the flow is advected inwards, thus balancing the heating generated by viscosity. Strong radiation driven winds can be easily produced; this can happen in BHXBs and NSXBs (e.g. Z-sources). Evidence exists that truncated inner disk (TID) is quite common in SEAF and outflow even dominates over inflow in SEAF [64]. The observed PL component in BHXBs cannot be explained by the SSD-like models and thus requires a hot accretion flow (HAF). At low accretion luminosity (e.g. the quiescent state), the HAF may be the advection dominated accretion flow (ADAF), the convection dominated accretion flow (CDAF), or the advection dominated accretion inflow/outflow solution (ADIOS). At higher luminosity (e.g. the hard state), the original hot and optically thin disk solution (i.e. the SLE solution) is unstable, because the electron heating efficiency by the Coulomb coupling between protons and electrons is too low. The luminous HAF (LHAF), however, has the advection as a heating source to electrons, so the heating efficiency increases and thus electrons can cool the flow more effectively. Outflows can be produced in ADAF/ADIOS; collimated jets can also be produced if magnetic fields are involved, so an accretion flow may even be jet dominated (i.e. JDAF). The interplays between the SSD and HAF may be responsible for the observed different states in BHXBs discussed above. If the PL component is produced by Compton up-scattering, then the combination of the optical depth τ and L h / L s , where L h is the heating power in electrons and L s is the cooling power in seed photons, can describe the observed variety of spectra. For example, the hard state has L h / L s glyph[greatermuch] 1, but the thermal and SPL state have L h / L s ≤ 1. The location of r in is proposed to be closely related to L h / L s , as illustrated in Figure 12. As discussed above, the steeper PL in the thermal and SPL states is mostly non-thermal in nature, so non-thermal Comptonization is required. Detailed comparisons with data suggest thermal Comptonization also cannot be ignored even in the thermal and SPL states. In Figure 12, the hot inner flow (HIF) and patchy corona are responsible for thermal and non-thermal Comptonization, respectively. When a source transits from hard to SPL state, r in decreases, HIF is reduced but the patchy corona becomes dominant. The transition from SPL to soft state is then marked by the disappearance of HIF and significant reduction of the patchy corona. The above scenario obviously depends on two fundamental assumptions: (1) slab-HIF produces the non-thermal PL component; (2) slab-HIF is mostly located between the TID and the BH. Both assumptions are examined exhaustively based on the existing observations and their spectral modeling. Slab-HIF is found to be consistent with essentially all data. An alternative to slab-HIF is that the PL is produced from the jet base and beamed away from the disk; however observations suggest the PL emission is quite isotropic, thus in conflict with this. TID is found also consistent with data when the PL component becomes important, since the observed L disk ∼ T 4 in deviates significantly from a linear relation, suggesting r in is larger when the PL component becomes important. However some studies showed that r in is unchanged in the initial hard state, if the Compton up-scattering process is treated properly in Monte-Carlo simulations to recover the lost disk photons [65]; this issue will be further discussed in Section IV. The above developed HIF/TID model based on the spectral evolutions of BHXBs can also be applied to explain the majority of time variability PDS of BHXBs consistently. The TID acts as a low pass filter as it cannot response effectively to variations in the HIF, so r in controls the low frequency break of the PDS: ν LFB ∼ 0 . 2( r/ 6) -3 / 2 ( m/ 10) -1 Hz, where r = r in /r g and m = M BH /M glyph[circledot] ; this predicts that ν LFB changes from 0.03 to 0.2 Hz as observed during transitions from the hard to SPL and soft states if r decreases from 20 to 6. The HIF/TID model can also explain the observed LFQPO variations with luminosity, if the LFQPOs are some kinds of characteristic frequencies related to r in . To some extend, the shape of the PDS can also be explained by this model. However the observed variability rms ∼ flux linear correlation and log-normal distribution of fluctuations may need additional ingredient, such as the proposed propagation fluctuation model. However I point out that such correlations and distributions are also observed in the light curves of gamma-ray bursts and solar flares [66, 67], which can be produced by the generic self-organized criticality mechanism [66]. Similar rms ∼ flux correlation has also been found for blazars, in agreement with the minijets-in-a-jet statistical model [68]. The spectral and timing properties of weakly magnetized NSXBs are known to have many similarities and differences from BHXBs; this is particularly true for the atoll sources that have similar ranges of L/L Edd to BHXBs. The essential distinction is that a NS has a solid surface, but a BH does not. Observationally the continuum spectra of atoll sources can be modeled as composed of SSD, PL emission, and blackbody emission from the NS surface (or the boundary layer between the TID and NS surface); the former two components are quite similar to BHXBs. The spectral evolutions of these NSXBs are thus driven similarly by the combinations of τ and L h / L s . However the blackbody emission is an additional source of L s , so the PL is not as hard as that in BHXBs and r in variations are less effective in changing L h / L s ; the latter means it is more difficult to find evidence of TID from modeling only the spectral evolutions in NSXBs. On the other hand, the TID/HIF model in NSXBs can produce timing behaviors in the same way as in BHXBs discussed above, consistent with observations; additional timing behaviors, such as coherent X-ray pulsations and kilo-Hz QPOs observed in these NSXBs, are caused by the rapid spins of the hard surfaces of the NSs.","pages":[9,10,11]},{"title":"IV. FURTHER DEVELOPMENTS ON BH SPIN MEASUREMENTS","content":"A BH predicted in GR can only possess three parameters, namely, mass, spin and electric charge, known as the so-called BH no hair theorem. Even if a BH was born with net electric charge, its electric charge can be rapidly neutralized by attracting the opposite charge around it in any astrophysical setting, because the strength of electromagnetic interaction is many orders of magnitude stronger than that of gravitational interaction. Therefore an astrophysical BH may only have two measurable properties, namely, mass and spin, making BHs the simplest macroscopic objects in the universe. Practically, only Newtonian gravity is needed in measuring the BH mass in a binary system. However, GR is needed in measuring the BH spin. The mass and spin of a BH has different astrophysical meanings. Its mass can be used to address the question of ' How much matter (and energy) has plunged into the BH?'. However its spin can be used to address the question of ' How did the matter (and energy) plunge into the BH?'. This is because matter and energy plunged into a BH can carry angular momentum, which is a vector with respect to the spin axis of the BH. In order to increase the total gravitating mass-energy from M i with zero spin to M f , the added rest-mass must be [69, 70] and its final spin becomes where J is the BH's angular momentum. Clearly we have a ∗ = 1 when M f /M i = 6 1 / 2 ; further accretion simply maintains this state [70]. Therefore the required additional rest-mass to spin a BH from zero to maximum spin is ∆ M glyph[similarequal] 1 . 85 M i = 0 . 75 M f ; this is a lower limit to the accreted mass [71]. Figure 13 shows a ∗ as a function of ∆ M . Ignoring Hawking radiation of a macroscopic BH, the only way to extract the energy of a BH and thus reducing its gravitating mass is by extracting its spin energy. Recently, evidence of BH spin energy extraction to power relativistic jets has been found, from the observed correlation between the maximum radio luminosity and its BH spin of a microquasar [72, 73]; however the average jet power is not correlated with BH spin [74]. This indicates that jets may be produced by both BlandfordPayne (BP) [75] and Blandford-Znajek (BZ) [76] mechanisms; but the BZ mechanism is more powerful and responsible for producing the peak radio luminosity. This provides another possible way to estimate a BH's spin [73], similar to a recent proposal of using the peak luminosity of the disk emission to estimate a BH's spin [77]. However there is so far no independent demonstration of validness of either of the above two new methods, which are thus not discussed further in this section. Figure 14 shows the mass reduction, ∆ m = ( M max -M f ) /M max ( M max and M f are the BH's maximum mass and final mass, respectively), as a function of extracting efficiency glyph[epsilon1] of an extreme Kerr BH. For a 10 M glyph[circledot] BH with glyph[epsilon1] = 1 (∆ m ∼ 0 . 3), the total extracted energy is ∆ mM f c 2 glyph[similarequal] 10 54 erg and its total gravitating mass is reduced to M max / √ 2 [71]; this energy could be sufficient to power gamma-ray bursts (GRBs). It is thus plausible that supercritical accretion onto a newly born BH may spin it up and extract its spin energy to power ultrarelativistic jets; multiple spin-up and spin-down cycles may also happen during one GRB, if the collapsing material is clumpy. As shown in Figure 15, the radius of the ISCO of the BH, R ISCO , is a monotonic function of the BH spin [78], beyond which radius a test particle will plunge into the BH under any perturbation; however, in Newtonian gravity a stable circular orbit can be found at any radius. It is thus reasonable to assume that the accretion disk around a BH terminates at this radius, i.e., r in = R ISCO . Therefore, a ∗ can be inferred if one can measure R ISCO in units of its gravitational radius r g = GM/c 2 . Currently three methods have been proposed to measure the BH spin in BHXBs, and all these methods rely essentially on measuring R ISCO . In case the radiative efficiency ( η ≡ L/ ˙ Mc 2 ) can be measured, a ∗ can also be determined this way, as shown in Figure 16. Actually η is a very simple function of R ISCO , i.e, η ∼ 1 /R ISCO , as shown in Figure 17.","pages":[11,12]},{"title":"A. The first black hole spin measurement with X-ray spectral continuum fitting","content":"In 1997, I and my colleagues proposed a method of measuring the BH spin in BHXBs. It started when we tried to measure the mass of the BH in GRO J1655-40 [11] by measuring r in from its X-ray continuum spectral fitting and assuming the BH is not spinning; we found the mass of the BH is around 4 M glyph[circledot] [79]. Coincidentally just when this paper was going to press, Orosz and Bailyn [80] announced their accurate measurement of the BH mass of GRO J1655-40, which is significantly larger than what we found. A co-author of our paper, Rashid Sunyaev urged me to resolve this apparent discrepancy. I immediately realized that a spinning BH of 7 M glyph[circledot] in GRO J1655-40 would be consistent with the inferred with Xray data, and added a note in proof in the paper suggesting this possibility [79]. This turns out to be the first BH spin measurement on record. I then invited two of my close friends and collaborators, Wei Cui and Wan Chen, to join me to apply this method systematically to other BHXBs; a new main conclusion in this work was that the first microquasar in the Milky Way, GRS 1915+105, also contains a spinning BH [47]. At the time DISKBB was the only available fitting model in the XSPEC package for determining r in from an observed X-ray continuum spectra, if the disk contin- uum is described by the SSD model: L disk = 4 πσr 2 in T 2 in , where T in in DISKBB is the disk temperature at r in (i.e. the peak disk temperature in SSD) and L disk can be calculated from the disk flux (after the correction to absorption), the distance to the source and disk inclination (an issue to be discussed later). To determine the physical inner disk radius from the DISKBB parameter r in , several effects must be considered: (1) electron scattering in the disk modifies the observed X-ray spectrum; (2) the temperature distribution in the disk is not accurately described by the Newtonian gravity as assumed in SSD; (3) the observed temperature distribution is different from the locally emitted one; (4) the observed flux is different from the locally emitted one. The latter three are all due to GR effects [52]. For each of the above effects, we introduced a correction factor, using the best available knowledge at the time. Since then, several improvements have been made to correct for these effects and this continuum fitting (CF) method is now quite mature in making accurate BH spin measurements, given sufficiently high quality X-ray continuum spectral measurements and accurate system parameters of the observed BHXB.","pages":[12,13]},{"title":"B. Further developments and applications of the continuum fitting method","content":"This CF method of measuring BH spin has since been applied widely to essentially every BHXB with a well measured X-ray continuum spectrum showing a prominent thermal accretion disk component. In particular, this method has been improved and incorporated into the widely used X-ray spectral fitting package XSPEC, e.g., KERRBB [81], BHSPEC [82], and KERRBB2 [35, 37]. Both the KERRBB and BHSPEC are relativistic models, but they have their own drawbacks and advantages (See Ref.[35] for a detailed comparison). KERRBB includes all the relativistic effects, but it requires to fix the spectral hardening factor. In contrast, BHSPEC could calculate the spectral hardening factor on its own; however, it does not include the returning radiation effect, which turns out to be an important factor in BH spin determination in BHXBs. KERRBB2 combines both models by generating the spectral hardening factor table from BHSPEC and using the table as the input for KERRBB . The research group led by Ramesh Narayan of Harvard University, Jeffrey McClintock at Smithsonian Astronomical Observatory (SAO), and Ronald Remillard of Massachusetts Institute of Technology (MIT) [83] has since applied this method and contributed to most of the BH spin measurements available in the community, as shown in Table II. The CF method relies on two fundamental assumptions: (1) The measured r in is uniquely related to R ISCO of the BH. (2) There is no or negligible X-ray radiation from the plunging matter onto the BH beyond R ISCO . The latter has been studied with numerical simulations that include the full physics of the magnetized flow, which predict that a small fraction of the disks total luminosity emanates from the plunging region [84]. However, in the context of BH spin estimation, it has been found that the neglected inner light in the CF method only has a modest effect, i.e., this bias is less than typical observational systematic errors [85, 86]. The first assumption above requires that the measured r in remains stable as a BHXB changes its spectral state and luminosity. However it was noticed that r in measured is usually much smaller, sometimes even smaller than R ISCO of a extreme Kerr BH in a prograde orbit, when the X-ray spectrum contains a significant hard PL component, which is believed to be produced by inverse Compton scattering of the thermal disk photons in a hot corona. We realized that the inferred smaller r in could be due to the lost thermal disk photons in the scattering process. We then investigated this problem and confirmed that the inferred r in can be made consistent with r in inferred from the thermal disk component dominated spectrum, if the scattered photons are recovered properly by doing detailed radiative transfer in the corona [65]; the same conclusion was also reached by the Harvard/SAO/MIT group independently without knowing our much earlier results [87]. Therefore the method of BH spin measurement by X-ray continuum fitting can also be applied to some SPL state with strong PL component. The stable nature of the measured r in is proven with the textbook case of LMC X-3, when its X-ray luminosity varied over more than one order of magnitude observed in nearly two decades with many different X-ray instruments, as shown in Figure 18 [88]. However, when ordered by the observed disk luminosity l D = L disk /L Edd , the measured r in shows a clear increasing trend when l D > 0 . 3, as shown in Figure 19 [88]. The similar trend has also been found in another BHXB GRS 1915+105 [35] and two NSXBs [64, 89]. It was found that r = r in /r g indeed increases physically when l D > 0 . 3, by comparing the evolution of r as a function of l D over a large range for several BHXBs and NSXBs. Using the blackbody surface emissions of the NSs in these NSXBs is critical in evaluating any possible disk thickening due to high luminosity that would block at least part of the NS surface emission, as well as determining the actual NS mass accretion rate, which turns out to be much less than the disk mass accretion rate; this suggests that the increased radiation pressure is responsible for the increase of r and significant outflow when l D > 0 . 3 [64]. The same trend is much more pronounced in the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2, as shown in Figure 20 together with the data from other BHXBs and NSXBs [90]. However the exact value of r obtained this way should be taken with caution, since the non-negligible energy advection at high accretion rate can modify the disk structure in non-trivial ways, thus making the SSD prescription inaccurate in this case [91]. Figure 20 also shows that as l D decreases, r again starts to increases. However r increases at higher l D and with a different slope for a NSXB than for a BHXB, which can be naturally explained as due to the 'propeller' effect of the interaction between the NS's magnetosphere and its accretion disk [92] and the 'no-hair' of the BH [90]. Figure 21 shows the radiative efficiencies of various systems; BHXBs may have either higher or lower efficiencies than NSXBs, because a BH has neither solid surface nor magnetic field [5]. It is interesting to compare the BH spin results in our first paper [47] and the most recent literature for the same BHXBs as listed in Table II: (1) For GRS 1915+105, A0620-003 and LMC X-3, both results are fully consistent; (2) For GRO J1655-40, the original result points to an extreme Kerr BH ( a ∗ ∼ 0 . 93) [47], somewhat different from the most recent result of mildly spinning BH ( a ∗ = 0 . 65 -0 . 75) [32]. However, the original result was obtained using the BH mass of 7 M glyph[circledot] , about 10% larger than the currently best estimate that was used to obtain an updated BH spin in the most recent literature [32]. From Figure 15, it can be seen that a ∗ would be decreased from 0.93 to 0.87, if the BH mass is decreased by about 10%; actually a ∗ = 0 . 85 is also allowed in the new estimate [32]; (3) For Cygnus X-1, our original conclusion that a ∗ = 0 . 75 in the high/soft state and a ∗ = -0 . 75 in the low/hard state was based on the assumption that r in decreased by a factor of two when the source made a transition from its normal low/hard state to the unusual high/soft state [94]. However the more realistic constraint is that r in changed by more than a factor of 1.8-3.2 during the state transition [94], which implies that a ∗ switched from | a ∗ | > 0 . 85, fully compatible with the latest result of a ∗ > 0 . 95 in the high/soft state [37]. The above would suggest that a very low temperature disk component exists in its hard state spectrum, if Cygnus X-1 indeed harbors a Kerr BH and the accretion disk switches from a retrograde mode to a prograde mode when it makes the hard-to-soft state transition. For a supermassive BH in the center of a galaxy, it is well understood that its mass is mostly gained through accretion in its AGN phase [95]. Therefore random accretion (between prograde and retrograde modes) tends to make its final BH spin close to zero, regardless its initial BH spin. For Cygnus X-1, its current BH mass of around 10-20 M glyph[circledot] cannot be gained through post-formation accretion, since its observed average mass accretion rate ˙ M ∼ 2 × 10 -9 M glyph[circledot] /yr and the age of its companion is much less that 10 8 yr; actually the age of the companion is estimated to range from 4.6 and 7.8 million years [96]. This means that its post-formation BH mass growth is much less than a fraction of its current mass and thus its current BH spin must be quite close to that at birth. Even if accreting at Eddington rate, to grow its BH spin from 0 and the final mass to be the current observed value, the timescale is around 3 . 1 × 10 7 yr and the accreted mass is roughly 7.3 M glyph[circledot] [37]. If its accretions al- tes between prograde and retrograde modes, then its current BH spin should be even closer to its initial spin than that in low mass BHXBs that only stay in one accretion mode due to roche-lobe overflows. Therefore the current high BH spin in Cyg X-1 must be natal; this conclusion is also true for the other highly spinning BHs in other BHXBs [48]. In summary, BH spin in BHXBs can now be measured reliably with the CF method, when the luminosity of a BHXB is between ∼ 0 . 02 and ∼ 0 . 3 in Eddington unit and their system parameters are well-known a prior . The observed thermal disk spectrum can be modeled directly to obtain the BH spin with the available KERRBB2 model in XSPEC when the X-ray spectrum in dominated by this component, i.e., the source is in the thermal state. When a significant power-law component is present, the inverse Compton scattering process must be taken into account to recover the disk photons scattered into this PL component, with for example the SIMPL/SIMPLR model [37, 97] in XSPEC. Recently another way to measure BH spin using the outburst properties of BHXBs has been proposed [77], which is mentioned briefly in Section VIII; however the effectiveness of this method needs to be tested.","pages":[13,14,15,16]},{"title":"C. Uncertainties of the continuum fitting method","content":"In spite of the tremendous progress made so far on BH spin measurements in BHXBs, most of these BH spin measurements suffer from considerable uncertainties. Actually the major source of these uncertainties comes primarily from the uncertainties in their BH masses, accretion disk inclination angles, and distances. The accurate BH mass measurement is required because R ISCO must be in units of r g = GM/c 2 in Figure 15. The disk inclination angle and distance are also required because the total luminosity of the disk emission L disk is needed, in order to estimate the absolute value of a ∗ . So far, all BH masses in X-ray binaries (XRBs) have been estimated using the Kepler's 3rd law of stellar motion, expressed in the so-called the mass function given in Equation (1). Since the only direct observables are P orb and K C , both M C and i have to be determined indirectly in order to obtain the BH mass estimate reliably. The companion's mass M C can be determined relatively reliably by the observed spectral type of the companion star and i can be estimated by modeling the observed ellipsoidal modulation of the companion's optical or infrared light curve. The observed ellipsoidal modulation is a consequence of exposing different parts of the pearshaped companion star to the observer at different orbital phases (see Figs. 1 and 5); the pear-shape is caused by the tidal force of the compact star, which also heats the side of the companion star facing it. For details of BH mass estimates using this method, please refer to [2]. However model dependence and other uncertainties (such as accretion disk contamination) cannot be circum- vented completely and thus systematic error may exist in determining their system parameters. For example, three optical states, namely 'passive', 'loop' and 'active' states, have been identified in the normally called 'quiescent' state of A0620-003 when its X-ray luminosity is very low; only during the passive state its optical light curve modulation is purely ellipsoidal, i.e., accretion disk contamination is completely negligible, as shown in Figure 22 [98]. This means that considerable systematic errors in determining its inclination angle may occur unless only the 'passive' state data are used. Unfortunately previous observations of BHXBs used to determine their inclinations did not always occur during the passive state, thus systematic errors may be common in previous results [99]. Even for GRO J1655-40, of which all previous observations were made during its passive state [99], its BH mass measured with different observations, or the same data analyzed by different groups are not exactly the same, and even not completely consistent between them, as shown in Figure 23, which show a scatter of about 20-30% to the estimated BH mass, much larger than its statistical error of a few percents. It should be pointed out that the inclination i in Equation (1) refers to that of the orbital plane of the binary system. Because the accreting material initially carries the angular momentum inherited from the companion star, the formed disk should be co-planar with the orbital plane of the binary system. However, the BH spin in a BHXB cannot be changed significantly by accretion [71] (and see discussion above on Cygnus X-1). Therefore its spin axis may not coincide with the normal direc- tion of the orbital plane of the binary system. In this case the Bardeen-Petterson effect [106], due to frame-dragging by the spinning BH, can rapidly align the normal axis of the inner disk region with the spin axis of the BH, making the inner disk and binary system mis-aligned. Circumstantial evidence already exists for mis-alignment between the two axes, because the orbital inclination of GRO J1655-40 shown in the upper panel of Figure 23 is significantly different from ∼ 80 · inferred from its relativistic jets [107], if the jets are powered by extracting the spin energy of the BH via the Blandford-Znajek mechanism [76]. Nevertheless, the orbital inclination is normally used in place of the disk inclination, which is needed in BH spin measurement using the CF method, in order to infer the total disk luminosity and calculate GR correction factors. However, the inner disk inclination is currently not available essentially for any of the known BHXBs. For example, a mis-alignment of 10 · from i = 70 · can cause nearly 50% error to the total disk luminosity, which will translate into nearly 30% error in r in . A Schwarzschild BH may be estimated to have a ∗ falling any where between [ -0 . 5 , 0 . 5], if r in is uncertain within about 30%, according to Fig.(15). Accurate determinations of distances of astrophysical objects in the Milky Way are difficult, e.g., for BHXBs that are not standard candles. Normally some absorption features in their spectra, in conjunction with their posi- tions in the galactic coordinates, are used to infer their distances. For example, the distance of GRO J1655-40 is commonly taken as 3 . 2 ± 0 . 2 kpc, based primarily on observed absorption lines and somewhat on the dynamics of the observed jets. Critical examinations of all available data related to its distance, however, are in favor of a much closer distance of less than 2 kpc and more likely just 1 kpc [108, 109]. Similar conclusion is also reached to the distance of A0620-003, revising its distance from the commonly accepted 1 . 2 ± 0 . 4 kpc to ∼ 0 . 4 kpc, making its possibly the closest BHXB known so far [109]; however, a distance of 1 . 06 ± 0 . 12 pc was preferred in a more recent study [30]. Similarly the currently adopted distances of many other BHXBs may also have considerable systematic errors. If true, this would change significantly the current estimates on their masses and spins. Therefore future improvements of the continuum fitting method depend upon the improved measurements on their BH masses, accretion disk inclination, and distances.","pages":[16,17]},{"title":"D. Future improvements of the continuum fitting method","content":"The mass ratio q can be determined directly according to the law of momentum conservation, i.e., if the semi-amplitude of the velocity curve of the BH K BH can be observed directly, as illustrated in Figure 1. The orbital inclination i can be then calculated using Equation (1), avoiding any systematics related to the ellipsoidal light curve modeling. Since a BH is not directly observable, we can only hope to observe any emission or absorption line feature comoving with it. The accretion disk certainly moves with the accreting BH. However any line feature of the inner accretion disk suffers from the broadening of disk's orbital motion and distortions by relativistic effects around the BH, thus making it practically impossible, or difficult at least, for detecting the binary orbital motion of the BH. Orbital motion of double-peaked disk emission lines were observed for NSXB Sco X-1 [110], the BHXBs A0620-003 [111, 112]) and GRS 1124-68 [112]. Unfortunately a significant phase offset of velocity modulation was found from that expected based on the observed orbital motion of the companion, though the velocity semi-amplitude is consistent with the expected mass ratio [112]. Soria et al. (1998) observed the orbital motion of the double-peaked disk emission line He II λ 4686 from GRO J1655-40, and found its velocity modulation phase and semi-amplitude in agreement with the kinematic and dynamical parameters of the system [113]. However one major problem in accurately measuring the orbital motion of the primary from the observed double-peaked emission lines is how to determine reliably the line cen- er, because the lines are typically asymmetric and also variable. We have recently proposed to observe the Doppler shifts of the absorption lines of the accretion disk winds co-rotating with the BH around its companion star [114], since in many XRBs accretion disk winds are ubiquitous and appear to be rather stable when observed (e.g., in Ref.[115]). We verified this method using Chandra and HST high resolution spectroscopic observations of GRO J1655-40 (shown in Figure 24) and LMC X-3. Unfortunately the currently available data only covered small portions of their orbital phases and thus do not allow better constraints to their system parameters. Future more observations of these two sources and other sources with detectable absorption lines from their accretion disk winds will allow reliable and precise measurements of the BH masses and orbital inclination angles in accreting BHXBs. Since the accretion disk has very high density and is ionized near the BH, where the majority of the observed disk emission is produced due to the very deep gravitational potential near the BH, scattering of the primary disk emission is inevitable. The scattered light is polarized and its polarization fraction and position angle depend on the viewing direction (inclination ), scattering optical depth and the radius where the scattering occurs [116-119]. Ignoring many details, it can be shown that the polarization fraction, P ( i ), of the observed disk photons (initial disk emission plus the scattered emission) is given by where A is a constant depending upon the scattering op- tical depth; detailed calculations made by Chandrasekhar [120] gives P (75 · ) = 0 . 04. Note that here the disk photons are from the Raleigh-Jeans part of the multicolor blackbody spectrum with a characteristic shape of f ( ν ) ∝ ν 1 / 3 , i.e., no GR effect is included. We can therefore find the disk inclination angle by measuring the polarization fraction of this part of the disk emission, as shown in Figure 25. At energies above the Raleigh-Jeans part of the multi-color blackbody spectrum, the polarization is strongly effected by both the inclination and BH spin, as shown in Figure 26. The continuum spectra are clearly degenerated for the different combinations of inclination and BH spin, but the polarization fraction and angle as functions of energy can clearly distinguish between them [118]. Therefore X-ray spectra-polarimetry observations of BHXBs will certainly make important progresses in measuring BH spin. Besides using polarization measurements to obtain inner disk inclination (and BH spin), the broad iron Kalpha measurements can also be used to do so [121-123], because iron K-alpha emission is believed to come from the fluorescent emission of the disk as a result of illumination by a hard X-ray source above the disk. Naturally this emission is sensitive to the disk inclination (and BH spin). However, compared to the polarization measurement method, this method is less straight forward and may suffer from systematic uncertainties in deriving the disk inclination (and BH spin), because complicated modeling of the hard X-ray component and line emissivity from the disk is required. Therefore it is essential to use all methods discussed above to measure both inclination angles (orbital plane and inner disk) and BH spin. Studying the relationship between the results obtained with different methods is also important in its own rights, in order to understand accretion disk physics, its interaction with the BH in its center, production of relativistic jets, the origin of the BH spin, and ultimately the formation mechanisms of BHs and BHXB systems. The recent dispute on the distances of GRO J1655-40 and A0620-003 exemplifies the difficulty of determining the distances of BHXBs using mostly absorption lines [108, 109]. We have recently suggested a method of using the delay time between the X-ray fluxes of an XRB and its X-ray scattering halo by interstellar dust to infer its distance [124, 125]. However this method may suffer from our incomplete knowledge of the distribution of interstellar medium. Ideally precise astrometry can determine their distances model-independently, by measuring their trigonometric parallaxes. Recently the distance to Cygnus X-1 was determined reliably and accurately this way (1 . 86 +0 . 12 -0 . 11 kpc; [126]); which is key to the consequent measurement of its BH mass and spin [36, 37]. Currently it remains challenging to measure the trigonometric parallaxes of objects at distances beyond several kpc where most BHXBs are located. Future high precision astrometry missions are expected to improve the distance estimates to these BHXBs significantly. Therefore we expect that the improved measurements discussed above on their BH masses, inner disk inclination, and distances will allow future improvements of BH spin measurement with the CF method.","pages":[17,18,19]},{"title":"E. Possible application of the continuum fitting method to AGNs","content":"More recently, the CF method is also applied to constrain the BH spin in an active galactic nucleus (AGN), which is powered by matter accretion onto the central supermassive BH in its center [127]. However the BH spin inferred this way for an supermassive BH is quite uncertain, because: (1) The peak temperature of the accretion disk inversely increases with the mass of a BH. Therefore, for a supermassive, its temperature would be in the ultraviolet energy range which will be strongly absorbed and hard to observe; (2) The system parameters of a supermassive BH (e.g., the mass of the BH and the inclination angle of the accretion disk) have larger uncertainties; (3) The uncertain mechanism for some components (e.g., the soft X-ray excess) in AGN spectra also increases the difficulty; (4) Some emission and absorption lines may distort the continuum spectrum substantially; and (5) In some cases the contribution of its host galaxy to the observed total continuum spectrum cannot be removed satisfactorily. The polarized continuum of an AGN should be a pure accretion disk continuum, at least in the optical to near infrared band [128]. One possible way to measure the BH spin in an AGN with the CF method is to combine the observed polarized optical to near infrared continuum spectrum with the observed total UV continuum spectrum to get a broad band continuum spectrum of an AGN [129]. In principle the broadened Balmer edge features and the total UV spectrum can be used to constrain the disk inclination angle and fraction of host galaxy contamination, respectively [129]. However the quality of the currently available data is still insufficient to allow accurate determination of BH spin in AGNs with this method. Nevertheless the principal method of measuring BH spin in AGNs should be using the reflected broad iron line and continuum components [121-123]. The main reason is that i can also be determined simultaneously and r in obtained this way is already in units of r g , thus avoiding naturally the uncertainties caused by the BH mass and inclination in the CF method. However cross calibration can be done between the CF and reflection fitting methods if both can be applied to the same supermassive BH.","pages":[19,20]},{"title":"V. FURTHER DEVELOPMENTS ON HOT ACCRETION FLOWS","content":"In this section I briefly summarize some further developments on hot accretion flows, which are believed to be responsible for the PL component of the spectra in BHXBs. The multi-waveband spectra of the hard state of the BHXB XTE J1118+480 was modeled with the TID with hot accretion flow (ADAF) geometry (panel (a) in Figure 27) and TID (with ADAF) plus jet model (panel (b) in Figure 27), with ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D [93]. The steep UV spectrum provides clear evidence for a large truncation radius for the SSD ( r = 600), and the radio to infrared spectrum dominates the jet emission, which also contribute to the hard PL component [93]. Such low accretion rate gives very low SSD luminosity l D ∼ 5 × 10 -4 (since the disk radiation efficiency η ∼ 1 /r ; see Figure 17), which is far below the turn-over luminosity of l D ∼ 10 -2 shown in Figure 20. Therefore the inferred truncation radius of XTE J1118+480 agrees with the extrapolation of data points of XTE J1817-330 down to very low disk luminosity. Large truncation radii are also reported from several other sources in the hard state (e.g., in Ref.[130]). However, the exact values of these truncation radii may have large uncertainties, since no direct detection of the inner disk peak emission was available, unlike the strong case of XTE J1118+480 [93]. For this reason I did not include these reported values in Figure 20. Panel (b) in Figure 27 also shows that the radiation from both the hot accretion flow and the jet contribute to the X-ray emission. However the former is roughly proportional to ˙ m 2 , whereas the latter to ˙ m . Thus with the decrease of ˙ m , the contribution from the jet becomes more and more important, thus the X-ray radiation will be dominated by the jet (when l glyph[lessorsimilar] 10 -5 -10 -6 ), as shown in Figure 28 [93]. The observational data of very low-luminosity AGNs clearly show a correlation between radio and X-ray with a correlation index of ∼ 1 . 2 [131], in excellent agreement with the prediction shown in Figure 28 [93]. It is well known that the highest luminosity an ADAF can produce is only about 3% L Edd . However the observed highest luminosity a hard state can reach can be 10% L Edd or even higher, which can be described by LHAF shown in Figure 29, where several previously known solutions of accretion flows are all unified in a single scheme [133]. The hard state spectrum of XTE J1550-564 with L ∼ 6% L Edd is explained by LHAF very well, including the X-ray spectral slope and the value of the cutoff energy [134].","pages":[20]},{"title":"VI. FURTHER DEVELOPMENTS ON CORONA FORMATION","content":"While the formation of the SSD in a BHXB is reasonably well understood, the formation of the corona, which is the hot accretion flow discussed in Section V, remains less understood. Over the last more than 10 years, Liu and her collaborators have developed a model to explain the formation and evolution of the corona in a BHXB or AGN, in which the mass accretion rate ˙ m in units of the Eddington ratio drives the variations of the complex accretion flows by the interaction between the cold SSD and the hot corona [60, 135-139]. Specifically, the coupling between the hot corona and the cold disk leads to mass exchange between them. The gas in the thin disk is heated up and evaporates into the corona as a consequence of thermal conduction from the hot corona, or the corona gas condenses into the disk as a result of overcooling by, for example, external inverse Compton scattering. If ˙ m is low, evaporation occurs and can completely remove the thin disk, leaving only the hot corona in the inner region and a truncated thin disk in outer region; this provides a mechanism for ADAF at low ˙ m . If ˙ m is high, the gas in the corona partially condenses to the disk due to strong Compton cooling, resulting in disk dominant accretion. The model naturally explains the different structures of accretion flow in different spectral states as shown in Figure 30 [60, 135, 137-144]. The hysteresis observed in spectral state transitions can also be explained by different irradiations from different evolution history under the same scenario [145-147]. Figure 30 is significantly different from the illustration of accretion flow structures in different spectral state in Figure 12 on three aspects: (1) At intermediate ˙ m , the SSD is broken by ADAF into two parts, an outer disk and an inner disk; (2) The inner disk boundary here is always located very close to the BH, except in the very low ˙ m hard or quiescent state, which is very different from the TID scenario depicted in Figure 12; (3) The corona here covers essentially the whole accretion disk, especially the inner disk region, whereas in Figure 12 the corona is mostly located inward from the inner disk boundary. The observed soft X-ray component in the low/hard state can be explained by the existence of the inner disk [139, 140, 142, 144]. As I have discussed above, the inner disk boundary radius inferred with the CF method in the presence of a strong PL component is actually consistent with that in the soft state when the PL component is weak, after taking into account the Compton scattering in the corona [65, 87]. Actually the essential assumption behind the broad iron line/reflection fitting method of determining BH spin is that the inner disk boundary is at the ISCO when the PL component is strong. All these tend to support the existence of the inner disk at intermediate ˙ m . However it remains to be demonstrated that if the whole SSD is indeed broken into the two parts at intermediate ˙ m , as illustrated in Figure 30. Recently it has been suggested that the ADAF and corona shown in Figure 30 may have clumpy structures, as shown in Figure 31 [148]. The 'clumpy' model (Figure 31) has mainly two different consequences from the 'uniform' model (Figure 30): (1) The inner disk is transient in the 'clumpy' model; (2) The 'clumpy' model can explain the variabilities observed in X-ray binaries (such as the state transitions discussed in Section VII) and radio-loud AGNs (such as BL Lac objects). Determining the structure of the corona in an BHXB observationally remains difficult; the fundamental issue on whether the corona covers mostly the accretion disk or the central compact object still remains unclear so far. Since the observed similar spectral and state transitions between some NS and BH XRBs are quite similar [149], it is reasonable to assume that they have similar coronae. Recently we have used type I X-ray bursts from low-mass NSXBs to show that X-ray bursts experience negligible Comptonization and that the corona cools rapidly during the rising phase of X-ray bursts and is then heated up rapidly during the rising phase of X-ray bursts, as shown in the upper panel of Figure 32 for IGR J1747721 [150]. These results suggest that the corona cannot cover the central compact object completely (lower panel of Figure 32) and that the destruction and formation time scales of the corona are as short as seconds; such short time scales are quite difficult to understand in the above discussed evaporation model, in which the time scales are related to the viscous time scales of the accretion disk. However, this short time scale is consistent with a corona produced by magnetic reconnections in the accretion disk, in a similar way to the solar corona heating [151]; this conclusion was based on the inferred accretion flow structure of a BHXB shown in panel (A) of Figure 33, in comparison with the atmospheric structure of the Sun shown in panel (B) of Figure 33. It is worth noticing that the purported coronae in both XRBs and AGNs have strikingly similar properties, e.g., they all have electron temperatures of the order of hun- dreds keV, in spite that the temperatures, inner disk radii and variability time scales of their cold accretion disks all scale with their BH masses (and accretion rates) as predicted in the SSD model. This means that their coronae are scale independent. It is perhaps not coincidental that the electrons' velocities in a corona are approximately the same as the Keplerian orbital velocities of the inner disk, which are also roughly the same as the launching velocities of jets. Of course these velocities are also the varialized velocities of the central BHs. It is plausible that turbulent small scale magnetic fields lifts the plasma in the 'warm layer' shown in Figure 33 to form the corona, which is then launched into the jets by the rotating large scale magnetic fields through either the BP or BZ mechanism; during these processes the magnetic fields are mostly responsible for changing the directions of motion of plasmas by the Lorentz force. If the corona is clumped as discussed in Section VI [148], then the plasma channeled into the jets should be clumpy and thus the ejected jets should be episodic and have knotted structures, in agreement with observations; this scenario is illustrated in Figure 34. Alternatively episodic ejections may also occur in the disk, similar to the coronal mass ejection on the Sun [152]. Interestingly, the soft X-ray excess (SXE) frequently observed in AGNs can be interpreted as the warm layer found in XRBs, since the inferred plasmas parameters, with electron temperature of 0.1 to 0.3 keV and Compton scattering optical depth of around 10, are similar to that of XRBs and universal among AGNs with different BHmasses, but seem to be related only with ˙ m [153-155]. Therefore their warm layers are also scale independent. Indeed, magnetic flares, perhaps caused by magnetic reconnections in the cold disk, can produce the warm layers in both XRBs and AGNs [156, 157].","pages":[20,21,22,23]},{"title":"VII. FURTHER DEVELOPMENTS ON STATE TRANSITIONS","content":"To further understand the mechanism of the spectral state transitions, it is important to study those spectral state transitions in individual BHXBs during different outbursts. The advantages are obvious - uncertainties in our estimates of black hole mass, the binary properties such as the orbital period, or the source distance will not play any role in producing the observed diverse transi- tion properties in individual BHXBs, which can only be driven by the accretion process under different initial conditions. The sources which firstly allowed such a study were the BH transients GX 339-4 and XTE J1550-564, the NS transient Aquila X-1, and the flaring NS low-mass XRB (LMXB) 4U 1705-44, in which a remarkable correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the following soft state was found [158-160]. More recently, a comprehensive study of the hard-to-soft spectral state transitions detected in all the bright XRBs in a period of about five years with simultaneous X-ray monitoring observations with the RXTE/ASM and the Swift/BAT confirmed the correlation between the hard-to-soft transition and the peak luminosity of the following soft state [161, 162], as shown in the upper panel of Figure 35 [161]. More important was the discovery of the correlation between the transition luminosity and the rate-of-change of the luminosity during the rising phase of an outburst of transients or a flare of persistent sources, as shown in the lower panel of Figure 35 [161]. The above correlation implies that in most cases it is the rate-of-increase of the mass accretion rate, rather than the mass accretion rate itself, determines the hardto-soft spectral transition; this is depicted in Figure 36 [161]. In addition, the discovery of the relation between the hard X-ray peak flux and the waiting time of transient outburst in the BH transient GX 339-4 (as shown in Figure 37) supports that the total mass in the disk determines the peak soft state luminosity of the following outburst [159]. Therefore spectral states should be understood in the non-stationary accretion regime, which would be described by both ˙ m and m , as well as the total mass in the disk before an outburst.","pages":[23]},{"title":"VIII. FURTHER DEVELOPMENTS ON THERMAL STABILITY OF SSD","content":"The SSD model predicts that when the accretion rate is over a small fraction of the Eddington rate, which corresponds to L glyph[greaterorsimilar] 0 . 06 L Edd , the inner region of the disk is radiation-pressure-dominated and then both secularly [163] thermally unstable [164, 165]. However, observations of the high/soft state of black hole X-ray binaries with luminosity well within this regime (0 . 01 L Edd glyph[lessorsimilar] L glyph[lessorsimilar] 0 . 5 L Edd ) indicate that the disk has very little variability, i.e., quite stable [166]. It has been well established that the accretion flow in this state is described by the SSD model [4, 167]. Radiation magnetohydrodynamic simulations of a vertically stratified shearing box have confirmed the absence of the thermal instability [168]. Recently, the thermal stability is revisited by linear analysis, by taking into account the role of magnetic field in the accretion flow [169]. By assuming that the field responds negatively to a positive temperature perturbation, it was found that the threshold of accretion rate above which the disk becomes thermally unstable increases signifi- cantly, compared with the case of not considering the role of magnetic field. This accounts for the stability of the observed sources with high luminosities, as shown in Figure 38. If the magnetic pressure is less than about 24% of the total pressure, then this model can explain the 'heart-beat' limit-cycle instability observed in GRS 1915+105 at its highest luminosity; this peculiar source holds the highest accretion rate (or luminosity) among BHXBs. Observations of GRS 1915+105 showed that t high (the duration of the outburst phase) is comparable to t low (duration of the quiescent phase) and L high is 3 to 20 times larger than L low [170]. However, numerical calculations showed that t high is less than 5 percent of t low , and L high is around two orders of magnitude larger than L low . Some efforts have been made to improve the theory in order to explain observations either by some artificial viscosity prescription [171] or by additional assumption of the energy exchange between the disk and corona [172]. Taking into account the stress evolution process, it was found that the growth rate of thermally unstable modes can decrease significantly owing to the stress delay, which may help to understand the 'heart-beat' limit-cycle variability of GRS 1915+105 [173]. The limit-cycle properties are found to be dominated by the mass-supply rate (accre- · · tion rate at the outer boundary) and the value of the α -viscosity parameter in the SSD model that assumes that the viscous torque is proportional to the total pressure [77]. It was also found that only the maximal outburst luminosity (in Eddington units) is positively correlated with the spin of a BH, providing another way to probe BH spin [77]; this is mainly due to the smaller inner disk radius and thus higher radiative efficiency for larger a ∗ as shown in Figures 15, 16 and 17.","pages":[23,24,25]},{"title":"IX. UNIFICATION AND OUTLOOK","content":"Despite of the many progresses made over the last decades on the study of BHXBs and microquasars, there are still many outstanding and unresolved issues, which can be pierced by two big pictures. One big picture is related to the astrophysics of BHXBs and microquasars, which is centered on understanding the accretion-outflow (wind or jet) connections at different accretion rates for different types of astrophysical systems involving BHs. The other big picture is related to the fundamental physics of BHXBs and microquasars, which is centered on identifying astrophysical BHs and testing theories of strong gravity. These two pictures are actually also entangled together, because a strong gravity theory, such as GR, is needed to describe the astrophysical aspects of these systems, and we need to understand the astrophysics of BHXBs and microquasars before we can start to test any strong gravity theory[174]. Figure 39 shows the relation between the collimationcorrected gamma-ray luminosity and the kinetic power for AGNs and gamma-ray bursts (GRBs)[175]. For at least some of GRBs, a super-Eddington accreting compact stellar object, probably a BH with around 10 M glyph[circledot] , is believed to be responsible for powering the highly relativistic jets, which produces intense gamma-rays through violent collisions of blobs in the jets[176]. The two types of AGNs shown here, flat-spectrum radio quasars (FS- RQs) and BL Lacs, have all been observed to have mildly relativistic jets. Their central engines are supermassive BHs with around 10 7 -9 M glyph[circledot] with accretion rates just below (for FSRQs) or far below (for BL Lacs) Eddington rate[177]. The good correlation over about 10 orders of magnitudes between these very different systems with very different accretion rates suggests that there must be some common mechanisms responsible for the accretionoutflow (wind/jet) connections for all accreting BHs[178]. Putting together all related phenomenologies and some theoretical modeling of BHXBs and microquasars discussed in this article, an unification scheme is illustrated in Figure 40 with the major elements in the accretionoutflow connections in different types of astrophysical systems harboring both stellar mass BHs and supermassive BHs with accretion rates over several orders of magnitudes. The types of accreting stellar mass BHs include BHXBs and microquasars in different spectral states, as well as ultra-luminous X-ray sources (ULXs) (some of which are most likely ultra-Eddington accreting BHs[180]), and GRBs. The types of accreting supermassive mass BHs include low-luminosity AGNs (LLAGNs), BL Lac objects, normal radio quiet quasars (RQQs), FSRQs and broad-line-less luminous quasars (BL3Qs) [179] (I create the acronym 'BL3Q' here just for fun). This scheme is not a theoretical model, or even toy model yet. It, however, can be used as one possible chain to pierce many observed phenomenologies together, providing a possible frame work for further theoretical developments on accretion-outflow connections. Future observations will scrutinize this unification scheme and","pages":[25]},{"title":"Some necessary elaborations and points:","content":"revise it inevitably. Of particular importance is the reliable determinations of the mass and spin of accreting BHs. The BH mass allows us to determine the accretion rate in units of Eddington rate, i.e., ˙ m , which is a key parameter to the unification scheme of accretion-outflow connections. The BH spin is of course another key parameter, because it can determine the radiative efficiency of the disk and jet power in the BZ mechanism. Once we have well determined BH parameters and a consistent and predictable theory describing the observed accretion-outflow connections, we are then ready to study the fundamental physics of BHs with BHXBs and microquasars, such as the properties of event horizon, spacetime around Schwarzschild and Kerr BHs, BH spin energy (and mass) extraction. For example, both the broad iron line and CF fitting methods can be used to measure a BH's spin. However the two methods are equivalent only if the metric is accurately described with Kerr metric. Different spin measurements for a BH with the two different methods would invalidate Kerr metric, provided that we understand the accretion disk physics thoroughly. On the other hand, a correct metric is of course required to describe accurately the accretion flow around BHs. This is one example of the entanglement or interplay between astrophysics and fundamental physics of accreting BHs. The stake is high, but the job is difficult. Acknowledgment. I appreciate inputs from Profs. Lijun Gou, Weimin Gu, Lixin Li, Bifang Liu, Dingxiong Wang, Wenfei Yu, Feng Yuan, and Shu Zhang. The editors of this book are thanked for inviting me to write this article, as well as their patience, persistency, proof reading it, and offering comments and suggestions to improve it in the end. This work is partially supported with funding by the 973 Program of China under grant 2009CB824800, by the National Natural Science Foundation of China under grant Nos. 11133002 and 10725313, and by the Qianren start-up grant 292012312D1117210. Orford, J L Osborne, M Ouchrif, M Panter, G Pelletier, S Pita, G Puhlhofer, M Punch, B C Raubenheimer, M Raue, J Raux, S M Rayner, A Reimer, O Reimer, J Ripken, L Rob, L Rolland, G Rowell, V Sahakian, L Saug'e, S Schlenker, R Schlickeiser, C Schuster, U Schwanke, M Siewert, H Sol, D Spangler, R Steenkamp, C Stegmann, J-P Tavernet, R Terrier, C G Th'eoret, M Tluczykont, G Vasileiadis, C Venter, P Vincent, H J Volk, and S J Wagner. Discovery of very high energy gamma rays associated with an x-ray binary. Science (New York, N.Y.) , 309(5735):746-9, July 2005. Flow above a Thin Disk. The Astrophysical Journal , 575(1):117-126, August 2002.","pages":[26,27,28,32]}],"string":"[\n {\n \"title\": \"Black Hole Binaries and Microquasars\",\n \"content\": \"Shuang-Nan Zhang 1 , 2 ∗ 1 Laboratory for Particle Astrophysics, Institute of High Energy Physics,Beijing 100049, China. 2 National Astronomical Observatories, Chinese Academy Of Sciences, Beijing 100012, China. (Dated: October 31, 2018) This is a general review on the observations and physics of black hole X-ray binaries and microquasars, with the emphasize on recent developments in the high energy regime. The focus is put on understanding the accretion flows and measuring the parameters of black holes in them. It includes mainly two parts: (1) Brief review of several recent review article on this subject; (2) Further development on several topics, including black hole spin measurements, hot accretion flows, corona formation, state transitions and thermal stability of standard think disk. This is thus not a regular bottom-up approach, which I feel not necessary at this stage. Major effort is made in making and incorporating from many sources useful plots and illustrations, in order to make this article more comprehensible to non-expert readers. In the end I attempt to make a unification scheme on the accretion-outflow (wind/jet) connections of all types of accreting BHs of all accretion rates and all BH mass scales, and finally provide a brief outlook.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"CONTENTS\",\n \"content\": \"VIII. Further developments on thermal stability of SSD 23\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"I. SYNOPSIS\",\n \"content\": \"I will start by defining what I mean by black hole binaries (BHBs) and microquasars in this article. I decide to restrict myself to only a subclass of BHBs, namely, BH X-ray binaries (BHXBs), since these are the only class of BHBs known observationally. I will then simply refer microquasars as BHXBs for reasons discussed in Section II. Since many excellent, comprehensive and quite up-todate review articles on BHXBs are readily available in literature, I feel it is not necessary to write another bottomup and comprehensive review article on the same subjects at this stage. I will thus take quite an unusual approach in this article. I will first give some concise guides on several representative review articles [1-4], with some necessary updates. I will then focus on the further developments on several topics I feel deserve more discussions, i.e., BH spin measurements (Section IV), hot accretion flows (Section V), corona formation (Section VI), state transitions (Section VII) and thermal stability of SSD (Section VIII). The emphasis is thus put on understanding the accretion flows and measuring the parameters of black holes in them. Some rather general issues on BH astrophysics, such as what astrophysical BHs are and how to identify them observationally, are not discussed here but can be found from my recent book chapter entitled 'Astrophysical Black Holes in the Physical Universe' [5]. The usual practice of writing a review article is to end by listing some outstanding issues and major unsolved problems, and then to propose some possible approaches to them. I initially did not do this in the first draft. The history of astronomy tells us that major progress is almost always made by unexpected discoveries and research results; unpredictability is an essential nature of astronomy. This article is not intended to be read by funding agencies or proposal reviewers, so I thought I did not have to do it. In astronomy, knowing what has happened, but looking and doing it differently are far more important and effective than following other people's advises. However, the editors of this book suggested me to write a brief 'outlook' in the end. I thus did it nevertheless.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"II. ACRONYMS AND TERMINOLOGY\",\n \"content\": \"In Table I, I list all acronyms used in this article; most of these are quite commonly used in this community. A BH binary (BHB) is a gravitationally bound binary system in which one of the objects is a stellar mass BH with mass from several to tens of solar masses ( M glyph[circledot] ); the other object, i.e. its companion, can be either a normal star, a white dwarf, or a neutron star (NS). In case a binary system consists of two BHs, it is is referred to as a binary BH system, which is not covered in this article. When the companion in a BHB is a normal star, the gas from the star may be accreted to the BH and X-rays are produced, as a consequence of the heating by converting the gravitational potential energy into the kinetic energy of the gas, and a BH X-ray binary (BHXB) is referred to as such a binary system, as shown in Figure 1. The possible existence of BHXBs was first suggested by Zel'dovich & Novikov [6, 7]. The first BHXB found is Cygnus X-1 [8], now a well-studied system among many others found subsequently in the Milky Way and nearby galaxies. The terminology of microquasar has some twisting in it. Historically it was first referred to the BHXB 1E1740.7 -2942 in the Galactic center region, because a double-sided jet was detected from it, mimicking some quasars with similar radio lobes, which have much larger scales [9]. Soon after, superluminal jets are observed from a BHXB GRS 1915+105 [10], which is also referred to as a microquasar. Nevertheless both BHXBs are quite unusual compared to many others, thus microquasars were considered quite unusual. However it became clear that microquasars may be quite common among BHXBs, since the discovery of a normal BHXB GRO J1655 -40 [11], whose superluminal jets were observed [12] and showed some correlations with its X-ray emission [13]. Subsequently several more BHXBs have been observed with superluminal jets. At this stage microquasars began to be referred to as BHXBs with relativistic jets (with bulk motion of about or larger than 90% of the speed of light), to distinguish them from NS X-ray binaries (NSXBs) that only have mildly relativistic jets (with bulk motion of about or smaller than 50% of the speed of light) [14]. However the discovery of relativistic jets from a NSXB Circinus X-1 made the situation complicated: relativistic jets are no longer uniquely linked to BHs [15]. Now in retrospect, a microquasar can be literally and easily understood as the micro version of a quasar; however a quasar may or may not be observed with collimated jets. A quasar has been already understood as a special galaxy centered by an actively accreting supermassive BH with a mass from millions to billions of M glyph[circledot] , and thus its total light output is dominated by the BH's accretion process, in a similar way as in BHXBs. The production or lack of relativistic jets may have similar or even the same underlying physical mechanisms in BHXBs and AGNs, though their surrounding environments may modify their observed morphologies [16]. It is therefore more natural to simply refer microquasars as BHXBs; in the rest of this article, microquasar and BHXB are used interchangeably. We therefore will focus on BHXBs and thus will not discuss binary systems producing gamma-rays and sometimes radio jets, which are most likely high-mass NSXBs and in which jets or pulsars' winds interact with the wind of its high-mass companion to produce the observed gamma-rays [17-23]. Such systems show very different observational characteristics, e.g. the long (1667 days) super-orbital modulation with phase offset (about 280 days) between its X-ray and radio light curves found in LS I+61 · 303 [24].\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"III. REVIEW OF REVIEWS\",\n \"content\": \"Here I attempt to review the four recent review articles [1-4] I consider most useful to readers. Additional information and updates are provided when necessary. Some overlaps exists between these review articles, as expected and inevitable in bottom-up review articles. To avoid repetitions as much as possible in this article, I thus put different emphasizes on different articles, with of course my personal tastes and perceptions.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"A. The most recent Science Collection\",\n \"content\": \"The recent collection of perspectives and reviews in the Science magazine provides excellent introductions to and concise summaries of the current state of our understanding of BH physics and astrophysics [1, 25-27]. To the subjects of this article, the most relevant article in this collection is the one by Fender and Belloni enti- tled 'Stellar-Mass Black Holes and Ultraluminous X-ray Sources', which is focused on the observational characteristics of BHXBs [1]. In particular, the article provides an excellent description of the general picture of spectral evolutions of BHXBs with the Hardness-IntensityDiagram (HID), which is found to be well correlated with the observed jets from BHXBs, as shown in Figure 2. Actually this cycle is also well tracked by their flux variability, represented by the measured root-mean-squares (rms) above its average flux, as shown by the RMSIntensity-Diagram (RID) [28] of the BHXB GX 339-4 in Figure 3, which also shows additional horizontal tracks at intermediate intensities. Sometimes, a full HID cycle does not go into the soft state at all (Figure 4), perhaps due to a failed outburst [29]. The basic scenario is as follows. During the initial stage of an X-ray outburst of a BHXB (A → B), which is triggered by a sudden increase of accretion rate onto the BH, its spectrum is normally hard and steady jets are always observed. After reaching its peak luminosity, its spectrum begins to soften in a chaotic way and transient jets are normally observed (B → C → D). After this transition, the system calms down with a soft spectrum and no jets are present (D → E). Finally the system returns to its quiescent state with a hard spectrum accompanied with the reappearance of jets (E → F). Throughout this cycle, the presence of hot accretion disk winds appears to be anti-correlated with its spectral hardness and jet ejection. This empirical pattern appears to be quite universal for all BHXBs with very few exceptions, though the underlying physics is still not well understood yet. Nevertheless putting together the above scenario is a very significant progress in this field over the last ten years. Due to the conciseness of this article, some other important subjects on BHXBs are not fully discussed and many original references are also missing. Other articles in this collection [25-27] are less relevant to the subjects of this article, but are still quite interesting to read, within the context of BH astrophysics and physics. The only other type of astrophysical BHs known to exist in the physical universe are supermassive BHs in the center of almost each galaxy. Volonteri [27] concisely summarized our current understanding on how they are formed and grow over the cosmic time; merging of two BHs is a key process here. Thorne [25] focused on what happens when BHs merge together to produce gravitational waves, which might be used as a new laboratory for studying gravitational physics and a new window for exploring the universe. Witten [26] then explained the quantum properties of BHs, in particular the basic ideas behind Hawking radiation, which might not be important at all for astrophysical BHs. Nevertheless the understanding gained through studying the quantum mechanics of BHs actually plays very important roles in developing other theories of physics, such as that in heavy ion collisions and high-temperature superconductors, as vividly described by Witten [26].\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"B. The most recent ARA&A article on BHXBs, with some updates\",\n \"content\": \"The most recent ARA&A article on BHXBs by Remillard and McClintock entitled 'X-Ray Properties of Black-Hole Binaries' [2] (referred to as RM06 hereafter) provides the most complete, comprehensive and accurate review of BHXBs, which actually covers subjects much beyond just the X-ray properties of BHXBs. The Introduction of RM06 highlights the initial theoretical and observational studies of BHXBs, followed by brief comments on several main review articles on BHXBs preceding this one and the basic properties of BHs within the context of general relativity (GR). All main properties of BHXBs known at the time are summarized in the first table and figure there. In Table II, I compile the most updated data on all BHXBs currently known, including three BHXBs outside the Galaxy and not in the Large Magellanic Cloud (LMC); the currently available spin measurements for these BHs are also included for completeness. In Table II, where P orb is the orbital period, K C is the semiamplitude of the velocity curve of the companion star, M BH is BH mass, i is the the orbital inclination angle, and q ≡ M C /M BH is the mass ratio. In Figure 5, I show the updated graphical representation of most of the BHXBs listed in Table II. The X-ray properties of BHXBs are characterized by their X-ray light curves, timing and spectra. Essentially all known BHXBs were discovered initially as bright Xray sources, and the majority of them were detected as transient X-ray sources with X-ray all-sky monitors. The transient properties of some of BHXBs can be interpreted by the disk instability model (DIM) (see Ref.[49] and references therein), which assumes a constant mass transfer rate from the mass donor to the accretion disk. However the accretion rate from the disk to the compact object, i.e. a white dwarf (WD), a neutron star (NS), or a BH, is normally lower than the mass supply rate in the disk, so mass is accumulated in the disk. When the accumulated mass exceeds a certain critical value, a sudden increase of accretion rate results in a nova-like outburst. DIM is most successful in explaining an outburst with fast rise and exponential decay (FRED). However as shown in RM06, many observed X-ray light curves of BHXBs a : Reported in the first paper on systematic BH spin measurements [47]; b : Reported in the most recent literature; c: Postulated to be extreme retrograde Kerr BH, due to the lack of the thermal disk component above 2 keV (GRO J1719-24 also belongs to this class) [47]; d : Postulated to be non-spinning BH, due to the observed low disk temperature; e : BH mass of 7 M glyph[circledot] assumed [47]; f : Based on the inner disk radius decrease by a factor of two from the hard state to the soft state transition [47]. are far more complicated than just FRED and their recurrent time scales are also not compatible with DIM. Perhaps disk truncation (a subject to be discussed extensively more later) and mass transfer instability are additional ingredients [50]. X-ray emission from a BHXB is variable at all time scales, from the free-fall or Keplerian orbital time scale of milliseconds near the BH, to the various oscillations (some are even related to GR) in the disk with time scales from milliseconds to minutes, to the viscous time scales of minutes, and to the various instabilities of different time scales. Therefore timing studies of BHXBs can probe the geometry and dynamics in BHXBs. However, lack of coherent signals, such as that observed from pulsars, makes it difficult to unambiguously identify the underlying mechanisms from the detected X-ray variabilities. Nevertheless power density spectra (PDS) and rms still allow us to make progress in understanding the general characteristics of a BHXB, in particular when combined with its spectral behaviors, as already discussed briefly above. The most successful understanding of BHXBs so far is the description of their thermal X-ray spectral component by the classical Shakura-Sunyaev Disk (SSD) model [51]. After applying GR to SSD, the temperature distribution in the disk can be obtained [52]. Applying this model to BHXBs, one can even measure the spin of their BHs [47], by assuming that the inner accretion disk boundary is the inner-most stable circular orbit (ISCO) of the BH (a subject to be discussed more in Section IV). However a power-law (PL) component is almost ubiquitous in the spectra of BHXBs. The interplay between these two components results in various spectral states, which are also found to be well correlated with their timing properties. Historically these spectral states have been named in many different ways, reflecting mostly how they were identified with the observations available at the times. In RM06, three states are defined, which show distinctively different spectral and timing behaviors, as shown in Figure 6 based on the RXTE data on the BHXB GRO J1655-40. The thermal state has its X-ray spectrum dominated by the thermal disk component and very little variability. The hard state has its X-ray spectrum dominated by the a PL component and strong variability. The steep power-law (SPL) state is almost a combination of the above two states, but the PL is steeper. I will keep using these definitions throughout this article for consis- tency. Please refer to Table 2 in RM06 for quantitative descriptions of these three states. When a BHXB is in quiescence, i.e. not in an outburst, its spectral shape is similar to the hard state spectrum. Referring back to Figure 2, RM06 found that the HID can be reasonably well understood in terms of transitions between the above three states; I summarize this in Table III. After examining the extensive RXTE data on spectral evolutions of six BHXBs, the above conclusion seems to be valid generally. Thus a coherent picture seems to emerge: jets are produced only when the PL component in the spectrum is strong and winds are present only when the PL component disappears; jets and winds appear to be mutually exclusive. This suggests that transient winds should appear in the SPL state, anti-phased with transient jets, if winds quench jets as suggested recently [53, 54]. The robust link between the PL component and jets provides an important clue to the jet production mechanism. The PL component is believed to be produced in an optically thin and geometrically thick corona, in which hot electrons up-scatter the thermal photons from the disk. As discussed in RM06, the inner region of the disk appears to be truncated quite far away from the BH in the hard state; this should be the primary reason for the low luminosity of this state, as evidence for the existence of BHs in BHXBs. However it is still not understood how the corona is formed; I will discuss this later in Section VI. In the final part, RM06 discussed the exciting possibilities of using BHXBs as probes of strong gravity, including confirming these BHXBs contain true BHs, measuring the spins of BHs, relating BH spin to the Penrose process and other phenomena, and finally carrying out tests of the Kerr Metric. They also discussed in length how to measure the spins of BHs and commented the various methods of doing so. In Section IV I will re-visit the BH spin measurements in details. Lacking of a solid surface, a BHXB is not expected to produce and actually never observed to have coherent pulses. However quasi-periodic oscillations (QPOs) are frequently detected in their X-ray light curves; a QPO is defined as a 'bump' feature in the PDS, if Q = ν/ ∆ ν > 2, where ν and ∆ ν are the peak frequency and FWHM of the bump, respectively. The observed QPOs are further divided into low frequency (LF) (0.1-30 Hz) and high frequency (HF) ( > 30 Hz) QPOs. Typical LFQPOs can be found as the peak around several Hz in the top and bottom PDS of Figure 6. In literature, LFQPOs are further divided into several subclasses and their underlying mechanisms are far from clear at this stage, though many models have been proposed and briefly discussed in RM06. HFQPOs have been detected from several sources at 40-450 Hz. Of particular interests are their stable nature when detected, which may be linked to either the mass or spin or both of the BH in a BHXB. In some cases a 3:2 frequency ratio is found. Although the 3:2 HFQPO pairs could be interpreted in some epicyclic resonance models [55, 56], there remain serious uncertainties as to whether epicyclic resonance could overcome the severe damping forces and emit Xrays with sufficient amplitude and coherence to produce the HFQPOs. A revised model applies epicyclic resonances to the magnetic coupling (MC) of a BH's accretion disk to interpret the HFQPOs [57]. This model naturally explains the association of the 3:2 HFQPO pairs with the steep power-law states and finds that the severe damping can be overcome by transferring energy and angular momentum from a spinning BH to the inner disk in the MC process.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7\n ]\n },\n {\n \"title\": \"C. Two comprehensive and long articles on modeling accretion flows in BHXBs\",\n \"content\": \"Page limitations to the above review articles did not allow in depth discussions on the detailed processes and models of accretion flows in BHXBs, which are responsible for the above described energy spectra, PDS, state transitions, jets, and wind. Here I introduce two comprehensive and long articles just on this, by Done [3] and Done, Gierli'nski & Kubota [4].\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"1. A beginner's guide\",\n \"content\": \"The first is intended to readers who just start research in this field [3]. It started with the basic tools in plotting spectra and variability, and then described the basic ideas and methods used to infer the inner accretion disk radius when a BHXB is in a thermal state, in order to measure the BH spin [47]. A useful introduction is made on how to make various corrections to account for various effects, including color correction, special and general relativistic effects, starting from the original work [47]. Several commonly use fitting models, i.e. DISKBB , BHSPEC and KERRBB , in the XSPEC package are also briefly introduced. An example is given to demonstrate that the expected relation L disk ∝ T 4 in [47] agrees with the data from the BHXB GX 339-4 [58], where L disk and T in are the disk's total luminosity and temperature at the inner disk boundary, respectively. The hard state is then briefly touched upon, using the Advection Dominated Accretion Flow (ADAF) model [59]. ADAF naturally explains the hot corona required by the observed PL component, especially at very low accretion rate. Evaporation at low accretion rate in the inner disk region has been proposed as the mechanism producing a geometry with a radially truncated disk and a hot inner flow; the latter might be the ADAF [60]. This also means that when a significant PL component is present in the spectrum, BH spin measurement cannot be done with the inferred inner disk boundary (see, however, counter evidence discussed in Sections IV and VI). This geometry is considered a paradigm that can account for many of the observed diverse phenomena from spectral evolution to timing properties. A brief, yet interesting discussion is given on scaling up the above models to Active Galactic Nuclei (AGNs), which host actively accreting supermassive BHs in the centers of galaxies. Applying the insights learnt from BHXBs on their spectral evolution, changing disktemperature with accretion rate and BH mass, and diskjet connections, one might be able to understand many phenomena beyond the simple AGN unification scheme, in which the different observational appearance is all attributed to an viewing angle difference. The continuum emissions of both the disk thermal and PL components are modified by absorptions and added by additional spectral features along the line of sight (LOS). Absorptions in neutral media produce various photoelectric absorption edges, but in ionized media result in both absorption edges and lines. In addition to recovering the original X-ray emission of a BHXB, modeling the absorption features is important in learning the physical properties of the absorption media, such as column density, ionization and velocity along LOS. Winds from BHXBs discussed above are always detected this way. Several XSPEC fitting models for various kinds of absorption edges and lines are also introduced here, i.e. TBABS , ZTHABS , TBVARABS , ZVPHABS , TBNEW , ABSORI and WARMABS . Figure 7 shows the calculated absorption structures with ABSORI and WARMABS ; note that ABSORI does not include absorption lines, but WARMABS does. X-rays interacting with the surrounding medium can ionize it and heat the free electrons up by Compton scattering; hot electrons can also loose energy by interacting with lower energy photons. A Compton temperature of the plasma is reached when the above two processes reaches an equilibrium. The temperature is determined by only the spectral shape of the continuum, so the heated plasma can escape as winds from the disk if the velocity of the ions exceed that of the escape velocity at that radius, which is usually very far away from the inner disk boundary. As the continuum luminosity approaches the Eddington limit, the radiation pressure reduces the escape velocity substantially in the inner disk region such that continuum driven winds can be launched almost everywhere in the disk of a BHXB, forming a radiation driven wind, which is also called thermal wind; alternatively winds may also be driven magnetically in the inner disk region, but this is much less understood yet. In contrast, in an AGN the peak continuum emission is in the UV band, which has a much larger opacity than Compton scattering in neutral or weakly ionized medium during both photoelectric and line absorptions. This means the effective Eddington luminosity is reduced by large factors and UV line driven wind is easily produced at high velocity. This explains the relatively lower velocity (hundreds km/s) and highly ionized winds from BHXBs, but much higher velocity (thousands km/s to 0.2 c ) and weakly ionized winds from AGNs. Material illuminated by X-rays produce both fluorescence lines and reflection features, which depends on both the continuum spectral shape and ionization state of the material, as shown in Figure 8 calculated with the XSPEC ATABLE model REFLIONX.MOD, which includes the self-consistent line and recombination continuum emission. Note that for highly ionized reflection, the Compton heated upper layers of the disk broadens the spectral features; these effects are not included in the simpler XSPEC model PEXRIV . Replacing the stationary slab by a disk around a BH, both the special and general relativistic effects will further smear (broaden) the spectral features, as shown in Figure 9 for the iron line region with different values of ξ , inner disk radius r in , and viewing angle i ; these are calculated with the REFLIONX.MOD and then convolved with KDBLUR in the XSPEC package. Other similar XSPEC fitting models are DISKLINE , LAOR , and KY . The inner disk radius can in principle be determined by modeling the observed broad iron line features, that in turn can be used to measure BH spins, as will be discussed briefly in Section IV. Figure 10 shows all components of a broad band spectrum of a BHXBs, including interstellar absorption. We thus have two ways discussed so far to determine r in of BHXBs by either fitting its thermal continuum or the fluorescent iron line feature. However results are not always consistent, obtained with different methods, even using the same data by different authors. The author believes that r in increases at low accrete rate, to make room for the corona occupying this space. Nevertheless a fairly good summary on the current conflicts and confrontations is made on this issue, which I will discuss further in Section IV.\",\n \"pages\": [\n 7,\n 8,\n 9\n ]\n },\n {\n \"title\": \"2. An expert's handbook\",\n \"content\": \"The second one can be considered as a handbook on accretion flows on BHXBs, and really lives up to its subtitle 'Everything you always wanted to know about accretion but were afraid to ask' [4] (DGK07 hereafter). Besides its much longer length of 66 pages, the main difference from the above review articles is that it is focused on confrontations between theories and observations and intends to depict a coherence picture of the accretion physics in BHXBs. In the following I will summarize briefly the main points and conclusions reached in DGK07; those I have reviewed above and will discuss more later will be skipped for brevity. The underlying physics of DIM for triggering the outbursts in BHXBs discussed above is the hydrogenionization instability, which produces the so-called 'S'curve, as shown in Figure 11; irradiation by the inner hot disk to keep the outer disk hot is required to produce the slow flux decays, e.g. the exponential decays, frequently observed in them. The outer disk radius is obviously another key parameter, which is determined by the tidal instability in a binary system. This mechanism can explain why BHXBs with high mass companions are all persistently bright, since their outer disks are always in the upper branch due to the combination of their higher average mass transfer rate and inner disk irradiation. Similarly it also explains the differences and similarities between the light curve properties of neutron star X-ray binaries (NSXBs) and BHXBs, since a NS has a lower mass. It should be noted that the additional surface emission from the NS may also help to maintain the outer disk hot and stay in the upper stable branch [62]. The SSD prescription assumes that the stress is proportional to pressure. Because the gas pressure is P gas ∝ T but radiation pressure is P rad ∝ T 4 , so a small temperature increase causes a large pressure increase when P rad ≥ P gas , and thus large stress increase, which in turn heats the disk even more. The opacity cannot decrease effectively to cool it down, so the disk becomes unstable when L ≥ 0 . 06 L Edd , where L Edd is the Eddington luminosity of a BHXB. However in most BHXBs their disk emissions appear to be stable up to around 0 . 5 L Edd . One way out is to assume that the stress is proportional to √ P rad P gas ∝ T 5 / 2 , so the stress increases slower in the radiation pressure dominated regime. Beyond this the disk should become unstable, as evidenced by the sometimes 'heart-beat' bursts of the super-Eddington BHXB GRS 1915+105; other BHXBs with L max > L Edd (e.g. V404 Cyg and V4641) were not observed with such instabilities, perhaps due to the combination of insufficient observational coverage and sensitivity. However similar 'heart-beat' bursts were also observed re- IGR J17091-3624, which is likely substantially sub-Eddington unless it is located much beyond 20 kpc and/or its BH mass is quite small [63]. Super-Eddington accretion flow (SEAF) can become stable again, if the radiation instability is overcome by an optically thick ADAF, i.e. the slim disk model, in which the trapped photons in the flow is advected inwards, thus balancing the heating generated by viscosity. Strong radiation driven winds can be easily produced; this can happen in BHXBs and NSXBs (e.g. Z-sources). Evidence exists that truncated inner disk (TID) is quite common in SEAF and outflow even dominates over inflow in SEAF [64]. The observed PL component in BHXBs cannot be explained by the SSD-like models and thus requires a hot accretion flow (HAF). At low accretion luminosity (e.g. the quiescent state), the HAF may be the advection dominated accretion flow (ADAF), the convection dominated accretion flow (CDAF), or the advection dominated accretion inflow/outflow solution (ADIOS). At higher luminosity (e.g. the hard state), the original hot and optically thin disk solution (i.e. the SLE solution) is unstable, because the electron heating efficiency by the Coulomb coupling between protons and electrons is too low. The luminous HAF (LHAF), however, has the advection as a heating source to electrons, so the heating efficiency increases and thus electrons can cool the flow more effectively. Outflows can be produced in ADAF/ADIOS; collimated jets can also be produced if magnetic fields are involved, so an accretion flow may even be jet dominated (i.e. JDAF). The interplays between the SSD and HAF may be responsible for the observed different states in BHXBs discussed above. If the PL component is produced by Compton up-scattering, then the combination of the optical depth τ and L h / L s , where L h is the heating power in electrons and L s is the cooling power in seed photons, can describe the observed variety of spectra. For example, the hard state has L h / L s glyph[greatermuch] 1, but the thermal and SPL state have L h / L s ≤ 1. The location of r in is proposed to be closely related to L h / L s , as illustrated in Figure 12. As discussed above, the steeper PL in the thermal and SPL states is mostly non-thermal in nature, so non-thermal Comptonization is required. Detailed comparisons with data suggest thermal Comptonization also cannot be ignored even in the thermal and SPL states. In Figure 12, the hot inner flow (HIF) and patchy corona are responsible for thermal and non-thermal Comptonization, respectively. When a source transits from hard to SPL state, r in decreases, HIF is reduced but the patchy corona becomes dominant. The transition from SPL to soft state is then marked by the disappearance of HIF and significant reduction of the patchy corona. The above scenario obviously depends on two fundamental assumptions: (1) slab-HIF produces the non-thermal PL component; (2) slab-HIF is mostly located between the TID and the BH. Both assumptions are examined exhaustively based on the existing observations and their spectral modeling. Slab-HIF is found to be consistent with essentially all data. An alternative to slab-HIF is that the PL is produced from the jet base and beamed away from the disk; however observations suggest the PL emission is quite isotropic, thus in conflict with this. TID is found also consistent with data when the PL component becomes important, since the observed L disk ∼ T 4 in deviates significantly from a linear relation, suggesting r in is larger when the PL component becomes important. However some studies showed that r in is unchanged in the initial hard state, if the Compton up-scattering process is treated properly in Monte-Carlo simulations to recover the lost disk photons [65]; this issue will be further discussed in Section IV. The above developed HIF/TID model based on the spectral evolutions of BHXBs can also be applied to explain the majority of time variability PDS of BHXBs consistently. The TID acts as a low pass filter as it cannot response effectively to variations in the HIF, so r in controls the low frequency break of the PDS: ν LFB ∼ 0 . 2( r/ 6) -3 / 2 ( m/ 10) -1 Hz, where r = r in /r g and m = M BH /M glyph[circledot] ; this predicts that ν LFB changes from 0.03 to 0.2 Hz as observed during transitions from the hard to SPL and soft states if r decreases from 20 to 6. The HIF/TID model can also explain the observed LFQPO variations with luminosity, if the LFQPOs are some kinds of characteristic frequencies related to r in . To some extend, the shape of the PDS can also be explained by this model. However the observed variability rms ∼ flux linear correlation and log-normal distribution of fluctuations may need additional ingredient, such as the proposed propagation fluctuation model. However I point out that such correlations and distributions are also observed in the light curves of gamma-ray bursts and solar flares [66, 67], which can be produced by the generic self-organized criticality mechanism [66]. Similar rms ∼ flux correlation has also been found for blazars, in agreement with the minijets-in-a-jet statistical model [68]. The spectral and timing properties of weakly magnetized NSXBs are known to have many similarities and differences from BHXBs; this is particularly true for the atoll sources that have similar ranges of L/L Edd to BHXBs. The essential distinction is that a NS has a solid surface, but a BH does not. Observationally the continuum spectra of atoll sources can be modeled as composed of SSD, PL emission, and blackbody emission from the NS surface (or the boundary layer between the TID and NS surface); the former two components are quite similar to BHXBs. The spectral evolutions of these NSXBs are thus driven similarly by the combinations of τ and L h / L s . However the blackbody emission is an additional source of L s , so the PL is not as hard as that in BHXBs and r in variations are less effective in changing L h / L s ; the latter means it is more difficult to find evidence of TID from modeling only the spectral evolutions in NSXBs. On the other hand, the TID/HIF model in NSXBs can produce timing behaviors in the same way as in BHXBs discussed above, consistent with observations; additional timing behaviors, such as coherent X-ray pulsations and kilo-Hz QPOs observed in these NSXBs, are caused by the rapid spins of the hard surfaces of the NSs.\",\n \"pages\": [\n 9,\n 10,\n 11\n ]\n },\n {\n \"title\": \"IV. FURTHER DEVELOPMENTS ON BH SPIN MEASUREMENTS\",\n \"content\": \"A BH predicted in GR can only possess three parameters, namely, mass, spin and electric charge, known as the so-called BH no hair theorem. Even if a BH was born with net electric charge, its electric charge can be rapidly neutralized by attracting the opposite charge around it in any astrophysical setting, because the strength of electromagnetic interaction is many orders of magnitude stronger than that of gravitational interaction. Therefore an astrophysical BH may only have two measurable properties, namely, mass and spin, making BHs the simplest macroscopic objects in the universe. Practically, only Newtonian gravity is needed in measuring the BH mass in a binary system. However, GR is needed in measuring the BH spin. The mass and spin of a BH has different astrophysical meanings. Its mass can be used to address the question of ' How much matter (and energy) has plunged into the BH?'. However its spin can be used to address the question of ' How did the matter (and energy) plunge into the BH?'. This is because matter and energy plunged into a BH can carry angular momentum, which is a vector with respect to the spin axis of the BH. In order to increase the total gravitating mass-energy from M i with zero spin to M f , the added rest-mass must be [69, 70] and its final spin becomes where J is the BH's angular momentum. Clearly we have a ∗ = 1 when M f /M i = 6 1 / 2 ; further accretion simply maintains this state [70]. Therefore the required additional rest-mass to spin a BH from zero to maximum spin is ∆ M glyph[similarequal] 1 . 85 M i = 0 . 75 M f ; this is a lower limit to the accreted mass [71]. Figure 13 shows a ∗ as a function of ∆ M . Ignoring Hawking radiation of a macroscopic BH, the only way to extract the energy of a BH and thus reducing its gravitating mass is by extracting its spin energy. Recently, evidence of BH spin energy extraction to power relativistic jets has been found, from the observed correlation between the maximum radio luminosity and its BH spin of a microquasar [72, 73]; however the average jet power is not correlated with BH spin [74]. This indicates that jets may be produced by both BlandfordPayne (BP) [75] and Blandford-Znajek (BZ) [76] mechanisms; but the BZ mechanism is more powerful and responsible for producing the peak radio luminosity. This provides another possible way to estimate a BH's spin [73], similar to a recent proposal of using the peak luminosity of the disk emission to estimate a BH's spin [77]. However there is so far no independent demonstration of validness of either of the above two new methods, which are thus not discussed further in this section. Figure 14 shows the mass reduction, ∆ m = ( M max -M f ) /M max ( M max and M f are the BH's maximum mass and final mass, respectively), as a function of extracting efficiency glyph[epsilon1] of an extreme Kerr BH. For a 10 M glyph[circledot] BH with glyph[epsilon1] = 1 (∆ m ∼ 0 . 3), the total extracted energy is ∆ mM f c 2 glyph[similarequal] 10 54 erg and its total gravitating mass is reduced to M max / √ 2 [71]; this energy could be sufficient to power gamma-ray bursts (GRBs). It is thus plausible that supercritical accretion onto a newly born BH may spin it up and extract its spin energy to power ultrarelativistic jets; multiple spin-up and spin-down cycles may also happen during one GRB, if the collapsing material is clumpy. As shown in Figure 15, the radius of the ISCO of the BH, R ISCO , is a monotonic function of the BH spin [78], beyond which radius a test particle will plunge into the BH under any perturbation; however, in Newtonian gravity a stable circular orbit can be found at any radius. It is thus reasonable to assume that the accretion disk around a BH terminates at this radius, i.e., r in = R ISCO . Therefore, a ∗ can be inferred if one can measure R ISCO in units of its gravitational radius r g = GM/c 2 . Currently three methods have been proposed to measure the BH spin in BHXBs, and all these methods rely essentially on measuring R ISCO . In case the radiative efficiency ( η ≡ L/ ˙ Mc 2 ) can be measured, a ∗ can also be determined this way, as shown in Figure 16. Actually η is a very simple function of R ISCO , i.e, η ∼ 1 /R ISCO , as shown in Figure 17.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"A. The first black hole spin measurement with X-ray spectral continuum fitting\",\n \"content\": \"In 1997, I and my colleagues proposed a method of measuring the BH spin in BHXBs. It started when we tried to measure the mass of the BH in GRO J1655-40 [11] by measuring r in from its X-ray continuum spectral fitting and assuming the BH is not spinning; we found the mass of the BH is around 4 M glyph[circledot] [79]. Coincidentally just when this paper was going to press, Orosz and Bailyn [80] announced their accurate measurement of the BH mass of GRO J1655-40, which is significantly larger than what we found. A co-author of our paper, Rashid Sunyaev urged me to resolve this apparent discrepancy. I immediately realized that a spinning BH of 7 M glyph[circledot] in GRO J1655-40 would be consistent with the inferred with Xray data, and added a note in proof in the paper suggesting this possibility [79]. This turns out to be the first BH spin measurement on record. I then invited two of my close friends and collaborators, Wei Cui and Wan Chen, to join me to apply this method systematically to other BHXBs; a new main conclusion in this work was that the first microquasar in the Milky Way, GRS 1915+105, also contains a spinning BH [47]. At the time DISKBB was the only available fitting model in the XSPEC package for determining r in from an observed X-ray continuum spectra, if the disk contin- uum is described by the SSD model: L disk = 4 πσr 2 in T 2 in , where T in in DISKBB is the disk temperature at r in (i.e. the peak disk temperature in SSD) and L disk can be calculated from the disk flux (after the correction to absorption), the distance to the source and disk inclination (an issue to be discussed later). To determine the physical inner disk radius from the DISKBB parameter r in , several effects must be considered: (1) electron scattering in the disk modifies the observed X-ray spectrum; (2) the temperature distribution in the disk is not accurately described by the Newtonian gravity as assumed in SSD; (3) the observed temperature distribution is different from the locally emitted one; (4) the observed flux is different from the locally emitted one. The latter three are all due to GR effects [52]. For each of the above effects, we introduced a correction factor, using the best available knowledge at the time. Since then, several improvements have been made to correct for these effects and this continuum fitting (CF) method is now quite mature in making accurate BH spin measurements, given sufficiently high quality X-ray continuum spectral measurements and accurate system parameters of the observed BHXB.\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"B. Further developments and applications of the continuum fitting method\",\n \"content\": \"This CF method of measuring BH spin has since been applied widely to essentially every BHXB with a well measured X-ray continuum spectrum showing a prominent thermal accretion disk component. In particular, this method has been improved and incorporated into the widely used X-ray spectral fitting package XSPEC, e.g., KERRBB [81], BHSPEC [82], and KERRBB2 [35, 37]. Both the KERRBB and BHSPEC are relativistic models, but they have their own drawbacks and advantages (See Ref.[35] for a detailed comparison). KERRBB includes all the relativistic effects, but it requires to fix the spectral hardening factor. In contrast, BHSPEC could calculate the spectral hardening factor on its own; however, it does not include the returning radiation effect, which turns out to be an important factor in BH spin determination in BHXBs. KERRBB2 combines both models by generating the spectral hardening factor table from BHSPEC and using the table as the input for KERRBB . The research group led by Ramesh Narayan of Harvard University, Jeffrey McClintock at Smithsonian Astronomical Observatory (SAO), and Ronald Remillard of Massachusetts Institute of Technology (MIT) [83] has since applied this method and contributed to most of the BH spin measurements available in the community, as shown in Table II. The CF method relies on two fundamental assumptions: (1) The measured r in is uniquely related to R ISCO of the BH. (2) There is no or negligible X-ray radiation from the plunging matter onto the BH beyond R ISCO . The latter has been studied with numerical simulations that include the full physics of the magnetized flow, which predict that a small fraction of the disks total luminosity emanates from the plunging region [84]. However, in the context of BH spin estimation, it has been found that the neglected inner light in the CF method only has a modest effect, i.e., this bias is less than typical observational systematic errors [85, 86]. The first assumption above requires that the measured r in remains stable as a BHXB changes its spectral state and luminosity. However it was noticed that r in measured is usually much smaller, sometimes even smaller than R ISCO of a extreme Kerr BH in a prograde orbit, when the X-ray spectrum contains a significant hard PL component, which is believed to be produced by inverse Compton scattering of the thermal disk photons in a hot corona. We realized that the inferred smaller r in could be due to the lost thermal disk photons in the scattering process. We then investigated this problem and confirmed that the inferred r in can be made consistent with r in inferred from the thermal disk component dominated spectrum, if the scattered photons are recovered properly by doing detailed radiative transfer in the corona [65]; the same conclusion was also reached by the Harvard/SAO/MIT group independently without knowing our much earlier results [87]. Therefore the method of BH spin measurement by X-ray continuum fitting can also be applied to some SPL state with strong PL component. The stable nature of the measured r in is proven with the textbook case of LMC X-3, when its X-ray luminosity varied over more than one order of magnitude observed in nearly two decades with many different X-ray instruments, as shown in Figure 18 [88]. However, when ordered by the observed disk luminosity l D = L disk /L Edd , the measured r in shows a clear increasing trend when l D > 0 . 3, as shown in Figure 19 [88]. The similar trend has also been found in another BHXB GRS 1915+105 [35] and two NSXBs [64, 89]. It was found that r = r in /r g indeed increases physically when l D > 0 . 3, by comparing the evolution of r as a function of l D over a large range for several BHXBs and NSXBs. Using the blackbody surface emissions of the NSs in these NSXBs is critical in evaluating any possible disk thickening due to high luminosity that would block at least part of the NS surface emission, as well as determining the actual NS mass accretion rate, which turns out to be much less than the disk mass accretion rate; this suggests that the increased radiation pressure is responsible for the increase of r and significant outflow when l D > 0 . 3 [64]. The same trend is much more pronounced in the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2, as shown in Figure 20 together with the data from other BHXBs and NSXBs [90]. However the exact value of r obtained this way should be taken with caution, since the non-negligible energy advection at high accretion rate can modify the disk structure in non-trivial ways, thus making the SSD prescription inaccurate in this case [91]. Figure 20 also shows that as l D decreases, r again starts to increases. However r increases at higher l D and with a different slope for a NSXB than for a BHXB, which can be naturally explained as due to the 'propeller' effect of the interaction between the NS's magnetosphere and its accretion disk [92] and the 'no-hair' of the BH [90]. Figure 21 shows the radiative efficiencies of various systems; BHXBs may have either higher or lower efficiencies than NSXBs, because a BH has neither solid surface nor magnetic field [5]. It is interesting to compare the BH spin results in our first paper [47] and the most recent literature for the same BHXBs as listed in Table II: (1) For GRS 1915+105, A0620-003 and LMC X-3, both results are fully consistent; (2) For GRO J1655-40, the original result points to an extreme Kerr BH ( a ∗ ∼ 0 . 93) [47], somewhat different from the most recent result of mildly spinning BH ( a ∗ = 0 . 65 -0 . 75) [32]. However, the original result was obtained using the BH mass of 7 M glyph[circledot] , about 10% larger than the currently best estimate that was used to obtain an updated BH spin in the most recent literature [32]. From Figure 15, it can be seen that a ∗ would be decreased from 0.93 to 0.87, if the BH mass is decreased by about 10%; actually a ∗ = 0 . 85 is also allowed in the new estimate [32]; (3) For Cygnus X-1, our original conclusion that a ∗ = 0 . 75 in the high/soft state and a ∗ = -0 . 75 in the low/hard state was based on the assumption that r in decreased by a factor of two when the source made a transition from its normal low/hard state to the unusual high/soft state [94]. However the more realistic constraint is that r in changed by more than a factor of 1.8-3.2 during the state transition [94], which implies that a ∗ switched from | a ∗ | > 0 . 85, fully compatible with the latest result of a ∗ > 0 . 95 in the high/soft state [37]. The above would suggest that a very low temperature disk component exists in its hard state spectrum, if Cygnus X-1 indeed harbors a Kerr BH and the accretion disk switches from a retrograde mode to a prograde mode when it makes the hard-to-soft state transition. For a supermassive BH in the center of a galaxy, it is well understood that its mass is mostly gained through accretion in its AGN phase [95]. Therefore random accretion (between prograde and retrograde modes) tends to make its final BH spin close to zero, regardless its initial BH spin. For Cygnus X-1, its current BH mass of around 10-20 M glyph[circledot] cannot be gained through post-formation accretion, since its observed average mass accretion rate ˙ M ∼ 2 × 10 -9 M glyph[circledot] /yr and the age of its companion is much less that 10 8 yr; actually the age of the companion is estimated to range from 4.6 and 7.8 million years [96]. This means that its post-formation BH mass growth is much less than a fraction of its current mass and thus its current BH spin must be quite close to that at birth. Even if accreting at Eddington rate, to grow its BH spin from 0 and the final mass to be the current observed value, the timescale is around 3 . 1 × 10 7 yr and the accreted mass is roughly 7.3 M glyph[circledot] [37]. If its accretions al- tes between prograde and retrograde modes, then its current BH spin should be even closer to its initial spin than that in low mass BHXBs that only stay in one accretion mode due to roche-lobe overflows. Therefore the current high BH spin in Cyg X-1 must be natal; this conclusion is also true for the other highly spinning BHs in other BHXBs [48]. In summary, BH spin in BHXBs can now be measured reliably with the CF method, when the luminosity of a BHXB is between ∼ 0 . 02 and ∼ 0 . 3 in Eddington unit and their system parameters are well-known a prior . The observed thermal disk spectrum can be modeled directly to obtain the BH spin with the available KERRBB2 model in XSPEC when the X-ray spectrum in dominated by this component, i.e., the source is in the thermal state. When a significant power-law component is present, the inverse Compton scattering process must be taken into account to recover the disk photons scattered into this PL component, with for example the SIMPL/SIMPLR model [37, 97] in XSPEC. Recently another way to measure BH spin using the outburst properties of BHXBs has been proposed [77], which is mentioned briefly in Section VIII; however the effectiveness of this method needs to be tested.\",\n \"pages\": [\n 13,\n 14,\n 15,\n 16\n ]\n },\n {\n \"title\": \"C. Uncertainties of the continuum fitting method\",\n \"content\": \"In spite of the tremendous progress made so far on BH spin measurements in BHXBs, most of these BH spin measurements suffer from considerable uncertainties. Actually the major source of these uncertainties comes primarily from the uncertainties in their BH masses, accretion disk inclination angles, and distances. The accurate BH mass measurement is required because R ISCO must be in units of r g = GM/c 2 in Figure 15. The disk inclination angle and distance are also required because the total luminosity of the disk emission L disk is needed, in order to estimate the absolute value of a ∗ . So far, all BH masses in X-ray binaries (XRBs) have been estimated using the Kepler's 3rd law of stellar motion, expressed in the so-called the mass function given in Equation (1). Since the only direct observables are P orb and K C , both M C and i have to be determined indirectly in order to obtain the BH mass estimate reliably. The companion's mass M C can be determined relatively reliably by the observed spectral type of the companion star and i can be estimated by modeling the observed ellipsoidal modulation of the companion's optical or infrared light curve. The observed ellipsoidal modulation is a consequence of exposing different parts of the pearshaped companion star to the observer at different orbital phases (see Figs. 1 and 5); the pear-shape is caused by the tidal force of the compact star, which also heats the side of the companion star facing it. For details of BH mass estimates using this method, please refer to [2]. However model dependence and other uncertainties (such as accretion disk contamination) cannot be circum- vented completely and thus systematic error may exist in determining their system parameters. For example, three optical states, namely 'passive', 'loop' and 'active' states, have been identified in the normally called 'quiescent' state of A0620-003 when its X-ray luminosity is very low; only during the passive state its optical light curve modulation is purely ellipsoidal, i.e., accretion disk contamination is completely negligible, as shown in Figure 22 [98]. This means that considerable systematic errors in determining its inclination angle may occur unless only the 'passive' state data are used. Unfortunately previous observations of BHXBs used to determine their inclinations did not always occur during the passive state, thus systematic errors may be common in previous results [99]. Even for GRO J1655-40, of which all previous observations were made during its passive state [99], its BH mass measured with different observations, or the same data analyzed by different groups are not exactly the same, and even not completely consistent between them, as shown in Figure 23, which show a scatter of about 20-30% to the estimated BH mass, much larger than its statistical error of a few percents. It should be pointed out that the inclination i in Equation (1) refers to that of the orbital plane of the binary system. Because the accreting material initially carries the angular momentum inherited from the companion star, the formed disk should be co-planar with the orbital plane of the binary system. However, the BH spin in a BHXB cannot be changed significantly by accretion [71] (and see discussion above on Cygnus X-1). Therefore its spin axis may not coincide with the normal direc- tion of the orbital plane of the binary system. In this case the Bardeen-Petterson effect [106], due to frame-dragging by the spinning BH, can rapidly align the normal axis of the inner disk region with the spin axis of the BH, making the inner disk and binary system mis-aligned. Circumstantial evidence already exists for mis-alignment between the two axes, because the orbital inclination of GRO J1655-40 shown in the upper panel of Figure 23 is significantly different from ∼ 80 · inferred from its relativistic jets [107], if the jets are powered by extracting the spin energy of the BH via the Blandford-Znajek mechanism [76]. Nevertheless, the orbital inclination is normally used in place of the disk inclination, which is needed in BH spin measurement using the CF method, in order to infer the total disk luminosity and calculate GR correction factors. However, the inner disk inclination is currently not available essentially for any of the known BHXBs. For example, a mis-alignment of 10 · from i = 70 · can cause nearly 50% error to the total disk luminosity, which will translate into nearly 30% error in r in . A Schwarzschild BH may be estimated to have a ∗ falling any where between [ -0 . 5 , 0 . 5], if r in is uncertain within about 30%, according to Fig.(15). Accurate determinations of distances of astrophysical objects in the Milky Way are difficult, e.g., for BHXBs that are not standard candles. Normally some absorption features in their spectra, in conjunction with their posi- tions in the galactic coordinates, are used to infer their distances. For example, the distance of GRO J1655-40 is commonly taken as 3 . 2 ± 0 . 2 kpc, based primarily on observed absorption lines and somewhat on the dynamics of the observed jets. Critical examinations of all available data related to its distance, however, are in favor of a much closer distance of less than 2 kpc and more likely just 1 kpc [108, 109]. Similar conclusion is also reached to the distance of A0620-003, revising its distance from the commonly accepted 1 . 2 ± 0 . 4 kpc to ∼ 0 . 4 kpc, making its possibly the closest BHXB known so far [109]; however, a distance of 1 . 06 ± 0 . 12 pc was preferred in a more recent study [30]. Similarly the currently adopted distances of many other BHXBs may also have considerable systematic errors. If true, this would change significantly the current estimates on their masses and spins. Therefore future improvements of the continuum fitting method depend upon the improved measurements on their BH masses, accretion disk inclination, and distances.\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"D. Future improvements of the continuum fitting method\",\n \"content\": \"The mass ratio q can be determined directly according to the law of momentum conservation, i.e., if the semi-amplitude of the velocity curve of the BH K BH can be observed directly, as illustrated in Figure 1. The orbital inclination i can be then calculated using Equation (1), avoiding any systematics related to the ellipsoidal light curve modeling. Since a BH is not directly observable, we can only hope to observe any emission or absorption line feature comoving with it. The accretion disk certainly moves with the accreting BH. However any line feature of the inner accretion disk suffers from the broadening of disk's orbital motion and distortions by relativistic effects around the BH, thus making it practically impossible, or difficult at least, for detecting the binary orbital motion of the BH. Orbital motion of double-peaked disk emission lines were observed for NSXB Sco X-1 [110], the BHXBs A0620-003 [111, 112]) and GRS 1124-68 [112]. Unfortunately a significant phase offset of velocity modulation was found from that expected based on the observed orbital motion of the companion, though the velocity semi-amplitude is consistent with the expected mass ratio [112]. Soria et al. (1998) observed the orbital motion of the double-peaked disk emission line He II λ 4686 from GRO J1655-40, and found its velocity modulation phase and semi-amplitude in agreement with the kinematic and dynamical parameters of the system [113]. However one major problem in accurately measuring the orbital motion of the primary from the observed double-peaked emission lines is how to determine reliably the line cen- er, because the lines are typically asymmetric and also variable. We have recently proposed to observe the Doppler shifts of the absorption lines of the accretion disk winds co-rotating with the BH around its companion star [114], since in many XRBs accretion disk winds are ubiquitous and appear to be rather stable when observed (e.g., in Ref.[115]). We verified this method using Chandra and HST high resolution spectroscopic observations of GRO J1655-40 (shown in Figure 24) and LMC X-3. Unfortunately the currently available data only covered small portions of their orbital phases and thus do not allow better constraints to their system parameters. Future more observations of these two sources and other sources with detectable absorption lines from their accretion disk winds will allow reliable and precise measurements of the BH masses and orbital inclination angles in accreting BHXBs. Since the accretion disk has very high density and is ionized near the BH, where the majority of the observed disk emission is produced due to the very deep gravitational potential near the BH, scattering of the primary disk emission is inevitable. The scattered light is polarized and its polarization fraction and position angle depend on the viewing direction (inclination ), scattering optical depth and the radius where the scattering occurs [116-119]. Ignoring many details, it can be shown that the polarization fraction, P ( i ), of the observed disk photons (initial disk emission plus the scattered emission) is given by where A is a constant depending upon the scattering op- tical depth; detailed calculations made by Chandrasekhar [120] gives P (75 · ) = 0 . 04. Note that here the disk photons are from the Raleigh-Jeans part of the multicolor blackbody spectrum with a characteristic shape of f ( ν ) ∝ ν 1 / 3 , i.e., no GR effect is included. We can therefore find the disk inclination angle by measuring the polarization fraction of this part of the disk emission, as shown in Figure 25. At energies above the Raleigh-Jeans part of the multi-color blackbody spectrum, the polarization is strongly effected by both the inclination and BH spin, as shown in Figure 26. The continuum spectra are clearly degenerated for the different combinations of inclination and BH spin, but the polarization fraction and angle as functions of energy can clearly distinguish between them [118]. Therefore X-ray spectra-polarimetry observations of BHXBs will certainly make important progresses in measuring BH spin. Besides using polarization measurements to obtain inner disk inclination (and BH spin), the broad iron Kalpha measurements can also be used to do so [121-123], because iron K-alpha emission is believed to come from the fluorescent emission of the disk as a result of illumination by a hard X-ray source above the disk. Naturally this emission is sensitive to the disk inclination (and BH spin). However, compared to the polarization measurement method, this method is less straight forward and may suffer from systematic uncertainties in deriving the disk inclination (and BH spin), because complicated modeling of the hard X-ray component and line emissivity from the disk is required. Therefore it is essential to use all methods discussed above to measure both inclination angles (orbital plane and inner disk) and BH spin. Studying the relationship between the results obtained with different methods is also important in its own rights, in order to understand accretion disk physics, its interaction with the BH in its center, production of relativistic jets, the origin of the BH spin, and ultimately the formation mechanisms of BHs and BHXB systems. The recent dispute on the distances of GRO J1655-40 and A0620-003 exemplifies the difficulty of determining the distances of BHXBs using mostly absorption lines [108, 109]. We have recently suggested a method of using the delay time between the X-ray fluxes of an XRB and its X-ray scattering halo by interstellar dust to infer its distance [124, 125]. However this method may suffer from our incomplete knowledge of the distribution of interstellar medium. Ideally precise astrometry can determine their distances model-independently, by measuring their trigonometric parallaxes. Recently the distance to Cygnus X-1 was determined reliably and accurately this way (1 . 86 +0 . 12 -0 . 11 kpc; [126]); which is key to the consequent measurement of its BH mass and spin [36, 37]. Currently it remains challenging to measure the trigonometric parallaxes of objects at distances beyond several kpc where most BHXBs are located. Future high precision astrometry missions are expected to improve the distance estimates to these BHXBs significantly. Therefore we expect that the improved measurements discussed above on their BH masses, inner disk inclination, and distances will allow future improvements of BH spin measurement with the CF method.\",\n \"pages\": [\n 17,\n 18,\n 19\n ]\n },\n {\n \"title\": \"E. Possible application of the continuum fitting method to AGNs\",\n \"content\": \"More recently, the CF method is also applied to constrain the BH spin in an active galactic nucleus (AGN), which is powered by matter accretion onto the central supermassive BH in its center [127]. However the BH spin inferred this way for an supermassive BH is quite uncertain, because: (1) The peak temperature of the accretion disk inversely increases with the mass of a BH. Therefore, for a supermassive, its temperature would be in the ultraviolet energy range which will be strongly absorbed and hard to observe; (2) The system parameters of a supermassive BH (e.g., the mass of the BH and the inclination angle of the accretion disk) have larger uncertainties; (3) The uncertain mechanism for some components (e.g., the soft X-ray excess) in AGN spectra also increases the difficulty; (4) Some emission and absorption lines may distort the continuum spectrum substantially; and (5) In some cases the contribution of its host galaxy to the observed total continuum spectrum cannot be removed satisfactorily. The polarized continuum of an AGN should be a pure accretion disk continuum, at least in the optical to near infrared band [128]. One possible way to measure the BH spin in an AGN with the CF method is to combine the observed polarized optical to near infrared continuum spectrum with the observed total UV continuum spectrum to get a broad band continuum spectrum of an AGN [129]. In principle the broadened Balmer edge features and the total UV spectrum can be used to constrain the disk inclination angle and fraction of host galaxy contamination, respectively [129]. However the quality of the currently available data is still insufficient to allow accurate determination of BH spin in AGNs with this method. Nevertheless the principal method of measuring BH spin in AGNs should be using the reflected broad iron line and continuum components [121-123]. The main reason is that i can also be determined simultaneously and r in obtained this way is already in units of r g , thus avoiding naturally the uncertainties caused by the BH mass and inclination in the CF method. However cross calibration can be done between the CF and reflection fitting methods if both can be applied to the same supermassive BH.\",\n \"pages\": [\n 19,\n 20\n ]\n },\n {\n \"title\": \"V. FURTHER DEVELOPMENTS ON HOT ACCRETION FLOWS\",\n \"content\": \"In this section I briefly summarize some further developments on hot accretion flows, which are believed to be responsible for the PL component of the spectra in BHXBs. The multi-waveband spectra of the hard state of the BHXB XTE J1118+480 was modeled with the TID with hot accretion flow (ADAF) geometry (panel (a) in Figure 27) and TID (with ADAF) plus jet model (panel (b) in Figure 27), with ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D [93]. The steep UV spectrum provides clear evidence for a large truncation radius for the SSD ( r = 600), and the radio to infrared spectrum dominates the jet emission, which also contribute to the hard PL component [93]. Such low accretion rate gives very low SSD luminosity l D ∼ 5 × 10 -4 (since the disk radiation efficiency η ∼ 1 /r ; see Figure 17), which is far below the turn-over luminosity of l D ∼ 10 -2 shown in Figure 20. Therefore the inferred truncation radius of XTE J1118+480 agrees with the extrapolation of data points of XTE J1817-330 down to very low disk luminosity. Large truncation radii are also reported from several other sources in the hard state (e.g., in Ref.[130]). However, the exact values of these truncation radii may have large uncertainties, since no direct detection of the inner disk peak emission was available, unlike the strong case of XTE J1118+480 [93]. For this reason I did not include these reported values in Figure 20. Panel (b) in Figure 27 also shows that the radiation from both the hot accretion flow and the jet contribute to the X-ray emission. However the former is roughly proportional to ˙ m 2 , whereas the latter to ˙ m . Thus with the decrease of ˙ m , the contribution from the jet becomes more and more important, thus the X-ray radiation will be dominated by the jet (when l glyph[lessorsimilar] 10 -5 -10 -6 ), as shown in Figure 28 [93]. The observational data of very low-luminosity AGNs clearly show a correlation between radio and X-ray with a correlation index of ∼ 1 . 2 [131], in excellent agreement with the prediction shown in Figure 28 [93]. It is well known that the highest luminosity an ADAF can produce is only about 3% L Edd . However the observed highest luminosity a hard state can reach can be 10% L Edd or even higher, which can be described by LHAF shown in Figure 29, where several previously known solutions of accretion flows are all unified in a single scheme [133]. The hard state spectrum of XTE J1550-564 with L ∼ 6% L Edd is explained by LHAF very well, including the X-ray spectral slope and the value of the cutoff energy [134].\",\n \"pages\": [\n 20\n ]\n },\n {\n \"title\": \"VI. FURTHER DEVELOPMENTS ON CORONA FORMATION\",\n \"content\": \"While the formation of the SSD in a BHXB is reasonably well understood, the formation of the corona, which is the hot accretion flow discussed in Section V, remains less understood. Over the last more than 10 years, Liu and her collaborators have developed a model to explain the formation and evolution of the corona in a BHXB or AGN, in which the mass accretion rate ˙ m in units of the Eddington ratio drives the variations of the complex accretion flows by the interaction between the cold SSD and the hot corona [60, 135-139]. Specifically, the coupling between the hot corona and the cold disk leads to mass exchange between them. The gas in the thin disk is heated up and evaporates into the corona as a consequence of thermal conduction from the hot corona, or the corona gas condenses into the disk as a result of overcooling by, for example, external inverse Compton scattering. If ˙ m is low, evaporation occurs and can completely remove the thin disk, leaving only the hot corona in the inner region and a truncated thin disk in outer region; this provides a mechanism for ADAF at low ˙ m . If ˙ m is high, the gas in the corona partially condenses to the disk due to strong Compton cooling, resulting in disk dominant accretion. The model naturally explains the different structures of accretion flow in different spectral states as shown in Figure 30 [60, 135, 137-144]. The hysteresis observed in spectral state transitions can also be explained by different irradiations from different evolution history under the same scenario [145-147]. Figure 30 is significantly different from the illustration of accretion flow structures in different spectral state in Figure 12 on three aspects: (1) At intermediate ˙ m , the SSD is broken by ADAF into two parts, an outer disk and an inner disk; (2) The inner disk boundary here is always located very close to the BH, except in the very low ˙ m hard or quiescent state, which is very different from the TID scenario depicted in Figure 12; (3) The corona here covers essentially the whole accretion disk, especially the inner disk region, whereas in Figure 12 the corona is mostly located inward from the inner disk boundary. The observed soft X-ray component in the low/hard state can be explained by the existence of the inner disk [139, 140, 142, 144]. As I have discussed above, the inner disk boundary radius inferred with the CF method in the presence of a strong PL component is actually consistent with that in the soft state when the PL component is weak, after taking into account the Compton scattering in the corona [65, 87]. Actually the essential assumption behind the broad iron line/reflection fitting method of determining BH spin is that the inner disk boundary is at the ISCO when the PL component is strong. All these tend to support the existence of the inner disk at intermediate ˙ m . However it remains to be demonstrated that if the whole SSD is indeed broken into the two parts at intermediate ˙ m , as illustrated in Figure 30. Recently it has been suggested that the ADAF and corona shown in Figure 30 may have clumpy structures, as shown in Figure 31 [148]. The 'clumpy' model (Figure 31) has mainly two different consequences from the 'uniform' model (Figure 30): (1) The inner disk is transient in the 'clumpy' model; (2) The 'clumpy' model can explain the variabilities observed in X-ray binaries (such as the state transitions discussed in Section VII) and radio-loud AGNs (such as BL Lac objects). Determining the structure of the corona in an BHXB observationally remains difficult; the fundamental issue on whether the corona covers mostly the accretion disk or the central compact object still remains unclear so far. Since the observed similar spectral and state transitions between some NS and BH XRBs are quite similar [149], it is reasonable to assume that they have similar coronae. Recently we have used type I X-ray bursts from low-mass NSXBs to show that X-ray bursts experience negligible Comptonization and that the corona cools rapidly during the rising phase of X-ray bursts and is then heated up rapidly during the rising phase of X-ray bursts, as shown in the upper panel of Figure 32 for IGR J1747721 [150]. These results suggest that the corona cannot cover the central compact object completely (lower panel of Figure 32) and that the destruction and formation time scales of the corona are as short as seconds; such short time scales are quite difficult to understand in the above discussed evaporation model, in which the time scales are related to the viscous time scales of the accretion disk. However, this short time scale is consistent with a corona produced by magnetic reconnections in the accretion disk, in a similar way to the solar corona heating [151]; this conclusion was based on the inferred accretion flow structure of a BHXB shown in panel (A) of Figure 33, in comparison with the atmospheric structure of the Sun shown in panel (B) of Figure 33. It is worth noticing that the purported coronae in both XRBs and AGNs have strikingly similar properties, e.g., they all have electron temperatures of the order of hun- dreds keV, in spite that the temperatures, inner disk radii and variability time scales of their cold accretion disks all scale with their BH masses (and accretion rates) as predicted in the SSD model. This means that their coronae are scale independent. It is perhaps not coincidental that the electrons' velocities in a corona are approximately the same as the Keplerian orbital velocities of the inner disk, which are also roughly the same as the launching velocities of jets. Of course these velocities are also the varialized velocities of the central BHs. It is plausible that turbulent small scale magnetic fields lifts the plasma in the 'warm layer' shown in Figure 33 to form the corona, which is then launched into the jets by the rotating large scale magnetic fields through either the BP or BZ mechanism; during these processes the magnetic fields are mostly responsible for changing the directions of motion of plasmas by the Lorentz force. If the corona is clumped as discussed in Section VI [148], then the plasma channeled into the jets should be clumpy and thus the ejected jets should be episodic and have knotted structures, in agreement with observations; this scenario is illustrated in Figure 34. Alternatively episodic ejections may also occur in the disk, similar to the coronal mass ejection on the Sun [152]. Interestingly, the soft X-ray excess (SXE) frequently observed in AGNs can be interpreted as the warm layer found in XRBs, since the inferred plasmas parameters, with electron temperature of 0.1 to 0.3 keV and Compton scattering optical depth of around 10, are similar to that of XRBs and universal among AGNs with different BHmasses, but seem to be related only with ˙ m [153-155]. Therefore their warm layers are also scale independent. Indeed, magnetic flares, perhaps caused by magnetic reconnections in the cold disk, can produce the warm layers in both XRBs and AGNs [156, 157].\",\n \"pages\": [\n 20,\n 21,\n 22,\n 23\n ]\n },\n {\n \"title\": \"VII. FURTHER DEVELOPMENTS ON STATE TRANSITIONS\",\n \"content\": \"To further understand the mechanism of the spectral state transitions, it is important to study those spectral state transitions in individual BHXBs during different outbursts. The advantages are obvious - uncertainties in our estimates of black hole mass, the binary properties such as the orbital period, or the source distance will not play any role in producing the observed diverse transi- tion properties in individual BHXBs, which can only be driven by the accretion process under different initial conditions. The sources which firstly allowed such a study were the BH transients GX 339-4 and XTE J1550-564, the NS transient Aquila X-1, and the flaring NS low-mass XRB (LMXB) 4U 1705-44, in which a remarkable correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the following soft state was found [158-160]. More recently, a comprehensive study of the hard-to-soft spectral state transitions detected in all the bright XRBs in a period of about five years with simultaneous X-ray monitoring observations with the RXTE/ASM and the Swift/BAT confirmed the correlation between the hard-to-soft transition and the peak luminosity of the following soft state [161, 162], as shown in the upper panel of Figure 35 [161]. More important was the discovery of the correlation between the transition luminosity and the rate-of-change of the luminosity during the rising phase of an outburst of transients or a flare of persistent sources, as shown in the lower panel of Figure 35 [161]. The above correlation implies that in most cases it is the rate-of-increase of the mass accretion rate, rather than the mass accretion rate itself, determines the hardto-soft spectral transition; this is depicted in Figure 36 [161]. In addition, the discovery of the relation between the hard X-ray peak flux and the waiting time of transient outburst in the BH transient GX 339-4 (as shown in Figure 37) supports that the total mass in the disk determines the peak soft state luminosity of the following outburst [159]. Therefore spectral states should be understood in the non-stationary accretion regime, which would be described by both ˙ m and m , as well as the total mass in the disk before an outburst.\",\n \"pages\": [\n 23\n ]\n },\n {\n \"title\": \"VIII. FURTHER DEVELOPMENTS ON THERMAL STABILITY OF SSD\",\n \"content\": \"The SSD model predicts that when the accretion rate is over a small fraction of the Eddington rate, which corresponds to L glyph[greaterorsimilar] 0 . 06 L Edd , the inner region of the disk is radiation-pressure-dominated and then both secularly [163] thermally unstable [164, 165]. However, observations of the high/soft state of black hole X-ray binaries with luminosity well within this regime (0 . 01 L Edd glyph[lessorsimilar] L glyph[lessorsimilar] 0 . 5 L Edd ) indicate that the disk has very little variability, i.e., quite stable [166]. It has been well established that the accretion flow in this state is described by the SSD model [4, 167]. Radiation magnetohydrodynamic simulations of a vertically stratified shearing box have confirmed the absence of the thermal instability [168]. Recently, the thermal stability is revisited by linear analysis, by taking into account the role of magnetic field in the accretion flow [169]. By assuming that the field responds negatively to a positive temperature perturbation, it was found that the threshold of accretion rate above which the disk becomes thermally unstable increases signifi- cantly, compared with the case of not considering the role of magnetic field. This accounts for the stability of the observed sources with high luminosities, as shown in Figure 38. If the magnetic pressure is less than about 24% of the total pressure, then this model can explain the 'heart-beat' limit-cycle instability observed in GRS 1915+105 at its highest luminosity; this peculiar source holds the highest accretion rate (or luminosity) among BHXBs. Observations of GRS 1915+105 showed that t high (the duration of the outburst phase) is comparable to t low (duration of the quiescent phase) and L high is 3 to 20 times larger than L low [170]. However, numerical calculations showed that t high is less than 5 percent of t low , and L high is around two orders of magnitude larger than L low . Some efforts have been made to improve the theory in order to explain observations either by some artificial viscosity prescription [171] or by additional assumption of the energy exchange between the disk and corona [172]. Taking into account the stress evolution process, it was found that the growth rate of thermally unstable modes can decrease significantly owing to the stress delay, which may help to understand the 'heart-beat' limit-cycle variability of GRS 1915+105 [173]. The limit-cycle properties are found to be dominated by the mass-supply rate (accre- · · tion rate at the outer boundary) and the value of the α -viscosity parameter in the SSD model that assumes that the viscous torque is proportional to the total pressure [77]. It was also found that only the maximal outburst luminosity (in Eddington units) is positively correlated with the spin of a BH, providing another way to probe BH spin [77]; this is mainly due to the smaller inner disk radius and thus higher radiative efficiency for larger a ∗ as shown in Figures 15, 16 and 17.\",\n \"pages\": [\n 23,\n 24,\n 25\n ]\n },\n {\n \"title\": \"IX. UNIFICATION AND OUTLOOK\",\n \"content\": \"Despite of the many progresses made over the last decades on the study of BHXBs and microquasars, there are still many outstanding and unresolved issues, which can be pierced by two big pictures. One big picture is related to the astrophysics of BHXBs and microquasars, which is centered on understanding the accretion-outflow (wind or jet) connections at different accretion rates for different types of astrophysical systems involving BHs. The other big picture is related to the fundamental physics of BHXBs and microquasars, which is centered on identifying astrophysical BHs and testing theories of strong gravity. These two pictures are actually also entangled together, because a strong gravity theory, such as GR, is needed to describe the astrophysical aspects of these systems, and we need to understand the astrophysics of BHXBs and microquasars before we can start to test any strong gravity theory[174]. Figure 39 shows the relation between the collimationcorrected gamma-ray luminosity and the kinetic power for AGNs and gamma-ray bursts (GRBs)[175]. For at least some of GRBs, a super-Eddington accreting compact stellar object, probably a BH with around 10 M glyph[circledot] , is believed to be responsible for powering the highly relativistic jets, which produces intense gamma-rays through violent collisions of blobs in the jets[176]. The two types of AGNs shown here, flat-spectrum radio quasars (FS- RQs) and BL Lacs, have all been observed to have mildly relativistic jets. Their central engines are supermassive BHs with around 10 7 -9 M glyph[circledot] with accretion rates just below (for FSRQs) or far below (for BL Lacs) Eddington rate[177]. The good correlation over about 10 orders of magnitudes between these very different systems with very different accretion rates suggests that there must be some common mechanisms responsible for the accretionoutflow (wind/jet) connections for all accreting BHs[178]. Putting together all related phenomenologies and some theoretical modeling of BHXBs and microquasars discussed in this article, an unification scheme is illustrated in Figure 40 with the major elements in the accretionoutflow connections in different types of astrophysical systems harboring both stellar mass BHs and supermassive BHs with accretion rates over several orders of magnitudes. The types of accreting stellar mass BHs include BHXBs and microquasars in different spectral states, as well as ultra-luminous X-ray sources (ULXs) (some of which are most likely ultra-Eddington accreting BHs[180]), and GRBs. The types of accreting supermassive mass BHs include low-luminosity AGNs (LLAGNs), BL Lac objects, normal radio quiet quasars (RQQs), FSRQs and broad-line-less luminous quasars (BL3Qs) [179] (I create the acronym 'BL3Q' here just for fun). This scheme is not a theoretical model, or even toy model yet. It, however, can be used as one possible chain to pierce many observed phenomenologies together, providing a possible frame work for further theoretical developments on accretion-outflow connections. Future observations will scrutinize this unification scheme and\",\n \"pages\": [\n 25\n ]\n },\n {\n \"title\": \"Some necessary elaborations and points:\",\n \"content\": \"revise it inevitably. Of particular importance is the reliable determinations of the mass and spin of accreting BHs. The BH mass allows us to determine the accretion rate in units of Eddington rate, i.e., ˙ m , which is a key parameter to the unification scheme of accretion-outflow connections. The BH spin is of course another key parameter, because it can determine the radiative efficiency of the disk and jet power in the BZ mechanism. Once we have well determined BH parameters and a consistent and predictable theory describing the observed accretion-outflow connections, we are then ready to study the fundamental physics of BHs with BHXBs and microquasars, such as the properties of event horizon, spacetime around Schwarzschild and Kerr BHs, BH spin energy (and mass) extraction. For example, both the broad iron line and CF fitting methods can be used to measure a BH's spin. However the two methods are equivalent only if the metric is accurately described with Kerr metric. Different spin measurements for a BH with the two different methods would invalidate Kerr metric, provided that we understand the accretion disk physics thoroughly. On the other hand, a correct metric is of course required to describe accurately the accretion flow around BHs. This is one example of the entanglement or interplay between astrophysics and fundamental physics of accreting BHs. The stake is high, but the job is difficult. Acknowledgment. I appreciate inputs from Profs. Lijun Gou, Weimin Gu, Lixin Li, Bifang Liu, Dingxiong Wang, Wenfei Yu, Feng Yuan, and Shu Zhang. The editors of this book are thanked for inviting me to write this article, as well as their patience, persistency, proof reading it, and offering comments and suggestions to improve it in the end. This work is partially supported with funding by the 973 Program of China under grant 2009CB824800, by the National Natural Science Foundation of China under grant Nos. 11133002 and 10725313, and by the Qianren start-up grant 292012312D1117210. Orford, J L Osborne, M Ouchrif, M Panter, G Pelletier, S Pita, G Puhlhofer, M Punch, B C Raubenheimer, M Raue, J Raux, S M Rayner, A Reimer, O Reimer, J Ripken, L Rob, L Rolland, G Rowell, V Sahakian, L Saug'e, S Schlenker, R Schlickeiser, C Schuster, U Schwanke, M Siewert, H Sol, D Spangler, R Steenkamp, C Stegmann, J-P Tavernet, R Terrier, C G Th'eoret, M Tluczykont, G Vasileiadis, C Venter, P Vincent, H J Volk, and S J Wagner. Discovery of very high energy gamma rays associated with an x-ray binary. Science (New York, N.Y.) , 309(5735):746-9, July 2005. Flow above a Thin Disk. The Astrophysical Journal , 575(1):117-126, August 2002.\",\n \"pages\": [\n 26,\n 27,\n 28,\n 32\n ]\n }\n]"}}},{"rowIdx":266,"cells":{"bibcode":{"kind":"string","value":"2013GApFD.107..185P"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1205.4835.pdf"},"content":{"kind":"string","value":"\nGeophysical and Astrophysical Fluid Dynamics Vol. 00, No. 00, Month 2013, 1-19\nHelicity-vorticity turbulent pumping of magnetic fields in the solar convection zone\nV. V. PIPIN\nInstitute Solar-Terrestrial Physics, Irkutsk, Russia\n( July 31, 2021, Revision: 1.20 )\nWe study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker-Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. We estimate the magnitude of the drift velocity and find that it can be several m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. We find that the helicity-vorticity pumping effect can have an influence on the features of the sunspot time-latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.\n1 Introduction\nIt is believed that the evolution of the large-scale magnetic field of the Sun is governed by the interplay between large-scale motions, like differential rotation and meridional circulation, turbulent convection flows and magnetic fields. One of the most important issues in solar dynamo theory is related to the origin of the equatorial drift of sunspot activity in the equatorial regions and, simultaneously at high latitudes, the poleward drift of the location of large-scale unipolar regions and quiet prominences. Parker (1955) and Yoshimura (1975) suggested that the evolution of large-scale magnetic activity of the Sun can be interpreted as dynamo waves propagating perpendicular to the direction of shear from the differential rotation. They found that the propagation can be considered as a diffusion process, which follows the iso-rotation surfaces of angular velocity in the solar convection zone. The direction of propagation can be modified by meridional circulation, anisotropic diffusion and the effects of turbulent pumping (see, e.g., Choudhuri et al. 1995, Kitchatinov 2002, Guerrero and de Gouveia Dal Pino 2008). The latter induces an effective drift of the large-scale magnetic field even though the mean flow of the turbulent medium may be zero.\nThe turbulent pumping effects can be equally important both for dynamos without meridional circulation and for the meridional circulation-dominated dynamo regimes. For the latter case the velocity of turbulent pumping has to be comparable to the meridional circulation speed. It is known that an effect of this magnitude can be produced by diamagnetic pumping and perhaps by so-called topological pumping. Both effects produce pumping in the radial direction and have not a direct impact on the latitudinal drift of the large-scale magnetic field.\nRecently (Pipin 2008, Mitra et al. 2009, Leprovost and Kim 2010), it has been found that the helical convective motions and the helical turbulent magnetic fields interacting with large-scale magnetic fields and differential rotation can produce effective pumping in the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. It is believed that these regions, namely the tachocline beneath the solar convection zone and the subsurface shear layer, are important for the solar dynamo. Figure 1 illustrates the principal processes that induce the helicity-vorticity pumping effect. It is suggested that this effect produces an anisotropic drift of the large-scale magnetic field, which means that the different components of the large-scale magnetic field drift in different directions. Earlier work, e.g. by Kichatinov (1991) and Kleeorin and Rogachevskii (2003), suggests that the effect of anisotropy in the\n2\n\n\n
Figure 1. The field lines of the large-scale magnetic field, B ( T ) , are transformed by the helical motions to a twisted Ω -like shape. This loop is folded by the large-scale shear, V ( T ) , into the direction of the background large-scale magnetic field, B ( T ) . The induced electromotive force has a component, E ( P ) , which is perpendicular to the field B ( T ) . The resulting effect is identical to the effective drift of the large-scale magnetic field along the x -axis, in the direction opposite to the large-scale vorticity vector W = ∇× V ( T ) , i.e., E ( P ) ∼ -W × B ( T ) .
\n\ntransport of mean-field is related to nonlinear effects of the global Coriolis force on the convection. Also, nonlinear effects of the large-scale magnetic field result in an anisotropy of turbulent pumping (Kleeorin et al. 1996). It is noteworthy, that the helicity-vorticity effect produces an anisotropy of the large-scale magnetic field drift already in the case of slow rotation and a weak magnetic field. A comprehensive study of the linear helicity-vorticity pumping effect for the case of weak shear and slow rotation was given by Rogachevskii et al. (2011) and their results were extended by DNS with a more general test-field method Brandenburg et al. (2012).\nIn this paper we analytically estimate the helicity-vorticity pumping effect taking into account the Coriolis force due to global rotation. The calculations were done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. The results are applied to mean field dynamo models, which are used to examine this effect on the dynamo. The paper is structured as follows. In the next section we briefly outline the basic equations and assumptions, and consider the results of calculations. Next, we apply the results to the solar dynamo. In Section 3 we summarize the main results of the paper. The details of analytical calculations are given in the Appendices A and B.\n2 Basic equations\nIn the spirit of mean-field magnetohydrodynamics, we split the physical quantities of the turbulent conducting fluid into mean and fluctuating parts where the mean part is defined as an ensemble average. One assumes the validity of the Reynolds rules. The magnetic field B and the velocity V are decomposed as B = B + b and V = V + u , respectively. Hereafter, we use small letters for the fluctuating parts and capital letters with an overbar for mean fields. Angle brackets are used for ensemble averages of products. We use the two-scale approximation (Roberts and Soward 1975, Krause and Radler 1980) and assume that mean fields vary over much larger scales (both in time and in space) than fluctuating fields. The average effect of MHD-turbulence on the large-scale magnetic field (LSMF) evolution is described by the mean-electromotive force (MEMF), E = 〈 u × b 〉 . The governing equations for fluctuating magnetic field\nand velocity are written in a rotating coordinate system as follows:\n∂ b ∂t = ∇× ( u × ¯ B + ¯ V × b ) + η ∇ 2 b + G , (1)\n∂u i ∂t +2( Ω × u ) i = -∇ i ( p + ( b · ¯ B ) µ ) + ν ∆ u i (2)\n+ 1 µ ∇ j ( ¯ B j b i + ¯ B i b j ) -∇ j ( ¯ V j u i + ¯ V i u j ) + f i + F i ,\nwhere G , F stand for nonlinear contributions to the fluctuating fields, p is the fluctuating pressure, Ω is the angular velocity responsible for the Coriolis force, ¯ V is mean flow which is a weakly variable in space, and f is the random force driving the turbulence. Equations (1) and (2) are used to compute the mean-electromotive force E = 〈 u × b 〉 . It was computed with the help of the equations for the second moments of fluctuating velocity and magnetic fields using the double-scale Fourier transformation and the minimal τ -approximations and for a given model of background turbulence. To simplify the estimation of nonlinear effects due to global rotation, we use scale-independent background turbulence spectra and correlation time. Details of the calculations are given in Appendix A. In what follows we discuss only those parts of the mean-electromotive force which are related to shear and the pumping effect.\n2.1 Results\nThe large-scale shear flow is described by the tensor V i,j = ∇ V i . It can be decomposed into a sum of strain and vorticity tensors, ∇ j V i = 1 2 ( V i,j + V j,i ) -1 2 ε ijp W p , where W = ∇ × V is the large-scale vorticity vector. The joint effect of large-scale shear, helical turbulent flows and magnetic fields can be expressed by the following contributions to the mean-electromotive force (omitting the α -effect):\nE ( H ) = ( W × B ) ( f ( γ ) 2 h C + f ( γ ) 1 h K ) τ 2 c + ˜ V ( B ) ( f ( γ ) 4 h C + f ( γ ) 3 h K ) τ 2 c (3) + e [( e × W ) · B ] ( f ( γ ) 6 h C + f ( γ ) 5 h K ) τ 2 c + ( e × W ) ( e · B ) ( f ( γ ) 8 h C + f ( γ ) 7 h K ) τ 2 c ,\nwhere ˜ V ( B ) = B j 2 ( V i,j + V j,i ) , e = Ω | Ω | is the unit vector along the rotation axis, τ c is the typical\nrelaxation time of turbulent flows and magnetic fields, h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇ × b (0) 〉 µρ are kinetic and current helicity of the background turbulence. These parameters are assumed to be known in advance. Functions f ( γ ) n ( Ω ∗ ) are given in Appendix B, they depend on the Coriolis number Ω ∗ = 2 Ω 0 τ c and describe the nonlinear effect due the Coriolis force, and Ω 0 is the global rotation rate.\nFor slow rotation, Ω ∗ glyph[lessmuch] 1, we perform a Taylor expansion of f ( γ ) n ( Ω ∗ ) and obtain\nE ( H ) = τ 2 c 2 ( W × B ) ( h C -h K ) + τ 2 c 5 ˜ V ( B ) ( 3 h K -13 3 h C ) . (4)\nThe coefficients in the kinetic part of Eq. (4) are two times larger than those found by Rogachevskii et al. (2011). This difference results from our assumption that the background turbulence spectra and the correlation time are scale-independent. The results for the magnetic part are in agreement with our earlier findings (see Pipin 2008). The first term in Eq. (4) describes turbulent pumping with an effective velocity τ 2 c W 2 ( h C -h K ) and the second term describes anisotropic turbulent pumping. Its structure depends on the geometry of the shear flow. For large Coriolis numbers, Ω ∗ glyph[greatermuch] 1, only the kinetic helicity contributions\n4\nV. V. Pipin\n\n\n
Figure 2. The dependence of the pumping effects on the Coriolis number. Solid lines show contributions from kinetic helicity and dashed lines the same for current helicity.
\n\nsurvive:\nE ( H ) = -τ 2 c 6 ( W × B ) h K + τ 2 c 5 ˜ V ( B ) h K . (5)\nFigure 2 show the dependence of the pumping effects on the Coriolis number. We observe that for the terms ( W × B ) and ˜ V ( B ) the effects of kinetic helicity are non-monotonic and have a maximum at Ω ∗ ≈ 1. The effects of current helicity for these terms are monotonically quenched with increasing values of Ω ∗ . The additional contributions in Eq. (3) are rather small in comparison with the main terms. Thus, we can conclude that the first line in Eq. (3) describes the leading effect of pumping due to the helicity of turbulent flows and magnetic field. Below, we drop the contributions from the second line in Eq. (3) from our analysis.\n2.2 Helicity-vorticity pumping in the solar convection zone\n2.2.1 The dynamo model. To estimate the impact of this pumping effect on the dynamo we consider the example of a dynamo model which takes into account contributions of the mean electromotive force given by Eq. (3). The dynamo model employed in this paper has been described in detail by Pipin and Kosovichev (2011a,c). This type of dynamo was proposed originally by Brandenburg (2005). The reader may find the discussion for different types of mean-field dynamos in Brandenburg and Subramanian (2005) and Tobias and Weiss (2007).\nWe study the standard mean-field induction equation in a perfectly conducting medium:\n∂ B ∂t = ∇ × ( E + U × B ) , (6)\nwhere E = u × b is the mean electromotive force, with u, b being fluctuating velocity and magnetic field, respectively, U is the mean velocity (differential rotation and meridional circulation), and the axisymmetric magnetic field is:\nB = e φ B + ∇× A e φ r sin θ ,\nwhere θ is the polar angle. The expression for the mean electromotive force E is given by Pipin (2008). It\nis expressed as follows:\nE i = ( α ij + γ ( Λ ) ij ) B -η ijk ∇ j B k + E ( H ) i . (7)\nThe new addition due to helicity and mean vorticity effects is marked by E H . The tensor α ij represents the α -effect. It includes hydrodynamic and magnetic helicity contributions,\nα ij = C α sin 2 θα ( H ) ij + α ( M ) ij , (8)\nα ( H ) ij = δ ij { 3 η T ( f ( a ) 10 ( e · Λ ( ρ ) ) + f ( a ) 11 ( e · Λ ( u ) ))} + (9)\n+ e i e j { 3 η T ( f ( a ) 5 ( e · Λ ( ρ ) ) + f ( a ) 4 ( e · Λ ( u ) ))} + (10)\n3 η T {( e i Λ ( ρ ) j + e j Λ ( ρ ) i ) f ( a ) 6 + ( e i Λ ( u ) j + e j Λ ( u ) i ) f ( a ) 8 } , (11)\nwhere the hydrodynamic part of the α -effect is defined by α ( H ) ij , Λ ( ρ ) = ∇ log ρ quantifies the density stratification, Λ ( u ) = ∇ log ( η (0) T ) quantifies the turbulent diffusivity variation, and e = Ω / | Ω | is a unit vector along the axis of rotation. The turbulent pumping, γ ( Λ ) ij , depends on mean density and turbulent diffusivity stratification, and on the Coriolis number Ω ∗ = 2 τ c Ω 0 where τ c is the typical convective turnover time and Ω 0 is the global angular velocity. Following the results of Pipin (2008), γ ( Λ ) ij is expressed as follows:\nγ ( Λ ) ij = 3 η T { f ( a ) 3 Λ ( ρ ) n + f ( a ) 1 ( e · Λ ( ρ ) ) e n } ε inj -3 η T f ( a ) 1 e j ε inm e n Λ ( ρ ) m , (12)\n-3 η T ( ε -1) { f ( a ) 2 Λ ( u ) n + f ( a ) 1 ( e · Λ ( u ) ) e n } ε inj . (13)\nThe effect of turbulent diffusivity, which is anisotropic due to the Coriolis force, is given by:\nη ijk = 3 η T {( 2 f ( a ) 1 -f ( d ) 2 ) ε ijk -2 f ( a ) 1 e i e n ε njk + εC ω f ( d ) 4 e j δ ik } . (14)\nThe last term in Eq. (14) describes Radler's Ω × J effect. The functions f ( a,d ) { 1 -11 } depend on the Coriolis number. They can be found in Pipin (2008); see also Pipin and Kosovichev (2011a) or Pipin and Sokoloff\nb 2 , which measures the ratio between magnetic and kinetic\n(2011)). In the model, the parameter ε = µ 0 ρ u 2\nenergies of the fluctuations in the background turbulence, is assumed to be equal to 1. This corresponds to perfect energy equipartition. The ε contribution in the second line of Eq. (12) describes the paramagnetic effect (Kleeorin and Rogachevskii 2003). In the state of perfect energy equipartition the effect of diamagnetic pumping is compensated by the paramagnetic effect. We can, formally, skip the second line in Eq. (12) from our consideration if ε = 1. To compare the magnitude of the helicity-vorticity pumping effect with the diamagnetic effect we will show results for the pumping velocity distribution with ε = 0.\nThe contribution of small-scale magnetic helicity χ = a · b ( a is the fluctuating vector-potential of the magnetic field) to the α -effect is defined as\nα ( M ) ij = 2 f ( a ) 2 δ ij χτ c µ 0 ρglyph[lscript] 2 -2 f ( a ) 1 e i e j χτ c µ 0 ρglyph[lscript] 2 . (15)\nThe nonlinear feedback of the large-scale magnetic field to the α -effect is described by a dynamical quenching due to the constraint of magnetic helicity conservation. The magnetic helicity, χ , subject to a conservation law, is described by the following equation (Kleeorin and Rogachevskii 1999, Subramanian and\n6\nV. V. Pipin\nBrandenburg 2004):\n∂χ ∂t = -2 ( E· B ) -χ R χ τ c + ∇· ( η χ ∇ ¯ χ ) , (16)\nwhere τ c is a typical convective turnover time. The parameter R χ controls the helicity dissipation rate without specifying the nature of the loss. The turnover time τ c decreases from about 2 months at the bottom of the integration domain, which is located at 0 . 71 R glyph[circledot] , to several hours at the top boundary located at 0 . 99 R glyph[circledot] . It seems reasonable that the helicity dissipation is most efficient near the surface. The last term in Eq. (16) describes a turbulent diffusive flux of magnetic helicity (Mitra et al. 2010).\nWe use the solar convection zone model computed by Stix (2002), in which the mixing-length is defined as glyph[lscript] = α MLT ∣ ∣ Λ ( p ) ∣ ∣ -1 , where Λ ( p ) = ∇ log p quantifies the pressure variation, and α MLT = 2. The turbulent diffusivity is parameterized in the form, η T = C η η (0) T , where η (0) T = u ' glyph[lscript] 3 is the characteristic mixinglength turbulent diffusivity, glyph[lscript] is the typical correlation length of the turbulence, and C η is a constant to control the efficiency of large-scale magnetic field dragging by the turbulent flow. Currently, this parameter cannot be introduced in the mean-field theory in a consistent way. In this paper we use C η = 0 . 05. The differential rotation profile, Ω = Ω 0 f Ω ( x, µ ) (shown in Fig.3a) is a slightly modified version of the analytic approximation proposed by Antia et al. (1998):\nf Ω ( x, µ ) = 1 Ω 0 [ Ω 0 +55( x -0 . 7) φ ( x, x 0 ) φ ( -x, -0 . 96) (17) -200 ( x -0 . 95) φ ( x, 0 . 96)) + (21 P 3 ( µ ) + 3 P 5 ( µ )] ( µ 2 j p ( x ) + 1 -µ 2 j e ( x ) ) /Ω 0 , j p = 1 1 + exp ( 0 . 709 -x 0 . 02 ) , j e = 1 1 + exp ( 0 . 692 -x 0 . 01 ) ,\nwhere Ω 0 = 2 . 87 · 10 -6 s -1 is the equatorial angular velocity of the Sun at the surface, x = r/R glyph[circledot] , φ ( x, x 0 ) = 0 . 5 [1 + tanh [100( x -x 0 )]], and x 0 = 0 . 71.\n2.2.2 Pumping effects in the solar convection zone. The components of the strain tensor ˜ V in a spherical coordinate system are given by the matrix:\n˜ V = 0 0 ˜ V ( r,ϕ ) 0 0 ˜ V ( θ,ϕ ) ˜ V ( r,ϕ ) ˜ V ( θ,ϕ ) 0 ,\nwhere we take into account only the azimuthal component of the large-scale flow, ˜ V ( r,ϕ ) = r sin θ∂ r Ω ( r, θ ), ˜ V ( θ,ϕ ) = sin θ∂ θ Ω ( r, θ ), so ˆ V ( B ) = ( B ˜ V ( r,ϕ ) , B ˜ V ( θ,ϕ ) , B p i ˜ V ( i,ϕ ) ) . Substituting this into Eq. (3) we find the\n\n\n
Figure 3. Distributions of the angular velocity and the turbulent parameters, and the kinetic helicity inside the solar convection zone. The bottom panel shows the patterns of the pumping velocity fields for the toroidal magnetic field(left) and for the poloidal field(right). They were computed on the basis of Eqs. (18,19,20).
\n\ncomponents of the mean-electromotive force for the helicity-vorticity pumping effect,\nE ( H ) r = Ω ∗ τ c 2 sin θ { [ h K ( f ( γ ) 3 -f ( γ ) 1 ) + h C ( f ( γ ) 4 -f ( γ ) 2 )] x ∂ ˜ Ω ∂x -2 ( ˜ Ω -1 )[ h K f ( γ ) 1 + h C f ( γ ) 2 ] } B,\n--( ˜ Ω -1 ) Ω ∗ τ c x sin θ [ h K f ( γ ) 1 + h C f ( γ ) 2 ] ( µ ∂A ∂x + sin 2 θ x ∂A ∂µ ) ,\n(18) E ( H ) θ = Ω ∗ τ c 2 { sin 2 θ [ h K ( f ( γ ) 3 -f ( γ ) 1 ) + h C ( f ( γ ) 4 -f ( γ ) 2 )] ∂ ˜ Ω ∂µ -2 µ ( ˜ Ω -1 )[ h K f ( γ ) 1 + h C f ( γ ) 2 ] } B, (19) E ( H ) φ = Ω ∗ τ c 2 sin θ x [ h K ( f ( γ ) 3 + f ( γ ) 1 ) + h C ( f ( γ ) 4 + f ( γ ) 2 )] ∂ ( ˜ Ω,A ) ∂ ( x, µ ) (20)\nwhere h C = C C χ µ 0 ρglyph[lscript] 2 . It remains to define the kinetic helicity distribution. We use a formula proposed in our earlier study (see Kuzanyan et al. 2006),\nh K = C η C K u (0)2 2 ∂ ∂r log ( ρ √ u (0)2 ) F 1 cos θ,\nwhere F 1 ( Ω ∗ ) was defined in the above cited paper. The radial profile of h K cos θ is shown in Figure 3. The radial profile of kinetic helicity is shown in Figure 3a of the above cited paper. The parameters C K , C are introduced to switch on/off the pumping effects in the model.\n\n\n
Figure 4. The patterns of the total (including the diamagnetic and the density gradient effects) pumping velocity fields for the toroidal magnetic field(left) and for the poloidal field(right).
\n\nThe expressions given by Eq. (3) are valid for the case of weak shear, when τ c max (∣ ∣ ∇ i V j ∣ ∣ ) glyph[lessmuch] 1. In terms of the strain tensor ˜ V this condition of weak shear implies Ω glyph[star] max (∣ ∣ ∣ r∂ r ˜ Ω ∣ ∣ ∣ , ∣ ∣ ∣ ∂ θ ˜ Ω ∣ ∣ ∣ ) glyph[lessmuch] 1. This is not valid at the bottom of the solar convection zone where the radial gradient of the angular velocity is strong and Ω glyph[star] glyph[greatermuch] 1 and τ c max (∣ ∣ ∇ i V j ∣ ∣ ) ≈ 2. Leprovost and Kim (2010) suggested that this pumping effect is quenched with increasing shear inversely proportional to ( τ c max (∣ ∣ ∇ i V j ∣ ∣ )) 1 ... 2 . Therefore, we introduce an ad-hoc quenching function for the pumping effect:\nf ( S ) = 1 1 + C S Ω ∗ s (∣ ∣ ∣ ∣ ∣ r ∂ ˜ Ω ∂r ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∂ ˜ Ω ∂θ ∣ ∣ ∣ ∣ ∣ ) s , (21)\nwhere C S is a constant to control the magnitude of the quenching, and s = 1. Results by Leprovost and Kim (2010) suggest 1 < s < 2 in relation to geometry of the large-scale shear. We find that for the solar convection zone the amplitude of the pumping effect does not change very much ( ∼ 1 m / s) with s varying in the range 1 . . . 2.\nFrom the given relations, using E ( H ) = U (eff) × B , we find the effective drift velocity, U (eff) , due to the helicity-vorticity pumping effect. Taking into account the variation of turbulence parameters in the solar convection zone we compute U (eff) . The bottom panel of Figure 3 shows the distribution of the velocity field U ( eff ) for the helicity-vorticity pumping effect for the toroidal and poloidal components of the large-scale magnetic field. The maximum velocity drift occurs in the middle and at the bottom of the convection zone. The direction of drift has equatorial and polar cells corresponding to two regions in the solar convection zone with different signs of the radial gradient of the angular velocity. The anisotropy in transport of the toroidal and poloidal components of the large-scale magnetic field is clearly seen.\nThe other important pumping effects are due to mean density and turbulence intensity gradients (Zeldovich 1957, Kichatinov 1991, Kichatinov and Rudiger 1992, Tobias et al. 2001). These effects were estimated using Eq. (12). For these calculations we put C η = 1, ε = 0, C K = 1 and χ = 0. Figure 4 shows the sum of the pumping effects for the toroidal and poloidal components of mean magnetic fields including the helicity-vorticity pumping effect. In agreement with previous studies, it is found that the radial direction is the principal direction of mean-field transport in the solar convection zone. In its upper part the transport is downward because of pumping due to the density gradient (Kichatinov 1991). At the bottom of the convection zone the diamagnetic pumping effect produces downward transport as well (Kichatinov 1991, Rudiger and Brandenburg 1995). The diamagnetic pumping is quenched inversely proportional to the Coriolis number (e.g., Kichatinov 1991, Pipin 2008) and it has the same order of magnitude as the helicity-vorticity pumping effect. The latter effect modifies the direction of effective drift of the toroidal magnetic field near the bottom of the convection zone. There is also upward drift of the toroidal field at\n·\n
\n\n
Table 1. The parameters of the models. Here, B max is the maximum of the toroidal magnetic field strength inside the convection zone, P is the dynamo period of the model.
\n
\nlow latitudes in the middle of the convection zone. It results from the combined effects of density gradient and global rotation (Kichatinov 1991, Krivodubskij 2004). For the poloidal magnetic field the transport is downward everywhere in the convection zone. At the bottom of the convection zone the action of the diamagnetic pumping on the meridional component of the large-scale magnetic field is amplified due to the helicity-vorticity pumping effect.\nThe obtained pattern of large-scale magnetic field drift in the solar convection zone does not take into account nonlinear effects, e.g., because of magnetic buoyancy. The effect of mean-field buoyancy is rather small compared with flux-tube buoyancy (Kichatinov and Pipin 1993, cf. Guerrero and Kapyla 2011).\nTo find out the current helicity counterpart of the pumping effect we analyze dynamo models by solving Eqs. (6, 16). The governing parameters of the model are C η = 0 . 05, C ω = 1 3 C α . We discuss the choice of the governing parameters later. The other parameters of the model are given in the Table 1. Because of the weakening factor C η the magnitude of the pumping velocity is about one order of magnitude smaller than what is shown in Figure 4.\nFollowing Pipin and Kosovichev (2011b), we use a combination of 'open' and 'closed' boundary conditions at the top, controlled by a parameter δ = 0 . 95, with\nδ η T r e B +(1 -δ ) E θ = 0 . (22)\nThis is similar to the boundary condition discussed by Kitchatinov et al. (2000). For the poloidal field we apply a combination of the local condition A = 0 and the requirement of a smooth transition from the internal poloidal field to the external potential (vacuum) field:\nδ ( ∂A ∂r ∣ ∣ ∣ ∣ r = r e -∂A ( vac ) ∂r ∣ ∣ ∣ ∣ ∣ r = r e ) +(1 -δ ) A = 0 . (23)\nWe assume perfect conductivity at the bottom boundary with standard boundary conditions. For the magnetic helicity, similar to Guerrero et al. (2010), we put ∇ r ¯ χ = 0 at the bottom of the domain and ¯ χ = 0 at the top of the convection zone.\nIn this paper we study dynamo models which include Radler's Ω × J dynamo effect due to a large-scale current and global rotation (Radler 1969). There is also a dynamo effect due to large-scale shear and current (Rogachevskii and Kleeorin 2003). The motivation to consider these addional turbulent sources in the mean-field dynamo comes from DNS dynamo experiments (Brandenburg and Kapyla 2007, Kapyla et al. 2008, Hughes and Proctor 2009, Kapyla et al. 2009) and from our earlier studies (Pipin and Seehafer 2009, Seehafer and Pipin 2009). The dynamo effect due to large-scale current gives an additional source of large-scale poloidal magnetic field. This can help to solve the issue with the dynamo period being otherwise too short. Also, in the models the large-scale current dynamo effect produces less overlapping cycles than dynamo models with α -effect alone. The choice of parameters in the dynamo is justified by our previous studies (Pipin and Seehafer 2009, Pipin and Kosovichev 2011c), where we showed that solar-types dynamos can be obtained for C α /C ω > 2. In those papers we find the approximate threshold to be C α ≈ 0 . 02 for a given diffusivity dilution factor of C η = 0 . 05.\nAs follows from the results given in Fig.4, the kinetic helicity-vorticity pumping effect has a negligible contribution in the near-surface layers, where downward pumping due to density stratification dominates. Therefore, it is expected that the surface dynamo waves are not affected if we discard magnetic helicity\n\n\n
Figure 5. The time-latitude diagrams for the toroidal and radial magnetic fields for the models D1 and D2: a) the model D1, the toroidal field (iso-contours, ± . 25 KG ) near the surface and the radial field (gray-scale density plot); b) the model D1, the toroidal field at the bottom of the solar convection zone, the contours drawn in the range ± . 5 KG ; c) the same as for item a) for the model D2; d) the same as for item b) for the model D2.
\n\nfrom the dynamo equations. Figure 5 shows time-latitude diagrams for toroidal and radial magnetic fields at the surface and for toroidal magnetic field at the bottom of the convection zone for two dynamo models D1 and D2 with and without the helicity-vorticity pumping effect, but magnetic helicity is taken into account as the main dynamo quenching effect. To compare with observational data from a time-latitude diagram of sunspot area (e.g., Hathaway 2011), we multiply the toroidal field component B by factor sin θ . This gives a quantity, which is proportional to the flux of large-scale toroidal field at colatitude θ . We further assume that the sunspot area is related to this flux.\nNear the surface, models D1 and D2 give similar patterns of magnetic field evolution. At the bottom of the convection zone model D1 shows both poleward and equatorward branches of the dynamo wave propagation that is in agreement with the Parker-Yoshimura rule. Both branches have nearly the same time scale that equals glyph[similarequal] 16 years. The results from model D2 show that at the bottom of the convection zone the poleward branch of the dynamo wave dominates. Thus we conclude that the helicity-vorticity pumping effect alters the propagation of the dynamo wave near the bottom of the solar convection zone. We find that models with magnetic helicity contributions to the pumping effect do not change this conclusion.\nFigure 6 shows a typical snapshot of the magnetic helicity distribution in the northern hemisphere for all our models. The helicity has a negative sign in the bulk of the solar convection zone. Regions with positive current helicity roughly correspond to domains of the negative large-scale current helicity concentration.\n\n\n
Figure 8 shows in more detail the latitudinal drift of the maximum of the toroidal magnetic field evolution during the cycle (left panel in the Figure 8),
\n\n\n\n\n×\nFigure 6. Snapshots for the mean magnetic field and the current helicity distributions at the north hemisphere in the model D4. Left panel shows the field lines of the poloidal component of the mean magnetic field. The right panel shows the toroidal magnetic field (iso-contours ± 500G) and the current helicity (gray scale density plot).\nThey are located in the middle of the solar convection zone and at the high and low latitudes near the top of the solar convection zone. As follows from Fig. 6, the pumping effect due to current helicity may be efficient in the upper part of the solar convection zone where it might intensify the equatorial drift of the dynamo wave along iso-surfaces of the angular velocity. We find that the pumping effect that results from magnetic helicity is rather small in our models. This may be due to the weakness of the magnetic field. Observations (Zhang et al. 2010) give about one order magnitude larger current helicity than what\nis shown in Fig. 6. In the model we estimate the current helicity as H C = χ µ 0 glyph[lscript] 2 . This result depends\nessentially on the mixing length parameter glyph[lscript] . The stronger helicity is concentrated to the surface, the larger H C . In observations, we do not know from were the helical magnetic structures come from. In view of the given uncertainties we estimate the probable effect of a larger magnitude of magnetic helicity in the model by increasing the parameter C C to 10 (model D3). In addition, we consider the results for the nonlinear model D4. It has a higher C α and a lower R χ to increase the nonlinear impact of the magnetic helicity on the large-scale magnetic field evolution.\nThe top panel of Figure 7 shows a time-latitude diagram of toroidal magnetic field and current helicity evolution near the surface for model D4. We find a positive sign of current helicity at the decay edges of the toroidal magnetic field butterfly diagram. There are also areas with positive magnetic helicity at high latitudes at the growing edges of the toroidal magnetic field butterfly diagram. The induced pumping velocity is about 1 cm s -1 . The increase of the magnetic helicity pumping effect by a factor of 10 (model D3) shifts the latitude of the maximum of the toroidal magnetic field by about 5 · toward the equator. The induced pumping velocity is about 5 cm s -1 .\nStronger nonlinearity (model D4) and a stronger magnetic helicity pumping effect (model D3) modify the butterfly diagram in different ways. Model D3 shows a simple shift of the maximum of toroidal magnetic field toward the equator. Model D4 shows a fast drift of large-scale toroidal field at the beginning of a cycle and a slow-down of the drift velocity as the cycle progresses.\nλ max ( t ) = 90 · -max θ> 45 · ( | B S ( θ ) | sin θ ) , (24)\nand the latitudinal drift of the centroid position of the toroidal magnetic field flux (cf. Hathaway 2011)\nλ C ( t ) = 90 · -∫ π/ 2 0 θB S ( θ ) sin θdθ ∫ π/ 2 0 B S ( θ ) sin θdθ , (25)\nwhere B S ( θ ) = 〈 B ( r, θ ) 〉 (0 . 9 , 0 . 99) R is the toroidal magnetic field, which is averaged over the surface layers. Note that the overlap between subsequent cycles influences the value of λ C more than the value of λ max . The behaviour of λ max in models D1,D2 and D3 reproduces qualitatively the exponential drift of maximum\n\n\n\n×\nC\n\n\n
Figure 7. Top, the near-surface time-latitude diagrams for the toroidal magnetic field and the current helicity for the models D4. Bottom, the near-surface time-latitude diagrams for the toroidal magnetic field and the latitudinal component of the drift velocity induced by the magnetic helicity for the model D3.Figure 8. The drift of the latitude of maximum (left) and the centroid position of the magnetic flux at the near-surface layer in the models D(1-4). The dash-dotted line shows results for the model D1, the red dashed line - for the model D2, the solid black line - for the model D3, the black dashed line - for the model D4 and the solid green line shows the exponential law of the sunspot area centroid drift, as suggested by Hathaway (2011).
\n\nlatitude as suggested by Hathaway (2011):\nλ C ( t ) = 28 · exp ( -12 t 90 ) ,\nwhere t is time measured in years. Model D4 shows a change between fast (nearly steady dynamo wave) drift at the beginning of the cycle to slow drift at the decaying phase of the cycle. The overlap between subsequent cycles is growing from model D1 to model D4. In all the models the highest latitude of the centroid position of the toroidal magnetic flux is below 30 · . Models D3 and D4 have nearly equal starting latitude of the centroid position. It is about 24 · . This means that a model with increased magnetic helicity pumping produces nearly the same effect for the shift of the centroid position as a model with a strong nonlinear effect of magnetic helicity.\n3 Discussion and conclusions\nWe have shown that the interaction of helical convective motions and differential rotation in the solar convection zone produces a turbulent drift of large-scale magnetic field. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. The large-scale vorticity vector roughly follows to iso-surfaces of angular velocity. Since the direction of the drift depends on the sign of helicity, the pumping effect is governed by the Parker-Yoshimura rule (Parker 1955, Yoshimura 1975).\nThe effect is computed within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. In the calculations, we have assumed that the turbulent kinetic and current helicities are given. The calculations were done for arbitrary Coriolis number. In agreement with Mitra et al. (2009) and Rogachevskii et al. (2011), the analytical calculations show that the leading effect of pumping is described by a large-scale magnetic drift in the direction of the large-scale vorticity vector and by anisotropic pumping which produces a drift of toroidal and poloidal components of the field in opposite directions. The component of the drift that is induced by global rotation and helicity (second line in Eq. (3)) is rather small compared to the main effect. The latter conclusion should be checked separately for a different model of background turbulence, taking into account the generation of kinetic helicity due to global rotation and stratification in a turbulent medium.\nWe have estimated the pumping effect for the solar convection zone and compared it with other turbulent pumping effects including diamagnetic pumping and turbulent pumping that results from magnetic fluctuations in stratified turbulence (Kichatinov 1991, Pipin 2008). The latter is sometimes referred to as 'density-gradient pumping effect' (Krivodubskij 2004). The diamagnetic pumping is upward in the upper part of the convection zone and downward near the bottom. The velocity field of density-gradient pumping is more complicated (see Figure 4). However, its major effect is concentrated near the surface. Both diamagnetic pumping and density-gradient pumping effects are quenched inversely proportional to the Coriolis number (Kichatinov 1991, Pipin 2008). The helicity-vorticity pumping effect modifies the direction of large-scale magnetic drift at the bottom of the convection zone. This effect was illustrated by a dynamo model that shows a dominant poleward branch of the dynamo wave at the bottom of the convection zone.\nIt is found that the magnetic helicity contribution of the pumping effect can be important for explaining the fine structure of the sunspot butterfly diagram. In particular, the magnetic helicity contribution results in a slow-down of equatorial propagation of the dynamo wave. The slow-down starts just before the maximum of the cycle. Observations indicate a similar behavior in sunspot activity (Ternullo 2007, Hathaway 2011). A behavior like this can be seen in flux-transport models as well (Rempel 2006). 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A new dynamo pattern revealed by solar helical magnetic fields. Mon. Not. Roy. Astron. Soc. , 402:L30-L33, 2010.\n\nAppendix A\nTo compute E it is convenient to write equations (1) and (2) in Fourier space:\n( ∂ ∂t + ηz 2 ) ˆ b j = i µ z l ˆ u j ( z ) ¯ B l + (26) + iz l ∫ [ ˆ b l ( z -q ) ̂ ¯ V j ( q ) -ˆ b j ( z -q ) ̂ ¯ V l ( q ) ] d q + ̂ G j . ( ∂ ∂t + νz 2 ) ˆ u i = ˆ f i + ̂ F i -2 ( Ω ̂ z ) ( ̂ z × ̂ u ) i (27) -iπ if ( z ) z l ∫ [ ˆ u l ( z -q ) ̂ ¯ V f ( q ) + ˆ u f ( z -q ) ̂ ¯ V l ( q ) ] d q + i µ ˆ b i ( z ) ( z · ¯ B ) ,\nwhere the turbulent pressure was excluded from (2) by convolution with tensor π ij ( z ) = δ ij -ˆ z i ˆ z j , δ ij is the Kronecker symbol and ̂ z is a unit wave vector. The equations for the second-order moments that make contributions to the mean-electro-motive force(MEMF) can be found directly from (26, 27). As the preliminary step we write the equations for the second-order products of the fluctuating fields, and make the ensemble averaging of them,\n∂ ∂t 〈 ˆ u i ( z ) ˆ b j ( z ' ) 〉 = i z ' l ¯ B l 〈 ˆ u i ( z ) ˆ u j ( z ' )〉 -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z ) ˆ b j ( z ' ) 〉 (28) + i z ' l ∫ [〈 ˆ u i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ u i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V l ( q ) ] d q + i µ ¯ B l z l 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 + Th κ ij ( z , z ' ) -( ηz ' 2 + νz 2 ) 〈 ˆ u i ( z ) ˆ b j ( z ' ) 〉 , ∂ ∂t 〈 ˆ u i ( z ) ˆ u j ( z ' )〉 = -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z )ˆ u j ( z ' ) 〉 -2 ( Ω ̂ z ' ) ε jln ̂ z ' l 〈 ˆ u i ( z )ˆ u n ( z ' ) 〉 (29) -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V l ( q ) ] d q -i π jf ( z ' ) z ' l ∫ [〈 ˆ u i ( z ) ˆ u l ( z -q ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u i ( z ) ˆ u f ( z -q ) 〉 ̂ ¯ V l ( q ) ] d q + Th v ij ( z , z ' ) -ν ( z ' 2 + z 2 ) 〈 ˆ u i ( z ) ˆ u j ( z ' )〉 , ∂ ∂t 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 = Th h ij ( z , z ' ) -( ηz ' 2 + ηz 2 ) 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 (30) + i z ' l ∫ [〈 ˆ b i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ b i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q ,\nwhere, the terms Th ( κ ,v,h ) ij involve the third-order moments of fluctuating fields and second-order moments of them with the forcing term. Next, we apply the τ -approximation, substituting the Th ( κ ,v,h ) ij -terms by the corresponding τ relaxation terms of the second-order contributions,\nTh ( κ ) ij →-〈 ˆ m i ( z ) ˆ b j ( z ' ) 〉 /τ c , (31)\nTh ( v ) ij →-〈 ˆ m i ( z ) ˆ m j ( z ' ) 〉 - 〈 ˆ m i ( z ) ˆ m j ( z ' ) 〉 (0) τ c , (32)\nTh ( h ) ij →-〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 -〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 (0) τ c , (33)\nwhere the superscript . . . (0) denotes the moments of the background turbulence. Approximating these complicated contributions by the simple relaxation terms has to be considered as a questionable assumption. It involves additional assumptions (see Radler and Rheinhardt 2007), e.g., it is assumed that the second-order correlations in Eq. (8) do not vary significantly on the time scale of τ c . This assumption is consistent with scale separation between the mean and fluctuating quantities in the mean-field magneto hydrodynamics. The reader can find a comprehensive discussion of the τ -approximation in the above cited papers. Furthermore, we restrict ourselves to the high Reynolds numbers limit and discard the microscopic diffusion terms. The contributions of the mean magnetic field in the turbulent stresses will be neglected because they give the nonlinear terms in the cross helicity tensor. Also, τ c is independent on k (cf, Radler\net al. 2003, Rogachevskii and Kleeorin 2003, Brandenburg and Subramanian 2005) and it is independent on the mean fields as well. This should be taken into account in considering the nonlinear effects due to rotation. Taking all the above assumptions into account, we get the system of equations for the moments for the stationary case:\n〈\n〈 ˆ u i ( z ) ˆ b j ( z ' ) 〉 τ c = -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z ) ˆ b j ( z ' ) 〉 + i z ' l ¯ B l ˆ u i ( z ) ˆ u j ( z ' ) 〉 (34) + i z ' l ∫ [〈 ˆ u i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ u i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V l ( q ) ] d q + i µ ¯ B l z l 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 , ˆ m i ( z ) ˆ m j ( z ' ) 〉 τ c = -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z )ˆ u j ( z ' ) 〉 -2 ( Ω ̂ z ' ) ε jln ̂ z ' l 〈 ˆ u i ( z )ˆ u n ( z ' ) 〉 (35) -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V l ( q ) ] d q -i π jf ( z ' ) z ' l ∫ [〈 ˆ u i ( z ) ˆ u l ( z -q ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u i ( z ) ˆ u f ( z -q ) 〉 ̂ ¯ V l ( q ) ] d q + 〈 ˆ m i ( z ) ˆ m j ( z ' ) 〉 (0) τ c 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 τ c = 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 (0) τ c (36) + i z ' l ∫ [〈 ˆ b i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ b i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q ,\nTo proceed further, we have to introduce some conventions and notations that are widely used in the literature. The double Fourier transformation of an ensemble average of two fluctuating quantities, say f and g , taken at equal times and at the different positions x , x ' , is given by\n〈 f ( x ) g ( x ' )〉 = ∫ ∫ 〈 ˆ f ( z ) ˆ g ( z ' ) 〉 e i ( z · x + z ' · x ' ) d 3 z d 3 z ' . (37)\nIn the spirit of the general formalism of the two-scale approximation (Roberts and Soward 1975) we introduce 'fast' and 'slow' variables. They are defined by the relative r = x -x ' and the mean R = 1 2 ( x + x ' ) coordinates, respectively. Then, Eq. (37) can be written in the form\n〈 f ( x ) g ( x ' )〉 = ∫ ∫ 〈 ˆ f ( k + 1 2 K ) ˆ g ( -k + 1 2 K )〉 e i ( K · R + k · r ) d 3 K d 3 k , (38)\nwhere we have introduced the wave vectors k = 1 2 ( z -z ' ) and K = z + z ' . Then, following to Subramanian and Brandenburg (2004), we define the correlation function of ̂ f and ̂ g obtained from (38) by integration with respect to K ,\nΦ ( ˆ f, ˆ g, k , R ) = ∫ 〈 ˆ f ( k + 1 2 K ) ˆ g ( -k + 1 2 K )〉 e i ( K · R ) d 3 K . (39)\nFor further convenience we define the second order correlations of momentum density, magnetic fluctuations and the cross-correlations of momentum and magnetic fluctuations via\nˆ v ij ( k , R ) = Φ(ˆ u i , ˆ u j , k , R ) , 〈 u 2 〉 ( R ) = ∫ ˆ v ii ( k , R ) d 3 k , (40)\nˆ h ij ( k , R ) = Φ( ˆ b i , ˆ b j , k , R ) , 〈 b 2 〉 ( R ) = ∫ ˆ h ii ( k , R ) d 3 k , (41)\nˆ κ ij ( k , R ) = Φ(ˆ u i , ˆ b j , k , R ) , E i ( R ) = ε ijk ∫ ˆ κ jk ( k , R ) d 3 k . (42)\nWe now return to equations (34), (35) and (36). As the first step, we solve these equations about Ω (nonlinear effects of the Coriolis force) and make the Taylor expansion with respect to the 'slow' variables and take the Fourier transformation, (39), about them. The details of this procedure can be found in (Subramanian and Brandenburg 2004). In result we get the following equations for the second moments\nˆ κ ij τ c = -ι D (0) if ( B · k ) ( v fj -m fj µ ) + D (0) if ¯ V j,l ˆ κ fl -D (0) if ¯ V f,l ˆ κ lj + + 2 D (0) ip ˆ k p ˆ k f ˆ κ lj ¯ V f,l + D (0) if k l ¯ V f,l ∂ ˆ κ fj ∂k f ,\nD (0) if = δ if + ψ Ω ˆ k p ε ifp + ψ 2 Ω ˆ k i ˆ k f 1 + ψ 2 Ω , ψ Ω = 2 ( Ω · ˆ k ) τ c\nˆ v ij τ c = T (0) ijnm ( ˆ v (0) nm τ c +2 ˆ k f ¯ V f,l ( ˆ k n ˆ v lm + ˆ k m ˆ v nl ) -¯ V n,l ˆ v lm -¯ V m,l ˆ v nl + k l ¯ V f,l ∂ ˆ v nm ∂k f ) ,\nΩ\n(44) T (0) ijnm = δ in δ jm + ψ Ω ˆ k p M ( ε inp δ jm + ε jmp δ in ) -ψ 2 Ω M ( δ ij π nm -δ nm ˆ k i ˆ k j + δ im ˆ k n ˆ k j + δ nj ˆ k i ˆ k m -2 δ n [ i δ j ] m ) , M = 1 + 4 ψ 2\nˆ h ij = ˆ h (0) ij + τ c ˆ h il ¯ V j,l + τ c ˆ h lj ¯ V i,l + τ c k l ¯ V f,l ∂ ˆ h ij ∂k f (45)\nThese equations were solved with respect to the shear tensor, V i,j = ∇ j V i , by means of perturbation procedure. One remains to define the spectra of the background turbulence. We will adopt the isotropic form of the spectra (Roberts and Soward 1975). Additionally, the background magnetic fluctuations are helical while there is no prescribed kinetic helicity in the background turbulence:\nˆ v (0) ij = { π ij ( k ) E ( k, R ) 8 πk 2 -i ε ijp k p H ( k, R ) 8 πk 4 } , (46)\nˆ h (0) ij = { π ij ( k ) B ( k, R ) 8 πk 2 -i ε ijp k p N ( k, R ) 8 πk 4 } , (47)\nwhere, the spectral functions E ( k, R ) , B ( k, R ) , N ( k, R ) define, respectively, the intensity of the velocity fluctuations, the intensity of the magnetic fluctuations and amount of current helicity in the background\n(43)\nturbulence. They are defined via\n〈 u (0)2 〉 = ∫ E ( k, R ) 4 πk 2 d 3 k , 〈 b (0)2 〉 = ∫ B ( k, R ) 4 πk 2 d 3 k , (48) h (0) K = ∫ H ( k, R ) 4 πk 2 d 3 k , h (0) C = 1 µρ ∫ N ( k, R ) 4 πk 2 d 3 k ,\nwhere h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇× b (0) 〉 µρ . In final results we use the relation between intensities of magnetic and kinetic fluctuations which is defined via B ( k, R ) = εµ ¯ ρE ( k, R ). The state with ε = 1 means equipartition between energies of magnetic an kinetic fluctuations in the background turbulence.\nAppendix B\nf ( γ ) 1 = 1 (24 Ω ∗ ) 2 ( ( 1300 Ω ∗ 2 +391 ) arctan (2 Ω ∗ ) 2 Ω ∗ -1456 ( Ω ∗ 2 +1 ) arctan ( Ω ∗ ) Ω ∗ -3(32 Ω ∗ 2 -355) ) f ( γ ) 2 = 3 4 Ω ∗ 2 ( ( Ω ∗ 2 +1) arctan ( Ω ∗ ) Ω ∗ -1 ) f ( γ ) 3 = -1 36540 Ω ∗ 4 ( 5 (( 10672 Ω ∗ 2 +3872 ) Ω ∗ 2 +337 ) arctan (2 Ω ∗ ) 2 Ω ∗ -320 (( 515 Ω ∗ 2 -128 ) Ω ∗ 2 -895 ) arctan ( Ω ∗ ) Ω ∗ +3 (( 2304 Ω ∗ 2 +3380 ) Ω ∗ 2 -48295 ) ) f ( γ ) 4 = -1 24 Ω ∗ 4 ( 3 (( 11 Ω ∗ 2 +8 ) Ω ∗ 2 -7 ) ) arctan ( Ω ∗ ) Ω ∗ -( 31 Ω ∗ 2 -21 ) ) f ( γ ) 5 = -1 55296 Ω ∗ 4 ( ( 7472 Ω ∗ 4 +5016 Ω ∗ 2 -1685 ) arctan (2 Ω ∗ ) 2 Ω ∗ -32(79 Ω ∗ 4 +1410 Ω ∗ 2 +4475) arctan ( Ω ∗ ) Ω ∗ + 3 ( 9872 Ω ∗ 6 +183632 Ω ∗ 4 +238183 Ω ∗ 2 +48295 ) ( Ω ∗ 2 +1)(4 Ω ∗ 2 +1) ) f ( γ ) 6 = 1 32 Ω ∗ 4 ( (( 9 Ω ∗ 2 +30 ) Ω ∗ 2 -35 ) arctan ( Ω ∗ ) Ω ∗ -(( 101 Ω ∗ 2 +20 ) Ω ∗ 2 -105 ) 3 ( Ω ∗ 2 +1) ) f ( γ ) 7 = -1 55296 Ω ∗ 4 ( ( 20528 Ω ∗ 4 +16536 Ω ∗ 2 -1685 ) arctan (2 Ω ∗ ) 2 Ω ∗ -32 ( 577 Ω ∗ 4 +3660 Ω ∗ 2 +4475 ) arctan ( Ω ∗ ) Ω ∗ + 3 ( 80528 Ω ∗ 6 +282512 Ω ∗ 4 +258343 Ω ∗ 2 +48295 ) ( Ω ∗ 2 +1)(4 Ω ∗ 2 +1) ) )\nf ( γ ) 8 = -( Ω ∗ 4 +7) 32 Ω ∗ 4 ( ( Ω ∗ 2 +5) arctan ( Ω ∗ ) Ω ∗ -( 13 Ω ∗ 2 +15 ) 3( Ω ∗ 2 +1)\n"},"sections":{"kind":"list like","value":[{"title":"ABSTRACT","content":"Geophysical and Astrophysical Fluid Dynamics Vol. 00, No. 00, Month 2013, 1-19","pages":[1]},{"title":"Institute Solar-Terrestrial Physics, Irkutsk, Russia","content":"( July 31, 2021, Revision: 1.20 ) We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker-Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. We estimate the magnitude of the drift velocity and find that it can be several m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. We find that the helicity-vorticity pumping effect can have an influence on the features of the sunspot time-latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.","pages":[1]},{"title":"1 Introduction","content":"It is believed that the evolution of the large-scale magnetic field of the Sun is governed by the interplay between large-scale motions, like differential rotation and meridional circulation, turbulent convection flows and magnetic fields. One of the most important issues in solar dynamo theory is related to the origin of the equatorial drift of sunspot activity in the equatorial regions and, simultaneously at high latitudes, the poleward drift of the location of large-scale unipolar regions and quiet prominences. Parker (1955) and Yoshimura (1975) suggested that the evolution of large-scale magnetic activity of the Sun can be interpreted as dynamo waves propagating perpendicular to the direction of shear from the differential rotation. They found that the propagation can be considered as a diffusion process, which follows the iso-rotation surfaces of angular velocity in the solar convection zone. The direction of propagation can be modified by meridional circulation, anisotropic diffusion and the effects of turbulent pumping (see, e.g., Choudhuri et al. 1995, Kitchatinov 2002, Guerrero and de Gouveia Dal Pino 2008). The latter induces an effective drift of the large-scale magnetic field even though the mean flow of the turbulent medium may be zero. The turbulent pumping effects can be equally important both for dynamos without meridional circulation and for the meridional circulation-dominated dynamo regimes. For the latter case the velocity of turbulent pumping has to be comparable to the meridional circulation speed. It is known that an effect of this magnitude can be produced by diamagnetic pumping and perhaps by so-called topological pumping. Both effects produce pumping in the radial direction and have not a direct impact on the latitudinal drift of the large-scale magnetic field. Recently (Pipin 2008, Mitra et al. 2009, Leprovost and Kim 2010), it has been found that the helical convective motions and the helical turbulent magnetic fields interacting with large-scale magnetic fields and differential rotation can produce effective pumping in the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. It is believed that these regions, namely the tachocline beneath the solar convection zone and the subsurface shear layer, are important for the solar dynamo. Figure 1 illustrates the principal processes that induce the helicity-vorticity pumping effect. It is suggested that this effect produces an anisotropic drift of the large-scale magnetic field, which means that the different components of the large-scale magnetic field drift in different directions. Earlier work, e.g. by Kichatinov (1991) and Kleeorin and Rogachevskii (2003), suggests that the effect of anisotropy in the 2 transport of mean-field is related to nonlinear effects of the global Coriolis force on the convection. Also, nonlinear effects of the large-scale magnetic field result in an anisotropy of turbulent pumping (Kleeorin et al. 1996). It is noteworthy, that the helicity-vorticity effect produces an anisotropy of the large-scale magnetic field drift already in the case of slow rotation and a weak magnetic field. A comprehensive study of the linear helicity-vorticity pumping effect for the case of weak shear and slow rotation was given by Rogachevskii et al. (2011) and their results were extended by DNS with a more general test-field method Brandenburg et al. (2012). In this paper we analytically estimate the helicity-vorticity pumping effect taking into account the Coriolis force due to global rotation. The calculations were done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. The results are applied to mean field dynamo models, which are used to examine this effect on the dynamo. The paper is structured as follows. In the next section we briefly outline the basic equations and assumptions, and consider the results of calculations. Next, we apply the results to the solar dynamo. In Section 3 we summarize the main results of the paper. The details of analytical calculations are given in the Appendices A and B.","pages":[1,2]},{"title":"2 Basic equations","content":"In the spirit of mean-field magnetohydrodynamics, we split the physical quantities of the turbulent conducting fluid into mean and fluctuating parts where the mean part is defined as an ensemble average. One assumes the validity of the Reynolds rules. The magnetic field B and the velocity V are decomposed as B = B + b and V = V + u , respectively. Hereafter, we use small letters for the fluctuating parts and capital letters with an overbar for mean fields. Angle brackets are used for ensemble averages of products. We use the two-scale approximation (Roberts and Soward 1975, Krause and Radler 1980) and assume that mean fields vary over much larger scales (both in time and in space) than fluctuating fields. The average effect of MHD-turbulence on the large-scale magnetic field (LSMF) evolution is described by the mean-electromotive force (MEMF), E = 〈 u × b 〉 . The governing equations for fluctuating magnetic field and velocity are written in a rotating coordinate system as follows: where G , F stand for nonlinear contributions to the fluctuating fields, p is the fluctuating pressure, Ω is the angular velocity responsible for the Coriolis force, ¯ V is mean flow which is a weakly variable in space, and f is the random force driving the turbulence. Equations (1) and (2) are used to compute the mean-electromotive force E = 〈 u × b 〉 . It was computed with the help of the equations for the second moments of fluctuating velocity and magnetic fields using the double-scale Fourier transformation and the minimal τ -approximations and for a given model of background turbulence. To simplify the estimation of nonlinear effects due to global rotation, we use scale-independent background turbulence spectra and correlation time. Details of the calculations are given in Appendix A. In what follows we discuss only those parts of the mean-electromotive force which are related to shear and the pumping effect.","pages":[2,3]},{"title":"2.1 Results","content":"The large-scale shear flow is described by the tensor V i,j = ∇ V i . It can be decomposed into a sum of strain and vorticity tensors, ∇ j V i = 1 2 ( V i,j + V j,i ) -1 2 ε ijp W p , where W = ∇ × V is the large-scale vorticity vector. The joint effect of large-scale shear, helical turbulent flows and magnetic fields can be expressed by the following contributions to the mean-electromotive force (omitting the α -effect): where ˜ V ( B ) = B j 2 ( V i,j + V j,i ) , e = Ω | Ω | is the unit vector along the rotation axis, τ c is the typical relaxation time of turbulent flows and magnetic fields, h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇ × b (0) 〉 µρ are kinetic and current helicity of the background turbulence. These parameters are assumed to be known in advance. Functions f ( γ ) n ( Ω ∗ ) are given in Appendix B, they depend on the Coriolis number Ω ∗ = 2 Ω 0 τ c and describe the nonlinear effect due the Coriolis force, and Ω 0 is the global rotation rate. For slow rotation, Ω ∗ glyph[lessmuch] 1, we perform a Taylor expansion of f ( γ ) n ( Ω ∗ ) and obtain The coefficients in the kinetic part of Eq. (4) are two times larger than those found by Rogachevskii et al. (2011). This difference results from our assumption that the background turbulence spectra and the correlation time are scale-independent. The results for the magnetic part are in agreement with our earlier findings (see Pipin 2008). The first term in Eq. (4) describes turbulent pumping with an effective velocity τ 2 c W 2 ( h C -h K ) and the second term describes anisotropic turbulent pumping. Its structure depends on the geometry of the shear flow. For large Coriolis numbers, Ω ∗ glyph[greatermuch] 1, only the kinetic helicity contributions 4 V. V. Pipin survive: Figure 2 show the dependence of the pumping effects on the Coriolis number. We observe that for the terms ( W × B ) and ˜ V ( B ) the effects of kinetic helicity are non-monotonic and have a maximum at Ω ∗ ≈ 1. The effects of current helicity for these terms are monotonically quenched with increasing values of Ω ∗ . The additional contributions in Eq. (3) are rather small in comparison with the main terms. Thus, we can conclude that the first line in Eq. (3) describes the leading effect of pumping due to the helicity of turbulent flows and magnetic field. Below, we drop the contributions from the second line in Eq. (3) from our analysis.","pages":[3,4]},{"title":"2.2 Helicity-vorticity pumping in the solar convection zone","content":"2.2.1 The dynamo model. To estimate the impact of this pumping effect on the dynamo we consider the example of a dynamo model which takes into account contributions of the mean electromotive force given by Eq. (3). The dynamo model employed in this paper has been described in detail by Pipin and Kosovichev (2011a,c). This type of dynamo was proposed originally by Brandenburg (2005). The reader may find the discussion for different types of mean-field dynamos in Brandenburg and Subramanian (2005) and Tobias and Weiss (2007). We study the standard mean-field induction equation in a perfectly conducting medium: where E = u × b is the mean electromotive force, with u, b being fluctuating velocity and magnetic field, respectively, U is the mean velocity (differential rotation and meridional circulation), and the axisymmetric magnetic field is: where θ is the polar angle. The expression for the mean electromotive force E is given by Pipin (2008). It is expressed as follows: The new addition due to helicity and mean vorticity effects is marked by E H . The tensor α ij represents the α -effect. It includes hydrodynamic and magnetic helicity contributions, where the hydrodynamic part of the α -effect is defined by α ( H ) ij , Λ ( ρ ) = ∇ log ρ quantifies the density stratification, Λ ( u ) = ∇ log ( η (0) T ) quantifies the turbulent diffusivity variation, and e = Ω / | Ω | is a unit vector along the axis of rotation. The turbulent pumping, γ ( Λ ) ij , depends on mean density and turbulent diffusivity stratification, and on the Coriolis number Ω ∗ = 2 τ c Ω 0 where τ c is the typical convective turnover time and Ω 0 is the global angular velocity. Following the results of Pipin (2008), γ ( Λ ) ij is expressed as follows: The effect of turbulent diffusivity, which is anisotropic due to the Coriolis force, is given by: The last term in Eq. (14) describes Radler's Ω × J effect. The functions f ( a,d ) { 1 -11 } depend on the Coriolis number. They can be found in Pipin (2008); see also Pipin and Kosovichev (2011a) or Pipin and Sokoloff b 2 , which measures the ratio between magnetic and kinetic (2011)). In the model, the parameter ε = µ 0 ρ u 2 energies of the fluctuations in the background turbulence, is assumed to be equal to 1. This corresponds to perfect energy equipartition. The ε contribution in the second line of Eq. (12) describes the paramagnetic effect (Kleeorin and Rogachevskii 2003). In the state of perfect energy equipartition the effect of diamagnetic pumping is compensated by the paramagnetic effect. We can, formally, skip the second line in Eq. (12) from our consideration if ε = 1. To compare the magnitude of the helicity-vorticity pumping effect with the diamagnetic effect we will show results for the pumping velocity distribution with ε = 0. The contribution of small-scale magnetic helicity χ = a · b ( a is the fluctuating vector-potential of the magnetic field) to the α -effect is defined as The nonlinear feedback of the large-scale magnetic field to the α -effect is described by a dynamical quenching due to the constraint of magnetic helicity conservation. The magnetic helicity, χ , subject to a conservation law, is described by the following equation (Kleeorin and Rogachevskii 1999, Subramanian and 6 V. V. Pipin Brandenburg 2004): where τ c is a typical convective turnover time. The parameter R χ controls the helicity dissipation rate without specifying the nature of the loss. The turnover time τ c decreases from about 2 months at the bottom of the integration domain, which is located at 0 . 71 R glyph[circledot] , to several hours at the top boundary located at 0 . 99 R glyph[circledot] . It seems reasonable that the helicity dissipation is most efficient near the surface. The last term in Eq. (16) describes a turbulent diffusive flux of magnetic helicity (Mitra et al. 2010). We use the solar convection zone model computed by Stix (2002), in which the mixing-length is defined as glyph[lscript] = α MLT ∣ ∣ Λ ( p ) ∣ ∣ -1 , where Λ ( p ) = ∇ log p quantifies the pressure variation, and α MLT = 2. The turbulent diffusivity is parameterized in the form, η T = C η η (0) T , where η (0) T = u ' glyph[lscript] 3 is the characteristic mixinglength turbulent diffusivity, glyph[lscript] is the typical correlation length of the turbulence, and C η is a constant to control the efficiency of large-scale magnetic field dragging by the turbulent flow. Currently, this parameter cannot be introduced in the mean-field theory in a consistent way. In this paper we use C η = 0 . 05. The differential rotation profile, Ω = Ω 0 f Ω ( x, µ ) (shown in Fig.3a) is a slightly modified version of the analytic approximation proposed by Antia et al. (1998): where Ω 0 = 2 . 87 · 10 -6 s -1 is the equatorial angular velocity of the Sun at the surface, x = r/R glyph[circledot] , φ ( x, x 0 ) = 0 . 5 [1 + tanh [100( x -x 0 )]], and x 0 = 0 . 71. 2.2.2 Pumping effects in the solar convection zone. The components of the strain tensor ˜ V in a spherical coordinate system are given by the matrix: where we take into account only the azimuthal component of the large-scale flow, ˜ V ( r,ϕ ) = r sin θ∂ r Ω ( r, θ ), ˜ V ( θ,ϕ ) = sin θ∂ θ Ω ( r, θ ), so ˆ V ( B ) = ( B ˜ V ( r,ϕ ) , B ˜ V ( θ,ϕ ) , B p i ˜ V ( i,ϕ ) ) . Substituting this into Eq. (3) we find the components of the mean-electromotive force for the helicity-vorticity pumping effect, where h C = C C χ µ 0 ρglyph[lscript] 2 . It remains to define the kinetic helicity distribution. We use a formula proposed in our earlier study (see Kuzanyan et al. 2006), where F 1 ( Ω ∗ ) was defined in the above cited paper. The radial profile of h K cos θ is shown in Figure 3. The radial profile of kinetic helicity is shown in Figure 3a of the above cited paper. The parameters C K , C are introduced to switch on/off the pumping effects in the model. The expressions given by Eq. (3) are valid for the case of weak shear, when τ c max (∣ ∣ ∇ i V j ∣ ∣ ) glyph[lessmuch] 1. In terms of the strain tensor ˜ V this condition of weak shear implies Ω glyph[star] max (∣ ∣ ∣ r∂ r ˜ Ω ∣ ∣ ∣ , ∣ ∣ ∣ ∂ θ ˜ Ω ∣ ∣ ∣ ) glyph[lessmuch] 1. This is not valid at the bottom of the solar convection zone where the radial gradient of the angular velocity is strong and Ω glyph[star] glyph[greatermuch] 1 and τ c max (∣ ∣ ∇ i V j ∣ ∣ ) ≈ 2. Leprovost and Kim (2010) suggested that this pumping effect is quenched with increasing shear inversely proportional to ( τ c max (∣ ∣ ∇ i V j ∣ ∣ )) 1 ... 2 . Therefore, we introduce an ad-hoc quenching function for the pumping effect: where C S is a constant to control the magnitude of the quenching, and s = 1. Results by Leprovost and Kim (2010) suggest 1 < s < 2 in relation to geometry of the large-scale shear. We find that for the solar convection zone the amplitude of the pumping effect does not change very much ( ∼ 1 m / s) with s varying in the range 1 . . . 2. From the given relations, using E ( H ) = U (eff) × B , we find the effective drift velocity, U (eff) , due to the helicity-vorticity pumping effect. Taking into account the variation of turbulence parameters in the solar convection zone we compute U (eff) . The bottom panel of Figure 3 shows the distribution of the velocity field U ( eff ) for the helicity-vorticity pumping effect for the toroidal and poloidal components of the large-scale magnetic field. The maximum velocity drift occurs in the middle and at the bottom of the convection zone. The direction of drift has equatorial and polar cells corresponding to two regions in the solar convection zone with different signs of the radial gradient of the angular velocity. The anisotropy in transport of the toroidal and poloidal components of the large-scale magnetic field is clearly seen. The other important pumping effects are due to mean density and turbulence intensity gradients (Zeldovich 1957, Kichatinov 1991, Kichatinov and Rudiger 1992, Tobias et al. 2001). These effects were estimated using Eq. (12). For these calculations we put C η = 1, ε = 0, C K = 1 and χ = 0. Figure 4 shows the sum of the pumping effects for the toroidal and poloidal components of mean magnetic fields including the helicity-vorticity pumping effect. In agreement with previous studies, it is found that the radial direction is the principal direction of mean-field transport in the solar convection zone. In its upper part the transport is downward because of pumping due to the density gradient (Kichatinov 1991). At the bottom of the convection zone the diamagnetic pumping effect produces downward transport as well (Kichatinov 1991, Rudiger and Brandenburg 1995). The diamagnetic pumping is quenched inversely proportional to the Coriolis number (e.g., Kichatinov 1991, Pipin 2008) and it has the same order of magnitude as the helicity-vorticity pumping effect. The latter effect modifies the direction of effective drift of the toroidal magnetic field near the bottom of the convection zone. There is also upward drift of the toroidal field at · low latitudes in the middle of the convection zone. It results from the combined effects of density gradient and global rotation (Kichatinov 1991, Krivodubskij 2004). For the poloidal magnetic field the transport is downward everywhere in the convection zone. At the bottom of the convection zone the action of the diamagnetic pumping on the meridional component of the large-scale magnetic field is amplified due to the helicity-vorticity pumping effect. The obtained pattern of large-scale magnetic field drift in the solar convection zone does not take into account nonlinear effects, e.g., because of magnetic buoyancy. The effect of mean-field buoyancy is rather small compared with flux-tube buoyancy (Kichatinov and Pipin 1993, cf. Guerrero and Kapyla 2011). To find out the current helicity counterpart of the pumping effect we analyze dynamo models by solving Eqs. (6, 16). The governing parameters of the model are C η = 0 . 05, C ω = 1 3 C α . We discuss the choice of the governing parameters later. The other parameters of the model are given in the Table 1. Because of the weakening factor C η the magnitude of the pumping velocity is about one order of magnitude smaller than what is shown in Figure 4. Following Pipin and Kosovichev (2011b), we use a combination of 'open' and 'closed' boundary conditions at the top, controlled by a parameter δ = 0 . 95, with This is similar to the boundary condition discussed by Kitchatinov et al. (2000). For the poloidal field we apply a combination of the local condition A = 0 and the requirement of a smooth transition from the internal poloidal field to the external potential (vacuum) field: We assume perfect conductivity at the bottom boundary with standard boundary conditions. For the magnetic helicity, similar to Guerrero et al. (2010), we put ∇ r ¯ χ = 0 at the bottom of the domain and ¯ χ = 0 at the top of the convection zone. In this paper we study dynamo models which include Radler's Ω × J dynamo effect due to a large-scale current and global rotation (Radler 1969). There is also a dynamo effect due to large-scale shear and current (Rogachevskii and Kleeorin 2003). The motivation to consider these addional turbulent sources in the mean-field dynamo comes from DNS dynamo experiments (Brandenburg and Kapyla 2007, Kapyla et al. 2008, Hughes and Proctor 2009, Kapyla et al. 2009) and from our earlier studies (Pipin and Seehafer 2009, Seehafer and Pipin 2009). The dynamo effect due to large-scale current gives an additional source of large-scale poloidal magnetic field. This can help to solve the issue with the dynamo period being otherwise too short. Also, in the models the large-scale current dynamo effect produces less overlapping cycles than dynamo models with α -effect alone. The choice of parameters in the dynamo is justified by our previous studies (Pipin and Seehafer 2009, Pipin and Kosovichev 2011c), where we showed that solar-types dynamos can be obtained for C α /C ω > 2. In those papers we find the approximate threshold to be C α ≈ 0 . 02 for a given diffusivity dilution factor of C η = 0 . 05. As follows from the results given in Fig.4, the kinetic helicity-vorticity pumping effect has a negligible contribution in the near-surface layers, where downward pumping due to density stratification dominates. Therefore, it is expected that the surface dynamo waves are not affected if we discard magnetic helicity from the dynamo equations. Figure 5 shows time-latitude diagrams for toroidal and radial magnetic fields at the surface and for toroidal magnetic field at the bottom of the convection zone for two dynamo models D1 and D2 with and without the helicity-vorticity pumping effect, but magnetic helicity is taken into account as the main dynamo quenching effect. To compare with observational data from a time-latitude diagram of sunspot area (e.g., Hathaway 2011), we multiply the toroidal field component B by factor sin θ . This gives a quantity, which is proportional to the flux of large-scale toroidal field at colatitude θ . We further assume that the sunspot area is related to this flux. Near the surface, models D1 and D2 give similar patterns of magnetic field evolution. At the bottom of the convection zone model D1 shows both poleward and equatorward branches of the dynamo wave propagation that is in agreement with the Parker-Yoshimura rule. Both branches have nearly the same time scale that equals glyph[similarequal] 16 years. The results from model D2 show that at the bottom of the convection zone the poleward branch of the dynamo wave dominates. Thus we conclude that the helicity-vorticity pumping effect alters the propagation of the dynamo wave near the bottom of the solar convection zone. We find that models with magnetic helicity contributions to the pumping effect do not change this conclusion. Figure 6 shows a typical snapshot of the magnetic helicity distribution in the northern hemisphere for all our models. The helicity has a negative sign in the bulk of the solar convection zone. Regions with positive current helicity roughly correspond to domains of the negative large-scale current helicity concentration. × They are located in the middle of the solar convection zone and at the high and low latitudes near the top of the solar convection zone. As follows from Fig. 6, the pumping effect due to current helicity may be efficient in the upper part of the solar convection zone where it might intensify the equatorial drift of the dynamo wave along iso-surfaces of the angular velocity. We find that the pumping effect that results from magnetic helicity is rather small in our models. This may be due to the weakness of the magnetic field. Observations (Zhang et al. 2010) give about one order magnitude larger current helicity than what is shown in Fig. 6. In the model we estimate the current helicity as H C = χ µ 0 glyph[lscript] 2 . This result depends essentially on the mixing length parameter glyph[lscript] . The stronger helicity is concentrated to the surface, the larger H C . In observations, we do not know from were the helical magnetic structures come from. In view of the given uncertainties we estimate the probable effect of a larger magnitude of magnetic helicity in the model by increasing the parameter C C to 10 (model D3). In addition, we consider the results for the nonlinear model D4. It has a higher C α and a lower R χ to increase the nonlinear impact of the magnetic helicity on the large-scale magnetic field evolution. The top panel of Figure 7 shows a time-latitude diagram of toroidal magnetic field and current helicity evolution near the surface for model D4. We find a positive sign of current helicity at the decay edges of the toroidal magnetic field butterfly diagram. There are also areas with positive magnetic helicity at high latitudes at the growing edges of the toroidal magnetic field butterfly diagram. The induced pumping velocity is about 1 cm s -1 . The increase of the magnetic helicity pumping effect by a factor of 10 (model D3) shifts the latitude of the maximum of the toroidal magnetic field by about 5 · toward the equator. The induced pumping velocity is about 5 cm s -1 . Stronger nonlinearity (model D4) and a stronger magnetic helicity pumping effect (model D3) modify the butterfly diagram in different ways. Model D3 shows a simple shift of the maximum of toroidal magnetic field toward the equator. Model D4 shows a fast drift of large-scale toroidal field at the beginning of a cycle and a slow-down of the drift velocity as the cycle progresses. and the latitudinal drift of the centroid position of the toroidal magnetic field flux (cf. Hathaway 2011) where B S ( θ ) = 〈 B ( r, θ ) 〉 (0 . 9 , 0 . 99) R is the toroidal magnetic field, which is averaged over the surface layers. Note that the overlap between subsequent cycles influences the value of λ C more than the value of λ max . The behaviour of λ max in models D1,D2 and D3 reproduces qualitatively the exponential drift of maximum × C latitude as suggested by Hathaway (2011): where t is time measured in years. Model D4 shows a change between fast (nearly steady dynamo wave) drift at the beginning of the cycle to slow drift at the decaying phase of the cycle. The overlap between subsequent cycles is growing from model D1 to model D4. In all the models the highest latitude of the centroid position of the toroidal magnetic flux is below 30 · . Models D3 and D4 have nearly equal starting latitude of the centroid position. It is about 24 · . This means that a model with increased magnetic helicity pumping produces nearly the same effect for the shift of the centroid position as a model with a strong nonlinear effect of magnetic helicity.","pages":[4,5,6,7,8,9,10,11,12]},{"title":"3 Discussion and conclusions","content":"We have shown that the interaction of helical convective motions and differential rotation in the solar convection zone produces a turbulent drift of large-scale magnetic field. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. The large-scale vorticity vector roughly follows to iso-surfaces of angular velocity. Since the direction of the drift depends on the sign of helicity, the pumping effect is governed by the Parker-Yoshimura rule (Parker 1955, Yoshimura 1975). The effect is computed within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. In the calculations, we have assumed that the turbulent kinetic and current helicities are given. The calculations were done for arbitrary Coriolis number. In agreement with Mitra et al. (2009) and Rogachevskii et al. (2011), the analytical calculations show that the leading effect of pumping is described by a large-scale magnetic drift in the direction of the large-scale vorticity vector and by anisotropic pumping which produces a drift of toroidal and poloidal components of the field in opposite directions. The component of the drift that is induced by global rotation and helicity (second line in Eq. (3)) is rather small compared to the main effect. The latter conclusion should be checked separately for a different model of background turbulence, taking into account the generation of kinetic helicity due to global rotation and stratification in a turbulent medium. We have estimated the pumping effect for the solar convection zone and compared it with other turbulent pumping effects including diamagnetic pumping and turbulent pumping that results from magnetic fluctuations in stratified turbulence (Kichatinov 1991, Pipin 2008). The latter is sometimes referred to as 'density-gradient pumping effect' (Krivodubskij 2004). The diamagnetic pumping is upward in the upper part of the convection zone and downward near the bottom. The velocity field of density-gradient pumping is more complicated (see Figure 4). However, its major effect is concentrated near the surface. Both diamagnetic pumping and density-gradient pumping effects are quenched inversely proportional to the Coriolis number (Kichatinov 1991, Pipin 2008). The helicity-vorticity pumping effect modifies the direction of large-scale magnetic drift at the bottom of the convection zone. This effect was illustrated by a dynamo model that shows a dominant poleward branch of the dynamo wave at the bottom of the convection zone. It is found that the magnetic helicity contribution of the pumping effect can be important for explaining the fine structure of the sunspot butterfly diagram. In particular, the magnetic helicity contribution results in a slow-down of equatorial propagation of the dynamo wave. The slow-down starts just before the maximum of the cycle. Observations indicate a similar behavior in sunspot activity (Ternullo 2007, Hathaway 2011). A behavior like this can be seen in flux-transport models as well (Rempel 2006). For the time being it is unclear what are the differences between different dynamo models and how well do they reproduce the observations. A more detailed analysis is needed.","pages":[13]},{"title":"REFERENCES","content":"065903 . Radler, K.-H.. On the electrodynamics of turbulent fields under the influence of corilois forces. Monats. Dt. Akad. Wiss., 11:, 194-201, 1969.","pages":[15]},{"title":"Appendix A","content":"To compute E it is convenient to write equations (1) and (2) in Fourier space: where the turbulent pressure was excluded from (2) by convolution with tensor π ij ( z ) = δ ij -ˆ z i ˆ z j , δ ij is the Kronecker symbol and ̂ z is a unit wave vector. The equations for the second-order moments that make contributions to the mean-electro-motive force(MEMF) can be found directly from (26, 27). As the preliminary step we write the equations for the second-order products of the fluctuating fields, and make the ensemble averaging of them, where, the terms Th ( κ ,v,h ) ij involve the third-order moments of fluctuating fields and second-order moments of them with the forcing term. Next, we apply the τ -approximation, substituting the Th ( κ ,v,h ) ij -terms by the corresponding τ relaxation terms of the second-order contributions, where the superscript . . . (0) denotes the moments of the background turbulence. Approximating these complicated contributions by the simple relaxation terms has to be considered as a questionable assumption. It involves additional assumptions (see Radler and Rheinhardt 2007), e.g., it is assumed that the second-order correlations in Eq. (8) do not vary significantly on the time scale of τ c . This assumption is consistent with scale separation between the mean and fluctuating quantities in the mean-field magneto hydrodynamics. The reader can find a comprehensive discussion of the τ -approximation in the above cited papers. Furthermore, we restrict ourselves to the high Reynolds numbers limit and discard the microscopic diffusion terms. The contributions of the mean magnetic field in the turbulent stresses will be neglected because they give the nonlinear terms in the cross helicity tensor. Also, τ c is independent on k (cf, Radler et al. 2003, Rogachevskii and Kleeorin 2003, Brandenburg and Subramanian 2005) and it is independent on the mean fields as well. This should be taken into account in considering the nonlinear effects due to rotation. Taking all the above assumptions into account, we get the system of equations for the moments for the stationary case: To proceed further, we have to introduce some conventions and notations that are widely used in the literature. The double Fourier transformation of an ensemble average of two fluctuating quantities, say f and g , taken at equal times and at the different positions x , x ' , is given by In the spirit of the general formalism of the two-scale approximation (Roberts and Soward 1975) we introduce 'fast' and 'slow' variables. They are defined by the relative r = x -x ' and the mean R = 1 2 ( x + x ' ) coordinates, respectively. Then, Eq. (37) can be written in the form where we have introduced the wave vectors k = 1 2 ( z -z ' ) and K = z + z ' . Then, following to Subramanian and Brandenburg (2004), we define the correlation function of ̂ f and ̂ g obtained from (38) by integration with respect to K , For further convenience we define the second order correlations of momentum density, magnetic fluctuations and the cross-correlations of momentum and magnetic fluctuations via We now return to equations (34), (35) and (36). As the first step, we solve these equations about Ω (nonlinear effects of the Coriolis force) and make the Taylor expansion with respect to the 'slow' variables and take the Fourier transformation, (39), about them. The details of this procedure can be found in (Subramanian and Brandenburg 2004). In result we get the following equations for the second moments These equations were solved with respect to the shear tensor, V i,j = ∇ j V i , by means of perturbation procedure. One remains to define the spectra of the background turbulence. We will adopt the isotropic form of the spectra (Roberts and Soward 1975). Additionally, the background magnetic fluctuations are helical while there is no prescribed kinetic helicity in the background turbulence: where, the spectral functions E ( k, R ) , B ( k, R ) , N ( k, R ) define, respectively, the intensity of the velocity fluctuations, the intensity of the magnetic fluctuations and amount of current helicity in the background turbulence. They are defined via where h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇× b (0) 〉 µρ . In final results we use the relation between intensities of magnetic and kinetic fluctuations which is defined via B ( k, R ) = εµ ¯ ρE ( k, R ). The state with ε = 1 means equipartition between energies of magnetic an kinetic fluctuations in the background turbulence.","pages":[15,16,17,18,19]}],"string":"[\n {\n \"title\": \"ABSTRACT\",\n \"content\": \"Geophysical and Astrophysical Fluid Dynamics Vol. 00, No. 00, Month 2013, 1-19\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Institute Solar-Terrestrial Physics, Irkutsk, Russia\",\n \"content\": \"( July 31, 2021, Revision: 1.20 ) We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker-Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. We estimate the magnitude of the drift velocity and find that it can be several m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. We find that the helicity-vorticity pumping effect can have an influence on the features of the sunspot time-latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"It is believed that the evolution of the large-scale magnetic field of the Sun is governed by the interplay between large-scale motions, like differential rotation and meridional circulation, turbulent convection flows and magnetic fields. One of the most important issues in solar dynamo theory is related to the origin of the equatorial drift of sunspot activity in the equatorial regions and, simultaneously at high latitudes, the poleward drift of the location of large-scale unipolar regions and quiet prominences. Parker (1955) and Yoshimura (1975) suggested that the evolution of large-scale magnetic activity of the Sun can be interpreted as dynamo waves propagating perpendicular to the direction of shear from the differential rotation. They found that the propagation can be considered as a diffusion process, which follows the iso-rotation surfaces of angular velocity in the solar convection zone. The direction of propagation can be modified by meridional circulation, anisotropic diffusion and the effects of turbulent pumping (see, e.g., Choudhuri et al. 1995, Kitchatinov 2002, Guerrero and de Gouveia Dal Pino 2008). The latter induces an effective drift of the large-scale magnetic field even though the mean flow of the turbulent medium may be zero. The turbulent pumping effects can be equally important both for dynamos without meridional circulation and for the meridional circulation-dominated dynamo regimes. For the latter case the velocity of turbulent pumping has to be comparable to the meridional circulation speed. It is known that an effect of this magnitude can be produced by diamagnetic pumping and perhaps by so-called topological pumping. Both effects produce pumping in the radial direction and have not a direct impact on the latitudinal drift of the large-scale magnetic field. Recently (Pipin 2008, Mitra et al. 2009, Leprovost and Kim 2010), it has been found that the helical convective motions and the helical turbulent magnetic fields interacting with large-scale magnetic fields and differential rotation can produce effective pumping in the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. It is believed that these regions, namely the tachocline beneath the solar convection zone and the subsurface shear layer, are important for the solar dynamo. Figure 1 illustrates the principal processes that induce the helicity-vorticity pumping effect. It is suggested that this effect produces an anisotropic drift of the large-scale magnetic field, which means that the different components of the large-scale magnetic field drift in different directions. Earlier work, e.g. by Kichatinov (1991) and Kleeorin and Rogachevskii (2003), suggests that the effect of anisotropy in the 2 transport of mean-field is related to nonlinear effects of the global Coriolis force on the convection. Also, nonlinear effects of the large-scale magnetic field result in an anisotropy of turbulent pumping (Kleeorin et al. 1996). It is noteworthy, that the helicity-vorticity effect produces an anisotropy of the large-scale magnetic field drift already in the case of slow rotation and a weak magnetic field. A comprehensive study of the linear helicity-vorticity pumping effect for the case of weak shear and slow rotation was given by Rogachevskii et al. (2011) and their results were extended by DNS with a more general test-field method Brandenburg et al. (2012). In this paper we analytically estimate the helicity-vorticity pumping effect taking into account the Coriolis force due to global rotation. The calculations were done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. The results are applied to mean field dynamo models, which are used to examine this effect on the dynamo. The paper is structured as follows. In the next section we briefly outline the basic equations and assumptions, and consider the results of calculations. Next, we apply the results to the solar dynamo. In Section 3 we summarize the main results of the paper. The details of analytical calculations are given in the Appendices A and B.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 Basic equations\",\n \"content\": \"In the spirit of mean-field magnetohydrodynamics, we split the physical quantities of the turbulent conducting fluid into mean and fluctuating parts where the mean part is defined as an ensemble average. One assumes the validity of the Reynolds rules. The magnetic field B and the velocity V are decomposed as B = B + b and V = V + u , respectively. Hereafter, we use small letters for the fluctuating parts and capital letters with an overbar for mean fields. Angle brackets are used for ensemble averages of products. We use the two-scale approximation (Roberts and Soward 1975, Krause and Radler 1980) and assume that mean fields vary over much larger scales (both in time and in space) than fluctuating fields. The average effect of MHD-turbulence on the large-scale magnetic field (LSMF) evolution is described by the mean-electromotive force (MEMF), E = 〈 u × b 〉 . The governing equations for fluctuating magnetic field and velocity are written in a rotating coordinate system as follows: where G , F stand for nonlinear contributions to the fluctuating fields, p is the fluctuating pressure, Ω is the angular velocity responsible for the Coriolis force, ¯ V is mean flow which is a weakly variable in space, and f is the random force driving the turbulence. Equations (1) and (2) are used to compute the mean-electromotive force E = 〈 u × b 〉 . It was computed with the help of the equations for the second moments of fluctuating velocity and magnetic fields using the double-scale Fourier transformation and the minimal τ -approximations and for a given model of background turbulence. To simplify the estimation of nonlinear effects due to global rotation, we use scale-independent background turbulence spectra and correlation time. Details of the calculations are given in Appendix A. In what follows we discuss only those parts of the mean-electromotive force which are related to shear and the pumping effect.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.1 Results\",\n \"content\": \"The large-scale shear flow is described by the tensor V i,j = ∇ V i . It can be decomposed into a sum of strain and vorticity tensors, ∇ j V i = 1 2 ( V i,j + V j,i ) -1 2 ε ijp W p , where W = ∇ × V is the large-scale vorticity vector. The joint effect of large-scale shear, helical turbulent flows and magnetic fields can be expressed by the following contributions to the mean-electromotive force (omitting the α -effect): where ˜ V ( B ) = B j 2 ( V i,j + V j,i ) , e = Ω | Ω | is the unit vector along the rotation axis, τ c is the typical relaxation time of turbulent flows and magnetic fields, h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇ × b (0) 〉 µρ are kinetic and current helicity of the background turbulence. These parameters are assumed to be known in advance. Functions f ( γ ) n ( Ω ∗ ) are given in Appendix B, they depend on the Coriolis number Ω ∗ = 2 Ω 0 τ c and describe the nonlinear effect due the Coriolis force, and Ω 0 is the global rotation rate. For slow rotation, Ω ∗ glyph[lessmuch] 1, we perform a Taylor expansion of f ( γ ) n ( Ω ∗ ) and obtain The coefficients in the kinetic part of Eq. (4) are two times larger than those found by Rogachevskii et al. (2011). This difference results from our assumption that the background turbulence spectra and the correlation time are scale-independent. The results for the magnetic part are in agreement with our earlier findings (see Pipin 2008). The first term in Eq. (4) describes turbulent pumping with an effective velocity τ 2 c W 2 ( h C -h K ) and the second term describes anisotropic turbulent pumping. Its structure depends on the geometry of the shear flow. For large Coriolis numbers, Ω ∗ glyph[greatermuch] 1, only the kinetic helicity contributions 4 V. V. Pipin survive: Figure 2 show the dependence of the pumping effects on the Coriolis number. We observe that for the terms ( W × B ) and ˜ V ( B ) the effects of kinetic helicity are non-monotonic and have a maximum at Ω ∗ ≈ 1. The effects of current helicity for these terms are monotonically quenched with increasing values of Ω ∗ . The additional contributions in Eq. (3) are rather small in comparison with the main terms. Thus, we can conclude that the first line in Eq. (3) describes the leading effect of pumping due to the helicity of turbulent flows and magnetic field. Below, we drop the contributions from the second line in Eq. (3) from our analysis.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.2 Helicity-vorticity pumping in the solar convection zone\",\n \"content\": \"2.2.1 The dynamo model. To estimate the impact of this pumping effect on the dynamo we consider the example of a dynamo model which takes into account contributions of the mean electromotive force given by Eq. (3). The dynamo model employed in this paper has been described in detail by Pipin and Kosovichev (2011a,c). This type of dynamo was proposed originally by Brandenburg (2005). The reader may find the discussion for different types of mean-field dynamos in Brandenburg and Subramanian (2005) and Tobias and Weiss (2007). We study the standard mean-field induction equation in a perfectly conducting medium: where E = u × b is the mean electromotive force, with u, b being fluctuating velocity and magnetic field, respectively, U is the mean velocity (differential rotation and meridional circulation), and the axisymmetric magnetic field is: where θ is the polar angle. The expression for the mean electromotive force E is given by Pipin (2008). It is expressed as follows: The new addition due to helicity and mean vorticity effects is marked by E H . The tensor α ij represents the α -effect. It includes hydrodynamic and magnetic helicity contributions, where the hydrodynamic part of the α -effect is defined by α ( H ) ij , Λ ( ρ ) = ∇ log ρ quantifies the density stratification, Λ ( u ) = ∇ log ( η (0) T ) quantifies the turbulent diffusivity variation, and e = Ω / | Ω | is a unit vector along the axis of rotation. The turbulent pumping, γ ( Λ ) ij , depends on mean density and turbulent diffusivity stratification, and on the Coriolis number Ω ∗ = 2 τ c Ω 0 where τ c is the typical convective turnover time and Ω 0 is the global angular velocity. Following the results of Pipin (2008), γ ( Λ ) ij is expressed as follows: The effect of turbulent diffusivity, which is anisotropic due to the Coriolis force, is given by: The last term in Eq. (14) describes Radler's Ω × J effect. The functions f ( a,d ) { 1 -11 } depend on the Coriolis number. They can be found in Pipin (2008); see also Pipin and Kosovichev (2011a) or Pipin and Sokoloff b 2 , which measures the ratio between magnetic and kinetic (2011)). In the model, the parameter ε = µ 0 ρ u 2 energies of the fluctuations in the background turbulence, is assumed to be equal to 1. This corresponds to perfect energy equipartition. The ε contribution in the second line of Eq. (12) describes the paramagnetic effect (Kleeorin and Rogachevskii 2003). In the state of perfect energy equipartition the effect of diamagnetic pumping is compensated by the paramagnetic effect. We can, formally, skip the second line in Eq. (12) from our consideration if ε = 1. To compare the magnitude of the helicity-vorticity pumping effect with the diamagnetic effect we will show results for the pumping velocity distribution with ε = 0. The contribution of small-scale magnetic helicity χ = a · b ( a is the fluctuating vector-potential of the magnetic field) to the α -effect is defined as The nonlinear feedback of the large-scale magnetic field to the α -effect is described by a dynamical quenching due to the constraint of magnetic helicity conservation. The magnetic helicity, χ , subject to a conservation law, is described by the following equation (Kleeorin and Rogachevskii 1999, Subramanian and 6 V. V. Pipin Brandenburg 2004): where τ c is a typical convective turnover time. The parameter R χ controls the helicity dissipation rate without specifying the nature of the loss. The turnover time τ c decreases from about 2 months at the bottom of the integration domain, which is located at 0 . 71 R glyph[circledot] , to several hours at the top boundary located at 0 . 99 R glyph[circledot] . It seems reasonable that the helicity dissipation is most efficient near the surface. The last term in Eq. (16) describes a turbulent diffusive flux of magnetic helicity (Mitra et al. 2010). We use the solar convection zone model computed by Stix (2002), in which the mixing-length is defined as glyph[lscript] = α MLT ∣ ∣ Λ ( p ) ∣ ∣ -1 , where Λ ( p ) = ∇ log p quantifies the pressure variation, and α MLT = 2. The turbulent diffusivity is parameterized in the form, η T = C η η (0) T , where η (0) T = u ' glyph[lscript] 3 is the characteristic mixinglength turbulent diffusivity, glyph[lscript] is the typical correlation length of the turbulence, and C η is a constant to control the efficiency of large-scale magnetic field dragging by the turbulent flow. Currently, this parameter cannot be introduced in the mean-field theory in a consistent way. In this paper we use C η = 0 . 05. The differential rotation profile, Ω = Ω 0 f Ω ( x, µ ) (shown in Fig.3a) is a slightly modified version of the analytic approximation proposed by Antia et al. (1998): where Ω 0 = 2 . 87 · 10 -6 s -1 is the equatorial angular velocity of the Sun at the surface, x = r/R glyph[circledot] , φ ( x, x 0 ) = 0 . 5 [1 + tanh [100( x -x 0 )]], and x 0 = 0 . 71. 2.2.2 Pumping effects in the solar convection zone. The components of the strain tensor ˜ V in a spherical coordinate system are given by the matrix: where we take into account only the azimuthal component of the large-scale flow, ˜ V ( r,ϕ ) = r sin θ∂ r Ω ( r, θ ), ˜ V ( θ,ϕ ) = sin θ∂ θ Ω ( r, θ ), so ˆ V ( B ) = ( B ˜ V ( r,ϕ ) , B ˜ V ( θ,ϕ ) , B p i ˜ V ( i,ϕ ) ) . Substituting this into Eq. (3) we find the components of the mean-electromotive force for the helicity-vorticity pumping effect, where h C = C C χ µ 0 ρglyph[lscript] 2 . It remains to define the kinetic helicity distribution. We use a formula proposed in our earlier study (see Kuzanyan et al. 2006), where F 1 ( Ω ∗ ) was defined in the above cited paper. The radial profile of h K cos θ is shown in Figure 3. The radial profile of kinetic helicity is shown in Figure 3a of the above cited paper. The parameters C K , C are introduced to switch on/off the pumping effects in the model. The expressions given by Eq. (3) are valid for the case of weak shear, when τ c max (∣ ∣ ∇ i V j ∣ ∣ ) glyph[lessmuch] 1. In terms of the strain tensor ˜ V this condition of weak shear implies Ω glyph[star] max (∣ ∣ ∣ r∂ r ˜ Ω ∣ ∣ ∣ , ∣ ∣ ∣ ∂ θ ˜ Ω ∣ ∣ ∣ ) glyph[lessmuch] 1. This is not valid at the bottom of the solar convection zone where the radial gradient of the angular velocity is strong and Ω glyph[star] glyph[greatermuch] 1 and τ c max (∣ ∣ ∇ i V j ∣ ∣ ) ≈ 2. Leprovost and Kim (2010) suggested that this pumping effect is quenched with increasing shear inversely proportional to ( τ c max (∣ ∣ ∇ i V j ∣ ∣ )) 1 ... 2 . Therefore, we introduce an ad-hoc quenching function for the pumping effect: where C S is a constant to control the magnitude of the quenching, and s = 1. Results by Leprovost and Kim (2010) suggest 1 < s < 2 in relation to geometry of the large-scale shear. We find that for the solar convection zone the amplitude of the pumping effect does not change very much ( ∼ 1 m / s) with s varying in the range 1 . . . 2. From the given relations, using E ( H ) = U (eff) × B , we find the effective drift velocity, U (eff) , due to the helicity-vorticity pumping effect. Taking into account the variation of turbulence parameters in the solar convection zone we compute U (eff) . The bottom panel of Figure 3 shows the distribution of the velocity field U ( eff ) for the helicity-vorticity pumping effect for the toroidal and poloidal components of the large-scale magnetic field. The maximum velocity drift occurs in the middle and at the bottom of the convection zone. The direction of drift has equatorial and polar cells corresponding to two regions in the solar convection zone with different signs of the radial gradient of the angular velocity. The anisotropy in transport of the toroidal and poloidal components of the large-scale magnetic field is clearly seen. The other important pumping effects are due to mean density and turbulence intensity gradients (Zeldovich 1957, Kichatinov 1991, Kichatinov and Rudiger 1992, Tobias et al. 2001). These effects were estimated using Eq. (12). For these calculations we put C η = 1, ε = 0, C K = 1 and χ = 0. Figure 4 shows the sum of the pumping effects for the toroidal and poloidal components of mean magnetic fields including the helicity-vorticity pumping effect. In agreement with previous studies, it is found that the radial direction is the principal direction of mean-field transport in the solar convection zone. In its upper part the transport is downward because of pumping due to the density gradient (Kichatinov 1991). At the bottom of the convection zone the diamagnetic pumping effect produces downward transport as well (Kichatinov 1991, Rudiger and Brandenburg 1995). The diamagnetic pumping is quenched inversely proportional to the Coriolis number (e.g., Kichatinov 1991, Pipin 2008) and it has the same order of magnitude as the helicity-vorticity pumping effect. The latter effect modifies the direction of effective drift of the toroidal magnetic field near the bottom of the convection zone. There is also upward drift of the toroidal field at · low latitudes in the middle of the convection zone. It results from the combined effects of density gradient and global rotation (Kichatinov 1991, Krivodubskij 2004). For the poloidal magnetic field the transport is downward everywhere in the convection zone. At the bottom of the convection zone the action of the diamagnetic pumping on the meridional component of the large-scale magnetic field is amplified due to the helicity-vorticity pumping effect. The obtained pattern of large-scale magnetic field drift in the solar convection zone does not take into account nonlinear effects, e.g., because of magnetic buoyancy. The effect of mean-field buoyancy is rather small compared with flux-tube buoyancy (Kichatinov and Pipin 1993, cf. Guerrero and Kapyla 2011). To find out the current helicity counterpart of the pumping effect we analyze dynamo models by solving Eqs. (6, 16). The governing parameters of the model are C η = 0 . 05, C ω = 1 3 C α . We discuss the choice of the governing parameters later. The other parameters of the model are given in the Table 1. Because of the weakening factor C η the magnitude of the pumping velocity is about one order of magnitude smaller than what is shown in Figure 4. Following Pipin and Kosovichev (2011b), we use a combination of 'open' and 'closed' boundary conditions at the top, controlled by a parameter δ = 0 . 95, with This is similar to the boundary condition discussed by Kitchatinov et al. (2000). For the poloidal field we apply a combination of the local condition A = 0 and the requirement of a smooth transition from the internal poloidal field to the external potential (vacuum) field: We assume perfect conductivity at the bottom boundary with standard boundary conditions. For the magnetic helicity, similar to Guerrero et al. (2010), we put ∇ r ¯ χ = 0 at the bottom of the domain and ¯ χ = 0 at the top of the convection zone. In this paper we study dynamo models which include Radler's Ω × J dynamo effect due to a large-scale current and global rotation (Radler 1969). There is also a dynamo effect due to large-scale shear and current (Rogachevskii and Kleeorin 2003). The motivation to consider these addional turbulent sources in the mean-field dynamo comes from DNS dynamo experiments (Brandenburg and Kapyla 2007, Kapyla et al. 2008, Hughes and Proctor 2009, Kapyla et al. 2009) and from our earlier studies (Pipin and Seehafer 2009, Seehafer and Pipin 2009). The dynamo effect due to large-scale current gives an additional source of large-scale poloidal magnetic field. This can help to solve the issue with the dynamo period being otherwise too short. Also, in the models the large-scale current dynamo effect produces less overlapping cycles than dynamo models with α -effect alone. The choice of parameters in the dynamo is justified by our previous studies (Pipin and Seehafer 2009, Pipin and Kosovichev 2011c), where we showed that solar-types dynamos can be obtained for C α /C ω > 2. In those papers we find the approximate threshold to be C α ≈ 0 . 02 for a given diffusivity dilution factor of C η = 0 . 05. As follows from the results given in Fig.4, the kinetic helicity-vorticity pumping effect has a negligible contribution in the near-surface layers, where downward pumping due to density stratification dominates. Therefore, it is expected that the surface dynamo waves are not affected if we discard magnetic helicity from the dynamo equations. Figure 5 shows time-latitude diagrams for toroidal and radial magnetic fields at the surface and for toroidal magnetic field at the bottom of the convection zone for two dynamo models D1 and D2 with and without the helicity-vorticity pumping effect, but magnetic helicity is taken into account as the main dynamo quenching effect. To compare with observational data from a time-latitude diagram of sunspot area (e.g., Hathaway 2011), we multiply the toroidal field component B by factor sin θ . This gives a quantity, which is proportional to the flux of large-scale toroidal field at colatitude θ . We further assume that the sunspot area is related to this flux. Near the surface, models D1 and D2 give similar patterns of magnetic field evolution. At the bottom of the convection zone model D1 shows both poleward and equatorward branches of the dynamo wave propagation that is in agreement with the Parker-Yoshimura rule. Both branches have nearly the same time scale that equals glyph[similarequal] 16 years. The results from model D2 show that at the bottom of the convection zone the poleward branch of the dynamo wave dominates. Thus we conclude that the helicity-vorticity pumping effect alters the propagation of the dynamo wave near the bottom of the solar convection zone. We find that models with magnetic helicity contributions to the pumping effect do not change this conclusion. Figure 6 shows a typical snapshot of the magnetic helicity distribution in the northern hemisphere for all our models. The helicity has a negative sign in the bulk of the solar convection zone. Regions with positive current helicity roughly correspond to domains of the negative large-scale current helicity concentration. × They are located in the middle of the solar convection zone and at the high and low latitudes near the top of the solar convection zone. As follows from Fig. 6, the pumping effect due to current helicity may be efficient in the upper part of the solar convection zone where it might intensify the equatorial drift of the dynamo wave along iso-surfaces of the angular velocity. We find that the pumping effect that results from magnetic helicity is rather small in our models. This may be due to the weakness of the magnetic field. Observations (Zhang et al. 2010) give about one order magnitude larger current helicity than what is shown in Fig. 6. In the model we estimate the current helicity as H C = χ µ 0 glyph[lscript] 2 . This result depends essentially on the mixing length parameter glyph[lscript] . The stronger helicity is concentrated to the surface, the larger H C . In observations, we do not know from were the helical magnetic structures come from. In view of the given uncertainties we estimate the probable effect of a larger magnitude of magnetic helicity in the model by increasing the parameter C C to 10 (model D3). In addition, we consider the results for the nonlinear model D4. It has a higher C α and a lower R χ to increase the nonlinear impact of the magnetic helicity on the large-scale magnetic field evolution. The top panel of Figure 7 shows a time-latitude diagram of toroidal magnetic field and current helicity evolution near the surface for model D4. We find a positive sign of current helicity at the decay edges of the toroidal magnetic field butterfly diagram. There are also areas with positive magnetic helicity at high latitudes at the growing edges of the toroidal magnetic field butterfly diagram. The induced pumping velocity is about 1 cm s -1 . The increase of the magnetic helicity pumping effect by a factor of 10 (model D3) shifts the latitude of the maximum of the toroidal magnetic field by about 5 · toward the equator. The induced pumping velocity is about 5 cm s -1 . Stronger nonlinearity (model D4) and a stronger magnetic helicity pumping effect (model D3) modify the butterfly diagram in different ways. Model D3 shows a simple shift of the maximum of toroidal magnetic field toward the equator. Model D4 shows a fast drift of large-scale toroidal field at the beginning of a cycle and a slow-down of the drift velocity as the cycle progresses. and the latitudinal drift of the centroid position of the toroidal magnetic field flux (cf. Hathaway 2011) where B S ( θ ) = 〈 B ( r, θ ) 〉 (0 . 9 , 0 . 99) R is the toroidal magnetic field, which is averaged over the surface layers. Note that the overlap between subsequent cycles influences the value of λ C more than the value of λ max . The behaviour of λ max in models D1,D2 and D3 reproduces qualitatively the exponential drift of maximum × C latitude as suggested by Hathaway (2011): where t is time measured in years. Model D4 shows a change between fast (nearly steady dynamo wave) drift at the beginning of the cycle to slow drift at the decaying phase of the cycle. The overlap between subsequent cycles is growing from model D1 to model D4. In all the models the highest latitude of the centroid position of the toroidal magnetic flux is below 30 · . Models D3 and D4 have nearly equal starting latitude of the centroid position. It is about 24 · . This means that a model with increased magnetic helicity pumping produces nearly the same effect for the shift of the centroid position as a model with a strong nonlinear effect of magnetic helicity.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12\n ]\n },\n {\n \"title\": \"3 Discussion and conclusions\",\n \"content\": \"We have shown that the interaction of helical convective motions and differential rotation in the solar convection zone produces a turbulent drift of large-scale magnetic field. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. The large-scale vorticity vector roughly follows to iso-surfaces of angular velocity. Since the direction of the drift depends on the sign of helicity, the pumping effect is governed by the Parker-Yoshimura rule (Parker 1955, Yoshimura 1975). The effect is computed within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. In the calculations, we have assumed that the turbulent kinetic and current helicities are given. The calculations were done for arbitrary Coriolis number. In agreement with Mitra et al. (2009) and Rogachevskii et al. (2011), the analytical calculations show that the leading effect of pumping is described by a large-scale magnetic drift in the direction of the large-scale vorticity vector and by anisotropic pumping which produces a drift of toroidal and poloidal components of the field in opposite directions. The component of the drift that is induced by global rotation and helicity (second line in Eq. (3)) is rather small compared to the main effect. The latter conclusion should be checked separately for a different model of background turbulence, taking into account the generation of kinetic helicity due to global rotation and stratification in a turbulent medium. We have estimated the pumping effect for the solar convection zone and compared it with other turbulent pumping effects including diamagnetic pumping and turbulent pumping that results from magnetic fluctuations in stratified turbulence (Kichatinov 1991, Pipin 2008). The latter is sometimes referred to as 'density-gradient pumping effect' (Krivodubskij 2004). The diamagnetic pumping is upward in the upper part of the convection zone and downward near the bottom. The velocity field of density-gradient pumping is more complicated (see Figure 4). However, its major effect is concentrated near the surface. Both diamagnetic pumping and density-gradient pumping effects are quenched inversely proportional to the Coriolis number (Kichatinov 1991, Pipin 2008). The helicity-vorticity pumping effect modifies the direction of large-scale magnetic drift at the bottom of the convection zone. This effect was illustrated by a dynamo model that shows a dominant poleward branch of the dynamo wave at the bottom of the convection zone. It is found that the magnetic helicity contribution of the pumping effect can be important for explaining the fine structure of the sunspot butterfly diagram. In particular, the magnetic helicity contribution results in a slow-down of equatorial propagation of the dynamo wave. The slow-down starts just before the maximum of the cycle. Observations indicate a similar behavior in sunspot activity (Ternullo 2007, Hathaway 2011). A behavior like this can be seen in flux-transport models as well (Rempel 2006). For the time being it is unclear what are the differences between different dynamo models and how well do they reproduce the observations. A more detailed analysis is needed.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"REFERENCES\",\n \"content\": \"065903 . Radler, K.-H.. On the electrodynamics of turbulent fields under the influence of corilois forces. Monats. Dt. Akad. Wiss., 11:, 194-201, 1969.\",\n \"pages\": [\n 15\n ]\n },\n {\n \"title\": \"Appendix A\",\n \"content\": \"To compute E it is convenient to write equations (1) and (2) in Fourier space: where the turbulent pressure was excluded from (2) by convolution with tensor π ij ( z ) = δ ij -ˆ z i ˆ z j , δ ij is the Kronecker symbol and ̂ z is a unit wave vector. The equations for the second-order moments that make contributions to the mean-electro-motive force(MEMF) can be found directly from (26, 27). As the preliminary step we write the equations for the second-order products of the fluctuating fields, and make the ensemble averaging of them, where, the terms Th ( κ ,v,h ) ij involve the third-order moments of fluctuating fields and second-order moments of them with the forcing term. Next, we apply the τ -approximation, substituting the Th ( κ ,v,h ) ij -terms by the corresponding τ relaxation terms of the second-order contributions, where the superscript . . . (0) denotes the moments of the background turbulence. Approximating these complicated contributions by the simple relaxation terms has to be considered as a questionable assumption. It involves additional assumptions (see Radler and Rheinhardt 2007), e.g., it is assumed that the second-order correlations in Eq. (8) do not vary significantly on the time scale of τ c . This assumption is consistent with scale separation between the mean and fluctuating quantities in the mean-field magneto hydrodynamics. The reader can find a comprehensive discussion of the τ -approximation in the above cited papers. Furthermore, we restrict ourselves to the high Reynolds numbers limit and discard the microscopic diffusion terms. The contributions of the mean magnetic field in the turbulent stresses will be neglected because they give the nonlinear terms in the cross helicity tensor. Also, τ c is independent on k (cf, Radler et al. 2003, Rogachevskii and Kleeorin 2003, Brandenburg and Subramanian 2005) and it is independent on the mean fields as well. This should be taken into account in considering the nonlinear effects due to rotation. Taking all the above assumptions into account, we get the system of equations for the moments for the stationary case: To proceed further, we have to introduce some conventions and notations that are widely used in the literature. The double Fourier transformation of an ensemble average of two fluctuating quantities, say f and g , taken at equal times and at the different positions x , x ' , is given by In the spirit of the general formalism of the two-scale approximation (Roberts and Soward 1975) we introduce 'fast' and 'slow' variables. They are defined by the relative r = x -x ' and the mean R = 1 2 ( x + x ' ) coordinates, respectively. Then, Eq. (37) can be written in the form where we have introduced the wave vectors k = 1 2 ( z -z ' ) and K = z + z ' . Then, following to Subramanian and Brandenburg (2004), we define the correlation function of ̂ f and ̂ g obtained from (38) by integration with respect to K , For further convenience we define the second order correlations of momentum density, magnetic fluctuations and the cross-correlations of momentum and magnetic fluctuations via We now return to equations (34), (35) and (36). As the first step, we solve these equations about Ω (nonlinear effects of the Coriolis force) and make the Taylor expansion with respect to the 'slow' variables and take the Fourier transformation, (39), about them. The details of this procedure can be found in (Subramanian and Brandenburg 2004). In result we get the following equations for the second moments These equations were solved with respect to the shear tensor, V i,j = ∇ j V i , by means of perturbation procedure. One remains to define the spectra of the background turbulence. We will adopt the isotropic form of the spectra (Roberts and Soward 1975). Additionally, the background magnetic fluctuations are helical while there is no prescribed kinetic helicity in the background turbulence: where, the spectral functions E ( k, R ) , B ( k, R ) , N ( k, R ) define, respectively, the intensity of the velocity fluctuations, the intensity of the magnetic fluctuations and amount of current helicity in the background turbulence. They are defined via where h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇× b (0) 〉 µρ . In final results we use the relation between intensities of magnetic and kinetic fluctuations which is defined via B ( k, R ) = εµ ¯ ρE ( k, R ). The state with ε = 1 means equipartition between energies of magnetic an kinetic fluctuations in the background turbulence.\",\n \"pages\": [\n 15,\n 16,\n 17,\n 18,\n 19\n ]\n }\n]"}}},{"rowIdx":267,"cells":{"bibcode":{"kind":"string","value":"2013GReGr..45.1005N"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1201.2806.pdf"},"content":{"kind":"string","value":"\nDragged metrics\nM. Novello ∗† and E. Bittencourt ‡\nInstituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: November 15, 2018)\nWe show that the path of any accelerated body in an arbitrary space-time geometry g µν can be described as geodesics in a dragged metric ˆ q µν that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kind of non-gravitational forces as modifications of the metric of space-time. This method of effective elimination of the forces by a change of the metric of the substratum can be understood as a generalization of the d'Alembert principle applied to all relativistic processes.\nPACS numbers: 04.20.-q\nI. INTRODUCTION\nIn 1923 Gordon [1] made a seminal suggestion to treat the propagation of electromagnetic waves in a moving dielectric as a modification of the metric structure of the background. He showed that the waves propagate as geodesics not in the geometry η µν but instead in the dragged metric\n̂ q µν = η µν +( /epsilon1µ -1) v µ v ν , (1) where /epsilon1 and µ are constant parameters that characterize the dielectric. Latter it was recognized that this interpretation can be used to describe non-linear structures when /epsilon1 and µ depends on the intensity of the field [2].\nIn recent years an intense activity concerning properties of Riemannian geometries similar to the one described by Gordon has been done [3]. In particular those that allows a binomial form for both the metric and its inverse, that is its covariant and the corresponding contravariant expressions\nand\n̂ q µν = η µν + b Φ µν , (2)\nThus, the tensor Φ µν must satisfy the condition\n̂ q µν = Aη µν + B Φ µν . (3)\nΦ µν Φ νλ = mδ λ µ + n Φ λ µ . (4)\nIn the present paper we limit our analysis only to the simplest dragged form by setting Φ µν = v µ v ν . In this case the coefficients of the covariant form of the metric are given by\nA = 1; B = -b 1 + b .\nThe origin of the dragging effect in the case of Gordon's metric is due to the modifications of the path of the electromagnetic waves inside the moving dielectric. Then we face the question: could such particular description of the electromagnetic waves in moving bodies be generalized for other cases, independently of the electromagnetic forces? In other words, could such geometrized paths be used to describe other kinds of forces? We shall see that the answer is yes. Indeed, we will show that it is possible to geometrize different kinds of forces by the introduction of a dragged metric ̂ q µν such that in this geometry the accelerated body follows the free path of geodesics.\nLet us emphasize that we deal here with any kind of force that has a non-gravitational character. It is precisely the consequences of such non-gravitational force that we describe in terms of a modified dragged metric. This means that the observable effects of any force can be interpreted as nothing but a modification of the geometry of space-time. In other words the motion of any accelerated body can be described as a free body following geodesics in a modified metric. This procedures generalizes d'Alembert principle of classical mechanics [4, 5] which states that it is possible to transform a dynamical problem into a static one, where the body is free of any force. Going from the background metric - where an accelerated body experiences a non-gravitational force - to a dragged metric where the body follows a geodesics and become free of non-gravitational forces is the relativistic expression of this principle. In this way we produce a geometric description of all kind of motion whatever the force that originates it.\nII. A SPECIAL CASE\nWe claim that accelerated bodies in a flat Minkowski space-time 1 can be equivalently described as free bodies\nfollowing geodesics in an associated dragged geometry. In order to simplify our calculation we restrict this section to the case in which the acceleration vector a µ is the gradient of a function, that is 2\na µ = ∂ µ Ψ . (5)\nThus the force acting on the body under observation comes from a potential V in the Lagrangian formalism, i.e., F µ = ∂ µ V.\nWe write the dragged metric in the form\n̂ q µν = η µν + b v µ v ν . The associated covariant derivative is defined by\nwhere the corresponding Christoffel symbol is given by\nv α ; µ = v α , µ + ̂ Γ α µν v ν ,\n̂ Γ /epsilon1 µν = 1 2 ( η /epsilon1α + b v /epsilon1 v α ) ( ̂ q αµ,ν + ̂ q αν,µ -̂ q µν,α ) , (6) where we are using a comma to denote simple derivative, that is A ,µ ≡ ∂ µ A. The description of an accelerated curve in a flat space-time as a geodesics in a dragged metric is possible if the following condition is satisfied\n( v µ,ν -̂ Γ /epsilon1 µν v /epsilon1 ) v ν = 0 , (7)\n( v µ,ν -̂ Γ /epsilon1 µν v /epsilon1 ) v ν = 0 . (8)\n̂ where we have used the dragged metric to write ̂ v µ ≡ ̂ q µν v ν . Or, equivalently,\nThen noting that the acceleration in the background is defined by a µ = v µ,ν v ν and using equation (5) the condition of geodesics in the dragged geometry takes the form\nWe have\n∂ µ Ψ = ̂ Γ /epsilon1 µν v /epsilon1 v ν . (9)\n̂ Γ /epsilon1 µν v /epsilon1 v ν = 1 + b 2 v α v ν ̂ q αν,µ . (10)\na µ + ∂ µ b 2(1 + b ) = 0 ,\nUsing the expression of ̂ q αν and combining with condition (8) it follows\nthat is, the expression of the coefficient b of the dragged metric is given in terms of the potential of the acceleration\n1 + b = e -2Ψ . (11)\nA. The curvature of the dragged metric\nIn the case the background metric is not flat or if we use an arbitrary coordinate system the connection is given by the sum of the corresponding background one and a tensor, that is\n̂ Γ /epsilon1 µν = Γ /epsilon1 µν + K /epsilon1 µν . (12) In the case of the Minkowski background a direct calculation gives for the connection Γ /epsilon1 µν the form\n̂ K /epsilon1 µν = v /epsilon1 ( a µ v ν + a ν v µ ) -a /epsilon1 v µ v ν . (13)\nThen,\nK /epsilon1 µ/epsilon1 = a µ .\nThe contracted Ricci curvature has the expression\n̂ R µν = a µ,ν -a µ a ν +( ω + a α ,α ) v µ v ν , (14) Noting that a µ = a ν q µν = a µ it follows that\nThe scalar of curvature ̂ R = ̂ R µν ̂ q µν is\n̂ ̂ ω ≡ a µ a µ = a µ a ν η µν = a µ a ν ̂ q µν = ̂ ω.\n̂ R = (2 + b ) a α ,α . (15) These expressions can be re-written in a covariant way by noting that\nwhich yields\na µ ; ν ≡ a µ,ν -̂ Γ /epsilon1 µν a /epsilon1 ,\n̂ R µν = a µ ; ν -a µ a ν -( ω -a α ; α ) v µ v ν , (16)\n̂ R = (2 + b ) ( a α ; α -ω ) . (17)\nand for the scalar ̂ R the form\nB. Analog gravity\nSuppose that an observer following a path with fourvelocity v µ and acceleration a µ in the flat Minkowski space-time background is not able to identify the origin of the force that is acting on him. In other words he is going to believe that only long-range gravitational forces are constraining his motion. Let us assume that he knows that gravity does not accelerate any curve but instead change the metric of the background according to the principles of general relativity. This means that if he is able to represent his motion as a geodesics in a dragged\nmetric ̂ q µν he will consider that the origin of such curved metric is nothing but a consequence of a distribution of energy which he will describe by using the equation\n̂ R µν -1 2 ̂ R ̂ q µν = -̂ T µν . (18)\nˆ u α = √ 1 + b v α .\nHe will identify the different terms of the source through his own motion. From his velocity v µ he defines the normalized four-velocity ˆ u α in the ̂ Q metric by setting\nHe then proceed to characterize the origin of the curved metric using the standard decomposition\n\n(a) density of energy\n(b) isotropic pressure\n(c) heat flux\n\n̂ ρ = ̂ T µν ˆ u µ ˆ u ν ; (19)\n̂ p = -1 3 ̂ T µν h µν ; (20)\n\n(d) anisotropic pressure\n\n̂ q λ = ̂ T αβ ˆ u β h α λ ; (21)\nIn these expressions we used\n̂ π µν = ̂ T αβ h α µ h β ν + ̂ p h µν . (22)\n̂ h µν = ̂ q µν -̂ u µ ̂ u ν . Note that h µν = h µν . Thus, he will write\n̂ ̂ ̂ ̂ ̂ ̂ From this decomposition, using equation (18) and the curvature (16) he will identify the energy-momentum distribution as\n̂ T µν = ρ ˆ u µ ˆ u ν -p h µν + q µ ˆ u ν + q ν ˆ u µ + π µν . (23)\nwhere\n̂ ρ = b 2 Q, ̂ p = ( 2 3 + b 2 ) Q, (24) ̂ q µ = 0 , ̂ π µν = -a µ ; ν + a µ a ν -Q 3 ̂ q µν + Q 3 ̂ u µ ˆ u ν ,\nQ = ω -a α ; α ,\nSummarizing, we can say that this observer will state that there is a gravitational field represented by the metric ̂ q µν produced by the distribution of energy given\nby equation (24). We note that this reduction of the dragged metric to the framework of general relativity is not mandatory. Indeed, we deal here precisely with some accelerated paths that are not reduced to the gravitational force in the standard theory. This will become more clear when we present examples of accelerated curves in specific solutions of general relativity in the next sections.\nIndeed, let us present some clarifying examples. The first one considers the motion of rotating bodies in Minkowski space-time. The other ones take into account the general relativity effects. We shall see that it is possible to produce what could be called double gravity , if the origin of the curvature of the dragged metric is identified to an effective energy-momentum tensor satisfying the equations of general relativity. However there is no reason for this restriction. We will come back to this question elsewhere.\nIII. ACCELERATION IN MINKOWSKI SPACE-TIME\nLet us consider a simple example concerning the acceleration of a body in flat Minkowski space-time written in non-stationary cylindrical coordinate system ( t, r, φ, z ) to express the following line element\nds 2 = a 2 [ dt 2 -dr 2 -dz 2 + g ( r ) dφ 2 +2 h ( r ) dφdt ] , (25)\nwhere a is a constant. We choose the following local tetrad frame given implicitly by the 1-forms\nθ 0 = a ( dt + hdφ ) , θ 1 = adr, θ 2 = a ∆ dφ, θ 3 = adz, (26)\nwhere we define ∆ = √ h 2 -g . The unique nonidentically null components of the Ricci tensor R AB in the tetrad frame are\nR 00 = 1 2 a 2 ( h ' ∆ ) , R 11 = 1 a 2 ( ∆ '' ∆ -1 2 h ' 2 ∆ 2 ) = R 22 = R 33 , R 02 = 1 2 a 2 ( -h '' ∆ + h ' ∆ ∆ 2 ) . (27)\nwhere a prime means derivative with respect to coordinate r. The equations of general relativity for this geometry have two simple solutions that we shall analyze below.\nIn the case of R AB =0, we get\nh ' = 0; ∆ '' = 0 .\nSolving these equations, we find\nh ≡ const ; ∆ ≡ ω 2 r 2 ,\nwhere ω is a constant. Therefore, Eq. (25) takes the form\nds 2 = a 2 [ dt 2 -dr 2 -dz 2 +( h 2 -ω 2 r 2 ) dφ 2 +2 h ( r ) dφdt ] . (28)\nIf we consider the observer field\nv µ = 1 a √ h 2 -ω 2 r 2 δ µ 2 .\nThis path corresponds to an acceleration given by\na µ = ( 0 , ω 2 r ( h 2 -ω 2 r 2 ) , 0 , 0 ) .\nThis means that a µ = ∂ µ Ψ , where\n2Ψ = -ln( h 2 -ω 2 r 2 ) .\nWe are in a situation similar to the previous section since the acceleration is a gradient. The parameter b of the dragged metric is given by the expression (11)\n1 + b = h 2 -ω 2 r 2 ,\nand for the dragged metric the form\nds 2 a 2 = ω 4 r 4 -ω 2 r 2 h 2 +1 ( h 2 -ω 2 r 2 ) 2 dt 2 + dφ 2 + 2 h h 2 -ω 2 r 2 dφ dt -dr 2 -dz 2 (29)\n̂ ω 2 r 2 ± = h 2 ± √ h 4 -4 2 .\nNote that we are dealing with the case in which h 2 -ω 2 r 2 > 0 . This allows the presence of accelerated closed time-like curves (CTC) in the original background that will be mapped into closed time-like geodesics (CTG) in the dragged geometry. Note that there exists a real singularity in r = h/ω and that q 00 change sign where\nIV. ACCELERATION IN CURVED SPACE-TIMES\nLet us present now how our dragged metric approach works in some solutions of the equations of general relativity. We choose three well-known geometries: Schwarzschild, Godel universe and the Kerr solution. In these Riemannian manifolds we analyze some examples of accelerated paths that are interpreted as geodesics in the associated dragged metrics.\nA. Schwarzschild geometry\nWe set the Schwarzschild metric in the ( t, r, θ, ϕ ) coordinate system\nds 2 = (1 -r H r ) dt 2 -1 (1 -r H r ) dr 2 -r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (30)\nChoose the path described by the four-velocity\nv µ = √ 1 -r H r δ µ 0 .\nThe corresponding acceleration is\na µ = ( 0 , -r H 2( r 2 -r r H ) , 0 , 0 ) .\nIn this case the acceleration is the gradient of function Ψ given by\nΨ = -1 2 ln(1 -r H r ) .\nThe factor b of the dragged metric is given by\nb = -r H r ,\nand the dragged metric takes the form\nds 2 = dt 2 -1 (1 -r H r ) dr 2 -r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (31)\nThe only non-null Ricci curvature of this ̂ q µν metric are\nR 1 1 = r H r 3 ;\nR 2 2 = R 3 3 = -1 2 R 1 1 .\nAll 14 Debever invariants are finite in all points except at the origin r = 0 .\nB. Godel's geometry\nLet us now turn our analysis to the Godel geometry. In the cylindrical coordinate system this metric is given by Eq. (25), where a is a constant related to the vorticity a = 2 /ω 2 and\nh ( r ) = √ 2 sinh 2 r ;\ng ( r ) = sinh 2 r (sinh 2 r -1) .\nFor completeness we note the non-trivial contravariant terms of the metric:\ng 00 = 1 -sinh 2 r a 2 cosh 2 r , g 02 = √ 2 a 2 cosh 2 r , g 22 = -1 a 2 sinh 2 r cosh 2 r . (32)\nIn [6] it was pointed out the acausal properties of a particle moving into a circular orbit around the z -axis with four-velocity\nv µ = ( 0 , 0 , 1 a sinh r √ sinh 2 r -1 , 0 ) .\nThis path corresponds to an acceleration given by\na µ = ( 0 , cosh r [2 sinh 2 r -1] a 2 sinh r [sinh 2 r -1] , 0 , 0 ) .\nThis means that a µ = ∂ µ Ψ , where\nΨ = -ln(sinh r √ sinh 2 r -1) .\nAgain, we are in a situation where the acceleration is a gradient. Therefore, the parameter b of the dragged metric is given by the expression (11) which in the Godel's background takes the form\n1 + b = sinh 2 r (sinh 2 r -1) ,\nand the dragged metric has the following form\nd ̂ s 2 a 2 = 3 -sinh 4 r (sinh 2 r -1) 2 dt 2 + dφ 2 +2 √ 2 sinh 2 r -1 dφ dt , -dr 2 -dz 2 (33)\nFrom the analysis of geodesics in Godel geometry the domain r < r c where sinh 2 r c = 1 separates causal from non-causal regions of the space-time. This is related to the fact that a geodesic that pass the value r = 0 will be confined within the domain Ω i defined by the values of coordinate r in the region 0 < r < r c . See [7] for details. However, the gravitational field is finite in the region r = r c . Nothing similar in the dragged metric, once at sinh 2 r = 1 there exists a real singularity in the dragged metric. Only the exterior domain is allowed. This means that for this kind of accelerated path in Godel geometry the allowed domain for the dragged metric is precisely the whole acausal region.\nC. Kerr metric\nLet us turn now to the dragged metric approach in the case the background is the Kerr metric. In the BoyerLindquist coordinate system this metric is given by\nds 2 = ( 1 -2 Mr ρ 2 ) dt 2 -ρ 2 Σ dr 2 -ρ 2 dθ 2 + + 4 Mra sin 2 θ ρ 2 dtdφ + -[ ( r 2 + a 2 ) sin 2 θ + 2 Mra 2 sin 4 θ ρ 2 ] dφ 2 , (34)\nwhere Σ = r 2 + a 2 -2 Mr and ρ 2 = r 2 + a 2 cos 2 θ . On equatorial plane ( θ = π/ 2) consider the following vector field\nv µ = ( 0 , 0 , 0 , r √ -( r 2 + a 2 ) 2 + a 2 Σ ) .\nThis path corresponds to an acceleration given by\na µ = ( 0 , -r 3 -Ma 2 r 4 + r 2 a 2 +2 Mra 2 , 0 , 0 ) .\nThis means that a µ = ∂ µ Ψ , where\n2Ψ = -ln [ -( r 2 + a 2 + 2 Ma 2 r )] .\nOnce more we choose an accelerated path that can be represented by a gradient. The parameter b of the dragged metric is given by the expression (11)\n1 + b = -( r 2 + a 2 + 2 Ma 2 r ) ,\nand for the dragged metric, on the equatorial plane, the form\nds 2 a 2 = 1 (1 + b ) 2 ( 1 -2 M r -b 4 M 2 a 2 r 2 ) dt 2 + dφ 2 + 4 Ma (1 + b ) r dtdφ -r 2 ∆ dr 2 , (35)\nThese two last cases (Godel and Kerr) show a very curious and intriguing property: the accelerated CTC's at their respective metrics) are transformed in curves that are geodesics, that is CTG's (at their corresponding dragged metric). Besides, in both cases, the dragged metrics display a real singularity excluding the causal domain.\nV. GENERAL CASE\nIn the precedent sections we limited our analysis to the case in which the acceleration is given by a unique function. Let us now pass to more general situation. In\norder to geometrize any kind of force we must deal with a larger class of geometries. The most general form of dragged metric that allows the description of accelerated bodies as true geodesics in a modified geometry has the form\n̂ q µν = g µν + b v µ v ν + ma µ a ν + n ( v µ a ν + a µ v ν ) . (36) The three parameters b, m, n are related to the three degrees of freedom of the acceleration vector. The corresponding covariant form of the metric is given by\n̂ q µν = g µν + Bv µ v ν + Ma µ a ν + N ( v µ a ν + a µ v ν ) . (37) in which B,M,N are given in terms of the parameters b, m, n by the relations\nB = -b (1 + mω ) -n 2 ω (1 + b ) (1 + mω ) -n 2 ω ,\nM = 1 1 + mω ( -m + n 2 (1 + b ) (1 + mω ) -n 2 ω ) ,\nN = -n (1 + b ) (1 + mω ) -n 2 ω .\nIn this case the equation that generalizes the condition of geodesics (8) has the form\na µ = 1 2 ( (1 + b ) v λ v ν + na λ a ν ) [ ̂ q λµ,ν + q λν , µ -q µν , λ ] . (38)\n̂ ̂ This equation can be cast in the following formal expression\na µ = Ψ 1 ∂ µ b +Ψ 2 ∂ µ m +Ψ 3 ∂ µ n, (39)\nwhere each term Ψ 1 , Ψ 2 and Ψ 3 depends on all three functions m,b and n.\nSolving this equation for these three functions provide the most general expression for any acceleration.\n\n[1] W. Gordon Ann. Phys. (Leipzig) 72 421 (1923);\n[2] M. Novello, V.A. De Lorenci, J. M. Salim and R. Klippert Phys. Rev. D 61 045001 (2000);\n[3] M. Novello and E. Goulart Class. Quantum Grav. 28 145022 (2011) and references therein;\n[4] C. Lanczos, The Variational Principles of Mechanics , Ed. 4, Dover, New York (1970);\n\nWith these results, we have transformed the path of any particle submitted to any kind of force as a geodetic motion in the dragged metric. This result is the extension of the d'Alembert principle, corresponding to all types of motion i.e. the acceleration is geometrized through the dragged metric.\nVI. CONCLUSION\nWe summarize the novelty of our analysis in the following steps:\n\n· Let v µ represent the four-vector that describes the kinematics of a body in an arbitrary space-time endowed with a geometry g µν ;\n· If the body experiences a non-gravitational force it acquires an acceleration a µ ;\n· It is always possible to define an associated dragged metric ̂ q µν given by (37) such that in this metric the acceleration is removed. That is, the path is represented as a free particle that follows geodesics in the dragged geometry.\n\nWe have shown by a constructive operation that the following conjecture is true:\nConjecture. For any accelerated path Γ described by four-vector velocity v µ and acceleration a µ in a given Riemannian geometry g µν we can always construct another geometry ̂ Q endowed with a dragged metric ̂ q µν which depends only on g µν , v µ and a µ such that the path Γ is a geodesic in ̂ Q.\nVII. ACKNOWLEDGEMENTS\nWe would like to thank Dr Ivano Dami˜ao Soares for his comments in a previous version of this paper. We would like to thank FINEP, CNPq and FAPERJ and EB CNPq for their financial support.\n\n[5] E. Mach, The Science of Mechanics , Ed. 6, The Open Court Publishing CO., Illinois (1960);\n[6] M. Novello, N.F. Svaiter and M. E. X. Guimares Mod. Phys. Lett. A 7 5 381 (1992);\n[7] M. Novello, I. Dami˜ao Soares and J. Tiomno, Phys. Rev. D 27 4 779 (1983).\n"},"sections":{"kind":"list like","value":[{"title":"Dragged metrics","content":"M. Novello ∗† and E. Bittencourt ‡ Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: November 15, 2018) We show that the path of any accelerated body in an arbitrary space-time geometry g µν can be described as geodesics in a dragged metric ˆ q µν that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kind of non-gravitational forces as modifications of the metric of space-time. This method of effective elimination of the forces by a change of the metric of the substratum can be understood as a generalization of the d'Alembert principle applied to all relativistic processes. PACS numbers: 04.20.-q","pages":[1]},{"title":"I. INTRODUCTION","content":"In 1923 Gordon [1] made a seminal suggestion to treat the propagation of electromagnetic waves in a moving dielectric as a modification of the metric structure of the background. He showed that the waves propagate as geodesics not in the geometry η µν but instead in the dragged metric ̂ q µν = η µν +( /epsilon1µ -1) v µ v ν , (1) where /epsilon1 and µ are constant parameters that characterize the dielectric. Latter it was recognized that this interpretation can be used to describe non-linear structures when /epsilon1 and µ depends on the intensity of the field [2]. In recent years an intense activity concerning properties of Riemannian geometries similar to the one described by Gordon has been done [3]. In particular those that allows a binomial form for both the metric and its inverse, that is its covariant and the corresponding contravariant expressions and Thus, the tensor Φ µν must satisfy the condition In the present paper we limit our analysis only to the simplest dragged form by setting Φ µν = v µ v ν . In this case the coefficients of the covariant form of the metric are given by The origin of the dragging effect in the case of Gordon's metric is due to the modifications of the path of the electromagnetic waves inside the moving dielectric. Then we face the question: could such particular description of the electromagnetic waves in moving bodies be generalized for other cases, independently of the electromagnetic forces? In other words, could such geometrized paths be used to describe other kinds of forces? We shall see that the answer is yes. Indeed, we will show that it is possible to geometrize different kinds of forces by the introduction of a dragged metric ̂ q µν such that in this geometry the accelerated body follows the free path of geodesics. Let us emphasize that we deal here with any kind of force that has a non-gravitational character. It is precisely the consequences of such non-gravitational force that we describe in terms of a modified dragged metric. This means that the observable effects of any force can be interpreted as nothing but a modification of the geometry of space-time. In other words the motion of any accelerated body can be described as a free body following geodesics in a modified metric. This procedures generalizes d'Alembert principle of classical mechanics [4, 5] which states that it is possible to transform a dynamical problem into a static one, where the body is free of any force. Going from the background metric - where an accelerated body experiences a non-gravitational force - to a dragged metric where the body follows a geodesics and become free of non-gravitational forces is the relativistic expression of this principle. In this way we produce a geometric description of all kind of motion whatever the force that originates it.","pages":[1]},{"title":"II. A SPECIAL CASE","content":"We claim that accelerated bodies in a flat Minkowski space-time 1 can be equivalently described as free bodies following geodesics in an associated dragged geometry. In order to simplify our calculation we restrict this section to the case in which the acceleration vector a µ is the gradient of a function, that is 2 Thus the force acting on the body under observation comes from a potential V in the Lagrangian formalism, i.e., F µ = ∂ µ V. We write the dragged metric in the form ̂ q µν = η µν + b v µ v ν . The associated covariant derivative is defined by where the corresponding Christoffel symbol is given by ̂ Γ /epsilon1 µν = 1 2 ( η /epsilon1α + b v /epsilon1 v α ) ( ̂ q αµ,ν + ̂ q αν,µ -̂ q µν,α ) , (6) where we are using a comma to denote simple derivative, that is A ,µ ≡ ∂ µ A. The description of an accelerated curve in a flat space-time as a geodesics in a dragged metric is possible if the following condition is satisfied ̂ where we have used the dragged metric to write ̂ v µ ≡ ̂ q µν v ν . Or, equivalently, Then noting that the acceleration in the background is defined by a µ = v µ,ν v ν and using equation (5) the condition of geodesics in the dragged geometry takes the form We have Using the expression of ̂ q αν and combining with condition (8) it follows that is, the expression of the coefficient b of the dragged metric is given in terms of the potential of the acceleration","pages":[1,2]},{"title":"A. The curvature of the dragged metric","content":"In the case the background metric is not flat or if we use an arbitrary coordinate system the connection is given by the sum of the corresponding background one and a tensor, that is ̂ Γ /epsilon1 µν = Γ /epsilon1 µν + K /epsilon1 µν . (12) In the case of the Minkowski background a direct calculation gives for the connection Γ /epsilon1 µν the form Then, The contracted Ricci curvature has the expression The scalar of curvature ̂ R = ̂ R µν ̂ q µν is ̂ R = (2 + b ) a α ,α . (15) These expressions can be re-written in a covariant way by noting that which yields and for the scalar ̂ R the form","pages":[2]},{"title":"B. Analog gravity","content":"Suppose that an observer following a path with fourvelocity v µ and acceleration a µ in the flat Minkowski space-time background is not able to identify the origin of the force that is acting on him. In other words he is going to believe that only long-range gravitational forces are constraining his motion. Let us assume that he knows that gravity does not accelerate any curve but instead change the metric of the background according to the principles of general relativity. This means that if he is able to represent his motion as a geodesics in a dragged metric ̂ q µν he will consider that the origin of such curved metric is nothing but a consequence of a distribution of energy which he will describe by using the equation He will identify the different terms of the source through his own motion. From his velocity v µ he defines the normalized four-velocity ˆ u α in the ̂ Q metric by setting He then proceed to characterize the origin of the curved metric using the standard decomposition In these expressions we used ̂ h µν = ̂ q µν -̂ u µ ̂ u ν . Note that h µν = h µν . Thus, he will write ̂ ̂ ̂ ̂ ̂ ̂ From this decomposition, using equation (18) and the curvature (16) he will identify the energy-momentum distribution as where Summarizing, we can say that this observer will state that there is a gravitational field represented by the metric ̂ q µν produced by the distribution of energy given by equation (24). We note that this reduction of the dragged metric to the framework of general relativity is not mandatory. Indeed, we deal here precisely with some accelerated paths that are not reduced to the gravitational force in the standard theory. This will become more clear when we present examples of accelerated curves in specific solutions of general relativity in the next sections. Indeed, let us present some clarifying examples. The first one considers the motion of rotating bodies in Minkowski space-time. The other ones take into account the general relativity effects. We shall see that it is possible to produce what could be called double gravity , if the origin of the curvature of the dragged metric is identified to an effective energy-momentum tensor satisfying the equations of general relativity. However there is no reason for this restriction. We will come back to this question elsewhere.","pages":[2,3]},{"title":"III. ACCELERATION IN MINKOWSKI SPACE-TIME","content":"Let us consider a simple example concerning the acceleration of a body in flat Minkowski space-time written in non-stationary cylindrical coordinate system ( t, r, φ, z ) to express the following line element where a is a constant. We choose the following local tetrad frame given implicitly by the 1-forms where we define ∆ = √ h 2 -g . The unique nonidentically null components of the Ricci tensor R AB in the tetrad frame are where a prime means derivative with respect to coordinate r. The equations of general relativity for this geometry have two simple solutions that we shall analyze below. In the case of R AB =0, we get Solving these equations, we find where ω is a constant. Therefore, Eq. (25) takes the form If we consider the observer field This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where We are in a situation similar to the previous section since the acceleration is a gradient. The parameter b of the dragged metric is given by the expression (11) and for the dragged metric the form Note that we are dealing with the case in which h 2 -ω 2 r 2 > 0 . This allows the presence of accelerated closed time-like curves (CTC) in the original background that will be mapped into closed time-like geodesics (CTG) in the dragged geometry. Note that there exists a real singularity in r = h/ω and that q 00 change sign where","pages":[3,4]},{"title":"IV. ACCELERATION IN CURVED SPACE-TIMES","content":"Let us present now how our dragged metric approach works in some solutions of the equations of general relativity. We choose three well-known geometries: Schwarzschild, Godel universe and the Kerr solution. In these Riemannian manifolds we analyze some examples of accelerated paths that are interpreted as geodesics in the associated dragged metrics.","pages":[4]},{"title":"A. Schwarzschild geometry","content":"We set the Schwarzschild metric in the ( t, r, θ, ϕ ) coordinate system Choose the path described by the four-velocity The corresponding acceleration is In this case the acceleration is the gradient of function Ψ given by The factor b of the dragged metric is given by and the dragged metric takes the form The only non-null Ricci curvature of this ̂ q µν metric are All 14 Debever invariants are finite in all points except at the origin r = 0 .","pages":[4]},{"title":"B. Godel's geometry","content":"Let us now turn our analysis to the Godel geometry. In the cylindrical coordinate system this metric is given by Eq. (25), where a is a constant related to the vorticity a = 2 /ω 2 and For completeness we note the non-trivial contravariant terms of the metric: In [6] it was pointed out the acausal properties of a particle moving into a circular orbit around the z -axis with four-velocity This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where Again, we are in a situation where the acceleration is a gradient. Therefore, the parameter b of the dragged metric is given by the expression (11) which in the Godel's background takes the form and the dragged metric has the following form From the analysis of geodesics in Godel geometry the domain r < r c where sinh 2 r c = 1 separates causal from non-causal regions of the space-time. This is related to the fact that a geodesic that pass the value r = 0 will be confined within the domain Ω i defined by the values of coordinate r in the region 0 < r < r c . See [7] for details. However, the gravitational field is finite in the region r = r c . Nothing similar in the dragged metric, once at sinh 2 r = 1 there exists a real singularity in the dragged metric. Only the exterior domain is allowed. This means that for this kind of accelerated path in Godel geometry the allowed domain for the dragged metric is precisely the whole acausal region.","pages":[4,5]},{"title":"C. Kerr metric","content":"Let us turn now to the dragged metric approach in the case the background is the Kerr metric. In the BoyerLindquist coordinate system this metric is given by where Σ = r 2 + a 2 -2 Mr and ρ 2 = r 2 + a 2 cos 2 θ . On equatorial plane ( θ = π/ 2) consider the following vector field This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where Once more we choose an accelerated path that can be represented by a gradient. The parameter b of the dragged metric is given by the expression (11) and for the dragged metric, on the equatorial plane, the form These two last cases (Godel and Kerr) show a very curious and intriguing property: the accelerated CTC's at their respective metrics) are transformed in curves that are geodesics, that is CTG's (at their corresponding dragged metric). Besides, in both cases, the dragged metrics display a real singularity excluding the causal domain.","pages":[5]},{"title":"V. GENERAL CASE","content":"In the precedent sections we limited our analysis to the case in which the acceleration is given by a unique function. Let us now pass to more general situation. In order to geometrize any kind of force we must deal with a larger class of geometries. The most general form of dragged metric that allows the description of accelerated bodies as true geodesics in a modified geometry has the form ̂ q µν = g µν + b v µ v ν + ma µ a ν + n ( v µ a ν + a µ v ν ) . (36) The three parameters b, m, n are related to the three degrees of freedom of the acceleration vector. The corresponding covariant form of the metric is given by ̂ q µν = g µν + Bv µ v ν + Ma µ a ν + N ( v µ a ν + a µ v ν ) . (37) in which B,M,N are given in terms of the parameters b, m, n by the relations In this case the equation that generalizes the condition of geodesics (8) has the form ̂ ̂ This equation can be cast in the following formal expression where each term Ψ 1 , Ψ 2 and Ψ 3 depends on all three functions m,b and n. Solving this equation for these three functions provide the most general expression for any acceleration. With these results, we have transformed the path of any particle submitted to any kind of force as a geodetic motion in the dragged metric. This result is the extension of the d'Alembert principle, corresponding to all types of motion i.e. the acceleration is geometrized through the dragged metric.","pages":[5,6]},{"title":"VI. CONCLUSION","content":"We summarize the novelty of our analysis in the following steps: We have shown by a constructive operation that the following conjecture is true: Conjecture. For any accelerated path Γ described by four-vector velocity v µ and acceleration a µ in a given Riemannian geometry g µν we can always construct another geometry ̂ Q endowed with a dragged metric ̂ q µν which depends only on g µν , v µ and a µ such that the path Γ is a geodesic in ̂ Q.","pages":[6]},{"title":"VII. ACKNOWLEDGEMENTS","content":"We would like to thank Dr Ivano Dami˜ao Soares for his comments in a previous version of this paper. We would like to thank FINEP, CNPq and FAPERJ and EB CNPq for their financial support.","pages":[6]}],"string":"[\n {\n \"title\": \"Dragged metrics\",\n \"content\": \"M. Novello ∗† and E. Bittencourt ‡ Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: November 15, 2018) We show that the path of any accelerated body in an arbitrary space-time geometry g µν can be described as geodesics in a dragged metric ˆ q µν that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kind of non-gravitational forces as modifications of the metric of space-time. This method of effective elimination of the forces by a change of the metric of the substratum can be understood as a generalization of the d'Alembert principle applied to all relativistic processes. PACS numbers: 04.20.-q\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"In 1923 Gordon [1] made a seminal suggestion to treat the propagation of electromagnetic waves in a moving dielectric as a modification of the metric structure of the background. He showed that the waves propagate as geodesics not in the geometry η µν but instead in the dragged metric ̂ q µν = η µν +( /epsilon1µ -1) v µ v ν , (1) where /epsilon1 and µ are constant parameters that characterize the dielectric. Latter it was recognized that this interpretation can be used to describe non-linear structures when /epsilon1 and µ depends on the intensity of the field [2]. In recent years an intense activity concerning properties of Riemannian geometries similar to the one described by Gordon has been done [3]. In particular those that allows a binomial form for both the metric and its inverse, that is its covariant and the corresponding contravariant expressions and Thus, the tensor Φ µν must satisfy the condition In the present paper we limit our analysis only to the simplest dragged form by setting Φ µν = v µ v ν . In this case the coefficients of the covariant form of the metric are given by The origin of the dragging effect in the case of Gordon's metric is due to the modifications of the path of the electromagnetic waves inside the moving dielectric. Then we face the question: could such particular description of the electromagnetic waves in moving bodies be generalized for other cases, independently of the electromagnetic forces? In other words, could such geometrized paths be used to describe other kinds of forces? We shall see that the answer is yes. Indeed, we will show that it is possible to geometrize different kinds of forces by the introduction of a dragged metric ̂ q µν such that in this geometry the accelerated body follows the free path of geodesics. Let us emphasize that we deal here with any kind of force that has a non-gravitational character. It is precisely the consequences of such non-gravitational force that we describe in terms of a modified dragged metric. This means that the observable effects of any force can be interpreted as nothing but a modification of the geometry of space-time. In other words the motion of any accelerated body can be described as a free body following geodesics in a modified metric. This procedures generalizes d'Alembert principle of classical mechanics [4, 5] which states that it is possible to transform a dynamical problem into a static one, where the body is free of any force. Going from the background metric - where an accelerated body experiences a non-gravitational force - to a dragged metric where the body follows a geodesics and become free of non-gravitational forces is the relativistic expression of this principle. In this way we produce a geometric description of all kind of motion whatever the force that originates it.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"II. A SPECIAL CASE\",\n \"content\": \"We claim that accelerated bodies in a flat Minkowski space-time 1 can be equivalently described as free bodies following geodesics in an associated dragged geometry. In order to simplify our calculation we restrict this section to the case in which the acceleration vector a µ is the gradient of a function, that is 2 Thus the force acting on the body under observation comes from a potential V in the Lagrangian formalism, i.e., F µ = ∂ µ V. We write the dragged metric in the form ̂ q µν = η µν + b v µ v ν . The associated covariant derivative is defined by where the corresponding Christoffel symbol is given by ̂ Γ /epsilon1 µν = 1 2 ( η /epsilon1α + b v /epsilon1 v α ) ( ̂ q αµ,ν + ̂ q αν,µ -̂ q µν,α ) , (6) where we are using a comma to denote simple derivative, that is A ,µ ≡ ∂ µ A. The description of an accelerated curve in a flat space-time as a geodesics in a dragged metric is possible if the following condition is satisfied ̂ where we have used the dragged metric to write ̂ v µ ≡ ̂ q µν v ν . Or, equivalently, Then noting that the acceleration in the background is defined by a µ = v µ,ν v ν and using equation (5) the condition of geodesics in the dragged geometry takes the form We have Using the expression of ̂ q αν and combining with condition (8) it follows that is, the expression of the coefficient b of the dragged metric is given in terms of the potential of the acceleration\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"A. The curvature of the dragged metric\",\n \"content\": \"In the case the background metric is not flat or if we use an arbitrary coordinate system the connection is given by the sum of the corresponding background one and a tensor, that is ̂ Γ /epsilon1 µν = Γ /epsilon1 µν + K /epsilon1 µν . (12) In the case of the Minkowski background a direct calculation gives for the connection Γ /epsilon1 µν the form Then, The contracted Ricci curvature has the expression The scalar of curvature ̂ R = ̂ R µν ̂ q µν is ̂ R = (2 + b ) a α ,α . (15) These expressions can be re-written in a covariant way by noting that which yields and for the scalar ̂ R the form\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"B. Analog gravity\",\n \"content\": \"Suppose that an observer following a path with fourvelocity v µ and acceleration a µ in the flat Minkowski space-time background is not able to identify the origin of the force that is acting on him. In other words he is going to believe that only long-range gravitational forces are constraining his motion. Let us assume that he knows that gravity does not accelerate any curve but instead change the metric of the background according to the principles of general relativity. This means that if he is able to represent his motion as a geodesics in a dragged metric ̂ q µν he will consider that the origin of such curved metric is nothing but a consequence of a distribution of energy which he will describe by using the equation He will identify the different terms of the source through his own motion. From his velocity v µ he defines the normalized four-velocity ˆ u α in the ̂ Q metric by setting He then proceed to characterize the origin of the curved metric using the standard decomposition In these expressions we used ̂ h µν = ̂ q µν -̂ u µ ̂ u ν . Note that h µν = h µν . Thus, he will write ̂ ̂ ̂ ̂ ̂ ̂ From this decomposition, using equation (18) and the curvature (16) he will identify the energy-momentum distribution as where Summarizing, we can say that this observer will state that there is a gravitational field represented by the metric ̂ q µν produced by the distribution of energy given by equation (24). We note that this reduction of the dragged metric to the framework of general relativity is not mandatory. Indeed, we deal here precisely with some accelerated paths that are not reduced to the gravitational force in the standard theory. This will become more clear when we present examples of accelerated curves in specific solutions of general relativity in the next sections. Indeed, let us present some clarifying examples. The first one considers the motion of rotating bodies in Minkowski space-time. The other ones take into account the general relativity effects. We shall see that it is possible to produce what could be called double gravity , if the origin of the curvature of the dragged metric is identified to an effective energy-momentum tensor satisfying the equations of general relativity. However there is no reason for this restriction. We will come back to this question elsewhere.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"III. ACCELERATION IN MINKOWSKI SPACE-TIME\",\n \"content\": \"Let us consider a simple example concerning the acceleration of a body in flat Minkowski space-time written in non-stationary cylindrical coordinate system ( t, r, φ, z ) to express the following line element where a is a constant. We choose the following local tetrad frame given implicitly by the 1-forms where we define ∆ = √ h 2 -g . The unique nonidentically null components of the Ricci tensor R AB in the tetrad frame are where a prime means derivative with respect to coordinate r. The equations of general relativity for this geometry have two simple solutions that we shall analyze below. In the case of R AB =0, we get Solving these equations, we find where ω is a constant. Therefore, Eq. (25) takes the form If we consider the observer field This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where We are in a situation similar to the previous section since the acceleration is a gradient. The parameter b of the dragged metric is given by the expression (11) and for the dragged metric the form Note that we are dealing with the case in which h 2 -ω 2 r 2 > 0 . This allows the presence of accelerated closed time-like curves (CTC) in the original background that will be mapped into closed time-like geodesics (CTG) in the dragged geometry. Note that there exists a real singularity in r = h/ω and that q 00 change sign where\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"IV. ACCELERATION IN CURVED SPACE-TIMES\",\n \"content\": \"Let us present now how our dragged metric approach works in some solutions of the equations of general relativity. We choose three well-known geometries: Schwarzschild, Godel universe and the Kerr solution. In these Riemannian manifolds we analyze some examples of accelerated paths that are interpreted as geodesics in the associated dragged metrics.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"A. Schwarzschild geometry\",\n \"content\": \"We set the Schwarzschild metric in the ( t, r, θ, ϕ ) coordinate system Choose the path described by the four-velocity The corresponding acceleration is In this case the acceleration is the gradient of function Ψ given by The factor b of the dragged metric is given by and the dragged metric takes the form The only non-null Ricci curvature of this ̂ q µν metric are All 14 Debever invariants are finite in all points except at the origin r = 0 .\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"B. Godel's geometry\",\n \"content\": \"Let us now turn our analysis to the Godel geometry. In the cylindrical coordinate system this metric is given by Eq. (25), where a is a constant related to the vorticity a = 2 /ω 2 and For completeness we note the non-trivial contravariant terms of the metric: In [6] it was pointed out the acausal properties of a particle moving into a circular orbit around the z -axis with four-velocity This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where Again, we are in a situation where the acceleration is a gradient. Therefore, the parameter b of the dragged metric is given by the expression (11) which in the Godel's background takes the form and the dragged metric has the following form From the analysis of geodesics in Godel geometry the domain r < r c where sinh 2 r c = 1 separates causal from non-causal regions of the space-time. This is related to the fact that a geodesic that pass the value r = 0 will be confined within the domain Ω i defined by the values of coordinate r in the region 0 < r < r c . See [7] for details. However, the gravitational field is finite in the region r = r c . Nothing similar in the dragged metric, once at sinh 2 r = 1 there exists a real singularity in the dragged metric. Only the exterior domain is allowed. This means that for this kind of accelerated path in Godel geometry the allowed domain for the dragged metric is precisely the whole acausal region.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"C. Kerr metric\",\n \"content\": \"Let us turn now to the dragged metric approach in the case the background is the Kerr metric. In the BoyerLindquist coordinate system this metric is given by where Σ = r 2 + a 2 -2 Mr and ρ 2 = r 2 + a 2 cos 2 θ . On equatorial plane ( θ = π/ 2) consider the following vector field This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where Once more we choose an accelerated path that can be represented by a gradient. The parameter b of the dragged metric is given by the expression (11) and for the dragged metric, on the equatorial plane, the form These two last cases (Godel and Kerr) show a very curious and intriguing property: the accelerated CTC's at their respective metrics) are transformed in curves that are geodesics, that is CTG's (at their corresponding dragged metric). Besides, in both cases, the dragged metrics display a real singularity excluding the causal domain.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"V. GENERAL CASE\",\n \"content\": \"In the precedent sections we limited our analysis to the case in which the acceleration is given by a unique function. Let us now pass to more general situation. In order to geometrize any kind of force we must deal with a larger class of geometries. The most general form of dragged metric that allows the description of accelerated bodies as true geodesics in a modified geometry has the form ̂ q µν = g µν + b v µ v ν + ma µ a ν + n ( v µ a ν + a µ v ν ) . (36) The three parameters b, m, n are related to the three degrees of freedom of the acceleration vector. The corresponding covariant form of the metric is given by ̂ q µν = g µν + Bv µ v ν + Ma µ a ν + N ( v µ a ν + a µ v ν ) . (37) in which B,M,N are given in terms of the parameters b, m, n by the relations In this case the equation that generalizes the condition of geodesics (8) has the form ̂ ̂ This equation can be cast in the following formal expression where each term Ψ 1 , Ψ 2 and Ψ 3 depends on all three functions m,b and n. Solving this equation for these three functions provide the most general expression for any acceleration. With these results, we have transformed the path of any particle submitted to any kind of force as a geodetic motion in the dragged metric. This result is the extension of the d'Alembert principle, corresponding to all types of motion i.e. the acceleration is geometrized through the dragged metric.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"VI. CONCLUSION\",\n \"content\": \"We summarize the novelty of our analysis in the following steps: We have shown by a constructive operation that the following conjecture is true: Conjecture. For any accelerated path Γ described by four-vector velocity v µ and acceleration a µ in a given Riemannian geometry g µν we can always construct another geometry ̂ Q endowed with a dragged metric ̂ q µν which depends only on g µν , v µ and a µ such that the path Γ is a geodesic in ̂ Q.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"VII. ACKNOWLEDGEMENTS\",\n \"content\": \"We would like to thank Dr Ivano Dami˜ao Soares for his comments in a previous version of this paper. We would like to thank FINEP, CNPq and FAPERJ and EB CNPq for their financial support.\",\n \"pages\": [\n 6\n ]\n }\n]"}}},{"rowIdx":268,"cells":{"bibcode":{"kind":"string","value":"2013GReGr..45.2441P"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1310.0159.pdf"},"content":{"kind":"string","value":"\nScalar filed evolution and Area Spectrum for Lovelock-AdS Black Holes\nC.B Prasobh\n· V.C Kuriakose\nReceived: 25 June 2013 / Accepted: 30 August 2013\nAbstract We study the modes of evolution of massless scalar fields in the asymptotically AdS spacetime surrounding maximally symmetric black holes of large and intermediate size in the Lovelock model. It is observed that all modes are purely damped at higher orders. Also, the rate of damping is seen to be independent of order at higher dimensions. The asymptotic form of these frequencies for the case of large black holes is found analytically. Finally, the area spectrum for such black holes is found from these asymptotic modes.\nKeywords Quasinormal Modes Lovelock AdS Area Spectrum Scalar Field\n· · · ·\nPACS PACS code1 · PACS code2 · more\n1 Introduction\nGauge-gravity dualities like the AdS/CFT correspondence [1] make it possible to study the properties of conformal fields in a particular dimension d by studying the evolution of fields in a black hole spacetime that is asymptotically AdS in ( d +1) dimensions and this has led to considerable interest in the study of asymptotically AdS black hole spacetimes. The main difficulty in studying field evolution in such spacetimes is that the stability of the spacetime against perturbations is not always guaranteed, unlike in the case of asymptotically flat spacetimes in first order theories in four dimensions. The instability of linear perturbations in higher-order and higher dimensional theories has already been investigated [2,3, 4]. The instability also extends to the thermodynamics of the black hole when we consider AdS spacetimes, with the well-known Hawking-Page phase transitions [5] signaling a transition between the black hole spacetime and thermal AdS spacetime. Recent studies [6,7,8] on the Lovelock model have confirmed the existence of\nC.B Prasobh\nDepartment of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: prasobhcb@outlook.com\nV.C Kuriakose\nDepartment of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: vck@cusat.ac.in\ndynamic instability against metric perturbations in asymptotically flat black hole spacetimes. Dynamic instability means that the solutions to the equation for the metric perturbation become unstable outside the event horizon for large values of their eigenvalue. This instability exists for all types of metric perturbations-tensor, vector and scalar. These instabilities occur when the mass of the black hole falls below a lower critical bound that depends on the coupling constants, the dimension of the spacetime and the order of the theory.\nQuasi normal modes are damped oscillations (having complex frequencies), known to dominate the intermediate stage of the evolution of small perturbations of a black hole spacetime and have been studied extensively. Detailed reviews and methods of calculation of quasi normal modes are found in numerous papers that include [9,10,11,12,13,14]. Quasi normal modes are known to depend only on the parameters of the black hole, such as mass, charge and angular momentum, and be completely independent of the type of the agent that caused it. These modes are obtained as the solution of the respective field equation, when solved with respect to the metric that describes the particular black hole spacetime of interest. In the case of asymptotically flat spacetimes, the effective potential that is perceived by the field in the spacetime often resembles a finite potential barrier such that the potential vanishes at infinity. This leads to the possibility of obtaining solutions that are purely ingoing at the event horizon of the black hole and purely outgoing at infinity, both resembling plane waves when expressed in terms of a scaled co-ordinate called the tortoise co-ordinate. These modes may be observed in the future with the aid of gravitational detectors. In the case of asymptotically AdS spacetimes, however, the potential grows indefinitely at infinity. Therefore, one usually looks for solutions that vanish at infinity, while the boundary condition at the event horizon unchanged [15]. This is motivated by the case of pure AdS spacetime, where the potential effectively confines the field as if 'in a box' and solutions exist only with a discrete spectrum of real frequencies. Even though black holes in asymptotically AdS spacetimes are not believed to exist in nature, interest in studying their quasi normal modes stems from the above-mentioned AdS/CFT correspondence [1]. According to the AdS/CFT correspondence, these perturbations correspond to perturbations of the thermal state of the strongly coupled conformal field at the boundary of the spacetime and the quasi normal modes correspond to the return to thermal equilibrium, so that the quasi normal frequencies give a measure of the time scale for the relaxation, which is difficult to compute directly. This provides the motivation to study the quasi normal modes of various fields in asymptotically AdS spacetimes. Earlier work the quasi normal modes of Schwarzschild-AdS black holes [15] have proved that the modes scale with the temperature of the event horizon.\nComplete quantization of gravity is one of the major goals of modern theoretical physics. Despite decades of research by physicists all over the world, it is yet to be achieved with complete success. One often takes clues from the classical theory of a system when attempting its quantization and gravity needs to be no different. Quantization of the black hole horizon area is expected to be a major feature of any successful quantum theory of gravity. The original 'classical' theory of gravity, namely the General Theory of Relativity (GTR) does provide us with the tools necessary to estimate the value of the area quantum. It is known\nfrom field theory that the presence of a periodicity in the classical theory of a system points to the existence of an adiabatic invariant with a discrete spectrum in the corresponding quantum theory. Interestingly, it has been observed in GTR that the numerical value of these frequencies, in the limit of 'large' frequencies, follow a distinct pattern, with the real part approaching a fixed value. These are termed asymptotic frequencies. Suggestions have been made that the fixed value of the real part can be viewed as a physically relevant periodicity in the (classical) black hole system which would then lead to the existence of certain adiabatic invariant quantity, which in turn would possess equally spaced spectrum according to Bohr-Sommerfeld quantization. Once we read it together with Bekenstein's original proposal [16] that the black hole entropy is an adiabatic invariant with a discrete, equally spaced spectrum, we come to the conclusion that the entropy spectrum (and, by extension, the area spectrum) can be deduced from the asymptotic value of the quasi normal frequencies. The connection between the fixed asymptotic frequencies and the quantized area spectrum was made by Hod [17]. Dreyer [18] recovered Hod's result in the Loop Quantum Gravity. A new interpretation for the quasi normal modes ω = ω R + iω I of perturbed black holes as equivalent to that of a collection of damped harmonic oscillators with real frequency ω 0 = √ ω 2 R + ω 2 I was introduced by Maggiore [19] and used in conjunction with Hod's method in order to compute the area spectrum of Schwarzschild black holes.\nAs with the case of quasi normal modes, it is the AdS/CFT correspondence that provides the motivation for studying the area spectrum of black holes in asymptotically AdS spacetimes. Recent studies ([20,21,22,23,24] and references therein) suggest that the gravitational dual of the holographic entanglement entropy in quantum field theories is the area of minimal-area surfaces in AdS spaces. The entropy can be used to study phase transitions between various states of the field. An area-entanglement entropy relation of the form S A = Area ( γ A ) 4 G ( d -2) N has been proposed [20,25] which is very similar to the familiar area-entropy relation in the General Theory of Relativity. Here, γ A is the d -dimensional minimal area surface in AdS d +2 and G ( d -2) N is the ( d +2)-dimensional gravitational constant of the AdS gravity. Although the above relation was originally proposed for AdS spaces, it is equally applicable to any asymptotically AdS space, including one containing a black hole. In that case, the minimal surface tends to wrap the horizon and thus we can use the area of the event horizon in order to compute the entanglement entropy in the conformal field theory.\nIn this work, we numerically compute the quasi normal frequencies for massless scalar field perturbations in asymptotically AdS, spherically symmetric spacetimes in Lovelock model using the metric derived in [26]. We analytically find out the asymptotic form of the frequencies following [27] and use it to deduce the area spectrum of large black holes in the model. A brief outline of the paper is as follows: in Sect. 2, we explain the maximally symmetric Lovelock model and the resulting metric for the spacetime as given in [26]. Details of the Horowitz-Hubeny numerical method of computing the quasi normal frequencies in AdS black hole spacetimes to the case of massless scalar fields in the vicinity of such a black hole in the Lovelock-AdS model are also given in the same section. The results of the numerical procedure are presented and analyzed in Sect. 3. In Sect. 4, we\nanalytically determine the asymptotic quasi normal frequencies of the field for the case of large back holes following [27]. The results of that analysis are used in order to find the area spectrum of the black hole using the Kunstatter's method [28] in Sect. 4.1. The results are summarized in Sect. 5.\n2 The metric and the numerical computation of Quasi normal Frequencies\nThe Lovelock model [29] is considered to be the most natural generalization of GTR. The Lovelock Lagrangian is a polynomial which consists of dimensionally continued higher order curvature terms. The most striking property of this Lagrangian is that it yields field equations that are in second order in the metric although the Lagrangian itself may contain higher order terms. Also, the theory is known to give solutions that are free of ghosts. The maximum order of terms in the action, k , is fixed by the number of dimensions of the spacetime d in Lovelock model according to the relation k = [ d -1 2 ] where [ x ] denotes the integer part of x . The action is written in terms of the Riemann curvature R ab = dω ab + ω a c ω cb and the vielbein e a as\nI G = κ ∫ k ∑ p =0 α p L ( p ) , (1)\nwhere α p are arbitrary (positive) coupling constants, and L ( p ) , given by\nL ( p ) = /epsilon1 a 1 ··· a d R a 1 a 2 ··· R a 2 p -1 a 2 p e a 2 p +1 ··· e a d , (2)\nis the p th order dimensionally continued term in the Lagrangian. The difficulty with the Lagrangian given above, with arbitrary values for α p , is that it becomes very difficult (if not impossible) to study the evolution of fields, since it is not at all clear whether the operator representing the evolution is Hermitian or not. As mentioned in the previous section, this problem, for the case of metric perturbations, has been analyzed in [6]. Although a general instability depending on the black hole mass has been established in that work, the numerical value for the critical mass has not been calculated. Also, it is very difficult to predict whether the presence of a cosmological constant raises or lowers the critical mass, as long as we consider a model with arbitrary α p . Moreover, for the same case, the the existence of negative energy solutions with horizons and positive energy solutions with naked singularities for (1) has been pointed out earlier [30,31]. These difficulties bring out the necessity of selecting suitable values for the coupling coefficients in order to have models that support maximally symmetric solutions and external perturbations. Maximally symmetric solutions to Lovelock model have long been known [26], which are derived by requiring that the theories must possess a unique cosmological constant (and consequently a unique AdS radius R ) for all orders. The resulting set of coupling constants are seen to be labeled by the order k and the gravitational constant G k . The metric describing the spherically symmetric black hole spacetime is derived from the action given in (1) with the choice\nα p = R 2( p -k ) ( d -2 p ) ( k p ) , p ≤ k 0 , p > k (3)\nwhere 1 ≤ k ≤ [ d -1 2 ]. The resulting field equations are of the form\n/epsilon1 ba 1 ··· a d -1 ¯ R a 1 a 2 ··· ¯ R a 2 k -1 a 2 k e a 2 k +1 ··· e a d -1 = 0 (4)\n/epsilon1 aba 3 ··· a d ¯ R a 3 a 4 ··· ¯ R a 2 k -1 a 2 k T a 2 k +1 e a 2 k +2 ··· e a d -1 = 0 (5)\nHere, ¯ R ab := R ab + 1 R 2 e a e b . Such theories are labeled by k and have two fundamental constants, κ and R , related to the gravitational constant G k and the cosmological constant Λ respectively through the relations\nκ = 1 2( d -2)! Ω d -2 G k , (6)\nΛ = -( d -1)( d -2) 2 R 2 , (7)\nΩ d -2 being the volume of the ( d -2) dimensional spherically symmetric tangent space. The static and spherically symmetric solutions to (4), written in Schwarzschild-like coordinates, take the form\nds 2 = f ( r ) dt 2 + dr 2 f ( r ) + r 2 dΩ 2 d -2 , (8)\nwhere f ( r ) is given by\nf ( r ) = 1 + r 2 R 2 -σ ( C 1 r d -2 k -1 ) 1 /k . (9)\nWe take σ = 1. The integration constant C 1 is written as\nC 1 = 2 G k ( M + C 0 ) , (10)\nwhere M stands for the mass of the black hole. The constant C 0 is chosen so that the horizon shrinks to a point for M → 0, as\nC 0 = 1 2 G k δ d -2 k, 1 . (11)\n/negationslash\nIt is interesting that the exponent of ( 1 r ) in (9) is proportional to ( d -2 k -1). Since k = [ d -1 2 ], ( d -2 k -1) = 0 in odd dimensions and ( d -2 k -1) = 1 in even dimensions. Thus the solution in even dimensions resemble the Schwarzschild-AdS solution. The ( d -2 k -1) = 0 cases correspond to Chern-Simmons theories which have a vacuum that is different from AdS [26]. Their quasi normal modes, mass and area spectra have already been computed [32]. What makes it interesting is the fact that recent studies [6] on the stability of metric perturbations in Lovelock model also point out that it is possible to predict the (in)stability of the perturbations only in even dimensions. The present work is limited to the cases where d -2 k -1 = 0. Then we have C 0 = 0 and C 1 = 2 G k M . Consider the scalar field Φ ( r, t, x i ) that obeys the Klein-Gordon equation given by\n1 √ g ∂ A √ gg AB ∂ B Φ = 0 , (12)\nx i being the co-ordinates in the spherically symmetric tangent space and g AB being components of the metric tensor. We impose the boundary conditions of ingoing plane wave solution at the event horizon and vanishing field at the boundary. The boundary condition at the horizon suggests the ansatz Φ = e -iω ( t + r ∗ ) where r ∗ is the tortoise coordinate defined by\ndr ∗ = dr f ( r ) . (13)\nIn the ( v = t + r ∗ , r ) system, the metric reads\nds 2 = -f ( r ) dv 2 +2 dvdr + r 2 dΩ 2 d -2 , (14)\nand we take the ansatz\nΦ ( v, r, x i ) = r 2 -d 2 ψ ( r ) Y ( x i ) e -iωv , (15)\nso that (12) becomes\nf ( r ) d 2 dr 2 ψ ( r ) + [ f ' ( r ) -2 iω ] d dr ψ ( r ) -V ( r ) ψ ( r ) = 0 , (16)\nwith the effective potential\nV ( r ) = ( d -2)( d -4) 4 r 2 f ( r ) + d -2 2 r f ' ( r ) + l ( l + d -3) r 2 . (17)\nHere, l represents the eigenvalue of the operator on the LHS of (12) acting on the functions Y ( x i ) in the spherically symmetric tangent space. In order to numerically calculate the quasi normal frequencies for (16), we expand the field Φ as a power series about the horizon and impose the vanishing boundary condition at infinity. We change the variable from r to x = 1 r in order to map the range r + < r < ∞ to a finite range. In terms of x , (16) becomes\ns ( x ) d 2 dx 2 ψ ( x ) + t ( x ) x -x + d dx ψ ( x ) + u ( x ) ( x -x + ) 2 ψ ( x ) = 0 , (18)\nwhere\nand\nV ( x ) = ( d -2)( d -4) x 2 f ( x ) 4 -( d -2) x 3 f ' ( x ) 2 + l ( l + d -3) x 2 . (20)\nIn the numerical procedure, we find out the coefficients of the expansion of the functions s ( x ) , t ( x ) and u ( x ) as power series in ( x -x + ) using a computer with s i , t i and u i denoting the coefficient for the i th term. Then an expansion for ψ ( x ) of the form\ns ( x ) = -x 4 f ( x ) , t ( x ) = -x 4 f ' ( x ) -2 x 3 f ( x ) -2 iωx 2 , u ( x ) = ( x -x + ) V ( x ) , (19)\nψ ( x ) = ∞ ∑ n =0 a n ( x -x + ) n , (21)\nis substituted into (18) which yields\na n = -1 P n n -1 ∑ k =0 [ k ( k -1) s n -k + kt n -k + u n -k ] a k , (22)\nwhere\nP n = n ( n -1) s 0 + nt 0 . (23)\nWe fix a 0 and numerically calculate the coefficients in (19) and (22) to different orders and compute the value of the filed ψ ( x ) as given in (21). Since we wish to impose the boundary condition of vanishing field at infinity, we solve the equation ψ (0) = 0 for ω . It is observed, by comparison with the values in [15], that the quasi normal frequencies are one of the solutions of the equation ψ (0) = 0, solved for ω , after ψ has been computed using (21) for some reasonably high value of n , which should be fixed by trial and error. We assume the same to be true in higher orders as well and look for the value of the quasi normal modes among the set of discrete values of ω that are obtained. For each value of ω obtained, we evaluate the absolute value of the LHS of (18) at a point close to the event horizon, since we have assumed plane wave solutions there. We choose that value of ω as the quasi normal frequency for which the absolute value comes closest to zero (the assumption here is that (21) is satisfied exactly only for the quasi normal frequency, which we seek, and not by other roots of ψ = 0.). We increase the number of terms to which ψ ( x ) and the coefficients are calculated until the required precision is attained.\n3 Discussion on Results of the Numerical Calculation\n/negationslash\nWe implement the procedure outlined above after fixing the value of the constants a 0 and R to 1. We investigate the quasi normal frequencies of large ( r h /greatermuch R ) and intermediate ( r + ∼ R ) black holes and set ω = ω R -iω I as done in [15] since we are interested in damped modes. As mentioned before, the analysis is limited to the cases where d -2 k -1 = 0. The results for the lowest ( l = 0) modes of the massless scalar field for first order theories have been summarized in TABLE 1. TABLE 2 contains the same for higher orders. Figures 1 to 6 show the results in detail. As evident from Figure 1 and Figure 2, both ω R and ω I show linear dependence on r + for the case of large black holes in first order theories. The linearity seems to be broken when we move to the intermediate-sized black holes, the results for which have been plotted in Figure 3 and Figure 4. Although the plots for intermediate-sized black holes look linear, the ω R -r + dependence for their case rather resembles an ( x, y = x + 1 x ) relation. The temperature of the event horizon for the metric (8), given by\nT = 1 4 πκ B k ( ( d -1) r + R 2 + d -2 k -1 r + ) , (24)\nd\n= 4\n, k\n= 1\nd\n= 5\n, k\n= 1\nd\n= 6\n, k\n= 1\n
\n\n
Table 2 Variation of the frequency of the lowest ( l = 0) massless ( m = 0) mode in higher orders.
\n
\n
\n\n
Table 1 Variation of the frequency of the lowest ( l = 0) massless ( m = 0) mode for first order theories.
\n
\nwhere κ B denotes the Boltzmann's constant, also depends on r + in the same manner. Since ( x, y = x + 1 x ) ∼ x for large x , we conclude that both ω R and ω I scale with the temperature for large as well as intermediate-sized black holes for first order theories, in agreement with earlier works [15]. Another observation is that ω I seems to be independent of dimension d in first order theories. When we consider higher order theories, the numerical results for which have been plotted in Figure 5 and Figure 6, we observe that all modes are purely damped ones. We have only plotted ω I vs r + in the case of higher order theories for this reason. There, we observe that, both for large and intermediate-sized black holes, ω I is independent of the order of the theory when the dimension d stays the same.\n4 Asymptotic quasi normal modes and area spectrum of large black holes\nWe analytically find the asymptotic form of the quasi normal frequencies in the large black hole limit following the method of perturbative expansion of the wave equation in the dimensionless parameter ω/T H that has earlier been employed in the case of d -dimensional SAdS black holes [27]. Here, T H is the Hawking temperature of the horizon and ω is the frequency of the mode. We take the metric to be of the form given in (8). For large black holes, the metric gets approximated as,\nω\n\n\n
Fig. 1 ω R vs r + plot for large black holes in first order theories
\n\n\n\n
Fig. 2 ω I vs r + plot for large black holes in first order theories
\n\nds 2 = ˆ f ( r ) dt 2 + dr 2 ˆ f ( r ) + r 2 ds 2 ( E d -2 ) , (25)\nwith\nˆ f ( r ) = r 2 R 2 -( 2 G k M r d -2 k -1 ) . (26)\nIt is easily seen that the event horizon is given by\nr h = R [ 2 G k M R d -2 k -1 ] 1 d -1 . (27)\nIn terms of the new metric (25), the Klein-Gordon field equation (12) for m = 0 becomes\nω\n\n\n
Fig. 3 ω R vs r + plot for intermediate black holes in first order theories
\n\nω\n\n\n
Fig. 4 ω I vs r + plot for intermediate black holes in first order theories
\n\n+\n1 r d -2 ∂ r ( r d A ( r ) ∂ r Φ ) -R 4 r 2 A ( r ) ∂ 2 t Φ -R 2 r 2 ∇ 2 Φ = 0 , (28)\nwhere\nWe write the field Φ as\nA ( r ) = 1 -( r h r ) d -1 k . (29)\nΦ ( t, r, x i ) = e i ( ωt -p · x ) Ψ ( r ) , (30)\nand change the variable from r to\ny = ( r r h ) d -1 2 k , (31)\nso that (28) becomes\nω\n\n\n
Fig. 5 ω I vs r + plot for large black holes in higher order theories
\n\nω\n\n\n
Fig. 6 ω I vs r + plot for intermediate black holes in first order theories
\n\ny Q ( y 2 -1) ( y 2 k -1 ( y 2 -1) Ψ ' ) ' + [ ˆ ω 2 A 2 y 2 -ˆ p 2 A 2 ( y 2 -1) ] Ψ = 0 , (32)\nwhere the parameters ˆ ω and ˆ p are defined as\nˆ ω = ωR 2 r h , ˆ p = | p | R r h , (33)\nand\nQ = 6 k -(2 k -1) d -1 d -1 , A = d -1 2 k . (34)\nWe investigate the behavior of (32) near the boundaries y → 1 and y →∞ and the point y →-1 in order to develop an ansatz for Ψ ( y ). The following solutions are obtained:\nΨ ∼ y -2 k , y →∞ ( y -1) ± i ˆ ω/ 2 A , y → 1 ( y +1) ± ˆ ω/ 2 A , y →-1 (35)\n Since we demand ingoing plane wave like solutions at the horizon, we take the form Ψ ∼ ( y -1) -i ˆ ω/ 2 A near the horizon ( y = 1). We isolate the solutions near y = ± 1 and write\nΨ ( y ) = ( y -1) -i ˆ ω/ 2 A ( y +1) ± ˆ ω/ 2 A N ( y ) . (36)\nSubstituting (36) in (32), we deduce the equation satisfied by N ( y ) as\ny ( y 2 -1) N '' -ˆ ω 2 y 2 A 2 ( y 2 -1) N + + { ˆ ω A ( ∓ i ˆ ω 2 A ± k -ik ) y -( i ± 1)(2 k -1) ˆ ω 2 A } N {( 2 k +1 -i ∓ 1 A ˆ ω ) y 2 -i ± 1 A ˆ ωy -(2 k -1) } N ' + 1 y Q +2 k -2 ( ˆ ω 2 y 2 A 2 ( y 2 -1) -ˆ p 2 A 2 ) N = 0 . (37)\nWe consider (37) in the range of large ˆ ω and large y , so that y 2 ≈ y 2 -1 and the terms proportional to 1 / ( y Q +2 k -2 ) may be dropped along with the constant terms. Then (37) reduces to\n( y 2 -1) N '' + ˆ ω A ( ∓ i ˆ ω 2 A ± k -ik ) N + {( (2 k +1) -i ∓ 1 A ˆ ω ) y -i ± 1 A ˆ ω } N ' = 0 , (38)\nwhich is the Hypergeometric equation with the solution\nN ( y ) = 2 F 1 ( a, b ; c ; ( y +1) / 2) , (39)\nwhere\na = k -i ∓ 1 2 A ˆ ω + k, b = -i ∓ 1 2 A ˆ ω, c = 2 k +1 2 ± ˆ ω A . (40)\nIn order to match the behavior of the solution (36) at infinity with that demanded by (35), we demand that N ( y ) be a polynomial as y →∞ . That condition is satisfied when\na = -n, n = 1 , 2 , ... (41)\nIf a = -n , then, according to the property of the hypergeometric equation, N ( y ) ∼ y n = y -a , so that, according to (36),\nΨ ∼ ( y -1) -i ˆ ω/ 2 A ( y +1) ± ˆ ω/ 2 A y -a ≈ y -i ˆ ω/ 2 A y ± ˆ ω/ 2 A y -a = y -2 k , (42)\nas required. We deduce the expression for the asymptotic form of quasi normal frequencies from (41) as follows:\na = -n ⇒ 2 k -i ∓ 1 2 A ˆ ω = -n ⇒ ˆ ω asy = A ( n +2 k )( ± 1 -i ) , (43)\nso that (33) implies\nω asy = A ( r h R 2 ) ( n +2 k )( ± 1 -i ) , (44)\nwhich gives the asymptotic form of the quasi normal frequencies for large maximally symmetric AdS black holes in the Lovelock model. We observe that the high-overtone quasi normal frequencies are equispaced, which is in agreement with earlier observations [15,9,12].\n4.1 Area Spectrum\nWe calculate the area spectrum of the large AdS black holes in Lovelock model using the new physical interpretation of the quasi normal modes proposed by Maggiore [19] and following Kunstatter's method [28]. According to the first law of black hole thermodynamics, for a black hole system with energy E (or M ) with Hawking temperature T H and horizon area A , the following relation holds:\ndM = 1 4 T H dA. (45)\nAccording to Kunstatter, if the frequency of oscillation of the system is ω ( E ), then the quantity\nI = ∫ dE ω ( E ) , (46)\nis to be taken as the corresponding adiabatic invariant. According to Maggiore [19], the black hole system has to be modeled by a collection of damped harmonic oscillators. If the system has a quasi normal frequency ω = ω R + iω I , then the corresponding vibrational frequency according to the model is to be taken as\nω 0 = √ ω 2 R + ω 2 I . (47)\nIn the highly damped and highly excited cases, ω 0 can be approximated by ω I and ω R respectively. Here, we see that for higher values of the number n , the both ω R and ω I increase. Therefore we consider transitions between two adjacent energy levels of the system and take the physical frequency as equal to the difference in ω 0 for the two systems; that is, we take\nω ( E ) = ∆ω = ( ω 0 ) n -( ω 0 ) n -1 . (48)\nWe deduce the area spectrum from (46) using the relations (27) and (44). The expression for the adiabatic invariant now reads\nI = ∫ dE ω ( E ) = ∫ dM ∆ω = ∫ ( dM dr h )( 1 ∆ω ) dr h . (49)\nUsing (27), (44), (47) and (48) in (49), it is easy to see that the Bohr-Sommerfeld quantization condition, namely I = n /planckover2pi1 now reads\n( d -1 d -2 )( 1 2 √ 2 AG k R 2 k -2 ) r d -2 h = n /planckover2pi1 , (50)\nwhich, in terms of the area A of the horizon, can be written in the form\nA = γn /planckover2pi1 , (51)\nwhere\nγ = [ Γ ( d -1 2 )( d -1 d -2 )( 1 4 √ 2 π ( d -1) / 2 AG k R 2 k -2 ) ] -1 . (52)\nThus the area spectrum, and consequently the entropy spectrum, of large black holes in asymptotically AdS Lovelock spacetimes is seen to depend on the parameters G k and the AdS radius R of the theory. The dependence of A on R is observed only when higher orders are considered. A is also dependent on the dimension d of the spacetime, and consequently on the order of the theory.\n5 Conclusion\nWe have analyzed the evolution of massless Klein-Gordon field in the maximally symmetric asymptotically AdS spacetime surrounding a black hole in the Lovelock model. We have used the form of the metric that has been derived in [26] in order to compute the quasi normal frequencies using the Horowitz-Hubeny method. The results of the numerical computation show that the modes in the case of higher order theories are purely damped. For the case of large as well as intermediate black holes, the frequencies are observed to scale linearly with the temperature of the event horizon. When we consider higher order theories, the imaginary part of the quasi normal frequencies is observed to be independent of the order of the theory in higher dimensions. They appear to be dependent on the dimension only.\nThe asymptotic form of the quasi normal frequencies for the case of very large black holes has been analytically determined using the method of perturbative expansion of the wave equation in terms of ω/T H , as developed in [27]. We find that the asymptotic modes are equispaced, in agreement with previous results. We have also calculated the area spectrum spacing and found it to be dependent on the value of R , d and k . This is also in contrast to the case of first order theories where we always obtain area spectra that are equidistant even when the parameters of the black hole spacetime change.\n6 Acknowledgement\nThe authors would like to acknowledge financial assistance from the Council of Scientific and Industrial Research (CSIR), India during the work under the CSIR Emeritus Scientistship Scheme sanctioned to one of the authors (VCK). VCK would also like to acknowledge Associateship of IUCAA, Pune, India.\nReferences\n\n1. J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).\n2. R. Gregory and R. Laflamme, Phys. Rev.Lett. 70, 2837 (1993).\n3. G. Dotti and R.J. Gleiser, Phys. Rev. D 72, 044018 (2005).\n4. G. Dotti and R.J. Gleiser, Phys. Rev. D 72, 124002 (2005).\n5. Hawking S.W and Page D.N, Comm. Math.Phys. 87, 577 (1983).\n6. T. Takahashi and J. Soda, Prog.Theor.Phys. 124:711-729 (2010).\n7. T. Takahashi and J. Soda, Phys.Rev.D 79, 104025 (2009).\n8. T. Takahashi and J. Soda, Phys. Rev. D 80, 104021 (2009).\n9. E. Berti, V. Cardoso and A.O. Starinets, Class. Quantum Grav. 26, 163001 (2009).\n10. H.P. Nollert, Class. Quantum Grav. 16, R159 (2009).\n11. K.D. Kokkotas and B. Schmidt, Living Rev. Relativity 2, (1999), 2.\n12. R.A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011).\n13. R.A. Konoplya, Phys. Rev. D 66, 044009 (2002).\n14. H.T. Cho et al., Advances in Mathematical Physics Volume 2012, Article ID 281705.\n15. G. T. Horowitz and V. E. Hubeny, Phys. Rev. D 62, 024027 (2000).\n16. J.D. Bekenstein, Lett. Nuovo Cimento Soc. Ital. Fis. 11, 467 (1974).\n17. S. Hod, Phys. Rev. Lett. 81, 4293 (1998).\n18. O. Dreyer, Phys. Rev. Lett. 90, 081301 (2003).\n19. M. Maggiore, Phys. Rev. Lett. 100, 141301 (2008).\n20. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006).\n21. S. Ryu and T. Takayanagi, JHEP 07 (2007) 62.\n22. S. Ryu and T. Takayanagi and T. Nishioka, J. Phys. A: Math. Theor. 42 (2009) 504008.\n23. T. Takayanagi, Class. Quantum Grav. 29, 153001 (2009).\n24. P. Dey and S. Roy, Phys. Rev. D 87, 066001 (2013).\n25. S. Ryu and T. Takayanagi, JHEP 08 (2006) 045.\n26. J. Crisostomo, R. Troncoso and J. Zanelli, Phys. Rev. D 62, 084013 (2000).\n27. S. Musiri and G. Siopsis, Phys. Lett. B 576, 309-313 (2003).\n28. G. Kunstatter, Phys. Rev. Lett. 90, 161301 (2003).\n29. D. Lovelock, J. Math. Phys. (N.Y.) 12, 498 (1971).\n30. D.G. Boulware and S. Deser, Phys.Rev.Lett. 55, 2656 (1985)\n31. J. T. Wheeler, Nucl. Phys. B268, 737 (1986).\n32. Pablo Gonzalez et al., JHEP 1008, 050 (2010)\n"},"sections":{"kind":"list like","value":[{"title":"Scalar filed evolution and Area Spectrum for Lovelock-AdS Black Holes","content":"C.B Prasobh · V.C Kuriakose Received: 25 June 2013 / Accepted: 30 August 2013 Abstract We study the modes of evolution of massless scalar fields in the asymptotically AdS spacetime surrounding maximally symmetric black holes of large and intermediate size in the Lovelock model. It is observed that all modes are purely damped at higher orders. Also, the rate of damping is seen to be independent of order at higher dimensions. The asymptotic form of these frequencies for the case of large black holes is found analytically. Finally, the area spectrum for such black holes is found from these asymptotic modes. Keywords Quasinormal Modes Lovelock AdS Area Spectrum Scalar Field · · · · PACS PACS code1 · PACS code2 · more","pages":[1]},{"title":"1 Introduction","content":"Gauge-gravity dualities like the AdS/CFT correspondence [1] make it possible to study the properties of conformal fields in a particular dimension d by studying the evolution of fields in a black hole spacetime that is asymptotically AdS in ( d +1) dimensions and this has led to considerable interest in the study of asymptotically AdS black hole spacetimes. The main difficulty in studying field evolution in such spacetimes is that the stability of the spacetime against perturbations is not always guaranteed, unlike in the case of asymptotically flat spacetimes in first order theories in four dimensions. The instability of linear perturbations in higher-order and higher dimensional theories has already been investigated [2,3, 4]. The instability also extends to the thermodynamics of the black hole when we consider AdS spacetimes, with the well-known Hawking-Page phase transitions [5] signaling a transition between the black hole spacetime and thermal AdS spacetime. Recent studies [6,7,8] on the Lovelock model have confirmed the existence of","pages":[1]},{"title":"C.B Prasobh","content":"Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: prasobhcb@outlook.com V.C Kuriakose Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: vck@cusat.ac.in dynamic instability against metric perturbations in asymptotically flat black hole spacetimes. Dynamic instability means that the solutions to the equation for the metric perturbation become unstable outside the event horizon for large values of their eigenvalue. This instability exists for all types of metric perturbations-tensor, vector and scalar. These instabilities occur when the mass of the black hole falls below a lower critical bound that depends on the coupling constants, the dimension of the spacetime and the order of the theory. Quasi normal modes are damped oscillations (having complex frequencies), known to dominate the intermediate stage of the evolution of small perturbations of a black hole spacetime and have been studied extensively. Detailed reviews and methods of calculation of quasi normal modes are found in numerous papers that include [9,10,11,12,13,14]. Quasi normal modes are known to depend only on the parameters of the black hole, such as mass, charge and angular momentum, and be completely independent of the type of the agent that caused it. These modes are obtained as the solution of the respective field equation, when solved with respect to the metric that describes the particular black hole spacetime of interest. In the case of asymptotically flat spacetimes, the effective potential that is perceived by the field in the spacetime often resembles a finite potential barrier such that the potential vanishes at infinity. This leads to the possibility of obtaining solutions that are purely ingoing at the event horizon of the black hole and purely outgoing at infinity, both resembling plane waves when expressed in terms of a scaled co-ordinate called the tortoise co-ordinate. These modes may be observed in the future with the aid of gravitational detectors. In the case of asymptotically AdS spacetimes, however, the potential grows indefinitely at infinity. Therefore, one usually looks for solutions that vanish at infinity, while the boundary condition at the event horizon unchanged [15]. This is motivated by the case of pure AdS spacetime, where the potential effectively confines the field as if 'in a box' and solutions exist only with a discrete spectrum of real frequencies. Even though black holes in asymptotically AdS spacetimes are not believed to exist in nature, interest in studying their quasi normal modes stems from the above-mentioned AdS/CFT correspondence [1]. According to the AdS/CFT correspondence, these perturbations correspond to perturbations of the thermal state of the strongly coupled conformal field at the boundary of the spacetime and the quasi normal modes correspond to the return to thermal equilibrium, so that the quasi normal frequencies give a measure of the time scale for the relaxation, which is difficult to compute directly. This provides the motivation to study the quasi normal modes of various fields in asymptotically AdS spacetimes. Earlier work the quasi normal modes of Schwarzschild-AdS black holes [15] have proved that the modes scale with the temperature of the event horizon. Complete quantization of gravity is one of the major goals of modern theoretical physics. Despite decades of research by physicists all over the world, it is yet to be achieved with complete success. One often takes clues from the classical theory of a system when attempting its quantization and gravity needs to be no different. Quantization of the black hole horizon area is expected to be a major feature of any successful quantum theory of gravity. The original 'classical' theory of gravity, namely the General Theory of Relativity (GTR) does provide us with the tools necessary to estimate the value of the area quantum. It is known from field theory that the presence of a periodicity in the classical theory of a system points to the existence of an adiabatic invariant with a discrete spectrum in the corresponding quantum theory. Interestingly, it has been observed in GTR that the numerical value of these frequencies, in the limit of 'large' frequencies, follow a distinct pattern, with the real part approaching a fixed value. These are termed asymptotic frequencies. Suggestions have been made that the fixed value of the real part can be viewed as a physically relevant periodicity in the (classical) black hole system which would then lead to the existence of certain adiabatic invariant quantity, which in turn would possess equally spaced spectrum according to Bohr-Sommerfeld quantization. Once we read it together with Bekenstein's original proposal [16] that the black hole entropy is an adiabatic invariant with a discrete, equally spaced spectrum, we come to the conclusion that the entropy spectrum (and, by extension, the area spectrum) can be deduced from the asymptotic value of the quasi normal frequencies. The connection between the fixed asymptotic frequencies and the quantized area spectrum was made by Hod [17]. Dreyer [18] recovered Hod's result in the Loop Quantum Gravity. A new interpretation for the quasi normal modes ω = ω R + iω I of perturbed black holes as equivalent to that of a collection of damped harmonic oscillators with real frequency ω 0 = √ ω 2 R + ω 2 I was introduced by Maggiore [19] and used in conjunction with Hod's method in order to compute the area spectrum of Schwarzschild black holes. As with the case of quasi normal modes, it is the AdS/CFT correspondence that provides the motivation for studying the area spectrum of black holes in asymptotically AdS spacetimes. Recent studies ([20,21,22,23,24] and references therein) suggest that the gravitational dual of the holographic entanglement entropy in quantum field theories is the area of minimal-area surfaces in AdS spaces. The entropy can be used to study phase transitions between various states of the field. An area-entanglement entropy relation of the form S A = Area ( γ A ) 4 G ( d -2) N has been proposed [20,25] which is very similar to the familiar area-entropy relation in the General Theory of Relativity. Here, γ A is the d -dimensional minimal area surface in AdS d +2 and G ( d -2) N is the ( d +2)-dimensional gravitational constant of the AdS gravity. Although the above relation was originally proposed for AdS spaces, it is equally applicable to any asymptotically AdS space, including one containing a black hole. In that case, the minimal surface tends to wrap the horizon and thus we can use the area of the event horizon in order to compute the entanglement entropy in the conformal field theory. In this work, we numerically compute the quasi normal frequencies for massless scalar field perturbations in asymptotically AdS, spherically symmetric spacetimes in Lovelock model using the metric derived in [26]. We analytically find out the asymptotic form of the frequencies following [27] and use it to deduce the area spectrum of large black holes in the model. A brief outline of the paper is as follows: in Sect. 2, we explain the maximally symmetric Lovelock model and the resulting metric for the spacetime as given in [26]. Details of the Horowitz-Hubeny numerical method of computing the quasi normal frequencies in AdS black hole spacetimes to the case of massless scalar fields in the vicinity of such a black hole in the Lovelock-AdS model are also given in the same section. The results of the numerical procedure are presented and analyzed in Sect. 3. In Sect. 4, we analytically determine the asymptotic quasi normal frequencies of the field for the case of large back holes following [27]. The results of that analysis are used in order to find the area spectrum of the black hole using the Kunstatter's method [28] in Sect. 4.1. The results are summarized in Sect. 5.","pages":[1,2,3,4]},{"title":"2 The metric and the numerical computation of Quasi normal Frequencies","content":"The Lovelock model [29] is considered to be the most natural generalization of GTR. The Lovelock Lagrangian is a polynomial which consists of dimensionally continued higher order curvature terms. The most striking property of this Lagrangian is that it yields field equations that are in second order in the metric although the Lagrangian itself may contain higher order terms. Also, the theory is known to give solutions that are free of ghosts. The maximum order of terms in the action, k , is fixed by the number of dimensions of the spacetime d in Lovelock model according to the relation k = [ d -1 2 ] where [ x ] denotes the integer part of x . The action is written in terms of the Riemann curvature R ab = dω ab + ω a c ω cb and the vielbein e a as where α p are arbitrary (positive) coupling constants, and L ( p ) , given by is the p th order dimensionally continued term in the Lagrangian. The difficulty with the Lagrangian given above, with arbitrary values for α p , is that it becomes very difficult (if not impossible) to study the evolution of fields, since it is not at all clear whether the operator representing the evolution is Hermitian or not. As mentioned in the previous section, this problem, for the case of metric perturbations, has been analyzed in [6]. Although a general instability depending on the black hole mass has been established in that work, the numerical value for the critical mass has not been calculated. Also, it is very difficult to predict whether the presence of a cosmological constant raises or lowers the critical mass, as long as we consider a model with arbitrary α p . Moreover, for the same case, the the existence of negative energy solutions with horizons and positive energy solutions with naked singularities for (1) has been pointed out earlier [30,31]. These difficulties bring out the necessity of selecting suitable values for the coupling coefficients in order to have models that support maximally symmetric solutions and external perturbations. Maximally symmetric solutions to Lovelock model have long been known [26], which are derived by requiring that the theories must possess a unique cosmological constant (and consequently a unique AdS radius R ) for all orders. The resulting set of coupling constants are seen to be labeled by the order k and the gravitational constant G k . The metric describing the spherically symmetric black hole spacetime is derived from the action given in (1) with the choice where 1 ≤ k ≤ [ d -1 2 ]. The resulting field equations are of the form Here, ¯ R ab := R ab + 1 R 2 e a e b . Such theories are labeled by k and have two fundamental constants, κ and R , related to the gravitational constant G k and the cosmological constant Λ respectively through the relations Ω d -2 being the volume of the ( d -2) dimensional spherically symmetric tangent space. The static and spherically symmetric solutions to (4), written in Schwarzschild-like coordinates, take the form where f ( r ) is given by We take σ = 1. The integration constant C 1 is written as where M stands for the mass of the black hole. The constant C 0 is chosen so that the horizon shrinks to a point for M → 0, as /negationslash It is interesting that the exponent of ( 1 r ) in (9) is proportional to ( d -2 k -1). Since k = [ d -1 2 ], ( d -2 k -1) = 0 in odd dimensions and ( d -2 k -1) = 1 in even dimensions. Thus the solution in even dimensions resemble the Schwarzschild-AdS solution. The ( d -2 k -1) = 0 cases correspond to Chern-Simmons theories which have a vacuum that is different from AdS [26]. Their quasi normal modes, mass and area spectra have already been computed [32]. What makes it interesting is the fact that recent studies [6] on the stability of metric perturbations in Lovelock model also point out that it is possible to predict the (in)stability of the perturbations only in even dimensions. The present work is limited to the cases where d -2 k -1 = 0. Then we have C 0 = 0 and C 1 = 2 G k M . Consider the scalar field Φ ( r, t, x i ) that obeys the Klein-Gordon equation given by x i being the co-ordinates in the spherically symmetric tangent space and g AB being components of the metric tensor. We impose the boundary conditions of ingoing plane wave solution at the event horizon and vanishing field at the boundary. The boundary condition at the horizon suggests the ansatz Φ = e -iω ( t + r ∗ ) where r ∗ is the tortoise coordinate defined by In the ( v = t + r ∗ , r ) system, the metric reads and we take the ansatz so that (12) becomes with the effective potential Here, l represents the eigenvalue of the operator on the LHS of (12) acting on the functions Y ( x i ) in the spherically symmetric tangent space. In order to numerically calculate the quasi normal frequencies for (16), we expand the field Φ as a power series about the horizon and impose the vanishing boundary condition at infinity. We change the variable from r to x = 1 r in order to map the range r + < r < ∞ to a finite range. In terms of x , (16) becomes where and In the numerical procedure, we find out the coefficients of the expansion of the functions s ( x ) , t ( x ) and u ( x ) as power series in ( x -x + ) using a computer with s i , t i and u i denoting the coefficient for the i th term. Then an expansion for ψ ( x ) of the form is substituted into (18) which yields where We fix a 0 and numerically calculate the coefficients in (19) and (22) to different orders and compute the value of the filed ψ ( x ) as given in (21). Since we wish to impose the boundary condition of vanishing field at infinity, we solve the equation ψ (0) = 0 for ω . It is observed, by comparison with the values in [15], that the quasi normal frequencies are one of the solutions of the equation ψ (0) = 0, solved for ω , after ψ has been computed using (21) for some reasonably high value of n , which should be fixed by trial and error. We assume the same to be true in higher orders as well and look for the value of the quasi normal modes among the set of discrete values of ω that are obtained. For each value of ω obtained, we evaluate the absolute value of the LHS of (18) at a point close to the event horizon, since we have assumed plane wave solutions there. We choose that value of ω as the quasi normal frequency for which the absolute value comes closest to zero (the assumption here is that (21) is satisfied exactly only for the quasi normal frequency, which we seek, and not by other roots of ψ = 0.). We increase the number of terms to which ψ ( x ) and the coefficients are calculated until the required precision is attained.","pages":[4,5,6,7]},{"title":"3 Discussion on Results of the Numerical Calculation","content":"/negationslash We implement the procedure outlined above after fixing the value of the constants a 0 and R to 1. We investigate the quasi normal frequencies of large ( r h /greatermuch R ) and intermediate ( r + ∼ R ) black holes and set ω = ω R -iω I as done in [15] since we are interested in damped modes. As mentioned before, the analysis is limited to the cases where d -2 k -1 = 0. The results for the lowest ( l = 0) modes of the massless scalar field for first order theories have been summarized in TABLE 1. TABLE 2 contains the same for higher orders. Figures 1 to 6 show the results in detail. As evident from Figure 1 and Figure 2, both ω R and ω I show linear dependence on r + for the case of large black holes in first order theories. The linearity seems to be broken when we move to the intermediate-sized black holes, the results for which have been plotted in Figure 3 and Figure 4. Although the plots for intermediate-sized black holes look linear, the ω R -r + dependence for their case rather resembles an ( x, y = x + 1 x ) relation. The temperature of the event horizon for the metric (8), given by d = 4 , k = 1 d = 5 , k = 1 d = 6 , k = 1 where κ B denotes the Boltzmann's constant, also depends on r + in the same manner. Since ( x, y = x + 1 x ) ∼ x for large x , we conclude that both ω R and ω I scale with the temperature for large as well as intermediate-sized black holes for first order theories, in agreement with earlier works [15]. Another observation is that ω I seems to be independent of dimension d in first order theories. When we consider higher order theories, the numerical results for which have been plotted in Figure 5 and Figure 6, we observe that all modes are purely damped ones. We have only plotted ω I vs r + in the case of higher order theories for this reason. There, we observe that, both for large and intermediate-sized black holes, ω I is independent of the order of the theory when the dimension d stays the same.","pages":[7,8]},{"title":"4 Asymptotic quasi normal modes and area spectrum of large black holes","content":"We analytically find the asymptotic form of the quasi normal frequencies in the large black hole limit following the method of perturbative expansion of the wave equation in the dimensionless parameter ω/T H that has earlier been employed in the case of d -dimensional SAdS black holes [27]. Here, T H is the Hawking temperature of the horizon and ω is the frequency of the mode. We take the metric to be of the form given in (8). For large black holes, the metric gets approximated as, ω with It is easily seen that the event horizon is given by In terms of the new metric (25), the Klein-Gordon field equation (12) for m = 0 becomes ω ω + where We write the field Φ as and change the variable from r to so that (28) becomes ω ω where the parameters ˆ ω and ˆ p are defined as and We investigate the behavior of (32) near the boundaries y → 1 and y →∞ and the point y →-1 in order to develop an ansatz for Ψ ( y ). The following solutions are obtained: Since we demand ingoing plane wave like solutions at the horizon, we take the form Ψ ∼ ( y -1) -i ˆ ω/ 2 A near the horizon ( y = 1). We isolate the solutions near y = ± 1 and write Substituting (36) in (32), we deduce the equation satisfied by N ( y ) as We consider (37) in the range of large ˆ ω and large y , so that y 2 ≈ y 2 -1 and the terms proportional to 1 / ( y Q +2 k -2 ) may be dropped along with the constant terms. Then (37) reduces to which is the Hypergeometric equation with the solution where In order to match the behavior of the solution (36) at infinity with that demanded by (35), we demand that N ( y ) be a polynomial as y →∞ . That condition is satisfied when If a = -n , then, according to the property of the hypergeometric equation, N ( y ) ∼ y n = y -a , so that, according to (36), as required. We deduce the expression for the asymptotic form of quasi normal frequencies from (41) as follows: so that (33) implies which gives the asymptotic form of the quasi normal frequencies for large maximally symmetric AdS black holes in the Lovelock model. We observe that the high-overtone quasi normal frequencies are equispaced, which is in agreement with earlier observations [15,9,12].","pages":[8,9,10,11,12,13]},{"title":"4.1 Area Spectrum","content":"We calculate the area spectrum of the large AdS black holes in Lovelock model using the new physical interpretation of the quasi normal modes proposed by Maggiore [19] and following Kunstatter's method [28]. According to the first law of black hole thermodynamics, for a black hole system with energy E (or M ) with Hawking temperature T H and horizon area A , the following relation holds: According to Kunstatter, if the frequency of oscillation of the system is ω ( E ), then the quantity is to be taken as the corresponding adiabatic invariant. According to Maggiore [19], the black hole system has to be modeled by a collection of damped harmonic oscillators. If the system has a quasi normal frequency ω = ω R + iω I , then the corresponding vibrational frequency according to the model is to be taken as In the highly damped and highly excited cases, ω 0 can be approximated by ω I and ω R respectively. Here, we see that for higher values of the number n , the both ω R and ω I increase. Therefore we consider transitions between two adjacent energy levels of the system and take the physical frequency as equal to the difference in ω 0 for the two systems; that is, we take We deduce the area spectrum from (46) using the relations (27) and (44). The expression for the adiabatic invariant now reads Using (27), (44), (47) and (48) in (49), it is easy to see that the Bohr-Sommerfeld quantization condition, namely I = n /planckover2pi1 now reads which, in terms of the area A of the horizon, can be written in the form where Thus the area spectrum, and consequently the entropy spectrum, of large black holes in asymptotically AdS Lovelock spacetimes is seen to depend on the parameters G k and the AdS radius R of the theory. The dependence of A on R is observed only when higher orders are considered. A is also dependent on the dimension d of the spacetime, and consequently on the order of the theory.","pages":[13,14]},{"title":"5 Conclusion","content":"We have analyzed the evolution of massless Klein-Gordon field in the maximally symmetric asymptotically AdS spacetime surrounding a black hole in the Lovelock model. We have used the form of the metric that has been derived in [26] in order to compute the quasi normal frequencies using the Horowitz-Hubeny method. The results of the numerical computation show that the modes in the case of higher order theories are purely damped. For the case of large as well as intermediate black holes, the frequencies are observed to scale linearly with the temperature of the event horizon. When we consider higher order theories, the imaginary part of the quasi normal frequencies is observed to be independent of the order of the theory in higher dimensions. They appear to be dependent on the dimension only. The asymptotic form of the quasi normal frequencies for the case of very large black holes has been analytically determined using the method of perturbative expansion of the wave equation in terms of ω/T H , as developed in [27]. We find that the asymptotic modes are equispaced, in agreement with previous results. We have also calculated the area spectrum spacing and found it to be dependent on the value of R , d and k . This is also in contrast to the case of first order theories where we always obtain area spectra that are equidistant even when the parameters of the black hole spacetime change.","pages":[14]},{"title":"6 Acknowledgement","content":"The authors would like to acknowledge financial assistance from the Council of Scientific and Industrial Research (CSIR), India during the work under the CSIR Emeritus Scientistship Scheme sanctioned to one of the authors (VCK). VCK would also like to acknowledge Associateship of IUCAA, Pune, India.","pages":[15]}],"string":"[\n {\n \"title\": \"Scalar filed evolution and Area Spectrum for Lovelock-AdS Black Holes\",\n \"content\": \"C.B Prasobh · V.C Kuriakose Received: 25 June 2013 / Accepted: 30 August 2013 Abstract We study the modes of evolution of massless scalar fields in the asymptotically AdS spacetime surrounding maximally symmetric black holes of large and intermediate size in the Lovelock model. It is observed that all modes are purely damped at higher orders. Also, the rate of damping is seen to be independent of order at higher dimensions. The asymptotic form of these frequencies for the case of large black holes is found analytically. Finally, the area spectrum for such black holes is found from these asymptotic modes. Keywords Quasinormal Modes Lovelock AdS Area Spectrum Scalar Field · · · · PACS PACS code1 · PACS code2 · more\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"Gauge-gravity dualities like the AdS/CFT correspondence [1] make it possible to study the properties of conformal fields in a particular dimension d by studying the evolution of fields in a black hole spacetime that is asymptotically AdS in ( d +1) dimensions and this has led to considerable interest in the study of asymptotically AdS black hole spacetimes. The main difficulty in studying field evolution in such spacetimes is that the stability of the spacetime against perturbations is not always guaranteed, unlike in the case of asymptotically flat spacetimes in first order theories in four dimensions. The instability of linear perturbations in higher-order and higher dimensional theories has already been investigated [2,3, 4]. The instability also extends to the thermodynamics of the black hole when we consider AdS spacetimes, with the well-known Hawking-Page phase transitions [5] signaling a transition between the black hole spacetime and thermal AdS spacetime. Recent studies [6,7,8] on the Lovelock model have confirmed the existence of\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"C.B Prasobh\",\n \"content\": \"Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: prasobhcb@outlook.com V.C Kuriakose Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: vck@cusat.ac.in dynamic instability against metric perturbations in asymptotically flat black hole spacetimes. Dynamic instability means that the solutions to the equation for the metric perturbation become unstable outside the event horizon for large values of their eigenvalue. This instability exists for all types of metric perturbations-tensor, vector and scalar. These instabilities occur when the mass of the black hole falls below a lower critical bound that depends on the coupling constants, the dimension of the spacetime and the order of the theory. Quasi normal modes are damped oscillations (having complex frequencies), known to dominate the intermediate stage of the evolution of small perturbations of a black hole spacetime and have been studied extensively. Detailed reviews and methods of calculation of quasi normal modes are found in numerous papers that include [9,10,11,12,13,14]. Quasi normal modes are known to depend only on the parameters of the black hole, such as mass, charge and angular momentum, and be completely independent of the type of the agent that caused it. These modes are obtained as the solution of the respective field equation, when solved with respect to the metric that describes the particular black hole spacetime of interest. In the case of asymptotically flat spacetimes, the effective potential that is perceived by the field in the spacetime often resembles a finite potential barrier such that the potential vanishes at infinity. This leads to the possibility of obtaining solutions that are purely ingoing at the event horizon of the black hole and purely outgoing at infinity, both resembling plane waves when expressed in terms of a scaled co-ordinate called the tortoise co-ordinate. These modes may be observed in the future with the aid of gravitational detectors. In the case of asymptotically AdS spacetimes, however, the potential grows indefinitely at infinity. Therefore, one usually looks for solutions that vanish at infinity, while the boundary condition at the event horizon unchanged [15]. This is motivated by the case of pure AdS spacetime, where the potential effectively confines the field as if 'in a box' and solutions exist only with a discrete spectrum of real frequencies. Even though black holes in asymptotically AdS spacetimes are not believed to exist in nature, interest in studying their quasi normal modes stems from the above-mentioned AdS/CFT correspondence [1]. According to the AdS/CFT correspondence, these perturbations correspond to perturbations of the thermal state of the strongly coupled conformal field at the boundary of the spacetime and the quasi normal modes correspond to the return to thermal equilibrium, so that the quasi normal frequencies give a measure of the time scale for the relaxation, which is difficult to compute directly. This provides the motivation to study the quasi normal modes of various fields in asymptotically AdS spacetimes. Earlier work the quasi normal modes of Schwarzschild-AdS black holes [15] have proved that the modes scale with the temperature of the event horizon. Complete quantization of gravity is one of the major goals of modern theoretical physics. Despite decades of research by physicists all over the world, it is yet to be achieved with complete success. One often takes clues from the classical theory of a system when attempting its quantization and gravity needs to be no different. Quantization of the black hole horizon area is expected to be a major feature of any successful quantum theory of gravity. The original 'classical' theory of gravity, namely the General Theory of Relativity (GTR) does provide us with the tools necessary to estimate the value of the area quantum. It is known from field theory that the presence of a periodicity in the classical theory of a system points to the existence of an adiabatic invariant with a discrete spectrum in the corresponding quantum theory. Interestingly, it has been observed in GTR that the numerical value of these frequencies, in the limit of 'large' frequencies, follow a distinct pattern, with the real part approaching a fixed value. These are termed asymptotic frequencies. Suggestions have been made that the fixed value of the real part can be viewed as a physically relevant periodicity in the (classical) black hole system which would then lead to the existence of certain adiabatic invariant quantity, which in turn would possess equally spaced spectrum according to Bohr-Sommerfeld quantization. Once we read it together with Bekenstein's original proposal [16] that the black hole entropy is an adiabatic invariant with a discrete, equally spaced spectrum, we come to the conclusion that the entropy spectrum (and, by extension, the area spectrum) can be deduced from the asymptotic value of the quasi normal frequencies. The connection between the fixed asymptotic frequencies and the quantized area spectrum was made by Hod [17]. Dreyer [18] recovered Hod's result in the Loop Quantum Gravity. A new interpretation for the quasi normal modes ω = ω R + iω I of perturbed black holes as equivalent to that of a collection of damped harmonic oscillators with real frequency ω 0 = √ ω 2 R + ω 2 I was introduced by Maggiore [19] and used in conjunction with Hod's method in order to compute the area spectrum of Schwarzschild black holes. As with the case of quasi normal modes, it is the AdS/CFT correspondence that provides the motivation for studying the area spectrum of black holes in asymptotically AdS spacetimes. Recent studies ([20,21,22,23,24] and references therein) suggest that the gravitational dual of the holographic entanglement entropy in quantum field theories is the area of minimal-area surfaces in AdS spaces. The entropy can be used to study phase transitions between various states of the field. An area-entanglement entropy relation of the form S A = Area ( γ A ) 4 G ( d -2) N has been proposed [20,25] which is very similar to the familiar area-entropy relation in the General Theory of Relativity. Here, γ A is the d -dimensional minimal area surface in AdS d +2 and G ( d -2) N is the ( d +2)-dimensional gravitational constant of the AdS gravity. Although the above relation was originally proposed for AdS spaces, it is equally applicable to any asymptotically AdS space, including one containing a black hole. In that case, the minimal surface tends to wrap the horizon and thus we can use the area of the event horizon in order to compute the entanglement entropy in the conformal field theory. In this work, we numerically compute the quasi normal frequencies for massless scalar field perturbations in asymptotically AdS, spherically symmetric spacetimes in Lovelock model using the metric derived in [26]. We analytically find out the asymptotic form of the frequencies following [27] and use it to deduce the area spectrum of large black holes in the model. A brief outline of the paper is as follows: in Sect. 2, we explain the maximally symmetric Lovelock model and the resulting metric for the spacetime as given in [26]. Details of the Horowitz-Hubeny numerical method of computing the quasi normal frequencies in AdS black hole spacetimes to the case of massless scalar fields in the vicinity of such a black hole in the Lovelock-AdS model are also given in the same section. The results of the numerical procedure are presented and analyzed in Sect. 3. In Sect. 4, we analytically determine the asymptotic quasi normal frequencies of the field for the case of large back holes following [27]. The results of that analysis are used in order to find the area spectrum of the black hole using the Kunstatter's method [28] in Sect. 4.1. The results are summarized in Sect. 5.\",\n \"pages\": [\n 1,\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"2 The metric and the numerical computation of Quasi normal Frequencies\",\n \"content\": \"The Lovelock model [29] is considered to be the most natural generalization of GTR. The Lovelock Lagrangian is a polynomial which consists of dimensionally continued higher order curvature terms. The most striking property of this Lagrangian is that it yields field equations that are in second order in the metric although the Lagrangian itself may contain higher order terms. Also, the theory is known to give solutions that are free of ghosts. The maximum order of terms in the action, k , is fixed by the number of dimensions of the spacetime d in Lovelock model according to the relation k = [ d -1 2 ] where [ x ] denotes the integer part of x . The action is written in terms of the Riemann curvature R ab = dω ab + ω a c ω cb and the vielbein e a as where α p are arbitrary (positive) coupling constants, and L ( p ) , given by is the p th order dimensionally continued term in the Lagrangian. The difficulty with the Lagrangian given above, with arbitrary values for α p , is that it becomes very difficult (if not impossible) to study the evolution of fields, since it is not at all clear whether the operator representing the evolution is Hermitian or not. As mentioned in the previous section, this problem, for the case of metric perturbations, has been analyzed in [6]. Although a general instability depending on the black hole mass has been established in that work, the numerical value for the critical mass has not been calculated. Also, it is very difficult to predict whether the presence of a cosmological constant raises or lowers the critical mass, as long as we consider a model with arbitrary α p . Moreover, for the same case, the the existence of negative energy solutions with horizons and positive energy solutions with naked singularities for (1) has been pointed out earlier [30,31]. These difficulties bring out the necessity of selecting suitable values for the coupling coefficients in order to have models that support maximally symmetric solutions and external perturbations. Maximally symmetric solutions to Lovelock model have long been known [26], which are derived by requiring that the theories must possess a unique cosmological constant (and consequently a unique AdS radius R ) for all orders. The resulting set of coupling constants are seen to be labeled by the order k and the gravitational constant G k . The metric describing the spherically symmetric black hole spacetime is derived from the action given in (1) with the choice where 1 ≤ k ≤ [ d -1 2 ]. The resulting field equations are of the form Here, ¯ R ab := R ab + 1 R 2 e a e b . Such theories are labeled by k and have two fundamental constants, κ and R , related to the gravitational constant G k and the cosmological constant Λ respectively through the relations Ω d -2 being the volume of the ( d -2) dimensional spherically symmetric tangent space. The static and spherically symmetric solutions to (4), written in Schwarzschild-like coordinates, take the form where f ( r ) is given by We take σ = 1. The integration constant C 1 is written as where M stands for the mass of the black hole. The constant C 0 is chosen so that the horizon shrinks to a point for M → 0, as /negationslash It is interesting that the exponent of ( 1 r ) in (9) is proportional to ( d -2 k -1). Since k = [ d -1 2 ], ( d -2 k -1) = 0 in odd dimensions and ( d -2 k -1) = 1 in even dimensions. Thus the solution in even dimensions resemble the Schwarzschild-AdS solution. The ( d -2 k -1) = 0 cases correspond to Chern-Simmons theories which have a vacuum that is different from AdS [26]. Their quasi normal modes, mass and area spectra have already been computed [32]. What makes it interesting is the fact that recent studies [6] on the stability of metric perturbations in Lovelock model also point out that it is possible to predict the (in)stability of the perturbations only in even dimensions. The present work is limited to the cases where d -2 k -1 = 0. Then we have C 0 = 0 and C 1 = 2 G k M . Consider the scalar field Φ ( r, t, x i ) that obeys the Klein-Gordon equation given by x i being the co-ordinates in the spherically symmetric tangent space and g AB being components of the metric tensor. We impose the boundary conditions of ingoing plane wave solution at the event horizon and vanishing field at the boundary. The boundary condition at the horizon suggests the ansatz Φ = e -iω ( t + r ∗ ) where r ∗ is the tortoise coordinate defined by In the ( v = t + r ∗ , r ) system, the metric reads and we take the ansatz so that (12) becomes with the effective potential Here, l represents the eigenvalue of the operator on the LHS of (12) acting on the functions Y ( x i ) in the spherically symmetric tangent space. In order to numerically calculate the quasi normal frequencies for (16), we expand the field Φ as a power series about the horizon and impose the vanishing boundary condition at infinity. We change the variable from r to x = 1 r in order to map the range r + < r < ∞ to a finite range. In terms of x , (16) becomes where and In the numerical procedure, we find out the coefficients of the expansion of the functions s ( x ) , t ( x ) and u ( x ) as power series in ( x -x + ) using a computer with s i , t i and u i denoting the coefficient for the i th term. Then an expansion for ψ ( x ) of the form is substituted into (18) which yields where We fix a 0 and numerically calculate the coefficients in (19) and (22) to different orders and compute the value of the filed ψ ( x ) as given in (21). Since we wish to impose the boundary condition of vanishing field at infinity, we solve the equation ψ (0) = 0 for ω . It is observed, by comparison with the values in [15], that the quasi normal frequencies are one of the solutions of the equation ψ (0) = 0, solved for ω , after ψ has been computed using (21) for some reasonably high value of n , which should be fixed by trial and error. We assume the same to be true in higher orders as well and look for the value of the quasi normal modes among the set of discrete values of ω that are obtained. For each value of ω obtained, we evaluate the absolute value of the LHS of (18) at a point close to the event horizon, since we have assumed plane wave solutions there. We choose that value of ω as the quasi normal frequency for which the absolute value comes closest to zero (the assumption here is that (21) is satisfied exactly only for the quasi normal frequency, which we seek, and not by other roots of ψ = 0.). We increase the number of terms to which ψ ( x ) and the coefficients are calculated until the required precision is attained.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7\n ]\n },\n {\n \"title\": \"3 Discussion on Results of the Numerical Calculation\",\n \"content\": \"/negationslash We implement the procedure outlined above after fixing the value of the constants a 0 and R to 1. We investigate the quasi normal frequencies of large ( r h /greatermuch R ) and intermediate ( r + ∼ R ) black holes and set ω = ω R -iω I as done in [15] since we are interested in damped modes. As mentioned before, the analysis is limited to the cases where d -2 k -1 = 0. The results for the lowest ( l = 0) modes of the massless scalar field for first order theories have been summarized in TABLE 1. TABLE 2 contains the same for higher orders. Figures 1 to 6 show the results in detail. As evident from Figure 1 and Figure 2, both ω R and ω I show linear dependence on r + for the case of large black holes in first order theories. The linearity seems to be broken when we move to the intermediate-sized black holes, the results for which have been plotted in Figure 3 and Figure 4. Although the plots for intermediate-sized black holes look linear, the ω R -r + dependence for their case rather resembles an ( x, y = x + 1 x ) relation. The temperature of the event horizon for the metric (8), given by d = 4 , k = 1 d = 5 , k = 1 d = 6 , k = 1 where κ B denotes the Boltzmann's constant, also depends on r + in the same manner. Since ( x, y = x + 1 x ) ∼ x for large x , we conclude that both ω R and ω I scale with the temperature for large as well as intermediate-sized black holes for first order theories, in agreement with earlier works [15]. Another observation is that ω I seems to be independent of dimension d in first order theories. When we consider higher order theories, the numerical results for which have been plotted in Figure 5 and Figure 6, we observe that all modes are purely damped ones. We have only plotted ω I vs r + in the case of higher order theories for this reason. There, we observe that, both for large and intermediate-sized black holes, ω I is independent of the order of the theory when the dimension d stays the same.\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"4 Asymptotic quasi normal modes and area spectrum of large black holes\",\n \"content\": \"We analytically find the asymptotic form of the quasi normal frequencies in the large black hole limit following the method of perturbative expansion of the wave equation in the dimensionless parameter ω/T H that has earlier been employed in the case of d -dimensional SAdS black holes [27]. Here, T H is the Hawking temperature of the horizon and ω is the frequency of the mode. We take the metric to be of the form given in (8). For large black holes, the metric gets approximated as, ω with It is easily seen that the event horizon is given by In terms of the new metric (25), the Klein-Gordon field equation (12) for m = 0 becomes ω ω + where We write the field Φ as and change the variable from r to so that (28) becomes ω ω where the parameters ˆ ω and ˆ p are defined as and We investigate the behavior of (32) near the boundaries y → 1 and y →∞ and the point y →-1 in order to develop an ansatz for Ψ ( y ). The following solutions are obtained: Since we demand ingoing plane wave like solutions at the horizon, we take the form Ψ ∼ ( y -1) -i ˆ ω/ 2 A near the horizon ( y = 1). We isolate the solutions near y = ± 1 and write Substituting (36) in (32), we deduce the equation satisfied by N ( y ) as We consider (37) in the range of large ˆ ω and large y , so that y 2 ≈ y 2 -1 and the terms proportional to 1 / ( y Q +2 k -2 ) may be dropped along with the constant terms. Then (37) reduces to which is the Hypergeometric equation with the solution where In order to match the behavior of the solution (36) at infinity with that demanded by (35), we demand that N ( y ) be a polynomial as y →∞ . That condition is satisfied when If a = -n , then, according to the property of the hypergeometric equation, N ( y ) ∼ y n = y -a , so that, according to (36), as required. We deduce the expression for the asymptotic form of quasi normal frequencies from (41) as follows: so that (33) implies which gives the asymptotic form of the quasi normal frequencies for large maximally symmetric AdS black holes in the Lovelock model. We observe that the high-overtone quasi normal frequencies are equispaced, which is in agreement with earlier observations [15,9,12].\",\n \"pages\": [\n 8,\n 9,\n 10,\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"4.1 Area Spectrum\",\n \"content\": \"We calculate the area spectrum of the large AdS black holes in Lovelock model using the new physical interpretation of the quasi normal modes proposed by Maggiore [19] and following Kunstatter's method [28]. According to the first law of black hole thermodynamics, for a black hole system with energy E (or M ) with Hawking temperature T H and horizon area A , the following relation holds: According to Kunstatter, if the frequency of oscillation of the system is ω ( E ), then the quantity is to be taken as the corresponding adiabatic invariant. According to Maggiore [19], the black hole system has to be modeled by a collection of damped harmonic oscillators. If the system has a quasi normal frequency ω = ω R + iω I , then the corresponding vibrational frequency according to the model is to be taken as In the highly damped and highly excited cases, ω 0 can be approximated by ω I and ω R respectively. Here, we see that for higher values of the number n , the both ω R and ω I increase. Therefore we consider transitions between two adjacent energy levels of the system and take the physical frequency as equal to the difference in ω 0 for the two systems; that is, we take We deduce the area spectrum from (46) using the relations (27) and (44). The expression for the adiabatic invariant now reads Using (27), (44), (47) and (48) in (49), it is easy to see that the Bohr-Sommerfeld quantization condition, namely I = n /planckover2pi1 now reads which, in terms of the area A of the horizon, can be written in the form where Thus the area spectrum, and consequently the entropy spectrum, of large black holes in asymptotically AdS Lovelock spacetimes is seen to depend on the parameters G k and the AdS radius R of the theory. The dependence of A on R is observed only when higher orders are considered. A is also dependent on the dimension d of the spacetime, and consequently on the order of the theory.\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"5 Conclusion\",\n \"content\": \"We have analyzed the evolution of massless Klein-Gordon field in the maximally symmetric asymptotically AdS spacetime surrounding a black hole in the Lovelock model. We have used the form of the metric that has been derived in [26] in order to compute the quasi normal frequencies using the Horowitz-Hubeny method. The results of the numerical computation show that the modes in the case of higher order theories are purely damped. For the case of large as well as intermediate black holes, the frequencies are observed to scale linearly with the temperature of the event horizon. When we consider higher order theories, the imaginary part of the quasi normal frequencies is observed to be independent of the order of the theory in higher dimensions. They appear to be dependent on the dimension only. The asymptotic form of the quasi normal frequencies for the case of very large black holes has been analytically determined using the method of perturbative expansion of the wave equation in terms of ω/T H , as developed in [27]. We find that the asymptotic modes are equispaced, in agreement with previous results. We have also calculated the area spectrum spacing and found it to be dependent on the value of R , d and k . This is also in contrast to the case of first order theories where we always obtain area spectra that are equidistant even when the parameters of the black hole spacetime change.\",\n \"pages\": [\n 14\n ]\n },\n {\n \"title\": \"6 Acknowledgement\",\n \"content\": \"The authors would like to acknowledge financial assistance from the Council of Scientific and Industrial Research (CSIR), India during the work under the CSIR Emeritus Scientistship Scheme sanctioned to one of the authors (VCK). VCK would also like to acknowledge Associateship of IUCAA, Pune, India.\",\n \"pages\": [\n 15\n ]\n }\n]"}}},{"rowIdx":269,"cells":{"bibcode":{"kind":"string","value":"2013Galax...2....1A"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1303.1394.pdf"},"content":{"kind":"string","value":"\nGalaxy Rotation Curves in Covariant Hoˇrava-Lifshitz Gravity\nJ. Alexandre 1 , M. Kostacinska 2\n\n1 King's College London, Department of Physics, WC2R 2LS\n2 Imperial College London, Theoretical Physics, SW7 2AZ\n\nAbstract\nUsing the multiplicity of solutions for the projectable case of the covariant extension of Hoˇrava-Lifshitz Gravity, we show that an appropriate choice for the auxiliary field allows for an effective description of galaxy rotation curves. This description is based on static and spherically symmetric solutions of covariant Hoˇrava-Lifshitz Gravity and does not require Dark Matter.\n1 Introduction\n/negationslash\nAlthough General Relativity (GR) has been accurately checked in the solar system, the small distance behaviour of gravity is still not understood, and several alternatives to GR have been proposed. Amongst these modified theories is Hoˇrava-Lifshitz (HL) gravity [1], which is based on the Lifshitz approach and consists of introducing an anisotropy between space and time: when space is rescaled as x → bx , time is rescaled as t → b z t , where z is an integer. In Lifshitz models, Lorentz invariance must be recovered for z = 1, and the situation z = 1 leads to new renormalizable interactions: the convergence of loop integrals is improved by inclusion of higher orders in space derivatives, without introducing ghosts, since the number of time derivatives remains the minimum. A review of Lifshitz-type quantum field theories in flat space time can be found in [2], and the HL power-counting renormalizable theory of gravity is reviewed in [3], [4] and [5].\nA fundamental problem of the original model of HL gravity is the existence of an additional scalar degree of freedom for the metric, which can be understood as a Goldstone mode arising from breaking of 4-dimensional diffeomorphism [6], [7]. A solution to this problem has been proposed in [8], where an auxiliary field A is introduced, such that its 'equation of motion' leads to an additional constraint, eliminating the unwanted scalar degree of freedom of the metric. The resulting theory is invariant under a new Abelian gauge symmetry U Σ (1) which involves the metric components, the auxiliary field A and an additional auxiliary field ν . This gauge symmetry, together with the 3-dimensional diffeomorphism of HL gravity, can be shown to be equivalent to a 4-dimensional diffeomorphism\nat the lowest order in a post-Newtonian expansion, showing the equivalence with GR at long distances. For this reason, this modified version of HL gravity is called covariant HL gravity. We note however, that the long-distance limit is not obviously recovered: it has been shown in [9] that the equivalence principle is not automatically retrieved in the infrared, and that the meaning of the auxiliary field A and its coupling to matter are still open questions.\nBecause of the anisotropy between space and time, HL gravity is naturally described in terms of the Arnowitt-Deser-Misner (ADM) decomposition of the metric, which expresses a space-time foliation. An important consequences of space-time anisotropy is the possibility of imposing the lapse function N to depend on time only. This situation is called the projectable case and leads to an interesting feature: the solutions of the equations of motion are not unique. Indeed, the Hamiltonian constraint, obtained by variation of the action with respect to N , leads to an integral equation, which does not have a unique solution, as will be seen in the present article. This multiplicity of solutions is independent of the above mentioned new gauge symmetry U Σ (1), and the different solutions for A in the projectable case belong to different gauge orbits. On the other hand, in the nonprojectable case where N depends on both space and time, the Hamiltonian constraint leads to a differential equation, which has a unique solution, after fixing the constants of integration. The non-projectable case has been studied in [10] for static spherically symmetric solutions of covariant HL gravity.\nIn this article we use the freedom of choice of the auxiliary field A in the projectable case to study the possibility of fitting galaxy rotations curves without the need for Dark Matter. One can argue that the ambiguity in the choice of A leaves its physical interpretation unclear. This may be understood as one of the incomplete aspects of covariant HL gravity, and one could conjecture the existence of an additional symmetry which would restrict the set of solutions for the auxiliary field A , but this still remains to be studied. We note that the multiplicity of solutions, arising from a Hamiltonian constraint expressed in terms of an integral over space, has been discussed in [11] as an alternative to Dark Matter, in the context of the original HL gravity. Fits to galaxy rotation curves have been studied in the framework of the original HL gravity in [12] and [13], where a detailed comparison with experimental data is presented. Also, an interesting analogy between HL gravity for z = 0 and Modified Newtonian Dynamics theories has been described in [14].\nIn section 2 we review the static and spherically symmetric solutions of covariant HL gravity, and in section 3 we show how galaxy rotation curves can be described by the covariant extension of HL gravity. Although the curves we use were derived from Dark Matter models, we use them as experimental data fits. It is interesting to note that we obtain exact solutions for the auxiliary field A . Another exact solution in the Lifshitz context has been derived in [15], where the exact effective potential for a Liouville scalar theory, renormalizable in 3+1 dimensions with anisotropic scaling z = 3, proves to be an exponential, as in the usual 1+1 dimensional relativistic Liouville theory. Finally, we conclude by discussing the possiblity to describe the solar system with our solution.\n2 Static and spherically symmetric solutions of covariant HL gravity\nThe static and spherically symmetric solutions of covariant HL gravity were derived in [25] and [26], and we review here the aspects relevant to our present study.\n2.1 Action\nThe ADM metric we consider is\nds 2 = -c 2 N 2 dt 2 + g ij ( dx i + N i dt ) ( dx j + N j dt ) , (1)\nwhere N and N i are the lapse and shift functions respectively, and g ij is the threedimensional space metric. We consider the anisotropic scaling with z = 3, for which the different operators, allowed for the model to be power-counting renormalizable, have the mass dimension of 6, at most. We note that in this case the mass dimension of the speed of light is [ c ] = z -1 = 2, and we will keep c in the different expression for the sake of clarity.\nThe covariant Hoˇrava-Lifshitz action, in the absence of cosmological constant, is\nS = ∫ dtd 3 x √ g ( N [ K ij K ij -K 2 -V + ν Θ ij (2 K ij + ∇ i ∇ j ν ) ] -AR (3) ) , (2)\nwhere g is the determinant of the three-dimensional metric g ij , with curvature scalar R (3) . A is an auxiliary field (without kinetic term) of mass dimension z + 1 = 4 and ν is an auxiliary field of mass dimension z -2 = 1. In the above expression, the extrinsic curvature is\nK ij = 1 2 N { ˙ g ij -∇ i N j -∇ j N i } , i, j = 1 , 2 , 3 , (3)\nwhere a dot denotes a time derivative, and\nΘ ij = R (3) ij -1 2 R (3) g ij , (4)\n/negationslash\nwhere where R (3) ij is the Ricci tensor corresponding to the three-dimensional space metric g ij . Note that a more general kinetic term K ij K ij -λK 2 can be considered, with λ a generic parameter. The latter should be equal to 1 if one wishes to recover GR in the infrared, but in the framework of HL gravity, no symmetry imposes this parameter to be equal to 1. Constraints on λ resulting from observations of Type Ia Supernovae, Baryon Acoustic Oscillations, CMB and the requirement of Big Bang Nucleosynthesis all point towards a value near the GR parameter λ = 1 [16], while it has also been shown that covariant HL gravity with λ = 1 leads to inconsistencies, if one compares the predictions of the model with solar system tests [17]. We therefore focus here on the situation where λ = 1 only. Independently of these tests, the mathematical consistency of covariant HL gravity for λ = 1 has been studied in [18], and the corresponding Hamiltonian structure in\n/negationslash\n[19]. We stress again here that the purpose of this covariant extension of HL gravity is to eliminate the unwanted additional scalar graviton from the spectrum. As shown in [8], no singularity arises in the limit λ → 1, and the strong coupling of the scalar graviton does not exist anymore, as it is the case for the original HL gravity. Besides [6] and [7], one can find detailed analysis of the strong coupling problem in [20], [21], [22], and also related discussions in [23] on a extension of the original HL gravity [24].\nThe potential term V contains up to six spatial derivatives of the metric g ij :\nV = -c 2 R (3) -α 1 ( R (3) ) 2 -α 2 R (3) ij R (3) ij -β 1 ( R (3) ) 3 -β 2 R (3) R (3) ij R (3) ij (5)\n-β 3 R (3) i j R (3) j k R (3) k i -β 4 R (3) ∇ 2 R (3) -β 5 ∇ i R (3) jk ∇ i R (3) jk , (6)\nwhere the mass dimensions of the different couplings are\n[ α 1 ] = [ α 2 ] = z -1 = 2 , [ β 1 ] = [ β 2 ] = [ β 3 ] = [ β 4 ] = [ β 5 ] = z -3 = 0 . (7)\nWe consider the most general, static, spherically symmetric metric, of the form:\nds 2 = -c 2 N 2 dt 2 + 1 f ( r ) ( dr + n ( r ) dt ) 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (8)\nwhere n ( r ) = N r ( r ) is the radial component of shift function, and N r = n ( r ) /f ( r ) since g rr = 1 /f ( r ). Note that n has mass dimension z -1 = 2.\n2.2 Constraints and equations of motion\nThe variation of the action (2) with respect to the different degrees of freedom leads to the following constraints, or equations of motion, where a prime denotes a derivative with respect to the radial coordinate r .\n\n· The variation with respect to A gives R (3) = 0, or equivalently\n\nrf ' + f -1 = 0 , (9)\nwhich imposes the function f ( r ) in the metric (8) to be\nf ( r ) = 1 -2 B r , (10)\nwhere B is a constant of integration.\n\n· The variation with respect to ν gives\n\nΘ ij ∇ i ∇ j ν +Θ ij K ij = 0 , (11)\nIn what follows we will assume the Gauge fixing of ν = 0, then the above constraint gives Θ ij K ij = 0, which is satisfied for spherically symmetric solutions.\n\n· The variation with respect to N gives the Hamiltonian constraint\n\n∫ ∞ 0 dr r 2 √ f ( r ) ( K ij K ij -K 2 + V ) = 0 , (12)\nwhich is an space-integral equation since N depends on time only.\n\n· The variation with respect to n gives\n\nf ' ( r ) n ( r ) = 0 , (13)\nsuch that either f is a constant ( B = 0) or n = 0.\n\n· The variation with respect to f gives\n\nA ' + A 2 r ( 1 -1 f ) +4 fn ( √ rn ) ' √ r = { r 4 f -√ f 2 r 3 ∑ n =0 ( -1) n d n dr n ( r 2 √ f ∂ ∂f ( n ) )} V, (14)\nwhere the time has been rescaled such that N = 1.\n2.3 Solutions\nThe different solutions of the previous set of equations were derived in [25], [26], and here we shortly review the different cases.\n\n· n = 0: the solution of the differential equation (14) gives short-distance corrections to the Schwarzschild solution, details of which can be found in [25], [26];\n\n/negationslash\n\n· n = 0 and A = 0: leads to the Schwarzschild solution in the Painleve-Gullstrand coordinates ( n ∝ r -1 / 2 ), and the Hamiltonian constraint is automatically satisfied;\n\n/negationslash\n/negationslash\n\n· n = 0 and A = 0: corresponds to the situation with multiple solutions for A and n , on which this article focuses. Since B = 0, we have f = 1 and V = 0, such that the equations (12) and (14) give respectively\n\n∫ ∞ 0 dr ( rn 2 ) ' = 0 rA ' +2( rn 2 ) ' = 0 , (15)\nwith solutions\nwhere C is a constant of integration and the auxiliary field A satisfies\nn 2 ( r ) = C r -1 2 A ( r ) + 1 2 r ∫ r 0 dρ A ( ρ ) , (16)\n∫ ∞ 0 drrA ' ( r ) = 0 . (17)\nThe flexibility in the choice of solution for A will help describe galaxy rotation curves, as explained in the next section.\n3 Galaxy rotation curves\n/negationslash\n/negationslash\nIn what follows, we consider each star in the spiral arms of the galaxy as a test particle, moving on a circular trajectory under the influence of the potential φ ( r ), generated by the centre of the galaxy, which is assumed to be static and spherically symmetric. Our approach is to start from a stellar velocity distribution and derive the corresponding expression for the auxiliary field A , in the above situation with n = 0 and A = 0 , where multiple solutions for A are allowed. We note the following few points:\n(i) We consider vacuum solutions of covariant HL gravity, for which one cannot in principle describe the region in the centre of the galaxy. Nevertheless, as we will see in case of the following velocity profiles, a consistent solution for A can be found for 0 ≤ r ≤ R , where R is a typical radial length describing the galaxy. To be consistent with the usual studies of galaxy rotations, we will choose here R = r 200 , corresponding to the virial radius of the galaxy. The latter is defined as the distance from the centre of the galaxy, where the density ρ 200 is 200 times the critical density ρ c of the Universe\nρ 200 = 200 ρ c = 200 3 H 2 8 πG , (18)\nwhere H is the Hubble constant;\n(ii) For r > r 200 , we assume the solution A = 0, which leads to the usual Schwarzschild solution outside the galaxy. Indeed, far from the galaxy, one expects to see the Newtonian potential. This solution implies that the Hamiltonian constraint is expressed in terms of an integral over the finite range [0 , r 200 ] of radial coordinate r , which we can impose to vanish by fixing the constants of integration;\n(iii) For r ≤ r 200 , since the shift A → A +constant does not have a physical implication, one can expect to find a consistent solution for A which is continuous at r = r 200 , i.e. A ( r 200 ) = 0. On the other hand, one cannot expect the auxiliary field to be differentiable at r = r 200 , but this is not necessary for the consistency of covariant HL gravity: the second derivative of A does not appear in any equation of motion or constraint, so the first derivative can be discontinuous.\n3.1 Gravitational potential\nOn a circular trajectory, the relation between the speed v of a star and the Newtonian potential φ is\nv 2 r = dφ dr , (19)\nwhere φ is obtained from the g 00 component of the low energy effective theory\n-c 2 + n 2 = -c 2 (1 + 2 φ ) , (20)\nand is thus given by\nφ ( r ) = -n 2 ( r ) 2 c 2 . (21)\nWe note here that, in the situation where the second auxiliary field ν does not vanish, the Newtonian potential is instead given by\nφ ( r ) = -( n ( r ) -∇ r ν ( r )) 2 2 c 2 , (22)\nand φ is independent of the U Σ (1) gauge choice for ν and A . Also, φ is dimensionless, so that the speed v we use in this article is also dimensionless, which corresponds to a usual definition of speed, for isotropic space-time. The corresponding 'Lifshitz velocity' is cv , with mass dimension z -1 = 2.\nFrom the relations (16) and (19), the speed v and the auxiliary field A are then related by\nrA ' ( r ) 4 c 2 = d dr ( r ∫ r dρ v 2 ( ρ ) ρ ) = v 2 ( r ) + ∫ r dρ v 2 ( ρ ) ρ , (23)\nand we have to check that the Hamiltonian constraint (17) is satisfied, in order to show the consistency of the approach.\nWe also note from eqs.(16,19) that the solution A = 0 gives,\nv ( r ) = √ C 2 c 2 r , (24)\nwhich is expected from Newtonian mechanics outside the galaxy, for r > r 200 . If one fixes the constant of integration C to the value\nC = 2 c 2 r 200 v 2 200 , (25)\nwhere v 200 = v ( r 200 ), one obtains a continuous speed at r = r 200 . Finally a different choice of the auxiliary field ν would modify the field A in such a way that physical results would not be changed.\n3.2 Navarro, Frenk and White profile\nThis profile originates from Cold Dark Matter halo models and is characterised by the 'cusp' shape of the density distribution. Its validity has been tested on variety of observational results, including Low Surface Brightness galaxies in [27] and spherical galaxies and clusters in [28]. For our purposes, we use the circular velocity profile derived from the mass density profile in [29], based on the assumption of a Dark Matter halo and ignoring the contribution of baryons,\nv ( r ) = v 0 √ r 200 r ln ( 1 + ar r 200 ) -a 1 + ar/r 200 , (26)\nwhere a is the concentration parameter and\nv 0 = v 200 √ ln (1 + a ) -a/ (1 + a ) (27)\nwhere v 200 , the virial velocity, is the circular velocity at the virial radius. Some typical values are [30]\n
\n\n
\n±\n±\n±\nWith the profile (26), eq.(23) leads to\nrA ' ( r ) 4 c 2 v 2 0 = φ 1 -a 1 + ar/r 200 , (28)\nwhere φ 1 is a constant of integration, and therefore\nA ( r ) 4 c 2 v 2 0 = ( φ 1 -a ) ln ( r r 200 ) + a ln ( 1 + a r r 200 ) + A 1 , (29)\nwhere A 1 is a constant of integration. Since we consider the solution A ( r ) = 0 for r > r 200 and impose A ( r ) to be continuous at r = r 200 , then A 1 = -a ln(1 + a ) and\nA ( r ) 4 c 2 v 2 0 = ( φ 1 -a ) ln ( r r 200 ) + a ln ( 1 + ar/r 200 1 + a ) if 0 ≤ r ≤ r 200 (30) A ( r ) = 0 if r > r 200 .\nFinally, the Hamiltonian constraint\n0 = ∫ ∞ 0 dr rA ' ( r ) = 4 c 2 v 2 0 ∫ r 200 0 dr ( φ 1 -a 1 + ar/r 200 ) , (31)\nis satisfied if\nφ 1 = ln(1 + a ) . (32)\nWe note that the solution (30) is consistent for any relevant values of the parameters a, v 0 , r 200 and therefore allows the description of a whole range of galaxies.\n3.3 Pseudoisothermal profile\nThe pseudoisothermal mass density profile assumes an existence of a 'cored' dark matter halo component in the galaxy [31] , with mass density of an approximately constant value in the central region of the galaxy, for r ≤ R c . Empirically motivated, the model is often contrasted with the aforementioned NFW 'cuspy' profile, while evidence shows that it provides better fit of the galaxy rotation velocities [27]. The velocity profile is\nv ( r ) = v 0 √ 1 -R C r arctan ( r R C ) , (33)\nwhere v 0 is given by\nand ρ 0 , the central density, is the density within R C . Typical values are [30]\nv 0 = √ 4 πGρ 0 R 2 C , (34)\n
\n\n
\n±\n±\n±\nWe note that in the case of NCG 2403, both the NFW and the present profile can be shown to produce fits of comparably good quality.\nWith the profile (33), eq.(23) leads to\nrA ' ( r ) 4 c 2 v 2 0 = φ 2 + 1 2 ln ( 1 + r 2 R 2 c ) , (35)\nwhere φ 2 is a constant of integration. The same steps as those describe for the previous profile lead to the solution\nA ( r ) 4 c 2 v 2 0 = φ 2 ln ( r r 200 ) + 1 2 ∫ r/R c 0 dt t ln(1 + t 2 ) + A 2 if 0 ≤ r ≤ r 200 (36) A ( r ) = 0 if r > r 200 ,\nwhere the constants of integration are\nA 2 = -1 2 ∫ r 200 /R c 0 dt t ln(1 + t 2 ) , (37)\nfor the continuity of A ( r ) at r = r 200 , and\nφ 2 = -1 2 ln ( 1 + r 2 200 R 2 c ) +1 -R c r 200 arctan ( r 200 R c ) , (38)\nfor the Hamiltonian constraint to be satisfied.\n3.4 Vacuum energy contribution of the auxiliary field\nSince the present model does not require Dark Matter to explain galaxy rotation curves, the resulting mass content of the Universe might therefore be reduced. But the static configuration of A actually results in an effective vacuum energy. This can be seen from the original action (2), after integrating by parts the term -√ gAR (3) , which involve secondorder space derivatives of the metric. This leads to the following contribution to the\nHamiltonian\nM ∝ 1 c ∫ d 3 x √ gg ij ∇ i ∇ j A = 4 π c ∫ r 200 0 r 2 dr ∂ 2 r A , (39)\nwhere the factor 1 /c arises from the absence of integration over time, and restores the correct dimension.\nIn the case of the Navarro, Frenk and White profile, the energy (39) reads\nM 1 = 16 πcv 2 0 r 200 [ ln(1 + a ) -a 1 + a ] , (40)\nwhich is always positive, for any value of the parameter a ≥ 0.\nFor the Pseudoisothermal case, we obtain from eq.(39)\nM 2 = 16 πcv 2 0 r 200 [ 1 -R c r 200 arctan ( r 200 R c )] , (41)\nwhich is also always positive, for any value of the ratio r 200 /R c . The auxilliary field A\nhas therefore a cosmological role, by contributing to the vacuum energy of the Universe, consistently with its observed acceleration.\n4 Conclusion\nThe multiplicity of solutions in covariant HL gravity has been used to describe galaxy rotation curves, without the need for Dark Matter. This multiplicity is a consequence of the breaking of 4-dimensional diffeomorphism, in the projectable case, where the lapse function is imposed to depend on time only. We considered two specific velocity profiles and showed in each case that a corresponding consistent solution can be derived for the auxiliary field A . Clearly, similar results can be expected with other profiles. A more detailed analysis would consist in taking into account the finite star density at the centre of the galaxy, as well as its rotation.\nOne should note that the solutions found, for the auxilliary field A , depend on parameters wich cannot appear in the Lagrangian in a universal way. These parameters arise as constant of integrations, as a result of the integral form of the Hamiltonian constraint, and they have to be fitted to each galaxy which is studied. This is a bit similar to the Schwazschild metric in General Relativity, involving a mass which is a constant of integration and depends on the star/black hole under consideration.\nIt is interesting to note that the solutions (30) and (36) actually do not only describe galaxy rotations curves, but also the usual Schwarzschild solutions for the Solar System, by taking the limit r 200 → 0 such that A = 0 for all r . In this sense these solutions provide an effective unified description of gravitational effects, from the solar system scale to the galaxy scale, where local Lorentz violation appears gradually, as the observational length increases from the the size of the solar system to the size of the galaxy.\nWe finally note that, although HL gravity was initially developed to modify ultraviolet behaviour of gravity, our present work uses another consequence of local Lorentz symmetry violation, which is the existence of a non-unique solution to the Hamiltonian constraint. Since it has been shown that canonical Hoˇrava - Lifshitz gravity provides a description of Cosmology which is consistent with constraints from observational data [32], it would be interesting to see if the non-unique time-dependent solutions of the covariant version of the theory could also allow for an effective description of Cosmology, including an alternative to Dark Energy.\nReferences\n\n[1] P. Horava, 'Quantum Gravity at a Lifshitz Point,' Phys. Rev. D 79 (2009) 084008084022 [arXiv:0901.3775 [hep-th]].\n[2] J. Alexandre, 'Lifshitz-type Quantum Field Theories in Particle Physics,' Int. J. Mod. Phys. A 26 (2011) 4523-4541 [arXiv:1109.5629 [hep-ph]].\n[3] A. Padilla, 'The good, the bad and the ugly .... of Horava gravity,' J. Phys. Conf. Ser. 259 (2010) 012033-012039 [arXiv:1009.4074 [hep-th]];\n[4] S. Mukohyama, 'Horava-Lifshitz Cosmology: A Review,' Class. Quant. Grav. 27 (2010) 223101-223125 [arXiv:1007.5199 [hep-th]];\n[5] T. P. Sotiriou, 'Horava-Lifshitz gravity: a status report,' J. Phys. Conf. Ser. 283 (2011) 012034-012050 [arXiv:1010.3218 [hep-th]].\n[6] T. P. Sotiriou, M. Visser and S. Weinfurtner, 'Quantum gravity without Lorentz invariance,' JHEP 0910 (2009) 033-065 [arXiv:0905.2798 [hep-th]];\n[7] C. Charmousis, G. Niz, A. Padilla and P. M. Saffin, 'Strong coupling in Horava gravity,' JHEP 0908 (2009) 070-086 [arXiv:0905.2579 [hep-th]].\n[8] P. Horava and C. M. Melby-Thompson, 'General Covariance in Quantum Gravity at a Lifshitz Point,' Phys. Rev. D 82 (2010) 064027-064047 [arXiv:1007.2410 [hep-th]].\n[9] E. Abdalla and A. M. da Silva, 'On the motion of particles in covariant HoravaLifshitz gravity and the meaning of the A-field,' Phys. Lett. B 707 (2012) 311-314 [arXiv:1111.2224 [hep-th]].\n[10] K. Lin and A. Wang, 'Static post-Newtonian limits in non-projectable Hoˇrava-Lifshitz gravity with an extra U(1) symmetry,' arXiv:1212.6794 [hep-th].\n[11] S. Mukohyama, 'Dark matter as integration constant in Horava-Lifshitz gravity,' Phys. Rev. D 80 (2009) 064005-064010 [arXiv:0905.3563 [hep-th]].\n[12] V. F. Cardone, N. Radicella, M. L. Ruggiero and M. Capone, 'The Milky Way rotation curve in Horava - Lifshitz theory,' arXiv:1003.2144 [astro-ph.CO];\n[13] V. F. Cardone, M. Capone, N. Radicella and M. L. Ruggiero, 'Spiral galaxies rotation curves in the Horava - Lifshitz gravity theory,' arXiv:1202.2233 [astro-ph.CO].\n[14] J. M. Romero, R. Bernal-Jaquez and O. Gonzalez-Gaxiola, 'Is it possible to relate MOND with Horava Gravity?,' Mod. Phys. Lett. A 25 (2010) 2501-2506 [arXiv:1003.0684 [hep-th]].\n[15] J. Alexandre, K. Farakos and A. Tsapalis, 'Liouville-Lifshitz theory in 3+1 dimensions,' Phys. Rev. D 81 (2010) 105029-105034 [arXiv:1004.4201 [hep-th]].\n\n
\n\n
\n\n[29] J. F. Navarro, C. S. Frenk and S. D. M. White, 'The Structure of cold dark matter halos,' Astrophys. J. 462 (1996) 563-575 [astro-ph/9508025].\n[30] W. J. G. de Blok, F. Walter, E. Brinks, C. Trachternach, S-H. Oh and R. C. Kennicutt, Jr., 'High-Resolution Rotation Curves and Galaxy Mass Models from THINGS,' Astron. J. 136 (2008) 2648-2719 [arXiv:0810.2100 [astro-ph]].\n[31] D. Lynden-Bell, 'Statistical mechanics of violent relaxation in stellar systems,' Mon. Not. Roy. Astron. Soc. 136 (1967) 101-121\n[32] S. Dutta and E. N. Saridakis, 'Observational constraints on Horava-Lifshitz cosmology,' JCAP 1001 (2010) 013-035 [arXiv:0911.1435 [hep-th]].\n\n"},"sections":{"kind":"list like","value":[{"title":"Abstract","content":"Using the multiplicity of solutions for the projectable case of the covariant extension of Hoˇrava-Lifshitz Gravity, we show that an appropriate choice for the auxiliary field allows for an effective description of galaxy rotation curves. This description is based on static and spherically symmetric solutions of covariant Hoˇrava-Lifshitz Gravity and does not require Dark Matter.","pages":[1]},{"title":"1 Introduction","content":"/negationslash Although General Relativity (GR) has been accurately checked in the solar system, the small distance behaviour of gravity is still not understood, and several alternatives to GR have been proposed. Amongst these modified theories is Hoˇrava-Lifshitz (HL) gravity [1], which is based on the Lifshitz approach and consists of introducing an anisotropy between space and time: when space is rescaled as x → bx , time is rescaled as t → b z t , where z is an integer. In Lifshitz models, Lorentz invariance must be recovered for z = 1, and the situation z = 1 leads to new renormalizable interactions: the convergence of loop integrals is improved by inclusion of higher orders in space derivatives, without introducing ghosts, since the number of time derivatives remains the minimum. A review of Lifshitz-type quantum field theories in flat space time can be found in [2], and the HL power-counting renormalizable theory of gravity is reviewed in [3], [4] and [5]. A fundamental problem of the original model of HL gravity is the existence of an additional scalar degree of freedom for the metric, which can be understood as a Goldstone mode arising from breaking of 4-dimensional diffeomorphism [6], [7]. A solution to this problem has been proposed in [8], where an auxiliary field A is introduced, such that its 'equation of motion' leads to an additional constraint, eliminating the unwanted scalar degree of freedom of the metric. The resulting theory is invariant under a new Abelian gauge symmetry U Σ (1) which involves the metric components, the auxiliary field A and an additional auxiliary field ν . This gauge symmetry, together with the 3-dimensional diffeomorphism of HL gravity, can be shown to be equivalent to a 4-dimensional diffeomorphism at the lowest order in a post-Newtonian expansion, showing the equivalence with GR at long distances. For this reason, this modified version of HL gravity is called covariant HL gravity. We note however, that the long-distance limit is not obviously recovered: it has been shown in [9] that the equivalence principle is not automatically retrieved in the infrared, and that the meaning of the auxiliary field A and its coupling to matter are still open questions. Because of the anisotropy between space and time, HL gravity is naturally described in terms of the Arnowitt-Deser-Misner (ADM) decomposition of the metric, which expresses a space-time foliation. An important consequences of space-time anisotropy is the possibility of imposing the lapse function N to depend on time only. This situation is called the projectable case and leads to an interesting feature: the solutions of the equations of motion are not unique. Indeed, the Hamiltonian constraint, obtained by variation of the action with respect to N , leads to an integral equation, which does not have a unique solution, as will be seen in the present article. This multiplicity of solutions is independent of the above mentioned new gauge symmetry U Σ (1), and the different solutions for A in the projectable case belong to different gauge orbits. On the other hand, in the nonprojectable case where N depends on both space and time, the Hamiltonian constraint leads to a differential equation, which has a unique solution, after fixing the constants of integration. The non-projectable case has been studied in [10] for static spherically symmetric solutions of covariant HL gravity. In this article we use the freedom of choice of the auxiliary field A in the projectable case to study the possibility of fitting galaxy rotations curves without the need for Dark Matter. One can argue that the ambiguity in the choice of A leaves its physical interpretation unclear. This may be understood as one of the incomplete aspects of covariant HL gravity, and one could conjecture the existence of an additional symmetry which would restrict the set of solutions for the auxiliary field A , but this still remains to be studied. We note that the multiplicity of solutions, arising from a Hamiltonian constraint expressed in terms of an integral over space, has been discussed in [11] as an alternative to Dark Matter, in the context of the original HL gravity. Fits to galaxy rotation curves have been studied in the framework of the original HL gravity in [12] and [13], where a detailed comparison with experimental data is presented. Also, an interesting analogy between HL gravity for z = 0 and Modified Newtonian Dynamics theories has been described in [14]. In section 2 we review the static and spherically symmetric solutions of covariant HL gravity, and in section 3 we show how galaxy rotation curves can be described by the covariant extension of HL gravity. Although the curves we use were derived from Dark Matter models, we use them as experimental data fits. It is interesting to note that we obtain exact solutions for the auxiliary field A . Another exact solution in the Lifshitz context has been derived in [15], where the exact effective potential for a Liouville scalar theory, renormalizable in 3+1 dimensions with anisotropic scaling z = 3, proves to be an exponential, as in the usual 1+1 dimensional relativistic Liouville theory. Finally, we conclude by discussing the possiblity to describe the solar system with our solution.","pages":[1,2]},{"title":"2 Static and spherically symmetric solutions of covariant HL gravity","content":"The static and spherically symmetric solutions of covariant HL gravity were derived in [25] and [26], and we review here the aspects relevant to our present study.","pages":[3]},{"title":"2.1 Action","content":"The ADM metric we consider is where N and N i are the lapse and shift functions respectively, and g ij is the threedimensional space metric. We consider the anisotropic scaling with z = 3, for which the different operators, allowed for the model to be power-counting renormalizable, have the mass dimension of 6, at most. We note that in this case the mass dimension of the speed of light is [ c ] = z -1 = 2, and we will keep c in the different expression for the sake of clarity. The covariant Hoˇrava-Lifshitz action, in the absence of cosmological constant, is where g is the determinant of the three-dimensional metric g ij , with curvature scalar R (3) . A is an auxiliary field (without kinetic term) of mass dimension z + 1 = 4 and ν is an auxiliary field of mass dimension z -2 = 1. In the above expression, the extrinsic curvature is where a dot denotes a time derivative, and /negationslash where where R (3) ij is the Ricci tensor corresponding to the three-dimensional space metric g ij . Note that a more general kinetic term K ij K ij -λK 2 can be considered, with λ a generic parameter. The latter should be equal to 1 if one wishes to recover GR in the infrared, but in the framework of HL gravity, no symmetry imposes this parameter to be equal to 1. Constraints on λ resulting from observations of Type Ia Supernovae, Baryon Acoustic Oscillations, CMB and the requirement of Big Bang Nucleosynthesis all point towards a value near the GR parameter λ = 1 [16], while it has also been shown that covariant HL gravity with λ = 1 leads to inconsistencies, if one compares the predictions of the model with solar system tests [17]. We therefore focus here on the situation where λ = 1 only. Independently of these tests, the mathematical consistency of covariant HL gravity for λ = 1 has been studied in [18], and the corresponding Hamiltonian structure in /negationslash [19]. We stress again here that the purpose of this covariant extension of HL gravity is to eliminate the unwanted additional scalar graviton from the spectrum. As shown in [8], no singularity arises in the limit λ → 1, and the strong coupling of the scalar graviton does not exist anymore, as it is the case for the original HL gravity. Besides [6] and [7], one can find detailed analysis of the strong coupling problem in [20], [21], [22], and also related discussions in [23] on a extension of the original HL gravity [24]. The potential term V contains up to six spatial derivatives of the metric g ij : where the mass dimensions of the different couplings are We consider the most general, static, spherically symmetric metric, of the form: where n ( r ) = N r ( r ) is the radial component of shift function, and N r = n ( r ) /f ( r ) since g rr = 1 /f ( r ). Note that n has mass dimension z -1 = 2.","pages":[3,4]},{"title":"2.2 Constraints and equations of motion","content":"The variation of the action (2) with respect to the different degrees of freedom leads to the following constraints, or equations of motion, where a prime denotes a derivative with respect to the radial coordinate r . which imposes the function f ( r ) in the metric (8) to be where B is a constant of integration. In what follows we will assume the Gauge fixing of ν = 0, then the above constraint gives Θ ij K ij = 0, which is satisfied for spherically symmetric solutions. which is an space-integral equation since N depends on time only. such that either f is a constant ( B = 0) or n = 0. where the time has been rescaled such that N = 1.","pages":[4,5]},{"title":"2.3 Solutions","content":"The different solutions of the previous set of equations were derived in [25], [26], and here we shortly review the different cases. /negationslash /negationslash /negationslash with solutions where C is a constant of integration and the auxiliary field A satisfies The flexibility in the choice of solution for A will help describe galaxy rotation curves, as explained in the next section.","pages":[5]},{"title":"3 Galaxy rotation curves","content":"/negationslash /negationslash In what follows, we consider each star in the spiral arms of the galaxy as a test particle, moving on a circular trajectory under the influence of the potential φ ( r ), generated by the centre of the galaxy, which is assumed to be static and spherically symmetric. Our approach is to start from a stellar velocity distribution and derive the corresponding expression for the auxiliary field A , in the above situation with n = 0 and A = 0 , where multiple solutions for A are allowed. We note the following few points: (i) We consider vacuum solutions of covariant HL gravity, for which one cannot in principle describe the region in the centre of the galaxy. Nevertheless, as we will see in case of the following velocity profiles, a consistent solution for A can be found for 0 ≤ r ≤ R , where R is a typical radial length describing the galaxy. To be consistent with the usual studies of galaxy rotations, we will choose here R = r 200 , corresponding to the virial radius of the galaxy. The latter is defined as the distance from the centre of the galaxy, where the density ρ 200 is 200 times the critical density ρ c of the Universe where H is the Hubble constant; (ii) For r > r 200 , we assume the solution A = 0, which leads to the usual Schwarzschild solution outside the galaxy. Indeed, far from the galaxy, one expects to see the Newtonian potential. This solution implies that the Hamiltonian constraint is expressed in terms of an integral over the finite range [0 , r 200 ] of radial coordinate r , which we can impose to vanish by fixing the constants of integration; (iii) For r ≤ r 200 , since the shift A → A +constant does not have a physical implication, one can expect to find a consistent solution for A which is continuous at r = r 200 , i.e. A ( r 200 ) = 0. On the other hand, one cannot expect the auxiliary field to be differentiable at r = r 200 , but this is not necessary for the consistency of covariant HL gravity: the second derivative of A does not appear in any equation of motion or constraint, so the first derivative can be discontinuous.","pages":[6]},{"title":"3.1 Gravitational potential","content":"On a circular trajectory, the relation between the speed v of a star and the Newtonian potential φ is where φ is obtained from the g 00 component of the low energy effective theory and is thus given by We note here that, in the situation where the second auxiliary field ν does not vanish, the Newtonian potential is instead given by and φ is independent of the U Σ (1) gauge choice for ν and A . Also, φ is dimensionless, so that the speed v we use in this article is also dimensionless, which corresponds to a usual definition of speed, for isotropic space-time. The corresponding 'Lifshitz velocity' is cv , with mass dimension z -1 = 2. From the relations (16) and (19), the speed v and the auxiliary field A are then related by and we have to check that the Hamiltonian constraint (17) is satisfied, in order to show the consistency of the approach. We also note from eqs.(16,19) that the solution A = 0 gives, which is expected from Newtonian mechanics outside the galaxy, for r > r 200 . If one fixes the constant of integration C to the value where v 200 = v ( r 200 ), one obtains a continuous speed at r = r 200 . Finally a different choice of the auxiliary field ν would modify the field A in such a way that physical results would not be changed.","pages":[6,7]},{"title":"3.2 Navarro, Frenk and White profile","content":"This profile originates from Cold Dark Matter halo models and is characterised by the 'cusp' shape of the density distribution. Its validity has been tested on variety of observational results, including Low Surface Brightness galaxies in [27] and spherical galaxies and clusters in [28]. For our purposes, we use the circular velocity profile derived from the mass density profile in [29], based on the assumption of a Dark Matter halo and ignoring the contribution of baryons, where a is the concentration parameter and where v 200 , the virial velocity, is the circular velocity at the virial radius. Some typical values are [30] ± ± ± With the profile (26), eq.(23) leads to where φ 1 is a constant of integration, and therefore where A 1 is a constant of integration. Since we consider the solution A ( r ) = 0 for r > r 200 and impose A ( r ) to be continuous at r = r 200 , then A 1 = -a ln(1 + a ) and Finally, the Hamiltonian constraint is satisfied if We note that the solution (30) is consistent for any relevant values of the parameters a, v 0 , r 200 and therefore allows the description of a whole range of galaxies.","pages":[7,8]},{"title":"3.3 Pseudoisothermal profile","content":"The pseudoisothermal mass density profile assumes an existence of a 'cored' dark matter halo component in the galaxy [31] , with mass density of an approximately constant value in the central region of the galaxy, for r ≤ R c . Empirically motivated, the model is often contrasted with the aforementioned NFW 'cuspy' profile, while evidence shows that it provides better fit of the galaxy rotation velocities [27]. The velocity profile is where v 0 is given by and ρ 0 , the central density, is the density within R C . Typical values are [30] ± ± ± We note that in the case of NCG 2403, both the NFW and the present profile can be shown to produce fits of comparably good quality. With the profile (33), eq.(23) leads to where φ 2 is a constant of integration. The same steps as those describe for the previous profile lead to the solution where the constants of integration are for the continuity of A ( r ) at r = r 200 , and for the Hamiltonian constraint to be satisfied.","pages":[8,9]},{"title":"3.4 Vacuum energy contribution of the auxiliary field","content":"Since the present model does not require Dark Matter to explain galaxy rotation curves, the resulting mass content of the Universe might therefore be reduced. But the static configuration of A actually results in an effective vacuum energy. This can be seen from the original action (2), after integrating by parts the term -√ gAR (3) , which involve secondorder space derivatives of the metric. This leads to the following contribution to the Hamiltonian where the factor 1 /c arises from the absence of integration over time, and restores the correct dimension. In the case of the Navarro, Frenk and White profile, the energy (39) reads which is always positive, for any value of the parameter a ≥ 0. For the Pseudoisothermal case, we obtain from eq.(39) which is also always positive, for any value of the ratio r 200 /R c . The auxilliary field A has therefore a cosmological role, by contributing to the vacuum energy of the Universe, consistently with its observed acceleration.","pages":[9,10]},{"title":"4 Conclusion","content":"The multiplicity of solutions in covariant HL gravity has been used to describe galaxy rotation curves, without the need for Dark Matter. This multiplicity is a consequence of the breaking of 4-dimensional diffeomorphism, in the projectable case, where the lapse function is imposed to depend on time only. We considered two specific velocity profiles and showed in each case that a corresponding consistent solution can be derived for the auxiliary field A . Clearly, similar results can be expected with other profiles. A more detailed analysis would consist in taking into account the finite star density at the centre of the galaxy, as well as its rotation. One should note that the solutions found, for the auxilliary field A , depend on parameters wich cannot appear in the Lagrangian in a universal way. These parameters arise as constant of integrations, as a result of the integral form of the Hamiltonian constraint, and they have to be fitted to each galaxy which is studied. This is a bit similar to the Schwazschild metric in General Relativity, involving a mass which is a constant of integration and depends on the star/black hole under consideration. It is interesting to note that the solutions (30) and (36) actually do not only describe galaxy rotations curves, but also the usual Schwarzschild solutions for the Solar System, by taking the limit r 200 → 0 such that A = 0 for all r . In this sense these solutions provide an effective unified description of gravitational effects, from the solar system scale to the galaxy scale, where local Lorentz violation appears gradually, as the observational length increases from the the size of the solar system to the size of the galaxy. We finally note that, although HL gravity was initially developed to modify ultraviolet behaviour of gravity, our present work uses another consequence of local Lorentz symmetry violation, which is the existence of a non-unique solution to the Hamiltonian constraint. Since it has been shown that canonical Hoˇrava - Lifshitz gravity provides a description of Cosmology which is consistent with constraints from observational data [32], it would be interesting to see if the non-unique time-dependent solutions of the covariant version of the theory could also allow for an effective description of Cosmology, including an alternative to Dark Energy.","pages":[10]}],"string":"[\n {\n \"title\": \"Abstract\",\n \"content\": \"Using the multiplicity of solutions for the projectable case of the covariant extension of Hoˇrava-Lifshitz Gravity, we show that an appropriate choice for the auxiliary field allows for an effective description of galaxy rotation curves. This description is based on static and spherically symmetric solutions of covariant Hoˇrava-Lifshitz Gravity and does not require Dark Matter.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"/negationslash Although General Relativity (GR) has been accurately checked in the solar system, the small distance behaviour of gravity is still not understood, and several alternatives to GR have been proposed. Amongst these modified theories is Hoˇrava-Lifshitz (HL) gravity [1], which is based on the Lifshitz approach and consists of introducing an anisotropy between space and time: when space is rescaled as x → bx , time is rescaled as t → b z t , where z is an integer. In Lifshitz models, Lorentz invariance must be recovered for z = 1, and the situation z = 1 leads to new renormalizable interactions: the convergence of loop integrals is improved by inclusion of higher orders in space derivatives, without introducing ghosts, since the number of time derivatives remains the minimum. A review of Lifshitz-type quantum field theories in flat space time can be found in [2], and the HL power-counting renormalizable theory of gravity is reviewed in [3], [4] and [5]. A fundamental problem of the original model of HL gravity is the existence of an additional scalar degree of freedom for the metric, which can be understood as a Goldstone mode arising from breaking of 4-dimensional diffeomorphism [6], [7]. A solution to this problem has been proposed in [8], where an auxiliary field A is introduced, such that its 'equation of motion' leads to an additional constraint, eliminating the unwanted scalar degree of freedom of the metric. The resulting theory is invariant under a new Abelian gauge symmetry U Σ (1) which involves the metric components, the auxiliary field A and an additional auxiliary field ν . This gauge symmetry, together with the 3-dimensional diffeomorphism of HL gravity, can be shown to be equivalent to a 4-dimensional diffeomorphism at the lowest order in a post-Newtonian expansion, showing the equivalence with GR at long distances. For this reason, this modified version of HL gravity is called covariant HL gravity. We note however, that the long-distance limit is not obviously recovered: it has been shown in [9] that the equivalence principle is not automatically retrieved in the infrared, and that the meaning of the auxiliary field A and its coupling to matter are still open questions. Because of the anisotropy between space and time, HL gravity is naturally described in terms of the Arnowitt-Deser-Misner (ADM) decomposition of the metric, which expresses a space-time foliation. An important consequences of space-time anisotropy is the possibility of imposing the lapse function N to depend on time only. This situation is called the projectable case and leads to an interesting feature: the solutions of the equations of motion are not unique. Indeed, the Hamiltonian constraint, obtained by variation of the action with respect to N , leads to an integral equation, which does not have a unique solution, as will be seen in the present article. This multiplicity of solutions is independent of the above mentioned new gauge symmetry U Σ (1), and the different solutions for A in the projectable case belong to different gauge orbits. On the other hand, in the nonprojectable case where N depends on both space and time, the Hamiltonian constraint leads to a differential equation, which has a unique solution, after fixing the constants of integration. The non-projectable case has been studied in [10] for static spherically symmetric solutions of covariant HL gravity. In this article we use the freedom of choice of the auxiliary field A in the projectable case to study the possibility of fitting galaxy rotations curves without the need for Dark Matter. One can argue that the ambiguity in the choice of A leaves its physical interpretation unclear. This may be understood as one of the incomplete aspects of covariant HL gravity, and one could conjecture the existence of an additional symmetry which would restrict the set of solutions for the auxiliary field A , but this still remains to be studied. We note that the multiplicity of solutions, arising from a Hamiltonian constraint expressed in terms of an integral over space, has been discussed in [11] as an alternative to Dark Matter, in the context of the original HL gravity. Fits to galaxy rotation curves have been studied in the framework of the original HL gravity in [12] and [13], where a detailed comparison with experimental data is presented. Also, an interesting analogy between HL gravity for z = 0 and Modified Newtonian Dynamics theories has been described in [14]. In section 2 we review the static and spherically symmetric solutions of covariant HL gravity, and in section 3 we show how galaxy rotation curves can be described by the covariant extension of HL gravity. Although the curves we use were derived from Dark Matter models, we use them as experimental data fits. It is interesting to note that we obtain exact solutions for the auxiliary field A . Another exact solution in the Lifshitz context has been derived in [15], where the exact effective potential for a Liouville scalar theory, renormalizable in 3+1 dimensions with anisotropic scaling z = 3, proves to be an exponential, as in the usual 1+1 dimensional relativistic Liouville theory. Finally, we conclude by discussing the possiblity to describe the solar system with our solution.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 Static and spherically symmetric solutions of covariant HL gravity\",\n \"content\": \"The static and spherically symmetric solutions of covariant HL gravity were derived in [25] and [26], and we review here the aspects relevant to our present study.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.1 Action\",\n \"content\": \"The ADM metric we consider is where N and N i are the lapse and shift functions respectively, and g ij is the threedimensional space metric. We consider the anisotropic scaling with z = 3, for which the different operators, allowed for the model to be power-counting renormalizable, have the mass dimension of 6, at most. We note that in this case the mass dimension of the speed of light is [ c ] = z -1 = 2, and we will keep c in the different expression for the sake of clarity. The covariant Hoˇrava-Lifshitz action, in the absence of cosmological constant, is where g is the determinant of the three-dimensional metric g ij , with curvature scalar R (3) . A is an auxiliary field (without kinetic term) of mass dimension z + 1 = 4 and ν is an auxiliary field of mass dimension z -2 = 1. In the above expression, the extrinsic curvature is where a dot denotes a time derivative, and /negationslash where where R (3) ij is the Ricci tensor corresponding to the three-dimensional space metric g ij . Note that a more general kinetic term K ij K ij -λK 2 can be considered, with λ a generic parameter. The latter should be equal to 1 if one wishes to recover GR in the infrared, but in the framework of HL gravity, no symmetry imposes this parameter to be equal to 1. Constraints on λ resulting from observations of Type Ia Supernovae, Baryon Acoustic Oscillations, CMB and the requirement of Big Bang Nucleosynthesis all point towards a value near the GR parameter λ = 1 [16], while it has also been shown that covariant HL gravity with λ = 1 leads to inconsistencies, if one compares the predictions of the model with solar system tests [17]. We therefore focus here on the situation where λ = 1 only. Independently of these tests, the mathematical consistency of covariant HL gravity for λ = 1 has been studied in [18], and the corresponding Hamiltonian structure in /negationslash [19]. We stress again here that the purpose of this covariant extension of HL gravity is to eliminate the unwanted additional scalar graviton from the spectrum. As shown in [8], no singularity arises in the limit λ → 1, and the strong coupling of the scalar graviton does not exist anymore, as it is the case for the original HL gravity. Besides [6] and [7], one can find detailed analysis of the strong coupling problem in [20], [21], [22], and also related discussions in [23] on a extension of the original HL gravity [24]. The potential term V contains up to six spatial derivatives of the metric g ij : where the mass dimensions of the different couplings are We consider the most general, static, spherically symmetric metric, of the form: where n ( r ) = N r ( r ) is the radial component of shift function, and N r = n ( r ) /f ( r ) since g rr = 1 /f ( r ). Note that n has mass dimension z -1 = 2.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.2 Constraints and equations of motion\",\n \"content\": \"The variation of the action (2) with respect to the different degrees of freedom leads to the following constraints, or equations of motion, where a prime denotes a derivative with respect to the radial coordinate r . which imposes the function f ( r ) in the metric (8) to be where B is a constant of integration. In what follows we will assume the Gauge fixing of ν = 0, then the above constraint gives Θ ij K ij = 0, which is satisfied for spherically symmetric solutions. which is an space-integral equation since N depends on time only. such that either f is a constant ( B = 0) or n = 0. where the time has been rescaled such that N = 1.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"2.3 Solutions\",\n \"content\": \"The different solutions of the previous set of equations were derived in [25], [26], and here we shortly review the different cases. /negationslash /negationslash /negationslash with solutions where C is a constant of integration and the auxiliary field A satisfies The flexibility in the choice of solution for A will help describe galaxy rotation curves, as explained in the next section.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"3 Galaxy rotation curves\",\n \"content\": \"/negationslash /negationslash In what follows, we consider each star in the spiral arms of the galaxy as a test particle, moving on a circular trajectory under the influence of the potential φ ( r ), generated by the centre of the galaxy, which is assumed to be static and spherically symmetric. Our approach is to start from a stellar velocity distribution and derive the corresponding expression for the auxiliary field A , in the above situation with n = 0 and A = 0 , where multiple solutions for A are allowed. We note the following few points: (i) We consider vacuum solutions of covariant HL gravity, for which one cannot in principle describe the region in the centre of the galaxy. Nevertheless, as we will see in case of the following velocity profiles, a consistent solution for A can be found for 0 ≤ r ≤ R , where R is a typical radial length describing the galaxy. To be consistent with the usual studies of galaxy rotations, we will choose here R = r 200 , corresponding to the virial radius of the galaxy. The latter is defined as the distance from the centre of the galaxy, where the density ρ 200 is 200 times the critical density ρ c of the Universe where H is the Hubble constant; (ii) For r > r 200 , we assume the solution A = 0, which leads to the usual Schwarzschild solution outside the galaxy. Indeed, far from the galaxy, one expects to see the Newtonian potential. This solution implies that the Hamiltonian constraint is expressed in terms of an integral over the finite range [0 , r 200 ] of radial coordinate r , which we can impose to vanish by fixing the constants of integration; (iii) For r ≤ r 200 , since the shift A → A +constant does not have a physical implication, one can expect to find a consistent solution for A which is continuous at r = r 200 , i.e. A ( r 200 ) = 0. On the other hand, one cannot expect the auxiliary field to be differentiable at r = r 200 , but this is not necessary for the consistency of covariant HL gravity: the second derivative of A does not appear in any equation of motion or constraint, so the first derivative can be discontinuous.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.1 Gravitational potential\",\n \"content\": \"On a circular trajectory, the relation between the speed v of a star and the Newtonian potential φ is where φ is obtained from the g 00 component of the low energy effective theory and is thus given by We note here that, in the situation where the second auxiliary field ν does not vanish, the Newtonian potential is instead given by and φ is independent of the U Σ (1) gauge choice for ν and A . Also, φ is dimensionless, so that the speed v we use in this article is also dimensionless, which corresponds to a usual definition of speed, for isotropic space-time. The corresponding 'Lifshitz velocity' is cv , with mass dimension z -1 = 2. From the relations (16) and (19), the speed v and the auxiliary field A are then related by and we have to check that the Hamiltonian constraint (17) is satisfied, in order to show the consistency of the approach. We also note from eqs.(16,19) that the solution A = 0 gives, which is expected from Newtonian mechanics outside the galaxy, for r > r 200 . If one fixes the constant of integration C to the value where v 200 = v ( r 200 ), one obtains a continuous speed at r = r 200 . Finally a different choice of the auxiliary field ν would modify the field A in such a way that physical results would not be changed.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"3.2 Navarro, Frenk and White profile\",\n \"content\": \"This profile originates from Cold Dark Matter halo models and is characterised by the 'cusp' shape of the density distribution. Its validity has been tested on variety of observational results, including Low Surface Brightness galaxies in [27] and spherical galaxies and clusters in [28]. For our purposes, we use the circular velocity profile derived from the mass density profile in [29], based on the assumption of a Dark Matter halo and ignoring the contribution of baryons, where a is the concentration parameter and where v 200 , the virial velocity, is the circular velocity at the virial radius. Some typical values are [30] ± ± ± With the profile (26), eq.(23) leads to where φ 1 is a constant of integration, and therefore where A 1 is a constant of integration. Since we consider the solution A ( r ) = 0 for r > r 200 and impose A ( r ) to be continuous at r = r 200 , then A 1 = -a ln(1 + a ) and Finally, the Hamiltonian constraint is satisfied if We note that the solution (30) is consistent for any relevant values of the parameters a, v 0 , r 200 and therefore allows the description of a whole range of galaxies.\",\n \"pages\": [\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.3 Pseudoisothermal profile\",\n \"content\": \"The pseudoisothermal mass density profile assumes an existence of a 'cored' dark matter halo component in the galaxy [31] , with mass density of an approximately constant value in the central region of the galaxy, for r ≤ R c . Empirically motivated, the model is often contrasted with the aforementioned NFW 'cuspy' profile, while evidence shows that it provides better fit of the galaxy rotation velocities [27]. The velocity profile is where v 0 is given by and ρ 0 , the central density, is the density within R C . Typical values are [30] ± ± ± We note that in the case of NCG 2403, both the NFW and the present profile can be shown to produce fits of comparably good quality. With the profile (33), eq.(23) leads to where φ 2 is a constant of integration. The same steps as those describe for the previous profile lead to the solution where the constants of integration are for the continuity of A ( r ) at r = r 200 , and for the Hamiltonian constraint to be satisfied.\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"3.4 Vacuum energy contribution of the auxiliary field\",\n \"content\": \"Since the present model does not require Dark Matter to explain galaxy rotation curves, the resulting mass content of the Universe might therefore be reduced. But the static configuration of A actually results in an effective vacuum energy. This can be seen from the original action (2), after integrating by parts the term -√ gAR (3) , which involve secondorder space derivatives of the metric. This leads to the following contribution to the Hamiltonian where the factor 1 /c arises from the absence of integration over time, and restores the correct dimension. In the case of the Navarro, Frenk and White profile, the energy (39) reads which is always positive, for any value of the parameter a ≥ 0. For the Pseudoisothermal case, we obtain from eq.(39) which is also always positive, for any value of the ratio r 200 /R c . The auxilliary field A has therefore a cosmological role, by contributing to the vacuum energy of the Universe, consistently with its observed acceleration.\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"4 Conclusion\",\n \"content\": \"The multiplicity of solutions in covariant HL gravity has been used to describe galaxy rotation curves, without the need for Dark Matter. This multiplicity is a consequence of the breaking of 4-dimensional diffeomorphism, in the projectable case, where the lapse function is imposed to depend on time only. We considered two specific velocity profiles and showed in each case that a corresponding consistent solution can be derived for the auxiliary field A . Clearly, similar results can be expected with other profiles. A more detailed analysis would consist in taking into account the finite star density at the centre of the galaxy, as well as its rotation. One should note that the solutions found, for the auxilliary field A , depend on parameters wich cannot appear in the Lagrangian in a universal way. These parameters arise as constant of integrations, as a result of the integral form of the Hamiltonian constraint, and they have to be fitted to each galaxy which is studied. This is a bit similar to the Schwazschild metric in General Relativity, involving a mass which is a constant of integration and depends on the star/black hole under consideration. It is interesting to note that the solutions (30) and (36) actually do not only describe galaxy rotations curves, but also the usual Schwarzschild solutions for the Solar System, by taking the limit r 200 → 0 such that A = 0 for all r . In this sense these solutions provide an effective unified description of gravitational effects, from the solar system scale to the galaxy scale, where local Lorentz violation appears gradually, as the observational length increases from the the size of the solar system to the size of the galaxy. We finally note that, although HL gravity was initially developed to modify ultraviolet behaviour of gravity, our present work uses another consequence of local Lorentz symmetry violation, which is the existence of a non-unique solution to the Hamiltonian constraint. Since it has been shown that canonical Hoˇrava - Lifshitz gravity provides a description of Cosmology which is consistent with constraints from observational data [32], it would be interesting to see if the non-unique time-dependent solutions of the covariant version of the theory could also allow for an effective description of Cosmology, including an alternative to Dark Energy.\",\n \"pages\": [\n 10\n ]\n }\n]"}}},{"rowIdx":270,"cells":{"bibcode":{"kind":"string","value":"2013GeCoA.101..285K"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1210.1674.pdf"},"content":{"kind":"string","value":"\nAb initio prediction of equilibrium boron isotope fractionation between minerals and aqueous fluids at high P and T\nPiotr M. Kowalski 1 , 2 ∗ , Bernd Wunder 1 and Sandro Jahn 1\n1 GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany\n2 Forschungszentrum Julich, Institute of Energy and Climate Research (IEK-6), Wilhelm-Johnen-Strasse, 52425 Julich, Germany\nAbstract\nOver the last decade experimental studies have shown a large B isotope fractionation between materials carrying boron incorporated in trigonally and tetrahedrally coordinated sites, but the mechanisms responsible for producing the observed isotopic signatures are poorly known. In order to understand the boron isotope fractionation processes and to obtain a better interpretation of the experimental data and isotopic signatures observed in natural samples, we use first principles calculations based on density functional theory in conjunction with ab initio molecular dynamics and a new pseudofrequency analysis method to investigate the B isotope fractionation between B-bearing minerals (such as tourmaline and micas) and aqueous fluids containing H3BO3 and H4BO -4 species. We confirm the experimental finding that the isotope fractionation is mainly driven by the coordination of the fractionating boron atoms and have found in addition that the strength of the produced isotopic signature is strongly correlated with the B-O bond length. We also demonstrate the ability of our computational scheme to predict the isotopic signatures of fluids at extreme pressures by showing the consistency of computed pressure-dependent β factors with the measured pressure shifts of the B-O vibrational frequencies of H3BO3 and H4BO -4 in aqueous fluid. The comparison of the predicted with measured fractionation factors between boromuscovite and neutral fluid confirms the existence of the admixture of tetrahedral boron species in neutral fluid at high P and T found experimentally, which also explains the inconsistency between the various measurements on the tourmaline-mica system reported in the literature. Our investigation shows that the calculated equilibrium isotope fractionation factors have an accuracy comparable to the experiments and give unique and valuable insight into the processes governing the isotope fractionation mechanisms on the atomic scale.\nKeywords: stable isotope fractionation, boron, fluid speciation, DFT, molecular dynamics\n1. Introduction\nThe use of isotopes as geochemical tracers depends upon the existence of reliable P -T -dependent equilibrium isotope fractionation data between solids, fluids and melts. The common method used for determination of such data is to conduct experiments, where the phases of interest are equilibrated at a range of di ff erent conditions and individually measured for their equilibrium isotopic compositions. Recently, with the development of computational methods, software and increase in hardware performance it is also possible to simulate and compute the isotope fractionation with ab initio\nmethods. These computational considerations complement the experimental e ff ort and provide information on the mechanisms governing the equilibrium isotope fractionation processes on the atomic scale. By establishing an e ffi cient computational approach for materials at high P and T and testing its reliability by computing Li isotope fractionation between minerals and aqueous fluids, Kowalski & Jahn (2011) have shown that ab initio methods can provide a reliable estimation of equilibrium isotope fractionation factors at an accuracy level comparable to experiments. Motivated by the encouraging results for the fractionation of lithium isotopes we applied our new method to study boron isotope fractionation. The main goals of the present study are to make theoretical predictions and obtain a better understanding of the B isotope fractionation process between tourmaline, B-bearing mica and flu-\nids at various pressures and temperatures, to compare these data to results from recent in situ experimental studies (Wunder et al., 2005; Meyer et al., 2008) and measurements of isotopic signatures in natural samples (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002) and to investigate the underlying mechanisms driving the boron isotope fractionation processes between the considered materials.\nWith two stable isotopes 10 B and 11 B, of relatively large mass di ff erence of about 10 %, boron isotopes strongly fractionate during geological processes, thus leading to natural δ 11 B-variations ranging from -30 to + 60 /permil (Barth, 1993). Therefore, B isotopes are ideal for the distinction of di ff erent geological environments and for quantifying mass transfer processes, e.g. in the range of subduction zones. First-order criteria driving isotope fractionation in Earth materials are di ff erences in coordination and in the bonding environments of coexisting phases. The lighter isotope usually preferentially occupies the higher coordinated site, which is generally accompanied with a longer cation-anion bond length and weaker bond strength (Schauble et al., 2009; Wunder et al., 2011).\nTourmaline has an extensive chemical variability and is the most widespread borosilicate in various rocks over a large range of bulk compositions. It has a large P -T -stability which ranges from surface conditions to high pressures and temperatures of at least 7.0 GPa and about 1000 o C as determined experimentally for dravitic tourmaline (Krosse, 1995). Most tourmaline minerals contain 3 boron atoms per formula unit (pfu) with B in trigonal planar coordination (B [3] ). High pressure combined with an Al-rich environment can lead to the formation of olenitic tourmaline with significant amounts of excess B substituting for Si at the tetrahedral site (B [4] ). The highest amounts of up to 1.2 B [4] pfu have been found in olenitic tourmaline from Koralpe, Austria (Ertl et al., 1997). Al-rich tourmaline with up to 2.2 B [4] pfu was synthesized experimentally at 2 . 5 GPa, 600 o C (Schreyer et al., 2000). The maximum possible amount of B [4] in olenitic tourmaline is limited to three B [4] pfu, due to structural and crystal chemical reasons.\nAs tourmaline is not stable at basic pH (Morgan & London, 1989), the dominant B-species of fluids coexisting with tourmaline is B(OH)3. Therefore, due to the absence of change in boron coordination, B-isotopic fractionation between B [4] -free tourmaline and fluid ( ∆ 11 B(tour-fl)) should be small. However, at 0 . 2 GPa, the experimentally determined ∆ 11 B(tour-fl)values of -2 . 5 ± 0 . 4 /permil at 400 o C and -0 . 4 ± 0 . 4 /permil at 700 o C (Meyer et al., 2008) suggest small but significant di ff erences in the B-O(H) bond strength between\ntourmaline and the neutral fluids. Incorporation of B [4] into tourmaline is expected to significantly increase the fractionation of boron isotopes between the mineral and aqueous fluid (i.e. increase of | ∆ 11 B | (tour-fl)). In this contribution we present calculated P -T -dependent data on B-isotope fractionation between B [4] -bearing tourmaline and fluid, which so far are not available from experiments.\nDespite of its low boron content of up to maximum values of 270 ppm (Domanik et al., 1993), potassic white mica is probably the main host of boron in metasedimentary and metabasaltic blueschists and eclogites, because of its high modal abundance and B-incompatibility in all other stable minerals of these rocks (Brenan et al., 1998). During subduction the modal amount of mica continuously decreases by dehydration reactions and the chemistry of residual micas is shifted towards phengitic compositions. Phengitic mica has an extended stability, extending as deep as 300 km within cold subduction zones (Schmidt, 1996). In contrast to most tourmalines, boron is tetrahedrally coordinated in mica, where it substitutes for aluminum. Due to the coordination change from mostly three-fold coordinated in near-neutral fluids (Schmidt et al., 2005) to B [4] in mica, B-isotopic fractionation between Bbearing mica and such fluids is much larger than for tourmaline - fluid. ∆ 11 B(mica-fl)-values determined experimentally at P = 3 . 0 GPa, are -10 . 9 ± 1 . 3 /permil at 500 o C and -7 . 1 ± 0 . 5 /permil at 700 o C (Wunder et al., 2005). Such a strong B-isotope fractionation and its pronounced T -dependence, in combination with the continuous dehydration of micas during ongoing subduction and boron transport via fluids into mantle wedge regions of arc magma-formation, probably determines the boron variations and B-isotopic signatures in volcanic arcs (Wunder et al., 2005).\nUsing in situ Raman spectroscopic measurements of near-neutral B-bearing fluids, Schmidt et al. (2005) observed a significant amount of 4-fold B-species at high P and T . The abundance of these species increases with temperature and pressure and at T = 800 K and P = 1 . 9 GPa there should be a considerable amount (15 -30%) of B [4] species in the fluid. Such a significant amount of tetrahedrally coordinated B-species in high temperature and pressure fluids should a ff ect solid-fluid B-isotopic fractionation, which we investigate in our calculations. In the light of this, we also discuss recently determined B-isotope data from coexisting natural tourmaline and B-bearing mica (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002), which show slight inconsistency with in situ measurements of Wunder et al. (2005) and Meyer et al. (2008).\nReliable ab initio computational methods to predict isotope fractionation factors have been established recently. Several groups have proved that such calculations can contribute towards understanding geochemical mechanisms responsible for production of isotope signatures (Driesner, 1997; Yamaji et al., 2001; Schauble, 2004; Domagal-Goldman et al., 2008; Hill & Schauble, 2008; Meheut et al., 2009, 2007; Schauble et al., 2009; Zeebe, 2009; Hill et al., 2010; Rustad et al., 2010a; Rustad et al, 2010b; Zeebe, 2009, 2010; Kowalski & Jahn, 2011). The majority of these works, however, concentrate on the computation of the stable isotope fractionation between various, mostly simple crystalline minerals, and the aqueous solutions are usually computed using an isolated cluster containing fractionating species and a hydration shell (e.g. Zeebe (2010, 2009); Hill et al. (2010); Rustad et al. (2010a); Domagal-Goldman et al. (2008); Hill & Schauble (2008); Rustad et al (2010b); Zeebe (2005)). However, aqueous solutions at high pressure and temperature must be computed with caution as discussed in Kowalski & Jahn (2011). This is because the distribution of cation coordination and cation-oxygen bond lengths that a ff ect the isotope fractionation (Bigeleisen & Mayer, 1947) is driven by the dynamics of the system and change under compression (Jahn & Wunder, 2009; Wunder et al., 2011; Kowalski & Jahn, 2011). The only recent ab initio work, besides Kowalski & Jahn (2011), that accounts for the dynamical e ff ects on the isotope fractionation in fluid is by Rustad & Bylaska (2007) who considered boron equilibrium isotope fractionation between B(OH)3 and B(OH) -4 species in aqueous solution. They performed ab initio molecular dynamics simulations of these fluids and attempted to use the vibrational density of states, derived through the Fourier transform of the velocity auto-correlation function, as an input for the calculation of the 11 B / 10 B isotope fractionation coe ffi -cient. However, the resulting fractionation factor α = 0 . 86 happened to be much lower than the experimental value of α = 1 . 028. Interestingly, the discrepancy between experiment and theory is cured by quenching the selected configurations along the molecular dynamics trajectory and computing the harmonic frequencies. The fractionation factor derived using these frequencies exactly reproduces the experimental value.\nIn our approach both solids and fluids are treated as extended systems by application of periodic boundary conditions in all three spatial directions, which is crucial to model high pressure materials. Large enough supercells are chosen to avoid significant interaction between atoms and their periodic images, as well as to reduce the\nnumber of k-points and q-vectors for sampling the Brillouin zones and the phonon spectra of the crystals, respectively (for a liquid or fluid, both the Brillouin zone and phonons are not defined). In our investigation we use cells of at least 7 Å width in each spatial dimension. A representative statistical sampling of the fluid structure is obtained by performing Car-Parrinello molecular dynamics simulations (Car & Parrinello, 1985). For the calculation of the isotope fractionation factors in fluids, several random snapshots from the simulation runs are chosen. The force constants acting on the fractionating element and the resulting fractionation factors are then obtained for each ionic configuration, and the relevant fractionation factor for the boron species in the fluid is computed as an average over the whole set of considered geometries. In Kowalski & Jahn (2011) it was shown that in line with the Bigeleisen & Mayer (1947) approximation, considering the force constants acting on the fractionating atom only leads to a satisfactory estimation of the Li isotope fractionation factors for high temperature fluids and minerals. Here, we will show that approximating the vibrational spectrum by the three pseudofrequencies derived from the force constants allows for further improvement of the accuracy of the predicted isotope fractionation factors, especially at lower temperatures.\nIn this contribution we present the theoretical prediction of B isotope fractionation factors between B bearing aqueous fluids and solids, specifically tourmaline and B-muscovite, for which experimental data and measurements on natural samples are available for comparison (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011). We will show that the application of ab initio methods to B-bearing crystalline solids and fluids not only provides unique insight into the mechanisms driving equilibrium B-isotope fractionation on the atomic scale, but helps in proper interpretation of the data.\n2. Computational approach\n2.1. Theoretical model\n2.1.1. The single atom approximation: &\nBigeleisen Mayer (1947) approach\nMass-dependent equilibrium isotope fractionation is driven by the change in molecular and crystalline vibration frequencies resulting from the di ff erent masses of the isotopes. The fractionation between species and an ideal monoatomic gas is called the β factor. In the harmonic approximation it is given by the formula\n
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Table 1: The computed and measured vibrational frequencies for H3BO3 and H4BO -4 molecules reported for two di ff erent boron isotopes B 11 / B 10 . Only the frequencies a ff ected by the isotopic substitution are reported. The units are cm -1 .
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\nReferences: 1 Zeebe (2005), 2 Liu & Tossell (2005), 3 Rustad et al (2010b), 4 Sanchez-Valle et al. (2005), 5 Gilson (1991), 6 Oi (2000); Liu & Tossell (2005), 7 Andrews & Burkholder (1992), 8 Ogden & Young (1988), 9 Zeebe (2005) and references hereafter.\n(Bigeleisen & Mayer, 1947; Urey, 1947; Chacko et al., 2001):\nβ = Ndof ∏ i = 1 u ∗ i ui exp [ ( ui -u ∗ i ) 2 ] 1 -exp( -ui ) 1 -exp( -u ∗ i ) , (1)\nwhere u = h ν i / kBT , h is the Planck constant, ν i is the vibrational frequency of the i -th degree of freedom, kB is the Boltzmann constant, Ndof is the number of degrees of freedom, which for N being the number of atoms in the considered system (molecule, mineral or fluid) is equal to 3 N -5 for a diatomic molecule, 3 N -6 for multiatomic molecules and 3 N for crystals, and a star symbol marks the heavier isotope. The fractionation factor between two substances A and B, α A -B is computed as the ratio of the relevant β factors, which is well approximated by the di ff erences in the β factors:\nα A -B = β A /β B , ∆ A -B /simequal 1000 ln β A -1000 ln β B [ /permil ] . (2)\nThe calculation of the β factor requires only knowledge of the vibrational properties of the considered system computed for the two di ff erent isotopes. However, computation of the whole vibrational spectra of complex, multiparticle minerals or fluids requires substantial computational resources and is currently limited to systems containing a few dozens of atoms or less. In our recent work (Kowalski & Jahn, 2011) we proposed\nto use an e ffi cient method for computing the high temperature isotope fractionation factors between complex materials such as fluids and crystalline solids, which requires the knowledge of the force constants acting upon the fractionating element only. The β factor (Eq. 1) can be then approximated by (Bigeleisen & Mayer, 1947; Kowalski & Jahn, 2011):\nβ /similarequal 1 + Ndof ∑ i = 1 u 2 i -u ∗ 2 i 24 = 1 + ∆ m mm ∗ /planckover2pi1 2 24 k 2 B T 2 3 ∑ i = 1 Ai , (3)\nwhere Ai are the force constants acting on the isotopic atom in the three perpendicular spatial directions (x, y and z), ∆ m = m ∗ -m , where m and m ∗ are the masses of the lighter and heavier isotopes of the fractionating element. As the computation of the β factors from formula 3 requires the knowledge of properties of the fractionating element only we will call such an approach the single atom approximation throughout the paper. The validity criteria restricts the usage of the formula to frequencies ν [cm -1 ] /lessorsimilar 1 . 39 T [K] (assuming u < 2, see Fig. 1 of Bigeleisen & Mayer (1947)). We are interested in temperature range 800-1000 K. The highest vibrational frequency of the modes involving movement of B atoms for H3BO3 is ∼ 1400 cm -1 and of H4BO -1 4 is < 1200 cm -1 (Table 1). In case of H3BO3, the single atom approximation may produce an error of 2 . 2 /permil in the β factor at T = 800 K. The relevant error for\n
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Table 2: 1000( β -1) factors for isolated H3BO3 and H4BO -4 molecules at T = 300K obtained using three methods: (1) the full frequency spectrum and Equation 1, (2) the single atom approximation of Kowalski & Jahn (2011), (3) the single atom approximation with the pseudofrequencies and equation 1. The units are /permil . The ∆ β/β BM is the check of condition given by Eq. 15 and indicates the improvement of the method (3) over method (2) expressed in %.
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\nH4BO -1 4 is 0 . 8 /permil (Table 2). A further improvement to the method is therefore desired.\n2.1.2. The single atom approximation with pseudofrequencies: our improvement\nWe will show that the error of the single atom approximation can be substantially reduced if one uses the three frequencies ¯ ν i derived from the force constants acting on the fractionating element (¯ ν 2 i = Ai / 4 π 2 m ). We call them 'pseudofrequencies' , and compute the β factors using formula 1. In the following we present the formal justification of such an approach. According to Bigeleisen & Mayer (1947), equation 1 for small ∆ ui = ui -u ∗ i reduces to (Bigeleisen & Mayer (1947), Eq. 11a):\nβ = 1 + Ndof ∑ i = 1 ( 1 2 -1 ui + 1 exp( ui ) -1 ) ∆ ui . (4)\nThe Taylor expansion of the function appearing under the summation sign is:\nG ( u ) = 1 2 -1 u + 1 exp( ui ) -1 = u 12 -u 3 720 + u 5 30240 -u 7 1209600 + .... (5)\nWhen we consider just the first term of the expansion the β factor is:\nβ = 1 + Ndof ∑ i ui 12 ∆ ui = 1 + Ndof ∑ i = 1 ∆ u 2 i 24 = 1 + Ndof ∑ i = 1 u 2 i -u ∗ 2 i 24 , (6)\n
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Table 3: The three pseudofrequencies of the B atom derived for two selected molecules and two crystalline solids. The units are cm -1 .
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\nwhich is exactly equation 3.\nLet us consider the Taylor expansion of the di ff erent estimations of β factors. Equation 4 then reads:\nβ = β exact = 1 + Ndof ∑ i = 1 ui 12 -u 3 i 720 + ... ∆ ui . (7)\nThe Bigeleisen & Mayer (1947) approximation given by equation 3 reads:\nβ ∼ β BM = 1 + Ndof ∑ i = 1 ui 12 ∆ ui , (8)\nand the proposed approximation based on pseudofrequencies is:\nβ ∼ β pseudo = 1 + 3 ∑ i = 1 ¯ ui 12 -¯ u 3 i 720 + ... ∆ ¯ ui . (9)\nIn the above equation ¯ ui = h ¯ ν i / kBT . Next we check how this approximation compares to the Bigeleisen & Mayer (1947) approximation given by equations 3 and 8. In order to make a comparison we derive the di ff erences between the two approximate expressions β BM , β pseudo and the exact one β exact (Eq. 7). In the case of the\n\n\n
Figure 1: β factors for isolated H3BO3 and H4BO -4 molecules as well as dravite and boromuscovite 1M crystalline solids. The lines represent the results obtained using: (1) the full frequency spectrum and Eq. 1 (solid), (2) the single atom approximation of Bigeleisen & Mayer (1947); Kowalski & Jahn (2011), Eq. 3 (dotted) and (3) the method described in the text with the pseudofrequencies and Eq. 1 (dashed lines).
\n\nBigeleisen & Mayer (1947) approximation we have:\n∆ β BM = β BM -β exact = Ndof ∑ i ui 12 -Ndof ∑ i ui 12 -u 3 i 720 + ... ∆ ui\nand having from equations 3 and 8 that\n= Ndof ∑ i u 3 i 720 -... ∆ ui (10)\nNdof ∑ i ui 12 ∆ ui = 3 ∑ i ¯ ui 12 ∆ ¯ ui (11)\nin the case of the proposed approximation we get:\n∆ β pseudo = β pseudo -β exact = 3 ∑ i = 1 -¯ u 3 i 720 + ... ∆ ¯ ui + Ndof ∑ i u 3 i 720 -... ∆ ui = = 3 ∑ i = 1 -¯ u 3 i 720 + ... ∆ ¯ ui + ∆ β BM (12)\nBecause relation G ( u ) < u 12 holds for any u (see Bigeleisen & Mayer (1947), Fig. 1), the function\n3 ∑ i = 1 -¯ u 3 i 720 + ... ∆ ui = 3 ∑ i = 1 ( G ( ¯ ui ) -¯ ui 12 ) ∆ ¯ ui < 0 (13)\nand ∆ β pseudo < ∆ β BM . On the other hand the expression for ∆ β pseudo is given by a di ff erence of the higher order terms of the Taylor expansions of the two expressions for the β factor. In the considered cases the values of pseudofrequencies are similar to the real frequencies that are a ff ected upon B isotope substitution in a given B-bearing system. This can be seen by comparing the pseudofrequencies computed for the selected cases of B-bearing molecules and crystalline solids considered here and reported in Table 3 with the real frequencies given in Table 1. This indicates that the two terms of opposite signs in Eq. 12 should be similar in value and cancel out to a great extent, so | ∆ β pseudo | << ∆ β BM . Therefore the approach proposed here to compute the β factor based on pseudofrequencies and Eq. 1 should give a better approximation to the exact β factors than equation 3, which we will show in section 3.1. The difference to the Bigeleisen & Mayer (1947) approxima-\n
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Table 4: The lattice parameters of the investigated B-bearing crystalline solids. Natoms is the number of atoms in the modeled supercell.
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\nReferences: 1 Liang et al. (1995), 2 Ertl et al. (2010), 3 Marler et al. (2002)\ntion is given by:\n∆ β = 1 + 3 ∑ i = 1 ¯ ui 12 ∆ ¯ ui -β pseudo (14)\nand can be easily computed for any considered system. We assume that the pseudofrequency-based approach to the computation of β factors is applicable if it is just a correction to equation 3, i.e. when:\n∆ β << 3 ∑ i = 1 ¯ ui 12 ∆ ¯ ui . (15)\nWe also note that the proposed approach satisfies the Redlich-Teller product rule (Redlich, 1935) when the mass of considered isotope m is much smaller than the mass of the whole considered system M , namely,\nNdof ∏ i = 1 u ∗ i ui = ( M ∗ M m m ∗ ) 3 / 2 /approxorequal ( m m ∗ ) 3 / 2 = 3 ∏ i = 1 ¯ u ∗ i ¯ ui . (16)\nWe notice that we could force the strict conservation of the Redlich-Teller product by just adjusting the ratios of ¯ ui / ¯ u ∗ i . However, such a modification would not preserve relation 11 and the pseudofrequency approach would not recover exactly the high temperature limit (Eq. 3) of the exact solution (Eq. 1), which is a more important constraint to fulfill strictly by the proposed approximation.\n2.2. Representation of solids\nIn this paper we investigate the boron isotope fractionation between dravite, olenite and boromuscovite minerals and aqueous fluids. The solids were represented by large cells containing at least 84 atoms. The number of atoms used in the crystal calculations together with the lattice parameters of modeled crystals are summarized in Table 4. The lattice parameters and chemical compositions of the modeled crystalline solids are the experimental values measured at\nambient conditions found in the literature. Dravite is the crystalline solid which was used in the experiments on tourmaline by Meyer et al. (2008). The chemical composition of the supercell used in the investigation is Na3Mg9Al18(Si18O54)(B [3] O3)9(OH)12 with structural data of Marler et al. (2002). Olenite can contain B in both trigonal and tetragonal sites. The modeled structure is that of Ertl et al. (2010). The chemical composition of the unit cell used in the investigation is NaAl3Al6(Si4B [4] 2 O18)(B [3] O3)3(OH)3O. For boromuscovite, the 1M and 2M1 crystal structures of Liang et al. (1995) were used. In the isotope fractionation experiments of Wunder et al. (2005) boromuscovite forms two polytypes, 1M and 2M1, with relative abundances of 10% and 90% respectively. In boromuscovite B occupies the 4-fold coordinated site occupied mainly by Si atoms. The constructed model constitutes a 2x1x1 supercell of elementary chemical composition KAl2(B [4] Si3O10)(OH)2.\n2.3. Representation of aqueous solution\nThe aqueous solution was represented by a periodically repeated box containing up to 64 water molecules and one H3BO3 or H4BO -4 molecule. The pressure and temperature conditions were chosen to be close to the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The pressure of the aqueous solution for a given temperature and volume was calculated according to the equation of state of Wagner & Pruss (2002). The ab initio molecular dynamics simulations (AIMD) of aqueous fluids were performed for fixed temperature and volume using the Car-Parrinello scheme (Car & Parrinello, 1985). The temperature during each run was controlled by a Nos'e-Hoover chain thermostat (Nos'e & Klein, 1983; Hoover, 1985). For each T -V conditions at least 10 ps long trajectories were generated with an integration step of 0 . 12 fs.\n\n\n\n\n\n
Figure 2: Left panel: β factors for H3BO3 (solid line) and H4BO -4 (dashed lines) in the gas phase. Right panel: fractionation factors between H3BO3 and H4BO -4 in gas phase. The thick lines represent our results, while the thin lines represent the results obtained using frequencies of Zeebe (2005).
\n\n2.4. Computational technique\nThe calculations of pseudofrequencies and β factors for solids and aqueous solutions were performed by applying density functional theory (DFT) methods, which are currently the most e ffi cient methods allowing for treating extended many particle systems quantummechanically. We used the planewave DFT code CPMD (Marx & Hutter, 2000), which is especially suited for ab initio simulations of fluids, the BLYP exchangecorrelation functional (Becke, 1988; Lee et al., 1988) and norm-conserving Goedecker pseudopotentials for the description of the core electrons (Goedecker et al., 1996). One advantage of using the BLYP functional is that it usually gives harmonic frequencies that most closely resemble the observed frequencies of benchmark chemical systems (Finley & Stephens, 1995; Alecu et al., 2010) 1 . The energy cut-o ff for the plane wave basis set was 70 Ryd for geometry relaxations and molecular dynamics simulations and 140 Ryd for computation of vibrational frequencies. Periodic boundary conditions were applied for both crystalline solids and aqueous solutions to preserve the continuity of the media.\nThe force constants and frequencies needed for the computation of the β factors were computed using the finite displacement scheme. Before performing the calculations of the crystal structures all atomic positions were relaxed to the equilibrium positions to minimize the forces acting on the atoms. We note that to compute the β factors for crystals one formally should account for phonon dispersion. Here we use large supercells and restrict our calculations to a single phonon wave-vector ( Γ ). Schauble (2011) has shown recently for 26 Mg / 24 Mg fractionation in Mg-bearing minerals that supercells containing more than 20 atoms are su ffi -cient to get very accurate β factors even at T = 300 K (error of 0.1 /permil ). At T = 1000 K the error is in the order of 0.01 /permil . The accuracy of the high temperature isotope fractionation factors computed on a single phonon wave-vector is also demonstrated for ironbearing minerals by Blanchard et al. (2009) and confirmed with good agreement of the predicted with the measured Li isotope fractionation factors between staurolite, spodumene, micas and aqueous fluid presented in our previous work (Kowalski & Jahn, 2011).\nPrior to the computation of the force constants and frequencies of boron atoms in the fluids the positions of all the atoms constituting the boron-carrying molecule (H3BO3 or H4BO -4 ) were relaxed to the equilibrium positions, while all other atomic positions remained unchanged. The full normal mode analyzes were per-\n\n\n\n\n\n
Figure 3: Left panel: β factors for H3BO3 (solid line) and H4BO -4 (dashed lines) in aqueous solution. Right panel: fractionation factors between H3BO3 and H4BO -4 in aqueous solution. The thick lines represent our results, while the thin lines represent the results of Sanchez-Valle et al. (2005) obtained using harmonic frequencies (their Table 2). The dotted line represents the corrected Sanchez-Valle et al. (2005) results. The correction is made by comparing the work of Rustad et al (2010b) and the correction derivation procedure is discussed in the text.
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Table 5: The 1000( β -1), α = β H3BO3 /β H4BO -4 and ∆ = 1000(ln β H3BO3 -ln β H4BO -4 ) factors computed for isolated H3BO3 and H4BO -4 at T = 300 K by Zeebe (2005) using di ff erent basis sets and our result obtained using the full normal mode spectrum and equation 1. The units are /permil .
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\nformed using the same method, but displacing all the atoms constituting the considered system. In the latter case the frequencies were obtained through the diagonalization of the full dynamical matrix (Schauble, 2004) as implemented in CPMD code. The e ff ect of the various approximations on the derived fractionation factors was studied by additional computations of H3BO3 and H4BO -4 isolated clusters. For that purpose we used a large, isolated simulation box with a cell length of 16 Å, forcing the charge density to be zero at the boundary, as implemented in CPMD code. In order to compute the β factors of boron species in the aqueous fluid we apply the same method as in our recent work on Li isotopes (Kowalski & Jahn, 2011), with the exception that we use the pseudofrequencies, i.e. the frequencies obtained from the three force constants acting on the fractionating element, and formula 1 for calculation of β factors, as discussed in section 2.1.2. In order to fully account for the spatial continuity of the fluid and\nits dynamical motion we produced 10 ps long molecular dynamics trajectories of systems consisting of 64 H2O molecules and one H3BO3 or H4BO -4 molecule for different T = 1000 K, 800 K and 600 K and pressure of 0 . 5 GPa, which closely resembles the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The corresponding simulation box length is 13 . 75 Å at T = 1000 K. The β factors were computed on the ionic configuration snapshots extracted uniformly in 0 . 1 ps intervals along the molecular dynamics trajectories.\n2.5. Error estimation technique\nThe errors in the computed value of the ( β -1) and ∆ fractionation factors were estimated from an average error of vibrational frequencies computed using the chosen DFT method. Finley & Stephens (1995), Menconi & Tozer (2002) and Alecu et al. (2010) estimated the errors made in calculations of vibrational frequencies of small molecules using di ff erent DFT functionals. According to these works the BLYP functional\nsystematically overestimates the harmonic frequencies by ∼ 3 . 5 %, with a deviation from the mean o ff set of ∼ 1 %. Therefore, we expect that using BLYP functional the ( β -1) and ∆ values are systematically overestimated by 7 % and that in addition there is a 2 % error in derived ( β -1) factors. Similar errors result from using other functionals or even more sophisticated and time consuming post-Hartree-Fock methods such as MP2 (Finley & Stephens, 1995; Alecu et al., 2010).\n3. Results and discussion\n3.1. Test of the computational method\nFirst, we illustrate the performance of the approximation proposed in section 2.1.2 by computing the β factors for the isolated H3BO3 and H4BO -4 molecules and selected crystalline solids. In Figure 1 we present three sets of calculations of β factors: (1) the 'exact' result obtained from a full normal mode analysis and formula 1, (2) the results obtained applying Kowalski & Jahn (2011) method based on Eq. 3, (3) the results obtained using pseudofrequencies computed for the fractionating element and Eq. 1 for the estimation of the β factor. The numerical values for selected temperatures are reported in Table 2. Approach (3) results in much better agreement with the 'exact' result. For H3BO3, the β factor is overestimated by only 0 . 5 /permil and 1 . 5 /permil for temperatures of 800 K and 600 K respectively. Applying method (2), the error is more pronounced, 2 . 2 /permil and 6 . 7 /permil respectively. In the case of molecular H4BO -4 , the errors using method (3) for the same temperatures are only 0 . 1 /permil and 0 . 5 /permil respectively. The same behavior is shown for dravite and boromuscovite crystalline solids that contain boron in the coordinative arrangement that resemble the configurations of aforementioned B-bearing molecules. For T > 600 K the proposed method represents only a few percent correction to the approximation given by equation (3), so the relation (15) is satisfied. It is evident that for B-bearing materials considered here the improvement made by using the pseudofrequencies based approach is substantial. It corrects for about 75% of error of the Bigeleisen & Mayer (1947) approximation (Eq. 3). However, the question of general applicability of the proposed method to other isotopic systems would require careful testing on a large set of materials, which is well beyond the scope of the current paper.\n3.2. B isotope fractionation in gas and fluid phases 3.2.1. B isotope fractionation between H3BO3 and H4BO -4 in the gas phase\nIn a first step of our investigation of boron-rich aqueous fluids we derived the full frequency spectra of\nmolecules in the gas phase (isolated molecules). The relevant β factors were computed using Equation 1. These studies were performed in order to compare our results with the published values of Zeebe (2005), both computed using the same DFT BLYP functional. In Table 1 we report the computed frequencies that are affected by the di ff erent B isotope substitutions along with other theoretical estimations and experimental measurements. The computed frequencies are in good agreement with earlier theoretical predictions and show similar agreement with the experimental measurements. The results in terms of computed β factors for the two considered species are reported in Figure 2, where we compare our results with the values computed using frequencies of Zeebe (2005). The comparison of the two sets of calculations reveals that our β factors for both species are smaller by ∼ 1 /permil at 600 K -1000 K than the values of Zeebe (2005). However, the di ff erence between β factors of H3BO3 and H4BO -4 remains nearly identical in both sets of calculations and the agreement is nearly perfect for higher temperatures. We note, that for the comparison we used the frequencies of Zeebe (2005) computed using 6-31 + G(d) basis set, as only these are provided by the authors. β and α factors obtained at T = 300 K using a more extended 6311 + G(d,p) basis set indicate that the β factors using 6-31 + G(d) basis set are not fully converged. In particular, the ( β -1) factor of H4BO -4 computed with 6311 + G(d,p) basis set is 3 . 9 /permil smaller than the one derived using 6-31 + G(d). In Table 5, we compare these results with the results of our calculation. It is clearly seen that for lower temperatures such as T = 300 K the values computed with 6-311 + G(d,p) basis set are in better agreement with our results indicating that planewave based DFT approach we use provides adequate vibrational frequencies and resulting isotope fractionation factors.\n3.2.2. B isotope fractionation between H3BO3 and H4BO -4 in aqueous fluid\nIn order to obtain the temperature dependent β factor for aqueous fluids we fitted the function 1 + A / T 2 + B / T 4 to the computed values using the least squares minimization procedure. The computed β values for H3BO3 in fluid are: 1 . 02366 ± 0 . 00012, 1 . 03624 ± 0 . 00018 and 1 . 06262 ± 0 . 00010and for H4BO -4 in fluid are: 1 . 01745 ± 0 . 00005, 1 . 02650 ± 0 . 00010and 1 . 04597 ± 0 . 00015, for the temperatures of 1000 K, 800 K and 600 K respectively. The resulting temperature dependent β factor for H3BO3 is β = 1 + 2 . 416 · 10 4 / T 2 -5 . 823 · 10 8 / T 4 and for H4BO -4 is β = 1 + 1 . 772 · 10 4 / T 2 -4 . 234 · 10 8 / T 4 .\nThe results for H3BO3 and H4BO -4 in aqueous so-\n\n\n\n\n\n
Figure 4: Left panel: pressure dependence of the β factor of neutral fluid (H3BO3) at T = 1000K. Dashed line is the linear regression fit to the calculated values (points): 1000( β -1) = 23 . 6 + 0 . 28 P (GPa); Right panel: The computed change of the β factor with pressure (symbols) in comparison to the increase in the β factor derived from the frequency shifts of the 666 cm -1 and 1454 cm -1 lines measured by Sanchez-Valle et al. (2005) (dashed line). The comparison is made assuming that ( β -1) ∼ ν 2 .
\n\nlutions are shown in Figure 3. As was observed for the isolated molecules, the β factor of H3BO3-bearing fluid is substantially larger than the one for the H4BO -4 . This can be understood in terms of the substantial difference in the B-O bond lengths exhibited by the two considered species. In case of isolated molecules our calculations indicate a B-O bond length of 1 . 40 Å for H3BO3 and 1 . 51 Å for H 4BO -4 . We compared our β factors with the values computed by Sanchez-Valle et al. (2005), which were derived by the combination of force field methods and experimental data to derive accurate vibrational frequencies. For H3BO3 we got a nearly identical result. In case of H4BO -4 our calculation predicts a value which is lower by 2 -4 /permil . However, Rustad et al (2010b) and Rustad & Bylaska (2007) revealed the improper assignment of a major fractionating vibrational mode of H4BO -4 in the force field by Sanchez-Valle et al. (2005). This leads to the underestimation of the fractionation factor between aqueous H3BO3 and H4BO -4 by Sanchez-Valle et al. (2005). Assuming that α ∝ T -2 and having the di ff erence between BLYP calculations of Rustad et al (2010b) and Sanchez-Valle et al. (2005) of ∆ α = 16 . 4 /permil at T = 300 K, the value reported by Sanchez-Valle et al. (2005) should be underestimated by ∆ α = 16 . 4 · 2(300 / T ) 2 /permil , which results in ∆ α ∼ 1 . 5 /permil at T = 1000 K. Corrected in such as way result of Sanchez-Valle et al. (2005) is\nalso plotted in Figure 3. It is now very consistent with our prediction.\n3.2.3. Discussion of computational errors\nMost previous computational studies of boron isotope fractionation in aqueous solutions concentrate on the computation of the isotope fractionation at ambient conditions (Rustad et al, 2010b; Rustad & Bylaska, 2007; Liu & Tossell, 2005; Zeebe, 2005). Rustad et al (2010b) performed detailed analysis of impact of the chosen computational method (HF, MP2, di ff erent DFT functionals) and size of the basis set on the calculated fractionation factors between H3BO3 and H4BO -4 . They found that DFT methods are not performing well for the borate system and concluded that DFT 'is of limited usefulness in chemically accurate predictions of isotope fractionation in aqueous systems' (Rustad et al, 2010b). The empirically derived error of the derived fractionation factor is of the order of 5 -10 /permil for a total fractionation of ∼ 30 /permil . We note that this is expected and clearly visible if we apply the error estimation procedure outlined in section 2.5. For instance, at room temperature the derived beta factors using the BLYP functional are 213 . 6 /permil and 173 . 3 /permil respectively (Rustad et al (2010b), Table 2). This gives a fractionation factor of 1.0343. Following our error estimation scheme, the absolute error of the fractionation factor is 10 . 5 /permil , and the properly reported computed value\n\n\n\n\n\n
Figure 5: Left panel: pressure dependence of the β factor of strongly basic fluid (H4BO -4 ) at T = 1000 K. Dashed line is the regression fit to the calculated values (points): 1000( β -1) = 17 . 15 + 0 . 754 P -0 . 027 P 2 , where pressure is given in GPa; Right panel: The computed change of the β factor with pressure in comparison to the increase in the β factor derived from the frequency shift of the 975 cm -1 line measured by Sanchez-Valle et al. (2005) (dashed line). The comparison is made assuming that ( β -1) ∼ ν 2 .
\n\nis α = 1 . 034 ± 0 . 011. When one corrects for the systematic error of 7% and assumes 2% of statistical error on β factors, then the value of α decreases and the error is slightly smaller, i.e. α = 1 . 032 ± 0 . 008. This is in good agreement with the experimental data reported in Rustad et al (2010b) and explains the spread of the values computed using di ff erent methods and reported in that paper. It is very di ffi cult to get the fractionation factors for ambient conditions, as the fractionation factor is often just a small fraction of the relevant ( β -1) factors, ( α -1) ∼ 0 . 15( β -1) in the considered case. Assuming that ( α -1) = 0 . 15( β -1), a 2% error in the ( β -1) factors leads to an absolute error in ( α -1) of 0 . 04( β -1) = 0 . 04( α -1) / 0 . 15 ∼ 0 . 27( α -1), i.e. ∼ 27% of relative error in the derived fractionation factor ( α -1). On the other hand, we note that such a big error is not substantially larger than the uncertainties in the experimental data reported by Rustad et al (2010b) in their Figure 2. Thus, the case of boron fractionation in aqueous fluid at ambient conditions does not necessarily show the limited usefulness of DFT in the prediction of isotope fractionation factors, but only reflects the fact that precise estimation or measurement of the B isotopes fractionation factors at ambient condition requires unprecedented accuracy of both experimental or computational techniques. For instance, in order to get the value of ( α -1) with a relative error of 5% (at ambient\nconditions) one needs to estimate the ( β -1) factors or measure relevant quantities with precision of less than 1%. At higher temperatures the situation is di ff erent. Looking just at the fractionation between H3BO3 and H4BO -4 in the gas phase or the aqueous solution one can see that for T > 600 K the fractionation factor between the two substances, ( α -1), is at least 25% of the ( β -1) factor. This results in smaller 0 . 04 / 0 . 25 ∼ 16% for T = 600 K and 0 . 04 / 0 . 36 ∼ 11% for T = 1000 K relative error, which is acceptable in our calculations. Nevertheless, this case shows the importance of proper error estimation on the computed fractionation factors. Such an estimation is usually omitted or not provided explicitly, which can lead to wrong conclusions when the theoretical prediction is confronted with the measured data.\n3.2.4. Pressure dependence of the fluid fractionation factor\nIn our recent paper (Kowalski & Jahn, 2011) we have shown that due to compression the β factor of Li in aqueous fluid increases with increase in pressure (for P > 2 GPa). The same should happen for H3BO3 and H4BO -4 aqueous fluid as the vibrational frequencies of boron species in aqueous fluid increase with increase in pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). Having the experimental data we checked whether the derived pressure-dependent β\nfactors are consistent with the pressure shifts of vibrational frequencies of considered boron species measured by Sanchez-Valle et al. (2005). For that purpose we performed a set of calculations using supercells containing 8 water molecules and the relevant boron species. We note that in line with our previous results for Li (Kowalski & Jahn, 2011), the obtained values of ( β -1) at P = 0 . 5 GPa are within 0 . 1 /permil in agreement with the values obtained for supercells containing 64 water molecules. The results are given in Figures 4 and 5. The computed ( β -1) values for H3BO3 fluid show a linear dependence in pressure, ( β -1) = 23 . 60 + 0 . 28 P (GPa) /permil . This is expected as ( β -1) ∝ ν 2 ∼ ν 2 0 + 2 ν 0 ∆ ν (Schauble, 2004) and ∆ ν is a linear function of pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). In case of H4BO -4 the pressure-dependence is linear up to P ∼ 2 -3 GPa and it becomes less steep at higher pressures. In order to quantitatively check the consistency of our prediction with the measured vibrational frequency shifts of Sanchez-Valle et al. (2005) we derived the relative shifts in the ( β -1) factor assuming that ( β -1) ∝ ν 2 and the measured pressure dependence of the frequency shifts: ∆ ν = 2 . 15 cm -1 · P (GPa) and ∆ ν = 3 . 50 cm -1 · P (GPa) for 1454 cm -1 and 666 cm -1 vibrational frequencies of H3BO3 and ∆ ν = 6 . 47 cm -1 · P (GPa) for the 975 cm -1 vibrational frequency of H4BO -4 . The chosen vibrational frequencies are the ones a ff ected by the di ff erent B isotope substitution. Our predicted shift of ( β -1) matches well the shifts derived from the measured frequency shifts. Such a good agreement with the experimental data validates further our computational approach and shows that ab initio calculations can be successfully used in derivation of the pressure dependence of the fractionation factors and pressure-induced vibrational frequencies shifts. Moreover, first principles calculations can be useful in extrapolation of the experimental values for β and ∆ ν to more extreme conditions, which otherwise are extremely di ffi cult to reach by experimental techniques.\n3.3. Fluid-mineral fractionation\nNext we present the results of the fractionation between boron bearing fluids and minerals such as dravite, olenite and boromuscovite. The aim of these studies is to investigate the mechanisms driving the fractionation process, the role of coordination and the B-O bond length. Below we discuss each case separately.\n3.3.1. Tourmaline-neutral fluid\nMeyer et al. (2008) measured the boron isotope fractionation between tourmaline and neutral fluid at T =\n\n\n
Figure 6: The fractionation factors between tourmaline and aqueous fluid. The solid line represents our prediction for the fractionation between dravite and H3BO3 neutral fluid and the shadowed region represent the computational uncertainty. The dotted line represents our prediction for the fractionation between olenite containing B [3] species only and neutral fluid. The computational error is similar in both cases. The data points are the values measured for dravite-fluid system by Meyer et al. (2008).
\n\n400 -700 o C and P = 0 . 2 GPa. In the experiment the tourmaline was represented by dravite. In contrast to the former measurements of Palmer et al. (1992) the measured fractionation is very small and does not exceed 2 . 5 /permil at 400 o C. Our calculated fractionation curve together with the experimental data are given in Figure 6. Our result correctly reproduces the experimental measurements within the computational accuracy. The dravite-fluid fractionation is small as the two materials contain boron in BO3 units. We also predict a small fractionation between olenite carrying 3-fold coordinated boron only and aqueous fluid, although the olenite-fluid fractionation is positive because of the shorter B-O bond lengths for olenite (1 . 378 Å vs. 1 . 397 Å).\n3.3.2. Boromuscovite-strongly basic fluid\nBoromuscovite synthesized in the experiments of Wunder et al. (2005) consisted of two type of polytypes, 1M ( ∼ 10 %) and 2M1 ( ∼ 90 %). In order to be consistent with the experimental conditions, we derived the β factors for both polytypes and computed their weighted average. We note that the β factors for both polytypes of mica are similar with the di ff erence in ( β -1) not larger than 3 %. This is consistent with similar B-O bond lengths (1 . 532 Å) found in both polytypes. Boromuscovite contains boron in tetrahedral sites. Therefore, in order to investigate the impact of the B-O bond length on the fractionation we first compare the fractionation between the mineral and a strongly basic fluid contain-\n\n\n
Figure 7: The fractionation factors between boromuscovite and basic aqueous fluid (fluid containing H4BO -4 ). The lines represent our results assuming the presence of boron species in form of H4BO -4 only (solid line) and mixture of 90% of H4BO -4 and 10% of H3BO3 (dotted line) in the fluid. The data points are the values measured by Wunder et al. (2005). The uncertainty of calculated values is indicated by shadowed area and is similar for both curves.
\n\nn in H4BO -4 . The result, together with the measurements of Wunder et al. (2005) of the fractionation between boromuscovite and strongly basic fluid, is summarized in Figure 7. Our calculations predict a negative fractionation between mica and the H4BO -4 fluid. The agreement of our prediction with the experimental measurements is relatively good; however, the experimental data indicate slightly stronger fractionation. We note that the experimental conditions of Wunder et al. (2005) do not assure that the measured basic fluid contained four-fold coordinated boron species, i.e. H4BO -4 , only. As we indicate in the Figure 7, the presence of as little as 10% of H3BO3 in the measured basic fluid brings the prediction and measurements into much better agreement. It makes sense that the β factor of boromuscovite is smaller than for aqueous H4BO -4 as the average B-O bond length in mica is 1 . 532 Å, while it is 1 . 513 Å and therefore shorter in case of aqueous H4BO -4 .\nWe note that in our derivation we assumed that the fluid consists mostly of [4] B species. As we already mentioned, although Wunder et al. (2005) call the fluid 'strongly basic' its exact composition, especially the amount of [3] B species is unknown. However, if for instance the [4] B to [3] B ratio was 1, then the predicted mica-basic fluid fractionation at 800 K would be 5 /permil larger than measured. This would result in a large inconsistency between the computed values and the experimental data. On the other hand, good agreement between the prediction and the measurements indicates that the strongly basic fluid was dominated by H4BO -4\n\n\n
Figure 8: The fractionation factors between boromuscovite and aqueous fluid. The data points are the values measured by Wunder et al. (2005). The solid line represents the result obtained for ambient pressure. The dashed line represents the result for fluid containing H3BO3 only obtained for P = 3GPa accounting for compression and thermal expansion, with the uncertainties in calculated values indicated by shadowed area. The dotted lines represent the results assuming different admixture of four-fold coordinated boron species (represented by H4BO -4 with abundance indicated in the figure) to the fluid. The computational error is comparable for all the results.
\n\nspecies, which is in line with previous studies (Zeebe, 2005; Sanchez-Valle et al., 2005).\n3.3.3. Boromuscovite-neutral fluid\nThe fractionation between boromuscovite and neutral fluid involves a change in coordination from B [4] in boromuscovite to B [3] in neutral fluid. Wunder et al. (2005) measured the fractionation between the two materials at 3 GPa, shown in Figure 8. The predicted fractionation is about 3 ± 2 . 5 /permil larger than the measured value. Looking for potential sources of this discrepancy, we have checked for the e ff ect of the change in lattice parameters due to combined thermal expansion and compression. For that purpose we applied the EOS of Holland and Powell (2011) for muscovite, which gives 4 . 4%, 3 . 8% and 3 . 1% decrease in volume for T = 600 K, 800 K and 1000 K respectively and P = 3 GPa. As muscovites show highly anisotropic compressibility patterns, in line with Comodi & Zanazzi (1995) we applied the T and P driven change in volume assuming the 16%, 19% and 65% contribution to compression along the /vector a , /vector b , /vector c lattice vectors. On the other hand ( β -1) factors of H3BO3 and H4BO -4 in aqueous solutions at P = 3 GPa increase by 3 . 5% and 11 . 8% respectively (Figures 4 and 5), leading to pressure-induced increase in the boromuscovite-aqueous fluid fractionation at the experimental pressure. The ∆ factor, corrected for e ff ects of thermal expansion and compression of\nboromuscovite and compression of fluid, is also given in Figure 8. Because the high T and P e ff ects result in similar increases in the β factors for both solid and aqueous fluid, the resulting fractionation factor between these two phases is close to the one derived at ambient conditions. Therefore, thermal expansion and compression e ff ects cannot explain the observed discrepancy between prediction and the measurements of Wunder et al. (2005).\nOn the other hand, the comparison of our results with the experimental data suggests that the fractionation between boromuscovite and fluid is the same as between H3BO3 and H4BO -4 fluids (see Figure 3), which is at odds with the non-negligible and negative fractionation between boromuscovite and a strongly basic fluid. In section 3.3.2 we have shown that we are able to correctly reproduce the fractionation between boromuscovite and strongly basic fluid, which indicates that our result for boromuscovite is reliable. This suggests that another, unaccounted e ff ect leads to the decrease of the boron isotope fractionation between mica and neutral fluid in the experiments of Wunder et al. (2005).\nOne possible solution for the discrepancy is a nonnegligible amount of boron residing in four-fold coordinated configurations in neutral solution. This is in line with the Raman spectroscopy measurements of Schmidt et al. (2005), who detected a broad peak in the Raman spectra of neutral H3BO3-dominated fluid and attributed it to B [4] species. The integrated area of this peak, compared to the peak of the Raman 877 cm -1 line of B [3] species, indicates the presence of at least 15 -30 % of B [4] species by mole fraction. Assuming that there is 15 -30 % of B [4] species present in the fluid and that the β factor of these species is similar to that of H4BO -4 , the fractionation factor between boromuscovite and H3BO3 aqueous fluid decreases bringing the theory and the experiment to better agreement, which is illustrated in Figure 8. If this interpretation is true, it suggests that boron isotope fractionation could be used to gather information on the speciation of B in aqueous fluids.\n3.4. B isotope fractionation between minerals\nThe boron isotope fractionation between B-bearing crystalline solids has received considerable attention recently (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002). We focus here on the investigation of boron isotope fractionation between mica and tourmaline as boron atoms in these minerals occupy sites of di ff erent coordination, which should result in a large B isotope fractionation between these two minerals. In mi-\n\n\n
Figure 9: The fractionation factors between mica and tourmaline. The solid line represents our value for fractionation between Bmuscovite and dravite. The dashed line is the experimental fractionation factor between tourmaline and mica determined by Wunder et al. (2005) and Meyer et al. (2008). The experimental error is 2 /permil . The dotted line is the experimental fractionation factor of Wunder et al. (2005) and Meyer et al. (2008) but corrected for the presence of B [4] species in the neutral fluid in the high P experiments of Wunder et al. (2005), as is discussed in the text. The diamonds are the data from natural samples taken from Klemme et al. (2011) and references herein. The uncertainties in calculated values are indicated by shadowed area.
\n\ncas boron substitutes for silicon in the four-fold coordinated site (Wunder et al., 2005), while in tourmaline (dravite) it is incorporated in the three fold coordinated site (Meyer et al., 2008). Comparing the B isotope fractionation between di ff erent minerals, melts and fluids Wunder et al. (2005) have shown that the fractionation between two materials of di ff erent B coordination is large, reaching 5 /permil at 1000 K and much higher values at lower temperatures. Klemme et al. (2011),Marschall (2005) and Hervig et al. (2002) measured the fractionation between coexisting phases of the two minerals in natural samples. The fractionation between these two minerals is also derived from experimental isotopic fractionation data of B-muscovite-fluid (Wunder et al., 2005) and tourmaline-fluid (Meyer et al., 2008) systems. The results of these measurements and our computed T -dependent fractionation curve are given in Figure 9. The first striking observation is that our predicted fractionation factors are much larger (taking the absolute value) than the experimental values (Wunder et al., 2005; Meyer et al., 2008). The latter are also incon-\nsistent with the natural samples data of Klemme et al. (2011) and previous studies discussed in that paper (Marschall, 2005; Hervig et al., 2002). On the other hand, the measurements on natural samples are consistent with our calculated values, which tends to validate our predictions. We notice that the most recent measurements of boron isotope signatures of tourmaline and white mica from the Broken Hill area in Australia by Klemme et al. (2011) indicate for the assumed temperature of 600 o C that the fractionation factor between the two phases is 10.4 ± 2 . 7 /permil , which is in good agreement with our computed value of 10 . 7 ± 1 . 8 /permil . The experimental mica-tourmaline B isotope fractionation factors of Wunder et al. (2005) and Meyer et al. (2008) are 2 /permil and 6 /permil smaller at temperatures of 1000 K and 800 K respectively, with an experimental uncertainty of 2 /permil . However, this discrepancy can be resolved by assuming that in the experiments of Wunder et al. (2005) the fluid contained a significant admixture of B [4] species, which leads to the underestimation of the experimental boromuscovite-fluid fractionation factor by ∼ 2 /permil at 1000 K and ∼ 3 . 5 /permil at 1000 K, as is seen in Figure 8. The experimental mica-tourmaline fractionation factor corrected for the presence of B [4] species is also plotted in Figure 9. It is now more consistent with the natural data. We note that this result independently supports the conclusion underlined in section 3.3.3 and result of Schmidt et al. (2005) that highP , B-bearing neutral fluids contain significant admixtures of B [4] species.\nOlenite is a mineral which can incorporate boron in both trigonal and tetrahedral sites as it substitutes for both Al and Si atoms. It is therefore interesting to check the fractionation of boron isotopes between the two differently coordinated sites in one mineral and compare it with the above result for mica and tourmaline. The computed fractionation between the trigonal and tetrahedral sites at 600 o C is 10 . 6 ± 1 . 9 /permil , which is consistent with the fractionation between tourmaline and mica, indicating that the coordination of the B atom is the driving factor for the fractionation of the B isotopes. Similarly, we computed the boron isotopes fractionation between trigonal and tetragonal boron sites in dravite. In order to create the tetragonal B site we replaced one Si atom with B and we added one H atom forming an additional OH group to compensate the charge. The computed fractionation between the sites at 600 o C is 8 . 9 ± 1 . 7 /permil , which is also in agreement with the aforementioned results. Next, we will show that the value of the β factor depends not only on coordination but is also strongly correlated with B-O bond length.\n\n\n
Figure 10: β factors for various considered materials as a function of temperature. The materials are indicated on the right side. Their order reflects the value of the β factor at 1000 K from the largest (top) to the smallest (bottom).
\n\n3.5. Fractionation between B [3] and B [4] materials\nThe β factors computed for all the considered materials are grouped together in Figure 10. β factors can be grouped into two sets, one that includes materials with boron in three-fold coordination and another one that includes materials having boron in four-fold coordination. For olenite and dravite we also computed the β factors with boron sitting on four-fold coordinated site. The β factor for these crystalline solids with given B [3] / B [4] ratio can be derived as a weighted average of the β factors obtained for boron sitting on the two di ff erently coordinated sites. The fractionation factor between materials of di ff erent boron coordination is ∼ 8 /permil on average at T = 1000 K. We note that it is ∼ 3 /permil larger than the one deduced by Wunder et al. (2005) from measurements performed on solids, silicate melts and fluids, but this can be attributed to the underestimation of the fractionation factors for boromuscovite-fluid system by Wunder et al. (2005) due to potential admixture of B [4] species in the investigated fluid.\nOur results show also a substantial spread of β factors of substances containing boron of a given coordination. The spread is at least 4 /permil and results from di ff erent B-O bond lengths. We illustrate this in Figure 11 by plotting together the β factors derived for all considered materials at T = 1000 K as a function of B-O bond length. It is clearly seen that there is a roughly linear correlation between the β factor and BO bond length, which is especially evident comparing the results for crystalline solids. For instance, out of the considered B [4] -bearing minerals boromuscovite has the\n\n\n
Figure 11: The β factor at T = 1000K for various considered materials as a function of B-O bond length. Filled circles represent the values obtained for boron in trigonal sites and open circles represent the values obtained for tetragonal sites.
\n\nlongest B-O bond length of 1 . 525 Å (1M) and 1 . 516 Å (2M1), followed by dravite 1 . 514 Å and olenite with BObond length of 1 . 502 Å. This tracks the di ff erences in the β factors derived for these materials. In addition the materials having B [3] species only exhibit shorter bond lengths of ∼ 1 . 37 Å and higher β factors, while the materials containing B [4] species having bond lengths of about ∼ 1 . 52 Å show much smaller β factors. Therefore, the tighter bonding of B [3] species likely explains why the heavy B isotope prefers the less coordinated phases. This clearly shows that the change in the B-O bond length during an isotope exchange is the leading factor driving the production of the boron equilibrium isotope signatures at high T .\n4. Conclusions\nIn this work we have presented a detailed analysis of boron isotope fractionation between boronbearing crystalline solids and aqueous fluids at high T and P conditions. In order to perform our investigation we have applied and extended a computationally e ffi cient approach for the computation of isotope fractionation factors for complex minerals and fluids at high temperatures and pressures presented by Kowalski & Jahn (2011). As an extension to the\nBigeleisen & Mayer (1947) 'single atom approximation' method we demonstrated that using the pseudofrequencies derived from the force constants acting on the fractionating element together with the full formula for computation of the reduced partition function ratios results in significant improvement in the accuracy of the computed fractionation factors, which is essential when lower temperature materials and high vibrational frequency complexes are considered.\nIn order to understand the fractionation between Bbearing crystalline solids and aqueous fluids we performed a set of calculations of β factors for dravite, olenite, boromuscovite and aqueous solutions of H3BO3 and H4BO -4 . In agreement with the experimental findings we show that the fractionation strongly correlates with coordination through the change in the B-O bond length. The lower trigonal coordination BO3 arrangement results in higher 11 B / 10 B (by ∼ 8 /permil at T = 1000 K) than the tetrahedrally coordinated boron complexes, which exhibit ∼ 0 . 15 Å longer B-O bonds. The computed fractionation between minerals and fluids of the same coordination are in good agreement with experiments. However, we predict larger isotope fractionation between boromuscovite and H3BO3 fluid (by at least a few /permil ) than was measured in situ at high P by Wunder et al. (2005) and Meyer et al. (2008), but that is consistent with measurements on natural samples. We note that the presence of B [4] in highP fluid could reconcile the in situ experimental results with our prediction and other measurements. This is expected from the experiments of Schmidt et al. (2005), but requires further experimental confirmation. If true, this would open the possibility for using the isotope fractionation techniques as a tool to measure the speciation of boron in fluids and crystalline solids. We have also demonstrated that with our computational approach we are able to correctly predict the pressure-induced isotope fractionation for compressed aqueous fluids, which indicates the ability of ab initio methods to predict the isotopic signatures of highly compressed materials, even those that are difficult to investigate experimentally.\nOur study confirms that ab initio computer simulations are a useful tool not only for prediction but also understanding the equilibrium stable isotope fractionation processes between various phases, including aqueous solutions, at high pressures and temperatures. They can nicely complement experimental e ff orts, provide unique insight into the isotope fractionation process on the atomic scale and deliver data for conditions that are inaccessible by the current experimental techniques.\nAcknowledgements\nThe authors wish to acknowledge financial support in the framework of DFG project no. JA 1469 / 4-1. Part of the calculations were performed on the IBM BlueGene / P JUGENE of the John von Neumann Institute for Computing (NIC). We are also grateful the associate editor Edwin A. Schauble and anonymous referees for constructive comments that helped improving the manuscript.\nReferences\nAlecu I. M., Zheng J. , Zhao Y. and Truhlar D. G. (2010) Computational Thermochemistry: Scale Factor Databases and Scale Factors for Vibrational Frequencies Obtained from Electronic Model Chemistries. J. Chem. Theory Comput. 6 , 2872-2887.\nAndrews, L., Burkholder, T.R., (1992) Infrared spectra of molecular B(OH)3 and HOBO in solid argon, J. Chem. 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A 105 , 602613.\n\nZeebe R. E., (2005) Stable boron isotope fractionation between dissolved B(OH)3 and B(OH) -4 Geochim. Cosmochim. Acta 69 , 27532766\nZeebe R. E. (2009) Hydration in solution is critical for stable oxygen isotope fractionation between carbonate ion and water. Geochim. Cosmochim. Acta 73 , 5283-5291.\nZeebe R. E. (2010) A new value for the stable oxygen isotope fractionation between dissolved sulfate ion and water. Geochim. Cosmochim. Acta 74 , 818-828.\n"},"sections":{"kind":"list like","value":[{"title":"Ab initio prediction of equilibrium boron isotope fractionation between minerals and aqueous fluids at high P and T","content":"Piotr M. Kowalski 1 , 2 ∗ , Bernd Wunder 1 and Sandro Jahn 1 1 GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany 2 Forschungszentrum Julich, Institute of Energy and Climate Research (IEK-6), Wilhelm-Johnen-Strasse, 52425 Julich, Germany","pages":[1]},{"title":"Abstract","content":"Over the last decade experimental studies have shown a large B isotope fractionation between materials carrying boron incorporated in trigonally and tetrahedrally coordinated sites, but the mechanisms responsible for producing the observed isotopic signatures are poorly known. In order to understand the boron isotope fractionation processes and to obtain a better interpretation of the experimental data and isotopic signatures observed in natural samples, we use first principles calculations based on density functional theory in conjunction with ab initio molecular dynamics and a new pseudofrequency analysis method to investigate the B isotope fractionation between B-bearing minerals (such as tourmaline and micas) and aqueous fluids containing H3BO3 and H4BO -4 species. We confirm the experimental finding that the isotope fractionation is mainly driven by the coordination of the fractionating boron atoms and have found in addition that the strength of the produced isotopic signature is strongly correlated with the B-O bond length. We also demonstrate the ability of our computational scheme to predict the isotopic signatures of fluids at extreme pressures by showing the consistency of computed pressure-dependent β factors with the measured pressure shifts of the B-O vibrational frequencies of H3BO3 and H4BO -4 in aqueous fluid. The comparison of the predicted with measured fractionation factors between boromuscovite and neutral fluid confirms the existence of the admixture of tetrahedral boron species in neutral fluid at high P and T found experimentally, which also explains the inconsistency between the various measurements on the tourmaline-mica system reported in the literature. Our investigation shows that the calculated equilibrium isotope fractionation factors have an accuracy comparable to the experiments and give unique and valuable insight into the processes governing the isotope fractionation mechanisms on the atomic scale. Keywords: stable isotope fractionation, boron, fluid speciation, DFT, molecular dynamics","pages":[1]},{"title":"1. Introduction","content":"The use of isotopes as geochemical tracers depends upon the existence of reliable P -T -dependent equilibrium isotope fractionation data between solids, fluids and melts. The common method used for determination of such data is to conduct experiments, where the phases of interest are equilibrated at a range of di ff erent conditions and individually measured for their equilibrium isotopic compositions. Recently, with the development of computational methods, software and increase in hardware performance it is also possible to simulate and compute the isotope fractionation with ab initio methods. These computational considerations complement the experimental e ff ort and provide information on the mechanisms governing the equilibrium isotope fractionation processes on the atomic scale. By establishing an e ffi cient computational approach for materials at high P and T and testing its reliability by computing Li isotope fractionation between minerals and aqueous fluids, Kowalski & Jahn (2011) have shown that ab initio methods can provide a reliable estimation of equilibrium isotope fractionation factors at an accuracy level comparable to experiments. Motivated by the encouraging results for the fractionation of lithium isotopes we applied our new method to study boron isotope fractionation. The main goals of the present study are to make theoretical predictions and obtain a better understanding of the B isotope fractionation process between tourmaline, B-bearing mica and flu- ids at various pressures and temperatures, to compare these data to results from recent in situ experimental studies (Wunder et al., 2005; Meyer et al., 2008) and measurements of isotopic signatures in natural samples (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002) and to investigate the underlying mechanisms driving the boron isotope fractionation processes between the considered materials. With two stable isotopes 10 B and 11 B, of relatively large mass di ff erence of about 10 %, boron isotopes strongly fractionate during geological processes, thus leading to natural δ 11 B-variations ranging from -30 to + 60 /permil (Barth, 1993). Therefore, B isotopes are ideal for the distinction of di ff erent geological environments and for quantifying mass transfer processes, e.g. in the range of subduction zones. First-order criteria driving isotope fractionation in Earth materials are di ff erences in coordination and in the bonding environments of coexisting phases. The lighter isotope usually preferentially occupies the higher coordinated site, which is generally accompanied with a longer cation-anion bond length and weaker bond strength (Schauble et al., 2009; Wunder et al., 2011). Tourmaline has an extensive chemical variability and is the most widespread borosilicate in various rocks over a large range of bulk compositions. It has a large P -T -stability which ranges from surface conditions to high pressures and temperatures of at least 7.0 GPa and about 1000 o C as determined experimentally for dravitic tourmaline (Krosse, 1995). Most tourmaline minerals contain 3 boron atoms per formula unit (pfu) with B in trigonal planar coordination (B [3] ). High pressure combined with an Al-rich environment can lead to the formation of olenitic tourmaline with significant amounts of excess B substituting for Si at the tetrahedral site (B [4] ). The highest amounts of up to 1.2 B [4] pfu have been found in olenitic tourmaline from Koralpe, Austria (Ertl et al., 1997). Al-rich tourmaline with up to 2.2 B [4] pfu was synthesized experimentally at 2 . 5 GPa, 600 o C (Schreyer et al., 2000). The maximum possible amount of B [4] in olenitic tourmaline is limited to three B [4] pfu, due to structural and crystal chemical reasons. As tourmaline is not stable at basic pH (Morgan & London, 1989), the dominant B-species of fluids coexisting with tourmaline is B(OH)3. Therefore, due to the absence of change in boron coordination, B-isotopic fractionation between B [4] -free tourmaline and fluid ( ∆ 11 B(tour-fl)) should be small. However, at 0 . 2 GPa, the experimentally determined ∆ 11 B(tour-fl)values of -2 . 5 ± 0 . 4 /permil at 400 o C and -0 . 4 ± 0 . 4 /permil at 700 o C (Meyer et al., 2008) suggest small but significant di ff erences in the B-O(H) bond strength between tourmaline and the neutral fluids. Incorporation of B [4] into tourmaline is expected to significantly increase the fractionation of boron isotopes between the mineral and aqueous fluid (i.e. increase of | ∆ 11 B | (tour-fl)). In this contribution we present calculated P -T -dependent data on B-isotope fractionation between B [4] -bearing tourmaline and fluid, which so far are not available from experiments. Despite of its low boron content of up to maximum values of 270 ppm (Domanik et al., 1993), potassic white mica is probably the main host of boron in metasedimentary and metabasaltic blueschists and eclogites, because of its high modal abundance and B-incompatibility in all other stable minerals of these rocks (Brenan et al., 1998). During subduction the modal amount of mica continuously decreases by dehydration reactions and the chemistry of residual micas is shifted towards phengitic compositions. Phengitic mica has an extended stability, extending as deep as 300 km within cold subduction zones (Schmidt, 1996). In contrast to most tourmalines, boron is tetrahedrally coordinated in mica, where it substitutes for aluminum. Due to the coordination change from mostly three-fold coordinated in near-neutral fluids (Schmidt et al., 2005) to B [4] in mica, B-isotopic fractionation between Bbearing mica and such fluids is much larger than for tourmaline - fluid. ∆ 11 B(mica-fl)-values determined experimentally at P = 3 . 0 GPa, are -10 . 9 ± 1 . 3 /permil at 500 o C and -7 . 1 ± 0 . 5 /permil at 700 o C (Wunder et al., 2005). Such a strong B-isotope fractionation and its pronounced T -dependence, in combination with the continuous dehydration of micas during ongoing subduction and boron transport via fluids into mantle wedge regions of arc magma-formation, probably determines the boron variations and B-isotopic signatures in volcanic arcs (Wunder et al., 2005). Using in situ Raman spectroscopic measurements of near-neutral B-bearing fluids, Schmidt et al. (2005) observed a significant amount of 4-fold B-species at high P and T . The abundance of these species increases with temperature and pressure and at T = 800 K and P = 1 . 9 GPa there should be a considerable amount (15 -30%) of B [4] species in the fluid. Such a significant amount of tetrahedrally coordinated B-species in high temperature and pressure fluids should a ff ect solid-fluid B-isotopic fractionation, which we investigate in our calculations. In the light of this, we also discuss recently determined B-isotope data from coexisting natural tourmaline and B-bearing mica (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002), which show slight inconsistency with in situ measurements of Wunder et al. (2005) and Meyer et al. (2008). Reliable ab initio computational methods to predict isotope fractionation factors have been established recently. Several groups have proved that such calculations can contribute towards understanding geochemical mechanisms responsible for production of isotope signatures (Driesner, 1997; Yamaji et al., 2001; Schauble, 2004; Domagal-Goldman et al., 2008; Hill & Schauble, 2008; Meheut et al., 2009, 2007; Schauble et al., 2009; Zeebe, 2009; Hill et al., 2010; Rustad et al., 2010a; Rustad et al, 2010b; Zeebe, 2009, 2010; Kowalski & Jahn, 2011). The majority of these works, however, concentrate on the computation of the stable isotope fractionation between various, mostly simple crystalline minerals, and the aqueous solutions are usually computed using an isolated cluster containing fractionating species and a hydration shell (e.g. Zeebe (2010, 2009); Hill et al. (2010); Rustad et al. (2010a); Domagal-Goldman et al. (2008); Hill & Schauble (2008); Rustad et al (2010b); Zeebe (2005)). However, aqueous solutions at high pressure and temperature must be computed with caution as discussed in Kowalski & Jahn (2011). This is because the distribution of cation coordination and cation-oxygen bond lengths that a ff ect the isotope fractionation (Bigeleisen & Mayer, 1947) is driven by the dynamics of the system and change under compression (Jahn & Wunder, 2009; Wunder et al., 2011; Kowalski & Jahn, 2011). The only recent ab initio work, besides Kowalski & Jahn (2011), that accounts for the dynamical e ff ects on the isotope fractionation in fluid is by Rustad & Bylaska (2007) who considered boron equilibrium isotope fractionation between B(OH)3 and B(OH) -4 species in aqueous solution. They performed ab initio molecular dynamics simulations of these fluids and attempted to use the vibrational density of states, derived through the Fourier transform of the velocity auto-correlation function, as an input for the calculation of the 11 B / 10 B isotope fractionation coe ffi -cient. However, the resulting fractionation factor α = 0 . 86 happened to be much lower than the experimental value of α = 1 . 028. Interestingly, the discrepancy between experiment and theory is cured by quenching the selected configurations along the molecular dynamics trajectory and computing the harmonic frequencies. The fractionation factor derived using these frequencies exactly reproduces the experimental value. In our approach both solids and fluids are treated as extended systems by application of periodic boundary conditions in all three spatial directions, which is crucial to model high pressure materials. Large enough supercells are chosen to avoid significant interaction between atoms and their periodic images, as well as to reduce the number of k-points and q-vectors for sampling the Brillouin zones and the phonon spectra of the crystals, respectively (for a liquid or fluid, both the Brillouin zone and phonons are not defined). In our investigation we use cells of at least 7 Å width in each spatial dimension. A representative statistical sampling of the fluid structure is obtained by performing Car-Parrinello molecular dynamics simulations (Car & Parrinello, 1985). For the calculation of the isotope fractionation factors in fluids, several random snapshots from the simulation runs are chosen. The force constants acting on the fractionating element and the resulting fractionation factors are then obtained for each ionic configuration, and the relevant fractionation factor for the boron species in the fluid is computed as an average over the whole set of considered geometries. In Kowalski & Jahn (2011) it was shown that in line with the Bigeleisen & Mayer (1947) approximation, considering the force constants acting on the fractionating atom only leads to a satisfactory estimation of the Li isotope fractionation factors for high temperature fluids and minerals. Here, we will show that approximating the vibrational spectrum by the three pseudofrequencies derived from the force constants allows for further improvement of the accuracy of the predicted isotope fractionation factors, especially at lower temperatures. In this contribution we present the theoretical prediction of B isotope fractionation factors between B bearing aqueous fluids and solids, specifically tourmaline and B-muscovite, for which experimental data and measurements on natural samples are available for comparison (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011). We will show that the application of ab initio methods to B-bearing crystalline solids and fluids not only provides unique insight into the mechanisms driving equilibrium B-isotope fractionation on the atomic scale, but helps in proper interpretation of the data.","pages":[1,2,3]},{"title":"2.1. Theoretical model","content":"2.1.1. The single atom approximation: & Bigeleisen Mayer (1947) approach Mass-dependent equilibrium isotope fractionation is driven by the change in molecular and crystalline vibration frequencies resulting from the di ff erent masses of the isotopes. The fractionation between species and an ideal monoatomic gas is called the β factor. In the harmonic approximation it is given by the formula References: 1 Zeebe (2005), 2 Liu & Tossell (2005), 3 Rustad et al (2010b), 4 Sanchez-Valle et al. (2005), 5 Gilson (1991), 6 Oi (2000); Liu & Tossell (2005), 7 Andrews & Burkholder (1992), 8 Ogden & Young (1988), 9 Zeebe (2005) and references hereafter. (Bigeleisen & Mayer, 1947; Urey, 1947; Chacko et al., 2001): where u = h ν i / kBT , h is the Planck constant, ν i is the vibrational frequency of the i -th degree of freedom, kB is the Boltzmann constant, Ndof is the number of degrees of freedom, which for N being the number of atoms in the considered system (molecule, mineral or fluid) is equal to 3 N -5 for a diatomic molecule, 3 N -6 for multiatomic molecules and 3 N for crystals, and a star symbol marks the heavier isotope. The fractionation factor between two substances A and B, α A -B is computed as the ratio of the relevant β factors, which is well approximated by the di ff erences in the β factors: The calculation of the β factor requires only knowledge of the vibrational properties of the considered system computed for the two di ff erent isotopes. However, computation of the whole vibrational spectra of complex, multiparticle minerals or fluids requires substantial computational resources and is currently limited to systems containing a few dozens of atoms or less. In our recent work (Kowalski & Jahn, 2011) we proposed to use an e ffi cient method for computing the high temperature isotope fractionation factors between complex materials such as fluids and crystalline solids, which requires the knowledge of the force constants acting upon the fractionating element only. The β factor (Eq. 1) can be then approximated by (Bigeleisen & Mayer, 1947; Kowalski & Jahn, 2011): where Ai are the force constants acting on the isotopic atom in the three perpendicular spatial directions (x, y and z), ∆ m = m ∗ -m , where m and m ∗ are the masses of the lighter and heavier isotopes of the fractionating element. As the computation of the β factors from formula 3 requires the knowledge of properties of the fractionating element only we will call such an approach the single atom approximation throughout the paper. The validity criteria restricts the usage of the formula to frequencies ν [cm -1 ] /lessorsimilar 1 . 39 T [K] (assuming u < 2, see Fig. 1 of Bigeleisen & Mayer (1947)). We are interested in temperature range 800-1000 K. The highest vibrational frequency of the modes involving movement of B atoms for H3BO3 is ∼ 1400 cm -1 and of H4BO -1 4 is < 1200 cm -1 (Table 1). In case of H3BO3, the single atom approximation may produce an error of 2 . 2 /permil in the β factor at T = 800 K. The relevant error for H4BO -1 4 is 0 . 8 /permil (Table 2). A further improvement to the method is therefore desired.","pages":[3,4,5]},{"title":"2.1.2. The single atom approximation with pseudofrequencies: our improvement","content":"We will show that the error of the single atom approximation can be substantially reduced if one uses the three frequencies ¯ ν i derived from the force constants acting on the fractionating element (¯ ν 2 i = Ai / 4 π 2 m ). We call them 'pseudofrequencies' , and compute the β factors using formula 1. In the following we present the formal justification of such an approach. According to Bigeleisen & Mayer (1947), equation 1 for small ∆ ui = ui -u ∗ i reduces to (Bigeleisen & Mayer (1947), Eq. 11a): The Taylor expansion of the function appearing under the summation sign is: When we consider just the first term of the expansion the β factor is: which is exactly equation 3. Let us consider the Taylor expansion of the di ff erent estimations of β factors. Equation 4 then reads: The Bigeleisen & Mayer (1947) approximation given by equation 3 reads: and the proposed approximation based on pseudofrequencies is: In the above equation ¯ ui = h ¯ ν i / kBT . Next we check how this approximation compares to the Bigeleisen & Mayer (1947) approximation given by equations 3 and 8. In order to make a comparison we derive the di ff erences between the two approximate expressions β BM , β pseudo and the exact one β exact (Eq. 7). In the case of the Bigeleisen & Mayer (1947) approximation we have: and having from equations 3 and 8 that in the case of the proposed approximation we get: Because relation G ( u ) < u 12 holds for any u (see Bigeleisen & Mayer (1947), Fig. 1), the function and ∆ β pseudo < ∆ β BM . On the other hand the expression for ∆ β pseudo is given by a di ff erence of the higher order terms of the Taylor expansions of the two expressions for the β factor. In the considered cases the values of pseudofrequencies are similar to the real frequencies that are a ff ected upon B isotope substitution in a given B-bearing system. This can be seen by comparing the pseudofrequencies computed for the selected cases of B-bearing molecules and crystalline solids considered here and reported in Table 3 with the real frequencies given in Table 1. This indicates that the two terms of opposite signs in Eq. 12 should be similar in value and cancel out to a great extent, so | ∆ β pseudo | << ∆ β BM . Therefore the approach proposed here to compute the β factor based on pseudofrequencies and Eq. 1 should give a better approximation to the exact β factors than equation 3, which we will show in section 3.1. The difference to the Bigeleisen & Mayer (1947) approxima- References: 1 Liang et al. (1995), 2 Ertl et al. (2010), 3 Marler et al. (2002) tion is given by: and can be easily computed for any considered system. We assume that the pseudofrequency-based approach to the computation of β factors is applicable if it is just a correction to equation 3, i.e. when: We also note that the proposed approach satisfies the Redlich-Teller product rule (Redlich, 1935) when the mass of considered isotope m is much smaller than the mass of the whole considered system M , namely, We notice that we could force the strict conservation of the Redlich-Teller product by just adjusting the ratios of ¯ ui / ¯ u ∗ i . However, such a modification would not preserve relation 11 and the pseudofrequency approach would not recover exactly the high temperature limit (Eq. 3) of the exact solution (Eq. 1), which is a more important constraint to fulfill strictly by the proposed approximation.","pages":[5,6,7]},{"title":"2.2. Representation of solids","content":"In this paper we investigate the boron isotope fractionation between dravite, olenite and boromuscovite minerals and aqueous fluids. The solids were represented by large cells containing at least 84 atoms. The number of atoms used in the crystal calculations together with the lattice parameters of modeled crystals are summarized in Table 4. The lattice parameters and chemical compositions of the modeled crystalline solids are the experimental values measured at ambient conditions found in the literature. Dravite is the crystalline solid which was used in the experiments on tourmaline by Meyer et al. (2008). The chemical composition of the supercell used in the investigation is Na3Mg9Al18(Si18O54)(B [3] O3)9(OH)12 with structural data of Marler et al. (2002). Olenite can contain B in both trigonal and tetragonal sites. The modeled structure is that of Ertl et al. (2010). The chemical composition of the unit cell used in the investigation is NaAl3Al6(Si4B [4] 2 O18)(B [3] O3)3(OH)3O. For boromuscovite, the 1M and 2M1 crystal structures of Liang et al. (1995) were used. In the isotope fractionation experiments of Wunder et al. (2005) boromuscovite forms two polytypes, 1M and 2M1, with relative abundances of 10% and 90% respectively. In boromuscovite B occupies the 4-fold coordinated site occupied mainly by Si atoms. The constructed model constitutes a 2x1x1 supercell of elementary chemical composition KAl2(B [4] Si3O10)(OH)2.","pages":[7]},{"title":"2.3. Representation of aqueous solution","content":"The aqueous solution was represented by a periodically repeated box containing up to 64 water molecules and one H3BO3 or H4BO -4 molecule. The pressure and temperature conditions were chosen to be close to the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The pressure of the aqueous solution for a given temperature and volume was calculated according to the equation of state of Wagner & Pruss (2002). The ab initio molecular dynamics simulations (AIMD) of aqueous fluids were performed for fixed temperature and volume using the Car-Parrinello scheme (Car & Parrinello, 1985). The temperature during each run was controlled by a Nos'e-Hoover chain thermostat (Nos'e & Klein, 1983; Hoover, 1985). For each T -V conditions at least 10 ps long trajectories were generated with an integration step of 0 . 12 fs.","pages":[7]},{"title":"2.4. Computational technique","content":"The calculations of pseudofrequencies and β factors for solids and aqueous solutions were performed by applying density functional theory (DFT) methods, which are currently the most e ffi cient methods allowing for treating extended many particle systems quantummechanically. We used the planewave DFT code CPMD (Marx & Hutter, 2000), which is especially suited for ab initio simulations of fluids, the BLYP exchangecorrelation functional (Becke, 1988; Lee et al., 1988) and norm-conserving Goedecker pseudopotentials for the description of the core electrons (Goedecker et al., 1996). One advantage of using the BLYP functional is that it usually gives harmonic frequencies that most closely resemble the observed frequencies of benchmark chemical systems (Finley & Stephens, 1995; Alecu et al., 2010) 1 . The energy cut-o ff for the plane wave basis set was 70 Ryd for geometry relaxations and molecular dynamics simulations and 140 Ryd for computation of vibrational frequencies. Periodic boundary conditions were applied for both crystalline solids and aqueous solutions to preserve the continuity of the media. The force constants and frequencies needed for the computation of the β factors were computed using the finite displacement scheme. Before performing the calculations of the crystal structures all atomic positions were relaxed to the equilibrium positions to minimize the forces acting on the atoms. We note that to compute the β factors for crystals one formally should account for phonon dispersion. Here we use large supercells and restrict our calculations to a single phonon wave-vector ( Γ ). Schauble (2011) has shown recently for 26 Mg / 24 Mg fractionation in Mg-bearing minerals that supercells containing more than 20 atoms are su ffi -cient to get very accurate β factors even at T = 300 K (error of 0.1 /permil ). At T = 1000 K the error is in the order of 0.01 /permil . The accuracy of the high temperature isotope fractionation factors computed on a single phonon wave-vector is also demonstrated for ironbearing minerals by Blanchard et al. (2009) and confirmed with good agreement of the predicted with the measured Li isotope fractionation factors between staurolite, spodumene, micas and aqueous fluid presented in our previous work (Kowalski & Jahn, 2011). Prior to the computation of the force constants and frequencies of boron atoms in the fluids the positions of all the atoms constituting the boron-carrying molecule (H3BO3 or H4BO -4 ) were relaxed to the equilibrium positions, while all other atomic positions remained unchanged. The full normal mode analyzes were per- formed using the same method, but displacing all the atoms constituting the considered system. In the latter case the frequencies were obtained through the diagonalization of the full dynamical matrix (Schauble, 2004) as implemented in CPMD code. The e ff ect of the various approximations on the derived fractionation factors was studied by additional computations of H3BO3 and H4BO -4 isolated clusters. For that purpose we used a large, isolated simulation box with a cell length of 16 Å, forcing the charge density to be zero at the boundary, as implemented in CPMD code. In order to compute the β factors of boron species in the aqueous fluid we apply the same method as in our recent work on Li isotopes (Kowalski & Jahn, 2011), with the exception that we use the pseudofrequencies, i.e. the frequencies obtained from the three force constants acting on the fractionating element, and formula 1 for calculation of β factors, as discussed in section 2.1.2. In order to fully account for the spatial continuity of the fluid and its dynamical motion we produced 10 ps long molecular dynamics trajectories of systems consisting of 64 H2O molecules and one H3BO3 or H4BO -4 molecule for different T = 1000 K, 800 K and 600 K and pressure of 0 . 5 GPa, which closely resembles the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The corresponding simulation box length is 13 . 75 Å at T = 1000 K. The β factors were computed on the ionic configuration snapshots extracted uniformly in 0 . 1 ps intervals along the molecular dynamics trajectories.","pages":[8,9]},{"title":"2.5. Error estimation technique","content":"The errors in the computed value of the ( β -1) and ∆ fractionation factors were estimated from an average error of vibrational frequencies computed using the chosen DFT method. Finley & Stephens (1995), Menconi & Tozer (2002) and Alecu et al. (2010) estimated the errors made in calculations of vibrational frequencies of small molecules using di ff erent DFT functionals. According to these works the BLYP functional systematically overestimates the harmonic frequencies by ∼ 3 . 5 %, with a deviation from the mean o ff set of ∼ 1 %. Therefore, we expect that using BLYP functional the ( β -1) and ∆ values are systematically overestimated by 7 % and that in addition there is a 2 % error in derived ( β -1) factors. Similar errors result from using other functionals or even more sophisticated and time consuming post-Hartree-Fock methods such as MP2 (Finley & Stephens, 1995; Alecu et al., 2010).","pages":[9,10]},{"title":"3.1. Test of the computational method","content":"First, we illustrate the performance of the approximation proposed in section 2.1.2 by computing the β factors for the isolated H3BO3 and H4BO -4 molecules and selected crystalline solids. In Figure 1 we present three sets of calculations of β factors: (1) the 'exact' result obtained from a full normal mode analysis and formula 1, (2) the results obtained applying Kowalski & Jahn (2011) method based on Eq. 3, (3) the results obtained using pseudofrequencies computed for the fractionating element and Eq. 1 for the estimation of the β factor. The numerical values for selected temperatures are reported in Table 2. Approach (3) results in much better agreement with the 'exact' result. For H3BO3, the β factor is overestimated by only 0 . 5 /permil and 1 . 5 /permil for temperatures of 800 K and 600 K respectively. Applying method (2), the error is more pronounced, 2 . 2 /permil and 6 . 7 /permil respectively. In the case of molecular H4BO -4 , the errors using method (3) for the same temperatures are only 0 . 1 /permil and 0 . 5 /permil respectively. The same behavior is shown for dravite and boromuscovite crystalline solids that contain boron in the coordinative arrangement that resemble the configurations of aforementioned B-bearing molecules. For T > 600 K the proposed method represents only a few percent correction to the approximation given by equation (3), so the relation (15) is satisfied. It is evident that for B-bearing materials considered here the improvement made by using the pseudofrequencies based approach is substantial. It corrects for about 75% of error of the Bigeleisen & Mayer (1947) approximation (Eq. 3). However, the question of general applicability of the proposed method to other isotopic systems would require careful testing on a large set of materials, which is well beyond the scope of the current paper.","pages":[10]},{"title":"3.2. B isotope fractionation in gas and fluid phases 3.2.1. B isotope fractionation between H3BO3 and H4BO -4 in the gas phase","content":"In a first step of our investigation of boron-rich aqueous fluids we derived the full frequency spectra of molecules in the gas phase (isolated molecules). The relevant β factors were computed using Equation 1. These studies were performed in order to compare our results with the published values of Zeebe (2005), both computed using the same DFT BLYP functional. In Table 1 we report the computed frequencies that are affected by the di ff erent B isotope substitutions along with other theoretical estimations and experimental measurements. The computed frequencies are in good agreement with earlier theoretical predictions and show similar agreement with the experimental measurements. The results in terms of computed β factors for the two considered species are reported in Figure 2, where we compare our results with the values computed using frequencies of Zeebe (2005). The comparison of the two sets of calculations reveals that our β factors for both species are smaller by ∼ 1 /permil at 600 K -1000 K than the values of Zeebe (2005). However, the di ff erence between β factors of H3BO3 and H4BO -4 remains nearly identical in both sets of calculations and the agreement is nearly perfect for higher temperatures. We note, that for the comparison we used the frequencies of Zeebe (2005) computed using 6-31 + G(d) basis set, as only these are provided by the authors. β and α factors obtained at T = 300 K using a more extended 6311 + G(d,p) basis set indicate that the β factors using 6-31 + G(d) basis set are not fully converged. In particular, the ( β -1) factor of H4BO -4 computed with 6311 + G(d,p) basis set is 3 . 9 /permil smaller than the one derived using 6-31 + G(d). In Table 5, we compare these results with the results of our calculation. It is clearly seen that for lower temperatures such as T = 300 K the values computed with 6-311 + G(d,p) basis set are in better agreement with our results indicating that planewave based DFT approach we use provides adequate vibrational frequencies and resulting isotope fractionation factors.","pages":[10]},{"title":"3.2.2. B isotope fractionation between H3BO3 and H4BO -4 in aqueous fluid","content":"In order to obtain the temperature dependent β factor for aqueous fluids we fitted the function 1 + A / T 2 + B / T 4 to the computed values using the least squares minimization procedure. The computed β values for H3BO3 in fluid are: 1 . 02366 ± 0 . 00012, 1 . 03624 ± 0 . 00018 and 1 . 06262 ± 0 . 00010and for H4BO -4 in fluid are: 1 . 01745 ± 0 . 00005, 1 . 02650 ± 0 . 00010and 1 . 04597 ± 0 . 00015, for the temperatures of 1000 K, 800 K and 600 K respectively. The resulting temperature dependent β factor for H3BO3 is β = 1 + 2 . 416 · 10 4 / T 2 -5 . 823 · 10 8 / T 4 and for H4BO -4 is β = 1 + 1 . 772 · 10 4 / T 2 -4 . 234 · 10 8 / T 4 . The results for H3BO3 and H4BO -4 in aqueous so- lutions are shown in Figure 3. As was observed for the isolated molecules, the β factor of H3BO3-bearing fluid is substantially larger than the one for the H4BO -4 . This can be understood in terms of the substantial difference in the B-O bond lengths exhibited by the two considered species. In case of isolated molecules our calculations indicate a B-O bond length of 1 . 40 Å for H3BO3 and 1 . 51 Å for H 4BO -4 . We compared our β factors with the values computed by Sanchez-Valle et al. (2005), which were derived by the combination of force field methods and experimental data to derive accurate vibrational frequencies. For H3BO3 we got a nearly identical result. In case of H4BO -4 our calculation predicts a value which is lower by 2 -4 /permil . However, Rustad et al (2010b) and Rustad & Bylaska (2007) revealed the improper assignment of a major fractionating vibrational mode of H4BO -4 in the force field by Sanchez-Valle et al. (2005). This leads to the underestimation of the fractionation factor between aqueous H3BO3 and H4BO -4 by Sanchez-Valle et al. (2005). Assuming that α ∝ T -2 and having the di ff erence between BLYP calculations of Rustad et al (2010b) and Sanchez-Valle et al. (2005) of ∆ α = 16 . 4 /permil at T = 300 K, the value reported by Sanchez-Valle et al. (2005) should be underestimated by ∆ α = 16 . 4 · 2(300 / T ) 2 /permil , which results in ∆ α ∼ 1 . 5 /permil at T = 1000 K. Corrected in such as way result of Sanchez-Valle et al. (2005) is also plotted in Figure 3. It is now very consistent with our prediction.","pages":[10,11]},{"title":"3.2.3. Discussion of computational errors","content":"Most previous computational studies of boron isotope fractionation in aqueous solutions concentrate on the computation of the isotope fractionation at ambient conditions (Rustad et al, 2010b; Rustad & Bylaska, 2007; Liu & Tossell, 2005; Zeebe, 2005). Rustad et al (2010b) performed detailed analysis of impact of the chosen computational method (HF, MP2, di ff erent DFT functionals) and size of the basis set on the calculated fractionation factors between H3BO3 and H4BO -4 . They found that DFT methods are not performing well for the borate system and concluded that DFT 'is of limited usefulness in chemically accurate predictions of isotope fractionation in aqueous systems' (Rustad et al, 2010b). The empirically derived error of the derived fractionation factor is of the order of 5 -10 /permil for a total fractionation of ∼ 30 /permil . We note that this is expected and clearly visible if we apply the error estimation procedure outlined in section 2.5. For instance, at room temperature the derived beta factors using the BLYP functional are 213 . 6 /permil and 173 . 3 /permil respectively (Rustad et al (2010b), Table 2). This gives a fractionation factor of 1.0343. Following our error estimation scheme, the absolute error of the fractionation factor is 10 . 5 /permil , and the properly reported computed value is α = 1 . 034 ± 0 . 011. When one corrects for the systematic error of 7% and assumes 2% of statistical error on β factors, then the value of α decreases and the error is slightly smaller, i.e. α = 1 . 032 ± 0 . 008. This is in good agreement with the experimental data reported in Rustad et al (2010b) and explains the spread of the values computed using di ff erent methods and reported in that paper. It is very di ffi cult to get the fractionation factors for ambient conditions, as the fractionation factor is often just a small fraction of the relevant ( β -1) factors, ( α -1) ∼ 0 . 15( β -1) in the considered case. Assuming that ( α -1) = 0 . 15( β -1), a 2% error in the ( β -1) factors leads to an absolute error in ( α -1) of 0 . 04( β -1) = 0 . 04( α -1) / 0 . 15 ∼ 0 . 27( α -1), i.e. ∼ 27% of relative error in the derived fractionation factor ( α -1). On the other hand, we note that such a big error is not substantially larger than the uncertainties in the experimental data reported by Rustad et al (2010b) in their Figure 2. Thus, the case of boron fractionation in aqueous fluid at ambient conditions does not necessarily show the limited usefulness of DFT in the prediction of isotope fractionation factors, but only reflects the fact that precise estimation or measurement of the B isotopes fractionation factors at ambient condition requires unprecedented accuracy of both experimental or computational techniques. For instance, in order to get the value of ( α -1) with a relative error of 5% (at ambient conditions) one needs to estimate the ( β -1) factors or measure relevant quantities with precision of less than 1%. At higher temperatures the situation is di ff erent. Looking just at the fractionation between H3BO3 and H4BO -4 in the gas phase or the aqueous solution one can see that for T > 600 K the fractionation factor between the two substances, ( α -1), is at least 25% of the ( β -1) factor. This results in smaller 0 . 04 / 0 . 25 ∼ 16% for T = 600 K and 0 . 04 / 0 . 36 ∼ 11% for T = 1000 K relative error, which is acceptable in our calculations. Nevertheless, this case shows the importance of proper error estimation on the computed fractionation factors. Such an estimation is usually omitted or not provided explicitly, which can lead to wrong conclusions when the theoretical prediction is confronted with the measured data.","pages":[11,12]},{"title":"3.2.4. Pressure dependence of the fluid fractionation factor","content":"In our recent paper (Kowalski & Jahn, 2011) we have shown that due to compression the β factor of Li in aqueous fluid increases with increase in pressure (for P > 2 GPa). The same should happen for H3BO3 and H4BO -4 aqueous fluid as the vibrational frequencies of boron species in aqueous fluid increase with increase in pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). Having the experimental data we checked whether the derived pressure-dependent β factors are consistent with the pressure shifts of vibrational frequencies of considered boron species measured by Sanchez-Valle et al. (2005). For that purpose we performed a set of calculations using supercells containing 8 water molecules and the relevant boron species. We note that in line with our previous results for Li (Kowalski & Jahn, 2011), the obtained values of ( β -1) at P = 0 . 5 GPa are within 0 . 1 /permil in agreement with the values obtained for supercells containing 64 water molecules. The results are given in Figures 4 and 5. The computed ( β -1) values for H3BO3 fluid show a linear dependence in pressure, ( β -1) = 23 . 60 + 0 . 28 P (GPa) /permil . This is expected as ( β -1) ∝ ν 2 ∼ ν 2 0 + 2 ν 0 ∆ ν (Schauble, 2004) and ∆ ν is a linear function of pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). In case of H4BO -4 the pressure-dependence is linear up to P ∼ 2 -3 GPa and it becomes less steep at higher pressures. In order to quantitatively check the consistency of our prediction with the measured vibrational frequency shifts of Sanchez-Valle et al. (2005) we derived the relative shifts in the ( β -1) factor assuming that ( β -1) ∝ ν 2 and the measured pressure dependence of the frequency shifts: ∆ ν = 2 . 15 cm -1 · P (GPa) and ∆ ν = 3 . 50 cm -1 · P (GPa) for 1454 cm -1 and 666 cm -1 vibrational frequencies of H3BO3 and ∆ ν = 6 . 47 cm -1 · P (GPa) for the 975 cm -1 vibrational frequency of H4BO -4 . The chosen vibrational frequencies are the ones a ff ected by the di ff erent B isotope substitution. Our predicted shift of ( β -1) matches well the shifts derived from the measured frequency shifts. Such a good agreement with the experimental data validates further our computational approach and shows that ab initio calculations can be successfully used in derivation of the pressure dependence of the fractionation factors and pressure-induced vibrational frequencies shifts. Moreover, first principles calculations can be useful in extrapolation of the experimental values for β and ∆ ν to more extreme conditions, which otherwise are extremely di ffi cult to reach by experimental techniques.","pages":[12,13]},{"title":"3.3. Fluid-mineral fractionation","content":"Next we present the results of the fractionation between boron bearing fluids and minerals such as dravite, olenite and boromuscovite. The aim of these studies is to investigate the mechanisms driving the fractionation process, the role of coordination and the B-O bond length. Below we discuss each case separately.","pages":[13]},{"title":"3.3.1. Tourmaline-neutral fluid","content":"Meyer et al. (2008) measured the boron isotope fractionation between tourmaline and neutral fluid at T = 400 -700 o C and P = 0 . 2 GPa. In the experiment the tourmaline was represented by dravite. In contrast to the former measurements of Palmer et al. (1992) the measured fractionation is very small and does not exceed 2 . 5 /permil at 400 o C. Our calculated fractionation curve together with the experimental data are given in Figure 6. Our result correctly reproduces the experimental measurements within the computational accuracy. The dravite-fluid fractionation is small as the two materials contain boron in BO3 units. We also predict a small fractionation between olenite carrying 3-fold coordinated boron only and aqueous fluid, although the olenite-fluid fractionation is positive because of the shorter B-O bond lengths for olenite (1 . 378 Å vs. 1 . 397 Å).","pages":[13]},{"title":"3.3.2. Boromuscovite-strongly basic fluid","content":"Boromuscovite synthesized in the experiments of Wunder et al. (2005) consisted of two type of polytypes, 1M ( ∼ 10 %) and 2M1 ( ∼ 90 %). In order to be consistent with the experimental conditions, we derived the β factors for both polytypes and computed their weighted average. We note that the β factors for both polytypes of mica are similar with the di ff erence in ( β -1) not larger than 3 %. This is consistent with similar B-O bond lengths (1 . 532 Å) found in both polytypes. Boromuscovite contains boron in tetrahedral sites. Therefore, in order to investigate the impact of the B-O bond length on the fractionation we first compare the fractionation between the mineral and a strongly basic fluid contain- n in H4BO -4 . The result, together with the measurements of Wunder et al. (2005) of the fractionation between boromuscovite and strongly basic fluid, is summarized in Figure 7. Our calculations predict a negative fractionation between mica and the H4BO -4 fluid. The agreement of our prediction with the experimental measurements is relatively good; however, the experimental data indicate slightly stronger fractionation. We note that the experimental conditions of Wunder et al. (2005) do not assure that the measured basic fluid contained four-fold coordinated boron species, i.e. H4BO -4 , only. As we indicate in the Figure 7, the presence of as little as 10% of H3BO3 in the measured basic fluid brings the prediction and measurements into much better agreement. It makes sense that the β factor of boromuscovite is smaller than for aqueous H4BO -4 as the average B-O bond length in mica is 1 . 532 Å, while it is 1 . 513 Å and therefore shorter in case of aqueous H4BO -4 . We note that in our derivation we assumed that the fluid consists mostly of [4] B species. As we already mentioned, although Wunder et al. (2005) call the fluid 'strongly basic' its exact composition, especially the amount of [3] B species is unknown. However, if for instance the [4] B to [3] B ratio was 1, then the predicted mica-basic fluid fractionation at 800 K would be 5 /permil larger than measured. This would result in a large inconsistency between the computed values and the experimental data. On the other hand, good agreement between the prediction and the measurements indicates that the strongly basic fluid was dominated by H4BO -4 species, which is in line with previous studies (Zeebe, 2005; Sanchez-Valle et al., 2005).","pages":[13,14]},{"title":"3.3.3. Boromuscovite-neutral fluid","content":"The fractionation between boromuscovite and neutral fluid involves a change in coordination from B [4] in boromuscovite to B [3] in neutral fluid. Wunder et al. (2005) measured the fractionation between the two materials at 3 GPa, shown in Figure 8. The predicted fractionation is about 3 ± 2 . 5 /permil larger than the measured value. Looking for potential sources of this discrepancy, we have checked for the e ff ect of the change in lattice parameters due to combined thermal expansion and compression. For that purpose we applied the EOS of Holland and Powell (2011) for muscovite, which gives 4 . 4%, 3 . 8% and 3 . 1% decrease in volume for T = 600 K, 800 K and 1000 K respectively and P = 3 GPa. As muscovites show highly anisotropic compressibility patterns, in line with Comodi & Zanazzi (1995) we applied the T and P driven change in volume assuming the 16%, 19% and 65% contribution to compression along the /vector a , /vector b , /vector c lattice vectors. On the other hand ( β -1) factors of H3BO3 and H4BO -4 in aqueous solutions at P = 3 GPa increase by 3 . 5% and 11 . 8% respectively (Figures 4 and 5), leading to pressure-induced increase in the boromuscovite-aqueous fluid fractionation at the experimental pressure. The ∆ factor, corrected for e ff ects of thermal expansion and compression of boromuscovite and compression of fluid, is also given in Figure 8. Because the high T and P e ff ects result in similar increases in the β factors for both solid and aqueous fluid, the resulting fractionation factor between these two phases is close to the one derived at ambient conditions. Therefore, thermal expansion and compression e ff ects cannot explain the observed discrepancy between prediction and the measurements of Wunder et al. (2005). On the other hand, the comparison of our results with the experimental data suggests that the fractionation between boromuscovite and fluid is the same as between H3BO3 and H4BO -4 fluids (see Figure 3), which is at odds with the non-negligible and negative fractionation between boromuscovite and a strongly basic fluid. In section 3.3.2 we have shown that we are able to correctly reproduce the fractionation between boromuscovite and strongly basic fluid, which indicates that our result for boromuscovite is reliable. This suggests that another, unaccounted e ff ect leads to the decrease of the boron isotope fractionation between mica and neutral fluid in the experiments of Wunder et al. (2005). One possible solution for the discrepancy is a nonnegligible amount of boron residing in four-fold coordinated configurations in neutral solution. This is in line with the Raman spectroscopy measurements of Schmidt et al. (2005), who detected a broad peak in the Raman spectra of neutral H3BO3-dominated fluid and attributed it to B [4] species. The integrated area of this peak, compared to the peak of the Raman 877 cm -1 line of B [3] species, indicates the presence of at least 15 -30 % of B [4] species by mole fraction. Assuming that there is 15 -30 % of B [4] species present in the fluid and that the β factor of these species is similar to that of H4BO -4 , the fractionation factor between boromuscovite and H3BO3 aqueous fluid decreases bringing the theory and the experiment to better agreement, which is illustrated in Figure 8. If this interpretation is true, it suggests that boron isotope fractionation could be used to gather information on the speciation of B in aqueous fluids.","pages":[14,15]},{"title":"3.4. B isotope fractionation between minerals","content":"The boron isotope fractionation between B-bearing crystalline solids has received considerable attention recently (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002). We focus here on the investigation of boron isotope fractionation between mica and tourmaline as boron atoms in these minerals occupy sites of di ff erent coordination, which should result in a large B isotope fractionation between these two minerals. In mi- cas boron substitutes for silicon in the four-fold coordinated site (Wunder et al., 2005), while in tourmaline (dravite) it is incorporated in the three fold coordinated site (Meyer et al., 2008). Comparing the B isotope fractionation between di ff erent minerals, melts and fluids Wunder et al. (2005) have shown that the fractionation between two materials of di ff erent B coordination is large, reaching 5 /permil at 1000 K and much higher values at lower temperatures. Klemme et al. (2011),Marschall (2005) and Hervig et al. (2002) measured the fractionation between coexisting phases of the two minerals in natural samples. The fractionation between these two minerals is also derived from experimental isotopic fractionation data of B-muscovite-fluid (Wunder et al., 2005) and tourmaline-fluid (Meyer et al., 2008) systems. The results of these measurements and our computed T -dependent fractionation curve are given in Figure 9. The first striking observation is that our predicted fractionation factors are much larger (taking the absolute value) than the experimental values (Wunder et al., 2005; Meyer et al., 2008). The latter are also incon- sistent with the natural samples data of Klemme et al. (2011) and previous studies discussed in that paper (Marschall, 2005; Hervig et al., 2002). On the other hand, the measurements on natural samples are consistent with our calculated values, which tends to validate our predictions. We notice that the most recent measurements of boron isotope signatures of tourmaline and white mica from the Broken Hill area in Australia by Klemme et al. (2011) indicate for the assumed temperature of 600 o C that the fractionation factor between the two phases is 10.4 ± 2 . 7 /permil , which is in good agreement with our computed value of 10 . 7 ± 1 . 8 /permil . The experimental mica-tourmaline B isotope fractionation factors of Wunder et al. (2005) and Meyer et al. (2008) are 2 /permil and 6 /permil smaller at temperatures of 1000 K and 800 K respectively, with an experimental uncertainty of 2 /permil . However, this discrepancy can be resolved by assuming that in the experiments of Wunder et al. (2005) the fluid contained a significant admixture of B [4] species, which leads to the underestimation of the experimental boromuscovite-fluid fractionation factor by ∼ 2 /permil at 1000 K and ∼ 3 . 5 /permil at 1000 K, as is seen in Figure 8. The experimental mica-tourmaline fractionation factor corrected for the presence of B [4] species is also plotted in Figure 9. It is now more consistent with the natural data. We note that this result independently supports the conclusion underlined in section 3.3.3 and result of Schmidt et al. (2005) that highP , B-bearing neutral fluids contain significant admixtures of B [4] species. Olenite is a mineral which can incorporate boron in both trigonal and tetrahedral sites as it substitutes for both Al and Si atoms. It is therefore interesting to check the fractionation of boron isotopes between the two differently coordinated sites in one mineral and compare it with the above result for mica and tourmaline. The computed fractionation between the trigonal and tetrahedral sites at 600 o C is 10 . 6 ± 1 . 9 /permil , which is consistent with the fractionation between tourmaline and mica, indicating that the coordination of the B atom is the driving factor for the fractionation of the B isotopes. Similarly, we computed the boron isotopes fractionation between trigonal and tetragonal boron sites in dravite. In order to create the tetragonal B site we replaced one Si atom with B and we added one H atom forming an additional OH group to compensate the charge. The computed fractionation between the sites at 600 o C is 8 . 9 ± 1 . 7 /permil , which is also in agreement with the aforementioned results. Next, we will show that the value of the β factor depends not only on coordination but is also strongly correlated with B-O bond length.","pages":[15,16]},{"title":"3.5. Fractionation between B [3] and B [4] materials","content":"The β factors computed for all the considered materials are grouped together in Figure 10. β factors can be grouped into two sets, one that includes materials with boron in three-fold coordination and another one that includes materials having boron in four-fold coordination. For olenite and dravite we also computed the β factors with boron sitting on four-fold coordinated site. The β factor for these crystalline solids with given B [3] / B [4] ratio can be derived as a weighted average of the β factors obtained for boron sitting on the two di ff erently coordinated sites. The fractionation factor between materials of di ff erent boron coordination is ∼ 8 /permil on average at T = 1000 K. We note that it is ∼ 3 /permil larger than the one deduced by Wunder et al. (2005) from measurements performed on solids, silicate melts and fluids, but this can be attributed to the underestimation of the fractionation factors for boromuscovite-fluid system by Wunder et al. (2005) due to potential admixture of B [4] species in the investigated fluid. Our results show also a substantial spread of β factors of substances containing boron of a given coordination. The spread is at least 4 /permil and results from di ff erent B-O bond lengths. We illustrate this in Figure 11 by plotting together the β factors derived for all considered materials at T = 1000 K as a function of B-O bond length. It is clearly seen that there is a roughly linear correlation between the β factor and BO bond length, which is especially evident comparing the results for crystalline solids. For instance, out of the considered B [4] -bearing minerals boromuscovite has the longest B-O bond length of 1 . 525 Å (1M) and 1 . 516 Å (2M1), followed by dravite 1 . 514 Å and olenite with BObond length of 1 . 502 Å. This tracks the di ff erences in the β factors derived for these materials. In addition the materials having B [3] species only exhibit shorter bond lengths of ∼ 1 . 37 Å and higher β factors, while the materials containing B [4] species having bond lengths of about ∼ 1 . 52 Å show much smaller β factors. Therefore, the tighter bonding of B [3] species likely explains why the heavy B isotope prefers the less coordinated phases. This clearly shows that the change in the B-O bond length during an isotope exchange is the leading factor driving the production of the boron equilibrium isotope signatures at high T .","pages":[16,17]},{"title":"4. Conclusions","content":"In this work we have presented a detailed analysis of boron isotope fractionation between boronbearing crystalline solids and aqueous fluids at high T and P conditions. In order to perform our investigation we have applied and extended a computationally e ffi cient approach for the computation of isotope fractionation factors for complex minerals and fluids at high temperatures and pressures presented by Kowalski & Jahn (2011). As an extension to the Bigeleisen & Mayer (1947) 'single atom approximation' method we demonstrated that using the pseudofrequencies derived from the force constants acting on the fractionating element together with the full formula for computation of the reduced partition function ratios results in significant improvement in the accuracy of the computed fractionation factors, which is essential when lower temperature materials and high vibrational frequency complexes are considered. In order to understand the fractionation between Bbearing crystalline solids and aqueous fluids we performed a set of calculations of β factors for dravite, olenite, boromuscovite and aqueous solutions of H3BO3 and H4BO -4 . In agreement with the experimental findings we show that the fractionation strongly correlates with coordination through the change in the B-O bond length. The lower trigonal coordination BO3 arrangement results in higher 11 B / 10 B (by ∼ 8 /permil at T = 1000 K) than the tetrahedrally coordinated boron complexes, which exhibit ∼ 0 . 15 Å longer B-O bonds. The computed fractionation between minerals and fluids of the same coordination are in good agreement with experiments. However, we predict larger isotope fractionation between boromuscovite and H3BO3 fluid (by at least a few /permil ) than was measured in situ at high P by Wunder et al. (2005) and Meyer et al. (2008), but that is consistent with measurements on natural samples. We note that the presence of B [4] in highP fluid could reconcile the in situ experimental results with our prediction and other measurements. This is expected from the experiments of Schmidt et al. (2005), but requires further experimental confirmation. If true, this would open the possibility for using the isotope fractionation techniques as a tool to measure the speciation of boron in fluids and crystalline solids. We have also demonstrated that with our computational approach we are able to correctly predict the pressure-induced isotope fractionation for compressed aqueous fluids, which indicates the ability of ab initio methods to predict the isotopic signatures of highly compressed materials, even those that are difficult to investigate experimentally. Our study confirms that ab initio computer simulations are a useful tool not only for prediction but also understanding the equilibrium stable isotope fractionation processes between various phases, including aqueous solutions, at high pressures and temperatures. They can nicely complement experimental e ff orts, provide unique insight into the isotope fractionation process on the atomic scale and deliver data for conditions that are inaccessible by the current experimental techniques.","pages":[17]},{"title":"Acknowledgements","content":"The authors wish to acknowledge financial support in the framework of DFG project no. JA 1469 / 4-1. Part of the calculations were performed on the IBM BlueGene / P JUGENE of the John von Neumann Institute for Computing (NIC). We are also grateful the associate editor Edwin A. Schauble and anonymous referees for constructive comments that helped improving the manuscript.","pages":[18]},{"title":"References","content":"Alecu I. M., Zheng J. , Zhao Y. and Truhlar D. G. (2010) Computational Thermochemistry: Scale Factor Databases and Scale Factors for Vibrational Frequencies Obtained from Electronic Model Chemistries. J. Chem. Theory Comput. 6 , 2872-2887. Andrews, L., Burkholder, T.R., (1992) Infrared spectra of molecular B(OH)3 and HOBO in solid argon, J. Chem. Phys. , 97 , 10, 72037210. Barth S. (1993) Boron isotope variations in nature: a synthesis. Geol. Rundsch. 82 , 640-651. Becke A. D. (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38 , 3098-3100. Bigeleisen J. and Mayer M. G. (1947) Calculation of Equilibrium Constants for Isotopic Exchange Reactions. J. Chem. Phys. 15 , 262-267. Blanchard M., Poitrasson F., Meheut M., Lazzeri M., Mauri F., and Balan E. (2009) Iron isotope fractionation between pyrite (FeS2), hematite (Fe2O3) and siderite (FeCO3): A first-principles density functional theory study. Geochim. Cosmochim. Acta 73 , 65656578. Brenan J.M., Ryerson F.J. and Shaw H.F. (1998) The role of aqueous fluids in the slab-to-mantle transfer of boron, beryllium and lithium during subduction: experiments and models. Geochim. Cosmochim. Acta 62 , 3337-3347. Domagal-Goldman S. D. and Kubicki J. D. (2008) Density functional theory predictions of equilibrium isotope fractionation of iron due to redox changes and organic complexation. Geochim. Cosmochim. Acta 72 , 5201-5216. Domanik K.J., Hervig R.L. and Peacock S.M. (1993) Beryllium and boron in subduction zone minerals: An ion microprobe study. Geochim. Cosmochim. Acta 57 , 4997-5010. Marler B., Borowski M., Wodara U., Schreyer W. (2002) Synthetic tourmaline (olenite) with excess boron replacing silicon in the tetrahedral site: II. Structure analysis Eur. J. Mineral. 14 , 763-771. Marschall H. R. (2005): Lithium, Beryllium, and Boron in High-Pressure Metamorphic Rocks from Syros (Greece). PhD thesis, Universitt Heidelberg, Heidelberg, Germany [http: // www.ub.uni-heidelberg.de / archiv / 5634]. Marx D. and Hutter J. (2000) Ab initio molecular dynamics: Theory and Implementation. Modern Methods and Algorithms of Quantum Chemistry 1 edited by J. Grotendorst, NIC, FZ Julich, 301-449 ; CPMD code: J. Hutter et al. , see: www.cpmd.org . Meheut M., Lazzeri M., Balan E. and Mauri F. (2007) Equilibrium isotopic fractionation in the kaolinite, quartz, water system: Prediction from first-principles density-functional theory. Geochim. Cosmochim. Acta 71 , 3170-3181. Chem. Geol. 258 , 28-37. Menconi G. and Tozer D. J. (2002) Diatomic bond lengths and vibrational frequencies: assesment of recently developed exchangecorrelation functionals. Chem. Phys. Lett. 360 , 38-46. Meyer Ch., Wunder B., Meixner A., Romer R. L., and Heinrich, W. (2008) Boron isotope fractionation between tourmaline and fluid; an experimental re-investigation. Contrib. Mineral. Petrol. 156 , 259-267. Morgan VI, G.B. and London, D. (1989) Experimental reactions of amphibolite with boron-bearing aqueous fluids at 200 MPa: implications for tourmaline stability and partial melting in mafic rocks. Contrib. Mineral. Petrol. 102 , 281-298. Redlich, O. (1935). Eine allgemeine Beziehung zwichen den Schwingungsfrequenzen isotoper Molekeln. Z. Physikal. Chem. B 28 , 371-382. Rustad J. R, Casey W. H., Yin Q.-Z., Bylaska E. J., Felmy, A. R., Bogatko S. A., Jackson V. E., Dixon D. A. (2010) Isotopic fractionation of Mg2 + (aq), Ca2 + (aq), and Fe2 + (aq) with carbonate minerals, Geochim. Cosmochim. Acta 74 , 6301-6323. Rustad J. R. and Bylaska E. J. (2007) Ab Initio Calculation of Isotopic Fractionation in B(OH)3(aq) and BOH -4 (aq). J. Am. Chem. Soc. (Communication) 129 , 2222-2223. Sanchez-Valle C., Reynard B., Daniel I., Lecuyer C., Martinez I., and Chervin J. C. (2005) Boron isotopic fractionation between minerals and fluids; new insights from in situ high pressure-high temperature vibrational spectroscopic data Geochim. Cosmochim. Acta 69 , 4301-4313 Schauble E. A. (2004) Applying Stable Isotope Fractionation Theory to New Systems. Rev. Mineral. Geochem. 55 , 65-111. Schreyer W., Wodara U., Marler B., van Aken P.A., Seifert F. and Robert J.-L. (2000) Synthetic tourmaline (olenite) with excess boron replacing silicon in the tetrahedral site: I. Synthesis conditions, chemical and spectroscopic evidence. Eur. J. Mineral. 12 , 529-541. Urey, H. C. (1947). The Thermodynamic Properties of Isotopic Substances. J. Chem. Soc. , 562581. and scientific use. J. Phys. Chem. Ref. Data 31 , 387-535. Wunder B., Meixner A., Romer R. L., Wirth R., and Heinrich W. (2005) The geochemical cycle of boron; constraints from boron isotope partitioning experiments between mica and fluid Lithos 84 , 206-216 Yamaji K., Makita Y., Watanabe H., Sonoda A., Kanoh H., Hirotsu T. and Ooi K. (2001) Theoretical Estimation of Lithium Isotopic Reduced Partition Function Ratio for Lithium Ions in Aqueous Solution. J. Phys. Chem. A 105 , 602613.","pages":[18,19]}],"string":"[\n {\n \"title\": \"Ab initio prediction of equilibrium boron isotope fractionation between minerals and aqueous fluids at high P and T\",\n \"content\": \"Piotr M. Kowalski 1 , 2 ∗ , Bernd Wunder 1 and Sandro Jahn 1 1 GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany 2 Forschungszentrum Julich, Institute of Energy and Climate Research (IEK-6), Wilhelm-Johnen-Strasse, 52425 Julich, Germany\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"Over the last decade experimental studies have shown a large B isotope fractionation between materials carrying boron incorporated in trigonally and tetrahedrally coordinated sites, but the mechanisms responsible for producing the observed isotopic signatures are poorly known. In order to understand the boron isotope fractionation processes and to obtain a better interpretation of the experimental data and isotopic signatures observed in natural samples, we use first principles calculations based on density functional theory in conjunction with ab initio molecular dynamics and a new pseudofrequency analysis method to investigate the B isotope fractionation between B-bearing minerals (such as tourmaline and micas) and aqueous fluids containing H3BO3 and H4BO -4 species. We confirm the experimental finding that the isotope fractionation is mainly driven by the coordination of the fractionating boron atoms and have found in addition that the strength of the produced isotopic signature is strongly correlated with the B-O bond length. We also demonstrate the ability of our computational scheme to predict the isotopic signatures of fluids at extreme pressures by showing the consistency of computed pressure-dependent β factors with the measured pressure shifts of the B-O vibrational frequencies of H3BO3 and H4BO -4 in aqueous fluid. The comparison of the predicted with measured fractionation factors between boromuscovite and neutral fluid confirms the existence of the admixture of tetrahedral boron species in neutral fluid at high P and T found experimentally, which also explains the inconsistency between the various measurements on the tourmaline-mica system reported in the literature. Our investigation shows that the calculated equilibrium isotope fractionation factors have an accuracy comparable to the experiments and give unique and valuable insight into the processes governing the isotope fractionation mechanisms on the atomic scale. Keywords: stable isotope fractionation, boron, fluid speciation, DFT, molecular dynamics\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"The use of isotopes as geochemical tracers depends upon the existence of reliable P -T -dependent equilibrium isotope fractionation data between solids, fluids and melts. The common method used for determination of such data is to conduct experiments, where the phases of interest are equilibrated at a range of di ff erent conditions and individually measured for their equilibrium isotopic compositions. Recently, with the development of computational methods, software and increase in hardware performance it is also possible to simulate and compute the isotope fractionation with ab initio methods. These computational considerations complement the experimental e ff ort and provide information on the mechanisms governing the equilibrium isotope fractionation processes on the atomic scale. By establishing an e ffi cient computational approach for materials at high P and T and testing its reliability by computing Li isotope fractionation between minerals and aqueous fluids, Kowalski & Jahn (2011) have shown that ab initio methods can provide a reliable estimation of equilibrium isotope fractionation factors at an accuracy level comparable to experiments. Motivated by the encouraging results for the fractionation of lithium isotopes we applied our new method to study boron isotope fractionation. The main goals of the present study are to make theoretical predictions and obtain a better understanding of the B isotope fractionation process between tourmaline, B-bearing mica and flu- ids at various pressures and temperatures, to compare these data to results from recent in situ experimental studies (Wunder et al., 2005; Meyer et al., 2008) and measurements of isotopic signatures in natural samples (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002) and to investigate the underlying mechanisms driving the boron isotope fractionation processes between the considered materials. With two stable isotopes 10 B and 11 B, of relatively large mass di ff erence of about 10 %, boron isotopes strongly fractionate during geological processes, thus leading to natural δ 11 B-variations ranging from -30 to + 60 /permil (Barth, 1993). Therefore, B isotopes are ideal for the distinction of di ff erent geological environments and for quantifying mass transfer processes, e.g. in the range of subduction zones. First-order criteria driving isotope fractionation in Earth materials are di ff erences in coordination and in the bonding environments of coexisting phases. The lighter isotope usually preferentially occupies the higher coordinated site, which is generally accompanied with a longer cation-anion bond length and weaker bond strength (Schauble et al., 2009; Wunder et al., 2011). Tourmaline has an extensive chemical variability and is the most widespread borosilicate in various rocks over a large range of bulk compositions. It has a large P -T -stability which ranges from surface conditions to high pressures and temperatures of at least 7.0 GPa and about 1000 o C as determined experimentally for dravitic tourmaline (Krosse, 1995). Most tourmaline minerals contain 3 boron atoms per formula unit (pfu) with B in trigonal planar coordination (B [3] ). High pressure combined with an Al-rich environment can lead to the formation of olenitic tourmaline with significant amounts of excess B substituting for Si at the tetrahedral site (B [4] ). The highest amounts of up to 1.2 B [4] pfu have been found in olenitic tourmaline from Koralpe, Austria (Ertl et al., 1997). Al-rich tourmaline with up to 2.2 B [4] pfu was synthesized experimentally at 2 . 5 GPa, 600 o C (Schreyer et al., 2000). The maximum possible amount of B [4] in olenitic tourmaline is limited to three B [4] pfu, due to structural and crystal chemical reasons. As tourmaline is not stable at basic pH (Morgan & London, 1989), the dominant B-species of fluids coexisting with tourmaline is B(OH)3. Therefore, due to the absence of change in boron coordination, B-isotopic fractionation between B [4] -free tourmaline and fluid ( ∆ 11 B(tour-fl)) should be small. However, at 0 . 2 GPa, the experimentally determined ∆ 11 B(tour-fl)values of -2 . 5 ± 0 . 4 /permil at 400 o C and -0 . 4 ± 0 . 4 /permil at 700 o C (Meyer et al., 2008) suggest small but significant di ff erences in the B-O(H) bond strength between tourmaline and the neutral fluids. Incorporation of B [4] into tourmaline is expected to significantly increase the fractionation of boron isotopes between the mineral and aqueous fluid (i.e. increase of | ∆ 11 B | (tour-fl)). In this contribution we present calculated P -T -dependent data on B-isotope fractionation between B [4] -bearing tourmaline and fluid, which so far are not available from experiments. Despite of its low boron content of up to maximum values of 270 ppm (Domanik et al., 1993), potassic white mica is probably the main host of boron in metasedimentary and metabasaltic blueschists and eclogites, because of its high modal abundance and B-incompatibility in all other stable minerals of these rocks (Brenan et al., 1998). During subduction the modal amount of mica continuously decreases by dehydration reactions and the chemistry of residual micas is shifted towards phengitic compositions. Phengitic mica has an extended stability, extending as deep as 300 km within cold subduction zones (Schmidt, 1996). In contrast to most tourmalines, boron is tetrahedrally coordinated in mica, where it substitutes for aluminum. Due to the coordination change from mostly three-fold coordinated in near-neutral fluids (Schmidt et al., 2005) to B [4] in mica, B-isotopic fractionation between Bbearing mica and such fluids is much larger than for tourmaline - fluid. ∆ 11 B(mica-fl)-values determined experimentally at P = 3 . 0 GPa, are -10 . 9 ± 1 . 3 /permil at 500 o C and -7 . 1 ± 0 . 5 /permil at 700 o C (Wunder et al., 2005). Such a strong B-isotope fractionation and its pronounced T -dependence, in combination with the continuous dehydration of micas during ongoing subduction and boron transport via fluids into mantle wedge regions of arc magma-formation, probably determines the boron variations and B-isotopic signatures in volcanic arcs (Wunder et al., 2005). Using in situ Raman spectroscopic measurements of near-neutral B-bearing fluids, Schmidt et al. (2005) observed a significant amount of 4-fold B-species at high P and T . The abundance of these species increases with temperature and pressure and at T = 800 K and P = 1 . 9 GPa there should be a considerable amount (15 -30%) of B [4] species in the fluid. Such a significant amount of tetrahedrally coordinated B-species in high temperature and pressure fluids should a ff ect solid-fluid B-isotopic fractionation, which we investigate in our calculations. In the light of this, we also discuss recently determined B-isotope data from coexisting natural tourmaline and B-bearing mica (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002), which show slight inconsistency with in situ measurements of Wunder et al. (2005) and Meyer et al. (2008). Reliable ab initio computational methods to predict isotope fractionation factors have been established recently. Several groups have proved that such calculations can contribute towards understanding geochemical mechanisms responsible for production of isotope signatures (Driesner, 1997; Yamaji et al., 2001; Schauble, 2004; Domagal-Goldman et al., 2008; Hill & Schauble, 2008; Meheut et al., 2009, 2007; Schauble et al., 2009; Zeebe, 2009; Hill et al., 2010; Rustad et al., 2010a; Rustad et al, 2010b; Zeebe, 2009, 2010; Kowalski & Jahn, 2011). The majority of these works, however, concentrate on the computation of the stable isotope fractionation between various, mostly simple crystalline minerals, and the aqueous solutions are usually computed using an isolated cluster containing fractionating species and a hydration shell (e.g. Zeebe (2010, 2009); Hill et al. (2010); Rustad et al. (2010a); Domagal-Goldman et al. (2008); Hill & Schauble (2008); Rustad et al (2010b); Zeebe (2005)). However, aqueous solutions at high pressure and temperature must be computed with caution as discussed in Kowalski & Jahn (2011). This is because the distribution of cation coordination and cation-oxygen bond lengths that a ff ect the isotope fractionation (Bigeleisen & Mayer, 1947) is driven by the dynamics of the system and change under compression (Jahn & Wunder, 2009; Wunder et al., 2011; Kowalski & Jahn, 2011). The only recent ab initio work, besides Kowalski & Jahn (2011), that accounts for the dynamical e ff ects on the isotope fractionation in fluid is by Rustad & Bylaska (2007) who considered boron equilibrium isotope fractionation between B(OH)3 and B(OH) -4 species in aqueous solution. They performed ab initio molecular dynamics simulations of these fluids and attempted to use the vibrational density of states, derived through the Fourier transform of the velocity auto-correlation function, as an input for the calculation of the 11 B / 10 B isotope fractionation coe ffi -cient. However, the resulting fractionation factor α = 0 . 86 happened to be much lower than the experimental value of α = 1 . 028. Interestingly, the discrepancy between experiment and theory is cured by quenching the selected configurations along the molecular dynamics trajectory and computing the harmonic frequencies. The fractionation factor derived using these frequencies exactly reproduces the experimental value. In our approach both solids and fluids are treated as extended systems by application of periodic boundary conditions in all three spatial directions, which is crucial to model high pressure materials. Large enough supercells are chosen to avoid significant interaction between atoms and their periodic images, as well as to reduce the number of k-points and q-vectors for sampling the Brillouin zones and the phonon spectra of the crystals, respectively (for a liquid or fluid, both the Brillouin zone and phonons are not defined). In our investigation we use cells of at least 7 Å width in each spatial dimension. A representative statistical sampling of the fluid structure is obtained by performing Car-Parrinello molecular dynamics simulations (Car & Parrinello, 1985). For the calculation of the isotope fractionation factors in fluids, several random snapshots from the simulation runs are chosen. The force constants acting on the fractionating element and the resulting fractionation factors are then obtained for each ionic configuration, and the relevant fractionation factor for the boron species in the fluid is computed as an average over the whole set of considered geometries. In Kowalski & Jahn (2011) it was shown that in line with the Bigeleisen & Mayer (1947) approximation, considering the force constants acting on the fractionating atom only leads to a satisfactory estimation of the Li isotope fractionation factors for high temperature fluids and minerals. Here, we will show that approximating the vibrational spectrum by the three pseudofrequencies derived from the force constants allows for further improvement of the accuracy of the predicted isotope fractionation factors, especially at lower temperatures. In this contribution we present the theoretical prediction of B isotope fractionation factors between B bearing aqueous fluids and solids, specifically tourmaline and B-muscovite, for which experimental data and measurements on natural samples are available for comparison (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011). We will show that the application of ab initio methods to B-bearing crystalline solids and fluids not only provides unique insight into the mechanisms driving equilibrium B-isotope fractionation on the atomic scale, but helps in proper interpretation of the data.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.1. Theoretical model\",\n \"content\": \"2.1.1. The single atom approximation: & Bigeleisen Mayer (1947) approach Mass-dependent equilibrium isotope fractionation is driven by the change in molecular and crystalline vibration frequencies resulting from the di ff erent masses of the isotopes. The fractionation between species and an ideal monoatomic gas is called the β factor. In the harmonic approximation it is given by the formula References: 1 Zeebe (2005), 2 Liu & Tossell (2005), 3 Rustad et al (2010b), 4 Sanchez-Valle et al. (2005), 5 Gilson (1991), 6 Oi (2000); Liu & Tossell (2005), 7 Andrews & Burkholder (1992), 8 Ogden & Young (1988), 9 Zeebe (2005) and references hereafter. (Bigeleisen & Mayer, 1947; Urey, 1947; Chacko et al., 2001): where u = h ν i / kBT , h is the Planck constant, ν i is the vibrational frequency of the i -th degree of freedom, kB is the Boltzmann constant, Ndof is the number of degrees of freedom, which for N being the number of atoms in the considered system (molecule, mineral or fluid) is equal to 3 N -5 for a diatomic molecule, 3 N -6 for multiatomic molecules and 3 N for crystals, and a star symbol marks the heavier isotope. The fractionation factor between two substances A and B, α A -B is computed as the ratio of the relevant β factors, which is well approximated by the di ff erences in the β factors: The calculation of the β factor requires only knowledge of the vibrational properties of the considered system computed for the two di ff erent isotopes. However, computation of the whole vibrational spectra of complex, multiparticle minerals or fluids requires substantial computational resources and is currently limited to systems containing a few dozens of atoms or less. In our recent work (Kowalski & Jahn, 2011) we proposed to use an e ffi cient method for computing the high temperature isotope fractionation factors between complex materials such as fluids and crystalline solids, which requires the knowledge of the force constants acting upon the fractionating element only. The β factor (Eq. 1) can be then approximated by (Bigeleisen & Mayer, 1947; Kowalski & Jahn, 2011): where Ai are the force constants acting on the isotopic atom in the three perpendicular spatial directions (x, y and z), ∆ m = m ∗ -m , where m and m ∗ are the masses of the lighter and heavier isotopes of the fractionating element. As the computation of the β factors from formula 3 requires the knowledge of properties of the fractionating element only we will call such an approach the single atom approximation throughout the paper. The validity criteria restricts the usage of the formula to frequencies ν [cm -1 ] /lessorsimilar 1 . 39 T [K] (assuming u < 2, see Fig. 1 of Bigeleisen & Mayer (1947)). We are interested in temperature range 800-1000 K. The highest vibrational frequency of the modes involving movement of B atoms for H3BO3 is ∼ 1400 cm -1 and of H4BO -1 4 is < 1200 cm -1 (Table 1). In case of H3BO3, the single atom approximation may produce an error of 2 . 2 /permil in the β factor at T = 800 K. The relevant error for H4BO -1 4 is 0 . 8 /permil (Table 2). A further improvement to the method is therefore desired.\",\n \"pages\": [\n 3,\n 4,\n 5\n ]\n },\n {\n \"title\": \"2.1.2. The single atom approximation with pseudofrequencies: our improvement\",\n \"content\": \"We will show that the error of the single atom approximation can be substantially reduced if one uses the three frequencies ¯ ν i derived from the force constants acting on the fractionating element (¯ ν 2 i = Ai / 4 π 2 m ). We call them 'pseudofrequencies' , and compute the β factors using formula 1. In the following we present the formal justification of such an approach. According to Bigeleisen & Mayer (1947), equation 1 for small ∆ ui = ui -u ∗ i reduces to (Bigeleisen & Mayer (1947), Eq. 11a): The Taylor expansion of the function appearing under the summation sign is: When we consider just the first term of the expansion the β factor is: which is exactly equation 3. Let us consider the Taylor expansion of the di ff erent estimations of β factors. Equation 4 then reads: The Bigeleisen & Mayer (1947) approximation given by equation 3 reads: and the proposed approximation based on pseudofrequencies is: In the above equation ¯ ui = h ¯ ν i / kBT . Next we check how this approximation compares to the Bigeleisen & Mayer (1947) approximation given by equations 3 and 8. In order to make a comparison we derive the di ff erences between the two approximate expressions β BM , β pseudo and the exact one β exact (Eq. 7). In the case of the Bigeleisen & Mayer (1947) approximation we have: and having from equations 3 and 8 that in the case of the proposed approximation we get: Because relation G ( u ) < u 12 holds for any u (see Bigeleisen & Mayer (1947), Fig. 1), the function and ∆ β pseudo < ∆ β BM . On the other hand the expression for ∆ β pseudo is given by a di ff erence of the higher order terms of the Taylor expansions of the two expressions for the β factor. In the considered cases the values of pseudofrequencies are similar to the real frequencies that are a ff ected upon B isotope substitution in a given B-bearing system. This can be seen by comparing the pseudofrequencies computed for the selected cases of B-bearing molecules and crystalline solids considered here and reported in Table 3 with the real frequencies given in Table 1. This indicates that the two terms of opposite signs in Eq. 12 should be similar in value and cancel out to a great extent, so | ∆ β pseudo | << ∆ β BM . Therefore the approach proposed here to compute the β factor based on pseudofrequencies and Eq. 1 should give a better approximation to the exact β factors than equation 3, which we will show in section 3.1. The difference to the Bigeleisen & Mayer (1947) approxima- References: 1 Liang et al. (1995), 2 Ertl et al. (2010), 3 Marler et al. (2002) tion is given by: and can be easily computed for any considered system. We assume that the pseudofrequency-based approach to the computation of β factors is applicable if it is just a correction to equation 3, i.e. when: We also note that the proposed approach satisfies the Redlich-Teller product rule (Redlich, 1935) when the mass of considered isotope m is much smaller than the mass of the whole considered system M , namely, We notice that we could force the strict conservation of the Redlich-Teller product by just adjusting the ratios of ¯ ui / ¯ u ∗ i . However, such a modification would not preserve relation 11 and the pseudofrequency approach would not recover exactly the high temperature limit (Eq. 3) of the exact solution (Eq. 1), which is a more important constraint to fulfill strictly by the proposed approximation.\",\n \"pages\": [\n 5,\n 6,\n 7\n ]\n },\n {\n \"title\": \"2.2. Representation of solids\",\n \"content\": \"In this paper we investigate the boron isotope fractionation between dravite, olenite and boromuscovite minerals and aqueous fluids. The solids were represented by large cells containing at least 84 atoms. The number of atoms used in the crystal calculations together with the lattice parameters of modeled crystals are summarized in Table 4. The lattice parameters and chemical compositions of the modeled crystalline solids are the experimental values measured at ambient conditions found in the literature. Dravite is the crystalline solid which was used in the experiments on tourmaline by Meyer et al. (2008). The chemical composition of the supercell used in the investigation is Na3Mg9Al18(Si18O54)(B [3] O3)9(OH)12 with structural data of Marler et al. (2002). Olenite can contain B in both trigonal and tetragonal sites. The modeled structure is that of Ertl et al. (2010). The chemical composition of the unit cell used in the investigation is NaAl3Al6(Si4B [4] 2 O18)(B [3] O3)3(OH)3O. For boromuscovite, the 1M and 2M1 crystal structures of Liang et al. (1995) were used. In the isotope fractionation experiments of Wunder et al. (2005) boromuscovite forms two polytypes, 1M and 2M1, with relative abundances of 10% and 90% respectively. In boromuscovite B occupies the 4-fold coordinated site occupied mainly by Si atoms. The constructed model constitutes a 2x1x1 supercell of elementary chemical composition KAl2(B [4] Si3O10)(OH)2.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"2.3. Representation of aqueous solution\",\n \"content\": \"The aqueous solution was represented by a periodically repeated box containing up to 64 water molecules and one H3BO3 or H4BO -4 molecule. The pressure and temperature conditions were chosen to be close to the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The pressure of the aqueous solution for a given temperature and volume was calculated according to the equation of state of Wagner & Pruss (2002). The ab initio molecular dynamics simulations (AIMD) of aqueous fluids were performed for fixed temperature and volume using the Car-Parrinello scheme (Car & Parrinello, 1985). The temperature during each run was controlled by a Nos'e-Hoover chain thermostat (Nos'e & Klein, 1983; Hoover, 1985). For each T -V conditions at least 10 ps long trajectories were generated with an integration step of 0 . 12 fs.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"2.4. Computational technique\",\n \"content\": \"The calculations of pseudofrequencies and β factors for solids and aqueous solutions were performed by applying density functional theory (DFT) methods, which are currently the most e ffi cient methods allowing for treating extended many particle systems quantummechanically. We used the planewave DFT code CPMD (Marx & Hutter, 2000), which is especially suited for ab initio simulations of fluids, the BLYP exchangecorrelation functional (Becke, 1988; Lee et al., 1988) and norm-conserving Goedecker pseudopotentials for the description of the core electrons (Goedecker et al., 1996). One advantage of using the BLYP functional is that it usually gives harmonic frequencies that most closely resemble the observed frequencies of benchmark chemical systems (Finley & Stephens, 1995; Alecu et al., 2010) 1 . The energy cut-o ff for the plane wave basis set was 70 Ryd for geometry relaxations and molecular dynamics simulations and 140 Ryd for computation of vibrational frequencies. Periodic boundary conditions were applied for both crystalline solids and aqueous solutions to preserve the continuity of the media. The force constants and frequencies needed for the computation of the β factors were computed using the finite displacement scheme. Before performing the calculations of the crystal structures all atomic positions were relaxed to the equilibrium positions to minimize the forces acting on the atoms. We note that to compute the β factors for crystals one formally should account for phonon dispersion. Here we use large supercells and restrict our calculations to a single phonon wave-vector ( Γ ). Schauble (2011) has shown recently for 26 Mg / 24 Mg fractionation in Mg-bearing minerals that supercells containing more than 20 atoms are su ffi -cient to get very accurate β factors even at T = 300 K (error of 0.1 /permil ). At T = 1000 K the error is in the order of 0.01 /permil . The accuracy of the high temperature isotope fractionation factors computed on a single phonon wave-vector is also demonstrated for ironbearing minerals by Blanchard et al. (2009) and confirmed with good agreement of the predicted with the measured Li isotope fractionation factors between staurolite, spodumene, micas and aqueous fluid presented in our previous work (Kowalski & Jahn, 2011). Prior to the computation of the force constants and frequencies of boron atoms in the fluids the positions of all the atoms constituting the boron-carrying molecule (H3BO3 or H4BO -4 ) were relaxed to the equilibrium positions, while all other atomic positions remained unchanged. The full normal mode analyzes were per- formed using the same method, but displacing all the atoms constituting the considered system. In the latter case the frequencies were obtained through the diagonalization of the full dynamical matrix (Schauble, 2004) as implemented in CPMD code. The e ff ect of the various approximations on the derived fractionation factors was studied by additional computations of H3BO3 and H4BO -4 isolated clusters. For that purpose we used a large, isolated simulation box with a cell length of 16 Å, forcing the charge density to be zero at the boundary, as implemented in CPMD code. In order to compute the β factors of boron species in the aqueous fluid we apply the same method as in our recent work on Li isotopes (Kowalski & Jahn, 2011), with the exception that we use the pseudofrequencies, i.e. the frequencies obtained from the three force constants acting on the fractionating element, and formula 1 for calculation of β factors, as discussed in section 2.1.2. In order to fully account for the spatial continuity of the fluid and its dynamical motion we produced 10 ps long molecular dynamics trajectories of systems consisting of 64 H2O molecules and one H3BO3 or H4BO -4 molecule for different T = 1000 K, 800 K and 600 K and pressure of 0 . 5 GPa, which closely resembles the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The corresponding simulation box length is 13 . 75 Å at T = 1000 K. The β factors were computed on the ionic configuration snapshots extracted uniformly in 0 . 1 ps intervals along the molecular dynamics trajectories.\",\n \"pages\": [\n 8,\n 9\n ]\n },\n {\n \"title\": \"2.5. Error estimation technique\",\n \"content\": \"The errors in the computed value of the ( β -1) and ∆ fractionation factors were estimated from an average error of vibrational frequencies computed using the chosen DFT method. Finley & Stephens (1995), Menconi & Tozer (2002) and Alecu et al. (2010) estimated the errors made in calculations of vibrational frequencies of small molecules using di ff erent DFT functionals. According to these works the BLYP functional systematically overestimates the harmonic frequencies by ∼ 3 . 5 %, with a deviation from the mean o ff set of ∼ 1 %. Therefore, we expect that using BLYP functional the ( β -1) and ∆ values are systematically overestimated by 7 % and that in addition there is a 2 % error in derived ( β -1) factors. Similar errors result from using other functionals or even more sophisticated and time consuming post-Hartree-Fock methods such as MP2 (Finley & Stephens, 1995; Alecu et al., 2010).\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"3.1. Test of the computational method\",\n \"content\": \"First, we illustrate the performance of the approximation proposed in section 2.1.2 by computing the β factors for the isolated H3BO3 and H4BO -4 molecules and selected crystalline solids. In Figure 1 we present three sets of calculations of β factors: (1) the 'exact' result obtained from a full normal mode analysis and formula 1, (2) the results obtained applying Kowalski & Jahn (2011) method based on Eq. 3, (3) the results obtained using pseudofrequencies computed for the fractionating element and Eq. 1 for the estimation of the β factor. The numerical values for selected temperatures are reported in Table 2. Approach (3) results in much better agreement with the 'exact' result. For H3BO3, the β factor is overestimated by only 0 . 5 /permil and 1 . 5 /permil for temperatures of 800 K and 600 K respectively. Applying method (2), the error is more pronounced, 2 . 2 /permil and 6 . 7 /permil respectively. In the case of molecular H4BO -4 , the errors using method (3) for the same temperatures are only 0 . 1 /permil and 0 . 5 /permil respectively. The same behavior is shown for dravite and boromuscovite crystalline solids that contain boron in the coordinative arrangement that resemble the configurations of aforementioned B-bearing molecules. For T > 600 K the proposed method represents only a few percent correction to the approximation given by equation (3), so the relation (15) is satisfied. It is evident that for B-bearing materials considered here the improvement made by using the pseudofrequencies based approach is substantial. It corrects for about 75% of error of the Bigeleisen & Mayer (1947) approximation (Eq. 3). However, the question of general applicability of the proposed method to other isotopic systems would require careful testing on a large set of materials, which is well beyond the scope of the current paper.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"3.2. B isotope fractionation in gas and fluid phases 3.2.1. B isotope fractionation between H3BO3 and H4BO -4 in the gas phase\",\n \"content\": \"In a first step of our investigation of boron-rich aqueous fluids we derived the full frequency spectra of molecules in the gas phase (isolated molecules). The relevant β factors were computed using Equation 1. These studies were performed in order to compare our results with the published values of Zeebe (2005), both computed using the same DFT BLYP functional. In Table 1 we report the computed frequencies that are affected by the di ff erent B isotope substitutions along with other theoretical estimations and experimental measurements. The computed frequencies are in good agreement with earlier theoretical predictions and show similar agreement with the experimental measurements. The results in terms of computed β factors for the two considered species are reported in Figure 2, where we compare our results with the values computed using frequencies of Zeebe (2005). The comparison of the two sets of calculations reveals that our β factors for both species are smaller by ∼ 1 /permil at 600 K -1000 K than the values of Zeebe (2005). However, the di ff erence between β factors of H3BO3 and H4BO -4 remains nearly identical in both sets of calculations and the agreement is nearly perfect for higher temperatures. We note, that for the comparison we used the frequencies of Zeebe (2005) computed using 6-31 + G(d) basis set, as only these are provided by the authors. β and α factors obtained at T = 300 K using a more extended 6311 + G(d,p) basis set indicate that the β factors using 6-31 + G(d) basis set are not fully converged. In particular, the ( β -1) factor of H4BO -4 computed with 6311 + G(d,p) basis set is 3 . 9 /permil smaller than the one derived using 6-31 + G(d). In Table 5, we compare these results with the results of our calculation. It is clearly seen that for lower temperatures such as T = 300 K the values computed with 6-311 + G(d,p) basis set are in better agreement with our results indicating that planewave based DFT approach we use provides adequate vibrational frequencies and resulting isotope fractionation factors.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"3.2.2. B isotope fractionation between H3BO3 and H4BO -4 in aqueous fluid\",\n \"content\": \"In order to obtain the temperature dependent β factor for aqueous fluids we fitted the function 1 + A / T 2 + B / T 4 to the computed values using the least squares minimization procedure. The computed β values for H3BO3 in fluid are: 1 . 02366 ± 0 . 00012, 1 . 03624 ± 0 . 00018 and 1 . 06262 ± 0 . 00010and for H4BO -4 in fluid are: 1 . 01745 ± 0 . 00005, 1 . 02650 ± 0 . 00010and 1 . 04597 ± 0 . 00015, for the temperatures of 1000 K, 800 K and 600 K respectively. The resulting temperature dependent β factor for H3BO3 is β = 1 + 2 . 416 · 10 4 / T 2 -5 . 823 · 10 8 / T 4 and for H4BO -4 is β = 1 + 1 . 772 · 10 4 / T 2 -4 . 234 · 10 8 / T 4 . The results for H3BO3 and H4BO -4 in aqueous so- lutions are shown in Figure 3. As was observed for the isolated molecules, the β factor of H3BO3-bearing fluid is substantially larger than the one for the H4BO -4 . This can be understood in terms of the substantial difference in the B-O bond lengths exhibited by the two considered species. In case of isolated molecules our calculations indicate a B-O bond length of 1 . 40 Å for H3BO3 and 1 . 51 Å for H 4BO -4 . We compared our β factors with the values computed by Sanchez-Valle et al. (2005), which were derived by the combination of force field methods and experimental data to derive accurate vibrational frequencies. For H3BO3 we got a nearly identical result. In case of H4BO -4 our calculation predicts a value which is lower by 2 -4 /permil . However, Rustad et al (2010b) and Rustad & Bylaska (2007) revealed the improper assignment of a major fractionating vibrational mode of H4BO -4 in the force field by Sanchez-Valle et al. (2005). This leads to the underestimation of the fractionation factor between aqueous H3BO3 and H4BO -4 by Sanchez-Valle et al. (2005). Assuming that α ∝ T -2 and having the di ff erence between BLYP calculations of Rustad et al (2010b) and Sanchez-Valle et al. (2005) of ∆ α = 16 . 4 /permil at T = 300 K, the value reported by Sanchez-Valle et al. (2005) should be underestimated by ∆ α = 16 . 4 · 2(300 / T ) 2 /permil , which results in ∆ α ∼ 1 . 5 /permil at T = 1000 K. Corrected in such as way result of Sanchez-Valle et al. (2005) is also plotted in Figure 3. It is now very consistent with our prediction.\",\n \"pages\": [\n 10,\n 11\n ]\n },\n {\n \"title\": \"3.2.3. Discussion of computational errors\",\n \"content\": \"Most previous computational studies of boron isotope fractionation in aqueous solutions concentrate on the computation of the isotope fractionation at ambient conditions (Rustad et al, 2010b; Rustad & Bylaska, 2007; Liu & Tossell, 2005; Zeebe, 2005). Rustad et al (2010b) performed detailed analysis of impact of the chosen computational method (HF, MP2, di ff erent DFT functionals) and size of the basis set on the calculated fractionation factors between H3BO3 and H4BO -4 . They found that DFT methods are not performing well for the borate system and concluded that DFT 'is of limited usefulness in chemically accurate predictions of isotope fractionation in aqueous systems' (Rustad et al, 2010b). The empirically derived error of the derived fractionation factor is of the order of 5 -10 /permil for a total fractionation of ∼ 30 /permil . We note that this is expected and clearly visible if we apply the error estimation procedure outlined in section 2.5. For instance, at room temperature the derived beta factors using the BLYP functional are 213 . 6 /permil and 173 . 3 /permil respectively (Rustad et al (2010b), Table 2). This gives a fractionation factor of 1.0343. Following our error estimation scheme, the absolute error of the fractionation factor is 10 . 5 /permil , and the properly reported computed value is α = 1 . 034 ± 0 . 011. When one corrects for the systematic error of 7% and assumes 2% of statistical error on β factors, then the value of α decreases and the error is slightly smaller, i.e. α = 1 . 032 ± 0 . 008. This is in good agreement with the experimental data reported in Rustad et al (2010b) and explains the spread of the values computed using di ff erent methods and reported in that paper. It is very di ffi cult to get the fractionation factors for ambient conditions, as the fractionation factor is often just a small fraction of the relevant ( β -1) factors, ( α -1) ∼ 0 . 15( β -1) in the considered case. Assuming that ( α -1) = 0 . 15( β -1), a 2% error in the ( β -1) factors leads to an absolute error in ( α -1) of 0 . 04( β -1) = 0 . 04( α -1) / 0 . 15 ∼ 0 . 27( α -1), i.e. ∼ 27% of relative error in the derived fractionation factor ( α -1). On the other hand, we note that such a big error is not substantially larger than the uncertainties in the experimental data reported by Rustad et al (2010b) in their Figure 2. Thus, the case of boron fractionation in aqueous fluid at ambient conditions does not necessarily show the limited usefulness of DFT in the prediction of isotope fractionation factors, but only reflects the fact that precise estimation or measurement of the B isotopes fractionation factors at ambient condition requires unprecedented accuracy of both experimental or computational techniques. For instance, in order to get the value of ( α -1) with a relative error of 5% (at ambient conditions) one needs to estimate the ( β -1) factors or measure relevant quantities with precision of less than 1%. At higher temperatures the situation is di ff erent. Looking just at the fractionation between H3BO3 and H4BO -4 in the gas phase or the aqueous solution one can see that for T > 600 K the fractionation factor between the two substances, ( α -1), is at least 25% of the ( β -1) factor. This results in smaller 0 . 04 / 0 . 25 ∼ 16% for T = 600 K and 0 . 04 / 0 . 36 ∼ 11% for T = 1000 K relative error, which is acceptable in our calculations. Nevertheless, this case shows the importance of proper error estimation on the computed fractionation factors. Such an estimation is usually omitted or not provided explicitly, which can lead to wrong conclusions when the theoretical prediction is confronted with the measured data.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"3.2.4. Pressure dependence of the fluid fractionation factor\",\n \"content\": \"In our recent paper (Kowalski & Jahn, 2011) we have shown that due to compression the β factor of Li in aqueous fluid increases with increase in pressure (for P > 2 GPa). The same should happen for H3BO3 and H4BO -4 aqueous fluid as the vibrational frequencies of boron species in aqueous fluid increase with increase in pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). Having the experimental data we checked whether the derived pressure-dependent β factors are consistent with the pressure shifts of vibrational frequencies of considered boron species measured by Sanchez-Valle et al. (2005). For that purpose we performed a set of calculations using supercells containing 8 water molecules and the relevant boron species. We note that in line with our previous results for Li (Kowalski & Jahn, 2011), the obtained values of ( β -1) at P = 0 . 5 GPa are within 0 . 1 /permil in agreement with the values obtained for supercells containing 64 water molecules. The results are given in Figures 4 and 5. The computed ( β -1) values for H3BO3 fluid show a linear dependence in pressure, ( β -1) = 23 . 60 + 0 . 28 P (GPa) /permil . This is expected as ( β -1) ∝ ν 2 ∼ ν 2 0 + 2 ν 0 ∆ ν (Schauble, 2004) and ∆ ν is a linear function of pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). In case of H4BO -4 the pressure-dependence is linear up to P ∼ 2 -3 GPa and it becomes less steep at higher pressures. In order to quantitatively check the consistency of our prediction with the measured vibrational frequency shifts of Sanchez-Valle et al. (2005) we derived the relative shifts in the ( β -1) factor assuming that ( β -1) ∝ ν 2 and the measured pressure dependence of the frequency shifts: ∆ ν = 2 . 15 cm -1 · P (GPa) and ∆ ν = 3 . 50 cm -1 · P (GPa) for 1454 cm -1 and 666 cm -1 vibrational frequencies of H3BO3 and ∆ ν = 6 . 47 cm -1 · P (GPa) for the 975 cm -1 vibrational frequency of H4BO -4 . The chosen vibrational frequencies are the ones a ff ected by the di ff erent B isotope substitution. Our predicted shift of ( β -1) matches well the shifts derived from the measured frequency shifts. Such a good agreement with the experimental data validates further our computational approach and shows that ab initio calculations can be successfully used in derivation of the pressure dependence of the fractionation factors and pressure-induced vibrational frequencies shifts. Moreover, first principles calculations can be useful in extrapolation of the experimental values for β and ∆ ν to more extreme conditions, which otherwise are extremely di ffi cult to reach by experimental techniques.\",\n \"pages\": [\n 12,\n 13\n ]\n },\n {\n \"title\": \"3.3. Fluid-mineral fractionation\",\n \"content\": \"Next we present the results of the fractionation between boron bearing fluids and minerals such as dravite, olenite and boromuscovite. The aim of these studies is to investigate the mechanisms driving the fractionation process, the role of coordination and the B-O bond length. Below we discuss each case separately.\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"3.3.1. Tourmaline-neutral fluid\",\n \"content\": \"Meyer et al. (2008) measured the boron isotope fractionation between tourmaline and neutral fluid at T = 400 -700 o C and P = 0 . 2 GPa. In the experiment the tourmaline was represented by dravite. In contrast to the former measurements of Palmer et al. (1992) the measured fractionation is very small and does not exceed 2 . 5 /permil at 400 o C. Our calculated fractionation curve together with the experimental data are given in Figure 6. Our result correctly reproduces the experimental measurements within the computational accuracy. The dravite-fluid fractionation is small as the two materials contain boron in BO3 units. We also predict a small fractionation between olenite carrying 3-fold coordinated boron only and aqueous fluid, although the olenite-fluid fractionation is positive because of the shorter B-O bond lengths for olenite (1 . 378 Å vs. 1 . 397 Å).\",\n \"pages\": [\n 13\n ]\n },\n {\n \"title\": \"3.3.2. Boromuscovite-strongly basic fluid\",\n \"content\": \"Boromuscovite synthesized in the experiments of Wunder et al. (2005) consisted of two type of polytypes, 1M ( ∼ 10 %) and 2M1 ( ∼ 90 %). In order to be consistent with the experimental conditions, we derived the β factors for both polytypes and computed their weighted average. We note that the β factors for both polytypes of mica are similar with the di ff erence in ( β -1) not larger than 3 %. This is consistent with similar B-O bond lengths (1 . 532 Å) found in both polytypes. Boromuscovite contains boron in tetrahedral sites. Therefore, in order to investigate the impact of the B-O bond length on the fractionation we first compare the fractionation between the mineral and a strongly basic fluid contain- n in H4BO -4 . The result, together with the measurements of Wunder et al. (2005) of the fractionation between boromuscovite and strongly basic fluid, is summarized in Figure 7. Our calculations predict a negative fractionation between mica and the H4BO -4 fluid. The agreement of our prediction with the experimental measurements is relatively good; however, the experimental data indicate slightly stronger fractionation. We note that the experimental conditions of Wunder et al. (2005) do not assure that the measured basic fluid contained four-fold coordinated boron species, i.e. H4BO -4 , only. As we indicate in the Figure 7, the presence of as little as 10% of H3BO3 in the measured basic fluid brings the prediction and measurements into much better agreement. It makes sense that the β factor of boromuscovite is smaller than for aqueous H4BO -4 as the average B-O bond length in mica is 1 . 532 Å, while it is 1 . 513 Å and therefore shorter in case of aqueous H4BO -4 . We note that in our derivation we assumed that the fluid consists mostly of [4] B species. As we already mentioned, although Wunder et al. (2005) call the fluid 'strongly basic' its exact composition, especially the amount of [3] B species is unknown. However, if for instance the [4] B to [3] B ratio was 1, then the predicted mica-basic fluid fractionation at 800 K would be 5 /permil larger than measured. This would result in a large inconsistency between the computed values and the experimental data. On the other hand, good agreement between the prediction and the measurements indicates that the strongly basic fluid was dominated by H4BO -4 species, which is in line with previous studies (Zeebe, 2005; Sanchez-Valle et al., 2005).\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"3.3.3. Boromuscovite-neutral fluid\",\n \"content\": \"The fractionation between boromuscovite and neutral fluid involves a change in coordination from B [4] in boromuscovite to B [3] in neutral fluid. Wunder et al. (2005) measured the fractionation between the two materials at 3 GPa, shown in Figure 8. The predicted fractionation is about 3 ± 2 . 5 /permil larger than the measured value. Looking for potential sources of this discrepancy, we have checked for the e ff ect of the change in lattice parameters due to combined thermal expansion and compression. For that purpose we applied the EOS of Holland and Powell (2011) for muscovite, which gives 4 . 4%, 3 . 8% and 3 . 1% decrease in volume for T = 600 K, 800 K and 1000 K respectively and P = 3 GPa. As muscovites show highly anisotropic compressibility patterns, in line with Comodi & Zanazzi (1995) we applied the T and P driven change in volume assuming the 16%, 19% and 65% contribution to compression along the /vector a , /vector b , /vector c lattice vectors. On the other hand ( β -1) factors of H3BO3 and H4BO -4 in aqueous solutions at P = 3 GPa increase by 3 . 5% and 11 . 8% respectively (Figures 4 and 5), leading to pressure-induced increase in the boromuscovite-aqueous fluid fractionation at the experimental pressure. The ∆ factor, corrected for e ff ects of thermal expansion and compression of boromuscovite and compression of fluid, is also given in Figure 8. Because the high T and P e ff ects result in similar increases in the β factors for both solid and aqueous fluid, the resulting fractionation factor between these two phases is close to the one derived at ambient conditions. Therefore, thermal expansion and compression e ff ects cannot explain the observed discrepancy between prediction and the measurements of Wunder et al. (2005). On the other hand, the comparison of our results with the experimental data suggests that the fractionation between boromuscovite and fluid is the same as between H3BO3 and H4BO -4 fluids (see Figure 3), which is at odds with the non-negligible and negative fractionation between boromuscovite and a strongly basic fluid. In section 3.3.2 we have shown that we are able to correctly reproduce the fractionation between boromuscovite and strongly basic fluid, which indicates that our result for boromuscovite is reliable. This suggests that another, unaccounted e ff ect leads to the decrease of the boron isotope fractionation between mica and neutral fluid in the experiments of Wunder et al. (2005). One possible solution for the discrepancy is a nonnegligible amount of boron residing in four-fold coordinated configurations in neutral solution. This is in line with the Raman spectroscopy measurements of Schmidt et al. (2005), who detected a broad peak in the Raman spectra of neutral H3BO3-dominated fluid and attributed it to B [4] species. The integrated area of this peak, compared to the peak of the Raman 877 cm -1 line of B [3] species, indicates the presence of at least 15 -30 % of B [4] species by mole fraction. Assuming that there is 15 -30 % of B [4] species present in the fluid and that the β factor of these species is similar to that of H4BO -4 , the fractionation factor between boromuscovite and H3BO3 aqueous fluid decreases bringing the theory and the experiment to better agreement, which is illustrated in Figure 8. If this interpretation is true, it suggests that boron isotope fractionation could be used to gather information on the speciation of B in aqueous fluids.\",\n \"pages\": [\n 14,\n 15\n ]\n },\n {\n \"title\": \"3.4. B isotope fractionation between minerals\",\n \"content\": \"The boron isotope fractionation between B-bearing crystalline solids has received considerable attention recently (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002). We focus here on the investigation of boron isotope fractionation between mica and tourmaline as boron atoms in these minerals occupy sites of di ff erent coordination, which should result in a large B isotope fractionation between these two minerals. In mi- cas boron substitutes for silicon in the four-fold coordinated site (Wunder et al., 2005), while in tourmaline (dravite) it is incorporated in the three fold coordinated site (Meyer et al., 2008). Comparing the B isotope fractionation between di ff erent minerals, melts and fluids Wunder et al. (2005) have shown that the fractionation between two materials of di ff erent B coordination is large, reaching 5 /permil at 1000 K and much higher values at lower temperatures. Klemme et al. (2011),Marschall (2005) and Hervig et al. (2002) measured the fractionation between coexisting phases of the two minerals in natural samples. The fractionation between these two minerals is also derived from experimental isotopic fractionation data of B-muscovite-fluid (Wunder et al., 2005) and tourmaline-fluid (Meyer et al., 2008) systems. The results of these measurements and our computed T -dependent fractionation curve are given in Figure 9. The first striking observation is that our predicted fractionation factors are much larger (taking the absolute value) than the experimental values (Wunder et al., 2005; Meyer et al., 2008). The latter are also incon- sistent with the natural samples data of Klemme et al. (2011) and previous studies discussed in that paper (Marschall, 2005; Hervig et al., 2002). On the other hand, the measurements on natural samples are consistent with our calculated values, which tends to validate our predictions. We notice that the most recent measurements of boron isotope signatures of tourmaline and white mica from the Broken Hill area in Australia by Klemme et al. (2011) indicate for the assumed temperature of 600 o C that the fractionation factor between the two phases is 10.4 ± 2 . 7 /permil , which is in good agreement with our computed value of 10 . 7 ± 1 . 8 /permil . The experimental mica-tourmaline B isotope fractionation factors of Wunder et al. (2005) and Meyer et al. (2008) are 2 /permil and 6 /permil smaller at temperatures of 1000 K and 800 K respectively, with an experimental uncertainty of 2 /permil . However, this discrepancy can be resolved by assuming that in the experiments of Wunder et al. (2005) the fluid contained a significant admixture of B [4] species, which leads to the underestimation of the experimental boromuscovite-fluid fractionation factor by ∼ 2 /permil at 1000 K and ∼ 3 . 5 /permil at 1000 K, as is seen in Figure 8. The experimental mica-tourmaline fractionation factor corrected for the presence of B [4] species is also plotted in Figure 9. It is now more consistent with the natural data. We note that this result independently supports the conclusion underlined in section 3.3.3 and result of Schmidt et al. (2005) that highP , B-bearing neutral fluids contain significant admixtures of B [4] species. Olenite is a mineral which can incorporate boron in both trigonal and tetrahedral sites as it substitutes for both Al and Si atoms. It is therefore interesting to check the fractionation of boron isotopes between the two differently coordinated sites in one mineral and compare it with the above result for mica and tourmaline. The computed fractionation between the trigonal and tetrahedral sites at 600 o C is 10 . 6 ± 1 . 9 /permil , which is consistent with the fractionation between tourmaline and mica, indicating that the coordination of the B atom is the driving factor for the fractionation of the B isotopes. Similarly, we computed the boron isotopes fractionation between trigonal and tetragonal boron sites in dravite. In order to create the tetragonal B site we replaced one Si atom with B and we added one H atom forming an additional OH group to compensate the charge. The computed fractionation between the sites at 600 o C is 8 . 9 ± 1 . 7 /permil , which is also in agreement with the aforementioned results. Next, we will show that the value of the β factor depends not only on coordination but is also strongly correlated with B-O bond length.\",\n \"pages\": [\n 15,\n 16\n ]\n },\n {\n \"title\": \"3.5. Fractionation between B [3] and B [4] materials\",\n \"content\": \"The β factors computed for all the considered materials are grouped together in Figure 10. β factors can be grouped into two sets, one that includes materials with boron in three-fold coordination and another one that includes materials having boron in four-fold coordination. For olenite and dravite we also computed the β factors with boron sitting on four-fold coordinated site. The β factor for these crystalline solids with given B [3] / B [4] ratio can be derived as a weighted average of the β factors obtained for boron sitting on the two di ff erently coordinated sites. The fractionation factor between materials of di ff erent boron coordination is ∼ 8 /permil on average at T = 1000 K. We note that it is ∼ 3 /permil larger than the one deduced by Wunder et al. (2005) from measurements performed on solids, silicate melts and fluids, but this can be attributed to the underestimation of the fractionation factors for boromuscovite-fluid system by Wunder et al. (2005) due to potential admixture of B [4] species in the investigated fluid. Our results show also a substantial spread of β factors of substances containing boron of a given coordination. The spread is at least 4 /permil and results from di ff erent B-O bond lengths. We illustrate this in Figure 11 by plotting together the β factors derived for all considered materials at T = 1000 K as a function of B-O bond length. It is clearly seen that there is a roughly linear correlation between the β factor and BO bond length, which is especially evident comparing the results for crystalline solids. For instance, out of the considered B [4] -bearing minerals boromuscovite has the longest B-O bond length of 1 . 525 Å (1M) and 1 . 516 Å (2M1), followed by dravite 1 . 514 Å and olenite with BObond length of 1 . 502 Å. This tracks the di ff erences in the β factors derived for these materials. In addition the materials having B [3] species only exhibit shorter bond lengths of ∼ 1 . 37 Å and higher β factors, while the materials containing B [4] species having bond lengths of about ∼ 1 . 52 Å show much smaller β factors. Therefore, the tighter bonding of B [3] species likely explains why the heavy B isotope prefers the less coordinated phases. This clearly shows that the change in the B-O bond length during an isotope exchange is the leading factor driving the production of the boron equilibrium isotope signatures at high T .\",\n \"pages\": [\n 16,\n 17\n ]\n },\n {\n \"title\": \"4. Conclusions\",\n \"content\": \"In this work we have presented a detailed analysis of boron isotope fractionation between boronbearing crystalline solids and aqueous fluids at high T and P conditions. In order to perform our investigation we have applied and extended a computationally e ffi cient approach for the computation of isotope fractionation factors for complex minerals and fluids at high temperatures and pressures presented by Kowalski & Jahn (2011). As an extension to the Bigeleisen & Mayer (1947) 'single atom approximation' method we demonstrated that using the pseudofrequencies derived from the force constants acting on the fractionating element together with the full formula for computation of the reduced partition function ratios results in significant improvement in the accuracy of the computed fractionation factors, which is essential when lower temperature materials and high vibrational frequency complexes are considered. In order to understand the fractionation between Bbearing crystalline solids and aqueous fluids we performed a set of calculations of β factors for dravite, olenite, boromuscovite and aqueous solutions of H3BO3 and H4BO -4 . In agreement with the experimental findings we show that the fractionation strongly correlates with coordination through the change in the B-O bond length. The lower trigonal coordination BO3 arrangement results in higher 11 B / 10 B (by ∼ 8 /permil at T = 1000 K) than the tetrahedrally coordinated boron complexes, which exhibit ∼ 0 . 15 Å longer B-O bonds. The computed fractionation between minerals and fluids of the same coordination are in good agreement with experiments. However, we predict larger isotope fractionation between boromuscovite and H3BO3 fluid (by at least a few /permil ) than was measured in situ at high P by Wunder et al. (2005) and Meyer et al. (2008), but that is consistent with measurements on natural samples. We note that the presence of B [4] in highP fluid could reconcile the in situ experimental results with our prediction and other measurements. This is expected from the experiments of Schmidt et al. (2005), but requires further experimental confirmation. If true, this would open the possibility for using the isotope fractionation techniques as a tool to measure the speciation of boron in fluids and crystalline solids. We have also demonstrated that with our computational approach we are able to correctly predict the pressure-induced isotope fractionation for compressed aqueous fluids, which indicates the ability of ab initio methods to predict the isotopic signatures of highly compressed materials, even those that are difficult to investigate experimentally. Our study confirms that ab initio computer simulations are a useful tool not only for prediction but also understanding the equilibrium stable isotope fractionation processes between various phases, including aqueous solutions, at high pressures and temperatures. They can nicely complement experimental e ff orts, provide unique insight into the isotope fractionation process on the atomic scale and deliver data for conditions that are inaccessible by the current experimental techniques.\",\n \"pages\": [\n 17\n ]\n },\n {\n \"title\": \"Acknowledgements\",\n \"content\": \"The authors wish to acknowledge financial support in the framework of DFG project no. JA 1469 / 4-1. Part of the calculations were performed on the IBM BlueGene / P JUGENE of the John von Neumann Institute for Computing (NIC). We are also grateful the associate editor Edwin A. Schauble and anonymous referees for constructive comments that helped improving the manuscript.\",\n \"pages\": [\n 18\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Alecu I. M., Zheng J. , Zhao Y. and Truhlar D. G. (2010) Computational Thermochemistry: Scale Factor Databases and Scale Factors for Vibrational Frequencies Obtained from Electronic Model Chemistries. J. Chem. 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Acta 62 , 3337-3347. Domagal-Goldman S. D. and Kubicki J. D. (2008) Density functional theory predictions of equilibrium isotope fractionation of iron due to redox changes and organic complexation. Geochim. Cosmochim. Acta 72 , 5201-5216. Domanik K.J., Hervig R.L. and Peacock S.M. (1993) Beryllium and boron in subduction zone minerals: An ion microprobe study. Geochim. Cosmochim. Acta 57 , 4997-5010. Marler B., Borowski M., Wodara U., Schreyer W. (2002) Synthetic tourmaline (olenite) with excess boron replacing silicon in the tetrahedral site: II. Structure analysis Eur. J. Mineral. 14 , 763-771. Marschall H. R. (2005): Lithium, Beryllium, and Boron in High-Pressure Metamorphic Rocks from Syros (Greece). PhD thesis, Universitt Heidelberg, Heidelberg, Germany [http: // www.ub.uni-heidelberg.de / archiv / 5634]. Marx D. and Hutter J. (2000) Ab initio molecular dynamics: Theory and Implementation. Modern Methods and Algorithms of Quantum Chemistry 1 edited by J. Grotendorst, NIC, FZ Julich, 301-449 ; CPMD code: J. Hutter et al. , see: www.cpmd.org . Meheut M., Lazzeri M., Balan E. and Mauri F. (2007) Equilibrium isotopic fractionation in the kaolinite, quartz, water system: Prediction from first-principles density-functional theory. Geochim. Cosmochim. Acta 71 , 3170-3181. Chem. Geol. 258 , 28-37. Menconi G. and Tozer D. J. (2002) Diatomic bond lengths and vibrational frequencies: assesment of recently developed exchangecorrelation functionals. Chem. Phys. Lett. 360 , 38-46. Meyer Ch., Wunder B., Meixner A., Romer R. L., and Heinrich, W. (2008) Boron isotope fractionation between tourmaline and fluid; an experimental re-investigation. Contrib. Mineral. Petrol. 156 , 259-267. Morgan VI, G.B. and London, D. (1989) Experimental reactions of amphibolite with boron-bearing aqueous fluids at 200 MPa: implications for tourmaline stability and partial melting in mafic rocks. Contrib. Mineral. Petrol. 102 , 281-298. Redlich, O. (1935). Eine allgemeine Beziehung zwichen den Schwingungsfrequenzen isotoper Molekeln. Z. Physikal. Chem. B 28 , 371-382. Rustad J. R, Casey W. H., Yin Q.-Z., Bylaska E. J., Felmy, A. R., Bogatko S. A., Jackson V. E., Dixon D. A. (2010) Isotopic fractionation of Mg2 + (aq), Ca2 + (aq), and Fe2 + (aq) with carbonate minerals, Geochim. Cosmochim. Acta 74 , 6301-6323. Rustad J. R. and Bylaska E. J. (2007) Ab Initio Calculation of Isotopic Fractionation in B(OH)3(aq) and BOH -4 (aq). J. Am. Chem. Soc. (Communication) 129 , 2222-2223. Sanchez-Valle C., Reynard B., Daniel I., Lecuyer C., Martinez I., and Chervin J. C. (2005) Boron isotopic fractionation between minerals and fluids; new insights from in situ high pressure-high temperature vibrational spectroscopic data Geochim. Cosmochim. Acta 69 , 4301-4313 Schauble E. A. (2004) Applying Stable Isotope Fractionation Theory to New Systems. Rev. Mineral. Geochem. 55 , 65-111. Schreyer W., Wodara U., Marler B., van Aken P.A., Seifert F. and Robert J.-L. (2000) Synthetic tourmaline (olenite) with excess boron replacing silicon in the tetrahedral site: I. Synthesis conditions, chemical and spectroscopic evidence. Eur. J. Mineral. 12 , 529-541. Urey, H. C. (1947). The Thermodynamic Properties of Isotopic Substances. J. Chem. Soc. , 562581. and scientific use. J. Phys. Chem. Ref. Data 31 , 387-535. Wunder B., Meixner A., Romer R. L., Wirth R., and Heinrich W. (2005) The geochemical cycle of boron; constraints from boron isotope partitioning experiments between mica and fluid Lithos 84 , 206-216 Yamaji K., Makita Y., Watanabe H., Sonoda A., Kanoh H., Hirotsu T. and Ooi K. (2001) Theoretical Estimation of Lithium Isotopic Reduced Partition Function Ratio for Lithium Ions in Aqueous Solution. J. Phys. Chem. A 105 , 602613.\",\n \"pages\": [\n 18,\n 19\n ]\n }\n]"}}},{"rowIdx":271,"cells":{"bibcode":{"kind":"string","value":"2013GrCo...19..275C"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1212.0484.pdf"},"content":{"kind":"string","value":"\nOn stability of power-law solution in multidimensional Gauss-Bonnet cosmology\nD.M.Chirkov 1,2 ∗ , A.V.Toporensky 1 †\nJuly 16, 2018\n1 Sternberg Astronomical Institute, Moscow State University, Universitetsky pr., 13, Moscow, 119991, Russia\n2\nFaculty of Physics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia\nAbstract\nWe consider dynamics of a flat anisotropic multidimensional cosmological model in Gauss-Bonnet gravity in the presence of a homogeneous magnetic field. In particular, we find conditions under which the known power-law vacuum solution can be an attractor for the case with non-zero magnetic field. We also describe a particular class of numerical solution in (5 + 1) -dimensional case which does not approach the power-law regime.\nI. Introduction.\nThe fact that structure of initial cosmological singularity can be rather complicated have been recognized since 60-th of the last century when the conception of BKL chaos have been presented [1]. It appears that Kasner solution being a general solution for a vacuum Bianchi I Universe becomes unstable in the case of metric with positive spatial curvature (belonging to Bianchi IX class) and is replaced by a complicated sequence of transient \"Kasner epochs\". Later it was found that some classes of an anisotropic matter can induce similar type of cosmological behavior even in flat Bianchi I case. This can be shown for a magnetic field by LeBlanc [2] and for a general vector field by Kirillov [3]. The BKL analogs for the flat magnetic Universe is a typical behavior when off-diagonal terms in the metric in the frame determined by magnetic field are present. For a diagonal case see for example [4].\nPossible generalisation of these results may involve studies in the framework of multidimensional cosmology as well as alternative gravity theories. Analogs of magnetic field for multidimensional space-times have been studied in [13, 14, 15]. If we want to introduce some corrections to Einstein gravity, the first goal is to find a solution which replaces Kasner solution for a flat anisotropic Universe. In the present paper\nwe consider Lovelock gravity as a generalization of General Relativity. The principal feature of Lovelock gravity is that this theory keeps the order of equations of motion the same as in GR, while other theories (like popular now f ( R ) -theory) results in increasing of the number of derivatives in equations of motion. Another important property of Lovelock gravity is that it gives corrections to GR only in higher-dimensional space-time, so it is natural to consider it in the framework of multidimensional cosmology. As the number of non-Einstein terms in Lovelock gravity is finite for any given dimensionality of space-time, it is possible to consider regimes when these terms are not only small corrections to Einstein gravity. In particular, it is reasonable to expect that the highest order Lovelock term, consisting of highest power of curvature invariants, dominates near a cosmological singularity. In the present paper we consider the second Lovelock term, which is the famous Gauss-Bonnet combination. It is the highest possible Lovelock term for (4 + 1) and (5 + 1) dimensional spacetimes. As cosmology in the (4 + 1) dimensional case has some pathological features, we restrict ourselves by the ( N +1) dimensional case with N /greaterorequalslant 5 in the present paper.\nIn late 80-th some vacuum solutions for a flat anisotropic Universe in Gauss-Bonnet gravity for N = 4 , 5 have been found [5, 6]. They replace Kasner solution of GR. Later these solutions have been rediscovered and verified for N = 6 , 7 in [7]; after that this solution have been generalized for all N and also to the general Lovelock gravity [8]. The main goal of this paper is to address question of its stability near the initial singularity. It is known that in the presence of an ordinary isotropic matter these solutions are stable to the past if the matter has the equation of state with w < 1 / 3 , otherwise the solution tends to isotropic one [9, 10]. For more interesting regimes near a singularity it is necessary to consider either anisotropic matter or curved geometries. As introduction of spatial curvature usually leads to very cumbersome equations of motion, we have chosen a magnetic fields as a possible source of instability. In this paper we restrict ourselves by the diagonal case.\nIt is also known that in contrast to Einstein gravity certain initial conditions lead to exponential solutions instead to those of power-law behavior [9, 11, 12], we do not consider them in the present paper.\nThe structure of the paper is as follows: In Sec. II. we describe the metric and matter content of the model studied, in Sec. III. the model is investigated in the framework of Einstein relativity. We present results on stability of Kasner solution in the presence of magnetic field. The same model in Gauss-Bonnet gravity is studied in Sec. IV. with presentation made as parallel as possible to presentation in the preceding section in order to compare Einstein and Gauss-Bonnet cases. In Sec. V. we describe a particular regime existing in the zone of instability of power-law regimes in (5+1) -dimensional Gauss-Bonnet gravity. Sec. VI. contains a brief summary of results obtained.\nII. Preliminaries.\nIn what follows we use a reference system chosen so that 1 :\ng 00 = 1 , g kk = -e 2 a k ( t ) , g ij = 0 , i = j (1)\n/negationslash\nIn this section we looking for the energy-momentum tensor of a pure magnetic field in ( N +1) -dimensional space-time. In the general case components of the energy-momentum tensor of the electromagnetic field are\nT µ ν = 1 4 π ( F νγ F γµ + 1 4 δ µ ν F αβ F αβ ) (2)\nwhere F αβ is the Faraday tensor. The components of the Faraday tensor obey the following equations:\n{ ∇ µ F µν = 0 , dF = 0 , (3)\nor, in more detail:\n{ D µ F µν +Γ µ µλ F λν +Γ ν µσ F µσ = 0 , ( dF ) αβγ = 0 , α < β < γ , (4)\nwhere D µ = ∂ ∂x µ . As it is known\n( dF ) α 1 α 2 α 3 = D α 1 F α 2 α 3 -D α 2 F α 1 α 3 + D α 3 F α 1 α 2 , (5)\ntherefore we may rewrite system (4) in the form\n{ D µ F µν +Γ µ µλ F λν +Γ ν µσ F µσ = 0 , D α F βγ -D β F αγ + D γ F αβ = 0 , α < β < γ (6)\nThe components of the Riemann connection have the form:\nΓ i i 0 = Γ i 0 i = ˙ a i ( t ) , Γ 0 jj = ˙ a j ( t ) e 2 a j ( t ) (7)\nIts other components are equal to zero. Since we consider the case of a homogenous space, Faraday tensor depends on time t only; then, in view of (7) the system (6) takes the form:\nIts solutions are\n ˙ F 0 i + F 0 i ∑ k ˙ a k = 0 , ˙ F ij = 0 (8)\nHereinafter we will be interested in the case of a pure magnetic field; so that φ m = 0 and, as a consequence,\n F 0 m = φ m e -∑ k a k , φ m = const , F ij = ψ ij , ψ ij = const (9)\nF 0 m = F 0 m = 0 (10)\nIn what follows we will deal with diagonal energy-momentum tensor; in view of (1),(2),(9) and (10) it implies that\n/negationslash\n/negationslash\nFunctions e -2 a l are linearly independent, therefore from (11) it follows that\nT µ ν = 0 , µ = ν ⇐⇒ F νγ F γµ = 0 , µ = ν ⇐⇒ ∑ l e -2 a l ψ il ψ lk = 0 (11)\nψ il ψ lk = 0 , i, k, l = 1 , N, i = k (12)\n/negationslash\nEach ψ ij is multiplied by all that ψ kl , which has one of the indices k, l coincident with one of the indices i, j .\n/negationslash\nLet us fix pair ( i, j ) ; the number of combinations, in which ψ ij is found, equals to 2( N -2) . Let ψ ij = 0 ; then the other 2( N -2) quantities ψ ik ( ψ jk ), which are multiplied by ψ ij , must be equals to zero; indeed, if ψ ik = 0 ( k = i, j ) , then ψ ij ψ ik = 0 , but that contradict (12).\nContinuing the argument, we find the number of zero magnetic field components:\n/negationslash\n/negationslash\n/negationslash\n2( N -2) + 2( N -4) + . . . = 2 ( N -2) + ( N -2 N -1 2 ) 2 N -1 2 = ( N -1) 2 2 for an odd dimensions (13)\n2( N -2) + 2( N -4) + . . . = 2 ( N -2) + ( N -2 N 2 ) 2 N 2 = N ( N -2) 2 for an even dimensions (14)\nWe took into account that the number of pairs of indices without the same elements equals to N -1 2 for an odd dimensions and to N 2 for an even dimensions.\nThe total number of components of the magnetic field is N ( N -1) 2 ; then the number χ of non-zero magnetic field components is\nχ = N ( N -1) 2 -( N -1) 2 2 = N -1 2 for an odd dimensions , (15)\nχ = N ( N -1) 2 -N ( N -2) 2 = N 2 for an even dimensions (16)\nIn what follows they are exactly ψ 12 , ψ 34 , . . . , ψ 2 n -1 , 2 n , n = 1 , χ that we set to be non-zero; other components of the Faraday tensor are assumed to be zero. Thus, components of the energy-momentum tensor of the electromagnetic field take the form:\nT µ ν = 0 , µ = ν, (17)\n/negationslash\nT 0 0 = ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( a 2 i -1 + a 2 i ) , T n n = -ψ 2 2 n -1 , 2 n e -2( a 2 n -1 + a 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (18)\n/negationslash\nHereinafter it will be convenient to use notations like this:\nWe assume that\n< n > = { 2 n -1 , 2 n } , n = 1 , χ (19)\ni = < n > ⇐⇒ i = 2 n -1 ∧ i = 2 n, k = < n > ⇐⇒ k = 2 n -1 ∨ k = 2 n (20)\n/negationslash\n/negationslash\n/negationslash\nIII. Stability of the Kasner solutions.\nIII.1. Field equations.\nThe action reads:\nS = 1 16 π ∫ d N +1 x √ | det( g ) | ( L E + L m ) (21)\nL E = R, L m = F αβ F αβ (22)\nwhere R and F αβ are scalar curvature and Faraday tensor respectively. Here and after we use Planck units:\nc = 1 , G = 1 m N -2 Pl = 1 , (23)\nm Pl is the Planck mass. The gravitational equations is given by\nG µ ν = 8 πT µ ν (24)\nwhere\nG µ ν = R µ ν -1 2 δ µ ν R (25)\nare the components of the Einstein tensor. Taking into account (1) we get:\nG µ ν = 0 , µ = ν (26)\n/negationslash\nG 0 0 = ∑ i\n/negationslash\nAs we consider the diagonal case here, then in view of (17),(18),(24),(27) for an even number of dimensions the gravitational equations can be written as\n/negationslash\n∑ i = ( a i + ˙ a 2 i ) + ∑ i ˙ a i ˙ a j = -ψ 2 2 n -1 , 2 n e -2( a 2 n -1 + a 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (28)\n/negationslash\n∑ i\nNote that each expression like (28) describes two equations with numbers 2 n -1 and 2 n simultaneously (we use the notation (19) here). In the case of an odd number of dimensions there is one more equation in addition to these:\n/negationslash\n∑ i = N ( a i + ˙ a 2 i ) + ∑ i\n/negationslash\nThe left side of the expression (29) is a first integral of (28).\nIII.2. Vacuum solution.\nWhen there is no any matter the equations (28)-(30) lead to:\n/negationslash\n∑ i = n ( a i + ˙ a 2 i ) + ∑ i\nThese equations has the following solutions [16]:\n/negationslash\n∑ i\na k = p k ln( t ) + C k , C k = const , k = 1 , N, ∑ n p 2 n = 1 , ∑ n p n = 1 (33)\nAs a consequence, components of the metric tensor follows a power law:\nThis is well-known multidimensional Kasner solution.\ng kk ( t ) = e 2 a k ( t ) = ˜ C k t 2 p k , ˜ C k = const , k = 1 , N (34)\n/negationslash\n/negationslash\nIII.3. Stability conditions.\nMetric ceases to obey a power law when there is a matter; but it turns out that the metric of the space filled with the magnetic field may get close to the Kasner metric and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (28) that get close to the Kasner solutions (33) and converge to them as time tends to the point t = 0 . Considering an influence of the magnetic field as a perturbation we will search for solutions of the equations (28) in the form:\na k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (35)\nwhere a 0 k is the Kasner solution (33), ϕ k ∈ C 2 ( R ) is a perturbation. We will say that solutions a 0 k are asymptotically stable if\nlim t → 0 a k ( t ) = a 0 k ( t ) , k = 1 , N (36)\nIn other words, solutions a 0 k are asymptotically stable if and only if\nlim t → 0 ϕ k ( t ) = 0 , k = 1 , N (37)\nNow we will find out conditions which specify asymptotically stable Kasner solutions.\nProposition 1. Kasner solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 1 , n = 1 , χ .\n\n/square We consider the space of an even dimension; that of an odd dimension is treated similar way.\n\n1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (31)-(32) given by (33); then there exists solution { a 1 , . . . , a N } of the equations (28),(29) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that\na k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (38)\nlim t → 0 ϕ k ( t ) = 0 , k = 1 , N (39)\nSubstitution (33),(38) to the equations (28) and (29) leads to:\n/negationslash\n/negationslash\n/negationslash\n∑ i = p i ( p i -1)+ ∑ j p j p k + t ∑ i = p i ˙ ϕ i + ∑ j ( p j ˙ ϕ k + p k ˙ ϕ j ) + t 2 ∑ i = ( ¨ ϕ i + ˙ ϕ 2 i ) + ∑ j ˙ ϕ j ˙ ϕ k = = -ψ 2 2 n -1 , 2 n e -2( ϕ 2 n -1 + ϕ 2 n ) t 2 -2( p 2 n -1 + p 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( ϕ 2 j -1 + ϕ 2 j ) t 2 -2( p 2 j -1 + p 2 j ) (40) ∑ i\n/negationslash\nIt follows from (39) that\nLet t → 0 ; in view of (33),(39),(42) equations (40)-(41) take the form:\n/negationslash\nlim t → 0 ( t ˙ ϕ k ( t ) ) = 0 , lim t → 0 ( t 2 ¨ ϕ k ( t ) ) = 0 , k = 1 , N (42)\n/negationslash\n0 = -ψ 2 2 n -1 , 2 n t 2 -2( p 2 n -1 + p 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 2 -2( p 2 j -1 + p 2 j ) (43)\n/negationslash\n/negationslash\n0 = ∑ i =1 ,χ ψ 2 2 i -1 , 2 i t 2 -2( p 2 i -1 + p 2 i ) (44)\nEqualities (43) and (44) are satisfied if and only if\np 2 n -1 + p 2 n < 1 , n = 1 , χ (45)\nq.e.d.\n2. Let { a 0 1 , . . . , a 0 N } be the Kasner solution given by (33) such that p 2 n -1 + p 2 n < 1 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have:\na k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (46)\nLet t = t 0 > 0 be the initial moment; we assume that\n| ϕ i ( t 0 ) | /lessmuch | a 0 k ( t 0 ) | , | ˙ ϕ i ( t 0 ) | /lessmuch ∣ ∣ ˙ a 0 k ( t 0 ) ∣ ∣ , i = 1 , N (47)\n/negationslash\nA Φ( t ) = R ( t ) , t ∈ ( t 0 -ε, t 0 + ε ) (49)\nA k k = 0 , A k j = 1 , k = j ; Φ k ( t ) = ¨ ϕ k ( t ) , j, k = 1 , N (50)\n/negationslash\nR 2 n -1 ( t ) = R 2 n ( t ) = ψ 2 2 n -1 , 2 n t -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t -2( p 2 j -1 + p 2 j ) , n = 1 , χ (51)\nIt is easy to check that matrix A is nonsingular, so we obtain:\nΦ( t ) = A -1 R ( t ) , (52)\nor,\n¨ ϕ k ( t ) = ∑ m =1 ,χ η km t λ m , η km = const , λ m = -2( p 2 m -1 + p 2 m ) , k = 1 , N (53)\nHence we deduce that\n˙ ϕ k ( t ) = ∑ m =1 ,χ η km λ m +1 t λ m +1 , ϕ k ( t ) = ∑ m =1 ,χ η km λ m +2 t λ m +2 , k = 1 , N (54)\nIt follows from p 2 m -1 + p 2 m < 1 , m = 1 , χ that λ m +2 > 0 ; therefore\nlim t → 0 ϕ k ( t ) = 0 , k = 1 , N (55)\nand solutions (33) with p 2 m -1 + p 2 m < 1 , m = 1 , χ are asymptotically stable.\n/squaresolid\n/negationslash\nSubstitution (33),(46) to the equations (28) and (29) give us equations (40)-(41). In view of (47) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j in the lhs of the equations (40)-(41) and terms contained ϕ i in the rhs of that equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (40) can be written in the form:\n/negationslash\nt 2 ∑ i = ¨ ϕ i = -ψ 2 2 n -1 , 2 n t 2 -2( p 2 n -1 + p 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 2 -2( p 2 j -1 + p 2 j ) , n = 1 , χ (48)\nor, in matrix notations,\nwhere\nExamples.\n\n· (3+1)-dimensional space-time; in view of (15) we have a single non-zero component of the Faraday tensor; according to our convention (see Sec.II.) we denote this component by ψ 12 ; under the Proposition 1 it follows that stable Kasner solutions are characterized by the condition p 1 + p 2 < 1 ; it implies that p 3 > 0 -the space expands along the magnetic field.\n· (4+1)-dimensional space-time; in that case we have a couple non-zero components of the Faraday tensor: ψ 12 and ψ 34 ; in accordance with the Proposition 1 it follows that stable Kasner solutions are specified by the criteria p 1 + p 2 < 1 and p 3 + p 4 < 1 .\n\nIII.4. Numerical calculations.\nThe result presented above have been verified numerically. We have got a few hundreds numerical solutions with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found a lot of solutions that converge to the Kasner ones. Example of such solution is presented on the Fig. 1 (a).\nIt should be note that looking for asymptotically stable solutions numerically using the synchronous time t has a number of drawbacks; particularly, numerical solutions of the equations (68)-(69) increase too rapidly impeding computations as t goes to zero. To avoid these problem we have introduced new time coordinate τ by means of the relation\ndτ = e -∑ j a j ( t ) dt (56)\nThe change of variables from t to τ Indeed, (33) and (56) leads to\nresults in transformation of logarithmic functions (33) into linear ones.\ndτ = C t dt, C = e -∑ j C j = const (57)\nConsequently,\nτ = C ln( t ) + ˜ C, ˜ C = const (58) a k ( τ ) = p k C τ + C k -p k ˜ C C , k = 1 , N (59)\nWithout lack of generality we can set\nThen, as was announced,\nC 1 = . . . = C N = ˜ C = 0 (60)\na k ( τ ) = p k τ, k = 1 , N (61)\nSuch a way, looking for asymptotically stable solutions numerically, we should expect to get quasi-linear solutions like ones that shown on the Fig. 1 (a).\n\n\n\n\n\n
Figure 1: Numerical solutions for (5+1)-dimension space-time (Einstein gravity). The figure a) illustrates stability of the Kasner solution near the initial singularity. The figure b) shows decreasing of the determinant of the metric; it implies that any given element of volume dV = √ | det( g ) | d N x tend to zero and therefore we go to the initial singularity.
\n\nIV. Stability of power-law solutions in Gauss-Bonnet models.\nIV.1. Field equations.\nAction of the theory under consideration is given by:\nS = 1 16 π ∫ d N +1 x √ | det( g ) | ( L GB + L m ) (62)\nL GB = R αβγδ R αβγδ -4 R αβ R αβ + R 2 , L m = F αβ F αβ (63)\nwhere R,R αβ , R αβγδ , F αβ are the ( N + 1) -dimensional scalar curvature, Ricci tensor, Riemann tensor and Faraday tensor respectively. The gravitational equations has the form:\nH µ ν = 8 πT µ ν , (64)\nwhere\nH µ ν = 2 RR µ ν -4 R µ γ R γ ν -4 R αβ R µ . ανβ +2 R µαβγ R ναβγ -1 2 δ µ ν ( R αβγδ R αβγδ -4 R αβ R αβ + R 2 ) (65)\nare the components of the Gauss-Bonnet tensor. Taking into account (1) we get:\nH µ ν = 0 , µ = ν (66)\n/negationslash\n/negationslash\n/negationslash\nH 0 0 = -12 ∑ i\nFor the diagonal case in view of (17),(18),(64),(67) for an even number of dimensions the gravitational equations can be written as:\n/negationslash\n∑ i = ( a i + ˙ a 2 i ) ∑ j ˙ a j ˙ a k +3 ∑ i ˙ a i ˙ a j ˙ a k ˙ a l =\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n= 1 4 ψ 2 2 n -1 , 2 n e -2( a 2 n -1 + a 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (68) ∑ i\n/negationslash\nThere is one more equation in addition to these in the case of an odd number of dimensions:\n/negationslash\n∑ i = N ( a i + ˙ a 2 i ) ∑ j\nThe left side of the expression (69) is a first integral of (68).\nIV.2. Vacuum solution.\nWhen there is no any matter the equations (68)-(70) lead to:\n/negationslash\n∑ i = n ( a i + ˙ a 2 i ) ∑ j\n/negationslash\n/negationslash\n∑ i\nThese equations has the following solutions [5, 6]:\na k = p k ln( t ) + C k , C k = const , k = 1 , N, ∑ i\nAs a consequence, components of the metric tensor follows a power law:\ng kk ( t ) = e 2 a k ( t ) = C k t 2 p k , C k = const , k = 1 , N (74)\nThus by analogy with (33)-(34) we will call solutions (73) and metric (74) Kasner-like for brevity.\nIV.3. Stability conditions.\nAs well as in the case of the Einstein gravity (see III.3.), metric ceases to obey a power law when there is a matter; it appears however that the metric of the space filled with the magnetic field may get close to the Kasner-like metric (74) and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (68) that get close to the Kasnerlike solutions (73) and converge to them as time goes to the point t = 0 . We will looking for solutions of the equations (68) in the form:\na k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (75)\nwhere a 0 k is the Kasner-like solution (73), ϕ k ∈ C 2 ( R ) . We will say that solutions a 0 k are asymptotically stable if\nlim t → 0 a k ( t ) = a 0 k ( t ) , k = 1 , N (76)\nIn other words, solutions a 0 k are asymptotically stable if and only if\nlim t → 0 ϕ k ( t ) = 0 , k = 1 , N (77)\nNow we will find out conditions which specify asymptotically stable Kasner-like solutions.\nProposition 2 . Kasner-like solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 2 , n = 1 , χ .\n\n/square We consider the space of an even dimension; that of an odd dimension is treated similar way. 1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (71)-(72) given by (73); then there exists solution { a 1 , . . . , a N } of the equations (68),(69) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that\n\na k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (78)\nlim t → 0 ϕ k ( t ) = 0 , k = 1 , N (79)\nLet us introduce the following notation:\nA ( i 1 B i 2 · . . . · B i N ) = A i 1 B i 2 · . . . · B i N + A i 2 B i 1 · . . . · B i N + . . . + A i N B i 2 · . . . · B i 1 (80)\nwhere A i 1 , . . . , A i N , B i 1 , . . . , B i N are any indexed mathematical objects. Substitution (73),(78) to the equations (68) and (69) leads to:\n/negationslash\n∑ ∑ ∑\n/negationslash\ni = p i ( p i -1) j p j p k +3 i p i p j p k p l + + t ∑ i = p i ( p i -1) ∑ j p ( j ˙ ϕ k ) +2 ∑ i = p i ˙ ϕ i ∑ j p j p k +3 ∑ i ˙ ϕ ( i p j p k p l ) + + t 2 ∑ i = p i ( p i -1) ∑ j ˙ ϕ j ˙ ϕ k +2 ∑ i = p i ˙ ϕ i ∑ j p ( j ˙ ϕ k ) + t 2 ∑ i = ( ¨ ϕ i + ˙ ϕ 2 i ) ∑ j p j p k +3 ∑ i ( ˙ ϕ ( i p k p l ) ˙ ϕ j + ˙ ϕ i ˙ ϕ ( j p k p l ) ) + t 3 2 ∑ i = p i ˙ ϕ i ∑ j ˙ ϕ j ˙ ϕ k + ∑ i = ( ¨ ϕ i + ˙ ϕ 2 i ) ∑ j p ( j ˙ ϕ k ) +3 ∑ i p ( i ˙ ϕ j ˙ ϕ k ˙ ϕ l ) + + t 4 ∑ i = ( ¨ ϕ i + ˙ ϕ 2 i ) ∑ j ˙ ϕ j ˙ ϕ k +3 ∑ i ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l = = 1 4 ψ 2 2 n -1 , 2 n e -2( ϕ 2 n -1 + ϕ 2 n ) t 4 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( ϕ 2 j -1 + ϕ 2 j ) t 4 -2( p 2 j -1 + p 2 j ) (81)\n/negationslash\n+\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\n/negationslash\nwhere\n/negationslash\n/negationslash\n∑ i p i p j p k p l + t ∑ i ˙ ϕ ( i p j p k p l ) + t 2 ∑ i ( ˙ ϕ ( i p k p l ) ˙ ϕ j + ˙ ϕ i ˙ ϕ ( j p k p l ) ) + + t 3 ∑ i p ( i ˙ ϕ j ˙ ϕ k ˙ ϕ l ) + t 4 ∑ i ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l = -1 12 ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( ϕ 2 j -1 + ϕ 2 j ) t 4 -2( p 2 i -1 + p 2 i ) (82)\nIt follows from (79) that\nIn the limit t → 0 in view of (73),(79),(83) equations (81)-(82) take the form:\nlim t → 0 ( t ˙ ϕ k ( t ) ) = 0 , lim t → 0 ( t 2 ¨ ϕ k ( t ) ) = 0 , k = 1 , N (83)\n/negationslash\n0 = -1 12 lim t → 0 ∑ i =1 ,χ ψ 2 2 i -1 , 2 i t 4 -2( p 2 i -1 + p 2 i ) (85)\n0 = 1 4 lim t → 0 ψ 2 2 n -1 , 2 n t 4 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 4 -2( p 2 j -1 + p 2 j ) (84)\nEqualities (84) and (85) are satisfied if and only if\np 2 n -1 + p 2 n < 2 , n = 1 , χ (86)\nq.e.d.\n2. Let { a 0 1 , . . . , a 0 N } be the Kasner-like solution given by (73) such that p 2 n -1 + p 2 n < 2 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have:\na k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (87)\nLet t = t 0 > 0 be the initial moment; we assume that\n| ϕ i ( t 0 ) | /lessmuch | a 0 k ( t 0 ) | , | ˙ ϕ i ( t 0 ) | /lessmuch ∣ ∣ ˙ a 0 k ( t 0 ) ∣ ∣ , i = 1 , N (88)\n/negationslash\nSubstitution (73),(87) to the equations (68) and (69) give us equations (81)-(82). In view of (88) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j , t 3 ˙ ϕ i ˙ ϕ j ˙ ϕ k , t 4 ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l in the lhs of the equations (81)-(82) and terms contained ϕ i in the rhs of these equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (81) can be written in the form:\n/negationslash\nt 2 ∑ i = ¨ ϕ i ∑ j p j p k = 1 4 ψ 2 2 n -1 , 2 n t 4 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 4 -2( p 2 j -1 + p 2 j ) , n = 1 , χ (89)\n/negationslash\nor, in matrix notations,\nA Φ( t ) = R ( t ) , t ∈ ( t 0 -ε, t 0 + ε ) (90)\n/negationslash\n/negationslash\n/negationslash\n/negationslash\nA k 2 n -1 = ∑ i\n/negationslash\n/negationslash\n/negationslash\nAssuming that matrix A is nonsingular 2 , we obtain:\nR 2 n -1 ( t ) = R 2 n ( t ) = 1 4 ψ 2 2 n -1 , 2 n t 2 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 2 -2( p 2 j -1 + p 2 j ) , n = 1 , χ (92)\nΦ( t ) = A -1 R ( t ) , (93)\n¨ ϕ k ( t ) = ∑ m =1 ,χ δ km t γ m , δ km = const , γ m = 2 -2( p 2 m -1 + p 2 m ) , k = 1 , N (94)\nor,\nHence we deduce that\n˙ ϕ k ( t ) = ∑ m =1 ,χ δ km γ m +1 t γ m +1 , ϕ k ( t ) = ∑ m =1 ,χ δ km γ m +2 t γ m +2 , k = 1 , N (95)\nIt follows from p 2 m -1 + p 2 m < 2 , m = 1 , χ that γ m +2 > 0 ; therefore\nlim t → 0 ϕ k ( t ) = 0 , k = 1 , N (96)\nand solutions (73) with p 2 m -1 + p 2 m < 2 , m = 1 , χ are asymptotically stable. /squaresolid\nExample. Let us consider (5+1)-dimensional space-time. According to (15) there are pair of non-zero components of the Faraday tensor; taking into account our convention (see II. for details) we denote them ψ 12 and ψ 34 . Then under the Proposition 2 it follows that stable Kasner-like solutions are described by the conditions p 1 + p 2 < 2 and p 3 + p 4 < 2 .\nIV.4. Numerical calculations.\nThe result obtained have been confirmed by numerical calculations. As well as in the study of the stability of the Kasner solutions (see III.3.) we have received a few hundreds solutions numerically with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found numerous solutions that converge to the Kasner-like ones. Example of such solution is presented on the Fig. 2 (a).\nFor the reasons mentioned above (see III.4.) one should use special time coordinate for numerical calculations. Let us introduce new time coordinate τ by the following way:\ndτ = e -1 3 ∑ j a j ( t ) dt (97)\nThe change of variables from t to τ results in transformation of logarithmic functions (73) into linear ones. Indeed, (73) and (97) give us:\ndτ = C t dt, C = e -1 3 ∑ j C j = const (98)\n/negationslash\n\n\n\n\n\n
Figure 2: Numerical solutions for (5+1)-dimension space-time (Gauss-Bonnet gravity). The figure a) illustrates stability of the Kasner-like regime near the initial singularity. The figure b) shows decreasing of the determinant of the metric; it implies that any given element of volume dV = √ | det( g ) | d N x tend to zero and, consequently, we go to the initial singularity.
\n\nConsequently,\nτ = C ln( t ) + ˜ C, ˜ C = const (99) a k ( τ ) = p k C τ + C k -p k ˜ C C , k = 1 , N (100)\nWithout loss of generality we can set\nThen, as was announced,\nC 1 = . . . = C N = ˜ C = 0 (101)\na k ( τ ) = p k τ, k = 1 , N (102)\nThus, searching for asymptotically stable solutions numerically we should expect to get quasi-linear solutions like ones that shown on the Fig. 2 (a).\nV. Oscillatory regime.\nResults of the preceding section pose an interesting question about the fate of trajectories in the instability zone. They can either reach stable Kasner-like attractor or represent some other type of dynamics different from Kasner-like behavior. In this section we show that the latter alternative can be realized. In our numerical calculations we have revealed a specific class of solutions of the equations (68). The discovered solutions possess the following properties: two of N functions a 1 , . . . , a N oscillate and has opposite signs, the other ones vary slowly; owing to phase difference between the oscillating functions volume element √ | det( g ) | d N x oscillates too. In view of limited computational capabilities we have obtained numerical solutions of the equations (68) only for the cases N = 5 , 6 , 7 , 8 ; for each of that dimensions we have found out sufficiently large number of oscillatory solutions. Example of such solution is presented on the Fig. 3 (a); the corresponding oscillations of the volume element is shown on the Fig. 3 (b).\nFor N = 5 we have been able to derive an analytical approximation to such solutions on a certain interval of time. Below we describe the procedure of derivation of this approximation. Henceforth we use\n\n\n
Figure 3: Numerical solutions for (5+1)-dimension space-time (Gauss-Bonnet gravity). The figure a) depicts oscillatory solution which does not tend to Kasner-like solutions both to the past and to the future. The figure b) shows the oscillatory behavior of the determinant of the metric; it means that any given volume element dV = √ | det( g ) | d N x oscillates in this regime.
\n\ntime coordinate τ introduced by (97). In the (5+1)-dimensional case equations (68) become:\n( a ' 2 [ a ' 3 a ' 4 + a ' 3 a ' 5 + a ' 4 a ' 5 ] + a ' 3 a ' 4 a ' 5 ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 12 e -2 a 1 -2 a 2 -2 ψ 2 34 e -2 a 3 -2 a 4 ) (103)\n( a ' 1 a ' 2 [ a ' 4 + a ' 5 ] + a ' 4 a ' 5 [ a ' 1 + a ' 2 ] ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 34 e -2 a 3 -2 a 4 -2 ψ 2 12 e -2 a 1 -2 a 2 ) (105)\n( a ' 1 [ a ' 3 a ' 4 + a ' 3 a ' 5 + a ' 4 a ' 5 ] + a ' 3 a ' 4 a ' 5 ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 12 e -2 a 1 -2 a 2 -2 ψ 2 34 e -2 a 3 -2 a 4 ) (104)\n( a ' 1 a ' 2 [ a ' 3 + a ' 5 ] + a ' 3 a ' 5 [ a ' 1 + a ' 2 ] ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 34 e -2 a 3 -2 a 4 -2 ψ 2 12 e -2 a 1 -2 a 2 ) (106)\nso that\nLet us decompose functions f, g, h with respect to ( 4 3 v -2 3 u ) and ( 4 3 u -2 3 v ) up to the first order:\n( a ' 1 a ' 2 [ a ' 3 + a ' 4 ] + a ' 3 a ' 4 [ a ' 1 + a ' 2 ] ) ' = -1 3 e 4 3 ∑ j a j ( ψ 2 12 e -2 a 1 -2 a 2 + ψ 2 34 e -2 a 3 -2 a 4 ) (107)\nwith the first integral\na ' 1 a ' 2 a ' 3 a ' 4 + a ' 1 a ' 2 a ' 3 a ' 5 + a ' 1 a ' 2 a ' 4 a ' 5 + a ' 1 a ' 3 a ' 4 a ' 5 + a ' 2 a ' 3 a ' 4 a ' 5 = -1 12 e 4 3 ∑ j a j ( ψ 2 12 e -2 a 1 -2 a 2 + ψ 2 34 e -2 a 3 -2 a 4 ) (108)\nThe dash denotes derivative with respect to τ . Analysis of numerical solutions shows that there exists τ 0 > 0 ( τ = 0 is accepted as the initial moment) such that\na ' 5 ( τ ) ≈ 0 , a ' 3 ( τ ) ≈ a ' 4 ( τ ) , τ > τ 0 (109)\n(see Fig. 6). In view of (109) equations (103)-(108) take the form:\n( a ' 1 a ' 3 a ' 4 ) ' = f 2 , ( a ' 2 a ' 3 a ' 4 ) ' = f 2 , ( a ' 1 a ' 2 a ' 3 ) ' = g 2 , ( a ' 1 a ' 2 a ' 4 ) ' = g 2 (110)\na ' 1 a ' 2 a ' 3 a ' 4 = h 4 (112)\n( a ' 1 a ' 2 [ a ' 3 + a ' 4 ] + a ' 3 a ' 4 [ a ' 1 + a ' 2 ] ) ' = h (111)\nwhere\nf = ( κ 1 e 4 3 v -2 3 u -2 κ 2 e 4 3 u -2 3 v ) , g = ( -2 κ 1 e 4 3 v -2 3 u + κ 2 e 4 3 u -2 3 v ) , h = -( κ 1 e 4 3 v -2 3 u + κ 2 e 4 3 u -2 3 v ) (113)\nv = a 3 + a 4 , u = a 1 + a 2 (114)\nκ 1 = ψ 2 12 3 , κ 2 = ψ 2 34 3 (115)\nEquation (111) is the sum of the equations (110); equations (110) can be replaced by their equivalent:\n( [ a ' 1 + a ' 2 ] a ' 3 a ' 4 ) ' = f, ( [ a ' 1 -a ' 2 ] a ' 3 a ' 4 ) ' = 0 , ( a ' 1 a ' 2 [ a ' 3 + a ' 4 ] ) ' = g, ( a ' 1 a ' 2 [ a ' 3 -a ' 4 ] ) ' = 0 (116)\nIn view of (109) the last equation from (116) is satisfied automatically, and the other equations become\n( u ' v ' 2 ) ' = 4 f, ( a ' 1 a ' 2 v ' ) ' = g, ( [ a ' 1 -a ' 2 ] v ' 2 ) ' = 0 (117)\nThe last of equations (117) gives:\na ' 1 -a ' 2 = C v ' 2 , C = const (118)\na ' 1 = 1 2 ( u ' + C v ' 2 ) , a ' 2 = 1 2 ( u ' -C v ' 2 ) (119)\nf ≈ [ κ 1 -2 κ 2 ] -2 3 [ κ 1 +4 κ 2 ] u + 4 3 [ κ 1 + κ 2 ] v (120)\ng ≈ [ κ 2 -2 κ 1 ] + 4 3 [ κ 1 + κ 2 ] u -2 3 [ κ 2 +4 κ 1 ] v (121)\nConsequently,\nh ≈ -[ κ 1 + κ 2 ] + 2 3 [ κ 1 -2 κ 2 ] u + 2 3 [ κ 2 -2 κ 1 ] u (122)\nNumerical results show that such assumption is valid in a certain interval of time. Taking into account (119)-\n(122) we get:\n( u ' v ' 2 ) ' = 4 [ κ 1 -2 κ 2 ] -8 3 [ κ 1 +4 κ 2 ] u + 16 3 [ κ 1 + κ 2 ] v (123)\nLet us consider the equation (124). Numerical calculations show that in the regime under consideration v '' / ( v ' ) 4 is the leading term in the lhs of the equation (124); then:\n([ ( u ' ) 2 -C 2 v ' 4 ] v ' ) ' = 4 [ κ 2 -2 κ 1 ] + 16 3 [ κ 1 + κ 2 ] u -8 3 [ κ 2 +4 κ 1 ] v (124)\nC 2 v '' v ' 4 = 4 [ κ 2 -2 κ 1 ] + 16 3 [ κ 1 + κ 2 ] u -8 3 [ κ 2 +4 κ 1 ] v (125)\nWe can obtain analytical solution of the equation (125) by omitting the term containing u in the rhs of (125):\nv '' v ' 4 = 4 [ κ 2 -2 κ 1 ] -8 3 [ κ 2 +4 κ 1 ] v (126)\nSolution of the equation (126) has the form\nv ( τ ) = η √ τ + λ, η, λ = const (127)\nLet us consider the equation (123). We omit the term, containing v in the rhs of (123):\n( u ' v ' 2 ) ' = 4 [ κ 1 -2 κ 2 ] -8 3 [ κ 1 +4 κ 2 ] u (128)\nUnder substitution of (127) into (128) we obtain:\nτu '' ( τ ) -u ' ( τ ) + ξ 2 τ 2 u ( τ ) = ζτ 2 (129)\nWe have used the following notations:\nξ 2 = 32 [ κ 1 +4 κ 2 ] 3 η 2 , ζ = 16 [ κ 1 -2 κ 2 ] η 2 (130)\nEquation (129) has the following solution:\nu ( τ ) = τC 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + τC 2 Y 2 3 ( 2 ξτ 3 / 2 3 ) + ζ ξ 2 , C 1 , C 2 = const (131)\nHere J 2 3 , Y 2 3 are Bessel functions of the first and the second kind respectively. Going back to the initial variables we get:\na ' 1 -a ' 2 = C v ' 2 , a 3 + a 4 = v, a ' 3 = a ' 4 , a ' 5 = 0 (132)\na 1 ( τ ) -a 2 ( τ ) = 2 C η 2 τ 2 + C, a 3 ( τ ) -˜ C = a 4 ( τ ) = η √ τ + λ, C, ˜ C = const (133)\nand, finally:\nwhere\na 1 ( τ ) ≈ -C 0 τ 2 + τ [ C 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + C 2 Y 2 3 ( 2 ξτ 3 / 2 3 )] + D 1 (134)\na 2 ( τ ) ≈ C 0 τ 2 + τ [ C 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + C 2 Y 2 3 ( 2 ξτ 3 / 2 3 )] + D 2 (135)\na 3 ( τ ) ≈ η √ τ + D 3 , a 4 ( τ ) ≈ η √ τ + λ, a 5 ( τ ) ≈ D, (136)\nC 0 , D 1 , D 2 , D 3 , D = const (137)\nTo demonstrate oscillations of volume element √ | det( g ) | d N x one should write down expression for the determinant of the metric:\n| det( g ) | = e 2( a 1 + ... + a 5 ) = exp { 4 ( η √ τ + τ [ C 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + C 2 Y 2 3 ( 2 ξτ 3 / 2 3 )] + D )} (138)\nD = D + D 1 + D 2 + D 3 + λ = const (139)\nDespite the fact that functions (134)-(136) are the solution of the simplified system (126),(128) they catch correctly qualitative features of solution of the full system (103)-(107) which can be shown by comparing functions (134)-(136) with numerical solution of the system (103)-(107). Figures 4,5 below depicts pairs ( a 2 , A 2 ); ( a 4 , A 4 ) of numerical and analytical solutions; pairs ( a 1 , A 1 ); ( a 3 , A 3 ) behave themselves analogously.\nThis regime is not similar to Kasner-like one (Kasner-like solution has linear behavior on chosen time coordinate τ (see IV.4. for details), the leading asymptotic of the oscillatory solution has quadratic behaviour w.r.t. τ ).\n\n\n
Figure 4: a) Comparison of numerical ( a 2 ) and analytical ( A 2 ) solutions. b) Relative difference of numerical ( a 2 ) and analytical ( A 2 ) solutions. A 1 ( τ ) = -τ 2 +0 . 7 τJ 2 3 ( 10 τ 3 2 ) +0 . 7 τY 2 3 ( 10 τ 3 2 ) +15 . 2 .
\n\n\n\n
Figure 5: a) Comparison of numerical ( a 4 ) and analytical ( A 4 ) solutions. b) Relative difference of numerical ( a 4 ) and analytical ( A 4 ) solutions. A 4 ( τ ) = 0 . 741 √ τ -0 . 749 .
\n\n\n\n
Figure 6: a) The figure shows that the derivative a ' 5 ( τ ) asymptotically tends to zero. b) The figure indicates that the difference a ' 3 ( τ ) -a ' 4 ( τ ) asymptotically tends to zero.
\n\nVI. Conclusions\nIn this paper we provided an analysis of stability of known power-law solutions in Gauss-Bonnet anisotropic cosmology in the presence of a homogeneous magnetic field in the case when metric is diagonal in the frame determined by magnetic field. The relative simplicity of equations of motion in this case allows us to find analytically a simple condition for the particular power-law solution to be stable. It appears that a set of initial conditions for which the power-law solution can not be an attractor near a cosmological singularity has non-zero measure (though does not coincide with the whole initial conditions space). The result has the same structure as in the Einstein gravity (particular sums of Kasner indices should be restricted by some number), the only difference is the exact value of this number.\nThe fate of trajectories near singularity from this unstable domain requires future investigations. In the present paper we describe one particular class of trajectories found by numerical studies. The volume element experiences oscillations in the presented regime, making it qualitatively different from BKL oscillations.\nWe can expect that some other regimes which do not tend to Kasner-like power-law regime exist in the system under investigation. Full description of them (the task which we leave for a future work) in this particular rather simple model can be useful in studies of cosmological singularity in Lovelock gravity in the presence of spatial curvature. The latter model usually lead to very cumbersome equations of motion, so understanding the nature of possible regimes in the instability zone for the Kasner-like solution in the present model would help in finding similar solution in more general and more technically complex situations of curved multidimensional geometries.\nAcknowledgments\nThe authors are grateful to Vladimir Ivashchuk for helpful discussion. The work is partially supported by the RFBR grant 11-02-00643.\nReferences\n[1] V.A.Belinskii, I.M.Khalatnikov and E.M.Lifshitz, Adv.Phys. 19, 525 (1970). 1 [2] Victor G. LeBlanc, Class.Quantum.Grav. 14, 2281 (1997). 1 [3] R. Benini, A.A. Kirillov, G. Montani, Class.Quant.Grav. 22, 1483-1491 (2005). 1 [4] G. Rosen, J.Math.Phys. 3, 313 (1962). 1 [5] N. Deruelle, Nucl.Phys. B 327, 253-266 (1989). 2, 10 [6] N. Deruelle, L. Farina Busto, Phys.Rev. D 41, 3696 (1990). 2, 10 [7] A. Toporensky, P.Tretyakov, Grav.Cosmol. 13, 207-210 (2007); arXive: 0705.1346. 2 [8] S.A. Pavluchenko, Phys.Rev. D 80, 107501 (2009); arXiv: 0906.0141. 2\n\n[9] I. Kirnos, A. Makarenko, S. Pavluchenko and A. Toporensky, Gen.Rel.Grav. 42, 2633 (2010) arXive: 0906.0140. 2\n[10] I.Kirnos, S. Pavluchenko and A. Toporensky, Grav. Cosmol., 16, 274-282 (2010) arXive: 1002.4488. 2\n[11] V. Ivashchuk, Grav.Cosmol., 16, 118-125 (2010) arXive: 0910.3426. 2\n[12] V. Ivashchuk, Int.J.Geom.Meth.Mod.Phys., 7, 797-819 (2010) arXive: 0910.3426. 2\n[13] N.V. Mitskievich, Rev.Mex.Fis., 49S2, 39-51 (2003) arXive: 0202032. 1\n[14] N.V. Mitskievich, Relativistic Physics in Arbitrary Reference Frames , Nova Science Publishers, 2006 arXive: 9606051 1\n[15] N.V. Mitskievich, Electromagnetism and perfect fluids interplay in multidimensional spacetimes , in Proc. MG11 (2006) arXive: 0707.3190 1\n[16] Kasner E., American Journal of Math., 43, 217-221 (1921). 5\n\n"},"sections":{"kind":"list like","value":[{"title":"On stability of power-law solution in multidimensional Gauss-Bonnet cosmology","content":"D.M.Chirkov 1,2 ∗ , A.V.Toporensky 1 † July 16, 2018 1 Sternberg Astronomical Institute, Moscow State University, Universitetsky pr., 13, Moscow, 119991, Russia 2 Faculty of Physics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia","pages":[1]},{"title":"Abstract","content":"We consider dynamics of a flat anisotropic multidimensional cosmological model in Gauss-Bonnet gravity in the presence of a homogeneous magnetic field. In particular, we find conditions under which the known power-law vacuum solution can be an attractor for the case with non-zero magnetic field. We also describe a particular class of numerical solution in (5 + 1) -dimensional case which does not approach the power-law regime.","pages":[1]},{"title":"I. Introduction.","content":"The fact that structure of initial cosmological singularity can be rather complicated have been recognized since 60-th of the last century when the conception of BKL chaos have been presented [1]. It appears that Kasner solution being a general solution for a vacuum Bianchi I Universe becomes unstable in the case of metric with positive spatial curvature (belonging to Bianchi IX class) and is replaced by a complicated sequence of transient \"Kasner epochs\". Later it was found that some classes of an anisotropic matter can induce similar type of cosmological behavior even in flat Bianchi I case. This can be shown for a magnetic field by LeBlanc [2] and for a general vector field by Kirillov [3]. The BKL analogs for the flat magnetic Universe is a typical behavior when off-diagonal terms in the metric in the frame determined by magnetic field are present. For a diagonal case see for example [4]. Possible generalisation of these results may involve studies in the framework of multidimensional cosmology as well as alternative gravity theories. Analogs of magnetic field for multidimensional space-times have been studied in [13, 14, 15]. If we want to introduce some corrections to Einstein gravity, the first goal is to find a solution which replaces Kasner solution for a flat anisotropic Universe. In the present paper we consider Lovelock gravity as a generalization of General Relativity. The principal feature of Lovelock gravity is that this theory keeps the order of equations of motion the same as in GR, while other theories (like popular now f ( R ) -theory) results in increasing of the number of derivatives in equations of motion. Another important property of Lovelock gravity is that it gives corrections to GR only in higher-dimensional space-time, so it is natural to consider it in the framework of multidimensional cosmology. As the number of non-Einstein terms in Lovelock gravity is finite for any given dimensionality of space-time, it is possible to consider regimes when these terms are not only small corrections to Einstein gravity. In particular, it is reasonable to expect that the highest order Lovelock term, consisting of highest power of curvature invariants, dominates near a cosmological singularity. In the present paper we consider the second Lovelock term, which is the famous Gauss-Bonnet combination. It is the highest possible Lovelock term for (4 + 1) and (5 + 1) dimensional spacetimes. As cosmology in the (4 + 1) dimensional case has some pathological features, we restrict ourselves by the ( N +1) dimensional case with N /greaterorequalslant 5 in the present paper. In late 80-th some vacuum solutions for a flat anisotropic Universe in Gauss-Bonnet gravity for N = 4 , 5 have been found [5, 6]. They replace Kasner solution of GR. Later these solutions have been rediscovered and verified for N = 6 , 7 in [7]; after that this solution have been generalized for all N and also to the general Lovelock gravity [8]. The main goal of this paper is to address question of its stability near the initial singularity. It is known that in the presence of an ordinary isotropic matter these solutions are stable to the past if the matter has the equation of state with w < 1 / 3 , otherwise the solution tends to isotropic one [9, 10]. For more interesting regimes near a singularity it is necessary to consider either anisotropic matter or curved geometries. As introduction of spatial curvature usually leads to very cumbersome equations of motion, we have chosen a magnetic fields as a possible source of instability. In this paper we restrict ourselves by the diagonal case. It is also known that in contrast to Einstein gravity certain initial conditions lead to exponential solutions instead to those of power-law behavior [9, 11, 12], we do not consider them in the present paper. The structure of the paper is as follows: In Sec. II. we describe the metric and matter content of the model studied, in Sec. III. the model is investigated in the framework of Einstein relativity. We present results on stability of Kasner solution in the presence of magnetic field. The same model in Gauss-Bonnet gravity is studied in Sec. IV. with presentation made as parallel as possible to presentation in the preceding section in order to compare Einstein and Gauss-Bonnet cases. In Sec. V. we describe a particular regime existing in the zone of instability of power-law regimes in (5+1) -dimensional Gauss-Bonnet gravity. Sec. VI. contains a brief summary of results obtained.","pages":[1,2]},{"title":"II. Preliminaries.","content":"In what follows we use a reference system chosen so that 1 : /negationslash In this section we looking for the energy-momentum tensor of a pure magnetic field in ( N +1) -dimensional space-time. In the general case components of the energy-momentum tensor of the electromagnetic field are where F αβ is the Faraday tensor. The components of the Faraday tensor obey the following equations: or, in more detail: where D µ = ∂ ∂x µ . As it is known therefore we may rewrite system (4) in the form The components of the Riemann connection have the form: Its other components are equal to zero. Since we consider the case of a homogenous space, Faraday tensor depends on time t only; then, in view of (7) the system (6) takes the form: Its solutions are Hereinafter we will be interested in the case of a pure magnetic field; so that φ m = 0 and, as a consequence, In what follows we will deal with diagonal energy-momentum tensor; in view of (1),(2),(9) and (10) it implies that /negationslash /negationslash Functions e -2 a l are linearly independent, therefore from (11) it follows that /negationslash Each ψ ij is multiplied by all that ψ kl , which has one of the indices k, l coincident with one of the indices i, j . /negationslash Let us fix pair ( i, j ) ; the number of combinations, in which ψ ij is found, equals to 2( N -2) . Let ψ ij = 0 ; then the other 2( N -2) quantities ψ ik ( ψ jk ), which are multiplied by ψ ij , must be equals to zero; indeed, if ψ ik = 0 ( k = i, j ) , then ψ ij ψ ik = 0 , but that contradict (12). Continuing the argument, we find the number of zero magnetic field components: /negationslash /negationslash /negationslash We took into account that the number of pairs of indices without the same elements equals to N -1 2 for an odd dimensions and to N 2 for an even dimensions. The total number of components of the magnetic field is N ( N -1) 2 ; then the number χ of non-zero magnetic field components is In what follows they are exactly ψ 12 , ψ 34 , . . . , ψ 2 n -1 , 2 n , n = 1 , χ that we set to be non-zero; other components of the Faraday tensor are assumed to be zero. Thus, components of the energy-momentum tensor of the electromagnetic field take the form: /negationslash /negationslash Hereinafter it will be convenient to use notations like this: We assume that /negationslash /negationslash /negationslash","pages":[2,3,4]},{"title":"III.1. Field equations.","content":"The action reads: where R and F αβ are scalar curvature and Faraday tensor respectively. Here and after we use Planck units: m Pl is the Planck mass. The gravitational equations is given by where are the components of the Einstein tensor. Taking into account (1) we get: /negationslash /negationslash As we consider the diagonal case here, then in view of (17),(18),(24),(27) for an even number of dimensions the gravitational equations can be written as /negationslash /negationslash Note that each expression like (28) describes two equations with numbers 2 n -1 and 2 n simultaneously (we use the notation (19) here). In the case of an odd number of dimensions there is one more equation in addition to these: /negationslash /negationslash The left side of the expression (29) is a first integral of (28).","pages":[4,5]},{"title":"III.2. Vacuum solution.","content":"When there is no any matter the equations (28)-(30) lead to: /negationslash These equations has the following solutions [16]: /negationslash As a consequence, components of the metric tensor follows a power law: This is well-known multidimensional Kasner solution. /negationslash /negationslash","pages":[5]},{"title":"III.3. Stability conditions.","content":"Metric ceases to obey a power law when there is a matter; but it turns out that the metric of the space filled with the magnetic field may get close to the Kasner metric and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (28) that get close to the Kasner solutions (33) and converge to them as time tends to the point t = 0 . Considering an influence of the magnetic field as a perturbation we will search for solutions of the equations (28) in the form: where a 0 k is the Kasner solution (33), ϕ k ∈ C 2 ( R ) is a perturbation. We will say that solutions a 0 k are asymptotically stable if In other words, solutions a 0 k are asymptotically stable if and only if Now we will find out conditions which specify asymptotically stable Kasner solutions. Proposition 1. Kasner solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 1 , n = 1 , χ . 1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (31)-(32) given by (33); then there exists solution { a 1 , . . . , a N } of the equations (28),(29) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that Substitution (33),(38) to the equations (28) and (29) leads to: /negationslash /negationslash /negationslash /negationslash It follows from (39) that Let t → 0 ; in view of (33),(39),(42) equations (40)-(41) take the form: /negationslash /negationslash /negationslash /negationslash Equalities (43) and (44) are satisfied if and only if q.e.d. 2. Let { a 0 1 , . . . , a 0 N } be the Kasner solution given by (33) such that p 2 n -1 + p 2 n < 1 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have: Let t = t 0 > 0 be the initial moment; we assume that /negationslash /negationslash It is easy to check that matrix A is nonsingular, so we obtain: or, Hence we deduce that It follows from p 2 m -1 + p 2 m < 1 , m = 1 , χ that λ m +2 > 0 ; therefore and solutions (33) with p 2 m -1 + p 2 m < 1 , m = 1 , χ are asymptotically stable. /squaresolid /negationslash Substitution (33),(46) to the equations (28) and (29) give us equations (40)-(41). In view of (47) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j in the lhs of the equations (40)-(41) and terms contained ϕ i in the rhs of that equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (40) can be written in the form: /negationslash or, in matrix notations, where","pages":[6,7]},{"title":"III.4. Numerical calculations.","content":"The result presented above have been verified numerically. We have got a few hundreds numerical solutions with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found a lot of solutions that converge to the Kasner ones. Example of such solution is presented on the Fig. 1 (a). It should be note that looking for asymptotically stable solutions numerically using the synchronous time t has a number of drawbacks; particularly, numerical solutions of the equations (68)-(69) increase too rapidly impeding computations as t goes to zero. To avoid these problem we have introduced new time coordinate τ by means of the relation The change of variables from t to τ Indeed, (33) and (56) leads to results in transformation of logarithmic functions (33) into linear ones. Consequently, Without lack of generality we can set Then, as was announced, Such a way, looking for asymptotically stable solutions numerically, we should expect to get quasi-linear solutions like ones that shown on the Fig. 1 (a).","pages":[8]},{"title":"IV.1. Field equations.","content":"Action of the theory under consideration is given by: where R,R αβ , R αβγδ , F αβ are the ( N + 1) -dimensional scalar curvature, Ricci tensor, Riemann tensor and Faraday tensor respectively. The gravitational equations has the form: where are the components of the Gauss-Bonnet tensor. Taking into account (1) we get: /negationslash /negationslash /negationslash For the diagonal case in view of (17),(18),(64),(67) for an even number of dimensions the gravitational equations can be written as: /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash There is one more equation in addition to these in the case of an odd number of dimensions: /negationslash The left side of the expression (69) is a first integral of (68).","pages":[9,10]},{"title":"IV.2. Vacuum solution.","content":"When there is no any matter the equations (68)-(70) lead to: /negationslash /negationslash /negationslash These equations has the following solutions [5, 6]: As a consequence, components of the metric tensor follows a power law: Thus by analogy with (33)-(34) we will call solutions (73) and metric (74) Kasner-like for brevity.","pages":[10]},{"title":"IV.3. Stability conditions.","content":"As well as in the case of the Einstein gravity (see III.3.), metric ceases to obey a power law when there is a matter; it appears however that the metric of the space filled with the magnetic field may get close to the Kasner-like metric (74) and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (68) that get close to the Kasnerlike solutions (73) and converge to them as time goes to the point t = 0 . We will looking for solutions of the equations (68) in the form: where a 0 k is the Kasner-like solution (73), ϕ k ∈ C 2 ( R ) . We will say that solutions a 0 k are asymptotically stable if In other words, solutions a 0 k are asymptotically stable if and only if Now we will find out conditions which specify asymptotically stable Kasner-like solutions. Proposition 2 . Kasner-like solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 2 , n = 1 , χ . Let us introduce the following notation: where A i 1 , . . . , A i N , B i 1 , . . . , B i N are any indexed mathematical objects. Substitution (73),(78) to the equations (68) and (69) leads to: /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash where /negationslash /negationslash It follows from (79) that In the limit t → 0 in view of (73),(79),(83) equations (81)-(82) take the form: /negationslash Equalities (84) and (85) are satisfied if and only if q.e.d. 2. Let { a 0 1 , . . . , a 0 N } be the Kasner-like solution given by (73) such that p 2 n -1 + p 2 n < 2 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have: Let t = t 0 > 0 be the initial moment; we assume that /negationslash Substitution (73),(87) to the equations (68) and (69) give us equations (81)-(82). In view of (88) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j , t 3 ˙ ϕ i ˙ ϕ j ˙ ϕ k , t 4 ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l in the lhs of the equations (81)-(82) and terms contained ϕ i in the rhs of these equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (81) can be written in the form: /negationslash /negationslash or, in matrix notations, /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash Assuming that matrix A is nonsingular 2 , we obtain: or, Hence we deduce that It follows from p 2 m -1 + p 2 m < 2 , m = 1 , χ that γ m +2 > 0 ; therefore and solutions (73) with p 2 m -1 + p 2 m < 2 , m = 1 , χ are asymptotically stable. /squaresolid Example. Let us consider (5+1)-dimensional space-time. According to (15) there are pair of non-zero components of the Faraday tensor; taking into account our convention (see II. for details) we denote them ψ 12 and ψ 34 . Then under the Proposition 2 it follows that stable Kasner-like solutions are described by the conditions p 1 + p 2 < 2 and p 3 + p 4 < 2 .","pages":[10,11,12,13]},{"title":"IV.4. Numerical calculations.","content":"The result obtained have been confirmed by numerical calculations. As well as in the study of the stability of the Kasner solutions (see III.3.) we have received a few hundreds solutions numerically with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found numerous solutions that converge to the Kasner-like ones. Example of such solution is presented on the Fig. 2 (a). For the reasons mentioned above (see III.4.) one should use special time coordinate for numerical calculations. Let us introduce new time coordinate τ by the following way: The change of variables from t to τ results in transformation of logarithmic functions (73) into linear ones. Indeed, (73) and (97) give us: /negationslash Consequently, Without loss of generality we can set Then, as was announced, Thus, searching for asymptotically stable solutions numerically we should expect to get quasi-linear solutions like ones that shown on the Fig. 2 (a).","pages":[13,14]},{"title":"V. Oscillatory regime.","content":"Results of the preceding section pose an interesting question about the fate of trajectories in the instability zone. They can either reach stable Kasner-like attractor or represent some other type of dynamics different from Kasner-like behavior. In this section we show that the latter alternative can be realized. In our numerical calculations we have revealed a specific class of solutions of the equations (68). The discovered solutions possess the following properties: two of N functions a 1 , . . . , a N oscillate and has opposite signs, the other ones vary slowly; owing to phase difference between the oscillating functions volume element √ | det( g ) | d N x oscillates too. In view of limited computational capabilities we have obtained numerical solutions of the equations (68) only for the cases N = 5 , 6 , 7 , 8 ; for each of that dimensions we have found out sufficiently large number of oscillatory solutions. Example of such solution is presented on the Fig. 3 (a); the corresponding oscillations of the volume element is shown on the Fig. 3 (b). For N = 5 we have been able to derive an analytical approximation to such solutions on a certain interval of time. Below we describe the procedure of derivation of this approximation. Henceforth we use time coordinate τ introduced by (97). In the (5+1)-dimensional case equations (68) become: so that Let us decompose functions f, g, h with respect to ( 4 3 v -2 3 u ) and ( 4 3 u -2 3 v ) up to the first order: with the first integral The dash denotes derivative with respect to τ . Analysis of numerical solutions shows that there exists τ 0 > 0 ( τ = 0 is accepted as the initial moment) such that (see Fig. 6). In view of (109) equations (103)-(108) take the form: where Equation (111) is the sum of the equations (110); equations (110) can be replaced by their equivalent: In view of (109) the last equation from (116) is satisfied automatically, and the other equations become The last of equations (117) gives: Consequently, Numerical results show that such assumption is valid in a certain interval of time. Taking into account (119)- (122) we get: Let us consider the equation (124). Numerical calculations show that in the regime under consideration v '' / ( v ' ) 4 is the leading term in the lhs of the equation (124); then: We can obtain analytical solution of the equation (125) by omitting the term containing u in the rhs of (125): Solution of the equation (126) has the form Let us consider the equation (123). We omit the term, containing v in the rhs of (123): Under substitution of (127) into (128) we obtain: We have used the following notations: Equation (129) has the following solution: Here J 2 3 , Y 2 3 are Bessel functions of the first and the second kind respectively. Going back to the initial variables we get: and, finally: where To demonstrate oscillations of volume element √ | det( g ) | d N x one should write down expression for the determinant of the metric: Despite the fact that functions (134)-(136) are the solution of the simplified system (126),(128) they catch correctly qualitative features of solution of the full system (103)-(107) which can be shown by comparing functions (134)-(136) with numerical solution of the system (103)-(107). Figures 4,5 below depicts pairs ( a 2 , A 2 ); ( a 4 , A 4 ) of numerical and analytical solutions; pairs ( a 1 , A 1 ); ( a 3 , A 3 ) behave themselves analogously. This regime is not similar to Kasner-like one (Kasner-like solution has linear behavior on chosen time coordinate τ (see IV.4. for details), the leading asymptotic of the oscillatory solution has quadratic behaviour w.r.t. τ ).","pages":[14,15,16,17,18]},{"title":"VI. Conclusions","content":"In this paper we provided an analysis of stability of known power-law solutions in Gauss-Bonnet anisotropic cosmology in the presence of a homogeneous magnetic field in the case when metric is diagonal in the frame determined by magnetic field. The relative simplicity of equations of motion in this case allows us to find analytically a simple condition for the particular power-law solution to be stable. It appears that a set of initial conditions for which the power-law solution can not be an attractor near a cosmological singularity has non-zero measure (though does not coincide with the whole initial conditions space). The result has the same structure as in the Einstein gravity (particular sums of Kasner indices should be restricted by some number), the only difference is the exact value of this number. The fate of trajectories near singularity from this unstable domain requires future investigations. In the present paper we describe one particular class of trajectories found by numerical studies. The volume element experiences oscillations in the presented regime, making it qualitatively different from BKL oscillations. We can expect that some other regimes which do not tend to Kasner-like power-law regime exist in the system under investigation. Full description of them (the task which we leave for a future work) in this particular rather simple model can be useful in studies of cosmological singularity in Lovelock gravity in the presence of spatial curvature. The latter model usually lead to very cumbersome equations of motion, so understanding the nature of possible regimes in the instability zone for the Kasner-like solution in the present model would help in finding similar solution in more general and more technically complex situations of curved multidimensional geometries.","pages":[22]},{"title":"Acknowledgments","content":"The authors are grateful to Vladimir Ivashchuk for helpful discussion. The work is partially supported by the RFBR grant 11-02-00643.","pages":[22]}],"string":"[\n {\n \"title\": \"On stability of power-law solution in multidimensional Gauss-Bonnet cosmology\",\n \"content\": \"D.M.Chirkov 1,2 ∗ , A.V.Toporensky 1 † July 16, 2018 1 Sternberg Astronomical Institute, Moscow State University, Universitetsky pr., 13, Moscow, 119991, Russia 2 Faculty of Physics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"We consider dynamics of a flat anisotropic multidimensional cosmological model in Gauss-Bonnet gravity in the presence of a homogeneous magnetic field. In particular, we find conditions under which the known power-law vacuum solution can be an attractor for the case with non-zero magnetic field. We also describe a particular class of numerical solution in (5 + 1) -dimensional case which does not approach the power-law regime.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"I. Introduction.\",\n \"content\": \"The fact that structure of initial cosmological singularity can be rather complicated have been recognized since 60-th of the last century when the conception of BKL chaos have been presented [1]. It appears that Kasner solution being a general solution for a vacuum Bianchi I Universe becomes unstable in the case of metric with positive spatial curvature (belonging to Bianchi IX class) and is replaced by a complicated sequence of transient \\\"Kasner epochs\\\". Later it was found that some classes of an anisotropic matter can induce similar type of cosmological behavior even in flat Bianchi I case. This can be shown for a magnetic field by LeBlanc [2] and for a general vector field by Kirillov [3]. The BKL analogs for the flat magnetic Universe is a typical behavior when off-diagonal terms in the metric in the frame determined by magnetic field are present. For a diagonal case see for example [4]. Possible generalisation of these results may involve studies in the framework of multidimensional cosmology as well as alternative gravity theories. Analogs of magnetic field for multidimensional space-times have been studied in [13, 14, 15]. If we want to introduce some corrections to Einstein gravity, the first goal is to find a solution which replaces Kasner solution for a flat anisotropic Universe. In the present paper we consider Lovelock gravity as a generalization of General Relativity. The principal feature of Lovelock gravity is that this theory keeps the order of equations of motion the same as in GR, while other theories (like popular now f ( R ) -theory) results in increasing of the number of derivatives in equations of motion. Another important property of Lovelock gravity is that it gives corrections to GR only in higher-dimensional space-time, so it is natural to consider it in the framework of multidimensional cosmology. As the number of non-Einstein terms in Lovelock gravity is finite for any given dimensionality of space-time, it is possible to consider regimes when these terms are not only small corrections to Einstein gravity. In particular, it is reasonable to expect that the highest order Lovelock term, consisting of highest power of curvature invariants, dominates near a cosmological singularity. In the present paper we consider the second Lovelock term, which is the famous Gauss-Bonnet combination. It is the highest possible Lovelock term for (4 + 1) and (5 + 1) dimensional spacetimes. As cosmology in the (4 + 1) dimensional case has some pathological features, we restrict ourselves by the ( N +1) dimensional case with N /greaterorequalslant 5 in the present paper. In late 80-th some vacuum solutions for a flat anisotropic Universe in Gauss-Bonnet gravity for N = 4 , 5 have been found [5, 6]. They replace Kasner solution of GR. Later these solutions have been rediscovered and verified for N = 6 , 7 in [7]; after that this solution have been generalized for all N and also to the general Lovelock gravity [8]. The main goal of this paper is to address question of its stability near the initial singularity. It is known that in the presence of an ordinary isotropic matter these solutions are stable to the past if the matter has the equation of state with w < 1 / 3 , otherwise the solution tends to isotropic one [9, 10]. For more interesting regimes near a singularity it is necessary to consider either anisotropic matter or curved geometries. As introduction of spatial curvature usually leads to very cumbersome equations of motion, we have chosen a magnetic fields as a possible source of instability. In this paper we restrict ourselves by the diagonal case. It is also known that in contrast to Einstein gravity certain initial conditions lead to exponential solutions instead to those of power-law behavior [9, 11, 12], we do not consider them in the present paper. The structure of the paper is as follows: In Sec. II. we describe the metric and matter content of the model studied, in Sec. III. the model is investigated in the framework of Einstein relativity. We present results on stability of Kasner solution in the presence of magnetic field. The same model in Gauss-Bonnet gravity is studied in Sec. IV. with presentation made as parallel as possible to presentation in the preceding section in order to compare Einstein and Gauss-Bonnet cases. In Sec. V. we describe a particular regime existing in the zone of instability of power-law regimes in (5+1) -dimensional Gauss-Bonnet gravity. Sec. VI. contains a brief summary of results obtained.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"II. Preliminaries.\",\n \"content\": \"In what follows we use a reference system chosen so that 1 : /negationslash In this section we looking for the energy-momentum tensor of a pure magnetic field in ( N +1) -dimensional space-time. In the general case components of the energy-momentum tensor of the electromagnetic field are where F αβ is the Faraday tensor. The components of the Faraday tensor obey the following equations: or, in more detail: where D µ = ∂ ∂x µ . As it is known therefore we may rewrite system (4) in the form The components of the Riemann connection have the form: Its other components are equal to zero. Since we consider the case of a homogenous space, Faraday tensor depends on time t only; then, in view of (7) the system (6) takes the form: Its solutions are Hereinafter we will be interested in the case of a pure magnetic field; so that φ m = 0 and, as a consequence, In what follows we will deal with diagonal energy-momentum tensor; in view of (1),(2),(9) and (10) it implies that /negationslash /negationslash Functions e -2 a l are linearly independent, therefore from (11) it follows that /negationslash Each ψ ij is multiplied by all that ψ kl , which has one of the indices k, l coincident with one of the indices i, j . /negationslash Let us fix pair ( i, j ) ; the number of combinations, in which ψ ij is found, equals to 2( N -2) . Let ψ ij = 0 ; then the other 2( N -2) quantities ψ ik ( ψ jk ), which are multiplied by ψ ij , must be equals to zero; indeed, if ψ ik = 0 ( k = i, j ) , then ψ ij ψ ik = 0 , but that contradict (12). Continuing the argument, we find the number of zero magnetic field components: /negationslash /negationslash /negationslash We took into account that the number of pairs of indices without the same elements equals to N -1 2 for an odd dimensions and to N 2 for an even dimensions. The total number of components of the magnetic field is N ( N -1) 2 ; then the number χ of non-zero magnetic field components is In what follows they are exactly ψ 12 , ψ 34 , . . . , ψ 2 n -1 , 2 n , n = 1 , χ that we set to be non-zero; other components of the Faraday tensor are assumed to be zero. Thus, components of the energy-momentum tensor of the electromagnetic field take the form: /negationslash /negationslash Hereinafter it will be convenient to use notations like this: We assume that /negationslash /negationslash /negationslash\",\n \"pages\": [\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"III.1. Field equations.\",\n \"content\": \"The action reads: where R and F αβ are scalar curvature and Faraday tensor respectively. Here and after we use Planck units: m Pl is the Planck mass. The gravitational equations is given by where are the components of the Einstein tensor. Taking into account (1) we get: /negationslash /negationslash As we consider the diagonal case here, then in view of (17),(18),(24),(27) for an even number of dimensions the gravitational equations can be written as /negationslash /negationslash Note that each expression like (28) describes two equations with numbers 2 n -1 and 2 n simultaneously (we use the notation (19) here). In the case of an odd number of dimensions there is one more equation in addition to these: /negationslash /negationslash The left side of the expression (29) is a first integral of (28).\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"III.2. Vacuum solution.\",\n \"content\": \"When there is no any matter the equations (28)-(30) lead to: /negationslash These equations has the following solutions [16]: /negationslash As a consequence, components of the metric tensor follows a power law: This is well-known multidimensional Kasner solution. /negationslash /negationslash\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"III.3. Stability conditions.\",\n \"content\": \"Metric ceases to obey a power law when there is a matter; but it turns out that the metric of the space filled with the magnetic field may get close to the Kasner metric and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (28) that get close to the Kasner solutions (33) and converge to them as time tends to the point t = 0 . Considering an influence of the magnetic field as a perturbation we will search for solutions of the equations (28) in the form: where a 0 k is the Kasner solution (33), ϕ k ∈ C 2 ( R ) is a perturbation. We will say that solutions a 0 k are asymptotically stable if In other words, solutions a 0 k are asymptotically stable if and only if Now we will find out conditions which specify asymptotically stable Kasner solutions. Proposition 1. Kasner solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 1 , n = 1 , χ . 1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (31)-(32) given by (33); then there exists solution { a 1 , . . . , a N } of the equations (28),(29) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that Substitution (33),(38) to the equations (28) and (29) leads to: /negationslash /negationslash /negationslash /negationslash It follows from (39) that Let t → 0 ; in view of (33),(39),(42) equations (40)-(41) take the form: /negationslash /negationslash /negationslash /negationslash Equalities (43) and (44) are satisfied if and only if q.e.d. 2. Let { a 0 1 , . . . , a 0 N } be the Kasner solution given by (33) such that p 2 n -1 + p 2 n < 1 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have: Let t = t 0 > 0 be the initial moment; we assume that /negationslash /negationslash It is easy to check that matrix A is nonsingular, so we obtain: or, Hence we deduce that It follows from p 2 m -1 + p 2 m < 1 , m = 1 , χ that λ m +2 > 0 ; therefore and solutions (33) with p 2 m -1 + p 2 m < 1 , m = 1 , χ are asymptotically stable. /squaresolid /negationslash Substitution (33),(46) to the equations (28) and (29) give us equations (40)-(41). In view of (47) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j in the lhs of the equations (40)-(41) and terms contained ϕ i in the rhs of that equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (40) can be written in the form: /negationslash or, in matrix notations, where\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"III.4. Numerical calculations.\",\n \"content\": \"The result presented above have been verified numerically. We have got a few hundreds numerical solutions with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found a lot of solutions that converge to the Kasner ones. Example of such solution is presented on the Fig. 1 (a). It should be note that looking for asymptotically stable solutions numerically using the synchronous time t has a number of drawbacks; particularly, numerical solutions of the equations (68)-(69) increase too rapidly impeding computations as t goes to zero. To avoid these problem we have introduced new time coordinate τ by means of the relation The change of variables from t to τ Indeed, (33) and (56) leads to results in transformation of logarithmic functions (33) into linear ones. Consequently, Without lack of generality we can set Then, as was announced, Such a way, looking for asymptotically stable solutions numerically, we should expect to get quasi-linear solutions like ones that shown on the Fig. 1 (a).\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"IV.1. Field equations.\",\n \"content\": \"Action of the theory under consideration is given by: where R,R αβ , R αβγδ , F αβ are the ( N + 1) -dimensional scalar curvature, Ricci tensor, Riemann tensor and Faraday tensor respectively. The gravitational equations has the form: where are the components of the Gauss-Bonnet tensor. Taking into account (1) we get: /negationslash /negationslash /negationslash For the diagonal case in view of (17),(18),(64),(67) for an even number of dimensions the gravitational equations can be written as: /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash There is one more equation in addition to these in the case of an odd number of dimensions: /negationslash The left side of the expression (69) is a first integral of (68).\",\n \"pages\": [\n 9,\n 10\n ]\n },\n {\n \"title\": \"IV.2. Vacuum solution.\",\n \"content\": \"When there is no any matter the equations (68)-(70) lead to: /negationslash /negationslash /negationslash These equations has the following solutions [5, 6]: As a consequence, components of the metric tensor follows a power law: Thus by analogy with (33)-(34) we will call solutions (73) and metric (74) Kasner-like for brevity.\",\n \"pages\": [\n 10\n ]\n },\n {\n \"title\": \"IV.3. Stability conditions.\",\n \"content\": \"As well as in the case of the Einstein gravity (see III.3.), metric ceases to obey a power law when there is a matter; it appears however that the metric of the space filled with the magnetic field may get close to the Kasner-like metric (74) and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (68) that get close to the Kasnerlike solutions (73) and converge to them as time goes to the point t = 0 . We will looking for solutions of the equations (68) in the form: where a 0 k is the Kasner-like solution (73), ϕ k ∈ C 2 ( R ) . We will say that solutions a 0 k are asymptotically stable if In other words, solutions a 0 k are asymptotically stable if and only if Now we will find out conditions which specify asymptotically stable Kasner-like solutions. Proposition 2 . Kasner-like solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 2 , n = 1 , χ . Let us introduce the following notation: where A i 1 , . . . , A i N , B i 1 , . . . , B i N are any indexed mathematical objects. Substitution (73),(78) to the equations (68) and (69) leads to: /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash where /negationslash /negationslash It follows from (79) that In the limit t → 0 in view of (73),(79),(83) equations (81)-(82) take the form: /negationslash Equalities (84) and (85) are satisfied if and only if q.e.d. 2. Let { a 0 1 , . . . , a 0 N } be the Kasner-like solution given by (73) such that p 2 n -1 + p 2 n < 2 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have: Let t = t 0 > 0 be the initial moment; we assume that /negationslash Substitution (73),(87) to the equations (68) and (69) give us equations (81)-(82). In view of (88) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j , t 3 ˙ ϕ i ˙ ϕ j ˙ ϕ k , t 4 ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l in the lhs of the equations (81)-(82) and terms contained ϕ i in the rhs of these equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (81) can be written in the form: /negationslash /negationslash or, in matrix notations, /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash Assuming that matrix A is nonsingular 2 , we obtain: or, Hence we deduce that It follows from p 2 m -1 + p 2 m < 2 , m = 1 , χ that γ m +2 > 0 ; therefore and solutions (73) with p 2 m -1 + p 2 m < 2 , m = 1 , χ are asymptotically stable. /squaresolid Example. Let us consider (5+1)-dimensional space-time. According to (15) there are pair of non-zero components of the Faraday tensor; taking into account our convention (see II. for details) we denote them ψ 12 and ψ 34 . Then under the Proposition 2 it follows that stable Kasner-like solutions are described by the conditions p 1 + p 2 < 2 and p 3 + p 4 < 2 .\",\n \"pages\": [\n 10,\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"IV.4. Numerical calculations.\",\n \"content\": \"The result obtained have been confirmed by numerical calculations. As well as in the study of the stability of the Kasner solutions (see III.3.) we have received a few hundreds solutions numerically with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found numerous solutions that converge to the Kasner-like ones. Example of such solution is presented on the Fig. 2 (a). For the reasons mentioned above (see III.4.) one should use special time coordinate for numerical calculations. Let us introduce new time coordinate τ by the following way: The change of variables from t to τ results in transformation of logarithmic functions (73) into linear ones. Indeed, (73) and (97) give us: /negationslash Consequently, Without loss of generality we can set Then, as was announced, Thus, searching for asymptotically stable solutions numerically we should expect to get quasi-linear solutions like ones that shown on the Fig. 2 (a).\",\n \"pages\": [\n 13,\n 14\n ]\n },\n {\n \"title\": \"V. Oscillatory regime.\",\n \"content\": \"Results of the preceding section pose an interesting question about the fate of trajectories in the instability zone. They can either reach stable Kasner-like attractor or represent some other type of dynamics different from Kasner-like behavior. In this section we show that the latter alternative can be realized. In our numerical calculations we have revealed a specific class of solutions of the equations (68). The discovered solutions possess the following properties: two of N functions a 1 , . . . , a N oscillate and has opposite signs, the other ones vary slowly; owing to phase difference between the oscillating functions volume element √ | det( g ) | d N x oscillates too. In view of limited computational capabilities we have obtained numerical solutions of the equations (68) only for the cases N = 5 , 6 , 7 , 8 ; for each of that dimensions we have found out sufficiently large number of oscillatory solutions. Example of such solution is presented on the Fig. 3 (a); the corresponding oscillations of the volume element is shown on the Fig. 3 (b). For N = 5 we have been able to derive an analytical approximation to such solutions on a certain interval of time. Below we describe the procedure of derivation of this approximation. Henceforth we use time coordinate τ introduced by (97). In the (5+1)-dimensional case equations (68) become: so that Let us decompose functions f, g, h with respect to ( 4 3 v -2 3 u ) and ( 4 3 u -2 3 v ) up to the first order: with the first integral The dash denotes derivative with respect to τ . Analysis of numerical solutions shows that there exists τ 0 > 0 ( τ = 0 is accepted as the initial moment) such that (see Fig. 6). In view of (109) equations (103)-(108) take the form: where Equation (111) is the sum of the equations (110); equations (110) can be replaced by their equivalent: In view of (109) the last equation from (116) is satisfied automatically, and the other equations become The last of equations (117) gives: Consequently, Numerical results show that such assumption is valid in a certain interval of time. Taking into account (119)- (122) we get: Let us consider the equation (124). Numerical calculations show that in the regime under consideration v '' / ( v ' ) 4 is the leading term in the lhs of the equation (124); then: We can obtain analytical solution of the equation (125) by omitting the term containing u in the rhs of (125): Solution of the equation (126) has the form Let us consider the equation (123). We omit the term, containing v in the rhs of (123): Under substitution of (127) into (128) we obtain: We have used the following notations: Equation (129) has the following solution: Here J 2 3 , Y 2 3 are Bessel functions of the first and the second kind respectively. Going back to the initial variables we get: and, finally: where To demonstrate oscillations of volume element √ | det( g ) | d N x one should write down expression for the determinant of the metric: Despite the fact that functions (134)-(136) are the solution of the simplified system (126),(128) they catch correctly qualitative features of solution of the full system (103)-(107) which can be shown by comparing functions (134)-(136) with numerical solution of the system (103)-(107). Figures 4,5 below depicts pairs ( a 2 , A 2 ); ( a 4 , A 4 ) of numerical and analytical solutions; pairs ( a 1 , A 1 ); ( a 3 , A 3 ) behave themselves analogously. This regime is not similar to Kasner-like one (Kasner-like solution has linear behavior on chosen time coordinate τ (see IV.4. for details), the leading asymptotic of the oscillatory solution has quadratic behaviour w.r.t. τ ).\",\n \"pages\": [\n 14,\n 15,\n 16,\n 17,\n 18\n ]\n },\n {\n \"title\": \"VI. Conclusions\",\n \"content\": \"In this paper we provided an analysis of stability of known power-law solutions in Gauss-Bonnet anisotropic cosmology in the presence of a homogeneous magnetic field in the case when metric is diagonal in the frame determined by magnetic field. The relative simplicity of equations of motion in this case allows us to find analytically a simple condition for the particular power-law solution to be stable. It appears that a set of initial conditions for which the power-law solution can not be an attractor near a cosmological singularity has non-zero measure (though does not coincide with the whole initial conditions space). The result has the same structure as in the Einstein gravity (particular sums of Kasner indices should be restricted by some number), the only difference is the exact value of this number. The fate of trajectories near singularity from this unstable domain requires future investigations. In the present paper we describe one particular class of trajectories found by numerical studies. The volume element experiences oscillations in the presented regime, making it qualitatively different from BKL oscillations. We can expect that some other regimes which do not tend to Kasner-like power-law regime exist in the system under investigation. Full description of them (the task which we leave for a future work) in this particular rather simple model can be useful in studies of cosmological singularity in Lovelock gravity in the presence of spatial curvature. The latter model usually lead to very cumbersome equations of motion, so understanding the nature of possible regimes in the instability zone for the Kasner-like solution in the present model would help in finding similar solution in more general and more technically complex situations of curved multidimensional geometries.\",\n \"pages\": [\n 22\n ]\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"The authors are grateful to Vladimir Ivashchuk for helpful discussion. The work is partially supported by the RFBR grant 11-02-00643.\",\n \"pages\": [\n 22\n ]\n }\n]"}}},{"rowIdx":272,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..289...74N"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1411.5999.pdf"},"content":{"kind":"string","value":"\nDetermining distances to stars statistically from photometry\nHeidi Jo Newberg 1\n1 Rensselaer Polytechnic Institute, Dept. of Physics, Applied Physics & Astronomy, 110 8 th St., Troy, NY 12180, USA\nemail: newbeh@rpi.edu\nAbstract. In determining the distances to stars within the Milky Way galaxy, one often uses photometric or spectroscopic parallax. In these methods, the type of each individual star is determined, and the absolute magnitude of that star type is compared with the measured apparent magnitude to determine individual distances. In this article, we define the term statistical photometric parallax , in which statistical knowledge of the absolute magnitudes of stellar populations is used to determine the underlying density distributions of those stars. This technique has been used to determine the density distribution of the Milky Way stellar halo and its component tidal streams, using very large samples of stars from the Sloan Digital Sky Survey. Most recently, the volunteer computing platform MilkyWay@home has been used to find the best fit model parameters for the density of these halo stars.\nKeywords. methods: data analysis, methods: statistical, stars: distances, stars: statistics, Galaxy: globular clusters: general, Galaxy: stellar content, Galaxy: structure\n1. Introduction\nWe often use the stars in the Milky Way to trace its structure. The brightest stars, which can be used to trace Galactic structure to the largest distances, include blue horizontal branch stars; O, B and A main sequence stars; RR Lyraes; Cepheid variables; red clump stars; K giants; and M giants (see Figure 1). All of these stellar types are reasonably well calibrated as distance indicators. K giant and red clump stars almost always require spectroscopy to accurately distinguish them from the more numerous main sequence stars of the same color, and RR Lyrae stars are usually identified from multiple epochs of photometry. All of these bright stars are relatively rare in any stellar population, and many of them are only observed in certain populations. For example, blue horizontal branch stars and RR Lyrae stars are only found in old populations, M giants are only found in relatively metal-rich populations, and O, B, and A main sequence stars are only found in very young populations (though A stars with main sequence gravities are found in some old populations as blue stragglers).\nDeep, large sky area, multicolor surveys with high accuracy calibrations, like the Sloan Digital Sky Survey (SDSS; York et al. 2000) enable new methods for studying the structure of the Milky Way. For example, Ivezi'c et al. (2008) derived a formula that allows us to estimate the temperature and metallicity of main sequence stars of type G and later from SDSS photometry. Since these lower mass main sequence stars do not evolve in the age of the Universe, their absolute magnitudes are independent of age. Photometry cannot tell us the surface gravities of these types of stars (Lenz et al. 1998), but since the vast majority of the red stars in the photometric survey are main sequence stars, this is not a major obstacle. Juri'c et al. (2009) used this technique to measure the distances to 48 million stars, and then used these distances to determine the density distributions\nof the disks and halo at 100 pc to 20 kpc from the Sun, over 6500 square degrees of sky. Though this analysis used a large number of stars, and resulted in the measurement of stellar density, the Juri'c analysis was using photometric parallax; they were determining the distance to each star individually using photometry.\nIn this article, we introduce the concept of statistical photometric parallax . We have used this technique most successfully to study the structure of the Galaxy using turnoff stars (Newberg et al. 2002). These stars are by definition brighter than main sequence stars such as those used by Juri'c et al., so they can trace the structure of the Milky Way 30 kpc or more from the Sun. However, the turnoff stars in a single stellar population can differ in absolute magnitude by two magnitudes (producing a distance error of a factor of 2.5). We do not have a way to determine the distance to a single turnoff star with reasonable accuracy. However, it has been shown that the absolute magnitude distribution of turnoff stars in halo globular clusters are surprisingly similar to each other (Newby et al. 2011), over a metallicity range -2.3 dex < [Fe/H] < -1.2 and ages ranging from 9 to 13.5 Gyr. Recently, Grabowski, Newby, & Newberg (2012) showed that this similarity even holds for the globular cluster Whiting 1, which is thought to be only 6.5 Gyrs old (Carraro, Zinn & Moni Bidin 2007).\nThis striking similarity between the absolute magnitude distributions of turnoff stars was not expected. One expects that younger globular clusters would have brighter, bluer turnoff stars. Also, one expects that more metal-rich clusters will have dimmer, redder turnoff stars. As it turns out, older stars in the Milky Way generally have lower metallicity, and the two effects cancel each other. This appears to be an unanticipated consequence of the Milky Way's Age-Metallicity Relationship (AMR - Muratov & Gnedin 2010, Dotter; Sarajedini & Anderson 2011). Apparently, the absolute magnitude distribution of turnoff stars is similar over the full age and metallicity range of typical stellar populations in the Milky Way halo.\nWe will describe here the general techique of statistical photometric parallax, which can be used to statistically account for the effects of a range of intrinsic brightnesses of the stellar population which is being used to trace Milky Way density structure, and can also statistically account for the observational biases in a survey such as the Sloan Digital Sky Survey. We have implemented this technique as a search for the density parameters with the highest likelihood of matching the observed data. Because this parameter search can be computationally expensive, we have employed supercomputers and a large volunteer computing platform, MilkyWay@home, that was built to solve this problem.\n2. Using Statistical Photometric Parallax\nWe are defining the term statistical photometric parallax here for the first time, and it is intended to apply in general for any case where the statistical distribution of absolute magnitudes is used to find the underlying density distribution of stars. However, we will describe here as an example the application of this technique to determine halo substructure using color-selected F turnoff stars, as used by Newberg et al. (2002), Cole et al. (2008), and Newby et al. (2012).\nIn Newberg et al. (2002), stars with colors 0 . 1 < ( g -r ) 0 < 0 . 3 and ( u -g ) 0 > 0 . 4 were selected as turnoff stars. The color range was chosen to be bluer than the turnoff of the thick disk, so that halo stars would preferentially be selected. In this paper, only the simplest form of statistical photometric parallax was employed. The distance to the Sagittarius dwarf tidal stream was determined by assuming the center of the absolute magnitude distribution of turnoff stars in the g filter was M g = 4 . 2. This number was calculated by comparing the apparent magnitude of the turnoff to the apparent\nmagnitude of RR Lyrae stars in the same stellar population. In this example, distances to single stars were not calculated; instead we made a more accurate determination of distance by looking at the distribution of apparent magnitudes of a particular set of stars.\nIn Cole et al. (2008), an algorithm was presented that allowed us to determine not just the distance to the center of a stream, but the three dimensional density of turnoff stars. The technique used maximum likelihood to find the model parameters /vector Q that make the observed star positions ( l i , b i , g i ) the most likely. The Likelihood L is given by the product of the probability density functions (PDFs) evaluated at all of the star positions:\nL = ∏ PDF( l i , b i , g i | /vector Q ) .\nThe PDF is constructed by the following steps:\n\n( a ) For each stellar component, one assumes a parameterized model (for example a double exponential, NFW, Hernquist, etc.) for the spatial density.\n( b ) This spatial density is transformed to ( l, b, g ) coordinates, assuming that the absolute magnitude of each of the stars is the average for the population.\n( c ) This density is convolved with the absolute magnitude distribution of the tracers, so that we produce the distribution that we expect to observe.\n( d ) This expected distribution is multiplied by the completeness for observing stars of a given apparent magnitude in a given survey, as a function of apparent magnitude.\n( e ) The resulting distribution is normalized so that the integrated probability of finding a star in the entire volume observed is one.\n( f ) The final PDF is the sum of the fraction of stars in each component times the normalized distribution, summed over the number of components in the model. The fraction of stars in each component are also parameters that are fit in the maximum likelihood optimization.\nOf these steps, the most time-consuming is the calculation of the integral over the volume. Contrary to first impressions, the time to calculate the likelihood depends more heavily on the number of sub-volumes into which the survey space needs to be divided to achieve an accurate result, than on the number of stars in the dataset.\n\nOne then uses an optimization technique to find the model parameters that produce the highest likelihood. When using a supercomputer, we usually use conjugate gradient descent. This algorithm is sequential; one evaluates the likelihood and the derivatives with respect to each parameter, chooses a direction, then decides how far in that direction to go before repeating that process.\nNewby et al. (2012) applied this technique to all of the data available in SDSS DR7 to find the density of the Sagittarius dwarf tidal stream in the North Galactic Cap, and in three SDSS stripes in the South Galactic Cap. One of the advantages of this probablistic technique is that we are able to extract from the sample of 1.7 million turnoff stars a set of 200 , 000 stars that have the spatial characteristics of the Sagittarius dwarf tidal stream. This is accomplished by generating a random mumber for each star, and using that random number to place it either in the Sagittarius dwarf tidal stream catalog, with the probability that a star at that position in the Galaxy is in the Sagittarius dwarf tidal stream; or in the catalog of non-Sagittarius halo stars, with the probability that a star at that position in the Galaxy is not in the Sagittarius dwaf tidal stream. Note that if you wanted to find actual stars in the Sagittarius dwarf tidal stream, say for spectroscopic follow-up, you should use the original catalog of stars and select those with the highest probability of being in the Sagittarius dwarf tidal stream. However, the statistically separated catalog we generated facilitates the study of density substructures in the halo. In particular, we can remove the Sagittarius stream from the original stellar\nsample so we can study the smaller tidal streams, and the density structure of the smooth component of the halo. An example stripe analyzed in Newby et al. (2012) is shown in Figure 2.\nIn Cole et al. (2008) and Newby et al. (2012) we modeled the distribution of turnoff star absolute magnitudes as a Gaussian centered at M g = 4 . 2 with a width of σ = 0 . 6 magnitudes. Since then, we have recognized that the distribution is asymmetric; there are more stars fainter than the maximum than there are brighter than the maximum. More importantly we have learned that, due to larger color errors at fainter magnitudes, the absolute magnitude distribution of color-selected stars is different near the survey limit than it is for the brighter stars. This effect is much larger than we expected it to be. Near the survey limit, the majority of the stars are not turnoff stars, but are fainter main sequence stars that have scattered into our color selection limits due to large measurement errors. Because we are using a statistical approach, this effect can be included in the analysis by varying the absolute magnitude distribution as a function of apparent magnitude. We plan to include this in future analyses.\n3. Processing time and the MilkyWay@home Volunteer Computing Platform\nWe originally tried to implement the search for maximumum likelihood on a single CPU. In a single 2 . 5 · -wide SDSS data stripe, we fit 2 parameters to a smooth halo with a Hernquist profile, and 6 parameters per tidal debris stream. If there were three tidal streams in a single stripe, there would be 20 parameters. Evaluating the likelihood for one guess for the model parameters currently takes about 4 hours, with most of the time spent integrating the PDF over the survey volume. In order to optimize 20 parameters requires about 50 likelihood evaluations per conjugate gradient descent step and 50 steps per maximum likelihood evaluation. This totals ten thousand hours per optimization (over a year).\nLuckily the optimization is embarrassingly parallel. It is possible to parallelize the integral, since each integral volume calculation is completely independent of the others. We are able to run this algorithm on a 256 node rack of a Blue Gene/L supercomputer. Parallelizing the integral over 256 nodes cuts the time per likelihood calculation down under a minute. A conjugate gradient step can then be accomplished in 47 minutes, and ten iterations can be accomplished in under eight hours (which is comfortably less the the maximum job size allowed in our queue). In practice, we need to try several conjugate gradient descents to approximate the best parameters. Once they are known approximately, we run of order ten conjugate gradient descents starting near the best values. Including submitting jobs a few at a time and waiting for queue time, this process can take a couple of weeks to validate the results for one SDSS stripe.\nCurrently our best method for computing the parameters is the volunteer computing platform MilkyWay@home. This project is part of the Berkeley Open Infrastructure for Network Computing (BOINC; Anderson, Korpela & Walton 2005) group of volunteer computing platforms. The first and most famous of these is SETI@home. BOINC offers us a template server and database application, and an infrastructure for volunteers to donate their time to our server. We implemented our own server, including the maximum likelihood algorithm, a set of optimization routines that will run in a heterogeneous, asynchronous parallel computing environment (Desell et al. 2010b), and a modified server application that sends out 'work units' to the volunteers, collects the results, and validates that the results are not in error (Desell et al. 2010a). One of the surprises in operating a BOINC server is that some of the results sent back from the\nvolunteers are not correct, either because their hardware is malfunctioning, they did not update the software correctly, or they purposely sent back a wrong answer quickly so that they can accumulate BOINC 'credit' more quickly. It is impossible to overestimate how important it is to our volunteers that they get credit for the work units their computers crunch, and that credit is apportioned fairly between the volunteers.\nMilkyWay@home is currently delivering 0.5 PetaFLOPS of computing power from 25,000 active volunteers giving us access to over 35,000 CPUs or GPUs. The majority of the computing power comes from the GPUs, the best of which can process our likelihood calculations about 100 times faster than the CPUs (Desell et al. 2009). It is not easy to parallelize the computation of the integral on BOINC, because that would require communication between the processors, which is not possible at this time. Instead, we parallelize the calculations by sending a single likelihood calculation (including the whole integral for a given /vector Q ) to each volunteer. These work units can take a couple minutes (if the work unit is sent to a GPU) or four or more hours (if it goes to a CPU), but it might take minutes, days, or weeks for the likelihood to be returned depending on how much the volunteer is using the computer for her own purposes, and whether she turns it off.\nIt takes far more computing power to calculate best fit parameters on the MilkyWay@home computing system than on a supercomputer. There are four factors responsible for this: (1) We cannot use sequential searches like conjugate gradient descent. Instead, we use 'particle swarm' or a genetic search algorithm. In the particle swarm technique, we send out a random set of guesses that span the parameter space. As the likelihood results from the volunteers come back, we send out more work units with guesses that are closer to the higher likelihoods. This search method requires many more steps, but produces more accurate results. (2) A fraction (about 10%) of the work units, selected at random, are sent out five times so that we can validate that they are correct. If three or more are returned with the same answer, then the result is validated. We also choose to validate the best likelihoods (since those influence our future guesses), and the likelihoods of users that have submitted previous results that did not validate correctly. (3) Some of the work units that are sent out are never returned. (4) Because we can, we run the searches for a longer period of time over a wider range of of parameter space and we get better global values for the parameters (Newby et al. 2012). To optimize one stripe takes 1-2 weeks, and hundreds of thousands of likelihood calculations. There are enough volunteers that we can optimize 4-5 stripes at the same time and still get the results within the same time scale. By putting more jobs on MilkyWay@home at the same time, we decrease the number of work units that are sent out at the same time. By waiting a little longer to send out more work units, the results of previous searches can be used to make better guesses of the parameters, so adding more jobs increases the computing time at a rate that is less than linear in the number of jobs.\n4. Discussion and Conclusion\nThe purpose of this conference contribution is to define the term statistical photometric parallax , which allows us to determine the density of a population of stars, even if we cannot determine the distance to each individual star in the population. We find the most likely parameters for the density distribution, given that we know the absolute magnitude distribution and the observational constraints of the observed sample of stars. If the population of stars is all at the same distance (for example in a globular cluster or tidal stream), then statistical photometric parallax can be used to determine the distance to the stellar population. Because the SDSS made available a large, well calibrated sample of stars with multi-color photometry, this technique has recently become feasible.\nThe example I present is the use of turnoff stars to determine the density distribution of stars in the stellar halo of the Milky Way. By apparent coincidence, the distribution of absolute magnitudes of turnoff stars is very similar for all stellar populations in the age and metallicity range of halo stars. This appears to be a result of the Milky Way age-metallicity relationship. We are in the process of using this technique to accurately map the density distribution of the entire Milky Way stellar halo.\nThe drawback to this techique is that to use it one must use a maximum likelihood algorithm that can in some cases require high performance parallel computing to get an accurate measurement of the parameters in the density function. In the process of learning to use this method, we created a large volunteer computing platform called MilkyWay@home.\nAcknowledgement\nI would like to thank Matthew Newby and Brian Yanny for helping me obtain the data for the figures in this publication. I also thank Matthew Newby and Jeff Carlin for their help in proofreading. This paper is based upon work supported by the National Science Foundation under Grant No. AST 10-09670. I also would like to thank the MilkyWay@home volunteers for providing us with computing power at no cost, and the Marvin Clan for their support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.\nReferences\nAnderson, D. P., Korpela, E., and Walton, R. 2005, The First International Conf. on e-Science\nand Grid Tech. (e-Science 2005), 196 Carraro, G., Zinn, R., & Moni Bidin, C. 2007, A&A , 466, 181 Cole, N., Newberg, H. J., Magdon-Ismail, M., et al. 2008, ApJ , 683, 750 Desell, T., Waters, A., Magdon-Ismail, M., et al. 2009, PPAM, Pt I, Lecture Notes in Computer Science, 6067, 276 Desell, T., Magdon-Ismail, M., Szymanski, B., Varela, C., Newberg, H., & Anderson, D. 2010a, DAIS, Lecture Notes in Computer Science, 6115, 29 Desell, T., Anderson, D., Magdon-Ismail, M., Szymanski, B., Newberg, H. J., Varela, C. 2010b, The 2010 IEEE congress on evolutionary computation (IEEE CEC) Grabowski, K., Newby, M., and Newberg, H. J. 2012, J. of Undergrad. Res. in Phys. , submittted Ivezi'c, ˇ Z., Sesar, B., Juri'c, M., et al. 2008, ApJ , 684, 287 Juri'c, M., Ivezi'c, ˇ Z., Brooks, A., et al. 2008, ApJ , 673, 864 Lenz, D. D., Newberg, J., Rosner, R., Richards, G. T., & Stoughton, C. 1998, ApJ (Supplement), 119, 121 Muratov, A. L., & Gnedin, O. Y. 2010, ApJ , 718, 1266 Newberg, H. J., Yanny, B., Rockosi, C., et al. 2002, ApJ , 569, 245 Newby, M., Newberg, H. J., Simones, J., Cole, N., & Monaco, M. 2011, ApJ , 743, 187 Newby, M., Cole, N., Newberg, H. J., Desell, T., Magdon-Ismail, M., Szymanski, B., Varela, C., Willett, B., & Yanny, B. 2012, ApJ , submitted Rave, H. A., Zhao, C., Newberg, H. J., et al. 2003, ApJ (Supplement), 145, 245 SDSS-III Collaboration, Ahn, C. P., et al. 2012, arXiv:1207.7137 Yanny, B., Newberg, H. J., Johnson, J. A., et al. 2009, ApJ , 700, 1282 York, D. G., Adelman, J., Anderson, J. E., Jr., et al. 2000, AJ , 120, 1579\n\n\n
Figure 1. Color-magnitude diagram of stars withing 4 arcminutes of the globular cluster Palomar 5. All stars within four arcminutes of Pal5 were selected from SDSS DR9 (SDSS III Collaboration et al. 2012). We illustrate how sparse the brighter stars are in a stellar population, compared to the main sequence and turnoff. Most of the selection boxes for each type of star were drawn to include the representative stars in Palomar 5. The RR Lyrae selection box was taken from Rave et al. (2003). The M giant selection box was estimated from Yanny et al. (2009) measurements of M giants in the Sagittarius dwarf tidal stream. There are no representatives of these types of stars in the Palomar 5 globular cluster. Although giant stars of all types can be observed at larger distances, many types of giant stars are specific to particular populations. K giants are seen in all but the youngest stellar populations, but these stars are more difficult to separate from dwarf K stars that have the same colors. Also, K giant stars are more difficult to use as distance indicators because their absolute magnitude is a strong function of color (temperature), and is also sensitive to metallicity and age. The drawback to using turnoff stars as distance indicators is that they vary in absolute magnitude by two magnitudes. Statistical photometric parallax allows us to used them effectively to determine the underlying density of stars even though we cannot accurately measure the distance to each individual star.
\n\n\n\n
Figure 2. The A and F star distribution on the Celestial Equator. All of the stars in this figure were selected within 0.1 degrees of the Celestial Equator, and 145 · < RA < 265 · . The top left panel shows photometrically selected A stars from SDSS DR9 (SDSS-III Collaboration et al. 2012). The stars were selected with -0 . 3 < ( g -r ) 0 < 0 . 0, 0 . 8 < ( u -g ) 0 < 1 . 5, and 14 < g 0 < 22 . 5, where the subscript indicates that the values have been corrected for reddening using the extinction calculated in the database. The top right panel shows photometrically selected turnoff stars, selected from SDSS DR9. The stars were selected with 0 . 2 < ( g -r ) 0 < 0 . 4 and 14 < g 0 < 23. Notice that even though we selected only the bluest turnoff stars, there are very many more blue turnoff stars than stars with the colors of A stars. However, the wide range of g 0 magnitudes observed in the globular cluster Palomar 5 at RA = 229 · indicates that the range of absolute magnitudes of turnoff stars is at least two magnitudes. Even though the range of absolute magnitudes is large, we can see the Palomar 5 globular cluster, the Sagittarius dwarf galaxy tidal stream at ( RA,g 0 ) = (210 · , 22 . 5), the Virgo Overdensity at ( RA,g 0 ) = (180 · , 21), and stars that are presumably part of the smooth spheroid near the Galactic center at ( RA,g 0 ) = (245 · , 19). The substructure in A-type stars is not as evident partially because there are many fewer A-type stars, partially because we have included both blue stragglers and blue horizontal branch stars, and partially because not all stellar populations have A-type stars. Blue horizontal branch stars are about 1.5 magnitudes brighter than turnoff stars, and blue stragglers are about 3.5 magnitudes brigher than turnoff stars. A small number of blue stragglers from the Sagittarius dwarf tidal stream are evident at g 0 = 21 in the A star panel, but Palomar 5 blue stragglers are not evident. We do not see blue horizontal branch stars from either Palomar 5 or the Sagittarius dwarf tidal stream in this plot. One can see the advantage of using turnoff stars in identifying the Sagittarius dwarf tidal stream. The lower two panels show the stars selected by Newby et al. (2012), from this small region of the sky, that have the density distribution of the Sagittarius dwarf tidal stream (left) and the density distribution over everything else (right). The region around the globular cluster Palomar 5 was removed from the data before fitting, so that this globular cluster would not affect the results.
\n\n"},"sections":{"kind":"list like","value":[{"title":"Heidi Jo Newberg 1","content":"1 Rensselaer Polytechnic Institute, Dept. of Physics, Applied Physics & Astronomy, 110 8 th St., Troy, NY 12180, USA email: newbeh@rpi.edu Abstract. In determining the distances to stars within the Milky Way galaxy, one often uses photometric or spectroscopic parallax. In these methods, the type of each individual star is determined, and the absolute magnitude of that star type is compared with the measured apparent magnitude to determine individual distances. In this article, we define the term statistical photometric parallax , in which statistical knowledge of the absolute magnitudes of stellar populations is used to determine the underlying density distributions of those stars. This technique has been used to determine the density distribution of the Milky Way stellar halo and its component tidal streams, using very large samples of stars from the Sloan Digital Sky Survey. Most recently, the volunteer computing platform MilkyWay@home has been used to find the best fit model parameters for the density of these halo stars. Keywords. methods: data analysis, methods: statistical, stars: distances, stars: statistics, Galaxy: globular clusters: general, Galaxy: stellar content, Galaxy: structure","pages":[1]},{"title":"1. Introduction","content":"We often use the stars in the Milky Way to trace its structure. The brightest stars, which can be used to trace Galactic structure to the largest distances, include blue horizontal branch stars; O, B and A main sequence stars; RR Lyraes; Cepheid variables; red clump stars; K giants; and M giants (see Figure 1). All of these stellar types are reasonably well calibrated as distance indicators. K giant and red clump stars almost always require spectroscopy to accurately distinguish them from the more numerous main sequence stars of the same color, and RR Lyrae stars are usually identified from multiple epochs of photometry. All of these bright stars are relatively rare in any stellar population, and many of them are only observed in certain populations. For example, blue horizontal branch stars and RR Lyrae stars are only found in old populations, M giants are only found in relatively metal-rich populations, and O, B, and A main sequence stars are only found in very young populations (though A stars with main sequence gravities are found in some old populations as blue stragglers). Deep, large sky area, multicolor surveys with high accuracy calibrations, like the Sloan Digital Sky Survey (SDSS; York et al. 2000) enable new methods for studying the structure of the Milky Way. For example, Ivezi'c et al. (2008) derived a formula that allows us to estimate the temperature and metallicity of main sequence stars of type G and later from SDSS photometry. Since these lower mass main sequence stars do not evolve in the age of the Universe, their absolute magnitudes are independent of age. Photometry cannot tell us the surface gravities of these types of stars (Lenz et al. 1998), but since the vast majority of the red stars in the photometric survey are main sequence stars, this is not a major obstacle. Juri'c et al. (2009) used this technique to measure the distances to 48 million stars, and then used these distances to determine the density distributions of the disks and halo at 100 pc to 20 kpc from the Sun, over 6500 square degrees of sky. Though this analysis used a large number of stars, and resulted in the measurement of stellar density, the Juri'c analysis was using photometric parallax; they were determining the distance to each star individually using photometry. In this article, we introduce the concept of statistical photometric parallax . We have used this technique most successfully to study the structure of the Galaxy using turnoff stars (Newberg et al. 2002). These stars are by definition brighter than main sequence stars such as those used by Juri'c et al., so they can trace the structure of the Milky Way 30 kpc or more from the Sun. However, the turnoff stars in a single stellar population can differ in absolute magnitude by two magnitudes (producing a distance error of a factor of 2.5). We do not have a way to determine the distance to a single turnoff star with reasonable accuracy. However, it has been shown that the absolute magnitude distribution of turnoff stars in halo globular clusters are surprisingly similar to each other (Newby et al. 2011), over a metallicity range -2.3 dex < [Fe/H] < -1.2 and ages ranging from 9 to 13.5 Gyr. Recently, Grabowski, Newby, & Newberg (2012) showed that this similarity even holds for the globular cluster Whiting 1, which is thought to be only 6.5 Gyrs old (Carraro, Zinn & Moni Bidin 2007). This striking similarity between the absolute magnitude distributions of turnoff stars was not expected. One expects that younger globular clusters would have brighter, bluer turnoff stars. Also, one expects that more metal-rich clusters will have dimmer, redder turnoff stars. As it turns out, older stars in the Milky Way generally have lower metallicity, and the two effects cancel each other. This appears to be an unanticipated consequence of the Milky Way's Age-Metallicity Relationship (AMR - Muratov & Gnedin 2010, Dotter; Sarajedini & Anderson 2011). Apparently, the absolute magnitude distribution of turnoff stars is similar over the full age and metallicity range of typical stellar populations in the Milky Way halo. We will describe here the general techique of statistical photometric parallax, which can be used to statistically account for the effects of a range of intrinsic brightnesses of the stellar population which is being used to trace Milky Way density structure, and can also statistically account for the observational biases in a survey such as the Sloan Digital Sky Survey. We have implemented this technique as a search for the density parameters with the highest likelihood of matching the observed data. Because this parameter search can be computationally expensive, we have employed supercomputers and a large volunteer computing platform, MilkyWay@home, that was built to solve this problem.","pages":[1,2]},{"title":"2. Using Statistical Photometric Parallax","content":"We are defining the term statistical photometric parallax here for the first time, and it is intended to apply in general for any case where the statistical distribution of absolute magnitudes is used to find the underlying density distribution of stars. However, we will describe here as an example the application of this technique to determine halo substructure using color-selected F turnoff stars, as used by Newberg et al. (2002), Cole et al. (2008), and Newby et al. (2012). In Newberg et al. (2002), stars with colors 0 . 1 < ( g -r ) 0 < 0 . 3 and ( u -g ) 0 > 0 . 4 were selected as turnoff stars. The color range was chosen to be bluer than the turnoff of the thick disk, so that halo stars would preferentially be selected. In this paper, only the simplest form of statistical photometric parallax was employed. The distance to the Sagittarius dwarf tidal stream was determined by assuming the center of the absolute magnitude distribution of turnoff stars in the g filter was M g = 4 . 2. This number was calculated by comparing the apparent magnitude of the turnoff to the apparent magnitude of RR Lyrae stars in the same stellar population. In this example, distances to single stars were not calculated; instead we made a more accurate determination of distance by looking at the distribution of apparent magnitudes of a particular set of stars. In Cole et al. (2008), an algorithm was presented that allowed us to determine not just the distance to the center of a stream, but the three dimensional density of turnoff stars. The technique used maximum likelihood to find the model parameters /vector Q that make the observed star positions ( l i , b i , g i ) the most likely. The Likelihood L is given by the product of the probability density functions (PDFs) evaluated at all of the star positions: The PDF is constructed by the following steps: One then uses an optimization technique to find the model parameters that produce the highest likelihood. When using a supercomputer, we usually use conjugate gradient descent. This algorithm is sequential; one evaluates the likelihood and the derivatives with respect to each parameter, chooses a direction, then decides how far in that direction to go before repeating that process. Newby et al. (2012) applied this technique to all of the data available in SDSS DR7 to find the density of the Sagittarius dwarf tidal stream in the North Galactic Cap, and in three SDSS stripes in the South Galactic Cap. One of the advantages of this probablistic technique is that we are able to extract from the sample of 1.7 million turnoff stars a set of 200 , 000 stars that have the spatial characteristics of the Sagittarius dwarf tidal stream. This is accomplished by generating a random mumber for each star, and using that random number to place it either in the Sagittarius dwarf tidal stream catalog, with the probability that a star at that position in the Galaxy is in the Sagittarius dwarf tidal stream; or in the catalog of non-Sagittarius halo stars, with the probability that a star at that position in the Galaxy is not in the Sagittarius dwaf tidal stream. Note that if you wanted to find actual stars in the Sagittarius dwarf tidal stream, say for spectroscopic follow-up, you should use the original catalog of stars and select those with the highest probability of being in the Sagittarius dwarf tidal stream. However, the statistically separated catalog we generated facilitates the study of density substructures in the halo. In particular, we can remove the Sagittarius stream from the original stellar sample so we can study the smaller tidal streams, and the density structure of the smooth component of the halo. An example stripe analyzed in Newby et al. (2012) is shown in Figure 2. In Cole et al. (2008) and Newby et al. (2012) we modeled the distribution of turnoff star absolute magnitudes as a Gaussian centered at M g = 4 . 2 with a width of σ = 0 . 6 magnitudes. Since then, we have recognized that the distribution is asymmetric; there are more stars fainter than the maximum than there are brighter than the maximum. More importantly we have learned that, due to larger color errors at fainter magnitudes, the absolute magnitude distribution of color-selected stars is different near the survey limit than it is for the brighter stars. This effect is much larger than we expected it to be. Near the survey limit, the majority of the stars are not turnoff stars, but are fainter main sequence stars that have scattered into our color selection limits due to large measurement errors. Because we are using a statistical approach, this effect can be included in the analysis by varying the absolute magnitude distribution as a function of apparent magnitude. We plan to include this in future analyses.","pages":[2,3,4]},{"title":"3. Processing time and the MilkyWay@home Volunteer Computing Platform","content":"We originally tried to implement the search for maximumum likelihood on a single CPU. In a single 2 . 5 · -wide SDSS data stripe, we fit 2 parameters to a smooth halo with a Hernquist profile, and 6 parameters per tidal debris stream. If there were three tidal streams in a single stripe, there would be 20 parameters. Evaluating the likelihood for one guess for the model parameters currently takes about 4 hours, with most of the time spent integrating the PDF over the survey volume. In order to optimize 20 parameters requires about 50 likelihood evaluations per conjugate gradient descent step and 50 steps per maximum likelihood evaluation. This totals ten thousand hours per optimization (over a year). Luckily the optimization is embarrassingly parallel. It is possible to parallelize the integral, since each integral volume calculation is completely independent of the others. We are able to run this algorithm on a 256 node rack of a Blue Gene/L supercomputer. Parallelizing the integral over 256 nodes cuts the time per likelihood calculation down under a minute. A conjugate gradient step can then be accomplished in 47 minutes, and ten iterations can be accomplished in under eight hours (which is comfortably less the the maximum job size allowed in our queue). In practice, we need to try several conjugate gradient descents to approximate the best parameters. Once they are known approximately, we run of order ten conjugate gradient descents starting near the best values. Including submitting jobs a few at a time and waiting for queue time, this process can take a couple of weeks to validate the results for one SDSS stripe. Currently our best method for computing the parameters is the volunteer computing platform MilkyWay@home. This project is part of the Berkeley Open Infrastructure for Network Computing (BOINC; Anderson, Korpela & Walton 2005) group of volunteer computing platforms. The first and most famous of these is SETI@home. BOINC offers us a template server and database application, and an infrastructure for volunteers to donate their time to our server. We implemented our own server, including the maximum likelihood algorithm, a set of optimization routines that will run in a heterogeneous, asynchronous parallel computing environment (Desell et al. 2010b), and a modified server application that sends out 'work units' to the volunteers, collects the results, and validates that the results are not in error (Desell et al. 2010a). One of the surprises in operating a BOINC server is that some of the results sent back from the volunteers are not correct, either because their hardware is malfunctioning, they did not update the software correctly, or they purposely sent back a wrong answer quickly so that they can accumulate BOINC 'credit' more quickly. It is impossible to overestimate how important it is to our volunteers that they get credit for the work units their computers crunch, and that credit is apportioned fairly between the volunteers. MilkyWay@home is currently delivering 0.5 PetaFLOPS of computing power from 25,000 active volunteers giving us access to over 35,000 CPUs or GPUs. The majority of the computing power comes from the GPUs, the best of which can process our likelihood calculations about 100 times faster than the CPUs (Desell et al. 2009). It is not easy to parallelize the computation of the integral on BOINC, because that would require communication between the processors, which is not possible at this time. Instead, we parallelize the calculations by sending a single likelihood calculation (including the whole integral for a given /vector Q ) to each volunteer. These work units can take a couple minutes (if the work unit is sent to a GPU) or four or more hours (if it goes to a CPU), but it might take minutes, days, or weeks for the likelihood to be returned depending on how much the volunteer is using the computer for her own purposes, and whether she turns it off. It takes far more computing power to calculate best fit parameters on the MilkyWay@home computing system than on a supercomputer. There are four factors responsible for this: (1) We cannot use sequential searches like conjugate gradient descent. Instead, we use 'particle swarm' or a genetic search algorithm. In the particle swarm technique, we send out a random set of guesses that span the parameter space. As the likelihood results from the volunteers come back, we send out more work units with guesses that are closer to the higher likelihoods. This search method requires many more steps, but produces more accurate results. (2) A fraction (about 10%) of the work units, selected at random, are sent out five times so that we can validate that they are correct. If three or more are returned with the same answer, then the result is validated. We also choose to validate the best likelihoods (since those influence our future guesses), and the likelihoods of users that have submitted previous results that did not validate correctly. (3) Some of the work units that are sent out are never returned. (4) Because we can, we run the searches for a longer period of time over a wider range of of parameter space and we get better global values for the parameters (Newby et al. 2012). To optimize one stripe takes 1-2 weeks, and hundreds of thousands of likelihood calculations. There are enough volunteers that we can optimize 4-5 stripes at the same time and still get the results within the same time scale. By putting more jobs on MilkyWay@home at the same time, we decrease the number of work units that are sent out at the same time. By waiting a little longer to send out more work units, the results of previous searches can be used to make better guesses of the parameters, so adding more jobs increases the computing time at a rate that is less than linear in the number of jobs.","pages":[4,5]},{"title":"4. Discussion and Conclusion","content":"The purpose of this conference contribution is to define the term statistical photometric parallax , which allows us to determine the density of a population of stars, even if we cannot determine the distance to each individual star in the population. We find the most likely parameters for the density distribution, given that we know the absolute magnitude distribution and the observational constraints of the observed sample of stars. If the population of stars is all at the same distance (for example in a globular cluster or tidal stream), then statistical photometric parallax can be used to determine the distance to the stellar population. Because the SDSS made available a large, well calibrated sample of stars with multi-color photometry, this technique has recently become feasible. The example I present is the use of turnoff stars to determine the density distribution of stars in the stellar halo of the Milky Way. By apparent coincidence, the distribution of absolute magnitudes of turnoff stars is very similar for all stellar populations in the age and metallicity range of halo stars. This appears to be a result of the Milky Way age-metallicity relationship. We are in the process of using this technique to accurately map the density distribution of the entire Milky Way stellar halo. The drawback to this techique is that to use it one must use a maximum likelihood algorithm that can in some cases require high performance parallel computing to get an accurate measurement of the parameters in the density function. In the process of learning to use this method, we created a large volunteer computing platform called MilkyWay@home.","pages":[5,6]},{"title":"Acknowledgement","content":"I would like to thank Matthew Newby and Brian Yanny for helping me obtain the data for the figures in this publication. I also thank Matthew Newby and Jeff Carlin for their help in proofreading. This paper is based upon work supported by the National Science Foundation under Grant No. AST 10-09670. I also would like to thank the MilkyWay@home volunteers for providing us with computing power at no cost, and the Marvin Clan for their support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.","pages":[6]},{"title":"References","content":"Anderson, D. P., Korpela, E., and Walton, R. 2005, The First International Conf. on e-Science","pages":[6]}],"string":"[\n {\n \"title\": \"Heidi Jo Newberg 1\",\n \"content\": \"1 Rensselaer Polytechnic Institute, Dept. of Physics, Applied Physics & Astronomy, 110 8 th St., Troy, NY 12180, USA email: newbeh@rpi.edu Abstract. In determining the distances to stars within the Milky Way galaxy, one often uses photometric or spectroscopic parallax. In these methods, the type of each individual star is determined, and the absolute magnitude of that star type is compared with the measured apparent magnitude to determine individual distances. In this article, we define the term statistical photometric parallax , in which statistical knowledge of the absolute magnitudes of stellar populations is used to determine the underlying density distributions of those stars. This technique has been used to determine the density distribution of the Milky Way stellar halo and its component tidal streams, using very large samples of stars from the Sloan Digital Sky Survey. Most recently, the volunteer computing platform MilkyWay@home has been used to find the best fit model parameters for the density of these halo stars. Keywords. methods: data analysis, methods: statistical, stars: distances, stars: statistics, Galaxy: globular clusters: general, Galaxy: stellar content, Galaxy: structure\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"We often use the stars in the Milky Way to trace its structure. The brightest stars, which can be used to trace Galactic structure to the largest distances, include blue horizontal branch stars; O, B and A main sequence stars; RR Lyraes; Cepheid variables; red clump stars; K giants; and M giants (see Figure 1). All of these stellar types are reasonably well calibrated as distance indicators. K giant and red clump stars almost always require spectroscopy to accurately distinguish them from the more numerous main sequence stars of the same color, and RR Lyrae stars are usually identified from multiple epochs of photometry. All of these bright stars are relatively rare in any stellar population, and many of them are only observed in certain populations. For example, blue horizontal branch stars and RR Lyrae stars are only found in old populations, M giants are only found in relatively metal-rich populations, and O, B, and A main sequence stars are only found in very young populations (though A stars with main sequence gravities are found in some old populations as blue stragglers). Deep, large sky area, multicolor surveys with high accuracy calibrations, like the Sloan Digital Sky Survey (SDSS; York et al. 2000) enable new methods for studying the structure of the Milky Way. For example, Ivezi'c et al. (2008) derived a formula that allows us to estimate the temperature and metallicity of main sequence stars of type G and later from SDSS photometry. Since these lower mass main sequence stars do not evolve in the age of the Universe, their absolute magnitudes are independent of age. Photometry cannot tell us the surface gravities of these types of stars (Lenz et al. 1998), but since the vast majority of the red stars in the photometric survey are main sequence stars, this is not a major obstacle. Juri'c et al. (2009) used this technique to measure the distances to 48 million stars, and then used these distances to determine the density distributions of the disks and halo at 100 pc to 20 kpc from the Sun, over 6500 square degrees of sky. Though this analysis used a large number of stars, and resulted in the measurement of stellar density, the Juri'c analysis was using photometric parallax; they were determining the distance to each star individually using photometry. In this article, we introduce the concept of statistical photometric parallax . We have used this technique most successfully to study the structure of the Galaxy using turnoff stars (Newberg et al. 2002). These stars are by definition brighter than main sequence stars such as those used by Juri'c et al., so they can trace the structure of the Milky Way 30 kpc or more from the Sun. However, the turnoff stars in a single stellar population can differ in absolute magnitude by two magnitudes (producing a distance error of a factor of 2.5). We do not have a way to determine the distance to a single turnoff star with reasonable accuracy. However, it has been shown that the absolute magnitude distribution of turnoff stars in halo globular clusters are surprisingly similar to each other (Newby et al. 2011), over a metallicity range -2.3 dex < [Fe/H] < -1.2 and ages ranging from 9 to 13.5 Gyr. Recently, Grabowski, Newby, & Newberg (2012) showed that this similarity even holds for the globular cluster Whiting 1, which is thought to be only 6.5 Gyrs old (Carraro, Zinn & Moni Bidin 2007). This striking similarity between the absolute magnitude distributions of turnoff stars was not expected. One expects that younger globular clusters would have brighter, bluer turnoff stars. Also, one expects that more metal-rich clusters will have dimmer, redder turnoff stars. As it turns out, older stars in the Milky Way generally have lower metallicity, and the two effects cancel each other. This appears to be an unanticipated consequence of the Milky Way's Age-Metallicity Relationship (AMR - Muratov & Gnedin 2010, Dotter; Sarajedini & Anderson 2011). Apparently, the absolute magnitude distribution of turnoff stars is similar over the full age and metallicity range of typical stellar populations in the Milky Way halo. We will describe here the general techique of statistical photometric parallax, which can be used to statistically account for the effects of a range of intrinsic brightnesses of the stellar population which is being used to trace Milky Way density structure, and can also statistically account for the observational biases in a survey such as the Sloan Digital Sky Survey. We have implemented this technique as a search for the density parameters with the highest likelihood of matching the observed data. Because this parameter search can be computationally expensive, we have employed supercomputers and a large volunteer computing platform, MilkyWay@home, that was built to solve this problem.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. Using Statistical Photometric Parallax\",\n \"content\": \"We are defining the term statistical photometric parallax here for the first time, and it is intended to apply in general for any case where the statistical distribution of absolute magnitudes is used to find the underlying density distribution of stars. However, we will describe here as an example the application of this technique to determine halo substructure using color-selected F turnoff stars, as used by Newberg et al. (2002), Cole et al. (2008), and Newby et al. (2012). In Newberg et al. (2002), stars with colors 0 . 1 < ( g -r ) 0 < 0 . 3 and ( u -g ) 0 > 0 . 4 were selected as turnoff stars. The color range was chosen to be bluer than the turnoff of the thick disk, so that halo stars would preferentially be selected. In this paper, only the simplest form of statistical photometric parallax was employed. The distance to the Sagittarius dwarf tidal stream was determined by assuming the center of the absolute magnitude distribution of turnoff stars in the g filter was M g = 4 . 2. This number was calculated by comparing the apparent magnitude of the turnoff to the apparent magnitude of RR Lyrae stars in the same stellar population. In this example, distances to single stars were not calculated; instead we made a more accurate determination of distance by looking at the distribution of apparent magnitudes of a particular set of stars. In Cole et al. (2008), an algorithm was presented that allowed us to determine not just the distance to the center of a stream, but the three dimensional density of turnoff stars. The technique used maximum likelihood to find the model parameters /vector Q that make the observed star positions ( l i , b i , g i ) the most likely. The Likelihood L is given by the product of the probability density functions (PDFs) evaluated at all of the star positions: The PDF is constructed by the following steps: One then uses an optimization technique to find the model parameters that produce the highest likelihood. When using a supercomputer, we usually use conjugate gradient descent. This algorithm is sequential; one evaluates the likelihood and the derivatives with respect to each parameter, chooses a direction, then decides how far in that direction to go before repeating that process. Newby et al. (2012) applied this technique to all of the data available in SDSS DR7 to find the density of the Sagittarius dwarf tidal stream in the North Galactic Cap, and in three SDSS stripes in the South Galactic Cap. One of the advantages of this probablistic technique is that we are able to extract from the sample of 1.7 million turnoff stars a set of 200 , 000 stars that have the spatial characteristics of the Sagittarius dwarf tidal stream. This is accomplished by generating a random mumber for each star, and using that random number to place it either in the Sagittarius dwarf tidal stream catalog, with the probability that a star at that position in the Galaxy is in the Sagittarius dwarf tidal stream; or in the catalog of non-Sagittarius halo stars, with the probability that a star at that position in the Galaxy is not in the Sagittarius dwaf tidal stream. Note that if you wanted to find actual stars in the Sagittarius dwarf tidal stream, say for spectroscopic follow-up, you should use the original catalog of stars and select those with the highest probability of being in the Sagittarius dwarf tidal stream. However, the statistically separated catalog we generated facilitates the study of density substructures in the halo. In particular, we can remove the Sagittarius stream from the original stellar sample so we can study the smaller tidal streams, and the density structure of the smooth component of the halo. An example stripe analyzed in Newby et al. (2012) is shown in Figure 2. In Cole et al. (2008) and Newby et al. (2012) we modeled the distribution of turnoff star absolute magnitudes as a Gaussian centered at M g = 4 . 2 with a width of σ = 0 . 6 magnitudes. Since then, we have recognized that the distribution is asymmetric; there are more stars fainter than the maximum than there are brighter than the maximum. More importantly we have learned that, due to larger color errors at fainter magnitudes, the absolute magnitude distribution of color-selected stars is different near the survey limit than it is for the brighter stars. This effect is much larger than we expected it to be. Near the survey limit, the majority of the stars are not turnoff stars, but are fainter main sequence stars that have scattered into our color selection limits due to large measurement errors. Because we are using a statistical approach, this effect can be included in the analysis by varying the absolute magnitude distribution as a function of apparent magnitude. We plan to include this in future analyses.\",\n \"pages\": [\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"3. Processing time and the MilkyWay@home Volunteer Computing Platform\",\n \"content\": \"We originally tried to implement the search for maximumum likelihood on a single CPU. In a single 2 . 5 · -wide SDSS data stripe, we fit 2 parameters to a smooth halo with a Hernquist profile, and 6 parameters per tidal debris stream. If there were three tidal streams in a single stripe, there would be 20 parameters. Evaluating the likelihood for one guess for the model parameters currently takes about 4 hours, with most of the time spent integrating the PDF over the survey volume. In order to optimize 20 parameters requires about 50 likelihood evaluations per conjugate gradient descent step and 50 steps per maximum likelihood evaluation. This totals ten thousand hours per optimization (over a year). Luckily the optimization is embarrassingly parallel. It is possible to parallelize the integral, since each integral volume calculation is completely independent of the others. We are able to run this algorithm on a 256 node rack of a Blue Gene/L supercomputer. Parallelizing the integral over 256 nodes cuts the time per likelihood calculation down under a minute. A conjugate gradient step can then be accomplished in 47 minutes, and ten iterations can be accomplished in under eight hours (which is comfortably less the the maximum job size allowed in our queue). In practice, we need to try several conjugate gradient descents to approximate the best parameters. Once they are known approximately, we run of order ten conjugate gradient descents starting near the best values. Including submitting jobs a few at a time and waiting for queue time, this process can take a couple of weeks to validate the results for one SDSS stripe. Currently our best method for computing the parameters is the volunteer computing platform MilkyWay@home. This project is part of the Berkeley Open Infrastructure for Network Computing (BOINC; Anderson, Korpela & Walton 2005) group of volunteer computing platforms. The first and most famous of these is SETI@home. BOINC offers us a template server and database application, and an infrastructure for volunteers to donate their time to our server. We implemented our own server, including the maximum likelihood algorithm, a set of optimization routines that will run in a heterogeneous, asynchronous parallel computing environment (Desell et al. 2010b), and a modified server application that sends out 'work units' to the volunteers, collects the results, and validates that the results are not in error (Desell et al. 2010a). One of the surprises in operating a BOINC server is that some of the results sent back from the volunteers are not correct, either because their hardware is malfunctioning, they did not update the software correctly, or they purposely sent back a wrong answer quickly so that they can accumulate BOINC 'credit' more quickly. It is impossible to overestimate how important it is to our volunteers that they get credit for the work units their computers crunch, and that credit is apportioned fairly between the volunteers. MilkyWay@home is currently delivering 0.5 PetaFLOPS of computing power from 25,000 active volunteers giving us access to over 35,000 CPUs or GPUs. The majority of the computing power comes from the GPUs, the best of which can process our likelihood calculations about 100 times faster than the CPUs (Desell et al. 2009). It is not easy to parallelize the computation of the integral on BOINC, because that would require communication between the processors, which is not possible at this time. Instead, we parallelize the calculations by sending a single likelihood calculation (including the whole integral for a given /vector Q ) to each volunteer. These work units can take a couple minutes (if the work unit is sent to a GPU) or four or more hours (if it goes to a CPU), but it might take minutes, days, or weeks for the likelihood to be returned depending on how much the volunteer is using the computer for her own purposes, and whether she turns it off. It takes far more computing power to calculate best fit parameters on the MilkyWay@home computing system than on a supercomputer. There are four factors responsible for this: (1) We cannot use sequential searches like conjugate gradient descent. Instead, we use 'particle swarm' or a genetic search algorithm. In the particle swarm technique, we send out a random set of guesses that span the parameter space. As the likelihood results from the volunteers come back, we send out more work units with guesses that are closer to the higher likelihoods. This search method requires many more steps, but produces more accurate results. (2) A fraction (about 10%) of the work units, selected at random, are sent out five times so that we can validate that they are correct. If three or more are returned with the same answer, then the result is validated. We also choose to validate the best likelihoods (since those influence our future guesses), and the likelihoods of users that have submitted previous results that did not validate correctly. (3) Some of the work units that are sent out are never returned. (4) Because we can, we run the searches for a longer period of time over a wider range of of parameter space and we get better global values for the parameters (Newby et al. 2012). To optimize one stripe takes 1-2 weeks, and hundreds of thousands of likelihood calculations. There are enough volunteers that we can optimize 4-5 stripes at the same time and still get the results within the same time scale. By putting more jobs on MilkyWay@home at the same time, we decrease the number of work units that are sent out at the same time. By waiting a little longer to send out more work units, the results of previous searches can be used to make better guesses of the parameters, so adding more jobs increases the computing time at a rate that is less than linear in the number of jobs.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"4. Discussion and Conclusion\",\n \"content\": \"The purpose of this conference contribution is to define the term statistical photometric parallax , which allows us to determine the density of a population of stars, even if we cannot determine the distance to each individual star in the population. We find the most likely parameters for the density distribution, given that we know the absolute magnitude distribution and the observational constraints of the observed sample of stars. If the population of stars is all at the same distance (for example in a globular cluster or tidal stream), then statistical photometric parallax can be used to determine the distance to the stellar population. Because the SDSS made available a large, well calibrated sample of stars with multi-color photometry, this technique has recently become feasible. The example I present is the use of turnoff stars to determine the density distribution of stars in the stellar halo of the Milky Way. By apparent coincidence, the distribution of absolute magnitudes of turnoff stars is very similar for all stellar populations in the age and metallicity range of halo stars. This appears to be a result of the Milky Way age-metallicity relationship. We are in the process of using this technique to accurately map the density distribution of the entire Milky Way stellar halo. The drawback to this techique is that to use it one must use a maximum likelihood algorithm that can in some cases require high performance parallel computing to get an accurate measurement of the parameters in the density function. In the process of learning to use this method, we created a large volunteer computing platform called MilkyWay@home.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"Acknowledgement\",\n \"content\": \"I would like to thank Matthew Newby and Brian Yanny for helping me obtain the data for the figures in this publication. I also thank Matthew Newby and Jeff Carlin for their help in proofreading. This paper is based upon work supported by the National Science Foundation under Grant No. AST 10-09670. I also would like to thank the MilkyWay@home volunteers for providing us with computing power at no cost, and the Marvin Clan for their support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Anderson, D. P., Korpela, E., and Walton, R. 2005, The First International Conf. on e-Science\",\n \"pages\": [\n 6\n ]\n }\n]"}}},{"rowIdx":273,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..289..209W"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1210.7307.pdf"},"content":{"kind":"string","value":"\nAsymptotic Giant Branch Variables as Extragalactic Distance Indicators\nPatricia A. Whitelock 1 , 2\n\n1\nSouth African Astronomical Observatory, P. O. Box 9, 7935 Observatory, South Africa 2 Astronomy, Cosmology and Gravity Centre, Astronomy Department, University of Cape Town, 7701 Rondebosch, South Africa\n\nAbstract. Large-amplitude asymptotic giant branch variables potentially rival Cepheid variables as fundamental calibrators of the distance scale, particularly if observations are made in the infrared, or where there is substantial interstellar obscuration. They are particularly useful for probing somewhat older populations, such as those found in dwarf spheroidal galaxies, elliptical galaxies or in the halos of spirals. Calibration data from the Galaxy and new observations of various Local Group galaxies are described and the outlook for the future, with a calibration from Gaia and observations from the next generation of infrared telescopes, is discussed.\nKeywords. stars: late-type, stars: distances, stars: carbon, stars: AGB and post-AGB, galaxies: distances and redshifts, galaxies: dwarf, galaxies: individual (NGC 6822, Fornax dSph, Leo i , Sculptor, Phoenix), galaxies: stellar content\n1. Introduction\nThe focus of this paper is on the large-amplitude, or Mira, asymptotic giant branch (AGB) variables. Because of their high luminosity, and particularly their high infrared luminosity, stars in this group have huge potential as extragalactic distance indicators. I briefly describe what we know about the Mira period-luminosity (PL) relationship, before going on to review recent work on Miras in Local Group galaxies, contrasting those found in dwarf spheroidals with those in the dwarf irregular galaxy NGC 6822.\nMiras are very large-amplitude (∆ V > 2 . 5, ∆ K > 0 . 4 mag), long-period ( P > 100 days) variables. They are close to the maximum bolometric luminosity that they will ever achieve and they are cool, with spectra dominated by molecular absorption. Stars with initial masses in the range from 0.8 to 8 M /circledot , possibly higher, are thought to go through the Mira evolutionary phase. For most of their AGB lifetime, their nuclear energy comes from burning of the hydrogen shell. However, from time to time stars on the upper AGB experience helium-shell flashes with far-reaching consequences for the stellar atmospheres.\nIt is following these flashes that dredge-up can occur, bringing material from the core into the convective regime, and the atmospheric abundance can change from oxygen (O)to carbon (C)-rich. In relatively high-mass AGB stars, hot-bottom burning can also affect abundances, turning carbon to nitrogen and producing lithium, among other things. The relative fraction of O- and C-rich Miras in any population is a function of both age and metallicity.\n2. Period-Luminosity Relations\nEarly work on the Mira PL relation (e.g., at K or M bol ) in the Large Magellanic Cloud (LMC) was published by Feast et al. (1989) and Hughes et al. (1990). However,\nour understanding of this was greatly advanced when Wood (2000) demonstrated that most AGB and upper-giant-branch variables followed various parallel PL sequences. The Mira PL relation, which had been known for some while, was just one of them. Wood showed that the Mira sequence corresponds to fundamental pulsation and includes most of the large-amplitude variables as well as a few of the low-amplitude, or semi-regular, variables. Some of the O-rich large-amplitude, long-period ( P > 400 days) Miras lay above the fundamental sequence, as had also been known for some while.\nThe early studies of the multiple sequences used periods determined from microlensing studies, e.g., from macho and ogle at V and I , and these results are now being refined as more systematic observations are made at infrared wavelengths. Among other things, the situation is obviously confused by the fact that many of the Miras have circumstellar shells. In some cases these are so thick that they affect the position of the star on the PL( K ) relation (e.g., Ita & Matsunaga 2011). In those cases, it is preferable to work with bolometric magnitudes, if they can be reliably estimated, although it is sometimes possible to correct for circumstellar extinction.\nThe Miras which lie above the PL relation are an interesting subgroup that may be experiencing hot-bottom burning; there is certainly evidence for lithium and other enrichment (Whitelock et al. 2003; Whitelock 2012; and references therein). It also seems possible that they are pulsating in the first overtone (Feast 2009).\nFrom the distance-scale perspective, the multiple PL sequences of the small-amplitude variables are not very useful, since it is unclear which sequence any particular star will be on. It is therefore only the Miras, i.e., the large-amplitude variables, that we consider useful for distance-scale studies and, where possible, Miras with short periods ( P < 400 days) and thin dust shells are easier to work with.\nWhitelock et al. (2008) discuss the Galactic calibration of the PL( K ) relation and show that it is consistent with the LMC relation. Rejkuba (2004) demonstrated that the PL( K ) relation for O-rich Miras in the LMC also fitted similar stars in NGC 5128. This, together with recent work on Miras in dwarf spheroidals and NGC 6822 (see below) is consistent with the same PL relation applying everywhere. It remains possible that there are metallicity effects, but these are unlikely to be significantly greater than 0.1 mag for stars from populations considered so far (see Matsunaga 2012).\n3. Which Mira PL relation for distance-scale studies?\nFor most purposes at the present time it is preferable to work with the PL( K ) relation, primarily because the K magnitude is easy to measure, the pulsation amplitude at K is lower than at shorter wavelengths and K is not often severely affected by circumstellar emission or absorption. However, pulsation drives mass loss, and there are Miras with thick shells and severe circumstellar reddening. This is particularly true of the very longperiod O-rich Miras, known as OH/IR stars, but it is also the case for a variety of C-rich stars.\nIn principle using bolometric magnitudes avoids the problem with circumstellar extinction and there are three related ways of deriving such magnitudes, all of which have drawbacks:\n\n( a ) Ideally, one obtains bolometric magnitudes derived from multi-epoch measurements across a wide wavelength range extending at least into the mid-infrared regime. Such data are rarely available for many stars.\n( b ) An alternative approach uses well-calibrated colour-dependent (e.g., J -K ) bolometric corrections, derived for stars that are similar to those of interest, and for which multi-wavelength observations are available. A potential problem is that the observed\n\n\n\n
Figure 1. PL( K ) relation for the Miras in the dwarf spheroidals, coded as follows: Fornax (cyan), Leo i (magenta), Sculptor (green), Phoenix (black). The line is at the slope of the LMC PL relation and assumes a distance modulus to the LMC of 18.39 mag (changing this distance modulus will not alter the loci of the points and the line, only their place relative to the magnitude scale).
\n\ncolour is a combination of the intrinsic colour of the star and the reddening. The same ( J -K ) colour can be due to different combinations of these two effects. Thus, ( J -K ) may not necessarily be uniquely related to the true bolometric correction.\n( c ) It is sometimes possible to make a correction for the circumstellar reddening (e.g., Matsunaga et al. 2009), which is similar to applying a colour-dependent bolometric correction. The weakness of this is that it requires adoption of an intrinsic colour for the star and we have little evidence of the way this colour correlates with period, particularly for long-period stars, almost all of which have shells.\nOf course, non-uniform shells, which are the obvious consequence of non-uniform mass loss, present obvious problems to any method for estimating the bolometric magnitude. Different methods of calculating the bolometric magnitude give very significantly different results. For example, Groenewegen et al. (2007) and Kamath et al. (2010) derive bolometric magnitudes for several pulsating stars in the Small Magellanic Cloud (SMC) cluster NGC 419 using slightly different JHKL values but the same Spitzer data. Their bolometric magnitudes differ by amounts that range from -0 . 1 to 0.4 mag for the same star (see also Kerschbaum et al. 2010).\nNoting the challenge of determining accurate bolometric magnitudes, estimated values can still provide useful distances, provided a very systematic approach is followed. In practice, this means determining the bolometric magnitude of the star for which the distance is required in exactly the same way as for the calibrators used to define the PL relation.\n\n\n
Figure 2. Bolometric PL relation for the Miras in the dwarf spheroidal galaxies, colours as in Fig. 1.
\n\n4. Globular Cluster: Lyng˚a 7\nMatsunaga (2006) discovered a Mira variable, V1, in Lyng˚a 7 (an old, metal-rich Galactic bulge cluster) with a period of 551 days, a large amplitude, ∆ K = 1 . 22 mag, and red colour, ( J -K ) = 4 . 1 mag. Sloan et al. (2010) showed, based on a Spitzer spectrum, that it is carbon-rich and Feast et al. (2012a) used a spectrum from the Southern African Large Telescope ( SALT ) to demonstrate that V1 is a radial-velocity member of the cluster.\nAssuming a distance modulus of 14.55 mag (Sarajedini et al. 2007), this Mira has M bol = -5 . 0 mag, in agreement with the PL-relation value of M bol = -5 . 2 mag. Such a luminous star must have had an initial mass M i ∼ 1 . 5M /circledot and cannot be a normal member of the cluster. It therefore must have formed from a stellar merger.\nTo the best of our knowledge, this is the first ever demonstration of a star that was produced by the merger of two others, but that nevertheless obeys a PL relation. It is an interesting result and we might well expect there to be remnants of other mergers in the dense environment of the Galactic bulge.\n5. Local Group Galaxies\nA group of us from South Africa and Japan have used the Infrared Survey Facility ( IRSF ) at the South African Astronomical Observatory to survey a variety of Local Group galaxies for AGB variables. The several-year survey uses observations made with the SIRIUS camera, which simultaneously gives J , H and K s photometry over a 7 × 7 arcmin 2 field.\n5.1. Dwarf Spheroidal Galaxies\nResults so far have been published for a total of 17 Miras from Fornax, Leo i , Sculptor and Phoenix (Menzies et al. 2008, 2010, 2011; Whitelock et al. 2009). Where spectral types are available for these stars, they show them to be C-rich and we assume that they are all C-type stars.\nFig. 1 shows the absolute K magnitudes on a PL( K ) relation for all the Miras in dwarf spheroidals. The large scatter is very striking and the distance below the LMC's PL( K ) relation is a function of the ( J -K ) colour, indicating that the stars below the line are there because of their thick circumstellar shells.\nBolometric magnitudes can be estimated using a ( J -K )-dependent bolometric correction (Whitelock et al. 2009) and the results are shown in a PL relation in Fig. 2. The scatter is vastly reduced, although there are still two stars which lie well below the mean relation. Two possible explanations have been offered for these faint points. It may be that the bolometric correction does not apply to these intrinsically relatively blue, shortperiod, stars (see point [ b ] in Sect. 3), since it was derived for significantly longer-period stars. Alternatively, they are undergoing obscuration events of the type that are common among C-rich Miras in the Galaxy and the LMC (e.g., Whitelock et al. 2006) and have non-uniform shells.\n5.2. NGC6822\nNGC6822 is an isolated barred dwarf galaxy, similar to the SMC, but with slightly higher metallicity. It has been examined for AGB variables in the same way as the dwarf spheroidals over 3.5 years and has had numerous Miras catalogued (Whitelock et al. 2012; see also Battinelli & Demers 2011). Fig. 3 shows the variables in a colour-magnitude diagram. Spectral types are available for only very few stars and the split into O- and C-rich assumes that all very red stars are C-rich. Several of the large-amplitude stars without measured periods are probably also Miras and all very red stars are variable.\nFig. 4 shows the NGC 6822 Miras on a PL( K ) relation. Most of the longer-period Orich stars fall above the PL relation and are probably similar to the stars in the LMC (mentioned above) that may be hot-bottom burning (Whitelock et al. 2003). Feast (2009) suggested that these stars may be pulsating in the first overtone. Many of the C-rich Miras fall well below the line as the result of thick circumstellar shells. Fig. 5 shows the same stars on a bolometric PL relation, and we see that the C stars scatter around a relation that is very similar to the one obeyed by LMC Miras. The slope is almost identical to the slope of the LMC line, within the uncertainties.\nUsing an LMC distance modulus of 18.5 mag, we determine from the C-rich Miras that ( m -M ) 0 = 23 . 56 ± 0 . 03 mag for NGC 6822. This may be compared to 23 . 40 ± 0 . 05 mag derived from Cepheid variables (Feast et al. 2012b) and 23 . 49 ± 0 . 03 mag from RR Lyrae variables (Clementini et al. 2003). Note that all errors quoted here are internal, but there are systematic uncertainties in all of the measurements. The agreement is reasonable and certainly shows that Miras offer a viable alternative to the more conventional distance indicators.\n5.3. Challenges\nHowever, I should note that there remain challenges in using Miras as distance indicators. One of the most serious of these is ensuring that measurements made with different photometric systems give the same result. Battinelli & Demers (2011) have 16 largeamplitude variables in common with Whitelock et al. (2012) in NGC 6822. The periods determined for these agree well, but the mean magnitudes differ by ∆ K = 0 . 25 mag. Both groups know that their photometry of normal stars, i.e., those with ( J -K ) < 1 . 0 mag,\n\n\n
Figure 3. Colour-magnitude diagram for the variables in NGC 6822, showing Cepheids (magenta), large-amplitude variables with measured periods, which are presumed O-rich (blue) or C-rich (red), and variables without measured periods that have large (yellow) or small (green) amplitudes.
\n\nis on the 2mass system. Dealing with very red stars, of which there are no non-variable examples, will require considerably more effort.\nIt also remains possible that we will find metallicity effects as the PL relationships become better defined.\n6. Cepheids and/or Miras?\nCepheids have long provided a vital step on the distance-scale ladder linking the Galaxy to distant supernovae. However, Mira variables offer a viable alternative which may well be preferable for the following reasons:\n\n· Miras are of comparable brightness to Cepheids at K and brighter at longer wavelengths (see Table 1 and Whitelock 2012).\n· Miras are found in galaxies which do not host Cepheids, such as dwarf spheroidals, and will be found in ellipticals.\n\n\n\n
Figure 4. PL( K ) relation for NGC 6822 showing O-rich (open circles) and C-rich Miras. The line has the slope of the LMC PL( K ) relation and has been fit to the four short-period O-rich Miras.
\n\n\n\n
Figure 5. Bolometric PL relation for NGC 6822, showing the same stars as in Fig. 5. The dashed line is the LMC PL relation, while the solid line is the relation fitted to these data.
\n\n\n· Miras are found in the haloes of spiral galaxies, where they may be less confused, and therefore more easily observable at large distances, than stars in the spiral arms.\n· Miras are best observed in the infrared and are therefore not severely affected by interstellar extinction.\n· The next generation of space telescopes, as well as large ground-based telescopes equipped with adaptive optics, will primarily work in the infrared. They will be ideally suited to observing distant Mira variables.\n\n7. Conclusion\nLarge-amplitude AGB variables offer a viable alternative to Cepheids for distance-scale studies, which will be particularly valuable when infrared observations are available. There remain, however, calibration issues that must be resolved if observations from different instruments are to be combined reliably. The Gaia satellite will provide a vital Galactic calibration that will put the Mira absolute-magnitude scale on a new footing (see Whitelock 2012).\n
\n\n
Table 1. Comparison of the absolute magnitudes of Cepheid and Mira variables at 2.2 and 8 µ m (Whitelock 2012 and Feast 2010, unpublished).
\n
\nAcknowledgements\nI am grateful to my colleagues for allowing me to discuss our results and particularly to Michael Feast and John Menzies for their comments on this manuscript. I acknowledge a grant from the South African National Research Foundation.\nReferences\nBattinelli, P., & Demers, S. 2011, A&A , 525, 69 Clementini, G., Held, E.V., Baldacci, L., & Rizzi, L. 2003, ApJ , 588, L85 Feast, M.W., Glass, I.S., Whitelock, P.A., & Catchpole, R.M. 1989, MNRAS , 241, 375 Feast, M.W. 2009, in: AGB stars and related phenomena (Ueta, T., Matsunaga, N., & Ita, Y., eds.), AGB stars and related phenomena , p. 48 Feast, M.W., Menzies, J.W., & Whitelock, P.A. 2012a, MNRAS , in press (arXiv:1210.0415) Feast, M.W., Whitelock, P.A., Menzies, J.W., & Matsunaga, N. 2012b, MNRAS , 421, 2998 Groenewegen M.A.T., et al. 2007, MNRAS , 376, 313 Hughes, S.M.G., & Wood, P.R 1990, AJ , 99, 784 Ita, Y., & Matsunaga, N. 2011, MNRAS , 412, 2345 Kamath, D., Wood, P.R., Soszy´nski, I., & Lebzelter, T. 2010, MNRAS , 408, 522 Kerschbaum, F., Lebzelter, T., & Makul, L. 2010, A&A , 524, A87 Matsunaga, N. 2012, Ap&SS , 341, 93 Matsunaga, N., 2006, Ph.D. Thesis , University of Tokyo (Japan) Matsunaga, N., Kawadu, T., Nishiyama, S., Nagayama, T., Hatano, H., Tamura, M., Glass, I.S., & Nagata, T. 2009, MNRAS , 399, 1709 Menzies, J.W., Feast, M.W., Whitelock, P.A., Olivier, E., Matsunaga, N., & da Costa, G. 2008, MNRAS , 385, 1045 Menzies, J.W., Whitelock, P.A., Feast, M.W., & Matsunaga, N. 2010, MNRAS , 406, 86 Menzies, J.W., Feast, M.W., Whitelock, P.A., & Matsunaga, N. 2011, MNRAS , 414, 3492 Rejkuba, M. 2004, A&A , 413, 903 Sarajedini, A., et al. 2007, AJ , 133, 1658 Sloan, G.C., Matsunaga, N., Matsuura, M., et al. 2010, ApJ , 719, 1274 Whitelock, P.A. 2012, Ap&SS , 341, 123 Whitelock, P.A., Feast, M.W., van Loon, J.Th., & Zijlstra, A.A. 2003, MNRAS , 342, 86 Whitelock, P.A., Feast, M.W., Marang, F., & Groenewegen, M.A.T. 2006, MNRAS , 369, 751 Whitelock, P.A., Feast, M.W., & van Leeuwen, F. 2008, MNRAS , 386, 313 Whitelock, P.A., Menzies, J.W., Feast, M.W., Matsunaga, N., Tanab´e, T., & Ita, Y. 2009, MNRAS , 394, 795\nWhitelock, P.A., Menzies, J.W., Feast, M.W., Nsengiyumva, F., & Matsunaga, N. 2012, MNRAS , in press (arXiv:1210.3695)\nWood, P.R. 2000, Publ. Astron. Soc. Aus. , 17, 18\n"},"sections":{"kind":"list like","value":[{"title":"Patricia A. Whitelock 1 , 2","content":"Abstract. Large-amplitude asymptotic giant branch variables potentially rival Cepheid variables as fundamental calibrators of the distance scale, particularly if observations are made in the infrared, or where there is substantial interstellar obscuration. They are particularly useful for probing somewhat older populations, such as those found in dwarf spheroidal galaxies, elliptical galaxies or in the halos of spirals. Calibration data from the Galaxy and new observations of various Local Group galaxies are described and the outlook for the future, with a calibration from Gaia and observations from the next generation of infrared telescopes, is discussed. Keywords. stars: late-type, stars: distances, stars: carbon, stars: AGB and post-AGB, galaxies: distances and redshifts, galaxies: dwarf, galaxies: individual (NGC 6822, Fornax dSph, Leo i , Sculptor, Phoenix), galaxies: stellar content","pages":[1]},{"title":"1. Introduction","content":"The focus of this paper is on the large-amplitude, or Mira, asymptotic giant branch (AGB) variables. Because of their high luminosity, and particularly their high infrared luminosity, stars in this group have huge potential as extragalactic distance indicators. I briefly describe what we know about the Mira period-luminosity (PL) relationship, before going on to review recent work on Miras in Local Group galaxies, contrasting those found in dwarf spheroidals with those in the dwarf irregular galaxy NGC 6822. Miras are very large-amplitude (∆ V > 2 . 5, ∆ K > 0 . 4 mag), long-period ( P > 100 days) variables. They are close to the maximum bolometric luminosity that they will ever achieve and they are cool, with spectra dominated by molecular absorption. Stars with initial masses in the range from 0.8 to 8 M /circledot , possibly higher, are thought to go through the Mira evolutionary phase. For most of their AGB lifetime, their nuclear energy comes from burning of the hydrogen shell. However, from time to time stars on the upper AGB experience helium-shell flashes with far-reaching consequences for the stellar atmospheres. It is following these flashes that dredge-up can occur, bringing material from the core into the convective regime, and the atmospheric abundance can change from oxygen (O)to carbon (C)-rich. In relatively high-mass AGB stars, hot-bottom burning can also affect abundances, turning carbon to nitrogen and producing lithium, among other things. The relative fraction of O- and C-rich Miras in any population is a function of both age and metallicity.","pages":[1]},{"title":"2. Period-Luminosity Relations","content":"Early work on the Mira PL relation (e.g., at K or M bol ) in the Large Magellanic Cloud (LMC) was published by Feast et al. (1989) and Hughes et al. (1990). However, our understanding of this was greatly advanced when Wood (2000) demonstrated that most AGB and upper-giant-branch variables followed various parallel PL sequences. The Mira PL relation, which had been known for some while, was just one of them. Wood showed that the Mira sequence corresponds to fundamental pulsation and includes most of the large-amplitude variables as well as a few of the low-amplitude, or semi-regular, variables. Some of the O-rich large-amplitude, long-period ( P > 400 days) Miras lay above the fundamental sequence, as had also been known for some while. The early studies of the multiple sequences used periods determined from microlensing studies, e.g., from macho and ogle at V and I , and these results are now being refined as more systematic observations are made at infrared wavelengths. Among other things, the situation is obviously confused by the fact that many of the Miras have circumstellar shells. In some cases these are so thick that they affect the position of the star on the PL( K ) relation (e.g., Ita & Matsunaga 2011). In those cases, it is preferable to work with bolometric magnitudes, if they can be reliably estimated, although it is sometimes possible to correct for circumstellar extinction. The Miras which lie above the PL relation are an interesting subgroup that may be experiencing hot-bottom burning; there is certainly evidence for lithium and other enrichment (Whitelock et al. 2003; Whitelock 2012; and references therein). It also seems possible that they are pulsating in the first overtone (Feast 2009). From the distance-scale perspective, the multiple PL sequences of the small-amplitude variables are not very useful, since it is unclear which sequence any particular star will be on. It is therefore only the Miras, i.e., the large-amplitude variables, that we consider useful for distance-scale studies and, where possible, Miras with short periods ( P < 400 days) and thin dust shells are easier to work with. Whitelock et al. (2008) discuss the Galactic calibration of the PL( K ) relation and show that it is consistent with the LMC relation. Rejkuba (2004) demonstrated that the PL( K ) relation for O-rich Miras in the LMC also fitted similar stars in NGC 5128. This, together with recent work on Miras in dwarf spheroidals and NGC 6822 (see below) is consistent with the same PL relation applying everywhere. It remains possible that there are metallicity effects, but these are unlikely to be significantly greater than 0.1 mag for stars from populations considered so far (see Matsunaga 2012).","pages":[1,2]},{"title":"3. Which Mira PL relation for distance-scale studies?","content":"For most purposes at the present time it is preferable to work with the PL( K ) relation, primarily because the K magnitude is easy to measure, the pulsation amplitude at K is lower than at shorter wavelengths and K is not often severely affected by circumstellar emission or absorption. However, pulsation drives mass loss, and there are Miras with thick shells and severe circumstellar reddening. This is particularly true of the very longperiod O-rich Miras, known as OH/IR stars, but it is also the case for a variety of C-rich stars. In principle using bolometric magnitudes avoids the problem with circumstellar extinction and there are three related ways of deriving such magnitudes, all of which have drawbacks: colour is a combination of the intrinsic colour of the star and the reddening. The same ( J -K ) colour can be due to different combinations of these two effects. Thus, ( J -K ) may not necessarily be uniquely related to the true bolometric correction. ( c ) It is sometimes possible to make a correction for the circumstellar reddening (e.g., Matsunaga et al. 2009), which is similar to applying a colour-dependent bolometric correction. The weakness of this is that it requires adoption of an intrinsic colour for the star and we have little evidence of the way this colour correlates with period, particularly for long-period stars, almost all of which have shells. Of course, non-uniform shells, which are the obvious consequence of non-uniform mass loss, present obvious problems to any method for estimating the bolometric magnitude. Different methods of calculating the bolometric magnitude give very significantly different results. For example, Groenewegen et al. (2007) and Kamath et al. (2010) derive bolometric magnitudes for several pulsating stars in the Small Magellanic Cloud (SMC) cluster NGC 419 using slightly different JHKL values but the same Spitzer data. Their bolometric magnitudes differ by amounts that range from -0 . 1 to 0.4 mag for the same star (see also Kerschbaum et al. 2010). Noting the challenge of determining accurate bolometric magnitudes, estimated values can still provide useful distances, provided a very systematic approach is followed. In practice, this means determining the bolometric magnitude of the star for which the distance is required in exactly the same way as for the calibrators used to define the PL relation.","pages":[2,3]},{"title":"4. Globular Cluster: Lyng˚a 7","content":"Matsunaga (2006) discovered a Mira variable, V1, in Lyng˚a 7 (an old, metal-rich Galactic bulge cluster) with a period of 551 days, a large amplitude, ∆ K = 1 . 22 mag, and red colour, ( J -K ) = 4 . 1 mag. Sloan et al. (2010) showed, based on a Spitzer spectrum, that it is carbon-rich and Feast et al. (2012a) used a spectrum from the Southern African Large Telescope ( SALT ) to demonstrate that V1 is a radial-velocity member of the cluster. Assuming a distance modulus of 14.55 mag (Sarajedini et al. 2007), this Mira has M bol = -5 . 0 mag, in agreement with the PL-relation value of M bol = -5 . 2 mag. Such a luminous star must have had an initial mass M i ∼ 1 . 5M /circledot and cannot be a normal member of the cluster. It therefore must have formed from a stellar merger. To the best of our knowledge, this is the first ever demonstration of a star that was produced by the merger of two others, but that nevertheless obeys a PL relation. It is an interesting result and we might well expect there to be remnants of other mergers in the dense environment of the Galactic bulge.","pages":[4]},{"title":"5. Local Group Galaxies","content":"A group of us from South Africa and Japan have used the Infrared Survey Facility ( IRSF ) at the South African Astronomical Observatory to survey a variety of Local Group galaxies for AGB variables. The several-year survey uses observations made with the SIRIUS camera, which simultaneously gives J , H and K s photometry over a 7 × 7 arcmin 2 field.","pages":[4]},{"title":"5.1. Dwarf Spheroidal Galaxies","content":"Results so far have been published for a total of 17 Miras from Fornax, Leo i , Sculptor and Phoenix (Menzies et al. 2008, 2010, 2011; Whitelock et al. 2009). Where spectral types are available for these stars, they show them to be C-rich and we assume that they are all C-type stars. Fig. 1 shows the absolute K magnitudes on a PL( K ) relation for all the Miras in dwarf spheroidals. The large scatter is very striking and the distance below the LMC's PL( K ) relation is a function of the ( J -K ) colour, indicating that the stars below the line are there because of their thick circumstellar shells. Bolometric magnitudes can be estimated using a ( J -K )-dependent bolometric correction (Whitelock et al. 2009) and the results are shown in a PL relation in Fig. 2. The scatter is vastly reduced, although there are still two stars which lie well below the mean relation. Two possible explanations have been offered for these faint points. It may be that the bolometric correction does not apply to these intrinsically relatively blue, shortperiod, stars (see point [ b ] in Sect. 3), since it was derived for significantly longer-period stars. Alternatively, they are undergoing obscuration events of the type that are common among C-rich Miras in the Galaxy and the LMC (e.g., Whitelock et al. 2006) and have non-uniform shells.","pages":[5]},{"title":"5.2. NGC6822","content":"NGC6822 is an isolated barred dwarf galaxy, similar to the SMC, but with slightly higher metallicity. It has been examined for AGB variables in the same way as the dwarf spheroidals over 3.5 years and has had numerous Miras catalogued (Whitelock et al. 2012; see also Battinelli & Demers 2011). Fig. 3 shows the variables in a colour-magnitude diagram. Spectral types are available for only very few stars and the split into O- and C-rich assumes that all very red stars are C-rich. Several of the large-amplitude stars without measured periods are probably also Miras and all very red stars are variable. Fig. 4 shows the NGC 6822 Miras on a PL( K ) relation. Most of the longer-period Orich stars fall above the PL relation and are probably similar to the stars in the LMC (mentioned above) that may be hot-bottom burning (Whitelock et al. 2003). Feast (2009) suggested that these stars may be pulsating in the first overtone. Many of the C-rich Miras fall well below the line as the result of thick circumstellar shells. Fig. 5 shows the same stars on a bolometric PL relation, and we see that the C stars scatter around a relation that is very similar to the one obeyed by LMC Miras. The slope is almost identical to the slope of the LMC line, within the uncertainties. Using an LMC distance modulus of 18.5 mag, we determine from the C-rich Miras that ( m -M ) 0 = 23 . 56 ± 0 . 03 mag for NGC 6822. This may be compared to 23 . 40 ± 0 . 05 mag derived from Cepheid variables (Feast et al. 2012b) and 23 . 49 ± 0 . 03 mag from RR Lyrae variables (Clementini et al. 2003). Note that all errors quoted here are internal, but there are systematic uncertainties in all of the measurements. The agreement is reasonable and certainly shows that Miras offer a viable alternative to the more conventional distance indicators.","pages":[5]},{"title":"5.3. Challenges","content":"However, I should note that there remain challenges in using Miras as distance indicators. One of the most serious of these is ensuring that measurements made with different photometric systems give the same result. Battinelli & Demers (2011) have 16 largeamplitude variables in common with Whitelock et al. (2012) in NGC 6822. The periods determined for these agree well, but the mean magnitudes differ by ∆ K = 0 . 25 mag. Both groups know that their photometry of normal stars, i.e., those with ( J -K ) < 1 . 0 mag, is on the 2mass system. Dealing with very red stars, of which there are no non-variable examples, will require considerably more effort. It also remains possible that we will find metallicity effects as the PL relationships become better defined.","pages":[5,6]},{"title":"6. Cepheids and/or Miras?","content":"Cepheids have long provided a vital step on the distance-scale ladder linking the Galaxy to distant supernovae. However, Mira variables offer a viable alternative which may well be preferable for the following reasons:","pages":[6]},{"title":"7. Conclusion","content":"Large-amplitude AGB variables offer a viable alternative to Cepheids for distance-scale studies, which will be particularly valuable when infrared observations are available. There remain, however, calibration issues that must be resolved if observations from different instruments are to be combined reliably. The Gaia satellite will provide a vital Galactic calibration that will put the Mira absolute-magnitude scale on a new footing (see Whitelock 2012).","pages":[7]},{"title":"Acknowledgements","content":"I am grateful to my colleagues for allowing me to discuss our results and particularly to Michael Feast and John Menzies for their comments on this manuscript. I acknowledge a grant from the South African National Research Foundation.","pages":[8]},{"title":"References","content":"Battinelli, P., & Demers, S. 2011, A&A , 525, 69 Clementini, G., Held, E.V., Baldacci, L., & Rizzi, L. 2003, ApJ , 588, L85 Feast, M.W., Glass, I.S., Whitelock, P.A., & Catchpole, R.M. 1989, MNRAS , 241, 375 Feast, M.W. 2009, in: AGB stars and related phenomena (Ueta, T., Matsunaga, N., & Ita, Y., eds.), AGB stars and related phenomena , p. 48 Feast, M.W., Menzies, J.W., & Whitelock, P.A. 2012a, MNRAS , in press (arXiv:1210.0415) Feast, M.W., Whitelock, P.A., Menzies, J.W., & Matsunaga, N. 2012b, MNRAS , 421, 2998 Groenewegen M.A.T., et al. 2007, MNRAS , 376, 313 Hughes, S.M.G., & Wood, P.R 1990, AJ , 99, 784 Ita, Y., & Matsunaga, N. 2011, MNRAS , 412, 2345 Kamath, D., Wood, P.R., Soszy´nski, I., & Lebzelter, T. 2010, MNRAS , 408, 522 Kerschbaum, F., Lebzelter, T., & Makul, L. 2010, A&A , 524, A87 Matsunaga, N. 2012, Ap&SS , 341, 93 Matsunaga, N., 2006, Ph.D. Thesis , University of Tokyo (Japan) Matsunaga, N., Kawadu, T., Nishiyama, S., Nagayama, T., Hatano, H., Tamura, M., Glass, I.S., & Nagata, T. 2009, MNRAS , 399, 1709 Menzies, J.W., Feast, M.W., Whitelock, P.A., Olivier, E., Matsunaga, N., & da Costa, G. 2008, MNRAS , 385, 1045 Menzies, J.W., Whitelock, P.A., Feast, M.W., & Matsunaga, N. 2010, MNRAS , 406, 86 Menzies, J.W., Feast, M.W., Whitelock, P.A., & Matsunaga, N. 2011, MNRAS , 414, 3492 Rejkuba, M. 2004, A&A , 413, 903 Sarajedini, A., et al. 2007, AJ , 133, 1658 Sloan, G.C., Matsunaga, N., Matsuura, M., et al. 2010, ApJ , 719, 1274 Whitelock, P.A. 2012, Ap&SS , 341, 123 Whitelock, P.A., Feast, M.W., van Loon, J.Th., & Zijlstra, A.A. 2003, MNRAS , 342, 86 Whitelock, P.A., Feast, M.W., Marang, F., & Groenewegen, M.A.T. 2006, MNRAS , 369, 751 Whitelock, P.A., Feast, M.W., & van Leeuwen, F. 2008, MNRAS , 386, 313 Whitelock, P.A., Menzies, J.W., Feast, M.W., Matsunaga, N., Tanab´e, T., & Ita, Y. 2009, MNRAS , 394, 795 Whitelock, P.A., Menzies, J.W., Feast, M.W., Nsengiyumva, F., & Matsunaga, N. 2012, MNRAS , in press (arXiv:1210.3695) Wood, P.R. 2000, Publ. Astron. Soc. Aus. , 17, 18","pages":[8]}],"string":"[\n {\n \"title\": \"Patricia A. Whitelock 1 , 2\",\n \"content\": \"Abstract. Large-amplitude asymptotic giant branch variables potentially rival Cepheid variables as fundamental calibrators of the distance scale, particularly if observations are made in the infrared, or where there is substantial interstellar obscuration. They are particularly useful for probing somewhat older populations, such as those found in dwarf spheroidal galaxies, elliptical galaxies or in the halos of spirals. Calibration data from the Galaxy and new observations of various Local Group galaxies are described and the outlook for the future, with a calibration from Gaia and observations from the next generation of infrared telescopes, is discussed. Keywords. stars: late-type, stars: distances, stars: carbon, stars: AGB and post-AGB, galaxies: distances and redshifts, galaxies: dwarf, galaxies: individual (NGC 6822, Fornax dSph, Leo i , Sculptor, Phoenix), galaxies: stellar content\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"The focus of this paper is on the large-amplitude, or Mira, asymptotic giant branch (AGB) variables. Because of their high luminosity, and particularly their high infrared luminosity, stars in this group have huge potential as extragalactic distance indicators. I briefly describe what we know about the Mira period-luminosity (PL) relationship, before going on to review recent work on Miras in Local Group galaxies, contrasting those found in dwarf spheroidals with those in the dwarf irregular galaxy NGC 6822. Miras are very large-amplitude (∆ V > 2 . 5, ∆ K > 0 . 4 mag), long-period ( P > 100 days) variables. They are close to the maximum bolometric luminosity that they will ever achieve and they are cool, with spectra dominated by molecular absorption. Stars with initial masses in the range from 0.8 to 8 M /circledot , possibly higher, are thought to go through the Mira evolutionary phase. For most of their AGB lifetime, their nuclear energy comes from burning of the hydrogen shell. However, from time to time stars on the upper AGB experience helium-shell flashes with far-reaching consequences for the stellar atmospheres. It is following these flashes that dredge-up can occur, bringing material from the core into the convective regime, and the atmospheric abundance can change from oxygen (O)to carbon (C)-rich. In relatively high-mass AGB stars, hot-bottom burning can also affect abundances, turning carbon to nitrogen and producing lithium, among other things. The relative fraction of O- and C-rich Miras in any population is a function of both age and metallicity.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2. Period-Luminosity Relations\",\n \"content\": \"Early work on the Mira PL relation (e.g., at K or M bol ) in the Large Magellanic Cloud (LMC) was published by Feast et al. (1989) and Hughes et al. (1990). However, our understanding of this was greatly advanced when Wood (2000) demonstrated that most AGB and upper-giant-branch variables followed various parallel PL sequences. The Mira PL relation, which had been known for some while, was just one of them. Wood showed that the Mira sequence corresponds to fundamental pulsation and includes most of the large-amplitude variables as well as a few of the low-amplitude, or semi-regular, variables. Some of the O-rich large-amplitude, long-period ( P > 400 days) Miras lay above the fundamental sequence, as had also been known for some while. The early studies of the multiple sequences used periods determined from microlensing studies, e.g., from macho and ogle at V and I , and these results are now being refined as more systematic observations are made at infrared wavelengths. Among other things, the situation is obviously confused by the fact that many of the Miras have circumstellar shells. In some cases these are so thick that they affect the position of the star on the PL( K ) relation (e.g., Ita & Matsunaga 2011). In those cases, it is preferable to work with bolometric magnitudes, if they can be reliably estimated, although it is sometimes possible to correct for circumstellar extinction. The Miras which lie above the PL relation are an interesting subgroup that may be experiencing hot-bottom burning; there is certainly evidence for lithium and other enrichment (Whitelock et al. 2003; Whitelock 2012; and references therein). It also seems possible that they are pulsating in the first overtone (Feast 2009). From the distance-scale perspective, the multiple PL sequences of the small-amplitude variables are not very useful, since it is unclear which sequence any particular star will be on. It is therefore only the Miras, i.e., the large-amplitude variables, that we consider useful for distance-scale studies and, where possible, Miras with short periods ( P < 400 days) and thin dust shells are easier to work with. Whitelock et al. (2008) discuss the Galactic calibration of the PL( K ) relation and show that it is consistent with the LMC relation. Rejkuba (2004) demonstrated that the PL( K ) relation for O-rich Miras in the LMC also fitted similar stars in NGC 5128. This, together with recent work on Miras in dwarf spheroidals and NGC 6822 (see below) is consistent with the same PL relation applying everywhere. It remains possible that there are metallicity effects, but these are unlikely to be significantly greater than 0.1 mag for stars from populations considered so far (see Matsunaga 2012).\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"3. Which Mira PL relation for distance-scale studies?\",\n \"content\": \"For most purposes at the present time it is preferable to work with the PL( K ) relation, primarily because the K magnitude is easy to measure, the pulsation amplitude at K is lower than at shorter wavelengths and K is not often severely affected by circumstellar emission or absorption. However, pulsation drives mass loss, and there are Miras with thick shells and severe circumstellar reddening. This is particularly true of the very longperiod O-rich Miras, known as OH/IR stars, but it is also the case for a variety of C-rich stars. In principle using bolometric magnitudes avoids the problem with circumstellar extinction and there are three related ways of deriving such magnitudes, all of which have drawbacks: colour is a combination of the intrinsic colour of the star and the reddening. The same ( J -K ) colour can be due to different combinations of these two effects. Thus, ( J -K ) may not necessarily be uniquely related to the true bolometric correction. ( c ) It is sometimes possible to make a correction for the circumstellar reddening (e.g., Matsunaga et al. 2009), which is similar to applying a colour-dependent bolometric correction. The weakness of this is that it requires adoption of an intrinsic colour for the star and we have little evidence of the way this colour correlates with period, particularly for long-period stars, almost all of which have shells. Of course, non-uniform shells, which are the obvious consequence of non-uniform mass loss, present obvious problems to any method for estimating the bolometric magnitude. Different methods of calculating the bolometric magnitude give very significantly different results. For example, Groenewegen et al. (2007) and Kamath et al. (2010) derive bolometric magnitudes for several pulsating stars in the Small Magellanic Cloud (SMC) cluster NGC 419 using slightly different JHKL values but the same Spitzer data. Their bolometric magnitudes differ by amounts that range from -0 . 1 to 0.4 mag for the same star (see also Kerschbaum et al. 2010). Noting the challenge of determining accurate bolometric magnitudes, estimated values can still provide useful distances, provided a very systematic approach is followed. In practice, this means determining the bolometric magnitude of the star for which the distance is required in exactly the same way as for the calibrators used to define the PL relation.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"4. Globular Cluster: Lyng˚a 7\",\n \"content\": \"Matsunaga (2006) discovered a Mira variable, V1, in Lyng˚a 7 (an old, metal-rich Galactic bulge cluster) with a period of 551 days, a large amplitude, ∆ K = 1 . 22 mag, and red colour, ( J -K ) = 4 . 1 mag. Sloan et al. (2010) showed, based on a Spitzer spectrum, that it is carbon-rich and Feast et al. (2012a) used a spectrum from the Southern African Large Telescope ( SALT ) to demonstrate that V1 is a radial-velocity member of the cluster. Assuming a distance modulus of 14.55 mag (Sarajedini et al. 2007), this Mira has M bol = -5 . 0 mag, in agreement with the PL-relation value of M bol = -5 . 2 mag. Such a luminous star must have had an initial mass M i ∼ 1 . 5M /circledot and cannot be a normal member of the cluster. It therefore must have formed from a stellar merger. To the best of our knowledge, this is the first ever demonstration of a star that was produced by the merger of two others, but that nevertheless obeys a PL relation. It is an interesting result and we might well expect there to be remnants of other mergers in the dense environment of the Galactic bulge.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"5. Local Group Galaxies\",\n \"content\": \"A group of us from South Africa and Japan have used the Infrared Survey Facility ( IRSF ) at the South African Astronomical Observatory to survey a variety of Local Group galaxies for AGB variables. The several-year survey uses observations made with the SIRIUS camera, which simultaneously gives J , H and K s photometry over a 7 × 7 arcmin 2 field.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"5.1. Dwarf Spheroidal Galaxies\",\n \"content\": \"Results so far have been published for a total of 17 Miras from Fornax, Leo i , Sculptor and Phoenix (Menzies et al. 2008, 2010, 2011; Whitelock et al. 2009). Where spectral types are available for these stars, they show them to be C-rich and we assume that they are all C-type stars. Fig. 1 shows the absolute K magnitudes on a PL( K ) relation for all the Miras in dwarf spheroidals. The large scatter is very striking and the distance below the LMC's PL( K ) relation is a function of the ( J -K ) colour, indicating that the stars below the line are there because of their thick circumstellar shells. Bolometric magnitudes can be estimated using a ( J -K )-dependent bolometric correction (Whitelock et al. 2009) and the results are shown in a PL relation in Fig. 2. The scatter is vastly reduced, although there are still two stars which lie well below the mean relation. Two possible explanations have been offered for these faint points. It may be that the bolometric correction does not apply to these intrinsically relatively blue, shortperiod, stars (see point [ b ] in Sect. 3), since it was derived for significantly longer-period stars. Alternatively, they are undergoing obscuration events of the type that are common among C-rich Miras in the Galaxy and the LMC (e.g., Whitelock et al. 2006) and have non-uniform shells.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"5.2. NGC6822\",\n \"content\": \"NGC6822 is an isolated barred dwarf galaxy, similar to the SMC, but with slightly higher metallicity. It has been examined for AGB variables in the same way as the dwarf spheroidals over 3.5 years and has had numerous Miras catalogued (Whitelock et al. 2012; see also Battinelli & Demers 2011). Fig. 3 shows the variables in a colour-magnitude diagram. Spectral types are available for only very few stars and the split into O- and C-rich assumes that all very red stars are C-rich. Several of the large-amplitude stars without measured periods are probably also Miras and all very red stars are variable. Fig. 4 shows the NGC 6822 Miras on a PL( K ) relation. Most of the longer-period Orich stars fall above the PL relation and are probably similar to the stars in the LMC (mentioned above) that may be hot-bottom burning (Whitelock et al. 2003). Feast (2009) suggested that these stars may be pulsating in the first overtone. Many of the C-rich Miras fall well below the line as the result of thick circumstellar shells. Fig. 5 shows the same stars on a bolometric PL relation, and we see that the C stars scatter around a relation that is very similar to the one obeyed by LMC Miras. The slope is almost identical to the slope of the LMC line, within the uncertainties. Using an LMC distance modulus of 18.5 mag, we determine from the C-rich Miras that ( m -M ) 0 = 23 . 56 ± 0 . 03 mag for NGC 6822. This may be compared to 23 . 40 ± 0 . 05 mag derived from Cepheid variables (Feast et al. 2012b) and 23 . 49 ± 0 . 03 mag from RR Lyrae variables (Clementini et al. 2003). Note that all errors quoted here are internal, but there are systematic uncertainties in all of the measurements. The agreement is reasonable and certainly shows that Miras offer a viable alternative to the more conventional distance indicators.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"5.3. Challenges\",\n \"content\": \"However, I should note that there remain challenges in using Miras as distance indicators. One of the most serious of these is ensuring that measurements made with different photometric systems give the same result. Battinelli & Demers (2011) have 16 largeamplitude variables in common with Whitelock et al. (2012) in NGC 6822. The periods determined for these agree well, but the mean magnitudes differ by ∆ K = 0 . 25 mag. Both groups know that their photometry of normal stars, i.e., those with ( J -K ) < 1 . 0 mag, is on the 2mass system. Dealing with very red stars, of which there are no non-variable examples, will require considerably more effort. It also remains possible that we will find metallicity effects as the PL relationships become better defined.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"6. Cepheids and/or Miras?\",\n \"content\": \"Cepheids have long provided a vital step on the distance-scale ladder linking the Galaxy to distant supernovae. However, Mira variables offer a viable alternative which may well be preferable for the following reasons:\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"7. Conclusion\",\n \"content\": \"Large-amplitude AGB variables offer a viable alternative to Cepheids for distance-scale studies, which will be particularly valuable when infrared observations are available. There remain, however, calibration issues that must be resolved if observations from different instruments are to be combined reliably. The Gaia satellite will provide a vital Galactic calibration that will put the Mira absolute-magnitude scale on a new footing (see Whitelock 2012).\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"Acknowledgements\",\n \"content\": \"I am grateful to my colleagues for allowing me to discuss our results and particularly to Michael Feast and John Menzies for their comments on this manuscript. I acknowledge a grant from the South African National Research Foundation.\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Battinelli, P., & Demers, S. 2011, A&A , 525, 69 Clementini, G., Held, E.V., Baldacci, L., & Rizzi, L. 2003, ApJ , 588, L85 Feast, M.W., Glass, I.S., Whitelock, P.A., & Catchpole, R.M. 1989, MNRAS , 241, 375 Feast, M.W. 2009, in: AGB stars and related phenomena (Ueta, T., Matsunaga, N., & Ita, Y., eds.), AGB stars and related phenomena , p. 48 Feast, M.W., Menzies, J.W., & Whitelock, P.A. 2012a, MNRAS , in press (arXiv:1210.0415) Feast, M.W., Whitelock, P.A., Menzies, J.W., & Matsunaga, N. 2012b, MNRAS , 421, 2998 Groenewegen M.A.T., et al. 2007, MNRAS , 376, 313 Hughes, S.M.G., & Wood, P.R 1990, AJ , 99, 784 Ita, Y., & Matsunaga, N. 2011, MNRAS , 412, 2345 Kamath, D., Wood, P.R., Soszy´nski, I., & Lebzelter, T. 2010, MNRAS , 408, 522 Kerschbaum, F., Lebzelter, T., & Makul, L. 2010, A&A , 524, A87 Matsunaga, N. 2012, Ap&SS , 341, 93 Matsunaga, N., 2006, Ph.D. Thesis , University of Tokyo (Japan) Matsunaga, N., Kawadu, T., Nishiyama, S., Nagayama, T., Hatano, H., Tamura, M., Glass, I.S., & Nagata, T. 2009, MNRAS , 399, 1709 Menzies, J.W., Feast, M.W., Whitelock, P.A., Olivier, E., Matsunaga, N., & da Costa, G. 2008, MNRAS , 385, 1045 Menzies, J.W., Whitelock, P.A., Feast, M.W., & Matsunaga, N. 2010, MNRAS , 406, 86 Menzies, J.W., Feast, M.W., Whitelock, P.A., & Matsunaga, N. 2011, MNRAS , 414, 3492 Rejkuba, M. 2004, A&A , 413, 903 Sarajedini, A., et al. 2007, AJ , 133, 1658 Sloan, G.C., Matsunaga, N., Matsuura, M., et al. 2010, ApJ , 719, 1274 Whitelock, P.A. 2012, Ap&SS , 341, 123 Whitelock, P.A., Feast, M.W., van Loon, J.Th., & Zijlstra, A.A. 2003, MNRAS , 342, 86 Whitelock, P.A., Feast, M.W., Marang, F., & Groenewegen, M.A.T. 2006, MNRAS , 369, 751 Whitelock, P.A., Feast, M.W., & van Leeuwen, F. 2008, MNRAS , 386, 313 Whitelock, P.A., Menzies, J.W., Feast, M.W., Matsunaga, N., Tanab´e, T., & Ita, Y. 2009, MNRAS , 394, 795 Whitelock, P.A., Menzies, J.W., Feast, M.W., Nsengiyumva, F., & Matsunaga, N. 2012, MNRAS , in press (arXiv:1210.3695) Wood, P.R. 2000, Publ. Astron. Soc. Aus. , 17, 18\",\n \"pages\": [\n 8\n ]\n }\n]"}}},{"rowIdx":274,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..290..163B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1211.1905.pdf"},"content":{"kind":"string","value":"\nLOFT: the Large Observatory For X-ray Timing\nTomaso M. Belloni 1 and Enrico Bozzo 2 (on behalf of the LOFT Consortium)\n1 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy\nemail: tomaso.belloni@brera.inaf.it email:\n2 ISDC, Data Center for Astrophysics of the University of Geneva chemin d' ´ Ecogia, 16 1290 Versoix, Switzerland enrico.bozzo@unige.ch\nAbstract. LOFT, the large observatory for X-ray timing, is a new mission concept competing with other four candidates for a launch opportunity in 2022-2024. LOFT will be performing high-time resolution X-ray observations of compact objects, combining for the first time an unprecedented large collecting area for X-ray photons and a spectral resolution approaching that of CCD-based X-ray instruments (down to 200 eV FWHM at 6 keV). The operating energy range is 2-80 keV. The main science goals of LOFT are the measurement of the neutron stars equation of states and the test of General Relativity in the strong field regime. The breakthrough capabilities of the instruments on-board LOFT will permit to open also new discovery windows for a wide range of Galactic and extragalactic X-ray sources.\nIn this contribution, we provide a general description of the mission concept and summarize its main scientific capabilities.\nKeywords. instrumentation: detectors, X-rays: binaries, X-rays: galaxies, relativity, equation of state\n1. Introduction\nLOFT, the Large Observatory for X-ray Timing, is one of the four mission candidates currently competing for a launch opportunity in 2022-2024. The four missions were selected by the European Space Agency (ESA) in February 2011 within the framework of the Cosmic Vision program. LOFT was specifically conceived to investigate the behavior of matter in presence of strong gravitational fields and at supranuclear densities by performing high-time-resolution X-ray observations of compact objects. Measurements of the rapid X-ray flux and spectral variability in these sources can indeed provide constraints on the equation of states of neutron stars (NS) and measurements of black holes masses and spins (see Sect. 3). A representation of the current view of the LOFT satellite is provided in Fig. 1.\nThe Large Area Detector (LAD) is the main instrument on board LOFT. It is a collimated experiment with a field-of-view (FOV) of about 1 degree, reaching a peak X-ray sensitive effective area of ∼ 10 m 2 at 8 keV (see Fig. 3). The expected scientific throughput is of ∼ 240,000 cts/s for a source with a flux of 1 Crab in the 2-80 keV energy range. The main operating energy band of the LAD is 2-30 keV, where the expected spectral resolution is < 260 eV (200 eV for single anode events, i.e. about 40% of the total counts). In the range 30-80 keV, only a coarse energy resolution of ∼ 2 keV will be available. The time resolution of the LAD is 10 µ s.\nThe second instrument completing the LOFT payload is the Wide Field Monitor\n\n\n
Figure 1. Left : an artist representation of the LOFT satellite. The LAD instrument comprises 6 panels connected to the optical bench of the spacecraft; the WFM is located on the top of the optical bench. Right : Breakdown of the LAD instrument. Each panel of the LAD (6 in total) comprises 21 modules and each module is equipped with 16 SDDs.
\n\n(WFM). This is a coded mask imager whose main goal is to detect new transient sources and state changes of any X-ray emitter that can be suitable for observations with the LAD. The WFM will be observing about 1/3 of the sky at once and will be capable to provide data for each source in this FOV with a spectral and timing resolution similar to that of the LAD.\nWe provide in Sect. 2 a detailed description of the LOFT payload, and describe in Sect. 3 some of the most recent improvements in the instruments design. All images and simulations used in the following sections are provided by the LOFT teams † .\n2. LOFT Spacecraft and Instrumentation\nIn the current configuration of the LOFT mission (see Fig. 1), the LAD instrument comprises six panels attached to the spacecraft optical bench. On the top of it, the five units (10 cameras) of the WFM are placed in a way to maximize the sky coverage (Brandt et al. 2012).\nThe LAD is the prime instrument onboard LOFT. Its geometric area is ∼ 15 m 2 and provides a peak effective area for X-ray observations that is a factor of 20 larger than that of any previously flown X-ray experiment (see left panel of Fig. 3). The LAD achieves an energy resolution (FWHM) better than 260 eV at 6 keV (end-of-life, EoL). The two enabling technologies of the LAD are the large area Silicon drift detectors developed originally for the ALICE experiment at CERN (Vacchi et al. 1991; Rashevski et al. 2002) and the capillary plates collimators designed for the ESA BepiColombo mission (Fraser et al. 2010). The SDDs have been optimized for LOFT to reach a configuration in which each detector is now 450 µ m thick and provides a total active area of 76 cm 2 , readout by two rows of anodes with a pitch of 970 µ m and a drift channel of 35 mm. The capillary-plate collimator was chosen to make advantage of the lead glass with millions of micro-pores, already proven to be efficient in collimating X-ray photons within a field of view (FOV) of ∼ 1 degree and energies up to 30-40 keV. In the current LOFT design,\n† See http://www.isdc.unige.ch/loft/index.php/preliminar-response-files-and-simulatedbackground and http://www.isdc.unige.ch/ loft/index.php/instruments-on-board-loft.\n\n\n\n\n\n
Figure 2. Left : detailed view of the WFM located at the top of the optical bench. The insert shows the concept of a WFM unit comprising two orthogonal cameras to achieve a full 2D imaging capable system. Right: The WFM FOV expected when the instrument points toward the Galactic center. The color bar at the bottom gives the effective area in units of cm 2 .
\n\neach of the SDD is equipped with a single collimator tile ( ∼ 80 cm 2 in area). There are in total 6 Detector Panels, each hosting 21 Detector Modules. 16 SDDs are allocated within each module (see also Zane et al. 2012). A total of 2016 SDDs are needed for the complete LAD instrument. The 6 LAD panels will be initially stowed in the launcher, and then deployed once in space (Feroci et al. 2011). A summary of the LAD capabilities is provided in Table 2.\nThe WFM is a wide field coded mask imager working mainly in the 2-50 keV energy range (see Fig. 2). The instrument is designed from the heritage of the SuperAGILE experiment (Feroci et al. 2007), but presents very notable improvements in the low energy threshold, energy resolution and imaging capabilities. These are provided by the usage of similar SDDs to that employed for the LAD but optimized for imaging purposes (Campana et al. 2011). Due to this optimization, the time resolution is kept of the order of ∼ 10 µ s, whereas the spectral resolution is somewhat lower than that achieved by the LAD (300 eV FWHM at 6 keV). The WFM SDDs are able to measure the impact point of a photon with an accuracy that corresponds to few arcmin along the anode direction and few degrees along the drift direction (Evangelista et al. (2012)). For this reason, each of the WFM unit (5 in total, see Fig. 2) comprises two orthogonal cameras to achieve a full 2D imaging capability (Brandt et al. (2012)). This design has also the advantage of providing a strong redundancy in case of failure of one camera. In that case, indeed, a unit will be left with a fine resolution in one direction and a coarse resolution in the other, still ensuring some coverage of the sky in the corresponding region). The imaging properties of the WFM are extensively discussed in Donnarumma et al. 2012. In Fig. 2 we show as an example the WFM FOV during a pointing toward the Galactic Center. The WFM will cover more than 1/3 of the sky at once, corresponding to 50% of the sky accessible to the LAD at any time. A summary of the WFM capabilities is provided in Table 2.\n2.1. Latest mission improvements\nDue to its large field of view, wide energy coverage and imaging capabilities, the WFM is expected to detect and localize a large number of gamma-ray bursts and other fast transients every year (Brandt et al. 2012). The orientation of one of the WFM unit in the anti-Sun direction makes also particularly favorable the follow-up of any interesting event\n
\n\n
WFM
\n
\na : Events in the 30-80 keV energy range are used\n\nto monitor contamination from bright sources.\n\nTable 1. Scientific requirements and goals of the LAD and WFM.\ndetected by this instrument with ground-based facility that will be operating at the time of LOFT. In order to optimize the synergy between LOFT and these facilities, the WFM onboard data processing is endowed with a triggering and imaging system to calculate the coordinates of the transient event in real-time on-board (LBAS, i.e. LOFT Burst Alert System). This information is broadcast to the ground through a VHF transmission system to within /lessorsimilar 30 s from the detection. Due to possible telemetry limitations, the maximum sustainable rate of triggers might be of one per orbit (i.e. one every 90 min.).\nSome effort has been recently devoted to increase the fraction of the sky accessible to the LAD at any time with respect to that defined in the original LOFT configuration † . This will improve the flexibility of the mission in response to requests for target of opportunity observations (ToOs). However, pointings performed outside the nominal sky region might be affected by a somewhat degraded spectral resolution (a factor 1.5 worse than the nominal value) due to the more challenging thermal environment at which the SDDs have to be operated. This is, however, not expected to be an issue for all those observations that do not deal with narrow spectral features and do not require the maximum achievable spectral resolution.\n3. LOFT Science\nThe main science goals of LOFT are the study of matter in presence of extreme densities and strong gravitational fields. Through its unique timing capabilities and fine spectral resolution, LOFT will be able to provide unprecedented constraints to the neutron star equation of state (EoS) and test general relativistic effects that are expected to affect X-rays emitted to within a few gravitational radii ( r g ) from a central black hole (so far General Relativity has been tested only in the weak-field regime, i.e. for r g ∼ 10 5 -10 6 ). Beside this, the unique capabilities of the LAD and WFM will also contribute to dramatically deepen our understanding of the physics of a wide range of Galactic and extra-galactic sources. The two instruments will permit spectroscopic and variability studies down to previously unexplored small timescales (few µ s), and provide prompt fine spectral and timing resolution data for any impulsive bright events originating from Galactic sources as well as cosmic gamma-ray bursts. About 50% of the total LOFT observational time will be devoted to this 'observatory science'.\n3.1. Main Science Goals\nObtaining sufficiently accurate (within a few %) measurements of neutron stars masses and radii is one of the most direct probes we have to test our currently understanding of\n† See http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=49448\n
\n\n
LAD
\n
\n\na : Refers to single-anode events.\n\n\n\n
Figure 3. Left : Plot of the LAD effective area as a function of the energy according to the instrument requirements (see Table 2). The effective area of some other instruments is reported for comparison. Right : A simulated power-spectrum of the black hole candidate XTE J1550-564 as observed by RXTE/PCA (Miller et al. 2001). For the simulation it is assumed that the two QPOs have frequencies of ν 1 =188 Hz, ν 2 =268 Hz and the fractional rms are 2.8% and 6.2%, respectively. The source flux is 1 Crab (3-20 keV). The exposure time used for the LAD simulation is 1 ks. The improvement in the S/N of this detection should be compared by looking at Fig. 4 in Miller et al. (2001), where the original RXTE/PCA exposure time of 54 ks permitted only a detection at 3-4 σ confidence level.
\n\nmatter at supranuclear densities, as this can constraint the different EoS models proposed for these objects. LOFT will be able to measure the neutron star mass and radius down to an accuracy of a few percent independently by using different techniques (e.g., pulse profile fitting in millisecond X-ray pulsars and spectral fitting of the type-I X-ray bursts observed from neutron star low mass X-ray binaries; Feroci et al. 2011). The top-level science goals of LOFT related to the determination of dense matter EOS are summarized in Feroci et al. (2012).\nBy making advantage of its unique throughput, LOFT will also be able to test General Relativistic effects produced by the strong gravitational field of a black hole onto accreting matter orbiting close-by (a few r g ). One of the most promising way to carry out such investigations is to observe quasi-periodic oscillations (QPOs) arising from the X-ray emission of the accretion flow. The dynamical time-scales of the inner accretion flow are typically of few milliseconds, and thus the interpretation of the highest frequency QPOs involves fundamental frequencies of the motion of matter orbiting in disk regions dominated by the strong black hole gravitational field (e.g., the relativistic radial and vertical epicyclic frequencies or the relativistic nodal and periastron precession). Distinguishing among these possibilities have been proved impossible so far, due to the limited observation capabilities of the previous generation of X-ray timing instruments. The enhanced capabilities of the LAD will permit to dramatically increase the S/N of any of the QPO feature (see right panel of Fig. 3), allowing to measure QPOs accurately within their coherence time and follow their evolution in time with the source X-ray flux (see Fig. 4). This will secure access to still untested General relativistic effects, such as frame-dragging, strong-field periastron precession, and the existence of an innermost stable orbit around black holes. An independent investigation of the motion of the accretion flow orbiting close to the central black hole can be carried out through the observation of the fluorescence Fe K line profile emitted by this material at different orbital phases, by making advantages of the fine resolution of the LAD. Variation in the line profile are expected as a consequence of the Lense-Thirring precession of the inner disk (at ∼ r g ),\n\n\n
Figure 4. The goal of the LAD is not only to detect QPOs with higher S/N, as shown in Fig. 3 (right panel). In this figure we show that the LAD will also be able to follow the evolution in time of the QPOs, removing the observational constraints of its predecessor the RXTE/PCA (represented with a blue obscuring rectangular in the upper panel). In the bottom two panels we show a simulation in which the same QPO in Fig. 3 is evolved following the prediction of the epicyclic resonant model (Abramowicz et al. 2001) and the relativistic precession model (Stella et al. 1999). In the simulations we assumed: flux of 1 Crab and fractional rms 2.8% and 6.2% for the blue QPOs, flux of 400 mCrab and fractional rms 1.4% and 3.1% for the green QPOs, flux of 300 mCrab and fractional rms 0.7% and 1.5% for the magenta QPOs. The exposure time is 16 ks in all cases.
\n\nand provide a tool to measure mass and spin of Galactic and extra-galactic black holes. In Fig. 5, we show a simulation of such a study in the Active Galactic Nuclei (AGN) MGC6-30. A summary of the top-level science goals of LOFT with respect to the measure of General Relativistic effects can be found in Feroci et al. (2012).\n3.2. Observatory Science\nAbout 50% of the total LOFT observational time will be made available for observatory science. The breakthrough capabilities of the LAD will revolutionize X-ray observations of any relatively bright source down to a limiting flux of few mCrab. Providing an overview of the entire potential discovery space of the LAD is not possible in this relatively short contribution. Instead we provide here some more examples of the capabilities of the WFM for the observatory science. This instrument combines a very wide FOV, covering more than 1/3 of the sky at once, together with a fine timing (few µ s) and spectral resolution ( < 300 eV). Through the LBAS system, the position and trigger time of bright events can be broadcast to the ground within a delay of /lessorsimilar 30 s, allowing for quick follow-up with ground-based facilities. Due to the large FOV, about 150 GRB and thousands typeI X-ray bursts are expected to be detected with the WFM per year. The low energy threshold and fine spectral resolution of the instrument will allow to investigate in detail any spectral feature expected during these events, including absorption edges. We show two specific simulations in Fig. 6. As for the LAD, the range of scientific investigations\n\n\n
Figure 5. Simulation of the detection with the LAD of the iron line in four different orbital phases of the accretion flow in the AGN MGC 6-30. The simulation assumes an average flux of 3 mCrab and an exposure time of 16 ks for each of the orbital phases (each phase is observed twice; courtesy of A. de Rosa).
\n\n\n\n
Figure 6. Left : Absorption edges in energetic type-I X-ray bursts were predicted theoretically by Weinberg et al. (2006) and the only observational evidence of them was reported so far by in't Zand et al. (2010). Following their results we simulated here an X-ray burst with a flux of 5 Crab with the WFM including a τ =1 absorption edge at 6 keV. The edge is clearly detected already with 1 s exposure (courtesy of WFM-SIM group). Right : The only example in the literature of narrow spectral features in the prompt emission of a GRB is reported by Amati et al. (2000). Here we simulated an observation of that GRB with the WFM using an exposure time of 13 s. The edge at 3.8 keV is clearly detected (courtesy of WFM-SIM group).
\n\navailable to the WFM cannot be sufficiently summarized here. As an example, we refer the reader to Romano et al. (2012) and Ferrigno et al. (2012) for a description of the WFM contribution to the improvement in our knowledge of the Be X-ray binaries and Supergiant Fast X-ray Transients.\nReferences\nAmati, L. et al. 2000, Science, 290, 953\nAbramowicz, M.A. & Kluzniak, W. 2001, A&A, 374, 19 Brandt, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-88, 2012 Campana, R. et al. 2011, NIMPA, 633, 22 Campana, R. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-209, 2012 Donnarumma, I. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-211, 2012 Evangelista, Y. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-85, 2012 Feroci, M. et al. 2007, NIMPA, 581, 728 Feroci, M. et al. 2011, ExA, in press (arXiv:1107.0436) Feroci, M. et al. 2012, Proceedings of SPIE, Vol. 8443, Paper No. 8443-210, 2012 Ferrigno, C. et al. 2012, Proceedings of COSPAR, in preparation in't Zand, J. et al. 2010, A&A, 520, 81 Miller, J.M. et al. 2001, ApJ, 563, 928 Romano, P. et al. 2012, AIP, in preparation Stella, L. & Vietri 1999, ApJ, 524, 63\nZane, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-87, 2012\nWeimberg, . et al., 2012 ApJ, 639, 1018\n"},"sections":{"kind":"list like","value":[{"title":"Tomaso M. Belloni 1 and Enrico Bozzo 2 (on behalf of the LOFT Consortium)","content":"1 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy email: tomaso.belloni@brera.inaf.it email: 2 ISDC, Data Center for Astrophysics of the University of Geneva chemin d' ´ Ecogia, 16 1290 Versoix, Switzerland enrico.bozzo@unige.ch Abstract. LOFT, the large observatory for X-ray timing, is a new mission concept competing with other four candidates for a launch opportunity in 2022-2024. LOFT will be performing high-time resolution X-ray observations of compact objects, combining for the first time an unprecedented large collecting area for X-ray photons and a spectral resolution approaching that of CCD-based X-ray instruments (down to 200 eV FWHM at 6 keV). The operating energy range is 2-80 keV. The main science goals of LOFT are the measurement of the neutron stars equation of states and the test of General Relativity in the strong field regime. The breakthrough capabilities of the instruments on-board LOFT will permit to open also new discovery windows for a wide range of Galactic and extragalactic X-ray sources. In this contribution, we provide a general description of the mission concept and summarize its main scientific capabilities. Keywords. instrumentation: detectors, X-rays: binaries, X-rays: galaxies, relativity, equation of state","pages":[1]},{"title":"1. Introduction","content":"LOFT, the Large Observatory for X-ray Timing, is one of the four mission candidates currently competing for a launch opportunity in 2022-2024. The four missions were selected by the European Space Agency (ESA) in February 2011 within the framework of the Cosmic Vision program. LOFT was specifically conceived to investigate the behavior of matter in presence of strong gravitational fields and at supranuclear densities by performing high-time-resolution X-ray observations of compact objects. Measurements of the rapid X-ray flux and spectral variability in these sources can indeed provide constraints on the equation of states of neutron stars (NS) and measurements of black holes masses and spins (see Sect. 3). A representation of the current view of the LOFT satellite is provided in Fig. 1. The Large Area Detector (LAD) is the main instrument on board LOFT. It is a collimated experiment with a field-of-view (FOV) of about 1 degree, reaching a peak X-ray sensitive effective area of ∼ 10 m 2 at 8 keV (see Fig. 3). The expected scientific throughput is of ∼ 240,000 cts/s for a source with a flux of 1 Crab in the 2-80 keV energy range. The main operating energy band of the LAD is 2-30 keV, where the expected spectral resolution is < 260 eV (200 eV for single anode events, i.e. about 40% of the total counts). In the range 30-80 keV, only a coarse energy resolution of ∼ 2 keV will be available. The time resolution of the LAD is 10 µ s. The second instrument completing the LOFT payload is the Wide Field Monitor (WFM). This is a coded mask imager whose main goal is to detect new transient sources and state changes of any X-ray emitter that can be suitable for observations with the LAD. The WFM will be observing about 1/3 of the sky at once and will be capable to provide data for each source in this FOV with a spectral and timing resolution similar to that of the LAD. We provide in Sect. 2 a detailed description of the LOFT payload, and describe in Sect. 3 some of the most recent improvements in the instruments design. All images and simulations used in the following sections are provided by the LOFT teams † .","pages":[1,2]},{"title":"2. LOFT Spacecraft and Instrumentation","content":"In the current configuration of the LOFT mission (see Fig. 1), the LAD instrument comprises six panels attached to the spacecraft optical bench. On the top of it, the five units (10 cameras) of the WFM are placed in a way to maximize the sky coverage (Brandt et al. 2012). The LAD is the prime instrument onboard LOFT. Its geometric area is ∼ 15 m 2 and provides a peak effective area for X-ray observations that is a factor of 20 larger than that of any previously flown X-ray experiment (see left panel of Fig. 3). The LAD achieves an energy resolution (FWHM) better than 260 eV at 6 keV (end-of-life, EoL). The two enabling technologies of the LAD are the large area Silicon drift detectors developed originally for the ALICE experiment at CERN (Vacchi et al. 1991; Rashevski et al. 2002) and the capillary plates collimators designed for the ESA BepiColombo mission (Fraser et al. 2010). The SDDs have been optimized for LOFT to reach a configuration in which each detector is now 450 µ m thick and provides a total active area of 76 cm 2 , readout by two rows of anodes with a pitch of 970 µ m and a drift channel of 35 mm. The capillary-plate collimator was chosen to make advantage of the lead glass with millions of micro-pores, already proven to be efficient in collimating X-ray photons within a field of view (FOV) of ∼ 1 degree and energies up to 30-40 keV. In the current LOFT design, † See http://www.isdc.unige.ch/loft/index.php/preliminar-response-files-and-simulatedbackground and http://www.isdc.unige.ch/ loft/index.php/instruments-on-board-loft. each of the SDD is equipped with a single collimator tile ( ∼ 80 cm 2 in area). There are in total 6 Detector Panels, each hosting 21 Detector Modules. 16 SDDs are allocated within each module (see also Zane et al. 2012). A total of 2016 SDDs are needed for the complete LAD instrument. The 6 LAD panels will be initially stowed in the launcher, and then deployed once in space (Feroci et al. 2011). A summary of the LAD capabilities is provided in Table 2. The WFM is a wide field coded mask imager working mainly in the 2-50 keV energy range (see Fig. 2). The instrument is designed from the heritage of the SuperAGILE experiment (Feroci et al. 2007), but presents very notable improvements in the low energy threshold, energy resolution and imaging capabilities. These are provided by the usage of similar SDDs to that employed for the LAD but optimized for imaging purposes (Campana et al. 2011). Due to this optimization, the time resolution is kept of the order of ∼ 10 µ s, whereas the spectral resolution is somewhat lower than that achieved by the LAD (300 eV FWHM at 6 keV). The WFM SDDs are able to measure the impact point of a photon with an accuracy that corresponds to few arcmin along the anode direction and few degrees along the drift direction (Evangelista et al. (2012)). For this reason, each of the WFM unit (5 in total, see Fig. 2) comprises two orthogonal cameras to achieve a full 2D imaging capability (Brandt et al. (2012)). This design has also the advantage of providing a strong redundancy in case of failure of one camera. In that case, indeed, a unit will be left with a fine resolution in one direction and a coarse resolution in the other, still ensuring some coverage of the sky in the corresponding region). The imaging properties of the WFM are extensively discussed in Donnarumma et al. 2012. In Fig. 2 we show as an example the WFM FOV during a pointing toward the Galactic Center. The WFM will cover more than 1/3 of the sky at once, corresponding to 50% of the sky accessible to the LAD at any time. A summary of the WFM capabilities is provided in Table 2.","pages":[2,3]},{"title":"2.1. Latest mission improvements","content":"Due to its large field of view, wide energy coverage and imaging capabilities, the WFM is expected to detect and localize a large number of gamma-ray bursts and other fast transients every year (Brandt et al. 2012). The orientation of one of the WFM unit in the anti-Sun direction makes also particularly favorable the follow-up of any interesting event a : Events in the 30-80 keV energy range are used detected by this instrument with ground-based facility that will be operating at the time of LOFT. In order to optimize the synergy between LOFT and these facilities, the WFM onboard data processing is endowed with a triggering and imaging system to calculate the coordinates of the transient event in real-time on-board (LBAS, i.e. LOFT Burst Alert System). This information is broadcast to the ground through a VHF transmission system to within /lessorsimilar 30 s from the detection. Due to possible telemetry limitations, the maximum sustainable rate of triggers might be of one per orbit (i.e. one every 90 min.). Some effort has been recently devoted to increase the fraction of the sky accessible to the LAD at any time with respect to that defined in the original LOFT configuration † . This will improve the flexibility of the mission in response to requests for target of opportunity observations (ToOs). However, pointings performed outside the nominal sky region might be affected by a somewhat degraded spectral resolution (a factor 1.5 worse than the nominal value) due to the more challenging thermal environment at which the SDDs have to be operated. This is, however, not expected to be an issue for all those observations that do not deal with narrow spectral features and do not require the maximum achievable spectral resolution.","pages":[3,4]},{"title":"3. LOFT Science","content":"The main science goals of LOFT are the study of matter in presence of extreme densities and strong gravitational fields. Through its unique timing capabilities and fine spectral resolution, LOFT will be able to provide unprecedented constraints to the neutron star equation of state (EoS) and test general relativistic effects that are expected to affect X-rays emitted to within a few gravitational radii ( r g ) from a central black hole (so far General Relativity has been tested only in the weak-field regime, i.e. for r g ∼ 10 5 -10 6 ). Beside this, the unique capabilities of the LAD and WFM will also contribute to dramatically deepen our understanding of the physics of a wide range of Galactic and extra-galactic sources. The two instruments will permit spectroscopic and variability studies down to previously unexplored small timescales (few µ s), and provide prompt fine spectral and timing resolution data for any impulsive bright events originating from Galactic sources as well as cosmic gamma-ray bursts. About 50% of the total LOFT observational time will be devoted to this 'observatory science'.","pages":[4]},{"title":"3.1. Main Science Goals","content":"Obtaining sufficiently accurate (within a few %) measurements of neutron stars masses and radii is one of the most direct probes we have to test our currently understanding of † See http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=49448 matter at supranuclear densities, as this can constraint the different EoS models proposed for these objects. LOFT will be able to measure the neutron star mass and radius down to an accuracy of a few percent independently by using different techniques (e.g., pulse profile fitting in millisecond X-ray pulsars and spectral fitting of the type-I X-ray bursts observed from neutron star low mass X-ray binaries; Feroci et al. 2011). The top-level science goals of LOFT related to the determination of dense matter EOS are summarized in Feroci et al. (2012). By making advantage of its unique throughput, LOFT will also be able to test General Relativistic effects produced by the strong gravitational field of a black hole onto accreting matter orbiting close-by (a few r g ). One of the most promising way to carry out such investigations is to observe quasi-periodic oscillations (QPOs) arising from the X-ray emission of the accretion flow. The dynamical time-scales of the inner accretion flow are typically of few milliseconds, and thus the interpretation of the highest frequency QPOs involves fundamental frequencies of the motion of matter orbiting in disk regions dominated by the strong black hole gravitational field (e.g., the relativistic radial and vertical epicyclic frequencies or the relativistic nodal and periastron precession). Distinguishing among these possibilities have been proved impossible so far, due to the limited observation capabilities of the previous generation of X-ray timing instruments. The enhanced capabilities of the LAD will permit to dramatically increase the S/N of any of the QPO feature (see right panel of Fig. 3), allowing to measure QPOs accurately within their coherence time and follow their evolution in time with the source X-ray flux (see Fig. 4). This will secure access to still untested General relativistic effects, such as frame-dragging, strong-field periastron precession, and the existence of an innermost stable orbit around black holes. An independent investigation of the motion of the accretion flow orbiting close to the central black hole can be carried out through the observation of the fluorescence Fe K line profile emitted by this material at different orbital phases, by making advantages of the fine resolution of the LAD. Variation in the line profile are expected as a consequence of the Lense-Thirring precession of the inner disk (at ∼ r g ), and provide a tool to measure mass and spin of Galactic and extra-galactic black holes. In Fig. 5, we show a simulation of such a study in the Active Galactic Nuclei (AGN) MGC6-30. A summary of the top-level science goals of LOFT with respect to the measure of General Relativistic effects can be found in Feroci et al. (2012).","pages":[4,5,6]},{"title":"3.2. Observatory Science","content":"About 50% of the total LOFT observational time will be made available for observatory science. The breakthrough capabilities of the LAD will revolutionize X-ray observations of any relatively bright source down to a limiting flux of few mCrab. Providing an overview of the entire potential discovery space of the LAD is not possible in this relatively short contribution. Instead we provide here some more examples of the capabilities of the WFM for the observatory science. This instrument combines a very wide FOV, covering more than 1/3 of the sky at once, together with a fine timing (few µ s) and spectral resolution ( < 300 eV). Through the LBAS system, the position and trigger time of bright events can be broadcast to the ground within a delay of /lessorsimilar 30 s, allowing for quick follow-up with ground-based facilities. Due to the large FOV, about 150 GRB and thousands typeI X-ray bursts are expected to be detected with the WFM per year. The low energy threshold and fine spectral resolution of the instrument will allow to investigate in detail any spectral feature expected during these events, including absorption edges. We show two specific simulations in Fig. 6. As for the LAD, the range of scientific investigations available to the WFM cannot be sufficiently summarized here. As an example, we refer the reader to Romano et al. (2012) and Ferrigno et al. (2012) for a description of the WFM contribution to the improvement in our knowledge of the Be X-ray binaries and Supergiant Fast X-ray Transients.","pages":[6,7]},{"title":"References","content":"Amati, L. et al. 2000, Science, 290, 953 Abramowicz, M.A. & Kluzniak, W. 2001, A&A, 374, 19 Brandt, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-88, 2012 Campana, R. et al. 2011, NIMPA, 633, 22 Campana, R. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-209, 2012 Donnarumma, I. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-211, 2012 Evangelista, Y. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-85, 2012 Feroci, M. et al. 2007, NIMPA, 581, 728 Feroci, M. et al. 2011, ExA, in press (arXiv:1107.0436) Feroci, M. et al. 2012, Proceedings of SPIE, Vol. 8443, Paper No. 8443-210, 2012 Ferrigno, C. et al. 2012, Proceedings of COSPAR, in preparation in't Zand, J. et al. 2010, A&A, 520, 81 Miller, J.M. et al. 2001, ApJ, 563, 928 Romano, P. et al. 2012, AIP, in preparation Stella, L. & Vietri 1999, ApJ, 524, 63 Zane, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-87, 2012 Weimberg, . et al., 2012 ApJ, 639, 1018","pages":[7,8]}],"string":"[\n {\n \"title\": \"Tomaso M. Belloni 1 and Enrico Bozzo 2 (on behalf of the LOFT Consortium)\",\n \"content\": \"1 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy email: tomaso.belloni@brera.inaf.it email: 2 ISDC, Data Center for Astrophysics of the University of Geneva chemin d' ´ Ecogia, 16 1290 Versoix, Switzerland enrico.bozzo@unige.ch Abstract. LOFT, the large observatory for X-ray timing, is a new mission concept competing with other four candidates for a launch opportunity in 2022-2024. LOFT will be performing high-time resolution X-ray observations of compact objects, combining for the first time an unprecedented large collecting area for X-ray photons and a spectral resolution approaching that of CCD-based X-ray instruments (down to 200 eV FWHM at 6 keV). The operating energy range is 2-80 keV. The main science goals of LOFT are the measurement of the neutron stars equation of states and the test of General Relativity in the strong field regime. The breakthrough capabilities of the instruments on-board LOFT will permit to open also new discovery windows for a wide range of Galactic and extragalactic X-ray sources. In this contribution, we provide a general description of the mission concept and summarize its main scientific capabilities. Keywords. instrumentation: detectors, X-rays: binaries, X-rays: galaxies, relativity, equation of state\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"LOFT, the Large Observatory for X-ray Timing, is one of the four mission candidates currently competing for a launch opportunity in 2022-2024. The four missions were selected by the European Space Agency (ESA) in February 2011 within the framework of the Cosmic Vision program. LOFT was specifically conceived to investigate the behavior of matter in presence of strong gravitational fields and at supranuclear densities by performing high-time-resolution X-ray observations of compact objects. Measurements of the rapid X-ray flux and spectral variability in these sources can indeed provide constraints on the equation of states of neutron stars (NS) and measurements of black holes masses and spins (see Sect. 3). A representation of the current view of the LOFT satellite is provided in Fig. 1. The Large Area Detector (LAD) is the main instrument on board LOFT. It is a collimated experiment with a field-of-view (FOV) of about 1 degree, reaching a peak X-ray sensitive effective area of ∼ 10 m 2 at 8 keV (see Fig. 3). The expected scientific throughput is of ∼ 240,000 cts/s for a source with a flux of 1 Crab in the 2-80 keV energy range. The main operating energy band of the LAD is 2-30 keV, where the expected spectral resolution is < 260 eV (200 eV for single anode events, i.e. about 40% of the total counts). In the range 30-80 keV, only a coarse energy resolution of ∼ 2 keV will be available. The time resolution of the LAD is 10 µ s. The second instrument completing the LOFT payload is the Wide Field Monitor (WFM). This is a coded mask imager whose main goal is to detect new transient sources and state changes of any X-ray emitter that can be suitable for observations with the LAD. The WFM will be observing about 1/3 of the sky at once and will be capable to provide data for each source in this FOV with a spectral and timing resolution similar to that of the LAD. We provide in Sect. 2 a detailed description of the LOFT payload, and describe in Sect. 3 some of the most recent improvements in the instruments design. All images and simulations used in the following sections are provided by the LOFT teams † .\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. LOFT Spacecraft and Instrumentation\",\n \"content\": \"In the current configuration of the LOFT mission (see Fig. 1), the LAD instrument comprises six panels attached to the spacecraft optical bench. On the top of it, the five units (10 cameras) of the WFM are placed in a way to maximize the sky coverage (Brandt et al. 2012). The LAD is the prime instrument onboard LOFT. Its geometric area is ∼ 15 m 2 and provides a peak effective area for X-ray observations that is a factor of 20 larger than that of any previously flown X-ray experiment (see left panel of Fig. 3). The LAD achieves an energy resolution (FWHM) better than 260 eV at 6 keV (end-of-life, EoL). The two enabling technologies of the LAD are the large area Silicon drift detectors developed originally for the ALICE experiment at CERN (Vacchi et al. 1991; Rashevski et al. 2002) and the capillary plates collimators designed for the ESA BepiColombo mission (Fraser et al. 2010). The SDDs have been optimized for LOFT to reach a configuration in which each detector is now 450 µ m thick and provides a total active area of 76 cm 2 , readout by two rows of anodes with a pitch of 970 µ m and a drift channel of 35 mm. The capillary-plate collimator was chosen to make advantage of the lead glass with millions of micro-pores, already proven to be efficient in collimating X-ray photons within a field of view (FOV) of ∼ 1 degree and energies up to 30-40 keV. In the current LOFT design, † See http://www.isdc.unige.ch/loft/index.php/preliminar-response-files-and-simulatedbackground and http://www.isdc.unige.ch/ loft/index.php/instruments-on-board-loft. each of the SDD is equipped with a single collimator tile ( ∼ 80 cm 2 in area). There are in total 6 Detector Panels, each hosting 21 Detector Modules. 16 SDDs are allocated within each module (see also Zane et al. 2012). A total of 2016 SDDs are needed for the complete LAD instrument. The 6 LAD panels will be initially stowed in the launcher, and then deployed once in space (Feroci et al. 2011). A summary of the LAD capabilities is provided in Table 2. The WFM is a wide field coded mask imager working mainly in the 2-50 keV energy range (see Fig. 2). The instrument is designed from the heritage of the SuperAGILE experiment (Feroci et al. 2007), but presents very notable improvements in the low energy threshold, energy resolution and imaging capabilities. These are provided by the usage of similar SDDs to that employed for the LAD but optimized for imaging purposes (Campana et al. 2011). Due to this optimization, the time resolution is kept of the order of ∼ 10 µ s, whereas the spectral resolution is somewhat lower than that achieved by the LAD (300 eV FWHM at 6 keV). The WFM SDDs are able to measure the impact point of a photon with an accuracy that corresponds to few arcmin along the anode direction and few degrees along the drift direction (Evangelista et al. (2012)). For this reason, each of the WFM unit (5 in total, see Fig. 2) comprises two orthogonal cameras to achieve a full 2D imaging capability (Brandt et al. (2012)). This design has also the advantage of providing a strong redundancy in case of failure of one camera. In that case, indeed, a unit will be left with a fine resolution in one direction and a coarse resolution in the other, still ensuring some coverage of the sky in the corresponding region). The imaging properties of the WFM are extensively discussed in Donnarumma et al. 2012. In Fig. 2 we show as an example the WFM FOV during a pointing toward the Galactic Center. The WFM will cover more than 1/3 of the sky at once, corresponding to 50% of the sky accessible to the LAD at any time. A summary of the WFM capabilities is provided in Table 2.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.1. Latest mission improvements\",\n \"content\": \"Due to its large field of view, wide energy coverage and imaging capabilities, the WFM is expected to detect and localize a large number of gamma-ray bursts and other fast transients every year (Brandt et al. 2012). The orientation of one of the WFM unit in the anti-Sun direction makes also particularly favorable the follow-up of any interesting event a : Events in the 30-80 keV energy range are used detected by this instrument with ground-based facility that will be operating at the time of LOFT. In order to optimize the synergy between LOFT and these facilities, the WFM onboard data processing is endowed with a triggering and imaging system to calculate the coordinates of the transient event in real-time on-board (LBAS, i.e. LOFT Burst Alert System). This information is broadcast to the ground through a VHF transmission system to within /lessorsimilar 30 s from the detection. Due to possible telemetry limitations, the maximum sustainable rate of triggers might be of one per orbit (i.e. one every 90 min.). Some effort has been recently devoted to increase the fraction of the sky accessible to the LAD at any time with respect to that defined in the original LOFT configuration † . This will improve the flexibility of the mission in response to requests for target of opportunity observations (ToOs). However, pointings performed outside the nominal sky region might be affected by a somewhat degraded spectral resolution (a factor 1.5 worse than the nominal value) due to the more challenging thermal environment at which the SDDs have to be operated. This is, however, not expected to be an issue for all those observations that do not deal with narrow spectral features and do not require the maximum achievable spectral resolution.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"3. LOFT Science\",\n \"content\": \"The main science goals of LOFT are the study of matter in presence of extreme densities and strong gravitational fields. Through its unique timing capabilities and fine spectral resolution, LOFT will be able to provide unprecedented constraints to the neutron star equation of state (EoS) and test general relativistic effects that are expected to affect X-rays emitted to within a few gravitational radii ( r g ) from a central black hole (so far General Relativity has been tested only in the weak-field regime, i.e. for r g ∼ 10 5 -10 6 ). Beside this, the unique capabilities of the LAD and WFM will also contribute to dramatically deepen our understanding of the physics of a wide range of Galactic and extra-galactic sources. The two instruments will permit spectroscopic and variability studies down to previously unexplored small timescales (few µ s), and provide prompt fine spectral and timing resolution data for any impulsive bright events originating from Galactic sources as well as cosmic gamma-ray bursts. About 50% of the total LOFT observational time will be devoted to this 'observatory science'.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3.1. Main Science Goals\",\n \"content\": \"Obtaining sufficiently accurate (within a few %) measurements of neutron stars masses and radii is one of the most direct probes we have to test our currently understanding of † See http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=49448 matter at supranuclear densities, as this can constraint the different EoS models proposed for these objects. LOFT will be able to measure the neutron star mass and radius down to an accuracy of a few percent independently by using different techniques (e.g., pulse profile fitting in millisecond X-ray pulsars and spectral fitting of the type-I X-ray bursts observed from neutron star low mass X-ray binaries; Feroci et al. 2011). The top-level science goals of LOFT related to the determination of dense matter EOS are summarized in Feroci et al. (2012). By making advantage of its unique throughput, LOFT will also be able to test General Relativistic effects produced by the strong gravitational field of a black hole onto accreting matter orbiting close-by (a few r g ). One of the most promising way to carry out such investigations is to observe quasi-periodic oscillations (QPOs) arising from the X-ray emission of the accretion flow. The dynamical time-scales of the inner accretion flow are typically of few milliseconds, and thus the interpretation of the highest frequency QPOs involves fundamental frequencies of the motion of matter orbiting in disk regions dominated by the strong black hole gravitational field (e.g., the relativistic radial and vertical epicyclic frequencies or the relativistic nodal and periastron precession). Distinguishing among these possibilities have been proved impossible so far, due to the limited observation capabilities of the previous generation of X-ray timing instruments. The enhanced capabilities of the LAD will permit to dramatically increase the S/N of any of the QPO feature (see right panel of Fig. 3), allowing to measure QPOs accurately within their coherence time and follow their evolution in time with the source X-ray flux (see Fig. 4). This will secure access to still untested General relativistic effects, such as frame-dragging, strong-field periastron precession, and the existence of an innermost stable orbit around black holes. An independent investigation of the motion of the accretion flow orbiting close to the central black hole can be carried out through the observation of the fluorescence Fe K line profile emitted by this material at different orbital phases, by making advantages of the fine resolution of the LAD. Variation in the line profile are expected as a consequence of the Lense-Thirring precession of the inner disk (at ∼ r g ), and provide a tool to measure mass and spin of Galactic and extra-galactic black holes. In Fig. 5, we show a simulation of such a study in the Active Galactic Nuclei (AGN) MGC6-30. A summary of the top-level science goals of LOFT with respect to the measure of General Relativistic effects can be found in Feroci et al. (2012).\",\n \"pages\": [\n 4,\n 5,\n 6\n ]\n },\n {\n \"title\": \"3.2. Observatory Science\",\n \"content\": \"About 50% of the total LOFT observational time will be made available for observatory science. The breakthrough capabilities of the LAD will revolutionize X-ray observations of any relatively bright source down to a limiting flux of few mCrab. Providing an overview of the entire potential discovery space of the LAD is not possible in this relatively short contribution. Instead we provide here some more examples of the capabilities of the WFM for the observatory science. This instrument combines a very wide FOV, covering more than 1/3 of the sky at once, together with a fine timing (few µ s) and spectral resolution ( < 300 eV). Through the LBAS system, the position and trigger time of bright events can be broadcast to the ground within a delay of /lessorsimilar 30 s, allowing for quick follow-up with ground-based facilities. Due to the large FOV, about 150 GRB and thousands typeI X-ray bursts are expected to be detected with the WFM per year. The low energy threshold and fine spectral resolution of the instrument will allow to investigate in detail any spectral feature expected during these events, including absorption edges. We show two specific simulations in Fig. 6. As for the LAD, the range of scientific investigations available to the WFM cannot be sufficiently summarized here. As an example, we refer the reader to Romano et al. (2012) and Ferrigno et al. (2012) for a description of the WFM contribution to the improvement in our knowledge of the Be X-ray binaries and Supergiant Fast X-ray Transients.\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Amati, L. et al. 2000, Science, 290, 953 Abramowicz, M.A. & Kluzniak, W. 2001, A&A, 374, 19 Brandt, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-88, 2012 Campana, R. et al. 2011, NIMPA, 633, 22 Campana, R. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-209, 2012 Donnarumma, I. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-211, 2012 Evangelista, Y. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-85, 2012 Feroci, M. et al. 2007, NIMPA, 581, 728 Feroci, M. et al. 2011, ExA, in press (arXiv:1107.0436) Feroci, M. et al. 2012, Proceedings of SPIE, Vol. 8443, Paper No. 8443-210, 2012 Ferrigno, C. et al. 2012, Proceedings of COSPAR, in preparation in't Zand, J. et al. 2010, A&A, 520, 81 Miller, J.M. et al. 2001, ApJ, 563, 928 Romano, P. et al. 2012, AIP, in preparation Stella, L. & Vietri 1999, ApJ, 524, 63 Zane, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-87, 2012 Weimberg, . et al., 2012 ApJ, 639, 1018\",\n \"pages\": [\n 7,\n 8\n ]\n }\n]"}}},{"rowIdx":275,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..290..237K"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1304.7581.pdf"},"content":{"kind":"string","value":"\n\n\n\n\n\n\nThe X-ray spectro-timing properties of a major radio flare episode in Cygnus X-3\nKarri Koljonen ∗ †\nAalto University Metsähovi Observatory, Finland\nE-mail: karri.koljonen@gmail.com\nMichael McCollough\nSmithsonian Astrophysical Observatory, USA\nE-mail: mmccollough@head.cfa.harvard.edu\nDiana Hannikainen\nFlorida Institute of Technology, USA\nE-mail: ddcarina@gmail.com\nRobert Droulans\nLycée classique d'Echternach, Luxemburg\nE-mail: robert.droulans@education.lu\nWe have performed a principal component analysis on the X-ray spectra of the microquasar Cygnus X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event in 2006 MayJuly. The analysis showed that there are two main variability components in play, i.e. two principal components explained almost all the variability in the X-ray lightcurves. According to the spectral shape of these components and spectral fits to the original data, the most probable emission components corresponding to the principal components are inverse-Compton scattering and bremsstrahlung. We find that these components form a double-peaked profile when phase-folded with the peaks occurring in opposite phases. This could be due to an asymmetrical wind around the companion star with which the compact object is interacting.\nAn INTEGRAL view of the high-energy sky (the first 10 years) - 9th INTEGRAL Workshop and celebration of the 10th anniversary of the launch\n15-19 October 2012\nBibliotheque Nationale de France, Paris, France\n1. Introduction\nModeling the X-ray spectra of X-ray binaries (XRBs) often leads to a problem of degeneracy, i.e. multiple distinct models fit the observed data equally well. A striking example can be seen e.g. in [1], where three very different models are fitted equally well to the same data set of Cygnus X-1, despite the excellent quality of the data that was obtained by all the X-ray satellites in orbit at the time. Similar spectral degeneracy was observed for Cygnus X-3 in [2]. Therefore, even if an apparently good fit is obtained between the data and the model, it does not necessarily imply a match between theory and physical reality.\nIn order to make sense of this degeneracy we need to take other data dimensions into account, namely timing and/or polarization. While we are just reaching the point where the polarization dimension can be explored with very long exposure observations [3], X-ray timing data is readily available, and several methods have been developed to combine spectral and timing analyses (e.g. [4]). These comparisons, however, do not reveal what the actual spectral components causing the variability are .\nOne way of revealing the variability components of X-ray spectra is to employ principal component analysis (PCA), one of the standard tools of time series analysis that has been introduced in the analysis of the X-ray data of XRBs by [5] and references therein, and further refined in [6] for Cyg X-3 to single out individual emission components causing the variability in the X-ray lightcurves.\n2. Principal component analysis\nBelow, we summarize the main points of the PCA. More detail can be found from [5] and [6]. The first step of the PCA is to arrange a number of spectra measured at different times, i.e. a time series of X-ray spectra, to a data matrix. Then the eigenvectors of the covariance matrix calculated from the original data matrix form the principal components that are responsible for the spectral variation. The accompanying eigenvalue states the proportion of variance of a particular eigenvector, i.e. the highest eigenvalue and accompanying eigenvector is the first principal component of the data set etc. One can then form a linear decomposition of the data set using these eigenvectors, ordered by the proportion of variance. The components producing only a small fraction of the overall variance can then be dropped, reducing the dimensionality, i.e. choosing a small number of eigenvectors for the linear composition. The PCA can be exploited in different ways using both spectral and timing information. In the spectral domain this includes producing the variability spectra and log-eigenvalue diagrams. In the timing domain we can follow the evolution of individual principal components by tracking the value of its normalization.\n3. Applying PCA to Cygnus X-3 data\nCyg X-3 is one of the most peculiar sources amongst microquasars. It is known for massive outbursts that emit radiation from radio to g -rays and produce major radio flaring episodes usually with multiple flares that peak up to 20 Jy [7], making it the most radio luminous single object in our Galaxy. The binary components of Cyg X-3 orbit each other in a tight 4.8-hour period [8], typical\n\n\n
Figure 1: The figure show the one-day integrated Ryle/AMI-LA 15 GHz lightcurves from the May 2006 major radio flare. The top part of the panel shows the epochs of the RXTE (black), Swift (orange) and INTEGRAL (magenta) data that are examined here.
\n\nfor XRBs with a low-mass companion. However, infrared spectral observations suggest that the mass-donating companion in the binary is a massive Wolf-Rayet star [9]. Due to this discrepancy Cyg X-3 is by definition a unique source and similar sources have only been found in other nearby galaxies [10]. The X-ray spectral and timing properties of Cyg X-3 show a disparity with other XRBs/microquasars increasing the difficulty of interpreting the nature of the system. This disparity could be due to the interaction of the strong stellar wind of the Wolf-Rayet companion with the compact object.\nWe have performed a PCA to the X-ray spectra of Cyg X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event. The X-ray observations were obtained during the 2006 May major radio flaring episode, that consisted of two major radio flares with peak flux densities of 13.8 Jy (15 GHz) and 11.2 Jy (11.2 GHz) at MJD 53865 and MJD 53942, respectively (see Fig. 1 for the May 2006 flare). For RXTE and Swift data we used the consecutive observations taken right after a flare and for INTEGRAL we used revolutions 437, 438 and 462. The radio data are from [11].\n3.1 PCA tools of the trade\nThe standard tool for deciding how many principal components one should retain is called the log-eigenvalue (LEV) diagram (upper panels in Fig. 2). If the principal components decay in geometric progression in the data, the corresponding eigenvalues will appear as a straight line in the LEV diagram and thus signal the start of the 'noise\" components. The variance spectrum (lower panels in Fig. 2) is a graph that shows the measured variance as a function of energy. It can be plotted for all the principal components, i.e. showing the overall variance across the energy range of the data, or for each principal component independently thus showing the spectral shape of the varying principal components. From Fig. 2 we see that there are two main variability components in play in the Cyg X-3 data, i.e. two principal components explain almost all the variability in the X-ray lightcurves.\n\n\n
Figure 2: Upper panels: The LEV diagrams of the principal components from all X-ray observatories (left: Swift , center: RXTE , right: INTEGRAL ). The panels show the proportion of variance attributed to each principal component. Lower panels: The variance spectra of all observations (black solid line with data points) from all observatories. The colored curves and areas in the figure show the contributions of the principal components a 1 (red), a 2 (blue), a 3 (green) and the remainder of the components totaling to noise and systematic errors (orange).
\n\n3.2 The nature of the components\nAccording to the variance spectral shape of the principal components, a number of models can be 'guessed' and then fitted to the time-averaged data. The first principal component resembles a Comptonized component and the second a thermal component 1 . Since most of the Cyg X-3 spectra are well fitted by hybrid Comptonization (e.g. [11, 2]) we assume that the first principal component is also coupled to this model. However, we need a second spectral component to fit all the spectra successfully and to satisfy the PCA. As the overall effect of the second component in the PCA and X-ray spectra is smaller, multiple components will fit the spectra. We found good fits when using reflection, multicolor disc blackbody, thermal bremsstrahlung, or other thermal Comptonization models in addition to hybrid Comptonization. These best-fitting models and their spectral component normalizations are then compared to the time-averaged evolution of the principal component normalizations. This imposes a second requirement for the X-ray spectral fits, so that in addition to fitting the spectra acceptably, the resulting fits also have to satisfy the spectral evolution inferred from the PCA. This extra requirement reduces greatly, if not completely, the degeneracy of simply using the results from the spectral fits in determining the emission components of the system. Thus, the most probable emission components are those producing the best correlations to the principal\ncomponent evolution and fits well the X-ray spectra.\nTo summarize, the best-fitting model in spectral and variability terms has optically thin, rather thermal, Comptonization dominating the variability throughout the X-ray regime and a thermal, rather hot ( ∼ 5 keV) plasma component producing variability in the ∼ 10-20 keV regime. We model the hybrid Comptonization component with BELM [12] and the thermal component as thermal bremsstrahlung (Fig. 3). We add to the final model a photoionized emission line model (Savolainen et al., in prep.) and multiply all with a simple absorption model from [11].\n\n\n\n\n\n
Figure 3: Left: An example of two energy spectra with the best-fitting model (black, solid line) and the individual components (Comptonization: blue; bremsstrahlung: green; iron line: orange) overlaid and labelled. Right: The scatter plot matrix of the best-fitting model satisfying the principal component evolution. The grid shows the spectral model normalizations NBELM and NBremss and the first two principal components ( a 1, a 2) with robust correlations drawn and written in the appropriate grid cells.
\n\n4. Discussion\nThe most plausible origin of the thermal component is a plasma cloud that forms as a result of the compact object colliding with the WR stellar wind [13]. This model was evoked to explain Cyg X-3's lack of high frequencies in the power spectra and the peculiar hard state X-ray spectra with ∼ 30 keV cut-off by Compton downscattering. For the plasma parameters found in [13] the thermal bremsstrahlung emission becomes a substantial source for photons which get upscattered by Comptonization in the plasma cloud.\nDue to the short period of the system we can track the principal components through phase. Both components show a double-peaked profile (Fig. 4). When relating the second principal component to the bremsstrahlung normalization (see Fig. 3) which is proportional to n e n i V , we see that a change in the density and/or the volume of the bremsstrahlung-emitting plasma is observed along the orbit. The peaks are formed opposite each other, which can be explained by a disk-like shape of the plasma and could arise if the stellar wind is asymmetric.\nSimilar thermal and hot components have also been found in other microquasars such as GRS 1915 + 105 [14, 15], SS 433 [16], and in several XRBs [17], thus raising questions such as could the thermal, hot component be something intrinsic to microquasars/XRBs. Furthermore, could the emission mechanism be the same and could this scenario be extended to disk winds.\n\n\n
Figure 4: Left: The first and second principal component phase-folded through all the RXTE data. Right: A sketch depicting the geometry of the system, with the Wolf-Rayet companion surrounded by a disc-like stellar wind and the companion object orbiting it (shown in four different orbital phases). Depending on the orbital phase the compact object is either inside (enhanced bremsstrahlung emission) or outside the stellar wind.
\n\nReferences\n\n[1] Nowak M. A., et al., 2011, ApJ, 728, 13\n[2] Hjalmarsdotter L., Zdziarski A. A., Larsson S., Beckmann V., McCollough M., Hannikainen D. C., Vilhu O., 2008, MNRAS, 384, 278\n[3] Laurent P., Rodriguez J., Wilms J., Cadolle Bel M., Pottschmidt K., Grinberg V., 2011, Sci, 332, 438\n[4] Vaughan B. A. & Nowak M. A, 1997, ApJ, 474, L43\n[5] Malzac J., et al., 2006, A&A, 448, 1125\n[6] Koljonen K. I. I., McCollough M. L., Hannikainen D. C., Droulans R., 2013, MNRAS, 429, 1173\n[7] Waltman E. B. et al., 1995, AJ, 110, 290\n[8] Parsignault D. R. et al., 1972, Nat. Phys. Sci., 239, 123\n[9] van Kerkwijk M. H. et al., 1992, Nat, 355, 703\n[10] Bulik T., Belczynski K., Prestwich A., 2011, ApJ, 730, 140\n[11] Koljonen K. I. I., Hannikainen D. C., McCollough M. L., Pooley G. G., Trushkin S. A., 2010, MNRAS, 406, 307\n[12] Belmont, R., Malzac, J., Marcowith, A., 2008, A&A, 491, 617\n[13] Zdziarski, A. A., Misra, R., Gierlinski, M., 2010, MNRAS, 402, 767\n[14] Titarchuk, L. & Seifina, E., 2009, ApJ, 706, 1463\n[15] Mineo, T. et al., 2012, A&A, 537, 18\n[16] Seifina, E., & Titarchuk, L., 2010, ApJ, 722, 586\n[17] Z˙ycki, P. T., Done, C., Smith, D. A., 2001, MNRAS, 326, 1367\n\n"},"sections":{"kind":"list like","value":[{"title":"Karri Koljonen ∗ †","content":"Aalto University Metsähovi Observatory, Finland E-mail: karri.koljonen@gmail.com","pages":[1]},{"title":"Michael McCollough","content":"Smithsonian Astrophysical Observatory, USA E-mail: mmccollough@head.cfa.harvard.edu","pages":[1]},{"title":"Diana Hannikainen","content":"Florida Institute of Technology, USA E-mail: ddcarina@gmail.com","pages":[1]},{"title":"Robert Droulans","content":"Lycée classique d'Echternach, Luxemburg E-mail: robert.droulans@education.lu We have performed a principal component analysis on the X-ray spectra of the microquasar Cygnus X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event in 2006 MayJuly. The analysis showed that there are two main variability components in play, i.e. two principal components explained almost all the variability in the X-ray lightcurves. According to the spectral shape of these components and spectral fits to the original data, the most probable emission components corresponding to the principal components are inverse-Compton scattering and bremsstrahlung. We find that these components form a double-peaked profile when phase-folded with the peaks occurring in opposite phases. This could be due to an asymmetrical wind around the companion star with which the compact object is interacting. An INTEGRAL view of the high-energy sky (the first 10 years) - 9th INTEGRAL Workshop and celebration of the 10th anniversary of the launch 15-19 October 2012 Bibliotheque Nationale de France, Paris, France","pages":[1]},{"title":"1. Introduction","content":"Modeling the X-ray spectra of X-ray binaries (XRBs) often leads to a problem of degeneracy, i.e. multiple distinct models fit the observed data equally well. A striking example can be seen e.g. in [1], where three very different models are fitted equally well to the same data set of Cygnus X-1, despite the excellent quality of the data that was obtained by all the X-ray satellites in orbit at the time. Similar spectral degeneracy was observed for Cygnus X-3 in [2]. Therefore, even if an apparently good fit is obtained between the data and the model, it does not necessarily imply a match between theory and physical reality. In order to make sense of this degeneracy we need to take other data dimensions into account, namely timing and/or polarization. While we are just reaching the point where the polarization dimension can be explored with very long exposure observations [3], X-ray timing data is readily available, and several methods have been developed to combine spectral and timing analyses (e.g. [4]). These comparisons, however, do not reveal what the actual spectral components causing the variability are . One way of revealing the variability components of X-ray spectra is to employ principal component analysis (PCA), one of the standard tools of time series analysis that has been introduced in the analysis of the X-ray data of XRBs by [5] and references therein, and further refined in [6] for Cyg X-3 to single out individual emission components causing the variability in the X-ray lightcurves.","pages":[2]},{"title":"2. Principal component analysis","content":"Below, we summarize the main points of the PCA. More detail can be found from [5] and [6]. The first step of the PCA is to arrange a number of spectra measured at different times, i.e. a time series of X-ray spectra, to a data matrix. Then the eigenvectors of the covariance matrix calculated from the original data matrix form the principal components that are responsible for the spectral variation. The accompanying eigenvalue states the proportion of variance of a particular eigenvector, i.e. the highest eigenvalue and accompanying eigenvector is the first principal component of the data set etc. One can then form a linear decomposition of the data set using these eigenvectors, ordered by the proportion of variance. The components producing only a small fraction of the overall variance can then be dropped, reducing the dimensionality, i.e. choosing a small number of eigenvectors for the linear composition. The PCA can be exploited in different ways using both spectral and timing information. In the spectral domain this includes producing the variability spectra and log-eigenvalue diagrams. In the timing domain we can follow the evolution of individual principal components by tracking the value of its normalization.","pages":[2]},{"title":"3. Applying PCA to Cygnus X-3 data","content":"Cyg X-3 is one of the most peculiar sources amongst microquasars. It is known for massive outbursts that emit radiation from radio to g -rays and produce major radio flaring episodes usually with multiple flares that peak up to 20 Jy [7], making it the most radio luminous single object in our Galaxy. The binary components of Cyg X-3 orbit each other in a tight 4.8-hour period [8], typical for XRBs with a low-mass companion. However, infrared spectral observations suggest that the mass-donating companion in the binary is a massive Wolf-Rayet star [9]. Due to this discrepancy Cyg X-3 is by definition a unique source and similar sources have only been found in other nearby galaxies [10]. The X-ray spectral and timing properties of Cyg X-3 show a disparity with other XRBs/microquasars increasing the difficulty of interpreting the nature of the system. This disparity could be due to the interaction of the strong stellar wind of the Wolf-Rayet companion with the compact object. We have performed a PCA to the X-ray spectra of Cyg X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event. The X-ray observations were obtained during the 2006 May major radio flaring episode, that consisted of two major radio flares with peak flux densities of 13.8 Jy (15 GHz) and 11.2 Jy (11.2 GHz) at MJD 53865 and MJD 53942, respectively (see Fig. 1 for the May 2006 flare). For RXTE and Swift data we used the consecutive observations taken right after a flare and for INTEGRAL we used revolutions 437, 438 and 462. The radio data are from [11].","pages":[2,3]},{"title":"3.1 PCA tools of the trade","content":"The standard tool for deciding how many principal components one should retain is called the log-eigenvalue (LEV) diagram (upper panels in Fig. 2). If the principal components decay in geometric progression in the data, the corresponding eigenvalues will appear as a straight line in the LEV diagram and thus signal the start of the 'noise\" components. The variance spectrum (lower panels in Fig. 2) is a graph that shows the measured variance as a function of energy. It can be plotted for all the principal components, i.e. showing the overall variance across the energy range of the data, or for each principal component independently thus showing the spectral shape of the varying principal components. From Fig. 2 we see that there are two main variability components in play in the Cyg X-3 data, i.e. two principal components explain almost all the variability in the X-ray lightcurves.","pages":[3]},{"title":"3.2 The nature of the components","content":"According to the variance spectral shape of the principal components, a number of models can be 'guessed' and then fitted to the time-averaged data. The first principal component resembles a Comptonized component and the second a thermal component 1 . Since most of the Cyg X-3 spectra are well fitted by hybrid Comptonization (e.g. [11, 2]) we assume that the first principal component is also coupled to this model. However, we need a second spectral component to fit all the spectra successfully and to satisfy the PCA. As the overall effect of the second component in the PCA and X-ray spectra is smaller, multiple components will fit the spectra. We found good fits when using reflection, multicolor disc blackbody, thermal bremsstrahlung, or other thermal Comptonization models in addition to hybrid Comptonization. These best-fitting models and their spectral component normalizations are then compared to the time-averaged evolution of the principal component normalizations. This imposes a second requirement for the X-ray spectral fits, so that in addition to fitting the spectra acceptably, the resulting fits also have to satisfy the spectral evolution inferred from the PCA. This extra requirement reduces greatly, if not completely, the degeneracy of simply using the results from the spectral fits in determining the emission components of the system. Thus, the most probable emission components are those producing the best correlations to the principal component evolution and fits well the X-ray spectra. To summarize, the best-fitting model in spectral and variability terms has optically thin, rather thermal, Comptonization dominating the variability throughout the X-ray regime and a thermal, rather hot ( ∼ 5 keV) plasma component producing variability in the ∼ 10-20 keV regime. We model the hybrid Comptonization component with BELM [12] and the thermal component as thermal bremsstrahlung (Fig. 3). We add to the final model a photoionized emission line model (Savolainen et al., in prep.) and multiply all with a simple absorption model from [11].","pages":[4,5]},{"title":"4. Discussion","content":"The most plausible origin of the thermal component is a plasma cloud that forms as a result of the compact object colliding with the WR stellar wind [13]. This model was evoked to explain Cyg X-3's lack of high frequencies in the power spectra and the peculiar hard state X-ray spectra with ∼ 30 keV cut-off by Compton downscattering. For the plasma parameters found in [13] the thermal bremsstrahlung emission becomes a substantial source for photons which get upscattered by Comptonization in the plasma cloud. Due to the short period of the system we can track the principal components through phase. Both components show a double-peaked profile (Fig. 4). When relating the second principal component to the bremsstrahlung normalization (see Fig. 3) which is proportional to n e n i V , we see that a change in the density and/or the volume of the bremsstrahlung-emitting plasma is observed along the orbit. The peaks are formed opposite each other, which can be explained by a disk-like shape of the plasma and could arise if the stellar wind is asymmetric. Similar thermal and hot components have also been found in other microquasars such as GRS 1915 + 105 [14, 15], SS 433 [16], and in several XRBs [17], thus raising questions such as could the thermal, hot component be something intrinsic to microquasars/XRBs. Furthermore, could the emission mechanism be the same and could this scenario be extended to disk winds.","pages":[5]}],"string":"[\n {\n \"title\": \"Karri Koljonen ∗ †\",\n \"content\": \"Aalto University Metsähovi Observatory, Finland E-mail: karri.koljonen@gmail.com\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Michael McCollough\",\n \"content\": \"Smithsonian Astrophysical Observatory, USA E-mail: mmccollough@head.cfa.harvard.edu\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Diana Hannikainen\",\n \"content\": \"Florida Institute of Technology, USA E-mail: ddcarina@gmail.com\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Robert Droulans\",\n \"content\": \"Lycée classique d'Echternach, Luxemburg E-mail: robert.droulans@education.lu We have performed a principal component analysis on the X-ray spectra of the microquasar Cygnus X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event in 2006 MayJuly. The analysis showed that there are two main variability components in play, i.e. two principal components explained almost all the variability in the X-ray lightcurves. According to the spectral shape of these components and spectral fits to the original data, the most probable emission components corresponding to the principal components are inverse-Compton scattering and bremsstrahlung. We find that these components form a double-peaked profile when phase-folded with the peaks occurring in opposite phases. This could be due to an asymmetrical wind around the companion star with which the compact object is interacting. An INTEGRAL view of the high-energy sky (the first 10 years) - 9th INTEGRAL Workshop and celebration of the 10th anniversary of the launch 15-19 October 2012 Bibliotheque Nationale de France, Paris, France\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Modeling the X-ray spectra of X-ray binaries (XRBs) often leads to a problem of degeneracy, i.e. multiple distinct models fit the observed data equally well. A striking example can be seen e.g. in [1], where three very different models are fitted equally well to the same data set of Cygnus X-1, despite the excellent quality of the data that was obtained by all the X-ray satellites in orbit at the time. Similar spectral degeneracy was observed for Cygnus X-3 in [2]. Therefore, even if an apparently good fit is obtained between the data and the model, it does not necessarily imply a match between theory and physical reality. In order to make sense of this degeneracy we need to take other data dimensions into account, namely timing and/or polarization. While we are just reaching the point where the polarization dimension can be explored with very long exposure observations [3], X-ray timing data is readily available, and several methods have been developed to combine spectral and timing analyses (e.g. [4]). These comparisons, however, do not reveal what the actual spectral components causing the variability are . One way of revealing the variability components of X-ray spectra is to employ principal component analysis (PCA), one of the standard tools of time series analysis that has been introduced in the analysis of the X-ray data of XRBs by [5] and references therein, and further refined in [6] for Cyg X-3 to single out individual emission components causing the variability in the X-ray lightcurves.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2. Principal component analysis\",\n \"content\": \"Below, we summarize the main points of the PCA. More detail can be found from [5] and [6]. The first step of the PCA is to arrange a number of spectra measured at different times, i.e. a time series of X-ray spectra, to a data matrix. Then the eigenvectors of the covariance matrix calculated from the original data matrix form the principal components that are responsible for the spectral variation. The accompanying eigenvalue states the proportion of variance of a particular eigenvector, i.e. the highest eigenvalue and accompanying eigenvector is the first principal component of the data set etc. One can then form a linear decomposition of the data set using these eigenvectors, ordered by the proportion of variance. The components producing only a small fraction of the overall variance can then be dropped, reducing the dimensionality, i.e. choosing a small number of eigenvectors for the linear composition. The PCA can be exploited in different ways using both spectral and timing information. In the spectral domain this includes producing the variability spectra and log-eigenvalue diagrams. In the timing domain we can follow the evolution of individual principal components by tracking the value of its normalization.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3. Applying PCA to Cygnus X-3 data\",\n \"content\": \"Cyg X-3 is one of the most peculiar sources amongst microquasars. It is known for massive outbursts that emit radiation from radio to g -rays and produce major radio flaring episodes usually with multiple flares that peak up to 20 Jy [7], making it the most radio luminous single object in our Galaxy. The binary components of Cyg X-3 orbit each other in a tight 4.8-hour period [8], typical for XRBs with a low-mass companion. However, infrared spectral observations suggest that the mass-donating companion in the binary is a massive Wolf-Rayet star [9]. Due to this discrepancy Cyg X-3 is by definition a unique source and similar sources have only been found in other nearby galaxies [10]. The X-ray spectral and timing properties of Cyg X-3 show a disparity with other XRBs/microquasars increasing the difficulty of interpreting the nature of the system. This disparity could be due to the interaction of the strong stellar wind of the Wolf-Rayet companion with the compact object. We have performed a PCA to the X-ray spectra of Cyg X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event. The X-ray observations were obtained during the 2006 May major radio flaring episode, that consisted of two major radio flares with peak flux densities of 13.8 Jy (15 GHz) and 11.2 Jy (11.2 GHz) at MJD 53865 and MJD 53942, respectively (see Fig. 1 for the May 2006 flare). For RXTE and Swift data we used the consecutive observations taken right after a flare and for INTEGRAL we used revolutions 437, 438 and 462. The radio data are from [11].\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3.1 PCA tools of the trade\",\n \"content\": \"The standard tool for deciding how many principal components one should retain is called the log-eigenvalue (LEV) diagram (upper panels in Fig. 2). If the principal components decay in geometric progression in the data, the corresponding eigenvalues will appear as a straight line in the LEV diagram and thus signal the start of the 'noise\\\" components. The variance spectrum (lower panels in Fig. 2) is a graph that shows the measured variance as a function of energy. It can be plotted for all the principal components, i.e. showing the overall variance across the energy range of the data, or for each principal component independently thus showing the spectral shape of the varying principal components. From Fig. 2 we see that there are two main variability components in play in the Cyg X-3 data, i.e. two principal components explain almost all the variability in the X-ray lightcurves.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.2 The nature of the components\",\n \"content\": \"According to the variance spectral shape of the principal components, a number of models can be 'guessed' and then fitted to the time-averaged data. The first principal component resembles a Comptonized component and the second a thermal component 1 . Since most of the Cyg X-3 spectra are well fitted by hybrid Comptonization (e.g. [11, 2]) we assume that the first principal component is also coupled to this model. However, we need a second spectral component to fit all the spectra successfully and to satisfy the PCA. As the overall effect of the second component in the PCA and X-ray spectra is smaller, multiple components will fit the spectra. We found good fits when using reflection, multicolor disc blackbody, thermal bremsstrahlung, or other thermal Comptonization models in addition to hybrid Comptonization. These best-fitting models and their spectral component normalizations are then compared to the time-averaged evolution of the principal component normalizations. This imposes a second requirement for the X-ray spectral fits, so that in addition to fitting the spectra acceptably, the resulting fits also have to satisfy the spectral evolution inferred from the PCA. This extra requirement reduces greatly, if not completely, the degeneracy of simply using the results from the spectral fits in determining the emission components of the system. Thus, the most probable emission components are those producing the best correlations to the principal component evolution and fits well the X-ray spectra. To summarize, the best-fitting model in spectral and variability terms has optically thin, rather thermal, Comptonization dominating the variability throughout the X-ray regime and a thermal, rather hot ( ∼ 5 keV) plasma component producing variability in the ∼ 10-20 keV regime. We model the hybrid Comptonization component with BELM [12] and the thermal component as thermal bremsstrahlung (Fig. 3). We add to the final model a photoionized emission line model (Savolainen et al., in prep.) and multiply all with a simple absorption model from [11].\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"4. Discussion\",\n \"content\": \"The most plausible origin of the thermal component is a plasma cloud that forms as a result of the compact object colliding with the WR stellar wind [13]. This model was evoked to explain Cyg X-3's lack of high frequencies in the power spectra and the peculiar hard state X-ray spectra with ∼ 30 keV cut-off by Compton downscattering. For the plasma parameters found in [13] the thermal bremsstrahlung emission becomes a substantial source for photons which get upscattered by Comptonization in the plasma cloud. Due to the short period of the system we can track the principal components through phase. Both components show a double-peaked profile (Fig. 4). When relating the second principal component to the bremsstrahlung normalization (see Fig. 3) which is proportional to n e n i V , we see that a change in the density and/or the volume of the bremsstrahlung-emitting plasma is observed along the orbit. The peaks are formed opposite each other, which can be explained by a disk-like shape of the plasma and could arise if the stellar wind is asymmetric. Similar thermal and hot components have also been found in other microquasars such as GRS 1915 + 105 [14, 15], SS 433 [16], and in several XRBs [17], thus raising questions such as could the thermal, hot component be something intrinsic to microquasars/XRBs. Furthermore, could the emission mechanism be the same and could this scenario be extended to disk winds.\",\n \"pages\": [\n 5\n ]\n }\n]"}}},{"rowIdx":276,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..291...61W"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1210.1910.pdf"},"content":{"kind":"string","value":"\nStructure of Quark Stars\nFridolin Weber 1 , Milva Orsaria 2 , † Hilario Rodrigues 3 ‡ and Shu-Hua Yang 4 ¶\nDepartment of Physics, San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA 1\nemail: fweber@mail.sdsu.edu\n2 email: morsaria@fcaglp.fcaglp.unlp.edu.ar 3 email: harg@cefet-rj.br 4 email: ysh@phy.ccnu.edu.cn\nAbstract. This paper gives an brief overview of the structure of hypothetical strange quarks stars (quark stars, for short), which are made of absolutely stable 3-flavor strange quark matter. Such objects can be either bare or enveloped in thin nuclear crusts, which consist of heavy ions immersed in an electron gas. In contrast to neutron stars, the structure of quark stars is determined by two (rather than one) parameters, the central star density and the density at the base of the crust. If bare, quark stars possess ultra-high electric fields on the order of 10 18 to 10 19 V/cm. These features render the properties of quark stars more multifaceted than those of neutron stars and may allow one to observationally distinguish quark stars from neutron stars.\nKeywords. neutron stars, quark stars, pulsars, strange quark matter, equation of state\n1. Introduction\nThe theoretical possibility that quark matter made of up, down and strange quarks (so-called strange quark matter (Farhi & Jaffe 1984)) may be more stable than ordinary nuclear matter has been pointed out by Bodmer (1971), Terazawa (1979), and Witten (1984). This so-called strange matter hypothesis constitutes one of the most startling possibilities regarding the behavior of superdense matter, which, if true, would have implications of fundamental importance for cosmology, the early universe, its evolution to the present day, and astrophysical compact objects such as neutron stars and white dwarfs (see Alcock & Farhi 1986, Alcock & Olinto 1988, Aarhus 1991, Weber 1999, Madsen 1999, Glendenning 2000, Weber 2005, Page & Reddy 2006, Sagert et al. 2006, and references therein). The properties of quark stars are compared with those of neutron stars in Table 1 and Fig. 1. Even to the present day there is no sound scientific basis on which one can either confirm or reject the hypothesis so that it is a serious possibility of fundamental significance for various (astro) physical phenomena.\nThe multifaceted properties of these objects are reviewed in this paper. Particular emphasis is is put on stellar properties such as rapid rotation, ultra-high electric surface fields, and rotational vortex expulsion, which may allow one to observationally discriminate between quark stars and neutron stars and-ultimately-prove or disprove the strange\n\n† Home address: CONICET, Rivadavia 1917, 1033 Buenos Aires; Gravitation, Astrophysics and Cosmology Group, Facultad de Ciencias Astron'omicas y Geofisicas, Paseo del Bosque S/N (1900), Universidad Nacional de La Plata UNLP, La Plata, Argentina.\n‡ Home address: Centro Federal de Educa¸c˜ao Tecnol'ogica do Rio de Janeiro, Av Maracan˜ a 249, 20271-110, Rio de Janeiro, RJ, Brazil.\n¶ Home address: Institute of Astrophysics, Huazhong Normal University, Wuhan, 430079, P. R. China.\n\n
\n\n
Table 1. Theoretical properties of quark stars and neutron stars compared.
\n
\nquark matter hypothesis. Futher information on the existence of qark stars may come from quark novae, hypothetical types of supernovae which could occur if neutron stars spontaneously collapse to quark stars (Ouyed et al. 2002).\n2. Quark-Lepton Composition of Quark Stars\nQuark star matter is composed of the three lightest quark flavor states (up, down, and strange quarks). Per hypothesis, the energy per baryon of such matter is lower than the energy per baryon of the most stable atomic nucleus, 56 Fe. Since stars in their lowest energy state are electrically charge neutral to very high precision, any net positive quark charge must be balanced by electrons. The concentration of electrons is largest at low densities due to the finite strange-quark mass, which leads to a deficit of net negative quark charge. If quark star matter forms a color superconductor (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008, and references therein) in the Color-Flavor-Locked (CFL) phase the interiors of quarks stars will be rigorously electrically neutral with no need for electrons, as shown by Rajagopal & Wilczek (2001). For sufficiently large strange quark masses, however, the low density regime of quark star matter is rather expected to form other condensation patterns (e.g. 2SC, CFLK 0 , CFLK + , CFLπ 0 , -) in which electrons will be present (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008). The presence of electrons in quark star matter is crucial for the possible existence of a nuclear crust on quark stars. As shown by Alcock et al. (1986), Kettner et al. (1995), and Alcock & Olinto (1988), the electrons, because they are bound to strange matter by the Coulomb force rather than the strong force, extend several hundred fermi beyond the surface of the strange star. Associated with this electron displacement is a electric dipole layer which can support, out of contact with the surface of the strange star, a crust\n\n\n
Figure 1. Schematic structures of quark stars and neutron stars.
\n\nof nuclear material, which it polarizes (Alcock et al. 1986, Alcock & Olinto 1988). The maximum possible density at the base of the crust (inner crust density) is determined by neutron drip, which occurs at about 4 . 3 × 10 11 g / cm 3 .\n3. Bare versus Dressed Quark Stars and Eddington Limit\nA bare quark star differs qualitatively from a neutron star which has a density at the surface of about 0.1 to 1 g / cm 3 . The thickness of the quark surface is just ∼ 1 fm, the length scale of the strong interaction. The electrons at the surface of a quark star are held to quark matter electrostatically, and the thickness of the electron surface is several hundred fermis. Since neither component, electrons and quark matter, is held in place gravitationally, the Eddington limit to the luminosity that a static surface may emit does not apply, so that bare quark stars may have photon luminosities much greater than 10 38 erg / s. It was shown by Usov (1998) that this value may be exceeded by many orders of magnitude by the luminosity of e + e -pairs produced by the Coulomb barrier at the surface of a hot strange star. For a surface temperature of ∼ 10 11 K, the luminosity in the outflowing pair plasma was calculated to be as high as ∼ 3 × 10 51 erg / s. Such an effect may be a good observational signature of bare strange stars (Usov 2001a, Usov 2001b, Usov 1998, and Cheng & Harko 2003). If the strange star is enveloped in a nuclear crust, however, which is gravitationally bound to the strange star, the surface, made of ordinary atomic matter, would be subject to the Eddington limit. Hence the photon emissivity of such a 'dressed' quark star would be the same as for an ordinary neutron star. If quark matter at the stellar surface is in the CFL phase the process of e + e -pair creation at the stellar quark matter surface may be turned off. This may be different for the early stages of a very hot CFL quark star (Vogt et al. 2004).\n4. Mass-Radius Relationship of Quark Stars\nThe mass-radius relationship of bare quark stars is shown in Fig. 2. In contrast to neutron stars, the radii of self-bound quark stars decrease the lighter the stars, according to M ∝ R 3 . The existence of nuclear crusts on quark stars changes the situation drastically (Glendenning et al. 1995, Weber 1999, and Weber 2005). Since the crust is bound gravitationally, the mass-radius relationship of quark stars with crusts is then qualitatively similar to neutron stars.\nIn general, quark stars with or without nuclear crusts possess smaller radii than neutron\n/s32\n\n\n
Figure 2. Mass-radius relationship of bare quark stars (from Orsaria et al. 2011).
\n\nstars. This feature implies that quark stars posses smaller mass shedding periods than neutron stars. Due to the smaller radii of quarks stars, the complete sequence of such objects-and not just those close to the mass peak, as it is the case for neutron stars-can sustain extremely rapid rotation (Glendenning et al. 1995, Weber 1999, and Weber 2005). In particular, a strange star with a typical pulsar mass of around 1 . 45 M /circledot can rotate at Kepler (mass shedding) periods as small as 0 . 55 < ∼ P K / msec < ∼ 0 . 8 (Glendenning & Weber 1992, and Glendenning et al. 1995). This range is to be compared with P K ∼ 1 msec obtained for neutron stars of the same mass (Weber 1999).\nAnother novelty of the strange quark matter hypothesis concerns the existence of a new class of white-dwarf-like objects, referred to as strange (quark matter) dwarfs (Glendenning et al. 1995). The mass-radius relationship of the latter may differs somewhat from the mass-radius relationship of ordinary white-dwarf, which may be testable in the future. Until recently, only rather vague tests of the theoretical mass-radius relation of white dwarfs were possible. This has changed dramatically because of the availability of new data emerging from the Hipparcos project (Provencal 1998). These data allow the first accurate measurements of white dwarf distances and, as a result, establishing the mass-radius relation of such objects empirically.\n5. Pulsar Glitches\nOf considerable relevance for the viability of the strange matter hypothesis is the question of whether strange stars can exhibit glitches in rotation frequency. From the study performed by Glendenning & Weber (1992) and Zdunik et al. (2001) it is known that the ratio of the crustal moment of inertia to the total moment of inertia varies between 10 -3 and ∼ 10 -5 . If the angular momentum of the pulsar is conserved in a stellar quake one obtains for the change of the star's frequency ∆Ω / Ω ∼ (10 -5 -10 -3 ) f , where 0 < f < 1 (Glendenning & Weber 1992). The factor f represents the fraction of the crustal moment of inertia that is altered in the quake. Since the observed glitches have relative frequency changes ∆Ω / Ω = (10 -9 -10 -6 ), a change in the crustal moment of inertia of f < ∼ 0 . 1 would cause a giant glitch (Glendenning & Weber 1992). Moreover it turns out that the observed range of the fractional change in the spin-down rate, ˙ Ω, is consistent with the crust having a small moment of inertia and the quake involving only a small fraction f of that. For this purpose we write ∆ ˙ Ω / ˙ Ω > (10 -1 to 10) f (Glendenning & Weber 1992). This relation yields a small f value, i.e., f < (10 -4 to 10 -1 ), in agree-\n/s32\nment with f < ∼ 0 . 1 established just above. For these estimates, the measured values of (∆Ω / Ω) / (∆ ˙ Ω / ˙ Ω) ∼ 10 -6 to 10 -4 for Crab and Vela, respectively, have been used.\n6. Possible Connection to CCOs\nOne of the most amazing features of quark stars concerns the possible existence of ultra-high electric fields on their surfaces, which, for ordinary quark matter, is around 10 18 V/cm. If strange matter forms a color superconductor, as expected for such matter, the strength of the electric field may increase to values that exceed 10 19 V/cm. The energy density associated with such huge electric fields is on the same order of magnitude as the energy density of strange matter itself, which alters the masses and radii of strange quark stars at the 15% and 5% level, respectively (Negreiros et al. 2009). Such mass increases facilitate the interpretation of massive compact stars, with masses of around 2 M /circledot , as strange quark stars (see also Rodrigues et al. 2011).\nThe electrons at the surface of a quark star are not necessarily in a fixed position but may rotate with respect to the quark matter star (Negreiros et al. 2010). In this event magnetic fields can be generated which, for moderate effective rotational frequencies between the electron layer and the stellar body, agree with the magnetic fields inferred for several Compact Central Objects (CCOs). These objects could thus be interpreted as quark stars whose electron atmospheres rotate at frequencies that are moderately different ( ∼ 10 Hz) from the rotational frequency of the quark star itself.\nLast but not least, we mention that the electron surface layer may be strongly affected by the magnetic field of a quark star in such a way that the electron layer performs vortex hydrodynamical oscillations (Xu et al. 2012). The frequency spectrum of these oscillations has been derived in analytic form by Xu et al. (2012). If the thermal Xray spectra of quark stars are modulated by vortex hydrodynamical oscillations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model (Xu et al. 2012).\nA study which looks at the thermal evolution of CCOs is presently being carried out by Yang et al. (2012). Preliminary results indicate that the observed temperatures of CCOs can be well reproduced if one assumes that these objects are small quark matter objects with radii less than around 3 km.\n7. Possible Connection to SGRs, AXPs, and XDINs\nIf quarks stars are made of color superconducting quark matter rather than normal non-superconducting quark matter. If rotating, superconducting quark stars ought to be threaded with rotational vortex lines, within which the star's interior magnetic field is at least partially confined. The vortices (and thus magnetic flux) would be expelled from the star during stellar spin-down, leading to magnetic reconnection at the surface of the star and the prolific production of thermal energy. Niebergal et al. (2010) have shown that this energy release can re-heat quark stars to exceptionally high temperatures, such as observed for Soft Gamma Repeaters (SGRs), Anomalous X-Ray pulsars (AXPs), and X-ray dim isolated neutron stars (XDINs). Moreover, numerical investigations of the temperature evolution, spin-down rate, and magnetic field behavior of such superconducting\nquark stars suggest that SGRs, AXPs, and XDINs may be linked ancestrally (Niebergal et al 2010).\nAcknowledgements\nM. Orsaria thanks CONICET for financial support. H. Rodrigues thanks CAPES for financial support under contract number BEX 6379/10-9. F. Weber is supported by the National Science Foundation (USA) under Grant PHY-0854699. S.-H. Yang is supported by the China Scholarship Council (CSC) and by NFSC under Grant No. 11147170.\nReferences\nProc. Int. Workshop on Strange Quark Matter in Physics and Astrophysics 1991, J. Madsen and P. Haensel (eds.), Nucl. Phys. B (Proc. Suppl.) , 24B\nAlcock, C., Farhi, E., & Olinto, A. V. 1986, ApJ , 310, 261 Alcock, C., & Olinto, A.V. 1988, Ann. Rev. Nucl. Part. Sci. , 38, 161 Alford, M. 2001, Ann. Rev. Nucl. Part. Sci. , 51, 131 Alford, M.G., Schmitt, A., Rajagopal, K., & Schafer, T. 2008, Rev. Mod. 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Sci. , 56, 327 Provencal, J.L., Shipman, H.L., Hog, E., and Thejll, P. 1998, ApJ , 494, 759 Rajagopal K. & Wilczek, F. 2001, The Condensed Matter Physics of QCD , At the Frontier of Particle Physics/Handbook of QCD, ed. M. Shifman (World Scientific) Rajagopal, K., & Wilczek, F. 2001, Phys. Rev. Lett. , 86, 3492 Rodrigues, H., Duarte, S.B., & de Oliveira, J.C.T. 2011, ApJ , 730, 31 Sagert, I., Wietoska, M., & Schaffner-Bielich, J. 2006 J. Phys. G. , 32, S241 Terazawa, H. 1979, INS-Report-338 (INS, Univ. of Tokyo ; 1989 J. Phys. Soc. Japan , 58, 3555; 1989 ibid. , 58, 4388; 1990 ibid. , 59, 1199 Usov, V.V. 1998, Phys. Rev. Lett. , 80, 230 Usov, V.V. 2001, ApJ , 550, L179 Usov, V.V. 2001, ApJ , 559, L137 Vogt, C., Rapp, R., & Ouyed, R. 2004, Nucl. Phys. , A735, 543 Weber, F. 1999, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics , (IOP Publishing, Bristol, Great Britain). Weber, F. 2005, Prog. Part. Nucl. Phys. , 54, 193 Witten, E. 1984, Phys. Rev. D , 30, 272 Xu, R.X., Bastrukov, S.I., Weber, F., Yu, J.W, & Molodtsova, I.V. 2012, Phys. Rev. D , 85, 023008\nYang, S.-H., Weber, F., Negreiros, R., & Becker, W. 2012, Cooling Simulations of CCOs (in preparation)\nZdunik, J. L., Haensel, E., & Gourgoulhon, E. 2001, A&A , 372, 535\n"},"sections":{"kind":"list like","value":[{"title":"Fridolin Weber 1 , Milva Orsaria 2 , † Hilario Rodrigues 3 ‡ and Shu-Hua Yang 4 ¶","content":"Department of Physics, San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA 1 email: fweber@mail.sdsu.edu 2 email: morsaria@fcaglp.fcaglp.unlp.edu.ar 3 email: harg@cefet-rj.br 4 email: ysh@phy.ccnu.edu.cn Abstract. This paper gives an brief overview of the structure of hypothetical strange quarks stars (quark stars, for short), which are made of absolutely stable 3-flavor strange quark matter. Such objects can be either bare or enveloped in thin nuclear crusts, which consist of heavy ions immersed in an electron gas. In contrast to neutron stars, the structure of quark stars is determined by two (rather than one) parameters, the central star density and the density at the base of the crust. If bare, quark stars possess ultra-high electric fields on the order of 10 18 to 10 19 V/cm. These features render the properties of quark stars more multifaceted than those of neutron stars and may allow one to observationally distinguish quark stars from neutron stars. Keywords. neutron stars, quark stars, pulsars, strange quark matter, equation of state","pages":[1]},{"title":"1. Introduction","content":"The theoretical possibility that quark matter made of up, down and strange quarks (so-called strange quark matter (Farhi & Jaffe 1984)) may be more stable than ordinary nuclear matter has been pointed out by Bodmer (1971), Terazawa (1979), and Witten (1984). This so-called strange matter hypothesis constitutes one of the most startling possibilities regarding the behavior of superdense matter, which, if true, would have implications of fundamental importance for cosmology, the early universe, its evolution to the present day, and astrophysical compact objects such as neutron stars and white dwarfs (see Alcock & Farhi 1986, Alcock & Olinto 1988, Aarhus 1991, Weber 1999, Madsen 1999, Glendenning 2000, Weber 2005, Page & Reddy 2006, Sagert et al. 2006, and references therein). The properties of quark stars are compared with those of neutron stars in Table 1 and Fig. 1. Even to the present day there is no sound scientific basis on which one can either confirm or reject the hypothesis so that it is a serious possibility of fundamental significance for various (astro) physical phenomena. The multifaceted properties of these objects are reviewed in this paper. Particular emphasis is is put on stellar properties such as rapid rotation, ultra-high electric surface fields, and rotational vortex expulsion, which may allow one to observationally discriminate between quark stars and neutron stars and-ultimately-prove or disprove the strange quark matter hypothesis. Futher information on the existence of qark stars may come from quark novae, hypothetical types of supernovae which could occur if neutron stars spontaneously collapse to quark stars (Ouyed et al. 2002).","pages":[1,2]},{"title":"2. Quark-Lepton Composition of Quark Stars","content":"Quark star matter is composed of the three lightest quark flavor states (up, down, and strange quarks). Per hypothesis, the energy per baryon of such matter is lower than the energy per baryon of the most stable atomic nucleus, 56 Fe. Since stars in their lowest energy state are electrically charge neutral to very high precision, any net positive quark charge must be balanced by electrons. The concentration of electrons is largest at low densities due to the finite strange-quark mass, which leads to a deficit of net negative quark charge. If quark star matter forms a color superconductor (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008, and references therein) in the Color-Flavor-Locked (CFL) phase the interiors of quarks stars will be rigorously electrically neutral with no need for electrons, as shown by Rajagopal & Wilczek (2001). For sufficiently large strange quark masses, however, the low density regime of quark star matter is rather expected to form other condensation patterns (e.g. 2SC, CFLK 0 , CFLK + , CFLπ 0 , -) in which electrons will be present (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008). The presence of electrons in quark star matter is crucial for the possible existence of a nuclear crust on quark stars. As shown by Alcock et al. (1986), Kettner et al. (1995), and Alcock & Olinto (1988), the electrons, because they are bound to strange matter by the Coulomb force rather than the strong force, extend several hundred fermi beyond the surface of the strange star. Associated with this electron displacement is a electric dipole layer which can support, out of contact with the surface of the strange star, a crust of nuclear material, which it polarizes (Alcock et al. 1986, Alcock & Olinto 1988). The maximum possible density at the base of the crust (inner crust density) is determined by neutron drip, which occurs at about 4 . 3 × 10 11 g / cm 3 .","pages":[2,3]},{"title":"3. Bare versus Dressed Quark Stars and Eddington Limit","content":"A bare quark star differs qualitatively from a neutron star which has a density at the surface of about 0.1 to 1 g / cm 3 . The thickness of the quark surface is just ∼ 1 fm, the length scale of the strong interaction. The electrons at the surface of a quark star are held to quark matter electrostatically, and the thickness of the electron surface is several hundred fermis. Since neither component, electrons and quark matter, is held in place gravitationally, the Eddington limit to the luminosity that a static surface may emit does not apply, so that bare quark stars may have photon luminosities much greater than 10 38 erg / s. It was shown by Usov (1998) that this value may be exceeded by many orders of magnitude by the luminosity of e + e -pairs produced by the Coulomb barrier at the surface of a hot strange star. For a surface temperature of ∼ 10 11 K, the luminosity in the outflowing pair plasma was calculated to be as high as ∼ 3 × 10 51 erg / s. Such an effect may be a good observational signature of bare strange stars (Usov 2001a, Usov 2001b, Usov 1998, and Cheng & Harko 2003). If the strange star is enveloped in a nuclear crust, however, which is gravitationally bound to the strange star, the surface, made of ordinary atomic matter, would be subject to the Eddington limit. Hence the photon emissivity of such a 'dressed' quark star would be the same as for an ordinary neutron star. If quark matter at the stellar surface is in the CFL phase the process of e + e -pair creation at the stellar quark matter surface may be turned off. This may be different for the early stages of a very hot CFL quark star (Vogt et al. 2004).","pages":[3]},{"title":"4. Mass-Radius Relationship of Quark Stars","content":"The mass-radius relationship of bare quark stars is shown in Fig. 2. In contrast to neutron stars, the radii of self-bound quark stars decrease the lighter the stars, according to M ∝ R 3 . The existence of nuclear crusts on quark stars changes the situation drastically (Glendenning et al. 1995, Weber 1999, and Weber 2005). Since the crust is bound gravitationally, the mass-radius relationship of quark stars with crusts is then qualitatively similar to neutron stars. In general, quark stars with or without nuclear crusts possess smaller radii than neutron /s32 stars. This feature implies that quark stars posses smaller mass shedding periods than neutron stars. Due to the smaller radii of quarks stars, the complete sequence of such objects-and not just those close to the mass peak, as it is the case for neutron stars-can sustain extremely rapid rotation (Glendenning et al. 1995, Weber 1999, and Weber 2005). In particular, a strange star with a typical pulsar mass of around 1 . 45 M /circledot can rotate at Kepler (mass shedding) periods as small as 0 . 55 < ∼ P K / msec < ∼ 0 . 8 (Glendenning & Weber 1992, and Glendenning et al. 1995). This range is to be compared with P K ∼ 1 msec obtained for neutron stars of the same mass (Weber 1999). Another novelty of the strange quark matter hypothesis concerns the existence of a new class of white-dwarf-like objects, referred to as strange (quark matter) dwarfs (Glendenning et al. 1995). The mass-radius relationship of the latter may differs somewhat from the mass-radius relationship of ordinary white-dwarf, which may be testable in the future. Until recently, only rather vague tests of the theoretical mass-radius relation of white dwarfs were possible. This has changed dramatically because of the availability of new data emerging from the Hipparcos project (Provencal 1998). These data allow the first accurate measurements of white dwarf distances and, as a result, establishing the mass-radius relation of such objects empirically.","pages":[3,4]},{"title":"5. Pulsar Glitches","content":"Of considerable relevance for the viability of the strange matter hypothesis is the question of whether strange stars can exhibit glitches in rotation frequency. From the study performed by Glendenning & Weber (1992) and Zdunik et al. (2001) it is known that the ratio of the crustal moment of inertia to the total moment of inertia varies between 10 -3 and ∼ 10 -5 . If the angular momentum of the pulsar is conserved in a stellar quake one obtains for the change of the star's frequency ∆Ω / Ω ∼ (10 -5 -10 -3 ) f , where 0 < f < 1 (Glendenning & Weber 1992). The factor f represents the fraction of the crustal moment of inertia that is altered in the quake. Since the observed glitches have relative frequency changes ∆Ω / Ω = (10 -9 -10 -6 ), a change in the crustal moment of inertia of f < ∼ 0 . 1 would cause a giant glitch (Glendenning & Weber 1992). Moreover it turns out that the observed range of the fractional change in the spin-down rate, ˙ Ω, is consistent with the crust having a small moment of inertia and the quake involving only a small fraction f of that. For this purpose we write ∆ ˙ Ω / ˙ Ω > (10 -1 to 10) f (Glendenning & Weber 1992). This relation yields a small f value, i.e., f < (10 -4 to 10 -1 ), in agree- /s32 ment with f < ∼ 0 . 1 established just above. For these estimates, the measured values of (∆Ω / Ω) / (∆ ˙ Ω / ˙ Ω) ∼ 10 -6 to 10 -4 for Crab and Vela, respectively, have been used.","pages":[4,5]},{"title":"6. Possible Connection to CCOs","content":"One of the most amazing features of quark stars concerns the possible existence of ultra-high electric fields on their surfaces, which, for ordinary quark matter, is around 10 18 V/cm. If strange matter forms a color superconductor, as expected for such matter, the strength of the electric field may increase to values that exceed 10 19 V/cm. The energy density associated with such huge electric fields is on the same order of magnitude as the energy density of strange matter itself, which alters the masses and radii of strange quark stars at the 15% and 5% level, respectively (Negreiros et al. 2009). Such mass increases facilitate the interpretation of massive compact stars, with masses of around 2 M /circledot , as strange quark stars (see also Rodrigues et al. 2011). The electrons at the surface of a quark star are not necessarily in a fixed position but may rotate with respect to the quark matter star (Negreiros et al. 2010). In this event magnetic fields can be generated which, for moderate effective rotational frequencies between the electron layer and the stellar body, agree with the magnetic fields inferred for several Compact Central Objects (CCOs). These objects could thus be interpreted as quark stars whose electron atmospheres rotate at frequencies that are moderately different ( ∼ 10 Hz) from the rotational frequency of the quark star itself. Last but not least, we mention that the electron surface layer may be strongly affected by the magnetic field of a quark star in such a way that the electron layer performs vortex hydrodynamical oscillations (Xu et al. 2012). The frequency spectrum of these oscillations has been derived in analytic form by Xu et al. (2012). If the thermal Xray spectra of quark stars are modulated by vortex hydrodynamical oscillations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model (Xu et al. 2012). A study which looks at the thermal evolution of CCOs is presently being carried out by Yang et al. (2012). Preliminary results indicate that the observed temperatures of CCOs can be well reproduced if one assumes that these objects are small quark matter objects with radii less than around 3 km.","pages":[5]},{"title":"7. Possible Connection to SGRs, AXPs, and XDINs","content":"If quarks stars are made of color superconducting quark matter rather than normal non-superconducting quark matter. If rotating, superconducting quark stars ought to be threaded with rotational vortex lines, within which the star's interior magnetic field is at least partially confined. The vortices (and thus magnetic flux) would be expelled from the star during stellar spin-down, leading to magnetic reconnection at the surface of the star and the prolific production of thermal energy. Niebergal et al. (2010) have shown that this energy release can re-heat quark stars to exceptionally high temperatures, such as observed for Soft Gamma Repeaters (SGRs), Anomalous X-Ray pulsars (AXPs), and X-ray dim isolated neutron stars (XDINs). Moreover, numerical investigations of the temperature evolution, spin-down rate, and magnetic field behavior of such superconducting quark stars suggest that SGRs, AXPs, and XDINs may be linked ancestrally (Niebergal et al 2010).","pages":[5,6]},{"title":"Acknowledgements","content":"M. Orsaria thanks CONICET for financial support. H. Rodrigues thanks CAPES for financial support under contract number BEX 6379/10-9. F. Weber is supported by the National Science Foundation (USA) under Grant PHY-0854699. S.-H. Yang is supported by the China Scholarship Council (CSC) and by NFSC under Grant No. 11147170.","pages":[6]},{"title":"References","content":"Proc. Int. Workshop on Strange Quark Matter in Physics and Astrophysics 1991, J. Madsen and P. Haensel (eds.), Nucl. Phys. B (Proc. Suppl.) , 24B Alcock, C., Farhi, E., & Olinto, A. V. 1986, ApJ , 310, 261 Alcock, C., & Olinto, A.V. 1988, Ann. Rev. Nucl. Part. Sci. , 38, 161 Alford, M. 2001, Ann. Rev. Nucl. Part. Sci. , 51, 131 Alford, M.G., Schmitt, A., Rajagopal, K., & Schafer, T. 2008, Rev. Mod. Phys. , 80, 1455 Bodmer, A.R. 1971, Phys. Rev. D , 4, 1601 Cheng, K.S. & Harko, T. 2003, ApJ , 596, 451 Farhi, E., & Jaffe, R. L. 1984, Phys. Rev. D , 30, 2379 Glendenning, N.K., & Weber, F. 1992, ApJ , 400, 647 Glendenning, N.K., Kettner, Ch., & Weber, F. 1995, ApJ , 450, 253 Glendenning, N.K. 2000, Compact Stars, Nuclear Physics, Particle Physics, and General Relativity , 2nd ed. (Springer-Verlag, New York) Kettner, Ch., Weber, F., Weigel, M.K., & Glendenning, N.K. 1995, Phys. Rev. D , 51, 1440 Madsen, J. 1999, Lecture Notes in Physics , 516, 162 Negreiros, R., Weber, F., Malheiro, M., & Usov, V. 2009, Phys. Rev. D , 80, 083006 Negreiros, R.P. , Mishustin, I.N., Schramm, S., & Weber, F. 2010, Phys. Rev. D , 82, 103010 Niebergal, B., Ouyed, R., Negreiros, R., & Weber, F. 2010, Phys. Rev. D , 81, 043005 Orsaria, M, Ranea-Sandoval, I.F., & Vucetich, H. 2011, ApJ , 734, 41 Ouyed, R., Dey, J., & Dey, M. 2002, A&A , 390, L39 Page D., & Reddy, S. 2006, Ann. Rev. Nucl. Part. Sci. , 56, 327 Provencal, J.L., Shipman, H.L., Hog, E., and Thejll, P. 1998, ApJ , 494, 759 Rajagopal K. & Wilczek, F. 2001, The Condensed Matter Physics of QCD , At the Frontier of Particle Physics/Handbook of QCD, ed. M. Shifman (World Scientific) Rajagopal, K., & Wilczek, F. 2001, Phys. Rev. Lett. , 86, 3492 Rodrigues, H., Duarte, S.B., & de Oliveira, J.C.T. 2011, ApJ , 730, 31 Sagert, I., Wietoska, M., & Schaffner-Bielich, J. 2006 J. Phys. G. , 32, S241 Terazawa, H. 1979, INS-Report-338 (INS, Univ. of Tokyo ; 1989 J. Phys. Soc. Japan , 58, 3555; 1989 ibid. , 58, 4388; 1990 ibid. , 59, 1199 Usov, V.V. 1998, Phys. Rev. Lett. , 80, 230 Usov, V.V. 2001, ApJ , 550, L179 Usov, V.V. 2001, ApJ , 559, L137 Vogt, C., Rapp, R., & Ouyed, R. 2004, Nucl. Phys. , A735, 543 Weber, F. 1999, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics , (IOP Publishing, Bristol, Great Britain). Weber, F. 2005, Prog. Part. Nucl. Phys. , 54, 193 Witten, E. 1984, Phys. Rev. D , 30, 272 Xu, R.X., Bastrukov, S.I., Weber, F., Yu, J.W, & Molodtsova, I.V. 2012, Phys. Rev. D , 85, 023008 Yang, S.-H., Weber, F., Negreiros, R., & Becker, W. 2012, Cooling Simulations of CCOs (in preparation) Zdunik, J. L., Haensel, E., & Gourgoulhon, E. 2001, A&A , 372, 535","pages":[6]}],"string":"[\n {\n \"title\": \"Fridolin Weber 1 , Milva Orsaria 2 , † Hilario Rodrigues 3 ‡ and Shu-Hua Yang 4 ¶\",\n \"content\": \"Department of Physics, San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA 1 email: fweber@mail.sdsu.edu 2 email: morsaria@fcaglp.fcaglp.unlp.edu.ar 3 email: harg@cefet-rj.br 4 email: ysh@phy.ccnu.edu.cn Abstract. This paper gives an brief overview of the structure of hypothetical strange quarks stars (quark stars, for short), which are made of absolutely stable 3-flavor strange quark matter. Such objects can be either bare or enveloped in thin nuclear crusts, which consist of heavy ions immersed in an electron gas. In contrast to neutron stars, the structure of quark stars is determined by two (rather than one) parameters, the central star density and the density at the base of the crust. If bare, quark stars possess ultra-high electric fields on the order of 10 18 to 10 19 V/cm. These features render the properties of quark stars more multifaceted than those of neutron stars and may allow one to observationally distinguish quark stars from neutron stars. Keywords. neutron stars, quark stars, pulsars, strange quark matter, equation of state\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"The theoretical possibility that quark matter made of up, down and strange quarks (so-called strange quark matter (Farhi & Jaffe 1984)) may be more stable than ordinary nuclear matter has been pointed out by Bodmer (1971), Terazawa (1979), and Witten (1984). This so-called strange matter hypothesis constitutes one of the most startling possibilities regarding the behavior of superdense matter, which, if true, would have implications of fundamental importance for cosmology, the early universe, its evolution to the present day, and astrophysical compact objects such as neutron stars and white dwarfs (see Alcock & Farhi 1986, Alcock & Olinto 1988, Aarhus 1991, Weber 1999, Madsen 1999, Glendenning 2000, Weber 2005, Page & Reddy 2006, Sagert et al. 2006, and references therein). The properties of quark stars are compared with those of neutron stars in Table 1 and Fig. 1. Even to the present day there is no sound scientific basis on which one can either confirm or reject the hypothesis so that it is a serious possibility of fundamental significance for various (astro) physical phenomena. The multifaceted properties of these objects are reviewed in this paper. Particular emphasis is is put on stellar properties such as rapid rotation, ultra-high electric surface fields, and rotational vortex expulsion, which may allow one to observationally discriminate between quark stars and neutron stars and-ultimately-prove or disprove the strange quark matter hypothesis. Futher information on the existence of qark stars may come from quark novae, hypothetical types of supernovae which could occur if neutron stars spontaneously collapse to quark stars (Ouyed et al. 2002).\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2. Quark-Lepton Composition of Quark Stars\",\n \"content\": \"Quark star matter is composed of the three lightest quark flavor states (up, down, and strange quarks). Per hypothesis, the energy per baryon of such matter is lower than the energy per baryon of the most stable atomic nucleus, 56 Fe. Since stars in their lowest energy state are electrically charge neutral to very high precision, any net positive quark charge must be balanced by electrons. The concentration of electrons is largest at low densities due to the finite strange-quark mass, which leads to a deficit of net negative quark charge. If quark star matter forms a color superconductor (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008, and references therein) in the Color-Flavor-Locked (CFL) phase the interiors of quarks stars will be rigorously electrically neutral with no need for electrons, as shown by Rajagopal & Wilczek (2001). For sufficiently large strange quark masses, however, the low density regime of quark star matter is rather expected to form other condensation patterns (e.g. 2SC, CFLK 0 , CFLK + , CFLπ 0 , -) in which electrons will be present (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008). The presence of electrons in quark star matter is crucial for the possible existence of a nuclear crust on quark stars. As shown by Alcock et al. (1986), Kettner et al. (1995), and Alcock & Olinto (1988), the electrons, because they are bound to strange matter by the Coulomb force rather than the strong force, extend several hundred fermi beyond the surface of the strange star. Associated with this electron displacement is a electric dipole layer which can support, out of contact with the surface of the strange star, a crust of nuclear material, which it polarizes (Alcock et al. 1986, Alcock & Olinto 1988). The maximum possible density at the base of the crust (inner crust density) is determined by neutron drip, which occurs at about 4 . 3 × 10 11 g / cm 3 .\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3. Bare versus Dressed Quark Stars and Eddington Limit\",\n \"content\": \"A bare quark star differs qualitatively from a neutron star which has a density at the surface of about 0.1 to 1 g / cm 3 . The thickness of the quark surface is just ∼ 1 fm, the length scale of the strong interaction. The electrons at the surface of a quark star are held to quark matter electrostatically, and the thickness of the electron surface is several hundred fermis. Since neither component, electrons and quark matter, is held in place gravitationally, the Eddington limit to the luminosity that a static surface may emit does not apply, so that bare quark stars may have photon luminosities much greater than 10 38 erg / s. It was shown by Usov (1998) that this value may be exceeded by many orders of magnitude by the luminosity of e + e -pairs produced by the Coulomb barrier at the surface of a hot strange star. For a surface temperature of ∼ 10 11 K, the luminosity in the outflowing pair plasma was calculated to be as high as ∼ 3 × 10 51 erg / s. Such an effect may be a good observational signature of bare strange stars (Usov 2001a, Usov 2001b, Usov 1998, and Cheng & Harko 2003). If the strange star is enveloped in a nuclear crust, however, which is gravitationally bound to the strange star, the surface, made of ordinary atomic matter, would be subject to the Eddington limit. Hence the photon emissivity of such a 'dressed' quark star would be the same as for an ordinary neutron star. If quark matter at the stellar surface is in the CFL phase the process of e + e -pair creation at the stellar quark matter surface may be turned off. This may be different for the early stages of a very hot CFL quark star (Vogt et al. 2004).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4. Mass-Radius Relationship of Quark Stars\",\n \"content\": \"The mass-radius relationship of bare quark stars is shown in Fig. 2. In contrast to neutron stars, the radii of self-bound quark stars decrease the lighter the stars, according to M ∝ R 3 . The existence of nuclear crusts on quark stars changes the situation drastically (Glendenning et al. 1995, Weber 1999, and Weber 2005). Since the crust is bound gravitationally, the mass-radius relationship of quark stars with crusts is then qualitatively similar to neutron stars. In general, quark stars with or without nuclear crusts possess smaller radii than neutron /s32 stars. This feature implies that quark stars posses smaller mass shedding periods than neutron stars. Due to the smaller radii of quarks stars, the complete sequence of such objects-and not just those close to the mass peak, as it is the case for neutron stars-can sustain extremely rapid rotation (Glendenning et al. 1995, Weber 1999, and Weber 2005). In particular, a strange star with a typical pulsar mass of around 1 . 45 M /circledot can rotate at Kepler (mass shedding) periods as small as 0 . 55 < ∼ P K / msec < ∼ 0 . 8 (Glendenning & Weber 1992, and Glendenning et al. 1995). This range is to be compared with P K ∼ 1 msec obtained for neutron stars of the same mass (Weber 1999). Another novelty of the strange quark matter hypothesis concerns the existence of a new class of white-dwarf-like objects, referred to as strange (quark matter) dwarfs (Glendenning et al. 1995). The mass-radius relationship of the latter may differs somewhat from the mass-radius relationship of ordinary white-dwarf, which may be testable in the future. Until recently, only rather vague tests of the theoretical mass-radius relation of white dwarfs were possible. This has changed dramatically because of the availability of new data emerging from the Hipparcos project (Provencal 1998). These data allow the first accurate measurements of white dwarf distances and, as a result, establishing the mass-radius relation of such objects empirically.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"5. Pulsar Glitches\",\n \"content\": \"Of considerable relevance for the viability of the strange matter hypothesis is the question of whether strange stars can exhibit glitches in rotation frequency. From the study performed by Glendenning & Weber (1992) and Zdunik et al. (2001) it is known that the ratio of the crustal moment of inertia to the total moment of inertia varies between 10 -3 and ∼ 10 -5 . If the angular momentum of the pulsar is conserved in a stellar quake one obtains for the change of the star's frequency ∆Ω / Ω ∼ (10 -5 -10 -3 ) f , where 0 < f < 1 (Glendenning & Weber 1992). The factor f represents the fraction of the crustal moment of inertia that is altered in the quake. Since the observed glitches have relative frequency changes ∆Ω / Ω = (10 -9 -10 -6 ), a change in the crustal moment of inertia of f < ∼ 0 . 1 would cause a giant glitch (Glendenning & Weber 1992). Moreover it turns out that the observed range of the fractional change in the spin-down rate, ˙ Ω, is consistent with the crust having a small moment of inertia and the quake involving only a small fraction f of that. For this purpose we write ∆ ˙ Ω / ˙ Ω > (10 -1 to 10) f (Glendenning & Weber 1992). This relation yields a small f value, i.e., f < (10 -4 to 10 -1 ), in agree- /s32 ment with f < ∼ 0 . 1 established just above. For these estimates, the measured values of (∆Ω / Ω) / (∆ ˙ Ω / ˙ Ω) ∼ 10 -6 to 10 -4 for Crab and Vela, respectively, have been used.\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"6. Possible Connection to CCOs\",\n \"content\": \"One of the most amazing features of quark stars concerns the possible existence of ultra-high electric fields on their surfaces, which, for ordinary quark matter, is around 10 18 V/cm. If strange matter forms a color superconductor, as expected for such matter, the strength of the electric field may increase to values that exceed 10 19 V/cm. The energy density associated with such huge electric fields is on the same order of magnitude as the energy density of strange matter itself, which alters the masses and radii of strange quark stars at the 15% and 5% level, respectively (Negreiros et al. 2009). Such mass increases facilitate the interpretation of massive compact stars, with masses of around 2 M /circledot , as strange quark stars (see also Rodrigues et al. 2011). The electrons at the surface of a quark star are not necessarily in a fixed position but may rotate with respect to the quark matter star (Negreiros et al. 2010). In this event magnetic fields can be generated which, for moderate effective rotational frequencies between the electron layer and the stellar body, agree with the magnetic fields inferred for several Compact Central Objects (CCOs). These objects could thus be interpreted as quark stars whose electron atmospheres rotate at frequencies that are moderately different ( ∼ 10 Hz) from the rotational frequency of the quark star itself. Last but not least, we mention that the electron surface layer may be strongly affected by the magnetic field of a quark star in such a way that the electron layer performs vortex hydrodynamical oscillations (Xu et al. 2012). The frequency spectrum of these oscillations has been derived in analytic form by Xu et al. (2012). If the thermal Xray spectra of quark stars are modulated by vortex hydrodynamical oscillations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model (Xu et al. 2012). A study which looks at the thermal evolution of CCOs is presently being carried out by Yang et al. (2012). Preliminary results indicate that the observed temperatures of CCOs can be well reproduced if one assumes that these objects are small quark matter objects with radii less than around 3 km.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"7. Possible Connection to SGRs, AXPs, and XDINs\",\n \"content\": \"If quarks stars are made of color superconducting quark matter rather than normal non-superconducting quark matter. If rotating, superconducting quark stars ought to be threaded with rotational vortex lines, within which the star's interior magnetic field is at least partially confined. The vortices (and thus magnetic flux) would be expelled from the star during stellar spin-down, leading to magnetic reconnection at the surface of the star and the prolific production of thermal energy. Niebergal et al. (2010) have shown that this energy release can re-heat quark stars to exceptionally high temperatures, such as observed for Soft Gamma Repeaters (SGRs), Anomalous X-Ray pulsars (AXPs), and X-ray dim isolated neutron stars (XDINs). Moreover, numerical investigations of the temperature evolution, spin-down rate, and magnetic field behavior of such superconducting quark stars suggest that SGRs, AXPs, and XDINs may be linked ancestrally (Niebergal et al 2010).\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"Acknowledgements\",\n \"content\": \"M. Orsaria thanks CONICET for financial support. H. Rodrigues thanks CAPES for financial support under contract number BEX 6379/10-9. F. Weber is supported by the National Science Foundation (USA) under Grant PHY-0854699. S.-H. Yang is supported by the China Scholarship Council (CSC) and by NFSC under Grant No. 11147170.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Proc. Int. Workshop on Strange Quark Matter in Physics and Astrophysics 1991, J. Madsen and P. Haensel (eds.), Nucl. Phys. B (Proc. Suppl.) , 24B Alcock, C., Farhi, E., & Olinto, A. V. 1986, ApJ , 310, 261 Alcock, C., & Olinto, A.V. 1988, Ann. Rev. Nucl. Part. Sci. , 38, 161 Alford, M. 2001, Ann. Rev. Nucl. Part. Sci. , 51, 131 Alford, M.G., Schmitt, A., Rajagopal, K., & Schafer, T. 2008, Rev. Mod. Phys. , 80, 1455 Bodmer, A.R. 1971, Phys. Rev. D , 4, 1601 Cheng, K.S. & Harko, T. 2003, ApJ , 596, 451 Farhi, E., & Jaffe, R. L. 1984, Phys. Rev. D , 30, 2379 Glendenning, N.K., & Weber, F. 1992, ApJ , 400, 647 Glendenning, N.K., Kettner, Ch., & Weber, F. 1995, ApJ , 450, 253 Glendenning, N.K. 2000, Compact Stars, Nuclear Physics, Particle Physics, and General Relativity , 2nd ed. (Springer-Verlag, New York) Kettner, Ch., Weber, F., Weigel, M.K., & Glendenning, N.K. 1995, Phys. Rev. D , 51, 1440 Madsen, J. 1999, Lecture Notes in Physics , 516, 162 Negreiros, R., Weber, F., Malheiro, M., & Usov, V. 2009, Phys. Rev. D , 80, 083006 Negreiros, R.P. , Mishustin, I.N., Schramm, S., & Weber, F. 2010, Phys. Rev. D , 82, 103010 Niebergal, B., Ouyed, R., Negreiros, R., & Weber, F. 2010, Phys. Rev. D , 81, 043005 Orsaria, M, Ranea-Sandoval, I.F., & Vucetich, H. 2011, ApJ , 734, 41 Ouyed, R., Dey, J., & Dey, M. 2002, A&A , 390, L39 Page D., & Reddy, S. 2006, Ann. Rev. Nucl. Part. Sci. , 56, 327 Provencal, J.L., Shipman, H.L., Hog, E., and Thejll, P. 1998, ApJ , 494, 759 Rajagopal K. & Wilczek, F. 2001, The Condensed Matter Physics of QCD , At the Frontier of Particle Physics/Handbook of QCD, ed. M. Shifman (World Scientific) Rajagopal, K., & Wilczek, F. 2001, Phys. Rev. Lett. , 86, 3492 Rodrigues, H., Duarte, S.B., & de Oliveira, J.C.T. 2011, ApJ , 730, 31 Sagert, I., Wietoska, M., & Schaffner-Bielich, J. 2006 J. Phys. G. , 32, S241 Terazawa, H. 1979, INS-Report-338 (INS, Univ. of Tokyo ; 1989 J. Phys. Soc. Japan , 58, 3555; 1989 ibid. , 58, 4388; 1990 ibid. , 59, 1199 Usov, V.V. 1998, Phys. Rev. Lett. , 80, 230 Usov, V.V. 2001, ApJ , 550, L179 Usov, V.V. 2001, ApJ , 559, L137 Vogt, C., Rapp, R., & Ouyed, R. 2004, Nucl. Phys. , A735, 543 Weber, F. 1999, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics , (IOP Publishing, Bristol, Great Britain). Weber, F. 2005, Prog. Part. Nucl. Phys. , 54, 193 Witten, E. 1984, Phys. Rev. D , 30, 272 Xu, R.X., Bastrukov, S.I., Weber, F., Yu, J.W, & Molodtsova, I.V. 2012, Phys. Rev. D , 85, 023008 Yang, S.-H., Weber, F., Negreiros, R., & Becker, W. 2012, Cooling Simulations of CCOs (in preparation) Zdunik, J. L., Haensel, E., & Gourgoulhon, E. 2001, A&A , 372, 535\",\n \"pages\": [\n 6\n ]\n }\n]"}}},{"rowIdx":277,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..291..356C"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1304.6907.pdf"},"content":{"kind":"string","value":"\nDoes a hadron-quark phase transition in dense matter preclude the existence of massive neutron stars ?\nN. Chamel 1 , A.F. Fantina 1 , J.M. Pearson 2 and S. Goriely 1\n1 Institut d'Astronomie et d'Astrophysique, CP-226, Universit'e Libre de Bruxelles, 1050 Brussels, Belgium\n2 D'ept. de Physique, Universit'e de Montr'eal, Montr'eal (Qu'ebec), H3C 3J7 Canada\nAbstract. We study the impact of a hadron-quark phase transition on the maximum neutronstar mass. The hadronic part of the equation of state relies on the most up-to-date Skyrme nuclear energy density functionals, fitted to essentially all experimental nuclear mass data and constrained to reproduce the properties of infinite nuclear matter as obtained from microscopic calculations using realistic forces. We show that the softening of the dense matter equation of state due to the phase transition is not necessarily incompatible with the existence of massive neutron stars like PSR J1614 -2230.\nKeywords. stars: neutron, dense matter, equation of state, stars: interiors, gravitation\n1. Introduction\nNeutron stars (NSs) result from the gravitational collapse of massive stars with M /greaterorsimilar 8 M /circledot at the end point of their evolution. They are among the most compact objects in the universe, with a central density which can reach several times the nuclear saturation density. At least, three different regions can be identified in the interior of a NS: (i) the 'outer crust', at densities above ∼ 10 4 g cm -3 , composed of fully ionized atoms, arranged in a Coulomb lattice of nuclei, neutralized by a degenerate electron gas, (ii) the 'inner crust', at densities above ∼ 4 × 10 11 g cm -3 , composed of neutron-proton clusters and unbound neutrons, neutralized by a degenerate electron gas, and (iii) the core, at densities above ∼ 10 14 g cm -3 . The precise measurement of the mass of pulsar PSR J1614 -2230 by Demorest et al. (2010) has revived the question of the composition of the core. Just below the crust, the matter consists of a mixture of neutrons, protons, electrons and possibly muons. The composition of the central region of a NS is still a matter of debate (see e.g. Haensel et al. 2007).\nIn the present work, we study the impact of a hadron-quark phase transition in dense matter on the maximum mass of cold isolated NSs (see Chamel et al. 2013 for a general discussion of the maximum mass of hybrid stars).\n2. Hadronic equation of state\nThe global structure of a NS is determined by the equation of state (EoS), i.e. the relation between the matter pressure P and the mass-energy density ρ . Before considering the possibility of a phase transition from hadronic to quark matter in the core of NSs, we will begin with the hadronic EoSs. A good starting point is the family of three EoSs, BSk19, BSk20 and BSk21, which have been developed to provide a unified treatment of all regions of a NS (see Pearson et al. 2011, Pearson et al. 2012). These EoSs are based on\nnuclear energy-density functionals derived from generalized Skyrme forces (in that they contain additional momentum- and density-dependent terms), which fit essentially all measured masses of atomic nuclei with an rms deviation of 0.58 MeV for all three models. Moreover, these functionals were constrained to reproduce three different neutron matter EoSs, as obtained from microscopic calculations (see Goriely et al. 2010). All three EoSs assume that the core of a NS is made of nucleons and leptons. The BSk19 EoS was found to be too soft to support NSs as massive as PSR J1614 -2230 (Chamel et al. 2011) and therefore, it will not be considered here.\n3. Hadron-quark phase transition\nGiven the uncertainties in the composition of dense matter in NSs, we will simply suppose that above the average baryon density n N , matter undergoes a first-order phase transition to deconfined quark matter subject to the following restrictions: (i) for the transition to occur the energy density of the quark phase must be lower than that of the hadronic phase, (ii) according to perturbative quantum chromodynamics (QCD) calculations (e.g. Kurkela et al. 2010), the speed of sound in quark matter cannot exceed c/ √ 3 where c is the speed of light. At densities below n N , matter is purely hadronic while a pure quark phase is found at densities above some density n X . In the intermediate region ( n N < n < n X ) where the two phases can coexist, the pressure and the chemical potential of the two phases are equal: P quark ( n ) = P hadron ( n N ) and µ quark ( n ) = µ hadron ( n N ). The EoS of the quark phase at n > n X is given by:\nP quark ( n ) = 1 3 ( E quark ( n ) -E quark ( n X )) + P hadron ( n N ) . (3.1)\nWe set the density n N to lie above the highest density found in nuclei as predicted by Hartree-Fock-Bogoliubov calculations, namely n N = 0 . 2 fm -3 (BRUSLIB). The density n X is adjusted to optimize the maximum mass under the conditions mentioned above. Eq. (3.1) turns out to be very similar to that obtained within the simple MIT bag model, which has been widely applied to describe quark matter in compact stars (see e.g. Haensel et al. 2007). The effective bag constant B associated with the BSk21 hadronic EoS is 56 . 7 MeV fm -3 .\n4. Maximum mass\nConsidering the stiffest hadronic EoS (BSk21), we have solved the Tolman-OppenheimerVolkoff equations (Tolman 1939, Oppenheimer & Volkoff 1939) in order to determine the global structure of a nonrotating NS. The effect of rotation on the maximum mass was found to be very small for stars with spin-periods comparable to that of PSR J1614 -2230 (Chamel et al. 2011); we therefore neglect it. The gravitational mass versus circumferential radius relation is shown in Fig. 1. We have considered two cases: a purely hadronic NS described by our BSk21 EoS (dashed line) and a hybrid star with a quark core (solid line). The corresponding maximum masses are 2.28 M /circledot and 2.02 M /circledot respectively. In both cases, the existence of two-solar mass NSs is therefore allowed.\n5. Conclusions\nThe presence of a deconfined quark-matter phase in NS cores leads to a maximum mass of about 2 M /circledot , which is still compatible with the mass measurement of PSR J1614 -2230 by Demorest et al. (2010), but which could be challenged by observations of significantly\n\n\n
Figure 1. Gravitational mass versus circumferential radius with (solid line) and without (dashed line) a quark-matter core. See the text for detail.
\n\nmore massive NSs (see Clark et al. 2002, Freire et al. 2008, van Kerkwijk et al. 2011) unless the sound speed in quark matter is significantly larger than that predicted by perturbative QCD calculations (Kurkela et al. 2010).\nAcknowledgements\nFNRS (Belgium), NSERC (Canada) and CompStar, a Research Networking Programme of the European Science Foundation are gratefully acknowledged.\nReferences\nBRUSLIB http://www.astro.ulb.ac.be/bruslib Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2011, Phys. Rev. C , 84, 062802 Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2013, Astron. Astrophys. , 553, A22 Clark, J. S., Goodwin, S. P., Crowther, P. A., Kaper, L, Fairbairn, M., Langer, N., Brocksopp, C. 2002, Astron. Astrophys. , 392, 909 Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Nature , 467, 1081 Freire, P. C. C., Ransom, S. M., B'egin, S., Stairs, I. H., Hessels, J. W. T., Frey, L. H., Camilo, F. 2008, Astrophys. J. , 675, 670 Goriely, S., Chamel, N., & Pearson, J. M. 2010, Phys. Rev. C , 82, 035804 Haensel, P., Potekhin, A. Y., & Yakovlev, D. G. 2007, Astrophys. Space Sc. L. , 326 Kurkela, A., Romatschke, P., & Vuorinen, A. 2010, Phys. Rev. D , 81, 105021 Oppenheimer, J. R., & Volkoff, G. M. 1939, Phys. Rev. , 55, 374 Pearson, J. M., Goriely, S., & Chamel, N. 2011, Phys. Rev. C , 83, 065810 Pearson, J. M., Chamel, N., Goriely, S., & Ducoin, C. 2012, Phys. Rev. C , 85, 065803 Tolman, R. C. 1939, Phys. Rev. , 55, 364\nvan Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, Astrophys. J. 728, 95\n"},"sections":{"kind":"list like","value":[{"title":"N. Chamel 1 , A.F. Fantina 1 , J.M. Pearson 2 and S. Goriely 1","content":"1 Institut d'Astronomie et d'Astrophysique, CP-226, Universit'e Libre de Bruxelles, 1050 Brussels, Belgium 2 D'ept. de Physique, Universit'e de Montr'eal, Montr'eal (Qu'ebec), H3C 3J7 Canada Abstract. We study the impact of a hadron-quark phase transition on the maximum neutronstar mass. The hadronic part of the equation of state relies on the most up-to-date Skyrme nuclear energy density functionals, fitted to essentially all experimental nuclear mass data and constrained to reproduce the properties of infinite nuclear matter as obtained from microscopic calculations using realistic forces. We show that the softening of the dense matter equation of state due to the phase transition is not necessarily incompatible with the existence of massive neutron stars like PSR J1614 -2230. Keywords. stars: neutron, dense matter, equation of state, stars: interiors, gravitation","pages":[1]},{"title":"1. Introduction","content":"Neutron stars (NSs) result from the gravitational collapse of massive stars with M /greaterorsimilar 8 M /circledot at the end point of their evolution. They are among the most compact objects in the universe, with a central density which can reach several times the nuclear saturation density. At least, three different regions can be identified in the interior of a NS: (i) the 'outer crust', at densities above ∼ 10 4 g cm -3 , composed of fully ionized atoms, arranged in a Coulomb lattice of nuclei, neutralized by a degenerate electron gas, (ii) the 'inner crust', at densities above ∼ 4 × 10 11 g cm -3 , composed of neutron-proton clusters and unbound neutrons, neutralized by a degenerate electron gas, and (iii) the core, at densities above ∼ 10 14 g cm -3 . The precise measurement of the mass of pulsar PSR J1614 -2230 by Demorest et al. (2010) has revived the question of the composition of the core. Just below the crust, the matter consists of a mixture of neutrons, protons, electrons and possibly muons. The composition of the central region of a NS is still a matter of debate (see e.g. Haensel et al. 2007). In the present work, we study the impact of a hadron-quark phase transition in dense matter on the maximum mass of cold isolated NSs (see Chamel et al. 2013 for a general discussion of the maximum mass of hybrid stars).","pages":[1]},{"title":"2. Hadronic equation of state","content":"The global structure of a NS is determined by the equation of state (EoS), i.e. the relation between the matter pressure P and the mass-energy density ρ . Before considering the possibility of a phase transition from hadronic to quark matter in the core of NSs, we will begin with the hadronic EoSs. A good starting point is the family of three EoSs, BSk19, BSk20 and BSk21, which have been developed to provide a unified treatment of all regions of a NS (see Pearson et al. 2011, Pearson et al. 2012). These EoSs are based on nuclear energy-density functionals derived from generalized Skyrme forces (in that they contain additional momentum- and density-dependent terms), which fit essentially all measured masses of atomic nuclei with an rms deviation of 0.58 MeV for all three models. Moreover, these functionals were constrained to reproduce three different neutron matter EoSs, as obtained from microscopic calculations (see Goriely et al. 2010). All three EoSs assume that the core of a NS is made of nucleons and leptons. The BSk19 EoS was found to be too soft to support NSs as massive as PSR J1614 -2230 (Chamel et al. 2011) and therefore, it will not be considered here.","pages":[1,2]},{"title":"3. Hadron-quark phase transition","content":"Given the uncertainties in the composition of dense matter in NSs, we will simply suppose that above the average baryon density n N , matter undergoes a first-order phase transition to deconfined quark matter subject to the following restrictions: (i) for the transition to occur the energy density of the quark phase must be lower than that of the hadronic phase, (ii) according to perturbative quantum chromodynamics (QCD) calculations (e.g. Kurkela et al. 2010), the speed of sound in quark matter cannot exceed c/ √ 3 where c is the speed of light. At densities below n N , matter is purely hadronic while a pure quark phase is found at densities above some density n X . In the intermediate region ( n N < n < n X ) where the two phases can coexist, the pressure and the chemical potential of the two phases are equal: P quark ( n ) = P hadron ( n N ) and µ quark ( n ) = µ hadron ( n N ). The EoS of the quark phase at n > n X is given by: We set the density n N to lie above the highest density found in nuclei as predicted by Hartree-Fock-Bogoliubov calculations, namely n N = 0 . 2 fm -3 (BRUSLIB). The density n X is adjusted to optimize the maximum mass under the conditions mentioned above. Eq. (3.1) turns out to be very similar to that obtained within the simple MIT bag model, which has been widely applied to describe quark matter in compact stars (see e.g. Haensel et al. 2007). The effective bag constant B associated with the BSk21 hadronic EoS is 56 . 7 MeV fm -3 .","pages":[2]},{"title":"4. Maximum mass","content":"Considering the stiffest hadronic EoS (BSk21), we have solved the Tolman-OppenheimerVolkoff equations (Tolman 1939, Oppenheimer & Volkoff 1939) in order to determine the global structure of a nonrotating NS. The effect of rotation on the maximum mass was found to be very small for stars with spin-periods comparable to that of PSR J1614 -2230 (Chamel et al. 2011); we therefore neglect it. The gravitational mass versus circumferential radius relation is shown in Fig. 1. We have considered two cases: a purely hadronic NS described by our BSk21 EoS (dashed line) and a hybrid star with a quark core (solid line). The corresponding maximum masses are 2.28 M /circledot and 2.02 M /circledot respectively. In both cases, the existence of two-solar mass NSs is therefore allowed.","pages":[2]},{"title":"5. Conclusions","content":"The presence of a deconfined quark-matter phase in NS cores leads to a maximum mass of about 2 M /circledot , which is still compatible with the mass measurement of PSR J1614 -2230 by Demorest et al. (2010), but which could be challenged by observations of significantly more massive NSs (see Clark et al. 2002, Freire et al. 2008, van Kerkwijk et al. 2011) unless the sound speed in quark matter is significantly larger than that predicted by perturbative QCD calculations (Kurkela et al. 2010).","pages":[2,3]},{"title":"Acknowledgements","content":"FNRS (Belgium), NSERC (Canada) and CompStar, a Research Networking Programme of the European Science Foundation are gratefully acknowledged.","pages":[3]},{"title":"References","content":"BRUSLIB http://www.astro.ulb.ac.be/bruslib Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2011, Phys. Rev. C , 84, 062802 Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2013, Astron. Astrophys. , 553, A22 Clark, J. S., Goodwin, S. P., Crowther, P. A., Kaper, L, Fairbairn, M., Langer, N., Brocksopp, C. 2002, Astron. Astrophys. , 392, 909 Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Nature , 467, 1081 Freire, P. C. C., Ransom, S. M., B'egin, S., Stairs, I. H., Hessels, J. W. T., Frey, L. H., Camilo, F. 2008, Astrophys. J. , 675, 670 Goriely, S., Chamel, N., & Pearson, J. M. 2010, Phys. Rev. C , 82, 035804 Haensel, P., Potekhin, A. Y., & Yakovlev, D. G. 2007, Astrophys. Space Sc. L. , 326 Kurkela, A., Romatschke, P., & Vuorinen, A. 2010, Phys. Rev. D , 81, 105021 Oppenheimer, J. R., & Volkoff, G. M. 1939, Phys. Rev. , 55, 374 Pearson, J. M., Goriely, S., & Chamel, N. 2011, Phys. Rev. C , 83, 065810 Pearson, J. M., Chamel, N., Goriely, S., & Ducoin, C. 2012, Phys. Rev. C , 85, 065803 Tolman, R. C. 1939, Phys. Rev. , 55, 364 van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, Astrophys. J. 728, 95","pages":[3]}],"string":"[\n {\n \"title\": \"N. Chamel 1 , A.F. Fantina 1 , J.M. Pearson 2 and S. Goriely 1\",\n \"content\": \"1 Institut d'Astronomie et d'Astrophysique, CP-226, Universit'e Libre de Bruxelles, 1050 Brussels, Belgium 2 D'ept. de Physique, Universit'e de Montr'eal, Montr'eal (Qu'ebec), H3C 3J7 Canada Abstract. We study the impact of a hadron-quark phase transition on the maximum neutronstar mass. The hadronic part of the equation of state relies on the most up-to-date Skyrme nuclear energy density functionals, fitted to essentially all experimental nuclear mass data and constrained to reproduce the properties of infinite nuclear matter as obtained from microscopic calculations using realistic forces. We show that the softening of the dense matter equation of state due to the phase transition is not necessarily incompatible with the existence of massive neutron stars like PSR J1614 -2230. Keywords. stars: neutron, dense matter, equation of state, stars: interiors, gravitation\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Neutron stars (NSs) result from the gravitational collapse of massive stars with M /greaterorsimilar 8 M /circledot at the end point of their evolution. They are among the most compact objects in the universe, with a central density which can reach several times the nuclear saturation density. At least, three different regions can be identified in the interior of a NS: (i) the 'outer crust', at densities above ∼ 10 4 g cm -3 , composed of fully ionized atoms, arranged in a Coulomb lattice of nuclei, neutralized by a degenerate electron gas, (ii) the 'inner crust', at densities above ∼ 4 × 10 11 g cm -3 , composed of neutron-proton clusters and unbound neutrons, neutralized by a degenerate electron gas, and (iii) the core, at densities above ∼ 10 14 g cm -3 . The precise measurement of the mass of pulsar PSR J1614 -2230 by Demorest et al. (2010) has revived the question of the composition of the core. Just below the crust, the matter consists of a mixture of neutrons, protons, electrons and possibly muons. The composition of the central region of a NS is still a matter of debate (see e.g. Haensel et al. 2007). In the present work, we study the impact of a hadron-quark phase transition in dense matter on the maximum mass of cold isolated NSs (see Chamel et al. 2013 for a general discussion of the maximum mass of hybrid stars).\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2. Hadronic equation of state\",\n \"content\": \"The global structure of a NS is determined by the equation of state (EoS), i.e. the relation between the matter pressure P and the mass-energy density ρ . Before considering the possibility of a phase transition from hadronic to quark matter in the core of NSs, we will begin with the hadronic EoSs. A good starting point is the family of three EoSs, BSk19, BSk20 and BSk21, which have been developed to provide a unified treatment of all regions of a NS (see Pearson et al. 2011, Pearson et al. 2012). These EoSs are based on nuclear energy-density functionals derived from generalized Skyrme forces (in that they contain additional momentum- and density-dependent terms), which fit essentially all measured masses of atomic nuclei with an rms deviation of 0.58 MeV for all three models. Moreover, these functionals were constrained to reproduce three different neutron matter EoSs, as obtained from microscopic calculations (see Goriely et al. 2010). All three EoSs assume that the core of a NS is made of nucleons and leptons. The BSk19 EoS was found to be too soft to support NSs as massive as PSR J1614 -2230 (Chamel et al. 2011) and therefore, it will not be considered here.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"3. Hadron-quark phase transition\",\n \"content\": \"Given the uncertainties in the composition of dense matter in NSs, we will simply suppose that above the average baryon density n N , matter undergoes a first-order phase transition to deconfined quark matter subject to the following restrictions: (i) for the transition to occur the energy density of the quark phase must be lower than that of the hadronic phase, (ii) according to perturbative quantum chromodynamics (QCD) calculations (e.g. Kurkela et al. 2010), the speed of sound in quark matter cannot exceed c/ √ 3 where c is the speed of light. At densities below n N , matter is purely hadronic while a pure quark phase is found at densities above some density n X . In the intermediate region ( n N < n < n X ) where the two phases can coexist, the pressure and the chemical potential of the two phases are equal: P quark ( n ) = P hadron ( n N ) and µ quark ( n ) = µ hadron ( n N ). The EoS of the quark phase at n > n X is given by: We set the density n N to lie above the highest density found in nuclei as predicted by Hartree-Fock-Bogoliubov calculations, namely n N = 0 . 2 fm -3 (BRUSLIB). The density n X is adjusted to optimize the maximum mass under the conditions mentioned above. Eq. (3.1) turns out to be very similar to that obtained within the simple MIT bag model, which has been widely applied to describe quark matter in compact stars (see e.g. Haensel et al. 2007). The effective bag constant B associated with the BSk21 hadronic EoS is 56 . 7 MeV fm -3 .\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"4. Maximum mass\",\n \"content\": \"Considering the stiffest hadronic EoS (BSk21), we have solved the Tolman-OppenheimerVolkoff equations (Tolman 1939, Oppenheimer & Volkoff 1939) in order to determine the global structure of a nonrotating NS. The effect of rotation on the maximum mass was found to be very small for stars with spin-periods comparable to that of PSR J1614 -2230 (Chamel et al. 2011); we therefore neglect it. The gravitational mass versus circumferential radius relation is shown in Fig. 1. We have considered two cases: a purely hadronic NS described by our BSk21 EoS (dashed line) and a hybrid star with a quark core (solid line). The corresponding maximum masses are 2.28 M /circledot and 2.02 M /circledot respectively. In both cases, the existence of two-solar mass NSs is therefore allowed.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"5. Conclusions\",\n \"content\": \"The presence of a deconfined quark-matter phase in NS cores leads to a maximum mass of about 2 M /circledot , which is still compatible with the mass measurement of PSR J1614 -2230 by Demorest et al. (2010), but which could be challenged by observations of significantly more massive NSs (see Clark et al. 2002, Freire et al. 2008, van Kerkwijk et al. 2011) unless the sound speed in quark matter is significantly larger than that predicted by perturbative QCD calculations (Kurkela et al. 2010).\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"Acknowledgements\",\n \"content\": \"FNRS (Belgium), NSERC (Canada) and CompStar, a Research Networking Programme of the European Science Foundation are gratefully acknowledged.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"BRUSLIB http://www.astro.ulb.ac.be/bruslib Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2011, Phys. Rev. C , 84, 062802 Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2013, Astron. Astrophys. , 553, A22 Clark, J. S., Goodwin, S. P., Crowther, P. A., Kaper, L, Fairbairn, M., Langer, N., Brocksopp, C. 2002, Astron. Astrophys. , 392, 909 Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Nature , 467, 1081 Freire, P. C. C., Ransom, S. M., B'egin, S., Stairs, I. H., Hessels, J. W. T., Frey, L. H., Camilo, F. 2008, Astrophys. J. , 675, 670 Goriely, S., Chamel, N., & Pearson, J. M. 2010, Phys. Rev. C , 82, 035804 Haensel, P., Potekhin, A. Y., & Yakovlev, D. G. 2007, Astrophys. Space Sc. L. , 326 Kurkela, A., Romatschke, P., & Vuorinen, A. 2010, Phys. Rev. D , 81, 105021 Oppenheimer, J. R., & Volkoff, G. M. 1939, Phys. Rev. , 55, 374 Pearson, J. M., Goriely, S., & Chamel, N. 2011, Phys. Rev. C , 83, 065810 Pearson, J. M., Chamel, N., Goriely, S., & Ducoin, C. 2012, Phys. Rev. C , 85, 065803 Tolman, R. C. 1939, Phys. Rev. , 55, 364 van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, Astrophys. J. 728, 95\",\n \"pages\": [\n 3\n ]\n }\n]"}}},{"rowIdx":278,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..291..496S"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1209.5171.pdf"},"content":{"kind":"string","value":"\nNew Constraints on Preferred Frame Effects from Binary Pulsars\nLijing Shao 1 , Norbert Wex 2 , Michael Kramer 3\n1 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany School of Physics, Peking University, Beijing 100871, China email: lshao@pku.edu.cn\n2 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany email: wex@mpifr-bonn.mpg.de\n3 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, M13 9PL, UK email: mkramer@mpifr-bonn.mpg.de\nAbstract. Preferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, ˆ α 1 and ˆ α 2 . In the limit of a small orbital eccentricity, ˆ α 1 and ˆ α 2 contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of ˆ α 1 and ˆ α 2 in the strong-field regime. The constraint on ˆ α 1 also surpasses its counterpart in the weak-field regime.\nKeywords. Gravitation, pulsars: general, pulsars: individual (J1012+5307, J1738+0333)\n1. Introduction\nPulsars are extremely stable electromagnetic emitters, and along with their extreme physical properties and surrounding environments, they provide useful astrophysical laboratories to study fundamental physics (Lorimer & Kramer 2005). Radio timing of binary pulsars maps out the binary orbital dynamics through recording the time-of-arrivals of the pulsar signals at the telescope. For millisecond pulsars this can be done with high precision, providing a powerful tool to probe gravity (see e.g., Stairs (2003) and Kramer et al. (2006)). In this work, we summarize new results on testing the local Lorentz invariance (LLI) of gravity from binary pulsars obtained by Shao & Wex (2012).\n2. New Limits on Preferred Frame Effects\nNon-gravitational LLI is an important ingredient of the Einstein equivalence principle (EEP) (Will 1993, 2006). But even metric gravity, which fulfills the EEP exactly, could still exhibit a violation of LLI in the gravitational sector (Will & Nordtvedt 1972, Nordtvedt & Will 1972, Damour & Esposito-Far'ese 1992, Will 1993). Such a violation of LLI induces preferred frame effects (PFEs) in the orbital dynamics of a binary system that moves with respect to the preferred frame. In the parametrized post-Newtonian (PPN) formalism, PFEs of a semi-conservative gravity theory are described by two parameters, ˆ α 1 and ˆ α 2 . †\n\n† To distinguish from their weak-field counterparts ( α 1 and α 2 ), here 'hat' indicates possible modifications by strong-field effects.\n\nThe orbital dynamics of binary pulsars with non-vanishing ˆ α 1 and ˆ α 2 are obtained from a generic semi-conservative Lagrangian (Damour & Esposito-Far'ese 1992). It is found that in the limit of small orbital eccentricity, PFEs induced by ˆ α 1 and ˆ α 2 decouple, and lead to separable effects in the timing observations. Hence they can be tested independently using observations of only one binary pulsar (Shao & Wex 2012).\n\n· A non-zero ˆ α 1 induces a polarization of the eccentricity vector towards a direction in the orbital plane perpendicular to the velocity of the binary system with respect to the preferred frame, w . † The observed eccentricity vector e ( t ) is a vectorial superposition of a 'relativistically rotating' eccentricity e R ( t ) (of constant length) and a 'fixed eccentricity' e F ∝ ˆ α 1 : e ( t ) = e R ( t ) + e F (Damour & Esposito-Far'ese 1992). The effect is graphically illustrated in the left panel of Fig. 1. Previous methods use the smallness of the observed eccentricity, combined with probabilistic considerations concerning the unknown angle θ (the angle between e R and e F ) to constrain ˆ α 1 (Damour & Esposito-Far'ese 1992, Wex 2000). The method developed in Shao & Wex (2012) is an extension of the method by Damour & Esposito-Far'ese (1992) that does not require any probabilistic considerations concerning θ . It is applicable to binary pulsars of short orbital period that have been observed for a long enough time, during which the periastron has advanced significantly. Even if the advance of periastron is not resolved in the timing observation, the constraints on the (observed) eccentricity vector can be converted into a limit on ˆ α 1 . From the 10 years of timing and the optical observations of PSR J1738+0333 (Antoniadis et al. 2012, Freire et al. 2012) one obtains the most constraining limit,\n\nˆ α 1 = -0 . 4 +3 . 7 -3 . 1 × 10 -5 (95% C.L.) , (2.1)\nwhich is significantly better than the current best limits from both weak field (Muller et al. 2008) and strong field (Wex 2000).\n\n· A non-vanishing ˆ α 2 induces a precession of the orbital angular momentum around the direction of w (see the right panel of Fig. 1), which changes the observed orbital inclination. Using the long-term timing results on PSRs J1012+5307 (Lazaridis et al. 2009) and J1738+0333 (Antoniadis et al. 2012, Freire et al. 2012), Shao & Wex (2012) find an upper limit\n\n| ˆ α 2 | < 1 . 8 × 10 -4 (95% C.L.) , (2.2)\nwhich is better than the current best limit for strongly self-gravitating bodies (Wex &\n\n† Here we choose the isotropic cosmic microwave background as the preferred frame; nevertheless, see Shao & Wex (2012) for constraints on other preferred frames.\n\n\n\n
Figure 1. Illustration of preferred frame effects in the orbital dynamics of small-eccentricity binary pulsars; see Shao & Wex (2012) for details. Left : ˆ α 1 tends to polarize the orbital eccentricity vector e ( t ) towards the direction perpendicular to w ⊥ (Damour & Esposito-Far'ese 1992); right : ˆ α 2 induces a precession of the orbital angular momentum around the direction of w .
\n\nKramer 2007) by more than three orders of magnitude, but still considerably weaker than the weak-field limit of α 2 by Nordtvedt (1987).\n3. Summary\nWe summarize results presented in Shao & Wex (2012) that proposed new tests of LLI. These yield improved constraints on PFEs from binary pulsar experiments. Specifically, limits on parameters ˆ α 1 and ˆ α 2 are obtained from long-term timing of two binary pulsars with short orbital period (see Eqs. (2.1) and (2.2)). Our extended ˆ α 1 test no longer requires probabilistic considerations related to unknown angles. The proposed tests have the advantage that they continuously improve with time, and will benefit greatly from the next generation of radio telescopes, like FAST (Nan et al. 2011) and SKA (Smits et al. 2009).\nAcknowledgment\nLijing Shao is supported by China Scholarship Council (CSC).\nReferences\nAntoniadis, J., van Kerkwijk, M. H., Koester, D., Freire, P. C. C., Wex, N., Tauris, T. M., Kramer, M., & Bassa, C. G. 2012, MNRAS , 423, 3316\nDamour, T. & Esposito-Far'ese, G. 1992, Phys. Rev. D , 46, 4128\nFreire, P. C. C., Wex, N., Esposito-Far'ese, G., Verbiest, J. P. W., Bailes, M., Jacoby, B. A., Kramer, M., Stairs, I. H., Antoniadis, J., & Janssen., G. H. 2012, MNRAS , 423, 3328\nKramer, M., Stairs, I. H., Manchester, R. N., McLaughlin, M. A., Lyne, A. G., Ferdman, R. D., Burgay, M., Lorimer, D. R., Possenti, A., D'Amico, N., Sarkissian, J. M., Hobbs, G. B., Reynolds, J. E., Freire, P. C. C., & Camilo, F. 2006, Science , 314, 97\nLazaridis, K., Wex, N., Jessner, A., Kramer, M., Stappers, B. W., Janssen, G. H., Desvignes, G., Purver, M. B., Cognard, I., Theureau, G., Lyne, A. G., Jordan, C. A., & Zensus, J. A. 2009, MNRAS , 400, 805\nLorimer, D. R. & Kramer, M. 2005, Handbook of Pulsar Astronomy (Cambridge University Press)\nMuller, J., Williams, J. G., & Turyshev, S. G. 2008, in: H. Dittus, C. Lammerzahl, & S. G. Turyshev (eds.), Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space 349 (Astrophysics and Space Science Library: Springer), p. 457\nNan, R., Li, D., Jin, C., Wang, Q., Zhu, L., Zhu, W., Zhang, H., Yue, Y., & Qian, L. 2011, International Journal of Modern Physics D , 20, 989\nNordtvedt, K. 1987, ApJ , 320, 871\nNordtvedt, K. & Will, C. M. 1972, ApJ , 177, 775\nShao, L., & Wex, N. 2012, Class. Quantum Grav. , accepted [arXiv:1209.4503]\nSmits, R., Kramer, M., Stappers, B., Lorimer, D. R., Cordes, J., & Faulkner, A. 2009, A&A , 493, 1161\nStairs, I. H. 2003, 'Testing General Relativity with Pulsar Timing' , Living Rev. Relativity 6, 5. URL (cited on 2012-09-11): http://www.livingreviews.org/lrr-2003-5\nWex, N. 2000, in: M. Kramer, N. Wex, & R. Wielebinski (eds.), IAU Colloq. 177: Pulsar Astronomy - 2000 and Beyond 202 (Astronomical Society of the Pacific Conference Series: San Francisco), p. 113\nWex, N., & Kramer, M. 2007, MNRAS , 380, 455\nWill, C. M. 1993, Theory and Experiment in Gravitational Physics (Cambridge University Press)\nWill, C. M. 2006, 'The Confrontation between General Relativity and Experiment' , Living Rev.\nRelativity 9, 3. URL (cited on 2012-09-06): http://www.livingreviews.org/lrr-2006-3\nWill, C. M., & Nordtvedt, K. 1972, ApJ , 177, 757\n"},"sections":{"kind":"list like","value":[{"title":"Lijing Shao 1 , Norbert Wex 2 , Michael Kramer 3","content":"1 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany School of Physics, Peking University, Beijing 100871, China email: lshao@pku.edu.cn 2 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany email: wex@mpifr-bonn.mpg.de 3 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, M13 9PL, UK email: mkramer@mpifr-bonn.mpg.de Abstract. Preferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, ˆ α 1 and ˆ α 2 . In the limit of a small orbital eccentricity, ˆ α 1 and ˆ α 2 contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of ˆ α 1 and ˆ α 2 in the strong-field regime. The constraint on ˆ α 1 also surpasses its counterpart in the weak-field regime. Keywords. Gravitation, pulsars: general, pulsars: individual (J1012+5307, J1738+0333)","pages":[1]},{"title":"1. Introduction","content":"Pulsars are extremely stable electromagnetic emitters, and along with their extreme physical properties and surrounding environments, they provide useful astrophysical laboratories to study fundamental physics (Lorimer & Kramer 2005). Radio timing of binary pulsars maps out the binary orbital dynamics through recording the time-of-arrivals of the pulsar signals at the telescope. For millisecond pulsars this can be done with high precision, providing a powerful tool to probe gravity (see e.g., Stairs (2003) and Kramer et al. (2006)). In this work, we summarize new results on testing the local Lorentz invariance (LLI) of gravity from binary pulsars obtained by Shao & Wex (2012).","pages":[1]},{"title":"2. New Limits on Preferred Frame Effects","content":"Non-gravitational LLI is an important ingredient of the Einstein equivalence principle (EEP) (Will 1993, 2006). But even metric gravity, which fulfills the EEP exactly, could still exhibit a violation of LLI in the gravitational sector (Will & Nordtvedt 1972, Nordtvedt & Will 1972, Damour & Esposito-Far'ese 1992, Will 1993). Such a violation of LLI induces preferred frame effects (PFEs) in the orbital dynamics of a binary system that moves with respect to the preferred frame. In the parametrized post-Newtonian (PPN) formalism, PFEs of a semi-conservative gravity theory are described by two parameters, ˆ α 1 and ˆ α 2 . † The orbital dynamics of binary pulsars with non-vanishing ˆ α 1 and ˆ α 2 are obtained from a generic semi-conservative Lagrangian (Damour & Esposito-Far'ese 1992). It is found that in the limit of small orbital eccentricity, PFEs induced by ˆ α 1 and ˆ α 2 decouple, and lead to separable effects in the timing observations. Hence they can be tested independently using observations of only one binary pulsar (Shao & Wex 2012). which is significantly better than the current best limits from both weak field (Muller et al. 2008) and strong field (Wex 2000). which is better than the current best limit for strongly self-gravitating bodies (Wex & Kramer 2007) by more than three orders of magnitude, but still considerably weaker than the weak-field limit of α 2 by Nordtvedt (1987).","pages":[1,2,3]},{"title":"3. Summary","content":"We summarize results presented in Shao & Wex (2012) that proposed new tests of LLI. These yield improved constraints on PFEs from binary pulsar experiments. Specifically, limits on parameters ˆ α 1 and ˆ α 2 are obtained from long-term timing of two binary pulsars with short orbital period (see Eqs. (2.1) and (2.2)). Our extended ˆ α 1 test no longer requires probabilistic considerations related to unknown angles. The proposed tests have the advantage that they continuously improve with time, and will benefit greatly from the next generation of radio telescopes, like FAST (Nan et al. 2011) and SKA (Smits et al. 2009).","pages":[3]},{"title":"Acknowledgment","content":"Lijing Shao is supported by China Scholarship Council (CSC).","pages":[3]},{"title":"References","content":"Antoniadis, J., van Kerkwijk, M. H., Koester, D., Freire, P. C. C., Wex, N., Tauris, T. M., Kramer, M., & Bassa, C. G. 2012, MNRAS , 423, 3316 Damour, T. & Esposito-Far'ese, G. 1992, Phys. Rev. D , 46, 4128 Freire, P. C. C., Wex, N., Esposito-Far'ese, G., Verbiest, J. P. W., Bailes, M., Jacoby, B. A., Kramer, M., Stairs, I. H., Antoniadis, J., & Janssen., G. H. 2012, MNRAS , 423, 3328 Kramer, M., Stairs, I. H., Manchester, R. N., McLaughlin, M. A., Lyne, A. G., Ferdman, R. D., Burgay, M., Lorimer, D. R., Possenti, A., D'Amico, N., Sarkissian, J. M., Hobbs, G. B., Reynolds, J. E., Freire, P. C. C., & Camilo, F. 2006, Science , 314, 97 Lazaridis, K., Wex, N., Jessner, A., Kramer, M., Stappers, B. W., Janssen, G. H., Desvignes, G., Purver, M. B., Cognard, I., Theureau, G., Lyne, A. G., Jordan, C. A., & Zensus, J. A. 2009, MNRAS , 400, 805 Lorimer, D. R. & Kramer, M. 2005, Handbook of Pulsar Astronomy (Cambridge University Press) Muller, J., Williams, J. G., & Turyshev, S. G. 2008, in: H. Dittus, C. Lammerzahl, & S. G. Turyshev (eds.), Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space 349 (Astrophysics and Space Science Library: Springer), p. 457 Nan, R., Li, D., Jin, C., Wang, Q., Zhu, L., Zhu, W., Zhang, H., Yue, Y., & Qian, L. 2011, International Journal of Modern Physics D , 20, 989 Nordtvedt, K. 1987, ApJ , 320, 871 Nordtvedt, K. & Will, C. M. 1972, ApJ , 177, 775 Shao, L., & Wex, N. 2012, Class. Quantum Grav. , accepted [arXiv:1209.4503] Smits, R., Kramer, M., Stappers, B., Lorimer, D. R., Cordes, J., & Faulkner, A. 2009, A&A , 493, 1161 Stairs, I. H. 2003, 'Testing General Relativity with Pulsar Timing' , Living Rev. Relativity 6, 5. URL (cited on 2012-09-11): http://www.livingreviews.org/lrr-2003-5 Wex, N. 2000, in: M. Kramer, N. Wex, & R. Wielebinski (eds.), IAU Colloq. 177: Pulsar Astronomy - 2000 and Beyond 202 (Astronomical Society of the Pacific Conference Series: San Francisco), p. 113 Wex, N., & Kramer, M. 2007, MNRAS , 380, 455 Will, C. M. 1993, Theory and Experiment in Gravitational Physics (Cambridge University Press) Will, C. M. 2006, 'The Confrontation between General Relativity and Experiment' , Living Rev. Relativity 9, 3. URL (cited on 2012-09-06): http://www.livingreviews.org/lrr-2006-3 Will, C. M., & Nordtvedt, K. 1972, ApJ , 177, 757","pages":[3]}],"string":"[\n {\n \"title\": \"Lijing Shao 1 , Norbert Wex 2 , Michael Kramer 3\",\n \"content\": \"1 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany School of Physics, Peking University, Beijing 100871, China email: lshao@pku.edu.cn 2 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany email: wex@mpifr-bonn.mpg.de 3 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, M13 9PL, UK email: mkramer@mpifr-bonn.mpg.de Abstract. Preferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, ˆ α 1 and ˆ α 2 . In the limit of a small orbital eccentricity, ˆ α 1 and ˆ α 2 contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of ˆ α 1 and ˆ α 2 in the strong-field regime. The constraint on ˆ α 1 also surpasses its counterpart in the weak-field regime. Keywords. Gravitation, pulsars: general, pulsars: individual (J1012+5307, J1738+0333)\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Pulsars are extremely stable electromagnetic emitters, and along with their extreme physical properties and surrounding environments, they provide useful astrophysical laboratories to study fundamental physics (Lorimer & Kramer 2005). Radio timing of binary pulsars maps out the binary orbital dynamics through recording the time-of-arrivals of the pulsar signals at the telescope. For millisecond pulsars this can be done with high precision, providing a powerful tool to probe gravity (see e.g., Stairs (2003) and Kramer et al. (2006)). In this work, we summarize new results on testing the local Lorentz invariance (LLI) of gravity from binary pulsars obtained by Shao & Wex (2012).\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2. New Limits on Preferred Frame Effects\",\n \"content\": \"Non-gravitational LLI is an important ingredient of the Einstein equivalence principle (EEP) (Will 1993, 2006). But even metric gravity, which fulfills the EEP exactly, could still exhibit a violation of LLI in the gravitational sector (Will & Nordtvedt 1972, Nordtvedt & Will 1972, Damour & Esposito-Far'ese 1992, Will 1993). Such a violation of LLI induces preferred frame effects (PFEs) in the orbital dynamics of a binary system that moves with respect to the preferred frame. In the parametrized post-Newtonian (PPN) formalism, PFEs of a semi-conservative gravity theory are described by two parameters, ˆ α 1 and ˆ α 2 . † The orbital dynamics of binary pulsars with non-vanishing ˆ α 1 and ˆ α 2 are obtained from a generic semi-conservative Lagrangian (Damour & Esposito-Far'ese 1992). It is found that in the limit of small orbital eccentricity, PFEs induced by ˆ α 1 and ˆ α 2 decouple, and lead to separable effects in the timing observations. Hence they can be tested independently using observations of only one binary pulsar (Shao & Wex 2012). which is significantly better than the current best limits from both weak field (Muller et al. 2008) and strong field (Wex 2000). which is better than the current best limit for strongly self-gravitating bodies (Wex & Kramer 2007) by more than three orders of magnitude, but still considerably weaker than the weak-field limit of α 2 by Nordtvedt (1987).\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"3. Summary\",\n \"content\": \"We summarize results presented in Shao & Wex (2012) that proposed new tests of LLI. These yield improved constraints on PFEs from binary pulsar experiments. Specifically, limits on parameters ˆ α 1 and ˆ α 2 are obtained from long-term timing of two binary pulsars with short orbital period (see Eqs. (2.1) and (2.2)). Our extended ˆ α 1 test no longer requires probabilistic considerations related to unknown angles. The proposed tests have the advantage that they continuously improve with time, and will benefit greatly from the next generation of radio telescopes, like FAST (Nan et al. 2011) and SKA (Smits et al. 2009).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"Acknowledgment\",\n \"content\": \"Lijing Shao is supported by China Scholarship Council (CSC).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Antoniadis, J., van Kerkwijk, M. H., Koester, D., Freire, P. C. C., Wex, N., Tauris, T. M., Kramer, M., & Bassa, C. G. 2012, MNRAS , 423, 3316 Damour, T. & Esposito-Far'ese, G. 1992, Phys. Rev. D , 46, 4128 Freire, P. C. C., Wex, N., Esposito-Far'ese, G., Verbiest, J. P. W., Bailes, M., Jacoby, B. A., Kramer, M., Stairs, I. H., Antoniadis, J., & Janssen., G. H. 2012, MNRAS , 423, 3328 Kramer, M., Stairs, I. H., Manchester, R. N., McLaughlin, M. A., Lyne, A. G., Ferdman, R. D., Burgay, M., Lorimer, D. R., Possenti, A., D'Amico, N., Sarkissian, J. M., Hobbs, G. B., Reynolds, J. E., Freire, P. C. C., & Camilo, F. 2006, Science , 314, 97 Lazaridis, K., Wex, N., Jessner, A., Kramer, M., Stappers, B. W., Janssen, G. H., Desvignes, G., Purver, M. B., Cognard, I., Theureau, G., Lyne, A. G., Jordan, C. A., & Zensus, J. A. 2009, MNRAS , 400, 805 Lorimer, D. R. & Kramer, M. 2005, Handbook of Pulsar Astronomy (Cambridge University Press) Muller, J., Williams, J. G., & Turyshev, S. G. 2008, in: H. Dittus, C. Lammerzahl, & S. G. Turyshev (eds.), Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space 349 (Astrophysics and Space Science Library: Springer), p. 457 Nan, R., Li, D., Jin, C., Wang, Q., Zhu, L., Zhu, W., Zhang, H., Yue, Y., & Qian, L. 2011, International Journal of Modern Physics D , 20, 989 Nordtvedt, K. 1987, ApJ , 320, 871 Nordtvedt, K. & Will, C. M. 1972, ApJ , 177, 775 Shao, L., & Wex, N. 2012, Class. Quantum Grav. , accepted [arXiv:1209.4503] Smits, R., Kramer, M., Stappers, B., Lorimer, D. R., Cordes, J., & Faulkner, A. 2009, A&A , 493, 1161 Stairs, I. H. 2003, 'Testing General Relativity with Pulsar Timing' , Living Rev. Relativity 6, 5. URL (cited on 2012-09-11): http://www.livingreviews.org/lrr-2003-5 Wex, N. 2000, in: M. Kramer, N. Wex, & R. Wielebinski (eds.), IAU Colloq. 177: Pulsar Astronomy - 2000 and Beyond 202 (Astronomical Society of the Pacific Conference Series: San Francisco), p. 113 Wex, N., & Kramer, M. 2007, MNRAS , 380, 455 Will, C. M. 1993, Theory and Experiment in Gravitational Physics (Cambridge University Press) Will, C. M. 2006, 'The Confrontation between General Relativity and Experiment' , Living Rev. Relativity 9, 3. URL (cited on 2012-09-06): http://www.livingreviews.org/lrr-2006-3 Will, C. M., & Nordtvedt, K. 1972, ApJ , 177, 757\",\n \"pages\": [\n 3\n ]\n }\n]"}}},{"rowIdx":279,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..291..521T"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1210.4310.pdf"},"content":{"kind":"string","value":"\nWhat can Fermi tell us about magnetars?\nH. Tong 1 and R. X. Xu 2\n1 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China Email: tonghao@xao.ac.cn\n2 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Email: r.x.xu@pku.edu.cn\nAbstract.\nWe have analyzed the physical implications of Fermi observations of magnetars. Observationally, no significant detection is reported in Fermi observations of all magnetars. Then there are conflicts between outer gap model in the case of magnetars and Fermi observations. One possible explanation is that magnetars are wind braking instead of magnetic dipole braking. In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. A magnetism-powered pulsar wind nebula and a braking index smaller than three are the two predictions of wind braking of magnetars. Future deeper Fermi observations will help us make clear whether they are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.\nKeywords. pulsars: general, stars: magnetars, stars: neutron\n1. Introduction\nAnomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are peculiar pulsar-like objects. They are commonly assumed to be magnetars, magnetism-powered neutron stars. The traditional model of magnetars is that they are young neutron stars with both strong dipole field and strong multipole field (Duncan & Thompson 1992; Thompson et al. 2002). The presence of strong dipole field ensures that they can also accelerate particles to very high energy via the outer gap mechanism. Therefore, magnetar may emit high-energy gamma-rays detectable by Fermi -LAT (Cheng & Zhang 2001). These high-energy gamma-rays are rotation-powered in nature (Zhang 2003). Detection of rotation-powered activities in mangetars will help bridge the gap between magnetars and normal pulsars.\nFermi -LAT has observed the whole magnetar class. No significant detection is reported (Sasmaz Mus & Gogus 2010; Abdo et al. 2010). Then they are conflicts between the outer gap model in the case of magnetars and Fermi observations (Tong, Song, & Xu 2010, 2011). Below we will show what can Fermi tell us about magnetars.\n2. Implications of Fermi observations of magentars\nWe have analyzed the implications of Fermi observation of AXP 4U 0142+61 (Sasmaz Mus, & Gogus 2010; Tong, Song, & Xu 2010). It is shown that there are conflicts between outer gap model in the case of AXP 4U 0142+61 and Fermi observations. The fallback disk model for AXPs and SGRs can still not be ruled out. In Fermi observations of the whole magnetar class, still no significant detection is reported. This is consistent with out previous analysis (Abdo et al. 2010; Tong, Song, & Xu 2011). The upper limit of AXP 4U 0142+62 lies already below the theoretical calculations for some parameter space (Tong,\nSong, & Xu 2011). Future deeper Fermi observations will help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.\n3. Solutions and predictions\nThere are two possible explanations to the non-detection in Fermi observations of magnetars. One possibility is that AXPs and SGRs are accretion-powered systems. Then it is natural that they are not high-energy gamam-ray emitters. Various observations of AXPs and SGRs can be explained naturally in the accretion scenario (Tong & Xu 2011).\nThe other possibility is: magnetars are wind braking. If magnetars are wind braking instead of mangetic dipole braking, then their magnetosphere structure is different from that of normal pulsars. Vacuum gaps (including outer gap) may not exist in magnetars. This may explain the non-detection in Fermi observations of magnetars (section 4.2 in Tong, Xu, Song, & Qiao 2012).\nIn the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. Figure 1 shows the correspond dipole field in the case of wind braking and that of magnetic dipole braking\n\n\n
Figure 1. Dipole magnetic field in the case of wind braking versus that in the case of magnetic dipole braking. A wind luminosity L p = 10 35 erg s -1 is assumed. The solid, dashed, and dotted lines are for B dip , w = B dip , d , 0 . 1 B dip , d , 0 . 01 B dip , d , respectively. The dot-dashed line marks the position of quantum critical magnetic field B QED = 4 . 4 × 10 13 G. See figure 2 and corresponding text in Tong, Xu, Song, & Qiao (2012) for details.
\n\nRecent challenging observations of magnetars may be explained naturally in the wind braking scenario: (1) The supernova energies of magnetars are of normal value; (2) The\nproblem posed by the low-magnetic field soft gamma-ray repeater; (3) The relation between magnetars and high magnetic field pulsars ; (4) A decreasing period derivative during magnetar outbursts etc. A magnetism-powered (instead of rotation-powered) pulsar wind nebula will be one of the consequences of wind braking. For a magnetismpowered pulsar wind nebula, we should see a correlation between the nebula luminosity and the magnetar luminosity. The extended emission around AXP 1E 1547.0-5408 may be a magnetism-powered pulsar wind nebula. Under the wind braking scenario, a braking index smaller than three is expected. More details are presented in Tong, Xu, Song, & Qiao (2012).\nConsidering that magnetars are wind braking (both a rotation-powered and a magnetismpowered particle wind), many aspects of magnetars can be reinterpreted. For example, the 'low magnetic field' magnetar SGR 0418+5729 may actually be a normal magnetar (Rea et al. 2010; Tong & Xu 2012). It is a little special since it has a special geometry, e.g. a small magnetic inclination angle. Another example is: low luminosity magnetars are more likely to have radio emissions. The reason is that low luminosity magnears may have similar magnetospheric structure to that of normal radio pulsars (Rea et al. 2012; Liu, Tong, & Yuan 2012).\n4. Conclusions\nFermi observations of magnetars tell us that magnetars may be wind braking instead of magnetic dipole braking. Future deeper Fermi -LAT observations will help us make clear whether magnetars are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Therefore, deeper Fermi -LAT observations of the magnetar class in the future are highly recommended.\nAcknowledgments\nThis work is supported by the National Basic Research Program of China (2009CB824800), the National Natural Science Foundation of China (11103021, 10935001, 10973002), and the John Templeton Foundation.\nReferences\nAbdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ , 725, L73\nCheng, K. S., & Zhang, L. 2001, ApJ , 562, 918\nDuncan, R. C., & Thompson, C. 1992, ApJ , 392, L9\nLiu, Z. Y., Tong, H., & Yuan, J. P. 2012, arXiv:1210.2799\nRea, N., Esposito, P., Turolla, R., et al. 2010, Science , 330, 944\nRea, N., Pons, J. A., Torres, D. F., et al. 2012, ApJ , 748, L12\nSasmaz Mus, S., & Gogus, E. 2010, ApJ , 723, 100\nThompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, ApJ , 574, 332\nTong, H., Song, L. M., & Xu, R. X. 2010, ApJ , 725, L196\nTong, H., Song L. M., & Xu, R. X. 2011, ApJ , 738, 31\nTong, H., & Xu, R. X. 2011, Int. Jour. Mod. Phys. E , 20, 15\nTong, H., & Xu, R. X. 2012, ApJ , 757, L10\nTong, H., Xu, R. X., Song, L. M., & Qiao, G. J. 2012, arXiv:1205.1626\nZhang, B. 2003, Astrophysics and Space Science Library , 298, 27 (astro-ph/0212016)\n"},"sections":{"kind":"list like","value":[{"title":"H. Tong 1 and R. X. Xu 2","content":"1 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China Email: tonghao@xao.ac.cn 2 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Email: r.x.xu@pku.edu.cn","pages":[1]},{"title":"Abstract.","content":"We have analyzed the physical implications of Fermi observations of magnetars. Observationally, no significant detection is reported in Fermi observations of all magnetars. Then there are conflicts between outer gap model in the case of magnetars and Fermi observations. One possible explanation is that magnetars are wind braking instead of magnetic dipole braking. In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. A magnetism-powered pulsar wind nebula and a braking index smaller than three are the two predictions of wind braking of magnetars. Future deeper Fermi observations will help us make clear whether they are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Keywords. pulsars: general, stars: magnetars, stars: neutron","pages":[1]},{"title":"1. Introduction","content":"Anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are peculiar pulsar-like objects. They are commonly assumed to be magnetars, magnetism-powered neutron stars. The traditional model of magnetars is that they are young neutron stars with both strong dipole field and strong multipole field (Duncan & Thompson 1992; Thompson et al. 2002). The presence of strong dipole field ensures that they can also accelerate particles to very high energy via the outer gap mechanism. Therefore, magnetar may emit high-energy gamma-rays detectable by Fermi -LAT (Cheng & Zhang 2001). These high-energy gamma-rays are rotation-powered in nature (Zhang 2003). Detection of rotation-powered activities in mangetars will help bridge the gap between magnetars and normal pulsars. Fermi -LAT has observed the whole magnetar class. No significant detection is reported (Sasmaz Mus & Gogus 2010; Abdo et al. 2010). Then they are conflicts between the outer gap model in the case of magnetars and Fermi observations (Tong, Song, & Xu 2010, 2011). Below we will show what can Fermi tell us about magnetars.","pages":[1]},{"title":"2. Implications of Fermi observations of magentars","content":"We have analyzed the implications of Fermi observation of AXP 4U 0142+61 (Sasmaz Mus, & Gogus 2010; Tong, Song, & Xu 2010). It is shown that there are conflicts between outer gap model in the case of AXP 4U 0142+61 and Fermi observations. The fallback disk model for AXPs and SGRs can still not be ruled out. In Fermi observations of the whole magnetar class, still no significant detection is reported. This is consistent with out previous analysis (Abdo et al. 2010; Tong, Song, & Xu 2011). The upper limit of AXP 4U 0142+62 lies already below the theoretical calculations for some parameter space (Tong, Song, & Xu 2011). Future deeper Fermi observations will help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.","pages":[1,2]},{"title":"3. Solutions and predictions","content":"There are two possible explanations to the non-detection in Fermi observations of magnetars. One possibility is that AXPs and SGRs are accretion-powered systems. Then it is natural that they are not high-energy gamam-ray emitters. Various observations of AXPs and SGRs can be explained naturally in the accretion scenario (Tong & Xu 2011). The other possibility is: magnetars are wind braking. If magnetars are wind braking instead of mangetic dipole braking, then their magnetosphere structure is different from that of normal pulsars. Vacuum gaps (including outer gap) may not exist in magnetars. This may explain the non-detection in Fermi observations of magnetars (section 4.2 in Tong, Xu, Song, & Qiao 2012). In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. Figure 1 shows the correspond dipole field in the case of wind braking and that of magnetic dipole braking Recent challenging observations of magnetars may be explained naturally in the wind braking scenario: (1) The supernova energies of magnetars are of normal value; (2) The problem posed by the low-magnetic field soft gamma-ray repeater; (3) The relation between magnetars and high magnetic field pulsars ; (4) A decreasing period derivative during magnetar outbursts etc. A magnetism-powered (instead of rotation-powered) pulsar wind nebula will be one of the consequences of wind braking. For a magnetismpowered pulsar wind nebula, we should see a correlation between the nebula luminosity and the magnetar luminosity. The extended emission around AXP 1E 1547.0-5408 may be a magnetism-powered pulsar wind nebula. Under the wind braking scenario, a braking index smaller than three is expected. More details are presented in Tong, Xu, Song, & Qiao (2012). Considering that magnetars are wind braking (both a rotation-powered and a magnetismpowered particle wind), many aspects of magnetars can be reinterpreted. For example, the 'low magnetic field' magnetar SGR 0418+5729 may actually be a normal magnetar (Rea et al. 2010; Tong & Xu 2012). It is a little special since it has a special geometry, e.g. a small magnetic inclination angle. Another example is: low luminosity magnetars are more likely to have radio emissions. The reason is that low luminosity magnears may have similar magnetospheric structure to that of normal radio pulsars (Rea et al. 2012; Liu, Tong, & Yuan 2012).","pages":[2,3]},{"title":"4. Conclusions","content":"Fermi observations of magnetars tell us that magnetars may be wind braking instead of magnetic dipole braking. Future deeper Fermi -LAT observations will help us make clear whether magnetars are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Therefore, deeper Fermi -LAT observations of the magnetar class in the future are highly recommended.","pages":[3]},{"title":"Acknowledgments","content":"This work is supported by the National Basic Research Program of China (2009CB824800), the National Natural Science Foundation of China (11103021, 10935001, 10973002), and the John Templeton Foundation.","pages":[3]},{"title":"References","content":"Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ , 725, L73 Cheng, K. S., & Zhang, L. 2001, ApJ , 562, 918 Duncan, R. C., & Thompson, C. 1992, ApJ , 392, L9 Liu, Z. Y., Tong, H., & Yuan, J. P. 2012, arXiv:1210.2799 Rea, N., Esposito, P., Turolla, R., et al. 2010, Science , 330, 944 Rea, N., Pons, J. A., Torres, D. F., et al. 2012, ApJ , 748, L12 Sasmaz Mus, S., & Gogus, E. 2010, ApJ , 723, 100 Thompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, ApJ , 574, 332 Tong, H., Song, L. M., & Xu, R. X. 2010, ApJ , 725, L196 Tong, H., Song L. M., & Xu, R. X. 2011, ApJ , 738, 31 Tong, H., & Xu, R. X. 2011, Int. Jour. Mod. Phys. E , 20, 15 Tong, H., & Xu, R. X. 2012, ApJ , 757, L10 Tong, H., Xu, R. X., Song, L. M., & Qiao, G. J. 2012, arXiv:1205.1626 Zhang, B. 2003, Astrophysics and Space Science Library , 298, 27 (astro-ph/0212016)","pages":[3]}],"string":"[\n {\n \"title\": \"H. Tong 1 and R. X. Xu 2\",\n \"content\": \"1 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China Email: tonghao@xao.ac.cn 2 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Email: r.x.xu@pku.edu.cn\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract.\",\n \"content\": \"We have analyzed the physical implications of Fermi observations of magnetars. Observationally, no significant detection is reported in Fermi observations of all magnetars. Then there are conflicts between outer gap model in the case of magnetars and Fermi observations. One possible explanation is that magnetars are wind braking instead of magnetic dipole braking. In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. A magnetism-powered pulsar wind nebula and a braking index smaller than three are the two predictions of wind braking of magnetars. Future deeper Fermi observations will help us make clear whether they are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Keywords. pulsars: general, stars: magnetars, stars: neutron\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are peculiar pulsar-like objects. They are commonly assumed to be magnetars, magnetism-powered neutron stars. The traditional model of magnetars is that they are young neutron stars with both strong dipole field and strong multipole field (Duncan & Thompson 1992; Thompson et al. 2002). The presence of strong dipole field ensures that they can also accelerate particles to very high energy via the outer gap mechanism. Therefore, magnetar may emit high-energy gamma-rays detectable by Fermi -LAT (Cheng & Zhang 2001). These high-energy gamma-rays are rotation-powered in nature (Zhang 2003). Detection of rotation-powered activities in mangetars will help bridge the gap between magnetars and normal pulsars. Fermi -LAT has observed the whole magnetar class. No significant detection is reported (Sasmaz Mus & Gogus 2010; Abdo et al. 2010). Then they are conflicts between the outer gap model in the case of magnetars and Fermi observations (Tong, Song, & Xu 2010, 2011). Below we will show what can Fermi tell us about magnetars.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2. Implications of Fermi observations of magentars\",\n \"content\": \"We have analyzed the implications of Fermi observation of AXP 4U 0142+61 (Sasmaz Mus, & Gogus 2010; Tong, Song, & Xu 2010). It is shown that there are conflicts between outer gap model in the case of AXP 4U 0142+61 and Fermi observations. The fallback disk model for AXPs and SGRs can still not be ruled out. In Fermi observations of the whole magnetar class, still no significant detection is reported. This is consistent with out previous analysis (Abdo et al. 2010; Tong, Song, & Xu 2011). The upper limit of AXP 4U 0142+62 lies already below the theoretical calculations for some parameter space (Tong, Song, & Xu 2011). Future deeper Fermi observations will help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"3. Solutions and predictions\",\n \"content\": \"There are two possible explanations to the non-detection in Fermi observations of magnetars. One possibility is that AXPs and SGRs are accretion-powered systems. Then it is natural that they are not high-energy gamam-ray emitters. Various observations of AXPs and SGRs can be explained naturally in the accretion scenario (Tong & Xu 2011). The other possibility is: magnetars are wind braking. If magnetars are wind braking instead of mangetic dipole braking, then their magnetosphere structure is different from that of normal pulsars. Vacuum gaps (including outer gap) may not exist in magnetars. This may explain the non-detection in Fermi observations of magnetars (section 4.2 in Tong, Xu, Song, & Qiao 2012). In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. Figure 1 shows the correspond dipole field in the case of wind braking and that of magnetic dipole braking Recent challenging observations of magnetars may be explained naturally in the wind braking scenario: (1) The supernova energies of magnetars are of normal value; (2) The problem posed by the low-magnetic field soft gamma-ray repeater; (3) The relation between magnetars and high magnetic field pulsars ; (4) A decreasing period derivative during magnetar outbursts etc. A magnetism-powered (instead of rotation-powered) pulsar wind nebula will be one of the consequences of wind braking. For a magnetismpowered pulsar wind nebula, we should see a correlation between the nebula luminosity and the magnetar luminosity. The extended emission around AXP 1E 1547.0-5408 may be a magnetism-powered pulsar wind nebula. Under the wind braking scenario, a braking index smaller than three is expected. More details are presented in Tong, Xu, Song, & Qiao (2012). Considering that magnetars are wind braking (both a rotation-powered and a magnetismpowered particle wind), many aspects of magnetars can be reinterpreted. For example, the 'low magnetic field' magnetar SGR 0418+5729 may actually be a normal magnetar (Rea et al. 2010; Tong & Xu 2012). It is a little special since it has a special geometry, e.g. a small magnetic inclination angle. Another example is: low luminosity magnetars are more likely to have radio emissions. The reason is that low luminosity magnears may have similar magnetospheric structure to that of normal radio pulsars (Rea et al. 2012; Liu, Tong, & Yuan 2012).\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"4. Conclusions\",\n \"content\": \"Fermi observations of magnetars tell us that magnetars may be wind braking instead of magnetic dipole braking. Future deeper Fermi -LAT observations will help us make clear whether magnetars are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Therefore, deeper Fermi -LAT observations of the magnetar class in the future are highly recommended.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"This work is supported by the National Basic Research Program of China (2009CB824800), the National Natural Science Foundation of China (11103021, 10935001, 10973002), and the John Templeton Foundation.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ , 725, L73 Cheng, K. S., & Zhang, L. 2001, ApJ , 562, 918 Duncan, R. C., & Thompson, C. 1992, ApJ , 392, L9 Liu, Z. Y., Tong, H., & Yuan, J. P. 2012, arXiv:1210.2799 Rea, N., Esposito, P., Turolla, R., et al. 2010, Science , 330, 944 Rea, N., Pons, J. A., Torres, D. F., et al. 2012, ApJ , 748, L12 Sasmaz Mus, S., & Gogus, E. 2010, ApJ , 723, 100 Thompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, ApJ , 574, 332 Tong, H., Song, L. M., & Xu, R. X. 2010, ApJ , 725, L196 Tong, H., Song L. M., & Xu, R. X. 2011, ApJ , 738, 31 Tong, H., & Xu, R. X. 2011, Int. Jour. Mod. Phys. E , 20, 15 Tong, H., & Xu, R. X. 2012, ApJ , 757, L10 Tong, H., Xu, R. X., Song, L. M., & Qiao, G. J. 2012, arXiv:1205.1626 Zhang, B. 2003, Astrophysics and Space Science Library , 298, 27 (astro-ph/0212016)\",\n \"pages\": [\n 3\n ]\n }\n]"}}},{"rowIdx":280,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..292...87T"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1211.2170.pdf"},"content":{"kind":"string","value":"\nModes of star formation from Herschel\nL. Testi 1 , 2 , E. Bressert 1 , 3 and S. Longmore 1\n1 ESO, Karl Schwarzschild srt. 2, D-85748 Garching, Germany, email: ltesti@eso.org 2 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 3 School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK\nAbstract. We summarize some of the results obtained from Herschel surveys of the nearby star forming regions and the Galactic plane. We show that in the nearby star forming regions the starless core spatial surface density distribution is very similar to that of the young stellar objects. This, taken together with the similarity between the core mass function and the initial mass function for stars and the relationship between the amount of dense gas and star formation rate, suggest that the cloud fragmentation process defines the global outcome of star formation. This 'simple' view of star formation may not hold on all scales. In particular dynamical interactions are expected to become important at the conditions required to form young massive clusters. We describe the successes of a simple criterion to identify young massive cluster precursors in our Galaxy based on (sub-)millimetre wide area surveys. We further show that in the location of our Galaxy where the best candidate for a precursor of a young massive cluster is found, the 'simple' scaling relationship between dense gas and star formation rate appear to break down. We suggest that in regions where the conditions approach those of the central molecular zone of our Galaxy it may be necessary to revise the scaling laws for star formation.\nKeywords. stars: formation, ISM: clouds, Galaxy: center\n1. Introduction\nThe Herschel Space Observatory multiband wide area surveys of our own Galaxy in the far infrared and submillimeter, when combined with surveys at other wavelengths allow us, for the first time, to obtain a detailed view of the relationship between the physical structure of gas and dust in molecular clouds and cores and the young stellar populations that are produced in these clouds. In this contribution we focus on three results using data from the Herschel Gould Belt Survey and the HIGAL survey. Detailed descriptions of these surveys are given elsewhere in this volume by Di Francesco and Molinari see also Molinari et al. (2010) and Andr'e et al. (2010).\nWe will focus on the origin of the observed ranges of young stellar objects (YSO) surface densities in the nearby star forming regions, on the possible identification of precursors of the most massive clusters in our own Galaxy and the special conditions of star formation in the Central Molecular Zone (CMZ) of our Galaxy as compared to the Galactic disk, and in particular the star forming regions in the solar neighborhood.\n2. The starless cores surface density distribution\nFollowing the extensive Spitzer surveys in nearby star forming regions, it has become clear that young stellar objects are found in a continuous range of stellar densities rather than showing a clear bimodal distribution as postulated by the paradigm of clustered vs. distributed star formation. In particular Bressert et al. (2010) showed that the distribution of surface YSO densities in the regions surveyed by Spitzer is scale-free and the\ndefinition of the percentage of stars forming in clusters is arbitrary if based on specific thresholds on the local surface density.\nThanks to the wide area surveyed and the large samples of starless cores detected, the Herschel surveys of nearby star forming regions allow us to investigate whether this continuous distribution of surface densities are an imprint of the molecular cloud fragmentation. In Bressert et al. (2012b) we combined the Herschel data in the Perseus West and Serpens clouds with infrared data from the Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010) to select samples of starless cores and compare their spatial distribution with that of the YSOs. We employed WISE color-color diagrams to identify YSOs and remove extragalactic contamination (following the method of Koenig et al. 2012), this method was cross checked, in the overlap region, with the YSOs classification in the Spitzer c2d survey (Evans et al. 2009).\nThe spatial distribution of cores and young stellar object are being compared using two separate methodologies. We first compare the cumulative distributions of surface densities of the neighbours of each YSO or core and we show that, as found by Bressert et al. (2010) for YSOs, the core distribution is scale free and matches closely that of the formed stars (see Fig. 1). In addition, we employed a minimum spanning tree method to identify groups of YSOs or cores, following the method of Gutermuth et al. (2009). As discussed above, given the lack of preferential clustering scales, the identification of groups is based on an arbitrary criterion, but if we use the same criterion for YSOs and cores, we find that the groups of the two populations tend to overlap spatially and the average densities that we derive are consistent within the uncertainties (albeit affected by very large errors).\nThese results strongly favors the view in which the density distribution of YSOs in these regions are the result of the imprint of the cloud fragmentation process, rather than dynamical evolution. Taken together with the fact that the core mass function in nearby star forming regions closely resembles the initial mass function for stars (Testi & Sargent 1998; Motte, Andr'e & Neri 1998; Konyves et al. 2010), these results strongly support the view that in these regions the global outcome of the star formation process is mainly determined by cloud fragmentation process rather than by dynamical interactions.\n3. Looking for Young Massive Clusters precursors\nNearby star forming regions do not have the conditions to form very massive and dense stellar clusters, and have very modest capabilities of forming high mass stars (if at all). Nevertheless, the Galaxy is known to host a number of very massive and dense clusters, which are also prominently seen in extragalactic starbursts. These clusters are normally called Young Massive Clusters (YMC) or, in the most extreme cases, Super Star Clusters and have total stellar masses and stellar densities that in some cases approaching those of globular clusters (see Portegies Zwart et al. 2010, for a recent review). The conditions to form these systems are expected to be very different than those found in nearby star forming regions and dynamical interactions during formation may be the dominant mechanism (Smith et al. 2009). Given the importance of these systems for extragalactic studies, constraining the physical conditions for their formation is especially important. The precursors of YMCs are expected to be very rare in our galaxy, but wide area surveys of our Galaxy at far infrared and millimetre wavelengths offer a unique possibility to search for these rare objects.\nIn Bressert et al. (2012a) we developed and tested a set of guidelines to identify good candidates of YMC precursors. The criteria that we developed are based on the fact that these precursors not only need to contain enough mass to form a YMC, but also need to\n\n\n
Fig. 3. The surface densities ( Σ ) of the cores and YSOs in western Perseus and Serpens combined. The red dotted line and the blue dashed line represents the cores and the WISE YSOs, respectively. The solid black line is the all the cores and WISE YSOs combined. The light grey solid line is the profile shown in follows: 1. The ratio of YSOs to cores in Perseus-W and Serpens is 1 . 4 and 4 . 6, respectively. While the ratio of Class II to Class I objects is 1.2 and 3.2. This implies that the Perseus-W region is younger than Serpens. Figure 1. Cumulative distribution of surface densities of cores and YSOs in Perseus and Serpens combined. Red, dotted line is for Herschel starless cores, blue, dashed line for the WISE YSOs. Solid black line is for the two samples combined. The thin grey line is the distribution for YSOs in nearby star forming regions (Bressert et al. 2010), this reaches higher surface densities but has a shape consistent with the other distributions. Figure adapted from Bressert et al. (2012b). - 8 -
\n\nBressert et al. (2010) for nearly all YSOs in the solar neighbour-\nThe similarity between the YSOs and cores spatial distri-\nbutions suggest that these are set by the cloud fragmentation\nprocess. This result, together with the similarity between the\ncores mass function and the stellar IMF in these regions (Testi &\nSargent 1998; Sandell & Knee 2001), suggest that the dynamical\nprocesses in early stellar evolution is globally less important than\ncloud fragmentation for low-mass star forming environments.\n4. Summary\nCombining Herschel, WISE, and Spitzer data we presented a\ncomparison of the spatial distribution of cores and YSOs in\nPerseus-W and Serpens star-forming regions. The results are as\nhood using several Spitzer Legacy surveys. The data is hinting\n/\n2\npc\n2. Using the minimum spanning tree method to identify groups\nof cores and YSOs, we found that they tend to be associated\nrather than not. In Perseus-W we find a few core groups that\nare not directly associated with YSO groups. This finding is\nconsistent with Perseus-W being younger than Serpens.\n3. Using the nearest neighbour method we found that the cu-\nmulative surface density distribution of the Herschel cores\nand WISE YSOs in the solar neighbourhood have similar\nprofiles. In agreement with Bressert et al. (2010), the cumu-\nlative density profiles are smooth and imply that there are no\nfixed scales for clustering (scale-free).\n4. The similarity with the cores spatial distribution suggests\nthat the spatial distribution of YSOs is already set during the\ncloud fragmentation process, and supports the idea that dy-\nother star forming regions are required to put our findings on\nsolid statistical grounds and to verify if the global densities of\nThis is a result of Spitzer having higher angular resolution ( ∼ 5) than WISE and Herschel and for the YSOs a higher sensitivity limit than WISE. The similarity between the WISE YSOs and the cores implies that the YSOs are likely tracing the substructured and filamentary natal gas that they formed from, like their earlier predecessors, and that, even at the earliest stages, star formation is scale-free. The velocity dispersion of cores and YSOs is typically low (e.g.. v core /lessorsimilar 0 . 4 km s -1 in Ophiuchus and v YSO ∼ 1 . 4 kms -1 , Andr'e et al. 2007; Covey et al. 2006), which is consistent with our results where the Σ YSO and Σ core distributions are similar. If there was significant dynamical evolution in the YSO population, the Σ distribution between the YSOs and cores should di ff er. Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); IFSI, OAP / AOT, OAA / CAISMI, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESAPRODEX (Belgium), CEA / CNES (France), DLR (Germany), ASI (Italy), and CICT / MCT (Spain). These criteria successfully confirm the earlier identifications of candidate YMC precursor identified combining ALMA and VLT data in the Antennae Galaxies (Herrera et al. 2012) as well as the candidate precursor G 0 . 25+0 . 02 identified in the HOPS and HIGAL surveys by Longmore et al. (2012a). A few more candidates, identified by Ginsburg et al. (2012), need further study. G 0 . 25 + 0 . 02 in our Galaxy offer a unique opportunity to study in detail the initial conditions for YMC formation. The initial results seem to suggest that this YMC precursor is a highly turbulent cloud (Rathborne et al. 2012). While the properties and internal structure of the cloud are still being investigated, the origin of the cloud is unclear. The cloud is located in the CMZ of our own Galaxy and the special conditions near the Galactic Centre may be responsible for its formation. How-\n±\n\n\n\nand 17\n.\n28\n13\n.\n85, respectively. Within the uncertainties the\ngroups of di ff erent populations have similar densities. The closer connection between cores groups and YSO groups observed in Serpens as compared to Perseus-W is an additional indication that star formation is more advanced in the Serpens cloud. This is in agreement and reinforces the conclusions based on the statistics of cores and YSO classes presented in Sect. 2.2. namical processes in early stellar evolution is globally less important than cloud fragmentation for low-mass star forming environments. The cores and YSO groups in Perseus-W and Serpens have similar mean densities within the error (see § 3.1). Due to the large spread in the values of the mean densities, observations of Fig. 1.- The mass-radius parameter-space for clumps partitioned by radii for r Ω (solid blue) and rvir (solid black). MPC candidates are defined with the following properties (green shaded region): a minimum mass of 3 × 10 4 M /circledot , r > r Ω for Mclump < 8 . 4 × 10 4 M /circledot , and r > rvir for Mclump > 8 . 4 × 10 4 M /circledot . Clump masses and sizes are plotted on top from three different data catalogs: IRDCs (Rathborne et al. 2006), HOPS clumps (Walsh et al. 2011), and YMCs (Portegies Zwart et al. 2010). The YMCs are converted to their possible clump progenitors by assuming that SFE is ∼ 30%, which boosts the mass of the systems by a factor of 10 / 3. The scaled YMC progenitors happen to lie near the critical r Ω line without any tweaking of parameters. Two published sources that have radii less than both their respective r Ω and rvir are G0.253+0.016 (L12; Longmore et al. 2012) and an extragalactic massive proto-cluster candidate reported in Herrera et al. (H12; 2012). Figure 2. Criteria to identify precursors of YMCs on the radius-mass plane. The values for r Ω and r vir are shown as a thin blue line and a thick black line respectively (see text for definitions). The open symbols represent candidates YMC progenitors: triangle is a YMC candidate progenitor in the Antennae Galaxies (Herrera et al. 2012); the star is G 0 . 25 + 0 . 02 near the Galactic Centre (Longmore et al. 2012a); the squares are additional candidates in our Galaxy from Ginsburg et al. (2012). Figure adapted from Bressert et al. (2012a)\nThe MPC candidates reported in Ginsburg et al. (GS; submitted) are shown as squares.\n3.2. Surface densities Bressert et al. (2010) showed that the cumulative surface density of nearly all the YSOs in the solar neighbourhood exhibits a smooth and scale-free profile. In Fig. 3 we similarly investigate how the core population in Perseus-W and Serpens compare to the WISE YSO populations. The cumulative surface density profiles of the YSOs and cores are similar, but reach lower surface densities as compared to the Bressert et al. (2010) profile. the groups are distinct or not. Acknowledgments SPIRE has been developed by a consortium of institutes led by Cardi ff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); be very compact to allow for rapid and continuous star formation even under the effect of strong feedback from forming very massive stars. The final criteria to form a bound YMC with M tot glyph[greaterorequalslant] 10 4 M glyph[circledot] , requires objects to have more than 3 × 10 4 M glyph[circledot] (assuming 30% star formation efficiency), and a radius smaller than the minimum between r Ω , the maximum radius needed to gravitationally bind a fully ionised core, and r vir , the maximum radius that would allow a crossing time of ∼ 1 Myr (the typical observed value for YMCs, Portegies Zwart et al. 2010).\n4\never, it should be noted that YMCs are not only found in the CMZ in our own Galaxy. Further studies of this and other candidates will help clarify their nature.\n4. Star formation and dense gas in the CMZ\nIn the previous section we have discussed a unique massive core in the CMZ of our Galaxy, and we speculated that in the CMZ the conditions may be favorable for the assembly of such extreme objects. In the solar neighborhood, several different arguments suggest that the process of star formation is controlled by the amount of dense molecular gas. Lada et al. (2012) and Krumholz et al. (2012) presented the latest incarnation of the views that the star formation rates observed in star forming regions and entire galaxies could be understood in terms of simple scaling relations with the amount of dense gas above a certain column density threshold (Lada's formulation) or the total local volumetric gas density (Krumholz's formulation).\nWe used data from HOPS and HIGAL Galactic plane surveys as well as some other datasets to show that the relationships between star formation rate and dense gas that appear to hold throughout the Galactic disk do appear to break down in the CMZ (Longmore et al. 2012b). More details on the methodology and results are given by Longmore in this volume, but the basic result is that while ∼ 80% of the dense gas in our own Galaxy is concentrated in the CMZ, there is no similar enhancement in the star formation rate. Similarly, the predictions of the volumetric gas density relations exceed by at least a factor of ten the observed star formation rate. These findings suggest that the prescriptions based on a single parameter (the total gas density or the dense gas column density) break down in the CMZ and a more complicated formulation is needed to reconcile this region with the rest of the Galaxy. The large shear and increased turbulence in the CMZ need to be looked at as the possible causes for the increased stability of cloud cores in that region. These processes may also favor the formation of more massive stable clumps that once they became unstable and collapse may form YMCs.\nReferences\nAndr´e, Ph., Menshchikov, A., Bontemps, S., et al. 2010, A&A , 518, L102\nBressert, E., Bastian, N., Gutermuth, R., et al. 2010, MNRAS , 409, L54\nBressert, E., Ginsburg, A., Bally, J., et al. 2012, ApJL , in press\nBressert, E., Testi, L., Facchini, A., et al. 2012, A&A , submitted\nEvans, N. J., Dunham, M. M., Jrgensen, J. K., et al. 2009, ApJS , 181, 321\nGinsburg, A., Bressert, E., Bally, J., Battersby, C., et al. 2012, ApJ , in press\nHerrera,C.N., Boulanger,F., Nesvadba,N.P.H., Falgarone,E. 2012, A&A , 538, L9\nKoenig, X. P., Leisawitz, D. T., Benford, D. J., et al. 2012, ApJ , 744, 130\nK¨onyves, V., Andr´e, Ph., Men'shchikov, A., et al. 2010, A&A , 518, L106\nKrumholz, M.R., Dekel, A., McKee, C.F. 2012, ApJ , 745, 69\nLada, C.J., Forbrich, J., Lombardi, M., Alves, J.F. 2012, ApJ , 745, 190\nLongmore,S.N., Bally, J., Testi, L., et al. 2012, MNRAS , arXiv:1208.4256\nLongmore,S.N., Rathborne,J., Bastian,N., et al. 2012, ApJ , 746, 117\nMolinari, S., Swinyard, B., Bally, J., et al. 2012, A&A , 518, L100\nMotte, F., Andr´e, Ph., & Neri, R. 1998, A&A , 336, 150\nPortegies Zwart, S.F., McMillan, S.L.W., & Gieles, M. 2010, ARA&A 48, 431\nRathborne, J.M., Longmore, S.N., Jackson, J.M., et al. 2012, ApJ , submitted\nSmith, R.J., Longmore, S.N., Bonnell, I. 2009, MNRAS , 400, 1775\nTesti, L. & Sargent, A.I. 1998, ApJ , 508, L91\nWright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ , 140, 1868\n"},"sections":{"kind":"list like","value":[{"title":"Modes of star formation from Herschel","content":"L. Testi 1 , 2 , E. Bressert 1 , 3 and S. Longmore 1 1 ESO, Karl Schwarzschild srt. 2, D-85748 Garching, Germany, email: ltesti@eso.org 2 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 3 School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Abstract. We summarize some of the results obtained from Herschel surveys of the nearby star forming regions and the Galactic plane. We show that in the nearby star forming regions the starless core spatial surface density distribution is very similar to that of the young stellar objects. This, taken together with the similarity between the core mass function and the initial mass function for stars and the relationship between the amount of dense gas and star formation rate, suggest that the cloud fragmentation process defines the global outcome of star formation. This 'simple' view of star formation may not hold on all scales. In particular dynamical interactions are expected to become important at the conditions required to form young massive clusters. We describe the successes of a simple criterion to identify young massive cluster precursors in our Galaxy based on (sub-)millimetre wide area surveys. We further show that in the location of our Galaxy where the best candidate for a precursor of a young massive cluster is found, the 'simple' scaling relationship between dense gas and star formation rate appear to break down. We suggest that in regions where the conditions approach those of the central molecular zone of our Galaxy it may be necessary to revise the scaling laws for star formation. Keywords. stars: formation, ISM: clouds, Galaxy: center","pages":[1]},{"title":"1. Introduction","content":"The Herschel Space Observatory multiband wide area surveys of our own Galaxy in the far infrared and submillimeter, when combined with surveys at other wavelengths allow us, for the first time, to obtain a detailed view of the relationship between the physical structure of gas and dust in molecular clouds and cores and the young stellar populations that are produced in these clouds. In this contribution we focus on three results using data from the Herschel Gould Belt Survey and the HIGAL survey. Detailed descriptions of these surveys are given elsewhere in this volume by Di Francesco and Molinari see also Molinari et al. (2010) and Andr'e et al. (2010). We will focus on the origin of the observed ranges of young stellar objects (YSO) surface densities in the nearby star forming regions, on the possible identification of precursors of the most massive clusters in our own Galaxy and the special conditions of star formation in the Central Molecular Zone (CMZ) of our Galaxy as compared to the Galactic disk, and in particular the star forming regions in the solar neighborhood.","pages":[1]},{"title":"2. The starless cores surface density distribution","content":"Following the extensive Spitzer surveys in nearby star forming regions, it has become clear that young stellar objects are found in a continuous range of stellar densities rather than showing a clear bimodal distribution as postulated by the paradigm of clustered vs. distributed star formation. In particular Bressert et al. (2010) showed that the distribution of surface YSO densities in the regions surveyed by Spitzer is scale-free and the definition of the percentage of stars forming in clusters is arbitrary if based on specific thresholds on the local surface density. Thanks to the wide area surveyed and the large samples of starless cores detected, the Herschel surveys of nearby star forming regions allow us to investigate whether this continuous distribution of surface densities are an imprint of the molecular cloud fragmentation. In Bressert et al. (2012b) we combined the Herschel data in the Perseus West and Serpens clouds with infrared data from the Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010) to select samples of starless cores and compare their spatial distribution with that of the YSOs. We employed WISE color-color diagrams to identify YSOs and remove extragalactic contamination (following the method of Koenig et al. 2012), this method was cross checked, in the overlap region, with the YSOs classification in the Spitzer c2d survey (Evans et al. 2009). The spatial distribution of cores and young stellar object are being compared using two separate methodologies. We first compare the cumulative distributions of surface densities of the neighbours of each YSO or core and we show that, as found by Bressert et al. (2010) for YSOs, the core distribution is scale free and matches closely that of the formed stars (see Fig. 1). In addition, we employed a minimum spanning tree method to identify groups of YSOs or cores, following the method of Gutermuth et al. (2009). As discussed above, given the lack of preferential clustering scales, the identification of groups is based on an arbitrary criterion, but if we use the same criterion for YSOs and cores, we find that the groups of the two populations tend to overlap spatially and the average densities that we derive are consistent within the uncertainties (albeit affected by very large errors). These results strongly favors the view in which the density distribution of YSOs in these regions are the result of the imprint of the cloud fragmentation process, rather than dynamical evolution. Taken together with the fact that the core mass function in nearby star forming regions closely resembles the initial mass function for stars (Testi & Sargent 1998; Motte, Andr'e & Neri 1998; Konyves et al. 2010), these results strongly support the view that in these regions the global outcome of the star formation process is mainly determined by cloud fragmentation process rather than by dynamical interactions.","pages":[1,2]},{"title":"3. Looking for Young Massive Clusters precursors","content":"Nearby star forming regions do not have the conditions to form very massive and dense stellar clusters, and have very modest capabilities of forming high mass stars (if at all). Nevertheless, the Galaxy is known to host a number of very massive and dense clusters, which are also prominently seen in extragalactic starbursts. These clusters are normally called Young Massive Clusters (YMC) or, in the most extreme cases, Super Star Clusters and have total stellar masses and stellar densities that in some cases approaching those of globular clusters (see Portegies Zwart et al. 2010, for a recent review). The conditions to form these systems are expected to be very different than those found in nearby star forming regions and dynamical interactions during formation may be the dominant mechanism (Smith et al. 2009). Given the importance of these systems for extragalactic studies, constraining the physical conditions for their formation is especially important. The precursors of YMCs are expected to be very rare in our galaxy, but wide area surveys of our Galaxy at far infrared and millimetre wavelengths offer a unique possibility to search for these rare objects. In Bressert et al. (2012a) we developed and tested a set of guidelines to identify good candidates of YMC precursors. The criteria that we developed are based on the fact that these precursors not only need to contain enough mass to form a YMC, but also need to Bressert et al. (2010) for nearly all YSOs in the solar neighbour- The similarity between the YSOs and cores spatial distri- butions suggest that these are set by the cloud fragmentation process. This result, together with the similarity between the cores mass function and the stellar IMF in these regions (Testi & Sargent 1998; Sandell & Knee 2001), suggest that the dynamical processes in early stellar evolution is globally less important than cloud fragmentation for low-mass star forming environments. 4. Summary Combining Herschel, WISE, and Spitzer data we presented a comparison of the spatial distribution of cores and YSOs in Perseus-W and Serpens star-forming regions. The results are as hood using several Spitzer Legacy surveys. The data is hinting / 2 pc 2. Using the minimum spanning tree method to identify groups of cores and YSOs, we found that they tend to be associated rather than not. In Perseus-W we find a few core groups that are not directly associated with YSO groups. This finding is consistent with Perseus-W being younger than Serpens. 3. Using the nearest neighbour method we found that the cu- mulative surface density distribution of the Herschel cores and WISE YSOs in the solar neighbourhood have similar profiles. In agreement with Bressert et al. (2010), the cumu- lative density profiles are smooth and imply that there are no fixed scales for clustering (scale-free). 4. The similarity with the cores spatial distribution suggests that the spatial distribution of YSOs is already set during the cloud fragmentation process, and supports the idea that dy- other star forming regions are required to put our findings on solid statistical grounds and to verify if the global densities of This is a result of Spitzer having higher angular resolution ( ∼ 5) than WISE and Herschel and for the YSOs a higher sensitivity limit than WISE. The similarity between the WISE YSOs and the cores implies that the YSOs are likely tracing the substructured and filamentary natal gas that they formed from, like their earlier predecessors, and that, even at the earliest stages, star formation is scale-free. The velocity dispersion of cores and YSOs is typically low (e.g.. v core /lessorsimilar 0 . 4 km s -1 in Ophiuchus and v YSO ∼ 1 . 4 kms -1 , Andr'e et al. 2007; Covey et al. 2006), which is consistent with our results where the Σ YSO and Σ core distributions are similar. If there was significant dynamical evolution in the YSO population, the Σ distribution between the YSOs and cores should di ff er. Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); IFSI, OAP / AOT, OAA / CAISMI, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESAPRODEX (Belgium), CEA / CNES (France), DLR (Germany), ASI (Italy), and CICT / MCT (Spain). These criteria successfully confirm the earlier identifications of candidate YMC precursor identified combining ALMA and VLT data in the Antennae Galaxies (Herrera et al. 2012) as well as the candidate precursor G 0 . 25+0 . 02 identified in the HOPS and HIGAL surveys by Longmore et al. (2012a). A few more candidates, identified by Ginsburg et al. (2012), need further study. G 0 . 25 + 0 . 02 in our Galaxy offer a unique opportunity to study in detail the initial conditions for YMC formation. The initial results seem to suggest that this YMC precursor is a highly turbulent cloud (Rathborne et al. 2012). While the properties and internal structure of the cloud are still being investigated, the origin of the cloud is unclear. The cloud is located in the CMZ of our own Galaxy and the special conditions near the Galactic Centre may be responsible for its formation. How- ± and 17 . 28 13 . 85, respectively. Within the uncertainties the The MPC candidates reported in Ginsburg et al. (GS; submitted) are shown as squares. 3.2. Surface densities Bressert et al. (2010) showed that the cumulative surface density of nearly all the YSOs in the solar neighbourhood exhibits a smooth and scale-free profile. In Fig. 3 we similarly investigate how the core population in Perseus-W and Serpens compare to the WISE YSO populations. The cumulative surface density profiles of the YSOs and cores are similar, but reach lower surface densities as compared to the Bressert et al. (2010) profile. the groups are distinct or not. Acknowledgments SPIRE has been developed by a consortium of institutes led by Cardi ff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); be very compact to allow for rapid and continuous star formation even under the effect of strong feedback from forming very massive stars. The final criteria to form a bound YMC with M tot glyph[greaterorequalslant] 10 4 M glyph[circledot] , requires objects to have more than 3 × 10 4 M glyph[circledot] (assuming 30% star formation efficiency), and a radius smaller than the minimum between r Ω , the maximum radius needed to gravitationally bind a fully ionised core, and r vir , the maximum radius that would allow a crossing time of ∼ 1 Myr (the typical observed value for YMCs, Portegies Zwart et al. 2010). 4 ever, it should be noted that YMCs are not only found in the CMZ in our own Galaxy. Further studies of this and other candidates will help clarify their nature.","pages":[2,3,4]},{"title":"4. Star formation and dense gas in the CMZ","content":"In the previous section we have discussed a unique massive core in the CMZ of our Galaxy, and we speculated that in the CMZ the conditions may be favorable for the assembly of such extreme objects. In the solar neighborhood, several different arguments suggest that the process of star formation is controlled by the amount of dense molecular gas. Lada et al. (2012) and Krumholz et al. (2012) presented the latest incarnation of the views that the star formation rates observed in star forming regions and entire galaxies could be understood in terms of simple scaling relations with the amount of dense gas above a certain column density threshold (Lada's formulation) or the total local volumetric gas density (Krumholz's formulation). We used data from HOPS and HIGAL Galactic plane surveys as well as some other datasets to show that the relationships between star formation rate and dense gas that appear to hold throughout the Galactic disk do appear to break down in the CMZ (Longmore et al. 2012b). More details on the methodology and results are given by Longmore in this volume, but the basic result is that while ∼ 80% of the dense gas in our own Galaxy is concentrated in the CMZ, there is no similar enhancement in the star formation rate. Similarly, the predictions of the volumetric gas density relations exceed by at least a factor of ten the observed star formation rate. These findings suggest that the prescriptions based on a single parameter (the total gas density or the dense gas column density) break down in the CMZ and a more complicated formulation is needed to reconcile this region with the rest of the Galaxy. The large shear and increased turbulence in the CMZ need to be looked at as the possible causes for the increased stability of cloud cores in that region. These processes may also favor the formation of more massive stable clumps that once they became unstable and collapse may form YMCs.","pages":[4]},{"title":"References","content":"Andr´e, Ph., Menshchikov, A., Bontemps, S., et al. 2010, A&A , 518, L102 Bressert, E., Bastian, N., Gutermuth, R., et al. 2010, MNRAS , 409, L54 Bressert, E., Ginsburg, A., Bally, J., et al. 2012, ApJL , in press Bressert, E., Testi, L., Facchini, A., et al. 2012, A&A , submitted Evans, N. J., Dunham, M. M., Jrgensen, J. K., et al. 2009, ApJS , 181, 321 Ginsburg, A., Bressert, E., Bally, J., Battersby, C., et al. 2012, ApJ , in press Herrera,C.N., Boulanger,F., Nesvadba,N.P.H., Falgarone,E. 2012, A&A , 538, L9 Koenig, X. P., Leisawitz, D. T., Benford, D. J., et al. 2012, ApJ , 744, 130 K¨onyves, V., Andr´e, Ph., Men'shchikov, A., et al. 2010, A&A , 518, L106 Krumholz, M.R., Dekel, A., McKee, C.F. 2012, ApJ , 745, 69 Lada, C.J., Forbrich, J., Lombardi, M., Alves, J.F. 2012, ApJ , 745, 190 Longmore,S.N., Bally, J., Testi, L., et al. 2012, MNRAS , arXiv:1208.4256 Longmore,S.N., Rathborne,J., Bastian,N., et al. 2012, ApJ , 746, 117 Molinari, S., Swinyard, B., Bally, J., et al. 2012, A&A , 518, L100 Motte, F., Andr´e, Ph., & Neri, R. 1998, A&A , 336, 150 Portegies Zwart, S.F., McMillan, S.L.W., & Gieles, M. 2010, ARA&A 48, 431 Rathborne, J.M., Longmore, S.N., Jackson, J.M., et al. 2012, ApJ , submitted Smith, R.J., Longmore, S.N., Bonnell, I. 2009, MNRAS , 400, 1775 Testi, L. & Sargent, A.I. 1998, ApJ , 508, L91 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ , 140, 1868","pages":[4]}],"string":"[\n {\n \"title\": \"Modes of star formation from Herschel\",\n \"content\": \"L. Testi 1 , 2 , E. Bressert 1 , 3 and S. Longmore 1 1 ESO, Karl Schwarzschild srt. 2, D-85748 Garching, Germany, email: ltesti@eso.org 2 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 3 School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Abstract. We summarize some of the results obtained from Herschel surveys of the nearby star forming regions and the Galactic plane. We show that in the nearby star forming regions the starless core spatial surface density distribution is very similar to that of the young stellar objects. This, taken together with the similarity between the core mass function and the initial mass function for stars and the relationship between the amount of dense gas and star formation rate, suggest that the cloud fragmentation process defines the global outcome of star formation. This 'simple' view of star formation may not hold on all scales. In particular dynamical interactions are expected to become important at the conditions required to form young massive clusters. We describe the successes of a simple criterion to identify young massive cluster precursors in our Galaxy based on (sub-)millimetre wide area surveys. We further show that in the location of our Galaxy where the best candidate for a precursor of a young massive cluster is found, the 'simple' scaling relationship between dense gas and star formation rate appear to break down. We suggest that in regions where the conditions approach those of the central molecular zone of our Galaxy it may be necessary to revise the scaling laws for star formation. Keywords. stars: formation, ISM: clouds, Galaxy: center\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"The Herschel Space Observatory multiband wide area surveys of our own Galaxy in the far infrared and submillimeter, when combined with surveys at other wavelengths allow us, for the first time, to obtain a detailed view of the relationship between the physical structure of gas and dust in molecular clouds and cores and the young stellar populations that are produced in these clouds. In this contribution we focus on three results using data from the Herschel Gould Belt Survey and the HIGAL survey. Detailed descriptions of these surveys are given elsewhere in this volume by Di Francesco and Molinari see also Molinari et al. (2010) and Andr'e et al. (2010). We will focus on the origin of the observed ranges of young stellar objects (YSO) surface densities in the nearby star forming regions, on the possible identification of precursors of the most massive clusters in our own Galaxy and the special conditions of star formation in the Central Molecular Zone (CMZ) of our Galaxy as compared to the Galactic disk, and in particular the star forming regions in the solar neighborhood.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2. The starless cores surface density distribution\",\n \"content\": \"Following the extensive Spitzer surveys in nearby star forming regions, it has become clear that young stellar objects are found in a continuous range of stellar densities rather than showing a clear bimodal distribution as postulated by the paradigm of clustered vs. distributed star formation. In particular Bressert et al. (2010) showed that the distribution of surface YSO densities in the regions surveyed by Spitzer is scale-free and the definition of the percentage of stars forming in clusters is arbitrary if based on specific thresholds on the local surface density. Thanks to the wide area surveyed and the large samples of starless cores detected, the Herschel surveys of nearby star forming regions allow us to investigate whether this continuous distribution of surface densities are an imprint of the molecular cloud fragmentation. In Bressert et al. (2012b) we combined the Herschel data in the Perseus West and Serpens clouds with infrared data from the Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010) to select samples of starless cores and compare their spatial distribution with that of the YSOs. We employed WISE color-color diagrams to identify YSOs and remove extragalactic contamination (following the method of Koenig et al. 2012), this method was cross checked, in the overlap region, with the YSOs classification in the Spitzer c2d survey (Evans et al. 2009). The spatial distribution of cores and young stellar object are being compared using two separate methodologies. We first compare the cumulative distributions of surface densities of the neighbours of each YSO or core and we show that, as found by Bressert et al. (2010) for YSOs, the core distribution is scale free and matches closely that of the formed stars (see Fig. 1). In addition, we employed a minimum spanning tree method to identify groups of YSOs or cores, following the method of Gutermuth et al. (2009). As discussed above, given the lack of preferential clustering scales, the identification of groups is based on an arbitrary criterion, but if we use the same criterion for YSOs and cores, we find that the groups of the two populations tend to overlap spatially and the average densities that we derive are consistent within the uncertainties (albeit affected by very large errors). These results strongly favors the view in which the density distribution of YSOs in these regions are the result of the imprint of the cloud fragmentation process, rather than dynamical evolution. Taken together with the fact that the core mass function in nearby star forming regions closely resembles the initial mass function for stars (Testi & Sargent 1998; Motte, Andr'e & Neri 1998; Konyves et al. 2010), these results strongly support the view that in these regions the global outcome of the star formation process is mainly determined by cloud fragmentation process rather than by dynamical interactions.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"3. Looking for Young Massive Clusters precursors\",\n \"content\": \"Nearby star forming regions do not have the conditions to form very massive and dense stellar clusters, and have very modest capabilities of forming high mass stars (if at all). Nevertheless, the Galaxy is known to host a number of very massive and dense clusters, which are also prominently seen in extragalactic starbursts. These clusters are normally called Young Massive Clusters (YMC) or, in the most extreme cases, Super Star Clusters and have total stellar masses and stellar densities that in some cases approaching those of globular clusters (see Portegies Zwart et al. 2010, for a recent review). The conditions to form these systems are expected to be very different than those found in nearby star forming regions and dynamical interactions during formation may be the dominant mechanism (Smith et al. 2009). Given the importance of these systems for extragalactic studies, constraining the physical conditions for their formation is especially important. The precursors of YMCs are expected to be very rare in our galaxy, but wide area surveys of our Galaxy at far infrared and millimetre wavelengths offer a unique possibility to search for these rare objects. In Bressert et al. (2012a) we developed and tested a set of guidelines to identify good candidates of YMC precursors. The criteria that we developed are based on the fact that these precursors not only need to contain enough mass to form a YMC, but also need to Bressert et al. (2010) for nearly all YSOs in the solar neighbour- The similarity between the YSOs and cores spatial distri- butions suggest that these are set by the cloud fragmentation process. This result, together with the similarity between the cores mass function and the stellar IMF in these regions (Testi & Sargent 1998; Sandell & Knee 2001), suggest that the dynamical processes in early stellar evolution is globally less important than cloud fragmentation for low-mass star forming environments. 4. Summary Combining Herschel, WISE, and Spitzer data we presented a comparison of the spatial distribution of cores and YSOs in Perseus-W and Serpens star-forming regions. The results are as hood using several Spitzer Legacy surveys. The data is hinting / 2 pc 2. Using the minimum spanning tree method to identify groups of cores and YSOs, we found that they tend to be associated rather than not. In Perseus-W we find a few core groups that are not directly associated with YSO groups. This finding is consistent with Perseus-W being younger than Serpens. 3. Using the nearest neighbour method we found that the cu- mulative surface density distribution of the Herschel cores and WISE YSOs in the solar neighbourhood have similar profiles. In agreement with Bressert et al. (2010), the cumu- lative density profiles are smooth and imply that there are no fixed scales for clustering (scale-free). 4. The similarity with the cores spatial distribution suggests that the spatial distribution of YSOs is already set during the cloud fragmentation process, and supports the idea that dy- other star forming regions are required to put our findings on solid statistical grounds and to verify if the global densities of This is a result of Spitzer having higher angular resolution ( ∼ 5) than WISE and Herschel and for the YSOs a higher sensitivity limit than WISE. The similarity between the WISE YSOs and the cores implies that the YSOs are likely tracing the substructured and filamentary natal gas that they formed from, like their earlier predecessors, and that, even at the earliest stages, star formation is scale-free. The velocity dispersion of cores and YSOs is typically low (e.g.. v core /lessorsimilar 0 . 4 km s -1 in Ophiuchus and v YSO ∼ 1 . 4 kms -1 , Andr'e et al. 2007; Covey et al. 2006), which is consistent with our results where the Σ YSO and Σ core distributions are similar. If there was significant dynamical evolution in the YSO population, the Σ distribution between the YSOs and cores should di ff er. Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); IFSI, OAP / AOT, OAA / CAISMI, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESAPRODEX (Belgium), CEA / CNES (France), DLR (Germany), ASI (Italy), and CICT / MCT (Spain). These criteria successfully confirm the earlier identifications of candidate YMC precursor identified combining ALMA and VLT data in the Antennae Galaxies (Herrera et al. 2012) as well as the candidate precursor G 0 . 25+0 . 02 identified in the HOPS and HIGAL surveys by Longmore et al. (2012a). A few more candidates, identified by Ginsburg et al. (2012), need further study. G 0 . 25 + 0 . 02 in our Galaxy offer a unique opportunity to study in detail the initial conditions for YMC formation. The initial results seem to suggest that this YMC precursor is a highly turbulent cloud (Rathborne et al. 2012). While the properties and internal structure of the cloud are still being investigated, the origin of the cloud is unclear. The cloud is located in the CMZ of our own Galaxy and the special conditions near the Galactic Centre may be responsible for its formation. How- ± and 17 . 28 13 . 85, respectively. Within the uncertainties the The MPC candidates reported in Ginsburg et al. (GS; submitted) are shown as squares. 3.2. Surface densities Bressert et al. (2010) showed that the cumulative surface density of nearly all the YSOs in the solar neighbourhood exhibits a smooth and scale-free profile. In Fig. 3 we similarly investigate how the core population in Perseus-W and Serpens compare to the WISE YSO populations. The cumulative surface density profiles of the YSOs and cores are similar, but reach lower surface densities as compared to the Bressert et al. (2010) profile. the groups are distinct or not. Acknowledgments SPIRE has been developed by a consortium of institutes led by Cardi ff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); be very compact to allow for rapid and continuous star formation even under the effect of strong feedback from forming very massive stars. The final criteria to form a bound YMC with M tot glyph[greaterorequalslant] 10 4 M glyph[circledot] , requires objects to have more than 3 × 10 4 M glyph[circledot] (assuming 30% star formation efficiency), and a radius smaller than the minimum between r Ω , the maximum radius needed to gravitationally bind a fully ionised core, and r vir , the maximum radius that would allow a crossing time of ∼ 1 Myr (the typical observed value for YMCs, Portegies Zwart et al. 2010). 4 ever, it should be noted that YMCs are not only found in the CMZ in our own Galaxy. Further studies of this and other candidates will help clarify their nature.\",\n \"pages\": [\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"4. Star formation and dense gas in the CMZ\",\n \"content\": \"In the previous section we have discussed a unique massive core in the CMZ of our Galaxy, and we speculated that in the CMZ the conditions may be favorable for the assembly of such extreme objects. In the solar neighborhood, several different arguments suggest that the process of star formation is controlled by the amount of dense molecular gas. Lada et al. (2012) and Krumholz et al. (2012) presented the latest incarnation of the views that the star formation rates observed in star forming regions and entire galaxies could be understood in terms of simple scaling relations with the amount of dense gas above a certain column density threshold (Lada's formulation) or the total local volumetric gas density (Krumholz's formulation). We used data from HOPS and HIGAL Galactic plane surveys as well as some other datasets to show that the relationships between star formation rate and dense gas that appear to hold throughout the Galactic disk do appear to break down in the CMZ (Longmore et al. 2012b). More details on the methodology and results are given by Longmore in this volume, but the basic result is that while ∼ 80% of the dense gas in our own Galaxy is concentrated in the CMZ, there is no similar enhancement in the star formation rate. Similarly, the predictions of the volumetric gas density relations exceed by at least a factor of ten the observed star formation rate. These findings suggest that the prescriptions based on a single parameter (the total gas density or the dense gas column density) break down in the CMZ and a more complicated formulation is needed to reconcile this region with the rest of the Galaxy. The large shear and increased turbulence in the CMZ need to be looked at as the possible causes for the increased stability of cloud cores in that region. These processes may also favor the formation of more massive stable clumps that once they became unstable and collapse may form YMCs.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Andr´e, Ph., Menshchikov, A., Bontemps, S., et al. 2010, A&A , 518, L102 Bressert, E., Bastian, N., Gutermuth, R., et al. 2010, MNRAS , 409, L54 Bressert, E., Ginsburg, A., Bally, J., et al. 2012, ApJL , in press Bressert, E., Testi, L., Facchini, A., et al. 2012, A&A , submitted Evans, N. J., Dunham, M. M., Jrgensen, J. K., et al. 2009, ApJS , 181, 321 Ginsburg, A., Bressert, E., Bally, J., Battersby, C., et al. 2012, ApJ , in press Herrera,C.N., Boulanger,F., Nesvadba,N.P.H., Falgarone,E. 2012, A&A , 538, L9 Koenig, X. P., Leisawitz, D. T., Benford, D. J., et al. 2012, ApJ , 744, 130 K¨onyves, V., Andr´e, Ph., Men'shchikov, A., et al. 2010, A&A , 518, L106 Krumholz, M.R., Dekel, A., McKee, C.F. 2012, ApJ , 745, 69 Lada, C.J., Forbrich, J., Lombardi, M., Alves, J.F. 2012, ApJ , 745, 190 Longmore,S.N., Bally, J., Testi, L., et al. 2012, MNRAS , arXiv:1208.4256 Longmore,S.N., Rathborne,J., Bastian,N., et al. 2012, ApJ , 746, 117 Molinari, S., Swinyard, B., Bally, J., et al. 2012, A&A , 518, L100 Motte, F., Andr´e, Ph., & Neri, R. 1998, A&A , 336, 150 Portegies Zwart, S.F., McMillan, S.L.W., & Gieles, M. 2010, ARA&A 48, 431 Rathborne, J.M., Longmore, S.N., Jackson, J.M., et al. 2012, ApJ , submitted Smith, R.J., Longmore, S.N., Bonnell, I. 2009, MNRAS , 400, 1775 Testi, L. & Sargent, A.I. 1998, ApJ , 508, L91 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ , 140, 1868\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":281,"cells":{"bibcode":{"kind":"string","value":"2013IAUS..294..577T"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1306.3116.pdf"},"content":{"kind":"string","value":"\nAn Peculiar Microwave Quasi-periodic Pulsations with Zigzag Pattern in a CME-related Flare on 2005-01-15\nBaolin Tan 1\n1 Key Laboratory of Solar Activity, National Astronomical Observatories of the Chinese Academy of Sciences, Beijing 100012, China. email: bltan@nao.cas.cn\nAbstract.\nA microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in a solar flare on 200501-15 is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at 1.10-1.34 GHz. The zigzag pulsation occurred just in the early rising phase of the flare with weakly right-handed circular polarization. Its period is only several decades millisecond. Particularly, before and after the pulsation, there are many spectral fine structures, such as zebra patterns, fibers, and millisecond spikes. The microwave Z-QPP can provide some kinematic information of the source region in the early rising phase of the flare, and the source width changes from ∼ 1000 km to 3300 km, even if we have no imaging observations. The abundant spectral fine structures possibly reflect the dynamic features of non-thermal particles.\nKeywords. solar flare, microwave bursts, fine structures\nSolar microwave spectral fine structures are the most important and interesting phenomena, which can provide many intrinsic features of solar eruptions. This work reports first time a peculiar fine structure: a microwave quasi-periodic pulsation (QPP) with zigzag pattern (abbreviated as Z-QPP, hereafter) in a flare on 2005-01-15.\n\n\n
Figure 1. Left panels are the profiles of the M8.6 long-duration flare on 2005-1-15. Soft X-ray intensity observed by GOES (upper), microwave emission of left-handed circular polarization at 1.20 GHz (middle), and the microwave emission of right-handed circular polarization at 1.20 GHz (bottom). Right panels are the spectrogram of microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in the early rising phase of the flare.
\n\nThe flare is an M8.6 class long-duration event accompanying with a powerful CME in active region AR10720 with location of N16E04, very close to the center of solar disk. It is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at frequency of 1.10-1.34 GHz (4.0 MHz resolution, 1.25 ms cadence). The left panels of Figure 1 show that the flare lasts from 05:54 UT to 07:17 UT, for about 83 minutes (Cheng et al, 2010). The Z-QPP occurred just in the early rising phase of the flare. The right panels of present the Z-QPP with left- and right-handed circular polarization spectrogram. The Z-QPP is weakly right-handed circular polarization, its period is only several decades of milliseconds, which belongs to very short period pulsation. We may partition the Z-QPP into three paragraphs. The first paragraph starts at 06:04:32 UT, ends at 06:04:43 UT, the global frequency drifting rate is about -16.5 MHz/s, the pulse frequency drifting rate is 2.30-9.60 GHz/s, the period is 40-85 ms with averaged value of 60 ms. The bandwidth increases slowly from 60 MHz to 160 MHz. The second paragraph starts at 06:04:43 UT and ends at 06:04:47 UT, its global frequency drifting rate is about 0 and the pulse frequency drifting rate is 9.60-14.40 GHz/s, the period is 48-106 ms, and the bandwidth 140-200 MHz, few variations. The third paragraph starts at 06:04:47 UT and ends at 06:04:47 UT, its global frequency drifting rate is about -5.6 MHz/s and the pulse frequency drifting rate is 6.7-20 GHz/s, the period is 38-87 ms, and the bandwidth increases slowly from 60 to 140 MHz, and possibly beyond frequency domain (1.34 GHz).\nThe frequency distance between the two adjacent strips of the concomitant zebra pattern is about 80 MHz, which implies that the magnetic field in the source region is about 71.5 Gs from DPR model. The weakly polarization and strongly emission intensity imply that plasma emission is most possibly the emission mechanism. The Z-QPP may reflect the variations of the physical conditions and the dynamic processes in source region (Aschwanden & Benz, 1986; Tan, 2008). The velocities of source motion or energetic particles ( v ), and the width of source region (L) can be estimated:\nv = 2 df fdt H n , L = 2 /triangle f f H n . (0.1)\nH n is the scale length of plasma density in the background for the source motion and in the source region for the energetic particles.\nThe above estimation of the Z-QPP indicates the source region moves in a speed from 275-93 km/s upwards, the source width expands from 1000 km to 3300 km, and the associated speeds of energetic particles is about 0.13 0.53 c , 0.53-0.8 c , and 0.36-0.9 c in the three paragraphs, respectively. Here, c is the light speed. Before and after the Z-QPP, there are many spectral fine structures, such as zebra patterns, fibers and millisecond spikes, etc. Some of them are marked in Figure 1. The abundant spectral fine structures reflect the dynamic features of the non-thermal particles (Huang & Tan, 2012).\nAcknowledgements\nThis work is supported by NSFC Grant No. 11273030, 10921303, MOST Grant No. 2011CB811401, and the National Major Scientific Equipment R&D Project ZDYZ2009-3.\nReferences\nAschwanden, M.J., & Benz A.O. 1986, Astron Astrophys , 158, 102 Cheng, X., Ding, M.D., & Guo, Y., et al. 2010, ApJ , 716, L68 Huang, J., & Tan, B.L. 2012, ApJ , 745, 186 Qin, Z.H., Li, C.S., & Fu, Q.J., et al 1996, Sol. Phys. , 163, 383\nTan, B.L. 2008, Sol. Phys. , 253, 117\n"},"sections":{"kind":"list like","value":[{"title":"Baolin Tan 1","content":"1 Key Laboratory of Solar Activity, National Astronomical Observatories of the Chinese Academy of Sciences, Beijing 100012, China. email: bltan@nao.cas.cn","pages":[1]},{"title":"Abstract.","content":"A microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in a solar flare on 200501-15 is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at 1.10-1.34 GHz. The zigzag pulsation occurred just in the early rising phase of the flare with weakly right-handed circular polarization. Its period is only several decades millisecond. Particularly, before and after the pulsation, there are many spectral fine structures, such as zebra patterns, fibers, and millisecond spikes. The microwave Z-QPP can provide some kinematic information of the source region in the early rising phase of the flare, and the source width changes from ∼ 1000 km to 3300 km, even if we have no imaging observations. The abundant spectral fine structures possibly reflect the dynamic features of non-thermal particles. Keywords. solar flare, microwave bursts, fine structures Solar microwave spectral fine structures are the most important and interesting phenomena, which can provide many intrinsic features of solar eruptions. This work reports first time a peculiar fine structure: a microwave quasi-periodic pulsation (QPP) with zigzag pattern (abbreviated as Z-QPP, hereafter) in a flare on 2005-01-15. The flare is an M8.6 class long-duration event accompanying with a powerful CME in active region AR10720 with location of N16E04, very close to the center of solar disk. It is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at frequency of 1.10-1.34 GHz (4.0 MHz resolution, 1.25 ms cadence). The left panels of Figure 1 show that the flare lasts from 05:54 UT to 07:17 UT, for about 83 minutes (Cheng et al, 2010). The Z-QPP occurred just in the early rising phase of the flare. The right panels of present the Z-QPP with left- and right-handed circular polarization spectrogram. The Z-QPP is weakly right-handed circular polarization, its period is only several decades of milliseconds, which belongs to very short period pulsation. We may partition the Z-QPP into three paragraphs. The first paragraph starts at 06:04:32 UT, ends at 06:04:43 UT, the global frequency drifting rate is about -16.5 MHz/s, the pulse frequency drifting rate is 2.30-9.60 GHz/s, the period is 40-85 ms with averaged value of 60 ms. The bandwidth increases slowly from 60 MHz to 160 MHz. The second paragraph starts at 06:04:43 UT and ends at 06:04:47 UT, its global frequency drifting rate is about 0 and the pulse frequency drifting rate is 9.60-14.40 GHz/s, the period is 48-106 ms, and the bandwidth 140-200 MHz, few variations. The third paragraph starts at 06:04:47 UT and ends at 06:04:47 UT, its global frequency drifting rate is about -5.6 MHz/s and the pulse frequency drifting rate is 6.7-20 GHz/s, the period is 38-87 ms, and the bandwidth increases slowly from 60 to 140 MHz, and possibly beyond frequency domain (1.34 GHz). The frequency distance between the two adjacent strips of the concomitant zebra pattern is about 80 MHz, which implies that the magnetic field in the source region is about 71.5 Gs from DPR model. The weakly polarization and strongly emission intensity imply that plasma emission is most possibly the emission mechanism. The Z-QPP may reflect the variations of the physical conditions and the dynamic processes in source region (Aschwanden & Benz, 1986; Tan, 2008). The velocities of source motion or energetic particles ( v ), and the width of source region (L) can be estimated: H n is the scale length of plasma density in the background for the source motion and in the source region for the energetic particles. The above estimation of the Z-QPP indicates the source region moves in a speed from 275-93 km/s upwards, the source width expands from 1000 km to 3300 km, and the associated speeds of energetic particles is about 0.13 0.53 c , 0.53-0.8 c , and 0.36-0.9 c in the three paragraphs, respectively. Here, c is the light speed. Before and after the Z-QPP, there are many spectral fine structures, such as zebra patterns, fibers and millisecond spikes, etc. Some of them are marked in Figure 1. The abundant spectral fine structures reflect the dynamic features of the non-thermal particles (Huang & Tan, 2012).","pages":[1,2]},{"title":"Acknowledgements","content":"This work is supported by NSFC Grant No. 11273030, 10921303, MOST Grant No. 2011CB811401, and the National Major Scientific Equipment R&D Project ZDYZ2009-3.","pages":[2]},{"title":"References","content":"Aschwanden, M.J., & Benz A.O. 1986, Astron Astrophys , 158, 102 Cheng, X., Ding, M.D., & Guo, Y., et al. 2010, ApJ , 716, L68 Huang, J., & Tan, B.L. 2012, ApJ , 745, 186 Qin, Z.H., Li, C.S., & Fu, Q.J., et al 1996, Sol. Phys. , 163, 383 Tan, B.L. 2008, Sol. Phys. , 253, 117","pages":[2]}],"string":"[\n {\n \"title\": \"Baolin Tan 1\",\n \"content\": \"1 Key Laboratory of Solar Activity, National Astronomical Observatories of the Chinese Academy of Sciences, Beijing 100012, China. email: bltan@nao.cas.cn\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract.\",\n \"content\": \"A microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in a solar flare on 200501-15 is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at 1.10-1.34 GHz. The zigzag pulsation occurred just in the early rising phase of the flare with weakly right-handed circular polarization. Its period is only several decades millisecond. Particularly, before and after the pulsation, there are many spectral fine structures, such as zebra patterns, fibers, and millisecond spikes. The microwave Z-QPP can provide some kinematic information of the source region in the early rising phase of the flare, and the source width changes from ∼ 1000 km to 3300 km, even if we have no imaging observations. The abundant spectral fine structures possibly reflect the dynamic features of non-thermal particles. Keywords. solar flare, microwave bursts, fine structures Solar microwave spectral fine structures are the most important and interesting phenomena, which can provide many intrinsic features of solar eruptions. This work reports first time a peculiar fine structure: a microwave quasi-periodic pulsation (QPP) with zigzag pattern (abbreviated as Z-QPP, hereafter) in a flare on 2005-01-15. The flare is an M8.6 class long-duration event accompanying with a powerful CME in active region AR10720 with location of N16E04, very close to the center of solar disk. It is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at frequency of 1.10-1.34 GHz (4.0 MHz resolution, 1.25 ms cadence). The left panels of Figure 1 show that the flare lasts from 05:54 UT to 07:17 UT, for about 83 minutes (Cheng et al, 2010). The Z-QPP occurred just in the early rising phase of the flare. The right panels of present the Z-QPP with left- and right-handed circular polarization spectrogram. The Z-QPP is weakly right-handed circular polarization, its period is only several decades of milliseconds, which belongs to very short period pulsation. We may partition the Z-QPP into three paragraphs. The first paragraph starts at 06:04:32 UT, ends at 06:04:43 UT, the global frequency drifting rate is about -16.5 MHz/s, the pulse frequency drifting rate is 2.30-9.60 GHz/s, the period is 40-85 ms with averaged value of 60 ms. The bandwidth increases slowly from 60 MHz to 160 MHz. The second paragraph starts at 06:04:43 UT and ends at 06:04:47 UT, its global frequency drifting rate is about 0 and the pulse frequency drifting rate is 9.60-14.40 GHz/s, the period is 48-106 ms, and the bandwidth 140-200 MHz, few variations. The third paragraph starts at 06:04:47 UT and ends at 06:04:47 UT, its global frequency drifting rate is about -5.6 MHz/s and the pulse frequency drifting rate is 6.7-20 GHz/s, the period is 38-87 ms, and the bandwidth increases slowly from 60 to 140 MHz, and possibly beyond frequency domain (1.34 GHz). The frequency distance between the two adjacent strips of the concomitant zebra pattern is about 80 MHz, which implies that the magnetic field in the source region is about 71.5 Gs from DPR model. The weakly polarization and strongly emission intensity imply that plasma emission is most possibly the emission mechanism. The Z-QPP may reflect the variations of the physical conditions and the dynamic processes in source region (Aschwanden & Benz, 1986; Tan, 2008). The velocities of source motion or energetic particles ( v ), and the width of source region (L) can be estimated: H n is the scale length of plasma density in the background for the source motion and in the source region for the energetic particles. The above estimation of the Z-QPP indicates the source region moves in a speed from 275-93 km/s upwards, the source width expands from 1000 km to 3300 km, and the associated speeds of energetic particles is about 0.13 0.53 c , 0.53-0.8 c , and 0.36-0.9 c in the three paragraphs, respectively. Here, c is the light speed. Before and after the Z-QPP, there are many spectral fine structures, such as zebra patterns, fibers and millisecond spikes, etc. Some of them are marked in Figure 1. The abundant spectral fine structures reflect the dynamic features of the non-thermal particles (Huang & Tan, 2012).\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"Acknowledgements\",\n \"content\": \"This work is supported by NSFC Grant No. 11273030, 10921303, MOST Grant No. 2011CB811401, and the National Major Scientific Equipment R&D Project ZDYZ2009-3.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"Aschwanden, M.J., & Benz A.O. 1986, Astron Astrophys , 158, 102 Cheng, X., Ding, M.D., & Guo, Y., et al. 2010, ApJ , 716, L68 Huang, J., & Tan, B.L. 2012, ApJ , 745, 186 Qin, Z.H., Li, C.S., & Fu, Q.J., et al 1996, Sol. Phys. , 163, 383 Tan, B.L. 2008, Sol. Phys. , 253, 117\",\n \"pages\": [\n 2\n ]\n }\n]"}}},{"rowIdx":282,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33..980E"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1308.0161.pdf"},"content":{"kind":"string","value":"\n\n\n\nDiffuse TeV Gamma-Ray Emission in the H.E.S.S. Galactic Plane Survey\nK. EGBERTS 1 , F. BRUN 2 , S. CASANOVA 3 , 2 , W. HOFMANN 2 , M. DE NAUROIS 4 , O. REIMER 1 , Q. WEITZEL 2 FOR THE H.E.S.S. COLLABORATION, AND Y. FUKUI 5\n\n1 Institut fur Astro- und Teilchenphysik, Leopold-Franzens-Universitat Innsbruck, A-6020 Innsbruck, Austria\n2 Max-Planck-Institut fur Kernphysik, P.O. Box 103980, D 69029 Heidelberg, Germany\n3 Unit for Space Physics, North-West University, Potchefstroom 2520, South Africa\n4 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France\n5 Department of Astrophysics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan\n\nkathrin.egberts@uibk.ac.at\nAbstract: Diffuse γ -ray emission has long been established as the most prominent feature in the GeV sky. Although the imaging atmospheric Cherenkov technique has been successful in revealing a large population of discrete TeV γ -ray sources, a thorough investigation of diffuse emission at TeV energies is still pending. Data from the Galactic Plane Survey (GPS) obtained by the High Energy Stereoscopic System (H.E.S.S.) have now achieved a sensitivity and coverage adequate for probing signatures of diffuse emission in the energy range of ∼ 100 GeV to a few TeV. γ -rays are produced in cosmic-ray interactions with the interstellar medium (aka 'sea of cosmic rays') and in inverse Compton scattering on cosmic photon fields. This inevitably leads to guaranteed γ -ray emission related to the gas content along the line-of-sight. Further contributions relate to those γ -ray sources that fall below the current detection threshold and the aforementioned inverse Compton emission. Based on the H.E.S.S. GPS, we present the first observational assessment of diffuse TeV γ -ray emission. The observation is compared with corresponding flux predictions based on the HI (LAB data) and CO (as a tracer of H2, NANTEN data) gas distributions. Consequences for unresolved source contributions and the anticipated level of inverse Compton emission are discussed.\nKeywords: diffuse γ -ray emission, imaging atmospheric Cherenkov telescopes\n1 Introduction\nThe Galactic diffuse γ -ray emission, observed by several space-born γ -ray experiments like SAS-2 [1], COS-B [2], EGRET[3], and Fermi-LAT [4], is the most prominent feature of the sky at GeV energies. The processes leading to this emission are cosmic-ray interactions: bremsstrahlung and π 0 production in the interstellar medium and inverse Compton scattering on radiation fields. The contributions differ in their relative amplitude on different energy scales and vary in their spatial extension depending on the interaction targets as the matter density and the respective radiation fields. When moving to higher energies, discrete γ -ray sources become more prominent compared to cosmicray induced processes (at TeV energies, inverse Compton and π 0 decay) and dominate the celestial γ -ray emission. The TeV energy regime is the realm of ground-based observations. Diffuse γ -ray emission has been reported by the Milagro experiment [5] at a median energy of 15 TeV, and by ARGO-YBJ [6]. Common to both experiments is that while providing a very good duty cycle and a large field of view, they suffer from a rather poor angular resolution, which challenges the identification of discrete γ -ray sources. With a lower energy threshold and arc-minute angular resolution, imaging atmospheric Cherenkov telescopes have the potential to improve on these measurements at energies starting at ∼ 100 GeV and thereby improve our understanding of the Galactic diffuse γ -ray emission and its underlying production mechanisms. The most suited experiment for a measurement of diffuse emission in the Galactic Plane is the High Energy Stereoscopic System (H.E.S.S.): with its location in Namibia it provides\nideal viewing conditions on the central part of the Galactic Plane and features a comparatively large field of view for imaging atmospheric Cherenkov telescopes of 5 · in diameter. The H.E.S.S. experiment has performed a survey of the Galactic Plane and has from 2004 to 2013 accumulated about 2800 observation hours of good-quality data. Although the H.E.S.S. Galactic Plane Survey (GPS) has revealed a wealth of new sources [7], and already resolved diffuse emission in the Galactic Center region on sub-degree scales [8], a measurement of the large-scale diffuse γ -ray emission is at the edge of current instrumental sensitivity and a challenge for analysis methodology.\n2 Methodology\n2.1 Background subtraction\nA measurement of diffuse γ -ray emission is challenging for imaging atmospheric Cherenkov telescopes because of their restricted capabilities of γ -hadron separation and the limited field of view of a few degrees. The limitations in the γ -hadron separation require a subtraction of the background. In order to minimize systematic effects due to changing atmospheric or instrument conditions, the background level is determined from measurement, usually within the field of view, in regions of no known γ -ray sources. This has proven to be the most reliable and failsafe background-subtraction technique [9] and is therefore used throughout the community. However, this method places severe constraints on the size of emission that can be probed. Furthermore, for emission larger than the field of view, the procedure results necessarily in a subtraction\n\n\n\n\n\n
Fig. 1 : The region in Galactic longitude and latitude used for the measurement of diffuse γ -ray emission. Red denotes exclusion from the analysis because of the presence of significant γ -ray sources, white is in the following denoted 'diffuse analysis region'.
\n\nof part of the signal together with the background. Therefore, special care needs to be taken in the treatment of the background and the interpretation of signals obtained after background subtraction.\nFor the current analysis a dataset of 1926 hours of deadtime corrected observations are used, covering a region of -75 · < l < 60 · in longitude and -2 · < b < 2 · in latitude. The data were analysed using the Model analysis technique [10] with standard cuts for event reconstruction and background reduction. The remaining background was measured in regions that do not meet any of the following exclusion criteria:\n\n1. Any region is excluded that contains a γ -ray signal with significance of > 4 σ in the chosen analysis bin and a significance in one neighbouring bin of > 4 . 5 σ or lies within 0 . 2 · of such a signal (in order to include also tails in the point spread function used to describe γ -ray sources). Significances are considered for standard and hard cuts [10] and regions are determined iteratively. The resulting regions are visualized in Fig. 1.\n2. Galactic longitudes within a latitude range of | b | < 1 . 2 · are excluded in order to study a diffuse signal close to the Galactic Plane.\n\nThe first criterion is a very conservative approach assuring a minimum of γ -ray-source contaminations. The choice of the latitude range in the second criterion is a compromise between a desired large excluded region in order to avoid a contamination of the background estimate on the one hand and the need for statistics and reduction of systematics in the background measurement on the other hand. It is further motivated by the scale height in the distribution of interstellar gas, which is expected to correlate to a certain level with p-p interaction induced γ -ray emission. An adaptive ring background subtraction method has been chosen [7] for an optimal treatment of these constraints in the choice of background regions.\nIt is worth noting here that a subtraction of the background measured at regions with | b | > 1 . 2 · also means a subtraction of any large-scale diffuse signal that extends signifcantly beyond | b | > 1 . 2 · . Therefore, any signal measured at | b | < 1 . 2 · is a signal with respect to the baseline of a potential flux outside the region.\nAfter background subtraction the γ -ray excess events are folded with the exposure to obtain a 2-dimensional representation of flux in the Galactic Plane [7].\n2.2 Discrete γ -ray sources\nAs the flux in the Galactic Plane is completely dominated by many, mostly extended γ -ray sources, a measurement of diffuse γ -ray emission needs to exclude these sources from the analysis. Due to their spatial extension and the limitations in properly characterizing their flux distributions, a modelling of the sources turns out to be presently not feasible. Therefore, source locations are identified and excluded from the analysis by applying a cut in the observed detection significance. The same criterion (1) as for the choice of the background regions is applied. The remaining regions in the sky are denoted as 'diffuse analysis region' in the following. Results are stable with respect to the details in the choice of the analysis region.\nThe procedure of spatially excluding any significant signal results by definition in fluxes (as function of position) that are individually not significant. However, by investigating profiles of the flux distribution, the cummulative projected signal results in a notable flux excess.\n3 The diffuse γ -ray signal\nThe latitude flux profile of the Galactic Plane for a longitude range of -75 · < l < 60 · is shown in Fig. 2 for both, fluxes including γ -ray sources and fluxes of the diffuse analysis region only. Errors are 1 σ statistical errors and do not account for systematics.\nBoth distributions are characterized by a clear excess in the proximity of the Galactic Plane. Notebly, also the excess of the diffuse analysis region is significant. It peaks not exactly at b = 0 · but slightly offset at around b = -0 . 25 · with a peak value of around 2 . 5 × 10 -9 cm -2 s -1 TeV -1 sr -1 . The signal accumulates over the considered longitude range and consists of larger contributions from longitude values of the Galactic centre region and smaller ones from the outskirts of the observed region.\nThe observed diffuse emission can be interpreted as a combination of contributions from\n\n· unresolved γ -ray sources\n· γ -rays resulting from cosmic-ray interactions with the interstellar medium ( π 0 decay)\n· γ -rays resulting from cosmic-ray interactions with radiation fields (inverse Compton scattering).\n\nThe estimation of the individual contributions to the observed signal is non-trivial and further complicated by the relatively small signal. A large part of the signal can be ex-\npected to stem from faint sources that are unresolved because of their low fluxes (avoiding significant detection) or very large extension not properly handled in the standard analysis.\nA minimum level of cosmic-ray induced contribution can be estimated from p-p interactions with interstellar matter. The γ -ray flux can be calculated following [11]\ndN γ dAdE γ dtd Ω = ∫ dl d ∫ d σ p -→ γ dEp n ( l , b , l d ) J ( Ep ) dEp (1)\nwith l d being the line-of-sight, Ep the proton energy, J ( Ep ) the cosmic-ray spectrum, d σ p -→ γ dEp the interaction cross section for the γ -ray producing interaction, and n ( l , b , l d ) being the column density of the gas of the interstellar medium. The cosmic-ray spectrum used for the calculation is the one measured locally at Earth, taken from [12]. The interaction cross section is a parametrization of the SIBYLL interaction code following [13]. The interstellar matter that constitutes the target material are HI and H2. HI data are measurements from the Leiden/Argentine/Bonn survey (LAB [14]) assuming a spin temperature of T = 125 K, the H2 column density is obtained using NANTEN CO data [15] as tracer of H2. A conversion factor of XCO = 2 × 10 20 cm -2 K -1 km -1 s [16] has been used to convert the velocity-integrated NANTEN data to H2 column density. The results of these calculations can be seen in the model curves of Fig. 2. In order to assure comparability, the same regions in the sky have been used for the calculation of the expected γ -ray signal and the analysis, i.e. positions of γ -ray sources have been excluded from the model calculation for the diffuse analysis region as well. Note that due to the poor angular resolution of the HI data sources are excluded on scales smaller than the HI bin size, which is justified only by the apparent lack of correlation between γ -ray sources and HI densities.\nThe calculated γ -ray emission from p-p interactions has to be treated as a minimal or guaranteed level of the anticipated diffuse emission signal. The calculation uses the locally measured cosmic-ray spectrum, while the flux is expected to vary throughout the Galaxy. The flux observed at Earth, in no immediate proximity of any cosmic-ray accelerator is assumed to be the minimum level of cosmic rays in the Galaxy ('sea of cosmic rays'), while close to accelerators the level can be significantly enhanced.\nFurther simplifying assumptions that are invoked (most of them reducing the estimated contribution) include a constant XCO (as opposed to some functional dependence on Galactocentric distance), the limitation to hydrogen contributions and usage of p-p cross section also for heavier cosmic rays.\nAs the contribution of neutral-pion-decay γ -rays is neatly localized along the Galactic Plane, a comparison of the contribution can be made with the H.E.S.S. data in a first approximation without considering the issue of background subtraction and the corresponding reduction of the signal. In the region of | b | < 1 · in the diffuse analysis region the integrated contribution of the calculated γ -ray signal originating from π 0 decay is ∼ 25% of the H.E.S.S. measurement (as seen in Fig. 2 bottom panel), thereby limiting the contribution of unresolved sources (also expected to concentrate close to the Plane) to less than 75%. In comparison, the calculated contribution from neutral-piondecay γ -rays to the total flux including γ -ray sources is for the region | b | < 1 · less than 10% (as in Fig. 2 top panel).\n\n\n\n\n\n\n\n\n
Fig. 2 : The latitude profile of γ -ray flux (shown is the differential flux at an energy of 1 TeV), covering a longitude range of -75 · < l < 60 · , for the total flux including γ -ray sources (top panel) and for the diffuse analysis region only as defined in Fig. 1 (bottom panel). H.E.S.S. measurements (black data points) are compared with the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 (solid red line, the individual components are blue dotted for H2 and green dot-dashed for HI interactions).
\n\nThe second cosmic-ray interaction component is inverse Compton scattering. Cosmic-ray electrons upscatter photons of optical starlight, infrared dust emission and the cosmic microwave background. The cosmic-ray electrons that are responsible for the inverse Compton emission at TeV energies have, depending on the radiation field they are interacting with, energies between a few and a few hundred TeV. At these energies, energy losses are severe and their lifetimes and propagation distances are limited. Therefore, the electrons are found close to their production sites, which are either homogeneously distributed in case of secondary production of electrons via hadronic cosmicray interactions, or highly inhomogeneously localized as for primary cosmic-ray electrons [17]. This makes an estimation of a contribution to the diffuse γ -ray emission at TeV energies challenging and dependent on source model assumptions.\nSince the radiation fields as target for inverse Compton scattering extend to higher latitudes than the bulk of the gas distribution, the inverse Compton component to be contained in the diffuse γ -ray signal will be subject to background subtraction and only gradients will be measurable in the observed signal.\n4 Conclusion\nWe present the first ever investigation of large-scale diffuse emission with imaging atmospheric Cherenkov telescopes. Imaging atmospheric Cherenkov telescopes have the advantage of a rather precise direction reconstruction, which puts them into advantage compared to other instruments that measure at TeV energies in the identification\nand exclusion of γ -ray sources as dominant 'background' of a diffuse γ -ray analysis. The H.E.S.S. data reveal an excess around a latitude of about b = -0 . 25 · over the baseline at latitude extensions larger 1 . 2 · in the Galactic Plane. This signal exceeds the guaranteed contribution stemming from the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 via π 0 decay. Additional contributions to the observed signal relate to unresolved γ -ray sources and inverse Compton scattering of cosmic-ray electrons on interstellar radiation fields.\nThe support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.\nReferences\n\n[1] G. F. Bignami, C. E. Fichtel, D. A. Kniffen, D. J. Thompson, 1975, ApJ, 199, 54\n[2] H. A. Mayer-Hasselwander, G. Kranbach, K Bennett, et al., 1982, A&A, 105, 164\n[3] S. D. Hunter, D. L. Bertsch, J. R. Catelli, et al., 1997, ApJ, 481, 205\n[4] M. Ackermann, M. Ajello, W. B. Atwood, et al., 2012, ApJ, 750, 3\n[5] A. A. Abdo, B. Allen, T. Aune, et al. 2008, ApJ, 688, 1078\n[6] L. Ma, 2011, ICRC, 7, 2\n[7] S. Carrigan, for the H.E.S.S. Collaboration, 2013, Proceedings of the 48th Rencontres de Moriond, La Thuile, Italy\n[8] F. Aharonian, A. G. Akhperjanian, A. R. Bazer-Bachi, et al. 2006, Nature, 439, 695\n[9] D. Berge, S. Funk, J. Hinton, 2007, A&A, 466, 1219 [10] M. de Naurois, L. Rolland, 2009, Astropart. Phys.,\n32, 231\n[11] S. Casanova, F. A. Aharonian, Y. Fukui, et al. 2010, PASJ, 62, 769\n[12] J. Beringer et al. (Particle Data Group), 2012, Phys. Rev. D, 86, 010001\n[13] S. R. Kelner, F. A. Aharonian, V. V. Bugayov, 2006, Phys. Rev. D, 74, 034018\n[14] P. M. W. Kalberla, W. B. Burton, D. Hartmann, D., et al. 2005, A&A, 440, 775\n[15] private communication Y. Fukui\n[16] A. D. Bolatto, M. Wolfire, A. K. Leroy, (2013) arXiv:1301.3498 [astro-ph.GA]\n[17] M. Pohl, J. A. Esposito, 1998, ApJ, 507, 327\n\n\n\n\n"},"sections":{"kind":"list like","value":[{"title":"Diffuse TeV Gamma-Ray Emission in the H.E.S.S. Galactic Plane Survey","content":"K. EGBERTS 1 , F. BRUN 2 , S. CASANOVA 3 , 2 , W. HOFMANN 2 , M. DE NAUROIS 4 , O. REIMER 1 , Q. WEITZEL 2 FOR THE H.E.S.S. COLLABORATION, AND Y. FUKUI 5 kathrin.egberts@uibk.ac.at Abstract: Diffuse γ -ray emission has long been established as the most prominent feature in the GeV sky. Although the imaging atmospheric Cherenkov technique has been successful in revealing a large population of discrete TeV γ -ray sources, a thorough investigation of diffuse emission at TeV energies is still pending. Data from the Galactic Plane Survey (GPS) obtained by the High Energy Stereoscopic System (H.E.S.S.) have now achieved a sensitivity and coverage adequate for probing signatures of diffuse emission in the energy range of ∼ 100 GeV to a few TeV. γ -rays are produced in cosmic-ray interactions with the interstellar medium (aka 'sea of cosmic rays') and in inverse Compton scattering on cosmic photon fields. This inevitably leads to guaranteed γ -ray emission related to the gas content along the line-of-sight. Further contributions relate to those γ -ray sources that fall below the current detection threshold and the aforementioned inverse Compton emission. Based on the H.E.S.S. GPS, we present the first observational assessment of diffuse TeV γ -ray emission. The observation is compared with corresponding flux predictions based on the HI (LAB data) and CO (as a tracer of H2, NANTEN data) gas distributions. Consequences for unresolved source contributions and the anticipated level of inverse Compton emission are discussed. Keywords: diffuse γ -ray emission, imaging atmospheric Cherenkov telescopes","pages":[1]},{"title":"1 Introduction","content":"The Galactic diffuse γ -ray emission, observed by several space-born γ -ray experiments like SAS-2 [1], COS-B [2], EGRET[3], and Fermi-LAT [4], is the most prominent feature of the sky at GeV energies. The processes leading to this emission are cosmic-ray interactions: bremsstrahlung and π 0 production in the interstellar medium and inverse Compton scattering on radiation fields. The contributions differ in their relative amplitude on different energy scales and vary in their spatial extension depending on the interaction targets as the matter density and the respective radiation fields. When moving to higher energies, discrete γ -ray sources become more prominent compared to cosmicray induced processes (at TeV energies, inverse Compton and π 0 decay) and dominate the celestial γ -ray emission. The TeV energy regime is the realm of ground-based observations. Diffuse γ -ray emission has been reported by the Milagro experiment [5] at a median energy of 15 TeV, and by ARGO-YBJ [6]. Common to both experiments is that while providing a very good duty cycle and a large field of view, they suffer from a rather poor angular resolution, which challenges the identification of discrete γ -ray sources. With a lower energy threshold and arc-minute angular resolution, imaging atmospheric Cherenkov telescopes have the potential to improve on these measurements at energies starting at ∼ 100 GeV and thereby improve our understanding of the Galactic diffuse γ -ray emission and its underlying production mechanisms. The most suited experiment for a measurement of diffuse emission in the Galactic Plane is the High Energy Stereoscopic System (H.E.S.S.): with its location in Namibia it provides ideal viewing conditions on the central part of the Galactic Plane and features a comparatively large field of view for imaging atmospheric Cherenkov telescopes of 5 · in diameter. The H.E.S.S. experiment has performed a survey of the Galactic Plane and has from 2004 to 2013 accumulated about 2800 observation hours of good-quality data. Although the H.E.S.S. Galactic Plane Survey (GPS) has revealed a wealth of new sources [7], and already resolved diffuse emission in the Galactic Center region on sub-degree scales [8], a measurement of the large-scale diffuse γ -ray emission is at the edge of current instrumental sensitivity and a challenge for analysis methodology.","pages":[1]},{"title":"2.1 Background subtraction","content":"A measurement of diffuse γ -ray emission is challenging for imaging atmospheric Cherenkov telescopes because of their restricted capabilities of γ -hadron separation and the limited field of view of a few degrees. The limitations in the γ -hadron separation require a subtraction of the background. In order to minimize systematic effects due to changing atmospheric or instrument conditions, the background level is determined from measurement, usually within the field of view, in regions of no known γ -ray sources. This has proven to be the most reliable and failsafe background-subtraction technique [9] and is therefore used throughout the community. However, this method places severe constraints on the size of emission that can be probed. Furthermore, for emission larger than the field of view, the procedure results necessarily in a subtraction of part of the signal together with the background. Therefore, special care needs to be taken in the treatment of the background and the interpretation of signals obtained after background subtraction. For the current analysis a dataset of 1926 hours of deadtime corrected observations are used, covering a region of -75 · < l < 60 · in longitude and -2 · < b < 2 · in latitude. The data were analysed using the Model analysis technique [10] with standard cuts for event reconstruction and background reduction. The remaining background was measured in regions that do not meet any of the following exclusion criteria: The first criterion is a very conservative approach assuring a minimum of γ -ray-source contaminations. The choice of the latitude range in the second criterion is a compromise between a desired large excluded region in order to avoid a contamination of the background estimate on the one hand and the need for statistics and reduction of systematics in the background measurement on the other hand. It is further motivated by the scale height in the distribution of interstellar gas, which is expected to correlate to a certain level with p-p interaction induced γ -ray emission. An adaptive ring background subtraction method has been chosen [7] for an optimal treatment of these constraints in the choice of background regions. It is worth noting here that a subtraction of the background measured at regions with | b | > 1 . 2 · also means a subtraction of any large-scale diffuse signal that extends signifcantly beyond | b | > 1 . 2 · . Therefore, any signal measured at | b | < 1 . 2 · is a signal with respect to the baseline of a potential flux outside the region. After background subtraction the γ -ray excess events are folded with the exposure to obtain a 2-dimensional representation of flux in the Galactic Plane [7].","pages":[1,2]},{"title":"2.2 Discrete γ -ray sources","content":"As the flux in the Galactic Plane is completely dominated by many, mostly extended γ -ray sources, a measurement of diffuse γ -ray emission needs to exclude these sources from the analysis. Due to their spatial extension and the limitations in properly characterizing their flux distributions, a modelling of the sources turns out to be presently not feasible. Therefore, source locations are identified and excluded from the analysis by applying a cut in the observed detection significance. The same criterion (1) as for the choice of the background regions is applied. The remaining regions in the sky are denoted as 'diffuse analysis region' in the following. Results are stable with respect to the details in the choice of the analysis region. The procedure of spatially excluding any significant signal results by definition in fluxes (as function of position) that are individually not significant. However, by investigating profiles of the flux distribution, the cummulative projected signal results in a notable flux excess.","pages":[2]},{"title":"3 The diffuse γ -ray signal","content":"The latitude flux profile of the Galactic Plane for a longitude range of -75 · < l < 60 · is shown in Fig. 2 for both, fluxes including γ -ray sources and fluxes of the diffuse analysis region only. Errors are 1 σ statistical errors and do not account for systematics. Both distributions are characterized by a clear excess in the proximity of the Galactic Plane. Notebly, also the excess of the diffuse analysis region is significant. It peaks not exactly at b = 0 · but slightly offset at around b = -0 . 25 · with a peak value of around 2 . 5 × 10 -9 cm -2 s -1 TeV -1 sr -1 . The signal accumulates over the considered longitude range and consists of larger contributions from longitude values of the Galactic centre region and smaller ones from the outskirts of the observed region. The observed diffuse emission can be interpreted as a combination of contributions from The estimation of the individual contributions to the observed signal is non-trivial and further complicated by the relatively small signal. A large part of the signal can be ex- pected to stem from faint sources that are unresolved because of their low fluxes (avoiding significant detection) or very large extension not properly handled in the standard analysis. A minimum level of cosmic-ray induced contribution can be estimated from p-p interactions with interstellar matter. The γ -ray flux can be calculated following [11] with l d being the line-of-sight, Ep the proton energy, J ( Ep ) the cosmic-ray spectrum, d σ p -→ γ dEp the interaction cross section for the γ -ray producing interaction, and n ( l , b , l d ) being the column density of the gas of the interstellar medium. The cosmic-ray spectrum used for the calculation is the one measured locally at Earth, taken from [12]. The interaction cross section is a parametrization of the SIBYLL interaction code following [13]. The interstellar matter that constitutes the target material are HI and H2. HI data are measurements from the Leiden/Argentine/Bonn survey (LAB [14]) assuming a spin temperature of T = 125 K, the H2 column density is obtained using NANTEN CO data [15] as tracer of H2. A conversion factor of XCO = 2 × 10 20 cm -2 K -1 km -1 s [16] has been used to convert the velocity-integrated NANTEN data to H2 column density. The results of these calculations can be seen in the model curves of Fig. 2. In order to assure comparability, the same regions in the sky have been used for the calculation of the expected γ -ray signal and the analysis, i.e. positions of γ -ray sources have been excluded from the model calculation for the diffuse analysis region as well. Note that due to the poor angular resolution of the HI data sources are excluded on scales smaller than the HI bin size, which is justified only by the apparent lack of correlation between γ -ray sources and HI densities. The calculated γ -ray emission from p-p interactions has to be treated as a minimal or guaranteed level of the anticipated diffuse emission signal. The calculation uses the locally measured cosmic-ray spectrum, while the flux is expected to vary throughout the Galaxy. The flux observed at Earth, in no immediate proximity of any cosmic-ray accelerator is assumed to be the minimum level of cosmic rays in the Galaxy ('sea of cosmic rays'), while close to accelerators the level can be significantly enhanced. Further simplifying assumptions that are invoked (most of them reducing the estimated contribution) include a constant XCO (as opposed to some functional dependence on Galactocentric distance), the limitation to hydrogen contributions and usage of p-p cross section also for heavier cosmic rays. As the contribution of neutral-pion-decay γ -rays is neatly localized along the Galactic Plane, a comparison of the contribution can be made with the H.E.S.S. data in a first approximation without considering the issue of background subtraction and the corresponding reduction of the signal. In the region of | b | < 1 · in the diffuse analysis region the integrated contribution of the calculated γ -ray signal originating from π 0 decay is ∼ 25% of the H.E.S.S. measurement (as seen in Fig. 2 bottom panel), thereby limiting the contribution of unresolved sources (also expected to concentrate close to the Plane) to less than 75%. In comparison, the calculated contribution from neutral-piondecay γ -rays to the total flux including γ -ray sources is for the region | b | < 1 · less than 10% (as in Fig. 2 top panel). The second cosmic-ray interaction component is inverse Compton scattering. Cosmic-ray electrons upscatter photons of optical starlight, infrared dust emission and the cosmic microwave background. The cosmic-ray electrons that are responsible for the inverse Compton emission at TeV energies have, depending on the radiation field they are interacting with, energies between a few and a few hundred TeV. At these energies, energy losses are severe and their lifetimes and propagation distances are limited. Therefore, the electrons are found close to their production sites, which are either homogeneously distributed in case of secondary production of electrons via hadronic cosmicray interactions, or highly inhomogeneously localized as for primary cosmic-ray electrons [17]. This makes an estimation of a contribution to the diffuse γ -ray emission at TeV energies challenging and dependent on source model assumptions. Since the radiation fields as target for inverse Compton scattering extend to higher latitudes than the bulk of the gas distribution, the inverse Compton component to be contained in the diffuse γ -ray signal will be subject to background subtraction and only gradients will be measurable in the observed signal.","pages":[2,3]},{"title":"4 Conclusion","content":"We present the first ever investigation of large-scale diffuse emission with imaging atmospheric Cherenkov telescopes. Imaging atmospheric Cherenkov telescopes have the advantage of a rather precise direction reconstruction, which puts them into advantage compared to other instruments that measure at TeV energies in the identification and exclusion of γ -ray sources as dominant 'background' of a diffuse γ -ray analysis. The H.E.S.S. data reveal an excess around a latitude of about b = -0 . 25 · over the baseline at latitude extensions larger 1 . 2 · in the Galactic Plane. This signal exceeds the guaranteed contribution stemming from the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 via π 0 decay. Additional contributions to the observed signal relate to unresolved γ -ray sources and inverse Compton scattering of cosmic-ray electrons on interstellar radiation fields. The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.","pages":[3,4]}],"string":"[\n {\n \"title\": \"Diffuse TeV Gamma-Ray Emission in the H.E.S.S. Galactic Plane Survey\",\n \"content\": \"K. EGBERTS 1 , F. BRUN 2 , S. CASANOVA 3 , 2 , W. HOFMANN 2 , M. DE NAUROIS 4 , O. REIMER 1 , Q. WEITZEL 2 FOR THE H.E.S.S. COLLABORATION, AND Y. FUKUI 5 kathrin.egberts@uibk.ac.at Abstract: Diffuse γ -ray emission has long been established as the most prominent feature in the GeV sky. Although the imaging atmospheric Cherenkov technique has been successful in revealing a large population of discrete TeV γ -ray sources, a thorough investigation of diffuse emission at TeV energies is still pending. Data from the Galactic Plane Survey (GPS) obtained by the High Energy Stereoscopic System (H.E.S.S.) have now achieved a sensitivity and coverage adequate for probing signatures of diffuse emission in the energy range of ∼ 100 GeV to a few TeV. γ -rays are produced in cosmic-ray interactions with the interstellar medium (aka 'sea of cosmic rays') and in inverse Compton scattering on cosmic photon fields. This inevitably leads to guaranteed γ -ray emission related to the gas content along the line-of-sight. Further contributions relate to those γ -ray sources that fall below the current detection threshold and the aforementioned inverse Compton emission. Based on the H.E.S.S. GPS, we present the first observational assessment of diffuse TeV γ -ray emission. The observation is compared with corresponding flux predictions based on the HI (LAB data) and CO (as a tracer of H2, NANTEN data) gas distributions. Consequences for unresolved source contributions and the anticipated level of inverse Compton emission are discussed. Keywords: diffuse γ -ray emission, imaging atmospheric Cherenkov telescopes\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"The Galactic diffuse γ -ray emission, observed by several space-born γ -ray experiments like SAS-2 [1], COS-B [2], EGRET[3], and Fermi-LAT [4], is the most prominent feature of the sky at GeV energies. The processes leading to this emission are cosmic-ray interactions: bremsstrahlung and π 0 production in the interstellar medium and inverse Compton scattering on radiation fields. The contributions differ in their relative amplitude on different energy scales and vary in their spatial extension depending on the interaction targets as the matter density and the respective radiation fields. When moving to higher energies, discrete γ -ray sources become more prominent compared to cosmicray induced processes (at TeV energies, inverse Compton and π 0 decay) and dominate the celestial γ -ray emission. The TeV energy regime is the realm of ground-based observations. Diffuse γ -ray emission has been reported by the Milagro experiment [5] at a median energy of 15 TeV, and by ARGO-YBJ [6]. Common to both experiments is that while providing a very good duty cycle and a large field of view, they suffer from a rather poor angular resolution, which challenges the identification of discrete γ -ray sources. With a lower energy threshold and arc-minute angular resolution, imaging atmospheric Cherenkov telescopes have the potential to improve on these measurements at energies starting at ∼ 100 GeV and thereby improve our understanding of the Galactic diffuse γ -ray emission and its underlying production mechanisms. The most suited experiment for a measurement of diffuse emission in the Galactic Plane is the High Energy Stereoscopic System (H.E.S.S.): with its location in Namibia it provides ideal viewing conditions on the central part of the Galactic Plane and features a comparatively large field of view for imaging atmospheric Cherenkov telescopes of 5 · in diameter. The H.E.S.S. experiment has performed a survey of the Galactic Plane and has from 2004 to 2013 accumulated about 2800 observation hours of good-quality data. Although the H.E.S.S. Galactic Plane Survey (GPS) has revealed a wealth of new sources [7], and already resolved diffuse emission in the Galactic Center region on sub-degree scales [8], a measurement of the large-scale diffuse γ -ray emission is at the edge of current instrumental sensitivity and a challenge for analysis methodology.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2.1 Background subtraction\",\n \"content\": \"A measurement of diffuse γ -ray emission is challenging for imaging atmospheric Cherenkov telescopes because of their restricted capabilities of γ -hadron separation and the limited field of view of a few degrees. The limitations in the γ -hadron separation require a subtraction of the background. In order to minimize systematic effects due to changing atmospheric or instrument conditions, the background level is determined from measurement, usually within the field of view, in regions of no known γ -ray sources. This has proven to be the most reliable and failsafe background-subtraction technique [9] and is therefore used throughout the community. However, this method places severe constraints on the size of emission that can be probed. Furthermore, for emission larger than the field of view, the procedure results necessarily in a subtraction of part of the signal together with the background. Therefore, special care needs to be taken in the treatment of the background and the interpretation of signals obtained after background subtraction. For the current analysis a dataset of 1926 hours of deadtime corrected observations are used, covering a region of -75 · < l < 60 · in longitude and -2 · < b < 2 · in latitude. The data were analysed using the Model analysis technique [10] with standard cuts for event reconstruction and background reduction. The remaining background was measured in regions that do not meet any of the following exclusion criteria: The first criterion is a very conservative approach assuring a minimum of γ -ray-source contaminations. The choice of the latitude range in the second criterion is a compromise between a desired large excluded region in order to avoid a contamination of the background estimate on the one hand and the need for statistics and reduction of systematics in the background measurement on the other hand. It is further motivated by the scale height in the distribution of interstellar gas, which is expected to correlate to a certain level with p-p interaction induced γ -ray emission. An adaptive ring background subtraction method has been chosen [7] for an optimal treatment of these constraints in the choice of background regions. It is worth noting here that a subtraction of the background measured at regions with | b | > 1 . 2 · also means a subtraction of any large-scale diffuse signal that extends signifcantly beyond | b | > 1 . 2 · . Therefore, any signal measured at | b | < 1 . 2 · is a signal with respect to the baseline of a potential flux outside the region. After background subtraction the γ -ray excess events are folded with the exposure to obtain a 2-dimensional representation of flux in the Galactic Plane [7].\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2.2 Discrete γ -ray sources\",\n \"content\": \"As the flux in the Galactic Plane is completely dominated by many, mostly extended γ -ray sources, a measurement of diffuse γ -ray emission needs to exclude these sources from the analysis. Due to their spatial extension and the limitations in properly characterizing their flux distributions, a modelling of the sources turns out to be presently not feasible. Therefore, source locations are identified and excluded from the analysis by applying a cut in the observed detection significance. The same criterion (1) as for the choice of the background regions is applied. The remaining regions in the sky are denoted as 'diffuse analysis region' in the following. Results are stable with respect to the details in the choice of the analysis region. The procedure of spatially excluding any significant signal results by definition in fluxes (as function of position) that are individually not significant. However, by investigating profiles of the flux distribution, the cummulative projected signal results in a notable flux excess.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3 The diffuse γ -ray signal\",\n \"content\": \"The latitude flux profile of the Galactic Plane for a longitude range of -75 · < l < 60 · is shown in Fig. 2 for both, fluxes including γ -ray sources and fluxes of the diffuse analysis region only. Errors are 1 σ statistical errors and do not account for systematics. Both distributions are characterized by a clear excess in the proximity of the Galactic Plane. Notebly, also the excess of the diffuse analysis region is significant. It peaks not exactly at b = 0 · but slightly offset at around b = -0 . 25 · with a peak value of around 2 . 5 × 10 -9 cm -2 s -1 TeV -1 sr -1 . The signal accumulates over the considered longitude range and consists of larger contributions from longitude values of the Galactic centre region and smaller ones from the outskirts of the observed region. The observed diffuse emission can be interpreted as a combination of contributions from The estimation of the individual contributions to the observed signal is non-trivial and further complicated by the relatively small signal. A large part of the signal can be ex- pected to stem from faint sources that are unresolved because of their low fluxes (avoiding significant detection) or very large extension not properly handled in the standard analysis. A minimum level of cosmic-ray induced contribution can be estimated from p-p interactions with interstellar matter. The γ -ray flux can be calculated following [11] with l d being the line-of-sight, Ep the proton energy, J ( Ep ) the cosmic-ray spectrum, d σ p -→ γ dEp the interaction cross section for the γ -ray producing interaction, and n ( l , b , l d ) being the column density of the gas of the interstellar medium. The cosmic-ray spectrum used for the calculation is the one measured locally at Earth, taken from [12]. The interaction cross section is a parametrization of the SIBYLL interaction code following [13]. The interstellar matter that constitutes the target material are HI and H2. HI data are measurements from the Leiden/Argentine/Bonn survey (LAB [14]) assuming a spin temperature of T = 125 K, the H2 column density is obtained using NANTEN CO data [15] as tracer of H2. A conversion factor of XCO = 2 × 10 20 cm -2 K -1 km -1 s [16] has been used to convert the velocity-integrated NANTEN data to H2 column density. The results of these calculations can be seen in the model curves of Fig. 2. In order to assure comparability, the same regions in the sky have been used for the calculation of the expected γ -ray signal and the analysis, i.e. positions of γ -ray sources have been excluded from the model calculation for the diffuse analysis region as well. Note that due to the poor angular resolution of the HI data sources are excluded on scales smaller than the HI bin size, which is justified only by the apparent lack of correlation between γ -ray sources and HI densities. The calculated γ -ray emission from p-p interactions has to be treated as a minimal or guaranteed level of the anticipated diffuse emission signal. The calculation uses the locally measured cosmic-ray spectrum, while the flux is expected to vary throughout the Galaxy. The flux observed at Earth, in no immediate proximity of any cosmic-ray accelerator is assumed to be the minimum level of cosmic rays in the Galaxy ('sea of cosmic rays'), while close to accelerators the level can be significantly enhanced. Further simplifying assumptions that are invoked (most of them reducing the estimated contribution) include a constant XCO (as opposed to some functional dependence on Galactocentric distance), the limitation to hydrogen contributions and usage of p-p cross section also for heavier cosmic rays. As the contribution of neutral-pion-decay γ -rays is neatly localized along the Galactic Plane, a comparison of the contribution can be made with the H.E.S.S. data in a first approximation without considering the issue of background subtraction and the corresponding reduction of the signal. In the region of | b | < 1 · in the diffuse analysis region the integrated contribution of the calculated γ -ray signal originating from π 0 decay is ∼ 25% of the H.E.S.S. measurement (as seen in Fig. 2 bottom panel), thereby limiting the contribution of unresolved sources (also expected to concentrate close to the Plane) to less than 75%. In comparison, the calculated contribution from neutral-piondecay γ -rays to the total flux including γ -ray sources is for the region | b | < 1 · less than 10% (as in Fig. 2 top panel). The second cosmic-ray interaction component is inverse Compton scattering. Cosmic-ray electrons upscatter photons of optical starlight, infrared dust emission and the cosmic microwave background. The cosmic-ray electrons that are responsible for the inverse Compton emission at TeV energies have, depending on the radiation field they are interacting with, energies between a few and a few hundred TeV. At these energies, energy losses are severe and their lifetimes and propagation distances are limited. Therefore, the electrons are found close to their production sites, which are either homogeneously distributed in case of secondary production of electrons via hadronic cosmicray interactions, or highly inhomogeneously localized as for primary cosmic-ray electrons [17]. This makes an estimation of a contribution to the diffuse γ -ray emission at TeV energies challenging and dependent on source model assumptions. Since the radiation fields as target for inverse Compton scattering extend to higher latitudes than the bulk of the gas distribution, the inverse Compton component to be contained in the diffuse γ -ray signal will be subject to background subtraction and only gradients will be measurable in the observed signal.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"4 Conclusion\",\n \"content\": \"We present the first ever investigation of large-scale diffuse emission with imaging atmospheric Cherenkov telescopes. Imaging atmospheric Cherenkov telescopes have the advantage of a rather precise direction reconstruction, which puts them into advantage compared to other instruments that measure at TeV energies in the identification and exclusion of γ -ray sources as dominant 'background' of a diffuse γ -ray analysis. The H.E.S.S. data reveal an excess around a latitude of about b = -0 . 25 · over the baseline at latitude extensions larger 1 . 2 · in the Galactic Plane. This signal exceeds the guaranteed contribution stemming from the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 via π 0 decay. Additional contributions to the observed signal relate to unresolved γ -ray sources and inverse Compton scattering of cosmic-ray electrons on interstellar radiation fields. The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.\",\n \"pages\": [\n 3,\n 4\n ]\n }\n]"}}},{"rowIdx":283,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33.1032L"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1308.0475.pdf"},"content":{"kind":"string","value":"\n\n\n\nDiscovery of the mysterious gamma-ray source HESSJ1832-093 in the vicinity of SNR G22.7-0.2\nH. LAFFON 1 , 2 , F. ACERO 3 , F. BRUN 4 , B. KH'ELIFI 2 , G. P UHLHOFER 5 , R. TERRIER 6 , FOR THE H.E.S.S. COLLABORATION.\n\n1 Centre d'Etudes Nucl'eaires de Bordeaux-Gradignan, CNRS/IN2P3, Universit'e Bordeaux 1, France\n2 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France\n3 NASA Goddard Space Flight Center, Greenbelt, USA\n4 Max-Planck-Institut fur Kernphysik, Heidelberg, Germany\n5 Institut fur Astronomie und Astrophysik, Universitat Tubingen, Germany\n6 Astroparticules et Cosmologie, CNRS/IN2P3, Universit'e Paris 7, Paris, France\n\nlaffon@cenbg.in2p3.fr\nAbstract: Thanks to the use of advanced analysis techniques, the H.E.S.S. imaging Cherenkov telescope array has now reached a sensitivity level allowing the detection of sources with fluxes around 1% of the Crab with a limited observation time (less than ∼ 100 hours). 67 hours of observations in the region of the supernova remnant G22.7-0.2 thus yielded the detection of the faint point-like source HESS J1832-093 in spatial coincidence with a part of the radio shell of the supernova remnant. A multi-wavelength search for counterparts was then performed and led to the elaboration of various scenarii in order to explain the origin of the very high energy excess. The presence of molecular clouds in the line of sight could indicate a hadronic origin through π 0 production and decay. However, the discovery of an X-ray point source and its potential infrared counterpart give rise to other possibilities such as a pulsar wind nebula nature or a binary system. The latest results on this source are presented as well as the different scenarii brought by the multi-wavelength observations.\nKeywords:\nHESS J1832-093, SNR G22.7-0.2, molecular clouds, pulsar wind nebula, binary system\n1 Introduction\n2 H.E.S.S. observations\nH.E.S.S. (High Energy Stereoscopic System) is an array of four imaging atmospheric Cherenkov telescopes located 1800 m above sea level in the Khomas Highland of Namibia and has been fully operational since 2004 [1]. The H.E.S.S. collaboration has been conducting a systematic scan of the Galactic Plane, which led to the discovery of a rich population of very high energy (VHE) gamma-ray sources such as in the region around the supernova remnant (SNR) W41.\nAs a consequence, a dedicated observation campaign was launched by H.E.S.S. to study those TeV sources in detail. These observations led to the discovery of HESS J1832 -093 in spatial coincidence with the edge of SNR G22.7 -0.2. The latter shows a non-thermal shell of 26 ' diameter in radio [2] and partially overlaps the neighbouring remnant W41. A search for multiwavelength counterparts was performed in order to identify the nature of HESS J1832 -093. A proposal for observations in X-rays at the position of HESS J1832 -093 was submitted to the XMMNewton satellite and allowed the discovery of a pointlike source in X-rays near the center of the TeV source. Apart from the radio emission of the SNR, no radio pointlike counterpart was found but measurements of the 13 CO (J=1 → 0) transition line show the presence of molecular structures on the line of sight. However, an infrared pointlike source lying less than 2 '' away from the center of the Xray source was found in the 2MASS catalog. The features of the TeV emission discovered by H.E.S.S. are described in section 2. Details on the X-ray, IR and radio counterparts are given in section 3 and different considered scenarios to explain the VHE gamma-ray emission are presented in section 4.\n2.1 Detection and morphology\nA standard run selection procedure is used to remove bad quality observations in order to study the field of view. This results in a data set comprising 67 hours live time of observations taken from 2004 to 2011.\nA standard H.E.S.S. analysis frame is adopted with the Hillas reconstruction method using the weighted intersections of the main axis of the images to reconstruct the direction of the events [1]. A recently developped multi-variate analysis is used to provide a better discrimination between hadrons and gamma-rays [3]. A minimum charge of 110 photo-electrons in the shower images is applied to the data, resulting in an energy threshold of about 450 GeV. The background is estimated with the ring-background model, as described in [1].\nUsing these techniques, HESS J1832 -093 is detected with a peak significance of 7 . 9 σ pre-trials, resulting in a post-trial detection significance of 5 . 6 σ , following the approach described in [4]. The corresponding excess map of the field of view showing the new detected source is presented in Fig. 1. The average angular resolution (r68) for this selected data set and assuming a power-law spectral index of 2.3 is 0 . 081 · at the source position. A twodimensional symmetrical Gaussian function is used to determine the position and size of the TeV emission with a χ 2 minimization. The best-fit position obtained is RA = 18 h 32 m 50 s ± 32 s stat ± 36 s syst , Dec = -9 · 22 ' 36 '' ± 32 '' stat ± 36 '' syst ( J2000 ) ( χ 2 /ndf=0.89). No significant extension was found for the source.\n\n\n
Figure 1 : H.E.S.S. excess map oversampled with the r68 value of 0.081 · (dashed circle) and smoothed with a 2Dgaussian with the same width in units of counts per integration area. The newly discovered source HESS J1832 -093 is shown with its best-fit position and corresponding statistical errors (black cross). The SNR G22.7 -0.2 observed in radio [5] is represented by the white contours. The emission seen on the upper left is a small part of HESS J1834 -087 [4], the TeV source in spatial coincidence with SNR W41.
\n\n2.2 Spectral analysis\nTo produce the energy spectrum, only the highest quality data are used, corresponding to a data set of 59 hours live time. In order to broaden the accessible energy range, the charge cut is lowered to a minimum of 80 photo-electrons, resulting in an energy threshold of ∼ 400 GeV. The background is estimated with the reflected background model, which is better suited for spectral studies [1]. The forwardfolding method described in [1] is applied to the data to derive the spectrum. Source counts are extracted from a circular region of 0.1 · radius around the best fit position of HESS J1832 -093, a size optimized for point source studies with the applied cuts. The corresponding excess is 152 ± 24 gamma-like events.\nThe spectrum obtained between 400 GeV and 5 TeV (displayed on Fig. 2) can be described by a power-law d Φ dE = Φ 0 ( E 1TeV ) -Γ , with an index Γ = 2 . 6 ± 0 . 3stat ± 0 . 3syst and a differential flux normalisation at 1 TeV of Φ 0 = ( 4 . 8 ± 0 . 8stat ± 0 . 9syst ) × 10 -13 cm -2 s -1 TeV -1 . A light curve was produced with the available observations and no significant temporal variability was detected in the H.E.S.S. data set.\n3 Multi-wavelength observations\n3.1 X-Ray observations with XMMNewton\nIn order to constrain the nature of the source HESS J1832 -093, a dedicated XMMNewton observation (ID: 0654480101) was performed in March 2011 for 17 ks, targeted at the position of the gamma-ray emission. After fil-\n\n\n\n\n\n
Figure 2 : Top panel: VHE gamma-ray spectrum observed from HESS J1832 -093 obtained with the forward-folding method. Bottom panel: corresponding residuals. The green contour represents the 1 σ confidence level of the fitted spectrum using a power-law hypothesis.
\n\ntering out proton flare contamination, 13 ks and 7 ks of exposure time remained for the two EPIC-MOS cameras and for the EPIC-pn camera respectively.\nThe data were processed using the XMMNewton Science Analysis System (v10.0). The instrumental background was derived from a compilation of blank sky observations [6], renormalized to the actual exposure using the count rate in the 10-12 keV energy band.\n\n\n
Figure 3 : X-ray count map in the 2-10 keV band smoothed with a Gaussian of 15 '' width (HPD of the EPIC cameras PSF) of the field of view around HESS J1832 -093 as seen by XMMNewton . The white cross symbolizes the best-fit position of HESS J1832 -093 with corresponding statistical errors and the white contours represent the H.E.S.S. excess.
\n\nThe brightest object in the XMMNewton field of view is a point-like source located at RA = 18 h 32 m 45 s . 04 , Dec = -09 · 21 ' 53 '' . 9, 1.4 ' away from the best-fit position of the H.E.S.S. excess, as shown in Fig. 3. This new source,\ndubbed XMMU J183245 -0921539, lies within the 2 σ uncertainty of the VHE gamma-ray centroid.\nSpectra from the three instruments were extracted from a 15 '' radius circular region centered on XMMU J183245 -0921539. As the statistics are low ( ∼ 500 counts when summing the three EPIC instruments), the data were fitted in Xspec (v12.5) using Cash statistics and the spectra were not rebinned for the fitting process. The pvalue corresponding to the null-hypothesis probability is derived using the goodness command in XSPEC which performs Monte Carlo simulations of the goodness-of-fit. The best fit parameters for a power-law assumption are a column density NH = 10 . 5 + 3 . 1 -2 . 7 × 10 22 cm -2 , a photon index Γ = 1 . 3 + 0 . 5 -0 . 4 and an unabsorbed energy flux Φ (2-10 keV) = 6 . 9 + 1 . 7 -2 . 8 × 10 -13 ergcm -2 s -1 with a p-value of 0.75. A search for pulsations was performed at the position of XMMUJ183245 -0921539, but no pulsations were found in the data.\nA comparison of the absorption along the line of sight obtained from the X-ray spectral model with the column depth derived from the atomic (HI) and molecular ( 12 CO, J=1 → 0 transition line) gas can be used to provide a lower limit on the distance to XMMU J183245 -0921539, assuming that all absorbing material is at the near distance allowed by the Galactic rotation curve. The Galactic rotation curve model of [7] is used to translate the measured velocities into distances. A lower limit of about 5 kpc on the distance to XMMU J183245 -0921539 is thus derived.\n3.2 IR counterparts to XMMU J183245 -0921539\nThe 2MASScatalog 1 shows one infrared source around the position of XMMU J183245 -0921539 within the systematic pointing error of XMMNewton ( ∼ 2 '' ). This source, 2MASS J18324516 -0921545, lies 1.9 '' away from the center of gravity of the X-ray source. No optical counterpart is found, likely due to strong extinction in the galactic plane. The apparent magnitudes observed in the J, H, K bands are mJ = 15 . 52 ± 0 . 06, mH = 13 . 26 ± 0 . 04 and mK = 12 . 17 ± 0 . 02, respectively. The IR emission could originate from a massive companion to the X-ray compact source, thus forming a binary system, as discussed in section 4.2.\n3.3 13 COobservations\nThe Galactic Ring Survey (GRS) performed with the Boston University FCRAO telescopes [8] provides measurements of the 13 CO (J=1 → 0) transition line covering the velocity range from -5 to 135 km s -1 in this region. The detection of this line is evidence for the presence of dense molecular clouds that are known to be targets for cosmic-rays and hence gamma-ray emitters via neutral pion production and decay or bremsstrahlung emission. A 0.2 · -side square region is defined around the source HESS J1832 -093 to look for molecular clouds traced by the 13 CO transition line. Several molecular clouds measured at different radial velocities are found in this region. The two structures that show the best spatial coincidence with the TeV excess are selected. Their velocity ranges are 26 to 31 kms -1 and 73 to 81 kms -1 respectively. The antenna temperature of each molecular cloud is integrated on\n\n\n\nthe corresponding range. The same Galactic rotation curve model as used in section 3.1 [7] is assumed to translate the measured velocities into distances, each velocity corresponding to two possible distances given our position in the Milky Way. However, in case of an association with the SNR, only the near distance (2.3 kpc and 4.5 kpc respectively) would be compatible with the distance estimates to the remnant. Following the approach described in [9], the integrated antenna temperatures are used to derive the gas mass of each structure and the corresponding gas densities (20 cm -3 and 62 cm -3 respectively).\n4 Discussion\n4.1 PWNscenario\nDedicated XMMNewton observations at the position of HESS J1832 -093 have revealed the presence of the pointlike source XMMU J183245 -0921539 showing a hard spectrum ( Γ = 1 . 3 + 0 . 5 -0 . 4 ). The source position in the galactic plane points towards a location inside our galaxy, at a distance ≥ 5 kpc as deduced from the X-ray power-law model. XMMU J183245 -0921539 is a serious counterpart candidate for HESS J1832 -093 because of its proximity with the best-fit position of the TeV emission and its hard spectrum. A likely scenario would be that both the X-ray and TeV sources stem from a pulsar wind nebula (PWN) powered by a yet unknown pulsar. Even if the non-thermal aspect of the X-ray emission is not well determined, its hard spectral index is indicative of an emission from the vicinity of a pulsar (magnetospheric, striped wind [10],...). Therefore, despite the lack of observed pulsations in the object, we will consider a pulsar origin for XMMU J183245 -0921539 in the following. It can be tested whether energetically a PWN scenario plausibly matches with the population of known TeVemitting PWNe, under the hypothesis that the X-ray emission comes from the pulsar's magnetosphere.\nThe unabsorbed flux Φ X (2-10 keV) of the point source for the power-law model is used to compute the corresponding luminosity in the same energy band for a distance of 5 kpc: LX ( 2 -10keV ) /similarequal 2 × 10 33 ergs -1 .\nIt can then be translated to an estimate of the ˙ E of the hypothetical pulsar using the relation provided by [11]. The estimated spin-down luminosity is about 1 . 5 × 10 37 ergs -1 for the same distance.\nIf we now compute the ˙ E/d 2 we obtain a value of 6 × 10 35 ergs -1 kpc -2 , corresponding to the band for which 70% of the PWNe are detected by H.E.S.S. [12]. Therefore, if the putative pulsar powers a TeV PWN, it should be detectable by the H.E.S.S. array. Together with the absence of detected X-ray pulsations, the PWN scenario remains unconfirmed but is energetically possible.\n4.2 Binary scenario\nThe infrared source 2MASS J18324516 -0921545 discovered in spatial coindidence with XMMU J183245 -0921539 could suggest that the X-ray source resides in a binary system around a massive star. The magnitude extinction number in the optical band expected from the X-ray column density is AV=59 + 17 -15 , using the NH -A V relation given by [13]. This would result in an absolute magnitude MJ ≤ -\n14.6 for a distance ≥ 5 kpc [14]. This value is excluded but the NH -A V relation cannot apply in case of strong local absorption. Therefore, a binary scenario with 2MASS J18324516 -0921545 as optical companion could only work if the X-ray absorption arises mainly locally around XMMU J183245 -0921539. In the absence of orbitally modulated X-ray or TeV emission, the binary possibility remains unconfirmed for the moment.\n4.3 SNR-molecular cloud interaction scenario\nDespite the lack of X-ray emission from the SNR shell that could be due to synchrotron radiation of high energy electrons, or thermal X-ray emission which could stem from shock-gas interactions seen frequently in middleaged SNRs, the observed VHE emission might still come from particles accelerated in the remnant. Those particles would then interact with localized target material via neutral pion production and decay or bremsstrahlung emission. Indeed, 13 CO measurements show the presence of structures around HESS J1832 -093, while at corresponding distance ranges lower gas densities are seen in other portions of the SNR shell. There is, however, no support of the association of these gas structures with G22.7 -0.2, e.g. through maser emission from shock-cloud interaction. An identification with the TeV source might therefore be due to chance coincidence. Further investigation on a possible CRpropagationto a nearby MC at the origin of the gammaray emission thus needs to be performed.\n5 Conclusion\nObservations in the field of view of SNR G22.7 -0.2 have led to the discovery of the VHE source HESS J1832 -093 lying on the edge of the SNR radio rim. The available multi-wavelength data, using archival radio and infrared data, as well as a dedicated XMMNewton X-ray pointing towards the source, do not permit to unambiguously determine the nature of the object that gives rise to the VHE emission. A compelling X-ray counterpart, XMMU J183245 -0921539, has been discovered. The nature of this X-ray source could, however, not be established from the X-ray data alone. Together with the TeV emission and the infrared point source 2MASS J18324516 -0921545, plausible object classifications are a pulsar wind nebula or a binary system. Cosmic-rays accelerated in the SNR G22.7 -0.2 interacting with dense gas material could also result in TeV emission but the interaction between the SNR and a molecular cloud is not supported by observational evidence such as maser emission.\nThe nature of the gamma-ray source HESS J1832 -093 presented here remains, therefore, undetermined. More data in X-rays and radio to look for variability and faint diffuse emission could help to constrain the nature of the source.\nAcknowledgments\nThe support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Tech-\n\n\n\nnology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.\nReferences\n\n[1] H.E.S.S. collaboration, AAP 457 (2006) 899-915 doi:10.1051/0004-6361:20065351\n[2] Shaver, P. A. and Goss, W. M., AJPAS 14 (1970) p.133 Bib. code:1970AuJPA..14..133S\n[3] Becherini, Y. et al., Astro-Ph. 34 (2011) 858-870 doi:10.1016/j.astropartphys.2011.03.005\n[4] H.E.S.S. collaboration, ApJ 636 (2006) 777-797 doi:10.1086/498013\n[5] Helfand, D. J. et al., AJ 131 (2006) 2525-2537 doi:10.1086/503253\n[6] Carter, J. A. and Read, A. M., AAP 464 (2007) 1155-1166 doi:10.1051/0004-6361:20065882\n[7] Hou, L. G. and Han, J. L. and Shi, W. B., AAP 499 (2009) 473-482 doi:10.1051/0004-6361/200809692\n[8] Jackson, J. M. et al., ApJS 163 (2006) 145-159 doi:10.1086/500091\n[9] Simon, R. et al., ApJ 551 (2001) 747-763 doi:10.1086/320230\n[10] P'etri, J. and Lyubarsky, Y., AAP 473 (2007) 683-700 doi:10.1051/0004-6361:20066981\n[11] Li, X.-H. and Lu, F.-J. and Li, Z., ApJ 682 (2008)\n\n1166-1176 doi:10.1086/589495\n\n[12] Carrigan, S. et al. for the H.E.S.S. collaboration, proc. ICRC (2007) eprint:arXiv:0709.4094v1\n[13] Predehl, P. and Schmitt, J. H. M. M., A&A 293 (1995) 889-905 Bib. code:1995A&A...293..889P\n\n[14] Cardelli, J. A. and Clayton, G. C. and Mathis, J. S., proc.\nIAU (1989) Bib. code:1989IAUS..135P...5C\n\n[15] Drury, L. O. and Aharonian, F. A. and Voelk, H. J., AAP 287 (1994) 959-971 Bib. code:1994A&A...287..959D\n"},"sections":{"kind":"list like","value":[{"title":"Discovery of the mysterious gamma-ray source HESSJ1832-093 in the vicinity of SNR G22.7-0.2","content":"H. LAFFON 1 , 2 , F. ACERO 3 , F. BRUN 4 , B. KH'ELIFI 2 , G. P UHLHOFER 5 , R. TERRIER 6 , FOR THE H.E.S.S. COLLABORATION. laffon@cenbg.in2p3.fr Abstract: Thanks to the use of advanced analysis techniques, the H.E.S.S. imaging Cherenkov telescope array has now reached a sensitivity level allowing the detection of sources with fluxes around 1% of the Crab with a limited observation time (less than ∼ 100 hours). 67 hours of observations in the region of the supernova remnant G22.7-0.2 thus yielded the detection of the faint point-like source HESS J1832-093 in spatial coincidence with a part of the radio shell of the supernova remnant. A multi-wavelength search for counterparts was then performed and led to the elaboration of various scenarii in order to explain the origin of the very high energy excess. The presence of molecular clouds in the line of sight could indicate a hadronic origin through π 0 production and decay. However, the discovery of an X-ray point source and its potential infrared counterpart give rise to other possibilities such as a pulsar wind nebula nature or a binary system. The latest results on this source are presented as well as the different scenarii brought by the multi-wavelength observations. Keywords: HESS J1832-093, SNR G22.7-0.2, molecular clouds, pulsar wind nebula, binary system","pages":[1]},{"title":"2 H.E.S.S. observations","content":"H.E.S.S. (High Energy Stereoscopic System) is an array of four imaging atmospheric Cherenkov telescopes located 1800 m above sea level in the Khomas Highland of Namibia and has been fully operational since 2004 [1]. The H.E.S.S. collaboration has been conducting a systematic scan of the Galactic Plane, which led to the discovery of a rich population of very high energy (VHE) gamma-ray sources such as in the region around the supernova remnant (SNR) W41. As a consequence, a dedicated observation campaign was launched by H.E.S.S. to study those TeV sources in detail. These observations led to the discovery of HESS J1832 -093 in spatial coincidence with the edge of SNR G22.7 -0.2. The latter shows a non-thermal shell of 26 ' diameter in radio [2] and partially overlaps the neighbouring remnant W41. A search for multiwavelength counterparts was performed in order to identify the nature of HESS J1832 -093. A proposal for observations in X-rays at the position of HESS J1832 -093 was submitted to the XMMNewton satellite and allowed the discovery of a pointlike source in X-rays near the center of the TeV source. Apart from the radio emission of the SNR, no radio pointlike counterpart was found but measurements of the 13 CO (J=1 → 0) transition line show the presence of molecular structures on the line of sight. However, an infrared pointlike source lying less than 2 '' away from the center of the Xray source was found in the 2MASS catalog. The features of the TeV emission discovered by H.E.S.S. are described in section 2. Details on the X-ray, IR and radio counterparts are given in section 3 and different considered scenarios to explain the VHE gamma-ray emission are presented in section 4.","pages":[1]},{"title":"2.1 Detection and morphology","content":"A standard run selection procedure is used to remove bad quality observations in order to study the field of view. This results in a data set comprising 67 hours live time of observations taken from 2004 to 2011. A standard H.E.S.S. analysis frame is adopted with the Hillas reconstruction method using the weighted intersections of the main axis of the images to reconstruct the direction of the events [1]. A recently developped multi-variate analysis is used to provide a better discrimination between hadrons and gamma-rays [3]. A minimum charge of 110 photo-electrons in the shower images is applied to the data, resulting in an energy threshold of about 450 GeV. The background is estimated with the ring-background model, as described in [1]. Using these techniques, HESS J1832 -093 is detected with a peak significance of 7 . 9 σ pre-trials, resulting in a post-trial detection significance of 5 . 6 σ , following the approach described in [4]. The corresponding excess map of the field of view showing the new detected source is presented in Fig. 1. The average angular resolution (r68) for this selected data set and assuming a power-law spectral index of 2.3 is 0 . 081 · at the source position. A twodimensional symmetrical Gaussian function is used to determine the position and size of the TeV emission with a χ 2 minimization. The best-fit position obtained is RA = 18 h 32 m 50 s ± 32 s stat ± 36 s syst , Dec = -9 · 22 ' 36 '' ± 32 '' stat ± 36 '' syst ( J2000 ) ( χ 2 /ndf=0.89). No significant extension was found for the source.","pages":[1]},{"title":"2.2 Spectral analysis","content":"To produce the energy spectrum, only the highest quality data are used, corresponding to a data set of 59 hours live time. In order to broaden the accessible energy range, the charge cut is lowered to a minimum of 80 photo-electrons, resulting in an energy threshold of ∼ 400 GeV. The background is estimated with the reflected background model, which is better suited for spectral studies [1]. The forwardfolding method described in [1] is applied to the data to derive the spectrum. Source counts are extracted from a circular region of 0.1 · radius around the best fit position of HESS J1832 -093, a size optimized for point source studies with the applied cuts. The corresponding excess is 152 ± 24 gamma-like events. The spectrum obtained between 400 GeV and 5 TeV (displayed on Fig. 2) can be described by a power-law d Φ dE = Φ 0 ( E 1TeV ) -Γ , with an index Γ = 2 . 6 ± 0 . 3stat ± 0 . 3syst and a differential flux normalisation at 1 TeV of Φ 0 = ( 4 . 8 ± 0 . 8stat ± 0 . 9syst ) × 10 -13 cm -2 s -1 TeV -1 . A light curve was produced with the available observations and no significant temporal variability was detected in the H.E.S.S. data set.","pages":[2]},{"title":"3.1 X-Ray observations with XMMNewton","content":"In order to constrain the nature of the source HESS J1832 -093, a dedicated XMMNewton observation (ID: 0654480101) was performed in March 2011 for 17 ks, targeted at the position of the gamma-ray emission. After fil- tering out proton flare contamination, 13 ks and 7 ks of exposure time remained for the two EPIC-MOS cameras and for the EPIC-pn camera respectively. The data were processed using the XMMNewton Science Analysis System (v10.0). The instrumental background was derived from a compilation of blank sky observations [6], renormalized to the actual exposure using the count rate in the 10-12 keV energy band. The brightest object in the XMMNewton field of view is a point-like source located at RA = 18 h 32 m 45 s . 04 , Dec = -09 · 21 ' 53 '' . 9, 1.4 ' away from the best-fit position of the H.E.S.S. excess, as shown in Fig. 3. This new source, dubbed XMMU J183245 -0921539, lies within the 2 σ uncertainty of the VHE gamma-ray centroid. Spectra from the three instruments were extracted from a 15 '' radius circular region centered on XMMU J183245 -0921539. As the statistics are low ( ∼ 500 counts when summing the three EPIC instruments), the data were fitted in Xspec (v12.5) using Cash statistics and the spectra were not rebinned for the fitting process. The pvalue corresponding to the null-hypothesis probability is derived using the goodness command in XSPEC which performs Monte Carlo simulations of the goodness-of-fit. The best fit parameters for a power-law assumption are a column density NH = 10 . 5 + 3 . 1 -2 . 7 × 10 22 cm -2 , a photon index Γ = 1 . 3 + 0 . 5 -0 . 4 and an unabsorbed energy flux Φ (2-10 keV) = 6 . 9 + 1 . 7 -2 . 8 × 10 -13 ergcm -2 s -1 with a p-value of 0.75. A search for pulsations was performed at the position of XMMUJ183245 -0921539, but no pulsations were found in the data. A comparison of the absorption along the line of sight obtained from the X-ray spectral model with the column depth derived from the atomic (HI) and molecular ( 12 CO, J=1 → 0 transition line) gas can be used to provide a lower limit on the distance to XMMU J183245 -0921539, assuming that all absorbing material is at the near distance allowed by the Galactic rotation curve. The Galactic rotation curve model of [7] is used to translate the measured velocities into distances. A lower limit of about 5 kpc on the distance to XMMU J183245 -0921539 is thus derived.","pages":[2,3]},{"title":"3.2 IR counterparts to XMMU J183245 -0921539","content":"The 2MASScatalog 1 shows one infrared source around the position of XMMU J183245 -0921539 within the systematic pointing error of XMMNewton ( ∼ 2 '' ). This source, 2MASS J18324516 -0921545, lies 1.9 '' away from the center of gravity of the X-ray source. No optical counterpart is found, likely due to strong extinction in the galactic plane. The apparent magnitudes observed in the J, H, K bands are mJ = 15 . 52 ± 0 . 06, mH = 13 . 26 ± 0 . 04 and mK = 12 . 17 ± 0 . 02, respectively. The IR emission could originate from a massive companion to the X-ray compact source, thus forming a binary system, as discussed in section 4.2.","pages":[3]},{"title":"3.3 13 COobservations","content":"The Galactic Ring Survey (GRS) performed with the Boston University FCRAO telescopes [8] provides measurements of the 13 CO (J=1 → 0) transition line covering the velocity range from -5 to 135 km s -1 in this region. The detection of this line is evidence for the presence of dense molecular clouds that are known to be targets for cosmic-rays and hence gamma-ray emitters via neutral pion production and decay or bremsstrahlung emission. A 0.2 · -side square region is defined around the source HESS J1832 -093 to look for molecular clouds traced by the 13 CO transition line. Several molecular clouds measured at different radial velocities are found in this region. The two structures that show the best spatial coincidence with the TeV excess are selected. Their velocity ranges are 26 to 31 kms -1 and 73 to 81 kms -1 respectively. The antenna temperature of each molecular cloud is integrated on the corresponding range. The same Galactic rotation curve model as used in section 3.1 [7] is assumed to translate the measured velocities into distances, each velocity corresponding to two possible distances given our position in the Milky Way. However, in case of an association with the SNR, only the near distance (2.3 kpc and 4.5 kpc respectively) would be compatible with the distance estimates to the remnant. Following the approach described in [9], the integrated antenna temperatures are used to derive the gas mass of each structure and the corresponding gas densities (20 cm -3 and 62 cm -3 respectively).","pages":[3]},{"title":"4.1 PWNscenario","content":"Dedicated XMMNewton observations at the position of HESS J1832 -093 have revealed the presence of the pointlike source XMMU J183245 -0921539 showing a hard spectrum ( Γ = 1 . 3 + 0 . 5 -0 . 4 ). The source position in the galactic plane points towards a location inside our galaxy, at a distance ≥ 5 kpc as deduced from the X-ray power-law model. XMMU J183245 -0921539 is a serious counterpart candidate for HESS J1832 -093 because of its proximity with the best-fit position of the TeV emission and its hard spectrum. A likely scenario would be that both the X-ray and TeV sources stem from a pulsar wind nebula (PWN) powered by a yet unknown pulsar. Even if the non-thermal aspect of the X-ray emission is not well determined, its hard spectral index is indicative of an emission from the vicinity of a pulsar (magnetospheric, striped wind [10],...). Therefore, despite the lack of observed pulsations in the object, we will consider a pulsar origin for XMMU J183245 -0921539 in the following. It can be tested whether energetically a PWN scenario plausibly matches with the population of known TeVemitting PWNe, under the hypothesis that the X-ray emission comes from the pulsar's magnetosphere. The unabsorbed flux Φ X (2-10 keV) of the point source for the power-law model is used to compute the corresponding luminosity in the same energy band for a distance of 5 kpc: LX ( 2 -10keV ) /similarequal 2 × 10 33 ergs -1 . It can then be translated to an estimate of the ˙ E of the hypothetical pulsar using the relation provided by [11]. The estimated spin-down luminosity is about 1 . 5 × 10 37 ergs -1 for the same distance. If we now compute the ˙ E/d 2 we obtain a value of 6 × 10 35 ergs -1 kpc -2 , corresponding to the band for which 70% of the PWNe are detected by H.E.S.S. [12]. Therefore, if the putative pulsar powers a TeV PWN, it should be detectable by the H.E.S.S. array. Together with the absence of detected X-ray pulsations, the PWN scenario remains unconfirmed but is energetically possible.","pages":[3]},{"title":"4.2 Binary scenario","content":"The infrared source 2MASS J18324516 -0921545 discovered in spatial coindidence with XMMU J183245 -0921539 could suggest that the X-ray source resides in a binary system around a massive star. The magnitude extinction number in the optical band expected from the X-ray column density is AV=59 + 17 -15 , using the NH -A V relation given by [13]. This would result in an absolute magnitude MJ ≤ - 14.6 for a distance ≥ 5 kpc [14]. This value is excluded but the NH -A V relation cannot apply in case of strong local absorption. Therefore, a binary scenario with 2MASS J18324516 -0921545 as optical companion could only work if the X-ray absorption arises mainly locally around XMMU J183245 -0921539. In the absence of orbitally modulated X-ray or TeV emission, the binary possibility remains unconfirmed for the moment.","pages":[3,4]},{"title":"4.3 SNR-molecular cloud interaction scenario","content":"Despite the lack of X-ray emission from the SNR shell that could be due to synchrotron radiation of high energy electrons, or thermal X-ray emission which could stem from shock-gas interactions seen frequently in middleaged SNRs, the observed VHE emission might still come from particles accelerated in the remnant. Those particles would then interact with localized target material via neutral pion production and decay or bremsstrahlung emission. Indeed, 13 CO measurements show the presence of structures around HESS J1832 -093, while at corresponding distance ranges lower gas densities are seen in other portions of the SNR shell. There is, however, no support of the association of these gas structures with G22.7 -0.2, e.g. through maser emission from shock-cloud interaction. An identification with the TeV source might therefore be due to chance coincidence. Further investigation on a possible CRpropagationto a nearby MC at the origin of the gammaray emission thus needs to be performed.","pages":[4]},{"title":"5 Conclusion","content":"Observations in the field of view of SNR G22.7 -0.2 have led to the discovery of the VHE source HESS J1832 -093 lying on the edge of the SNR radio rim. The available multi-wavelength data, using archival radio and infrared data, as well as a dedicated XMMNewton X-ray pointing towards the source, do not permit to unambiguously determine the nature of the object that gives rise to the VHE emission. A compelling X-ray counterpart, XMMU J183245 -0921539, has been discovered. The nature of this X-ray source could, however, not be established from the X-ray data alone. Together with the TeV emission and the infrared point source 2MASS J18324516 -0921545, plausible object classifications are a pulsar wind nebula or a binary system. Cosmic-rays accelerated in the SNR G22.7 -0.2 interacting with dense gas material could also result in TeV emission but the interaction between the SNR and a molecular cloud is not supported by observational evidence such as maser emission. The nature of the gamma-ray source HESS J1832 -093 presented here remains, therefore, undetermined. More data in X-rays and radio to look for variability and faint diffuse emission could help to constrain the nature of the source.","pages":[4]},{"title":"Acknowledgments","content":"The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Tech- nology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.","pages":[4]},{"title":"References","content":"1166-1176 doi:10.1086/589495 [14] Cardelli, J. A. and Clayton, G. C. and Mathis, J. S., proc. IAU (1989) Bib. code:1989IAUS..135P...5C","pages":[4]}],"string":"[\n {\n \"title\": \"Discovery of the mysterious gamma-ray source HESSJ1832-093 in the vicinity of SNR G22.7-0.2\",\n \"content\": \"H. LAFFON 1 , 2 , F. ACERO 3 , F. BRUN 4 , B. KH'ELIFI 2 , G. P UHLHOFER 5 , R. TERRIER 6 , FOR THE H.E.S.S. COLLABORATION. laffon@cenbg.in2p3.fr Abstract: Thanks to the use of advanced analysis techniques, the H.E.S.S. imaging Cherenkov telescope array has now reached a sensitivity level allowing the detection of sources with fluxes around 1% of the Crab with a limited observation time (less than ∼ 100 hours). 67 hours of observations in the region of the supernova remnant G22.7-0.2 thus yielded the detection of the faint point-like source HESS J1832-093 in spatial coincidence with a part of the radio shell of the supernova remnant. A multi-wavelength search for counterparts was then performed and led to the elaboration of various scenarii in order to explain the origin of the very high energy excess. The presence of molecular clouds in the line of sight could indicate a hadronic origin through π 0 production and decay. However, the discovery of an X-ray point source and its potential infrared counterpart give rise to other possibilities such as a pulsar wind nebula nature or a binary system. The latest results on this source are presented as well as the different scenarii brought by the multi-wavelength observations. Keywords: HESS J1832-093, SNR G22.7-0.2, molecular clouds, pulsar wind nebula, binary system\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2 H.E.S.S. observations\",\n \"content\": \"H.E.S.S. (High Energy Stereoscopic System) is an array of four imaging atmospheric Cherenkov telescopes located 1800 m above sea level in the Khomas Highland of Namibia and has been fully operational since 2004 [1]. The H.E.S.S. collaboration has been conducting a systematic scan of the Galactic Plane, which led to the discovery of a rich population of very high energy (VHE) gamma-ray sources such as in the region around the supernova remnant (SNR) W41. As a consequence, a dedicated observation campaign was launched by H.E.S.S. to study those TeV sources in detail. These observations led to the discovery of HESS J1832 -093 in spatial coincidence with the edge of SNR G22.7 -0.2. The latter shows a non-thermal shell of 26 ' diameter in radio [2] and partially overlaps the neighbouring remnant W41. A search for multiwavelength counterparts was performed in order to identify the nature of HESS J1832 -093. A proposal for observations in X-rays at the position of HESS J1832 -093 was submitted to the XMMNewton satellite and allowed the discovery of a pointlike source in X-rays near the center of the TeV source. Apart from the radio emission of the SNR, no radio pointlike counterpart was found but measurements of the 13 CO (J=1 → 0) transition line show the presence of molecular structures on the line of sight. However, an infrared pointlike source lying less than 2 '' away from the center of the Xray source was found in the 2MASS catalog. The features of the TeV emission discovered by H.E.S.S. are described in section 2. Details on the X-ray, IR and radio counterparts are given in section 3 and different considered scenarios to explain the VHE gamma-ray emission are presented in section 4.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2.1 Detection and morphology\",\n \"content\": \"A standard run selection procedure is used to remove bad quality observations in order to study the field of view. This results in a data set comprising 67 hours live time of observations taken from 2004 to 2011. A standard H.E.S.S. analysis frame is adopted with the Hillas reconstruction method using the weighted intersections of the main axis of the images to reconstruct the direction of the events [1]. A recently developped multi-variate analysis is used to provide a better discrimination between hadrons and gamma-rays [3]. A minimum charge of 110 photo-electrons in the shower images is applied to the data, resulting in an energy threshold of about 450 GeV. The background is estimated with the ring-background model, as described in [1]. Using these techniques, HESS J1832 -093 is detected with a peak significance of 7 . 9 σ pre-trials, resulting in a post-trial detection significance of 5 . 6 σ , following the approach described in [4]. The corresponding excess map of the field of view showing the new detected source is presented in Fig. 1. The average angular resolution (r68) for this selected data set and assuming a power-law spectral index of 2.3 is 0 . 081 · at the source position. A twodimensional symmetrical Gaussian function is used to determine the position and size of the TeV emission with a χ 2 minimization. The best-fit position obtained is RA = 18 h 32 m 50 s ± 32 s stat ± 36 s syst , Dec = -9 · 22 ' 36 '' ± 32 '' stat ± 36 '' syst ( J2000 ) ( χ 2 /ndf=0.89). No significant extension was found for the source.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2.2 Spectral analysis\",\n \"content\": \"To produce the energy spectrum, only the highest quality data are used, corresponding to a data set of 59 hours live time. In order to broaden the accessible energy range, the charge cut is lowered to a minimum of 80 photo-electrons, resulting in an energy threshold of ∼ 400 GeV. The background is estimated with the reflected background model, which is better suited for spectral studies [1]. The forwardfolding method described in [1] is applied to the data to derive the spectrum. Source counts are extracted from a circular region of 0.1 · radius around the best fit position of HESS J1832 -093, a size optimized for point source studies with the applied cuts. The corresponding excess is 152 ± 24 gamma-like events. The spectrum obtained between 400 GeV and 5 TeV (displayed on Fig. 2) can be described by a power-law d Φ dE = Φ 0 ( E 1TeV ) -Γ , with an index Γ = 2 . 6 ± 0 . 3stat ± 0 . 3syst and a differential flux normalisation at 1 TeV of Φ 0 = ( 4 . 8 ± 0 . 8stat ± 0 . 9syst ) × 10 -13 cm -2 s -1 TeV -1 . A light curve was produced with the available observations and no significant temporal variability was detected in the H.E.S.S. data set.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3.1 X-Ray observations with XMMNewton\",\n \"content\": \"In order to constrain the nature of the source HESS J1832 -093, a dedicated XMMNewton observation (ID: 0654480101) was performed in March 2011 for 17 ks, targeted at the position of the gamma-ray emission. After fil- tering out proton flare contamination, 13 ks and 7 ks of exposure time remained for the two EPIC-MOS cameras and for the EPIC-pn camera respectively. The data were processed using the XMMNewton Science Analysis System (v10.0). The instrumental background was derived from a compilation of blank sky observations [6], renormalized to the actual exposure using the count rate in the 10-12 keV energy band. The brightest object in the XMMNewton field of view is a point-like source located at RA = 18 h 32 m 45 s . 04 , Dec = -09 · 21 ' 53 '' . 9, 1.4 ' away from the best-fit position of the H.E.S.S. excess, as shown in Fig. 3. This new source, dubbed XMMU J183245 -0921539, lies within the 2 σ uncertainty of the VHE gamma-ray centroid. Spectra from the three instruments were extracted from a 15 '' radius circular region centered on XMMU J183245 -0921539. As the statistics are low ( ∼ 500 counts when summing the three EPIC instruments), the data were fitted in Xspec (v12.5) using Cash statistics and the spectra were not rebinned for the fitting process. The pvalue corresponding to the null-hypothesis probability is derived using the goodness command in XSPEC which performs Monte Carlo simulations of the goodness-of-fit. The best fit parameters for a power-law assumption are a column density NH = 10 . 5 + 3 . 1 -2 . 7 × 10 22 cm -2 , a photon index Γ = 1 . 3 + 0 . 5 -0 . 4 and an unabsorbed energy flux Φ (2-10 keV) = 6 . 9 + 1 . 7 -2 . 8 × 10 -13 ergcm -2 s -1 with a p-value of 0.75. A search for pulsations was performed at the position of XMMUJ183245 -0921539, but no pulsations were found in the data. A comparison of the absorption along the line of sight obtained from the X-ray spectral model with the column depth derived from the atomic (HI) and molecular ( 12 CO, J=1 → 0 transition line) gas can be used to provide a lower limit on the distance to XMMU J183245 -0921539, assuming that all absorbing material is at the near distance allowed by the Galactic rotation curve. The Galactic rotation curve model of [7] is used to translate the measured velocities into distances. A lower limit of about 5 kpc on the distance to XMMU J183245 -0921539 is thus derived.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3.2 IR counterparts to XMMU J183245 -0921539\",\n \"content\": \"The 2MASScatalog 1 shows one infrared source around the position of XMMU J183245 -0921539 within the systematic pointing error of XMMNewton ( ∼ 2 '' ). This source, 2MASS J18324516 -0921545, lies 1.9 '' away from the center of gravity of the X-ray source. No optical counterpart is found, likely due to strong extinction in the galactic plane. The apparent magnitudes observed in the J, H, K bands are mJ = 15 . 52 ± 0 . 06, mH = 13 . 26 ± 0 . 04 and mK = 12 . 17 ± 0 . 02, respectively. The IR emission could originate from a massive companion to the X-ray compact source, thus forming a binary system, as discussed in section 4.2.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.3 13 COobservations\",\n \"content\": \"The Galactic Ring Survey (GRS) performed with the Boston University FCRAO telescopes [8] provides measurements of the 13 CO (J=1 → 0) transition line covering the velocity range from -5 to 135 km s -1 in this region. The detection of this line is evidence for the presence of dense molecular clouds that are known to be targets for cosmic-rays and hence gamma-ray emitters via neutral pion production and decay or bremsstrahlung emission. A 0.2 · -side square region is defined around the source HESS J1832 -093 to look for molecular clouds traced by the 13 CO transition line. Several molecular clouds measured at different radial velocities are found in this region. The two structures that show the best spatial coincidence with the TeV excess are selected. Their velocity ranges are 26 to 31 kms -1 and 73 to 81 kms -1 respectively. The antenna temperature of each molecular cloud is integrated on the corresponding range. The same Galactic rotation curve model as used in section 3.1 [7] is assumed to translate the measured velocities into distances, each velocity corresponding to two possible distances given our position in the Milky Way. However, in case of an association with the SNR, only the near distance (2.3 kpc and 4.5 kpc respectively) would be compatible with the distance estimates to the remnant. Following the approach described in [9], the integrated antenna temperatures are used to derive the gas mass of each structure and the corresponding gas densities (20 cm -3 and 62 cm -3 respectively).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4.1 PWNscenario\",\n \"content\": \"Dedicated XMMNewton observations at the position of HESS J1832 -093 have revealed the presence of the pointlike source XMMU J183245 -0921539 showing a hard spectrum ( Γ = 1 . 3 + 0 . 5 -0 . 4 ). The source position in the galactic plane points towards a location inside our galaxy, at a distance ≥ 5 kpc as deduced from the X-ray power-law model. XMMU J183245 -0921539 is a serious counterpart candidate for HESS J1832 -093 because of its proximity with the best-fit position of the TeV emission and its hard spectrum. A likely scenario would be that both the X-ray and TeV sources stem from a pulsar wind nebula (PWN) powered by a yet unknown pulsar. Even if the non-thermal aspect of the X-ray emission is not well determined, its hard spectral index is indicative of an emission from the vicinity of a pulsar (magnetospheric, striped wind [10],...). Therefore, despite the lack of observed pulsations in the object, we will consider a pulsar origin for XMMU J183245 -0921539 in the following. It can be tested whether energetically a PWN scenario plausibly matches with the population of known TeVemitting PWNe, under the hypothesis that the X-ray emission comes from the pulsar's magnetosphere. The unabsorbed flux Φ X (2-10 keV) of the point source for the power-law model is used to compute the corresponding luminosity in the same energy band for a distance of 5 kpc: LX ( 2 -10keV ) /similarequal 2 × 10 33 ergs -1 . It can then be translated to an estimate of the ˙ E of the hypothetical pulsar using the relation provided by [11]. The estimated spin-down luminosity is about 1 . 5 × 10 37 ergs -1 for the same distance. If we now compute the ˙ E/d 2 we obtain a value of 6 × 10 35 ergs -1 kpc -2 , corresponding to the band for which 70% of the PWNe are detected by H.E.S.S. [12]. Therefore, if the putative pulsar powers a TeV PWN, it should be detectable by the H.E.S.S. array. Together with the absence of detected X-ray pulsations, the PWN scenario remains unconfirmed but is energetically possible.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4.2 Binary scenario\",\n \"content\": \"The infrared source 2MASS J18324516 -0921545 discovered in spatial coindidence with XMMU J183245 -0921539 could suggest that the X-ray source resides in a binary system around a massive star. The magnitude extinction number in the optical band expected from the X-ray column density is AV=59 + 17 -15 , using the NH -A V relation given by [13]. This would result in an absolute magnitude MJ ≤ - 14.6 for a distance ≥ 5 kpc [14]. This value is excluded but the NH -A V relation cannot apply in case of strong local absorption. Therefore, a binary scenario with 2MASS J18324516 -0921545 as optical companion could only work if the X-ray absorption arises mainly locally around XMMU J183245 -0921539. In the absence of orbitally modulated X-ray or TeV emission, the binary possibility remains unconfirmed for the moment.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"4.3 SNR-molecular cloud interaction scenario\",\n \"content\": \"Despite the lack of X-ray emission from the SNR shell that could be due to synchrotron radiation of high energy electrons, or thermal X-ray emission which could stem from shock-gas interactions seen frequently in middleaged SNRs, the observed VHE emission might still come from particles accelerated in the remnant. Those particles would then interact with localized target material via neutral pion production and decay or bremsstrahlung emission. Indeed, 13 CO measurements show the presence of structures around HESS J1832 -093, while at corresponding distance ranges lower gas densities are seen in other portions of the SNR shell. There is, however, no support of the association of these gas structures with G22.7 -0.2, e.g. through maser emission from shock-cloud interaction. An identification with the TeV source might therefore be due to chance coincidence. Further investigation on a possible CRpropagationto a nearby MC at the origin of the gammaray emission thus needs to be performed.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"5 Conclusion\",\n \"content\": \"Observations in the field of view of SNR G22.7 -0.2 have led to the discovery of the VHE source HESS J1832 -093 lying on the edge of the SNR radio rim. The available multi-wavelength data, using archival radio and infrared data, as well as a dedicated XMMNewton X-ray pointing towards the source, do not permit to unambiguously determine the nature of the object that gives rise to the VHE emission. A compelling X-ray counterpart, XMMU J183245 -0921539, has been discovered. The nature of this X-ray source could, however, not be established from the X-ray data alone. Together with the TeV emission and the infrared point source 2MASS J18324516 -0921545, plausible object classifications are a pulsar wind nebula or a binary system. Cosmic-rays accelerated in the SNR G22.7 -0.2 interacting with dense gas material could also result in TeV emission but the interaction between the SNR and a molecular cloud is not supported by observational evidence such as maser emission. The nature of the gamma-ray source HESS J1832 -093 presented here remains, therefore, undetermined. More data in X-rays and radio to look for variability and faint diffuse emission could help to constrain the nature of the source.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Tech- nology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"1166-1176 doi:10.1086/589495 [14] Cardelli, J. A. and Clayton, G. C. and Mathis, J. S., proc. IAU (1989) Bib. code:1989IAUS..135P...5C\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":284,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33.1128B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1308.1512.pdf"},"content":{"kind":"string","value":"\n\n\n\nFACT - The First G-APD Cherenkov Telescope: Status and Results\nT. BRETZ 1 , H. ANDERHUB 1 , M. BACKES 2 , A. BILAND 1 ,A V. BOCCONE 3 , I. BRAUN 1 , J. BUSS 2 , F. CADOUX 3 , V. COMMICHAU 1 , L. DJAMBAZOV 1 , D. DORNER 4 , S. EINECKE 2 , D. EISENACHER 4 , A. GENDOTTI 1 , O. GRIMM 1 , H. VON GUNTEN 1 , C. HALLER 1 , D. HILDEBRAND 1 , U. HORISBERGER 1 , B. HUBER 1 a , K.-S. KIM 1 b , M. L. KNOETIG 1 , J.-H. K OHNE 2 , T. KR AHENB UHL 1 , B. KRUMM 2 , M. LEE 1 b , E. LORENZ 1 c , W. LUSTERMANN 1 , E. LYARD 3 , K. MANNHEIM 4 , M. MEHARGA 3 , K. MEIER 4 , T. MONTARULI 3 , D. NEISE 2 , F. NESSI-TEDALDI 1 , A.-K. OVERKEMPING 2 , A. PARAVAC 4 , F. PAUSS 1 , D. RENKER 1 d , W. RHODE 2 , M. RIBORDY 5 , U. R OSER 1 , J.-P. STUCKI 1 , J. SCHNEIDER 1 , T. STEINBRING 4 , F. TEMME 2 , J. THAELE 2 , S. TOBLER 1 , G. VIERTEL 1 , P. VOGLER 1 , R. WALTER 3 , K. WARDA 2 , Q. WEITZEL 1 , M. Z ANGLEIN 4 (FACT COLLABORATION)\n1 ETH Zurich, Switzerland -Institute for Particle Physics, Schafmattstr. 20, 8093 Zurich\n\n2 Technische Universitat Dortmund, Germany -Experimental Physics 5, Otto-Hahn-Str. 4, 44221 Dortmund\n3 University of Geneva, Switzerland -ISDC, Chemin d'Ecogia 16, 1290 Versoix -DPNC, Quai Ernest-Ansermet 24, 1211 Geneva\n4 Universitat Wurzburg, Germany -Institute for Theoretical Physics and Astrophysics, Emil-Fischer-Str. 31, 97074 Wurzburg\n5 EPF Lausanne, Switzerland -Laboratory for High Energy Physics, 1015 Lausanne\na Also at: University of Zurich, Physik-Institut, 8057 Zurich, Switzerland\nb Also at: Kyungpook National University, Center for High Energy Physics, 702-701 Daegu, Korea\nc Also at: Max-Planck-Institut fur Physik, D-80805 Munich, Germany\n\nd Also at: Technische Universitat Munchen, D-85748 Garching, Germany\nthomas.bretz@phys.ethz.ch\nAbstract: The First G-APD Cherenkov telescope (FACT) is the first telescope using silicon photon detectors (G-APD aka. SiPM). It is built on the mount of the HEGRA CT3 telescope, still located at the Observatorio del Roque de los Muchachos, and it is successfully in operation since Oct. 2011. The use of Silicon devices promises a higher photon detection efficiency, more robustness and higher precision than photo-multiplier tubes. The FACT collaboration is investigating with which precision these devices can be operated on the long-term. Currently, the telescope is successfully operated from remote and robotic operation is under development. During the past months of operation, the foreseen monitoring program of the brightest known TeV blazars has been carried out, and first physics results have been obtained including a strong flare of Mrk501. An instantaneous flare alert system is already in a testing phase. This presentation will give an overview of the project and summarize its goals, status and first results.\nKeywords: FACT, G-APD, silicon photo sensor, focal plane\n1 Introduction\nSince Oct. 2011, the FACT Collaboration is operating the First G-APD Cherenkov telescope [1] at the Observatorio del Roque de los Muchachos on the Canary Island of La Palma. The telescope's camera is the first focal plane installation using Geiger-mode avalanche photo-diodes (G-APD) for photo detection. Comprising 1440 channels individually read out, each pixel has a field-of-view of 0.11° yielding a total field-of-view of 4.5°. The camera is installed on the mount of the former HEGRA CT 3 telescope. After replacement of the old disc shaped mirrors with refurbished hexagonal mirrors, it has now a total reflective surface of 9.5 m 2 . A picture of the telescope is shown in Fig. 1.\nWith this novel camera, silicon photo sensors have started to replace photo multiplier tubes (PMT) still widely used for photo detection. They offer a high gain (10 5 to 10 6 ) and are robust enough to be operated under moon light conditions. Their single photon counting capability, their compactness and their low operational voltage ( < 100 V) simplifies the camera design. The photon detection efficiency (PDE) of commercially available G-APDs is already at the level of the best PMTs, and will still significantly improve in the future.\nThe challenge of the project is to understand and operate the silicon based photo sensors under changing environ-\nmental conditions, such as changing auxiliary temperature or changing photon flux from the diffuse night-sky background. A system to keep the G-APD response stable under these conditions has been developed.\nThe telescope is dedicated to long-term monitoring of the brightest known TeV blazars. These highly violent objects show variability on time scales of seconds to years. To study their behavior on the longer time scales, a complete data sample obtained from continuous measurements on long time scales is necessary.\n2 System overview\nThe camera of the telescope is compiled from 1440 channels. Each channel is equipped with a G-APD (Hamamatsu MPPC S10362-33-50C) and a solid light concentrator. The solid light concentrators have the advantage, as compared to hollow ones, to feature a better compression ratio between entrance window and exit and thus balance the relatively small size of the sensors (9 mm 2 ). Sensors, cones and a protective window were glued together using optical glue. All channels are individually read out by the Domino Ring Sampling chip (DRS 4). The data is transferred by a TCP/IP connection via Ethernet to a data acquisition PC. The summed signal of nine channels form a trigger signal which\n\n\n
Figure 1 : Picture of the telescope
\n\nis discriminated by a comparator. Each trigger patch is divided in two bias voltage channels with four and five GAPDs, resp. Although all read out electronics was integrated into the camera, the bias voltage supply is located in the counting hut. Each bias voltage channel has its own current readout. The precision of the voltage application is 12 bit up to 90 V and 12 bit up to 5 mA for the current readout.\nThe total power consumption of the camera electronics during operation is about 570 W. About 100 W are dissipated in the supply lines and another 100 W due to the limited efficiency of the DC-DC converters, yielding a total of 370 W consumed by the electronics, ≈ 260 mW per channel.\nTo get rid of the waste heat produced by the camera electronics, a passive water cooling is applied. To avoid the sensors being heat up by the electronics, a thermal insulation is installed in between. In total, 31 temperature sensors measure the temperature of the sensor plane.\nAll details about the camera construction, the electronics and control software can be found in [1].\n3 Status and achievements\n3.1 Operation\nFor a high duty cycle and consistent and stable data taking, a robust slow control and data acquisition software is needed. To avoid the need of shift crew on site, full remote operation must be possible. The camera is now operated since 20 months and its auxiliary hardware has been upgraded gradually, to allow for full remote control. This includes an interlock system, which purpose is to protect the camera from a cooling failure, Ethernet switchable 220 V plugs and the Ethernet access to all power supplies and a remote controllable lids. More details on that can be found in [2]. In parallel, a fully automatic control system has been developed. While all important parameters of the system are available via Web interface (see Fig. 2), the control of the system is achieved from a JavaScript engine reading the schedule from a database. Since a couple of weeks, this system is operating, and user interaction is usually only required in case of unexpected weather conditions.\nCurrently, a fully automatic analysis is also in the testing phase. It is applied on the data immediately after the data has been recorded and the file has been closed which typically happens every 5 min for physics data. The analysis takes\n\n\n\nabout the same time than data taking and will allow prompt flare alerts for other telescopes in the near future.\n3.2 Stability\nOne of the main goals of the project was the prove of the applicability of G-APDs during standard operation conditions. Since the gain and all other properties of GAPDs depend on their temperature and the voltage applied, a correcting element is necessary. By design, the temperature gradient in the focal plane is small enough allowing several G-APDs to be connected to a single bias voltage channel. In total, 31 temperature sensors installed in the focal plane allow to counteract temperature changes by adapting the applied bias voltage accordingly. The power supply of each sensor is equipped with a filter network of resistors. Due to the changing light conditions induced by changes of the atmospheric conditions or moon light, the current drawn by the G-APDs is varying inducing a voltage drop in the resistances. To correct for this voltage drop, the current of each channel is measured and the voltage corrected accordingly.\nTo measure the stability of the gain with these two feedback loops applied, three different methods are available:\nExternal light pulser An external temperature stabilized light pulser triggers the camera. Since its light yield is stable on average, the gain can directly be deduced from the average pulse height. However, the precision of this system measuring the gain of the G-APD is limited by the fact that also the transmission of the entrance window and the cones could change with time.\nDark count spectrum A direct and mostly unbiased method to extract the gain, is the extraction of the dark count spectrum. It only depends on the performance of the G-APDs and the readout chain. Its drawback is that it can only be determined under light conditions, i.e. count rates, which still allow for extraction of single pulses. This is the case with closed lids or at dark time conditions.\nRatescans To measure the stability of the gain in moon lit nights, so called ratescans are used. Ratescans measure the trigger response as a function of the applied trigger threshold. While at low thresholds, the very high rates are dominated by random coincidences of background photons, at high thresholds, the rate represents the number of triggers from coincident light flashed, induced mainly by hadronic showers. Although this method depends on the performance of the whole system, it allows to measure the response of the system even at the brightest light conditions. To ensure an unbiased measurement, only ratescans taken during good atmospheric conditions and weather conditions must be taken into account. A more detailed discussion of this can be found in [9].\nAll three methods have been applied almost daily since the beginning of operation. This allows to compare the response at any possible light condition and all temperatures occurring over the seasons. All methods have shown consistently that with the applied feedback system, a stable gain up to full moon conditions can be achieved. From the dark count spectrum, a stability of better than ± 3% over time and 4% pixel-to-pixel variations was deduced. Currently, the limiting factor of the system is the calibration of the\napplied bias voltage which is under investigation and will be improved soon.\nMore details about the stability can be found in [3] and will be available soon in [4].\n\n\n\n\n\n
Figure 2 : Web interface. To achieve low network traffic, only the average signal is displayed for four and five pixels. http://www.fact-project.org/smartfact/
\n\n3.3 Data acquisition\nThe Ethernet connection between the forty readout boards and the data acquisition PC is routed over four Ethernet switches, two located in the camera and two located in the control room. In total, four Ethernet lines are available, one Ethernet line per ten boards. A total maximum transfer rate of about 1.9 Gbps is achieved. This is limited by the maximum throughput of 50 Mbps of each of the Ethernetchips (Wiznet 5300) on-board of the forty data acquisition boards. The transfer rate corresponds to a trigger rate of ∼ 250 Hz for a readout window of 300 samples and ∼ 80 Hz for the readout of the full DRS pipeline (1024 samples).\n3.4 Trigger threshold setting\nOne of the most important aspects during data taking is the correct setting of the trigger thresholds. To avoid high rates from random coincidences of background photons, the need to be set above the noise level. Since the noise level, mainly defined by the diffuse light from the night sky background, is not necessarily constant, some contingency is added to avoid strong rate fluctuations during data taking. At the same time, they must be set to as low as possible to achieve the lowest energy threshold and thus the highest integral sensitivity. Since the number of breakdowns in the G-APD, i.e. the number of detected photons, is proportional to the measured current, the rate induced from random coincidences as a function of the trigger threshold can directly be determined from the current. From ratescans taken at different light conditions over more than a year, a parametrization for the rate induced from background light and for the rate induced by hadronic showers under best conditions, was derived. Since several weeks, the trigger threshold of the system is now directly deduced from the measured current resulting in very stable data taking conditions. To allow for easy analysis, it is kept constant during every 5 min run.\n3.5 Current and energy threshold prediction\nTo allow for an efficient schedule, the prediction of the system response under changing light conditions is mandatory. Therefore, the measured current has been correlated with moon conditions, i.e. moon brightness and zenith angle, which leads to a very precise prediction of the measured current. Since the trigger threshold is in the first order proportional to the light yield of the triggered shower, the change in energy threshold can directly be deduced. Including the change of the light yield with zenith angle, gives a very good estimate of the raise in energy threshold, i.e. loss in sensitivity, for each observation.\nThese topics are discussed in more details in [5, 6] and will be available soon in [4].\n4 Physics results\nVery soon after the first observations, first results could be presented [7]. From the measurements of the Crab Nebula, a sensitivity of 8% Crab in 50 h could be derived. This is based on the analysis of data taking during the first months of operation and based on an analysis primarily developed for MAGIC not yet necessarily optimized for the current system. From the rate of excess events an energy threshold between 400 GeV and 700 GeV can be derived taking the known spectrum of the source into account\nThe analysis, especially the automatic analysis, and first results are discussed in [8].\n5 Conclusions\nThe camera of the First G-APD Cherenkov Telescope (FACT) has proven the applicability of silicon photo sensors (G-APD) in focal plane installations, in particular as detectors in Cherenkov telescopes. The camera is now in operation since more than one and a half years, and although there were hardware problems during that time, none of the was related to the G-APDs at all. So far, no decrease in performance could be detected. Neither any hint for aging of\n\n\n\n\n\n
Figure 3 : An exemplary event. The color scale is proportional to the signal amplitude detected in each pixel.
\n\nthe G-APDs nor of the transmission of cones and window has been found.\nWith different types of measurements, it was shown that a stability to on the few percent level, independent of temperature and light conditions can be achieved. Further improvements are expected by an improvement of the bias voltage calibration.\nAn important result derived from the current study is that the application of a feedback system which keeps the G-APD's bias voltage constant, renders the need for an external calibration device obsolete. The combination of the extraction of the dark count spectrum and the current measurement, is enough for the operation of such a device.\nRecently, Hamamatsu presented a new generation of G-APD sensors with significantly lower dark count rates, significantly reduced optical crosstalk and significantly reduced afterpulse probability, expected to be available on the market within the next weeks [10].\nAlthough, it was shown that these improvements are not obligatory for the operation in a Cherenkov telescope, especially the reduction in optical crosstalk guarantees that the influence on image reconstruction is negligible.\nNot only, can the FACT telescope serve as an ideal instrument for long-term monitoring of bright blazars, the application of G-APDs in focal plane installations will significantly reduce construction and installation costs and improve stability of operation. This will yield a significantly impact on future projects like The Cherenkov Telescope Array (CTA) and makes small telescopes for monitoring purposes affordable.\nAcknowledgment The important contributions from ETH Zurich grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) are gratefully acknowledged. We thank the Instituto de Astrofisica de Canarias allowing us to operate the telescope at the Observatorio Roque de los Muchachos in La Palma, the Max-Planck-Institut fur Physik for providing us with the mount of the former HEGRA CT 3 telescope, and the MAGIC collaboration for their support. We also thank the group of Marinella Tose from the College of Engineering and Technology at Western\n\n\n\nMindanao State University, Philippines, for providing us with the scheduling web-interface.\nReferences\n\n[1] H. Anderhub et al. (FACT Collaboration), 2013, JINST 8 P06008 [arXiv:1304.1710].\n[2] A. Biland et al. (FACT Collaboration), these proc., ID 708.\n[3] T. Bretz et al. (FACT Collaboration), these proc., ID 683.\n[4] FACT Collaboration, in prep.\n[5] T. Bretz et al. (FACT Collaboration), these proc., ID 720.\n[6] M. Koetig et al. (FACT Collaboration), these proc., ID 695.\n[7] Bretz, T. et al. (FACT Collaboration) 2012, AIPC, 1505, 773.\n[8] D. Dorner et al. (FACT Collaboration), these proc., ID 686.\n[9] D. Hildebrand et al. (FACT Collaboration), these proc., ID709.\n\n[10] http://kicp-workshops.uchicago.edu/ieu2013 .\n"},"sections":{"kind":"list like","value":[{"title":"FACT - The First G-APD Cherenkov Telescope: Status and Results","content":"T. BRETZ 1 , H. ANDERHUB 1 , M. BACKES 2 , A. BILAND 1 ,A V. BOCCONE 3 , I. BRAUN 1 , J. BUSS 2 , F. CADOUX 3 , V. COMMICHAU 1 , L. DJAMBAZOV 1 , D. DORNER 4 , S. EINECKE 2 , D. EISENACHER 4 , A. GENDOTTI 1 , O. GRIMM 1 , H. VON GUNTEN 1 , C. HALLER 1 , D. HILDEBRAND 1 , U. HORISBERGER 1 , B. HUBER 1 a , K.-S. KIM 1 b , M. L. KNOETIG 1 , J.-H. K OHNE 2 , T. KR AHENB UHL 1 , B. KRUMM 2 , M. LEE 1 b , E. LORENZ 1 c , W. LUSTERMANN 1 , E. LYARD 3 , K. MANNHEIM 4 , M. MEHARGA 3 , K. MEIER 4 , T. MONTARULI 3 , D. NEISE 2 , F. NESSI-TEDALDI 1 , A.-K. OVERKEMPING 2 , A. PARAVAC 4 , F. PAUSS 1 , D. RENKER 1 d , W. RHODE 2 , M. RIBORDY 5 , U. R OSER 1 , J.-P. STUCKI 1 , J. SCHNEIDER 1 , T. STEINBRING 4 , F. TEMME 2 , J. THAELE 2 , S. TOBLER 1 , G. VIERTEL 1 , P. VOGLER 1 , R. WALTER 3 , K. WARDA 2 , Q. WEITZEL 1 , M. Z ANGLEIN 4 (FACT COLLABORATION) 1 ETH Zurich, Switzerland -Institute for Particle Physics, Schafmattstr. 20, 8093 Zurich d Also at: Technische Universitat Munchen, D-85748 Garching, Germany thomas.bretz@phys.ethz.ch Abstract: The First G-APD Cherenkov telescope (FACT) is the first telescope using silicon photon detectors (G-APD aka. SiPM). It is built on the mount of the HEGRA CT3 telescope, still located at the Observatorio del Roque de los Muchachos, and it is successfully in operation since Oct. 2011. The use of Silicon devices promises a higher photon detection efficiency, more robustness and higher precision than photo-multiplier tubes. The FACT collaboration is investigating with which precision these devices can be operated on the long-term. Currently, the telescope is successfully operated from remote and robotic operation is under development. During the past months of operation, the foreseen monitoring program of the brightest known TeV blazars has been carried out, and first physics results have been obtained including a strong flare of Mrk501. An instantaneous flare alert system is already in a testing phase. This presentation will give an overview of the project and summarize its goals, status and first results. Keywords: FACT, G-APD, silicon photo sensor, focal plane","pages":[1]},{"title":"1 Introduction","content":"Since Oct. 2011, the FACT Collaboration is operating the First G-APD Cherenkov telescope [1] at the Observatorio del Roque de los Muchachos on the Canary Island of La Palma. The telescope's camera is the first focal plane installation using Geiger-mode avalanche photo-diodes (G-APD) for photo detection. Comprising 1440 channels individually read out, each pixel has a field-of-view of 0.11° yielding a total field-of-view of 4.5°. The camera is installed on the mount of the former HEGRA CT 3 telescope. After replacement of the old disc shaped mirrors with refurbished hexagonal mirrors, it has now a total reflective surface of 9.5 m 2 . A picture of the telescope is shown in Fig. 1. With this novel camera, silicon photo sensors have started to replace photo multiplier tubes (PMT) still widely used for photo detection. They offer a high gain (10 5 to 10 6 ) and are robust enough to be operated under moon light conditions. Their single photon counting capability, their compactness and their low operational voltage ( < 100 V) simplifies the camera design. The photon detection efficiency (PDE) of commercially available G-APDs is already at the level of the best PMTs, and will still significantly improve in the future. The challenge of the project is to understand and operate the silicon based photo sensors under changing environ- mental conditions, such as changing auxiliary temperature or changing photon flux from the diffuse night-sky background. A system to keep the G-APD response stable under these conditions has been developed. The telescope is dedicated to long-term monitoring of the brightest known TeV blazars. These highly violent objects show variability on time scales of seconds to years. To study their behavior on the longer time scales, a complete data sample obtained from continuous measurements on long time scales is necessary.","pages":[1]},{"title":"2 System overview","content":"The camera of the telescope is compiled from 1440 channels. Each channel is equipped with a G-APD (Hamamatsu MPPC S10362-33-50C) and a solid light concentrator. The solid light concentrators have the advantage, as compared to hollow ones, to feature a better compression ratio between entrance window and exit and thus balance the relatively small size of the sensors (9 mm 2 ). Sensors, cones and a protective window were glued together using optical glue. All channels are individually read out by the Domino Ring Sampling chip (DRS 4). The data is transferred by a TCP/IP connection via Ethernet to a data acquisition PC. The summed signal of nine channels form a trigger signal which is discriminated by a comparator. Each trigger patch is divided in two bias voltage channels with four and five GAPDs, resp. Although all read out electronics was integrated into the camera, the bias voltage supply is located in the counting hut. Each bias voltage channel has its own current readout. The precision of the voltage application is 12 bit up to 90 V and 12 bit up to 5 mA for the current readout. The total power consumption of the camera electronics during operation is about 570 W. About 100 W are dissipated in the supply lines and another 100 W due to the limited efficiency of the DC-DC converters, yielding a total of 370 W consumed by the electronics, ≈ 260 mW per channel. To get rid of the waste heat produced by the camera electronics, a passive water cooling is applied. To avoid the sensors being heat up by the electronics, a thermal insulation is installed in between. In total, 31 temperature sensors measure the temperature of the sensor plane. All details about the camera construction, the electronics and control software can be found in [1].","pages":[1,2]},{"title":"3.1 Operation","content":"For a high duty cycle and consistent and stable data taking, a robust slow control and data acquisition software is needed. To avoid the need of shift crew on site, full remote operation must be possible. The camera is now operated since 20 months and its auxiliary hardware has been upgraded gradually, to allow for full remote control. This includes an interlock system, which purpose is to protect the camera from a cooling failure, Ethernet switchable 220 V plugs and the Ethernet access to all power supplies and a remote controllable lids. More details on that can be found in [2]. In parallel, a fully automatic control system has been developed. While all important parameters of the system are available via Web interface (see Fig. 2), the control of the system is achieved from a JavaScript engine reading the schedule from a database. Since a couple of weeks, this system is operating, and user interaction is usually only required in case of unexpected weather conditions. Currently, a fully automatic analysis is also in the testing phase. It is applied on the data immediately after the data has been recorded and the file has been closed which typically happens every 5 min for physics data. The analysis takes about the same time than data taking and will allow prompt flare alerts for other telescopes in the near future.","pages":[2]},{"title":"3.2 Stability","content":"One of the main goals of the project was the prove of the applicability of G-APDs during standard operation conditions. Since the gain and all other properties of GAPDs depend on their temperature and the voltage applied, a correcting element is necessary. By design, the temperature gradient in the focal plane is small enough allowing several G-APDs to be connected to a single bias voltage channel. In total, 31 temperature sensors installed in the focal plane allow to counteract temperature changes by adapting the applied bias voltage accordingly. The power supply of each sensor is equipped with a filter network of resistors. Due to the changing light conditions induced by changes of the atmospheric conditions or moon light, the current drawn by the G-APDs is varying inducing a voltage drop in the resistances. To correct for this voltage drop, the current of each channel is measured and the voltage corrected accordingly. To measure the stability of the gain with these two feedback loops applied, three different methods are available: External light pulser An external temperature stabilized light pulser triggers the camera. Since its light yield is stable on average, the gain can directly be deduced from the average pulse height. However, the precision of this system measuring the gain of the G-APD is limited by the fact that also the transmission of the entrance window and the cones could change with time. Dark count spectrum A direct and mostly unbiased method to extract the gain, is the extraction of the dark count spectrum. It only depends on the performance of the G-APDs and the readout chain. Its drawback is that it can only be determined under light conditions, i.e. count rates, which still allow for extraction of single pulses. This is the case with closed lids or at dark time conditions. Ratescans To measure the stability of the gain in moon lit nights, so called ratescans are used. Ratescans measure the trigger response as a function of the applied trigger threshold. While at low thresholds, the very high rates are dominated by random coincidences of background photons, at high thresholds, the rate represents the number of triggers from coincident light flashed, induced mainly by hadronic showers. Although this method depends on the performance of the whole system, it allows to measure the response of the system even at the brightest light conditions. To ensure an unbiased measurement, only ratescans taken during good atmospheric conditions and weather conditions must be taken into account. A more detailed discussion of this can be found in [9]. All three methods have been applied almost daily since the beginning of operation. This allows to compare the response at any possible light condition and all temperatures occurring over the seasons. All methods have shown consistently that with the applied feedback system, a stable gain up to full moon conditions can be achieved. From the dark count spectrum, a stability of better than ± 3% over time and 4% pixel-to-pixel variations was deduced. Currently, the limiting factor of the system is the calibration of the applied bias voltage which is under investigation and will be improved soon. More details about the stability can be found in [3] and will be available soon in [4].","pages":[2,3]},{"title":"3.3 Data acquisition","content":"The Ethernet connection between the forty readout boards and the data acquisition PC is routed over four Ethernet switches, two located in the camera and two located in the control room. In total, four Ethernet lines are available, one Ethernet line per ten boards. A total maximum transfer rate of about 1.9 Gbps is achieved. This is limited by the maximum throughput of 50 Mbps of each of the Ethernetchips (Wiznet 5300) on-board of the forty data acquisition boards. The transfer rate corresponds to a trigger rate of ∼ 250 Hz for a readout window of 300 samples and ∼ 80 Hz for the readout of the full DRS pipeline (1024 samples).","pages":[3]},{"title":"3.4 Trigger threshold setting","content":"One of the most important aspects during data taking is the correct setting of the trigger thresholds. To avoid high rates from random coincidences of background photons, the need to be set above the noise level. Since the noise level, mainly defined by the diffuse light from the night sky background, is not necessarily constant, some contingency is added to avoid strong rate fluctuations during data taking. At the same time, they must be set to as low as possible to achieve the lowest energy threshold and thus the highest integral sensitivity. Since the number of breakdowns in the G-APD, i.e. the number of detected photons, is proportional to the measured current, the rate induced from random coincidences as a function of the trigger threshold can directly be determined from the current. From ratescans taken at different light conditions over more than a year, a parametrization for the rate induced from background light and for the rate induced by hadronic showers under best conditions, was derived. Since several weeks, the trigger threshold of the system is now directly deduced from the measured current resulting in very stable data taking conditions. To allow for easy analysis, it is kept constant during every 5 min run.","pages":[3]},{"title":"3.5 Current and energy threshold prediction","content":"To allow for an efficient schedule, the prediction of the system response under changing light conditions is mandatory. Therefore, the measured current has been correlated with moon conditions, i.e. moon brightness and zenith angle, which leads to a very precise prediction of the measured current. Since the trigger threshold is in the first order proportional to the light yield of the triggered shower, the change in energy threshold can directly be deduced. Including the change of the light yield with zenith angle, gives a very good estimate of the raise in energy threshold, i.e. loss in sensitivity, for each observation. These topics are discussed in more details in [5, 6] and will be available soon in [4].","pages":[3]},{"title":"4 Physics results","content":"Very soon after the first observations, first results could be presented [7]. From the measurements of the Crab Nebula, a sensitivity of 8% Crab in 50 h could be derived. This is based on the analysis of data taking during the first months of operation and based on an analysis primarily developed for MAGIC not yet necessarily optimized for the current system. From the rate of excess events an energy threshold between 400 GeV and 700 GeV can be derived taking the known spectrum of the source into account The analysis, especially the automatic analysis, and first results are discussed in [8].","pages":[3]},{"title":"5 Conclusions","content":"The camera of the First G-APD Cherenkov Telescope (FACT) has proven the applicability of silicon photo sensors (G-APD) in focal plane installations, in particular as detectors in Cherenkov telescopes. The camera is now in operation since more than one and a half years, and although there were hardware problems during that time, none of the was related to the G-APDs at all. So far, no decrease in performance could be detected. Neither any hint for aging of the G-APDs nor of the transmission of cones and window has been found. With different types of measurements, it was shown that a stability to on the few percent level, independent of temperature and light conditions can be achieved. Further improvements are expected by an improvement of the bias voltage calibration. An important result derived from the current study is that the application of a feedback system which keeps the G-APD's bias voltage constant, renders the need for an external calibration device obsolete. The combination of the extraction of the dark count spectrum and the current measurement, is enough for the operation of such a device. Recently, Hamamatsu presented a new generation of G-APD sensors with significantly lower dark count rates, significantly reduced optical crosstalk and significantly reduced afterpulse probability, expected to be available on the market within the next weeks [10]. Although, it was shown that these improvements are not obligatory for the operation in a Cherenkov telescope, especially the reduction in optical crosstalk guarantees that the influence on image reconstruction is negligible. Not only, can the FACT telescope serve as an ideal instrument for long-term monitoring of bright blazars, the application of G-APDs in focal plane installations will significantly reduce construction and installation costs and improve stability of operation. This will yield a significantly impact on future projects like The Cherenkov Telescope Array (CTA) and makes small telescopes for monitoring purposes affordable. Acknowledgment The important contributions from ETH Zurich grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) are gratefully acknowledged. We thank the Instituto de Astrofisica de Canarias allowing us to operate the telescope at the Observatorio Roque de los Muchachos in La Palma, the Max-Planck-Institut fur Physik for providing us with the mount of the former HEGRA CT 3 telescope, and the MAGIC collaboration for their support. We also thank the group of Marinella Tose from the College of Engineering and Technology at Western Mindanao State University, Philippines, for providing us with the scheduling web-interface.","pages":[3,4]},{"title":"References","content":"[10] http://kicp-workshops.uchicago.edu/ieu2013 .","pages":[4]}],"string":"[\n {\n \"title\": \"FACT - The First G-APD Cherenkov Telescope: Status and Results\",\n \"content\": \"T. BRETZ 1 , H. ANDERHUB 1 , M. BACKES 2 , A. BILAND 1 ,A V. BOCCONE 3 , I. BRAUN 1 , J. BUSS 2 , F. CADOUX 3 , V. COMMICHAU 1 , L. DJAMBAZOV 1 , D. DORNER 4 , S. EINECKE 2 , D. EISENACHER 4 , A. GENDOTTI 1 , O. GRIMM 1 , H. VON GUNTEN 1 , C. HALLER 1 , D. HILDEBRAND 1 , U. HORISBERGER 1 , B. HUBER 1 a , K.-S. KIM 1 b , M. L. KNOETIG 1 , J.-H. K OHNE 2 , T. KR AHENB UHL 1 , B. KRUMM 2 , M. LEE 1 b , E. LORENZ 1 c , W. LUSTERMANN 1 , E. LYARD 3 , K. MANNHEIM 4 , M. MEHARGA 3 , K. MEIER 4 , T. MONTARULI 3 , D. NEISE 2 , F. NESSI-TEDALDI 1 , A.-K. OVERKEMPING 2 , A. PARAVAC 4 , F. PAUSS 1 , D. RENKER 1 d , W. RHODE 2 , M. RIBORDY 5 , U. R OSER 1 , J.-P. STUCKI 1 , J. SCHNEIDER 1 , T. STEINBRING 4 , F. TEMME 2 , J. THAELE 2 , S. TOBLER 1 , G. VIERTEL 1 , P. VOGLER 1 , R. WALTER 3 , K. WARDA 2 , Q. WEITZEL 1 , M. Z ANGLEIN 4 (FACT COLLABORATION) 1 ETH Zurich, Switzerland -Institute for Particle Physics, Schafmattstr. 20, 8093 Zurich d Also at: Technische Universitat Munchen, D-85748 Garching, Germany thomas.bretz@phys.ethz.ch Abstract: The First G-APD Cherenkov telescope (FACT) is the first telescope using silicon photon detectors (G-APD aka. SiPM). It is built on the mount of the HEGRA CT3 telescope, still located at the Observatorio del Roque de los Muchachos, and it is successfully in operation since Oct. 2011. The use of Silicon devices promises a higher photon detection efficiency, more robustness and higher precision than photo-multiplier tubes. The FACT collaboration is investigating with which precision these devices can be operated on the long-term. Currently, the telescope is successfully operated from remote and robotic operation is under development. During the past months of operation, the foreseen monitoring program of the brightest known TeV blazars has been carried out, and first physics results have been obtained including a strong flare of Mrk501. An instantaneous flare alert system is already in a testing phase. This presentation will give an overview of the project and summarize its goals, status and first results. Keywords: FACT, G-APD, silicon photo sensor, focal plane\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"Since Oct. 2011, the FACT Collaboration is operating the First G-APD Cherenkov telescope [1] at the Observatorio del Roque de los Muchachos on the Canary Island of La Palma. The telescope's camera is the first focal plane installation using Geiger-mode avalanche photo-diodes (G-APD) for photo detection. Comprising 1440 channels individually read out, each pixel has a field-of-view of 0.11° yielding a total field-of-view of 4.5°. The camera is installed on the mount of the former HEGRA CT 3 telescope. After replacement of the old disc shaped mirrors with refurbished hexagonal mirrors, it has now a total reflective surface of 9.5 m 2 . A picture of the telescope is shown in Fig. 1. With this novel camera, silicon photo sensors have started to replace photo multiplier tubes (PMT) still widely used for photo detection. They offer a high gain (10 5 to 10 6 ) and are robust enough to be operated under moon light conditions. Their single photon counting capability, their compactness and their low operational voltage ( < 100 V) simplifies the camera design. The photon detection efficiency (PDE) of commercially available G-APDs is already at the level of the best PMTs, and will still significantly improve in the future. The challenge of the project is to understand and operate the silicon based photo sensors under changing environ- mental conditions, such as changing auxiliary temperature or changing photon flux from the diffuse night-sky background. A system to keep the G-APD response stable under these conditions has been developed. The telescope is dedicated to long-term monitoring of the brightest known TeV blazars. These highly violent objects show variability on time scales of seconds to years. To study their behavior on the longer time scales, a complete data sample obtained from continuous measurements on long time scales is necessary.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2 System overview\",\n \"content\": \"The camera of the telescope is compiled from 1440 channels. Each channel is equipped with a G-APD (Hamamatsu MPPC S10362-33-50C) and a solid light concentrator. The solid light concentrators have the advantage, as compared to hollow ones, to feature a better compression ratio between entrance window and exit and thus balance the relatively small size of the sensors (9 mm 2 ). Sensors, cones and a protective window were glued together using optical glue. All channels are individually read out by the Domino Ring Sampling chip (DRS 4). The data is transferred by a TCP/IP connection via Ethernet to a data acquisition PC. The summed signal of nine channels form a trigger signal which is discriminated by a comparator. Each trigger patch is divided in two bias voltage channels with four and five GAPDs, resp. Although all read out electronics was integrated into the camera, the bias voltage supply is located in the counting hut. Each bias voltage channel has its own current readout. The precision of the voltage application is 12 bit up to 90 V and 12 bit up to 5 mA for the current readout. The total power consumption of the camera electronics during operation is about 570 W. About 100 W are dissipated in the supply lines and another 100 W due to the limited efficiency of the DC-DC converters, yielding a total of 370 W consumed by the electronics, ≈ 260 mW per channel. To get rid of the waste heat produced by the camera electronics, a passive water cooling is applied. To avoid the sensors being heat up by the electronics, a thermal insulation is installed in between. In total, 31 temperature sensors measure the temperature of the sensor plane. All details about the camera construction, the electronics and control software can be found in [1].\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"3.1 Operation\",\n \"content\": \"For a high duty cycle and consistent and stable data taking, a robust slow control and data acquisition software is needed. To avoid the need of shift crew on site, full remote operation must be possible. The camera is now operated since 20 months and its auxiliary hardware has been upgraded gradually, to allow for full remote control. This includes an interlock system, which purpose is to protect the camera from a cooling failure, Ethernet switchable 220 V plugs and the Ethernet access to all power supplies and a remote controllable lids. More details on that can be found in [2]. In parallel, a fully automatic control system has been developed. While all important parameters of the system are available via Web interface (see Fig. 2), the control of the system is achieved from a JavaScript engine reading the schedule from a database. Since a couple of weeks, this system is operating, and user interaction is usually only required in case of unexpected weather conditions. Currently, a fully automatic analysis is also in the testing phase. It is applied on the data immediately after the data has been recorded and the file has been closed which typically happens every 5 min for physics data. The analysis takes about the same time than data taking and will allow prompt flare alerts for other telescopes in the near future.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3.2 Stability\",\n \"content\": \"One of the main goals of the project was the prove of the applicability of G-APDs during standard operation conditions. Since the gain and all other properties of GAPDs depend on their temperature and the voltage applied, a correcting element is necessary. By design, the temperature gradient in the focal plane is small enough allowing several G-APDs to be connected to a single bias voltage channel. In total, 31 temperature sensors installed in the focal plane allow to counteract temperature changes by adapting the applied bias voltage accordingly. The power supply of each sensor is equipped with a filter network of resistors. Due to the changing light conditions induced by changes of the atmospheric conditions or moon light, the current drawn by the G-APDs is varying inducing a voltage drop in the resistances. To correct for this voltage drop, the current of each channel is measured and the voltage corrected accordingly. To measure the stability of the gain with these two feedback loops applied, three different methods are available: External light pulser An external temperature stabilized light pulser triggers the camera. Since its light yield is stable on average, the gain can directly be deduced from the average pulse height. However, the precision of this system measuring the gain of the G-APD is limited by the fact that also the transmission of the entrance window and the cones could change with time. Dark count spectrum A direct and mostly unbiased method to extract the gain, is the extraction of the dark count spectrum. It only depends on the performance of the G-APDs and the readout chain. Its drawback is that it can only be determined under light conditions, i.e. count rates, which still allow for extraction of single pulses. This is the case with closed lids or at dark time conditions. Ratescans To measure the stability of the gain in moon lit nights, so called ratescans are used. Ratescans measure the trigger response as a function of the applied trigger threshold. While at low thresholds, the very high rates are dominated by random coincidences of background photons, at high thresholds, the rate represents the number of triggers from coincident light flashed, induced mainly by hadronic showers. Although this method depends on the performance of the whole system, it allows to measure the response of the system even at the brightest light conditions. To ensure an unbiased measurement, only ratescans taken during good atmospheric conditions and weather conditions must be taken into account. A more detailed discussion of this can be found in [9]. All three methods have been applied almost daily since the beginning of operation. This allows to compare the response at any possible light condition and all temperatures occurring over the seasons. All methods have shown consistently that with the applied feedback system, a stable gain up to full moon conditions can be achieved. From the dark count spectrum, a stability of better than ± 3% over time and 4% pixel-to-pixel variations was deduced. Currently, the limiting factor of the system is the calibration of the applied bias voltage which is under investigation and will be improved soon. More details about the stability can be found in [3] and will be available soon in [4].\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"3.3 Data acquisition\",\n \"content\": \"The Ethernet connection between the forty readout boards and the data acquisition PC is routed over four Ethernet switches, two located in the camera and two located in the control room. In total, four Ethernet lines are available, one Ethernet line per ten boards. A total maximum transfer rate of about 1.9 Gbps is achieved. This is limited by the maximum throughput of 50 Mbps of each of the Ethernetchips (Wiznet 5300) on-board of the forty data acquisition boards. The transfer rate corresponds to a trigger rate of ∼ 250 Hz for a readout window of 300 samples and ∼ 80 Hz for the readout of the full DRS pipeline (1024 samples).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.4 Trigger threshold setting\",\n \"content\": \"One of the most important aspects during data taking is the correct setting of the trigger thresholds. To avoid high rates from random coincidences of background photons, the need to be set above the noise level. Since the noise level, mainly defined by the diffuse light from the night sky background, is not necessarily constant, some contingency is added to avoid strong rate fluctuations during data taking. At the same time, they must be set to as low as possible to achieve the lowest energy threshold and thus the highest integral sensitivity. Since the number of breakdowns in the G-APD, i.e. the number of detected photons, is proportional to the measured current, the rate induced from random coincidences as a function of the trigger threshold can directly be determined from the current. From ratescans taken at different light conditions over more than a year, a parametrization for the rate induced from background light and for the rate induced by hadronic showers under best conditions, was derived. Since several weeks, the trigger threshold of the system is now directly deduced from the measured current resulting in very stable data taking conditions. To allow for easy analysis, it is kept constant during every 5 min run.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3.5 Current and energy threshold prediction\",\n \"content\": \"To allow for an efficient schedule, the prediction of the system response under changing light conditions is mandatory. Therefore, the measured current has been correlated with moon conditions, i.e. moon brightness and zenith angle, which leads to a very precise prediction of the measured current. Since the trigger threshold is in the first order proportional to the light yield of the triggered shower, the change in energy threshold can directly be deduced. Including the change of the light yield with zenith angle, gives a very good estimate of the raise in energy threshold, i.e. loss in sensitivity, for each observation. These topics are discussed in more details in [5, 6] and will be available soon in [4].\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4 Physics results\",\n \"content\": \"Very soon after the first observations, first results could be presented [7]. From the measurements of the Crab Nebula, a sensitivity of 8% Crab in 50 h could be derived. This is based on the analysis of data taking during the first months of operation and based on an analysis primarily developed for MAGIC not yet necessarily optimized for the current system. From the rate of excess events an energy threshold between 400 GeV and 700 GeV can be derived taking the known spectrum of the source into account The analysis, especially the automatic analysis, and first results are discussed in [8].\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"5 Conclusions\",\n \"content\": \"The camera of the First G-APD Cherenkov Telescope (FACT) has proven the applicability of silicon photo sensors (G-APD) in focal plane installations, in particular as detectors in Cherenkov telescopes. The camera is now in operation since more than one and a half years, and although there were hardware problems during that time, none of the was related to the G-APDs at all. So far, no decrease in performance could be detected. Neither any hint for aging of the G-APDs nor of the transmission of cones and window has been found. With different types of measurements, it was shown that a stability to on the few percent level, independent of temperature and light conditions can be achieved. Further improvements are expected by an improvement of the bias voltage calibration. An important result derived from the current study is that the application of a feedback system which keeps the G-APD's bias voltage constant, renders the need for an external calibration device obsolete. The combination of the extraction of the dark count spectrum and the current measurement, is enough for the operation of such a device. Recently, Hamamatsu presented a new generation of G-APD sensors with significantly lower dark count rates, significantly reduced optical crosstalk and significantly reduced afterpulse probability, expected to be available on the market within the next weeks [10]. Although, it was shown that these improvements are not obligatory for the operation in a Cherenkov telescope, especially the reduction in optical crosstalk guarantees that the influence on image reconstruction is negligible. Not only, can the FACT telescope serve as an ideal instrument for long-term monitoring of bright blazars, the application of G-APDs in focal plane installations will significantly reduce construction and installation costs and improve stability of operation. This will yield a significantly impact on future projects like The Cherenkov Telescope Array (CTA) and makes small telescopes for monitoring purposes affordable. Acknowledgment The important contributions from ETH Zurich grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) are gratefully acknowledged. We thank the Instituto de Astrofisica de Canarias allowing us to operate the telescope at the Observatorio Roque de los Muchachos in La Palma, the Max-Planck-Institut fur Physik for providing us with the mount of the former HEGRA CT 3 telescope, and the MAGIC collaboration for their support. We also thank the group of Marinella Tose from the College of Engineering and Technology at Western Mindanao State University, Philippines, for providing us with the scheduling web-interface.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"References\",\n \"content\": \"[10] http://kicp-workshops.uchicago.edu/ieu2013 .\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":285,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33.1856M"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1307.3880.pdf"},"content":{"kind":"string","value":"\n\n\n\nAll Sky Cameras for the characterization of the Cherenkov Telescope Array candidate sites\nDUSAN MANDAT 1 , MIROSLAV PECH 1 , JAN EBR 1 , MIROSLAV HRABOVSKY 1 , MICHAEL PROUZA 1 , TOMASZ BULIK 2 , INGOMAR ALLEKOTTE 3 , FOR THE CTA CONSORTIUM.\n\n1 Institute of Physics of Academy of Science of The Czech Republic, Czech Republic.\n2 Astronomical Observatory University of Warsaw, Poland.\n3 Centro At'omico Bariloche, Comisi'on Nacional de Energ'ıa At'omica, San Carlos de Bariloche, Argentina.\n\nmandat@fzu.cz\nAbstract: The All Sky Camera (ASC) was developed as a universal device for the monitoring of the night sky quality. Eight ASCs are already installed and measure night sky parameters at eight of the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The ACS system consists of an astronomical CCD camera, a fish eye lens, a control computer and associated electronics. The measurement is carried out during astronomical night. The images are automatically taken every 5 minutes and automatically processed using the control computer of the device. The analysis results are the cloud fraction (the percentage of the sky covered by clouds) and night sky brightness (in mag/arcsec 2 ).\nKeywords: All Sky Camera, CTA, candidate sites, cloud fraction.\n1 Cloudiness characterization of CTA candidate sites\nThe future CTA observatory [1] will include two observatories. One on the Southern and one on the Northern Hemisphere. There are nine candidate sites for the future CTA observatories - five at the Southern and four at the Northern Hemisphere. The goal of Site Selection Work Package [2] (SITE WP) is to characterize the candidate sites and produce relevant data for the Site Selection Committee. One of the important characteristic of the candidate site is the cloud fraction of the night sky and the amount of cloudless nights or more specifically cloudless time in a year. The ASC is an automated instrument for measurement of this parameter. Starting in November 2011 the following sites are being characterized using ASCs :\n
\n\n
Table 1 : Candidate sites for CTA. Aar - Aar farm, Namibia, SAC - San Antonio de Los Cobres, Argentina, SPM SAN Pedro Martir, Mexico, AZE - Meteor Crater, USA, AZW- Yavapai ranch, USA, TEN - Tenerife Spain, CAS CASLEO observatory, Argentina, ARM - Cerro Armazones, Chile .
\n
\n\n\n\n\n\n
Fig. 1 : Electronics and the inside of the camera box.
\n\n1.1 System Description\nThe basis of the system is an astronomical camera G1 -2000 (Moravian Instruments a.s., Czech Republic, www.mii.cz), which is equipped with a CCD chip ICX274AL by SONY company. This chip has quantum efficiency higher than 50% at 450-550 nm and its spectral sensitivy covers the range from approximately 400 to 900 nm. It has low read-out noise, a 16-bit ADC and a resolution of 1600 × 1200 pixels. The camera is equipped with a fish-eye varifocal lens (the field of view is 185 degrees) and an electronic iris control. This setup is capable of detecting a star with visual magnitude 6.3 in zenith.\nA miniPC computer with USB I / O controls the system and processes the data. The I / O card controls the iris and the power switch of the camera. All electronics and a camera body (see Figure 1) are weatherproof. The system power is supplied by a solar power system. The switching electronics turns the system ON after sunset and keeps it going during astronomical night. The system is OFF during the day (to save energy). Temperature of the electronics is controlled and stabilized using internal heating system during winter time to protect the ASC system. The camera is a part of an Atmoscope instrument. The Atmoscope measures local weather conditions and night sky brightness parameters. The Atmoscope is connected to central data storage using GPRS or WIFI connection depending on\n\n\n
Fig. 2 : Image of clear night sky from San Pedro Martir.
\n\nAtmoscope local position and nearby facilities. The quality of the link limits the amount and speed of the data transfer.\n1.2 Calibration of the camera\nThe fish-eye varifocal lens has 185 degrees field of view (FOV) and its aberration distorts the image. The correct transformation of the incident angle to the imaged pixel position is very important. The mechanism of the calibration is as follows: the light source (a spot as small as possible) is placed at a defined distance (at least 10 m), then the body of camera is rotated around the axis such that the incident light angle (corresponding to the FOV angle) varies between 0 and 90 degrees. The position of the light spot is analyzed for each anglular step. This procedure is repeated for different plane cuts of the lens FOV. The final data are analyzed and fitted with a polynomial function. This calibration is regularly checked on-site by comparing the position of detected stars with the stars in a catalogue. Onsite measurements are also used to determine the variation of the sensitivty of the system with the zenith angle.\n1.3 Data acquisition and principles of analysis\nThe ASC system takes full sky images each 5 minutes (resp. 10 minutes - Aar and SAC). Examples of clear and partly cloudy night sky are shown in Figure 2 and 6. The onsite analysis (raw cloudiness analysis) follows immediately, the result is saved to two files. The first file consists of a compressed image and online analysis results, the second one consists of a RAW image and relevant data (UTC time, temperature of CCD, Moon and Sun position).\nThe measurement process consists of the following steps. Initially, every exposure of a night sky is followed by a dark frame image so that subtraction of one from the other enables significant reduction of noise in the image. The horizontal coordinates of each image pixel are then calculated with the calibration data and calibration constants used to determine the horizontal coordinates, azimuth and elevation of each image pixel. The total charge of each\n\n\n\n\n\n
Fig. 3 : Fog or haze affects the visibility of night sky, the nearby tower is not visible (the visibility fades with the fog/haze).
\n\npixel is proportional to the light intensity, but the intensity of the final image varies with respect to zenith angle due to vignetting and distortion of the lens. The calibration of this zenith angle intensity dependence is obtained by considering extinction in air masses, optical distortions and vignetting variation as a function of zenith angle. The resulting attenuation at, for example, 60 degrees zenith angle corresponds to 2 mag.\nTo estimate the cloud fraction, we investigate the presence of stars in the field of view of the camera. We separate the full sky to approximately 70 segments and analyze each segment of the sky individually to obtain the cloudiness. As a star catalogue we use Yale Bright Star Catalog BSC5\n\n\n
Fig. 4 : Lens of the ASC covered with water drops.
\n\n\n\n\n\n\n
Fig. 5 : An example of analysis of partly cloudy night sky. The image on the left shows RAW image of the sky. The image on the right shows the analysis results, the yellow crosses indicate the detected stars, green circles catalogue stars and red circles represent catalogue stars without detected pairs - the region covered by cloudiness. The RED circle indicates the limiting zenith angle (60 · ).
\n\n[3], particularly the visual magnitues of the stars. First we estimate the position of each segment on the sky using the data from the calibration. This information is then further improved using the brightest stars from the catalogue in each segement. Once the position of a segment on the sky is fixed, the stars for the segment are read from the catalogue and selected according to the expected sensitivty at the given zenith angle. For every star in the catalogue, we\n\n\n
Fig. 6 : Image of partly cloudy night sky from San Pedro Martir. Moon and clouds are visible close to horizon.
\n\nlook for a detected star within the angular limit of 1 deg. If a star that fits this criteria is found, then the catalogue and detected stars are flagged as paired. Figure 5 shows a result from cloudiness analysis of partly cloudy night sky. The ratio of paired / unpaired stars (for all segments) gives final cloudiness of the night sky. Typically, we check about 900 catalogue stars for the whole observed sky (up to 60 · which is the maximal operating zenith angle of Cherenkov telescopes). The error of the algorithm was calculated using artificial cloud simulations and the total error of the cloudiness calculation is ± 5%. The cloudiness of the night sky is computed up to the limiting angle with 10 · steps.\nBrightness of the night sky is evaluated directly from a light flux. The system has been calibrated using a darkness sky measurement tool Unihedron SQM-LE [4]. The night sky was scanned overnight and recorded data both from the camera and SQM instrument were compared to each other. Subsequently the conversion formula is obtained. The data represent the brightness of the night sky in mag / arcsec 2 and in visible light spectrum.\n1.4 Limits of the cloudiness analysis\nThe measurement using the ASC is limited by local weather conditions, camera misting and Moon position. The local weather limitations mean for example rain or snow, fog, haze and water condensation on the camera. Examples of rain, fog and snow are shown in Figures 4, 3 and 7 respectively. The algorithm usually fails if the lens of the ASC is obscured in some way, for example by snow, water resulting from rain fall, haze or fog surrounding the Atmoscope, or even bird droppings from birds found locally at the Aar and Tenerife sites. The presence of the Moon saturates the image and increases the background in the image, and consequently the algorithm for cloud detection could not be used at such times.\n\n\n
Fig. 7 : Snow covers the whole FOV of the ASC and no star is visible in this case.
\n\n2 Applications, operation and advantages\nThe main advantage of the all-sky camera is that it is a fully autonomous and self-reliant tool which can be used to monitor the night sky in even the world's most remote places, which happen to be just the ones most promising for astronomically related applications. When using solar power, the only requirements for the deployment of the system are the ability to transport the device and the solar power system to the location. The availability of internet connection is useful in order to ensure fast response to possible issues, but is by no means a requirement. The autonomy of the camera significantly contributes to the extremely low cost for its operation. The costs involved are concentrated mainly in the deployment phase, namely the costs of the parts and assembly of the camera, shipping (varies depending on destination, the shippable weight being 10 kg), the solar power system, if there is no pre-existing power on-site (the 180 W required by the system) and installation (two people working for two or three days). After installation, the camera can operate without maintenance for years. However, it is advantageous to have local support for the case of unexpected external hindrances, but in most cases, a nonspecialist local personnel is sufficient to fix most of the possible issues.\n3 Conclusions\nASC camera for night sky measurement was presented as a versatile remote system for cloudiness and night sky brightness characterization. Eight ASC's are being used to measure night sky parameters at the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The data will be used to characterize the candidate site and to use the data in the selection process for Southern and Northern CTA site.\nAcknowledgment: This work is supported by Ministry of Education of the Czech Republic, under the projects EUPROII LE13012 and Mobility 7AMB12AR013. We gratefully acknowl-\n\n\n\nedge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22.\nReferences\n\n[1] CTA consortium, Astropart. Phys. 43 , (2013), 3, doi:10.1016/j.astropartphys.2013.01.007.\n[2] T. Bulik, I. Puerto-Gimenez, 32nd INTERNATIONAL COSMIC RAY CONFERENCE,BEIJING 2011, doi:10.7529/ICRC2011/V09/1257.\n[3] http://tdc-www.harvard.edu/catalogs/bsc5.html.\n[4] http://www.unihedron.com/projects/darksky/.\n"},"sections":{"kind":"list like","value":[{"title":"All Sky Cameras for the characterization of the Cherenkov Telescope Array candidate sites","content":"DUSAN MANDAT 1 , MIROSLAV PECH 1 , JAN EBR 1 , MIROSLAV HRABOVSKY 1 , MICHAEL PROUZA 1 , TOMASZ BULIK 2 , INGOMAR ALLEKOTTE 3 , FOR THE CTA CONSORTIUM. mandat@fzu.cz Abstract: The All Sky Camera (ASC) was developed as a universal device for the monitoring of the night sky quality. Eight ASCs are already installed and measure night sky parameters at eight of the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The ACS system consists of an astronomical CCD camera, a fish eye lens, a control computer and associated electronics. The measurement is carried out during astronomical night. The images are automatically taken every 5 minutes and automatically processed using the control computer of the device. The analysis results are the cloud fraction (the percentage of the sky covered by clouds) and night sky brightness (in mag/arcsec 2 ). Keywords: All Sky Camera, CTA, candidate sites, cloud fraction.","pages":[1]},{"title":"1 Cloudiness characterization of CTA candidate sites","content":"The future CTA observatory [1] will include two observatories. One on the Southern and one on the Northern Hemisphere. There are nine candidate sites for the future CTA observatories - five at the Southern and four at the Northern Hemisphere. The goal of Site Selection Work Package [2] (SITE WP) is to characterize the candidate sites and produce relevant data for the Site Selection Committee. One of the important characteristic of the candidate site is the cloud fraction of the night sky and the amount of cloudless nights or more specifically cloudless time in a year. The ASC is an automated instrument for measurement of this parameter. Starting in November 2011 the following sites are being characterized using ASCs :","pages":[1]},{"title":"1.1 System Description","content":"The basis of the system is an astronomical camera G1 -2000 (Moravian Instruments a.s., Czech Republic, www.mii.cz), which is equipped with a CCD chip ICX274AL by SONY company. This chip has quantum efficiency higher than 50% at 450-550 nm and its spectral sensitivy covers the range from approximately 400 to 900 nm. It has low read-out noise, a 16-bit ADC and a resolution of 1600 × 1200 pixels. The camera is equipped with a fish-eye varifocal lens (the field of view is 185 degrees) and an electronic iris control. This setup is capable of detecting a star with visual magnitude 6.3 in zenith. A miniPC computer with USB I / O controls the system and processes the data. The I / O card controls the iris and the power switch of the camera. All electronics and a camera body (see Figure 1) are weatherproof. The system power is supplied by a solar power system. The switching electronics turns the system ON after sunset and keeps it going during astronomical night. The system is OFF during the day (to save energy). Temperature of the electronics is controlled and stabilized using internal heating system during winter time to protect the ASC system. The camera is a part of an Atmoscope instrument. The Atmoscope measures local weather conditions and night sky brightness parameters. The Atmoscope is connected to central data storage using GPRS or WIFI connection depending on Atmoscope local position and nearby facilities. The quality of the link limits the amount and speed of the data transfer.","pages":[1,2]},{"title":"1.2 Calibration of the camera","content":"The fish-eye varifocal lens has 185 degrees field of view (FOV) and its aberration distorts the image. The correct transformation of the incident angle to the imaged pixel position is very important. The mechanism of the calibration is as follows: the light source (a spot as small as possible) is placed at a defined distance (at least 10 m), then the body of camera is rotated around the axis such that the incident light angle (corresponding to the FOV angle) varies between 0 and 90 degrees. The position of the light spot is analyzed for each anglular step. This procedure is repeated for different plane cuts of the lens FOV. The final data are analyzed and fitted with a polynomial function. This calibration is regularly checked on-site by comparing the position of detected stars with the stars in a catalogue. Onsite measurements are also used to determine the variation of the sensitivty of the system with the zenith angle.","pages":[2]},{"title":"1.3 Data acquisition and principles of analysis","content":"The ASC system takes full sky images each 5 minutes (resp. 10 minutes - Aar and SAC). Examples of clear and partly cloudy night sky are shown in Figure 2 and 6. The onsite analysis (raw cloudiness analysis) follows immediately, the result is saved to two files. The first file consists of a compressed image and online analysis results, the second one consists of a RAW image and relevant data (UTC time, temperature of CCD, Moon and Sun position). The measurement process consists of the following steps. Initially, every exposure of a night sky is followed by a dark frame image so that subtraction of one from the other enables significant reduction of noise in the image. The horizontal coordinates of each image pixel are then calculated with the calibration data and calibration constants used to determine the horizontal coordinates, azimuth and elevation of each image pixel. The total charge of each pixel is proportional to the light intensity, but the intensity of the final image varies with respect to zenith angle due to vignetting and distortion of the lens. The calibration of this zenith angle intensity dependence is obtained by considering extinction in air masses, optical distortions and vignetting variation as a function of zenith angle. The resulting attenuation at, for example, 60 degrees zenith angle corresponds to 2 mag. To estimate the cloud fraction, we investigate the presence of stars in the field of view of the camera. We separate the full sky to approximately 70 segments and analyze each segment of the sky individually to obtain the cloudiness. As a star catalogue we use Yale Bright Star Catalog BSC5 [3], particularly the visual magnitues of the stars. First we estimate the position of each segment on the sky using the data from the calibration. This information is then further improved using the brightest stars from the catalogue in each segement. Once the position of a segment on the sky is fixed, the stars for the segment are read from the catalogue and selected according to the expected sensitivty at the given zenith angle. For every star in the catalogue, we look for a detected star within the angular limit of 1 deg. If a star that fits this criteria is found, then the catalogue and detected stars are flagged as paired. Figure 5 shows a result from cloudiness analysis of partly cloudy night sky. The ratio of paired / unpaired stars (for all segments) gives final cloudiness of the night sky. Typically, we check about 900 catalogue stars for the whole observed sky (up to 60 · which is the maximal operating zenith angle of Cherenkov telescopes). The error of the algorithm was calculated using artificial cloud simulations and the total error of the cloudiness calculation is ± 5%. The cloudiness of the night sky is computed up to the limiting angle with 10 · steps. Brightness of the night sky is evaluated directly from a light flux. The system has been calibrated using a darkness sky measurement tool Unihedron SQM-LE [4]. The night sky was scanned overnight and recorded data both from the camera and SQM instrument were compared to each other. Subsequently the conversion formula is obtained. The data represent the brightness of the night sky in mag / arcsec 2 and in visible light spectrum.","pages":[2,3]},{"title":"1.4 Limits of the cloudiness analysis","content":"The measurement using the ASC is limited by local weather conditions, camera misting and Moon position. The local weather limitations mean for example rain or snow, fog, haze and water condensation on the camera. Examples of rain, fog and snow are shown in Figures 4, 3 and 7 respectively. The algorithm usually fails if the lens of the ASC is obscured in some way, for example by snow, water resulting from rain fall, haze or fog surrounding the Atmoscope, or even bird droppings from birds found locally at the Aar and Tenerife sites. The presence of the Moon saturates the image and increases the background in the image, and consequently the algorithm for cloud detection could not be used at such times.","pages":[3]},{"title":"2 Applications, operation and advantages","content":"The main advantage of the all-sky camera is that it is a fully autonomous and self-reliant tool which can be used to monitor the night sky in even the world's most remote places, which happen to be just the ones most promising for astronomically related applications. When using solar power, the only requirements for the deployment of the system are the ability to transport the device and the solar power system to the location. The availability of internet connection is useful in order to ensure fast response to possible issues, but is by no means a requirement. The autonomy of the camera significantly contributes to the extremely low cost for its operation. The costs involved are concentrated mainly in the deployment phase, namely the costs of the parts and assembly of the camera, shipping (varies depending on destination, the shippable weight being 10 kg), the solar power system, if there is no pre-existing power on-site (the 180 W required by the system) and installation (two people working for two or three days). After installation, the camera can operate without maintenance for years. However, it is advantageous to have local support for the case of unexpected external hindrances, but in most cases, a nonspecialist local personnel is sufficient to fix most of the possible issues.","pages":[4]},{"title":"3 Conclusions","content":"ASC camera for night sky measurement was presented as a versatile remote system for cloudiness and night sky brightness characterization. Eight ASC's are being used to measure night sky parameters at the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The data will be used to characterize the candidate site and to use the data in the selection process for Southern and Northern CTA site. Acknowledgment: This work is supported by Ministry of Education of the Czech Republic, under the projects EUPROII LE13012 and Mobility 7AMB12AR013. We gratefully acknowl- edge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22.","pages":[4]}],"string":"[\n {\n \"title\": \"All Sky Cameras for the characterization of the Cherenkov Telescope Array candidate sites\",\n \"content\": \"DUSAN MANDAT 1 , MIROSLAV PECH 1 , JAN EBR 1 , MIROSLAV HRABOVSKY 1 , MICHAEL PROUZA 1 , TOMASZ BULIK 2 , INGOMAR ALLEKOTTE 3 , FOR THE CTA CONSORTIUM. mandat@fzu.cz Abstract: The All Sky Camera (ASC) was developed as a universal device for the monitoring of the night sky quality. Eight ASCs are already installed and measure night sky parameters at eight of the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The ACS system consists of an astronomical CCD camera, a fish eye lens, a control computer and associated electronics. The measurement is carried out during astronomical night. The images are automatically taken every 5 minutes and automatically processed using the control computer of the device. The analysis results are the cloud fraction (the percentage of the sky covered by clouds) and night sky brightness (in mag/arcsec 2 ). Keywords: All Sky Camera, CTA, candidate sites, cloud fraction.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Cloudiness characterization of CTA candidate sites\",\n \"content\": \"The future CTA observatory [1] will include two observatories. One on the Southern and one on the Northern Hemisphere. There are nine candidate sites for the future CTA observatories - five at the Southern and four at the Northern Hemisphere. The goal of Site Selection Work Package [2] (SITE WP) is to characterize the candidate sites and produce relevant data for the Site Selection Committee. One of the important characteristic of the candidate site is the cloud fraction of the night sky and the amount of cloudless nights or more specifically cloudless time in a year. The ASC is an automated instrument for measurement of this parameter. Starting in November 2011 the following sites are being characterized using ASCs :\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1.1 System Description\",\n \"content\": \"The basis of the system is an astronomical camera G1 -2000 (Moravian Instruments a.s., Czech Republic, www.mii.cz), which is equipped with a CCD chip ICX274AL by SONY company. This chip has quantum efficiency higher than 50% at 450-550 nm and its spectral sensitivy covers the range from approximately 400 to 900 nm. It has low read-out noise, a 16-bit ADC and a resolution of 1600 × 1200 pixels. The camera is equipped with a fish-eye varifocal lens (the field of view is 185 degrees) and an electronic iris control. This setup is capable of detecting a star with visual magnitude 6.3 in zenith. A miniPC computer with USB I / O controls the system and processes the data. The I / O card controls the iris and the power switch of the camera. All electronics and a camera body (see Figure 1) are weatherproof. The system power is supplied by a solar power system. The switching electronics turns the system ON after sunset and keeps it going during astronomical night. The system is OFF during the day (to save energy). Temperature of the electronics is controlled and stabilized using internal heating system during winter time to protect the ASC system. The camera is a part of an Atmoscope instrument. The Atmoscope measures local weather conditions and night sky brightness parameters. The Atmoscope is connected to central data storage using GPRS or WIFI connection depending on Atmoscope local position and nearby facilities. The quality of the link limits the amount and speed of the data transfer.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"1.2 Calibration of the camera\",\n \"content\": \"The fish-eye varifocal lens has 185 degrees field of view (FOV) and its aberration distorts the image. The correct transformation of the incident angle to the imaged pixel position is very important. The mechanism of the calibration is as follows: the light source (a spot as small as possible) is placed at a defined distance (at least 10 m), then the body of camera is rotated around the axis such that the incident light angle (corresponding to the FOV angle) varies between 0 and 90 degrees. The position of the light spot is analyzed for each anglular step. This procedure is repeated for different plane cuts of the lens FOV. The final data are analyzed and fitted with a polynomial function. This calibration is regularly checked on-site by comparing the position of detected stars with the stars in a catalogue. Onsite measurements are also used to determine the variation of the sensitivty of the system with the zenith angle.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"1.3 Data acquisition and principles of analysis\",\n \"content\": \"The ASC system takes full sky images each 5 minutes (resp. 10 minutes - Aar and SAC). Examples of clear and partly cloudy night sky are shown in Figure 2 and 6. The onsite analysis (raw cloudiness analysis) follows immediately, the result is saved to two files. The first file consists of a compressed image and online analysis results, the second one consists of a RAW image and relevant data (UTC time, temperature of CCD, Moon and Sun position). The measurement process consists of the following steps. Initially, every exposure of a night sky is followed by a dark frame image so that subtraction of one from the other enables significant reduction of noise in the image. The horizontal coordinates of each image pixel are then calculated with the calibration data and calibration constants used to determine the horizontal coordinates, azimuth and elevation of each image pixel. The total charge of each pixel is proportional to the light intensity, but the intensity of the final image varies with respect to zenith angle due to vignetting and distortion of the lens. The calibration of this zenith angle intensity dependence is obtained by considering extinction in air masses, optical distortions and vignetting variation as a function of zenith angle. The resulting attenuation at, for example, 60 degrees zenith angle corresponds to 2 mag. To estimate the cloud fraction, we investigate the presence of stars in the field of view of the camera. We separate the full sky to approximately 70 segments and analyze each segment of the sky individually to obtain the cloudiness. As a star catalogue we use Yale Bright Star Catalog BSC5 [3], particularly the visual magnitues of the stars. First we estimate the position of each segment on the sky using the data from the calibration. This information is then further improved using the brightest stars from the catalogue in each segement. Once the position of a segment on the sky is fixed, the stars for the segment are read from the catalogue and selected according to the expected sensitivty at the given zenith angle. For every star in the catalogue, we look for a detected star within the angular limit of 1 deg. If a star that fits this criteria is found, then the catalogue and detected stars are flagged as paired. Figure 5 shows a result from cloudiness analysis of partly cloudy night sky. The ratio of paired / unpaired stars (for all segments) gives final cloudiness of the night sky. Typically, we check about 900 catalogue stars for the whole observed sky (up to 60 · which is the maximal operating zenith angle of Cherenkov telescopes). The error of the algorithm was calculated using artificial cloud simulations and the total error of the cloudiness calculation is ± 5%. The cloudiness of the night sky is computed up to the limiting angle with 10 · steps. Brightness of the night sky is evaluated directly from a light flux. The system has been calibrated using a darkness sky measurement tool Unihedron SQM-LE [4]. The night sky was scanned overnight and recorded data both from the camera and SQM instrument were compared to each other. Subsequently the conversion formula is obtained. The data represent the brightness of the night sky in mag / arcsec 2 and in visible light spectrum.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"1.4 Limits of the cloudiness analysis\",\n \"content\": \"The measurement using the ASC is limited by local weather conditions, camera misting and Moon position. The local weather limitations mean for example rain or snow, fog, haze and water condensation on the camera. Examples of rain, fog and snow are shown in Figures 4, 3 and 7 respectively. The algorithm usually fails if the lens of the ASC is obscured in some way, for example by snow, water resulting from rain fall, haze or fog surrounding the Atmoscope, or even bird droppings from birds found locally at the Aar and Tenerife sites. The presence of the Moon saturates the image and increases the background in the image, and consequently the algorithm for cloud detection could not be used at such times.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2 Applications, operation and advantages\",\n \"content\": \"The main advantage of the all-sky camera is that it is a fully autonomous and self-reliant tool which can be used to monitor the night sky in even the world's most remote places, which happen to be just the ones most promising for astronomically related applications. When using solar power, the only requirements for the deployment of the system are the ability to transport the device and the solar power system to the location. The availability of internet connection is useful in order to ensure fast response to possible issues, but is by no means a requirement. The autonomy of the camera significantly contributes to the extremely low cost for its operation. The costs involved are concentrated mainly in the deployment phase, namely the costs of the parts and assembly of the camera, shipping (varies depending on destination, the shippable weight being 10 kg), the solar power system, if there is no pre-existing power on-site (the 180 W required by the system) and installation (two people working for two or three days). After installation, the camera can operate without maintenance for years. However, it is advantageous to have local support for the case of unexpected external hindrances, but in most cases, a nonspecialist local personnel is sufficient to fix most of the possible issues.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3 Conclusions\",\n \"content\": \"ASC camera for night sky measurement was presented as a versatile remote system for cloudiness and night sky brightness characterization. Eight ASC's are being used to measure night sky parameters at the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The data will be used to characterize the candidate site and to use the data in the selection process for Southern and Northern CTA site. Acknowledgment: This work is supported by Ministry of Education of the Czech Republic, under the projects EUPROII LE13012 and Mobility 7AMB12AR013. We gratefully acknowl- edge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22.\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":286,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33.2823K"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1307.3386.pdf"},"content":{"kind":"string","value":"\n\n\n\nDevelopment of the Photomultiplier-Tube Readout System for the CTA Large Size Telescope\n\nH. KUBO 1 , R. PAOLETTI 2 , Y. AWANE 1 , A. BAMBA 3 , M. BARCELO 4 , J.A. BARRIO 5 , O. BLANCH 4 , J. BOIX 4 ,\n\nC. DELGADO 6 , D. FINK 7 , D. GASCON 8 , S. GUNJI 9 , R. HAGIWARA 9 , Y. HANABATA 10 , K. HATANAKA 1 , M. HAYASHIDA 10 , M. IKENO 11 , S. KABUKI 12 , H. KATAGIRI 13 , J. KATAOKA 14 , Y. KONNO 1 , S. KOYAMA 15 , T. KISHIMOTO 1 , J. KUSHIDA 16 , G. MART'INEZ 6 , S. MASUDA 1 , J.M. MIRANDA 17 , R. MIRZOYAN 7 , T. MIZUNO 18 , T. NAGAYOSHI 15 , D. NAKAJIMA 7 , T. NAKAMORI 9 , H. OHOKA 10 , A. OKUMURA 19 , R. ORITO 20 , T. SAITO 1 , A. SANUY 8 , H. SASAKI 21 , M. SAWADA 3 , T. SCHWEIZER 7 , R. SUGAWARA 20 , K.-H. SULANKE 22 , H. TAJIMA 19 , M. TANAKA 11 , S. TANAKA 13 , L.A. TEJEDOR 5 , Y. TERADA 15 , M. TESHIMA 7 , 10 , F. TOKANAI 9 , Y. TSUCHIYA 1 , 11 15 13 21\n\nT. UCHIDA , H. UENO , K. UMEHARA , T. YAMAMOTO FOR THE CTA CONSORTIUM.\n\n1\nDepartment of Physics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan\n\n2 Universit'a di Siena, and INFN Pisa, I-53100 Siena, Italy\n3 Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa, 252-5258, Japan\n4 Institut de Fsica d'Altes Energies, Edifici Cn, Campus UAB, 08193 Bellaterra, Spain\n5 Grupo de Altas Energas, Universidad Complutense de Madrid, Av Complutense s/n, 28040 Madrid, Spain\n6 CIEMAT, Avda. Complutense 22, 28040 Madrid, Spain\n7 Max-Planck-Institut fur Physik, Fohringer Ring 6, D 80805 Munchen, Germany\n8 Universitat de Barcelona (ICC-UB), Mart i Franqus, 1, 08028, Barcelona, Spain\n9 Department of Physics, Faculty of Science, Yamagata University, Yamagata, Yamagata 990-8560, Japan\n10 Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan\n11 Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki 305-0801, Japan\n12 Department of Radiation Oncology, Tokai University, 143 Shimokasuya, Isehara-shi, Kanagawa 259-1193, Japan\n13 College of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan\n14 Research Institute for Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan\n15 Graduate School of Science and Engineering, Saitama University, 255 Simo-Ohkubo, Sakura-ku, Saitama 338-8570, Japan\n16 Department of Physics, Tokai University, 4-1-1 Kita-Kaname, Hiratsuka, Kanagawa 259-1292, Japan\n17 Grupo de Electronica, Universidad Complutense de Madrid, Av. Complutense s/n, 28040 Madrid, Spain\n18 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-hiroshima, 739-8526 Japan\n19 Solar-Terrestrial Environment Laboratory, Nagoya University, Chikusa, Nagoya 464-8601, Japan\n20 Faculty of Integrated Arts and Sciences, The University of Tokushima, Tokushima 770-8502, Japan\n21 Department of Physics, Konan University, 8-9-1 Okamoto Higashinada-ku Kobe, Hyogo, 658-8501 Japan\n22 Deutsches Elektronen-Synchrotron, Platanenallee 6, 15738 Zeuthen, Germany\n\nkubo@cr.scphys.kyoto-u.ac.jp\nAbstract: We have developed a prototype of the photomultiplier tube (PMT) readout system for the Cherenkov Telescope Array (CTA) Large Size Telescope (LST). Two thousand PMTs along with their readout systems are arranged on the focal plane of each telescope, with one readout system per 7-PMT cluster. The Cherenkov light pulses generated by the air showers are detected by the PMTs and amplified in a compact, low noise and wide dynamic range gain block. The output of this block is then digitized at a sampling rate of the order of GHz using the Domino Ring Sampler DRS4, an analog memory ASIC developed at Paul Scherrer Institute. The sampler has 1,024 capacitors per channel and four channels are cascaded for increased depth. After a trigger is generated in the system, the charges stored in the capacitors are digitized by an external slow sampling ADC and then transmitted via Gigabit Ethernet. An onboard FPGA controls the DRS4, trigger threshold, and Ethernet transfer. In addition, the control and monitoring of the Cockcroft-Walton circuit that provides high voltage for the 7-PMT cluster are performed by the same FPGA. A prototype named Dragon has been developed that has successfully sampled PMT signals at a rate of 2 GHz, and generated single photoelectron spectra.\nKeywords: Imaging Atmospheric Cherenkov Telescope, Gamma-rays, Electronics\n1 Introduction\nA ground-based imaging atmospheric Cherenkov telescope (IACT) measures Cherenkov light from an extended air shower (EAS) generated by the interaction between very high energy (VHE) gamma rays and the upper atmosphere. Night sky background (NSB) also enters a pixel photon sensor of the focal plane of the IACT with a rate of the order of 10-100 MHz, depending on the mirror size\nand the pixel size. In some region of the Galactic plane, the NSB can reach on average up to several hundred MHz. The NSB therefore becomes noise that affects the sensitivity and the energy threshold. Given this NSB pollution, and the fact that the duration of Cherenkov light from EAS is a few nanoseconds, a fast digitization speed of the readout system coupled to a fast photosensor like a photomultiplier tube (PMT) is beneficial to increase the pixels' signal-\nto-noise ratio. In addition, this system should be compact and have low cost and low power consumption because each IACT possesses several thousand photon sensor pixels and the readout system attached to the sensors is in a camera container located at the focal position. Furthermore, a wide dynamic range of more than 8-10 bits is required to resolve a single photoelectron and have a wider energy range.\nA commercial flash analog-to-digital converter (FADC) satisfies the requirement of a wide dynamic range. However, it is costly and consumes relatively high power of a few watts per channel. On the other hand, an analog memory application specific integrated circuit (ASIC), that consists of several hundred to several thousand switched capacitor arrays (SCA) per channel, can sample a signal at the order of GHz, with a wide dynamic range and lower power consumption. Several types of analog memory ASICs have been developed for applications in particle physics and cosmic ray physics. With respect to IACTs, a modified version of ARS0 [1], Swift Analog Memory (SAM) [2], and Domino Ring Sampler (DRS) [3] chips are used in the H.E.S.S.-I, H.E.S.S.-II, and MAGIC experiments, respectively.\nCherenkov Telescope Array (CTA) [4] is the next generation VHE gamma-ray observatory, which improves the sensitivity by a factor of 10 in the range 100 GeV-10 TeV and an extension to energies well below 100 GeV and above 100 TeV. CTA consists of telescopes having mirrors with size 23 m, 12 m, 3.5-7 m, and ∼ 50 m 2 , which are called large size telescope (LST), medium size telescope (MST), small size telescope (SST), and SchwarzschildCouder telescope (SCT), respectively. Several types of readout systems are developed for CTA with different analog memories for the requirements of LST, MST, SST, and SCT. At the same time, there has been progress in the development of photon sensors such as PMT and a Geigermode avalanche photodiode (or silicon photomultiplier) for CTA. At this time, the primary candidate for LST is a PMT.\nUsing the analog memory DRS version 4 (hereafter DRS4), we have so far developed three versions of prototypes for the PMT readout system for CTA, named Dragon . Using the first version of the prototype, we demonstrated that the waveform of a PMT signal can be well digitized with the DRS4 chip. The second version was developed with improvements made to the sampling depth of DRS4 and a trigger [5]. The third version was developed to be downsized and reduce the cost for mass production. In this paper, we report the design and performance of the third version of the prototype.\n2 Design of Readout System\n2.1 Overview\nTwo thousand PMTs and their readout systems are arranged on the focal plane of each LST telescope [6], with one readout system per 7-PMT cluster. We have developed the third prototype of the PMT readout system. Figures 1 and 2 show a photograph and block diagram of the prototype, respectively. The prototype consists of a 7-PMT cluster, a slow control board and a DRS4 readout board. The total size is 14 cm × 48 cm. The 7-PMT cluster and the slow control board are described in detail in reference [7]. In this paper, a brief description is provided.\n\n\n\nOur 7-PMT cluster consists of seven head-on type PMTs with a super-bialkali photocathode and 8-stage dynodes (Hamamatsu Photonics, R11920-100 with a diameter of 38 mm) [8], a Cockcroft-Walton (CW) circuit for high voltage supply to the PMTs (designed by Hamamatsu Photonics), and a preamplifier board. A signal from the PMT is amplified by a preamplifier, and fed to the DRS4 readout board. A commercial preamplifier, Mini-circuits LEE-39+, is used in the current Dragon (Fig. 3). A preamplifier board with an ASIC designed for CTA, PACTA [9], has been recently developed and will be used in the next Dragon .\n\n\n
Figure 1 : Photograph of the 7-PMT cluster and readout system (ver. 3).
\n\n\n\n
Figure 2 : Block diagram of the 7-PMT cluster and readout system (ver. 3).
\n\nOnthe DRS4 readout board the preamplified signal is divided into three lines: a high gain channel, a low gain channel, and a trigger channel. The high and low gain channels are connected to DRS4 chips. The signal is sampled at a rate of the order of GHz and the waveform is stored in a SCA in DRS4. When a trigger is generated in the trigger circuit, the voltages stored in the capacitor array are sequentially output and then digitized by an external slow sampling ( ∼ 30 MHz) ADC. The digitized data is sent to a field programmable gate array (FPGA) and then transmitted to a Gigabit Ethernet transceiver and a backplane via a data input/output (I/O) connector. The FPGA controls a static random access memory (SRAM) that stores large amounts of data before transmission and a digital-toanalog converter (DAC) used for thresholding in the trigger circuit.\nThe slow control board (Fig.1) is equipped with a generator for test pulses that are fed to the preamplifier, a temperature and humidity sensor with I2C interface, a DAC for setting the voltage of the CW high voltage circuit, and an ADC for monitoring both the CW circuit and the DC anode current. These devices on the slow control board are controlled by a complex programmable logic device\n\n\n
Figure 5 : Block diagram of the main amplifiers.
\n\n\n\n
Figure 3 : Preamplifier board with a commercial amplifier (left) and the PACTA (right).
\n\n(CPLD). Since the CPLD communicates with the FPGA on the DRS4 readout board, the data to and from the CPLD is sent via the Ethernet. The power supply to the DRS4 readout board and the slow control board is ± 3.3V and +5V. The total power consumption is ∼ 2W per readout channel.\n2.2 DRS4 Readout Board\nFigure 4 shows a photograph of the DRS4 readout board with a size of 14 cm × 30 cm, 10 cm shorter than the version 2. The slow control board attached to the 7-PMT cluster is connected to the DRS4 readout board from the right side via two card-edge connectors. The DRS4 board has main amplifiers, eight DRS4 chips, ADCs for digitizing a signal stored in the capacitor array in DRS4 at a sampling frequency of 33 MHz, a DAC to control the DRS4, an FPGA, a 18Mbit SRAM, a Gigabit Ethernet transceiver, and a data I/O connector to the backplane. A Xilinx Vertex4 FPGA was adopted in the version 2. In the version 3, a Xilinx Spartan-6 FPGA is used to reduce the cost for mass production. In addition, analog level 0 (L0) and 1 (L1) trigger mezzanines or a digital level 0 (L0) trigger mezzanine are mounted on the DRS4 board. Details of these parts are described in the following subsections.\n\n\n
Figure 4 : Photograph of the DRS4 readout board (ver. 3).
\n\n2.3 Main Amplifier\nFigure 5 shows a block diagram of the main amplifiers. It is designed to have a bandwidth greater than 250 MHz for the high gain channel, lower power consumption, and a dynamic range of 0.2-1000 photoelectrons for LST. A preamplified signal from a PMT with a typical gain of 4 × 10 4 is fed to the main amplifier. The signal is amplified using two differential amplifiers to meet the requirement for bandwidth and power consumption. For the high gain channel, it is amplified by a gain of 9 using Analog Devices, ADA4927. For the trigger, it is amplified by a gain of 4 using Analog Devices, ADA4927 and ADA4950. For the low gain channel, the preamplified signal is attenuated by 1/4 and then buffered with a differential amplifier using Analog Devices, ADA4950.\n2.4 Trigger\nThe DRS4 readout board has been designed to host an analog trigger [10] or digital trigger [5]. The different elements of the trigger system will be placed at different locations on the readout electronics, as shown in Fig. 4.\nFor the analog trigger, the decision boards (L0/L1) are placed on the readout board itself, while for the digital trigger the L0 board is placed in the frontend and the L1 in the backplane, keeping both analog and digital L0 as close as possible after the signals from PMTs. The distribution boards are placed at the backplane of the readout board, in order to send and receive trigger signals efficiently among neighbouring clusters. The choice of using trigger mezzanines and sharing of connectors has allowed the independent study of both analog and digital solution in the development phase.\n2.5 DRS4 Chip\nThe DRS chip is being developed at the Paul Scherrer Institute (PSI), Switzerland, for the MEG experiment [3]. The latest version, DRS4, is also used in the MAGIC experiment. The DRS4 chip includes nine differential input channels at a sampling speed of 700 MS/s-5 GS/s, with a bandwidth of 950 MHz, and a low noise of 0.35 mV after offset correction. The analog waveform is stored in 1,024 sampling capacitors per channel and the waveform can be read out after sampling via a shift register that is clocked at a maximum of 33 MHz for digitization using an external ADC. The power consumption of the DRS4 chip is 17.5 mWper channel at 2 GS/s sampling rate. The chip is fabricated using the 0.25 µ mCMOStechnology.\nIt is possible to cascade two or more channels to obtain deeper sampling depth. In the Dragon , four channels were cascaded, leading to a sampling depth of 4,096 for each PMT signal, which corresponds to a depth of 4 µ s at 1 GS/s sampling rate. It enables continuous sampling until the stereo coincidence trigger confirms the local telescope trigger, and initializes the camera readout.\n2.6 FPGA-based Gigabit Ethernet (SiTCP)\nThe digitized waveform data and the monitor/control data are transmitted via Ethernet with only two devices: FPGA and Gigabit Ethernet transceiver (PHY). This simple composition is available on a hardware-based TCP processor, SiTCP [11]. The circuit size of SiTCP in the FPGA is ∼ 3000 slices, which is small enough to allow implementation with user circuits on a single FPGA. In addition, the SiTCP has an advantage in that the throughputs in both directions can simultaneously reach the theoretical upper limits of Gigabit Ethernet.\n2.7 Mini-Camera with Three Clusters\nA prototype with three clusters of PMTs and their readout boards was constructed to develop the trigger system between multiple clusters (Fig. 6) [10].\n\n\n\n\n\n
Figure 8 shows normalized gains for high and low gain channels as a function of frequency. The bandwidths for high and low gain channels are 260 MHz and 190 MHz at -3dB, respectively while those of the DRS4 readout board combined with a preamplifier (LEE-39+) were 250 MHz and 180 MHz at -3dB for high and low gain channels, respectively.
\n\n\n\n\n\n\n
Figure 6 : Prototype with three clusters of PMTs and their readout boards.
\n\n3 Performance of Readout System\nThe performance of the DRS4 readout system Dragon was measured. Figure 7 shows that dynamic ranges of high and low gain channels are 60 p.e. and 2500 p.e., which satisfied the requirement of > 1000 photoelectrons. Measured noise level for the high gain channel is 0.1 p.e. (rms).\nAfter the above measurements, the performance of the DRS4 readout board attached to a PMT was investigated. Figure 9 shows a pulse shape of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with a UV laser and the DRS4 readout system at a sampling rate of 2 GS/s. The PMT signal having a width of ∼ 3 ns (FWHM) and a height corresponding to ∼ 3 photoelectrons was successfully digitized. Measured dead time for readout of 60 cells in the DRS4 chip is 0.9% and 5.4% at 1 kHz and 7 kHz, respectively. Figure 10 shows a single photoelectron spectrum of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with Dragon and an LED at a sampling rate of 2 GS/s. In the figure, a single photoelectron peak is clearly seen. The signal to noise ratio defined as (single-photoelectron mean - pedestal mean) / pedestal r.m.s. is 3.6.\n\n\n
Figure7 : Dynamic ranges of the DRS4 readout board.
\n\n4 Conclusion\nWe have developed a prototype 7-PMT cluster readout system for the large-size telescope of the next generation VHE gammaray observatory,CTA. In the readout system named Dragon , a PMT signal is amplified, and its waveform is then digitized at a sampling rate of the order of GHz using an analog memory ASIC DRS4 that has 4,096 capacitors per readout channel. Using the prototype system at-\n\n\n
Figure 8 : Bandwidths of high-(square) and low-(triangle) gain channels.Figure 9 : Pulse shape of the PMT signal measured with the DRS4 readout system at a sampling rate of 2 GS/s.
\n\n\n\n
Figure 10 : Single photoelectron spectrum of the PMT signal measured with the DRS4 readout system. The horizontal axis is in units of mV × ns.
\n\ntached to a PMT with a Cockcroft-Walton circuit, we successfully obtained a pulse shape of the signal of the PMT detecting a UV light at a sampling rate of 2 GS/s, and also a single photoelectron spectrum. Evaluation of the readout system's performance is ongoing, and a prototype with several dozen clusters and their readout boards will be constructed as the next step for the development of a full telescope camera.\nAcknowledgment: We gratefully acknowledge support from the agencies and organisations listed in this page: http://www.ctaobservatory.org/?q=node/22\nReferences\n\n[1] D. Lachartre, F. Feinstein, NIMA 442 (2000) 99-104\n[2] E. Delagnes et. al., NIMA 567 (2006) 21-26\n[3] S. Ritt, R. Dinapoli, U. Hartmann, NIMA 623 (2010) 486-488\n[4] The CTA Consortium, Experimental Astronomy 32 (2011) 193-316\n[5] H. Kubo et al., proceeding of ICRC2011 9 (2011) 179-182\n[6] O. Blanch et al., these proceedings, ID-776\n[7] R. Orito et. al., proceeding of ICRC2011 9 (2011) 171-174\n[8] T. Toyama et al., these proceedings, ID-684\n[9] A. Sanuy et al., JINST 7 (2012) C01100\n[10] J.A. Barrio et al., these proceedings, ID-396\n[11] T. Uchida, IEEE TNS 55 (2008) 1631-1637\n\n\n\n\n"},"sections":{"kind":"list like","value":[{"title":"Development of the Photomultiplier-Tube Readout System for the CTA Large Size Telescope","content":"C. DELGADO 6 , D. FINK 7 , D. GASCON 8 , S. GUNJI 9 , R. HAGIWARA 9 , Y. HANABATA 10 , K. HATANAKA 1 , M. HAYASHIDA 10 , M. IKENO 11 , S. KABUKI 12 , H. KATAGIRI 13 , J. KATAOKA 14 , Y. KONNO 1 , S. KOYAMA 15 , T. KISHIMOTO 1 , J. KUSHIDA 16 , G. MART'INEZ 6 , S. MASUDA 1 , J.M. MIRANDA 17 , R. MIRZOYAN 7 , T. MIZUNO 18 , T. NAGAYOSHI 15 , D. NAKAJIMA 7 , T. NAKAMORI 9 , H. OHOKA 10 , A. OKUMURA 19 , R. ORITO 20 , T. SAITO 1 , A. SANUY 8 , H. SASAKI 21 , M. SAWADA 3 , T. SCHWEIZER 7 , R. SUGAWARA 20 , K.-H. SULANKE 22 , H. TAJIMA 19 , M. TANAKA 11 , S. TANAKA 13 , L.A. TEJEDOR 5 , Y. TERADA 15 , M. TESHIMA 7 , 10 , F. TOKANAI 9 , Y. TSUCHIYA 1 , 11 15 13 21 1 Department of Physics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan kubo@cr.scphys.kyoto-u.ac.jp Abstract: We have developed a prototype of the photomultiplier tube (PMT) readout system for the Cherenkov Telescope Array (CTA) Large Size Telescope (LST). Two thousand PMTs along with their readout systems are arranged on the focal plane of each telescope, with one readout system per 7-PMT cluster. The Cherenkov light pulses generated by the air showers are detected by the PMTs and amplified in a compact, low noise and wide dynamic range gain block. The output of this block is then digitized at a sampling rate of the order of GHz using the Domino Ring Sampler DRS4, an analog memory ASIC developed at Paul Scherrer Institute. The sampler has 1,024 capacitors per channel and four channels are cascaded for increased depth. After a trigger is generated in the system, the charges stored in the capacitors are digitized by an external slow sampling ADC and then transmitted via Gigabit Ethernet. An onboard FPGA controls the DRS4, trigger threshold, and Ethernet transfer. In addition, the control and monitoring of the Cockcroft-Walton circuit that provides high voltage for the 7-PMT cluster are performed by the same FPGA. A prototype named Dragon has been developed that has successfully sampled PMT signals at a rate of 2 GHz, and generated single photoelectron spectra. Keywords: Imaging Atmospheric Cherenkov Telescope, Gamma-rays, Electronics","pages":[1]},{"title":"1 Introduction","content":"A ground-based imaging atmospheric Cherenkov telescope (IACT) measures Cherenkov light from an extended air shower (EAS) generated by the interaction between very high energy (VHE) gamma rays and the upper atmosphere. Night sky background (NSB) also enters a pixel photon sensor of the focal plane of the IACT with a rate of the order of 10-100 MHz, depending on the mirror size and the pixel size. In some region of the Galactic plane, the NSB can reach on average up to several hundred MHz. The NSB therefore becomes noise that affects the sensitivity and the energy threshold. Given this NSB pollution, and the fact that the duration of Cherenkov light from EAS is a few nanoseconds, a fast digitization speed of the readout system coupled to a fast photosensor like a photomultiplier tube (PMT) is beneficial to increase the pixels' signal- to-noise ratio. In addition, this system should be compact and have low cost and low power consumption because each IACT possesses several thousand photon sensor pixels and the readout system attached to the sensors is in a camera container located at the focal position. Furthermore, a wide dynamic range of more than 8-10 bits is required to resolve a single photoelectron and have a wider energy range. A commercial flash analog-to-digital converter (FADC) satisfies the requirement of a wide dynamic range. However, it is costly and consumes relatively high power of a few watts per channel. On the other hand, an analog memory application specific integrated circuit (ASIC), that consists of several hundred to several thousand switched capacitor arrays (SCA) per channel, can sample a signal at the order of GHz, with a wide dynamic range and lower power consumption. Several types of analog memory ASICs have been developed for applications in particle physics and cosmic ray physics. With respect to IACTs, a modified version of ARS0 [1], Swift Analog Memory (SAM) [2], and Domino Ring Sampler (DRS) [3] chips are used in the H.E.S.S.-I, H.E.S.S.-II, and MAGIC experiments, respectively. Cherenkov Telescope Array (CTA) [4] is the next generation VHE gamma-ray observatory, which improves the sensitivity by a factor of 10 in the range 100 GeV-10 TeV and an extension to energies well below 100 GeV and above 100 TeV. CTA consists of telescopes having mirrors with size 23 m, 12 m, 3.5-7 m, and ∼ 50 m 2 , which are called large size telescope (LST), medium size telescope (MST), small size telescope (SST), and SchwarzschildCouder telescope (SCT), respectively. Several types of readout systems are developed for CTA with different analog memories for the requirements of LST, MST, SST, and SCT. At the same time, there has been progress in the development of photon sensors such as PMT and a Geigermode avalanche photodiode (or silicon photomultiplier) for CTA. At this time, the primary candidate for LST is a PMT. Using the analog memory DRS version 4 (hereafter DRS4), we have so far developed three versions of prototypes for the PMT readout system for CTA, named Dragon . Using the first version of the prototype, we demonstrated that the waveform of a PMT signal can be well digitized with the DRS4 chip. The second version was developed with improvements made to the sampling depth of DRS4 and a trigger [5]. The third version was developed to be downsized and reduce the cost for mass production. In this paper, we report the design and performance of the third version of the prototype.","pages":[1,2]},{"title":"2.1 Overview","content":"Two thousand PMTs and their readout systems are arranged on the focal plane of each LST telescope [6], with one readout system per 7-PMT cluster. We have developed the third prototype of the PMT readout system. Figures 1 and 2 show a photograph and block diagram of the prototype, respectively. The prototype consists of a 7-PMT cluster, a slow control board and a DRS4 readout board. The total size is 14 cm × 48 cm. The 7-PMT cluster and the slow control board are described in detail in reference [7]. In this paper, a brief description is provided. Our 7-PMT cluster consists of seven head-on type PMTs with a super-bialkali photocathode and 8-stage dynodes (Hamamatsu Photonics, R11920-100 with a diameter of 38 mm) [8], a Cockcroft-Walton (CW) circuit for high voltage supply to the PMTs (designed by Hamamatsu Photonics), and a preamplifier board. A signal from the PMT is amplified by a preamplifier, and fed to the DRS4 readout board. A commercial preamplifier, Mini-circuits LEE-39+, is used in the current Dragon (Fig. 3). A preamplifier board with an ASIC designed for CTA, PACTA [9], has been recently developed and will be used in the next Dragon . Onthe DRS4 readout board the preamplified signal is divided into three lines: a high gain channel, a low gain channel, and a trigger channel. The high and low gain channels are connected to DRS4 chips. The signal is sampled at a rate of the order of GHz and the waveform is stored in a SCA in DRS4. When a trigger is generated in the trigger circuit, the voltages stored in the capacitor array are sequentially output and then digitized by an external slow sampling ( ∼ 30 MHz) ADC. The digitized data is sent to a field programmable gate array (FPGA) and then transmitted to a Gigabit Ethernet transceiver and a backplane via a data input/output (I/O) connector. The FPGA controls a static random access memory (SRAM) that stores large amounts of data before transmission and a digital-toanalog converter (DAC) used for thresholding in the trigger circuit. The slow control board (Fig.1) is equipped with a generator for test pulses that are fed to the preamplifier, a temperature and humidity sensor with I2C interface, a DAC for setting the voltage of the CW high voltage circuit, and an ADC for monitoring both the CW circuit and the DC anode current. These devices on the slow control board are controlled by a complex programmable logic device (CPLD). Since the CPLD communicates with the FPGA on the DRS4 readout board, the data to and from the CPLD is sent via the Ethernet. The power supply to the DRS4 readout board and the slow control board is ± 3.3V and +5V. The total power consumption is ∼ 2W per readout channel.","pages":[2,3]},{"title":"2.2 DRS4 Readout Board","content":"Figure 4 shows a photograph of the DRS4 readout board with a size of 14 cm × 30 cm, 10 cm shorter than the version 2. The slow control board attached to the 7-PMT cluster is connected to the DRS4 readout board from the right side via two card-edge connectors. The DRS4 board has main amplifiers, eight DRS4 chips, ADCs for digitizing a signal stored in the capacitor array in DRS4 at a sampling frequency of 33 MHz, a DAC to control the DRS4, an FPGA, a 18Mbit SRAM, a Gigabit Ethernet transceiver, and a data I/O connector to the backplane. A Xilinx Vertex4 FPGA was adopted in the version 2. In the version 3, a Xilinx Spartan-6 FPGA is used to reduce the cost for mass production. In addition, analog level 0 (L0) and 1 (L1) trigger mezzanines or a digital level 0 (L0) trigger mezzanine are mounted on the DRS4 board. Details of these parts are described in the following subsections.","pages":[3]},{"title":"2.3 Main Amplifier","content":"Figure 5 shows a block diagram of the main amplifiers. It is designed to have a bandwidth greater than 250 MHz for the high gain channel, lower power consumption, and a dynamic range of 0.2-1000 photoelectrons for LST. A preamplified signal from a PMT with a typical gain of 4 × 10 4 is fed to the main amplifier. The signal is amplified using two differential amplifiers to meet the requirement for bandwidth and power consumption. For the high gain channel, it is amplified by a gain of 9 using Analog Devices, ADA4927. For the trigger, it is amplified by a gain of 4 using Analog Devices, ADA4927 and ADA4950. For the low gain channel, the preamplified signal is attenuated by 1/4 and then buffered with a differential amplifier using Analog Devices, ADA4950.","pages":[3]},{"title":"2.4 Trigger","content":"The DRS4 readout board has been designed to host an analog trigger [10] or digital trigger [5]. The different elements of the trigger system will be placed at different locations on the readout electronics, as shown in Fig. 4. For the analog trigger, the decision boards (L0/L1) are placed on the readout board itself, while for the digital trigger the L0 board is placed in the frontend and the L1 in the backplane, keeping both analog and digital L0 as close as possible after the signals from PMTs. The distribution boards are placed at the backplane of the readout board, in order to send and receive trigger signals efficiently among neighbouring clusters. The choice of using trigger mezzanines and sharing of connectors has allowed the independent study of both analog and digital solution in the development phase.","pages":[3]},{"title":"2.5 DRS4 Chip","content":"The DRS chip is being developed at the Paul Scherrer Institute (PSI), Switzerland, for the MEG experiment [3]. The latest version, DRS4, is also used in the MAGIC experiment. The DRS4 chip includes nine differential input channels at a sampling speed of 700 MS/s-5 GS/s, with a bandwidth of 950 MHz, and a low noise of 0.35 mV after offset correction. The analog waveform is stored in 1,024 sampling capacitors per channel and the waveform can be read out after sampling via a shift register that is clocked at a maximum of 33 MHz for digitization using an external ADC. The power consumption of the DRS4 chip is 17.5 mWper channel at 2 GS/s sampling rate. The chip is fabricated using the 0.25 µ mCMOStechnology. It is possible to cascade two or more channels to obtain deeper sampling depth. In the Dragon , four channels were cascaded, leading to a sampling depth of 4,096 for each PMT signal, which corresponds to a depth of 4 µ s at 1 GS/s sampling rate. It enables continuous sampling until the stereo coincidence trigger confirms the local telescope trigger, and initializes the camera readout.","pages":[3]},{"title":"2.6 FPGA-based Gigabit Ethernet (SiTCP)","content":"The digitized waveform data and the monitor/control data are transmitted via Ethernet with only two devices: FPGA and Gigabit Ethernet transceiver (PHY). This simple composition is available on a hardware-based TCP processor, SiTCP [11]. The circuit size of SiTCP in the FPGA is ∼ 3000 slices, which is small enough to allow implementation with user circuits on a single FPGA. In addition, the SiTCP has an advantage in that the throughputs in both directions can simultaneously reach the theoretical upper limits of Gigabit Ethernet.","pages":[3]},{"title":"2.7 Mini-Camera with Three Clusters","content":"A prototype with three clusters of PMTs and their readout boards was constructed to develop the trigger system between multiple clusters (Fig. 6) [10].","pages":[3]},{"title":"3 Performance of Readout System","content":"The performance of the DRS4 readout system Dragon was measured. Figure 7 shows that dynamic ranges of high and low gain channels are 60 p.e. and 2500 p.e., which satisfied the requirement of > 1000 photoelectrons. Measured noise level for the high gain channel is 0.1 p.e. (rms). After the above measurements, the performance of the DRS4 readout board attached to a PMT was investigated. Figure 9 shows a pulse shape of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with a UV laser and the DRS4 readout system at a sampling rate of 2 GS/s. The PMT signal having a width of ∼ 3 ns (FWHM) and a height corresponding to ∼ 3 photoelectrons was successfully digitized. Measured dead time for readout of 60 cells in the DRS4 chip is 0.9% and 5.4% at 1 kHz and 7 kHz, respectively. Figure 10 shows a single photoelectron spectrum of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with Dragon and an LED at a sampling rate of 2 GS/s. In the figure, a single photoelectron peak is clearly seen. The signal to noise ratio defined as (single-photoelectron mean - pedestal mean) / pedestal r.m.s. is 3.6.","pages":[4]},{"title":"4 Conclusion","content":"We have developed a prototype 7-PMT cluster readout system for the large-size telescope of the next generation VHE gammaray observatory,CTA. In the readout system named Dragon , a PMT signal is amplified, and its waveform is then digitized at a sampling rate of the order of GHz using an analog memory ASIC DRS4 that has 4,096 capacitors per readout channel. Using the prototype system at- tached to a PMT with a Cockcroft-Walton circuit, we successfully obtained a pulse shape of the signal of the PMT detecting a UV light at a sampling rate of 2 GS/s, and also a single photoelectron spectrum. Evaluation of the readout system's performance is ongoing, and a prototype with several dozen clusters and their readout boards will be constructed as the next step for the development of a full telescope camera. Acknowledgment: We gratefully acknowledge support from the agencies and organisations listed in this page: http://www.ctaobservatory.org/?q=node/22","pages":[4]}],"string":"[\n {\n \"title\": \"Development of the Photomultiplier-Tube Readout System for the CTA Large Size Telescope\",\n \"content\": \"C. DELGADO 6 , D. FINK 7 , D. GASCON 8 , S. GUNJI 9 , R. HAGIWARA 9 , Y. HANABATA 10 , K. HATANAKA 1 , M. HAYASHIDA 10 , M. IKENO 11 , S. KABUKI 12 , H. KATAGIRI 13 , J. KATAOKA 14 , Y. KONNO 1 , S. KOYAMA 15 , T. KISHIMOTO 1 , J. KUSHIDA 16 , G. MART'INEZ 6 , S. MASUDA 1 , J.M. MIRANDA 17 , R. MIRZOYAN 7 , T. MIZUNO 18 , T. NAGAYOSHI 15 , D. NAKAJIMA 7 , T. NAKAMORI 9 , H. OHOKA 10 , A. OKUMURA 19 , R. ORITO 20 , T. SAITO 1 , A. SANUY 8 , H. SASAKI 21 , M. SAWADA 3 , T. SCHWEIZER 7 , R. SUGAWARA 20 , K.-H. SULANKE 22 , H. TAJIMA 19 , M. TANAKA 11 , S. TANAKA 13 , L.A. TEJEDOR 5 , Y. TERADA 15 , M. TESHIMA 7 , 10 , F. TOKANAI 9 , Y. TSUCHIYA 1 , 11 15 13 21 1 Department of Physics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan kubo@cr.scphys.kyoto-u.ac.jp Abstract: We have developed a prototype of the photomultiplier tube (PMT) readout system for the Cherenkov Telescope Array (CTA) Large Size Telescope (LST). Two thousand PMTs along with their readout systems are arranged on the focal plane of each telescope, with one readout system per 7-PMT cluster. The Cherenkov light pulses generated by the air showers are detected by the PMTs and amplified in a compact, low noise and wide dynamic range gain block. The output of this block is then digitized at a sampling rate of the order of GHz using the Domino Ring Sampler DRS4, an analog memory ASIC developed at Paul Scherrer Institute. The sampler has 1,024 capacitors per channel and four channels are cascaded for increased depth. After a trigger is generated in the system, the charges stored in the capacitors are digitized by an external slow sampling ADC and then transmitted via Gigabit Ethernet. An onboard FPGA controls the DRS4, trigger threshold, and Ethernet transfer. In addition, the control and monitoring of the Cockcroft-Walton circuit that provides high voltage for the 7-PMT cluster are performed by the same FPGA. A prototype named Dragon has been developed that has successfully sampled PMT signals at a rate of 2 GHz, and generated single photoelectron spectra. Keywords: Imaging Atmospheric Cherenkov Telescope, Gamma-rays, Electronics\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"A ground-based imaging atmospheric Cherenkov telescope (IACT) measures Cherenkov light from an extended air shower (EAS) generated by the interaction between very high energy (VHE) gamma rays and the upper atmosphere. Night sky background (NSB) also enters a pixel photon sensor of the focal plane of the IACT with a rate of the order of 10-100 MHz, depending on the mirror size and the pixel size. In some region of the Galactic plane, the NSB can reach on average up to several hundred MHz. The NSB therefore becomes noise that affects the sensitivity and the energy threshold. Given this NSB pollution, and the fact that the duration of Cherenkov light from EAS is a few nanoseconds, a fast digitization speed of the readout system coupled to a fast photosensor like a photomultiplier tube (PMT) is beneficial to increase the pixels' signal- to-noise ratio. In addition, this system should be compact and have low cost and low power consumption because each IACT possesses several thousand photon sensor pixels and the readout system attached to the sensors is in a camera container located at the focal position. Furthermore, a wide dynamic range of more than 8-10 bits is required to resolve a single photoelectron and have a wider energy range. A commercial flash analog-to-digital converter (FADC) satisfies the requirement of a wide dynamic range. However, it is costly and consumes relatively high power of a few watts per channel. On the other hand, an analog memory application specific integrated circuit (ASIC), that consists of several hundred to several thousand switched capacitor arrays (SCA) per channel, can sample a signal at the order of GHz, with a wide dynamic range and lower power consumption. Several types of analog memory ASICs have been developed for applications in particle physics and cosmic ray physics. With respect to IACTs, a modified version of ARS0 [1], Swift Analog Memory (SAM) [2], and Domino Ring Sampler (DRS) [3] chips are used in the H.E.S.S.-I, H.E.S.S.-II, and MAGIC experiments, respectively. Cherenkov Telescope Array (CTA) [4] is the next generation VHE gamma-ray observatory, which improves the sensitivity by a factor of 10 in the range 100 GeV-10 TeV and an extension to energies well below 100 GeV and above 100 TeV. CTA consists of telescopes having mirrors with size 23 m, 12 m, 3.5-7 m, and ∼ 50 m 2 , which are called large size telescope (LST), medium size telescope (MST), small size telescope (SST), and SchwarzschildCouder telescope (SCT), respectively. Several types of readout systems are developed for CTA with different analog memories for the requirements of LST, MST, SST, and SCT. At the same time, there has been progress in the development of photon sensors such as PMT and a Geigermode avalanche photodiode (or silicon photomultiplier) for CTA. At this time, the primary candidate for LST is a PMT. Using the analog memory DRS version 4 (hereafter DRS4), we have so far developed three versions of prototypes for the PMT readout system for CTA, named Dragon . Using the first version of the prototype, we demonstrated that the waveform of a PMT signal can be well digitized with the DRS4 chip. The second version was developed with improvements made to the sampling depth of DRS4 and a trigger [5]. The third version was developed to be downsized and reduce the cost for mass production. In this paper, we report the design and performance of the third version of the prototype.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2.1 Overview\",\n \"content\": \"Two thousand PMTs and their readout systems are arranged on the focal plane of each LST telescope [6], with one readout system per 7-PMT cluster. We have developed the third prototype of the PMT readout system. Figures 1 and 2 show a photograph and block diagram of the prototype, respectively. The prototype consists of a 7-PMT cluster, a slow control board and a DRS4 readout board. The total size is 14 cm × 48 cm. The 7-PMT cluster and the slow control board are described in detail in reference [7]. In this paper, a brief description is provided. Our 7-PMT cluster consists of seven head-on type PMTs with a super-bialkali photocathode and 8-stage dynodes (Hamamatsu Photonics, R11920-100 with a diameter of 38 mm) [8], a Cockcroft-Walton (CW) circuit for high voltage supply to the PMTs (designed by Hamamatsu Photonics), and a preamplifier board. A signal from the PMT is amplified by a preamplifier, and fed to the DRS4 readout board. A commercial preamplifier, Mini-circuits LEE-39+, is used in the current Dragon (Fig. 3). A preamplifier board with an ASIC designed for CTA, PACTA [9], has been recently developed and will be used in the next Dragon . Onthe DRS4 readout board the preamplified signal is divided into three lines: a high gain channel, a low gain channel, and a trigger channel. The high and low gain channels are connected to DRS4 chips. The signal is sampled at a rate of the order of GHz and the waveform is stored in a SCA in DRS4. When a trigger is generated in the trigger circuit, the voltages stored in the capacitor array are sequentially output and then digitized by an external slow sampling ( ∼ 30 MHz) ADC. The digitized data is sent to a field programmable gate array (FPGA) and then transmitted to a Gigabit Ethernet transceiver and a backplane via a data input/output (I/O) connector. The FPGA controls a static random access memory (SRAM) that stores large amounts of data before transmission and a digital-toanalog converter (DAC) used for thresholding in the trigger circuit. The slow control board (Fig.1) is equipped with a generator for test pulses that are fed to the preamplifier, a temperature and humidity sensor with I2C interface, a DAC for setting the voltage of the CW high voltage circuit, and an ADC for monitoring both the CW circuit and the DC anode current. These devices on the slow control board are controlled by a complex programmable logic device (CPLD). Since the CPLD communicates with the FPGA on the DRS4 readout board, the data to and from the CPLD is sent via the Ethernet. The power supply to the DRS4 readout board and the slow control board is ± 3.3V and +5V. The total power consumption is ∼ 2W per readout channel.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.2 DRS4 Readout Board\",\n \"content\": \"Figure 4 shows a photograph of the DRS4 readout board with a size of 14 cm × 30 cm, 10 cm shorter than the version 2. The slow control board attached to the 7-PMT cluster is connected to the DRS4 readout board from the right side via two card-edge connectors. The DRS4 board has main amplifiers, eight DRS4 chips, ADCs for digitizing a signal stored in the capacitor array in DRS4 at a sampling frequency of 33 MHz, a DAC to control the DRS4, an FPGA, a 18Mbit SRAM, a Gigabit Ethernet transceiver, and a data I/O connector to the backplane. A Xilinx Vertex4 FPGA was adopted in the version 2. In the version 3, a Xilinx Spartan-6 FPGA is used to reduce the cost for mass production. In addition, analog level 0 (L0) and 1 (L1) trigger mezzanines or a digital level 0 (L0) trigger mezzanine are mounted on the DRS4 board. Details of these parts are described in the following subsections.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.3 Main Amplifier\",\n \"content\": \"Figure 5 shows a block diagram of the main amplifiers. It is designed to have a bandwidth greater than 250 MHz for the high gain channel, lower power consumption, and a dynamic range of 0.2-1000 photoelectrons for LST. A preamplified signal from a PMT with a typical gain of 4 × 10 4 is fed to the main amplifier. The signal is amplified using two differential amplifiers to meet the requirement for bandwidth and power consumption. For the high gain channel, it is amplified by a gain of 9 using Analog Devices, ADA4927. For the trigger, it is amplified by a gain of 4 using Analog Devices, ADA4927 and ADA4950. For the low gain channel, the preamplified signal is attenuated by 1/4 and then buffered with a differential amplifier using Analog Devices, ADA4950.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.4 Trigger\",\n \"content\": \"The DRS4 readout board has been designed to host an analog trigger [10] or digital trigger [5]. The different elements of the trigger system will be placed at different locations on the readout electronics, as shown in Fig. 4. For the analog trigger, the decision boards (L0/L1) are placed on the readout board itself, while for the digital trigger the L0 board is placed in the frontend and the L1 in the backplane, keeping both analog and digital L0 as close as possible after the signals from PMTs. The distribution boards are placed at the backplane of the readout board, in order to send and receive trigger signals efficiently among neighbouring clusters. The choice of using trigger mezzanines and sharing of connectors has allowed the independent study of both analog and digital solution in the development phase.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.5 DRS4 Chip\",\n \"content\": \"The DRS chip is being developed at the Paul Scherrer Institute (PSI), Switzerland, for the MEG experiment [3]. The latest version, DRS4, is also used in the MAGIC experiment. The DRS4 chip includes nine differential input channels at a sampling speed of 700 MS/s-5 GS/s, with a bandwidth of 950 MHz, and a low noise of 0.35 mV after offset correction. The analog waveform is stored in 1,024 sampling capacitors per channel and the waveform can be read out after sampling via a shift register that is clocked at a maximum of 33 MHz for digitization using an external ADC. The power consumption of the DRS4 chip is 17.5 mWper channel at 2 GS/s sampling rate. The chip is fabricated using the 0.25 µ mCMOStechnology. It is possible to cascade two or more channels to obtain deeper sampling depth. In the Dragon , four channels were cascaded, leading to a sampling depth of 4,096 for each PMT signal, which corresponds to a depth of 4 µ s at 1 GS/s sampling rate. It enables continuous sampling until the stereo coincidence trigger confirms the local telescope trigger, and initializes the camera readout.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.6 FPGA-based Gigabit Ethernet (SiTCP)\",\n \"content\": \"The digitized waveform data and the monitor/control data are transmitted via Ethernet with only two devices: FPGA and Gigabit Ethernet transceiver (PHY). This simple composition is available on a hardware-based TCP processor, SiTCP [11]. The circuit size of SiTCP in the FPGA is ∼ 3000 slices, which is small enough to allow implementation with user circuits on a single FPGA. In addition, the SiTCP has an advantage in that the throughputs in both directions can simultaneously reach the theoretical upper limits of Gigabit Ethernet.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.7 Mini-Camera with Three Clusters\",\n \"content\": \"A prototype with three clusters of PMTs and their readout boards was constructed to develop the trigger system between multiple clusters (Fig. 6) [10].\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3 Performance of Readout System\",\n \"content\": \"The performance of the DRS4 readout system Dragon was measured. Figure 7 shows that dynamic ranges of high and low gain channels are 60 p.e. and 2500 p.e., which satisfied the requirement of > 1000 photoelectrons. Measured noise level for the high gain channel is 0.1 p.e. (rms). After the above measurements, the performance of the DRS4 readout board attached to a PMT was investigated. Figure 9 shows a pulse shape of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with a UV laser and the DRS4 readout system at a sampling rate of 2 GS/s. The PMT signal having a width of ∼ 3 ns (FWHM) and a height corresponding to ∼ 3 photoelectrons was successfully digitized. Measured dead time for readout of 60 cells in the DRS4 chip is 0.9% and 5.4% at 1 kHz and 7 kHz, respectively. Figure 10 shows a single photoelectron spectrum of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with Dragon and an LED at a sampling rate of 2 GS/s. In the figure, a single photoelectron peak is clearly seen. The signal to noise ratio defined as (single-photoelectron mean - pedestal mean) / pedestal r.m.s. is 3.6.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"4 Conclusion\",\n \"content\": \"We have developed a prototype 7-PMT cluster readout system for the large-size telescope of the next generation VHE gammaray observatory,CTA. In the readout system named Dragon , a PMT signal is amplified, and its waveform is then digitized at a sampling rate of the order of GHz using an analog memory ASIC DRS4 that has 4,096 capacitors per readout channel. Using the prototype system at- tached to a PMT with a Cockcroft-Walton circuit, we successfully obtained a pulse shape of the signal of the PMT detecting a UV light at a sampling rate of 2 GS/s, and also a single photoelectron spectrum. Evaluation of the readout system's performance is ongoing, and a prototype with several dozen clusters and their readout boards will be constructed as the next step for the development of a full telescope camera. Acknowledgment: We gratefully acknowledge support from the agencies and organisations listed in this page: http://www.ctaobservatory.org/?q=node/22\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":287,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33.2840N"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1307.4189.pdf"},"content":{"kind":"string","value":"\n\n\n\nSingle-Mirror Small-Size Telescope Structure for the Cherenkov Telescope Array\nJACEK NIEMIEC, JERZY MICHAŁOWSKI, MICHAŁ DYRDA, WOJCIECH KOCHA'NSKI, JAROMIR LUDWIN, MAREK STODULSKI, PAWEŁ ZI'OŁKOWSKI, PAWEŁ ˙ ZYCHOWSKI, FOR THE CTA CONSORTIUM.\nInstitute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krak'ow, Poland\njacek.niemiec@ifj.edu.pl\nAbstract: A single-mirror small-size (1M-SST) Davies-Cotton telescope has been proposed for the southern observatory of the Cherenkov Telescope Array (CTA) by a consortium of scientific institutions from Poland, Switzerland, and Germany. The telescope has a 4 m diameter reflector and will be equipped with a fully digital camera based on Geiger avalanche photodiodes (APDs). Such a design is particularly interesting for CTA because it represents a very simple, reliable, and cheap solution for a SST. Here we present the design and the characteristics of the mechanical structure of the 1M-SST telescope and its drive system. We also discuss the results of a finite element method analysis in order to demonstrate the conformance of the design with the CTA specifications and scientific objectives. In addition, we report on the current status of the construction of a aprototype telescope structure at the Institute of Nuclear Physics PAS in Krakow.\nKeywords: CTA, imaging atmospheric Cherenkov telescopes, FEM analysis\n1 Introduction\nA consortium of scientific institutions from Poland, Switzerland, and Germany has recently proposed a singlemirror small-size (1M-SST) telescope for the southern observatory of the Cherenkov Telescope Array (CTA) [1]. The telescope utilizes a Davies-Cotton design - a classic telescope type used in very high-energy ground-based astronomy. 1M-SST will be equipped with a fully digital and lightweight camera based on Geiger avalanche photodiodes (APDs) with the physical plate scale size of 24 mm (see [2]). With the assumed angular pixel pitch p = 0 . 25 deg, the telescope focal length, i.e., the distance from the center of the dish to the camera pixel plane, is thus f = 5 . 6 m. To assure that the optical point-spread function of the telescope is smaller than 0.25 deg at 4 deg off-axis, a reflective dish with diameter of 3.98 m is needed. This results in the camera field-of-view of 9 deg. The telescope's effective mirror area, corrected for the effect of shadowing by the camera and the mast, is 7.6 m 2 .\nThe 1M-SST telescope thus represents a compact structure. The use of silicon photomultipliers (PMTs) results in a low weight of the photodetector plane of 30 kg and the full camera weight of less than 300 kg. Hence, a number of engineering problems, challenged in earlier designs (e.g., H.E.S.S. or VERITAS telescopes [3, 4]) are largely relaxed. The latter were caused by a necessity of supporting a heavy and large camera based on analog PMTs on the mast of a telescope with a large focal length ( f > 10 m). A design of a low-cost solution for a 1M-SST that can be easily replicated to form a high-energy sub-array of the CTA is therefore feasible.\nThe design of the 1M-SST structure is described in this work. In sections 2.1 and 2.2 the telescope's mechanical structure and drive system are presented, respectively. The results of the finite element method (FEM) analysis are shown in section 2.3. Finally, in section 3 a current status of the construction of a prototype telescope structure at the Institute of Nuclear Physics PAS in Krakow is reported.\n2 Telescope Structure\n2.1 Frame\nThe telescope frame consists of several sub-systems as shown in figure 1. The mast (1) is directly connected to the dish support structure (2), to which the counterweight (3) is also attached. The rigidity of the mast is increased by the use of thin pre-tensioned steel rods. The mast positions the camera (4) with respect to the mirrors (5) mounted on the dish (6). The latter is fixed to the dish support structure mounted on the telescope support (tower, 7). Finally, the dock station (8) locks the telescope in a parking position.\nThe telescope is a compact and lightweight structure. Its hight is about 5 m, measured from the base of the tower, its width is 3.3 m, and the total length is 9.3 m. To cut the transportation costs a division of the telescope structure into smaller units that fit the standard size of a shipping container is applied. With steel used as a basic construction material, the telescope weighs less than 9 tons, including counterweights and the components of the drive system. The low weight, the fact that all steel profiles and tubes can be obtained as off-the-shelf products from the industry and that the design uses as many as possible identical components lowers down the cost of manufacturing and guarantees a low price of the telescope frame.\nThe layout of 18 mirror tiles that make up the reflecting dish of 4 m diameter is shown in figure 2 (left). Hexagonal mirrors of size 0.78 m flat-to-flat are used, with space between the tiles of 2 cm. The radius of curvature of a single mirror is 11.2 m. The dish structure is built of square steel profiles. It is to be connected to the dish support structure at four points, visible in figure 2 as pads. The required spherical shape of the dish is obtained with two hexagonal sections welded to eight straight star-forming profiles. A precise positioning of the mirrors with respect to the camera is accomplished through a use of dedicated mirror fixtures (see figures 2 and 3).\nThe mirror fixture design is presented in figure 3. The fixture consists of an aluminium triangular frame with\n\n\n
Figure 2 : Layout of 18 mirror tiles on the dish in the rear/back view (left). The dish connected to the dish support structure (right). /g4/g19/g2 /g6/g5/g11/g16/g1/g29/g24/g1/g47 /g6/g24/g29/g39/g29/g34/g33/g1/g47/g1/g64/g48/g65 /g5/g21/g39/g25 /g14/g37/g25/g31/g29/g32/g29/g33/g21/g37/g44/g1/g5/g21/g41/g29/g25/g38 /g4/g34/g39/g39/g34/g33 /g14/g25/g37/g26/g34/g37/g32/g21/g33/g23/g25/g1/g15/g25/g36/g40/g29/g37/g25/g32/g25/g33/g39/g38/g1/g37/g25/g41/g29/g25/g42
\n\n\n\n
Figure 1 : 1M-SST sub-systems. Drawings in figures 2, 4, and 5 are based on a technical documentation of the telescope structure. See text for the explanation of figure markings.
\n\nthree fixing points and a steel threaded rod with a sphere at one end and a threaded flange at the other end. The flanges are bolted to the dish. The use of the threaded rod enables\n\n\n\n\n\n\n/g4/g17/g2/g1/g15/g25/g26/g1/g47\n/g14/g21/g27/g25/g1/g47/g1/g65/g72\n/g68/g64\n\n\n
Figure 7.1.9: Mirror fixture with interfaces (left) and mirror attached (right) Mast and dish support structure Figure 3 : Mirror fixture with interfaces (left) and the mirror attached (right).
\n\na precise positioning of a mirror at the focal length distance from the telescope's optical center. The sphere of the threaded rod is tightened in the central fixing point of the triangular frame to create a hinge that allows for additional degrees of freedom during a preadjustment of a mirror orientation.\nThe final precise alignment of the mirrors is to be performed by means of the automatic motor controllers (AMCs), called actuators. Each mirror fixture houses three interfaces that provide 3-point isostatic support to a mirror. One interface is a so-called fixed point support (1, figure 3). Two other interfaces are actuators (2), which give a mirror facet two rotational degrees of freedom. The mirrors are fixed to these interfaces via steel pads (3) glued on the mirrors' rear surfaces. Note, that the design of the mirror fixtures applied here does not only enable an arcminutelevel stable alignment of the mirror facets relative to the camera, but it also allows for a simple and easy mounting/dismounting of the mirrors on/from the dish.\nThe design of the mast (see (1) in figure 1) guarantees a proper setup of the camera with respect to the reflecting mirror surface at the telescope's focal distance of 5.6 m. The mast is built of eight circular tubes that are bolted together into four longer beams. The beams are connected with the camera housing at one end, and with the dish support structure at the other end. In this way the mast does not incur any mechanical stresses to the dish so that all mast deformations are practically disconnected from the deformations of the dish. A rectangular frame roughly in the middle of the mast increases the stiffness of the mast structure and also facilitates its transport. The rigidity of the mast is additionally increased by the use of sixteen steel rods. The rods are pre-stressed with a force of 2000 N and fixed to the long tubes by means of turnbuckles. To assure a proper geometry of the mast, eight special-design interfaces between the rectangular frame and the beams must be used. Their shape is identical but requires computer numerical control (CNC) machining. Note, that such interfaces are not needed at points in which the mast is connected to the dish support structure.\nThe dish support structure is shown in figure 4. It is com-\nThe design of the mast guarantees a proper setup of the camera with respect to reflecting mirror surface at\nthe focal distance of 5.6 m. The main components of the mast, Fig. 7.1.10, are eight circular steel tubes (1)\nOD= 0.06 m and\nm thick bolted together into four longer beams (2) to provide the required focal length.\nAt one end the four tubes are connected with the camera interface (3) that is made of aluminium sheet. At\nthe second end the tubes are bolted to the dish support structure (4). The applied solution practically\n/g4/g19/g2\n\n\n
Figure 4 : Design of the dish support structure.
\n\nposed of various components. Two forks/Y-shaped frames (1) are box structures welded out of steel sheets and tubes. The forks are spaced with four tubes (2). As mentioned above, the beams of the mast are bolted to the four ends of the forks. Counterweights are fixed at their two remaining ends. Four struts (3) and steel sheets (wings) (4) provide the necessary stiffness of the structure. The latter also improve the resistance to the wind blows. There are also two circular tubes with pads (5) incorporated in the forks.\n\n\n
Figure 8.2.1: The drive system movement range in elevation (left) and azimuth (right) 8.2.1 Drive system requirements Figure 6 : 1M-SST drive system movement range in elevation (left) and azimuth (right).
\n\nThe drive system parameters have been determined according to the CTA requirements for 1M SST\navailable at the time the calculations were made. The assumed environmental conditions specify the\ntelescope site altitude at 2500 m above sea level and the operation temperature range between 20 C and\n+30 C. The observation mode assumes the average maximum wind velocity of 50 km/h and the emergency mode allows for a safe parking of the telescope in conditions with the maximum wind velocity reaching 100 km/h, including gusts. The emergency mode also includes the possibility of reaching the parking position at the decreased angular acceleration in the case of a failure of one of the drives. It is assumed that in the operation mode the telescope can fast slew to any defined point in the sky within 1 minute, including 6 seconds for both the acceleration at a constant rate from the start out position and the deceleration. Resulting maximum angular velocities are max,a=6 /sec and max,e=2 /sec with respect The pads represent the interfaces to which the elevation drive system components (slew drive and the ball-bearing mounted on the sides of the head of the telescope support) are bolted.\nto the azimuth and the elevation axis, respectively. The positioning accuracy is taken within 7 arcsec and the tracking accuracy is assumed not to exceed 5 arcmin 8.2.2. Drive mechanical torque calculations For the conditions thus specified mechanical torques and power requirements have been calculated that allowed for the selection of the drive system components. The calculations follow the guidelines of the 'Drive Power Estimates for CTA Telescopes' document, provided by the Project Office. The method uses a simplified approach that assumes the mechanical shaft power (the power required to move the telescope at the elevation and azimuth axes) to comprise of three components: the friction power, the wind power and the power required to accelerate the telescope. The total mechanical power is calculated as the product of the required total mechanical torque and the telescope angular velocity. The maximum values of the latter are given above. Tables 8.2.1 and 8.2.2 below present the maximum total torques about each axis and their components. Details of the calculations are provided in a separate document. The 1M-SST telescope support is built of two main components: the tower (1) and the head (2), figure 5. The elements of the azimuth drive system (3) are incorporated into the tower, while these of the elevation drive system (4) into the head. The tower must be bolted to a foundation in which a steel anchor grid structure with reinforcements is embedded. The slew drive (drawn in grey in figure 5) realizing the azimuth rotation of the telescope is fixed to the conical part of the tower (the tower cap). Inside the tower, there is a radial ball-bearing connected to the azimuth slew drive with the tube, that stabilizes the whole structure. The housing of the azimuth ball bearing inside the tower can be accessed for installation and maintenance purposes via two openings.\nThe head is rotated by the azimuth slew drive. The elevation drive system components, including the slew drive and the ball-bearing, figure 5, are bolted to the pads of the dish support structure (see figure 4). The slew drives and the bearings of both the azimuth and elevation drive systems are identical.\n2.2 Drive System\nAs introduced in section 2.1, a positioning and tracking system of the telescope is realized with two independent drive axes: the azimuth axis and the elevation axis. The range for the telescope movement in elevation and azimuth is shown in figure 6. The drive system of each axis is based on an IMO [5] slew drive and a radial ball-bearing. An IMO slew drive is a compact system that combines a warm gear with a motor and also a roller bearing, thus enabling transmission of both radial and axial forces. It has a fully enclosed and self-supporting housing. Rotating components are fixed to the housing with bolts. Each IMO slew drive selected for the 1M-SST contains a twin warm gear with two servo-motors. Such a solution helps to increase the torque capacity and to eliminate backlash that is needed to achieve a required pointing accuracy of 7 arcsec and the tracking accuracy less than 5 arcmin. The drive system of the medium-size telescope (MST) for CTA is also based on the IMO slew drives and the choice of this type of drives for the 1M-SST has been partially done for the sake of the uniformity among the various CTA telescope array subcomponents. For this reason the concepts for control software and structure and safety will also be elaborated based on the MST solutions.\n/g88/g85/g1\n2.3 FEMAnalysis\nThe 1M-SST telescope structure design has been optimized based on the static FEM analysis. In order to perform such analysis a simplified computer-aided design (CAD) model of the telescope structure was built with ANSYS [6]. The model allows for an investigation of the structure deformations and mechanical stresses under gravity, wind, snow, and ice loads, and also for the earthquake conditions. The results show that the design of 1M-SST meets CTA optical requirement that the maximum displacement of the camera with respect to the dish is smaller than 1/3 of the pixel size for the extreme observing mode conditions. The latter assume the maximum wind velocity with gusts of 50 km/h (the average velocity of 36 km/h) during which observation runs can still be taken. The resulting maximum displacement is 8.2 mm for the back-wind conditions and the telescope at 60 deg in the elevation angle. The maximum equivalent stresses in the structure are also well below the elasticity limit for the steel. With Re = 240 MPa the safety factor is about 3. Similar FEM analysis was conducted for the survival mode conditions. They allow for the wind velocity to reach 200 km/h while the telescope is locked in the parking position. Also in this case all stresses are well below the plasticity of the materials used for the mechanical construction. Preliminary FEM calculations show that the same is true if the earthquake loads are considered. Finally, the modal analysis, figure 7, demonstrates that the lowest eigenfrequency of the structure is 3.8 Hz. Thus it is well above the minimum value of 2.5 Hz specified for CTA telescopes.\nNote, that the FEM calculations performed demonstrate a full conformance of the design with the CTA specications and scientic objectives. /g7/g6/g11/g1/g21/g19/g31/g21/g42/g31/g19/g40/g34/g33/g38/g1\n\n\n
Figure 7 : Results of the modal analysis showing the three lowest eigenfrequencies and their corresponding modes. The elevation angle is given in parantheses.
\n\n/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g10/g10/g10/g1/g76/g1/g86/g86/g70/g89/g1/g9/g63/g1/g77/g88/g85/g82/g78/g1\n3 Towards 1M-SST Prototype\nBased on the design presented in section 2 a prototype telescope structure with drive system will be constructed at the Institute of Nuclear Physics PAS in Krak'ow. The elements of the drive system have already been ordered. The technical design of the telescope frame has recently been prepared by an industrial partner. A call for tender for manufacturing of the telescope components and preinstallation at the INP PAS site will soon be placed. It is\n\n\n\nforeseen that the 1M-SST prototype will be ready for tests by the end of 2013.\nAcknowledgment: This work was supported by The National Centre for Research and Development through project ERANET-ASPERA/01/10 and The National Science Centre through project DEC-2011/01/M/ST9/01891. We also gratefully acknowledge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22 .\nReferences\n\n[1] R. Moderski, et al., contribution no. 0840, these proceedings.\n[2] V. Boccone, et al., contribution no. 0234, these proceedings. [3] K. Bernloehr, et al., Astroparticle Physics 20 (2003)\n111-128.\n[4] J. Holder, et al., Astroparticle Physics 25 (2006) 391-401.\n[5] http://www.goimo.eu/Home.217.0.html\n[6] http://www.ansys.com/\n"},"sections":{"kind":"list like","value":[{"title":"Single-Mirror Small-Size Telescope Structure for the Cherenkov Telescope Array","content":"JACEK NIEMIEC, JERZY MICHAŁOWSKI, MICHAŁ DYRDA, WOJCIECH KOCHA'NSKI, JAROMIR LUDWIN, MAREK STODULSKI, PAWEŁ ZI'OŁKOWSKI, PAWEŁ ˙ ZYCHOWSKI, FOR THE CTA CONSORTIUM. Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krak'ow, Poland jacek.niemiec@ifj.edu.pl Abstract: A single-mirror small-size (1M-SST) Davies-Cotton telescope has been proposed for the southern observatory of the Cherenkov Telescope Array (CTA) by a consortium of scientific institutions from Poland, Switzerland, and Germany. The telescope has a 4 m diameter reflector and will be equipped with a fully digital camera based on Geiger avalanche photodiodes (APDs). Such a design is particularly interesting for CTA because it represents a very simple, reliable, and cheap solution for a SST. Here we present the design and the characteristics of the mechanical structure of the 1M-SST telescope and its drive system. We also discuss the results of a finite element method analysis in order to demonstrate the conformance of the design with the CTA specifications and scientific objectives. In addition, we report on the current status of the construction of a aprototype telescope structure at the Institute of Nuclear Physics PAS in Krakow. Keywords: CTA, imaging atmospheric Cherenkov telescopes, FEM analysis","pages":[1]},{"title":"1 Introduction","content":"A consortium of scientific institutions from Poland, Switzerland, and Germany has recently proposed a singlemirror small-size (1M-SST) telescope for the southern observatory of the Cherenkov Telescope Array (CTA) [1]. The telescope utilizes a Davies-Cotton design - a classic telescope type used in very high-energy ground-based astronomy. 1M-SST will be equipped with a fully digital and lightweight camera based on Geiger avalanche photodiodes (APDs) with the physical plate scale size of 24 mm (see [2]). With the assumed angular pixel pitch p = 0 . 25 deg, the telescope focal length, i.e., the distance from the center of the dish to the camera pixel plane, is thus f = 5 . 6 m. To assure that the optical point-spread function of the telescope is smaller than 0.25 deg at 4 deg off-axis, a reflective dish with diameter of 3.98 m is needed. This results in the camera field-of-view of 9 deg. The telescope's effective mirror area, corrected for the effect of shadowing by the camera and the mast, is 7.6 m 2 . The 1M-SST telescope thus represents a compact structure. The use of silicon photomultipliers (PMTs) results in a low weight of the photodetector plane of 30 kg and the full camera weight of less than 300 kg. Hence, a number of engineering problems, challenged in earlier designs (e.g., H.E.S.S. or VERITAS telescopes [3, 4]) are largely relaxed. The latter were caused by a necessity of supporting a heavy and large camera based on analog PMTs on the mast of a telescope with a large focal length ( f > 10 m). A design of a low-cost solution for a 1M-SST that can be easily replicated to form a high-energy sub-array of the CTA is therefore feasible. The design of the 1M-SST structure is described in this work. In sections 2.1 and 2.2 the telescope's mechanical structure and drive system are presented, respectively. The results of the finite element method (FEM) analysis are shown in section 2.3. Finally, in section 3 a current status of the construction of a prototype telescope structure at the Institute of Nuclear Physics PAS in Krakow is reported.","pages":[1]},{"title":"2.1 Frame","content":"The telescope frame consists of several sub-systems as shown in figure 1. The mast (1) is directly connected to the dish support structure (2), to which the counterweight (3) is also attached. The rigidity of the mast is increased by the use of thin pre-tensioned steel rods. The mast positions the camera (4) with respect to the mirrors (5) mounted on the dish (6). The latter is fixed to the dish support structure mounted on the telescope support (tower, 7). Finally, the dock station (8) locks the telescope in a parking position. The telescope is a compact and lightweight structure. Its hight is about 5 m, measured from the base of the tower, its width is 3.3 m, and the total length is 9.3 m. To cut the transportation costs a division of the telescope structure into smaller units that fit the standard size of a shipping container is applied. With steel used as a basic construction material, the telescope weighs less than 9 tons, including counterweights and the components of the drive system. The low weight, the fact that all steel profiles and tubes can be obtained as off-the-shelf products from the industry and that the design uses as many as possible identical components lowers down the cost of manufacturing and guarantees a low price of the telescope frame. The layout of 18 mirror tiles that make up the reflecting dish of 4 m diameter is shown in figure 2 (left). Hexagonal mirrors of size 0.78 m flat-to-flat are used, with space between the tiles of 2 cm. The radius of curvature of a single mirror is 11.2 m. The dish structure is built of square steel profiles. It is to be connected to the dish support structure at four points, visible in figure 2 as pads. The required spherical shape of the dish is obtained with two hexagonal sections welded to eight straight star-forming profiles. A precise positioning of the mirrors with respect to the camera is accomplished through a use of dedicated mirror fixtures (see figures 2 and 3). The mirror fixture design is presented in figure 3. The fixture consists of an aluminium triangular frame with three fixing points and a steel threaded rod with a sphere at one end and a threaded flange at the other end. The flanges are bolted to the dish. The use of the threaded rod enables /g4/g17/g2/g1/g15/g25/g26/g1/g47 /g14/g21/g27/g25/g1/g47/g1/g65/g72 /g68/g64 a precise positioning of a mirror at the focal length distance from the telescope's optical center. The sphere of the threaded rod is tightened in the central fixing point of the triangular frame to create a hinge that allows for additional degrees of freedom during a preadjustment of a mirror orientation. The final precise alignment of the mirrors is to be performed by means of the automatic motor controllers (AMCs), called actuators. Each mirror fixture houses three interfaces that provide 3-point isostatic support to a mirror. One interface is a so-called fixed point support (1, figure 3). Two other interfaces are actuators (2), which give a mirror facet two rotational degrees of freedom. The mirrors are fixed to these interfaces via steel pads (3) glued on the mirrors' rear surfaces. Note, that the design of the mirror fixtures applied here does not only enable an arcminutelevel stable alignment of the mirror facets relative to the camera, but it also allows for a simple and easy mounting/dismounting of the mirrors on/from the dish. The design of the mast (see (1) in figure 1) guarantees a proper setup of the camera with respect to the reflecting mirror surface at the telescope's focal distance of 5.6 m. The mast is built of eight circular tubes that are bolted together into four longer beams. The beams are connected with the camera housing at one end, and with the dish support structure at the other end. In this way the mast does not incur any mechanical stresses to the dish so that all mast deformations are practically disconnected from the deformations of the dish. A rectangular frame roughly in the middle of the mast increases the stiffness of the mast structure and also facilitates its transport. The rigidity of the mast is additionally increased by the use of sixteen steel rods. The rods are pre-stressed with a force of 2000 N and fixed to the long tubes by means of turnbuckles. To assure a proper geometry of the mast, eight special-design interfaces between the rectangular frame and the beams must be used. Their shape is identical but requires computer numerical control (CNC) machining. Note, that such interfaces are not needed at points in which the mast is connected to the dish support structure. The dish support structure is shown in figure 4. It is com- The design of the mast guarantees a proper setup of the camera with respect to reflecting mirror surface at the focal distance of 5.6 m. The main components of the mast, Fig. 7.1.10, are eight circular steel tubes (1) OD= 0.06 m and m thick bolted together into four longer beams (2) to provide the required focal length. At one end the four tubes are connected with the camera interface (3) that is made of aluminium sheet. At the second end the tubes are bolted to the dish support structure (4). The applied solution practically /g4/g19/g2 posed of various components. Two forks/Y-shaped frames (1) are box structures welded out of steel sheets and tubes. The forks are spaced with four tubes (2). As mentioned above, the beams of the mast are bolted to the four ends of the forks. Counterweights are fixed at their two remaining ends. Four struts (3) and steel sheets (wings) (4) provide the necessary stiffness of the structure. The latter also improve the resistance to the wind blows. There are also two circular tubes with pads (5) incorporated in the forks. /g4/g17/g2/g1/g15/g25/g26/g1/g47","pages":[1,2,3]},{"title":"/g6/g5/g11/g16/g1/g29/g24/g1/g47 /g6/g24/g29/g39/g29/g34/g33/g1/g47/g1/g64/g48/g65 /g5/g21/g39/g25 /g14/g21/g27/g25/g1/g47/g1/g67/g64 /g68/g64","content":"The drive system parameters have been determined according to the CTA requirements for 1M SST available at the time the calculations were made. The assumed environmental conditions specify the telescope site altitude at 2500 m above sea level and the operation temperature range between 20 C and +30 C. The observation mode assumes the average maximum wind velocity of 50 km/h and the emergency mode allows for a safe parking of the telescope in conditions with the maximum wind velocity reaching 100 km/h, including gusts. The emergency mode also includes the possibility of reaching the parking position at the decreased angular acceleration in the case of a failure of one of the drives. It is assumed that in the operation mode the telescope can fast slew to any defined point in the sky within 1 minute, including 6 seconds for both the acceleration at a constant rate from the start out position and the deceleration. Resulting maximum angular velocities are max,a=6 /sec and max,e=2 /sec with respect The pads represent the interfaces to which the elevation drive system components (slew drive and the ball-bearing mounted on the sides of the head of the telescope support) are bolted. to the azimuth and the elevation axis, respectively. The positioning accuracy is taken within 7 arcsec and the tracking accuracy is assumed not to exceed 5 arcmin 8.2.2. Drive mechanical torque calculations For the conditions thus specified mechanical torques and power requirements have been calculated that allowed for the selection of the drive system components. The calculations follow the guidelines of the 'Drive Power Estimates for CTA Telescopes' document, provided by the Project Office. The method uses a simplified approach that assumes the mechanical shaft power (the power required to move the telescope at the elevation and azimuth axes) to comprise of three components: the friction power, the wind power and the power required to accelerate the telescope. The total mechanical power is calculated as the product of the required total mechanical torque and the telescope angular velocity. The maximum values of the latter are given above. Tables 8.2.1 and 8.2.2 below present the maximum total torques about each axis and their components. Details of the calculations are provided in a separate document. The 1M-SST telescope support is built of two main components: the tower (1) and the head (2), figure 5. The elements of the azimuth drive system (3) are incorporated into the tower, while these of the elevation drive system (4) into the head. The tower must be bolted to a foundation in which a steel anchor grid structure with reinforcements is embedded. The slew drive (drawn in grey in figure 5) realizing the azimuth rotation of the telescope is fixed to the conical part of the tower (the tower cap). Inside the tower, there is a radial ball-bearing connected to the azimuth slew drive with the tube, that stabilizes the whole structure. The housing of the azimuth ball bearing inside the tower can be accessed for installation and maintenance purposes via two openings. The head is rotated by the azimuth slew drive. The elevation drive system components, including the slew drive and the ball-bearing, figure 5, are bolted to the pads of the dish support structure (see figure 4). The slew drives and the bearings of both the azimuth and elevation drive systems are identical.","pages":[3]},{"title":"2.2 Drive System","content":"As introduced in section 2.1, a positioning and tracking system of the telescope is realized with two independent drive axes: the azimuth axis and the elevation axis. The range for the telescope movement in elevation and azimuth is shown in figure 6. The drive system of each axis is based on an IMO [5] slew drive and a radial ball-bearing. An IMO slew drive is a compact system that combines a warm gear with a motor and also a roller bearing, thus enabling transmission of both radial and axial forces. It has a fully enclosed and self-supporting housing. Rotating components are fixed to the housing with bolts. Each IMO slew drive selected for the 1M-SST contains a twin warm gear with two servo-motors. Such a solution helps to increase the torque capacity and to eliminate backlash that is needed to achieve a required pointing accuracy of 7 arcsec and the tracking accuracy less than 5 arcmin. The drive system of the medium-size telescope (MST) for CTA is also based on the IMO slew drives and the choice of this type of drives for the 1M-SST has been partially done for the sake of the uniformity among the various CTA telescope array subcomponents. For this reason the concepts for control software and structure and safety will also be elaborated based on the MST solutions. /g88/g85/g1","pages":[3,4]},{"title":"2.3 FEMAnalysis","content":"The 1M-SST telescope structure design has been optimized based on the static FEM analysis. In order to perform such analysis a simplified computer-aided design (CAD) model of the telescope structure was built with ANSYS [6]. The model allows for an investigation of the structure deformations and mechanical stresses under gravity, wind, snow, and ice loads, and also for the earthquake conditions. The results show that the design of 1M-SST meets CTA optical requirement that the maximum displacement of the camera with respect to the dish is smaller than 1/3 of the pixel size for the extreme observing mode conditions. The latter assume the maximum wind velocity with gusts of 50 km/h (the average velocity of 36 km/h) during which observation runs can still be taken. The resulting maximum displacement is 8.2 mm for the back-wind conditions and the telescope at 60 deg in the elevation angle. The maximum equivalent stresses in the structure are also well below the elasticity limit for the steel. With Re = 240 MPa the safety factor is about 3. Similar FEM analysis was conducted for the survival mode conditions. They allow for the wind velocity to reach 200 km/h while the telescope is locked in the parking position. Also in this case all stresses are well below the plasticity of the materials used for the mechanical construction. Preliminary FEM calculations show that the same is true if the earthquake loads are considered. Finally, the modal analysis, figure 7, demonstrates that the lowest eigenfrequency of the structure is 3.8 Hz. Thus it is well above the minimum value of 2.5 Hz specified for CTA telescopes. Note, that the FEM calculations performed demonstrate a full conformance of the design with the CTA specications and scientic objectives. /g7/g6/g11/g1/g21/g19/g31/g21/g42/g31/g19/g40/g34/g33/g38/g1 /g20/g23/g21/g20/g29/g40/g16/g26/g39/g20/g36/g1","pages":[4]},{"title":"3 Towards 1M-SST Prototype","content":"Based on the design presented in section 2 a prototype telescope structure with drive system will be constructed at the Institute of Nuclear Physics PAS in Krak'ow. The elements of the drive system have already been ordered. The technical design of the telescope frame has recently been prepared by an industrial partner. A call for tender for manufacturing of the telescope components and preinstallation at the INP PAS site will soon be placed. It is foreseen that the 1M-SST prototype will be ready for tests by the end of 2013. Acknowledgment: This work was supported by The National Centre for Research and Development through project ERANET-ASPERA/01/10 and The National Science Centre through project DEC-2011/01/M/ST9/01891. We also gratefully acknowledge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22 .","pages":[4]}],"string":"[\n {\n \"title\": \"Single-Mirror Small-Size Telescope Structure for the Cherenkov Telescope Array\",\n \"content\": \"JACEK NIEMIEC, JERZY MICHAŁOWSKI, MICHAŁ DYRDA, WOJCIECH KOCHA'NSKI, JAROMIR LUDWIN, MAREK STODULSKI, PAWEŁ ZI'OŁKOWSKI, PAWEŁ ˙ ZYCHOWSKI, FOR THE CTA CONSORTIUM. Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krak'ow, Poland jacek.niemiec@ifj.edu.pl Abstract: A single-mirror small-size (1M-SST) Davies-Cotton telescope has been proposed for the southern observatory of the Cherenkov Telescope Array (CTA) by a consortium of scientific institutions from Poland, Switzerland, and Germany. The telescope has a 4 m diameter reflector and will be equipped with a fully digital camera based on Geiger avalanche photodiodes (APDs). Such a design is particularly interesting for CTA because it represents a very simple, reliable, and cheap solution for a SST. Here we present the design and the characteristics of the mechanical structure of the 1M-SST telescope and its drive system. We also discuss the results of a finite element method analysis in order to demonstrate the conformance of the design with the CTA specifications and scientific objectives. In addition, we report on the current status of the construction of a aprototype telescope structure at the Institute of Nuclear Physics PAS in Krakow. Keywords: CTA, imaging atmospheric Cherenkov telescopes, FEM analysis\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"A consortium of scientific institutions from Poland, Switzerland, and Germany has recently proposed a singlemirror small-size (1M-SST) telescope for the southern observatory of the Cherenkov Telescope Array (CTA) [1]. The telescope utilizes a Davies-Cotton design - a classic telescope type used in very high-energy ground-based astronomy. 1M-SST will be equipped with a fully digital and lightweight camera based on Geiger avalanche photodiodes (APDs) with the physical plate scale size of 24 mm (see [2]). With the assumed angular pixel pitch p = 0 . 25 deg, the telescope focal length, i.e., the distance from the center of the dish to the camera pixel plane, is thus f = 5 . 6 m. To assure that the optical point-spread function of the telescope is smaller than 0.25 deg at 4 deg off-axis, a reflective dish with diameter of 3.98 m is needed. This results in the camera field-of-view of 9 deg. The telescope's effective mirror area, corrected for the effect of shadowing by the camera and the mast, is 7.6 m 2 . The 1M-SST telescope thus represents a compact structure. The use of silicon photomultipliers (PMTs) results in a low weight of the photodetector plane of 30 kg and the full camera weight of less than 300 kg. Hence, a number of engineering problems, challenged in earlier designs (e.g., H.E.S.S. or VERITAS telescopes [3, 4]) are largely relaxed. The latter were caused by a necessity of supporting a heavy and large camera based on analog PMTs on the mast of a telescope with a large focal length ( f > 10 m). A design of a low-cost solution for a 1M-SST that can be easily replicated to form a high-energy sub-array of the CTA is therefore feasible. The design of the 1M-SST structure is described in this work. In sections 2.1 and 2.2 the telescope's mechanical structure and drive system are presented, respectively. The results of the finite element method (FEM) analysis are shown in section 2.3. Finally, in section 3 a current status of the construction of a prototype telescope structure at the Institute of Nuclear Physics PAS in Krakow is reported.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"2.1 Frame\",\n \"content\": \"The telescope frame consists of several sub-systems as shown in figure 1. The mast (1) is directly connected to the dish support structure (2), to which the counterweight (3) is also attached. The rigidity of the mast is increased by the use of thin pre-tensioned steel rods. The mast positions the camera (4) with respect to the mirrors (5) mounted on the dish (6). The latter is fixed to the dish support structure mounted on the telescope support (tower, 7). Finally, the dock station (8) locks the telescope in a parking position. The telescope is a compact and lightweight structure. Its hight is about 5 m, measured from the base of the tower, its width is 3.3 m, and the total length is 9.3 m. To cut the transportation costs a division of the telescope structure into smaller units that fit the standard size of a shipping container is applied. With steel used as a basic construction material, the telescope weighs less than 9 tons, including counterweights and the components of the drive system. The low weight, the fact that all steel profiles and tubes can be obtained as off-the-shelf products from the industry and that the design uses as many as possible identical components lowers down the cost of manufacturing and guarantees a low price of the telescope frame. The layout of 18 mirror tiles that make up the reflecting dish of 4 m diameter is shown in figure 2 (left). Hexagonal mirrors of size 0.78 m flat-to-flat are used, with space between the tiles of 2 cm. The radius of curvature of a single mirror is 11.2 m. The dish structure is built of square steel profiles. It is to be connected to the dish support structure at four points, visible in figure 2 as pads. The required spherical shape of the dish is obtained with two hexagonal sections welded to eight straight star-forming profiles. A precise positioning of the mirrors with respect to the camera is accomplished through a use of dedicated mirror fixtures (see figures 2 and 3). The mirror fixture design is presented in figure 3. The fixture consists of an aluminium triangular frame with three fixing points and a steel threaded rod with a sphere at one end and a threaded flange at the other end. The flanges are bolted to the dish. The use of the threaded rod enables /g4/g17/g2/g1/g15/g25/g26/g1/g47 /g14/g21/g27/g25/g1/g47/g1/g65/g72 /g68/g64 a precise positioning of a mirror at the focal length distance from the telescope's optical center. The sphere of the threaded rod is tightened in the central fixing point of the triangular frame to create a hinge that allows for additional degrees of freedom during a preadjustment of a mirror orientation. The final precise alignment of the mirrors is to be performed by means of the automatic motor controllers (AMCs), called actuators. Each mirror fixture houses three interfaces that provide 3-point isostatic support to a mirror. One interface is a so-called fixed point support (1, figure 3). Two other interfaces are actuators (2), which give a mirror facet two rotational degrees of freedom. The mirrors are fixed to these interfaces via steel pads (3) glued on the mirrors' rear surfaces. Note, that the design of the mirror fixtures applied here does not only enable an arcminutelevel stable alignment of the mirror facets relative to the camera, but it also allows for a simple and easy mounting/dismounting of the mirrors on/from the dish. The design of the mast (see (1) in figure 1) guarantees a proper setup of the camera with respect to the reflecting mirror surface at the telescope's focal distance of 5.6 m. The mast is built of eight circular tubes that are bolted together into four longer beams. The beams are connected with the camera housing at one end, and with the dish support structure at the other end. In this way the mast does not incur any mechanical stresses to the dish so that all mast deformations are practically disconnected from the deformations of the dish. A rectangular frame roughly in the middle of the mast increases the stiffness of the mast structure and also facilitates its transport. The rigidity of the mast is additionally increased by the use of sixteen steel rods. The rods are pre-stressed with a force of 2000 N and fixed to the long tubes by means of turnbuckles. To assure a proper geometry of the mast, eight special-design interfaces between the rectangular frame and the beams must be used. Their shape is identical but requires computer numerical control (CNC) machining. Note, that such interfaces are not needed at points in which the mast is connected to the dish support structure. The dish support structure is shown in figure 4. It is com- The design of the mast guarantees a proper setup of the camera with respect to reflecting mirror surface at the focal distance of 5.6 m. The main components of the mast, Fig. 7.1.10, are eight circular steel tubes (1) OD= 0.06 m and m thick bolted together into four longer beams (2) to provide the required focal length. At one end the four tubes are connected with the camera interface (3) that is made of aluminium sheet. At the second end the tubes are bolted to the dish support structure (4). The applied solution practically /g4/g19/g2 posed of various components. Two forks/Y-shaped frames (1) are box structures welded out of steel sheets and tubes. The forks are spaced with four tubes (2). As mentioned above, the beams of the mast are bolted to the four ends of the forks. Counterweights are fixed at their two remaining ends. Four struts (3) and steel sheets (wings) (4) provide the necessary stiffness of the structure. The latter also improve the resistance to the wind blows. There are also two circular tubes with pads (5) incorporated in the forks. /g4/g17/g2/g1/g15/g25/g26/g1/g47\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"/g6/g5/g11/g16/g1/g29/g24/g1/g47 /g6/g24/g29/g39/g29/g34/g33/g1/g47/g1/g64/g48/g65 /g5/g21/g39/g25 /g14/g21/g27/g25/g1/g47/g1/g67/g64 /g68/g64\",\n \"content\": \"The drive system parameters have been determined according to the CTA requirements for 1M SST available at the time the calculations were made. The assumed environmental conditions specify the telescope site altitude at 2500 m above sea level and the operation temperature range between 20 C and +30 C. The observation mode assumes the average maximum wind velocity of 50 km/h and the emergency mode allows for a safe parking of the telescope in conditions with the maximum wind velocity reaching 100 km/h, including gusts. The emergency mode also includes the possibility of reaching the parking position at the decreased angular acceleration in the case of a failure of one of the drives. It is assumed that in the operation mode the telescope can fast slew to any defined point in the sky within 1 minute, including 6 seconds for both the acceleration at a constant rate from the start out position and the deceleration. Resulting maximum angular velocities are max,a=6 /sec and max,e=2 /sec with respect The pads represent the interfaces to which the elevation drive system components (slew drive and the ball-bearing mounted on the sides of the head of the telescope support) are bolted. to the azimuth and the elevation axis, respectively. The positioning accuracy is taken within 7 arcsec and the tracking accuracy is assumed not to exceed 5 arcmin 8.2.2. Drive mechanical torque calculations For the conditions thus specified mechanical torques and power requirements have been calculated that allowed for the selection of the drive system components. The calculations follow the guidelines of the 'Drive Power Estimates for CTA Telescopes' document, provided by the Project Office. The method uses a simplified approach that assumes the mechanical shaft power (the power required to move the telescope at the elevation and azimuth axes) to comprise of three components: the friction power, the wind power and the power required to accelerate the telescope. The total mechanical power is calculated as the product of the required total mechanical torque and the telescope angular velocity. The maximum values of the latter are given above. Tables 8.2.1 and 8.2.2 below present the maximum total torques about each axis and their components. Details of the calculations are provided in a separate document. The 1M-SST telescope support is built of two main components: the tower (1) and the head (2), figure 5. The elements of the azimuth drive system (3) are incorporated into the tower, while these of the elevation drive system (4) into the head. The tower must be bolted to a foundation in which a steel anchor grid structure with reinforcements is embedded. The slew drive (drawn in grey in figure 5) realizing the azimuth rotation of the telescope is fixed to the conical part of the tower (the tower cap). Inside the tower, there is a radial ball-bearing connected to the azimuth slew drive with the tube, that stabilizes the whole structure. The housing of the azimuth ball bearing inside the tower can be accessed for installation and maintenance purposes via two openings. The head is rotated by the azimuth slew drive. The elevation drive system components, including the slew drive and the ball-bearing, figure 5, are bolted to the pads of the dish support structure (see figure 4). The slew drives and the bearings of both the azimuth and elevation drive systems are identical.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"2.2 Drive System\",\n \"content\": \"As introduced in section 2.1, a positioning and tracking system of the telescope is realized with two independent drive axes: the azimuth axis and the elevation axis. The range for the telescope movement in elevation and azimuth is shown in figure 6. The drive system of each axis is based on an IMO [5] slew drive and a radial ball-bearing. An IMO slew drive is a compact system that combines a warm gear with a motor and also a roller bearing, thus enabling transmission of both radial and axial forces. It has a fully enclosed and self-supporting housing. Rotating components are fixed to the housing with bolts. Each IMO slew drive selected for the 1M-SST contains a twin warm gear with two servo-motors. Such a solution helps to increase the torque capacity and to eliminate backlash that is needed to achieve a required pointing accuracy of 7 arcsec and the tracking accuracy less than 5 arcmin. The drive system of the medium-size telescope (MST) for CTA is also based on the IMO slew drives and the choice of this type of drives for the 1M-SST has been partially done for the sake of the uniformity among the various CTA telescope array subcomponents. For this reason the concepts for control software and structure and safety will also be elaborated based on the MST solutions. /g88/g85/g1\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"2.3 FEMAnalysis\",\n \"content\": \"The 1M-SST telescope structure design has been optimized based on the static FEM analysis. In order to perform such analysis a simplified computer-aided design (CAD) model of the telescope structure was built with ANSYS [6]. The model allows for an investigation of the structure deformations and mechanical stresses under gravity, wind, snow, and ice loads, and also for the earthquake conditions. The results show that the design of 1M-SST meets CTA optical requirement that the maximum displacement of the camera with respect to the dish is smaller than 1/3 of the pixel size for the extreme observing mode conditions. The latter assume the maximum wind velocity with gusts of 50 km/h (the average velocity of 36 km/h) during which observation runs can still be taken. The resulting maximum displacement is 8.2 mm for the back-wind conditions and the telescope at 60 deg in the elevation angle. The maximum equivalent stresses in the structure are also well below the elasticity limit for the steel. With Re = 240 MPa the safety factor is about 3. Similar FEM analysis was conducted for the survival mode conditions. They allow for the wind velocity to reach 200 km/h while the telescope is locked in the parking position. Also in this case all stresses are well below the plasticity of the materials used for the mechanical construction. Preliminary FEM calculations show that the same is true if the earthquake loads are considered. Finally, the modal analysis, figure 7, demonstrates that the lowest eigenfrequency of the structure is 3.8 Hz. Thus it is well above the minimum value of 2.5 Hz specified for CTA telescopes. Note, that the FEM calculations performed demonstrate a full conformance of the design with the CTA specications and scientic objectives. /g7/g6/g11/g1/g21/g19/g31/g21/g42/g31/g19/g40/g34/g33/g38/g1 /g20/g23/g21/g20/g29/g40/g16/g26/g39/g20/g36/g1\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"3 Towards 1M-SST Prototype\",\n \"content\": \"Based on the design presented in section 2 a prototype telescope structure with drive system will be constructed at the Institute of Nuclear Physics PAS in Krak'ow. The elements of the drive system have already been ordered. The technical design of the telescope frame has recently been prepared by an industrial partner. A call for tender for manufacturing of the telescope components and preinstallation at the INP PAS site will soon be placed. It is foreseen that the 1M-SST prototype will be ready for tests by the end of 2013. Acknowledgment: This work was supported by The National Centre for Research and Development through project ERANET-ASPERA/01/10 and The National Science Centre through project DEC-2011/01/M/ST9/01891. We also gratefully acknowledge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22 .\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":288,"cells":{"bibcode":{"kind":"string","value":"2013ICRC...33.3036G"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1307.4545.pdf"},"content":{"kind":"string","value":"\n\n\n\nThe NectarCAM camera project\nJ-F.GLICENSTEIN 1 , M.BARCELO 11 , J-A. BARRIO 12 , O.BLANCH 11 , J.BOIX 11 , J.BOLMONT 4 , C.BOUTONNET 2 S.CAZAUX 1 , E.CHABANNE 7 , C.CHAMPION 2 , F.CHATEAU 1 , S.COLONGES 2 , P.CORONA 4 , S.COUTURIER 5 , B.COURTY 2 , E.DELAGNES 1 , C.DELGADO 10 , J-P.ERNENWEIN 6 , S.FEGAN 5 , O.FERREIRA 5 , M.FESQUET 1 , G.FONTAINE 5 , N.FOUQUE 7 , F.HENAULT 8 , D.GASC 'ON 13 , D.HERRANZ 12 , R.HERMEL 7 , D.HOFFMANN 6 , J.HOULES 6 , S.KARKAR 4 , B.KHELIFI 5 , J.KN ODLSEDER 3 , G.MARTINEZ 10 , K.LACOMBE 3 , G.LAMANNA 7 , T.LEFLOUR 7 , R.LOPEZCOTO 11 , F.LOUIS 1 , A.MATHIEU 5 , E.MOULIN 1 , P.NAYMAN 4 , F.NUNIO 1 , J-F. OLIVE 3 , J-L. PANAZOL 7 , P-O. PETRUCCI 8 , M.PUNCH 2 , J.PRAST 7 , P.RAMON 3 , M.RIALLOT 1 , M.RIB 'O 13 , S.ROSIER-LEES 7 , A.SANUY 13 , J.SIERO 13 , J-P.TAVERNET 4 , L.A.TEJEDOR 12 , F.TOUSSENEL 455 , G.VASILEIADIS 9 , V.VOISIN 4 , V.WAEGEBERT 3 , C.ZURBACH 9 , FOR THE CTA CONSORTIUM.\n1 DSM/IRFU, CEA-Saclay, F91191 Gif-sur-Yvette, France\n2 APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France\n\n3 Institut de Recherche en Astrophysique et Planetologie, 9 av. colonel Roche, BP 44 346, 31028 Toulouse Cedex 4, France\n4 LPNHE, Universite Pierre et Marie Curie Paris 6, Universite Denis Diderot Paris 7, CNRS/IN2P3, 4 Place Jussieu, F-75252, Paris Cedex 5, France\n5 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France\n6 Centre de Physique des Particules de Marseille, CNRS/IN2P3, 168 Avenue de Luminy, 13009 Marseille, France\n7 LAPP, Universite de Savoie, CNRS/IN2P3, 9 Chemin de Bellevue - BP 110, 74941 Annecy-le-Vieux Cedex, France\n8 UJF-Grenoble 1 CNRS/INSU, Institut de Planetologie et d'Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble, F-38041, France\n9 Laboratoire Univers et Particules de Montpellier, Universite Montpellier 2, CNRS/IN2P3, CC 72, Place Eugene Bataillon, F-34095 Montpellier Cedex 5, France\n10 CIEMAT, Avda. Complutense 22, 28040 Madrid, Spain\n11 Institut de Fisica d'Altes Energies (IFAE) Edifici Cn, Campus UAB, 08193 Bellaterra, Spain\n12 Grupo de Altas Energias, Universidad Complutense de Madrid, Av Complutenses, 28040 Madrid, Spain\n13 Departament d'Astronomia i Meteorologia (ICC-UB), Universitat de Barcelona, Marti i Franques, 1, 08028, Barcelona, Spain\n\nglicens@cea.fr\nAbstract: In the framework of the next generation of Cherenkov telescopes, the Cherenkov Telescope Array (CTA), NectarCAM is a camera designed for the medium size telescopes covering the central energy range of 100 GeV to 30 TeV. NectarCAM will be finely pixelated ( ∼ 1800 pixels for a 8 o field of view, FoV) in order to image atmospheric Cherenkov showers by measuring the charge deposited within a few nanoseconds time-window. It will have additional features like the capacity to record the full waveform with GHz sampling for every pixel and to measure event times with nanosecond accuracy. An array of a few tens of medium size telescopes, equipped with NectarCAMs, will achieve up to a factor of ten improvement in sensitivity over existing instruments in the energy range of 100 GeV to 10 TeV. The camera is made of roughly 250 independent read-out modules, each composed of seven photo-multipliers, with their associated high voltage base and control, a read-out board and a multi-service backplane board. The read-out boards use NECTAr (New Electronics for the Cherenkov Telescope Array) ASICs which have the dual functionality of analogue memories and Analogue to Digital Converter (ADC). The camera trigger to be used will be flexible so as to minimize the read-out dead-time of the NECTAr chips. We present the camera concept and the design and tests of the various subcomponents. The design includes the mechanical parts, the cooling of the electronics, the readout, the data acquisition, the trigger, the monitoring and services.\nKeywords: Methods, techniques and instrumentation\n1 Introduction\nNectarCAM is a new camera design for the medium size telescopes (MST) of the planned CTA 1 array of Imaging Atmospheric Cerenkov telescopes (IACTs). The CTA will have several sizes of single or dual-mirror telescopes. The single-mirror MST has a 12-meter diameter dish. Its camera will face new challenges:\n\n1. The fields of view of camera are expected to be larger (e.g. at least 7 o compared to 5 o for the cameras of H.E.S.S.-1, an previous generation array of IACTs). As a consequence more distant showers will be seen\n\nin the cameras. The arrival of the photons from a distant shower can last more than 100 nanoseconds. However, the signal to noise ratio in a pixel is optimized by recording and later integrating the signal in a much smaller window ( ∼ 10 ns). If the camera trigger is not flexible enough (e.g is similar to the H.E.S.S. camera trigger), the event energies would be systematically underestimated. Special trigger strategies have been designed to trigger different pixels at different moments (see Figure 16 of [1]). These\n\n\n
Figure 1 : Early version (V0) of the NECTAr readout module used in the NectarCAM camera.
\n\nstrategies still need to be tested. The trigger rate is also expected to increase by a factor of two compared to the previous generation of Cherenkov telescopes, as result of the larger field of view, potetially leading to a larger dead time of the instrument.\n\n2. Large timing gradients are expected in high-energy gamma-ray events. Measuring these gradients improves the reconstruction of the events. Analyzing the whole waveform also helps improve the hadron rejection (see e.g., Aliu et al. [3]). It has been estimated that a convenient reconstruction of the time gradient could be achieved with a time resolution of 2 ns per pixel. To achieve a proper time resolution, it may be necessary to analyze the whole waveform sampled at 1 GHz. In the H.E.S.S. camera, an average charge, time and time spread per pixel was calculated on-board for every triggered event. Sending the whole waveform of the event to the ground implies a data rate per event roughly 5 times larger than that of H.E.S.S. Taking into account the increase in trigger rate, the data rate will increase by an order of magnitude compared to H.E.S.S.\n3. To achieve the aforementionned 2 ns time resolution per pixel, the systematics on the time measurement such as the analogue bandwidth of the read-out system, or the delay between pixels due to the transit time spread of phototubes have to be carefully evaluated and corrected whenever possible.\n\nAs a result of the larger field of view and better timing capabilities of the future Cherenkov cameras, it will be possible to improve the reconstruction of the events and to have a better hadron background rejection. However, these new capabilities present new challenges for Cherenkov telescopes.\nTwo starting points for the NectarCAM are the architecture of the H.E.S.S.-2 camera and a read-out module: the NECTAr module [1]. Figure 1 shows an early version of the NECTAr module. This module converts the light of a set of photomultiplier tubes (PMTs) into a digital signal when some triggering conditions are fulfilled. The modular architecture of the H.E.S.S.-2 camera has the advantage of avoiding costly cables and allowing for easy maintenance. The modular mechanical structure is described in Section 2. For the NectarCAM, modules are groups of seven pixels, with the associated front-end board, which are completely autonomous in what concerns the power and control electronics. The only cables that go to the ground are the input power voltage and optical fibers for network communications and array trigger connections. Preliminary versions of the readout module exist. The photo-sensors, the readout\n\n\n\n\n\n
Figure 2 : Exploded view of the NectarCAM camera mechanics
\n\nand trigger system, the acquisition system, the monitoring and the power supplies are described in Sections 3, 4, 5 and 6 respectively. The tests of a camera demonstrator, composed of 7 modules, and the future plans are presented in Section 7.\n2 Global structure and mechanics\nAll the scientific equipment and internal mechanical pieces are contained within a global structure, called the skeleton, that also provides the rigidity for the structure (Figure 2). The camera sealing is realized by aluminium honeycomb plates, creating the so-called skin, enclosing the camera equipment. The front part of the camera is sealed against rain by motorized lids controlled either remotely or locally. These lids have a secondary function to hold equipment for the electronics calibration (e.g. a Mylar plate for the single photo-electron calibration) and for the pointing calibration (positioning LEDs, reflecting screen). An additional plexiglass plate is placed between the lid and the entrance of the focal plane to avoid dust contamination. The camera will have ∼ 1800 PMTs grouped in ∼ 250 modules. These modules are inserted into a global structure called the sandwich. The rest of the internal equipment (services, communication and electrical interface with the exterior, cables) are held by mechanical fixtures such as small racks, wiring ducts and electrical cabinets. The weight of the camera is less than 2 tons. The total power consumption is of the order of 7.4 kW.\nSince the camera is sealed, proper care has to be taken to avoid overheating of the camera. Air-based cooling systems have been studied (Figure 3), and merging these with a water-cooling system is under study.\n3 Photodetector and detector unit\nThe focal plane of the NectarCAM camera is equipped with detector units . These detector units are composed of a photo-detector, the associated high voltage power supply, and an amplifier. CTA has decided to use PMTs for the\n\n\n
Figure 3 : Simulation of a cooling airflow in the NectarCAM sandwich. The sandwich hosts ∼ 250 NECTAr modules.
\n\n\n\n
Figure 4 : Principle of NECTAr readout modules.
\n\nphotodetectors of their single mirror telescopes. A candidate PMT is the R11920-100 developed by Hamamatsu 2 . The R11920-100 PMT has a single photon electron signal FWHMof 2.5 to 3 ns. The light from the dead spaces between PMTs will be collected by custom designed Winston cones or lenses. The high voltage of the PMTs will be obtained from an external 24-V supply with an ASIC, similar to a chip designed for the KM3Net experiment [2]. The output from the PMTs is amplified by a wideband (450 MHz) 16-bit amplifier called PACTA [4]. The large bandwidth of the PACTA allows a minimal distortion of the PMT signal, and thus a better measurement of the arrival time of detected photons.\n4 Readout\nAs illustrated in Figure 4, NECTAr modules are composed of 3 blocks. The block on the left hand side is composed of 7 PMTs, each associated with its detector unit, as described in Section 3. The central block is the front-end board. Signals from the detector units are amplified again in an ACTA[5] ASIC. At this level, the signal is divided into a low-gain, high-gain (relative gain 16) and a trigger channel. The trigger strategy is described in Section 5. The two first levels\n\n\n\n\n\n
Figure 5 : Single photoelectron spectrum obtained with the NectarCAM V1 module. The PMT signal is amplified by a PACTA and an ACTA amplifier.
\n\nof the trigger (L0 and L1) are temporarily implemented as mezzanines on the read-out board. The outputs of the low-gain and high-gain channels of the ACTA are sent to a NECTAr chip[6]. The NECTAr chip has the dual functionality of switched capacitor array and analog-todigital convertor. The switched capacitor array, which acts as a circular buffer, has a depth of 1024 samples and can be operated between 500 MHz and 3.2 GHz. The NECTAr chip has a bandwidth of more than 400 MHz and a dynamic range of 11.3 bits. The power consumption is 210 mW. The dead time is 2 m s for the readout of 16 cells. The NECTAr chip is thus a low power, cheap alternative to the use of a Flash Analogue to Digital Convert (FADC). Once a L1 trigger signal is emitted, the data from a 1620 ns time interval around the date of trigger are read out. They are retrieved from the NECTAr chip and sent to a FPGA located on the front end board. This FPGA has several functions: interface to the ethernet, control of the NECTAr chip, of the L0, L1 trigger configurations, and the HV power supply. High level quantites such as the integrated charge over the readout window and the arrival time of the signal can be calculated inside the FPGA. It is then possible to send these quantities over the ethernet connection instead of the full event sample. The last block, on the right hand side of Figure 4 is a backplane board. It is used for the clock and synchronisation signal distribution, the low voltage (24 V) power distribution, and the L0/L1 trigger distribution. In some trigger schemes, one of the backplane boards could be used to make the L1 (camera) trigger decision. The readout board has been tested with photon and electrical signals. Its charge response is linear. It allows the measurement of signals between 1 and 3000 photo-electrons. The single photoelectron peak can be measured for calibration purposes (see Figure 5).\n5 Data acquisition and trigger\nThe NectarCAM camera is triggered in a multilevel scheme, shown in Figure 6. The first level trigger (L0) is a modulelevel trigger. The information from several modules is combined to realize a camera-level trigger. The L0 and L1 trigger can be implemented with an 'analogue'[7] or a\n\n\n
Figure 6 : Triggering scheme of the NectarCAM camera.
\n\n'digital' solution. The latency of the camera trigger is less than 400 nanoseconds. This is much less than the depth of the switched capacitor array in the NECTAr chip, so that trigged events can be read back from the past. The trigged events are time-stamped and sent to a camera server by ethernet. Events on the camera server are accepted only if they are coincident with triggers from one or several other telescopes. The typical trigger rate of a single telescope is 5 kHz. The latency of the array trigger is a few m s. Single telescope events can be time stamped with an accuracy of a few ( ∼ 2) nanoseconds. The data acquisition follows the model described in [9]. Before being sent to the camera server, data from NECTAr modules are concentrated in ∼ 7 switches. The switches are linked to the camera server through three 10 Gbit connections. The data rate between a NECTAr module and the switches is 2 Mbit/s, if only the total charge and arrival time of pixels are transferred. It is ∼ 40 Mbit/s if all the samples in the region of interest are transferred.\n6 Slow control and services\nNectarCAM will have sensors to monitor the temperature, pressure and humidity inside the camera. Its safety will be ensured with ambient light sensors, smoke detectors and by tracking the position of the lids and back doors. Most subsystems of the camera are controlled remotely. This is the case for the calibration-related hardware (positionning leds), for the cooling system, for the DAQ crate and the clock distribution board. The monitoring/slow control of the NectarCAM will use either industrial solutions such as Programmable Logic Controllers, or custom made boards with FPGAs. The latter solution, where the sensors are accessed by industrial buses such as I2C, has been used for the H.E.S.S-2 telescope.\nNectarCAM is powered by industrial low voltage supplies, since the High Voltage for operating the PMTs is created in the detector units (Section 3).\n7 Camera demonstrators\nThe performance of the NectarCAM, especially the timing and triggering performances, the cooling and data acquisition will be tested with demonstrators. A first demonstrator with 7 NECTAr modules (Figure 7) is readily available.\n\n\n\n\n\n
Figure 7 : A 7-module demonstrator of the NectarCAM camera.
\n\nIt will enable to test the performance of the analogue and digital triggers. The construction of a second demonstrator with 19 modules should start by the end of the year, with the aim of testing the integration of the components and the industrial processes.\n8 Acknowledgements\nWe gratefully acknowledge support from the agencies and organizations listed in this page http: http://www.ctaobservatory.org/?q=node/22.\nReferences\n\n[1] C. Naumann et al (the NECTAr consortium), Nuclear Instruments and Methods in Physics Research A 695 (2012), 44-51 doi:10.1016/j.nima.2011.11.008\n[2] D. Gajanana, V. Gromov and P. Timmer, Journal of Instrumentation 8 (2013), C02030\ndoi:10.1088/1748-0221/8/02/C02030\n[3] E.Aliu et al (the MAGIC collaboration), Astroparticle Physics 30 (2009), 293-305 doi:10.1016/j.astropartphys.2008.10.003\n[4] A. Sanuy, D. Gascon, J-M. Paredes, L. Garrido, M. Rib and J. Sieiro, Journal of Instrumentation 7 (2012), C01100 doi:10.1088/1748-0221/7/01/C01100\n[5] A. Sanuy, E.Delagnes, D.Gascon, X. Sieiro et al (the NECTAr consortium), Nuclear Instruments and Methods in Physics Research A 695 (2012), 385-389 doi:10.1016/j.nima.2011.12.025\n[6] E.Delagnes et al. (the NECTAr consortium), Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (2011), 1457-1462 doi:10.1109/NSSMIC.2011.6154348\n[7] M.Barcelo et al, these proceedings, ID-0396\n[8] K.H.Sulanke et al, these proceedings, ID-0XXX\n[9] D.Hoffmann and J.Houles, Journal of Physics Conference Series 396 (2012), 2024 doi:10.1088/1742-6596/396/1/012024\n"},"sections":{"kind":"list like","value":[{"title":"The NectarCAM camera project","content":"J-F.GLICENSTEIN 1 , M.BARCELO 11 , J-A. BARRIO 12 , O.BLANCH 11 , J.BOIX 11 , J.BOLMONT 4 , C.BOUTONNET 2 S.CAZAUX 1 , E.CHABANNE 7 , C.CHAMPION 2 , F.CHATEAU 1 , S.COLONGES 2 , P.CORONA 4 , S.COUTURIER 5 , B.COURTY 2 , E.DELAGNES 1 , C.DELGADO 10 , J-P.ERNENWEIN 6 , S.FEGAN 5 , O.FERREIRA 5 , M.FESQUET 1 , G.FONTAINE 5 , N.FOUQUE 7 , F.HENAULT 8 , D.GASC 'ON 13 , D.HERRANZ 12 , R.HERMEL 7 , D.HOFFMANN 6 , J.HOULES 6 , S.KARKAR 4 , B.KHELIFI 5 , J.KN ODLSEDER 3 , G.MARTINEZ 10 , K.LACOMBE 3 , G.LAMANNA 7 , T.LEFLOUR 7 , R.LOPEZCOTO 11 , F.LOUIS 1 , A.MATHIEU 5 , E.MOULIN 1 , P.NAYMAN 4 , F.NUNIO 1 , J-F. OLIVE 3 , J-L. PANAZOL 7 , P-O. PETRUCCI 8 , M.PUNCH 2 , J.PRAST 7 , P.RAMON 3 , M.RIALLOT 1 , M.RIB 'O 13 , S.ROSIER-LEES 7 , A.SANUY 13 , J.SIERO 13 , J-P.TAVERNET 4 , L.A.TEJEDOR 12 , F.TOUSSENEL 455 , G.VASILEIADIS 9 , V.VOISIN 4 , V.WAEGEBERT 3 , C.ZURBACH 9 , FOR THE CTA CONSORTIUM. 1 DSM/IRFU, CEA-Saclay, F91191 Gif-sur-Yvette, France 2 APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France glicens@cea.fr Abstract: In the framework of the next generation of Cherenkov telescopes, the Cherenkov Telescope Array (CTA), NectarCAM is a camera designed for the medium size telescopes covering the central energy range of 100 GeV to 30 TeV. NectarCAM will be finely pixelated ( ∼ 1800 pixels for a 8 o field of view, FoV) in order to image atmospheric Cherenkov showers by measuring the charge deposited within a few nanoseconds time-window. It will have additional features like the capacity to record the full waveform with GHz sampling for every pixel and to measure event times with nanosecond accuracy. An array of a few tens of medium size telescopes, equipped with NectarCAMs, will achieve up to a factor of ten improvement in sensitivity over existing instruments in the energy range of 100 GeV to 10 TeV. The camera is made of roughly 250 independent read-out modules, each composed of seven photo-multipliers, with their associated high voltage base and control, a read-out board and a multi-service backplane board. The read-out boards use NECTAr (New Electronics for the Cherenkov Telescope Array) ASICs which have the dual functionality of analogue memories and Analogue to Digital Converter (ADC). The camera trigger to be used will be flexible so as to minimize the read-out dead-time of the NECTAr chips. We present the camera concept and the design and tests of the various subcomponents. The design includes the mechanical parts, the cooling of the electronics, the readout, the data acquisition, the trigger, the monitoring and services. Keywords: Methods, techniques and instrumentation","pages":[1]},{"title":"1 Introduction","content":"NectarCAM is a new camera design for the medium size telescopes (MST) of the planned CTA 1 array of Imaging Atmospheric Cerenkov telescopes (IACTs). The CTA will have several sizes of single or dual-mirror telescopes. The single-mirror MST has a 12-meter diameter dish. Its camera will face new challenges: in the cameras. The arrival of the photons from a distant shower can last more than 100 nanoseconds. However, the signal to noise ratio in a pixel is optimized by recording and later integrating the signal in a much smaller window ( ∼ 10 ns). If the camera trigger is not flexible enough (e.g is similar to the H.E.S.S. camera trigger), the event energies would be systematically underestimated. Special trigger strategies have been designed to trigger different pixels at different moments (see Figure 16 of [1]). These strategies still need to be tested. The trigger rate is also expected to increase by a factor of two compared to the previous generation of Cherenkov telescopes, as result of the larger field of view, potetially leading to a larger dead time of the instrument. As a result of the larger field of view and better timing capabilities of the future Cherenkov cameras, it will be possible to improve the reconstruction of the events and to have a better hadron background rejection. However, these new capabilities present new challenges for Cherenkov telescopes. Two starting points for the NectarCAM are the architecture of the H.E.S.S.-2 camera and a read-out module: the NECTAr module [1]. Figure 1 shows an early version of the NECTAr module. This module converts the light of a set of photomultiplier tubes (PMTs) into a digital signal when some triggering conditions are fulfilled. The modular architecture of the H.E.S.S.-2 camera has the advantage of avoiding costly cables and allowing for easy maintenance. The modular mechanical structure is described in Section 2. For the NectarCAM, modules are groups of seven pixels, with the associated front-end board, which are completely autonomous in what concerns the power and control electronics. The only cables that go to the ground are the input power voltage and optical fibers for network communications and array trigger connections. Preliminary versions of the readout module exist. The photo-sensors, the readout and trigger system, the acquisition system, the monitoring and the power supplies are described in Sections 3, 4, 5 and 6 respectively. The tests of a camera demonstrator, composed of 7 modules, and the future plans are presented in Section 7.","pages":[1,2]},{"title":"2 Global structure and mechanics","content":"All the scientific equipment and internal mechanical pieces are contained within a global structure, called the skeleton, that also provides the rigidity for the structure (Figure 2). The camera sealing is realized by aluminium honeycomb plates, creating the so-called skin, enclosing the camera equipment. The front part of the camera is sealed against rain by motorized lids controlled either remotely or locally. These lids have a secondary function to hold equipment for the electronics calibration (e.g. a Mylar plate for the single photo-electron calibration) and for the pointing calibration (positioning LEDs, reflecting screen). An additional plexiglass plate is placed between the lid and the entrance of the focal plane to avoid dust contamination. The camera will have ∼ 1800 PMTs grouped in ∼ 250 modules. These modules are inserted into a global structure called the sandwich. The rest of the internal equipment (services, communication and electrical interface with the exterior, cables) are held by mechanical fixtures such as small racks, wiring ducts and electrical cabinets. The weight of the camera is less than 2 tons. The total power consumption is of the order of 7.4 kW. Since the camera is sealed, proper care has to be taken to avoid overheating of the camera. Air-based cooling systems have been studied (Figure 3), and merging these with a water-cooling system is under study.","pages":[2]},{"title":"3 Photodetector and detector unit","content":"The focal plane of the NectarCAM camera is equipped with detector units . These detector units are composed of a photo-detector, the associated high voltage power supply, and an amplifier. CTA has decided to use PMTs for the photodetectors of their single mirror telescopes. A candidate PMT is the R11920-100 developed by Hamamatsu 2 . The R11920-100 PMT has a single photon electron signal FWHMof 2.5 to 3 ns. The light from the dead spaces between PMTs will be collected by custom designed Winston cones or lenses. The high voltage of the PMTs will be obtained from an external 24-V supply with an ASIC, similar to a chip designed for the KM3Net experiment [2]. The output from the PMTs is amplified by a wideband (450 MHz) 16-bit amplifier called PACTA [4]. The large bandwidth of the PACTA allows a minimal distortion of the PMT signal, and thus a better measurement of the arrival time of detected photons.","pages":[2,3]},{"title":"4 Readout","content":"As illustrated in Figure 4, NECTAr modules are composed of 3 blocks. The block on the left hand side is composed of 7 PMTs, each associated with its detector unit, as described in Section 3. The central block is the front-end board. Signals from the detector units are amplified again in an ACTA[5] ASIC. At this level, the signal is divided into a low-gain, high-gain (relative gain 16) and a trigger channel. The trigger strategy is described in Section 5. The two first levels of the trigger (L0 and L1) are temporarily implemented as mezzanines on the read-out board. The outputs of the low-gain and high-gain channels of the ACTA are sent to a NECTAr chip[6]. The NECTAr chip has the dual functionality of switched capacitor array and analog-todigital convertor. The switched capacitor array, which acts as a circular buffer, has a depth of 1024 samples and can be operated between 500 MHz and 3.2 GHz. The NECTAr chip has a bandwidth of more than 400 MHz and a dynamic range of 11.3 bits. The power consumption is 210 mW. The dead time is 2 m s for the readout of 16 cells. The NECTAr chip is thus a low power, cheap alternative to the use of a Flash Analogue to Digital Convert (FADC). Once a L1 trigger signal is emitted, the data from a 1620 ns time interval around the date of trigger are read out. They are retrieved from the NECTAr chip and sent to a FPGA located on the front end board. This FPGA has several functions: interface to the ethernet, control of the NECTAr chip, of the L0, L1 trigger configurations, and the HV power supply. High level quantites such as the integrated charge over the readout window and the arrival time of the signal can be calculated inside the FPGA. It is then possible to send these quantities over the ethernet connection instead of the full event sample. The last block, on the right hand side of Figure 4 is a backplane board. It is used for the clock and synchronisation signal distribution, the low voltage (24 V) power distribution, and the L0/L1 trigger distribution. In some trigger schemes, one of the backplane boards could be used to make the L1 (camera) trigger decision. The readout board has been tested with photon and electrical signals. Its charge response is linear. It allows the measurement of signals between 1 and 3000 photo-electrons. The single photoelectron peak can be measured for calibration purposes (see Figure 5).","pages":[3]},{"title":"5 Data acquisition and trigger","content":"The NectarCAM camera is triggered in a multilevel scheme, shown in Figure 6. The first level trigger (L0) is a modulelevel trigger. The information from several modules is combined to realize a camera-level trigger. The L0 and L1 trigger can be implemented with an 'analogue'[7] or a 'digital' solution. The latency of the camera trigger is less than 400 nanoseconds. This is much less than the depth of the switched capacitor array in the NECTAr chip, so that trigged events can be read back from the past. The trigged events are time-stamped and sent to a camera server by ethernet. Events on the camera server are accepted only if they are coincident with triggers from one or several other telescopes. The typical trigger rate of a single telescope is 5 kHz. The latency of the array trigger is a few m s. Single telescope events can be time stamped with an accuracy of a few ( ∼ 2) nanoseconds. The data acquisition follows the model described in [9]. Before being sent to the camera server, data from NECTAr modules are concentrated in ∼ 7 switches. The switches are linked to the camera server through three 10 Gbit connections. The data rate between a NECTAr module and the switches is 2 Mbit/s, if only the total charge and arrival time of pixels are transferred. It is ∼ 40 Mbit/s if all the samples in the region of interest are transferred.","pages":[3,4]},{"title":"6 Slow control and services","content":"NectarCAM will have sensors to monitor the temperature, pressure and humidity inside the camera. Its safety will be ensured with ambient light sensors, smoke detectors and by tracking the position of the lids and back doors. Most subsystems of the camera are controlled remotely. This is the case for the calibration-related hardware (positionning leds), for the cooling system, for the DAQ crate and the clock distribution board. The monitoring/slow control of the NectarCAM will use either industrial solutions such as Programmable Logic Controllers, or custom made boards with FPGAs. The latter solution, where the sensors are accessed by industrial buses such as I2C, has been used for the H.E.S.S-2 telescope. NectarCAM is powered by industrial low voltage supplies, since the High Voltage for operating the PMTs is created in the detector units (Section 3).","pages":[4]},{"title":"7 Camera demonstrators","content":"The performance of the NectarCAM, especially the timing and triggering performances, the cooling and data acquisition will be tested with demonstrators. A first demonstrator with 7 NECTAr modules (Figure 7) is readily available. It will enable to test the performance of the analogue and digital triggers. The construction of a second demonstrator with 19 modules should start by the end of the year, with the aim of testing the integration of the components and the industrial processes.","pages":[4]},{"title":"8 Acknowledgements","content":"We gratefully acknowledge support from the agencies and organizations listed in this page http: http://www.ctaobservatory.org/?q=node/22.","pages":[4]}],"string":"[\n {\n \"title\": \"The NectarCAM camera project\",\n \"content\": \"J-F.GLICENSTEIN 1 , M.BARCELO 11 , J-A. BARRIO 12 , O.BLANCH 11 , J.BOIX 11 , J.BOLMONT 4 , C.BOUTONNET 2 S.CAZAUX 1 , E.CHABANNE 7 , C.CHAMPION 2 , F.CHATEAU 1 , S.COLONGES 2 , P.CORONA 4 , S.COUTURIER 5 , B.COURTY 2 , E.DELAGNES 1 , C.DELGADO 10 , J-P.ERNENWEIN 6 , S.FEGAN 5 , O.FERREIRA 5 , M.FESQUET 1 , G.FONTAINE 5 , N.FOUQUE 7 , F.HENAULT 8 , D.GASC 'ON 13 , D.HERRANZ 12 , R.HERMEL 7 , D.HOFFMANN 6 , J.HOULES 6 , S.KARKAR 4 , B.KHELIFI 5 , J.KN ODLSEDER 3 , G.MARTINEZ 10 , K.LACOMBE 3 , G.LAMANNA 7 , T.LEFLOUR 7 , R.LOPEZCOTO 11 , F.LOUIS 1 , A.MATHIEU 5 , E.MOULIN 1 , P.NAYMAN 4 , F.NUNIO 1 , J-F. OLIVE 3 , J-L. PANAZOL 7 , P-O. PETRUCCI 8 , M.PUNCH 2 , J.PRAST 7 , P.RAMON 3 , M.RIALLOT 1 , M.RIB 'O 13 , S.ROSIER-LEES 7 , A.SANUY 13 , J.SIERO 13 , J-P.TAVERNET 4 , L.A.TEJEDOR 12 , F.TOUSSENEL 455 , G.VASILEIADIS 9 , V.VOISIN 4 , V.WAEGEBERT 3 , C.ZURBACH 9 , FOR THE CTA CONSORTIUM. 1 DSM/IRFU, CEA-Saclay, F91191 Gif-sur-Yvette, France 2 APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France glicens@cea.fr Abstract: In the framework of the next generation of Cherenkov telescopes, the Cherenkov Telescope Array (CTA), NectarCAM is a camera designed for the medium size telescopes covering the central energy range of 100 GeV to 30 TeV. NectarCAM will be finely pixelated ( ∼ 1800 pixels for a 8 o field of view, FoV) in order to image atmospheric Cherenkov showers by measuring the charge deposited within a few nanoseconds time-window. It will have additional features like the capacity to record the full waveform with GHz sampling for every pixel and to measure event times with nanosecond accuracy. An array of a few tens of medium size telescopes, equipped with NectarCAMs, will achieve up to a factor of ten improvement in sensitivity over existing instruments in the energy range of 100 GeV to 10 TeV. The camera is made of roughly 250 independent read-out modules, each composed of seven photo-multipliers, with their associated high voltage base and control, a read-out board and a multi-service backplane board. The read-out boards use NECTAr (New Electronics for the Cherenkov Telescope Array) ASICs which have the dual functionality of analogue memories and Analogue to Digital Converter (ADC). The camera trigger to be used will be flexible so as to minimize the read-out dead-time of the NECTAr chips. We present the camera concept and the design and tests of the various subcomponents. The design includes the mechanical parts, the cooling of the electronics, the readout, the data acquisition, the trigger, the monitoring and services. Keywords: Methods, techniques and instrumentation\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"NectarCAM is a new camera design for the medium size telescopes (MST) of the planned CTA 1 array of Imaging Atmospheric Cerenkov telescopes (IACTs). The CTA will have several sizes of single or dual-mirror telescopes. The single-mirror MST has a 12-meter diameter dish. Its camera will face new challenges: in the cameras. The arrival of the photons from a distant shower can last more than 100 nanoseconds. However, the signal to noise ratio in a pixel is optimized by recording and later integrating the signal in a much smaller window ( ∼ 10 ns). If the camera trigger is not flexible enough (e.g is similar to the H.E.S.S. camera trigger), the event energies would be systematically underestimated. Special trigger strategies have been designed to trigger different pixels at different moments (see Figure 16 of [1]). These strategies still need to be tested. The trigger rate is also expected to increase by a factor of two compared to the previous generation of Cherenkov telescopes, as result of the larger field of view, potetially leading to a larger dead time of the instrument. As a result of the larger field of view and better timing capabilities of the future Cherenkov cameras, it will be possible to improve the reconstruction of the events and to have a better hadron background rejection. However, these new capabilities present new challenges for Cherenkov telescopes. Two starting points for the NectarCAM are the architecture of the H.E.S.S.-2 camera and a read-out module: the NECTAr module [1]. Figure 1 shows an early version of the NECTAr module. This module converts the light of a set of photomultiplier tubes (PMTs) into a digital signal when some triggering conditions are fulfilled. The modular architecture of the H.E.S.S.-2 camera has the advantage of avoiding costly cables and allowing for easy maintenance. The modular mechanical structure is described in Section 2. For the NectarCAM, modules are groups of seven pixels, with the associated front-end board, which are completely autonomous in what concerns the power and control electronics. The only cables that go to the ground are the input power voltage and optical fibers for network communications and array trigger connections. Preliminary versions of the readout module exist. The photo-sensors, the readout and trigger system, the acquisition system, the monitoring and the power supplies are described in Sections 3, 4, 5 and 6 respectively. The tests of a camera demonstrator, composed of 7 modules, and the future plans are presented in Section 7.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 Global structure and mechanics\",\n \"content\": \"All the scientific equipment and internal mechanical pieces are contained within a global structure, called the skeleton, that also provides the rigidity for the structure (Figure 2). The camera sealing is realized by aluminium honeycomb plates, creating the so-called skin, enclosing the camera equipment. The front part of the camera is sealed against rain by motorized lids controlled either remotely or locally. These lids have a secondary function to hold equipment for the electronics calibration (e.g. a Mylar plate for the single photo-electron calibration) and for the pointing calibration (positioning LEDs, reflecting screen). An additional plexiglass plate is placed between the lid and the entrance of the focal plane to avoid dust contamination. The camera will have ∼ 1800 PMTs grouped in ∼ 250 modules. These modules are inserted into a global structure called the sandwich. The rest of the internal equipment (services, communication and electrical interface with the exterior, cables) are held by mechanical fixtures such as small racks, wiring ducts and electrical cabinets. The weight of the camera is less than 2 tons. The total power consumption is of the order of 7.4 kW. Since the camera is sealed, proper care has to be taken to avoid overheating of the camera. Air-based cooling systems have been studied (Figure 3), and merging these with a water-cooling system is under study.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"3 Photodetector and detector unit\",\n \"content\": \"The focal plane of the NectarCAM camera is equipped with detector units . These detector units are composed of a photo-detector, the associated high voltage power supply, and an amplifier. CTA has decided to use PMTs for the photodetectors of their single mirror telescopes. A candidate PMT is the R11920-100 developed by Hamamatsu 2 . The R11920-100 PMT has a single photon electron signal FWHMof 2.5 to 3 ns. The light from the dead spaces between PMTs will be collected by custom designed Winston cones or lenses. The high voltage of the PMTs will be obtained from an external 24-V supply with an ASIC, similar to a chip designed for the KM3Net experiment [2]. The output from the PMTs is amplified by a wideband (450 MHz) 16-bit amplifier called PACTA [4]. The large bandwidth of the PACTA allows a minimal distortion of the PMT signal, and thus a better measurement of the arrival time of detected photons.\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"4 Readout\",\n \"content\": \"As illustrated in Figure 4, NECTAr modules are composed of 3 blocks. The block on the left hand side is composed of 7 PMTs, each associated with its detector unit, as described in Section 3. The central block is the front-end board. Signals from the detector units are amplified again in an ACTA[5] ASIC. At this level, the signal is divided into a low-gain, high-gain (relative gain 16) and a trigger channel. The trigger strategy is described in Section 5. The two first levels of the trigger (L0 and L1) are temporarily implemented as mezzanines on the read-out board. The outputs of the low-gain and high-gain channels of the ACTA are sent to a NECTAr chip[6]. The NECTAr chip has the dual functionality of switched capacitor array and analog-todigital convertor. The switched capacitor array, which acts as a circular buffer, has a depth of 1024 samples and can be operated between 500 MHz and 3.2 GHz. The NECTAr chip has a bandwidth of more than 400 MHz and a dynamic range of 11.3 bits. The power consumption is 210 mW. The dead time is 2 m s for the readout of 16 cells. The NECTAr chip is thus a low power, cheap alternative to the use of a Flash Analogue to Digital Convert (FADC). Once a L1 trigger signal is emitted, the data from a 1620 ns time interval around the date of trigger are read out. They are retrieved from the NECTAr chip and sent to a FPGA located on the front end board. This FPGA has several functions: interface to the ethernet, control of the NECTAr chip, of the L0, L1 trigger configurations, and the HV power supply. High level quantites such as the integrated charge over the readout window and the arrival time of the signal can be calculated inside the FPGA. It is then possible to send these quantities over the ethernet connection instead of the full event sample. The last block, on the right hand side of Figure 4 is a backplane board. It is used for the clock and synchronisation signal distribution, the low voltage (24 V) power distribution, and the L0/L1 trigger distribution. In some trigger schemes, one of the backplane boards could be used to make the L1 (camera) trigger decision. The readout board has been tested with photon and electrical signals. Its charge response is linear. It allows the measurement of signals between 1 and 3000 photo-electrons. The single photoelectron peak can be measured for calibration purposes (see Figure 5).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"5 Data acquisition and trigger\",\n \"content\": \"The NectarCAM camera is triggered in a multilevel scheme, shown in Figure 6. The first level trigger (L0) is a modulelevel trigger. The information from several modules is combined to realize a camera-level trigger. The L0 and L1 trigger can be implemented with an 'analogue'[7] or a 'digital' solution. The latency of the camera trigger is less than 400 nanoseconds. This is much less than the depth of the switched capacitor array in the NECTAr chip, so that trigged events can be read back from the past. The trigged events are time-stamped and sent to a camera server by ethernet. Events on the camera server are accepted only if they are coincident with triggers from one or several other telescopes. The typical trigger rate of a single telescope is 5 kHz. The latency of the array trigger is a few m s. Single telescope events can be time stamped with an accuracy of a few ( ∼ 2) nanoseconds. The data acquisition follows the model described in [9]. Before being sent to the camera server, data from NECTAr modules are concentrated in ∼ 7 switches. The switches are linked to the camera server through three 10 Gbit connections. The data rate between a NECTAr module and the switches is 2 Mbit/s, if only the total charge and arrival time of pixels are transferred. It is ∼ 40 Mbit/s if all the samples in the region of interest are transferred.\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"6 Slow control and services\",\n \"content\": \"NectarCAM will have sensors to monitor the temperature, pressure and humidity inside the camera. Its safety will be ensured with ambient light sensors, smoke detectors and by tracking the position of the lids and back doors. Most subsystems of the camera are controlled remotely. This is the case for the calibration-related hardware (positionning leds), for the cooling system, for the DAQ crate and the clock distribution board. The monitoring/slow control of the NectarCAM will use either industrial solutions such as Programmable Logic Controllers, or custom made boards with FPGAs. The latter solution, where the sensors are accessed by industrial buses such as I2C, has been used for the H.E.S.S-2 telescope. NectarCAM is powered by industrial low voltage supplies, since the High Voltage for operating the PMTs is created in the detector units (Section 3).\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"7 Camera demonstrators\",\n \"content\": \"The performance of the NectarCAM, especially the timing and triggering performances, the cooling and data acquisition will be tested with demonstrators. A first demonstrator with 7 NECTAr modules (Figure 7) is readily available. It will enable to test the performance of the analogue and digital triggers. The construction of a second demonstrator with 19 modules should start by the end of the year, with the aim of testing the integration of the components and the industrial processes.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"8 Acknowledgements\",\n \"content\": \"We gratefully acknowledge support from the agencies and organizations listed in this page http: http://www.ctaobservatory.org/?q=node/22.\",\n \"pages\": [\n 4\n ]\n }\n]"}}},{"rowIdx":289,"cells":{"bibcode":{"kind":"string","value":"2013IJAA....3....1N"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1207.6846.pdf"},"content":{"kind":"string","value":"\nWave optics and image formation in gravitational lensing\nYasusada Nambu\nDepartment of Physics, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan\nE-mail:\nnambu@gravity.phys.nagoya-u.ac.jp\nAbstract. We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing.\nPACS numbers: 04.20.-q, 42.25.Fx\nwave optics; image formation; gravitational lens\n1. Introduction\nGravitational lensing is one of the prediction of Einstein's general theory of relativity and many samples of images caused by gravitational lensing have been obtained observationally [1]. Light rays obey null geodesics in curved spacetime and they are deflected by gravitational potential of lens objects. In weak gravitational field with thin lens approximation, a path of a light ray obeys so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing effect are carried out based on this equation. Especially, we can obtain images of source objects by solving the lens equation using a ray tracing method. As a path of light ray is derived as the high frequency limit of electromagnetic wave, wave effects of gravitational lensing become important when the wavelength is not so much smaller than the size of lens objects and in such a situation, we must take into account of wave effects. For example, when we consider gravitational wave is scattered by gravitational lens objects, the wave effect gives significant impact on the amplification factor of intensity for waves [2, 3, 4]. Another example is direct detection of black holes via imaging their shadows [5, 6]. The apparent angular size of black hole shadows are so small that their detectability depends on angular resolution of telescopes which is determined by diffraction limit of image formation system. Thus it is important to investigate wave effects on images for successful detection of black hole shadows.\nAlthough interference and diffraction of waves by gravitational lensing has been discussed in connection with amplification of waves, a little was discussed about how images by gravitational lensing are obtained based on wave optics. For electromagnetic wave, E. Herlt and H. Stephani [7] discussed the position of images by a spherical gravitational lens evaluating the Poynting flux of scattered wave at an observer. They claimed that there is a disagreement between wave optics and geometrical optics concerning the position of double images of a point source. But they have not presented complete understanding of image formation. In this paper, we consider image formation in gravitational lensing using wave optics and aim to understand how images by gravitational lensing are obtained in terms of waves. For this purpose, we adopt the diffraction theory of image formation in wave optics [8], which explains image formation in optical systems in terms of diffraction of waves.\n2. Wave optics in gravitational lensing\nWe review the basic formalism of gravitational lensing based on wave optics [1]. In this paper, we does not consider polarization of waves and treat scalar waves as a model for electromagnetic waves.\n\n\n
Figure 1. The gravitational lens geometry of the source, the lens and the observer. α /lessmuch 1 is the deflection angle. r L is the distance from the observer to the lens object and r S is the distance from the observer to the source. r LS = r S -r L .
\n\nLet us consider waves propagating under the influence of the gravitational potential of a lens object. The background metric is assumed to be\nds 2 = g µν dx µ dx ν = -(1 + 2 U ( r )) dt 2 +(1 -2 U ( r )) d r 2 , (1)\nwhere U ( r ) is the gravitational potential of the lens object with the condition | U | /lessmuch 1. The scalar wave propagation in this curved spacetime is described by the following wave equation:\nand for a monochromatic wave with the angular frequency ω ,\n∂ µ ( √ -gg µν ∂ ν Φ ) = 0 , (2)\n( ∇ 2 + ω 2 )Φ = 4 ω 2 U ( r )Φ , (3)\nwhere ∇ 2 is the flat space Laplacian.\nWe show the configuration of the gravitational lensing system considering here (Figure 1). The wave is emitted by a point source, scattered by the gravitational potential of the lens object and reaches the observer. We assume the wave scattering occurs in a small spatial region around the lens object and outside of this region, the wave propagates in a flat space. With the assumptions of the eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula provides the following amplitude of the wave at the observer [1, 3]\nΦ( η , ∆ ) = ωa 0 2 πir LS r L ∫ d 2 ξ exp [ iωS ( η , ξ , ∆ )] (4)\nwhere S ( η , ξ , ∆ ) is the effective path length (eikonal) along a path from the source position η to the observer position ∆ via a point ξ on the lens plane\nS ( η , ξ , ∆ ) = [ ( ξ -η ) 2 + r 2 LS ] 1 / 2 + [ ( ξ -∆ ) 2 + r 2 L ] 1 / 2 -ˆ ψ ( ξ ) ≈ ( η -∆ ) 2 2 r S + r S + r L r S 2 r LS ( ξ -∆ r L -η -∆ r S ) 2 -ˆ ψ ( ξ ) (5)\nand we assume that | η -∆ | /lessmuch r S and | ξ -∆ | /lessmuch r L . A constant a 0 represents the intensity of a point source. The two dimensional gravitational potential is introduced by\nˆ ψ ( ξ ) = 2 ∫ ∞ -∞ dzU ( ξ , z ) . (6)\nThen the wave amplitude at the observer can be written as [1, 2]\nΦ( η , ∆ ) = Φ 0 ( η , ∆ ) F ( η , ∆ ) (7)\nwhere Φ 0 is the wave amplitude at the observer in the absence of the gravitational potential U :\nΦ 0 ( η , ∆ ) = a 0 r S exp [ iωS 0 ( η , ∆ )] , S 0 ( η , ∆ ) = ( η -∆ ) 2 2 r S + r S . (8)\nS 0 ( η , ∆ ) is the path length along a straight path from η to ∆ . The amplification factor F is given by the following form of a diffraction integral\nF ( η , ∆ ) = r S r L r LS ω 2 πi ∫ d 2 ξ exp [ iωS 1 ( η , ξ , ∆ )] , (9)\nwhere S 1 ( η , ξ , ∆ ) is the Fermat potential along a path from the source position η to the observer position ∆ via a point ξ on the lens plane. The first term in S 1 is the difference of the geometric time delay between a straight path from the source to the observer and a deflected path. The second term is the time delay due to the gravitational potential of the lens object. Now we introduce the following dimensionless variables:\nS 1 ( η , ξ , ∆ ) = r L r S 2 r LS ( ξ -∆ r L -η -∆ r S ) 2 -ˆ ψ ( ξ ) (10)\nx = ξ ξ 0 , y = r L r S η ξ 0 , d = ( 1 -r L r S ) ∆ ξ 0 , w = r S ξ 2 0 r LS r L ω, ψ = r L r LS r S ξ 2 0 ˆ ψ (11)\nwhere we choose ξ 0 as\nξ 0 = r L θ E , θ E = √ 4 Mr LS r L r S . (12)\nM is the mass of the gravitational source, ξ 0 and θ E represent the Einstein radius and the Einstein angle, respectively. Using these dimensionless variables,\nF ( y , d ) = w 2 πi ∫ d 2 x exp [ iw ( 1 2 ( x -y -d ) 2 -ψ ( x ) )] , (13)\nΦ 0 ( y , d ) = a 0 r S exp [ iw ( r LS 2 r L ( y -r L r LS d ) 2 + r LS r L ξ 2 0 )] . (14)\nIn the geometrical optics limit w /greatermuch 1, the diffraction integral (13) can be evaluated around the stationary points of the phase function in the integrand. The stationary points are determined by the solution of the following equation:\nx -y -d -∇ x ψ ( x ) = 0 . (15)\nThis is the lens equation for gravitational lensing and determines the location of the image x for given source position y . As the specific model of gravitational lensing,\nwe consider a point mass as a gravitational source. In this case, the two dimensional gravitational potential is\nψ ( x ) = ln | x | (16)\nand the deflection angle is given by ‡\nα = |∇ ξ ˆ ψ | = 4 M ξ . (17)\nFor y = d = 0, the solution of the lens equation (15) is\n| x | = 1 (18)\nand represents the Einstein ring with the apparent angular radius θ E defined by (12). We show an example of images obtained as solutions of the lens equation (15) (Figure 2). To produce these images, we have assumed an extended source with Gaussian distribution of intensity.\n\n\n\n\n\n\n\n\n\n\n\n
Figure 2. Images of gravitational lensing by a point mass. The source is assumed to have the intensity with Gaussian distribution. From the left to the right panels, the source positions are y = 0 . 0 , 0 . 5 , 1 . 0 , 1 . 5.
\n\nThe wave property is obtained by evaluating the diffraction integral (13). For a point mass lens potential (16), the integral can be obtained exactly\nF ( y ) = e ( i/ 2) w ( | y | 2 +ln( w/ 2)) e π/ 4 Γ ( 1 -i 2 w ) 1 F 1 ( 1 -i 2 w, 1 , -i 2 w | y | 2 ) . (19)\nOn the observer plane, an interference pattern appears (Figure 3).\n\n\n
Figure 3. Amplification factor for w = 10.
\n\n‡ Using the original variables, the lens equation is ξ -∆ r L -η -∆ r S -r LS r S ∇ ξ ˆ ψ = 0 .\nNear y = 0, the distance between adjacent fringes of the interference pattern is\n∆ y ∼ √ 2 π 4 Mω . (20)\nThis fringe pattern is interpreted as interference between double images of a point source by the gravitational lensing. The question we aim to raise in this paper is how the interference pattern on the observer plane is related to the images of gravitational lensing in the geometrical optics limit. The wave amplitude on the observer plane does not make the image of the source and we have to transform the wave function to extract images. To answer this question, we introduce a 'telescope' in the gravitational lensing system and simulate observation of a star (a point source) using the telescope. With this setup, it is possible to understand how images of a source are formed in the framework of wave optics.\n3. Image formation in wave optics\nTo establish relation between the interference pattern of the wave and the images of the source in the gravitational lensing system, we first consider an image formation system composed of a single convex lens and review how images of source objects appear in the framework of wave optics [8].\n3.1. Image formation by a convex lens\nLet us Φ in ( x ) is the incident wave from a point source in front of a thin convex lens and Φ t ( x ) is the transmitted wave by the lens (Figure 4). They are connected by the following relation\nΦ t ( x ) = T ( x )Φ in ( x ) , T ( x ) = e -iω | x | 2 2 f (21)\nwhere T ( x ) is called a lens transformation function. The action of a convex lens is to modify the phase of the incident wave. For a point source placed at z = -f (front focal point), the incident wave and the transmitted wave are\nΦ in ( x ) = e iωr r ≈ e iω ( f + | x | 2 2 f ) f , Φ t ( x ) = T Φ in = e iωf ,\nwhere we have used r = √ f 2 + | x | 2 ≈ f + | x | 2 / (2 f ) assuming | x | /lessmuch f . Thus, a convex lens converts a spherical wave front to a plane wave front.\n\n\n
Figure 4. Wave front modification by a convex lens.
\n\nUsing this action of a convex lens for the incident wave and the transmitted wave, we can demonstrate the image formation by a convex lens in the framework of wave optics. Let us consider the configuration of the lens system shown in Figure 5.\n\n\n
Figure 5. One lens image formation system.
\n\nWe assume the distribution of the source field on the object plane z = -a as Φ 0 ( x 0 ). Using the Fresnel-Kirchhoff diffraction formula, the amplitude of the wave in front of the lens is given by\nΦ 1 ( x ) ∝ ∫ d 2 x 0 Φ 0 ( x 0 ) e iωr 1 ≈ ∫ d 2 x 0 Φ 0 ( x 0 ) e iω ( a + | x 1 -x 0 | 2 2 a )\nwhere r 1 is the path length from a point on the object plane to a point on the lens plane and we have assumed | x 1 -x 0 | /lessmuch a . The amplitude of the wave just behind the lens is given by the relation (21)\nΦ 1 ' ( x 1 ) = t L ( x 1 ) e -iω | x 1 | 2 2 f Φ 1 ( x 1 )\nwhere t L is the aperture function of the lens defined by t L ( x ) = 1 for 0 ≤ | x | ≤ D and t L ( x ) = 0 for D < | x | . D represents a radius of the lens. With the assumption | x 1 -x | /lessmuch b , the amplitude of the wave on the z = b plane behind the lens is\nΦ 2 ( x ) ∝ ∫ d 2 x 1 Φ 1 ' ( x 1 ) e iωr 2\n∝ ∫ d 2 x 0 d 2 x 1 Φ 0 ( x 0 ) t L ( x 1 ) e i ω 2 a | x 1 -x 0 | 2 e -i ω 2 f | x 1 | 2 e i ω 2 b | x 1 -x | 2 = ∫ d 2 x 0 d 2 x 1 Φ 0 ( x 0 ) t L ( x 1 ) exp [ iω { 1 2 ( 1 a + 1 b -1 f ) | x 1 | 2 -( x 0 a + x b ) · x 1 }] × exp [ i ω 2 ( | x 0 | 2 a + | x | 2 b )] . (22)\nFor a value of b satisfying the following relation (the lens equation for a convex thin lens),\n1 a + 1 b = 1 f , (23)\nthe wave amplitude becomes\nΦ 2 ( x ) ∝ ∫ d 2 x 0 d 2 x 1 Φ 0 ( x 0 ) t L ( x 1 ) exp [ -iω ( x 0 a + x b ) · x 1 ] exp [ iω | x 0 | 2 2 a ] ∝ ∫ d 2 x 0 Φ 0 ( x 0 ) ( 2 J 1 ( ωD | x /b + x 0 /a | ) ωD | x /b + x 0 /a | ) exp [ iω | x 0 | 2 2 a ] . (24)\nFor ωD → ∞ limit, the Bessel function in (24) becomes the delta function and we obtains the following wave amplitude on z = b :\nΦ 2 ( x ) ∝ ∫ d 2 x 0 Φ 0 ( x 0 ) × δ 2 [ x 0 a + x b ] = Φ 0 ( -a b x ) . (25)\nThus, a magnified image of the source field appears on the z = b plane. This reproduces the result of image formation in geometric optics; we have shown that an inverted images with magnification b/a of a source object appears on z = b satisfying the lens equation (23).\nIf we do not take ωD →∞ limit, due to the diffraction effect, an image of a point source has finite size on the image plane called the Airy disk [8]. Its size is given by\n∆ x Airy ∼ bλ D , λ = 2 π ω . (26)\nThis value determines the resolving power of image formation system. For two point sources at x 0 = -d / 2 , d / 2, their separation on the image plane is bd/a . To resolve them, their separation must be larger than the size of the Airy disk:\nd a > λ D ≡ θ 0 . (27)\nThe lefthand side of this inequality is the angular separation of the sources and θ 0 determines the resolving power of the image formation system.\n3.2. Image formation in gravitational lens system\nAs we have observed that a convex lens can be a device for image formation in wave optics, we combine it with a gravitational lensing system and obtain images by gravitational lensing. We consider a configuration of the gravitational lens system shown in Figure 6 and examine how the images of the source object appear using wave optics.\n\n\n
Figure 6. Configuration of a gravitational lens with a convex lens system. Thin orange lines represent paths that contribute to diffraction integrals.
\n\nAs the source object, we assume a point source of wave. The amplitude of the wave just in front of a convex lens is\nΦ L ( y , d ' ) = Φ 0 ( y , d ' ) F ( y , d ' ) . (28)\nThis equation is the same as (7). After passing through the convex lens, the wave amplitude on the image plane z 2 is given by\nΦ I ( η , ∆ ) = ∫ | ∆ ' |≤ ∆ 0 d 2 ∆ ' Φ L ( η , ∆ ' ) exp [ -iω 2 f ∆ ' 2 ] exp [ iω 2 z 2 ( ∆ -∆ ' ) 2 ] (29)\nwhere ∆ 0 denotes the aperture of the convex lens. Using dimensionless variables, the wave amplitude on the image plane is\nΦ I ( y , d ) = a 0 r S ∫ | d ' |≤ d 0 d 2 d ' F ( y + d ' ) (30) × exp [ iw { r L r S 2 r LS ( 1 r S + 1 z 2 -1 f ) d ' 2 -( y + r L r S r LS z 2 d ) · d ' + 1 2 ( r LS r L y 2 + r L r S r LS z 2 d 2 )}] .\nIf we choose the location of the image plane z 2 to satisfy the following 'lens equation' for a convex lens,\n1 r S + 1 z 2 = 1 f ,\nthen the wave amplitude on the image plane becomes\nΦ I ( y , d ) = a 0 r S ∫ | d ' |≤ d 0 d 2 d ' F ( y + d ' ) exp [ -iw ( y + r L r S r LS f d ) · d ' ] × e i ( w/ 2) g ( y , d ) (31)\nwhere g = ( r LS /r L ) y 2 +( r L r S /r LS z 2 ) d 2 . Thus the wave amplitude on the image plane is the Fourier transform of the amplification factor F which gives the interference fringe pattern. Under the geometrical optics limit w /greatermuch 1, x integral in the amplification factor (13) can be approximated by the WKB form\nF ( y + d ' ) ≈ Ae iw [ 1 2 ( x ∗ -y -d ' ) 2 -ψ ( x ∗ ) ] ,\nwhere x ∗ ( y , d ' ) is the solution of the lens equation\n0 = x -y -d ' -∇ x ψ ( x ) ≈ x -y -∇ x ψ ( x ) . (32)\nWe have assumed that the aperture of the convex lens is sufficiently smaller than the size of the gravitational lensing system and | d ' | ≤ d 0 /lessmuch 1 holds. Then, the wave amplitude on the image plane is\nΦ I ( y , d ) ∝ Ae iw [ 1 2 ( x ∗ -y ) 2 -ψ ( x ∗ ) ] e iwg ( y , d ) / 2 ∫ | d ' |≤ d 0 d 2 d ' exp [ -iw ( x ∗ + r L r S r LS f d ) · d ' ] = Ae iw [ 1 2 ( x ∗ -y ) 2 -ψ ( x ∗ ) ] e iwg ( y , d ) / 2 × 2 πd 2 0 J 1 ( w | x ∗ + β d | d 0 ) w | x ∗ + β d | d 0 , β ≡ r L r S r LS f . (33)\nΦ I ( y , d ) ∝ δ 2 [ x ∗ ( y ) + r L r S r LS f d ] (34)\nFor wd 0 →∞ limit (large lens aperture limit or high frequency limit), we obtains\nand the image of the point source appears at the following location on the image plane determined by the lens equation (32):\nd = -r LS f r L r S × x ∗ ( y ) . (35)\n(34) and (35) reproduce the same result of image formation in the geometrical optics (ray tracing) in terms of the wave optics. This is what we aim to clarify in this paper. If the lens equation (32) has multiple solutions x ( j ) ∗ , j = 1 , 2 , · · · , the wave amplitude on the image plane becomes\nΦ I ( y , d ) ∝ ∑ j A j 2 J 1 ( w | x ( j ) ∗ + β d | d 0 ) w | x ( j ) ∗ + β d | d 0 (36)\nwhere A j are constants.\nAs an example of image formation in a gravitational lensing system using wave optics, we present the wave optical images of a point source by the gravitational lensing of a point mass (Figure 7). They are obtained by Fourier transformation of the amplification factor F (equation (31)) and teh lens equation for gravitational lensing (15) has not been used. This procedure corresponds to image formation by a convex lens. These images correspond to images obtained by geometric optics (Figure 2). We can observe wave effect in these images.\n\n\n\n\n\n\n\n\n\n\n\n
Figure 7. Wave optical images of a point source by the gravitational lensing of a point mass. Parameters are w = 40 , d 0 = 0 . 5(aperture of a convex lens), y = 0 , 0 . 5 , 1 , 1 . 5.
\n\nIn each images, we can observe concentric interference pattern which is caused by finite size of the lens aperture and this is not intrinsic feature of the gravitational lensing system. We can also observe radial non-concentric patterns. They are caused by interference between double images and represent the intrinsic feature of the gravitational lensing system. For y = 0 case which corresponds to the Einstein ring in the geometrical optics limit, we can observe a bright spot at the center of the ring, which is the result of constructive interference and does not appear in geometric optics. For sufficiently large values of wd 0 , the wave amplitude at the observer coincides with the result obtained by geometric optics.\n4. Summary\nWe investigated image formation in gravitational lensing system based on wave optics. Instead of using a ray tracing method, we obtained images directly from wave functions at the observer without using a lens equation of gravitational lensing. For this purpose, we introduced a 'telescope' with a single convex thin lens, which acts as a Fourier transformer for the interference pattern formed at an observer. The analysis in this paper relates the wave amplitude and images of the gravitational lensing directly. In the geometric optics limit of waves, images by lens systems are obtained by a lens equation which determines paths of each light rays. As light rays are trajectories of massless test particles (photon), expressing image in terms of wave is to express particle motion in terms of waves.\nAs an application and extension of analysis presented in this paper, we plan to investigate gravitational lensing by a black hole and obtain wave optical images of black holes. This subject is related to observation of black hole shadows [5, 6]. As the apparent angular sizes of black hole shadows are so small, the diffraction effect on images are crucial to resolve black hole shadows in observation using radio interferometer. For SgrA ∗ , which is the black hole candidate at Galactic center, the apparent angular size of its shadow is estimated to be ∼ 30 µ arc seconds and this value is the largest among black hole candidates. For a sub-mm VLBI with a baseline length D , using (27), the condition to resolve the shadow becomes\nD > 1000 km ,\nand this requirement shows the possibility to detect the black hole shadow of SgrA ∗ using the present day technology of VLBI telescope. Thus, analysis of black hole shadows based on wave optics is an important task to evaluate detectability of shadows and determination of black hole parameters via imaging of black holes.\nThe topic of wave optical image formation in black hole spacetimes belongs to a classical problem of wave scattering in black hole spacetimes [9]. As is well known, waves incident to a rotating black hole are amplified by the superradiance [10] due to dragging of spacetimes. This effect enables waves to extract the rotation energy of black holes. On the other hand, it is known that particles can also extract the rotation energy of\nblack holes via so called Penrose process. By investigating images of scattered waves by a rotating black hole, we expect to find out new aspect or interpretation of phenomena associated with superradiance in connection with the Penrose process.\nAcknowledgments\nThis work was supported in part by the JSPS Grant-In-Aid for Scientific Research (C) (23540297). The author thanks all member of 'black hole horizon project meeting' in which the preliminary version of this paper was presented.\nReferences\n\n[1] Schneider P, Ehlers J and Falco E E Gravitational Lenses (Springer-Verlag, 1992)\n[2] Nakamura T T and Deguchi S 1999 Wave Optics in Gravitational Lensing Prog. Theor. Phys. Suppl. 133 137-153\n[3] Baraldo C and Hosoya A 1999 Gravitationally induced interference of gravitational waves by a rotating massive object Phys. Rev. D 59 083001\n[4] Matsunaga N and Yamamoto K 2006 The finite source size effect and wave optics in gravitational lensing JCAP 01 023\n[5] Falcke H, Melia F and Agol E 2000 Viewing the shadow of the black hole at the galactic center Astrophys. J. 528 L13-L16\n[6] Miyoshi M, Ishituska K, Kameno S and Shen Z Q 2004 Direct imaging of the black hole, SgrA ∗ Prog. Theor. Phys. Suppl. 155 186-189\n[7] Herlt E and Stephani H 1976 Wave Optics of the Spherical Gravitational Lens Part I: Diffraction of a Plane Electromagnetic Wave by a Large Star Int. J. of Theor. Phys. 15 45-65\n[8] Sharma K K Optics: principles and applications (Academic Press, 2006)\n[9] Futterman J A H, Handler F A and Matzner R A Scattering from black holes (Cambridge Univ. Press, 1988)\n[10] Frolov V P and Novikov I D Black Hole Physics (Kluwer Academic Publishers, 1998)\n"},"sections":{"kind":"list like","value":[{"title":"Yasusada Nambu","content":"Department of Physics, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan E-mail: nambu@gravity.phys.nagoya-u.ac.jp Abstract. We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing. PACS numbers: 04.20.-q, 42.25.Fx wave optics; image formation; gravitational lens","pages":[1]},{"title":"1. Introduction","content":"Gravitational lensing is one of the prediction of Einstein's general theory of relativity and many samples of images caused by gravitational lensing have been obtained observationally [1]. Light rays obey null geodesics in curved spacetime and they are deflected by gravitational potential of lens objects. In weak gravitational field with thin lens approximation, a path of a light ray obeys so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing effect are carried out based on this equation. Especially, we can obtain images of source objects by solving the lens equation using a ray tracing method. As a path of light ray is derived as the high frequency limit of electromagnetic wave, wave effects of gravitational lensing become important when the wavelength is not so much smaller than the size of lens objects and in such a situation, we must take into account of wave effects. For example, when we consider gravitational wave is scattered by gravitational lens objects, the wave effect gives significant impact on the amplification factor of intensity for waves [2, 3, 4]. Another example is direct detection of black holes via imaging their shadows [5, 6]. The apparent angular size of black hole shadows are so small that their detectability depends on angular resolution of telescopes which is determined by diffraction limit of image formation system. Thus it is important to investigate wave effects on images for successful detection of black hole shadows. Although interference and diffraction of waves by gravitational lensing has been discussed in connection with amplification of waves, a little was discussed about how images by gravitational lensing are obtained based on wave optics. For electromagnetic wave, E. Herlt and H. Stephani [7] discussed the position of images by a spherical gravitational lens evaluating the Poynting flux of scattered wave at an observer. They claimed that there is a disagreement between wave optics and geometrical optics concerning the position of double images of a point source. But they have not presented complete understanding of image formation. In this paper, we consider image formation in gravitational lensing using wave optics and aim to understand how images by gravitational lensing are obtained in terms of waves. For this purpose, we adopt the diffraction theory of image formation in wave optics [8], which explains image formation in optical systems in terms of diffraction of waves.","pages":[2]},{"title":"2. Wave optics in gravitational lensing","content":"We review the basic formalism of gravitational lensing based on wave optics [1]. In this paper, we does not consider polarization of waves and treat scalar waves as a model for electromagnetic waves. Let us consider waves propagating under the influence of the gravitational potential of a lens object. The background metric is assumed to be where U ( r ) is the gravitational potential of the lens object with the condition | U | /lessmuch 1. The scalar wave propagation in this curved spacetime is described by the following wave equation: and for a monochromatic wave with the angular frequency ω , where ∇ 2 is the flat space Laplacian. We show the configuration of the gravitational lensing system considering here (Figure 1). The wave is emitted by a point source, scattered by the gravitational potential of the lens object and reaches the observer. We assume the wave scattering occurs in a small spatial region around the lens object and outside of this region, the wave propagates in a flat space. With the assumptions of the eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula provides the following amplitude of the wave at the observer [1, 3] where S ( η , ξ , ∆ ) is the effective path length (eikonal) along a path from the source position η to the observer position ∆ via a point ξ on the lens plane and we assume that | η -∆ | /lessmuch r S and | ξ -∆ | /lessmuch r L . A constant a 0 represents the intensity of a point source. The two dimensional gravitational potential is introduced by Then the wave amplitude at the observer can be written as [1, 2] where Φ 0 is the wave amplitude at the observer in the absence of the gravitational potential U : S 0 ( η , ∆ ) is the path length along a straight path from η to ∆ . The amplification factor F is given by the following form of a diffraction integral where S 1 ( η , ξ , ∆ ) is the Fermat potential along a path from the source position η to the observer position ∆ via a point ξ on the lens plane. The first term in S 1 is the difference of the geometric time delay between a straight path from the source to the observer and a deflected path. The second term is the time delay due to the gravitational potential of the lens object. Now we introduce the following dimensionless variables: where we choose ξ 0 as M is the mass of the gravitational source, ξ 0 and θ E represent the Einstein radius and the Einstein angle, respectively. Using these dimensionless variables, In the geometrical optics limit w /greatermuch 1, the diffraction integral (13) can be evaluated around the stationary points of the phase function in the integrand. The stationary points are determined by the solution of the following equation: This is the lens equation for gravitational lensing and determines the location of the image x for given source position y . As the specific model of gravitational lensing, we consider a point mass as a gravitational source. In this case, the two dimensional gravitational potential is and the deflection angle is given by ‡ For y = d = 0, the solution of the lens equation (15) is and represents the Einstein ring with the apparent angular radius θ E defined by (12). We show an example of images obtained as solutions of the lens equation (15) (Figure 2). To produce these images, we have assumed an extended source with Gaussian distribution of intensity. The wave property is obtained by evaluating the diffraction integral (13). For a point mass lens potential (16), the integral can be obtained exactly On the observer plane, an interference pattern appears (Figure 3). ‡ Using the original variables, the lens equation is ξ -∆ r L -η -∆ r S -r LS r S ∇ ξ ˆ ψ = 0 . Near y = 0, the distance between adjacent fringes of the interference pattern is This fringe pattern is interpreted as interference between double images of a point source by the gravitational lensing. The question we aim to raise in this paper is how the interference pattern on the observer plane is related to the images of gravitational lensing in the geometrical optics limit. The wave amplitude on the observer plane does not make the image of the source and we have to transform the wave function to extract images. To answer this question, we introduce a 'telescope' in the gravitational lensing system and simulate observation of a star (a point source) using the telescope. With this setup, it is possible to understand how images of a source are formed in the framework of wave optics.","pages":[2,3,4,5,6]},{"title":"3. Image formation in wave optics","content":"To establish relation between the interference pattern of the wave and the images of the source in the gravitational lensing system, we first consider an image formation system composed of a single convex lens and review how images of source objects appear in the framework of wave optics [8].","pages":[6]},{"title":"3.1. Image formation by a convex lens","content":"Let us Φ in ( x ) is the incident wave from a point source in front of a thin convex lens and Φ t ( x ) is the transmitted wave by the lens (Figure 4). They are connected by the following relation where T ( x ) is called a lens transformation function. The action of a convex lens is to modify the phase of the incident wave. For a point source placed at z = -f (front focal point), the incident wave and the transmitted wave are where we have used r = √ f 2 + | x | 2 ≈ f + | x | 2 / (2 f ) assuming | x | /lessmuch f . Thus, a convex lens converts a spherical wave front to a plane wave front. Using this action of a convex lens for the incident wave and the transmitted wave, we can demonstrate the image formation by a convex lens in the framework of wave optics. Let us consider the configuration of the lens system shown in Figure 5. We assume the distribution of the source field on the object plane z = -a as Φ 0 ( x 0 ). Using the Fresnel-Kirchhoff diffraction formula, the amplitude of the wave in front of the lens is given by where r 1 is the path length from a point on the object plane to a point on the lens plane and we have assumed | x 1 -x 0 | /lessmuch a . The amplitude of the wave just behind the lens is given by the relation (21) where t L is the aperture function of the lens defined by t L ( x ) = 1 for 0 ≤ | x | ≤ D and t L ( x ) = 0 for D < | x | . D represents a radius of the lens. With the assumption | x 1 -x | /lessmuch b , the amplitude of the wave on the z = b plane behind the lens is For a value of b satisfying the following relation (the lens equation for a convex thin lens), the wave amplitude becomes For ωD → ∞ limit, the Bessel function in (24) becomes the delta function and we obtains the following wave amplitude on z = b : Thus, a magnified image of the source field appears on the z = b plane. This reproduces the result of image formation in geometric optics; we have shown that an inverted images with magnification b/a of a source object appears on z = b satisfying the lens equation (23). If we do not take ωD →∞ limit, due to the diffraction effect, an image of a point source has finite size on the image plane called the Airy disk [8]. Its size is given by This value determines the resolving power of image formation system. For two point sources at x 0 = -d / 2 , d / 2, their separation on the image plane is bd/a . To resolve them, their separation must be larger than the size of the Airy disk: The lefthand side of this inequality is the angular separation of the sources and θ 0 determines the resolving power of the image formation system.","pages":[6,7,8]},{"title":"3.2. Image formation in gravitational lens system","content":"As we have observed that a convex lens can be a device for image formation in wave optics, we combine it with a gravitational lensing system and obtain images by gravitational lensing. We consider a configuration of the gravitational lens system shown in Figure 6 and examine how the images of the source object appear using wave optics. As the source object, we assume a point source of wave. The amplitude of the wave just in front of a convex lens is This equation is the same as (7). After passing through the convex lens, the wave amplitude on the image plane z 2 is given by where ∆ 0 denotes the aperture of the convex lens. Using dimensionless variables, the wave amplitude on the image plane is If we choose the location of the image plane z 2 to satisfy the following 'lens equation' for a convex lens, then the wave amplitude on the image plane becomes where g = ( r LS /r L ) y 2 +( r L r S /r LS z 2 ) d 2 . Thus the wave amplitude on the image plane is the Fourier transform of the amplification factor F which gives the interference fringe pattern. Under the geometrical optics limit w /greatermuch 1, x integral in the amplification factor (13) can be approximated by the WKB form where x ∗ ( y , d ' ) is the solution of the lens equation We have assumed that the aperture of the convex lens is sufficiently smaller than the size of the gravitational lensing system and | d ' | ≤ d 0 /lessmuch 1 holds. Then, the wave amplitude on the image plane is For wd 0 →∞ limit (large lens aperture limit or high frequency limit), we obtains and the image of the point source appears at the following location on the image plane determined by the lens equation (32): (34) and (35) reproduce the same result of image formation in the geometrical optics (ray tracing) in terms of the wave optics. This is what we aim to clarify in this paper. If the lens equation (32) has multiple solutions x ( j ) ∗ , j = 1 , 2 , · · · , the wave amplitude on the image plane becomes where A j are constants. As an example of image formation in a gravitational lensing system using wave optics, we present the wave optical images of a point source by the gravitational lensing of a point mass (Figure 7). They are obtained by Fourier transformation of the amplification factor F (equation (31)) and teh lens equation for gravitational lensing (15) has not been used. This procedure corresponds to image formation by a convex lens. These images correspond to images obtained by geometric optics (Figure 2). We can observe wave effect in these images. In each images, we can observe concentric interference pattern which is caused by finite size of the lens aperture and this is not intrinsic feature of the gravitational lensing system. We can also observe radial non-concentric patterns. They are caused by interference between double images and represent the intrinsic feature of the gravitational lensing system. For y = 0 case which corresponds to the Einstein ring in the geometrical optics limit, we can observe a bright spot at the center of the ring, which is the result of constructive interference and does not appear in geometric optics. For sufficiently large values of wd 0 , the wave amplitude at the observer coincides with the result obtained by geometric optics.","pages":[8,9,10,11]},{"title":"4. Summary","content":"We investigated image formation in gravitational lensing system based on wave optics. Instead of using a ray tracing method, we obtained images directly from wave functions at the observer without using a lens equation of gravitational lensing. For this purpose, we introduced a 'telescope' with a single convex thin lens, which acts as a Fourier transformer for the interference pattern formed at an observer. The analysis in this paper relates the wave amplitude and images of the gravitational lensing directly. In the geometric optics limit of waves, images by lens systems are obtained by a lens equation which determines paths of each light rays. As light rays are trajectories of massless test particles (photon), expressing image in terms of wave is to express particle motion in terms of waves. As an application and extension of analysis presented in this paper, we plan to investigate gravitational lensing by a black hole and obtain wave optical images of black holes. This subject is related to observation of black hole shadows [5, 6]. As the apparent angular sizes of black hole shadows are so small, the diffraction effect on images are crucial to resolve black hole shadows in observation using radio interferometer. For SgrA ∗ , which is the black hole candidate at Galactic center, the apparent angular size of its shadow is estimated to be ∼ 30 µ arc seconds and this value is the largest among black hole candidates. For a sub-mm VLBI with a baseline length D , using (27), the condition to resolve the shadow becomes","pages":[11]},{"title":"D > 1000 km ,","content":"and this requirement shows the possibility to detect the black hole shadow of SgrA ∗ using the present day technology of VLBI telescope. Thus, analysis of black hole shadows based on wave optics is an important task to evaluate detectability of shadows and determination of black hole parameters via imaging of black holes. The topic of wave optical image formation in black hole spacetimes belongs to a classical problem of wave scattering in black hole spacetimes [9]. As is well known, waves incident to a rotating black hole are amplified by the superradiance [10] due to dragging of spacetimes. This effect enables waves to extract the rotation energy of black holes. On the other hand, it is known that particles can also extract the rotation energy of black holes via so called Penrose process. By investigating images of scattered waves by a rotating black hole, we expect to find out new aspect or interpretation of phenomena associated with superradiance in connection with the Penrose process.","pages":[11,12]},{"title":"Acknowledgments","content":"This work was supported in part by the JSPS Grant-In-Aid for Scientific Research (C) (23540297). The author thanks all member of 'black hole horizon project meeting' in which the preliminary version of this paper was presented.","pages":[12]}],"string":"[\n {\n \"title\": \"Yasusada Nambu\",\n \"content\": \"Department of Physics, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan E-mail: nambu@gravity.phys.nagoya-u.ac.jp Abstract. We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing. PACS numbers: 04.20.-q, 42.25.Fx wave optics; image formation; gravitational lens\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Gravitational lensing is one of the prediction of Einstein's general theory of relativity and many samples of images caused by gravitational lensing have been obtained observationally [1]. Light rays obey null geodesics in curved spacetime and they are deflected by gravitational potential of lens objects. In weak gravitational field with thin lens approximation, a path of a light ray obeys so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing effect are carried out based on this equation. Especially, we can obtain images of source objects by solving the lens equation using a ray tracing method. As a path of light ray is derived as the high frequency limit of electromagnetic wave, wave effects of gravitational lensing become important when the wavelength is not so much smaller than the size of lens objects and in such a situation, we must take into account of wave effects. For example, when we consider gravitational wave is scattered by gravitational lens objects, the wave effect gives significant impact on the amplification factor of intensity for waves [2, 3, 4]. Another example is direct detection of black holes via imaging their shadows [5, 6]. The apparent angular size of black hole shadows are so small that their detectability depends on angular resolution of telescopes which is determined by diffraction limit of image formation system. Thus it is important to investigate wave effects on images for successful detection of black hole shadows. Although interference and diffraction of waves by gravitational lensing has been discussed in connection with amplification of waves, a little was discussed about how images by gravitational lensing are obtained based on wave optics. For electromagnetic wave, E. Herlt and H. Stephani [7] discussed the position of images by a spherical gravitational lens evaluating the Poynting flux of scattered wave at an observer. They claimed that there is a disagreement between wave optics and geometrical optics concerning the position of double images of a point source. But they have not presented complete understanding of image formation. In this paper, we consider image formation in gravitational lensing using wave optics and aim to understand how images by gravitational lensing are obtained in terms of waves. For this purpose, we adopt the diffraction theory of image formation in wave optics [8], which explains image formation in optical systems in terms of diffraction of waves.\",\n \"pages\": [\n 2\n ]\n },\n {\n \"title\": \"2. Wave optics in gravitational lensing\",\n \"content\": \"We review the basic formalism of gravitational lensing based on wave optics [1]. In this paper, we does not consider polarization of waves and treat scalar waves as a model for electromagnetic waves. Let us consider waves propagating under the influence of the gravitational potential of a lens object. The background metric is assumed to be where U ( r ) is the gravitational potential of the lens object with the condition | U | /lessmuch 1. The scalar wave propagation in this curved spacetime is described by the following wave equation: and for a monochromatic wave with the angular frequency ω , where ∇ 2 is the flat space Laplacian. We show the configuration of the gravitational lensing system considering here (Figure 1). The wave is emitted by a point source, scattered by the gravitational potential of the lens object and reaches the observer. We assume the wave scattering occurs in a small spatial region around the lens object and outside of this region, the wave propagates in a flat space. With the assumptions of the eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula provides the following amplitude of the wave at the observer [1, 3] where S ( η , ξ , ∆ ) is the effective path length (eikonal) along a path from the source position η to the observer position ∆ via a point ξ on the lens plane and we assume that | η -∆ | /lessmuch r S and | ξ -∆ | /lessmuch r L . A constant a 0 represents the intensity of a point source. The two dimensional gravitational potential is introduced by Then the wave amplitude at the observer can be written as [1, 2] where Φ 0 is the wave amplitude at the observer in the absence of the gravitational potential U : S 0 ( η , ∆ ) is the path length along a straight path from η to ∆ . The amplification factor F is given by the following form of a diffraction integral where S 1 ( η , ξ , ∆ ) is the Fermat potential along a path from the source position η to the observer position ∆ via a point ξ on the lens plane. The first term in S 1 is the difference of the geometric time delay between a straight path from the source to the observer and a deflected path. The second term is the time delay due to the gravitational potential of the lens object. Now we introduce the following dimensionless variables: where we choose ξ 0 as M is the mass of the gravitational source, ξ 0 and θ E represent the Einstein radius and the Einstein angle, respectively. Using these dimensionless variables, In the geometrical optics limit w /greatermuch 1, the diffraction integral (13) can be evaluated around the stationary points of the phase function in the integrand. The stationary points are determined by the solution of the following equation: This is the lens equation for gravitational lensing and determines the location of the image x for given source position y . As the specific model of gravitational lensing, we consider a point mass as a gravitational source. In this case, the two dimensional gravitational potential is and the deflection angle is given by ‡ For y = d = 0, the solution of the lens equation (15) is and represents the Einstein ring with the apparent angular radius θ E defined by (12). We show an example of images obtained as solutions of the lens equation (15) (Figure 2). To produce these images, we have assumed an extended source with Gaussian distribution of intensity. The wave property is obtained by evaluating the diffraction integral (13). For a point mass lens potential (16), the integral can be obtained exactly On the observer plane, an interference pattern appears (Figure 3). ‡ Using the original variables, the lens equation is ξ -∆ r L -η -∆ r S -r LS r S ∇ ξ ˆ ψ = 0 . Near y = 0, the distance between adjacent fringes of the interference pattern is This fringe pattern is interpreted as interference between double images of a point source by the gravitational lensing. The question we aim to raise in this paper is how the interference pattern on the observer plane is related to the images of gravitational lensing in the geometrical optics limit. The wave amplitude on the observer plane does not make the image of the source and we have to transform the wave function to extract images. To answer this question, we introduce a 'telescope' in the gravitational lensing system and simulate observation of a star (a point source) using the telescope. With this setup, it is possible to understand how images of a source are formed in the framework of wave optics.\",\n \"pages\": [\n 2,\n 3,\n 4,\n 5,\n 6\n ]\n },\n {\n \"title\": \"3. Image formation in wave optics\",\n \"content\": \"To establish relation between the interference pattern of the wave and the images of the source in the gravitational lensing system, we first consider an image formation system composed of a single convex lens and review how images of source objects appear in the framework of wave optics [8].\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3.1. Image formation by a convex lens\",\n \"content\": \"Let us Φ in ( x ) is the incident wave from a point source in front of a thin convex lens and Φ t ( x ) is the transmitted wave by the lens (Figure 4). They are connected by the following relation where T ( x ) is called a lens transformation function. The action of a convex lens is to modify the phase of the incident wave. For a point source placed at z = -f (front focal point), the incident wave and the transmitted wave are where we have used r = √ f 2 + | x | 2 ≈ f + | x | 2 / (2 f ) assuming | x | /lessmuch f . Thus, a convex lens converts a spherical wave front to a plane wave front. Using this action of a convex lens for the incident wave and the transmitted wave, we can demonstrate the image formation by a convex lens in the framework of wave optics. Let us consider the configuration of the lens system shown in Figure 5. We assume the distribution of the source field on the object plane z = -a as Φ 0 ( x 0 ). Using the Fresnel-Kirchhoff diffraction formula, the amplitude of the wave in front of the lens is given by where r 1 is the path length from a point on the object plane to a point on the lens plane and we have assumed | x 1 -x 0 | /lessmuch a . The amplitude of the wave just behind the lens is given by the relation (21) where t L is the aperture function of the lens defined by t L ( x ) = 1 for 0 ≤ | x | ≤ D and t L ( x ) = 0 for D < | x | . D represents a radius of the lens. With the assumption | x 1 -x | /lessmuch b , the amplitude of the wave on the z = b plane behind the lens is For a value of b satisfying the following relation (the lens equation for a convex thin lens), the wave amplitude becomes For ωD → ∞ limit, the Bessel function in (24) becomes the delta function and we obtains the following wave amplitude on z = b : Thus, a magnified image of the source field appears on the z = b plane. This reproduces the result of image formation in geometric optics; we have shown that an inverted images with magnification b/a of a source object appears on z = b satisfying the lens equation (23). If we do not take ωD →∞ limit, due to the diffraction effect, an image of a point source has finite size on the image plane called the Airy disk [8]. Its size is given by This value determines the resolving power of image formation system. For two point sources at x 0 = -d / 2 , d / 2, their separation on the image plane is bd/a . To resolve them, their separation must be larger than the size of the Airy disk: The lefthand side of this inequality is the angular separation of the sources and θ 0 determines the resolving power of the image formation system.\",\n \"pages\": [\n 6,\n 7,\n 8\n ]\n },\n {\n \"title\": \"3.2. Image formation in gravitational lens system\",\n \"content\": \"As we have observed that a convex lens can be a device for image formation in wave optics, we combine it with a gravitational lensing system and obtain images by gravitational lensing. We consider a configuration of the gravitational lens system shown in Figure 6 and examine how the images of the source object appear using wave optics. As the source object, we assume a point source of wave. The amplitude of the wave just in front of a convex lens is This equation is the same as (7). After passing through the convex lens, the wave amplitude on the image plane z 2 is given by where ∆ 0 denotes the aperture of the convex lens. Using dimensionless variables, the wave amplitude on the image plane is If we choose the location of the image plane z 2 to satisfy the following 'lens equation' for a convex lens, then the wave amplitude on the image plane becomes where g = ( r LS /r L ) y 2 +( r L r S /r LS z 2 ) d 2 . Thus the wave amplitude on the image plane is the Fourier transform of the amplification factor F which gives the interference fringe pattern. Under the geometrical optics limit w /greatermuch 1, x integral in the amplification factor (13) can be approximated by the WKB form where x ∗ ( y , d ' ) is the solution of the lens equation We have assumed that the aperture of the convex lens is sufficiently smaller than the size of the gravitational lensing system and | d ' | ≤ d 0 /lessmuch 1 holds. Then, the wave amplitude on the image plane is For wd 0 →∞ limit (large lens aperture limit or high frequency limit), we obtains and the image of the point source appears at the following location on the image plane determined by the lens equation (32): (34) and (35) reproduce the same result of image formation in the geometrical optics (ray tracing) in terms of the wave optics. This is what we aim to clarify in this paper. If the lens equation (32) has multiple solutions x ( j ) ∗ , j = 1 , 2 , · · · , the wave amplitude on the image plane becomes where A j are constants. As an example of image formation in a gravitational lensing system using wave optics, we present the wave optical images of a point source by the gravitational lensing of a point mass (Figure 7). They are obtained by Fourier transformation of the amplification factor F (equation (31)) and teh lens equation for gravitational lensing (15) has not been used. This procedure corresponds to image formation by a convex lens. These images correspond to images obtained by geometric optics (Figure 2). We can observe wave effect in these images. In each images, we can observe concentric interference pattern which is caused by finite size of the lens aperture and this is not intrinsic feature of the gravitational lensing system. We can also observe radial non-concentric patterns. They are caused by interference between double images and represent the intrinsic feature of the gravitational lensing system. For y = 0 case which corresponds to the Einstein ring in the geometrical optics limit, we can observe a bright spot at the center of the ring, which is the result of constructive interference and does not appear in geometric optics. For sufficiently large values of wd 0 , the wave amplitude at the observer coincides with the result obtained by geometric optics.\",\n \"pages\": [\n 8,\n 9,\n 10,\n 11\n ]\n },\n {\n \"title\": \"4. Summary\",\n \"content\": \"We investigated image formation in gravitational lensing system based on wave optics. Instead of using a ray tracing method, we obtained images directly from wave functions at the observer without using a lens equation of gravitational lensing. For this purpose, we introduced a 'telescope' with a single convex thin lens, which acts as a Fourier transformer for the interference pattern formed at an observer. The analysis in this paper relates the wave amplitude and images of the gravitational lensing directly. In the geometric optics limit of waves, images by lens systems are obtained by a lens equation which determines paths of each light rays. As light rays are trajectories of massless test particles (photon), expressing image in terms of wave is to express particle motion in terms of waves. As an application and extension of analysis presented in this paper, we plan to investigate gravitational lensing by a black hole and obtain wave optical images of black holes. This subject is related to observation of black hole shadows [5, 6]. As the apparent angular sizes of black hole shadows are so small, the diffraction effect on images are crucial to resolve black hole shadows in observation using radio interferometer. For SgrA ∗ , which is the black hole candidate at Galactic center, the apparent angular size of its shadow is estimated to be ∼ 30 µ arc seconds and this value is the largest among black hole candidates. For a sub-mm VLBI with a baseline length D , using (27), the condition to resolve the shadow becomes\",\n \"pages\": [\n 11\n ]\n },\n {\n \"title\": \"D > 1000 km ,\",\n \"content\": \"and this requirement shows the possibility to detect the black hole shadow of SgrA ∗ using the present day technology of VLBI telescope. Thus, analysis of black hole shadows based on wave optics is an important task to evaluate detectability of shadows and determination of black hole parameters via imaging of black holes. The topic of wave optical image formation in black hole spacetimes belongs to a classical problem of wave scattering in black hole spacetimes [9]. As is well known, waves incident to a rotating black hole are amplified by the superradiance [10] due to dragging of spacetimes. This effect enables waves to extract the rotation energy of black holes. On the other hand, it is known that particles can also extract the rotation energy of black holes via so called Penrose process. By investigating images of scattered waves by a rotating black hole, we expect to find out new aspect or interpretation of phenomena associated with superradiance in connection with the Penrose process.\",\n \"pages\": [\n 11,\n 12\n ]\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"This work was supported in part by the JSPS Grant-In-Aid for Scientific Research (C) (23540297). The author thanks all member of 'black hole horizon project meeting' in which the preliminary version of this paper was presented.\",\n \"pages\": [\n 12\n ]\n }\n]"}}},{"rowIdx":290,"cells":{"bibcode":{"kind":"string","value":"2013IJAsB..12...63B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1206.0953.pdf"},"content":{"kind":"string","value":"\nGalactic exploration by directed self-replicating probes, and its implications for the Fermi paradox\nMartin T. Barlow 1\nDepartment of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. Email: barlow@math.ubc.ca\nAbstract\nThis paper proposes a long term scheme for robotic exploration of the galaxy, and then considers the implications in terms of the 'Fermi paradox' and our search for ETI. We discuss the 'galactic ecology' of civilizations in terms of the parameters T (time between ET civilizations arising) and L , the lifetime of these civilizations. Six di ff erent regions are described.\nKeywords: Self-Replicating Probes, Galactic Exploration, Search for Extraterrestrial Life, Fermi Paradox\n1. Introduction\nThe proposal to explore the galaxy by selfreplicating probes goes back at least as far as Freitas 1980a and Tipler 1980. Numerous criticisms have been made, notably by Sagan and Newman 1983. A recent paper (Wiley 2012) reconsiders the topic, and finds some of these criticisms without much merit. This paper outlines in more detail one exploration scheme and its benefits, and then considers the implication for the galactic 'ecology' of intelligent species.\n2. Exploration of the galaxy by self replicating probes\nOur starting point is as in Freitas 1980b and Tipler 1980. This relies on two technologies which we do not have at present, but which it is reasonable to suppose we will attain within the next few hundred years:\n\n(T1) A propulsion system capable of sending probes to nearby stars, at say 0.01 c.\n(T2) An AI system which, in total, is able, with the resources found in most star systems, of replicating itself.\n\nPropulsion systems. While no existing system can reach these speeds, various proposals for 100 year probes to Alpha Centauri have been made, such as Project Daedalus. A speed of 0.01c does not seem unduly optimistic, and in fact many proposals for galactic\nexploration, such Bjørk (2007), consider probes which travel at 0.1c.\nAIsystemsAgain, we do not have self-replicating systems at present, and what such a system would be like must rely largely on conjecture. In this context note the following remark in Wiley (2012):\nOne point we take issue with is an inherent and frequently unconscious biological bias that pervades consideration of computerized intelligence, including SRPs.\nA common tendency has been to imagine a selfreplicating machine as being rather like a bacterium: that is a single machine which (somehow, almost magically) is able to move around in its environment and replicate itself. If we start with present technology, we are forced to imagine something rather di ff erent. The system as a whole might consist of 3 parts:\n\n(A) A number of robots and probes, of several di ff erent types, which are together capable of exploring a solar system, and gathering resources (metals, volatiles etc). (B) A 'slow assembler' which would be able refine these materials into components which would make the final factory (C).\n(C) A large scale factory, or collection of factories, which would be able to manufacture copies of (A) and (B), as well as additional surveying and communication devices.\n\nThe payload of the probe would consist of (A + B), together with enough raw materials (fuel etc.) to get started in the new system. Once (C) was made, resources would be gathered for as long as was necessary,\nand a number of probes would then be sent to nearby stars. If we take this view, then a 'self-replicating space probe' (SRP) would not be a single machine, but rather a collection of di ff erent machines with an overall capability of replication.\nSee Freitas 1980b for a much more detailed description of such a probe, with the probe (A + B) plus the fuel for the voyage having a mass of of around 10 10 kg. The factory described there only makes 1 new probe every 500 years, but (see Section 7.1) using a longer period for the initial construction gives a larger factory which can create 1000 new probes in 1500-200 years. For simplicity I will take the reproduction time, between the arrival of the initial probe in a star system, and the completed factory (C) sending out new probes, to be TR = 1000.\nThe AI needed for such a system far exceeds what is possible at present. But while the kinds of decisions necessary for the AI (e.g. 'what kind of material is present in this asteroid?', 'can it be transported to the factory?') would require a very high level of skill, this would be within fairly narrow parameters, and a human level of overall initiative and judgement would not be required. Even if machines with a human intelligence can be constructed, it might be desirable to limit the intelligence of the SRPs.\nAs Tipler 1980 notes, there are reasons other than stellar exploration to develop these technologies. Progress in AI has been far slower than supposed by early optimists, but it still seems reasonable to suppose that, within a few hundred years, we will be able to build such SRPs. The development of such machines would, at least for a while, introduce an age of plenty, since it would open up the resources of the solar system for our exploitation. Some idealistic individuals or groups might then be willing to invest the resources in making a number of the probes (A + B), and send them to nearby stars. (See Mathews 2011 for another proposal to explore the galaxy by SRPs.)\n2.1. Exploration strategy\nI now propose a scheme for galactic exploration, assuming that we do develop the technologies (T1) and (T2) described above. The first step would be to send probes to the 10-100 nearest stars, which are all within about 20 ly of Earth. (Long before the start of such a mission we will have very good data on the planetary systems of these stars.) The probes would arrive at their destinations 400-2000 years after the mission start. They would remain in radio contact with Earth, (with a time lag of 40 years or less), would report on their discoveries, and would be able to receive updates on strategy. (Among the exploration devices (A) would\nbe systems able to transmit and receive narrow band radio or laser communication over a distance of say 100 ly.)\nI will call the initial star systems Level 1 'colonies', though there is no suggestion that they would have a human population. After the construction of the factory (C) on a Level 1 colony, the colony would send out SRPs (let us say about 1,000 - 10,000) to create colonies at 'Level 2'. I have suggested an initial 'hop size' of 10-20 ly, since the number of probes that could be sent out from our Solar System might be limited by resource constraints. However once a colony at Level 1 or higher had a working factory, there would be no such limit on the number of probes that could be sent out, and it would be sensible to send as many as was necessary to explore every star system within the second 'maximum hop size'. There are about 15,000 stars within 100 ly of Earth, so with some useful duplication the Level 1 systems would together be able to send probes to every star in this region.\nThe maximum hop size hm would be the greatest distance such a probe could be sent with a probability greater than 90% of arriving. I will take hm = 100; this also needs to be less than the maximum distance for radio or laser communication, but this is much greater than 100 ly.\nThe Probes from the Level 2 colonies would then establish Level 3 colonies, and so on. Each colony at Level n would report back to its Level n -1 ancestor, and receive updating instructions from it. While it would be desirable for the Level 1 colonies to produce manyprobes, as the radius of the exploration sphere became larger, and so the curvature of its surface became less, fewer new probes per colony would be needed.\nWithin a few thousand years of the mission start our descendants on earth (if they still existed) would be receiving a flood of information from the exploration of hundreds of star systems. The Great Pyramid was built around 4500 years ago; 4500 years after its start the mission would be well under way, and have given us detailed data on every star system within about 30 ly of Earth.\nThe overall mission would continue until the planetary system of every star in the galaxy had been explored. Let vp = 0 . 01 c be the speed of the probes, and ve be the propagation speed of the exploration front. Then\nve = hm TR + hm / vp = vp 1 1 + vpTR / hm .\nSo TR = 1000 and hm = 100 give ve / vp = 10 / 11, the exploration front travels at nearly the same speed as the probes, and the total time to explore the galaxy\nis around 10 7 years. This compares with exploration times of the order of 10 8 years given by Bjørk 2007, using probes which travel at 0.1c, but do not replicate.\n2.2. Refinements\nWhile the basic strategy is as above, it is necessary to consider a number of refinements.\n(a)Resourceusewithinsystem. The best place for the construction of the factory (C) might be the moons of a planet in the outer reaches of the star system. Assuming a mass of 10 10 kg for the probes (A + B) and fuel, the construction of (C) plus say 10,000 probes would use at most a handful of minor planets and comets. This would leave plenty of material behind, even on the Level 1 colonies, and it would not be necessary to to 'strip mine' the galaxy in order to complete this exploration. One would only need a few probes per star one plus a margin for accidents.\n\n(b) Systems with planets with life. In systems with planets with complex life, a di ff erent procedure should be followed. Two possibilities would be:\n(i) Report, build a factory (C), explore the system thor-\noughly, and then await instructions from Earth, (ii) Report, do nothing, and await instructions from Earth.\n\nThe first and second level probes would provide enough data to refine this strategy, at an early stage of the overall mission. In the very unlikely event that more than 90% of systems have planets with complex life, a modification of (ii) would be needed so that a reasonable proportion of colonies did send out probes.\n(c) Extinctionof the humanrace. Once set in motion, the exploration could continue without any further human intervention. However, this proposal envisions continued interaction and direction: Earth would receive data from the probes, and based on this revised instructions on exploration (as well as possible system upgrades) would be sent out. What however if humanity becomes extinct, or just loses interest in the mission? There are many possible procedures which could be followed, of which the simplest are: (i) Continue anyway, (ii) Abandon further exploration. For simplicity I propose (ii), and suggest that every 100 years the Level 1 colonies would ask Earth 'Shall we continue?' If 1000 years went by with no positive response, the project would be mothballed, and instructions would be sent through the communication tree that no further SRPs were to be built.\n(d) Communication and direction. A key part of this exploration scheme is that the SRPs are not autonomous, but that the whole exploration process is directed, ultimately from Earth. The first requirement\nhere is that Level n colonies be able to communicate (over a distance of say 100 ly) with their Level n -1 ancestors. Even with present technology, we could build transmitters and receivers capable of working over these distances.\nWhile all the Level 1 colonies would send out probes, this would not be necessary for higher level colonies, and stars could be divided into two groups. For the first, 'end nodes', a factory C would be built, the star system explored, but no probes would be sent out. The second, 'branch nodes' would send out probes. Among the pieces of infrastructure built in each colony would be telescopes to survey the stellar neighbourhood, and using this data, nearby Level n colonies would coordinate the exploration of their neighbourhoods. (Nearby colonies would be 10-100 ly apart, so communication time would be small compared with the time to build probes, or for the journeys.) While the algorithm to coordinate this process may appear complicated, it is well within our current capabilities - unlike the AI needed for robotic exploration of a planetary system\n(e) Mission creep and machine mutation. A widely voiced concern with SRPs has been that they might mutate, run amok, and eat up the galaxy - see Sagan and Newman 1983. However Wiley 2012 argues that it should be possible to build in su ffi cient reliability to avoid this outcome - note also his comment above on inappropriate biological analogies. In the exploration scheme proposed here about 1000 generations would be needed to explore the galaxy - fewer if hops longer than 100 ly are feasible. The total number of replicators (C) built would be of the same order as the number of stars in the galaxy, that is about 4 × 10 11 . Wiley 2012 points out that this is much less than the number of cell divisions within a human lifetime, which is of the order of 10 16 .\nIf necessary, further steps could be taken to reduce the overall risk. In the initial stages of the exploration we would want every planetary system to be explored carefully. However, it seems likely that after the first million or so systems had been explored, we would have a good understanding of the processes underlying the formation and development of planetary systems, and might only be interested in those systems which had life, complex life, or other exceptional features. The later phases of the exploration could therefore proceed as follows. Each 'branch node' would first send out about 10 new full probes (A + B) to establish the next generation of colonies. Next, it would send out reduced probes, just consisting of (A), to all the stars in its exploration patch. These would explore the target system, and report back to the send-\ning branch node. Without the reproductive capacity (B) these probes would ultimately run out of fuel and become inactive. A full probe (A + B) would then be sent to any system that merited further attention. Assuming that 'interesting' systems are rare, this modification would reduce the number of full replications by a factor of 1000 or so. Further safety mechanisms could also be built in, such as deeply embedded software constraints on the total number of probes that the factories (C) could make, or on the total number of permitted generations.\n(f)Crossinglargespacesandpercolation. Landis 1998 has suggested a percolation model for the spread of a species through the galaxy, and shown that in some cases this leads to large vacant (unexplored) regions. However bond percolation on the lattice is a poor model for the type of exploration proposed above, since each 'branch node' would send out rather more than 5 probes. Further, the communication envisaged between colonies would mean that colonies would become aware of interstellar voids (with no or few stars), and regions that, perhaps because of the failure of a number of probes, were remaining unexplored. They could then send additional probes to explore these regions. If we consider the mathematical graph whose vertices are the stars, and join by edges all pairs of stars within 100 ly., then the exploration scheme proposed here will explore all stars in the connected component containing our sun, and it seems overwhelmingly likely that this spans most of the galaxy.\n(g) System updates. We would want to be able to incorporate updates into the systems (A,B,C). It is possible that this could be done by radio, but the available bandwidth might be too small for the necessary amount of data. One can imagine a system of 'fast packets' - small probes carrying data, which travel at say 0.1c between colonies with the infrastructure to send and receive them. However one disadvantage of allowing such updates is that it would make the colonies more vulnerable to mutations or computer viruses.\n(h)Contactwithextraterrestrialintelligence(ETI).Detailed thought would need to be given on what course of action should be taken if either ETI were found, or traces of them. There would be time to refine strategies in the first few millenia of the mission, as data on frequency and type of life in other star systems accumulated. The number of possible actions the probes could take is large, and a full discussion of this is beyond the scope of this article. The simplest (but not the quickest) option would be for the the probe to report back, and take no further action until instructed.\n3. Implications for SETI, and Fermi's question\nLet us now make the hypothesis (H) that the technolgies (T1) and (T2) can be attained, and explore the consequences. The exploration scheme outlined above, using these technologies is one which, if it survives long enough, the human race might adopt - no doubt with a number of improvements. The payo ff is that with a relatively low initial cost our descendants would obtain detailed data about every star system in the galaxy. In particular they would learn how many planets support life, what kind of life it is, and just how rare complex or intelligent life is.\nIf there have been technological ETI in the galaxy, then they would also have had this option. So - this is Fermi's question 'Where are they?' (This is often called the 'Fermi paradox', but it is only a paradox if one begins with the assumption that intelligent life is common. In fact we have no information on this.)\nLet us recall the Drake equation, slightly modified for our purposes:\nN = R ∗ · fp · ne · f /lscript · fci · L ;\nhere N is the number of existing civilizations sending out SRPs, R ∗ is the rate of star formation per year in the galaxy, fp is the fraction of those stars that have planets, ne is the average number of planets that can potentially support life per star that has planets, fl is the fraction of these that develop life, fci is the fraction of these that develop civilizations that send out SRPs, and L is the average lifetime of such civilizations. This lifetime is the time that either the civilization itself, or its SRPs, remain active. (From now on, I will use the term 'civilization' for 'civilizations that send out SRPs').\nWe do have estimates of at least the order of magnitude of some of the early terms in this expression: for example R ∗ /similarequal 7, and data from the Kepler satellite suggests that fp /similarequal 0 . 5, while ne is quite small. (Out of about 10,000 systems surveyed, only a handful have planets which look really promising from the point of finding earthlike life.) At present fl and fci are utterly unknown, though estimates of fl may at some point become available via spectroscopic search for oxygen.\nI have given the Drake equation in a simple form. A more realistic equation would take account of randomness, and the fact that these factors are not constant in time - see for example Glade et. al. 2012. However, the uncertainty in our knowledge of the parameters in the equation is so great that these refinements seem to the author of this article to add little to what can be achieved with a simple 'back of an envelope' calculation.\nPSfrag replacements\n\n\n
\n1. Introduction\nUnveiling the nature of dark matter (DM) 1 is a quest in both Astroparticle and Particle Physics. Among the list of well-motivated candidates, the most popular particles are cold and weakly interacting, and typically predict missing-energy signals at particle colliders. Supersymmetry 2-10 and models with extra dimensions 11 are theoretical scenarios that inherently provide such a dark matter candidate. Highenergy colliders, such as the CERN Large Hadron Collider, 12 are ideal machines for producing and eventually detecting DM. Experiments in upcoming colliders, such as the ILC 13 and CLIC, 14 are expected to further constraint such models, should they are materialized in Nature, and subsequently make a key step in understanding dark matter.\nIn parallel, the exploration of dark matter is being pursued through other types of detection methods: direct detection in low-background underground experiments 15,16 and indirect detection of neutrinos, γ -rays and antimatter with terrestrial and space-borne detectors. 17 A recent review on the status and results of these instruments is given in Ref. 18.\nThe structure of this paper is as follows. Section 1 provides a brief introduction to the properties of dark matter as defined by the current cosmological data and its implications for physics in colliders. Section 2 highlights the features of the LHC experiments that play a central role in exploring DM. In Section 3, the strategy, methods, and results of the LHC experiments as far as model-independent DMproduction is concerned are discussed. In Sections 4 and 5, the latest results in searches for supersymmetry and for extra dimensions, respectively, are presented. The prospects for exploring dark matter and possibly measuring its properties at future colliders are given in Section 6. The paper concludes with a summary and an outlook in Section 7.\n1.1. Dark matter evidence\nThe nature of the dark sector of the Universe constitutes one of the major mysteries of fundamental physics. According to the latest observations by the Planck mission team, 19 most of our Universe energy budget consists of unknown entities: ∼ 26 . 8% is dark matter and ∼ 68 . 3% is dark energy, a form of ground-state energy. Dark energy is believed to be responsible for the current-era acceleration of the Universe. Dark matter, on the other hand, is matter inferred to exist from gravitational effects on visible matter, being undetectable by emitted or scattered electromagnetic radiation. A possible explanation other than the introduction of one or more yet-unknown particles is to ascribe the observed effects to modified Newtonian dynamics. 20,21\nThe energy budget of the Cosmos (Fig. 1) has been obtained by combining a variety of astrophysical data, such as type-Ia supernovae, 22-24 cosmic microwave background (CMB), 25,26 baryonic acoustic oscillations 27,28 and weak-lensing data. 29 The most precise measurement comes from anisotropies of the cosmic microwave background, as reflected in the its power spectrum, shown in Fig. 2.\n\n\n
Fig. 1. The energy budget of the Universe according to recent cosmological evidence 19 and assuming the ΛCDM model. 30
\n\n\n\n
Fig. 2. Temperature power spectrum from Planck. The densities of baryoninc and dark matter are measured from the relative heights of the acoustic peaks. The third acoustic peak is sensitive to the dark matter density. From Ref. 26.
\n\nEvidence from the formation of large-scale structure (galaxies and their clusters) strongly favor cosmologies where non-baryonic DM is entirely composed of cold dark matter (CDM), i.e. non-relativistic particles. CDM particles, in turn, may be axions, 31 superheavy non-thermal relics (wimpzillas, cryptons) 32 or weakly interacting massive particles (WIMPs). The latter class of DM candidates arises naturally in models which attempt to explain the origin of electroweak symmetry breaking and this is precisely where the connection between Cosmology and Particle Physics lies. Furthermore, the typical (weak-scale) cross sections characterizing these models are of the same order of magnitude as the WIMP annihilation cross section, thus establishing the so-called WIMP miracle . A review on the interplay between (string-inspired) Cosmology and the LHC with an emphasis on the dark sector is given in Ref. 33 and in references therein.\n1.2. Connection between WIMPs and colliders\nWIMP dark matter candidates include the lightest neutralino in models with weakscale supersymmetry, 2-10 Kaluza-Klein photons arise in scenarios with universal extra dimensions (UED), 11 while lightest T -odd particles are predicted in Little Higgs models 34 with a conserved T -parity. The common denominator in these theories is that they all predict the existence of an electrically neutral, colorless and stable particle, whose decay is prevented by a kind of symmetry: R -parity, connected to baryon and lepton number conservation in SUSY models; KK-parity, the four-dimensional remnant of momentum conservation in extra dimension scenarios; and a Z 2 discrete symmetry called T -parity in Little Higgs models.\nWeakly interacting massive particles do not interact neither electromagnetically nor strongly with matter and thus, once produced, they traverse the various detectors layers without leaving a trace, just like neutrinos. However by exploiting the hermeticity of the experiments, we can get a hint of the WIMP presence through\nthe balance of the energy/momentum measured in the various detector components, the so-called missing energy . In hadron colliders, in particular, since the longitudinal momenta of the colliding partons are unknown, only the transverse missing energy , E miss T , can be reliably used to 'detect' DM particles. In this paper, we focus on generic DM searches, on supersymmetric signatures, and on possible signals from extra dimensions, all based on E miss T , performed by the two main LHC experiments, ATLAS 35 and CMS. 36\n2. The ATLAS and CMS Experiments at the LHC\nThe Large Hadron Collider (LHC), 12 situated at CERN, the European Laboratory for Particle Physics, outside Geneva, Switzerland, started its physics program in 2010 colliding two counter-rotating beams of protons or heavy ions. Before the scheduled 2013-2015 long shutdown, the LHC succeeded in delivering ∼ 5 fb -1 of integrated luminosity at center-of-mass energy of 7 TeV during 2010-2011 and another ∼ 23 fb -1 at √ s = 8 TeV in 2012. The LHC has already extended considerably the reach of its predecessor hadron machine, the Fermilab Tevatron, both in terms of instantaneous luminosity and energy, despite the fact that it has not arrived yet to its design capabilities.\nThe two general-purpose experiments, ATLAS (A Toroidal LHC ApparatuS) 35 and CMS (Compact Muon Solenoid), 36 have been constructed and operate with the aim of exploring a wide range of possible signals of New Physics that LHC renders accessible, on one hand, and performing precision measurements of Standard Model (SM) 37-39 parameters, on the other. Two other large experiments, namely LHCb 40 and ALICE, 41 are dedicated to B -physics and heavy ions, respectively, also run at the LHC. It is worth mentioning that the MoEDAL 42 experiment is specifically designed to explore high-ionization signatures that may also arise in some dark matter theoretical scenarios. 43\nThe ATLAS and CMS detectors are designed to overcome difficult experimental challenges: high radiation levels, large interaction rate and extremely small production cross sections of New Physics signals with respect to the well-known SM processes. To this end, both experiments feature separate subsystems to measure charged particle momentum, energy deposited by electromagnetic showers from photons and electrons, energy from hadronic showers of strongly-interacting particles and muon-track momentum. Complete descriptions of the CMS and ATLAS detectors are available in Refs. 36 and 35, respectively.\nThe most remarkable highlight of ATLAS and CMS operation so far is undoubtedly the discovery 44,45 of a new particle that so far seems to have all the features pinpointing to a SM(-like) Higgs boson. 46-48 The observation of this new boson has strong impact not only on our understanding of the fundamental interactions of Nature, as encoded in the SM, but on the proposed theoretical scenarios of Physics beyond the SM (BSM), as we shall see in the following.\n3. Model-Independent DM Production at the LHC\nCollider searches for dark matter are highly complementary to direct 2-10,49-53 and indirect 2-10,51-54 DM detection methods. The main advantage of collider searches is that they do not suffer from astrophysical uncertainties and that there is no lower limit to the DM masses to which they are sensitive.\nThe leading generic diagrams responsible for DM production 55-57 at hadron colliders, as shown in Fig. 3, involve the pair-production of WIMPs plus the initialor final-state radiation (ISR/FSR) of a gluon, photon or a weak gauge boson Z, W . The ISR/FSR particle is necessary to balance the two WIMPs' momentum, so that they are not produced back-to-back resulting in negligible E miss T . Therefore the search is based on selecting events highE miss T events, due to the WIMPs, and a single jet, photon or boson candidate. A single-jet event from the CMS experiment is visible in Fig. 4.\n\n\n\n→\n\n\n\n→\n\n\n
Fig. 3. WIMP production at hadron colliders in association with (a) a jet or (b) a photon or a Z or W boson.Fig. 4. The cylindrical view of a monojet candidate event ( p jet T = 574 . 2 GeV, E miss T = 598 . 3 GeV) from the CMS experiment. 58
\n\nThe 'blobs' in the diagrams of Fig. 3 represent a largely model-independent effective-field-theory framework, in which the interactions between a DM Dirac fermion χ and SM fermions f are described by contact operators of the form\nO V = (¯ χγ µ χ )( ¯ fγ µ f ) M 2 , ( s -channel, vector) (1)\nO A = (¯ χγ µ γ 5 χ )( ¯ fγ µ γ 5 f ) M 2 ∗ , ( s -channel, axial vector) (3)\n∗ O S = (¯ χχ )( ¯ ff ) M 2 ∗ , ( s -channel, scalar) (2)\nO t = (¯ χf )( ¯ fχ ) M 2 ∗ . ( t -channel, scalar) (4)\nPrevious work relating collider searches to direct and indirect searches for dark matter has focused on the Tevatron 59-61 and the LEP. 62,63 While the hadronic machine probes the dark matter couplings to light quarks, the LEP data are sensitive to the DM-electron coupling. In general, Tevatron constraints are very strong for lighter dark matter and fall off when the dark matter mass exceeds the typical energy reach of the collider. 64 The constraints also depend on the coupling of the dark matter; if the dark matter primarily couples to gluons, the constraints from colliders become especially strong. 55 Moreover, if the possibility of light mediators is taken into account, as motivated by cosmic-ray excesses, 65,66 the introduction of a light mediator of mass glyph[lessorsimilar] 10 GeV alleviates the monojet bounds completely for most cases. This leads to the conclusion that if a direct dark matter signal is established in a region that is in conflict with collider bounds, a new light state should be introduced to reconcile the data. 57 One mode in which dark matter may be searched for at LEP, with relatively little model dependence, is its pair production in association with a hard photon. The LEP experiments have searched for anomalous monophoton events in their data sets, but have found no discrepancy from the prediction of the standard model. 67\nWhile this set of operators is not exhaustive, it encompasses the essential phenomenologically distinct scenarios: spin-dependent and spin-independent dark matter-nucleus scattering, as well as s - and p -wave annihilation. The classification of the effective operators as s -channel or t -channel refers to the renormalizable model from which they typically arise: (1)-(3) are most straightforwardly obtained if dark matter pair production is mediated by a new neutral particle propagating in the s -channel, while Eq. (4) arises naturally if the mediator is a charged scalar exchanged in the t -channel (for instance a squark or slepton). The suppression scale M ∗ can be interpreted as the mass of the mediator M , divided by the geometric mean of its couplings to SM fermions, g f , and dark matter, g χ : M ∗ = M/ √ g f g χ . It is worth noting that the DM particles are not explicitly assumed to interact via the weak force; they may as well couple to the SM through a new force. Within this framework, interactions of SM particles and WIMPs are described by only two parameters, the suppression scale M ∗ and the WIMP mass m χ .\n3.1. Monojet searches\nEvent topologies with a single highE T jet and large E miss T , henceforth referred to as monojets , are important probes of physics beyond the SM at the LHC. The ATLAS 68 and CMS 58 experiments have performed searches for an excess of monojet events over SM expectations. The analyses outlined here use the full 2011 pp LHC dataset at a center-of-mass energy of √ s = 7 TeV. The primary SM process yielding a true monojet final state is Z -boson production in association with a jet, where Z → νν . Other known processes acting as background in this search are Z ( → glyph[lscript]glyph[lscript] )+jets, W +jets, t ¯ t and single-top events, with glyph[lscript] = e, µ . All electroweak backgrounds and multijet events passing the selections criteria, as well as non-collision backrounds, are determined by data-driven methods. Top and diboson backgrounds are determined solely from Monte Carlo (MC) simulation.\nThe monojet analyses for ATLAS and CMS are based on some more-or-less common requirements: large E miss T , with thresholds typically ranging from 120 GeV to 500 GeV and a energetic jet with a variable p T threshold higher than 110 GeV that fulfills high jet-reconstruction quality criteria. In addition, events with at least one electron or muon or a third jet are rejected. Back-to-back dijet events are suppressed by requiring the subleading jet not to point in the direction of p miss T . The selected data are required to pass a trigger based on high E miss T (ATLAS) or large E miss T plus one highE T jet (CMS).\nThe data, amounting to ∼ 5 fb -1 , are found to be in agreement with the SM expectations. The results are interpreted in a framework of WIMP production with the simulated WIMP-signal MC samples corresponding to various assumptions of the effective field theory, as discussed previously; some of the options are listed in Table 1. The derived limits are independent of the theory behind the WIMP (SUSY, extra dimensions, etc), yet it has been assumed that other hypothetical particles are two heavy to be produced directly in pp collisions. In the presented limits, a Dirac DM fermion is considered, however conclusions for Majorana fermions can also be drawn, since their production cross section only differs by a factor of two. In this framework, the interaction between SM and DM particles are defined by only two parameters, namely the DM-particle mass, m χ , and the suppression scale, M ∗ , which is related to the mediator mass and to its coupling to SM and DM particles.\nExperimental and theoretical systematic uncertainties are considered when setting limits on the model parameters M ∗ and m χ . The experimental uncertainties on jet energy scale and resolution and on E miss T range from 1 -20% of the WIMP event yield, depending on the E miss T and p T thresholds and the considered interaction operator. Other experimental uncertainties include the ones associated with the trigger efficiency and the luminosity measurement. On the other hand, the partondistribution-function set, the amount of ISR/FSR, and the factorization and renormalization scales assumed lead to theoretical uncertainties on the simulated WIMP signal.\nFrom the limit on the visible cross section of new physics processes BSM, lower\n
\n\n
Table 1. Effective interaction operators of WIMP pair production considered in the monojet and monophoton analyses, following the formalism of Ref. 55.
\n
\nlimits on the suppression scale as a function of the WIMP mass have been derived by the ATLAS Collaboration. 68 The 90% confidence level (CL) lower limits for the D5 and D11 operators are shown in Fig. 5. The observed limit on M ∗ includes experimental uncertainties; the effect of theoretical uncertainties is indicated by dotted ± 1 σ lines above and below it. Around the expected limit, ± 1 σ variations due to statistical and systematic uncertainties are shown as a gray band. The lower limits are flat up to m χ glyph[lessorsimilar] 100 GeV and become weaker at higher mass due to the collision energy. In the bottom-right corner of the m χ -M ∗ plane (light-gray shaded area), the effective field theory approach is no longer valid. The rising lines correspond to couplings consistent with the measured thermal relic density, 55 assuming annihilation in the early universe proceeded exclusively via the given operator. Similar exclusion limits for all operators listed in Table 1 are given in Ref. 68. For the operator D1, the limits are much weaker ( ∼ 30 GeV) than for other operators. Nevertheless, if heavy-quark loops are included in the analysis, much stronger bounds on M ∗ can be obtained. 69\n\n\n\n\n\n
Fig. 5. ATLAS lower limits at 90% CL on the suppression scale, M ∗ , for different masses of χ obtained with the monojet analysis. The region below the limit lines is excluded. All shown curves and areas are explained in the text. From Ref. 68.
\n\nThe observed limit on the dark matter-nucleon scattering cross section depends on the mass of the dark matter particle and the nature of its interaction with the SM particles. The limits on the suppression scale as a function of m χ can be translated into a limit on the cross section using the reduced mass of χ -nucleon system, 57 which can be compared with the constraints from direct and indirect detection experiments. Figure 6 shows the 90% CL upper limits on the dark matternucleon scattering cross section as a function of the mass of DM particle for the spin-independent (left) and spin-dependent (right) models obtained by CMS. 58 Limits from CDF, 61 XENON100, 70 CoGent, 71 CDMS II, 72,73 SIMPLE, 74 COUPP, 75 Picasso, 76 IceCube, 77 Super-K, 78 as well as the CMS monophoton 79 analysis are superimposed for comparison. Similar limits have been obtained by the ATLAS experiment. 68\n\n\n\n\n\n
Fig. 6. 90% CL upper limits on the dark matter-nucleon scattering cross section versus DM particle mass for the spin-independent (left) and spin-dependent (right) obtained with the CMS monojet analysis. Explanation of the shown curves is given in the text. From Ref. 58.
\n\nThe spin-dependent limits, derived from the operator D8, give a smaller, hence better, bound on the WIMP-nucleon cross section throughout the range of m χ , compared to direct DM experiments. In the spin-independent case the bounds from direct detection experiments are stronger for m χ glyph[greaterorsimilar] 10 GeV, whereas the collider bounds, acquired with the operator D5, get important for the region of low DM masses.\nThe ATLAS collider limits on vector (D5) and axial-vector (D8) interactions are also interpreted in terms of the relic abundance of WIMPs, using the same effective theory approach. 55 The upper limits on the annihilation rate of WIMPs into light quarks, defined as the product of the annihilation cross section σ and the relative WIMP velocity v averaged over the WIMP velocity distribution 〈 σv 〉 , are shown in Fig. 7. The results are compared to limits on WIMP annihilation to b ¯ b , obtained from galactic high-energy gamma-ray observations, measured by the Fermi-LAT telescope. 80 Gamma-ray spectra and yields from WIMPs annihilating to b ¯ b , where photons are produced in the hadronization of the quarks, are expected\nto be very similar to those from WIMPs annihilating to light quarks. 81,82 Under this assumption, the ATLAS and Fermi-LAT limits can be compared, after scaling up the Fermi-LAT values by a factor of two to account for the Majorana-to-Dirac fermion adaptation. Again, the ATLAS bounds are especially important for small WIMP masses: below 10 GeV for vector couplings and below about 100 GeV for axial-vector ones. In this region, the ATLAS limits are below the annihilation cross section needed to be consistent with the thermic relic value, keeping the assumption that WIMPs have annihilated to SM quarks only via the particular operator in question. For masses of m χ glyph[greaterorsimilar] 200 GeV the ATLAS sensitivity becomes worse than the Fermi-LAT one. In this region, improvements can be expected when going to larger center-of-mass energies at the LHC.\n\n\n
Fig. 7. Inferred ATLAS 95% CL limits on WIMP annihilation rates 〈 σv 〉 versus mass m χ . Explanation of the shown curves is given in the text. From Ref. 68.
\n\n3.2. Monophoton-based probes\nSimilarly to the monojet searches, the monophoton analyses aim at probing dark matter requiring large E miss T -from the χ -pair production- and at least one ISR/FSR photon. Searches in monophoton events by ATLAS 83 and CMS 79 also show an agreement with the SM expectations. The limits are derived in a similar fashion as for monojet search, however the monophoton search is found to be somewhat less sensitive with respect to the monojet topology.\nThe primary background for a γ + E miss T signal is the irreducible SM background from Zγ → ν ¯ νγ production. This and other SM backgrounds, including Wγ , W → eν , γ +jet multijet, diphoton and diboson events, as well as backgrounds from beam halo and cosmic-ray muons, are taken into account in the analyses. The CMS analysis is based on singe-photon triggers, whilst ATLAS relies on highE miss T trig-\nThe photon candidate is required to pass tight quality and isolation criteria, in particular in order to reject events with electrons faking photons. The missing transverse momentum of the selected events should be as high as 150 GeV (130 GeV) in the ATLAS (CMS) search. In CMS, events with a reconstructed jet are vetoed, while the ATLAS analysis rejects events with an electron, a muon or a second jet.\nBoth analyses, observe no significant excess of events over the expected background when applied on ∼ 5 fb -1 of pp collision data at √ s = 7 TeV. Hence they set lower limits on the suppression scale, M ∗ versus the DM fermion mass, m χ , which in turn they are translated into upper limits on the nucleon-WIMP interaction cross section applying the prescription in Ref. 55. Figure 8 shows the 90% CL upper limits on the nucleon-WIMP cross section as a function of m χ derived from the ATLAS search. 83 The results are compared with previous CDF, 61 CMS 58,79 and direct WIMP detection experiments 70-74,84 results. The CMS limit curve generally overlaps the ATLAS curve.\n\n\n
Fig. 8. ATLAS-derived 90% CL upper limits on the nucleon-WIMP cross section as a function of m χ for spin-dependent (left) and spin-independent (right) interactions, corresponding to D8, D9, D1, and D5 operators. Explanation of the shown curves is given in the text. From Ref. 83.
\n\nThe observed limits on M ∗ typically decrease by 2% to 10% if the -1 σ theoretical uncertainty, resulting from the same sources as the one cited in the monojet analysis, is considered. This translates into a 10% to 50% increase of the quoted nucleon-WIMP cross section limits. To recapitulate, the exclusion in the region 1 GeV < m χ < 1 GeV (1 GeV < m χ < 3 . 5 GeV) for spin-dependent (spinindependent) nucleon-WIMP interactions is driven by the results from collider ex-\nents, always under the assumption of the validity of the effective theory, and is still dominated by the monojet results.\n3.3. MonoW and monoZ final states\nAs demonstrated in the previous sections, searches for monojet or monophoton signatures have yielded powerful constraints on dark matter interactions with SM particles. Other studies propose probing DM at LHC through a pp → χ ¯ χ + W/Z , with a leptonically decaying W 85 or Z . 86,87 The final state in this case would be large E miss T and a single charged lepton (electron or muon) for the mono-W signature ( monolepton ) or a pair of charged leptons that reconstruct to the Z mass for the mono-Z signature. In either case, the gauge boson radiations off a q ¯ q initial state and an effective field theory is deployed to describe the contact interactions that couple the SM particle with the WIMP.\nIn Ref. 85, the existing W ' searches from CMS 88,89 -which share a similar final state with monoW searches- are used to place a bound on monoW production at LHC, which for some choices of couplings are better than monojet bounds. This is illustrated in the left panel of Fig. 9, where the spin-independent WIMP-proton cross section limits are drawn. The parameter ξ parametrizes the relative strength of the coupling to down-quarks with respect to up-quarks: ξ = +1 for equal couplings; ξ = -1 for opposite-sign ones; and ξ = 0 when there is no coupling to down quarks. Even in cases where the monoleptons do not provide the most stringent constraints, they provide an interesting mechanism to disentangle WIMP couplings to up-type versus down-type quarks. This analysis has been followed up by CMS 90 with the full 2012 dataset yielding similar to limits based on W ' searches.\n/RParen1\n/LParen1\n\n\n
50 100 /LParen1 /RParen1 Fig. 9. Left: Monolepton bounds and bounds from direct detection projected into the plane of the WIMP mass and the spin-independent cross section with protons. The red and blue lines are from CMS W ' searches: 5 fb -1 data 88 at 7 TeV and 20 fb -1 data 89 at 8 TeV, respectively. From Ref. 85. Right: MonoZ exclusion regions at 90% CL in the DM mass versus WIMP-nucleon spindependent cross section plane obtained from the ATLAS ZZ measurement. 91 Existing collider limits are also shown for comparison. From Ref. 87.
\n\nFurthermore a (leptonic) monoZ signal has been considered in the literature, highlighting the kinematic features 86 and recasting LHC results to constrain models. 87 Specifically, the ATLAS measurement 91 of ZZ → glyph[lscript]glyph[lscript]ν ν carried out with ∼ 5 fb -1 of 7 TeV data has been reinterpreted into a bound on production of dark matter in association with a Z boson. The obtained bounds for the spin-dependent WIMP-nucleon cross section is depicted in the right panel of Fig. 9 along with other bounds from colliders. 58,61,68 The monoZ signature yields limits which are somewhat weaker than those from monojets or monophotons. Nevertheless, leptonic monoZ searches are less subject to systematic uncertainties from jet energy scales and photon identification, and hence may scale better at large integrated luminosities.\nThe ATLAS Collaboration has recently 92 extended the range of possible monoX probes by looking for pp → χ ¯ χ + W/Z , when the gauge boson decays to two quarks, as opposed to the leptonic signatures discussed so far. The analysis searches for the production of W or Z bosons decaying hadronically and reconstructed as a single massive jet in association with large E miss T from the undetected χ ¯ χ particles. For this analysis, the jet candidates are reconstructed using a filtering procedure referred to as large-radius jets . 93 This search, the first of its kind, is sensitive to WIMP pair production, as well as to other DM-related models, such as invisible Higgs boson decays ( WH or ZH production with H → χ ¯ χ ).\n\n\n
Fig. 10. ATLAS-derived limits on χ -nucleon cross sections as a function of m χ at 90% CL for spin-independent (left) and spin-dependent (right) cases, obtained with the monoW/Z analysis and compared to previous limits. From Ref. 92.
\n\nAs shown in Fig. 10, this search for dark matter pair production in association with a W or Z boson extends the limits on the dark matter-nucleon scattering cross\nsection in the low mass region m χ < 10 GeV where the direct detection experiments have less sensitivity. The new limits are also compared to the limits set by ATLAS in the 7 TeV monojet analysis 68 and by direct detection experiments. 71-74,84,94-96 For the spin-independent case with the opposite-sign up-type and down-type couplings, the limits are improved by about three orders of magnitude. For other cases, the limits are similar.\nTo complement the effective-field-theory models, limits are calculated for an UV-complete theory with a light mediator, the Higgs boson. The upper limit on the cross section of the Higgs boson production through WH and ZH modes and decay to invisible particles is 1.3 pb at 95% CL for m H = 125 GeV. Figure 11 shows the upper limit of the total cross section of WH and ZH processes with H → χ ¯ χ , normalized to the SM next-to-leading order prediction for the WH and ZH production cross section (0.8 pb for m H = 125 GeV), 97 which is 1.6 at 95% CL for m H = 125 GeV.\n\n\n
Fig. 11. ATLAS-derived limit on the Higgs boson cross section for decay to invisible particles divided by the cross section for decays to Standard Model particles as a function of m H at 95% CL, derived from the monoW/Z analysis with E miss T > 350 GeV. From Ref. 92.
\n\n3.4. Heavy-quark signatures\nFor operators generated by the exchange of a scalar mediator, couplings to light quarks are suppressed and the prospect of probing such interactions through the inclusive monojet channel at the LHC is limited. Dedicated searches focusing on bottom and top quark final states, occurring in processes as the ones shown in Fig. 12, have been proposed 98,99 to constrain this class of operators. A search in mono b -jets can significantly improve the current limits. The monob signal arises partly from direct production of b -quarks in association with dark matter (Fig. 12a), but the dominant component is from top-quark pair production in the kinematic\nregime where one top is boosted (Fig. 12c). A search for a top-quark pair + E miss T can strengthen the bounds even more; in this case signal and background would have very different missing energy distributions, providing a handle to disentangle one from the other.\n\n\n
Fig. 12. Some of the dominant diagrams contributing to WIMP associated production with (a) a bottom, (b) two bottom quarks and (c) a top quark pair. From Ref. 98.
\n\n→\n→\n→\n4. Searches for Supersymmetry\nSupersymmetry (SUSY) 100 is an extension of the Standard Model which assigns to each SM field a superpartner field with a spin differing by a half unit. SUSY provides elegant solutions to several open issues in the SM, such as the hierarchy problem and the grand unification. In particular, SUSY predicts the existence of a stable weakly interacting particle -the lightest supersymmetric particle (LSP)that has the pertinent properties to be a dark matter particle, thus providing a very compelling argument in favor of SUSY.\nSUSY searches in ATLAS 35 and CMS 36 experiments typically focus on events with high transverse missing energy, which can arise from (weakly interacting) LSPs, in the case of R -parity conserving SUSY, or from neutrinos produced in LSP decays, if R -parity is broken (c.f. Section 4.4). Hence, the event selection criteria of inclusive channels are based on large E miss T , no or few leptons ( e , µ ), many jets and/or b -jets, τ -leptons and photons. In addition, kinematical variables such as the transverse mass, M T , and the effective mass, M eff , assist in distinguishing further SUSY from SM events, whilst the effective transverse energy 101 can be useful to cross-check results, allowing a better and more robust identification of the SUSY mass scale, should a positive signal is found. Although the majority of the analysis simply look for an excess of events over the SM background, there is an increasing application of distribution shape fitting techniques. 102\nTypical SM backgrounds are top-quark production -including single-top-, W / Z in association with jets, dibosons and QCD multijet events. These are estimated using semi- or fully data-driven techniques. Although the various analyses are optimized for a specific SUSY scenario, the interpretation of the results are extended to various SUSY models or topologies.\nAnalyses exploring R -parity conserving SUSY models at LHC are roughly divided into inclusive searches for squarks and gluinos, for third-generation fermions, and for electroweak production (pairs of ˜ χ 0 , ˜ χ ± , ˜ glyph[lscript] ). Although these searches are designed and optimised to look for R -parity conserving SUSY, interpretation in terms of R -parity violating (RPV) models is also possible. Other analyses are purely motivated by oriented by RPV scenarios and/or by the expectation of long-lived sparticles. Recent summary results from each category of ATLAS and CMS searches are presented in this section.\n4.1. Gluinos and first two generations of quarks\nAt the LHC, supersymmetric particles are expected to be predominantly produced hadronically, i.e. through gluino-pair, squark-pair and squark-gluino production. Each of these (heavy) sparticles is going to decay into lighter ones in a cascade decay that finally leads to an LSP, which in most of the scenarios considered is the lightest neutralino ˜ χ 0 1 . The two LSPs would escape detection giving rise to high transverse missing energy, hence the search strategy followed is based on the detection of high E miss T , many jets and possibly energetic leptons. The analyses make extensive use of data-driven Standard Model background measurements.\nThe most powerful of the existing searches are based on all-hadronic final states with large missing transverse momentum. 103-105 In the 0-lepton search, events are selected based on a jet+ E miss T trigger, applying a lepton veto, requiring a minimum number of jets, high E miss T , and large azimuthal separation between the E miss T and reconstructed jets, in order to reject multijet background. In addition, searches for squark and gluino production in a final state with one or two leptons have been performed. 106-108 The events are categorized by whether the leptons have higher or lower momentum and are referred to as the hard and soft lepton channels respectively. The soft-lepton analysis which enhances the sensitivity of the search in the difficult kinematic region where the neutralino and gluino masses are close to each other forming the so-called compressed spectrum. 109,110 Leptons in the soft category are characterized by low leptonp T thresholds (6 -10 GeV) and such events are triggered by sufficient E miss T . Hard leptons pass a threshold of ∼ 25 GeV and are seeded with both lepton and E miss T triggers. Analyses based on the razor 111 variable have also been carried out by both experiments. 112,113\nThe major backgrounds ( t ¯ t , W +jets, Z +jets) are estimated by isolating each of them in a dedicated control region, normalizing the simulation to data in that control region, and then using the simulation to extrapolate the background expectations into the signal region. The multijet background is determined from the data by a matrix method. All other (smaller) backgrounds are estimated entirely from the simulation, using the most accurate theoretical cross sections available. To account for the cross-contamination of physics processes across control regions, the final estimate of the background is obtained with a simultaneous, combined fit to all control regions.\nIn the absence of deviations from SM predictions, limits for squark and gluino production are set. Figure 13 (left) illustrates the 95% CL limits set by ATLAS under the minimal Supergravity (mSUGRA) model in the ( m 0 , m 1 / 2 ) plane with the 0-lepton plus E miss T plus multijets analysis. 103 The remaining parameters are set to tan β = 30, A 0 = -2 m 0 , µ > 0, so as to acquire parameter-space points where the predicted mass of the lightest Higgs boson, h 0 , is near 125 GeV, i.e. compatible with the recently observed Higgs-like boson. 44-48 Exclusion limits are obtained by using the signal region with the best expected sensitivity at each point. By assumption, the mSUGRA model avoids both flavor-changing neutral currents and extra sources of CP violation. For masses in the TeV range, it typically predicts too much cold dark matter, however these predictions depend of the presence of stringy effects that may dilute 114 or enhance 115,116 the predicted relic dark matter density. In the mSUGRA case, the limit on squark mass reaches 1750 GeV and on gluino mass is 1400 GeV if the results of various analyses are deployed. 103,117-122\n\n\n
Fig. 13. Left: Exclusion limits at 95% CL for 8 TeV ATLAS multijets plus E miss T analysis in the ( m 0 , m 1 / 2 ) plane for the mSUGRA model. From Ref. 103. Right: Summary of observed and expected limits 112,123-126 for gluino pair production with gluino decaying via a 3-body decay to a top, an anti-top and a neutralino. From Ref. 127.
\n\n\n\n\n~\n4.2. Third-generation squarks\nThe previously presented limits from inclusive channels indicate that the masses of gluinos and first/second generation squarks are expected to be above 1 TeV. Nevertheless, in order to solve the hierarchy problem in a natural way, the masses of the stops, sbottoms, higgsinos and gluinos need to be below the TeV-scale to properly cancel the divergences in the Higgs mass radiative corrections. Despite their production cross sections being smaller than for the first and second generation\nsquarks, stop and sbottom may well be directly produced at the LHC and could provide the only direct observation of SUSY at the LHC in case the other sparticles are outside of the LHC energy reach. The lightest mass eigenstates of the sbottom and stop particles, ˜ b 1 and ˜ t 1 , could hence be produced either directly in pairs or through ˜ g pair production followed by ˜ g → ˜ b 1 b or ˜ g → ˜ t 1 t decays. Both cases will be discussed in the following.\nFor the aforementioned reasons, direct searches for third generation squarks have become a priority in both ATLAS and CMS. Such events are characterized by several energetic jets (some of them b -jets), possibly accompanied by light leptons, as well as high E miss T . A suite of channels have been considered, depending on the topologies allowed and the exclusions generally come with some assumptions driven by the shortcomings of the techniques and variables used, such as the requirement of 100% branching ratios into particular decay modes.\nIn the case of the gluino-mediated production of stops, a simplified scenario ('Gtt model'), where ˜ t 1 is the lightest squark but m ˜ g < m ˜ t 1 , has been considered. Pair production of gluinos is the only process taken into account since the mass of all other sparticles apart from the ˜ χ 0 1 are above the TeV scale. A three-body decay via off-shell stop is assumed for the gluino, yielding a 100% branching ratio for the decay ˜ g → t ¯ t ˜ χ 0 1 . The stop mass has no impact on the kinematics of the decay and the exclusion limits 112,123-126 set by the CMS experiment are presented in the ( m ˜ g , m ˜ χ 0 1 ) plane in the right panel of Fig. 13. For a massless LSP, gluinos with masses from 560 GeV to 1320 GeV are excluded.\nIf the gluino is also too heavy to be produced at the LHC, the only remaining possibility is the direct ˜ t 1 ˜ t 1 and ˜ b 1 ˜ b 1 production. If stop pairs are considered, two decay channels can be distinguished depending on the mass of the stop: ˜ t 1 → b ˜ χ ± 1 and ˜ t 1 → t ˜ χ 0 1 , as shown in the diagrams in Fig. 14. CMS and ATLAS carried out a wide range of different analyses in each of these modes at both 7 TeV and 8 TeV center-of-mass energy. In all these searches, the number of observed events has been found to be consistent with the SM expectation. Limits have been set by ATLAS on the mass of the scalar top for different assumptions on the mass hierarchy scalar topchargino-lightest neutralino. 128-138 A scalar top quark of mass of up to 480 GeV is excluded at 95% CL for a massless neutralino and a 150 GeV chargino. For a 300 GeV scalar top quark and a 290 GeV chargino, models with a neutralino with mass lower than 175 GeV are excluded at 95% CL.\nFor the case of a high-mass stop decaying to a top and neutralino ( ˜ t 1 → t ˜ χ 0 1 ), analyses requiring one, two or three isolated leptons, jets and large E miss T have been carried out. No significant excess of events above the rate predicted by the SM is observed and 95% CL upper limits are set on the stop mass in the stopneutralino mass plane. The region of excluded stop and neutralino masses is shown on the right panel of Fig. 14 for the CMS analyses. 139,140 Stop masses are excluded between 200 GeV and 750 GeV for massless neutralinos, and stop masses around 500 GeV are excluded along a line which approximately corresponds to neutralino masses up to 250 GeV.\n\n\n\nb\n~\n~\n\n\n
Fig. 14. Left: Diagrams of ˜ t 1 ˜ t 1 direct production with decays ˜ t 1 → b ˜ χ ± 1 (top) and ˜ t 1 → t ˜ χ 0 1 (bottom). Right: Summary of the dedicated CMS searches 139,140 for stop pair production based on pp collision data taken at √ s = 8 TeV. Exclusion limits at 95% CL are shown in the ( ˜ t 1 , ˜ χ 0 1 ) mass plane for channels targeting ˜ t 1 → b ˜ χ ± 1 and ˜ t 1 → t ˜ χ 0 1 . The dashed and solid lines show the expected and observed limits, respectively. From Ref. 127.
\n\n4.3. Electroweak gaugino production\nIf all squarks and gluinos are above the TeV scale, weak gauginos with masses of few hundred GeV may be the only sparticles accessible at the LHC. As an example, at √ s = 7 TeV, the cross-section of the associated production ˜ χ ± 1 ˜ χ 0 2 with degenerate masses of 200 GeV is above the 1-TeV gluino-gluino production cross section by one order of magnitude. Chargino pair production is searched for in events with two opposite-sign leptons and E miss T using a jet veto, through the decay ˜ χ ± 1 → glyph[lscript] ± ν ˜ χ 0 1 . A summary of related analyses 141,142 performed by CMS is shown in Fig. 15. Charginos with masses between 140 and 560 GeV are excluded for a massless LSP in the chargino-pair production with an intermediate slepton/sneutrino between the ˜ χ ± 1 and the ˜ χ 0 1 . If ˜ χ ± 1 ˜ χ 0 2 production is assumed instead, the limits range from 11 to 760 GeV. The corresponding limits involving intermediate W , Z and/or H are significantly weaker.\n3 In several analyses the EW sector of the MSSM has been studied for parameter choices that yield the correct DM relic density. In Ref. 143, the constraints coming from the trilepton/dilepton search by ATLAS and CMS from direct pair production of chargino and neutralino or slepton pair production have been considered and the implication on DM and collider searches have been investigated, while in Ref. 144 we have examined the search prospects of DM-allowed SUSY signals with several models in the light of LHC data.\n\n\n
Fig. 15. Summary of observed limits for electroweak-gaugino production from CMS. 141,142 From Ref. 127.
\n\n2\n1\n4.4. R -parity violating SUSY and meta-stable sparticles\nR -parity is defined as: R = ( -1) 3( B -L )+2 S , where B , L and S are the baryon number, lepton number and spin, respectively. Hence R = +1 for all Standard Model particles and R = -1 for all SUSY particles. It is stressed that the conservation of R -parity is an ad-hoc assumption. The only firm restriction comes from the proton lifetime: non-conservation of both B and L leads to rapid proton decay. R -parity conservation has serious consequences in SUSY phenomenology in colliders: the SUSY particles are produced in pairs and the lightest SUSY particle is absolutely stable, thus providing a WIMP candidate. Here we highlight the status of RPV supersymmetry 145 searches at the LHC.\nBoth ATLAS and CMS experiments have probed RPV SUSY through various channels, either by exclusively searching for specific decay chains, or by inclusively searching for multilepton events. ATLAS has looked for resonant production of eµ , eτ and µτ , 146-148 for multijets, 149 for events with at least four leptons 150 and for excesses in the eµ continuum. 151 Null inclusive searches in the one-lepton channel 152,153 have also been interpreted in the context of a model where RPV is induced through bilinear terms. 154-158\nglyph[negationslash]\nRecent CMS analyses are focused on studying the lepton number violating terms λ ijk L i L j ¯ e k and λ ' ijk L i Q j ¯ d k , which result in specific signatures involving leptons in events produced in pp collisions at LHC. A search for resonant production and the following decay of ˜ µ which is caused by λ ' 211 = 0 has been conducted. 159 Multilepton signatures caused by LSP decays due to various λ and λ ' terms in stop production have been probed. 160 Ref. 161 discusses the possibility of the generic model independent search for RPV SUSY in 4-lepton events. A summary of the limits set by several CMS analyses 126,160,162-164 are listed in Fig. 16.\n\n\n
Fig. 16. Best exclusion limits for the masses of the mother particles, for RPV scenarios, for each topology, for all CMS results. 126,160,162-164 In this plot, the lowest mass range is m mother = 0, but results are available starting from a certain mass depending on the analyses and topologies. Branching ratios of 100% are assumed, values shown in plot are to be interpreted as upper bounds on the mass limits. From Ref. 127.
\n\nIn view of the null results in other SUSY searches, it became mandatory to fully explore the SUSY scenario predicting meta-stable or long-lived particles. These particles, not present in the Standard Model, would provide striking signatures in the detector and rely heavily on a detailed understanding of its performance. In SUSY, non-prompt particle decay can be caused by (i) very weak RPV, 165 (ii) low mass difference between a SUSY particle and the LSP, 166 or (iii) very weak coupling to the gravitino in GMSB models. 167-170 Asmall part of these possibilities have been explored by the ATLAS 122 and CMS 127 experiments covering specific cases, difficult to summarize here. There is still a wide panorama of signatures to be explored, in view of various proposed SUSY scenarios pointing towards this direction.\nAs a last remark, we address the issue of (not necessarily cold) dark matter in RPV SUSY models. These seemingly incompatible concepts can be reconciled in models with a gravitino 171-173 or an axino 174 LSP with a lifetime exceeding the age of the Universe. In both cases, RPV is induced by bilinear terms in the superpotential that can also explain current data on neutrino masses and mixings without invoking any GUT-scale physics. 154-158 Decays of the next-to-lightest superparticle occur rapidly via RPV interaction, and thus they do not upset the Big-Bang nucleosynthesis, unlike the R -parity conserving case. Such gravitino DM is proposed\nin the context of µν SSM 175-177 with profound prospects for detecting γ rays from their decay. 178\n5. Looking for Extra Dimensions\nTheories with universal extra dimensions (UED) 179 are very promising for solving shortcomings of the Standard Model, such as explaining the three fermion generations in terms of anomaly cancellations and providing a mechanism for an efficient suppression of the proton decay. In the UED framework, unlike in other proposed extra-dimensional models, all SM particles are postulated to propagate in a TeV -1 -sized bulk , i.e. normal space plus the extra compactified dimensions. In addition, UED models can naturally incorporate a Z 2 symmetry called KK parity, analogous to R parity in supersymmetry, leading to a well-motivated dark matter a candidate, the lightest KK particle. 11\nIndirect constraints on the compactification radius R from electroweak precision tests and the dark matter relic density favor a mass scale for the first KK modes of O (1 TeV). Therefore UED models can be directly probed at the LHC, either through E miss T -based signatures or via searches for resonances near the TeV scale. Since the mass scale of the KK resonances is rather compressed, UED is only accessible through analyses based on soft leptons/jets and moderately-high missing transverse momentum. 180,181\nThe rich LHC phenomenology of UED models has been exploited to study the discovery reach or set limits based on already performed searches in leptonic 182-186 final states, photon 187 channels and through the Higgs sector. 188,189 In particular, several limits have been set on the minimal UED model (mUED), 190 in which only the 5D extensions of the SM operators are present at the cutoff scale Λ, whereas boundary operators and other higher-dimensional bulk operators are assumed to vanish at Λ. Existing CMS limits on the ratio R , defined as\nobtained by searching for resonances in the dilepton spectrum, 191 have been reinterpreted 186 to set bounds on the mass of the A (2) mode, as shown in Fig. 17 (left). This way, lower limits on m A (2) have been set at ∼ 1400 GeV.\nR = σ ( pp → Z ' + X → glyph[lscript]glyph[lscript] + X ) σ ( pp → Z + X → glyph[lscript]glyph[lscript] + X ) , (5)\nIn another analysis, 189 the Higgs sector of mUED is exploited to test this model at the LHC, by using combined ATLAS and CMS limits in the gg → h → γγ , gg → h → W + W -→ glyph[lscript] + ¯ νglyph[lscript] -ν and gg → h → ZZ → glyph[lscript] + glyph[lscript] -glyph[lscript] + glyph[lscript] -channels, based on 7 TeV and 8 TeV data. These limits lead to bounds on the mUED model in the ( m h , R -1 ) plane, m h being the Higgs mass, as shown in Fig. 17 (right). It is found that R -1 < 550 GeV is excluded at 95% CL, while for larger R -1 only a very narrow ( ± 1 -4 GeV) mass window around m h = 125 GeV, i.e. the mass of the recently\na Other species of extra dimension models have been probed thoroughly with LHC data, however they do not provide a viable DM candidate, hence they are beyond the scope of this article.\n\n\n\n\n\n
Fig. 17. Left: Ratio R defined in Eq. (5) for different benchmark points of the mUED model, as a function of the resonance mass M Res ≡ m A (2) . From Ref. 186. Right: 95% CL exclusion limits in the mUED ( m h , R -1 ) plane from Higgs boson searches at the LHC. The allowed region is in light green (light gray) and the excluded region is in red (medium gray). From Ref. 189.
\n\nobserved boson, 44,45 and another short window ∼ 118 GeV (for R -1 > 1 TeV) remain unconstrained.\n6. The Future: e + e -Colliders\nLinear e + e -accelerators of the next generation, namely the ILC 192 and the CLIC, 193 may have enough energy to produce and study WIMPs. The International Linear Collider (ILC) 13 is a 200 -500 GeV -extendable to 1 TeV- centerof-mass high-luminosity linear e + e -collider, based on 1 . 3 GHz superconducting radio-frequency accelerating technology. The Compact Linear Collider (CLIC), on the other hand, is a TeV-scale high-luminosity linear electron-positron collider based on a novel two-beam technique providing acceleration gradients at the level of 100 MV / m.\nPositron-electron colliders can play a major role in providing precision data for understanding dark matter, should it be discovered in colliders, among other measurements, due to three characteristics: (i) all energy of incoming particles is transferred to the final-state particles, allowing the setting of severe constraints on the mass of invisible particles; (ii) the cross sections of all production processes are of the same order of magnitude, thus making the decays of the BSM particles clearly visible; and (iii) the energy, projectile and polarization of the beam can be tuned to choose the optimal configuration for the physics of interest. All these features are instrumental in pinning down the properties of DM in such colliders.\nThe study of model-independent production of WIMP pairs at the linear collider through the monophoton channel, e + e -→ χ ¯ χγ , has shown that a WIMP in the mass range of ∼ 60 -200 GeV can be discovered with a 5 σ significance for an annihilation fraction of unity. 194 In terms of the effective dark matter model, it is found that the ILC should be able to probe couplings of 10 -7 GeV -2 or 10 -4 GeV -1 , depending on the mass dimension of the theory. 195 In model predicting vector dark\nmatter, the ILC may be able to probe even weaker couplings in the case of low DM mass.\nOnce DM is detected through a non-gravitational interaction, the new-particle mass may be constrained through methods based on matching specific decay chains to measurements of kinematic edges in invariant-mass distributions of two or three reconstructed objects. 196,197 This is one way to overcome the unconstrained kinematics of the production of two invisible particles in conjunction with the measurement of the momentum spectrum of the final-state leptons and the scanning of the particle pair production thresholds. 198\nThe determination of the spin of the new particle will play a major role in the identification of the DM nature. This issue has been studied thoroughly in the case of SUSY versus UED. 198 Both models feature a stable particle that is a viable DM candidate: the lightest neutralino, ˜ χ 0 1 , in SUSY, and the lightest KK excitation of the photon, γ (1) in UED. The fact that similar decay chains lead to those WIMP candidates, while their spins are different, can be exploited to distinguish them. In that case, the difference in the distribution shape of the muon polar angle, θ µ , for e + e -→ ˜ µ + ˜ µ -→ µ + µ -˜ χ 0 1 ˜ χ 0 1 and e + e -→ µ (1) µ (1) → µ + µ -γ (1) γ (1) is shown in Fig. 18 (left) for a study carried out for CLIC.\n\n\n\n\n\n
Fig. 18. Left: Differential cross section d σ/ dcos θ µ for UED (left) and supersymmetry (right) as a function of the muon scattering angle θ µ , including the effects of event selection, beamstrahlung and detector resolution and acceptance. The data points represent the sum of background and signal events, while the yellow (light grey) shaded area is the signal only. From Ref. 198. Right: Relic density for benchmark point LCC2. The three curves show the results for expected measurements from LHC (red), up-to-500 GeV ILC (magenta) and up-to-1 TeV ILC (blue). There are two overlapping very high peaks at Ω χ h 2 < 0 . 01, due to the wino and higgsino solutions to the LHC constraints. From Ref. 197.
FIG. 1: An asymptotically inertial trajectory in a Penrose diagram. In this example, z ( t ) = -1 2 sinh -1 ( e t ).
\n\nA moving mirror in flat spacetime reveals a connection between statistics and dynamics that is not present in Minkowski spacetime without acceleration. Statistics can be derived from the dynamics of the moving mirror as an accelerated mirror gives a beta Bogoliubov transformation coefficient which is non-zero. The statistics are determined by the algebra of the creation operators. The associated relations of the creation operators are commuting for Bose-Einstein statistics and anticommuting for Fermi-Dirac statistics. The following derivation shows that for the spin-zero field, only Bose-Einstein statistics are invariable with the flat spacetime dynamics that include an accelerated external boundary condition.\nThe field equation of motion is the Klein-Gordon equation /square ψ = 0. The moving mirror is imposed such that at the position of the mirror the field is zero, ψ | z = 0. This system has modes that can be used to expand the field,\nψ = ∫ ∞ 0 dω ' [ a ω ' φ ω ' + a † ω ' φ ∗ ω ' ] , (1)\nψ = ∫ ∞ 0 dω [ b ω χ ω + b † ω χ ∗ ω ] . (2)\n\n\n
FIG. 2: An asymptotically inertial trajectory with a final coasting speed of half the speed of light displayed in the usual spacetime diagram. The gray dashed lines represent the light cone, and the black dotted-dashed line shows the asymptote of the mirror trajectory. The trajectory example here is the same as Figure 1.
\n\nThe modes are orthonormal and complete. They take the null coordinate form,\nφ ω ' = (4 πω ' ) -1 / 2 [ e -iω ' v -e -iω ' p ( u ) ] , (3)\nχ ω = (4 πω ) -1 / 2 [ e -iωf ( v ) -e -iωu ] . (4)\nThe modes can be expanded in terms of each other,\nφ ω ' = ∫ ∞ 0 dω [ α ω ' ω χ ω + β ω ' ω χ ∗ ω ] , (5)\nχ ω = ∫ ∞ 0 dω ' [ α ∗ ω ' ω φ ω ' -β ω ' ω φ ∗ ω ' ] . (6)\nby introduction of Bogoliubov coefficients,\nα ω ' ω = ( φ ω ' , χ ω ) , β ω ' ω = -( φ ω ' , χ ∗ ω ) , (7)\ndefined by the flat-space scalar product, in spacetime coordinates,\n( φ ω ' , χ ω ) = i ∫ ∞ -∞ dx φ ∗ ω ' ↔ ∂ t χ ω , (8)\nor in null coordinates,\n( φ ω ' , χ ω ) = i ∫ ∞ -∞ du φ ∗ ω ' ↔ ∂ u χ ω + i ∫ ∞ -∞ dv φ ∗ ω ' ↔ ∂ v χ ω . (9)\nThe field equation combined with the moving mirror boundary condition imply that the Bogoliubov coefficients α ω ' ω and β ω ' ω relate the operators a ω ' and a † ω ' to the operators b ω and b † ω :\na ω ' = ∫ dω [ α ∗ ω ' ω b ω -β ∗ ω ' ω b † ω ] , (10)\nb ω = ∫ dω ' [ α ω ' ω a ω ' + β ∗ ω ' ω a † ω ' ] . (11)\nThe modes are orthonormal such that,\n( φ ω , φ ω ' ) = -( φ ∗ ω , φ ∗ ω ' ) = δ ( ω -ω ' ) , ( φ ω , φ ∗ ω ' ) = 0 , (12)\n( χ ω , χ ω ' ) = -( χ ∗ ω , χ ∗ ω ' ) = δ ( ω -ω ' ) , ( χ ω , χ ∗ ω ' ) = 0 . (13)\nSince the modes are related by a Bogoliubov transformation, the above scalar products imply several identities. Using (12), (5), and (13) the following Wronskian identity holds,\n∫ dω '' [ α ωω '' α ∗ ω ' ω '' -β ωω '' β ∗ ω ' ω '' ] = δ ( ω -ω ' ) . (14)\nNote that this relation is a derived consequence only of the field equation and the imposed asymptotically inertial mirror. This relation is a result of the dynamics.\nConsider now, the possible commutation relations of the creation operators, a ω ' , a † ω ' and b ω , b † ω (which are associated with the φ ω ' and χ ω modes, respectively),\n[ a ω , a † ω ' ] ± = δ ( ω ' -ω ) , [ a ω , a ω ' ] ± = [ a † ω , a † ω ' ] ± = 0 , (15)\n[ b ω , b † ω ' ] ± = δ ( ω ' -ω ) , [ b ω , b ω ' ] ± = [ b † ω , b † ω ' ] ± = 0 , (16)\nHere the + sign corresponds to Fermi-Dirac statistics from the anticommutator, while the -sign corresponds to Bose-Einstein statistics from the commutator. The relations associated with the φ ω ' modes may be expressed in terms of Bogoliubov coefficients using (10),\n[ a ω , a † ω ' ] ± = ∫ dω '' [ α ω '' ω α ∗ ω '' ω ' ± β ω '' ω ' β ∗ ω '' ω ] . (17)\nThe relations associated with the χ ω modes may be expressed in terms of Bogoliubov coefficients using (11),\n[ b ω , b † ω ' ] ± = ∫ dω '' [ α ω '' ω α ∗ ω '' ω ' ± β ω '' ω ' β ∗ ω '' ω ] . (18)\nThe dynamics-only result (14) dictates which sign to use in (17) and (18). Since the dynamics indicates the choice of the -sign, the election of Bose-Einstein statistics is straightforward. Similar identities to (14), using (12) and (13), dictate the correct sign to be used in (15) and (16) which give, in total,\n[ a ω , a † ω ' ] -= δ ( ω ' -ω ) , [ a ω , a ω ' ] -= [ a † ω , a † ω ' ] -= 0 , (19)\n[ b ω , b † ω ' ] -= δ ( ω ' -ω ) , [ b ω , b ω ' ] -= [ b † ω , b † ω ' ] -= 0 . (20)\nParticles at different times, (those associated with b † ω b ω and a † ω ' a ω ' ) obey the same statistics only in the BoseEinstein case. The Bose-Einstein case is then fixed with the dynamics of the field and mirror. When the mirror has acceleration, the β Bogoliubov transformation coefficient is not zero. If the mirror is not accelerating, then β = 0, and this connection between dynamics and statistics is absent, despite the presence of a boundary and the abandonment of Minkowski spacetime. The spinstatistics-dynamics connection in flat spacetime relies on a non-zero acceleration. The derivation presented above is an 'accelerated boundary condition in flat spacetime' -moving mirror- derivation of the relationship between spin and statistics.\nB. Infinitely Accelerated Mirror\nConsider a moving mirror which accelerates for all time. When a mirror in flat spacetime accelerates forever, a horizon singularity is formed. The horizon can be seen most easily in a Penrose diagram, (see Figure 3 or in a spacetime diagram in Figure 4). It is reasonable to ask if the physically pathological singularity breaks the dynamics-statistics connection. As it turns out, the acceleration singularity complicates the mathematics but does not ruin the dynamics-statistics link. The spin-statistics connection is apparent in the case of an infinitely accelerated boundary.\n\n\n
FIG. 3: Asymptotically infinite accelerating trajectories in a Penrose diagram. In this figure, all mirrors emit thermal radiation at all times with T = κ/ 2 π along the trajectory z ( t ) = -t -1 κ W ( e -2 κt ), where κ = π/ 8 , π/ 4 , π/ 2 , π, 2 π .
\n\nIn this section only, we adopt Carlitz-Willey's normalization, transformation conventions and left-right con-\n\n\n
FIG. 4: The asymptotically infinite accelerating trajectory of the Carlitz-Willey mirror in a spacetime diagram. The mirror emits thermal radiation at all times with T = κ/ 2 π along the trajectory z ( t ) = -t -1 κ W ( e -2 κt ). Notice the Carlitz-Willey mirror does not start statically, and also approaches the speed of light in the future with an acceleration singularity, resulting in a horizon.
\n\nstruction [13]. The advantage here is the simplicity of the explicit mode forms. The field motion is the KleinGordon equation /square ψ ( u, v ) = 0, and the field is written as\nψ = 1 4 π ∫ ∞ 0 dω ' ω ' [ a ω ' φ ω ' + a † ω ' φ ∗ ω ' ] (21)\nψ = 1 4 π ∑ I ∫ ∞ 0 dω ω [ b I ω χ I ω + b I † ω χ I ∗ ω ] (22)\nwhere I = R,L . The modes take the simplified form,\nφ ω ' = e -iω ' v -e -iω ' p ( u ) (23)\n{ χ R ω = e -iωf R ( v ) Θ( -v ) -e -iωu χ L ω = e -iωf L ( v ) Θ( v ) (24)\nwhile the ray-tracing functions for the specified CarlitzWilley trajectory assume\np ( u ) = -e -κu κ (25)\n{ f R ( v ) = -κ -1 ln( -κv ) v < 0 f L ( v ) = + κ -1 ln(+ κv ) v > 0 . (26)\nCarlitz and Willey point out that a constant energy flux emitted by the mirror, κ 2 48 π , is obtained by the ray tracing function of (25). Substituting p ( u ) = t + z ( t ) into (25) gives the resulting condition on the trajectory,\nt + z ( t ) = -1 κ e -κt + κz ( t ) . (27)\nCarlitz and Willey did not provide an analytic solution to this equation. However one can be obtained using the Lambert W function (the product logarithm) with the result that\nz ( t ) = -t -1 κ W ( e -2 κt ) . (28)\nFigure 3 and Figure 4 are plots of this trajectory. It is straightforward to show that ˙ z → ∓ 1 in the limits t → ±∞ and that z < 0 for all time. The proper acceleration is\nα ( t ) = -κ 2 √ W ( e -2 κt ) . (29)\nNote that the proper acceleration is not constant even though the energy flux is. The acceleration 29 is required to have thermal emission, for all times, in the moving mirror model. Contrast this to the Unruh effect, where time-independent acceleration, κ , is responsible for thermal emission.\nThe dynamics of the field-mirror system give rise to the Bogoliubov coefficients α I ω ' ω and β I ω ' ω which relate the operators a ω ' and a † ω ' to the operators b I ω and b I † ω :\na ω ' = 1 4 π ∑ I ∫ dω ω [ α I ω ' ω b I ω + β I ω ' ω b I † ω ] (30)\nSince the operators are related by a Bogoliubov transformation, it now pays to examine the the conserved scalar products for the two distinct modes φ and χ , (here J = L, R ). Specifically,\ni ∫ ∞ -∞ dvφ ∗ ω ' ↔ ∂ v φ ω = 4 πωδ ( ω -ω ' ) (31)\ni ∫ ∞ 0 dvχ I ∗ ω ' ↔ ∂ v χ J ω + i ∫ ∞ -∞ duχ I ∗ ω ' ↔ ∂ u χ J ω = 4 πωδ ( ω -ω ' ) δ IJ (32)\nThe conserved scalar products imply several identities, as first shown by [13], in particular:\n1 4 π ∑ I ∫ dω ω [ α I ω ' ω α I ∗ ω '' ω -β I ω ' ω β I ∗ ω '' ω ] = 4 πω ' δ ( ω '' -ω ' ) (33)\n1 4 π ∑ I ∫ dω ω [ α I ω ' ω β I ω '' ω -β I ω ' ω α I ω '' ω ] = 0 (34)\n1 4 π ∫ dω ' ω ' [ α I ∗ ω ' ω α J ω ' ω '' -β I ω ' ω β J ∗ ω ' ω '' ] = 4 πωδ ( ω -ω '' ) δ IJ (35)\n1 4 π ∫ dω ' ω ' [ α I ∗ ω ' ω β J ω ' ω '' -β I ω ' ω α J ∗ ωω '' ] = 0 (36)\nThese relations are derived consequences of the field equation and the imposed Carlitz-Willey mirror. Now consider the possible commutations relations\n[ a ω , a † ω ' ] ± = 4 πωδ ( ω -ω ' ) (37)\n[ a ω , a ω ' ] ± = [ a † ω , a † ω ' ] ± = 0 (38)\nAs before, the + sign corresponds to FD statistics, while the -sign corresponds to BE statistics. The possible commutation relations in terms of the transformation coefficients are\n[ a ω , a † ω ' ] ± = 1 4 π ∫ dω '' ω '' [ α I ∗ ω '' ω α J ω '' ω ' -β I ω '' ω β J ∗ ω '' ω ' ] (39)\nThe dynamics result (35) can be used to determine which of the possible commutation relations are to be used in (39). The dynamics dictates the commutation relations use the -sign,\n[ a ω , a † ω ' ] -= 4 πωδ ( ω -ω ' ) (40)\nThe acceleration singularity does not corrupt the spinstatistics-dynamics connection.\nC. Unruh Effect\nConsider the spin-statistics connection in the setting of the Unruh effect. We specify to the right movers in the right quadrant of the Rindler wedge, x > 0 and x > | t | . The left movers follow suit. Further consideration to the left Rindler wedge is similar to the right Rindler wedge [14]. To satisfy completeness on the full Cauchy surface, both the left and right movers, and both the left and right Rindler wedges must be treated. It is found that correct commutations relations are given due to the presence of acceleration for the Rindler observer. We start with the Minkowski metric, ds 2 = -dt 2 + dx 2 , and KleinGordon field equation ( -∂ 2 t + ∂ 2 x ) ψ = 0. Transformation to Rindler coordinates,\nt = ρ sinh κτ, x = ρ cosh κτ (41)\nwhere ρ > 0 and -∞ < τ < ∞ , gives the metric and field equation,\nds 2 = -ρ 2 κ 2 dτ 2 + dρ 2 , (42)\n[ -1 κ 2 ∂ 2 ( ∂τ ) 2 + ∂ 2 [ ∂ (ln κρ )] 2 ] ψ = 0 . (43)\nThe mode functions that are right-movers, (dependent only on U ) and the right quadrant, R, are shown graphically in Figure 5. The left-movers are dependent on V and follow the analogous forthcoming procedure. Specifying to these right-moving eigenmodes and the R\n\n\n
FIG. 5: Right movers in the right-quadrant Rindler wedge. In this shaded spacetime region, U < 0, and V > 0. The hyperbolas are constant τ values, while the radial lines are constant ρ values.
\n\nspacetime region is clarifying. This is done by the use of null coordinates,\nu v } = τ ∓ ln κρ κ , (44)\nand\nU V } = t ∓ x = { -κ -1 e -κu κ -1 e κv . (45)\nConsider now, the two representations of the field:\nψ ( U ) = ψ R ( U ) = ∫ ∞ 0 dω ' [ a ω ' φ ω ' + a † ω ' φ ∗ ω ' ] (46)\nand\nψ R ( U ) = θ ( -U ) ∫ ∞ 0 dω [ b R ω χ R ω + b R † ω χ R ∗ ω ] (47)\nNotice that we have specified to the right quadrant by the use of θ ( -U ) and restricted our concern to right movers by utilizing the single U coordinate dependence. The\nadvantage here is that left and right movers in the null coordinates U and V do not mix under the Bogoliubov transformation.\nThe modes take the form,\nφ ω ' = 1 √ 4 πω ' e -iω ' U , (48)\nχ R ω = θ ( -U ) 1 √ 4 πω ( -κU ) iω κ . (49)\nThe Bogoliubov transformation for the operator associated with the φ ω ' takes the R-form,\na ω ' = ∫ ∞ 0 dw [ α R ∗ ω ' ω b R ω -β R ∗ ω ' ω b R † ω ] (50)\nand the operator associated with the χ R ω is\nb R ω = ∫ ∞ 0 dw ' [ α R ω ' ω a ω ' + β R ∗ ω ' ω a † ω ' ] (51)\nThe modes are non-trivially expanded in terms of each other,\nφ ω ' = ∫ ∞ 0 dω [ α R ω ' ω χ R ω + β R ω ' ω χ R ∗ ω ] , (52)\nχ R ω = ∫ ∞ 0 dω ' [ α R ∗ ω ' ω φ ω ' -β R ω ' ω φ ∗ ω ' ] . (53)\nby introduction of Bogoliubov coefficients,\nα R ω ' ω = ( φ ω ' , χ R ω ) , β R ω ' ω = -( φ ω ' , χ R ∗ ω ) , (54)\ndefined by the flat-space scalar product, in spacetime coordinates,\n( φ ω ' , χ R ω ) = i ∫ ∞ 0 dx φ ∗ ω ' ↔ ∂ t χ R ω , (55)\nor in null coordinates,\n( φ ω ' , χ R ω ) = i ∫ 0 -∞ dU φ ∗ ω ' ↔ ∂ U χ R ω (56)\nExpanding ( χ R ω , χ R ω ' ) in terms of φ ω ' using the transformation (53) and utilizing the orthonormality of the φ ω ' modes, the R-completeness relation holds\n∫ ∞ 0 dω '' [ α R ωω '' α R ∗ ω ' ω '' -β R ωω '' β R ∗ ω ' ω '' ] = δ ( ω -ω ' ) . (57)\nThis is contingent on the orthonormality of both sets of distinct modes by the conserved inner product,\n( φ ω , φ ω ' ) = -( φ ∗ ω , φ ∗ ω ' ) = δ ( ω -ω ' ) , (58)\n( χ R ω , χ R ω ' ) = -( χ R ∗ ω , χ R ∗ ω ' ) = δ ( ω -ω ' ) . (59)\nand where ( φ ω , φ ∗ ω ' ) = ( χ R ω , χ R ∗ ω ' ) = 0. As we have seen in a complementary fashion in Section II A, the possible commutation relations are\n[ a ω , a † ω ' ] ± = ∫ ∞ 0 dω '' [ α R ωω '' α R ∗ ω ' ω '' ± β R ωω '' β R ∗ ω ' ω '' ] , (60)\n[ b R ω , b R † ω ' ] ± = ∫ ∞ 0 dω '' [ α R ωω '' α R ∗ ω ' ω '' ± β R ωω '' β R ∗ ω ' ω '' ] . (61)\nThe Wronskian R-completeness relation (57) from the acceleration dynamics connects to the spin-statistics by necessitating the use of the negative sign in the commutation relations (60) and (61). It is dictated that\n[ a ω , a † ω ' ] -= [ b R ω , b R † ω ' ] -= δ ( ω -ω ' ) (62)\nand similarly, the zero commuting relations stem from the analogous identities to (57). Therefore, in totality, the accelerated Rindler observer entails the spin-statistics connection.\nIII. CONCLUSIONS\nThe crux of these results are dependent on the presence of acceleration. In the cases presented, acceleration arises as an asymptotically inertial mirror, an infinitely accelerated mirror and the constant acceleration Unruh effect. That is, in the far future these examples correspond to a proper acceleration α → 0 , ∞ , κ , respectively. These new results where found: 1) The assumption of particles at early times obeying late time statistics is un-necessary, as only commutation relations are acceptable in either region, as shown in the asymptotically inertial mirror case. 2) In a non-trivial system, the presence of external acceleration conditions is enough to provide the link to commutation relations. This is in contrast to the specification to dynamic curved spacetime. Flat-spacetime acceleration conditions allowing non-trivial Bogoliubov transformations establishes the spin-statistics connection. 3) With regards to thermal emission, the time-dependent expression for acceleration in the moving mirror case was explicitly revealed. This clarifies a salient difference between the Unruh effect and the dynamical Casimir effect.\nAn acceleration in these above situations gives rise to distinct eigenmode solutions that can be used to represent the field. One set of eigenmodes can be expressed in mixed positive and negative frequencies pieces of the other set of eigenmodes. The spin-statistics connection is possible to obtain because of the properties of the conserved inner product and of the existence of these sets of eigenmodes in the first place. If there was zero acceleration, the Bogoliubov transformation would be trivial and the distinct sets of eigenmodes would not exist.\nAcknowledgments\nMRRG is grateful for discussions with Paul R. Anderson, Charles R. Evans and Xiong Chi. MRRG appre-\n\n[1] Leonard Parker. Quantized fields and particle creation in expanding universes. 1. Phys.Rev. , 183:1057-1068, 1969.\n[2] Leonard Parker. Quantized fields and particle creation in expanding universes. ii. Phys. Rev. D , 3:346-356, Jan 1971.\n[3] Leonard Parker and Yi Wang. Statistics from dynamics in curved spacetime. Phys. Rev. D , 39:3596-3605, Jun 1989.\n[4] J. W. Goodison and D. J. Toms. No generalized statistics from dynamics in curved spacetime. Phys. Rev. Lett. , 71:3240-3242, Nov 1993.\n[5] V. Bardek, S. Meljanac, and A. Perica. Generalized statistics and dynamics in curved space-time. Phys.Lett. , B338:20-22, 1994.\n[6] J.W. Goodison. Calogero-Vasilev oscillator in dynamically evolving curved space-time. Phys.Lett. , B350:17-21, 1995.\n[7] Atsushi Higuchi, Leonard Parker, and Yi Wang. Consistency of faddeev-popov ghost statistics with gravitationally induced pair creation. Phys. Rev. D , 42:4078-4081, Dec 1990.\n[8] Leonard E Parker and David J Toms. Quantum field\n\nciates the financial support and hospitality of the IASSingapore.\n\ntheory in curved spacetime: quantized fields and gravity . Cambridge monographs on mathematical physics. Cambridge Univ. Press, New York, NY, 2009.\n[9] C.M. et al. Wilson. Observation of the Dynamical Casimir Effect in a Superconducting Circuit, arXiv:1105.4714v1[quant-ph]. 2011.\n[10] Robert M. Wald. Existence of the S Matrix in Quantum Field Theory In Curved Space-time. Annals Phys. , 118:490-510, 1979.\n[11] P. C. W. Davies and S. A. Fulling. Radiation from a moving mirror in two-dimensional space- time conformal anomaly. Proc. Roy. Soc. Lond. , A348:393-414, 1976.\n[12] P. C. W. Davies and S. A. Fulling. Radiation from Moving Mirrors and from Black Holes. Proc. Roy. Soc. Lond. , A356:237, 1977.\n[13] Robert D. Carlitz and Raymond S. Willey. Reflections on Moving Mirrors. Phys. Rev. , D36:2327, 1987.\n[14] R. Brout, S. Massar, R. Parentani, and Ph. Spindel. A Primer for black hole quantum physics. Phys.Rept. , 260:329-454, 1995.\n"},"sections":{"kind":"list like","value":[{"title":"On Spin-Statistics and Bogoliubov Transformations in Flat Spacetime With Acceleration Conditions","content":"Michael R.R. Good ∗ Institute of Advanced Studies, Singapore (Dated: September 26, 2018) A single real scalar field of spin zero obeying the Klein-Gordon equation in flat spacetime under external conditions is considered in the context of the spin-statistics connection. An imposed accelerated boundary on the field is made to become, in the far future, (1) asymptotically inertial and (2) asymptotically non-inertial (with an infinite acceleration). The constant acceleration Unruh effect is also considered. The systems involving non-trivial Bogoliubov transformations contain dynamics which point to commutation relations. Particles described by in-modes obey the same statistics as particles described by out-modes. It is found in the non-trivial systems that the spin-statistics connection can be manifest from the acceleration. The equation of motion for the boundary which forever emits thermal radiation is revealed. PACS numbers: 03.70.+k, 04.62.+v, 04.60.-m Keywords: dynamical Casimir effect, Unruh effect, external conditions, moving mirrors","pages":[1]},{"title":"I. INTRODUCTION","content":"Parker demonstrated that the spin-statistics connection is evident from the dynamics of curved spacetime.[1] [2] [3] It was first shown using a quantized real scalar field of spin zero which obeys the curved spacetime KleinGordon equation[1]. It was then found that consistency resulted when commutation relations imposed on the creation and annihilation operators for the in-region, were also imposed for the creation and annihilation operators for the out-region. It is not consistent with the dynamics to impose anti-commutation relations for the out-region. Parker's derivation was extended to fermions of spin1/2 which obey the Dirac equation and satisfy anticommutation relations [2]. Higher spin and parastatistics was treated [3], the statistics were generalized [4], [5], [6], and ghost fields [7] were examined. It is assumed [8] that a connection between statistics and dynamics is not present in Minkowski spacetime. It is also generally accepted that particles at late times should obey the same statistics as at early times. Recently, researchers may have presented the first experimental evidence for observation of the dynamical Casimir effect [9]. Within the context of an imposed boundary in flat spacetime, the question arises whether the connection between spin and statistics is revealed from the dynamics of a field-boundary system in the same way as from the dynamics of curved spacetime. In this note, it is shown that if a quantized real scalar field of spin-0 obeys the flat spacetime Klein-Gordon equation with an imposed accelerated boundary condition, then commutation relations are the admissible algebra that may be imposed upon the creation and annihilation operators on the in-region and the out-region. It will not be consistent with the dynamics to establish anti-commutation relations upon this field-boundary system. As is pointed out by Wald [10], the spin-statistics results from curved spacetime depend on the conserved inner product. Wald emphasized that no reasonable quantum field theory in curved spacetime (or in flat spacetime with an external potential), can have Bose-Einstein statistics if the inner product is positive definite for all positive and negative frequency solutions. Likewise, it's unreasonable to have Fermi-Dirac statistics if the inner product is positive definite only on positive frequency solutions. In this note, it is emphasized that acceleration induces the appearance of distinct mode solutions with their respective positive and negative frequency pieces. These distinct modes give rise to a non-trivial Bogoliubov transformation which dictates the commutation relations. In the following three sections, it is shown that an acceleration ultimately gives rise to the spin-statistics connection in three salient examples: an asymptotically inertial mirror in Section II A (the most simple case), an infinitely accelerated mirror in Section II B and the Unruh effect in Section II C.","pages":[1]},{"title":"A. Asymptotically Inertial Mirror","content":"The moving mirror model is the most simple example of the dynamical Casimir effect [11] [12]. Consider a moving mirror (an external boundary condition in 1+1 dimensions) which does not accelerate forever. This mirror starts (ends) at time-like past (future) infinity, asymptotically having zero acceleration in the far past (future). As such, an asymptotically inertial mirror will contain no horizons or pathological acceleration singularities, as seen in Figure 1 and Figure 2. The use of an asymptotically inertial mirror is chosen for clarity of derivation. Relaxing the condition of asymptotically inertial motion will alter the form of the derivation and the complexity of the relevant identities. The connection between acceleration and the spin-statistics connection remains. The use of the forever accelerating observer in the Unruh effect, or a forever accelerating mirror that results in a null horizon, requires the appropriate left-right construction (for example, [13]) to incorporate complete Cauchy surface information. A moving mirror in flat spacetime reveals a connection between statistics and dynamics that is not present in Minkowski spacetime without acceleration. Statistics can be derived from the dynamics of the moving mirror as an accelerated mirror gives a beta Bogoliubov transformation coefficient which is non-zero. The statistics are determined by the algebra of the creation operators. The associated relations of the creation operators are commuting for Bose-Einstein statistics and anticommuting for Fermi-Dirac statistics. The following derivation shows that for the spin-zero field, only Bose-Einstein statistics are invariable with the flat spacetime dynamics that include an accelerated external boundary condition. The field equation of motion is the Klein-Gordon equation /square ψ = 0. The moving mirror is imposed such that at the position of the mirror the field is zero, ψ | z = 0. This system has modes that can be used to expand the field, The modes are orthonormal and complete. They take the null coordinate form, The modes can be expanded in terms of each other, by introduction of Bogoliubov coefficients, defined by the flat-space scalar product, in spacetime coordinates, or in null coordinates, The field equation combined with the moving mirror boundary condition imply that the Bogoliubov coefficients α ω ' ω and β ω ' ω relate the operators a ω ' and a † ω ' to the operators b ω and b † ω : The modes are orthonormal such that, Since the modes are related by a Bogoliubov transformation, the above scalar products imply several identities. Using (12), (5), and (13) the following Wronskian identity holds, Note that this relation is a derived consequence only of the field equation and the imposed asymptotically inertial mirror. This relation is a result of the dynamics. Consider now, the possible commutation relations of the creation operators, a ω ' , a † ω ' and b ω , b † ω (which are associated with the φ ω ' and χ ω modes, respectively), Here the + sign corresponds to Fermi-Dirac statistics from the anticommutator, while the -sign corresponds to Bose-Einstein statistics from the commutator. The relations associated with the φ ω ' modes may be expressed in terms of Bogoliubov coefficients using (10), The relations associated with the χ ω modes may be expressed in terms of Bogoliubov coefficients using (11), The dynamics-only result (14) dictates which sign to use in (17) and (18). Since the dynamics indicates the choice of the -sign, the election of Bose-Einstein statistics is straightforward. Similar identities to (14), using (12) and (13), dictate the correct sign to be used in (15) and (16) which give, in total, Particles at different times, (those associated with b † ω b ω and a † ω ' a ω ' ) obey the same statistics only in the BoseEinstein case. The Bose-Einstein case is then fixed with the dynamics of the field and mirror. When the mirror has acceleration, the β Bogoliubov transformation coefficient is not zero. If the mirror is not accelerating, then β = 0, and this connection between dynamics and statistics is absent, despite the presence of a boundary and the abandonment of Minkowski spacetime. The spinstatistics-dynamics connection in flat spacetime relies on a non-zero acceleration. The derivation presented above is an 'accelerated boundary condition in flat spacetime' -moving mirror- derivation of the relationship between spin and statistics.","pages":[1,2,3]},{"title":"B. Infinitely Accelerated Mirror","content":"Consider a moving mirror which accelerates for all time. When a mirror in flat spacetime accelerates forever, a horizon singularity is formed. The horizon can be seen most easily in a Penrose diagram, (see Figure 3 or in a spacetime diagram in Figure 4). It is reasonable to ask if the physically pathological singularity breaks the dynamics-statistics connection. As it turns out, the acceleration singularity complicates the mathematics but does not ruin the dynamics-statistics link. The spin-statistics connection is apparent in the case of an infinitely accelerated boundary. In this section only, we adopt Carlitz-Willey's normalization, transformation conventions and left-right con- struction [13]. The advantage here is the simplicity of the explicit mode forms. The field motion is the KleinGordon equation /square ψ ( u, v ) = 0, and the field is written as where I = R,L . The modes take the simplified form, while the ray-tracing functions for the specified CarlitzWilley trajectory assume Carlitz and Willey point out that a constant energy flux emitted by the mirror, κ 2 48 π , is obtained by the ray tracing function of (25). Substituting p ( u ) = t + z ( t ) into (25) gives the resulting condition on the trajectory, Carlitz and Willey did not provide an analytic solution to this equation. However one can be obtained using the Lambert W function (the product logarithm) with the result that Figure 3 and Figure 4 are plots of this trajectory. It is straightforward to show that ˙ z → ∓ 1 in the limits t → ±∞ and that z < 0 for all time. The proper acceleration is Note that the proper acceleration is not constant even though the energy flux is. The acceleration 29 is required to have thermal emission, for all times, in the moving mirror model. Contrast this to the Unruh effect, where time-independent acceleration, κ , is responsible for thermal emission. The dynamics of the field-mirror system give rise to the Bogoliubov coefficients α I ω ' ω and β I ω ' ω which relate the operators a ω ' and a † ω ' to the operators b I ω and b I † ω : Since the operators are related by a Bogoliubov transformation, it now pays to examine the the conserved scalar products for the two distinct modes φ and χ , (here J = L, R ). Specifically, The conserved scalar products imply several identities, as first shown by [13], in particular: These relations are derived consequences of the field equation and the imposed Carlitz-Willey mirror. Now consider the possible commutations relations As before, the + sign corresponds to FD statistics, while the -sign corresponds to BE statistics. The possible commutation relations in terms of the transformation coefficients are The dynamics result (35) can be used to determine which of the possible commutation relations are to be used in (39). The dynamics dictates the commutation relations use the -sign, The acceleration singularity does not corrupt the spinstatistics-dynamics connection.","pages":[3,4,5]},{"title":"C. Unruh Effect","content":"Consider the spin-statistics connection in the setting of the Unruh effect. We specify to the right movers in the right quadrant of the Rindler wedge, x > 0 and x > | t | . The left movers follow suit. Further consideration to the left Rindler wedge is similar to the right Rindler wedge [14]. To satisfy completeness on the full Cauchy surface, both the left and right movers, and both the left and right Rindler wedges must be treated. It is found that correct commutations relations are given due to the presence of acceleration for the Rindler observer. We start with the Minkowski metric, ds 2 = -dt 2 + dx 2 , and KleinGordon field equation ( -∂ 2 t + ∂ 2 x ) ψ = 0. Transformation to Rindler coordinates, where ρ > 0 and -∞ < τ < ∞ , gives the metric and field equation, The mode functions that are right-movers, (dependent only on U ) and the right quadrant, R, are shown graphically in Figure 5. The left-movers are dependent on V and follow the analogous forthcoming procedure. Specifying to these right-moving eigenmodes and the R spacetime region is clarifying. This is done by the use of null coordinates, and Consider now, the two representations of the field: and Notice that we have specified to the right quadrant by the use of θ ( -U ) and restricted our concern to right movers by utilizing the single U coordinate dependence. The advantage here is that left and right movers in the null coordinates U and V do not mix under the Bogoliubov transformation. The modes take the form, The Bogoliubov transformation for the operator associated with the φ ω ' takes the R-form, and the operator associated with the χ R ω is The modes are non-trivially expanded in terms of each other, by introduction of Bogoliubov coefficients, defined by the flat-space scalar product, in spacetime coordinates, or in null coordinates, Expanding ( χ R ω , χ R ω ' ) in terms of φ ω ' using the transformation (53) and utilizing the orthonormality of the φ ω ' modes, the R-completeness relation holds This is contingent on the orthonormality of both sets of distinct modes by the conserved inner product, and where ( φ ω , φ ∗ ω ' ) = ( χ R ω , χ R ∗ ω ' ) = 0. As we have seen in a complementary fashion in Section II A, the possible commutation relations are The Wronskian R-completeness relation (57) from the acceleration dynamics connects to the spin-statistics by necessitating the use of the negative sign in the commutation relations (60) and (61). It is dictated that and similarly, the zero commuting relations stem from the analogous identities to (57). Therefore, in totality, the accelerated Rindler observer entails the spin-statistics connection.","pages":[5,6]},{"title":"III. CONCLUSIONS","content":"The crux of these results are dependent on the presence of acceleration. In the cases presented, acceleration arises as an asymptotically inertial mirror, an infinitely accelerated mirror and the constant acceleration Unruh effect. That is, in the far future these examples correspond to a proper acceleration α → 0 , ∞ , κ , respectively. These new results where found: 1) The assumption of particles at early times obeying late time statistics is un-necessary, as only commutation relations are acceptable in either region, as shown in the asymptotically inertial mirror case. 2) In a non-trivial system, the presence of external acceleration conditions is enough to provide the link to commutation relations. This is in contrast to the specification to dynamic curved spacetime. Flat-spacetime acceleration conditions allowing non-trivial Bogoliubov transformations establishes the spin-statistics connection. 3) With regards to thermal emission, the time-dependent expression for acceleration in the moving mirror case was explicitly revealed. This clarifies a salient difference between the Unruh effect and the dynamical Casimir effect. An acceleration in these above situations gives rise to distinct eigenmode solutions that can be used to represent the field. One set of eigenmodes can be expressed in mixed positive and negative frequencies pieces of the other set of eigenmodes. The spin-statistics connection is possible to obtain because of the properties of the conserved inner product and of the existence of these sets of eigenmodes in the first place. If there was zero acceleration, the Bogoliubov transformation would be trivial and the distinct sets of eigenmodes would not exist.","pages":[6]},{"title":"Acknowledgments","content":"MRRG is grateful for discussions with Paul R. Anderson, Charles R. Evans and Xiong Chi. MRRG appre- ciates the financial support and hospitality of the IASSingapore.","pages":[7]}],"string":"[\n {\n \"title\": \"On Spin-Statistics and Bogoliubov Transformations in Flat Spacetime With Acceleration Conditions\",\n \"content\": \"Michael R.R. Good ∗ Institute of Advanced Studies, Singapore (Dated: September 26, 2018) A single real scalar field of spin zero obeying the Klein-Gordon equation in flat spacetime under external conditions is considered in the context of the spin-statistics connection. An imposed accelerated boundary on the field is made to become, in the far future, (1) asymptotically inertial and (2) asymptotically non-inertial (with an infinite acceleration). The constant acceleration Unruh effect is also considered. The systems involving non-trivial Bogoliubov transformations contain dynamics which point to commutation relations. Particles described by in-modes obey the same statistics as particles described by out-modes. It is found in the non-trivial systems that the spin-statistics connection can be manifest from the acceleration. The equation of motion for the boundary which forever emits thermal radiation is revealed. PACS numbers: 03.70.+k, 04.62.+v, 04.60.-m Keywords: dynamical Casimir effect, Unruh effect, external conditions, moving mirrors\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Parker demonstrated that the spin-statistics connection is evident from the dynamics of curved spacetime.[1] [2] [3] It was first shown using a quantized real scalar field of spin zero which obeys the curved spacetime KleinGordon equation[1]. It was then found that consistency resulted when commutation relations imposed on the creation and annihilation operators for the in-region, were also imposed for the creation and annihilation operators for the out-region. It is not consistent with the dynamics to impose anti-commutation relations for the out-region. Parker's derivation was extended to fermions of spin1/2 which obey the Dirac equation and satisfy anticommutation relations [2]. Higher spin and parastatistics was treated [3], the statistics were generalized [4], [5], [6], and ghost fields [7] were examined. It is assumed [8] that a connection between statistics and dynamics is not present in Minkowski spacetime. It is also generally accepted that particles at late times should obey the same statistics as at early times. Recently, researchers may have presented the first experimental evidence for observation of the dynamical Casimir effect [9]. Within the context of an imposed boundary in flat spacetime, the question arises whether the connection between spin and statistics is revealed from the dynamics of a field-boundary system in the same way as from the dynamics of curved spacetime. In this note, it is shown that if a quantized real scalar field of spin-0 obeys the flat spacetime Klein-Gordon equation with an imposed accelerated boundary condition, then commutation relations are the admissible algebra that may be imposed upon the creation and annihilation operators on the in-region and the out-region. It will not be consistent with the dynamics to establish anti-commutation relations upon this field-boundary system. As is pointed out by Wald [10], the spin-statistics results from curved spacetime depend on the conserved inner product. Wald emphasized that no reasonable quantum field theory in curved spacetime (or in flat spacetime with an external potential), can have Bose-Einstein statistics if the inner product is positive definite for all positive and negative frequency solutions. Likewise, it's unreasonable to have Fermi-Dirac statistics if the inner product is positive definite only on positive frequency solutions. In this note, it is emphasized that acceleration induces the appearance of distinct mode solutions with their respective positive and negative frequency pieces. These distinct modes give rise to a non-trivial Bogoliubov transformation which dictates the commutation relations. In the following three sections, it is shown that an acceleration ultimately gives rise to the spin-statistics connection in three salient examples: an asymptotically inertial mirror in Section II A (the most simple case), an infinitely accelerated mirror in Section II B and the Unruh effect in Section II C.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"A. Asymptotically Inertial Mirror\",\n \"content\": \"The moving mirror model is the most simple example of the dynamical Casimir effect [11] [12]. Consider a moving mirror (an external boundary condition in 1+1 dimensions) which does not accelerate forever. This mirror starts (ends) at time-like past (future) infinity, asymptotically having zero acceleration in the far past (future). As such, an asymptotically inertial mirror will contain no horizons or pathological acceleration singularities, as seen in Figure 1 and Figure 2. The use of an asymptotically inertial mirror is chosen for clarity of derivation. Relaxing the condition of asymptotically inertial motion will alter the form of the derivation and the complexity of the relevant identities. The connection between acceleration and the spin-statistics connection remains. The use of the forever accelerating observer in the Unruh effect, or a forever accelerating mirror that results in a null horizon, requires the appropriate left-right construction (for example, [13]) to incorporate complete Cauchy surface information. A moving mirror in flat spacetime reveals a connection between statistics and dynamics that is not present in Minkowski spacetime without acceleration. Statistics can be derived from the dynamics of the moving mirror as an accelerated mirror gives a beta Bogoliubov transformation coefficient which is non-zero. The statistics are determined by the algebra of the creation operators. The associated relations of the creation operators are commuting for Bose-Einstein statistics and anticommuting for Fermi-Dirac statistics. The following derivation shows that for the spin-zero field, only Bose-Einstein statistics are invariable with the flat spacetime dynamics that include an accelerated external boundary condition. The field equation of motion is the Klein-Gordon equation /square ψ = 0. The moving mirror is imposed such that at the position of the mirror the field is zero, ψ | z = 0. This system has modes that can be used to expand the field, The modes are orthonormal and complete. They take the null coordinate form, The modes can be expanded in terms of each other, by introduction of Bogoliubov coefficients, defined by the flat-space scalar product, in spacetime coordinates, or in null coordinates, The field equation combined with the moving mirror boundary condition imply that the Bogoliubov coefficients α ω ' ω and β ω ' ω relate the operators a ω ' and a † ω ' to the operators b ω and b † ω : The modes are orthonormal such that, Since the modes are related by a Bogoliubov transformation, the above scalar products imply several identities. Using (12), (5), and (13) the following Wronskian identity holds, Note that this relation is a derived consequence only of the field equation and the imposed asymptotically inertial mirror. This relation is a result of the dynamics. Consider now, the possible commutation relations of the creation operators, a ω ' , a † ω ' and b ω , b † ω (which are associated with the φ ω ' and χ ω modes, respectively), Here the + sign corresponds to Fermi-Dirac statistics from the anticommutator, while the -sign corresponds to Bose-Einstein statistics from the commutator. The relations associated with the φ ω ' modes may be expressed in terms of Bogoliubov coefficients using (10), The relations associated with the χ ω modes may be expressed in terms of Bogoliubov coefficients using (11), The dynamics-only result (14) dictates which sign to use in (17) and (18). Since the dynamics indicates the choice of the -sign, the election of Bose-Einstein statistics is straightforward. Similar identities to (14), using (12) and (13), dictate the correct sign to be used in (15) and (16) which give, in total, Particles at different times, (those associated with b † ω b ω and a † ω ' a ω ' ) obey the same statistics only in the BoseEinstein case. The Bose-Einstein case is then fixed with the dynamics of the field and mirror. When the mirror has acceleration, the β Bogoliubov transformation coefficient is not zero. If the mirror is not accelerating, then β = 0, and this connection between dynamics and statistics is absent, despite the presence of a boundary and the abandonment of Minkowski spacetime. The spinstatistics-dynamics connection in flat spacetime relies on a non-zero acceleration. The derivation presented above is an 'accelerated boundary condition in flat spacetime' -moving mirror- derivation of the relationship between spin and statistics.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"B. Infinitely Accelerated Mirror\",\n \"content\": \"Consider a moving mirror which accelerates for all time. When a mirror in flat spacetime accelerates forever, a horizon singularity is formed. The horizon can be seen most easily in a Penrose diagram, (see Figure 3 or in a spacetime diagram in Figure 4). It is reasonable to ask if the physically pathological singularity breaks the dynamics-statistics connection. As it turns out, the acceleration singularity complicates the mathematics but does not ruin the dynamics-statistics link. The spin-statistics connection is apparent in the case of an infinitely accelerated boundary. In this section only, we adopt Carlitz-Willey's normalization, transformation conventions and left-right con- struction [13]. The advantage here is the simplicity of the explicit mode forms. The field motion is the KleinGordon equation /square ψ ( u, v ) = 0, and the field is written as where I = R,L . The modes take the simplified form, while the ray-tracing functions for the specified CarlitzWilley trajectory assume Carlitz and Willey point out that a constant energy flux emitted by the mirror, κ 2 48 π , is obtained by the ray tracing function of (25). Substituting p ( u ) = t + z ( t ) into (25) gives the resulting condition on the trajectory, Carlitz and Willey did not provide an analytic solution to this equation. However one can be obtained using the Lambert W function (the product logarithm) with the result that Figure 3 and Figure 4 are plots of this trajectory. It is straightforward to show that ˙ z → ∓ 1 in the limits t → ±∞ and that z < 0 for all time. The proper acceleration is Note that the proper acceleration is not constant even though the energy flux is. The acceleration 29 is required to have thermal emission, for all times, in the moving mirror model. Contrast this to the Unruh effect, where time-independent acceleration, κ , is responsible for thermal emission. The dynamics of the field-mirror system give rise to the Bogoliubov coefficients α I ω ' ω and β I ω ' ω which relate the operators a ω ' and a † ω ' to the operators b I ω and b I † ω : Since the operators are related by a Bogoliubov transformation, it now pays to examine the the conserved scalar products for the two distinct modes φ and χ , (here J = L, R ). Specifically, The conserved scalar products imply several identities, as first shown by [13], in particular: These relations are derived consequences of the field equation and the imposed Carlitz-Willey mirror. Now consider the possible commutations relations As before, the + sign corresponds to FD statistics, while the -sign corresponds to BE statistics. The possible commutation relations in terms of the transformation coefficients are The dynamics result (35) can be used to determine which of the possible commutation relations are to be used in (39). The dynamics dictates the commutation relations use the -sign, The acceleration singularity does not corrupt the spinstatistics-dynamics connection.\",\n \"pages\": [\n 3,\n 4,\n 5\n ]\n },\n {\n \"title\": \"C. Unruh Effect\",\n \"content\": \"Consider the spin-statistics connection in the setting of the Unruh effect. We specify to the right movers in the right quadrant of the Rindler wedge, x > 0 and x > | t | . The left movers follow suit. Further consideration to the left Rindler wedge is similar to the right Rindler wedge [14]. To satisfy completeness on the full Cauchy surface, both the left and right movers, and both the left and right Rindler wedges must be treated. It is found that correct commutations relations are given due to the presence of acceleration for the Rindler observer. We start with the Minkowski metric, ds 2 = -dt 2 + dx 2 , and KleinGordon field equation ( -∂ 2 t + ∂ 2 x ) ψ = 0. Transformation to Rindler coordinates, where ρ > 0 and -∞ < τ < ∞ , gives the metric and field equation, The mode functions that are right-movers, (dependent only on U ) and the right quadrant, R, are shown graphically in Figure 5. The left-movers are dependent on V and follow the analogous forthcoming procedure. Specifying to these right-moving eigenmodes and the R spacetime region is clarifying. This is done by the use of null coordinates, and Consider now, the two representations of the field: and Notice that we have specified to the right quadrant by the use of θ ( -U ) and restricted our concern to right movers by utilizing the single U coordinate dependence. The advantage here is that left and right movers in the null coordinates U and V do not mix under the Bogoliubov transformation. The modes take the form, The Bogoliubov transformation for the operator associated with the φ ω ' takes the R-form, and the operator associated with the χ R ω is The modes are non-trivially expanded in terms of each other, by introduction of Bogoliubov coefficients, defined by the flat-space scalar product, in spacetime coordinates, or in null coordinates, Expanding ( χ R ω , χ R ω ' ) in terms of φ ω ' using the transformation (53) and utilizing the orthonormality of the φ ω ' modes, the R-completeness relation holds This is contingent on the orthonormality of both sets of distinct modes by the conserved inner product, and where ( φ ω , φ ∗ ω ' ) = ( χ R ω , χ R ∗ ω ' ) = 0. As we have seen in a complementary fashion in Section II A, the possible commutation relations are The Wronskian R-completeness relation (57) from the acceleration dynamics connects to the spin-statistics by necessitating the use of the negative sign in the commutation relations (60) and (61). It is dictated that and similarly, the zero commuting relations stem from the analogous identities to (57). Therefore, in totality, the accelerated Rindler observer entails the spin-statistics connection.\",\n \"pages\": [\n 5,\n 6\n ]\n },\n {\n \"title\": \"III. CONCLUSIONS\",\n \"content\": \"The crux of these results are dependent on the presence of acceleration. In the cases presented, acceleration arises as an asymptotically inertial mirror, an infinitely accelerated mirror and the constant acceleration Unruh effect. That is, in the far future these examples correspond to a proper acceleration α → 0 , ∞ , κ , respectively. These new results where found: 1) The assumption of particles at early times obeying late time statistics is un-necessary, as only commutation relations are acceptable in either region, as shown in the asymptotically inertial mirror case. 2) In a non-trivial system, the presence of external acceleration conditions is enough to provide the link to commutation relations. This is in contrast to the specification to dynamic curved spacetime. Flat-spacetime acceleration conditions allowing non-trivial Bogoliubov transformations establishes the spin-statistics connection. 3) With regards to thermal emission, the time-dependent expression for acceleration in the moving mirror case was explicitly revealed. This clarifies a salient difference between the Unruh effect and the dynamical Casimir effect. An acceleration in these above situations gives rise to distinct eigenmode solutions that can be used to represent the field. One set of eigenmodes can be expressed in mixed positive and negative frequencies pieces of the other set of eigenmodes. The spin-statistics connection is possible to obtain because of the properties of the conserved inner product and of the existence of these sets of eigenmodes in the first place. If there was zero acceleration, the Bogoliubov transformation would be trivial and the distinct sets of eigenmodes would not exist.\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"MRRG is grateful for discussions with Paul R. Anderson, Charles R. Evans and Xiong Chi. MRRG appre- ciates the financial support and hospitality of the IASSingapore.\",\n \"pages\": [\n 7\n ]\n }\n]"}}},{"rowIdx":293,"cells":{"bibcode":{"kind":"string","value":"2013IJMPA..2850148C"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1205.3138.pdf"},"content":{"kind":"string","value":"\nAn Alternative String Landscape Cosmology: Eliminating Bizarreness\nL. Clavelli ∗† and Gary R. Goldstein ‡\nDept. of Physics and Astronomy, Tufts University, Medford MA 02155\nAbstract\nIn what has become a standard eternal inflation picture of the string landscape there are many problematic consequences and a difficulty defining probabilities for the occurrence of each type of universe. One feature in particular that might be philosophically disconcerting is the infinite cloning of each individual and each civilization in infinite numbers of separated regions of the multiverse. Even if this is not ruled out due to causal separation one should ask whether the infinite cloning is a universal prediction of string landscape models or whether there are scenarios in which it is avoided. If a viable alternative cosmology can be constructed one might search for predictions that might allow one to discriminate experimentally between the models. We present one such scenario although, in doing so, we are forced to give up several popular presuppositions including the absence of a preferred frame and the homogeneity of matter in the universe. The model also has several ancillary advantages. We also consider the future lifetime of the current universe before becoming a light trapping region.\nkeywords: String Landscape, String cosmology, Multiverse, Supersymmetry, preferred frame\n1 Introduction\nBy now it is widely known that Einstein's greatest blunder was not, as he thought, the introduction of the cosmological constant but rather his uncritical presupposition that the universe was static and eternal into the past and into the future. It might be that the eternal inflation (or chaotic inflation) model [1] rests on a similar untested presupposition that the universe has neither beginning nor end and is an isotropic, chaotically bubbling but otherwise homogeneous vat of varying vacuum energies and matter content.\nIn the standard picture it is difficult to define probabilities [2] and difficult to prevent the occurrence of infinite cloning of each individual as well as the production of disembodied brains (Boltzmann brains [3]) and other bizarre entities. In a homogeneous and isotropic multiverse, everything that can exist does exist in infinite numbers. There is also an entropy problem in which, if the universe had an infinite past, one might wonder why our universe is not totally featureless. We ask whether one can construct a viable landscape scenario that avoids the paradoxes and bizarre features of the eternal inflation scenario. For example, can one avoid an infinite number of human, quasi-human, and monster populations in the string landscape and can one avoid the other problems mentioned above?\nIf, as is commonly assumed, the discovery of dark energy is due to our existence in a De Sitter space-time, the universe is homogeneous and isotropic with a scale factor growing at large times as an exponential of t. In this world the matter density is homogeneous and varies as the inverse cube of the scale factor a ( t ).\nlim t →∞ a ( t ) = e Ht (1.1)\nρ m ( t ) ≈ a ( t ) -3 . (1.2)\nA homogeneous De Sitter space, of course, is not compatible with the notion that each universe within the multiverse is a spherical bubble with a center. We seek to construct a bubble universe that retains, in an adequate approximation, the results of eq. 1.2. This necessarily involves a preferred frame, the bubble center, and implies some amount of matter inhomogeneity.\nIf our model, like the standard cosmology, is not in conflict with observations, each physicist is entitled to assume as a presupposition either model based on perceived aesthetic differences keeping in mind that the alternative is not ruled out. In addition to being consistent with current observations we point out several phenomenological advantages of our picture including the banishing of infinite numbers of monsters from the multiverse. In this we may be pursuing a time-honored role of science.\nNext we briefly enumerate nine assumptions. The number of assumptions may seem too great to be justified by the mere avoidance of infinite cloning and other landscape paradoxes. However, if these assumptions are not ruled out and can be distinguished in their consequences from those of the standard inflation picture, we feel that they should be critically considered.\nIn a subsequent section we elaborate on the assumptions and examine their consequences relative to the more standard string landscape picture. Finally, in a concluding section, we summarize the advantages of our proposed model.\nWe assume:\n\n1. The multiverse is not infinite into the past but originated at some time t = -t 0 .\n2. There is a finite rest energy of matter Mc 2 in the multiverse centered on an origin.\n3. At time t = -t 0 the matter density is proportional to a delta function at the spatial origin.\n4. The multiverse begins at t = -t 0 in a state of high vacuum energy density. It then rapidly cascades down to the present state of vacuum energy\n\nρ v = 4 . 0 ± 0 . 2 GeV/c 2 /m 3 . (1.3)\n\n5. These transitions are governed by the Coleman-De Luccia thin wall formulae and transitions up to higher vacuum energy are of negligible probability.\n6. Since such a phase transition does not create particles only a transition to our ρ v near the peak of the matter distribution ( r = 0 ) will lead to galaxies, planets, and people.\n7. We are protected from being enveloped in a black hole like region by a future phase transition to zero vacuum energy.\n8. Eventually, a bubble of zero vacuum energy should form within our universe and expand to convert the entire universe to this presumably supersymmetric state.\n9. This zero vacuum energy state is the ground state of the multiverse.\n\n2 Discussion\nWe now reiterate the assumptions and explain their relevance and effect to compare with the standard eternal inflation picture.\n\n1. The multiverse is not infinite into the past but originated at some time t = -t 0 . There is a growing consensus that the alternative leads to numerous thorny paradoxes [4]. The maintainance of a universe from the infinite past requires that there is an equilibrium between transitions to lower vacuum energy and transitions to higher vacuum energy. However, in a thermodynamic system, transitions to lower energy must be enormously favored over the entropy decreasing transitions to higher states and asymptotically in time the system should preferentially settle into the lowest available state.\n2. There is a finite rest energy of matter Mc 2 in the multiverse centered on an origin. This assumption may be initially repugnant to most cosmologists but is not ruled out. It suggests, contrary to what physics has held for a century, that there is a preferred frame in the multiverse. The assumption suggests that inertial frames are\n\nthose that are unaccelerated relative to this origin. This allows rotating coordinating systems to be inequivalent. In classical physics and in general relativity it is taught that the laws of physics hold in inertial frames but no prescription is given a priori for defining an inertial frame. The best that can be done is to state that, if a frame is found in which Newton's laws hold, they will also hold in any frame traveling relative to that frame with constant velocity. Newton, himself, was aware of the puzzle and posed the famous 'bucket problem'. A rotating bucket of water has a concave surface even when viewed from a co-rotating frame. The current assumption is analogous to Mach's Principle which states that the matter distribution of the universe defines a preferred frame, in his case the rest frame of the distant stars. Given a finite total mass, there can be no infinite cloning of individuals. Similarly, the Boltzmann Brain problem of standard string cosmology is largely due to the infinite number of universes like ours. Thermal fluctuations are a property of material substances. Given that only a vanishing fraction of the multiverse contains appreciable matter in our model, thermal fluctuations are unlikely to create Boltzmann Brains and quantum fluctuations are unlikely to produce such macroscopic complex objects in the short time available. Given the normal laws of evolution, it is easier to create a complete human being than a disconnected brain and this preference may become absolute if there is only a limited amount of matter available. The number of Boltzmann Brains with macroscopic baryon number produced in our model in matter-containing regions due to quantum fluctuations in the short time since the big bang should not be expected to be large compared to the vanishing number we have already observed.\n\n3. In the limit t →-t 0 the matter density is proportional to a delta function at the spatial origin. With two free parameters, M and R , we may write as a non-unique example\n\nρ m ( r, t ) = M ( R √ π a ( t )) 3 e -r 2 / ( Ra ( t )) 2 . (2.4)\nThe scale factor a ( t ) is an increasing function of time that vanishes at t = -t 0 and that can be taken equal to unity at the current time, t = 0. The equation of continuity\n/vector ∇· ( ρ m /vectorv ( r )) + ∂ρ m ∂t = 0 (2.5)\nhas, independent of R , the solution\n/vectorv ( r ) = /vectorr ˙ a a (2.6)\nin agreement with Hubble's law. This justifies the identification of a ( t ) with the scale factor. An important open question left to future study is the effect of curvature corrections to these equations and other results of this work. In classical and quantum physics the initial conditions do not follow from physical law although they must be among the possible states of the system. Thus one cannot require a justification for initial conditions such as we propose in the preceeding assumptions. In standard cosmology there is no comparably clear statement of initial conditions.\nAs t approaches -t 0 the matter density of 2.4 approaches a spatial delta function at the origin\nρ m ( /vectorr , -t 0 )= Mδ 3 ( /vectorr ) (2.7)\nand it integrates to a time independent mass\n∫ d 3 r ρ m ( r, t ) = M . (2.8)\nρ m can be taken to be a neutral, flavor and color singlet density of quarks, leptons, and gauge bosons together with their broken susy partners. Apart from pair production at positive times, there is no anti-matter in the universe. This possible solution to the baryon asymmetry problem is not available in the standard picture where matter is pair produced at the end of the inflationary era. Of course, one is free to consider, in our model also, the possibility that the matter density at the origin of time is a CP symmetric state, thus preserving a possible role for CP violation at later times and the interesting problem of insuring an asymmetric survival of a sufficient number of baryons.\nThe matter distribution is isotropic relative to its origin. At r/R << a ( t ) and at r/R >> a ( t ) the matter density is also homogeneous so the metric approaches a Friedmann, Robertson, Walker (FRW) metric and the matter density satisfies eq. 1.2. At sufficiently large t , the density is homogeneous (and for large r negligible) around all fixed positions. A prediction of this picture is that significant deviations from homogeneity should exist near r/R ≈ a ( t ). Observations from Earth cannot probe inhomogeneities at scales larger than the Hubble length, the distance light has travelled since the big bang. Thus if R , is much greater than the Hubble length, the proposed matter distribution is indistinguishable from the homogeneous distribution of the standard cosmological model. This will be our default assumption although, in future work, one might ask whether smaller values of R might also be observationally viable. Of course if we choose R to be beyond current limits of observability, the attractiveness of the model, though not its viability, must depend on other conceptual advantages.\nThe cosmic background radiation is isotropic in its rest frame. Relative to this frame the earth frame is presently moving with a speed of 369 ± 1 km/s. At present the origins of these two frames are 5 . 1 Mpc apart. One way to subtract the dipole term from the CBR is to cite all measurements as they would appear in the CBR rest frame. If the CBR rest frame is the r = 0 position of our model the universe is isotropic relative to this frame but not homogeneous except on scales small compared to our R parameter.\nOn the other hand if the r = 0 position is far from the center of our sphere of last scattering, the universe would still appear isotropic and homogeneous if R is much greater than the Hubble length, 4 . 2 Gpc. A direct measure of how homogeneous the universe is on large scales is a complicated issue [5].\nIf the sun is close to the r = 0 position, it could provide a basis for understanding the remarkable large angle correlations [6] between the ecliptic plane and the CBR.\n(The great statistical significance of these correlations, however, could be somewhat weakened by systematic effects [7]). Then, if R is in the Gpc range one would have to wonder whether some selection effect led to the closeness of the solar system to the origin or, equivalently, to the smallness of the solar velocity relative to the CBR. E.g. perhaps the larger collision rate of fast moving inhomogeneities with the matter background inhibits the rise of life. Alternatively it could be related to galactic merger rates as a function of velocity. These could be residual effects related to the anthropic discrimination against high inflation rates [8].\nGiven a small velocity, the parameters, M and R , in eq. 2.4 are constrained by the observed current matter density and degree of matter homogeneity. If at the current time the averaged matter distribution is homogeneous out to a Hubble length and falls off rapidly thereafter, the current value of the matter density, ρ m (0) = Ω m (0) ρ c = 1 . 4 ± 0 . 2 GeV/c 2 /m 3 then implies that the rest energy of the universe is\nMc 2 ≈ ρ m (0) c 2 π 3 / 2 R 3 > 2 . 6 · 10 76 GeV . (2.9)\nNewton's laws hold to an excellent approximation implying, in the current context, that the earth and the milky way are presently moving with negligible acceleration and with some velocity, /vectorv relative to the origin. The magnitude of /vectorv could be related to the dipole term that is evident in the cosmic background radiation (CBR) or could be some other speed. The matter distribution relative to earth is then\nρ m ( /vectorr , t ) = M ( R √ π a ( t )) 3 e -( /vectorr + /vectorv ( t + t 0 )) 2 / ( Ra ( t )) 2 . (2.10)\nHere /vectorv is not to be confused with the Hubble flow of eq. 2.6. Small deviations from isotropy beyond the dipole term can be obtained by multiplying this equation or that of eq. 2.4 by the spherical harmonic expansion\nF ( r, t ) = 1 + ∑ l,m f l,m ( r, t ) Y l,m ( θ, φ ) . (2.11)\nThe acoustic peak analysis suggests that the lowest values of l for which f l,m are appreciable are near l ≈ 100. Expanding eq. 2.10 in the velocity /vectorv produces correlated low multipole moments suppressed by inverse powers of R . The quadrupole moment, for instance, is suppressed in agreement with observation whereas this suppression is problematic in the standard cosmology.\nIf the f l,m vanish at r = 0 and are sufficiently well behaved in t and at large r , the properties of eqs. 2.7 and 2.8 are preserved.\n\n4. The multiverse begins at t = -t 0 in a state of high vacuum energy density, ( M I c 2 ) 4 / (¯ hc ) 3 . It then rapidly cascades down to the present state of vacuum energy\n\nρ v = 4 . 0 ± 0 . 2 GeV/c 2 /m 3 . (2.12)\n\n5. These transitions are governed by the Coleman-De Luccia [9] thin wall formulae and transitions up to higher vacuum energy are of negligible probability. In the thin wall approximation, the energy released in the transition to a lower vacuum energy does not create particles but goes into the surface energy. In the words of Coleman [10] 'This refutes the naive expectation that the decay of the false vacuum would leave behind it a roiling sea of mesons'. The thin wall assumption thus implies that galaxies, planets, and people come not from vacuum decay but from a primordial matter distribution such as that of eq. 2.4.\n\nThe probability per unit time per unit volume to nucleate a bubble of critical size destined to take over the universe is\ndP dt d 3 r = Ae -B (2.13)\nwith\nB = 27 π 2 S 4 2¯ hc ( c 2 ∆ ρ ) 3 (2.14)\nwhere S is the energy per unit area on the surface.\nThe two parameters A and S must at this point be considered free since they are determined by the exact shape of the unknown effective potential. One can speculate about various values for these parameters. For example, if supernovae Ia are triggered by a transition [11] to exact supersymmetry (zero vacuum energy) within a white dwarf one arrives at the estimates\nS ≈ 4 . 76 · 10 34 GeV/m 2 A -1 ≈ 7 . 89 · 10 36 m 3 s . (2.15)\nHowever, it has been suggested [12] that A -1 depends on ∆ ρ . The wall thickness in the transition out of inflation can be estimated by equating the energy released in a shell of thickness δr to the increase of energy stored in the wall.\n4 πr 2 ∆ ρ c 2 δr = 4 πr 2 S . (2.16)\nIf ∆ ρ is anywhere near the GUT density M G 4 c 3 / ¯ h 3 and S is near the estimate of eq. 2.15, the wall thickness is extremely small.\nδr ≈ 3 . 64 · 10 -77 m . (2.17)\nAt the time of bubble nucleation this is 1 / 3 of the bubble radius and becomes rapidly negligible in comparison as the bubble grows.\nAlso, in the case of the estimates of [11], B is small for the initial transition. If B remains small until the vacuum approaches our vacuum, the situation is one of rapid cascading with each intermediate bubble carried away by the higher vacuum energy background. Thus the inflating multiverse may decay explosively everywhere to our current low-lying metastable positive vacuum energy universe.\n\n6. Since such a phase transition does not create particles only a transition to our ρ v near the peak of the pre-existing matter distribution ( r = 0 ) will lead to galaxies, planets, and people. Bubbles nucleated at r significantly different from zero will have insufficient matter to generate life. It is possible that only one or only very few bubbles contain intelligent life.\n\nEven at r = 0 the matter distribution falls off as a ( t ) -3 so there is, perhaps, an anthropic understanding of an early end to inflation. Neighboring bubbles have very close to the same vacuum energy so there is little effect from bubble collisions contrary to the prediction of the chaotic inflation picture [13].\nOur assumption is contrary to a common untested presupposition (often referred to as the cosmological principle) that our universe does not occupy a privileged position in the multiverse.\nThe root mean square radius of the matter distribution from eq. 2.4 is\n√ < r 2 > = 3 2 a ( t ) R . (2.18)\n\n7. We are protected from being enveloped in a black hole like region by a future phase transition to zero vacuum energy As each bubble grows the energy enclosed approaches the value at which light is trapped. If there were no energy outside, this critical mass would be the mass of a black hole of that radius and the metric outside would be that of Schwarzschild. In the standard cosmology every trajectory through the multiverse, except for a set of measure zero, ends on a black hole. This and other problematic features discussed in [14] derive primarily from the asymptotic features of eternal inflation. Banks has shown that transitions between symmetric (constant curvature) spaces of differing vacuum energy do not occur contrary to the usual multiverse picture. This leaves open the possibility that inhomogeneous models such as ours could support vacuum transitions although bubble nucleation in regions of negligible matter density could be suppressed.\n\nThe finite age of the multiverse allows that the limiting radius has not yet been reached.\n∫ r 0 4 πr ' 2 dr ' ( ρ m ( r ' , t ) + ρ v ) + 4 πr 2 S/c 2 < rc 2 2 G N . (2.19)\nHere S is the energy per unit area on the bubble surface. If it is as small as the estimate of eq. 2.15 the surface term can be neglected. The time at which the visible universe saturates this inequality is given by putting r = ct max . The effect of pressure may be important but has not yet been analyzed. The condition that this limit has not yet been reached puts a tight limit on the current matter density of the universe. Whether reaching the limit is incompatible with the continuation of life in the universe is not clear.\n4 π 3 ρ c t 3 max = t max / (2 G N ) (2.20)\nor\nt max = √ 3 8 πρ c G N = 1 H 0 = (13 . 58 ± 0 . 27 ) Gyr . (2.21)\nHere H 0 is the current value of Hubble's constant which is determined observationally by the ratio of a galaxy's recession speed to its distance from us. The critical density, ρ c , is determined by the central equation of eq. 2.21. The numerical value for H 0 given in eq. 2.21 is from the latest WMAP compilation including other relevant data. t max should be compared to the current age of the universe, t 0 .\nt 0 = 1 H 0 F (Ω m (0) , Ω Λ (0) , Ω γ (0) , .. ) . (2.22)\nThe correction factor, F , which depends on the current values of various densities is obtained [15] by integrating back from the current time t = 0 to the point at which the scale factor goes to zero. Since the scale factor is only determined up to a constant multiplicative factor we may take a (0) = 1. The current values of the Hubble constant and the matter density of the universe taken from the 2011 Particle Data Group compilation [16] are\nh =0 . 702 ± 0 . 014 (2.23)\nH 0 -1 = 9 . 777752 Gyr h = 13 . 58 ± 0 . 27 Gyr (2.24)\nΩ m (0)=0 . 26 ± 0 . 02 . (2.25)\nFor r very small and very large we have approximately a De Sitter space so we can use, as an approximation, the usual equations for the scale factor. Since\n˙ a a = H ( t ) (2.26)\nand, with a sum over species,\na a = -4 πG N 3 ∑ ( ρ i +3 p i ) = H 0 2 ( -ρ m 2 ρ c + ρ v ρ c ) (2.27)\nwe have\nWe may write\nda aH ( t ) = dt . (2.29)\nThe age of the universe, t 0 , is therefore\nt 0 = ∫ 1 0 da aH (2.30)\nwhere we have written H parametrically as a function of a . Equivalently, t 0 can also be defined by\n∫ 0 -t 0 dtH ( t ) = ∫ 1 0 da/a (2.31)\ndH dt = -3 2 H 0 2 Ω m ( t ) . (2.28)\nwhere H ( -t 0 ) = ∞ . If we are not already living in a light trapping region t 0 must be less than the t max of eq. 2.21, i.e. the correction factor F in eq. 2.22 must be less than unity. The time from now at which our universe becomes light trapping is\n∆ t = 1 H 0 (1 -F (Ω m (0) , Ω Λ (0) , Ω γ (0) , .. )) . (2.32)\nAssuming that Ω Λ (0) = 1 -Ω m (0) and that the photon and neutrino contributions to the energy content of the universe can be neglected, ∆ t is plotted against Ω m (0) in fig. 1. To be consistent with the experimental value, Ω m (0) must also be less than 0 . 28 and the future lifetime of the universe in the present phase must be less than 0 . 23 Gyr. This estimate is two orders of magnitude lower than another estimate [17] which suggests that the future lifetime against vacuum decay could be of order 20 Gyr.\n\n8. Eventually, a bubble of zero vacuum energy should form within our universe and expand to convert the entire universe to this presumably supersymmetric state. At this point the interior of the bubble may never reach the light trapping limit since the first term in eq. 2.19 is limited by M and the vacuum energy term drops to zero. The surface term can never contribute to light trapping since it is itself moving out at the speed of light. Other regions devoid of significant matter densities will undergo the transition to zero vacuum energy without producing matter. If we are not destined to be enveloped in a light trapping region, the current model predicts a cosmologically proximate end to the current era in a transition to a universe of zero vacuum energy.\n9. The zero vacuum energy state is the ground state of the universe. If there are local minima of negative vacuum energy with the same number of degrees of freedom as our universe, there might eventually be a transition to one or more of these leading to a big crunch. However, Banks [14] has given strong arguments that this will not happen. In the standard picture where the mass is not finite, the eventual formation of a light trapping region is then unavoidable. Assumption 9 and the previous one are motivated by avoiding the light trapping limit although it is not clear whether such a limit would be inimical to life or pose other problems to the theory.\n\n3 Conclusion\nIn the current model we have introduced several free parameters for which we have only given preliminary estimates. The model can be falsified if inconsistencies develop and could be confirmed if systematic matter inhomogeneities are found within the Hubble length. The thesis of this paper is that what is called the cosmological principle may be an overgeneralization of observations at lower distance scales. Our proposal is anti-Copernican in the sense that our inhabited bubble universe is at the center of the multiverse. As R → ∞ , the model approaches the standard landscape cosmology. However, for any finite R , the model has the following advantages over the standard picture.\n\n\n
Figure 1: The time from now before our universe becomes light trapping is plotted against the current matter density, Ω m (0). The experimental range of this quantity is shown by the dashed line. As indicated, if the universe is not already light trapping, (∆ t > 0), Ω m (0) must be greater than 0 . 2645.
\n\nm\n\n· we can avoid an infinite number of human, quasi-human, and monster populations in the string landscape and we have, at least, a reduced production of Boltzmann Brains and other bizarres entities,\n· with a finite amount of matter in the multiverse we have no measure problem,\n· we can unify an inhomogeneous matter distribution with an apparent FRW metric,\n· a simple solution to the baryon asymmetry problem is possible,\n· a clear definition of inertial frames is obtained,\n· the earth rest frame is approximately inertial,\n· we can understand the current order of magnitude equality between matter and dark energy densities, and\n· we can at least potentially avoid becoming enveloped by a black hole like state.\n\nAt present it may be considered a matter of individual taste as to whether these advantages outweigh the required paradigm shift. In particular, many cosmologists seem at peace with the prospect that we and the entire human history of life on earth may be playing out an infinite number of times in other parts of the multiverse. At a minimum the present paper could be considered a challenge to cosmologists to prove the uniqueness of the infinite cloning prediction or to find other alternative scenarios which avoid the infinite cloning and preserve some or all of the conceptional advantages of our model while also preserving the string landscape explanation for the smallness of the vacuum energy.\nIt is left to future work to consider the general relativistic corrections to the picture presented here and to incorporate the effects of non-zero pressure. In the process, modifications to the matter density of eq. 2.4 that preserve the properties of eqs. 2.7 and 2.8 may become evident.\nAcknowledgements We acknowledge useful discussions of the matter presented here with Larry Ford at Tufts University. The research of LC was supported in part by the DOE under grant DE-FG02-10ER41714 and that of GG under DOE grant DE-FG02-92ER40702.\nReferences\n\nRaphael Bousso and Leonard Susskind, Phys.Rev.D85 (2012) 045007,\n[1] A. Linde, hep-th/0611043, JCAP 0701:022 (2007). arXiv:1105.3796 .\n[2] A. Guth, V. Vanchurin, . MIT-CTP-4284, SU-ITP-11-36, arXiv:1108.0665 .\n[3] L. Dyson, M. Kleban, and L. Susskind, Journal of High Energy Physics 0210 (2002) 011 (hep-th/0208013).\n[4] A. Borde, A. Guth, and A. Vilenkin, Phys.Rev.Lett.90 (2003) 151301, gr-qc/0110012 . A. Mithani and A. Vilenkin, JCAP 1201:028 (2012), arXiv:1110.4096.\n[5] P. P'apai and I. Szapudi, ApJ 725 (2010) 2078, arXiv:1009.0754.\nP. Wesson, Astroph Lett, 21, (1981) 97.\nR. Maartens, arXiv:1104.1300.\n[6] D. Schwarz et al., astro-ph/0403353\nC. Copi et al., Mon. Not. R. Astron. Soc., 418 (2011) 505.\n[7] K. Inoue and J. Silk, Ap J 664 (2006) 650, astro-ph/0612347.\n[8] S. Weinberg, Phys. Rev. Lett. 59 (22): (1987) 2607.\n[9] S. Coleman and F. De Luccia, Phys.Rev.D21 (1980) 3305.\nC. Callan, Jr. and S. Coleman, Phys.Rev.D16 (1977) 1762-1768.\n[10] S. Coleman, Phys.Rev.D15 (1977) 2929, Erratum-ibid.D16 (1977) 1248.\n[11] Peter Biermann and L. Clavelli, Phys. Rev. D84 (2011) 023001, arXiv:1011.1687.\n[12] P. Frampton, Mod.Phys.Lett.A19 (2004) 801, hep-th/0302007.\n[13] M. Johnson, H. Peiris, and L. Lehner, arXiv:1112.4487 .\n[14] T. Banks, Lectures given at Theoretical and Advanced Study Institute in Elementary Particle Physics: (TASI 2010): String theory and its Applications: From meV to the Planck Scale, Boulder, Colorado, 1-25 Jun 2010. arXiv:1007.4001 .\nT. Banks, arXiv:1208.5715 .\n[15] N. Jarosik et al, The Astrophysical Journal Supplement Series 192: (2011) 14. arXiv:1001.4744.\n[16] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, (2010) 075021 and 2011 partial update for the 2012 edition.\n[17] D. Page, Phys. Rev. D78, 063535, 2008, hep-th/0610079.\n"},"sections":{"kind":"list like","value":[{"title":"An Alternative String Landscape Cosmology: Eliminating Bizarreness","content":"L. Clavelli ∗† and Gary R. Goldstein ‡ Dept. of Physics and Astronomy, Tufts University, Medford MA 02155","pages":[1]},{"title":"Abstract","content":"In what has become a standard eternal inflation picture of the string landscape there are many problematic consequences and a difficulty defining probabilities for the occurrence of each type of universe. One feature in particular that might be philosophically disconcerting is the infinite cloning of each individual and each civilization in infinite numbers of separated regions of the multiverse. Even if this is not ruled out due to causal separation one should ask whether the infinite cloning is a universal prediction of string landscape models or whether there are scenarios in which it is avoided. If a viable alternative cosmology can be constructed one might search for predictions that might allow one to discriminate experimentally between the models. We present one such scenario although, in doing so, we are forced to give up several popular presuppositions including the absence of a preferred frame and the homogeneity of matter in the universe. The model also has several ancillary advantages. We also consider the future lifetime of the current universe before becoming a light trapping region. keywords: String Landscape, String cosmology, Multiverse, Supersymmetry, preferred frame","pages":[1]},{"title":"1 Introduction","content":"By now it is widely known that Einstein's greatest blunder was not, as he thought, the introduction of the cosmological constant but rather his uncritical presupposition that the universe was static and eternal into the past and into the future. It might be that the eternal inflation (or chaotic inflation) model [1] rests on a similar untested presupposition that the universe has neither beginning nor end and is an isotropic, chaotically bubbling but otherwise homogeneous vat of varying vacuum energies and matter content. In the standard picture it is difficult to define probabilities [2] and difficult to prevent the occurrence of infinite cloning of each individual as well as the production of disembodied brains (Boltzmann brains [3]) and other bizarre entities. In a homogeneous and isotropic multiverse, everything that can exist does exist in infinite numbers. There is also an entropy problem in which, if the universe had an infinite past, one might wonder why our universe is not totally featureless. We ask whether one can construct a viable landscape scenario that avoids the paradoxes and bizarre features of the eternal inflation scenario. For example, can one avoid an infinite number of human, quasi-human, and monster populations in the string landscape and can one avoid the other problems mentioned above? If, as is commonly assumed, the discovery of dark energy is due to our existence in a De Sitter space-time, the universe is homogeneous and isotropic with a scale factor growing at large times as an exponential of t. In this world the matter density is homogeneous and varies as the inverse cube of the scale factor a ( t ). A homogeneous De Sitter space, of course, is not compatible with the notion that each universe within the multiverse is a spherical bubble with a center. We seek to construct a bubble universe that retains, in an adequate approximation, the results of eq. 1.2. This necessarily involves a preferred frame, the bubble center, and implies some amount of matter inhomogeneity. If our model, like the standard cosmology, is not in conflict with observations, each physicist is entitled to assume as a presupposition either model based on perceived aesthetic differences keeping in mind that the alternative is not ruled out. In addition to being consistent with current observations we point out several phenomenological advantages of our picture including the banishing of infinite numbers of monsters from the multiverse. In this we may be pursuing a time-honored role of science. Next we briefly enumerate nine assumptions. The number of assumptions may seem too great to be justified by the mere avoidance of infinite cloning and other landscape paradoxes. However, if these assumptions are not ruled out and can be distinguished in their consequences from those of the standard inflation picture, we feel that they should be critically considered. In a subsequent section we elaborate on the assumptions and examine their consequences relative to the more standard string landscape picture. Finally, in a concluding section, we summarize the advantages of our proposed model.","pages":[1,2]},{"title":"2 Discussion","content":"We now reiterate the assumptions and explain their relevance and effect to compare with the standard eternal inflation picture. those that are unaccelerated relative to this origin. This allows rotating coordinating systems to be inequivalent. In classical physics and in general relativity it is taught that the laws of physics hold in inertial frames but no prescription is given a priori for defining an inertial frame. The best that can be done is to state that, if a frame is found in which Newton's laws hold, they will also hold in any frame traveling relative to that frame with constant velocity. Newton, himself, was aware of the puzzle and posed the famous 'bucket problem'. A rotating bucket of water has a concave surface even when viewed from a co-rotating frame. The current assumption is analogous to Mach's Principle which states that the matter distribution of the universe defines a preferred frame, in his case the rest frame of the distant stars. Given a finite total mass, there can be no infinite cloning of individuals. Similarly, the Boltzmann Brain problem of standard string cosmology is largely due to the infinite number of universes like ours. Thermal fluctuations are a property of material substances. Given that only a vanishing fraction of the multiverse contains appreciable matter in our model, thermal fluctuations are unlikely to create Boltzmann Brains and quantum fluctuations are unlikely to produce such macroscopic complex objects in the short time available. Given the normal laws of evolution, it is easier to create a complete human being than a disconnected brain and this preference may become absolute if there is only a limited amount of matter available. The number of Boltzmann Brains with macroscopic baryon number produced in our model in matter-containing regions due to quantum fluctuations in the short time since the big bang should not be expected to be large compared to the vanishing number we have already observed. The scale factor a ( t ) is an increasing function of time that vanishes at t = -t 0 and that can be taken equal to unity at the current time, t = 0. The equation of continuity has, independent of R , the solution in agreement with Hubble's law. This justifies the identification of a ( t ) with the scale factor. An important open question left to future study is the effect of curvature corrections to these equations and other results of this work. In classical and quantum physics the initial conditions do not follow from physical law although they must be among the possible states of the system. Thus one cannot require a justification for initial conditions such as we propose in the preceeding assumptions. In standard cosmology there is no comparably clear statement of initial conditions. As t approaches -t 0 the matter density of 2.4 approaches a spatial delta function at the origin and it integrates to a time independent mass ρ m can be taken to be a neutral, flavor and color singlet density of quarks, leptons, and gauge bosons together with their broken susy partners. Apart from pair production at positive times, there is no anti-matter in the universe. This possible solution to the baryon asymmetry problem is not available in the standard picture where matter is pair produced at the end of the inflationary era. Of course, one is free to consider, in our model also, the possibility that the matter density at the origin of time is a CP symmetric state, thus preserving a possible role for CP violation at later times and the interesting problem of insuring an asymmetric survival of a sufficient number of baryons. The matter distribution is isotropic relative to its origin. At r/R << a ( t ) and at r/R >> a ( t ) the matter density is also homogeneous so the metric approaches a Friedmann, Robertson, Walker (FRW) metric and the matter density satisfies eq. 1.2. At sufficiently large t , the density is homogeneous (and for large r negligible) around all fixed positions. A prediction of this picture is that significant deviations from homogeneity should exist near r/R ≈ a ( t ). Observations from Earth cannot probe inhomogeneities at scales larger than the Hubble length, the distance light has travelled since the big bang. Thus if R , is much greater than the Hubble length, the proposed matter distribution is indistinguishable from the homogeneous distribution of the standard cosmological model. This will be our default assumption although, in future work, one might ask whether smaller values of R might also be observationally viable. Of course if we choose R to be beyond current limits of observability, the attractiveness of the model, though not its viability, must depend on other conceptual advantages. The cosmic background radiation is isotropic in its rest frame. Relative to this frame the earth frame is presently moving with a speed of 369 ± 1 km/s. At present the origins of these two frames are 5 . 1 Mpc apart. One way to subtract the dipole term from the CBR is to cite all measurements as they would appear in the CBR rest frame. If the CBR rest frame is the r = 0 position of our model the universe is isotropic relative to this frame but not homogeneous except on scales small compared to our R parameter. On the other hand if the r = 0 position is far from the center of our sphere of last scattering, the universe would still appear isotropic and homogeneous if R is much greater than the Hubble length, 4 . 2 Gpc. A direct measure of how homogeneous the universe is on large scales is a complicated issue [5]. If the sun is close to the r = 0 position, it could provide a basis for understanding the remarkable large angle correlations [6] between the ecliptic plane and the CBR. (The great statistical significance of these correlations, however, could be somewhat weakened by systematic effects [7]). Then, if R is in the Gpc range one would have to wonder whether some selection effect led to the closeness of the solar system to the origin or, equivalently, to the smallness of the solar velocity relative to the CBR. E.g. perhaps the larger collision rate of fast moving inhomogeneities with the matter background inhibits the rise of life. Alternatively it could be related to galactic merger rates as a function of velocity. These could be residual effects related to the anthropic discrimination against high inflation rates [8]. Given a small velocity, the parameters, M and R , in eq. 2.4 are constrained by the observed current matter density and degree of matter homogeneity. If at the current time the averaged matter distribution is homogeneous out to a Hubble length and falls off rapidly thereafter, the current value of the matter density, ρ m (0) = Ω m (0) ρ c = 1 . 4 ± 0 . 2 GeV/c 2 /m 3 then implies that the rest energy of the universe is Newton's laws hold to an excellent approximation implying, in the current context, that the earth and the milky way are presently moving with negligible acceleration and with some velocity, /vectorv relative to the origin. The magnitude of /vectorv could be related to the dipole term that is evident in the cosmic background radiation (CBR) or could be some other speed. The matter distribution relative to earth is then Here /vectorv is not to be confused with the Hubble flow of eq. 2.6. Small deviations from isotropy beyond the dipole term can be obtained by multiplying this equation or that of eq. 2.4 by the spherical harmonic expansion The acoustic peak analysis suggests that the lowest values of l for which f l,m are appreciable are near l ≈ 100. Expanding eq. 2.10 in the velocity /vectorv produces correlated low multipole moments suppressed by inverse powers of R . The quadrupole moment, for instance, is suppressed in agreement with observation whereas this suppression is problematic in the standard cosmology. If the f l,m vanish at r = 0 and are sufficiently well behaved in t and at large r , the properties of eqs. 2.7 and 2.8 are preserved. The probability per unit time per unit volume to nucleate a bubble of critical size destined to take over the universe is with where S is the energy per unit area on the surface. The two parameters A and S must at this point be considered free since they are determined by the exact shape of the unknown effective potential. One can speculate about various values for these parameters. For example, if supernovae Ia are triggered by a transition [11] to exact supersymmetry (zero vacuum energy) within a white dwarf one arrives at the estimates However, it has been suggested [12] that A -1 depends on ∆ ρ . The wall thickness in the transition out of inflation can be estimated by equating the energy released in a shell of thickness δr to the increase of energy stored in the wall. If ∆ ρ is anywhere near the GUT density M G 4 c 3 / ¯ h 3 and S is near the estimate of eq. 2.15, the wall thickness is extremely small. At the time of bubble nucleation this is 1 / 3 of the bubble radius and becomes rapidly negligible in comparison as the bubble grows. Also, in the case of the estimates of [11], B is small for the initial transition. If B remains small until the vacuum approaches our vacuum, the situation is one of rapid cascading with each intermediate bubble carried away by the higher vacuum energy background. Thus the inflating multiverse may decay explosively everywhere to our current low-lying metastable positive vacuum energy universe. Even at r = 0 the matter distribution falls off as a ( t ) -3 so there is, perhaps, an anthropic understanding of an early end to inflation. Neighboring bubbles have very close to the same vacuum energy so there is little effect from bubble collisions contrary to the prediction of the chaotic inflation picture [13]. Our assumption is contrary to a common untested presupposition (often referred to as the cosmological principle) that our universe does not occupy a privileged position in the multiverse. The root mean square radius of the matter distribution from eq. 2.4 is The finite age of the multiverse allows that the limiting radius has not yet been reached. Here S is the energy per unit area on the bubble surface. If it is as small as the estimate of eq. 2.15 the surface term can be neglected. The time at which the visible universe saturates this inequality is given by putting r = ct max . The effect of pressure may be important but has not yet been analyzed. The condition that this limit has not yet been reached puts a tight limit on the current matter density of the universe. Whether reaching the limit is incompatible with the continuation of life in the universe is not clear. or Here H 0 is the current value of Hubble's constant which is determined observationally by the ratio of a galaxy's recession speed to its distance from us. The critical density, ρ c , is determined by the central equation of eq. 2.21. The numerical value for H 0 given in eq. 2.21 is from the latest WMAP compilation including other relevant data. t max should be compared to the current age of the universe, t 0 . The correction factor, F , which depends on the current values of various densities is obtained [15] by integrating back from the current time t = 0 to the point at which the scale factor goes to zero. Since the scale factor is only determined up to a constant multiplicative factor we may take a (0) = 1. The current values of the Hubble constant and the matter density of the universe taken from the 2011 Particle Data Group compilation [16] are For r very small and very large we have approximately a De Sitter space so we can use, as an approximation, the usual equations for the scale factor. Since and, with a sum over species, we have We may write The age of the universe, t 0 , is therefore where we have written H parametrically as a function of a . Equivalently, t 0 can also be defined by where H ( -t 0 ) = ∞ . If we are not already living in a light trapping region t 0 must be less than the t max of eq. 2.21, i.e. the correction factor F in eq. 2.22 must be less than unity. The time from now at which our universe becomes light trapping is Assuming that Ω Λ (0) = 1 -Ω m (0) and that the photon and neutrino contributions to the energy content of the universe can be neglected, ∆ t is plotted against Ω m (0) in fig. 1. To be consistent with the experimental value, Ω m (0) must also be less than 0 . 28 and the future lifetime of the universe in the present phase must be less than 0 . 23 Gyr. This estimate is two orders of magnitude lower than another estimate [17] which suggests that the future lifetime against vacuum decay could be of order 20 Gyr.","pages":[3,4,5,6,7,8,9,10]},{"title":"3 Conclusion","content":"In the current model we have introduced several free parameters for which we have only given preliminary estimates. The model can be falsified if inconsistencies develop and could be confirmed if systematic matter inhomogeneities are found within the Hubble length. The thesis of this paper is that what is called the cosmological principle may be an overgeneralization of observations at lower distance scales. Our proposal is anti-Copernican in the sense that our inhabited bubble universe is at the center of the multiverse. As R → ∞ , the model approaches the standard landscape cosmology. However, for any finite R , the model has the following advantages over the standard picture. m At present it may be considered a matter of individual taste as to whether these advantages outweigh the required paradigm shift. In particular, many cosmologists seem at peace with the prospect that we and the entire human history of life on earth may be playing out an infinite number of times in other parts of the multiverse. At a minimum the present paper could be considered a challenge to cosmologists to prove the uniqueness of the infinite cloning prediction or to find other alternative scenarios which avoid the infinite cloning and preserve some or all of the conceptional advantages of our model while also preserving the string landscape explanation for the smallness of the vacuum energy. It is left to future work to consider the general relativistic corrections to the picture presented here and to incorporate the effects of non-zero pressure. In the process, modifications to the matter density of eq. 2.4 that preserve the properties of eqs. 2.7 and 2.8 may become evident. Acknowledgements We acknowledge useful discussions of the matter presented here with Larry Ford at Tufts University. The research of LC was supported in part by the DOE under grant DE-FG02-10ER41714 and that of GG under DOE grant DE-FG02-92ER40702.","pages":[10,11,12]}],"string":"[\n {\n \"title\": \"An Alternative String Landscape Cosmology: Eliminating Bizarreness\",\n \"content\": \"L. Clavelli ∗† and Gary R. Goldstein ‡ Dept. of Physics and Astronomy, Tufts University, Medford MA 02155\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"In what has become a standard eternal inflation picture of the string landscape there are many problematic consequences and a difficulty defining probabilities for the occurrence of each type of universe. One feature in particular that might be philosophically disconcerting is the infinite cloning of each individual and each civilization in infinite numbers of separated regions of the multiverse. Even if this is not ruled out due to causal separation one should ask whether the infinite cloning is a universal prediction of string landscape models or whether there are scenarios in which it is avoided. If a viable alternative cosmology can be constructed one might search for predictions that might allow one to discriminate experimentally between the models. We present one such scenario although, in doing so, we are forced to give up several popular presuppositions including the absence of a preferred frame and the homogeneity of matter in the universe. The model also has several ancillary advantages. We also consider the future lifetime of the current universe before becoming a light trapping region. keywords: String Landscape, String cosmology, Multiverse, Supersymmetry, preferred frame\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 Introduction\",\n \"content\": \"By now it is widely known that Einstein's greatest blunder was not, as he thought, the introduction of the cosmological constant but rather his uncritical presupposition that the universe was static and eternal into the past and into the future. It might be that the eternal inflation (or chaotic inflation) model [1] rests on a similar untested presupposition that the universe has neither beginning nor end and is an isotropic, chaotically bubbling but otherwise homogeneous vat of varying vacuum energies and matter content. In the standard picture it is difficult to define probabilities [2] and difficult to prevent the occurrence of infinite cloning of each individual as well as the production of disembodied brains (Boltzmann brains [3]) and other bizarre entities. In a homogeneous and isotropic multiverse, everything that can exist does exist in infinite numbers. There is also an entropy problem in which, if the universe had an infinite past, one might wonder why our universe is not totally featureless. We ask whether one can construct a viable landscape scenario that avoids the paradoxes and bizarre features of the eternal inflation scenario. For example, can one avoid an infinite number of human, quasi-human, and monster populations in the string landscape and can one avoid the other problems mentioned above? If, as is commonly assumed, the discovery of dark energy is due to our existence in a De Sitter space-time, the universe is homogeneous and isotropic with a scale factor growing at large times as an exponential of t. In this world the matter density is homogeneous and varies as the inverse cube of the scale factor a ( t ). A homogeneous De Sitter space, of course, is not compatible with the notion that each universe within the multiverse is a spherical bubble with a center. We seek to construct a bubble universe that retains, in an adequate approximation, the results of eq. 1.2. This necessarily involves a preferred frame, the bubble center, and implies some amount of matter inhomogeneity. If our model, like the standard cosmology, is not in conflict with observations, each physicist is entitled to assume as a presupposition either model based on perceived aesthetic differences keeping in mind that the alternative is not ruled out. In addition to being consistent with current observations we point out several phenomenological advantages of our picture including the banishing of infinite numbers of monsters from the multiverse. In this we may be pursuing a time-honored role of science. Next we briefly enumerate nine assumptions. The number of assumptions may seem too great to be justified by the mere avoidance of infinite cloning and other landscape paradoxes. However, if these assumptions are not ruled out and can be distinguished in their consequences from those of the standard inflation picture, we feel that they should be critically considered. In a subsequent section we elaborate on the assumptions and examine their consequences relative to the more standard string landscape picture. Finally, in a concluding section, we summarize the advantages of our proposed model.\",\n \"pages\": [\n 1,\n 2\n ]\n },\n {\n \"title\": \"2 Discussion\",\n \"content\": \"We now reiterate the assumptions and explain their relevance and effect to compare with the standard eternal inflation picture. those that are unaccelerated relative to this origin. This allows rotating coordinating systems to be inequivalent. In classical physics and in general relativity it is taught that the laws of physics hold in inertial frames but no prescription is given a priori for defining an inertial frame. The best that can be done is to state that, if a frame is found in which Newton's laws hold, they will also hold in any frame traveling relative to that frame with constant velocity. Newton, himself, was aware of the puzzle and posed the famous 'bucket problem'. A rotating bucket of water has a concave surface even when viewed from a co-rotating frame. The current assumption is analogous to Mach's Principle which states that the matter distribution of the universe defines a preferred frame, in his case the rest frame of the distant stars. Given a finite total mass, there can be no infinite cloning of individuals. Similarly, the Boltzmann Brain problem of standard string cosmology is largely due to the infinite number of universes like ours. Thermal fluctuations are a property of material substances. Given that only a vanishing fraction of the multiverse contains appreciable matter in our model, thermal fluctuations are unlikely to create Boltzmann Brains and quantum fluctuations are unlikely to produce such macroscopic complex objects in the short time available. Given the normal laws of evolution, it is easier to create a complete human being than a disconnected brain and this preference may become absolute if there is only a limited amount of matter available. The number of Boltzmann Brains with macroscopic baryon number produced in our model in matter-containing regions due to quantum fluctuations in the short time since the big bang should not be expected to be large compared to the vanishing number we have already observed. The scale factor a ( t ) is an increasing function of time that vanishes at t = -t 0 and that can be taken equal to unity at the current time, t = 0. The equation of continuity has, independent of R , the solution in agreement with Hubble's law. This justifies the identification of a ( t ) with the scale factor. An important open question left to future study is the effect of curvature corrections to these equations and other results of this work. In classical and quantum physics the initial conditions do not follow from physical law although they must be among the possible states of the system. Thus one cannot require a justification for initial conditions such as we propose in the preceeding assumptions. In standard cosmology there is no comparably clear statement of initial conditions. As t approaches -t 0 the matter density of 2.4 approaches a spatial delta function at the origin and it integrates to a time independent mass ρ m can be taken to be a neutral, flavor and color singlet density of quarks, leptons, and gauge bosons together with their broken susy partners. Apart from pair production at positive times, there is no anti-matter in the universe. This possible solution to the baryon asymmetry problem is not available in the standard picture where matter is pair produced at the end of the inflationary era. Of course, one is free to consider, in our model also, the possibility that the matter density at the origin of time is a CP symmetric state, thus preserving a possible role for CP violation at later times and the interesting problem of insuring an asymmetric survival of a sufficient number of baryons. The matter distribution is isotropic relative to its origin. At r/R << a ( t ) and at r/R >> a ( t ) the matter density is also homogeneous so the metric approaches a Friedmann, Robertson, Walker (FRW) metric and the matter density satisfies eq. 1.2. At sufficiently large t , the density is homogeneous (and for large r negligible) around all fixed positions. A prediction of this picture is that significant deviations from homogeneity should exist near r/R ≈ a ( t ). Observations from Earth cannot probe inhomogeneities at scales larger than the Hubble length, the distance light has travelled since the big bang. Thus if R , is much greater than the Hubble length, the proposed matter distribution is indistinguishable from the homogeneous distribution of the standard cosmological model. This will be our default assumption although, in future work, one might ask whether smaller values of R might also be observationally viable. Of course if we choose R to be beyond current limits of observability, the attractiveness of the model, though not its viability, must depend on other conceptual advantages. The cosmic background radiation is isotropic in its rest frame. Relative to this frame the earth frame is presently moving with a speed of 369 ± 1 km/s. At present the origins of these two frames are 5 . 1 Mpc apart. One way to subtract the dipole term from the CBR is to cite all measurements as they would appear in the CBR rest frame. If the CBR rest frame is the r = 0 position of our model the universe is isotropic relative to this frame but not homogeneous except on scales small compared to our R parameter. On the other hand if the r = 0 position is far from the center of our sphere of last scattering, the universe would still appear isotropic and homogeneous if R is much greater than the Hubble length, 4 . 2 Gpc. A direct measure of how homogeneous the universe is on large scales is a complicated issue [5]. If the sun is close to the r = 0 position, it could provide a basis for understanding the remarkable large angle correlations [6] between the ecliptic plane and the CBR. (The great statistical significance of these correlations, however, could be somewhat weakened by systematic effects [7]). Then, if R is in the Gpc range one would have to wonder whether some selection effect led to the closeness of the solar system to the origin or, equivalently, to the smallness of the solar velocity relative to the CBR. E.g. perhaps the larger collision rate of fast moving inhomogeneities with the matter background inhibits the rise of life. Alternatively it could be related to galactic merger rates as a function of velocity. These could be residual effects related to the anthropic discrimination against high inflation rates [8]. Given a small velocity, the parameters, M and R , in eq. 2.4 are constrained by the observed current matter density and degree of matter homogeneity. If at the current time the averaged matter distribution is homogeneous out to a Hubble length and falls off rapidly thereafter, the current value of the matter density, ρ m (0) = Ω m (0) ρ c = 1 . 4 ± 0 . 2 GeV/c 2 /m 3 then implies that the rest energy of the universe is Newton's laws hold to an excellent approximation implying, in the current context, that the earth and the milky way are presently moving with negligible acceleration and with some velocity, /vectorv relative to the origin. The magnitude of /vectorv could be related to the dipole term that is evident in the cosmic background radiation (CBR) or could be some other speed. The matter distribution relative to earth is then Here /vectorv is not to be confused with the Hubble flow of eq. 2.6. Small deviations from isotropy beyond the dipole term can be obtained by multiplying this equation or that of eq. 2.4 by the spherical harmonic expansion The acoustic peak analysis suggests that the lowest values of l for which f l,m are appreciable are near l ≈ 100. Expanding eq. 2.10 in the velocity /vectorv produces correlated low multipole moments suppressed by inverse powers of R . The quadrupole moment, for instance, is suppressed in agreement with observation whereas this suppression is problematic in the standard cosmology. If the f l,m vanish at r = 0 and are sufficiently well behaved in t and at large r , the properties of eqs. 2.7 and 2.8 are preserved. The probability per unit time per unit volume to nucleate a bubble of critical size destined to take over the universe is with where S is the energy per unit area on the surface. The two parameters A and S must at this point be considered free since they are determined by the exact shape of the unknown effective potential. One can speculate about various values for these parameters. For example, if supernovae Ia are triggered by a transition [11] to exact supersymmetry (zero vacuum energy) within a white dwarf one arrives at the estimates However, it has been suggested [12] that A -1 depends on ∆ ρ . The wall thickness in the transition out of inflation can be estimated by equating the energy released in a shell of thickness δr to the increase of energy stored in the wall. If ∆ ρ is anywhere near the GUT density M G 4 c 3 / ¯ h 3 and S is near the estimate of eq. 2.15, the wall thickness is extremely small. At the time of bubble nucleation this is 1 / 3 of the bubble radius and becomes rapidly negligible in comparison as the bubble grows. Also, in the case of the estimates of [11], B is small for the initial transition. If B remains small until the vacuum approaches our vacuum, the situation is one of rapid cascading with each intermediate bubble carried away by the higher vacuum energy background. Thus the inflating multiverse may decay explosively everywhere to our current low-lying metastable positive vacuum energy universe. Even at r = 0 the matter distribution falls off as a ( t ) -3 so there is, perhaps, an anthropic understanding of an early end to inflation. Neighboring bubbles have very close to the same vacuum energy so there is little effect from bubble collisions contrary to the prediction of the chaotic inflation picture [13]. Our assumption is contrary to a common untested presupposition (often referred to as the cosmological principle) that our universe does not occupy a privileged position in the multiverse. The root mean square radius of the matter distribution from eq. 2.4 is The finite age of the multiverse allows that the limiting radius has not yet been reached. Here S is the energy per unit area on the bubble surface. If it is as small as the estimate of eq. 2.15 the surface term can be neglected. The time at which the visible universe saturates this inequality is given by putting r = ct max . The effect of pressure may be important but has not yet been analyzed. The condition that this limit has not yet been reached puts a tight limit on the current matter density of the universe. Whether reaching the limit is incompatible with the continuation of life in the universe is not clear. or Here H 0 is the current value of Hubble's constant which is determined observationally by the ratio of a galaxy's recession speed to its distance from us. The critical density, ρ c , is determined by the central equation of eq. 2.21. The numerical value for H 0 given in eq. 2.21 is from the latest WMAP compilation including other relevant data. t max should be compared to the current age of the universe, t 0 . The correction factor, F , which depends on the current values of various densities is obtained [15] by integrating back from the current time t = 0 to the point at which the scale factor goes to zero. Since the scale factor is only determined up to a constant multiplicative factor we may take a (0) = 1. The current values of the Hubble constant and the matter density of the universe taken from the 2011 Particle Data Group compilation [16] are For r very small and very large we have approximately a De Sitter space so we can use, as an approximation, the usual equations for the scale factor. Since and, with a sum over species, we have We may write The age of the universe, t 0 , is therefore where we have written H parametrically as a function of a . Equivalently, t 0 can also be defined by where H ( -t 0 ) = ∞ . If we are not already living in a light trapping region t 0 must be less than the t max of eq. 2.21, i.e. the correction factor F in eq. 2.22 must be less than unity. The time from now at which our universe becomes light trapping is Assuming that Ω Λ (0) = 1 -Ω m (0) and that the photon and neutrino contributions to the energy content of the universe can be neglected, ∆ t is plotted against Ω m (0) in fig. 1. To be consistent with the experimental value, Ω m (0) must also be less than 0 . 28 and the future lifetime of the universe in the present phase must be less than 0 . 23 Gyr. This estimate is two orders of magnitude lower than another estimate [17] which suggests that the future lifetime against vacuum decay could be of order 20 Gyr.\",\n \"pages\": [\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10\n ]\n },\n {\n \"title\": \"3 Conclusion\",\n \"content\": \"In the current model we have introduced several free parameters for which we have only given preliminary estimates. The model can be falsified if inconsistencies develop and could be confirmed if systematic matter inhomogeneities are found within the Hubble length. The thesis of this paper is that what is called the cosmological principle may be an overgeneralization of observations at lower distance scales. Our proposal is anti-Copernican in the sense that our inhabited bubble universe is at the center of the multiverse. As R → ∞ , the model approaches the standard landscape cosmology. However, for any finite R , the model has the following advantages over the standard picture. m At present it may be considered a matter of individual taste as to whether these advantages outweigh the required paradigm shift. In particular, many cosmologists seem at peace with the prospect that we and the entire human history of life on earth may be playing out an infinite number of times in other parts of the multiverse. At a minimum the present paper could be considered a challenge to cosmologists to prove the uniqueness of the infinite cloning prediction or to find other alternative scenarios which avoid the infinite cloning and preserve some or all of the conceptional advantages of our model while also preserving the string landscape explanation for the smallness of the vacuum energy. It is left to future work to consider the general relativistic corrections to the picture presented here and to incorporate the effects of non-zero pressure. In the process, modifications to the matter density of eq. 2.4 that preserve the properties of eqs. 2.7 and 2.8 may become evident. Acknowledgements We acknowledge useful discussions of the matter presented here with Larry Ford at Tufts University. The research of LC was supported in part by the DOE under grant DE-FG02-10ER41714 and that of GG under DOE grant DE-FG02-92ER40702.\",\n \"pages\": [\n 10,\n 11,\n 12\n ]\n }\n]"}}},{"rowIdx":294,"cells":{"bibcode":{"kind":"string","value":"2013IJMPA..2850160K"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1209.3612.pdf"},"content":{"kind":"string","value":"\nNote About Hamiltonian Formalism for General Non-Linear Massive Gravity Action in Stuckelberg Formalism\nJ. Klusoˇn\nDepartment of Theoretical Physics and Astrophysics Faculty of Science, Masaryk University Kotl´aˇrsk´a 2, 611 37, Brno Czech Republic E-mail: klu@physics.muni.cz\nAbstract: In this note we try to prove the absence of the ghosts in case of the general non-linear massive gravity action in Stuckelberg formalism. We argue that in order to find the explicit form of the Hamiltonian it is natural to start with the general non-linear massive gravity action found in arXiv:1106.3344 [hep-th]. We perform the complete Hamiltonian analysis of the Stuckelberg form of the minimal the non-linear gravity action in this formulation and show that the constraint structure is so rich that it is possible to eliminate non-physical modes. Then we extend this analysis to the case of the general non-linear massive gravity action. We find the corresponding Hamiltonian and collection of the primary constraints. Unfortunately we are not able to finish the complete analysis of the stability of all constraints due to the complex form of one primary constraint so that we are not able to determine the conditions under which given constraint is preserved during the time evolution of the system.\nKeywords: Massive Gravity.\nContents\n
\n\n
\n1. Introduction and Summary\nThe first formulation of the massive gravity was performed by [1] at least in its linearized level as the propagation of the massive graviton above the flat background 1 . Even if the theory seems to be well defined at the linearized level there is a Boulware-Deser ghost [3] in the naive non-linear extension of the Fierz-Pauli formulation. On the other hand recently there has been much progress in the non-linear formulation of the massive gravity without the Boulware-Deser ghost [4, 5] and also [37, 38] 2 .\nThe Hamiltonian treatment of non-linear massive gravity theory was performed in many papers with emphasis on the general proof of the absence of the ghosts in given theory. The first attempt for the analysis of the constraint structure of the non-linear massive gravity was performed in [47]. However it turned out that this analysis was not complete and the wrong conclusion was reached as was then shown in the fundamental paper [40] where the complete Hamiltonian analysis of the gauge fixed form of the general non-linear massive gravity was performed. The fundamental result of given paper is the proof of the existence of two additional constraints in the theory which are crucial for the elimination of non physical modes and hence for the consistency of the non-linear massive gravity at least at the classical level.\nThen the Hamiltonian analysis of the non-linear massive gravity in the Stuckelberg formulation was performed in [48, 49]. Unfortunately the wrong conclusion was again reached in the first versions of given papers as was then shown in [50] where the absence of the ghosts in the minimal version of non-linear massive gravity was proven for the first time. Then an independent proof of the absence of the ghosts in the minimal version of non-linear massive gravity in the Stuckelberg formulation was presented in [51].\nHowever the proof of the absence of the ghosts in the general form of the non-linear massive gravity in the Stuckelberg formulation is still lacking. The difficulty with the possible Hamiltonian formulation of given theory is that the action depends on the kinetic\nterms of the Stuckelberg fields in a highly non-linear way so that it seems to be impossible to find an explicit relation between canonical conjugate momenta and the time derivatives of the Stuckelberg fields. On the other hand there exists the formulation of the non-linear massive gravity action with the linear dependence on the kinetic term for the Stuckelberg fields. This is the form of the non-linear massive gravity action that arises from the original one when the redefinition of the shift functions is performed [38, 37, 41, 42]. The goal of this paper is to perform the Hamiltonian analysis of given action and try to identify all constraints.\nAs we argued above the main advantage of the formulation of the non-linear massive gravity with redefined shift function is that the kinetic term for the Stuckelberg fields appears linearly and hence it is easy to find the corresponding Hamiltonian even for the general form of the non-linear massive gravity action. Further it is possible to identify four primary constraints of the theory where the three ones are parts of the generators of the spatial diffeomorphism. Note that the presence of the diffeomorphism constraints is the reflection of the fact that we have manifestly diffeomorphism invariant theory. On the other hand the fourth primary constraint could be responsible for the elimination of the additional non physical mode. However this claim is only true when the requirement of the preservation of given constraint during the time development of the system generates another additional constraint. Unfortunately we find that the original primary constraint cannot provide such additional constraint due to the fact that the Poisson bracket between the primary constraints defined at different space points is non zero. For that reason we should find another constraint that obeys the property that Poisson bracket between these constraints defined at different space points is zero. We find such a constraint in the case of the minimal non-linear massive gravity action and we show that this constraint has the same form as the primary constraint found in [51]. Then we will be able to show that the requirement of the preservation of given constraint during the time evolution of the system implies the additional constraint and these two constraints together allow to eliminate two non-physical modes. This result agrees with previous two independent analysis performed in [50] and in [51].\nUnfortunately we are not able to reach the main goal of this paper which is the proof of the absence of the ghosts for the general non-linear massive gravity theory in Stuckelberg formalism. The reason is that we are not able to find the primary constraint that has vanishing Poisson bracket between these constraints defined at different space points and that has the Poisson brackets with another constraints that vanish on the constraint surface. This is crucial condition for the existence of the additional constraint. It is rather worrying that we are not able to finish the Hamiltonian analysis for the general non-linear massive gravity action especially in the light of the very nice proof of the absence of the ghosts in case of the gauge fixed non-linear massive gravity action [40]. However there is a possibility that the proof of the absence of the ghosts for general non-linear massive gravity action in Stuckelberg formalism could be found in the very elegant formulation of the massive and multi metric theories of gravity presented in [23]. We hope to return to this problem in future.\nThe structure of this paper is as follows. In the next section (2) we introduce the\nnon-linear massive gravity action and perform the field redefinition of the shift function.\nWe also we find the Hamiltonian formulation of the minimal form of the non-lineal massive gravity in Stuckelberg formulation with redefined shift functions and we find that given theory is free from the ghosts. Then we extend this approach to the case of the general non-linear massive gravity theory in the section ( ?? ). We identify all primary constraints and discuss the difficulties that prevent us to finish the complete Hamiltonian analysis.\n2. Non-linear Massive Gravity with Redefined Shift Functions\nAs we stressed in the introduction section the goal of this paper is to perform the Hamiltonian analysis of the general non-linear massive gravity with presence of the Stuckelberg fields. It turns out that it is useful to consider this action with redefined shift functions [38, 37, 41, 42]. More explicitly, let us begin with following general form of the non-linear massive gravity action\nS = M 2 p ∫ d 4 x √ -ˆ g [ (4) R +2 m 2 3 ∑ n =0 β n e n ( √ ˆ g -1 f )] , (2.1)\nwhere e k ( A ) are elementary symmetric polynomials of the eigenvalues of A . For generic 4 × 4 matrix they are given by\ne 0 ( A ) = 1 , e 1 ( A ) = [ A ] , e 2 ( A ) = 1 2 ([ A ] 2 -[ A 2 ]) , e 3 ( A ) = 1 6 ( [ A ] 3 -3[ A ][ A 2 ] + 2[ A 3 ] ) , e 4 ( A ) = 1 24 ( [ A ] 4 -6[ A ] 2 [ A 2 ] + 3[ A 2 ] 2 +8[ A ][ A 3 ] -6[ A 4 ] ) , e k ( A ) = 0 , for k > 4 , (2.2)\nwhere A µ ν is 4 × 4 matrix and where\n[ A ] = Tr A µ µ . (2.3)\nOf the four β n two combinations are related to the mass and the cosmological constant while the remaining two combinations are free parameters. If we consider the case when the cosmological constant is zero and the parameter m is mass, the four β n are parameterized in terms of the α 3 and α 4 as [5]\nβ n = ( -1) n ( 1 2 (4 -n )(3 -n ) -(4 -n ) α 3 + α 4 ) . (2.4)\nThe minimal action corresponds to β 2 = β 3 = 0 that implies α 3 = α 4 = 1 and consequently β 0 = 3 , β 1 = -1.\nWe consider the massive gravity with that is manifestly diffeomorphism invariant. This can be ensured with the help of 4 scalar fields φ A , A = 0 , 1 , 2 , 3 so that\nˆ g µν f νρ = ˆ g µν ∂ ν φ A ∂ ρ φ A . (2.5)\nThen we have\nN 2 ˆ g -1 f = ( -f 00 + N l f l 0 -f 0 j + N l f lj N 2 g il f l 0 -N i ( -f 00 + N l f l 0 ) N 2 g il f lj -N i ( -f 0 j + N l f lj ) ) , (2.6)\nwhere we also used 3 + 1 decomposition of the four dimensional metric ˆ g µν [45, 46]\nˆ g 00 = -N 2 + N i g ij N j , ˆ g 0 i = N i , ˆ g ij = g ij , ˆ g 00 = -1 N 2 , ˆ g 0 i = N i N 2 , ˆ g ij = g ij -N i N j N 2 . (2.7)\nLet us now perform the redefinition of the shift function N i that was introduced in [37, 38, 41, 42]\nN i = M ˜ n i + f ik f 0 k + N ˜ D i j ˜ n j , (2.8)\nwhere\n˜ x = 1 -˜ n i f ij ˜ n j , M 2 = -f 00 + f 0 k f kl f l 0 (2.9)\nand where we defined f ij as the inverse to f ij in the sense 3\nf ik f kj = δ j i . (2.10)\nFinally note that the matrix ˜ D i j obeys the equation\n√ ˜ x ˜ D i j = √ ( g ik -˜ D i m ˜ n m ˜ D k n ˜ n n ) f kj (2.11)\nand also following important identity\nf ik ˜ D k j = f jk ˜ D k i . (2.12)\nLet us now concentrate on the minimal form of the non-linear massive gravity action. Using the redefinition (2.8) we find that it takes the form\nS = M 2 p ∫ d 3 x dt [ N √ g ˜ K ij G ijkl ˜ K kl + N √ gR -√ gMU -2 m 2 ( N √ g √ ˜ xD i i -3 N √ g )] , (2.13)\nwhere\nU = 2 m 2 √ ˜ x , (2.14)\nand where we used the 3 + 1 decomposition of the four dimensional scalar curvature\n(4) R = ˜ K ij G ijkl ˜ K kl + R , (2.15)\nwhere R is three dimensional scalar curvature and where\nG ijkl = 1 2 ( g ik g jl + g il g jk ) -g ij g kl (2.16)\nwith inverse\nG ijkl = 1 2 ( g ik g jl + g il g jk ) -1 2 g ij g kl , G ijkl G klmn = 1 2 ( δ m i δ n j + δ n i δ m j ) . (2.17)\nNote that in (2.15) we ignored the terms containing total derivatives. Finally note that ˜ K ij is defined as\n˜ K ij = 1 2 N ( ∂ t g ij -∇ i N j (˜ n, g ) -∇ j N i (˜ n, g )) , (2.18)\nwhere N i depends on ˜ n i and g through the relation (2.8).\nAt this point we should stress the reason why we consider the non-linear massive gravity action in the form (2.13). The reason is that we want to perform the Hamiltonian analysis for the general non-linear massive gravity action written in the Stuckelberg formalism. It turns out that the action (2.13) has formally the same form as in case of the general nonlinear massive gravity action when we replace U with more general form whose explicit form was determined in [37, 38, 41, 42]. On the other hand the main advantage of the action (2.13) is that it depends on the time derivatives of φ A through the term M and that this term appears linearly in the action (2.13). We should compare this fact with the original form of the non-linear massive gravity action where the dependence on the time derivatives of φ A is highly non-linear and hence it is very difficult to find corresponding Hamiltonian.\nExplicitly, from (2.13) we find the momenta conjugate to N, ˜ n i and g ij\nπ N ≈ 0 , π i ≈ 0 , π ij = M 2 p √ g G ijkl ˜ K kl (2.19)\nand the momentum conjugate to φ A\np A = -( δM δ∂ t φ A ˜ n i + f ij ∂ j φ A ) R i -M 2 p √ g δM ∂ t φ A U ,\n(2.20)\nwhere\nR i = -2 g ik ∇ j π kj . (2.21)\nIt turns out that it is useful to write M 2 in the form\nM 2 = -∂ t φ A M AB ∂ t φ B , M AB = η AB -∂ i φ A f ij ∂ j φ B ,\n(2.22)\nwhere by definition the matrix M AB obeys following relations\nBC A\nM AB η M CD = M AD , det M B = 1 (2.23)\ntogether with\n∂ i φ A M AB = ∂ i φ B -∂ i φ A ∂ k φ A f kl ∂ l φ B = 0 . (2.24)\nWith the help of these results we find\np A + R i f ij ∂ j φ A = (˜ n i R i + M 2 p √ gU ) 1 M M AB ∂ t φ B (2.25)\nand consequently\nM 2 = -∂ t φ A M AB ∂ t φ B = -M 2 (˜ n i R i + M 2 p √ gU ) 2 ( p A + R i f ij ∂ j φ A ) η AB ( p B + R i f ij ∂ j φ B ) (2.26)\nwhich however implies following primary constraint\nΣ p = (˜ n i R i + M 2 p √ gU ) 2 +( p A + R i f ij ∂ j φ A )( p A + R i f ij ∂ j φ A ) ≈ 0 . (2.27)\nNote that using (2.24) we obtain another set of the primary constraints\n∂ i φ A Π A = ∂ i φ A p A + R i = Σ i ≈ 0 .\n(2.28)\nObserve that using (2.28) we can write\np A + R i f ij ∂ j φ A = M AC η CB p B +Σ i f ij ∂ j φ A (2.29)\nso that we can rewrite Σ p into the form\nΣ p = (˜ n i R i + M 2 p √ gU ) 2 + p A M AB p B + H i Σ i , (2.30)\nwhere H i are functions of the phase space variables. As a result we see that it is natural to consider following independent constraint Σ p\nΣ p = (˜ n i R i + M 2 p √ gU ) 2 + p A M AB p B ≈ 0 . (2.31)\nWe return to the analysis of the constraint Σ p below.\nNow we are ready to write the extended Hamiltonian which includes all the primary constraints\nwhere\nH E = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p Σ p +Ω i ˜ Σ i ) , (2.32)\nC 0 = 1 √ gM 2 p π ij G ijkl π kl -M 2 p √ gR +2 m 2 M 2 p √ g √ ˜ x ˜ D i i -6 m 2 M 2 p √ g + ˜ D i j ˜ n j R i (2.33)\nand where we introduced the constraints ˜ Σ i defined as\n˜ Σ i = Σ i + ∂ i ˜ n i π i + ∂ j (˜ n j π i ) . (2.34)\nNote that ˜ Σ i is defined as linear combination of the constraints Σ i ≈ 0 together with the constraints π i ≈ 0.\nTo proceed further we have to check the stability of all constraints. To do this we have to calculate the Poisson brackets between all constraints and the Hamiltonian H E . Note that we have following set of the canonical variables g ij , π ij , φ A , p A , ˜ n i , π i and N,π N with non-zero Poisson brackets\n{ g ij ( x ) , π kl ( y ) } = 1 2 ( δ k i δ l j + δ l i δ k j ) δ ( x -y ) , { φ A ( x ) , p B ( y ) } = δ A B δ ( x -y ) , { N ( x ) , π N ( y ) } = δ ( x -y ) , { ˜ n i ( x ) , π j ( y ) } = δ i j δ ( x -y ) . (2.35)\nNow we show that the smeared form of the constraint ˜ Σ i ≈ 0\nT S ( ζ i ) = ∫ d 3 x ζ i ˜ Σ i (2.36)\nis the generator of the spatial diffeomorphism. First of all using (2.35) we find\n{ T S ( ζ i ) , ˜ n k } = -ζ i ∂ i ˜ n k + ˜ n j ∂ j ζ k (2.37)\nwhich is the correct transformation rule for ˜ n i . Then using (2.35) we find\n{ T S ( N i ) , R j } = -∂ i N i R j -N i ∂ i R j -R i ∂ j N i , { T S ( N i ) , p A } = -N i ∂ i p A -∂ i N i p A , { T S ( N i ) , φ A } = -N i ∂ i φ A , { T S ( N i ) , g ij } = -N k ∂ k g ij -∂ i N k g kj -g ik ∂ j N k , { T S ( N i ) , π ij } = -∂ k ( N k π ij ) + ∂ k N i π kj + π ik ∂ k N j , { T S ( N i ) , f ij } = -N k ∂ k f ij -∂ i N k f kj -f ik ∂ j N k , { T S ( N i ) , π i } = -∂ i N i π j -N i ∂ i π j + ∂ j N i π j , (2.38)\nthat are the correct transformation rules of the canonical variables under spatial diffeomorphism. To proceed further we need the Poisson bracket between T S ( N i ) and ˜ D i j . It turns out that it is convenient to know the explicit form of the matrix ˜ D i j [37, 38, 41, 42]\nwhere\n˜ D i j = √ g im f mn Q n p ( Q -1 ) p j , (2.39)\nQ i j = ˜ xδ i j + ˜ n i ˜ n k f kj , ( Q -1 ) p q = 1 ˜ x ( δ p q -˜ n p ˜ n m f mq ) . (2.40)\nUsing this expression we can easily determine the Poisson brackets between T S ( N i ) and ˜ D k j . In fact, by definition we have\n{ T S ( N i ) , Q k l } = -N m ∂ m Q k l + ∂ m N k Q m l -Q k n ∂ l N n , { T S ( N i ) , Q i j Q j k } = -N m ∂ m ( Q i j Q j k ) + ∂ m N i Q i j Q j k -Q i j Q j m ∂ k N m . (2.41)\nUsing (2.39) and the results derived above we find\n{ T S ( N i ) , ˜ D i j } = -N m ∂ m ˜ D i j + ∂ m N i ˜ D m j -˜ D i m ∂ j N m . (2.42)\nCollecting all these results and after some calculations we find\n{ T S ( N i ) , C 0 } = -N m ∂ m C 0 -∂ m N m C 0 , { T S ( N i ) , Σ p } = -N m ∂ m Σ p -∂ m N m Σ p . (2.43)\nNote also that it is easy to show that following Poisson bracket holds\n{ T S ( N i ) , T S ( M j ) } = T S ( N j ∂ j M i -M j ∂ j N i ) . (2.44)\nNow we are ready to analyze the stability of all primary constraints. As usual the requirement of the preservation of the constraint π N ≈ 0 implies an existence of the secondary constraint C 0 ≈ 0. However the fact that C 0 is the constraint immediately implies that the constraint ˜ Σ i ≈ 0 is preserved during the time evolution of the system, using (2.43) and (2.44). Now we analyze the requirement of the preservation of the constraints π i ≈ 0 during the time evolution of the system\n∂ t π i = { π i , H E } = -( Ω p δ k i + ∂ ( ˜ D k j ˜ n j ) ∂ ˜ n i ) ( R k -2 m 2 M 2 p √ g √ ˜ x f km ˜ n m ) = 0 . (2.45)\nIt turns out that the following matrix\nΩ p δ k i + ∂ ( ˜ D k j ˜ n j ) ∂ ˜ n i = 0 (2.46)\ncannot be solved for Ω p and hence we have to demand the existence of following secondary constraints [37, 38, 41, 42]\nC i ≡ R i -2 m 2 M 2 p √ g √ ˜ x f ij ˜ n j ≈ 0 . (2.47)\nFinally we have to proceed to the analysis of the time development of the constraint Σ p ≈ 0. However it turns out that it is very difficult to perform this analysis for Σ p due to the presence of the terms that contain the spatial derivatives of φ A . Then the explicit\ncalculation gives { Σ p ( x ) , Σ p ( y ) } /negationslash = 0. For that reason we proceed in a different way when we try to simplify the constraint Σ p . Using C i and Σ i we find that\nA = 4 m 4 M 4 p g ˜ n i f ij ˜ n j √ ˜ x + F i Σ i + G i C i , (2.48)\nwhere\nA = p A ∂ i φ A f ij ∂ j φ B p B , (2.49)\nand where F i , G i are the phase space functions whose explicit form is not important for us. Then with the help of (2.48) we express ˜ n i f ij ˜ n j as a function of the phase space variables p A , φ A and g ij , π ij\n˜ n i f ij ˜ n j = A -F i Σ i -G i C i ( A -F i Σ i -G i C i ) + 4 m 4 M 4 p g . (2.50)\nIn the same way we obtain\n˜ n i R i = A -F i Σ i -G i C i √ ( A -F i Σ i -G i C i ) + 4 m 4 M 4 p g + ˜ n i C i , (2.51)\n˜ n i = -∂ j φ A p A f ji √ A +4 m 4 M 4 p g + ˜ F i Σ i + ˜ G i C i , (2.52)\nand\nwhere again ˜ F i , ˜ G i are phase space functions whose explicit form is not needed for us. Now using these results we find that the constraint Σ p takes the form\nΣ p = ( A -F i Σ i -G i C i +4 m 4 M 4 p g )4 m 4 M 4 p g A -F i Σ i -G i C i +4 m 4 M 4 p g + H i Σ i + p A p A = = p A p A +4 m 4 M 4 p g + H i Σ i ≡ 4 m 4 M 4 p g ˜ Σ p + H i Σ i , (2.53)\nwhere we introduced new independent constraint ˜ Σ p\n˜ Σ p = p A p A 4 m 4 M 4 p g +1 = 0 (2.54)\nthat has precisely the same form as in [51]. Note that the constraint ˜ Σ p has the desired property that { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. As a result we see that it is more natural to consider ˜ Σ p instead of Σ p as an independent constraint. Then the total Hamiltonian, where we include all constraints, takes the form\nH T = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p ˜ Σ p +Ω i ˜ Σ i +Γ i C i ) . (2.55)\nNow we are ready to analyze the stability of all constraints that appear in (2.55). First of all we find that π N ≈ 0 is automatically preserved while the preservation of the constraint π i ≈ 0 gives\n∂ t π i = { π i , H T } ≈ ∫ d 3 x Γ j ( x ) { π i , C j ( x ) } = = -2 m 2 Γ j 1 √ ˜ x ( f ij -f ik ˜ n k f jl ˜ n l ) ≡ -/triangle π i , C j Γ j . (2.56)\nBy definition\n/negationslash\ndet( f ij -f ik ˜ n k f il ˜ n l ) = ˜ x det f ij = 0 (2.57)\nand hence the matrix /triangle π i , C j is non-singular. Then the only solution of the equation (2.56) is Γ i = 0.\nAs the next step we proceed to the analysis of the stability of the constraint ˜ Σ p . As is clear from (2.54) we have\n{ ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0 . (2.58)\nThen the time evolution of given constraint takes the form\n∂ t ˜ Σ p = { ˜ Σ p , H T } ≈ ∫ d 3 x N ( x ) { Σ p , C 0 ( x ) } (2.59)\nusing the fact that ˜ Σ p does not depend on ˜ n i together with Γ i = 0 and also the fact that ˜ Σ p is manifestly diffeomorphism invariant.\nIn order to explicitly determine (2.59) we need following expression\nδ ( √ ˜ x ˜ D k k ) δf ij = √ ˜ x 2 ˜ D j p f pi -1 √ ˜ x ˜ n l f lm δ ( ˜ D m p ˜ n p ) δf ij . (2.60)\nThen after some calculations we obtain\n{ ˜ Σ p , ∫ d 3 x N C 0 } = 2 ∂ i [ N ˜ D i j ]˜ n j Σ p + + 1 M 4 p m 4 g p A ∂ i [ N δ ( ˜ D k l ˜ n l ) δf ij C k ∂ j φ A ] -1 M 4 p m 4 g p A ∂ j φ A ∂ i [ N δ ( ˜ D k l ˜ n l ) δf ij ] Σ k + + N ( -˜ D i j ∂ i [˜ n j p A ] p A 2 m 4 p m 4 g + 2 m 2 M 2 p M 4 p m 4 g ˜ D i k p A ∂ i [ √ g √ ˜ xf kj ∂ j φ A ] ) ≈ N Σ II p . (2.61)\nIn order to simplify Σ II p further we use (2.50), (2.51) together with (2.52) to make Σ II p independent on ˜ n i . In fact, using (2.39)(2.40) we obtain\nQ m p = 1 A +4 M 4 p m 4 g (4 m 4 M 4 p gδ m p + ∂ j φ A p A f jm ∂ p φ B p B ) , ( Q -1 ) m p = A +4 M 4 p m 4 g 4 m 4 M 4 p g ( δ m p -1 A +4 m 4 M 4 p g ∂ j φ A p A f jm ∂ p φ B p B ) (2.62)\nup to terms proportional to the constraints C i , Σ i . With the help of these results we obtain\nΣ II p = -˜ D i j ∂ i [˜ n j p A ] p A 2 m 4 p m 4 g + 2 m 2 M 2 p M 4 p m 4 g ˜ D i p p A ∂ i [ √ g √ ˜ xf pj ∂ j φ A ] + F ' i Σ i + G ' i C i ≡ ≡ ˜ Σ II p + F ' i Σ i + G ' i C i , (2.63)\n/negationslash\nwhere ˜ x and ˜ D i j are functions of p A , ∂ j φ A and g through the relations (2.50),(2.51) and (2.52). Now we see from (2.59) that the time evolution of the constraint ˜ Σ p ≈ 0 is obeyed on condition when either N = 0 or when ˜ Σ II p = 0. Note that we should interpreted N as the Lagrange multiplier so that it is possible to demand that N = 0 on condition when ˜ Σ II p = 0 on the whole phase space. Of course such a condition is too strong so that it is more natural to demand that ˜ Σ II p ≈ 0 and N = 0. In other words ˜ Σ II p ≈ 0 is the new secondary constraint.\n/negationslash\nIn summary we have following collection of constraints: π N ≈ 0 , π i ≈ 0 , C 0 ≈ 0 , C i ≈ 0 , ˜ Σ i ≈ 0 , ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The dynamics of these constraints is governed by the total Hamiltonian\nH T = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p ˜ Σ p +Ω II p ˜ Σ II p +Ω i ˜ Σ i +Γ i C i ) . (2.64)\nAs the final step we have to analyze the preservation of all constraints. The case of π N ≈ 0 is trivial. For π i ≈ 0 we obtain\n∂ i π i ( x ) = { π i ( x ) , H T } = ∫ d 3 y (Γ j ( y ) { π i ( x ) , C j ( y ) } +Ω II p ( y ) { π i ( x ) , ˜ Σ II p ( y ) } ) = = Γ j /triangle π i , C j ( x ) = 0 (2.65)\ndue to the crucial fact that ˜ Σ II p does not depend on ˜ n i . This is the main reason why we introduced ˜ Σ II p instead of Σ II p . Then as we argued above the only solution of the equation is Γ i = 0. Now the time development of C i is given by the equation\n∂ t C i ( x ) = {C i ( x ) , H T } ≈ ≈ ∫ d 3 x ( N ( y ) {C i ( x ) , C 0 ( y ) } + v j ( y ) {C i ( x ) , π j ( y ) } + + Ω p ( y ) { C i ( x ) , ˜ Σ p ( y ) } +Ω II p ( y ) { C i ( x ) , ˜ Σ II p ( y ) }) (2.66)\nand the time development of the constraint ˜ Σ p is governed by the equation\n∂ t ˜ Σ p ( x ) = { ˜ Σ p ( x ) , H T } ≈ ∫ d 3 x Ω II p ( y ) { ˜ Σ p ( x ) , ˜ Σ II p ( y ) } . (2.67)\nAs follows from the explicit form of the constraint ˜ Σ II p we see that { ˜ Σ II p ( x ) , ˜ Σ p ( y ) } is non-zero and proportional also to the higher order derivatives of the delta functions. As a consequence we find that the only solution of the equation above is Ω II p = 0. Further we analyze the time evolution of the constraint ˜ Σ II p\n∂ t ˜ Σ II p ( x ) = { ˜ Σ II p ( x ) , H T } = = ∫ d 3 x ( N ( y ) { ˜ Σ II p ( x ) , C 0 ( y ) } +Ω p ( y ) { ˜ Σ II p ( x ) , ˜ Σ p ( y ) }) = 0 . (2.68)\nNow from the last equation we obtain Ω p as a function of the phase space variables and N , at least in principle. Then inserting this result into the equation for the preservation of C i (2.66) we determine v j as functions of the phase space variables. Finally note also that the constraint C 0 is automatically preserved due to the fact that Γ i = Ω II p = 0 and also the fact that {C 0 ( x ) , C 0 ( y ) } ≈ 0 as was shown in [40].\nIn summary we obtain following picture. We have five the first class constraints π N ≈ 0 , C 0 ≈ 0 , ˜ Σ i ≈ 0 together with eight the second class constraints π i ≈ 0 , C i ≈ 0 and ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The constraints π i ≈ 0 together with C i ≈ 0 can be solved for π i and ˜ n i . Then the constraint ˜ Σ p can be solved for one of the four momenta p A while the constraint ˜ Σ II p can be solved for one of the four φ A . As a result we have 12 gravitational degrees of freedom g ij , π ij ,6 scalars degrees of freedom together with 4 first class constraints C 0 ≈ 0 , ˜ Σ i ≈ 0. Then we find that the number of physical degrees of freedom is 10 which is the correct number of physical degrees of freedom of the massive gravity.\n3. General Non-Linear Massive Gravity Action\nLet us try to apply the procedure performed in previous section to the case of the general non-linear massive gravity whose action takes the form\nS = M 2 p ∫ d 3 x dt [ N √ g ˜ K ij G ijkl ˜ K kl + √ gNR +2 m 2 √ gMU +2 m 2 N √ gV ] , (3.1)\nwhere\n√ √\nU = β 1 ˜ x + β 2 [( ˜ x ) 2 ˜ D i i + ˜ n i f ij ˜ D j k ˜ n k ] + + β 3 [ √ ˜ x ( ˜ D l l ˜ n i f ij ˜ D j k ˜ n k -˜ D i k ˜ n k f ij ˜ D j l ˜ n l ) + 1 2 √ ˜ x 3 ( ˜ D i i ˜ D j j -˜ D i j ˜ D j i )] , V = β 0 + β 1 √ ˜ x ˜ D i i + 1 2 √ ˜ x 2 [ ˜ D i i ˜ D j j + ˜ D i j ˜ D j i ] + + 1 6 β 3 √ ˜ x 3 [ ˜ D i i ˜ D j j ˜ D k k -3 ˜ D i i ˜ D j k ˜ D k j +2 ˜ D i j ˜ D j k ˜ D k i ]\n(3.2)\nFollowing the analysis performed in the previous section we find the extended Hamiltonian in the form\nwhere\nH E = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p Σ p +Ω i ˜ Σ i ) , (3.3)\nC 0 = 1 √ gM 2 p π ij G ijkl π kl -M 2 p √ gR -2 m 2 M 2 p V + R i ˜ D i j ˜ n j , (3.4)\nand where the primary constraint Σ p takes the form\nΣ p : (˜ n i R i +2 m 2 M 2 p √ gU ) 2 + p A M AB p B ≈ 0 . (3.5)\nAs the next step we should analyze the stability of all primary constraints. As in previous section we find that the stability of the constraint π N ≈ 0 implies the secondary constraint C 0 ≈ 0 while the stability of the constraints π i implies set of the secondary constraints C i [37, 38, 41, 42]\nC i = R i -2 m 2 √ g ˜ n l f lj √ ˜ x [ β 1 δ j i + β 2 √ ˜ x ( δ j i ˜ D m m -˜ D j i )+ + β 3 √ ˜ x 2 ( 1 2 δ j i ( ˜ D m m ˜ D n n -˜ D m n ˜ D n m ) + ˜ D j m ˜ D m i -˜ D j i ˜ D m m )] . (3.6)\nNow we come to the key point of the analysis which is the requirement of the preservation of the constraint Σ p ≈ 0 during the time evolution of the system. This is very complicated expression which depends on the all phase space variables. Remember that in the minimal case we expressed ˜ n i as functions of g ij , p A and φ A . As a result we found that Σ p can be expressed as a linear combination of C i , Σ i and ˜ Σ p where ˜ Σ p obeys an important property { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0.It would be certainly nice to repeat the same procedure in the case of the constraint Σ p given in (3.5). Unfortunately we are not able to solve the constraint C i in order to express ˜ n i as a function of R i . Consequently we are not able to express the constraint Σ p as a linear combination of the constraints C i and possibly ˜ Σ i , C 0 and the new constraint ˜ Σ p where { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. In other words, despite of the fact we are able to find the primary constraint Σ p ≈ 0 we are not able to determine the additional secondary constraint which is necessary for the elimination of two non-physical phase space modes. 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\n\n
\n\n[41] S. F. Hassan and R. A. Rosen, 'Bimetric Gravity from Ghost-free Massive Gravity,' JHEP 1202 (2012) 126 [arXiv:1109.3515 [hep-th]].\n[42] S. F. Hassan, R. A. Rosen and A. Schmidt-May, 'Ghost-free Massive Gravity with a General Reference Metric,' JHEP 1202 (2012) 026 [arXiv:1109.3230 [hep-th]].\n[43] Q. -G. Huang, Y. -S. Piao and S. -Y. Zhou, 'Mass-Varying Massive Gravity,' arXiv:1206.5678 [hep-th].\n[44] Y. Akrami, T. S. Koivisto and M. Sandstad, 'Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality,' arXiv:1209.0457 [astro-ph.CO].\n[45] E. Gourgoulhon, '3+1 formalism and bases of numerical relativity,' gr-qc/0703035 [GR-QC].\n[46] R. L. Arnowitt, S. Deser, C. W. Misner, 'The Dynamics of general relativity,' [gr-qc/0405109].\n[47] J. Kluson, 'Note About Hamiltonian Structure of Non-Linear Massive Gravity,' JHEP 1201 (2012) 013 [arXiv:1109.3052 [hep-th]].\n[48] J. Kluson, 'Comments About Hamiltonian Formulation of Non-Linear Massive Gravity with Stuckelberg Fields,' JHEP 1206 (2012) 170 [arXiv:1112.5267 [hep-th]].\n[49] J. Kluson, 'Remark About Hamiltonian Formulation of Non-Linear Massive Gravity in Stuckelberg Formalism,' arXiv:1202.5899 [hep-th].\n[50] S. F. Hassan, A. Schmidt-May and M. von Strauss, 'Proof of Consistency of Nonlinear Massive Gravity in the St'uckelberg Formulation,' Phys. Lett. B 715 (2012) 335 [arXiv:1203.5283 [hep-th]].\n[51] J. Kluson, 'Non-Linear Massive Gravity with Additional Primary Constraint and Absence of Ghosts,' Phys. Rev. D 86 (2012) 044024 [arXiv:1204.2957 [hep-th]].\n\n"},"sections":{"kind":"list like","value":[{"title":"J. Klusoˇn","content":"Department of Theoretical Physics and Astrophysics Faculty of Science, Masaryk University Kotl´aˇrsk´a 2, 611 37, Brno Czech Republic E-mail: klu@physics.muni.cz Abstract: In this note we try to prove the absence of the ghosts in case of the general non-linear massive gravity action in Stuckelberg formalism. We argue that in order to find the explicit form of the Hamiltonian it is natural to start with the general non-linear massive gravity action found in arXiv:1106.3344 [hep-th]. We perform the complete Hamiltonian analysis of the Stuckelberg form of the minimal the non-linear gravity action in this formulation and show that the constraint structure is so rich that it is possible to eliminate non-physical modes. Then we extend this analysis to the case of the general non-linear massive gravity action. We find the corresponding Hamiltonian and collection of the primary constraints. Unfortunately we are not able to finish the complete analysis of the stability of all constraints due to the complex form of one primary constraint so that we are not able to determine the conditions under which given constraint is preserved during the time evolution of the system. Keywords: Massive Gravity.","pages":[1]},{"title":"1. Introduction and Summary","content":"The first formulation of the massive gravity was performed by [1] at least in its linearized level as the propagation of the massive graviton above the flat background 1 . Even if the theory seems to be well defined at the linearized level there is a Boulware-Deser ghost [3] in the naive non-linear extension of the Fierz-Pauli formulation. On the other hand recently there has been much progress in the non-linear formulation of the massive gravity without the Boulware-Deser ghost [4, 5] and also [37, 38] 2 . The Hamiltonian treatment of non-linear massive gravity theory was performed in many papers with emphasis on the general proof of the absence of the ghosts in given theory. The first attempt for the analysis of the constraint structure of the non-linear massive gravity was performed in [47]. However it turned out that this analysis was not complete and the wrong conclusion was reached as was then shown in the fundamental paper [40] where the complete Hamiltonian analysis of the gauge fixed form of the general non-linear massive gravity was performed. The fundamental result of given paper is the proof of the existence of two additional constraints in the theory which are crucial for the elimination of non physical modes and hence for the consistency of the non-linear massive gravity at least at the classical level. Then the Hamiltonian analysis of the non-linear massive gravity in the Stuckelberg formulation was performed in [48, 49]. Unfortunately the wrong conclusion was again reached in the first versions of given papers as was then shown in [50] where the absence of the ghosts in the minimal version of non-linear massive gravity was proven for the first time. Then an independent proof of the absence of the ghosts in the minimal version of non-linear massive gravity in the Stuckelberg formulation was presented in [51]. However the proof of the absence of the ghosts in the general form of the non-linear massive gravity in the Stuckelberg formulation is still lacking. The difficulty with the possible Hamiltonian formulation of given theory is that the action depends on the kinetic terms of the Stuckelberg fields in a highly non-linear way so that it seems to be impossible to find an explicit relation between canonical conjugate momenta and the time derivatives of the Stuckelberg fields. On the other hand there exists the formulation of the non-linear massive gravity action with the linear dependence on the kinetic term for the Stuckelberg fields. This is the form of the non-linear massive gravity action that arises from the original one when the redefinition of the shift functions is performed [38, 37, 41, 42]. The goal of this paper is to perform the Hamiltonian analysis of given action and try to identify all constraints. As we argued above the main advantage of the formulation of the non-linear massive gravity with redefined shift function is that the kinetic term for the Stuckelberg fields appears linearly and hence it is easy to find the corresponding Hamiltonian even for the general form of the non-linear massive gravity action. Further it is possible to identify four primary constraints of the theory where the three ones are parts of the generators of the spatial diffeomorphism. Note that the presence of the diffeomorphism constraints is the reflection of the fact that we have manifestly diffeomorphism invariant theory. On the other hand the fourth primary constraint could be responsible for the elimination of the additional non physical mode. However this claim is only true when the requirement of the preservation of given constraint during the time development of the system generates another additional constraint. Unfortunately we find that the original primary constraint cannot provide such additional constraint due to the fact that the Poisson bracket between the primary constraints defined at different space points is non zero. For that reason we should find another constraint that obeys the property that Poisson bracket between these constraints defined at different space points is zero. We find such a constraint in the case of the minimal non-linear massive gravity action and we show that this constraint has the same form as the primary constraint found in [51]. Then we will be able to show that the requirement of the preservation of given constraint during the time evolution of the system implies the additional constraint and these two constraints together allow to eliminate two non-physical modes. This result agrees with previous two independent analysis performed in [50] and in [51]. Unfortunately we are not able to reach the main goal of this paper which is the proof of the absence of the ghosts for the general non-linear massive gravity theory in Stuckelberg formalism. The reason is that we are not able to find the primary constraint that has vanishing Poisson bracket between these constraints defined at different space points and that has the Poisson brackets with another constraints that vanish on the constraint surface. This is crucial condition for the existence of the additional constraint. It is rather worrying that we are not able to finish the Hamiltonian analysis for the general non-linear massive gravity action especially in the light of the very nice proof of the absence of the ghosts in case of the gauge fixed non-linear massive gravity action [40]. However there is a possibility that the proof of the absence of the ghosts for general non-linear massive gravity action in Stuckelberg formalism could be found in the very elegant formulation of the massive and multi metric theories of gravity presented in [23]. We hope to return to this problem in future. The structure of this paper is as follows. In the next section (2) we introduce the non-linear massive gravity action and perform the field redefinition of the shift function. We also we find the Hamiltonian formulation of the minimal form of the non-lineal massive gravity in Stuckelberg formulation with redefined shift functions and we find that given theory is free from the ghosts. Then we extend this approach to the case of the general non-linear massive gravity theory in the section ( ?? ). We identify all primary constraints and discuss the difficulties that prevent us to finish the complete Hamiltonian analysis.","pages":[2,3,4]},{"title":"2. Non-linear Massive Gravity with Redefined Shift Functions","content":"As we stressed in the introduction section the goal of this paper is to perform the Hamiltonian analysis of the general non-linear massive gravity with presence of the Stuckelberg fields. It turns out that it is useful to consider this action with redefined shift functions [38, 37, 41, 42]. More explicitly, let us begin with following general form of the non-linear massive gravity action where e k ( A ) are elementary symmetric polynomials of the eigenvalues of A . For generic 4 × 4 matrix they are given by where A µ ν is 4 × 4 matrix and where Of the four β n two combinations are related to the mass and the cosmological constant while the remaining two combinations are free parameters. If we consider the case when the cosmological constant is zero and the parameter m is mass, the four β n are parameterized in terms of the α 3 and α 4 as [5] The minimal action corresponds to β 2 = β 3 = 0 that implies α 3 = α 4 = 1 and consequently β 0 = 3 , β 1 = -1. We consider the massive gravity with that is manifestly diffeomorphism invariant. This can be ensured with the help of 4 scalar fields φ A , A = 0 , 1 , 2 , 3 so that Then we have where we also used 3 + 1 decomposition of the four dimensional metric ˆ g µν [45, 46] Let us now perform the redefinition of the shift function N i that was introduced in [37, 38, 41, 42] where and where we defined f ij as the inverse to f ij in the sense 3 Finally note that the matrix ˜ D i j obeys the equation and also following important identity Let us now concentrate on the minimal form of the non-linear massive gravity action. Using the redefinition (2.8) we find that it takes the form where and where we used the 3 + 1 decomposition of the four dimensional scalar curvature where R is three dimensional scalar curvature and where with inverse Note that in (2.15) we ignored the terms containing total derivatives. Finally note that ˜ K ij is defined as where N i depends on ˜ n i and g through the relation (2.8). At this point we should stress the reason why we consider the non-linear massive gravity action in the form (2.13). The reason is that we want to perform the Hamiltonian analysis for the general non-linear massive gravity action written in the Stuckelberg formalism. It turns out that the action (2.13) has formally the same form as in case of the general nonlinear massive gravity action when we replace U with more general form whose explicit form was determined in [37, 38, 41, 42]. On the other hand the main advantage of the action (2.13) is that it depends on the time derivatives of φ A through the term M and that this term appears linearly in the action (2.13). We should compare this fact with the original form of the non-linear massive gravity action where the dependence on the time derivatives of φ A is highly non-linear and hence it is very difficult to find corresponding Hamiltonian. Explicitly, from (2.13) we find the momenta conjugate to N, ˜ n i and g ij and the momentum conjugate to φ A (2.20) where It turns out that it is useful to write M 2 in the form (2.22) where by definition the matrix M AB obeys following relations together with With the help of these results we find and consequently which however implies following primary constraint Note that using (2.24) we obtain another set of the primary constraints (2.28) Observe that using (2.28) we can write so that we can rewrite Σ p into the form where H i are functions of the phase space variables. As a result we see that it is natural to consider following independent constraint Σ p We return to the analysis of the constraint Σ p below. Now we are ready to write the extended Hamiltonian which includes all the primary constraints where and where we introduced the constraints ˜ Σ i defined as Note that ˜ Σ i is defined as linear combination of the constraints Σ i ≈ 0 together with the constraints π i ≈ 0. To proceed further we have to check the stability of all constraints. To do this we have to calculate the Poisson brackets between all constraints and the Hamiltonian H E . Note that we have following set of the canonical variables g ij , π ij , φ A , p A , ˜ n i , π i and N,π N with non-zero Poisson brackets Now we show that the smeared form of the constraint ˜ Σ i ≈ 0 is the generator of the spatial diffeomorphism. First of all using (2.35) we find which is the correct transformation rule for ˜ n i . Then using (2.35) we find that are the correct transformation rules of the canonical variables under spatial diffeomorphism. To proceed further we need the Poisson bracket between T S ( N i ) and ˜ D i j . It turns out that it is convenient to know the explicit form of the matrix ˜ D i j [37, 38, 41, 42] where Using this expression we can easily determine the Poisson brackets between T S ( N i ) and ˜ D k j . In fact, by definition we have Using (2.39) and the results derived above we find Collecting all these results and after some calculations we find Note also that it is easy to show that following Poisson bracket holds Now we are ready to analyze the stability of all primary constraints. As usual the requirement of the preservation of the constraint π N ≈ 0 implies an existence of the secondary constraint C 0 ≈ 0. However the fact that C 0 is the constraint immediately implies that the constraint ˜ Σ i ≈ 0 is preserved during the time evolution of the system, using (2.43) and (2.44). Now we analyze the requirement of the preservation of the constraints π i ≈ 0 during the time evolution of the system It turns out that the following matrix cannot be solved for Ω p and hence we have to demand the existence of following secondary constraints [37, 38, 41, 42] Finally we have to proceed to the analysis of the time development of the constraint Σ p ≈ 0. However it turns out that it is very difficult to perform this analysis for Σ p due to the presence of the terms that contain the spatial derivatives of φ A . Then the explicit calculation gives { Σ p ( x ) , Σ p ( y ) } /negationslash = 0. For that reason we proceed in a different way when we try to simplify the constraint Σ p . Using C i and Σ i we find that where and where F i , G i are the phase space functions whose explicit form is not important for us. Then with the help of (2.48) we express ˜ n i f ij ˜ n j as a function of the phase space variables p A , φ A and g ij , π ij In the same way we obtain and where again ˜ F i , ˜ G i are phase space functions whose explicit form is not needed for us. Now using these results we find that the constraint Σ p takes the form where we introduced new independent constraint ˜ Σ p that has precisely the same form as in [51]. Note that the constraint ˜ Σ p has the desired property that { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. As a result we see that it is more natural to consider ˜ Σ p instead of Σ p as an independent constraint. Then the total Hamiltonian, where we include all constraints, takes the form Now we are ready to analyze the stability of all constraints that appear in (2.55). First of all we find that π N ≈ 0 is automatically preserved while the preservation of the constraint π i ≈ 0 gives By definition /negationslash and hence the matrix /triangle π i , C j is non-singular. Then the only solution of the equation (2.56) is Γ i = 0. As the next step we proceed to the analysis of the stability of the constraint ˜ Σ p . As is clear from (2.54) we have Then the time evolution of given constraint takes the form using the fact that ˜ Σ p does not depend on ˜ n i together with Γ i = 0 and also the fact that ˜ Σ p is manifestly diffeomorphism invariant. In order to explicitly determine (2.59) we need following expression Then after some calculations we obtain In order to simplify Σ II p further we use (2.50), (2.51) together with (2.52) to make Σ II p independent on ˜ n i . In fact, using (2.39)(2.40) we obtain up to terms proportional to the constraints C i , Σ i . With the help of these results we obtain /negationslash where ˜ x and ˜ D i j are functions of p A , ∂ j φ A and g through the relations (2.50),(2.51) and (2.52). Now we see from (2.59) that the time evolution of the constraint ˜ Σ p ≈ 0 is obeyed on condition when either N = 0 or when ˜ Σ II p = 0. Note that we should interpreted N as the Lagrange multiplier so that it is possible to demand that N = 0 on condition when ˜ Σ II p = 0 on the whole phase space. Of course such a condition is too strong so that it is more natural to demand that ˜ Σ II p ≈ 0 and N = 0. In other words ˜ Σ II p ≈ 0 is the new secondary constraint. /negationslash In summary we have following collection of constraints: π N ≈ 0 , π i ≈ 0 , C 0 ≈ 0 , C i ≈ 0 , ˜ Σ i ≈ 0 , ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The dynamics of these constraints is governed by the total Hamiltonian As the final step we have to analyze the preservation of all constraints. The case of π N ≈ 0 is trivial. For π i ≈ 0 we obtain due to the crucial fact that ˜ Σ II p does not depend on ˜ n i . This is the main reason why we introduced ˜ Σ II p instead of Σ II p . Then as we argued above the only solution of the equation is Γ i = 0. Now the time development of C i is given by the equation and the time development of the constraint ˜ Σ p is governed by the equation As follows from the explicit form of the constraint ˜ Σ II p we see that { ˜ Σ II p ( x ) , ˜ Σ p ( y ) } is non-zero and proportional also to the higher order derivatives of the delta functions. As a consequence we find that the only solution of the equation above is Ω II p = 0. Further we analyze the time evolution of the constraint ˜ Σ II p Now from the last equation we obtain Ω p as a function of the phase space variables and N , at least in principle. Then inserting this result into the equation for the preservation of C i (2.66) we determine v j as functions of the phase space variables. Finally note also that the constraint C 0 is automatically preserved due to the fact that Γ i = Ω II p = 0 and also the fact that {C 0 ( x ) , C 0 ( y ) } ≈ 0 as was shown in [40]. In summary we obtain following picture. We have five the first class constraints π N ≈ 0 , C 0 ≈ 0 , ˜ Σ i ≈ 0 together with eight the second class constraints π i ≈ 0 , C i ≈ 0 and ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The constraints π i ≈ 0 together with C i ≈ 0 can be solved for π i and ˜ n i . Then the constraint ˜ Σ p can be solved for one of the four momenta p A while the constraint ˜ Σ II p can be solved for one of the four φ A . As a result we have 12 gravitational degrees of freedom g ij , π ij ,6 scalars degrees of freedom together with 4 first class constraints C 0 ≈ 0 , ˜ Σ i ≈ 0. Then we find that the number of physical degrees of freedom is 10 which is the correct number of physical degrees of freedom of the massive gravity.","pages":[4,5,6,7,8,9,10,11,12,13]},{"title":"3. General Non-Linear Massive Gravity Action","content":"Let us try to apply the procedure performed in previous section to the case of the general non-linear massive gravity whose action takes the form where Following the analysis performed in the previous section we find the extended Hamiltonian in the form where and where the primary constraint Σ p takes the form As the next step we should analyze the stability of all primary constraints. As in previous section we find that the stability of the constraint π N ≈ 0 implies the secondary constraint C 0 ≈ 0 while the stability of the constraints π i implies set of the secondary constraints C i [37, 38, 41, 42] Now we come to the key point of the analysis which is the requirement of the preservation of the constraint Σ p ≈ 0 during the time evolution of the system. This is very complicated expression which depends on the all phase space variables. Remember that in the minimal case we expressed ˜ n i as functions of g ij , p A and φ A . As a result we found that Σ p can be expressed as a linear combination of C i , Σ i and ˜ Σ p where ˜ Σ p obeys an important property { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0.It would be certainly nice to repeat the same procedure in the case of the constraint Σ p given in (3.5). Unfortunately we are not able to solve the constraint C i in order to express ˜ n i as a function of R i . Consequently we are not able to express the constraint Σ p as a linear combination of the constraints C i and possibly ˜ Σ i , C 0 and the new constraint ˜ Σ p where { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. In other words, despite of the fact we are able to find the primary constraint Σ p ≈ 0 we are not able to determine the additional secondary constraint which is necessary for the elimination of two non-physical phase space modes. As a result the main goal of this paper which was the proof of the absence of the ghosts in the general non-linear massive gravity action in Stuckelberg formalism cannot be completed. Acknowledgement: This work was supported by the Grant agency of the Czech republic under the grant P201/12/G028.","pages":[13,14]}],"string":"[\n {\n \"title\": \"J. Klusoˇn\",\n \"content\": \"Department of Theoretical Physics and Astrophysics Faculty of Science, Masaryk University Kotl´aˇrsk´a 2, 611 37, Brno Czech Republic E-mail: klu@physics.muni.cz Abstract: In this note we try to prove the absence of the ghosts in case of the general non-linear massive gravity action in Stuckelberg formalism. We argue that in order to find the explicit form of the Hamiltonian it is natural to start with the general non-linear massive gravity action found in arXiv:1106.3344 [hep-th]. We perform the complete Hamiltonian analysis of the Stuckelberg form of the minimal the non-linear gravity action in this formulation and show that the constraint structure is so rich that it is possible to eliminate non-physical modes. Then we extend this analysis to the case of the general non-linear massive gravity action. We find the corresponding Hamiltonian and collection of the primary constraints. Unfortunately we are not able to finish the complete analysis of the stability of all constraints due to the complex form of one primary constraint so that we are not able to determine the conditions under which given constraint is preserved during the time evolution of the system. Keywords: Massive Gravity.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction and Summary\",\n \"content\": \"The first formulation of the massive gravity was performed by [1] at least in its linearized level as the propagation of the massive graviton above the flat background 1 . Even if the theory seems to be well defined at the linearized level there is a Boulware-Deser ghost [3] in the naive non-linear extension of the Fierz-Pauli formulation. On the other hand recently there has been much progress in the non-linear formulation of the massive gravity without the Boulware-Deser ghost [4, 5] and also [37, 38] 2 . The Hamiltonian treatment of non-linear massive gravity theory was performed in many papers with emphasis on the general proof of the absence of the ghosts in given theory. The first attempt for the analysis of the constraint structure of the non-linear massive gravity was performed in [47]. However it turned out that this analysis was not complete and the wrong conclusion was reached as was then shown in the fundamental paper [40] where the complete Hamiltonian analysis of the gauge fixed form of the general non-linear massive gravity was performed. The fundamental result of given paper is the proof of the existence of two additional constraints in the theory which are crucial for the elimination of non physical modes and hence for the consistency of the non-linear massive gravity at least at the classical level. Then the Hamiltonian analysis of the non-linear massive gravity in the Stuckelberg formulation was performed in [48, 49]. Unfortunately the wrong conclusion was again reached in the first versions of given papers as was then shown in [50] where the absence of the ghosts in the minimal version of non-linear massive gravity was proven for the first time. Then an independent proof of the absence of the ghosts in the minimal version of non-linear massive gravity in the Stuckelberg formulation was presented in [51]. However the proof of the absence of the ghosts in the general form of the non-linear massive gravity in the Stuckelberg formulation is still lacking. The difficulty with the possible Hamiltonian formulation of given theory is that the action depends on the kinetic terms of the Stuckelberg fields in a highly non-linear way so that it seems to be impossible to find an explicit relation between canonical conjugate momenta and the time derivatives of the Stuckelberg fields. On the other hand there exists the formulation of the non-linear massive gravity action with the linear dependence on the kinetic term for the Stuckelberg fields. This is the form of the non-linear massive gravity action that arises from the original one when the redefinition of the shift functions is performed [38, 37, 41, 42]. The goal of this paper is to perform the Hamiltonian analysis of given action and try to identify all constraints. As we argued above the main advantage of the formulation of the non-linear massive gravity with redefined shift function is that the kinetic term for the Stuckelberg fields appears linearly and hence it is easy to find the corresponding Hamiltonian even for the general form of the non-linear massive gravity action. Further it is possible to identify four primary constraints of the theory where the three ones are parts of the generators of the spatial diffeomorphism. Note that the presence of the diffeomorphism constraints is the reflection of the fact that we have manifestly diffeomorphism invariant theory. On the other hand the fourth primary constraint could be responsible for the elimination of the additional non physical mode. However this claim is only true when the requirement of the preservation of given constraint during the time development of the system generates another additional constraint. Unfortunately we find that the original primary constraint cannot provide such additional constraint due to the fact that the Poisson bracket between the primary constraints defined at different space points is non zero. For that reason we should find another constraint that obeys the property that Poisson bracket between these constraints defined at different space points is zero. We find such a constraint in the case of the minimal non-linear massive gravity action and we show that this constraint has the same form as the primary constraint found in [51]. Then we will be able to show that the requirement of the preservation of given constraint during the time evolution of the system implies the additional constraint and these two constraints together allow to eliminate two non-physical modes. This result agrees with previous two independent analysis performed in [50] and in [51]. Unfortunately we are not able to reach the main goal of this paper which is the proof of the absence of the ghosts for the general non-linear massive gravity theory in Stuckelberg formalism. The reason is that we are not able to find the primary constraint that has vanishing Poisson bracket between these constraints defined at different space points and that has the Poisson brackets with another constraints that vanish on the constraint surface. This is crucial condition for the existence of the additional constraint. It is rather worrying that we are not able to finish the Hamiltonian analysis for the general non-linear massive gravity action especially in the light of the very nice proof of the absence of the ghosts in case of the gauge fixed non-linear massive gravity action [40]. However there is a possibility that the proof of the absence of the ghosts for general non-linear massive gravity action in Stuckelberg formalism could be found in the very elegant formulation of the massive and multi metric theories of gravity presented in [23]. We hope to return to this problem in future. The structure of this paper is as follows. In the next section (2) we introduce the non-linear massive gravity action and perform the field redefinition of the shift function. We also we find the Hamiltonian formulation of the minimal form of the non-lineal massive gravity in Stuckelberg formulation with redefined shift functions and we find that given theory is free from the ghosts. Then we extend this approach to the case of the general non-linear massive gravity theory in the section ( ?? ). We identify all primary constraints and discuss the difficulties that prevent us to finish the complete Hamiltonian analysis.\",\n \"pages\": [\n 2,\n 3,\n 4\n ]\n },\n {\n \"title\": \"2. Non-linear Massive Gravity with Redefined Shift Functions\",\n \"content\": \"As we stressed in the introduction section the goal of this paper is to perform the Hamiltonian analysis of the general non-linear massive gravity with presence of the Stuckelberg fields. It turns out that it is useful to consider this action with redefined shift functions [38, 37, 41, 42]. More explicitly, let us begin with following general form of the non-linear massive gravity action where e k ( A ) are elementary symmetric polynomials of the eigenvalues of A . For generic 4 × 4 matrix they are given by where A µ ν is 4 × 4 matrix and where Of the four β n two combinations are related to the mass and the cosmological constant while the remaining two combinations are free parameters. If we consider the case when the cosmological constant is zero and the parameter m is mass, the four β n are parameterized in terms of the α 3 and α 4 as [5] The minimal action corresponds to β 2 = β 3 = 0 that implies α 3 = α 4 = 1 and consequently β 0 = 3 , β 1 = -1. We consider the massive gravity with that is manifestly diffeomorphism invariant. This can be ensured with the help of 4 scalar fields φ A , A = 0 , 1 , 2 , 3 so that Then we have where we also used 3 + 1 decomposition of the four dimensional metric ˆ g µν [45, 46] Let us now perform the redefinition of the shift function N i that was introduced in [37, 38, 41, 42] where and where we defined f ij as the inverse to f ij in the sense 3 Finally note that the matrix ˜ D i j obeys the equation and also following important identity Let us now concentrate on the minimal form of the non-linear massive gravity action. Using the redefinition (2.8) we find that it takes the form where and where we used the 3 + 1 decomposition of the four dimensional scalar curvature where R is three dimensional scalar curvature and where with inverse Note that in (2.15) we ignored the terms containing total derivatives. Finally note that ˜ K ij is defined as where N i depends on ˜ n i and g through the relation (2.8). At this point we should stress the reason why we consider the non-linear massive gravity action in the form (2.13). The reason is that we want to perform the Hamiltonian analysis for the general non-linear massive gravity action written in the Stuckelberg formalism. It turns out that the action (2.13) has formally the same form as in case of the general nonlinear massive gravity action when we replace U with more general form whose explicit form was determined in [37, 38, 41, 42]. On the other hand the main advantage of the action (2.13) is that it depends on the time derivatives of φ A through the term M and that this term appears linearly in the action (2.13). We should compare this fact with the original form of the non-linear massive gravity action where the dependence on the time derivatives of φ A is highly non-linear and hence it is very difficult to find corresponding Hamiltonian. Explicitly, from (2.13) we find the momenta conjugate to N, ˜ n i and g ij and the momentum conjugate to φ A (2.20) where It turns out that it is useful to write M 2 in the form (2.22) where by definition the matrix M AB obeys following relations together with With the help of these results we find and consequently which however implies following primary constraint Note that using (2.24) we obtain another set of the primary constraints (2.28) Observe that using (2.28) we can write so that we can rewrite Σ p into the form where H i are functions of the phase space variables. As a result we see that it is natural to consider following independent constraint Σ p We return to the analysis of the constraint Σ p below. Now we are ready to write the extended Hamiltonian which includes all the primary constraints where and where we introduced the constraints ˜ Σ i defined as Note that ˜ Σ i is defined as linear combination of the constraints Σ i ≈ 0 together with the constraints π i ≈ 0. To proceed further we have to check the stability of all constraints. To do this we have to calculate the Poisson brackets between all constraints and the Hamiltonian H E . Note that we have following set of the canonical variables g ij , π ij , φ A , p A , ˜ n i , π i and N,π N with non-zero Poisson brackets Now we show that the smeared form of the constraint ˜ Σ i ≈ 0 is the generator of the spatial diffeomorphism. First of all using (2.35) we find which is the correct transformation rule for ˜ n i . Then using (2.35) we find that are the correct transformation rules of the canonical variables under spatial diffeomorphism. To proceed further we need the Poisson bracket between T S ( N i ) and ˜ D i j . It turns out that it is convenient to know the explicit form of the matrix ˜ D i j [37, 38, 41, 42] where Using this expression we can easily determine the Poisson brackets between T S ( N i ) and ˜ D k j . In fact, by definition we have Using (2.39) and the results derived above we find Collecting all these results and after some calculations we find Note also that it is easy to show that following Poisson bracket holds Now we are ready to analyze the stability of all primary constraints. As usual the requirement of the preservation of the constraint π N ≈ 0 implies an existence of the secondary constraint C 0 ≈ 0. However the fact that C 0 is the constraint immediately implies that the constraint ˜ Σ i ≈ 0 is preserved during the time evolution of the system, using (2.43) and (2.44). Now we analyze the requirement of the preservation of the constraints π i ≈ 0 during the time evolution of the system It turns out that the following matrix cannot be solved for Ω p and hence we have to demand the existence of following secondary constraints [37, 38, 41, 42] Finally we have to proceed to the analysis of the time development of the constraint Σ p ≈ 0. However it turns out that it is very difficult to perform this analysis for Σ p due to the presence of the terms that contain the spatial derivatives of φ A . Then the explicit calculation gives { Σ p ( x ) , Σ p ( y ) } /negationslash = 0. For that reason we proceed in a different way when we try to simplify the constraint Σ p . Using C i and Σ i we find that where and where F i , G i are the phase space functions whose explicit form is not important for us. Then with the help of (2.48) we express ˜ n i f ij ˜ n j as a function of the phase space variables p A , φ A and g ij , π ij In the same way we obtain and where again ˜ F i , ˜ G i are phase space functions whose explicit form is not needed for us. Now using these results we find that the constraint Σ p takes the form where we introduced new independent constraint ˜ Σ p that has precisely the same form as in [51]. Note that the constraint ˜ Σ p has the desired property that { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. As a result we see that it is more natural to consider ˜ Σ p instead of Σ p as an independent constraint. Then the total Hamiltonian, where we include all constraints, takes the form Now we are ready to analyze the stability of all constraints that appear in (2.55). First of all we find that π N ≈ 0 is automatically preserved while the preservation of the constraint π i ≈ 0 gives By definition /negationslash and hence the matrix /triangle π i , C j is non-singular. Then the only solution of the equation (2.56) is Γ i = 0. As the next step we proceed to the analysis of the stability of the constraint ˜ Σ p . As is clear from (2.54) we have Then the time evolution of given constraint takes the form using the fact that ˜ Σ p does not depend on ˜ n i together with Γ i = 0 and also the fact that ˜ Σ p is manifestly diffeomorphism invariant. In order to explicitly determine (2.59) we need following expression Then after some calculations we obtain In order to simplify Σ II p further we use (2.50), (2.51) together with (2.52) to make Σ II p independent on ˜ n i . In fact, using (2.39)(2.40) we obtain up to terms proportional to the constraints C i , Σ i . With the help of these results we obtain /negationslash where ˜ x and ˜ D i j are functions of p A , ∂ j φ A and g through the relations (2.50),(2.51) and (2.52). Now we see from (2.59) that the time evolution of the constraint ˜ Σ p ≈ 0 is obeyed on condition when either N = 0 or when ˜ Σ II p = 0. Note that we should interpreted N as the Lagrange multiplier so that it is possible to demand that N = 0 on condition when ˜ Σ II p = 0 on the whole phase space. Of course such a condition is too strong so that it is more natural to demand that ˜ Σ II p ≈ 0 and N = 0. In other words ˜ Σ II p ≈ 0 is the new secondary constraint. /negationslash In summary we have following collection of constraints: π N ≈ 0 , π i ≈ 0 , C 0 ≈ 0 , C i ≈ 0 , ˜ Σ i ≈ 0 , ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The dynamics of these constraints is governed by the total Hamiltonian As the final step we have to analyze the preservation of all constraints. The case of π N ≈ 0 is trivial. For π i ≈ 0 we obtain due to the crucial fact that ˜ Σ II p does not depend on ˜ n i . This is the main reason why we introduced ˜ Σ II p instead of Σ II p . Then as we argued above the only solution of the equation is Γ i = 0. Now the time development of C i is given by the equation and the time development of the constraint ˜ Σ p is governed by the equation As follows from the explicit form of the constraint ˜ Σ II p we see that { ˜ Σ II p ( x ) , ˜ Σ p ( y ) } is non-zero and proportional also to the higher order derivatives of the delta functions. As a consequence we find that the only solution of the equation above is Ω II p = 0. Further we analyze the time evolution of the constraint ˜ Σ II p Now from the last equation we obtain Ω p as a function of the phase space variables and N , at least in principle. Then inserting this result into the equation for the preservation of C i (2.66) we determine v j as functions of the phase space variables. Finally note also that the constraint C 0 is automatically preserved due to the fact that Γ i = Ω II p = 0 and also the fact that {C 0 ( x ) , C 0 ( y ) } ≈ 0 as was shown in [40]. In summary we obtain following picture. We have five the first class constraints π N ≈ 0 , C 0 ≈ 0 , ˜ Σ i ≈ 0 together with eight the second class constraints π i ≈ 0 , C i ≈ 0 and ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The constraints π i ≈ 0 together with C i ≈ 0 can be solved for π i and ˜ n i . Then the constraint ˜ Σ p can be solved for one of the four momenta p A while the constraint ˜ Σ II p can be solved for one of the four φ A . As a result we have 12 gravitational degrees of freedom g ij , π ij ,6 scalars degrees of freedom together with 4 first class constraints C 0 ≈ 0 , ˜ Σ i ≈ 0. Then we find that the number of physical degrees of freedom is 10 which is the correct number of physical degrees of freedom of the massive gravity.\",\n \"pages\": [\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13\n ]\n },\n {\n \"title\": \"3. General Non-Linear Massive Gravity Action\",\n \"content\": \"Let us try to apply the procedure performed in previous section to the case of the general non-linear massive gravity whose action takes the form where Following the analysis performed in the previous section we find the extended Hamiltonian in the form where and where the primary constraint Σ p takes the form As the next step we should analyze the stability of all primary constraints. As in previous section we find that the stability of the constraint π N ≈ 0 implies the secondary constraint C 0 ≈ 0 while the stability of the constraints π i implies set of the secondary constraints C i [37, 38, 41, 42] Now we come to the key point of the analysis which is the requirement of the preservation of the constraint Σ p ≈ 0 during the time evolution of the system. This is very complicated expression which depends on the all phase space variables. Remember that in the minimal case we expressed ˜ n i as functions of g ij , p A and φ A . As a result we found that Σ p can be expressed as a linear combination of C i , Σ i and ˜ Σ p where ˜ Σ p obeys an important property { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0.It would be certainly nice to repeat the same procedure in the case of the constraint Σ p given in (3.5). Unfortunately we are not able to solve the constraint C i in order to express ˜ n i as a function of R i . Consequently we are not able to express the constraint Σ p as a linear combination of the constraints C i and possibly ˜ Σ i , C 0 and the new constraint ˜ Σ p where { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. In other words, despite of the fact we are able to find the primary constraint Σ p ≈ 0 we are not able to determine the additional secondary constraint which is necessary for the elimination of two non-physical phase space modes. As a result the main goal of this paper which was the proof of the absence of the ghosts in the general non-linear massive gravity action in Stuckelberg formalism cannot be completed. Acknowledgement: This work was supported by the Grant agency of the Czech republic under the grant P201/12/G028.\",\n \"pages\": [\n 13,\n 14\n ]\n }\n]"}}},{"rowIdx":295,"cells":{"bibcode":{"kind":"string","value":"2013IJMPB..2750119B"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1212.0368.pdf"},"content":{"kind":"string","value":"\nMEASUREMENT OF THE THERMAL EXPANSION COEFFICIENT OF AN Al-Mg ALLOY AT ULTRA-LOW TEMPERATURES\nM. BASSAN b,c , B. BUONOMO a , G. CAVALLARI e , E. COCCIA b,c , S. D'ANTONIO b , V. FAFONE b,c , L.G. FOGGETTA a , C. LIGI a ∗ , A. MARINI a , G. MAZZITELLI a , G. MODESTINO a , G. PIZZELLA c,a , L. QUINTIERI a † , F. RONGA a , P. VALENTE d\na Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Via E. Fermi, 40 - 00044 Frascati, Italy b Istituto Nazionale di Fisica Nucleare - Sezione Roma2, Via della Ricerca Scientifica - 00133 Rome, Italy c Dipartimento di Fisica, Universit'a di Tor Vergata, Via della Ricerca Scientifica - 00133 Rome, Italy d Istituto Nazionale di Fisica Nucleare - Sezione Roma1, Piazzale Aldo Moro 2 - 00185 Rome, Italy e CERN, CH1211, Gen'eve, Switzerland carlo.ligi@lnf.infn.it\nWe describe a result coming from an experiment based on an Al-Mg alloy ( ∼ 5% Mg) suspended bar hit by an electron beam and operated above and below the termperature of transition from superconducting to normal state of the material. The amplitude of the bar first longitudinal mode of oscillation, excited by the beam interacting with the bulk, and the energy deposited by the beam in the bar are the quantities measured by the experiment. These quantities, inserted in the equations describing the mechanism of the mode excitation and complemented by an independent measurement of the specific heat, allow us to determine the linear expansion coefficient α of the material. We obtain α = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 for the normal state of conduction in the temperature interval 0 . 9 < T < 2 K and α = [( -2 . 45 ± 0 . 60)+( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 for the superconducting state in the interval 0 . 3 < T < 0 . 8 K.\nKeywords : Thermal expansion; Low temperature; Aluminum alloy.\n1. Introduction\nWrought aluminum-magnesium alloys (International Alloy Designation System: 5XXX, Mg content: 5.6% maximum) are commonly used in applications where good workability, very good resistance to corrosion, high fatigue strength are desidered. Example of applications are: oil, fuel lines and tanks, pressure cryogenic vessels, marine structures and fittings, automotive and architectural components.\nThe alloy Al 5056 (5.2% Mg, 0.1% Mn, 0.1% Cr) is the material of the test mass of gravitational wave (GW) resonant antennas AURIGA 1 , EXPLORER 2 , NAUTILUS 3 in Italy and ALLEGRO 4 in the United States, operated at liquid Helium temperature and below. During the last decade, all these detectors took\npart, as a network, in coordinated searches for impulsive GW excitations (no GW events were detected). 5 , 6\nThe test mass of the GW resonant antennas is a suspended cylinder ( ∼ 2300 kg mass), acting as a mechanical resonator whose resonances are excited by the incoming GW. The elastic vibrations are converted to electrical signals by a transducer system. The minimum detectable energy of a GW resonant antenna inversely depends on Q , the acoustic quality factor of the material also known as the inverse of the internal friction. Al 5056 was chosen as the material for the GW antennas due to the high values of Q , 7 exhibited at very low temperatures.\nB.L. Baron and R. Hofstadter first measured 8 mechanical oscillations in piezoelectric disks penetrated by high energy electron beams and they outlined the possibility that cosmic ray (CR) events could excite mechanical vibrations in a metallic cylinder at its resonant frequency and that these events could mimic GW events detected by resonant antennas. NAUTILUS has been equipped with a CR detector 9 to study the interactions caused by CRs and to provide a veto against the CR induced events in the antenna.\nCorrelating NAUTILUS data with CR observations, the following results were obtained: 1) the rate of high energy signals due to CR showers was larger than expectations 10 , 11 when the antenna was operated at a temperature T = 0 . 14 K, i.e. well below the transition temperature from normal-conducting ( n ) to superconducting ( s ) states of Al 5056; 12 2) this feature was not observed when the antenna was operated at T = 1 . 5 K, i.e. above the transition temperature. 13\nSince these findings had not a straightforward interpretation, we performed an experiment (RAP) aimed at measuring the longitudinal oscillations of a small suspended Al 5056 cylinder hit by an electron beam of known energy. The RAP experiment, performed at the Beam Test Facility (BTF) 14 of the DAFNE Φ-factory complex in the INFN Frascati Laboratory, has shown that 1) the amplitude of the longitudinal oscillations of the cylinder hit by ionizing particles depends on the state of conduction of the material 15 , 16 and that 2) the observed amplitudes are consistent with the amplitudes measured by NAUTILUS in CR events both in ( n ) and ( s ) states. 17 Moreover, the amplitude can be evaluated in the framework of the Thermo-Acoustic model: in the ( n ) state it depends on the ratio of the thermal expansion coefficient to the specific heat, this ratio being part of the definition of the Gruneisen parameter. In the ( s ) state the amplitude also depends on the fractional volume change between the ( s ) and ( n ) states. In principle, the relevant thermophysical properties of the Al 5056 at low temperature are not known with sufficient precision to allow the comparison between experimental and theoretical values of the amplitude predicted by the model. However, our confidence in the model is also based on the small discrepancy, of the order of 15%, between the amplitudes measured in the alloy and those computed by using the data for pure aluminum in the region 4.5 ≤ T ≤ 264 K. 18\nDirect measurements of the Gruneisen parameter of elastic materials have been\nreported: 19 the authors used pulsed electron beams hitting the front surface of a thin slab, and measured the induced motion on the rear face surface. The equation of motion describing thermoelastic effects depends on the Gruneisen parameter and on the energy deposited by the beam in the slab. The measurement of the displacement, the knowledge of the deposited energy and the use of the equation of motion allowed them to directly determine the Gruneisen parameter.\nIn a similar way this paper presents experimental results on the linear expansion coefficient of the aluminum alloy above and below the temperature of transition between the ( s ) and ( n ) state. In order to determine the coefficient we use: a) the oscillation amplitude measurements, b) the equations describing the amplitude in the framework of the Thermo-Acoustic model, c) the measurements of the deposited energy by the electron beam, obtained by the product of the measured beam multiplicity and the calculated energy loss in the material by each particle, and d) an independent measurement of the specific heat.\n2. The Thermo-Acoustic model\n2.1. ( n ) state\nWe consider a cylinder (radius R , length L , mass M ) that is suspended in correspondence of the middle section, with its axis horizontal. A ionizing particle, after hitting the lateral surface of the cylinder and interacting with the material, generates a pressure pulse in the bulk. This sonic pulse is the result of the local thermal expansion of the material caused by the warming up due to the energy lost by the particle in the material. The sonic pulse determines the excitation of the vibrational oscillation modes of the suspended bar. By introducing a vector field u ( x , t ) to describe the local displacements from equilibrium, we express the amplitude of the k -th longitudinal mode of oscillation as proportional to the quantity: 20 , 21 , 22\ng k,therm = ∆ P therm ρ A ' I k = γ ρ ∣ ∣ ∣ dW dl ∣ ∣ ∣ I k , (1)\n∣ ∣ where ∆ P therm is the pressure pulse due to the sonic source described above, ρ is the mass density, dW/dl is the energy loss per unit length of the interacting particle, A ' is the cross section of the tubular zone centered on the particle path in which the effects are generated and I k = ∫ dl ( ∇· u k ( x )) is a line integral over the particle path involving the normal mode of oscillation u k ( x ). In the previous relation γ is the adimensional Gruneisen parameter:\nγ = βK T ρc V , (2)\nwith β the volume thermal expansion coefficient ( β = 3 α for cubic elements), c V the isochoric specific heat and K T the isothermal bulk module.\n4 C. Ligi et al.\nA thin bar ( R/L /lessmuch 1) hit by a ionizing particle at the center of the lateral surface is a particular case of the more general one that leads to relation (1). In this case, the fundamental longitudinal mode of oscillation is excited to a maximum amplitude given by: 23\nX = 2 αLW πc V M , (3)\nand then exponentially decays due to internal friction. Here W is the total energy released in the bar by the particle beam. The experimental conditions of RAP are close to this particular case and the amplitude maximum of the fundamental longitudinal mode of oscillation is the observable measured by the experiment. However, in order to model the amplitude in the most realistic way, we have performed a Monte Carlo (MC) simulation, 18 which takes into account the corrections O [( R/L ) 2 ] for the modes of oscillation of a cylinder, the transverse dimension of the electron beam at the impact point and the trajectories of the secondary particles generated by the electron interactions in the bar. All these effects are summarized in a corrective parameter /epsilon1 and the amplitude maximum of the fundamental longitudinal mode of oscillation is modeled by:\nX ( n ) = X (1 + /epsilon1 ) (4)\nThe value of /epsilon1 for the aluminum alloy bar used in the experiment is estimated by MC to be -0.04.\n2.2. ( s ) state\nThe energy released by the ionizing particles to the material determines the suppression of the superconductivity in a region (Hot Spot) centered around the particle path. The maximum value of the Hot Spot radius, r HS , is obtained by equating the energy lost per unit length by the particle, dW/dl , to the enthalpy variation in the volume undergoing the ( s ) → ( n ) transition at temperature T : 24 , 25 , 26\nr HS = √ | dW/dl | π ∆ h = √ A '' π , (5)\nwhere ∆ h is the enthalpy change per unit volume and A '' is the cross section of the zone switched to the ( n ) state.\nThe amplitude of the longitudinal oscillations is the sum of two terms, one related to thermodynamic effects in the ( s ) → ( n ) transition and the other to the thermal effects in the region switched to the ( n ) state, which have been already described in the Section 2.1. The contribution to the amplitude of the cylinder k -th oscillation\nmode due to the ( s ) → ( n ) transition is proportional to: 20 , 21\ng k,trans = ∆ P trans ρ A '' I k = 1 ρ [ K T ∆ V V + γT ∆ S V ] A '' I k ,\nwhere V = V ( s ) and ∆ V , ∆ S are the differences of the volume and the entropy in the two states of conduction. According to the thermodynamics of volume and pressure effects for Type-II superconductors, these differences can be written as a : 27\n∆ V = V ( n ) -V ( s ) = V H c 4 π ∂H c ∂P + { H 2 c 8 π ∂V ∂P } (6)\nand\n∆ S = S ( n ) -S ( s ) = -V H c 4 π ∂H c ∂T -{ H 2 c 8 π ∂V ∂T } (7)\nHere H c is the superconducting critical magnetic field that is supposed to have the parabolic behavior H c ( t ) = H c (0)(1 -t 2 ), where t = T/T c and T c is the transition temperature. Moreover, the terms in { } brackets are expected to be smaller than the preceding terms 28 and are usually ignored for practical purposes.\nFinally, the knowledge of the specific heat of the material for the ( s ) state, c ( s ) , allows us to compute the variation of the entalpy per unit volume in the form: 29\n∆ h = C int ( T ) + T ∆ S V (8)\nwith:\nC int ( T ) = ∫ T c T c ( s ) ( T ' ) dT '\nThe amplitude maximum of the fundamental longitudinal mode of oscillation in the ( s ) state, as already indicated, is the sum of two terms, one related to the transition and the other to the ( n ) contribution from the switched region: X ( s ) = X trans + X ( n ) . By observing that X trans /X ( n ) = g k =0 ,trans /g k =0 ,therm we can describe the amplitude maximum according to the following relation:\nX ( s ) = X ( n ) { 1 + X trans X ( n ) } = X ( n ) { 1 + [ Π ∆ V V + T ∆ S V ] [∆ h ] -1 } , (9)\n6 C. Ligi et al.\nwhere the relation (4) and the definition (2) of γ are used to define:\nΠ = 2 ρL (1 + /epsilon1 ) 3 πM ( X ( n ) /W )\nWe note that the term contained in { } brackets in the relation (9) is independent from W , the energy released by the particle to the bar, and that X ( s ) linearly depends on W through X ( n ) .\n3. Experimental setup and procedures\nThe experiment mechanical layout, its cryogenic setup and operations, the electron beam characteristics, the instrumentation and the procedures for calibrations, data taking and analysis have been fully described in Refs. 17-18. Briefly, the dimensions and the mass of the Al 5056 cylindrical bar are R = 0 . 091 m, L = 0 . 500 m, M = 34 . 1 kg, respectively. The bar hangs from the cryostat top by means of a multistage suspension system providing a 150 dB attenuation of the external mechanical noise in the 1700-6500 Hz frequency window. The frequency of the fundamental longitudinal mode of oscillation of the bar is f 0 = 5413 . 6 Hz below T = 4 K. The cryostat is equipped with a dilution refrigerator. The temperatures are measured inside the cryostat by 11 thermometers controlled by two multi-channel resistance bridges and, among them, a calibrated RuO 2 resistor measures the temperature of one of the bar end faces with an accuracy of 0.01 K below T = 4 K. Two piezoelectric ceramics (Pz), electrically connected in parallel, are inserted in a slot cut in the position opposite to the bar suspension point and are squeezed when the bar shrinks. In this Pz arrangement the strain measured at the bar center is proportional to the displacement of the bar end faces. The Pz output is first amplified and then sampled at 100 kHz by an ADC embedded in a VME system dedicated to the data acquisition. A Pz calibration procedure, performed before each run of data taking, provides the factor converting the ADC samples into the displacements of the bar end faces. A software filtering algorithm, known as 'digital lock-in', extracts the Fourier component at the frequency f 0 from the time sequence formed by the ADC samples before and after the beam hit, determining the amplitude of the induced fundamental oscillation. The sign of the amplitude is taken positive or negative according to the sign of the first sample raising above the threshold after the beam hit.\nThe BTF beam line delivers to the bar single pulses of ∼ 10 ns duration, containing N e electrons of 510 ± 2 MeV energy. N e ranges from about 5 × 10 7 to 10 9 and is measured with an accuracy of ∼ 3% (for N e > 5 × 10 8 ) by an integrating current transformer placed close to the beam line exit point. The MC simulation, discussed in Section 2, estimates an average energy lost 〈 ∆ E 〉 ± σ ∆ E = 195 . 2 ± 70 . 6 MeV for a 512 MeV electron interacting in the bar and, consequently, the total energy loss per beam pulse is given by W = N e 〈 ∆ E 〉 , σ W = √ N e σ ∆ E . Two sources of error affect the vibration maximum amplitude: the first is an overall systematic error of\nthe order of ± 6%, that accounts for the slightly different set-up and calibration procedures implemented in the runs over two years and the second is related to the noise in the measurement of the oscillation amplitude ( ± 1 . 3 × 10 -13 m).\n4. Linear expansion coefficient measurements\nThe coefficient α is determined by inserting the measured values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) , X ( s ) ) and the corresponding measurements of the deposited energy ( W ) into the relations modeling the amplitudes (equations (4),(9)). The measurement 30 at very low temperatures of the specific heat of an Al 5056 sample, of the same production batch of the RAP bar, is a fundamental ingredient for the coefficient determination both in ( n ) and ( s ) states. Moreover, the superconducting characterization of the material has shown that T c = 0 . 845 ± 0 . 002 K with a total transition width of about 0.1 K. 30 Thus we determine α for the two states of conduction ( α ( n ) , α ( s ) ) according to this temperature value.\n4.1. ( n ) state\nValues of the amplitude maximum of the fundamental longitudinal mode of oscillation and the related values of the deposited energies by the beam have been measured and analyzed in the temperature interval 0 . 9 ≤ T ≤ 2 K. 17 The values of X ( n ) show a strict linear correlation with the values of W (Fig. 1) and they have been fitted, including the errors, by a line constrained to the origin ( X ( n ) = p 0 W ). The slope of the fit, which has a χ 2 per degree of freedom equal to 0 . 45 / 15, is p 0 = (2 . 24 ± 0 . 05) × 10 -10 m J -1 . From the relation (4) it follows that:\nα c V = πMp 0 2 L (1 + /epsilon1 ) (10)\nIn the same temperature interval the parametrization c V = CT + DT 3 , where C is the electronic specific heat coefficient per unit volume and D is the lattice contribution, is determined by 30 C = (4 . 382 ± 0 . 117) × 10 -2 J kg -1 K -2 and D = (5 . 20 ± 0 . 37) × 10 -3 J kg -1 K -4 . Inserting this c V parametrization in relation (10) directly gives:\nα ( n ) = α ( n ) ,e + α ( n ) ,/lscript = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 (11)\nfor the linear expansion coefficient (Fig. 2) expressed in terms of the electronic and lattice components in the temperature interval 0 . 9 ≤ T ≤ 2 K.\nWe note that the value of α ( n ) for pure aluminum is reported 31 to be 2 . 7 × 10 -9 K -1 at T = 2 K. This value, which is the one associated to the lowest temperature in the previously cited work, can be compared with α ( n ) = (3 . 2 ± 0 . 2) × 10 -9 K -1 obtained in the present work for Al 5056 at the same temperature.\n\n\n
Fig. 1. Normal state. Values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) ) vs. the deposited energies (W) measured in the temperature interval 0 . 9 ≤ T ≤ 2 K. Data have been fitted by a straight line, constrained to zero, with slope (2 . 24 ± 0 . 05) × 10 -10 m J -1 .
\n\n4.2. ( s ) state\nFrom eq. (9) we can derive the volume variation among the two states of conduction:\n∆ V V = 1 Π [ R ∆ h p 0 -T ∆ S V ] (12)\nwhere R = X ( s ) /W -p 0 , with p 0 introduced in Section 4.1. The volume variation depends on measured quantities ( X ( s ) , W, p 0 , c V , T, T c ) and calculated ones. The latter are related to the critical field H c ( T = 0) and to its parabolic dependance on the temperature, which also allows us to compute the derivative ∂H c /∂T . If the superconducting properties of Al 5056 can be described by the BCS theory, then H c ( T = 0) ≈ 2 . 42 √ CT c ≈ 70 Oe, C being the electronic specific heat coefficient\nMeasurement of the thermal expansion 9\n\n\n
Fig. 2. Normal state. Measured values of the linear expansion coefficient ( α ( n ) ) vs. temperature (T). The thinner lines enclose the region compatible with the uncertainties.
\n\nper unit volume (Section 4.1), here expressed in suitable cgs units (erg cm -3 K -2 ). As already mentioned, the model predicts that the amplitude maximum of the fundamental longitudinal mode of oscillation for both states of conduction ( X ( n ) , X ( s ) ) linearly depends on the energy deposited by the e -beam in the bar. On the contrary, the X ( s ) values for Al 5056 15 , 17 show an increasing deviation from linearity with the increase of the deposited energy W , due to saturation effects discussed in Ref. 17. However, if we restrict our analysis to the data gathered at the lowest deposited energies, these non linearities can safely be neglected, and we can use the relation (12) to compute the volume variation. Fig. 3 shows the values of ∆ V/V versus the temperature T for the data with released energy: 1 . 5 × 10 -3 < W < 9 × 10 -3 J. ∆ V/V data have been fitted by a 2nd order polynomial q 0 + q 1 T + q 2 T 2 obtaining q 0 = ( -1 . 737 ± 0 . 054) × 10 -8 , q 1 = (7 . 334 ± 1 . 785) × 10 -9 K -1 , q 2 = (1 . 766 ± 0 . 187) × 10 -8 K -2 with a χ 2 = 1 . 38\nnormalized over 94 degrees of freedom.\n\n\n
Fig. 3. Superconducting state. Measured values of ∆ V/V vs. temperature for beam hits deposing in the bar an energy W < 9 mJ. The thick dotted line is a fit to the data with a 2nd order polynomial. The thinner lines enclose a region compatible with the uncertainties in the fit parameters.
\n\nThe use of q 0 and of the relation (6) gives ∂H c /∂P = ( -3 . 12 ± 0 . 10) × 10 -12 T Pa -1 at T = 0 K, a value that can be compared with ( -2 . 67 ± 0 . 06) × 10 -12 T Pa -1 obtained by Harris and Mapother in their experimental study of the critical field of pure aluminum as a function of pressure and temperature. 32 The difference of the thermal expansion coefficients in the normal and superconducting phases is obtained by taking the derivative with respect to T of the difference of volumes: 27 ∆ α = α ( n ) -α ( s ) = 1 3 d dT ∆ V V .\nUnder the hypothesis that α ( n ) is still represented by the relation (11) in the inter-\nal 0 . 3 K < T < T c and by using the polynomial fitting for ∆ V/V , the following expression is obtained for that temperature interval:\nα ( s ) = α ( n ) -∆ α = [( -2 . 45 ± 0 . 60) + ( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 (13)\n\n\n
Fig. 4. Superconducting state. Values of α ( s ) vs. reduced temperature ( t = T/T c ) for Al 5056 (solid line). For reference, we also show the result for pure Al (dotted line) derived from data of Ref. 32. The thinner lines enclose the regions compatible with the uncertainties in the α ( s ) values.
\n\nThis result can be compared with the expansion coefficient of pure aluminum in the ( s ) state, using for pure aluminum the relation α ( s ) = α ( s ) ,e + α /lscript , in which it is assumed that the lattice component does not depend on the conduction state and that its values are given in Ref. 33. The definition (2) gives α ( s ) ,e = ρc V,e ( s ) γ ( s ) ,e / (3 K T ), where the electronic component of c V in the ( s ) state ( c V,e ( s ) ) is taken from the\nwork of Phillips, 34 the electronic component of the Gruneisen parameter in the ( s ) state ( γ ( s ) ,e ) from Ref. 35 and K ( s ) T ∼ K ( n ) T = 79 . 4 × 10 9 Pa at low temperatures. 36 The comparison is shown in Fig. 4 as a function of the reduced temperature t .\n5. Conclusions\nIn this paper we present an evaluation of the linear expansion coefficient of an AlMg alloy obtained by measuring both the amplitude of the fundamental mode of the longitudinal oscillation excited by electrons interacting in a suspended bar and the energy released in the bar by the electron pulse. The use of this method to determine α is new, although similarities exist with the direct measurement of the the Gruneisen parameter performed by hitting thin slabs with particle beams to generate thermoelastic pulses. No expansion data for aluminum alloys were previously available in the literature for the temperature range 0 . 3 < T < 2 K explored by this experiment. The expansion coefficient is negative in the superconducting state and its absolute value just below T c is larger by an order of magnitude than the value above this temperature. This feature is expected also for pure aluminum, according to an analysis 35 of the measured values of H c ( P, T ) available in Ref. 32.\nThis work was partially supported by the EU FP-6 Project ILIAS (RII3-CT-2004506222).\nReferences\n\n1. M. Cerdonio et al. , Class. Quantum Gravity 14 (1997) 1491\n2. P. Astone et al. , Phys. Rev. D 47 (1993) 362.\n3. P. Astone et al. , Astropart. Phys. 7 (1997) 231\n4. M. P. McHugh et al. , Class. Quantum Gravity 22 (2005) S965\n5. P. Astone et al. , Phys. Rev. D 76 (2007) 102001\n6. P. Astone et al. , Phys. Rev. D 82 (2010) 022003\n7. T. Suzuki et al. , Phys. Lett. A 67 (1978) 2\nE. Coccia and T.O. Niinikoski, Lett. Nuovo Cimento 41 (1984) 242\nW. Duffy, J. Appl. Phys. 68 (1990) 5601\nB. Kubler et al. , Physica B 316-317 (2002) 558\n8. B.L. Beron and R. Hofstadter, Phys. Rev. Lett. 23 (1969) 184\n9. E. Coccia et al. , Nucl. Instrum. Meth. A 355 (1995) 624\n10. P. Astone et al. , Phys. Rev. Lett. 84 (2000) 14\n11. P. Astone et al. , Phys. Lett. B 499 (2001) 16\n12. E. Coccia and T.O. Niinikoski, J. Phys. E: Sci. Instrum. 16 (1983) 695\n13. P. Astone et al. , Phys. Lett. B 540 (2002) 179\n14. G. Mazzitelli et al. , Nucl. Instrum. Meth. A 515 (2003) 524\n15. M. Barucci et al. , Phys. Lett. A 373 (2009) 1801\n16. M. Bassan et al. , Europhys. Lett. 76 (2006) 987\n17. M. Bassan et al. , Nucl. Instrum. Meth. A 659 (2011) 289\n18. B. Buonomo et al. , Astropart. Phys. 24 (2005) 65\n\n\n19. More information on the method and the related references are in T.H.K. Barron, J.C. Collins and G.K. White, Adv. Phys. 29 (1980) 609\n20. A.M. Allega and N. Cabibbo, Lett. Nuovo Cimento 38 (1983) 263\n21. C. Bernard, A. De Rujula and B. Lautrup, Nucl. Phys. B 242 (1984) 93\n22. G. Liu and B. Barish, Phys. Rev. Lett. 61 (1988) 271\n23. A.M. Grassi Strini, G. Strini and G. Tagliaferri, J. Appl. Phys. 51 (1980) 948\n24. K.E. Gray in Proceedings of the Superconductive Particle Detectors , 26-29 October 1987, edited by A. Barone (World Scientific, Singapore, 1988)\n25. B. Strehl et al., Phys. Lett. B 242 (1990) 285\n26. M.P. Lisitskiy, J. Appl. Phys. 106 (2009) 103927\n27. R.R. Hake, Phys. Rev. 166 (1968) 471\n28. D. Shoenberg, Superconductivity , 2nd Edition, Cambridge University Press, England (1956) 73-77\n29. D. White et al. , Phys. Rev. 109 (1958) 797\n30. M. Barucci et al. , Physica B 405 (2010) 1452\n31. F.R. Kroeger and C.A. Swenson, J. Appl. Phys. 48 (1977) 853\n32. E.F. Harris and D.E. Mapother, Phys. Rev. 165 (1968) 522\n33. J.G. Collins, G.K. White, and C.A. Swenson, J. Low Temp. Phys. 10 (1973) 69\n34. N.E. Phillips, Phys. Rev. 114 (1959) 676\n35. A. Marini, J. Low Temp. Phys. 133 (2003) 313\n36. G.N. Kamm and G.A. Alers, J. Appl. Phys. 35 (1964) 327\n"},"sections":{"kind":"list like","value":[{"title":"MEASUREMENT OF THE THERMAL EXPANSION COEFFICIENT OF AN Al-Mg ALLOY AT ULTRA-LOW TEMPERATURES","content":"M. BASSAN b,c , B. BUONOMO a , G. CAVALLARI e , E. COCCIA b,c , S. D'ANTONIO b , V. FAFONE b,c , L.G. FOGGETTA a , C. LIGI a ∗ , A. MARINI a , G. MAZZITELLI a , G. MODESTINO a , G. PIZZELLA c,a , L. QUINTIERI a † , F. RONGA a , P. VALENTE d a Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Via E. Fermi, 40 - 00044 Frascati, Italy b Istituto Nazionale di Fisica Nucleare - Sezione Roma2, Via della Ricerca Scientifica - 00133 Rome, Italy c Dipartimento di Fisica, Universit'a di Tor Vergata, Via della Ricerca Scientifica - 00133 Rome, Italy d Istituto Nazionale di Fisica Nucleare - Sezione Roma1, Piazzale Aldo Moro 2 - 00185 Rome, Italy e CERN, CH1211, Gen'eve, Switzerland carlo.ligi@lnf.infn.it We describe a result coming from an experiment based on an Al-Mg alloy ( ∼ 5% Mg) suspended bar hit by an electron beam and operated above and below the termperature of transition from superconducting to normal state of the material. The amplitude of the bar first longitudinal mode of oscillation, excited by the beam interacting with the bulk, and the energy deposited by the beam in the bar are the quantities measured by the experiment. These quantities, inserted in the equations describing the mechanism of the mode excitation and complemented by an independent measurement of the specific heat, allow us to determine the linear expansion coefficient α of the material. We obtain α = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 for the normal state of conduction in the temperature interval 0 . 9 < T < 2 K and α = [( -2 . 45 ± 0 . 60)+( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 for the superconducting state in the interval 0 . 3 < T < 0 . 8 K. Keywords : Thermal expansion; Low temperature; Aluminum alloy.","pages":[1]},{"title":"1. Introduction","content":"Wrought aluminum-magnesium alloys (International Alloy Designation System: 5XXX, Mg content: 5.6% maximum) are commonly used in applications where good workability, very good resistance to corrosion, high fatigue strength are desidered. Example of applications are: oil, fuel lines and tanks, pressure cryogenic vessels, marine structures and fittings, automotive and architectural components. The alloy Al 5056 (5.2% Mg, 0.1% Mn, 0.1% Cr) is the material of the test mass of gravitational wave (GW) resonant antennas AURIGA 1 , EXPLORER 2 , NAUTILUS 3 in Italy and ALLEGRO 4 in the United States, operated at liquid Helium temperature and below. During the last decade, all these detectors took part, as a network, in coordinated searches for impulsive GW excitations (no GW events were detected). 5 , 6 The test mass of the GW resonant antennas is a suspended cylinder ( ∼ 2300 kg mass), acting as a mechanical resonator whose resonances are excited by the incoming GW. The elastic vibrations are converted to electrical signals by a transducer system. The minimum detectable energy of a GW resonant antenna inversely depends on Q , the acoustic quality factor of the material also known as the inverse of the internal friction. Al 5056 was chosen as the material for the GW antennas due to the high values of Q , 7 exhibited at very low temperatures. B.L. Baron and R. Hofstadter first measured 8 mechanical oscillations in piezoelectric disks penetrated by high energy electron beams and they outlined the possibility that cosmic ray (CR) events could excite mechanical vibrations in a metallic cylinder at its resonant frequency and that these events could mimic GW events detected by resonant antennas. NAUTILUS has been equipped with a CR detector 9 to study the interactions caused by CRs and to provide a veto against the CR induced events in the antenna. Correlating NAUTILUS data with CR observations, the following results were obtained: 1) the rate of high energy signals due to CR showers was larger than expectations 10 , 11 when the antenna was operated at a temperature T = 0 . 14 K, i.e. well below the transition temperature from normal-conducting ( n ) to superconducting ( s ) states of Al 5056; 12 2) this feature was not observed when the antenna was operated at T = 1 . 5 K, i.e. above the transition temperature. 13 Since these findings had not a straightforward interpretation, we performed an experiment (RAP) aimed at measuring the longitudinal oscillations of a small suspended Al 5056 cylinder hit by an electron beam of known energy. The RAP experiment, performed at the Beam Test Facility (BTF) 14 of the DAFNE Φ-factory complex in the INFN Frascati Laboratory, has shown that 1) the amplitude of the longitudinal oscillations of the cylinder hit by ionizing particles depends on the state of conduction of the material 15 , 16 and that 2) the observed amplitudes are consistent with the amplitudes measured by NAUTILUS in CR events both in ( n ) and ( s ) states. 17 Moreover, the amplitude can be evaluated in the framework of the Thermo-Acoustic model: in the ( n ) state it depends on the ratio of the thermal expansion coefficient to the specific heat, this ratio being part of the definition of the Gruneisen parameter. In the ( s ) state the amplitude also depends on the fractional volume change between the ( s ) and ( n ) states. In principle, the relevant thermophysical properties of the Al 5056 at low temperature are not known with sufficient precision to allow the comparison between experimental and theoretical values of the amplitude predicted by the model. However, our confidence in the model is also based on the small discrepancy, of the order of 15%, between the amplitudes measured in the alloy and those computed by using the data for pure aluminum in the region 4.5 ≤ T ≤ 264 K. 18 Direct measurements of the Gruneisen parameter of elastic materials have been reported: 19 the authors used pulsed electron beams hitting the front surface of a thin slab, and measured the induced motion on the rear face surface. The equation of motion describing thermoelastic effects depends on the Gruneisen parameter and on the energy deposited by the beam in the slab. The measurement of the displacement, the knowledge of the deposited energy and the use of the equation of motion allowed them to directly determine the Gruneisen parameter. In a similar way this paper presents experimental results on the linear expansion coefficient of the aluminum alloy above and below the temperature of transition between the ( s ) and ( n ) state. In order to determine the coefficient we use: a) the oscillation amplitude measurements, b) the equations describing the amplitude in the framework of the Thermo-Acoustic model, c) the measurements of the deposited energy by the electron beam, obtained by the product of the measured beam multiplicity and the calculated energy loss in the material by each particle, and d) an independent measurement of the specific heat.","pages":[1,2,3]},{"title":"2.1. ( n ) state","content":"We consider a cylinder (radius R , length L , mass M ) that is suspended in correspondence of the middle section, with its axis horizontal. A ionizing particle, after hitting the lateral surface of the cylinder and interacting with the material, generates a pressure pulse in the bulk. This sonic pulse is the result of the local thermal expansion of the material caused by the warming up due to the energy lost by the particle in the material. The sonic pulse determines the excitation of the vibrational oscillation modes of the suspended bar. By introducing a vector field u ( x , t ) to describe the local displacements from equilibrium, we express the amplitude of the k -th longitudinal mode of oscillation as proportional to the quantity: 20 , 21 , 22 ∣ ∣ where ∆ P therm is the pressure pulse due to the sonic source described above, ρ is the mass density, dW/dl is the energy loss per unit length of the interacting particle, A ' is the cross section of the tubular zone centered on the particle path in which the effects are generated and I k = ∫ dl ( ∇· u k ( x )) is a line integral over the particle path involving the normal mode of oscillation u k ( x ). In the previous relation γ is the adimensional Gruneisen parameter: with β the volume thermal expansion coefficient ( β = 3 α for cubic elements), c V the isochoric specific heat and K T the isothermal bulk module.","pages":[3]},{"title":"4 C. Ligi et al.","content":"A thin bar ( R/L /lessmuch 1) hit by a ionizing particle at the center of the lateral surface is a particular case of the more general one that leads to relation (1). In this case, the fundamental longitudinal mode of oscillation is excited to a maximum amplitude given by: 23 and then exponentially decays due to internal friction. Here W is the total energy released in the bar by the particle beam. The experimental conditions of RAP are close to this particular case and the amplitude maximum of the fundamental longitudinal mode of oscillation is the observable measured by the experiment. However, in order to model the amplitude in the most realistic way, we have performed a Monte Carlo (MC) simulation, 18 which takes into account the corrections O [( R/L ) 2 ] for the modes of oscillation of a cylinder, the transverse dimension of the electron beam at the impact point and the trajectories of the secondary particles generated by the electron interactions in the bar. All these effects are summarized in a corrective parameter /epsilon1 and the amplitude maximum of the fundamental longitudinal mode of oscillation is modeled by: The value of /epsilon1 for the aluminum alloy bar used in the experiment is estimated by MC to be -0.04.","pages":[4]},{"title":"2.2. ( s ) state","content":"The energy released by the ionizing particles to the material determines the suppression of the superconductivity in a region (Hot Spot) centered around the particle path. The maximum value of the Hot Spot radius, r HS , is obtained by equating the energy lost per unit length by the particle, dW/dl , to the enthalpy variation in the volume undergoing the ( s ) → ( n ) transition at temperature T : 24 , 25 , 26 where ∆ h is the enthalpy change per unit volume and A '' is the cross section of the zone switched to the ( n ) state. The amplitude of the longitudinal oscillations is the sum of two terms, one related to thermodynamic effects in the ( s ) → ( n ) transition and the other to the thermal effects in the region switched to the ( n ) state, which have been already described in the Section 2.1. The contribution to the amplitude of the cylinder k -th oscillation mode due to the ( s ) → ( n ) transition is proportional to: 20 , 21 where V = V ( s ) and ∆ V , ∆ S are the differences of the volume and the entropy in the two states of conduction. According to the thermodynamics of volume and pressure effects for Type-II superconductors, these differences can be written as a : 27 and Here H c is the superconducting critical magnetic field that is supposed to have the parabolic behavior H c ( t ) = H c (0)(1 -t 2 ), where t = T/T c and T c is the transition temperature. Moreover, the terms in { } brackets are expected to be smaller than the preceding terms 28 and are usually ignored for practical purposes. Finally, the knowledge of the specific heat of the material for the ( s ) state, c ( s ) , allows us to compute the variation of the entalpy per unit volume in the form: 29 with: The amplitude maximum of the fundamental longitudinal mode of oscillation in the ( s ) state, as already indicated, is the sum of two terms, one related to the transition and the other to the ( n ) contribution from the switched region: X ( s ) = X trans + X ( n ) . By observing that X trans /X ( n ) = g k =0 ,trans /g k =0 ,therm we can describe the amplitude maximum according to the following relation:","pages":[4,5]},{"title":"6 C. Ligi et al.","content":"where the relation (4) and the definition (2) of γ are used to define: We note that the term contained in { } brackets in the relation (9) is independent from W , the energy released by the particle to the bar, and that X ( s ) linearly depends on W through X ( n ) .","pages":[6]},{"title":"3. Experimental setup and procedures","content":"The experiment mechanical layout, its cryogenic setup and operations, the electron beam characteristics, the instrumentation and the procedures for calibrations, data taking and analysis have been fully described in Refs. 17-18. Briefly, the dimensions and the mass of the Al 5056 cylindrical bar are R = 0 . 091 m, L = 0 . 500 m, M = 34 . 1 kg, respectively. The bar hangs from the cryostat top by means of a multistage suspension system providing a 150 dB attenuation of the external mechanical noise in the 1700-6500 Hz frequency window. The frequency of the fundamental longitudinal mode of oscillation of the bar is f 0 = 5413 . 6 Hz below T = 4 K. The cryostat is equipped with a dilution refrigerator. The temperatures are measured inside the cryostat by 11 thermometers controlled by two multi-channel resistance bridges and, among them, a calibrated RuO 2 resistor measures the temperature of one of the bar end faces with an accuracy of 0.01 K below T = 4 K. Two piezoelectric ceramics (Pz), electrically connected in parallel, are inserted in a slot cut in the position opposite to the bar suspension point and are squeezed when the bar shrinks. In this Pz arrangement the strain measured at the bar center is proportional to the displacement of the bar end faces. The Pz output is first amplified and then sampled at 100 kHz by an ADC embedded in a VME system dedicated to the data acquisition. A Pz calibration procedure, performed before each run of data taking, provides the factor converting the ADC samples into the displacements of the bar end faces. A software filtering algorithm, known as 'digital lock-in', extracts the Fourier component at the frequency f 0 from the time sequence formed by the ADC samples before and after the beam hit, determining the amplitude of the induced fundamental oscillation. The sign of the amplitude is taken positive or negative according to the sign of the first sample raising above the threshold after the beam hit. The BTF beam line delivers to the bar single pulses of ∼ 10 ns duration, containing N e electrons of 510 ± 2 MeV energy. N e ranges from about 5 × 10 7 to 10 9 and is measured with an accuracy of ∼ 3% (for N e > 5 × 10 8 ) by an integrating current transformer placed close to the beam line exit point. The MC simulation, discussed in Section 2, estimates an average energy lost 〈 ∆ E 〉 ± σ ∆ E = 195 . 2 ± 70 . 6 MeV for a 512 MeV electron interacting in the bar and, consequently, the total energy loss per beam pulse is given by W = N e 〈 ∆ E 〉 , σ W = √ N e σ ∆ E . Two sources of error affect the vibration maximum amplitude: the first is an overall systematic error of the order of ± 6%, that accounts for the slightly different set-up and calibration procedures implemented in the runs over two years and the second is related to the noise in the measurement of the oscillation amplitude ( ± 1 . 3 × 10 -13 m).","pages":[6,7]},{"title":"4. Linear expansion coefficient measurements","content":"The coefficient α is determined by inserting the measured values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) , X ( s ) ) and the corresponding measurements of the deposited energy ( W ) into the relations modeling the amplitudes (equations (4),(9)). The measurement 30 at very low temperatures of the specific heat of an Al 5056 sample, of the same production batch of the RAP bar, is a fundamental ingredient for the coefficient determination both in ( n ) and ( s ) states. Moreover, the superconducting characterization of the material has shown that T c = 0 . 845 ± 0 . 002 K with a total transition width of about 0.1 K. 30 Thus we determine α for the two states of conduction ( α ( n ) , α ( s ) ) according to this temperature value.","pages":[7]},{"title":"4.1. ( n ) state","content":"Values of the amplitude maximum of the fundamental longitudinal mode of oscillation and the related values of the deposited energies by the beam have been measured and analyzed in the temperature interval 0 . 9 ≤ T ≤ 2 K. 17 The values of X ( n ) show a strict linear correlation with the values of W (Fig. 1) and they have been fitted, including the errors, by a line constrained to the origin ( X ( n ) = p 0 W ). The slope of the fit, which has a χ 2 per degree of freedom equal to 0 . 45 / 15, is p 0 = (2 . 24 ± 0 . 05) × 10 -10 m J -1 . From the relation (4) it follows that: In the same temperature interval the parametrization c V = CT + DT 3 , where C is the electronic specific heat coefficient per unit volume and D is the lattice contribution, is determined by 30 C = (4 . 382 ± 0 . 117) × 10 -2 J kg -1 K -2 and D = (5 . 20 ± 0 . 37) × 10 -3 J kg -1 K -4 . Inserting this c V parametrization in relation (10) directly gives: for the linear expansion coefficient (Fig. 2) expressed in terms of the electronic and lattice components in the temperature interval 0 . 9 ≤ T ≤ 2 K. We note that the value of α ( n ) for pure aluminum is reported 31 to be 2 . 7 × 10 -9 K -1 at T = 2 K. This value, which is the one associated to the lowest temperature in the previously cited work, can be compared with α ( n ) = (3 . 2 ± 0 . 2) × 10 -9 K -1 obtained in the present work for Al 5056 at the same temperature.","pages":[7]},{"title":"4.2. ( s ) state","content":"From eq. (9) we can derive the volume variation among the two states of conduction: where R = X ( s ) /W -p 0 , with p 0 introduced in Section 4.1. The volume variation depends on measured quantities ( X ( s ) , W, p 0 , c V , T, T c ) and calculated ones. The latter are related to the critical field H c ( T = 0) and to its parabolic dependance on the temperature, which also allows us to compute the derivative ∂H c /∂T . If the superconducting properties of Al 5056 can be described by the BCS theory, then H c ( T = 0) ≈ 2 . 42 √ CT c ≈ 70 Oe, C being the electronic specific heat coefficient","pages":[8]},{"title":"Measurement of the thermal expansion 9","content":"per unit volume (Section 4.1), here expressed in suitable cgs units (erg cm -3 K -2 ). As already mentioned, the model predicts that the amplitude maximum of the fundamental longitudinal mode of oscillation for both states of conduction ( X ( n ) , X ( s ) ) linearly depends on the energy deposited by the e -beam in the bar. On the contrary, the X ( s ) values for Al 5056 15 , 17 show an increasing deviation from linearity with the increase of the deposited energy W , due to saturation effects discussed in Ref. 17. However, if we restrict our analysis to the data gathered at the lowest deposited energies, these non linearities can safely be neglected, and we can use the relation (12) to compute the volume variation. Fig. 3 shows the values of ∆ V/V versus the temperature T for the data with released energy: 1 . 5 × 10 -3 < W < 9 × 10 -3 J. ∆ V/V data have been fitted by a 2nd order polynomial q 0 + q 1 T + q 2 T 2 obtaining q 0 = ( -1 . 737 ± 0 . 054) × 10 -8 , q 1 = (7 . 334 ± 1 . 785) × 10 -9 K -1 , q 2 = (1 . 766 ± 0 . 187) × 10 -8 K -2 with a χ 2 = 1 . 38 normalized over 94 degrees of freedom. The use of q 0 and of the relation (6) gives ∂H c /∂P = ( -3 . 12 ± 0 . 10) × 10 -12 T Pa -1 at T = 0 K, a value that can be compared with ( -2 . 67 ± 0 . 06) × 10 -12 T Pa -1 obtained by Harris and Mapother in their experimental study of the critical field of pure aluminum as a function of pressure and temperature. 32 The difference of the thermal expansion coefficients in the normal and superconducting phases is obtained by taking the derivative with respect to T of the difference of volumes: 27 ∆ α = α ( n ) -α ( s ) = 1 3 d dT ∆ V V . Under the hypothesis that α ( n ) is still represented by the relation (11) in the inter- al 0 . 3 K < T < T c and by using the polynomial fitting for ∆ V/V , the following expression is obtained for that temperature interval: This result can be compared with the expansion coefficient of pure aluminum in the ( s ) state, using for pure aluminum the relation α ( s ) = α ( s ) ,e + α /lscript , in which it is assumed that the lattice component does not depend on the conduction state and that its values are given in Ref. 33. The definition (2) gives α ( s ) ,e = ρc V,e ( s ) γ ( s ) ,e / (3 K T ), where the electronic component of c V in the ( s ) state ( c V,e ( s ) ) is taken from the work of Phillips, 34 the electronic component of the Gruneisen parameter in the ( s ) state ( γ ( s ) ,e ) from Ref. 35 and K ( s ) T ∼ K ( n ) T = 79 . 4 × 10 9 Pa at low temperatures. 36 The comparison is shown in Fig. 4 as a function of the reduced temperature t .","pages":[9,10,11,12]},{"title":"5. Conclusions","content":"In this paper we present an evaluation of the linear expansion coefficient of an AlMg alloy obtained by measuring both the amplitude of the fundamental mode of the longitudinal oscillation excited by electrons interacting in a suspended bar and the energy released in the bar by the electron pulse. The use of this method to determine α is new, although similarities exist with the direct measurement of the the Gruneisen parameter performed by hitting thin slabs with particle beams to generate thermoelastic pulses. No expansion data for aluminum alloys were previously available in the literature for the temperature range 0 . 3 < T < 2 K explored by this experiment. The expansion coefficient is negative in the superconducting state and its absolute value just below T c is larger by an order of magnitude than the value above this temperature. This feature is expected also for pure aluminum, according to an analysis 35 of the measured values of H c ( P, T ) available in Ref. 32. This work was partially supported by the EU FP-6 Project ILIAS (RII3-CT-2004506222).","pages":[12]}],"string":"[\n {\n \"title\": \"MEASUREMENT OF THE THERMAL EXPANSION COEFFICIENT OF AN Al-Mg ALLOY AT ULTRA-LOW TEMPERATURES\",\n \"content\": \"M. BASSAN b,c , B. BUONOMO a , G. CAVALLARI e , E. COCCIA b,c , S. D'ANTONIO b , V. FAFONE b,c , L.G. FOGGETTA a , C. LIGI a ∗ , A. MARINI a , G. MAZZITELLI a , G. MODESTINO a , G. PIZZELLA c,a , L. QUINTIERI a † , F. RONGA a , P. VALENTE d a Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Via E. Fermi, 40 - 00044 Frascati, Italy b Istituto Nazionale di Fisica Nucleare - Sezione Roma2, Via della Ricerca Scientifica - 00133 Rome, Italy c Dipartimento di Fisica, Universit'a di Tor Vergata, Via della Ricerca Scientifica - 00133 Rome, Italy d Istituto Nazionale di Fisica Nucleare - Sezione Roma1, Piazzale Aldo Moro 2 - 00185 Rome, Italy e CERN, CH1211, Gen'eve, Switzerland carlo.ligi@lnf.infn.it We describe a result coming from an experiment based on an Al-Mg alloy ( ∼ 5% Mg) suspended bar hit by an electron beam and operated above and below the termperature of transition from superconducting to normal state of the material. The amplitude of the bar first longitudinal mode of oscillation, excited by the beam interacting with the bulk, and the energy deposited by the beam in the bar are the quantities measured by the experiment. These quantities, inserted in the equations describing the mechanism of the mode excitation and complemented by an independent measurement of the specific heat, allow us to determine the linear expansion coefficient α of the material. We obtain α = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 for the normal state of conduction in the temperature interval 0 . 9 < T < 2 K and α = [( -2 . 45 ± 0 . 60)+( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 for the superconducting state in the interval 0 . 3 < T < 0 . 8 K. Keywords : Thermal expansion; Low temperature; Aluminum alloy.\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1. Introduction\",\n \"content\": \"Wrought aluminum-magnesium alloys (International Alloy Designation System: 5XXX, Mg content: 5.6% maximum) are commonly used in applications where good workability, very good resistance to corrosion, high fatigue strength are desidered. Example of applications are: oil, fuel lines and tanks, pressure cryogenic vessels, marine structures and fittings, automotive and architectural components. The alloy Al 5056 (5.2% Mg, 0.1% Mn, 0.1% Cr) is the material of the test mass of gravitational wave (GW) resonant antennas AURIGA 1 , EXPLORER 2 , NAUTILUS 3 in Italy and ALLEGRO 4 in the United States, operated at liquid Helium temperature and below. During the last decade, all these detectors took part, as a network, in coordinated searches for impulsive GW excitations (no GW events were detected). 5 , 6 The test mass of the GW resonant antennas is a suspended cylinder ( ∼ 2300 kg mass), acting as a mechanical resonator whose resonances are excited by the incoming GW. The elastic vibrations are converted to electrical signals by a transducer system. The minimum detectable energy of a GW resonant antenna inversely depends on Q , the acoustic quality factor of the material also known as the inverse of the internal friction. Al 5056 was chosen as the material for the GW antennas due to the high values of Q , 7 exhibited at very low temperatures. B.L. Baron and R. Hofstadter first measured 8 mechanical oscillations in piezoelectric disks penetrated by high energy electron beams and they outlined the possibility that cosmic ray (CR) events could excite mechanical vibrations in a metallic cylinder at its resonant frequency and that these events could mimic GW events detected by resonant antennas. NAUTILUS has been equipped with a CR detector 9 to study the interactions caused by CRs and to provide a veto against the CR induced events in the antenna. Correlating NAUTILUS data with CR observations, the following results were obtained: 1) the rate of high energy signals due to CR showers was larger than expectations 10 , 11 when the antenna was operated at a temperature T = 0 . 14 K, i.e. well below the transition temperature from normal-conducting ( n ) to superconducting ( s ) states of Al 5056; 12 2) this feature was not observed when the antenna was operated at T = 1 . 5 K, i.e. above the transition temperature. 13 Since these findings had not a straightforward interpretation, we performed an experiment (RAP) aimed at measuring the longitudinal oscillations of a small suspended Al 5056 cylinder hit by an electron beam of known energy. The RAP experiment, performed at the Beam Test Facility (BTF) 14 of the DAFNE Φ-factory complex in the INFN Frascati Laboratory, has shown that 1) the amplitude of the longitudinal oscillations of the cylinder hit by ionizing particles depends on the state of conduction of the material 15 , 16 and that 2) the observed amplitudes are consistent with the amplitudes measured by NAUTILUS in CR events both in ( n ) and ( s ) states. 17 Moreover, the amplitude can be evaluated in the framework of the Thermo-Acoustic model: in the ( n ) state it depends on the ratio of the thermal expansion coefficient to the specific heat, this ratio being part of the definition of the Gruneisen parameter. In the ( s ) state the amplitude also depends on the fractional volume change between the ( s ) and ( n ) states. In principle, the relevant thermophysical properties of the Al 5056 at low temperature are not known with sufficient precision to allow the comparison between experimental and theoretical values of the amplitude predicted by the model. However, our confidence in the model is also based on the small discrepancy, of the order of 15%, between the amplitudes measured in the alloy and those computed by using the data for pure aluminum in the region 4.5 ≤ T ≤ 264 K. 18 Direct measurements of the Gruneisen parameter of elastic materials have been reported: 19 the authors used pulsed electron beams hitting the front surface of a thin slab, and measured the induced motion on the rear face surface. The equation of motion describing thermoelastic effects depends on the Gruneisen parameter and on the energy deposited by the beam in the slab. The measurement of the displacement, the knowledge of the deposited energy and the use of the equation of motion allowed them to directly determine the Gruneisen parameter. In a similar way this paper presents experimental results on the linear expansion coefficient of the aluminum alloy above and below the temperature of transition between the ( s ) and ( n ) state. In order to determine the coefficient we use: a) the oscillation amplitude measurements, b) the equations describing the amplitude in the framework of the Thermo-Acoustic model, c) the measurements of the deposited energy by the electron beam, obtained by the product of the measured beam multiplicity and the calculated energy loss in the material by each particle, and d) an independent measurement of the specific heat.\",\n \"pages\": [\n 1,\n 2,\n 3\n ]\n },\n {\n \"title\": \"2.1. ( n ) state\",\n \"content\": \"We consider a cylinder (radius R , length L , mass M ) that is suspended in correspondence of the middle section, with its axis horizontal. A ionizing particle, after hitting the lateral surface of the cylinder and interacting with the material, generates a pressure pulse in the bulk. This sonic pulse is the result of the local thermal expansion of the material caused by the warming up due to the energy lost by the particle in the material. The sonic pulse determines the excitation of the vibrational oscillation modes of the suspended bar. By introducing a vector field u ( x , t ) to describe the local displacements from equilibrium, we express the amplitude of the k -th longitudinal mode of oscillation as proportional to the quantity: 20 , 21 , 22 ∣ ∣ where ∆ P therm is the pressure pulse due to the sonic source described above, ρ is the mass density, dW/dl is the energy loss per unit length of the interacting particle, A ' is the cross section of the tubular zone centered on the particle path in which the effects are generated and I k = ∫ dl ( ∇· u k ( x )) is a line integral over the particle path involving the normal mode of oscillation u k ( x ). In the previous relation γ is the adimensional Gruneisen parameter: with β the volume thermal expansion coefficient ( β = 3 α for cubic elements), c V the isochoric specific heat and K T the isothermal bulk module.\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"4 C. Ligi et al.\",\n \"content\": \"A thin bar ( R/L /lessmuch 1) hit by a ionizing particle at the center of the lateral surface is a particular case of the more general one that leads to relation (1). In this case, the fundamental longitudinal mode of oscillation is excited to a maximum amplitude given by: 23 and then exponentially decays due to internal friction. Here W is the total energy released in the bar by the particle beam. The experimental conditions of RAP are close to this particular case and the amplitude maximum of the fundamental longitudinal mode of oscillation is the observable measured by the experiment. However, in order to model the amplitude in the most realistic way, we have performed a Monte Carlo (MC) simulation, 18 which takes into account the corrections O [( R/L ) 2 ] for the modes of oscillation of a cylinder, the transverse dimension of the electron beam at the impact point and the trajectories of the secondary particles generated by the electron interactions in the bar. All these effects are summarized in a corrective parameter /epsilon1 and the amplitude maximum of the fundamental longitudinal mode of oscillation is modeled by: The value of /epsilon1 for the aluminum alloy bar used in the experiment is estimated by MC to be -0.04.\",\n \"pages\": [\n 4\n ]\n },\n {\n \"title\": \"2.2. ( s ) state\",\n \"content\": \"The energy released by the ionizing particles to the material determines the suppression of the superconductivity in a region (Hot Spot) centered around the particle path. The maximum value of the Hot Spot radius, r HS , is obtained by equating the energy lost per unit length by the particle, dW/dl , to the enthalpy variation in the volume undergoing the ( s ) → ( n ) transition at temperature T : 24 , 25 , 26 where ∆ h is the enthalpy change per unit volume and A '' is the cross section of the zone switched to the ( n ) state. The amplitude of the longitudinal oscillations is the sum of two terms, one related to thermodynamic effects in the ( s ) → ( n ) transition and the other to the thermal effects in the region switched to the ( n ) state, which have been already described in the Section 2.1. The contribution to the amplitude of the cylinder k -th oscillation mode due to the ( s ) → ( n ) transition is proportional to: 20 , 21 where V = V ( s ) and ∆ V , ∆ S are the differences of the volume and the entropy in the two states of conduction. According to the thermodynamics of volume and pressure effects for Type-II superconductors, these differences can be written as a : 27 and Here H c is the superconducting critical magnetic field that is supposed to have the parabolic behavior H c ( t ) = H c (0)(1 -t 2 ), where t = T/T c and T c is the transition temperature. Moreover, the terms in { } brackets are expected to be smaller than the preceding terms 28 and are usually ignored for practical purposes. Finally, the knowledge of the specific heat of the material for the ( s ) state, c ( s ) , allows us to compute the variation of the entalpy per unit volume in the form: 29 with: The amplitude maximum of the fundamental longitudinal mode of oscillation in the ( s ) state, as already indicated, is the sum of two terms, one related to the transition and the other to the ( n ) contribution from the switched region: X ( s ) = X trans + X ( n ) . By observing that X trans /X ( n ) = g k =0 ,trans /g k =0 ,therm we can describe the amplitude maximum according to the following relation:\",\n \"pages\": [\n 4,\n 5\n ]\n },\n {\n \"title\": \"6 C. Ligi et al.\",\n \"content\": \"where the relation (4) and the definition (2) of γ are used to define: We note that the term contained in { } brackets in the relation (9) is independent from W , the energy released by the particle to the bar, and that X ( s ) linearly depends on W through X ( n ) .\",\n \"pages\": [\n 6\n ]\n },\n {\n \"title\": \"3. Experimental setup and procedures\",\n \"content\": \"The experiment mechanical layout, its cryogenic setup and operations, the electron beam characteristics, the instrumentation and the procedures for calibrations, data taking and analysis have been fully described in Refs. 17-18. Briefly, the dimensions and the mass of the Al 5056 cylindrical bar are R = 0 . 091 m, L = 0 . 500 m, M = 34 . 1 kg, respectively. The bar hangs from the cryostat top by means of a multistage suspension system providing a 150 dB attenuation of the external mechanical noise in the 1700-6500 Hz frequency window. The frequency of the fundamental longitudinal mode of oscillation of the bar is f 0 = 5413 . 6 Hz below T = 4 K. The cryostat is equipped with a dilution refrigerator. The temperatures are measured inside the cryostat by 11 thermometers controlled by two multi-channel resistance bridges and, among them, a calibrated RuO 2 resistor measures the temperature of one of the bar end faces with an accuracy of 0.01 K below T = 4 K. Two piezoelectric ceramics (Pz), electrically connected in parallel, are inserted in a slot cut in the position opposite to the bar suspension point and are squeezed when the bar shrinks. In this Pz arrangement the strain measured at the bar center is proportional to the displacement of the bar end faces. The Pz output is first amplified and then sampled at 100 kHz by an ADC embedded in a VME system dedicated to the data acquisition. A Pz calibration procedure, performed before each run of data taking, provides the factor converting the ADC samples into the displacements of the bar end faces. A software filtering algorithm, known as 'digital lock-in', extracts the Fourier component at the frequency f 0 from the time sequence formed by the ADC samples before and after the beam hit, determining the amplitude of the induced fundamental oscillation. The sign of the amplitude is taken positive or negative according to the sign of the first sample raising above the threshold after the beam hit. The BTF beam line delivers to the bar single pulses of ∼ 10 ns duration, containing N e electrons of 510 ± 2 MeV energy. N e ranges from about 5 × 10 7 to 10 9 and is measured with an accuracy of ∼ 3% (for N e > 5 × 10 8 ) by an integrating current transformer placed close to the beam line exit point. The MC simulation, discussed in Section 2, estimates an average energy lost 〈 ∆ E 〉 ± σ ∆ E = 195 . 2 ± 70 . 6 MeV for a 512 MeV electron interacting in the bar and, consequently, the total energy loss per beam pulse is given by W = N e 〈 ∆ E 〉 , σ W = √ N e σ ∆ E . Two sources of error affect the vibration maximum amplitude: the first is an overall systematic error of the order of ± 6%, that accounts for the slightly different set-up and calibration procedures implemented in the runs over two years and the second is related to the noise in the measurement of the oscillation amplitude ( ± 1 . 3 × 10 -13 m).\",\n \"pages\": [\n 6,\n 7\n ]\n },\n {\n \"title\": \"4. Linear expansion coefficient measurements\",\n \"content\": \"The coefficient α is determined by inserting the measured values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) , X ( s ) ) and the corresponding measurements of the deposited energy ( W ) into the relations modeling the amplitudes (equations (4),(9)). The measurement 30 at very low temperatures of the specific heat of an Al 5056 sample, of the same production batch of the RAP bar, is a fundamental ingredient for the coefficient determination both in ( n ) and ( s ) states. Moreover, the superconducting characterization of the material has shown that T c = 0 . 845 ± 0 . 002 K with a total transition width of about 0.1 K. 30 Thus we determine α for the two states of conduction ( α ( n ) , α ( s ) ) according to this temperature value.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"4.1. ( n ) state\",\n \"content\": \"Values of the amplitude maximum of the fundamental longitudinal mode of oscillation and the related values of the deposited energies by the beam have been measured and analyzed in the temperature interval 0 . 9 ≤ T ≤ 2 K. 17 The values of X ( n ) show a strict linear correlation with the values of W (Fig. 1) and they have been fitted, including the errors, by a line constrained to the origin ( X ( n ) = p 0 W ). The slope of the fit, which has a χ 2 per degree of freedom equal to 0 . 45 / 15, is p 0 = (2 . 24 ± 0 . 05) × 10 -10 m J -1 . From the relation (4) it follows that: In the same temperature interval the parametrization c V = CT + DT 3 , where C is the electronic specific heat coefficient per unit volume and D is the lattice contribution, is determined by 30 C = (4 . 382 ± 0 . 117) × 10 -2 J kg -1 K -2 and D = (5 . 20 ± 0 . 37) × 10 -3 J kg -1 K -4 . Inserting this c V parametrization in relation (10) directly gives: for the linear expansion coefficient (Fig. 2) expressed in terms of the electronic and lattice components in the temperature interval 0 . 9 ≤ T ≤ 2 K. We note that the value of α ( n ) for pure aluminum is reported 31 to be 2 . 7 × 10 -9 K -1 at T = 2 K. This value, which is the one associated to the lowest temperature in the previously cited work, can be compared with α ( n ) = (3 . 2 ± 0 . 2) × 10 -9 K -1 obtained in the present work for Al 5056 at the same temperature.\",\n \"pages\": [\n 7\n ]\n },\n {\n \"title\": \"4.2. ( s ) state\",\n \"content\": \"From eq. (9) we can derive the volume variation among the two states of conduction: where R = X ( s ) /W -p 0 , with p 0 introduced in Section 4.1. The volume variation depends on measured quantities ( X ( s ) , W, p 0 , c V , T, T c ) and calculated ones. The latter are related to the critical field H c ( T = 0) and to its parabolic dependance on the temperature, which also allows us to compute the derivative ∂H c /∂T . If the superconducting properties of Al 5056 can be described by the BCS theory, then H c ( T = 0) ≈ 2 . 42 √ CT c ≈ 70 Oe, C being the electronic specific heat coefficient\",\n \"pages\": [\n 8\n ]\n },\n {\n \"title\": \"Measurement of the thermal expansion 9\",\n \"content\": \"per unit volume (Section 4.1), here expressed in suitable cgs units (erg cm -3 K -2 ). As already mentioned, the model predicts that the amplitude maximum of the fundamental longitudinal mode of oscillation for both states of conduction ( X ( n ) , X ( s ) ) linearly depends on the energy deposited by the e -beam in the bar. On the contrary, the X ( s ) values for Al 5056 15 , 17 show an increasing deviation from linearity with the increase of the deposited energy W , due to saturation effects discussed in Ref. 17. However, if we restrict our analysis to the data gathered at the lowest deposited energies, these non linearities can safely be neglected, and we can use the relation (12) to compute the volume variation. Fig. 3 shows the values of ∆ V/V versus the temperature T for the data with released energy: 1 . 5 × 10 -3 < W < 9 × 10 -3 J. ∆ V/V data have been fitted by a 2nd order polynomial q 0 + q 1 T + q 2 T 2 obtaining q 0 = ( -1 . 737 ± 0 . 054) × 10 -8 , q 1 = (7 . 334 ± 1 . 785) × 10 -9 K -1 , q 2 = (1 . 766 ± 0 . 187) × 10 -8 K -2 with a χ 2 = 1 . 38 normalized over 94 degrees of freedom. The use of q 0 and of the relation (6) gives ∂H c /∂P = ( -3 . 12 ± 0 . 10) × 10 -12 T Pa -1 at T = 0 K, a value that can be compared with ( -2 . 67 ± 0 . 06) × 10 -12 T Pa -1 obtained by Harris and Mapother in their experimental study of the critical field of pure aluminum as a function of pressure and temperature. 32 The difference of the thermal expansion coefficients in the normal and superconducting phases is obtained by taking the derivative with respect to T of the difference of volumes: 27 ∆ α = α ( n ) -α ( s ) = 1 3 d dT ∆ V V . Under the hypothesis that α ( n ) is still represented by the relation (11) in the inter- al 0 . 3 K < T < T c and by using the polynomial fitting for ∆ V/V , the following expression is obtained for that temperature interval: This result can be compared with the expansion coefficient of pure aluminum in the ( s ) state, using for pure aluminum the relation α ( s ) = α ( s ) ,e + α /lscript , in which it is assumed that the lattice component does not depend on the conduction state and that its values are given in Ref. 33. The definition (2) gives α ( s ) ,e = ρc V,e ( s ) γ ( s ) ,e / (3 K T ), where the electronic component of c V in the ( s ) state ( c V,e ( s ) ) is taken from the work of Phillips, 34 the electronic component of the Gruneisen parameter in the ( s ) state ( γ ( s ) ,e ) from Ref. 35 and K ( s ) T ∼ K ( n ) T = 79 . 4 × 10 9 Pa at low temperatures. 36 The comparison is shown in Fig. 4 as a function of the reduced temperature t .\",\n \"pages\": [\n 9,\n 10,\n 11,\n 12\n ]\n },\n {\n \"title\": \"5. Conclusions\",\n \"content\": \"In this paper we present an evaluation of the linear expansion coefficient of an AlMg alloy obtained by measuring both the amplitude of the fundamental mode of the longitudinal oscillation excited by electrons interacting in a suspended bar and the energy released in the bar by the electron pulse. The use of this method to determine α is new, although similarities exist with the direct measurement of the the Gruneisen parameter performed by hitting thin slabs with particle beams to generate thermoelastic pulses. No expansion data for aluminum alloys were previously available in the literature for the temperature range 0 . 3 < T < 2 K explored by this experiment. The expansion coefficient is negative in the superconducting state and its absolute value just below T c is larger by an order of magnitude than the value above this temperature. This feature is expected also for pure aluminum, according to an analysis 35 of the measured values of H c ( P, T ) available in Ref. 32. This work was partially supported by the EU FP-6 Project ILIAS (RII3-CT-2004506222).\",\n \"pages\": [\n 12\n ]\n }\n]"}}},{"rowIdx":296,"cells":{"bibcode":{"kind":"string","value":"2013IJMPD..2230030L"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1310.8459.pdf"},"content":{"kind":"string","value":"\nNeutrino coupling to cosmological background: A review on gravitational Baryo/Leptogenesis\nGaetano Lambiase a,b , Subhendra Mohanty c , and A.R. Prasanna c\na b\nUniversity of Salerno, Baronisi, Italy.\nINFN , Sezione di Napoli, Italy. and\nc Physical Research Laboratory, Ahmedabad 380009, India.\nAbstract\nIn this work we review the theories of origin of matter-antimatter asymmetry in the Universe. The general conditions for achieving baryogenesis and leptogenesis in a CPT conserving field theory have been laid down by Sakharov. In this review we discuss scenarios where a background scalar or gravitational field spontaneously breaks the CPT symmetry and splits the energy levels between particles and anti-particles. Baryon or Lepton number violating processes in proceeding at thermal equilibrium in such backgrounds gives rise to Baryon or Lepton number asymmetry.\n1. INTRODUCTION\nIt is largely accepted that General Relativity is the best (self-consistent) theory of gravity. It dynamically describes the space-time evolution and matter content in the Universe and is able to explain several gravitational phenomena ranging from laboratory and solar system scales (where it has been mainly tested) to astrophysical and cosmological scales. On cosmological scales, the cornerstone of General Relativity is represented by Hubble expansion, the Big Bang Nucleosynthesis, i.e. the formation of light elements in the early Universe, and cosmic microwave background (CMB) radiation. Despite these crucial predictions, Einstein's theory of gravity is in disagreement with the increasingly high number of observational data, such as those coming for example from SNIA-type, large scale structure ranging from galaxies up to galaxy super-clusters, provided by the advent of the Precision Cosmology and the achievement of high sensitivity of experiments. The experimental evidences that the observable Universe is at the moment expanding in an accelerating phase [1, 2] represents without any doubts the most exciting discovery of the modern Cosmology. As a consequence of this discovery, there has been in the last years more and more interest to understand the evolution not only of the early Universe, but also of the present Universe, and for this formidable task new ideas and theories beyond the standard Cosmology and particle physics have been proposed. Attempts to explain the recent observational data and at the same time try to preserve the conceptual structure of General Relativity, lead cosmologists to introduce two new fundamental concepts: Dark Matter (DM) and Dark Energy (DE). Observational data indicate that a huge amounts of DM and DE are indeed needed to explain the observed cosmic acceleration of the Universe in expansion (as well as all new observational data), and at the moment there are no experimental and theoretical evidence that definitively shed some light on such mysterious components (see [3-16] for DE reviews and [17-26] for DM reviews, and references therein). Moreover, recent data suggest that also in the very early epoch, the Universe was in an accelerated phase. This era is called Inflation, and is able to to solve the problems that affect the standard cosmological model (the Cosmology based on General Relativity): 1) The flatness of the Universe, that is why Ω = ρ/ρ c /similarequal O (1). Here ρ is the average cosmological energy density, and ρ c = 3 H 2 m 2 P / 8 π the critical density. 2) The problems of homogeneity, isotropy, and horizon (which created headache in the frameworks of the standard Friedman-Robertson-Walker (FRW cosmology).\nInflation provides a natural mechanism of generation of small density perturbations with almost flat spectrum[203]. This agrees with observations. In order to solve all problems of standard FRW cosmology, it is required that the duration of Inflation is\nN ≡ Ht ∼ 70 -100 . (1)\nAt the end of the Inflationary epoch, according to the standard cosmological model, the Universe is in a cold, low entropy state, and appears baryon symmetric, that is the same amount of matter and antimatter. On the contrary, the present Universe looks baryon asymmetric. The issue that one has to solve is about the physical mechanism occurred during the Universe evolution for which it ends up being matter dominated. Equivalently, what about anti-matter? Theories that try to explain how the asymmetry between baryon and antibaryon was generated in the early phases of the Universe evolution are called baryogenesis . They represent a perfect interplay between particle physics and cosmology.\nThe parameter characterizing the baryon asymmetry is defined as\nη ' ≡ n B -n ¯ B n γ ,\nwhere n B ( n ¯ B ) is the number of baryons (antibaryon) per unit volume, and n γ = 2 ξ (3) T 3 π 2 is the photon number density at temperature T . A different definition of the parameter η that refers to the entropy density\ns = 2 π 2 g eff T 3 = 7 . 04 n γ , (2)\n45\nη ≡ n B -n ¯ B s = 1 7 . 04 η ' .\nFinally, the baryon asymmetry can be also expressed in term of the baryonic fraction Ω B = ρ B /ρ c , i.e.\nη = 2 . 74 × 10 -8 Ω B h 2 ,\nwhere h = 0 . 701 ± 0 . 013 is the present Hubble parameter.\nThe physics of the CMB temperature anisotropies, which are related to the acoustic oscillations of baryon-photon fluid around the decoupling of photons, provides a strong probe of the baryon asymmetry. In fact, the observation of the acoustic peaks in CMB measured\ng eff = g γ + T 3 ν T 3 γ g ν , is given by\nby WMAP satellite [27], when combined with measurements of large scale structures, leads to following estimation of the parameter η\nη ( CMB ) ∼ (6 . 3 ± 0 . 3) × 10 -10 0 . 0215 ≤ Ω B h 2 ≤ 0 . 0239 (3)\nAn independent measurement of η can be carried out in the framework of the BBN [28], that gives\nη ( BBN ) ∼ (3 . 4 -6 . 9) × 10 -10 0 . 017 ≤ Ω B h 2 ≤ 0 . 024 . (4)\nIt is remarkable that two completely different probes of the baryon content of the Universe (the synthesis of light elements occurred during the first 3 minutes of the Universe evolution, and the the photons decoupling occurred when the Universe was 400 thousand years old) give compatible results. This represents one of the great success of modern Cosmology.\nAlthough many mechanisms have been proposed, the explanation of the asymmetry between matter and antimatter is still an open problem of the modern Cosmology and Particle Physics. In this review we discuss some general topics related to the baryogenesis. The work is divided in two parts. In the first part we recall some models of baryogenesis/leptogenesis, which are mainly based on particle physics (essentially GUT and SUSY). In the second part we discuss different approaches to the baryon asymmetry which rely on the coupling of baryon/lepton currents with the gravitational background. Particular attention will be devoted to the mechanism based on the spin-gravity coupling of neutrinos with the gravitational waves of the cosmological background, which are generated by quantum fluctuations of the inflation field during the inflationary era.\n2. SOME TOPICS OF BARYOGENESIS AND LEPTOGENESIS\nIn this Section, we recall some general topics of Baryogenesis and Leptogenesis. More details can be found in [29-34].\nStandard cosmological model is unable to explain the so small magic value of the baryon asymmetry (Eqs. (3) and (4)) and why the Universe starting from an initial baryon symmetry ( n B = n ¯ B ) evolves in a final state such that matter dominates over antimatter ( n B /greatermuch n ¯ B ).\n2.1. The Sakharov conditions\nAs pointed out by Sakharov [36], in a CPT conserving theory a baryon asymmetry B may be dynamically generated in the early Universe provided that:\n/negationslash\n\n1) There exist interactions that violate the baryon number B . Baryon number violating interactions are required because one starts from a baryon symmetric Universe ( B = 0) to end in a baryon asymmetric Universe ( B = 0). Direct experimental proofs that baryons are not conserved are still missing. From a theoretical point of view, both GUT and the standard electroweak theory (via sphaleron processes) give non conservation of baryon number (notice that another possibility to break all global charges and in particular the baryon charge is related to gravity, as discussed in Ref. [37]).\n2) The discrete symmetries C and CP must be violated. This condition is necessary in order that matter and antimatter can be differentiated, as otherwise B non-conserving interactions would produce baryons and antibaryons at the same rate thus maintaining the net baryon number to be zero. In contrast to non-conservation of baryons, the breaking of CP -symmetry was discovered in direct experiment ( CP violation has been indeed observed in the kaon system).\n\nThe C and CP violation imply that B/L violating reactions in the forward and reverse channels do not cancel ( L stands for Lepton number). To see this, consider the process: X → Y + B , where X is the initial state with B = 0, Y the final state with B = 0, and B the excess baryon produced. Suppose that C is a symmetry. Then the C -conjugate process is characterized by the fact that\nΓ( X → Y + B ) = Γ( ¯ X → ¯ Y + ¯ B ) .\nThe net rate of baryon production evolves in time as\ndB dt ∝ Γ( ¯ X → ¯ Y + ¯ B ) -Γ( X → Y + B ) = 0 if C is a symmetry\nSimilar arguments hold for CP symmetry. Therefore both C and CP discrete symmetry violation are required to generate a net baryon asymmetry.\n\n3) Departure from thermal equilibrium: This condition is required because the statistical distribution of particles and anti-particles is the same if the Hamiltonian commutes\n\nwith CPT , i.e. [ H,CPT ] = 0, which implies n B = n ¯ B . Hence only a departure from thermal equilibrium, which means that the form of n B, ¯ B has to be modified, can allow for a finite baryon excess (so that n B -n ¯ B = 0).\n/negationslash\nMore specifically, consider again the process X → Y + B . If the process is in thermal equilibrium , then by definition\nΓ( X → Y + B ) = Γ( Y + B → X ) ,\nso that no net baryon asymmetry can be produced since the inverse process destroy B as fast as the forward process creates it (see also Appendix A).\n/negationslash\nHowever, as discussed in the seminal paper by Cohen and Kaplan [38], it is possible to generate lepton/baryon asymmetry at thermal equilibrium (without requiring CP violation). The reason is due to the general result that in an expanding Universe at finite temperature, CPT is not a good symmetry, i.e. CPT can be (spontaneously) violated. The third Sakharov criterion is therefore violated. As before pointed out, in fact, CPT invariance requires that the baryon number must be generated out of thermal equilibrium, but CPT invariance requires that the thermal distribution of baryon and anti-baryon will be identical. This condition fails if there is a spontaneous CPT violation in the theory which modifies the baryon-antibaryon spectrum. As a consequence one obtains E particle = E antiparticle which implies n B -n ¯ B = 0.\n/negationslash\nThe CPT violation allows for the generation of the baryon asymmetry during an era when baryon(lepton) violating interaction are still in thermal equilibrium. The asymmetry gets frozen at the decoupling temperature T d when the baryon(lepton) violation goes out of equilibrium. The decoupling temperature is calculated by equating the interaction rate of processes Γ and the expansion rate of the Universe represented by the Hubble constant H , Γ( T d ) /similarequal H ( T d ). The scenario underlying these processes in an expanding Universe can be schematized as follows: in the regime Γ /greatermuch H , or T > T d B-asymmetry is generated by B-violating processes at thermal equilibrium; at T = T d , i.e. Γ /similarequal H , the decoupling occurs, and finally when Γ < H , or T < T d the B-asymmetry gets frozen.\n2.2. Leptogenesis scenario\nLeptogenesis is a mechanism, proposed by Fukugita and Yanagida, that allows to convert the lepton asymmetry to baryon asymmetry via electroweak (EW) effects. Even if the baryon number is conserved at high scales, it is possible to generate the baryon asymmetry in the present Universe if lepton asymmetry is generated at either GUT or intermediate scales. This idea attracted much attention in view of discovery of a possible lepton number violation in the neutrino sector [39, 40]. For a recent review see [41, 42] (the role of neutrinos in cosmology has been recently treated in [43]).\nThe Leptogenesis scenario is the simplest extension of the standard model able to realize the Sakharov conditions for explaining the matter antimatter asymmetry in the Universe. In this model, the standard model is modified by adding right handed neutrinos which permit the implementation of the see-saw mechanism and provide the explanation of light mass of the standard model neutrinos. At the same time, the augmented model is able to spontaneously generate leptons from the decays of right handed neutrinos.\nFor later convenience, we shall discuss in a nutshell the Leptogenesis scenario. The leptonic Lagrangian density is given by (here we follow [44])\nL = h ∗ β ( ¯ L β φ c ∗ ) E β -λ ∗ αk ( ¯ L α φ ∗ ) N k -1 2 ¯ N j M j N c j + h.c. (1)\nIn this expression L is the Standard Model left-handed doublet, E is the right-handed singlet, N j are the singlet fermions (Majorana neutrinos), α, β are the flavor indices of the Standard Model, i.e. α, β = e, µτ , M is mass matrix and λ Yakawa matrices. Equation (1 is written in a basis where the coupling h and the mass matrix M are diagonal and real, whereas λ is complex. This Lagrangian leads, once the heavy fermions N i are integrated out, to the effective light neutrino masses (see-saw mechanism): m ν αβ = λ αk M -1 k λ βk . The Lagrangian (1) satisfies the Sakharov conditions. In fact it violates the leptonic number L due to λ - and M -terms; CP is violated through the complex Yukawa coupling λ αk ; Since the interactions are only determined by Yukawa's interaction terms, the smallness of these couplings may provide the right conditions for which the interaction rates are smaller than the expansion rate of the Universe, establishing in such a way the condition for the out of equilibrium (i.e. the heavy Majorana fermions can decay out of equilibrium).\nFor simplicity consider the lightest Majorana singlet N 1 . It may decay in two channels\nN 1 → L α φ, N 1 → ¯ L α φ † . (2)\nAs a consequence of the N 1 decay, the net baryon asymmetry can be generated. The parameter η turns out to be\nη /similarequal 135 ζ (3) 4 π 4 g ∗ C sph η eff /epsilon1 . (3)\nHere η eff is the efficiency factor which assumes the value in the range 0 < η eff < 1 (owing to inverse decays, washout processes and inefficiency in N 1 production). Below to the freeout temperature T F , the temperature for which Γ( φL → N 1 ) < H , where Γ( φL → N 1 ) /similarequal 1 2 Γ D e -M 1 /T , with\nΓ D = ( λ † λ ) 11 M 1 8 π , (4)\nand H is the expansion rate of the Universe during the radiation dominated era,\nH = 1 . 66 g 1 / 2 ∗ T 2 M 2 P , (5)\nthe density of the fermion N 1 is Boltzmann suppressed ( N 1 ∼ e -M 1 /T ). Therefore, below T F the decay of N 1 contribute to the lepton asymmetry, and the efficiency factor is\nη eff /similarequal n N 1 ( T F ) n N 1 ( T /greatermuch M 1 ) /similarequal e -M 1 /T F /similarequal m ∗ ˜ m , ˜ m\nwhere\n˜ m ≡ 8 πv 2 M 2 1 Γ D = ( λ † λ ) 11 v 2 M 1 , m ∗ ≡ 8 πv 2 M 2 1 H ( T = M 1 ) /similarequal 1 . 1 × 10 -3 eV . (7)\nC sph is a factor that takes into account the dilution of the asymmetry due to fast processes. Finally /epsilon1 is the CP parameter related to the asymmetry in the N 1 decays and defined as\n/epsilon1 = Γ( N 1 → φL ) -Γ( N 1 → φ † ¯ L ) Γ( N 1 → φL ) + Γ( N 1 → φ † ¯ L ) . (8)\nIts non vanishing value arises from the interference of three level and one loop amplitudes (complex Yakawa couplings). One gets\n/epsilon1 = 1 8 π 1 ( λ † λ ) 11 ∑ j I/arrowdblbothv { [ ( λ † λ ) 1 j ] 2 } g ( x j ) , (9)\nwhere\ng ( x ) = √ x [ 2 -x 1 -x -(1 + x ) ln 1 + x x ] , x j ≡ M 2 j M 2 1 .\nTaking into account Eqs. (6), (9) and (7), the net lepton asymmetry is\nη /similarequal 10 -3 10 -3 eV ˜ m /epsilon1 . (10)\nLeptogenesis is then related to Baryogenesis by a phenomenon that happens in the currently accepted Standard Model. Indeed, certain non-perturbative configurations of gauge fields, the sphalerons, can convert leptons into baryons and vice versa. These processes that violate B + L and conserve B -L occur at the electroweak scale. Under normal conditions sphalerons processes are unobservably rare due to the fact that the transition rates are extremely small Γ ∼ e -16 π 2 /g 2 ∼ O (10 -165 ), hence are completely negligible in the Standard Model (at T = 0). However as emphasized by Kuzmin, Rubakov, and Shaposhnikov [40], in the thermal bah provided by the expanding Universe, thermal fluctuations becomes important and B + L violating processes can occur at a significant rate and these processes can be in equilibrium in the expanding Universe (see B).\nFinally from (10) it follows that by requiring η ∼ 10 -10 the lower bound of the mass of the Majorana neutrinos N 1 is\nM 1 /greaterorsimilar 10 11 GeV , (11)\nwhere the light value of the neutrino mass has been used: ˜ m /similarequal (10 -3 -10 -1 )eV.\n2.3. Models of Baryogenesis\nMany models aimed to explain the generation of the baryon asymmetry have been proposed in literature. These are GUT Baryogenesis, Affleck-Dine Baryogenesis and AffleckDine Leptogenesis, Leptogenesis from heavy Majorana neutrinos, Leptogensis from ν R oscillation, Thermal baryogenesis, Electroweak baryogenesis, Spontaneous baryogenesis, Baryogenesis through evaporation of primordial black holes. Details of such mechanisms can be found in [29-33, 45] and references therein. Here we list some of them in Table I.\n\n· GUT-Baryogenesis or decay of heavy particles:\n\n/negationslash\nConsider the X -boson decays in two channels X → 2 q and X → 2¯ q , with the probabilities given by P X → 2 q = P X → 2¯ q due to CP violation. This implies the excess of baryons over anti-baryons. In the original scenario of GUT baryogenesis one uses the heavy gauge bosons X and Y (leptoquarks), which decay while they decouple from\nequilibrium. This is called delayed decay scenario . It was soon realized that this boson gauge decay does not produce the required baryon asymmetry because that the X and Y boson masses predicted are too low to satisfy the out-of-equilibrium condition (in non-SUSY GUT). The alternative scenario was to use decays of coloured Higgs particles. If more than two Higgs particles exist, sufficiently large baryon asymmetry can be generated (provided that the Kuzmin, Rubakov, Shaposhnikov effect is switched off).\n\n· SUSY: Supersymmetry actually opens a number of options. Since Supersymmetry extends the particle content of the theory near the EW scale, the possibility of a strong EW phase transition cannot yet be completely excluded. This revives the hope of explaining baryon asymmetry entirely within the MSSM.\n\nAffleck-Dine scenario (1985): This scenario is based on the observation that in SUSY theories ordinary quarks and leptons are accompanied by supersymmetric partners - s-quarks and s-leptons - which are scalars. The corresponding scalar fields carry baryon and lepton number, which can in principle be very large in the case of a scalar condensate (classical scalar field). An important feature of SUSY theories is the existence of flat directions in the superpotential, along which the relevant components of the complex scalar fields ϕ can be considered as massless. The condensate is frozen until supersymmetry breaking takes place. Supersymmetry breaking lifts the flat directions and the scalar fields acquire mass. When the Hubble constant becomes of the order of this mass, the scalar fields starts to oscillate and decays. At this time, B , L , and CP violating terms (for example, quartic couplings λ 1 ϕ 3 ϕ ∗ + c.c. and λ 2 ϕ 4 + c.c. , with complex λ 1 , 2 ) becomes important and a substantial baryon asymmetry can be produced. The scalar particles decay into ordinary quarks and leptons transferring to them the generated baryon asymmetry.\nThe Affleck-Dine mechanism can be implemented at nearly any energy scale, even below 200 GeV. By suitable choice of the parameters one can explain almost any amount of baryon asymmetry and this lack of a falsifiable prediction is an unattractive feature of the Affleck-Dine mechanism.\n\n· Electroweak baryogenesis ( ∼ TeV ): The asymmetry is generated by phase transitions involving SU (2) × U (1) breaking. The EWBG is assumed to occur during the radiation\n\ndominated era of the early Universe, a period in which the SU (2) L × U (1) Y electroweak symmetry is manifest. As the temperature falls down the EW scale ( T EW ∼ 100GeV), the Higgs field acquires an expectation value and the electroweak symmetry is spontaneously broken to the subgroup U (1). The EWBG occurs during this phase transition, and in order that it could be an available mechanism, it is required that the transition is of the first order. Remarkably the EWBG satisfies all the three Sakharov's conditions: 1) The rapid sphaleron transitions in the symmetric phase provide the required violation of the baryon number; 2) The scattering of plasma with bubble walls generates the C and CP asymmetry of the number of particles if the underlying theory does contain terms that violate these discrete symmetries (these processes bias the sphalerons to create more baryon than anti-baryons); 3) The rapid expansion of bubble walls through the plasma induces the departure from the thermal equilibrium. All these conditions are fulfilled by the Standard Model. However, EWBG is unable to explain the observed baryon asymmetry of the Universe if it is only based on the Standard Model. The reason is due to the fact that EW phase transition in the Standard Model is of the first order if the Higgs mass is constrained by m H /lessorsimilar 70GeV, in disagreement with experimental lower bound obtained from LEP II experiment, i.e. m H /greaterorsimilar 114GeV, as well as, from recent LHC results that give a value of the Higgs mass near to 125GeV. Recent studies however, open the possibilities to reconsider EWBG as an available candidate for the generation of baryon asymmetry[46]. Moreover, the EWBG mechanism is also affected by the problem related to the CP violation because the latter generated by the Cabibbo-Kobayashi-Maskawa phase is unable to generate large enough chiral asymmetry. For a recent review on EWBG see[47].\n\n· An interesting idea to mention is the Baryogenesis generated through evaporation of primordial BHs[48, 49].\n\n3. BARYOGENESIS GENERATED BY COUPLING OF BARYON CURRENTS AND GRAVITATIONAL BACKGROUND\nAs we have seen in the previous Section, the Baryo/Leptogenesis is generated in the framework of particle interactions (essentially GUT and SUSY). The gravitational field\n/negationslash\n
\n\n
TABLE I: Models of baryogenesis
\n
\nenters marginally in these mechanisms. In the last years, however, many mechanisms have been proposed in which gravity plays a fundamental role in generating the baryo/leptogenesis (see Table II). In these models matter or hadron/lepton currents are coupled with some physical quantity characterizing the gravitational background, such as Ricci curvature or its derivative, Riemann tensor, gravitational waves (GW),\nL int ∼ J · F , J → ¯ ψγ µ ψ, ¯ ψγ µ γ 5 ψ, φ∂ µ φ ∗ , .... , F → R , ∂R , ∂φ , ... .\nTypically, the background is the FRW geometry, but there are also models in which the gravitational background is described by black holes physics.\nGravitational baryogenesis share some basic features of the spontaneous spontaneous (or quintessential) baryogenesis[38]. In this mechanism scalar fields (or their derivatives) couple to matter or hadron/lepton current. To illustrate in some detail the spontaneous baryogenesis, consider a neutral scalar field φ . The interaction between a baryon current J µ B and ∂ µ φ is\nL = 1 M s J µ B ∂ µ φ, (1)\n/negationslash\nwhere M s characterize is a cut-off scale. In a isotropic and homogenous Universe, like FRW Universe, φ does only depend on cosmic time, In such a case only the zero component of the baryon current ( J 0 B = n B with n B the number density of baryons) contribute in (1), L = µn B , where µ ≡ ˙ φ/M s for baryons and µ ≡ -˙ φ/M s for antibaryons. Here is assumed that the current J B is not conserved and that, of course, ˙ φ = 0. The coupling (1) therefore gives rise to an effective chemical potential with opposite sign for B and ¯ B leading to a generation of a net baryon asymmetry even at thermal equilibrium. The latter point bypass the third Sakharov condition because CPT violation occurs owing the Universe expansion. The scalar field could also play the role of DE or DM. Models based on spontaneous (quintessence) baryogenesis are studied in[50-57].\n3.1. Gravitational Baryogenesis\nThe key ingredient for the gravitational baryo/leptogenesis is a CP-violating interaction between the derivative of the Ricci scalar curvature R and the B(aryon)/L(epton) current J µ [58, 59]\nL int = 1 M 2 ∗ √ -gJ µ ∂ µ R (2)\n/negationslash\n
Fig. 1. Constellation scheme for the measurement of the Shapiro time delay for the determination of the PPN-parameter γ (Eddington parameter).
\n\nprogram by an international researcher team. ASTROD I was selected as one of the final 14 candidates out of nearly 50 proposals which shows the genaral interest in the mission concept. Finally ASTROD I was not selected yet.\n2. Mission goals overview\n
\n\n
Table 1. Mission goals for the ASTROD I mission (for 1200 days).
\n
\nThere are three main scientific goals. The first goal is to test relativistic gravity and the fundamental laws of spacetime with an improvement of three orders of magnitude in sensitivity, specifically, to measure the PPN parameters γ (Eddington light-deflection parameter; for general relativity, it is 1) via measurements of the Shapiro time delay to 3 × 10 -8 , β (relativistic nonlinear-gravity parameter; for general relativity, it is 1) to 6 × 10 -6 and others with significant improvement; and to measure the fractional time rate of change of the gravitational constant (dG/dt)/G with two orders of magnitude improvement. The Pioneer Anomaly could meanwhile be explained as an anisotropic thermal radiation effect. 7 Therefor a further investigation on the Pioneer Anomaly is not of strong interest anymore. However, relativistic MOND (MOdified Newtonian Dynamics) theories 8 are of current interests as they are alternatives to dark matter theories. MOND theories have effects similar to the Pioneer effect in the solar system. Testing MOND using LISA Pathfinder 9 has been considered and analyzed. 10, 11 With ASTROD I precision, MOND effects on ASTROD I need to be studied, and possible tests on MOND theories would then be evaluated a .\nThe second goal is to initiate a revolution of astrodynamics with laser ranging in the solar system. With mm precision, the astrodynamics and ephemeris of the solar-system bodies would be improved by orders of magnitude.\nThe third goal is to increase the sensitivity of the determination of solar, planetary and asteroid parameters by 1 - 3 orders of magnitude. In this context, the\nmeasurement of the solar quadrupole moment parameter J 2 will be improved by two orders of magnitude, i.e., to 10 -9 . The mission goals for ASTROD I are listed in Table 1). These goals are based on orbit simulation and assumptions for a timing uncertainty of 3 pico-seconds (ps) and a drag-free uncertainty of 3 × 10 -14 ms -2 Hz -1 / 2 at 0.1 mHz. The timing uncertainty has been achieved in T2L2 on board Jason 2. 12-14 The drag-free requirement is comparable to the actual goal of LISA Pathfinder. 9 See section 5 for more discussions.\nIn the area of solar and galactic physics ASTROD I can also be used for measuring solar energetic particles (SEPs) and galactic cosmic rays (GCRs), with corresponding applications to space weather. 5, 15, 16 The radiation detectors onboard the spacecraft for monitoring test-mass charging of the inertial drag-free sensor can be used for this purpose.\nLike LLR, the scientific data are ranges between ASTROD I and laser stations on earth as functions of time. All goals can (must) be extracted from these ranges with fittings. No payload system could be omitted without affecting the range measurement except the possibility of omitting the radiation monitor which is only a small part of the ASTROD I payload related to the inertial sensor and which is good for space weather applications.\n3. Spacecraft orbit selection\n\n\n
Fig. 2. A typical orbit in heliocentric ecliptic coordinate system.
\n\nBesides the technical details of ASTROD I (spacecraft and payload) the success of the mission depends on the selection of an appropriate orbit to reach the mission\n\n\n
Fig. 3. The 2021 orbit in the Sun-Earth fixed frame.
\n\ngoals. Some effort has been made to simulate possible orbits including planetary swing-bys in order to find adequate solutions. 5 The swing-bys are intended to be geodesic and can be used for mapping Venus gravity. The fuel requirement is minimal with only drag-free thrusters needed. One simulation with a launch in 2021 and a Venus encounter 112 days after launch and three oppositions (spacecraft, Sun and Earth in one line of sight) - after 250, 613 and 976 days after launch - gives a good example for such an orbit. 17 For the choice of the best orbit a trade-off is needed between minimising the time to reach the far side of the Sun because of the risk to the mission from aging of equipment.\nThe simulated launch-2021 orbit for the ASTROD I spacecraft is shown in Fig. 2, the corresponding orbit in the Sun-Earth fixed frame in Fig. 3. The Venus swing-by strategy is also useful for ASTROD-GW mission design and optimization. 18\n4. The ASTROD I spacecraft\n\n\n
Fig. 4. Preliminary design of the ASTROD I spacecraft (design by H. Selig, ZARM).
\n\nASTROD I is planned as a cylindrical spacecraft with a diameter of 2.5 meters\nand a hight of 2 meters. The total mass including a 100 kg apogee motor with liquid fuel is 490 kg. The spacecraft is stabilized around 3 axes by a drag free system. The total electrical power (including payload) is estimated to be around 390 W. In order to protect the payload from thermal load of sunlight - especially at close distances to the Sun - the spacecraft will have to be equipped with passive thermal insulation systems - e.g.: multi-layer-insulation (MLI), and perhaps even with active thermal control devices like thermel Louvers. The minimum requirement for the thermal stability of the payload is a temperature variation of less than ± 1 K. Due to the variation of the distance between the spacecraft and the Sun (0.4 to 0.8 AU) the thermal influx varies by a factor of 4. 5 This will have to be taken into account for the detailed spacecraft design in terms of thermal control. The needed electrical power of 390 W will be generated by solar cells. One possibility is to cover the spacecraft surface with the needed area of solar cells. This could create a thermal problem due to the limited efficency of solar cells. Assuming an efficency of 0.2, most of the lost power will be transformed into heat. In order to prevent the spacecraft from heating up, the solar cells could also be placed on solar panels (see Fig. 4). For the radio communication with the Earth the spacecraft is equipped with a 1.3 m diameter X-band antenna and low gain S-band antennas. During the whole mission the ASTROD I spacecraft will have to be pointed towards the earth to keep the laser link running. Therefore the high gain antenna does not need to be tracked as long as the high gain antenna is pointed parallel to the laser optics which simplifies the setup.\n5. The ASTROD I payload\nThe ASTROD I payload consists of the following components:\n\n· 300 mm Cassegrain telescope\n· Optical bench\n· Two color pulsed Nd:YAG-laser system (532 nm, 1064 nm)\n· Single photon detecting photodiodes\n· Event timer\n· Cesium/Rubidium clock\n· Drag-free inertial sensor\n· Drag-free micro-Newton-thrusters\n\nThe Cassegrain telescope is used for transmitting and receiving the laser pulses from/to Earth/spacecraft. The optical bench consits of several optical components for processing the laser light inside the spacecraft. The main purpose of the optical bench is to seperate the incoming/outgoing laser pulses by polarisation- and color-filtering. Fig. 5 shows the preliminary design of the optical bench. Detailed descriptions are given in Refs. 5, 6. The laser system onboard the spacecraft generates laser pulses with two wavelengths (532 and 1064 nm). The reason for this two-color setup is the need for a correction of the effect of the air column of the\n\n\n
Figure 1: Virialisation fraction f vir ( z ) in Virgo Consortium 256 3 -particle simulations [1, 2] with 240 h -1 Mpc and 85 h -1 Mpc box sizes, shown as continuous thick curves for the Einstein-de Sitter model, and dark energy parameter Ω Λ ( z ) evolution for Ω Λ0 = 0 . 72. Thin curve: 240 h -1 Mpc ΛCDM simulation.
\n\n1 'Dark energy' traces inhomogeneity\nThe standard model of cosmology is generally accepted to be a spacetime with a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric [3, 4, 5, 6, 7], solving the Einstein equation rather simply thanks to the assumption of homogeneous density on any spatial slice, and measured to have the Concordance Model [8] values of the matter density and dark energy parameters, Ω m0 ≈ 0 . 32 (e.g. [9]) and Ω Λ0 := 1 -Ω m0 , respectively (hereafter, ΛCDM). However, we live in the inhomogeneous epoch: galaxies and voids certainly exist today. As found by [10], in an Einstein-de Sitter (EdS) FLRW model (i.e. with Ω m0 = 1 , Ω Λ0 = 0), the fraction of matter in a large region that is virialised, f vir , evolves in a very similar way to that of the dark energy parameter in a flat FLRW model with negligible radiation density,\nΩ Λ ( z ) = 1 -Ω m0 a 3 Ω Λ0 +Ω m0 . (1)\nThis is shown in Fig. 1, where Ω Λ0 = 0 . 68.\nThis seems like an extraordinary coincidence. Over the same redshift range during which one expects that the Universe is inhomogeneous, the degree of inhomogeneity, as expressed by f vir ( z ) in an EdS model, approximately follows the proportion of the critical density represented by dark energy, if the dark energy is inferred from forcing a homogeneous model on the observational data. To first order, f vir is not sensitive to the choice of FLRW model (see Fig. 1), so the coincidence also exists for inhomogeneity in a ΛCDM model.\nThe simplest inference is that a homogeneous-model-inferred non-zero dark energy parameter is really just a measurement of inhomogeneity .\nWhat physical link could there be between this inhomogeneity parameter and homogeneous-model-inferred 'dark energy'?\n2 Void dominance: low matter density, high critical density\nThe most obvious physical link between inhomogeneity and homogeneousinferred dark energy is the volume dominance of voids compared to virialised regions at recent epochs [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. This is because gravitational collapse implies an increase in density, i.e., a reduction in volume, by a factor of about δ vir ∼ 100-200 (e.g., 8-18 π 2 [21]), so that, to first order, the collapsed matter occupies a negligible fraction of the spatial volume. Thus, recent-epoch spatial volume is overwhelmingly dominated by low-density regions.\nMoreover, there are velocity flows out of the voids, since otherwise, the voids couldn't be nearly empty. Thus, the critical density defining spatial flatness in the voids is higher than it would be in a homogeneous calculation. The standard, FLRW approach insists that space expands uniformly with spatially constant curvature, i.e. space is rigid in comoving coordinates-it is forbidden from bending under the influence of gravity. N -body simulations are typically used to study the Newtonian formation of overdense structures and voids within this rigid, comoving background model. However, both the low density of the voids and the velocity flow out of them imply that the matter density parameter in the voids is sub-critical. Thus, the voids are hyperbolic. Geometrically, this hyperbolicity should also be taken into account, implying an even lower matter density parameter (conservatively, let us ignore this: the effect is small).\n3 Volume-weighted averaged metric\nThese arguments can be formalised using the volume-weighted averaging approach to modelling inhomogeneous spatial slices [22, 23, 11, 24, 16], in which the Friedmann equation is generalised to (12) of [13]. For simplicity, let us (i) set the dark energy term to zero, (ii) neglect the kinematic backreaction as much smaller than the curvature backreaction (see [16] for numerical justification), and (iii) combine the curvature parameters into a single full curvature parameter as suggested in [13]. Writing 'k' instead of ' R ', this gives the domain-averaged, effective Friedmann equation\nΩ eff k ( z ) = 1 -Ω eff m ( z ) . (2)\nAlong a typical, large-scale, random, spacelike or null geodesic over recent epochs, what proportions of the geodesic lie in the emptied and virialised\nregions? The proportions at a given z are, on average, (1 -f vir /δ vir ) : f vir /δ vir , respectively. Given that δ vir ∼ 100-200 and\n0 ≤ f vir ≤ 1 , (3)\nless than about 1% of the geodesic falls within the virialised regions. Thus, by starting with a large-scale, high-redshift, 'background' FLRW model-in this case, an EdS model-an effective metric can be written by assuming that the virialised matter contributes negligibly. As in [10], the effective expansion rate is\nH eff ( z ) = H ( z ) + H pec ( z ) , (4)\ncombining the FLRW expansion H ( z ) with the peculiar velocity gradient (physical, not comoving) across voids, H pec ( z ), estimated numerically from an N -body simulation ([1, 2], a 240 h -1 Mpc box-size EdS simulation; see [10]) The background Hubble constant H bg is set to make the present value consistent with low redshift estimates [25, 26], i.e.\nH bg := 74km / s / Mpc -H pec (0) ∼ 50km / s / Mpc . (5)\nThus, the loss of matter from the voids and the higher critical density in voids both decrease the EdS background value from Ω m , bg = 1 to an effective value of\nΩ eff m ( z ) ≈ (1 -f vir ) ( H H eff ) 2 Ω m = (1 -f vir ) ( H bg H eff ) 2 Ω m , bg a -3 . (6)\nThe effective radius of curvature is\nR eff C ( z ) = c aH eff ( z ) √ Ω eff k ( z ) . (7)\nThe effective metric (cf [17]) is\nd s 2 = -d t 2 + a 2 ( t ) [ d χ eff 2 + R eff C 2 ( sinh 2 χ eff R eff C ) (d θ 2 +cos 2 θ d φ 2 ) ] , (8)\nwhere the radial comoving component is\nd χ eff ( z ) := c a 2 H eff ( z ) d a. (9)\n\n\n
Figure 2: Distance modulus normalised to the Milne model (Ω m = 0 , Ω Λ = 0 , ∀ z ) for the homogeneous ΛCDM model (top, black), the uncorrected, homogeneous EdS model (bottom, red), and the void-corrected EdS model (middle, thick, green; 'VA' = virialisation approximation).
\n\n4 Matter density parameter and luminosity distances\nWithout any attempt to fit this approximation to observational data , apart from (5) above, the correction of the EdS model as presented above gives an effective matter density parameter (6) that drops slowly from its background value of unity at high redshift down to Ω eff m = 0 . 27 at the present epoch z = 0, remarkably close to the last two decades' local estimates of the matter density parameter.\nThe effective luminosity distance follows directly from the radial comoving distance and hyperbolicity,\nd eff L = (1 + z ) R eff C sinh χ eff R eff C . (10)\nFigure 2 shows that despite the rough nature of the virialisation approximation, it shifts the homogeneous EdS magnitude-redshift relation by a substantial fraction towards the homogeneous ΛCDM relation, and thus, towards the observational supernovae type Ia relation.\n5 Conclusion\nA handful of simple formulae, lying at the heart of homogeneous, spatially rigid cosmology, remain approximately valid when generalised to inhomogeneous, spatially flexible cosmology [13] and applied to what observationally\nand theoretically dominate the present-day spatial volume-the voids. The result is a correction to a large-scale, high-redshift, background Einstein-de Sitter cosmological model. The correction approximately gives the observed lowredshift matter density parameter and nearly matches the type Ia supernovae luminosity distance relation. The amplitude of the correction is unlikely to be much smaller than estimated here. At the present epoch, direct observations [27], N -body simulations [1, 2], and the existence of the cosmic web itself establish inhomogeneous peculiar velocity gradients of ∼ 20-30 km/s/Mpc, forcing, at least, a factor of ∼ (75 / 50) 2 reduction of the Einstein-de Sitter matter density to Ω eff m < 0 . 5, via Eqs (4), (5), and (6). A virialisation fraction of the order of ∼ 50% reduces this to ∼ 0 . 25. Even when forcing the homogeneous FLRW models onto the data, the initial analysis of the Planck Surveyor cosmic microwave background data finds H bg = 67 . 3 ± 1 . 2 km/s/Mpc at z ≈ 1100 [9]-not as low as the velocity gradients imply, but still significantly lower than the low redshift estimates of H eff (0) = 74 . 0 ± 1 . 6 km/s/Mpc ([26, 25], standard error in the mean).\nHow could such a simple, back-of-the-envelope calculation have been missed for so long? While the volume-averaging approach to cosmology has been developed over many years (see e.g. [13] for a review), possibly the answer lies, ironically, in confusion between the spacelike, unobserved , comoving, present time slice and the past light cone. A gigaparsec-scale void in the former should have a very weak ( δ ∼ 10 -5 ) underdensity, and our would-be location at its centre would be uncomfortably anti-Copernican. But these are both moot points! On the past light cone, an average , gigaparsec-scale, sub-critical (0 < Ω eff m < 0 . 8) void is perfectly natural, since it is defined by the onset of the virialisation epoch at z < ∼ 3. Moreover, we are naturally located at this pseudo-void's centre, by the nature of the past light cone.\nWhat is simpler: relativistic, hyperbolic voids, observed by an observer at the tip of the past light cone, with no dark energy parameter? Or rigid, Newtonian voids together with a dark energy parameter that traces the virialisation fraction?\nAcknowledgments: This essay is in part based on work conducted within the 'Lyon Institute of Origins' under grant ANR-10-LABX-66, and on Program Oblicze'n Wielkich Wyzwa'n nauki i techniki (POWIEW) computational resources (grant 87) at the Pozna'n Supercomputing and Networking Center (PCSS).\nReferences\n\n[1] Jenkins A, Frenk C S, Pearce F R, Thomas P A, Colberg J M, White S D M, Couchman H M P, Peacock J A, Efstathiou G and Nelson A H 1998 Astrophys.J. 499 , 20, [arXiv:astro-ph/9709010].\n\n\n[2] Thomas P A, Colberg J M, Couchman H M P, Efstathiou G P, Frenk C S, Jenkins A R, Nelson A H, Hutchings R M, Peacock J A, Pearce F R, White S D M and Virgo Consortium 1998 MNRAS 296 , 1061-1071.\n[3] de Sitter W 1917 MNRAS 78 , 3-28.\n[4] Friedmann A 1923 Mir kak prostranstvo i vremya (The Universe as Space and Time) Petrograd: Academia.\n[5] Friedmann A 1924 Zeitschr. fur Phys. 21 , 326.\n[6] Lemaˆıtre G 1927 Annales de la Soci'et'e Scientifique de Bruxelles 47 , 4959.\n[7] Robertson H P 1935 Astrophys.J. 82 , 284.\n[8] Ostriker J P and Steinhardt P J 1995 ArXiv Astrophysics e-prints , [arXiv:astro-ph/9505066].\n[9] Planck Collaboration, Ade P A R, Aghanim N, Armitage-Caplan C, Arnaud M, Ashdown M, Atrio-Barandela F, Aumont J, Baccigalupi C, Banday A J and et al. 2013 ArXiv e-prints , [arXiv:1303.5076].\n[10] Roukema B F, Ostrowski J J and Buchert T 2013 ArXiv e-prints , [arXiv:1303.4444].\n[11] Buchert T and Carfora M 2008 Classical and Quantum Gravity 25 (19), 195001-+, [arXiv:0803.1401].\n[12] Rasanen S 2011 Classical and Quantum Gravity 28 (16), 164008, [arXiv:1102.0408].\n[13] Buchert T 2011 Classical and Quantum Gravity 28 (16), 164007, [arXiv:1103.2016].\n[14] Buchert T and Rasanen S 2012 Annual Review of Nuclear and Particle Science 62 , 57-79, [arXiv:1112.5335].\n[15] Wiltshire D L 2011 Classical and Quantum Gravity 28 (16), 164006, [arXiv:1106.1693].\n[16] Buchert T, Nayet C and Wiegand A 2013 Phys.Rev.D in press , [arXiv:1303.6193].\n[17] Larena J, Alimi J M, Buchert T, Kunz M and Corasaniti P S 2009 Phys.Rev.D 79 (8), 083011, [arXiv:0808.1161].\n[18] Roy X and Buchert T 2010 Classical and Quantum Gravity 27 (17), 175013, [arXiv:0909.4155].\n\n\n[19] Clarkson C, Ellis G, Larena J and Umeh O 2011 ArXiv e-prints , [arXiv:1109.2314].\n[20] Wiegand A and Buchert T 2010 Phys.Rev.D 82 (2), 023523-+, [arXiv:1002.3912].\n[21] Lacey C and Cole S 1993 MNRAS 262 , 627-649.\n[22] Buchert T, Kerscher M and Sicka C 2000 Phys.Rev.D 62 (4), 043525, [arXiv:astro-ph/9912347].\n[23] Buchert T 2008 General Relativity and Gravitation 40 , 467-527, [arXiv:0707.2153].\n[24] Buchert T and Ostermann M 2012 Phys.Rev.D 86 (2), 023520, [arXiv:1203.6263].\n[25] Riess A G, Macri L, Casertano S, Lampeitl H, Ferguson H C, Filippenko A V, Jha S W, Li W, Chornock R and Silverman J M 2011 Astrophys.J. 730 , 119, [arXiv:1103.2976].\n[26] Freedman W L, Madore B F, Scowcroft V, Burns C, Monson A, Persson S E, Seibert M and Rigby J 2012 Astrophys.J. 758 , 24, [arXiv:1208.3281].\n[27] Tully R B, Shaya E J, Karachentsev I D, Courtois H M, Kocevski D D, Rizzi L and Peel A 2008 Astrophys.J. 676 , 184-205, [arXiv:0705.4139].\n\n"},"sections":{"kind":"list like","value":[{"title":"Dark energy with rigid voids versus relativistic voids alone","content":"Boudewijn F. Roukema 1 , 2 1 Toru´n Centre for Astronomy Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University ul. Gagarina 11, 87-100 Toru´n, Poland 2 Universit´e de Lyon, Observatoire de Lyon Centre de Recherche Astrophysique de Lyon ´ CNRS UMR 5574: Universit´e Lyon 1 and Ecole Normale Sup´erieure de Lyon 9 avenue Charles Andr´e, F-69230 Saint-Genis-Laval, France ∗ boud@astro.uni.torun.pl","pages":[1]},{"title":"30 March 2013","content":"Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation","pages":[1]},{"title":"Abstract","content":"The standard model of cosmology is dominated-at the present epochby dark energy. Its voids are rigid and Newtonian within a relativistic background. The model prevents them from becoming hyperbolic. Observations of rapid velocity flows out of voids are normally interpreted within the standard model that is rigid in comoving coordinates, instead of allowing the voids' density parameter to drop below critical and their curvature to become negative. Isn't it time to advance beyond nineteenth century physics and relegate dark energy back to the 'no significant evidence' box?","pages":[1]},{"title":"1 'Dark energy' traces inhomogeneity","content":"The standard model of cosmology is generally accepted to be a spacetime with a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric [3, 4, 5, 6, 7], solving the Einstein equation rather simply thanks to the assumption of homogeneous density on any spatial slice, and measured to have the Concordance Model [8] values of the matter density and dark energy parameters, Ω m0 ≈ 0 . 32 (e.g. [9]) and Ω Λ0 := 1 -Ω m0 , respectively (hereafter, ΛCDM). However, we live in the inhomogeneous epoch: galaxies and voids certainly exist today. As found by [10], in an Einstein-de Sitter (EdS) FLRW model (i.e. with Ω m0 = 1 , Ω Λ0 = 0), the fraction of matter in a large region that is virialised, f vir , evolves in a very similar way to that of the dark energy parameter in a flat FLRW model with negligible radiation density, This is shown in Fig. 1, where Ω Λ0 = 0 . 68. This seems like an extraordinary coincidence. Over the same redshift range during which one expects that the Universe is inhomogeneous, the degree of inhomogeneity, as expressed by f vir ( z ) in an EdS model, approximately follows the proportion of the critical density represented by dark energy, if the dark energy is inferred from forcing a homogeneous model on the observational data. To first order, f vir is not sensitive to the choice of FLRW model (see Fig. 1), so the coincidence also exists for inhomogeneity in a ΛCDM model. The simplest inference is that a homogeneous-model-inferred non-zero dark energy parameter is really just a measurement of inhomogeneity . What physical link could there be between this inhomogeneity parameter and homogeneous-model-inferred 'dark energy'?","pages":[2,3]},{"title":"2 Void dominance: low matter density, high critical density","content":"The most obvious physical link between inhomogeneity and homogeneousinferred dark energy is the volume dominance of voids compared to virialised regions at recent epochs [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. This is because gravitational collapse implies an increase in density, i.e., a reduction in volume, by a factor of about δ vir ∼ 100-200 (e.g., 8-18 π 2 [21]), so that, to first order, the collapsed matter occupies a negligible fraction of the spatial volume. Thus, recent-epoch spatial volume is overwhelmingly dominated by low-density regions. Moreover, there are velocity flows out of the voids, since otherwise, the voids couldn't be nearly empty. Thus, the critical density defining spatial flatness in the voids is higher than it would be in a homogeneous calculation. The standard, FLRW approach insists that space expands uniformly with spatially constant curvature, i.e. space is rigid in comoving coordinates-it is forbidden from bending under the influence of gravity. N -body simulations are typically used to study the Newtonian formation of overdense structures and voids within this rigid, comoving background model. However, both the low density of the voids and the velocity flow out of them imply that the matter density parameter in the voids is sub-critical. Thus, the voids are hyperbolic. Geometrically, this hyperbolicity should also be taken into account, implying an even lower matter density parameter (conservatively, let us ignore this: the effect is small).","pages":[3]},{"title":"3 Volume-weighted averaged metric","content":"These arguments can be formalised using the volume-weighted averaging approach to modelling inhomogeneous spatial slices [22, 23, 11, 24, 16], in which the Friedmann equation is generalised to (12) of [13]. For simplicity, let us (i) set the dark energy term to zero, (ii) neglect the kinematic backreaction as much smaller than the curvature backreaction (see [16] for numerical justification), and (iii) combine the curvature parameters into a single full curvature parameter as suggested in [13]. Writing 'k' instead of ' R ', this gives the domain-averaged, effective Friedmann equation Along a typical, large-scale, random, spacelike or null geodesic over recent epochs, what proportions of the geodesic lie in the emptied and virialised regions? The proportions at a given z are, on average, (1 -f vir /δ vir ) : f vir /δ vir , respectively. Given that δ vir ∼ 100-200 and less than about 1% of the geodesic falls within the virialised regions. Thus, by starting with a large-scale, high-redshift, 'background' FLRW model-in this case, an EdS model-an effective metric can be written by assuming that the virialised matter contributes negligibly. As in [10], the effective expansion rate is combining the FLRW expansion H ( z ) with the peculiar velocity gradient (physical, not comoving) across voids, H pec ( z ), estimated numerically from an N -body simulation ([1, 2], a 240 h -1 Mpc box-size EdS simulation; see [10]) The background Hubble constant H bg is set to make the present value consistent with low redshift estimates [25, 26], i.e. Thus, the loss of matter from the voids and the higher critical density in voids both decrease the EdS background value from Ω m , bg = 1 to an effective value of The effective radius of curvature is The effective metric (cf [17]) is where the radial comoving component is","pages":[3,4]},{"title":"4 Matter density parameter and luminosity distances","content":"Without any attempt to fit this approximation to observational data , apart from (5) above, the correction of the EdS model as presented above gives an effective matter density parameter (6) that drops slowly from its background value of unity at high redshift down to Ω eff m = 0 . 27 at the present epoch z = 0, remarkably close to the last two decades' local estimates of the matter density parameter. The effective luminosity distance follows directly from the radial comoving distance and hyperbolicity, Figure 2 shows that despite the rough nature of the virialisation approximation, it shifts the homogeneous EdS magnitude-redshift relation by a substantial fraction towards the homogeneous ΛCDM relation, and thus, towards the observational supernovae type Ia relation.","pages":[5]},{"title":"5 Conclusion","content":"A handful of simple formulae, lying at the heart of homogeneous, spatially rigid cosmology, remain approximately valid when generalised to inhomogeneous, spatially flexible cosmology [13] and applied to what observationally and theoretically dominate the present-day spatial volume-the voids. The result is a correction to a large-scale, high-redshift, background Einstein-de Sitter cosmological model. The correction approximately gives the observed lowredshift matter density parameter and nearly matches the type Ia supernovae luminosity distance relation. The amplitude of the correction is unlikely to be much smaller than estimated here. At the present epoch, direct observations [27], N -body simulations [1, 2], and the existence of the cosmic web itself establish inhomogeneous peculiar velocity gradients of ∼ 20-30 km/s/Mpc, forcing, at least, a factor of ∼ (75 / 50) 2 reduction of the Einstein-de Sitter matter density to Ω eff m < 0 . 5, via Eqs (4), (5), and (6). A virialisation fraction of the order of ∼ 50% reduces this to ∼ 0 . 25. Even when forcing the homogeneous FLRW models onto the data, the initial analysis of the Planck Surveyor cosmic microwave background data finds H bg = 67 . 3 ± 1 . 2 km/s/Mpc at z ≈ 1100 [9]-not as low as the velocity gradients imply, but still significantly lower than the low redshift estimates of H eff (0) = 74 . 0 ± 1 . 6 km/s/Mpc ([26, 25], standard error in the mean). How could such a simple, back-of-the-envelope calculation have been missed for so long? While the volume-averaging approach to cosmology has been developed over many years (see e.g. [13] for a review), possibly the answer lies, ironically, in confusion between the spacelike, unobserved , comoving, present time slice and the past light cone. A gigaparsec-scale void in the former should have a very weak ( δ ∼ 10 -5 ) underdensity, and our would-be location at its centre would be uncomfortably anti-Copernican. But these are both moot points! On the past light cone, an average , gigaparsec-scale, sub-critical (0 < Ω eff m < 0 . 8) void is perfectly natural, since it is defined by the onset of the virialisation epoch at z < ∼ 3. Moreover, we are naturally located at this pseudo-void's centre, by the nature of the past light cone. What is simpler: relativistic, hyperbolic voids, observed by an observer at the tip of the past light cone, with no dark energy parameter? Or rigid, Newtonian voids together with a dark energy parameter that traces the virialisation fraction? Acknowledgments: This essay is in part based on work conducted within the 'Lyon Institute of Origins' under grant ANR-10-LABX-66, and on Program Oblicze'n Wielkich Wyzwa'n nauki i techniki (POWIEW) computational resources (grant 87) at the Pozna'n Supercomputing and Networking Center (PCSS).","pages":[5,6]}],"string":"[\n {\n \"title\": \"Dark energy with rigid voids versus relativistic voids alone\",\n \"content\": \"Boudewijn F. Roukema 1 , 2 1 Toru´n Centre for Astronomy Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University ul. Gagarina 11, 87-100 Toru´n, Poland 2 Universit´e de Lyon, Observatoire de Lyon Centre de Recherche Astrophysique de Lyon ´ CNRS UMR 5574: Universit´e Lyon 1 and Ecole Normale Sup´erieure de Lyon 9 avenue Charles Andr´e, F-69230 Saint-Genis-Laval, France ∗ boud@astro.uni.torun.pl\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"30 March 2013\",\n \"content\": \"Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"The standard model of cosmology is dominated-at the present epochby dark energy. Its voids are rigid and Newtonian within a relativistic background. The model prevents them from becoming hyperbolic. Observations of rapid velocity flows out of voids are normally interpreted within the standard model that is rigid in comoving coordinates, instead of allowing the voids' density parameter to drop below critical and their curvature to become negative. Isn't it time to advance beyond nineteenth century physics and relegate dark energy back to the 'no significant evidence' box?\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"1 'Dark energy' traces inhomogeneity\",\n \"content\": \"The standard model of cosmology is generally accepted to be a spacetime with a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric [3, 4, 5, 6, 7], solving the Einstein equation rather simply thanks to the assumption of homogeneous density on any spatial slice, and measured to have the Concordance Model [8] values of the matter density and dark energy parameters, Ω m0 ≈ 0 . 32 (e.g. [9]) and Ω Λ0 := 1 -Ω m0 , respectively (hereafter, ΛCDM). However, we live in the inhomogeneous epoch: galaxies and voids certainly exist today. As found by [10], in an Einstein-de Sitter (EdS) FLRW model (i.e. with Ω m0 = 1 , Ω Λ0 = 0), the fraction of matter in a large region that is virialised, f vir , evolves in a very similar way to that of the dark energy parameter in a flat FLRW model with negligible radiation density, This is shown in Fig. 1, where Ω Λ0 = 0 . 68. This seems like an extraordinary coincidence. Over the same redshift range during which one expects that the Universe is inhomogeneous, the degree of inhomogeneity, as expressed by f vir ( z ) in an EdS model, approximately follows the proportion of the critical density represented by dark energy, if the dark energy is inferred from forcing a homogeneous model on the observational data. To first order, f vir is not sensitive to the choice of FLRW model (see Fig. 1), so the coincidence also exists for inhomogeneity in a ΛCDM model. The simplest inference is that a homogeneous-model-inferred non-zero dark energy parameter is really just a measurement of inhomogeneity . What physical link could there be between this inhomogeneity parameter and homogeneous-model-inferred 'dark energy'?\",\n \"pages\": [\n 2,\n 3\n ]\n },\n {\n \"title\": \"2 Void dominance: low matter density, high critical density\",\n \"content\": \"The most obvious physical link between inhomogeneity and homogeneousinferred dark energy is the volume dominance of voids compared to virialised regions at recent epochs [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. This is because gravitational collapse implies an increase in density, i.e., a reduction in volume, by a factor of about δ vir ∼ 100-200 (e.g., 8-18 π 2 [21]), so that, to first order, the collapsed matter occupies a negligible fraction of the spatial volume. Thus, recent-epoch spatial volume is overwhelmingly dominated by low-density regions. Moreover, there are velocity flows out of the voids, since otherwise, the voids couldn't be nearly empty. Thus, the critical density defining spatial flatness in the voids is higher than it would be in a homogeneous calculation. The standard, FLRW approach insists that space expands uniformly with spatially constant curvature, i.e. space is rigid in comoving coordinates-it is forbidden from bending under the influence of gravity. N -body simulations are typically used to study the Newtonian formation of overdense structures and voids within this rigid, comoving background model. However, both the low density of the voids and the velocity flow out of them imply that the matter density parameter in the voids is sub-critical. Thus, the voids are hyperbolic. Geometrically, this hyperbolicity should also be taken into account, implying an even lower matter density parameter (conservatively, let us ignore this: the effect is small).\",\n \"pages\": [\n 3\n ]\n },\n {\n \"title\": \"3 Volume-weighted averaged metric\",\n \"content\": \"These arguments can be formalised using the volume-weighted averaging approach to modelling inhomogeneous spatial slices [22, 23, 11, 24, 16], in which the Friedmann equation is generalised to (12) of [13]. For simplicity, let us (i) set the dark energy term to zero, (ii) neglect the kinematic backreaction as much smaller than the curvature backreaction (see [16] for numerical justification), and (iii) combine the curvature parameters into a single full curvature parameter as suggested in [13]. Writing 'k' instead of ' R ', this gives the domain-averaged, effective Friedmann equation Along a typical, large-scale, random, spacelike or null geodesic over recent epochs, what proportions of the geodesic lie in the emptied and virialised regions? The proportions at a given z are, on average, (1 -f vir /δ vir ) : f vir /δ vir , respectively. Given that δ vir ∼ 100-200 and less than about 1% of the geodesic falls within the virialised regions. Thus, by starting with a large-scale, high-redshift, 'background' FLRW model-in this case, an EdS model-an effective metric can be written by assuming that the virialised matter contributes negligibly. As in [10], the effective expansion rate is combining the FLRW expansion H ( z ) with the peculiar velocity gradient (physical, not comoving) across voids, H pec ( z ), estimated numerically from an N -body simulation ([1, 2], a 240 h -1 Mpc box-size EdS simulation; see [10]) The background Hubble constant H bg is set to make the present value consistent with low redshift estimates [25, 26], i.e. Thus, the loss of matter from the voids and the higher critical density in voids both decrease the EdS background value from Ω m , bg = 1 to an effective value of The effective radius of curvature is The effective metric (cf [17]) is where the radial comoving component is\",\n \"pages\": [\n 3,\n 4\n ]\n },\n {\n \"title\": \"4 Matter density parameter and luminosity distances\",\n \"content\": \"Without any attempt to fit this approximation to observational data , apart from (5) above, the correction of the EdS model as presented above gives an effective matter density parameter (6) that drops slowly from its background value of unity at high redshift down to Ω eff m = 0 . 27 at the present epoch z = 0, remarkably close to the last two decades' local estimates of the matter density parameter. The effective luminosity distance follows directly from the radial comoving distance and hyperbolicity, Figure 2 shows that despite the rough nature of the virialisation approximation, it shifts the homogeneous EdS magnitude-redshift relation by a substantial fraction towards the homogeneous ΛCDM relation, and thus, towards the observational supernovae type Ia relation.\",\n \"pages\": [\n 5\n ]\n },\n {\n \"title\": \"5 Conclusion\",\n \"content\": \"A handful of simple formulae, lying at the heart of homogeneous, spatially rigid cosmology, remain approximately valid when generalised to inhomogeneous, spatially flexible cosmology [13] and applied to what observationally and theoretically dominate the present-day spatial volume-the voids. The result is a correction to a large-scale, high-redshift, background Einstein-de Sitter cosmological model. The correction approximately gives the observed lowredshift matter density parameter and nearly matches the type Ia supernovae luminosity distance relation. The amplitude of the correction is unlikely to be much smaller than estimated here. At the present epoch, direct observations [27], N -body simulations [1, 2], and the existence of the cosmic web itself establish inhomogeneous peculiar velocity gradients of ∼ 20-30 km/s/Mpc, forcing, at least, a factor of ∼ (75 / 50) 2 reduction of the Einstein-de Sitter matter density to Ω eff m < 0 . 5, via Eqs (4), (5), and (6). A virialisation fraction of the order of ∼ 50% reduces this to ∼ 0 . 25. Even when forcing the homogeneous FLRW models onto the data, the initial analysis of the Planck Surveyor cosmic microwave background data finds H bg = 67 . 3 ± 1 . 2 km/s/Mpc at z ≈ 1100 [9]-not as low as the velocity gradients imply, but still significantly lower than the low redshift estimates of H eff (0) = 74 . 0 ± 1 . 6 km/s/Mpc ([26, 25], standard error in the mean). How could such a simple, back-of-the-envelope calculation have been missed for so long? While the volume-averaging approach to cosmology has been developed over many years (see e.g. [13] for a review), possibly the answer lies, ironically, in confusion between the spacelike, unobserved , comoving, present time slice and the past light cone. A gigaparsec-scale void in the former should have a very weak ( δ ∼ 10 -5 ) underdensity, and our would-be location at its centre would be uncomfortably anti-Copernican. But these are both moot points! On the past light cone, an average , gigaparsec-scale, sub-critical (0 < Ω eff m < 0 . 8) void is perfectly natural, since it is defined by the onset of the virialisation epoch at z < ∼ 3. Moreover, we are naturally located at this pseudo-void's centre, by the nature of the past light cone. What is simpler: relativistic, hyperbolic voids, observed by an observer at the tip of the past light cone, with no dark energy parameter? Or rigid, Newtonian voids together with a dark energy parameter that traces the virialisation fraction? Acknowledgments: This essay is in part based on work conducted within the 'Lyon Institute of Origins' under grant ANR-10-LABX-66, and on Program Oblicze'n Wielkich Wyzwa'n nauki i techniki (POWIEW) computational resources (grant 87) at the Pozna'n Supercomputing and Networking Center (PCSS).\",\n \"pages\": [\n 5,\n 6\n ]\n }\n]"}}},{"rowIdx":299,"cells":{"bibcode":{"kind":"string","value":"2013IJMPD..2242024K"},"pdf_url":{"kind":"string","value":"https://arxiv.org/pdf/1312.3462.pdf"},"content":{"kind":"string","value":"\nMatters on a moving brane\nTomi Sebastian Koivisto 1, ∗ and Danielle Elizabeth Wills 2, †\n1\nInstitute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway 2 Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, UK\n(Dated: November 13, 2018)\nAbstract\n'An idle brane is the devil's workshop.'\nA novel generalisation of the Dirac-Born-Infeld string scenario is described. It is shown that matter residing on the moving brane is dark and has the so-called disformal coupling to gravity. This gives rise to cosmologies where dark matter stems from the oscillations of the open strings along the brane and the transverse oscillations result in dark energy. Furthermore, due to a new screening mechanism that conceals the fifth force from local experiments, one may even entertain the possibility that the visible sector is also moving along the extra dimensions.\nEssay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation, Submitted 31.3.2013\nThe relation between physical and gravitational geometry is a basic issue of fundamental importance to both classical and quantum theory. Gravity is the geometry of space-time described by the metric g µν , which in General Relativity is prescribed dynamics by the Einstein-Hilbert action. The geometry governing the movement of matter fields in the space-time is called the physical geometry ¯ g µν ,\nS = ∫ d 4 x √ -g R 16 πG + S (matter , ¯ g µν ) . (1)\nDeviations from the postulate of minimal coupling, ¯ g µν = g µν , have to be constrained experimentally [1]. For simplicity, one may consider such deviations to be given by a single scalar field φ . It can then be argued that the most general physically meaningful relation between the two metrics has the form [2]\n¯ g µν = C ( φ, X ) g µν + D ( φ, X ) φ ,µ φ ,ν , (2)\nwhere X = ( ∂φ ) 2 is the kinetic term of the field. The function C gives the very well known conformal transformation. In this essay we discuss the role of the function D that gives the so-called disformal transformation.\nIt is easy to argue that though largely neglected in the past literature, this relation is generic. The Brans-Dicke class of scalar-tensor theories is described by a nontrivial function C ; a very extensively studied example is the f ( R ) type of gravity theory. However, when one considers any more general scalar-tensor theory, or writes a covariant action for the metric involving any other invariant besides R , the function D must appear in the Einstein frame formulation of the theory 1 . Here our starting point, instead of an ad hoc modification of gravity, is a higher dimensional theory with pure Einstein gravity minimally coupled to matter: we will see that the resulting four-dimensional description generically involves a nontrivial function D .\nThe primary novel contribution of this essay is to present a generalisation of the DiracBorn-Infeld (DBI) string scenario, that predicts matter couplings of the form (2), with both functions C and D robustly derivable from first principles. It is shown that any matter residing upon the moving brane will have this coupling to our gravitational geometry, when the scalar field φ is then identified with the radial coordinate of the brane in a warped\n\n\n
FIG. 1: Dark matter emerges from oscillations of the open strings along the brane, visualised here as ripples on the surface sourced by the open string endpoints. Dark energy emerges from the oscillations of the open strings transverse to the brane, visualised here as the movement of the brane as a single wave-front or wave-pulse in the radial direction.
\n\nx\nbackground, the DBI radion. If the brane doesn't intersect the Standard Model stack of branes 2 , the disformally coupled matter would interact only gravitationally with baryonic matter. In particular we propose to associate the U (1) gauge field upon the moving brane with this invisible matter.\nAn appealing unification of the cosmological dark sector has then emerged: dark matter stems from the oscillations of the open strings in the directions along the brane, which propagate as a vector field on the world-volume, and the dynamical dark energy field is the scalar DBI radion, which encodes the oscillations of open strings transverse to the brane, see Fig. 1. Gravity emerges from the oscillations of the closed strings in the bulk spacetime. In generic compactifications, one obtains both massive and massless vector fields. In cosmology, the former manifest as dark matter and the latter as 'dark radiation', both of which are non-minimally coupled to the dark energy field via the disformal relation.\nIn a warped flux compactification of Type IIB string theory, the ten-dimensional metric takes the form\nG MN dx M dx N = h -1 / 2 g µν dx µ dx ν + h 1 / 2 g ab dx a dx b , (3)\nwhere h = h ( x a ) is the warp factor which depends only on the compact coordinates indexed by a, b = 4 , ..., 9. These are the wrapped dimensions of the Calabi-Yau space. The capital indices M,N = 0 , ..., 9 run over all space-time dimensions, the greek µ, ν = 0 , ..., 3 over the large 4 dimensions. For simplicity we consider a probe D3-brane embedded in this background, as the set-up can readily be generalised to branes of lower codimension. In the Einstein frame the Dirac-Born-Infeld (DBI) action describing the dynamics of a D3-brane is given by\nS DBI = -(2 π ) -3 glyph[lscript] -4 s ∫ d 4 ξ √ -det( γ µν + e -ϕ 2 F µν ) , (4)\nwhere the integration is over the brane coordinates on the world-volume, ξ µ , and the string scale glyph[lscript] 2 s gives the tension of the brane. The dilaton ϕ we assume to be stabilized as usual. The induced metric on the brane is denoted by γ µν . Finally, there appears\nF µν = B µν +2 πglyph[lscript] 2 s F µν , (5)\nthe gauge invariant combination of the pullback of the NSNS 2-form field B 2 and the field strength F µν of the world-volume U (1) gauge field. Below we will describe how the induced metric γ µν gives rise to the disformal relation in gravity, and the field F µν to disformally coupled matter.\nLet us first focus on the geometry, see Fig. 2. The pullback of the ten dimensional metric onto the brane world-volume is given by\nγ µν = G MN ∂ µ x M ∂ ν x N . (6)\nAdopting the static gauge, the D3-brane fills the large dimensions such that ξ µ = x µ . We allow the compact spacetime coordinates which are transverse to the brane to be functions of the four-dimensional world-volume coordinates, x a = x a ( ξ µ ). This entails that the brane may move in the internal space. With these choices, the four dimensional components of the induced metric on the brane world-volume take the form\nγ µν = G µν + ∂x a ∂ξ µ ∂x b ∂ξ v G ab . (7)\nIt can be justified to consider purely one-dimensional brane trajectories in the radial direction of a warped throat [5]. Returning to Eq. (2), we may identify the scalar field φ with the canonically normalised radial coordinate of the brane, one of the x a 's in Eq. (7). Thus we have obtained the relation (2), where γ µν → ¯ g µν and the functions C ∼ h 1 2 ( φ ) and D ∼ h -1 2 ( φ ) are given by the warp factor.\n\n\n
FIG. 2: A D3-brane moving in the radial direction of a warped throat in a compact Calabi-Yau threefold.
\n\nIt is straightforward to compute the determinant in the DBI action explicitly to separate the geometric part into the contributions from the gravitational metric and the scalar field. To complete the picture, we include the D3-brane charge C 4 which is given by the warp factor as ∼ 1 /h , as well as the potential V ( φ ) that may emerge from the coupling of the brane to other sectors and is in principle computable given any explicit set-up. We arrive at the following description of our four-dimensional gravity:\nS = ∫ d 4 x √ -g [ R 16 πG -1 h ( φ ) ( √ 1 + h ( φ ) X -1 ) -V ( φ ) ] + S ( F µν , ¯ g µν ) , (8)\nwhere all the F -dependent terms are collected into an effective matter action. Here we have written γ µν = ¯ g µν , to emphasize that what we have is precisely a nontrivial realization of our starting point (1), supplemented with the Lagrangian for the scalar field that governs the relation (2).\nLet us then consider the matter sources. The vector fields are associated with the open string endpoints on the brane, and may acquire masses via Stuckelberg couplings to bulk two-forms. The masses will depend on the geometry and are inherently very large, such that for the late-time universe these fields will have long decayed into lighter particles, unless they are stable. In the end, for either case, we do expect massive particles on the brane, and by construction, they will be both dark and disformally coupled to gravity.\nThus we have arrived at a novel generalisation of the coupled quintessence cosmology. In such models, the fine-tuning problem of dark energy could be alleviated by the fact that the energy density is promoted to a dynamical field, and the coincidence problem could be addressed by the direct coupling between the dark matter and dark energy components. A crucial feature is that the coupling will affect the formation of structure in dark matter and thus will modify the predictions for the observed large-scale structure. In the equations governing the evolution of cosmological perturbations, the disformal coupling has been shown to have a considerably richer structure than the purely conformal interaction term [4], and we expect that the precise model presented here can be very efficiently constrained by studying the matter power spectrum at both linear and nonlinear regimes.\nAn even more radical possibility emerges due to the recently discovered disformal screening mechanism [6], which can conceal the coupling from local experiments even while it has drastic consequences at cosmological scales. In the DBI string scenario presented above this suggests that our Standard Model could be residing on a moving stack of branes 3 without introducing a cosmological moduli problem. This could potentially bring new classes of string cosmologies with novel late-time phenomenology into serious consideration.\nThe rich phenomenology of disformal couplings has very recently attracted growing interest. Whereas the function C in (2) is a local scale transformation that leaves the causal structure untouched, the function D affects angles and thus distorts the light cones. Therefore coupling the electromagnetic field disformally results in a varying-speed-of light theory. Constraints on such couplings have been derived from both high-precision laboratory experiments of low-energy photons [7] and from the cosmological evolution of the cosmic microwave background black-body radiation [8]. Bounds can also be derived by considering the coupling of baryonic fluids in the radiation dominated epoch of the universe evolution [9]. As\nwe have shown here, such phenomenology can in fact be considered as a probe of extra dimensional brane movement in a robust string theoretical setting.\nA note added: The theoretical underpinnings and cosmological implications of the scenario outlined here are explored much further in Ref. [10] where we refer the reader for more details. See also [11-13] for related recent studies of disformal couplings.\nAcknowledgments\nTK is supported by the Research Council of Norway and DW by an STFC studentship. We would like to thank Ruth Gregory, David Mota, Ivonne Zavala and Miguel Zumalac'arregui for useful discussions.\n\n[1] C. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, 1993), ISBN 9780521439732.\n[2] J. D. Bekenstein, Phys.Rev. D48 , 3641 (1993).\n[3] L. Amendola, K. Enqvist, and T. Koivisto, Phys.Rev. D83 , 044016 (2011), 1010.4776.\n[4] M. Zumalacarregui, T. S. Koivisto, and D. F. Mota, Phys.Rev. D87 , 083010 (2013), 1210.8016.\n[5] D. A. Easson, R. Gregory, D. F. Mota, G. Tasinato, and I. Zavala, JCAP 0802 , 010 (2008), 0709.2666.\n[6] T. S. Koivisto, D. F. Mota, and M. Zumalacarregui, Phys.Rev.Lett. 109 , 241102 (2012), 1205.3167.\n[7] P. Brax, C. Burrage, and A.-C. Davis, JCAP 1210 , 016 (2012), 1206.1809.\n[8] C. van de Bruck, J. Morrice, and S. Vu (2013), 1303.1773.\n[9] C. van de Bruck and G. Sculthorpe, Phys.Rev. D87 , 044004 (2013), 1210.2168.\n[10] T. Koivisto, D. Wills, and I. Zavala (2013), 1312.2597.\n[11] D. Bettoni and S. Liberati, Phys.Rev. D88 , 084020 (2013), 1306.6724.\n[12] M. Zumalacrregui and J. Garca-Bellido (2013), 1308.4685.\n[13] P. Brax, C. Burrage, A.-C. Davis, and G. Gubitosi, JCAP 1311 , 001 (2013), 1306.4168.\n\n"},"sections":{"kind":"list like","value":[{"title":"Matters on a moving brane","content":"Tomi Sebastian Koivisto 1, ∗ and Danielle Elizabeth Wills 2, † 1 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway 2 Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, UK (Dated: November 13, 2018)","pages":[1]},{"title":"Abstract","content":"'An idle brane is the devil's workshop.' A novel generalisation of the Dirac-Born-Infeld string scenario is described. It is shown that matter residing on the moving brane is dark and has the so-called disformal coupling to gravity. This gives rise to cosmologies where dark matter stems from the oscillations of the open strings along the brane and the transverse oscillations result in dark energy. Furthermore, due to a new screening mechanism that conceals the fifth force from local experiments, one may even entertain the possibility that the visible sector is also moving along the extra dimensions. Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation, Submitted 31.3.2013 The relation between physical and gravitational geometry is a basic issue of fundamental importance to both classical and quantum theory. Gravity is the geometry of space-time described by the metric g µν , which in General Relativity is prescribed dynamics by the Einstein-Hilbert action. The geometry governing the movement of matter fields in the space-time is called the physical geometry ¯ g µν , Deviations from the postulate of minimal coupling, ¯ g µν = g µν , have to be constrained experimentally [1]. For simplicity, one may consider such deviations to be given by a single scalar field φ . It can then be argued that the most general physically meaningful relation between the two metrics has the form [2] where X = ( ∂φ ) 2 is the kinetic term of the field. The function C gives the very well known conformal transformation. In this essay we discuss the role of the function D that gives the so-called disformal transformation. It is easy to argue that though largely neglected in the past literature, this relation is generic. The Brans-Dicke class of scalar-tensor theories is described by a nontrivial function C ; a very extensively studied example is the f ( R ) type of gravity theory. However, when one considers any more general scalar-tensor theory, or writes a covariant action for the metric involving any other invariant besides R , the function D must appear in the Einstein frame formulation of the theory 1 . Here our starting point, instead of an ad hoc modification of gravity, is a higher dimensional theory with pure Einstein gravity minimally coupled to matter: we will see that the resulting four-dimensional description generically involves a nontrivial function D . The primary novel contribution of this essay is to present a generalisation of the DiracBorn-Infeld (DBI) string scenario, that predicts matter couplings of the form (2), with both functions C and D robustly derivable from first principles. It is shown that any matter residing upon the moving brane will have this coupling to our gravitational geometry, when the scalar field φ is then identified with the radial coordinate of the brane in a warped x background, the DBI radion. If the brane doesn't intersect the Standard Model stack of branes 2 , the disformally coupled matter would interact only gravitationally with baryonic matter. In particular we propose to associate the U (1) gauge field upon the moving brane with this invisible matter. An appealing unification of the cosmological dark sector has then emerged: dark matter stems from the oscillations of the open strings in the directions along the brane, which propagate as a vector field on the world-volume, and the dynamical dark energy field is the scalar DBI radion, which encodes the oscillations of open strings transverse to the brane, see Fig. 1. Gravity emerges from the oscillations of the closed strings in the bulk spacetime. In generic compactifications, one obtains both massive and massless vector fields. In cosmology, the former manifest as dark matter and the latter as 'dark radiation', both of which are non-minimally coupled to the dark energy field via the disformal relation. In a warped flux compactification of Type IIB string theory, the ten-dimensional metric takes the form where h = h ( x a ) is the warp factor which depends only on the compact coordinates indexed by a, b = 4 , ..., 9. These are the wrapped dimensions of the Calabi-Yau space. The capital indices M,N = 0 , ..., 9 run over all space-time dimensions, the greek µ, ν = 0 , ..., 3 over the large 4 dimensions. For simplicity we consider a probe D3-brane embedded in this background, as the set-up can readily be generalised to branes of lower codimension. In the Einstein frame the Dirac-Born-Infeld (DBI) action describing the dynamics of a D3-brane is given by where the integration is over the brane coordinates on the world-volume, ξ µ , and the string scale glyph[lscript] 2 s gives the tension of the brane. The dilaton ϕ we assume to be stabilized as usual. The induced metric on the brane is denoted by γ µν . Finally, there appears the gauge invariant combination of the pullback of the NSNS 2-form field B 2 and the field strength F µν of the world-volume U (1) gauge field. Below we will describe how the induced metric γ µν gives rise to the disformal relation in gravity, and the field F µν to disformally coupled matter. Let us first focus on the geometry, see Fig. 2. The pullback of the ten dimensional metric onto the brane world-volume is given by Adopting the static gauge, the D3-brane fills the large dimensions such that ξ µ = x µ . We allow the compact spacetime coordinates which are transverse to the brane to be functions of the four-dimensional world-volume coordinates, x a = x a ( ξ µ ). This entails that the brane may move in the internal space. With these choices, the four dimensional components of the induced metric on the brane world-volume take the form It can be justified to consider purely one-dimensional brane trajectories in the radial direction of a warped throat [5]. Returning to Eq. (2), we may identify the scalar field φ with the canonically normalised radial coordinate of the brane, one of the x a 's in Eq. (7). Thus we have obtained the relation (2), where γ µν → ¯ g µν and the functions C ∼ h 1 2 ( φ ) and D ∼ h -1 2 ( φ ) are given by the warp factor. It is straightforward to compute the determinant in the DBI action explicitly to separate the geometric part into the contributions from the gravitational metric and the scalar field. To complete the picture, we include the D3-brane charge C 4 which is given by the warp factor as ∼ 1 /h , as well as the potential V ( φ ) that may emerge from the coupling of the brane to other sectors and is in principle computable given any explicit set-up. We arrive at the following description of our four-dimensional gravity: where all the F -dependent terms are collected into an effective matter action. Here we have written γ µν = ¯ g µν , to emphasize that what we have is precisely a nontrivial realization of our starting point (1), supplemented with the Lagrangian for the scalar field that governs the relation (2). Let us then consider the matter sources. The vector fields are associated with the open string endpoints on the brane, and may acquire masses via Stuckelberg couplings to bulk two-forms. The masses will depend on the geometry and are inherently very large, such that for the late-time universe these fields will have long decayed into lighter particles, unless they are stable. In the end, for either case, we do expect massive particles on the brane, and by construction, they will be both dark and disformally coupled to gravity. Thus we have arrived at a novel generalisation of the coupled quintessence cosmology. In such models, the fine-tuning problem of dark energy could be alleviated by the fact that the energy density is promoted to a dynamical field, and the coincidence problem could be addressed by the direct coupling between the dark matter and dark energy components. A crucial feature is that the coupling will affect the formation of structure in dark matter and thus will modify the predictions for the observed large-scale structure. In the equations governing the evolution of cosmological perturbations, the disformal coupling has been shown to have a considerably richer structure than the purely conformal interaction term [4], and we expect that the precise model presented here can be very efficiently constrained by studying the matter power spectrum at both linear and nonlinear regimes. An even more radical possibility emerges due to the recently discovered disformal screening mechanism [6], which can conceal the coupling from local experiments even while it has drastic consequences at cosmological scales. In the DBI string scenario presented above this suggests that our Standard Model could be residing on a moving stack of branes 3 without introducing a cosmological moduli problem. This could potentially bring new classes of string cosmologies with novel late-time phenomenology into serious consideration. The rich phenomenology of disformal couplings has very recently attracted growing interest. Whereas the function C in (2) is a local scale transformation that leaves the causal structure untouched, the function D affects angles and thus distorts the light cones. Therefore coupling the electromagnetic field disformally results in a varying-speed-of light theory. Constraints on such couplings have been derived from both high-precision laboratory experiments of low-energy photons [7] and from the cosmological evolution of the cosmic microwave background black-body radiation [8]. Bounds can also be derived by considering the coupling of baryonic fluids in the radiation dominated epoch of the universe evolution [9]. As we have shown here, such phenomenology can in fact be considered as a probe of extra dimensional brane movement in a robust string theoretical setting. A note added: The theoretical underpinnings and cosmological implications of the scenario outlined here are explored much further in Ref. [10] where we refer the reader for more details. See also [11-13] for related recent studies of disformal couplings.","pages":[1,2,3,4,5,6,7]},{"title":"Acknowledgments","content":"TK is supported by the Research Council of Norway and DW by an STFC studentship. We would like to thank Ruth Gregory, David Mota, Ivonne Zavala and Miguel Zumalac'arregui for useful discussions.","pages":[7]}],"string":"[\n {\n \"title\": \"Matters on a moving brane\",\n \"content\": \"Tomi Sebastian Koivisto 1, ∗ and Danielle Elizabeth Wills 2, † 1 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway 2 Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, UK (Dated: November 13, 2018)\",\n \"pages\": [\n 1\n ]\n },\n {\n \"title\": \"Abstract\",\n \"content\": \"'An idle brane is the devil's workshop.' A novel generalisation of the Dirac-Born-Infeld string scenario is described. It is shown that matter residing on the moving brane is dark and has the so-called disformal coupling to gravity. This gives rise to cosmologies where dark matter stems from the oscillations of the open strings along the brane and the transverse oscillations result in dark energy. Furthermore, due to a new screening mechanism that conceals the fifth force from local experiments, one may even entertain the possibility that the visible sector is also moving along the extra dimensions. Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation, Submitted 31.3.2013 The relation between physical and gravitational geometry is a basic issue of fundamental importance to both classical and quantum theory. Gravity is the geometry of space-time described by the metric g µν , which in General Relativity is prescribed dynamics by the Einstein-Hilbert action. The geometry governing the movement of matter fields in the space-time is called the physical geometry ¯ g µν , Deviations from the postulate of minimal coupling, ¯ g µν = g µν , have to be constrained experimentally [1]. For simplicity, one may consider such deviations to be given by a single scalar field φ . It can then be argued that the most general physically meaningful relation between the two metrics has the form [2] where X = ( ∂φ ) 2 is the kinetic term of the field. The function C gives the very well known conformal transformation. In this essay we discuss the role of the function D that gives the so-called disformal transformation. It is easy to argue that though largely neglected in the past literature, this relation is generic. The Brans-Dicke class of scalar-tensor theories is described by a nontrivial function C ; a very extensively studied example is the f ( R ) type of gravity theory. However, when one considers any more general scalar-tensor theory, or writes a covariant action for the metric involving any other invariant besides R , the function D must appear in the Einstein frame formulation of the theory 1 . Here our starting point, instead of an ad hoc modification of gravity, is a higher dimensional theory with pure Einstein gravity minimally coupled to matter: we will see that the resulting four-dimensional description generically involves a nontrivial function D . The primary novel contribution of this essay is to present a generalisation of the DiracBorn-Infeld (DBI) string scenario, that predicts matter couplings of the form (2), with both functions C and D robustly derivable from first principles. It is shown that any matter residing upon the moving brane will have this coupling to our gravitational geometry, when the scalar field φ is then identified with the radial coordinate of the brane in a warped x background, the DBI radion. If the brane doesn't intersect the Standard Model stack of branes 2 , the disformally coupled matter would interact only gravitationally with baryonic matter. In particular we propose to associate the U (1) gauge field upon the moving brane with this invisible matter. An appealing unification of the cosmological dark sector has then emerged: dark matter stems from the oscillations of the open strings in the directions along the brane, which propagate as a vector field on the world-volume, and the dynamical dark energy field is the scalar DBI radion, which encodes the oscillations of open strings transverse to the brane, see Fig. 1. Gravity emerges from the oscillations of the closed strings in the bulk spacetime. In generic compactifications, one obtains both massive and massless vector fields. In cosmology, the former manifest as dark matter and the latter as 'dark radiation', both of which are non-minimally coupled to the dark energy field via the disformal relation. In a warped flux compactification of Type IIB string theory, the ten-dimensional metric takes the form where h = h ( x a ) is the warp factor which depends only on the compact coordinates indexed by a, b = 4 , ..., 9. These are the wrapped dimensions of the Calabi-Yau space. The capital indices M,N = 0 , ..., 9 run over all space-time dimensions, the greek µ, ν = 0 , ..., 3 over the large 4 dimensions. For simplicity we consider a probe D3-brane embedded in this background, as the set-up can readily be generalised to branes of lower codimension. In the Einstein frame the Dirac-Born-Infeld (DBI) action describing the dynamics of a D3-brane is given by where the integration is over the brane coordinates on the world-volume, ξ µ , and the string scale glyph[lscript] 2 s gives the tension of the brane. The dilaton ϕ we assume to be stabilized as usual. The induced metric on the brane is denoted by γ µν . Finally, there appears the gauge invariant combination of the pullback of the NSNS 2-form field B 2 and the field strength F µν of the world-volume U (1) gauge field. Below we will describe how the induced metric γ µν gives rise to the disformal relation in gravity, and the field F µν to disformally coupled matter. Let us first focus on the geometry, see Fig. 2. The pullback of the ten dimensional metric onto the brane world-volume is given by Adopting the static gauge, the D3-brane fills the large dimensions such that ξ µ = x µ . We allow the compact spacetime coordinates which are transverse to the brane to be functions of the four-dimensional world-volume coordinates, x a = x a ( ξ µ ). This entails that the brane may move in the internal space. With these choices, the four dimensional components of the induced metric on the brane world-volume take the form It can be justified to consider purely one-dimensional brane trajectories in the radial direction of a warped throat [5]. Returning to Eq. (2), we may identify the scalar field φ with the canonically normalised radial coordinate of the brane, one of the x a 's in Eq. (7). Thus we have obtained the relation (2), where γ µν → ¯ g µν and the functions C ∼ h 1 2 ( φ ) and D ∼ h -1 2 ( φ ) are given by the warp factor. It is straightforward to compute the determinant in the DBI action explicitly to separate the geometric part into the contributions from the gravitational metric and the scalar field. To complete the picture, we include the D3-brane charge C 4 which is given by the warp factor as ∼ 1 /h , as well as the potential V ( φ ) that may emerge from the coupling of the brane to other sectors and is in principle computable given any explicit set-up. We arrive at the following description of our four-dimensional gravity: where all the F -dependent terms are collected into an effective matter action. Here we have written γ µν = ¯ g µν , to emphasize that what we have is precisely a nontrivial realization of our starting point (1), supplemented with the Lagrangian for the scalar field that governs the relation (2). Let us then consider the matter sources. The vector fields are associated with the open string endpoints on the brane, and may acquire masses via Stuckelberg couplings to bulk two-forms. The masses will depend on the geometry and are inherently very large, such that for the late-time universe these fields will have long decayed into lighter particles, unless they are stable. In the end, for either case, we do expect massive particles on the brane, and by construction, they will be both dark and disformally coupled to gravity. Thus we have arrived at a novel generalisation of the coupled quintessence cosmology. In such models, the fine-tuning problem of dark energy could be alleviated by the fact that the energy density is promoted to a dynamical field, and the coincidence problem could be addressed by the direct coupling between the dark matter and dark energy components. A crucial feature is that the coupling will affect the formation of structure in dark matter and thus will modify the predictions for the observed large-scale structure. In the equations governing the evolution of cosmological perturbations, the disformal coupling has been shown to have a considerably richer structure than the purely conformal interaction term [4], and we expect that the precise model presented here can be very efficiently constrained by studying the matter power spectrum at both linear and nonlinear regimes. An even more radical possibility emerges due to the recently discovered disformal screening mechanism [6], which can conceal the coupling from local experiments even while it has drastic consequences at cosmological scales. In the DBI string scenario presented above this suggests that our Standard Model could be residing on a moving stack of branes 3 without introducing a cosmological moduli problem. This could potentially bring new classes of string cosmologies with novel late-time phenomenology into serious consideration. The rich phenomenology of disformal couplings has very recently attracted growing interest. Whereas the function C in (2) is a local scale transformation that leaves the causal structure untouched, the function D affects angles and thus distorts the light cones. Therefore coupling the electromagnetic field disformally results in a varying-speed-of light theory. Constraints on such couplings have been derived from both high-precision laboratory experiments of low-energy photons [7] and from the cosmological evolution of the cosmic microwave background black-body radiation [8]. Bounds can also be derived by considering the coupling of baryonic fluids in the radiation dominated epoch of the universe evolution [9]. As we have shown here, such phenomenology can in fact be considered as a probe of extra dimensional brane movement in a robust string theoretical setting. A note added: The theoretical underpinnings and cosmological implications of the scenario outlined here are explored much further in Ref. [10] where we refer the reader for more details. See also [11-13] for related recent studies of disformal couplings.\",\n \"pages\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7\n ]\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"TK is supported by the Research Council of Norway and DW by an STFC studentship. We would like to thank Ruth Gregory, David Mota, Ivonne Zavala and Miguel Zumalac'arregui for useful discussions.\",\n \"pages\": [\n 7\n ]\n }\n]"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":2,"numItemsPerPage":100,"numTotalItems":3891,"offset":200,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTczODUyNTg3NCwic3ViIjoiL2RhdGFzZXRzL2FzaGlzaGtncGlhbi9hZHNwYXBlcnNfNzVfODAiLCJleHAiOjE3Mzg1Mjk0NzQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.roIjNjFxJSxZ50GPjhjhvS8tOv5mgK6x7kW0GzuN-yz96iXVsW0VpjeP-WWJ87s5usKnxea8934HlXwrHq8SDw","displayUrls":true},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false}">
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<section_header_level_1><location><page_1><loc_13><loc_86><loc_87><loc_88></location>NUSTAR SPECTROSCOPY OF GRS 1915 + 105: DISK REFLECTION, SPIN, AND CONNECTIONS TO JETS</section_header_level_1>
<text><location><page_1><loc_8><loc_81><loc_92><loc_85></location>J. M. MILLER 1 , M. L. PARKER 2 , F. FUERST 3 , M. BACHETTI 4,5 , F. A. HARRISON 3 , D. BARRET 5 , S. E. BOGGS 6 , D. CHAKRABARTY 7 , F. E. CHRISTENSEN 8 , W. W. CRAIG 9,10 , A. C. FABIAN 2 , B. W. GREFENSTETTE 3 , C. J. HAILEY 10 , A. L. KING 1 , D. K. STERN 11 , J. A. TOMSICK 6 , D. J. WALTON 3 , W. W. ZHANG 12</text>
<text><location><page_1><loc_46><loc_80><loc_54><loc_81></location>Submitted to ApJ</text>
<section_header_level_1><location><page_1><loc_46><loc_77><loc_54><loc_79></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_60><loc_86><loc_77></location>We report on the results of spectral fits made to a NuSTAR observation of the black hole GRS 1915 + 105 in a 'plateau' state. This state is of special interest because it is similar to the 'low/hard' state seen in other black holes, especially in that compact, steady jets are launched in this phase. The 3-79 keV bandpass of NuSTAR , and its ability to obtain moderate-resolution spectra free from distortions such as photon pile-up, are extremely well suited to studies of disk reflection in X-ray binaries. In only 15 ks of net exposure, an extraordinarily sensitive spectrum of GRS 1915 + 105 was measured across the full bandpass. Ionized reflection from a disk around a rapidly-spinning black hole is clearly required to fit the spectra; even hybrid Comptonization models including ionized reflection from a disk around a Schwarzschild black hole proved inadequate. A spin parameter of a = 0 . 98 ± 0 . 01 (1 σ statistical error) is measured via the best-fit model; low spins are ruled out at a high level of confidence. This result suggests that jets can be launched from a disk extending to the innermost stable circular orbit. A very steep inner disk emissivity profile is also measured, consistent with models of compact coronae above Kerr black holes. These results support an emerging association between the hard X-ray corona and the base of the relativistic jet.</text>
<text><location><page_1><loc_14><loc_58><loc_86><loc_59></location>Subject headings: Black hole physics - relativity - stars: binaries - physical data and processes: accretion disks</text>
<section_header_level_1><location><page_1><loc_22><loc_55><loc_34><loc_56></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_33><loc_48><loc_54></location>Reflection of hard X-ray emission from a 'corona' onto the accretion disk can measure black hole spin, and can also serve as a powerful probe of the geometry of black hole accretion flows. Disk reflection spectra excited near to black holes will bear the imprints of gravitational red-shifts and strong Doppler shifts (e.g. Fabian et al. 1989). As long as the accretion disk extends to the innermost stable circular orbit (ISCO; Bardeen, Press, & Teukolsky 1972), the degree of the distortions imposed by these shifts can be used to infer the spin of the black hole; efforts to exploit disk reflection as a spin diagnostic in X-ray binaries began in earnest over a decade ago. Owing to the fact that the effects on Fe K emission lines are especially pronounced features, and owing to the high flux levels observed in Galactic X-ray binaries, spin measurements have been made in a number of systems using this technique (e.g. Miller 2007; Miller et al. 2009).</text>
<text><location><page_1><loc_8><loc_30><loc_48><loc_33></location>In cases where the disk extends to the ISCO and the continuum is known to be fairly simple, not only can spin be in-</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_26><loc_48><loc_29></location>1 Department of Astronomy, The University of Michigan, 500 Church Street, Ann Arbor, MI 48109-1046, jonmm@umich.edu</list_item>
<list_item><location><page_1><loc_10><loc_24><loc_48><loc_26></location>2 Institute of Astronomy, The University of Cambridge, Madingley Road, Cambridge, CB3 OHA, UK</list_item>
<list_item><location><page_1><loc_10><loc_22><loc_48><loc_24></location>3 Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA, 91125</list_item>
<list_item><location><page_1><loc_11><loc_21><loc_39><loc_22></location>4 Universite de Toulouse, UPS-OMP, Touluse, France</list_item>
<list_item><location><page_1><loc_10><loc_19><loc_48><loc_21></location>5 CNRS, Institut de Recherche en Astrophyique et Planetologie, 9 Av. colonel Roche, BP 44346, F-31028, Toulouse cedex 4, France</list_item>
<list_item><location><page_1><loc_10><loc_16><loc_48><loc_19></location>6 Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720-7450</list_item>
<list_item><location><page_1><loc_10><loc_14><loc_48><loc_16></location>7 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 70 Vassar Street, Cambridge, MA 02139</list_item>
<list_item><location><page_1><loc_11><loc_13><loc_34><loc_14></location>8 Danish Technical University, Lyngby, DK</list_item>
<list_item><location><page_1><loc_11><loc_12><loc_11><loc_13></location>9</list_item>
<list_item><location><page_1><loc_12><loc_12><loc_42><loc_13></location>Lawrence Livermore National Laboratory, Livermore, CA</list_item>
<list_item><location><page_1><loc_11><loc_11><loc_38><loc_12></location>10 Columbia University, New York, NY 10027, USA</list_item>
<list_item><location><page_1><loc_10><loc_8><loc_48><loc_10></location>11 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109</list_item>
<list_item><location><page_1><loc_11><loc_7><loc_43><loc_8></location>12 NASA Goddard Space Flight Center, Greenbelt, MD 20771</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_45><loc_92><loc_56></location>ferred, the geometry of the corona can also be discerned. The best spectra and variability studies appear to point toward a very compact central corona ( r ≤ 10-20 GM / c 2 ; e.g. Reis & Miller 2013), consistent with prior results suggesting that hard X-ray emission may arise in the base of a relativistic jet (e.g. Fender et al. 1999; Markoff, Nowak, & Wilms 2005; Miller et al. 2012). However, this is not yet clear, and it also unclear that this geometry holds universally.</text>
<text><location><page_1><loc_52><loc_30><loc_92><loc_45></location>Extremely high sensitivity - especially over a broad spectral band - provides a path forward in situations where the continuum and reflection spectrum may be more difficult to parse. NuSTAR detectors have a triggered read-out; unlike CCD spectrometers, they are not subject to pile-up distortions (Harrison et al. 2013). In this respect, NuSTAR is especially well-suited to disk reflection studies of bright Galactic compact objects. Moreover, NuSTAR offers unprecedented sensitivity out to almost 80 keV, giving an excellent view of the Compton back-scattering hump (typically peaking in the 2030 keV), and any additional curvature or breaks.</text>
<text><location><page_1><loc_52><loc_12><loc_92><loc_31></location>GRS1915 + 105 is a particularly important source for understanding black hole spin, disk-jet connections in all accreting systems, and how accretion flows evolve with the mass accretion rate. Prior efforts to measure the spin of GRS 1915 + 105 have not come to a clear consensus. Moreover, a multiplicity of states are observed in GRS 1915 + 105 (Belloni et al. 2000). The most intriguing of these may be the so-called 'plateau' state, because it bears the closest analogy with the 'low/hard' state in other black hole transients. Notably, radio emission consistent with compact jet production and strong lowfrequency quasi-periodic oscillations (QPOs) are observed in this state (e.g. Muno et al. 2001); when combined with sensitive spectroscopy, these features may offer unique insights into the inner accretion flow.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_12></location>In Section 2, we describe the NuSTAR observation of GRS 1915 + 105 and our reduction of the data. Section 3 describes our analysis of the FPMA and FPMB specta. In Section 4, we</text>
<text><location><page_2><loc_8><loc_91><loc_44><loc_92></location>discuss the results of our spectral fits and their impacts.</text>
<section_header_level_1><location><page_2><loc_15><loc_88><loc_41><loc_90></location>2. OBSERVATIONS AND DATA REDUCTION</section_header_level_1>
<text><location><page_2><loc_8><loc_68><loc_48><loc_88></location>NuSTAR observed GRS 1915 + 105 on 03 July 2012, over a span of 59.8 ks. The data were screened and processed using NuSTARDAS version 1.1.1. Spectra from the FPMA and FPMB detectors were extracted from 90' regions centered on the source position. Background spectra were extracted from regions of equivalent size on each detector; however, the background is negligible. Response files appropriate for the pointing (on-axis), source type (point, not extended) and region size were automatically created by the NuSTARDAS software. After all efficiencies and screening, the net exposure time for the resultant spectra was 14.7 ks for the FPMA, and 15.2 ksec for the FPMB. The net observing time is small compared to the total observing due to the source flux, and in part because the observation occurred very early in the mission, and in part owing to detector dead-time.</text>
<text><location><page_2><loc_8><loc_59><loc_48><loc_68></location>The spectra were analyzed using XSPEC version 12.6 (Arnaud & Dorman 2000). The χ 2 statistic was used to assess the relative quality of different spectral models. We used 'Churazov' weighting for all fits to govern the influence of bins with progressively less signal at high energy (Churazov et al. 1996). All errors reported in this work reflect the 1 σ confidence interval on a given parameter.</text>
<section_header_level_1><location><page_2><loc_20><loc_57><loc_37><loc_58></location>3. ANALYSIS AND RESULTS</section_header_level_1>
<text><location><page_2><loc_8><loc_31><loc_48><loc_56></location>Examination of the Swift /BAT light curve of GRS 1915 + 105 shows that our observation was made at the start of an ∼ 100-day interval with sustained hard flux and only moderate variability. Intervals before and after have much stronger day-to-day variability. The lightcurve of our observation shows significant source variability on short time scales, typical of GRS 1915 + 105, as well as moderately strong QPOs between 0.5-3.0 Hz. A full timing analysis will be reported in a separate paper (Bachetti et al. 2013, in preparation), but the fact of these variability properties helps us to make a secure identification of the source state. These timing properties, as well as the source flux observed by the Swift /BAT, are typical of the 'plateau' state of GRS 1915 + 105 (e.g., Muno et al. 2001, Trudolyubov 2001, Fender & Belloni 2004). Observations with the RATAN-600 radio telescope found that GRS 1915 + 105 varied between 12 ± 3 and 6 ± 3 mJy at 4.8 GHz (Trushkin, private communication) during the NuSTAR observation, consistent with relatively radio-faint 'plateau' states.</text>
<text><location><page_2><loc_8><loc_16><loc_48><loc_31></location>Version 1.1.1 of the NuSTARDAS software and calibration has verified the detector response over the 3-79 keV band, partly through careful comparisons to the Crab. In all cases, the FPMA and FPMB spectra of GRS 1915 + 105 were jointly fit over the 3-79 keV band. An overall constant was allowed to float between the detectors to account for any mismatch in their absolute flux calibration; in all cases, the value of this constant was found to be 1.02 or less. In all fits, absorption in the ISM was fit using the 'tbabs' model (Wilms, Allen, & McCray 2000), using corresponding abundances ('wilm') and cross sections ('vern'; Verner et al. 1996).</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_16></location>Figure 1 shows the FPMA and FPMB spectra of GRS 1915 + 105, fit with a basic power-law model. The sensitivity of both spectra is excellent. Simple, broken, and cut-off power-law models all fail to fit the data. However, they approximate the continuum, and the prominence of the remaining disk reflection features in the spectra is readily discerned in the ratio plots in Figure 2. The power-law indices ob-</text>
<text><location><page_2><loc_52><loc_88><loc_92><loc_92></location>tained in these simple fits are broadly consistent with values measured in fits to Suzaku spectra of GRS 1915 + 105 in the 'plateau' state (Blum et al. 2009).</text>
<text><location><page_2><loc_52><loc_73><loc_92><loc_88></location>The 'comptt' model describes thermal Comptonization (Titarchuk 1994). It also leaves strong reflection-like residuals, and does not provide an acceptable fit. The 'nthcomp' model is essentially a more physical means of obtaining a cutoff power-law continuum by mixing thermal and non-thermal electron distributions (Zycki, Done, & Smith 1999). Importantly, 'nthcomp' is capable of accounting for curvature that might otherwise be mistaken for disk reflection. However, the data/model ratio and fit statistic in Figure 2 show that even 'nthcomp' is unable to account for the strong, broad Fe K line and the Compton back-scattering hump.</text>
<text><location><page_2><loc_52><loc_35><loc_92><loc_73></location>The 'eqpair' model describes Compton scattering in a sophisticated way, allowing mixtures of thermal and nonthermal electron distributions (Coppi 1999). 'Eqpair' also explicitly includes blurred disk reflection. However, the reflection spectrum (described via the 'pexriv' model, Magdziarz & Zdziarski 1995) is blurred with the 'rdblur' function (Fabian et al. 1989), which only describes the Schwarzschild metric and does not permit spin measurements. The internal reflection was coupled to an external 'diskline' model, which is the kernel of 'rdblur', in order to account for the emission line. In our fits, we fixed the cosine of the inclination angle to 0.3, the elemental abundances to solar values, the inner disk radius to minimum possible rin = 6 GM / c 2 , the outer radius to rout = 1000 GM / c 2 , the emissivity to the Euclidian value of q = 3 (recall that J ∝ r -q ) and the disk temperature to T = 10 6 K (the maximum allowed). The reflection fraction and disk ionization were allowed to vary. Numerous parameters control the hybrid thermal and non-thermal continuum. For simplicity, we fixed the disk blackbody temperature from which photons are up-scattered to kT = 0 . 2 keV, and varied the soft photon compactness ( lbb ), the ratio of the hard to soft compactness ( lh / ls ), the fraction of the power supplied to energetic particles that goes into accelerating non-thermal particles ( lnt / lh ), and the Thomson scattering depth ( τ ). The radius of the scattering region could not be constrained and was fixed at 1 . 5 × 10 6 cm. Default values were assumed for all other parameters. Fitting 'eqpair' in this way, a large improvement is achieved ( χ 2 /ν = 5529 / 3784 (see Figure 2).</text>
<text><location><page_2><loc_52><loc_8><loc_92><loc_35></location>Given these results, models focused on ionized disk reflection were next pursued. Our best-fit spectral model is constant × tbabs × ((kerrconv × reflionx_hc) + cutoffpl) (see Table 1, and Figures 3 and 4). 'Kerrconv' is a relativistic blurring function, based on ray-tracing simulations (Brenneman & Reynolds 2006). It includes inner and outer disk emissivity indices (following Wilkins & Fabian 2012, q 1 floated freely but q 2 ≥ 0 was required), an emissivity break radius, the black hole spin parameter, the inner disk inclination (bounded between 65 · < i < 80 · , based on jet studies by Fender et al. 1999), and inner and outer disk radii (in units of the ISCO radius; values of rin = 1 . 0 and rout = 400 were frozen in all fits). 'Reflionx_hc' is a new version of the well-known 'reflionx' model that describes reflection from an ionized accretion disk of constant density (Ross & Fabian 2005), assuming an incident power-law with a cut-off. These models capture important effects by solving the ionization balance within the disk, and scatter-broadening photoelectric absorption edges. That is, 'reflionx_hc' includes broadening due to scattering, and this effect is balanced against dynamical and gravitational broadening when the model is convolved with</text>
<text><location><page_3><loc_8><loc_82><loc_48><loc_92></location>'kerrconv'. The power-law index of the hard emission in the 'reflionx_hc' and 'cutoffpl' models was linked in our fits, as was the characteristic exponential cut-off energy. The abundance of Fe within 'reflionx_hc' was allowed to vary in the 1 . 0 ≤ AFe ≤ 2 . 0 range, and the ionization parameter was allowed to float freely ( ξ = L / nr 2 ). Flux normalizations for the 'reflionx_hc' and 'cutoffpl' models were also measured.</text>
<text><location><page_3><loc_8><loc_70><loc_48><loc_82></location>As shown in Table 1, the best blurred reflection model gives a fit statistic of χ 2 /ν = 4070 . 6 / 3785. This model returns a precise spin measurement: a = 0 . 98 ± 0 . 01. The quoted error is only the statistical error. Systematic errors are likely much larger, and related to the assumption that the opticallythick disk truncates at the ISCO (see, e.g., Shafee et al. 2008; Reynolds & Fabian 2008; Noble, Krolik, & Hawley 2010), and different methods and physics captured in different spectral models.</text>
<text><location><page_3><loc_8><loc_53><loc_48><loc_70></location>To obtain a broader view of the spin measurment and its uncertainty, we scanned the 0 ≤ a ≤ 0 . 998 range using the 'steppar' command in XSPEC. We made an initial scan with 100 points across the full band, and a second scan with 50 points in the 0 . 95 ≤ a ≤ 0 . 998 range. Figure 4 shows the results of this error scan. There is a clear minimum at a = 0 . 98; a maximal spin of a = 0 . 998 is rejected at very high confidence, and so too are low spin values. It is notable that the χ 2 versus a contour shows local fluctuations, especially between 0 . 8 ≤ a ≤ 0 . 95, although all χ 2 values are significantly higher than achieved for the best-fit spin of a = 0 . 98(1). The fluctuations may indicate deficiencies in the spectral model, or could be partly due to the limited energy resolution of the spectra.</text>
<text><location><page_3><loc_8><loc_33><loc_48><loc_53></location>We also explored a number of fits with key parameters fixed at particular values (see Table 1). The data clearly prefer a solar abundance of Fe, and the very steep inner emissivity index, for instance. The data strongly exclude a model with a much higher cut-off energy. As also indicated in Figure 4, the data rule out reflection from a black hole with zero spin at very high confidence (the model considered in Table 1 fixes a = 0 and the emissitivity indices at q = 3, appropriate for a 'lamp post' model in a Schwarzschild regime). Importantly, a plausible model for the low/hard state is also ruled out. The 'truncated' model in Table 1 changed the best-fit model to require an inner disk radius fixed at 20 times the ISCO, and q = 3. We also fit the 'eqpair' model again, fixing q = 10; this returned χ 2 /ν = 8059 / 3782, potentially indicating the importance of spin effects.</text>
<text><location><page_3><loc_8><loc_23><loc_48><loc_33></location>The best-fit model in Table 1 gives a flux of F = 2 . 07(1) × 10 -8 erg cm -2 s -1 (0.1-100 keV). Adopting the mass and distance values favored by Steeghs et al. (2013), MBH = 10 . 1 ± 0 . 6 M /circledot and d = 11 kpc, this flux gives a luminosity of L = 3 . 0(5) × 10 38 erg s -1 (where the error is based on an assumed distance uncertainty of ∆ d = ± 1 kpc), or an Eddington fraction of λ = 0 . 23 ± 0 . 04.</text>
<section_header_level_1><location><page_3><loc_24><loc_21><loc_33><loc_22></location>4. DISCUSSION</section_header_level_1>
<text><location><page_3><loc_8><loc_8><loc_48><loc_20></location>We have fit numerous models to an early broad-band NuSTAR spectrum of GRS 1915 + 105, obtained in a 'plateau' state. The sensitivity of the spectrum is extraordinary, in that the effects of continuum curvature and disk reflection can clealy be distinguished. Models that predict continuum curvature but which do not include reflection are unable to provide satisfactory fits. The data require a continuum with an exponential cut-off, and reflection from an ionized accretion disk around a black hole with a spin of a = 0 . 98(1).</text>
<text><location><page_3><loc_10><loc_7><loc_48><loc_8></location>Evidence of a relativistic disk line in GRS 1915 + 105 was</text>
<text><location><page_3><loc_52><loc_79><loc_92><loc_92></location>first detected with BeppoSAX (Martocchia et al. 2002). Fits to the line detected in archival ASCA spectra recorded a steep emissivity and small inner radius ( r = 1 . 8 rg ) commensurate with a spin approaching a /similarequal 0 . 9 (Miller et al. 2005; similar values were subsequently found by McClintock et al. 2006 and Middleton et al. 2006 using the disk continuum). Two observations with XMM-Newton also detected broad lines but were inconclusive with respect to spin (Martocchia et al. 2006), as was a deep spectrum of GRS 1915 + 105 in the 'plateau' obtained with Suzaku (Blum et al. 2009).</text>
<text><location><page_3><loc_52><loc_64><loc_92><loc_78></location>The measurement of a high spin parameter in a source known for jet production is interesting in that it may indicate that spin powers jet production, as predicted by e.g. Blandford & Znajek (1977). It is possible that the jet is powered partly by tapping the spin (Miller et al. 2009; Fender, Gallo, & Russell 2010; Narayan & McClintock 2012, Steiner, McClintock, & Narayan 2013; Russell, Fender, & Gallo 2013). However, the broadest survey of available data suggests that the mass accretion rate and/or magnetic field may act as a kind of 'throttle' (King et al. 2013a, 2013b) and do more to affect jet power.</text>
<text><location><page_3><loc_52><loc_43><loc_92><loc_64></location>The spectral fits presented in this paper also offer some potential insights into the geometry of the inner accretion flow, and into jet production. Compared to an Euclidean emissivity of q = 3, the inner emissivity index is extremely steep ( q /similarequal 10, see Table 1). This may ultimately be unphysical or incorrect; however, the same spin is obtained when q =5is fixed (see Table 1). Our results appear to broadly confirm the predictions of independent ray-tracing studies that find steep and broken emissivity profiles for compact, on-axis, hard X-ray sources emitting close to rapidly-spinning black holes (Wilkins & Fabian 2011, 2012; Dauser et al. 2013). The emissivity is also predicted to flatten at moderate radii, again consistent with our results. Given that GRS 1915 + 105 launches compact radio jets in the 'plateau' state (e.g. Muno et al. 2001; Trudolyubov 2001; Fender & Belloni 2004), the hard X-ray region may plausibly associated with the base of the jet.</text>
<text><location><page_3><loc_52><loc_28><loc_92><loc_43></location>A very steep inner emissivity profile was recently reported in fits to the Suzaku spectrum of Cygnus X-1 in the 'low/hard' state (Fabian et al. 2012). Joint Suzaku and radio monitoring of Cygnus X-1 in the 'low/hard' state also concluded that the hard X-ray continuum is likely produced in the base of the relativistic jet (Miller et al. 2012). More broadly, similar emissivity profiles have been seen in massive black holes acceting at relatively high Eddington fractions, notably 1H 0707 -495 (Fabian et al. 2009). Studies of time lags in Seyferts and microlensing in quasars suggest that very compact coronae may be common (Reis & Miller 2013).</text>
<text><location><page_3><loc_52><loc_11><loc_92><loc_28></location>Advection-dominated accretion flow models predict that the inner disk should be truncated at ˙ mEdd /similarequal 0 . 08, or λ /similarequal 0 . 008 (assuming an efficiency of 10%; Esin, McClintock, & Narayan 1997). This is broadly consistent with the luminosity at which many sources transition into the 'low/hard' state, wherein jet production is ubiquitous. Our results indicate that a steady jet can potentially be launched from a disk that extends to the ISCO. The disk, corona, and jet are undoubtedly a complex, coupled system, but jet production in black holes may be more closely tied to the nature of the corona than the inner disk radius. This may support a new model for jets and QPOs in accreting black holes (McKinney, Tchekhovskoy, & Blandford 2012).</text>
<text><location><page_3><loc_52><loc_7><loc_92><loc_11></location>This NuSTAR observation has offered new insights into nature of the accretion flow in the 'plateau' state, owing to its extraordinary sensitivity. Similarly, it has also provided the</text>
<text><location><page_4><loc_8><loc_83><loc_48><loc_92></location>first strong spin constraint based on disk reflection modeling. However, additional modeling using developing disk reflection codes, and a deeper observation in the 'plateau' state, are likely required in order to confirm these initial modeldependent results. A NuSTAR observation in a softer, more luminous state is likely also required in order to rigidly test and verify the spin measurement.</text>
<text><location><page_4><loc_52><loc_83><loc_92><loc_90></location>This work was supported under NASA Contract No. NNG08FD60C, and made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by NASA. JMM thanks Sergei Trushkin for communicating radio results.</text>
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</unordered_list>
<figure>
<location><page_5><loc_11><loc_62><loc_67><loc_93></location>
<caption>FIG. 1.- The 3-79 keV NuSTAR FPMA (black) and FPMB (red) spectra of GRS 1915 + 105, fit with a simple power-law assuming NH = 6 × 10 22 cm -2 . The 4.0-8.0 keV and 15.0-45.0 keV bands were ignored in order to portray the curvature in the spectrum. A strong, skewed Fe K line is visible in the 4-8 keV band. The curvature in the 20-30 keV band is due to a combination of a spectral cut-off and disk reflection.</caption>
</figure>
<figure>
<location><page_6><loc_6><loc_24><loc_65><loc_89></location>
<caption>FIG. 2.- The data/model ratios obtained when the 3-79 keV FPMA (black) and FPMB (red) spectra of GRS 1915 + 105 are jointly fit with common spectral models. In each panel, the name of the spectral model is given on the vertical axis. The key parameters derived from each spectral fit, including the χ 2 statistic, are given in each panel (in XSPEC parlance). In each case, 'K' is the flux normalization of the model. Note that even 'nthcomp' and 'eqpair' (sophisticated Comptonization models) fail to describe the spectra owing to the strong, blurred reflection features that are present.</caption>
</figure>
<section_header_level_1><location><page_7><loc_35><loc_89><loc_68><loc_90></location>RELATIVISTICALLY-BLURRED DISK REFLECTION MODELS</section_header_level_1>
<table>
<location><page_7><loc_8><loc_75><loc_95><loc_88></location>
<caption>TABLE 1</caption>
</table>
<text><location><page_7><loc_8><loc_72><loc_92><loc_75></location>NOTE. - The parameters obtained for the best-fit relativistically-blurred reflection model, tbabs ∗ kerrconv ∗ ( re f lionx _ hc + cuto f f pl ). The cut-off power-law normalization, Kpow , has units of photons cm -2 s -1 keV -1 at 1 keV. Please see the text for additional details. The table also lists the results obtained for various models wherein parameters were fixed in order to explore the sensitivity of the fit statistic to plausible variations. Errors were only calculated for the best-fit model; the reported errors are 1 σ confidence limits. Parameters marked with an asterisk denote those fixed at a particular trial value in the rejected models. In the 'truncation' model, the inner radius of the disk was fixed at 20 × r ISCO .</text>
<figure>
<location><page_8><loc_11><loc_61><loc_67><loc_94></location>
<caption>FIG. 3.- The FPMA (black) and FPMB (red) spectra of GRS 1915 + 105, fit with a relativisitically-blurred disk reflection model. The continuum and reflection model include an exponential cut-off, as indicated by the simple fits, and consistent with prior results obtained in the 'plateau' state. Using this model, a black hole spin parameter of a = 0 . 98(1) (statistical error only) is measured (see Table 1). The spectra were rebinned for visual clarity.</caption>
</figure>
<figure>
<location><page_9><loc_8><loc_62><loc_77><loc_90></location>
<caption>FIG. 4.- The ∆ χ 2 fitting statistic, plotted versus different values of the black hole spin parameter a = cJ / GM 2 . The panel at left shows the full range, while the panel at right shows the 0 . 9 ≤ a ≤ 1 . 0 range, for clarity. The spin measurement is based on relativistically-blurred disk reflection modeling of the NuSTAR spectrum of GRS 1915 + 105 in the 'plateau' state (see Table 1). The error range was scanned using the XSPEC tool 'steppar', which allows all parameters to vary during the scan. The horizontal confidence levels indicate the Gaussian equivalent σ value for the indicated change in χ 2 , assuming one interesting parameter.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "We report on the results of spectral fits made to a NuSTAR observation of the black hole GRS 1915 + 105 in a 'plateau' state. This state is of special interest because it is similar to the 'low/hard' state seen in other black holes, especially in that compact, steady jets are launched in this phase. The 3-79 keV bandpass of NuSTAR , and its ability to obtain moderate-resolution spectra free from distortions such as photon pile-up, are extremely well suited to studies of disk reflection in X-ray binaries. In only 15 ks of net exposure, an extraordinarily sensitive spectrum of GRS 1915 + 105 was measured across the full bandpass. Ionized reflection from a disk around a rapidly-spinning black hole is clearly required to fit the spectra; even hybrid Comptonization models including ionized reflection from a disk around a Schwarzschild black hole proved inadequate. A spin parameter of a = 0 . 98 ± 0 . 01 (1 σ statistical error) is measured via the best-fit model; low spins are ruled out at a high level of confidence. This result suggests that jets can be launched from a disk extending to the innermost stable circular orbit. A very steep inner disk emissivity profile is also measured, consistent with models of compact coronae above Kerr black holes. These results support an emerging association between the hard X-ray corona and the base of the relativistic jet. Subject headings: Black hole physics - relativity - stars: binaries - physical data and processes: accretion disks",
"pages": [
1
]
},
{
"title": "NUSTAR SPECTROSCOPY OF GRS 1915 + 105: DISK REFLECTION, SPIN, AND CONNECTIONS TO JETS",
"content": "J. M. MILLER 1 , M. L. PARKER 2 , F. FUERST 3 , M. BACHETTI 4,5 , F. A. HARRISON 3 , D. BARRET 5 , S. E. BOGGS 6 , D. CHAKRABARTY 7 , F. E. CHRISTENSEN 8 , W. W. CRAIG 9,10 , A. C. FABIAN 2 , B. W. GREFENSTETTE 3 , C. J. HAILEY 10 , A. L. KING 1 , D. K. STERN 11 , J. A. TOMSICK 6 , D. J. WALTON 3 , W. W. ZHANG 12 Submitted to ApJ",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Reflection of hard X-ray emission from a 'corona' onto the accretion disk can measure black hole spin, and can also serve as a powerful probe of the geometry of black hole accretion flows. Disk reflection spectra excited near to black holes will bear the imprints of gravitational red-shifts and strong Doppler shifts (e.g. Fabian et al. 1989). As long as the accretion disk extends to the innermost stable circular orbit (ISCO; Bardeen, Press, & Teukolsky 1972), the degree of the distortions imposed by these shifts can be used to infer the spin of the black hole; efforts to exploit disk reflection as a spin diagnostic in X-ray binaries began in earnest over a decade ago. Owing to the fact that the effects on Fe K emission lines are especially pronounced features, and owing to the high flux levels observed in Galactic X-ray binaries, spin measurements have been made in a number of systems using this technique (e.g. Miller 2007; Miller et al. 2009). In cases where the disk extends to the ISCO and the continuum is known to be fairly simple, not only can spin be in- ferred, the geometry of the corona can also be discerned. The best spectra and variability studies appear to point toward a very compact central corona ( r ≤ 10-20 GM / c 2 ; e.g. Reis & Miller 2013), consistent with prior results suggesting that hard X-ray emission may arise in the base of a relativistic jet (e.g. Fender et al. 1999; Markoff, Nowak, & Wilms 2005; Miller et al. 2012). However, this is not yet clear, and it also unclear that this geometry holds universally. Extremely high sensitivity - especially over a broad spectral band - provides a path forward in situations where the continuum and reflection spectrum may be more difficult to parse. NuSTAR detectors have a triggered read-out; unlike CCD spectrometers, they are not subject to pile-up distortions (Harrison et al. 2013). In this respect, NuSTAR is especially well-suited to disk reflection studies of bright Galactic compact objects. Moreover, NuSTAR offers unprecedented sensitivity out to almost 80 keV, giving an excellent view of the Compton back-scattering hump (typically peaking in the 2030 keV), and any additional curvature or breaks. GRS1915 + 105 is a particularly important source for understanding black hole spin, disk-jet connections in all accreting systems, and how accretion flows evolve with the mass accretion rate. Prior efforts to measure the spin of GRS 1915 + 105 have not come to a clear consensus. Moreover, a multiplicity of states are observed in GRS 1915 + 105 (Belloni et al. 2000). The most intriguing of these may be the so-called 'plateau' state, because it bears the closest analogy with the 'low/hard' state in other black hole transients. Notably, radio emission consistent with compact jet production and strong lowfrequency quasi-periodic oscillations (QPOs) are observed in this state (e.g. Muno et al. 2001); when combined with sensitive spectroscopy, these features may offer unique insights into the inner accretion flow. In Section 2, we describe the NuSTAR observation of GRS 1915 + 105 and our reduction of the data. Section 3 describes our analysis of the FPMA and FPMB specta. In Section 4, we discuss the results of our spectral fits and their impacts.",
"pages": [
1,
2
]
},
{
"title": "2. OBSERVATIONS AND DATA REDUCTION",
"content": "NuSTAR observed GRS 1915 + 105 on 03 July 2012, over a span of 59.8 ks. The data were screened and processed using NuSTARDAS version 1.1.1. Spectra from the FPMA and FPMB detectors were extracted from 90' regions centered on the source position. Background spectra were extracted from regions of equivalent size on each detector; however, the background is negligible. Response files appropriate for the pointing (on-axis), source type (point, not extended) and region size were automatically created by the NuSTARDAS software. After all efficiencies and screening, the net exposure time for the resultant spectra was 14.7 ks for the FPMA, and 15.2 ksec for the FPMB. The net observing time is small compared to the total observing due to the source flux, and in part because the observation occurred very early in the mission, and in part owing to detector dead-time. The spectra were analyzed using XSPEC version 12.6 (Arnaud & Dorman 2000). The χ 2 statistic was used to assess the relative quality of different spectral models. We used 'Churazov' weighting for all fits to govern the influence of bins with progressively less signal at high energy (Churazov et al. 1996). All errors reported in this work reflect the 1 σ confidence interval on a given parameter.",
"pages": [
2
]
},
{
"title": "3. ANALYSIS AND RESULTS",
"content": "Examination of the Swift /BAT light curve of GRS 1915 + 105 shows that our observation was made at the start of an ∼ 100-day interval with sustained hard flux and only moderate variability. Intervals before and after have much stronger day-to-day variability. The lightcurve of our observation shows significant source variability on short time scales, typical of GRS 1915 + 105, as well as moderately strong QPOs between 0.5-3.0 Hz. A full timing analysis will be reported in a separate paper (Bachetti et al. 2013, in preparation), but the fact of these variability properties helps us to make a secure identification of the source state. These timing properties, as well as the source flux observed by the Swift /BAT, are typical of the 'plateau' state of GRS 1915 + 105 (e.g., Muno et al. 2001, Trudolyubov 2001, Fender & Belloni 2004). Observations with the RATAN-600 radio telescope found that GRS 1915 + 105 varied between 12 ± 3 and 6 ± 3 mJy at 4.8 GHz (Trushkin, private communication) during the NuSTAR observation, consistent with relatively radio-faint 'plateau' states. Version 1.1.1 of the NuSTARDAS software and calibration has verified the detector response over the 3-79 keV band, partly through careful comparisons to the Crab. In all cases, the FPMA and FPMB spectra of GRS 1915 + 105 were jointly fit over the 3-79 keV band. An overall constant was allowed to float between the detectors to account for any mismatch in their absolute flux calibration; in all cases, the value of this constant was found to be 1.02 or less. In all fits, absorption in the ISM was fit using the 'tbabs' model (Wilms, Allen, & McCray 2000), using corresponding abundances ('wilm') and cross sections ('vern'; Verner et al. 1996). Figure 1 shows the FPMA and FPMB spectra of GRS 1915 + 105, fit with a basic power-law model. The sensitivity of both spectra is excellent. Simple, broken, and cut-off power-law models all fail to fit the data. However, they approximate the continuum, and the prominence of the remaining disk reflection features in the spectra is readily discerned in the ratio plots in Figure 2. The power-law indices ob- tained in these simple fits are broadly consistent with values measured in fits to Suzaku spectra of GRS 1915 + 105 in the 'plateau' state (Blum et al. 2009). The 'comptt' model describes thermal Comptonization (Titarchuk 1994). It also leaves strong reflection-like residuals, and does not provide an acceptable fit. The 'nthcomp' model is essentially a more physical means of obtaining a cutoff power-law continuum by mixing thermal and non-thermal electron distributions (Zycki, Done, & Smith 1999). Importantly, 'nthcomp' is capable of accounting for curvature that might otherwise be mistaken for disk reflection. However, the data/model ratio and fit statistic in Figure 2 show that even 'nthcomp' is unable to account for the strong, broad Fe K line and the Compton back-scattering hump. The 'eqpair' model describes Compton scattering in a sophisticated way, allowing mixtures of thermal and nonthermal electron distributions (Coppi 1999). 'Eqpair' also explicitly includes blurred disk reflection. However, the reflection spectrum (described via the 'pexriv' model, Magdziarz & Zdziarski 1995) is blurred with the 'rdblur' function (Fabian et al. 1989), which only describes the Schwarzschild metric and does not permit spin measurements. The internal reflection was coupled to an external 'diskline' model, which is the kernel of 'rdblur', in order to account for the emission line. In our fits, we fixed the cosine of the inclination angle to 0.3, the elemental abundances to solar values, the inner disk radius to minimum possible rin = 6 GM / c 2 , the outer radius to rout = 1000 GM / c 2 , the emissivity to the Euclidian value of q = 3 (recall that J ∝ r -q ) and the disk temperature to T = 10 6 K (the maximum allowed). The reflection fraction and disk ionization were allowed to vary. Numerous parameters control the hybrid thermal and non-thermal continuum. For simplicity, we fixed the disk blackbody temperature from which photons are up-scattered to kT = 0 . 2 keV, and varied the soft photon compactness ( lbb ), the ratio of the hard to soft compactness ( lh / ls ), the fraction of the power supplied to energetic particles that goes into accelerating non-thermal particles ( lnt / lh ), and the Thomson scattering depth ( τ ). The radius of the scattering region could not be constrained and was fixed at 1 . 5 × 10 6 cm. Default values were assumed for all other parameters. Fitting 'eqpair' in this way, a large improvement is achieved ( χ 2 /ν = 5529 / 3784 (see Figure 2). Given these results, models focused on ionized disk reflection were next pursued. Our best-fit spectral model is constant × tbabs × ((kerrconv × reflionx_hc) + cutoffpl) (see Table 1, and Figures 3 and 4). 'Kerrconv' is a relativistic blurring function, based on ray-tracing simulations (Brenneman & Reynolds 2006). It includes inner and outer disk emissivity indices (following Wilkins & Fabian 2012, q 1 floated freely but q 2 ≥ 0 was required), an emissivity break radius, the black hole spin parameter, the inner disk inclination (bounded between 65 · < i < 80 · , based on jet studies by Fender et al. 1999), and inner and outer disk radii (in units of the ISCO radius; values of rin = 1 . 0 and rout = 400 were frozen in all fits). 'Reflionx_hc' is a new version of the well-known 'reflionx' model that describes reflection from an ionized accretion disk of constant density (Ross & Fabian 2005), assuming an incident power-law with a cut-off. These models capture important effects by solving the ionization balance within the disk, and scatter-broadening photoelectric absorption edges. That is, 'reflionx_hc' includes broadening due to scattering, and this effect is balanced against dynamical and gravitational broadening when the model is convolved with 'kerrconv'. The power-law index of the hard emission in the 'reflionx_hc' and 'cutoffpl' models was linked in our fits, as was the characteristic exponential cut-off energy. The abundance of Fe within 'reflionx_hc' was allowed to vary in the 1 . 0 ≤ AFe ≤ 2 . 0 range, and the ionization parameter was allowed to float freely ( ξ = L / nr 2 ). Flux normalizations for the 'reflionx_hc' and 'cutoffpl' models were also measured. As shown in Table 1, the best blurred reflection model gives a fit statistic of χ 2 /ν = 4070 . 6 / 3785. This model returns a precise spin measurement: a = 0 . 98 ± 0 . 01. The quoted error is only the statistical error. Systematic errors are likely much larger, and related to the assumption that the opticallythick disk truncates at the ISCO (see, e.g., Shafee et al. 2008; Reynolds & Fabian 2008; Noble, Krolik, & Hawley 2010), and different methods and physics captured in different spectral models. To obtain a broader view of the spin measurment and its uncertainty, we scanned the 0 ≤ a ≤ 0 . 998 range using the 'steppar' command in XSPEC. We made an initial scan with 100 points across the full band, and a second scan with 50 points in the 0 . 95 ≤ a ≤ 0 . 998 range. Figure 4 shows the results of this error scan. There is a clear minimum at a = 0 . 98; a maximal spin of a = 0 . 998 is rejected at very high confidence, and so too are low spin values. It is notable that the χ 2 versus a contour shows local fluctuations, especially between 0 . 8 ≤ a ≤ 0 . 95, although all χ 2 values are significantly higher than achieved for the best-fit spin of a = 0 . 98(1). The fluctuations may indicate deficiencies in the spectral model, or could be partly due to the limited energy resolution of the spectra. We also explored a number of fits with key parameters fixed at particular values (see Table 1). The data clearly prefer a solar abundance of Fe, and the very steep inner emissivity index, for instance. The data strongly exclude a model with a much higher cut-off energy. As also indicated in Figure 4, the data rule out reflection from a black hole with zero spin at very high confidence (the model considered in Table 1 fixes a = 0 and the emissitivity indices at q = 3, appropriate for a 'lamp post' model in a Schwarzschild regime). Importantly, a plausible model for the low/hard state is also ruled out. The 'truncated' model in Table 1 changed the best-fit model to require an inner disk radius fixed at 20 times the ISCO, and q = 3. We also fit the 'eqpair' model again, fixing q = 10; this returned χ 2 /ν = 8059 / 3782, potentially indicating the importance of spin effects. The best-fit model in Table 1 gives a flux of F = 2 . 07(1) × 10 -8 erg cm -2 s -1 (0.1-100 keV). Adopting the mass and distance values favored by Steeghs et al. (2013), MBH = 10 . 1 ± 0 . 6 M /circledot and d = 11 kpc, this flux gives a luminosity of L = 3 . 0(5) × 10 38 erg s -1 (where the error is based on an assumed distance uncertainty of ∆ d = ± 1 kpc), or an Eddington fraction of λ = 0 . 23 ± 0 . 04.",
"pages": [
2,
3
]
},
{
"title": "4. DISCUSSION",
"content": "We have fit numerous models to an early broad-band NuSTAR spectrum of GRS 1915 + 105, obtained in a 'plateau' state. The sensitivity of the spectrum is extraordinary, in that the effects of continuum curvature and disk reflection can clealy be distinguished. Models that predict continuum curvature but which do not include reflection are unable to provide satisfactory fits. The data require a continuum with an exponential cut-off, and reflection from an ionized accretion disk around a black hole with a spin of a = 0 . 98(1). Evidence of a relativistic disk line in GRS 1915 + 105 was first detected with BeppoSAX (Martocchia et al. 2002). Fits to the line detected in archival ASCA spectra recorded a steep emissivity and small inner radius ( r = 1 . 8 rg ) commensurate with a spin approaching a /similarequal 0 . 9 (Miller et al. 2005; similar values were subsequently found by McClintock et al. 2006 and Middleton et al. 2006 using the disk continuum). Two observations with XMM-Newton also detected broad lines but were inconclusive with respect to spin (Martocchia et al. 2006), as was a deep spectrum of GRS 1915 + 105 in the 'plateau' obtained with Suzaku (Blum et al. 2009). The measurement of a high spin parameter in a source known for jet production is interesting in that it may indicate that spin powers jet production, as predicted by e.g. Blandford & Znajek (1977). It is possible that the jet is powered partly by tapping the spin (Miller et al. 2009; Fender, Gallo, & Russell 2010; Narayan & McClintock 2012, Steiner, McClintock, & Narayan 2013; Russell, Fender, & Gallo 2013). However, the broadest survey of available data suggests that the mass accretion rate and/or magnetic field may act as a kind of 'throttle' (King et al. 2013a, 2013b) and do more to affect jet power. The spectral fits presented in this paper also offer some potential insights into the geometry of the inner accretion flow, and into jet production. Compared to an Euclidean emissivity of q = 3, the inner emissivity index is extremely steep ( q /similarequal 10, see Table 1). This may ultimately be unphysical or incorrect; however, the same spin is obtained when q =5is fixed (see Table 1). Our results appear to broadly confirm the predictions of independent ray-tracing studies that find steep and broken emissivity profiles for compact, on-axis, hard X-ray sources emitting close to rapidly-spinning black holes (Wilkins & Fabian 2011, 2012; Dauser et al. 2013). The emissivity is also predicted to flatten at moderate radii, again consistent with our results. Given that GRS 1915 + 105 launches compact radio jets in the 'plateau' state (e.g. Muno et al. 2001; Trudolyubov 2001; Fender & Belloni 2004), the hard X-ray region may plausibly associated with the base of the jet. A very steep inner emissivity profile was recently reported in fits to the Suzaku spectrum of Cygnus X-1 in the 'low/hard' state (Fabian et al. 2012). Joint Suzaku and radio monitoring of Cygnus X-1 in the 'low/hard' state also concluded that the hard X-ray continuum is likely produced in the base of the relativistic jet (Miller et al. 2012). More broadly, similar emissivity profiles have been seen in massive black holes acceting at relatively high Eddington fractions, notably 1H 0707 -495 (Fabian et al. 2009). Studies of time lags in Seyferts and microlensing in quasars suggest that very compact coronae may be common (Reis & Miller 2013). Advection-dominated accretion flow models predict that the inner disk should be truncated at ˙ mEdd /similarequal 0 . 08, or λ /similarequal 0 . 008 (assuming an efficiency of 10%; Esin, McClintock, & Narayan 1997). This is broadly consistent with the luminosity at which many sources transition into the 'low/hard' state, wherein jet production is ubiquitous. Our results indicate that a steady jet can potentially be launched from a disk that extends to the ISCO. The disk, corona, and jet are undoubtedly a complex, coupled system, but jet production in black holes may be more closely tied to the nature of the corona than the inner disk radius. This may support a new model for jets and QPOs in accreting black holes (McKinney, Tchekhovskoy, & Blandford 2012). This NuSTAR observation has offered new insights into nature of the accretion flow in the 'plateau' state, owing to its extraordinary sensitivity. Similarly, it has also provided the first strong spin constraint based on disk reflection modeling. However, additional modeling using developing disk reflection codes, and a deeper observation in the 'plateau' state, are likely required in order to confirm these initial modeldependent results. A NuSTAR observation in a softer, more luminous state is likely also required in order to rigidly test and verify the spin measurement. This work was supported under NASA Contract No. NNG08FD60C, and made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by NASA. JMM thanks Sergei Trushkin for communicating radio results.",
"pages": [
3,
4
]
},
{
"title": "REFERENCES",
"content": "Arnaud, K. A., and Dorman, B., 2000, XSPEC is available via the HEASARC",
"pages": [
4
]
},
{
"title": "RELATIVISTICALLY-BLURRED DISK REFLECTION MODELS",
"content": "NOTE. - The parameters obtained for the best-fit relativistically-blurred reflection model, tbabs ∗ kerrconv ∗ ( re f lionx _ hc + cuto f f pl ). The cut-off power-law normalization, Kpow , has units of photons cm -2 s -1 keV -1 at 1 keV. Please see the text for additional details. The table also lists the results obtained for various models wherein parameters were fixed in order to explore the sensitivity of the fit statistic to plausible variations. Errors were only calculated for the best-fit model; the reported errors are 1 σ confidence limits. Parameters marked with an asterisk denote those fixed at a particular trial value in the rejected models. In the 'truncation' model, the inner radius of the disk was fixed at 20 × r ISCO .",
"pages": [
7
]
}
]
2013ApJ...775L..48D
https://arxiv.org/pdf/1309.2948.pdf
<document>
<section_header_level_1><location><page_1><loc_10><loc_85><loc_91><loc_88></location>DISCOVERY OF NINE INTERMEDIATE REDSHIFT COMPACT QUIESCENT GALAXIES IN THE SLOAN DIGITAL SKY SURVEY</section_header_level_1>
<text><location><page_1><loc_18><loc_83><loc_81><loc_85></location>Ivana Damjanov 1 , Igor Chilingarian 2,3 , Ho Seong Hwang 2 , and Margaret J. Geller 2</text>
<section_header_level_1><location><page_1><loc_45><loc_80><loc_55><loc_81></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_65><loc_86><loc_80></location>We identify nine galaxies with dynamical masses of M dyn /greaterorsimilar 10 10 M /circledot as photometric point sources, but with redshifts between z = 0 . 2 and z = 0 . 6, in the Sloan Digital Sky Survey (SDSS) spectrophotometric database. All nine galaxies have archival Hubble Space Telescope (HST) images. Surface brightness profile fitting confirms that all nine galaxies are extremely compact (0 . 4 < R e,c < 6 . 6 kpc with the median R e,c = 0 . 74 kpc) for their velocity dispersion (110 < σ < 340 km s -1 ; median σ = 178 km s -1 ). From the SDSS spectra, three systems are dominated by very young stars; the other six are older than ∼ 1 Gyr (two are E+A galaxies). The three young galaxies have disturbed morphologies and the older systems have smooth profiles consistent with a single S'ersic function. All nine lie below the z ∼ 0 velocity dispersion-half-light radius relation. The most massive system SDSSJ123657.44+631115.4 - lies right within the locus for massive compact z > 1 galaxies and the other eight objects follow the high-redshift dynamical size-mass relation.</text>
<text><location><page_1><loc_14><loc_62><loc_86><loc_64></location>Subject headings: galaxies: evolution - galaxies: fundamental parameters - galaxies: stellar content - galaxies: structure</text>
<section_header_level_1><location><page_1><loc_22><loc_58><loc_35><loc_59></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_34><loc_48><loc_58></location>The observed strong size evolution of massive quiescent galaxies is a fascinating challenge to our understanding of galaxy formation and evolution (e.g., Khochfar & Silk 2006; Fan et al. 2008; Nipoti et al. 2009; Naab et al. 2009; Hopkins et al. 2010; Oser et al. 2010; Ragone-Figueroa & Granato 2011; Shankar et al. 2012). Daddi et al. (2005) first discovered extremely compact passively evolving systems with half-light radii R e < 1 kpc at redshift z > 1 . 4. Further HST observations show that at a fixed stellar mass, galaxies at zero redshift are generally a factor of 3 -5 larger than their high redshift counterparts (e.g., Trujillo et al. 2007; Toft et al. 2007; Zirm et al. 2007; Buitrago et al. 2008; Cimatti et al. 2008; van Dokkum et al. 2008; Bezanson et al. 2009; Damjanov et al. 2009, 2011; Carrasco et al. 2010; Strazzullo et al. 2010; Saracco et al. 2011; Cassata et al. 2011; Szomoru et al. 2012; Bruce et al. 2012).</text>
<text><location><page_1><loc_8><loc_21><loc_48><loc_34></location>Velocity dispersions measured for small samples of quiescent systems at high redshift confirm that their dynamical masses agree well with the stellar masses derived from SED-fitting (see van de Sande et al. 2013, and the references therein). There are thus two coexisting populations of massive quiescent systems at z /greaterorsimilar 1: 1) very dense compact systems and 2) systems with sizes comparable with typical z ∼ 0 quiescent galaxies (Mancini et al. 2010; Saracco et al. 2011; Cassata et al. 2011; Newman et al. 2012; Onodera et al. 2012).</text>
<text><location><page_1><loc_8><loc_15><loc_48><loc_20></location>Compact massive systems seem to disappear by z ∼ 0, but there are conflicting observations. Trujillo et al. (2009) and Taylor et al. (2010) use the SDSS to suggest that the number density of compact massive sys-</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_11><loc_48><loc_14></location>1 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138</list_item>
<list_item><location><page_1><loc_10><loc_9><loc_48><loc_11></location>2 Smithsonian Astrophysical Observatory, 60 Garden St., Cambridge, MA 02138</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_9></location>3 Sternberg Astronomical Institute, Moscow State University, 13 Universitetsky prospect, 119992 Moscow Russia</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_42><loc_92><loc_59></location>tems at z < 0 . 2 is more than three orders of magnitude below the comoving density at z ∼ 2. In contrast, from ground-based imaging combined with spectroscopy, Valentinuzzi et al. (2010b) find a significant fraction of compact massive galaxies in the WINGS cluster sample at z ∼ 0 . 05; they derive a lower limit on the number density of n ∼ 1 . 3 × 10 -5 Mpc -3 , comparable with the number density of the high-redshift analogs. A similar study of the field population at 0 . 03 /lessorequalslant z /lessorequalslant 0 . 11 (Poggianti et al. 2013) suggests that compact dense galaxies exist in this redshift range but their fraction is three times smaller than in the WINGS cluster environment sample.</text>
<text><location><page_1><loc_52><loc_24><loc_92><loc_42></location>There are few observational tests of the existence of compact galaxies at redshifts 0 . 1 /lessorsimilar z /lessorsimilar 1 (Saglia et al. 2010; Carollo et al. 2013). Valentinuzzi et al. (2010a) identified compact systems with stellar masses M ∗ > 4 × 10 10 M /circledot among spectroscopically confirmed members of rich galaxy clusters with 0 . 5 < z < 0 . 8. Here we carry out an environment independent search for compact objects in the redshift range 0 . 2 < z < 0 . 6 by combining the photometric and spectroscopic SDSS databases with high-resolution images in the Mikulski Archive for Space Telescopes (MAST). Discovery of dense galaxies in this redshift range is important because larger samples with well-defined selection criteria potentially constrain models of galaxy evolution.</text>
<text><location><page_1><loc_52><loc_20><loc_92><loc_24></location>We adopt a Ω Λ = 0 . 73, Ω M = 0 . 27, and H 0 = 70 km s -1 Mpc -1 cosmology, and quote magnitudes in the AB system.</text>
<section_header_level_1><location><page_1><loc_54><loc_18><loc_90><loc_19></location>2. IDENTIFYING COMPACT GALAXY CANDIDATES</section_header_level_1>
<text><location><page_1><loc_52><loc_7><loc_92><loc_17></location>We use the SDSS (Release 7; SDSS DR 7) to initiate the search for candidate compact quiescent galaxies in the redshift range 0 . 2 < z < 0 . 6, where the SDSS main sample combined with the BOSS Survey contain spectra for large numbers of objects (Ahn et al. 2013). To identify compact systems we search for objects identified photometrically as stars (in the PhotoObj view) but with a redshift in our target range (from the SpecObj view).</text>
<text><location><page_2><loc_8><loc_84><loc_48><loc_92></location>Thus we obtain a list of object with sizes less than the SDSS PSF ( ∼ 1 . '' 5). We check that the photometric and spectroscopic objects actually have the same center and that the objects are visually compact in the SDSS images. Additionally we eliminate objects classified spectroscopically as quasars.</text>
<text><location><page_2><loc_8><loc_74><loc_48><loc_84></location>To restrict the list to quiescent galaxies, we require that the equivalent width of the [OII] λλ 3726 , 3729 emission line doublet is EW[O II] < 5 ˚ A. We check visually that each spectrum has a clear 4000 ˚ A break along with several absorption features (e.g. Balmer series, Ca H+K and G-band). The final list of SDSS DR7 compact system candidates at 0 . 2 < z < 0 . 6 includes 635 galaxies.</text>
<text><location><page_2><loc_8><loc_66><loc_48><loc_74></location>The SDSS photometric dataset provides only an upper limit to the angular size of these systems, corresponding to a physical radius between 2.5 kpc (at z = 0 . 2) and 5 kpc (at z = 0 . 6). To obtain direct size measurement we searched the HST archive. Nine of our 635 candidates have HST images.</text>
<text><location><page_2><loc_8><loc_60><loc_48><loc_66></location>Table 2 lists the camera and the filter for each HST observational program together with the corresponding pixel scale. The exquisite HST resolution of ∼ 0 . '' 15 allows analysis of the structure of these systems on a spatial scale of a few hundred pc ( /lessorsimilar 500 pc at z = 0 . 6).</text>
<section_header_level_1><location><page_2><loc_16><loc_57><loc_40><loc_58></location>3. SPECTROSCOPY AND IMAGING</section_header_level_1>
<section_header_level_1><location><page_2><loc_20><loc_55><loc_36><loc_57></location>3.1. The SDSS Spectra</section_header_level_1>
<text><location><page_2><loc_8><loc_40><loc_48><loc_55></location>We reanalyze the SDSS spectrum for each of our nine objects to obtain radial velocity, velocity dispersion, mean age and metallicity. We first fit SDSS DR7 spectra of galaxy candidates against a grid of pegase.hr (Le Borgne et al. 2004) simple stellar population (SSP) models based on the MILES stellar library (S'anchez-Bl'azquez et al. 2006) using the nbursts pixel space fitting technique (Chilingarian et al. 2007b,a). For every spectrum we first convolve the SSP model grid covering a wide range of ages and metallicities with the instrumental response of the SDSS spectrograph.</text>
<text><location><page_2><loc_8><loc_31><loc_48><loc_40></location>Next our minimization procedure convolves the SSP models again with a Gaussian line-of-sight velocity distribution, and multiplies the models by a smooth low-order continuum polynomial aimed at absorbing flux calibration errors in both models and the data. We choose the best-fitting SSP (or a linear combination of two) by interpolating a grid in age and metallicity.</text>
<text><location><page_2><loc_8><loc_20><loc_48><loc_31></location>We repeat the fitting procedure to the entire spectrum and to a spectrum with the regions of known emission lines blocked out. We analyze the full available spectral range (from the rest-frame 3640 ˚ Ato 6800 ˚ Aor the red end of the spectrum if it occurs at shorter rest-frame wavelength). If the reduced χ 2 value is significantly lower for the emission-line clipped case, the galaxy has significant emission lines in its spectrum.</text>
<text><location><page_2><loc_8><loc_13><loc_48><loc_20></location>In the sample of candidate galaxies, we identify two populations (Table 1). Six galaxies resemble classical elliptical galaxies with ages t /greaterorsimilar 1 Gyr, high metallicities (from slightly sub-solar to super-solar), and absence of ongoing star formation. Two of these objects are E+As.</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_13></location>Three blue galaxies have young stellar populations ( t < 50 Myr), high-velocity outflows, and sometimes residual star formation (Diamond-Stanic et al. 2012). In one case (SDSSJ112518.89-014532.4, Table 1 and Figure 1) we identify an 'extreme post-starburst galaxy'</text>
<text><location><page_2><loc_52><loc_81><loc_92><loc_92></location>which we apparently observe immediately after the cessation of a strong and short star formation episode: this object has no prominent emission lines. However, the mean stellar age is ∼ 30 Myr, close to the lowest limit covered by our SSP models. This object is considerably younger than typical E+A galaxies with luminosity weighted ages of 500-800 Myr (Chilingarian et al. 2009; Du et al. 2010).</text>
<section_header_level_1><location><page_2><loc_58><loc_78><loc_85><loc_80></location>3.2. HST Imaging Structural Analysis</section_header_level_1>
<text><location><page_2><loc_52><loc_67><loc_92><loc_78></location>We process dithered images of the nine galaxies in Table 1 using AstroDrizzle 4 . This image processing step combines individual exposures and rejects spurious pixels (cosmic rays and hot pixels) without changing the native pixel scale. In the special case of SDSSJ123657.44+631115.4, we process the WFC3 IR image to produce a science mosaic with a pixel scale corresponding to the scale of the ACS images.</text>
<text><location><page_2><loc_52><loc_54><loc_92><loc_67></location>We characterize the HST surface brightness profiles of our candidates by fitting a set of 2D R 1 n S'ersic profiles (GALFIT; Peng et al. 2010). Before fitting, we convolve the models with the Tiny Tim PSF (Krist et al. 2011). For many of our candidates there are not enough suitable stars to construct a high signal-to-noise PSF. For targets where a number of stars are available in the image, using the Tiny Tim PSF and the stellar PSF produce very similar results. For consistency, we use the Tiny Tim PSF for all images.</text>
<text><location><page_2><loc_52><loc_32><loc_92><loc_54></location>To fit the surface brightness profiles, we start with a single S'ersic profile and simultaneously fit all objects and the sky background in the ∼ 10 '' × 10 '' FoV around our target. With each fitting iteration we enhance the complexity of the model by adding one more S'ersic profile. We repeat fitting until the sky background estimates reach a plateau. Huang et al. (2013) show that this procedure is reliable for large enough regions. If the residual image does not show any prominent structure, we adopt the multiple S'ersic profile as the best fit. With the sky background fixed, we then fit a single S'ersic profile to extract structural parameters for direct comparison with the highz compact passive systems. The resolution at high redshift limits the fit to a single S'ersic profile. Table 2 lists the parameters of the best-fit multi-component and single-component models.</text>
<text><location><page_2><loc_52><loc_23><loc_92><loc_32></location>Figures 1 and 2 show the HST images, the best-fit 2D GALFIT models and residuals, the 1D observed and modeled profiles, and the SDSS spectra and fits for a representative young and old object, respectively. We have applied K plus evolutionary corrections to shift the surface brightness (density) µ profiles to the rest-frame at z = 0.</text>
<text><location><page_2><loc_52><loc_14><loc_92><loc_23></location>Young Objects: Three systems in this group have composite structure, visible only in the HST images. SDSSJ112518.89-014532.4 is a peculiar young object with only a weak [O II] line (section 3.1) observed with the HST/WFC3 program targeting recently quenched galaxies with high-velocity gas outflows (Proposal ID 12272: C.Tremonti).</text>
<text><location><page_2><loc_52><loc_10><loc_92><loc_14></location>The best-fit model surface brightness profile combines two functions: a compact and an extended S'ersic profile. The largest half-light radius exceeds the one for the</text>
<figure>
<location><page_3><loc_15><loc_24><loc_82><loc_82></location>
<caption>Fig. 1.A compact intermediate-redshift galaxy dominated by a young stellar population. We show the HST/WFC3 F814W image (top left), the 2D fit (top center), and the residual (top right). The white bars show relevant scales. The central panel shows the 1D observed surface profile (black), the best single (red) and double (green) S'ersic fits, and corresponding half-light radii. We also show the PSF (blue). The bottom panel shows the smoothed SDSS spectrum (black), the best-fit SSP model (red), and the regions excluded from the fit (blue). The bottom axis of this panel shows observed wavelength and the top axis gives galaxy rest-frame wavelength. Note the strong Balmer absorption and absence of prominent emission lines.</caption>
</figure>
<text><location><page_4><loc_8><loc_78><loc_48><loc_92></location>best-fit single S'ersic model by a factor of nine (R multi e,c = 6 . 54 kpc vs. R single e,c = 0 . 74 kpc, Table 2). The residual map in Fig. 1 shows spiral structure that we do not try to model. Based on the difference between the observed and model surface brightness profiles (Fig 1), at galactocentric distances R e /greaterorsimilar 3 kpc ≈ 4 . 5 × R single e,c the multi-profile model describes the observed light profile at low surface brightness levels considerably better than the single S'ersic model. The second component makes no significant difference in the core.</text>
<text><location><page_4><loc_8><loc_67><loc_48><loc_78></location>HST images of the other two young systems exhibit similarly complex rest-frame optical morphology. In the most extreme case (SDSSJ150603.69+613148.1) two objects appear in the SDSS images as a single point source. We attribute the SDSS spectrum of this double system, dominated by a young-age SSP, to the larger object containing most of the observed light with the best-fit surface brightness profile parameters in Table 2.</text>
<text><location><page_4><loc_8><loc_60><loc_48><loc_67></location>SDSSJ112518.89-014532.4 (Fig 1) has the highest velocity dispersion in this subsample ( σ (SDSSJ112518 . 89 -014532 . 4) > 200 km s -1 , Table 1) and its morphology is bulge-dominated. The light profiles for the other two young galaxies are disk-like.</text>
<text><location><page_4><loc_8><loc_50><loc_48><loc_60></location>Old Objects : These galaxies generally display much smoother surface brightness profiles. SDSSJ123657.44+631115.4 resides in one of the X-ray luminous clusters targeted by multi-wavelength HST ACS/WFC3 snapshot survey (Proposal ID 12166: H. Ebeling). The abundant HST imaging allows construction of the best-fit 2D models in three filters (F606W, F814W and F110W).</text>
<text><location><page_4><loc_8><loc_31><loc_48><loc_50></location>The light profiles in all three bands have very similar S'ersic parameters (R multi e,c ≈ 3 kpc, R single e,c ≈ 1 . 7 kpc). We obtain a good fit with only two S'ersic profiles (Table 2). The first row of Figure 2 shows an RGB image composed of two HST ACS images and one rescaled HST WFC3 image. We also plot the corresponding best-fit 2D models and residuals. Radial surface brightness profiles of the models in all three wavelength bands closely follow the observed profile out to 3 -4 R e,c . We note that modeling light profiles for galaxies like this target that are close to the brightest cluster galaxy (BCG) depends critically on the simultaneous modeling of the light profiles for all neighboring systems, including the two-component BCG (e.g., Gonzalez et al. 2005).</text>
<text><location><page_4><loc_8><loc_25><loc_48><loc_31></location>The HST morphology of the other five old systems have similar structure. Most of these systems have round shapes and spheroid-like single S'ersic profiles with n single S'ersic > 2 . 5.</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_25></location>Like their highz counterparts (e.g. Bruce et al. 2012; Buitrago et al. 2012), three of these objects have some disk component. One of them, SDSSJ001619.07003358.8, appears to be an old system with a faceon disk. Based on the best-fit multi-S'ersic model, the surface brightness profile of this system is a combination of two disk profiles. The observed profiles of two other galaxies, SDSSJ123130.98+123224.2 and SDSSJ132950.58+285254.8, display edge-on disk components. The visible light of SDSSJ123130.98+123224.2 is mostly distributed within a disk-like profile (its bulgeto-total ratio is B/T < 50%, based on the best twocomponent fit). SDSSJ132950.58+285254.8 is a bulgelike object with a weak extended disk ( B/T > 60%).</text>
<section_header_level_1><location><page_4><loc_54><loc_90><loc_91><loc_92></location>4. STRUCTURAL AND DYNAMICAL PROPERTIES OF COMPACT SYSTEMS AT INTERMEDIATE REDSHIFTS</section_header_level_1>
<text><location><page_4><loc_52><loc_84><loc_92><loc_89></location>We combine the parameters of the best-fit single S'ersic model for each galaxy (from Table 2) with measured velocity dispersions (from Table 1) to derive dynamical masses:</text>
<formula><location><page_4><loc_62><loc_79><loc_92><loc_83></location>M dyn = β ( n single S'ersic ) σ 2 R single e,c G , (1)</formula>
<text><location><page_4><loc_52><loc_76><loc_92><loc_79></location>where β ( n single S'ersic ) is a function of the S'ersic index (Cappellari et al. 2006):</text>
<formula><location><page_4><loc_54><loc_72><loc_92><loc_74></location>β ( n single S'ersic ) = 8 . 87 -0 . 831 n single S'ersic +0 . 024( n single S'ersic ) 2 . (2)</formula>
<text><location><page_4><loc_52><loc_42><loc_92><loc_71></location>We compare the relations between structural and dynamical properties of our 0 . 2 < z < 0 . 6 sample with the results obtained in two different redshift regimes: z /lessorsimilar 0 . 3 and 0 . 8 < z < 2 . 2. The structural parameters of single-S'ersic models for the low-redshift sample (Table 3; Simard et al. 2011) represent 2D decompositions of the g -and r -band surface brightness profiles for resolved systems in SDSS DR7. Again we require EW[O II] < 5 ˚ A (see section 2). For the z ∼ 0 sample retrieved from the SDSS DR7 database we also require 5 : i) an average signal-to-noise ratio per pixel of S/N > 10 and ii) a measured velocity dispersions in the range 70 km s -1 < z < 420 km s -1 . We include small samples of compact z ∼ 0 galaxies described in Trujillo et al. (2009) and Taylor et al. (2010). The high-redshift comparison sample is a collection of highresolution HST imaging and spectroscopic data compiled by van de Sande et al. (2013). All systems in this sample are quiescent galaxies with dynamical masses of M dyn > 2 × 10 10 M /circledot . We correct measured velocity dispersions for all three samples using the model provided in van de Sande et al. (2013).</text>
<text><location><page_4><loc_52><loc_24><loc_92><loc_42></location>The left-hand panel of Figure 3 clearly demonstrates the difference in size between galaxies with similar velocity dispersions at z ∼ 0 and at z > 0 . 2. For the main z ∼ 0 comparison sample (gray histogram), the median velocity dispersion is 185 km s -1 . For our sample (cyan; z ∼ 0 . 4) the median is 178 km s -1 and for z > 1 (red) the median is 260 km s -1 . The corresponding median sizes are 5.9 kpc, 0.74 kpc, and 2.2 kpc, respectively. The median size of z ∼ 0 sample selected to be compact (gray points) is 1.5 kpc. Although the sizes of our intermediate-redshift compact galaxies are several times smaller than for the z ∼ 0 systems, velocity dispersions of two samples span the same range of values.</text>
<text><location><page_4><loc_52><loc_16><loc_92><loc_24></location>Most of our sample lies in the lower portion of the velocity dispersion range covered by the z > 1 sample, but they follow the same trend in size-velocity dispersion parameter space. Furthermore, the intermediate-redshift object with the highest velocity dispersion (Fig. 2) falls very close to the locus of the high-redshift sample.</text>
<text><location><page_4><loc_52><loc_10><loc_92><loc_16></location>We note that multiple-profile models tend to overestimate the size of the young morphologically disturbed systems in order to fit their extended asymmetric lowsurface-brightness features. Thus the upper limits on the half-light radii derived from the best multi-S'ersic fits</text>
<figure>
<location><page_5><loc_16><loc_25><loc_82><loc_83></location>
<caption>Fig. 2.A compact massive intermediate-redshift galaxy dominated by an old stellar population. The RGB image (top left) is composed of HST/WFC3 F110W (red), HST/ACS F814W (blue), and HST/ACS F606W (green) light profiles. We also show the RGB image of the best-fit 2D multi-S'ersic profile models in these three filters (middle), and the three-color residual between the observed surface brightness profiles and the models (right). White bars show relevant scales. The central panel shows 1D radial surface brightness profiles (solid lines), the best-fit composite models (dashed lines), and the PSF (dashed-dotted lines). Green arrows denote the half-light radii. The bottom panel shows the smoothed SDSS spectrum (black), the best-fit SSP model (red), and the regions excluded from the fit (blue). The bottom axis of this panel shows observed wavelength and the top axis gives galaxy rest-frame wavelength.</caption>
</figure>
<text><location><page_6><loc_8><loc_88><loc_48><loc_92></location>can bring three young compact galaxies in our sample very close to the locus of low-redshift galaxies, but they may be misleading (see the 2D profile in Figure 1).</text>
<text><location><page_6><loc_8><loc_58><loc_48><loc_88></location>The right-hand panel of Figure 3 shows a tight sizedynamical mass relation in all three redshift regimes. This relation includes additional information about the S'ersic index of the best-fit profiles (equations 1 and 2). As noted by e.g. van de Sande et al. (2013), there is a clear offset between the loci of z ∼ 0 and z > 1 galaxies in the range of dynamical masses where the two samples overlap (2 × 10 10 M /circledot /lessorequalslant M dyn /lessorequalslant 1 . 86 × 10 12 M /circledot ). In contrast, the sample of compact z ∼ 0 galaxies overlaps with high-redshift systems. Although our intermediate-redshift galaxies have an average dynamical mass lower than high-redshift quiescent systems (7 . 95 × 10 9 M /circledot /lessorequalslant M dyn ( z ∼ 0 . 4) /lessorequalslant 6 . 31 × 10 11 M /circledot ), the two samples follow the same size-dynamical mass relation. The half-light radius of our most extreme target with σ > 300 km s -1 (Fig. 2) is very similar to or even smaller than the average size of similarly massive high-redshift systems. This result suggests that M dyn ≈ 10 10 M /circledot compact systems should also exist at z > 1, but with half-light radii of R e,c ≈ 0 . 5 kpc (or 0 . '' 05 at z = 1) and with currently undetectable extended low surface brightness features.</text>
<section_header_level_1><location><page_6><loc_22><loc_56><loc_34><loc_57></location>5. CONCLUSIONS</section_header_level_1>
<text><location><page_6><loc_8><loc_48><loc_48><loc_56></location>We identify nine galaxies with dynamical masses of M dyn /greaterorsimilar 10 10 M /circledot as photometric point sources, but with redshifts between 0 . 2 < z < 0 . 6 in the SDSS spectrophotometric database. These nine galaxies have archival HST images demonstrating that they are indeed extremely compact.</text>
<text><location><page_6><loc_8><loc_37><loc_48><loc_48></location>It is imperative to track the change in number density of compact systems with redshift, but no meaningful constraint can be derived from our inhomogeneous, serendipitous sample (see Tables 1 and 2). Our sample, however, demonstrates existence: larger samples of intermediate redshift compact quiescent galaxies based on well-defined selection criteria should provide number density estimates.</text>
<text><location><page_6><loc_8><loc_25><loc_48><loc_37></location>In size-dynamical-mass parameter space our nine compact galaxies lie away from the typical z ∼ 0 SDSS galaxies of similar mass. The most massive system in our sample - SDSSJ123657.44+631115.4 - lies right within the locus of massive compact z > 1 galaxies. The existence of these serendipitously discovered intermediate redshift compact galaxies provide clues to uncovering larger samples for determining the evolution of dense systems routinely observed at high redshift.</text>
<text><location><page_6><loc_8><loc_11><loc_48><loc_23></location>We acknowledge the use of the SDSS DR7 data ( http://www.sdss.org/dr7/ ) and the MAST HST database ( http://archive.stsci.edu/hst/ ). We thank the referee for prompt, helpful comments. ID is supported by the Harvard College Observatory Menzel Fellowship and NSERC (PDF-421224-2012). The Smithsonian Institution supports the research of IC, HSH, and MJG. IC acknowledges support from grant MD3288.2012.2.</text>
<figure>
<location><page_7><loc_11><loc_41><loc_89><loc_66></location>
<caption>Fig. 3.The size-velocity dispersion relation ( left) and the size-dynamical mass relation ( right) for quiescent galaxies in three redshift ranges: a) z /lessorsimilar 0 . 3 (grey 2D histogram), b) 0 . 2 < z < 0 . 6 (this study, stars), and c) 0 . 8 < z < 2 . 2 (triangles). The symbol color indicates galaxy redshifts according to the color bars. Two points connected by a solid line represent each target in our sample: the smaller circularized half-light radii R e,c (larger star) corresponds to the single-S'ersic profile and the larger R e,c (smaller star) denotes the upper limit from the largest R e,c in the multi-S'ersic profile. The red bar shows the average error for the high-redshift sample. The grey points denote the z ∼ 0 compact sample (Trujillo et al. 2009; Taylor et al. 2010).</caption>
</figure>
<section_header_level_1><location><page_8><loc_45><loc_91><loc_55><loc_92></location>REFERENCES</section_header_level_1>
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</unordered_list>
<table>
<location><page_9><loc_8><loc_60><loc_92><loc_88></location>
<caption>TABLE 1 Spectroscopic properties</caption>
</table>
<text><location><page_9><loc_8><loc_58><loc_92><loc_60></location>Note . - Columns: (1) SDSS designation; (2) Redshift; (3) Observed velocity dispersion; (4) Aperture correction for single-/multi-S'ersic profile (see Table 2); (5) Age of the best-fit SSP model; (6) Metallicity of the best-fit SSP model; (7) SDSS target flag; (8) Classification</text>
<table>
<location><page_9><loc_8><loc_33><loc_95><loc_51></location>
<caption>TABLE 2 Structural properties</caption>
</table>
<text><location><page_9><loc_8><loc_28><loc_92><loc_32></location>Note . - Columns: (1) SDSS designation (abridged); (2) Circularized half-light radius of the single-profile model (R single e,c = R single e × √ b/a , where R e is the major axis half-light radius and b/a is the axial ratio); (3) S'ersic index of the single-profile model; (4) Axial ratio of the single-profile model; (5) The largest (circularized) half-light radius of the composite model; (6) S'ersic index corresponding to (5); (7) Axial ratio corresponding to (5); (8) Number of profiles in the composite model; (9) HST program ID, camera and filter (corresponding pixel scale); (10) Morphology.</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We identify nine galaxies with dynamical masses of M dyn /greaterorsimilar 10 10 M /circledot as photometric point sources, but with redshifts between z = 0 . 2 and z = 0 . 6, in the Sloan Digital Sky Survey (SDSS) spectrophotometric database. All nine galaxies have archival Hubble Space Telescope (HST) images. Surface brightness profile fitting confirms that all nine galaxies are extremely compact (0 . 4 < R e,c < 6 . 6 kpc with the median R e,c = 0 . 74 kpc) for their velocity dispersion (110 < σ < 340 km s -1 ; median σ = 178 km s -1 ). From the SDSS spectra, three systems are dominated by very young stars; the other six are older than ∼ 1 Gyr (two are E+A galaxies). The three young galaxies have disturbed morphologies and the older systems have smooth profiles consistent with a single S'ersic function. All nine lie below the z ∼ 0 velocity dispersion-half-light radius relation. The most massive system SDSSJ123657.44+631115.4 - lies right within the locus for massive compact z > 1 galaxies and the other eight objects follow the high-redshift dynamical size-mass relation. Subject headings: galaxies: evolution - galaxies: fundamental parameters - galaxies: stellar content - galaxies: structure",
"pages": [
1
]
},
{
"title": "DISCOVERY OF NINE INTERMEDIATE REDSHIFT COMPACT QUIESCENT GALAXIES IN THE SLOAN DIGITAL SKY SURVEY",
"content": "Ivana Damjanov 1 , Igor Chilingarian 2,3 , Ho Seong Hwang 2 , and Margaret J. Geller 2",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The observed strong size evolution of massive quiescent galaxies is a fascinating challenge to our understanding of galaxy formation and evolution (e.g., Khochfar & Silk 2006; Fan et al. 2008; Nipoti et al. 2009; Naab et al. 2009; Hopkins et al. 2010; Oser et al. 2010; Ragone-Figueroa & Granato 2011; Shankar et al. 2012). Daddi et al. (2005) first discovered extremely compact passively evolving systems with half-light radii R e < 1 kpc at redshift z > 1 . 4. Further HST observations show that at a fixed stellar mass, galaxies at zero redshift are generally a factor of 3 -5 larger than their high redshift counterparts (e.g., Trujillo et al. 2007; Toft et al. 2007; Zirm et al. 2007; Buitrago et al. 2008; Cimatti et al. 2008; van Dokkum et al. 2008; Bezanson et al. 2009; Damjanov et al. 2009, 2011; Carrasco et al. 2010; Strazzullo et al. 2010; Saracco et al. 2011; Cassata et al. 2011; Szomoru et al. 2012; Bruce et al. 2012). Velocity dispersions measured for small samples of quiescent systems at high redshift confirm that their dynamical masses agree well with the stellar masses derived from SED-fitting (see van de Sande et al. 2013, and the references therein). There are thus two coexisting populations of massive quiescent systems at z /greaterorsimilar 1: 1) very dense compact systems and 2) systems with sizes comparable with typical z ∼ 0 quiescent galaxies (Mancini et al. 2010; Saracco et al. 2011; Cassata et al. 2011; Newman et al. 2012; Onodera et al. 2012). Compact massive systems seem to disappear by z ∼ 0, but there are conflicting observations. Trujillo et al. (2009) and Taylor et al. (2010) use the SDSS to suggest that the number density of compact massive sys- tems at z < 0 . 2 is more than three orders of magnitude below the comoving density at z ∼ 2. In contrast, from ground-based imaging combined with spectroscopy, Valentinuzzi et al. (2010b) find a significant fraction of compact massive galaxies in the WINGS cluster sample at z ∼ 0 . 05; they derive a lower limit on the number density of n ∼ 1 . 3 × 10 -5 Mpc -3 , comparable with the number density of the high-redshift analogs. A similar study of the field population at 0 . 03 /lessorequalslant z /lessorequalslant 0 . 11 (Poggianti et al. 2013) suggests that compact dense galaxies exist in this redshift range but their fraction is three times smaller than in the WINGS cluster environment sample. There are few observational tests of the existence of compact galaxies at redshifts 0 . 1 /lessorsimilar z /lessorsimilar 1 (Saglia et al. 2010; Carollo et al. 2013). Valentinuzzi et al. (2010a) identified compact systems with stellar masses M ∗ > 4 × 10 10 M /circledot among spectroscopically confirmed members of rich galaxy clusters with 0 . 5 < z < 0 . 8. Here we carry out an environment independent search for compact objects in the redshift range 0 . 2 < z < 0 . 6 by combining the photometric and spectroscopic SDSS databases with high-resolution images in the Mikulski Archive for Space Telescopes (MAST). Discovery of dense galaxies in this redshift range is important because larger samples with well-defined selection criteria potentially constrain models of galaxy evolution. We adopt a Ω Λ = 0 . 73, Ω M = 0 . 27, and H 0 = 70 km s -1 Mpc -1 cosmology, and quote magnitudes in the AB system.",
"pages": [
1
]
},
{
"title": "2. IDENTIFYING COMPACT GALAXY CANDIDATES",
"content": "We use the SDSS (Release 7; SDSS DR 7) to initiate the search for candidate compact quiescent galaxies in the redshift range 0 . 2 < z < 0 . 6, where the SDSS main sample combined with the BOSS Survey contain spectra for large numbers of objects (Ahn et al. 2013). To identify compact systems we search for objects identified photometrically as stars (in the PhotoObj view) but with a redshift in our target range (from the SpecObj view). Thus we obtain a list of object with sizes less than the SDSS PSF ( ∼ 1 . '' 5). We check that the photometric and spectroscopic objects actually have the same center and that the objects are visually compact in the SDSS images. Additionally we eliminate objects classified spectroscopically as quasars. To restrict the list to quiescent galaxies, we require that the equivalent width of the [OII] λλ 3726 , 3729 emission line doublet is EW[O II] < 5 ˚ A. We check visually that each spectrum has a clear 4000 ˚ A break along with several absorption features (e.g. Balmer series, Ca H+K and G-band). The final list of SDSS DR7 compact system candidates at 0 . 2 < z < 0 . 6 includes 635 galaxies. The SDSS photometric dataset provides only an upper limit to the angular size of these systems, corresponding to a physical radius between 2.5 kpc (at z = 0 . 2) and 5 kpc (at z = 0 . 6). To obtain direct size measurement we searched the HST archive. Nine of our 635 candidates have HST images. Table 2 lists the camera and the filter for each HST observational program together with the corresponding pixel scale. The exquisite HST resolution of ∼ 0 . '' 15 allows analysis of the structure of these systems on a spatial scale of a few hundred pc ( /lessorsimilar 500 pc at z = 0 . 6).",
"pages": [
1,
2
]
},
{
"title": "3.1. The SDSS Spectra",
"content": "We reanalyze the SDSS spectrum for each of our nine objects to obtain radial velocity, velocity dispersion, mean age and metallicity. We first fit SDSS DR7 spectra of galaxy candidates against a grid of pegase.hr (Le Borgne et al. 2004) simple stellar population (SSP) models based on the MILES stellar library (S'anchez-Bl'azquez et al. 2006) using the nbursts pixel space fitting technique (Chilingarian et al. 2007b,a). For every spectrum we first convolve the SSP model grid covering a wide range of ages and metallicities with the instrumental response of the SDSS spectrograph. Next our minimization procedure convolves the SSP models again with a Gaussian line-of-sight velocity distribution, and multiplies the models by a smooth low-order continuum polynomial aimed at absorbing flux calibration errors in both models and the data. We choose the best-fitting SSP (or a linear combination of two) by interpolating a grid in age and metallicity. We repeat the fitting procedure to the entire spectrum and to a spectrum with the regions of known emission lines blocked out. We analyze the full available spectral range (from the rest-frame 3640 ˚ Ato 6800 ˚ Aor the red end of the spectrum if it occurs at shorter rest-frame wavelength). If the reduced χ 2 value is significantly lower for the emission-line clipped case, the galaxy has significant emission lines in its spectrum. In the sample of candidate galaxies, we identify two populations (Table 1). Six galaxies resemble classical elliptical galaxies with ages t /greaterorsimilar 1 Gyr, high metallicities (from slightly sub-solar to super-solar), and absence of ongoing star formation. Two of these objects are E+As. Three blue galaxies have young stellar populations ( t < 50 Myr), high-velocity outflows, and sometimes residual star formation (Diamond-Stanic et al. 2012). In one case (SDSSJ112518.89-014532.4, Table 1 and Figure 1) we identify an 'extreme post-starburst galaxy' which we apparently observe immediately after the cessation of a strong and short star formation episode: this object has no prominent emission lines. However, the mean stellar age is ∼ 30 Myr, close to the lowest limit covered by our SSP models. This object is considerably younger than typical E+A galaxies with luminosity weighted ages of 500-800 Myr (Chilingarian et al. 2009; Du et al. 2010).",
"pages": [
2
]
},
{
"title": "3.2. HST Imaging Structural Analysis",
"content": "We process dithered images of the nine galaxies in Table 1 using AstroDrizzle 4 . This image processing step combines individual exposures and rejects spurious pixels (cosmic rays and hot pixels) without changing the native pixel scale. In the special case of SDSSJ123657.44+631115.4, we process the WFC3 IR image to produce a science mosaic with a pixel scale corresponding to the scale of the ACS images. We characterize the HST surface brightness profiles of our candidates by fitting a set of 2D R 1 n S'ersic profiles (GALFIT; Peng et al. 2010). Before fitting, we convolve the models with the Tiny Tim PSF (Krist et al. 2011). For many of our candidates there are not enough suitable stars to construct a high signal-to-noise PSF. For targets where a number of stars are available in the image, using the Tiny Tim PSF and the stellar PSF produce very similar results. For consistency, we use the Tiny Tim PSF for all images. To fit the surface brightness profiles, we start with a single S'ersic profile and simultaneously fit all objects and the sky background in the ∼ 10 '' × 10 '' FoV around our target. With each fitting iteration we enhance the complexity of the model by adding one more S'ersic profile. We repeat fitting until the sky background estimates reach a plateau. Huang et al. (2013) show that this procedure is reliable for large enough regions. If the residual image does not show any prominent structure, we adopt the multiple S'ersic profile as the best fit. With the sky background fixed, we then fit a single S'ersic profile to extract structural parameters for direct comparison with the highz compact passive systems. The resolution at high redshift limits the fit to a single S'ersic profile. Table 2 lists the parameters of the best-fit multi-component and single-component models. Figures 1 and 2 show the HST images, the best-fit 2D GALFIT models and residuals, the 1D observed and modeled profiles, and the SDSS spectra and fits for a representative young and old object, respectively. We have applied K plus evolutionary corrections to shift the surface brightness (density) µ profiles to the rest-frame at z = 0. Young Objects: Three systems in this group have composite structure, visible only in the HST images. SDSSJ112518.89-014532.4 is a peculiar young object with only a weak [O II] line (section 3.1) observed with the HST/WFC3 program targeting recently quenched galaxies with high-velocity gas outflows (Proposal ID 12272: C.Tremonti). The best-fit model surface brightness profile combines two functions: a compact and an extended S'ersic profile. The largest half-light radius exceeds the one for the best-fit single S'ersic model by a factor of nine (R multi e,c = 6 . 54 kpc vs. R single e,c = 0 . 74 kpc, Table 2). The residual map in Fig. 1 shows spiral structure that we do not try to model. Based on the difference between the observed and model surface brightness profiles (Fig 1), at galactocentric distances R e /greaterorsimilar 3 kpc ≈ 4 . 5 × R single e,c the multi-profile model describes the observed light profile at low surface brightness levels considerably better than the single S'ersic model. The second component makes no significant difference in the core. HST images of the other two young systems exhibit similarly complex rest-frame optical morphology. In the most extreme case (SDSSJ150603.69+613148.1) two objects appear in the SDSS images as a single point source. We attribute the SDSS spectrum of this double system, dominated by a young-age SSP, to the larger object containing most of the observed light with the best-fit surface brightness profile parameters in Table 2. SDSSJ112518.89-014532.4 (Fig 1) has the highest velocity dispersion in this subsample ( σ (SDSSJ112518 . 89 -014532 . 4) > 200 km s -1 , Table 1) and its morphology is bulge-dominated. The light profiles for the other two young galaxies are disk-like. Old Objects : These galaxies generally display much smoother surface brightness profiles. SDSSJ123657.44+631115.4 resides in one of the X-ray luminous clusters targeted by multi-wavelength HST ACS/WFC3 snapshot survey (Proposal ID 12166: H. Ebeling). The abundant HST imaging allows construction of the best-fit 2D models in three filters (F606W, F814W and F110W). The light profiles in all three bands have very similar S'ersic parameters (R multi e,c ≈ 3 kpc, R single e,c ≈ 1 . 7 kpc). We obtain a good fit with only two S'ersic profiles (Table 2). The first row of Figure 2 shows an RGB image composed of two HST ACS images and one rescaled HST WFC3 image. We also plot the corresponding best-fit 2D models and residuals. Radial surface brightness profiles of the models in all three wavelength bands closely follow the observed profile out to 3 -4 R e,c . We note that modeling light profiles for galaxies like this target that are close to the brightest cluster galaxy (BCG) depends critically on the simultaneous modeling of the light profiles for all neighboring systems, including the two-component BCG (e.g., Gonzalez et al. 2005). The HST morphology of the other five old systems have similar structure. Most of these systems have round shapes and spheroid-like single S'ersic profiles with n single S'ersic > 2 . 5. Like their highz counterparts (e.g. Bruce et al. 2012; Buitrago et al. 2012), three of these objects have some disk component. One of them, SDSSJ001619.07003358.8, appears to be an old system with a faceon disk. Based on the best-fit multi-S'ersic model, the surface brightness profile of this system is a combination of two disk profiles. The observed profiles of two other galaxies, SDSSJ123130.98+123224.2 and SDSSJ132950.58+285254.8, display edge-on disk components. The visible light of SDSSJ123130.98+123224.2 is mostly distributed within a disk-like profile (its bulgeto-total ratio is B/T < 50%, based on the best twocomponent fit). SDSSJ132950.58+285254.8 is a bulgelike object with a weak extended disk ( B/T > 60%).",
"pages": [
2,
4
]
},
{
"title": "4. STRUCTURAL AND DYNAMICAL PROPERTIES OF COMPACT SYSTEMS AT INTERMEDIATE REDSHIFTS",
"content": "We combine the parameters of the best-fit single S'ersic model for each galaxy (from Table 2) with measured velocity dispersions (from Table 1) to derive dynamical masses: where β ( n single S'ersic ) is a function of the S'ersic index (Cappellari et al. 2006): We compare the relations between structural and dynamical properties of our 0 . 2 < z < 0 . 6 sample with the results obtained in two different redshift regimes: z /lessorsimilar 0 . 3 and 0 . 8 < z < 2 . 2. The structural parameters of single-S'ersic models for the low-redshift sample (Table 3; Simard et al. 2011) represent 2D decompositions of the g -and r -band surface brightness profiles for resolved systems in SDSS DR7. Again we require EW[O II] < 5 ˚ A (see section 2). For the z ∼ 0 sample retrieved from the SDSS DR7 database we also require 5 : i) an average signal-to-noise ratio per pixel of S/N > 10 and ii) a measured velocity dispersions in the range 70 km s -1 < z < 420 km s -1 . We include small samples of compact z ∼ 0 galaxies described in Trujillo et al. (2009) and Taylor et al. (2010). The high-redshift comparison sample is a collection of highresolution HST imaging and spectroscopic data compiled by van de Sande et al. (2013). All systems in this sample are quiescent galaxies with dynamical masses of M dyn > 2 × 10 10 M /circledot . We correct measured velocity dispersions for all three samples using the model provided in van de Sande et al. (2013). The left-hand panel of Figure 3 clearly demonstrates the difference in size between galaxies with similar velocity dispersions at z ∼ 0 and at z > 0 . 2. For the main z ∼ 0 comparison sample (gray histogram), the median velocity dispersion is 185 km s -1 . For our sample (cyan; z ∼ 0 . 4) the median is 178 km s -1 and for z > 1 (red) the median is 260 km s -1 . The corresponding median sizes are 5.9 kpc, 0.74 kpc, and 2.2 kpc, respectively. The median size of z ∼ 0 sample selected to be compact (gray points) is 1.5 kpc. Although the sizes of our intermediate-redshift compact galaxies are several times smaller than for the z ∼ 0 systems, velocity dispersions of two samples span the same range of values. Most of our sample lies in the lower portion of the velocity dispersion range covered by the z > 1 sample, but they follow the same trend in size-velocity dispersion parameter space. Furthermore, the intermediate-redshift object with the highest velocity dispersion (Fig. 2) falls very close to the locus of the high-redshift sample. We note that multiple-profile models tend to overestimate the size of the young morphologically disturbed systems in order to fit their extended asymmetric lowsurface-brightness features. Thus the upper limits on the half-light radii derived from the best multi-S'ersic fits can bring three young compact galaxies in our sample very close to the locus of low-redshift galaxies, but they may be misleading (see the 2D profile in Figure 1). The right-hand panel of Figure 3 shows a tight sizedynamical mass relation in all three redshift regimes. This relation includes additional information about the S'ersic index of the best-fit profiles (equations 1 and 2). As noted by e.g. van de Sande et al. (2013), there is a clear offset between the loci of z ∼ 0 and z > 1 galaxies in the range of dynamical masses where the two samples overlap (2 × 10 10 M /circledot /lessorequalslant M dyn /lessorequalslant 1 . 86 × 10 12 M /circledot ). In contrast, the sample of compact z ∼ 0 galaxies overlaps with high-redshift systems. Although our intermediate-redshift galaxies have an average dynamical mass lower than high-redshift quiescent systems (7 . 95 × 10 9 M /circledot /lessorequalslant M dyn ( z ∼ 0 . 4) /lessorequalslant 6 . 31 × 10 11 M /circledot ), the two samples follow the same size-dynamical mass relation. The half-light radius of our most extreme target with σ > 300 km s -1 (Fig. 2) is very similar to or even smaller than the average size of similarly massive high-redshift systems. This result suggests that M dyn ≈ 10 10 M /circledot compact systems should also exist at z > 1, but with half-light radii of R e,c ≈ 0 . 5 kpc (or 0 . '' 05 at z = 1) and with currently undetectable extended low surface brightness features.",
"pages": [
4,
6
]
},
{
"title": "5. CONCLUSIONS",
"content": "We identify nine galaxies with dynamical masses of M dyn /greaterorsimilar 10 10 M /circledot as photometric point sources, but with redshifts between 0 . 2 < z < 0 . 6 in the SDSS spectrophotometric database. These nine galaxies have archival HST images demonstrating that they are indeed extremely compact. It is imperative to track the change in number density of compact systems with redshift, but no meaningful constraint can be derived from our inhomogeneous, serendipitous sample (see Tables 1 and 2). Our sample, however, demonstrates existence: larger samples of intermediate redshift compact quiescent galaxies based on well-defined selection criteria should provide number density estimates. In size-dynamical-mass parameter space our nine compact galaxies lie away from the typical z ∼ 0 SDSS galaxies of similar mass. The most massive system in our sample - SDSSJ123657.44+631115.4 - lies right within the locus of massive compact z > 1 galaxies. The existence of these serendipitously discovered intermediate redshift compact galaxies provide clues to uncovering larger samples for determining the evolution of dense systems routinely observed at high redshift. We acknowledge the use of the SDSS DR7 data ( http://www.sdss.org/dr7/ ) and the MAST HST database ( http://archive.stsci.edu/hst/ ). We thank the referee for prompt, helpful comments. ID is supported by the Harvard College Observatory Menzel Fellowship and NSERC (PDF-421224-2012). The Smithsonian Institution supports the research of IC, HSH, and MJG. IC acknowledges support from grant MD3288.2012.2.",
"pages": [
6
]
},
{
"title": "REFERENCES",
"content": "Note . - Columns: (1) SDSS designation; (2) Redshift; (3) Observed velocity dispersion; (4) Aperture correction for single-/multi-S'ersic profile (see Table 2); (5) Age of the best-fit SSP model; (6) Metallicity of the best-fit SSP model; (7) SDSS target flag; (8) Classification Note . - Columns: (1) SDSS designation (abridged); (2) Circularized half-light radius of the single-profile model (R single e,c = R single e × √ b/a , where R e is the major axis half-light radius and b/a is the axial ratio); (3) S'ersic index of the single-profile model; (4) Axial ratio of the single-profile model; (5) The largest (circularized) half-light radius of the composite model; (6) S'ersic index corresponding to (5); (7) Axial ratio corresponding to (5); (8) Number of profiles in the composite model; (9) HST program ID, camera and filter (corresponding pixel scale); (10) Morphology.",
"pages": [
9
]
}
]
2013ApJ...776...38F
https://arxiv.org/pdf/1308.4165.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_87><loc_91><loc_88></location>FAR-INFRARED FINE-STRUCTURE LINE DIAGNOSTICS OF ULTRALUMINOUS INFRARED GALAXIES</section_header_level_1>
<text><location><page_1><loc_9><loc_82><loc_91><loc_86></location>D. Farrah 1 , V. Lebouteiller 2,3 , H. W. W. Spoon 2 , J. Bernard-Salas 4 , C. Pearson 4,5 , D. Rigopoulou 6,5 , H. A. Smith 7 , E. Gonz'alez-Alfonso 8 , D. L. Clements 9 , A. Efstathiou 10 , D. Cormier 11 , J. Afonso 12,13 , S. M. Petty 1 , K. Harris 1 , P. Hurley 14 , C. Borys 15 , A. Verma 6 , A. Cooray 16 , and V. Salvatelli 17,16</text>
<section_header_level_1><location><page_1><loc_45><loc_79><loc_55><loc_80></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_56><loc_86><loc_79></location>We present Herschel observations of six fine-structure lines in 25 Ultraluminous Infrared Galaxies at z < 0 . 27. The lines, [O III ]52 µ m, [N III ]57 µ m, [O I ]63 µ m, [N II ]122 µ m, [O I ]145 µ m, and [C II ]158 µ m, are mostly single gaussians with widths < 600 km s -1 and luminosities of 10 7 -10 9 L /circledot . There are deficits in the [O I ]63/L IR , [N II ]/L IR , [O I ]145/L IR , and [C II ]/L IR ratios compared to lower luminosity systems. The majority of the line deficits are consistent with dustier H II regions, but part of the [C II ] deficit may arise from an additional mechanism, plausibly charged dust grains. This is consistent with some of the [C II ] originating from PDRs or the ISM. We derive relations between far-IR line luminosities and both IR luminosity and star formation rate. We find that [N II ] and both [O I ] lines are good tracers of IR luminosity and star formation rate. In contrast, [C II ] is a poor tracer of IR luminosity and star formation rate, and does not improve as a tracer of either quantity if the [C II ] deficit is accounted for. The continuum luminosity densities also correlate with IR luminosity and star formation rate. We derive ranges for the gas density and ultraviolet radiation intensity of 10 1 < n < 10 2 . 5 and 10 2 . 2 < G 0 < 10 3 . 6 , respectively. These ranges depend on optical type, the importance of star formation, and merger stage. We do not find relationships between far-IR line properties and several other parameters; AGN activity, merger stage, mid-IR excitation, and SMBH mass. We conclude that these far-IR lines arise from gas heated by starlight, and that they are not strongly influenced by AGN activity.</text>
<text><location><page_1><loc_14><loc_54><loc_82><loc_56></location>Subject headings: galaxies: starburst - infrared: galaxies - galaxies: evolution - galaxies: active</text>
<section_header_level_1><location><page_1><loc_22><loc_51><loc_35><loc_52></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_42><loc_48><loc_50></location>Ultraluminous Infrared Galaxies (ULIRGS, objects with L IR > 10 12 L /circledot , Sanders & Mirabel 1996; Lonsdale et al. 2006) are a cosmologically important population whose nature changes substantially with redshift. At z < 0 . 3 ULIRGs are rare (e.g. Soifer & Neugebauer 1991; Vaccari et al. 2010), with</text>
<unordered_list>
<list_item><location><page_1><loc_11><loc_39><loc_45><loc_40></location>1 Department of Physics, Virginia Tech, VA 24061, USA</list_item>
<list_item><location><page_1><loc_10><loc_37><loc_48><loc_39></location>2 Cornell University, Space Sciences Building, Ithaca, NY 14853, USA</list_item>
<list_item><location><page_1><loc_11><loc_36><loc_39><loc_37></location>3 CEA-Saclay, F-91191 Gif-sur-Yvette, France</list_item>
<list_item><location><page_1><loc_10><loc_33><loc_48><loc_36></location>4 Dept. of Physics & Astronomy, The Open University, MK7 6AA, UK.</list_item>
<list_item><location><page_1><loc_10><loc_31><loc_48><loc_34></location>5 Rutherford Appleton Laboratory, Harwell, Oxford, OX11 0QX, UK.</list_item>
<list_item><location><page_1><loc_10><loc_29><loc_48><loc_31></location>6 Denys Wilkinson Building, University of Oxford, Keble Rd, Oxford OX1 3RH, UK</list_item>
<list_item><location><page_1><loc_10><loc_27><loc_48><loc_29></location>7 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA</list_item>
<list_item><location><page_1><loc_10><loc_25><loc_48><loc_27></location>8 Universidad de Alcal'a, Departamento de F'ısica y Matem'aticas, Campus Universitario, Madrid, Spain</list_item>
<list_item><location><page_1><loc_10><loc_23><loc_48><loc_25></location>9 Physics Department, Imperial College London, London, SW7 2AZ, UK</list_item>
<list_item><location><page_1><loc_10><loc_21><loc_48><loc_23></location>10 School of Sciences, European University Cyprus, Diogenes Street, Engomi, 1516 Nicosia, Cyprus</list_item>
<list_item><location><page_1><loc_10><loc_19><loc_48><loc_21></location>11 Institut fur theoretische Astrophysik, Zentrum fur Astronomie der Universitat Heidelberg, Heidelberg, Germany</list_item>
<list_item><location><page_1><loc_10><loc_15><loc_48><loc_19></location>12 Centro de Astronomia e Astrof'ısica da Universidade de Lisboa, Observat'orio Astron'omico de Lisboa, Lisbon, Portugal 13 Department of Physics, University of Lisbon, 1749-016 Lisbon, Portugal</list_item>
<list_item><location><page_1><loc_11><loc_14><loc_40><loc_15></location>14 University of Sussex, Brighton BN1 9QH, UK</list_item>
<list_item><location><page_1><loc_10><loc_11><loc_48><loc_14></location>15 Infrared Processing and Analysis Center, MS220-6, California Institute of Technology, Pasadena, CA 91125, USA</list_item>
<list_item><location><page_1><loc_10><loc_9><loc_48><loc_12></location>16 Department of Physics & Astronomy, University of California, Irvine,CA 92697, USA</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_9></location>17 Physics Department and INFN, Universit'a di Roma 'La Sapienza', Ple Aldo Moro 2, 00185, Rome, Italy</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_20><loc_92><loc_52></location>less than one per ∼ hundred square degrees. They are invariably mergers between approximately equal mass galaxies (Clements et al. 1996; Surace et al. 2000; Cui et al. 2001; Farrah et al. 2001; Bushouse et al. 2002; Veilleux et al. 2002, 2006). Evidence suggests that their IR emission arises mainly from high rates of star formation (Genzel et al 1998; Tran et al. 2001; Franceschini et al. 2003; Nardini et al. 2010; Wang et al. 2011), though of order half also contain a luminous AGN (Rigopoulou et al. 1999; Farrah et al. 2003; Imanishi et al. 2007; Vega et al. 2008; Nardini & Risaliti 2011). The AGN in ULIRGs may become more important with increasing IR luminosity, and advancing merger stage (Teng & Veilleux 2010; Yuan et al. 2010; Stierwalt et al. 2013), and sometimes initiate powerful outflows (Spoon et al. 2009; Fischer et al. 2010; Rupke & Veilleux 2011; Sturm et al. 2011; Westmoquette et al. 2012; Rodr'ıguez Zaur'ın et al. 2013). A small fraction of (low-redshift) ULIRGs become optical QSOs (Tacconi et al. 2002; Kawakatu et al. 2006, 2007; Farrah et al. 2007b; Meng et al. 2010; Hou et al. 2011) and a large fraction end up as early-type galaxies (Genzel et al. 2001; Dasyra et al. 2006; Rothberg et al. 2013; Wang et al. 2013).</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_20></location>Over 0 . 3 /lessorsimilar z < 1 the number of ULIRGs rises rapidly (e.g. Le Floc'h et al. 2005), reaching a density on the sky of several hundred per square degree at z /greaterorsimilar 1 (Rowan-Robinson et al. 1997; Dole et al. 2001; Borys et al. 2003; Mortier et al. 2005; Austermann et al. 2010; Goto et al. 2011). The fraction of z /greaterorsimilar 1 ULIRGs that are starburst dominated mergers is high (Farrah et al. 2002; Chapman et al. 2003; Smail et al. 2004; Takata et al. 2006; Borys et al. 2006; Valiante et al. 2007; Berta et al. 2007; Bridge et al.</text>
<table>
<location><page_2><loc_22><loc_57><loc_78><loc_88></location>
<caption>Table 1 The Sample.</caption>
</table>
<text><location><page_2><loc_22><loc_44><loc_78><loc_46></location>c SMBH mass, in units of 10 8 M /circledot (Zheng et al. 2002; Dasyra et al. 2006; Greene & Ho 2007; Zhang et al. 2008; Veilleux et al. 2009).</text>
<text><location><page_2><loc_8><loc_26><loc_48><loc_43></location>2007; Lonsdale et al. 2009; Huang et al. 2009; Magnelli et al. 2012; Lo Faro et al. 2013; Johnson et al. 2013) but the merger fraction may be lower than locally (Melbourne et al. 2008; Kartaltepe et al. 2010; Draper & Ballantyne 2012, but see also Xu et al. 2012). High redshift ULIRGs may also have a wider range in dust temperature (Magdis et al. 2010; Rowan-Robinson et al. 2010; Symeonidis et al. 2011, 2013; Bridge et al. 2013) and SED shapes (Farrah et al. 2008; Sajina et al. 2012; Nordon et al. 2012), and a greater star formation efficiency (Iglesias-P'aramo et al. 2007; Combes et al. 2011, 2013; Hanami et al. 2012; Geach et al. 2013) compared to local examples.</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_25></location>Determining why the number and properties of ULIRGs change so markedly with redshift may provide insight into the history of stellar and SMBH mass assembly in /greaterorsimilar L ∗ galaxies. ULIRGs at z < 0 . 3 are central to this endeavour, as they establish a baseline from which to measure evolution with redshift in the ULIRG population. The far-infrared ( /similarequal 50 -500 µ m) is a powerful tool for studying ULIRGs, as demonstrated by the Infrared Space Observatory (ISO, e.g. Fischer et al. 1999; Negishi et al. 2001; Luhman et al. 2003; Spinoglio et al. 2005; Brauher et al. 2008). The Herschel Space Observatory (Pilbratt et al. 2010) offers dramatic advances in far-infrared observing capability over ISO. Its instruments, the Photodetector Array</text>
<text><location><page_2><loc_52><loc_33><loc_92><loc_43></location>Camera and Spectrometer (PACS, Poglitsch et al. 2010), Spectral and Photometric Imaging REceiver (SPIRE, Griffin et al. 2010) and Heterodyne Instrument for the Far Infrared (de Graauw et al. 2010) can observe wavelength ranges that are inaccessible from the ground, and have improved sensitivity and resolution over previous space-based facilities.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_33></location>We have used Herschel to conduct the Herschel ULIRG Survey (HERUS), which assembles PACS and SPIRE observations of nearly all ULIRGs with a 60 µ mflux greater than ∼ 1 . 7Jy. In this paper we present observations of fine-structure lines for 24 of the sample. Analysis of the SPIRE FTS spectra is presented in Pearson et al 2013, in preparation. Observations of the OH 119 µ m and 79 µ m profiles are presented in Spoon et al. 2013, while modelling of these profiles is presented in Smith et al, in preparation. Finally, a detailed study of the ULIRG IRAS 08572+3915 is presented in Efstathiou et al. 2013. We define infrared luminosity, L IR , to be the luminosity integrated over 8-1000 µ m in the rest frame. We quote luminosities and masses in units of Solar (L /circledot = 3 . 839 × 10 26 Watts, M /circledot = 1 . 99 × 10 30 Kg, respectively). We assume a spatially flat cosmology with H 0 = 67 . 3km s -1 Mpc -1 , Ω = 1 and Ω m = 0 . 315 (Planck Collaboration et al. 2013).</text>
<table>
<location><page_3><loc_27><loc_75><loc_73><loc_88></location>
<caption>Table 2 Properties of the lines observed</caption>
</table>
<text><location><page_3><loc_28><loc_74><loc_72><loc_75></location>Note . - Electron and Hydrogen critical densities are given for n=500cm -3</text>
<section_header_level_1><location><page_3><loc_21><loc_72><loc_36><loc_73></location>2.1. Sample Selection</section_header_level_1>
<text><location><page_3><loc_8><loc_45><loc_48><loc_71></location>HERUS is a photometric and spectroscopic atlas of the z < 0 . 27 ULIRG population. The sample comprises all 40 ULIRGs from the IRAS PSC-z survey (Saunders et al. 2000) with 60 µ m fluxes greater than 2Jy, together with three randomly selected ULIRGs with lower 60 µ m fluxes; IRAS 00397-1312 (1.8 Jy), IRAS 07598+6508 (1.7 Jy) and IRAS 13451+1232 (1.9 Jy). All objects have been observed with the Infrared Spectrograph (IRS, Houck et al. 2004) onboard Spitzer (Armus et al. 2007; Farrah et al. 2007a; Desai et al. 2007). The SHINING survey (Fischer et al. 2010; Sturm et al. 2011; Hailey-Dunsheath et al. 2012; Gonz'alez-Alfonso et al. 2013) obtained PACS spectroscopy for 19/43 sources, so we observed, and present here, the remaining 24 objects. We also include Mrk 231 (Fischer et al. 2010) to give a final sample of 25 objects (Table 1). This sample is not flux limited, but does include nearly all ULIRGs at z < 0 . 27 with 60 µ m fluxes between 1.7 Jy and 6 Jy, together with Mrk 231. The sample therefore gives an almost unbiased view of z < 0 . 3 ULIRGs.</text>
<section_header_level_1><location><page_3><loc_22><loc_42><loc_34><loc_43></location>2.2. Observations</section_header_level_1>
<text><location><page_3><loc_8><loc_23><loc_48><loc_41></location>The PACS observations were performed between March 18, 2011 and April 8, 2012 (Operational Day 6731060). The PACS integral-field spectrometer samples the spatial direction with 25 pixels and the spectral direction with 16 pixels. Each spectral pixel scans a distinct wavelength range by varying the grating angle. The combination of the 16 ranges makes the final spectrum. The resulting projection of the PACS array on the sky is a footprint of 5 × 5 spatial pixels (spaxels), corresponding to a 47 '' × 47 '' field-of-view. The point spread function full width at half maximum (FWHM) is ≈ 9 . 5 '' between 55 µ m and 110 µ m, and increases to about 14 '' by 200 µ m. A spaxel at the mean redshift of the sample is ∼ 3 kpc in extent.</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_23></location>A single footprint observation was performed for each object as they are all smaller than the footprint size. The coordinates were chosen to place the optical centroids in the central spaxel. We observed the sample in the following lines: [O III ]52 µ m, [N III ]57 µ m, [O I ]63 µ m, [N II ]122 µ m, [O I ]145 µ m, and [C II ]158 µ m (Table 2). Observations were done in range spectroscopy mode. We used optical narrow-line redshifts to set the central wavelengths of each range scan. For one object, IRAS07598+6508, the input coordinates were incorrect, placing the source near the edge of the PACS array, thus making the flux determination uncertain. We therefore</text>
<text><location><page_3><loc_52><loc_68><loc_92><loc_73></location>substituted observations of this source from other programs. For [C II ], we used the dataset 1342243534 (PI Weedman), and for [N II ] we used the dataset 1342231959 (PI Veilleux).</text>
<text><location><page_3><loc_52><loc_54><loc_92><loc_68></location>We set the wavelength range of each range scan to accomodate uncertainties such as offsets between optical and far-IR line redshifts, and asymmetric or broadened lines. The chop/nod observation mode was used, in which the source is observed by alternating between the on-source position and a clean off-source position. Since the extent of the targets is always < 1 ' , the smallest throw ( ± 1 . 5 ' ) was used to reduce the effect of field-rotation between the two chop positions. Two nod positions were used in order to eliminate the telescope background emission.</text>
<text><location><page_3><loc_52><loc_45><loc_92><loc_54></location>The data reduction was performed in the Herschel Interactive Processing Environment (HIPE) version 8.0 (Ott 2010) using the default chop/nod pipeline script. The level one products (calibrated in flux and in wavelength, with bad pixel masks from HIPE) were then exported, and processed by the PACSman tool (Lebouteiller et al. 2012).</text>
<section_header_level_1><location><page_3><loc_63><loc_43><loc_80><loc_44></location>2.3. Line Measurements</section_header_level_1>
<text><location><page_3><loc_52><loc_28><loc_92><loc_42></location>The best method to determine a line flux depends on the position and morphology of the line-emitting regions within the PACS footprint. If their combined spatial extent is significantly smaller than a single spaxel, are well centered on the central spaxel, and Herschel maintains accurate pointing (the pointing accuracy of Herschel is ∼ 2 . 5 '' ) then the best flux measurement is that of the central spaxel, scaled by an appropriate point-source correction. We call this method ' M 1'. If these conditions are not satisfied then M 1 will give a lower limit on the flux.</text>
<text><location><page_3><loc_52><loc_12><loc_92><loc_28></location>We do not know, a priori , the morphologies of the farIR line emitting regions, since high spatial resolution images of this emission do not exist. Moreover, we cannot assume that the morphologies of these regions are traced reliably by emission at other wavelengths. We therefore are unable to straightforwardly distinguish between scenarios such as the source being centered and spatially extended, against the source being off-center and pointlike. Finally, we cannot assume that the morphologies of different lines in the same object are similar, since the line strengths are governed by different excitation temperatures and critical densities.</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_12></location>Given these caveats, there are five further methods to determine a line flux:</text>
<unordered_list>
<list_item><location><page_3><loc_54><loc_7><loc_92><loc_8></location>· M 2 - fit line profiles to the central 3 × 3 spaxels</list_item>
</unordered_list>
<figure>
<location><page_4><loc_27><loc_46><loc_74><loc_91></location>
<caption>Figure 1. The one-dimensional continuum-subtracted spectra for the first eight objects in Table 1. Each row shows the spectra for one object, while each column shows the spectra for one line. The vertical red line is the (optical) redshift. If present, a red number shows the factor by which that spectrum has been scaled to fit within the y axis range. We plot the rebinned spectra rather than the full data cloud, and show data only from the central spaxel.</caption>
</figure>
<text><location><page_4><loc_12><loc_23><loc_48><loc_38></location>individually, sum the resulting fluxes, and apply a point-source correction (of order 15% or less of the total flux) that accounts for the additional areal sampling of the PSF. This method is suitable if the source is spatially extended or shifted by at most a significant fraction of a spaxel. The point-source correction is wavelength-dependent, but since our range scans span wavelength ranges of order 1 µ m it is equivalent to either calculate the flux and then apply the correction, or apply the correction to the spectrum and then calculate the flux.</text>
<unordered_list>
<list_item><location><page_4><loc_11><loc_13><loc_48><loc_22></location>· M 3 - Co-add the spectra of the central 3 × 3 spaxels, apply a point-source correction, and fit a line profile to the combined profile. This method is identical to M 2, except for additional uncertainties from combining spectra with different shaped continua. We mention this method for completeness but do not use it.</list_item>
<list_item><location><page_4><loc_11><loc_7><loc_48><loc_12></location>· M 4 - fit line profiles to those spaxels with a line detection, then sum the resulting fluxes. This is appropriate if the source is extended in any fashion, but suffers from uncertainties due to imperfect</list_item>
</unordered_list>
<text><location><page_4><loc_56><loc_37><loc_82><loc_38></location>knowledge of the source morphology.</text>
<unordered_list>
<list_item><location><page_4><loc_54><loc_26><loc_92><loc_36></location>· M 5 - fit line profiles to every spaxel in the PACS array and sum them. The point-source correction for this method is negligible. This method will capture all of the line flux, but will have overestimated uncertainties unless the source is both bright and extended across at least most of the PACS field-ofview.</list_item>
<list_item><location><page_4><loc_54><loc_19><loc_92><loc_25></location>· M 0 - fit a PSF, as a function of position and intensity, across the whole PACS array and adopt the best fit. This is a photometric equivalent of 'optimal' extraction as described in Lebouteiller et al. 2010.</list_item>
</unordered_list>
<text><location><page_4><loc_52><loc_7><loc_92><loc_17></location>To choose the method for the lines in each source we proceed as follows. First, we determine all six measurements for each line. If the emission is consistent with a well-centered point source, then the fluxes from all six methods will agree with each other, with larger errors for the methods that include more spaxels. We found this to be so in the majority of the sample. For these, we adopted M 1. For the others, the measurements from</text>
<figure>
<location><page_5><loc_27><loc_46><loc_74><loc_91></location>
<caption>Figure 2. The one-dimensional spectra of each line, for the second eight objects in Table 1, excluding Mrk 231 (see Fischer et al. 2010). The labelling of the plot is as given for Figure 1.</caption>
</figure>
<text><location><page_5><loc_8><loc_27><loc_48><loc_40></location>M 2 through M 6 were higher than M 1, but were consistent with each other. This indicates that the line emission is mostly confined to the central 3 × 3 spaxels. We therefore discarded M 5. To avoid uncertainties in using a simulated PSF, we then used in most cases the measurements from M 2 rather than M 4 or M 6, even though the M 2 errors are larger. In most cases, for each object, the same measurement method was used for all lines. In a few cases though [C II ] is extended while the other lines are consistent with point sources.</text>
<text><location><page_5><loc_8><loc_16><loc_48><loc_27></location>For one object, IRAS 00397-1312, [C II ] is redshifted such that it lies in a part of the PACS wavelength range that suffers from significant flux leakage. In this wavelength range, which spans approximately 190 µ m to 210 µ m, there is superimposed emission from the second order, at 95-110 µ m. Since [C II ] in IRAS 00397-1312 was observed with SPIRE, we substitute the SPIRE-FTS measurement for this line.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_16></location>Except for [C II ] in IRAS 00397-1312, line fits were performed with PACSman for each spaxel at each raster position, using all points in the data cloud. Errors were estimated from the dispersion of the cloud in each wavelength bin. The fitting function was a Gaussian profile, adjusted simultaneously with a polynomial continuum of degrees one to three. The instrument spectral resolution</text>
<text><location><page_5><loc_52><loc_33><loc_92><loc_40></location>ranges from ∼ 55 kms -1 to ∼ 320kms -1 depending on the band, order and wavelength. The intrinsic line broadening FWHM was calculated by calculating the quadratic difference between the observed FWHM and the instrumental FWHM, assuming gaussian profiles.</text>
<section_header_level_1><location><page_5><loc_62><loc_31><loc_82><loc_32></location>3. RESULTS AND ANALYSIS</section_header_level_1>
<text><location><page_5><loc_52><loc_13><loc_92><loc_30></location>We tabulate the far-infrared line properties in Table 4, and present their profiles in Figures 1, 2, and 3. In the following analysis we frequently compare the far-IR line properties to those of the mid-IR fine-structure lines (Farrah et al. 2007a), the 6.2 µ m and 11.2 µ m Polycyclic Aromatic Hydrocarbon (PAH) features, and the 9.7 µ m silicate feature (Table 3). The PAH and silicate feature data were measured from spectra taken from the CASSIS v4 database (Lebouteiller et al. 2011). The PAH luminosities were measured by integrating over 5.9-6.6 µ m and 10.8-11.8 µ m in the continuum subtracted spectra, respectively (Spoon et al. 2007). For the silicate feature, we define its strength, S Sil , as:</text>
<formula><location><page_5><loc_66><loc_9><loc_92><loc_12></location>S Sil = ln [ f obs f cont ] (1)</formula>
<text><location><page_5><loc_52><loc_7><loc_92><loc_8></location>where f obs is the observed flux at rest-frame 9.7 µ m and</text>
<figure>
<location><page_6><loc_27><loc_46><loc_74><loc_91></location>
<caption>Figure 4. Distribution of line widths for each line, color coded by optical class (blue: H II , green: LINER, orange: Sy2, red: Sy1). See § 3.1.1 for discussion. Objects with errors on their widths exceeding 30% are not plotted. The open black circles show the (weighted) mean and error for each line. The line widths are corrected for instrumental broadening.</caption>
</figure>
<text><location><page_6><loc_55><loc_46><loc_56><loc_46></location>rest)</text>
<figure>
<location><page_6><loc_10><loc_18><loc_46><loc_40></location>
<caption>Figure 3. The one-dimensional spectra of each line, for the last eight objects in Table 1. The labelling of the plot is as given for Figure 1.</caption>
</figure>
<text><location><page_6><loc_29><loc_18><loc_31><loc_18></location>Line</text>
<text><location><page_6><loc_52><loc_34><loc_92><loc_41></location>f cont is the continuum flux at rest-frame 9.7 µ m in the absence of silicate absorption, inferred from a spline fit to the continuum at 5-7 µ m and 14-14.5 µ m (Spoon et al. 2007; Levenson et al. 2007). A positive value corresponds to silicates in absorption.</text>
<section_header_level_1><location><page_6><loc_65><loc_32><loc_79><loc_33></location>3.1. Line Properties</section_header_level_1>
<section_header_level_1><location><page_6><loc_65><loc_30><loc_79><loc_31></location>3.1.1. Profile Shapes</section_header_level_1>
<text><location><page_6><loc_52><loc_7><loc_92><loc_29></location>In most cases the lines are reproducible by single gaussians with widths between 250km s -1 and 600kms -1 . We do not see greater widths in the higher ionization lines. We also do not see strong asymmetries, or systemic offsets in velocity (compared to the optical redshift) of any line. In a few cases the line profiles are not reproducible with single symmetric profiles. The [O I ]63 lines in two objects, IRAS 06206 -6315 and IRAS 20414-1651 are consistent with significant self-absorption (see also Graci'a-Carpio et al. 2011). There is weaker evidence for [O I ]63 self absorption in IRAS 00188-0856, IRAS 110950238, and IRAS 19254-7245. Self-absorption in [O I ]63 can occur when cool oxygen in foreground clouds reduces the [O I ]63 flux (Poglitsch et al. 1996; Fischer et al. 1999; Vastel et al. 2002; Luhman et al. 2003). The [C II ] profiles in IRAS 20087-0308, IRAS 20414-4651 and IRAS 23253-5415 may show subtle asymmetries. Finally, in</text>
<figure>
<location><page_7><loc_27><loc_54><loc_74><loc_91></location>
<caption>Figure 5. Selected far-IR line luminosities, plotted against each other ( § 3.1.2). The label in each panel identifies which line is plotted on the x and y axes, respectively. The solid, dashed and dot-dashed lines in the [O I ]63 vs. [O I ]145 plot indicate ratios of 1:1, 10:1, and 20:1, respectively. The solid and dotted lines in the [N II ] vs. [C II ] plot indicate ratios of 1:1, 1:4, respectively. Objects with green annuli may show self absorption in [O I ]63. Objects with red cores may show an additional component in [C II ].</caption>
</figure>
<text><location><page_7><loc_8><loc_36><loc_48><loc_46></location>four cases (IRAS 06035-7102, Mrk 463, IRAS 110950238, and IRAS 20100-4156), [C II ] may show an additional, broad emission component with widths between 600kms -1 and 1200kms -1 ˙ For consistency with the other lines we do not include the broad component in the line fluxes in Table 4, but discuss it instead in a future paper.</text>
<text><location><page_7><loc_8><loc_10><loc_48><loc_36></location>We examine the distribution of line widths, using only the narrow components in those cases where an additional broad one exists, in Figure 4. The range in widths of all six lines are consistent with each other. We see no dependence of the range in widths on optical class. For individual objects though there are sometimes substantial differences between individual line widths. In Mrk 1014 for example the [C II ] and [N II ] line widths differ by nearly a factor of two. We speculate that these differences are due to one or more of (a) differences in the critical densities of the lines, (b) that an individually unresolved broad component in one line is brighter than the equivalent component in the other line, and (c) in the case of [C II ] and [N II ], that the [N II ] emission arises mostly from H II regions, while at least some of the [C II ] emission arises from photodissociation regions (PDRs) or the diffuse ISM. It is however also possible that the [N II ] profile is affected by the 4 32 -4 23 transition of o-H 2 O at 121.72 µ m(Fischer et al. 2010; Gonz'alez-Alfonso et al. 2010). We discuss this in Spoon et al. 2013.</text>
<text><location><page_7><loc_52><loc_27><loc_92><loc_46></location>We plot luminosities of selected individual lines against each other in Figure 5. The line luminosities range from just under 10 7 L /circledot to ∼ 3 × 10 9 L /circledot . The [C II ] or [O I ]63 lines are usually the most luminous lines, though the [O III ] line may be the second most luminous in 3C273 and IRAS 00397-1312. The least luminous two lines are in most cases [N II ] and [O I ]145. In some cases [N III ] is consistent with being the least luminous, and in a few cases (e.g. IRAS 07598+6508 IRAS 23253-5415) [O III ] is consistent with being the least luminous. The range in luminosities decreases approximately with increasing wavelength; the [O III ] and [N III ] line luminosities span a factor of ∼ 50, whereas the [C II ] luminosity spans only a factor of five, less than the range in L IR of the sample.</text>
<text><location><page_7><loc_52><loc_13><loc_92><loc_27></location>To determine if the line luminosities correlate with each other, we fit the relation in Equation 2 (see § 3.2.1 for methodology). We find positive correlations in all cases, with β values mostly between 0.5 and 1.2. In no case however is the correlation particularly strong (in terms of the S/N on β ). We find that, among these six lines, the luminosity of one cannot be used to predict the luminosity of another to better than ∼ 0 . 2dex. If we instead plot line luminosities normalized by L IR (Figure 6) then the correlations do not change significantly. Neither do they depend on optical spectral type.</text>
<text><location><page_7><loc_52><loc_7><loc_92><loc_13></location>One object, IRAS 01003-2238, may have an unusually low [N II ] luminosity. The six line measurement methods for [N II ] in this object are in reasonable agreement, despite the low S/N. We therefore do not have a good</text>
<figure>
<location><page_8><loc_23><loc_50><loc_77><loc_91></location>
<caption>Figure 6. Selected far-IR line pairs normalized by L IR , plotted against each other ( § 3.1.2, see also Figure 5).</caption>
</figure>
<text><location><page_8><loc_50><loc_49><loc_52><loc_50></location>Line</text>
<text><location><page_8><loc_53><loc_49><loc_53><loc_50></location>IR</text>
<text><location><page_8><loc_8><loc_40><loc_48><loc_45></location>explanation for this. The only other way in which IRAS 01003-2238 is atypical is that its optical spectrum contains Wolf-Rayet star features (Farrah et al. 2005) but we do not see why this would affect the [N II ] luminosity.</text>
<text><location><page_8><loc_8><loc_23><loc_48><loc_40></location>Finally, we comment briefly on two line ratios. First is [O I ]145/[O I ]63, which is related to the physical conditions in PDRs. This ratio (note the inverse is plotted in Figure 5) depends on both the gas temperature and density. It decreases with decreasing gas temperature, and decreases with increasing gas density if the density is above the critical density. In a diffuse PDR illuminated by intense UV radiation, the expected [O I ] ratio lies below ∼ 0 . 1 (e.g., Kaufman et al. 2006). From Figure 5, several of our sample have [O I ] ratios above 0.1. A likely reason for this is self absorption of [O I ]63, or a higher optical depth in [O I ]63 than [O I]145 (e.g., Abel et al. 2007).</text>
<text><location><page_8><loc_8><loc_8><loc_48><loc_23></location>Second is the [N II ]/[C II ] ratio, which can be used to estimate the relative contribution of [C II ] arising in PDRs and in the ionized gas. While [N II ] arises almost entirely from H II regions, [C II ] can additionally arise from PDRs and the diffuse ISM (cold neutral, warm neutral, and warm diffuse ionized). The [N II ]/[C II ] ratio can therefore quantify the contribution to [C II ] from the ionized gas, by comparing the observed [N II ]/[C II ] ratio with theoretical expectations from a photoionized nebula. The expected ratio in the ionized gas lies between 0.2 and 2.0 depending on density (e.g., figure 7</text>
<text><location><page_8><loc_52><loc_37><loc_92><loc_45></location>of Bernard-Salas et al. 2012). The ratios in our sample are however nearly all below 0.25. This indicates that a significant fraction of the [C II ] emission originates from photodissociation regions (PDRs). A precise estimate is however hampered by lack of knowledge on the ionized gas density.</text>
<section_header_level_1><location><page_8><loc_63><loc_35><loc_81><loc_36></location>3.2. Infrared Luminosity</section_header_level_1>
<section_header_level_1><location><page_8><loc_64><loc_33><loc_80><loc_34></location>3.2.1. Line Luminosities</section_header_level_1>
<text><location><page_8><loc_52><loc_26><loc_92><loc_33></location>We compare far-IR line luminosities to L IR in Figure 7. Qualitatively, some of the line luminosities crudely correlate with L IR , with greater scatter among the Sy1s. To determine if correlations exist between L IR and L Line , we assume that L IR and L Line are related by:</text>
<formula><location><page_8><loc_60><loc_22><loc_92><loc_25></location>log ( L IR L /circledot ) = α + β log ( L Line L /circledot ) (2)</formula>
<text><location><page_8><loc_52><loc_12><loc_92><loc_21></location>where α and β are free parameters. We fit this relation following the method of Tremaine et al. (2002). For a correlation to exist, we require that β > 0 at > 3 . 5 σ significance. We do not claim that Equation 2 codifies an underlying physical relation. Neither do we claim that Equation 2 is the only relation that applies over the luminosity range of our sample 18 . Finally, we do not</text>
<text><location><page_8><loc_52><loc_7><loc_92><loc_11></location>18 Indeed, in most cases through this paper where fit results are quoted, a linear ( Y = α + βX ) or exponential ( Y = αX β ) model with appropriate choices for α and β both serve equally well</text>
<figure>
<location><page_9><loc_26><loc_43><loc_74><loc_91></location>
<caption>Figure 7. Far-IR line luminosities vs L IR ( § 3.2.1). The lines show the fits in Equations 3 to 8, if the fit is significant. Here and elsewhere, we use the same x and y axis ranges for each line, while still showing most of the objects and all significant trends.</caption>
</figure>
<text><location><page_9><loc_8><loc_35><loc_48><loc_38></location>consider more complex models as our data do not prefer them.</text>
<text><location><page_9><loc_8><loc_33><loc_48><loc_35></location>Excluding objects with Sy1 spectra yields the following relations between L IR and L Line :</text>
<unordered_list>
<list_item><location><page_9><loc_11><loc_29><loc_48><loc_30></location>logL IR =(4 . 46 ± 1 . 77) + (0 . 92 ± 0 . 20)log(L [OIII] ) (3)</list_item>
<list_item><location><page_9><loc_16><loc_27><loc_48><loc_29></location>=(4 . 74 ± 1 . 59) + (0 . 94 ± 0 . 19)log(L [NIII] ) (4)</list_item>
<list_item><location><page_9><loc_16><loc_25><loc_48><loc_27></location>=(4 . 28 ± 1 . 89) + (0 . 91 ± 0 . 21)log(L [OI]63 ) (5)</list_item>
<list_item><location><page_9><loc_16><loc_23><loc_48><loc_25></location>=(5 . 29 ± 1 . 57) + (0 . 89 ± 0 . 20)log(L [NII] ) (6)</list_item>
<list_item><location><page_9><loc_16><loc_21><loc_48><loc_23></location>=(1 . 75 ± 2 . 11) + (1 . 34 ± 0 . 27)log(L [OI]145 ) (7)</list_item>
</unordered_list>
<formula><location><page_9><loc_16><loc_20><loc_48><loc_21></location>=(6 . 73 ± 2 . 44) + (0 . 64 ± 0 . 27)log(L [CII] ) (8)</formula>
<text><location><page_9><loc_8><loc_7><loc_48><loc_17></location>which are significant (in terms of β ) for all the lines except [C II ]. There is the caveat though that some of the relations are based on a small number of formal detections (Table 4). Including the Sy1s yields significant correlations only for [O III ] and [O I ]145, both with flatter slopes than the above. The insignificant correlation for [C II ] suggests that it traces L IR to an accuracy of about an order of magnitude at best. This result is consistent</text>
<text><location><page_9><loc_52><loc_35><loc_92><loc_38></location>with previous authors, who find a crude correlation over a wider luminosity range (Sargsyan et al. 2012).</text>
<text><location><page_9><loc_52><loc_29><loc_92><loc_35></location>We also investigated whether sums of the lines in Equations 3 to 8 show stronger relations with L IR than individual lines. We found correlations in several cases, but none that improved significantly on those for the individual lines.</text>
<text><location><page_9><loc_52><loc_13><loc_92><loc_29></location>The weaker correlations between L IR and L Line when Sy1s are included in the fits is consistent with far-IR lines being better tracers of L IR in ULIRGs without an optical AGN. If this is true then we may see a similar result if we include in the fits only those objects with prominent PAH features, since PAHs are probably exclusively associated with star formation (see § 3.3.1). We test this hypothesis by repeating the fits, but this time including only those objects with 11.2 µ mPAHEWsgreater than 0.05 µ m. We chose the 11.2 µ m PAH because (a) it is bright, and (b) its value correlates well with the evolutionary paradigm for ULIRGs in Farrah et al. (2009) 19 . We find:</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_10></location>19 Using the 11.2 µ m PAH may however lead to more bias than using the 6.2 µ m PAH if absorption from ices and silicate dust is primarily associated with the background continuum source</text>
<figure>
<location><page_10><loc_26><loc_43><loc_74><loc_91></location>
<caption>Figure 8. Continuum luminosity densities near the wavelength of the indicated line, vs total IR luminosity ( § 3.2.2). The lines show fits with the Sy1s removed (Equation 15 through 20). The Sy1 object with low flux densities near all six lines but a high IR luminosity is 3C 273.</caption>
</figure>
<unordered_list>
<list_item><location><page_10><loc_11><loc_34><loc_48><loc_35></location>logL IR =(7 . 01 ± 1 . 07) + (0 . 63 ± 0 . 12)log(L [OIII] ) (9)</list_item>
<list_item><location><page_10><loc_16><loc_32><loc_48><loc_33></location>=(6 . 52 ± 1 . 33) + (0 . 71 ± 0 . 16)log(L [NIII] ) (10)</list_item>
<list_item><location><page_10><loc_16><loc_30><loc_48><loc_31></location>=(6 . 71 ± 1 . 30) + (0 . 64 ± 0 . 15)log(L [OI]63 )(11)</list_item>
<list_item><location><page_10><loc_16><loc_28><loc_48><loc_29></location>=(6 . 43 ± 1 . 51) + (0 . 74 ± 0 . 19)log(L [NII] ) (12)</list_item>
<list_item><location><page_10><loc_16><loc_26><loc_48><loc_27></location>=(2 . 77 ± 2 . 12) + (1 . 20 ± 0 . 26)log(L [OI]145 )(13)</list_item>
</unordered_list>
<formula><location><page_10><loc_16><loc_24><loc_48><loc_26></location>=(9 . 44 ± 2 . 20) + (0 . 33 ± 0 . 25)log(L [CII] ) (14)</formula>
<text><location><page_10><loc_8><loc_14><loc_48><loc_22></location>These relations are somewhat flatter than those in Equations 3 to 8. Again the [C II ] line shows no significant correlation. We conclude that these far-IR lines are better tracers of L IR in systems without type 1 AGN, but that it is unclear whether they are better tracers of L IR in systems with more prominent star formation.</text>
<section_header_level_1><location><page_10><loc_18><loc_12><loc_39><loc_13></location>3.2.2. Continuum Luminosities</section_header_level_1>
<text><location><page_10><loc_8><loc_7><loc_48><loc_11></location>We compare L IR to continuum luminosity densities (in units of L /circledot Hz -1 ) near the wavelengths of the far-IR lines in Figure 8. We find a significant correlation at</text>
<text><location><page_10><loc_52><loc_31><loc_92><loc_37></location>all six wavelengths, irrespective of whether Sy1s are included (though the continua of the Sy1s are usually not detected). For consistency with the line comparisons, we exclude the Sy1s and fit relations of the form in Equation 2, obtaining:</text>
<formula><location><page_10><loc_55><loc_19><loc_92><loc_28></location>logL IR =(12 . 87 ± 0 . 09) + (0 . 88 ± 0 . 17)log(L 52 ) (15) =(12 . 86 ± 0 . 05) + (0 . 91 ± 0 . 11)log(L 57 ) (16) =(12 . 86 ± 0 . 08) + (0 . 98 ± 0 . 18)log(L 63 ) (17) =(12 . 95 ± 0 . 07) + (0 . 87 ± 0 . 11)log(L 122 ) (18) =(13 . 05 ± 0 . 11) + (0 . 90 ± 0 . 16)log(L 145 ) (19) =(13 . 08 ± 0 . 11) + (0 . 83 ± 0 . 11)log(L 158 ) (20)</formula>
<text><location><page_10><loc_52><loc_13><loc_92><loc_16></location>Including the Sy1s yields comparable relations. If we instead exclude objects with PAH 11.2 µ m EW /lessorsimilar 0 . 05 then we obtain consistent relations.</text>
<section_header_level_1><location><page_10><loc_65><loc_10><loc_78><loc_11></location>3.2.3. Line Deficits</section_header_level_1>
<text><location><page_10><loc_52><loc_7><loc_92><loc_9></location>Several far-IR lines in ULIRGs show a deficit in their L Line /L IR ratios compared to the ratios expected from</text>
<figure>
<location><page_11><loc_27><loc_54><loc_74><loc_91></location>
<caption>Figure 9. Comparison of the [O I ]63/L IR , [N II ]/L IR , [O I ]145/L IR , and [C II ]/L IR ratios with L IR ( § 3.2.3). The colored points are our sample, coded by optical class. ULIRGs with a grey annulus have a [Ne V ]14.32 detection. The black points are taken from Brauher et al. (2008). The yellow and light blue open symbols show the means and dispersions for the Brauher and our samples, respectively. The ULIRGs show a deficit compared to the lower luminosity systems in all four cases. There is however no clear dependence of the deficit on optical class, or the detection of [Ne V ]14.32.</caption>
</figure>
<text><location><page_11><loc_8><loc_30><loc_48><loc_45></location>systems with lower values of L IR (e.g. Luhman et al. 2003). In contrast, high redshift ULIRGs, at least for [O I ]63 and [C II ], do not show such pronounced deficits (Stacey et al. 2010; Coppin et al. 2012; Rigopoulou et al. 2013). We examine if the [O I ], [N II ] and [C II ] lines in our sample show such deficits in Figure 9 (we do not test the other lines as we lack archival data to compare to). We find a deficit in all four lines. Comparing the mean ratios at 10 11 L /circledot and 10 12 . 2 L /circledot we find differences of factors of 2.75, 4.46, 1.50, and 4.95 for [O I ]63, [N II ], [O I ]145, and [C II ], respectively.</text>
<text><location><page_11><loc_8><loc_23><loc_48><loc_30></location>The four line deficits show no clear dependence on optical spectral type, the presence of an obscured AGN (as diagnosed from the detection of [Ne V ]14.32, see § 3.5), or on PAH 11.2 µ m EW, for any line. There are however trends with both S Sil and merger stage:</text>
<unordered_list>
<list_item><location><page_11><loc_11><loc_14><loc_48><loc_22></location>· If S Sil /greaterorsimilar 1 . 4 then the [C II ] and [N II ] deficits increase with increasing S Sil . There is however no obvious trend of the [C II ] and [N II ] deficits with S Sil at S Sil /lessorsimilar 1 . 4 (top left panel of Figure 10 & Figure 13). Conversely the [O I ] deficits show no trends with S Sil .</list_item>
<list_item><location><page_11><loc_11><loc_7><loc_48><loc_12></location>· We find no evidence that the [N II ] and [O I ] deficits depend on merger stage, but the [C II ] deficit is stronger, on average, in advanced mergers (classes IVb and V) than in early-stage mergers (classes</list_item>
</unordered_list>
<text><location><page_11><loc_56><loc_42><loc_92><loc_45></location>IIIa through IVa, top right panel of Figure 10 20 , see also Diaz-Santos et al. 2013).</text>
<text><location><page_11><loc_52><loc_31><loc_92><loc_41></location>Finally, we plot in the bottom row of Figure 10 L [CII] /L IR against L PAH /L IR as a function of merger stage and S Sil . We see in both plots consistent trends; ULIRGs in advanced mergers and with S Sil /greaterorsimilar 2 have lower L [CII] /L IR and L PAH /L IR ratios, compared to ULIRGs in early-stage stage mergers and with S Sil /similarequal 1 . 4 -2.</text>
<text><location><page_11><loc_52><loc_13><loc_92><loc_31></location>We do not believe that the line deficits arise from missing an asymmetric or broad component (see § 3.1.1) since such components are rare. Neither do we believe the deficits arise due to self-absorption, since we see no self absorption in [C II ], which shows the strongest deficit. We do however see self absorption in [O I ]63, which shows a weaker deficit. Instead, the stronger [C II ] and [N II ] deficits in sources with higher S Sil (at S Sil /greaterorsimilar 1 . 4) are consistent with the H II regions in ULIRGs being dustier than H II regions in lower luminosity systems. In this scenario (Luhman et al. 2003; Gonz'alez-Alfonso et al. 2008; Abel et al. 2009; Graci'a-Carpio et al. 2011), a higher fraction of the UV photons are absorbed by dust rather than neutral Hydrogen, thus contributing more to L IR</text>
<text><location><page_11><loc_52><loc_7><loc_92><loc_12></location>20 We also constructed the deficit plots as a function of merger stage using only those sources with S Sil < 1 . 4. The [C II ] deficit still strengthens with advancing merger stage, while no trends emerge with merger stage for the other lines.</text>
<figure>
<location><page_12><loc_27><loc_49><loc_75><loc_91></location>
<caption>Figure 10. Top row: Zoom in of the [C II ]/L IR deficit in ULIRGs, as a function of merger stage and silicate depth ( § 3.2.3). Other details are as in Figure 9. We include in this plot additional ULIRGs with [C II ] detections; Arp220, Mrk 273, NGC 6240, IRAS 04103-2838, IRAS 05189-2524, IRAS 10565+2448, IRAS 12018+1941, IRAS 13342+3932, IRAS 15001+1433, IRAS 17208-0014, IRAS 20037-1547, IRAS 20100-4156, IRAS 20551-4250, IRAS 23128-5919 (Brauher et al. 2008). Bottom row: [C II ]/L IR plotted against L PAH /L IR .</caption>
</figure>
<text><location><page_12><loc_51><loc_48><loc_53><loc_49></location>[CII]</text>
<text><location><page_12><loc_54><loc_48><loc_55><loc_49></location>LIR</text>
<text><location><page_12><loc_8><loc_27><loc_48><loc_40></location>but less to the photoionization heating of gas in the H II regions, thus decreasing line emission relative to L IR . This mechanism would also produce a deficit in [C II ] and the 'deficit' in the PAH emission, even if the bulk of the [C II ] and PAHs are in the PDRs, since there would also be fewer UV photons for photoelectric heating of the PDRs. Furthermore, this mechanism is consistent with the [O I ] deficits. From figure 3 of Graci'a-Carpio et al. 2011 the conditions consistent with the observed deficits of all four lines are n h /lessorsimilar 300cm -3 and 0 . 01 /lessorsimilar 〈 U 〉 /lessorsimilar 0 . 1.</text>
<text><location><page_12><loc_8><loc_12><loc_48><loc_27></location>We propose though that dustier H II regions are not the sole origin of the line deficits. This is based on three observations. First, it is puzzling that we see no strong dependence of any of the line deficits on S Sil when S Sil /lessorsimilar 1 . 4; if the deficit arises entirely in H II regions, and if S Sil is a proxy for the dust column in these H II regions at S Sil /greaterorsimilar 0, then we should see a dependence. We note though that we have only a small sample, so we could be missing a weak dependence, and that the assumption that S Sil is a proxy for dust column in H II regions is not proven 21 . Second, we see significant line deficits for some</text>
<text><location><page_12><loc_8><loc_7><loc_48><loc_10></location>21 It is also (arguably) puzzling that we see dependences on any line deficit at S Sil /greaterorsimilar 1 . 4, since (some) models demand that silicate strengths greater than about this value require smooth rather than</text>
<text><location><page_12><loc_52><loc_34><loc_92><loc_40></location>sources with S Sil < 0, i.e. a silicate emission feature. Third, only the [C II ] deficit gets stronger with advancing merger stage. If more advanced mergers host dustier H II regions, then we would also see a dependence of the [N II ] deficit on merger stage.</text>
<text><location><page_12><loc_52><loc_22><loc_92><loc_34></location>These observations suggest that some fraction of the [C II ] deficit is not driven by increased dust in H II regions. We lack the data to investigate this in detail, so we only briefly discuss this further. We consider three possible origins; (1) increased charging of dust grains (e.g. Malhotra et al. 2001), leading to a lower gas heating efficiency, (2) a softer radiation field in the diffuse ISM (e.g. Spaans et al. 1994), and (3) dense gas in the PDRs, making [O I ] rather than [C II ] the primary coolant.</text>
<text><location><page_12><loc_52><loc_10><loc_92><loc_22></location>The third of these possibilities is feasible, but we do not have the data to confirm or refute it as a mechanism. The second possibility is unlikely, due to the energetic nature of star formation in ULIRGs and from the arguments in Luhman et al. (2003). For the first possibility; if the origin of the additional deficit in [C II ] is grain charging, then we would see a higher G 0 in advanced mergers compared to early stage mergers. From Figure 18 (see § 3.6) the advanced mergers have a roughly order of mag-</text>
<text><location><page_12><loc_52><loc_7><loc_81><loc_8></location>clumpy dust distributions (Nenkova et al. 2008).</text>
<figure>
<location><page_13><loc_25><loc_40><loc_76><loc_90></location>
<caption>Figure 11. Far-IR line luminosities vs L PAH ( § 3.3.1). The black lines show the fits in Equation 21 through 26.</caption>
</figure>
<text><location><page_13><loc_8><loc_23><loc_48><loc_35></location>nitude higher value of G 0 for about the same n 22 . We therefore infer, cautiously, and with the caveat that we cannot rule out [O I ] being a major cooling line, that part of the [C II ] deficit arises due to grain charging in PDRs or in the diffuse ISM. We further propose that this increase in grain charge is not driven mainly by a luminous AGN, obscured or otherwise, since we see no dependence of the [C II ] deficit on either optical spectral type or the presence of [Ne V ]14.32.</text>
<section_header_level_1><location><page_13><loc_19><loc_21><loc_22><loc_21></location>3.3.</section_header_level_1>
<section_header_level_1><location><page_13><loc_20><loc_19><loc_37><loc_21></location>Star Formation Rate 3.3.1.</section_header_level_1>
<text><location><page_13><loc_25><loc_19><loc_36><loc_20></location>Line Luminosities</text>
<text><location><page_13><loc_8><loc_13><loc_48><loc_18></location>We examine far-IR fine-structure lines as star formation rate indicators by comparing their luminosities to those of PAHs. PAHs are thought to originate from short-lived Asymptotic Giant Branch stars</text>
<text><location><page_13><loc_8><loc_7><loc_48><loc_12></location>22 We note though that this also holds for the trends discussed with S Sil ; dividing into two samples with 1 . 4 < S Sil < 2 . 1 and S Sil > 2 . 1 yields a G 0 /n ratio about a factor of three higher in the latter sample</text>
<text><location><page_13><loc_52><loc_9><loc_92><loc_34></location>(Gehrz 1989; Habing 1996; Blommaert et al. 2005; Bernard-Salas et al. 2006), and therefore to be associated with star-forming regions (Tielens 2008). They are prominent in starburst galaxies but weak in AGNs (Laurent et al. 2000; Weedman et al. 2005). While uncertainty remains over how to calibrate PAHs as star formation rate measures (Peeters et al. 2004; Forster Schreiber et al. 2004; Sargsyan et al. 2012), their luminosities are likely reasonable proxies for the instantaneous rate of star formation. Conversely, for far-IR fine-structure lines the relation between line luminosity and star formation rate is less clear. We may expect a correlation, as the inter-stellar radiation field (ISRF) from young stars may be an important excitation mechanism. Moreover, correlations between far-IR line luminosities and star formation rate have been observed previously (e.g. Boselli et al. 2002; de Looze et al. 2011; Sargsyan et al. 2012; Zhao et al. 2013). The extent to which correlations exist is however poorly constrained.</text>
<text><location><page_13><loc_53><loc_8><loc_92><loc_9></location>With the caveats that PAH luminosity depends both</text>
<figure>
<location><page_14><loc_26><loc_43><loc_74><loc_91></location>
<caption>Figure 12. Continuum luminosity densities at the wavelength of the indicated line, vs L PAH ( § 3.3.2). The fits are given in Equations 28 to 33.</caption>
</figure>
<text><location><page_14><loc_8><loc_24><loc_48><loc_38></location>on metallicity (Madden et al. 2006; Calzetti et al. 2007; Khramtsova et al. 2013) and dust obscuration, neither of which we can correct for, we assume the PAH luminosities give 'true' star formation rate measures, which we compare to the far-IR line luminosities. In doing so, we further assume that there is negligible differential extinction between the PAHs and the far-IR lines. To mitigate effects from variation of individual PAH features (Smith et al. 2007), we sum the luminosities of the PAH 6.2 µ mand 11.2 µ mfeatures into a single luminosity, L PAH (Farrah et al. 2007a).</text>
<text><location><page_14><loc_8><loc_13><loc_48><loc_24></location>We plot far-IR line luminosities against L PAH in Figure 11. Unlike the plots with L IR , there is no significant difference between the Sy1s and H II /LINERs. This is consistent with both PAHs and the far-IR lines primarily tracing star formation. Fitting relations of the form in Equation 2 to all the objects, and converting to star formation rate by using equation 5 of Farrah et al. (2007a), yields:</text>
<formula><location><page_14><loc_10><loc_9><loc_48><loc_10></location>log( ˙ M)=( -7 . 02 ± 1 . 25) + (1 . 07 ± 0 . 14)log(L [OIII] )(21)</formula>
<formula><location><page_14><loc_15><loc_7><loc_48><loc_8></location>=( -5 . 13 ± 0 . 72) + (0 . 91 ± 0 . 09)log(L [NIII] )(22)</formula>
<formula><location><page_14><loc_58><loc_37><loc_92><loc_38></location>=( -5 . 44 ± 1 . 79) + (0 . 86 ± 0 . 20)log(L [OI]63 )(23)</formula>
<formula><location><page_14><loc_58><loc_35><loc_92><loc_36></location>=( -7 . 30 ± 0 . 87) + (1 . 19 ± 0 . 11)log(L [NII] ) (24)</formula>
<text><location><page_14><loc_58><loc_33><loc_92><loc_34></location>=( -10 . 04 ± 1 . 34) + (1 . 55 ± 0 . 17)log(L [OI]145 ) (25)</text>
<formula><location><page_14><loc_58><loc_31><loc_92><loc_33></location>=( -6 . 24 ± 1 . 72) + (0 . 95 ± 0 . 19)log(L [CII] ) (26)</formula>
<text><location><page_14><loc_52><loc_15><loc_92><loc_28></location>Using the criteria from § 3.2.1 then all the lines show a significant correlation. The trends with [O I ]63 and [C II ] are only barely significant (see also Diaz-Santos et al. 2013). Excluding the Sy1s and Sy2s from the fits yields consistent slopes and intercepts in all cases, though the fits are now no longer significant for [O I ]63 and [C II ]. Considering only those lines detected at > 3 σ , then the derived star formation rates are, for a given object, consistent to within a factor of three in nearly all cases. We present the mean star formation rates in Table 3.</text>
<text><location><page_14><loc_52><loc_7><loc_92><loc_15></location>An alternative to PAH luminosity as a tracer of star formation rate is the sum of the [Ne III ]15.56 µ m and [Ne II ]12.81 µ m line luminosities, L Neon (Thornley et al. 2000; Ho & Keto 2007; Shipley et al. 2013). We thus compared L Neon to L Line . For the whole sample, we find relations mostly in agreement with Equations 21 through</text>
<figure>
<location><page_15><loc_26><loc_43><loc_74><loc_91></location>
<caption>Figure 13. Normalized far-IR line luminosities, plotted against [Ne III ]15.56/[Ne II ]12.81. There are no clear correlations ( § 3.4). We code the points by S Sil to highlight the trends described in § 3.2.3.</caption>
</figure>
<text><location><page_15><loc_51><loc_43><loc_52><loc_44></location>Line</text>
<text><location><page_15><loc_53><loc_43><loc_54><loc_44></location>IR</text>
<text><location><page_15><loc_8><loc_34><loc_48><loc_38></location>26, though the relation with [C II ] is now formally insignificant. Excluding the Seyferts and comparing L Neon to L Line yields similar results to those with L PAH .</text>
<text><location><page_15><loc_8><loc_17><loc_48><loc_34></location>We note four further points. First, we tested sums of far-IR lines as tracers of star formation rate but found no improvement on the individual relations. Second, if we instead consider L Line / L IR vs. L PAH / L IR then we see correlations with comparable scatter to those in Figure 11. Third, we investigated whether using L PAH is better than using a single PAH luminosity, by reproducing Figure 11 using only the PAH 6.2 µ m luminosity. We found consistent relations in all cases, albeit with larger scatter for the L 6 . 2 PAH plots. Fourth, if we instead code the points in Figure 11 by the 11.2 µ m PAH EW then we see no dependence of the relations on the energetic importance of star formation.</text>
<text><location><page_15><loc_8><loc_7><loc_48><loc_17></location>Finally, we note two points about the relation between star formation rate and L [CII] . First, we investigated whether L [CII] shows an improved correlation with star formation rate if objects with a strong [C II ] deficit are excluded. Adopting a (somewhat arbitrary) boundary of L [CII] /L IR = 2 × 10 -4 yields only a marginally different relation:</text>
<formula><location><page_15><loc_53><loc_34><loc_92><loc_36></location>log( ˙ M) = ( -6 . 59 ± 1 . 58) + (0 . 99 ± 0 . 18)log(L [CII] ) (27)</formula>
<text><location><page_15><loc_52><loc_25><loc_92><loc_34></location>suggesting that the correlation does not depend strongly on the [C II ] deficit, though there is the caveat that star formation rate is derived from L PAH (see § 3.2.3 & Figure 10). Second, the relation between ˙ M /circledot and L [CII] is consistent with that given by Sargsyan et al. (2012), though their sample mostly consists of lower luminosity systems.</text>
<section_header_level_1><location><page_15><loc_61><loc_23><loc_82><loc_24></location>3.3.2. Continuum Luminosities</section_header_level_1>
<text><location><page_15><loc_52><loc_15><loc_92><loc_22></location>We examine continuum luminosity densities as star formation rate tracers in Figure 12. Employing the same method as in § 3.2 we find that the continua near all the lines provide acceptable fits. Converting to relations with star formation rate (see § 3.3) we find:</text>
<formula><location><page_15><loc_55><loc_12><loc_92><loc_14></location>log( ˙ M)=(3 . 24 ± 0 . 24) + (1 . 89 ± 0 . 46)log(L 52 ) (28)</formula>
<unordered_list>
<list_item><location><page_15><loc_60><loc_10><loc_92><loc_12></location>=(2 . 95 ± 0 . 10) + (1 . 38 ± 0 . 19)log(L 57 ) (29)</list_item>
<list_item><location><page_15><loc_60><loc_8><loc_92><loc_10></location>=(3 . 04 ± 0 . 17) + (1 . 79 ± 0 . 39)log(L 63 ) (30)</list_item>
</unordered_list>
<formula><location><page_15><loc_60><loc_7><loc_92><loc_8></location>=(2 . 83 ± 0 . 11) + (0 . 93 ± 0 . 16)log(L 122 ) (31)</formula>
<figure>
<location><page_16><loc_27><loc_53><loc_74><loc_91></location>
<caption>Figure 14. Normalized far-IR line luminosities vs optical class ( § 3.5).</caption>
</figure>
<text><location><page_16><loc_51><loc_53><loc_52><loc_53></location>line</text>
<text><location><page_16><loc_50><loc_53><loc_51><loc_54></location>L</text>
<text><location><page_16><loc_52><loc_53><loc_52><loc_54></location>/L</text>
<text><location><page_16><loc_52><loc_53><loc_53><loc_53></location>LIR</text>
<table>
<location><page_16><loc_20><loc_12><loc_80><loc_43></location>
<caption>Table 3 Mid-Infrared Properties and Far-IR derived star formation rates</caption>
</table>
<text><location><page_16><loc_20><loc_8><loc_79><loc_12></location>Note . - Flux units are × 10 -20 W cm -2 . PAH data are taken from the IRS spectra in the CASSIS database (Lebouteiller et al. 2011, see also e.g. Spoon et al. 2007; Desai et al. 2007). The PAH 6.2 µ m EWs (but not the fluxes) have been corrected for ice absorption. The last two columns give mean star formation rates from lines and continua detected at > 3 σ , using Equations 21-26 and 28-33.</text>
<figure>
<location><page_17><loc_26><loc_28><loc_74><loc_75></location>
<caption>Figure 15. Far-IR line luminosities vs [Ne V ]14.32/[Ne II ]12.88 ( § 3.5). The lines in the lower right panel are AGN contribution as a function of [Ne V ]14.32/[Ne II ]12.88, from Sturm et al. 2002 (we plot these on only one panel for clarity).</caption>
</figure>
<text><location><page_17><loc_57><loc_28><loc_58><loc_28></location>O</text>
<unordered_list>
<list_item><location><page_18><loc_17><loc_90><loc_48><loc_92></location>=(3 . 02 ± 0 . 18) + (1 . 09 ± 0 . 25)log(L 145 ) (32)</list_item>
</unordered_list>
<formula><location><page_18><loc_17><loc_89><loc_48><loc_90></location>=(3 . 36 ± 0 . 22) + (1 . 42 ± 0 . 30)log(L 158 ) (33)</formula>
<text><location><page_18><loc_8><loc_75><loc_48><loc_87></location>which are all significant using the criteria from § 3.2.1. The correlations with continua at ≤ 63 µ m may however be stronger, consistent with stronger correlations with warmer (star formation heated) dust. This is consistent with findings by previous authors (Brandl et al. 2006; Calzetti et al. 2007). We present the mean star formation rates derived from these relations in Table 3. In most cases they are consistent with the line-derived star formation rates.</text>
<section_header_level_1><location><page_18><loc_19><loc_72><loc_37><loc_73></location>3.4. Gas Photoionization</section_header_level_1>
<text><location><page_18><loc_8><loc_61><loc_48><loc_72></location>If electron densities are below the critical density in the narrow-line region, then the hardness of the radiation field ionizing an element can be estimated via flux ratios of adjacent ionization states of that element; f X i +1 /f X i . For a fixed U this ratio will be approximately proportional to the number of photons producing the observed X i flux relative to the number of Lyman continuum photons.</text>
<text><location><page_18><loc_8><loc_45><loc_48><loc_61></location>For the mid-IR line-emitting gas, two diagnostic ratios of this type can be used; [Ne III ]15.56/[Ne II ]12.81 and [S IV ]10.51/[S III ]18.71. The photon energies required to produce these four ions are all < 50eV, meaning that they can be produced in star-forming regions (Smith & Houck 2001; Bernard-Salas et al. 2001; Peeters et al. 2002; Verma et al. 2003). As the Neon lines are detected in all of our sample, we use the Neon ratio as a proxy for mid-IR gas excitation 23 . We find no trend of this ratio with individual far-IR line luminosities (e.g. Figure 13), either for the whole sample or for subsamples divided by optical type or PAH 11.2 µ m EW.</text>
<text><location><page_18><loc_8><loc_36><loc_48><loc_45></location>We also examined three mid-to-far and far-IR line ratios to try fashioning an excitation plane diagram, in a similar manner to Dale et al. (2006); [O IV ]26/[O III ], [N III ]/[N II ], and [O III ]/[N II ]. In no case did we find any trends. The large uncertainties on the [O IV ]26, [O III ], and [N III ] lines means though that we cannot conclude that such trends do not exist.</text>
<section_header_level_1><location><page_18><loc_22><loc_33><loc_35><loc_35></location>3.5. AGN Activity</section_header_level_1>
<text><location><page_18><loc_8><loc_16><loc_48><loc_33></location>We first compare far-IR line luminosities to optical spectral classification. We see no trends. If we normalize the line luminosities by L IR then no trends emerge, either for individual lines or sums of lines (Figure 14). We also see no trends if we compare optical class to line ratios, or normalized ratios. Moreover, the five objects with an additional broad component in [C II ] ( § 3.1.1) do not have an unusually high incidence of Seyfert spectra. We conclude that optical class cannot be inferred from far-IR line luminosities or ratios. This is consistent with the gas producing the optical emission not being strongly associated (in terms of heating mechanism) with the farIR line emitting gas, at least in the majority of cases.</text>
<text><location><page_18><loc_8><loc_12><loc_48><loc_16></location>Optical spectra may however misclassify AGN in obscured systems as H II or LINERs. We therefore employ the [Ne V ]14.32/[Ne II ]12.88 line ratio as an AGN</text>
<figure>
<location><page_18><loc_58><loc_66><loc_84><loc_92></location>
<caption>Figure 16. Results from simultaneous modelling of the [O I ]63/[C II ], [OI145/CII158], and ([O I ]63 +[C II ])/L FIR ratios, using PDRToolbox (Kaufman et al. 2006; Pound & Wolfire 2008), to constrain the electron density n and incident far-UV radiation field intensity G 0 ( § 3.6). The image is the median-combined stacked n vs. G 0 plane for the whole sample. Units of n are cm -3 and units of G 0 are the local Galactic interstellar FUV field found by Habing 1968 (1 . 6 × 10 -3 ergs cm -2 s -1 )). The x and y axis ranges are fixed by PDRToolbox.</caption>
</figure>
<text><location><page_18><loc_52><loc_39><loc_92><loc_53></location>diagnostic. Both these lines are less affected by extinction than are optical lines. The [Ne V ]14.32 line can arise in planetary nebulae and supernova remnants (Oliva et al. 1999). For extragalactic sources though, it is weak or absent in star forming regions (e.g. Lutz et al. 1998; Sturm et al. 2002; Bernard-Salas et al. 2009), but strong in spectra of AGN (e.g. Spinoglio et al. 2009). The [Ne II ]12.88 line on the other hand is seen almost universally in galaxies. Their ratio should therefore be a reasonable proxy for the presence of an AGN.</text>
<text><location><page_18><loc_52><loc_26><loc_92><loc_39></location>We plot [Ne V ]14.32/[Ne II ]12.88 against far-IR line luminosities in Figure 15. There is a correlation between the Neon line ratio and optical classification, but no correlations with far-IR line luminosity. If we substitute optical class for PAH 11.2 µ m EW, then no trends emerge. Considering the Sturm et al mixing lines (bottom right panel of Figure 15) then we see no trends among objects classified either as weak or strong AGN. Finally, if we instead plot line luminosity normalized by L IR on the x axis, then we still see no trends.</text>
<text><location><page_18><loc_52><loc_10><loc_92><loc_26></location>We searched for trends with far-IR line ratios, normalized ratios, sums and normalized (by L IR ) sums, but found nothing convincing, though the small number of sources with [Ne V ]14.32 detections means we are not certain that no trends exist. We conclude, cautiously, that for ULIRGs there is no reliable diagnostic of AGN luminosity using only simple combinations of far-IR line luminosities. This result is consistent with the weaker correlation observed between L Line and L IR if Sy1s are included (see § 3.2.1), if the AGN supplies an effectively random additional contribution to L IR , thus increasing the scatter in the relation.</text>
<text><location><page_19><loc_8><loc_70><loc_48><loc_92></location>From § 3.3 & 3.5, it is plausible that at least the plurality of the [C II ] emission arises from PDRs. We defer rigorous modelling to a future paper, and here only estimate the beam-averaged PDR hydrogen nucleus density, n (cm -3 ), and incident far-ultraviolet (FUV; 6 eV < E < 13.6 eV) radiation field intensity G 0 (in units of the local Galactic interstellar FUV field found by Habing 1968; 1 . 6 × 10 -3 erg cm -2 s -1 ) using the web-based tool PDR Toolbox 24 (Kaufman et al. 2006; Pound & Wolfire 2008). We set constraints using three line ratios; [O I ]63/[C II ], [O I ]145/[C II ], and ([O I ]63 +[C II ])/L FIR , where L FIR is the IR luminosity longward of 30 µ m. We assume that all three lines trace PDRs, and that there are no differential extinction effects. We estimate L FIR using the same methods as for L IR (Table 1).</text>
<text><location><page_19><loc_8><loc_50><loc_48><loc_70></location>For the whole sample (Figure 16) we find (taking a conservative cut of χ 2 red < 5) ranges of 10 1 < n < 10 2 . 5 and 10 2 . 2 < G 0 < 10 3 . 6 , with a power-law dependence between the two. The ranges of both n and G 0 depend on optical class (Figure 17). For H II objects we find 10 1 . 1 < n < 10 2 . 2 and 10 2 . 4 < G 0 < 10 3 . 3 . For LINERS and Seyferts however the ranges widen; for LINERS we find 10 0 . 8 < n < 10 2 . 5 and 10 2 . 4 < G 0 < 10 4 . 1 , and for Sy2s we find 10 0 . 7 < n < 10 3 and 10 1 . 9 < G 0 < 10 3 . 9 . For Sy1s the range for n is comparable to that of LINERs and Sy2s but the range for G 0 increases to 10 2 . 5 < G 0 < 10 4 . 7 . For Sy1s there is a secondary solution that is close to acceptable, which has G 0 and n values approximately four orders of magnitude lower and higher, respectively, than the primary solution.</text>
<text><location><page_19><loc_8><loc_30><loc_48><loc_50></location>If we divide the sample in two by PAH 11.2 EW (top row of Figure 18) then we see a difference, also. The range in n for both samples is comparable, at about 10 0 . 8 < n < 10 2 . 5 . The ranges for G 0 are however different; for objects with prominent PAHs it is 10 2 . 1 < G 0 < 10 3 . 7 while for objects with weak PAHs it is 10 2 . 6 < G 0 < 10 4 . 3 . We obtain similar ranges for both parameters if we instead divide the sample on merger stage (bottom row of Figure 18). This is consistent with a more intense ISRF destroying PAH molecules (see also e.g. Hern'an-Caballero et al. 2009). It is however also consistent with a luminous AGN (with a harder UV radiation field) arising after the star formation has faded. This would give the same observation but with no direct relation between the two phenomena.</text>
<text><location><page_19><loc_8><loc_10><loc_48><loc_30></location>There are three caveats in using these models to estimate G 0 and n for our sample. First, we cannot account for different beam filling factors for different lines. This is potentially a significant problem for [C II ] (see § 3.1.2 and § 3.2.3). Second, these models have difficulty in predicting PAH emission strengths (Luhman et al. 2003; Abel et al. 2009), suggesting an incomplete description of the dependence of far-IR line strengths on dust-grain size distribution, PAH properties and ISRF spectral shape (see also e.g. Okada et al. 2013). Since our targets are dusty, it is surprising that we obtain reasonable solutions, indicating that the line ratios and the adopted IR luminosities are compatible with each other. Our derived parameter ranges for G 0 and n should however be viewed with caution.</text>
<text><location><page_19><loc_52><loc_69><loc_92><loc_92></location>Third is that the [O I ] lines are complex to model. Assuming emission in PDRs then, like [C II ], the [O I ] lines are expected to form within A v ∼ 3 magnitudes of the PDR surfaces. It is in these regions that all of the carbon and oxygen should be ionized and atomic, respectively, with gas temperatures between about 250 and 700K (e.g. Kaufman et al. 1999). The dust in these regions has only a small effect on [C II ], but can have a large impact on the [O I ] levels, which are affected by both radiative and collisional processes. Such processes can alter the power of [O I ]63 via the absorption of 63 µ m line-emitted photons by dust grains, or by pumping of oxygen atoms by 63 µ m continuum dust emitted photons. The effect of dust should not be neglected when modeling [O I ]63, or the [O I ] line ratio, in sources that are optically thick at wavelengths shorter than 100 µ m (Gonz'alez-Alfonso et al. 2008).</text>
<section_header_level_1><location><page_19><loc_66><loc_67><loc_78><loc_68></location>3.7. Merger Stage</section_header_level_1>
<text><location><page_19><loc_52><loc_42><loc_92><loc_66></location>There is evidence that the power source in ULIRGs evolves as a function of merger stage, with star formation dominating, on average, until the progenitors coalesce, whereupon an AGN sometimes becomes energetically important (e.g. Rigopoulou et al. 1999; Farrah et al. 2009). We may therefore see a correlation between far-IR line properties and merger stage. We find however no correlations between merger stage and far-IR line luminosities. We also see no trend with any far-IR line ratio. Comparing merger stage to normalized far-IR line luminosities (Figure 19) there may be a weak trend; for [O I ]63 and longer lines, the advanced mergers might show a smaller normalized line luminosity than earlier stage mergers. This is consistent with the line luminosities tracing star formation, and with star formation becoming less important as merger stage advances. However, the trend is not strong, and does not depend on optical classification or PAH 11.2 µ m EW.</text>
<section_header_level_1><location><page_19><loc_66><loc_40><loc_78><loc_41></location>3.8. SMBH Mass</section_header_level_1>
<text><location><page_19><loc_52><loc_22><loc_92><loc_39></location>Scaling relations have been derived between the masses of supermassive black holes and the FWHM and continuum luminosities of several UV, optical, and midIR emission lines (Kaspi et al. 2000; Vestergaard 2002; Vestergaard & Peterson 2006; Dasyra et al. 2008). We here explore, using SMBH masses derived from optical lines (Table 1), whether there exist correlations between far-IR line properties and SMBH mass. While the absolute uncertainties on the SMBH masses from these studies are of order 0.4 dex, the relative uncertainties within the sample are likely smaller as we focus on one class of object and use H β derived masses in nearly all cases. We therefore assume an error on SMBH mass of 20%.</text>
<text><location><page_19><loc_52><loc_14><loc_92><loc_22></location>We compare the SMBH masses to line luminosities in Figure 20. For the whole sample, no line shows a trend with SMBH mass. Considering only the Sy1 and Sy2s, and excluding 3C273, then some line luminosities show, qualitatively, a positive trend with SMBH mass. The trend is only significant for L [NIII] , for which we derive:</text>
<formula><location><page_19><loc_53><loc_10><loc_92><loc_12></location>log(M BH ) = (1 . 09 ± 1 . 43)+(0 . 82 ± 0 . 18)log(L [NIII] ) (34)</formula>
<text><location><page_19><loc_52><loc_7><loc_92><loc_9></location>We see similar results if we instead compare M SMBH / L IR to L Line / L IR .</text>
<figure>
<location><page_20><loc_19><loc_36><loc_36><loc_90></location>
<caption>Figure 17. Results from PDRToolbox modelling, see Figure 16 for details. The panels show samples divided on optical spectral type.</caption>
</figure>
<text><location><page_20><loc_8><loc_10><loc_48><loc_28></location>It is plausible to exclude 3C273, since it is the only Blazar in the sample. We do not, however, claim that this relation is real, for four reasons. First, if we assume a (still reasonable) error on the SMBH masses in excess of 30% then the relation is no longer significant. Second, there is no trend of L [NIII] with the AGN diagnostics considered in § 3.5. Third, if this relation is real then it is strange that we do not see a correlation of SMBH mass with L [OIII] (see Table 2, though there is a potentially important difference; [N III ] is a ground-state transition whereas [O III ] is not). Fourth, we searched for correlations between SMBH mass and far-IR continuum luminosities near 57 µ m, but did not find any clear relations.</text>
<figure>
<location><page_20><loc_62><loc_36><loc_79><loc_90></location>
<caption>Figure 18. Results from PDRToolbox modelling, see Figure 16 for details. The panels show samples divided on PAH 11.2 µ m EW ( < 0 . 05 µ m vs. ≥ 0 . 05 µ m) and merger stage (Table 1).</caption>
</figure>
<text><location><page_20><loc_52><loc_21><loc_92><loc_29></location>We have presented observations with PACS onboard Herschel of 25 ULIRGs at z < 0 . 27. We observed each ULIRG in six lines: [O III ]52 µ m, [N III ]57 µ m, [O I ]63 µ m, [N II ]122 µ m, [O I ]145 µ m, and [C II ]158 µ m. We used the properties of these lines, together with diagnostics at other wavelengths, to draw the following conclusions:</text>
<text><location><page_20><loc_52><loc_7><loc_92><loc_21></location>1 - In most cases the line profiles are reproducible by single gaussians, with widths between 250 km s -1 and 600 km s -1 . The exceptions are [O I ]63 and [C II ], which occasionally show self absorption and a second, broad component, respectively. We do not see significant systemic offsets of the far-IR lines compared to the optical redshifts. The line luminosities range from just under 10 7 L /circledot to just over 2 × 10 9 L /circledot . The [O I ]63 and [C II ] lines are usually the most luminous, while [O I ]145 and [N II ] are usually the least luminous. The line luminosities correlate with each other, though in no case is the</text>
<figure>
<location><page_21><loc_27><loc_54><loc_74><loc_91></location>
<caption>Figure 19. Line luminosities normalized by L IR , vs Merger Stage ( § 3.7).</caption>
</figure>
<text><location><page_21><loc_51><loc_54><loc_53><loc_54></location>Line</text>
<text><location><page_21><loc_54><loc_54><loc_55><loc_54></location>IR</text>
<text><location><page_21><loc_8><loc_44><loc_48><loc_49></location>correlation particularly strong. Simple line ratio diagnostics suggest relatively low gas densities, on average, and that a significant fraction of the [C II ] emission originates from outside H II regions.</text>
<text><location><page_21><loc_8><loc_22><loc_48><loc_44></location>2 - There is a deficit in the [O I ]63/L IR , [N II ]/L IR , [O I ]145/L IR , and [C II ]/L IR ratios compared to lower luminosity systems, of factors of 2.75, 4.46, 1.50, and 4.95, respectively. There is evidence that the [C II ] and [N II ] deficits correlate with 9.7 µ m silicate feature strength (S Sil ); if S Sil /greaterorsimilar 1 . 4 then the [C II ] and [N II ] deficits rise with increasing S Sil . We also see a correlation between [L PAH ]/L IR and S Sil . Furthermore, the [C II ] deficit correlates with merger stage; objects in advanced mergers show a greater deficit than objects in early stage mergers. These results are consistent with the majority of the line deficits arising due to increased levels of dust in H II regions. We propose though that a significant fraction of the [C II ] deficit arises from an additional mechanism, plausibly grain charging in PDRs and/or the diffuse ISM.</text>
<text><location><page_21><loc_8><loc_17><loc_48><loc_22></location>3 - The line luminosities only weakly correlate with IR luminosity. The correlations improve if Sy1 objects are excluded. Doing so, and fitting a relation of the form log(L IR ) = α + β log(L Line ), yields:</text>
<formula><location><page_21><loc_10><loc_7><loc_48><loc_14></location>log(L IR ) = (4 . 46 ± 1 . 77) + (0 . 92 ± 0 . 20)log(L [OIII] )(35) =(4 . 74 ± 1 . 59) + (0 . 94 ± 0 . 19)log(L [NIII] )(36) =(4 . 28 ± 1 . 89) + (0 . 91 ± 0 . 21)log(L [OI]63 )(37) =(5 . 29 ± 1 . 57) + (0 . 89 ± 0 . 20)log(L [NII] ) (38)</formula>
<formula><location><page_21><loc_60><loc_46><loc_92><loc_49></location>=(1 . 75 ± 2 . 11) + (1 . 34 ± 0 . 27)log(L [OI]145 ) (39) =(6 . 73 ± 2 . 44) + (0 . 64 ± 0 . 27)log(L [CII] ) (40)</formula>
<text><location><page_21><loc_52><loc_38><loc_92><loc_45></location>The best tracers of L IR are thus the five shorter wavelength lines. The [C II ] line is a poor tracer of L IR , accurate to about an order of magnitude at best. Its accuracy does not noticeably improve if objects with a strong [C II ] deficit are excluded.</text>
<text><location><page_21><loc_52><loc_34><loc_92><loc_38></location>4 - The continuum luminosity densities near the wavelengths of the lines correlate with L IR , irrespective of the presence of Sy1s. We derive:</text>
<formula><location><page_21><loc_54><loc_22><loc_92><loc_32></location>log(L IR ) = (12 . 87 ± 0 . 09) + (0 . 88 ± 0 . 17)log(L 52 ) (41) =(12 . 86 ± 0 . 05) + (0 . 91 ± 0 . 11)log(L 57 ) (42) =(12 . 86 ± 0 . 08) + (0 . 98 ± 0 . 18)log(L 63 ) (43) =(12 . 95 ± 0 . 07) + (0 . 87 ± 0 . 11)log(L 122 )(44) =(13 . 05 ± 0 . 11) + (0 . 90 ± 0 . 16)log(L 145 )(45) =(13 . 08 ± 0 . 11) + (0 . 83 ± 0 . 11)log(L 158 )(46)</formula>
<text><location><page_21><loc_52><loc_16><loc_92><loc_21></location>5 - We find correlations between star formation rate, estimated using L PAH , and both line luminosities and continuum luminosity densities. For line luminosities we derive:</text>
<formula><location><page_21><loc_53><loc_7><loc_92><loc_14></location>log( ˙ M /circledot ) = ( -7 . 02 ± 1 . 25) + (1 . 07 ± 0 . 14)log(L [OIII] )(47) =( -5 . 13 ± 0 . 72) + (0 . 91 ± 0 . 09)log(L [NIII] )(48) =( -5 . 44 ± 1 . 79) + (0 . 86 ± 0 . 20)log(L [OI]63 ) (49) =( -7 . 30 ± 0 . 87) + (1 . 19 ± 0 . 11)log(L [NII] )(50)</formula>
<figure>
<location><page_22><loc_27><loc_44><loc_74><loc_91></location>
<caption>Figure 20. Far-IR line luminosities vs SMBH Mass ( § 3.8). The fit in the [N III ] panel is Equation 34, and is to the Seyferts, excluding 3C273.</caption>
</figure>
<formula><location><page_22><loc_15><loc_35><loc_48><loc_38></location>=( -10 . 04 ± 1 . 34) + (1 . 55 ± 0 . 17)log(L [OI]145 ) (51) =( -6 . 24 ± 1 . 72) + (0 . 95 ± 0 . 19)log(L [CII] )(52)</formula>
<text><location><page_22><loc_8><loc_33><loc_48><loc_34></location>while for the continuum luminosity densities we derive:</text>
<formula><location><page_22><loc_11><loc_21><loc_48><loc_31></location>log( ˙ M /circledot ) = (3 . 24 ± 0 . 24) + (1 . 89 ± 0 . 46)log(L 52 ) (53) =(2 . 95 ± 0 . 10) + (1 . 38 ± 0 . 19)log(L 57 ) (54) =(3 . 04 ± 0 . 17) + (1 . 79 ± 0 . 39)log(L 63 ) (55) =(2 . 83 ± 0 . 11) + (0 . 93 ± 0 . 16)log(L 122 )(56) =(3 . 02 ± 0 . 18) + (1 . 09 ± 0 . 25)log(L 145 )(57) =(3 . 36 ± 0 . 22) + (1 . 42 ± 0 . 30)log(L 158 )(58)</formula>
<text><location><page_22><loc_8><loc_17><loc_48><loc_20></location>On average, the shorter wavelength continua show stronger correlations.</text>
<text><location><page_22><loc_8><loc_7><loc_48><loc_17></location>6 - Assuming the [O I ] and [C II ] lines arise mainly in PDRs, we use a simple model to extract estimates for the hydrogen nucleus density, n , and incident far-ultraviolet radiation field G 0 , in the far-IR line emitting gas. We find 10 1 < n < 10 2 . 5 and 10 2 . 2 < G 0 < 10 3 . 6 for the whole sample, with a power-law dependence between the two. The ranges depend on optical spectral class; for H II -like objects we find 10 1 . 1 < n < 10 2 . 2 and 10 2 . 4 < G 0 < 10 3 . 3 ,</text>
<text><location><page_22><loc_52><loc_28><loc_92><loc_38></location>while for Sy1s we find 10 0 . 8 < n < 10 2 . 7 and 10 2 . 5 < G 0 < 10 4 . 7 . There is also a dependence of G 0 on the importance of star formation; objects with weak PAHs have 10 2 . 6 < G 0 < 10 4 . 3 while objects with prominent PAHs have 10 2 . 1 < G 0 < 10 3 . 7 . We find similar ranges for early- vs. late-stage mergers. This is consistent with, but not exclusively supportive of, a more intense ISRF destroying PAH molecules.</text>
<text><location><page_22><loc_52><loc_10><loc_92><loc_28></location>7 - We searched for relations between far-IR line luminosities and ratios, and several other parameters; AGN activity (either from optical spectral class or the detection of [Ne V ]14.32), merger stage, mid-IR excitation, and SMBH mass. For the first three parameters we found no relations. We conclude that the far-IR lines do not arise primarily due to AGN activity, and that the properties of the far-IR line emitting gas do not strongly depend on either mid-IR excitation or merger stage. For SMBH mass we found one superficially striking correlation, with L [NIII] , but subsequent tests were not supportive. We conclude that far-IR line luminosities do not straightforwardly trace SMBH mass.</text>
<text><location><page_22><loc_53><loc_7><loc_92><loc_8></location>We thank the staff of the Herschel helpdesk for many</text>
<text><location><page_23><loc_8><loc_60><loc_48><loc_92></location>valuable discussions, and the referee for a very helpful report. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. This work is based on observations made with the Spitzer Space Telescope. Spitzer is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research has made extensive use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA, and of NASA's Astrophysics Data System. This research has also made use of Ned Wrights online cosmology calculator (Wright 2006). V.L. is supported by a CEA/Marie Curie Eurotalents fellowship. J.A. acknowledges support from the Science and Technology Foundation (FCT, Portugal) through the research grants PTDC/CTE-AST/105287/2008, PEstOE/FIS/UI2751/2011 and PTDC/FIS-AST/2194/2012. E.G-A is a Research Associate at the HarvardSmithsonian Center for Astrophysics, and thanks the support by the Spanish Ministerio de Econom'ıa y Competitividad under projects AYA2010-21697-C05-0 and FIS2012-39162-C06-01.</text>
<text><location><page_23><loc_10><loc_59><loc_30><loc_60></location>Facilities: Herschel, Spitzer.</text>
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<text><location><page_25><loc_30><loc_82><loc_32><loc_83></location>]</text>
<text><location><page_25><loc_30><loc_82><loc_32><loc_82></location>I</text>
<text><location><page_25><loc_30><loc_82><loc_32><loc_82></location>I</text>
<text><location><page_25><loc_30><loc_81><loc_32><loc_81></location>C</text>
<text><location><page_25><loc_30><loc_80><loc_32><loc_81></location>[</text>
<text><location><page_25><loc_30><loc_72><loc_32><loc_72></location>5</text>
<text><location><page_25><loc_30><loc_71><loc_32><loc_72></location>4</text>
<text><location><page_25><loc_30><loc_70><loc_32><loc_71></location>1</text>
<text><location><page_25><loc_30><loc_16><loc_32><loc_16></location>y</text>
<text><location><page_25><loc_30><loc_15><loc_32><loc_16></location>x</text>
<text><location><page_25><loc_30><loc_15><loc_32><loc_15></location>a</text>
<text><location><page_25><loc_30><loc_15><loc_32><loc_15></location>l</text>
<text><location><page_25><loc_30><loc_14><loc_32><loc_15></location>a</text>
<text><location><page_25><loc_30><loc_13><loc_32><loc_14></location>G</text>
<text><location><page_25><loc_33><loc_87><loc_34><loc_87></location>)</text>
<text><location><page_25><loc_33><loc_86><loc_34><loc_87></location>2</text>
<text><location><page_25><loc_33><loc_86><loc_34><loc_86></location>(</text>
<text><location><page_25><loc_33><loc_85><loc_34><loc_85></location>3</text>
<text><location><page_25><loc_33><loc_84><loc_34><loc_85></location>1</text>
<text><location><page_25><loc_33><loc_83><loc_34><loc_84></location>±</text>
<text><location><page_25><loc_33><loc_83><loc_34><loc_84></location>9</text>
<text><location><page_25><loc_33><loc_82><loc_34><loc_83></location>7</text>
<text><location><page_25><loc_33><loc_82><loc_34><loc_82></location>2</text>
<text><location><page_25><loc_34><loc_85><loc_35><loc_85></location>1</text>
<text><location><page_25><loc_34><loc_84><loc_35><loc_85></location>6</text>
<text><location><page_25><loc_34><loc_84><loc_35><loc_84></location>±</text>
<text><location><page_25><loc_34><loc_83><loc_35><loc_84></location>7</text>
<text><location><page_25><loc_34><loc_82><loc_35><loc_83></location>2</text>
<text><location><page_25><loc_34><loc_82><loc_35><loc_82></location>4</text>
<text><location><page_25><loc_34><loc_81><loc_34><loc_81></location>a</text>
<text><location><page_25><loc_33><loc_81><loc_34><loc_81></location>2</text>
<text><location><page_25><loc_33><loc_80><loc_34><loc_81></location>0</text>
<text><location><page_25><loc_33><loc_80><loc_34><loc_80></location>.</text>
<text><location><page_25><loc_33><loc_79><loc_34><loc_80></location>1</text>
<text><location><page_25><loc_33><loc_78><loc_34><loc_79></location>±</text>
<text><location><page_25><loc_33><loc_78><loc_34><loc_78></location>5</text>
<text><location><page_25><loc_33><loc_77><loc_34><loc_78></location>8</text>
<text><location><page_25><loc_33><loc_77><loc_34><loc_77></location>.</text>
<text><location><page_25><loc_33><loc_76><loc_34><loc_77></location>3</text>
<text><location><page_25><loc_33><loc_75><loc_34><loc_76></location>)</text>
<text><location><page_25><loc_33><loc_75><loc_34><loc_75></location>2</text>
<text><location><page_25><loc_33><loc_74><loc_34><loc_75></location>(</text>
<text><location><page_25><loc_33><loc_73><loc_34><loc_74></location>3</text>
<text><location><page_25><loc_33><loc_73><loc_34><loc_73></location>5</text>
<text><location><page_25><loc_33><loc_72><loc_34><loc_73></location>±</text>
<text><location><page_25><loc_33><loc_71><loc_34><loc_72></location>0</text>
<text><location><page_25><loc_33><loc_71><loc_34><loc_71></location>5</text>
<text><location><page_25><loc_34><loc_80><loc_35><loc_81></location>5</text>
<text><location><page_25><loc_34><loc_80><loc_35><loc_80></location>2</text>
<text><location><page_25><loc_34><loc_79><loc_35><loc_80></location>.</text>
<text><location><page_25><loc_34><loc_79><loc_35><loc_79></location>0</text>
<text><location><page_25><loc_34><loc_78><loc_35><loc_79></location>±</text>
<text><location><page_25><loc_34><loc_78><loc_35><loc_78></location>1</text>
<text><location><page_25><loc_34><loc_77><loc_35><loc_78></location>6</text>
<text><location><page_25><loc_34><loc_77><loc_35><loc_77></location>.</text>
<text><location><page_25><loc_34><loc_76><loc_35><loc_77></location>1</text>
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<text><location><page_25><loc_34><loc_75><loc_35><loc_75></location>1</text>
<text><location><page_25><loc_34><loc_74><loc_35><loc_75></location>(</text>
<text><location><page_25><loc_34><loc_73><loc_35><loc_74></location>4</text>
<text><location><page_25><loc_34><loc_73><loc_35><loc_73></location>9</text>
<text><location><page_25><loc_34><loc_72><loc_35><loc_73></location>±</text>
<text><location><page_25><loc_34><loc_71><loc_35><loc_72></location>0</text>
<text><location><page_25><loc_34><loc_71><loc_35><loc_71></location>5</text>
<text><location><page_25><loc_36><loc_93><loc_63><loc_94></location>Far-infrared spectroscopy of ULIRGs</text>
<text><location><page_25><loc_90><loc_93><loc_92><loc_94></location>25</text>
<text><location><page_25><loc_35><loc_87><loc_37><loc_87></location>)</text>
<text><location><page_25><loc_35><loc_86><loc_37><loc_87></location>2</text>
<text><location><page_25><loc_35><loc_86><loc_37><loc_86></location>(</text>
<text><location><page_25><loc_35><loc_85><loc_37><loc_85></location>7</text>
<text><location><page_25><loc_35><loc_84><loc_37><loc_85></location>±</text>
<text><location><page_25><loc_35><loc_83><loc_37><loc_84></location>0</text>
<text><location><page_25><loc_35><loc_83><loc_37><loc_83></location>5</text>
<text><location><page_25><loc_35><loc_82><loc_37><loc_83></location>1</text>
<text><location><page_25><loc_35><loc_81><loc_36><loc_81></location>d</text>
<text><location><page_25><loc_35><loc_80><loc_37><loc_81></location>3</text>
<text><location><page_25><loc_35><loc_80><loc_37><loc_80></location>2</text>
<text><location><page_25><loc_35><loc_79><loc_37><loc_80></location>.</text>
<text><location><page_25><loc_35><loc_79><loc_37><loc_79></location>1</text>
<text><location><page_25><loc_35><loc_78><loc_37><loc_79></location>±</text>
<text><location><page_25><loc_35><loc_77><loc_37><loc_78></location>6</text>
<text><location><page_25><loc_35><loc_77><loc_37><loc_77></location>3</text>
<text><location><page_25><loc_35><loc_77><loc_37><loc_77></location>.</text>
<text><location><page_25><loc_35><loc_76><loc_37><loc_77></location>5</text>
<text><location><page_25><loc_35><loc_75><loc_37><loc_76></location>)</text>
<text><location><page_25><loc_35><loc_75><loc_37><loc_75></location>1</text>
<text><location><page_25><loc_35><loc_74><loc_37><loc_75></location>(</text>
<text><location><page_25><loc_35><loc_73><loc_37><loc_74></location>9</text>
<text><location><page_25><loc_35><loc_73><loc_37><loc_73></location>3</text>
<text><location><page_25><loc_35><loc_72><loc_37><loc_73></location>±</text>
<text><location><page_25><loc_35><loc_71><loc_37><loc_72></location>5</text>
<text><location><page_25><loc_35><loc_71><loc_37><loc_71></location>6</text>
<text><location><page_25><loc_37><loc_87><loc_38><loc_87></location>)</text>
<text><location><page_25><loc_37><loc_86><loc_38><loc_87></location>2</text>
<text><location><page_25><loc_37><loc_86><loc_38><loc_86></location>(</text>
<text><location><page_25><loc_37><loc_85><loc_38><loc_85></location>7</text>
<text><location><page_25><loc_37><loc_84><loc_38><loc_85></location>1</text>
<text><location><page_25><loc_37><loc_84><loc_38><loc_84></location>±</text>
<text><location><page_25><loc_37><loc_83><loc_38><loc_84></location>6</text>
<text><location><page_25><loc_37><loc_82><loc_38><loc_83></location>1</text>
<text><location><page_25><loc_37><loc_82><loc_38><loc_82></location>3</text>
<text><location><page_25><loc_37><loc_81><loc_38><loc_81></location>7</text>
<text><location><page_25><loc_37><loc_80><loc_38><loc_81></location>0</text>
<text><location><page_25><loc_37><loc_80><loc_38><loc_80></location>.</text>
<text><location><page_25><loc_37><loc_79><loc_38><loc_80></location>1</text>
<text><location><page_25><loc_37><loc_78><loc_38><loc_79></location>±</text>
<text><location><page_25><loc_37><loc_78><loc_38><loc_78></location>3</text>
<text><location><page_25><loc_37><loc_77><loc_38><loc_78></location>3</text>
<text><location><page_25><loc_37><loc_77><loc_38><loc_77></location>.</text>
<text><location><page_25><loc_37><loc_76><loc_38><loc_77></location>4</text>
<text><location><page_25><loc_37><loc_75><loc_38><loc_76></location>)</text>
<text><location><page_25><loc_37><loc_75><loc_38><loc_75></location>2</text>
<text><location><page_25><loc_37><loc_74><loc_38><loc_75></location>(</text>
<text><location><page_25><loc_37><loc_73><loc_38><loc_74></location>9</text>
<text><location><page_25><loc_37><loc_73><loc_38><loc_73></location>6</text>
<text><location><page_25><loc_37><loc_72><loc_38><loc_73></location>±</text>
<text><location><page_25><loc_37><loc_71><loc_38><loc_72></location>8</text>
<text><location><page_25><loc_37><loc_71><loc_38><loc_71></location>5</text>
<text><location><page_25><loc_38><loc_87><loc_40><loc_87></location>)</text>
<text><location><page_25><loc_38><loc_86><loc_40><loc_87></location>2</text>
<text><location><page_25><loc_38><loc_86><loc_40><loc_86></location>(</text>
<text><location><page_25><loc_38><loc_85><loc_40><loc_85></location>4</text>
<text><location><page_25><loc_38><loc_84><loc_40><loc_85></location>±</text>
<text><location><page_25><loc_38><loc_83><loc_40><loc_84></location>3</text>
<text><location><page_25><loc_38><loc_83><loc_40><loc_83></location>4</text>
<text><location><page_25><loc_38><loc_82><loc_40><loc_83></location>2</text>
<text><location><page_25><loc_38><loc_81><loc_40><loc_81></location>1</text>
<text><location><page_25><loc_38><loc_80><loc_40><loc_81></location>1</text>
<text><location><page_25><loc_38><loc_80><loc_40><loc_80></location>.</text>
<text><location><page_25><loc_38><loc_79><loc_40><loc_80></location>0</text>
<text><location><page_25><loc_38><loc_78><loc_40><loc_79></location>±</text>
<text><location><page_25><loc_38><loc_78><loc_40><loc_78></location>3</text>
<text><location><page_25><loc_38><loc_77><loc_40><loc_78></location>6</text>
<text><location><page_25><loc_38><loc_77><loc_40><loc_77></location>.</text>
<text><location><page_25><loc_38><loc_76><loc_40><loc_77></location>5</text>
<text><location><page_25><loc_38><loc_75><loc_40><loc_76></location>)</text>
<text><location><page_25><loc_38><loc_75><loc_40><loc_75></location>2</text>
<text><location><page_25><loc_38><loc_74><loc_40><loc_75></location>(</text>
<text><location><page_25><loc_38><loc_73><loc_40><loc_74></location>8</text>
<text><location><page_25><loc_38><loc_73><loc_40><loc_73></location>4</text>
<text><location><page_25><loc_38><loc_72><loc_40><loc_73></location>±</text>
<text><location><page_25><loc_38><loc_71><loc_40><loc_72></location>4</text>
<text><location><page_25><loc_38><loc_71><loc_40><loc_71></location>1</text>
<text><location><page_25><loc_40><loc_87><loc_41><loc_87></location>)</text>
<text><location><page_25><loc_40><loc_86><loc_41><loc_87></location>2</text>
<text><location><page_25><loc_40><loc_86><loc_41><loc_86></location>(</text>
<text><location><page_25><loc_40><loc_85><loc_41><loc_85></location>7</text>
<text><location><page_25><loc_40><loc_84><loc_41><loc_85></location>2</text>
<text><location><page_25><loc_40><loc_83><loc_41><loc_84></location>±</text>
<text><location><page_25><loc_40><loc_83><loc_41><loc_84></location>2</text>
<text><location><page_25><loc_40><loc_82><loc_41><loc_83></location>4</text>
<text><location><page_25><loc_40><loc_82><loc_41><loc_82></location>4</text>
<text><location><page_25><loc_40><loc_80><loc_41><loc_81></location>8</text>
<text><location><page_25><loc_40><loc_80><loc_41><loc_81></location>0</text>
<text><location><page_25><loc_40><loc_80><loc_41><loc_80></location>.</text>
<text><location><page_25><loc_40><loc_79><loc_41><loc_80></location>1</text>
<text><location><page_25><loc_40><loc_78><loc_41><loc_79></location>±</text>
<text><location><page_25><loc_40><loc_78><loc_41><loc_78></location>5</text>
<text><location><page_25><loc_40><loc_77><loc_41><loc_78></location>7</text>
<text><location><page_25><loc_40><loc_77><loc_41><loc_77></location>.</text>
<text><location><page_25><loc_40><loc_76><loc_41><loc_77></location>3</text>
<text><location><page_25><loc_40><loc_75><loc_41><loc_76></location>)</text>
<text><location><page_25><loc_40><loc_75><loc_41><loc_75></location>2</text>
<text><location><page_25><loc_40><loc_74><loc_41><loc_75></location>(</text>
<text><location><page_25><loc_40><loc_73><loc_41><loc_74></location>4</text>
<text><location><page_25><loc_40><loc_73><loc_41><loc_73></location>1</text>
<text><location><page_25><loc_40><loc_72><loc_41><loc_73></location>1</text>
<text><location><page_25><loc_40><loc_71><loc_41><loc_72></location>±</text>
<text><location><page_25><loc_40><loc_71><loc_41><loc_71></location>0</text>
<text><location><page_25><loc_40><loc_70><loc_41><loc_71></location>5</text>
<text><location><page_25><loc_41><loc_87><loc_42><loc_87></location>)</text>
<text><location><page_25><loc_41><loc_86><loc_42><loc_87></location>4</text>
<text><location><page_25><loc_41><loc_86><loc_42><loc_86></location>(</text>
<text><location><page_25><loc_41><loc_85><loc_42><loc_85></location>2</text>
<text><location><page_25><loc_41><loc_84><loc_42><loc_85></location>±</text>
<text><location><page_25><loc_41><loc_83><loc_42><loc_84></location>7</text>
<text><location><page_25><loc_41><loc_83><loc_42><loc_83></location>7</text>
<text><location><page_25><loc_41><loc_82><loc_42><loc_83></location>2</text>
<text><location><page_25><loc_41><loc_81><loc_42><loc_82></location>d</text>
<text><location><page_25><loc_41><loc_80><loc_42><loc_81></location>4</text>
<text><location><page_25><loc_41><loc_80><loc_42><loc_80></location>0</text>
<text><location><page_25><loc_41><loc_80><loc_42><loc_80></location>.</text>
<text><location><page_25><loc_41><loc_79><loc_42><loc_80></location>2</text>
<text><location><page_25><loc_41><loc_78><loc_42><loc_79></location>±</text>
<text><location><page_25><loc_41><loc_78><loc_42><loc_78></location>8</text>
<text><location><page_25><loc_41><loc_77><loc_42><loc_78></location>5</text>
<text><location><page_25><loc_41><loc_77><loc_42><loc_77></location>.</text>
<text><location><page_25><loc_41><loc_76><loc_42><loc_77></location>0</text>
<text><location><page_25><loc_41><loc_76><loc_42><loc_76></location>3</text>
<text><location><page_25><loc_41><loc_75><loc_42><loc_76></location>)</text>
<text><location><page_25><loc_41><loc_75><loc_42><loc_75></location>2</text>
<text><location><page_25><loc_41><loc_74><loc_42><loc_75></location>(</text>
<text><location><page_25><loc_41><loc_73><loc_42><loc_74></location>6</text>
<text><location><page_25><loc_41><loc_73><loc_42><loc_73></location>1</text>
<text><location><page_25><loc_41><loc_72><loc_42><loc_73></location>±</text>
<text><location><page_25><loc_41><loc_71><loc_42><loc_72></location>2</text>
<text><location><page_25><loc_41><loc_71><loc_42><loc_71></location>4</text>
<text><location><page_25><loc_43><loc_87><loc_44><loc_87></location>)</text>
<text><location><page_25><loc_43><loc_86><loc_44><loc_87></location>2</text>
<text><location><page_25><loc_43><loc_86><loc_44><loc_86></location>(</text>
<text><location><page_25><loc_43><loc_85><loc_44><loc_85></location>0</text>
<text><location><page_25><loc_43><loc_84><loc_44><loc_85></location>1</text>
<text><location><page_25><loc_43><loc_83><loc_44><loc_84></location>±</text>
<text><location><page_25><loc_43><loc_83><loc_44><loc_84></location>9</text>
<text><location><page_25><loc_43><loc_82><loc_44><loc_83></location>6</text>
<text><location><page_25><loc_43><loc_82><loc_44><loc_82></location>3</text>
<text><location><page_25><loc_43><loc_81><loc_44><loc_81></location>9</text>
<text><location><page_25><loc_43><loc_80><loc_44><loc_81></location>1</text>
<text><location><page_25><loc_43><loc_80><loc_44><loc_80></location>.</text>
<text><location><page_25><loc_43><loc_79><loc_44><loc_80></location>1</text>
<text><location><page_25><loc_43><loc_78><loc_44><loc_79></location>±</text>
<text><location><page_25><loc_43><loc_78><loc_44><loc_78></location>5</text>
<text><location><page_25><loc_43><loc_77><loc_44><loc_78></location>2</text>
<text><location><page_25><loc_43><loc_77><loc_44><loc_77></location>.</text>
<text><location><page_25><loc_43><loc_76><loc_44><loc_77></location>9</text>
<text><location><page_25><loc_43><loc_75><loc_44><loc_76></location>)</text>
<text><location><page_25><loc_43><loc_75><loc_44><loc_75></location>2</text>
<text><location><page_25><loc_43><loc_74><loc_44><loc_75></location>(</text>
<text><location><page_25><loc_43><loc_73><loc_44><loc_74></location>6</text>
<text><location><page_25><loc_43><loc_73><loc_44><loc_73></location>6</text>
<text><location><page_25><loc_43><loc_72><loc_44><loc_73></location>±</text>
<text><location><page_25><loc_43><loc_71><loc_44><loc_72></location>0</text>
<text><location><page_25><loc_43><loc_71><loc_44><loc_71></location>5</text>
<text><location><page_25><loc_44><loc_87><loc_45><loc_87></location>)</text>
<text><location><page_25><loc_44><loc_86><loc_45><loc_87></location>1</text>
<text><location><page_25><loc_44><loc_86><loc_45><loc_86></location>(</text>
<text><location><page_25><loc_44><loc_85><loc_45><loc_85></location>5</text>
<text><location><page_25><loc_44><loc_84><loc_45><loc_85></location>3</text>
<text><location><page_25><loc_44><loc_83><loc_45><loc_84></location>±</text>
<text><location><page_25><loc_44><loc_83><loc_45><loc_84></location>7</text>
<text><location><page_25><loc_44><loc_82><loc_45><loc_83></location>9</text>
<text><location><page_25><loc_44><loc_82><loc_45><loc_82></location>1</text>
<text><location><page_25><loc_44><loc_80><loc_45><loc_81></location>8</text>
<text><location><page_25><loc_44><loc_80><loc_45><loc_81></location>3</text>
<text><location><page_25><loc_44><loc_80><loc_45><loc_80></location>.</text>
<text><location><page_25><loc_44><loc_79><loc_45><loc_80></location>0</text>
<text><location><page_25><loc_44><loc_78><loc_45><loc_79></location>±</text>
<text><location><page_25><loc_44><loc_78><loc_45><loc_78></location>0</text>
<text><location><page_25><loc_44><loc_77><loc_45><loc_78></location>1</text>
<text><location><page_25><loc_44><loc_77><loc_45><loc_77></location>.</text>
<text><location><page_25><loc_44><loc_76><loc_45><loc_77></location>2</text>
<text><location><page_25><loc_44><loc_75><loc_45><loc_76></location>)</text>
<text><location><page_25><loc_44><loc_75><loc_45><loc_75></location>1</text>
<text><location><page_25><loc_45><loc_87><loc_47><loc_87></location>)</text>
<text><location><page_25><loc_45><loc_86><loc_47><loc_87></location>1</text>
<text><location><page_25><loc_45><loc_86><loc_47><loc_86></location>(</text>
<text><location><page_25><loc_45><loc_85><loc_47><loc_85></location>3</text>
<text><location><page_25><loc_45><loc_84><loc_47><loc_85></location>±</text>
<text><location><page_25><loc_45><loc_83><loc_47><loc_84></location>2</text>
<text><location><page_25><loc_45><loc_83><loc_47><loc_83></location>7</text>
<text><location><page_25><loc_45><loc_82><loc_47><loc_83></location>2</text>
<text><location><page_25><loc_45><loc_81><loc_47><loc_81></location>5</text>
<text><location><page_25><loc_45><loc_80><loc_47><loc_81></location>3</text>
<text><location><page_25><loc_45><loc_80><loc_47><loc_80></location>.</text>
<text><location><page_25><loc_45><loc_79><loc_47><loc_80></location>0</text>
<text><location><page_25><loc_45><loc_79><loc_47><loc_79></location>±</text>
<text><location><page_25><loc_45><loc_78><loc_47><loc_79></location>2</text>
<text><location><page_25><loc_45><loc_78><loc_47><loc_78></location>3</text>
<text><location><page_25><loc_45><loc_77><loc_47><loc_78></location>.</text>
<text><location><page_25><loc_45><loc_77><loc_47><loc_77></location>4</text>
<text><location><page_25><loc_45><loc_76><loc_47><loc_77></location>2</text>
<text><location><page_25><loc_45><loc_75><loc_47><loc_76></location>)</text>
<text><location><page_25><loc_45><loc_75><loc_47><loc_75></location>1</text>
<text><location><page_25><loc_47><loc_87><loc_48><loc_87></location>)</text>
<text><location><page_25><loc_47><loc_86><loc_48><loc_87></location>1</text>
<text><location><page_25><loc_47><loc_86><loc_48><loc_86></location>(</text>
<text><location><page_25><loc_47><loc_85><loc_48><loc_85></location>0</text>
<text><location><page_25><loc_47><loc_84><loc_48><loc_85></location>2</text>
<text><location><page_25><loc_47><loc_83><loc_48><loc_84></location>±</text>
<text><location><page_25><loc_47><loc_83><loc_48><loc_84></location>9</text>
<text><location><page_25><loc_47><loc_82><loc_48><loc_83></location>0</text>
<text><location><page_25><loc_47><loc_82><loc_48><loc_82></location>4</text>
<text><location><page_25><loc_47><loc_81><loc_48><loc_81></location>0</text>
<text><location><page_25><loc_47><loc_80><loc_48><loc_81></location>2</text>
<text><location><page_25><loc_47><loc_80><loc_48><loc_80></location>.</text>
<text><location><page_25><loc_47><loc_79><loc_48><loc_80></location>0</text>
<text><location><page_25><loc_47><loc_78><loc_48><loc_79></location>±</text>
<text><location><page_25><loc_47><loc_78><loc_48><loc_78></location>3</text>
<text><location><page_25><loc_47><loc_77><loc_48><loc_78></location>5</text>
<text><location><page_25><loc_47><loc_77><loc_48><loc_77></location>.</text>
<text><location><page_25><loc_47><loc_76><loc_48><loc_77></location>3</text>
<text><location><page_25><loc_47><loc_75><loc_48><loc_76></location>)</text>
<text><location><page_25><loc_47><loc_75><loc_48><loc_75></location>1</text>
<text><location><page_25><loc_48><loc_87><loc_50><loc_87></location>)</text>
<text><location><page_25><loc_48><loc_86><loc_50><loc_87></location>2</text>
<text><location><page_25><loc_48><loc_86><loc_50><loc_86></location>(</text>
<text><location><page_25><loc_48><loc_85><loc_50><loc_85></location>9</text>
<text><location><page_25><loc_48><loc_84><loc_50><loc_85></location>±</text>
<text><location><page_25><loc_48><loc_83><loc_50><loc_84></location>6</text>
<text><location><page_25><loc_48><loc_83><loc_50><loc_83></location>9</text>
<text><location><page_25><loc_48><loc_82><loc_50><loc_83></location>2</text>
<text><location><page_25><loc_48><loc_81><loc_49><loc_81></location>d</text>
<text><location><page_25><loc_48><loc_80><loc_50><loc_81></location>8</text>
<text><location><page_25><loc_48><loc_80><loc_50><loc_80></location>1</text>
<text><location><page_25><loc_48><loc_79><loc_50><loc_80></location>.</text>
<text><location><page_25><loc_48><loc_79><loc_50><loc_79></location>1</text>
<text><location><page_25><loc_48><loc_78><loc_50><loc_79></location>±</text>
<text><location><page_25><loc_48><loc_77><loc_50><loc_78></location>1</text>
<text><location><page_25><loc_48><loc_77><loc_50><loc_77></location>5</text>
<text><location><page_25><loc_48><loc_77><loc_50><loc_77></location>.</text>
<text><location><page_25><loc_48><loc_76><loc_50><loc_77></location>7</text>
<text><location><page_25><loc_48><loc_75><loc_50><loc_76></location>)</text>
<text><location><page_25><loc_48><loc_75><loc_50><loc_75></location>2</text>
<text><location><page_25><loc_50><loc_87><loc_51><loc_87></location>)</text>
<text><location><page_25><loc_50><loc_86><loc_51><loc_87></location>2</text>
<text><location><page_25><loc_50><loc_86><loc_51><loc_86></location>(</text>
<text><location><page_25><loc_50><loc_85><loc_51><loc_85></location>6</text>
<text><location><page_25><loc_50><loc_84><loc_51><loc_85></location>±</text>
<text><location><page_25><loc_50><loc_83><loc_51><loc_84></location>0</text>
<text><location><page_25><loc_50><loc_83><loc_51><loc_83></location>5</text>
<text><location><page_25><loc_50><loc_82><loc_51><loc_83></location>2</text>
<text><location><page_25><loc_50><loc_81><loc_50><loc_81></location>d</text>
<text><location><page_25><loc_50><loc_80><loc_51><loc_81></location>0</text>
<text><location><page_25><loc_50><loc_80><loc_51><loc_80></location>9</text>
<text><location><page_25><loc_50><loc_79><loc_51><loc_80></location>.</text>
<text><location><page_25><loc_50><loc_79><loc_51><loc_79></location>0</text>
<text><location><page_25><loc_50><loc_78><loc_51><loc_79></location>±</text>
<text><location><page_25><loc_50><loc_77><loc_51><loc_78></location>8</text>
<text><location><page_25><loc_50><loc_77><loc_51><loc_77></location>4</text>
<text><location><page_25><loc_50><loc_77><loc_51><loc_77></location>.</text>
<text><location><page_25><loc_50><loc_76><loc_51><loc_77></location>6</text>
<text><location><page_25><loc_50><loc_75><loc_51><loc_76></location>)</text>
<text><location><page_25><loc_50><loc_75><loc_51><loc_75></location>1</text>
<text><location><page_25><loc_51><loc_87><loc_53><loc_87></location>)</text>
<text><location><page_25><loc_51><loc_86><loc_53><loc_87></location>1</text>
<text><location><page_25><loc_51><loc_86><loc_53><loc_86></location>(</text>
<text><location><page_25><loc_51><loc_85><loc_53><loc_85></location>6</text>
<text><location><page_25><loc_51><loc_84><loc_53><loc_85></location>6</text>
<text><location><page_25><loc_51><loc_84><loc_53><loc_84></location>±</text>
<text><location><page_25><loc_51><loc_83><loc_53><loc_84></location>5</text>
<text><location><page_25><loc_51><loc_82><loc_53><loc_83></location>6</text>
<text><location><page_25><loc_51><loc_82><loc_53><loc_82></location>4</text>
<text><location><page_25><loc_51><loc_81><loc_53><loc_81></location>4</text>
<text><location><page_25><loc_51><loc_80><loc_53><loc_81></location>2</text>
<text><location><page_25><loc_51><loc_80><loc_53><loc_80></location>.</text>
<text><location><page_25><loc_51><loc_79><loc_53><loc_80></location>0</text>
<text><location><page_25><loc_51><loc_78><loc_53><loc_79></location>±</text>
<text><location><page_25><loc_51><loc_78><loc_53><loc_78></location>7</text>
<text><location><page_25><loc_51><loc_77><loc_53><loc_78></location>4</text>
<text><location><page_25><loc_51><loc_77><loc_53><loc_77></location>.</text>
<text><location><page_25><loc_51><loc_76><loc_53><loc_77></location>1</text>
<text><location><page_25><loc_51><loc_75><loc_53><loc_76></location>)</text>
<text><location><page_25><loc_51><loc_75><loc_53><loc_75></location>1</text>
<text><location><page_25><loc_54><loc_87><loc_55><loc_87></location>)</text>
<text><location><page_25><loc_54><loc_86><loc_55><loc_87></location>1</text>
<text><location><page_25><loc_54><loc_86><loc_55><loc_86></location>(</text>
<text><location><page_25><loc_54><loc_85><loc_55><loc_85></location>8</text>
<text><location><page_25><loc_54><loc_84><loc_55><loc_85></location>1</text>
<text><location><page_25><loc_54><loc_84><loc_55><loc_84></location>±</text>
<text><location><page_25><loc_54><loc_83><loc_55><loc_84></location>5</text>
<text><location><page_25><loc_54><loc_82><loc_55><loc_83></location>6</text>
<text><location><page_25><loc_54><loc_82><loc_55><loc_82></location>4</text>
<text><location><page_25><loc_54><loc_81><loc_55><loc_81></location>1</text>
<text><location><page_25><loc_54><loc_80><loc_55><loc_81></location>2</text>
<text><location><page_25><loc_54><loc_80><loc_55><loc_80></location>.</text>
<text><location><page_25><loc_54><loc_79><loc_55><loc_80></location>0</text>
<text><location><page_25><loc_54><loc_78><loc_55><loc_79></location>±</text>
<text><location><page_25><loc_54><loc_78><loc_55><loc_78></location>8</text>
<text><location><page_25><loc_54><loc_77><loc_55><loc_78></location>4</text>
<text><location><page_25><loc_54><loc_77><loc_55><loc_77></location>.</text>
<text><location><page_25><loc_54><loc_76><loc_55><loc_77></location>4</text>
<text><location><page_25><loc_54><loc_75><loc_55><loc_76></location>)</text>
<text><location><page_25><loc_54><loc_75><loc_55><loc_75></location>1</text>
<paragraph><location><page_25><loc_44><loc_71><loc_55><loc_75></location>± 7 4 ( ± 2 0 ( ± 1 0 1 ( ± 4 3 ( ± 4 3 ( ± 3 2 1 ( 0 8 ± 5 3 (</paragraph>
<text><location><page_25><loc_44><loc_71><loc_45><loc_72></location>4</text>
<text><location><page_25><loc_44><loc_71><loc_45><loc_71></location>8</text>
<text><location><page_25><loc_45><loc_71><loc_47><loc_72></location>8</text>
<text><location><page_25><loc_45><loc_71><loc_47><loc_71></location>4</text>
<text><location><page_25><loc_47><loc_71><loc_48><loc_71></location>5</text>
<text><location><page_25><loc_47><loc_70><loc_48><loc_71></location>7</text>
<text><location><page_25><loc_48><loc_71><loc_50><loc_72></location>8</text>
<text><location><page_25><loc_48><loc_71><loc_50><loc_71></location>5</text>
<text><location><page_25><loc_50><loc_71><loc_51><loc_72></location>5</text>
<text><location><page_25><loc_50><loc_71><loc_51><loc_71></location>9</text>
<text><location><page_25><loc_51><loc_71><loc_53><loc_71></location>0</text>
<text><location><page_25><loc_51><loc_70><loc_53><loc_71></location>0</text>
<text><location><page_25><loc_54><loc_71><loc_55><loc_72></location>0</text>
<text><location><page_25><loc_54><loc_71><loc_55><loc_71></location>0</text>
<text><location><page_25><loc_56><loc_87><loc_57><loc_87></location>)</text>
<text><location><page_25><loc_56><loc_86><loc_57><loc_87></location>2</text>
<text><location><page_25><loc_56><loc_86><loc_57><loc_86></location>(</text>
<text><location><page_25><loc_56><loc_85><loc_57><loc_85></location>2</text>
<text><location><page_25><loc_56><loc_84><loc_57><loc_85></location>±</text>
<text><location><page_25><loc_56><loc_83><loc_57><loc_84></location>0</text>
<text><location><page_25><loc_56><loc_83><loc_57><loc_83></location>5</text>
<text><location><page_25><loc_56><loc_82><loc_57><loc_83></location>2</text>
<text><location><page_25><loc_55><loc_81><loc_56><loc_82></location>d</text>
<text><location><page_25><loc_56><loc_80><loc_57><loc_81></location>8</text>
<text><location><page_25><loc_56><loc_80><loc_57><loc_80></location>8</text>
<text><location><page_25><loc_56><loc_80><loc_57><loc_80></location>.</text>
<text><location><page_25><loc_56><loc_79><loc_57><loc_80></location>0</text>
<text><location><page_25><loc_56><loc_78><loc_57><loc_79></location>±</text>
<text><location><page_25><loc_56><loc_78><loc_57><loc_78></location>3</text>
<text><location><page_25><loc_56><loc_77><loc_57><loc_78></location>3</text>
<text><location><page_25><loc_56><loc_77><loc_57><loc_77></location>.</text>
<text><location><page_25><loc_56><loc_76><loc_57><loc_77></location>1</text>
<text><location><page_25><loc_56><loc_76><loc_57><loc_76></location>2</text>
<text><location><page_25><loc_56><loc_75><loc_57><loc_76></location>)</text>
<text><location><page_25><loc_56><loc_75><loc_57><loc_75></location>2</text>
<text><location><page_25><loc_56><loc_74><loc_57><loc_75></location>(</text>
<text><location><page_25><loc_56><loc_73><loc_57><loc_74></location>1</text>
<text><location><page_25><loc_56><loc_73><loc_57><loc_73></location>1</text>
<text><location><page_25><loc_56><loc_72><loc_57><loc_73></location>±</text>
<text><location><page_25><loc_56><loc_71><loc_57><loc_72></location>0</text>
<text><location><page_25><loc_56><loc_71><loc_57><loc_71></location>5</text>
<text><location><page_25><loc_57><loc_87><loc_58><loc_87></location>)</text>
<text><location><page_25><loc_57><loc_86><loc_58><loc_87></location>4</text>
<text><location><page_25><loc_57><loc_86><loc_58><loc_86></location>(</text>
<text><location><page_25><loc_57><loc_85><loc_58><loc_85></location>4</text>
<text><location><page_25><loc_57><loc_84><loc_58><loc_85></location>±</text>
<text><location><page_25><loc_57><loc_83><loc_58><loc_84></location>3</text>
<text><location><page_25><loc_57><loc_83><loc_58><loc_83></location>6</text>
<text><location><page_25><loc_57><loc_82><loc_58><loc_83></location>1</text>
<text><location><page_25><loc_57><loc_81><loc_58><loc_81></location>9</text>
<text><location><page_25><loc_57><loc_80><loc_58><loc_81></location>7</text>
<text><location><page_25><loc_57><loc_80><loc_58><loc_80></location>.</text>
<text><location><page_25><loc_57><loc_79><loc_58><loc_80></location>0</text>
<text><location><page_25><loc_57><loc_78><loc_58><loc_79></location>±</text>
<text><location><page_25><loc_57><loc_78><loc_58><loc_78></location>0</text>
<text><location><page_25><loc_57><loc_77><loc_58><loc_78></location>4</text>
<text><location><page_25><loc_57><loc_77><loc_58><loc_77></location>.</text>
<text><location><page_25><loc_57><loc_76><loc_58><loc_77></location>7</text>
<text><location><page_25><loc_57><loc_75><loc_58><loc_76></location>)</text>
<text><location><page_25><loc_57><loc_75><loc_58><loc_75></location>2</text>
<text><location><page_25><loc_57><loc_74><loc_58><loc_75></location>(</text>
<text><location><page_25><loc_57><loc_73><loc_58><loc_74></location>5</text>
<text><location><page_25><loc_57><loc_73><loc_58><loc_73></location>3</text>
<text><location><page_25><loc_57><loc_72><loc_58><loc_73></location>±</text>
<text><location><page_25><loc_57><loc_71><loc_58><loc_72></location>2</text>
<text><location><page_25><loc_57><loc_71><loc_58><loc_71></location>5</text>
<text><location><page_25><loc_58><loc_87><loc_60><loc_87></location>)</text>
<text><location><page_25><loc_58><loc_86><loc_60><loc_87></location>2</text>
<text><location><page_25><loc_58><loc_86><loc_60><loc_86></location>(</text>
<text><location><page_25><loc_58><loc_85><loc_60><loc_85></location>8</text>
<text><location><page_25><loc_58><loc_84><loc_60><loc_85></location>±</text>
<text><location><page_25><loc_58><loc_83><loc_60><loc_84></location>2</text>
<text><location><page_25><loc_58><loc_83><loc_60><loc_83></location>5</text>
<text><location><page_25><loc_58><loc_82><loc_60><loc_83></location>3</text>
<text><location><page_25><loc_58><loc_81><loc_60><loc_81></location>0</text>
<text><location><page_25><loc_58><loc_80><loc_60><loc_81></location>2</text>
<text><location><page_25><loc_58><loc_80><loc_60><loc_80></location>.</text>
<text><location><page_25><loc_58><loc_79><loc_60><loc_80></location>1</text>
<text><location><page_25><loc_58><loc_78><loc_60><loc_79></location>±</text>
<text><location><page_25><loc_58><loc_78><loc_60><loc_78></location>4</text>
<text><location><page_25><loc_58><loc_77><loc_60><loc_78></location>5</text>
<text><location><page_25><loc_58><loc_77><loc_60><loc_77></location>.</text>
<text><location><page_25><loc_58><loc_76><loc_60><loc_77></location>9</text>
<text><location><page_25><loc_58><loc_75><loc_60><loc_76></location>)</text>
<text><location><page_25><loc_58><loc_75><loc_60><loc_75></location>2</text>
<text><location><page_25><loc_58><loc_74><loc_60><loc_75></location>(</text>
<text><location><page_25><loc_58><loc_73><loc_60><loc_74></location>8</text>
<text><location><page_25><loc_58><loc_73><loc_60><loc_73></location>4</text>
<text><location><page_25><loc_58><loc_72><loc_60><loc_73></location>±</text>
<text><location><page_25><loc_58><loc_71><loc_60><loc_72></location>7</text>
<text><location><page_25><loc_58><loc_71><loc_60><loc_71></location>3</text>
<text><location><page_25><loc_60><loc_87><loc_61><loc_87></location>)</text>
<text><location><page_25><loc_60><loc_86><loc_61><loc_87></location>2</text>
<text><location><page_25><loc_60><loc_86><loc_61><loc_86></location>(</text>
<text><location><page_25><loc_60><loc_85><loc_61><loc_85></location>2</text>
<text><location><page_25><loc_60><loc_84><loc_61><loc_85></location>1</text>
<text><location><page_25><loc_60><loc_84><loc_61><loc_84></location>±</text>
<text><location><page_25><loc_60><loc_83><loc_61><loc_84></location>4</text>
<text><location><page_25><loc_60><loc_82><loc_61><loc_83></location>1</text>
<text><location><page_25><loc_60><loc_82><loc_61><loc_82></location>5</text>
<text><location><page_25><loc_60><loc_81><loc_61><loc_81></location>2</text>
<text><location><page_25><loc_60><loc_80><loc_61><loc_81></location>9</text>
<text><location><page_25><loc_60><loc_80><loc_61><loc_80></location>.</text>
<text><location><page_25><loc_60><loc_79><loc_61><loc_80></location>1</text>
<text><location><page_25><loc_60><loc_79><loc_61><loc_79></location>±</text>
<text><location><page_25><loc_60><loc_78><loc_61><loc_79></location>7</text>
<text><location><page_25><loc_60><loc_78><loc_61><loc_78></location>3</text>
<text><location><page_25><loc_60><loc_77><loc_61><loc_78></location>.</text>
<text><location><page_25><loc_60><loc_77><loc_61><loc_77></location>1</text>
<text><location><page_25><loc_60><loc_76><loc_61><loc_77></location>3</text>
<text><location><page_25><loc_60><loc_75><loc_61><loc_76></location>)</text>
<text><location><page_25><loc_60><loc_75><loc_61><loc_75></location>2</text>
<text><location><page_25><loc_60><loc_74><loc_61><loc_75></location>(</text>
<text><location><page_25><loc_60><loc_73><loc_61><loc_74></location>9</text>
<text><location><page_25><loc_60><loc_73><loc_61><loc_73></location>5</text>
<text><location><page_25><loc_60><loc_72><loc_61><loc_73></location>±</text>
<text><location><page_25><loc_60><loc_71><loc_61><loc_72></location>0</text>
<text><location><page_25><loc_60><loc_71><loc_61><loc_71></location>2</text>
<text><location><page_25><loc_61><loc_87><loc_63><loc_87></location>)</text>
<text><location><page_25><loc_61><loc_86><loc_63><loc_87></location>2</text>
<text><location><page_25><loc_61><loc_86><loc_63><loc_86></location>(</text>
<text><location><page_25><loc_61><loc_85><loc_63><loc_85></location>5</text>
<text><location><page_25><loc_61><loc_84><loc_63><loc_85></location>±</text>
<text><location><page_25><loc_61><loc_83><loc_63><loc_84></location>0</text>
<text><location><page_25><loc_61><loc_83><loc_63><loc_83></location>5</text>
<text><location><page_25><loc_61><loc_82><loc_63><loc_83></location>5</text>
<text><location><page_25><loc_61><loc_81><loc_63><loc_81></location>8</text>
<text><location><page_25><loc_61><loc_80><loc_63><loc_81></location>1</text>
<text><location><page_25><loc_61><loc_80><loc_63><loc_80></location>.</text>
<text><location><page_25><loc_61><loc_79><loc_63><loc_80></location>0</text>
<text><location><page_25><loc_61><loc_79><loc_63><loc_79></location>±</text>
<text><location><page_25><loc_61><loc_78><loc_63><loc_79></location>4</text>
<text><location><page_25><loc_61><loc_78><loc_63><loc_78></location>1</text>
<text><location><page_25><loc_61><loc_77><loc_63><loc_78></location>.</text>
<text><location><page_25><loc_61><loc_77><loc_63><loc_77></location>7</text>
<text><location><page_25><loc_61><loc_76><loc_63><loc_77></location>1</text>
<text><location><page_25><loc_61><loc_75><loc_63><loc_76></location>)</text>
<text><location><page_25><loc_61><loc_75><loc_63><loc_75></location>2</text>
<text><location><page_25><loc_61><loc_74><loc_63><loc_75></location>(</text>
<text><location><page_25><loc_61><loc_73><loc_63><loc_74></location>5</text>
<text><location><page_25><loc_61><loc_73><loc_63><loc_73></location>5</text>
<text><location><page_25><loc_61><loc_72><loc_63><loc_73></location>±</text>
<text><location><page_25><loc_61><loc_71><loc_63><loc_72></location>9</text>
<text><location><page_25><loc_61><loc_71><loc_63><loc_71></location>8</text>
<text><location><page_25><loc_63><loc_87><loc_64><loc_87></location>)</text>
<text><location><page_25><loc_63><loc_86><loc_64><loc_87></location>1</text>
<text><location><page_25><loc_63><loc_86><loc_64><loc_86></location>(</text>
<text><location><page_25><loc_63><loc_85><loc_64><loc_85></location>4</text>
<text><location><page_25><loc_63><loc_84><loc_64><loc_85></location>±</text>
<text><location><page_25><loc_63><loc_83><loc_64><loc_84></location>7</text>
<text><location><page_25><loc_63><loc_83><loc_64><loc_83></location>4</text>
<text><location><page_25><loc_63><loc_82><loc_64><loc_83></location>2</text>
<text><location><page_25><loc_63><loc_81><loc_63><loc_81></location>d</text>
<text><location><page_25><loc_63><loc_80><loc_64><loc_81></location>9</text>
<text><location><page_25><loc_63><loc_80><loc_64><loc_80></location>1</text>
<text><location><page_25><loc_63><loc_79><loc_64><loc_80></location>.</text>
<text><location><page_25><loc_63><loc_79><loc_64><loc_79></location>0</text>
<text><location><page_25><loc_63><loc_78><loc_64><loc_79></location>±</text>
<text><location><page_25><loc_63><loc_77><loc_64><loc_78></location>5</text>
<text><location><page_25><loc_63><loc_77><loc_64><loc_77></location>6</text>
<text><location><page_25><loc_63><loc_77><loc_64><loc_77></location>.</text>
<text><location><page_25><loc_63><loc_76><loc_64><loc_77></location>9</text>
<text><location><page_25><loc_63><loc_75><loc_64><loc_76></location>)</text>
<text><location><page_25><loc_63><loc_75><loc_64><loc_75></location>1</text>
<text><location><page_25><loc_63><loc_74><loc_64><loc_75></location>(</text>
<text><location><page_25><loc_63><loc_73><loc_64><loc_74></location>7</text>
<text><location><page_25><loc_63><loc_73><loc_64><loc_73></location>2</text>
<text><location><page_25><loc_63><loc_72><loc_64><loc_73></location>±</text>
<text><location><page_25><loc_63><loc_71><loc_64><loc_72></location>0</text>
<text><location><page_25><loc_63><loc_71><loc_64><loc_71></location>0</text>
<text><location><page_25><loc_64><loc_87><loc_65><loc_87></location>)</text>
<text><location><page_25><loc_64><loc_86><loc_65><loc_87></location>2</text>
<text><location><page_25><loc_64><loc_86><loc_65><loc_86></location>(</text>
<text><location><page_25><loc_64><loc_85><loc_65><loc_85></location>2</text>
<text><location><page_25><loc_64><loc_84><loc_65><loc_85></location>1</text>
<text><location><page_25><loc_64><loc_84><loc_65><loc_84></location>±</text>
<text><location><page_25><loc_64><loc_83><loc_65><loc_84></location>5</text>
<text><location><page_25><loc_64><loc_82><loc_65><loc_83></location>7</text>
<text><location><page_25><loc_64><loc_82><loc_65><loc_82></location>4</text>
<text><location><page_25><loc_64><loc_81><loc_65><loc_81></location>6</text>
<text><location><page_25><loc_64><loc_80><loc_65><loc_81></location>4</text>
<text><location><page_25><loc_64><loc_80><loc_65><loc_80></location>.</text>
<text><location><page_25><loc_64><loc_79><loc_65><loc_80></location>0</text>
<text><location><page_25><loc_64><loc_78><loc_65><loc_79></location>±</text>
<text><location><page_25><loc_64><loc_78><loc_65><loc_78></location>3</text>
<text><location><page_25><loc_64><loc_77><loc_65><loc_78></location>1</text>
<text><location><page_25><loc_64><loc_77><loc_65><loc_77></location>.</text>
<text><location><page_25><loc_64><loc_76><loc_65><loc_77></location>9</text>
<text><location><page_25><loc_64><loc_75><loc_65><loc_76></location>)</text>
<text><location><page_25><loc_64><loc_75><loc_65><loc_75></location>1</text>
<text><location><page_25><loc_64><loc_74><loc_65><loc_75></location>(</text>
<text><location><page_25><loc_64><loc_73><loc_65><loc_74></location>0</text>
<text><location><page_25><loc_64><loc_73><loc_65><loc_73></location>5</text>
<text><location><page_25><loc_64><loc_72><loc_65><loc_73></location>±</text>
<text><location><page_25><loc_64><loc_71><loc_65><loc_72></location>0</text>
<text><location><page_25><loc_64><loc_71><loc_65><loc_71></location>5</text>
<text><location><page_25><loc_66><loc_87><loc_67><loc_87></location>)</text>
<text><location><page_25><loc_66><loc_86><loc_67><loc_87></location>1</text>
<text><location><page_25><loc_66><loc_86><loc_67><loc_86></location>(</text>
<text><location><page_25><loc_66><loc_85><loc_67><loc_85></location>5</text>
<text><location><page_25><loc_66><loc_84><loc_67><loc_85></location>±</text>
<text><location><page_25><loc_66><loc_83><loc_67><loc_84></location>7</text>
<text><location><page_25><loc_66><loc_83><loc_67><loc_83></location>0</text>
<text><location><page_25><loc_66><loc_82><loc_67><loc_83></location>3</text>
<text><location><page_25><loc_66><loc_81><loc_67><loc_81></location>0</text>
<text><location><page_25><loc_66><loc_80><loc_67><loc_81></location>2</text>
<text><location><page_25><loc_66><loc_80><loc_67><loc_80></location>.</text>
<text><location><page_25><loc_66><loc_79><loc_67><loc_80></location>0</text>
<text><location><page_25><loc_66><loc_78><loc_67><loc_79></location>±</text>
<text><location><page_25><loc_66><loc_78><loc_67><loc_78></location>9</text>
<text><location><page_25><loc_66><loc_77><loc_67><loc_78></location>9</text>
<text><location><page_25><loc_66><loc_77><loc_67><loc_77></location>.</text>
<text><location><page_25><loc_66><loc_76><loc_67><loc_77></location>9</text>
<text><location><page_25><loc_66><loc_75><loc_67><loc_76></location>)</text>
<text><location><page_25><loc_66><loc_75><loc_67><loc_75></location>1</text>
<text><location><page_25><loc_66><loc_74><loc_67><loc_75></location>(</text>
<text><location><page_25><loc_66><loc_73><loc_67><loc_74></location>7</text>
<text><location><page_25><loc_66><loc_73><loc_67><loc_73></location>2</text>
<text><location><page_25><loc_66><loc_72><loc_67><loc_73></location>±</text>
<text><location><page_25><loc_66><loc_71><loc_67><loc_72></location>9</text>
<text><location><page_25><loc_66><loc_71><loc_67><loc_71></location>7</text>
<text><location><page_25><loc_67><loc_87><loc_68><loc_87></location>)</text>
<text><location><page_25><loc_67><loc_86><loc_68><loc_87></location>2</text>
<text><location><page_25><loc_67><loc_86><loc_68><loc_86></location>(</text>
<text><location><page_25><loc_67><loc_85><loc_68><loc_85></location>8</text>
<text><location><page_25><loc_67><loc_84><loc_68><loc_85></location>±</text>
<text><location><page_25><loc_67><loc_83><loc_68><loc_84></location>3</text>
<text><location><page_25><loc_67><loc_83><loc_68><loc_83></location>7</text>
<text><location><page_25><loc_67><loc_82><loc_68><loc_83></location>4</text>
<text><location><page_25><loc_67><loc_81><loc_68><loc_82></location>d</text>
<text><location><page_25><loc_67><loc_80><loc_68><loc_81></location>8</text>
<text><location><page_25><loc_67><loc_80><loc_68><loc_80></location>0</text>
<text><location><page_25><loc_67><loc_80><loc_68><loc_80></location>.</text>
<text><location><page_25><loc_67><loc_79><loc_68><loc_80></location>1</text>
<text><location><page_25><loc_67><loc_78><loc_68><loc_79></location>±</text>
<text><location><page_25><loc_67><loc_78><loc_68><loc_78></location>7</text>
<text><location><page_25><loc_67><loc_77><loc_68><loc_78></location>5</text>
<text><location><page_25><loc_67><loc_77><loc_68><loc_77></location>.</text>
<text><location><page_25><loc_67><loc_76><loc_68><loc_77></location>2</text>
<text><location><page_25><loc_67><loc_76><loc_68><loc_76></location>1</text>
<text><location><page_25><loc_67><loc_75><loc_68><loc_76></location>)</text>
<text><location><page_25><loc_67><loc_75><loc_68><loc_75></location>1</text>
<text><location><page_25><loc_67><loc_74><loc_68><loc_75></location>(</text>
<text><location><page_25><loc_67><loc_73><loc_68><loc_74></location>9</text>
<text><location><page_25><loc_67><loc_73><loc_68><loc_73></location>6</text>
<text><location><page_25><loc_67><loc_72><loc_68><loc_73></location>±</text>
<text><location><page_25><loc_67><loc_71><loc_68><loc_72></location>5</text>
<text><location><page_25><loc_67><loc_71><loc_68><loc_71></location>7</text>
<text><location><page_25><loc_71><loc_86><loc_72><loc_87></location>.</text>
<text><location><page_25><loc_71><loc_86><loc_72><loc_86></location>)</text>
<text><location><page_25><loc_71><loc_86><loc_72><loc_86></location>3</text>
<text><location><page_25><loc_71><loc_85><loc_72><loc_86></location>.</text>
<text><location><page_25><loc_71><loc_85><loc_72><loc_85></location>2</text>
<text><location><page_25><loc_71><loc_84><loc_72><loc_85></location>§</text>
<text><location><page_25><loc_71><loc_84><loc_72><loc_84></location>(</text>
<text><location><page_25><loc_71><loc_83><loc_72><loc_84></location>x</text>
<text><location><page_25><loc_71><loc_82><loc_72><loc_83></location>u</text>
<text><location><page_25><loc_71><loc_82><loc_72><loc_82></location>fl</text>
<text><location><page_25><loc_71><loc_81><loc_72><loc_82></location>e</text>
<text><location><page_25><loc_71><loc_81><loc_72><loc_81></location>h</text>
<text><location><page_25><loc_71><loc_80><loc_72><loc_81></location>t</text>
<text><location><page_25><loc_71><loc_79><loc_72><loc_80></location>e</text>
<text><location><page_25><loc_71><loc_79><loc_72><loc_79></location>r</text>
<text><location><page_25><loc_71><loc_78><loc_72><loc_79></location>u</text>
<text><location><page_25><loc_71><loc_78><loc_72><loc_78></location>s</text>
<text><location><page_25><loc_71><loc_78><loc_72><loc_78></location>a</text>
<text><location><page_25><loc_71><loc_77><loc_72><loc_78></location>e</text>
<text><location><page_25><loc_71><loc_76><loc_72><loc_77></location>m</text>
<text><location><page_25><loc_71><loc_75><loc_72><loc_76></location>o</text>
<text><location><page_25><loc_71><loc_75><loc_72><loc_75></location>t</text>
<text><location><page_25><loc_71><loc_74><loc_72><loc_75></location>d</text>
<text><location><page_25><loc_71><loc_74><loc_72><loc_74></location>e</text>
<text><location><page_25><loc_71><loc_73><loc_72><loc_74></location>s</text>
<text><location><page_25><loc_71><loc_73><loc_72><loc_73></location>u</text>
<text><location><page_25><loc_71><loc_72><loc_72><loc_72></location>d</text>
<text><location><page_25><loc_71><loc_71><loc_72><loc_72></location>o</text>
<text><location><page_25><loc_71><loc_71><loc_72><loc_71></location>h</text>
<table>
<location><page_25><loc_12><loc_30><loc_87><loc_71></location>
</table>
<text><location><page_25><loc_33><loc_29><loc_34><loc_30></location>)</text>
<text><location><page_25><loc_34><loc_29><loc_35><loc_30></location>)</text>
<text><location><page_25><loc_35><loc_29><loc_37><loc_30></location>)</text>
<text><location><page_25><loc_37><loc_29><loc_38><loc_30></location>)</text>
<text><location><page_25><loc_38><loc_29><loc_40><loc_30></location>)</text>
<text><location><page_25><loc_40><loc_29><loc_41><loc_30></location>)</text>
<text><location><page_25><loc_41><loc_29><loc_42><loc_30></location>)</text>
<text><location><page_25><loc_43><loc_29><loc_44><loc_30></location>)</text>
<text><location><page_25><loc_44><loc_29><loc_45><loc_30></location>)</text>
<text><location><page_25><loc_45><loc_29><loc_47><loc_30></location>)</text>
<text><location><page_25><loc_47><loc_29><loc_48><loc_30></location>)</text>
<text><location><page_25><loc_48><loc_29><loc_50><loc_30></location>)</text>
<text><location><page_25><loc_50><loc_29><loc_51><loc_30></location>)</text>
<text><location><page_25><loc_51><loc_29><loc_53><loc_30></location>)</text>
<text><location><page_25><loc_54><loc_29><loc_55><loc_30></location>)</text>
<text><location><page_25><loc_56><loc_29><loc_57><loc_30></location>)</text>
<text><location><page_25><loc_57><loc_29><loc_58><loc_30></location>)</text>
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<text><location><page_25><loc_73><loc_23><loc_74><loc_24></location>3</text>
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<text><location><page_25><loc_75><loc_25><loc_77><loc_26></location>1</text>
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<text><location><page_25><loc_78><loc_26><loc_79><loc_26></location>1</text>
<text><location><page_25><loc_78><loc_25><loc_79><loc_26></location>.</text>
<text><location><page_25><loc_78><loc_25><loc_79><loc_26></location>3</text>
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<text><location><page_25><loc_78><loc_20><loc_79><loc_20></location>y</text>
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<text><location><page_25><loc_71><loc_21><loc_72><loc_21></location>h</text>
<text><location><page_25><loc_71><loc_20><loc_72><loc_21></location>c</text>
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<text><location><page_25><loc_71><loc_18><loc_72><loc_18></location>o</text>
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<text><location><page_25><loc_71><loc_13><loc_72><loc_14></location>N</text>
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<text><location><page_25><loc_67><loc_25><loc_68><loc_26></location>5</text>
<text><location><page_25><loc_67><loc_24><loc_68><loc_25></location>7</text>
<text><location><page_25><loc_67><loc_24><loc_68><loc_24></location>3</text>
<text><location><page_25><loc_67><loc_22><loc_68><loc_22></location>6</text>
<text><location><page_25><loc_67><loc_21><loc_68><loc_22></location>7</text>
<text><location><page_25><loc_67><loc_21><loc_68><loc_21></location>.</text>
<text><location><page_25><loc_67><loc_20><loc_68><loc_21></location>0</text>
<text><location><page_25><loc_67><loc_19><loc_68><loc_20></location><</text>
<text><location><page_25><loc_67><loc_17><loc_68><loc_17></location>5</text>
<text><location><page_25><loc_67><loc_16><loc_68><loc_17></location>1</text>
<text><location><page_25><loc_67><loc_16><loc_68><loc_16></location>4</text>
<text><location><page_25><loc_67><loc_15><loc_68><loc_16></location>5</text>
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<text><location><page_25><loc_67><loc_14><loc_68><loc_15></location>3</text>
<text><location><page_25><loc_67><loc_14><loc_68><loc_14></location>5</text>
<text><location><page_25><loc_67><loc_13><loc_68><loc_14></location>2</text>
<text><location><page_25><loc_67><loc_13><loc_68><loc_13></location>3</text>
<text><location><page_25><loc_67><loc_12><loc_68><loc_13></location>2</text>
<text><location><page_25><loc_66><loc_26><loc_67><loc_27></location>±</text>
<text><location><page_25><loc_66><loc_25><loc_67><loc_26></location>0</text>
<text><location><page_25><loc_66><loc_25><loc_67><loc_25></location>5</text>
<text><location><page_25><loc_66><loc_24><loc_67><loc_25></location>2</text>
<text><location><page_25><loc_66><loc_23><loc_67><loc_23></location>1</text>
<text><location><page_25><loc_66><loc_22><loc_67><loc_23></location>9</text>
<text><location><page_25><loc_66><loc_22><loc_67><loc_22></location>.</text>
<text><location><page_25><loc_66><loc_21><loc_67><loc_22></location>0</text>
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<text><location><page_25><loc_66><loc_20><loc_67><loc_20></location>2</text>
<text><location><page_25><loc_66><loc_19><loc_67><loc_20></location>0</text>
<text><location><page_25><loc_66><loc_19><loc_67><loc_19></location>.</text>
<text><location><page_25><loc_66><loc_18><loc_67><loc_19></location>4</text>
<text><location><page_25><loc_66><loc_17><loc_67><loc_17></location>6</text>
<text><location><page_25><loc_66><loc_16><loc_67><loc_17></location>2</text>
<text><location><page_25><loc_66><loc_16><loc_67><loc_16></location>9</text>
<text><location><page_25><loc_66><loc_15><loc_67><loc_16></location>6</text>
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<text><location><page_25><loc_66><loc_14><loc_67><loc_15></location>0</text>
<text><location><page_25><loc_66><loc_14><loc_67><loc_14></location>3</text>
<text><location><page_25><loc_66><loc_13><loc_67><loc_14></location>2</text>
<text><location><page_25><loc_66><loc_13><loc_67><loc_13></location>3</text>
<text><location><page_25><loc_66><loc_12><loc_67><loc_13></location>2</text>
<text><location><page_25><loc_64><loc_26><loc_65><loc_26></location>±</text>
<text><location><page_25><loc_64><loc_25><loc_65><loc_26></location>0</text>
<text><location><page_25><loc_64><loc_24><loc_65><loc_25></location>5</text>
<text><location><page_25><loc_64><loc_24><loc_65><loc_24></location>5</text>
<text><location><page_25><loc_64><loc_23><loc_65><loc_23></location>4</text>
<text><location><page_25><loc_64><loc_22><loc_65><loc_23></location>1</text>
<text><location><page_25><loc_64><loc_22><loc_65><loc_22></location>.</text>
<text><location><page_25><loc_64><loc_21><loc_65><loc_22></location>1</text>
<text><location><page_25><loc_64><loc_20><loc_65><loc_21></location>±</text>
<text><location><page_25><loc_64><loc_20><loc_65><loc_20></location>4</text>
<text><location><page_25><loc_64><loc_19><loc_65><loc_20></location>2</text>
<text><location><page_25><loc_64><loc_19><loc_65><loc_19></location>.</text>
<text><location><page_25><loc_64><loc_18><loc_65><loc_19></location>1</text>
<text><location><page_25><loc_64><loc_17><loc_65><loc_17></location>1</text>
<text><location><page_25><loc_64><loc_16><loc_65><loc_17></location>5</text>
<text><location><page_25><loc_64><loc_16><loc_65><loc_16></location>6</text>
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<text><location><page_25><loc_64><loc_14><loc_65><loc_15></location>4</text>
<text><location><page_25><loc_64><loc_14><loc_65><loc_14></location>1</text>
<text><location><page_25><loc_64><loc_13><loc_65><loc_14></location>4</text>
<text><location><page_25><loc_64><loc_13><loc_65><loc_13></location>0</text>
<text><location><page_25><loc_64><loc_12><loc_65><loc_13></location>2</text>
<text><location><page_25><loc_63><loc_26><loc_64><loc_27></location>±</text>
<text><location><page_25><loc_63><loc_25><loc_64><loc_26></location>2</text>
<text><location><page_25><loc_63><loc_25><loc_64><loc_25></location>0</text>
<text><location><page_25><loc_63><loc_24><loc_64><loc_25></location>2</text>
<text><location><page_25><loc_63><loc_23><loc_64><loc_23></location>1</text>
<text><location><page_25><loc_63><loc_22><loc_64><loc_23></location>8</text>
<text><location><page_25><loc_63><loc_22><loc_64><loc_22></location>.</text>
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<text><location><page_25><loc_63><loc_20><loc_64><loc_20></location>0</text>
<text><location><page_25><loc_63><loc_19><loc_64><loc_20></location>9</text>
<text><location><page_25><loc_63><loc_19><loc_64><loc_19></location>.</text>
<text><location><page_25><loc_63><loc_18><loc_64><loc_19></location>4</text>
<text><location><page_25><loc_63><loc_17><loc_64><loc_17></location>6</text>
<text><location><page_25><loc_63><loc_16><loc_64><loc_17></location>5</text>
<text><location><page_25><loc_63><loc_16><loc_64><loc_16></location>1</text>
<text><location><page_25><loc_63><loc_15><loc_64><loc_16></location>4</text>
<text><location><page_25><loc_63><loc_15><loc_64><loc_15></location>-</text>
<text><location><page_25><loc_63><loc_14><loc_64><loc_15></location>0</text>
<text><location><page_25><loc_63><loc_14><loc_64><loc_14></location>0</text>
<text><location><page_25><loc_63><loc_13><loc_64><loc_14></location>1</text>
<text><location><page_25><loc_63><loc_13><loc_64><loc_13></location>0</text>
<text><location><page_25><loc_63><loc_12><loc_64><loc_13></location>2</text>
<text><location><page_25><loc_61><loc_26><loc_63><loc_26></location>±</text>
<text><location><page_25><loc_61><loc_25><loc_63><loc_26></location>5</text>
<text><location><page_25><loc_61><loc_24><loc_63><loc_25></location>7</text>
<text><location><page_25><loc_61><loc_24><loc_63><loc_24></location>4</text>
<text><location><page_25><loc_61><loc_23><loc_63><loc_23></location>6</text>
<text><location><page_25><loc_61><loc_22><loc_63><loc_23></location>2</text>
<text><location><page_25><loc_61><loc_22><loc_63><loc_22></location>.</text>
<text><location><page_25><loc_61><loc_21><loc_63><loc_22></location>1</text>
<text><location><page_25><loc_61><loc_21><loc_63><loc_21></location>1</text>
<text><location><page_25><loc_61><loc_20><loc_63><loc_21></location>±</text>
<text><location><page_25><loc_61><loc_20><loc_63><loc_20></location>2</text>
<text><location><page_25><loc_61><loc_19><loc_63><loc_20></location>9</text>
<text><location><page_25><loc_61><loc_19><loc_63><loc_19></location>.</text>
<text><location><page_25><loc_61><loc_18><loc_63><loc_19></location>2</text>
<text><location><page_25><loc_61><loc_17><loc_63><loc_17></location>8</text>
<text><location><page_25><loc_61><loc_16><loc_63><loc_17></location>0</text>
<text><location><page_25><loc_61><loc_16><loc_63><loc_16></location>3</text>
<text><location><page_25><loc_61><loc_15><loc_63><loc_16></location>0</text>
<text><location><page_25><loc_61><loc_15><loc_63><loc_15></location>-</text>
<text><location><page_25><loc_61><loc_14><loc_63><loc_15></location>7</text>
<text><location><page_25><loc_61><loc_14><loc_63><loc_14></location>8</text>
<text><location><page_25><loc_61><loc_13><loc_63><loc_14></location>0</text>
<text><location><page_25><loc_61><loc_13><loc_63><loc_13></location>0</text>
<text><location><page_25><loc_61><loc_12><loc_63><loc_13></location>2</text>
<text><location><page_25><loc_60><loc_26><loc_61><loc_26></location>±</text>
<text><location><page_25><loc_60><loc_25><loc_61><loc_26></location>5</text>
<text><location><page_25><loc_60><loc_24><loc_61><loc_25></location>7</text>
<text><location><page_25><loc_60><loc_24><loc_61><loc_24></location>4</text>
<text><location><page_25><loc_60><loc_23><loc_61><loc_23></location>1</text>
<text><location><page_25><loc_60><loc_22><loc_61><loc_23></location>2</text>
<text><location><page_25><loc_60><loc_22><loc_61><loc_22></location>.</text>
<text><location><page_25><loc_60><loc_21><loc_61><loc_22></location>5</text>
<text><location><page_25><loc_60><loc_21><loc_61><loc_21></location>1</text>
<text><location><page_25><loc_60><loc_20><loc_61><loc_21></location>±</text>
<text><location><page_25><loc_60><loc_20><loc_61><loc_20></location>6</text>
<text><location><page_25><loc_60><loc_19><loc_61><loc_20></location>8</text>
<text><location><page_25><loc_60><loc_19><loc_61><loc_19></location>.</text>
<text><location><page_25><loc_60><loc_18><loc_61><loc_19></location>8</text>
<text><location><page_25><loc_60><loc_17><loc_61><loc_17></location>5</text>
<text><location><page_25><loc_60><loc_16><loc_61><loc_17></location>4</text>
<text><location><page_25><loc_60><loc_16><loc_61><loc_16></location>2</text>
<text><location><page_25><loc_60><loc_15><loc_61><loc_16></location>7</text>
<text><location><page_25><loc_60><loc_15><loc_61><loc_15></location>-</text>
<text><location><page_25><loc_60><loc_14><loc_61><loc_15></location>4</text>
<text><location><page_25><loc_60><loc_14><loc_61><loc_14></location>5</text>
<text><location><page_25><loc_60><loc_13><loc_61><loc_14></location>2</text>
<text><location><page_25><loc_60><loc_13><loc_61><loc_13></location>9</text>
<text><location><page_25><loc_60><loc_12><loc_61><loc_13></location>1</text>
<text><location><page_25><loc_58><loc_26><loc_60><loc_26></location>±</text>
<text><location><page_25><loc_58><loc_25><loc_60><loc_26></location>0</text>
<text><location><page_25><loc_58><loc_24><loc_60><loc_25></location>0</text>
<text><location><page_25><loc_58><loc_24><loc_60><loc_24></location>5</text>
<text><location><page_25><loc_58><loc_23><loc_60><loc_23></location>1</text>
<text><location><page_25><loc_58><loc_22><loc_60><loc_23></location>2</text>
<text><location><page_25><loc_58><loc_22><loc_60><loc_22></location>.</text>
<text><location><page_25><loc_58><loc_21><loc_60><loc_22></location>5</text>
<text><location><page_25><loc_58><loc_21><loc_60><loc_21></location>±</text>
<text><location><page_25><loc_58><loc_20><loc_60><loc_21></location>1</text>
<text><location><page_25><loc_58><loc_20><loc_60><loc_20></location>1</text>
<text><location><page_25><loc_58><loc_19><loc_60><loc_20></location>.</text>
<text><location><page_25><loc_58><loc_19><loc_60><loc_19></location>0</text>
<text><location><page_25><loc_58><loc_18><loc_60><loc_19></location>1</text>
<text><location><page_25><loc_58><loc_17><loc_60><loc_17></location>9</text>
<text><location><page_25><loc_58><loc_16><loc_60><loc_17></location>3</text>
<text><location><page_25><loc_58><loc_16><loc_60><loc_16></location>1</text>
<text><location><page_25><loc_58><loc_15><loc_60><loc_16></location>0</text>
<text><location><page_25><loc_58><loc_15><loc_60><loc_15></location>-</text>
<text><location><page_25><loc_58><loc_14><loc_60><loc_15></location>0</text>
<text><location><page_25><loc_58><loc_14><loc_60><loc_14></location>9</text>
<text><location><page_25><loc_58><loc_13><loc_60><loc_14></location>0</text>
<text><location><page_25><loc_58><loc_13><loc_60><loc_13></location>6</text>
<text><location><page_25><loc_58><loc_12><loc_60><loc_13></location>1</text>
<text><location><page_25><loc_57><loc_26><loc_58><loc_27></location>±</text>
<text><location><page_25><loc_57><loc_25><loc_58><loc_26></location>5</text>
<text><location><page_25><loc_57><loc_25><loc_58><loc_25></location>2</text>
<text><location><page_25><loc_57><loc_24><loc_58><loc_25></location>1</text>
<text><location><page_25><loc_57><loc_23><loc_58><loc_23></location>7</text>
<text><location><page_25><loc_57><loc_22><loc_58><loc_23></location>2</text>
<text><location><page_25><loc_57><loc_22><loc_58><loc_22></location>.</text>
<text><location><page_25><loc_57><loc_21><loc_58><loc_22></location>5</text>
<text><location><page_25><loc_57><loc_20><loc_58><loc_21></location>±</text>
<text><location><page_25><loc_57><loc_20><loc_58><loc_20></location>5</text>
<text><location><page_25><loc_57><loc_19><loc_58><loc_20></location>4</text>
<text><location><page_25><loc_57><loc_19><loc_58><loc_19></location>.</text>
<text><location><page_25><loc_57><loc_18><loc_58><loc_19></location>8</text>
<text><location><page_25><loc_57><loc_17><loc_58><loc_17></location>0</text>
<text><location><page_25><loc_57><loc_16><loc_58><loc_17></location>5</text>
<text><location><page_25><loc_57><loc_16><loc_58><loc_16></location>4</text>
<text><location><page_25><loc_57><loc_15><loc_58><loc_16></location>0</text>
<text><location><page_25><loc_57><loc_15><loc_58><loc_15></location>-</text>
<text><location><page_25><loc_57><loc_14><loc_58><loc_15></location>2</text>
<text><location><page_25><loc_57><loc_14><loc_58><loc_14></location>6</text>
<text><location><page_25><loc_57><loc_13><loc_58><loc_14></location>4</text>
<text><location><page_25><loc_57><loc_13><loc_58><loc_13></location>5</text>
<text><location><page_25><loc_57><loc_12><loc_58><loc_13></location>1</text>
<text><location><page_25><loc_56><loc_26><loc_57><loc_27></location>±</text>
<text><location><page_25><loc_56><loc_25><loc_57><loc_26></location>2</text>
<text><location><page_25><loc_56><loc_25><loc_57><loc_25></location>7</text>
<text><location><page_25><loc_56><loc_24><loc_57><loc_25></location>2</text>
<text><location><page_25><loc_56><loc_23><loc_57><loc_24></location>4</text>
<text><location><page_25><loc_56><loc_23><loc_57><loc_23></location>8</text>
<text><location><page_25><loc_56><loc_22><loc_57><loc_23></location>.</text>
<text><location><page_25><loc_56><loc_22><loc_57><loc_22></location>3</text>
<text><location><page_25><loc_56><loc_21><loc_57><loc_22></location>1</text>
<text><location><page_25><loc_56><loc_20><loc_57><loc_21></location>±</text>
<text><location><page_25><loc_56><loc_20><loc_57><loc_20></location>3</text>
<text><location><page_25><loc_56><loc_19><loc_57><loc_20></location>7</text>
<text><location><page_25><loc_56><loc_19><loc_57><loc_19></location>.</text>
<text><location><page_25><loc_56><loc_18><loc_57><loc_19></location>8</text>
<text><location><page_25><loc_56><loc_18><loc_57><loc_18></location>2</text>
<text><location><page_25><loc_56><loc_15><loc_57><loc_16></location>3</text>
<text><location><page_25><loc_56><loc_15><loc_57><loc_15></location>6</text>
<text><location><page_25><loc_56><loc_14><loc_57><loc_15></location>4</text>
<text><location><page_25><loc_56><loc_13><loc_57><loc_14></location>k</text>
<text><location><page_25><loc_56><loc_13><loc_57><loc_13></location>r</text>
<text><location><page_25><loc_56><loc_12><loc_57><loc_13></location>M</text>
<text><location><page_25><loc_54><loc_26><loc_55><loc_26></location>±</text>
<text><location><page_25><loc_54><loc_25><loc_55><loc_26></location>0</text>
<text><location><page_25><loc_54><loc_24><loc_55><loc_25></location>0</text>
<text><location><page_25><loc_54><loc_24><loc_55><loc_24></location>3</text>
<text><location><page_25><loc_54><loc_22><loc_55><loc_22></location>8</text>
<text><location><page_25><loc_54><loc_21><loc_55><loc_22></location>3</text>
<text><location><page_25><loc_54><loc_21><loc_55><loc_21></location>.</text>
<text><location><page_25><loc_54><loc_20><loc_55><loc_21></location>0</text>
<text><location><page_25><loc_54><loc_19><loc_55><loc_20></location><</text>
<text><location><page_25><loc_54><loc_17><loc_55><loc_18></location>2</text>
<text><location><page_25><loc_54><loc_17><loc_55><loc_17></location>3</text>
<text><location><page_25><loc_54><loc_16><loc_55><loc_17></location>2</text>
<text><location><page_25><loc_54><loc_15><loc_55><loc_16></location>1</text>
<text><location><page_25><loc_54><loc_15><loc_55><loc_15></location>+</text>
<text><location><page_25><loc_54><loc_14><loc_55><loc_15></location>1</text>
<text><location><page_25><loc_54><loc_14><loc_55><loc_14></location>5</text>
<text><location><page_25><loc_54><loc_13><loc_55><loc_14></location>4</text>
<text><location><page_25><loc_54><loc_13><loc_55><loc_13></location>3</text>
<text><location><page_25><loc_54><loc_12><loc_55><loc_13></location>1</text>
<text><location><page_25><loc_51><loc_26><loc_53><loc_26></location>±</text>
<text><location><page_25><loc_51><loc_25><loc_53><loc_26></location>1</text>
<text><location><page_25><loc_51><loc_24><loc_53><loc_25></location>6</text>
<text><location><page_25><loc_51><loc_24><loc_53><loc_24></location>4</text>
<text><location><page_25><loc_51><loc_23><loc_53><loc_23></location>3</text>
<text><location><page_25><loc_51><loc_22><loc_53><loc_23></location>8</text>
<text><location><page_25><loc_51><loc_22><loc_53><loc_22></location>.</text>
<text><location><page_25><loc_51><loc_21><loc_53><loc_22></location>0</text>
<text><location><page_25><loc_51><loc_20><loc_53><loc_21></location>±</text>
<text><location><page_25><loc_51><loc_20><loc_53><loc_20></location>4</text>
<text><location><page_25><loc_51><loc_19><loc_53><loc_20></location>8</text>
<text><location><page_25><loc_51><loc_19><loc_53><loc_19></location>.</text>
<text><location><page_25><loc_51><loc_18><loc_53><loc_19></location>2</text>
<text><location><page_25><loc_51><loc_15><loc_53><loc_15></location>3</text>
<text><location><page_25><loc_51><loc_14><loc_53><loc_15></location>7</text>
<text><location><page_25><loc_51><loc_14><loc_53><loc_14></location>2</text>
<text><location><page_25><loc_51><loc_13><loc_53><loc_13></location>C</text>
<text><location><page_25><loc_51><loc_12><loc_53><loc_13></location>3</text>
<text><location><page_25><loc_50><loc_26><loc_51><loc_27></location>±</text>
<text><location><page_25><loc_50><loc_25><loc_51><loc_26></location>0</text>
<text><location><page_25><loc_50><loc_25><loc_51><loc_25></location>0</text>
<text><location><page_25><loc_50><loc_24><loc_51><loc_25></location>3</text>
<text><location><page_25><loc_50><loc_23><loc_51><loc_23></location>1</text>
<text><location><page_25><loc_50><loc_22><loc_51><loc_23></location>7</text>
<text><location><page_25><loc_50><loc_22><loc_51><loc_22></location>.</text>
<text><location><page_25><loc_50><loc_21><loc_51><loc_22></location>0</text>
<text><location><page_25><loc_50><loc_20><loc_51><loc_21></location>±</text>
<text><location><page_25><loc_50><loc_20><loc_51><loc_20></location>1</text>
<text><location><page_25><loc_50><loc_19><loc_51><loc_20></location>0</text>
<text><location><page_25><loc_50><loc_19><loc_51><loc_19></location>.</text>
<text><location><page_25><loc_50><loc_18><loc_51><loc_19></location>2</text>
<text><location><page_25><loc_50><loc_17><loc_51><loc_17></location>4</text>
<text><location><page_25><loc_50><loc_16><loc_51><loc_17></location>4</text>
<text><location><page_25><loc_50><loc_16><loc_51><loc_16></location>4</text>
<text><location><page_25><loc_50><loc_15><loc_51><loc_16></location>0</text>
<text><location><page_25><loc_50><loc_15><loc_51><loc_15></location>-</text>
<text><location><page_25><loc_50><loc_14><loc_51><loc_15></location>1</text>
<text><location><page_25><loc_50><loc_14><loc_51><loc_14></location>7</text>
<text><location><page_25><loc_50><loc_13><loc_51><loc_14></location>0</text>
<text><location><page_25><loc_50><loc_13><loc_51><loc_13></location>2</text>
<text><location><page_25><loc_50><loc_12><loc_51><loc_13></location>1</text>
<text><location><page_25><loc_48><loc_26><loc_50><loc_27></location>±</text>
<text><location><page_25><loc_48><loc_25><loc_50><loc_26></location>3</text>
<text><location><page_25><loc_48><loc_25><loc_50><loc_25></location>2</text>
<text><location><page_25><loc_48><loc_24><loc_50><loc_25></location>3</text>
<text><location><page_25><loc_48><loc_23><loc_50><loc_23></location>5</text>
<text><location><page_25><loc_48><loc_22><loc_50><loc_23></location>2</text>
<text><location><page_25><loc_48><loc_22><loc_50><loc_22></location>.</text>
<text><location><page_25><loc_48><loc_21><loc_50><loc_22></location>1</text>
<text><location><page_25><loc_48><loc_20><loc_50><loc_21></location>±</text>
<text><location><page_25><loc_48><loc_20><loc_50><loc_20></location>5</text>
<text><location><page_25><loc_48><loc_19><loc_50><loc_20></location>4</text>
<text><location><page_25><loc_48><loc_19><loc_50><loc_19></location>.</text>
<text><location><page_25><loc_48><loc_18><loc_50><loc_19></location>4</text>
<text><location><page_25><loc_48><loc_17><loc_50><loc_17></location>8</text>
<text><location><page_25><loc_48><loc_16><loc_50><loc_17></location>3</text>
<text><location><page_25><loc_48><loc_16><loc_50><loc_16></location>2</text>
<text><location><page_25><loc_48><loc_15><loc_50><loc_16></location>0</text>
<text><location><page_25><loc_48><loc_15><loc_50><loc_15></location>-</text>
<text><location><page_25><loc_48><loc_14><loc_50><loc_15></location>5</text>
<text><location><page_25><loc_48><loc_14><loc_50><loc_14></location>9</text>
<text><location><page_25><loc_48><loc_13><loc_50><loc_14></location>0</text>
<text><location><page_25><loc_48><loc_13><loc_50><loc_13></location>1</text>
<text><location><page_25><loc_48><loc_12><loc_50><loc_13></location>1</text>
<text><location><page_25><loc_47><loc_26><loc_48><loc_26></location>±</text>
<text><location><page_25><loc_47><loc_25><loc_48><loc_26></location>1</text>
<text><location><page_25><loc_47><loc_24><loc_48><loc_25></location>5</text>
<text><location><page_25><loc_47><loc_24><loc_48><loc_24></location>3</text>
<text><location><page_25><loc_47><loc_23><loc_48><loc_23></location>0</text>
<text><location><page_25><loc_47><loc_22><loc_48><loc_23></location>8</text>
<text><location><page_25><loc_47><loc_22><loc_48><loc_22></location>.</text>
<text><location><page_25><loc_47><loc_21><loc_48><loc_22></location>0</text>
<text><location><page_25><loc_47><loc_20><loc_48><loc_21></location>±</text>
<text><location><page_25><loc_47><loc_20><loc_48><loc_20></location>9</text>
<text><location><page_25><loc_47><loc_19><loc_48><loc_20></location>0</text>
<text><location><page_25><loc_47><loc_19><loc_48><loc_19></location>.</text>
<text><location><page_25><loc_47><loc_18><loc_48><loc_19></location>1</text>
<text><location><page_25><loc_47><loc_17><loc_48><loc_18></location>9</text>
<text><location><page_25><loc_47><loc_17><loc_48><loc_17></location>0</text>
<text><location><page_25><loc_47><loc_16><loc_48><loc_17></location>1</text>
<text><location><page_25><loc_47><loc_15><loc_48><loc_16></location>1</text>
<text><location><page_25><loc_47><loc_15><loc_48><loc_15></location>+</text>
<text><location><page_25><loc_47><loc_14><loc_48><loc_15></location>8</text>
<text><location><page_25><loc_47><loc_14><loc_48><loc_14></location>7</text>
<text><location><page_25><loc_47><loc_13><loc_48><loc_14></location>3</text>
<text><location><page_25><loc_47><loc_13><loc_48><loc_13></location>0</text>
<text><location><page_25><loc_47><loc_12><loc_48><loc_13></location>1</text>
<text><location><page_25><loc_45><loc_26><loc_47><loc_27></location>±</text>
<text><location><page_25><loc_45><loc_25><loc_47><loc_26></location>9</text>
<text><location><page_25><loc_45><loc_25><loc_47><loc_25></location>7</text>
<text><location><page_25><loc_45><loc_24><loc_47><loc_25></location>3</text>
<text><location><page_25><loc_45><loc_23><loc_47><loc_23></location>4</text>
<text><location><page_25><loc_45><loc_22><loc_47><loc_23></location>5</text>
<text><location><page_25><loc_45><loc_22><loc_47><loc_22></location>.</text>
<text><location><page_25><loc_45><loc_21><loc_47><loc_22></location>1</text>
<text><location><page_25><loc_45><loc_21><loc_47><loc_21></location>±</text>
<text><location><page_25><loc_45><loc_20><loc_47><loc_21></location>2</text>
<text><location><page_25><loc_45><loc_20><loc_47><loc_20></location>4</text>
<text><location><page_25><loc_45><loc_19><loc_47><loc_20></location>.</text>
<text><location><page_25><loc_45><loc_19><loc_47><loc_19></location>4</text>
<text><location><page_25><loc_45><loc_18><loc_47><loc_19></location>1</text>
<text><location><page_25><loc_45><loc_17><loc_47><loc_17></location>9</text>
<text><location><page_25><loc_45><loc_16><loc_47><loc_17></location>5</text>
<text><location><page_25><loc_45><loc_16><loc_47><loc_16></location>4</text>
<text><location><page_25><loc_45><loc_15><loc_47><loc_16></location>2</text>
<text><location><page_25><loc_45><loc_15><loc_47><loc_15></location>-</text>
<text><location><page_25><loc_45><loc_14><loc_47><loc_15></location>1</text>
<text><location><page_25><loc_45><loc_14><loc_47><loc_14></location>1</text>
<text><location><page_25><loc_45><loc_13><loc_47><loc_14></location>3</text>
<text><location><page_25><loc_45><loc_13><loc_47><loc_13></location>8</text>
<text><location><page_25><loc_45><loc_12><loc_47><loc_13></location>0</text>
<text><location><page_25><loc_44><loc_26><loc_45><loc_26></location>±</text>
<text><location><page_25><loc_44><loc_25><loc_45><loc_26></location>0</text>
<text><location><page_25><loc_44><loc_24><loc_45><loc_25></location>5</text>
<text><location><page_25><loc_44><loc_24><loc_45><loc_24></location>1</text>
<text><location><page_25><loc_44><loc_23><loc_45><loc_23></location>0</text>
<text><location><page_25><loc_44><loc_22><loc_45><loc_23></location>2</text>
<text><location><page_25><loc_44><loc_22><loc_45><loc_22></location>.</text>
<text><location><page_25><loc_44><loc_21><loc_45><loc_22></location>2</text>
<text><location><page_25><loc_44><loc_20><loc_45><loc_21></location>±</text>
<text><location><page_25><loc_44><loc_20><loc_45><loc_20></location>4</text>
<text><location><page_25><loc_44><loc_19><loc_45><loc_20></location>3</text>
<text><location><page_25><loc_44><loc_19><loc_45><loc_19></location>.</text>
<text><location><page_25><loc_44><loc_18><loc_45><loc_19></location>0</text>
<text><location><page_25><loc_44><loc_17><loc_45><loc_18></location>8</text>
<text><location><page_25><loc_44><loc_17><loc_45><loc_17></location>0</text>
<text><location><page_25><loc_44><loc_16><loc_45><loc_17></location>5</text>
<text><location><page_25><loc_44><loc_15><loc_45><loc_16></location>6</text>
<text><location><page_25><loc_44><loc_15><loc_45><loc_15></location>+</text>
<text><location><page_25><loc_44><loc_14><loc_45><loc_15></location>8</text>
<text><location><page_25><loc_44><loc_14><loc_45><loc_14></location>9</text>
<text><location><page_25><loc_44><loc_13><loc_45><loc_14></location>5</text>
<text><location><page_25><loc_44><loc_13><loc_45><loc_13></location>7</text>
<text><location><page_25><loc_44><loc_12><loc_45><loc_13></location>0</text>
<text><location><page_25><loc_43><loc_26><loc_44><loc_26></location>±</text>
<text><location><page_25><loc_43><loc_25><loc_44><loc_26></location>0</text>
<text><location><page_25><loc_43><loc_24><loc_44><loc_25></location>5</text>
<text><location><page_25><loc_43><loc_24><loc_44><loc_24></location>3</text>
<text><location><page_25><loc_43><loc_23><loc_44><loc_23></location>4</text>
<text><location><page_25><loc_43><loc_22><loc_44><loc_23></location>5</text>
<text><location><page_25><loc_43><loc_22><loc_44><loc_22></location>.</text>
<text><location><page_25><loc_43><loc_21><loc_44><loc_22></location>5</text>
<text><location><page_25><loc_43><loc_20><loc_44><loc_21></location>±</text>
<text><location><page_25><loc_43><loc_20><loc_44><loc_20></location>6</text>
<text><location><page_25><loc_43><loc_19><loc_44><loc_20></location>5</text>
<text><location><page_25><loc_43><loc_19><loc_44><loc_19></location>.</text>
<text><location><page_25><loc_43><loc_18><loc_44><loc_19></location>4</text>
<text><location><page_25><loc_43><loc_17><loc_44><loc_17></location>5</text>
<text><location><page_25><loc_43><loc_16><loc_44><loc_17></location>1</text>
<text><location><page_25><loc_43><loc_16><loc_44><loc_16></location>3</text>
<text><location><page_25><loc_43><loc_15><loc_44><loc_16></location>6</text>
<text><location><page_25><loc_43><loc_15><loc_44><loc_15></location>-</text>
<text><location><page_25><loc_43><loc_14><loc_44><loc_15></location>6</text>
<text><location><page_25><loc_43><loc_14><loc_44><loc_14></location>0</text>
<text><location><page_25><loc_43><loc_13><loc_44><loc_14></location>2</text>
<text><location><page_25><loc_43><loc_13><loc_44><loc_13></location>6</text>
<text><location><page_25><loc_43><loc_12><loc_44><loc_13></location>0</text>
<text><location><page_25><loc_41><loc_26><loc_42><loc_26></location>±</text>
<text><location><page_25><loc_41><loc_25><loc_42><loc_26></location>5</text>
<text><location><page_25><loc_41><loc_24><loc_42><loc_25></location>1</text>
<text><location><page_25><loc_41><loc_24><loc_42><loc_24></location>2</text>
<text><location><page_25><loc_33><loc_26><loc_34><loc_26></location>±</text>
<text><location><page_25><loc_30><loc_24><loc_32><loc_25></location>I</text>
<text><location><page_25><loc_30><loc_24><loc_32><loc_24></location>I</text>
<text><location><page_25><loc_33><loc_24><loc_34><loc_25></location>5</text>
<text><location><page_25><loc_33><loc_24><loc_34><loc_24></location>3</text>
<text><location><page_25><loc_38><loc_26><loc_40><loc_26></location>±</text>
<text><location><page_25><loc_38><loc_25><loc_40><loc_26></location>0</text>
<text><location><page_25><loc_38><loc_24><loc_40><loc_25></location>0</text>
<text><location><page_25><loc_38><loc_24><loc_40><loc_24></location>3</text>
<text><location><page_25><loc_40><loc_26><loc_41><loc_26></location>±</text>
<text><location><page_25><loc_40><loc_25><loc_41><loc_26></location>0</text>
<text><location><page_25><loc_40><loc_24><loc_41><loc_25></location>5</text>
<text><location><page_25><loc_40><loc_24><loc_41><loc_24></location>3</text>
<text><location><page_25><loc_33><loc_22><loc_34><loc_22></location>.</text>
<text><location><page_25><loc_33><loc_21><loc_34><loc_22></location>0</text>
<text><location><page_25><loc_33><loc_20><loc_34><loc_21></location>±</text>
<text><location><page_25><loc_33><loc_20><loc_34><loc_20></location>4</text>
<text><location><page_25><loc_33><loc_19><loc_34><loc_20></location>6</text>
<text><location><page_25><loc_33><loc_19><loc_34><loc_19></location>.</text>
<text><location><page_25><loc_33><loc_18><loc_34><loc_19></location>0</text>
<text><location><page_25><loc_33><loc_17><loc_34><loc_17></location>6</text>
<text><location><page_25><loc_33><loc_16><loc_34><loc_17></location>5</text>
<text><location><page_25><loc_33><loc_16><loc_34><loc_16></location>8</text>
<text><location><page_25><loc_33><loc_15><loc_34><loc_16></location>0</text>
<text><location><page_25><loc_33><loc_15><loc_34><loc_15></location>-</text>
<text><location><page_25><loc_33><loc_14><loc_34><loc_15></location>8</text>
<text><location><page_25><loc_33><loc_14><loc_34><loc_14></location>8</text>
<text><location><page_25><loc_33><loc_13><loc_34><loc_14></location>1</text>
<text><location><page_25><loc_33><loc_13><loc_34><loc_13></location>0</text>
<text><location><page_25><loc_33><loc_12><loc_34><loc_13></location>0</text>
<text><location><page_25><loc_34><loc_22><loc_35><loc_22></location>.</text>
<text><location><page_25><loc_34><loc_21><loc_35><loc_22></location>0</text>
<text><location><page_25><loc_34><loc_20><loc_35><loc_21></location>±</text>
<text><location><page_25><loc_34><loc_20><loc_35><loc_20></location>5</text>
<text><location><page_25><loc_34><loc_19><loc_35><loc_20></location>0</text>
<text><location><page_25><loc_34><loc_19><loc_35><loc_19></location>.</text>
<text><location><page_25><loc_34><loc_18><loc_35><loc_19></location>3</text>
<text><location><page_25><loc_34><loc_17><loc_35><loc_17></location>2</text>
<text><location><page_25><loc_34><loc_16><loc_35><loc_17></location>1</text>
<text><location><page_25><loc_34><loc_16><loc_35><loc_16></location>3</text>
<text><location><page_25><loc_34><loc_15><loc_35><loc_16></location>1</text>
<text><location><page_25><loc_34><loc_15><loc_35><loc_15></location>-</text>
<text><location><page_25><loc_34><loc_14><loc_35><loc_15></location>7</text>
<text><location><page_25><loc_34><loc_14><loc_35><loc_14></location>9</text>
<text><location><page_25><loc_34><loc_13><loc_35><loc_14></location>3</text>
<text><location><page_25><loc_34><loc_13><loc_35><loc_13></location>0</text>
<text><location><page_25><loc_34><loc_12><loc_35><loc_13></location>0</text>
<text><location><page_25><loc_35><loc_22><loc_37><loc_22></location>.</text>
<text><location><page_25><loc_35><loc_21><loc_37><loc_22></location>1</text>
<text><location><page_25><loc_35><loc_20><loc_37><loc_21></location>±</text>
<text><location><page_25><loc_35><loc_20><loc_37><loc_20></location>0</text>
<text><location><page_25><loc_35><loc_19><loc_37><loc_20></location>2</text>
<text><location><page_25><loc_35><loc_19><loc_37><loc_19></location>.</text>
<text><location><page_25><loc_35><loc_18><loc_37><loc_19></location>3</text>
<text><location><page_25><loc_35><loc_17><loc_37><loc_17></location>8</text>
<text><location><page_25><loc_35><loc_16><loc_37><loc_17></location>3</text>
<text><location><page_25><loc_35><loc_16><loc_37><loc_16></location>2</text>
<text><location><page_25><loc_35><loc_15><loc_37><loc_16></location>2</text>
<text><location><page_25><loc_35><loc_15><loc_37><loc_15></location>-</text>
<text><location><page_25><loc_35><loc_14><loc_37><loc_15></location>3</text>
<text><location><page_25><loc_35><loc_14><loc_37><loc_14></location>0</text>
<text><location><page_25><loc_35><loc_13><loc_37><loc_14></location>0</text>
<text><location><page_25><loc_35><loc_13><loc_37><loc_13></location>1</text>
<text><location><page_25><loc_35><loc_12><loc_37><loc_13></location>0</text>
<text><location><page_25><loc_37><loc_22><loc_38><loc_22></location>.</text>
<text><location><page_25><loc_37><loc_21><loc_38><loc_22></location>4</text>
<text><location><page_25><loc_37><loc_20><loc_38><loc_21></location>±</text>
<text><location><page_25><loc_37><loc_20><loc_38><loc_20></location>3</text>
<text><location><page_25><loc_37><loc_19><loc_38><loc_20></location>4</text>
<text><location><page_25><loc_37><loc_19><loc_38><loc_19></location>.</text>
<text><location><page_25><loc_37><loc_18><loc_38><loc_19></location>4</text>
<text><location><page_25><loc_37><loc_16><loc_38><loc_16></location>4</text>
<text><location><page_25><loc_37><loc_15><loc_38><loc_16></location>1</text>
<text><location><page_25><loc_37><loc_15><loc_38><loc_15></location>0</text>
<text><location><page_25><loc_37><loc_14><loc_38><loc_15></location>1</text>
<text><location><page_25><loc_37><loc_13><loc_38><loc_14></location>k</text>
<text><location><page_25><loc_37><loc_13><loc_38><loc_13></location>r</text>
<text><location><page_25><loc_37><loc_12><loc_38><loc_13></location>M</text>
<text><location><page_25><loc_38><loc_22><loc_40><loc_22></location>.</text>
<text><location><page_25><loc_38><loc_21><loc_40><loc_22></location>4</text>
<text><location><page_25><loc_38><loc_20><loc_40><loc_21></location>±</text>
<text><location><page_25><loc_38><loc_20><loc_40><loc_20></location>8</text>
<text><location><page_25><loc_38><loc_19><loc_40><loc_20></location>3</text>
<text><location><page_25><loc_38><loc_19><loc_40><loc_19></location>.</text>
<text><location><page_25><loc_38><loc_18><loc_40><loc_19></location>5</text>
<text><location><page_25><loc_38><loc_17><loc_40><loc_18></location>7</text>
<text><location><page_25><loc_38><loc_17><loc_40><loc_17></location>2</text>
<text><location><page_25><loc_38><loc_16><loc_40><loc_17></location>2</text>
<text><location><page_25><loc_38><loc_15><loc_40><loc_16></location>4</text>
<text><location><page_25><loc_38><loc_15><loc_40><loc_15></location>+</text>
<text><location><page_25><loc_38><loc_14><loc_40><loc_15></location>8</text>
<text><location><page_25><loc_38><loc_14><loc_40><loc_14></location>5</text>
<text><location><page_25><loc_38><loc_13><loc_40><loc_14></location>1</text>
<text><location><page_25><loc_38><loc_13><loc_40><loc_13></location>3</text>
<text><location><page_25><loc_38><loc_12><loc_40><loc_13></location>0</text>
<text><location><page_25><loc_40><loc_22><loc_41><loc_22></location>.</text>
<text><location><page_25><loc_40><loc_21><loc_41><loc_22></location>4</text>
<text><location><page_25><loc_40><loc_20><loc_41><loc_21></location>±</text>
<text><location><page_25><loc_40><loc_20><loc_41><loc_20></location>4</text>
<text><location><page_25><loc_40><loc_19><loc_41><loc_20></location>7</text>
<text><location><page_25><loc_40><loc_19><loc_41><loc_19></location>.</text>
<text><location><page_25><loc_40><loc_18><loc_41><loc_19></location>4</text>
<text><location><page_25><loc_40><loc_17><loc_41><loc_18></location>8</text>
<text><location><page_25><loc_40><loc_17><loc_41><loc_17></location>2</text>
<text><location><page_25><loc_40><loc_16><loc_41><loc_17></location>0</text>
<text><location><page_25><loc_40><loc_15><loc_41><loc_16></location>0</text>
<text><location><page_25><loc_40><loc_15><loc_41><loc_15></location>+</text>
<text><location><page_25><loc_40><loc_14><loc_41><loc_15></location>1</text>
<text><location><page_25><loc_40><loc_14><loc_41><loc_14></location>2</text>
<text><location><page_25><loc_40><loc_13><loc_41><loc_14></location>5</text>
<text><location><page_25><loc_40><loc_13><loc_41><loc_13></location>3</text>
<text><location><page_25><loc_40><loc_12><loc_41><loc_13></location>0</text>
<text><location><page_25><loc_41><loc_21><loc_42><loc_22></location>0</text>
<text><location><page_25><loc_41><loc_21><loc_42><loc_21></location>1</text>
<text><location><page_25><loc_41><loc_20><loc_42><loc_21></location>±</text>
<text><location><page_25><loc_41><loc_20><loc_42><loc_20></location>2</text>
<text><location><page_25><loc_41><loc_19><loc_42><loc_20></location>5</text>
<text><location><page_25><loc_41><loc_19><loc_42><loc_19></location>.</text>
<text><location><page_25><loc_41><loc_18><loc_42><loc_19></location>9</text>
<text><location><page_25><loc_41><loc_17><loc_42><loc_17></location>2</text>
<text><location><page_25><loc_41><loc_16><loc_42><loc_17></location>0</text>
<text><location><page_25><loc_41><loc_16><loc_42><loc_16></location>1</text>
<text><location><page_25><loc_41><loc_15><loc_42><loc_16></location>7</text>
<text><location><page_25><loc_41><loc_15><loc_42><loc_15></location>-</text>
<text><location><page_25><loc_41><loc_14><loc_42><loc_15></location>5</text>
<text><location><page_25><loc_41><loc_14><loc_42><loc_14></location>3</text>
<text><location><page_25><loc_41><loc_13><loc_42><loc_14></location>0</text>
<text><location><page_25><loc_41><loc_13><loc_42><loc_13></location>6</text>
<text><location><page_25><loc_41><loc_12><loc_42><loc_13></location>0</text>
<text><location><page_25><loc_37><loc_26><loc_38><loc_27></location>±</text>
<text><location><page_25><loc_37><loc_25><loc_38><loc_26></location>0</text>
<text><location><page_25><loc_37><loc_25><loc_38><loc_25></location>7</text>
<text><location><page_25><loc_37><loc_24><loc_38><loc_25></location>2</text>
<text><location><page_25><loc_35><loc_26><loc_37><loc_27></location>±</text>
<text><location><page_25><loc_35><loc_25><loc_37><loc_26></location>0</text>
<text><location><page_25><loc_35><loc_25><loc_37><loc_25></location>0</text>
<text><location><page_25><loc_35><loc_24><loc_37><loc_25></location>4</text>
<text><location><page_25><loc_34><loc_26><loc_35><loc_27></location>±</text>
<text><location><page_25><loc_34><loc_25><loc_35><loc_26></location>9</text>
<text><location><page_25><loc_34><loc_25><loc_35><loc_25></location>5</text>
<text><location><page_25><loc_34><loc_24><loc_35><loc_25></location>2</text>
<text><location><page_25><loc_53><loc_84><loc_54><loc_84></location>7</text>
<text><location><page_25><loc_53><loc_83><loc_54><loc_84></location>4</text>
<text><location><page_25><loc_53><loc_83><loc_54><loc_83></location>2</text>
<text><location><page_25><loc_53><loc_81><loc_54><loc_81></location>0</text>
<text><location><page_25><loc_53><loc_80><loc_54><loc_81></location>3</text>
<text><location><page_25><loc_53><loc_80><loc_54><loc_80></location>.</text>
<text><location><page_25><loc_53><loc_79><loc_54><loc_80></location>1</text>
<text><location><page_25><loc_53><loc_79><loc_54><loc_79></location>±</text>
<text><location><page_25><loc_53><loc_78><loc_54><loc_79></location>0</text>
<text><location><page_25><loc_53><loc_78><loc_54><loc_78></location>3</text>
<text><location><page_25><loc_53><loc_77><loc_54><loc_78></location>.</text>
<text><location><page_25><loc_53><loc_77><loc_54><loc_77></location>8</text>
<text><location><page_25><loc_53><loc_76><loc_54><loc_77></location>3</text>
<text><location><page_25><loc_53><loc_71><loc_54><loc_72></location>2</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We present Herschel observations of six fine-structure lines in 25 Ultraluminous Infrared Galaxies at z < 0 . 27. The lines, [O III ]52 µ m, [N III ]57 µ m, [O I ]63 µ m, [N II ]122 µ m, [O I ]145 µ m, and [C II ]158 µ m, are mostly single gaussians with widths < 600 km s -1 and luminosities of 10 7 -10 9 L /circledot . There are deficits in the [O I ]63/L IR , [N II ]/L IR , [O I ]145/L IR , and [C II ]/L IR ratios compared to lower luminosity systems. The majority of the line deficits are consistent with dustier H II regions, but part of the [C II ] deficit may arise from an additional mechanism, plausibly charged dust grains. This is consistent with some of the [C II ] originating from PDRs or the ISM. We derive relations between far-IR line luminosities and both IR luminosity and star formation rate. We find that [N II ] and both [O I ] lines are good tracers of IR luminosity and star formation rate. In contrast, [C II ] is a poor tracer of IR luminosity and star formation rate, and does not improve as a tracer of either quantity if the [C II ] deficit is accounted for. The continuum luminosity densities also correlate with IR luminosity and star formation rate. We derive ranges for the gas density and ultraviolet radiation intensity of 10 1 < n < 10 2 . 5 and 10 2 . 2 < G 0 < 10 3 . 6 , respectively. These ranges depend on optical type, the importance of star formation, and merger stage. We do not find relationships between far-IR line properties and several other parameters; AGN activity, merger stage, mid-IR excitation, and SMBH mass. We conclude that these far-IR lines arise from gas heated by starlight, and that they are not strongly influenced by AGN activity. Subject headings: galaxies: starburst - infrared: galaxies - galaxies: evolution - galaxies: active",
"pages": [
1
]
},
{
"title": "FAR-INFRARED FINE-STRUCTURE LINE DIAGNOSTICS OF ULTRALUMINOUS INFRARED GALAXIES",
"content": "D. Farrah 1 , V. Lebouteiller 2,3 , H. W. W. Spoon 2 , J. Bernard-Salas 4 , C. Pearson 4,5 , D. Rigopoulou 6,5 , H. A. Smith 7 , E. Gonz'alez-Alfonso 8 , D. L. Clements 9 , A. Efstathiou 10 , D. Cormier 11 , J. Afonso 12,13 , S. M. Petty 1 , K. Harris 1 , P. Hurley 14 , C. Borys 15 , A. Verma 6 , A. Cooray 16 , and V. Salvatelli 17,16",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Ultraluminous Infrared Galaxies (ULIRGS, objects with L IR > 10 12 L /circledot , Sanders & Mirabel 1996; Lonsdale et al. 2006) are a cosmologically important population whose nature changes substantially with redshift. At z < 0 . 3 ULIRGs are rare (e.g. Soifer & Neugebauer 1991; Vaccari et al. 2010), with less than one per ∼ hundred square degrees. They are invariably mergers between approximately equal mass galaxies (Clements et al. 1996; Surace et al. 2000; Cui et al. 2001; Farrah et al. 2001; Bushouse et al. 2002; Veilleux et al. 2002, 2006). Evidence suggests that their IR emission arises mainly from high rates of star formation (Genzel et al 1998; Tran et al. 2001; Franceschini et al. 2003; Nardini et al. 2010; Wang et al. 2011), though of order half also contain a luminous AGN (Rigopoulou et al. 1999; Farrah et al. 2003; Imanishi et al. 2007; Vega et al. 2008; Nardini & Risaliti 2011). The AGN in ULIRGs may become more important with increasing IR luminosity, and advancing merger stage (Teng & Veilleux 2010; Yuan et al. 2010; Stierwalt et al. 2013), and sometimes initiate powerful outflows (Spoon et al. 2009; Fischer et al. 2010; Rupke & Veilleux 2011; Sturm et al. 2011; Westmoquette et al. 2012; Rodr'ıguez Zaur'ın et al. 2013). A small fraction of (low-redshift) ULIRGs become optical QSOs (Tacconi et al. 2002; Kawakatu et al. 2006, 2007; Farrah et al. 2007b; Meng et al. 2010; Hou et al. 2011) and a large fraction end up as early-type galaxies (Genzel et al. 2001; Dasyra et al. 2006; Rothberg et al. 2013; Wang et al. 2013). Over 0 . 3 /lessorsimilar z < 1 the number of ULIRGs rises rapidly (e.g. Le Floc'h et al. 2005), reaching a density on the sky of several hundred per square degree at z /greaterorsimilar 1 (Rowan-Robinson et al. 1997; Dole et al. 2001; Borys et al. 2003; Mortier et al. 2005; Austermann et al. 2010; Goto et al. 2011). The fraction of z /greaterorsimilar 1 ULIRGs that are starburst dominated mergers is high (Farrah et al. 2002; Chapman et al. 2003; Smail et al. 2004; Takata et al. 2006; Borys et al. 2006; Valiante et al. 2007; Berta et al. 2007; Bridge et al. c SMBH mass, in units of 10 8 M /circledot (Zheng et al. 2002; Dasyra et al. 2006; Greene & Ho 2007; Zhang et al. 2008; Veilleux et al. 2009). 2007; Lonsdale et al. 2009; Huang et al. 2009; Magnelli et al. 2012; Lo Faro et al. 2013; Johnson et al. 2013) but the merger fraction may be lower than locally (Melbourne et al. 2008; Kartaltepe et al. 2010; Draper & Ballantyne 2012, but see also Xu et al. 2012). High redshift ULIRGs may also have a wider range in dust temperature (Magdis et al. 2010; Rowan-Robinson et al. 2010; Symeonidis et al. 2011, 2013; Bridge et al. 2013) and SED shapes (Farrah et al. 2008; Sajina et al. 2012; Nordon et al. 2012), and a greater star formation efficiency (Iglesias-P'aramo et al. 2007; Combes et al. 2011, 2013; Hanami et al. 2012; Geach et al. 2013) compared to local examples. Determining why the number and properties of ULIRGs change so markedly with redshift may provide insight into the history of stellar and SMBH mass assembly in /greaterorsimilar L ∗ galaxies. ULIRGs at z < 0 . 3 are central to this endeavour, as they establish a baseline from which to measure evolution with redshift in the ULIRG population. The far-infrared ( /similarequal 50 -500 µ m) is a powerful tool for studying ULIRGs, as demonstrated by the Infrared Space Observatory (ISO, e.g. Fischer et al. 1999; Negishi et al. 2001; Luhman et al. 2003; Spinoglio et al. 2005; Brauher et al. 2008). The Herschel Space Observatory (Pilbratt et al. 2010) offers dramatic advances in far-infrared observing capability over ISO. Its instruments, the Photodetector Array Camera and Spectrometer (PACS, Poglitsch et al. 2010), Spectral and Photometric Imaging REceiver (SPIRE, Griffin et al. 2010) and Heterodyne Instrument for the Far Infrared (de Graauw et al. 2010) can observe wavelength ranges that are inaccessible from the ground, and have improved sensitivity and resolution over previous space-based facilities. We have used Herschel to conduct the Herschel ULIRG Survey (HERUS), which assembles PACS and SPIRE observations of nearly all ULIRGs with a 60 µ mflux greater than ∼ 1 . 7Jy. In this paper we present observations of fine-structure lines for 24 of the sample. Analysis of the SPIRE FTS spectra is presented in Pearson et al 2013, in preparation. Observations of the OH 119 µ m and 79 µ m profiles are presented in Spoon et al. 2013, while modelling of these profiles is presented in Smith et al, in preparation. Finally, a detailed study of the ULIRG IRAS 08572+3915 is presented in Efstathiou et al. 2013. We define infrared luminosity, L IR , to be the luminosity integrated over 8-1000 µ m in the rest frame. We quote luminosities and masses in units of Solar (L /circledot = 3 . 839 × 10 26 Watts, M /circledot = 1 . 99 × 10 30 Kg, respectively). We assume a spatially flat cosmology with H 0 = 67 . 3km s -1 Mpc -1 , Ω = 1 and Ω m = 0 . 315 (Planck Collaboration et al. 2013). Note . - Electron and Hydrogen critical densities are given for n=500cm -3",
"pages": [
1,
2,
3
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},
{
"title": "2.1. Sample Selection",
"content": "HERUS is a photometric and spectroscopic atlas of the z < 0 . 27 ULIRG population. The sample comprises all 40 ULIRGs from the IRAS PSC-z survey (Saunders et al. 2000) with 60 µ m fluxes greater than 2Jy, together with three randomly selected ULIRGs with lower 60 µ m fluxes; IRAS 00397-1312 (1.8 Jy), IRAS 07598+6508 (1.7 Jy) and IRAS 13451+1232 (1.9 Jy). All objects have been observed with the Infrared Spectrograph (IRS, Houck et al. 2004) onboard Spitzer (Armus et al. 2007; Farrah et al. 2007a; Desai et al. 2007). The SHINING survey (Fischer et al. 2010; Sturm et al. 2011; Hailey-Dunsheath et al. 2012; Gonz'alez-Alfonso et al. 2013) obtained PACS spectroscopy for 19/43 sources, so we observed, and present here, the remaining 24 objects. We also include Mrk 231 (Fischer et al. 2010) to give a final sample of 25 objects (Table 1). This sample is not flux limited, but does include nearly all ULIRGs at z < 0 . 27 with 60 µ m fluxes between 1.7 Jy and 6 Jy, together with Mrk 231. The sample therefore gives an almost unbiased view of z < 0 . 3 ULIRGs.",
"pages": [
3
]
},
{
"title": "2.2. Observations",
"content": "The PACS observations were performed between March 18, 2011 and April 8, 2012 (Operational Day 6731060). The PACS integral-field spectrometer samples the spatial direction with 25 pixels and the spectral direction with 16 pixels. Each spectral pixel scans a distinct wavelength range by varying the grating angle. The combination of the 16 ranges makes the final spectrum. The resulting projection of the PACS array on the sky is a footprint of 5 × 5 spatial pixels (spaxels), corresponding to a 47 '' × 47 '' field-of-view. The point spread function full width at half maximum (FWHM) is ≈ 9 . 5 '' between 55 µ m and 110 µ m, and increases to about 14 '' by 200 µ m. A spaxel at the mean redshift of the sample is ∼ 3 kpc in extent. A single footprint observation was performed for each object as they are all smaller than the footprint size. The coordinates were chosen to place the optical centroids in the central spaxel. We observed the sample in the following lines: [O III ]52 µ m, [N III ]57 µ m, [O I ]63 µ m, [N II ]122 µ m, [O I ]145 µ m, and [C II ]158 µ m (Table 2). Observations were done in range spectroscopy mode. We used optical narrow-line redshifts to set the central wavelengths of each range scan. For one object, IRAS07598+6508, the input coordinates were incorrect, placing the source near the edge of the PACS array, thus making the flux determination uncertain. We therefore substituted observations of this source from other programs. For [C II ], we used the dataset 1342243534 (PI Weedman), and for [N II ] we used the dataset 1342231959 (PI Veilleux). We set the wavelength range of each range scan to accomodate uncertainties such as offsets between optical and far-IR line redshifts, and asymmetric or broadened lines. The chop/nod observation mode was used, in which the source is observed by alternating between the on-source position and a clean off-source position. Since the extent of the targets is always < 1 ' , the smallest throw ( ± 1 . 5 ' ) was used to reduce the effect of field-rotation between the two chop positions. Two nod positions were used in order to eliminate the telescope background emission. The data reduction was performed in the Herschel Interactive Processing Environment (HIPE) version 8.0 (Ott 2010) using the default chop/nod pipeline script. The level one products (calibrated in flux and in wavelength, with bad pixel masks from HIPE) were then exported, and processed by the PACSman tool (Lebouteiller et al. 2012).",
"pages": [
3
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},
{
"title": "2.3. Line Measurements",
"content": "The best method to determine a line flux depends on the position and morphology of the line-emitting regions within the PACS footprint. If their combined spatial extent is significantly smaller than a single spaxel, are well centered on the central spaxel, and Herschel maintains accurate pointing (the pointing accuracy of Herschel is ∼ 2 . 5 '' ) then the best flux measurement is that of the central spaxel, scaled by an appropriate point-source correction. We call this method ' M 1'. If these conditions are not satisfied then M 1 will give a lower limit on the flux. We do not know, a priori , the morphologies of the farIR line emitting regions, since high spatial resolution images of this emission do not exist. Moreover, we cannot assume that the morphologies of these regions are traced reliably by emission at other wavelengths. We therefore are unable to straightforwardly distinguish between scenarios such as the source being centered and spatially extended, against the source being off-center and pointlike. Finally, we cannot assume that the morphologies of different lines in the same object are similar, since the line strengths are governed by different excitation temperatures and critical densities. Given these caveats, there are five further methods to determine a line flux: individually, sum the resulting fluxes, and apply a point-source correction (of order 15% or less of the total flux) that accounts for the additional areal sampling of the PSF. This method is suitable if the source is spatially extended or shifted by at most a significant fraction of a spaxel. The point-source correction is wavelength-dependent, but since our range scans span wavelength ranges of order 1 µ m it is equivalent to either calculate the flux and then apply the correction, or apply the correction to the spectrum and then calculate the flux. knowledge of the source morphology. To choose the method for the lines in each source we proceed as follows. First, we determine all six measurements for each line. If the emission is consistent with a well-centered point source, then the fluxes from all six methods will agree with each other, with larger errors for the methods that include more spaxels. We found this to be so in the majority of the sample. For these, we adopted M 1. For the others, the measurements from M 2 through M 6 were higher than M 1, but were consistent with each other. This indicates that the line emission is mostly confined to the central 3 × 3 spaxels. We therefore discarded M 5. To avoid uncertainties in using a simulated PSF, we then used in most cases the measurements from M 2 rather than M 4 or M 6, even though the M 2 errors are larger. In most cases, for each object, the same measurement method was used for all lines. In a few cases though [C II ] is extended while the other lines are consistent with point sources. For one object, IRAS 00397-1312, [C II ] is redshifted such that it lies in a part of the PACS wavelength range that suffers from significant flux leakage. In this wavelength range, which spans approximately 190 µ m to 210 µ m, there is superimposed emission from the second order, at 95-110 µ m. Since [C II ] in IRAS 00397-1312 was observed with SPIRE, we substitute the SPIRE-FTS measurement for this line. Except for [C II ] in IRAS 00397-1312, line fits were performed with PACSman for each spaxel at each raster position, using all points in the data cloud. Errors were estimated from the dispersion of the cloud in each wavelength bin. The fitting function was a Gaussian profile, adjusted simultaneously with a polynomial continuum of degrees one to three. The instrument spectral resolution ranges from ∼ 55 kms -1 to ∼ 320kms -1 depending on the band, order and wavelength. The intrinsic line broadening FWHM was calculated by calculating the quadratic difference between the observed FWHM and the instrumental FWHM, assuming gaussian profiles.",
"pages": [
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},
{
"title": "3. RESULTS AND ANALYSIS",
"content": "We tabulate the far-infrared line properties in Table 4, and present their profiles in Figures 1, 2, and 3. In the following analysis we frequently compare the far-IR line properties to those of the mid-IR fine-structure lines (Farrah et al. 2007a), the 6.2 µ m and 11.2 µ m Polycyclic Aromatic Hydrocarbon (PAH) features, and the 9.7 µ m silicate feature (Table 3). The PAH and silicate feature data were measured from spectra taken from the CASSIS v4 database (Lebouteiller et al. 2011). The PAH luminosities were measured by integrating over 5.9-6.6 µ m and 10.8-11.8 µ m in the continuum subtracted spectra, respectively (Spoon et al. 2007). For the silicate feature, we define its strength, S Sil , as: where f obs is the observed flux at rest-frame 9.7 µ m and rest) Line f cont is the continuum flux at rest-frame 9.7 µ m in the absence of silicate absorption, inferred from a spline fit to the continuum at 5-7 µ m and 14-14.5 µ m (Spoon et al. 2007; Levenson et al. 2007). A positive value corresponds to silicates in absorption.",
"pages": [
5,
6
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},
{
"title": "3.1.1. Profile Shapes",
"content": "In most cases the lines are reproducible by single gaussians with widths between 250km s -1 and 600kms -1 . We do not see greater widths in the higher ionization lines. We also do not see strong asymmetries, or systemic offsets in velocity (compared to the optical redshift) of any line. In a few cases the line profiles are not reproducible with single symmetric profiles. The [O I ]63 lines in two objects, IRAS 06206 -6315 and IRAS 20414-1651 are consistent with significant self-absorption (see also Graci'a-Carpio et al. 2011). There is weaker evidence for [O I ]63 self absorption in IRAS 00188-0856, IRAS 110950238, and IRAS 19254-7245. Self-absorption in [O I ]63 can occur when cool oxygen in foreground clouds reduces the [O I ]63 flux (Poglitsch et al. 1996; Fischer et al. 1999; Vastel et al. 2002; Luhman et al. 2003). The [C II ] profiles in IRAS 20087-0308, IRAS 20414-4651 and IRAS 23253-5415 may show subtle asymmetries. Finally, in four cases (IRAS 06035-7102, Mrk 463, IRAS 110950238, and IRAS 20100-4156), [C II ] may show an additional, broad emission component with widths between 600kms -1 and 1200kms -1 ˙ For consistency with the other lines we do not include the broad component in the line fluxes in Table 4, but discuss it instead in a future paper. We examine the distribution of line widths, using only the narrow components in those cases where an additional broad one exists, in Figure 4. The range in widths of all six lines are consistent with each other. We see no dependence of the range in widths on optical class. For individual objects though there are sometimes substantial differences between individual line widths. In Mrk 1014 for example the [C II ] and [N II ] line widths differ by nearly a factor of two. We speculate that these differences are due to one or more of (a) differences in the critical densities of the lines, (b) that an individually unresolved broad component in one line is brighter than the equivalent component in the other line, and (c) in the case of [C II ] and [N II ], that the [N II ] emission arises mostly from H II regions, while at least some of the [C II ] emission arises from photodissociation regions (PDRs) or the diffuse ISM. It is however also possible that the [N II ] profile is affected by the 4 32 -4 23 transition of o-H 2 O at 121.72 µ m(Fischer et al. 2010; Gonz'alez-Alfonso et al. 2010). We discuss this in Spoon et al. 2013. We plot luminosities of selected individual lines against each other in Figure 5. The line luminosities range from just under 10 7 L /circledot to ∼ 3 × 10 9 L /circledot . The [C II ] or [O I ]63 lines are usually the most luminous lines, though the [O III ] line may be the second most luminous in 3C273 and IRAS 00397-1312. The least luminous two lines are in most cases [N II ] and [O I ]145. In some cases [N III ] is consistent with being the least luminous, and in a few cases (e.g. IRAS 07598+6508 IRAS 23253-5415) [O III ] is consistent with being the least luminous. The range in luminosities decreases approximately with increasing wavelength; the [O III ] and [N III ] line luminosities span a factor of ∼ 50, whereas the [C II ] luminosity spans only a factor of five, less than the range in L IR of the sample. To determine if the line luminosities correlate with each other, we fit the relation in Equation 2 (see § 3.2.1 for methodology). We find positive correlations in all cases, with β values mostly between 0.5 and 1.2. In no case however is the correlation particularly strong (in terms of the S/N on β ). We find that, among these six lines, the luminosity of one cannot be used to predict the luminosity of another to better than ∼ 0 . 2dex. If we instead plot line luminosities normalized by L IR (Figure 6) then the correlations do not change significantly. Neither do they depend on optical spectral type. One object, IRAS 01003-2238, may have an unusually low [N II ] luminosity. The six line measurement methods for [N II ] in this object are in reasonable agreement, despite the low S/N. We therefore do not have a good Line IR explanation for this. The only other way in which IRAS 01003-2238 is atypical is that its optical spectrum contains Wolf-Rayet star features (Farrah et al. 2005) but we do not see why this would affect the [N II ] luminosity. Finally, we comment briefly on two line ratios. First is [O I ]145/[O I ]63, which is related to the physical conditions in PDRs. This ratio (note the inverse is plotted in Figure 5) depends on both the gas temperature and density. It decreases with decreasing gas temperature, and decreases with increasing gas density if the density is above the critical density. In a diffuse PDR illuminated by intense UV radiation, the expected [O I ] ratio lies below ∼ 0 . 1 (e.g., Kaufman et al. 2006). From Figure 5, several of our sample have [O I ] ratios above 0.1. A likely reason for this is self absorption of [O I ]63, or a higher optical depth in [O I ]63 than [O I]145 (e.g., Abel et al. 2007). Second is the [N II ]/[C II ] ratio, which can be used to estimate the relative contribution of [C II ] arising in PDRs and in the ionized gas. While [N II ] arises almost entirely from H II regions, [C II ] can additionally arise from PDRs and the diffuse ISM (cold neutral, warm neutral, and warm diffuse ionized). The [N II ]/[C II ] ratio can therefore quantify the contribution to [C II ] from the ionized gas, by comparing the observed [N II ]/[C II ] ratio with theoretical expectations from a photoionized nebula. The expected ratio in the ionized gas lies between 0.2 and 2.0 depending on density (e.g., figure 7 of Bernard-Salas et al. 2012). The ratios in our sample are however nearly all below 0.25. This indicates that a significant fraction of the [C II ] emission originates from photodissociation regions (PDRs). A precise estimate is however hampered by lack of knowledge on the ionized gas density.",
"pages": [
6,
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},
{
"title": "3.2.1. Line Luminosities",
"content": "We compare far-IR line luminosities to L IR in Figure 7. Qualitatively, some of the line luminosities crudely correlate with L IR , with greater scatter among the Sy1s. To determine if correlations exist between L IR and L Line , we assume that L IR and L Line are related by: where α and β are free parameters. We fit this relation following the method of Tremaine et al. (2002). For a correlation to exist, we require that β > 0 at > 3 . 5 σ significance. We do not claim that Equation 2 codifies an underlying physical relation. Neither do we claim that Equation 2 is the only relation that applies over the luminosity range of our sample 18 . Finally, we do not 18 Indeed, in most cases through this paper where fit results are quoted, a linear ( Y = α + βX ) or exponential ( Y = αX β ) model with appropriate choices for α and β both serve equally well consider more complex models as our data do not prefer them. Excluding objects with Sy1 spectra yields the following relations between L IR and L Line : which are significant (in terms of β ) for all the lines except [C II ]. There is the caveat though that some of the relations are based on a small number of formal detections (Table 4). Including the Sy1s yields significant correlations only for [O III ] and [O I ]145, both with flatter slopes than the above. The insignificant correlation for [C II ] suggests that it traces L IR to an accuracy of about an order of magnitude at best. This result is consistent with previous authors, who find a crude correlation over a wider luminosity range (Sargsyan et al. 2012). We also investigated whether sums of the lines in Equations 3 to 8 show stronger relations with L IR than individual lines. We found correlations in several cases, but none that improved significantly on those for the individual lines. The weaker correlations between L IR and L Line when Sy1s are included in the fits is consistent with far-IR lines being better tracers of L IR in ULIRGs without an optical AGN. If this is true then we may see a similar result if we include in the fits only those objects with prominent PAH features, since PAHs are probably exclusively associated with star formation (see § 3.3.1). We test this hypothesis by repeating the fits, but this time including only those objects with 11.2 µ mPAHEWsgreater than 0.05 µ m. We chose the 11.2 µ m PAH because (a) it is bright, and (b) its value correlates well with the evolutionary paradigm for ULIRGs in Farrah et al. (2009) 19 . We find: 19 Using the 11.2 µ m PAH may however lead to more bias than using the 6.2 µ m PAH if absorption from ices and silicate dust is primarily associated with the background continuum source These relations are somewhat flatter than those in Equations 3 to 8. Again the [C II ] line shows no significant correlation. We conclude that these far-IR lines are better tracers of L IR in systems without type 1 AGN, but that it is unclear whether they are better tracers of L IR in systems with more prominent star formation.",
"pages": [
8,
9,
10
]
},
{
"title": "3.2.2. Continuum Luminosities",
"content": "We compare L IR to continuum luminosity densities (in units of L /circledot Hz -1 ) near the wavelengths of the far-IR lines in Figure 8. We find a significant correlation at all six wavelengths, irrespective of whether Sy1s are included (though the continua of the Sy1s are usually not detected). For consistency with the line comparisons, we exclude the Sy1s and fit relations of the form in Equation 2, obtaining: Including the Sy1s yields comparable relations. If we instead exclude objects with PAH 11.2 µ m EW /lessorsimilar 0 . 05 then we obtain consistent relations.",
"pages": [
10
]
},
{
"title": "3.2.3. Line Deficits",
"content": "Several far-IR lines in ULIRGs show a deficit in their L Line /L IR ratios compared to the ratios expected from systems with lower values of L IR (e.g. Luhman et al. 2003). In contrast, high redshift ULIRGs, at least for [O I ]63 and [C II ], do not show such pronounced deficits (Stacey et al. 2010; Coppin et al. 2012; Rigopoulou et al. 2013). We examine if the [O I ], [N II ] and [C II ] lines in our sample show such deficits in Figure 9 (we do not test the other lines as we lack archival data to compare to). We find a deficit in all four lines. Comparing the mean ratios at 10 11 L /circledot and 10 12 . 2 L /circledot we find differences of factors of 2.75, 4.46, 1.50, and 4.95 for [O I ]63, [N II ], [O I ]145, and [C II ], respectively. The four line deficits show no clear dependence on optical spectral type, the presence of an obscured AGN (as diagnosed from the detection of [Ne V ]14.32, see § 3.5), or on PAH 11.2 µ m EW, for any line. There are however trends with both S Sil and merger stage: IIIa through IVa, top right panel of Figure 10 20 , see also Diaz-Santos et al. 2013). Finally, we plot in the bottom row of Figure 10 L [CII] /L IR against L PAH /L IR as a function of merger stage and S Sil . We see in both plots consistent trends; ULIRGs in advanced mergers and with S Sil /greaterorsimilar 2 have lower L [CII] /L IR and L PAH /L IR ratios, compared to ULIRGs in early-stage stage mergers and with S Sil /similarequal 1 . 4 -2. We do not believe that the line deficits arise from missing an asymmetric or broad component (see § 3.1.1) since such components are rare. Neither do we believe the deficits arise due to self-absorption, since we see no self absorption in [C II ], which shows the strongest deficit. We do however see self absorption in [O I ]63, which shows a weaker deficit. Instead, the stronger [C II ] and [N II ] deficits in sources with higher S Sil (at S Sil /greaterorsimilar 1 . 4) are consistent with the H II regions in ULIRGs being dustier than H II regions in lower luminosity systems. In this scenario (Luhman et al. 2003; Gonz'alez-Alfonso et al. 2008; Abel et al. 2009; Graci'a-Carpio et al. 2011), a higher fraction of the UV photons are absorbed by dust rather than neutral Hydrogen, thus contributing more to L IR 20 We also constructed the deficit plots as a function of merger stage using only those sources with S Sil < 1 . 4. The [C II ] deficit still strengthens with advancing merger stage, while no trends emerge with merger stage for the other lines. [CII] LIR but less to the photoionization heating of gas in the H II regions, thus decreasing line emission relative to L IR . This mechanism would also produce a deficit in [C II ] and the 'deficit' in the PAH emission, even if the bulk of the [C II ] and PAHs are in the PDRs, since there would also be fewer UV photons for photoelectric heating of the PDRs. Furthermore, this mechanism is consistent with the [O I ] deficits. From figure 3 of Graci'a-Carpio et al. 2011 the conditions consistent with the observed deficits of all four lines are n h /lessorsimilar 300cm -3 and 0 . 01 /lessorsimilar 〈 U 〉 /lessorsimilar 0 . 1. We propose though that dustier H II regions are not the sole origin of the line deficits. This is based on three observations. First, it is puzzling that we see no strong dependence of any of the line deficits on S Sil when S Sil /lessorsimilar 1 . 4; if the deficit arises entirely in H II regions, and if S Sil is a proxy for the dust column in these H II regions at S Sil /greaterorsimilar 0, then we should see a dependence. We note though that we have only a small sample, so we could be missing a weak dependence, and that the assumption that S Sil is a proxy for dust column in H II regions is not proven 21 . Second, we see significant line deficits for some 21 It is also (arguably) puzzling that we see dependences on any line deficit at S Sil /greaterorsimilar 1 . 4, since (some) models demand that silicate strengths greater than about this value require smooth rather than sources with S Sil < 0, i.e. a silicate emission feature. Third, only the [C II ] deficit gets stronger with advancing merger stage. If more advanced mergers host dustier H II regions, then we would also see a dependence of the [N II ] deficit on merger stage. These observations suggest that some fraction of the [C II ] deficit is not driven by increased dust in H II regions. We lack the data to investigate this in detail, so we only briefly discuss this further. We consider three possible origins; (1) increased charging of dust grains (e.g. Malhotra et al. 2001), leading to a lower gas heating efficiency, (2) a softer radiation field in the diffuse ISM (e.g. Spaans et al. 1994), and (3) dense gas in the PDRs, making [O I ] rather than [C II ] the primary coolant. The third of these possibilities is feasible, but we do not have the data to confirm or refute it as a mechanism. The second possibility is unlikely, due to the energetic nature of star formation in ULIRGs and from the arguments in Luhman et al. (2003). For the first possibility; if the origin of the additional deficit in [C II ] is grain charging, then we would see a higher G 0 in advanced mergers compared to early stage mergers. From Figure 18 (see § 3.6) the advanced mergers have a roughly order of mag- clumpy dust distributions (Nenkova et al. 2008). nitude higher value of G 0 for about the same n 22 . We therefore infer, cautiously, and with the caveat that we cannot rule out [O I ] being a major cooling line, that part of the [C II ] deficit arises due to grain charging in PDRs or in the diffuse ISM. We further propose that this increase in grain charge is not driven mainly by a luminous AGN, obscured or otherwise, since we see no dependence of the [C II ] deficit on either optical spectral type or the presence of [Ne V ]14.32.",
"pages": [
10,
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]
},
{
"title": "Star Formation Rate 3.3.1.",
"content": "Line Luminosities We examine far-IR fine-structure lines as star formation rate indicators by comparing their luminosities to those of PAHs. PAHs are thought to originate from short-lived Asymptotic Giant Branch stars 22 We note though that this also holds for the trends discussed with S Sil ; dividing into two samples with 1 . 4 < S Sil < 2 . 1 and S Sil > 2 . 1 yields a G 0 /n ratio about a factor of three higher in the latter sample (Gehrz 1989; Habing 1996; Blommaert et al. 2005; Bernard-Salas et al. 2006), and therefore to be associated with star-forming regions (Tielens 2008). They are prominent in starburst galaxies but weak in AGNs (Laurent et al. 2000; Weedman et al. 2005). While uncertainty remains over how to calibrate PAHs as star formation rate measures (Peeters et al. 2004; Forster Schreiber et al. 2004; Sargsyan et al. 2012), their luminosities are likely reasonable proxies for the instantaneous rate of star formation. Conversely, for far-IR fine-structure lines the relation between line luminosity and star formation rate is less clear. We may expect a correlation, as the inter-stellar radiation field (ISRF) from young stars may be an important excitation mechanism. Moreover, correlations between far-IR line luminosities and star formation rate have been observed previously (e.g. Boselli et al. 2002; de Looze et al. 2011; Sargsyan et al. 2012; Zhao et al. 2013). The extent to which correlations exist is however poorly constrained. With the caveats that PAH luminosity depends both on metallicity (Madden et al. 2006; Calzetti et al. 2007; Khramtsova et al. 2013) and dust obscuration, neither of which we can correct for, we assume the PAH luminosities give 'true' star formation rate measures, which we compare to the far-IR line luminosities. In doing so, we further assume that there is negligible differential extinction between the PAHs and the far-IR lines. To mitigate effects from variation of individual PAH features (Smith et al. 2007), we sum the luminosities of the PAH 6.2 µ mand 11.2 µ mfeatures into a single luminosity, L PAH (Farrah et al. 2007a). We plot far-IR line luminosities against L PAH in Figure 11. Unlike the plots with L IR , there is no significant difference between the Sy1s and H II /LINERs. This is consistent with both PAHs and the far-IR lines primarily tracing star formation. Fitting relations of the form in Equation 2 to all the objects, and converting to star formation rate by using equation 5 of Farrah et al. (2007a), yields: =( -10 . 04 ± 1 . 34) + (1 . 55 ± 0 . 17)log(L [OI]145 ) (25) Using the criteria from § 3.2.1 then all the lines show a significant correlation. The trends with [O I ]63 and [C II ] are only barely significant (see also Diaz-Santos et al. 2013). Excluding the Sy1s and Sy2s from the fits yields consistent slopes and intercepts in all cases, though the fits are now no longer significant for [O I ]63 and [C II ]. Considering only those lines detected at > 3 σ , then the derived star formation rates are, for a given object, consistent to within a factor of three in nearly all cases. We present the mean star formation rates in Table 3. An alternative to PAH luminosity as a tracer of star formation rate is the sum of the [Ne III ]15.56 µ m and [Ne II ]12.81 µ m line luminosities, L Neon (Thornley et al. 2000; Ho & Keto 2007; Shipley et al. 2013). We thus compared L Neon to L Line . For the whole sample, we find relations mostly in agreement with Equations 21 through Line IR 26, though the relation with [C II ] is now formally insignificant. Excluding the Seyferts and comparing L Neon to L Line yields similar results to those with L PAH . We note four further points. First, we tested sums of far-IR lines as tracers of star formation rate but found no improvement on the individual relations. Second, if we instead consider L Line / L IR vs. L PAH / L IR then we see correlations with comparable scatter to those in Figure 11. Third, we investigated whether using L PAH is better than using a single PAH luminosity, by reproducing Figure 11 using only the PAH 6.2 µ m luminosity. We found consistent relations in all cases, albeit with larger scatter for the L 6 . 2 PAH plots. Fourth, if we instead code the points in Figure 11 by the 11.2 µ m PAH EW then we see no dependence of the relations on the energetic importance of star formation. Finally, we note two points about the relation between star formation rate and L [CII] . First, we investigated whether L [CII] shows an improved correlation with star formation rate if objects with a strong [C II ] deficit are excluded. Adopting a (somewhat arbitrary) boundary of L [CII] /L IR = 2 × 10 -4 yields only a marginally different relation: suggesting that the correlation does not depend strongly on the [C II ] deficit, though there is the caveat that star formation rate is derived from L PAH (see § 3.2.3 & Figure 10). Second, the relation between ˙ M /circledot and L [CII] is consistent with that given by Sargsyan et al. (2012), though their sample mostly consists of lower luminosity systems.",
"pages": [
13,
14,
15
]
},
{
"title": "3.3.2. Continuum Luminosities",
"content": "We examine continuum luminosity densities as star formation rate tracers in Figure 12. Employing the same method as in § 3.2 we find that the continua near all the lines provide acceptable fits. Converting to relations with star formation rate (see § 3.3) we find: line L /L LIR Note . - Flux units are × 10 -20 W cm -2 . PAH data are taken from the IRS spectra in the CASSIS database (Lebouteiller et al. 2011, see also e.g. Spoon et al. 2007; Desai et al. 2007). The PAH 6.2 µ m EWs (but not the fluxes) have been corrected for ice absorption. The last two columns give mean star formation rates from lines and continua detected at > 3 σ , using Equations 21-26 and 28-33. O which are all significant using the criteria from § 3.2.1. The correlations with continua at ≤ 63 µ m may however be stronger, consistent with stronger correlations with warmer (star formation heated) dust. This is consistent with findings by previous authors (Brandl et al. 2006; Calzetti et al. 2007). We present the mean star formation rates derived from these relations in Table 3. In most cases they are consistent with the line-derived star formation rates.",
"pages": [
15,
16,
17,
18
]
},
{
"title": "3.4. Gas Photoionization",
"content": "If electron densities are below the critical density in the narrow-line region, then the hardness of the radiation field ionizing an element can be estimated via flux ratios of adjacent ionization states of that element; f X i +1 /f X i . For a fixed U this ratio will be approximately proportional to the number of photons producing the observed X i flux relative to the number of Lyman continuum photons. For the mid-IR line-emitting gas, two diagnostic ratios of this type can be used; [Ne III ]15.56/[Ne II ]12.81 and [S IV ]10.51/[S III ]18.71. The photon energies required to produce these four ions are all < 50eV, meaning that they can be produced in star-forming regions (Smith & Houck 2001; Bernard-Salas et al. 2001; Peeters et al. 2002; Verma et al. 2003). As the Neon lines are detected in all of our sample, we use the Neon ratio as a proxy for mid-IR gas excitation 23 . We find no trend of this ratio with individual far-IR line luminosities (e.g. Figure 13), either for the whole sample or for subsamples divided by optical type or PAH 11.2 µ m EW. We also examined three mid-to-far and far-IR line ratios to try fashioning an excitation plane diagram, in a similar manner to Dale et al. (2006); [O IV ]26/[O III ], [N III ]/[N II ], and [O III ]/[N II ]. In no case did we find any trends. The large uncertainties on the [O IV ]26, [O III ], and [N III ] lines means though that we cannot conclude that such trends do not exist.",
"pages": [
18
]
},
{
"title": "3.5. AGN Activity",
"content": "We first compare far-IR line luminosities to optical spectral classification. We see no trends. If we normalize the line luminosities by L IR then no trends emerge, either for individual lines or sums of lines (Figure 14). We also see no trends if we compare optical class to line ratios, or normalized ratios. Moreover, the five objects with an additional broad component in [C II ] ( § 3.1.1) do not have an unusually high incidence of Seyfert spectra. We conclude that optical class cannot be inferred from far-IR line luminosities or ratios. This is consistent with the gas producing the optical emission not being strongly associated (in terms of heating mechanism) with the farIR line emitting gas, at least in the majority of cases. Optical spectra may however misclassify AGN in obscured systems as H II or LINERs. We therefore employ the [Ne V ]14.32/[Ne II ]12.88 line ratio as an AGN diagnostic. Both these lines are less affected by extinction than are optical lines. The [Ne V ]14.32 line can arise in planetary nebulae and supernova remnants (Oliva et al. 1999). For extragalactic sources though, it is weak or absent in star forming regions (e.g. Lutz et al. 1998; Sturm et al. 2002; Bernard-Salas et al. 2009), but strong in spectra of AGN (e.g. Spinoglio et al. 2009). The [Ne II ]12.88 line on the other hand is seen almost universally in galaxies. Their ratio should therefore be a reasonable proxy for the presence of an AGN. We plot [Ne V ]14.32/[Ne II ]12.88 against far-IR line luminosities in Figure 15. There is a correlation between the Neon line ratio and optical classification, but no correlations with far-IR line luminosity. If we substitute optical class for PAH 11.2 µ m EW, then no trends emerge. Considering the Sturm et al mixing lines (bottom right panel of Figure 15) then we see no trends among objects classified either as weak or strong AGN. Finally, if we instead plot line luminosity normalized by L IR on the x axis, then we still see no trends. We searched for trends with far-IR line ratios, normalized ratios, sums and normalized (by L IR ) sums, but found nothing convincing, though the small number of sources with [Ne V ]14.32 detections means we are not certain that no trends exist. We conclude, cautiously, that for ULIRGs there is no reliable diagnostic of AGN luminosity using only simple combinations of far-IR line luminosities. This result is consistent with the weaker correlation observed between L Line and L IR if Sy1s are included (see § 3.2.1), if the AGN supplies an effectively random additional contribution to L IR , thus increasing the scatter in the relation. From § 3.3 & 3.5, it is plausible that at least the plurality of the [C II ] emission arises from PDRs. We defer rigorous modelling to a future paper, and here only estimate the beam-averaged PDR hydrogen nucleus density, n (cm -3 ), and incident far-ultraviolet (FUV; 6 eV < E < 13.6 eV) radiation field intensity G 0 (in units of the local Galactic interstellar FUV field found by Habing 1968; 1 . 6 × 10 -3 erg cm -2 s -1 ) using the web-based tool PDR Toolbox 24 (Kaufman et al. 2006; Pound & Wolfire 2008). We set constraints using three line ratios; [O I ]63/[C II ], [O I ]145/[C II ], and ([O I ]63 +[C II ])/L FIR , where L FIR is the IR luminosity longward of 30 µ m. We assume that all three lines trace PDRs, and that there are no differential extinction effects. We estimate L FIR using the same methods as for L IR (Table 1). For the whole sample (Figure 16) we find (taking a conservative cut of χ 2 red < 5) ranges of 10 1 < n < 10 2 . 5 and 10 2 . 2 < G 0 < 10 3 . 6 , with a power-law dependence between the two. The ranges of both n and G 0 depend on optical class (Figure 17). For H II objects we find 10 1 . 1 < n < 10 2 . 2 and 10 2 . 4 < G 0 < 10 3 . 3 . For LINERS and Seyferts however the ranges widen; for LINERS we find 10 0 . 8 < n < 10 2 . 5 and 10 2 . 4 < G 0 < 10 4 . 1 , and for Sy2s we find 10 0 . 7 < n < 10 3 and 10 1 . 9 < G 0 < 10 3 . 9 . For Sy1s the range for n is comparable to that of LINERs and Sy2s but the range for G 0 increases to 10 2 . 5 < G 0 < 10 4 . 7 . For Sy1s there is a secondary solution that is close to acceptable, which has G 0 and n values approximately four orders of magnitude lower and higher, respectively, than the primary solution. If we divide the sample in two by PAH 11.2 EW (top row of Figure 18) then we see a difference, also. The range in n for both samples is comparable, at about 10 0 . 8 < n < 10 2 . 5 . The ranges for G 0 are however different; for objects with prominent PAHs it is 10 2 . 1 < G 0 < 10 3 . 7 while for objects with weak PAHs it is 10 2 . 6 < G 0 < 10 4 . 3 . We obtain similar ranges for both parameters if we instead divide the sample on merger stage (bottom row of Figure 18). This is consistent with a more intense ISRF destroying PAH molecules (see also e.g. Hern'an-Caballero et al. 2009). It is however also consistent with a luminous AGN (with a harder UV radiation field) arising after the star formation has faded. This would give the same observation but with no direct relation between the two phenomena. There are three caveats in using these models to estimate G 0 and n for our sample. First, we cannot account for different beam filling factors for different lines. This is potentially a significant problem for [C II ] (see § 3.1.2 and § 3.2.3). Second, these models have difficulty in predicting PAH emission strengths (Luhman et al. 2003; Abel et al. 2009), suggesting an incomplete description of the dependence of far-IR line strengths on dust-grain size distribution, PAH properties and ISRF spectral shape (see also e.g. Okada et al. 2013). Since our targets are dusty, it is surprising that we obtain reasonable solutions, indicating that the line ratios and the adopted IR luminosities are compatible with each other. Our derived parameter ranges for G 0 and n should however be viewed with caution. Third is that the [O I ] lines are complex to model. Assuming emission in PDRs then, like [C II ], the [O I ] lines are expected to form within A v ∼ 3 magnitudes of the PDR surfaces. It is in these regions that all of the carbon and oxygen should be ionized and atomic, respectively, with gas temperatures between about 250 and 700K (e.g. Kaufman et al. 1999). The dust in these regions has only a small effect on [C II ], but can have a large impact on the [O I ] levels, which are affected by both radiative and collisional processes. Such processes can alter the power of [O I ]63 via the absorption of 63 µ m line-emitted photons by dust grains, or by pumping of oxygen atoms by 63 µ m continuum dust emitted photons. The effect of dust should not be neglected when modeling [O I ]63, or the [O I ] line ratio, in sources that are optically thick at wavelengths shorter than 100 µ m (Gonz'alez-Alfonso et al. 2008).",
"pages": [
18,
19
]
},
{
"title": "3.7. Merger Stage",
"content": "There is evidence that the power source in ULIRGs evolves as a function of merger stage, with star formation dominating, on average, until the progenitors coalesce, whereupon an AGN sometimes becomes energetically important (e.g. Rigopoulou et al. 1999; Farrah et al. 2009). We may therefore see a correlation between far-IR line properties and merger stage. We find however no correlations between merger stage and far-IR line luminosities. We also see no trend with any far-IR line ratio. Comparing merger stage to normalized far-IR line luminosities (Figure 19) there may be a weak trend; for [O I ]63 and longer lines, the advanced mergers might show a smaller normalized line luminosity than earlier stage mergers. This is consistent with the line luminosities tracing star formation, and with star formation becoming less important as merger stage advances. However, the trend is not strong, and does not depend on optical classification or PAH 11.2 µ m EW.",
"pages": [
19
]
},
{
"title": "3.8. SMBH Mass",
"content": "Scaling relations have been derived between the masses of supermassive black holes and the FWHM and continuum luminosities of several UV, optical, and midIR emission lines (Kaspi et al. 2000; Vestergaard 2002; Vestergaard & Peterson 2006; Dasyra et al. 2008). We here explore, using SMBH masses derived from optical lines (Table 1), whether there exist correlations between far-IR line properties and SMBH mass. While the absolute uncertainties on the SMBH masses from these studies are of order 0.4 dex, the relative uncertainties within the sample are likely smaller as we focus on one class of object and use H β derived masses in nearly all cases. We therefore assume an error on SMBH mass of 20%. We compare the SMBH masses to line luminosities in Figure 20. For the whole sample, no line shows a trend with SMBH mass. Considering only the Sy1 and Sy2s, and excluding 3C273, then some line luminosities show, qualitatively, a positive trend with SMBH mass. The trend is only significant for L [NIII] , for which we derive: We see similar results if we instead compare M SMBH / L IR to L Line / L IR . It is plausible to exclude 3C273, since it is the only Blazar in the sample. We do not, however, claim that this relation is real, for four reasons. First, if we assume a (still reasonable) error on the SMBH masses in excess of 30% then the relation is no longer significant. Second, there is no trend of L [NIII] with the AGN diagnostics considered in § 3.5. Third, if this relation is real then it is strange that we do not see a correlation of SMBH mass with L [OIII] (see Table 2, though there is a potentially important difference; [N III ] is a ground-state transition whereas [O III ] is not). Fourth, we searched for correlations between SMBH mass and far-IR continuum luminosities near 57 µ m, but did not find any clear relations. We have presented observations with PACS onboard Herschel of 25 ULIRGs at z < 0 . 27. We observed each ULIRG in six lines: [O III ]52 µ m, [N III ]57 µ m, [O I ]63 µ m, [N II ]122 µ m, [O I ]145 µ m, and [C II ]158 µ m. We used the properties of these lines, together with diagnostics at other wavelengths, to draw the following conclusions: 1 - In most cases the line profiles are reproducible by single gaussians, with widths between 250 km s -1 and 600 km s -1 . The exceptions are [O I ]63 and [C II ], which occasionally show self absorption and a second, broad component, respectively. We do not see significant systemic offsets of the far-IR lines compared to the optical redshifts. The line luminosities range from just under 10 7 L /circledot to just over 2 × 10 9 L /circledot . The [O I ]63 and [C II ] lines are usually the most luminous, while [O I ]145 and [N II ] are usually the least luminous. The line luminosities correlate with each other, though in no case is the Line IR correlation particularly strong. Simple line ratio diagnostics suggest relatively low gas densities, on average, and that a significant fraction of the [C II ] emission originates from outside H II regions. 2 - There is a deficit in the [O I ]63/L IR , [N II ]/L IR , [O I ]145/L IR , and [C II ]/L IR ratios compared to lower luminosity systems, of factors of 2.75, 4.46, 1.50, and 4.95, respectively. There is evidence that the [C II ] and [N II ] deficits correlate with 9.7 µ m silicate feature strength (S Sil ); if S Sil /greaterorsimilar 1 . 4 then the [C II ] and [N II ] deficits rise with increasing S Sil . We also see a correlation between [L PAH ]/L IR and S Sil . Furthermore, the [C II ] deficit correlates with merger stage; objects in advanced mergers show a greater deficit than objects in early stage mergers. These results are consistent with the majority of the line deficits arising due to increased levels of dust in H II regions. We propose though that a significant fraction of the [C II ] deficit arises from an additional mechanism, plausibly grain charging in PDRs and/or the diffuse ISM. 3 - The line luminosities only weakly correlate with IR luminosity. The correlations improve if Sy1 objects are excluded. Doing so, and fitting a relation of the form log(L IR ) = α + β log(L Line ), yields: The best tracers of L IR are thus the five shorter wavelength lines. The [C II ] line is a poor tracer of L IR , accurate to about an order of magnitude at best. Its accuracy does not noticeably improve if objects with a strong [C II ] deficit are excluded. 4 - The continuum luminosity densities near the wavelengths of the lines correlate with L IR , irrespective of the presence of Sy1s. We derive: 5 - We find correlations between star formation rate, estimated using L PAH , and both line luminosities and continuum luminosity densities. For line luminosities we derive: while for the continuum luminosity densities we derive: On average, the shorter wavelength continua show stronger correlations. 6 - Assuming the [O I ] and [C II ] lines arise mainly in PDRs, we use a simple model to extract estimates for the hydrogen nucleus density, n , and incident far-ultraviolet radiation field G 0 , in the far-IR line emitting gas. We find 10 1 < n < 10 2 . 5 and 10 2 . 2 < G 0 < 10 3 . 6 for the whole sample, with a power-law dependence between the two. The ranges depend on optical spectral class; for H II -like objects we find 10 1 . 1 < n < 10 2 . 2 and 10 2 . 4 < G 0 < 10 3 . 3 , while for Sy1s we find 10 0 . 8 < n < 10 2 . 7 and 10 2 . 5 < G 0 < 10 4 . 7 . There is also a dependence of G 0 on the importance of star formation; objects with weak PAHs have 10 2 . 6 < G 0 < 10 4 . 3 while objects with prominent PAHs have 10 2 . 1 < G 0 < 10 3 . 7 . We find similar ranges for early- vs. late-stage mergers. This is consistent with, but not exclusively supportive of, a more intense ISRF destroying PAH molecules. 7 - We searched for relations between far-IR line luminosities and ratios, and several other parameters; AGN activity (either from optical spectral class or the detection of [Ne V ]14.32), merger stage, mid-IR excitation, and SMBH mass. For the first three parameters we found no relations. We conclude that the far-IR lines do not arise primarily due to AGN activity, and that the properties of the far-IR line emitting gas do not strongly depend on either mid-IR excitation or merger stage. For SMBH mass we found one superficially striking correlation, with L [NIII] , but subsequent tests were not supportive. We conclude that far-IR line luminosities do not straightforwardly trace SMBH mass. We thank the staff of the Herschel helpdesk for many valuable discussions, and the referee for a very helpful report. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. This work is based on observations made with the Spitzer Space Telescope. Spitzer is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research has made extensive use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA, and of NASA's Astrophysics Data System. This research has also made use of Ned Wrights online cosmology calculator (Wright 2006). V.L. is supported by a CEA/Marie Curie Eurotalents fellowship. J.A. acknowledges support from the Science and Technology Foundation (FCT, Portugal) through the research grants PTDC/CTE-AST/105287/2008, PEstOE/FIS/UI2751/2011 and PTDC/FIS-AST/2194/2012. E.G-A is a Research Associate at the HarvardSmithsonian Center for Astrophysics, and thanks the support by the Spanish Ministerio de Econom'ıa y Competitividad under projects AYA2010-21697-C05-0 and FIS2012-39162-C06-01. Facilities: Herschel, Spitzer.",
"pages": [
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23
]
},
{
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Z., et al. 2002, AJ, 124, 18 ] I I C [ 5 4 1 y x a l a G ) 2 ( 3 1 ± 9 7 2 1 6 ± 7 2 4 a 2 0 . 1 ± 5 8 . 3 ) 2 ( 3 5 ± 0 5 5 2 . 0 ± 1 6 . 1 ) 1 ( 4 9 ± 0 5 Far-infrared spectroscopy of ULIRGs 25 ) 2 ( 7 ± 0 5 1 d 3 2 . 1 ± 6 3 . 5 ) 1 ( 9 3 ± 5 6 ) 2 ( 7 1 ± 6 1 3 7 0 . 1 ± 3 3 . 4 ) 2 ( 9 6 ± 8 5 ) 2 ( 4 ± 3 4 2 1 1 . 0 ± 3 6 . 5 ) 2 ( 8 4 ± 4 1 ) 2 ( 7 2 ± 2 4 4 8 0 . 1 ± 5 7 . 3 ) 2 ( 4 1 1 ± 0 5 ) 4 ( 2 ± 7 7 2 d 4 0 . 2 ± 8 5 . 0 3 ) 2 ( 6 1 ± 2 4 ) 2 ( 0 1 ± 9 6 3 9 1 . 1 ± 5 2 . 9 ) 2 ( 6 6 ± 0 5 ) 1 ( 5 3 ± 7 9 1 8 3 . 0 ± 0 1 . 2 ) 1 ) 1 ( 3 ± 2 7 2 5 3 . 0 ± 2 3 . 4 2 ) 1 ) 1 ( 0 2 ± 9 0 4 0 2 . 0 ± 3 5 . 3 ) 1 ) 2 ( 9 ± 6 9 2 d 8 1 . 1 ± 1 5 . 7 ) 2 ) 2 ( 6 ± 0 5 2 d 0 9 . 0 ± 8 4 . 6 ) 1 ) 1 ( 6 6 ± 5 6 4 4 2 . 0 ± 7 4 . 1 ) 1 ) 1 ( 8 1 ± 5 6 4 1 2 . 0 ± 8 4 . 4 ) 1 4 8 8 4 5 7 8 5 5 9 0 0 0 0 ) 2 ( 2 ± 0 5 2 d 8 8 . 0 ± 3 3 . 1 2 ) 2 ( 1 1 ± 0 5 ) 4 ( 4 ± 3 6 1 9 7 . 0 ± 0 4 . 7 ) 2 ( 5 3 ± 2 5 ) 2 ( 8 ± 2 5 3 0 2 . 1 ± 4 5 . 9 ) 2 ( 8 4 ± 7 3 ) 2 ( 2 1 ± 4 1 5 2 9 . 1 ± 7 3 . 1 3 ) 2 ( 9 5 ± 0 2 ) 2 ( 5 ± 0 5 5 8 1 . 0 ± 4 1 . 7 1 ) 2 ( 5 5 ± 9 8 ) 1 ( 4 ± 7 4 2 d 9 1 . 0 ± 5 6 . 9 ) 1 ( 7 2 ± 0 0 ) 2 ( 2 1 ± 5 7 4 6 4 . 0 ± 3 1 . 9 ) 1 ( 0 5 ± 0 5 ) 1 ( 5 ± 7 0 3 0 2 . 0 ± 9 9 . 9 ) 1 ( 7 2 ± 9 7 ) 2 ( 8 ± 3 7 4 d 8 0 . 1 ± 7 5 . 2 1 ) 1 ( 9 6 ± 5 7 . ) 3 . 2 § ( x u fl e h t e r u s a e m o t d e s u d o h ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) i l a l a - - b 1 3 2 k r M c s i F m o r f n e k a t s t n e m e r u s a e M . 3 . 2 § e e s , x u fl S T F - E R I P S a b . 1 . 3 § e e s , d e b r o s b a - f l e s e b y a M c d a n a r o y r t e m m y s a w o h s y a M d . 1 . 3 § e e s , w o l y l l a u s u n u e b y a M e h t t s r fi , e n i l h c a e r o F - . e t o N ± 5 7 3 6 7 . 0 < 5 1 4 5 - 3 5 2 3 2 ± 0 5 2 1 9 . 0 ± 2 0 . 4 6 2 9 6 - 0 3 2 3 2 ± 0 5 5 4 1 . 1 ± 4 2 . 1 1 5 6 1 - 4 1 4 0 2 ± 2 0 2 1 8 . 0 ± 0 9 . 4 6 5 1 4 - 0 0 1 0 2 ± 5 7 4 6 2 . 1 1 ± 2 9 . 2 8 0 3 0 - 7 8 0 0 2 ± 5 7 4 1 2 . 5 1 ± 6 8 . 8 5 4 2 7 - 4 5 2 9 1 ± 0 0 5 1 2 . 5 ± 1 1 . 0 1 9 3 1 0 - 0 9 0 6 1 ± 5 2 1 7 2 . 5 ± 5 4 . 8 0 5 4 0 - 2 6 4 5 1 ± 2 7 2 4 8 . 3 1 ± 3 7 . 8 2 3 6 4 k r M ± 0 0 3 8 3 . 0 < 2 3 2 1 + 1 5 4 3 1 ± 1 6 4 3 8 . 0 ± 4 8 . 2 3 7 2 C 3 ± 0 0 3 1 7 . 0 ± 1 0 . 2 4 4 4 0 - 1 7 0 2 1 ± 3 2 3 5 2 . 1 ± 5 4 . 4 8 3 2 0 - 5 9 0 1 1 ± 1 5 3 0 8 . 0 ± 9 0 . 1 9 0 1 1 + 8 7 3 0 1 ± 9 7 3 4 5 . 1 ± 2 4 . 4 1 9 5 4 2 - 1 1 3 8 0 ± 0 5 1 0 2 . 2 ± 4 3 . 0 8 0 5 6 + 8 9 5 7 0 ± 0 5 3 4 5 . 5 ± 6 5 . 4 5 1 3 6 - 6 0 2 6 0 ± 5 1 2 ± I I 5 3 ± 0 0 3 ± 0 5 3 . 0 ± 4 6 . 0 6 5 8 0 - 8 8 1 0 0 . 0 ± 5 0 . 3 2 1 3 1 - 7 9 3 0 0 . 1 ± 0 2 . 3 8 3 2 2 - 3 0 0 1 0 . 4 ± 3 4 . 4 4 1 0 1 k r M . 4 ± 8 3 . 5 7 2 2 4 + 8 5 1 3 0 . 4 ± 4 7 . 4 8 2 0 0 + 1 2 5 3 0 0 1 ± 2 5 . 9 2 0 1 7 - 5 3 0 6 0 ± 0 7 2 ± 0 0 4 ± 9 5 2 7 4 2 0 3 . 1 ± 0 3 . 8 3 2",
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2013ApJ...776...44M
https://arxiv.org/pdf/1308.3248.pdf
<document>
<text><location><page_1><loc_10><loc_90><loc_42><loc_92></location>Draft version October 10, 2018 Preprint typeset using L A T E X style emulateapj v. 5/2/11</text>
<section_header_level_1><location><page_1><loc_17><loc_83><loc_83><loc_85></location>ACCRETION VARIABILITY OF HERBIG AE/BE STARS OBSERVED BY X-SHOOTER HD 31648 AND HD 163296</section_header_level_1>
<section_header_level_1><location><page_1><loc_45><loc_81><loc_55><loc_82></location>I. Mendigut'ıa</section_header_level_1>
<text><location><page_1><loc_22><loc_80><loc_78><loc_81></location>Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0978, USA</text>
<section_header_level_1><location><page_1><loc_45><loc_76><loc_55><loc_77></location>S.D. Brittain</section_header_level_1>
<text><location><page_1><loc_23><loc_75><loc_78><loc_76></location>Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0978, USA</text>
<section_header_level_1><location><page_1><loc_47><loc_72><loc_53><loc_73></location>C. Eiroa</section_header_level_1>
<text><location><page_1><loc_11><loc_69><loc_90><loc_71></location>Departamento de F'ısica Te'orica, M'odulo 15, Facultad de Ciencias, Universidad Aut'onoma de Madrid, PO Box 28049, Cantoblanco, Madrid, Spain.</text>
<section_header_level_1><location><page_1><loc_47><loc_66><loc_54><loc_67></location>G. Meeus</section_header_level_1>
<text><location><page_1><loc_11><loc_64><loc_90><loc_66></location>Departamento de F'ısica Te'orica, M'odulo 15, Facultad de Ciencias, Universidad Aut'onoma de Madrid, PO Box 28049, Cantoblanco, Madrid, Spain.</text>
<section_header_level_1><location><page_1><loc_45><loc_60><loc_55><loc_61></location>B. Montesinos</section_header_level_1>
<text><location><page_1><loc_11><loc_58><loc_90><loc_60></location>Centro de Astrobiolog'ıa, Departamento de Astrof'ısica (CSIC-INTA), ESAC Campus, P.O. Box 78, 28691 Villanueva de la Ca˜nada, Madrid, Spain.</text>
<section_header_level_1><location><page_1><loc_47><loc_54><loc_53><loc_55></location>A. Mora</section_header_level_1>
<text><location><page_1><loc_9><loc_53><loc_92><loc_54></location>GAIA Science Operations Centre, ESA, European Space Astronomy Centre, PO Box 78, 28691, Villanueva de la Ca˜nada, Madrid, Spain.</text>
<section_header_level_1><location><page_1><loc_45><loc_50><loc_55><loc_51></location>J. Muzerolle</section_header_level_1>
<text><location><page_1><loc_25><loc_48><loc_76><loc_49></location>Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD, 21218, USA</text>
<section_header_level_1><location><page_1><loc_44><loc_45><loc_56><loc_46></location>R.D. Oudmaijer</section_header_level_1>
<text><location><page_1><loc_23><loc_44><loc_78><loc_45></location>School of Physics & Astronomy, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</text>
<section_header_level_1><location><page_1><loc_46><loc_40><loc_54><loc_41></location>E. Rigliaco</section_header_level_1>
<text><location><page_1><loc_10><loc_38><loc_91><loc_40></location>Department of Planetary Science, Lunar and Planetary Lab, University of Arizona, 1629, E. University Blvd, 85719, Tucson, AZ, USA Draft version October 10, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_35><loc_55><loc_36></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_15><loc_86><loc_35></location>This work presents X-Shooter/VLT spectra of the prototypical, isolated Herbig Ae stars HD 31648 (MWC 480) and HD 163296 over five epochs separated by timescales ranging from days to months. Each spectrum spans over a wide wavelength range covering from 310 to 2475 nm. We have monitored the continuum excess in the Balmer region of the spectra and the luminosity of twelve ultraviolet, optical and near infrared spectral lines that are commonly used as accretion tracers for T Tauri stars. The observed strengths of the Balmer excesses have been reproduced from a magnetospheric accretion shock model, providing a mean mass accretion rate of 1.11 × 10 -7 and 4.50 × 10 -7 M /circledot yr -1 for HD 31648 and HD 163296, respectively. Accretion rate variations are observed, being more pronounced for HD 31648 (up to 0.5 dex). However, from the comparison with previous results it is found that the accretion rate of HD 163296 has increased by more than 1 dex, on a timescale of ∼ 15 years. Averaged accretion luminosities derived from the Balmer excess are consistent with the ones inferred from the empirical calibrations with the emission line luminosities, indicating that those can be extrapolated to HAe stars. In spite of that, the accretion rate variations do not generally coincide with those estimated from the line luminosities, suggesting that the empirical calibrations are not useful to accurately quantify accretion rate variability.</text>
<text><location><page_1><loc_14><loc_11><loc_86><loc_14></location>Subject headings: Accretion, accretion disks - circumstellar matter - Line: formation - Protoplanetary disks - Stars: pre-main sequence - Stars: variables: T Tauri, Herbig Ae/Be</text>
<section_header_level_1><location><page_2><loc_22><loc_91><loc_35><loc_92></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_8><loc_64><loc_48><loc_90></location>The star formation process can be studied from the evolution of the accretion rate (see e.g. Hartmann 1998; Sicilia-Aguilar et al. 2004; Fedele et al. 2010). At the initial stages, protostars show both envelope-to-disk and disk-to-star accretion, which can show variations of several orders of magnitude on relatively short timescales (FUor and EXor outbursts, e.g. Hartmann & Kenyon 1996; Herbig 2008; Liskowsky 2010). During the pre-main sequence (PMS) phase, once the envelope dissipates and the central object becomes optically visible, the mass accretion rate ( ˙ M acc ) decreases to typical values of ∼ 10 -8 and 10 -7 M /circledot yr -1 for T Tauri (TT) and Herbig Ae/Be (HAeBe) stars, respectively (Hartmann et al. 1998; Mendigut'ıa et al. 2011b). The accretion rate variability in the PMS phase is also expected to be lower than in previous stages. However, the precise strength of those variations is not well known, which can be partly attributed to discrepancies measuring accretion.</text>
<text><location><page_2><loc_8><loc_27><loc_48><loc_63></location>Primary signatures of accretion are the excess of UV/optical continuum emission and the veiling in the spectroscopic absorption lines that are observed in accreting stars. Both can be modeled from hot accretion shocks whose emission superimposes to that of the stellar atmosphere (Calvet & Gullbring 1998). The way that material from the inner disk reaches the stellar surface is understood from the magnetospheric accretion (MA) scenario, according to which gas does not fall directly from the disk to the star but follows the magnetic field lines that connect them (Uchida & Shibata 1985; Koenigl 1991; Shu et al. 1994). This is the accepted view for TT stars and seems to be valid at least for several HAeBes (see e.g. Vink et al. 2002; Muzerolle et al. 2004; Mottram et al. 2007; Donehew & Brittain 2011; Mendigut'ıa 2013). Accretion rates obtained from MA shock modeling correlate with the luminosity of several emission lines that span from the ultraviolet (UV) to the infrared (IR) (see e.g. the compilation in Rigliaco et al. 2011). These types of empirical calibrations have been extended to late-type HAeBes for the [O I ]6300, H α (Mendigut'ıa et al. 2011b) and Br γ (Donehew & Brittain 2011; Mendigut'ıa et al. 2011b) lines. The origin of the correlations is not fully understood, but these spectral lines can be considered as secondary signatures of accretion, being useful to easily derive accretion rates for wide samples of stars.</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_26></location>Although some works are based on observed variations of primary accretion signatures in TT stars (e.g. Herbst et al. 1994; Fernandez & Eiroa 1996; Batalha et al. 2001), studies of accretion rate variability are mainly based on secondary ones (e.g. Nguyen et al. 2009; Eisner et al. 2010; Costigan et al. 2012; Chou et al. 2013). These analyses provide different results depending on the spectroscopic line used as accretion tracer. As an example, Nguyen et al. (2009) studied a sample of 40 TT stars, reporting typical accretion rate variations of 0.35 dex from the CaII8862 line flux and 0.65 dex from the H α 10%. Similar results were more recently obtained by Costigan et al. (2012). In fact, differences in the variability shown by several lines</text>
<text><location><page_2><loc_52><loc_59><loc_92><loc_92></location>used as accretion tracers are reported for both TTs and HAeBes (Johns & Basri 1995; Lago & Gameiro 1998; Mendigut'ıa et al. 2011a). However, accretion variability from a primary accretion signature was found to be typically lower than 0.5 dex for the HAeBes, on timescales from days to months (Mendigut'ıa et al. 2011b), a result confirmed by Pogodin et al. (2012). Accretion variability of specific TT stars, determined from line veiling, is also expected to be low (Alencar & Batalha 2002). However, the simultaneous analysis of primary and secondary accretion signatures is necessary to test the reliability of the empirical calibrations between the accretion rate and different spectroscopic lines (Rigliaco et al. 2012), and to study the validity of MA in early-type HAeBe stars (Donehew & Brittain 2011; Mendigut'ıa et al. 2011b). These types of studies have been boosted by the advent of the X-Shooter spectrograph at the Very Large Telescope (VLT; Vernet et al. 2011), which allows simultaneous coverage of the accretion signatures from the UV to the near-IR. Simultaneity is crucial when dealing with PMS stars, given their variable nature. Several published and ongoing works face those type of questions, by 'X-Shooting' wide samples of PMS stars (Oudmaijer et al. 2011; Rigliaco et al. 2012; Manara et al. 2013).</text>
<text><location><page_2><loc_52><loc_32><loc_92><loc_57></location>This work aims to contribute to the understanding of the spectroscopic accretion tracers in HAeBe stars. Instead of relating primary and secondary accretion signatures from single-epoch spectra of large samples, we will analyze multi-epoch spectra of two relevant objects. This approach brings the additional possibility of analyzing the variability of the accretion rate. This paper focuses on the variability behavior of two HAe stars: HD 31648 (MWC 480) and HD 163296. The specific objective is twofold. First, to measure the accretion rate and its variability from the Balmer excess (the primary accretion signature for HAeBe stars, see Garrison 1978; Muzerolle et al. 2004). Second, to compare the previous results with several UV, optical and NIR line luminosities commonly used as secondary accretion tracers in low-mass stars. This comparison will provide relevant information about the applicability of the spectroscopic accretion tracers in late-type HAeBe stars.</text>
<text><location><page_2><loc_52><loc_19><loc_92><loc_31></location>The paper is organized as follows: Sect. 2 describes the observations and data reduction, Sect. 3 presents our observational results on the Balmer excess (Sect. 3.1) and the spectroscopic lines (Sect. 3.2), Sect. 4 includes the analysis of the Balmer excesses in terms of a MA model (Sect. 4.1), and discusses possible relations between the accretion luminosity derived from the Balmer excess and from the emission line luminosities (Sects. 4.2 and 4.3). Finally, Sect. 5 summarizes our main conclusions.</text>
<section_header_level_1><location><page_2><loc_54><loc_17><loc_90><loc_18></location>2. STARS, OBSERVATIONS AND DATA REDUCTION</section_header_level_1>
<text><location><page_2><loc_52><loc_7><loc_92><loc_16></location>HD 31648 and HD 163296 are HAe stars (spectral types A5 and A1, ages of ∼ 7 Myr and 5 Myr, respectively; Mora et al. 2001; Montesinos et al. 2009) with no reported stellar companions in the literature. We assume the parameters derived by Montesinos et al. (2009) for the surface temperature, stellar mass and radius (surface gravity), as well as for the distances (T ∗ = 8250 K , 9250</text>
<text><location><page_3><loc_8><loc_33><loc_48><loc_92></location>K ; M ∗ = 2.0 M /circledot , 2.2 M /circledot ; R ∗ = 2.3 R /circledot , 2.3 R /circledot ; d = 146 pc, 130 pc; where the first and second values refer to HD31648 and HD 163296, respectively). The stellar luminosities derived in that work are 22 L /circledot (HD 31648) and 34.5L /circledot (HD 163296). Both objects show continuum and line variations (Sitko et al. 2008), and have magnetically driven accretion signatures (Swartz et al. 2005; Hubrig et al. 2006). We obtained multi-epoch X-Shooter spectra of both objects. Table 1 shows the log of the observations. A total of 10 science spectra were taken, 5 spectra per star on a timescale from days to months. X-Shooter covers a wide wavelength region throughout its three arms: UVB (311-558 nm), VIS (558-1012 nm) and NIR (1012-2475 nm). The slit widths were 1.6, 0.9 and 0.9 '' , which translates into spectral resolutions of 3300, 8800 and 5600 for the UVB, VIS and NIR arms. The spectra were bias and flat field-corrected, flux and wavelength calibrated from standard procedures using the X-Shooter pipeline v1.5.0. (Modigliani et al. 2010). Atmospheric emission was subtracted from an ABBA dithering pattern. Additional telluric absorption correction was applied for specific lines in the NIR arm (Pa γ and Br γ ; see Sect. 3.2), by using telluric standards that were observed before and after the target stars. Flux calibration was tested from low-resolution spectra -i.e. obtained with the widest slit available for all arms (5 '' )-, taken consecutively for the target stars and several telluric standard main-sequence objects with similar air masses. Synthetic photometry obtained by convolving the spectra with broadband UBVRIJHK photometric filters were compared to published values for the telluric standard stars. In this way, the typical uncertainty is in-between 0.05 and 0.2 magnitudes (flux relative errors ∼ 5 and 25 %), where the specific uncertainties for each star depend on the X-Shooter arm and observing run (see below). Further accuracy could be obtained from spectro-photometric standards, which were not provided. Synthetic photometry extracted from our spectra is included in Table 1, which is consistent with previous measurements (de Winter et al. 2001; Oudmaijer et al. 2001; Eiroa et al. 2001). Flux calibrated spectra are shown in Fig. 1. Values from broadband photometry published in the literature (Oudmaijer et al. 2001; Eiroa et al. 2001) are over plotted for comparison.</text>
<section_header_level_1><location><page_3><loc_18><loc_30><loc_39><loc_31></location>3. OBSERVATIONAL RESULTS</section_header_level_1>
<section_header_level_1><location><page_3><loc_21><loc_28><loc_36><loc_29></location>3.1. Balmer excesses</section_header_level_1>
<text><location><page_3><loc_8><loc_7><loc_48><loc_27></location>The Balmer excess is defined from the UB Johnson's photometric bands as ∆D B = ( U -B ) phot -( U -B ) dered , where ( U -B ) phot is the photospheric color, and ( U -B ) dered the observed, dereddened color (Mendigut'ıa et al. 2011b). This can be derived from the extinction (in magnitudes), which for a given wavelength can be expressed from that in the V band, or from the B -V color excess, A λ = A V (k λ /k V ) = R V E ( B -V )(k λ /k V ). The opacity ratio k λ /k V is known from an extinction law, and R V is the total-to-selective extinction ratio. The X-Shooter UVB spectra were used to measure continuum excesses in the Balmer region, along the different observing runs. Therefore, the expression for the Balmer excess has been converted into fluxes. If the observed ones are normalized to the pho-</text>
<text><location><page_3><loc_52><loc_89><loc_92><loc_92></location>c emission at the B band (hereafter represented by the superscript 'norm, B'), then:</text>
<formula><location><page_3><loc_58><loc_79><loc_92><loc_87></location>∆ D B = 2 . 5 log ( F norm , B U F phot U ) + +2 . 5 R V ( k U k V -k B k V ) log ( F norm , B V F phot V ) , (1)</formula>
<text><location><page_3><loc_52><loc_7><loc_92><loc_78></location>which is the expression that is used in this work. The first term in Eq. 1 is essentially the same one as used by Donehew & Brittain (2011), with F norm,B U /F phot U the ratio between the normalized and the photospheric fluxes, measured at wavelengths corresponding to the U -band. The second term reflects the contribution of extinction, and can be estimated from an extinction law and from the ratio between the normalized and photospheric fluxes at wavelengths corresponding to the V band. Apart from being independent of absolute flux calibration (Donehew & Brittain 2011), the main advantage of the above expression for ∆D B is that it accounts for continuum excesses that can be ascribed mainly to hot shocks caused by accretion (Sect. 4.1), given that the possible contribution of extinction is ruled out by the second term. This contribution is typically an order of magnitude lower than the first term for the stars studied here -their E ( B -V ) values are low, see Sect. 3.2-, but it should not be neglected for objects with more pronounced extinctions (i.e. UXOr-like, see e.g. Grinin et al. 1994, and references therein). It is assumed that the third possible cause argued to be in the origin of photometric variability in PMS stars, photospheric variations caused by cold spots (Herbst et al. 1994), is negligible for the stellar temperatures and masses of both stars analyzed. The photospheric fluxes are represented by Kurucz models (Kurucz 1993) with the corresponding values of T ∗ and log g (Sect. 2). These were derived from the SYNTHE code, re-binning to a resolution similar to the X-Shooter spectra, as they were also used to estimate the photospheric contribution to the spectral lines (Sect. 3.2). Observed fluxes were normalized to the Kurucz ones, making their averaged continuum fluxes in the 400-460 nm band (representing the rough range covered by the B photometric filter) to coincide. The ratios F norm,B U /F phot U and F norm,B V /F phot V were then derived considering averaged continuum fluxes in the wavelength regions 350-370 nm and 540-560 nm, respectively. Balmer excesses were finally obtained from Eq. 1, assuming R V = 5 (Hern'andez et al. 2004), k U /k V = 1.6, and k B /k V = 1.3 (Rieke & Lebofsky 1985; Robitaille et al. 2007). The use of a larger R V value, compared with the typical for the interstellar medium (R V = 3.1), has been extensively justified for HAeBe stars (see e.g. Herbst et al. 1982; Gorti & Bhatt 1993; Hern'andez et al. 2004; Manoj et al. 2006, and references therein). In addition, interstellar extinction is most likely negligible compared with the circumstellar one for objects located closer than 200 pc (Fitzgerald 1968), as it is the case for HD 31648 and HD 163296. A different R V value than assumed here affects the second term in Eq. 1. In particular, R V = 3.1 provides Balmer excesses than can be lower by 0.08 magnitudes, being</text>
<table>
<location><page_4><loc_26><loc_64><loc_74><loc_88></location>
<caption>TABLE 1 Log of the observations and synthetic photometry</caption>
</table>
<text><location><page_4><loc_8><loc_62><loc_91><loc_64></location>Notes. Observing dates (dd/mm/yy) and synthetic photometry, in magnitudes, obtained by convolving Johnson-Cousins UBVRI (Bessell 1979) and JHK (Cohen et al. 1990) filter passbands with each flux-calibrated spectrum.</text>
<figure>
<location><page_4><loc_10><loc_12><loc_91><loc_59></location>
<caption>Fig. 1.X-Shooter spectra of HD 31648 and HD 163296 in red, green, yellow, blue and magenta, representing observing runs A, B, C, D and E, respectively. Selected broadband photometry and uncertainties taken from the literature are plotted with solid circles and vertical bars. The horizontal bars represent each total filter passband.</caption>
</figure>
<text><location><page_5><loc_8><loc_85><loc_48><loc_92></location>this upper limit obtained from specific observing runs of HD 163296. For simplicity, possible variations of R V for a given star, or strong dependences of the extinction law on R V , are not considered for the low-extinctions and the wavelength range covered in this work (Mathis 1990).</text>
<text><location><page_5><loc_8><loc_41><loc_48><loc_84></location>Figure 2 shows the observed spectra for the two stars studied here, normalized to the Kurucz ones at wavelengths corresponding to the B photometric filter. Balmer excesses with respect the photospheric spectra are apparent in the 350-370 nm region, and those can vary from one epoch to another. Table 2 shows the measured ∆D B values. Apart from the extinction law, the uncertainty of the Balmer excesses is only dependent of relative flux ratios in wavelengths corresponding to the U and V photometric filters (see Eq. 1). In other words, the accuracy of the Balmer excess would only decrease if errors in flux calibration are strongly wavelength-dependent in the 350-560 nm region, which is not the case for our spectra. A conservative uncertainty of 0.01 magnitudes was estimated by measuring -the theoretically null- ∆D B in consecutive X-Shooter spectra of main sequence stars in different observing dates and with different air masses, for which neither accretion nor changes in extinction were present. The last column in Table 2 shows the mean Balmer excess and its relative variability, σ (∆D B )/ < ∆D B > , defined as the ratio between the standard deviation and the average Balmer excess from the different observing runs. While HD 163296 shows the largest mean Balmer excess (0.41 magnitudes), HD 31648 shows the largest Balmer excess variations (up to a factor 2.5 in a few days). It is noted again the importance of the second term in Eq. 1: despite of the similar variability shown in the Balmer region of the spectra in both panels of Fig. 2, the contribution of -variable- extinction to this variability is a bit more pronounced in HD 163296 (its normalized flux shows larger differences with respect the Kurucz model at longer wavelengths).</text>
<section_header_level_1><location><page_5><loc_22><loc_39><loc_35><loc_40></location>3.2. Spectral lines</section_header_level_1>
<text><location><page_5><loc_8><loc_11><loc_48><loc_39></location>This work focuses on twelve emission lines for which there are previous empirical calibrations with the accretion luminosity in the literature (Sect. 4.2). These lines span over the whole UVB-VIS-NIR wavelength region covered by the X-Shooter spectra. In particular, we focused on transitions of neutral hydrogen, helium, sodium, oxygen, also including transitions of ionized calcium. Figure 3 shows the observed line profiles of HD 31648 and HD 163296, along the five observing runs. Forbidden emission lines most probably tracing outflow processes ([O I ], [S II ], [Fe I ]) were not detected in any of the spectra. However, both stars show variable blueshifted self-absorptions in several lines (H α , Ca II 8542, Pa β ), as well as variable red-shifted self-absorptions in others (Ca II K, H β , O I 8446). Several works deal with the physical origin of some of these lines (see e.g. Hartmann et al. 1994; Muzerolle et al. 1998, 2001, 2004; Tambovtseva et al. 1999; Kurosawa et al. 2006), which have been associated mainly with accretion and/or outflow processes. The study of the physical origin of the spectral lines is beyond the scope of this work.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_9></location>The circumstellar contribution to the equivalent width (EW) of a line was determined from EW cs = EW obs</text>
<text><location><page_5><loc_52><loc_57><loc_92><loc_92></location>-EW phot . The observed equivalent width, EW obs , was directly measured on the normalized spectra 1 , whereas the photospheric absorption, EW phot , was estimated from the Kurucz spectra with the corresponding stellar parameters (see Sect. 3.1). These profiles were broadened by stellar rotation, asuming projected rotational velocities of 133 and 102 km s -1 for HD 163296 and HD 31648, respectively (Mora et al. 2001). For the resolution of our X-Shooter spectra, variations of 500 K in the stellar temperature and changes of 0.50 dex in the surface gravity of the Kurucz spectra (uncertainties of ± 200 K and ± 0.2 dex were quoted in Montesinos et al. 2009) translate into typical relative errors of 10 % and 5 % in EW phot (the specific values depending on the specific star and line). However, since we are mainly interested on the variability of the lines and the photospheric contribution is expected to be constant, it is assumed that the uncertainty in EW cs is dominated by that in EW obs . This was determined for each individual line and observing run by measuring additional EWs from two levels located at ± 1.5 σ from the normalized continuum, fixed at unity. Table 3 shows the equivalent widths and the errors estimated in this way. The relative variability, σ (EW cs )/ < EW cs > (see e.g. Mendigut'ıa et al. 2011a), is indicated in those cases where the EW changes above the uncertainties.</text>
<text><location><page_5><loc_52><loc_33><loc_92><loc_56></location>Dereddened line luminosities (L dered λ ) were obtained from the values for EW cs , the observed continuum fluxes at wavelengths close to the lines (F obs cont,λ ), and from the distances to the stars (d). We used the expression L dered λ = 4 π d 2 × F dered λ , where F dered λ = EW cs × F dered cont,λ = EW cs × F obs cont,λ × 10 0 . 4 A λ . The extinction, A λ = R V E ( B -V )(k λ /k V ), was derived assuming again R V = 5, and from the k λ /k V values in Rieke & Lebofsky (1985) and Robitaille et al. (2007). ( B -V ) color excesses were obtained subtracting the corresponding magnitudes extracted from the Kurucz spectra from the ones included in Table 1. These are shown in Table 4, along with the dereddened line luminosities and their uncertainties (which consider those in the EW and in F obs cont,λ ), as well as the relative variability when a line shows luminosity variations above the uncertainties.</text>
<text><location><page_5><loc_52><loc_26><loc_92><loc_32></location>Although line luminosities tend to be larger for HD 163296, the variability of the lines, both in terms of the EWs and the line luminosities, are comparable for HD 31648 and HD 163296.</text>
<section_header_level_1><location><page_5><loc_61><loc_23><loc_83><loc_24></location>4. ANALYSIS AND DISCUSSION</section_header_level_1>
<section_header_level_1><location><page_5><loc_57><loc_21><loc_87><loc_22></location>4.1. Accretion rates from Balmer excesses</section_header_level_1>
<text><location><page_5><loc_52><loc_11><loc_92><loc_20></location>It is assumed that accretion is magnetically channeled for the two stars studied in this paper. Therefore, the observed Balmer excesses and their variations (Sect. 3.1) are analyzed in terms of a MA shock model. We follow the nomenclature and methodology described in Mendigut'ıa et al. (2011b), where several assumptions of the MA shock model in Calvet & Gullbring (1998) and</text>
<figure>
<location><page_6><loc_9><loc_70><loc_92><loc_93></location>
<caption>Fig. 2.X-Shooter UVB spectra of HD 31648 and HD 163296, normalized to Kurucz photospheric spectra at the stellar surface in the 400-460 nm band. Normalized spectra are plotted in different colors representing different observing runs, as in Fig. 1. Kurucz spectra are plotted with black solid lines.</caption>
</figure>
<table>
<location><page_6><loc_18><loc_55><loc_81><loc_61></location>
<caption>TABLE 2 Observed Balmer excesses.</caption>
</table>
<text><location><page_6><loc_8><loc_53><loc_91><loc_55></location>Notes. Observed Balmer excesses are shown in Cols. 2 to 6. The typical uncertainty is 0.01 magnitudes. The last Col. shows the mean Balmer excess and a number in brackets quantifying the relative variability.</text>
<text><location><page_6><loc_8><loc_46><loc_48><loc_53></location>Muzerolle et al. (2004) were applied to HAeBe stars. According to that, the total flux per wavelength unit emerging from the star ( F λ ) is composed of the emission from the naked photosphere ( F phot λ , represented by a Kurucz synthetic spectrum) plus the accretion contribution:</text>
<formula><location><page_6><loc_19><loc_43><loc_48><loc_45></location>F λ = fF col λ +(1 -f ) F phot λ (2)</formula>
<text><location><page_6><loc_8><loc_37><loc_48><loc_42></location>f being the filling factor that reflects the stellar surface coverage of the accretion columns, and F col λ the flux from the column. This parameter is modeled as a blackbody at a temperature:</text>
<formula><location><page_6><loc_21><loc_35><loc_48><loc_36></location>T col = [ F /σ +T 4 ∗ ] 1 / 4 (3)</formula>
<text><location><page_6><loc_8><loc_27><loc_48><loc_34></location>with σ the Stefan-Boltzmann constant and F the inward flux of energy carried by the accretion columns, which ranges typically between 10 11 and 10 12 erg cm -2 s -1 (Muzerolle et al. 2004). The filling factor can be estimated from:</text>
<formula><location><page_6><loc_19><loc_23><loc_48><loc_27></location>f = ( 1 -R ∗ R i ) GM ∗ ˙ M acc 4 π F R 3 ∗ , (4)</formula>
<text><location><page_6><loc_10><loc_21><loc_35><loc_23></location>with R i the disk truncation radius.</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_20></location>Table 5 summarizes the fixed input parameters that are used in this work. Figure 4 shows the average UV spectrum of HD 163296 normalized to the Kurucz photospheric spectrum in the B band. For a typical accretion rate of ∼ 10 -7 M /circledot yr -1 , fixed stellar parameters and disk truncation radius (see below), different values of F provide different blackbody temperatures (Eq. 3) and filling factors (Eq. 4). The corresponding total fluxes (Eq. 2) reproduce the observed spectra with different degrees of accuracy. For F ∼ 10 11 erg</text>
<text><location><page_6><loc_52><loc_7><loc_92><loc_53></location>cm -2 s -1 , the only part of the observed spectrum that can be reproduced is the bump between 365 and 370 nm, mismatching at shorter and longer wavelenghts ( λ > 380 nm). Values of F ∼ 10 12 erg cm -2 s -1 are able to reasonably reproduce both the short and long wavelength regions. Larger values ( F ∼ 10 13 erg cm -2 s -1 ) are not able to reproduce any part of the observed Balmer excesses. As it is described in the following, our modelling aims to reproduce mainly the strength of the observed Balmer excess, instead of its overall shape. We assume a 'classical' aproach using a single blackbody obtained by fixing F to 10 12 erg cm -2 s -1 , which not only provides reasonable agreement with the observed excesses and filling factors (Valenti et al. 1993), but is in turn the typical value for HAe stars (Muzerolle et al. 2004). More complex models using several values of F have been used to simultaneously reproduce wider wavelength ranges covering the near-UV and optical regions of T Tauri stars (see e.g. Ingleby et al. 2013). Disk truncation radii assumed in this work (Table 5) are also typical (Muzerolle et al. 2004), being lower than the corresponding co-rotation radii (Shu et al. 1994) by taking into account the projected rotational velocities of the stars (Mendigut'ıa et al. 2011b). We refer the reader to this paper for further details on the shock modeling and its dependences on the stellar parameters. Although already discussed there, further analysis on the influence of the adopted values for F and R i is included below. Total fluxes were synthesized from Eq. 2 for each star, considering different values for ˙ M acc . Modeled Balmer excesses were then derived by normalizing F λ to F phot λ in the the 400-460 nm band, and applying the same procedure as described for the observed Balmer excesses (Sect. 3.1). In this way, couples of values</text>
<text><location><page_7><loc_49><loc_86><loc_54><loc_86></location>HD 31648</text>
<text><location><page_7><loc_49><loc_18><loc_54><loc_19></location>v (km s</text>
<text><location><page_7><loc_54><loc_18><loc_55><loc_19></location>)</text>
<figure>
<location><page_7><loc_21><loc_19><loc_79><loc_86></location>
<caption>Fig. 3.Observed line profiles (normalized to unity) of HD 31648 and HD 163296 are shown with solid lines. Different colors represent different observing runs, as in Fig. 1. For the UV lines with a purely absorption profile (H η , Ca II K, H γ and H β ), the emission profiles (dashed lines) that result from the subtraction of the Kurucz synthetic spectra (solid black lines) from the observed ones are also shown (see text).</caption>
</figure>
<table>
<location><page_8><loc_18><loc_50><loc_82><loc_70></location>
<caption>TABLE 3 Line equivalent widths</caption>
</table>
<section_header_level_1><location><page_8><loc_43><loc_49><loc_50><loc_50></location>HD 163296</section_header_level_1>
<table>
<location><page_8><loc_18><loc_29><loc_82><loc_49></location>
</table>
<text><location><page_8><loc_8><loc_25><loc_91><loc_29></location>Notes. Photospheric and circumstellar equivalent widths are shown in Cols. 2 to 7. The last Col. shows the mean equivalent width and its uncertainty from the propagation of the individual ones. For the stars showing both absorptions and emissions, only the values of the most frequent absorption/emission behavior are considered to derive the mean values. A number in brackets quantifying the relative variability ( σ (EW cs )/ < EW cs > ) is provided when the equivalent width varies above the uncertainties.</text>
<table>
<location><page_9><loc_19><loc_68><loc_81><loc_89></location>
<caption>TABLE 4 Dereddened line luminosities</caption>
</table>
<table>
<location><page_9><loc_19><loc_47><loc_82><loc_67></location>
<caption>TABLE 6</caption>
</table>
<text><location><page_9><loc_8><loc_42><loc_91><loc_47></location>Notes. Emission line luminosities in Cols. 2 to 6 are dereddened using the color excesses indicated on the top of each column (in magnitudes), and a distance of 146 and 130 pc for HD 31648 and HD 163296, respectively. The last Col. shows the mean line luminosity and its uncertainty from the propagation of the individual ones. A number in brackets quantifying the relative variability ( σ (L)/ < L > ) is provided when the line luminosity varies above the uncertainties. Upper limits are not considered to derive mean and relative variability values. Blank spaces refer to lines shown in absorption.</text>
<table>
<location><page_9><loc_8><loc_30><loc_48><loc_38></location>
<caption>TABLE 5 Fixed model parameters</caption>
</table>
<text><location><page_9><loc_8><loc_9><loc_48><loc_29></location>( ˙ M acc , ∆D B ) were obtained for each star, which were used to construct the curves represented in Fig. 5. These ∆D B -˙ M acc calibrations were used to assign a mass accretion rate to each observed Balmer excess. Accretion luminosities were then derived considering L acc = GM ∗ ˙ M acc /R ∗ . The model parameters that best reproduce the values of the observed Balmer excesses are shown in Table 6. The uncertainties for ˙ M acc were estimated considering the typical uncertainty for the observed Balmer excess ( ± 0.01 magnitudes, see Sect. 3.1) in the ∆D B -˙ M acc calibration (Fig. 5). Errors in M ∗ /R ∗ ratios (Montesinos et al. 2009) add another 0.01 dex uncertainty to the accretion luminosities (see Mendigut'ıa et al. 2011b, and the caption of Table 6).</text>
<text><location><page_9><loc_10><loc_7><loc_48><loc_8></location>From the method described before, the mean mass</text>
<text><location><page_9><loc_52><loc_38><loc_92><loc_39></location>Accretion parameters from the observed Balmer excesses</text>
<table>
<location><page_9><loc_54><loc_22><loc_90><loc_38></location>
</table>
<text><location><page_9><loc_52><loc_19><loc_91><loc_22></location>Notes. Uncertainties for log ˙ M acc and log L acc are ± 0.03 dex and ± 0.04 dex (for HD 31648); ± 0.02 dex and ± 0.03 dex (for HD 163296).</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_18></location>accretion rate for HD 31648 was 1.11 × 10 -7 M /circledot yr -1 . The accretion rate changes are above the uncertainties, between 5.24 × 10 -8 M /circledot yr -1 and 1.46 × 10 -7 M /circledot yr -1 . This represents a mass accretion rate variation of almost 0.5 dex. The accretion luminosity showed an average value of 3.03 L /circledot , and changes by a factor of almost 3; from 1.43 to 3.97 L /circledot . This represents accretion luminosity variations between 6.5 and 18 %</text>
<figure>
<location><page_10><loc_9><loc_68><loc_48><loc_92></location>
<caption>Fig. 4.Average UV spectrum of HD 163296 (black dotted line) normalized to the kurucz synthetic spectrum (black solid line) at the B band. The coloured dotted lines refer to modelled spectra with fixed mass accretion rate (10 -7 M /circledot yr -1 ), stellar parameters and disk truncation radius (see text), but different values for the inward flux energy, F , as indicated in the legend. Those values are associated to different blackbody temperatures and filling factors by Eqs. 3 and 4: ∼ 9700 K (f ∼ 20 %), 12500 K (f ∼ 2 %) and 25000 K (f ∼ 0.1 %) for the red, green and blue lines, respectively.</caption>
</figure>
<figure>
<location><page_10><loc_9><loc_31><loc_48><loc_56></location>
<caption>Fig. 5.Calibration between the Balmer excess and the mass accretion rate for HD 31648 (solid line) and HD 163296 (dashed line). The observed Balmer excesses are over plotted with filled and open circles. The uncertainty in the observed Balmer excesses is represented by the black vertical bar.</caption>
</figure>
<text><location><page_10><loc_8><loc_12><loc_48><loc_24></location>of the stellar one. Regarding HD 163296, the mass accretion rate was typically larger -∼ 4.50 × 10 -7 M /circledot yr -1 - and roughly constant -variations lower than 0.1 dex-. The accretion luminosity was typically ∼ 40 % of the stellar one. Despite of this high ratio, significant veiling in photospheric absorption lines is not detected, which is consistent with a temperature of the accretion shocks similar to the effective temperature of the star (Muzerolle et al. 2004).</text>
<text><location><page_10><loc_8><loc_7><loc_48><loc_11></location>The above discussion assumes that the fraction of the star covered by the shocks, f , is the only model parameter that varies along the observations. This</text>
<text><location><page_10><loc_52><loc_67><loc_92><loc_92></location>assumption provides upper limits for the accretion rate changes. In other words, more complex scenarios assuming additional variations in F and R i , would explain a given Balmer excess change from a lower accretion rate variation than provided here. As an example, in the case of HD 31648 a change of 20% in F and R i would account for Balmer excess variations of 0.2 and 0.4 magnitudes, respectively, for a fixed accretion rate of 1.11 × 10 -7 M /circledot yr -1 . Further analysis in this sense is out of the scope of this work. However, it should be mentioned that changes in the disk truncation radius are considered in 3D simulations of MA (Kulkarni & Romanova 2013). In addition, given that there are observed variations in the ratio of the Balmer lines (which is a probe of the density ρ ), especially in the case of HD31648, it may be likely that the density of material in the columns is varying. Given that F ∝ ρv 3 , with v the velocity of the infalling material (see e.g. Calvet & Gullbring 1998), a change in the Balmer ratio could be indicating variations in F .</text>
<section_header_level_1><location><page_10><loc_57><loc_61><loc_87><loc_62></location>4.2. Accretion rates and line luminosities</section_header_level_1>
<text><location><page_10><loc_52><loc_21><loc_92><loc_60></location>The accretion luminosities derived from the Balmer excesses are plotted against the dereddened line luminosities in the different panels of Fig 6. Empirical calibrations relating both parameters were taken from the literature (see below), and are over plotted with solid lines. These have the form log L acc /L /circledot = a log L line /L /circledot + b, a and b being constants that depend on the spectral line. These relations were calibrated by relating primary accretion signatures (i.e. line veiling or continuum excess) with the line luminosities, and show a typical maximum scatter of ∼ ± 1 dex, which is represented by the dashed lines. The calibration obtained from HAeBe stars in Mendigut'ıa et al. (2011b) is over plotted in the H α panel. This is very similar to the corresponding relation for lower mass T Tauri stars and brown dwarfs (Dahm 2008; Herczeg & Hillenbrand 2008). The relation derived by Calvet et al. (2004) (see also Donehew & Brittain 2011; Mendigut'ıa et al. 2011b) is used for the Br γ panel. That was based on intermediate-mass TTs (F, G spectral types and masses up to 4 M /circledot ). For the remaining lines there are no L acc -L line relations calibrated from HAeBes. The empirical relations for the He I 5876, Ca II 8542, and Pa β are from Dahm (2008), based on objects with masses up to 2M /circledot . The calibrations with the rest of the lines were derived from objects with masses below 1M /circledot , and were taken from Herczeg & Hillenbrand (2008) (H η , Ca II K, H γ , Na I D, O I 8446), Gatti et al. (2008) (Pa γ ) and Fang et al. (2009) (H β ).</text>
<text><location><page_10><loc_52><loc_7><loc_92><loc_20></location>The accretion and line luminosities plotted in Fig 6 fall within the range expected from previous L acc -L line calibrations. This suggests that these relations, mostly derived from low-mass stars, can be extrapolated to provide accretion rate estimates for HAe stars. However, the fact that HD 31648 and HD 163296 show different Balmer excess -accretion rate- variability (Sects. 3.1 and 4.1) but comparable line luminosities changes (Sect. 3.2) constitutes a first hint suggesting that both variations are not directly related. In order to provide a more quantita-</text>
<figure>
<location><page_11><loc_8><loc_35><loc_91><loc_92></location>
<caption>Fig. 6.Accretion luminosities from the Balmer excess versus emission line luminosities for HD 31648 (filled symbols) and HD 163296 (open symbols). Left-triangles represent upper limits for the line luminosities. Upper limits for the accretion luminosity obtained ∼ 15 years ago (see text) are plotted with big triangles (filled and open for HD 31648 and HD 163296, respectively), with respect contemporaneous He I 5876, Na I D and H α luminosities. Empirical calibrations from previous works (Calvet et al. 2004; Dahm 2008; Gatti et al. 2008; Herczeg & Hillenbrand 2008; Fang et al. 2009; Mendigut'ıa et al. 2011b) and ± 1 dex uncertainties are indicated with solid and dashed lines, respectively.</caption>
</figure>
<text><location><page_11><loc_52><loc_34><loc_54><loc_35></location>line</text>
<text><location><page_11><loc_56><loc_34><loc_58><loc_35></location>sun</text>
<text><location><page_11><loc_8><loc_8><loc_48><loc_27></location>tive description, we computed the accretion luminosities from the above referred empirical calibrations with the spectral lines. Given that the scatter of these calibrations can reach ± 1 dex for a given value of a line luminosity, it is unrealistic to expect that they could trace accretion rate variations lower than 0.5 dex, as the ones reported in this work. However, it is possible to compare the accretion variability with the variations from the spectroscopic tracers if residuals with respect the averages are used (e.g. Pogodin et al. 2012). Fig. 7 shows the difference between single-epoch accretion luminosities derived from the calibrations with each emission line and the average accretion luminosity obtained from the same line. Error bars consider those in the line luminosi-</text>
<text><location><page_11><loc_52><loc_21><loc_92><loc_27></location>ies. The differences between the single-epoch and the mean accretion luminosities derived from the Balmer excesses are also included in Fig. 7 (squares), from which we extract the following:</text>
<unordered_list>
<list_item><location><page_11><loc_54><loc_7><loc_92><loc_20></location>· The mean accretion luminosity from the Balmer excess agrees with that derived from previous empirical calibrations within ± 0.5 dex, for most spectroscopic lines. The only exceptions are the Ca II K, Pa γ and H β lines, that provide mean accretion luminosities that can differ up to ∼ ± 1 dex. Not surprisingly, the latter calibrations were derived from objects with stellar masses well below the ones covered in this work (Gatti et al. 2008; Herczeg & Hillenbrand 2008; Fang et al. 2009).</list_item>
</unordered_list>
<figure>
<location><page_12><loc_9><loc_67><loc_92><loc_92></location>
<caption>Fig. 7.Residuals for each single epoch accretion luminosity derived from the indicated tracer with respect the mean accretion luminosity from the same tracer (horizontal dashed lines). Squares refer to the values derived from the Balmer excesses, and triangles represent upper limits. These were not considered to derive the mean values, except for the H γ line of HD 163296. For each accretion tracer, the observing runs are ordered chronologically from left to right, spaced by an arbitrary scale.</caption>
</figure>
<text><location><page_12><loc_20><loc_67><loc_21><loc_68></location>B</text>
<unordered_list>
<list_item><location><page_12><loc_11><loc_40><loc_48><loc_61></location>· The largest residual for the Balmer excess differs from that for the spectral lines, which in turn provide different values depending on the line considered. For instance, the Na I D line suggests accretion luminosity variations up to ∼ 0.30 dex larger than the largest change observed from the Balmer excess in HD 163296. However, the same line provides a similar upper limit for the accretion luminosity changes of HD 31648. On the contrary, the highest variability from the Pa β line roughly coincides with that from the Balmer excess for HD 163296, whereas for HD 31648 suggests accretion rate variations up to ∼ 0.30 dex lower. The largest residual from the Ca II K line is ∼ 0.40 and ∼ 0.20 dex higher than the corresponding from the Balmer excess, for HD 31648 and HD 163296, respectively.</list_item>
<list_item><location><page_12><loc_11><loc_11><loc_48><loc_39></location>· In spite of the last point, the residuals derived from some lines show a similar evolution pattern than that from the Balmer excess, which could be supporting a direct link between the physical origin of the lines and the accretion process. In particular, the residuals from the Ca II K, Ca II 8542 and Br γ lines increase/decrease when those from the Balmer excess increase/decrease, for HD 31648. The Br γ line also shows a similar behavior for HD 163296. In addition, although different from the residuals from the Balmer excess, the variability behavior is roughly similar for several other lines. For instance, the similar behavior of the H β , H α , Pa β and Pa γ lines could be suggesting a common physical origin explaining their variability, not necessarily connected with the accretion flow, or connected but with a certain time-delay (see Dupree 2013, and references therein). However, it is noted that these lines provide roughly constant estimates, considering the error bars, and more accurate measurements become necessary.</list_item>
</unordered_list>
<text><location><page_12><loc_8><loc_7><loc_48><loc_9></location>In summary, our results provide evidence that the L acc -L line calibrations can be extrapolated to HAe stars</text>
<text><location><page_12><loc_52><loc_44><loc_92><loc_61></location>to estimate typical, mean accretion rates, although the relations for some specific lines should most probably include minor modifications in order to achieve a ± 0.5 dex accuracy. However, the L acc -L line calibrations cannot generally be used to quantify the accretion rate variability. In addition, the analysis of accretion variability in terms of residuals provides evolution patterns that, if confirmed by follow-up observations, could be useful to understand the influence of accretion on the origin of the lines, and possible physical relations between the lines themselves. In this respect, accurate spectro-photometry should be performed in order to detect the smallest possible changes in the line luminosities.</text>
<section_header_level_1><location><page_12><loc_59><loc_41><loc_85><loc_42></location>4.3. Variability on a longer timescale</section_header_level_1>
<text><location><page_12><loc_52><loc_7><loc_92><loc_40></location>The methodology to obtain the calibration between the Balmer excess and the accretion rate described in Sect. 4.1 is the same one than that applied in Mendigut'ıa et al. (2011b), as well as the stellar and modeling parameters used for HD 31648 and HD 163296. Therefore, a direct comparison is feasible. UB photometry taken more than 15 years ago by the EXPORT consortium (Eiroa et al. 2000), was analyzed in that paper to provide accretion luminosities of HAeBe stars. Only upper limits could be derived for the two objects studied in this work, due to the lower accuracy of broad band photometry compared with intermediateresolution spectra. Mendigut'ıa et al. (2011b) provided L acc < 1.58 L /circledot (HD 31648) and L acc < 0.93 L /circledot (HD 163296). The accretion luminosity of HD 31648 during the X-Shooter observing run D is consistent with that work. Contemporaneous H α and Na I D emission line luminosities were also similar to the ones obtained here, although the He I 5876 luminosity was lower in the EXPORT campaigns by ∼ 0.5 dex. In the case of HD 163296, the averaged accretion luminosity inferred here is at least 1 dex larger than in the EXPORT campaigns. This strong change in the accretion luminosity from the Balmer excess is accompanied by changes in all optical line luminosities analyzed from the EXPORT spectra.</text>
<text><location><page_13><loc_8><loc_66><loc_48><loc_92></location>The He I 5876 line was then dominated by absorption whereas it is shown in emission in the X-Shooter data. The Na I D emission lines increased their luminosity by ∼ 0.5 dex from the EXPORT to the X-Shooter campaigns. Similarly, the H α EW changed from an averaged value of ∼ -23 ˚ A to -30 ˚ A in the current X-Shooter data, implying a line luminosity increase of almost 0.2 dex. Unfortunately, the EXPORT spectra analyzed in Mendigut'ıa et al. (2011b) did not covered the X-Shooter UVB and NIR regions, so it is not possible to explore the variations for the lines in those wavelengths. EXPORT results are over plotted in Fig. 6 for comparison. The analysis in a longer timescale suggests similar conclusions than those obtained from the study of the X-Shooter spectra in the previous section, at least for the optical lines: the L acc -L line calibrations are useful to estimate typical accretion rates with an accuracy of 0.5-1 dex, but are not reliable to accurately quantify accretion rate variability.</text>
<section_header_level_1><location><page_13><loc_16><loc_61><loc_40><loc_62></location>5. SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_13><loc_8><loc_47><loc_48><loc_60></location>We presented five-epoch X-Shooter spectra of the prototypical Herbig Ae stars HD 31648 and HD 163296. The strength of the excess shown in the Balmer region of the spectra, and its variations, have been reproduced from magnetospheric accretion shock modeling. The accretion rate of HD 31648 varied in-between 5.24 × 10 -8 and 1.46 × 10 -7 M /circledot s -1 on timescales of days to months, whereas that for HD 163296 remained roughly constant, with a typical value of 4.50 × 10 -7 M /circledot yr -1 . Higher accretion rate variations, exceeding 1 dex, are</text>
<text><location><page_13><loc_52><loc_77><loc_92><loc_92></location>found for HD 163296 on timescales of several years. The mean accretion rates derived from the Balmer excess are consistent with those inferred from the empirical calibrations with twelve UV, optical and NIR emission line luminosities, previously derived from lower-mass stars. This demonstrates that those calibrations can be used also for HAe stars. However, the variability of the accretion rate from the Balmer excess is not generally reflected by that from the line luminosities, suggesting that the empirical calibrations are not useful to derive accurate accretion rate variations.</text>
<text><location><page_13><loc_52><loc_69><loc_92><loc_76></location>Our results suggest that follow-up multi-epoch observations of these and other pre-main sequence stars will be useful to further constrain the underlying origin of the empirical calibrations with the line luminosities, and the origin of the lines themselves.</text>
<unordered_list>
<list_item><location><page_13><loc_52><loc_50><loc_92><loc_66></location>The authors thank the anonymous referee for his/her useful comments on the original manuscript, which helped us to improve the paper. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 088.C-0218. S.D. Brittain acknowledges support for this work from the National Science Foundation under grant number AST-0954811 C. Eiroa, G. Meeus and B. Montesinos are supported by AYA 2011-26202. G. Meeus is supported by RYC-2011-07920.</list_item>
</unordered_list>
<section_header_level_1><location><page_13><loc_53><loc_47><loc_70><loc_49></location>Facilities: VLT:Kueyen</section_header_level_1>
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</document>
[
{
"title": "ABSTRACT",
"content": "This work presents X-Shooter/VLT spectra of the prototypical, isolated Herbig Ae stars HD 31648 (MWC 480) and HD 163296 over five epochs separated by timescales ranging from days to months. Each spectrum spans over a wide wavelength range covering from 310 to 2475 nm. We have monitored the continuum excess in the Balmer region of the spectra and the luminosity of twelve ultraviolet, optical and near infrared spectral lines that are commonly used as accretion tracers for T Tauri stars. The observed strengths of the Balmer excesses have been reproduced from a magnetospheric accretion shock model, providing a mean mass accretion rate of 1.11 × 10 -7 and 4.50 × 10 -7 M /circledot yr -1 for HD 31648 and HD 163296, respectively. Accretion rate variations are observed, being more pronounced for HD 31648 (up to 0.5 dex). However, from the comparison with previous results it is found that the accretion rate of HD 163296 has increased by more than 1 dex, on a timescale of ∼ 15 years. Averaged accretion luminosities derived from the Balmer excess are consistent with the ones inferred from the empirical calibrations with the emission line luminosities, indicating that those can be extrapolated to HAe stars. In spite of that, the accretion rate variations do not generally coincide with those estimated from the line luminosities, suggesting that the empirical calibrations are not useful to accurately quantify accretion rate variability. Subject headings: Accretion, accretion disks - circumstellar matter - Line: formation - Protoplanetary disks - Stars: pre-main sequence - Stars: variables: T Tauri, Herbig Ae/Be",
"pages": [
1
]
},
{
"title": "I. Mendigut'ıa",
"content": "Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0978, USA",
"pages": [
1
]
},
{
"title": "S.D. Brittain",
"content": "Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0978, USA",
"pages": [
1
]
},
{
"title": "C. Eiroa",
"content": "Departamento de F'ısica Te'orica, M'odulo 15, Facultad de Ciencias, Universidad Aut'onoma de Madrid, PO Box 28049, Cantoblanco, Madrid, Spain.",
"pages": [
1
]
},
{
"title": "G. Meeus",
"content": "Departamento de F'ısica Te'orica, M'odulo 15, Facultad de Ciencias, Universidad Aut'onoma de Madrid, PO Box 28049, Cantoblanco, Madrid, Spain.",
"pages": [
1
]
},
{
"title": "B. Montesinos",
"content": "Centro de Astrobiolog'ıa, Departamento de Astrof'ısica (CSIC-INTA), ESAC Campus, P.O. Box 78, 28691 Villanueva de la Ca˜nada, Madrid, Spain.",
"pages": [
1
]
},
{
"title": "A. Mora",
"content": "GAIA Science Operations Centre, ESA, European Space Astronomy Centre, PO Box 78, 28691, Villanueva de la Ca˜nada, Madrid, Spain.",
"pages": [
1
]
},
{
"title": "J. Muzerolle",
"content": "Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD, 21218, USA",
"pages": [
1
]
},
{
"title": "R.D. Oudmaijer",
"content": "School of Physics & Astronomy, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK",
"pages": [
1
]
},
{
"title": "E. Rigliaco",
"content": "Department of Planetary Science, Lunar and Planetary Lab, University of Arizona, 1629, E. University Blvd, 85719, Tucson, AZ, USA Draft version October 10, 2018",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The star formation process can be studied from the evolution of the accretion rate (see e.g. Hartmann 1998; Sicilia-Aguilar et al. 2004; Fedele et al. 2010). At the initial stages, protostars show both envelope-to-disk and disk-to-star accretion, which can show variations of several orders of magnitude on relatively short timescales (FUor and EXor outbursts, e.g. Hartmann & Kenyon 1996; Herbig 2008; Liskowsky 2010). During the pre-main sequence (PMS) phase, once the envelope dissipates and the central object becomes optically visible, the mass accretion rate ( ˙ M acc ) decreases to typical values of ∼ 10 -8 and 10 -7 M /circledot yr -1 for T Tauri (TT) and Herbig Ae/Be (HAeBe) stars, respectively (Hartmann et al. 1998; Mendigut'ıa et al. 2011b). The accretion rate variability in the PMS phase is also expected to be lower than in previous stages. However, the precise strength of those variations is not well known, which can be partly attributed to discrepancies measuring accretion. Primary signatures of accretion are the excess of UV/optical continuum emission and the veiling in the spectroscopic absorption lines that are observed in accreting stars. Both can be modeled from hot accretion shocks whose emission superimposes to that of the stellar atmosphere (Calvet & Gullbring 1998). The way that material from the inner disk reaches the stellar surface is understood from the magnetospheric accretion (MA) scenario, according to which gas does not fall directly from the disk to the star but follows the magnetic field lines that connect them (Uchida & Shibata 1985; Koenigl 1991; Shu et al. 1994). This is the accepted view for TT stars and seems to be valid at least for several HAeBes (see e.g. Vink et al. 2002; Muzerolle et al. 2004; Mottram et al. 2007; Donehew & Brittain 2011; Mendigut'ıa 2013). Accretion rates obtained from MA shock modeling correlate with the luminosity of several emission lines that span from the ultraviolet (UV) to the infrared (IR) (see e.g. the compilation in Rigliaco et al. 2011). These types of empirical calibrations have been extended to late-type HAeBes for the [O I ]6300, H α (Mendigut'ıa et al. 2011b) and Br γ (Donehew & Brittain 2011; Mendigut'ıa et al. 2011b) lines. The origin of the correlations is not fully understood, but these spectral lines can be considered as secondary signatures of accretion, being useful to easily derive accretion rates for wide samples of stars. Although some works are based on observed variations of primary accretion signatures in TT stars (e.g. Herbst et al. 1994; Fernandez & Eiroa 1996; Batalha et al. 2001), studies of accretion rate variability are mainly based on secondary ones (e.g. Nguyen et al. 2009; Eisner et al. 2010; Costigan et al. 2012; Chou et al. 2013). These analyses provide different results depending on the spectroscopic line used as accretion tracer. As an example, Nguyen et al. (2009) studied a sample of 40 TT stars, reporting typical accretion rate variations of 0.35 dex from the CaII8862 line flux and 0.65 dex from the H α 10%. Similar results were more recently obtained by Costigan et al. (2012). In fact, differences in the variability shown by several lines used as accretion tracers are reported for both TTs and HAeBes (Johns & Basri 1995; Lago & Gameiro 1998; Mendigut'ıa et al. 2011a). However, accretion variability from a primary accretion signature was found to be typically lower than 0.5 dex for the HAeBes, on timescales from days to months (Mendigut'ıa et al. 2011b), a result confirmed by Pogodin et al. (2012). Accretion variability of specific TT stars, determined from line veiling, is also expected to be low (Alencar & Batalha 2002). However, the simultaneous analysis of primary and secondary accretion signatures is necessary to test the reliability of the empirical calibrations between the accretion rate and different spectroscopic lines (Rigliaco et al. 2012), and to study the validity of MA in early-type HAeBe stars (Donehew & Brittain 2011; Mendigut'ıa et al. 2011b). These types of studies have been boosted by the advent of the X-Shooter spectrograph at the Very Large Telescope (VLT; Vernet et al. 2011), which allows simultaneous coverage of the accretion signatures from the UV to the near-IR. Simultaneity is crucial when dealing with PMS stars, given their variable nature. Several published and ongoing works face those type of questions, by 'X-Shooting' wide samples of PMS stars (Oudmaijer et al. 2011; Rigliaco et al. 2012; Manara et al. 2013). This work aims to contribute to the understanding of the spectroscopic accretion tracers in HAeBe stars. Instead of relating primary and secondary accretion signatures from single-epoch spectra of large samples, we will analyze multi-epoch spectra of two relevant objects. This approach brings the additional possibility of analyzing the variability of the accretion rate. This paper focuses on the variability behavior of two HAe stars: HD 31648 (MWC 480) and HD 163296. The specific objective is twofold. First, to measure the accretion rate and its variability from the Balmer excess (the primary accretion signature for HAeBe stars, see Garrison 1978; Muzerolle et al. 2004). Second, to compare the previous results with several UV, optical and NIR line luminosities commonly used as secondary accretion tracers in low-mass stars. This comparison will provide relevant information about the applicability of the spectroscopic accretion tracers in late-type HAeBe stars. The paper is organized as follows: Sect. 2 describes the observations and data reduction, Sect. 3 presents our observational results on the Balmer excess (Sect. 3.1) and the spectroscopic lines (Sect. 3.2), Sect. 4 includes the analysis of the Balmer excesses in terms of a MA model (Sect. 4.1), and discusses possible relations between the accretion luminosity derived from the Balmer excess and from the emission line luminosities (Sects. 4.2 and 4.3). Finally, Sect. 5 summarizes our main conclusions.",
"pages": [
2
]
},
{
"title": "2. STARS, OBSERVATIONS AND DATA REDUCTION",
"content": "HD 31648 and HD 163296 are HAe stars (spectral types A5 and A1, ages of ∼ 7 Myr and 5 Myr, respectively; Mora et al. 2001; Montesinos et al. 2009) with no reported stellar companions in the literature. We assume the parameters derived by Montesinos et al. (2009) for the surface temperature, stellar mass and radius (surface gravity), as well as for the distances (T ∗ = 8250 K , 9250 K ; M ∗ = 2.0 M /circledot , 2.2 M /circledot ; R ∗ = 2.3 R /circledot , 2.3 R /circledot ; d = 146 pc, 130 pc; where the first and second values refer to HD31648 and HD 163296, respectively). The stellar luminosities derived in that work are 22 L /circledot (HD 31648) and 34.5L /circledot (HD 163296). Both objects show continuum and line variations (Sitko et al. 2008), and have magnetically driven accretion signatures (Swartz et al. 2005; Hubrig et al. 2006). We obtained multi-epoch X-Shooter spectra of both objects. Table 1 shows the log of the observations. A total of 10 science spectra were taken, 5 spectra per star on a timescale from days to months. X-Shooter covers a wide wavelength region throughout its three arms: UVB (311-558 nm), VIS (558-1012 nm) and NIR (1012-2475 nm). The slit widths were 1.6, 0.9 and 0.9 '' , which translates into spectral resolutions of 3300, 8800 and 5600 for the UVB, VIS and NIR arms. The spectra were bias and flat field-corrected, flux and wavelength calibrated from standard procedures using the X-Shooter pipeline v1.5.0. (Modigliani et al. 2010). Atmospheric emission was subtracted from an ABBA dithering pattern. Additional telluric absorption correction was applied for specific lines in the NIR arm (Pa γ and Br γ ; see Sect. 3.2), by using telluric standards that were observed before and after the target stars. Flux calibration was tested from low-resolution spectra -i.e. obtained with the widest slit available for all arms (5 '' )-, taken consecutively for the target stars and several telluric standard main-sequence objects with similar air masses. Synthetic photometry obtained by convolving the spectra with broadband UBVRIJHK photometric filters were compared to published values for the telluric standard stars. In this way, the typical uncertainty is in-between 0.05 and 0.2 magnitudes (flux relative errors ∼ 5 and 25 %), where the specific uncertainties for each star depend on the X-Shooter arm and observing run (see below). Further accuracy could be obtained from spectro-photometric standards, which were not provided. Synthetic photometry extracted from our spectra is included in Table 1, which is consistent with previous measurements (de Winter et al. 2001; Oudmaijer et al. 2001; Eiroa et al. 2001). Flux calibrated spectra are shown in Fig. 1. Values from broadband photometry published in the literature (Oudmaijer et al. 2001; Eiroa et al. 2001) are over plotted for comparison.",
"pages": [
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},
{
"title": "3.1. Balmer excesses",
"content": "The Balmer excess is defined from the UB Johnson's photometric bands as ∆D B = ( U -B ) phot -( U -B ) dered , where ( U -B ) phot is the photospheric color, and ( U -B ) dered the observed, dereddened color (Mendigut'ıa et al. 2011b). This can be derived from the extinction (in magnitudes), which for a given wavelength can be expressed from that in the V band, or from the B -V color excess, A λ = A V (k λ /k V ) = R V E ( B -V )(k λ /k V ). The opacity ratio k λ /k V is known from an extinction law, and R V is the total-to-selective extinction ratio. The X-Shooter UVB spectra were used to measure continuum excesses in the Balmer region, along the different observing runs. Therefore, the expression for the Balmer excess has been converted into fluxes. If the observed ones are normalized to the pho- c emission at the B band (hereafter represented by the superscript 'norm, B'), then: which is the expression that is used in this work. The first term in Eq. 1 is essentially the same one as used by Donehew & Brittain (2011), with F norm,B U /F phot U the ratio between the normalized and the photospheric fluxes, measured at wavelengths corresponding to the U -band. The second term reflects the contribution of extinction, and can be estimated from an extinction law and from the ratio between the normalized and photospheric fluxes at wavelengths corresponding to the V band. Apart from being independent of absolute flux calibration (Donehew & Brittain 2011), the main advantage of the above expression for ∆D B is that it accounts for continuum excesses that can be ascribed mainly to hot shocks caused by accretion (Sect. 4.1), given that the possible contribution of extinction is ruled out by the second term. This contribution is typically an order of magnitude lower than the first term for the stars studied here -their E ( B -V ) values are low, see Sect. 3.2-, but it should not be neglected for objects with more pronounced extinctions (i.e. UXOr-like, see e.g. Grinin et al. 1994, and references therein). It is assumed that the third possible cause argued to be in the origin of photometric variability in PMS stars, photospheric variations caused by cold spots (Herbst et al. 1994), is negligible for the stellar temperatures and masses of both stars analyzed. The photospheric fluxes are represented by Kurucz models (Kurucz 1993) with the corresponding values of T ∗ and log g (Sect. 2). These were derived from the SYNTHE code, re-binning to a resolution similar to the X-Shooter spectra, as they were also used to estimate the photospheric contribution to the spectral lines (Sect. 3.2). Observed fluxes were normalized to the Kurucz ones, making their averaged continuum fluxes in the 400-460 nm band (representing the rough range covered by the B photometric filter) to coincide. The ratios F norm,B U /F phot U and F norm,B V /F phot V were then derived considering averaged continuum fluxes in the wavelength regions 350-370 nm and 540-560 nm, respectively. Balmer excesses were finally obtained from Eq. 1, assuming R V = 5 (Hern'andez et al. 2004), k U /k V = 1.6, and k B /k V = 1.3 (Rieke & Lebofsky 1985; Robitaille et al. 2007). The use of a larger R V value, compared with the typical for the interstellar medium (R V = 3.1), has been extensively justified for HAeBe stars (see e.g. Herbst et al. 1982; Gorti & Bhatt 1993; Hern'andez et al. 2004; Manoj et al. 2006, and references therein). In addition, interstellar extinction is most likely negligible compared with the circumstellar one for objects located closer than 200 pc (Fitzgerald 1968), as it is the case for HD 31648 and HD 163296. A different R V value than assumed here affects the second term in Eq. 1. In particular, R V = 3.1 provides Balmer excesses than can be lower by 0.08 magnitudes, being Notes. Observing dates (dd/mm/yy) and synthetic photometry, in magnitudes, obtained by convolving Johnson-Cousins UBVRI (Bessell 1979) and JHK (Cohen et al. 1990) filter passbands with each flux-calibrated spectrum. this upper limit obtained from specific observing runs of HD 163296. For simplicity, possible variations of R V for a given star, or strong dependences of the extinction law on R V , are not considered for the low-extinctions and the wavelength range covered in this work (Mathis 1990). Figure 2 shows the observed spectra for the two stars studied here, normalized to the Kurucz ones at wavelengths corresponding to the B photometric filter. Balmer excesses with respect the photospheric spectra are apparent in the 350-370 nm region, and those can vary from one epoch to another. Table 2 shows the measured ∆D B values. Apart from the extinction law, the uncertainty of the Balmer excesses is only dependent of relative flux ratios in wavelengths corresponding to the U and V photometric filters (see Eq. 1). In other words, the accuracy of the Balmer excess would only decrease if errors in flux calibration are strongly wavelength-dependent in the 350-560 nm region, which is not the case for our spectra. A conservative uncertainty of 0.01 magnitudes was estimated by measuring -the theoretically null- ∆D B in consecutive X-Shooter spectra of main sequence stars in different observing dates and with different air masses, for which neither accretion nor changes in extinction were present. The last column in Table 2 shows the mean Balmer excess and its relative variability, σ (∆D B )/ < ∆D B > , defined as the ratio between the standard deviation and the average Balmer excess from the different observing runs. While HD 163296 shows the largest mean Balmer excess (0.41 magnitudes), HD 31648 shows the largest Balmer excess variations (up to a factor 2.5 in a few days). It is noted again the importance of the second term in Eq. 1: despite of the similar variability shown in the Balmer region of the spectra in both panels of Fig. 2, the contribution of -variable- extinction to this variability is a bit more pronounced in HD 163296 (its normalized flux shows larger differences with respect the Kurucz model at longer wavelengths).",
"pages": [
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},
{
"title": "3.2. Spectral lines",
"content": "This work focuses on twelve emission lines for which there are previous empirical calibrations with the accretion luminosity in the literature (Sect. 4.2). These lines span over the whole UVB-VIS-NIR wavelength region covered by the X-Shooter spectra. In particular, we focused on transitions of neutral hydrogen, helium, sodium, oxygen, also including transitions of ionized calcium. Figure 3 shows the observed line profiles of HD 31648 and HD 163296, along the five observing runs. Forbidden emission lines most probably tracing outflow processes ([O I ], [S II ], [Fe I ]) were not detected in any of the spectra. However, both stars show variable blueshifted self-absorptions in several lines (H α , Ca II 8542, Pa β ), as well as variable red-shifted self-absorptions in others (Ca II K, H β , O I 8446). Several works deal with the physical origin of some of these lines (see e.g. Hartmann et al. 1994; Muzerolle et al. 1998, 2001, 2004; Tambovtseva et al. 1999; Kurosawa et al. 2006), which have been associated mainly with accretion and/or outflow processes. The study of the physical origin of the spectral lines is beyond the scope of this work. The circumstellar contribution to the equivalent width (EW) of a line was determined from EW cs = EW obs -EW phot . The observed equivalent width, EW obs , was directly measured on the normalized spectra 1 , whereas the photospheric absorption, EW phot , was estimated from the Kurucz spectra with the corresponding stellar parameters (see Sect. 3.1). These profiles were broadened by stellar rotation, asuming projected rotational velocities of 133 and 102 km s -1 for HD 163296 and HD 31648, respectively (Mora et al. 2001). For the resolution of our X-Shooter spectra, variations of 500 K in the stellar temperature and changes of 0.50 dex in the surface gravity of the Kurucz spectra (uncertainties of ± 200 K and ± 0.2 dex were quoted in Montesinos et al. 2009) translate into typical relative errors of 10 % and 5 % in EW phot (the specific values depending on the specific star and line). However, since we are mainly interested on the variability of the lines and the photospheric contribution is expected to be constant, it is assumed that the uncertainty in EW cs is dominated by that in EW obs . This was determined for each individual line and observing run by measuring additional EWs from two levels located at ± 1.5 σ from the normalized continuum, fixed at unity. Table 3 shows the equivalent widths and the errors estimated in this way. The relative variability, σ (EW cs )/ < EW cs > (see e.g. Mendigut'ıa et al. 2011a), is indicated in those cases where the EW changes above the uncertainties. Dereddened line luminosities (L dered λ ) were obtained from the values for EW cs , the observed continuum fluxes at wavelengths close to the lines (F obs cont,λ ), and from the distances to the stars (d). We used the expression L dered λ = 4 π d 2 × F dered λ , where F dered λ = EW cs × F dered cont,λ = EW cs × F obs cont,λ × 10 0 . 4 A λ . The extinction, A λ = R V E ( B -V )(k λ /k V ), was derived assuming again R V = 5, and from the k λ /k V values in Rieke & Lebofsky (1985) and Robitaille et al. (2007). ( B -V ) color excesses were obtained subtracting the corresponding magnitudes extracted from the Kurucz spectra from the ones included in Table 1. These are shown in Table 4, along with the dereddened line luminosities and their uncertainties (which consider those in the EW and in F obs cont,λ ), as well as the relative variability when a line shows luminosity variations above the uncertainties. Although line luminosities tend to be larger for HD 163296, the variability of the lines, both in terms of the EWs and the line luminosities, are comparable for HD 31648 and HD 163296.",
"pages": [
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},
{
"title": "4.1. Accretion rates from Balmer excesses",
"content": "It is assumed that accretion is magnetically channeled for the two stars studied in this paper. Therefore, the observed Balmer excesses and their variations (Sect. 3.1) are analyzed in terms of a MA shock model. We follow the nomenclature and methodology described in Mendigut'ıa et al. (2011b), where several assumptions of the MA shock model in Calvet & Gullbring (1998) and Notes. Observed Balmer excesses are shown in Cols. 2 to 6. The typical uncertainty is 0.01 magnitudes. The last Col. shows the mean Balmer excess and a number in brackets quantifying the relative variability. Muzerolle et al. (2004) were applied to HAeBe stars. According to that, the total flux per wavelength unit emerging from the star ( F λ ) is composed of the emission from the naked photosphere ( F phot λ , represented by a Kurucz synthetic spectrum) plus the accretion contribution: f being the filling factor that reflects the stellar surface coverage of the accretion columns, and F col λ the flux from the column. This parameter is modeled as a blackbody at a temperature: with σ the Stefan-Boltzmann constant and F the inward flux of energy carried by the accretion columns, which ranges typically between 10 11 and 10 12 erg cm -2 s -1 (Muzerolle et al. 2004). The filling factor can be estimated from: with R i the disk truncation radius. Table 5 summarizes the fixed input parameters that are used in this work. Figure 4 shows the average UV spectrum of HD 163296 normalized to the Kurucz photospheric spectrum in the B band. For a typical accretion rate of ∼ 10 -7 M /circledot yr -1 , fixed stellar parameters and disk truncation radius (see below), different values of F provide different blackbody temperatures (Eq. 3) and filling factors (Eq. 4). The corresponding total fluxes (Eq. 2) reproduce the observed spectra with different degrees of accuracy. For F ∼ 10 11 erg cm -2 s -1 , the only part of the observed spectrum that can be reproduced is the bump between 365 and 370 nm, mismatching at shorter and longer wavelenghts ( λ > 380 nm). Values of F ∼ 10 12 erg cm -2 s -1 are able to reasonably reproduce both the short and long wavelength regions. Larger values ( F ∼ 10 13 erg cm -2 s -1 ) are not able to reproduce any part of the observed Balmer excesses. As it is described in the following, our modelling aims to reproduce mainly the strength of the observed Balmer excess, instead of its overall shape. We assume a 'classical' aproach using a single blackbody obtained by fixing F to 10 12 erg cm -2 s -1 , which not only provides reasonable agreement with the observed excesses and filling factors (Valenti et al. 1993), but is in turn the typical value for HAe stars (Muzerolle et al. 2004). More complex models using several values of F have been used to simultaneously reproduce wider wavelength ranges covering the near-UV and optical regions of T Tauri stars (see e.g. Ingleby et al. 2013). Disk truncation radii assumed in this work (Table 5) are also typical (Muzerolle et al. 2004), being lower than the corresponding co-rotation radii (Shu et al. 1994) by taking into account the projected rotational velocities of the stars (Mendigut'ıa et al. 2011b). We refer the reader to this paper for further details on the shock modeling and its dependences on the stellar parameters. Although already discussed there, further analysis on the influence of the adopted values for F and R i is included below. Total fluxes were synthesized from Eq. 2 for each star, considering different values for ˙ M acc . Modeled Balmer excesses were then derived by normalizing F λ to F phot λ in the the 400-460 nm band, and applying the same procedure as described for the observed Balmer excesses (Sect. 3.1). In this way, couples of values HD 31648 v (km s )",
"pages": [
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},
{
"title": "HD 163296",
"content": "Notes. Photospheric and circumstellar equivalent widths are shown in Cols. 2 to 7. The last Col. shows the mean equivalent width and its uncertainty from the propagation of the individual ones. For the stars showing both absorptions and emissions, only the values of the most frequent absorption/emission behavior are considered to derive the mean values. A number in brackets quantifying the relative variability ( σ (EW cs )/ < EW cs > ) is provided when the equivalent width varies above the uncertainties. Notes. Emission line luminosities in Cols. 2 to 6 are dereddened using the color excesses indicated on the top of each column (in magnitudes), and a distance of 146 and 130 pc for HD 31648 and HD 163296, respectively. The last Col. shows the mean line luminosity and its uncertainty from the propagation of the individual ones. A number in brackets quantifying the relative variability ( σ (L)/ < L > ) is provided when the line luminosity varies above the uncertainties. Upper limits are not considered to derive mean and relative variability values. Blank spaces refer to lines shown in absorption. ( ˙ M acc , ∆D B ) were obtained for each star, which were used to construct the curves represented in Fig. 5. These ∆D B -˙ M acc calibrations were used to assign a mass accretion rate to each observed Balmer excess. Accretion luminosities were then derived considering L acc = GM ∗ ˙ M acc /R ∗ . The model parameters that best reproduce the values of the observed Balmer excesses are shown in Table 6. The uncertainties for ˙ M acc were estimated considering the typical uncertainty for the observed Balmer excess ( ± 0.01 magnitudes, see Sect. 3.1) in the ∆D B -˙ M acc calibration (Fig. 5). Errors in M ∗ /R ∗ ratios (Montesinos et al. 2009) add another 0.01 dex uncertainty to the accretion luminosities (see Mendigut'ıa et al. 2011b, and the caption of Table 6). From the method described before, the mean mass Accretion parameters from the observed Balmer excesses Notes. Uncertainties for log ˙ M acc and log L acc are ± 0.03 dex and ± 0.04 dex (for HD 31648); ± 0.02 dex and ± 0.03 dex (for HD 163296). accretion rate for HD 31648 was 1.11 × 10 -7 M /circledot yr -1 . The accretion rate changes are above the uncertainties, between 5.24 × 10 -8 M /circledot yr -1 and 1.46 × 10 -7 M /circledot yr -1 . This represents a mass accretion rate variation of almost 0.5 dex. The accretion luminosity showed an average value of 3.03 L /circledot , and changes by a factor of almost 3; from 1.43 to 3.97 L /circledot . This represents accretion luminosity variations between 6.5 and 18 % of the stellar one. Regarding HD 163296, the mass accretion rate was typically larger -∼ 4.50 × 10 -7 M /circledot yr -1 - and roughly constant -variations lower than 0.1 dex-. The accretion luminosity was typically ∼ 40 % of the stellar one. Despite of this high ratio, significant veiling in photospheric absorption lines is not detected, which is consistent with a temperature of the accretion shocks similar to the effective temperature of the star (Muzerolle et al. 2004). The above discussion assumes that the fraction of the star covered by the shocks, f , is the only model parameter that varies along the observations. This assumption provides upper limits for the accretion rate changes. In other words, more complex scenarios assuming additional variations in F and R i , would explain a given Balmer excess change from a lower accretion rate variation than provided here. As an example, in the case of HD 31648 a change of 20% in F and R i would account for Balmer excess variations of 0.2 and 0.4 magnitudes, respectively, for a fixed accretion rate of 1.11 × 10 -7 M /circledot yr -1 . Further analysis in this sense is out of the scope of this work. However, it should be mentioned that changes in the disk truncation radius are considered in 3D simulations of MA (Kulkarni & Romanova 2013). In addition, given that there are observed variations in the ratio of the Balmer lines (which is a probe of the density ρ ), especially in the case of HD31648, it may be likely that the density of material in the columns is varying. Given that F ∝ ρv 3 , with v the velocity of the infalling material (see e.g. Calvet & Gullbring 1998), a change in the Balmer ratio could be indicating variations in F .",
"pages": [
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},
{
"title": "4.2. Accretion rates and line luminosities",
"content": "The accretion luminosities derived from the Balmer excesses are plotted against the dereddened line luminosities in the different panels of Fig 6. Empirical calibrations relating both parameters were taken from the literature (see below), and are over plotted with solid lines. These have the form log L acc /L /circledot = a log L line /L /circledot + b, a and b being constants that depend on the spectral line. These relations were calibrated by relating primary accretion signatures (i.e. line veiling or continuum excess) with the line luminosities, and show a typical maximum scatter of ∼ ± 1 dex, which is represented by the dashed lines. The calibration obtained from HAeBe stars in Mendigut'ıa et al. (2011b) is over plotted in the H α panel. This is very similar to the corresponding relation for lower mass T Tauri stars and brown dwarfs (Dahm 2008; Herczeg & Hillenbrand 2008). The relation derived by Calvet et al. (2004) (see also Donehew & Brittain 2011; Mendigut'ıa et al. 2011b) is used for the Br γ panel. That was based on intermediate-mass TTs (F, G spectral types and masses up to 4 M /circledot ). For the remaining lines there are no L acc -L line relations calibrated from HAeBes. The empirical relations for the He I 5876, Ca II 8542, and Pa β are from Dahm (2008), based on objects with masses up to 2M /circledot . The calibrations with the rest of the lines were derived from objects with masses below 1M /circledot , and were taken from Herczeg & Hillenbrand (2008) (H η , Ca II K, H γ , Na I D, O I 8446), Gatti et al. (2008) (Pa γ ) and Fang et al. (2009) (H β ). The accretion and line luminosities plotted in Fig 6 fall within the range expected from previous L acc -L line calibrations. This suggests that these relations, mostly derived from low-mass stars, can be extrapolated to provide accretion rate estimates for HAe stars. However, the fact that HD 31648 and HD 163296 show different Balmer excess -accretion rate- variability (Sects. 3.1 and 4.1) but comparable line luminosities changes (Sect. 3.2) constitutes a first hint suggesting that both variations are not directly related. In order to provide a more quantita- line sun tive description, we computed the accretion luminosities from the above referred empirical calibrations with the spectral lines. Given that the scatter of these calibrations can reach ± 1 dex for a given value of a line luminosity, it is unrealistic to expect that they could trace accretion rate variations lower than 0.5 dex, as the ones reported in this work. However, it is possible to compare the accretion variability with the variations from the spectroscopic tracers if residuals with respect the averages are used (e.g. Pogodin et al. 2012). Fig. 7 shows the difference between single-epoch accretion luminosities derived from the calibrations with each emission line and the average accretion luminosity obtained from the same line. Error bars consider those in the line luminosi- ies. The differences between the single-epoch and the mean accretion luminosities derived from the Balmer excesses are also included in Fig. 7 (squares), from which we extract the following: B In summary, our results provide evidence that the L acc -L line calibrations can be extrapolated to HAe stars to estimate typical, mean accretion rates, although the relations for some specific lines should most probably include minor modifications in order to achieve a ± 0.5 dex accuracy. However, the L acc -L line calibrations cannot generally be used to quantify the accretion rate variability. In addition, the analysis of accretion variability in terms of residuals provides evolution patterns that, if confirmed by follow-up observations, could be useful to understand the influence of accretion on the origin of the lines, and possible physical relations between the lines themselves. In this respect, accurate spectro-photometry should be performed in order to detect the smallest possible changes in the line luminosities.",
"pages": [
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},
{
"title": "4.3. Variability on a longer timescale",
"content": "The methodology to obtain the calibration between the Balmer excess and the accretion rate described in Sect. 4.1 is the same one than that applied in Mendigut'ıa et al. (2011b), as well as the stellar and modeling parameters used for HD 31648 and HD 163296. Therefore, a direct comparison is feasible. UB photometry taken more than 15 years ago by the EXPORT consortium (Eiroa et al. 2000), was analyzed in that paper to provide accretion luminosities of HAeBe stars. Only upper limits could be derived for the two objects studied in this work, due to the lower accuracy of broad band photometry compared with intermediateresolution spectra. Mendigut'ıa et al. (2011b) provided L acc < 1.58 L /circledot (HD 31648) and L acc < 0.93 L /circledot (HD 163296). The accretion luminosity of HD 31648 during the X-Shooter observing run D is consistent with that work. Contemporaneous H α and Na I D emission line luminosities were also similar to the ones obtained here, although the He I 5876 luminosity was lower in the EXPORT campaigns by ∼ 0.5 dex. In the case of HD 163296, the averaged accretion luminosity inferred here is at least 1 dex larger than in the EXPORT campaigns. This strong change in the accretion luminosity from the Balmer excess is accompanied by changes in all optical line luminosities analyzed from the EXPORT spectra. The He I 5876 line was then dominated by absorption whereas it is shown in emission in the X-Shooter data. The Na I D emission lines increased their luminosity by ∼ 0.5 dex from the EXPORT to the X-Shooter campaigns. Similarly, the H α EW changed from an averaged value of ∼ -23 ˚ A to -30 ˚ A in the current X-Shooter data, implying a line luminosity increase of almost 0.2 dex. Unfortunately, the EXPORT spectra analyzed in Mendigut'ıa et al. (2011b) did not covered the X-Shooter UVB and NIR regions, so it is not possible to explore the variations for the lines in those wavelengths. EXPORT results are over plotted in Fig. 6 for comparison. The analysis in a longer timescale suggests similar conclusions than those obtained from the study of the X-Shooter spectra in the previous section, at least for the optical lines: the L acc -L line calibrations are useful to estimate typical accretion rates with an accuracy of 0.5-1 dex, but are not reliable to accurately quantify accretion rate variability.",
"pages": [
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},
{
"title": "5. SUMMARY AND CONCLUSIONS",
"content": "We presented five-epoch X-Shooter spectra of the prototypical Herbig Ae stars HD 31648 and HD 163296. The strength of the excess shown in the Balmer region of the spectra, and its variations, have been reproduced from magnetospheric accretion shock modeling. The accretion rate of HD 31648 varied in-between 5.24 × 10 -8 and 1.46 × 10 -7 M /circledot s -1 on timescales of days to months, whereas that for HD 163296 remained roughly constant, with a typical value of 4.50 × 10 -7 M /circledot yr -1 . Higher accretion rate variations, exceeding 1 dex, are found for HD 163296 on timescales of several years. The mean accretion rates derived from the Balmer excess are consistent with those inferred from the empirical calibrations with twelve UV, optical and NIR emission line luminosities, previously derived from lower-mass stars. This demonstrates that those calibrations can be used also for HAe stars. However, the variability of the accretion rate from the Balmer excess is not generally reflected by that from the line luminosities, suggesting that the empirical calibrations are not useful to derive accurate accretion rate variations. Our results suggest that follow-up multi-epoch observations of these and other pre-main sequence stars will be useful to further constrain the underlying origin of the empirical calibrations with the line luminosities, and the origin of the lines themselves.",
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},
{
"title": "REFERENCES",
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"pages": [
13,
14
]
}
]
2013ApJ...776...49C
https://arxiv.org/pdf/1308.4404.pdf
<document>
<text><location><page_1><loc_10><loc_85><loc_10><loc_85></location>1</text>
<text><location><page_1><loc_10><loc_81><loc_10><loc_82></location>2</text>
<text><location><page_1><loc_10><loc_76><loc_10><loc_77></location>3</text>
<text><location><page_1><loc_10><loc_72><loc_10><loc_72></location>4</text>
<section_header_level_1><location><page_1><loc_19><loc_84><loc_81><loc_86></location>NarrowK -Band Observations of the GJ 1214 System</section_header_level_1>
<text><location><page_1><loc_37><loc_81><loc_63><loc_83></location>Knicole D. Col´on 1 , Eric Gaidos 1</text>
<section_header_level_1><location><page_1><loc_44><loc_76><loc_56><loc_77></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_31><loc_83><loc_73></location>GJ 1214 is a nearby M dwarf star that hosts a transiting super-Earth-size planet, making this system an excellent target for atmospheric studies. Most studies find that the transmission spectrum of GJ 1214b is flat, which favors either a high mean molecular weight or cloudy/hazy hydrogen (H) rich atmosphere model. Photometry at short wavelengths ( < 0.7 µ m) and in the K -band can discriminate the most between these different atmosphere models for GJ 1214b, but current observations do not have sufficiently high precision. We present photometry of seven transits of GJ 1214b through a narrow K -band (2.141 µ m) filter with the Wide Field Camera on the 3.8 m United Kingdom Infrared Telescope. Our photometric precision is typically 1.7 × 10 -3 (for a single transit), comparable with other ground-based observations of GJ 1214b. We measure a planet-star radius ratio of 0.1158 ± 0.0013, which, along with other studies, also supports a flat transmission spectrum for GJ 1214b. Since this does not exclude a scenario where GJ 1214b has a H-rich envelope with heavy elements that are sequestered below a cloud/haze layer, we compare K -band observations with models of H 2 collisioninduced absorption in an atmosphere for a range of temperatures. While we find no evidence for deviation from a flat spectrum (slope s = 0.0016 ± 0.0038), an H 2 dominated upper atmosphere ( < 60 mbar) cannot be excluded. More precise observations at < 0.7 µ m and in the K -band as well as a uniform analysis of all published data would be useful for establishing more robust limits on atmosphere models for GJ 1214b.</text>
<text><location><page_1><loc_10><loc_25><loc_83><loc_28></location>Subject headings: planetary systems - planets and satellites: atmospheres 5 techniques: photometric 6</text>
<text><location><page_2><loc_10><loc_85><loc_10><loc_85></location>7</text>
<section_header_level_1><location><page_2><loc_42><loc_85><loc_58><loc_86></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_9><loc_65><loc_88><loc_82></location>To date, approximately 300 planets have been confirmed to transit their host star(s), 8 and the Kepler 1 mission has discovered over 3500 additional transiting planet candidates. 9 About 65 transiting planets (which have both a measured mass and radius) are considered 10 to be in either the 'super-Earth' (1 /lessorsimilar R /lessorsimilar 2 R ⊕ ) or 'mini-Neptune' (2 /lessorsimilar R /lessorsimilar 4 R ⊕ ) 11 regime, but less than a handful of these orbit nearby stars (Schneider et al. 2011). Planets 12 like GJ 1214b, a ∼ 2.7 R ⊕ transiting planet discovered around a nearby ∼ 0.2 R /circledot M dwarf 13 star by the MEarth ground-based transit survey (Charbonneau et al. 2009), are therefore of 14 great interest for understanding the difference between Earth-like planets, 'super-Earths', 15 and 'mini-Neptunes.' 16</text>
<text><location><page_2><loc_9><loc_34><loc_88><loc_63></location>Planets in the super-Earth/mini-Neptune regime occupy a location in the planetary 17 mass-radius diagram that allows for diverse interior compositions (Rogers & Seager 2010). 18 However, it may be possible to constrain the overall bulk composition of these planets by 19 characterizing the planet's atmosphere. Models suggest that GJ 1214b contains a significant 20 amount of hydrogen (H) and helium (He) based on its low mean density (approximately 21 one-third that of Earth; Miller-Ricci & Fortney 2010). Valencia et al. (2013) constrained 22 the fraction of H and He in GJ 1214b using a model of its interior and evolution. They 23 conclude that there is some amount of H/He present and that the bulk amount of H/He 24 may be up to 7% by mass (similar to Neptune). While this is only a small fraction, it 25 suggests that GJ 1214b likely has some amount of H/He in its atmosphere. Since H has a 26 large scale height due to its low molecular weight, even with a small amount of H in the 27 atmosphere the upper atmosphere (i.e. a small distance above the homopause) can easily 28 be H-dominated (Pierrehumbert 2010). Thus, if GJ 1214b is differentiated, then ice and 29 rock will be concentrated in the interior and the concentration of H/He in the surrounding 30 atmosphere could be much higher than 7%. 31</text>
<text><location><page_2><loc_9><loc_15><loc_91><loc_32></location>Many groups have studied its atmosphere by transmission spectroscopy or spectropho32 tometry (e.g., Bean et al. 2010, 2011; Berta et al. 2011; Croll et al. 2011; Crossfield et al. 33 2011; D'esert et al. 2011; Berta et al. 2012; de Mooij et al. 2012; Murgas et al. 2012; Narita et al. 34 2012; Fraine et al. 2013; de Mooij et al. 2013; Narita et al. 2013; Teske et al. 2013) while 35 others have continued improving models of its atmosphere (e.g., Benneke & Seager 2012; 36 Heng & Kopparla 2012; Howe & Burrows 2012; Menou 2012; Miller-Ricci Kempton et al. 37 2012; Morley et al. 2013; Valencia et al. 2013). Figure 1 presents measurements of the 38 planet-star radius ratio (or simply, the radius ratio, R p / R /star ) of GJ 1214b, along with two 39 representative atmosphere models from Howe & Burrows (2012). GJ 1214b has a largely 40</text>
<unordered_list>
<list_item><location><page_3><loc_9><loc_57><loc_88><loc_86></location>flat, featureless spectrum, which supports an atmosphere with a high molecular weight and 41 small scale height, and/or a strongly scattering layer (clouds or aerosols) (e.g., Berta et al. 42 2012). However, published observations at < 0.7 µ m and in the K -band ( ∼ 1.9 -2.5 µ m) as 43 yet cannot discriminate between these two scenarios. Different observations from Bean et al. 44 (2011), Carter et al. (2011), Kundurthy et al. (2011), de Mooij et al. (2012), Murgas et al. 45 (2012), Narita et al. (2013), and de Mooij et al. (2013) disagree as to whether or not there 46 is a rise in the spectrum at < 0.7 µ m due to Rayleigh scattering. In the K s -band, obser47 vations from Croll et al. (2011) have been interpreted as showing a deviation from a flat 48 spectrum, suggesting a lower mean molecular weight atmosphere, while Bean et al. (2011), 49 de Mooij et al. (2012), Narita et al. (2012), and Narita et al. (2013) find a flat spectrum. 50 The disagreement could be a result of the use of slightly different bandpasses combined with 51 telluric effects at the edges of the K passband, so a K -band filter that avoids the edges of the 52 bandpass could help resolve the disagreement between these observations and help determine 53 whether GJ 1214b has a high mean molecular weight atmosphere or a cloudy/hazy H-rich 54 55</list_item>
<list_item><location><page_3><loc_12><loc_56><loc_22><loc_58></location>atmosphere.</list_item>
<list_item><location><page_3><loc_9><loc_53><loc_88><loc_55></location>In one of the latest of many model analyses of the data, Howe & Burrows (2012) pre56</list_item>
<list_item><location><page_3><loc_9><loc_25><loc_88><loc_53></location>sented a suite of atmosphere models for GJ 1214b. They ultimately selected five models 57 that they deemed to best fit the published data, with three models of a solar-abundance 58 atmosphere (two with hazes with different particle sizes and densities and one with a uni59 formly opaque cloud layer) and two of an atmosphere of 1% H 2 O and 99% N 2 plus either 60 haze or no haze. They ruled out several models including a H-rich atmosphere with no haze, 61 a H-rich atmosphere with a haze of smaller ( ∼ 0.01 µ m) tholin particles, as well as a H-poor 62 atmosphere with major sources of absorption other than water. The model that best fits 63 the short-wavelength rise (at < 0.7 µ m; Figure 1) is a solar composition atmosphere with a 64 (somewhat arbitrary) tholin haze layer having a particle size of 0.1 µ m and extending over 65 pressures of 10-0.1 mbar. This model is also the best fit to the K -band data, but only if 66 the observations from Croll et al. (2011) are correct over the observations from Bean et al. 67 (2011), de Mooij et al. (2012), Narita et al. (2012), and Narita et al. (2013), since a flat 68 spectrum in the K -band is inconsistent with a rise at short wavelengths. Howe & Burrows 69 (2012) conclude that if the rise at short wavelengths is valid, GJ 1214b should have a H-rich 70</list_item>
<list_item><location><page_3><loc_9><loc_23><loc_88><loc_25></location>atmosphere (albeit with some cloud or haze layer) rather than being composed primarily of 71</list_item>
<list_item><location><page_3><loc_9><loc_21><loc_46><loc_23></location>heavier molecules like water or nitrogen. 72</list_item>
<list_item><location><page_3><loc_16><loc_18><loc_90><loc_20></location>In this paper, we investigated the scenario of a H-rich atmosphere with a high cloud/aerosol</list_item>
<list_item><location><page_3><loc_9><loc_10><loc_88><loc_19></location>73 layer using narrowK -band observations of transits of GJ 1214b. We present observations 74 of seven transits of GJ 1214b through a narrow-band filter centered at the 2.141 µ m 1-0 75 S(1) vibrational line of H 2 (hereafter, referred to simply as the H 2 filter). We used our 76 observations to (1) help resolve the disagreement between K -band measurements published 77</list_item>
</unordered_list>
<text><location><page_4><loc_9><loc_73><loc_88><loc_86></location>to date and (2) test if there is any spectral structure in the K -band that was missed by 78 the broad-band observations. Specifically, we (1) compared published K -band data to test 79 if the disagreement between K -band measurements is the result of subtle differences in the 80 bandpasses used convolved with not-so-subtle differences in the spectrum of the planet and 81 telluric absorption/emission that were not previously appreciated, and (2) compared K -82 band data with a model of H 2 collision-induced absorption in an atmosphere for a range of 83 84</text>
<unordered_list>
<list_item><location><page_4><loc_12><loc_72><loc_23><loc_74></location>temperatures.</list_item>
</unordered_list>
<text><location><page_4><loc_9><loc_60><loc_88><loc_71></location>We describe our observations and data reduction in Section 2. In Section 3, we present 85 our light curve analysis and models. We present our results in Section 4, and in Section 5 we 86 compare our results with published K -band observations, compare the data with models of a 87 H-rich atmosphere, investigate variability in the stellar spectrum due to H 2 , and discuss the 88 effects of different systematics on our results. In Section 6, we summarize our conclusions 89 90</text>
<text><location><page_4><loc_9><loc_54><loc_10><loc_54></location>91</text>
<unordered_list>
<list_item><location><page_4><loc_12><loc_59><loc_43><loc_61></location>and offer suggestions for future work.</list_item>
</unordered_list>
<section_header_level_1><location><page_4><loc_32><loc_53><loc_68><loc_55></location>2. Observations and Data Reduction</section_header_level_1>
<text><location><page_4><loc_9><loc_23><loc_88><loc_51></location>We acquired photometry of seven transits of GJ 1214b between August 2011 and July 92 2012 with the Wide Field CAMera (WFCAM) on the 3.8 m United Kingdom InfraRed 93 Telescope (UKIRT) (Casali et al. 2001), located at the Mauna Kea Observatory in Hawaii. 94 WFCAM is a near-infrared wide field imager consisting of 4 Rockwell Hawaii-II (HgCdTe 95 2048 × 2048) 0.4 arcsec pixel arrays. Each individual camera covers a field of 13.65 ' × 13.65 ' , 96 and the total field of view (FOV) is 0.75 square degrees. A narrow-band filter centered on 97 the fundamental H 2 vibrational line [S(1) ν = 1 → 0] at 2.141 µ m (FWHM = 0.021 µ m) was 98 used for all observations. This filter probes wavelengths at which previous observations are 99 in disagreement (Section 1) and minimizes systematic variations from Earth's atmosphere. 100 In Figure 2, we compare the transmission profile of the H 2 filter to the broad-band K s 101 filter [the same filter used by Croll et al. (2011) on the Wide-field InfraRed Camera on the 102 Canada-France-Hawaii Telescope] and the atmosphere above Mauna Kea Observatory (at 103 an airmass of 1 and with a water vapor column of 1.2 mm). 2 Because it is so narrow, the H 2 104 filter is an optimal filter to avoid atmospheric effects (compared to the broader K s filter). 105</text>
<text><location><page_4><loc_9><loc_15><loc_88><loc_22></location>In Table 1, we present details of each transit observation. All observations were per106 formed in service mode. An exposure time of 60 s was used for each integration, with a 107 typical dead time between exposures of ∼ 2-3 s. The observation epochs (at mid-exposure) 108</text>
<text><location><page_5><loc_9><loc_70><loc_88><loc_86></location>were extracted from the FITS header for each image. The telescope was intentionally de109 focused to avoid saturation and to spread the stellar PSF over many pixels to minimize 110 error from an imperfect flat-field correction of the detector response. Due to the amount 111 of defocus, the telescope auto-guider was not able to function properly. As a result, a drift 112 of 45-52 pixels in the position of the target centroid occurred during the 2011 observations. 113 For the 2012 observations, the telescope operator routinely adjusted the guider to keep the 114 defocused guide star centered in the guider acquisition window. A centroid drift of less than 115 9 pixels was maintained during the 2012 observations. 116</text>
<text><location><page_5><loc_9><loc_68><loc_10><loc_68></location>117</text>
<text><location><page_5><loc_9><loc_66><loc_10><loc_66></location>118</text>
<text><location><page_5><loc_9><loc_64><loc_10><loc_64></location>119</text>
<text><location><page_5><loc_9><loc_62><loc_10><loc_62></location>120</text>
<text><location><page_5><loc_9><loc_60><loc_10><loc_60></location>121</text>
<text><location><page_5><loc_9><loc_58><loc_10><loc_58></location>122</text>
<text><location><page_5><loc_12><loc_57><loc_88><loc_69></location>We reduced all images using software written in GDL. 3 While all images taken with WFCAM are processed through a pipeline operated by the Cambridge Astronomical Survey Unit (CASU), we opted to process our images separately to ensure accurate calibration and photometry that was as precise as possible. The procedures were illumination correction, dark current correction, and flat fielding prior to performing circular aperture photometry, sky subtraction, and finally a radial distortion correction.</text>
<text><location><page_5><loc_9><loc_34><loc_88><loc_56></location>The illumination correction rectified each image for residual systematics, which are most 123 likely caused by either low-level non-linearity in the detectors, scattered light in the camera, 124 and/or spatially dependent PSF corrections. 4 Illumination correction tables are measured 125 monthly as a function of spatial location in the array. Since illumination correction tables 126 are only available for broad-band filters, we used the K -band tables as a proxy for the H 2 127 filter. For the dark current and flat-field correction, we used darks taken the night of each 128 observation and monthly twilight flats taken with the H 2 filter. The appropriate flats from 129 a given month were used. Circular aperture photometry was performed with radii = 8, 10, 130 12, 14, 16, 18 pixels (3.2, 4.0, 4.8, 5.6, 6.4, 7.2 arcsec). In our analysis we ultimately used 131 the aperture that gave the best photometric performance (Section 3.2). An annulus of 25 132 30 pixels (10 - 12 arcsec) was used for sky subtraction. 133</text>
<text><location><page_5><loc_9><loc_25><loc_88><loc_33></location>Finally, a radial distortion correction was applied to the sky-subtracted flux measured 134 within a given aperture to account for non-negligible field distortion in WFCAM, a result of 135 its extremely large FOV. Specifically, photometry of sources near the edge of the FOV have 136 a systematic error of up to 0.02 mag. We computed the corrected flux, F corr , from 137</text>
<formula><location><page_5><loc_41><loc_19><loc_88><loc_23></location>F corr = F (1 + k 3 r 2 ) 1 + 3 k 3 r 2 , (1)</formula>
<text><location><page_6><loc_9><loc_76><loc_88><loc_86></location>where F is the sky-subtracted flux measured within a given aperture, k 3 is the coefficient 138 of the third order polynomial term in the radial distortion equation and is approximately -60 139 radian -2 in the K -band (Hodgkin et al. 2009, and references therein), and r is the distance 140 of a star relative to the center of the optical system in radians. The corrected flux for the 141 target and each comparison star was used to generate the light curves (Section 3). 142</text>
<text><location><page_6><loc_9><loc_71><loc_10><loc_71></location>143</text>
<text><location><page_6><loc_9><loc_67><loc_10><loc_68></location>144</text>
<text><location><page_6><loc_9><loc_63><loc_10><loc_64></location>145</text>
<text><location><page_6><loc_9><loc_61><loc_10><loc_62></location>146</text>
<text><location><page_6><loc_9><loc_59><loc_10><loc_60></location>147</text>
<text><location><page_6><loc_9><loc_57><loc_10><loc_58></location>148</text>
<text><location><page_6><loc_9><loc_54><loc_10><loc_55></location>149</text>
<text><location><page_6><loc_9><loc_52><loc_10><loc_53></location>150</text>
<text><location><page_6><loc_9><loc_50><loc_10><loc_51></location>151</text>
<text><location><page_6><loc_9><loc_48><loc_10><loc_49></location>152</text>
<text><location><page_6><loc_9><loc_46><loc_10><loc_47></location>153</text>
<text><location><page_6><loc_9><loc_44><loc_10><loc_45></location>154</text>
<text><location><page_6><loc_9><loc_42><loc_10><loc_43></location>155</text>
<text><location><page_6><loc_9><loc_40><loc_10><loc_41></location>156</text>
<section_header_level_1><location><page_6><loc_44><loc_70><loc_56><loc_72></location>3. Analysis</section_header_level_1>
<section_header_level_1><location><page_6><loc_33><loc_67><loc_67><loc_68></location>3.1. Selection of Comparison Stars</section_header_level_1>
<text><location><page_6><loc_12><loc_57><loc_88><loc_65></location>Thanks to the large FOV covered by WFCAM, we could select among numerous comparison stars for relative photometry. We selected comparison stars that are of similar brightness to GJ 1214 and that do not appear to be intrinsically variable. We also used colors to select dwarf stars over giants and that are as close as possible to GJ 1214 in spectral type.</text>
<text><location><page_6><loc_12><loc_40><loc_88><loc_56></location>We downloaded J , H , and K magnitudes from the 2MASS catalog (Skrutskie et al. 2006) and proper motions from the PPMXL catalog (Roeser et al. 2010) for all stars in the vicinity (within 30 arcmin) of GJ 1214. We also downloaded V magnitudes from the Fourth U.S. Naval Observatory CCD Astrograph Catalog (Zacharias et al. 2013). We imposed a magnitude cut in K -band so that the reference stars were no more than 0.5 mag brighter and 2 mag fainter than GJ 1214 ( K = 8.782). Figure 3 is a color-color diagram of all stars that meet our magnitude criteria. Not all these stars are actually located in the FOV, since WFCAMhas four cameras, and there are gaps of 12.83 arcmin between the different cameras.</text>
<text><location><page_6><loc_9><loc_19><loc_88><loc_38></location>We selected reference stars from those shown in Figure 3 based on three different crite157 ria. We first selected eight stars that were more likely to be M dwarf stars based on color 158 criteria from L'epine & Gaidos (2011). Additional reference stars were selected based on 159 their proximity to a 2MASS main-sequence locus from Stead & Hoare (2011) and a locus for 160 K7 - M9.5 spectral types (Cutri et al. 2003). 5 Specifically, we selected stars in two regions 161 of color space, marked by the two boxes in Figure 3. These regions were also selected to 162 avoid the giant locus. 6 We identified three additional stars that fit this color criteria (and 163 that also were in the FOV). Finally, we computed the magnitude of the proper motion for 164 each star shown in Figure 3 (using proper motions from the PPMXL catalog). Following 165 L'epine & Gaidos (2011), we computed the reduced proper motion, H V , as 166</text>
<formula><location><page_7><loc_41><loc_82><loc_88><loc_84></location>H V = V +5log µ +5 . (2)</formula>
<text><location><page_7><loc_9><loc_77><loc_88><loc_80></location>We then applied the constraint from L'epine & Gaidos (2011) to separate M dwarfs from 167 giants based on 168</text>
<formula><location><page_7><loc_40><loc_71><loc_88><loc_73></location>H V > 2 . 2( V -J ) + 2 . 0 . (3)</formula>
<text><location><page_7><loc_9><loc_58><loc_88><loc_69></location>We found that all stars in Figure 3 met this criterion. Therefore, to maximize the 169 number of optimal comparison stars, we chose to select additional stars with the highest 170 proper motions of the sample. Of the five stars with the highest proper motions (excluding 171 GJ 1214), only one was actually located within our FOV (i.e. not in a gap between cameras). 172 We added this star to our reference ensemble, bringing the total number of comparison stars 173 to 12. 174</text>
<text><location><page_7><loc_9><loc_43><loc_88><loc_56></location>A preliminary visual examination of the light curve of each reference star (the flux of 175 a given reference star divided by the total flux of the remaining reference stars) revealed 176 that four of the twelve stars were potentially variable (i.e. the light curves displayed possible 177 periodic fluctuations). For completeness, we considered all 12 reference stars when generating 178 light curves for GJ 1214. The number of reference stars ultimately included in the light curve 179 analysis varied between two and ten depending on which combination of references produced 180 the lowest scatter in each set of baseline (out-of-transit) data. 181</text>
<text><location><page_7><loc_9><loc_37><loc_10><loc_38></location>182</text>
<section_header_level_1><location><page_7><loc_33><loc_37><loc_67><loc_38></location>3.2. Generating the Light Curves</section_header_level_1>
<text><location><page_7><loc_9><loc_15><loc_88><loc_35></location>Light curves were generated from each data set by computing the relative flux ratio, i.e. 183 the target signal divided by the total weighted signal of an ensemble of reference stars. Due 184 to varying weather conditions and the intrinsic variability of some of the reference stars, the 185 reference ensemble was composed of different stars for each set of observations. To determine 186 which reference stars would be included in each ensemble, the rms of each reference star's 187 signal was compared to an imposed photometric precision threshold (set to slightly different 188 values for each night to include an adequate number of reference stars, with a typical value 189 of 8 × 10 -3 ). The stars that had an rms that exceeded the threshold were excluded from the 190 reference signal, and the remaining stars had their signals weighted (based on the rms of 191 each individual reference signal) when computing the combined ensemble reference signal. 192</text>
<text><location><page_7><loc_9><loc_10><loc_88><loc_13></location>Normalized light curves were generated for each photometric aperture by dividing the 193 relative flux ratio by the median relative flux ratio measured in the out-of-transit data. We 194</text>
<text><location><page_8><loc_9><loc_80><loc_88><loc_86></location>considered the resulting out-of-transit rms scatter as measured in each aperture and for each 195 night of observations, and we chose the aperture that gave the smallest scatter in a given 196 night. 197</text>
<text><location><page_8><loc_9><loc_65><loc_88><loc_79></location>Each light curve was then regressed against target centroid position, airmass, and peak 198 counts (per pixel) and a linear trend removed. In Figure 4, we show the light curve from 199 August 24, 2011 prior to detrending, along with the parameters used for detrending. This 200 light curve has a notable negative deviation after mid-transit, which we discuss in Section 201 5.4. We also tested correcting against variations in the absolute flux of the target and the 202 variance of the flux of the target, but we found that these additional corrections resulted in 203 a negligible change in the flux ratios. 204</text>
<text><location><page_8><loc_9><loc_52><loc_88><loc_64></location>We discarded some June 16, 2012 data due to an incorrect defocus setting and saturation 205 (Section 2), as well as some points that were unexplained extreme outliers ( > 3 σ from the 206 mean of either the baseline data or the data at the bottom of the transit). We discarded 207 one data point from the August 24, 2011 observations and 28 points from the June 16, 2012 208 observations. We then fit transit models to the light curve data, as described in the next 209 section. 210</text>
<text><location><page_8><loc_9><loc_47><loc_10><loc_47></location>211</text>
<section_header_level_1><location><page_8><loc_40><loc_46><loc_60><loc_48></location>3.3. Transit Models</section_header_level_1>
<text><location><page_8><loc_9><loc_27><loc_88><loc_44></location>To measure R p / R /star we fit transit models to our seven transit light curves using the 212 Transit Analysis Package (TAP), a publicly available IDL code (Gazak et al. 2012). 7 TAP 213 fits limb-darkened transit light curves using EXOFAST (Eastman et al. 2013) to calculate 214 the model of Mandel & Agol (2002), along with a combination of Bayesian and Markov 215 Chain Monte Carlo (MCMC) techniques. TAP also employs a wavelet-based likelihood 216 function (Carter & Winn 2009) to robustly estimate parameter uncertainties by fitting both 217 uncorrelated ('white') and correlated ('red') noise. A quadratic limb darkening law was 218 used, so in the models described below, linear and quadratic ( µ 1 and µ 2 ) limb darkening 219 coefficients are included. 220</text>
<text><location><page_8><loc_9><loc_24><loc_10><loc_24></location>221</text>
<text><location><page_8><loc_9><loc_22><loc_10><loc_22></location>222</text>
<text><location><page_8><loc_9><loc_20><loc_10><loc_20></location>223</text>
<text><location><page_8><loc_9><loc_18><loc_10><loc_18></location>224</text>
<text><location><page_8><loc_9><loc_16><loc_10><loc_16></location>225</text>
<text><location><page_8><loc_12><loc_15><loc_88><loc_25></location>We modeled all seven transits simultaneously, using five MCMC chains with lengths of 100,000 links each, and keeping the following parameters fixed at a single value for all transits: orbital period ( P ), inclination ( i ), scaled semi-major axis ( a / R ∗ ), and the limb darkening coefficients ( µ 1 and µ 2 ). (We assumed a circular orbit.) The mid-transit time, as well as the white and red noise, was individually fitted for each transit. To obtain R p / R ∗</text>
<text><location><page_9><loc_9><loc_78><loc_88><loc_86></location>we (i) fit individual transit light curves and (ii) fit the combined light curves of all transits. 226 The fixed parameters we used are given in Table 2. The K -band limb darkening parameters 227 are from Claret & Bloemen (2011) for a star with T eff = 3000 K and log( g ) = 5 and are the 228 same as those used by Narita et al. (2012). 229</text>
<text><location><page_9><loc_9><loc_65><loc_88><loc_77></location>In Figure 5, we present the seven individual light curves. Figure 6 shows the combined 230 light curve (composed by phasing each of the individual light curves) and the best-fit model 231 for case (ii), where we fit a single R p / R ∗ to all transits. We base our primary conclusions 232 on the results from this case. However, we discuss the individually measured radius ratios 233 in Section 4 and the effects of fixing versus fitting the limb darkening parameters and a / R ∗ 234 in further detail in Section 5.5. 235</text>
<text><location><page_9><loc_9><loc_60><loc_10><loc_60></location>236</text>
<text><location><page_9><loc_9><loc_56><loc_10><loc_57></location>237</text>
<text><location><page_9><loc_9><loc_54><loc_10><loc_54></location>238</text>
<text><location><page_9><loc_9><loc_52><loc_10><loc_52></location>239</text>
<text><location><page_9><loc_9><loc_50><loc_10><loc_50></location>240</text>
<text><location><page_9><loc_9><loc_48><loc_10><loc_48></location>241</text>
<text><location><page_9><loc_9><loc_46><loc_10><loc_46></location>242</text>
<section_header_level_1><location><page_9><loc_45><loc_59><loc_55><loc_61></location>4. Results</section_header_level_1>
<text><location><page_9><loc_12><loc_46><loc_88><loc_57></location>In Table 3 we present the best-fit radius ratio, mid-transit time, white noise, and red noise measured for each individual transit. We also include the best-fit radius ratio based on fitting all seven transits together. Since TAP fits for correlated (red) noise, the uncertainties on the fitted parameters should be conservative. This was also pointed out by Teske et al. (2013), who compared results from TAP with results from other light curve fitting software. The correlated noise is discussed in further detail in Section 5.4.</text>
<text><location><page_9><loc_9><loc_25><loc_88><loc_44></location>We find that all transit times deviate from the predicted ephemeris (from Berta et al. 243 2012) by less than 29 s and are consistent with a linear ephemeris. We also find that the 244 fitted values for the white noise are generally consistent with the rms scatter in each transit 245 light curve (Section 5.3), which suggests that the red noise may be smaller than the values 246 measured by TAP. We present the individual best-fit radius ratios in Figure 7. They are 247 all consistent with the combined best-fit radius ratio (0.1158 ± 0.0013). The radius ratio 248 measured for the fifth transit observation (June 16, 2012) has the largest uncertainty, but it 249 is still consistent with those measured from the other individual transits. The photometry of 250 this particular transit was of inferior quality than the other transit observations (Figure 5), 251 likely due in part to the presence of mixed thin and thick clouds throughout the observations. 252</text>
<text><location><page_9><loc_9><loc_14><loc_88><loc_23></location>In Figure 8, we present the results of our analysis along with other K -band measurements 253 of the radius ratio for GJ 1214b. We find that our best-fit radius ratio is consistent with 254 the majority of other published values and supports a flat absorption spectrum for the 255 atmosphere of GJ 1214b with a slope s = 0.0016 ± 0.0038 [excluding 'outlying' points from 256 Croll et al. (2011) and de Mooij et al. (2012)]. We discuss this result further in Section 5.1. 257</text>
<text><location><page_10><loc_9><loc_81><loc_10><loc_82></location>259</text>
<section_header_level_1><location><page_10><loc_43><loc_85><loc_57><loc_86></location>5. Discussion</section_header_level_1>
<section_header_level_1><location><page_10><loc_29><loc_81><loc_71><loc_82></location>5.1. Comparison of K -Band Observations</section_header_level_1>
<text><location><page_10><loc_9><loc_41><loc_88><loc_79></location>Our measurements are inconsistent with the conclusion of Croll et al. (2011), that GJ 260 1214b has a low mean molecular weight atmosphere. One K s -band measurement from 261 de Mooij et al. (2012) also supports the data from Croll et al. (2011) and hints at a de262 viation from a flat spectrum (Figure 8). However, the points from Croll et al. (2011) and 263 de Mooij et al. (2012) only differ from our derived radius ratio by 1.5 σ and 1.1 σ , suggesting 264 that their data is in fact consistent with other published K -band observations. Differences 265 between the transmission measurements could be a result of the use of different filters (i.e. 266 our narrow-band H 2 filter versus their broad-band K s filter) and the correspondingly dif267 ferent wavelength coverage by the various groups. Our narrow-band filter probes a small 268 wavelength range (FWHM = 0.021 µ m) and is free of telluric features (Figure 2). The K s 269 transmission curve shown in Figure 2 is for the filter specifically used by Croll et al. (2011) 270 and contains some telluric absorption features. While telluric effects should be largely re271 moved by relative photometry, the choice of reference stars and their locations on the sky 272 (relative to the target) can result in imperfect removal of telluric features. The K s -band also 273 contains some stellar lines, and we discuss this further in Section 5.3. Broad-band photome274 try therefore can be subject to additional systematics and yield less accurate measurements 275 than narrow-band photometry. Ultimately, we find that K -band observations continue to 276 support a high molecular weight atmosphere or a H-rich atmosphere with a cloud or haze 277 layer over a cloud/haze-free H-rich atmosphere. 278</text>
<text><location><page_10><loc_9><loc_35><loc_10><loc_36></location>279</text>
<section_header_level_1><location><page_10><loc_22><loc_35><loc_78><loc_36></location>5.2. Constraints on a Cloudy/Hazy H-Rich Atmosphere</section_header_level_1>
<text><location><page_10><loc_9><loc_15><loc_88><loc_33></location>In the previous section, we concluded that K -band measurements are consistent with 280 a flat transmission spectrum, which favors either a high molecular weight atmosphere or 281 a H-rich atmosphere with a cloud or haze layer (Howe & Burrows 2012). However, the 282 available K -band data does not have sufficiently high precision to discern between these two 283 atmosphere models. Published B -band data (e.g., de Mooij et al. 2012, 2013; Narita et al. 284 2013; Teske et al. 2013) cannot definitively discriminate between these two scenarios either. 285 In this section, we assume that the atmosphere is H-dominated [following the results from 286 Valencia et al. (2013)], and we consider what constraints can be placed on a cloudy/hazy 287 H-rich atmosphere. 288</text>
<text><location><page_10><loc_9><loc_10><loc_88><loc_14></location>For a H-rich atmosphere with relatively high number densities where collisions are fre289 quent, collision-induced absorption (CIA) is a dominant source of opacity (Borysow 2002, 290</text>
<text><location><page_11><loc_9><loc_66><loc_88><loc_86></location>and references therein). Here, we consider a H-rich upper atmosphere that includes an 291 opaque cloud or haze layer at some low altitude/high pressure. We assume that any molec292 ular absorbers in the atmosphere are confined beneath a cloud/haze layer, and that the 293 atmosphere above the clouds/haze is metal-free as a result of some (photo)chemical or mix294 ing boundary (chemopause or homopause). If the H 2 envelope is sufficiently dense, it could 295 be detected by CIA in high-precision K -band spectrophotometry. This depends on tem296 perature, so the presence or absence of CIA features is also a constraint on temperature 297 (provided other conditions are met; see below). The equilibrium temperature of GJ 1214b 298 is ∼ 500 K (Charbonneau et al. 2009), but the upper atmosphere could be much hotter due 299 to XUV heating from the star (Lammer et al. 2013). 300</text>
<text><location><page_11><loc_9><loc_55><loc_88><loc_65></location>A condition for this model to hold is that the CIA opacity of H 2 dominates above the 301 pressure altitude of any cloud/haze layer present in the atmosphere (i.e. τ H 2 /greaterorsimilar 1 above the 302 chemopause or homopause). Under this condition, we computed the minimum pressure ( P 0 ) 303 at the top of a cloud/haze layer located in an isothermal atmosphere using the following 304 equation: 305</text>
<formula><location><page_11><loc_38><loc_48><loc_88><loc_53></location>P 0 = k B T √ 1 σ ( T, λ ) √ πR 0 H . (4)</formula>
<text><location><page_11><loc_9><loc_45><loc_58><loc_46></location>H is the scale height of the atmosphere, defined as 306</text>
<formula><location><page_11><loc_45><loc_39><loc_88><loc_42></location>H = k B T µg , (5)</formula>
<text><location><page_11><loc_9><loc_21><loc_88><loc_37></location>k B is the Boltzmann constant, T is the atmospheric temperature, µ is the mean molec307 ular weight of the atmosphere, and g is the planet's surface gravity. We assumed T = 500 308 K, and we set the reference planetary radius R 0 = 2.68 R ⊕ . We used H 2 -H 2 CIA opacities 309 ( σ ) from Borysow (2002) and computed the scale height H using a mean molecular weight 310 µ = 2 and a surface gravity g = 8.95 m s -2 . From this, we computed the minimum pressure 311 P 0 at the top of the cloud or aerosol layer to be 60 mbar (at λ ∼ 2.14 µ m). This minimum 312 pressure falls within the pressure range of 0.001 -100 mbar considered by Howe & Burrows 313 (2012). 314</text>
<text><location><page_11><loc_9><loc_16><loc_88><loc_20></location>We then calculated the effective R p / R /star versus wavelength, following a similar procedure 315 as in Howe & Burrows (2012): 316</text>
<formula><location><page_11><loc_38><loc_9><loc_88><loc_14></location>R p R /star = R 0 R /star + H 2 R /star ln [ σ ( T, λ ) σ 0 ] . (6)</formula>
<text><location><page_12><loc_9><loc_85><loc_10><loc_85></location>317</text>
<text><location><page_12><loc_9><loc_83><loc_10><loc_83></location>318</text>
<text><location><page_12><loc_9><loc_81><loc_10><loc_81></location>319</text>
<text><location><page_12><loc_9><loc_79><loc_10><loc_79></location>320</text>
<text><location><page_12><loc_9><loc_77><loc_10><loc_77></location>321</text>
<text><location><page_12><loc_9><loc_74><loc_10><loc_74></location>322</text>
<text><location><page_12><loc_9><loc_72><loc_10><loc_72></location>323</text>
<text><location><page_12><loc_9><loc_70><loc_10><loc_70></location>324</text>
<text><location><page_12><loc_9><loc_68><loc_10><loc_68></location>325</text>
<text><location><page_12><loc_9><loc_66><loc_10><loc_66></location>326</text>
<text><location><page_12><loc_9><loc_64><loc_10><loc_64></location>327</text>
<text><location><page_12><loc_9><loc_62><loc_10><loc_62></location>328</text>
<text><location><page_12><loc_12><loc_76><loc_88><loc_86></location>Here, R 0 is the radius at the wavelength where σ = σ 0 = 2.78 × 10 -6 cm -1 amagat -2 at λ ∼ 2.14 µ m [from the 500 K table from Borysow (2002)]. Because H 2 absorption is collisionally-induced, it depends on the square of the number density ( n 2 , where n = number density of absorbers) and this leads to an additional factor of 1/2 in Eqn. 6 [compared to the equation defined in Howe & Burrows (2012)].</text>
<text><location><page_12><loc_12><loc_61><loc_88><loc_75></location>Under the same conditions assumed above and using R /star =0.211 R /circledot (Charbonneau et al. 2009), we calculated the radius ratio for temperatures of 400, 500, 600, 700, 800, 900, and 1000 K. We compared the observed radius ratios to the predicted radius ratios by calculating χ 2 and identified the best-fit model (based on the minimum χ 2 ). Although the K s -band data from Croll et al. (2011) and de Mooij et al. (2012) are marginally consistent with other K -band data within measurement uncertainties, we still considered them to be unexplained outliers and excluded them from the analysis (Section 5.1).</text>
<text><location><page_12><loc_9><loc_42><loc_88><loc_60></location>We find the 400 K model had the smallest χ 2 , while the 1000 K model yielded the largest 329 χ 2 . We show the 400 and 1000 K models in Figure 8. As the temperature increases, the 330 scale height increases, leading to additional absorption and a larger apparent planet radius; 331 however, the features also become washed out at higher temperatures. We computed ∆ χ 2 332 (relative to the minimum χ 2 at 400 K) and found that the deviation between the data and 333 models increases with increasing temperature. However, ∆ χ 2 between the 400 and 1000 334 K models is only ∼ 0.72, since χ 2 is just 2 -3 for all models. We conclude that from the 335 available data, we cannot exclude higher temperature atmospheres ( T > 400 K) with any 336 confidence ( p < 0.01). 337</text>
<text><location><page_12><loc_9><loc_15><loc_88><loc_41></location>Considering the capabilities of future missions like the James Webb Space Telescope 338 ( JWST ) for high precision infrared spectroscopy, the atmosphere models were compared to 339 both the real data (using the actual measurement uncertainties) as well as an artificial data 340 set, consisting of the actual measurements with reduced errors. We defined artificial errors 341 so that the median error over the K -band data = 1 × 10 -4 [based on the precision achieved 342 by Fraine et al. (2013) from Spitzer measurements of GJ 1214b at 4.5 µ m]. For comparison, 343 the median value of the actual errors is 9.9 × 10 -4 . Based on the artificial high-precision 344 data, we again find that the 400 K model has the smallest χ 2 , and that ∆ χ 2 between the 345 400 and 1000 K models is 70.2, which would allow us to exclude atmospheres with T ≥ 800 346 K with > 99.7% confidence (3 σ ) assuming that we had such high precision data. We find 347 that a temperature of ≤ 400 K is preferred for a pure H 2 upper atmosphere, but significantly 348 higher precision data (as well as more data in general) is needed to confidently exclude higher 349 temperature atmospheres. 350</text>
<text><location><page_12><loc_9><loc_9><loc_88><loc_14></location>Ultimately, while we find no evidence for deviation from a flat spectrum, a thin upper 351 atmosphere ( ≤ 60 mbar) dominated by H 2 cannot be excluded. The possibility also remains 352</text>
<text><location><page_13><loc_9><loc_85><loc_69><loc_86></location>that there is simply no (or very little) H in the (upper) atmosphere. 353</text>
<text><location><page_13><loc_9><loc_79><loc_10><loc_79></location>354</text>
<section_header_level_1><location><page_13><loc_30><loc_78><loc_70><loc_80></location>5.3. Variability in the Stellar Spectrum</section_header_level_1>
<text><location><page_13><loc_9><loc_55><loc_88><loc_76></location>Fluorescent H 2 emission lines have been observed in the spectra of four planet-hosting 355 M dwarf stars, but not in GJ 1214 (France et al. 2013). These lines are produced by pho356 toexcitation by Ly α photons, and their detection indicates the presence of a 2000 -4000 K 357 molecular gas (France et al. 2013, and references therein). The H 2 filter is designed specifi358 cally to observe such lines, making it possible for us to also probe the stellar atmosphere with 359 our observations. While France et al. (2013) do not detect Ly α emission in GJ 1214, they 360 speculate that this is because the neutral H in GJ 1214's atmosphere is instead contained 361 in H 2 rather than H. Notably, GJ 1214 is the coolest star ( T eff ∼ 3250 K; Anglada-Escud'e 362 et al. 2013) in their sample, and H recombines at cool temperatures, so it is plausible that 363 molecular H is present in GJ 1214's atmosphere. This motivated us to look for H 2 emission 364 in GJ 1214. 365</text>
<text><location><page_13><loc_9><loc_38><loc_88><loc_53></location>We examined a K -band spectrum of GJ 1214 from Rojas-Ayala et al. (2012), shown in 366 Figure 9. We find no evidence of an H 2 emission feature around ∼ 2.14 µ m in GJ 1214's 367 spectrum. The H 2 line is close to the Brackett γ line (2.16 µ m), which is produced in T 368 Tauri stars by recombining magnetospheric gas (e.g., Hamann et al. 1988). However, we see 369 no evidence of the Brackett γ line in the spectrum of GJ 1214. Indeed, the H 2 bandpass is 370 free of any obvious stellar lines. The K s filter does include several stellar absorption lines, 371 but assuming that these features are only present in the star (and are not telluric) and that 372 they do not vary, these will not affect the photometry. 373</text>
<text><location><page_13><loc_9><loc_17><loc_88><loc_36></location>Due to the lower temperatures H 2 may predominate in star spots and may play an im374 portant role in the formation and evolution of spots (e.g., Jaeggli et al. 2012). Therefore, we 375 also looked for evidence of H 2 variability, or patchiness in the stellar disk in the H 2 bandpass 376 (either bright emission or dark absorption) which produces variability when transited by the 377 planet (i.e. 'H 2 -spots'). 8 We compared observations taken in-transit (when the time-varying 378 part of the star is blocked by the planet's disk) with those taken out-of-transit (when only 379 the star light is visible). Specifically, we compared the rms scatter (of the light curve resid380 uals, computed by subtracting the best-fit model from the data; Section 3.3) between the 381 in-transit and out-of-transit windows for each of the seven individual transits. The resulting 382 rms values are presented in Figure 10. We computed r 2 = 0.80 between the in- and out-of383</text>
<text><location><page_14><loc_9><loc_70><loc_88><loc_86></location>transit rms, which indicates a strong linear correlation between the two parameters. That 384 a strong linear correlation exists suggests that the in- and out-of-transit measurements are 385 dominated by the same source of variation, i.e. photometric errors as opposed to a source of 386 variation associated with the transit alone. From this, we conclude that if H 2 is present and 387 absorbing or emitting in the stellar and/or planetary atmosphere, it is either not variable 388 or its variations are not detectable in our data. It is likely that the vibrational transitions 389 such as the 1-0 S(1) transition are too weak to be seen in a standard K -band spectrum (but 390 instead might be seen through a linearly polarized spectrum; White & Kuhn 2011). 391</text>
<text><location><page_14><loc_9><loc_59><loc_88><loc_69></location>The H 2 1-0 S(1) line is also used to study molecular outflows from protostars (e.g., 392 Garcia Lopez et al. 2013). We considered the possibility of detecting H 2 'outflows' from 393 the atmosphere of the planet (driven by UV heating), but we see no persistent deviations 394 from the standard transit light curve models that require explanation by a shock or wind 395 (i.e. from some massive atmospheric escape from the planet). 396</text>
<text><location><page_14><loc_9><loc_54><loc_10><loc_54></location>397</text>
<section_header_level_1><location><page_14><loc_37><loc_53><loc_63><loc_55></location>5.4. Effects of Systematics</section_header_level_1>
<text><location><page_14><loc_9><loc_46><loc_88><loc_51></location>Systematics, whether astrophysical or instrumental in nature, can significantly affect 398 the derivation of light curve parameters. In this section, we describe potential sources of 399 systematics in our light curves and the effect they have on our measured radius ratios. 400</text>
<text><location><page_14><loc_9><loc_40><loc_10><loc_40></location>401</text>
<section_header_level_1><location><page_14><loc_38><loc_39><loc_62><loc_41></location>5.4.1. Astrophysical Sources</section_header_level_1>
<text><location><page_14><loc_9><loc_20><loc_88><loc_37></location>Intrinsic variability of the host star can affect measurements of the radius ratio, and GJ 402 1214 has been shown to be variable at red ( ∼ 0.8 µ m) wavelengths at the 1-2% level over ∼ 403 1-2 year timescales (Carter et al. 2011). While we see no obvious visual evidence of stellar 404 flares or star spots 9 in our light curves (Figure 5), in principle it is possible to use the radius 405 ratios measured for the individual transits to estimate how much the spot coverage changed 406 with time. Theoretically, the shallowest observed transit should correspond to the stellar 407 surface being nearly free of spots, or at least having minimal spot coverage. As presented 408 in Table 3 and shown in Figure 7, the measured radius ratios do not vary significantly with 409 transit epoch. Between the photometric quality of the data and the magnitude of the errors 410</text>
<text><location><page_15><loc_9><loc_64><loc_88><loc_86></location>derived for the radius ratios, we cannot robustly identify a minimum epoch of stellar activity 411 based on the measured radius ratios. Based on the rms values presented in Figure 10, our 412 data suggest variability at a level of ∼ 1.5 × 10 -3 (albeit only on hour-long timescales). We 413 also find that GJ 1214 is variable in the K -band at a level of ∼ 0.5% over longer timescales 414 (from August 2011 to July 2012 or ∼ 1 year), based on the target-to-reference flux ratio for 415 the most photometrically stable reference star in our sample. This is consistent with the 416 results from Carter et al. (2011), since stellar variability in our passband should be less than 417 the 1-2% measured by Carter et al. (2011) due to observing in a redder passband. Although 418 such variability does affect the measurement of radius ratios, since our radius ratios did not 419 vary significantly (within errors), we consider stellar variability to have a negligible effect on 420 our conclusions. 421</text>
<text><location><page_15><loc_9><loc_35><loc_88><loc_63></location>There is the possibility that spot-crossing events did occur in one or more of our observed 422 transits, but we simply cannot identify them by eye because they cause brightenings at or 423 below the level of our photometric precision. If we consider the red noise (Table 3) as a 424 measure of the effects of spot-crossing events, we find that the August 24, 2011 and June 425 16, 2012 transits have the highest levels of red noise. The June 16, 2012 transit had poorer 426 weather conditions, which is likely the cause of the higher red noise in that light curve, 427 as no evidence of a spot-crossing event is seen in that light curve. However, the August 428 24, 2011 transit, which had photometric conditions, does have an anomalous feature in its 429 light curve (Figure 11) where the transit depth appears to increase significantly immediately 430 before egress. The anomalous feature in the August 24, 2011 transit does not appear to be a 431 (dark) spot crossing event, since the transit depth before the feature occurred is consistent 432 (i.e. within 1 σ ) with the transit depth measured for the combined light curve (over all seven 433 transits). This suggests that the true transit depth should be based on the first part of the 434 transit, rather than the first part of the transit being a spot-crossing event. 435</text>
<text><location><page_15><loc_9><loc_20><loc_88><loc_34></location>To determine if this anomalous feature was produced by reference star variability, we 436 constructed alternative light curves using different reference stars. We found that the feature 437 was present in all cases. We also searched the literature to see if any other observations of 438 this specific transit had been published, but we only found an observation of the transit 439 following this one by Harpsøe et al. (2013). Their light curves do not appear to contain any 440 particularly anomalous features, which supports the idea of minimal (or at least un-transited) 441 spots at the time of the observations. 442</text>
<text><location><page_15><loc_9><loc_11><loc_88><loc_19></location>We also considered that the feature may instead be due to the planet passing over a 443 bright spot on the star, which would result in an increase in the transit depth, consistent 444 with our observations. However, the presence of a bright spot will also cause a decrease in 445 the unocculted transit depth. As illustrated in Figure 7, the transit depths measured for 446</text>
<text><location><page_16><loc_9><loc_76><loc_88><loc_86></location>all transits are consistent within 1 σ . This suggests that if the anomalous feature is a result 447 of a spot-crossing event, the spot is sufficiently compact/faint that it does not produce an 448 unocculted spot effect that we would detect. We do not find evidence of spot-crossing events 449 in our other transit observations, but we explore the bright spot hypothesis in more detail 450 here. 451</text>
<text><location><page_16><loc_9><loc_71><loc_88><loc_75></location>Assuming a circular spot with radius R spot , we estimated the size of such a spot based 452 on the duration of the spot crossing, 453</text>
<formula><location><page_16><loc_41><loc_65><loc_88><loc_69></location>R spot R ∗ = t spot τ √ 1 -b 2 . (7)</formula>
<text><location><page_16><loc_9><loc_54><loc_88><loc_64></location>The transit duration ( τ ) is 52.73 min (Carter et al. 2011), and the spot-crossing time 454 ( t spot ) was estimated from the light curve to be 15 min (Figure 11). Using an impact 455 parameter ( b ) of 0.28 (Bean et al. 2011), we computed a spot-star radius ratio of 0.27 (notably 456 over two times larger than R p / R /star ). We then computed how much the transit depth would 457 increase due to a bright spot (relative to the no spot case) from 458</text>
<formula><location><page_16><loc_42><loc_47><loc_88><loc_51></location>δ f = B ( R spot R ∗ ) 2 . (8)</formula>
<text><location><page_16><loc_9><loc_20><loc_88><loc_46></location>This equation is applicable because our hypothetical spot is much larger than the planet. 459 The increase in the transit depth during the anomalous event relative to the depth prior to 460 the anomalous event is ∆ δ = 0.0059 (Figure 11). The transit depth ( δ p ) is 0.0135. B is 461 the brightness enhancement of the spot relative to the rest of the stellar disk, which we 462 computed based on the relative depth of the anomalous feature, or B = ∆ δ / δ p = 0.44. 463 From this, we calculate δ f = 0.033. This translates to a change in the transit depth due 464 to the unocculted bright spot of δ f × δ p = 4.4 × 10 -4 , which is much smaller than our typical 465 photometric precision of 1.7 × 10 -3 (for a single transit). It is also consistent with GJ 1214 466 being variable on the order of 1-2% (Carter et al. 2011). We conclude that a single transient 467 bright spot about twice the size of the planet's disk could explain the transit of August 24, 468 2011. Regardless of the source of the feature, TAP recognized it as a systematic and derived 469 a radius ratio that is consistent with the radius ratios measured from the other transits 470 (Figure 7). 471</text>
<section_header_level_1><location><page_17><loc_38><loc_85><loc_62><loc_86></location>5.4.2. Instrumental Sources</section_header_level_1>
<text><location><page_17><loc_9><loc_67><loc_88><loc_82></location>Besides astrophysical systematics, we also considered the effect of instrumental system473 atics such as a nonlinear detector on our measured radius ratios. The WFCAM detector is 474 linear to < 1% up to about 40,000 counts per pixel. 10 To ensure that we avoided the defined 475 non-linear regime, we measured the peak counts in the target and each reference star. We 476 found that all stars remained below ∼ 22,000 counts (per pixel) in all observations. However, 477 the possibility remains that lower non-linearity is present at lower counts (Section 2). To 478 check this, we derived a 'variance coefficient' ( k ) for each reference star and for each night 479 based on the following equation, 480</text>
<formula><location><page_17><loc_38><loc_61><loc_88><loc_64></location>F observed = F actual -kF 2 observed . (9)</formula>
<text><location><page_17><loc_9><loc_34><loc_88><loc_60></location>Here, F observed is the normalized flux ratio we measured for each reference star, F actual 481 is what the ideal flux ratio should be in the absence of systematics (i.e. F actual ≡ 1), k is 482 defined as the variance coefficient, and F 2 observed is the variance of the observed flux ratio. 483 After deriving k for each reference star and for each night, we found that on most nights, 484 k << 1.1 × 10 -3 . Thus, we conclude that the effect of non-linearity is smaller than our 485 photometric precision (typically 1.7 × 10 -3 for a single transit). Only on two nights did the 486 flux ratios have significant deviations: during the fifth transit (June 16, 2012) and sixth 487 transit (June 27, 2012). That the fifth transit shows signs of non-linearity is consistent with 488 the observations, since the stellar image was insufficiently defocused to avoid high counts. 489 We have no explanation as to why the sixth transit is potentially affected by non-linearity. 490 Given that the measured transit depth is consistent with those measured from the other 491 individual transits, we conclude that any low-level non-linearity that might be present and 492 a source of systematics has a minimal effect on our photometry. 493</text>
<text><location><page_17><loc_9><loc_28><loc_10><loc_29></location>494</text>
<section_header_level_1><location><page_17><loc_28><loc_28><loc_72><loc_29></location>5.5. Treatment of Transit Model Parameters</section_header_level_1>
<text><location><page_17><loc_9><loc_16><loc_88><loc_26></location>Finally, we considered how limitations in our knowledge of stellar properties affect the 495 derived light curve parameters. In particular, there is a degeneracy between limb darkening 496 and a / R ∗ , which in turn affects estimates of the stellar density, orbital eccentricity, and 497 impact parameter. Furthermore, as Berta et al. (2012) point out, inaccurate treatment of 498 limb darkening could introduce false absorption features into the transmission spectrum. 499</text>
<text><location><page_18><loc_9><loc_82><loc_88><loc_86></location>Therefore, the accurate treatment of limb darkening is critical for precisely measuring light 500 curve parameters. 501</text>
<text><location><page_18><loc_9><loc_31><loc_88><loc_81></location>In the analyses described above, we held limb darkening coefficients and a / R ∗ fixed. 502 This decision was based on the fact that GJ 1214b's transit duration is only 52 min and 503 the ingress/egress events last 6 min, compared to our cadence of ∼ 1 min. Having a rela504 tively small number of data points during ingress and egress makes it difficult to accurately 505 fit the limb darkening. However, Csizmadia et al. (2013) argue that stellar limb darken506 ing parameters should be fitted and not fixed in order to derive high-precision light curve 507 parameters. Thus, we used TAP to fit additional models to our data, with (1) limb dark508 ening as a free parameter and a / R ∗ held fixed, (2) limb darkening held fixed and a / R ∗ as 509 a free parameter, and (3) both limb darkening and a / R ∗ as free parameters. In all cases, 510 we fit a single radius ratio over all seven transits. For the models where limb darkening 511 was a free parameter (1 and 3), we derive linear and quadratic limb darkening coefficients 512 of (0.089 ± 0.018, -0.131 ± 0.018) and (0.090 ± 0.018, -0.130 ± 0.018). Compared to the fixed 513 values that we used, (0.0475, 0.3502), we find that the fitted linear coefficient is consistent 514 with the fixed coefficient within 3 σ , but the fitted quadratic coefficient differs from the fixed 515 coefficient by more than 26 σ . The derived value for a / R ∗ from model 2 (15.111 +0 . 081 -0 . 080 ) is 516 consistent with the fixed value we used (14.975) within 1.7 σ . For model 3, we found a / R ∗ = 517 15.410 ± 0.080, which differs from the fixed value by 5.5 σ . Despite differences between some 518 of the fixed and fitted parameters, we find that for all models, the derived radius ratios are 519 consistent within 1 σ . Specifically, the measured radius ratios for the three cases described 520 above are: (1) R p /R /star = 0.1174 ± 0.0015 when limb darkening is free and a / R ∗ is fixed, (2) 521 R p /R /star = 0.1157 ± 0.0013 when limb darkening is fixed and a / R ∗ is free, and (3) R p /R /star = 522 0.1175 ± 0.0013 when both limb darkening and a / R ∗ are free. Recall that our base model 523 yielded R p /R /star = 0.1158 ± 0.0013 when keeping both limb darkening and a / R ∗ fixed. Since 524 all radius ratios are consistent within 1 σ , this indicates that fixing limb darkening as well as 525 a / R ∗ did not significantly affect our results or conclusions. 526</text>
<text><location><page_18><loc_9><loc_25><loc_10><loc_26></location>527</text>
<section_header_level_1><location><page_18><loc_42><loc_25><loc_58><loc_27></location>6. Conclusions</section_header_level_1>
<text><location><page_18><loc_9><loc_11><loc_88><loc_23></location>In this paper, we presented results from seven transit observations of GJ 1214b acquired 528 in a narrow-band H 2 filter. Our analysis included a thorough technique for selecting reference 529 stars and incorporated light curve fits with red (correlated) noise. We measured a radius ratio 530 of 0.1158 ± 0.0013 when fitting the data from all seven transits together. This radius ratio is 531 consistent with previous K -band measurements, including those from Croll et al. (2011) and 532 de Mooij et al. (2012) which differ from ours by only 1.5 σ and 1.1 σ . We conclude that all 533</text>
<text><location><page_19><loc_9><loc_80><loc_88><loc_86></location>K -band data support a flat absorption spectrum for GJ 1214b, which suggests that either a 534 high mean molecular weight atmosphere or a H-rich atmosphere with a cloud or haze layer 535 is favored (Howe & Burrows 2012). 536</text>
<text><location><page_19><loc_9><loc_55><loc_88><loc_79></location>Since Valencia et al. (2013) find that the bulk amount of H/He in GJ 1214b's volatile 537 envelope is nonzero and could be as much as 7% by mass, we explored the scenario where GJ 538 1214b has a H 2 -rich envelope and heavy elements are sequestered below a cloud or aerosol 539 layer. After comparing models of a pure H 2 atmosphere with K -band observations, we find 540 that we cannot exclude the possibility of a H 2 -rich upper atmosphere. It is difficult to disen541 tangle different plausible atmosphere models in the K -band given the precision of available 542 data and that the temperature of the upper atmosphere is otherwise unconstrained. We sug543 gest that additional high-precision spectroscopic observations (from space) across the K -band 544 would be most useful. High-precision measurements at short optical wavelengths ( < 0.7 µ m) 545 would also be helpful in searching for evidence of Rayleigh scattering (e.g. de Mooij et al. 546 2013; Narita et al. 2013), which would support a (cloudy/hazy) H-rich atmosphere for GJ 547 1214b (Howe & Burrows 2012). 548</text>
<text><location><page_19><loc_9><loc_53><loc_10><loc_53></location>549</text>
<text><location><page_19><loc_9><loc_51><loc_10><loc_51></location>550</text>
<text><location><page_19><loc_9><loc_49><loc_10><loc_49></location>551</text>
<text><location><page_19><loc_9><loc_47><loc_10><loc_47></location>552</text>
<text><location><page_19><loc_9><loc_44><loc_10><loc_44></location>553</text>
<text><location><page_19><loc_9><loc_42><loc_10><loc_42></location>554</text>
<text><location><page_19><loc_9><loc_40><loc_10><loc_40></location>555</text>
<text><location><page_19><loc_9><loc_38><loc_10><loc_38></location>556</text>
<text><location><page_19><loc_9><loc_36><loc_10><loc_36></location>557</text>
<text><location><page_19><loc_9><loc_34><loc_10><loc_34></location>558</text>
<text><location><page_19><loc_12><loc_46><loc_88><loc_54></location>Finally, we explored variability due to H 2 in the stellar spectrum and investigated the effects of different systematics on our results, such as star spots, CCD non-linearity, and limitations in our knowledge of light curve parameters. Overall, we conclude that systematics did not significantly affect our photometry and that our results are robust to these effects.</text>
<text><location><page_19><loc_12><loc_33><loc_88><loc_45></location>While there is ample spectroscopic and photometric data available for this planet, much of the data comes from different instruments as well as analyses in addition to suffering from poor precision. This in itself introduces an additional systematic when comparing the data to atmosphere models. While beyond the scope of this paper, we conclude that a uniform analysis of all public data for GJ 1214b would be useful for establishing more robust limits on atmosphere models.</text>
<text><location><page_19><loc_9><loc_14><loc_88><loc_30></location>We are grateful to Mike Irwin and Mike Read for assisting us in accessing the WF559 CAM data and illumination tables. We thank Norio Narita for helpful discussions on GJ 560 1214b. We also thank Paul Wilson for providing data prior to publication. We acknowl561 edge the anonymous referee for helping us improve this paper. This research was supported 562 by NASA grants NNX10AI90G (Astrobiology: Exobiology & Evolutionary Biology) and 563 NNX11AC33G (Origins of Solar Systems) to EG. The United Kingdom Infrared Telescope 564 is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities 565 Council of the U.K. This research has made use of the VizieR catalogue access tool, CDS, 566</text>
<text><location><page_19><loc_9><loc_13><loc_10><loc_13></location>567</text>
<text><location><page_19><loc_12><loc_12><loc_28><loc_14></location>Strasbourg, France.</text>
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<table>
<location><page_23><loc_13><loc_45><loc_87><loc_63></location>
<caption>Table 1. Transit Observations</caption>
</table>
<table>
<location><page_24><loc_28><loc_44><loc_72><loc_59></location>
<caption>Table 2. Fixed Transit Model Parameters a</caption>
</table>
<table>
<location><page_25><loc_16><loc_42><loc_84><loc_61></location>
<caption>Table 3. Best-Fit Model Parameters</caption>
</table>
<text><location><page_25><loc_16><loc_35><loc_84><loc_40></location>( a ) The best-fit radius ratio from fitting all seven transits together is 0.1158 ± 0.0013. The best-fit radius ratios from fitting the seven transits separately are shown in the fifth column and are also shown in Figure 7. See text for further details.</text>
<figure>
<location><page_26><loc_17><loc_41><loc_83><loc_77></location>
<caption>Fig. 1.R p / R /star from published observations of GJ 1214b. The horizontal error bars on each point indicate the approximate bandpass for each observation. The data points are colorcoded according to the source they were retrieved from and are shown in order of publication date. Also shown are two atmosphere models from Howe & Burrows (2012) that best fit a majority of the data published by 2012. The light gray line is a model for a solar composition atmosphere with a tholin haze at 10-0.1 mbar composed of 0.1 µ m particles with a particle density of 100 cm -3 . The dark gray line is a model for an atmosphere with a composition of 1% H 2 O and 99% N 2 and with a tholin haze at 0.1-0.001 mbar composed of 0.01 µ m particles with a density of 10 6 cm -3 .</caption>
</figure>
<figure>
<location><page_27><loc_17><loc_35><loc_83><loc_70></location>
<caption>Fig. 2.- Transmission profiles of the narrow-band H 2 filter (solid line), broad-band K s filter (dashed line), and atmosphere above Mauna Kea Observatory at an airmass of 1 and with a water vapor column of 1.2 mm (solid gray line).</caption>
</figure>
<figure>
<location><page_28><loc_18><loc_46><loc_83><loc_81></location>
<caption>Fig. 3.- Infrared color-color diagram of stars in the vicinity of GJ 1214 (on the plane of the sky) and with K -band magnitudes between 8.282 and 10.782 ( K = 8.782 for GJ 1214). The horizontal and vertical black lines indicate the typical error in the colors, and the black arrow indicates the interstellar reddening vector. The dashed green curve is the 2MASS main-sequence locus from Stead & Hoare (2011), the solid green curve is a main-sequence locus for later type stars, and the dashed red curve is the 2MASS giant locus. GJ 1214 is marked with a small green diamond. The two regions marked by black rectangles are where some reference stars were selected based on the location of the main-sequence locus. The locations of the boxes were chosen to avoid the giant locus. The upper right corner of the selection box on the left side of the figure is located at the colors of a G5 III star, the bluest common giant. The twelve comparison stars that were used for relative photometry based on color and reduced proper motion criteria from L'epine & Gaidos (2011), their proximity to the main-sequence locus, and their location in the sky (i.e. in the WFCAM FOV) are marked with small blue diamonds. See text for further details.</caption>
</figure>
<figure>
<location><page_29><loc_22><loc_37><loc_78><loc_71></location>
<caption>Fig. 4.- Plots of the raw light curve (relative flux versus time) for the August 24, 2011 transit of GJ 1214b (top panel), along with parameters used for detrending the light curve (bottom panels). The centroid position and peak counts shown are for the target. The detrended light curve is plotted in Figure 5.</caption>
</figure>
<figure>
<location><page_30><loc_18><loc_37><loc_82><loc_71></location>
<caption>Fig. 5.- Detrended light curves for each of the seven observed transits of GJ 1214b. The light curves have been offset for clarity. Each light curve has been corrected against linear trends in the target centroid position, airmass, and the peak counts in the target (per pixel). See Figure 4 for an example of the parameters used to detrend the light curves.</caption>
</figure>
<figure>
<location><page_31><loc_18><loc_39><loc_82><loc_73></location>
<caption>Fig. 6.- Combined light curve (black points), best-fit model (solid red curve), and light curve residuals (offset black points) of transits of GJ 1214b. The light curve shown is a combination of the data from all seven transits shown in Figure 5. The solid red curve is a best-fit model based on fitting a single radius ratio over all transits. The light curve residuals after removing the model from the data are offset for clarity and have an rms of 2.0 × 10 -3 . See text for further details.</caption>
</figure>
<figure>
<location><page_32><loc_17><loc_37><loc_83><loc_73></location>
<caption>Fig. 7.- Best-fit radius ratios derived from TAP models fit to the data. We show the best-fit radius ratios and their corresponding 1 σ uncertainties from fitting the seven transits separately versus transit epoch based on the ephemeris from Berta et al. (2012). The solid line is the best-fit radius ratio from fitting all seven transits together (with the ± 1 σ uncertainties shown as dashed lines).</caption>
</figure>
<figure>
<location><page_33><loc_17><loc_45><loc_83><loc_81></location>
<caption>Fig. 8.R p / R /star from our analysis (red circle) compared to others published in the literature. The symbols are the same as in Figure 1: the blue triangle is from Croll et al. (2011), blue squares are from Bean et al. (2011), blue circles are from de Mooij et al. (2012), the green square is from Narita et al. (2012), and the yellow square is from Narita et al. (2013). Vertical error bars are one standard deviation. The horizontal error bars on each point indicate the approximate bandpass of the filter used for each observation. The solid black and gray curves are 400 and 1000 K pure H 2 atmosphere models. The models have been offset by a reference radius ratio, R 0 /R /star , derived from fitting the models to the data. The 400 and 1000 K models were found to have the lowest and highest χ 2 values [after comparing atmosphere models with different temperatures with the K -band data, excluding the outlying Croll et al. (2011) and de Mooij et al. (2012) K s -band data]. The 470 K 1% H 2 O and 99% N 2 plus haze atmosphere model from Howe & Burrows (2012) shown in Figure 1 as the dark gray curve is also shown here as a dashed gray curve.</caption>
</figure>
<figure>
<location><page_34><loc_17><loc_35><loc_83><loc_71></location>
<caption>Fig. 9.K -band spectrum of GJ 1214 from Rojas-Ayala et al. (2012). Pronounced stellar absorption lines are labeled. The profiles of the narrow-band H 2 filter and the K s filter are illustrated by the dashed curves.</caption>
</figure>
<figure>
<location><page_35><loc_16><loc_34><loc_84><loc_70></location>
<caption>Fig. 10.- The in-transit rms versus out-of-transit rms for each observed transit. The solid line illustrated equality between the in- and out-of-transit rms values.</caption>
</figure>
<figure>
<location><page_36><loc_18><loc_37><loc_82><loc_71></location>
<caption>Fig. 11.- Light curve for the August 24, 2011 transit of GJ 1214b. The intervals for the transit duration τ and the crossing time of a hypothetical bright spot t spot are marked with horizontal black lines. The vertical black lines indicate the depth of the planetary transit ( δ p ) and the change in the transit depth due to the anomalous feature (∆ δ ).</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "GJ 1214 is a nearby M dwarf star that hosts a transiting super-Earth-size planet, making this system an excellent target for atmospheric studies. Most studies find that the transmission spectrum of GJ 1214b is flat, which favors either a high mean molecular weight or cloudy/hazy hydrogen (H) rich atmosphere model. Photometry at short wavelengths ( < 0.7 µ m) and in the K -band can discriminate the most between these different atmosphere models for GJ 1214b, but current observations do not have sufficiently high precision. We present photometry of seven transits of GJ 1214b through a narrow K -band (2.141 µ m) filter with the Wide Field Camera on the 3.8 m United Kingdom Infrared Telescope. Our photometric precision is typically 1.7 × 10 -3 (for a single transit), comparable with other ground-based observations of GJ 1214b. We measure a planet-star radius ratio of 0.1158 ± 0.0013, which, along with other studies, also supports a flat transmission spectrum for GJ 1214b. Since this does not exclude a scenario where GJ 1214b has a H-rich envelope with heavy elements that are sequestered below a cloud/haze layer, we compare K -band observations with models of H 2 collisioninduced absorption in an atmosphere for a range of temperatures. While we find no evidence for deviation from a flat spectrum (slope s = 0.0016 ± 0.0038), an H 2 dominated upper atmosphere ( < 60 mbar) cannot be excluded. More precise observations at < 0.7 µ m and in the K -band as well as a uniform analysis of all published data would be useful for establishing more robust limits on atmosphere models for GJ 1214b. Subject headings: planetary systems - planets and satellites: atmospheres 5 techniques: photometric 6 7",
"pages": [
1,
2
]
},
{
"title": "NarrowK -Band Observations of the GJ 1214 System",
"content": "Knicole D. Col´on 1 , Eric Gaidos 1",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "To date, approximately 300 planets have been confirmed to transit their host star(s), 8 and the Kepler 1 mission has discovered over 3500 additional transiting planet candidates. 9 About 65 transiting planets (which have both a measured mass and radius) are considered 10 to be in either the 'super-Earth' (1 /lessorsimilar R /lessorsimilar 2 R ⊕ ) or 'mini-Neptune' (2 /lessorsimilar R /lessorsimilar 4 R ⊕ ) 11 regime, but less than a handful of these orbit nearby stars (Schneider et al. 2011). Planets 12 like GJ 1214b, a ∼ 2.7 R ⊕ transiting planet discovered around a nearby ∼ 0.2 R /circledot M dwarf 13 star by the MEarth ground-based transit survey (Charbonneau et al. 2009), are therefore of 14 great interest for understanding the difference between Earth-like planets, 'super-Earths', 15 and 'mini-Neptunes.' 16 Planets in the super-Earth/mini-Neptune regime occupy a location in the planetary 17 mass-radius diagram that allows for diverse interior compositions (Rogers & Seager 2010). 18 However, it may be possible to constrain the overall bulk composition of these planets by 19 characterizing the planet's atmosphere. Models suggest that GJ 1214b contains a significant 20 amount of hydrogen (H) and helium (He) based on its low mean density (approximately 21 one-third that of Earth; Miller-Ricci & Fortney 2010). Valencia et al. (2013) constrained 22 the fraction of H and He in GJ 1214b using a model of its interior and evolution. They 23 conclude that there is some amount of H/He present and that the bulk amount of H/He 24 may be up to 7% by mass (similar to Neptune). While this is only a small fraction, it 25 suggests that GJ 1214b likely has some amount of H/He in its atmosphere. Since H has a 26 large scale height due to its low molecular weight, even with a small amount of H in the 27 atmosphere the upper atmosphere (i.e. a small distance above the homopause) can easily 28 be H-dominated (Pierrehumbert 2010). Thus, if GJ 1214b is differentiated, then ice and 29 rock will be concentrated in the interior and the concentration of H/He in the surrounding 30 atmosphere could be much higher than 7%. 31 Many groups have studied its atmosphere by transmission spectroscopy or spectropho32 tometry (e.g., Bean et al. 2010, 2011; Berta et al. 2011; Croll et al. 2011; Crossfield et al. 33 2011; D'esert et al. 2011; Berta et al. 2012; de Mooij et al. 2012; Murgas et al. 2012; Narita et al. 34 2012; Fraine et al. 2013; de Mooij et al. 2013; Narita et al. 2013; Teske et al. 2013) while 35 others have continued improving models of its atmosphere (e.g., Benneke & Seager 2012; 36 Heng & Kopparla 2012; Howe & Burrows 2012; Menou 2012; Miller-Ricci Kempton et al. 37 2012; Morley et al. 2013; Valencia et al. 2013). Figure 1 presents measurements of the 38 planet-star radius ratio (or simply, the radius ratio, R p / R /star ) of GJ 1214b, along with two 39 representative atmosphere models from Howe & Burrows (2012). GJ 1214b has a largely 40 to date and (2) test if there is any spectral structure in the K -band that was missed by 78 the broad-band observations. Specifically, we (1) compared published K -band data to test 79 if the disagreement between K -band measurements is the result of subtle differences in the 80 bandpasses used convolved with not-so-subtle differences in the spectrum of the planet and 81 telluric absorption/emission that were not previously appreciated, and (2) compared K -82 band data with a model of H 2 collision-induced absorption in an atmosphere for a range of 83 84 We describe our observations and data reduction in Section 2. In Section 3, we present 85 our light curve analysis and models. We present our results in Section 4, and in Section 5 we 86 compare our results with published K -band observations, compare the data with models of a 87 H-rich atmosphere, investigate variability in the stellar spectrum due to H 2 , and discuss the 88 effects of different systematics on our results. In Section 6, we summarize our conclusions 89 90 91",
"pages": [
2,
4
]
},
{
"title": "2. Observations and Data Reduction",
"content": "We acquired photometry of seven transits of GJ 1214b between August 2011 and July 92 2012 with the Wide Field CAMera (WFCAM) on the 3.8 m United Kingdom InfraRed 93 Telescope (UKIRT) (Casali et al. 2001), located at the Mauna Kea Observatory in Hawaii. 94 WFCAM is a near-infrared wide field imager consisting of 4 Rockwell Hawaii-II (HgCdTe 95 2048 × 2048) 0.4 arcsec pixel arrays. Each individual camera covers a field of 13.65 ' × 13.65 ' , 96 and the total field of view (FOV) is 0.75 square degrees. A narrow-band filter centered on 97 the fundamental H 2 vibrational line [S(1) ν = 1 → 0] at 2.141 µ m (FWHM = 0.021 µ m) was 98 used for all observations. This filter probes wavelengths at which previous observations are 99 in disagreement (Section 1) and minimizes systematic variations from Earth's atmosphere. 100 In Figure 2, we compare the transmission profile of the H 2 filter to the broad-band K s 101 filter [the same filter used by Croll et al. (2011) on the Wide-field InfraRed Camera on the 102 Canada-France-Hawaii Telescope] and the atmosphere above Mauna Kea Observatory (at 103 an airmass of 1 and with a water vapor column of 1.2 mm). 2 Because it is so narrow, the H 2 104 filter is an optimal filter to avoid atmospheric effects (compared to the broader K s filter). 105 In Table 1, we present details of each transit observation. All observations were per106 formed in service mode. An exposure time of 60 s was used for each integration, with a 107 typical dead time between exposures of ∼ 2-3 s. The observation epochs (at mid-exposure) 108 were extracted from the FITS header for each image. The telescope was intentionally de109 focused to avoid saturation and to spread the stellar PSF over many pixels to minimize 110 error from an imperfect flat-field correction of the detector response. Due to the amount 111 of defocus, the telescope auto-guider was not able to function properly. As a result, a drift 112 of 45-52 pixels in the position of the target centroid occurred during the 2011 observations. 113 For the 2012 observations, the telescope operator routinely adjusted the guider to keep the 114 defocused guide star centered in the guider acquisition window. A centroid drift of less than 115 9 pixels was maintained during the 2012 observations. 116 117 118 119 120 121 122 We reduced all images using software written in GDL. 3 While all images taken with WFCAM are processed through a pipeline operated by the Cambridge Astronomical Survey Unit (CASU), we opted to process our images separately to ensure accurate calibration and photometry that was as precise as possible. The procedures were illumination correction, dark current correction, and flat fielding prior to performing circular aperture photometry, sky subtraction, and finally a radial distortion correction. The illumination correction rectified each image for residual systematics, which are most 123 likely caused by either low-level non-linearity in the detectors, scattered light in the camera, 124 and/or spatially dependent PSF corrections. 4 Illumination correction tables are measured 125 monthly as a function of spatial location in the array. Since illumination correction tables 126 are only available for broad-band filters, we used the K -band tables as a proxy for the H 2 127 filter. For the dark current and flat-field correction, we used darks taken the night of each 128 observation and monthly twilight flats taken with the H 2 filter. The appropriate flats from 129 a given month were used. Circular aperture photometry was performed with radii = 8, 10, 130 12, 14, 16, 18 pixels (3.2, 4.0, 4.8, 5.6, 6.4, 7.2 arcsec). In our analysis we ultimately used 131 the aperture that gave the best photometric performance (Section 3.2). An annulus of 25 132 30 pixels (10 - 12 arcsec) was used for sky subtraction. 133 Finally, a radial distortion correction was applied to the sky-subtracted flux measured 134 within a given aperture to account for non-negligible field distortion in WFCAM, a result of 135 its extremely large FOV. Specifically, photometry of sources near the edge of the FOV have 136 a systematic error of up to 0.02 mag. We computed the corrected flux, F corr , from 137 where F is the sky-subtracted flux measured within a given aperture, k 3 is the coefficient 138 of the third order polynomial term in the radial distortion equation and is approximately -60 139 radian -2 in the K -band (Hodgkin et al. 2009, and references therein), and r is the distance 140 of a star relative to the center of the optical system in radians. The corrected flux for the 141 target and each comparison star was used to generate the light curves (Section 3). 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156",
"pages": [
4,
5,
6
]
},
{
"title": "3.1. Selection of Comparison Stars",
"content": "Thanks to the large FOV covered by WFCAM, we could select among numerous comparison stars for relative photometry. We selected comparison stars that are of similar brightness to GJ 1214 and that do not appear to be intrinsically variable. We also used colors to select dwarf stars over giants and that are as close as possible to GJ 1214 in spectral type. We downloaded J , H , and K magnitudes from the 2MASS catalog (Skrutskie et al. 2006) and proper motions from the PPMXL catalog (Roeser et al. 2010) for all stars in the vicinity (within 30 arcmin) of GJ 1214. We also downloaded V magnitudes from the Fourth U.S. Naval Observatory CCD Astrograph Catalog (Zacharias et al. 2013). We imposed a magnitude cut in K -band so that the reference stars were no more than 0.5 mag brighter and 2 mag fainter than GJ 1214 ( K = 8.782). Figure 3 is a color-color diagram of all stars that meet our magnitude criteria. Not all these stars are actually located in the FOV, since WFCAMhas four cameras, and there are gaps of 12.83 arcmin between the different cameras. We selected reference stars from those shown in Figure 3 based on three different crite157 ria. We first selected eight stars that were more likely to be M dwarf stars based on color 158 criteria from L'epine & Gaidos (2011). Additional reference stars were selected based on 159 their proximity to a 2MASS main-sequence locus from Stead & Hoare (2011) and a locus for 160 K7 - M9.5 spectral types (Cutri et al. 2003). 5 Specifically, we selected stars in two regions 161 of color space, marked by the two boxes in Figure 3. These regions were also selected to 162 avoid the giant locus. 6 We identified three additional stars that fit this color criteria (and 163 that also were in the FOV). Finally, we computed the magnitude of the proper motion for 164 each star shown in Figure 3 (using proper motions from the PPMXL catalog). Following 165 L'epine & Gaidos (2011), we computed the reduced proper motion, H V , as 166 We then applied the constraint from L'epine & Gaidos (2011) to separate M dwarfs from 167 giants based on 168 We found that all stars in Figure 3 met this criterion. Therefore, to maximize the 169 number of optimal comparison stars, we chose to select additional stars with the highest 170 proper motions of the sample. Of the five stars with the highest proper motions (excluding 171 GJ 1214), only one was actually located within our FOV (i.e. not in a gap between cameras). 172 We added this star to our reference ensemble, bringing the total number of comparison stars 173 to 12. 174 A preliminary visual examination of the light curve of each reference star (the flux of 175 a given reference star divided by the total flux of the remaining reference stars) revealed 176 that four of the twelve stars were potentially variable (i.e. the light curves displayed possible 177 periodic fluctuations). For completeness, we considered all 12 reference stars when generating 178 light curves for GJ 1214. The number of reference stars ultimately included in the light curve 179 analysis varied between two and ten depending on which combination of references produced 180 the lowest scatter in each set of baseline (out-of-transit) data. 181 182",
"pages": [
6,
7
]
},
{
"title": "3.2. Generating the Light Curves",
"content": "Light curves were generated from each data set by computing the relative flux ratio, i.e. 183 the target signal divided by the total weighted signal of an ensemble of reference stars. Due 184 to varying weather conditions and the intrinsic variability of some of the reference stars, the 185 reference ensemble was composed of different stars for each set of observations. To determine 186 which reference stars would be included in each ensemble, the rms of each reference star's 187 signal was compared to an imposed photometric precision threshold (set to slightly different 188 values for each night to include an adequate number of reference stars, with a typical value 189 of 8 × 10 -3 ). The stars that had an rms that exceeded the threshold were excluded from the 190 reference signal, and the remaining stars had their signals weighted (based on the rms of 191 each individual reference signal) when computing the combined ensemble reference signal. 192 Normalized light curves were generated for each photometric aperture by dividing the 193 relative flux ratio by the median relative flux ratio measured in the out-of-transit data. We 194 considered the resulting out-of-transit rms scatter as measured in each aperture and for each 195 night of observations, and we chose the aperture that gave the smallest scatter in a given 196 night. 197 Each light curve was then regressed against target centroid position, airmass, and peak 198 counts (per pixel) and a linear trend removed. In Figure 4, we show the light curve from 199 August 24, 2011 prior to detrending, along with the parameters used for detrending. This 200 light curve has a notable negative deviation after mid-transit, which we discuss in Section 201 5.4. We also tested correcting against variations in the absolute flux of the target and the 202 variance of the flux of the target, but we found that these additional corrections resulted in 203 a negligible change in the flux ratios. 204 We discarded some June 16, 2012 data due to an incorrect defocus setting and saturation 205 (Section 2), as well as some points that were unexplained extreme outliers ( > 3 σ from the 206 mean of either the baseline data or the data at the bottom of the transit). We discarded 207 one data point from the August 24, 2011 observations and 28 points from the June 16, 2012 208 observations. We then fit transit models to the light curve data, as described in the next 209 section. 210 211",
"pages": [
7,
8
]
},
{
"title": "3.3. Transit Models",
"content": "To measure R p / R /star we fit transit models to our seven transit light curves using the 212 Transit Analysis Package (TAP), a publicly available IDL code (Gazak et al. 2012). 7 TAP 213 fits limb-darkened transit light curves using EXOFAST (Eastman et al. 2013) to calculate 214 the model of Mandel & Agol (2002), along with a combination of Bayesian and Markov 215 Chain Monte Carlo (MCMC) techniques. TAP also employs a wavelet-based likelihood 216 function (Carter & Winn 2009) to robustly estimate parameter uncertainties by fitting both 217 uncorrelated ('white') and correlated ('red') noise. A quadratic limb darkening law was 218 used, so in the models described below, linear and quadratic ( µ 1 and µ 2 ) limb darkening 219 coefficients are included. 220 221 222 223 224 225 We modeled all seven transits simultaneously, using five MCMC chains with lengths of 100,000 links each, and keeping the following parameters fixed at a single value for all transits: orbital period ( P ), inclination ( i ), scaled semi-major axis ( a / R ∗ ), and the limb darkening coefficients ( µ 1 and µ 2 ). (We assumed a circular orbit.) The mid-transit time, as well as the white and red noise, was individually fitted for each transit. To obtain R p / R ∗ we (i) fit individual transit light curves and (ii) fit the combined light curves of all transits. 226 The fixed parameters we used are given in Table 2. The K -band limb darkening parameters 227 are from Claret & Bloemen (2011) for a star with T eff = 3000 K and log( g ) = 5 and are the 228 same as those used by Narita et al. (2012). 229 In Figure 5, we present the seven individual light curves. Figure 6 shows the combined 230 light curve (composed by phasing each of the individual light curves) and the best-fit model 231 for case (ii), where we fit a single R p / R ∗ to all transits. We base our primary conclusions 232 on the results from this case. However, we discuss the individually measured radius ratios 233 in Section 4 and the effects of fixing versus fitting the limb darkening parameters and a / R ∗ 234 in further detail in Section 5.5. 235 236 237 238 239 240 241 242",
"pages": [
8,
9
]
},
{
"title": "4. Results",
"content": "In Table 3 we present the best-fit radius ratio, mid-transit time, white noise, and red noise measured for each individual transit. We also include the best-fit radius ratio based on fitting all seven transits together. Since TAP fits for correlated (red) noise, the uncertainties on the fitted parameters should be conservative. This was also pointed out by Teske et al. (2013), who compared results from TAP with results from other light curve fitting software. The correlated noise is discussed in further detail in Section 5.4. We find that all transit times deviate from the predicted ephemeris (from Berta et al. 243 2012) by less than 29 s and are consistent with a linear ephemeris. We also find that the 244 fitted values for the white noise are generally consistent with the rms scatter in each transit 245 light curve (Section 5.3), which suggests that the red noise may be smaller than the values 246 measured by TAP. We present the individual best-fit radius ratios in Figure 7. They are 247 all consistent with the combined best-fit radius ratio (0.1158 ± 0.0013). The radius ratio 248 measured for the fifth transit observation (June 16, 2012) has the largest uncertainty, but it 249 is still consistent with those measured from the other individual transits. The photometry of 250 this particular transit was of inferior quality than the other transit observations (Figure 5), 251 likely due in part to the presence of mixed thin and thick clouds throughout the observations. 252 In Figure 8, we present the results of our analysis along with other K -band measurements 253 of the radius ratio for GJ 1214b. We find that our best-fit radius ratio is consistent with 254 the majority of other published values and supports a flat absorption spectrum for the 255 atmosphere of GJ 1214b with a slope s = 0.0016 ± 0.0038 [excluding 'outlying' points from 256 Croll et al. (2011) and de Mooij et al. (2012)]. We discuss this result further in Section 5.1. 257 259",
"pages": [
9,
10
]
},
{
"title": "5.1. Comparison of K -Band Observations",
"content": "Our measurements are inconsistent with the conclusion of Croll et al. (2011), that GJ 260 1214b has a low mean molecular weight atmosphere. One K s -band measurement from 261 de Mooij et al. (2012) also supports the data from Croll et al. (2011) and hints at a de262 viation from a flat spectrum (Figure 8). However, the points from Croll et al. (2011) and 263 de Mooij et al. (2012) only differ from our derived radius ratio by 1.5 σ and 1.1 σ , suggesting 264 that their data is in fact consistent with other published K -band observations. Differences 265 between the transmission measurements could be a result of the use of different filters (i.e. 266 our narrow-band H 2 filter versus their broad-band K s filter) and the correspondingly dif267 ferent wavelength coverage by the various groups. Our narrow-band filter probes a small 268 wavelength range (FWHM = 0.021 µ m) and is free of telluric features (Figure 2). The K s 269 transmission curve shown in Figure 2 is for the filter specifically used by Croll et al. (2011) 270 and contains some telluric absorption features. While telluric effects should be largely re271 moved by relative photometry, the choice of reference stars and their locations on the sky 272 (relative to the target) can result in imperfect removal of telluric features. The K s -band also 273 contains some stellar lines, and we discuss this further in Section 5.3. Broad-band photome274 try therefore can be subject to additional systematics and yield less accurate measurements 275 than narrow-band photometry. Ultimately, we find that K -band observations continue to 276 support a high molecular weight atmosphere or a H-rich atmosphere with a cloud or haze 277 layer over a cloud/haze-free H-rich atmosphere. 278 279",
"pages": [
10
]
},
{
"title": "5.2. Constraints on a Cloudy/Hazy H-Rich Atmosphere",
"content": "In the previous section, we concluded that K -band measurements are consistent with 280 a flat transmission spectrum, which favors either a high molecular weight atmosphere or 281 a H-rich atmosphere with a cloud or haze layer (Howe & Burrows 2012). However, the 282 available K -band data does not have sufficiently high precision to discern between these two 283 atmosphere models. Published B -band data (e.g., de Mooij et al. 2012, 2013; Narita et al. 284 2013; Teske et al. 2013) cannot definitively discriminate between these two scenarios either. 285 In this section, we assume that the atmosphere is H-dominated [following the results from 286 Valencia et al. (2013)], and we consider what constraints can be placed on a cloudy/hazy 287 H-rich atmosphere. 288 For a H-rich atmosphere with relatively high number densities where collisions are fre289 quent, collision-induced absorption (CIA) is a dominant source of opacity (Borysow 2002, 290 and references therein). Here, we consider a H-rich upper atmosphere that includes an 291 opaque cloud or haze layer at some low altitude/high pressure. We assume that any molec292 ular absorbers in the atmosphere are confined beneath a cloud/haze layer, and that the 293 atmosphere above the clouds/haze is metal-free as a result of some (photo)chemical or mix294 ing boundary (chemopause or homopause). If the H 2 envelope is sufficiently dense, it could 295 be detected by CIA in high-precision K -band spectrophotometry. This depends on tem296 perature, so the presence or absence of CIA features is also a constraint on temperature 297 (provided other conditions are met; see below). The equilibrium temperature of GJ 1214b 298 is ∼ 500 K (Charbonneau et al. 2009), but the upper atmosphere could be much hotter due 299 to XUV heating from the star (Lammer et al. 2013). 300 A condition for this model to hold is that the CIA opacity of H 2 dominates above the 301 pressure altitude of any cloud/haze layer present in the atmosphere (i.e. τ H 2 /greaterorsimilar 1 above the 302 chemopause or homopause). Under this condition, we computed the minimum pressure ( P 0 ) 303 at the top of a cloud/haze layer located in an isothermal atmosphere using the following 304 equation: 305 H is the scale height of the atmosphere, defined as 306 k B is the Boltzmann constant, T is the atmospheric temperature, µ is the mean molec307 ular weight of the atmosphere, and g is the planet's surface gravity. We assumed T = 500 308 K, and we set the reference planetary radius R 0 = 2.68 R ⊕ . We used H 2 -H 2 CIA opacities 309 ( σ ) from Borysow (2002) and computed the scale height H using a mean molecular weight 310 µ = 2 and a surface gravity g = 8.95 m s -2 . From this, we computed the minimum pressure 311 P 0 at the top of the cloud or aerosol layer to be 60 mbar (at λ ∼ 2.14 µ m). This minimum 312 pressure falls within the pressure range of 0.001 -100 mbar considered by Howe & Burrows 313 (2012). 314 We then calculated the effective R p / R /star versus wavelength, following a similar procedure 315 as in Howe & Burrows (2012): 316 317 318 319 320 321 322 323 324 325 326 327 328 Here, R 0 is the radius at the wavelength where σ = σ 0 = 2.78 × 10 -6 cm -1 amagat -2 at λ ∼ 2.14 µ m [from the 500 K table from Borysow (2002)]. Because H 2 absorption is collisionally-induced, it depends on the square of the number density ( n 2 , where n = number density of absorbers) and this leads to an additional factor of 1/2 in Eqn. 6 [compared to the equation defined in Howe & Burrows (2012)]. Under the same conditions assumed above and using R /star =0.211 R /circledot (Charbonneau et al. 2009), we calculated the radius ratio for temperatures of 400, 500, 600, 700, 800, 900, and 1000 K. We compared the observed radius ratios to the predicted radius ratios by calculating χ 2 and identified the best-fit model (based on the minimum χ 2 ). Although the K s -band data from Croll et al. (2011) and de Mooij et al. (2012) are marginally consistent with other K -band data within measurement uncertainties, we still considered them to be unexplained outliers and excluded them from the analysis (Section 5.1). We find the 400 K model had the smallest χ 2 , while the 1000 K model yielded the largest 329 χ 2 . We show the 400 and 1000 K models in Figure 8. As the temperature increases, the 330 scale height increases, leading to additional absorption and a larger apparent planet radius; 331 however, the features also become washed out at higher temperatures. We computed ∆ χ 2 332 (relative to the minimum χ 2 at 400 K) and found that the deviation between the data and 333 models increases with increasing temperature. However, ∆ χ 2 between the 400 and 1000 334 K models is only ∼ 0.72, since χ 2 is just 2 -3 for all models. We conclude that from the 335 available data, we cannot exclude higher temperature atmospheres ( T > 400 K) with any 336 confidence ( p < 0.01). 337 Considering the capabilities of future missions like the James Webb Space Telescope 338 ( JWST ) for high precision infrared spectroscopy, the atmosphere models were compared to 339 both the real data (using the actual measurement uncertainties) as well as an artificial data 340 set, consisting of the actual measurements with reduced errors. We defined artificial errors 341 so that the median error over the K -band data = 1 × 10 -4 [based on the precision achieved 342 by Fraine et al. (2013) from Spitzer measurements of GJ 1214b at 4.5 µ m]. For comparison, 343 the median value of the actual errors is 9.9 × 10 -4 . Based on the artificial high-precision 344 data, we again find that the 400 K model has the smallest χ 2 , and that ∆ χ 2 between the 345 400 and 1000 K models is 70.2, which would allow us to exclude atmospheres with T ≥ 800 346 K with > 99.7% confidence (3 σ ) assuming that we had such high precision data. We find 347 that a temperature of ≤ 400 K is preferred for a pure H 2 upper atmosphere, but significantly 348 higher precision data (as well as more data in general) is needed to confidently exclude higher 349 temperature atmospheres. 350 Ultimately, while we find no evidence for deviation from a flat spectrum, a thin upper 351 atmosphere ( ≤ 60 mbar) dominated by H 2 cannot be excluded. The possibility also remains 352 that there is simply no (or very little) H in the (upper) atmosphere. 353 354",
"pages": [
10,
11,
12,
13
]
},
{
"title": "5.3. Variability in the Stellar Spectrum",
"content": "Fluorescent H 2 emission lines have been observed in the spectra of four planet-hosting 355 M dwarf stars, but not in GJ 1214 (France et al. 2013). These lines are produced by pho356 toexcitation by Ly α photons, and their detection indicates the presence of a 2000 -4000 K 357 molecular gas (France et al. 2013, and references therein). The H 2 filter is designed specifi358 cally to observe such lines, making it possible for us to also probe the stellar atmosphere with 359 our observations. While France et al. (2013) do not detect Ly α emission in GJ 1214, they 360 speculate that this is because the neutral H in GJ 1214's atmosphere is instead contained 361 in H 2 rather than H. Notably, GJ 1214 is the coolest star ( T eff ∼ 3250 K; Anglada-Escud'e 362 et al. 2013) in their sample, and H recombines at cool temperatures, so it is plausible that 363 molecular H is present in GJ 1214's atmosphere. This motivated us to look for H 2 emission 364 in GJ 1214. 365 We examined a K -band spectrum of GJ 1214 from Rojas-Ayala et al. (2012), shown in 366 Figure 9. We find no evidence of an H 2 emission feature around ∼ 2.14 µ m in GJ 1214's 367 spectrum. The H 2 line is close to the Brackett γ line (2.16 µ m), which is produced in T 368 Tauri stars by recombining magnetospheric gas (e.g., Hamann et al. 1988). However, we see 369 no evidence of the Brackett γ line in the spectrum of GJ 1214. Indeed, the H 2 bandpass is 370 free of any obvious stellar lines. The K s filter does include several stellar absorption lines, 371 but assuming that these features are only present in the star (and are not telluric) and that 372 they do not vary, these will not affect the photometry. 373 Due to the lower temperatures H 2 may predominate in star spots and may play an im374 portant role in the formation and evolution of spots (e.g., Jaeggli et al. 2012). Therefore, we 375 also looked for evidence of H 2 variability, or patchiness in the stellar disk in the H 2 bandpass 376 (either bright emission or dark absorption) which produces variability when transited by the 377 planet (i.e. 'H 2 -spots'). 8 We compared observations taken in-transit (when the time-varying 378 part of the star is blocked by the planet's disk) with those taken out-of-transit (when only 379 the star light is visible). Specifically, we compared the rms scatter (of the light curve resid380 uals, computed by subtracting the best-fit model from the data; Section 3.3) between the 381 in-transit and out-of-transit windows for each of the seven individual transits. The resulting 382 rms values are presented in Figure 10. We computed r 2 = 0.80 between the in- and out-of383 transit rms, which indicates a strong linear correlation between the two parameters. That 384 a strong linear correlation exists suggests that the in- and out-of-transit measurements are 385 dominated by the same source of variation, i.e. photometric errors as opposed to a source of 386 variation associated with the transit alone. From this, we conclude that if H 2 is present and 387 absorbing or emitting in the stellar and/or planetary atmosphere, it is either not variable 388 or its variations are not detectable in our data. It is likely that the vibrational transitions 389 such as the 1-0 S(1) transition are too weak to be seen in a standard K -band spectrum (but 390 instead might be seen through a linearly polarized spectrum; White & Kuhn 2011). 391 The H 2 1-0 S(1) line is also used to study molecular outflows from protostars (e.g., 392 Garcia Lopez et al. 2013). We considered the possibility of detecting H 2 'outflows' from 393 the atmosphere of the planet (driven by UV heating), but we see no persistent deviations 394 from the standard transit light curve models that require explanation by a shock or wind 395 (i.e. from some massive atmospheric escape from the planet). 396 397",
"pages": [
13,
14
]
},
{
"title": "5.4. Effects of Systematics",
"content": "Systematics, whether astrophysical or instrumental in nature, can significantly affect 398 the derivation of light curve parameters. In this section, we describe potential sources of 399 systematics in our light curves and the effect they have on our measured radius ratios. 400 401",
"pages": [
14
]
},
{
"title": "5.4.1. Astrophysical Sources",
"content": "Intrinsic variability of the host star can affect measurements of the radius ratio, and GJ 402 1214 has been shown to be variable at red ( ∼ 0.8 µ m) wavelengths at the 1-2% level over ∼ 403 1-2 year timescales (Carter et al. 2011). While we see no obvious visual evidence of stellar 404 flares or star spots 9 in our light curves (Figure 5), in principle it is possible to use the radius 405 ratios measured for the individual transits to estimate how much the spot coverage changed 406 with time. Theoretically, the shallowest observed transit should correspond to the stellar 407 surface being nearly free of spots, or at least having minimal spot coverage. As presented 408 in Table 3 and shown in Figure 7, the measured radius ratios do not vary significantly with 409 transit epoch. Between the photometric quality of the data and the magnitude of the errors 410 derived for the radius ratios, we cannot robustly identify a minimum epoch of stellar activity 411 based on the measured radius ratios. Based on the rms values presented in Figure 10, our 412 data suggest variability at a level of ∼ 1.5 × 10 -3 (albeit only on hour-long timescales). We 413 also find that GJ 1214 is variable in the K -band at a level of ∼ 0.5% over longer timescales 414 (from August 2011 to July 2012 or ∼ 1 year), based on the target-to-reference flux ratio for 415 the most photometrically stable reference star in our sample. This is consistent with the 416 results from Carter et al. (2011), since stellar variability in our passband should be less than 417 the 1-2% measured by Carter et al. (2011) due to observing in a redder passband. Although 418 such variability does affect the measurement of radius ratios, since our radius ratios did not 419 vary significantly (within errors), we consider stellar variability to have a negligible effect on 420 our conclusions. 421 There is the possibility that spot-crossing events did occur in one or more of our observed 422 transits, but we simply cannot identify them by eye because they cause brightenings at or 423 below the level of our photometric precision. If we consider the red noise (Table 3) as a 424 measure of the effects of spot-crossing events, we find that the August 24, 2011 and June 425 16, 2012 transits have the highest levels of red noise. The June 16, 2012 transit had poorer 426 weather conditions, which is likely the cause of the higher red noise in that light curve, 427 as no evidence of a spot-crossing event is seen in that light curve. However, the August 428 24, 2011 transit, which had photometric conditions, does have an anomalous feature in its 429 light curve (Figure 11) where the transit depth appears to increase significantly immediately 430 before egress. The anomalous feature in the August 24, 2011 transit does not appear to be a 431 (dark) spot crossing event, since the transit depth before the feature occurred is consistent 432 (i.e. within 1 σ ) with the transit depth measured for the combined light curve (over all seven 433 transits). This suggests that the true transit depth should be based on the first part of the 434 transit, rather than the first part of the transit being a spot-crossing event. 435 To determine if this anomalous feature was produced by reference star variability, we 436 constructed alternative light curves using different reference stars. We found that the feature 437 was present in all cases. We also searched the literature to see if any other observations of 438 this specific transit had been published, but we only found an observation of the transit 439 following this one by Harpsøe et al. (2013). Their light curves do not appear to contain any 440 particularly anomalous features, which supports the idea of minimal (or at least un-transited) 441 spots at the time of the observations. 442 We also considered that the feature may instead be due to the planet passing over a 443 bright spot on the star, which would result in an increase in the transit depth, consistent 444 with our observations. However, the presence of a bright spot will also cause a decrease in 445 the unocculted transit depth. As illustrated in Figure 7, the transit depths measured for 446 all transits are consistent within 1 σ . This suggests that if the anomalous feature is a result 447 of a spot-crossing event, the spot is sufficiently compact/faint that it does not produce an 448 unocculted spot effect that we would detect. We do not find evidence of spot-crossing events 449 in our other transit observations, but we explore the bright spot hypothesis in more detail 450 here. 451 Assuming a circular spot with radius R spot , we estimated the size of such a spot based 452 on the duration of the spot crossing, 453 The transit duration ( τ ) is 52.73 min (Carter et al. 2011), and the spot-crossing time 454 ( t spot ) was estimated from the light curve to be 15 min (Figure 11). Using an impact 455 parameter ( b ) of 0.28 (Bean et al. 2011), we computed a spot-star radius ratio of 0.27 (notably 456 over two times larger than R p / R /star ). We then computed how much the transit depth would 457 increase due to a bright spot (relative to the no spot case) from 458 This equation is applicable because our hypothetical spot is much larger than the planet. 459 The increase in the transit depth during the anomalous event relative to the depth prior to 460 the anomalous event is ∆ δ = 0.0059 (Figure 11). The transit depth ( δ p ) is 0.0135. B is 461 the brightness enhancement of the spot relative to the rest of the stellar disk, which we 462 computed based on the relative depth of the anomalous feature, or B = ∆ δ / δ p = 0.44. 463 From this, we calculate δ f = 0.033. This translates to a change in the transit depth due 464 to the unocculted bright spot of δ f × δ p = 4.4 × 10 -4 , which is much smaller than our typical 465 photometric precision of 1.7 × 10 -3 (for a single transit). It is also consistent with GJ 1214 466 being variable on the order of 1-2% (Carter et al. 2011). We conclude that a single transient 467 bright spot about twice the size of the planet's disk could explain the transit of August 24, 468 2011. Regardless of the source of the feature, TAP recognized it as a systematic and derived 469 a radius ratio that is consistent with the radius ratios measured from the other transits 470 (Figure 7). 471",
"pages": [
14,
15,
16
]
},
{
"title": "5.4.2. Instrumental Sources",
"content": "Besides astrophysical systematics, we also considered the effect of instrumental system473 atics such as a nonlinear detector on our measured radius ratios. The WFCAM detector is 474 linear to < 1% up to about 40,000 counts per pixel. 10 To ensure that we avoided the defined 475 non-linear regime, we measured the peak counts in the target and each reference star. We 476 found that all stars remained below ∼ 22,000 counts (per pixel) in all observations. However, 477 the possibility remains that lower non-linearity is present at lower counts (Section 2). To 478 check this, we derived a 'variance coefficient' ( k ) for each reference star and for each night 479 based on the following equation, 480 Here, F observed is the normalized flux ratio we measured for each reference star, F actual 481 is what the ideal flux ratio should be in the absence of systematics (i.e. F actual ≡ 1), k is 482 defined as the variance coefficient, and F 2 observed is the variance of the observed flux ratio. 483 After deriving k for each reference star and for each night, we found that on most nights, 484 k << 1.1 × 10 -3 . Thus, we conclude that the effect of non-linearity is smaller than our 485 photometric precision (typically 1.7 × 10 -3 for a single transit). Only on two nights did the 486 flux ratios have significant deviations: during the fifth transit (June 16, 2012) and sixth 487 transit (June 27, 2012). That the fifth transit shows signs of non-linearity is consistent with 488 the observations, since the stellar image was insufficiently defocused to avoid high counts. 489 We have no explanation as to why the sixth transit is potentially affected by non-linearity. 490 Given that the measured transit depth is consistent with those measured from the other 491 individual transits, we conclude that any low-level non-linearity that might be present and 492 a source of systematics has a minimal effect on our photometry. 493 494",
"pages": [
17
]
},
{
"title": "5.5. Treatment of Transit Model Parameters",
"content": "Finally, we considered how limitations in our knowledge of stellar properties affect the 495 derived light curve parameters. In particular, there is a degeneracy between limb darkening 496 and a / R ∗ , which in turn affects estimates of the stellar density, orbital eccentricity, and 497 impact parameter. Furthermore, as Berta et al. (2012) point out, inaccurate treatment of 498 limb darkening could introduce false absorption features into the transmission spectrum. 499 Therefore, the accurate treatment of limb darkening is critical for precisely measuring light 500 curve parameters. 501 In the analyses described above, we held limb darkening coefficients and a / R ∗ fixed. 502 This decision was based on the fact that GJ 1214b's transit duration is only 52 min and 503 the ingress/egress events last 6 min, compared to our cadence of ∼ 1 min. Having a rela504 tively small number of data points during ingress and egress makes it difficult to accurately 505 fit the limb darkening. However, Csizmadia et al. (2013) argue that stellar limb darken506 ing parameters should be fitted and not fixed in order to derive high-precision light curve 507 parameters. Thus, we used TAP to fit additional models to our data, with (1) limb dark508 ening as a free parameter and a / R ∗ held fixed, (2) limb darkening held fixed and a / R ∗ as 509 a free parameter, and (3) both limb darkening and a / R ∗ as free parameters. In all cases, 510 we fit a single radius ratio over all seven transits. For the models where limb darkening 511 was a free parameter (1 and 3), we derive linear and quadratic limb darkening coefficients 512 of (0.089 ± 0.018, -0.131 ± 0.018) and (0.090 ± 0.018, -0.130 ± 0.018). Compared to the fixed 513 values that we used, (0.0475, 0.3502), we find that the fitted linear coefficient is consistent 514 with the fixed coefficient within 3 σ , but the fitted quadratic coefficient differs from the fixed 515 coefficient by more than 26 σ . The derived value for a / R ∗ from model 2 (15.111 +0 . 081 -0 . 080 ) is 516 consistent with the fixed value we used (14.975) within 1.7 σ . For model 3, we found a / R ∗ = 517 15.410 ± 0.080, which differs from the fixed value by 5.5 σ . Despite differences between some 518 of the fixed and fitted parameters, we find that for all models, the derived radius ratios are 519 consistent within 1 σ . Specifically, the measured radius ratios for the three cases described 520 above are: (1) R p /R /star = 0.1174 ± 0.0015 when limb darkening is free and a / R ∗ is fixed, (2) 521 R p /R /star = 0.1157 ± 0.0013 when limb darkening is fixed and a / R ∗ is free, and (3) R p /R /star = 522 0.1175 ± 0.0013 when both limb darkening and a / R ∗ are free. Recall that our base model 523 yielded R p /R /star = 0.1158 ± 0.0013 when keeping both limb darkening and a / R ∗ fixed. Since 524 all radius ratios are consistent within 1 σ , this indicates that fixing limb darkening as well as 525 a / R ∗ did not significantly affect our results or conclusions. 526 527",
"pages": [
17,
18
]
},
{
"title": "6. Conclusions",
"content": "In this paper, we presented results from seven transit observations of GJ 1214b acquired 528 in a narrow-band H 2 filter. Our analysis included a thorough technique for selecting reference 529 stars and incorporated light curve fits with red (correlated) noise. We measured a radius ratio 530 of 0.1158 ± 0.0013 when fitting the data from all seven transits together. This radius ratio is 531 consistent with previous K -band measurements, including those from Croll et al. (2011) and 532 de Mooij et al. (2012) which differ from ours by only 1.5 σ and 1.1 σ . We conclude that all 533 K -band data support a flat absorption spectrum for GJ 1214b, which suggests that either a 534 high mean molecular weight atmosphere or a H-rich atmosphere with a cloud or haze layer 535 is favored (Howe & Burrows 2012). 536 Since Valencia et al. (2013) find that the bulk amount of H/He in GJ 1214b's volatile 537 envelope is nonzero and could be as much as 7% by mass, we explored the scenario where GJ 538 1214b has a H 2 -rich envelope and heavy elements are sequestered below a cloud or aerosol 539 layer. After comparing models of a pure H 2 atmosphere with K -band observations, we find 540 that we cannot exclude the possibility of a H 2 -rich upper atmosphere. It is difficult to disen541 tangle different plausible atmosphere models in the K -band given the precision of available 542 data and that the temperature of the upper atmosphere is otherwise unconstrained. We sug543 gest that additional high-precision spectroscopic observations (from space) across the K -band 544 would be most useful. High-precision measurements at short optical wavelengths ( < 0.7 µ m) 545 would also be helpful in searching for evidence of Rayleigh scattering (e.g. de Mooij et al. 546 2013; Narita et al. 2013), which would support a (cloudy/hazy) H-rich atmosphere for GJ 547 1214b (Howe & Burrows 2012). 548 549 550 551 552 553 554 555 556 557 558 Finally, we explored variability due to H 2 in the stellar spectrum and investigated the effects of different systematics on our results, such as star spots, CCD non-linearity, and limitations in our knowledge of light curve parameters. Overall, we conclude that systematics did not significantly affect our photometry and that our results are robust to these effects. While there is ample spectroscopic and photometric data available for this planet, much of the data comes from different instruments as well as analyses in addition to suffering from poor precision. This in itself introduces an additional systematic when comparing the data to atmosphere models. While beyond the scope of this paper, we conclude that a uniform analysis of all public data for GJ 1214b would be useful for establishing more robust limits on atmosphere models. We are grateful to Mike Irwin and Mike Read for assisting us in accessing the WF559 CAM data and illumination tables. We thank Norio Narita for helpful discussions on GJ 560 1214b. We also thank Paul Wilson for providing data prior to publication. We acknowl561 edge the anonymous referee for helping us improve this paper. This research was supported 562 by NASA grants NNX10AI90G (Astrobiology: Exobiology & Evolutionary Biology) and 563 NNX11AC33G (Origins of Solar Systems) to EG. The United Kingdom Infrared Telescope 564 is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities 565 Council of the U.K. This research has made use of the VizieR catalogue access tool, CDS, 566 567 Strasbourg, France.",
"pages": [
18,
19
]
},
{
"title": "REFERENCES",
"content": "( a ) The best-fit radius ratio from fitting all seven transits together is 0.1158 ± 0.0013. The best-fit radius ratios from fitting the seven transits separately are shown in the fifth column and are also shown in Figure 7. See text for further details.",
"pages": [
25
]
}
]
2013ApJ...776...54S
https://arxiv.org/pdf/1308.5026.pdf
<document>
<section_header_level_1><location><page_1><loc_18><loc_85><loc_82><loc_86></location>Peristaltic Pumping near Post-CME Supra-Arcade Current Sheets</section_header_level_1>
<text><location><page_1><loc_45><loc_82><loc_55><loc_83></location>Roger B. Scott</text>
<text><location><page_1><loc_43><loc_79><loc_57><loc_81></location>Dana W. Longcope</text>
<text><location><page_1><loc_43><loc_77><loc_57><loc_78></location>David E. McKenzie</text>
<section_header_level_1><location><page_1><loc_45><loc_74><loc_55><loc_74></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_16><loc_54><loc_84><loc_71></location>Measurements of temperature and density near supra-arcade current sheets suggest that plasma on unreconnected field lines may experience some degree of 'pre-heating' and 'pre-densification' prior to their reconnection. Models of patchy reconnection allow for heating and acceleration of plasma along reconnected field lines but do not offer a mechanism for transport of thermal energy across field lines. Here we present a model in which a reconnected flux tube retracts, deforming the surrounding layer of unreconnected field. The deformation creates constrictions that act as peristaltic pumps, driving plasma flow along affected field lines. Under certain circumstances these flows lead to shocks that can extend far out into the unreconnected field, altering the plasma properties in the affected region. These findings have direct implications for observations in the solar corona, particularly in regard to such phenomena as high temperatures near current sheets in eruptive solar flares and wakes seen in the form of descending regions of density depletion or supra-arcade downflows.</text>
<section_header_level_1><location><page_1><loc_43><loc_49><loc_57><loc_50></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_33><loc_88><loc_47></location>Since the development of X-ray and EUV solar imaging, observations of evolving arcade structures have become a ubiquitous signature of magnetic reconnection in solar flares. Many of these structures also exhibit vertical fans with highly emissive coronal plasma and what is presumed to be a nearly vertical magnetic field rising above the apex of the arcade ( ˇ Svestka et al. 1998; McKenzie & Hudson 1999; Webb et al. 2003). This picture is consistent with the standard flare model in which a current sheet separates antiparallel layers of magnetic field between an arcade of reconnected flux and a rising coronal mass ejection (Cliver & Hudson 2002). But while the general properties of these structures are well established, the mechanism responsible for increased emission from plasma in the supra-arcade fan remains unclear (Seaton & Forbes 2009; Ko et al. 2010; Reeves et al. 2010).</text>
<text><location><page_1><loc_12><loc_16><loc_88><loc_32></location>One possibility is that the emitting plasma is within the current sheet itself and that its temperature has been increased as a result of ohmic heating. This explanation relies on the assumption that the lineof-sight depth of the current sheet is large enough to allow for a non-negligible emission measure. In cases where the current sheet is observed edge-on (or nearly so), the line-of-sight depth can easily exceed 10 5 km. These edge-on observations (e.g., Ciaravella & Raymond 2008; Savage et al. 2010) also enable upper limits to be placed on the thickness of the current sheet: the measurements indicate thicknesses of no more than 5-50 × 10 3 km. Conversely, Tucker (1973) used theoretical arguments and estimated that post-CME current sheets should have a thickness of roughly 10 -1 km. Such thickness estimates become crucial in cases where the current sheets are observed face-on, as in ˇ Svestka et al. (1998), McKenzie & Hudson (1999), Innes et al. (2003), Warren et al. (2011), and McKenzie (2013).</text>
<text><location><page_1><loc_12><loc_10><loc_88><loc_14></location>An alternative explanation is that the emission comes not from within the current sheet itself but rather from a thermal halo that surrounds the current sheet. The thermal halo could be orders of magnitude thicker than the current sheet and thus provide sufficient line-of-sight depth for observed emission. However, this</text>
<text><location><page_2><loc_12><loc_80><loc_88><loc_86></location>scenario requires some mechanism for increasing the local plasma density above that of the surrounding corona. Chromospheric evaporation is a likely candidate for this pre-densification, but one must still justify the timeliness of the evaporation, which is usually attributed to thermal conduction into the chromosphere (Cargill et al. 1995).</text>
<text><location><page_2><loc_12><loc_64><loc_88><loc_79></location>Reconnection within the current sheet is a likely source of energy both for the heating of plasma and for evaporation-driven pre-densification because it efficiently converts magnetic free energy into thermal and kinetic energy (Guidoni & Longcope 2010; Priest 1999). But while reconnection may provide sufficient energy to heat the surrounding plasma, thermal conductivity transverse to the magnetic field is very weak (Choudhuri 1998). Even if we assume that there exists a well of thermal energy in the reconnected field, it remains unclear what mechanism could be responsible for transporting the energy across field lines. And while radiative transfer is not limited by thermal conduction it is also far too weak given the low optical depth that is typical of the corona. Nonlinear mode-coupling could play a role if the reconnection event somehow excited magnetosonic waves that then dissipated energy in the surrounding plasma.</text>
<text><location><page_2><loc_12><loc_43><loc_88><loc_63></location>Recent observations in EUV (Savage et al. 2012; Warren et al. 2011; Savage & McKenzie 2011) have resolved what appear to be magnetic loops that descend through the supra-arcade fan. The loops seem to form wake-like structures that appear as density depletions or voids in the surrounding plasma. The nature of these voids was studied by Verwichte et al. (2005), who characterized the apparent wave motion of their edges. They found that the boundary between the low density voids and the surrounding plasma exhibited transverse oscillating wavelets that propagated sunward at speeds in the range of 50 km s -1 to 500 km s -1 . Costa et al. (2009) simulated the formation of these dark lanes from an initial pressure perturbation. They found that the lanes could be interpreted as an interference pattern resulting from the reflection of magnetosonic shocks and rarefaction waves. More recently, Cassak et al. (2013) simulated the formation of dark lanes as 'flow channels carved by sunward-directed outflow jets from reconnection.' The applicability of this last interpretation, which places the voids below the arcade itself, must be carefully considered when placed in the context of observations of supra-arcade features.</text>
<text><location><page_2><loc_12><loc_29><loc_88><loc_42></location>Another possibility is that these features are the result of patchy reconnection in which flux tubes retract toward the arcade under the influence of magnetic tension and are drawn through the surrounding, unreconnected field as depicted in Figure 1. Previous authors have modeled the dynamics of the retracting flux tube (Guidoni & Longcope 2010; Longcope et al. 2009; Linton & Longcope 2006), but have not yet considered its effect on the surrounding, unreconnected flux. Cargill et al. (1996) modeled the interaction of a magnetic cloud and the surrounding magnetic field, but their work focused on the highβ regime ( β = 8 πp/B 2 /greatermuch 1). For our analysis we will consider the consequences of an extremely lowβ scenario in which the magnetic field dominates all other energy contributions.</text>
<text><location><page_2><loc_12><loc_12><loc_88><loc_28></location>Our focus will be to consider how the plasma and unreconnected flux that surround the current sheet behave in response to a reconnection event. Toward this end we assume that a localized reconnection event has already occurred within a supra-arcade current sheet and has created a bundle of newly closed magnetic field lines, a flux tube, which retracts through the current sheet (Linton & Longcope 2006; Longcope et al. 2009) as depicted in Figure 1. The retracting flux tube is a prescribed element whose radius and motion are parameters of the model. The primary effect we consider is the deformation it creates in the surrounding field. The deflection of a given field line is bounded by the radius of the retracting tube and is smaller farther away. Due to this smallness the deformation is typically dismissed as a minor effect, though Linton & Longcope (2006) did consider the possibility that the work required to the deform the external field might contribute to a drag force on the retracting tube.</text>
<figure>
<location><page_3><loc_23><loc_68><loc_77><loc_85></location>
<caption>Fig. 1.- (1a) A reconnected flux tube piercing normally through the current sheet as in Savage et al. (2012). (1b) A reconnected flux tube embedded within the current sheet as in Linton & Longcope (2006). Taking ˆ z to point vertically away from the limb with ˆ x pointing along the reconnected flux tube, ˆ y is either normal to or in the plane of the current sheet, depending on the configuration.</caption>
</figure>
<text><location><page_3><loc_12><loc_46><loc_88><loc_57></location>The retracting flux tube could have two possible orientations relative to the current sheet. The theoretical work of Linton & Longcope (2006) assumes that a section of the tube lies within the plane of the current sheet, as shown Figure 1b. On the other hand imaging observations have been interpreted assuming the flux tube pierces the current sheet normally, as in Figure 1a (McKenzie 2000; Savage et al. 2012). Our modeling will be applicable to either scenario since both create identical deformations in the surrounding field. We hereafter focus on the surrounding field which is roughly vertical, and refer to the retracting flux as an intrusion .</text>
<text><location><page_3><loc_12><loc_30><loc_88><loc_45></location>In the present work we show that the deformation takes the form of a constriction, which moves downward through the surrounding field at the same speed as the retracting flux tube. Observations clearly show this speed to be some fraction of the local Alfv'en speed (Savage & McKenzie 2011), and often in excess of the local sound speed. We observe that the moving constriction behaves as a peristaltic pump, resulting in field-aligned plasma flows, which we dub peristaltic flows . We show below that there are regimes in which these flows lead to slow magnetosonic shocks that develop in the surrounding field. These manifest in our model as hydrodynamic shocks and rarefaction waves, which travel along the field at speeds comparable to the sound speed. The existence of such features leads to several dramatic effects, including significant heating and changes to the density and emission measure of plasma in the unreconnected field.</text>
<section_header_level_1><location><page_3><loc_44><loc_25><loc_56><loc_26></location>2. The Model</section_header_level_1>
<text><location><page_3><loc_12><loc_10><loc_88><loc_23></location>The model that we present here treats the unreconnected flux as current-free field along which plasma is constrained to move. We begin by determining the magnetic field subject to the influence of an intruding, reconnected flux tube. We assume that β is extremely small so that the magnetic field may be determined independent of the plasma. This dictates both the plasma flow trajectory and the cross section of parallel flow. Steady solutions are then found for plasma flow along each field line. Points where the flow is ill-defined are avoided through the introduction of rarefaction waves and acoustic shocks, which are a limiting form of slow magnetosonic shocks in very low β . The result is a piecewise continuous adiabatic series of solutions that evolve in time as the fluid jumps propagate. The 2D behavior is ultimately found through interpolating</text>
<text><location><page_4><loc_12><loc_85><loc_47><loc_86></location>between solutions along representative field lines.</text>
<text><location><page_4><loc_12><loc_76><loc_88><loc_84></location>Our analysis will invoke two distinct reference frames. The limb-frame is stationary with respect to the solar surface and in this frame the undisturbed plasma is at rest. Alternatively, in the comoving frame it is the descending intrusion that is at rest and the plasma is taken to be rising uniformally at large distances from the intrusion. It is in the comoving frame that the magnetic field is most easily determined because in this frame the boundary conditions are steady in time and therefore, so too is the field.</text>
<section_header_level_1><location><page_4><loc_37><loc_71><loc_63><loc_72></location>2.1. Deformed Potential Field</section_header_level_1>
<text><location><page_4><loc_12><loc_66><loc_88><loc_69></location>In the comoving frame the unreconnected magnetic field is a sum of the original magnetic field prior to distortion ( B 0 ) and a second field ( B ' ) representing the influence of the intruding flux tube</text>
<formula><location><page_4><loc_45><loc_63><loc_88><loc_64></location>B = B 0 + B ' . (1)</formula>
<text><location><page_4><loc_12><loc_55><loc_88><loc_61></location>Since the reconnected and unreconnected fields are topologically distinct we will impose the simplifying constraint that no unreconnected field lines may intersect the surface of the reconnected flux tube. Furthermore, as information about the reconnection event cannot have influenced the field at arbitrarily large distances, B must reduce to B 0 far from the flux tube, so B ' must vanish there.</text>
<figure>
<location><page_4><loc_34><loc_31><loc_66><loc_54></location>
<caption>Fig. 2.- An initially uniform field ( B 0 ) is altered by the intruding surface ( S ) with B ' introduced so that the net field ( B = B 0 + B ' ) satisfies the appropriate boundary conditions on S .</caption>
</figure>
<text><location><page_4><loc_12><loc_18><loc_88><loc_23></location>We take the background field to be uniform and vertical ( B 0 = ± ˆ z B 0 ) while the reconnected flux defines a uniform cylinder ( S ), centered at the origin, with radius R and symmetry axis pointing in the ˆ x direction. 1 B ' depends only on y and z . The total field is assumed to be potential with boundary conditions given by</text>
<formula><location><page_4><loc_40><loc_15><loc_88><loc_16></location>ˆ z × B | r →∞ = ˆ r · B | r ∈ S = 0 . (2)</formula>
<text><location><page_5><loc_12><loc_55><loc_13><loc_57></location>as</text>
<text><location><page_5><loc_12><loc_83><loc_88><loc_86></location>The first constraint ensures that the magnetic field is unaffected far from S while the latter ensures that no field lines intersect S .</text>
<text><location><page_5><loc_12><loc_79><loc_88><loc_82></location>The potential magnetic field, constrained by these boundary conditions, may be determined in terms of a flux function such that</text>
<formula><location><page_5><loc_38><loc_77><loc_88><loc_79></location>B = ˆ x × B 0 ∇ f = -B 0 ∇ × (ˆ x f ) , (3)</formula>
<text><location><page_5><loc_12><loc_72><loc_88><loc_76></location>where ˆ x , being the axis of symmetry of the intruding flux, is an ignorable direction. In the far field, setting f → y ensures that B → B 0 ˆ z , while on the surface of the feature ˆ r · B = 0 so ∂ θ f | S = 0. Thus, in terms of f the boundary conditions become</text>
<formula><location><page_5><loc_44><loc_70><loc_88><loc_71></location>∇ f × ˆ y | r →∞ = 0 (4)</formula>
<text><location><page_5><loc_12><loc_68><loc_14><loc_69></location>and</text>
<formula><location><page_5><loc_44><loc_66><loc_88><loc_67></location>∇ f × ˆ r | r → R = 0 . (5)</formula>
<text><location><page_5><loc_12><loc_64><loc_50><loc_65></location>And, since ∇ × B = 0, f satisfies Laplace's equation;</text>
<formula><location><page_5><loc_47><loc_61><loc_88><loc_62></location>∇ 2 f = 0 . (6)</formula>
<text><location><page_5><loc_15><loc_57><loc_88><loc_58></location>With these conditions and the choice that f be symmetric in y , the flux function is uniquely specified</text>
<formula><location><page_5><loc_35><loc_52><loc_88><loc_56></location>f = sin( θ ) ( R 2 r -r ) = y ( 1 -R 2 y 2 + z 2 ) , (7)</formula>
<text><location><page_5><loc_12><loc_47><loc_88><loc_52></location>with θ measured from -ˆ z . Since the magnetic field is everywhere orthogonal to the gradient of f , contours of f are themselves field lines, denoted X f , which can be parameterized by solving for z in terms of f and y so that</text>
<formula><location><page_5><loc_41><loc_44><loc_88><loc_47></location>z f ( y ) 2 = y y 2 -fy -R 2 f -y , (8)</formula>
<text><location><page_5><loc_12><loc_42><loc_63><loc_43></location>where, for a given f , the y coordinate along the field line is bounded by</text>
<formula><location><page_5><loc_39><loc_37><loc_88><loc_41></location>| f | < | y | < √ f 2 / 4 + 1 + | f | / 2 . (9)</formula>
<text><location><page_5><loc_12><loc_31><loc_88><loc_38></location>Figure 3 shows a contour plot of f ( y, z ) that traces a representative set of field lines. For each field line, y → f as | z | → ∞ . Values of f are therefore the lateral positions of the field lines in the far field. The deflection of each field line is largest abreast of the intrusion, where z = 0. For | f | /greatermuch R this deflection goes to zero while for the most strongly deflected field line ( f = 0) the deflection is | ∆ y | = R .</text>
<text><location><page_5><loc_12><loc_24><loc_88><loc_30></location>Ultimately, we will be interested in the parameterized cross section of an arbitrary, unreconnected flux tube. From ∇ · B = 0 it follows that the cross section scales inversely with the field strength. Let α ( X f ) be the inverse of the dimensionless field strength of an infinitesimal flux tube, normalized to unity in the far field and parameterized along an arbitrary field line, X f , so that 1 /α ≡ | B | /B o . In terms of f</text>
<formula><location><page_5><loc_40><loc_20><loc_88><loc_23></location>1 α 2 =( ∂ y f ) 2 +( ∂ z f ) 2 (10)</formula>
<formula><location><page_5><loc_42><loc_17><loc_88><loc_20></location>= 2 R 2 ( y 2 -z 2 ) + R 4 ( y 2 + z 2 ) 2 +1 , (11)</formula>
<text><location><page_5><loc_12><loc_14><loc_53><loc_15></location>which, after substituting in for z f ( y ), can be expressed as</text>
<formula><location><page_5><loc_41><loc_9><loc_88><loc_13></location>α 2 = y 2 R 2 f 2 R 2 +4 y 2 ( f -y ) 2 . (12)</formula>
<figure>
<location><page_6><loc_32><loc_67><loc_67><loc_86></location>
<caption>Fig. 3.- The potential magnetic field is deformed by the expansion of the origin onto a cylindrical surface of radius R. Field lines are deflected around the intrusion. The inverse normalized field strength ( α ) is shown on the left panel for the f = -R/ 2 field line, parameterized in z . The associated flux tube is traced in purple on the main panel. In the bottom panel the minimum inverse normalized field strength ( α z =0 ) is plotted first as a function of f (solid red) and then as a function of y (dashed red).</caption>
</figure>
<text><location><page_6><loc_12><loc_50><loc_88><loc_53></location>Holding f fixed, we define α f ( y ) to be the cross section of an infinitesimal flux tube centered on a field line X f , parameterized by the lateral deflection of the field line.</text>
<text><location><page_6><loc_12><loc_46><loc_88><loc_49></location>α f achieves a minimum value at z = 0, where the field line passes abreast of the intrusion. This location is referred to as the throat of the flux tube and has a cross section of</text>
<formula><location><page_6><loc_39><loc_39><loc_88><loc_45></location>α min ( f ) = 1 2 √ f 2 +4 R 2 + f √ f 2 +4 R 2 , (13)</formula>
<text><location><page_6><loc_12><loc_30><loc_88><loc_40></location>which is necessarily less than one. Moving away from the throat along the field line the flux tube expands until it reaches a maximum value, which is necessarily greater than one, and then slowly contracts toward unity as | z | → ∞ . In general, field lines that pass close to the intrusion have the smallest minimum cross section and greatest overall variability while for large | f | values α is nearly uniform along z . The f = 0 field line is both the most and least constricted with a cross section that diverges at y = 0 , z = ± R and achieves the global minimum of α min ( f = 0) = min[ α ( y, z )] = 0 . 5 at its throat.</text>
<section_header_level_1><location><page_6><loc_41><loc_25><loc_59><loc_26></location>2.2. Peristaltic Flow</section_header_level_1>
<text><location><page_6><loc_12><loc_16><loc_88><loc_23></location>Under the assumption of ideal magnetic induction, as a fluid element moves it must remain on the same field line and its cross section for flow parallel to the field must be the same as that of the associated flux tube. Since the magnetic field is stationary with respect to the descending intrusion, the flow will be steady in the co-moving frame. The steady version of the continuity equation, ∇ · ( ρ u ) = 0, is satisfied by a constant mass flux</text>
<formula><location><page_6><loc_47><loc_14><loc_88><loc_15></location>˙ m = ρuα, (14)</formula>
<text><location><page_6><loc_12><loc_10><loc_88><loc_13></location>where ρ is the density, u is the speed of the fluid, α is the cross section of the flux tube defined by f and ˙ m is a constant of integration that is conserved along X f . The steady flow must also satisfy the momentum</text>
<text><location><page_7><loc_12><loc_85><loc_18><loc_86></location>equation</text>
<formula><location><page_7><loc_34><loc_82><loc_88><loc_85></location>ρ ( u · ∇ ) u = 1 4 π ( ∇ × B ) × B + ∇ · σ -∇ p, (15)</formula>
<text><location><page_7><loc_12><loc_77><loc_88><loc_81></location>where gravity is omitted for simplicity. Here p is the plasma pressure and σ is the viscous stress tensor. Since the flow must be parallel to the magnetic field, the Lorentz force makes no contribution to the parallel momentum equation</text>
<formula><location><page_7><loc_36><loc_74><loc_88><loc_77></location>( u · ∇ ) 1 2 u 2 + 1 ρ ( u · ∇ ) p = 1 ρ u · ( ∇ · σ ) . (16)</formula>
<text><location><page_7><loc_12><loc_68><loc_88><loc_73></location>This equation is the same as that of a neutral fluid passing though a nozzle. Together with an energy equation relating ρ and p , Eqs. (16) and (14) fully specify the spatial variation of the fluid along the length of the flux tube.</text>
<text><location><page_7><loc_15><loc_66><loc_55><loc_67></location>For simplicity we adopt the isothermal equation of state</text>
<formula><location><page_7><loc_47><loc_63><loc_88><loc_65></location>p = C 2 s ρ, (17)</formula>
<text><location><page_7><loc_12><loc_57><loc_88><loc_61></location>where C s is the sound speed. This assumption is motivated by the very high thermal conduction along field lines. Combining Eq. (17) with Eq. (16) and integrating over the volume of a fluid element with parameterized length l leads to</text>
<formula><location><page_7><loc_35><loc_50><loc_88><loc_56></location>[ 1 2 u 2 + C 2 s ln ρ ]∣ ∣ ∣ l 2 l 1 = ∫ l 2 l 1 d l ˆ u · [ 1 ρ ∇ · σ ] , (18)</formula>
<text><location><page_7><loc_12><loc_46><loc_88><loc_53></location>∣ where l 1 and l 2 are two arbitrary locations along the flux tube. Under strong magnetization the viscous force is dominated by the parallel contribution (Guidoni & Longcope 2010). Using a field-aligned coordinate system it can be shown that this contribution takes the form</text>
<formula><location><page_7><loc_33><loc_41><loc_88><loc_45></location>ˆ u · [ ∇ · σ ] = µ (0) α 4 ( 4 3 ∂ 2 l ( u α ) +2 ∂ l ( u ∂ l 1 α )) , (19)</formula>
<text><location><page_7><loc_12><loc_32><loc_88><loc_40></location>where µ (0) is the dominant coefficient of dynamic viscosity, which is proportional to ρλC s , and λ is ion mean free path. When the flow is sufficiently smooth for the ion mean-free path to be negligible, the viscous contribution to the momentum equation may be neglected and the left hand side of Eq. (18) is conserved along the length of the flux tube. Only in the case of a shock, where fluid variation cascades to shorter length scales, is the viscous contribution significant.</text>
<text><location><page_7><loc_12><loc_28><loc_88><loc_31></location>After neglecting viscocity, Eqs. (18) and (14) lead to a relationship between the flux-tube cross section α and the Mach number ( M = u/C s ) that is equivalent de Laval's equation for steady flow through a nozzle;</text>
<formula><location><page_7><loc_41><loc_25><loc_88><loc_27></location>M 2 -ln M 2 -ln α 2 = B . (20)</formula>
<text><location><page_7><loc_12><loc_16><loc_88><loc_23></location>B is effectively Bernoulli's constant and is a conserved quantity along any time-independent, inviscid flow. B can, in principle, assume any real, positive value, and will generally be determined by evaluating M and α at a particular point of interest. For real values of M the quantity M 2 -ln M 2 has a minimum value of unity at M = 1 and diverges monotonically as M goes to either zero or infinity. There are two solutions to Eq. (20) corresponding to any value of B - one subsonic and one supersonic.</text>
<text><location><page_7><loc_12><loc_11><loc_88><loc_14></location>The behavior of this system can be visualized by plotting contours of B in M -z phase space. Figure 4 shows a representative set of solutions for the f = R/ 2 field line with B ranging from approximately 1.1 for</text>
<figure>
<location><page_8><loc_23><loc_61><loc_77><loc_87></location>
<caption>Fig. 4.- Contours of Eq. (20) are plotted for the f = R/ 2 field line. Each color represents a different value for B . Supercritical solutions (black) are well-behaved. Subcritical solutions (purple, red) are ill-behaved where M → 1 and ill-defined in the range -z crit < z < z crit . The two regimes are separated by the transonic contour (blue), which passes through M = 1 , z = 0. Fluid flow is from left to right.</caption>
</figure>
<text><location><page_8><loc_12><loc_46><loc_88><loc_49></location>the dashed purple contour up to nearly 3 for the black contour. The qualitative behavior of these solutions is dictated by how B relates to the critical value of</text>
<formula><location><page_8><loc_41><loc_43><loc_88><loc_45></location>B ts ( f ) = 1 -2 ln α min ( f ) , (21)</formula>
<text><location><page_8><loc_12><loc_34><loc_88><loc_42></location>which defines the transonic flow contour for which M → 1 exactly at the throat of the constriction where α → α min . The transonic contour separates the so called supercritical solutions, given by B ∈ B + > B ts , from the subcritical solutions, given by B ∈ B -< B ts . As an example, consider the f = R/ 2 field line depicted in Figure 4. The minimum cross section is α min ( R/ 2) ≈ 0 . 62 and so B ts ( R/ 2) ≈ 2. Inverting Eq. (20) for α = 1 (in the far field) we find the two transonic inflow values are M ts ( R/ 2) ≈ 1 . 75 and 0 . 41.</text>
<text><location><page_8><loc_12><loc_25><loc_88><loc_33></location>Supercritical solutions have the property that M + ( z ) 2 -ln M + ( z ) 2 > 1 for all values of α f ( z ) so that M ( B + , f, z ) is well defined along the entire flux tube. The black contours in Figure 4 represent the supersonic and subsonic solutions for a particular value of B + . Note that these contours are everywhere either supersonic or subsonic and are well defined as z → 0. Subcritical solutions do not have this property and are ill-defined at any location where the cross section is smaller than the so called critical cross section , given by</text>
<formula><location><page_8><loc_44><loc_22><loc_88><loc_23></location>α crit = e (1 -B -) / 2 . (22)</formula>
<text><location><page_8><loc_12><loc_11><loc_88><loc_20></location>Subcritical solutions are defined by the existence of a set of critical points , given by the two locations, z = ± z crit ( f ), that satisfy Eq. (22). At these critical points M ( B -, f, z crit ) = 1, while over the interval -z crit < z < z crit the Mach number is ill-defined. The dashed red and purple contours of Figure 4 represent subcritical solutions for two different values of B -. In both cases the Mach number goes to unity at z = ± z crit ( f ) and is ill-defined over the interval between the two critical points. A third solution is visible as the blue line in Figure 4 and corresponds to the transonic contour with B = B ts . This solution has the</text>
<text><location><page_9><loc_12><loc_83><loc_88><loc_86></location>unique property that α crit = α min ( f ) = α f ( z = 0) so that the two critical points occur exactly at the throat of the flux tube.</text>
<text><location><page_9><loc_12><loc_69><loc_88><loc_82></location>The two branches of the transonic contour separate the M -z phase space into the subcritical region, which is located between the transonic contours, and the supercritical region, which is located above and below the supersonic and subsonic branches of the transonic solution, respectively. As with subcritical solutions, the supersonic and subsonic branches of the transonic solution eventually intersect as the Mach number goes to unity. But, unlike the subcritical contours, the transonic contour is well defined over the entire length of the flux tube. And since ∂ z α f ( z ) = 0 at z = 0, the fluid is well-behaved at this point even as the Mach number goes to unity. This contour is therefore the only viable solution that allows for a fluid to smoothly pass between supersonic and subsonic flows while conserving the value of B .</text>
<section_header_level_1><location><page_9><loc_40><loc_64><loc_60><loc_65></location>2.3. Transitional Flows</section_header_level_1>
<text><location><page_9><loc_12><loc_41><loc_88><loc_62></location>If we were free to choose the value of B to be always equal to or greater than B ts the steady solutions described in 2.2 would be sufficient. Since the fluid is at rest in the limb frame, in the intrusion frame the fluid velocity in the far field is given by u far = u in ˆ z = ˆ z C s M in , where u in is the speed at which the intrusion descends and M in is its Mach number. Since ( M in ) must be allowed to assume any real value we are forced to consider that the far field boundary condition might correspond to a subcritical flow. The inadmissibility, between -z crit and z crit , of a solution with the value of B fixed by the boundary condition demands that the overall solution be one in which B is not conserved. This solution will take the form of several regions of constant B , each connected by a transition in which Bernoulli's equation does not hold. The transitions are either shocks or rarefaction waves whose locations change with time. The complete flow combines two shocks, both propagating upstream, enclosing a transonic flow on which B = B ts , and then a rarefaction wave propagating downstream away from the intrusion. For a careful discussion of shocks and rarefaction waves see Chapters IX and X of Landau & Lifshitz (1959). The following is a more specific discussion, aimed at our particular problem.</text>
<section_header_level_1><location><page_9><loc_45><loc_36><loc_55><loc_37></location>2.3.1. Shocks</section_header_level_1>
<text><location><page_9><loc_12><loc_24><loc_88><loc_34></location>In cases where the length scale of the fluid becomes comparable to the ion mean free path the viscosity has a non-negligible contribution to the momentum equation and cannot be ignored as it was leading to Eq. (20). The resulting behavior is referred to as a shock, which is a thin transition from one value of B to another. In a reference frame co-moving with the shock the flow must be steady and conserve mass and momentum. In the isothermal case these conditions lead to a version of the Rankine-Hugoniot condition, amounting to conservation of</text>
<formula><location><page_9><loc_46><loc_22><loc_88><loc_25></location>M ' + 1 M ' (23)</formula>
<text><location><page_9><loc_12><loc_15><loc_88><loc_21></location>across the jump, where M ' = ( u -u s ) /C s is the Mach number viewed from a frame moving at the shock speed u s . This conservation law differs from Eq. (20) because the jump is assumed so thin that viscosity cannot be ignored and α is approximately constant across it. Discounting the trivial case where the Mach number is unchanged, leads to the relation</text>
<formula><location><page_9><loc_46><loc_12><loc_88><loc_15></location>M ' 2 = 1 M ' 1 (24)</formula>
<text><location><page_10><loc_12><loc_83><loc_88><loc_86></location>between upstream and downstream Mach numbers, M ' 1 and M ' 2 . It is evident that one of these will be subsonic while the other is supersonic.</text>
<text><location><page_10><loc_15><loc_81><loc_75><loc_82></location>In terms of Mach numbers M j in the frame of the intrusion, Eq. (24) takes the form</text>
<formula><location><page_10><loc_40><loc_76><loc_88><loc_79></location>M 2 -M s = 1 M 1 -M s , (25)</formula>
<text><location><page_10><loc_12><loc_72><loc_88><loc_75></location>where M s = u s /C s is the Mach number of the shock. Knowing the upstream and downstream Mach numbers then gives the shock Mach number as</text>
<formula><location><page_10><loc_35><loc_67><loc_88><loc_71></location>M s = M 1 + M 2 2 -√ ( M 1 -M 2 2 ) 2 +1 , (26)</formula>
<text><location><page_10><loc_12><loc_63><loc_88><loc_66></location>assuming M 1 > M 2 > 0. The shock will move leftward ( M s < 0) if M 1 < 1 /M 2 , and rightward if M 1 > 1 /M 2 . Mass conservation, in the shock reference frame, then leads to the relation</text>
<formula><location><page_10><loc_28><loc_56><loc_88><loc_61></location>ρ 2 = ρ 1 M 1 -M s M 2 -M s = ρ 1 √ ( M 1 -M 2 ) 2 +4+( M 1 -M 2 ) √ ( M 1 -M 2 ) 2 +4 -( M 1 -M 2 ) , (27)</formula>
<text><location><page_10><loc_12><loc_54><loc_88><loc_57></location>between pre-shock and post-shock density. A shock must have M ' 1 > 1 > M ' 2 , and therefore ρ 2 > ρ 1 : it is compressive.</text>
<section_header_level_1><location><page_10><loc_41><loc_49><loc_59><loc_50></location>2.3.2. Rarefaction waves</section_header_level_1>
<text><location><page_10><loc_12><loc_34><loc_88><loc_47></location>A jump to lower density, not possible in a shock, must occur in a rarefaction wave. In cases without externally defined length scale the rarefaction wave will be self-similar (Landau & Lifshitz 1959, § 92), depending on space and time through a single similarity variable ( z -z 0 ) /t . A rarefaction wave is inherently time-dependent and so Bernoulli's equation is again invalid. In our solution, a shock and rarefaction wave will be generated simultaneously at t = 0 from the single point z = z crit . This initial state lacks a length scale and we may take the downstream rarefaction wave to be of the self-similar form. It will be bounded by weak discontinuities at its edges. The leading edge, at z = z 2 , propagates into the (unperturbed) downstream plasma at u 2 + C s . Upstream of this the velocity, and thus Mach number, is linear (Landau & Lifshitz 1959)</text>
<formula><location><page_10><loc_43><loc_30><loc_88><loc_33></location>M = M 2 -z 2 -z C s t . (28)</formula>
<text><location><page_10><loc_12><loc_22><loc_88><loc_29></location>Upstream of the trailing edge, at z = z 1 , the flow is again constant with M = M 1 < M 2 . Thus the extent of the rarefaction wave grows in time as ∆ z = ( M 2 -M 1 ) C s t , beginning as a discontinuity at t = 0. The initial discontinuity at z = + z crit decomposes into this rarefaction wave and a shock, in the manner of a Riemann problem (Landau & Lifshitz 1959).</text>
<text><location><page_10><loc_12><loc_18><loc_88><loc_21></location>Within the rarefaction wave the density is an explicit function of velocity (see Landau & Lifshitz 1959, § 92)</text>
<formula><location><page_10><loc_37><loc_14><loc_88><loc_18></location>ρ = ρ 2 e M -M 2 = ρ 2 exp ( z -z 2 C s t ) . (29)</formula>
<text><location><page_10><loc_12><loc_13><loc_58><loc_14></location>The upstream and downstream densities are therefore related by</text>
<formula><location><page_10><loc_44><loc_10><loc_88><loc_12></location>ρ 2 = ρ 1 e M 2 -M 1 , (30)</formula>
<text><location><page_11><loc_12><loc_80><loc_88><loc_86></location>across the ever-expanding rarefaction wave. In order for this solution to apply the interior size of the rarefaction wave must be much smaller than the length scale of variation of the fluid cross-section, α . Fortunately, the rarefaction wave, while growing in time, propagates vary quickly into the far field so that no matter how large it becomes, the scale over which α varies is always larger still.</text>
<section_header_level_1><location><page_11><loc_41><loc_75><loc_59><loc_76></location>2.4. Composite Flow</section_header_level_1>
<text><location><page_11><loc_12><loc_60><loc_88><loc_73></location>The complete solution, defined over the entire length of the affected flux tube, will be piecewise continuous using shocks, rarefaction waves and regions of pertistaltic flow so that the fluid velocity and density are treated in an internally consistent manner. The locations of the transition flows will travel along the length of the flux tubes in order to satisfy their governing equations and will therefore introduce time variations into the system despite our previous assumption of time independence. In letting this system evolve we are assuming that it can be treated as an adiabatic series of time-independent solutions. This assumption will be valid so long as the timescale over which a given feature evolves is long compared to its fluid crossing time.</text>
<text><location><page_11><loc_12><loc_49><loc_88><loc_59></location>In order to form a solution we use physical consideration to motivate the choice of initial conditions in the region between the critical points. Far above and below the intrusion we demand that the plasma density and velocity be unchanged and continuity demands that every jump in velocity have a corresponding jump in density. We therefore require at least two jumps with at least one unspecified intermediate value of B in order to have sufficient degrees of freedom to satisfy the boundary conditions on both velocity and pressure, which is equivalent to density in the isothermal limit.</text>
<text><location><page_11><loc_12><loc_37><loc_88><loc_48></location>If the fluid velocity in the far field is supersonic and lies below the transonic contour, then B in is subcritical and M will be ill-defined between the two critical points. 2 In order for the fluid to avoid an infinite acceleration at the upstream critical point there must be a transition away from B in and onto some well-behaved flow, B ∈ B + . In order for solution to remain well defined the transition must propagate upstream, away from the critical point. This is only possible in the case of a shock. A rarefaction wave would not propagate upstream with sufficient speed to escape the critical region. The downstream flow must therefore be subsonic in order for the system to be well defined at the upstream critical point.</text>
<text><location><page_11><loc_12><loc_23><loc_88><loc_36></location>At the downstream critical point there must again be a transition to connect the flow that resulted from the upstream shock back to the original contour B in . If the interior flow were everywhere subsonic the jump would again have to be a rarefaction wave and would propagate at Mach 1 into the higher density, subsonic fluid, ultimately making its way into the critical region and leaving the downstream critical point again ill-behaved. Thus, in the downstream region, the flow that resulted from the upstream shock must be supersonic at the critical point. Only the transonic contour B ts can satisfy this condition without introducing yet another jump within the critical region. We therefore reach the conclusion, well known in nozzle problems, that the flow must cross from subsonic to supersonic at the throat.</text>
<text><location><page_11><loc_12><loc_15><loc_88><loc_22></location>The density change across the leading shock, which connects the subcritical inflow to the transonic interior flow, is fixed by the relative speeds of the fluid on either side of the shock. At the downstream critical point the speed of the fluid is given by the transonic solution and the density is fixed through the continuity equation. Any transition from the transonic flow back to the original flow must therefore</text>
<figure>
<location><page_12><loc_23><loc_56><loc_75><loc_86></location>
<caption>Fig. 5.- The piecewise continuous, composite flow is formed by connecting the subcritical flow from the far field with the transonic solution in the interior and an unspecified subcritical solution in the intermediate downstream region. The unshocked density and Mach number indicate the far field subcritical solution in the absence of shocks. As in Figure 4 fluid flows from left to right.</caption>
</figure>
<text><location><page_12><loc_12><loc_34><loc_88><loc_43></location>satisfy the disparity in both speed and density at this point, a feat not achievable for either a shock or a rarefaction wave. The jump at the downstream critical point must therefore decompose into both a shock and a rarefaction wave, just as in an asymmetric Riemann problem (Landau & Lifshitz 1959). The rarefaction wave then propagates at Mach 1 into the downstream fluid and therefore moves away from the intrusion at speeds in excess of Mach 2. The shock propagates subsonically upstream into the transonic flow, which is itself supersonic, and therefore moves more slowly away from the intrusion.</text>
<text><location><page_12><loc_12><loc_15><loc_88><loc_33></location>Between the downstream shock and the rarefaction wave there is an initially infinitesimal intermediate region in which the fluid lies on an unspecified contour of B , which will be determined such that the net effect of the two downstream transitions exactly compensates for the upstream shock and transonic interior. The intermediate flow is supersonic but also slower than the initial, subcritical flow, so it too has critical points and these must be accounted for when the initial locations of the three transitions are chosen. The whole system is evolved by using the current velocity of each feature to determine its location at some future time and then constructing the new velocity and density profiles for the whole system at that time. This construct is shown in Figure 5 for the f = 0 . 5 field line with an inflow condition of M in = 1 . 5. The system is shown a short time after launch so that the transitions are spatially separated and can be easily distinguished. The original, subcritical flow is unphysical at z ≈ -R but the composite flow is transonic at this point and thus exhibits no critical phenomena.</text>
<text><location><page_12><loc_12><loc_11><loc_88><loc_13></location>Relative to the intrusion, the upstream transonic flow and downstream intermediate flow are both slower than the subcritical flow in the far field. In the limb frame the plasma in these regions is actually descending</text>
<text><location><page_13><loc_12><loc_77><loc_88><loc_86></location>toward the limb along with the intrusion. The two shocks similarly descend toward the limb with the leading shock pushing ahead and the trailing shock lagging ever farther behind while the rarefaction wave is sufficiently fast that it is not entrained with the intrusion and escapes rapidly upward. Note however that these shocks are not standoff shocks. They evolve in time and move steadily away from the intrusion, ultimately finding their way into the far field where their evolution slows and the assumption of timeindependence becomes increasingly exact.</text>
<section_header_level_1><location><page_13><loc_45><loc_71><loc_55><loc_73></location>3. Results</section_header_level_1>
<text><location><page_13><loc_12><loc_55><loc_88><loc_70></location>In order to gain insight into solar dynamics from this model we must construct synthetic observables which can be compared to actual observations. To this end we begin by constructing 2D maps of density and velocity, made by interpolating between a representative sample of field lines, each determined with the same boundary conditions. Features of the 1D fluid solution manifest in the 2D maps as broad fronts, and regions of high or low density. Figure 6 shows one such map of plasma density given at three successive times. From this example several features are visible. The leading shock, trailing shock and rarefaction wave are all distinguishable as abrupt changes in the plasma density. The high and low density 'head' and 'tail' grow in time and descend toward the limb while a slightly less rarefied region between the trailing shock and the rarefaction wave grows quickly upward as the rarefaction wave escapes away from the limb.</text>
<figure>
<location><page_13><loc_24><loc_27><loc_74><loc_54></location>
<caption>Fig. 6.- A M in = 1 . 5 descending intrusion is shown in the limb frame at times t = { 0 , 3 , 6 } R/C s . The high and low density regions are seen in red scale and the plots to the right of each panel show the exact density profile for the field line traced in white. The dashed lines indicate a normalized density of 1, as in the far field.</caption>
</figure>
<section_header_level_1><location><page_14><loc_40><loc_85><loc_60><loc_86></location>3.1. Emission Measure</section_header_level_1>
<text><location><page_14><loc_12><loc_62><loc_88><loc_83></location>As a proxy for synthetic images of the optically thin corona we choose the emission measure density ( /epsilon1 ∝ ρ 2 ). The emission measure profile will depend on the viewing angle. To begin with we consider a line of sight that is normal to the current sheet, consistent with many imaging observations of sheetlike structures above post-CME solar arcades ( ˇ Svestka et al. 1998; Gallagher et al. 2002; Innes et al. 2003; Savage & McKenzie 2011; Savage et al. 2012; McKenzie 2013). If the intrusion pierces normally through the current sheet as in Figure 1a then /epsilon1 can be constructed simply by squaring the 2D density maps such as in Figure 6. This viewing angle also applies to cases where the intrustion is imbedded within the current sheet (as in Figure 1b), which is itself viewed edge on. The resulting emission measure maps will exhibit the same features as Figure 6. A collection of four such emission measure maps is shown in Figure 7 for four different descending intrusions, each depicted at the instant of launch and then again after the shocks have propagated into the far field. At t = 0 the shocks trace out the loci of critical points for each field line. Then, as t →∞ the shock fronts move into the far field so that the shock column has infinite vertical extent both above and below the intrusion.</text>
<figure>
<location><page_14><loc_24><loc_46><loc_33><loc_61></location>
<caption>Fig. 7.- Four peristaltic shocks are shown for M in = { 1 . 4 , 1 . 5 , 1 . 6 , 1 . 7 } (left to right) and for times t = { 0 , 100 } R/C s (top to bottom). The edges of the shocked column in the lower row trace out the field lines that are transonic for each value of M in , which separate the shocked and unshocked regions.</caption>
</figure>
<figure>
<location><page_14><loc_66><loc_46><loc_76><loc_61></location>
<caption>M in = 1 . 4 , t = 0</caption>
</figure>
<paragraph><location><page_14><loc_37><loc_45><loc_47><loc_46></location>M in = 1 . 5 , t = 0</paragraph>
<paragraph><location><page_14><loc_51><loc_45><loc_61><loc_46></location>M in = 1 . 6 , t = 0</paragraph>
<figure>
<location><page_14><loc_24><loc_30><loc_34><loc_44></location>
<caption>M in = 1 . 7 , t = 0</caption>
</figure>
<figure>
<location><page_14><loc_66><loc_30><loc_76><loc_44></location>
<caption>M in = 1 . 4 , t = 100</caption>
</figure>
<paragraph><location><page_14><loc_37><loc_28><loc_49><loc_29></location>M in = 1 . 5 , t = 100</paragraph>
<paragraph><location><page_14><loc_51><loc_28><loc_63><loc_29></location>M in = 1 . 6 , t = 100</paragraph>
<paragraph><location><page_14><loc_66><loc_28><loc_77><loc_29></location>M in = 1 . 7 , t = 100</paragraph>
<text><location><page_14><loc_12><loc_11><loc_88><loc_19></location>Relative to the diameter of the intrusion, the width of the shocked column depends only on the speed of the intrusion M in , which dictates the fluid velocity in the far field. For intermediate speeds (between Mach 1.4 and and 1.7) the column width is of the same order as the intrusion diameter. The upper limit occurs as M in → 1 . 92 at which point all field lines become non-critical so the shocked column vanishes. In the opposite limit, as M in → 1 all field lines exhibit critical behavior and the shocked column becomes infinite</text>
<figure>
<location><page_14><loc_38><loc_30><loc_48><loc_45></location>
</figure>
<figure>
<location><page_14><loc_52><loc_30><loc_62><loc_44></location>
</figure>
<figure>
<location><page_14><loc_38><loc_46><loc_47><loc_61></location>
</figure>
<figure>
<location><page_14><loc_52><loc_46><loc_62><loc_61></location>
</figure>
<text><location><page_15><loc_12><loc_85><loc_50><loc_86></location>in width but with vanishing amplitude in the far field.</text>
<text><location><page_15><loc_12><loc_76><loc_88><loc_84></location>As an alternative we consider the system viewed from the side, such as if the line of sight were along ˆ y in Figure 1b. In this case the emission measure is constructed by integrating transversely across the 2D domain. The resulting profile will resemble that of an individual field line but will be somewhat smoother, having effectively averaged over all shocked field lines. We define the background-subtracted, normalized column emission measure as</text>
<formula><location><page_15><loc_39><loc_72><loc_88><loc_76></location>/epsilon1 ( z ) = ∫ L -L d y ( ρ ( y, z ) 2 ρ 2 0 -1 ) , (31)</formula>
<text><location><page_15><loc_12><loc_67><loc_88><loc_72></location>so that /epsilon1 ( z ) goes to zero for ρ ( y, z ) = ρ 0 and is negative or positive where ρ is depleted or enhanced with respect to the ambient plasma. Figure 8 shows a stack-plot of successive time-steps of /epsilon1 ( z, t ) for M in = 1 . 3. The propagating shocks and rarefaction wave can be seen as abrupt jumps in the red scale emission.</text>
<figure>
<location><page_15><loc_33><loc_49><loc_67><loc_65></location>
<caption>Fig. 8.- The line-of-sight integrated emission measure depicted here as a stack plot. Higher emission is indicated in yellow. Time increases to the right with the vertical profile at any given time corresponding to /epsilon1 ( z, t ) .</caption>
</figure>
<figure>
<location><page_15><loc_33><loc_24><loc_67><loc_39></location>
<caption>Fig. 9.- A sheared magnetic field results in a reconnected, horizontal field line which is drug downward through the adjacent layers of unreconnected field, forming a plateau.</caption>
</figure>
<text><location><page_15><loc_12><loc_10><loc_88><loc_16></location>To visualize how this kind of structure might manifest in the current sheet consider the case of a slightly sheared supra-arcade magnetic field. According to Guidoni & Longcope (2010), a local reconnection event will result in a growing, descending trapezoidal plateau that leads to something like Figures 1b and 9. The horizontal segment of the reconnected field, i.e. the intrusion, is embedded in the current sheet and drives</text>
<text><location><page_16><loc_12><loc_83><loc_88><loc_86></location>peristaltic flows in the surrounding layers of field. As the plateau descends the bends move outward so that more of the reconnected flux is embedded in the current sheet.</text>
<text><location><page_16><loc_12><loc_72><loc_88><loc_82></location>Because the field is sheared, we treat the two respective layers of magnetic field independently. They both exhibit peristaltic flows which result in an emission such as in Figure 8, but in one field the flow is slanted slightly to the right while the other is slanted to the left. The composite flows launch first on the field lines closest to the initial reconnection point. Then, as the plateau descends, those same field lines continue to evolve while field lines that are newly exposed to the growing plateau are initiated each in turn. The net result is a locus of shocked flows that grows as the plateau grows.</text>
<text><location><page_16><loc_12><loc_58><loc_88><loc_71></location>Such a system is depicted in Figure 10. The unreconnected field is angled up and to the right in the foreground and down and to the right in the background. The layers of field that pass close to the horizontal segment of reconnected field exhibit peristaltic pumping. The emission measure on each field line is given by /epsilon1 (˜ z, t -t f ), where ˜ z is the distance along the angled field line and t f is the time at which that field line was initiated. The collection of lower shocks leads to a vaguely arch shaped high density region while the upper shock leads to a similar rarefied region. The antisunward rarefaction waves lead to a nearly vertical column of low density owing to the fact that these waves propagate supersonically outward along field lines at a rate comparable to the growth of the plateau.</text>
<figure>
<location><page_16><loc_33><loc_33><loc_67><loc_57></location>
<caption>Fig. 10.- A high contrast plot of emission measure for a sheared peristaltic event with two sets of field lines, dashed and dotted. The locus of shock features are visible above and below the plateau created by the embedded segment of reconnected flux, depicted as a solid white line.</caption>
</figure>
<section_header_level_1><location><page_16><loc_38><loc_20><loc_62><loc_21></location>3.2. Momentum in the Fluid</section_header_level_1>
<text><location><page_16><loc_12><loc_12><loc_88><loc_18></location>Since the descending intrusion generates plasma motion in the surrounding field, there should be associated energy and momentum transfer into that fluid. For any finite domain we can find the momentum in the plasma through numerical integration of the plasma density and velocity. Figure 11 displays a representative plot of momentum density.</text>
<figure>
<location><page_17><loc_32><loc_64><loc_68><loc_86></location>
<caption>Fig. 11.- A time series of fluid momentum density. Red indicates sunward momentum while blue is antisunward.</caption>
</figure>
<text><location><page_17><loc_12><loc_42><loc_88><loc_55></location>Due to the fact that the normalized cross section in the far field is asymptotic to but always greater than unity, the supersonic fluid far from the intrusion always propagates slightly faster than M in in the rest frame of the intrusion. In the limb frame this fluid is slowly rising so the momentum density far from the intrusion is always anti-sunward. Closer to the intrusion it is directed sunward as the cross section becomes constricted and the fluid is slowed below M in . Immediately above the lower shock the momentum density is strongly sunward and then becomes anti-sunward as the transonic flow passes abreast of the intrusion and again exceeds M in . It then becomes sunward again across the second shock before finally returning to the far-field limit across the rarefaction wave.</text>
<text><location><page_17><loc_12><loc_38><loc_88><loc_41></location>To explore this more carefully we observe that the force per unit fluid cross section on a shocked flux tube may be found explicitly through</text>
<formula><location><page_17><loc_37><loc_36><loc_38><loc_36></location>∞</formula>
<formula><location><page_17><loc_32><loc_29><loc_88><loc_33></location>= ∂ t ( ∫ z 1 + δ z 1 -δ + ∫ z 2 + δ z 2 -δ + ∫ z 3 + δ z 3 -δ ) d z · ( ρ u α ) (33)</formula>
<formula><location><page_17><loc_31><loc_32><loc_88><loc_36></location>f = ∂ t ∫ -∞ d z · ( ρ u α ) (32)</formula>
<formula><location><page_17><loc_32><loc_27><loc_88><loc_29></location>= v z 1 [ ρuα ] | z 1 + δ 1 z 1 + v z 2 [ ρuα ] | z 2 + δ 2 z 2 + v z 3 [ ρuα ] | z 3 + δ 3 z 3 , (34)</formula>
<text><location><page_17><loc_12><loc_12><loc_88><loc_25></location>where δ i represents the width of each jump. This can be calculated numerically for every field line within a finite domain under the assumption that all three jumps have propagated into the far field where the shock speeds become steady and α → 1. In Figure 12 we see that the net force on the fluid appears to be finite even in the limit of M → 1, where the shock becomes infinitely wide. In order to confirm this the contribution from the far field may be approximated analytically. This calculation is not included in the present work but it can be shown that the force contributed by the shocks in the far field vanishes with an inverse power of distance greater than unity so that, indeed, the net force on the fluid remains finite even as the shocks fill all of space.</text>
<figure>
<location><page_18><loc_33><loc_71><loc_67><loc_86></location>
<caption>Fig. 12.- The time rate of change of momentum in the fluid yields a net force on the fluid that vanishes when the width of the shocked column goes to zero as M in → 1 . 92 and remains finite as the width expands and M in → 1.</caption>
</figure>
<section_header_level_1><location><page_18><loc_43><loc_60><loc_57><loc_61></location>3.3. Drag Force</section_header_level_1>
<text><location><page_18><loc_12><loc_48><loc_88><loc_58></location>If the total momentum in our model contained only two contributing terms we could use Figure 12 as a proxy for the drag force on the intrusion. In actuality the far-field boundary conditions also contribute momentum to the system and the drag force must be calculated explicitly by integrating the plasma pressure over the surface of the intrusion. The pressure is related to the density, which can be found explicitly by calculating the behavior of fluid on the f = 0 streamline. The net vertical force due to the pressure p s ( θ ) on the surface S is then</text>
<formula><location><page_18><loc_33><loc_45><loc_88><loc_49></location>F z = 2 ∫ π 0 r d θ p s ˆ n · ˆ z = 2 ∫ π 0 p s ( θ ) r cos( θ )d θ . (35)</formula>
<figure>
<location><page_18><loc_33><loc_27><loc_67><loc_43></location>
<caption>Fig. 13.- Pressure along the surface of the intrusion, parameterized by polar angle. ρ ts and M ts are the transonic density and Mach number far below the intrusion. The integrated pressure gives the force on the intrusion (per unit inserted length) as F ≈ 4 . 8 × ρ 0 M 0 RC 2 s .</caption>
</figure>
<text><location><page_18><loc_12><loc_10><loc_88><loc_18></location>The plasma on the f = 0 field line is always on the transonic contour. The f = 0 streamline intersects the surface of the intrusion at the two magnetic null points θ = { 0 , π } , with θ measured here from -ˆ z . At these points the fluid cross section diverges as the magnetic field strength vanishes. For θ = π the fluid Mach number also diverges so ρ must be zero by continuity. For θ = 0 the plasma is subsonic so M goes to zero as α →∞ . To ensure that the density is well-behaved at this point we perform a series expansion around</text>
<text><location><page_19><loc_12><loc_80><loc_88><loc_86></location>θ = 0 and find that ρ ( θ → 0) ≈ 3 . 3 ρ ts , where ρ ts is the plasma density on the transonic contour far from the intrusion. p ( θ ) is therefore well-behaved and can be calculated numerically as in Figure 13. The pressure decreases monotonically indicating an upward net force on the intrusion. The z -component of this force is given explicitly by the area under the dashed curve.</text>
<text><location><page_19><loc_12><loc_64><loc_88><loc_79></location>In order to find how the drag force depends on M in we must find the pressure jump across the leading shock, which will dictate ρ ts and hence p ts . The shock velocity M s is determined by M in and M ts . But, in the far field M ts → 0 . 319 for f = 0, α = 1. Thus, ρ ts depends only on M in and, when multiplied by the integration factor from Figure 13, the resulting drag force can be found as depicted in Figure 14. The drag is lowest for the Mach 1 limit and increases almost linearly up to M in ≈ 1 . 92, at which point the column disappears. At and above Mach 1.92 the drag is zero due to the symmetry of the de Laval flow solutions, just as in D'Alembert's paradox. Below Mach 1 we have not calculated the drag profile but we expect, given the extent of the subsonic critical regime, that the drag will remain finite down to the minimum critical value of M min ≈ 0 . 32, at which point the shocked column again vanishes.</text>
<figure>
<location><page_19><loc_33><loc_47><loc_67><loc_63></location>
<caption>Fig. 14.- The drag force is found by evaluating the density jump across the lower shock for a given intrusion Mach number and then combining the result with the integration factor from Figure 13.</caption>
</figure>
<text><location><page_19><loc_12><loc_30><loc_88><loc_39></location>As indicated, the drag curve in Figure 14 does not match the net force on the fluid from Figure 12. This is not surprising since the transonic flow is assymetric along the vertical direction and thus the centripetal force on each fluid element is unbalanced. Thus the magnetic field must deform asymmetrically in order to balance the fluid pressure. It follows that there must be some infinitesimal asymmetry between the far field magnetic field above and below the intrusion which in turn leads to a net force exerted on the system by the far field boundary conditions.</text>
<section_header_level_1><location><page_19><loc_44><loc_25><loc_56><loc_26></location>4. Discussion</section_header_level_1>
<text><location><page_19><loc_12><loc_10><loc_88><loc_23></location>In this work we have shown how field line retraction following a local reconnection event can manifest as a descending constriction in the nearby unreconnected field. This constriction behaves in many respects as a peristaltic pump, which leads to peristaltic flows and ultimately to shocks and rarefaction waves that alter the velocity and density of plasma on affected field lines. These shocks are not to be confused with standoff shocks, which form at a fixed distance in front of a traveling obstacle and are thereafter stationary in time. The fluid jumps that we have described cannot exist as time-independent solutions and must necessarily propagate away from the intrusion. The region between these jumps therefore grows in time and is, in and of itself, a dynamic feature.</text>
<text><location><page_20><loc_12><loc_75><loc_88><loc_86></location>The composite flows that form in this model are restricted to a column whose width is defined by the field-line that exhibits transonic flow for a particular boundary condition, M in . This width increases monotonically as the speed of the intrusion decreases toward the sound speed. The minimum width of zero occurs when the speed of the descending intrusion reaches M max = 1 . 92 while the maximum width is arbitrarily large as M in → 1. In this analysis we considered only supersonic values for M in but we acknowledge that the peristaltic flows will continue to exhibit critical behavior even for subsonic values of M in down to the point where B is again larger than B ts on the f = 0 field line.</text>
<text><location><page_20><loc_12><loc_61><loc_88><loc_74></location>Our profiles for the 2D density and emission measure maps bear striking resemblance to observations of voids and Supra-Arcade Downflows (SADs) in post-CME flares (McKenzie 2000; Savage et al. 2012). In our analysis we considered an isothermal plasma in order to make the development more tractable. We offer, without proof, that an adiabatic plasma would exhibit the same qualitative behavior with the addition that plasma in the region between the lower shock and the rarefaction wave would exhibit an increase in temperature. The rarefied tail and high-emission leading edge may even be useful as thermal diagnostics since a temperature increase in the rarefied column could move the emission outside of a particular observation band-pass, thereby increasing the contrast in these features.</text>
<text><location><page_20><loc_12><loc_52><loc_88><loc_60></location>We also considered an alternate geometry in which the particular shape of the emission profile is exchanged for a column integrated emission measure which occurs everywhere along the length of an embedded flux tube. This geometry also offers an interpretation for down-flowing features but may be more accurately used to describe how reconnection events and the contraction of reconnected flux can lead to heating of plasma along a broadly distributed volume of unreconnected field.</text>
<text><location><page_20><loc_12><loc_39><loc_88><loc_50></location>Our model also helps to explain how a thermal halo (Seaton & Forbes 2009) might form around the current sheet. Here we have described only the creation of shocks along constricted field lines. But these shocks could very well travel down the unreconnected field all the way to the chromosphere where they would then drive evaporation exactly as in conduction dominated flare loops (Cargill et al. 1995). This evaporation might then increase the density on 'post-peristaltic' field lines, which could then undergo their own reconnection or even experience another 'peristaltic process' in the event of a second nearby reconnection event.</text>
<text><location><page_20><loc_12><loc_25><loc_88><loc_38></location>We further describe how the alterations to velocity and density relate to the momentum density in the fluid and the subsequent net force (per unit embedded length) on the fluid. This force is related to but not equal to the net force on the retracting flux tube since a third contribution comes from the boundary conditions which maintain the field profile in the far field. The net force on the intrusion is found from a direct calculation of pressure integrated over its surface. This force points in the direction opposite the motion of the intrusion and is of order RC 2 s ρ in . It increases almost linearly by nearly a factor of two over the range 1 < M in < 2. Larger descent speeds correspond to a larger drag force so that, if this force is sufficient to influence the kinematics of the descending intrusion, the drag will decrease as the intrusion slows.</text>
<text><location><page_20><loc_12><loc_11><loc_88><loc_24></location>This drag force offers a possible explanation to the fact that reconnection outflows appear to move subAlfv'enically despite the predictions of reconnection models such as described by Seaton & Forbes (2009). If reconnection outflows originate in locations where peristaltic shocks can form then this could lead to a drag force that would keep their velocities below the Alfv'en speed. However, if the outflow velocity ever exceeds the maximum shock velocity then the drag force should vanish according to our model. These loops would descend much more rapidly in its absence. This line of reasoning suggests there may be a bimodal distribution in the velocity of retracting magnetic loops. Loops that move fast enough to avoid launching peristaltic shocks would remain fast-moving while slower moving loops would be damped by the momentum</text>
<text><location><page_21><loc_12><loc_85><loc_32><loc_86></location>transferred into the plasma.</text>
<text><location><page_21><loc_12><loc_72><loc_88><loc_84></location>In order for the aforementioned drag force to have a non-negligible influence, the plasma pressure must be comparable to the magnetic energy density in the retracting flux tube. But pressure balance between the intrusion and its surroundings demands that the field strength must be comparable between the retracting flux and the unreconnected field. Thus, the drag force will only be significant if the plasma pressure is comparable to the magnetic energy density in the unreconnected field. While this is not unlikely in reality it is in conflict with the zeroβ assumption and therefore cannot be reconciled with our model in its current form.</text>
<text><location><page_21><loc_12><loc_57><loc_88><loc_71></location>Our model assumes an extremely low β value in order to invoke the rigid magnetic field. However, observations suggest that this may not be an accurate assumption in the supra-arcade region (McKenzie 2013). It may be that by the time supra-arcade downflows become visibile in observations the local plasma β has already been increased due to previous instances of peristaltic pumping and that our model only applies to the early stages of flare activity, when the plasma density and temperature are still relatively low. Future work will therefore require the relaxation of the low β approximation, which will necessitate a numerical simulation. Another issue with the model is that we have been forced to stitch together time-independent solutions in an adiabatic fashion. The validity of this approximation, as well as those used in deriving the 1D MHD simplifications, will likewise need to be tested through simulations.</text>
<text><location><page_21><loc_12><loc_33><loc_88><loc_55></location>When comparing to observations, some key differences are also apparent. Our model cannot reproduce the oscillatory behavior on the edges of voids as seen in Verwichte et al. (2005), although it does predict a discontinuity in plasma density transverse to the field, which could support surface modes if the zeroβ assumption were relaxed. Also, while we predict that these features should occur for M in /lessorapproxeql 2, Savage & McKenzie (2011) measured a typical downflow speed of on the order of 10 2 km s -1 with some instances of much higher values. Depending on the local sound speed these velocities may fall above our Mach 2 prediction. A more careful study of SAD speeds and the associated local sound speed will need to be conducted in order to refine this estimate. Moreover, the upper limit yielded by our simple model can be relaxed by generalizations to non-circular intrusions. As a first attempt we calculated that for elliptical intrusions the value of M max could be increased by a factor of nearly two before the aspect ratio of the ellipse became unrealistic. It may be that an appropriate choice of intrusion cross section could reconcile any lingering disparities between the model and observations of flare loops. Ultimately we intend to further the investigation with a regimen of numerical simulations. The ultimate success of this model will be in providing a theoretical framework for interpreting features seen in more complex numerical simulations.</text>
<section_header_level_1><location><page_21><loc_41><loc_27><loc_59><loc_29></location>5. Acknowledgements</section_header_level_1>
<text><location><page_21><loc_12><loc_21><loc_88><loc_26></location>The authors are grateful to K. Reeves and S. Savage for valuable comments on early drafts of the manuscript. This work was supported in part by NASA under contract NNM07AB07C with the Smithsonian Astrophysical Observatory and in part by a grant form the NSF/DOE Partnership in Basic Plasma Physics.</text>
<section_header_level_1><location><page_21><loc_44><loc_16><loc_56><loc_17></location>REFERENCES</section_header_level_1>
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<text><location><page_21><loc_12><loc_11><loc_62><loc_12></location>Cargill, P. J., Mariska, J. T., & Antiochos, S. K. 1995, ApJ, 439, 1034</text>
<text><location><page_22><loc_12><loc_85><loc_74><loc_86></location>Cassak, P. A., Drake, J. F., Gosling, J. T., et al. 2013, ArXiv e-prints, arXiv:1307.3946</text>
<text><location><page_22><loc_12><loc_82><loc_88><loc_83></location>Choudhuri, A. R. 1998, The Physics of Fluids and Plasmas, An Introduction for Astrophysicists (New York:</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "Measurements of temperature and density near supra-arcade current sheets suggest that plasma on unreconnected field lines may experience some degree of 'pre-heating' and 'pre-densification' prior to their reconnection. Models of patchy reconnection allow for heating and acceleration of plasma along reconnected field lines but do not offer a mechanism for transport of thermal energy across field lines. Here we present a model in which a reconnected flux tube retracts, deforming the surrounding layer of unreconnected field. The deformation creates constrictions that act as peristaltic pumps, driving plasma flow along affected field lines. Under certain circumstances these flows lead to shocks that can extend far out into the unreconnected field, altering the plasma properties in the affected region. These findings have direct implications for observations in the solar corona, particularly in regard to such phenomena as high temperatures near current sheets in eruptive solar flares and wakes seen in the form of descending regions of density depletion or supra-arcade downflows.",
"pages": [
1
]
},
{
"title": "Peristaltic Pumping near Post-CME Supra-Arcade Current Sheets",
"content": "Roger B. Scott Dana W. Longcope David E. McKenzie",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Since the development of X-ray and EUV solar imaging, observations of evolving arcade structures have become a ubiquitous signature of magnetic reconnection in solar flares. Many of these structures also exhibit vertical fans with highly emissive coronal plasma and what is presumed to be a nearly vertical magnetic field rising above the apex of the arcade ( ˇ Svestka et al. 1998; McKenzie & Hudson 1999; Webb et al. 2003). This picture is consistent with the standard flare model in which a current sheet separates antiparallel layers of magnetic field between an arcade of reconnected flux and a rising coronal mass ejection (Cliver & Hudson 2002). But while the general properties of these structures are well established, the mechanism responsible for increased emission from plasma in the supra-arcade fan remains unclear (Seaton & Forbes 2009; Ko et al. 2010; Reeves et al. 2010). One possibility is that the emitting plasma is within the current sheet itself and that its temperature has been increased as a result of ohmic heating. This explanation relies on the assumption that the lineof-sight depth of the current sheet is large enough to allow for a non-negligible emission measure. In cases where the current sheet is observed edge-on (or nearly so), the line-of-sight depth can easily exceed 10 5 km. These edge-on observations (e.g., Ciaravella & Raymond 2008; Savage et al. 2010) also enable upper limits to be placed on the thickness of the current sheet: the measurements indicate thicknesses of no more than 5-50 × 10 3 km. Conversely, Tucker (1973) used theoretical arguments and estimated that post-CME current sheets should have a thickness of roughly 10 -1 km. Such thickness estimates become crucial in cases where the current sheets are observed face-on, as in ˇ Svestka et al. (1998), McKenzie & Hudson (1999), Innes et al. (2003), Warren et al. (2011), and McKenzie (2013). An alternative explanation is that the emission comes not from within the current sheet itself but rather from a thermal halo that surrounds the current sheet. The thermal halo could be orders of magnitude thicker than the current sheet and thus provide sufficient line-of-sight depth for observed emission. However, this scenario requires some mechanism for increasing the local plasma density above that of the surrounding corona. Chromospheric evaporation is a likely candidate for this pre-densification, but one must still justify the timeliness of the evaporation, which is usually attributed to thermal conduction into the chromosphere (Cargill et al. 1995). Reconnection within the current sheet is a likely source of energy both for the heating of plasma and for evaporation-driven pre-densification because it efficiently converts magnetic free energy into thermal and kinetic energy (Guidoni & Longcope 2010; Priest 1999). But while reconnection may provide sufficient energy to heat the surrounding plasma, thermal conductivity transverse to the magnetic field is very weak (Choudhuri 1998). Even if we assume that there exists a well of thermal energy in the reconnected field, it remains unclear what mechanism could be responsible for transporting the energy across field lines. And while radiative transfer is not limited by thermal conduction it is also far too weak given the low optical depth that is typical of the corona. Nonlinear mode-coupling could play a role if the reconnection event somehow excited magnetosonic waves that then dissipated energy in the surrounding plasma. Recent observations in EUV (Savage et al. 2012; Warren et al. 2011; Savage & McKenzie 2011) have resolved what appear to be magnetic loops that descend through the supra-arcade fan. The loops seem to form wake-like structures that appear as density depletions or voids in the surrounding plasma. The nature of these voids was studied by Verwichte et al. (2005), who characterized the apparent wave motion of their edges. They found that the boundary between the low density voids and the surrounding plasma exhibited transverse oscillating wavelets that propagated sunward at speeds in the range of 50 km s -1 to 500 km s -1 . Costa et al. (2009) simulated the formation of these dark lanes from an initial pressure perturbation. They found that the lanes could be interpreted as an interference pattern resulting from the reflection of magnetosonic shocks and rarefaction waves. More recently, Cassak et al. (2013) simulated the formation of dark lanes as 'flow channels carved by sunward-directed outflow jets from reconnection.' The applicability of this last interpretation, which places the voids below the arcade itself, must be carefully considered when placed in the context of observations of supra-arcade features. Another possibility is that these features are the result of patchy reconnection in which flux tubes retract toward the arcade under the influence of magnetic tension and are drawn through the surrounding, unreconnected field as depicted in Figure 1. Previous authors have modeled the dynamics of the retracting flux tube (Guidoni & Longcope 2010; Longcope et al. 2009; Linton & Longcope 2006), but have not yet considered its effect on the surrounding, unreconnected flux. Cargill et al. (1996) modeled the interaction of a magnetic cloud and the surrounding magnetic field, but their work focused on the highβ regime ( β = 8 πp/B 2 /greatermuch 1). For our analysis we will consider the consequences of an extremely lowβ scenario in which the magnetic field dominates all other energy contributions. Our focus will be to consider how the plasma and unreconnected flux that surround the current sheet behave in response to a reconnection event. Toward this end we assume that a localized reconnection event has already occurred within a supra-arcade current sheet and has created a bundle of newly closed magnetic field lines, a flux tube, which retracts through the current sheet (Linton & Longcope 2006; Longcope et al. 2009) as depicted in Figure 1. The retracting flux tube is a prescribed element whose radius and motion are parameters of the model. The primary effect we consider is the deformation it creates in the surrounding field. The deflection of a given field line is bounded by the radius of the retracting tube and is smaller farther away. Due to this smallness the deformation is typically dismissed as a minor effect, though Linton & Longcope (2006) did consider the possibility that the work required to the deform the external field might contribute to a drag force on the retracting tube. The retracting flux tube could have two possible orientations relative to the current sheet. The theoretical work of Linton & Longcope (2006) assumes that a section of the tube lies within the plane of the current sheet, as shown Figure 1b. On the other hand imaging observations have been interpreted assuming the flux tube pierces the current sheet normally, as in Figure 1a (McKenzie 2000; Savage et al. 2012). Our modeling will be applicable to either scenario since both create identical deformations in the surrounding field. We hereafter focus on the surrounding field which is roughly vertical, and refer to the retracting flux as an intrusion . In the present work we show that the deformation takes the form of a constriction, which moves downward through the surrounding field at the same speed as the retracting flux tube. Observations clearly show this speed to be some fraction of the local Alfv'en speed (Savage & McKenzie 2011), and often in excess of the local sound speed. We observe that the moving constriction behaves as a peristaltic pump, resulting in field-aligned plasma flows, which we dub peristaltic flows . We show below that there are regimes in which these flows lead to slow magnetosonic shocks that develop in the surrounding field. These manifest in our model as hydrodynamic shocks and rarefaction waves, which travel along the field at speeds comparable to the sound speed. The existence of such features leads to several dramatic effects, including significant heating and changes to the density and emission measure of plasma in the unreconnected field.",
"pages": [
1,
2,
3
]
},
{
"title": "2. The Model",
"content": "The model that we present here treats the unreconnected flux as current-free field along which plasma is constrained to move. We begin by determining the magnetic field subject to the influence of an intruding, reconnected flux tube. We assume that β is extremely small so that the magnetic field may be determined independent of the plasma. This dictates both the plasma flow trajectory and the cross section of parallel flow. Steady solutions are then found for plasma flow along each field line. Points where the flow is ill-defined are avoided through the introduction of rarefaction waves and acoustic shocks, which are a limiting form of slow magnetosonic shocks in very low β . The result is a piecewise continuous adiabatic series of solutions that evolve in time as the fluid jumps propagate. The 2D behavior is ultimately found through interpolating between solutions along representative field lines. Our analysis will invoke two distinct reference frames. The limb-frame is stationary with respect to the solar surface and in this frame the undisturbed plasma is at rest. Alternatively, in the comoving frame it is the descending intrusion that is at rest and the plasma is taken to be rising uniformally at large distances from the intrusion. It is in the comoving frame that the magnetic field is most easily determined because in this frame the boundary conditions are steady in time and therefore, so too is the field.",
"pages": [
3,
4
]
},
{
"title": "2.1. Deformed Potential Field",
"content": "In the comoving frame the unreconnected magnetic field is a sum of the original magnetic field prior to distortion ( B 0 ) and a second field ( B ' ) representing the influence of the intruding flux tube Since the reconnected and unreconnected fields are topologically distinct we will impose the simplifying constraint that no unreconnected field lines may intersect the surface of the reconnected flux tube. Furthermore, as information about the reconnection event cannot have influenced the field at arbitrarily large distances, B must reduce to B 0 far from the flux tube, so B ' must vanish there. We take the background field to be uniform and vertical ( B 0 = ± ˆ z B 0 ) while the reconnected flux defines a uniform cylinder ( S ), centered at the origin, with radius R and symmetry axis pointing in the ˆ x direction. 1 B ' depends only on y and z . The total field is assumed to be potential with boundary conditions given by as The first constraint ensures that the magnetic field is unaffected far from S while the latter ensures that no field lines intersect S . The potential magnetic field, constrained by these boundary conditions, may be determined in terms of a flux function such that where ˆ x , being the axis of symmetry of the intruding flux, is an ignorable direction. In the far field, setting f → y ensures that B → B 0 ˆ z , while on the surface of the feature ˆ r · B = 0 so ∂ θ f | S = 0. Thus, in terms of f the boundary conditions become and And, since ∇ × B = 0, f satisfies Laplace's equation; With these conditions and the choice that f be symmetric in y , the flux function is uniquely specified with θ measured from -ˆ z . Since the magnetic field is everywhere orthogonal to the gradient of f , contours of f are themselves field lines, denoted X f , which can be parameterized by solving for z in terms of f and y so that where, for a given f , the y coordinate along the field line is bounded by Figure 3 shows a contour plot of f ( y, z ) that traces a representative set of field lines. For each field line, y → f as | z | → ∞ . Values of f are therefore the lateral positions of the field lines in the far field. The deflection of each field line is largest abreast of the intrusion, where z = 0. For | f | /greatermuch R this deflection goes to zero while for the most strongly deflected field line ( f = 0) the deflection is | ∆ y | = R . Ultimately, we will be interested in the parameterized cross section of an arbitrary, unreconnected flux tube. From ∇ · B = 0 it follows that the cross section scales inversely with the field strength. Let α ( X f ) be the inverse of the dimensionless field strength of an infinitesimal flux tube, normalized to unity in the far field and parameterized along an arbitrary field line, X f , so that 1 /α ≡ | B | /B o . In terms of f which, after substituting in for z f ( y ), can be expressed as Holding f fixed, we define α f ( y ) to be the cross section of an infinitesimal flux tube centered on a field line X f , parameterized by the lateral deflection of the field line. α f achieves a minimum value at z = 0, where the field line passes abreast of the intrusion. This location is referred to as the throat of the flux tube and has a cross section of which is necessarily less than one. Moving away from the throat along the field line the flux tube expands until it reaches a maximum value, which is necessarily greater than one, and then slowly contracts toward unity as | z | → ∞ . In general, field lines that pass close to the intrusion have the smallest minimum cross section and greatest overall variability while for large | f | values α is nearly uniform along z . The f = 0 field line is both the most and least constricted with a cross section that diverges at y = 0 , z = ± R and achieves the global minimum of α min ( f = 0) = min[ α ( y, z )] = 0 . 5 at its throat.",
"pages": [
4,
5,
6
]
},
{
"title": "2.2. Peristaltic Flow",
"content": "Under the assumption of ideal magnetic induction, as a fluid element moves it must remain on the same field line and its cross section for flow parallel to the field must be the same as that of the associated flux tube. Since the magnetic field is stationary with respect to the descending intrusion, the flow will be steady in the co-moving frame. The steady version of the continuity equation, ∇ · ( ρ u ) = 0, is satisfied by a constant mass flux where ρ is the density, u is the speed of the fluid, α is the cross section of the flux tube defined by f and ˙ m is a constant of integration that is conserved along X f . The steady flow must also satisfy the momentum equation where gravity is omitted for simplicity. Here p is the plasma pressure and σ is the viscous stress tensor. Since the flow must be parallel to the magnetic field, the Lorentz force makes no contribution to the parallel momentum equation This equation is the same as that of a neutral fluid passing though a nozzle. Together with an energy equation relating ρ and p , Eqs. (16) and (14) fully specify the spatial variation of the fluid along the length of the flux tube. For simplicity we adopt the isothermal equation of state where C s is the sound speed. This assumption is motivated by the very high thermal conduction along field lines. Combining Eq. (17) with Eq. (16) and integrating over the volume of a fluid element with parameterized length l leads to ∣ where l 1 and l 2 are two arbitrary locations along the flux tube. Under strong magnetization the viscous force is dominated by the parallel contribution (Guidoni & Longcope 2010). Using a field-aligned coordinate system it can be shown that this contribution takes the form where µ (0) is the dominant coefficient of dynamic viscosity, which is proportional to ρλC s , and λ is ion mean free path. When the flow is sufficiently smooth for the ion mean-free path to be negligible, the viscous contribution to the momentum equation may be neglected and the left hand side of Eq. (18) is conserved along the length of the flux tube. Only in the case of a shock, where fluid variation cascades to shorter length scales, is the viscous contribution significant. After neglecting viscocity, Eqs. (18) and (14) lead to a relationship between the flux-tube cross section α and the Mach number ( M = u/C s ) that is equivalent de Laval's equation for steady flow through a nozzle; B is effectively Bernoulli's constant and is a conserved quantity along any time-independent, inviscid flow. B can, in principle, assume any real, positive value, and will generally be determined by evaluating M and α at a particular point of interest. For real values of M the quantity M 2 -ln M 2 has a minimum value of unity at M = 1 and diverges monotonically as M goes to either zero or infinity. There are two solutions to Eq. (20) corresponding to any value of B - one subsonic and one supersonic. The behavior of this system can be visualized by plotting contours of B in M -z phase space. Figure 4 shows a representative set of solutions for the f = R/ 2 field line with B ranging from approximately 1.1 for the dashed purple contour up to nearly 3 for the black contour. The qualitative behavior of these solutions is dictated by how B relates to the critical value of which defines the transonic flow contour for which M → 1 exactly at the throat of the constriction where α → α min . The transonic contour separates the so called supercritical solutions, given by B ∈ B + > B ts , from the subcritical solutions, given by B ∈ B -< B ts . As an example, consider the f = R/ 2 field line depicted in Figure 4. The minimum cross section is α min ( R/ 2) ≈ 0 . 62 and so B ts ( R/ 2) ≈ 2. Inverting Eq. (20) for α = 1 (in the far field) we find the two transonic inflow values are M ts ( R/ 2) ≈ 1 . 75 and 0 . 41. Supercritical solutions have the property that M + ( z ) 2 -ln M + ( z ) 2 > 1 for all values of α f ( z ) so that M ( B + , f, z ) is well defined along the entire flux tube. The black contours in Figure 4 represent the supersonic and subsonic solutions for a particular value of B + . Note that these contours are everywhere either supersonic or subsonic and are well defined as z → 0. Subcritical solutions do not have this property and are ill-defined at any location where the cross section is smaller than the so called critical cross section , given by Subcritical solutions are defined by the existence of a set of critical points , given by the two locations, z = ± z crit ( f ), that satisfy Eq. (22). At these critical points M ( B -, f, z crit ) = 1, while over the interval -z crit < z < z crit the Mach number is ill-defined. The dashed red and purple contours of Figure 4 represent subcritical solutions for two different values of B -. In both cases the Mach number goes to unity at z = ± z crit ( f ) and is ill-defined over the interval between the two critical points. A third solution is visible as the blue line in Figure 4 and corresponds to the transonic contour with B = B ts . This solution has the unique property that α crit = α min ( f ) = α f ( z = 0) so that the two critical points occur exactly at the throat of the flux tube. The two branches of the transonic contour separate the M -z phase space into the subcritical region, which is located between the transonic contours, and the supercritical region, which is located above and below the supersonic and subsonic branches of the transonic solution, respectively. As with subcritical solutions, the supersonic and subsonic branches of the transonic solution eventually intersect as the Mach number goes to unity. But, unlike the subcritical contours, the transonic contour is well defined over the entire length of the flux tube. And since ∂ z α f ( z ) = 0 at z = 0, the fluid is well-behaved at this point even as the Mach number goes to unity. This contour is therefore the only viable solution that allows for a fluid to smoothly pass between supersonic and subsonic flows while conserving the value of B .",
"pages": [
6,
7,
8,
9
]
},
{
"title": "2.3. Transitional Flows",
"content": "If we were free to choose the value of B to be always equal to or greater than B ts the steady solutions described in 2.2 would be sufficient. Since the fluid is at rest in the limb frame, in the intrusion frame the fluid velocity in the far field is given by u far = u in ˆ z = ˆ z C s M in , where u in is the speed at which the intrusion descends and M in is its Mach number. Since ( M in ) must be allowed to assume any real value we are forced to consider that the far field boundary condition might correspond to a subcritical flow. The inadmissibility, between -z crit and z crit , of a solution with the value of B fixed by the boundary condition demands that the overall solution be one in which B is not conserved. This solution will take the form of several regions of constant B , each connected by a transition in which Bernoulli's equation does not hold. The transitions are either shocks or rarefaction waves whose locations change with time. The complete flow combines two shocks, both propagating upstream, enclosing a transonic flow on which B = B ts , and then a rarefaction wave propagating downstream away from the intrusion. For a careful discussion of shocks and rarefaction waves see Chapters IX and X of Landau & Lifshitz (1959). The following is a more specific discussion, aimed at our particular problem.",
"pages": [
9
]
},
{
"title": "2.3.1. Shocks",
"content": "In cases where the length scale of the fluid becomes comparable to the ion mean free path the viscosity has a non-negligible contribution to the momentum equation and cannot be ignored as it was leading to Eq. (20). The resulting behavior is referred to as a shock, which is a thin transition from one value of B to another. In a reference frame co-moving with the shock the flow must be steady and conserve mass and momentum. In the isothermal case these conditions lead to a version of the Rankine-Hugoniot condition, amounting to conservation of across the jump, where M ' = ( u -u s ) /C s is the Mach number viewed from a frame moving at the shock speed u s . This conservation law differs from Eq. (20) because the jump is assumed so thin that viscosity cannot be ignored and α is approximately constant across it. Discounting the trivial case where the Mach number is unchanged, leads to the relation between upstream and downstream Mach numbers, M ' 1 and M ' 2 . It is evident that one of these will be subsonic while the other is supersonic. In terms of Mach numbers M j in the frame of the intrusion, Eq. (24) takes the form where M s = u s /C s is the Mach number of the shock. Knowing the upstream and downstream Mach numbers then gives the shock Mach number as assuming M 1 > M 2 > 0. The shock will move leftward ( M s < 0) if M 1 < 1 /M 2 , and rightward if M 1 > 1 /M 2 . Mass conservation, in the shock reference frame, then leads to the relation between pre-shock and post-shock density. A shock must have M ' 1 > 1 > M ' 2 , and therefore ρ 2 > ρ 1 : it is compressive.",
"pages": [
9,
10
]
},
{
"title": "2.3.2. Rarefaction waves",
"content": "A jump to lower density, not possible in a shock, must occur in a rarefaction wave. In cases without externally defined length scale the rarefaction wave will be self-similar (Landau & Lifshitz 1959, § 92), depending on space and time through a single similarity variable ( z -z 0 ) /t . A rarefaction wave is inherently time-dependent and so Bernoulli's equation is again invalid. In our solution, a shock and rarefaction wave will be generated simultaneously at t = 0 from the single point z = z crit . This initial state lacks a length scale and we may take the downstream rarefaction wave to be of the self-similar form. It will be bounded by weak discontinuities at its edges. The leading edge, at z = z 2 , propagates into the (unperturbed) downstream plasma at u 2 + C s . Upstream of this the velocity, and thus Mach number, is linear (Landau & Lifshitz 1959) Upstream of the trailing edge, at z = z 1 , the flow is again constant with M = M 1 < M 2 . Thus the extent of the rarefaction wave grows in time as ∆ z = ( M 2 -M 1 ) C s t , beginning as a discontinuity at t = 0. The initial discontinuity at z = + z crit decomposes into this rarefaction wave and a shock, in the manner of a Riemann problem (Landau & Lifshitz 1959). Within the rarefaction wave the density is an explicit function of velocity (see Landau & Lifshitz 1959, § 92) The upstream and downstream densities are therefore related by across the ever-expanding rarefaction wave. In order for this solution to apply the interior size of the rarefaction wave must be much smaller than the length scale of variation of the fluid cross-section, α . Fortunately, the rarefaction wave, while growing in time, propagates vary quickly into the far field so that no matter how large it becomes, the scale over which α varies is always larger still.",
"pages": [
10,
11
]
},
{
"title": "2.4. Composite Flow",
"content": "The complete solution, defined over the entire length of the affected flux tube, will be piecewise continuous using shocks, rarefaction waves and regions of pertistaltic flow so that the fluid velocity and density are treated in an internally consistent manner. The locations of the transition flows will travel along the length of the flux tubes in order to satisfy their governing equations and will therefore introduce time variations into the system despite our previous assumption of time independence. In letting this system evolve we are assuming that it can be treated as an adiabatic series of time-independent solutions. This assumption will be valid so long as the timescale over which a given feature evolves is long compared to its fluid crossing time. In order to form a solution we use physical consideration to motivate the choice of initial conditions in the region between the critical points. Far above and below the intrusion we demand that the plasma density and velocity be unchanged and continuity demands that every jump in velocity have a corresponding jump in density. We therefore require at least two jumps with at least one unspecified intermediate value of B in order to have sufficient degrees of freedom to satisfy the boundary conditions on both velocity and pressure, which is equivalent to density in the isothermal limit. If the fluid velocity in the far field is supersonic and lies below the transonic contour, then B in is subcritical and M will be ill-defined between the two critical points. 2 In order for the fluid to avoid an infinite acceleration at the upstream critical point there must be a transition away from B in and onto some well-behaved flow, B ∈ B + . In order for solution to remain well defined the transition must propagate upstream, away from the critical point. This is only possible in the case of a shock. A rarefaction wave would not propagate upstream with sufficient speed to escape the critical region. The downstream flow must therefore be subsonic in order for the system to be well defined at the upstream critical point. At the downstream critical point there must again be a transition to connect the flow that resulted from the upstream shock back to the original contour B in . If the interior flow were everywhere subsonic the jump would again have to be a rarefaction wave and would propagate at Mach 1 into the higher density, subsonic fluid, ultimately making its way into the critical region and leaving the downstream critical point again ill-behaved. Thus, in the downstream region, the flow that resulted from the upstream shock must be supersonic at the critical point. Only the transonic contour B ts can satisfy this condition without introducing yet another jump within the critical region. We therefore reach the conclusion, well known in nozzle problems, that the flow must cross from subsonic to supersonic at the throat. The density change across the leading shock, which connects the subcritical inflow to the transonic interior flow, is fixed by the relative speeds of the fluid on either side of the shock. At the downstream critical point the speed of the fluid is given by the transonic solution and the density is fixed through the continuity equation. Any transition from the transonic flow back to the original flow must therefore satisfy the disparity in both speed and density at this point, a feat not achievable for either a shock or a rarefaction wave. The jump at the downstream critical point must therefore decompose into both a shock and a rarefaction wave, just as in an asymmetric Riemann problem (Landau & Lifshitz 1959). The rarefaction wave then propagates at Mach 1 into the downstream fluid and therefore moves away from the intrusion at speeds in excess of Mach 2. The shock propagates subsonically upstream into the transonic flow, which is itself supersonic, and therefore moves more slowly away from the intrusion. Between the downstream shock and the rarefaction wave there is an initially infinitesimal intermediate region in which the fluid lies on an unspecified contour of B , which will be determined such that the net effect of the two downstream transitions exactly compensates for the upstream shock and transonic interior. The intermediate flow is supersonic but also slower than the initial, subcritical flow, so it too has critical points and these must be accounted for when the initial locations of the three transitions are chosen. The whole system is evolved by using the current velocity of each feature to determine its location at some future time and then constructing the new velocity and density profiles for the whole system at that time. This construct is shown in Figure 5 for the f = 0 . 5 field line with an inflow condition of M in = 1 . 5. The system is shown a short time after launch so that the transitions are spatially separated and can be easily distinguished. The original, subcritical flow is unphysical at z ≈ -R but the composite flow is transonic at this point and thus exhibits no critical phenomena. Relative to the intrusion, the upstream transonic flow and downstream intermediate flow are both slower than the subcritical flow in the far field. In the limb frame the plasma in these regions is actually descending toward the limb along with the intrusion. The two shocks similarly descend toward the limb with the leading shock pushing ahead and the trailing shock lagging ever farther behind while the rarefaction wave is sufficiently fast that it is not entrained with the intrusion and escapes rapidly upward. Note however that these shocks are not standoff shocks. They evolve in time and move steadily away from the intrusion, ultimately finding their way into the far field where their evolution slows and the assumption of timeindependence becomes increasingly exact.",
"pages": [
11,
12,
13
]
},
{
"title": "3. Results",
"content": "In order to gain insight into solar dynamics from this model we must construct synthetic observables which can be compared to actual observations. To this end we begin by constructing 2D maps of density and velocity, made by interpolating between a representative sample of field lines, each determined with the same boundary conditions. Features of the 1D fluid solution manifest in the 2D maps as broad fronts, and regions of high or low density. Figure 6 shows one such map of plasma density given at three successive times. From this example several features are visible. The leading shock, trailing shock and rarefaction wave are all distinguishable as abrupt changes in the plasma density. The high and low density 'head' and 'tail' grow in time and descend toward the limb while a slightly less rarefied region between the trailing shock and the rarefaction wave grows quickly upward as the rarefaction wave escapes away from the limb.",
"pages": [
13
]
},
{
"title": "3.1. Emission Measure",
"content": "As a proxy for synthetic images of the optically thin corona we choose the emission measure density ( /epsilon1 ∝ ρ 2 ). The emission measure profile will depend on the viewing angle. To begin with we consider a line of sight that is normal to the current sheet, consistent with many imaging observations of sheetlike structures above post-CME solar arcades ( ˇ Svestka et al. 1998; Gallagher et al. 2002; Innes et al. 2003; Savage & McKenzie 2011; Savage et al. 2012; McKenzie 2013). If the intrusion pierces normally through the current sheet as in Figure 1a then /epsilon1 can be constructed simply by squaring the 2D density maps such as in Figure 6. This viewing angle also applies to cases where the intrustion is imbedded within the current sheet (as in Figure 1b), which is itself viewed edge on. The resulting emission measure maps will exhibit the same features as Figure 6. A collection of four such emission measure maps is shown in Figure 7 for four different descending intrusions, each depicted at the instant of launch and then again after the shocks have propagated into the far field. At t = 0 the shocks trace out the loci of critical points for each field line. Then, as t →∞ the shock fronts move into the far field so that the shock column has infinite vertical extent both above and below the intrusion. Relative to the diameter of the intrusion, the width of the shocked column depends only on the speed of the intrusion M in , which dictates the fluid velocity in the far field. For intermediate speeds (between Mach 1.4 and and 1.7) the column width is of the same order as the intrusion diameter. The upper limit occurs as M in → 1 . 92 at which point all field lines become non-critical so the shocked column vanishes. In the opposite limit, as M in → 1 all field lines exhibit critical behavior and the shocked column becomes infinite in width but with vanishing amplitude in the far field. As an alternative we consider the system viewed from the side, such as if the line of sight were along ˆ y in Figure 1b. In this case the emission measure is constructed by integrating transversely across the 2D domain. The resulting profile will resemble that of an individual field line but will be somewhat smoother, having effectively averaged over all shocked field lines. We define the background-subtracted, normalized column emission measure as so that /epsilon1 ( z ) goes to zero for ρ ( y, z ) = ρ 0 and is negative or positive where ρ is depleted or enhanced with respect to the ambient plasma. Figure 8 shows a stack-plot of successive time-steps of /epsilon1 ( z, t ) for M in = 1 . 3. The propagating shocks and rarefaction wave can be seen as abrupt jumps in the red scale emission. To visualize how this kind of structure might manifest in the current sheet consider the case of a slightly sheared supra-arcade magnetic field. According to Guidoni & Longcope (2010), a local reconnection event will result in a growing, descending trapezoidal plateau that leads to something like Figures 1b and 9. The horizontal segment of the reconnected field, i.e. the intrusion, is embedded in the current sheet and drives peristaltic flows in the surrounding layers of field. As the plateau descends the bends move outward so that more of the reconnected flux is embedded in the current sheet. Because the field is sheared, we treat the two respective layers of magnetic field independently. They both exhibit peristaltic flows which result in an emission such as in Figure 8, but in one field the flow is slanted slightly to the right while the other is slanted to the left. The composite flows launch first on the field lines closest to the initial reconnection point. Then, as the plateau descends, those same field lines continue to evolve while field lines that are newly exposed to the growing plateau are initiated each in turn. The net result is a locus of shocked flows that grows as the plateau grows. Such a system is depicted in Figure 10. The unreconnected field is angled up and to the right in the foreground and down and to the right in the background. The layers of field that pass close to the horizontal segment of reconnected field exhibit peristaltic pumping. The emission measure on each field line is given by /epsilon1 (˜ z, t -t f ), where ˜ z is the distance along the angled field line and t f is the time at which that field line was initiated. The collection of lower shocks leads to a vaguely arch shaped high density region while the upper shock leads to a similar rarefied region. The antisunward rarefaction waves lead to a nearly vertical column of low density owing to the fact that these waves propagate supersonically outward along field lines at a rate comparable to the growth of the plateau.",
"pages": [
14,
15,
16
]
},
{
"title": "3.2. Momentum in the Fluid",
"content": "Since the descending intrusion generates plasma motion in the surrounding field, there should be associated energy and momentum transfer into that fluid. For any finite domain we can find the momentum in the plasma through numerical integration of the plasma density and velocity. Figure 11 displays a representative plot of momentum density. Due to the fact that the normalized cross section in the far field is asymptotic to but always greater than unity, the supersonic fluid far from the intrusion always propagates slightly faster than M in in the rest frame of the intrusion. In the limb frame this fluid is slowly rising so the momentum density far from the intrusion is always anti-sunward. Closer to the intrusion it is directed sunward as the cross section becomes constricted and the fluid is slowed below M in . Immediately above the lower shock the momentum density is strongly sunward and then becomes anti-sunward as the transonic flow passes abreast of the intrusion and again exceeds M in . It then becomes sunward again across the second shock before finally returning to the far-field limit across the rarefaction wave. To explore this more carefully we observe that the force per unit fluid cross section on a shocked flux tube may be found explicitly through where δ i represents the width of each jump. This can be calculated numerically for every field line within a finite domain under the assumption that all three jumps have propagated into the far field where the shock speeds become steady and α → 1. In Figure 12 we see that the net force on the fluid appears to be finite even in the limit of M → 1, where the shock becomes infinitely wide. In order to confirm this the contribution from the far field may be approximated analytically. This calculation is not included in the present work but it can be shown that the force contributed by the shocks in the far field vanishes with an inverse power of distance greater than unity so that, indeed, the net force on the fluid remains finite even as the shocks fill all of space.",
"pages": [
16,
17
]
},
{
"title": "3.3. Drag Force",
"content": "If the total momentum in our model contained only two contributing terms we could use Figure 12 as a proxy for the drag force on the intrusion. In actuality the far-field boundary conditions also contribute momentum to the system and the drag force must be calculated explicitly by integrating the plasma pressure over the surface of the intrusion. The pressure is related to the density, which can be found explicitly by calculating the behavior of fluid on the f = 0 streamline. The net vertical force due to the pressure p s ( θ ) on the surface S is then The plasma on the f = 0 field line is always on the transonic contour. The f = 0 streamline intersects the surface of the intrusion at the two magnetic null points θ = { 0 , π } , with θ measured here from -ˆ z . At these points the fluid cross section diverges as the magnetic field strength vanishes. For θ = π the fluid Mach number also diverges so ρ must be zero by continuity. For θ = 0 the plasma is subsonic so M goes to zero as α →∞ . To ensure that the density is well-behaved at this point we perform a series expansion around θ = 0 and find that ρ ( θ → 0) ≈ 3 . 3 ρ ts , where ρ ts is the plasma density on the transonic contour far from the intrusion. p ( θ ) is therefore well-behaved and can be calculated numerically as in Figure 13. The pressure decreases monotonically indicating an upward net force on the intrusion. The z -component of this force is given explicitly by the area under the dashed curve. In order to find how the drag force depends on M in we must find the pressure jump across the leading shock, which will dictate ρ ts and hence p ts . The shock velocity M s is determined by M in and M ts . But, in the far field M ts → 0 . 319 for f = 0, α = 1. Thus, ρ ts depends only on M in and, when multiplied by the integration factor from Figure 13, the resulting drag force can be found as depicted in Figure 14. The drag is lowest for the Mach 1 limit and increases almost linearly up to M in ≈ 1 . 92, at which point the column disappears. At and above Mach 1.92 the drag is zero due to the symmetry of the de Laval flow solutions, just as in D'Alembert's paradox. Below Mach 1 we have not calculated the drag profile but we expect, given the extent of the subsonic critical regime, that the drag will remain finite down to the minimum critical value of M min ≈ 0 . 32, at which point the shocked column again vanishes. As indicated, the drag curve in Figure 14 does not match the net force on the fluid from Figure 12. This is not surprising since the transonic flow is assymetric along the vertical direction and thus the centripetal force on each fluid element is unbalanced. Thus the magnetic field must deform asymmetrically in order to balance the fluid pressure. It follows that there must be some infinitesimal asymmetry between the far field magnetic field above and below the intrusion which in turn leads to a net force exerted on the system by the far field boundary conditions.",
"pages": [
18,
19
]
},
{
"title": "4. Discussion",
"content": "In this work we have shown how field line retraction following a local reconnection event can manifest as a descending constriction in the nearby unreconnected field. This constriction behaves in many respects as a peristaltic pump, which leads to peristaltic flows and ultimately to shocks and rarefaction waves that alter the velocity and density of plasma on affected field lines. These shocks are not to be confused with standoff shocks, which form at a fixed distance in front of a traveling obstacle and are thereafter stationary in time. The fluid jumps that we have described cannot exist as time-independent solutions and must necessarily propagate away from the intrusion. The region between these jumps therefore grows in time and is, in and of itself, a dynamic feature. The composite flows that form in this model are restricted to a column whose width is defined by the field-line that exhibits transonic flow for a particular boundary condition, M in . This width increases monotonically as the speed of the intrusion decreases toward the sound speed. The minimum width of zero occurs when the speed of the descending intrusion reaches M max = 1 . 92 while the maximum width is arbitrarily large as M in → 1. In this analysis we considered only supersonic values for M in but we acknowledge that the peristaltic flows will continue to exhibit critical behavior even for subsonic values of M in down to the point where B is again larger than B ts on the f = 0 field line. Our profiles for the 2D density and emission measure maps bear striking resemblance to observations of voids and Supra-Arcade Downflows (SADs) in post-CME flares (McKenzie 2000; Savage et al. 2012). In our analysis we considered an isothermal plasma in order to make the development more tractable. We offer, without proof, that an adiabatic plasma would exhibit the same qualitative behavior with the addition that plasma in the region between the lower shock and the rarefaction wave would exhibit an increase in temperature. The rarefied tail and high-emission leading edge may even be useful as thermal diagnostics since a temperature increase in the rarefied column could move the emission outside of a particular observation band-pass, thereby increasing the contrast in these features. We also considered an alternate geometry in which the particular shape of the emission profile is exchanged for a column integrated emission measure which occurs everywhere along the length of an embedded flux tube. This geometry also offers an interpretation for down-flowing features but may be more accurately used to describe how reconnection events and the contraction of reconnected flux can lead to heating of plasma along a broadly distributed volume of unreconnected field. Our model also helps to explain how a thermal halo (Seaton & Forbes 2009) might form around the current sheet. Here we have described only the creation of shocks along constricted field lines. But these shocks could very well travel down the unreconnected field all the way to the chromosphere where they would then drive evaporation exactly as in conduction dominated flare loops (Cargill et al. 1995). This evaporation might then increase the density on 'post-peristaltic' field lines, which could then undergo their own reconnection or even experience another 'peristaltic process' in the event of a second nearby reconnection event. We further describe how the alterations to velocity and density relate to the momentum density in the fluid and the subsequent net force (per unit embedded length) on the fluid. This force is related to but not equal to the net force on the retracting flux tube since a third contribution comes from the boundary conditions which maintain the field profile in the far field. The net force on the intrusion is found from a direct calculation of pressure integrated over its surface. This force points in the direction opposite the motion of the intrusion and is of order RC 2 s ρ in . It increases almost linearly by nearly a factor of two over the range 1 < M in < 2. Larger descent speeds correspond to a larger drag force so that, if this force is sufficient to influence the kinematics of the descending intrusion, the drag will decrease as the intrusion slows. This drag force offers a possible explanation to the fact that reconnection outflows appear to move subAlfv'enically despite the predictions of reconnection models such as described by Seaton & Forbes (2009). If reconnection outflows originate in locations where peristaltic shocks can form then this could lead to a drag force that would keep their velocities below the Alfv'en speed. However, if the outflow velocity ever exceeds the maximum shock velocity then the drag force should vanish according to our model. These loops would descend much more rapidly in its absence. This line of reasoning suggests there may be a bimodal distribution in the velocity of retracting magnetic loops. Loops that move fast enough to avoid launching peristaltic shocks would remain fast-moving while slower moving loops would be damped by the momentum transferred into the plasma. In order for the aforementioned drag force to have a non-negligible influence, the plasma pressure must be comparable to the magnetic energy density in the retracting flux tube. But pressure balance between the intrusion and its surroundings demands that the field strength must be comparable between the retracting flux and the unreconnected field. Thus, the drag force will only be significant if the plasma pressure is comparable to the magnetic energy density in the unreconnected field. While this is not unlikely in reality it is in conflict with the zeroβ assumption and therefore cannot be reconciled with our model in its current form. Our model assumes an extremely low β value in order to invoke the rigid magnetic field. However, observations suggest that this may not be an accurate assumption in the supra-arcade region (McKenzie 2013). It may be that by the time supra-arcade downflows become visibile in observations the local plasma β has already been increased due to previous instances of peristaltic pumping and that our model only applies to the early stages of flare activity, when the plasma density and temperature are still relatively low. Future work will therefore require the relaxation of the low β approximation, which will necessitate a numerical simulation. Another issue with the model is that we have been forced to stitch together time-independent solutions in an adiabatic fashion. The validity of this approximation, as well as those used in deriving the 1D MHD simplifications, will likewise need to be tested through simulations. When comparing to observations, some key differences are also apparent. Our model cannot reproduce the oscillatory behavior on the edges of voids as seen in Verwichte et al. (2005), although it does predict a discontinuity in plasma density transverse to the field, which could support surface modes if the zeroβ assumption were relaxed. Also, while we predict that these features should occur for M in /lessorapproxeql 2, Savage & McKenzie (2011) measured a typical downflow speed of on the order of 10 2 km s -1 with some instances of much higher values. Depending on the local sound speed these velocities may fall above our Mach 2 prediction. A more careful study of SAD speeds and the associated local sound speed will need to be conducted in order to refine this estimate. Moreover, the upper limit yielded by our simple model can be relaxed by generalizations to non-circular intrusions. As a first attempt we calculated that for elliptical intrusions the value of M max could be increased by a factor of nearly two before the aspect ratio of the ellipse became unrealistic. It may be that an appropriate choice of intrusion cross section could reconcile any lingering disparities between the model and observations of flare loops. Ultimately we intend to further the investigation with a regimen of numerical simulations. The ultimate success of this model will be in providing a theoretical framework for interpreting features seen in more complex numerical simulations.",
"pages": [
19,
20,
21
]
},
{
"title": "5. Acknowledgements",
"content": "The authors are grateful to K. Reeves and S. Savage for valuable comments on early drafts of the manuscript. This work was supported in part by NASA under contract NNM07AB07C with the Smithsonian Astrophysical Observatory and in part by a grant form the NSF/DOE Partnership in Basic Plasma Physics.",
"pages": [
21
]
},
{
"title": "REFERENCES",
"content": "Cargill, P. J., Chen, J., Spicer, D. S., & Zalesak, S. T. 1996, J. Geophys. Res., 101, 4855 Cargill, P. J., Mariska, J. T., & Antiochos, S. K. 1995, ApJ, 439, 1034 Cassak, P. A., Drake, J. F., Gosling, J. T., et al. 2013, ArXiv e-prints, arXiv:1307.3946 Choudhuri, A. R. 1998, The Physics of Fluids and Plasmas, An Introduction for Astrophysicists (New York: Cambridge University Press) Ciaravella, A., & Raymond, J. C. 2008, ApJ, 686, 1372 Cliver, E. W., & Hudson, H. S. 2002, Journal of Atmospheric and Solar-Terrestrial Physics, 64, 231 Costa, A., Elaskar, S., Fern'andez, C. A., & Mart'ınez, G. 2009, MNRAS, 400, L85 Gallagher, P. T., Dennis, B. R., Krucker, S., Schwartz, R. A., & Tolbert, A. K. 2002, Sol. Phys., 210, 341 Guidoni, S. E., & Longcope, D. W. 2010, ApJ, 718, 1476 Innes, D. E., McKenzie, D. E., & Wang, T. 2003, Sol. Phys., 217, 247 Ko, Y.-K., Raymond, J. C., Vrˇsnak, B., & Vuji'c, E. 2010, ApJ, 722, 625 Landau, L. D., & Lifshitz, E. M. 1959, Fluid Mechanics, Landau and Lifshitz Course of Theoretical Physics, Volume 6 (New York: Pergamon Press Ltd) Linton, M. G., & Longcope, D. W. 2006, ApJ, 642, 1177 Longcope, D. W., Guidoni, S. E., & Linton, M. G. 2009, ApJ, 690, L18 McKenzie, D. E. 2000, Sol. Phys., 195, 381 -. 2013, ApJ, 766, 39 McKenzie, D. E., & Hudson, H. S. 1999, ApJ, 519, L93 Priest, E. R. 1999, Ap&SS, 264, 77 Reeves, K. K., Linker, J. A., Miki'c, Z., & Forbes, T. G. 2010, ApJ, 721, 1547 Savage, S. L., & McKenzie, D. E. 2011, ApJ, 730, 98 Savage, S. L., McKenzie, D. E., & Reeves, K. K. 2012, ApJ, 747, L40 Savage, S. L., McKenzie, D. E., Reeves, K. K., Forbes, T. G., & Longcope, D. W. 2010, ApJ, 722, 329 Seaton, D. B., & Forbes, T. G. 2009, ApJ, 701, 348 Tucker, W. H. 1973, ApJ, 186, 285 ˇ Svestka, Z., F'arn'ık, F., Hudson, H. S., & Hick, P. 1998, Sol. Phys., 182, 179 Verwichte, E., Nakariakov, V. M., & Cooper, F. C. 2005, A&A, 430, L65 Warren, H. P., O'Brien, C. M., & Sheeley, Jr., N. R. 2011, ApJ, 742, 92 Webb, D. F., Burkepile, J., Forbes, T. G., & Riley, P. 2003, Journal of Geophysical Research (Space Physics), 108, 1440",
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2013ApJ...776...60B
https://arxiv.org/pdf/1308.5810.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_87><loc_88><loc_88></location>DEEP MULTI-TELESCOPE PHOTOMETRY OF NGC 5466. I. BLUE STRAGGLERS AND BINARY SYSTEMS.</section_header_level_1>
<text><location><page_1><loc_15><loc_81><loc_85><loc_86></location>G. BECCARI 1 , E. DALESSANDRO 2 , B. LANZONI 2 , F. R. FERRARO 2 , A. SOLLIMA 3 , M. BELLAZZINI 3 , P. MIOCCHI 2 1 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany, gbeccari@eso.org 2 Dipartimento di Fisica e Astronomia, Università degli Studi di Bologna, via Ranzani 1, I-40127 Bologna, Italy and 3 INAF-Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy</text>
<section_header_level_1><location><page_1><loc_46><loc_78><loc_54><loc_79></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_59><loc_86><loc_77></location>We present a detailed investigation of the radial distribution of blue straggler star and binary populations in the Galactic globular cluster NGC 5466, over the entire extension of the system. We used a combination of data acquired with the ACS on board the Hubble Space Telescope, the LBC-blue mounted on the Large Binocular Telescope, and MEGACAM on the Canadian-France-Hawaii Telescope. Blue straggler stars show a bimodal distribution with a mild central peak and a quite internal minimum. This feature is interpreted in terms of a relatively young dynamical age in the framework of the 'dynamical clock' concept proposed by Ferraro et al. (2012). The estimated fraction of binaries is ∼ 6 -7% in the central region ( r < 90 '' ) and slightly lower ( ∼ 5 . 5%) in the outskirts, at r > 200 '' . Quite interestingly, the comparison with the results of Milone et al. (2012) suggests that also binary systems may display a bimodal radial distribution, with the position of the minimum consistent with that of blue straggler stars. If confirmed, this feature would give additional support to the scenario where the radial distribution of objects more massive than the average cluster stars is primarily shaped by the effect of dynamical friction. Moreover, this would also be consistent with the idea that the unperturbed evolution of primordial binaries could be the dominant BSS formation process in low-density environments.</text>
<text><location><page_1><loc_14><loc_56><loc_86><loc_59></location>Subject headings: blue stragglers - binaries: general - globular clusters: general - globular clusters: individual(NGC 5466)</text>
<section_header_level_1><location><page_1><loc_22><loc_53><loc_34><loc_54></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_7><loc_48><loc_52></location>Galactic globular clusters (GCs) are dynamically active systems, where stellar interactions and collisions, especially those involving binaries, are quite frequent (e.g. Hut et al. 1992; Meylan & Heggie 1997) and generate populations of exotic objects, like X-ray binaries, millisecond pulsars, and Blue straggler stars (BSSs; e.g., Paresce et al. 1992; Bailyn 1995; Bellazzini et al. 1995; Ferraro et al. 1995, 2009;Ransom et al. 2005; Pooley & Hut 2006). Among these objects, BSSs are the most abundant and therefore may act as a crucial probe of the interplay between stellar evolution and stellar dynamics (e.g. Bailyn 1995). They are brighter and bluer (hotter) than main sequence (MS) turnoff (TO) stars and are located along an extrapolation of the MS in the optical color-magnitude diagram (CMD). Thus, they mimic a rejuvenated stellar population, with masses larger than those of normal cluster stars ( M /greaterorsimilar 1 -1 . 2 M /circledot ; this is also confirmed by direct measurements; Shara et al. 1997; Gilliland et al. 1998). This clearly indicates that some process which increases the initial mass of single stars must be at work. The most probable formation mechanisms of these peculiar objects are thought to be either the mass transfer (MT) activity between binary companions, even up to the complete coalescence of the system (McCrea 1964), or the merger of two single or binary stars driven by stellar collisions (COL; Zinn & Searle 1976; Hills & Day 1976). Unfortunately, it is still a quite hard task to distinguish BSSs formed by either channel. The most promising route seems to be high resolution spectroscopy, able to identify chemical anomalies (as a significant depletion of carbon and oxygen) on the BSS surface (see Ferraro et al. 2006a), as it is expected in the MT scenario. Also the recent discovery of a double sequence of BSSs in M30 (Ferraro et al. 2009) seems to indicate that, at least in GCs that recently suffered the core collapse, COL-BSSs can be distinguished from MTBSSs on the basis of their position in the color-magnitude di-</text>
<text><location><page_1><loc_52><loc_53><loc_62><loc_54></location>agram (CMD).</text>
<text><location><page_1><loc_52><loc_21><loc_92><loc_53></location>In general, since collisions are most frequent in high density regions, BSSs in different environments could have different origins: those in loose GCs might arise from the coalescence/mass-transfer of primordial binaries, while those in high density clusters might form mostly from stellar collisions (e.g., Bailyn 1992; Ferraro et al. 1995). However, Knigge et al. (2009) suggested that most BSSs, even those observed in the cores of high density GCs, formed from binary systems, although no strong correlation between the number or specific frequency of BSSs and that of binaries is found if all clusters (including the post-core collapse ones) are taken into account (Milone et al. 2012; Leigh et al. 2013). This is likely due to the fact that dynamical processes significantly modify the binary and the BSS content of GCs during their evolution. An exception seems to be represented by low density environments, where the efficiency of dynamical interactions is expected to be negligible. Very interestingly, indeed, a clear correlation between the binary and the BSS frequencies has been found in a sample of 13 low density GCs (Sollima et al. 2008). This remains the cleanest evidence that the unperturbed evolution of primordial binaries is the dominant BSS formation process in low-density environments (also consistently with the results obtained in open clusters; e.g. Mathieu &Geller 2009).</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_21></location>One of the most notable effects of stellar dynamics on BSSs is the shaping of their radial distribution within the host cluster. In most cases a clear bimodality has been observed in the radial distribution of the N BSS / N ref ratio ( N BSS and N ref being, respectively, the number of BSSs and the number of stars belonging to a reference population, as red giant or horizontal branch stars): such a ratio shows a high peak at the cluster centre, a dip at intermediate radii and a rising branch in the external regions (e.g. Beccari et al. 2008; Dalessandro et al. 2009). In a few other GCs (see, for example, NGC 1904 and</text>
<text><location><page_2><loc_8><loc_55><loc_48><loc_92></location>M75; Lanzoni et al. 2007a; Contreras Ramos et al. 2012, respectively) the radial distribution shows a clear central peak but no upturn in the external cluster regions. In the case of ω Centauri (Ferraro et al. 2006b), NGC 2419 (Dalessandro et al. 2008) and Palomar 14 (Beccari et al. 2011) the BSS radial distribution is flat all over the entire cluster extension. Since BSSs are more massive than the average cluster stars (and therefore suffer from the effects of dynamical friction), these different observed shapes have been interpreted as due to different 'dynamical ages' of the host clusters (Ferraro et al. 2012, hereafter F12): a flat BSS distribution is found in dynamically young GCs ( Family I ), where two-body relaxation has been ineffective in establishing energy-equipartition in a Hubble time and dynamical friction has not segregated the BSS population yet. In clusters of intermediate ages ( Family II ), the innermost BSSs have already sunk to the center, while the outermost ones are still evolving in isolation (this would produce the observed bimodality); finally, in the most evolved GCs ( Family III ), dynamical friction has already segregated toward the centre even the most remote BSSs, thus erasing the external rising branch of the distribution. These results demonstrate that the BSS radial distribution can be used as powerful dynamical clock. Of course, the same should hold for binary systems with comparable masses, but the identification of clean and statistically significant samples of binaries all over the entire cluster extension is a quite hard task and no investigations have been performed to date to confirm such an expectation.</text>
<text><location><page_2><loc_8><loc_31><loc_48><loc_55></location>NGC5466 is a high galactic latitude GC ( l = 42 . 2 · and b = 73 . 6 · ), located in the constellation of Boötes at a distance of D = 16 kpc (Harris 1996). The cluster has been recently found to be surrounded by huge tidal tails (Belokurov et al. 2006; Grillmair & Johnson 2006). The BSS stars in this cluster were first studied by Nemec & Harris (1987), and, more recently, Fekadu et al. (2007) analyzed the BSS population inside the first r ∼ 800 '' from the cluster center. Since the cluster tidal radius is 1580 '' (Miocchi et al. 2013), a survey of the BSS population over the whole body of the cluster is still missing. In the framework of the study of the origin of the BSS, it is important to mention that NGC 5466 is the first GCwhere eclipsing binaries were found among the BSS (Mateo et al. 1990, see also Kryachko et al.(2011)). In their paper Mateo et al. (1990) find indications that all the BSS in NGC 5466 were formed from the merger of the components of primordial, detached binaries evolved into compact binary systems.</text>
<text><location><page_2><loc_8><loc_25><loc_48><loc_31></location>In this paper we intend to study the BSS and binary populations of NGC 5466, focussing in particular on their radial distribution within the whole cluster to investigate its dynamical age.</text>
<text><location><page_2><loc_8><loc_16><loc_48><loc_25></location>The paper is organized as follows. In Section 2 we describe the observations and data analysis. Section 3 is devoted to the definition of the BSS population and the study of its radial distribution. In Section 4 we describe how the fraction of binaries and its variation with radius are investigated. The summary and discussion of the obtained results are presented in Section 5.</text>
<section_header_level_1><location><page_2><loc_16><loc_14><loc_41><loc_15></location>2. OBSERVATIONS AND DATA ANALYSIS</section_header_level_1>
<text><location><page_2><loc_8><loc_7><loc_48><loc_13></location>The photometric data used here consist of a combination of shallow and deep images (see Table 1) acquired through the blue channel of the Large Binocular Camera (LBC-blue) mounted on the Large Binocular Telescope (LBT), MEGACAMonthe Canadian-France-Hawaii Telescope (CFHT) and</text>
<text><location><page_2><loc_52><loc_89><loc_92><loc_92></location>the Advanced Camera for Survey (ACS) on board the Hubble Space Telescope (HST).</text>
<text><location><page_2><loc_52><loc_61><loc_92><loc_89></location>(i)-The LBC data-set : both short and long exposures were secured in 2010 under excellent seeing conditions (0 . '' 5-0 . '' 7) with the LBC-blue (Ragazzoni et al. 2006; Giallongo et al. 2008) mounted on the LBT (Hill et al. 2006) at Mount Graham, Arizona. The LBC is a double wide-field imager which provides an effective 23 ' × 23 ' field of view (FoV), sampled at 0 . 224 arcsec/pixel over four chips of 2048 × 4608 pixels each. LBC-blue is optimized for the UV-blue wavelengths, from 320 to 500 nm, and is equipped with the U -BESSEL , B -BESSEL , V -BESSEL , g -SLOAN and r -SLOAN filters (hereafter U , B , V , g , r , respectively). A set of short exposures (of 5, 60 and 90 seconds) was secured in the B and V filters with the cluster center positioned in the central chip (namely #2) of the LBC-blue CCD mosaic. Deep images (of 400 and 200 s in the B and V bands, respectively) were obtained with the LBC-blue FoV positioned ∼ 100 '' south from the cluster centre. A dithering pattern was adopted in both cases thus to fill the gaps of the CCD mosaic. The raw LBC images were corrected for bias and flat field, and the overscan region was trimmed using a pipeline specifically developed for LBC image pre-reduction from the LBC Survey data center 1 .</text>
<text><location><page_2><loc_52><loc_47><loc_92><loc_61></location>(ii)-The MEGACAM data-set : g and r wide-field MEGACAM images (Prop ID: 04AK03; PI: Jang-Hyun Park), acquired with excellent seeing conditions (0 . '' 6-0 . '' 8) were retrieved from the Canadian Astronomy Data Centre (CADC 2 ). Mounted at CFHT (Hawaii), the wide-field imager MEGACAM (built by CEA, France), consists of 36 2048 × 4612 pixel CCDs (a total of 340 megapixels), fully covering a 1 degree × 1 degree FoV with a resolution of 0.187 arcsecond/pixel. The data were preprocessed (removal of the instrumental signature) and calibrated (photometry and astrometry) by Elixir pipeline 3 .</text>
<text><location><page_2><loc_52><loc_36><loc_92><loc_47></location>(iii)-The ACS data-set consists of a set of high-resolution, deep images obtained with the ACS on board the HST through the F606W and F814W filters (GO-10775;PI: Sarajedini). The cluster was centered in the ACS field and observed with one short and five long-exposures per filter, for a total of two orbits. We retrieved and used only the deep exposures (see Table 5). All images were passed through the standard ACS/WFC reduction pipeline.</text>
<text><location><page_2><loc_52><loc_20><loc_92><loc_36></location>As shown in the left panel of Figure 1, the shallow LBC data-set is used to sample a region of 600 '' around the cluster center, and for r > 600 '' it is complemented with the MEGACAMobservations, which extend beyond the tidal radius, out to r ∼ 30 ' . In the following the combination of these two datasets will be referred to as the 'Shallow Sample' and adopted to study the BSS population over the entire cluster extension (see Sect. 3). The ACS observations sample the innermost ∼ 200 '' of the cluster and they are complemented with the deep LBC observations, extending out to rt (see right panel of Figure 1): this will be called the 'Deep Sample' and used to investigate the binary fraction of NGC 5466 (see Sect. 4).</text>
<section_header_level_1><location><page_2><loc_61><loc_18><loc_82><loc_19></location>2.1. Photometry and Astrometry</section_header_level_1>
<text><location><page_2><loc_52><loc_12><loc_92><loc_17></location>Given the very low crowding conditions in the external cluster regions, we performed aperture photometry on the two g and r MEGACAM images, by using SExtractor (Bertin & Arnouts 1996) with an aperture diameter of 0 . 9 '' (correspond-</text>
<text><location><page_3><loc_8><loc_72><loc_48><loc_92></location>to a FWHM of 5 pixels). Each of the 36 chips was reduced separately, and we finally obtained a catalog listing the relative positions and magnitudes of all the stars in common between the g and r catalogs. The LBC and ACS data-sets were reduced through a standard point spread function (PSF) fitting procedure, by using DAOPHOTII/ALLSTAR (Stetson 1987, 1994). The PSF was independently modeled on each image using more than 50 isolated and well sampled stars in the field. The photometric catalogs of each pass-band were combined, and the instrumental magnitudes were reported to a reference frame following the standard procedure (see, e.g., Ferraro et al. 1991, 1992). The magnitudes of stars successfully measured in at least three images per filter were then averaged, and the error about the mean was adopted as the associated photometric uncertainty.</text>
<text><location><page_3><loc_8><loc_55><loc_48><loc_72></location>In order to obtain an absolute astrometric solution for each of the 36 MEGACAM chips, we used more than 15000 stars in common with the Sloan Digital Sky Survey (SDSS) catalog 4 covering an area of 1 square degree centered on the cluster. The final r.m.s. global accuracy is of 0 . 3 '' , both in right ascension ( α ) and declination ( δ ). The same technique applied to the LBC sample gave an astrometric solution with similar accuracy ( σ < 0 . 3 '' r.m.s.). Considering that no standard astrometric stars can be found in the very central regions of the cluster, we used the stars in the LBC catalog as secondary astrometric standards and we thus determined an absolute astrometric solution for the ACS sample in the core region with the same accuracy obtained in the previous cases.</text>
<text><location><page_3><loc_8><loc_29><loc_48><loc_54></location>A sample of bright isolated stars in the ACS data-set was used to transform the ACS instrumental magnitudes to a fixed aperture of 0 . '' 5. The extrapolation to infinite and the transformation into the VEGAMAG photometric system was then performed using the updated values listed in Tables 5 and 10 of Sirianni et al. (2005) 5 . In order to transform the instrumental B and V magnitudes of the LBC sample into the Johnson/Kron-Cousins standard system, we used the stars in common with a photometric catalog previously published by Fekadu et al. (2007). These authors, in section 2.3 of their paper, provide a thorough comparison of their photometry with existing literature datasets, finding a satisfactory agreement. In particular they found an excellent agreement in the photometric zero-points (within less than 0.02 mag) and no residual trends with colors with the set of secondary standards by Stetson (2000), that should be considered as the most reliable reference. We verified, by direct comparison with Stetson (2000), that the quality of the agreement shown by Fekadu et al. (2007) is fully preserved in our final catalog.</text>
<text><location><page_3><loc_8><loc_14><loc_48><loc_29></location>Instead, the g and r magnitudes in the MEGACAM sample were calibrated using the stars in common with the SDSS catalog and we then adopted the equation that Robert Lupton derived by matching DR4 photometry to Peter Stetson's published photometry for stars 6 to transform the calibrated g and r into a V magnitude. We finally used more than 2000 stars in common between the LBC and the MEGACAM data-sets to search for any possible off-sets between the V magnitude in the two catalogs, finding an average systematic difference of 0.014 mag. We therefore applied this small correction to the MEGACAM catalog. Similarly, we used more than 200 stars in common between the ACS data-set and the LBC cata-</text>
<unordered_list>
<list_item><location><page_3><loc_10><loc_11><loc_47><loc_12></location>4 Available at web-site http://cas.sdss.org/dr6/en/tools/search/radial.asp</list_item>
<list_item><location><page_3><loc_8><loc_8><loc_48><loc_11></location>5 The new values are available at the STScI web pages: http://www.stsci.edu/hst/acs/analysis/zeropoints</list_item>
</unordered_list>
<text><location><page_3><loc_52><loc_87><loc_92><loc_92></location>log to calibrate the ACS F606W magnitude to the Johnson V magnitude. Such a procedure therefore provided us with a homogenous V magnitude scale in common to all the available data-sets.</text>
<section_header_level_1><location><page_3><loc_60><loc_84><loc_84><loc_85></location>2.2. The Color-Magnitude Diagrams</section_header_level_1>
<text><location><page_3><loc_52><loc_73><loc_92><loc_83></location>The CMD obtained from the shallow LBC exposures for the innermost 600 '' from the cluster center is shown in Figure 2. Typical photometric errors (magnitudes and colors) are indicated by black crosses. The high quality of the images allows us to accurately sample all the bright evolutionary sequences, from the Tip of the red giant branch (RGB), down to ∼ 2 magnitudes below the MS-TO, with a large population of BSSs well visible at 0 < ( B -V ) < 0 . 4 and 20 . 5 < V < 18.</text>
<text><location><page_3><loc_52><loc_54><loc_92><loc_73></location>In Figure 3 we show the CMDs obtained from the ACS and the deep LBC observations. As in Figure 2, the typical photometric errors (magnitudes and colors) for the two sample are indicated by black crosses. A well populated MS is visible in the ACS CMD, extending from the MS-TO ( V ∼ 20 . 5) down to V ∼ 27. Once again, the collecting power of the LBT combined with the LBC-blue imaging capabilities allowed us to sample the MS down to similar magnitudes in an area extending out to the cluster's tidal radius. For ( B -V ) /greaterorsimilar 1 . 5 the plume of Galactic M dwarfs is clearly visible, while for ( B -V ) /lessorsimilar 1 and V /greaterorsimilar 24 a population of unresolved background (possibly extended) objects is apparent. In both the CMDs a well defined sequence of unresolved binary systems is observed on the red side of the MS.</text>
<section_header_level_1><location><page_3><loc_55><loc_52><loc_89><loc_53></location>3. THE SHALLOW SAMPLE AND THE BSS POPULATION</section_header_level_1>
<text><location><page_3><loc_52><loc_39><loc_92><loc_51></location>The Shallow Sample was used to determine the cluster center of gravity ( C grav) and the projected density profile from accurate star counts 7 , following the procedure already adopted and described, e.g., in Beccari et al. (2012). The resulting value of C grav is α J2000 = 14 h 05 m 27 s . 2, δ J2000 = + 28 · 32 ' 01 . '' 8, in full agreement with that quoted by Goldsbury et al. (2010). The best fit King model to the observed density profile has concentration parameter c = 1 . 31, core radius rc = 72 '' and tidal radius rt = 1580 '' = 26 . 3 ' (Miocchi et al. 2013). 8</text>
<text><location><page_3><loc_52><loc_22><loc_92><loc_39></location>Fekadu et al. (2007) studied the radial distribution of the BSS population in NGC 5466, from the cluster centre out to r ∼ 800 '' (see also Nemec & Harris 1987). They selected 89 BSSs and showed that they are significantly more centrally concentrated than RGB and horizontal branch (HB) stars, but they did not find any evidence of upturn in the external regions. However, that study did not sample the entire radial extension of the cluster, while our previous experience demonstrates that even a few BSSs in the outer regions can generate the external rising branch. Hence, a detailed investigation even in the most remote regions is necessary before solidly excluding the presence of an upturn in the BSS radial distribution and confirming the dynamical age of the cluster.</text>
<section_header_level_1><location><page_3><loc_58><loc_19><loc_86><loc_20></location>3.1. BSS and reference population selection</section_header_level_1>
<text><location><page_3><loc_52><loc_16><loc_92><loc_19></location>The selection of the BSS population was primarily performed in the ( V , B -V ) plane (see open circles in Figure 4,</text>
<text><location><page_3><loc_52><loc_12><loc_92><loc_15></location>7 The observed profile is shown in Miocchi et al. (2013), as part of a catalog of 26 Galactic GCs for which also the best-fit King (1966) and Wilson (1975) model fits are discussed.</text>
<text><location><page_4><loc_8><loc_68><loc_48><loc_92></location>left panel), by adopting conservative limits in magnitude and color in order to avoid the possible contamination from stellar blends. The BSS selection has been then 'exported' to the MEGACAM ( V , V -r ) CMD by using the BSSs in common between the two data-sets in the region at r < 600 '' (see right hand panel). We finally count a population of 88 stars in the LBC sample and 9 stars in the MEGACAM one, for a total of 97 BSSs spanning the entire radial extent of the cluster. The cross-correlation of our sample with that of Fekadu et al. (2007) confirms that the majority of the stars selected in the previous work are real BSSs, but 18 of them are blended sources. In order to further test the completeness of our LBC shallow sample we performed a detailed comparison with the ACS photometric catalogue of Sarajedini et al. (2007) 9 . The completeness has been quantified as the fraction of stars in common between the two catalogs in a given magnitude interval: we find 93%, 95%, and 100% completness for stars with V < 22, 21.5, and 21, respectively.</text>
<text><location><page_4><loc_8><loc_56><loc_48><loc_68></location>For a meaningful study of the BSS population, a reference sample of 'normal' cluster stars has to be defined. Here we choose a post-MS population composed of sub-giant (SGB) and red giant branch stars (RGB; see grey triangles in Fig. 4). This has been selected in the same magnitude range (18 . 4 < V < 20) of BSSs, thus guaranteeing that the photometric completeness is the same in both samples. We count 1358 and 96 stars in the LBC and MEGACAM data-sets, respectively.</text>
<section_header_level_1><location><page_4><loc_18><loc_54><loc_39><loc_55></location>3.2. The BSS radial distribution</section_header_level_1>
<text><location><page_4><loc_8><loc_46><loc_48><loc_53></location>As extensively illustrated in literature (see for example Bailyn 1995; Ferraro et al. 2003, 2012, and references therein), the study of the BSS radial distribution is a very powerful tool to shed light on the internal dynamical evolution of GCs and possibly gets some hints on the BSS formation mechanisms.</text>
<text><location><page_4><loc_8><loc_28><loc_48><loc_46></location>As a first step, we compared the cumulative radial distribution of the selected BSS population to that of reference stars (see Figure 5). Clearly, BSSs are more centrally concentrated, and a Kolmogorov-Smirnov test yields a probability of 99 . 9% that the two samples are not extracted from the same parent population. This cumulative BSS distribution closely resembles that already observed in this cluster by Fekadu et al. (2007) out to ∼ 800 '' (see their Figure 19), with the only difference that we sample the entire cluster extension. Moreover, the distributions shown in Figure 5, closely resembles that observed in all GCs characterised by a bimodal radial distribution of BSSs (see, e.g., the cases of M3, Ferraro et al. 1997, 47Tuc, Ferraro et al. 2004; M53, Rey et al. 1998; M5, Lanzoni et al. 2007b).</text>
<text><location><page_4><loc_8><loc_11><loc_48><loc_28></location>Following the same procedure described in Lanzoni et al. (2007b), as a second step we divided the surveyed area in a series of concentric annuli centred on C grav. In Table 2 we report the values of the inner and outer radii of each annulus, together with the number of BSSs ( N BSS) and reference stars ( N RGB) counted in each radial bin. We also report the fraction of light sampled in the same area, with the luminosity being calculated as the sum of the luminosities of all the stars measured in the TO region at 19 . 5 /lessorequalslant V /lessorequalslant 20 . 5. The contamination of the cluster BSS and RGB populations due to background and foreground field stars can be severe, especially in the external regions where number counts are low. We therefore exploited the wide area covered by MEGACAM</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_9></location>9 The catalogue is available for download at http://www.astro.ufl.edu/ ∼ ata/public_hstgc/</text>
<text><location><page_4><loc_52><loc_81><loc_92><loc_92></location>(see Figure 1) to estimate the field contamination by simply counting the number of stars located beyond the cluster tidal radius (at r > 1580 '' ) that populate the BSS and RGB selection boxes in the CMD. We count 2 BSSs and 13 RGB stars in a ∼ 0 . 43 deg 2 area. The corresponding densities are used to calculate the number of field stars expected to contaminate each population in every considered radial bin (see numbers in parenthesis in Table 2).</text>
<text><location><page_4><loc_52><loc_69><loc_92><loc_81></location>In the upper panel of Figure 6 we show the resulting trend of N BSS / N RGB as a function of radius. Clearly it is bimodal, with a relatively low peak of BSSs in the inner region, a distinct dip at intermediate radii, and a constant value slightly smaller than the central one, in the outskirts. We identify the minimum of the radial distribution at r min ∼ 180 '' . This behavior is further confirmed by the radial distribution of the BSS double normalized ratio, R BSS (see the black circles in the lower panel of Figure 6).</text>
<text><location><page_4><loc_52><loc_52><loc_92><loc_69></location>For each radial bin, R BSS is defined as the ratio between the fraction of BSSs in the annulus and the fraction of light sampled by the observations in the same bin (see Ferraro et al. 2003). The same quantity computed for the reference population turns out to be equal to one at any radial distance (see grey regions in the figure). This is indeed expected from the stellar evolution theory (Renzini & Buzzoni 1986) and it confirms that the distribution of RGB stars follows that of the cluster sampled light. Instead, R BSS shows a bimodal behavior, confirming the result obtained above. Notice that NGC 5466 is one of the most sparsely populated Galactic GCs, and so the number of cluster's stars and the fraction of sampled light in the external regions is quite low.</text>
<section_header_level_1><location><page_4><loc_53><loc_49><loc_91><loc_50></location>4. THE DEEP SAMPLE AND THE CLUSTER BINARY FRACTION</section_header_level_1>
<text><location><page_4><loc_52><loc_14><loc_92><loc_49></location>Binaries play an important role in the formation and dynamical evolution of GCs (McMillan 1991). Indeed these systems secure a enormous energy reservoir able to quench masssegregation and delay or even prevent the collapse of the core. Up to now three main techniques have been used to measure the binary fraction: (i) radial velocity variability (e.g., Mathieu & Geller 2009), (ii) searches for eclipsing binaries (e.g. Mateo 1996; Cote et al. 1996) and (iii) studies of the distribution of stars along the cluster MS in CMD (e.g. Romani & Weinberg 1991; Bellazzini et al. 2002; Milone et al. 2012). In this work we adopt the latter approach, which has the advantage of being more efficient for a statistical investigation, and unbiased against the orbital parameters of the systems. We followed the prescriptions extensively described in Bellazzini et al. (2002) and Sollima et al. (2007, 2010, see also Dalessandro et al. 2011). This method relies on the fact that the luminosity of a binary system is the result of the contributions of both (unresolved) companions. The total luminosity is therefore given by the luminosity of the primary (having mass M 1) increased by that of the companion (with mass M 2). Along the MS, where stars obey a mass-luminosity relation, the magnitude increase depends on the mass ratio q = M 2 / M 1, reaching a maximum value of 0.75 mag for equal mass binaries ( q = 1). Taking advantage of the large FoV covered by our data-sets, we analyzed the binary fraction at different radial distances from the cluster center, out to r ∼ 800 '' .</text>
<section_header_level_1><location><page_4><loc_64><loc_11><loc_79><loc_12></location>4.1. Completeness tests</section_header_level_1>
<text><location><page_4><loc_52><loc_7><loc_92><loc_11></location>In order to estimate the fraction of binaries from the 'secondary' MS in the CMD, it is crucial to have a realistic and robust measure of other possible sources of broadening of the</text>
<text><location><page_5><loc_8><loc_85><loc_48><loc_92></location>MS (e.g. blending, photometric errors). These factors are related to the quality of the data and can be properly studied through artificial star experiments. We therefore produced a catalog of artificial stars for the ACS and LBC-deep data-sets, following the procedure described in Bellazzini et al. (2002).</text>
<text><location><page_5><loc_8><loc_69><loc_48><loc_85></location>As a first step, a list of input positions and magnitudes of artificial stars (X in ,Y in , Vin , F 814 Win and Bin ) is produced. Once these stars are added to the original frames at the X in and Y in positions, we repeat the measure of stellar magnitudes in the same way as described in Section 2.1. At the end of this procedure we obtain a list of stars including both real and artificial stars. Since we know precisely the positions and magnitudes of the artificial stars, the comparison of the latter with the measured Vout , F 814 Wout and Bout magnitudes, allow us to estimate the capability of our data-reduction strategy to properly detect stars in the images, including the effect of blending.</text>
<text><location><page_5><loc_8><loc_45><loc_48><loc_69></location>In practice, we first generated a catalog of simulated stars with a Vin magnitude randomly extracted from a luminosity function modeled to reproduce the observed one. Then the F 814 Win and Bin magnitudes were assigned to each sampled Vin magnitude by interpolating the mean ridge line of the cluster (solid grey lines in the left and right panels of Figure 3, for the ACS and the LBC-deep samples, respectively). The artificial stars were then added on each frame using the same PSF model calculated during the data reduction phase, and were spatially distributed on a grid of cells of fixed width (5 times larger than the mean FWHM of the stars in the frames). Only one star was randomly placed in such a box in each run, in order not to generate artificial stellar crowding from the simulated stars on the images. The artificial stars were added to the real images using the DAOPHOT/ADDSTAR routine. At the end of this step, we have a number of 'artificial' frames (which include the real stars and the artificial stars with know position in the frame) equal to the number of observed frames.</text>
<text><location><page_5><loc_8><loc_37><loc_48><loc_45></location>The reduction process was repeated on the artificial images in exactly the same way as for the scientific ones. A total of 100,000 stars were simulated on each data-set and photometric completeness ( φ comp) was then calculated as the ratio between the number of stars recovered by the photometric reduction ( N out) and the number of simulated stars ( N in).</text>
<text><location><page_5><loc_8><loc_20><loc_48><loc_37></location>In Figure 7, we show the photometric completeness estimated for the ACS and the LBC-deep data-sets (upper and lower panels, respectively) as a function of the V magnitude and for different radial regions around the cluster center. The completeness of the ACS sample is above 80% for all magnitudes down to V ∼ 26 (i.e., ∼ 5 . 5 . magnitudes below the cluster TO), even in the innermost region. On the other hand, a completeness above 50% down to V ∼ 24 is guaranteed only at r > 200 '' in the LBC-deep data-set. Instead, stellar crowding is so strong at lower radial distances (region D) that the completeness stays below ∼ 80%at all magnitudes. We therefore exclude this region (120 '' < r < 200 '' ) from our study of the cluster binary fraction.</text>
<section_header_level_1><location><page_5><loc_20><loc_18><loc_36><loc_19></location>4.2. The binary fractions</section_header_level_1>
<text><location><page_5><loc_8><loc_11><loc_48><loc_17></location>As mentioned at the beginning of Section 4, our estimation of the binary fraction in NGC 5466 is performed by determining the fraction of binaries populating the area between the MS and a secondary MS, blue-ward shifted in colour by ∼ 0 . 75 (Hurley & Tout 1998).</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_11></location>As a first step of our analysis, we measured the minimum binary fraction, i.e. the fraction of binaries with mass ratios large enough ( q > q min) that they can be clearly distinguished</text>
<text><location><page_5><loc_52><loc_79><loc_92><loc_92></location>from single MS stars in the CMD. When q approaches the null value, the shift in magnitude of the primary star (i.e. the more massive one) is negligible. For this reason binary systems with small values of q becomes indistinguishable from MS stars when photometric errors are present. Clearly, this is a lower limit to the total binary fraction of the cluster. However, it has the advantage of being a pure observational quantity, since it is calculated without any assumption on the distribution of mass ratios (Sollima et al. 2007; Dalessandro et al. 2011).</text>
<text><location><page_5><loc_52><loc_67><loc_92><loc_78></location>The magnitude interval in which the minimum binary fraction has been estimated is 21 < V < 24 for both the ACS and the LBC samples. According to a 12 Gyr model from Dotter et al. (2007) 10 , this corresponds to a mass range 0 . 5 M /circledot /lessorsimilar M /lessorsimilar 0 . 74 M /circledot for a single star on the MS. In this interval the photometric completeness φ comp is larger than 50% for both samples. For the magnitude and radial intervals just defined, we divided the entire FoV in four annuli (see Table 3 for details).</text>
<text><location><page_5><loc_52><loc_43><loc_92><loc_66></location>Then, the location of q min was estimated on the CMD as the value corresponding to a color difference of three times the photometric error from the MS ridge mean line (grey solid lines in the cluster's CMDs shown in Figure 3). This approach allows us to define an area on the CMD safe from contamination from single mass stars. We notice that for each of the four annuli, this color difference corresponds to q min = 0 . 5. The contamination from blended sources was calculated on the same region of the CMD by adopting the catalogue of artificial stars produced with the completeness tests (see Section 4.1). A catalogue of stars from the Galaxymodel of Robin et al. (2003) was used to estimate the field contamination. We thus estimated ξ q ≥ 0 . 5 independently for the four radial intervals using the same approach described in Sollima et al. (2007). As shown in Figure 8 (upper panel), we find that ξ q ≥ 0 . 5 is almost constant and equal to ∼ 6 . 5%for r /lessorsimilar 90 '' (i.e., within the ACS sample), then it drops at ∼ 5% at distances larger than 200 '' (in the LBC sample).</text>
<text><location><page_5><loc_52><loc_12><loc_92><loc_43></location>As described above, the minimum binary fraction has the advantage of being measured with no arbitrary assumptions about the mass ratio distribution f ( q ). It depends, however, on the photometric accuracy of the used data-sets, and caution must be exercised when comparing the minimum binary fractions estimated for different GCs and in different works. An alternative approach consists in measuring the global fraction of binaries ( ξ | rmTOT ), that can be obtained by simulating a binary population which follows a given distribution f ( q ), and by comparing the resulting synthetic CMD with the observed one. We already adopted this technique in Bellazzini et al. (2002), Sollima et al. (2007) and Dalessandro et al. (2011). Hence, we refer to these papers for details. In brief, in order to simulate the binary population we randomly extracted N bin values of the mass of the primary component from a Kroupa (2002) initial mass function, and N bin values of the binary mass ratio from the f ( q ) distribution observed by Fisher et al. (2005) in the solar neighborhood, thus also obtaining the mass of the secondary. We considered the same magnitude limits and radial intervals used before. For each adopted value of the input binary fraction ξ in (assumed to range between 0 and 15% with a step of 1%), we produced 100 synthetic CMDs. Then, for each value of ξ in the penalty function χ 2 ( ξ in) was</text>
<text><location><page_6><loc_8><loc_83><loc_48><loc_92></location>computed and a probability P ( ξ in) was associated to it (see Sollima et al. 2007). The mean of the best fitting Gaussian is the global binary fraction ξ TOT. The values obtained in the four considered radial bins are listed in Table 3. The radial distribution of ξ TOT well resembles that obtained for the minimum binary fraction, remaining flat at ∼ 8% in the innermost ∼ 90 '' and decreasing to ∼ 6% in the two outermost annuli.</text>
<section_header_level_1><location><page_6><loc_18><loc_80><loc_38><loc_81></location>5. SUMMARY AND DISCUSSION</section_header_level_1>
<text><location><page_6><loc_8><loc_73><loc_48><loc_80></location>We have investigated the radial distribution of BSSs and binary systems in the Galactic GC NGC 5466, sampling the entire cluster extension. This has been possible thanks to a proper combination of LBC-blue and MEGACAM observations, and deep ACS and LBC-blue data, respectively.</text>
<text><location><page_6><loc_8><loc_25><loc_48><loc_73></location>The BSS radial distribution has been found to be bimodal, as in the majority of GCs investigated to date (see F12 and references therein). In fact, our data-set allowed us to detect a 'rising branch' in the external portion of the BSS distribution that was not identified in previous investigations because of a too limited sample in radius ( r < 800 '' Fekadu et al. 2007). Following the F12 classification, NGC 5466 therefore belongs to Family II . The minimum of the distribution is a quite prominent, purely observational feature, that, it in the light of the results published by F12, can be used to investigate the dynamical age of NGC 5466. It is located at r min ∼ 180 '' , corresponding to only ∼ 2 . 5 rc . This is the second most internal location of the minimum, after that of M53. Hence, NGC 5466 is one of the dynamically youngest clusters of Family II , where BSSs have just started to sink toward the center of the system. The relatively low central peak of the BSS distribution ( R BSS /similarequal 1 . 7; see Fig. 6) is also consistent with this interpretation. As a further check, we calculated the core relaxation time ( t rc). F12 shows that log ( t rc) is proportional to log ( r min) (see their figure 4). The value of t rc has been computed by using equation (10) of Djorgovski (1993), adopting the new cluster structural parameters reported in Section 2.2. In Figure 9 we show the position of NGC 5466 (black solid circle) on the 'dynamical clock plane' ( t rc / t H as a function of r min / rc ; Figure 4 in F12), where t H is the age of the Universe ( t H = 13.7 Gyr). Clearly, this cluster nicely follows the same relation defined by the sample analyzed in F12 (see also Dalessandro et al. 2013), further confirming that the shape of the observed BSS distribution is a good measure of GC dynamical ages. This figure also confirms that NGC 5466 is one of the youngest cluster of Family II , meaning that only recently the action of dynamical friction started to segregate BSSs (and primordial binary systems of similar total mass) toward the cluster centre. In this scenario, the most remote BSSs are thought to be still evolving in isolation in the outer cluster regions.</text>
<text><location><page_6><loc_8><loc_15><loc_48><loc_25></location>The central values of the derived binary fractions ( ξ q ≥ 0 . 5 = 6 . 5% and ξ TOT = 8%) are slightly smaller than those quoted by Sollima et al. (2007). This is due to the differences in the adopted data reduction procedures and the consequent completeness analysis results, as well as the assumed luminosity functions. The value of ξ q ≥ 0 . 5 is in very good agreement with the estimate obtained in the same radial range by Milone et al. (2012) from the same ACS data-sets: ξ q ≥ 0 . 5 = (7 . 1 ± 0 . 4)%</text>
<text><location><page_6><loc_52><loc_77><loc_92><loc_92></location>(see their Figure 34 and the innermost open triangle in Fig. 8). Instead, our total binary fraction is significantly lower than that quoted by Milone et al. (2012, ξ TOT = 14 . 2%) because of the different assumptions made about the shape of f ( q ). In fact, these authors assumed a constant mass-ratio distribution for all q values, and their total binary fraction is simply twice the value of ξ q ≥ 0 . 5. Indeed, the measure of ξ TOT is quite sensitive to the assumed mass-ratio distribution, as also demonstrated by the value (11.7%) obtained by simulating binaries formed by random associations between stars of different masses (Sollima et al. 2007).</text>
<text><location><page_6><loc_52><loc_39><loc_92><loc_77></location>Milone et al. (2012) also provides the value of the minmum binary fraction abetween 2 . 35 ' and 2 . 45 ' from the cluster center: ξ q ≥ 0 . 5 = (1 . 6 ± 3 . 5)% . We emphasize, however, that this estimate has been obtained by using only the most external fragments of the ACS FoV, corresponding to less than 5% of the total sampled area (this likely explains the large uncertainty quoted by the authors). Keeping this caveat in mind, in light of the good agreement between our central value of ξ q ≥ 0 . 5 and that measured by Milone et al. (2012), we include their estimate in our analysis, thus sampling the intermediate region which is not covered by our data. Very interestingly, the value computed by Milone et al. (2012) defines a minimum in the binary fraction radial distribution (see the outer open triangle in Fig. 8), thus implying that also the radial distribution of binaries seems to be bimodal in NGC 5466. We stress that the investigated binary systems (having primary star masses between 0.5 and 0 . 74 M /circledot ) are in a mass range consistent with that of BSSs. Indeed, within the uncertainties, also the position of the minimum of the binary radial distribution is in good agreement with that of the BSS population. This result urges a confirmation through dedicated observations. In fact, we emphasize that this would be the first time that a similar feature is observed in a GC. It would further strengthen the interpretation proposed by F12 that the shape of the BSS radial distribution (and simialr mass objects) is primarily sculpted by dynamical friction. Moreover, it adds further support to the conclusions that the unperturbed evolution of primordial binaries could be the dominant BSS formation process in low-density environments (Sollima et al. 2008).</text>
<text><location><page_6><loc_52><loc_15><loc_92><loc_36></location>This research is part of the project COSMIC-LAB funded by the European Research Council (under contract ERC-2010-AdG-267675). GB acknowledges the European Community's Seventh Framework Programme under grant agreement no. 229517. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. Also based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute. STScI is operated by the association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555.</text>
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<figure>
<location><page_8><loc_9><loc_61><loc_48><loc_91></location>
</figure>
<figure>
<location><page_8><loc_50><loc_61><loc_89><loc_91></location>
<caption>FIG. 1.Left panel: map of the 'Shallow Sample' used to study the BSS population of NGC 5466. The dashed circle defines the cluster area probed by the shallow LBC-blue data, while the outer square corresponds to the MEGACAM field of view. The solid circle corresponds to the location of the cluster's tidal radius. Right panel: map of the 'Deep Sample' used to study the cluster binary fraction. The solid square corresponds to the ACS FoV, while the four rectangles mark the FoV of the deep LBC-blue data-set. The solid circle indicates the location of the cluster's tidal radius. The annulus within the two dashed circles shows the area excluded from this study because of the low photometric completeness (see Section 4.1).</caption>
</figure>
<figure>
<location><page_9><loc_17><loc_40><loc_83><loc_89></location>
<caption>FIG. 2.- CMD of the stars sampled with the shallow LBC data-set in an area of 10 ' around the cluster's center.</caption>
</figure>
<figure>
<location><page_10><loc_17><loc_40><loc_82><loc_89></location>
<caption>FIG. 3.- CMDs of the ACS and the deep LBC samples (left and right panels, respectively). The mean ridge line is shown as a solid line.</caption>
</figure>
<figure>
<location><page_11><loc_17><loc_40><loc_82><loc_89></location>
<caption>FIG. 4.- CMDs showing the selection of the BSS population (open circles) in the shallow LBC and the MEGACAM samples (left and right panels, respectively). The grey open triangles marks the population of sub-giant and RGB stars used as reference in the study of the BSS radial distribution.</caption>
</figure>
<figure>
<location><page_12><loc_16><loc_39><loc_82><loc_89></location>
<caption>FIG. 5.- Cumulative radial distribution of BSSs (solid line) and sub-giant+RGB stars (dashed line), as a function of the projected distance from the cluster center.</caption>
</figure>
<figure>
<location><page_13><loc_15><loc_39><loc_82><loc_89></location>
<caption>FIG. 6.- BSS rardial distribution. Upper panel : number of BSSs with respect to that of reference (SGB+RGB) stars, plotted as a function of the distance from the cluster center. Lower panel double normalized ratio (see text) of BSSs (dots and error bars) and reference stars (gray regions). The distribution is clearly bimodal, with a minimum at r min /similarequal 180 '' .</caption>
</figure>
<figure>
<location><page_14><loc_16><loc_40><loc_82><loc_89></location>
<caption>FIG. 7.- Photometric completeness φ as a function of V for the two data sets. The ACS and deep LBC data-sets have been divided into three and four concentric annuli areas of same completeness, respectively. The solid horizontal line shows the limits of 50% of completeness.</caption>
</figure>
<figure>
<location><page_15><loc_15><loc_38><loc_82><loc_89></location>
<caption>FIG. 8.- Minimum and total binary fractions (solid circles in the upper and lower panels, respectively) as a function of radial distance from the cluster center. The grey area show the region excluded from the analysis because of the very low photometric completeness. The empty triangles in the upper panel mark the estimate from Milone et al. (2012).</caption>
</figure>
<figure>
<location><page_16><loc_16><loc_60><loc_85><loc_90></location>
<caption>FIG. 9.- Core relaxation time ( t rc) normalized to the age of the Universe ( t H = 13.7 Gyr), as a function of r min / rc (with r min = 180 '' being the position of the minimum of the BSS radial distribution). The figure is the same as Figure 4 in F12: the arrows indicate the location of the dynamically young clusters ( Family I ); grey circles are GCs of intermediate age ( Family II in F12, plus M10 studied by Dalessandro et al. 2013); grey triangles are dynamically old clusters ( Family III ). The location of NGC 5466 in this plane is marked with the black circle and clearly shows that this is the second youngest member of Family III .</caption>
</figure>
<section_header_level_1><location><page_17><loc_42><loc_89><loc_58><loc_90></location>LOG OF THE OBSERVATIONS</section_header_level_1>
<table>
<location><page_17><loc_27><loc_70><loc_73><loc_88></location>
<caption>TABLE 2</caption>
</table>
<section_header_level_1><location><page_17><loc_38><loc_64><loc_62><loc_65></location>BSS RADIAL DISTRIBUTION IN NGC 5466</section_header_level_1>
<table>
<location><page_17><loc_30><loc_55><loc_69><loc_63></location>
<caption>TABLE 1</caption>
</table>
<text><location><page_17><loc_8><loc_53><loc_92><loc_55></location>NOTE. - Number of BSSs and RGB stars, and fraction of sampled light in the radial annuli considered to study the BSS radial distribution. The field contamination expected for each population is reported in parentheses.</text>
<table>
<location><page_17><loc_38><loc_39><loc_61><loc_46></location>
<caption>TABLE 3 BINARY FRACTION OF NGC 5466</caption>
</table>
<text><location><page_17><loc_10><loc_38><loc_59><loc_39></location>NOTE. - Minimum ( ξ min ) and total ( ξ TOT ) binary fraction as calculated in 4 concentric annuli.</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We present a detailed investigation of the radial distribution of blue straggler star and binary populations in the Galactic globular cluster NGC 5466, over the entire extension of the system. We used a combination of data acquired with the ACS on board the Hubble Space Telescope, the LBC-blue mounted on the Large Binocular Telescope, and MEGACAM on the Canadian-France-Hawaii Telescope. Blue straggler stars show a bimodal distribution with a mild central peak and a quite internal minimum. This feature is interpreted in terms of a relatively young dynamical age in the framework of the 'dynamical clock' concept proposed by Ferraro et al. (2012). The estimated fraction of binaries is ∼ 6 -7% in the central region ( r < 90 '' ) and slightly lower ( ∼ 5 . 5%) in the outskirts, at r > 200 '' . Quite interestingly, the comparison with the results of Milone et al. (2012) suggests that also binary systems may display a bimodal radial distribution, with the position of the minimum consistent with that of blue straggler stars. If confirmed, this feature would give additional support to the scenario where the radial distribution of objects more massive than the average cluster stars is primarily shaped by the effect of dynamical friction. Moreover, this would also be consistent with the idea that the unperturbed evolution of primordial binaries could be the dominant BSS formation process in low-density environments. Subject headings: blue stragglers - binaries: general - globular clusters: general - globular clusters: individual(NGC 5466)",
"pages": [
1
]
},
{
"title": "DEEP MULTI-TELESCOPE PHOTOMETRY OF NGC 5466. I. BLUE STRAGGLERS AND BINARY SYSTEMS.",
"content": "G. BECCARI 1 , E. DALESSANDRO 2 , B. LANZONI 2 , F. R. FERRARO 2 , A. SOLLIMA 3 , M. BELLAZZINI 3 , P. MIOCCHI 2 1 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany, gbeccari@eso.org 2 Dipartimento di Fisica e Astronomia, Università degli Studi di Bologna, via Ranzani 1, I-40127 Bologna, Italy and 3 INAF-Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Galactic globular clusters (GCs) are dynamically active systems, where stellar interactions and collisions, especially those involving binaries, are quite frequent (e.g. Hut et al. 1992; Meylan & Heggie 1997) and generate populations of exotic objects, like X-ray binaries, millisecond pulsars, and Blue straggler stars (BSSs; e.g., Paresce et al. 1992; Bailyn 1995; Bellazzini et al. 1995; Ferraro et al. 1995, 2009;Ransom et al. 2005; Pooley & Hut 2006). Among these objects, BSSs are the most abundant and therefore may act as a crucial probe of the interplay between stellar evolution and stellar dynamics (e.g. Bailyn 1995). They are brighter and bluer (hotter) than main sequence (MS) turnoff (TO) stars and are located along an extrapolation of the MS in the optical color-magnitude diagram (CMD). Thus, they mimic a rejuvenated stellar population, with masses larger than those of normal cluster stars ( M /greaterorsimilar 1 -1 . 2 M /circledot ; this is also confirmed by direct measurements; Shara et al. 1997; Gilliland et al. 1998). This clearly indicates that some process which increases the initial mass of single stars must be at work. The most probable formation mechanisms of these peculiar objects are thought to be either the mass transfer (MT) activity between binary companions, even up to the complete coalescence of the system (McCrea 1964), or the merger of two single or binary stars driven by stellar collisions (COL; Zinn & Searle 1976; Hills & Day 1976). Unfortunately, it is still a quite hard task to distinguish BSSs formed by either channel. The most promising route seems to be high resolution spectroscopy, able to identify chemical anomalies (as a significant depletion of carbon and oxygen) on the BSS surface (see Ferraro et al. 2006a), as it is expected in the MT scenario. Also the recent discovery of a double sequence of BSSs in M30 (Ferraro et al. 2009) seems to indicate that, at least in GCs that recently suffered the core collapse, COL-BSSs can be distinguished from MTBSSs on the basis of their position in the color-magnitude di- agram (CMD). In general, since collisions are most frequent in high density regions, BSSs in different environments could have different origins: those in loose GCs might arise from the coalescence/mass-transfer of primordial binaries, while those in high density clusters might form mostly from stellar collisions (e.g., Bailyn 1992; Ferraro et al. 1995). However, Knigge et al. (2009) suggested that most BSSs, even those observed in the cores of high density GCs, formed from binary systems, although no strong correlation between the number or specific frequency of BSSs and that of binaries is found if all clusters (including the post-core collapse ones) are taken into account (Milone et al. 2012; Leigh et al. 2013). This is likely due to the fact that dynamical processes significantly modify the binary and the BSS content of GCs during their evolution. An exception seems to be represented by low density environments, where the efficiency of dynamical interactions is expected to be negligible. Very interestingly, indeed, a clear correlation between the binary and the BSS frequencies has been found in a sample of 13 low density GCs (Sollima et al. 2008). This remains the cleanest evidence that the unperturbed evolution of primordial binaries is the dominant BSS formation process in low-density environments (also consistently with the results obtained in open clusters; e.g. Mathieu &Geller 2009). One of the most notable effects of stellar dynamics on BSSs is the shaping of their radial distribution within the host cluster. In most cases a clear bimodality has been observed in the radial distribution of the N BSS / N ref ratio ( N BSS and N ref being, respectively, the number of BSSs and the number of stars belonging to a reference population, as red giant or horizontal branch stars): such a ratio shows a high peak at the cluster centre, a dip at intermediate radii and a rising branch in the external regions (e.g. Beccari et al. 2008; Dalessandro et al. 2009). In a few other GCs (see, for example, NGC 1904 and M75; Lanzoni et al. 2007a; Contreras Ramos et al. 2012, respectively) the radial distribution shows a clear central peak but no upturn in the external cluster regions. In the case of ω Centauri (Ferraro et al. 2006b), NGC 2419 (Dalessandro et al. 2008) and Palomar 14 (Beccari et al. 2011) the BSS radial distribution is flat all over the entire cluster extension. Since BSSs are more massive than the average cluster stars (and therefore suffer from the effects of dynamical friction), these different observed shapes have been interpreted as due to different 'dynamical ages' of the host clusters (Ferraro et al. 2012, hereafter F12): a flat BSS distribution is found in dynamically young GCs ( Family I ), where two-body relaxation has been ineffective in establishing energy-equipartition in a Hubble time and dynamical friction has not segregated the BSS population yet. In clusters of intermediate ages ( Family II ), the innermost BSSs have already sunk to the center, while the outermost ones are still evolving in isolation (this would produce the observed bimodality); finally, in the most evolved GCs ( Family III ), dynamical friction has already segregated toward the centre even the most remote BSSs, thus erasing the external rising branch of the distribution. These results demonstrate that the BSS radial distribution can be used as powerful dynamical clock. Of course, the same should hold for binary systems with comparable masses, but the identification of clean and statistically significant samples of binaries all over the entire cluster extension is a quite hard task and no investigations have been performed to date to confirm such an expectation. NGC5466 is a high galactic latitude GC ( l = 42 . 2 · and b = 73 . 6 · ), located in the constellation of Boötes at a distance of D = 16 kpc (Harris 1996). The cluster has been recently found to be surrounded by huge tidal tails (Belokurov et al. 2006; Grillmair & Johnson 2006). The BSS stars in this cluster were first studied by Nemec & Harris (1987), and, more recently, Fekadu et al. (2007) analyzed the BSS population inside the first r ∼ 800 '' from the cluster center. Since the cluster tidal radius is 1580 '' (Miocchi et al. 2013), a survey of the BSS population over the whole body of the cluster is still missing. In the framework of the study of the origin of the BSS, it is important to mention that NGC 5466 is the first GCwhere eclipsing binaries were found among the BSS (Mateo et al. 1990, see also Kryachko et al.(2011)). In their paper Mateo et al. (1990) find indications that all the BSS in NGC 5466 were formed from the merger of the components of primordial, detached binaries evolved into compact binary systems. In this paper we intend to study the BSS and binary populations of NGC 5466, focussing in particular on their radial distribution within the whole cluster to investigate its dynamical age. The paper is organized as follows. In Section 2 we describe the observations and data analysis. Section 3 is devoted to the definition of the BSS population and the study of its radial distribution. In Section 4 we describe how the fraction of binaries and its variation with radius are investigated. The summary and discussion of the obtained results are presented in Section 5.",
"pages": [
1,
2
]
},
{
"title": "2. OBSERVATIONS AND DATA ANALYSIS",
"content": "The photometric data used here consist of a combination of shallow and deep images (see Table 1) acquired through the blue channel of the Large Binocular Camera (LBC-blue) mounted on the Large Binocular Telescope (LBT), MEGACAMonthe Canadian-France-Hawaii Telescope (CFHT) and the Advanced Camera for Survey (ACS) on board the Hubble Space Telescope (HST). (i)-The LBC data-set : both short and long exposures were secured in 2010 under excellent seeing conditions (0 . '' 5-0 . '' 7) with the LBC-blue (Ragazzoni et al. 2006; Giallongo et al. 2008) mounted on the LBT (Hill et al. 2006) at Mount Graham, Arizona. The LBC is a double wide-field imager which provides an effective 23 ' × 23 ' field of view (FoV), sampled at 0 . 224 arcsec/pixel over four chips of 2048 × 4608 pixels each. LBC-blue is optimized for the UV-blue wavelengths, from 320 to 500 nm, and is equipped with the U -BESSEL , B -BESSEL , V -BESSEL , g -SLOAN and r -SLOAN filters (hereafter U , B , V , g , r , respectively). A set of short exposures (of 5, 60 and 90 seconds) was secured in the B and V filters with the cluster center positioned in the central chip (namely #2) of the LBC-blue CCD mosaic. Deep images (of 400 and 200 s in the B and V bands, respectively) were obtained with the LBC-blue FoV positioned ∼ 100 '' south from the cluster centre. A dithering pattern was adopted in both cases thus to fill the gaps of the CCD mosaic. The raw LBC images were corrected for bias and flat field, and the overscan region was trimmed using a pipeline specifically developed for LBC image pre-reduction from the LBC Survey data center 1 . (ii)-The MEGACAM data-set : g and r wide-field MEGACAM images (Prop ID: 04AK03; PI: Jang-Hyun Park), acquired with excellent seeing conditions (0 . '' 6-0 . '' 8) were retrieved from the Canadian Astronomy Data Centre (CADC 2 ). Mounted at CFHT (Hawaii), the wide-field imager MEGACAM (built by CEA, France), consists of 36 2048 × 4612 pixel CCDs (a total of 340 megapixels), fully covering a 1 degree × 1 degree FoV with a resolution of 0.187 arcsecond/pixel. The data were preprocessed (removal of the instrumental signature) and calibrated (photometry and astrometry) by Elixir pipeline 3 . (iii)-The ACS data-set consists of a set of high-resolution, deep images obtained with the ACS on board the HST through the F606W and F814W filters (GO-10775;PI: Sarajedini). The cluster was centered in the ACS field and observed with one short and five long-exposures per filter, for a total of two orbits. We retrieved and used only the deep exposures (see Table 5). All images were passed through the standard ACS/WFC reduction pipeline. As shown in the left panel of Figure 1, the shallow LBC data-set is used to sample a region of 600 '' around the cluster center, and for r > 600 '' it is complemented with the MEGACAMobservations, which extend beyond the tidal radius, out to r ∼ 30 ' . In the following the combination of these two datasets will be referred to as the 'Shallow Sample' and adopted to study the BSS population over the entire cluster extension (see Sect. 3). The ACS observations sample the innermost ∼ 200 '' of the cluster and they are complemented with the deep LBC observations, extending out to rt (see right panel of Figure 1): this will be called the 'Deep Sample' and used to investigate the binary fraction of NGC 5466 (see Sect. 4).",
"pages": [
2
]
},
{
"title": "2.1. Photometry and Astrometry",
"content": "Given the very low crowding conditions in the external cluster regions, we performed aperture photometry on the two g and r MEGACAM images, by using SExtractor (Bertin & Arnouts 1996) with an aperture diameter of 0 . 9 '' (correspond- to a FWHM of 5 pixels). Each of the 36 chips was reduced separately, and we finally obtained a catalog listing the relative positions and magnitudes of all the stars in common between the g and r catalogs. The LBC and ACS data-sets were reduced through a standard point spread function (PSF) fitting procedure, by using DAOPHOTII/ALLSTAR (Stetson 1987, 1994). The PSF was independently modeled on each image using more than 50 isolated and well sampled stars in the field. The photometric catalogs of each pass-band were combined, and the instrumental magnitudes were reported to a reference frame following the standard procedure (see, e.g., Ferraro et al. 1991, 1992). The magnitudes of stars successfully measured in at least three images per filter were then averaged, and the error about the mean was adopted as the associated photometric uncertainty. In order to obtain an absolute astrometric solution for each of the 36 MEGACAM chips, we used more than 15000 stars in common with the Sloan Digital Sky Survey (SDSS) catalog 4 covering an area of 1 square degree centered on the cluster. The final r.m.s. global accuracy is of 0 . 3 '' , both in right ascension ( α ) and declination ( δ ). The same technique applied to the LBC sample gave an astrometric solution with similar accuracy ( σ < 0 . 3 '' r.m.s.). Considering that no standard astrometric stars can be found in the very central regions of the cluster, we used the stars in the LBC catalog as secondary astrometric standards and we thus determined an absolute astrometric solution for the ACS sample in the core region with the same accuracy obtained in the previous cases. A sample of bright isolated stars in the ACS data-set was used to transform the ACS instrumental magnitudes to a fixed aperture of 0 . '' 5. The extrapolation to infinite and the transformation into the VEGAMAG photometric system was then performed using the updated values listed in Tables 5 and 10 of Sirianni et al. (2005) 5 . In order to transform the instrumental B and V magnitudes of the LBC sample into the Johnson/Kron-Cousins standard system, we used the stars in common with a photometric catalog previously published by Fekadu et al. (2007). These authors, in section 2.3 of their paper, provide a thorough comparison of their photometry with existing literature datasets, finding a satisfactory agreement. In particular they found an excellent agreement in the photometric zero-points (within less than 0.02 mag) and no residual trends with colors with the set of secondary standards by Stetson (2000), that should be considered as the most reliable reference. We verified, by direct comparison with Stetson (2000), that the quality of the agreement shown by Fekadu et al. (2007) is fully preserved in our final catalog. Instead, the g and r magnitudes in the MEGACAM sample were calibrated using the stars in common with the SDSS catalog and we then adopted the equation that Robert Lupton derived by matching DR4 photometry to Peter Stetson's published photometry for stars 6 to transform the calibrated g and r into a V magnitude. We finally used more than 2000 stars in common between the LBC and the MEGACAM data-sets to search for any possible off-sets between the V magnitude in the two catalogs, finding an average systematic difference of 0.014 mag. We therefore applied this small correction to the MEGACAM catalog. Similarly, we used more than 200 stars in common between the ACS data-set and the LBC cata- log to calibrate the ACS F606W magnitude to the Johnson V magnitude. Such a procedure therefore provided us with a homogenous V magnitude scale in common to all the available data-sets.",
"pages": [
2,
3
]
},
{
"title": "2.2. The Color-Magnitude Diagrams",
"content": "The CMD obtained from the shallow LBC exposures for the innermost 600 '' from the cluster center is shown in Figure 2. Typical photometric errors (magnitudes and colors) are indicated by black crosses. The high quality of the images allows us to accurately sample all the bright evolutionary sequences, from the Tip of the red giant branch (RGB), down to ∼ 2 magnitudes below the MS-TO, with a large population of BSSs well visible at 0 < ( B -V ) < 0 . 4 and 20 . 5 < V < 18. In Figure 3 we show the CMDs obtained from the ACS and the deep LBC observations. As in Figure 2, the typical photometric errors (magnitudes and colors) for the two sample are indicated by black crosses. A well populated MS is visible in the ACS CMD, extending from the MS-TO ( V ∼ 20 . 5) down to V ∼ 27. Once again, the collecting power of the LBT combined with the LBC-blue imaging capabilities allowed us to sample the MS down to similar magnitudes in an area extending out to the cluster's tidal radius. For ( B -V ) /greaterorsimilar 1 . 5 the plume of Galactic M dwarfs is clearly visible, while for ( B -V ) /lessorsimilar 1 and V /greaterorsimilar 24 a population of unresolved background (possibly extended) objects is apparent. In both the CMDs a well defined sequence of unresolved binary systems is observed on the red side of the MS.",
"pages": [
3
]
},
{
"title": "3. THE SHALLOW SAMPLE AND THE BSS POPULATION",
"content": "The Shallow Sample was used to determine the cluster center of gravity ( C grav) and the projected density profile from accurate star counts 7 , following the procedure already adopted and described, e.g., in Beccari et al. (2012). The resulting value of C grav is α J2000 = 14 h 05 m 27 s . 2, δ J2000 = + 28 · 32 ' 01 . '' 8, in full agreement with that quoted by Goldsbury et al. (2010). The best fit King model to the observed density profile has concentration parameter c = 1 . 31, core radius rc = 72 '' and tidal radius rt = 1580 '' = 26 . 3 ' (Miocchi et al. 2013). 8 Fekadu et al. (2007) studied the radial distribution of the BSS population in NGC 5466, from the cluster centre out to r ∼ 800 '' (see also Nemec & Harris 1987). They selected 89 BSSs and showed that they are significantly more centrally concentrated than RGB and horizontal branch (HB) stars, but they did not find any evidence of upturn in the external regions. However, that study did not sample the entire radial extension of the cluster, while our previous experience demonstrates that even a few BSSs in the outer regions can generate the external rising branch. Hence, a detailed investigation even in the most remote regions is necessary before solidly excluding the presence of an upturn in the BSS radial distribution and confirming the dynamical age of the cluster.",
"pages": [
3
]
},
{
"title": "3.1. BSS and reference population selection",
"content": "The selection of the BSS population was primarily performed in the ( V , B -V ) plane (see open circles in Figure 4, 7 The observed profile is shown in Miocchi et al. (2013), as part of a catalog of 26 Galactic GCs for which also the best-fit King (1966) and Wilson (1975) model fits are discussed. left panel), by adopting conservative limits in magnitude and color in order to avoid the possible contamination from stellar blends. The BSS selection has been then 'exported' to the MEGACAM ( V , V -r ) CMD by using the BSSs in common between the two data-sets in the region at r < 600 '' (see right hand panel). We finally count a population of 88 stars in the LBC sample and 9 stars in the MEGACAM one, for a total of 97 BSSs spanning the entire radial extent of the cluster. The cross-correlation of our sample with that of Fekadu et al. (2007) confirms that the majority of the stars selected in the previous work are real BSSs, but 18 of them are blended sources. In order to further test the completeness of our LBC shallow sample we performed a detailed comparison with the ACS photometric catalogue of Sarajedini et al. (2007) 9 . The completeness has been quantified as the fraction of stars in common between the two catalogs in a given magnitude interval: we find 93%, 95%, and 100% completness for stars with V < 22, 21.5, and 21, respectively. For a meaningful study of the BSS population, a reference sample of 'normal' cluster stars has to be defined. Here we choose a post-MS population composed of sub-giant (SGB) and red giant branch stars (RGB; see grey triangles in Fig. 4). This has been selected in the same magnitude range (18 . 4 < V < 20) of BSSs, thus guaranteeing that the photometric completeness is the same in both samples. We count 1358 and 96 stars in the LBC and MEGACAM data-sets, respectively.",
"pages": [
3,
4
]
},
{
"title": "3.2. The BSS radial distribution",
"content": "As extensively illustrated in literature (see for example Bailyn 1995; Ferraro et al. 2003, 2012, and references therein), the study of the BSS radial distribution is a very powerful tool to shed light on the internal dynamical evolution of GCs and possibly gets some hints on the BSS formation mechanisms. As a first step, we compared the cumulative radial distribution of the selected BSS population to that of reference stars (see Figure 5). Clearly, BSSs are more centrally concentrated, and a Kolmogorov-Smirnov test yields a probability of 99 . 9% that the two samples are not extracted from the same parent population. This cumulative BSS distribution closely resembles that already observed in this cluster by Fekadu et al. (2007) out to ∼ 800 '' (see their Figure 19), with the only difference that we sample the entire cluster extension. Moreover, the distributions shown in Figure 5, closely resembles that observed in all GCs characterised by a bimodal radial distribution of BSSs (see, e.g., the cases of M3, Ferraro et al. 1997, 47Tuc, Ferraro et al. 2004; M53, Rey et al. 1998; M5, Lanzoni et al. 2007b). Following the same procedure described in Lanzoni et al. (2007b), as a second step we divided the surveyed area in a series of concentric annuli centred on C grav. In Table 2 we report the values of the inner and outer radii of each annulus, together with the number of BSSs ( N BSS) and reference stars ( N RGB) counted in each radial bin. We also report the fraction of light sampled in the same area, with the luminosity being calculated as the sum of the luminosities of all the stars measured in the TO region at 19 . 5 /lessorequalslant V /lessorequalslant 20 . 5. The contamination of the cluster BSS and RGB populations due to background and foreground field stars can be severe, especially in the external regions where number counts are low. We therefore exploited the wide area covered by MEGACAM 9 The catalogue is available for download at http://www.astro.ufl.edu/ ∼ ata/public_hstgc/ (see Figure 1) to estimate the field contamination by simply counting the number of stars located beyond the cluster tidal radius (at r > 1580 '' ) that populate the BSS and RGB selection boxes in the CMD. We count 2 BSSs and 13 RGB stars in a ∼ 0 . 43 deg 2 area. The corresponding densities are used to calculate the number of field stars expected to contaminate each population in every considered radial bin (see numbers in parenthesis in Table 2). In the upper panel of Figure 6 we show the resulting trend of N BSS / N RGB as a function of radius. Clearly it is bimodal, with a relatively low peak of BSSs in the inner region, a distinct dip at intermediate radii, and a constant value slightly smaller than the central one, in the outskirts. We identify the minimum of the radial distribution at r min ∼ 180 '' . This behavior is further confirmed by the radial distribution of the BSS double normalized ratio, R BSS (see the black circles in the lower panel of Figure 6). For each radial bin, R BSS is defined as the ratio between the fraction of BSSs in the annulus and the fraction of light sampled by the observations in the same bin (see Ferraro et al. 2003). The same quantity computed for the reference population turns out to be equal to one at any radial distance (see grey regions in the figure). This is indeed expected from the stellar evolution theory (Renzini & Buzzoni 1986) and it confirms that the distribution of RGB stars follows that of the cluster sampled light. Instead, R BSS shows a bimodal behavior, confirming the result obtained above. Notice that NGC 5466 is one of the most sparsely populated Galactic GCs, and so the number of cluster's stars and the fraction of sampled light in the external regions is quite low.",
"pages": [
4
]
},
{
"title": "4. THE DEEP SAMPLE AND THE CLUSTER BINARY FRACTION",
"content": "Binaries play an important role in the formation and dynamical evolution of GCs (McMillan 1991). Indeed these systems secure a enormous energy reservoir able to quench masssegregation and delay or even prevent the collapse of the core. Up to now three main techniques have been used to measure the binary fraction: (i) radial velocity variability (e.g., Mathieu & Geller 2009), (ii) searches for eclipsing binaries (e.g. Mateo 1996; Cote et al. 1996) and (iii) studies of the distribution of stars along the cluster MS in CMD (e.g. Romani & Weinberg 1991; Bellazzini et al. 2002; Milone et al. 2012). In this work we adopt the latter approach, which has the advantage of being more efficient for a statistical investigation, and unbiased against the orbital parameters of the systems. We followed the prescriptions extensively described in Bellazzini et al. (2002) and Sollima et al. (2007, 2010, see also Dalessandro et al. 2011). This method relies on the fact that the luminosity of a binary system is the result of the contributions of both (unresolved) companions. The total luminosity is therefore given by the luminosity of the primary (having mass M 1) increased by that of the companion (with mass M 2). Along the MS, where stars obey a mass-luminosity relation, the magnitude increase depends on the mass ratio q = M 2 / M 1, reaching a maximum value of 0.75 mag for equal mass binaries ( q = 1). Taking advantage of the large FoV covered by our data-sets, we analyzed the binary fraction at different radial distances from the cluster center, out to r ∼ 800 '' .",
"pages": [
4
]
},
{
"title": "4.1. Completeness tests",
"content": "In order to estimate the fraction of binaries from the 'secondary' MS in the CMD, it is crucial to have a realistic and robust measure of other possible sources of broadening of the MS (e.g. blending, photometric errors). These factors are related to the quality of the data and can be properly studied through artificial star experiments. We therefore produced a catalog of artificial stars for the ACS and LBC-deep data-sets, following the procedure described in Bellazzini et al. (2002). As a first step, a list of input positions and magnitudes of artificial stars (X in ,Y in , Vin , F 814 Win and Bin ) is produced. Once these stars are added to the original frames at the X in and Y in positions, we repeat the measure of stellar magnitudes in the same way as described in Section 2.1. At the end of this procedure we obtain a list of stars including both real and artificial stars. Since we know precisely the positions and magnitudes of the artificial stars, the comparison of the latter with the measured Vout , F 814 Wout and Bout magnitudes, allow us to estimate the capability of our data-reduction strategy to properly detect stars in the images, including the effect of blending. In practice, we first generated a catalog of simulated stars with a Vin magnitude randomly extracted from a luminosity function modeled to reproduce the observed one. Then the F 814 Win and Bin magnitudes were assigned to each sampled Vin magnitude by interpolating the mean ridge line of the cluster (solid grey lines in the left and right panels of Figure 3, for the ACS and the LBC-deep samples, respectively). The artificial stars were then added on each frame using the same PSF model calculated during the data reduction phase, and were spatially distributed on a grid of cells of fixed width (5 times larger than the mean FWHM of the stars in the frames). Only one star was randomly placed in such a box in each run, in order not to generate artificial stellar crowding from the simulated stars on the images. The artificial stars were added to the real images using the DAOPHOT/ADDSTAR routine. At the end of this step, we have a number of 'artificial' frames (which include the real stars and the artificial stars with know position in the frame) equal to the number of observed frames. The reduction process was repeated on the artificial images in exactly the same way as for the scientific ones. A total of 100,000 stars were simulated on each data-set and photometric completeness ( φ comp) was then calculated as the ratio between the number of stars recovered by the photometric reduction ( N out) and the number of simulated stars ( N in). In Figure 7, we show the photometric completeness estimated for the ACS and the LBC-deep data-sets (upper and lower panels, respectively) as a function of the V magnitude and for different radial regions around the cluster center. The completeness of the ACS sample is above 80% for all magnitudes down to V ∼ 26 (i.e., ∼ 5 . 5 . magnitudes below the cluster TO), even in the innermost region. On the other hand, a completeness above 50% down to V ∼ 24 is guaranteed only at r > 200 '' in the LBC-deep data-set. Instead, stellar crowding is so strong at lower radial distances (region D) that the completeness stays below ∼ 80%at all magnitudes. We therefore exclude this region (120 '' < r < 200 '' ) from our study of the cluster binary fraction.",
"pages": [
4,
5
]
},
{
"title": "4.2. The binary fractions",
"content": "As mentioned at the beginning of Section 4, our estimation of the binary fraction in NGC 5466 is performed by determining the fraction of binaries populating the area between the MS and a secondary MS, blue-ward shifted in colour by ∼ 0 . 75 (Hurley & Tout 1998). As a first step of our analysis, we measured the minimum binary fraction, i.e. the fraction of binaries with mass ratios large enough ( q > q min) that they can be clearly distinguished from single MS stars in the CMD. When q approaches the null value, the shift in magnitude of the primary star (i.e. the more massive one) is negligible. For this reason binary systems with small values of q becomes indistinguishable from MS stars when photometric errors are present. Clearly, this is a lower limit to the total binary fraction of the cluster. However, it has the advantage of being a pure observational quantity, since it is calculated without any assumption on the distribution of mass ratios (Sollima et al. 2007; Dalessandro et al. 2011). The magnitude interval in which the minimum binary fraction has been estimated is 21 < V < 24 for both the ACS and the LBC samples. According to a 12 Gyr model from Dotter et al. (2007) 10 , this corresponds to a mass range 0 . 5 M /circledot /lessorsimilar M /lessorsimilar 0 . 74 M /circledot for a single star on the MS. In this interval the photometric completeness φ comp is larger than 50% for both samples. For the magnitude and radial intervals just defined, we divided the entire FoV in four annuli (see Table 3 for details). Then, the location of q min was estimated on the CMD as the value corresponding to a color difference of three times the photometric error from the MS ridge mean line (grey solid lines in the cluster's CMDs shown in Figure 3). This approach allows us to define an area on the CMD safe from contamination from single mass stars. We notice that for each of the four annuli, this color difference corresponds to q min = 0 . 5. The contamination from blended sources was calculated on the same region of the CMD by adopting the catalogue of artificial stars produced with the completeness tests (see Section 4.1). A catalogue of stars from the Galaxymodel of Robin et al. (2003) was used to estimate the field contamination. We thus estimated ξ q ≥ 0 . 5 independently for the four radial intervals using the same approach described in Sollima et al. (2007). As shown in Figure 8 (upper panel), we find that ξ q ≥ 0 . 5 is almost constant and equal to ∼ 6 . 5%for r /lessorsimilar 90 '' (i.e., within the ACS sample), then it drops at ∼ 5% at distances larger than 200 '' (in the LBC sample). As described above, the minimum binary fraction has the advantage of being measured with no arbitrary assumptions about the mass ratio distribution f ( q ). It depends, however, on the photometric accuracy of the used data-sets, and caution must be exercised when comparing the minimum binary fractions estimated for different GCs and in different works. An alternative approach consists in measuring the global fraction of binaries ( ξ | rmTOT ), that can be obtained by simulating a binary population which follows a given distribution f ( q ), and by comparing the resulting synthetic CMD with the observed one. We already adopted this technique in Bellazzini et al. (2002), Sollima et al. (2007) and Dalessandro et al. (2011). Hence, we refer to these papers for details. In brief, in order to simulate the binary population we randomly extracted N bin values of the mass of the primary component from a Kroupa (2002) initial mass function, and N bin values of the binary mass ratio from the f ( q ) distribution observed by Fisher et al. (2005) in the solar neighborhood, thus also obtaining the mass of the secondary. We considered the same magnitude limits and radial intervals used before. For each adopted value of the input binary fraction ξ in (assumed to range between 0 and 15% with a step of 1%), we produced 100 synthetic CMDs. Then, for each value of ξ in the penalty function χ 2 ( ξ in) was computed and a probability P ( ξ in) was associated to it (see Sollima et al. 2007). The mean of the best fitting Gaussian is the global binary fraction ξ TOT. The values obtained in the four considered radial bins are listed in Table 3. The radial distribution of ξ TOT well resembles that obtained for the minimum binary fraction, remaining flat at ∼ 8% in the innermost ∼ 90 '' and decreasing to ∼ 6% in the two outermost annuli.",
"pages": [
5,
6
]
},
{
"title": "5. SUMMARY AND DISCUSSION",
"content": "We have investigated the radial distribution of BSSs and binary systems in the Galactic GC NGC 5466, sampling the entire cluster extension. This has been possible thanks to a proper combination of LBC-blue and MEGACAM observations, and deep ACS and LBC-blue data, respectively. The BSS radial distribution has been found to be bimodal, as in the majority of GCs investigated to date (see F12 and references therein). In fact, our data-set allowed us to detect a 'rising branch' in the external portion of the BSS distribution that was not identified in previous investigations because of a too limited sample in radius ( r < 800 '' Fekadu et al. 2007). Following the F12 classification, NGC 5466 therefore belongs to Family II . The minimum of the distribution is a quite prominent, purely observational feature, that, it in the light of the results published by F12, can be used to investigate the dynamical age of NGC 5466. It is located at r min ∼ 180 '' , corresponding to only ∼ 2 . 5 rc . This is the second most internal location of the minimum, after that of M53. Hence, NGC 5466 is one of the dynamically youngest clusters of Family II , where BSSs have just started to sink toward the center of the system. The relatively low central peak of the BSS distribution ( R BSS /similarequal 1 . 7; see Fig. 6) is also consistent with this interpretation. As a further check, we calculated the core relaxation time ( t rc). F12 shows that log ( t rc) is proportional to log ( r min) (see their figure 4). The value of t rc has been computed by using equation (10) of Djorgovski (1993), adopting the new cluster structural parameters reported in Section 2.2. In Figure 9 we show the position of NGC 5466 (black solid circle) on the 'dynamical clock plane' ( t rc / t H as a function of r min / rc ; Figure 4 in F12), where t H is the age of the Universe ( t H = 13.7 Gyr). Clearly, this cluster nicely follows the same relation defined by the sample analyzed in F12 (see also Dalessandro et al. 2013), further confirming that the shape of the observed BSS distribution is a good measure of GC dynamical ages. This figure also confirms that NGC 5466 is one of the youngest cluster of Family II , meaning that only recently the action of dynamical friction started to segregate BSSs (and primordial binary systems of similar total mass) toward the cluster centre. In this scenario, the most remote BSSs are thought to be still evolving in isolation in the outer cluster regions. The central values of the derived binary fractions ( ξ q ≥ 0 . 5 = 6 . 5% and ξ TOT = 8%) are slightly smaller than those quoted by Sollima et al. (2007). This is due to the differences in the adopted data reduction procedures and the consequent completeness analysis results, as well as the assumed luminosity functions. The value of ξ q ≥ 0 . 5 is in very good agreement with the estimate obtained in the same radial range by Milone et al. (2012) from the same ACS data-sets: ξ q ≥ 0 . 5 = (7 . 1 ± 0 . 4)% (see their Figure 34 and the innermost open triangle in Fig. 8). Instead, our total binary fraction is significantly lower than that quoted by Milone et al. (2012, ξ TOT = 14 . 2%) because of the different assumptions made about the shape of f ( q ). In fact, these authors assumed a constant mass-ratio distribution for all q values, and their total binary fraction is simply twice the value of ξ q ≥ 0 . 5. Indeed, the measure of ξ TOT is quite sensitive to the assumed mass-ratio distribution, as also demonstrated by the value (11.7%) obtained by simulating binaries formed by random associations between stars of different masses (Sollima et al. 2007). Milone et al. (2012) also provides the value of the minmum binary fraction abetween 2 . 35 ' and 2 . 45 ' from the cluster center: ξ q ≥ 0 . 5 = (1 . 6 ± 3 . 5)% . We emphasize, however, that this estimate has been obtained by using only the most external fragments of the ACS FoV, corresponding to less than 5% of the total sampled area (this likely explains the large uncertainty quoted by the authors). Keeping this caveat in mind, in light of the good agreement between our central value of ξ q ≥ 0 . 5 and that measured by Milone et al. (2012), we include their estimate in our analysis, thus sampling the intermediate region which is not covered by our data. Very interestingly, the value computed by Milone et al. (2012) defines a minimum in the binary fraction radial distribution (see the outer open triangle in Fig. 8), thus implying that also the radial distribution of binaries seems to be bimodal in NGC 5466. We stress that the investigated binary systems (having primary star masses between 0.5 and 0 . 74 M /circledot ) are in a mass range consistent with that of BSSs. Indeed, within the uncertainties, also the position of the minimum of the binary radial distribution is in good agreement with that of the BSS population. This result urges a confirmation through dedicated observations. In fact, we emphasize that this would be the first time that a similar feature is observed in a GC. It would further strengthen the interpretation proposed by F12 that the shape of the BSS radial distribution (and simialr mass objects) is primarily sculpted by dynamical friction. Moreover, it adds further support to the conclusions that the unperturbed evolution of primordial binaries could be the dominant BSS formation process in low-density environments (Sollima et al. 2008). This research is part of the project COSMIC-LAB funded by the European Research Council (under contract ERC-2010-AdG-267675). GB acknowledges the European Community's Seventh Framework Programme under grant agreement no. 229517. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. Also based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute. STScI is operated by the association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555.",
"pages": [
6
]
},
{
"title": "REFERENCES",
"content": "Beccari, G., Lützgendorf, N., Olczak, C., et al. 2012, ApJ, 754, 108 Bellazzini, M., Pasquali, A., Federici, L., Ferraro, F. R., & Pecci, F. F. 1995, ApJ, 439, 687 Bailyn, C. D. 1992, ApJ, 392, 519 Bailyn, C. D. 1995, ARA&A, 33, 133 Beccari, G., Lanzoni, B., Ferraro, F. R., et al. 2008, ApJ, 679, 712 Beccari, G., Sollima, A., Ferraro, F. R., et al. 2011, ApJ, 737, L3 Ferraro, F. R., Lanzoni, B., Dalessandro, E., et al. 2012, Nature, 492, 393 Fisher, J., Schröder, K.-P., & Smith, R. C. 2005, MNRAS, 361, 495 Giallongo, E., Ragazzoni, R., Grazian, A., et al. 2008, A&A, 482, 349 Gilliland, R. L., Bono, G., Edmonds, P. D., et al. 1998, ApJ, 507, 818 Goldsbury, R., Richer, H. B., Anderson, J., et al. 2010, AJ, 140, 1830 Grillmair, C. J., & Johnson, R. 2006, ApJ, 639, L17 Harris, W. E. 1996, AJ, 112, 1487 Hill, J. M., Green, R. F., & Slagle, J. H. 2006, Proc. SPIE, 6267, 62670Y Hills, J. G., & Day, C. A. 1976, Astrophys. Lett., 17, 87 Hut, P., McMillan, S., Goodman, J., et al. 1992, PASP, 104, 981 Zinn, R., & Searle, L. 1976, ApJ, 209, 734",
"pages": [
6,
7
]
},
{
"title": "BSS RADIAL DISTRIBUTION IN NGC 5466",
"content": "NOTE. - Number of BSSs and RGB stars, and fraction of sampled light in the radial annuli considered to study the BSS radial distribution. The field contamination expected for each population is reported in parentheses. NOTE. - Minimum ( ξ min ) and total ( ξ TOT ) binary fraction as calculated in 4 concentric annuli.",
"pages": [
17
]
}
]
2013ApJ...776...68K
https://arxiv.org/pdf/1309.6893.pdf
<document>
<section_header_level_1><location><page_1><loc_22><loc_85><loc_78><loc_86></location>Discovery of an extended X-ray jet in AP Librae</section_header_level_1>
<section_header_level_1><location><page_1><loc_45><loc_80><loc_55><loc_81></location>S. Kaufmann</section_header_level_1>
<text><location><page_1><loc_14><loc_77><loc_86><loc_78></location>Landessternwarte, Universit¨at Heidelberg, K¨onigstuhl, D-69117 Heidelberg, Germany</text>
<text><location><page_1><loc_34><loc_73><loc_66><loc_74></location>S.Kaufmann@lsw.uni-heidelberg.de</text>
<text><location><page_1><loc_45><loc_68><loc_55><loc_70></location>S.J. Wagner</text>
<text><location><page_1><loc_14><loc_65><loc_86><loc_67></location>Landessternwarte, Universit¨at Heidelberg, K¨onigstuhl, D-69117 Heidelberg, Germany</text>
<text><location><page_1><loc_46><loc_61><loc_54><loc_63></location>O. Tibolla</text>
<text><location><page_1><loc_12><loc_55><loc_88><loc_60></location>Institut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, D-97074 W¨urzburg, Germany</text>
<text><location><page_1><loc_20><loc_51><loc_27><loc_52></location>Received</text>
<text><location><page_1><loc_48><loc_51><loc_49><loc_52></location>;</text>
<text><location><page_1><loc_52><loc_51><loc_59><loc_52></location>accepted</text>
<section_header_level_1><location><page_2><loc_44><loc_85><loc_56><loc_86></location>ABSTRACT</section_header_level_1>
<text><location><page_2><loc_17><loc_44><loc_83><loc_80></location>Chandra observations of the low-energy peaked BL Lac object AP Librae revealed the clear discovery of a non-thermal X-ray jet. AP Lib is the first low energy peaked BL Lac object with an extended non-thermal X-ray jet that shows emission into the VHE range. The X-ray jet has an extension of ∼ 15 '' ( ≈ 14 kpc). The X-ray jet morphology is similar to the radio jet observed with VLA at 1.36 GHz emerging in south-east direction and bends by 50 · at a distance of 12 '' towards north-east. The intensity profiles of the X-ray emission are studied consistent with those found in the radio range. The spectral analysis reveals that the X-ray spectra of the core and jet region are both inverse Compton dominated. This adds to a still small sample of BL Lac objects whose X-ray jets are IC dominated and thus more similar to the high luminosity FRII sources than to the low luminosity FRI objects, which are usually considered to be the parent population of the BL Lac objects.</text>
<text><location><page_2><loc_17><loc_36><loc_81><loc_40></location>Subject headings: BL Lac objects: individual (AP Lib), galaxies: active, galaxies: jets, X-rays: galaxies</text>
<section_header_level_1><location><page_3><loc_42><loc_85><loc_58><loc_86></location>1. Introduction</section_header_level_1>
<text><location><page_3><loc_12><loc_68><loc_88><loc_81></location>The low-energy peaked BL Lac object AP Librae (AP Lib, PKS 1514-241) has a redshift of z=0.0486 (Disney et al. 1974) and is located at α J2000 = 15 h 17 m 41 . 81313 s ± 0 . 00002 s , δ J2000 = -24 · 22 ' 19 . 4759 '' ± 0 . 0003 '' as determined from VLBI observations by Lambert & Gontier (2009). It has been classified as a BL Lac object by Strittmatter et al. (1972) and Bond (1973).</text>
<text><location><page_3><loc_12><loc_56><loc_88><loc_66></location>AP Lib is well known as one of the most active blazars in the optical band. In data from 1989, intra-day variability was detected with a very high rate of change of 0 . 06 ± 0 . 01 mag/hr (Carini et al. 1991). Even on shorter time scales of 20min, variation of up to 0 . 5 mag have been detected in 1973 (Miller et al. 1974).</text>
<text><location><page_3><loc_12><loc_11><loc_88><loc_53></location>AP Lib was historically classified as a so-called radio selected BL Lac objects (RBL). In the 1990s, BL Lac objects were found mainly in radio or X-ray surveys and therefore classified as radio or X-ray selected BL Lac objects. The spectral energy distributions (SED) of BL Lac objects show two prominent peaks which are commonly described in a leptonic model as synchrotron and inverse Compton (IC) emission, respectively. The peak energy of the low-energy (synchrotron) component is used to classify BL Lac objects as low-energy peaked BL Lac object (LBL) and high-energy peaked (HBL), with occasional references to a group of intermediate-energy peaked BL Lac objects (IBL). This classification is frequently based on the slope of the X-ray spectrum. Ciliegi et al. (1995) found that the X-ray spectrum of AP Lib can be described by a power law with photon index of 1 . 5 -1 . 7. AP Lib is thus classified as low-energy peaked BL Lac object (LBL) and it is assumed that the X-ray emission of the core is due to IC scattering of the synchrotron emission, and, possibly, external radiation. The peak energy of the gamma-ray component of LBL objects is expected to arise in the keV-GeV range and therefore a rather low flux in the TeV γ -ray range is expected (close to or below the detection limit of current Cherenkov</text>
<text><location><page_4><loc_12><loc_79><loc_88><loc_86></location>telescopes). Hence, it is rather unexpected to detect very high energy (VHE,E > 100 GeV) γ -ray emission from an LBL. Therefore it was remarkable, that in June/July 2010, VHE γ -ray emission was detected from the position of AP Lib (Hofmann 2010).</text>
<text><location><page_4><loc_12><loc_60><loc_88><loc_76></location>The standard picture of AGN, e.g. Blandford & Rees (1974), explains the different observational characteristics of AGN with the orientation of the jet axis to the line of sight to the observer. BL Lac objects are interpreted as aligned (beamed) versions of Fanaroff-Riley I (FRI, Fanaroff & Riley (1974)) radio galaxies while steep and flat spectrum radio quasars (SSRQ and FSRQ) are aligned versions of FRII galaxies. BL Lac objects are known to have very energetic jets pointing under small viewing angle towards the observer.</text>
<text><location><page_4><loc_12><loc_42><loc_88><loc_58></location>Radio observations of AP Lib at 1.4 GHz reveal the detection of one-sided, diffuse radio emission at the arcmin scale (Condon et al. 1998). Observations with the Very Large Array (VLA) at 1.36 GHz and 4.6 GHz result in the detection of a one-sided radio jet emerging to the south-east direction and bending towards north-east at ∼ 12 arcsec distance from the core (Cassaro et al. 1999). Very Long Baseling Array (VLBA) observations detected the radio jet at milli-arcsec scale which emerges to the south (MOJAVE 1 , Lister et al. (2009)).</text>
<text><location><page_4><loc_12><loc_20><loc_88><loc_39></location>The inhomogeneous collection of published X-ray jets (status March 2013: 113 X-ray jets), the XJET database 2 , contains 77% high luminosity sources (FRII, quasars) and ∼ 23% low luminosity sources (20 FRI, 5 BL Lac objects and one Seyfert 1 galaxy). The X-ray emission detected from the jets of FRI sources are dominated by synchrotron emission (Harris & Krawczynski 2006). Most high luminosity sources (quasars, FRII) have X-ray jets with flat ( α < 1) spectra, suggesting this part of the spectrum to be dominated by inverse Compton emission (e.g. Harris & Krawczynski (2006), Worrall (2009)). While inverse</text>
<text><location><page_5><loc_12><loc_79><loc_87><loc_86></location>Compton scattering of CMB photons is an explanation for flat X-ray spectra preferred by many, problems with this interpretation have been discussed by Harris & Krawczynski (2006) and Worrall (2009).</text>
<text><location><page_5><loc_12><loc_66><loc_87><loc_76></location>Among the sources in this database, only discovered in the radio galaxies Cen A and M87 and the FSRQ PKS 1222+216 and PKS 1510+089 have been traced up to TeV energies. AP Lib is the first BL Lac object with an extended X-ray jet that has been detected in the TeV energy range.</text>
<text><location><page_5><loc_12><loc_59><loc_87><loc_64></location>The luminosity distance of AP Lib, using H 0 = 70 km s -1 Mpc -1 , is d L = 210 Mpc. The scale is 1 '' = 0 . 95 kpc.</text>
<section_header_level_1><location><page_5><loc_39><loc_52><loc_61><loc_54></location>2. Extended X-ray jet</section_header_level_1>
<text><location><page_5><loc_12><loc_33><loc_88><loc_49></location>X-ray observations of AP Lib were analysed to search for high-energy emission of the one-sided radio jet. The source was observed with Chandra on July 4, 2003 (ObsID: 3971) with an exposure of 12.8 ks. The data were taken in timed exposure mode using a subarray of 1/8 of the chip (128 rows). This mode decreases the frame time to 0.4 s and reduces pileup. The data were re-calibrated using the calibration database CALDB. The Chandra data have bee analysed with the software CIAO v4.1 .</text>
<text><location><page_5><loc_12><loc_23><loc_88><loc_31></location>A clear X-ray jet was detected (Kaufmann et al. 2011) in this Chandra observation (see Fig. 1). The jet extends toward the south-east direction of AP Lib. Unfortunately the exposure is rather low, so that the real extension of the jet cannot be measured.</text>
<text><location><page_5><loc_12><loc_11><loc_88><loc_21></location>Despite the timed exposure mode, the brightness of the core causes a faint 'readout artefact' visible in the column in which the bright core region is located. For the spectral analysis, the signal of the influenced columns is replaced by a typical background level to correct for this artefact. The tool acisreadcorr has been used to correct the data. For the</text>
<figure>
<location><page_6><loc_12><loc_45><loc_88><loc_85></location>
<caption>Fig. 1.- X-ray count map of AP Lib from 0.2 to 8 keV extracted from the 12.8 ks observation by Chandra . The jet is clearly visible. Due to the used subarray, the observed frame is cut at the left part of the image, indicated by the line.</caption>
</figure>
<text><location><page_6><loc_12><loc_30><loc_67><loc_31></location>used subarray, the BACKSCAL header keyword had to be modified 3 .</text>
<text><location><page_6><loc_12><loc_20><loc_88><loc_27></location>As can be seen in Fig. 1, the X-ray jet emerges in the South-East direction up to an extension of ∼ 12 '' and bends with an angle of ≈ 50 · to the North-East. The jet broadens towards the outer regions and becomes fainter.</text>
<text><location><page_6><loc_16><loc_16><loc_86><loc_18></location>In order to compare the morphology of the X-ray jet to the radio jet studied with</text>
<text><location><page_7><loc_12><loc_61><loc_87><loc_86></location>VLA by Cassaro et al. (1999), the X-ray count map has been smoothed (see Fig. 2) with an elliptical Gaussian with major axis 3 '' and minor axis 2 '' using the ftool fgauss . The shape for the elliptical Gaussian matches the beam profile of the VLA data. The VLA radio contours (described in section 3.1.1) are overlayed onto the smoothed X-ray map (see Fig. 2). The jet in the radio and X-ray bands have very similar morphologies. They extend in the same direction, bend at the same distance by the same amount, and have comparable spatial profiles. Neither the X-ray nor the radio jet displays knots, hotspots or other features of high contrast. No feature on the counter-jet side could be detected in either band. A quantitative comparison is presented in section 2.1.</text>
<section_header_level_1><location><page_7><loc_39><loc_54><loc_61><loc_55></location>2.1. Intensity profiles</section_header_level_1>
<text><location><page_7><loc_12><loc_29><loc_88><loc_51></location>Radial profiles have been extracted with the tool dmextract from the whole source (core and jet) using 20 equidistant circular annuli of width ∼ 1 '' . In addition, a fraction of the annuli (wedge) with opening angle 100 · in the direction of the jet has been used. A circular region close to the source in North-West direction with radius ∼ 9 '' was used to determine the background for the radial profile.The two methods (complete annuli and a fraction of the annuli) used to obtain the radial profile do not show significant differences. This radial profile and its comparison to the PSF are used to identify the extension of the jet and to find the best regions to obtain the core and jet spectra.</text>
<text><location><page_7><loc_12><loc_16><loc_88><loc_27></location>The radial profiles have been extracted for two different energy sub-bands (0.2 - 1.5 keV (S) and 1.5 - 8 keV (H)) and the hardness ratio profile has been calculated to study the spectral trends in the jet region. The hardness ratio profile has no significant trend within 15 '' .</text>
<text><location><page_7><loc_16><loc_13><loc_88><loc_14></location>The PSF for the specific on-axis angle of the source was derived using the Chandra Ray</text>
<text><location><page_8><loc_12><loc_70><loc_88><loc_86></location>Tracer (ChaRT 4 ) which simulates the High Resolution Mirror Assembly (HRMA) based on an input energy spectrum of the core and the exposure of the observation. The output from ChaRT can be modeled, taking into account instrument effects of the various detectors, using the software MARX 5 to obtain the image of the simulated PSF on the detector. The radial profile for this PSF has been created with wedge annuli with an opening angle of 100 · .</text>
<text><location><page_8><loc_12><loc_60><loc_88><loc_68></location>The radial profile of the X-ray source deviates significantly from the PSF at radii > 2 '' (see Fig. 3). The jet at radii > 3 '' has a linear structure and therefore the intensity profile for the jet has been created using rectangular regions along the jet.</text>
<text><location><page_8><loc_12><loc_36><loc_88><loc_58></location>The intensity profile of the X-ray and radio jet has been extracted by integrating the counts at equidistant steps of 1 . 5 '' along the jet and perpendicular to the direction of the jet to compare the X-ray and radio morphology of the jet (see Fig. 4). In order to account for the jet bend, profiles along and across the jet are determined from 3 '' to 13 . 5 '' along PA = 120 · and from 13 . 5 '' to 22 . 5 '' along PA = 70 · . In order to avoid contributions from the core, a minimum distance of 3 '' was chosen. As can be seen in Fig. 4, the same morphology is detected of the X-ray and radio emission with a slightly shallower gradient of radio flux in the outer region of the jet.</text>
<text><location><page_8><loc_12><loc_26><loc_87><loc_33></location>The transversal profile (see Fig. 4) of the jet from 3 '' up to 13 . 5 '' ( ≈ 2 . 9 -12 . 8 kpc) was determined along PA = 120 · . The transversal profile of the X-ray and radio emission is comparable and the jet width is 5 '' ≈ 4 . 8 kpc in both energy bands.</text>
<text><location><page_8><loc_12><loc_19><loc_88><loc_24></location>The profiles of both energy bands are compatible. This suggests that the same particles are responsible for the radio and X-ray jet.</text>
<section_header_level_1><location><page_9><loc_44><loc_85><loc_56><loc_86></location>2.2. Spectra</section_header_level_1>
<text><location><page_9><loc_12><loc_74><loc_87><loc_81></location>The spectra of the core and the jet of AP Lib and of a background region are determined with the tool dmextract . The response files are obtained with the tool mkrmf and the ARF are created using asphist and mkarf .</text>
<text><location><page_9><loc_12><loc_47><loc_88><loc_72></location>The X-ray spectrum of the core was extracted using a circular region with radius 3 '' . This extraction radius was chosen, since the radial profile matches the simulated PSF out to this radius. The background spectrum was determined within a region of radius 25 '' close-by to the source in the North direction. To obtain the spectrum of the extended emission, a circular region with radius 15 '' was used in which the region around the core was excluded (see Fig. 5). As can be seen from the radial profile, the core exclusion region is large enough to avoid any influence of the core photons. Based on the radial profile of the core and jet (Fig. 3) and the shown PSF, the wing of the PSF of the core contributes < 10% to the jet spectrum obtained in the range > 3 '' .</text>
<text><location><page_9><loc_12><loc_31><loc_88><loc_45></location>The X-ray spectra have been binned with the tool grppha to obtain at least 25 counts per bin to reach the necessary significance for the χ 2 statistics. The program xspec v12 was used to fit the X-ray spectra in the energy range 0.2 to 8 keV. The uncertainties on the model parameters are given as confidence intervals. The fit parameter is changed by ∆ χ 2 = 2 . 71. This represents the 90% confidence interval.</text>
<text><location><page_9><loc_12><loc_16><loc_88><loc_29></location>The spectrum of the core can be well ( χ 2 /dof = 165 / 147) described by a power law of the form N ( E ) = N 0 × E -Γ with Γ = 1 . 58 ± 0 . 04 taking into account the Galactic absorption of N H = 8 . 36 × 10 20 cm -2 (LAB survey 6 , Kalberla et al. (2005)). The resulting flux is F core , 2 -10keV = (2 . 9 ± 0 . 1) × 10 -12 erg cm -2 s -1 . Although no hints for pileup appear in the residuals of the power law fit, a test for pileup has been performed. Therefore, the</text>
<text><location><page_10><loc_12><loc_76><loc_88><loc_86></location>spectrum of the core region has been extracted from an annulus region of the same size, in which the innermost pixels (inner radius of ∼ 1 '' ) are excluded. The spectral slope of the determined spectrum (Γ = 1 . 6 ± 0 . 1) is comparable to the core spectrum of the circular region and therefore no pileup was detected.</text>
<text><location><page_10><loc_12><loc_60><loc_88><loc_73></location>To quantify and to search for spectral differences, three different regions have been used as illustrated in Fig. 5. The extended emission region is a circular region with radius 15 '' excluding the core region with radius 3 '' . The wide jet region is a wedge region with radius 20 '' with opening angle 100 · along the jet and the inner jet region is a wedge region with opening angle of 30 · and radius of 10 '' .</text>
<text><location><page_10><loc_12><loc_50><loc_87><loc_58></location>The spectrum of the jet obtained from the extended emission region can be described by Γ = 1 . 8 ± 0 . 1 taking into account the Galactic absorption ( χ 2 /dof = 13 / 18). The resulting flux is F jet , 2 -10keV = (2 . 3 ± 0 . 3) × 10 -13 erg cm -2 s -1 .</text>
<text><location><page_10><loc_12><loc_26><loc_87><loc_48></location>The spectrum from the wide jet region can be fit with a photon index of Γ = 1 . 8 ± 0 . 2 ( χ 2 /dof = 6 / 13) comparable to the one above. The resulting flux is (1 . 6 ± 0 . 3) × 10 -13 erg cm -2 s -1 . An even smaller region, the inner jet region was used to determine the spectrum of the inner parts of the jet. Since the region is very small, the spectrum consist of only a few photons and therefore a reduced flux of (6 . 0 ± 1 . 7) × 10 -14 erg cm -2 s -1 resulted. The slope (Γ = 1 . 9 ± 0 . 5) is comparable to the above determined spectral fits. For all spectral fits, the Galactic absorption was used as fixed parameter and no hint for additional absorption was found.</text>
<text><location><page_10><loc_12><loc_11><loc_88><loc_24></location>The X-ray spectra for the core and the jet are shown in Fig. 6. No significant difference of the slope between the jet and the core spectrum was determined. The jet spectrum can be well described with a power law model and no emission or absorption features were detected. A fit with the thermal model apec in Xspec , considering the Galactic absorption, resulted in a worse fit and a high gas temperature of kT = 6 ± 3 keV. Any acceptable fit</text>
<text><location><page_11><loc_12><loc_76><loc_88><loc_86></location>of a combined model using the combination of the thermal model apec (Smith et al. 2001) and a power law is dominated by the power law. Therefore the power law model is the favoured description for the X-ray spectrum of the jet and hence the jet is considered to be dominated by non-thermal emission.</text>
<text><location><page_11><loc_12><loc_63><loc_88><loc_73></location>The core and the jet spectrum have both photon indices of Γ < 2 and are thus interpreted to be inverse Compton dominated. The core spectrum is comparable with the original definition of AP Lib being a low-energy peaked BL Lac object with IC dominance in the X-ray spectrum.</text>
<text><location><page_11><loc_12><loc_56><loc_88><loc_61></location>The luminosity of the X-ray core is L core , 2 -10keV = (1 . 53 ± 0 . 05) × 10 43 erg s -1 and the jet is L jet , 2 -10keV = (5 . 6 ± 0 . 7) × 10 41 erg s -1 .</text>
<section_header_level_1><location><page_11><loc_31><loc_49><loc_69><loc_51></location>2.3. Variability of the X-ray emission</section_header_level_1>
<text><location><page_11><loc_12><loc_15><loc_88><loc_46></location>The Chandra light curve of the core region of AP Lib has been created by extracting the counts from the source region and the background region used for the spectral analysis. The energy range of 0.2 to 8 keV was used.The light curve has been studied in different binning, optimized for the exposure and frame time (80 . 08 s, 160 . 165 s, 320 . 33 s, 200 s , 1000 s). A periodic signal was found in the extracted light curve, that can be explained by the dithering of Chandra . The dither period for the ACIS detector in the X direction is 1000s (Y direction 707s) . As mentioned in 7 , the dither period becomes visible when the source pass a node boundary, bad pixels or the chip edges. In the case of AP Lib, the node between the detector coordinate in X (CHIPX) of 511 to 512 is located in the dither direction. This cause lower count rates with a period of 1000 seconds. Except for this instrumental effect, the X-ray emission of the core is not variable. The fit of a</text>
<text><location><page_12><loc_12><loc_79><loc_88><loc_86></location>constant to the light curve with a binning of ∼ 200 s results in an average count rate of 0 . 497 ± 0 . 007 counts / sec and a fit probability of p χ 2 ∼ 30% ( χ 2 /dof = 61 / 65) and therefore no significant variation could be detected.</text>
<text><location><page_12><loc_12><loc_40><loc_88><loc_76></location>On longer time scales, variation of the X-ray emission was determined with Swift observations conducted between 2007 and 2011. 10 Swift observations (obsID: 00036341001 to 00036341010) were conducted in 2007, 2008, 2010 and 2011 with a total exposure of 28.2 ks. The observations on May 14, 2007 (00036341002) and Feb. 16, 2010 (00036341009) were not taken into account due to their low exposure. For the Swift analysis, XRT exposure maps were generated with the xrtpipeline to account for some bad CCD columns that are masked out on-board. The masked hot columns appeared when the XRT CCD was hit by a micro meteoroid. Spectra of the Swift data in PC-mode have been extracted with xselect from a circular region with a radius of 20 pixel ≈ 0 . 8 ' at the position of AP Lib, which contains 90% of the PSF at 1.5 keV. The background was extracted from a circular region with radius of 80 pixel ≈ 3 ' near the source. The auxiliary response files were created with xrtmkarf and the response matrices were taken from the Swift package of the calibration database caldb .</text>
<text><location><page_12><loc_12><loc_15><loc_88><loc_37></location>The flux in the energy range 2 - 10 keV has two different level of F 2 -10keV = (3 . 2 ± 0 . 4) × 10 -12 erg cm -2 s -1 in 2007/2008 and F 2 -10keV = (4 . 9 ± 0 . 5) × 10 -12 erg cm -2 s -1 in 2010/2011. A fit of a constant results in a flux of F 2 -10keV = (3 . 9 ± 0 . 2) × 10 -12 erg cm -2 s -1 and a probability of p χ 2 = 6 . 6 × 10 -5 . During the full time period of the Swift observations no spectral change appeared (power law with average photon index of Γ = 1 . 6 ± 0 . 1 taking into account the Galactic absorption). To check for short term variability, light curves with a binning of 200 s from each single observation were created and no significant variation was detected on this short time scales.</text>
<text><location><page_12><loc_16><loc_12><loc_87><loc_13></location>The variation is assumed to result from the core region; the jet cannot be resolved in</text>
<text><location><page_13><loc_12><loc_85><loc_71><loc_86></location>the Swift XRT observations and the core dominates the measured flux.</text>
<text><location><page_13><loc_12><loc_63><loc_88><loc_82></location>In the sum of the available Swift observations in photon counting mode (total exposure of ∼ 28 ks), the X-ray jet is not visible. Since the angular resolution of Swift is 18 '' compared to 1 '' of Chandra , AP Lib appears point-like and the Chandra jet is fully contained in the XRT PSF. Only possible extension at larger scale of the X-ray jet, which could not be seen with Chandra due to the used subarray, could be determined with Swift . But the exposure of the available Swift observation is too low to determine any very faint extension of the jet beyond 18 '' distance to the core.</text>
<section_header_level_1><location><page_13><loc_33><loc_56><loc_67><loc_57></location>3. Multi-wavelength observations</section_header_level_1>
<section_header_level_1><location><page_13><loc_35><loc_51><loc_65><loc_53></location>3.1. Jet synchrotron emission</section_header_level_1>
<section_header_level_1><location><page_13><loc_40><loc_47><loc_60><loc_48></location>3.1.1. Radio emission</section_header_level_1>
<text><location><page_13><loc_12><loc_16><loc_88><loc_44></location>Observations with the Very Large Array (VLA) on AP Lib (Cassaro et al. 1999) show the clear detection of the radio jet (see Fig. 2). In VLA observations at 1.36 GHz and 4.88 GHz, the radio jet emerges along the south-east direction and bends towards north-east after ∼ 12 '' , for a total extent of ∼ 55 '' (Cassaro et al. 1999). The flux density of the jet at 1.36 GHz is given as 210 mJy and the core has a flux density of ∼ 1 . 6 Jy (Cassaro et al. 1999). The observations with the D array at 1.4 GHz show a diffuse emission on arcmin scale on the same side of the jet (Condon et al. (1998); Cassaro et al. (1999)). Instead, the jet on milli-arcsec scale emerges to the south with a position angle of ∼ 180 · (the VLBA monitoring program MOJAVE 8 , Lister et al. (2009)). The jet on arcsec scales determined with VLA is pointing at PA ∼ 120 · . The further bend with an angle of ∼ 50 · result in a</text>
<text><location><page_14><loc_12><loc_85><loc_73><loc_86></location>total change of direction of ∼ 110 · compared to the milli-arcsec scale jet.</text>
<text><location><page_14><loc_12><loc_69><loc_88><loc_82></location>The comparison of the kpc jet in radio and X-rays reveals the same jet morphology of the emission along the SE direction (see Fig. 2). An IC emission model for the X-ray jet assumes that radio synchrotron radiation is emitted by the same electrons that give rise to the X-ray radiation via Compton scattering. This is supported independently by the X-ray spectral data and the similarity of the jet morphology in the radio and X-ray bands.</text>
<section_header_level_1><location><page_14><loc_40><loc_62><loc_60><loc_63></location>3.1.2. Optical emission</section_header_level_1>
<text><location><page_14><loc_12><loc_51><loc_88><loc_59></location>In previous works, there have been no indication for a jet in the high energy synchrotron range. In HST observations, the elliptical host galaxy is seen, but no significant excess emission associated with the X-ray/radio jets is visible.</text>
<text><location><page_14><loc_12><loc_27><loc_88><loc_49></location>The core properties were studied by Westerlund et al. (1982) who used UBV measurements to separate a nucleus with a non-thermal continuum spectrum and an extended component identified as an elliptical galaxy. The optical flux of the unresolved nucleus varied between 14.6 and 17 mag in the V-band during the observations mentioned in Westerlund et al. (1982) and the colours are given as ( B -V ) = +0 . 55 ± 0 . 01 and ( U -B ) = -0 . 57 ± 0 . 01. For the extended component they extracted an optical flux of 14 . 7 ± 0 . 05 mag in the V-band after correction for reddening and determined the color of the host galaxy as ( B -V ) galaxy = 1 . 02.</text>
<text><location><page_14><loc_12><loc_12><loc_88><loc_25></location>In 1993, Stickel et al. (1993) found that the host galaxy of AP Lib appears asymmetric and elongated towards a nearby galaxy ( ≈ 65 '' to the north east). They suggested that AP Lib is an interacting system. The obtained spectra by Pesce et al. (1994) show several absorption lines for the second galaxy resulting in a redshift of z = 0 . 048. This indicates that the host galaxy of the BL Lac object and the nearby galaxy are associated and the</text>
<text><location><page_15><loc_12><loc_85><loc_53><loc_86></location>projected separation is 83 kpc (Pesce et al. 1994).</text>
<section_header_level_1><location><page_15><loc_31><loc_77><loc_69><loc_79></location>3.2. Core and jet spectral distribution</section_header_level_1>
<text><location><page_15><loc_12><loc_41><loc_88><loc_74></location>The spectra obtained from the Chandra observation for the core and jet region as well as radio and optical fluxes are summarized in Figure 7. For the core distribution, the radio core flux from Cassaro et al. (1999) and the unresolved radio emission from PLANCK and Kuhr et al. (1981) dominated by the core emission, are shown. The optical emission in the R and B band are values averaged over the time range of the 2FGL catalogue (Nolan et al. 2012) obtained with the ATOM telescope. It has to be considered that AP Lib is very variable in the optical band. The optical emission has been corrected for the influence of the host galaxy (Pursimo et al. 2002) and the Galactic extinction (using E ( B -V ) = 0 . 138, Schlegel et al. (1998)). The radio emission from the jet is taken from Cassaro et al. (1999). The spectral shape of the core and jet emission is rather similar and the radio and optical emission can be described by synchrotron photons. Instead, the X-ray spectra are clearly dominated by inverse Compton emission due to their hard spectral index.</text>
<text><location><page_15><loc_12><loc_28><loc_88><loc_38></location>The information for the spectral energy distribution of the extended emission is still limited and is difficult to derive a detailed emission model. The hard spectral index of the X-ray jet suggests that the jet is dominated by inverse Compton emission in a similar way as the core.</text>
<text><location><page_15><loc_12><loc_13><loc_87><loc_26></location>The high energy γ -ray spectrum from the 2FGL (Nolan et al. 2012) and the 1FHL (Ackermann et al. 2013) catalogues are also shown in Figure 7. It is not possible to discriminate between jet and core emission, due to the limited angular resolution of the Fermi -LAT instrument. In the 2FGL catalogue, the data are obtained over two years and no strong variation was found for AP Lib. The 1FHL catalogue considers one additional</text>
<text><location><page_16><loc_12><loc_79><loc_87><loc_86></location>year of Fermi -LAT observations, and is therefore not contemporaneous with the 2FGL. The butterfly spectrum for the 1FHL data represents the 1 σ uncertainty range of the flux density and photon index.</text>
<section_header_level_1><location><page_16><loc_35><loc_72><loc_65><loc_73></location>3.3. High energy IC emission</section_header_level_1>
<text><location><page_16><loc_12><loc_64><loc_87><loc_68></location>The instruments which can be used to detect the high energy IC emission above 100 MeV do not have the angular resolution to discriminate between jet and core emission.</text>
<text><location><page_16><loc_12><loc_51><loc_88><loc_62></location>A GeV source, 2FGL J1517.7-2421 (Nolan et al. 2012) was detected with the Fermi Gamma-ray space telescope and can be associated with AP Lib. The flat spectrum and its flux in the Fermi energy range indicates that the maximum of the high energy component of the emission is located at > 100 MeV.</text>
<text><location><page_16><loc_12><loc_45><loc_87><loc_49></location>Very high energy (E > 100 GeV) γ -ray emission was detected from AP Lib with the H.E.S.S. Cherenkov telescope array in June/July 2010 (Hofmann 2010).</text>
<text><location><page_16><loc_12><loc_29><loc_88><loc_42></location>Among all TeV BL Lac objects, AP Lib is the object with the lowest synchrotron peak frequency and the most extreme spectral indices for the radio-optical and optical-X-ray range (Fortin et al. 2011). Following the standard SSC model, the expected emission for AP Lib in the very high energy band is well below the sensitivity of current Cherenkov telescopes and it is remarkable to detect TeV emission from this object.</text>
<section_header_level_1><location><page_16><loc_36><loc_22><loc_64><loc_23></location>4. Summary and conclusion</section_header_level_1>
<text><location><page_16><loc_12><loc_12><loc_88><loc_19></location>AP Lib is classified as a low frequency BL Lac object. Unexpected for this subclass, the object has been detected in the TeV band (Hofmann 2010; Fortin et al. 2011). Generally, the emission model for explaining the TeV γ -ray emission at the base of a synchrotron</text>
<text><location><page_17><loc_12><loc_82><loc_85><loc_86></location>self-Compton (SSC) model requires rather high Doppler factors. The values found in spectral modelling significantly exceed the values determined from VLBI monitoring.</text>
<text><location><page_17><loc_12><loc_72><loc_87><loc_79></location>AP Lib shows the lowest peak frequency of the synchrotron emission and the most extreme spectral indices of all TeV BL Lac objects. This raises the question whether the jet properties of AP Lib are unusual as well.</text>
<text><location><page_17><loc_12><loc_27><loc_88><loc_69></location>A clearly visible extended, non-thermal X-ray jet was discovered in AP Lib in the analysis of the Chandra observation of July 4, 2003. The X-ray jet is located in the south-east direction of the source and bends towards north-east comparable with the jet visible in the VLA radio observation. The radial profile of the X-ray jet reveals an extension up to ∼ 15 '' , while no counter jet could be detected. The detailed study of different profiles of the X-ray jet show that the X-ray emission morphology is comparable with the radio emission at 1.36 GHz. Therefore it is reasonable to assume, that the same electron population produces the radio and X-ray emission in the jet. The X-ray and radio jet emerges in south-east direction and turns by an angle of ∼ 50 · between 2 '' and 20 '' and by 110 · between few milli-arcseconds and few arcseconds. This strong bending of the jet could be explained by a small angle of the jet axis to the line of sight of the observer. This would match the assumption of a high Doppler factor as expected from the detection of the TeV emission for which beaming is necessary. Rector et al. (2003) concluded that LBL are seen closer to the jet axis than HBLs since they have a wide distribution of parsec- and kpc-scale jet alignment angles.</text>
<text><location><page_17><loc_12><loc_12><loc_88><loc_25></location>BL Lac objects represent FRI galaxies in which the jet points under a small angle of θ < 15 · (Marscher & Jorstad 2011) to the line of sight. Considering this maximum angle, the de-projected length ( d jet ) of the visible X-ray jet would be d jet ≈ 54 kpc. The possible range for the de-projected length would be up to d jet ≈ 802 kpc assuming a much smaller angle of θ ≈ 1 · . This maximum value for the de-projected length is comparable to the value</text>
<text><location><page_18><loc_12><loc_84><loc_80><loc_86></location>determined by Marscher & Jorstad (2011) for OJ287 of d jet ≈ 640 kpc for θ = 3 . 2.</text>
<text><location><page_18><loc_12><loc_72><loc_88><loc_82></location>The X-ray spectrum of the core region is described by a power law with Γ = 1 . 58 ± 0 . 04 taking into account the Galactic absorption of N H = 8 . 36 × 10 20 cm -2 . The jet spectrum is described by a power law with photon index of Γ = 1 . 8 ± 0 . 1 and has a flux of ∼ 10% of the core flux.</text>
<text><location><page_18><loc_12><loc_53><loc_88><loc_69></location>Interestingly the core and jet spectra both have a hard spectral index indicating IC dominance. As described in e.g. Harris & Krawczynski (2006), BL Lac objects are interpreted as beamed versions of the low-luminosity FRI radio galaxies. The IC dominance of the X-ray jet is unusual for low luminosity AGN, as described in Harris & Krawczynski (2006) and Worrall (2009). Generally, the X-ray jet spectra for FRI are synchrotron dominated and IC spectra only appear in X-ray jets of high luminosity sources.</text>
<text><location><page_18><loc_12><loc_41><loc_88><loc_51></location>All six BL Lac objects with extended X-ray jets (AP Lib (this work), S5 2007+777 (Sambruna et al. 2008), OJ287 (Marscher & Jorstad 2011), PKS 0521-365 (Birkinshaw et al. 2002), 3C371 and PKS 2201+044 (Sambruna et al. 2007)) show jet luminosities intermediate between FRI and FRII.</text>
<text><location><page_18><loc_12><loc_11><loc_87><loc_38></location>The spectral shape of the X-ray emission of their jets is comparable to be flat or IC dominated. AP Lib, OJ 287 and S5 2007+777 show stronger IC dominance. Therefore the BL Lac objects behave more similar to FRII which have mostly IC dominated X-ray jets, although BL Lac objects are generally classified as beamed versions of FRI galaxies. The X-ray jets of BL Lac objects are also more luminous than the X-ray jets of FRI galaxies. Compared to the sample of FRI X-ray jets of Harwood & Hardcastle (2012), the X-ray luminosity of the jets of 3C371, PKS 2201+044 (Harwood & Hardcastle 2012) and PKS 0521-365 (Birkinshaw et al. 2002) are at the high end of the luminosity distribution of FRI with 2 × 10 16 W / Hz < L ν < 3 × 10 17 W / Hz. Their X-ray spectra are rather flat and the photon indices are comparable to 2. The jet of OJ 287, S5</text>
<text><location><page_19><loc_12><loc_79><loc_88><loc_86></location>2007+777 (Harwood & Hardcastle 2012) and AP Lib (this work) have highest luminosities of 6 × 10 17 W / Hz < L ν < 8 × 10 18 W / Hz and IC dominated X-ray spectra. The X-ray jet spectra become IC dominated with increasing luminosity.</text>
<text><location><page_19><loc_12><loc_49><loc_88><loc_76></location>Within the group of BL Lac objects with X-ray jets, the spectral indices of the jets also correlate with the spectral indices of the cores. The LBL have hard core spectra and show also harder spectra in the jet. The LBLs S5 2007+777 (Sambruna et al. 2008) and OJ287 (Marscher & Jorstad 2011) have, like AP Lib, X-ray jets with IC dominated spectra. The IBL have softer spectra in the core and their jet spectra also appear softer than the ones for LBLs. Hence, BL Lac objects with low synchrotron peak energies have higher probability to have IC dominance in the X-ray jet. AP Lib has the lowest luminosity in the X-ray jet among the jets of BL Lac objects with a clear IC dominated X-ray spectrum. This raises the question, if the explanation for IC jets as IC scattering of CMB photons is still working or whether other external photon fields become important.</text>
<text><location><page_19><loc_12><loc_24><loc_88><loc_46></location>Since also the core components of the X-ray emission of the BL Lac objects are IC dominated (only the core emission of S5 2007+777 has Γ ∼ 2), this component is expected to reach high energies. Furthermore, the IC spectrum emerges in the X-ray regime of the core emission and the fact that the peak of the IC emission is in the GeV range (all six BL Lac objects are detected by Fermi have 2 . 0 < Γ < 2 . 4 in the 2nd Fermi catalog (Nolan et al. 2012)) reveal a broad IC component. This behaviour of the IC scattering could also exist in the jet component. For AP Lib it is already known that the IC component reaches TeV energies (Hofmann 2010).</text>
<text><location><page_19><loc_12><loc_12><loc_88><loc_22></location>The discovery of an extended X-ray jet in the TeV BL Lac object AP Lib, which is IC dominated, provides therefore the opportunity to test the interpretation of the most common emission models as summarized in the review of X-ray jets, e.g. Harris & Krawczynski (2006).</text>
<text><location><page_20><loc_12><loc_73><loc_87><loc_86></location>The scientific results reported in this article are based on data obtained from the Chandra Data Archive. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application package CIAO. The use of the public HEASARC software packages is also acknowledged. S.K. and S.W. acknowledge support from the BMBF through grant DLR 50OR0906.</text>
<figure>
<location><page_21><loc_12><loc_34><loc_88><loc_74></location>
<caption>Fig. 2.X-ray count map of the energy 0.2 to 8 keV with adaptive smoothing using an elliptical Gaussian. The X-ray jet is clearly visible. Contours of the VLA observation of AP Lib in A+B configuration at 1.36 GHz and the restoring beam in the position angle (PA) 28 · are taken from Cassaro et al. (1999).</caption>
</figure>
<figure>
<location><page_22><loc_12><loc_35><loc_88><loc_75></location>
<caption>Fig. 3.- Background subtracted radial profile of AP Lib, extracted from wedge annuli regions for the energy range 0.2 - 8 keV in the direction of the jet with an opening angle of 100 · . The line represents the PSF simulated with MARX based on the spectrum of the core region. The dashed lines represent the uncertainty range for the PSF.</caption>
</figure>
<figure>
<location><page_23><loc_11><loc_54><loc_89><loc_67></location>
<caption>Fig. 4.left : Jet intensity profile of the X-ray (circles) and radio emission (dashed line) from box regions along the jet. The brightness has been normalized to the first value with a surface brightness of 5 . 2 counts / arcsec 2 for the X-ray emission and 16 . 15 mJy / beam for the radio emission. The dotted vertical line indicates the region of the bending of the jet. right : Transverse jet profile of the X-ray emission shown as circles and the radio emission indicated as dashed line. The brightness has been normalized to the highest value with a surface brightness of 7 . 6 counts / arcsec 2 for the X-ray emission and 26 . 1 mJy / beam for the radio emission.</caption>
</figure>
<figure>
<location><page_24><loc_20><loc_28><loc_88><loc_75></location>
<caption>Fig. 5.- X-ray count map of the Chandra observation with the different regions used for the spectral analysis of the jet.</caption>
</figure>
<figure>
<location><page_25><loc_12><loc_15><loc_73><loc_86></location>
<caption>Fig. 6.- X-ray spectra extracted from the core and the jet. The shown jet spectra are obtained from the extended emission region (black) and wide jet region (grey). The line represent a power law fit taking into account the Galactic absorption as described in the text. The lower panels show the residuals of each fit.</caption>
</figure>
<figure>
<location><page_26><loc_12><loc_49><loc_58><loc_72></location>
<caption>Fig. 7.- Historical radio (Kuhr et al. (1981); Cassaro et al. (1999) and PLANCK), optical (ATOM) and X-ray spectra ( Chandra ) of the core (black) of AP Librae. The optical fluxes are corrected for the influence of the host galaxy and the Galactic extinction and the X-ray spectra are corrected for Galactic absorption. In grey triangles, the radio (Cassaro et al. 1999) and X-ray spectra ( Chandra ) of the jet are shown. At γ -ray energies, the spectra from the 2FGL (open squares, Nolan et al. (2012)) and the 1FHL (butterfly, Ackermann et al. (2013)) catalogues are shown. At these energies, no discrimination between jet and core emission is possible.</caption>
</figure>
<section_header_level_1><location><page_27><loc_43><loc_85><loc_58><loc_86></location>REFERENCES</section_header_level_1>
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</document>
[
{
"title": "ABSTRACT",
"content": "Chandra observations of the low-energy peaked BL Lac object AP Librae revealed the clear discovery of a non-thermal X-ray jet. AP Lib is the first low energy peaked BL Lac object with an extended non-thermal X-ray jet that shows emission into the VHE range. The X-ray jet has an extension of ∼ 15 '' ( ≈ 14 kpc). The X-ray jet morphology is similar to the radio jet observed with VLA at 1.36 GHz emerging in south-east direction and bends by 50 · at a distance of 12 '' towards north-east. The intensity profiles of the X-ray emission are studied consistent with those found in the radio range. The spectral analysis reveals that the X-ray spectra of the core and jet region are both inverse Compton dominated. This adds to a still small sample of BL Lac objects whose X-ray jets are IC dominated and thus more similar to the high luminosity FRII sources than to the low luminosity FRI objects, which are usually considered to be the parent population of the BL Lac objects. Subject headings: BL Lac objects: individual (AP Lib), galaxies: active, galaxies: jets, X-rays: galaxies",
"pages": [
2
]
},
{
"title": "S. Kaufmann",
"content": "Landessternwarte, Universit¨at Heidelberg, K¨onigstuhl, D-69117 Heidelberg, Germany S.Kaufmann@lsw.uni-heidelberg.de S.J. Wagner Landessternwarte, Universit¨at Heidelberg, K¨onigstuhl, D-69117 Heidelberg, Germany O. Tibolla Institut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, D-97074 W¨urzburg, Germany Received ; accepted",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The low-energy peaked BL Lac object AP Librae (AP Lib, PKS 1514-241) has a redshift of z=0.0486 (Disney et al. 1974) and is located at α J2000 = 15 h 17 m 41 . 81313 s ± 0 . 00002 s , δ J2000 = -24 · 22 ' 19 . 4759 '' ± 0 . 0003 '' as determined from VLBI observations by Lambert & Gontier (2009). It has been classified as a BL Lac object by Strittmatter et al. (1972) and Bond (1973). AP Lib is well known as one of the most active blazars in the optical band. In data from 1989, intra-day variability was detected with a very high rate of change of 0 . 06 ± 0 . 01 mag/hr (Carini et al. 1991). Even on shorter time scales of 20min, variation of up to 0 . 5 mag have been detected in 1973 (Miller et al. 1974). AP Lib was historically classified as a so-called radio selected BL Lac objects (RBL). In the 1990s, BL Lac objects were found mainly in radio or X-ray surveys and therefore classified as radio or X-ray selected BL Lac objects. The spectral energy distributions (SED) of BL Lac objects show two prominent peaks which are commonly described in a leptonic model as synchrotron and inverse Compton (IC) emission, respectively. The peak energy of the low-energy (synchrotron) component is used to classify BL Lac objects as low-energy peaked BL Lac object (LBL) and high-energy peaked (HBL), with occasional references to a group of intermediate-energy peaked BL Lac objects (IBL). This classification is frequently based on the slope of the X-ray spectrum. Ciliegi et al. (1995) found that the X-ray spectrum of AP Lib can be described by a power law with photon index of 1 . 5 -1 . 7. AP Lib is thus classified as low-energy peaked BL Lac object (LBL) and it is assumed that the X-ray emission of the core is due to IC scattering of the synchrotron emission, and, possibly, external radiation. The peak energy of the gamma-ray component of LBL objects is expected to arise in the keV-GeV range and therefore a rather low flux in the TeV γ -ray range is expected (close to or below the detection limit of current Cherenkov telescopes). Hence, it is rather unexpected to detect very high energy (VHE,E > 100 GeV) γ -ray emission from an LBL. Therefore it was remarkable, that in June/July 2010, VHE γ -ray emission was detected from the position of AP Lib (Hofmann 2010). The standard picture of AGN, e.g. Blandford & Rees (1974), explains the different observational characteristics of AGN with the orientation of the jet axis to the line of sight to the observer. BL Lac objects are interpreted as aligned (beamed) versions of Fanaroff-Riley I (FRI, Fanaroff & Riley (1974)) radio galaxies while steep and flat spectrum radio quasars (SSRQ and FSRQ) are aligned versions of FRII galaxies. BL Lac objects are known to have very energetic jets pointing under small viewing angle towards the observer. Radio observations of AP Lib at 1.4 GHz reveal the detection of one-sided, diffuse radio emission at the arcmin scale (Condon et al. 1998). Observations with the Very Large Array (VLA) at 1.36 GHz and 4.6 GHz result in the detection of a one-sided radio jet emerging to the south-east direction and bending towards north-east at ∼ 12 arcsec distance from the core (Cassaro et al. 1999). Very Long Baseling Array (VLBA) observations detected the radio jet at milli-arcsec scale which emerges to the south (MOJAVE 1 , Lister et al. (2009)). The inhomogeneous collection of published X-ray jets (status March 2013: 113 X-ray jets), the XJET database 2 , contains 77% high luminosity sources (FRII, quasars) and ∼ 23% low luminosity sources (20 FRI, 5 BL Lac objects and one Seyfert 1 galaxy). The X-ray emission detected from the jets of FRI sources are dominated by synchrotron emission (Harris & Krawczynski 2006). Most high luminosity sources (quasars, FRII) have X-ray jets with flat ( α < 1) spectra, suggesting this part of the spectrum to be dominated by inverse Compton emission (e.g. Harris & Krawczynski (2006), Worrall (2009)). While inverse Compton scattering of CMB photons is an explanation for flat X-ray spectra preferred by many, problems with this interpretation have been discussed by Harris & Krawczynski (2006) and Worrall (2009). Among the sources in this database, only discovered in the radio galaxies Cen A and M87 and the FSRQ PKS 1222+216 and PKS 1510+089 have been traced up to TeV energies. AP Lib is the first BL Lac object with an extended X-ray jet that has been detected in the TeV energy range. The luminosity distance of AP Lib, using H 0 = 70 km s -1 Mpc -1 , is d L = 210 Mpc. The scale is 1 '' = 0 . 95 kpc.",
"pages": [
3,
4,
5
]
},
{
"title": "2. Extended X-ray jet",
"content": "X-ray observations of AP Lib were analysed to search for high-energy emission of the one-sided radio jet. The source was observed with Chandra on July 4, 2003 (ObsID: 3971) with an exposure of 12.8 ks. The data were taken in timed exposure mode using a subarray of 1/8 of the chip (128 rows). This mode decreases the frame time to 0.4 s and reduces pileup. The data were re-calibrated using the calibration database CALDB. The Chandra data have bee analysed with the software CIAO v4.1 . A clear X-ray jet was detected (Kaufmann et al. 2011) in this Chandra observation (see Fig. 1). The jet extends toward the south-east direction of AP Lib. Unfortunately the exposure is rather low, so that the real extension of the jet cannot be measured. Despite the timed exposure mode, the brightness of the core causes a faint 'readout artefact' visible in the column in which the bright core region is located. For the spectral analysis, the signal of the influenced columns is replaced by a typical background level to correct for this artefact. The tool acisreadcorr has been used to correct the data. For the used subarray, the BACKSCAL header keyword had to be modified 3 . As can be seen in Fig. 1, the X-ray jet emerges in the South-East direction up to an extension of ∼ 12 '' and bends with an angle of ≈ 50 · to the North-East. The jet broadens towards the outer regions and becomes fainter. In order to compare the morphology of the X-ray jet to the radio jet studied with VLA by Cassaro et al. (1999), the X-ray count map has been smoothed (see Fig. 2) with an elliptical Gaussian with major axis 3 '' and minor axis 2 '' using the ftool fgauss . The shape for the elliptical Gaussian matches the beam profile of the VLA data. The VLA radio contours (described in section 3.1.1) are overlayed onto the smoothed X-ray map (see Fig. 2). The jet in the radio and X-ray bands have very similar morphologies. They extend in the same direction, bend at the same distance by the same amount, and have comparable spatial profiles. Neither the X-ray nor the radio jet displays knots, hotspots or other features of high contrast. No feature on the counter-jet side could be detected in either band. A quantitative comparison is presented in section 2.1.",
"pages": [
5,
6,
7
]
},
{
"title": "2.1. Intensity profiles",
"content": "Radial profiles have been extracted with the tool dmextract from the whole source (core and jet) using 20 equidistant circular annuli of width ∼ 1 '' . In addition, a fraction of the annuli (wedge) with opening angle 100 · in the direction of the jet has been used. A circular region close to the source in North-West direction with radius ∼ 9 '' was used to determine the background for the radial profile.The two methods (complete annuli and a fraction of the annuli) used to obtain the radial profile do not show significant differences. This radial profile and its comparison to the PSF are used to identify the extension of the jet and to find the best regions to obtain the core and jet spectra. The radial profiles have been extracted for two different energy sub-bands (0.2 - 1.5 keV (S) and 1.5 - 8 keV (H)) and the hardness ratio profile has been calculated to study the spectral trends in the jet region. The hardness ratio profile has no significant trend within 15 '' . The PSF for the specific on-axis angle of the source was derived using the Chandra Ray Tracer (ChaRT 4 ) which simulates the High Resolution Mirror Assembly (HRMA) based on an input energy spectrum of the core and the exposure of the observation. The output from ChaRT can be modeled, taking into account instrument effects of the various detectors, using the software MARX 5 to obtain the image of the simulated PSF on the detector. The radial profile for this PSF has been created with wedge annuli with an opening angle of 100 · . The radial profile of the X-ray source deviates significantly from the PSF at radii > 2 '' (see Fig. 3). The jet at radii > 3 '' has a linear structure and therefore the intensity profile for the jet has been created using rectangular regions along the jet. The intensity profile of the X-ray and radio jet has been extracted by integrating the counts at equidistant steps of 1 . 5 '' along the jet and perpendicular to the direction of the jet to compare the X-ray and radio morphology of the jet (see Fig. 4). In order to account for the jet bend, profiles along and across the jet are determined from 3 '' to 13 . 5 '' along PA = 120 · and from 13 . 5 '' to 22 . 5 '' along PA = 70 · . In order to avoid contributions from the core, a minimum distance of 3 '' was chosen. As can be seen in Fig. 4, the same morphology is detected of the X-ray and radio emission with a slightly shallower gradient of radio flux in the outer region of the jet. The transversal profile (see Fig. 4) of the jet from 3 '' up to 13 . 5 '' ( ≈ 2 . 9 -12 . 8 kpc) was determined along PA = 120 · . The transversal profile of the X-ray and radio emission is comparable and the jet width is 5 '' ≈ 4 . 8 kpc in both energy bands. The profiles of both energy bands are compatible. This suggests that the same particles are responsible for the radio and X-ray jet.",
"pages": [
7,
8
]
},
{
"title": "2.2. Spectra",
"content": "The spectra of the core and the jet of AP Lib and of a background region are determined with the tool dmextract . The response files are obtained with the tool mkrmf and the ARF are created using asphist and mkarf . The X-ray spectrum of the core was extracted using a circular region with radius 3 '' . This extraction radius was chosen, since the radial profile matches the simulated PSF out to this radius. The background spectrum was determined within a region of radius 25 '' close-by to the source in the North direction. To obtain the spectrum of the extended emission, a circular region with radius 15 '' was used in which the region around the core was excluded (see Fig. 5). As can be seen from the radial profile, the core exclusion region is large enough to avoid any influence of the core photons. Based on the radial profile of the core and jet (Fig. 3) and the shown PSF, the wing of the PSF of the core contributes < 10% to the jet spectrum obtained in the range > 3 '' . The X-ray spectra have been binned with the tool grppha to obtain at least 25 counts per bin to reach the necessary significance for the χ 2 statistics. The program xspec v12 was used to fit the X-ray spectra in the energy range 0.2 to 8 keV. The uncertainties on the model parameters are given as confidence intervals. The fit parameter is changed by ∆ χ 2 = 2 . 71. This represents the 90% confidence interval. The spectrum of the core can be well ( χ 2 /dof = 165 / 147) described by a power law of the form N ( E ) = N 0 × E -Γ with Γ = 1 . 58 ± 0 . 04 taking into account the Galactic absorption of N H = 8 . 36 × 10 20 cm -2 (LAB survey 6 , Kalberla et al. (2005)). The resulting flux is F core , 2 -10keV = (2 . 9 ± 0 . 1) × 10 -12 erg cm -2 s -1 . Although no hints for pileup appear in the residuals of the power law fit, a test for pileup has been performed. Therefore, the spectrum of the core region has been extracted from an annulus region of the same size, in which the innermost pixels (inner radius of ∼ 1 '' ) are excluded. The spectral slope of the determined spectrum (Γ = 1 . 6 ± 0 . 1) is comparable to the core spectrum of the circular region and therefore no pileup was detected. To quantify and to search for spectral differences, three different regions have been used as illustrated in Fig. 5. The extended emission region is a circular region with radius 15 '' excluding the core region with radius 3 '' . The wide jet region is a wedge region with radius 20 '' with opening angle 100 · along the jet and the inner jet region is a wedge region with opening angle of 30 · and radius of 10 '' . The spectrum of the jet obtained from the extended emission region can be described by Γ = 1 . 8 ± 0 . 1 taking into account the Galactic absorption ( χ 2 /dof = 13 / 18). The resulting flux is F jet , 2 -10keV = (2 . 3 ± 0 . 3) × 10 -13 erg cm -2 s -1 . The spectrum from the wide jet region can be fit with a photon index of Γ = 1 . 8 ± 0 . 2 ( χ 2 /dof = 6 / 13) comparable to the one above. The resulting flux is (1 . 6 ± 0 . 3) × 10 -13 erg cm -2 s -1 . An even smaller region, the inner jet region was used to determine the spectrum of the inner parts of the jet. Since the region is very small, the spectrum consist of only a few photons and therefore a reduced flux of (6 . 0 ± 1 . 7) × 10 -14 erg cm -2 s -1 resulted. The slope (Γ = 1 . 9 ± 0 . 5) is comparable to the above determined spectral fits. For all spectral fits, the Galactic absorption was used as fixed parameter and no hint for additional absorption was found. The X-ray spectra for the core and the jet are shown in Fig. 6. No significant difference of the slope between the jet and the core spectrum was determined. The jet spectrum can be well described with a power law model and no emission or absorption features were detected. A fit with the thermal model apec in Xspec , considering the Galactic absorption, resulted in a worse fit and a high gas temperature of kT = 6 ± 3 keV. Any acceptable fit of a combined model using the combination of the thermal model apec (Smith et al. 2001) and a power law is dominated by the power law. Therefore the power law model is the favoured description for the X-ray spectrum of the jet and hence the jet is considered to be dominated by non-thermal emission. The core and the jet spectrum have both photon indices of Γ < 2 and are thus interpreted to be inverse Compton dominated. The core spectrum is comparable with the original definition of AP Lib being a low-energy peaked BL Lac object with IC dominance in the X-ray spectrum. The luminosity of the X-ray core is L core , 2 -10keV = (1 . 53 ± 0 . 05) × 10 43 erg s -1 and the jet is L jet , 2 -10keV = (5 . 6 ± 0 . 7) × 10 41 erg s -1 .",
"pages": [
9,
10,
11
]
},
{
"title": "2.3. Variability of the X-ray emission",
"content": "The Chandra light curve of the core region of AP Lib has been created by extracting the counts from the source region and the background region used for the spectral analysis. The energy range of 0.2 to 8 keV was used.The light curve has been studied in different binning, optimized for the exposure and frame time (80 . 08 s, 160 . 165 s, 320 . 33 s, 200 s , 1000 s). A periodic signal was found in the extracted light curve, that can be explained by the dithering of Chandra . The dither period for the ACIS detector in the X direction is 1000s (Y direction 707s) . As mentioned in 7 , the dither period becomes visible when the source pass a node boundary, bad pixels or the chip edges. In the case of AP Lib, the node between the detector coordinate in X (CHIPX) of 511 to 512 is located in the dither direction. This cause lower count rates with a period of 1000 seconds. Except for this instrumental effect, the X-ray emission of the core is not variable. The fit of a constant to the light curve with a binning of ∼ 200 s results in an average count rate of 0 . 497 ± 0 . 007 counts / sec and a fit probability of p χ 2 ∼ 30% ( χ 2 /dof = 61 / 65) and therefore no significant variation could be detected. On longer time scales, variation of the X-ray emission was determined with Swift observations conducted between 2007 and 2011. 10 Swift observations (obsID: 00036341001 to 00036341010) were conducted in 2007, 2008, 2010 and 2011 with a total exposure of 28.2 ks. The observations on May 14, 2007 (00036341002) and Feb. 16, 2010 (00036341009) were not taken into account due to their low exposure. For the Swift analysis, XRT exposure maps were generated with the xrtpipeline to account for some bad CCD columns that are masked out on-board. The masked hot columns appeared when the XRT CCD was hit by a micro meteoroid. Spectra of the Swift data in PC-mode have been extracted with xselect from a circular region with a radius of 20 pixel ≈ 0 . 8 ' at the position of AP Lib, which contains 90% of the PSF at 1.5 keV. The background was extracted from a circular region with radius of 80 pixel ≈ 3 ' near the source. The auxiliary response files were created with xrtmkarf and the response matrices were taken from the Swift package of the calibration database caldb . The flux in the energy range 2 - 10 keV has two different level of F 2 -10keV = (3 . 2 ± 0 . 4) × 10 -12 erg cm -2 s -1 in 2007/2008 and F 2 -10keV = (4 . 9 ± 0 . 5) × 10 -12 erg cm -2 s -1 in 2010/2011. A fit of a constant results in a flux of F 2 -10keV = (3 . 9 ± 0 . 2) × 10 -12 erg cm -2 s -1 and a probability of p χ 2 = 6 . 6 × 10 -5 . During the full time period of the Swift observations no spectral change appeared (power law with average photon index of Γ = 1 . 6 ± 0 . 1 taking into account the Galactic absorption). To check for short term variability, light curves with a binning of 200 s from each single observation were created and no significant variation was detected on this short time scales. The variation is assumed to result from the core region; the jet cannot be resolved in the Swift XRT observations and the core dominates the measured flux. In the sum of the available Swift observations in photon counting mode (total exposure of ∼ 28 ks), the X-ray jet is not visible. Since the angular resolution of Swift is 18 '' compared to 1 '' of Chandra , AP Lib appears point-like and the Chandra jet is fully contained in the XRT PSF. Only possible extension at larger scale of the X-ray jet, which could not be seen with Chandra due to the used subarray, could be determined with Swift . But the exposure of the available Swift observation is too low to determine any very faint extension of the jet beyond 18 '' distance to the core.",
"pages": [
11,
12,
13
]
},
{
"title": "3.1.1. Radio emission",
"content": "Observations with the Very Large Array (VLA) on AP Lib (Cassaro et al. 1999) show the clear detection of the radio jet (see Fig. 2). In VLA observations at 1.36 GHz and 4.88 GHz, the radio jet emerges along the south-east direction and bends towards north-east after ∼ 12 '' , for a total extent of ∼ 55 '' (Cassaro et al. 1999). The flux density of the jet at 1.36 GHz is given as 210 mJy and the core has a flux density of ∼ 1 . 6 Jy (Cassaro et al. 1999). The observations with the D array at 1.4 GHz show a diffuse emission on arcmin scale on the same side of the jet (Condon et al. (1998); Cassaro et al. (1999)). Instead, the jet on milli-arcsec scale emerges to the south with a position angle of ∼ 180 · (the VLBA monitoring program MOJAVE 8 , Lister et al. (2009)). The jet on arcsec scales determined with VLA is pointing at PA ∼ 120 · . The further bend with an angle of ∼ 50 · result in a total change of direction of ∼ 110 · compared to the milli-arcsec scale jet. The comparison of the kpc jet in radio and X-rays reveals the same jet morphology of the emission along the SE direction (see Fig. 2). An IC emission model for the X-ray jet assumes that radio synchrotron radiation is emitted by the same electrons that give rise to the X-ray radiation via Compton scattering. This is supported independently by the X-ray spectral data and the similarity of the jet morphology in the radio and X-ray bands.",
"pages": [
13,
14
]
},
{
"title": "3.1.2. Optical emission",
"content": "In previous works, there have been no indication for a jet in the high energy synchrotron range. In HST observations, the elliptical host galaxy is seen, but no significant excess emission associated with the X-ray/radio jets is visible. The core properties were studied by Westerlund et al. (1982) who used UBV measurements to separate a nucleus with a non-thermal continuum spectrum and an extended component identified as an elliptical galaxy. The optical flux of the unresolved nucleus varied between 14.6 and 17 mag in the V-band during the observations mentioned in Westerlund et al. (1982) and the colours are given as ( B -V ) = +0 . 55 ± 0 . 01 and ( U -B ) = -0 . 57 ± 0 . 01. For the extended component they extracted an optical flux of 14 . 7 ± 0 . 05 mag in the V-band after correction for reddening and determined the color of the host galaxy as ( B -V ) galaxy = 1 . 02. In 1993, Stickel et al. (1993) found that the host galaxy of AP Lib appears asymmetric and elongated towards a nearby galaxy ( ≈ 65 '' to the north east). They suggested that AP Lib is an interacting system. The obtained spectra by Pesce et al. (1994) show several absorption lines for the second galaxy resulting in a redshift of z = 0 . 048. This indicates that the host galaxy of the BL Lac object and the nearby galaxy are associated and the projected separation is 83 kpc (Pesce et al. 1994).",
"pages": [
14,
15
]
},
{
"title": "3.2. Core and jet spectral distribution",
"content": "The spectra obtained from the Chandra observation for the core and jet region as well as radio and optical fluxes are summarized in Figure 7. For the core distribution, the radio core flux from Cassaro et al. (1999) and the unresolved radio emission from PLANCK and Kuhr et al. (1981) dominated by the core emission, are shown. The optical emission in the R and B band are values averaged over the time range of the 2FGL catalogue (Nolan et al. 2012) obtained with the ATOM telescope. It has to be considered that AP Lib is very variable in the optical band. The optical emission has been corrected for the influence of the host galaxy (Pursimo et al. 2002) and the Galactic extinction (using E ( B -V ) = 0 . 138, Schlegel et al. (1998)). The radio emission from the jet is taken from Cassaro et al. (1999). The spectral shape of the core and jet emission is rather similar and the radio and optical emission can be described by synchrotron photons. Instead, the X-ray spectra are clearly dominated by inverse Compton emission due to their hard spectral index. The information for the spectral energy distribution of the extended emission is still limited and is difficult to derive a detailed emission model. The hard spectral index of the X-ray jet suggests that the jet is dominated by inverse Compton emission in a similar way as the core. The high energy γ -ray spectrum from the 2FGL (Nolan et al. 2012) and the 1FHL (Ackermann et al. 2013) catalogues are also shown in Figure 7. It is not possible to discriminate between jet and core emission, due to the limited angular resolution of the Fermi -LAT instrument. In the 2FGL catalogue, the data are obtained over two years and no strong variation was found for AP Lib. The 1FHL catalogue considers one additional year of Fermi -LAT observations, and is therefore not contemporaneous with the 2FGL. The butterfly spectrum for the 1FHL data represents the 1 σ uncertainty range of the flux density and photon index.",
"pages": [
15,
16
]
},
{
"title": "3.3. High energy IC emission",
"content": "The instruments which can be used to detect the high energy IC emission above 100 MeV do not have the angular resolution to discriminate between jet and core emission. A GeV source, 2FGL J1517.7-2421 (Nolan et al. 2012) was detected with the Fermi Gamma-ray space telescope and can be associated with AP Lib. The flat spectrum and its flux in the Fermi energy range indicates that the maximum of the high energy component of the emission is located at > 100 MeV. Very high energy (E > 100 GeV) γ -ray emission was detected from AP Lib with the H.E.S.S. Cherenkov telescope array in June/July 2010 (Hofmann 2010). Among all TeV BL Lac objects, AP Lib is the object with the lowest synchrotron peak frequency and the most extreme spectral indices for the radio-optical and optical-X-ray range (Fortin et al. 2011). Following the standard SSC model, the expected emission for AP Lib in the very high energy band is well below the sensitivity of current Cherenkov telescopes and it is remarkable to detect TeV emission from this object.",
"pages": [
16
]
},
{
"title": "4. Summary and conclusion",
"content": "AP Lib is classified as a low frequency BL Lac object. Unexpected for this subclass, the object has been detected in the TeV band (Hofmann 2010; Fortin et al. 2011). Generally, the emission model for explaining the TeV γ -ray emission at the base of a synchrotron self-Compton (SSC) model requires rather high Doppler factors. The values found in spectral modelling significantly exceed the values determined from VLBI monitoring. AP Lib shows the lowest peak frequency of the synchrotron emission and the most extreme spectral indices of all TeV BL Lac objects. This raises the question whether the jet properties of AP Lib are unusual as well. A clearly visible extended, non-thermal X-ray jet was discovered in AP Lib in the analysis of the Chandra observation of July 4, 2003. The X-ray jet is located in the south-east direction of the source and bends towards north-east comparable with the jet visible in the VLA radio observation. The radial profile of the X-ray jet reveals an extension up to ∼ 15 '' , while no counter jet could be detected. The detailed study of different profiles of the X-ray jet show that the X-ray emission morphology is comparable with the radio emission at 1.36 GHz. Therefore it is reasonable to assume, that the same electron population produces the radio and X-ray emission in the jet. The X-ray and radio jet emerges in south-east direction and turns by an angle of ∼ 50 · between 2 '' and 20 '' and by 110 · between few milli-arcseconds and few arcseconds. This strong bending of the jet could be explained by a small angle of the jet axis to the line of sight of the observer. This would match the assumption of a high Doppler factor as expected from the detection of the TeV emission for which beaming is necessary. Rector et al. (2003) concluded that LBL are seen closer to the jet axis than HBLs since they have a wide distribution of parsec- and kpc-scale jet alignment angles. BL Lac objects represent FRI galaxies in which the jet points under a small angle of θ < 15 · (Marscher & Jorstad 2011) to the line of sight. Considering this maximum angle, the de-projected length ( d jet ) of the visible X-ray jet would be d jet ≈ 54 kpc. The possible range for the de-projected length would be up to d jet ≈ 802 kpc assuming a much smaller angle of θ ≈ 1 · . This maximum value for the de-projected length is comparable to the value determined by Marscher & Jorstad (2011) for OJ287 of d jet ≈ 640 kpc for θ = 3 . 2. The X-ray spectrum of the core region is described by a power law with Γ = 1 . 58 ± 0 . 04 taking into account the Galactic absorption of N H = 8 . 36 × 10 20 cm -2 . The jet spectrum is described by a power law with photon index of Γ = 1 . 8 ± 0 . 1 and has a flux of ∼ 10% of the core flux. Interestingly the core and jet spectra both have a hard spectral index indicating IC dominance. As described in e.g. Harris & Krawczynski (2006), BL Lac objects are interpreted as beamed versions of the low-luminosity FRI radio galaxies. The IC dominance of the X-ray jet is unusual for low luminosity AGN, as described in Harris & Krawczynski (2006) and Worrall (2009). Generally, the X-ray jet spectra for FRI are synchrotron dominated and IC spectra only appear in X-ray jets of high luminosity sources. All six BL Lac objects with extended X-ray jets (AP Lib (this work), S5 2007+777 (Sambruna et al. 2008), OJ287 (Marscher & Jorstad 2011), PKS 0521-365 (Birkinshaw et al. 2002), 3C371 and PKS 2201+044 (Sambruna et al. 2007)) show jet luminosities intermediate between FRI and FRII. The spectral shape of the X-ray emission of their jets is comparable to be flat or IC dominated. AP Lib, OJ 287 and S5 2007+777 show stronger IC dominance. Therefore the BL Lac objects behave more similar to FRII which have mostly IC dominated X-ray jets, although BL Lac objects are generally classified as beamed versions of FRI galaxies. The X-ray jets of BL Lac objects are also more luminous than the X-ray jets of FRI galaxies. Compared to the sample of FRI X-ray jets of Harwood & Hardcastle (2012), the X-ray luminosity of the jets of 3C371, PKS 2201+044 (Harwood & Hardcastle 2012) and PKS 0521-365 (Birkinshaw et al. 2002) are at the high end of the luminosity distribution of FRI with 2 × 10 16 W / Hz < L ν < 3 × 10 17 W / Hz. Their X-ray spectra are rather flat and the photon indices are comparable to 2. The jet of OJ 287, S5 2007+777 (Harwood & Hardcastle 2012) and AP Lib (this work) have highest luminosities of 6 × 10 17 W / Hz < L ν < 8 × 10 18 W / Hz and IC dominated X-ray spectra. The X-ray jet spectra become IC dominated with increasing luminosity. Within the group of BL Lac objects with X-ray jets, the spectral indices of the jets also correlate with the spectral indices of the cores. The LBL have hard core spectra and show also harder spectra in the jet. The LBLs S5 2007+777 (Sambruna et al. 2008) and OJ287 (Marscher & Jorstad 2011) have, like AP Lib, X-ray jets with IC dominated spectra. The IBL have softer spectra in the core and their jet spectra also appear softer than the ones for LBLs. Hence, BL Lac objects with low synchrotron peak energies have higher probability to have IC dominance in the X-ray jet. AP Lib has the lowest luminosity in the X-ray jet among the jets of BL Lac objects with a clear IC dominated X-ray spectrum. This raises the question, if the explanation for IC jets as IC scattering of CMB photons is still working or whether other external photon fields become important. Since also the core components of the X-ray emission of the BL Lac objects are IC dominated (only the core emission of S5 2007+777 has Γ ∼ 2), this component is expected to reach high energies. Furthermore, the IC spectrum emerges in the X-ray regime of the core emission and the fact that the peak of the IC emission is in the GeV range (all six BL Lac objects are detected by Fermi have 2 . 0 < Γ < 2 . 4 in the 2nd Fermi catalog (Nolan et al. 2012)) reveal a broad IC component. This behaviour of the IC scattering could also exist in the jet component. For AP Lib it is already known that the IC component reaches TeV energies (Hofmann 2010). The discovery of an extended X-ray jet in the TeV BL Lac object AP Lib, which is IC dominated, provides therefore the opportunity to test the interpretation of the most common emission models as summarized in the review of X-ray jets, e.g. Harris & Krawczynski (2006). The scientific results reported in this article are based on data obtained from the Chandra Data Archive. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application package CIAO. The use of the public HEASARC software packages is also acknowledged. S.K. and S.W. acknowledge support from the BMBF through grant DLR 50OR0906.",
"pages": [
16,
17,
18,
19,
20
]
},
{
"title": "REFERENCES",
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"pages": [
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}
]
2013ApJ...776..133Y
https://arxiv.org/pdf/1308.5974.pdf
<document>
<section_header_level_1><location><page_1><loc_28><loc_85><loc_72><loc_86></location>Update on the Cetus Polar Stream and its Progenitor</section_header_level_1>
<text><location><page_1><loc_12><loc_79><loc_87><loc_83></location>William Yam 1 , Jeffrey L. Carlin 1 , Heidi Jo Newberg 1 , Julie Dumas 1 , Erin O'Malley 1 , 2 , Matthew Newby 1 Charles Martin 1</text>
<section_header_level_1><location><page_1><loc_45><loc_75><loc_55><loc_76></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_16><loc_35><loc_84><loc_72></location>We trace the Cetus Polar Stream (CPS) with blue horizontal branch (BHB) and red giant stars (RGBs) from Data Release 8 of the Sloan Digital Sky Survey (SDSS DR8). Using a larger dataset than was available previously, we are able to refine the measured distance and velocity to this tidal debris star stream in the south Galactic cap. Assuming the tidal debris traces the progenitor's orbit, we fit an orbit to the CPS and find that the stream is confined between ∼ 24 -36 kpc on a rather polar orbit inclined 87 · to the Galactic plane. The eccentricity of the orbit is 0.20, and the period ∼ 700 Myr. If we instead matched N -body simulations to the observed tidal debris, these orbital parameters would change by 10% or less. The CPS stars travel in the opposite direction to those from the Sagittarius tidal stream in the same region of the sky. Through N -body models of satellites on the best-fitting orbit, and assuming that mass follows light, we show that the stream width, line-of-sight depth, and velocity dispersion imply a progenitor of /greaterorsimilar 10 8 M /circledot . However, the density of stars along the stream requires either a disruption time on the order of one orbit, or a stellar population that is more centrally concentrated than the dark matter. We suggest that an ultra-faint dwarf galaxy progenitor could reproduce a large stream width and velocity dispersion without requiring a very recent deflection of the progenitor into its current orbit. We find that most Cetus stars have metallicities of -2 . 5 < [Fe/H] < -2 . 0, similar to the observed metallicities of the ultra-faint dwarfs. Our simulations suggest that the parameters of the dwarf galaxy progenitors, including their dark matter content, could be constrained by observations of their tidal tails through comparison of the debris with N -body simulations.</text>
<text><location><page_1><loc_16><loc_29><loc_84><loc_32></location>Subject headings: Galaxy: structure - Galaxy: kinematics and dynamics - Galaxy: stellar content - stars: kinematics - stars: abundances - galaxies: dwarf - Local Group</text>
<section_header_level_1><location><page_1><loc_44><loc_23><loc_56><loc_24></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_18><loc_89><loc_21></location>Our current view of galaxy formation on Milky Way (MW) scales, initially put forward by Searle & Zinn (1978), is based on the idea of hierarchical merging, the gradual agglomeration of tidally disrupted dwarf</text>
<text><location><page_2><loc_12><loc_47><loc_92><loc_86></location>galaxies and globular clusters onto larger host galaxies. The prevailing Λ -cold dark matter model of structure formation predicts frequent satellite disruption events continuing even to late times (e.g., Johnston 1998; Moore et al. 1999; Abadi et al. 2003; Bullock & Johnston 2005). With the advent of large-scale photometric (and, to a lesser extent, spectroscopic) surveys covering large volumes of the Milky Way, the Galactic halo has been revealed to be coursed with remnant streams from tidally disrupted late-infalling satellites (e.g., Sagittarius: Ibata et al. 2001a; Majewski et al. 2003; the Monoceros/Anticenter Stream complex: Newberg et al. 2002; Ibata et al. 2003; Yanny et al. 2003; Grillmair 2006b; Li et al. 2012; Cetus Polar Stream: Newberg et al. 2009; Virgo substructure: Vivas et al. 2001; Newberg et al. 2002; Duffau et al. 2006; Carlin et al. 2012; and various other SDSS streams: Belokurov et al. 2006; Grillmair 2006a; Grillmair & Dionatos 2006b,a; Grillmair & Johnson 2006; Belokurov et al. 2007; Grillmair 2009; for a summary of known Milky Way stellar streams, see Grillmair 2010). The stars in tidal streams are extremely sensitive probes of the underlying Galactic gravitational potential in which they orbit; thus the measurement of kinematics in a number of tidal streams traversing different regions of the Galaxy is an essential tool to be used in mapping the Galactic (dark matter) halo. This has been done for a handful of streams, the most prominent of which are the extensively-studied tidal streams emanating from the Sagittarius (Sgr) dwarf spheroidal galaxy. The Sgr streams have been used to argue for a MW dark matter halo that is nearly-spherical (e.g., Ibata et al. 2001b; Fellhauer et al. 2006), oblate (e.g., Johnston et al. 2005; Martínez-Delgado et al. 2007), prolate (e.g., Helmi 2004), and finally triaxial (Law et al. 2009; Law & Majewski 2010). Koposov et al. (2010) used orbits fit to kinematical data of stars in the "GD-1" stream (originally discovered by Grillmair & Dionatos 2006b) to show that the Galactic potential is slightly oblate within the narrow range of Galactic radii probed by the GD-1 stream.</text>
<text><location><page_2><loc_12><loc_33><loc_88><loc_46></location>In this work, we focus on the Cetus Polar Stream (CPS). This substructure was originally noted by Yanny et al. (2009) as a low-metallicity group of blue horizontal branch (BHB) stars near the Sgr trailing tidal tail among data from the Sloan Extension for Galactic Understanding and Exploration (SEGUE). The stream was found to be spatially coincident with and at a similar distance to the trailing tidal tail of the Sagittarius dwarf spheroidal near the south Galactic cap. The authors noted that the stream is more metal poor than Sgr, with [Fe/H] ∼ -2.0, and has a different Galactocentric radial velocity trend than Sgr stars in the same region of sky.</text>
<text><location><page_2><loc_12><loc_11><loc_88><loc_31></location>Newberg et al. (2009) used data from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) to confirm this discovery of a new stream in the south Galactic cap. The authors showed that this new stream of BHB stars crosses the Sagittarius trailing tidal tail at ( l , b ) ∼ (140 · , -70 · ), but is separated from Sgr by about 30 · in Galactic longitude at b ∼ -30 · . Because this newfound stream is located mostly in the constellation Cetus and is roughly distributed along constant Galactic longitude (i.e., the orbit is nearly polar), Newberg et al. (2009) dubbed it the Cetus Polar Stream. A slight gradient in the distance to the stream was detected, from ∼ 36 kpc at b ∼ -71 · to ∼ 30 kpc at b ∼ -46 · , where the distance nearer the South Galactic Pole places the CPS at approximately the same distance as the Sgr stream at that position. The authors examined BHB, red giant branch (RGB), and lower RGB (LRGB) stars identified by stellar parameters in the SEGUE spectroscopic database, and found a mean metallicity of [Fe/H] = -2.1 for the CPS. The ratio of blue straggler (BS) to BHB stars was shown to be much higher in Sgr than in the CPS,</text>
<text><location><page_3><loc_12><loc_66><loc_88><loc_86></location>suggesting that most of the BHB stars in this region of the sky may be associated with Cetus rather than the Sgr stream, as previous studies had assumed. From the limited SDSS data available in this region of the sky, this work concluded that the spatial distribution of the CPS is noticeably different than Sgr in the Southern Cap. Indeed, it was pointed out that this solved a mystery seen by Yanny et al. (2000), where the BHB stars in that earlier work did not seem to spatially coincide with the Sgr blue stragglers at ∼ 2 magnitudes fainter in the same SDSS stripe; the majority of these BHB stars must not be Sgr members, but CPS debris instead. The velocity signature of low- metallicity stars clearly differs from the Sgr velocity trend in this region of the sky, as was seen in the Yanny et al. (2009) study, further differentiating the CPS from Sgr. The velocity trend, positions, and distance estimates were used by Newberg et al. (2009) to fit an orbit to the CPS. However, because SDSS data available at the time only covered narrow stripes in the south Galactic cap, the orbit derived by Newberg et al. in this study was rather uncertain.</text>
<text><location><page_3><loc_12><loc_42><loc_88><loc_64></location>In a study of the portion of the Sagittarius trailing stream in the Southern Galactic hemisphere using SDSS Data Release 8 (DR8) data, Koposov et al. (2012) noted that the BS and BHB stars in the region near the Sgr stream are not at a constant magnitude offset from each other as a function of position, as would be expected if they were part of the same population. The magnitude difference between color-selected blue stragglers and BHBs is ∼ 2 magnitudes at Λ ∼ 130 · (where Λ is longitude in a coordinate system rotated into the Sagittarius orbital plane, such that larger Λ is increasingly further from the Sgr core along the trailing tail; Majewski et al. 2003), and decreases at lower Sgr longitudes. Thus, the BHBs at Λ well away from 130 · must not be associated with Sagittarius. The distance modulus of these Cetus BHBs shows a clear trend over more than 40 · along the stream. The authors show that BS and BHB stars with SDSS DR8 spectra enable kinematical separation of Sgr and the CPS, as had already been seen in the Newberg et al. (2009) data from DR7. The kinematics of Cetus are suggested by Koposov et al. (2012) to imply a counter-rotating orbit with respect to Sgr.</text>
<text><location><page_3><loc_12><loc_19><loc_88><loc_41></location>In this paper we follow up on the work by Newberg et al. (2009) using the extensive new imaging data made available in SDSS DR8, which now provides complete coverage of much of the south Galactic cap region. Additional spectra are available in DR8 as well, though they are limited to the DR7 footprint consisting of a few SDSS stripes only. The goal of this work is to measure the distances, positions, and velocities of CPS stars along the stream to sufficient accuracies that we can derive a reliable orbit for the Cetus Polar Stream. This will enable new constraints on the shape and strength of the Milky Way gravitational potential over the regions probed by Cetus, as well as illuminating another of the merger events in the hierarchical merging history of the Galactic halo. We then use this orbit to generate N -body models of the stream, and show that a ∼ 10 8 M /circledot satellite is required to reproduce the kinematics of the stream, but the distribution of stars along the stream requires either a short disruption time or a dwarf galaxy in which mass does not follow light. We argue that the CPS progenitor was likely a dark matter-dominated dwarf galaxy similar to the ultra-faint dwarfs in the Milky Way.</text>
<section_header_level_1><location><page_4><loc_28><loc_85><loc_72><loc_86></location>2. Finding CPS from Photometrically Selected BHB stars</section_header_level_1>
<text><location><page_4><loc_12><loc_64><loc_90><loc_83></location>Because blue horizontal branch (BHB) stars are prevalent in the CPS (Newberg et al. 2009; Koposov et al. 2012), we first select this population of stars in the larger SDSS DR8 dataset. BHBs are color-selected with -0 . 25 < ( g -r ) 0 < -0 . 05 and 0 . 8 < ( u -g ) 0 < 1 . 5, where the latter cut removes most of the QSOs from the sample. Here and throughout this paper, we use the subscript ' 0 ' to indicate that the magnitudes have been corrected by the Schlegel et al. (1998) extinction maps, as implemented in SDSS DR8. BHB stars were selected in the Galactic longitude range 120 · < l < 165 · , bracketing the nearly constant Galactic longitude of l ∼ 143 · found by Newberg et al. (2009) for the CPS in the south Galactic cap. We further separate stars within the initial color selections that are more likely to be higher surface gravity blue straggler (BS) stars from those more likely to be lower surface gravity BHBs using the ugr color-color cuts outlined in Yanny et al. (2000).</text>
<text><location><page_4><loc_12><loc_39><loc_88><loc_63></location>The extinction-corrected apparent magnitudes of the BHB stars are converted to absolute magnitude, M g 0 , using the relation between absolute g -band magnitude and ( g -r ) 0 color for BHB stars given by Equation 7 of Deason et al. (2011). The distance modulus for each BHB star then constitutes the difference between the apparent and absolute magnitudes. The distance moduli of the color-color selected BHB stars are shown as a function of Galactic latitude in Figure 1. Because the stream is extended along nearly constant Galactic longitude, Galactic latitude is nearly the same as angular distance along the stream. In this graph there is a clear concentration of BHB stars with an approximately linear relationship between distance modulus (which is between 0 . 45 /lessorsimilar M g 0 /lessorsimilar 0 . 7 magnitudes offset from the measured apparent magnitudes of these stars, depending on color) and Galactic latitude, as was found in Newberg et al. (2009) and Koposov et al. (2012). However, there is a hint that the relationship is not quite linear. Therefore, we chose to fit a parabola to the BHB data between 17 . 0 < g 0 -M g 0 < 18 . 0 and b < -30 · , iteratively rejecting outliers (beginning with 3 σ rejection, then reducing this to 2 σ , and finally 1 . 5 σ ) until the fit converged to the one overlaid in Figure 1 as a solid line. This fit is given by</text>
<formula><location><page_4><loc_23><loc_34><loc_88><loc_36></location>D mod ( b ) ≡ ( g 0 -M g 0 ) = 16 . 974 -(3 . 710 × 10 -3 × b ) + (9 . 1686 × 10 -5 × b 2 ) . (1)</formula>
<text><location><page_4><loc_12><loc_29><loc_88><loc_32></location>The dashed lines in the figure show ± 0 . 1-magnitude ranges about the fit for distance modulus as a function of latitude.</text>
<text><location><page_4><loc_12><loc_22><loc_88><loc_27></location>Figure 2 shows the same data as the previous figure, but with apparent magnitude corrected as a function of Galactic latitude for the trend fit in Figure 1 to place all stars at the BHB distance corresponding to CPS stars at b = -50 · . These new 'corrected' magnitudes, g corr , are defined as:</text>
<formula><location><page_4><loc_38><loc_18><loc_88><loc_19></location>g corr ( b ) = g 0 -D mod ( b ) + 17 . 389 , (2)</formula>
<text><location><page_4><loc_12><loc_10><loc_88><loc_15></location>where D mod ( b ) is the distance modulus fit as a function of latitude given by Equation 1, and 17.389 is the distance modulus of a star on the polynomial fit at b = -50 · . The CPS stars cluster tightly about the fit in Figure 2 at high latitudes ( b < -40 · ), then drop off near the Galactic plane (i.e., at b > -40 · ). This may be</text>
<figure>
<location><page_5><loc_15><loc_46><loc_83><loc_83></location>
<caption>Fig. 1.- Distance modulus to BHB stars, calculated by applying Equation 7 of Deason et al. (2011) to find M g , BHB as a function of ( g -r ) 0 color, as a function of Galactic latitude. Because the stream is at roughly constant longitude, latitude corresponds approximately to angle along the stream. The BHBs were selected with 120 · < l < 165 · , -0 . 25 < ( g -r ) 0 < -0 . 05, 0 . 8 < ( u -g ) 0 < 1 . 5, and using the ugr color-color selection found by Yanny et al. (2000) to favor low surface gravity BHB stars while reducing the population of higher surface gravity BS stars. We fit a quadratic function to the clear overdensity of BHB stars and found ( g 0 -M g 0 ) = 16 . 974 -(0 . 003710 × b ) + (0 . 000091686 × b 2 ). The solid green line shows this trendline. Upper and lower bounds for selecting CPS BHB candidates are shown as blue and red dashed lines 0.1 magnitudes above and below this trendline. The gap around a latitude of b ∼ -40 · is a result of the fact that the SDSS footprint does not cover the full Galactic longitude at this latitude (see Figure 4). The data were cut off at an upper bound of b = -20 · because the data at regions nearer the Galactic plane include primarily disk stars. The 'blob' at b ∼ -30 · , D mod ∼ 18 . 2 is actually made up of extremely luminous blue supergiants in the galaxy M33, and not BHB stars as assumed when deriving the distance moduli.</caption>
</figure>
<text><location><page_5><loc_12><loc_11><loc_88><loc_17></location>a real effect of the CPS BHB star density dropping off as one moves away from the Galactic pole along the stream (though also partly due to the non-uniform coverage of SDSS; see Figure 4 and further discussion in Sections 3 and 8).</text>
<figure>
<location><page_6><loc_15><loc_46><loc_83><loc_83></location>
<caption>Fig. 2.- Corrected magnitudes for BHB stars from Figure 1 as a function of Galactic latitude. These are defined as g corr ( b ) ≡ g 0 -D mod ( b ) + 17 . 389, using the relationships given in Equations 1 and 2. This corrects all of the BHB star magnitudes for the distance modulus trend of CPS stars, placing them at the CPS distance as measured at b = -50 · . Note that the density of CPS BHB stars seems to drop off at latitudes above b > -40 · (i.e., closer to the disk).</caption>
</figure>
<text><location><page_6><loc_12><loc_10><loc_88><loc_32></location>Wetake advantage of this relationship between latitude and distance for the CPS to study the properties of the BHB stars in the stream. The left panel of Figure 3 shows the observed Hess diagram for blue ( -0 . 3 < ( g -r ) 0 < 0 . 0) stars in the vicinity of the CPS (specifically, 120 · < l < 165 · , b < -40 · ), with an additional color-color selection from Yanny et al. (2000) to more prominently select BHB stars from among the more numerous blue stragglers. The BHB of Cetus stars is clearly visible at g 0 ∼ 18, with an additional large blob of stars at redder ( -0 . 15 /lessorsimilar ( g -r ) 0 < 0 . 0) colors and fainter ( g 0 ∼ 19 -20) magnitudes. As noted by Koposov et al. (2012), this feature is likely made up of BS stars from the Sgr dSph tidal stream, which intersects the CPS in this region of the sky. As shown in the previous paragraph and Figure 2, the latitude dependence of CPS BHB stars' magnitudes can be eliminated. The right panel of Figure 3 shows the same stars as the left panel, but with corrected magnitudes, g corr , instead of the measured magnitudes. The BHB locus in the 'corrected' panel is noticeably narrower than the original, and the absolute magnitude of BHB stars in the CPS appears to be only weakly color-dependent. The BS feature is relatively unchanged,</text>
<text><location><page_7><loc_17><loc_69><loc_17><loc_69></location>g</text>
<figure>
<location><page_7><loc_17><loc_54><loc_40><loc_83></location>
<caption>Fig. 3.Left panel : Hess diagram depicting the initial photometric selection of BHB stars: 120 · < l < 165 · , b < -40 · , 0 . 8 < ( u -g ) 0 < 1 . 5, and -0 . 3 < ( g -r ) 0 < 0 with a color-color selection from Yanny et al. (2000) to separate BHBs from BS stars. Darker shading indicates higher density of stars. The selection was centered around l ∼ 143 · to contain the spatial location of CPS debris as found by Newberg et al. (2009). A population of likely BS stars is evident at redder colors (( g -r ) 0 > -0 . 15) at fainter magnitudes ( g 0 ∼ 19 -20) than the main BHB population. These BS stars are likely associated with the Sgr stream, as was noted by Koposov et al. (2012). Right panel: Hess diagram of the same stars seen in the left panel, but with magnitudes corrected by Equation 2 for the distance modulus trend with longitude, as discussed in the text. The BHB is much narrower in this panel, with little dependence of magnitude on the color.</caption>
</figure>
<text><location><page_7><loc_50><loc_55><loc_50><loc_56></location>.</text>
<text><location><page_7><loc_50><loc_55><loc_51><loc_56></location>20</text>
<text><location><page_7><loc_52><loc_55><loc_53><loc_56></location>-</text>
<text><location><page_7><loc_53><loc_55><loc_53><loc_56></location>0</text>
<text><location><page_7><loc_51><loc_54><loc_52><loc_55></location>(</text>
<text><location><page_7><loc_52><loc_54><loc_52><loc_55></location>g</text>
<text><location><page_7><loc_53><loc_55><loc_54><loc_56></location>.</text>
<text><location><page_7><loc_53><loc_54><loc_54><loc_55></location>-</text>
<text><location><page_7><loc_54><loc_55><loc_55><loc_56></location>15</text>
<text><location><page_7><loc_55><loc_55><loc_56><loc_56></location>-</text>
<text><location><page_7><loc_56><loc_55><loc_57><loc_56></location>0</text>
<text><location><page_7><loc_54><loc_54><loc_54><loc_55></location>r</text>
<text><location><page_7><loc_54><loc_54><loc_55><loc_55></location>)</text>
<text><location><page_7><loc_55><loc_54><loc_55><loc_54></location>0</text>
<text><location><page_7><loc_12><loc_28><loc_88><loc_31></location>or perhaps even more diffuse, as would be expected to happen for a population with a different distance distribution than the CPS when applying the distance modulus correction.</text>
<section_header_level_1><location><page_7><loc_36><loc_22><loc_64><loc_23></location>3. The position of the CPS in the sky</section_header_level_1>
<text><location><page_7><loc_12><loc_11><loc_88><loc_20></location>We now examine the positions of photometrically-selected BHB stars to trace the path of the Cetus stream on the sky. In Figure 4 we show a polar plot in Galactic coordinates, centered on the south Galactic cap, of the sky positions of BHB stars selected within 0.1 magnitudes of the trendline we fit to the concentration of BHBs in Figure 1 (except we have removed the Galactic longitude constraint). There is a clear concentration of stars between l ∼ 120 · and ∼ 160 · relative to the number of BHBs in adjacent longitude</text>
<text><location><page_7><loc_41><loc_70><loc_42><loc_70></location>r</text>
<text><location><page_7><loc_41><loc_70><loc_42><loc_70></location>r</text>
<text><location><page_7><loc_41><loc_69><loc_42><loc_70></location>o</text>
<text><location><page_7><loc_41><loc_69><loc_42><loc_69></location>c</text>
<text><location><page_7><loc_41><loc_69><loc_42><loc_69></location>g</text>
<text><location><page_7><loc_42><loc_82><loc_43><loc_83></location>15</text>
<text><location><page_7><loc_42><loc_79><loc_43><loc_80></location>16</text>
<text><location><page_7><loc_42><loc_75><loc_43><loc_76></location>17</text>
<text><location><page_7><loc_42><loc_71><loc_43><loc_72></location>18</text>
<text><location><page_7><loc_42><loc_67><loc_43><loc_68></location>19</text>
<text><location><page_7><loc_42><loc_63><loc_43><loc_64></location>20</text>
<text><location><page_7><loc_42><loc_59><loc_43><loc_60></location>21</text>
<text><location><page_7><loc_42><loc_55><loc_43><loc_56></location>22</text>
<text><location><page_7><loc_42><loc_55><loc_43><loc_56></location>-</text>
<text><location><page_7><loc_43><loc_55><loc_43><loc_56></location>0</text>
<text><location><page_7><loc_43><loc_55><loc_44><loc_56></location>.</text>
<text><location><page_7><loc_44><loc_55><loc_45><loc_56></location>30</text>
<text><location><page_7><loc_45><loc_55><loc_46><loc_56></location>-</text>
<text><location><page_7><loc_46><loc_55><loc_47><loc_56></location>0</text>
<text><location><page_7><loc_47><loc_55><loc_47><loc_56></location>.</text>
<text><location><page_7><loc_47><loc_55><loc_48><loc_56></location>25</text>
<text><location><page_7><loc_48><loc_55><loc_49><loc_56></location>-</text>
<text><location><page_7><loc_49><loc_55><loc_50><loc_56></location>0</text>
<text><location><page_7><loc_57><loc_55><loc_57><loc_56></location>.</text>
<text><location><page_7><loc_57><loc_55><loc_58><loc_56></location>10</text>
<text><location><page_7><loc_59><loc_55><loc_59><loc_56></location>-</text>
<text><location><page_7><loc_59><loc_55><loc_60><loc_56></location>0</text>
<text><location><page_7><loc_60><loc_55><loc_60><loc_56></location>.</text>
<text><location><page_7><loc_60><loc_55><loc_61><loc_56></location>05</text>
<text><location><page_7><loc_62><loc_55><loc_63><loc_56></location>0</text>
<text><location><page_7><loc_63><loc_55><loc_63><loc_56></location>.</text>
<text><location><page_7><loc_63><loc_55><loc_64><loc_56></location>00</text>
<figure>
<location><page_8><loc_28><loc_46><loc_74><loc_82></location>
<caption>Fig. 4.- Galactic coordinates of color-selected BHB stars (black dots) that are within 0.1 magnitudes of the quadratic relationship in D mod vs. b , as shown in Figure 1 and Equation 1, are shown in a polar plot of the south Galactic cap. Dashed lines show the region in l , b space from which the BHB stars in Figure 3 were selected. The southern footprint of the SDSS DR8 photometric catalog is plotted in the background in grey. It is clear that BHB stars cluster around a region centered at l ∼ 143 · , in agreement with Newberg et al. (2009), though the incomplete coverage of DR8 for l > 150 · makes it difficult to assess whether the CPS extends to higher Galactic longitudes. Again, the CPS appears to drop off rapidly at latitudes above b = -40 · .</caption>
</figure>
<text><location><page_8><loc_12><loc_13><loc_88><loc_24></location>regions. In Newberg et al. (2009) the CPS was found to have a nearly constant longitude of l ∼ 143 · along its entire length. We aim to use the additional information that is available in this region from DR8 to revise the positional measurements of the CPS. However, two complications are obvious in Figure 4: first, the Sagittarius stream intersects the CPS at b ∼ -70 · at nearly the same distance as Cetus, and secondly, the DR8 footprint cuts off at l ∼ 150 · , making it difficult to determine whether the CPS extends beyond this longitude.</text>
<text><location><page_8><loc_15><loc_11><loc_88><loc_12></location>To measure the position of the CPS, we slice the BHB sample into five latitude bands, then plot</text>
<figure>
<location><page_9><loc_17><loc_43><loc_82><loc_82></location>
<caption>Fig. 5.- The color and distance modulus-selected BHB stars from Figure 4 were divided into the following latitude bins: -70 · < b < -62 · , -62 · < b < -56 · , -56 · < b < -50 · , -50 · < b < -42 · , and -42 · < b < -30 · . Number counts of BHB stars in each latitude strip were calculated in 10 · bins of longitude between 80 · < l < 180 · . Bin heights (N) for incomplete bins were scaled upward by the corresponding fraction of area covered by DR8 photometry such that the new bin height is N/(Fractional Area). Bins that have been corrected for incompleteness are denoted with diagonal crosshatching.</caption>
</figure>
<text><location><page_9><loc_12><loc_10><loc_88><loc_26></location>histograms of the longitude distribution in each strip. These histograms are seen in Figure 5 for BHB candidates within 0.1 magnitudes of the distance modulus trend fit previously (Equation 1). The latitude ranges were selected to include roughly the same number of stars in each range, with strips centered on b = [ -66 · , -59 · , -53 · , -46 · , -36 · ]. Bins with incomplete photometric coverage in DR8 were corrected upward by dividing the bin height by the fractional sky area covered by the bin in SDSS, with uncertainties in the bin heights corrected appropriately. A peak is evident in each panel of the figure, which we attribute to the CPS. We fit a Gaussian to the peak in each panel. The fitting was performed using a standard Gaussian function with an additional constant offset in N as a fit parameter. This parameter represents the unknown background level present in the data. A simple, coarse grid-search was performed over the expected pa-</text>
<figure>
<location><page_10><loc_17><loc_43><loc_81><loc_82></location>
<caption>Fig. 6.- Distance modulus of BHB stars between 120 · < l < 165 · in five latitude bins. A narrow peak is seen in each latitude bin that likely corresponds to CPS stars. A Gaussian fit to the peak is shown in each panel; the results of these fits are given in Table 1, and provide the distances used throughout this work.</caption>
</figure>
<text><location><page_10><loc_12><loc_12><loc_88><loc_32></location>nges to determine the general location of the global best-fit of each dataset; a gradient descent search was then started near the global best-fit parameters (as indicated by the grid search). The final parameters are those determined by the gradient descent, with the model errors at those parameters given by the square-roots of twice the diagonal elements of an inverted Hessian matrix. We find best-fit central Galactic longitudes of l = [140 . 7 ± 3 . 5 · , 136 . 7 ± 4 . 9 · , 150 . 0 ± 4 . 6 · , 142 . 2 ± 4 . 9 · , 144 . 7 ± 4 . 7 · ], respectively, for each of the latitude strips. In Table 1 we give the best fit positions with their associated errors as l . We note that the uncertainties in these positions are larger than those given by Newberg et al. (2009), but consistent within the error bars. The width of the stream (in degrees) is the best-fitting Gaussian σ ; these values are tabulated as σ l in Table 1 along with the positions and other stream parameters. Because the BHB stars are rather sparse tracers of the stream, the stream widths are easily biased by background fluctuations toward large (5 · -15 · ) values that are likely overestimates of the true stream width.</text>
<text><location><page_11><loc_12><loc_62><loc_88><loc_86></location>Distances to the CPS at each of these measured positions were determined in a similar manner to the positions. In each of the five latitude bins, we selected all BHB candidates between 120 · < l < 165 · . In each latitude bin, the distance moduli D mod (calculated using Equation 1) of the BHB stars were histogrammed, and we fit a Gaussian to their distribution using the gradient descent method described above. We restricted the fits to 16 < D mod < 19 in each bin, within which a peak was clearly visible in all five ranges considered. These histograms and the Gaussians fit to the peaks in each latitude bin are shown in Figure 6. The distance moduli and their Gaussian spread are given as D mod , fit and σ D mod , fit in Table 1, with the associated distances and line-of-sight depths reported as d fit and σ d , fit . These are likely much more robust measurements of the stream's physical width than the spatial distributions on the sky ( σ l and the associated width σ in kpc), as the distinct peaks in distance modulus are much less prone to contamination by non-stream stars than the positions on the sky. The line-of-sight depths are narrower than the widths of the stream on the sky in all cases. The stream width will be discussed further in Section 8, where we compare the measurements to N-body model results.</text>
<text><location><page_11><loc_12><loc_48><loc_88><loc_61></location>It might seem obvious, once the position and distance to CPS are well known, to look for the much more numerous F turnoff stars associated with it. We did make some unsuccessful attempts to do this. The F turnoff stars are expected to be 3.5 magnitudes fainter than the horizontal branch, at g corr ∼ 21 . 5. The difficulty is that there is a very large background of F turnoff stars in the halo and particularly in the Sgr dwarf tidal stream, which dominates the sky when we attempt to extract F turnoff stars. We were successful in tracing the CPS in BHB stars because there are relatively few BHB stars in the Sgr stream compared to CPS. This is apparently not the case for F turnoff stars.</text>
<section_header_level_1><location><page_11><loc_38><loc_42><loc_62><loc_43></location>4. The Metallicity of the CPS</section_header_level_1>
<text><location><page_11><loc_12><loc_35><loc_88><loc_40></location>We now use the CPS distance estimates and the velocities from Newberg et al. (2009), to select luminous stars from the Cetus Polar Stream. From these stars, we will study the range of metallicities in the stream.</text>
<text><location><page_11><loc_12><loc_12><loc_88><loc_34></location>We select stars with spectra in SDSS DR8 that are in the south Galactic cap. With a horizontal branch at g 0 = 18, the turnoff of the CPS is at approximately g 0 = 21 . 5. Since this is fainter than the spectroscopic limit of SDSS, we do not expect to find any main sequence CPS stars in the dataset. Therefore, we select only luminous giant stars, using the surface gravity criterion 1 . 0 < log g < 4 . 0. Here, log g was determined from the ELODIELOGG value in the SDSS database. At distances of about 30 kpc, we do not expect CPS stars to have a significant tangential velocity, so we also used the proper motion cut | µ | < 6 mas yr -1 , which selects SDSS objects whose proper motions are consistent with zero. We also eliminated nearby Milky Way stars with high metallicity by selecting only those with -4 . 0 < [Fe/H] < -1 . 0. The metallicity of the sample was calculated differently for stars with ( g -r ) 0 < 0 . 25 and ( g -r ) 0 > 0 . 25. This was because Newberg et al. (2009) find the WBG classification (FEHWBG; Wilhelm et al. 1999) to be a better measure of metallicity for BHB stars, and thus the WBG metallicity was used for ( g -r ) 0 < 0 . 25 while the adopted SDSS metallicity (FEHA; consisting of a combination of a number of different measurement techniques)</text>
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</figure>
<figure>
<location><page_12><loc_46><loc_60><loc_89><loc_85></location>
<caption>Fig. 7.Left panel: Hess diagram of all stars having SDSS DR8 spectra in the south Galactic cap ( b < 0 · ) that have surface gravity measurements consistent with classification as giant stars (1 . 0 < log g < 4 . 0), proper motions consistent with zero ( | µ b | < 6 mas yr -1 , | µ l | < 6 mas yr -1 ; to select against nearby stars with large tangential motions), and have low metallicity ( -4 . 0 < [Fe/H] < -1 . 0). The apparent magnitudes in this CMD have been latitude corrected, as defined in Figure 2 and Equation 2. Center panel: Stars with the same selection criteria as in the left panel that also have velocities consistent with the CPS (selected using the CPS velocity-selection criteria from Newberg et al. 2009). The BHB, RGB, and possibly the asymptotic giant branch of the CPS stand out among stars with Cetus-like velocities in this diagram. Right panel: As in the center panel, but for stars with Sgr tidal stream velocities (using selection criteria from Newberg et al. 2009). Because apparent magnitudes in this CMD have been latitude corrected for the distance to the CPS (not Sgr), we do not necessarily expect the Sgr stars to form clear sequences in this diagram. However, we can see that there are generally a large number of blue straggler and giant branch stars at approximately the right apparent magnitudes, and there are a few BHB stars that are likely members of the Sgr trailing tidal tail.</caption>
</figure>
<text><location><page_12><loc_12><loc_22><loc_88><loc_29></location>was used for ( g -r ) 0 > 0 . 25. The DR8 data we downloaded contained multiple entries for some stars. For these stars, we combined multiple measurements by calculating a weighted mean velocity (weighted by the SDSS velocity errors). Their stellar parameters ([Fe/H] and log g ) were set to the values of the highest S / N measurement.</text>
<text><location><page_12><loc_12><loc_11><loc_88><loc_20></location>The left panel of Figure 7 shows the latitude-corrected Hess diagram of log g -selected metal-poor giant star candidates in the south Galactic cap. The greyscale represents the density of stars satisfying the surface gravity, metallicity, and proper motion criteria. In the center panel, we show a CMD of those stars from the left panel that also have the expected velocity of the CPS. To select stars with CPS-like velocities, we used V gsr < ( -0 . 1818 × b -33 . 63) km s -1 and V gsr > ( -1 . 205 × b -130 . 36) km s -1 and V gsr > (2 . 91 × b + 13 . 67) km</text>
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<location><page_13><loc_12><loc_54><loc_59><loc_83></location>
<caption>Fig. 8.- Metallicity distribution of BHB candidates selected with colors of -0 . 25 < ( g -r ) 0 < -0 . 05, low surface gravity, and low proper motions. The hashed (dashed-line) histogram shows stars that are likely members of the CPS, selected within 0.1 magnitudes of the trendline defined in Figure 1, with CPS velocities, and with 120 · < l < 165 · . A narrow peak is visible between -2 . 5 /lessorsimilar [Fe / H] /lessorsimilar -2 . 0. The open (solid-line) histogram represents stars outside the spatial and velocity cuts of CPS stars, which provide a background comparison. The Cetus candidates have a lower mean metallicity than typical field BHBs in this area of the sky.</caption>
</figure>
<text><location><page_13><loc_12><loc_27><loc_88><loc_36></location>s -1 , where the constraints are those used by Newberg et al. (2009) to select Cetus members (note: here, and throughout this work, the subscript 'gsr' means that the velocities are along the line-of-sight, and relative to the Galactic standard of rest). One sees among the velocity-selected CPS candidates in this figure a strikingly narrow BHB (centered around ( g -r ) 0 ∼ -0 . 15), red giant branch (at 0 . 45 /lessorsimilar ( g -r ) 0 /lessorsimilar 0 . 7 and 16 . 5 /lessorsimilar g corr /lessorsimilar 19), and what appears to be an asymptotic giant branch at g corr ∼ 17 and ( g -r ) 0 ∼ 0 . 5.</text>
<text><location><page_13><loc_12><loc_11><loc_88><loc_26></location>To assess the possible contamination of the CPS sample by Sgr stars, we plot a similar color-magnitude diagram in the right panel of Figure 7. The points in this CMD were selected from the box chosen by Newberg et al. (2009) to highlight stars with Sgr velocities. Sgr velocities were chosen using V gsr < ( -3 . 01 × b -271) km s -1 and V gsr > ( -4 . 12 × b -395 . 25) km s -1 for b < -52 · and -180 < V gsr < -114 . 48 km s -1 for b > -52 · . Neither a clear BHB, nor any other obvious feature, is visible in this figure. There are possibly excess red giants among the velocity-selected Sgr candidates, but they do not form a clear sequence in the CMD. This is as expected if Sgr is not at the same distance as the CPS, since we have shifted things to CPS distances by the use of g corr ; this should smear out any Sgr features that are present in such a CMD.</text>
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<location><page_14><loc_12><loc_47><loc_83><loc_83></location>
<caption>Fig. 9.- Illustration of the effectiveness of selecting CPS BHB stars by metallicity. Dots represent BHB stars selected by surface gravity and low proper motion, whose distance is within 0.1 magnitudes of the (Galactic latitude-dependent) distance modulus of the CPS. Larger circles highlight stars with metallicities consistent with the CPS ( -2 . 5 < [Fe/H] WBG < -2 . 0). A trend is visible that stretches from V gsr ∼ -30 km s -1 at b = -70 · to V gsr ∼ -65 km s -1 at b = -45 · and perhaps beyond. The handful of stars at V gsr < -100 km s -1 correspond to Sgr velocities.</caption>
</figure>
<text><location><page_14><loc_12><loc_17><loc_88><loc_32></location>In Figure 8, we show a histogram of metallicities of the BHB stars from Figure 7. The hashed histogram is made up of CPS candidates that have -0 . 25 < ( g -r ) 0 < -0 . 05, distance moduli within 0.1 magnitude of the CPS trendline, V gsr within 20 km s -1 of the velocity trend used to select candidates in Figure 7, and are between 120 · < l < 165 · . The open (solid line) histogram is a background sample, selected from outside both the CPS spatial region and velocity criteria. The CPS BHB stars occupy a narrow metallicity range, with a peak around [Fe/H]= -2 . 2, that is clearly unlike the metallicity distribution of the background sample (typical uncertainties on the metallicity for each star are ∼ 0 . 25 dex). Thus, later in this paper, we will use -2 . 5 < [Fe/H] < -2 . 0 to preferentially select stars that are in the CPS.</text>
<text><location><page_14><loc_12><loc_11><loc_88><loc_16></location>To illustrate the effect of metallicity selection, we show in Figure 9 the gsr-frame line-of-sight velocity as a function of Galactic latitude for BHB stars (selected by low surface gravity and low proper motion) that have -0 . 25 < ( g -r ) 0 < -0 . 05 and are within 0.1 magnitudes of the correct distance modulus to be</text>
<text><location><page_15><loc_12><loc_77><loc_88><loc_86></location>members of the CPS. We did not select the stars based on Galactic longitude. The larger dots in this figure are the stars with -2 . 5 < [Fe/H] < -2 . 0. Note that the CPS stands out much more clearly in the metallicityselected sample, though we probably lose a few bonafide CPS stars. Using the low metallicity sample and our knowledge of the distance to the CPS, we have very little contamination from Sgr and other BHB stars in the Milky Way, even though we accept stars from all observed Galactic longitudes.</text>
<section_header_level_1><location><page_15><loc_35><loc_71><loc_65><loc_73></location>5. Red Giant Branch Stars in the CPS</section_header_level_1>
<text><location><page_15><loc_12><loc_51><loc_88><loc_69></location>In the center panel of Figure 7 there are apprently many Red Giant Branch (RGB) stars in the CPS, that we would like to add to our sample. Using the knowledge that CPS stars are metal-poor ( -2 . 5 < [Fe/H] < -2 . 0), we choose to select candidates using a fiducial sequence from the globular cluster NGC 5466, which has a metallicity ([Fe/H]) of -2.22, shifted to a distance modulus of 17.39 (corresponding to a distance of 30.06 kpc, which is the distance at b = -50 · from the fit in Figure 3). The fiducial sequence of NGC 5466 is from SDSS data, and is taken from An et al. (2008). A third order polynomial was fitted to the region of interest on the fiducial sequence of NGC 5466, and stars were selected using the criteria of -0 . 0081006 g 3 corr + 0 . 46944 g 2 corr -9 . 1207 g corr + 59 . 9130 -0 . 04 < ( g -r ) 0 < -0 . 0081006 g 3 corr + 0 . 46944 g 2 corr -9 . 1207 g corr + 59 . 9130 + 0 . 04 and 16 . 9 < g corr < 19 . 9, where g corr is the latitude corrected magnitude as defined in Equation 2.</text>
<text><location><page_15><loc_12><loc_39><loc_88><loc_49></location>Figure 10 shows a CMD similar to those in Figure 7 for spectroscopically selected giant stars between 120 · < l < 165 · with low proper motion. Colored points show those with metallicities of -2 . 5 < [Fe/H] < -2 . 0 and CPS velocities. We note that the velocity selection shown here for CPS is actually the final selection we arrive at later in this paper; however, the final result differs little from the previous measurement of the velocity trend by Newberg et al. (2009). The color-magnitude box described above for selecting CPS RGB stars, along with the NGC 5466 ridgeline upon which it is based, is shown on the diagram.</text>
<text><location><page_15><loc_12><loc_30><loc_88><loc_37></location>To determine the metallicity of the CPS, we used BHB stars, which are by their nature all low metallicity. We then selected red giant branch stars with the same metallicity range as the BHBs before we matched the RGB fiducial. It is natural to wonder, then, whether there are any higher metallicity RGB stars in the CPS.</text>
<text><location><page_15><loc_12><loc_16><loc_88><loc_29></location>In Figure 11, we show the line-of-sight, Galactic standard of rest velocities as a function of Galactic latitude for all of the low surface gravity, low proper motion stars with spectra in SDSS DR8, that are within the Galactic longitude limits 120 · < l < 165 · and within the box 0 . 4 < ( g -r ) 0 < 0 . 8 and 16 . 9 < g corr < 19 . 9. The three panels of this figure show all stars satisfying these criteria as black points, with large red dots representing stars with metallicities between -3 . 0 < [Fe/H] < -2 . 5, -2 . 5 < [Fe/H] < -2 . 0, and -2 . 0 < [Fe/H] < -1 . 5 in panels from left to right, respectively. Notice that we have opened up the range of colors and apparent magnitudes very wide, to accept all types of giant branch stars at a range of distances.</text>
<text><location><page_15><loc_12><loc_10><loc_88><loc_15></location>We see from this figure that the much more populated Sgr dwarf tidal stream, in the lower left corner of each panel, includes stars of many different metallicities, and is most pronounced in the metallicity range -2 . 0 < [Fe/H] < -1 . 5. On the other hand, the CPS, with velocities near V gsr ∼ -50 km s -1 , is most</text>
<figure>
<location><page_16><loc_15><loc_42><loc_65><loc_83></location>
<caption>Fig. 10.- Color-magnitude diagram of stars in the south Galactic cap region (black points) with low surface gravity, low proper motions consistent with their being Milky Way halo giants, and low metallicity ( -4 . 0 < [Fe/H] < -1 . 0). The colored (blue and red) circles represent low metallicity stars likely to be CPS members. These stars have -2 . 5 < [Fe/H] < -2 . 0, velocities within 20 km s -1 of the final fit of V gsr as a function of latitude for the CPS (which we show later in the paper, in Figure 12), and are within longitudes of 120 · < l < 165 · .</caption>
</figure>
<text><location><page_16><loc_12><loc_15><loc_88><loc_26></location>apparent in the center panel, with -2 . 5 < [Fe/H] < -2 . 0 (with perhaps a few slightly more metal-rich stars). The population of stars we see in the CPS includes predominantly metal-poor stars in a narrow range of metallicity. We also tried selecting stars with CPS velocities in color-magnitude fiducial sequences with a range of metallicities, but the stars in each fiducial sequence were dominated by stars in the same metallicity range of -2 . 5 < [Fe/H] < -2 . 0. Therefore our initial fiducial sequence of NGC 5466 as well as the selections in the CMD and in metallicity are justified.</text>
<text><location><page_16><loc_12><loc_10><loc_88><loc_13></location>If the CPS is the remnant of a dwarf galaxy, then either we are looking at the outer portions of the dwarf galaxy, and not the inner portion with more recent star formation, or we are looking at the remains</text>
<figure>
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<caption>Fig. 11.- To show that the metallicity range of -2 . 5 < [Fe/H] < -2 . 0 captures the most significant portion of the CPS, three graphs of V gsr vs. b for RGB stars are shown here highlighting different metallicity ranges. These stars are all selected between 0 . 4 < ( g -r ) 0 < 0 . 8 and 16 . 9 < g corr < 19 . 9 to encompass giant branch stars at a large range of distances. Black points represent all stars within these ranges satisfying all of the CMD and spatial cuts as used before, while large red circles in each panel show stars with -3 . 0 < [Fe/H] < -2 . 5, -2 . 5 < [Fe/H] < -2 . 0, and -2 . 0 < [Fe/H] < -1 . 5, respectively, from left to right. The left panel shows very few stars with [Fe/H] < -2 . 5. In the middle panel, the metallicity range we found (via Figure 8) to best represent CPS BHB stars seems to select predominantly CPS stars following the stream velocity trend, with very few Sgr stars in the -2 . 5 < [Fe/H] < -2 . 0 range. The right panel, in contrast, shows that most Sgr RGB stars are at higher metallicity than those in the CPS. Clearly, the initial metallicity selection brought forth by the BHB data highlights the strongest signal of a stream in the expected region.</caption>
</figure>
<text><location><page_17><loc_12><loc_39><loc_88><loc_42></location>of a smaller, possibly gas-stripped galaxy that never had a later epoch of star formation at all (similar to the ultra-faint dwarfs discovered recently that seem to have had only a single epoch of star formation).</text>
<section_header_level_1><location><page_17><loc_33><loc_33><loc_67><loc_34></location>6. The line-of-sight velocities along the CPS</section_header_level_1>
<text><location><page_17><loc_12><loc_13><loc_88><loc_31></location>We have now identified samples of BHB and RGB stars in the CPS that lie within the appropriate loci in Figure 10. In Figure 12 we show the line-of-sight, Galactic standard of rest velocities for these stars (removing the selection in V gsr ) as a function of Galactic latitude. To review, the stars in this plot are low surface gravity, low proper motion stars with spectroscopy in the SDSS DR8. They additionally are in the region of the sky inhabited by the CPS ( b < 0 · and 120 · < l < 165 · ). The colors of the symbols tell which CMDselection box each star is from: blue points are BHB candidates, and red: RGB. The larger symbols in the figure have the metallicity we have shown to preferentially select CPS stars: -2 . 5 < [Fe/H] < -2 . 0; large blue circles are BHB stars, and red squares are RGB candidates. The majority of the points in Figure 12 that have larger-sized symbols follow the expected velocity trend of the CPS, with only a handful of stars near the Sgr velocities, and a few scattered elsewhere.</text>
<text><location><page_17><loc_15><loc_10><loc_88><loc_11></location>Wefitapolynomial to the CPS velocities beginning with the entire metallicity-selected ( -2 . 5 < [Fe/H] <</text>
<figure>
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<caption>Fig. 12.- Galactic standard of rest velocities of giant stars (shown as colored points in Figure 10) as a function of Galactic latitude. These stars are selected to have low surface gravity and low proper motion, with metallicities in the range of -4 . 0 < [Fe/H] < -1 . 0, and lie in the region b < 0 · and 120 · < l < 165 · . Blue and red points represent BHB and RGB candidates, respectively. Larger symbols are stars with metallicities representative of CPS stars ( -2 . 5 < [Fe/H] < -2 . 0); BHBs are large blue circles, and RGB: red squares. The central solid green line represents the polynomial that we fit to the highlighted stars: V gsr = -41 . 67 -(0 . 84 × b ) -(0 . 014 × b 2 ). Upper and lower bounds (also shown as solid green lines) of 20 km s -1 on either side of this trendline were used to select CPS stars by their V gsr . The region enclosed within the blue lines represents the selection of Sgr velocities, as discussed in Newberg et al. (2009). The clump of stars at b = -38 · and V gsr = 95 km s -1 belong to the Pisces Stellar Stream (Bonaca et al. 2012; Martin et al. 2013).</caption>
</figure>
<text><location><page_18><loc_12><loc_10><loc_88><loc_19></location>-2 . 0) sample. The fitting was done iteratively, rejecting outliers at each iteration using the same technique as in Section 2, until it converged to a solution of V gsr = -41 . 67 -(0 . 84 × b ) -(0 . 014 × b 2 ). This fit is shown as the green line in Figure 12, with the limits we chose for CPS velocity selection at ± 20 km s -1 on either side of this fit. This is the velocity cut that was used to select the red and blue points in Figure 10, which constitute our 'best' sample of CPS stars.</text>
<figure>
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<caption>Fig. 13.- Polar plot in Galactic coordinates, centered on the south Galactic cap, of metal-poor ( -2 . 5 < [Fe/H] < -2 . 0) CPS candidates selected from SDSS DR8 to lie within the BHB and RGB colormagnitude selections, as well as to have V gsr within 20 km s -1 of the trendline fit in Figure 12. Symbol colors and shapes are the same as in Figure 12. Nearly all of the stars thus selected are in the region centered on l ∼ 145 · where CPS is known to be located. The shaded area shows the footprint of SDSS DR8, from which the spectroscopic targets were selected.</caption>
</figure>
<text><location><page_19><loc_12><loc_20><loc_88><loc_30></location>We now check to make sure the spectroscopically selected stars match our expectations for position on the sky. Using this newly found relationship, we select all of the stars from within the BHB and RGB colorg corr selection boxes, with velocities within 20 km s -1 of the trendline, and that are also metal-poor ( -2 . 5 < [Fe/H] < -2 . 0), and plot them in a polar plot of the south Galactic cap (Figure 13). Here, we can see that the stars match spatially with the photometric selection in Figure 4, with very few stars matching the selection criteria outside of the CPS region.</text>
<text><location><page_19><loc_12><loc_11><loc_88><loc_18></location>Because the stars with measured velocities are confined to 'clumps' at the positions where SEGUE plates were located, a selection in the same bins used to derive positions and distances would skew the velocity measurement in each bin toward positions of highest concentrations of data. (See, for example, Figure 12. The clump of stars at b ∼ -71 · would not fall within any of the bins from Table 1, and the</text>
<text><location><page_20><loc_12><loc_66><loc_88><loc_86></location>-70 · < b < -62 · bin would be skewed heavily toward the b = -62 · end rather than reflecting the velocity at b = -66 · .) Thus we chose to create separate bins from the data in Figure 12 with which to measure the mean velocities, centered on the highest concentrations of CPS candidates. Mean velocities and intrinsic velocity dispersions (accounting for the measurement errors) in each bin were calculated using a maximum likelihood method (e.g., Pryor & Meylan 1993; Hargreaves et al. 1994; Kleyna et al. 2002). The mean latitude, V gsr , intrinsic velocity dispersion, and number of stars in each of these bins are given in Table 2. We note that these velocity dispersions of ∼ 4 -8 km s -1 are typical of dwarf galaxies in the Local Group (see McConnachie 2012 and references therein), and higher than typical dispersions in globular clusters. To derive the velocities in Table 1, we interpolate between the values in Table 2; for example, the velocity at b = -66 · in Table 1 and its error were linearly interpolated from the b ∼ -71 · and b ∼ -61 · values in Table 2. Note also that all of the values given in each of these tables are consistent with the polynomial fit of V gsr vs. b seen in Figure 12.</text>
<text><location><page_20><loc_12><loc_52><loc_88><loc_64></location>Finally, we note that we often find halo substructure in plots of line-of-sight velocity for restricted volumes of space. There is a previously unidentified clump of stars at b = -38 · and V gsr = 95 km s -1 in Figures 11 and 12. We investigated this clump, and discovered eight low surface gravity, low proper motion stars that are within a degree of ( l , b ) = (136 · , -38 · ), have metallicities in the range -2 . 5 < [Fe/H] < -1 . 5, and follow a giant branch very similar to the CPS giant branch. This structure (also seen photometrically by Grillmair 2012 and Bonaca et al. 2012) is the subject of a separate publication (Martin et al. 2013), in which we show that these stars are part of a narrow tidal stream that we dub the 'Pisces Stellar Stream.'</text>
<section_header_level_1><location><page_20><loc_33><loc_46><loc_67><loc_47></location>7. Fitting an orbit to the Cetus Polar Stream</section_header_level_1>
<text><location><page_20><loc_12><loc_31><loc_88><loc_44></location>Having determined the position, velocity, and distance trends along the stream, we wish to use these to constrain the orbit of the Cetus Polar Stream progenitor. The orbit fitting routine requires a set of discrete data points, rather than the general trends we have fit as a function of latitude for the CPS. Note that we fit an orbit assuming that the debris we have measured follows the orbit of the progenitor. This assumption that the tidal debris trace the progenitor's orbit is not strictly true, and the stream-orbit misalignment is essentially independent of progenitor mass (Sanders & Binney 2013). We test and discuss this assumption that the stream follows the orbit in Section 8, which describes N -body simulations of this tidal stream.</text>
<text><location><page_20><loc_12><loc_17><loc_88><loc_30></location>The orbit was fit using the data at five positions given in Table 1 following the techniques described by Willett et al. (2009). The fit assumed a fixed Galactic gravitational potential of the same form used by Willett et al. (2009), which was in turn modeled after the potential of Law et al. (2005) and Johnston et al. (1999). This model contains a three-component potential made up of disk, bulge, and halo components. All parameters in the model were fixed at the same values given in Table 3 of Willett et al. (2009), and the Sun was taken to be 8 kpc from the Galactic center. The technique was slightly improved so that the orbit is not constrained to pass through the sky position of any of the data points.</text>
<text><location><page_20><loc_12><loc_11><loc_88><loc_16></location>Orbits are uniquely defined by a gravitational potential, a point on the orbit, and a velocity at that point. Because it doesn't matter where along the orbit the velocity is specified, we are free to arbitrarily choose one of the three spatial parameters. For fitting, we fixed the Galactic latitude of the point on the</text>
<table>
<location><page_21><loc_12><loc_62><loc_88><loc_74></location>
<caption>Table 1. Cetus Polar Stream Kinematics and Spatial Positions</caption>
</table>
<text><location><page_21><loc_13><loc_59><loc_53><loc_60></location>a This column represents a latitude range between which stars were selected.</text>
<table>
<location><page_21><loc_12><loc_21><loc_88><loc_35></location>
<caption>Table 2. Measured Velocities and Velocity Dispersions</caption>
</table>
<figure>
<location><page_22><loc_13><loc_46><loc_63><loc_85></location>
<caption>Fig. 14.- Position, V gsr , and heliocentric distance for the orbit fit to the five data points from Table 1, which are shown as filled squares with their associated error bars (note that the error bars in the middle and bottom panels are smaller than the point size). Because the stream is at roughly constant Galactic longitude, the data are shown as a function of Galactic latitude, which corresponds roughly to position along the stream. The best-fit orbit is given by the line in each panel, with the forward integration from our selected fiducial point at ( l , b ) = (138 . 2 · , -71 . 0 · ) shown as a solid line, and the backward integration as a dashed line. The open circles (and their associated error bars) represent the four data points used by Newberg et al. (2009) to fit the CPS orbit; the point at b ∼ -46 · is obscured by the solid square. The blue (dot-dashed) line is the best fit orbit from Newberg et al. (2009).</caption>
</figure>
<text><location><page_22><loc_12><loc_13><loc_88><loc_25></location>orbit at b = -71 · , and fit five orbital parameters: the heliocentric distance ( R ), Galactic longitude ( l ), and the components V X , V Y , and V Z of Galactic Cartesian space velocity. We evolve the test particle orbit both forward and backward from the starting position, and perform a goodness-of-fit calculation comparing the derived orbit to the data points in Table 1. The best-fit parameters are optimized using a gradient descent method to search parameter space. We find best-fit parameters at b = -71 · of l = 138 . 2 ± 3 . 8 · , R =32 . 9 ± 0 . 3 kpc, and ( V X , V Y , V Z ) = ( -118 . 1 , 64 . 8 , 76 . 3) ± (7 . 2 , 10 . 3 , 3 . 0) km s -1 .</text>
<figure>
<location><page_23><loc_14><loc_46><loc_65><loc_85></location>
<caption>Fig. 15.- Path of the best-fitting Cetus Polar Stream orbit in Galactocentric XYZ GC coordinates (in a right-handed system, with the Sun at ( X , Y , Z ) GC = ( -8 , 0 , 0) kpc). The large dot represents our fiducial point, with forward integration shown as a solid line, and the backward integration as a dashed curve. The nearly polar path of the low-eccentricity ( e ∼ 0 . 2) orbit is evident in the lower left panel, which shows the XY GC projection. The orbit is confined to a narrow range in this plane. The lower right panel depicts the Galactocentric distance of the orbit as a function of time. Evidently the selected fiducial point is very near the apocenter of the orbit, which occurred at R GC ≈ 36 kpc only ∼ 2 Myr prior to the anchor point of our fitting.</caption>
</figure>
<text><location><page_23><loc_12><loc_19><loc_88><loc_26></location>The errors in the measured orbit parameters 1 are smaller if the minimum in the χ 2 surface is narrower. These errors were calculated from the square root of the diagonal elements of twice the inverse Hessian matrix. The Hessian matrix consists of the second derivatives of χ 2 (not the reduced χ 2 ) with respect to the measured parameters, evaluated at the minimum.</text>
<text><location><page_24><loc_12><loc_73><loc_88><loc_86></location>The position, velocity, and heliocentric distance as a function of Galactic latitude predicted by the integrated best-fitting orbit is shown in Figure 14, with the data points constraining the fit shown as large filled squares. This fit had a formal reduced χ 2 of 0.48. This χ 2 is rather low, and likely arises because we have conservatively estimated our uncertainties. Note also that we have not discussed the effects of possible systematic offsets in the distance scale. We tried an additional orbit fit that included a multiplicative factor to scale the distances as a free parameter; this fit did not improve the χ 2 , so we retained the fit without the scale factor.</text>
<text><location><page_24><loc_12><loc_42><loc_88><loc_72></location>We find that the Cetus Polar Stream is on a rather polar orbit inclined to the Galactic plane by i ∼ 87 . 0 + 2 . 0 -1 . 3 degrees. This orbit has an apogalactic distance from the Galactic center of 35 . 67 ± 0 . 01 kpc and is at 23 . 9 + 3 . 4 -3 . 0 kpc when passing through perigalacticon. This results in an orbital eccentricity of e =0 . 20 ± 0 . 07. The period of our derived CPS orbit is ∼ 0 . 694 ± 0 . 04 Gyr ( ∼ 694 Myr), measured between consecutive pericentric passages. Errors on all of these quantities were derived by integrating the orbits using the maximum and minimum possible total velocities from the orbit-fitting errors in ( V X , V Y , V Z ) and comparing these to the best fit orbit. The orbit is shown in Galactic Cartesian XYZ GC coordinates 2 in Figure 15, integrated for a total of 1.5 Gyr (0.75 Gyr each in the 'forward' and 'backward' directions from our chosen position). The forward integration is given by the solid lines, and the backward integration is the dashed lines, as in Figure 14. The bottom right panel shows the Galactocentric distance as a function of time; from this panel it is clear that our chosen position to anchor the orbit is very near the apocenter of the orbit. Indeed, we find that the nearest apogalacticon was only ∼ 2 Myr prior to this fiducial point, at a position of ( l , b ) = (137 . 8 · , -71 . 6 · ). The most recent pericentric passage was ∼ 350 Myr ago, at a position of ( l , b ) = (337 . 4 · , 7 . 0 · ), and a distance of ∼ 27 . 2 kpc from the Galactic center. Note that our assumption that the debris trace the orbit likely leads to differences in the derived orbital parameters at the ∼ 10% level, especially for a massive progenitor. This will be discussed further in the next section.</text>
<section_header_level_1><location><page_24><loc_43><loc_37><loc_57><loc_38></location>8. N-body model</section_header_level_1>
<text><location><page_24><loc_12><loc_20><loc_88><loc_34></location>We now use the orbit we have derived in combination with information about the velocities, velocity dispersions, stellar density, and three-dimensional positions of observed CPS debris to explore the nature of the stream's (unknown) progenitor. To do so, we use N -body simulations of satellites disrupting in a Milky Way-like gravitational potential, on the well-defined orbit we have measured in this work. The density distribution of debris along the stream, as well as the velocity dispersion, stream width, and line-of-sight depth, are sensitive to the mass and size of the progenitor, as well as the time that the progenitor has been orbiting the Milky Way. A comprehensive modeling effort is beyond the scope of this work; here we aim to gain insight into the nature of the progenitor that will inform future, more detailed, modeling efforts.</text>
<text><location><page_25><loc_12><loc_79><loc_88><loc_86></location>The N -body simulations were run using the gyrfalcON tool (Dehnen 2002) of the NEMO Stellar Dynamics Toolbox (Teuben 1995). Satellites with masses configured using a Plummer model (Plummer 1911) were evolved in the same Galactic gravitational potential used to derive orbits. Each simulation has 10 4 bodies.</text>
<text><location><page_25><loc_12><loc_61><loc_89><loc_78></location>We started out with the assumption that mass follows light, so that the distribution of masses from the N -body simulation has the same density and velocity profile as the observed stars. We note that this assumption has been shown to work in modeling dwarf spheroidals by Muñoz et al. (2008). With this assuption, the Plummer scale radius of each satellite was chosen, roughly following the scaling relations from Tollerud et al. (2011; see, e.g., their Figure 7). A Plummer radius of 1 kpc was used for the 10 8 M /circledot satellite, with the remaining radii scaled using the empirical relation (Tollerud et al. 2011) between the half-light radius and the dark matter mass within that radius: M DM 1 / 2 ∝ r 2 . 32 1 / 2 (note that we assumed that the Plummer radius is roughly equal to the half-light radius). Since we assume mass follows light, the radius of the dark matter and the radius of the luminous matter are the same.</text>
<text><location><page_25><loc_12><loc_53><loc_88><loc_60></location>The total masses of the Plummer spheres were varied from 10 5 to 10 9 solar masses, with Plummer scale radii of a p ∼ (50 , 150 , 400 , 1000 , 2500) pc corresponding to (10 5 , 10 6 , 10 7 , 10 8 , 10 9 ) M /circledot models, respectively (note that our choice of scaling differs little from a simple choice of r ∝ M 1 / 3 , which would produce satellites with the same mass density).</text>
<text><location><page_25><loc_12><loc_37><loc_88><loc_51></location>Wehave seen that the density of stars in the CPS falls as one approaches the Galactic plane. This is also along the direction of our measured orbital motion, so that the observed stars near the south Galactic cap must be the leading tidal tail of a satellite (which could be completely disrupted) that is not too far behind the observed stars on the orbit. We place a point mass at the position ( l , b ) = (138 . 2 · , -71 · ) on the orbit at a distance of 32.9 kpc, then integrate the point backwards on the orbit for 3 Gyr. At that new position, we place a Plummer sphere of bodies representing the dwarf galaxy with the bulk velocity of the forward orbit at that position. We then integrate the N -body forward for three Gyr, just over four full orbits, so the dwarf galaxy ends up approximately at our initial position.</text>
<text><location><page_25><loc_12><loc_19><loc_88><loc_35></location>We measure the width across the stream ( σ l , which is the σ of a Gaussian fit to the profile), the line of sight depth of the stream ( σ d ; again from a Gaussian fit), and the stream velocity dispersion ( σ V ), from the results of each N -body model. Table 3 shows the measured values for the same five latitude bins we used to analyze the data, which should be compared with the observed properties of the stream stars in the same table. The stream widths, σ l and σ d , both come from photometrically selected BHB stars, while the velocity dispersion includes BHBs and RGB stars. The width in Galactic longitude, σ l , is large for the BHB stars, ranging from 5 . 0 · to 15 . 1 · , and has large uncertainties. This likely arises because of the low numbers of BHB stars in the samples, such that the poorly-defined background BHB density broadens the wings in some of these fits.</text>
<text><location><page_25><loc_12><loc_11><loc_88><loc_18></location>The 10 5 M /circledot , a p = 50 pc progenitor, which has short, narrow debris tails resembling those of the Pal 5 globular cluster (e.g., Grillmair & Dionatos 2006; Odenkirchen et al. 2001, 2003; Rockosi et al. 2002), gives unreliable results for many of our measurements simply because there is little debris to be measured. Nevertheless, it is clear that the low-mass ( M /circledot < 10 7 ) progenitors produce streams much narrower in both</text>
<figure>
<location><page_26><loc_17><loc_62><loc_63><loc_85></location>
<caption>Fig. 16.- Density of candidate photometrically-selected Cetus Polar Stream BHB stars selected using the photometric cuts discussed earlier, and limited to a distance modulus range of 16 . 5 < D mod < 18 . 0 to bracket the CPS debris distances. Each bin represents the difference in number counts of BHB stars between an 'onstream' field between 130 · < l < 152 · and an 'off-stream,' background field selected from 86 · < l < 130 · (scaled down by the factor of two difference in area). Error bars are derived from the Poisson errors in the number counts from each sample. A clear trend of decreasing BHB density along the stream can be seen between -70 · < b /lessorsimilar -40 · . Above b ∼ -40 · , the residuals are consistent with zero (i.e., the stream does not contain excess BHB stars above the background level), though from Figure 5 we expect there are at least a few stream stars with b > -42 · .</caption>
</figure>
<text><location><page_26><loc_12><loc_29><loc_88><loc_38></location>width on the sky and line-of-sight depth, than observed in the CPS. The low mass progenitors also have velocity dispersions that are too small. The measured width and depth of the stream stars suggest a 10 8 -10 9 M /circledot progenitor mass, while the velocity dispersions suggest a 10 7 -10 8 M /circledot progenitor mass. Thus it seems that the physical widths (both on the sky and along the line of sight) and the velocity dispersions we have measured are telling us that the progenitor of the Cetus Polar Stream had a mass of ∼ 10 8 M /circledot .</text>
<text><location><page_26><loc_12><loc_11><loc_88><loc_28></location>However, we now show that the density of stars along the stream is inconsistent with the mass-followslight progenitor in our model. We use the BHB stars selected photometrically by the methods described earlier in this work. The BHB sample is restricted to distance moduli of 16 . 5 < D mod < 18 . 0 to choose only the distance range containing CPS debris. The entire available range of Galactic latitude is chosen ( -72 · < b < -28 · ). We select an 'on-stream' and 'off-stream,' or background, sample. The on-stream BHB stars are selected in the longitude range of 130 · < l < 152 · to include only the regions completely sampled by SDSS DR8. The off-stream sample spans 86 · < l < 130 · over the same latitudes. We bin the number counts in 4-degree bins of latitude for each sample, then scale down the number counts in the off-stream regions by a factor of two to account for the additional area sampled.</text>
<figure>
<location><page_27><loc_12><loc_51><loc_44><loc_86></location>
</figure>
<figure>
<location><page_27><loc_46><loc_51><loc_77><loc_86></location>
<caption>Fig. 17.- Normalized density of tidal debris from each of the N -body models of satellites on the best-fitting Cetus Polar Stream orbit. The debris are selected and binned in the same way as the BHB stars in Figure 16 to facilitate direct comparison. The normalized density of BHB stars from Figure 16 is overplotted in each panel as a dashed line with associated error bars. Each bin (for both the observational data and the model) has been normalized by the total number of points in the sample. From top to bottom, the left panel shows models with progenitor masses of 10 5 , 10 6 , 10 7 , 10 8 , and 10 9 M /circledot evolved for 3 Gyr. The progenitor (or its remains) is just off the plot near the south Galactic pole. The right panels show the 10 8 M /circledot , 1 kpc radius model evolved for 0.5, 0.75, 1.0, 1.25, and 1.5 Gyr (top to bottom). In the 3 Gyr models, the least massive (10 5 M /circledot ) satellite has suffered little tidal disruption. Its debris is not spread far along the orbit because the satellite is compact and has very low velocity dispersion; this stream is thus more sharply peaked than the observed density. The 10 7 -10 9 M /circledot , 3 Gyr satellites show an almost constant density of debris along the entire length of the region plotted, while in the 10 6 M /circledot progenitor, the fall-off in stream density is more rapid as a function of latitude (essentially as a function of angular distance along the stream, since the stream runs along nearly constant longitude). The 10 6 M /circledot model is most consistent with the stellar density of the CPS along latitude, while the additional information on stream stars suggests a more massive progenitor. In the right panels, the 10 8 M /circledot , 1 Gyr model matches the data best. This would require that a fairly massive satellite had its (previously different) orbit perturbed 1 Gyr ago to place it on the nearly circular CPS orbit for only ∼ 1 . 5 orbits.</caption>
</figure>
<text><location><page_28><loc_12><loc_70><loc_88><loc_86></location>Figure 16 shows the density profile of BHB stars along the CPS after subtraction of the 'off-stream' background density. There is clearly an excess population of BHB stars in the CPS compared to the background at latitudes -70 · < b /lessorsimilar -40 · . Above b ∼ -40 · , the excess seems to disappear, with the BHB numbers consistent with the adjacent background level. However, recall from Figure 5 that there must be at least a few stream stars with -42 · < b < -30 · . This density gradient starting from highest density near the south Galactic cap and decreasing toward the Galactic plane must be reproduced by any mass-follows-light model of the stream. Note that the first bin in the figure is low because the area of sky between two longitudes near the south Galactic pole is small, and the stream does not go exactly through the pole; the density is actually highest in the bin nearest the pole, since that is close to the final position of the satellite.</text>
<text><location><page_28><loc_12><loc_42><loc_88><loc_68></location>We measure the density profile as a function of latitude for the set of five N -body models that have the progenitor ending up near the south Galactic cap. The debris was selected from the same longitude and distance ranges used for the BHB density in Figure 16 and histogrammed in 4-degree bins as in the BHB figure. These density profiles are displayed in the left panels of Figure 17 as solid lines for each of the five progenitor masses. Number counts in each bin were normalized by the total number of points in each panel to facilitate comparison with the BHB density profile, which is overlaid as a dashed histogram (with error bars). The 10 5 M /circledot progenitor has suffered little tidal disruption, and is thus strongly peaked near the satellite; this is likely because of the small (50 pc) radius of the satellite. The largest satellites (0.4-2.5 kpc, 10 7 -10 9 M /circledot ) produce nearly constant density along latitude, indicating that they are strongly disrupted and that debris has spread extensively along the orbit. Note that although our data is near the apogalacticon of the orbit, the orbit is not very eccentric so highly disrupted satellites have only a small increase in stellar density at the position in the stream that is most distant from the Galactic center. The stellar density along the stream in the simulations with a more massive dwarf galaxy does not agree with the observed gradient in the measured BHB density.</text>
<text><location><page_28><loc_12><loc_34><loc_88><loc_41></location>On the other hand, the 10 6 M /circledot model density appears to agree very well with the observations, with all five of the bins between -64 · < b < -44 · in agreemeent with the BHB number counts within 1 σ . However, we have already shown that a satellite with 10 6 M /circledot is not consistent with the width, depth and velocity dispersion of the stream.</text>
<text><location><page_28><loc_12><loc_11><loc_88><loc_33></location>Since the 10 8 solar mass progenitor seemed a reasonable fit to the stream width and velocity dispersion, but the distribution of stars along the stream was too dispersed, we tried reducing the time the stream has been disrupting in an attempt to fit all of the measured quantities. The right panels of Figure 17 show the distribution along the stream for disruption times of 0.5 to 1.5 Gyr, and Table 3 shows the velocity dispersion and stream widths for these simulations. We find that a 10 8 solar mass progenitor with a disruption time of 1-1.25 Gyr is a reasonable but not perfect fit to all of our data. It is a bit difficult to fit the large apparent width of the stream with N -body simulations, even with a relatively large mass progenitor of 10 8 M /circledot . Note that the best-matching disruption time of ∼ 1 Gyr is about one and a half orbits of the satellite, and would require that the dwarf galaxy was perturbed into its present, nearly circular orbit fairly recently; we did not model the effects of a recent deflection of the satellite in the simulation. It might be possible to adjust the disruption time slightly by varying the final position of the progenitor, but this would only produce a slight change to the best fit time.</text>
<table>
<location><page_29><loc_12><loc_22><loc_88><loc_80></location>
<caption>Table 3. Cetus Polar Stream N-body results (with measured values for comparison)</caption>
</table>
<text><location><page_29><loc_12><loc_15><loc_88><loc_17></location>Note. - For the 3-Gyr N -body models, particles were configured in Plummer spheres with scale radii of (50 , 150 , 400 , 1000 , 2500) pc, respectively, for the models with masses of (10 5 , 10 6 , 10 7 , 10 8 , 10 9 ) M /circledot .</text>
<figure>
<location><page_30><loc_12><loc_57><loc_49><loc_86></location>
<caption>Fig. 18.- Results of an N -body model of a satellite on the best-fitting Cetus Polar Stream orbit. The satellite was intialized as a Plummer sphere of 10,000 particles, with a total mass of 10 8 M /circledot . Its evolution was evolved for 1.0 Gyr in a model Milky Way potential. The black dots above are the model results. The panels show the distance from the Sun, V GSR , and position (right ascension and declination in the left column, and Galactic coordinates in the right column) of model particles. Overlaid solid red lines are the forward-integrated orbit, and dashed (blue) lines the reverse integration from our chosen fiducial position. Large blue points show the five data points derived in this work.</caption>
</figure>
<text><location><page_30><loc_12><loc_27><loc_88><loc_39></location>The results of the 1 Gyr N -body simulation for a 10 8 M /circledot satellite are shown in Figure 18. This figure shows the distribution of model debris in distance, velocity, and position (right ascension and declination in the left column, and Galactic coordinates in the right panels). One difficulty that is illuminated in Figure 18 is that our orbit, which was fit to the tidal debris, does not lead to an N -body simulation that fits the measured stream distances for a massive (10 8 M /circledot ) progenitor (in particular, see the upper left panel of Figure 18). The discrepancy between orbits fit to streams and the actual orbits of massive progenitors has been highlighted by Binney (2008) and Sanders & Binney (2013).</text>
<text><location><page_30><loc_12><loc_11><loc_88><loc_25></location>To address this discrepancy, we tried adjusting the orbit by hand to match the data. We were able to plausibly match the N -body simulations by increasing the distance of the orbit from the Galactic center. The new orbit has an apogalacticon of 38 kpc, a perigalacticon of 26 kpc, an inclination of 88 · , eccentricity of 0.19, and a period of 750 Myr. All of these values are within 10% of the orbit that was fit to the stream, and other than a shift to ∼ 2 kpc larger Galactocentric distances, all of the parameters are within the uncertainties of the orbital parameters from our original fit. The evolution time of the simulation also affects the results; this particular orbit was a better fit to the data with a 3 Gyr evolution time than a 1 Gyr evolution time. It is a much more difficult problem, and beyond the scope of this current effort, to fit orbital parameters by</text>
<text><location><page_31><loc_12><loc_79><loc_88><loc_86></location>matching data directly to N -body simulations. Thus we cannot verify that our solution is unique or is in fact the best solution. However, we note that this slightly altered orbit has very similar width, depth, velocity dispersion, and density along the stream as the original N -body simulation. The best fit model is still a 10 8 M /circledot satellite that may have been deflected into its current orbit ∼ 1 Gyr ago.</text>
<text><location><page_31><loc_12><loc_59><loc_88><loc_78></location>Another more plausible solution is that mass does not follow light in the stream progenitor. So far, our simulations have assumed that the N -body points, which are simply massive particles (and thus more representative of the dark matter in the satellite), will be distributed similarly to the luminous matter (i.e., the stars). There is little reason to believe this is the case, and in fact the observed properties of many dwarf galaxies are often interpreted as showing evidence of significant dark matter halos in these objects (see, e.g., McConnachie 2012 and references therein). Our N -body comparisons seem to suggest that the CPS progenitor was a dwarf galaxy with a large mass-to-light ratio and a dark matter halo that extends beyond the distribution of stars. A model of this type might explain the apparently small number of stars given the large spatial and velocity dispersion of the stream, and the relatively recent disruption of the stellar component of the progenitor as determined by the relatively steep gradient in star density along the stream.</text>
<text><location><page_31><loc_12><loc_43><loc_88><loc_58></location>Another argument in favor of an ultrafaint dwarf progenitor comes from the measured metallicity. We showed in Sections 4 and 5 that the metallicity of CPS stars is sharply peaked between -2 . 5 < [Fe/H] < -2 . 0. Assuming that we are looking at a disrupted dwarf galaxy, the 〈 [Fe/H] 〉 vs. M V relation for Local Group dwarf galaxies of Fig. 12 of McConnachie (2012) suggests that a dwarf spheroidal with 〈 [Fe/H] 〉 ∼ -2 . 25 should have M V ∼ -6. Likewise, the metallicity-luminosity relationship for Local Group dwarf galaxies from Simon & Geha (2007) predicts ultra-faint dwarf spheroidals with -4 > M V > -9 should have mean metallicities between -2 . 5 < [Fe/H] < -2 . 0, lower than most globular clusters at similar luminosities. Thus the metallicity of the stars in the CPS is also consistent with its progenitor being an ultra-faint dwarf galaxy.</text>
<text><location><page_31><loc_12><loc_33><loc_88><loc_42></location>The velocity dispersion and stream width alone (without the N -body models for comparison) are too large for a globular cluster progenitor. We therefore conclude that the progenitor was a dwarf galaxy. One possible solution is a mass-follows-light progenitor of about 10 8 solar masses, that was deflected into a nearly circular orbit of order one Gyr ago. A more plausible solution is that the progenitor was a dark matter dominated, ultrafaint dwarf galaxy.</text>
<text><location><page_31><loc_12><loc_25><loc_88><loc_32></location>Modeling a two component (dark matter plus stars) dwarf galaxy progenitor for the Cetus Polar Stream is beyond the scope of this work, but the prospect that the dark matter distribution of the progenitor dwarf galaxy could be encoded in the distribution of stars in a tidal tail is tantalizing, and will be pursued in a future paper.</text>
<text><location><page_31><loc_12><loc_14><loc_99><loc_23></location>We have been building the capacity to fit many parameters in an N -body simulation of dwarf galaxy tidal disruption using the Milkyway@home volunteer computing platform ( http://milkyway.cs.rpi.edu/milky We plan to eventually use this platform to find the best fit model parameters for the dwarf galaxy progenitor of the CPS given the available data for the tidal stream, and also to model other tidal streams in the Milky Way.</text>
<section_header_level_1><location><page_32><loc_44><loc_85><loc_56><loc_86></location>9. Conclusion</section_header_level_1>
<text><location><page_32><loc_12><loc_78><loc_88><loc_83></location>In this work, we use BHB and RGB stars from SDSS DR8 to improve the determination of the Cetus Polar Stream orbit near the south Galactic cap. We then model the evolution of satellites on this orbit to assess the nature of the progenitor that produced the CPS.</text>
<text><location><page_32><loc_12><loc_64><loc_88><loc_76></location>Much of the region of sky in which the CPS is located is dominated by the Sagittarius stream. However, we show that stars with the velocities of the Sgr tidal stream in the region near the CPS are predominantly blue stragglers (BS), and that Sgr is virtually devoid of BHB stars. This is in stark contrast with the CPS, which has a well populated BHB and very few, if any, observed BS stars. We are able to trace the CPS with BHBstars over more than 30 degrees in Galactic latitude along its nearly constant path in Galactic longitude. We were also able to disentangle CPS from Sgr based on the different velocities and mean metallicities of the two structures (see, e.g., Figure 11).</text>
<text><location><page_32><loc_12><loc_42><loc_88><loc_62></location>We fit an orbit to the Cetus Polar Stream tidal debris using the positions, distances, and velocities we have measured over at least 30 -40 · of the stream's extent. Under the assumption that the orbit follows the tidal debris, the best fit CPS orbit has an eccentricity of e = 0 . 20 ± 0 . 07, extending to an apogalactic distance from the Galactic center of R GC = 35 . 7 kpc at a point very near the lowest-latitude measurement in our study. At its perigalactic passage, the orbit is ∼ 24 kpc from the Galactic center. This orbit is inclined by 87 · to the Galactic plane, and has a period of ∼ 700 Myr. Our N -body models indicate a high mass progenitor ( > 10 8 M /circledot ), and that the orbit of the progenitor does not follow the tidal debris, particularly in distance. However, small tweaks to the orbit that make the simulations more consistent with the data change the orbit parameters by 10% or less. With the data presented here, we are unable to constrain the Galactic gravitational potential; further data on the CPS (or combining the data with other tidal streams) will be necessary to provide insight into the shape and strength of the halo potential probed by the stream.</text>
<text><location><page_32><loc_12><loc_16><loc_88><loc_40></location>In Section 3 we noted that we were unable to separate the CPS from Sagittarius using SDSS photometry of F turnoff stars. In RGB stars, which are common to both populations, we can only see the CPS if we have spectra, and can separate the CPS stars by velocity and metallicity. To study halo substructure in more detail, we require spectroscopic surveys that reach F turnoff stars at g magnitudes approaching 22. Alternatively, accurate proper motions could be used to separate the CPS from the Sgr tidal stream, because the two streams are orbiting in opposite directions, as was pointed out by Koposov et al. (2012). Our orbit fit predicts mean proper motions in the longitude bins with 〈 b 〉 ≈ -66 · , -59 · , -53 · , -46 · , and -36 · (see Tables 1 and 3) of ( µ α cos δ,µ δ ) ≈ (1 . 2 , -0 . 2) , (1 . 3 , -0 . 2) , (1 . 3 , -0 . 1) , (1 . 4 , -0 . 1) , (1 . 5 , 0 . 1) mas yr -1 , respectively. We were unable to distinguish the F turnoff stars in the two streams using currently available proper motions, though it might be possible to separate larger populations of faint F turnoff stars using future proper motion surveys. It is unlikely that the proper motion measurement of individual stars will accurate enough, but it has been shown (Carlin et al. 2012; Koposov et al. 2013) that the ensemble proper motion of turnoff stars in the Sgr stream at distances of ∼ 30 -40 kpc can be statistically determined from currently available data.</text>
<text><location><page_32><loc_12><loc_10><loc_88><loc_15></location>We simulate the evolution of satellites on the CPS orbit in a Milky Way-like potential via N -body modeling. From these mass-follows-light models, we show that with a mass-follows-light assumption, a 10 8 M /circledot satellite can produce a stream width, line-of-sight velocity, and velocity dispersion similar to our</text>
<text><location><page_33><loc_12><loc_64><loc_88><loc_86></location>observations. However, the distribution of stars along the stream and the apparent number of stars observed favor a lower mass satellite. It is possible to reconcile these observations if the dwarf galaxy was deflected into its current orbit on the order of one Gyr ago (less than one and a half orbits). However, a less contrived solution is that mass does not (or did not) follow light in the CPS progenitor, which may have been a dark matter-dominated satellite similar to the recently-discovered class of satellites known as 'ultra-faint dwarf spheroidals" (e.g., Simon & Geha 2007, McConnachie 2012). These are low luminosity, highly dark matterdominated satellites. The mean metallicity, 〈 [Fe / H] 〉 ≈ -2 . 2, that we find for CPS debris is consistent with the typical metallicities of ultra-faint dwarfs. More detailed modeling should be carried out in the future to confirm that dual-component (dark matter + stars) satellites reproduce the observed properties of the Cetus Polar Stream, and that the properties of the progenitor can be determined uniquely. Our results suggest that the structure of tidal streams could be used to constrain the properties of the stream progenitors, including their dark matter components.</text>
<text><location><page_33><loc_12><loc_54><loc_88><loc_61></location>Wethank the anonymous referee for thoughtful and detailed comments that greatly improved this work. This work was supported by the National Science Foundation grant AST 10-09670. JLC was supported by NSF grant AST 09-37523. WY was supported by NSF grant AST 11-15146. EO was supported by NSF grant DMR 08-50934. JD was supported by the NASA/NY Space Grant fellowship.</text>
<text><location><page_33><loc_12><loc_47><loc_88><loc_52></location>Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.</text>
<text><location><page_33><loc_12><loc_28><loc_88><loc_46></location>SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "We trace the Cetus Polar Stream (CPS) with blue horizontal branch (BHB) and red giant stars (RGBs) from Data Release 8 of the Sloan Digital Sky Survey (SDSS DR8). Using a larger dataset than was available previously, we are able to refine the measured distance and velocity to this tidal debris star stream in the south Galactic cap. Assuming the tidal debris traces the progenitor's orbit, we fit an orbit to the CPS and find that the stream is confined between ∼ 24 -36 kpc on a rather polar orbit inclined 87 · to the Galactic plane. The eccentricity of the orbit is 0.20, and the period ∼ 700 Myr. If we instead matched N -body simulations to the observed tidal debris, these orbital parameters would change by 10% or less. The CPS stars travel in the opposite direction to those from the Sagittarius tidal stream in the same region of the sky. Through N -body models of satellites on the best-fitting orbit, and assuming that mass follows light, we show that the stream width, line-of-sight depth, and velocity dispersion imply a progenitor of /greaterorsimilar 10 8 M /circledot . However, the density of stars along the stream requires either a disruption time on the order of one orbit, or a stellar population that is more centrally concentrated than the dark matter. We suggest that an ultra-faint dwarf galaxy progenitor could reproduce a large stream width and velocity dispersion without requiring a very recent deflection of the progenitor into its current orbit. We find that most Cetus stars have metallicities of -2 . 5 < [Fe/H] < -2 . 0, similar to the observed metallicities of the ultra-faint dwarfs. Our simulations suggest that the parameters of the dwarf galaxy progenitors, including their dark matter content, could be constrained by observations of their tidal tails through comparison of the debris with N -body simulations. Subject headings: Galaxy: structure - Galaxy: kinematics and dynamics - Galaxy: stellar content - stars: kinematics - stars: abundances - galaxies: dwarf - Local Group",
"pages": [
1
]
},
{
"title": "Update on the Cetus Polar Stream and its Progenitor",
"content": "William Yam 1 , Jeffrey L. Carlin 1 , Heidi Jo Newberg 1 , Julie Dumas 1 , Erin O'Malley 1 , 2 , Matthew Newby 1 Charles Martin 1",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Our current view of galaxy formation on Milky Way (MW) scales, initially put forward by Searle & Zinn (1978), is based on the idea of hierarchical merging, the gradual agglomeration of tidally disrupted dwarf galaxies and globular clusters onto larger host galaxies. The prevailing Λ -cold dark matter model of structure formation predicts frequent satellite disruption events continuing even to late times (e.g., Johnston 1998; Moore et al. 1999; Abadi et al. 2003; Bullock & Johnston 2005). With the advent of large-scale photometric (and, to a lesser extent, spectroscopic) surveys covering large volumes of the Milky Way, the Galactic halo has been revealed to be coursed with remnant streams from tidally disrupted late-infalling satellites (e.g., Sagittarius: Ibata et al. 2001a; Majewski et al. 2003; the Monoceros/Anticenter Stream complex: Newberg et al. 2002; Ibata et al. 2003; Yanny et al. 2003; Grillmair 2006b; Li et al. 2012; Cetus Polar Stream: Newberg et al. 2009; Virgo substructure: Vivas et al. 2001; Newberg et al. 2002; Duffau et al. 2006; Carlin et al. 2012; and various other SDSS streams: Belokurov et al. 2006; Grillmair 2006a; Grillmair & Dionatos 2006b,a; Grillmair & Johnson 2006; Belokurov et al. 2007; Grillmair 2009; for a summary of known Milky Way stellar streams, see Grillmair 2010). The stars in tidal streams are extremely sensitive probes of the underlying Galactic gravitational potential in which they orbit; thus the measurement of kinematics in a number of tidal streams traversing different regions of the Galaxy is an essential tool to be used in mapping the Galactic (dark matter) halo. This has been done for a handful of streams, the most prominent of which are the extensively-studied tidal streams emanating from the Sagittarius (Sgr) dwarf spheroidal galaxy. The Sgr streams have been used to argue for a MW dark matter halo that is nearly-spherical (e.g., Ibata et al. 2001b; Fellhauer et al. 2006), oblate (e.g., Johnston et al. 2005; Martínez-Delgado et al. 2007), prolate (e.g., Helmi 2004), and finally triaxial (Law et al. 2009; Law & Majewski 2010). Koposov et al. (2010) used orbits fit to kinematical data of stars in the \"GD-1\" stream (originally discovered by Grillmair & Dionatos 2006b) to show that the Galactic potential is slightly oblate within the narrow range of Galactic radii probed by the GD-1 stream. In this work, we focus on the Cetus Polar Stream (CPS). This substructure was originally noted by Yanny et al. (2009) as a low-metallicity group of blue horizontal branch (BHB) stars near the Sgr trailing tidal tail among data from the Sloan Extension for Galactic Understanding and Exploration (SEGUE). The stream was found to be spatially coincident with and at a similar distance to the trailing tidal tail of the Sagittarius dwarf spheroidal near the south Galactic cap. The authors noted that the stream is more metal poor than Sgr, with [Fe/H] ∼ -2.0, and has a different Galactocentric radial velocity trend than Sgr stars in the same region of sky. Newberg et al. (2009) used data from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) to confirm this discovery of a new stream in the south Galactic cap. The authors showed that this new stream of BHB stars crosses the Sagittarius trailing tidal tail at ( l , b ) ∼ (140 · , -70 · ), but is separated from Sgr by about 30 · in Galactic longitude at b ∼ -30 · . Because this newfound stream is located mostly in the constellation Cetus and is roughly distributed along constant Galactic longitude (i.e., the orbit is nearly polar), Newberg et al. (2009) dubbed it the Cetus Polar Stream. A slight gradient in the distance to the stream was detected, from ∼ 36 kpc at b ∼ -71 · to ∼ 30 kpc at b ∼ -46 · , where the distance nearer the South Galactic Pole places the CPS at approximately the same distance as the Sgr stream at that position. The authors examined BHB, red giant branch (RGB), and lower RGB (LRGB) stars identified by stellar parameters in the SEGUE spectroscopic database, and found a mean metallicity of [Fe/H] = -2.1 for the CPS. The ratio of blue straggler (BS) to BHB stars was shown to be much higher in Sgr than in the CPS, suggesting that most of the BHB stars in this region of the sky may be associated with Cetus rather than the Sgr stream, as previous studies had assumed. From the limited SDSS data available in this region of the sky, this work concluded that the spatial distribution of the CPS is noticeably different than Sgr in the Southern Cap. Indeed, it was pointed out that this solved a mystery seen by Yanny et al. (2000), where the BHB stars in that earlier work did not seem to spatially coincide with the Sgr blue stragglers at ∼ 2 magnitudes fainter in the same SDSS stripe; the majority of these BHB stars must not be Sgr members, but CPS debris instead. The velocity signature of low- metallicity stars clearly differs from the Sgr velocity trend in this region of the sky, as was seen in the Yanny et al. (2009) study, further differentiating the CPS from Sgr. The velocity trend, positions, and distance estimates were used by Newberg et al. (2009) to fit an orbit to the CPS. However, because SDSS data available at the time only covered narrow stripes in the south Galactic cap, the orbit derived by Newberg et al. in this study was rather uncertain. In a study of the portion of the Sagittarius trailing stream in the Southern Galactic hemisphere using SDSS Data Release 8 (DR8) data, Koposov et al. (2012) noted that the BS and BHB stars in the region near the Sgr stream are not at a constant magnitude offset from each other as a function of position, as would be expected if they were part of the same population. The magnitude difference between color-selected blue stragglers and BHBs is ∼ 2 magnitudes at Λ ∼ 130 · (where Λ is longitude in a coordinate system rotated into the Sagittarius orbital plane, such that larger Λ is increasingly further from the Sgr core along the trailing tail; Majewski et al. 2003), and decreases at lower Sgr longitudes. Thus, the BHBs at Λ well away from 130 · must not be associated with Sagittarius. The distance modulus of these Cetus BHBs shows a clear trend over more than 40 · along the stream. The authors show that BS and BHB stars with SDSS DR8 spectra enable kinematical separation of Sgr and the CPS, as had already been seen in the Newberg et al. (2009) data from DR7. The kinematics of Cetus are suggested by Koposov et al. (2012) to imply a counter-rotating orbit with respect to Sgr. In this paper we follow up on the work by Newberg et al. (2009) using the extensive new imaging data made available in SDSS DR8, which now provides complete coverage of much of the south Galactic cap region. Additional spectra are available in DR8 as well, though they are limited to the DR7 footprint consisting of a few SDSS stripes only. The goal of this work is to measure the distances, positions, and velocities of CPS stars along the stream to sufficient accuracies that we can derive a reliable orbit for the Cetus Polar Stream. This will enable new constraints on the shape and strength of the Milky Way gravitational potential over the regions probed by Cetus, as well as illuminating another of the merger events in the hierarchical merging history of the Galactic halo. We then use this orbit to generate N -body models of the stream, and show that a ∼ 10 8 M /circledot satellite is required to reproduce the kinematics of the stream, but the distribution of stars along the stream requires either a short disruption time or a dwarf galaxy in which mass does not follow light. We argue that the CPS progenitor was likely a dark matter-dominated dwarf galaxy similar to the ultra-faint dwarfs in the Milky Way.",
"pages": [
1,
2,
3
]
},
{
"title": "2. Finding CPS from Photometrically Selected BHB stars",
"content": "Because blue horizontal branch (BHB) stars are prevalent in the CPS (Newberg et al. 2009; Koposov et al. 2012), we first select this population of stars in the larger SDSS DR8 dataset. BHBs are color-selected with -0 . 25 < ( g -r ) 0 < -0 . 05 and 0 . 8 < ( u -g ) 0 < 1 . 5, where the latter cut removes most of the QSOs from the sample. Here and throughout this paper, we use the subscript ' 0 ' to indicate that the magnitudes have been corrected by the Schlegel et al. (1998) extinction maps, as implemented in SDSS DR8. BHB stars were selected in the Galactic longitude range 120 · < l < 165 · , bracketing the nearly constant Galactic longitude of l ∼ 143 · found by Newberg et al. (2009) for the CPS in the south Galactic cap. We further separate stars within the initial color selections that are more likely to be higher surface gravity blue straggler (BS) stars from those more likely to be lower surface gravity BHBs using the ugr color-color cuts outlined in Yanny et al. (2000). The extinction-corrected apparent magnitudes of the BHB stars are converted to absolute magnitude, M g 0 , using the relation between absolute g -band magnitude and ( g -r ) 0 color for BHB stars given by Equation 7 of Deason et al. (2011). The distance modulus for each BHB star then constitutes the difference between the apparent and absolute magnitudes. The distance moduli of the color-color selected BHB stars are shown as a function of Galactic latitude in Figure 1. Because the stream is extended along nearly constant Galactic longitude, Galactic latitude is nearly the same as angular distance along the stream. In this graph there is a clear concentration of BHB stars with an approximately linear relationship between distance modulus (which is between 0 . 45 /lessorsimilar M g 0 /lessorsimilar 0 . 7 magnitudes offset from the measured apparent magnitudes of these stars, depending on color) and Galactic latitude, as was found in Newberg et al. (2009) and Koposov et al. (2012). However, there is a hint that the relationship is not quite linear. Therefore, we chose to fit a parabola to the BHB data between 17 . 0 < g 0 -M g 0 < 18 . 0 and b < -30 · , iteratively rejecting outliers (beginning with 3 σ rejection, then reducing this to 2 σ , and finally 1 . 5 σ ) until the fit converged to the one overlaid in Figure 1 as a solid line. This fit is given by The dashed lines in the figure show ± 0 . 1-magnitude ranges about the fit for distance modulus as a function of latitude. Figure 2 shows the same data as the previous figure, but with apparent magnitude corrected as a function of Galactic latitude for the trend fit in Figure 1 to place all stars at the BHB distance corresponding to CPS stars at b = -50 · . These new 'corrected' magnitudes, g corr , are defined as: where D mod ( b ) is the distance modulus fit as a function of latitude given by Equation 1, and 17.389 is the distance modulus of a star on the polynomial fit at b = -50 · . The CPS stars cluster tightly about the fit in Figure 2 at high latitudes ( b < -40 · ), then drop off near the Galactic plane (i.e., at b > -40 · ). This may be a real effect of the CPS BHB star density dropping off as one moves away from the Galactic pole along the stream (though also partly due to the non-uniform coverage of SDSS; see Figure 4 and further discussion in Sections 3 and 8). Wetake advantage of this relationship between latitude and distance for the CPS to study the properties of the BHB stars in the stream. The left panel of Figure 3 shows the observed Hess diagram for blue ( -0 . 3 < ( g -r ) 0 < 0 . 0) stars in the vicinity of the CPS (specifically, 120 · < l < 165 · , b < -40 · ), with an additional color-color selection from Yanny et al. (2000) to more prominently select BHB stars from among the more numerous blue stragglers. The BHB of Cetus stars is clearly visible at g 0 ∼ 18, with an additional large blob of stars at redder ( -0 . 15 /lessorsimilar ( g -r ) 0 < 0 . 0) colors and fainter ( g 0 ∼ 19 -20) magnitudes. As noted by Koposov et al. (2012), this feature is likely made up of BS stars from the Sgr dSph tidal stream, which intersects the CPS in this region of the sky. As shown in the previous paragraph and Figure 2, the latitude dependence of CPS BHB stars' magnitudes can be eliminated. The right panel of Figure 3 shows the same stars as the left panel, but with corrected magnitudes, g corr , instead of the measured magnitudes. The BHB locus in the 'corrected' panel is noticeably narrower than the original, and the absolute magnitude of BHB stars in the CPS appears to be only weakly color-dependent. The BS feature is relatively unchanged, g . 20 - 0 ( g . - 15 - 0 r ) 0 or perhaps even more diffuse, as would be expected to happen for a population with a different distance distribution than the CPS when applying the distance modulus correction.",
"pages": [
4,
5,
6,
7
]
},
{
"title": "3. The position of the CPS in the sky",
"content": "We now examine the positions of photometrically-selected BHB stars to trace the path of the Cetus stream on the sky. In Figure 4 we show a polar plot in Galactic coordinates, centered on the south Galactic cap, of the sky positions of BHB stars selected within 0.1 magnitudes of the trendline we fit to the concentration of BHBs in Figure 1 (except we have removed the Galactic longitude constraint). There is a clear concentration of stars between l ∼ 120 · and ∼ 160 · relative to the number of BHBs in adjacent longitude r r o c g 15 16 17 18 19 20 21 22 - 0 . 30 - 0 . 25 - 0 . 10 - 0 . 05 0 . 00 regions. In Newberg et al. (2009) the CPS was found to have a nearly constant longitude of l ∼ 143 · along its entire length. We aim to use the additional information that is available in this region from DR8 to revise the positional measurements of the CPS. However, two complications are obvious in Figure 4: first, the Sagittarius stream intersects the CPS at b ∼ -70 · at nearly the same distance as Cetus, and secondly, the DR8 footprint cuts off at l ∼ 150 · , making it difficult to determine whether the CPS extends beyond this longitude. To measure the position of the CPS, we slice the BHB sample into five latitude bands, then plot histograms of the longitude distribution in each strip. These histograms are seen in Figure 5 for BHB candidates within 0.1 magnitudes of the distance modulus trend fit previously (Equation 1). The latitude ranges were selected to include roughly the same number of stars in each range, with strips centered on b = [ -66 · , -59 · , -53 · , -46 · , -36 · ]. Bins with incomplete photometric coverage in DR8 were corrected upward by dividing the bin height by the fractional sky area covered by the bin in SDSS, with uncertainties in the bin heights corrected appropriately. A peak is evident in each panel of the figure, which we attribute to the CPS. We fit a Gaussian to the peak in each panel. The fitting was performed using a standard Gaussian function with an additional constant offset in N as a fit parameter. This parameter represents the unknown background level present in the data. A simple, coarse grid-search was performed over the expected pa- nges to determine the general location of the global best-fit of each dataset; a gradient descent search was then started near the global best-fit parameters (as indicated by the grid search). The final parameters are those determined by the gradient descent, with the model errors at those parameters given by the square-roots of twice the diagonal elements of an inverted Hessian matrix. We find best-fit central Galactic longitudes of l = [140 . 7 ± 3 . 5 · , 136 . 7 ± 4 . 9 · , 150 . 0 ± 4 . 6 · , 142 . 2 ± 4 . 9 · , 144 . 7 ± 4 . 7 · ], respectively, for each of the latitude strips. In Table 1 we give the best fit positions with their associated errors as l . We note that the uncertainties in these positions are larger than those given by Newberg et al. (2009), but consistent within the error bars. The width of the stream (in degrees) is the best-fitting Gaussian σ ; these values are tabulated as σ l in Table 1 along with the positions and other stream parameters. Because the BHB stars are rather sparse tracers of the stream, the stream widths are easily biased by background fluctuations toward large (5 · -15 · ) values that are likely overestimates of the true stream width. Distances to the CPS at each of these measured positions were determined in a similar manner to the positions. In each of the five latitude bins, we selected all BHB candidates between 120 · < l < 165 · . In each latitude bin, the distance moduli D mod (calculated using Equation 1) of the BHB stars were histogrammed, and we fit a Gaussian to their distribution using the gradient descent method described above. We restricted the fits to 16 < D mod < 19 in each bin, within which a peak was clearly visible in all five ranges considered. These histograms and the Gaussians fit to the peaks in each latitude bin are shown in Figure 6. The distance moduli and their Gaussian spread are given as D mod , fit and σ D mod , fit in Table 1, with the associated distances and line-of-sight depths reported as d fit and σ d , fit . These are likely much more robust measurements of the stream's physical width than the spatial distributions on the sky ( σ l and the associated width σ in kpc), as the distinct peaks in distance modulus are much less prone to contamination by non-stream stars than the positions on the sky. The line-of-sight depths are narrower than the widths of the stream on the sky in all cases. The stream width will be discussed further in Section 8, where we compare the measurements to N-body model results. It might seem obvious, once the position and distance to CPS are well known, to look for the much more numerous F turnoff stars associated with it. We did make some unsuccessful attempts to do this. The F turnoff stars are expected to be 3.5 magnitudes fainter than the horizontal branch, at g corr ∼ 21 . 5. The difficulty is that there is a very large background of F turnoff stars in the halo and particularly in the Sgr dwarf tidal stream, which dominates the sky when we attempt to extract F turnoff stars. We were successful in tracing the CPS in BHB stars because there are relatively few BHB stars in the Sgr stream compared to CPS. This is apparently not the case for F turnoff stars.",
"pages": [
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},
{
"title": "4. The Metallicity of the CPS",
"content": "We now use the CPS distance estimates and the velocities from Newberg et al. (2009), to select luminous stars from the Cetus Polar Stream. From these stars, we will study the range of metallicities in the stream. We select stars with spectra in SDSS DR8 that are in the south Galactic cap. With a horizontal branch at g 0 = 18, the turnoff of the CPS is at approximately g 0 = 21 . 5. Since this is fainter than the spectroscopic limit of SDSS, we do not expect to find any main sequence CPS stars in the dataset. Therefore, we select only luminous giant stars, using the surface gravity criterion 1 . 0 < log g < 4 . 0. Here, log g was determined from the ELODIELOGG value in the SDSS database. At distances of about 30 kpc, we do not expect CPS stars to have a significant tangential velocity, so we also used the proper motion cut | µ | < 6 mas yr -1 , which selects SDSS objects whose proper motions are consistent with zero. We also eliminated nearby Milky Way stars with high metallicity by selecting only those with -4 . 0 < [Fe/H] < -1 . 0. The metallicity of the sample was calculated differently for stars with ( g -r ) 0 < 0 . 25 and ( g -r ) 0 > 0 . 25. This was because Newberg et al. (2009) find the WBG classification (FEHWBG; Wilhelm et al. 1999) to be a better measure of metallicity for BHB stars, and thus the WBG metallicity was used for ( g -r ) 0 < 0 . 25 while the adopted SDSS metallicity (FEHA; consisting of a combination of a number of different measurement techniques) was used for ( g -r ) 0 > 0 . 25. The DR8 data we downloaded contained multiple entries for some stars. For these stars, we combined multiple measurements by calculating a weighted mean velocity (weighted by the SDSS velocity errors). Their stellar parameters ([Fe/H] and log g ) were set to the values of the highest S / N measurement. The left panel of Figure 7 shows the latitude-corrected Hess diagram of log g -selected metal-poor giant star candidates in the south Galactic cap. The greyscale represents the density of stars satisfying the surface gravity, metallicity, and proper motion criteria. In the center panel, we show a CMD of those stars from the left panel that also have the expected velocity of the CPS. To select stars with CPS-like velocities, we used V gsr < ( -0 . 1818 × b -33 . 63) km s -1 and V gsr > ( -1 . 205 × b -130 . 36) km s -1 and V gsr > (2 . 91 × b + 13 . 67) km s -1 , where the constraints are those used by Newberg et al. (2009) to select Cetus members (note: here, and throughout this work, the subscript 'gsr' means that the velocities are along the line-of-sight, and relative to the Galactic standard of rest). One sees among the velocity-selected CPS candidates in this figure a strikingly narrow BHB (centered around ( g -r ) 0 ∼ -0 . 15), red giant branch (at 0 . 45 /lessorsimilar ( g -r ) 0 /lessorsimilar 0 . 7 and 16 . 5 /lessorsimilar g corr /lessorsimilar 19), and what appears to be an asymptotic giant branch at g corr ∼ 17 and ( g -r ) 0 ∼ 0 . 5. To assess the possible contamination of the CPS sample by Sgr stars, we plot a similar color-magnitude diagram in the right panel of Figure 7. The points in this CMD were selected from the box chosen by Newberg et al. (2009) to highlight stars with Sgr velocities. Sgr velocities were chosen using V gsr < ( -3 . 01 × b -271) km s -1 and V gsr > ( -4 . 12 × b -395 . 25) km s -1 for b < -52 · and -180 < V gsr < -114 . 48 km s -1 for b > -52 · . Neither a clear BHB, nor any other obvious feature, is visible in this figure. There are possibly excess red giants among the velocity-selected Sgr candidates, but they do not form a clear sequence in the CMD. This is as expected if Sgr is not at the same distance as the CPS, since we have shifted things to CPS distances by the use of g corr ; this should smear out any Sgr features that are present in such a CMD. In Figure 8, we show a histogram of metallicities of the BHB stars from Figure 7. The hashed histogram is made up of CPS candidates that have -0 . 25 < ( g -r ) 0 < -0 . 05, distance moduli within 0.1 magnitude of the CPS trendline, V gsr within 20 km s -1 of the velocity trend used to select candidates in Figure 7, and are between 120 · < l < 165 · . The open (solid line) histogram is a background sample, selected from outside both the CPS spatial region and velocity criteria. The CPS BHB stars occupy a narrow metallicity range, with a peak around [Fe/H]= -2 . 2, that is clearly unlike the metallicity distribution of the background sample (typical uncertainties on the metallicity for each star are ∼ 0 . 25 dex). Thus, later in this paper, we will use -2 . 5 < [Fe/H] < -2 . 0 to preferentially select stars that are in the CPS. To illustrate the effect of metallicity selection, we show in Figure 9 the gsr-frame line-of-sight velocity as a function of Galactic latitude for BHB stars (selected by low surface gravity and low proper motion) that have -0 . 25 < ( g -r ) 0 < -0 . 05 and are within 0.1 magnitudes of the correct distance modulus to be members of the CPS. We did not select the stars based on Galactic longitude. The larger dots in this figure are the stars with -2 . 5 < [Fe/H] < -2 . 0. Note that the CPS stands out much more clearly in the metallicityselected sample, though we probably lose a few bonafide CPS stars. Using the low metallicity sample and our knowledge of the distance to the CPS, we have very little contamination from Sgr and other BHB stars in the Milky Way, even though we accept stars from all observed Galactic longitudes.",
"pages": [
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},
{
"title": "5. Red Giant Branch Stars in the CPS",
"content": "In the center panel of Figure 7 there are apprently many Red Giant Branch (RGB) stars in the CPS, that we would like to add to our sample. Using the knowledge that CPS stars are metal-poor ( -2 . 5 < [Fe/H] < -2 . 0), we choose to select candidates using a fiducial sequence from the globular cluster NGC 5466, which has a metallicity ([Fe/H]) of -2.22, shifted to a distance modulus of 17.39 (corresponding to a distance of 30.06 kpc, which is the distance at b = -50 · from the fit in Figure 3). The fiducial sequence of NGC 5466 is from SDSS data, and is taken from An et al. (2008). A third order polynomial was fitted to the region of interest on the fiducial sequence of NGC 5466, and stars were selected using the criteria of -0 . 0081006 g 3 corr + 0 . 46944 g 2 corr -9 . 1207 g corr + 59 . 9130 -0 . 04 < ( g -r ) 0 < -0 . 0081006 g 3 corr + 0 . 46944 g 2 corr -9 . 1207 g corr + 59 . 9130 + 0 . 04 and 16 . 9 < g corr < 19 . 9, where g corr is the latitude corrected magnitude as defined in Equation 2. Figure 10 shows a CMD similar to those in Figure 7 for spectroscopically selected giant stars between 120 · < l < 165 · with low proper motion. Colored points show those with metallicities of -2 . 5 < [Fe/H] < -2 . 0 and CPS velocities. We note that the velocity selection shown here for CPS is actually the final selection we arrive at later in this paper; however, the final result differs little from the previous measurement of the velocity trend by Newberg et al. (2009). The color-magnitude box described above for selecting CPS RGB stars, along with the NGC 5466 ridgeline upon which it is based, is shown on the diagram. To determine the metallicity of the CPS, we used BHB stars, which are by their nature all low metallicity. We then selected red giant branch stars with the same metallicity range as the BHBs before we matched the RGB fiducial. It is natural to wonder, then, whether there are any higher metallicity RGB stars in the CPS. In Figure 11, we show the line-of-sight, Galactic standard of rest velocities as a function of Galactic latitude for all of the low surface gravity, low proper motion stars with spectra in SDSS DR8, that are within the Galactic longitude limits 120 · < l < 165 · and within the box 0 . 4 < ( g -r ) 0 < 0 . 8 and 16 . 9 < g corr < 19 . 9. The three panels of this figure show all stars satisfying these criteria as black points, with large red dots representing stars with metallicities between -3 . 0 < [Fe/H] < -2 . 5, -2 . 5 < [Fe/H] < -2 . 0, and -2 . 0 < [Fe/H] < -1 . 5 in panels from left to right, respectively. Notice that we have opened up the range of colors and apparent magnitudes very wide, to accept all types of giant branch stars at a range of distances. We see from this figure that the much more populated Sgr dwarf tidal stream, in the lower left corner of each panel, includes stars of many different metallicities, and is most pronounced in the metallicity range -2 . 0 < [Fe/H] < -1 . 5. On the other hand, the CPS, with velocities near V gsr ∼ -50 km s -1 , is most apparent in the center panel, with -2 . 5 < [Fe/H] < -2 . 0 (with perhaps a few slightly more metal-rich stars). The population of stars we see in the CPS includes predominantly metal-poor stars in a narrow range of metallicity. We also tried selecting stars with CPS velocities in color-magnitude fiducial sequences with a range of metallicities, but the stars in each fiducial sequence were dominated by stars in the same metallicity range of -2 . 5 < [Fe/H] < -2 . 0. Therefore our initial fiducial sequence of NGC 5466 as well as the selections in the CMD and in metallicity are justified. If the CPS is the remnant of a dwarf galaxy, then either we are looking at the outer portions of the dwarf galaxy, and not the inner portion with more recent star formation, or we are looking at the remains of a smaller, possibly gas-stripped galaxy that never had a later epoch of star formation at all (similar to the ultra-faint dwarfs discovered recently that seem to have had only a single epoch of star formation).",
"pages": [
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},
{
"title": "6. The line-of-sight velocities along the CPS",
"content": "We have now identified samples of BHB and RGB stars in the CPS that lie within the appropriate loci in Figure 10. In Figure 12 we show the line-of-sight, Galactic standard of rest velocities for these stars (removing the selection in V gsr ) as a function of Galactic latitude. To review, the stars in this plot are low surface gravity, low proper motion stars with spectroscopy in the SDSS DR8. They additionally are in the region of the sky inhabited by the CPS ( b < 0 · and 120 · < l < 165 · ). The colors of the symbols tell which CMDselection box each star is from: blue points are BHB candidates, and red: RGB. The larger symbols in the figure have the metallicity we have shown to preferentially select CPS stars: -2 . 5 < [Fe/H] < -2 . 0; large blue circles are BHB stars, and red squares are RGB candidates. The majority of the points in Figure 12 that have larger-sized symbols follow the expected velocity trend of the CPS, with only a handful of stars near the Sgr velocities, and a few scattered elsewhere. Wefitapolynomial to the CPS velocities beginning with the entire metallicity-selected ( -2 . 5 < [Fe/H] < -2 . 0) sample. The fitting was done iteratively, rejecting outliers at each iteration using the same technique as in Section 2, until it converged to a solution of V gsr = -41 . 67 -(0 . 84 × b ) -(0 . 014 × b 2 ). This fit is shown as the green line in Figure 12, with the limits we chose for CPS velocity selection at ± 20 km s -1 on either side of this fit. This is the velocity cut that was used to select the red and blue points in Figure 10, which constitute our 'best' sample of CPS stars. We now check to make sure the spectroscopically selected stars match our expectations for position on the sky. Using this newly found relationship, we select all of the stars from within the BHB and RGB colorg corr selection boxes, with velocities within 20 km s -1 of the trendline, and that are also metal-poor ( -2 . 5 < [Fe/H] < -2 . 0), and plot them in a polar plot of the south Galactic cap (Figure 13). Here, we can see that the stars match spatially with the photometric selection in Figure 4, with very few stars matching the selection criteria outside of the CPS region. Because the stars with measured velocities are confined to 'clumps' at the positions where SEGUE plates were located, a selection in the same bins used to derive positions and distances would skew the velocity measurement in each bin toward positions of highest concentrations of data. (See, for example, Figure 12. The clump of stars at b ∼ -71 · would not fall within any of the bins from Table 1, and the -70 · < b < -62 · bin would be skewed heavily toward the b = -62 · end rather than reflecting the velocity at b = -66 · .) Thus we chose to create separate bins from the data in Figure 12 with which to measure the mean velocities, centered on the highest concentrations of CPS candidates. Mean velocities and intrinsic velocity dispersions (accounting for the measurement errors) in each bin were calculated using a maximum likelihood method (e.g., Pryor & Meylan 1993; Hargreaves et al. 1994; Kleyna et al. 2002). The mean latitude, V gsr , intrinsic velocity dispersion, and number of stars in each of these bins are given in Table 2. We note that these velocity dispersions of ∼ 4 -8 km s -1 are typical of dwarf galaxies in the Local Group (see McConnachie 2012 and references therein), and higher than typical dispersions in globular clusters. To derive the velocities in Table 1, we interpolate between the values in Table 2; for example, the velocity at b = -66 · in Table 1 and its error were linearly interpolated from the b ∼ -71 · and b ∼ -61 · values in Table 2. Note also that all of the values given in each of these tables are consistent with the polynomial fit of V gsr vs. b seen in Figure 12. Finally, we note that we often find halo substructure in plots of line-of-sight velocity for restricted volumes of space. There is a previously unidentified clump of stars at b = -38 · and V gsr = 95 km s -1 in Figures 11 and 12. We investigated this clump, and discovered eight low surface gravity, low proper motion stars that are within a degree of ( l , b ) = (136 · , -38 · ), have metallicities in the range -2 . 5 < [Fe/H] < -1 . 5, and follow a giant branch very similar to the CPS giant branch. This structure (also seen photometrically by Grillmair 2012 and Bonaca et al. 2012) is the subject of a separate publication (Martin et al. 2013), in which we show that these stars are part of a narrow tidal stream that we dub the 'Pisces Stellar Stream.'",
"pages": [
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},
{
"title": "7. Fitting an orbit to the Cetus Polar Stream",
"content": "Having determined the position, velocity, and distance trends along the stream, we wish to use these to constrain the orbit of the Cetus Polar Stream progenitor. The orbit fitting routine requires a set of discrete data points, rather than the general trends we have fit as a function of latitude for the CPS. Note that we fit an orbit assuming that the debris we have measured follows the orbit of the progenitor. This assumption that the tidal debris trace the progenitor's orbit is not strictly true, and the stream-orbit misalignment is essentially independent of progenitor mass (Sanders & Binney 2013). We test and discuss this assumption that the stream follows the orbit in Section 8, which describes N -body simulations of this tidal stream. The orbit was fit using the data at five positions given in Table 1 following the techniques described by Willett et al. (2009). The fit assumed a fixed Galactic gravitational potential of the same form used by Willett et al. (2009), which was in turn modeled after the potential of Law et al. (2005) and Johnston et al. (1999). This model contains a three-component potential made up of disk, bulge, and halo components. All parameters in the model were fixed at the same values given in Table 3 of Willett et al. (2009), and the Sun was taken to be 8 kpc from the Galactic center. The technique was slightly improved so that the orbit is not constrained to pass through the sky position of any of the data points. Orbits are uniquely defined by a gravitational potential, a point on the orbit, and a velocity at that point. Because it doesn't matter where along the orbit the velocity is specified, we are free to arbitrarily choose one of the three spatial parameters. For fitting, we fixed the Galactic latitude of the point on the a This column represents a latitude range between which stars were selected. orbit at b = -71 · , and fit five orbital parameters: the heliocentric distance ( R ), Galactic longitude ( l ), and the components V X , V Y , and V Z of Galactic Cartesian space velocity. We evolve the test particle orbit both forward and backward from the starting position, and perform a goodness-of-fit calculation comparing the derived orbit to the data points in Table 1. The best-fit parameters are optimized using a gradient descent method to search parameter space. We find best-fit parameters at b = -71 · of l = 138 . 2 ± 3 . 8 · , R =32 . 9 ± 0 . 3 kpc, and ( V X , V Y , V Z ) = ( -118 . 1 , 64 . 8 , 76 . 3) ± (7 . 2 , 10 . 3 , 3 . 0) km s -1 . The errors in the measured orbit parameters 1 are smaller if the minimum in the χ 2 surface is narrower. These errors were calculated from the square root of the diagonal elements of twice the inverse Hessian matrix. The Hessian matrix consists of the second derivatives of χ 2 (not the reduced χ 2 ) with respect to the measured parameters, evaluated at the minimum. The position, velocity, and heliocentric distance as a function of Galactic latitude predicted by the integrated best-fitting orbit is shown in Figure 14, with the data points constraining the fit shown as large filled squares. This fit had a formal reduced χ 2 of 0.48. This χ 2 is rather low, and likely arises because we have conservatively estimated our uncertainties. Note also that we have not discussed the effects of possible systematic offsets in the distance scale. We tried an additional orbit fit that included a multiplicative factor to scale the distances as a free parameter; this fit did not improve the χ 2 , so we retained the fit without the scale factor. We find that the Cetus Polar Stream is on a rather polar orbit inclined to the Galactic plane by i ∼ 87 . 0 + 2 . 0 -1 . 3 degrees. This orbit has an apogalactic distance from the Galactic center of 35 . 67 ± 0 . 01 kpc and is at 23 . 9 + 3 . 4 -3 . 0 kpc when passing through perigalacticon. This results in an orbital eccentricity of e =0 . 20 ± 0 . 07. The period of our derived CPS orbit is ∼ 0 . 694 ± 0 . 04 Gyr ( ∼ 694 Myr), measured between consecutive pericentric passages. Errors on all of these quantities were derived by integrating the orbits using the maximum and minimum possible total velocities from the orbit-fitting errors in ( V X , V Y , V Z ) and comparing these to the best fit orbit. The orbit is shown in Galactic Cartesian XYZ GC coordinates 2 in Figure 15, integrated for a total of 1.5 Gyr (0.75 Gyr each in the 'forward' and 'backward' directions from our chosen position). The forward integration is given by the solid lines, and the backward integration is the dashed lines, as in Figure 14. The bottom right panel shows the Galactocentric distance as a function of time; from this panel it is clear that our chosen position to anchor the orbit is very near the apocenter of the orbit. Indeed, we find that the nearest apogalacticon was only ∼ 2 Myr prior to this fiducial point, at a position of ( l , b ) = (137 . 8 · , -71 . 6 · ). The most recent pericentric passage was ∼ 350 Myr ago, at a position of ( l , b ) = (337 . 4 · , 7 . 0 · ), and a distance of ∼ 27 . 2 kpc from the Galactic center. Note that our assumption that the debris trace the orbit likely leads to differences in the derived orbital parameters at the ∼ 10% level, especially for a massive progenitor. This will be discussed further in the next section.",
"pages": [
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},
{
"title": "8. N-body model",
"content": "We now use the orbit we have derived in combination with information about the velocities, velocity dispersions, stellar density, and three-dimensional positions of observed CPS debris to explore the nature of the stream's (unknown) progenitor. To do so, we use N -body simulations of satellites disrupting in a Milky Way-like gravitational potential, on the well-defined orbit we have measured in this work. The density distribution of debris along the stream, as well as the velocity dispersion, stream width, and line-of-sight depth, are sensitive to the mass and size of the progenitor, as well as the time that the progenitor has been orbiting the Milky Way. A comprehensive modeling effort is beyond the scope of this work; here we aim to gain insight into the nature of the progenitor that will inform future, more detailed, modeling efforts. The N -body simulations were run using the gyrfalcON tool (Dehnen 2002) of the NEMO Stellar Dynamics Toolbox (Teuben 1995). Satellites with masses configured using a Plummer model (Plummer 1911) were evolved in the same Galactic gravitational potential used to derive orbits. Each simulation has 10 4 bodies. We started out with the assumption that mass follows light, so that the distribution of masses from the N -body simulation has the same density and velocity profile as the observed stars. We note that this assumption has been shown to work in modeling dwarf spheroidals by Muñoz et al. (2008). With this assuption, the Plummer scale radius of each satellite was chosen, roughly following the scaling relations from Tollerud et al. (2011; see, e.g., their Figure 7). A Plummer radius of 1 kpc was used for the 10 8 M /circledot satellite, with the remaining radii scaled using the empirical relation (Tollerud et al. 2011) between the half-light radius and the dark matter mass within that radius: M DM 1 / 2 ∝ r 2 . 32 1 / 2 (note that we assumed that the Plummer radius is roughly equal to the half-light radius). Since we assume mass follows light, the radius of the dark matter and the radius of the luminous matter are the same. The total masses of the Plummer spheres were varied from 10 5 to 10 9 solar masses, with Plummer scale radii of a p ∼ (50 , 150 , 400 , 1000 , 2500) pc corresponding to (10 5 , 10 6 , 10 7 , 10 8 , 10 9 ) M /circledot models, respectively (note that our choice of scaling differs little from a simple choice of r ∝ M 1 / 3 , which would produce satellites with the same mass density). Wehave seen that the density of stars in the CPS falls as one approaches the Galactic plane. This is also along the direction of our measured orbital motion, so that the observed stars near the south Galactic cap must be the leading tidal tail of a satellite (which could be completely disrupted) that is not too far behind the observed stars on the orbit. We place a point mass at the position ( l , b ) = (138 . 2 · , -71 · ) on the orbit at a distance of 32.9 kpc, then integrate the point backwards on the orbit for 3 Gyr. At that new position, we place a Plummer sphere of bodies representing the dwarf galaxy with the bulk velocity of the forward orbit at that position. We then integrate the N -body forward for three Gyr, just over four full orbits, so the dwarf galaxy ends up approximately at our initial position. We measure the width across the stream ( σ l , which is the σ of a Gaussian fit to the profile), the line of sight depth of the stream ( σ d ; again from a Gaussian fit), and the stream velocity dispersion ( σ V ), from the results of each N -body model. Table 3 shows the measured values for the same five latitude bins we used to analyze the data, which should be compared with the observed properties of the stream stars in the same table. The stream widths, σ l and σ d , both come from photometrically selected BHB stars, while the velocity dispersion includes BHBs and RGB stars. The width in Galactic longitude, σ l , is large for the BHB stars, ranging from 5 . 0 · to 15 . 1 · , and has large uncertainties. This likely arises because of the low numbers of BHB stars in the samples, such that the poorly-defined background BHB density broadens the wings in some of these fits. The 10 5 M /circledot , a p = 50 pc progenitor, which has short, narrow debris tails resembling those of the Pal 5 globular cluster (e.g., Grillmair & Dionatos 2006; Odenkirchen et al. 2001, 2003; Rockosi et al. 2002), gives unreliable results for many of our measurements simply because there is little debris to be measured. Nevertheless, it is clear that the low-mass ( M /circledot < 10 7 ) progenitors produce streams much narrower in both width on the sky and line-of-sight depth, than observed in the CPS. The low mass progenitors also have velocity dispersions that are too small. The measured width and depth of the stream stars suggest a 10 8 -10 9 M /circledot progenitor mass, while the velocity dispersions suggest a 10 7 -10 8 M /circledot progenitor mass. Thus it seems that the physical widths (both on the sky and along the line of sight) and the velocity dispersions we have measured are telling us that the progenitor of the Cetus Polar Stream had a mass of ∼ 10 8 M /circledot . However, we now show that the density of stars along the stream is inconsistent with the mass-followslight progenitor in our model. We use the BHB stars selected photometrically by the methods described earlier in this work. The BHB sample is restricted to distance moduli of 16 . 5 < D mod < 18 . 0 to choose only the distance range containing CPS debris. The entire available range of Galactic latitude is chosen ( -72 · < b < -28 · ). We select an 'on-stream' and 'off-stream,' or background, sample. The on-stream BHB stars are selected in the longitude range of 130 · < l < 152 · to include only the regions completely sampled by SDSS DR8. The off-stream sample spans 86 · < l < 130 · over the same latitudes. We bin the number counts in 4-degree bins of latitude for each sample, then scale down the number counts in the off-stream regions by a factor of two to account for the additional area sampled. Figure 16 shows the density profile of BHB stars along the CPS after subtraction of the 'off-stream' background density. There is clearly an excess population of BHB stars in the CPS compared to the background at latitudes -70 · < b /lessorsimilar -40 · . Above b ∼ -40 · , the excess seems to disappear, with the BHB numbers consistent with the adjacent background level. However, recall from Figure 5 that there must be at least a few stream stars with -42 · < b < -30 · . This density gradient starting from highest density near the south Galactic cap and decreasing toward the Galactic plane must be reproduced by any mass-follows-light model of the stream. Note that the first bin in the figure is low because the area of sky between two longitudes near the south Galactic pole is small, and the stream does not go exactly through the pole; the density is actually highest in the bin nearest the pole, since that is close to the final position of the satellite. We measure the density profile as a function of latitude for the set of five N -body models that have the progenitor ending up near the south Galactic cap. The debris was selected from the same longitude and distance ranges used for the BHB density in Figure 16 and histogrammed in 4-degree bins as in the BHB figure. These density profiles are displayed in the left panels of Figure 17 as solid lines for each of the five progenitor masses. Number counts in each bin were normalized by the total number of points in each panel to facilitate comparison with the BHB density profile, which is overlaid as a dashed histogram (with error bars). The 10 5 M /circledot progenitor has suffered little tidal disruption, and is thus strongly peaked near the satellite; this is likely because of the small (50 pc) radius of the satellite. The largest satellites (0.4-2.5 kpc, 10 7 -10 9 M /circledot ) produce nearly constant density along latitude, indicating that they are strongly disrupted and that debris has spread extensively along the orbit. Note that although our data is near the apogalacticon of the orbit, the orbit is not very eccentric so highly disrupted satellites have only a small increase in stellar density at the position in the stream that is most distant from the Galactic center. The stellar density along the stream in the simulations with a more massive dwarf galaxy does not agree with the observed gradient in the measured BHB density. On the other hand, the 10 6 M /circledot model density appears to agree very well with the observations, with all five of the bins between -64 · < b < -44 · in agreemeent with the BHB number counts within 1 σ . However, we have already shown that a satellite with 10 6 M /circledot is not consistent with the width, depth and velocity dispersion of the stream. Since the 10 8 solar mass progenitor seemed a reasonable fit to the stream width and velocity dispersion, but the distribution of stars along the stream was too dispersed, we tried reducing the time the stream has been disrupting in an attempt to fit all of the measured quantities. The right panels of Figure 17 show the distribution along the stream for disruption times of 0.5 to 1.5 Gyr, and Table 3 shows the velocity dispersion and stream widths for these simulations. We find that a 10 8 solar mass progenitor with a disruption time of 1-1.25 Gyr is a reasonable but not perfect fit to all of our data. It is a bit difficult to fit the large apparent width of the stream with N -body simulations, even with a relatively large mass progenitor of 10 8 M /circledot . Note that the best-matching disruption time of ∼ 1 Gyr is about one and a half orbits of the satellite, and would require that the dwarf galaxy was perturbed into its present, nearly circular orbit fairly recently; we did not model the effects of a recent deflection of the satellite in the simulation. It might be possible to adjust the disruption time slightly by varying the final position of the progenitor, but this would only produce a slight change to the best fit time. Note. - For the 3-Gyr N -body models, particles were configured in Plummer spheres with scale radii of (50 , 150 , 400 , 1000 , 2500) pc, respectively, for the models with masses of (10 5 , 10 6 , 10 7 , 10 8 , 10 9 ) M /circledot . The results of the 1 Gyr N -body simulation for a 10 8 M /circledot satellite are shown in Figure 18. This figure shows the distribution of model debris in distance, velocity, and position (right ascension and declination in the left column, and Galactic coordinates in the right panels). One difficulty that is illuminated in Figure 18 is that our orbit, which was fit to the tidal debris, does not lead to an N -body simulation that fits the measured stream distances for a massive (10 8 M /circledot ) progenitor (in particular, see the upper left panel of Figure 18). The discrepancy between orbits fit to streams and the actual orbits of massive progenitors has been highlighted by Binney (2008) and Sanders & Binney (2013). To address this discrepancy, we tried adjusting the orbit by hand to match the data. We were able to plausibly match the N -body simulations by increasing the distance of the orbit from the Galactic center. The new orbit has an apogalacticon of 38 kpc, a perigalacticon of 26 kpc, an inclination of 88 · , eccentricity of 0.19, and a period of 750 Myr. All of these values are within 10% of the orbit that was fit to the stream, and other than a shift to ∼ 2 kpc larger Galactocentric distances, all of the parameters are within the uncertainties of the orbital parameters from our original fit. The evolution time of the simulation also affects the results; this particular orbit was a better fit to the data with a 3 Gyr evolution time than a 1 Gyr evolution time. It is a much more difficult problem, and beyond the scope of this current effort, to fit orbital parameters by matching data directly to N -body simulations. Thus we cannot verify that our solution is unique or is in fact the best solution. However, we note that this slightly altered orbit has very similar width, depth, velocity dispersion, and density along the stream as the original N -body simulation. The best fit model is still a 10 8 M /circledot satellite that may have been deflected into its current orbit ∼ 1 Gyr ago. Another more plausible solution is that mass does not follow light in the stream progenitor. So far, our simulations have assumed that the N -body points, which are simply massive particles (and thus more representative of the dark matter in the satellite), will be distributed similarly to the luminous matter (i.e., the stars). There is little reason to believe this is the case, and in fact the observed properties of many dwarf galaxies are often interpreted as showing evidence of significant dark matter halos in these objects (see, e.g., McConnachie 2012 and references therein). Our N -body comparisons seem to suggest that the CPS progenitor was a dwarf galaxy with a large mass-to-light ratio and a dark matter halo that extends beyond the distribution of stars. A model of this type might explain the apparently small number of stars given the large spatial and velocity dispersion of the stream, and the relatively recent disruption of the stellar component of the progenitor as determined by the relatively steep gradient in star density along the stream. Another argument in favor of an ultrafaint dwarf progenitor comes from the measured metallicity. We showed in Sections 4 and 5 that the metallicity of CPS stars is sharply peaked between -2 . 5 < [Fe/H] < -2 . 0. Assuming that we are looking at a disrupted dwarf galaxy, the 〈 [Fe/H] 〉 vs. M V relation for Local Group dwarf galaxies of Fig. 12 of McConnachie (2012) suggests that a dwarf spheroidal with 〈 [Fe/H] 〉 ∼ -2 . 25 should have M V ∼ -6. Likewise, the metallicity-luminosity relationship for Local Group dwarf galaxies from Simon & Geha (2007) predicts ultra-faint dwarf spheroidals with -4 > M V > -9 should have mean metallicities between -2 . 5 < [Fe/H] < -2 . 0, lower than most globular clusters at similar luminosities. Thus the metallicity of the stars in the CPS is also consistent with its progenitor being an ultra-faint dwarf galaxy. The velocity dispersion and stream width alone (without the N -body models for comparison) are too large for a globular cluster progenitor. We therefore conclude that the progenitor was a dwarf galaxy. One possible solution is a mass-follows-light progenitor of about 10 8 solar masses, that was deflected into a nearly circular orbit of order one Gyr ago. A more plausible solution is that the progenitor was a dark matter dominated, ultrafaint dwarf galaxy. Modeling a two component (dark matter plus stars) dwarf galaxy progenitor for the Cetus Polar Stream is beyond the scope of this work, but the prospect that the dark matter distribution of the progenitor dwarf galaxy could be encoded in the distribution of stars in a tidal tail is tantalizing, and will be pursued in a future paper. We have been building the capacity to fit many parameters in an N -body simulation of dwarf galaxy tidal disruption using the Milkyway@home volunteer computing platform ( http://milkyway.cs.rpi.edu/milky We plan to eventually use this platform to find the best fit model parameters for the dwarf galaxy progenitor of the CPS given the available data for the tidal stream, and also to model other tidal streams in the Milky Way.",
"pages": [
24,
25,
26,
28,
29,
30,
31
]
},
{
"title": "9. Conclusion",
"content": "In this work, we use BHB and RGB stars from SDSS DR8 to improve the determination of the Cetus Polar Stream orbit near the south Galactic cap. We then model the evolution of satellites on this orbit to assess the nature of the progenitor that produced the CPS. Much of the region of sky in which the CPS is located is dominated by the Sagittarius stream. However, we show that stars with the velocities of the Sgr tidal stream in the region near the CPS are predominantly blue stragglers (BS), and that Sgr is virtually devoid of BHB stars. This is in stark contrast with the CPS, which has a well populated BHB and very few, if any, observed BS stars. We are able to trace the CPS with BHBstars over more than 30 degrees in Galactic latitude along its nearly constant path in Galactic longitude. We were also able to disentangle CPS from Sgr based on the different velocities and mean metallicities of the two structures (see, e.g., Figure 11). We fit an orbit to the Cetus Polar Stream tidal debris using the positions, distances, and velocities we have measured over at least 30 -40 · of the stream's extent. Under the assumption that the orbit follows the tidal debris, the best fit CPS orbit has an eccentricity of e = 0 . 20 ± 0 . 07, extending to an apogalactic distance from the Galactic center of R GC = 35 . 7 kpc at a point very near the lowest-latitude measurement in our study. At its perigalactic passage, the orbit is ∼ 24 kpc from the Galactic center. This orbit is inclined by 87 · to the Galactic plane, and has a period of ∼ 700 Myr. Our N -body models indicate a high mass progenitor ( > 10 8 M /circledot ), and that the orbit of the progenitor does not follow the tidal debris, particularly in distance. However, small tweaks to the orbit that make the simulations more consistent with the data change the orbit parameters by 10% or less. With the data presented here, we are unable to constrain the Galactic gravitational potential; further data on the CPS (or combining the data with other tidal streams) will be necessary to provide insight into the shape and strength of the halo potential probed by the stream. In Section 3 we noted that we were unable to separate the CPS from Sagittarius using SDSS photometry of F turnoff stars. In RGB stars, which are common to both populations, we can only see the CPS if we have spectra, and can separate the CPS stars by velocity and metallicity. To study halo substructure in more detail, we require spectroscopic surveys that reach F turnoff stars at g magnitudes approaching 22. Alternatively, accurate proper motions could be used to separate the CPS from the Sgr tidal stream, because the two streams are orbiting in opposite directions, as was pointed out by Koposov et al. (2012). Our orbit fit predicts mean proper motions in the longitude bins with 〈 b 〉 ≈ -66 · , -59 · , -53 · , -46 · , and -36 · (see Tables 1 and 3) of ( µ α cos δ,µ δ ) ≈ (1 . 2 , -0 . 2) , (1 . 3 , -0 . 2) , (1 . 3 , -0 . 1) , (1 . 4 , -0 . 1) , (1 . 5 , 0 . 1) mas yr -1 , respectively. We were unable to distinguish the F turnoff stars in the two streams using currently available proper motions, though it might be possible to separate larger populations of faint F turnoff stars using future proper motion surveys. It is unlikely that the proper motion measurement of individual stars will accurate enough, but it has been shown (Carlin et al. 2012; Koposov et al. 2013) that the ensemble proper motion of turnoff stars in the Sgr stream at distances of ∼ 30 -40 kpc can be statistically determined from currently available data. We simulate the evolution of satellites on the CPS orbit in a Milky Way-like potential via N -body modeling. From these mass-follows-light models, we show that with a mass-follows-light assumption, a 10 8 M /circledot satellite can produce a stream width, line-of-sight velocity, and velocity dispersion similar to our observations. However, the distribution of stars along the stream and the apparent number of stars observed favor a lower mass satellite. It is possible to reconcile these observations if the dwarf galaxy was deflected into its current orbit on the order of one Gyr ago (less than one and a half orbits). However, a less contrived solution is that mass does not (or did not) follow light in the CPS progenitor, which may have been a dark matter-dominated satellite similar to the recently-discovered class of satellites known as 'ultra-faint dwarf spheroidals\" (e.g., Simon & Geha 2007, McConnachie 2012). These are low luminosity, highly dark matterdominated satellites. The mean metallicity, 〈 [Fe / H] 〉 ≈ -2 . 2, that we find for CPS debris is consistent with the typical metallicities of ultra-faint dwarfs. More detailed modeling should be carried out in the future to confirm that dual-component (dark matter + stars) satellites reproduce the observed properties of the Cetus Polar Stream, and that the properties of the progenitor can be determined uniquely. Our results suggest that the structure of tidal streams could be used to constrain the properties of the stream progenitors, including their dark matter components. Wethank the anonymous referee for thoughtful and detailed comments that greatly improved this work. This work was supported by the National Science Foundation grant AST 10-09670. JLC was supported by NSF grant AST 09-37523. WY was supported by NSF grant AST 11-15146. EO was supported by NSF grant DMR 08-50934. JD was supported by the NASA/NY Space Grant fellowship. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.",
"pages": [
32,
33
]
},
{
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2013ApJ...776L..19D
https://arxiv.org/pdf/1309.5933.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_80><loc_85><loc_86></location>VVV Survey Near-Infrared Photometry of Known Bulge RR Lyrae stars: the Distance to the Galactic Center and Absence of a Barred</section_header_level_1>
<section_header_level_1><location><page_1><loc_25><loc_77><loc_75><loc_79></location>Distribution of the Metal-Poor Population</section_header_level_1>
<text><location><page_1><loc_14><loc_71><loc_86><loc_75></location>I. D'ek'any 1 , 4 , D. Minniti 1 , 4 , 5 , M. Catelan 1 , 4 , M. Zoccali 1 , 4 , R. K. Saito 2 , M. Hempel 1 , 4 , O. A. Gonzalez 3</text>
<text><location><page_1><loc_12><loc_66><loc_88><loc_70></location>1 Instituto de Astrof'ısica, Facultad de F'ısica, Pontificia Universidad Cat'olica de Chile, Av. Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, Chile</text>
<text><location><page_1><loc_14><loc_61><loc_86><loc_65></location>2 Departamento de F'ısica, Universidade Federal de Sergipe, Av. Marechal Rondon s/n, 49100-000, S˜ao Crist'ov˜ao, SE, Brazil</text>
<section_header_level_1><location><page_1><loc_44><loc_53><loc_56><loc_54></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_24><loc_83><loc_50></location>We have combined optical and near-infrared data of known RR Lyrae (RRL) stars in the bulge in order to study the spatial distribution of its metal-poor component by measuring precise reddening values and distances of 7663 fundamentalmode RRL stars with high-quality photometry. We obtain a distance to the Galactic center of R 0 = 8 . 33 ± 0 . 05 ± 0 . 14 kpc. We find that the spatial distribution of the RRL stars differs from the structures traced by the predominantly metal-rich red clump (RC) stars. Unlike the RC stars, the RRL stars do not trace a strong bar, but have a more spheroidal, centrally concentrated distribution, showing only a slight elongation in its very center. We find a hint of bimodality in the density distribution at high southern latitudes ( b < -5 · ), which needs to be confirmed by extending the areal coverage of the current census. The different spatial distributions of the metal-rich and metal-poor stellar populations suggest that the Milky Way has a composite bulge.</text>
<text><location><page_2><loc_17><loc_83><loc_83><loc_86></location>Subject headings: stars: variables: RR Lyrae - Galaxy: bulge - Galaxy: stellar content - Galaxy: structure</text>
<section_header_level_1><location><page_2><loc_42><loc_76><loc_58><loc_78></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_56><loc_88><loc_74></location>RR Lyrae stars are easily identifiable primary distance indicators, which can be used to trace the structure of the Milky Way. They sample an old and metal-poor population, and are particularly useful in the reddened regions of the Galactic bulge because of several reasons: (i) they have a relatively high number density in the bulge (e.g., Soszy'nski et al. 2011); (ii) they follow precise period-luminosity (PL) relations in the near-infrared (NIR; e.g., Catelan et al. 2004); (iii) their PL relations allow the determination of the interstellar extinction on a star-by-star basis; (iv) their metallicities can be estimated using photometric data alone (e.g., Jurcsik & Kov'acs 1996; Smolec 2005); and (v) their brightness and amplitude (0 . 1 glyph[lessorsimilar] a K s glyph[lessorsimilar] 0 . 5 mag) make them detectable out to large distances.</text>
<text><location><page_2><loc_12><loc_26><loc_88><loc_55></location>The VISTA Variables in the V'ıa L'actea (VVV) ESO Public Survey (Minniti et al. 2010; Catelan et al. 2011; Saito et al. 2012) aims to probe the 3-dimensional (3D) structure of the Galactic bulge using various distance indicators. In a series of papers, the VVV ZY JHK s color atlas was used to trace the spatial distribution of red clump (RC) giants (Saito et al. 2011; Minniti et al. 2011; Gonzalez et al. 2011), and to provide photometric reddening and metallicity maps 1 of the bulge (Gonzalez et al. 2011, 2012, 2013; Chen et al. 2013). The RC stars clearly trace a complex, X-shaped distribution, as was first suggested by McWilliam & Zoccali (2010) and Nataf et al. (2010). They are predominantly metal-rich, with the mode of their [Fe/H] distribution falling between -0 . 5 and 0 dex, following a radial gradient (Zoccali et al. 2008; Johnson et al. 2011, 2013; Gonzalez et al. 2012). However, there is a more metal-poor population of stars present in the bulge. More than 16,000 RR Lyrae stars have been found in the bulge so far by the OGLE-III survey (Soszy'nski et al. 2011), and the census is still limited to a fraction of the complete bulge area. Their metallicity distribution is centered around [Fe / H] = -1 (Pietrukowicz et al. 2012) and has a very narrow spread, suggesting that they trace a more ancient stellar population with respect to the RC stars.</text>
<text><location><page_2><loc_12><loc_15><loc_88><loc_24></location>In this Letter, we analyze the spatial distribution of RR Lyrae stars from the OGLE-III catalog (Soszy'nski et al. 2011) by combining their I -band optical data with the available VVV K s -band photometry. Our optical-NIR data set has some important advantages over optical studies. First, the PL relations have increasing precision and decreasing metallicity dependence towards longer NIR wavelengths (Catelan et al. 2004). The combined optical-</text>
<text><location><page_3><loc_12><loc_65><loc_88><loc_86></location>NIR data allow us to rely primarily on the highly precise K s -band relation - with the I -band one appearing only in the reddening estimation - and to leave out the very rough correlation between the absolute V -band magnitude and the metallicity from the analysis. This yields more precise distances by increasing the precision of both the absolute magnitudes and individual reddening values through the intrinsic color indices. Secondly, since NIR wavebands are much less sensitive to interstellar reddening than optical ones, and due to the wide wavelength separation between the I and K s bands, the uncertainties in the intrinsic colors of the stars will cause much smaller errors in the reddening values ( R K s ,I -K s << R I,V -I ), and in turn, the distances. Finally, the VVV K s -band average magnitudes are more accurate than the OGLE V -band ones because of the higher phase coverage and smaller amplitude of the light-curves - increasing the accuracy of the distance determination.</text>
<section_header_level_1><location><page_3><loc_20><loc_58><loc_80><loc_60></location>2. The VVV NIR Photometry of the OGLE RR Lyrae stars</section_header_level_1>
<text><location><page_3><loc_12><loc_39><loc_88><loc_56></location>Our study is based on VVV K s photometric data provided by the VISTA Data Flow System (VDFS, see Emerson et al. 2004), acquired between October 19, 2009, and September 28, 2012. We used magnitudes measured on single detector frame stacks (a.k.a. ' pawprints ') by aperture photometry, using small, flux-corrected apertures, suitable for moderately crowded stellar fields (Irwin et al. 2004). The number of observational epochs varies between ∼ 20 and 60, depending on the brightness and the location of the object. The limiting K s magnitude ranges from ∼ 18 . 0 to ∼ 16 . 5 mag (depending on the Galactic latitude), which is enough to detect RR Lyrae stars all the way through the Galaxy, even with high interstellar extinction.</text>
<text><location><page_3><loc_12><loc_20><loc_88><loc_37></location>We performed a positional cross-matching procedure between sources extracted from VVV images and the complete catalog of 11756 fundamental-mode RR Lyrae (type RRab) stars from the OGLE-III survey (Soszy'nski et al. 2011) using the STILTS 2 software. In order to rule out photometric contamination by seeing-dependent source merging, nearby saturated stars, spurious signals, etc., we excluded all data points with cross-matching separations larger than 0 . 35 '' ∼ 1 pixel (less than 10% of all data) or lying closer to the detector edges than the jitter offset, and only considered sources with morphological classifications of -1 , -2 or 1 (see Saito et al. 2012). The procedure resulted in our initial database of 9694 K s -band light-curves with OGLE-III I -band counterparts.</text>
<text><location><page_3><loc_12><loc_15><loc_88><loc_18></location>We computed precise mean K s -band magnitudes for each star by employing the template Fourier fitting method of Kov'acs & Kupi (2007), using linear scaling of 6th-order Fourier-</text>
<text><location><page_4><loc_12><loc_81><loc_88><loc_86></location>ums fitted to the template light-curves of Jones et al. (1996), and iteratively omitting outliers during the procedure. Fig. 1 illustrates our data quality showing a typical light-curve with the fitted template.</text>
<text><location><page_4><loc_12><loc_48><loc_88><loc_79></location>In order to reject light-curves dominated by noise or systematic variations, we rejected all light-curves with a √ N/σ < 25, where a is the total amplitude of the template fit, σ is the standard deviation of the residual, and N is the number of data points. The above limit was set by a visual inspection of ∼ 10% of all fitted light-curves, and resulted in the rejection of ∼ 20% of light-curves in our initial sample . For minimizing the presence of systematic errors in the computed distances, we required the light-curves to comply with the following set of further selection criteria. We rejected all light-curves having a phase coverage 3 less than 0 . 6, and σ > 0 . 1 mag. Furthermore, to exclude sources suspected to be affected by blending or fitting runaways, we omitted stars outside the amplitude threshold of 0 . 08 < a K s < 0 . 6. By applying these cuts, we obtained our final sample of 7663 RRab light-curves. Figure 2 shows the location of the these sources in comparison with the bulge area covered by the VVV Survey. The various cuts in the sample does not result in any bias in the ( l, b ) distribution of the objects, and yields an increased relative precision in the traced spatial distribution out to high distances. The cost of this gain is a slightly increased incompleteness towards the far side of the bulge, due to the preference for rejecting faint stars, since their light-curves are more noisy and have less data.</text>
<section_header_level_1><location><page_4><loc_32><loc_41><loc_68><loc_43></location>3. Distances to the RR Lyrae stars</section_header_level_1>
<text><location><page_4><loc_12><loc_22><loc_88><loc_39></location>We derived the absolute K s and I magnitudes of the stars using the theoretical periodluminosity-metallicity (PLZ) relations from Catelan et al. (2004), estimating the apparent equilibrium brightnesses of the stars with their intensity-averaged mean magnitudes obtained from the template-fitting procedure (see Sect. 2). Since the original K -band relation in the above study is given in the Glass system, first we transformed the K magnitudes for each and every individual synthetic star in the Catelan et al. (2004) synthetic horizontal-branch populations into the VISTA filter system. For this purpose, we used Eqs. (A1)-(A3) from Carpenter (2001) to transform first into the 2MASS system, and then the equations provided by CASU 4 for transforming these into the VISTA system. We then recomputed all the linear</text>
<text><location><page_5><loc_12><loc_84><loc_70><loc_86></location>regressions, leading to the following relation for the VISTA K s band:</text>
<formula><location><page_5><loc_31><loc_81><loc_88><loc_83></location>M K s = -0 . 6365 -2 . 347 log P +0 . 1747 log Z . (1)</formula>
<text><location><page_5><loc_12><loc_72><loc_88><loc_79></location>In estimating the log Z values of the RRab stars, we used the exact same approach as Pietrukowicz et al. (2012), applying the empirical P, φ 31 → [Fe / H] relation of Smolec (2005) with the periods and phase differences derived from the OGLE-III I -band light-curves (as provided by Soszy'nski et al. 2011).</text>
<text><location><page_5><loc_12><loc_67><loc_88><loc_70></location>We determined the color excess of each star by comparing its observed color index to the intrinsic one predicted by the theoretical PLZ relations:</text>
<formula><location><page_5><loc_34><loc_63><loc_88><loc_65></location>E( I -K s ) = ( I -K s ) -( M I -M K s ) . (2)</formula>
<text><location><page_5><loc_12><loc_53><loc_88><loc_61></location>For converting the color excess values into absolute extinction in the K s band, we adopted the reddening law of Cardelli et al. (1989) with R V = 3 . 1. Using their Eqs. (2a-b) and (3a-b) with an effective wavelength of λ eff = 0 . 798 µ m for the Cousins I -band and λ eff = 2 . 149 µ m for the VISTA K s -band, we obtained the following relation:</text>
<formula><location><page_5><loc_40><loc_50><loc_88><loc_52></location>A K s = 0 . 164 E ( I -K s ) . (3)</formula>
<text><location><page_5><loc_12><loc_43><loc_88><loc_48></location>We note that due to the large wavelength difference between the bands, the coefficient in this relation changes only marginally (to the value of 0 . 160) if we adopt R V = 2 . 5, as recently suggested by Nataf et al. (2013).</text>
<text><location><page_5><loc_12><loc_38><loc_88><loc_41></location>Finally, we calculated the individual distances to our bulge RRab sample using the relation</text>
<formula><location><page_5><loc_38><loc_35><loc_88><loc_37></location>log R = 1 + 0 . 2 ( K s, 0 -M K s ) , (4)</formula>
<text><location><page_5><loc_12><loc_33><loc_61><loc_34></location>where K s, 0 is the extinction-corrected K s -band magnitude.</text>
<text><location><page_5><loc_12><loc_10><loc_88><loc_31></location>The precision of the resulting distances to individual stars is significantly higher than in earlier studies due to the advantages of NIR photometry mentioned in Sect. 1. The typical accuracy of the average K s magnitudes obtained from the template fitting procedure is between 0 . 01 -0 . 02 mag, depending on the amplitude and the brightness of the star which is similar to the 0.02 mag nominal accuracy of the I -band average magnitudes. The internal dispersion of the photometric metallicity estimator is 0.18 dex (Smolec 2005), and its observational error is dominated by the error in φ 31 , which is determined by the quality of the I -band light-curves, and which is negligible for most of the stars in our sample, according to Pietrukowicz et al. (2012). The error in log Z propagates into the distance through both M K s and A ( K s ), but the latter gives only a marginal contribution due to the very low ratio of absolute to selective extinction. Assuming a median statistical error of 0.18 dex in log Z ,</text>
<text><location><page_6><loc_12><loc_75><loc_88><loc_86></location>and 0.02 mag for the error in the average magnitudes in both bands, we get a statistical error of only 0.01 mag in A ( K s ). The errors propagate into a statistical uncertainty of only 0.053 mag in the extinction-corrected distance moduli, which converts into a distance error of 0.2 kpc for a star at a distance of 8.5 kpc. This is less than half the median statistical error in the distances obtained from optical photometry, as quoted by Pietrukowicz et al. (2012).</text>
<text><location><page_6><loc_12><loc_48><loc_88><loc_73></location>In addition to the statistical errors discussed above, there might be a significant net systematic error in the distances, originating from various different sources. A detailed exploration of the possible sources of systematic errors is out of the scope of this Letter; here we merely use a relevant globular cluster study as reference. Coppola et al. (2011) provide a detailed comparison of various recent PL/PLZ relations through the distance determination of the globular cluster M5, based on a sizable sample of RR Lyrae stars; and the resulting distance moduli from different empirical and theoretical relations are within 1 σ agreement, showing a high reliability of the method. Since this cluster is located at a similar distance as the Galactic Center, the spread of 0.07 mag in the distance moduli obtained from 4 different relations for RRab stars in their study is highly indicative of the possible mean systematic error in our analysis. Therefore, we adopted a mean systematic distance error of 0.14 kpc, corresponding to the same error in the distance modulus at the distance of the GC (see Sect 4).</text>
<section_header_level_1><location><page_6><loc_30><loc_41><loc_70><loc_43></location>4. The Distance to the Galactic Center</section_header_level_1>
<text><location><page_6><loc_12><loc_16><loc_88><loc_39></location>We determined the center of the RR Lyrae distribution by first projecting the individual distances to the Galactic plane, and then correcting the distance distribution for the distorting effect of our sample being distributed in a solid angle, thus observing more objects at larger distances (i.e., we scaled the distribution by R -2 ). Figure 3 shows the histogram of the projected distances of all RRab stars in our final sample based on the combined IK s photometry, before and after correcting for the aforementioned 'cone-effect'. We measured the center of the RR Lyrae spatial distribution (assumed to coincide with the Galactic Center, R 0 ) by determining the center of the density function of the best-fitting analytical distribution. We found that the distance distribution shows a very significant non-normality due to its heavy tails, and can be best represented by the Student's t -distribution. This is consistent with the result of Pietrukowicz et al. (2012), who found that the Cauchy-Lorentz 5 distribution fits best the distances obtained from the optical data.</text>
<figure>
<location><page_7><loc_19><loc_62><loc_80><loc_86></location>
<caption>Fig. 1.- VVV K s -band light-curve of a bulge RRab star with the fitted template. The identifier and period of the star are shown in the header.</caption>
</figure>
<figure>
<location><page_7><loc_20><loc_19><loc_80><loc_53></location>
<caption>Fig. 2.- Galactic coordinates of our final sample of 7663 OGLE-III RRab stars (dots) in the bulge area of the VVV Survey. The VVV bulge subfields are shown by an overlaying grid. The grayscale background shows the interstellar extinction map of Schlegel et al. (1998).</caption>
</figure>
<text><location><page_8><loc_12><loc_50><loc_88><loc_86></location>The distribution of RR Lyrae distances shows a natural asymmetry towards higher distances due to the 'cone-effect'. In our case, the distribution is further perturbed by our rejection procedure (see Sect. 2), resulting in a slight incompleteness at higher distances (since the light-curves of fainter stars have generally lower quality). The non-symmetry of the observed distance distribution has to be taken into account in the distribution fitting, in order not to bias the measurement of R 0 . We fitted a skewed Student's t -distribution to the projected distances of our final sample by likelihood inference (see Azzalini & Capitanio 2003). Using the fitted parameters, we evaluated the corresponding density function on a dense grid of quantiles, and corrected for the 'cone-effect' by scaling the obtained synthetic density distribution with R -2 . Then we determined R 0 by numerically computing the location of the maximum of the corrected synthetic distribution. The fitted density curves before and after scaling are shown in Fig. 3. We obtained a value of R 0 = 8 . 33 ± 0 . 05 ± 0 . 14 kpc for the distance to the Galactic Center, where the first error is purely statistical, representing the standard error in the location parameter of the fitted distribution, and the second one is the estimated systematic error discussed in Sect. 3. Our measurement of R 0 is consistent with recent literature values obtained from various other methods (e.g., Genzel et al. 2010), and we note the remarkable agreement with the recent result of 8 . 33 ± 0 . 35 kpc 6 , obtained by Gillessen et al. (2009) from monitoring stellar orbits around the central black hole.</text>
<section_header_level_1><location><page_8><loc_28><loc_44><loc_72><loc_46></location>5. The spatial distribution of RR Lyrae stars</section_header_level_1>
<text><location><page_8><loc_12><loc_17><loc_88><loc_42></location>The distribution of RC giants in the bulge is known to be barred (Stanek et al. 1997; Cao et al. 2013). Figure 4 shows the spatial distribution of the RRab stars obtained in the present study at various latitudinal stripes at negative latitudes, where our sample has sufficiently high number density. The stars show a centrally concentrated distribution at all latitudes. Figure 4a shows all stars in our final sample - note that the elongation along the line of sight (LOS) is an observational bias due to our incomplete longitudinal coverage. It already implies, however, that the RR Lyrae stars do not trace an inclined prominent bar like the RC stars do. Figures 4b-e show latitudinal subsets of the distribution. The longitudinal coverage between l = ± 5 · is very close to contiguous at -4 . 5 · < b < -2 . 5 · and fully contiguous at -4 · < b < -3 · (cf. Fig. 2), therefore the distribution is free of any significant bias between these longitudes. The longitudinal ranges of contiguous coverage are more confined at higher southern latitudes, but should still be free from a significant bias between l = +2 . 5 · and -5 · approximately (Figs. 4d,e).</text>
<figure>
<location><page_9><loc_20><loc_36><loc_80><loc_71></location>
<caption>Fig. 3.- Histogram of the projected distances of RRab stars in our final sample . Empty bars show the histogram of the measured distances, gray bars denote the histogram corrected for the 'cone-effect' (see text). The thin and thick lines show the density function of the skewed Student's t -distribution fitted to the observed distances, before and after applying the correction for the 'cone-effect', respectively.</caption>
</figure>
<text><location><page_10><loc_12><loc_61><loc_88><loc_86></location>In Figs. 4b-e, we compare the spatial distributions with the observed orientation of the Galactic bar traced by RC stars at different southern latitudes (data are taken from Gonzalez et al. 2011, 2012, see their Figs. 3 and 10, respectively). It is immediately visible that the spatial distribution of RR Lyrae stars is not significantly barred at any of these latitudes. At latitudes -4 . 5 · < b < -2 . 5 · , the distribution shows a slight elongation within the inner 1 kpc. This small substructure is inclined by only -12 . 5 · ± 0 . 5 · with respect to the LOS towards the GC, obtained by fitting elliptical centroids to the distribution with 0.2 kpc binning over a grid of small (X,Y) offsets of the bin positions. Our findings are in contrast with the results of Pietrukowicz et al. (2012), that showed a more inclined and longer bar-like substructure at certain negative latitudes based on the study of V, I optical data from the OGLE-III survey. We note again however, that our combined optical-NIR data has superior statistical quality due to the various arguments discussed in Sect. 3, providing extinction and distance values with significantly higher relative precision.</text>
<text><location><page_10><loc_12><loc_41><loc_88><loc_59></location>The spatial distribution at higher southern latitudes ( b < -5 · ) has a significantly wider peak between distances of 7.7 and 9.7 kpc, approximately. It also shows a hint of bimodality, having a secondary maximum in the number density distribution at a distance of about 9 . 6 kpc. It is somewhat resemblant of the split distribution we see in the case of RC stars due to the split arms of the X-shaped structure towards more negative latitudes. However, the dip test of Hartigan & Hartigan (1985), indicates only marginally significant bimodality (2 σ ). Unfortunately, the current census does not extend to sufficiently high southern latitudes, preventing us from getting a true picture on this possible substructure; a larger sample of RR Lyrae from a more extended bulge area is required.</text>
<section_header_level_1><location><page_10><loc_42><loc_35><loc_58><loc_37></location>6. Conclusions</section_header_level_1>
<text><location><page_10><loc_12><loc_16><loc_88><loc_33></location>We combined NIR (VVV) and optical (OGLE-III) photometric data in order to measure precise reddenings and distances for bulge RR Lyrae stars with high-quality light-curves. Our data are only little affected by the large and non-uniform interstellar extinction in these regions, allowing us to reach higher precision with respect to optical studies. We derive a distance to the Galactic Center of R 0 = 8 . 33 ± 0 . 05 ± 0 . 14 kpc. We find that the distribution of fundamental-mode RR Lyrae stars is different from the known distribution of RC giants. The most striking fact is that the RR Lyrae stars do not show a strong bar in the direction of the Galactic bulge as do the clump giants, but their distribution is more spherical, with only the central ∼ 1 kpc showing a bar-like substructure.</text>
<text><location><page_10><loc_12><loc_11><loc_88><loc_14></location>In recent years, the structure of the bulge has been discovered to be more complex than just a simple bar. It is now well-established that the predominantly metal-rich bulge RC</text>
<figure>
<location><page_11><loc_31><loc_34><loc_67><loc_83></location>
<caption>Fig. 4.- Number density diagrams showing the spatial distribution of the RRab stars in our final sample in various latitudinal stripes, projected onto the Galactic plane. The corresponding latitudinal ranges (in degrees) and the number of stars within them are shown in the bottom left and top right corners of each panel, respectively. Dotted lines denote lines of sight corresponding to l = ± 5 · and ± 10 · , the solid lines in panels d and e mark the upper limit of contiguous areal coverage in the corresponding stripes (see text). Crosses connected with broken lines in panels b-e show the observed location of the long bar at b = -1 · ( b, c ) and b = -5 · ( d, e ) as found by Gonzalez et al. (2011, 2012) based on RC distances. The bin size is 100 kpc in panel a and 200 kpc in panels b-e .</caption>
</figure>
<text><location><page_12><loc_12><loc_67><loc_88><loc_86></location>stars trace an X-shaped structure (McWilliam & Zoccali 2010; Nataf et al. 2010; Saito et al. 2011; Gonzalez et al. 2012). Recently, Ness et al. (2012, 2013) found that this structure disappears from the RC distribution at low metallicities. The observed prominent difference between the 3-D structures traced by the RC and metal-poor RR Lyrae stars is in line with their findings, and suggests that the Milky Way has a composite bulge, retaining an older, more spheroidal bulge component (see, e.g. Samland & Gerhard 2003; Obreja et al. 2013). The existence of a small, central bar-like structure in such a component like the one traced by the RR Lyrae stars is consistent with current numerical simulations of composite bulges, and is a result of the angular momentum transfer between the bar and the initial classical bulge during their co-evolution (Saha & Gerhard 2013).</text>
<text><location><page_12><loc_12><loc_50><loc_88><loc_63></location>We gratefully acknowledge the use of data from the ESO Public Survey program ID 179.B-2002 taken with the VISTA telescope, data products from the Cambridge Astronomical Survey Unit, funding from the BASAL CATA Center for Astrophysics and Associated Technologies through grant PFB-06, and the Ministry for the Economy, Development, and Tourism's Programa Iniciativa Cient'ıfica Milenio through grant P07-021-F, awarded to The Milky Way Millennium Nucleus. We acknowledge the support of FONDECYT Regular grants No. 1130196 (to D.M.), 1110326 (to I.D. and M.C.), and 1110393 (to M.Z.).</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "We have combined optical and near-infrared data of known RR Lyrae (RRL) stars in the bulge in order to study the spatial distribution of its metal-poor component by measuring precise reddening values and distances of 7663 fundamentalmode RRL stars with high-quality photometry. We obtain a distance to the Galactic center of R 0 = 8 . 33 ± 0 . 05 ± 0 . 14 kpc. We find that the spatial distribution of the RRL stars differs from the structures traced by the predominantly metal-rich red clump (RC) stars. Unlike the RC stars, the RRL stars do not trace a strong bar, but have a more spheroidal, centrally concentrated distribution, showing only a slight elongation in its very center. We find a hint of bimodality in the density distribution at high southern latitudes ( b < -5 · ), which needs to be confirmed by extending the areal coverage of the current census. The different spatial distributions of the metal-rich and metal-poor stellar populations suggest that the Milky Way has a composite bulge. Subject headings: stars: variables: RR Lyrae - Galaxy: bulge - Galaxy: stellar content - Galaxy: structure",
"pages": [
1,
2
]
},
{
"title": "Distribution of the Metal-Poor Population",
"content": "I. D'ek'any 1 , 4 , D. Minniti 1 , 4 , 5 , M. Catelan 1 , 4 , M. Zoccali 1 , 4 , R. K. Saito 2 , M. Hempel 1 , 4 , O. A. Gonzalez 3 1 Instituto de Astrof'ısica, Facultad de F'ısica, Pontificia Universidad Cat'olica de Chile, Av. Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, Chile 2 Departamento de F'ısica, Universidade Federal de Sergipe, Av. Marechal Rondon s/n, 49100-000, S˜ao Crist'ov˜ao, SE, Brazil",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "RR Lyrae stars are easily identifiable primary distance indicators, which can be used to trace the structure of the Milky Way. They sample an old and metal-poor population, and are particularly useful in the reddened regions of the Galactic bulge because of several reasons: (i) they have a relatively high number density in the bulge (e.g., Soszy'nski et al. 2011); (ii) they follow precise period-luminosity (PL) relations in the near-infrared (NIR; e.g., Catelan et al. 2004); (iii) their PL relations allow the determination of the interstellar extinction on a star-by-star basis; (iv) their metallicities can be estimated using photometric data alone (e.g., Jurcsik & Kov'acs 1996; Smolec 2005); and (v) their brightness and amplitude (0 . 1 glyph[lessorsimilar] a K s glyph[lessorsimilar] 0 . 5 mag) make them detectable out to large distances. The VISTA Variables in the V'ıa L'actea (VVV) ESO Public Survey (Minniti et al. 2010; Catelan et al. 2011; Saito et al. 2012) aims to probe the 3-dimensional (3D) structure of the Galactic bulge using various distance indicators. In a series of papers, the VVV ZY JHK s color atlas was used to trace the spatial distribution of red clump (RC) giants (Saito et al. 2011; Minniti et al. 2011; Gonzalez et al. 2011), and to provide photometric reddening and metallicity maps 1 of the bulge (Gonzalez et al. 2011, 2012, 2013; Chen et al. 2013). The RC stars clearly trace a complex, X-shaped distribution, as was first suggested by McWilliam & Zoccali (2010) and Nataf et al. (2010). They are predominantly metal-rich, with the mode of their [Fe/H] distribution falling between -0 . 5 and 0 dex, following a radial gradient (Zoccali et al. 2008; Johnson et al. 2011, 2013; Gonzalez et al. 2012). However, there is a more metal-poor population of stars present in the bulge. More than 16,000 RR Lyrae stars have been found in the bulge so far by the OGLE-III survey (Soszy'nski et al. 2011), and the census is still limited to a fraction of the complete bulge area. Their metallicity distribution is centered around [Fe / H] = -1 (Pietrukowicz et al. 2012) and has a very narrow spread, suggesting that they trace a more ancient stellar population with respect to the RC stars. In this Letter, we analyze the spatial distribution of RR Lyrae stars from the OGLE-III catalog (Soszy'nski et al. 2011) by combining their I -band optical data with the available VVV K s -band photometry. Our optical-NIR data set has some important advantages over optical studies. First, the PL relations have increasing precision and decreasing metallicity dependence towards longer NIR wavelengths (Catelan et al. 2004). The combined optical- NIR data allow us to rely primarily on the highly precise K s -band relation - with the I -band one appearing only in the reddening estimation - and to leave out the very rough correlation between the absolute V -band magnitude and the metallicity from the analysis. This yields more precise distances by increasing the precision of both the absolute magnitudes and individual reddening values through the intrinsic color indices. Secondly, since NIR wavebands are much less sensitive to interstellar reddening than optical ones, and due to the wide wavelength separation between the I and K s bands, the uncertainties in the intrinsic colors of the stars will cause much smaller errors in the reddening values ( R K s ,I -K s << R I,V -I ), and in turn, the distances. Finally, the VVV K s -band average magnitudes are more accurate than the OGLE V -band ones because of the higher phase coverage and smaller amplitude of the light-curves - increasing the accuracy of the distance determination.",
"pages": [
2,
3
]
},
{
"title": "2. The VVV NIR Photometry of the OGLE RR Lyrae stars",
"content": "Our study is based on VVV K s photometric data provided by the VISTA Data Flow System (VDFS, see Emerson et al. 2004), acquired between October 19, 2009, and September 28, 2012. We used magnitudes measured on single detector frame stacks (a.k.a. ' pawprints ') by aperture photometry, using small, flux-corrected apertures, suitable for moderately crowded stellar fields (Irwin et al. 2004). The number of observational epochs varies between ∼ 20 and 60, depending on the brightness and the location of the object. The limiting K s magnitude ranges from ∼ 18 . 0 to ∼ 16 . 5 mag (depending on the Galactic latitude), which is enough to detect RR Lyrae stars all the way through the Galaxy, even with high interstellar extinction. We performed a positional cross-matching procedure between sources extracted from VVV images and the complete catalog of 11756 fundamental-mode RR Lyrae (type RRab) stars from the OGLE-III survey (Soszy'nski et al. 2011) using the STILTS 2 software. In order to rule out photometric contamination by seeing-dependent source merging, nearby saturated stars, spurious signals, etc., we excluded all data points with cross-matching separations larger than 0 . 35 '' ∼ 1 pixel (less than 10% of all data) or lying closer to the detector edges than the jitter offset, and only considered sources with morphological classifications of -1 , -2 or 1 (see Saito et al. 2012). The procedure resulted in our initial database of 9694 K s -band light-curves with OGLE-III I -band counterparts. We computed precise mean K s -band magnitudes for each star by employing the template Fourier fitting method of Kov'acs & Kupi (2007), using linear scaling of 6th-order Fourier- ums fitted to the template light-curves of Jones et al. (1996), and iteratively omitting outliers during the procedure. Fig. 1 illustrates our data quality showing a typical light-curve with the fitted template. In order to reject light-curves dominated by noise or systematic variations, we rejected all light-curves with a √ N/σ < 25, where a is the total amplitude of the template fit, σ is the standard deviation of the residual, and N is the number of data points. The above limit was set by a visual inspection of ∼ 10% of all fitted light-curves, and resulted in the rejection of ∼ 20% of light-curves in our initial sample . For minimizing the presence of systematic errors in the computed distances, we required the light-curves to comply with the following set of further selection criteria. We rejected all light-curves having a phase coverage 3 less than 0 . 6, and σ > 0 . 1 mag. Furthermore, to exclude sources suspected to be affected by blending or fitting runaways, we omitted stars outside the amplitude threshold of 0 . 08 < a K s < 0 . 6. By applying these cuts, we obtained our final sample of 7663 RRab light-curves. Figure 2 shows the location of the these sources in comparison with the bulge area covered by the VVV Survey. The various cuts in the sample does not result in any bias in the ( l, b ) distribution of the objects, and yields an increased relative precision in the traced spatial distribution out to high distances. The cost of this gain is a slightly increased incompleteness towards the far side of the bulge, due to the preference for rejecting faint stars, since their light-curves are more noisy and have less data.",
"pages": [
3,
4
]
},
{
"title": "3. Distances to the RR Lyrae stars",
"content": "We derived the absolute K s and I magnitudes of the stars using the theoretical periodluminosity-metallicity (PLZ) relations from Catelan et al. (2004), estimating the apparent equilibrium brightnesses of the stars with their intensity-averaged mean magnitudes obtained from the template-fitting procedure (see Sect. 2). Since the original K -band relation in the above study is given in the Glass system, first we transformed the K magnitudes for each and every individual synthetic star in the Catelan et al. (2004) synthetic horizontal-branch populations into the VISTA filter system. For this purpose, we used Eqs. (A1)-(A3) from Carpenter (2001) to transform first into the 2MASS system, and then the equations provided by CASU 4 for transforming these into the VISTA system. We then recomputed all the linear regressions, leading to the following relation for the VISTA K s band: In estimating the log Z values of the RRab stars, we used the exact same approach as Pietrukowicz et al. (2012), applying the empirical P, φ 31 → [Fe / H] relation of Smolec (2005) with the periods and phase differences derived from the OGLE-III I -band light-curves (as provided by Soszy'nski et al. 2011). We determined the color excess of each star by comparing its observed color index to the intrinsic one predicted by the theoretical PLZ relations: For converting the color excess values into absolute extinction in the K s band, we adopted the reddening law of Cardelli et al. (1989) with R V = 3 . 1. Using their Eqs. (2a-b) and (3a-b) with an effective wavelength of λ eff = 0 . 798 µ m for the Cousins I -band and λ eff = 2 . 149 µ m for the VISTA K s -band, we obtained the following relation: We note that due to the large wavelength difference between the bands, the coefficient in this relation changes only marginally (to the value of 0 . 160) if we adopt R V = 2 . 5, as recently suggested by Nataf et al. (2013). Finally, we calculated the individual distances to our bulge RRab sample using the relation where K s, 0 is the extinction-corrected K s -band magnitude. The precision of the resulting distances to individual stars is significantly higher than in earlier studies due to the advantages of NIR photometry mentioned in Sect. 1. The typical accuracy of the average K s magnitudes obtained from the template fitting procedure is between 0 . 01 -0 . 02 mag, depending on the amplitude and the brightness of the star which is similar to the 0.02 mag nominal accuracy of the I -band average magnitudes. The internal dispersion of the photometric metallicity estimator is 0.18 dex (Smolec 2005), and its observational error is dominated by the error in φ 31 , which is determined by the quality of the I -band light-curves, and which is negligible for most of the stars in our sample, according to Pietrukowicz et al. (2012). The error in log Z propagates into the distance through both M K s and A ( K s ), but the latter gives only a marginal contribution due to the very low ratio of absolute to selective extinction. Assuming a median statistical error of 0.18 dex in log Z , and 0.02 mag for the error in the average magnitudes in both bands, we get a statistical error of only 0.01 mag in A ( K s ). The errors propagate into a statistical uncertainty of only 0.053 mag in the extinction-corrected distance moduli, which converts into a distance error of 0.2 kpc for a star at a distance of 8.5 kpc. This is less than half the median statistical error in the distances obtained from optical photometry, as quoted by Pietrukowicz et al. (2012). In addition to the statistical errors discussed above, there might be a significant net systematic error in the distances, originating from various different sources. A detailed exploration of the possible sources of systematic errors is out of the scope of this Letter; here we merely use a relevant globular cluster study as reference. Coppola et al. (2011) provide a detailed comparison of various recent PL/PLZ relations through the distance determination of the globular cluster M5, based on a sizable sample of RR Lyrae stars; and the resulting distance moduli from different empirical and theoretical relations are within 1 σ agreement, showing a high reliability of the method. Since this cluster is located at a similar distance as the Galactic Center, the spread of 0.07 mag in the distance moduli obtained from 4 different relations for RRab stars in their study is highly indicative of the possible mean systematic error in our analysis. Therefore, we adopted a mean systematic distance error of 0.14 kpc, corresponding to the same error in the distance modulus at the distance of the GC (see Sect 4).",
"pages": [
4,
5,
6
]
},
{
"title": "4. The Distance to the Galactic Center",
"content": "We determined the center of the RR Lyrae distribution by first projecting the individual distances to the Galactic plane, and then correcting the distance distribution for the distorting effect of our sample being distributed in a solid angle, thus observing more objects at larger distances (i.e., we scaled the distribution by R -2 ). Figure 3 shows the histogram of the projected distances of all RRab stars in our final sample based on the combined IK s photometry, before and after correcting for the aforementioned 'cone-effect'. We measured the center of the RR Lyrae spatial distribution (assumed to coincide with the Galactic Center, R 0 ) by determining the center of the density function of the best-fitting analytical distribution. We found that the distance distribution shows a very significant non-normality due to its heavy tails, and can be best represented by the Student's t -distribution. This is consistent with the result of Pietrukowicz et al. (2012), who found that the Cauchy-Lorentz 5 distribution fits best the distances obtained from the optical data. The distribution of RR Lyrae distances shows a natural asymmetry towards higher distances due to the 'cone-effect'. In our case, the distribution is further perturbed by our rejection procedure (see Sect. 2), resulting in a slight incompleteness at higher distances (since the light-curves of fainter stars have generally lower quality). The non-symmetry of the observed distance distribution has to be taken into account in the distribution fitting, in order not to bias the measurement of R 0 . We fitted a skewed Student's t -distribution to the projected distances of our final sample by likelihood inference (see Azzalini & Capitanio 2003). Using the fitted parameters, we evaluated the corresponding density function on a dense grid of quantiles, and corrected for the 'cone-effect' by scaling the obtained synthetic density distribution with R -2 . Then we determined R 0 by numerically computing the location of the maximum of the corrected synthetic distribution. The fitted density curves before and after scaling are shown in Fig. 3. We obtained a value of R 0 = 8 . 33 ± 0 . 05 ± 0 . 14 kpc for the distance to the Galactic Center, where the first error is purely statistical, representing the standard error in the location parameter of the fitted distribution, and the second one is the estimated systematic error discussed in Sect. 3. Our measurement of R 0 is consistent with recent literature values obtained from various other methods (e.g., Genzel et al. 2010), and we note the remarkable agreement with the recent result of 8 . 33 ± 0 . 35 kpc 6 , obtained by Gillessen et al. (2009) from monitoring stellar orbits around the central black hole.",
"pages": [
6,
8
]
},
{
"title": "5. The spatial distribution of RR Lyrae stars",
"content": "The distribution of RC giants in the bulge is known to be barred (Stanek et al. 1997; Cao et al. 2013). Figure 4 shows the spatial distribution of the RRab stars obtained in the present study at various latitudinal stripes at negative latitudes, where our sample has sufficiently high number density. The stars show a centrally concentrated distribution at all latitudes. Figure 4a shows all stars in our final sample - note that the elongation along the line of sight (LOS) is an observational bias due to our incomplete longitudinal coverage. It already implies, however, that the RR Lyrae stars do not trace an inclined prominent bar like the RC stars do. Figures 4b-e show latitudinal subsets of the distribution. The longitudinal coverage between l = ± 5 · is very close to contiguous at -4 . 5 · < b < -2 . 5 · and fully contiguous at -4 · < b < -3 · (cf. Fig. 2), therefore the distribution is free of any significant bias between these longitudes. The longitudinal ranges of contiguous coverage are more confined at higher southern latitudes, but should still be free from a significant bias between l = +2 . 5 · and -5 · approximately (Figs. 4d,e). In Figs. 4b-e, we compare the spatial distributions with the observed orientation of the Galactic bar traced by RC stars at different southern latitudes (data are taken from Gonzalez et al. 2011, 2012, see their Figs. 3 and 10, respectively). It is immediately visible that the spatial distribution of RR Lyrae stars is not significantly barred at any of these latitudes. At latitudes -4 . 5 · < b < -2 . 5 · , the distribution shows a slight elongation within the inner 1 kpc. This small substructure is inclined by only -12 . 5 · ± 0 . 5 · with respect to the LOS towards the GC, obtained by fitting elliptical centroids to the distribution with 0.2 kpc binning over a grid of small (X,Y) offsets of the bin positions. Our findings are in contrast with the results of Pietrukowicz et al. (2012), that showed a more inclined and longer bar-like substructure at certain negative latitudes based on the study of V, I optical data from the OGLE-III survey. We note again however, that our combined optical-NIR data has superior statistical quality due to the various arguments discussed in Sect. 3, providing extinction and distance values with significantly higher relative precision. The spatial distribution at higher southern latitudes ( b < -5 · ) has a significantly wider peak between distances of 7.7 and 9.7 kpc, approximately. It also shows a hint of bimodality, having a secondary maximum in the number density distribution at a distance of about 9 . 6 kpc. It is somewhat resemblant of the split distribution we see in the case of RC stars due to the split arms of the X-shaped structure towards more negative latitudes. However, the dip test of Hartigan & Hartigan (1985), indicates only marginally significant bimodality (2 σ ). Unfortunately, the current census does not extend to sufficiently high southern latitudes, preventing us from getting a true picture on this possible substructure; a larger sample of RR Lyrae from a more extended bulge area is required.",
"pages": [
8,
10
]
},
{
"title": "6. Conclusions",
"content": "We combined NIR (VVV) and optical (OGLE-III) photometric data in order to measure precise reddenings and distances for bulge RR Lyrae stars with high-quality light-curves. Our data are only little affected by the large and non-uniform interstellar extinction in these regions, allowing us to reach higher precision with respect to optical studies. We derive a distance to the Galactic Center of R 0 = 8 . 33 ± 0 . 05 ± 0 . 14 kpc. We find that the distribution of fundamental-mode RR Lyrae stars is different from the known distribution of RC giants. The most striking fact is that the RR Lyrae stars do not show a strong bar in the direction of the Galactic bulge as do the clump giants, but their distribution is more spherical, with only the central ∼ 1 kpc showing a bar-like substructure. In recent years, the structure of the bulge has been discovered to be more complex than just a simple bar. It is now well-established that the predominantly metal-rich bulge RC stars trace an X-shaped structure (McWilliam & Zoccali 2010; Nataf et al. 2010; Saito et al. 2011; Gonzalez et al. 2012). Recently, Ness et al. (2012, 2013) found that this structure disappears from the RC distribution at low metallicities. The observed prominent difference between the 3-D structures traced by the RC and metal-poor RR Lyrae stars is in line with their findings, and suggests that the Milky Way has a composite bulge, retaining an older, more spheroidal bulge component (see, e.g. Samland & Gerhard 2003; Obreja et al. 2013). The existence of a small, central bar-like structure in such a component like the one traced by the RR Lyrae stars is consistent with current numerical simulations of composite bulges, and is a result of the angular momentum transfer between the bar and the initial classical bulge during their co-evolution (Saha & Gerhard 2013). We gratefully acknowledge the use of data from the ESO Public Survey program ID 179.B-2002 taken with the VISTA telescope, data products from the Cambridge Astronomical Survey Unit, funding from the BASAL CATA Center for Astrophysics and Associated Technologies through grant PFB-06, and the Ministry for the Economy, Development, and Tourism's Programa Iniciativa Cient'ıfica Milenio through grant P07-021-F, awarded to The Milky Way Millennium Nucleus. We acknowledge the support of FONDECYT Regular grants No. 1130196 (to D.M.), 1110326 (to I.D. and M.C.), and 1110393 (to M.Z.).",
"pages": [
10,
12
]
},
{
"title": "REFERENCES",
"content": "Azzalini, A., & Capitanio, A. 2003, J. R. Stat. Soc. Series B, 65, 367 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245 Carpenter, J. M. 2001, AJ, 121, 2851 Cao, L., Mao, S., Nataf, D., Rattenbury, N. J., & Gould, A. 2013, arXiv:1303.6430 Catelan, M., Pritzl, B. J., & Smith, H. A. 2004, ApJS, 154, 633 Catelan, M., Minniti, D., Lucas, P. W., et al. 2011, RR Lyrae Stars, Metal-Poor Stars, and the Galaxy, 145 Chen, B. Q., Schultheis, M., Jiang, B. W., et al. 2013, A&A, 550, A42 Coppola, G., Dall'Ora, M., Ripepi, V., et al. 2011, MNRAS, 416, 1056 Emerson, J. P., Irwin, M. J., Lewis, J., et al. 2004, Proc. SPIE, 5493, 401 Zoccali, M., Hill, V., Lecureur, A., et al. 2008, A&A, 486, 177",
"pages": [
12,
14
]
}
]
2013ApJ...776L..38F
https://arxiv.org/pdf/1309.5370.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_81><loc_82><loc_85></location>A Significantly Low CO Abundance Toward the TW Hya Protoplanetary Disk: A Path to Active Carbon Chemistry?</section_header_level_1>
<text><location><page_1><loc_15><loc_73><loc_80><loc_78></location>C'ecile Favre, L. Ilsedore Cleeves, Edwin A. Bergin, Department of Astronomy, University of Michigan, 500 Church St., Ann Arbor, MI 48109 cfavre@umich.edu</text>
<section_header_level_1><location><page_1><loc_42><loc_69><loc_53><loc_70></location>Chunhua Qi,</section_header_level_1>
<text><location><page_1><loc_16><loc_65><loc_79><loc_68></location>Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 and</text>
<section_header_level_1><location><page_1><loc_40><loc_61><loc_55><loc_62></location>Geoffrey A. Blake</section_header_level_1>
<text><location><page_1><loc_9><loc_58><loc_86><loc_61></location>California Institute of Technology, Division of Geological & Planetary Sciences, MS 150-21, Pasadena, CA 91125, USA</text>
<section_header_level_1><location><page_1><loc_41><loc_51><loc_54><loc_52></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_13><loc_32><loc_82><loc_50></location>In this Letter we report the CO abundance relative to H 2 derived toward the circumstellar disk of the T-Tauri star TW Hya from the HD (1 -0) and C 18 O (2 -1) emission lines. The HD (1 -0) line was observed by the Herschel Space Observatory Photodetector Array Camera and Spectrometer whereas C 18 O (2 -1) observations were carried out with the Submillimeter Array at a spatial resolution of 2 . '' 8 × 1 . '' 9 (corresponding to ∼ 142 × 97 AU). In the disk's warm molecular layer ( T > 20 K) we measure a disk-averaged gas-phase CO abundance relative to H 2 of χ (CO) = (0 . 1 -3) × 10 -5 , substantially lower than the canonical value of χ (CO) = 10 -4 . We infer that the best explanation of this low χ (CO) is the chemical destruction of CO followed by rapid formation of carbon chains, or perhaps CO 2 , that can subsequently freeze-out, resulting in the bulk mass of carbon locked up in ice grain mantles and oxygen in water. As a consequence of this likely time-dependent carbon sink mechanism, CO may be an unreliable tracer of H 2 gas mass.</text>
<text><location><page_1><loc_13><loc_30><loc_78><loc_31></location>Subject headings: protoplanetary disks - astrochemistry - ISM: abundances - stars: formation</text>
<section_header_level_1><location><page_1><loc_9><loc_27><loc_23><loc_29></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_10><loc_45><loc_26></location>Molecular hydrogen is the main gas-phase constituent in star-forming gas. However, it does not appreciably emit for typical gas conditions. Consequently carbon monoxide is widely used as a proxy for H 2 in the molecular interstellar medium (e.g. Dickman 1978) and protoplanetary disks (Koerner & Sargent 1995; Dutrey et al. 1996). With a suite of transitions at millimeter/submillimeter wavelengths, the optically thick and thermalized 12 CO lines trace gas temperature while optically thin CO isotopologues (namely</text>
<text><location><page_1><loc_51><loc_12><loc_86><loc_29></location>13 CO and C 18 O) probe the CO column and hence molecular mass. A key component of the latter calculation is the calibration of CO to H 2 , assuming an abundance of carbon monoxide, χ (CO). In the ISM this factor can be constrained via comparisons of dust extinction to measurements of optically thin isotopologue lines. Ripple et al. (2013) showed that typical 13 CO abundances range from ∼ 1 -3 × 10 -6 in several clouds. This corresponds to a CO abundance of ∼ 0 . 6 -2 × 10 -4 , assuming an isotopic ratio of 60.</text>
<text><location><page_1><loc_53><loc_11><loc_86><loc_12></location>Since the dense ISM provides CO to the pro-</text>
<text><location><page_2><loc_9><loc_70><loc_45><loc_86></location>toplanetary disk during its formation, it is reasonable to assume that χ (CO) in disks is similar to its interstellar value. Furthermore, at such high abundances, CO would represent the main gas-phase reservoir of carbon in disks. Spatially resolved observations of CO could thus be used to determine the distribution and abundance of volatile carbon, which has implications for the inclusion of carbon into planetary systems (Lee et al. 2010; Bond et al. 2010; Oberg et al. 2011).</text>
<text><location><page_2><loc_9><loc_51><loc_45><loc_69></location>In this paper we combine spatially integrated observations of optically thin C 18 O emission with a detection of the fundamental rotational transition of hydrogen deuteride, HD, towards the closest T-Tauri system, TW Hya, at 51 pc (Mamajek 2005). HD emission provides a separate probe of H 2 (Bergin et al. 2013, hereafter B13), with which we measure the χ (CO) in this system. We show that the main reservoir of gas-phase carbon, CO, is substantially reduced ( < 10% remaining) in the warm ( > 20 K) molecular layers of the disk and discuss implications of this result.</text>
<section_header_level_1><location><page_2><loc_9><loc_48><loc_40><loc_50></location>2. Observations and data reduction</section_header_level_1>
<text><location><page_2><loc_9><loc_19><loc_45><loc_47></location>The observations of TW Hya were made on 2005 February 27 and April 10 using the Submillimeter Array 1 (SMA, Ho et al. 2004) located atop Mauna Kea, Hawaii. The SMA receivers operated in a double-sideband mode with an intermediate frequency (IF) band of 4-6 GHz from the local oscillator frequency, sent over fiber optic transmission lines to 24 overlapping 'chunks' of the digital correlator. The correlator was configured to include CO, 13 CO and C 18 O, in one setting: the tuning was centered on the CO (2 -1) line at 230.538 GHz in chunk S15, while the 13 CO/C 18 O (2 -1) transitions at 220.399/219.560 GHz were simultaneously observed in chunks 12 and 22, respectively. CO (2 -1) data were reported in Qi et al. (2006). Combinations of two array configurations (compact and extended) were used to obtain projected baselines ranging from 6 to 180 m. The observing</text>
<text><location><page_2><loc_51><loc_64><loc_86><loc_86></location>loops used J1037-295 as the gain calibrator, with bandpass calibration using observations of 3C279. Flux calibration was done using observations of Titan and Callisto. Routine calibration tasks were performed using the MIR software package 2 , imaging and deconvolution were accomplished in the MIRIAD software package. The resulting synthesized beam sizes were 2 . '' 8 × 1 . '' 9 (PA = -1 . 3 · ) and 2 . '' 7 × 1 . '' 8 (PA=-3.0 · ) for C 18 O and 13 CO, respectively. HD observations toward TW Hya were been carried out with the Herschel Space Observatory Photodetector Array Camera and Spectrometer (Poglitsch et al. 2010; Pilbratt et al. 2010). Further informations concerning both reduction and line analysis are presented in B13.</text>
<text><location><page_2><loc_51><loc_51><loc_86><loc_63></location>In the present work we focus on the integrated line fluxes from HD, 13 CO, and C 18 O. Spectroscopic parameters of these molecules and measured spectrally integrated fluxes within an 8 '' box (or 408 AU assuming a distance of 51 pc; Mamajek 2005) are given in Table 1. The spatially integrated spectra of C 18 O (2 -1) and 13 CO (2 -1) are presented in Figure 1.</text>
<section_header_level_1><location><page_2><loc_51><loc_48><loc_61><loc_50></location>3. Analysis</section_header_level_1>
<text><location><page_2><loc_51><loc_28><loc_86><loc_47></location>In the present study, we derive TW Hya's diskaveraged gas-phase CO abundance from the observed C 18 O (2 -1) and HD (1 -0) lines. The conversion from integrated line intensity to physical column density is dependent on optical depth and temperature. In the following sections we explore a range of physically motivated parameter space assuming the emission is co-spatial and in LTE. Based upon these assumptions we calculate a range of χ (CO) in the warm (T > 20 K) disk using HD as our gas mass tracer. Caveats of this approach and their implications for our measurement will be discussed in Section 4.</text>
<section_header_level_1><location><page_2><loc_51><loc_25><loc_66><loc_26></location>3.1. Line Opacity</section_header_level_1>
<text><location><page_2><loc_51><loc_13><loc_86><loc_24></location>The determination of the CO mass from the C 18 O emission relies on the assumption that C 18 O(2 -1) is optically thin and an 16 O / 18 Oratio. To estimate the disk-averaged opacity of C 18 O, we compare C 18 O (2 -1) to 13 CO (2 -1) and find the disk-averaged 13 CO/C 18 O flux ratio is ∼ 3.3 ± 0.9. This measurement is strongly affected by the opac-</text>
<table>
<location><page_3><loc_19><loc_74><loc_76><loc_83></location>
<caption>Table 1: C 18 O, 13 CO and HD spectroscopic line parameters a and total integrated fluxes observed towards TW Hya.</caption>
</table>
<figure>
<location><page_3><loc_12><loc_24><loc_43><loc_55></location>
<caption>Fig. 1.- Spatially integrated spectra of C 18 O (2 -1) (top) and 13 CO (2 -1) (bottom) in a 8 '' square box centered on TW Hya. The vertical dashed line indicates the LSR systemic velocity of the source (2.86 km s -1 ).</caption>
</figure>
<text><location><page_3><loc_51><loc_52><loc_86><loc_60></location>of 13 CO (2 -1), where τ ( 13 CO) ∼ 2 . 9 assuming isotope ratios of 12 C / 13 C = 70 and 16 O / 18 O = 557 for the local ISM (Wilson 1999). This ratio suggests that the spatially integrated C 18 O emission is thin, τ (C 18 O) ∼ 0 . 36.</text>
<section_header_level_1><location><page_3><loc_51><loc_49><loc_75><loc_51></location>3.2. Hints from Disk Models</section_header_level_1>
<text><location><page_3><loc_51><loc_17><loc_89><loc_48></location>The mismatch between the normal CO abundance and mass needed to match HD can be understood by computing the optically thin C 18 O emission predicted by the sophisticated Gorti et al. (2011) model. For this purpose we adopt the non-LTE code LIME (Brinch & Hogerheijde 2010) with the Gorti et al. (2011) physical structure employed in the original modeling effort of Bergin et al. (2013), which best matched the HD emission. In these calculations we include CO freeze-out assuming a binding energy of 855 K ( Oberg et al. 2005). The disk model natively assumes χ (CO)= 2 . 5 × 10 -4 and if one adopts 16 O / 18 O = 500, over-predicts the C 18 O (2 -1) flux by ∼ 10 × . Furthermore, the C 18 O (2 -1) emission is predicted to be optically thick and, to match the observed flux, χ (C 18 O) needs to be reduced to ∼ 7 × 10 -9 , i.e., χ (CO) = 4 × 10 -6 . We note that this abundance is dependent on the assumed binding energy, discussed further in Section 4.2.</text>
<section_header_level_1><location><page_4><loc_9><loc_83><loc_45><loc_86></location>3.3. Mass and Model Independent χ (CO) Determination</section_header_level_1>
<text><location><page_4><loc_9><loc_75><loc_45><loc_82></location>Under the assumption of optically thin HD (1 -0) and C 18 O (2 -1) emission, we can define the observable R obs as the ratio between the observed number (denoted N ) of C 18 O and HD molecules in their respective upper states,</text>
<formula><location><page_4><loc_11><loc_70><loc_45><loc_74></location>R obs = N (C 18 O , J u = 2) N (HD , J u = 1) = F C 18 O A HD ν HD F HD A C 18 O ν C 18 O , (1)</formula>
<text><location><page_4><loc_9><loc_50><loc_45><loc_69></location>where ν X , A X and F X are the frequency, Einstein A coefficient and total integrated flux of the measured transition, respectively (see Table 1). To determine the total CO abundance in LTE, we must calculate the fractional population in the upper state, f u (X), and assume isotopic ratios. We adopt the isotopic oxygen ratio described in Section 3.1 and an isotopic ratio of HD relative to H 2 of χ (HD) = 3 × 10 -5 , based on a D/H elemental abundance of (1 . 50 ± 0 . 10) × 10 -5 (Linsky 1998). Assuming LTE and inserting the measured fluxes, the 12 CO abundance relative to H 2 can be written as:</text>
<formula><location><page_4><loc_9><loc_43><loc_45><loc_49></location>χ (CO) = 1 . 76 × 10 -5 ( 16 O / 18 O 557 )( R obs 1 . 05 × 10 -3 ) × ( χ (HD) 3 × 10 -5 ) f u (HD , J u = 1) f u (C 18 O , J u = 2) . (2)</formula>
<text><location><page_4><loc_9><loc_21><loc_45><loc_41></location>It is important to note that the above analysis hinges upon the assumption that HD (1 -0) and C 18 O (2 -1) are in LTE. Based on the Gorti et al. (2011) model, at radii between R ∼ 50 -150 AU the typical H 2 density at gas temperatures near T g = 30 K ranges between ∼ 10 6 -10 8 cm -3 . Critical densities for the HD (1 -0) and C 18 O (2 -1) transitions are respectively 2 . 7 × 10 3 cm -3 and 10 4 cm -3 at T g ∼ 30 K, respectively, which assumes collision rate coefficients with H 2 at 30 K of HD (2 × 10 -11 cm 3 s -1 ; Flower et al. 2000) and C 18 O (6 × 10 -11 cm 3 s -1 ; Yang et al. 2010). Under these conditions both lines are thermalized and the assumption of LTE is reasonable.</text>
<text><location><page_4><loc_9><loc_10><loc_45><loc_20></location>The measured gas-phase disk-averaged χ (CO), Eq. 2, depends sensitively on the temperature of the emitting material, viz., the upper state fraction, f u . Formally, gas temperatures vary by orders of magnitude throughout the disk. However, to first order, as a result of the abundance distribution and excitation of a given rotational transition,</text>
<text><location><page_4><loc_51><loc_76><loc_86><loc_86></location>emission generally arises from a narrower range of temperatures. There are two ways temperatures can be estimated: 1) by observing optically thick lines originating from the same gas and measuring an average kinetic temperature of the emitting gas within the beam, or 2) by inferring temperatures from disk thermochemical models.</text>
<text><location><page_4><loc_51><loc_51><loc_86><loc_75></location>In the latter case we estimate a characteristic temperature of CO by dividing up the emissive mass of the Gorti et al. (2011) model into temperature bins for both C 18 O and HD, Figure 2. To compute the emissive mass we: following the Gorti et al. (2011) TW Hya model, for each temperature bin integrate the mass in HD ( J = 1) and in C 18 O ( J = 2) in the upper state within the specified temperature range, and normalize this to the total mass throughout the disk in the corresponding upper state for each species, i.e., M upper (HD) = 4 π ∫ n HD J =1 rdrdz , where n HD J=1 is the upper state volume density calculated from the LIME excitation models (Brinch & Hogerheijde 2010) performed for HD in B13.</text>
<text><location><page_4><loc_51><loc_39><loc_86><loc_51></location>One notable feature of Figure 2 is that the two lines have slightly different peak maximally emissive temperatures, ∼ 20 K for C 18 O and ∼ 40 -60 K HD. However, over the temperature range expected for HD, the difference in the ratio of fractional populations for HD and C 18 O is not enough to bring the CO abundance close to 10 -4 using Equation 2.</text>
<text><location><page_4><loc_51><loc_16><loc_86><loc_38></location>Guided by this range of temperatures, we compute the χ (CO) from Eq. 2 assuming a C 18 O gas temperature of T g = 20 K and varying the HD emitting temperature T g (HD) between 20 and 60 K, accounting for the possibility of HD emitting from warmer gas than the C 18 O. The obtained χ (CO) is provided in Fig. 3. In all cases, χ (CO) in the gas is lower than the canonical value of χ (CO) ∼ 10 -4 ; ranging between (0 . 1 -3) × 10 -5 . From the modeled mass distribution shown in Fig. 2, the center of the gas temperature distribution probed by HD is T g ∼ 40 K, while C 18 O mostly emits from 20 K. With this value the resulting CO abundance is only χ (CO) = 7 × 10 -6 , over 10 × less than the canonical value.</text>
<text><location><page_4><loc_51><loc_10><loc_86><loc_16></location>Therefore, to get χ (CO) up to the 10 -4 range, significant corrections to the upper state fraction of each species is required. Concerning C 18 O, that requires the gas to be either significantly colder or</text>
<figure>
<location><page_5><loc_15><loc_64><loc_44><loc_85></location>
</figure>
<figure>
<location><page_5><loc_45><loc_64><loc_79><loc_85></location>
<caption>Fig. 2.Left: Image details the radial and vertical distribution of the C 18 O J = 2 volume density, n C 18 O J=2 , predicted in the Gorti et al. (2011) model with M gas = 0 . 06 M /circledot and χ (C 18 O) = 7 × 10 -9 . Contours are the gas temperature structure at 10, 20 (dashed line), 50, 75, 100, 150, 200, 250, and 300 K, respectively. Right: Mass fraction of HD (gray) and C 18 O (unfilled) in their respective upper states arising from gas at the specified temperature from the non-LTE calculation. The mass is normalized to the total mass in the upper state J u . For further details see Section 3.3</caption>
</figure>
<text><location><page_5><loc_47><loc_51><loc_48><loc_52></location>.</text>
<figure>
<location><page_5><loc_11><loc_28><loc_43><loc_44></location>
<caption>Fig. 3.- CO Abundance with respect to H 2 as a function of emitting temperature within the warm molecular layer. Light gray histograms show χ (CO) for fixed T ex (C 18 O) = 20 K and T ex (HD) in the range 20-60K. Dark gray histograms show χ (CO) for T ex (C 18 O) = T ex (HD). One sigma error bars taking into account the calibration uncertainty are shown.</caption>
</figure>
<text><location><page_5><loc_51><loc_44><loc_86><loc_48></location>hotter such that the J = 2 becomes depopulated. Both scenarios are unlikely (see Fig. 2) and unsupported by the 12 CO data (Qi et al. 2006).</text>
<text><location><page_5><loc_51><loc_18><loc_86><loc_43></location>Alternatively, 12 CO emission can constrain the temperature in the layers where its emission becomes optically thick. Using the resolved Band 6 TWHya ALMA Science Verification (S.V.) observations of 12 CO (2 -1), the peak beam temperature is 24.5 K within a 2 . '' 83 × 2 . '' 39 (P . A . = 44 · ) beam. This temperature represents the beam averaged kinetic temperature of the CO emitting gas within the inner R ∼ 70 AU, in agreement with values reported by B13 for the Band 7 S.V. data of the CO (3 -2) line and the observations of Qi et al. (2006) for CO (6 -5) ( T R ∼ 29 . 7 K and ∼ 30 . 6 K respectively). Under these conditions, the CO abundance traced is less than 3 × 10 -6 . We conclude that it is difficult for excitation alone to reconcile the emission with a CO abundance of 10 -4 .</text>
<section_header_level_1><location><page_5><loc_51><loc_15><loc_78><loc_16></location>4. χ (CO) Measurement Caveats</section_header_level_1>
<text><location><page_5><loc_51><loc_11><loc_86><loc_14></location>The analysis above assumes HD and CO emit from similar regions and therefore trace the gas-</text>
<text><location><page_6><loc_9><loc_80><loc_45><loc_86></location>phase χ (CO) directly. In the following section we relax this assumption and discuss various physical mechanisms that could modify the interpretation of the measured χ (CO).</text>
<section_header_level_1><location><page_6><loc_9><loc_77><loc_36><loc_79></location>4.1. Different Emitting Regions</section_header_level_1>
<text><location><page_6><loc_9><loc_40><loc_45><loc_76></location>In Fig. 4 we illustrate some of the key issues concerning the above discussion. First, while HD is spatially distributed broadly, gas-phase C 18 O is not, freezing onto dust grains with T dust < 20 K. Because of the strong temperature dependence in the Boltzmann factor for the J = 1 state, we would expect the HD emission to be sharply curtailed below T g /lessorsimilar 20 K. For a massive midplane, some HD emission could arise from dense gas directly behind the CO snow-line (shown as magenta), but the HD emissivity from such cold gas is lessened by the fact that adding more mass (or enriching the dust) would increase the dust optical depth at 112 µ m, hiding some fraction of the HD emission. Furthermore, this emission cannot contribute significantly to the observations as it would drive the H 2 mass to unrealistically high levels. For example, if ∼ 20% of the HD (1 -0) emission arises from gas at 15 K, the H 2 mass at this temperature is 0.05 M /circledot in addition to the contribution from the rest of the disk. Therefore it is difficult for the 15 K mass to add appreciably to the emission without driving the disk to extremely high masses.</text>
<text><location><page_6><loc_9><loc_17><loc_45><loc_40></location>Another likely scenario is where the HD gas emits from primarily warm gas in the innermost disk, while CO and C 18 O trace cooler emitting regions and thus larger physical radii. As a result, CO would trace more gas (full disk) than HD (warm inner disk). Consequently, HD (1 -0) would miss H 2 mass in the outer disk, resulting in a lower limit to the disk mass estimation and in turn an overestimate of χ (CO). The CO abundance could hence be lower. In addition, it is important to note that in B13 the authors find the outer disk does not emit appreciably, with only ∼ 10% of the HD flux coming from outside of 100 AU (see their Fig. 2c) based upon the model of Gorti et al. (2011).</text>
<section_header_level_1><location><page_6><loc_9><loc_15><loc_23><loc_16></location>4.2. Freeze-out</section_header_level_1>
<text><location><page_6><loc_9><loc_11><loc_45><loc_14></location>Previous studies have attributed measured low CO abundances to gas-phase depletion by adsorp-</text>
<text><location><page_6><loc_51><loc_77><loc_86><loc_86></location>tion onto grains (Aikawa et al. 1996; Dartois et al. 2003). Under normal conditions CO freezes-out at low temperatures present in the midplane, T /lessorsimilar 20 K, where HD does not strongly emit, and therefore the reduced measured χ (CO) in the gas-phase is unlikely to be the result of freeze-out.</text>
<text><location><page_6><loc_51><loc_66><loc_87><loc_77></location>In fact a number of studies find the measured CO antenna temperatures of T < 17 K (Pi'etu et al. 2007; Dartois et al. 2003; Hersant et al. 2009). If these estimates are correct, then the total volume of gas traced by the C 18 O line exceeds that traced by the HD line, leading to an overprediction of the true χ (CO).</text>
<text><location><page_6><loc_51><loc_47><loc_86><loc_66></location>There is, however, uncertainty in the freeze-out temperatures, which depend formally on the binding energies assumed. The binding energies are a function of the binding-surface, often assumed to be CO ice. Alternatively, if the grain surface is water ice or bare dust, the binding energy can be significantly higher (Bergin et al. 1995; Fraser et al. 2004). If this is the case, CO can freeze-out at higher temperatures T > 25 K, and therefore the CO emitting region would be smaller than the HD emitting region. In this instance the measured CO abundance would be lower than the true CO abundance.</text>
<section_header_level_1><location><page_6><loc_51><loc_44><loc_62><loc_45></location>4.3. Opacity</section_header_level_1>
<text><location><page_6><loc_51><loc_28><loc_86><loc_43></location>Another caveat of our χ (CO) estimates are the opacities of the HD (1 -0) and C 18 O (2 -1) lines. In this study, we assume that emission of both species is optically thin. Although we show in Sec. 3.1 that the C 18 O (2 -1) emission is thin in the disk-averaged data, the possibility of optically thick HD emission still remains. However, if τ HD /greaterorsimilar 1, the derived HD mass should be a lower limit and therefore the measured χ (CO) is an upper limit on the true CO abundance.</text>
<section_header_level_1><location><page_6><loc_51><loc_25><loc_85><loc_26></location>4.4. Photodissociation and Self-shielding</section_header_level_1>
<text><location><page_6><loc_51><loc_11><loc_91><loc_24></location>Photodissociation by UV is a major CO destruction mechanism in disks that regulates the molecular abundance of species in the gas. Photodissociation models for HD and CO isotopologues have been investigated by Roueff & Node-Langlois (1999); Le Petit et al. (2002); Visser et al. (2009). Roueff & Node-Langlois (1999) finds HD should self-shield at smaller A V than CO. Therefore, in the absence of dust shielding and selective iso-</text>
<text><location><page_7><loc_9><loc_65><loc_45><loc_86></location>topologue photodissociation, HD could emit from warm layers where C 18 O is destroyed. If those surface layers are essential contributors to the HD emission, χ (CO) would be underestimated. However, the modeling of B13 suggests that the high surface layers do not dominate the emissive mass of HD, and therefore, even if photodissociation cannot be ruled out, it only minimally affects the measured χ (CO). Alternatively, if selective isotopologue photodissociation operates for C 18 O from external UV irradiation, we may be missing CO mass from the outer disk edge. As discussed in Section 4.1, however, the outer disk does not significantly contribute to the HD emission.</text>
<section_header_level_1><location><page_7><loc_9><loc_62><loc_42><loc_63></location>5. Implications: Where is the Carbon?</section_header_level_1>
<text><location><page_7><loc_9><loc_19><loc_45><loc_61></location>Our study shows that the main reservoir of gas-phase carbon, CO, is reduced by at least an order of magnitude in the TW Hya disk compared to dense clouds. In both T-Tauri and Herbig Ae disks similarly low CO abundances have been inferred and attributed to photodissociation and freeze-out (e.g., Dutrey et al. 2003; Chapillon et al. 2008; Qi et al. 2011). The difference between the previous studies and the results reported here is the use of HD to probe H 2 above 20 K and hence provide stronger constraints on χ (CO) in the warm molecular layer. It is important to state that both C 18 O and HD do not trace the midplane of the disk because of freezeout (C 18 O) and low excitation (HD). Thus it is possible that the χ (CO ice ) is 'normal' in the midplane, which would be consistent with the similarity between interstellar ices and cometary volatiles (Mumma & Charnley 2011). We argue differences in photodissociation of C 18 O and HD are unlikely to account for the low χ (CO). This would argue against the possibility that the carbon is sequestered in atomic form either neutral or ionized. Bruderer et al. (2012) supports this assertion with observations of all primary forms of carbon in a Be star disk (HD 100547). They argue the total carbon abundance is depleted in the warm atmosphere, which is consistent with our conclusion.</text>
<text><location><page_7><loc_9><loc_11><loc_45><loc_18></location>This finding leads one to ask where the missing carbon might be found. One possibility is suggested by the modeling of kinetic chemistry in disks by Aikawa et al. (1997). The deep disk layers are exposed to X-rays from the central star</text>
<text><location><page_7><loc_51><loc_61><loc_86><loc_86></location>(Glassgold et al. 1997), though likely not cosmic rays (Cleeves et al. 2013). In these layers CO can exist in the gas via thermal- or photo-desorption from grains. X-rays produce He + and, with sufficient time, carbon can be extracted from CO via reactions with He + . CO reforms, but a portion of the carbon is placed into hydrocarbons (C X H X ) or CO 2 . Many of these species have freeze-out temperatures higher than CO and trap the carbon in ices. In a sense the chemistry works towards the first carbon-bearing molecule that freezes-out, creating a carbon sink (Aikawa et al. 1997). Therefore we suggest that the low measured gas-phase CO abundance in the TW Hya disk is a result of this chemical mechanism, and the use of CO as a mass tracer has very significant, and likely timedependent, uncertainty.</text>
<text><location><page_7><loc_51><loc_35><loc_86><loc_59></location>We thank the anonymous referee for raising interesting issues. This work was supported by the National Science Foundation under grant 1008800. This paper makes use of the following ALMA data: ADS/JAO.ALMA2011.0.00001.SV and SMA data. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.</text>
<text><location><page_7><loc_51><loc_30><loc_86><loc_34></location>Facilities: SMA, Herschel Space Observatory Photodetector Array Camera and Spectrometer, ALMA</text>
<section_header_level_1><location><page_7><loc_51><loc_27><loc_63><loc_28></location>REFERENCES</section_header_level_1>
<text><location><page_7><loc_51><loc_23><loc_86><loc_26></location>Aikawa, Y., Miyama, S. M., Nakano, T., & Umebayashi, T. 1996, ApJ, 467, 684</text>
<text><location><page_7><loc_51><loc_19><loc_86><loc_22></location>Aikawa, Y., Umebayashi, T., Nakano, T., & Miyama, S. M. 1997, ApJ, 486, L51</text>
<text><location><page_7><loc_51><loc_15><loc_86><loc_18></location>Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481</text>
<text><location><page_7><loc_51><loc_11><loc_86><loc_14></location>Bergin, E. A., Langer, W. D., & Goldsmith, P. F. 1995, ApJ, 441, 222</text>
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<text><location><page_9><loc_9><loc_83><loc_45><loc_86></location>Visser, R., van Dishoeck, E. F., & Black, J. H. 2009, A&A, 503, 323</text>
<text><location><page_9><loc_9><loc_79><loc_45><loc_82></location>Wilson, T. L. 1999, Reports on Progress in Physics, 62, 143</text>
<text><location><page_9><loc_9><loc_75><loc_45><loc_78></location>Yang, B., Stancil, P. C., Balakrishnan, N., & Forrey, R. C. 2010, ApJ, 718, 1062</text>
<figure>
<location><page_9><loc_51><loc_52><loc_86><loc_67></location>
<caption>Fig. 4.- Schematic illustrating the regions that contribute to HD and C 18 O emission. Horizontal black lines denote C 18 O emitting region, T g > 20 K. We indicate the warm molecular layer (W.M.L.), the zone where CO is present in the gas. The yellow-dotted region denotes the HD (1 -0) emitting region, generally restricted to T g > 20 K because of excitation considerations. The magenta region denotes layers where HD could emit below 20 K, provided the midplane is sufficiently massive. However, the midplane dust can become optically thick at 112 µ m in a portion of this layer (denoted as the white dashed line), blocking HD emission from below.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "In this Letter we report the CO abundance relative to H 2 derived toward the circumstellar disk of the T-Tauri star TW Hya from the HD (1 -0) and C 18 O (2 -1) emission lines. The HD (1 -0) line was observed by the Herschel Space Observatory Photodetector Array Camera and Spectrometer whereas C 18 O (2 -1) observations were carried out with the Submillimeter Array at a spatial resolution of 2 . '' 8 × 1 . '' 9 (corresponding to ∼ 142 × 97 AU). In the disk's warm molecular layer ( T > 20 K) we measure a disk-averaged gas-phase CO abundance relative to H 2 of χ (CO) = (0 . 1 -3) × 10 -5 , substantially lower than the canonical value of χ (CO) = 10 -4 . We infer that the best explanation of this low χ (CO) is the chemical destruction of CO followed by rapid formation of carbon chains, or perhaps CO 2 , that can subsequently freeze-out, resulting in the bulk mass of carbon locked up in ice grain mantles and oxygen in water. As a consequence of this likely time-dependent carbon sink mechanism, CO may be an unreliable tracer of H 2 gas mass. Subject headings: protoplanetary disks - astrochemistry - ISM: abundances - stars: formation",
"pages": [
1
]
},
{
"title": "A Significantly Low CO Abundance Toward the TW Hya Protoplanetary Disk: A Path to Active Carbon Chemistry?",
"content": "C'ecile Favre, L. Ilsedore Cleeves, Edwin A. Bergin, Department of Astronomy, University of Michigan, 500 Church St., Ann Arbor, MI 48109 cfavre@umich.edu",
"pages": [
1
]
},
{
"title": "Chunhua Qi,",
"content": "Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 and",
"pages": [
1
]
},
{
"title": "Geoffrey A. Blake",
"content": "California Institute of Technology, Division of Geological & Planetary Sciences, MS 150-21, Pasadena, CA 91125, USA",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Molecular hydrogen is the main gas-phase constituent in star-forming gas. However, it does not appreciably emit for typical gas conditions. Consequently carbon monoxide is widely used as a proxy for H 2 in the molecular interstellar medium (e.g. Dickman 1978) and protoplanetary disks (Koerner & Sargent 1995; Dutrey et al. 1996). With a suite of transitions at millimeter/submillimeter wavelengths, the optically thick and thermalized 12 CO lines trace gas temperature while optically thin CO isotopologues (namely 13 CO and C 18 O) probe the CO column and hence molecular mass. A key component of the latter calculation is the calibration of CO to H 2 , assuming an abundance of carbon monoxide, χ (CO). In the ISM this factor can be constrained via comparisons of dust extinction to measurements of optically thin isotopologue lines. Ripple et al. (2013) showed that typical 13 CO abundances range from ∼ 1 -3 × 10 -6 in several clouds. This corresponds to a CO abundance of ∼ 0 . 6 -2 × 10 -4 , assuming an isotopic ratio of 60. Since the dense ISM provides CO to the pro- toplanetary disk during its formation, it is reasonable to assume that χ (CO) in disks is similar to its interstellar value. Furthermore, at such high abundances, CO would represent the main gas-phase reservoir of carbon in disks. Spatially resolved observations of CO could thus be used to determine the distribution and abundance of volatile carbon, which has implications for the inclusion of carbon into planetary systems (Lee et al. 2010; Bond et al. 2010; Oberg et al. 2011). In this paper we combine spatially integrated observations of optically thin C 18 O emission with a detection of the fundamental rotational transition of hydrogen deuteride, HD, towards the closest T-Tauri system, TW Hya, at 51 pc (Mamajek 2005). HD emission provides a separate probe of H 2 (Bergin et al. 2013, hereafter B13), with which we measure the χ (CO) in this system. We show that the main reservoir of gas-phase carbon, CO, is substantially reduced ( < 10% remaining) in the warm ( > 20 K) molecular layers of the disk and discuss implications of this result.",
"pages": [
1,
2
]
},
{
"title": "2. Observations and data reduction",
"content": "The observations of TW Hya were made on 2005 February 27 and April 10 using the Submillimeter Array 1 (SMA, Ho et al. 2004) located atop Mauna Kea, Hawaii. The SMA receivers operated in a double-sideband mode with an intermediate frequency (IF) band of 4-6 GHz from the local oscillator frequency, sent over fiber optic transmission lines to 24 overlapping 'chunks' of the digital correlator. The correlator was configured to include CO, 13 CO and C 18 O, in one setting: the tuning was centered on the CO (2 -1) line at 230.538 GHz in chunk S15, while the 13 CO/C 18 O (2 -1) transitions at 220.399/219.560 GHz were simultaneously observed in chunks 12 and 22, respectively. CO (2 -1) data were reported in Qi et al. (2006). Combinations of two array configurations (compact and extended) were used to obtain projected baselines ranging from 6 to 180 m. The observing loops used J1037-295 as the gain calibrator, with bandpass calibration using observations of 3C279. Flux calibration was done using observations of Titan and Callisto. Routine calibration tasks were performed using the MIR software package 2 , imaging and deconvolution were accomplished in the MIRIAD software package. The resulting synthesized beam sizes were 2 . '' 8 × 1 . '' 9 (PA = -1 . 3 · ) and 2 . '' 7 × 1 . '' 8 (PA=-3.0 · ) for C 18 O and 13 CO, respectively. HD observations toward TW Hya were been carried out with the Herschel Space Observatory Photodetector Array Camera and Spectrometer (Poglitsch et al. 2010; Pilbratt et al. 2010). Further informations concerning both reduction and line analysis are presented in B13. In the present work we focus on the integrated line fluxes from HD, 13 CO, and C 18 O. Spectroscopic parameters of these molecules and measured spectrally integrated fluxes within an 8 '' box (or 408 AU assuming a distance of 51 pc; Mamajek 2005) are given in Table 1. The spatially integrated spectra of C 18 O (2 -1) and 13 CO (2 -1) are presented in Figure 1.",
"pages": [
2
]
},
{
"title": "3. Analysis",
"content": "In the present study, we derive TW Hya's diskaveraged gas-phase CO abundance from the observed C 18 O (2 -1) and HD (1 -0) lines. The conversion from integrated line intensity to physical column density is dependent on optical depth and temperature. In the following sections we explore a range of physically motivated parameter space assuming the emission is co-spatial and in LTE. Based upon these assumptions we calculate a range of χ (CO) in the warm (T > 20 K) disk using HD as our gas mass tracer. Caveats of this approach and their implications for our measurement will be discussed in Section 4.",
"pages": [
2
]
},
{
"title": "3.1. Line Opacity",
"content": "The determination of the CO mass from the C 18 O emission relies on the assumption that C 18 O(2 -1) is optically thin and an 16 O / 18 Oratio. To estimate the disk-averaged opacity of C 18 O, we compare C 18 O (2 -1) to 13 CO (2 -1) and find the disk-averaged 13 CO/C 18 O flux ratio is ∼ 3.3 ± 0.9. This measurement is strongly affected by the opac- of 13 CO (2 -1), where τ ( 13 CO) ∼ 2 . 9 assuming isotope ratios of 12 C / 13 C = 70 and 16 O / 18 O = 557 for the local ISM (Wilson 1999). This ratio suggests that the spatially integrated C 18 O emission is thin, τ (C 18 O) ∼ 0 . 36.",
"pages": [
2,
3
]
},
{
"title": "3.2. Hints from Disk Models",
"content": "The mismatch between the normal CO abundance and mass needed to match HD can be understood by computing the optically thin C 18 O emission predicted by the sophisticated Gorti et al. (2011) model. For this purpose we adopt the non-LTE code LIME (Brinch & Hogerheijde 2010) with the Gorti et al. (2011) physical structure employed in the original modeling effort of Bergin et al. (2013), which best matched the HD emission. In these calculations we include CO freeze-out assuming a binding energy of 855 K ( Oberg et al. 2005). The disk model natively assumes χ (CO)= 2 . 5 × 10 -4 and if one adopts 16 O / 18 O = 500, over-predicts the C 18 O (2 -1) flux by ∼ 10 × . Furthermore, the C 18 O (2 -1) emission is predicted to be optically thick and, to match the observed flux, χ (C 18 O) needs to be reduced to ∼ 7 × 10 -9 , i.e., χ (CO) = 4 × 10 -6 . We note that this abundance is dependent on the assumed binding energy, discussed further in Section 4.2.",
"pages": [
3
]
},
{
"title": "3.3. Mass and Model Independent χ (CO) Determination",
"content": "Under the assumption of optically thin HD (1 -0) and C 18 O (2 -1) emission, we can define the observable R obs as the ratio between the observed number (denoted N ) of C 18 O and HD molecules in their respective upper states, where ν X , A X and F X are the frequency, Einstein A coefficient and total integrated flux of the measured transition, respectively (see Table 1). To determine the total CO abundance in LTE, we must calculate the fractional population in the upper state, f u (X), and assume isotopic ratios. We adopt the isotopic oxygen ratio described in Section 3.1 and an isotopic ratio of HD relative to H 2 of χ (HD) = 3 × 10 -5 , based on a D/H elemental abundance of (1 . 50 ± 0 . 10) × 10 -5 (Linsky 1998). Assuming LTE and inserting the measured fluxes, the 12 CO abundance relative to H 2 can be written as: It is important to note that the above analysis hinges upon the assumption that HD (1 -0) and C 18 O (2 -1) are in LTE. Based on the Gorti et al. (2011) model, at radii between R ∼ 50 -150 AU the typical H 2 density at gas temperatures near T g = 30 K ranges between ∼ 10 6 -10 8 cm -3 . Critical densities for the HD (1 -0) and C 18 O (2 -1) transitions are respectively 2 . 7 × 10 3 cm -3 and 10 4 cm -3 at T g ∼ 30 K, respectively, which assumes collision rate coefficients with H 2 at 30 K of HD (2 × 10 -11 cm 3 s -1 ; Flower et al. 2000) and C 18 O (6 × 10 -11 cm 3 s -1 ; Yang et al. 2010). Under these conditions both lines are thermalized and the assumption of LTE is reasonable. The measured gas-phase disk-averaged χ (CO), Eq. 2, depends sensitively on the temperature of the emitting material, viz., the upper state fraction, f u . Formally, gas temperatures vary by orders of magnitude throughout the disk. However, to first order, as a result of the abundance distribution and excitation of a given rotational transition, emission generally arises from a narrower range of temperatures. There are two ways temperatures can be estimated: 1) by observing optically thick lines originating from the same gas and measuring an average kinetic temperature of the emitting gas within the beam, or 2) by inferring temperatures from disk thermochemical models. In the latter case we estimate a characteristic temperature of CO by dividing up the emissive mass of the Gorti et al. (2011) model into temperature bins for both C 18 O and HD, Figure 2. To compute the emissive mass we: following the Gorti et al. (2011) TW Hya model, for each temperature bin integrate the mass in HD ( J = 1) and in C 18 O ( J = 2) in the upper state within the specified temperature range, and normalize this to the total mass throughout the disk in the corresponding upper state for each species, i.e., M upper (HD) = 4 π ∫ n HD J =1 rdrdz , where n HD J=1 is the upper state volume density calculated from the LIME excitation models (Brinch & Hogerheijde 2010) performed for HD in B13. One notable feature of Figure 2 is that the two lines have slightly different peak maximally emissive temperatures, ∼ 20 K for C 18 O and ∼ 40 -60 K HD. However, over the temperature range expected for HD, the difference in the ratio of fractional populations for HD and C 18 O is not enough to bring the CO abundance close to 10 -4 using Equation 2. Guided by this range of temperatures, we compute the χ (CO) from Eq. 2 assuming a C 18 O gas temperature of T g = 20 K and varying the HD emitting temperature T g (HD) between 20 and 60 K, accounting for the possibility of HD emitting from warmer gas than the C 18 O. The obtained χ (CO) is provided in Fig. 3. In all cases, χ (CO) in the gas is lower than the canonical value of χ (CO) ∼ 10 -4 ; ranging between (0 . 1 -3) × 10 -5 . From the modeled mass distribution shown in Fig. 2, the center of the gas temperature distribution probed by HD is T g ∼ 40 K, while C 18 O mostly emits from 20 K. With this value the resulting CO abundance is only χ (CO) = 7 × 10 -6 , over 10 × less than the canonical value. Therefore, to get χ (CO) up to the 10 -4 range, significant corrections to the upper state fraction of each species is required. Concerning C 18 O, that requires the gas to be either significantly colder or . hotter such that the J = 2 becomes depopulated. Both scenarios are unlikely (see Fig. 2) and unsupported by the 12 CO data (Qi et al. 2006). Alternatively, 12 CO emission can constrain the temperature in the layers where its emission becomes optically thick. Using the resolved Band 6 TWHya ALMA Science Verification (S.V.) observations of 12 CO (2 -1), the peak beam temperature is 24.5 K within a 2 . '' 83 × 2 . '' 39 (P . A . = 44 · ) beam. This temperature represents the beam averaged kinetic temperature of the CO emitting gas within the inner R ∼ 70 AU, in agreement with values reported by B13 for the Band 7 S.V. data of the CO (3 -2) line and the observations of Qi et al. (2006) for CO (6 -5) ( T R ∼ 29 . 7 K and ∼ 30 . 6 K respectively). Under these conditions, the CO abundance traced is less than 3 × 10 -6 . We conclude that it is difficult for excitation alone to reconcile the emission with a CO abundance of 10 -4 .",
"pages": [
4,
5
]
},
{
"title": "4. χ (CO) Measurement Caveats",
"content": "The analysis above assumes HD and CO emit from similar regions and therefore trace the gas- phase χ (CO) directly. In the following section we relax this assumption and discuss various physical mechanisms that could modify the interpretation of the measured χ (CO).",
"pages": [
5,
6
]
},
{
"title": "4.1. Different Emitting Regions",
"content": "In Fig. 4 we illustrate some of the key issues concerning the above discussion. First, while HD is spatially distributed broadly, gas-phase C 18 O is not, freezing onto dust grains with T dust < 20 K. Because of the strong temperature dependence in the Boltzmann factor for the J = 1 state, we would expect the HD emission to be sharply curtailed below T g /lessorsimilar 20 K. For a massive midplane, some HD emission could arise from dense gas directly behind the CO snow-line (shown as magenta), but the HD emissivity from such cold gas is lessened by the fact that adding more mass (or enriching the dust) would increase the dust optical depth at 112 µ m, hiding some fraction of the HD emission. Furthermore, this emission cannot contribute significantly to the observations as it would drive the H 2 mass to unrealistically high levels. For example, if ∼ 20% of the HD (1 -0) emission arises from gas at 15 K, the H 2 mass at this temperature is 0.05 M /circledot in addition to the contribution from the rest of the disk. Therefore it is difficult for the 15 K mass to add appreciably to the emission without driving the disk to extremely high masses. Another likely scenario is where the HD gas emits from primarily warm gas in the innermost disk, while CO and C 18 O trace cooler emitting regions and thus larger physical radii. As a result, CO would trace more gas (full disk) than HD (warm inner disk). Consequently, HD (1 -0) would miss H 2 mass in the outer disk, resulting in a lower limit to the disk mass estimation and in turn an overestimate of χ (CO). The CO abundance could hence be lower. In addition, it is important to note that in B13 the authors find the outer disk does not emit appreciably, with only ∼ 10% of the HD flux coming from outside of 100 AU (see their Fig. 2c) based upon the model of Gorti et al. (2011).",
"pages": [
6
]
},
{
"title": "4.2. Freeze-out",
"content": "Previous studies have attributed measured low CO abundances to gas-phase depletion by adsorp- tion onto grains (Aikawa et al. 1996; Dartois et al. 2003). Under normal conditions CO freezes-out at low temperatures present in the midplane, T /lessorsimilar 20 K, where HD does not strongly emit, and therefore the reduced measured χ (CO) in the gas-phase is unlikely to be the result of freeze-out. In fact a number of studies find the measured CO antenna temperatures of T < 17 K (Pi'etu et al. 2007; Dartois et al. 2003; Hersant et al. 2009). If these estimates are correct, then the total volume of gas traced by the C 18 O line exceeds that traced by the HD line, leading to an overprediction of the true χ (CO). There is, however, uncertainty in the freeze-out temperatures, which depend formally on the binding energies assumed. The binding energies are a function of the binding-surface, often assumed to be CO ice. Alternatively, if the grain surface is water ice or bare dust, the binding energy can be significantly higher (Bergin et al. 1995; Fraser et al. 2004). If this is the case, CO can freeze-out at higher temperatures T > 25 K, and therefore the CO emitting region would be smaller than the HD emitting region. In this instance the measured CO abundance would be lower than the true CO abundance.",
"pages": [
6
]
},
{
"title": "4.3. Opacity",
"content": "Another caveat of our χ (CO) estimates are the opacities of the HD (1 -0) and C 18 O (2 -1) lines. In this study, we assume that emission of both species is optically thin. Although we show in Sec. 3.1 that the C 18 O (2 -1) emission is thin in the disk-averaged data, the possibility of optically thick HD emission still remains. However, if τ HD /greaterorsimilar 1, the derived HD mass should be a lower limit and therefore the measured χ (CO) is an upper limit on the true CO abundance.",
"pages": [
6
]
},
{
"title": "4.4. Photodissociation and Self-shielding",
"content": "Photodissociation by UV is a major CO destruction mechanism in disks that regulates the molecular abundance of species in the gas. Photodissociation models for HD and CO isotopologues have been investigated by Roueff & Node-Langlois (1999); Le Petit et al. (2002); Visser et al. (2009). Roueff & Node-Langlois (1999) finds HD should self-shield at smaller A V than CO. Therefore, in the absence of dust shielding and selective iso- topologue photodissociation, HD could emit from warm layers where C 18 O is destroyed. If those surface layers are essential contributors to the HD emission, χ (CO) would be underestimated. However, the modeling of B13 suggests that the high surface layers do not dominate the emissive mass of HD, and therefore, even if photodissociation cannot be ruled out, it only minimally affects the measured χ (CO). Alternatively, if selective isotopologue photodissociation operates for C 18 O from external UV irradiation, we may be missing CO mass from the outer disk edge. As discussed in Section 4.1, however, the outer disk does not significantly contribute to the HD emission.",
"pages": [
6,
7
]
},
{
"title": "5. Implications: Where is the Carbon?",
"content": "Our study shows that the main reservoir of gas-phase carbon, CO, is reduced by at least an order of magnitude in the TW Hya disk compared to dense clouds. In both T-Tauri and Herbig Ae disks similarly low CO abundances have been inferred and attributed to photodissociation and freeze-out (e.g., Dutrey et al. 2003; Chapillon et al. 2008; Qi et al. 2011). The difference between the previous studies and the results reported here is the use of HD to probe H 2 above 20 K and hence provide stronger constraints on χ (CO) in the warm molecular layer. It is important to state that both C 18 O and HD do not trace the midplane of the disk because of freezeout (C 18 O) and low excitation (HD). Thus it is possible that the χ (CO ice ) is 'normal' in the midplane, which would be consistent with the similarity between interstellar ices and cometary volatiles (Mumma & Charnley 2011). We argue differences in photodissociation of C 18 O and HD are unlikely to account for the low χ (CO). This would argue against the possibility that the carbon is sequestered in atomic form either neutral or ionized. Bruderer et al. (2012) supports this assertion with observations of all primary forms of carbon in a Be star disk (HD 100547). They argue the total carbon abundance is depleted in the warm atmosphere, which is consistent with our conclusion. This finding leads one to ask where the missing carbon might be found. One possibility is suggested by the modeling of kinetic chemistry in disks by Aikawa et al. (1997). The deep disk layers are exposed to X-rays from the central star (Glassgold et al. 1997), though likely not cosmic rays (Cleeves et al. 2013). In these layers CO can exist in the gas via thermal- or photo-desorption from grains. X-rays produce He + and, with sufficient time, carbon can be extracted from CO via reactions with He + . CO reforms, but a portion of the carbon is placed into hydrocarbons (C X H X ) or CO 2 . Many of these species have freeze-out temperatures higher than CO and trap the carbon in ices. In a sense the chemistry works towards the first carbon-bearing molecule that freezes-out, creating a carbon sink (Aikawa et al. 1997). Therefore we suggest that the low measured gas-phase CO abundance in the TW Hya disk is a result of this chemical mechanism, and the use of CO as a mass tracer has very significant, and likely timedependent, uncertainty. We thank the anonymous referee for raising interesting issues. This work was supported by the National Science Foundation under grant 1008800. This paper makes use of the following ALMA data: ADS/JAO.ALMA2011.0.00001.SV and SMA data. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. Facilities: SMA, Herschel Space Observatory Photodetector Array Camera and Spectrometer, ALMA",
"pages": [
7
]
},
{
"title": "REFERENCES",
"content": "Aikawa, Y., Miyama, S. M., Nakano, T., & Umebayashi, T. 1996, ApJ, 467, 684 Aikawa, Y., Umebayashi, T., Nakano, T., & Miyama, S. M. 1997, ApJ, 486, L51 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481 Bergin, E. A., Langer, W. D., & Goldsmith, P. F. 1995, ApJ, 441, 222 Visser, R., van Dishoeck, E. F., & Black, J. H. 2009, A&A, 503, 323 Wilson, T. L. 1999, Reports on Progress in Physics, 62, 143 Yang, B., Stancil, P. C., Balakrishnan, N., & Forrey, R. C. 2010, ApJ, 718, 1062",
"pages": [
7,
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]
}
]
2013ApJ...777...97S
https://arxiv.org/pdf/1307.4764.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_85><loc_92><loc_87></location>THE SL2S GALAXY-SCALE LENS SAMPLE. III. LENS MODELS, SURFACE PHOTOMETRY AND STELLAR MASSES FOR THE FINAL SAMPLE</section_header_level_1>
<text><location><page_1><loc_9><loc_81><loc_90><loc_84></location>Alessandro Sonnenfeld 1 ∗ , Raphael Gavazzi 2 , Sherry H. Suyu 1,3,4 , Tommaso Treu 1 † , and Philip J. Marshall 3,5 Draft version July 19, 2013</text>
<section_header_level_1><location><page_1><loc_45><loc_79><loc_55><loc_80></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_49><loc_86><loc_78></location>We present Hubble Space Telescope ( HST ) imaging data and CFHT Near IR ground-based images for the final sample of 56 candidate galaxy-scale lenses uncovered in the CFHT Legacy Survey as part of the Strong Lensing in the Legacy Survey (SL2S) project. The new images are used to perform lens modeling, measure surface photometry, and estimate stellar masses of the deflector early-type galaxies. Lens modeling is performed on the HST images (or CFHT when HST is not available) by fitting the spatially extended light distribution of the lensed features assuming a singular isothermal ellipsoid mass profile and by reconstructing the intrinsic source light distribution on a pixelized grid. Based on the analysis of systematic uncertainties and comparison with inference based on different methods we estimate that our Einstein Radii are accurate to ∼ 3%. HST imaging provides a much higher success rate in confirming gravitational lenses and measuring their Einstein radii than CFHT imaging does. Lens modeling with ground-based images however, when successful, yields Einstein radius measurements that are competitive with spaced-based images. Information from the lens models is used together with spectroscopic information from the companion paper IV to classify the systems, resulting in a final sample of 39 confirmed (grade-A) lenses and 17 promising candidates (grade-B,C). This represents an increase of half an order of magnitude in sample size with respect to the sample of confirmed lenses studied in papers I and II. The Einstein radii of the confirmed lenses in our sample span the range 5 -15 kpc and are typically larger than those of other surveys, probing the mass in regions where the dark matter contribution is more important. Stellar masses are in the range 10 11 -10 12 M glyph[circledot] , covering the range of massive ETGs. The redshifts of the main deflector span a range 0 . 3 ≤ zd ≤ 0 . 8, which nicely complements low-redshift samples like the SLACS and thus provides an excellent sample for the study of the cosmic evolution of the mass distribution of early-type galaxies over the second half of the history of the Universe.</text>
<text><location><page_1><loc_14><loc_48><loc_70><loc_49></location>Subject headings: galaxies: fundamental parameters - gravitational lensing -</text>
<section_header_level_1><location><page_1><loc_22><loc_44><loc_35><loc_45></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_24><loc_48><loc_43></location>Strong gravitational lensing is a powerful and consolidated technique for measuring the distribution of matter in massive galaxies at cosmological distances. Strong lensing provides, with very few assumptions, a measurement of the projected mass of a galaxy integrated within an aperture to better than a few percent. Early-type galaxy (ETG) lenses in particular have allowed for a number of studies covering relevant topics of cosmology such as the density profile of ETGs (e.g., Rusin et al. 2003a,b; Koopmans & Treu 2004; Barnab'e et al. 2011), the value of the Hubble constant and other cosmological parameters (e.g., Suyu et al. 2010, 2013; Gavazzi et al. 2008), the abundance of mass substructure in galaxies (e.g., Vegetti & Koopmans 2009), the stellar initial mass function (e.g., Treu et al. 2010; Ferreras et al.</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_19><loc_48><loc_21></location>1 Physics Department, University of California, Santa Barbara, CA 93106, USA</list_item>
<list_item><location><page_1><loc_10><loc_16><loc_48><loc_19></location>2 Institut d'Astrophysique de Paris, UMR7095 CNRS - Universit'e Pierre et Marie Curie, 98bis bd Arago, 75014 Paris, France</list_item>
<list_item><location><page_1><loc_11><loc_15><loc_12><loc_16></location>3</list_item>
<list_item><location><page_1><loc_10><loc_12><loc_48><loc_16></location>Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305, USA 4 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan</list_item>
<list_item><location><page_1><loc_10><loc_10><loc_48><loc_12></location>5 Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK</list_item>
<list_item><location><page_1><loc_11><loc_8><loc_28><loc_10></location>* sonnen@physics.ucsb.edu</list_item>
<list_item><location><page_1><loc_11><loc_7><loc_27><loc_8></location>† Packard Research Fellow</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_15><loc_92><loc_45></location>2010) and the shape of dark matter halos (e.g., Sonnenfeld et al. 2012; Grillo 2012). The current number of known early-type galaxy lenses is avobe two hundred. While some of these lenses were serendipitous findings, most of them were discovered in the context of dedicated surveys. The largest such survey to date is the Sloan Lens ACS (SLACS) survey (Bolton et al. 2004), which provided about 80 lenses. Although this sample has yielded interesting results on the properties of ETGs, there are many astrophysical questions that can be better answered with a larger number of strong lenses spanning a larger volume in the space of relevant physical parameters. For instance, quantities like the dark matter fraction or the density slope of ETGs, measurable with lensing and stellar kinematics information, might be correlated with other observables such as the stellar mass or the effective radius. Moreover, the mass structure of ETGs could be evolving in time as a result of the mass accretion history. In order to test this scenario, a statistically significant number of lenses covering a range of redshift is needed. However, most of the galaxy-scale lenses known today are limited at a redshift z < 0 . 3, corresponding to a lookback time of about 3.4 Gyr.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_15></location>One of the goals of the Strong Lensing Legacy Survey (SL2S) collaboration is to extend to higher redshifts the sample of known galaxy-scale gravitational lenses. In Papers I and II (Gavazzi et al. 2012; Ruff et al. 2011) we presented the pilot sample of 16 lenses. Here we extend</text>
<text><location><page_2><loc_8><loc_77><loc_48><loc_92></location>our study to a sample of 56 objects at redshifts up to z = 0 . 8. In this paper we present the lensing models of the new systems along with revisited models of the old ones. Furthermore, we make more conservative assumptions about the intrinsic shape of the lensed sources by reconstructing them on a pixelized grid (Warren & Dye 2003; Suyu et al. 2006; Koopmans & Treu 2004). In a companion paper (Sonnenfeld et al. 2013, hereafter Paper IV) we include the stellar kinematic measurements and address the issue of the time evolution of the density profile of ETGs.</text>
<text><location><page_2><loc_8><loc_49><loc_48><loc_77></location>The goal of this paper is to present our new sample of lenses, characterize it in terms of Einstein radii and stellar masses, and to compare the effectiveness of ground-based versus space-based images for the purpose of confirming gravitational lens candidates. This paper, the third in the series, is organized as follows. Section 2 summarizes the SL2S and the associated Canada-FranceHawaii-Telescope Legacy Survey (CFHTLS) data, the lens detection method and the sample selection. In Section 3 we present all the photometric data set of the SL2S lenses, either coming from the CFHTLS parent photometry or from additional Hubble Space Telescope ( HST ) and Near infrared (IR) follow-up imaging. In Section 4 we describe the lens models of the 56 systems. In Section 5 we show measurements of the stellar mass of our lenses from stellar population synthesis fitting. We discuss and summarize our results in Section 6. Throughout this paper, magnitudes are given in the AB system. When computing distances, we assume a ΛCDM cosmology with matter and dark energy density Ω m = 0 . 3, Ω Λ = 0 . 7, and Hubble constant H 0 =70 km s -1 Mpc -1 .</text>
<section_header_level_1><location><page_2><loc_13><loc_47><loc_44><loc_48></location>2. THE STRONG LENSING LEGACY SURVEY</section_header_level_1>
<text><location><page_2><loc_8><loc_29><loc_48><loc_47></location>SL2S (Cabanac et al. 2007) is a project dedicated to finding and studying galaxy-scale and group-scale strong gravitational lenses in the Canada France Hawaii Telescope Legacy Survey (CFHTLS). The main targets of this paper are massive red galaxies. The galaxy-scale SL2S lenses are found with a procedure described in detail in Paper I (Gavazzi et al. 2012) that can be summarized as follows. We scan the 170 square degrees of the CFHTLS with the automated software RingFinder (Gavazzi et al., in prep.) looking for tangentially elongated blue features around red galaxies. The lens candidates are then visually inspected and the most promising systems are followed up with HST and/or spectroscopy.</text>
<text><location><page_2><loc_8><loc_10><loc_48><loc_29></location>Previous papers have demonstrated the success of this technique. In Paper I (Gavazzi et al. 2012), we obtained lens models for a pilot sample of 16 lenses and in Paper II (Ruff et al. 2011), we combined this information with spectroscopic data to investigate the total mass density profile of the lens galaxies, and its evolution. Here we complete the sample by presenting all the new systems that have been followed-up with either high-resolution imaging or spectroscopy since the start of the campaign. We also re-analyze the pilot sample to ensure consistency. This paper is focused on the sample's photometric data and lens models, while in Paper IV we present the corresponding spectroscopic observations, model the mass density profile of our lenses, and explore the population's evolution with time.</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_9></location>SL2S is by no means the only systematic survey of massive galaxy lenses: two other large strong-lens samples</text>
<table>
<location><page_2><loc_52><loc_80><loc_93><loc_88></location>
<caption>TABLE 1 Census of SL2S lenses.</caption>
</table>
<text><location><page_2><loc_52><loc_73><loc_92><loc_79></location>Note . - Number of SL2S candidates for which we obtained follow-up observations in each quality bin. Grade A: definite lenses, B: probable lenses, C: possible lenses, X: non-lenses. We differentiate between lenses with spectroscopic follow-up, highresolution imaging follow-up or any of the two. In bold font we give the numbers that add up to our overall sample size of 56.</text>
<text><location><page_2><loc_52><loc_28><loc_92><loc_71></location>are those of the SLACS (Bolton et al. 2004) and BELLS (BOSS Emission-Line Lensing Survey; Brownstein et al. 2012) survey. SL2S differs from SLACS and BELLS in the way lenses are found. While we look for lenses in wide-field imaging data, the SLACS and BELLS teams selected candidates by looking for spectroscopic signatures coming from two objects at different redshifts on the same line of sight in the Sloan Digital Sky Survey (SDSS) spectra. These two different techniques correspond to differences in the population of lenses in the respective samples. Given the relatively small fiber used in SDSS spectroscopic observations (1 . '' 5 and 1 '' in radius, for SLACS and BELLS respectively), the spectroscopic surveys tend to select preferentially lenses with small Einstein radii, where both the arc from the lensed source and the deflector can be captured within the fiber. SL2S, on the other hand, finds with higher frequency lenses with Einstein radii above 1 '' , since they can be more clearly resolved in ground-based images (even after the lensed sources have been deblended from the light of the central deflector). At a given redshift, different values of the Einstein radius correspond to different physical radii at which masses can be measured with lensing. For a quantitative estimate of the range of physical radii probed by the different surveys, we plot in Figure 1 the distribution of Einstein radii and the effective radii for lenses from SL2S (determined in Sections 3.1 and 4.1), BELLS (Brownstein et al. 2012) and SLACS (Auger et al. 2010), together with 5 lenses from the LSD study (Treu & Koopmans 2004). The different surveys complement each other nicely, each one providing independent information that cannot be easily gathered from the others.</text>
<text><location><page_2><loc_52><loc_16><loc_92><loc_28></location>In Table 1 we provide a census of SL2S targets that have been followed up so far. The systems are graded according to their subjective likelihood of being strong lenses: grade A are definite lenses, B are probable lenses, C are possible lenses or, more conservatively, systems for which the additional data set does not lead to conclusive answers about their actual strong lensing nature, and, grade X are non-lenses. Grades for individual systems are shown in Table 6 and discussed in Section 4.2.</text>
<text><location><page_2><loc_52><loc_10><loc_92><loc_16></location>In this paper we show detailed measurements of photometric properties, lens models and stellar masses for all grade A lenses and for all grade B and C systems with spectroscopic follow-up. The same selection criterion is applied in Paper IV.</text>
<section_header_level_1><location><page_2><loc_60><loc_7><loc_84><loc_8></location>3. PHOTOMETRIC OBSERVATIONS</section_header_level_1>
<figure>
<location><page_3><loc_10><loc_69><loc_44><loc_90></location>
<caption>Fig. 1.Top Panel: Distribution of Einstein radii, scaled by the effective radius, of lenses from SLACS (Auger et al. 2010), BELLS (Brownstein et al. 2012), LSD (Treu & Koopmans 2004) and grade-A SL2S. Bottom Panel: Same samples shown in the R Ein -R eff plane.</caption>
</figure>
<figure>
<location><page_3><loc_10><loc_46><loc_45><loc_67></location>
</figure>
<text><location><page_3><loc_8><loc_29><loc_48><loc_39></location>All the SL2S lens candidates are first imaged by the CFHT as part of the CFHT Legacy Survey. CFHT optical images are taken with the instrument Megacam in the u, g, r, i, z filters under excellent seeing conditions. The typical FWHM in the g and i bands is 0 . '' 7. We refer to Gavazzi et al. (2012) for a more detailed description of ground-based optical observations.</text>
<text><location><page_3><loc_8><loc_13><loc_48><loc_29></location>The WIRCam (Puget et al. 2004) mounted on the CFHT was used to get Near IR follow-up photometry for some of the SL2S lens galaxies (Programs 11BF01, PI Gavazzi, and 07BF15 PI Soucail) or from existing surveys like WIRDS (Bielby et al. 2010, 2012) 8 or from an ongoing one, called Miracles that is gathering a deep Near IR counter-part to a subset of the CFHTLS in the W1 and W4 fields (Arnouts et al., in prep). All these data were kindly reduced by the Terapix team. 9 Ks (and sometimes also J and H ) band is used for the systems listed in Table 2 to estimate more accurate stellar masses (see 5).</text>
<text><location><page_3><loc_10><loc_12><loc_48><loc_13></location>In addition to ground-based photometry, 33 of the</text>
<text><location><page_3><loc_8><loc_8><loc_48><loc_11></location>8 see also http://terapix.iap.fr/article.php?id_article= 832</text>
<table>
<location><page_3><loc_52><loc_57><loc_94><loc_88></location>
<caption>TABLE 2 Summary of NIR observationsTABLE 3 Summary of HST observations</caption>
</table>
<table>
<location><page_3><loc_53><loc_42><loc_91><loc_51></location>
</table>
<text><location><page_3><loc_52><loc_34><loc_92><loc_42></location>56 lens systems presented here have been observed with HST as part of programs 10876, 11289 (PI Kneib) and 11588 (PI Gavazzi), over the course of cycles 15, 16 and 17 respectively. A summary of HST observations is given in Table 3. The standard data reduction described in Paper I was performed.</text>
<section_header_level_1><location><page_3><loc_61><loc_31><loc_83><loc_32></location>3.1. Properties of lens galaxies</section_header_level_1>
<text><location><page_3><loc_52><loc_7><loc_92><loc_31></location>We wish to measure magnitudes, colors, effective radii, ellipticities and orientations of the stellar components of our lenses. This is done first by using the CFHT data, for all systems. We simultaneously fit for all the above parameters to the full set of images in the 5 optical filters, and NIR bands when available, by using the software spasmoid , developed by M. W. Auger and described in Bennert et al. (2011). Results are reported in Table 4. For systems with available HST data we repeat the fit using HST images alone. The measured parameters are reported in Table 5. Uncertainties on CFHT lens galaxy magnitudes are dominated by contamination from the background source and are estimated to be 0 . 3 in u band, 0 . 2 in g and r , 0 . 1 in all redder bands, while HST magnitudes have an uncertainty of 0.1. Although we used the same data, some of the CFHT magnitudes previously reported for the lenses studied in Paper I and Paper II are slightly inconsistent with the values measured</text>
<figure>
<location><page_4><loc_9><loc_69><loc_45><loc_90></location>
<caption>Fig. 2.Comparison between effective radii measured from ground-based versus space-based photometry. Error bars on HST effective radii represent the assigned 10% systematic uncertainty due to fixing the light profile to a de Vaucouleurs model. The relative scatter between the best fit values of the two measurements is 30% and is used to quantify uncertainties in CFHT effective radii.</caption>
</figure>
<text><location><page_4><loc_8><loc_28><loc_48><loc_60></location>here. This difference is partly due to a different procedure in the masking of the lensed arcs. In Paper I and II, the lensed features were masked out automatically by clipping all the pixels more than 4 σ above the best fit de Vaucouleurs profile obtained by fitting the deflector light distribution with Galfit (Peng et al. 2002, 2010), while here the masks are defined manually for every lens. We verified that this different approach is sufficient for causing the observed mismatch. The masking procedure adopted here is more robust and therefore we consider the new magnitudes more reliable. In addition, the measurements reported in Paper I and Paper II were allowing for different effective radii in different bands and the resulting magnitudes depend on the extrapolation of the light profile at large radii where the signal-to-noise ratio is extremely low. Here we fit for a unique effective radius in all bands, resulting in a more robust determination of relative fluxes, i.e. colors, important for the determination of stellar masses from photometry fitting. We note that this corresponds to an assumption of negligible intrinsic color gradient in the lens galaxies. However, asserting an effective radius that is constant across bandpasses mitigates against the much larger contamination from the lensed source.</text>
<text><location><page_4><loc_8><loc_10><loc_48><loc_28></location>Uncertainties on the HST effective radii are dominated by the choice of the model light profile: different models can fit the data equally well but give different estimates of R eff . The dispersion is ∼ 10%, estimated by repeating the fit with a different surface brightness model, a Hernquist profile, and comparing the newly obtained values of R eff with the de Vaucouleurs ones. Uncertainties on the CFHT effective radii are instead dominated by contamination from the background sources. Effective radii measured from CFHT images are in good agreement with those measured from HST data, when present, as shown in Figure 2. The scatter on the quantity R eff , CFHT -R eff , HST is ∼ 30%; we take this as our uncertainty on CFHT effective radii.</text>
<text><location><page_4><loc_52><loc_77><loc_92><loc_92></location>The main goal is to measure Einstein radii of our lenses. We define the Einstein radius R Ein to be the radius within which the mean surface mass density ¯ Σ( < R Ein ) equals the critical density Σ cr of the lensing configuration. While the critical density depends on the lens and source redshifts, the ratio of ¯ Σ( < R Ein ) / Σ cr (i.e., the convergence) does not: in practice then, the deflection angles and lensed image positions can all be predicted given a model with its Einstein radius in angular units. We only consider Einstein radii in angular units throughout this paper.</text>
<section_header_level_1><location><page_4><loc_66><loc_75><loc_78><loc_76></location>4.1. The method</section_header_level_1>
<text><location><page_4><loc_52><loc_69><loc_92><loc_74></location>We measure Einstein radii by fitting model mass distributions to the lensing data. We describe our lenses as singular isothermal ellipsoids (SIE), with convergence κ given by</text>
<formula><location><page_4><loc_66><loc_66><loc_92><loc_69></location>κ ( x, y ) = R Ein 2 r , (1)</formula>
<text><location><page_4><loc_52><loc_45><loc_92><loc_65></location>where r 2 ≡ qx 2 + y 2 /q and q is the axis ratio of the elliptical isodensity contours. The free parameters of the lens model are therefore R Ein , the axis ratio q , the position angle (PA) of the major axis, and the x and y positions of the centroid. In principle, more degrees of freedom could be introduced. In some cases, lens models are found to require a constant external shear, with strength γ ext and position angle PA ext , in order to describe the lensing effect of massive objects (such as groups or clusters) close to the optical axis. However, this external shear is highly degenerate with the mass orientation of the main lens, and our data are not detailed enough to distinguish between the two. For this reason we only include a shear component for the lenses that we cannot otherwise find a working model.</text>
<text><location><page_4><loc_52><loc_27><loc_92><loc_45></location>The fit is performed by generating model lensed images and comparing them to the observed images that have the lens light subtracted. For fixed lens parameters, light from the image plane is mapped back to a grid on the source plane and the source light distribution is then reconstructed following Suyu et al. (2006). This source reconstruction, as well as the entire lensing analysis, follows a Bayesian approach. For a given model lens, the Bayesian evidence of the source reconstruction is computed, which then defines the quality of the lens model. The lens parameter space is then explored with a MonteCarlo Markov Chain (MCMC) sampler, propagating the source reconstruction evidence as the likelihood of the lens model parameters.</text>
<text><location><page_4><loc_52><loc_14><loc_92><loc_27></location>The practical realization of this procedure is done by using the lens modeling software GLEE , developed by Suyu & Halkola (2010). This approach differs slightly from the one adopted in Paper I, in that a pixelized source reconstruction is used instead of fitting S'ersic components. To make sure that our analysis is robust, we repeat the fit for the systems previously analyzed in Paper I. This allows us to gauge the importance of systematic effects related to the choice of modeling technique.</text>
<text><location><page_4><loc_52><loc_7><loc_92><loc_13></location>For systems with HST imaging in more than one band, only the bluest band image is used for the analysis as the signal from the blue star-forming lensed sources is highest. The g band image is used when modeling CFHT data. Typically we only attempt to model a small re-</text>
<table>
<location><page_5><loc_15><loc_15><loc_86><loc_86></location>
<caption>TABLE 4 Lens light parameters, CFHT photometry.</caption>
</table>
<text><location><page_5><loc_15><loc_9><loc_85><loc_15></location>Note . -Best fit parameters for de Vaucouleurs models of the surface brightness profile of the lens galaxies, after careful manual masking of the lensed images. Columns 2-4 correspond to the effective radius ( R eff ), the axis ratio of the elliptical isophotes ( q ), and the position angle measured east of north (PA). The system SL2SJ140533+550231 has two lens galaxies of comparable magnitude, and the parameters of both galaxies are given. The typical uncertainties are a few degrees for the position angle, ∆ q ∼ 0 . 03 for the axis ratio, 0 . 3 for u -band magnitudes, 0 . 2 for g and r -band magnitudes, 0 . 1 for magnitudes in the remaining bands, 30% on the effective radii.</text>
<table>
<location><page_6><loc_18><loc_28><loc_82><loc_73></location>
<caption>TABLE 5 Lens light parameters, HST photometry.</caption>
</table>
<text><location><page_6><loc_18><loc_22><loc_81><loc_28></location>Note . - Best fit parameters for de Vaucouleurs models of the surface brightness profile of the lens galaxies, after careful manual masking of the lensed images. Columns 2-4 correspond to the effective radius ( R eff ), the axis ratio of the elliptical isophotes ( q ), and the position angle measured east of north (PA). The system SL2SJ140533+550231 has two lens galaxies of comparable magnitude, and the parameters of both galaxies are given. The last column indicates the set of observations used, from the list in Table 3.</text>
<text><location><page_7><loc_8><loc_75><loc_48><loc_92></location>gion of the image around the identified lensed sources, then check that our lens models do not predict detectable lensed sources in areas outside the data region. We assume uniform priors on all the lens parameters except the centroid, for which we use a Gaussian PDF centered on the observed light distribution and with a dispersion of 1 pixel. For systems with only ground-based imaging, for which the lensing signal is diluted by the large PSF, we keep the centroid fixed to that of the optimal light profile. In some cases we also adopt a Gaussian prior on the mass PA, centered on the PA of the light, or we keep the PA fixed. Those cases are individually discussed below.</text>
<text><location><page_7><loc_8><loc_65><loc_48><loc_74></location>Our analysis also allows us to determine the brightness of the lensed sources. This is important information as it allows us to constrain their distance in cases where their spectroscopic redshift is unknown (Ruff et al. 2011). The unlensed magnitude of the background object is recovered by fitting S'ersic components to the reconstructed source.</text>
<text><location><page_7><loc_8><loc_49><loc_48><loc_65></location>The values of the measured lens parameters with 68% credible intervals (1 -σ uncertainties) derived from the posterior probability distribution function marginalized over the remaining parameters are reported in Table 6. Cutouts of the lens systems with the most-probable image and source reconstruction are shown in Figure 3. All images are orientated north up and east left, with the exception of lens models based on WFPC2 data. Those models are performed in the native detector frame in order to avoid degrading further the quality of the WFPC2 images, as they typically have a low S/N. In such situations a compass is displayed to guide the eye.</text>
<text><location><page_7><loc_8><loc_7><loc_48><loc_49></location>The formal uncertainties on the Einstein radius given by the MCMC sampling are typically very small: the 1σ uncertainty is for most lenses smaller than 1%. However, our measurements of the Einstein radius rely partly on the assumption of an SIE profile for the lens mass distribution: in principle, mass models with density slope different from isothermal or isodensity contours different from ellipses can produce different Einstein radii. Perhaps more significantly, some systematic effects can be introduced at various points in our analysis: in particular, the assertions of a specific model PSF, a specific arc mask, and a specific lens light subtraction procedure all induce uncertainty in the final prepared data image ( e.g. Marshall et al. 2007; Suyu et al. 2009). Bolton et al. (2008) estimated the systematic uncertainty on typical Einstein radius measurements to be about 2%. We can further verify this result by comparing Einstein radius measurements from paper I with the new values found here. The analysis of Paper I differs from the present one in the lens light subtraction, choice of the arc mask and lens model technique (S'ersic component fitting versus pixelized source reconstruction), so a comparison of the two different measurements should reflect systematics from most of the effects listed above. For a few of the systems already analyzed in Paper I, the current lens models are qualitatively different from the ones presented in Paper I and the measured values of the Einstein radii are correspondingly different. In most cases this is in virtue of the collection of new data with HST WFC3 that revealed features on the lensed arcs, previously undetected, that helped improve the lens model. After excluding those systems, the relative scatter between the</text>
<text><location><page_7><loc_52><loc_80><loc_92><loc_92></location>most probable values of R Ein measured in the two different approaches (current and previous) is 3%. We thus take 3% as our estimate of the systematic uncertainty on the measurement of the Einstein radius with the technique used here, and convolve the posterior probability distribution for the Einstein radius obtained from the MCMC with a Gaussian with 3% dispersion. All the uncertainties on R Ein quoted in this paper reflect this choice.</text>
<section_header_level_1><location><page_7><loc_66><loc_78><loc_77><loc_79></location>4.2. The lenses</section_header_level_1>
<text><location><page_7><loc_52><loc_71><loc_92><loc_77></location>Although the lens modeling procedure is the same for all lenses, each system has its own peculiarities that need to be taken care of. In what follows we describe briefly and case by case the relevant aspects of those lens models that deserve some discussion.</text>
<text><location><page_7><loc_52><loc_36><loc_92><loc_71></location>Lens grades are also discussed in this subsection, when explanation is needed, and are reported in Table 6. In general we apply the following guidelines. For a system with HST imaging we require, in order for it to be a grade A, that at least a pair of multiple images of the same source is visible and that we can describe it with a robust lens mass model compatible with the light distribution of the lens galaxy (i.e. similar centroid, orientation and axis ratio). For systems with only ground-based imaging we impose the additional requirement of having a spectroscopic detection of the background source, in order to be sure that the blue arcs that we observe are not part of the foreground galaxy. Spectroscopic data therefore enters the lens classification process. We refer to our companion paper (Paper IV) when discussing spectroscopic measurements. Furthermore, systems with a reliable ground-based lens model but no source spectroscopy are given grade B, as well as systems with secure spectroscopic detection of the source but no robust lens model. Systems lacking both, or for which we suspect that strong lensing might not be present are instead given grade C. We stress that a grade is not necessarily a statement on the quality or usefulness of a system as a lens, but rather its likelihood of being a strong lens given the available data. Consequently, grades are subject to change as new data become available.</text>
<unordered_list>
<list_item><location><page_7><loc_54><loc_16><loc_92><loc_35></location>· SL2SJ020833-071414. The lensed features of this system consist of a double image of a bright compact component and a low surface brightness ring. The model cannot fully reproduce the image of the bright double but this is probably due to the presence of a compact unresolved component, like an AGN. AGNs in the source plane are difficult to model with a pixelized reconstruction technique, because the image regularization process smoothes our model images. This effect is present in other lenses with sharp peaks in the source surface brightness distribution. Since the signal-tonoise ratio of the HST image is low and no additional information comes from spectroscopy, this lens is given a grade B.</list_item>
<list_item><location><page_7><loc_54><loc_7><loc_92><loc_15></location>· SL2SJ021325-074355. The source lensed by this high redshift galaxy ( z d = 0 . 717) appears to have two separate bright components. Our source reconstruction confirms this picture. There is a massive elliptical galaxy in the foreground that may be providing extra deflection to the light coming from the</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_12><loc_72><loc_48><loc_92></location>source, thus perturbing the image. This perturber is very close to the observer ( z = 0 . 0161, from SDSS) and therefore its lensing power is greatly reduced with respect to the main deflector. We model the mass distribution from this galaxy with an additional SIE with centroid and PA fixed to the light distribution and R Ein and q as free parameters. To quickly calculate the deflection angles from this perturber we make the approximation that it lies at the same redshift as the main lens. While this is not formally correct, the model still describes qualitatively the presence of an extra source of deflection towards the direction of the foreground galaxy. The impact of this perturber on the lensing model is in any case small.</list_item>
<list_item><location><page_8><loc_11><loc_58><loc_48><loc_71></location>· SL2SJ021411-040502. The source has two bright components, one of which is lensed into the big arc and its fainter couter-image. The second component forms a double of smaller magnification. This lens was modeled in Paper I where we explained how there are two lens models, with different Einstein radii, that match the image configuration. The pixelized source reconstruction technique adopted here to model the lens favors the solution alternative to the one chosen in Paper I.</list_item>
<list_item><location><page_8><loc_11><loc_45><loc_48><loc_57></location>· SL2SJ021737-051329. This lens system is in a cusp configuration, meaning that the source lies just within one of the four cusps of the astroid caustic of the lens. Either a mass centroid offset from the light center or a large shear is required to match the curvature of the big arc. This was also needed in Paper I and previously found by Tu et al. (2009). Here, we find the amount of external shear to be γ ext = 0 . 11 ± 0 . 01</list_item>
<list_item><location><page_8><loc_11><loc_36><loc_48><loc_44></location>· SL2SJ021801-010606. This system shows a nearly complete ring. The redshift of the blue component is 2.06 but we were not able to measure the redshift of the deflector, therefore we label this system as a grade B lens, needing follow-up with deeper spectroscopy.</list_item>
<list_item><location><page_8><loc_11><loc_22><loc_48><loc_35></location>· SL2SJ022346-053418. The CFHT image shows an extended arc and a bright knot at the opposite side with respect to the lens. Although this latter component might be the counter-image to the arc, its color is different and it is not detected spectroscopically. Therefore only the arc is used for the lensing analysis. The main arc has a higher redshift than the lens, however the lens model is not definitive in assessing whether this system is a strong lens. This is therefore a grade B lens.</list_item>
<list_item><location><page_8><loc_11><loc_16><loc_48><loc_21></location>· SL2SJ022357-065142. The lensed source appears to have a complex morphology. We identify three distinct components, each of which is doubly imaged.</list_item>
<list_item><location><page_8><loc_11><loc_7><loc_48><loc_15></location>· SL2SJ084934-043352. Only one arc is visible in the CFHT image. In order to obtain a meaningful lens model we need to fix the PA of the mass profile to that of the light. This system is a grade B due to the lack of spectroscopic detection of the background source.</list_item>
<list_item><location><page_8><loc_54><loc_85><loc_92><loc_92></location>· SL2SJ084959-025142 is a double-like lens system. Part of the light close to the smaller arc is masked out in our analysis, as it is probably a contamination from objects not associated with the lensed source.</list_item>
<list_item><location><page_8><loc_54><loc_75><loc_92><loc_84></location>· SL2SJ085019-034710. The CFHT image shows a bright arc produced by the lensing effect of a disk galaxy. The lens model predicts the presence of a counter-image opposite to the arc, but it is not bright enough to be distinguished from the disk of the lens. In addition, such a counterimage could suffer from substantial extinction.</list_item>
<list_item><location><page_8><loc_54><loc_64><loc_92><loc_74></location>· SL2SJ085559-040917. The main blue arc of this system is at redshift 2.95. The other blue features seen in CFHT data however are too faint for us to establish an unambiguous interpretation of the lens configuration. Therefore we conservatively assign grade B to this system. Higher resolution photometry is needed to confirm this lens.</list_item>
<list_item><location><page_8><loc_54><loc_53><loc_92><loc_63></location>· SL2SJ090106-025906. The WFPC2 image of this system is contaminated with a cosmic ray, which has been masked out in our analysis. Our lens model predicts an image at the position of the cosmic ray, the presence of which cannot be verified with our data. The model however appears to be convincing and the background source is spectroscopically detected, therefore this is a grade A lens.</list_item>
<list_item><location><page_8><loc_54><loc_41><loc_92><loc_51></location>· SL2SJ095921+020638. This system, belonging to the COSMOS survey had previously been modeled by Anguita et al. (2009). These authors report a source redshift of 3 . 14 ± 0 . 05 whereas we find a slightly greater value of 3 . 35 ± 0 . 01 based on our own XSHOOTER data (Paper IV). They report an Einstein radius R Ein ∼ 0 . '' 71 in close agreement with our 0 . '' 74 ± 0 . '' 04 estimate.</list_item>
<list_item><location><page_8><loc_54><loc_36><loc_92><loc_40></location>· SL2SJ135847+545913. We identify two distinct bright components in the source: one forms the big arc, the other one is only doubly-imaged.</list_item>
<list_item><location><page_8><loc_54><loc_19><loc_92><loc_34></location>· SL2SJ140123+555705 is a cusp-like system: three images of a single bright knot can be identified on the arc. The counter-image however is too faint to be detected in the WFC3 snapshot. This lens was already modeled in Paper I. The Einstein radius that we obtain here is not consistent with the value reported then. This is because the current model is obtained by analyzing newly obtained WPC3 data, which reveal more details on the arc. The lack of a counterimage does not prevent an accurate lens modeling because the main arc is very thin, curved and extended.</list_item>
<list_item><location><page_8><loc_54><loc_7><loc_92><loc_17></location>· SL2SJ140533+550231. This is a particular system, in that there are two lens galaxies of comparable brightness. The lensed image shows four images of a bright knot. We model the system with two SIE components, centered in correspondence with the two light components. Our inference shows a substantial degeneracy between the Einstein radii of the two lenses.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_9><loc_11><loc_85><loc_48><loc_92></location>· SL2SJ140546+524311. This system shows a quadruply-imaged bright compact component. Two of the images are almost merged. A relatively large shear is required to match the position and shape of the counter-image opposite to the arcs.</list_item>
<list_item><location><page_9><loc_11><loc_71><loc_48><loc_84></location>· SL2SJ140614+520253. A few different blue blobs are visible in the CFHT image, but there is no working lens model that can associate them with the same source. As done in Paper I, we model only the bright extended arc. The resulting Einstein radius differs from the value of Paper I. This reflects the fact that the interpretation of this system as a lens is not straightforward. This is Grade B until future HST data shed more light on the actual nature of this system.</list_item>
<list_item><location><page_9><loc_11><loc_63><loc_48><loc_69></location>· SL2SJ140650+522619 has a cusp configuration. Even though the source appears to have two separate components, the compact structure outside of the main arc is actually a foreground object, as revealed by our spectroscopic observations.</list_item>
<list_item><location><page_9><loc_11><loc_53><loc_48><loc_61></location>· SL2SJ141137+565119 shows a complete ring. Our lens model cannot account for all the flux in one bright knot on the arc, North of the lens. This could be the result of the presence of substructure close to the highly magnified unresolved knot that requires a minute knowledge of the PSF.</list_item>
<list_item><location><page_9><loc_11><loc_45><loc_48><loc_52></location>· SL2SJ141917+511729. Only two bright points can be identified on the arc of this system, while no counter-image is visible. The Einstein radius of this lens is rather large ( ∼ 4 '' ), which puts this system in the category of group-scale lenses.</list_item>
<list_item><location><page_9><loc_11><loc_37><loc_48><loc_44></location>· SL2SJ142003+523137. This disk galaxy is producing a lensed arc. The reconstructed source is compact and difficult to resolve. The predicted counter-image of the arc is too faint to be detected and possibly affected by extinction.</list_item>
<list_item><location><page_9><loc_11><loc_29><loc_48><loc_36></location>· SL2SJ142059+563007. The WFC3 image of this lens offers a detailed view on the source structure. We identify three separate bright components, two quads and one double, which allow us to constrain robustly the Einstein radius.</list_item>
<list_item><location><page_9><loc_11><loc_24><loc_48><loc_28></location>· SL2SJ142731+551645. The source lensed by this disky galaxy is in a typical fold-like configuration, with two of its four images merging into an arc.</list_item>
<list_item><location><page_9><loc_11><loc_7><loc_48><loc_23></location>· SL2SJ220329+020518. This system shows a bright arc, and a possible counter-image close to the center. However, we are not able to fit both the light from the arc and the candidate counter-image. On the other hand, our spectroscopic analysis reveals OII emission at the redshift of the lens (Paper IV), which suggests that the blue bright spot close to the center might be a substructure associated with the lens and not the source. We model the system using light from the arc only. Our model predicts the existence of a faint counter-image that cannot be ruled out by our snapshot observation.</list_item>
<list_item><location><page_9><loc_54><loc_85><loc_92><loc_92></location>· SL2SJ220506+014703. The spectroscopic followup revealed emission from the bright arc at z = 2 . 52. No emission is detected from its counterimage, but since the lens model is robust we give this lens a grade A.</list_item>
<list_item><location><page_9><loc_54><loc_75><loc_92><loc_84></location>· SL2SJ220629+005728. The image shows a secondary component with a color similar to the main lens, in the proximity of one of the arcs. This component could contribute to the overall lensing effect. We modeled it with a singular isothermal sphere. The fit yielded a small value for its Einstein radius as in Paper I.</list_item>
<list_item><location><page_9><loc_54><loc_69><loc_92><loc_74></location>· SL2SJ221326-000946 is a disky lens. A merging pair and a third image of the same bright component are identified on the arc. No counter-image is visible in our images.</list_item>
<list_item><location><page_9><loc_54><loc_61><loc_92><loc_68></location>· SL2SJ221407-180712. Analogous to other systems with CFHT data only, we need to fix the PA of the mass distribution in order to constrain accurately the Einstein radius. It is a grade B because of the lack of source spectroscopy.</list_item>
<list_item><location><page_9><loc_54><loc_51><loc_92><loc_60></location>· SL2SJ221929-001743. Only one source component, at a spectroscopic redshift of z = 1 . 02, is visible in the CFHT image. The constraints that this image provides on the lens model are not good enough and we need to fix the position angle of the mass to that of the light. The best-fit model does not predict multiple images. Grade B.</list_item>
<list_item><location><page_9><loc_54><loc_30><loc_92><loc_50></location>· SL2SJ222012+010606. The CFHT image shows two blue components on opposite sides of the lens. The brighter arc is measured to be at a higher redshift than the lens, while we have no spectroscopic information on the fainter blob. The lens model that we obtain is only partly satisfactory, because it predicts a mass centroid off by ∼ 1 . 5 pixels from the light centroid. Moreover, the stellar mass and velocity dispersion of the foreground galaxy are unusually low in relation to the measured Einstein radius. It seems then plausible that the secondary source component is not a counterimage to the main arc. The foreground galaxy is definitely providing some lensing, but probably not strong. Grade C.</list_item>
<list_item><location><page_9><loc_54><loc_25><loc_92><loc_29></location>· SL2SJ222148+011542. Two arcs are visible both in photometry and in spectroscopy, making this a grade A lens.</list_item>
<list_item><location><page_9><loc_54><loc_16><loc_92><loc_24></location>· SL2SJ222217+001202. An arc with no clear counter-image is visible in the ground-based image of this lens. We put a Gaussian prior on the lens PA, centered on the light PA and with a 10 degree dispersion, in order to obtain a meaningful model of this lens. Grade B.</list_item>
</unordered_list>
<section_header_level_1><location><page_9><loc_65><loc_13><loc_79><loc_14></location>5. STELLAR MASSES</section_header_level_1>
<text><location><page_9><loc_52><loc_7><loc_92><loc_12></location>One of the goals of our study is to better understand the mass assembly of early-type galaxies over cosmic time. While gravitational lensing provides us with a precise measurement of the total mass enclosed within</text>
<table>
<location><page_10><loc_19><loc_12><loc_81><loc_88></location>
<caption>TABLE 6 Lens model parameters</caption>
</table>
<text><location><page_10><loc_19><loc_4><loc_80><loc_11></location>Note . - Peak value and 68% confidence interval of the posterior probability distribution of each lens parameter, marginalized over the other parameters. Columns 2-4 correspond to the Einstein radius ( R Ein ), the axis ratio of the elliptical isodensity contours ( q ), and the position angle measured east of north (PA) of the SIE lens model. Column 5 shows the magnitude of the de-lensed source in the band used for the lensing analysis: the bluest available band for HST data, or g band for CFHT data. The typical uncertainty on the source magnitude is ∼ 0 . 5. Column 6 lists notes on the lens morphology, while column 7 indicates whether the lens has HST imaging.</text>
<figure>
<location><page_11><loc_12><loc_16><loc_87><loc_88></location>
<caption>Fig. 3.Lens modeling results showing, on each row, from left to right, a color cutout image, the input science imaged used for the modeling with uninteresting areas cropped out, the reconstructed lensed image, the reconstructed intrinsic source and the difference image (data -model) normalized in units of the estimated pixel uncertainties.</caption>
</figure>
<figure>
<location><page_12><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_13><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_14><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_15><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_16><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_17><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_18><loc_12><loc_14><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_19><loc_12><loc_15><loc_88><loc_87></location>
<caption>Fig. 3.continued.</caption>
</figure>
<figure>
<location><page_20><loc_12><loc_68><loc_87><loc_92></location>
<caption>Fig. 3.continued.</caption>
</figure>
<text><location><page_20><loc_8><loc_27><loc_48><loc_64></location>the Einstein radius of our lenses, measurements of the stellar mass are needed to separate the contribution of baryonic and dark matter to the total mass balance. In this paper we estimate stellar masses through stellar population synthesis (SPS) fitting of our photometric measurements: we create stellar populations assuming a simply-parametrized star formation history and stellar initial mass function (IMF), calculate magnitudes in the observed bands and fit to the measurements. The implementation of this procedure is the same as the one in Auger et al. (2009) and is based on a code written by M. W. Auger. We create composite stellar populations from stellar templates by Bruzual & Charlot (2003), with both a Salpeter and a Chabrier IMF. We assume an exponentially declining star formation history, appropriate given the old age of the red galaxies in our sample. In order to obtain robust stellar masses, measurements in a few different bands are needed. Although HST images provide better spatial resolution, useful to deblend the lens light from that of the background source, our objects have HST data in at most two bands which are not enough for the purpose of fitting SPS models. CFHT images on the other hand are deep and available consistently in five different bands for all of the targets. The inclusion of the HST photometry to the overall SED fitting would not bring much new information and we therefore discard it. The fit is based on an MCMC sampling. The measured values of the stellar masses are reported in Table 7.</text>
<text><location><page_20><loc_8><loc_7><loc_48><loc_27></location>For the systems with additional NIR observations the fit is repeated including those data. The addition of NIR fluxes produces stellar masses consistent with the values measured with optical data only, but with smaller uncertainty (see Figure 4). The relative scatter between stellar masses obtained from optical photometry alone and with the addition of NIR data is 0 . 06 dex in log M ∗ and the bias is 0 . 01. This gives us an estimate of the systematic error coming from the stellar templates being not a perfect description of the data over all photometric bands; in Paper IV, this systematic uncertainty is added to the statistical uncertainty on M ∗ when dealing with stellar masses. On the one hand the tight agreement between optical and optical+NIR stellar masses should not come as a surprise since the two data sets differ in most</text>
<figure>
<location><page_20><loc_53><loc_42><loc_88><loc_62></location>
<caption>Fig. 4.-. Comparison of stellar masses obtained with either optical ugriz bands only or with optical + near IR bands, for a Salpeter IMF. We observe no significant differences in the recovered masses.</caption>
</figure>
<text><location><page_20><loc_52><loc_27><loc_92><loc_36></location>cases only by the addition of one band. On the other hand, if the optical data were contaminated with poor subtraction of light from the blue arcs the resulting stellar masses could be biased. The fact that NIR data, with little to no contamination from the background source, does not change the inference is reassuring on the quality of our photometric measurements.</text>
<text><location><page_20><loc_52><loc_19><loc_92><loc_27></location>Some of the stellar masses measured here are not consistent with previous measurements from Paper II. This reflects the difference in the measured magnitudes due to the different source masking strategy discussed in Section 3.1. The values reported here are to be considered more robust.</text>
<text><location><page_20><loc_52><loc_10><loc_92><loc_19></location>The median stellar mass of the sub-sample of grade A SL2S lenses is 10 11 . 53 M glyph[circledot] , if a Salpeter IMF is assumed, and the standard deviation of the sample is 0.3 dex in log M ∗ . The distribution in stellar mass of SL2S galaxies is very similar to that of SLACS galaxies, as shown in Figure 5. This is important in view of analyses that combine data from both samples, as we do in Paper IV.</text>
<table>
<location><page_21><loc_14><loc_14><loc_87><loc_83></location>
<caption>TABLE 7 Stellar mass measurements</caption>
</table>
<text><location><page_21><loc_14><loc_12><loc_86><loc_14></location>Note . - Stellar masses from the fit of stellar population synthesis models to photometric data. The redshift of the lens galaxies is reported in column (2) and extensively discussed in Paper IV.</text>
<figure>
<location><page_22><loc_9><loc_69><loc_46><loc_92></location>
<caption>Fig. 5.Distribution in stellar mass of the grade A SL2S, SLACS and LSD lenses. SLACS stellar masses are from Auger et al. (2010) and LSD masses are taken from Ruff et al. (2011). Stellar masses are obtained assuming a Salpeter IMF.</caption>
</figure>
<text><location><page_22><loc_8><loc_35><loc_48><loc_59></location>We presented photometric measurements, lens models and stellar mass measurements for a sample of 56 systems, of which 39 are grade A (definite lenses) and 15 are grade B (probable lenses). We find that HST imaging, even in snapshot mode, offers a clear-cut way to determine whether SL2S candidates are actual lenses. Not surprisingly, most grade A lenses are found for systems with HST data. 13 of the systems with high-resolution imaging are labeled as grade C lenses, meaning that their nature is undetermined. The data for these systems, not shown in this paper, come largely from WFPC2 snapshot observations. The signal-to-noise ratio of these WFPC2 images is low compared to images taken with ACS or WFC3 despite the longer exposure times. Most of the remaining grade C systems are targets observed with NIR photometry and adaptive optics, which proved not to be a very useful technique for the follow-up of our candidates.</text>
<text><location><page_22><loc_8><loc_13><loc_48><loc_35></location>Ground-based data can be used in some cases to construct lens models and measure precise Einstein radii: 9 out of 23 lenses with only CFHT photometry are grade A lenses. The uncertainty on R Ein for those lenses is still dominated by the 3% systematic error, meaning that ground based photometry can sometimes be as good as space based imaging for the purpose of measuring Einstein radii. For most systems however the information is not enough to draw definite conclusions on their nature, and in a few cases the data does not offer enough constraints to measure Einstein radii, mostly because of the difficulty in detecting and exploiting the counterimage as seen from the ground. The range in Einstein radii covered by the grade A lenses in our sample is 5 -15 kpc, typically larger than those of other surveys such as SLACS, probing the mass in regions where the contribution of dark matter is larger.</text>
<text><location><page_22><loc_8><loc_10><loc_48><loc_13></location>Stellar masses of lens galaxies can be measured from ground-based data. Measurements of M ∗ are robust to</text>
<text><location><page_22><loc_52><loc_81><loc_92><loc_92></location>the inclusion of NIR data. NIR should give more reliable stellar masses, since the blue background sources contribute very little to the infrared flux. Our result suggests that our measurements of the optical photometry of our lenses have little contamination from the background sources, and that we effectively deblended lens and source light. Stellar masses of SL2S lenses cover the range 10 11 -10 12 M glyph[circledot] , corresponding to massive ETGs.</text>
<text><location><page_22><loc_52><loc_77><loc_92><loc_81></location>In Paper IV we use all these measurements to put constraints on the mass profile of massive early-type galaxies and its evolution in the redshfit range 0 . 1 < z < 0 . 8.</text>
<text><location><page_22><loc_52><loc_10><loc_92><loc_75></location>We thank our friends of the SLACS and SL2S collaborations for many useful and insightful discussions over the course of the past years. We thank V.N. Bennert and M. Bradac for their help in our observational campaign. TT thanks S.W. Allen and B. Poggianti for useful discussions. RG acknowledges support from the Centre National des Etudes Spatiales (CNES). PJM acknowledges support from the Royal Society in the form of a research fellowship. TT acknowledges support from the NSF through CAREER award NSF-0642621, and from the Packard Foundation through a Packard Research Fellowship. This research is based on XSHOOTER observations made with ESO Telescopes at the Paranal Observatory under programme IDs 086.B-0407(A) and 089.B-0057(A). This research is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, and with WIRCam, a joint project of CFHT, Taiwan, Korea, Canada and France, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre. The authors would like to thank S. Arnouts, L. Van waerbeke and G. Morrison for giving access to the WIRCam data collected in W1 and W4 as part of additional CFHT programs. We are particularly thankful to Terapix for the data reduction of this dataset. This research is supported by NASA through Hubble Space Telescope programs GO10876, GO-11289, GO-11588 and in part by the National Science Foundation under Grant No. PHY99-07949, and is based on observations made with the NASA/ESA Hubble Space Telescope and obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555, and at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "We present Hubble Space Telescope ( HST ) imaging data and CFHT Near IR ground-based images for the final sample of 56 candidate galaxy-scale lenses uncovered in the CFHT Legacy Survey as part of the Strong Lensing in the Legacy Survey (SL2S) project. The new images are used to perform lens modeling, measure surface photometry, and estimate stellar masses of the deflector early-type galaxies. Lens modeling is performed on the HST images (or CFHT when HST is not available) by fitting the spatially extended light distribution of the lensed features assuming a singular isothermal ellipsoid mass profile and by reconstructing the intrinsic source light distribution on a pixelized grid. Based on the analysis of systematic uncertainties and comparison with inference based on different methods we estimate that our Einstein Radii are accurate to ∼ 3%. HST imaging provides a much higher success rate in confirming gravitational lenses and measuring their Einstein radii than CFHT imaging does. Lens modeling with ground-based images however, when successful, yields Einstein radius measurements that are competitive with spaced-based images. Information from the lens models is used together with spectroscopic information from the companion paper IV to classify the systems, resulting in a final sample of 39 confirmed (grade-A) lenses and 17 promising candidates (grade-B,C). This represents an increase of half an order of magnitude in sample size with respect to the sample of confirmed lenses studied in papers I and II. The Einstein radii of the confirmed lenses in our sample span the range 5 -15 kpc and are typically larger than those of other surveys, probing the mass in regions where the dark matter contribution is more important. Stellar masses are in the range 10 11 -10 12 M glyph[circledot] , covering the range of massive ETGs. The redshifts of the main deflector span a range 0 . 3 ≤ zd ≤ 0 . 8, which nicely complements low-redshift samples like the SLACS and thus provides an excellent sample for the study of the cosmic evolution of the mass distribution of early-type galaxies over the second half of the history of the Universe. Subject headings: galaxies: fundamental parameters - gravitational lensing -",
"pages": [
1
]
},
{
"title": "THE SL2S GALAXY-SCALE LENS SAMPLE. III. LENS MODELS, SURFACE PHOTOMETRY AND STELLAR MASSES FOR THE FINAL SAMPLE",
"content": "Alessandro Sonnenfeld 1 ∗ , Raphael Gavazzi 2 , Sherry H. Suyu 1,3,4 , Tommaso Treu 1 † , and Philip J. Marshall 3,5 Draft version July 19, 2013",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Strong gravitational lensing is a powerful and consolidated technique for measuring the distribution of matter in massive galaxies at cosmological distances. Strong lensing provides, with very few assumptions, a measurement of the projected mass of a galaxy integrated within an aperture to better than a few percent. Early-type galaxy (ETG) lenses in particular have allowed for a number of studies covering relevant topics of cosmology such as the density profile of ETGs (e.g., Rusin et al. 2003a,b; Koopmans & Treu 2004; Barnab'e et al. 2011), the value of the Hubble constant and other cosmological parameters (e.g., Suyu et al. 2010, 2013; Gavazzi et al. 2008), the abundance of mass substructure in galaxies (e.g., Vegetti & Koopmans 2009), the stellar initial mass function (e.g., Treu et al. 2010; Ferreras et al. 2010) and the shape of dark matter halos (e.g., Sonnenfeld et al. 2012; Grillo 2012). The current number of known early-type galaxy lenses is avobe two hundred. While some of these lenses were serendipitous findings, most of them were discovered in the context of dedicated surveys. The largest such survey to date is the Sloan Lens ACS (SLACS) survey (Bolton et al. 2004), which provided about 80 lenses. Although this sample has yielded interesting results on the properties of ETGs, there are many astrophysical questions that can be better answered with a larger number of strong lenses spanning a larger volume in the space of relevant physical parameters. For instance, quantities like the dark matter fraction or the density slope of ETGs, measurable with lensing and stellar kinematics information, might be correlated with other observables such as the stellar mass or the effective radius. Moreover, the mass structure of ETGs could be evolving in time as a result of the mass accretion history. In order to test this scenario, a statistically significant number of lenses covering a range of redshift is needed. However, most of the galaxy-scale lenses known today are limited at a redshift z < 0 . 3, corresponding to a lookback time of about 3.4 Gyr. One of the goals of the Strong Lensing Legacy Survey (SL2S) collaboration is to extend to higher redshifts the sample of known galaxy-scale gravitational lenses. In Papers I and II (Gavazzi et al. 2012; Ruff et al. 2011) we presented the pilot sample of 16 lenses. Here we extend our study to a sample of 56 objects at redshifts up to z = 0 . 8. In this paper we present the lensing models of the new systems along with revisited models of the old ones. Furthermore, we make more conservative assumptions about the intrinsic shape of the lensed sources by reconstructing them on a pixelized grid (Warren & Dye 2003; Suyu et al. 2006; Koopmans & Treu 2004). In a companion paper (Sonnenfeld et al. 2013, hereafter Paper IV) we include the stellar kinematic measurements and address the issue of the time evolution of the density profile of ETGs. The goal of this paper is to present our new sample of lenses, characterize it in terms of Einstein radii and stellar masses, and to compare the effectiveness of ground-based versus space-based images for the purpose of confirming gravitational lens candidates. This paper, the third in the series, is organized as follows. Section 2 summarizes the SL2S and the associated Canada-FranceHawaii-Telescope Legacy Survey (CFHTLS) data, the lens detection method and the sample selection. In Section 3 we present all the photometric data set of the SL2S lenses, either coming from the CFHTLS parent photometry or from additional Hubble Space Telescope ( HST ) and Near infrared (IR) follow-up imaging. In Section 4 we describe the lens models of the 56 systems. In Section 5 we show measurements of the stellar mass of our lenses from stellar population synthesis fitting. We discuss and summarize our results in Section 6. Throughout this paper, magnitudes are given in the AB system. When computing distances, we assume a ΛCDM cosmology with matter and dark energy density Ω m = 0 . 3, Ω Λ = 0 . 7, and Hubble constant H 0 =70 km s -1 Mpc -1 .",
"pages": [
1,
2
]
},
{
"title": "2. THE STRONG LENSING LEGACY SURVEY",
"content": "SL2S (Cabanac et al. 2007) is a project dedicated to finding and studying galaxy-scale and group-scale strong gravitational lenses in the Canada France Hawaii Telescope Legacy Survey (CFHTLS). The main targets of this paper are massive red galaxies. The galaxy-scale SL2S lenses are found with a procedure described in detail in Paper I (Gavazzi et al. 2012) that can be summarized as follows. We scan the 170 square degrees of the CFHTLS with the automated software RingFinder (Gavazzi et al., in prep.) looking for tangentially elongated blue features around red galaxies. The lens candidates are then visually inspected and the most promising systems are followed up with HST and/or spectroscopy. Previous papers have demonstrated the success of this technique. In Paper I (Gavazzi et al. 2012), we obtained lens models for a pilot sample of 16 lenses and in Paper II (Ruff et al. 2011), we combined this information with spectroscopic data to investigate the total mass density profile of the lens galaxies, and its evolution. Here we complete the sample by presenting all the new systems that have been followed-up with either high-resolution imaging or spectroscopy since the start of the campaign. We also re-analyze the pilot sample to ensure consistency. This paper is focused on the sample's photometric data and lens models, while in Paper IV we present the corresponding spectroscopic observations, model the mass density profile of our lenses, and explore the population's evolution with time. SL2S is by no means the only systematic survey of massive galaxy lenses: two other large strong-lens samples Note . - Number of SL2S candidates for which we obtained follow-up observations in each quality bin. Grade A: definite lenses, B: probable lenses, C: possible lenses, X: non-lenses. We differentiate between lenses with spectroscopic follow-up, highresolution imaging follow-up or any of the two. In bold font we give the numbers that add up to our overall sample size of 56. are those of the SLACS (Bolton et al. 2004) and BELLS (BOSS Emission-Line Lensing Survey; Brownstein et al. 2012) survey. SL2S differs from SLACS and BELLS in the way lenses are found. While we look for lenses in wide-field imaging data, the SLACS and BELLS teams selected candidates by looking for spectroscopic signatures coming from two objects at different redshifts on the same line of sight in the Sloan Digital Sky Survey (SDSS) spectra. These two different techniques correspond to differences in the population of lenses in the respective samples. Given the relatively small fiber used in SDSS spectroscopic observations (1 . '' 5 and 1 '' in radius, for SLACS and BELLS respectively), the spectroscopic surveys tend to select preferentially lenses with small Einstein radii, where both the arc from the lensed source and the deflector can be captured within the fiber. SL2S, on the other hand, finds with higher frequency lenses with Einstein radii above 1 '' , since they can be more clearly resolved in ground-based images (even after the lensed sources have been deblended from the light of the central deflector). At a given redshift, different values of the Einstein radius correspond to different physical radii at which masses can be measured with lensing. For a quantitative estimate of the range of physical radii probed by the different surveys, we plot in Figure 1 the distribution of Einstein radii and the effective radii for lenses from SL2S (determined in Sections 3.1 and 4.1), BELLS (Brownstein et al. 2012) and SLACS (Auger et al. 2010), together with 5 lenses from the LSD study (Treu & Koopmans 2004). The different surveys complement each other nicely, each one providing independent information that cannot be easily gathered from the others. In Table 1 we provide a census of SL2S targets that have been followed up so far. The systems are graded according to their subjective likelihood of being strong lenses: grade A are definite lenses, B are probable lenses, C are possible lenses or, more conservatively, systems for which the additional data set does not lead to conclusive answers about their actual strong lensing nature, and, grade X are non-lenses. Grades for individual systems are shown in Table 6 and discussed in Section 4.2. In this paper we show detailed measurements of photometric properties, lens models and stellar masses for all grade A lenses and for all grade B and C systems with spectroscopic follow-up. The same selection criterion is applied in Paper IV.",
"pages": [
2
]
},
{
"title": "3. PHOTOMETRIC OBSERVATIONS",
"content": "All the SL2S lens candidates are first imaged by the CFHT as part of the CFHT Legacy Survey. CFHT optical images are taken with the instrument Megacam in the u, g, r, i, z filters under excellent seeing conditions. The typical FWHM in the g and i bands is 0 . '' 7. We refer to Gavazzi et al. (2012) for a more detailed description of ground-based optical observations. The WIRCam (Puget et al. 2004) mounted on the CFHT was used to get Near IR follow-up photometry for some of the SL2S lens galaxies (Programs 11BF01, PI Gavazzi, and 07BF15 PI Soucail) or from existing surveys like WIRDS (Bielby et al. 2010, 2012) 8 or from an ongoing one, called Miracles that is gathering a deep Near IR counter-part to a subset of the CFHTLS in the W1 and W4 fields (Arnouts et al., in prep). All these data were kindly reduced by the Terapix team. 9 Ks (and sometimes also J and H ) band is used for the systems listed in Table 2 to estimate more accurate stellar masses (see 5). In addition to ground-based photometry, 33 of the 8 see also http://terapix.iap.fr/article.php?id_article= 832 56 lens systems presented here have been observed with HST as part of programs 10876, 11289 (PI Kneib) and 11588 (PI Gavazzi), over the course of cycles 15, 16 and 17 respectively. A summary of HST observations is given in Table 3. The standard data reduction described in Paper I was performed.",
"pages": [
3
]
},
{
"title": "3.1. Properties of lens galaxies",
"content": "We wish to measure magnitudes, colors, effective radii, ellipticities and orientations of the stellar components of our lenses. This is done first by using the CFHT data, for all systems. We simultaneously fit for all the above parameters to the full set of images in the 5 optical filters, and NIR bands when available, by using the software spasmoid , developed by M. W. Auger and described in Bennert et al. (2011). Results are reported in Table 4. For systems with available HST data we repeat the fit using HST images alone. The measured parameters are reported in Table 5. Uncertainties on CFHT lens galaxy magnitudes are dominated by contamination from the background source and are estimated to be 0 . 3 in u band, 0 . 2 in g and r , 0 . 1 in all redder bands, while HST magnitudes have an uncertainty of 0.1. Although we used the same data, some of the CFHT magnitudes previously reported for the lenses studied in Paper I and Paper II are slightly inconsistent with the values measured here. This difference is partly due to a different procedure in the masking of the lensed arcs. In Paper I and II, the lensed features were masked out automatically by clipping all the pixels more than 4 σ above the best fit de Vaucouleurs profile obtained by fitting the deflector light distribution with Galfit (Peng et al. 2002, 2010), while here the masks are defined manually for every lens. We verified that this different approach is sufficient for causing the observed mismatch. The masking procedure adopted here is more robust and therefore we consider the new magnitudes more reliable. In addition, the measurements reported in Paper I and Paper II were allowing for different effective radii in different bands and the resulting magnitudes depend on the extrapolation of the light profile at large radii where the signal-to-noise ratio is extremely low. Here we fit for a unique effective radius in all bands, resulting in a more robust determination of relative fluxes, i.e. colors, important for the determination of stellar masses from photometry fitting. We note that this corresponds to an assumption of negligible intrinsic color gradient in the lens galaxies. However, asserting an effective radius that is constant across bandpasses mitigates against the much larger contamination from the lensed source. Uncertainties on the HST effective radii are dominated by the choice of the model light profile: different models can fit the data equally well but give different estimates of R eff . The dispersion is ∼ 10%, estimated by repeating the fit with a different surface brightness model, a Hernquist profile, and comparing the newly obtained values of R eff with the de Vaucouleurs ones. Uncertainties on the CFHT effective radii are instead dominated by contamination from the background sources. Effective radii measured from CFHT images are in good agreement with those measured from HST data, when present, as shown in Figure 2. The scatter on the quantity R eff , CFHT -R eff , HST is ∼ 30%; we take this as our uncertainty on CFHT effective radii. The main goal is to measure Einstein radii of our lenses. We define the Einstein radius R Ein to be the radius within which the mean surface mass density ¯ Σ( < R Ein ) equals the critical density Σ cr of the lensing configuration. While the critical density depends on the lens and source redshifts, the ratio of ¯ Σ( < R Ein ) / Σ cr (i.e., the convergence) does not: in practice then, the deflection angles and lensed image positions can all be predicted given a model with its Einstein radius in angular units. We only consider Einstein radii in angular units throughout this paper.",
"pages": [
3,
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]
},
{
"title": "4.1. The method",
"content": "We measure Einstein radii by fitting model mass distributions to the lensing data. We describe our lenses as singular isothermal ellipsoids (SIE), with convergence κ given by where r 2 ≡ qx 2 + y 2 /q and q is the axis ratio of the elliptical isodensity contours. The free parameters of the lens model are therefore R Ein , the axis ratio q , the position angle (PA) of the major axis, and the x and y positions of the centroid. In principle, more degrees of freedom could be introduced. In some cases, lens models are found to require a constant external shear, with strength γ ext and position angle PA ext , in order to describe the lensing effect of massive objects (such as groups or clusters) close to the optical axis. However, this external shear is highly degenerate with the mass orientation of the main lens, and our data are not detailed enough to distinguish between the two. For this reason we only include a shear component for the lenses that we cannot otherwise find a working model. The fit is performed by generating model lensed images and comparing them to the observed images that have the lens light subtracted. For fixed lens parameters, light from the image plane is mapped back to a grid on the source plane and the source light distribution is then reconstructed following Suyu et al. (2006). This source reconstruction, as well as the entire lensing analysis, follows a Bayesian approach. For a given model lens, the Bayesian evidence of the source reconstruction is computed, which then defines the quality of the lens model. The lens parameter space is then explored with a MonteCarlo Markov Chain (MCMC) sampler, propagating the source reconstruction evidence as the likelihood of the lens model parameters. The practical realization of this procedure is done by using the lens modeling software GLEE , developed by Suyu & Halkola (2010). This approach differs slightly from the one adopted in Paper I, in that a pixelized source reconstruction is used instead of fitting S'ersic components. To make sure that our analysis is robust, we repeat the fit for the systems previously analyzed in Paper I. This allows us to gauge the importance of systematic effects related to the choice of modeling technique. For systems with HST imaging in more than one band, only the bluest band image is used for the analysis as the signal from the blue star-forming lensed sources is highest. The g band image is used when modeling CFHT data. Typically we only attempt to model a small re- Note . -Best fit parameters for de Vaucouleurs models of the surface brightness profile of the lens galaxies, after careful manual masking of the lensed images. Columns 2-4 correspond to the effective radius ( R eff ), the axis ratio of the elliptical isophotes ( q ), and the position angle measured east of north (PA). The system SL2SJ140533+550231 has two lens galaxies of comparable magnitude, and the parameters of both galaxies are given. The typical uncertainties are a few degrees for the position angle, ∆ q ∼ 0 . 03 for the axis ratio, 0 . 3 for u -band magnitudes, 0 . 2 for g and r -band magnitudes, 0 . 1 for magnitudes in the remaining bands, 30% on the effective radii. Note . - Best fit parameters for de Vaucouleurs models of the surface brightness profile of the lens galaxies, after careful manual masking of the lensed images. Columns 2-4 correspond to the effective radius ( R eff ), the axis ratio of the elliptical isophotes ( q ), and the position angle measured east of north (PA). The system SL2SJ140533+550231 has two lens galaxies of comparable magnitude, and the parameters of both galaxies are given. The last column indicates the set of observations used, from the list in Table 3. gion of the image around the identified lensed sources, then check that our lens models do not predict detectable lensed sources in areas outside the data region. We assume uniform priors on all the lens parameters except the centroid, for which we use a Gaussian PDF centered on the observed light distribution and with a dispersion of 1 pixel. For systems with only ground-based imaging, for which the lensing signal is diluted by the large PSF, we keep the centroid fixed to that of the optimal light profile. In some cases we also adopt a Gaussian prior on the mass PA, centered on the PA of the light, or we keep the PA fixed. Those cases are individually discussed below. Our analysis also allows us to determine the brightness of the lensed sources. This is important information as it allows us to constrain their distance in cases where their spectroscopic redshift is unknown (Ruff et al. 2011). The unlensed magnitude of the background object is recovered by fitting S'ersic components to the reconstructed source. The values of the measured lens parameters with 68% credible intervals (1 -σ uncertainties) derived from the posterior probability distribution function marginalized over the remaining parameters are reported in Table 6. Cutouts of the lens systems with the most-probable image and source reconstruction are shown in Figure 3. All images are orientated north up and east left, with the exception of lens models based on WFPC2 data. Those models are performed in the native detector frame in order to avoid degrading further the quality of the WFPC2 images, as they typically have a low S/N. In such situations a compass is displayed to guide the eye. The formal uncertainties on the Einstein radius given by the MCMC sampling are typically very small: the 1σ uncertainty is for most lenses smaller than 1%. However, our measurements of the Einstein radius rely partly on the assumption of an SIE profile for the lens mass distribution: in principle, mass models with density slope different from isothermal or isodensity contours different from ellipses can produce different Einstein radii. Perhaps more significantly, some systematic effects can be introduced at various points in our analysis: in particular, the assertions of a specific model PSF, a specific arc mask, and a specific lens light subtraction procedure all induce uncertainty in the final prepared data image ( e.g. Marshall et al. 2007; Suyu et al. 2009). Bolton et al. (2008) estimated the systematic uncertainty on typical Einstein radius measurements to be about 2%. We can further verify this result by comparing Einstein radius measurements from paper I with the new values found here. The analysis of Paper I differs from the present one in the lens light subtraction, choice of the arc mask and lens model technique (S'ersic component fitting versus pixelized source reconstruction), so a comparison of the two different measurements should reflect systematics from most of the effects listed above. For a few of the systems already analyzed in Paper I, the current lens models are qualitatively different from the ones presented in Paper I and the measured values of the Einstein radii are correspondingly different. In most cases this is in virtue of the collection of new data with HST WFC3 that revealed features on the lensed arcs, previously undetected, that helped improve the lens model. After excluding those systems, the relative scatter between the most probable values of R Ein measured in the two different approaches (current and previous) is 3%. We thus take 3% as our estimate of the systematic uncertainty on the measurement of the Einstein radius with the technique used here, and convolve the posterior probability distribution for the Einstein radius obtained from the MCMC with a Gaussian with 3% dispersion. All the uncertainties on R Ein quoted in this paper reflect this choice.",
"pages": [
4,
5,
6,
7
]
},
{
"title": "4.2. The lenses",
"content": "Although the lens modeling procedure is the same for all lenses, each system has its own peculiarities that need to be taken care of. In what follows we describe briefly and case by case the relevant aspects of those lens models that deserve some discussion. Lens grades are also discussed in this subsection, when explanation is needed, and are reported in Table 6. In general we apply the following guidelines. For a system with HST imaging we require, in order for it to be a grade A, that at least a pair of multiple images of the same source is visible and that we can describe it with a robust lens mass model compatible with the light distribution of the lens galaxy (i.e. similar centroid, orientation and axis ratio). For systems with only ground-based imaging we impose the additional requirement of having a spectroscopic detection of the background source, in order to be sure that the blue arcs that we observe are not part of the foreground galaxy. Spectroscopic data therefore enters the lens classification process. We refer to our companion paper (Paper IV) when discussing spectroscopic measurements. Furthermore, systems with a reliable ground-based lens model but no source spectroscopy are given grade B, as well as systems with secure spectroscopic detection of the source but no robust lens model. Systems lacking both, or for which we suspect that strong lensing might not be present are instead given grade C. We stress that a grade is not necessarily a statement on the quality or usefulness of a system as a lens, but rather its likelihood of being a strong lens given the available data. Consequently, grades are subject to change as new data become available.",
"pages": [
7
]
},
{
"title": "5. STELLAR MASSES",
"content": "One of the goals of our study is to better understand the mass assembly of early-type galaxies over cosmic time. While gravitational lensing provides us with a precise measurement of the total mass enclosed within Note . - Peak value and 68% confidence interval of the posterior probability distribution of each lens parameter, marginalized over the other parameters. Columns 2-4 correspond to the Einstein radius ( R Ein ), the axis ratio of the elliptical isodensity contours ( q ), and the position angle measured east of north (PA) of the SIE lens model. Column 5 shows the magnitude of the de-lensed source in the band used for the lensing analysis: the bluest available band for HST data, or g band for CFHT data. The typical uncertainty on the source magnitude is ∼ 0 . 5. Column 6 lists notes on the lens morphology, while column 7 indicates whether the lens has HST imaging. the Einstein radius of our lenses, measurements of the stellar mass are needed to separate the contribution of baryonic and dark matter to the total mass balance. In this paper we estimate stellar masses through stellar population synthesis (SPS) fitting of our photometric measurements: we create stellar populations assuming a simply-parametrized star formation history and stellar initial mass function (IMF), calculate magnitudes in the observed bands and fit to the measurements. The implementation of this procedure is the same as the one in Auger et al. (2009) and is based on a code written by M. W. Auger. We create composite stellar populations from stellar templates by Bruzual & Charlot (2003), with both a Salpeter and a Chabrier IMF. We assume an exponentially declining star formation history, appropriate given the old age of the red galaxies in our sample. In order to obtain robust stellar masses, measurements in a few different bands are needed. Although HST images provide better spatial resolution, useful to deblend the lens light from that of the background source, our objects have HST data in at most two bands which are not enough for the purpose of fitting SPS models. CFHT images on the other hand are deep and available consistently in five different bands for all of the targets. The inclusion of the HST photometry to the overall SED fitting would not bring much new information and we therefore discard it. The fit is based on an MCMC sampling. The measured values of the stellar masses are reported in Table 7. For the systems with additional NIR observations the fit is repeated including those data. The addition of NIR fluxes produces stellar masses consistent with the values measured with optical data only, but with smaller uncertainty (see Figure 4). The relative scatter between stellar masses obtained from optical photometry alone and with the addition of NIR data is 0 . 06 dex in log M ∗ and the bias is 0 . 01. This gives us an estimate of the systematic error coming from the stellar templates being not a perfect description of the data over all photometric bands; in Paper IV, this systematic uncertainty is added to the statistical uncertainty on M ∗ when dealing with stellar masses. On the one hand the tight agreement between optical and optical+NIR stellar masses should not come as a surprise since the two data sets differ in most cases only by the addition of one band. On the other hand, if the optical data were contaminated with poor subtraction of light from the blue arcs the resulting stellar masses could be biased. The fact that NIR data, with little to no contamination from the background source, does not change the inference is reassuring on the quality of our photometric measurements. Some of the stellar masses measured here are not consistent with previous measurements from Paper II. This reflects the difference in the measured magnitudes due to the different source masking strategy discussed in Section 3.1. The values reported here are to be considered more robust. The median stellar mass of the sub-sample of grade A SL2S lenses is 10 11 . 53 M glyph[circledot] , if a Salpeter IMF is assumed, and the standard deviation of the sample is 0.3 dex in log M ∗ . The distribution in stellar mass of SL2S galaxies is very similar to that of SLACS galaxies, as shown in Figure 5. This is important in view of analyses that combine data from both samples, as we do in Paper IV. Note . - Stellar masses from the fit of stellar population synthesis models to photometric data. The redshift of the lens galaxies is reported in column (2) and extensively discussed in Paper IV. We presented photometric measurements, lens models and stellar mass measurements for a sample of 56 systems, of which 39 are grade A (definite lenses) and 15 are grade B (probable lenses). We find that HST imaging, even in snapshot mode, offers a clear-cut way to determine whether SL2S candidates are actual lenses. Not surprisingly, most grade A lenses are found for systems with HST data. 13 of the systems with high-resolution imaging are labeled as grade C lenses, meaning that their nature is undetermined. The data for these systems, not shown in this paper, come largely from WFPC2 snapshot observations. The signal-to-noise ratio of these WFPC2 images is low compared to images taken with ACS or WFC3 despite the longer exposure times. Most of the remaining grade C systems are targets observed with NIR photometry and adaptive optics, which proved not to be a very useful technique for the follow-up of our candidates. Ground-based data can be used in some cases to construct lens models and measure precise Einstein radii: 9 out of 23 lenses with only CFHT photometry are grade A lenses. The uncertainty on R Ein for those lenses is still dominated by the 3% systematic error, meaning that ground based photometry can sometimes be as good as space based imaging for the purpose of measuring Einstein radii. For most systems however the information is not enough to draw definite conclusions on their nature, and in a few cases the data does not offer enough constraints to measure Einstein radii, mostly because of the difficulty in detecting and exploiting the counterimage as seen from the ground. The range in Einstein radii covered by the grade A lenses in our sample is 5 -15 kpc, typically larger than those of other surveys such as SLACS, probing the mass in regions where the contribution of dark matter is larger. Stellar masses of lens galaxies can be measured from ground-based data. Measurements of M ∗ are robust to the inclusion of NIR data. NIR should give more reliable stellar masses, since the blue background sources contribute very little to the infrared flux. Our result suggests that our measurements of the optical photometry of our lenses have little contamination from the background sources, and that we effectively deblended lens and source light. Stellar masses of SL2S lenses cover the range 10 11 -10 12 M glyph[circledot] , corresponding to massive ETGs. In Paper IV we use all these measurements to put constraints on the mass profile of massive early-type galaxies and its evolution in the redshfit range 0 . 1 < z < 0 . 8. We thank our friends of the SLACS and SL2S collaborations for many useful and insightful discussions over the course of the past years. We thank V.N. Bennert and M. Bradac for their help in our observational campaign. TT thanks S.W. Allen and B. Poggianti for useful discussions. RG acknowledges support from the Centre National des Etudes Spatiales (CNES). PJM acknowledges support from the Royal Society in the form of a research fellowship. TT acknowledges support from the NSF through CAREER award NSF-0642621, and from the Packard Foundation through a Packard Research Fellowship. This research is based on XSHOOTER observations made with ESO Telescopes at the Paranal Observatory under programme IDs 086.B-0407(A) and 089.B-0057(A). This research is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, and with WIRCam, a joint project of CFHT, Taiwan, Korea, Canada and France, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre. The authors would like to thank S. Arnouts, L. Van waerbeke and G. Morrison for giving access to the WIRCam data collected in W1 and W4 as part of additional CFHT programs. We are particularly thankful to Terapix for the data reduction of this dataset. This research is supported by NASA through Hubble Space Telescope programs GO10876, GO-11289, GO-11588 and in part by the National Science Foundation under Grant No. PHY99-07949, and is based on observations made with the NASA/ESA Hubble Space Telescope and obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555, and at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.",
"pages": [
9,
10,
20,
21,
22
]
},
{
"title": "REFERENCES",
"content": "Suyu, S. H., Auger, M. W., Hilbert, S., Marshall, P. J., Tewes, M., Treu, T., Fassnacht, C. D., Koopmans, L. V. E., Sluse, D., Blandford, R. D., Courbin, F., & Meylan, G. 2013, ApJ, 766, 70 Suyu, S. H., & Halkola, A. 2010, A&A, 524, A94",
"pages": [
23
]
}
]
2013ApJ...777..102Q
https://arxiv.org/pdf/1309.0090.pdf
<document>
<section_header_level_1><location><page_1><loc_18><loc_77><loc_82><loc_81></location>The disk evaporation model for the spectral features of low-luminosity active galactic nuclei</section_header_level_1>
<text><location><page_1><loc_24><loc_73><loc_75><loc_75></location>Erlin Qiao 1 , B. F. Liu 1 , Francesca Panessa 2 and J. Y. Liu 3</text>
<text><location><page_1><loc_41><loc_70><loc_58><loc_71></location>qiaoel@nao.cas.cn</text>
<section_header_level_1><location><page_1><loc_44><loc_65><loc_56><loc_67></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_23><loc_83><loc_62></location>Observations show that the accretion flows in low-luminosity active galactic nuclei (LLAGNs) probably have a two-component structure with an inner hot, optically thin, advection dominated accretion flow (ADAF) and an outer truncated cool, optically thick accretion disk. As shown by Taam et al. (2012), the truncation radius as a function of mass accretion rate is strongly affected by including the magnetic field within the framework of disk evaporation model, i.e., an increase of the magnetic field results in a smaller truncation radius of the accretion disk. In this work, we calculate the emergent spectrum of an inner ADAF + an outer truncated accretion disk around a supermassive black hole based on the prediction by Taam et al. (2012). It is found that an increase of the magnetic field from β = 0 . 8 to β = 0 . 5 (with magnetic pressure p m = B 2 / 8 π = (1 -β ) p tot , p tot = p gas + p m ) results in an increase of ∼ 8 . 7 times of the luminosity from the truncated accretion disk, meanwhile results in the peak emission of the truncated accretion disk shifting towards a higher frequency by a factor of ∼ 5 times. We found that the equipartition of gas pressure to magnetic pressure, i.e., β = 0 . 5, failed to explain the observed anticorrelation between L 2 -10keV /L Edd and the bolometric correction κ 2 -10keV (with κ 2 -10keV = L bol /L 2 -10keV ). The emergent spectra for larger value β = 0 . 8 or β = 0 . 95 can well explain the observed L 2 -10keV /L Edd -κ 2 -10keV correlation. We argue that in the disk evaporation model, the electrons in the corona are assumed</text>
<text><location><page_2><loc_17><loc_76><loc_83><loc_86></location>to be heated only by a transfer of energy from the ions to electrons via Coulomb collisions, which is reasonable for the accretion with a lower mass accretion rate. Coulomb heating is the dominated heating mechanism for the electrons only if the magnetic field is strongly sub-equipartition, which is roughly consistent with observations.</text>
<text><location><page_2><loc_17><loc_71><loc_83><loc_74></location>Subject headings: accretion, accretion disks - Black hole physics - galaxies: active - X-rays: galaxies</text>
<section_header_level_1><location><page_2><loc_42><loc_64><loc_58><loc_66></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_17><loc_88><loc_62></location>Active galactic nuclei (AGNs) are believed to be powered by accretion onto supermassive black holes, which are strong sources emitting from radio to X-rays. The luminous AGNs, mainly including radio quiet quasars and bright Seyfert galaxies, are believed to be powered dominantly by a geometrically thin, optically thick, accretion disk extending down to the innermost stable circular orbits (ISCO) of a black hole (Shakura & Sunyaev 1973; Shields 1978; Malkan & Sargent 1982; Kishimoto, Antouncci & Blaes 2005; Shang et al. 2005; Liu et al. 2012). Observations show that accretion flows in low-luminosity active galactic nuclei (LLAGNs) are very different. A two-component structure of the accretion flow with an inner hot, optically thin, advection dominated accretion flow (ADAF), an outer cool, optically thick truncated accretion disk, and a jet are supported by a number of observational evidence for LLAGNs (Ho 2008 and the references therein). The observational evidence for such a structure in LLAGNs comes mainly from the broadband emission from the radio to X-rays (Lasota et al. 1996; Quataert et al. 1999; Di Matteo et al.2000, 2003; Yuan et al. 2009; Li et al. 2009). For example, by fitting the spectral energy distribution (SED) of NGC 1097, Nemmen et al. (2006) found that the optical and X-ray portion of the SED can be well fitted by an inner ADAF and an outer accretion disk truncated at 225 Schwarzschild radii, and the observed radio emission can be very well interpreted by the synchrotron emission of a relativistic jet modeled within the framework of the internal shock scenario (Yuan et al. 2005). The presence of a jet in LLAGNs has been clearly detected by the VLBI radio observations (Falcke et al. 2000; Nagar et al. 2001). Meanwhile, theoretically, because of the positive Bernoulli parameter of the hot ADAF, it is probably that the formation of the bipolar jets are driven by the ADAF (Narayan & Yi 1995b; Blandford & Begelman 1999; Meier et al. 2001; Yuan et al. 2012a, b).</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_15></location>The evidence for the presence of an inner ADAF in LLAGNs is inferred from the very low luminosity and the very lower radiative efficiency estimated from the available mass supply rate (Ho 2009). In terms of Eddington luminosity, LLAGNs often have λ /lessorsimilar 0 . 01 contrary to</text>
<text><location><page_3><loc_12><loc_70><loc_88><loc_86></location>the luminous AGNs with λ /greaterorsimilar 0 . 01 (with λ = L bol /L Edd , L Edd = 1 . 26 × 10 38 M/M /circledot erg s -1 ) (Panessa et al. 2006). The observed anti-correlation between the hard X-ray index Γ 2 -10keV and λ implies that LLAGNs may be dominated by ADAF (Gu & Cao 2009; Younes et al. 2011, 2012). Xu (2011) collected a LLAGN sample composed of 49 sources including 28 local Seyfert galaxies and 21 low-ionization nuclear emission-line regions (LINERs) with optical/UV and X-ray observations. It is found that there is a strong anti-correlation between α ox and Eddington ratio λ for the sources with λ /lessorsimilar 10 -3 , which supports that ADAF model is a very promising candidate in LLAGNs.</text>
<text><location><page_3><loc_12><loc_31><loc_88><loc_69></location>The evidence for the truncation of the accretion disk in LLAGNs is inferred from the lack of a 'big blue bump', instead of a 'big red bump' (Nemmen et al. 2012). The very weak or absent broad iron K α line, which is attributed to X-ray fluorescence off of an optically thick accretion disk extending down to a few Schwarzschild radii of luminous AGNs, also supports that the accretion disk truncates at a larger radius ( ∼ 100 -1000 Schwarzschild radii) off the black hole in LLAGNs (Nandra et al. 2007). Quataert et al. (1999) fitted the optical/UV spectrum of M81 with a mass accretion rate of 0 . 01 Eddington accretion rate and an accretion disk truncated at 100 Schwarzschild radii. A similar fit to the optical/UV spectrum of NGC 4579 yielded a higher mass accretion rate of 0 . 03 Eddington accretion rate, while the truncation radius of the accretion disk is still 100 Schwarzschild radii (Quataert et al. 1999). The relationship between the truncation radius and the mass accretion rate is still controversial. By fitting the SEDs of a population of LLAGNs, Yuan & Narayan (2004) found that the lower Eddington ratio L bol /L Edd was associated with a larger truncation radius. By studying 33 PG quasars with Fe K α emission line detected by the XMM-Newton survey, Inoue, Terashima & Ho (2007) found that the Fe K α line systematically becomes narrow with decreasing L bol /L Edd , which probably means that the truncation radius of the accretion disk increases with decreasing mass accretion rate. It is interesting that the inverse correlation between the truncation radius of the accretion disk and the mass accretion rate is also found in the low/hard spectral state of black hole X-ray binaries (Cabanac et al. 2009).</text>
<text><location><page_3><loc_12><loc_10><loc_88><loc_30></location>The physical mechanism for determining the truncation of the accretion disk has been proposed by many authors (Honma 1996; Manmoto & Kato 2000; Lu et al. 2004; Spruit & Deufel 2002; Dullemond & Spruit 2005). One of the promising model among them is the disk evaporation model, which was first proposed by Meyer & Meyer-Hofmeister (1994) for dwarf novae, established for black holes by Meyer et al. (2000a, b) and modified by Liu et al. (2002), where the decoupling of ions and electrons and Compton cooling effect are taken into account. The disk evaporation model has been applied to explain the spectral state transition in stellar-mass black holes (e.g., Meyer-Hofmeister & Meyer 2005; Qiao & Liu 2009; Qian et al. 2007). The spectral features of an inner ADAF and an outer truncated accretion disk predicted by the disk evaporation model in stellar-mass black holes were investigated</text>
<text><location><page_4><loc_12><loc_64><loc_88><loc_86></location>by Qiao & Liu (2010, 2012). The first application of disk evaporation model to interpret the truncation of the accretion disk in LLAGNs was from Liu et al. (1999), in which a theoretical relation between the truncation radius and the mass accretion rate was given. Taam et al. (2012) generalized the disk evaporation model in black hole X-ray binaries by including the effect of a magnetic field in accretion disk to apply the model to the truncation of the accretion disk in a large number of LLAGNs. It is found that the truncation radius of the accretion disk can be very strongly affected by the magnetic parameters β , which is defined as p m = B 2 / 8 π = (1 -β ) p tot , (where p tot = p gas + p m , p gas is gas pressure and p m is magnetic pressure), describing the strength of the magnetic field in the accretion flows. Taam et al. (2012) found that the inclusion of magnetic field results in a smaller truncation radius compared to the case without magnetic field.</text>
<text><location><page_4><loc_12><loc_33><loc_88><loc_63></location>In this work, based on the prediction by Taam et al. (2012) for the truncation of the accretion disk, the emergent spectrum of a two-component structure with an inner ADAF and an outer truncated accretion disk around a supermassive black hole is calculated. It is found that the disk evaporation model can roughly reproduce the observed anti-correlation between the hard X-ray index Γ 2 -10keV and the Eddington ratio λ in LLAGNs (e.g., Gu & Cao 2009). As shown by Taam et al. (2012), the truncation radius of the accretion disk is very sensitive to β , which consequently affects the emergent spectrum. Our calculations show that the equipartition of gas pressure to magnetic pressure, i.e., β = 0 . 5, failed to explain the observed anti-correlation between L 2 -10keV /L Edd and the bolometric correction κ 2 -10keV (with κ 2 -10keV = L bol /L 2 -10keV ). The emergent spectra for larger value β = 0 . 8 or β = 0 . 95 can well explain the observed L 2 -10keV /L Edd -κ 2 -10keV correlation. It is argued that the sub-equipartition of the magnetic field is reasonable in the case with a low mass accretion. In section 2, we briefly introduce the disk evaporation model. The emergent spectra of disk evaporation model are presented in section 3. Some applications of the disk evaporation model to LLAGNs are presented in section 4. Section 5 is the conclusion.</text>
<section_header_level_1><location><page_4><loc_43><loc_27><loc_57><loc_29></location>2. The Model</section_header_level_1>
<section_header_level_1><location><page_4><loc_32><loc_24><loc_68><loc_25></location>2.1. Truncation of the accretion disk</section_header_level_1>
<text><location><page_4><loc_12><loc_10><loc_88><loc_21></location>We consider a hot corona above a geometrically thin standard disk around a central black hole. In the corona, viscous dissipation leads to ion heating, which is partially transferred to the electrons by means of Coulomb collisions. This energy is then conducted down into lower, cooler, and denser corona. If the density in this layer is sufficiently high, the conductive flux is radiated away. If the density is too low to efficiently radiate energy, cool matter is heated up and evaporation into the corona takes place. The mass evaporation goes on</text>
<text><location><page_5><loc_12><loc_44><loc_88><loc_86></location>until an equilibrium density is established. The gas evaporating into the corona still retains angular momentum and with the role of viscosity will differentially rotate around the central object. By friction the gas loses angular momentum and drifts inward, and thus continuously drains mass from the corona towards the central object. This is compensated by a steady mass evaporation flow from the underlying disk. The process is driven by the gravitational potential energy released by friction in the form of heat in the corona. Therefore, mass accretes to the central object partially through the corona (evaporated part) and partially through the disk (the left part of the supplying mass). Such a model for black holes was established by Meyer et al. (2000a, b). The inflow and outflow of mass, energy, and angular momentum between neighboring zones were included by Meyer-Hofmeister & Meyer (2003). The effect of the viscosity parameter α is investigated by (Qiao & Liu 2009), and the effect of the magnetic parameter β is studied by (Qian et al. 2007). The model we used here is based on Liu & Taam (2009), in which the structure of the corona and the evaporation features are determined by the equation of state, equation of continuity, and equations of momentum and energy. The calculations show that the evaporation rate increases with decreasing distance in the outer region of the accretion disk, reaching a maximum value and then dropping towards the central black hole. Taam et al. (2012) generalized the results of disk evaporation model for black hole X-ray binaries by including the effect of viscosity parameters α and magnetic parameter β in accretion disks around a supermassive black hole. The maximum evaporation rate and the corresponding radius from the black hole as functions of α and β are given as (Taam et al. 2012),</text>
<formula><location><page_5><loc_39><loc_41><loc_88><loc_43></location>˙ m max ≈ 0 . 38 α 2 . 34 β -0 . 41 , (1)</formula>
<formula><location><page_5><loc_39><loc_36><loc_88><loc_38></location>r min ≈ 18 . 80 α -2 . 00 β 4 . 97 , (2)</formula>
<text><location><page_5><loc_12><loc_28><loc_88><loc_34></location>where the evaporation rate is in units of Eddington accretion rate ˙ M Edd ( ˙ M Edd = 1 . 39 × 10 18 M/M /circledot g s -1 ), and the radius is in units of Schwarzschild radius R S ( R S = 2 . 95 × 10 5 M/M /circledot cm).</text>
<text><location><page_5><loc_12><loc_13><loc_88><loc_27></location>If the accretion rate is higher than the maximum evaporation rate, the evaporation can only make a fraction of the disk accretion flow go to the corona and the optically thick disk is never completely truncated by evaporation. However, if the mass supply rate is less than the maximum evaporation rate, the matter in the inner region of the disk will be fully evaporated to form a geometrically thick, optically thin accretion inner region, which is generally called advection-dominated accretion flow (ADAF). The truncation radius of the disk can be generalized as (Taam et al. 2012),</text>
<formula><location><page_5><loc_37><loc_10><loc_88><loc_12></location>r tr ≈ 17 . 3 ˙ m -0 . 886 α 0 . 07 β 4 . 61 , (3)</formula>
<text><location><page_6><loc_12><loc_46><loc_88><loc_86></location>where ˙ m is the mass supply rate from the most outer region of the accretion disk. If α , β and ˙ m are specified, we can self-consistently get a two-component structure of the accretion flow with an inner ADAF and an outer truncated accretion disk. It can be seen that from Equation (3), the truncation radius is very weakly dependent on α , but strongly dependent on β . It is argued that the effect of magnetic fields on evaporation rate is a competition between its tendency to increase the evaporation as a result of the energy balance and to decrease the evaporation as a result of the pressure balance. The additional pressure contributed by the magnetic fields results in a greater heating via the shear stress. This effect is similar to an increase viscosity parameter and leads to an increase of evaporation rate in the inner region with little effect in the outer region. In this paper, because we focus on the truncation of the accretion disk in the outer region, the additional pressure contribution suppresses the evaporation at all distances as a result of force balance. The effect of the magnetic field results in little change in the value of the maximal evaporation rate, but make the evaporate curve move systematically inward, which means the truncation radius will decrease for a given mass accretion rate. Meanwhile, we address that Equation (3) is a good fit only when the mass accretion rate is less than half of the maximum evaporation rate, i.e., ˙ m /lessorsimilar (1 / 2) ˙ m max = 0 . 19 α 2 . 34 β -0 . 41 ≈ 0 . 01 (assuming a standard viscosity parameter α = 0 . 3). When the mass accretion rate is close to the maximum evaporation rate, the truncation radius deviates from the power-law expression of Equation (3). In this case, the truncation radius will be determined by detailed numerical calculations (Taam et al. 2012).</text>
<section_header_level_1><location><page_6><loc_39><loc_40><loc_61><loc_42></location>2.2. The ADAF model</section_header_level_1>
<text><location><page_6><loc_12><loc_10><loc_88><loc_38></location>Inside the truncation radius, the accretion flows are in the form of advection dominated accretion flow (ADAF) or radiatively inefficient accretion flow (RIAF) from our diskevaporation model (Rees et al. 1982; Narayan & Yi 1994; Narayan et al. 1998; Quataert 2001; Narayan & McClintock 2008, and the references therein). The self-similar solution of ADAF was first proposed by Narayan & Yi (1994, 1995b), with which the spectrum of transient sources A0620-00 and V404 Cyg with lower luminosity were well fitted (Narayan et al. 1996). Later, the global solution of ADAF are conducted by several authors (Narayan et al. 1997; Manmoto 1997, 2000; Yuan et al. 1999, 2000; Zhang et al. 2010). The applications of ADAF to interpret the broadband spectrum of supermassive black holes can be seen, e.g., in Yuan et al. (2003) for the Galactic center, and by Quataert et al. (1999) for the LLAGN M81 and NGC 4579 and so on. All of them show that the self-similar solution is a good approximation at a radius far enough from the ISCO. For simplicity, in this paper, the self-similar solution of ADAF is adopted (Narayan & Yi 1995a, b). The structure of an ADAF surrounding a black hole with mass M can be calculated if the parameters including</text>
<text><location><page_7><loc_12><loc_85><loc_33><loc_86></location>˙ m , α and β are specified.</text>
<section_header_level_1><location><page_7><loc_40><loc_78><loc_60><loc_80></location>3. Numerical result</section_header_level_1>
<text><location><page_7><loc_12><loc_65><loc_88><loc_76></location>We calculate the emergent spectra predicted by the disk evaporation model for a twocomponent structure composted of an inner ADAF and a truncated accretion disk around a supermassive black hole with mass M when the parameters including ˙ m , α and β are specified. In the calculation, we fix central black hole mass at M = 10 8 M /circledot , assuming a viscosity parameter of α = 0 . 3, as adopted by Taam et al. (2012) for the spectral fits to LLAGNs.</text>
<text><location><page_7><loc_12><loc_24><loc_88><loc_63></location>The emergent spectra with mass accretion rates for β = 0 . 8 are plotted in the left panel of Figure 1 for mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001. The black-solid line is the total emergent spectrum for ˙ m = 0 . 01, and the black-dashed line is the emission from the accretion disk with a truncation radius r tr = 310 predicted by Equation (3). The maximum effective temperature of the truncated accretion disk is ∼ 2599K, which has a typical emission peaking at ∼ 1 . 1 µm (in L ν vs. ν ). The hard X-ray emission between 2-10 keV for ˙ m = 0 . 01 can be described by a power law with a hard X-ray index Γ = 1 . 65, which is produced by the self-Compton scattering of the synchrotron and bremsstrahlung photons of the ADAF itself. With a decrease of the mass accretion rate to ˙ m = 0 . 005, the predicted truncation radius of the accretion disk is r tr = 573 , which has a maximum effective temperature ∼ 1388K with a peak emission at ∼ 2 . 1 µm . Meanwhile, the hard X-ray index between 2-10 keV is Γ = 1 . 75. The blue-solid line in the left panel of Figure 1 is the total emergent spectrum for ˙ m = 0 . 005, and the blue-dashed line is the emission from the truncated accretion disk. With a further decrease of the mass accretion rates to ˙ m = 0 . 003 and ˙ m = 0 . 001, the truncation radii of the accretion disk are r tr = 901 and r tr = 2386, which correspond to an effective temperature of the accretion disk of ∼ 874K and ∼ 321K with peak emissions at ∼ 3 . 3 µm and ∼ 9 µm respectively. Meanwhile, the hard X-ray indices are Γ = 1 . 81 for ˙ m = 0 . 003 and Γ = 1 . 98 for ˙ m = 0 . 001 respectively. The purple-solid line and the red-solid line in Figure 1 are the total emergent spectra for ˙ m = 0 . 003 and ˙ m = 0 . 001, and the purple-dashed and the red-dashed lines are the emissions from the truncated accretion disks respectively.</text>
<text><location><page_7><loc_12><loc_11><loc_88><loc_22></location>We plot the hard X-ray index Γ 2 -10keV as a function of Eddington ratio L bol /L Edd for β = 0 . 8 in Figure 2 with a red line. Here the bolometric luminosity L bol is calculated by integrating the emergent spectrum. It is found that there is an anti-correlation between Γ 2 -10keV and L bol /L Edd , which is qualitatively consistent with the observations in LLAGNs (Constantin et al. 2009; Wu & Gu 2009; Younes et al. 2011, 2012; Xu 2011). This is because, with the decrease of the mass accretion rate, the electron temperature of ADAF</text>
<text><location><page_8><loc_12><loc_74><loc_88><loc_86></location>T e changes only slightly, and it is always around 10 9 K (Mehadevan 1997). However, the decrease of the mass accretion rate will result in a direct decrease of the Compton scattering optical depth τ es . So the Compton parameter y = 4 kT e /m e c 2 Max( τ es , τ 2 es ) of the ADAF decreases with decreasing mass accretion rate, consequently resulting in a softer spectrum, as also discovered in the low/hard spectral state of black hole X-ray binaries (Qiao & Liu 2010; 2013; Wu & Cao 2008; Yuan et al. 2005).</text>
<text><location><page_8><loc_12><loc_47><loc_88><loc_73></location>The emergent spectra with mass accretion rates for β = 0 . 5 are plotted in the right panel of Figure 1 for mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001. The black-solid line is the emergent spectrum for ˙ m = 0 . 01, and the black-dashed line is the emission from the accretion disk with a truncation radius r tr = 30 predicted by Equation 3. The maximum effective temperature of the truncated accretion disk is ∼ 1 . 4 × 10 4 K, which has a typical UV emission peaking at ∼ 2075 ˚ A. The hard X-ray emission for ˙ m = 0 . 01 in 2-10 keV can be described by a power law with a hard X-ray index Γ = 1 . 78. For ˙ m = 0 . 005 , 0 . 003 , 0 . 001, the truncation radii of the accretion disk are r tr = 55 . 4 , 87 , 231 respectively. The maximum effective temperature of the truncated accretion disk is ∼ 7635K , ∼ 4862K and ∼ 1814K respectively, with the emission peaking at 3798 ˚ A, 5964 ˚ A and 1 . 6 µ m respectively. The hard X-ray indices between 2-10 keV are Γ 2 -10keV = 1 . 79 , 1 . 85 , 2 . 04 respectively. We plot Γ 2 -10keV as a function of L bol /L Edd for β = 0 . 5 in Figure 2 with the black line. It is also clear that there is an anti-correlation between Γ 2 -10keV and L bol /L Edd .</text>
<text><location><page_8><loc_12><loc_14><loc_88><loc_46></location>In order to clearly show the effect of the magnetic parameter β on the spectra, we fix the central black hole mass at M = 10 8 M /circledot , α = 0 . 3 and ˙ m = 0 . 01 to calculate the emergent spectra for β = 0 . 8 and β = 0 . 5. The emergent spectrum for β = 0 . 8 is plotted in Figure 3 with a red-solid line, and the emergent spectrum for β = 0 . 5 is plotted in Figure 3 with a black-solid line. The dashed lines are the emissions from the truncated accretion disk. The bolometric luminosity for β = 0 . 8 is L bol = 9 . 6 × 10 42 erg s -1 , which corresponds to 7 . 6 × 10 -4 L Edd . The radiative efficiency η (defined as η = L bol / ˙ Mc 2 ) of the accretion flow for β = 0 . 8 is η = 7 . 7 × 10 -3 , which is much lower than the radiative efficiency η ≈ 0 . 1 predicted by the standard accretion disk extending down to the ISCO of a non-rotating black hole. The bolometric luminosity for β = 0 . 5 is L bol = 3 . 1 × 10 43 erg s -1 , which corresponds to 2 . 5 × 10 -3 L Edd . The radiative efficiency of the accretion flow for β = 0 . 5 is η = 0 . 025. For comparison, we also plot the emergent spectrum for ˙ m = 0 . 01 with the standard accretion disk extending down to the ISCO of a non-rotating black hole (the blue-solid line in Figure 3). The maximum temperature of the accretion disk is ∼ 4 . 2 × 10 4 K, which has a emission peaking at ∼ 870 . 5 ˚ A compared to the peak emission at ∼ 2075 ˚ A for β = 0 . 5 and at ∼ 1 . 1 µm for β = 0 . 8.</text>
<text><location><page_8><loc_16><loc_11><loc_88><loc_13></location>It has been demonstrated that the emergent spectra predicted by the disk evaporation</text>
<text><location><page_9><loc_12><loc_58><loc_88><loc_86></location>model can be strongly affected by the magnetic parameter β . The effect of β to the emergent spectrum is mainly from the emission of the truncated accretion disk. From Equation (3), an increase of β from 0.5 to 0.8 will result in a (0 . 8 / 0 . 5) 4 . 61 ≈ 8 . 7 times increase of the truncation radius of the accretion disk. Because the luminosity of the truncated accretion disk L disk ∝ r -1 tr , the truncated accretion disk luminosity will decrease by a factor of 8.7. Meanwhile, because the maximum effective temperature of the truncated accretion disk T eff , max ∝ r -3 / 4 tr , an increase of β from 0.5 to 0.8 will result in the peak emission of the truncated accretion disk shifting towards a lower frequency by a factor of ∼ 5 times, e.g., taking M = 10 8 M /circledot , assuming viscosity parameter α = 0 . 3, for ˙ m = 0 . 01, the peak emission of the truncated accretion disk is at ∼ 2075 ˚ A for β = 0 . 5 and at ∼ 1 . 1 µm for β = 0 . 8. For the inner ADAF, a change of β mainly affects the radio emission, while it has only very little effect on the optical/UV and X-ray emission. The luminosity of ADAF L ADAF ∝ β , so a change of β from 0.5 to 0.8 will also imply a little change in the luminosity of the ADAF (Taam et al. 2012; Mahadevan 1997).</text>
<text><location><page_9><loc_12><loc_45><loc_88><loc_57></location>In order to show the effect of the black hole mass on the emergent spectra, we plot the emergent spectra for different black hole masses as comparisons. The emergent spectra for M = 10 6 M /circledot with mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001 are plotted in Figure 4 for the left panel with β = 0 . 8, and for the right panel with β = 0 . 5. The emergent spectra for M = 10 9 M /circledot with mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001 are plotted in Figure 5 for the left panel with β = 0 . 8, and for the right panel with β = 0 . 5.</text>
<section_header_level_1><location><page_9><loc_18><loc_39><loc_82><loc_41></location>4. Comparison with observations-Bolometric Correction κ 2 -10keV</section_header_level_1>
<text><location><page_9><loc_12><loc_11><loc_88><loc_37></location>Generally, the bolometric luminosity of AGNs is estimated by multiplying a suitable bolometric correction at a given band. So far the most secure measurements to the bolometric luminosity is from X-ray observations. The X-ray bolometric correction κ 2 -10keV is determined from the mean energy distribution calculated from 47 luminous, mostly luminous quasars (Elvis et al. 1994), in which κ 2 -10keV ≈ 30. However, since the SED of AGNs can change much with mass accretion rate, it is not a good approximation to correct the bolometric luminosity with a single correction factor. A correlation between κ 2 -10keV and Eddington ratio λ is found by Vasudevan & Fabian (2007; 2009), in which κ 2 -10keV is calculated for a sample with simultaneous X-ray/Optical-UV observations. It is found that κ 2 -10keV ≈ 15 -30 for λ /lessorsimilar 0 . 1, κ 2 -10keV ≈ 20 -70 for 0 . 1 /lessorsimilar λ /lessorsimilar 0 . 2, and κ 2 -10keV ≈ 70 -150 for λ /greaterorsimilar 0 . 2 respectively. By compiling the SEDs of a small LLAGN sample, Ho (1999b) found that the median bolometric correction is κ 2 -10keV ≈ 8. A more extensive data set suggested a value larger by a factor of 2 (Ho 2000; 2009), having a median value κ 2 -10keV ≈ 15 . 8.</text>
<text><location><page_10><loc_12><loc_80><loc_88><loc_86></location>Because LLAGNs tend to be 'X-ray-loud', they have smaller values of κ 2 -10keV compared with the luminous sources. This is consistent with the results of vasudevan & Fabian (2007; 2009) and Elvis et al. (1994).</text>
<text><location><page_10><loc_12><loc_61><loc_88><loc_79></location>We collect a sample composed of 10 LLAGNs, including NGC 1097, NGC 3031, NGC 4203, NGC 4261, NGC 4374, NGC 4450, NGC 4486, NGC 4579, NGC 4594 and NGC 6251 with 2-10 keV luminosity L 2 -10keV measurement and bolometric luminosity measurement from Ho (2009). The bolometric luminosity is obtained by integrating the interpolated SEDs shown in Ho (1999b) and Ho et al. (2000). The black hole masses are also collected by Ho (1999b) and Ho (2000). The bolometric correction κ 2 -10keV as a function of L 2 -10keV / L Edd is plotted with the sign ♦ in Figure 6. Ho (2009) conservatively estimated that the errors of L bol /L 2 -10keV of the sources in the sample should be within 0.3 dex. The best-fitting linear regression for the correlation between κ 2 -10keV and L 2 -10keV / L Edd is as follows:</text>
<formula><location><page_10><loc_29><loc_58><loc_88><loc_60></location>κ 2 -10keV = -4 . 8 -3 . 0 × log 10 ( L 2 -10keV /L Edd ) , (4)</formula>
<text><location><page_10><loc_12><loc_55><loc_42><loc_56></location>plotted as a dotted-line in Figure 6.</text>
<text><location><page_10><loc_12><loc_42><loc_88><loc_53></location>In order to compare with observations, we calculate the emergent spectra predicted by the disk evaporation model. As shown in section 3.1, magnetic parameter β has significant effects on the emergent spectrum, we calculate the emergent spectra for β = 0 . 8, β = 0 . 95, and β = 0 . 5 for comparisons. By integrating the emergent spectrum, we calculate the 2-10 keV luminosity L 2 -10keV and the bolometric luminosity L bol , with which we can calculate κ 2 -10keV and L 2 -10keV / L Edd .</text>
<text><location><page_10><loc_12><loc_15><loc_88><loc_40></location>Fixing β = 0 . 8, taking M = 10 8 M /circledot and assuming α = 0 . 3, the ratio of 2-10keV luminosity L 2 -10keV to Eddington luminosity L Edd is L 2 -10keV /L Edd =8 . 45 × 10 -5 , 1 . 40 × 10 -5 , 4 . 13 × 10 -6 , 2 . 90 × 10 -7 , and the bolometric correction is κ 2 -10keV = 9 . 0 , 12 . 2 , 14 . 2 , 22 . 6 for ˙ m = 0 . 01 . 0 . 005 , 0 . 003 , 0 . 001 respectively. κ 2 -10keV as a function of L 2 -10keV /L Edd is plotted with a red-solid line in Figure 6. It can be seen that there is an anti-correlation between L 2 -10keV /L Edd and κ 2 -10keV . In order to check the effect of the black hole mass on L 2 -10keV /L bol -κ 2 -10keV correlation, we take different black hole masses for comparisons. For M = 10 6 M /circledot , κ 2 -10keV as a function of L 2 -10keV /L bol is plotted with a black-solid line in Figure 6. For M = 10 9 M /circledot , κ 2 -10keV as a function of L 2 -10keV /L bol is plotted with a bluesolid line in Figure 6. It can be seen that the effect of the black hole mass on L 2 -10keV /L Edd -κ 2 -10keV correlation is very weak. As an extreme example, we also plot L 2 -10keV /L Edd -κ 2 -10keV correlation for β = 0 . 95 in Figure 6. The red long-dashed, black long-dashed, blue long-dashed lines are for M = 10 8 M /circledot , M = 10 6 M /circledot and M = 10 9 M /circledot respectively.</text>
<text><location><page_10><loc_12><loc_10><loc_88><loc_13></location>Fixing β = 0 . 5, taking M = 10 8 M /circledot and assuming α = 0 . 3, the ratio of 2-10keV luminosity L 2 -10keV to Eddington luminosity L Edd is L 2 -10keV /L Edd =8 . 94 × 10 -5 , 1 . 87 × 10 -5 ,</text>
<text><location><page_11><loc_12><loc_76><loc_88><loc_86></location>5 . 32 × 10 -6 , 3 . 04 × 10 -7 , and the bolometric correction is κ 2 -10keV = 27 . 5 , 37 . 5 , 50 . 3 , 112 . 6 respectively. κ 2 -10keV as a function of L 2 -10keV /L Edd is plotted with a red short-dashed line in Figure 6. The bolometric correction κ 2 -10keV as a function of L 2 -10keV /L Edd for M = 10 6 M /circledot is plotted with a black short-dashed line in Figure 6, and for M = 10 9 M /circledot is plotted with a blue short-dashed line in Figure 6.</text>
<text><location><page_11><loc_12><loc_47><loc_88><loc_75></location>From Figure 6, it is clear that the bolometric correction strongly depends on β . L 2 -10keV /L bol -κ 2 -10keV correlation predicted by β = 0 . 8 or β = 0 . 95 can well explain the observations, while the prediction by β = 0 . 5 severely deviates from the observations. Although our model for larger β can roughly interpret the L 2 -10keV /L bol -κ 2 -10keV correlation, it is still a few sources lay below the model line. This may be because, in this work, for simplicity, we assume that, outside the truncation radius, the accretion flow exists in the form of a pure standard accretion disk. However, according to the disk evaporation model, outside the truncation radius, the matter should be in the form of 'disk+corona', but not the pure stand accretion disk. The emission from the corona will result in an increase of the X-ray emission, which will make the bolometric correction decrease. Consequently, the model-predicted κ 2 -10keV will systematically shift downward. We also need to keep in mind that, due to the inner ADAF dominating the X-ray emission, the emission contribution from the outer corona to the bolometric correction will be very small, so still the data support a bigger value of β = 0 . 8 or β = 0 . 95.</text>
<text><location><page_11><loc_12><loc_10><loc_88><loc_46></location>Theoretically, it would be useful to understand how the magnetic parameter is determined and how it is constrained. The value of β can be approximately known from the numerical simulations (e.g., Balbus & Hawley 1998). Detailed MHD numerical simulations for the formation of a magnetized corona have shown that a strongly magnetized corona can form above an initially weakly magnetized disk (e.g., Miller & Stone 2000; Machida et al. 2000; Hawley & Balbus 2002). However, in the disk evaporation model, we consider a slab corona in a large vertical extent, in which the corona is vertically stratified, in contrast to an isothermal torus as from MHD simulations. Furthermore, thermal conduction in vertical direction is taken into account, which will make an efficient mass evaporation from the disk to the corona, resulting in a higher mass density than in the coronal envelope seen in MHD simulations. More importantly, in the disk evaporation model, the electrons in the corona are assumed to be heated only by a transfer of energy from the ions to electrons via Coulomb collisions, which is reasonable for the accretion with a lower mass accretion rate (Narayan 1995b). Coulomb heating is the dominated heating mechanism for the electrons only if the magnetic field is strongly sub-equipartition (i.e., ratio of gas pressure to magnetic pressure > 10) (Malzac & Belmont 2009). The spectral modeling of Sgr A ∗ showed that an additional direct heating to electrons is required, which is probably produced by magnetohydrodynamics turbulence, magnetic reconnection and weak shocks (Yuan et al. 2003). However, the</text>
<text><location><page_12><loc_12><loc_80><loc_88><loc_86></location>magnetic heating mechanism to the electrons is very unclear, consequently still a weak magnetic field, i.e., a high value of β is assured in the case of a lower mass accretion rate within the framework of disk evaporation model.</text>
<section_header_level_1><location><page_12><loc_43><loc_74><loc_57><loc_76></location>5. Conclusion</section_header_level_1>
<text><location><page_12><loc_12><loc_41><loc_88><loc_72></location>Based on the prediction by Taam et al. (2012) of a truncation radius of the accretion disk on the mass accretion rate by including the magnetic field, we have calculated the emergent spectrum of an inner ADAF + an outer truncated accretion disk around a supermassive black hole. It is found that the disk evaporation model can roughly reproduce the observed anti-correlation between hard X-ray index Γ 2 -10keV and Eddington ratio λ for λ /lessorsimilar 10 -3 in LLAGNs. As shown by Taam et al. (2012), the truncation radius of the accretion disk is sensitive to the magnetic parameter β , which consequently affects the emergent spectrum. Our calculations show that the equipartition of gas pressure to magnetic pressure, i.e., β = 0 . 5, failed to explain the observed anti-correlation between L 2 -10keV /L Edd and the bolometric correction κ 2 -10keV . The resulted spectra for larger value β = 0 . 8 or β = 0 . 95 can better explain the observed L 2 -10keV /L Edd -κ 2 -10keV correlation. It is argued that in the disk evaporation model, the electrons in the corona are assumed to be heated only by a transfer of energy from the ions to electrons via Coulomb collisions, which is reasonable for the accretion with a lower mass accretion rate. Coulomb heating is the dominated heating mechanism for the electrons only if the magnetic field is strongly sub-equipartition, which is roughly consistent with observations.</text>
<text><location><page_12><loc_12><loc_32><loc_88><loc_39></location>We thank the very useful discussions with Prof. R. E. Taam from Northwestern University. This work is supported by the National Natural Science Foundation of China (grants 11033007 and 11173029), by the National Basic Re-search Program of China-973 Program 2009CB824800.</text>
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<figure>
<location><page_16><loc_14><loc_45><loc_50><loc_69></location>
</figure>
<figure>
<location><page_16><loc_54><loc_45><loc_90><loc_69></location>
<caption>Fig. 1.-</caption>
</figure>
<text><location><page_16><loc_12><loc_27><loc_88><loc_40></location>Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk evaporation model with M = 10 8 M /circledot assuming α = 0 . 3. Left panel: β = 0 . 8 is adopted. From bottom to top, the solid lines are the combined emergent spectra from an inner ADAF plus an outer truncated accretion disk for ˙ m = 10 -3 , 3 × 10 -3 , 5 × 10 -3 and 0 . 01 respectively. The dashed line are the emergent spectra from the truncated accretion disk. Right panel: β = 0 . 8 is adopted, and the meaning of the line style is same with the left panel.</text>
<figure>
<location><page_17><loc_14><loc_41><loc_50><loc_64></location>
<caption>Fig. 2.-</caption>
</figure>
<paragraph><location><page_17><loc_12><loc_31><loc_88><loc_36></location>Hard X-ray index Γ 2 -10keV as a function of Eddington ratio L bol /L Edd . In the calculations, we fix the black hole mass at M = 10 8 M /circledot , assuming a viscosity parameter α = 0 . 3. The red line is for β = 0 . 8 and the black line is for β = 0 . 5.</paragraph>
<figure>
<location><page_18><loc_14><loc_45><loc_50><loc_69></location>
<caption>Fig. 3.-</caption>
</figure>
<text><location><page_18><loc_12><loc_27><loc_88><loc_40></location>Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk evaporation model with M = 10 8 M /circledot assuming α = 0 . 3. The solid-black line is for β = 0 . 5 and ˙ m = 0 . 01, where the disk is truncated at 30 R S . The red-solid line is for β = 0 . 8 and ˙ m = 0 . 01, where the disk is truncated at 310 R S . The dashed lines are the emergent spectra from the truncated accretion disk. The blue line is the emergent spectrum for ˙ m = 0 . 01 with the standard accretion disk extending down to the ISCO of a non-rotating black hole.</text>
<figure>
<location><page_19><loc_14><loc_45><loc_50><loc_69></location>
</figure>
<figure>
<location><page_19><loc_54><loc_45><loc_90><loc_69></location>
<caption>Fig. 4.-</caption>
</figure>
<text><location><page_19><loc_12><loc_27><loc_88><loc_40></location>Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk evaporation model with M = 10 6 M /circledot assuming α = 0 . 3. Left panel: β = 0 . 8 is adopted. From bottom to top, the solid lines are the combined emergent spectra from an inner ADAF plus an outer truncated accretion disk for ˙ m = 10 -3 , 3 × 10 -3 , 5 × 10 -3 and 0 . 01 respectively. The dashed lines are the emergent spectra from the truncated accretion disk. Right panel: β = 0 . 5 is adopted, and the meaning of the line style is same with the left panel.</text>
<figure>
<location><page_20><loc_14><loc_45><loc_50><loc_69></location>
</figure>
<figure>
<location><page_20><loc_54><loc_45><loc_90><loc_69></location>
<caption>Fig. 5.-</caption>
</figure>
<text><location><page_20><loc_12><loc_27><loc_88><loc_40></location>Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk-evaporation model with M = 10 9 M /circledot assuming α = 0 . 3. Left panel: β = 0 . 8 is adopted. From bottom to top, the solid lines are the combined emergent spectra from an inner ADAF plus an outer truncated accretion disk for ˙ m = 10 -3 , 3 × 10 -3 , 5 × 10 -3 and 0 . 01 respectively. The dashed lines are the emergent spectra from the truncated accretion disk. Right panel: β = 0 . 5 is adopted, and the meaning of the line style is same with the left panel.</text>
<figure>
<location><page_21><loc_13><loc_45><loc_50><loc_68></location>
<caption>Fig. 6.-</caption>
</figure>
<text><location><page_21><loc_12><loc_27><loc_88><loc_40></location>Bolometric correction κ 2 -10keV as a function of L 2 -10keV /L Edd . The short-dashed line is for β = 0 . 5, the solid line is for β = 0 . 8 and the long-dashed line is for β = 0 . 95. The black line is for M = 10 6 M /circledot , red line is M = 10 8 M /circledot and the blue line is for M = 10 9 M /circledot . In all the calculation, α = 0 . 3 is adopted. The sign ♦ is the observed data, including NGC 1097, NGC 3031, NGC 4203, NGC 4261, NGC 4374, NGC 4450, NGC 4486, NGC 4579, NGC 4594 and NGC 6251 with 2-10 keV luminosity L 2 -10keV measurement and bolometric luminosity measurement from Ho (2009).</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Observations show that the accretion flows in low-luminosity active galactic nuclei (LLAGNs) probably have a two-component structure with an inner hot, optically thin, advection dominated accretion flow (ADAF) and an outer truncated cool, optically thick accretion disk. As shown by Taam et al. (2012), the truncation radius as a function of mass accretion rate is strongly affected by including the magnetic field within the framework of disk evaporation model, i.e., an increase of the magnetic field results in a smaller truncation radius of the accretion disk. In this work, we calculate the emergent spectrum of an inner ADAF + an outer truncated accretion disk around a supermassive black hole based on the prediction by Taam et al. (2012). It is found that an increase of the magnetic field from β = 0 . 8 to β = 0 . 5 (with magnetic pressure p m = B 2 / 8 π = (1 -β ) p tot , p tot = p gas + p m ) results in an increase of ∼ 8 . 7 times of the luminosity from the truncated accretion disk, meanwhile results in the peak emission of the truncated accretion disk shifting towards a higher frequency by a factor of ∼ 5 times. We found that the equipartition of gas pressure to magnetic pressure, i.e., β = 0 . 5, failed to explain the observed anticorrelation between L 2 -10keV /L Edd and the bolometric correction κ 2 -10keV (with κ 2 -10keV = L bol /L 2 -10keV ). The emergent spectra for larger value β = 0 . 8 or β = 0 . 95 can well explain the observed L 2 -10keV /L Edd -κ 2 -10keV correlation. We argue that in the disk evaporation model, the electrons in the corona are assumed to be heated only by a transfer of energy from the ions to electrons via Coulomb collisions, which is reasonable for the accretion with a lower mass accretion rate. Coulomb heating is the dominated heating mechanism for the electrons only if the magnetic field is strongly sub-equipartition, which is roughly consistent with observations. Subject headings: accretion, accretion disks - Black hole physics - galaxies: active - X-rays: galaxies",
"pages": [
1,
2
]
},
{
"title": "The disk evaporation model for the spectral features of low-luminosity active galactic nuclei",
"content": "Erlin Qiao 1 , B. F. Liu 1 , Francesca Panessa 2 and J. Y. Liu 3 qiaoel@nao.cas.cn",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Active galactic nuclei (AGNs) are believed to be powered by accretion onto supermassive black holes, which are strong sources emitting from radio to X-rays. The luminous AGNs, mainly including radio quiet quasars and bright Seyfert galaxies, are believed to be powered dominantly by a geometrically thin, optically thick, accretion disk extending down to the innermost stable circular orbits (ISCO) of a black hole (Shakura & Sunyaev 1973; Shields 1978; Malkan & Sargent 1982; Kishimoto, Antouncci & Blaes 2005; Shang et al. 2005; Liu et al. 2012). Observations show that accretion flows in low-luminosity active galactic nuclei (LLAGNs) are very different. A two-component structure of the accretion flow with an inner hot, optically thin, advection dominated accretion flow (ADAF), an outer cool, optically thick truncated accretion disk, and a jet are supported by a number of observational evidence for LLAGNs (Ho 2008 and the references therein). The observational evidence for such a structure in LLAGNs comes mainly from the broadband emission from the radio to X-rays (Lasota et al. 1996; Quataert et al. 1999; Di Matteo et al.2000, 2003; Yuan et al. 2009; Li et al. 2009). For example, by fitting the spectral energy distribution (SED) of NGC 1097, Nemmen et al. (2006) found that the optical and X-ray portion of the SED can be well fitted by an inner ADAF and an outer accretion disk truncated at 225 Schwarzschild radii, and the observed radio emission can be very well interpreted by the synchrotron emission of a relativistic jet modeled within the framework of the internal shock scenario (Yuan et al. 2005). The presence of a jet in LLAGNs has been clearly detected by the VLBI radio observations (Falcke et al. 2000; Nagar et al. 2001). Meanwhile, theoretically, because of the positive Bernoulli parameter of the hot ADAF, it is probably that the formation of the bipolar jets are driven by the ADAF (Narayan & Yi 1995b; Blandford & Begelman 1999; Meier et al. 2001; Yuan et al. 2012a, b). The evidence for the presence of an inner ADAF in LLAGNs is inferred from the very low luminosity and the very lower radiative efficiency estimated from the available mass supply rate (Ho 2009). In terms of Eddington luminosity, LLAGNs often have λ /lessorsimilar 0 . 01 contrary to the luminous AGNs with λ /greaterorsimilar 0 . 01 (with λ = L bol /L Edd , L Edd = 1 . 26 × 10 38 M/M /circledot erg s -1 ) (Panessa et al. 2006). The observed anti-correlation between the hard X-ray index Γ 2 -10keV and λ implies that LLAGNs may be dominated by ADAF (Gu & Cao 2009; Younes et al. 2011, 2012). Xu (2011) collected a LLAGN sample composed of 49 sources including 28 local Seyfert galaxies and 21 low-ionization nuclear emission-line regions (LINERs) with optical/UV and X-ray observations. It is found that there is a strong anti-correlation between α ox and Eddington ratio λ for the sources with λ /lessorsimilar 10 -3 , which supports that ADAF model is a very promising candidate in LLAGNs. The evidence for the truncation of the accretion disk in LLAGNs is inferred from the lack of a 'big blue bump', instead of a 'big red bump' (Nemmen et al. 2012). The very weak or absent broad iron K α line, which is attributed to X-ray fluorescence off of an optically thick accretion disk extending down to a few Schwarzschild radii of luminous AGNs, also supports that the accretion disk truncates at a larger radius ( ∼ 100 -1000 Schwarzschild radii) off the black hole in LLAGNs (Nandra et al. 2007). Quataert et al. (1999) fitted the optical/UV spectrum of M81 with a mass accretion rate of 0 . 01 Eddington accretion rate and an accretion disk truncated at 100 Schwarzschild radii. A similar fit to the optical/UV spectrum of NGC 4579 yielded a higher mass accretion rate of 0 . 03 Eddington accretion rate, while the truncation radius of the accretion disk is still 100 Schwarzschild radii (Quataert et al. 1999). The relationship between the truncation radius and the mass accretion rate is still controversial. By fitting the SEDs of a population of LLAGNs, Yuan & Narayan (2004) found that the lower Eddington ratio L bol /L Edd was associated with a larger truncation radius. By studying 33 PG quasars with Fe K α emission line detected by the XMM-Newton survey, Inoue, Terashima & Ho (2007) found that the Fe K α line systematically becomes narrow with decreasing L bol /L Edd , which probably means that the truncation radius of the accretion disk increases with decreasing mass accretion rate. It is interesting that the inverse correlation between the truncation radius of the accretion disk and the mass accretion rate is also found in the low/hard spectral state of black hole X-ray binaries (Cabanac et al. 2009). The physical mechanism for determining the truncation of the accretion disk has been proposed by many authors (Honma 1996; Manmoto & Kato 2000; Lu et al. 2004; Spruit & Deufel 2002; Dullemond & Spruit 2005). One of the promising model among them is the disk evaporation model, which was first proposed by Meyer & Meyer-Hofmeister (1994) for dwarf novae, established for black holes by Meyer et al. (2000a, b) and modified by Liu et al. (2002), where the decoupling of ions and electrons and Compton cooling effect are taken into account. The disk evaporation model has been applied to explain the spectral state transition in stellar-mass black holes (e.g., Meyer-Hofmeister & Meyer 2005; Qiao & Liu 2009; Qian et al. 2007). The spectral features of an inner ADAF and an outer truncated accretion disk predicted by the disk evaporation model in stellar-mass black holes were investigated by Qiao & Liu (2010, 2012). The first application of disk evaporation model to interpret the truncation of the accretion disk in LLAGNs was from Liu et al. (1999), in which a theoretical relation between the truncation radius and the mass accretion rate was given. Taam et al. (2012) generalized the disk evaporation model in black hole X-ray binaries by including the effect of a magnetic field in accretion disk to apply the model to the truncation of the accretion disk in a large number of LLAGNs. It is found that the truncation radius of the accretion disk can be very strongly affected by the magnetic parameters β , which is defined as p m = B 2 / 8 π = (1 -β ) p tot , (where p tot = p gas + p m , p gas is gas pressure and p m is magnetic pressure), describing the strength of the magnetic field in the accretion flows. Taam et al. (2012) found that the inclusion of magnetic field results in a smaller truncation radius compared to the case without magnetic field. In this work, based on the prediction by Taam et al. (2012) for the truncation of the accretion disk, the emergent spectrum of a two-component structure with an inner ADAF and an outer truncated accretion disk around a supermassive black hole is calculated. It is found that the disk evaporation model can roughly reproduce the observed anti-correlation between the hard X-ray index Γ 2 -10keV and the Eddington ratio λ in LLAGNs (e.g., Gu & Cao 2009). As shown by Taam et al. (2012), the truncation radius of the accretion disk is very sensitive to β , which consequently affects the emergent spectrum. Our calculations show that the equipartition of gas pressure to magnetic pressure, i.e., β = 0 . 5, failed to explain the observed anti-correlation between L 2 -10keV /L Edd and the bolometric correction κ 2 -10keV (with κ 2 -10keV = L bol /L 2 -10keV ). The emergent spectra for larger value β = 0 . 8 or β = 0 . 95 can well explain the observed L 2 -10keV /L Edd -κ 2 -10keV correlation. It is argued that the sub-equipartition of the magnetic field is reasonable in the case with a low mass accretion. In section 2, we briefly introduce the disk evaporation model. The emergent spectra of disk evaporation model are presented in section 3. Some applications of the disk evaporation model to LLAGNs are presented in section 4. Section 5 is the conclusion.",
"pages": [
2,
3,
4
]
},
{
"title": "2.1. Truncation of the accretion disk",
"content": "We consider a hot corona above a geometrically thin standard disk around a central black hole. In the corona, viscous dissipation leads to ion heating, which is partially transferred to the electrons by means of Coulomb collisions. This energy is then conducted down into lower, cooler, and denser corona. If the density in this layer is sufficiently high, the conductive flux is radiated away. If the density is too low to efficiently radiate energy, cool matter is heated up and evaporation into the corona takes place. The mass evaporation goes on until an equilibrium density is established. The gas evaporating into the corona still retains angular momentum and with the role of viscosity will differentially rotate around the central object. By friction the gas loses angular momentum and drifts inward, and thus continuously drains mass from the corona towards the central object. This is compensated by a steady mass evaporation flow from the underlying disk. The process is driven by the gravitational potential energy released by friction in the form of heat in the corona. Therefore, mass accretes to the central object partially through the corona (evaporated part) and partially through the disk (the left part of the supplying mass). Such a model for black holes was established by Meyer et al. (2000a, b). The inflow and outflow of mass, energy, and angular momentum between neighboring zones were included by Meyer-Hofmeister & Meyer (2003). The effect of the viscosity parameter α is investigated by (Qiao & Liu 2009), and the effect of the magnetic parameter β is studied by (Qian et al. 2007). The model we used here is based on Liu & Taam (2009), in which the structure of the corona and the evaporation features are determined by the equation of state, equation of continuity, and equations of momentum and energy. The calculations show that the evaporation rate increases with decreasing distance in the outer region of the accretion disk, reaching a maximum value and then dropping towards the central black hole. Taam et al. (2012) generalized the results of disk evaporation model for black hole X-ray binaries by including the effect of viscosity parameters α and magnetic parameter β in accretion disks around a supermassive black hole. The maximum evaporation rate and the corresponding radius from the black hole as functions of α and β are given as (Taam et al. 2012), where the evaporation rate is in units of Eddington accretion rate ˙ M Edd ( ˙ M Edd = 1 . 39 × 10 18 M/M /circledot g s -1 ), and the radius is in units of Schwarzschild radius R S ( R S = 2 . 95 × 10 5 M/M /circledot cm). If the accretion rate is higher than the maximum evaporation rate, the evaporation can only make a fraction of the disk accretion flow go to the corona and the optically thick disk is never completely truncated by evaporation. However, if the mass supply rate is less than the maximum evaporation rate, the matter in the inner region of the disk will be fully evaporated to form a geometrically thick, optically thin accretion inner region, which is generally called advection-dominated accretion flow (ADAF). The truncation radius of the disk can be generalized as (Taam et al. 2012), where ˙ m is the mass supply rate from the most outer region of the accretion disk. If α , β and ˙ m are specified, we can self-consistently get a two-component structure of the accretion flow with an inner ADAF and an outer truncated accretion disk. It can be seen that from Equation (3), the truncation radius is very weakly dependent on α , but strongly dependent on β . It is argued that the effect of magnetic fields on evaporation rate is a competition between its tendency to increase the evaporation as a result of the energy balance and to decrease the evaporation as a result of the pressure balance. The additional pressure contributed by the magnetic fields results in a greater heating via the shear stress. This effect is similar to an increase viscosity parameter and leads to an increase of evaporation rate in the inner region with little effect in the outer region. In this paper, because we focus on the truncation of the accretion disk in the outer region, the additional pressure contribution suppresses the evaporation at all distances as a result of force balance. The effect of the magnetic field results in little change in the value of the maximal evaporation rate, but make the evaporate curve move systematically inward, which means the truncation radius will decrease for a given mass accretion rate. Meanwhile, we address that Equation (3) is a good fit only when the mass accretion rate is less than half of the maximum evaporation rate, i.e., ˙ m /lessorsimilar (1 / 2) ˙ m max = 0 . 19 α 2 . 34 β -0 . 41 ≈ 0 . 01 (assuming a standard viscosity parameter α = 0 . 3). When the mass accretion rate is close to the maximum evaporation rate, the truncation radius deviates from the power-law expression of Equation (3). In this case, the truncation radius will be determined by detailed numerical calculations (Taam et al. 2012).",
"pages": [
4,
5,
6
]
},
{
"title": "2.2. The ADAF model",
"content": "Inside the truncation radius, the accretion flows are in the form of advection dominated accretion flow (ADAF) or radiatively inefficient accretion flow (RIAF) from our diskevaporation model (Rees et al. 1982; Narayan & Yi 1994; Narayan et al. 1998; Quataert 2001; Narayan & McClintock 2008, and the references therein). The self-similar solution of ADAF was first proposed by Narayan & Yi (1994, 1995b), with which the spectrum of transient sources A0620-00 and V404 Cyg with lower luminosity were well fitted (Narayan et al. 1996). Later, the global solution of ADAF are conducted by several authors (Narayan et al. 1997; Manmoto 1997, 2000; Yuan et al. 1999, 2000; Zhang et al. 2010). The applications of ADAF to interpret the broadband spectrum of supermassive black holes can be seen, e.g., in Yuan et al. (2003) for the Galactic center, and by Quataert et al. (1999) for the LLAGN M81 and NGC 4579 and so on. All of them show that the self-similar solution is a good approximation at a radius far enough from the ISCO. For simplicity, in this paper, the self-similar solution of ADAF is adopted (Narayan & Yi 1995a, b). The structure of an ADAF surrounding a black hole with mass M can be calculated if the parameters including ˙ m , α and β are specified.",
"pages": [
6,
7
]
},
{
"title": "3. Numerical result",
"content": "We calculate the emergent spectra predicted by the disk evaporation model for a twocomponent structure composted of an inner ADAF and a truncated accretion disk around a supermassive black hole with mass M when the parameters including ˙ m , α and β are specified. In the calculation, we fix central black hole mass at M = 10 8 M /circledot , assuming a viscosity parameter of α = 0 . 3, as adopted by Taam et al. (2012) for the spectral fits to LLAGNs. The emergent spectra with mass accretion rates for β = 0 . 8 are plotted in the left panel of Figure 1 for mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001. The black-solid line is the total emergent spectrum for ˙ m = 0 . 01, and the black-dashed line is the emission from the accretion disk with a truncation radius r tr = 310 predicted by Equation (3). The maximum effective temperature of the truncated accretion disk is ∼ 2599K, which has a typical emission peaking at ∼ 1 . 1 µm (in L ν vs. ν ). The hard X-ray emission between 2-10 keV for ˙ m = 0 . 01 can be described by a power law with a hard X-ray index Γ = 1 . 65, which is produced by the self-Compton scattering of the synchrotron and bremsstrahlung photons of the ADAF itself. With a decrease of the mass accretion rate to ˙ m = 0 . 005, the predicted truncation radius of the accretion disk is r tr = 573 , which has a maximum effective temperature ∼ 1388K with a peak emission at ∼ 2 . 1 µm . Meanwhile, the hard X-ray index between 2-10 keV is Γ = 1 . 75. The blue-solid line in the left panel of Figure 1 is the total emergent spectrum for ˙ m = 0 . 005, and the blue-dashed line is the emission from the truncated accretion disk. With a further decrease of the mass accretion rates to ˙ m = 0 . 003 and ˙ m = 0 . 001, the truncation radii of the accretion disk are r tr = 901 and r tr = 2386, which correspond to an effective temperature of the accretion disk of ∼ 874K and ∼ 321K with peak emissions at ∼ 3 . 3 µm and ∼ 9 µm respectively. Meanwhile, the hard X-ray indices are Γ = 1 . 81 for ˙ m = 0 . 003 and Γ = 1 . 98 for ˙ m = 0 . 001 respectively. The purple-solid line and the red-solid line in Figure 1 are the total emergent spectra for ˙ m = 0 . 003 and ˙ m = 0 . 001, and the purple-dashed and the red-dashed lines are the emissions from the truncated accretion disks respectively. We plot the hard X-ray index Γ 2 -10keV as a function of Eddington ratio L bol /L Edd for β = 0 . 8 in Figure 2 with a red line. Here the bolometric luminosity L bol is calculated by integrating the emergent spectrum. It is found that there is an anti-correlation between Γ 2 -10keV and L bol /L Edd , which is qualitatively consistent with the observations in LLAGNs (Constantin et al. 2009; Wu & Gu 2009; Younes et al. 2011, 2012; Xu 2011). This is because, with the decrease of the mass accretion rate, the electron temperature of ADAF T e changes only slightly, and it is always around 10 9 K (Mehadevan 1997). However, the decrease of the mass accretion rate will result in a direct decrease of the Compton scattering optical depth τ es . So the Compton parameter y = 4 kT e /m e c 2 Max( τ es , τ 2 es ) of the ADAF decreases with decreasing mass accretion rate, consequently resulting in a softer spectrum, as also discovered in the low/hard spectral state of black hole X-ray binaries (Qiao & Liu 2010; 2013; Wu & Cao 2008; Yuan et al. 2005). The emergent spectra with mass accretion rates for β = 0 . 5 are plotted in the right panel of Figure 1 for mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001. The black-solid line is the emergent spectrum for ˙ m = 0 . 01, and the black-dashed line is the emission from the accretion disk with a truncation radius r tr = 30 predicted by Equation 3. The maximum effective temperature of the truncated accretion disk is ∼ 1 . 4 × 10 4 K, which has a typical UV emission peaking at ∼ 2075 ˚ A. The hard X-ray emission for ˙ m = 0 . 01 in 2-10 keV can be described by a power law with a hard X-ray index Γ = 1 . 78. For ˙ m = 0 . 005 , 0 . 003 , 0 . 001, the truncation radii of the accretion disk are r tr = 55 . 4 , 87 , 231 respectively. The maximum effective temperature of the truncated accretion disk is ∼ 7635K , ∼ 4862K and ∼ 1814K respectively, with the emission peaking at 3798 ˚ A, 5964 ˚ A and 1 . 6 µ m respectively. The hard X-ray indices between 2-10 keV are Γ 2 -10keV = 1 . 79 , 1 . 85 , 2 . 04 respectively. We plot Γ 2 -10keV as a function of L bol /L Edd for β = 0 . 5 in Figure 2 with the black line. It is also clear that there is an anti-correlation between Γ 2 -10keV and L bol /L Edd . In order to clearly show the effect of the magnetic parameter β on the spectra, we fix the central black hole mass at M = 10 8 M /circledot , α = 0 . 3 and ˙ m = 0 . 01 to calculate the emergent spectra for β = 0 . 8 and β = 0 . 5. The emergent spectrum for β = 0 . 8 is plotted in Figure 3 with a red-solid line, and the emergent spectrum for β = 0 . 5 is plotted in Figure 3 with a black-solid line. The dashed lines are the emissions from the truncated accretion disk. The bolometric luminosity for β = 0 . 8 is L bol = 9 . 6 × 10 42 erg s -1 , which corresponds to 7 . 6 × 10 -4 L Edd . The radiative efficiency η (defined as η = L bol / ˙ Mc 2 ) of the accretion flow for β = 0 . 8 is η = 7 . 7 × 10 -3 , which is much lower than the radiative efficiency η ≈ 0 . 1 predicted by the standard accretion disk extending down to the ISCO of a non-rotating black hole. The bolometric luminosity for β = 0 . 5 is L bol = 3 . 1 × 10 43 erg s -1 , which corresponds to 2 . 5 × 10 -3 L Edd . The radiative efficiency of the accretion flow for β = 0 . 5 is η = 0 . 025. For comparison, we also plot the emergent spectrum for ˙ m = 0 . 01 with the standard accretion disk extending down to the ISCO of a non-rotating black hole (the blue-solid line in Figure 3). The maximum temperature of the accretion disk is ∼ 4 . 2 × 10 4 K, which has a emission peaking at ∼ 870 . 5 ˚ A compared to the peak emission at ∼ 2075 ˚ A for β = 0 . 5 and at ∼ 1 . 1 µm for β = 0 . 8. It has been demonstrated that the emergent spectra predicted by the disk evaporation model can be strongly affected by the magnetic parameter β . The effect of β to the emergent spectrum is mainly from the emission of the truncated accretion disk. From Equation (3), an increase of β from 0.5 to 0.8 will result in a (0 . 8 / 0 . 5) 4 . 61 ≈ 8 . 7 times increase of the truncation radius of the accretion disk. Because the luminosity of the truncated accretion disk L disk ∝ r -1 tr , the truncated accretion disk luminosity will decrease by a factor of 8.7. Meanwhile, because the maximum effective temperature of the truncated accretion disk T eff , max ∝ r -3 / 4 tr , an increase of β from 0.5 to 0.8 will result in the peak emission of the truncated accretion disk shifting towards a lower frequency by a factor of ∼ 5 times, e.g., taking M = 10 8 M /circledot , assuming viscosity parameter α = 0 . 3, for ˙ m = 0 . 01, the peak emission of the truncated accretion disk is at ∼ 2075 ˚ A for β = 0 . 5 and at ∼ 1 . 1 µm for β = 0 . 8. For the inner ADAF, a change of β mainly affects the radio emission, while it has only very little effect on the optical/UV and X-ray emission. The luminosity of ADAF L ADAF ∝ β , so a change of β from 0.5 to 0.8 will also imply a little change in the luminosity of the ADAF (Taam et al. 2012; Mahadevan 1997). In order to show the effect of the black hole mass on the emergent spectra, we plot the emergent spectra for different black hole masses as comparisons. The emergent spectra for M = 10 6 M /circledot with mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001 are plotted in Figure 4 for the left panel with β = 0 . 8, and for the right panel with β = 0 . 5. The emergent spectra for M = 10 9 M /circledot with mass accretion rates ˙ m = 0 . 01, 0 . 005, 0 . 003, 0 . 001 are plotted in Figure 5 for the left panel with β = 0 . 8, and for the right panel with β = 0 . 5.",
"pages": [
7,
8,
9
]
},
{
"title": "4. Comparison with observations-Bolometric Correction κ 2 -10keV",
"content": "Generally, the bolometric luminosity of AGNs is estimated by multiplying a suitable bolometric correction at a given band. So far the most secure measurements to the bolometric luminosity is from X-ray observations. The X-ray bolometric correction κ 2 -10keV is determined from the mean energy distribution calculated from 47 luminous, mostly luminous quasars (Elvis et al. 1994), in which κ 2 -10keV ≈ 30. However, since the SED of AGNs can change much with mass accretion rate, it is not a good approximation to correct the bolometric luminosity with a single correction factor. A correlation between κ 2 -10keV and Eddington ratio λ is found by Vasudevan & Fabian (2007; 2009), in which κ 2 -10keV is calculated for a sample with simultaneous X-ray/Optical-UV observations. It is found that κ 2 -10keV ≈ 15 -30 for λ /lessorsimilar 0 . 1, κ 2 -10keV ≈ 20 -70 for 0 . 1 /lessorsimilar λ /lessorsimilar 0 . 2, and κ 2 -10keV ≈ 70 -150 for λ /greaterorsimilar 0 . 2 respectively. By compiling the SEDs of a small LLAGN sample, Ho (1999b) found that the median bolometric correction is κ 2 -10keV ≈ 8. A more extensive data set suggested a value larger by a factor of 2 (Ho 2000; 2009), having a median value κ 2 -10keV ≈ 15 . 8. Because LLAGNs tend to be 'X-ray-loud', they have smaller values of κ 2 -10keV compared with the luminous sources. This is consistent with the results of vasudevan & Fabian (2007; 2009) and Elvis et al. (1994). We collect a sample composed of 10 LLAGNs, including NGC 1097, NGC 3031, NGC 4203, NGC 4261, NGC 4374, NGC 4450, NGC 4486, NGC 4579, NGC 4594 and NGC 6251 with 2-10 keV luminosity L 2 -10keV measurement and bolometric luminosity measurement from Ho (2009). The bolometric luminosity is obtained by integrating the interpolated SEDs shown in Ho (1999b) and Ho et al. (2000). The black hole masses are also collected by Ho (1999b) and Ho (2000). The bolometric correction κ 2 -10keV as a function of L 2 -10keV / L Edd is plotted with the sign ♦ in Figure 6. Ho (2009) conservatively estimated that the errors of L bol /L 2 -10keV of the sources in the sample should be within 0.3 dex. The best-fitting linear regression for the correlation between κ 2 -10keV and L 2 -10keV / L Edd is as follows: plotted as a dotted-line in Figure 6. In order to compare with observations, we calculate the emergent spectra predicted by the disk evaporation model. As shown in section 3.1, magnetic parameter β has significant effects on the emergent spectrum, we calculate the emergent spectra for β = 0 . 8, β = 0 . 95, and β = 0 . 5 for comparisons. By integrating the emergent spectrum, we calculate the 2-10 keV luminosity L 2 -10keV and the bolometric luminosity L bol , with which we can calculate κ 2 -10keV and L 2 -10keV / L Edd . Fixing β = 0 . 8, taking M = 10 8 M /circledot and assuming α = 0 . 3, the ratio of 2-10keV luminosity L 2 -10keV to Eddington luminosity L Edd is L 2 -10keV /L Edd =8 . 45 × 10 -5 , 1 . 40 × 10 -5 , 4 . 13 × 10 -6 , 2 . 90 × 10 -7 , and the bolometric correction is κ 2 -10keV = 9 . 0 , 12 . 2 , 14 . 2 , 22 . 6 for ˙ m = 0 . 01 . 0 . 005 , 0 . 003 , 0 . 001 respectively. κ 2 -10keV as a function of L 2 -10keV /L Edd is plotted with a red-solid line in Figure 6. It can be seen that there is an anti-correlation between L 2 -10keV /L Edd and κ 2 -10keV . In order to check the effect of the black hole mass on L 2 -10keV /L bol -κ 2 -10keV correlation, we take different black hole masses for comparisons. For M = 10 6 M /circledot , κ 2 -10keV as a function of L 2 -10keV /L bol is plotted with a black-solid line in Figure 6. For M = 10 9 M /circledot , κ 2 -10keV as a function of L 2 -10keV /L bol is plotted with a bluesolid line in Figure 6. It can be seen that the effect of the black hole mass on L 2 -10keV /L Edd -κ 2 -10keV correlation is very weak. As an extreme example, we also plot L 2 -10keV /L Edd -κ 2 -10keV correlation for β = 0 . 95 in Figure 6. The red long-dashed, black long-dashed, blue long-dashed lines are for M = 10 8 M /circledot , M = 10 6 M /circledot and M = 10 9 M /circledot respectively. Fixing β = 0 . 5, taking M = 10 8 M /circledot and assuming α = 0 . 3, the ratio of 2-10keV luminosity L 2 -10keV to Eddington luminosity L Edd is L 2 -10keV /L Edd =8 . 94 × 10 -5 , 1 . 87 × 10 -5 , 5 . 32 × 10 -6 , 3 . 04 × 10 -7 , and the bolometric correction is κ 2 -10keV = 27 . 5 , 37 . 5 , 50 . 3 , 112 . 6 respectively. κ 2 -10keV as a function of L 2 -10keV /L Edd is plotted with a red short-dashed line in Figure 6. The bolometric correction κ 2 -10keV as a function of L 2 -10keV /L Edd for M = 10 6 M /circledot is plotted with a black short-dashed line in Figure 6, and for M = 10 9 M /circledot is plotted with a blue short-dashed line in Figure 6. From Figure 6, it is clear that the bolometric correction strongly depends on β . L 2 -10keV /L bol -κ 2 -10keV correlation predicted by β = 0 . 8 or β = 0 . 95 can well explain the observations, while the prediction by β = 0 . 5 severely deviates from the observations. Although our model for larger β can roughly interpret the L 2 -10keV /L bol -κ 2 -10keV correlation, it is still a few sources lay below the model line. This may be because, in this work, for simplicity, we assume that, outside the truncation radius, the accretion flow exists in the form of a pure standard accretion disk. However, according to the disk evaporation model, outside the truncation radius, the matter should be in the form of 'disk+corona', but not the pure stand accretion disk. The emission from the corona will result in an increase of the X-ray emission, which will make the bolometric correction decrease. Consequently, the model-predicted κ 2 -10keV will systematically shift downward. We also need to keep in mind that, due to the inner ADAF dominating the X-ray emission, the emission contribution from the outer corona to the bolometric correction will be very small, so still the data support a bigger value of β = 0 . 8 or β = 0 . 95. Theoretically, it would be useful to understand how the magnetic parameter is determined and how it is constrained. The value of β can be approximately known from the numerical simulations (e.g., Balbus & Hawley 1998). Detailed MHD numerical simulations for the formation of a magnetized corona have shown that a strongly magnetized corona can form above an initially weakly magnetized disk (e.g., Miller & Stone 2000; Machida et al. 2000; Hawley & Balbus 2002). However, in the disk evaporation model, we consider a slab corona in a large vertical extent, in which the corona is vertically stratified, in contrast to an isothermal torus as from MHD simulations. Furthermore, thermal conduction in vertical direction is taken into account, which will make an efficient mass evaporation from the disk to the corona, resulting in a higher mass density than in the coronal envelope seen in MHD simulations. More importantly, in the disk evaporation model, the electrons in the corona are assumed to be heated only by a transfer of energy from the ions to electrons via Coulomb collisions, which is reasonable for the accretion with a lower mass accretion rate (Narayan 1995b). Coulomb heating is the dominated heating mechanism for the electrons only if the magnetic field is strongly sub-equipartition (i.e., ratio of gas pressure to magnetic pressure > 10) (Malzac & Belmont 2009). The spectral modeling of Sgr A ∗ showed that an additional direct heating to electrons is required, which is probably produced by magnetohydrodynamics turbulence, magnetic reconnection and weak shocks (Yuan et al. 2003). However, the magnetic heating mechanism to the electrons is very unclear, consequently still a weak magnetic field, i.e., a high value of β is assured in the case of a lower mass accretion rate within the framework of disk evaporation model.",
"pages": [
9,
10,
11,
12
]
},
{
"title": "5. Conclusion",
"content": "Based on the prediction by Taam et al. (2012) of a truncation radius of the accretion disk on the mass accretion rate by including the magnetic field, we have calculated the emergent spectrum of an inner ADAF + an outer truncated accretion disk around a supermassive black hole. It is found that the disk evaporation model can roughly reproduce the observed anti-correlation between hard X-ray index Γ 2 -10keV and Eddington ratio λ for λ /lessorsimilar 10 -3 in LLAGNs. As shown by Taam et al. (2012), the truncation radius of the accretion disk is sensitive to the magnetic parameter β , which consequently affects the emergent spectrum. Our calculations show that the equipartition of gas pressure to magnetic pressure, i.e., β = 0 . 5, failed to explain the observed anti-correlation between L 2 -10keV /L Edd and the bolometric correction κ 2 -10keV . The resulted spectra for larger value β = 0 . 8 or β = 0 . 95 can better explain the observed L 2 -10keV /L Edd -κ 2 -10keV correlation. It is argued that in the disk evaporation model, the electrons in the corona are assumed to be heated only by a transfer of energy from the ions to electrons via Coulomb collisions, which is reasonable for the accretion with a lower mass accretion rate. Coulomb heating is the dominated heating mechanism for the electrons only if the magnetic field is strongly sub-equipartition, which is roughly consistent with observations. We thank the very useful discussions with Prof. R. E. Taam from Northwestern University. This work is supported by the National Natural Science Foundation of China (grants 11033007 and 11173029), by the National Basic Re-search Program of China-973 Program 2009CB824800.",
"pages": [
12
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},
{
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C. 2009, ApJ, 703, 1034 Yuan, F., Bu, Defu, & Wu, Maochun 2012a, ApJ, 761, 130 Yuan, F., Wu, Maochun, & Bu, Defu 2012b, ApJ, 761, 129 Zhang, H., Yuan, F., & Chaty, S. 2010, ApJ, 717, 929 Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk evaporation model with M = 10 8 M /circledot assuming α = 0 . 3. Left panel: β = 0 . 8 is adopted. From bottom to top, the solid lines are the combined emergent spectra from an inner ADAF plus an outer truncated accretion disk for ˙ m = 10 -3 , 3 × 10 -3 , 5 × 10 -3 and 0 . 01 respectively. The dashed line are the emergent spectra from the truncated accretion disk. Right panel: β = 0 . 8 is adopted, and the meaning of the line style is same with the left panel. Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk evaporation model with M = 10 8 M /circledot assuming α = 0 . 3. The solid-black line is for β = 0 . 5 and ˙ m = 0 . 01, where the disk is truncated at 30 R S . The red-solid line is for β = 0 . 8 and ˙ m = 0 . 01, where the disk is truncated at 310 R S . The dashed lines are the emergent spectra from the truncated accretion disk. The blue line is the emergent spectrum for ˙ m = 0 . 01 with the standard accretion disk extending down to the ISCO of a non-rotating black hole. Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk evaporation model with M = 10 6 M /circledot assuming α = 0 . 3. Left panel: β = 0 . 8 is adopted. From bottom to top, the solid lines are the combined emergent spectra from an inner ADAF plus an outer truncated accretion disk for ˙ m = 10 -3 , 3 × 10 -3 , 5 × 10 -3 and 0 . 01 respectively. The dashed lines are the emergent spectra from the truncated accretion disk. Right panel: β = 0 . 5 is adopted, and the meaning of the line style is same with the left panel. Emergent spectra of an inner ADAF and an outer truncated accretion disk around a black hole predicted by the disk-evaporation model with M = 10 9 M /circledot assuming α = 0 . 3. Left panel: β = 0 . 8 is adopted. From bottom to top, the solid lines are the combined emergent spectra from an inner ADAF plus an outer truncated accretion disk for ˙ m = 10 -3 , 3 × 10 -3 , 5 × 10 -3 and 0 . 01 respectively. The dashed lines are the emergent spectra from the truncated accretion disk. Right panel: β = 0 . 5 is adopted, and the meaning of the line style is same with the left panel. Bolometric correction κ 2 -10keV as a function of L 2 -10keV /L Edd . The short-dashed line is for β = 0 . 5, the solid line is for β = 0 . 8 and the long-dashed line is for β = 0 . 95. The black line is for M = 10 6 M /circledot , red line is M = 10 8 M /circledot and the blue line is for M = 10 9 M /circledot . In all the calculation, α = 0 . 3 is adopted. The sign ♦ is the observed data, including NGC 1097, NGC 3031, NGC 4203, NGC 4261, NGC 4374, NGC 4450, NGC 4486, NGC 4579, NGC 4594 and NGC 6251 with 2-10 keV luminosity L 2 -10keV measurement and bolometric luminosity measurement from Ho (2009).",
"pages": [
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}
]
2013ApJ...778....2S
https://arxiv.org/pdf/1309.3281.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_85><loc_88><loc_87></location>VALIDATION OF THE EQUILIBRIUM MODEL FOR GALAXY EVOLUTION TO Z ∼ 3 THROUGH MOLECULAR GAS AND DUST OBSERVATIONS OF LENSED STAR-FORMING GALAXIES 1</section_header_level_1>
<text><location><page_1><loc_9><loc_80><loc_92><loc_84></location>Am'elie Saintonge 2 , Dieter Lutz 2 , Reinhard Genzel 2 , Benjamin Magnelli 3 , Raanan Nordon 4 , Linda J. Tacconi 2 , Andrew J. Baker 5 , Kaushala Bandara 6,2 , Stefano Berta 2 , Natascha M. Forster Schreiber 2 , Albrecht Poglitsch 2 , Eckhard Sturm 2 , Eva Wuyts 2 & Stijn Wuyts 2</text>
<text><location><page_1><loc_40><loc_79><loc_60><loc_80></location>Draft version September 28, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_76><loc_55><loc_78></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_51><loc_86><loc_76></location>We combine IRAM Plateau de Bure Interferometer and Herschel PACS and SPIRE measurements to study the dust and gas contents of high-redshift star forming galaxies. We present new observations for a sample of 17 lensed galaxies at z = 1 . 4 -3 . 1, which allow us to directly probe the cold ISM of normal star-forming galaxies with stellar masses of ∼ 10 10 M /circledot , a regime otherwise not (yet) accessible by individual detections in Herschel and molecular gas studies. The lensed galaxies are combined with reference samples of sub-millimeter and normal z ∼ 1 -2 star-forming galaxies with similar far-infrared photometry to study the gas and dust properties of galaxies in the SFRM ∗ -redshift parameter space. The mean gas depletion timescale of main sequence galaxies at z > 2 is measured to be only ∼ 450Myr, a factor of ∼ 1 . 5 ( ∼ 5) shorter than at z = 1 ( z = 0), in agreement with a (1 + z ) -1 scaling. The mean gas mass fraction at z = 2 . 8 is 40 ± 15% (44% after incompleteness correction), suggesting a flattening or even a reversal of the trend of increasing gas fractions with redshift recently observed up to z ∼ 2. The depletion timescale and gas fractions of the z > 2 normal star-forming galaxies can be explained under the 'equilibrium model' for galaxy evolution, in which the gas reservoir of galaxies is the primary driver of the redshift evolution of specific star formation rates. Due to their high star formation efficiencies and low metallicities, the z > 2 lensed galaxies have warm dust despite being located on the star formation main sequence. At fixed metallicity, they also have a gas-to-dust ratio 1.7 times larger than observed locally when using the same standard techniques, suggesting that applying the local calibration of the δ GDR -metallicity relation to infer the molecular gas mass of high redshift galaxies may lead to systematic differences with CO-based estimates.</text>
<text><location><page_1><loc_14><loc_48><loc_86><loc_50></location>Subject headings: galaxies: evolution - galaxies: high-redshift - infrared: ISM - ISM: dust - ISM: molecules</text>
<section_header_level_1><location><page_1><loc_22><loc_44><loc_35><loc_46></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_27><loc_48><loc_44></location>Most star-forming galaxies follow a relation between their stellar masses and star formation rates that has been well characterized up to z ∼ 2 . 5 (e.g. Noeske et al. 2007; Salim et al. 2007; Elbaz et al. 2011; Whitaker et al. 2012). The very existence and tightness of this relation suggests that these galaxies live in a state of equilibrium where their ability to form stars is regulated by the availability of gas and the amount of material they return to the circum-galactic medium through outflows (e.g. Genel et al. 2008; Bouch'e et al. 2010; Dav'e et al. 2011, 2012; Krumholz & Dekel 2012; Lilly et al. 2013). Simultaneously, it downplays the importance of galaxy mergers in the global star for-</text>
<text><location><page_1><loc_10><loc_19><loc_48><loc_25></location>1 Based on observations carried out with the IRAM Plateau de Bure Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). Based also on observations from Herschel , an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.</text>
<text><location><page_1><loc_10><loc_17><loc_48><loc_19></location>2 Max-Planck Institut fur extraterrestrische Physik, 85741 Garching, Germany</text>
<text><location><page_1><loc_10><loc_15><loc_48><loc_17></location>3 Argelander-Institut fur Astronomy, Universitat Bonn, 53121 Bonn, Germany</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_11><loc_48><loc_15></location>4 School of Physics and Astronomy, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel</list_item>
<list_item><location><page_1><loc_10><loc_9><loc_48><loc_12></location>5 Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ 08854-8019 USA</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_9></location>6 Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_35><loc_92><loc_46></location>mation budget of the Universe (Robaina et al. 2009; Rodighiero et al. 2011; Kaviraj et al. 2013) and highlights the influence of secular processes with longer duty cycles such as gas accretion, bar formation and bulge growth (Genzel et al. 2008). We refer to this general framework as the 'equilibrium model' for galaxy evolution, the formalism of which is detailed in recent work (e.g. Dav'e et al. 2012; Lilly et al. 2013).</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_35></location>This paradigm was ushered in by a combination of large imaging surveys, detailed kinematics studies, molecular gas measurements, and theoretical efforts. For example, large optical and infrared surveys have contributed by allowing for the accurate measurement of stellar masses and star formation rates in very large galaxy samples, often supported by significant spectroscopic observing campaigns. Recently, Herschel observations in the wavelength range of 70-500 µ m with the PACS and SPIRE instruments have further contributed, by providing direct calorimetric SFRs that have been used to improve and recalibrate other indicators (e.g. Nordon et al. 2010, 2012; Elbaz et al. 2010; Wuyts et al. 2011a).</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_16></location>At the same time, near-infrared integral field spectroscopy measurements at z ∼ 2 have started revealing that a significant fraction of high redshift galaxies are rotation-dominated discs (e.g. Forster Schreiber et al. 2006; Wright et al. 2007; Genzel et al. 2006, 2008; Forster Schreiber et al. 2009; Shapiro et al. 2008; Jones et al. 2010b; Gnerucci et al. 2011; Wisnioski et al.</text>
<text><location><page_2><loc_8><loc_77><loc_48><loc_92></location>2011; Epinat et al. 2012; Newman et al. 2013). These observations convincingly demonstrate that the high SFRs measured in these galaxies are generally not caused by major mergers, as was previously assumed by analogy with the local ULIRGs which have comparable SFRs and are all major mergers (Sanders & Mirabel 1996; Veilleux et al. 2002). Instead, the high SFRs of z ∼ 2 galaxies are caused by high molecular gas fractions, well above the 5 -10% typically observed at z = 0 (Tacconi et al. 2010; Daddi et al. 2010a; Tacconi et al. 2013).</text>
<text><location><page_2><loc_8><loc_47><loc_48><loc_77></location>Some of the strongest direct evidence in favor of the equilibrium model above indeed comes from molecular gas observations. In the local Universe, with the exception of ULIRGSs, it has now been directly observed that the location of a galaxy in the SFRM ∗ plane is mostly determined by its supply of molecular gas, with variations in star formation efficiency playing a second order role (Saintonge et al. 2012). Similar conclusions have also been reached based on high-redshift galaxy samples, whether directly using CO data (Tacconi et al. 2013), or indirectly using far-infrared photometry to estimate gas masses (Magdis et al. 2012a). Also, outflows of molecular material, which are an important element in setting the balance between gas and star formation in the models, have now been directly observed in a range of objects (e.g. Sturm et al. 2011). But mostly, it is now possible to detect CO line emission in normal star-forming galaxies at z > 1 (e.g. Tacconi et al. 2010, 2013; Daddi et al. 2010a). These observations convincingly show that the rapid decline in the specific SFR of galaxies since z ∼ 2 can be explained by the measured gas fractions and a slowly varying depletion timescale ( t dep ∝ (1 + z ) -1 , Tacconi et al. 2013).</text>
<text><location><page_2><loc_8><loc_11><loc_48><loc_47></location>Out to z ∼ 1 . 5 it is possible to detect far-infrared emission and CO lines in individual objects with current instrumentation. However, even in the deepest Herschel fields and with long integrations at the IRAM PdBI, it is still not possible to measure directly the dust and gas contents of individual normal star-forming galaxies with masses ∼ 10 10 M /circledot at z /greaterorsimilar 2. A proven way to study molecular gas in high redshift galaxies with lower masses and lower SFRs is instead to target objects that are gravitationally lensed (Baker et al. 2004; Coppin et al. 2007; Danielson et al. 2011). Samples of lensed starforming galaxies have for example also been used to extend to higher redshifts and lower stellar masses studies of the kinematics and stellar populations of disc galaxies, of the mass-metallicity relation (e.g. Jones et al. 2010b; Richard et al. 2011; Wuyts et al. 2012a,b), and of the origin of metallicity gradients in galaxy discs (Yuan et al. 2011; Jones et al. 2013). In this paper, we explore the relation between dust, gas and star formation at z = 2 -3 using a sample of 17 UV-bright lensed galaxies targeted for deep Herschel PACS and SPIRE observations and having SFRs and stellar masses characteristic of mainsequence objects at their redshifts. We analyze the results of these observations in the context of the equilibrium model. In addition, we present new IRAM PdBI observations that more than double the number of published lensed galaxies with CO line measurements.</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_11></location>After we describe the sample in § 2, the multiwavelength observations are presented in § 3, including the new Herschel and IRAM PdBI observations. In § 4,</text>
<text><location><page_2><loc_52><loc_77><loc_92><loc_92></location>we describe how key quantities such as stellar masses, star formation rates, dust masses, dust temperatures, gas masses and metallicities were calculated homogeneously. The key results of this study are presented in § 5 and summarized in § 6; in short, we find that the z > 2 lensed galaxies have low dust and gas masses, but high dust temperatures as a consequence of an efficient conversion of their gas into stars. Gas mass fractions and depletion times follow a redshift evolution out to z = 3 that is consistent with the expected scaling relations under the equilibrium model.</text>
<text><location><page_2><loc_52><loc_68><loc_92><loc_77></location>All rest-frame and derived quantities in this work assume a Chabrier (2003) IMF, and a cosmology with H 0 = 70km s -1 Mpc -1 , Ω m = 0 . 3 and Ω Λ = 0 . 7. All molecular gas masses ( M H 2 ) and derived quantities such as gas fractions and depletion times, presented and plotted include a factor of 1.36 to account for the presence of helium.</text>
<section_header_level_1><location><page_2><loc_68><loc_63><loc_76><loc_65></location>2. SAMPLE</section_header_level_1>
<section_header_level_1><location><page_2><loc_62><loc_62><loc_82><loc_63></location>2.1. Lensed galaxies sample</section_header_level_1>
<text><location><page_2><loc_52><loc_36><loc_92><loc_61></location>The main sample consists of 17 lensed galaxies that were selected for deep Herschel PACS/SPIRE imaging (details of these observations are presented in § 3.1). These galaxies, selected from the literature, have been discovered as bright blue arcs of conspicuous morphology and then spectroscopically confirmed to be high redshift lensed objects. As shown in § 2.1.1, their intrinsic properties are similar to UV-selected Lyman Break Galaxies (LBGs), or their z ∼ 2 BX/BM analogs, but the observed fluxes are strongly amplified. The sources come both from the traditional method of searching for galaxies lensed by massive clusters, and from the recent searches for bright blue arcs in the Sloan Digital Sky Survey (SDSS), with lenses that typically are individual luminous red galaxies (e.g. Allam et al. 2007). In addition, we also consider the submm-identified Eyelash galaxy (Swinbank et al. 2010), as it is located within the field of view of our Herschelobservations of the Cosmic Eye.</text>
<text><location><page_2><loc_52><loc_20><loc_92><loc_36></location>The sample is by no means a complete census of UVbright galaxies, but instead samples some of the bestknown objects with rich multi-wavelength observations, including near-infrared imaging and spectroscopy and millimeter continuum and line measurements. In Table 1 the basic properties of the sample are given (coordinates, redshifts and amplification factors). The galaxies have redshifts in the range 2 < z < 3, with the exception of four galaxies with z ∼ 1 . 5 (median redshift is 2.3), and most have stellar masses in the range 9 . 5 < log M ∗ /M /circledot < 11 . 0 (median, 1 . 6 × 10 10 M /circledot ), as will be presented in § 4.1.</text>
<section_header_level_1><location><page_2><loc_53><loc_17><loc_90><loc_19></location>2.1.1. Is the lensed galaxies sample representative of the z > 2 star-forming population?</section_header_level_1>
<text><location><page_2><loc_52><loc_7><loc_92><loc_16></location>As outlined in the introduction, the power of gravitational lensing is used here to extend the galaxy population where direct measurements of both molecular gas and dust masses can be performed. In particular, most galaxies in the sample have a stellar mass of ∼ 10 10 M /circledot and modest SFRs, as determined from IR and UV photometry, of ∼ 50 M /circledot yr -1 , given their median redshift</text>
<table>
<location><page_3><loc_13><loc_66><loc_87><loc_88></location>
<caption>Table 1 Lensed galaxy sample</caption>
</table>
<text><location><page_3><loc_13><loc_65><loc_53><loc_66></location>a Assuming a 20% uncertainty on the magnification, see § A.</text>
<text><location><page_3><loc_8><loc_52><loc_48><loc_64></location>of z = 2 . 4. We stress that the only way to directly detect the far-infrared emission of such galaxies across multiple Herschel bands is to target systems that are gravitationally lensed. Even in the deepest PACS and SPIRE blank fields, this mass/redshift/SFR regime is barely accessible through stacking of tens of galaxies (Reddy et al. 2012; Magnelli et al. 2013). Therefore, it is impossible to directly compare the FIR properties of our lensed galaxies to those of a well-matched, un-lensed reference sample.</text>
<text><location><page_3><loc_8><loc_42><loc_48><loc_52></location>Such a comparison is however desirable. Since the lensed galaxies were selected visually based on their restframe UV light, there is a possibility that they represent a biased sub-sample of the high-redshift star-forming galaxy population. For example, we could expect the selection technique to preferentially isolate objects that are particularly UV-bright and therefore dust- and metalpoor.</text>
<text><location><page_3><loc_8><loc_11><loc_48><loc_42></location>To answer this question, we create a control sample of un-lensed galaxies, matching on mass, redshift, and IR luminosity, where the latter is obtained either from PACS or MIPS 24 µ mphotometry when available, or else derived indirectly from the optical/UV photometry. We then compare the UV-to-IR ratio between the lensed and control samples. More exactly, for each lensed galaxy, we extract a control galaxy from the GOODS fields catalog (Wuyts et al. 2011b), within 0.2 dex in M ∗ , 0.2 dex in L IR and 0.2 in z . To check whether the lensed galaxies are less extincted than the control sample, the distributions of log(SFR IR / SFR UV ) are compared. As described in § 4.3, for the lensed galaxies, we calculate SFR IR from the 160 µ mfluxes and SFR UV from the B- or V-band photometry. For the GOODS sample, Wuyts et al. (2011a) used a ladder of star formation indicators to infer the respective contributions of obscured and un-obscured starformation, even in the absence of flux measurements at FIR wavelengths, therefore also allowing a measurement of log(SFR IR / SFR UV ). A KS test gives a probability > 90% that the distributions of log(SFR IR / SFR UV ) in the lensed and control samples are representative of the same parent population.</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_11></location>As a second test, we study the UV slope β of the lensed galaxies. In Figure 1 the lensed galaxies are shown in the A IRX -β plane, where A IRX is the effective UV</text>
<text><location><page_3><loc_52><loc_63><loc_61><loc_64></location>attenuation,</text>
<formula><location><page_3><loc_61><loc_58><loc_92><loc_62></location>A IRX = 2 . 5 log ( SFR IR SFR UV +1 ) . (1)</formula>
<text><location><page_3><loc_52><loc_7><loc_92><loc_57></location>The UV slope is measured from HST optical photometry using bands corresponding to rest-frame wavelengths redwards of Lyman α . The observed values are interpolated to derive the magnitudes at 1600 ˚ A and 2800 ˚ A (rest-frame) which are then used to compute β . As a comparison, we use again the GOODS fields sample for which Nordon et al. (2013) have derived values of the UV slope β . This reference sample is shown in Fig. 1 as gray circles. As these are PACS-detected galaxies only in the GOODS fields, the sample is biased towards large values of A IRX . To circumvent the necessity of detecting individual objects in several Herschel bands, Reddy et al. (2012) used stacks of UV-selected galaxies to probe galaxies with typically lower masses and SFRs. Their results are shown in Fig. 1 for two different bins of β . The lensed galaxies are located in the same region of the A IRX -β as the BX/BM galaxies of Reddy et al. (2012), and are also seen to follow the Meurer et al. (1999) extinction law (solid line). The only exception is the Eyelash, which is an outlier with high UV attenuation as expected since it is the only submillimeter-selected galaxy in the sample. Using the sample of Nordon et al. (2013) (both Herschel-detected and undetected galaxies) we can also define a control sample for each of our lensed galaxies by matching on M ∗ , SFR and redshift. In 10 of the 12 galaxies where we have sufficient data to conduct this experiment the median of the β distribution in the control sample agrees with the measured value of β for the lensed galaxy within its uncertainty. The two exceptions are J0900 and J1527 where measured values of β are -1.8 while the control samples suggest -0.6 and -1.2, respectively. We also note that while previous studies of the Cosmic Eye and cB58 suggested that these objects were better represented by an SMC extinction law than by the Meurer et al. (1999) relation (see e.g. Wuyts et al. 2012a), the new Herschel data revise the IR luminosities upward and find these galaxies in agreement with the Meurer/Calzetti relation.</text>
<figure>
<location><page_4><loc_11><loc_64><loc_46><loc_91></location>
<caption>Figure 1. Relation between the effective UV attenuation ( A IRX ) and the UV spectral slope ( β ), showing the lensed galaxies (blue circles) and stacked points from Reddy et al. (2012) for UVselected BX/BM galaxies in two bins of β (red squares). The β values for the lensed galaxies are computed from HST photometry, or taken from Richard et al. (2011) and Wuyts et al. (2012a). As a comparison, the relations of Calzetti et al. (2000) (dotted line), Meurer et al. (1999) (solid line) are shown, as well as the IRselected sample of Nordon et al. (2013) (gray circles). The lensed galaxy in this figure with the very high value of A IRX is the Eyelash.</caption>
</figure>
<text><location><page_4><loc_8><loc_40><loc_48><loc_51></location>The lack of significant differences in either the effective UV attenuation ( A IRX ) or the UV slope ( β ) between the lensed galaxies of this study and their respective matched control samples suggests that they are not a biased subsample of the underlying population. We therefore proceed under the assumption that they are representative of the bulk of the star-forming galaxy population at their specific redshifts and modest masses.</text>
<section_header_level_1><location><page_4><loc_10><loc_37><loc_46><loc_39></location>2.1.2. Does differential lensing introduce biases in the measurements?</section_header_level_1>
<text><location><page_4><loc_8><loc_21><loc_48><loc_36></location>In Table 1, we give the magnification factors used to correct measured quantities for the lensing effect. The magnification factors are derived from optical imaging (typically from HST), and represent the average factor across the discs of these extended galaxies. However, in reality, different regions of the galaxies may be lensed differentially, with regions falling on or near caustics being more strongly magnified. Therefore, components of the galaxy having different spatial extents and compactness may not be magnified uniformly. This effect is known as differential lensing (e.g. Blandford & Narayan 1992).</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_21></location>In practice, this could impact our analysis since both warm dust and high -J COtransitions are typically emitted in denser, more compact regions than cold dust or CO (1-0). Both the far-infrared SED and the CO spectral line energy distribution (SLED) might be biased, and standard calibrations to measure dust masses and temperatures, as well as excitation corrections to extrapolate total gas masses from a (3-2) line flux, may not apply (Downes et al. 1995; Serjeant 2012). For the specific case of galaxy lenses selected in large area, flux-limited submm/FIR surveys, there is a bias towards galaxies</text>
<text><location><page_4><loc_52><loc_75><loc_92><loc_92></location>with very compact dusty star-forming regions, as these are most likely to benefit from the strongest magnifications (Hezaveh et al. 2012). With the exception of the Eyelash, our lenses are not submm/FIR-identified, but rather selected from cluster fields and SDSS imaging. In the discovery papers, SDSS galaxy lenses were typically identified visually as blue arcs. This includes the arc length, driven by the basic geometric configuration of lens and background object, as well as flux. They will therefore not suffer the specific biases discussed by Hezaveh et al. (2012), but the rest-UV emission used to derive the lensing model may still differ in extent or centroid from that at longer wavelengths.</text>
<text><location><page_4><loc_52><loc_53><loc_92><loc_75></location>It is difficult to directly assess the impact of differential lensing, as the exercise requires detailed magnification maps, and high resolution imaging of the different components (optical/UV continuum, FIR continuum, CO line emission,...). The effect has therefore been so far quantified only based on simulations and very few, special galaxies. In particular, most of the literature on this concentrates on the special case of submm- or FIRselected galaxies such as the Eyelash (e.g. Hezaveh et al. 2012; Serjeant 2012; Fu et al. 2012; Wardlow et al. 2013). As can be seen in Figure 1, the Eyelash is a very special object, the only one in our sample to be submm-selected, and any conclusions drawn from such extreme dusty systems may not apply to all normal star-forming galaxies. We can however use these studies to get a sense of the amplitude of the issue in the most extreme cases.</text>
<text><location><page_4><loc_52><loc_20><loc_92><loc_53></location>Using a submillimeter galaxy as a model, Serjeant (2012) estimate the impact of differential lensing on FIRselected galaxy samples. They find that even when comparing quantities measured from rest-frame optical/NIR wavelengths (e.g. stellar masses) and FIR observations (e.g. SFRs), the median differential magnification ratio is ∼ 0 . 8 with small dispersion. Therefore even in these cases, differential lensing does not affect the position of a galaxies in the SFR-mass plane significantly. Serjeant (2012) find a stronger effect of differential lensing on the CO SLED, but the effect manifests itself mostly for transitions with J upper > 4. Indeed, for the few galaxies in our sample where CO(1-0) measurements are also available, a typical (3-2)/(1-0) line ratio of ∼ 0 . 7 is retrieved (Riechers et al. 2010; Danielson et al. 2011), similar to what is measured in unlensed galaxies (e.g. Harris et al. 2010; Ivison et al. 2011; Bothwell et al. 2013). A common explanation for this ubiquitous value of the (32)/(1-0) line ratio is that the lines are emitted from the same moderately excited component of the ISM, but Harris et al. (2010) argue that it may instead reflect a generic feature of multi-component star-forming ISMs, in which the different lines are emitted by different but well mixed, optically thick and thermalized components with different but characteristic filling factors.</text>
<text><location><page_4><loc_52><loc_7><loc_92><loc_20></location>Since our Herschel and IRAM observations are mostly unresolved, it is not possible to directly assess the impact of differential lensing on our measurements. Generally, this effect has not yet been extensively studied for normal star-forming high-redshift discs (as opposed to submm- or FIR-selected galaxies). However, based on the arguments above and the hypothesis that the FIR continuum and the CO line emission originate from similar physical regions (see however Fu et al. 2012), we can conclude that if occurring, differential lensing is un-</text>
<text><location><page_5><loc_8><loc_83><loc_48><loc_92></location>likely to significantly affect the results of this study. It may be biasing our gas masses high if the CO SLEDs are affected, but by no more than 40%, comparing our adopted excitation correction R 13 = 2 . 0 and the value of 1.4 indicated by the observations of the Eye, Eyelash and cB58 (Riechers et al. 2010; Danielson et al. 2011, see also § 4.5).</text>
<text><location><page_5><loc_8><loc_67><loc_48><loc_82></location>Similarly, differential lensing could be biasing the inferred dust properties. If the more compact, hotter regions of the galaxies are more strongly lensed than the diffuse component, the inferred dust temperatures could be too high (although Hezaveh et al. 2012, show that the effect can be reversed in systems with lower magnification factors of µ ∼ 10). This would result in under-estimated dust masses as well. However, the fact that we do not observe a strong effect on the CO SLED argues against a strong effect, but we cannot rule out the possibility of a small bias and include this caveat in the discussion in § 5.3.</text>
<section_header_level_1><location><page_5><loc_19><loc_64><loc_37><loc_65></location>2.2. Comparison samples</section_header_level_1>
<text><location><page_5><loc_8><loc_38><loc_48><loc_63></location>We add to the z ∼ 2 lensed galaxy sample additional objects from the literature to serve as a comparison point. First, we use a compilation of 16 galaxies in the GOODS-North field for which CO line fluxes as well as deep Herschel photometry are available (Magnelli et al. 2012b). These galaxies are located mostly at 1 . 0 < z < 1 . 5 and sample well the SFR-stellar mass plane. We also use the compilation of submillimeter galaxies (SMGs) from Magnelli et al. (2012a), which expand the sample toward higher SFRs at fixed stellar mass and redshift. All the SMGs from the Magnelli et al. (2012a) compilation are used in the analysis when only dust measurements are involved, and the subset of these also found in Bothwell et al. (2013) whenever CO measurements are also needed. All the galaxies in the comparison sample have Herschel PACS and SPIRE photometry, which was processed identically as for the lensed galaxy sample in order to derive infrared luminosities, SFRs, dust temperatures and dust masses (see details in § 4.2).</text>
<text><location><page_5><loc_8><loc_26><loc_48><loc_38></location>The lensed and comparison galaxy samples are shown in Figure 2 in the SFRM ∗ plane for two redshift intervals, 1 . 0 < z < 1 . 6 and 2 . 0 < z < 3 . 0, using the GOODS and EGS catalogs of Wuyts et al. (2011b) to provide a reference and define the main sequence (MS). These are the samples onto which our measurements of stellar mass and star formation rate are calibrated (sections 4.1 and 4.3), making them the best matched reference catalogs. The MS we derive from these is</text>
<formula><location><page_5><loc_17><loc_24><loc_48><loc_25></location>log SFR MS = a + b log( M ∗ /M /circledot ) , (2)</formula>
<text><location><page_5><loc_8><loc_19><loc_48><loc_23></location>where a = [ -6 . 102 , -6 . 704 , -6 . 923] and b = [0 . 728 , 0 . 807 , 0 . 834] for the three redshift intervals of [1 . 0 -1 . 6], [2 . 0 -2 . 5] and [2 . 5 -3 . 0], respectively.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_19></location>When discussing the molecular gas properties of the lensed galaxies, we also use additional references. In particular, we make use of the data from the COLD GASS survey (Saintonge et al. 2011a), which includes CO(1-0) measurements for a representative sample of 365 SDSS-selected galaxies at 0 . 025 < z < 0 . 050 with M ∗ > 10 10 M /circledot . For high redshift galaxies, the PHIBSS sample (Tacconi et al. 2013) is by far the largest compilation of CO measurements for normal star-forming galax-</text>
<text><location><page_5><loc_52><loc_89><loc_92><loc_92></location>s, and is the perfect reference to study e.g. the redshift evolution of the gas contents of galaxies (see section 5.3).</text>
<section_header_level_1><location><page_5><loc_69><loc_87><loc_75><loc_88></location>3. DATA</section_header_level_1>
<section_header_level_1><location><page_5><loc_63><loc_85><loc_81><loc_86></location>3.1. Herschel photometry</section_header_level_1>
<section_header_level_1><location><page_5><loc_57><loc_84><loc_87><loc_85></location>3.1.1. PACS observations and data reduction</section_header_level_1>
<text><location><page_5><loc_52><loc_56><loc_92><loc_83></location>We have obtained 70, 100 and 160 µ m 'miniscanmaps' of our targets using the PACS instrument (Poglitsch et al. 2010) on board the Herschel Space Observatory (Pilbratt et al. 2010). Total observing time per source was 1 hour at 70 and 100 µ m each, and 2 hours at 160 µ m wavelength, which is observed in parallel to both of the shorter wavelengths. The resulting PACS maps cover an area of ∼ 3 ' × 1 . 5 ' at more than half of the peak coverage, and have useful information over an area ∼ 6 ' × 2 . 5 ' . We processed the PACS data to maps using standard procedures similar to those described for the PEPproject in Lutz et al. (2011), in build 7.0.1786 of the Herschel HIPE software (Ott 2010). The Herschel blind pointing accuracy is ∼ 2 '' RMS (Pilbratt et al. 2010). To secure the astrometry, we therefore inspected the PACS maps to identify far-infrared sources clearly associated with counterparts having accurate astrometry (typically from SDSS). These reference positions were used to correct the astrometry of the PACS data to sub-arcsecond accuracy.</text>
<section_header_level_1><location><page_5><loc_57><loc_54><loc_87><loc_55></location>3.1.2. SPIRE observations and data reduction</section_header_level_1>
<text><location><page_5><loc_52><loc_34><loc_92><loc_53></location>We used the SPIRE instrument (Griffin et al. 2010) to simultaneously obtain 250, 350, and 500 µ m 'small maps' of our sources, using 14 repetitions and a total observing time of 35 minutes per source. Maps were produced with the standard reduction pipeline in HIPE (version 4.0.1349). Following the recommendation in the SPIRE Photometer Instrument Description, the maps are scaled with the appropriate flux correction factors of 1.02, 1.05, and 0.94 at 250 µ m, 350 µ m, and 500 µ m, respectively. The typical calibration accuracy of SPIRE maps is 15%. A preliminary source extraction at 250 µ m was performed, and the sources with counterparts in the PACS images (shifted as described above to the appropriate coordinate zero point) were used to correct the astrometry in all SPIRE bands.</text>
<section_header_level_1><location><page_5><loc_59><loc_31><loc_84><loc_32></location>3.1.3. Far-infrared flux measurements</section_header_level_1>
<text><location><page_5><loc_52><loc_23><loc_92><loc_31></location>We developed a measurement technique that combines aspects of blind source extraction, and guided extraction using prior information on the position of the sources. The procedure is aimed at measuring reliable fluxes for the lensed sources across the six Herschel bands, rather than produce complete catalogs in any given band.</text>
<text><location><page_5><loc_52><loc_7><loc_92><loc_23></location>The first step is to perform a blind extraction with StarFinder (Diolaiti et al. 2000) on the 100 µ m, 160 µ m and 250 µ m maps. The resulting catalogs are used as prior information on the position of sources brighter than 3 σ , which is used to extract accurate fluxes. The main advantages of this PSF-fitting technique are that it accounts for most of the blending which could be an issue in these typically crowded fields at the longer wavelengths, and allows for photometry on any specific object that is reliable across the different bands (Magnelli et al. 2009, 2011; Lutz et al. 2011). For the PACS images, we use the merged blind 100 µ m and 160 µ m catalogs as a prior,</text>
<figure>
<location><page_6><loc_12><loc_72><loc_91><loc_92></location>
<caption>Figure 2. Distribution of the sample in the SFRM ∗ plane for the redshift ranges of (a) 1 . 0 < z < 1 . 6 and (b) 2 . 0 < z < 3 . 0. The lensed galaxy sample is represented in red, while the GOODS-N sample is in orange and the SMG compilation in blue (see sample description in § 2). Additionally, we show the position in this plane of the PHIBSS galaxies (Tacconi et al. 2013), which are used in the analysis in § 5.3. The gray contours represent the distribution of galaxies from the GOODS and EGS fields, with SFRs and stellar masses on the same calibration scheme as our main galaxy sample (Wuyts et al. 2011b). The star formation main sequence (dashed line) is derived at each redshift from this large reference sample, and given by eq. 2. In panel (b), we differentiate between the main sequence at z = 2 . 0 -2 . 5 (dashed line) and at z = 2 . 5 -3 . 0 (dashed-dotted line). In panel (c) we show the distribution of offsets from the star formation main sequence for all samples (same color scheme as in panels a and b), defined at fixed stellar mass as ∆ log( sSFR ) = log (SFR) -log (SFR) MS .</caption>
</figure>
<table>
<location><page_6><loc_19><loc_34><loc_82><loc_55></location>
<caption>Table 2 PACS, SPIRE and MAMBO photometry</caption>
</table>
<unordered_list>
<list_item><location><page_6><loc_19><loc_32><loc_67><loc_33></location>a Including a factor 2.4 scaling to account for aperture effects (see § 3.2).</list_item>
<list_item><location><page_6><loc_19><loc_31><loc_33><loc_32></location>b Baker et al. (2001)</list_item>
<list_item><location><page_6><loc_19><loc_30><loc_52><loc_31></location>c SMA measurement from Swinbank et al. (2010)</list_item>
</unordered_list>
<text><location><page_6><loc_8><loc_12><loc_48><loc_29></location>while the 250 µ m blind catalog is used as the prior for the SPIRE bands. In cases where the lensed galaxy of interest is detected at less than 3 σ , the position of this galaxy is added in the prior catalogs based on its coordinates at shorter wavelengths. In most cases, the closest object with a 3 σ detection in the priors catalog, be it the lens or another object, is located 20-40 '' from the lensed galaxy and therefore easily separable at both 160 and 250 µ m. Only in the cases of cB58, the Cosmic Eye and J1133 is the closest neighboring object closer, at a distance of 10-15 '' . Even in these cases the objects are well separated in the PACS imaging, which we use as a guide for the SPIRE priors.</text>
<text><location><page_6><loc_8><loc_8><loc_48><loc_12></location>The Herschel fluxes obtained from the PSF-fitting are finally aperture-corrected using factors of (0.883, 0.866, 0.811, 0.835, 0.848, 0.898), derived specifically for the</text>
<text><location><page_6><loc_52><loc_18><loc_92><loc_29></location>PSFs used at (70, 100, 160, 250, 350, 500) µ m. Unless the lens is detected in the PACS/SPIRE images with > 3 σ , it is assumed that its FIR emission is negligible and does not contaminate the measurement for the lensed galaxy of interest. The final Herschel fluxes for the 17 lensed galaxies are given in Table 2. Values < 2 σ should be interpreted as upper limits, which occurs for 5 galaxies at both PACS 160 µ m and SPIRE 250 µ m.</text>
<section_header_level_1><location><page_6><loc_60><loc_15><loc_84><loc_17></location>3.2. IRAM-MAMBO photometry</section_header_level_1>
<text><location><page_6><loc_52><loc_7><loc_92><loc_15></location>Photometric observations in the 1.2mm continuum were obtained for part of our sample during the pool observing sessions at the IRAM 30m telescope in the winters 2006/2007 and 2007/2008. We used the 117 element version of the Max Planck Millimeter Bolometer (MAMBO) array (Kreysa et al. 1998). On-off observa-</text>
<text><location><page_7><loc_8><loc_65><loc_48><loc_92></location>tions were typically obtained in blocks of 6 scans of 20 subscans each, and repeated in later observing nights unless a detection was already reached. The data were reduced with standard procedures in the MOPSIC package developed by R. Zylka, using the default calibration files for the applicable pool periods. Table 2 lists the measured 1.2mm fluxes and their statistical uncertainties. We add the MAMBO detection which we already obtained for cB58 (Baker et al. 2001) and the SMA flux obtained by Swinbank et al. (2010) for the Eyelash. For J0901, we obtained a clear MAMBO detection at the target position centered on the southern bright lensed component but both the PACS maps and IRAM-PdB CO maps clearly indicate that the 11 '' MAMBO beam is missing flux. Based on the CO and PACS 100 µ m maps, the ratio between the total flux of J0901 and the flux in the southern component is independently measured to be 2.4 and 2.5, respectively. We hence scale the MAMBO flux from the southern component by a factor of 2.4 to infer the total flux before further use.</text>
<section_header_level_1><location><page_7><loc_17><loc_63><loc_39><loc_64></location>3.3. IRAM-PdBI CO mapping</section_header_level_1>
<text><location><page_7><loc_8><loc_52><loc_48><loc_62></location>Molecular gas mass measurements through observations of the CO(3-2) line have been performed for 10 of the 17 galaxies in the Herschel sample, and we report here on both the previously published measurements and those coming from new PdBI observations. In sections 5 and 6, the full sample of 17 galaxies is used for the elements of the analysis that do not invoke molecular gas, and this subset of 10 galaxies for the rest of the analysis.</text>
<section_header_level_1><location><page_7><loc_18><loc_49><loc_39><loc_50></location>3.3.1. Previous CO observations</section_header_level_1>
<text><location><page_7><loc_8><loc_43><loc_48><loc_48></location>The CO line has been previously observed in seven galaxies from our Herschel sample. We briefly review the specifics of the observations of each object below, while a summary of the CO measurements is given in Table 6.</text>
<text><location><page_7><loc_8><loc_23><loc_48><loc_42></location>Eye -The Cosmic Eye was observed in the CO(3-2) line with the IRAM PdBI (Coppin et al. 2007). The line width is 190 ± 24 km s -1 and the total line flux integrated over the line is 0 . 50 ± 0 . 07 Jy km s -1 . The CO emission appears to be spatially associated with component B 1 of the system, and Coppin et al. (2007) therefore suggest that the appropriate magnification correction factor for the CO line flux is 8 rather than the value of 30 found for the entire system (Dye et al. 2007). However, Riechers et al. (2010) report a detection of the CO(1-0) with the VLA that is spatially consistent with the bulk of the rest-frame UV emission, leading them to conclude that the total magnification value of the system should be used. This is the approach we adopt here.</text>
<text><location><page_7><loc_8><loc_14><loc_48><loc_23></location>Eyelash -Multiple CO lines of both 12 CO and 13 CO were observed for this object (Danielson et al. 2011). Although a 12 CO(1-0) flux has been measured, for uniformity with the rest of the sample we adopt for the Eyelash the flux of 13 . 20 ± 0 . 10 Jy km s -1 measured in the (3-2) line.</text>
<text><location><page_7><loc_8><loc_7><loc_48><loc_14></location>J0901 -Sharon (2013) reports on both EVLA CO(1-0) and PdBI CO(3-2) observations. For consistency with the rest of the dataset, we adopt the CO(3-2) flux of 19 . 8 ± 2 . 0 Jy km s -1 , which is obtained after primary beam correction given the large angular size of the source.</text>
<text><location><page_7><loc_52><loc_84><loc_92><loc_92></location>cB58 -We adopt for this object the CO(3-2) flux of 0.37 ± 0.08 Jy km s -1 measured by Baker et al. (2004), which itself is consistent with an upper limit previously set by Frayer et al. (1997) and with the VLA CO(1-0) detection by Riechers et al. (2010) for the value of R 13 we adopt for the bulk of our sample (see § 4.5).</text>
<text><location><page_7><loc_52><loc_72><loc_92><loc_83></location>8:00arc -The 'eight o'clock arc' (thereafter the 8:00arc) was observed in 2007 May with the IRAM PdBI in compact configuration, using the 3mm SIS receivers to target the redshifted CO(3-2) line (Baker et al. in prep.). Following a standard reduction process, a 0.45 mJy continuum source associated with the lens was subtracted (see also Volino et al. 2010), and a final line flux of 0.85 ± 0.24 Jy km s -1 is measured.</text>
<text><location><page_7><loc_52><loc_62><loc_92><loc_71></location>Clone, Horseshoe -Baker et al. (in prep.) report on IRAM PdBI compact configuration observations of these two sources, combining datasets taken in July to October 2007 and April 2009. Using standard reductions and integrating over the spatially extended CO emission, total line fluxes for the redshifted CO(3-2) line of 0 . 48 ± 0 . 15 (Clone) and 0 . 44 ± 0 . 18 (Horseshoe) are derived.</text>
<section_header_level_1><location><page_7><loc_53><loc_59><loc_90><loc_61></location>3.3.2. New PdBI observations: J0900, J1137 and J1226</section_header_level_1>
<text><location><page_7><loc_52><loc_46><loc_92><loc_59></location>In July-October 2011 we obtained CO(3-2) maps (rest frequency of 345.998 GHz) for three additional UV-bright lensed objects with the IRAM PdBI (Guilloteau et al. 1992). The three targets, J0900, J1137 and J1226, were chosen to extend the parameter space of objects with molecular gas measurements. Specifically, J0900 has the lowest metallicity of the sources with reliable Herschel flux measurements, and J1137 and J1226 are at the very low and very high ends of the redshift range of the sample, respectively.</text>
<text><location><page_7><loc_52><loc_31><loc_92><loc_45></location>The lower redshift source, J1137, was observed in the 2mm band, while for the other two objects the CO(3-2) line was visible in the 3mm band. Observations were carried out under average to mediocre summer conditions with five of the six 15-m antennae in operation and in compact configuration. A total of 6-10 tracks of various duration per object were necessary to reach the required line sensitivities. Data were recorded with the dual polarization, large bandwidth WideX correlators, providing spectral resolution of 1.95 MHz over a total bandwidth of 3.6 GHz.</text>
<text><location><page_7><loc_52><loc_16><loc_92><loc_31></location>The data were calibrated using the CLIC package and maps produced with MAPPING, within the IRAM GILDAS 7 software environment. A standard passband calibration scheme was first applied, followed by phase and amplitude calibration. Due to the poor observing conditions during some of the runs (e.g. high precipitable water vapour, strong winds, low elevation), particular care was taken to flag data with high phase noise. The absolute flux calibration was done using observation of reference bright quasars, and is typically accurate to better than 20% (Tacconi et al. 2010).</text>
<text><location><page_7><loc_52><loc_10><loc_92><loc_16></location>The data cubes were examined for sources at the expected spatial and spectral positions. The CO(3-2) line is clearly detected in J1137, but not in J0900 and J1226. Assuming 200 km s -1 line widths, the 3 σ upper limit on the line flux is 0.16 Jy km s -1 for J1226 and 0.44</text>
<text><location><page_8><loc_8><loc_85><loc_48><loc_92></location>Jy km s -1 for J0900. The measured CO(3-2) integrated line flux of J1137 is 1 . 16 ± 0 . 12 Jy km s -1 , and the line FWHM is 137 km s -1 as determined by a gaussian fit. The integrated CO(3-2) line map and spectrum of J1137 are shown in Figure 3.</text>
<section_header_level_1><location><page_8><loc_19><loc_82><loc_37><loc_83></location>4. DERIVED QUANTITIES</section_header_level_1>
<section_header_level_1><location><page_8><loc_22><loc_80><loc_35><loc_82></location>4.1. Stellar masses</section_header_level_1>
<text><location><page_8><loc_8><loc_63><loc_48><loc_80></location>Stellar masses found in the literature can vary significantly for the same object depending on the measurement technique, and specific assumptions made regarding star formation histories, metallicities, and stellar population ages. For example there is an order of magnitude difference for cB58 between Siana et al. (2008) and Wuyts et al. (2012a), for the Eye between Richard et al. (2011) and Sommariva et al. (2012), and for the 8:00arc between Finkelstein et al. (2009) and Dessauges-Zavadsky et al. (2011) or Richard et al. (2011). Since homogeneity is paramount for the analysis we conduct here, we derive new stellar masses consistently for all the lensed galaxies in our sample.</text>
<text><location><page_8><loc_8><loc_47><loc_48><loc_63></location>This is done with Spitzer /IRAC imaging, which is available for all the lensed galaxies with the exception of J1441. The calibrated images at 3.6 and 4.5 µ m were retrieved from the Spitzer archive, and fluxes measured using a custom-made pipeline. Since most of the lensed galaxies appear as resolved arcs in the IRAC images, and since they are also often situated in the wings of bright sources (generally, the lensing galaxy), standard photometric tools are not adequate. The details of the extraction technique are given in Appendix B, and the measured 3.6 and 4.5 µ m fluxes (un-corrected for lensing) are given in Table 3.</text>
<text><location><page_8><loc_8><loc_8><loc_48><loc_47></location>To compute stellar masses from these IRAC fluxes that will be consistent with the masses of the comparison sample, we use the catalog of Wuyts et al. (2011b) for the GOODS fields as a calibration set. For each of our lensed galaxies, we extract from the calibration set all galaxies within ∆ z = ± 0 . 2 that are star forming based on their location in the SFRM ∗ plane. This subset of galaxies is used to determine the relation between observed 3.6 and 4.5 µ m fluxes and stellar mass. The scatter in these empirical relations varies from 0.13 dex for the lensed galaxies at z ∼ 1 . 5 to 0.19 dex at the highest redshifts. These uncertainties are comparable to the typical variations in stellar masses derived though SED modeling under varying assumptions (e.g. Forster Schreiber et al. 2004; Shapley et al. 2005; Maraston et al. 2010). Stellar masses for the lensed galaxies are then obtained by taking the observed IRAC fluxes (Table 3), correcting them for the lensing magnification, and then applying the empirical calibration determined for each object. The masses derived from the 3.6 and 4.5 µ m images are consistent within the errors, and we adopt as our stellar masses the mean between the two values. These values are summarized in Table 3 with the errors quoted obtained by propagating the uncertainties on the parameters of the fit to the calibration data set, the measurement errors on the IRAC fluxes, and the uncertainty on the magnification factor. The measurement and calibration uncertainties account for ∼ 10 -30% of the error budget, the rest coming from the uncertainty on the magnification.</text>
<text><location><page_8><loc_10><loc_7><loc_48><loc_8></location>When available, we compare in Figure 4 previously</text>
<text><location><page_8><loc_52><loc_77><loc_92><loc_92></location>published stellar masses (typically from SED-fitting) to our IRAC-derived masses. For cB58, the Cosmic Eye and the 8:00arc, two previously published values differing by ∼ one order of magnitude are shown. The IRAC-derived masses are always consistent with the higher of the two values. Adopting these higher estimates for these three galaxies, the scatter between the IRAC- and SED-derived stellar masses is 0.23 dex. This scatter and the outlier points are caused by the uncertainty on the calibration of our IRAC-based stellar masses, and by different assumptions about star formation histories in the SED modeling.</text>
<section_header_level_1><location><page_8><loc_56><loc_73><loc_87><loc_76></location>4.2. Dust masses, dust temperatures and IR luminosities</section_header_level_1>
<text><location><page_8><loc_52><loc_61><loc_92><loc_73></location>The far-infrared SEDs from the PACS, SPIRE and MAMBOphotometry are shown in Figure 5. They represent some of the highest fidelity individual Herschel SEDs of star-forming galaxies at z > 1 . 5. These SEDs are used to derive dust temperatures, dust masses, and total infrared luminosities. The procedure is identical to the one applied on the comparison samples. We summarize the key elements here, with the full details given in Magnelli et al. (2012b).</text>
<text><location><page_8><loc_52><loc_42><loc_92><loc_61></location>Dust masses are calculated using the models of Draine & Li (2007) (DL07). A grid of models is created to sample the expected values of PAH abundances, radiation field intensities and dust fractions in the diffuse ISM. At each grid point, the model SED is compared with the Herschel photometry, with the dust mass given by the normalization of the SED minimizing the χ 2 . For each galaxy, the final dust mass assigned is the mean value over the grid points where χ 2 < χ 2 min + 1. To derive dust temperatures, a single modified black-body model with dust emissivity β = 1 . 5 is then fit to all the SED points with λ rest > 50 µ m. Dust temperatures and masses, along with their measured uncertainties, are given in Table 4.</text>
<text><location><page_8><loc_52><loc_18><loc_92><loc_42></location>While infrared luminosities can be derived from the DL07 modeling at the same time as the dust masses, we adopt here the values that are derived from the 160 µ m fluxes (although for completeness we also give in Tab. 4 the values of L IR obtained by integrating in the wavelength range 8-1000 µ m the best fitting DL07 model SED). Nordon et al. (2012) have demonstrated that for our z ∼ 2 galaxies this method is robust, as the uncertainties on the 160 µ m-toL IR conversion factors provided by the Chary & Elbaz (2001) template library are /lessorsimilar 0 . 1 dex. Even in the cases where the lensed galaxies are not detected in the SPIRE bands, making the fitting of DL07 model templates difficult, there is always a /greaterorsimilar 3 σ detection at 160 µ m. For the galaxies with good detections in all PACS and SPIRE bands, there is an excellent agreement between the infrared luminosities derived by the DL07 modeling and by the extrapolation from 160 µ m flux (see also Elbaz et al. 2010).</text>
<section_header_level_1><location><page_8><loc_63><loc_15><loc_81><loc_17></location>4.3. Star formation rates</section_header_level_1>
<text><location><page_8><loc_52><loc_7><loc_92><loc_15></location>Dust-obscured SFRs can be obtained simply from the infrared luminosities as SFR IR = 10 -10 L IR , with L IR in units of solar luminosity and SFR IR in M /circledot yr -1 , assuming a Chabrier IMF. At high redshifts and at high total SFR (SFR tot ), SFR IR is the dominant contribution (e.g. Pannella et al. 2009; Reddy et al. 2010;</text>
<figure>
<location><page_9><loc_10><loc_74><loc_32><loc_91></location>
</figure>
<figure>
<location><page_9><loc_38><loc_74><loc_61><loc_91></location>
</figure>
<figure>
<location><page_9><loc_62><loc_70><loc_91><loc_91></location>
<caption>Figure 3. Results of the IRAM-PdBI observations of J1137. The velocity-integrated map is shown in the middle panel, with contours showing the 2,3,5 and 8 σ levels. The shape of the PdBI beam for these data (3.75 '' × 3.26 '' ) is shown in the bottom-left corner, and the crosses show the position of the lens and of the main arc. On the left panel, the CO(3-2) contours are overlaid on top of the three-color HST ACS image. Both images are centered on α J 2000 = 11 : 37 : 40 . 1, δ J 2000 = +49 : 36 : 36 . 1 and have a size of 20 '' × 20 '' . Finally, the CO(3-2) line spectrum is shown in the left panel, with the best-fitting gaussian model shown. The line has a FWHM of 137 km s -1 and a total integrated flux of 1 . 16 ± 0 . 12 Jy km s -1 .</caption>
</figure>
<table>
<location><page_9><loc_21><loc_35><loc_80><loc_57></location>
<caption>Table 3 IRAC photometry and derived stellar masses</caption>
</table>
<text><location><page_9><loc_21><loc_31><loc_79><loc_35></location>Note . - Alternative published values of stellar masses (in log M /circledot units) are 10 . 02 ± 0 . 36 for the 8:00arc (Richard et al. 2011), 9 . 55 ± 0 . 14 for the Cosmic Eye (Sommariva et al. 2012), and 8 . 94 ± 0 . 15 for cB58 (Siana et al. 2008).</text>
<text><location><page_9><loc_8><loc_14><loc_48><loc_30></location>Wuyts et al. 2011b; Whitaker et al. 2012; Nordon et al. 2013; Heinis et al. 2013) and it is commonly assumed that SFR tot ∼ SFR IR , neglecting the un-obscured component (SFR UV ). Since the lensed galaxies in the sample were mostly selected for being bright blue arcs in optical images (i.e. bright in rest frame UV), and since due to their large magnification factors their total intrinsic SFRs are modest, it is important in this case to estimate the contribution of SFR UV to SFR tot . Following Kennicutt (1998), we measure this for a Chabrier IMF as SFR UV = 8 . 2 × 10 -29 L ν, 1600 . The rest-frame 1600 ˚ A luminosity (in erg/s/Hz) is taken to be:</text>
<formula><location><page_9><loc_14><loc_10><loc_48><loc_13></location>L ν, 1600 = 4 πD 2 L µ (1 + z ) 10 (48 . 6+ m 1600 ) / ( -2 . 5) , (3)</formula>
<text><location><page_9><loc_8><loc_7><loc_48><loc_9></location>where D L is the luminosity distance in cm, µ is the magnification factor from Table 1, and m 1600 the apparent</text>
<text><location><page_9><loc_52><loc_8><loc_92><loc_31></location>AB-magnitude at a rest wavelength 1600 ˚ A. The most accurate approach to obtain m 1600 would be to interpolate between all available bands as was done for example by Nordon et al. (2013), but its main drawback for the lensed galaxy sample is in the lack of homogenous, multiwavelength photometry. While some galaxies have published photometry in several HST bands, some have been observed only with a single HST filter, while some others only have ground based photometry available. Given the nature of the available data and the fact that the SEDs of galaxies are typically relatively flat at these wavelengths, we directly adopt for m 1600 the observed magnitude in the available optical band the closest to rest-frame 1600 ˚ A. Given the typical UV slopes β in the range from -2.0 to -1.0 (see Fig. 1), the conversion factor from AB magnitude to SFR UV has only a very weak wavelength dependence, and therefore this approxima-</text>
<table>
<location><page_10><loc_21><loc_60><loc_80><loc_88></location>
<caption>Table 4 Dust temperatures, dust masses and star formation rates</caption>
</table>
<text><location><page_10><loc_21><loc_58><loc_79><loc_60></location>Note . -SFR IR = L IR, 160 µm × 10 -10 , with L IR, 160 µm the infrared luminosity derived from the PACS 160 µ m flux.</text>
<figure>
<location><page_10><loc_12><loc_32><loc_45><loc_56></location>
<caption>Figure 4. Comparison between the stellar masses derived from the IRAC photometry and the values recovered from the literature and converted to a Chabrier IMF when necessary (see Table 3). Alternative published values for the 8:00arc, the Eye and cB58 are shown as open symbols.</caption>
</figure>
<text><location><page_10><loc_8><loc_22><loc_48><loc_24></location>tion of adopting the closest band leads to systematic uncertainties of no more than 10%.</text>
<text><location><page_10><loc_8><loc_10><loc_48><loc_21></location>We therefore adopt for the lensed galaxies a total SFR given by SFR UV + SFR IR . For the GOODSN comparison sample, we retrieve the ACS photometry from the multi-wavelength catalog of Berta et al. (2011) 8 and compute SFR UV from Eq. 3 using either B- or V-band magnitudes, depending on the redshift of each galaxy. For the SMG sample, we assume that SFR tot = SFR IR , based on their high IR luminosities and location in the SFRM ∗ plane, indicating</text>
<text><location><page_10><loc_52><loc_51><loc_92><loc_57></location>a very high attenuation of the UV light and therefore a negligible contribution of SFR UV (Buat et al. 2005; Chapman et al. 2005; Wuyts et al. 2011a; Nordon et al. 2013; Casey et al. 2013).</text>
<section_header_level_1><location><page_10><loc_66><loc_47><loc_78><loc_48></location>4.4. Metallicities</section_header_level_1>
<text><location><page_10><loc_52><loc_31><loc_92><loc_47></location>In order to obtain a homogenous set of metallicities for all galaxies in our Herschel/IRAM sample, we have compiled from the literature [NII] and H α line fluxes (Table 5). Such fluxes were available for 11 of the 17 lensed galaxies. Using the observed [NII] /Hα ratio, we compute the nebular abundance using the calibration from Denicol'o et al. (2002). This is the calibration that produces the least amount of scatter in the mass-metallicity relation (Kewley & Ellison 2008) and that was used by Genzel et al. (2012) to derive a prescription for the metallicity-dependence of the CO-to-H 2 conversion factor.</text>
<text><location><page_10><loc_52><loc_11><loc_92><loc_31></location>In the absence of a measured [NII] /Hα ratio, metallicities measured from the strong line indicator R 23 are adopted for the Eye, J1226 (see Appendix C.2) and J1441. These individual measurements are converted to the Denicol'o et al. (2002) scale using the appropriate relation given in Kewley & Ellison (2008). For the remaining three galaxies (Eyelash, J1133 and J1137), we use the mass-metallicity (MZ) relation as given in Genzel et al. (2012) to infer a metallicity. For the special case of J0901, we adopt the metallicity value derived from the MZ relation even in the presence of a measured [NII]/H α ratio as this value is likely affected by the presence of an AGN (details in § C.1). For galaxies in the comparison sample, the latter method is used as explained in Magnelli et al. (2012b).</text>
<text><location><page_10><loc_52><loc_7><loc_92><loc_11></location>All the metallicities, as well as the H α luminosities and [NII]/H α line ratios when available, are summarized in Table 5.</text>
<figure>
<location><page_11><loc_13><loc_36><loc_87><loc_90></location>
<caption>Figure 5. Far-infrared spectral energy distributions. The fluxes have been corrected for the magnification. The red line is the best-fitting modified black body function, and the black line is from the Draine & Li (2007) models.</caption>
</figure>
<section_header_level_1><location><page_11><loc_19><loc_29><loc_38><loc_30></location>4.5. Molecular gas masses</section_header_level_1>
<text><location><page_11><loc_8><loc_23><loc_48><loc_28></location>Starting from I CO (3 -2) , the integrated CO(3-2) line fluxes in Jy km s -1 presented in Table 6, we calculate the CO luminosity of the lensed galaxies, L ' CO , in units of (K km s -1 pc 2 ) following Solomon et al. (1997):</text>
<formula><location><page_11><loc_13><loc_18><loc_48><loc_22></location>L ' CO = 3 . 25 × 10 7 I CO (3 -2) R 13 µ D 2 L ν 2 obs (1 + z ) 3 , (4)</formula>
<text><location><page_11><loc_8><loc_7><loc_48><loc_18></location>where µ is the magnification factor as given in Table 1, D L is the luminosity distance in Mpc and ν obs the observed frequency in GHz. The factor R 13 ≡ I CO (1 -0) /I CO (3 -2) is the excitation correction to extrapolate the CO(1-0) line flux from our CO(3-2) observations. As in Tacconi et al. (2013), we adopt a value of R 13 = 2 based on recent studies of the CO spectral line energy distribution (e.g. Weiss et al. 2007;</text>
<text><location><page_11><loc_52><loc_17><loc_92><loc_30></location>Dannerbauer et al. 2009; Harris et al. 2010; Ivison et al. 2011; Bauermeister et al. 2013; Bothwell et al. 2013). These studies have targeted both normal highz starforming galaxies and SMGs, with similar results pointing to a characteristic value of R 13 ∼ 2 that may be due to low excitation or to the typical filling factors of the two lines. We therefore apply this value of R 13 uniformly across our sample, and assume a conservative 20% uncertainty on this value compared to the typical errors presented in these individual studies.</text>
<text><location><page_11><loc_52><loc_8><loc_92><loc_17></location>Total molecular gas masses are inferred from L ' CO using the CO-to-H 2 conversion factor, α CO ( M H 2 = α CO L ' CO ). The specific value of α CO to be applied for each galaxy must be determined with care, as most have sub-solar metallicities ((12 + log O / H) /circledot = 8 . 69 ± 0 . 05; Asplund et al. 2009). Under such low metallicity conditions, both observations and models suggest that the</text>
<table>
<location><page_12><loc_19><loc_64><loc_82><loc_88></location>
<caption>Table 5 [NII] and H α line fluxes and derived gas-phase metallicites</caption>
</table>
<text><location><page_12><loc_19><loc_59><loc_81><loc_64></location>Note . - The metallicities in this table are either from the N2 indicator using the values of [NII]/H α from Column (3), the R 23 indicator, or derived from the mass-metallicity relation (MZ). In all cases, the metallicities are converted to the scale of the Denicol'o et al. (2002) calibration.</text>
<unordered_list>
<list_item><location><page_12><loc_19><loc_58><loc_73><loc_59></location>a See Appendix C.1 for a discussion about the presence of an AGN in this galaxy.</list_item>
</unordered_list>
<figure>
<location><page_12><loc_22><loc_34><loc_79><loc_56></location>
<caption>Figure 6. Comparison between the conversion factor α CO derived either using the 'inverse KS relation' method (left) or using the gasto-dust ratio method (right) as a function of metallicity. Only the lensed galaxies with a direct measurement of metallicity are represented (filled circles). The error bars are obtained by propagating the measurement errors on L CO , M dust and SFR, but do not include the systematic uncertainty in the calibration of the two techniques. The z = 0 data points from Leroy et al. (2011) are shown as filled squares for the gas-to-dust method. Finally, the empirical relation for the metallicity-dependence of α CO from Genzel et al. (2012) is shown as a solid line, and the value of 4.35 M /circledot (K km/s pc 2 ) -1 derived from Milky Way clouds and typically assumed for solar-metallicity star forming galaxies is highlighted with the horizontal dotted line.</caption>
</figure>
<text><location><page_12><loc_8><loc_17><loc_48><loc_25></location>value of α CO increases, as the CO molecule becomes a poor proxy for H 2 (e.g. Israel 1997; Dame et al. 2001; Rosolowsky et al. 2003; Blitz et al. 2007; Leroy et al. 2011; Glover & Mac Low 2011; Shetty et al. 2011; Feldmann et al. 2012; Genzel et al. 2012). Here, different methods to estimate α CO are investigated.</text>
<text><location><page_12><loc_8><loc_8><loc_48><loc_17></location>The first is the 'inverse Kennicutt-Schmidt relation' method. Under the assumption that a tight relation exists between the surface density of molecular gas and the SFR surface density, the value of α CO can be estimated knowing Σ SFR and the CO luminosity. Using a compilation of local and high redshift galaxies, Genzel et al. (2012) calibrated a relation between α CO and metallicity</text>
<text><location><page_12><loc_52><loc_17><loc_92><loc_25></location>using this approach. Since most recent studies suggest that the KS relation is near-linear (e.g. Leroy et al. 2008; Blanc et al. 2009; Genzel et al. 2010; Daddi et al. 2010b; Bigiel et al. 2011; Rahman et al. 2012; Saintonge et al. 2012; Shetty et al. 2013; Feldmann 2013), the problem is further simplified and α CO can be estimated as:</text>
<formula><location><page_12><loc_63><loc_12><loc_92><loc_16></location>α CO,KS = SFR t dep ( H 2 ) L ' CO , (5)</formula>
<text><location><page_12><loc_52><loc_7><loc_92><loc_11></location>where L ' CO is the CO(1-0) line luminosity in (K km s -1 pc 2 ), and α CO the CO-to-H 2 conversion factor in M /circledot (K km s -1 pc 2 ) -1 . In applying eq. 5, we adopt a redshift-</text>
<text><location><page_13><loc_8><loc_83><loc_48><loc_92></location>endent depletion time, t dep (H 2 )= 1 . 5(1 + z ) -1 , as suggested by Tacconi et al. (2013) and further supported here to z = 2 -3 in § 5.3. In Figure 6, left panel, the value of α CO,KS is plotted against metallicity, for the lensed galaxies. The values of α CO scatter around the relation derived by Genzel et al. (2012) using this method and a sample of both local and high redshift galaxies,</text>
<formula><location><page_13><loc_14><loc_80><loc_48><loc_82></location>log α CO = -1 . 27[12 + log (O / H)] + 11 . 8 . (6)</formula>
<text><location><page_13><loc_8><loc_68><loc_48><loc_79></location>The second approach, the 'gas-to-dust ratio method', relies on a measurement of the dust mass and a motivated choice of a gas-to-dust ratio. This method has been used successfully in the local Universe (e.g. Israel 1997; Gratier et al. 2010; Leroy et al. 2011; Bolatto et al. 2011; Sandstrom et al. 2012), and shown to be applicable also at high redshifts (Magdis et al. 2011, 2012b; Magnelli et al. 2012b). In the case of high redshift galaxies, the conversion factor can be estimated simply as:</text>
<formula><location><page_13><loc_18><loc_63><loc_48><loc_67></location>α CO,dust = δ GDR ( Z ) M dust L ' CO , (7)</formula>
<text><location><page_13><loc_8><loc_38><loc_48><loc_63></location>where M dust is in solar masses, and we adopt a metallicity-dependent gas-to-dust ratio, δ GDR ( Z ), from Leroy et al. (2011). All the assumptions required to apply equation 7 to high redshift galaxies are extensively described in § 5.1 of Magnelli et al. (2012b). In particular, it needs to be assumed that the CO lines and the FIR continuum are emitted from the same physical regions of the galaxies given that Herschel does not resolve them, and that at high redshift M H 2 /greatermuch M HI and therefore that the atomic component of the ISM can be neglected in Eq. 7 (see § 5.3.1 for a justification of this assumption). A significant additional uncertainty lies in the assumption that the δ GDR ( Z ) relation of Leroy et al. (2011), calibrated on a handful of very nearby galaxies, applies directly at z ∼ 1 -3. Furthermore, other studies of nearby galaxies indicate that the scatter in the δ GDR -Z relation may be larger than suggested by this specific z = 0 sample, especially at low metallicities (Draine et al. 2007; Galametz et al. 2011).</text>
<text><location><page_13><loc_8><loc_28><loc_48><loc_38></location>In Figure 6, right panel, α CO,dust measured from eq. 7 is shown as a function of metallicity. The z > 2 galaxies follow the inverse relation between α CO and metallicity seen in the z = 0 data and in the empirical relation of Genzel et al. (2012), although with a small systematic offset to lower values of α CO at fixed metallicity, the implications of which are discussed in § 5.2.</text>
<text><location><page_13><loc_8><loc_10><loc_48><loc_28></location>Although affected by different sets of uncertainties and assumptions, the two methods of estimating α CO produce consistent results, which in the mean and within their errors reproduce the metallicity-dependence of α CO recently calibrated at z > 1 by Genzel et al. (2012), and previously also observed locally (e.g. Wilson 1995; Israel 1997; Boselli et al. 2002; Bolatto et al. 2011; Leroy et al. 2011). Throughout the rest of this paper, we therefore adopt as a consensus between α CO,KS and α CO,dust the value of α CO obtained from the prescription of Genzel et al. (2012), as given in eq. 6, using our best estimates of metallicities given in Table 5. These values of α CO , as well as the CO luminosities and derived molecular gas masses are presented in Table 6.</text>
<text><location><page_13><loc_52><loc_85><loc_92><loc_92></location>Having derived accurate and homogeneous measures of M ∗ , SFR, T dust , M dust , M H 2 and 12+logO / Hacross the lensed galaxies and comparison samples, we now investigate the gas and dust properties of galaxies as a function of their position in the SFRM ∗ -z parameter space.</text>
<section_header_level_1><location><page_13><loc_54><loc_82><loc_90><loc_83></location>5.1. High dust temperatures in z > 2 lensed galaxies</section_header_level_1>
<text><location><page_13><loc_52><loc_60><loc_92><loc_81></location>In Figure 7, we plot the dust temperature against the offset from the star formation main sequence and the total infrared luminosity, both for the lensed and comparison galaxy samples. While we recover with the comparison sample the known trend between T dust and these two quantities (Dale et al. 2001; Chapman et al. 2003; Hwang et al. 2010; Magnelli et al. 2012b; Symeonidis et al. 2013), the z > 2 lensed galaxies occupy a significantly different region of the plot. Dust temperatures in these galaxies are very high ( T dust ∼ 50K), even though they are located on the main sequence and have modest infrared luminosities ( L IR ∼ 10 11 -10 12 L /circledot ) given their redshifts and stellar masses. Plotting dust temperatures against specific star formation rate rather than main sequence offset produces qualitatively equivalent results.</text>
<text><location><page_13><loc_52><loc_43><loc_92><loc_60></location>There is evidence that the tightness of the T dust -L IR relation observed in classical samples of SMGs (e.g. Chapman et al. 2005, see also Fig. 7b) is due to selection biases (Casey et al. 2009; Magnelli et al. 2010, 2012a). In particular, the sub-millimeter selection technique favors colder objects, especially at low L IR . When also considering samples of dusty high-redshift star-forming galaxies selected through other techniques (e.g. the optically-faint radio galaxies of Chapman et al. 2004), dust temperatures can be significantly higher at fixed L IR . The lensed galaxies shown in Fig. 7b extend these results to galaxies with even lower IR luminosities and higher dust temperatures.</text>
<text><location><page_13><loc_52><loc_28><loc_92><loc_43></location>To confirm the high T dust among the lensed galaxies, we also show in Figure 7c-d similar relations using the infrared color, defined as the ratio between rest-frame 60 µ m and 100 µ m fluxes ( S 60 /S 100 , where S 60 and S 100 are determined by linear interpolation of the Herschel photometry). This quantity, commonly used in IRASbased studies, is a proxy for T dust but is independent of any model assumptions. The same behavior is observed for the z > 2 lensed galaxies, with these objects having unusually high S 60 /S 100 ratios given their modest infrared luminosities.</text>
<text><location><page_13><loc_52><loc_7><loc_92><loc_28></location>Among the z > 2 lensed galaxies, there are however two exceptions, J0901 and the Eyelash. With T dust ∼ 36K and S 60 /S 100 ∼ 0 . 5, these two galaxies follow the general trend between T dust and MS offset (or L IR ) traced by the various comparison samples (Magnelli et al. 2012a; Roseboom et al. 2012). These two galaxies also differ from the rest of the z > 2 lensed galaxy sample in other regards. For example, they have the largest dust masses ( M dust ∼ 10 9 M /circledot as compared to ∼ 1 -3 × 10 7 M /circledot ), and although no direct metallicity measurement is available for the Eyelash, J0901 has the highest metallicity of all the galaxies in the sample (but see Appendix C.1). The high dust temperatures in the rest of the z > 2 lensed galaxies therefore seem to be linked to their low dust contents compared to their SFRs. As we could not find evidence in 2.1.1 that the z > 2</text>
<table>
<location><page_14><loc_8><loc_73><loc_93><loc_88></location>
<caption>Table 6 Lensed galaxies CO line fluxes and molecular gas masses</caption>
</table>
<unordered_list>
<list_item><location><page_14><loc_8><loc_69><loc_92><loc_71></location>b CO line luminosity, including the magnification correction and an excitation correction to bring this measurement on the CO(1-0) scale (see eq. 4).</list_item>
<list_item><location><page_14><loc_8><loc_67><loc_92><loc_69></location>c Measured from L ' CO and α CO,Z , the metallicity-dependent conversion factor of Genzel et al. (2012). These values of M H 2 do include the contribution of helium and are therefore total molecular gas masses.</list_item>
</unordered_list>
<text><location><page_14><loc_8><loc_47><loc_48><loc_64></location>lensed galaxies form a biased subsample of the underlying galaxy population, this observations suggests that ∼ 10 10 M /circledot galaxies at z = 2 -3 have higher dust temperatures and lower dust masses than similar galaxies at lower redshifts. Using deep Herschel PACS and SPIRE blind fields and a stacking technique, the mean T dust of normal star-forming galaxies with M ∗ ∼ 10 10 M /circledot at z > 2 is just barely measurable. Such studies indeed suggest a rise in temperature and intensity of the radiation field on the main sequence in the mass/redshift regime that we have now probed directly with the lensed galaxies (Reddy et al. 2012; Magdis et al. 2012a; Magnelli et al. 2013).</text>
<section_header_level_1><location><page_14><loc_21><loc_45><loc_36><loc_46></location>5.2. Gas-to-dust ratio</section_header_level_1>
<text><location><page_14><loc_8><loc_29><loc_48><loc_44></location>In the local Universe, the typical gas-to-dust ratio, δ GDR , for star-forming galaxies with solar metallicities is of order 100 (Draine et al. 2007). This ratio has been shown to increase in low metallicity environments (e.g. Hunt et al. 2005; Engelbracht et al. 2008; Leroy et al. 2011), as predicted by dust formation models (e.g. Edmunds 2001). The value of δ GDR is also expected to vary in high density environments, such as the nuclear regions of starbursts, but observations suggest only a mild decrease of no more than a factor of 2 (Wilson et al. 2008; Clements et al. 2010; Santini et al. 2010).</text>
<text><location><page_14><loc_8><loc_7><loc_48><loc_29></location>In the previous section, we inferred that the high dust temperatures of the z > 2 lensed galaxies were due to their low dust masses and metallicities. To understand whether the dust content of these galaxies is abnormally low for their other properties we plot in Figure 8 the measured value of δ GDR as a function of metallicity for the galaxies within 0.5 dex of the star formation main sequence. The gas-to-dust ratio is measured as δ GDR = M gas /M dust , where M gas is the molecular gas mass from Table 6 calculated from the CO luminosity using the metallicity-dependent conversion factor of Genzel et al. (2012), and M dust is derived from the Herschel photometry using the DL07 models (see § 4.2). Figure 8 reveals that the high redshift galaxies have a gas-to-dust ratio that scales inversely with metallicity like in the local Universe. However, fitting the δ GDR -Z relation of the z > 2 galaxies while keeping the slope</text>
<text><location><page_14><loc_52><loc_62><loc_92><loc_64></location>fixed to the z = 0 reference relation reveals an increase in the mean δ GDR at fixed metallicity by a factor of 1.7.</text>
<text><location><page_14><loc_52><loc_46><loc_92><loc_62></location>Since measuring CO in normal star-forming galaxies at high redshifts is challenging, there has been significant interest recently in using dust masses as a proxy for the total molecular gas contents (e.g. Magdis et al. 2011, 2012a; Scoville 2012). The method consists in measuring M dust using far-infrared and/or sub-mm photometry, and then applying the estimated gas-to-dust ratio ( δ GDR ( Z ), see § 4.5) to arrive at the gas mass. The systematic offset between the z > 2 normal star-forming galaxies and the reference relation of δ GDR ( Z ) therefore needs to be taken into consideration. There are at least three possible explanations:</text>
<text><location><page_14><loc_52><loc_22><loc_92><loc_45></location>Are dust properties different at z > 2 ? -As detailed in § 4.2, dust masses for the lensed galaxies are computed using the model of DL07. In this model, dust is considered to be a combination of carbonaceous and amorphous silicate grains with a specific size distribution set to reproduce the Milky Way extinction curve. The model also assumes that a large fraction of the dust is found in the diffuse ISM, the rest in photodissociation regions, with the two components exposed to different radiation fields, the parameters of which are set to reproduce Milky Way conditions or left as free parameters. As argued by Bolatto et al. (2013), not only are dust grain properties poorly understood even in the local universe, the lack of a clear understanding of how dust is exactly produced and destroyed leads to significant uncertainties in applying the DL07 model to the denser, hotter interstellar medium of high redshift galaxies.</text>
<text><location><page_14><loc_52><loc_7><loc_92><loc_21></location>Is α CO for the lensed galaxies smaller than expected? -If we instead assume that the dust masses are accurately measured in the lensed galaxies, the offset in δ GDR ( Z ) compared to z = 0 could be explained by a conversion factor α CO smaller on average by a factor of ∼ 1 . 7. The lensed galaxies are located on the main-sequence, which is interpreted as being the locus of galaxies where star formation takes place in virialized GMCs rather than in a centrallyconcentrated, dense starburst mode. Under such conditions, α CO should scale as ρ 0 . 5 /T (Tacconi et al. 2008; Bolatto et al. 2013). Assuming that gas and dust are</text>
<figure>
<location><page_15><loc_23><loc_47><loc_80><loc_90></location>
<caption>Figure 7. Dust temperature as a function of (a) offset from the star formation main sequence, as defined using the GOODS and EGS catalogs of Wuyts et al. (2011b) as a reference, and (b) infrared luminosity. The infrared color ( S 60 /S 100 ), a proxy for dust temperature, is also shown as a function of (c) ∆ log( sSFR ) and (d) L IR . The lensed galaxies are shown in colored symbols (red: z > 2, orange: z < 2), and the GOODS-N and SMG reference sample (see § 2) is in gray (dark: z > 2, light: z < 2). The T dust -∆(MS) relations of Magnelli et al. (2013) between z ∼ 1 and z ∼ 2 is shown as the shaded region on the left-hand panels (note that this relation is derived using a sample of galaxies more massive than our typical lensed galaxies). On the right-hand panels, we show as a comparison the local relation of Chapman et al. (2003) (filled dashed region), the locus of high-redshift SMGs (Chapman et al. 2005) (solid filled region), and the relation of Roseboom et al. (2012) (dashed line). To convert the different reference relations from T dust to S 60 /S 100 (or vice-versa), we have assumed a simple modified black body with β = 1 . 5.</caption>
</figure>
<text><location><page_15><loc_8><loc_20><loc_48><loc_32></location>thermalized, the large measured values of T dust may indicate a reduction of α CO , unless the gas density ρ also increases in proportion. The two z > 2 lensed galaxies with normal temperatures of ∼ 35K (J0901 and the Eyelash, see Fig. 7) also have gas-to-dust ratios that are elevated compared to the z = 0 relation, arguing that the ISM in the rest of the sample is not only warm but also denser such that α CO is not affected by the temperature variations.</text>
<text><location><page_15><loc_8><loc_7><loc_48><loc_19></location>Is the gas-to-dust ratio really higher at z > 2 ? -As a last step, we can assume that none of these concerns apply and that both M gas and M dust are accurately measured in the lensed galaxies, leaving us with the conclusion that the gas-to-dust ratio is higher at fixed metallicity than in the local universe. This could happen for example if a smaller fraction of the metals are locked up in dust grains under the specific conditions prevailing in the ISM of the high redshift galaxies, or if the far-infrared emission</text>
<text><location><page_15><loc_52><loc_28><loc_92><loc_32></location>used to compute M dust and the rest-frame optical line emission used to compute the metallicity are not emitted from the same physical regions.</text>
<text><location><page_15><loc_52><loc_15><loc_92><loc_28></location>However, irrespectively of which one of these possible explanations is valid, we can conclude that when measuring dust and gas masses though standard techniques, an offset of 0.23 dex is obtained between the measured gas-to-dust ratio and the standard z = 0 prescription of Leroy et al. (2011). Care must therefore be taken in applying the gas-to-dust ratio method to estimate molecular gas masses of high redshift galaxies, as small but systematic differences with CO-based measurements may otherwise occur.</text>
<section_header_level_1><location><page_15><loc_53><loc_11><loc_90><loc_14></location>5.3. Star formation efficiencies and gas fractions in z > 2 normal star-forming galaxies</section_header_level_1>
<text><location><page_15><loc_52><loc_7><loc_92><loc_11></location>In § 5.1 above, we suggested that the high T dust of the lensed galaxies is caused by their low dust contents. To further illustrate this, we repeat the analy-</text>
<figure>
<location><page_16><loc_10><loc_63><loc_47><loc_91></location>
<caption>Figure 8. Gas-to-dust ratio as a function of gas-phase metallicity for the z > 2 lensed galaxies (red filled circles) and the z ∼ 1 . 5 reference sample in GOODS-N (black filled square; the mean value in the sample is given since no direct metallicities are available only estimates from the mass-metallicity relation). Only galaxies within 0.5 dex of the star formation main sequence are considered. For all high-redshift galaxies, M dust is computed from the Herschel photometry using the DL07 models, and M gas is the molecular gas mass computed from L ' CO and a metallicity-dependent conversion factor α CO (Genzel et al. 2012). As a comparison, the open squares and the dashed line are from the study of Leroy et al. (2011) of local galaxies. Fitting the z > 2 galaxies while keeping the slope fixed (solid orange line and shaded region) shows a mean increase of δ GDR by a factor of 1.7.</caption>
</figure>
<text><location><page_16><loc_8><loc_16><loc_48><loc_44></location>is done in Figure 7, but this time normalizing L IR by the dust or the gas mass. We first show in Figure 9 how both T dust and the S 60 /S 100 ratio depend on the star formation efficiency (SFE ≡ SFR/M H 2 ). A correlation between these quantities is expected as T dust increases as a function of ∆ log( sSFR ) and L IR (Fig. 7), and there is also a positive correlation between SFE and ∆log( sSFR ) (Saintonge et al. 2012; Magdis et al. 2012a; Sargent et al. 2013). Figure 9 reveals that when the gas mass in each galaxy is taken into account, most of the offset between the z > 2 lensed galaxies and the comparison samples disappears, although the z > 2 normal star-forming galaxies still appear to have dust temperatures higher by ∼ 5 -10K at fixed SFE compared to the reference sample (gray symbols). The offset is strongest for those galaxies with the lowest metallicities. While metallicity may account for part of the remaining offset in T dust at fixed SFE, there could also be a contribution from differential lensing (see § 2.1.2), which if present at all, could manifest itself in an overestimation of the dust temperatures.</text>
<text><location><page_16><loc_8><loc_7><loc_48><loc_16></location>In Figure 10 we then show a similar relation, but using the dust rather than the gas mass as the normalization factor, allowing us to include a larger number of galaxies for which dust masses are measured, but gas masses are not available. In this case again, some of the lensed galaxies appear to be systematically offset to higher T dust at fixed value of SFR /M dust . The z > 2 lensed galaxies</text>
<text><location><page_16><loc_52><loc_75><loc_92><loc_92></location>having typically low metallicities, each dust grain will be exposed to more UV photons for a fixed SFR per unit gas mass, and therefore will have a hotter temperature. There is evidence for this in the fact that the galaxies with ∼ solar metallicities follow more closely the relation traced by the reference sample, in particular when using the S 60 /S 100 ratio (see Fig. 10). The combination of Figures 8-10 thus argues that in these galaxies, the balance between M dust , M gas and Z follows expected relations (modulo the possible change of δ GDR with redshift), and that the high dust temperatures are the result of the high SFRs per unit gas and dust mass, i.e. the high star formation efficiencies.</text>
<text><location><page_16><loc_52><loc_53><loc_92><loc_75></location>This assertion that the z > 2 lensed galaxies have high star formation efficiencies (or put differently, short molecular gas depletion timescales, t dep (H 2 )) needs to be investigated in more detail. In the local Universe, normal starforming galaxies have t dep (H 2 ) ∼ 1 . 5 Gyr (Leroy et al. 2008; Bigiel et al. 2011; Saintonge et al. 2011b, 2012). In the PHIBSS survey, Tacconi et al. (2013) show that at z = 1 -1 . 5 the depletion time for main-sequence galaxies is reduced to ∼ 700 Myr (with 0.24 dex scatter in the log t dep (H 2 ) distribution, and assuming a Galactic conversion factor α CO ). Based on these observations, Tacconi et al. (2013) infer a redshift-dependence of the form t dep (H 2 )= 1 . 5(1 + z ) -1 . This redshift evolution is only slightly slower than the dependence of (1 + z ) -1 . 5 expected if t dep (H 2 ) is proportional to the dynamical timescale (Dav'e et al. 2012).</text>
<text><location><page_16><loc_52><loc_22><loc_92><loc_53></location>The lensed and reference galaxies at z > 2 can be used to track the behavior of the depletion time at even higher redshifts. In Figure 11, left panel, the redshift evolution of t dep (H 2 ) is shown. The gray band shows the expected trend based on the empirical (1 + z ) -1 dependence (upper envelope), and the analytical expectation of a (1 + z ) -1 . 5 dependence (lower envelope). The mean values in four different redshift intervals are shown. For the highest redshift interval (2 < z < 3), we combine galaxies from the lensed and comparison samples as well as from PHIBSS. While the mean depletion time is ∼ 700 Myr at z ∼ 1 . 2, it decreases further to ∼ 450 Myr at z ∼ 2 . 2, consistently with the predicted redshift evolution. We caution however that the sample at z > 2 is by no means complete or representative of all normal star forming galaxies, but in terms of stellar masses and main sequence offset it is at least comparable to the z ∼ 1 -1 . 5 PHIBSS sample and the z = 0 reference sample from COLD GASS. This measurement at z > 2 suggests that the redshift-dependence of the molecular gas depletion time for main sequence galaxies, as predicted analytically (e.g. Dav'e et al. 2011, 2012) and recently observed up to z ∼ 1 . 5 (Tacconi et al. 2013), extends to z ∼ 3.</text>
<text><location><page_16><loc_52><loc_18><loc_92><loc_22></location>Using the lensed galaxies and the various comparison samples, we can also trace the redshift evolution of the molecular gas mass fraction,</text>
<formula><location><page_16><loc_65><loc_14><loc_92><loc_17></location>f gas = M H 2 M H 2 + M ∗ . (8)</formula>
<text><location><page_16><loc_52><loc_7><loc_92><loc_13></location>In what follows, we restrict the samples only to galaxies found within 0.5 dex of the star formation main sequence at the appropriate redshift, i.e. we study the gas contents of normal, star-forming discs as a function of redshift. Several studies have now reported a rapid increase of f gas</text>
<figure>
<location><page_17><loc_22><loc_70><loc_82><loc_92></location>
<caption>Figure 9. Dust temperature (left) and S 60 /S 100 ratio (right) as a function of star formation efficiency ( ≡ SFR/M H 2 ). The lensed galaxies are shown in colored symbols (red: z > 2, orange: z < 2), and the reference sample in gray (dark: z > 2, light: z < 2). The z > 2 lensed galaxies with ∼ solar metallicities (12 + logO/H > 8 . 6) are marked with open green circles. The blue band is the T dust -∆log( sSFR ) relation of Fig. 7 converted to a T dust -SFE relation using the relation of Magdis et al. (2012a) between main sequence offset and star formation efficiency, SFE ∝ ∆log( sSFR ) 1 . 34 ± 0 . 13 (see also Saintonge et al. 2012; Sargent et al. 2013).</caption>
</figure>
<figure>
<location><page_17><loc_22><loc_39><loc_80><loc_61></location>
<caption>Figure 10. Same as Figure 9, both with the SFR on the x -axis normalized by M dust rather than M H 2 . The location of the subset of the most metal-rich lensed galaxies in the figure, closer the locus of the reference samples, suggests that the remaining offset in T dust and S 60 /S 100 ratio is a consequence of the low metallicities (see e.g. Fig. 8).</caption>
</figure>
<text><location><page_17><loc_8><loc_13><loc_48><loc_34></location>with redshift (Tacconi et al. 2010; Daddi et al. 2010a; Geach et al. 2011; Magdis et al. 2012a). The most robust analysis so far was performed by Tacconi et al. (2013). In that work, the PHIBSS data at z ∼ 1 and z ∼ 2 were corrected for incompleteness and compared to a matched local control sample extracted from the COLD GASS catalog, revealing an increase of f gas from 8% at z = 0 to 33% at z ∼ 1 and 47% at z ∼ 2. These three secure measurements are reproduced in Figure 11 (right panel). There are very few galaxies at z ∼ 0 . 5 with published CO measurements, but in Fig. 11 the few systems found in Geach et al. (2011) and Bauermeister et al. (2013) are compiled (CO measurements for several galaxies with 0 . 6 < z < 1 . 0 are published also in Combes et al. (2012), but we do not include them here as they are above-main sequence objects).</text>
<text><location><page_17><loc_8><loc_8><loc_48><loc_13></location>As the PHIBSS sample extends to z = 2 . 4, we combine all the main-sequence galaxies in our lensed and comparison samples above that redshift to derive a mean gas fraction of 40 ± 15% at < z > = 2 . 8. We then apply</text>
<text><location><page_17><loc_52><loc_25><loc_92><loc_34></location>the methodology of Tacconi et al. (2013) to correct for sample incompleteness. As the sample of lensed galaxies with z > 2 . 4 is richer in on- and below-main sequence galaxies, accounting for this bias rises the mean gas fraction to 44%. Our observations therefore suggest that the trend for increasing gas fraction with redshift does not extend beyond z ∼ 2, and may even be reversing.</text>
<text><location><page_17><loc_52><loc_20><loc_92><loc_25></location>Can this flattening of the relation between gas fractions and redshift at z > 2 be expected under the equilibrium model? The definition of the gas fraction (eq. 9) can be re-expressed as:</text>
<formula><location><page_17><loc_62><loc_16><loc_92><loc_19></location>f gas = 1 1 + ( t dep sSFR) -1 , (9)</formula>
<text><location><page_17><loc_52><loc_10><loc_92><loc_15></location>and the best predictions available for the redshift evolution of t dep and sSFR used to compute the expected behavior of f gas ( z ). As explained in § 5.3, it is estimated that</text>
<formula><location><page_17><loc_62><loc_9><loc_92><loc_10></location>t dep ( z ) = 1 . 5(1 + z ) α [Gyr] , (10)</formula>
<text><location><page_17><loc_52><loc_7><loc_92><loc_8></location>with α measured to be -1.0 by Tacconi et al. (2013)</text>
<figure>
<location><page_18><loc_52><loc_68><loc_83><loc_91></location>
<caption>Figure 11. Redshift evolution of the mean gas depletion time (left) and gas mass fraction (right) for main-sequence galaxies from our lensed and reference samples for which a CO-based measurement of M H 2 is available. Error bars show the standard deviation within each redshift bin. The different colors represent the following datasets: blue : representative z = 0 sample from COLD GASS, green : galaxies from Geach et al. (2011) and Bauermeister et al. (2013) at z ∼ 0 . 4, orange : the incompleteness-corrected mean values from Tacconi et al. (2013). The red points show the contribution from this study and include all the galaxies in the specified redshift intervals from the lensed and comparison samples as well as from PHIBSS, and corrects for sample incompleteness. On the left plot, the gray shaded region shows the expected trend between between t dep (H 2 ) and z described by eq. 10, for α = [ -1 . 0 , -1 . 5]. On the right panel, the gray shaded region is the expected redshift dependence of f gas derived from equations 9-11, assuming that α = -1 . 0 (Tacconi et al. 2013) and sSFR follows eq. 11 (Lilly et al. 2013). Alternative relations for f gas ( z ) are obtained by assuming that sSFR ∝ (1 + z ) 2 . 8 at all redshifts (dotted dark blue line), or else reaches a plateau at z = 2 (dashed light blue line).</caption>
</figure>
<text><location><page_18><loc_8><loc_45><loc_48><loc_56></location>and predicted to be -1.5 in the analytic model of Dav'e et al. (2012). The relation is normalized to the typical depletion time of 1.5 Gyr observed in local galaxies (Leroy et al. 2008; Bigiel et al. 2011; Saintonge et al. 2011b, 2012). Based on studies of the slope and redshift evolution of the star formation main sequence, the typical sSFR (in Gyr -1 ) of a star-forming galaxy of mass M ∗ at redshift z is</text>
<formula><location><page_18><loc_8><loc_37><loc_49><loc_44></location>sSFR( M ∗ , z ) = { 0 . 07 ( M ∗ 10 10 . 5 M /circledot ) -0 . 1 (1 + z ) 3 if z < 2 0 . 30 ( M ∗ 10 10 . 5 M /circledot ) -0 . 1 (1 + z ) 5 / 3 if z > 2 . (11)</formula>
<text><location><page_18><loc_8><loc_17><loc_48><loc_37></location>The above equation is presentd by Lilly et al. (2013) based on results from a number of recent highredshift imaging surveys (Noeske et al. 2007; Elbaz et al. 2007; Daddi et al. 2007; Pannella et al. 2009; Stark et al. 2012). The expected redshift evolution of the gas fraction for galaxies of a given stellar mass can then be obtained by combining equations 9-11. For galaxies in the mass range 10 10 -5 × 10 11 and for α = -1 . 0 in eq. 10, the expected trend is shown in Figure 11 as the light gray band. At z > 2, f gas flattens out because of the shallower evolution of sSFR with redshift mostly canceling out the (1 + z ) -1 term from the t dep (H 2 ) relation (eq. 10). This model predicts a very modest evolution of the mean gas fraction from 47% at z = 2 . 2 to 49% at z = 2 . 8, consistently with our measurement.</text>
<text><location><page_18><loc_8><loc_7><loc_48><loc_17></location>A different behavior for f gas ( z ) can be obtained by modifying the redshift-dependence of either t dep or sSFR (eq. 9). For example, if a value of α = -1 . 5 is used to set the evolution of t dep , the mean gas fraction expected at any redshift is lower than in the first model. However, since there is no observational evidence for a redshift evolution of t dep different from what has been assumed so far, we turn instead our attention to sSFR.</text>
<text><location><page_18><loc_52><loc_45><loc_92><loc_56></location>Equation 11 assumes that at fixed stellar mass, the specific star formation rate increases as (1+ z ) 3 out to z = 2, and that this evolution then slows down to (1 + z ) 5 / 3 . If instead we took sSFR to keep increasing steadily with redshift, for example as sSFR ∝ M ∗ -0 . 3 (1 + z ) 2 . 8 (Tacconi et al. 2013), the predicted mean gas fraction at z = 2 . 8 would become 57%, in tension with our measured value of 40 ± 15%.</text>
<text><location><page_18><loc_52><loc_19><loc_92><loc_45></location>Alternatively, it has been reported that the characteristic sSFR of main-sequence galaxies reaches a plateau at z ∼ 2 (Stark et al. 2009; Gonz'alez et al. 2010; Rodighiero et al. 2010; Weinmann et al. 2011; Reddy et al. 2012). If we assume that sSFR increases up to z = 2 according to eq. 11 and then remains constant at z > 2, the mean gas fraction is predicted to drop to 41% at z = 2 . 8. While this behavior is consistent with the observations presented here, the existence of such a plateau in sSFR at z ∼ 2 -7 has recently been challenged. After accounting for contamination by strong line emission, different authors suggest an increase of the mean sSFR by a factor of 2-5 between z = 2 and z = 7 (e.g. Bouwens et al. 2012; Stark et al. 2012; Gonzalez et al. 2012). This decrease in f gas at z > 2 should therefore be considered as a lower limit, with the relation based on eq. 11 providing the most realistic prediction of the redshift evolution of the gas fraction, as supported by our observations.</text>
<section_header_level_1><location><page_18><loc_61><loc_17><loc_82><loc_18></location>5.3.1. Missing cold gas at z > 2 ?</section_header_level_1>
<text><location><page_18><loc_52><loc_7><loc_92><loc_16></location>Another factor that could contribute to the redshift evolution of the gas fraction is the relative contribution of atomic hydrogen to the cold gas budget of the galaxies (M HI +M H 2 ) at the different epochs. For star forming galaxies with 10 . 5 < log M ∗ /M /circledot < 11 . 5 at z = 1 -2 such as those in the PHIBSS sample, it is generally assumed that M H 2 >> M HI , at least within the parts of the</text>
<text><location><page_19><loc_8><loc_47><loc_48><loc_92></location>discs where star formation is actively taking place. This assumption is based on the observed high surface densities, above the characteristic threshold for the atomicto-molecular conversion. On the other hand, for nearby galaxies with log M ∗ /M /circledot > 10 . 0, it is observed that M HI ∼ 3 M H 2 , albeit with large galaxy-galaxy variations (Saintonge et al. 2011a). An important fraction of this atomic gas is located in the outer regions of galaxies, outside of the actively star-forming disks, but even within the central regions a significant fraction of the cold gas is in atomic form as the HI-to-H 2 transition typically occurs at a radius of ∼ 0 . 4 r 25 (Bigiel et al. 2008; Leroy et al. 2008). We can therefore estimate that on average within r 25 , M HI ∼ M H 2 , raising the cold gas mass fraction from 8% when only the molecular phase was considered to ∼ 15%. We cannot directly quantify the fraction of atomic gas in our high redshift lensed galaxies, but it could be significant given the low stellar masses. Not only do we know locally that the HI fractions increase as stellar masses decrease (Catinella et al. 2010, 2012; Cortese et al. 2011; Huang et al. 2012), it can also be expected that the gas surface densities are lower in the z > 2, log M ∗ /M /circledot ∼ 10 lensed galaxies than in others at the high-mass end of the star-formation main sequence. Therefore, an additional interpretation for the low mean value of f H 2 at < z > = 2 . 8 is that a higher fraction of the cold gas mass is in neutral form than in the sample at < z > = 1 . 2 and 2.2. However, were we missing a significant cold gas component, we would have measured lower than expected gas-to-dust ratios, as any dust should be mixed with both the molecular and atomic phases of the ISM. Figure 8 shows the opposite behavior, suggesting that the lower gas fraction measured at z = 2 . 8 is not the result of neglecting the atomic component.</text>
<section_header_level_1><location><page_19><loc_11><loc_44><loc_46><loc_46></location>6. SUMMARY OF OBSERVATIONAL RESULTS AND CONCLUSIONS</section_header_level_1>
<text><location><page_19><loc_8><loc_40><loc_48><loc_43></location>The main observational results presented in this paper can be summarized as:</text>
<unordered_list>
<list_item><location><page_19><loc_10><loc_25><loc_48><loc_39></location>1. We measure a mean molecular gas depletion time of 450 Myr at < z > = 2 . 5. This decrease of t dep by a factor of 5 since z = 0 and 1.5 since z = 1 is consistent with the expected scaling of t dep = 1 . 5(1+ z ) α ( α = -1 . 0 , -1 . 5) if the depletion time is linked to the dynamical timescale. Our results validate up to z ∼ 3 and down to M ∗ ∼ 10 10 M /circledot the calibration of this relation established by Tacconi et al. (2013) out to z = 1 . 5 using data from the PHIBSS and COLD GASS surveys and a metallicity-dependent conversion factor.</list_item>
<list_item><location><page_19><loc_10><loc_12><loc_48><loc_24></location>2. The mean gas fraction measured at < z > = 2 . 8 is 40 ± 15% (44% after accounting for sample incompleteness), suggesting that the trend of increasing gas fraction with redshift ( < f gas > = [0 . 08 , 0 . 33 , 0 . 47] at z = [0 , 1 , 2]) does not extend beyond z ∼ 2. This observation is consistent with recent studies suggesting that the evolution of the mean sSFR of main-sequence galaxies slows down or even reaches a plateau beyond z = 2.</list_item>
<list_item><location><page_19><loc_10><loc_7><loc_48><loc_11></location>3. The lensed galaxies at z > 2 exhibit high dust temperatures, with values of ∼ 50K such as found typically only in galaxies with extreme IR luminosities.</list_item>
</unordered_list>
<text><location><page_19><loc_56><loc_87><loc_92><loc_92></location>The high values of T dust are a consequence of the fact that the lensed galaxies have low metallicities and short gas depletion times, as expected for their high redshifts and low stellar masses.</text>
<unordered_list>
<list_item><location><page_19><loc_54><loc_76><loc_92><loc_85></location>4. Using CO line luminosities, dust masses, and direct metallicity measurements, the conversion factor α CO for the lensed galaxies is estimated using the 'inverse Kennicutt-Schmidt' and the gasto-dust ratio methods, and in both cases agrees with a scaling of α CO ∝ Z -1 . 3 as parametrized by Genzel et al. (2012).</list_item>
<list_item><location><page_19><loc_54><loc_63><loc_92><loc_75></location>5. The gas-to-dust ratios in the z > 2 lensed galaxies exhibit the same metallicity dependence as observed in the local Universe, but with a systematic offset (Fig. 8). At fixed metallicity, we observe δ GDR to be larger by a factor of 1.7 at z > 2, suggesting that applying the local calibration of the δ GDR -metallicity relation to infer the molecular gas mass of high redshift galaxies may lead to systematic differences with CO-based estimates.</list_item>
</unordered_list>
<text><location><page_19><loc_52><loc_41><loc_92><loc_61></location>Most of these results could be explained by assuming that our lensed galaxies sample, mostly selected in SDSS imaging based on high luminosities at rest-UV wavelengths, is heavily biased toward dust-poor objects. This could serve to explain for example the high dust temperatures, the low dust and gas masses, and the short depletion times. However, we have conducted extensive tests (see § 2.1.1), and could not find evidence of any such bias. Instead, the sample appears to be representative of the overall population of main sequence galaxies at < z > = 2 . 5 with M /circledot ∼ 10 10 M /circledot . While it should be kept in mind that sample biases may be present, in the absence of evidence to the contrary we have proceeded with the analysis under the assumption that our results apply to the bulk of the high redshift galaxy population.</text>
<text><location><page_19><loc_52><loc_23><loc_92><loc_41></location>The combined Herschel and IRAM observations presented in this paper then suggest that main sequence galaxies with modest stellar masses (9 . 5 < log M ∗ /M /circledot < 10 . 5) at < z > = 2 . 5 have high star formation efficiencies and a molecular gas mass fraction no larger than measured at z = 1 -2, consistently with a simple model where the redshift evolution of the characteristic sSFR of mainsequence galaxies can be explained by a slowly varying gas depletion time and the measured gas fractions. The short depletion times and the possible redshift evolution of the gas-to-dust ratio imply that these high redshift galaxies have less dust at fixed SFR, producing the high T dust values we measure as each dust grain is exposed to more radiation (Fig. 7).</text>
<text><location><page_19><loc_52><loc_7><loc_92><loc_23></location>Before concluding, we wish to point out that the equilibrium model on which this analysis relies requires that the gas depletion timescale is shorter than the accretion timescale. This balance is reached at a redshift z eq . At z > z eq , the star formation process cannot keep up with the accretion of new gas, and this is thus the epoch where the gas reservoirs of the galaxies are filling up with gas fractions expected to be high. The exact value of z eq is however still debated. For example, Dav'e et al. (2012) found that z eq ∼ 2 in the absence of outflows, but that even modest outflows raised the threshold to z eq ∼ 7, the latter value being in agreement with the</text>
<text><location><page_20><loc_8><loc_85><loc_48><loc_92></location>analytical work of Bouch'e et al. (2010). On the other hand, Krumholz & Dekel (2012) suggest that z eq ∼ 2, and Papovich et al. (2011) estimate that z eq ∼ 4, using a sample of Lyman break galaxies and an indirect measurement of gas fractions.</text>
<text><location><page_20><loc_8><loc_57><loc_48><loc_85></location>In this study, we have used direct measurements of CO line fluxes and the power of gravitational lensing to push to higher redshifts the study of the redshift evolution of gas fractions. After correcting for sample incompleteness, Tacconi et al. (2013) measure a mean gas fraction of 47% at z ∼ 2 . 2. In this study, we directly measure a mean gas fraction of 40% at < z > = 2 . 8, which is then corrected upward to 44% after accounting for sample incompleteness. This observation suggests that f gas does not increase significantly between z = 2 and 3. A similar conclusion was reached by Magdis et al. (2012b) based on CO observations of two z ∼ 3 LBGs. Since gas fractions are expected to be high during the gas accretion phase, these results may indicate that we have not yet reached observationally the regime where galaxies are out of equilibrium, and therefore that z eq > 3. The improved determination of stellar mass and star formation rates in large samples of galaxies beyond z = 2, as well as the direct measurement of gas masses in normal starforming galaxies at z > 3 with ALMA and NOEMA will be essential to refine this picture.</text>
<text><location><page_20><loc_8><loc_49><loc_48><loc_55></location>We thank C. Feruglio for assistance with the reduction of the IRAM data and J. Richard for sharing the magnification factor for J1133 ahead of publication. We also thank T. Jones and C. Sharon for useful input, as well as the anonymous referee for constructive comments.</text>
<text><location><page_20><loc_8><loc_22><loc_48><loc_49></location>PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAFIFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA).</text>
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<section_header_level_1><location><page_23><loc_46><loc_24><loc_54><loc_25></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_23><loc_40><loc_22><loc_61><loc_23></location>SDSS J1137+4936 LENS MODEL</section_header_level_1>
<text><location><page_23><loc_8><loc_11><loc_92><loc_21></location>J1137 is a galaxy-scale gravitational lens system, where a luminous red galaxy (i.e. the lens galaxy) at z = 0 . 45 is forming a bright blue arc of the background source (i.e. the source galaxy). Follow-up spectroscopy of the bright blue arc by Kubo et al. (2009) confirms a primary source galaxy redshift of z = 1 . 41 and shows evidence of a secondary source galaxy at z = 1 . 38. As shown in Figure 2 of Kubo et al. (2009), SDSS imaging of the blue arc resembles two split knots and does not distinguish between the two background sources that are very close in redshift, forming nearly overlapping multiple images. However, multiwavelength HST -WFPC2 imaging of SDSS J1137 indicates that the bright blue arc is comprised of several images, as shown in Figure 12( left ). Figure 12( right ) demonstrates the subtraction of the lens galaxy light profile, showing the lensed features more clearly:</text>
<unordered_list>
<list_item><location><page_23><loc_11><loc_7><loc_92><loc_9></location>· features A through C , which resemble three small knots merging together to form a faint secondary arc, result from the lensing of the source galaxy at z = 1 . 38</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_24><loc_11><loc_91><loc_85><loc_92></location>· feature D is an extraneous, non-lensing object that is masked during the initial lens modeling process</list_item>
<list_item><location><page_24><loc_11><loc_88><loc_90><loc_89></location>· feature E corresponds to the primary lensed arc, resulting from the lensing of the source galaxy at z = 1 . 41.</list_item>
</unordered_list>
<text><location><page_24><loc_8><loc_86><loc_60><loc_87></location>Therefore, an accurate lens model must account for all images except D .</text>
<text><location><page_24><loc_8><loc_77><loc_92><loc_86></location>We use LENSFIT (Peng et al. 2006), an extension of the galaxy decomposition software GALFIT (Peng et al. 2002, 2010) for strong gravitational lens analysis, to derive the lens model. We refer the reader to Peng et al. (2006) and Bandara et al. (2013, ApJ submitted), for a detailed overview of LENSFIT and its application for the analysis of galaxy-scale gravitational lenses discovered in the Sloan Lens ACS (SLACS) Survey, and summarize our methodology below. To describe the mass distribution of the lens galaxy, we assume a singular isothermal ellipsoid mass model (SIE, Kormann et al. 1994) with an external shear field to model the tidal effects by nearby objects. The projected mass density of the SIE model is,</text>
<formula><location><page_24><loc_36><loc_71><loc_92><loc_76></location>κ ( x, y ) = b 2 [ 2 q 2 1 + q 2 ( x 2 + y 2 /q 2 ) ] -1 / 2 (A1)</formula>
<text><location><page_24><loc_8><loc_68><loc_92><loc_72></location>where b is the mass scale and q is the axis ratio of the mass model. The mass scale parameter ( b ) is approximately the Einstein radius of the lens (denoted as b SIE ); however, this relation is only exact at the q = 1 limit (Kochanek et al. 2001; Peng et al. 2006). At this limit, b SIE is related to the physical quantities of the mass model by,</text>
<formula><location><page_24><loc_41><loc_64><loc_92><loc_67></location>b SIE = 4 π σ SIE 2 c 2 D LS D S (A2)</formula>
<text><location><page_24><loc_8><loc_54><loc_92><loc_63></location>where σ SIE is the velocity dispersion of the mass model and D LS and D S are angular diameter distances from the lens to the source galaxy and from the observer to the source galaxy respectively (Kochanek et al. 2001). The SIE mass model of LENSFIT is characterized by the following parameters: mass model centroid ( x SIE , y SIE ), Einstein radius ( b SIE ), axis ratio of the mass model ( q SIE ), position angle of the major axis measured E from N ( PA SIE ), external shear ( γ SIE ) and the position angle of the external shear component measured E from N ( PA γ ). Furthermore, LENSFIT is a parametric lensing code that describes the unlensed source galaxy light profile through a set of parametric functions. To model the source galaxies of J1137, we use S'ersic profiles (Sersic 1968),</text>
<formula><location><page_24><loc_36><loc_51><loc_92><loc_53></location>Σ( r ) = Σ e exp( -κ ( n ) [( r/r hl ) 1 / n -1]) (A3)</formula>
<text><location><page_24><loc_8><loc_44><loc_92><loc_51></location>where Σ( r ) is the surface brightness at a given radius r , r hl is the half-light radius (i.e. also referred to as the effective radius, r e ), Σ e is the pixel surface brightness at the effective radius and n is the concentration parameter. The elliptically symmetric S'ersic profile is characterized by the following parameters: position of the S'ersic component ( x s , y s ), half-light radius ( r hl ), apparent magnitude ( m ), the S'ersic index ( n ), axis ratio of the elliptical profile ( q ) and the position angle of the major axis measured E from N ( PA SIE ).</text>
<text><location><page_24><loc_8><loc_29><loc_92><loc_44></location>We perform the lens modeling of J1137 in the HST -WFPC2 filters F814W, F606W and F450W. Since the gravitational lensing phenomenon is achromatic, the mass-model should be identical across multiple filters within the systematic uncertainties. Therefore, when modeling a gravitational lens in multiple filters, the mass-model parameters are typically fixed to those determined by the highest signal-to-noise filter (in this case, the F814W filter). However, we allow the SIE mass model parameters to vary freely for the F606W filter (which also has high signal-to-noise), such that we can test whether the lens modeling process is sufficiently robust to converge to the same mass model parameters for different filters. We find that the SIE mass model parameters from F814W and F606W imaging are virtually indistinguishable, thus confirming that the lens model solution is robust. Since F450W imaging has a lower signal-to-noise, we initially fix the SIE mass model parameters to those constrained by F814W and F606W imaging. During the final iteration of the F450W lens model, we allow the mass model to vary freely and find that the fractional difference between F450W parameters and those obtained from F814W/F606W imaging is less than ∼ 2%.</text>
<text><location><page_24><loc_8><loc_20><loc_92><loc_29></location>The morphology of the z = 1 . 41 unlensed source galaxy (which forms the lensed arc E on the image plane) is best described by three S'ersic components, in the F814W and F606W filters, and two S'ersic components in the F450W filter. In addition, the morphology of the z = 1 . 38 unlensed source galaxy (which forms features A through C on the image plane) is best described by a single S'ersic component in all three filters. Figure 13 shows the results of lens modeling of SDSS J1137 in the F814W filter. As indicated by the 'double residual' image of SDSS J1137, shown in the fourth panel of Figure 13, the complete lens model predicts images A through C and E but not D (which was unmasked during the final steps of the lensing analysis).</text>
<text><location><page_24><loc_8><loc_7><loc_92><loc_20></location>LENSFIT output parameters include the unlensed flux of each S'ersic component of the source galaxy light profile. Due to the circular feedback mechanism between the mass model and the source galaxy light profile, the use of multiple S'ersic components to describe the unlensed source galaxy yields the best-fit mass model. In other words, it is important to minimize the residuals of the source galaxy light profile to fully constrain the mass model, using multiple S'ersic components if necessary. However, from the standpoint of computing the overall intrinsic properties of the source galaxy (e.g. for comparison with non-lensed galaxy samples), we require a simplified representation that mimics the analysis techniques of high-redshift studies which typically use a single S'ersic component in galaxy fitting. Therefore, we compute the global properties of the unlensed z = 1 . 41 source galaxy (i.e. the flux of a single S'ersic component that best minimizes the overall residuals) by performing an additional LENSFIT iteration using single S'ersic components. The SIE mass model, which was fully constrained by the use of multiple S'ersic components to define the z = 1 . 41</text>
<text><location><page_25><loc_8><loc_89><loc_92><loc_92></location>source galaxy, and the z = 1 . 38 source galaxy light profile parameters are fixed to those implied by the best-fit lens model.</text>
<text><location><page_25><loc_8><loc_87><loc_92><loc_89></location>For this study, the quantity of interest is the magnification factor of the source galaxy at z = 1 . 41 resulting in lensed image E , defined as</text>
<formula><location><page_25><loc_32><loc_83><loc_92><loc_86></location>magnification factor = flux of the lensed image flux of the unlensed galaxy . (A4)</formula>
<text><location><page_25><loc_8><loc_67><loc_92><loc_82></location>We use the SExtractor photometry package (Bertin & Arnouts 1996) to deblend and measure the flux within the lensed image E in all three filters. We then combine the total flux of the lensed image, obtained from the SExtractor photometry, and the unlensed flux of the global source galaxy profile to derive the magnification factor, according to equation A4. The total magnification factor of SDSS J1137 is the mean value of F814W, F606W and F450W filters and corresponds to ∼ 17 × . To estimate the uncertainty on this measurement, we propagate the errors coming from the determination of both the unlensed and lensed galaxy fluxes, of which the magnification factor is the ratio. The former includes the error on the slope of the mass model (Marshall et al. 2007), the error caused by the lens galaxy subtraction, and the error associated with the PSF model, while the lensed galaxy flux uncertainty comes only from the SExtractor photometry. Including all these sources of uncertainty, we arrive at a total error of 18% for the magnification factor of J1137. This also justifies our choice of a blanket 20% uncertainty on the magnification factors taken from the literature when published without an uncertainty value (see Table 1).</text>
<figure>
<location><page_25><loc_9><loc_35><loc_49><loc_66></location>
</figure>
<figure>
<location><page_25><loc_50><loc_35><loc_91><loc_66></location>
<caption>Figure 12. HST -WFPC2 observations of SDSS J1137+4936. Left: HST -WFPC2 image of SDSS J1137+4936 taken in the F814W filter (7 . '' 5 × 7 . '' 5). Right: The lens galaxy subtracted F814W image (5 . '' 0 × 5 . '' 0) showing the lensed features. The lens galaxy light profile was modeled using multiple S'ersic (Sersic 1968) components. Images A , B and C correspond to a single source on the source plane (at z = 1 . 38). Image D corresponds to an extraneous feature that is commonly mistaken as a counter-image of the lensed arc. This feature was masked during the initial lens modeling. Arc E corresponds to the primary source galaxy at z = 1 . 41.</caption>
</figure>
<section_header_level_1><location><page_25><loc_33><loc_24><loc_68><loc_24></location>IRAC PHOTOMETRY OF THE LENSED GALAXIES</section_header_level_1>
<text><location><page_25><loc_8><loc_15><loc_92><loc_23></location>The Spitzer archive was queried for observations of the lensed galaxies in the 3.6 and 4.5 µ mbands, and the calibrated images retrieved. The only lensed galaxy in our sample without IRAC coverage is J1441. The 3.6 µ m images of the other 16 galaxies are shown in Figure 14. Most of the lensed galaxies appear as extended arcs in the IRAC images, and they are generally located within the wings of the foreground lenses, which are typically bright at these wavelengths. For these two main reasons, standard photometry packages are not optimal and instead a custom IDL script was written to measure accurate fluxes.</text>
<text><location><page_25><loc_8><loc_7><loc_92><loc_15></location>We define an annulus, typically centered on the foreground lens, that encompasses fully the arc. The section of this annulus containing the arc is used to integrate the flux of the object, and the remaining section is used to estimate the background, as indicated n Figure 14. The advantages of this approach are twofold: (1) the shape of the aperture naturally matches well that of the lensed arcs, and (2) the background is estimated over a region with equivalent noise properties and contamination from the bright lens. The exact parameters defining the aperture (inner and outer radii, position angle and opening angle) are fixed using curve of growth arguments. In Figure 14, the curves of</text>
<figure>
<location><page_26><loc_10><loc_77><loc_29><loc_92></location>
</figure>
<figure>
<location><page_26><loc_71><loc_77><loc_90><loc_92></location>
</figure>
<figure>
<location><page_26><loc_50><loc_77><loc_70><loc_92></location>
</figure>
<figure>
<location><page_26><loc_30><loc_77><loc_50><loc_92></location>
<caption>Figure 13. LENSFIT model of SDSS J1137+4936 in the F814W filter. From left to right: HST -WFPC2 image of SDSS J1137+4936; The lens galaxy subtracted image showing the lensed features; The complete lens model on the image plane including the light profile of the lens galaxy and the lensed features from the best-fit SIE mass-model; The 'double residual' image after subtracting the PSF convolved lens model from the F814W image. All images are 7 . '' 5 × 7 . '' 5 in size.</caption>
</figure>
<text><location><page_26><loc_8><loc_60><loc_92><loc_69></location>growth obtained by changing the outer radius of the apertures are shown. The background level within the aperture is estimated by measuring the mode of the sky pixels in the 'background aperture' and multiplying it by the area of the aperture. The mode is calculated with the MMM task in IDL (based of the DAOPHOT task of the same name), allowing an accurate measure of the sky properties even in the presence of bright point sources within the 'background aperture'. The error on the total flux of the lensed galaxy is obtained by adding in quadrature the uncertainty on the flux in the aperture from the IRAC error map, the scatter in the sky values, and the uncertainty in the mean sky brightness.</text>
<text><location><page_26><loc_8><loc_51><loc_92><loc_60></location>Some of the lensed galaxies are however best modeled by a simple circular aperture than with an arc aperture. This is the case of the Eye, the Eyelash, cB58, J0744, J1149 and J1226. In those cases, a simple circular aperture is used, with a radius determined from the curve of growth (see Fig. 14). The background is determined from an annulus around this aperture, using the same technique described above. For the lensed galaxy J0712, neither technique produces reliable results, since the faint blue arc is heavily blended with the brighter lens, as shown in Fig. 14. For that galaxy, we adopt the stellar mass of log M ∗ /M /circledot = 10 . 23 ± 0 . 48 derived by Richard et al. (2011) from SED fitting to the HST photometry.</text>
<text><location><page_26><loc_8><loc_48><loc_92><loc_51></location>The final, background subtracted fluxes at 3.6 and 4.5 µ m are summarized in Table 3, and our methodology to derive stellar masses from these fluxes is described in § 4.1.</text>
<section_header_level_1><location><page_26><loc_38><loc_46><loc_62><loc_47></location>NOTES ON INDIVIDUAL OBJECTS</section_header_level_1>
<section_header_level_1><location><page_26><loc_39><loc_44><loc_61><loc_45></location>AGN contamination in J0901?</section_header_level_1>
<text><location><page_26><loc_8><loc_23><loc_92><loc_43></location>The lensed galaxy J0901 has been reported by Hainline et al. (2009) to harbor an AGN, based on a high [NII]/H α ratio, above the range expected from star forming galaxies, suggesting excitation of the lines through AGN and/or shocks (e.g. Kewley et al. 2001; Levesque et al. 2010). Care must therefore be taken that the AGN emission does not affect our estimates of stellar mass and star formation rate. Since our stellar masses are derived from the IRAC photometry at 3.6 and 4.5 µ m, we compare in Figure 15 the shape of the observed near-infrared spectral energy distribution to templates from the SWIRE library (Polletta et al. 2007). The emission observed in the IRAC bands is fully consistent with the starburst templates, suggesting that the 3.6 and 4.5 µ m fluxes (and therefore the stellar mass derived from them) are not affected by the presence of an AGN. Fadely et al. (2010) also concluded from Spitzer /IRS spectroscopy that the contribution of the AGN to the mid-infrared emission is insignificant and that the system is instead starburst-driven. In the absence of an AGN signature at near- and mid-infrared wavelengths, it can be safely concluded that there is no contamination at even longer wavelengths and therefore that the SFR and dust mass/temperature derived from the Herschel photometry are secure (see Rosario et al. 2012, for a discussion of AGN contamination in the far-infrared). However, given the un-physically high [NII]/H α ratio for star-forming regions, we adopt for this galaxy the metallicity given by the MZ relation, 12 + log O / H = 8 . 91. This value is consistent with the lower limit of 1.3 Z /circledot reported by Fadely et al. (2010) based on Argon line fluxes from Spitzer mid-infrared spectroscopy.</text>
<section_header_level_1><location><page_26><loc_38><loc_21><loc_63><loc_22></location>Metallicity measurement for J1226</section_header_level_1>
<text><location><page_26><loc_8><loc_7><loc_92><loc_20></location>In the absence of a [NII]/H α ratio measurement, we computed the metallicity for J1226 using the R 23 and O 23 indicators, using H β , [O III ] and [O II ] line fluxes from the spectrum of Wuyts et al. (2012a), further corrected for extinction (J.R. Rigby, private communication). The values derived are log R 23 = 1 . 00 ± 0 . 13 and log O 32 = 0 . 09 ± 0 . 18. To compute a metallicity from these line ratios, we use the calibration of McGaugh (1991) as parametrized by Kuzio de Naray et al. (2004). The measured values of R 23 and O 32 put this galaxy at the transition between the two branches of the relation, with 12+ logO/H=8.46 and 8.45 for the lower and upper branches, respectively. We can therefore obtain a reliable metallicity estimate, even without [NII]/H α to discriminate between the two branches. Finally, we use the relation of Kewley & Ellison (2008) to convert this metallicity on the Denicol'o et al. (2002) scale, for consistency with the rest of the sample (Section 4.4). The final value we adopt for J1226 is therefore 12+ logO/H= 8 . 27 ± 0 . 19, where the uncertainty is the sum in quadrature of the propagated measurement errors, and the average</text>
<figure>
<location><page_27><loc_16><loc_20><loc_86><loc_92></location>
<caption>Figure 14. IRAC 3.6 µ m images, each 40 '' × 40 '' , for all lensed galaxies in the sample with the exception of J1441 where no such data exist. The apertures used to extract flux measurements are shown with solid blue lines, and the areas used for background estimation are delineated by dashed orange lines. In the case of J0712, no reliable flux measurement could be made due to the severe blending between the foreground lens (red cross) and the faint arc (blue line). For all other galaxies, the right panel shows the curve of growth, as was used to determine the maximum size of the aperture (vertical blue lines).</caption>
</figure>
<text><location><page_28><loc_8><loc_91><loc_44><loc_92></location>scatter of 0.15 in the McGaugh (1991) calibration.</text>
<figure>
<location><page_28><loc_28><loc_57><loc_70><loc_88></location>
<caption>Figure 15. Fluxes in the IRAC bands for J0901, compared with template SEDs from the library of Polletta et al. (2007) normalized to the observed wavelength of 4.5 µ m. Shown are spectra of both starbursts (yellow, orange and red lines) and AGN-dominated objects (blue and green lines), confirming the conclusion of Fadely et al. (2010) that the AGN contribution to the mid-infrared emission is negligible.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "We combine IRAM Plateau de Bure Interferometer and Herschel PACS and SPIRE measurements to study the dust and gas contents of high-redshift star forming galaxies. We present new observations for a sample of 17 lensed galaxies at z = 1 . 4 -3 . 1, which allow us to directly probe the cold ISM of normal star-forming galaxies with stellar masses of ∼ 10 10 M /circledot , a regime otherwise not (yet) accessible by individual detections in Herschel and molecular gas studies. The lensed galaxies are combined with reference samples of sub-millimeter and normal z ∼ 1 -2 star-forming galaxies with similar far-infrared photometry to study the gas and dust properties of galaxies in the SFRM ∗ -redshift parameter space. The mean gas depletion timescale of main sequence galaxies at z > 2 is measured to be only ∼ 450Myr, a factor of ∼ 1 . 5 ( ∼ 5) shorter than at z = 1 ( z = 0), in agreement with a (1 + z ) -1 scaling. The mean gas mass fraction at z = 2 . 8 is 40 ± 15% (44% after incompleteness correction), suggesting a flattening or even a reversal of the trend of increasing gas fractions with redshift recently observed up to z ∼ 2. The depletion timescale and gas fractions of the z > 2 normal star-forming galaxies can be explained under the 'equilibrium model' for galaxy evolution, in which the gas reservoir of galaxies is the primary driver of the redshift evolution of specific star formation rates. Due to their high star formation efficiencies and low metallicities, the z > 2 lensed galaxies have warm dust despite being located on the star formation main sequence. At fixed metallicity, they also have a gas-to-dust ratio 1.7 times larger than observed locally when using the same standard techniques, suggesting that applying the local calibration of the δ GDR -metallicity relation to infer the molecular gas mass of high redshift galaxies may lead to systematic differences with CO-based estimates. Subject headings: galaxies: evolution - galaxies: high-redshift - infrared: ISM - ISM: dust - ISM: molecules",
"pages": [
1
]
},
{
"title": "VALIDATION OF THE EQUILIBRIUM MODEL FOR GALAXY EVOLUTION TO Z ∼ 3 THROUGH MOLECULAR GAS AND DUST OBSERVATIONS OF LENSED STAR-FORMING GALAXIES 1",
"content": "Am'elie Saintonge 2 , Dieter Lutz 2 , Reinhard Genzel 2 , Benjamin Magnelli 3 , Raanan Nordon 4 , Linda J. Tacconi 2 , Andrew J. Baker 5 , Kaushala Bandara 6,2 , Stefano Berta 2 , Natascha M. Forster Schreiber 2 , Albrecht Poglitsch 2 , Eckhard Sturm 2 , Eva Wuyts 2 & Stijn Wuyts 2 Draft version September 28, 2018",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Most star-forming galaxies follow a relation between their stellar masses and star formation rates that has been well characterized up to z ∼ 2 . 5 (e.g. Noeske et al. 2007; Salim et al. 2007; Elbaz et al. 2011; Whitaker et al. 2012). The very existence and tightness of this relation suggests that these galaxies live in a state of equilibrium where their ability to form stars is regulated by the availability of gas and the amount of material they return to the circum-galactic medium through outflows (e.g. Genel et al. 2008; Bouch'e et al. 2010; Dav'e et al. 2011, 2012; Krumholz & Dekel 2012; Lilly et al. 2013). Simultaneously, it downplays the importance of galaxy mergers in the global star for- 1 Based on observations carried out with the IRAM Plateau de Bure Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). Based also on observations from Herschel , an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. 2 Max-Planck Institut fur extraterrestrische Physik, 85741 Garching, Germany 3 Argelander-Institut fur Astronomy, Universitat Bonn, 53121 Bonn, Germany mation budget of the Universe (Robaina et al. 2009; Rodighiero et al. 2011; Kaviraj et al. 2013) and highlights the influence of secular processes with longer duty cycles such as gas accretion, bar formation and bulge growth (Genzel et al. 2008). We refer to this general framework as the 'equilibrium model' for galaxy evolution, the formalism of which is detailed in recent work (e.g. Dav'e et al. 2012; Lilly et al. 2013). This paradigm was ushered in by a combination of large imaging surveys, detailed kinematics studies, molecular gas measurements, and theoretical efforts. For example, large optical and infrared surveys have contributed by allowing for the accurate measurement of stellar masses and star formation rates in very large galaxy samples, often supported by significant spectroscopic observing campaigns. Recently, Herschel observations in the wavelength range of 70-500 µ m with the PACS and SPIRE instruments have further contributed, by providing direct calorimetric SFRs that have been used to improve and recalibrate other indicators (e.g. Nordon et al. 2010, 2012; Elbaz et al. 2010; Wuyts et al. 2011a). At the same time, near-infrared integral field spectroscopy measurements at z ∼ 2 have started revealing that a significant fraction of high redshift galaxies are rotation-dominated discs (e.g. Forster Schreiber et al. 2006; Wright et al. 2007; Genzel et al. 2006, 2008; Forster Schreiber et al. 2009; Shapiro et al. 2008; Jones et al. 2010b; Gnerucci et al. 2011; Wisnioski et al. 2011; Epinat et al. 2012; Newman et al. 2013). These observations convincingly demonstrate that the high SFRs measured in these galaxies are generally not caused by major mergers, as was previously assumed by analogy with the local ULIRGs which have comparable SFRs and are all major mergers (Sanders & Mirabel 1996; Veilleux et al. 2002). Instead, the high SFRs of z ∼ 2 galaxies are caused by high molecular gas fractions, well above the 5 -10% typically observed at z = 0 (Tacconi et al. 2010; Daddi et al. 2010a; Tacconi et al. 2013). Some of the strongest direct evidence in favor of the equilibrium model above indeed comes from molecular gas observations. In the local Universe, with the exception of ULIRGSs, it has now been directly observed that the location of a galaxy in the SFRM ∗ plane is mostly determined by its supply of molecular gas, with variations in star formation efficiency playing a second order role (Saintonge et al. 2012). Similar conclusions have also been reached based on high-redshift galaxy samples, whether directly using CO data (Tacconi et al. 2013), or indirectly using far-infrared photometry to estimate gas masses (Magdis et al. 2012a). Also, outflows of molecular material, which are an important element in setting the balance between gas and star formation in the models, have now been directly observed in a range of objects (e.g. Sturm et al. 2011). But mostly, it is now possible to detect CO line emission in normal star-forming galaxies at z > 1 (e.g. Tacconi et al. 2010, 2013; Daddi et al. 2010a). These observations convincingly show that the rapid decline in the specific SFR of galaxies since z ∼ 2 can be explained by the measured gas fractions and a slowly varying depletion timescale ( t dep ∝ (1 + z ) -1 , Tacconi et al. 2013). Out to z ∼ 1 . 5 it is possible to detect far-infrared emission and CO lines in individual objects with current instrumentation. However, even in the deepest Herschel fields and with long integrations at the IRAM PdBI, it is still not possible to measure directly the dust and gas contents of individual normal star-forming galaxies with masses ∼ 10 10 M /circledot at z /greaterorsimilar 2. A proven way to study molecular gas in high redshift galaxies with lower masses and lower SFRs is instead to target objects that are gravitationally lensed (Baker et al. 2004; Coppin et al. 2007; Danielson et al. 2011). Samples of lensed starforming galaxies have for example also been used to extend to higher redshifts and lower stellar masses studies of the kinematics and stellar populations of disc galaxies, of the mass-metallicity relation (e.g. Jones et al. 2010b; Richard et al. 2011; Wuyts et al. 2012a,b), and of the origin of metallicity gradients in galaxy discs (Yuan et al. 2011; Jones et al. 2013). In this paper, we explore the relation between dust, gas and star formation at z = 2 -3 using a sample of 17 UV-bright lensed galaxies targeted for deep Herschel PACS and SPIRE observations and having SFRs and stellar masses characteristic of mainsequence objects at their redshifts. We analyze the results of these observations in the context of the equilibrium model. In addition, we present new IRAM PdBI observations that more than double the number of published lensed galaxies with CO line measurements. After we describe the sample in § 2, the multiwavelength observations are presented in § 3, including the new Herschel and IRAM PdBI observations. In § 4, we describe how key quantities such as stellar masses, star formation rates, dust masses, dust temperatures, gas masses and metallicities were calculated homogeneously. The key results of this study are presented in § 5 and summarized in § 6; in short, we find that the z > 2 lensed galaxies have low dust and gas masses, but high dust temperatures as a consequence of an efficient conversion of their gas into stars. Gas mass fractions and depletion times follow a redshift evolution out to z = 3 that is consistent with the expected scaling relations under the equilibrium model. All rest-frame and derived quantities in this work assume a Chabrier (2003) IMF, and a cosmology with H 0 = 70km s -1 Mpc -1 , Ω m = 0 . 3 and Ω Λ = 0 . 7. All molecular gas masses ( M H 2 ) and derived quantities such as gas fractions and depletion times, presented and plotted include a factor of 1.36 to account for the presence of helium.",
"pages": [
1,
2
]
},
{
"title": "2.1. Lensed galaxies sample",
"content": "The main sample consists of 17 lensed galaxies that were selected for deep Herschel PACS/SPIRE imaging (details of these observations are presented in § 3.1). These galaxies, selected from the literature, have been discovered as bright blue arcs of conspicuous morphology and then spectroscopically confirmed to be high redshift lensed objects. As shown in § 2.1.1, their intrinsic properties are similar to UV-selected Lyman Break Galaxies (LBGs), or their z ∼ 2 BX/BM analogs, but the observed fluxes are strongly amplified. The sources come both from the traditional method of searching for galaxies lensed by massive clusters, and from the recent searches for bright blue arcs in the Sloan Digital Sky Survey (SDSS), with lenses that typically are individual luminous red galaxies (e.g. Allam et al. 2007). In addition, we also consider the submm-identified Eyelash galaxy (Swinbank et al. 2010), as it is located within the field of view of our Herschelobservations of the Cosmic Eye. The sample is by no means a complete census of UVbright galaxies, but instead samples some of the bestknown objects with rich multi-wavelength observations, including near-infrared imaging and spectroscopy and millimeter continuum and line measurements. In Table 1 the basic properties of the sample are given (coordinates, redshifts and amplification factors). The galaxies have redshifts in the range 2 < z < 3, with the exception of four galaxies with z ∼ 1 . 5 (median redshift is 2.3), and most have stellar masses in the range 9 . 5 < log M ∗ /M /circledot < 11 . 0 (median, 1 . 6 × 10 10 M /circledot ), as will be presented in § 4.1.",
"pages": [
2
]
},
{
"title": "2.1.1. Is the lensed galaxies sample representative of the z > 2 star-forming population?",
"content": "As outlined in the introduction, the power of gravitational lensing is used here to extend the galaxy population where direct measurements of both molecular gas and dust masses can be performed. In particular, most galaxies in the sample have a stellar mass of ∼ 10 10 M /circledot and modest SFRs, as determined from IR and UV photometry, of ∼ 50 M /circledot yr -1 , given their median redshift a Assuming a 20% uncertainty on the magnification, see § A. of z = 2 . 4. We stress that the only way to directly detect the far-infrared emission of such galaxies across multiple Herschel bands is to target systems that are gravitationally lensed. Even in the deepest PACS and SPIRE blank fields, this mass/redshift/SFR regime is barely accessible through stacking of tens of galaxies (Reddy et al. 2012; Magnelli et al. 2013). Therefore, it is impossible to directly compare the FIR properties of our lensed galaxies to those of a well-matched, un-lensed reference sample. Such a comparison is however desirable. Since the lensed galaxies were selected visually based on their restframe UV light, there is a possibility that they represent a biased sub-sample of the high-redshift star-forming galaxy population. For example, we could expect the selection technique to preferentially isolate objects that are particularly UV-bright and therefore dust- and metalpoor. To answer this question, we create a control sample of un-lensed galaxies, matching on mass, redshift, and IR luminosity, where the latter is obtained either from PACS or MIPS 24 µ mphotometry when available, or else derived indirectly from the optical/UV photometry. We then compare the UV-to-IR ratio between the lensed and control samples. More exactly, for each lensed galaxy, we extract a control galaxy from the GOODS fields catalog (Wuyts et al. 2011b), within 0.2 dex in M ∗ , 0.2 dex in L IR and 0.2 in z . To check whether the lensed galaxies are less extincted than the control sample, the distributions of log(SFR IR / SFR UV ) are compared. As described in § 4.3, for the lensed galaxies, we calculate SFR IR from the 160 µ mfluxes and SFR UV from the B- or V-band photometry. For the GOODS sample, Wuyts et al. (2011a) used a ladder of star formation indicators to infer the respective contributions of obscured and un-obscured starformation, even in the absence of flux measurements at FIR wavelengths, therefore also allowing a measurement of log(SFR IR / SFR UV ). A KS test gives a probability > 90% that the distributions of log(SFR IR / SFR UV ) in the lensed and control samples are representative of the same parent population. As a second test, we study the UV slope β of the lensed galaxies. In Figure 1 the lensed galaxies are shown in the A IRX -β plane, where A IRX is the effective UV attenuation, The UV slope is measured from HST optical photometry using bands corresponding to rest-frame wavelengths redwards of Lyman α . The observed values are interpolated to derive the magnitudes at 1600 ˚ A and 2800 ˚ A (rest-frame) which are then used to compute β . As a comparison, we use again the GOODS fields sample for which Nordon et al. (2013) have derived values of the UV slope β . This reference sample is shown in Fig. 1 as gray circles. As these are PACS-detected galaxies only in the GOODS fields, the sample is biased towards large values of A IRX . To circumvent the necessity of detecting individual objects in several Herschel bands, Reddy et al. (2012) used stacks of UV-selected galaxies to probe galaxies with typically lower masses and SFRs. Their results are shown in Fig. 1 for two different bins of β . The lensed galaxies are located in the same region of the A IRX -β as the BX/BM galaxies of Reddy et al. (2012), and are also seen to follow the Meurer et al. (1999) extinction law (solid line). The only exception is the Eyelash, which is an outlier with high UV attenuation as expected since it is the only submillimeter-selected galaxy in the sample. Using the sample of Nordon et al. (2013) (both Herschel-detected and undetected galaxies) we can also define a control sample for each of our lensed galaxies by matching on M ∗ , SFR and redshift. In 10 of the 12 galaxies where we have sufficient data to conduct this experiment the median of the β distribution in the control sample agrees with the measured value of β for the lensed galaxy within its uncertainty. The two exceptions are J0900 and J1527 where measured values of β are -1.8 while the control samples suggest -0.6 and -1.2, respectively. We also note that while previous studies of the Cosmic Eye and cB58 suggested that these objects were better represented by an SMC extinction law than by the Meurer et al. (1999) relation (see e.g. Wuyts et al. 2012a), the new Herschel data revise the IR luminosities upward and find these galaxies in agreement with the Meurer/Calzetti relation. The lack of significant differences in either the effective UV attenuation ( A IRX ) or the UV slope ( β ) between the lensed galaxies of this study and their respective matched control samples suggests that they are not a biased subsample of the underlying population. We therefore proceed under the assumption that they are representative of the bulk of the star-forming galaxy population at their specific redshifts and modest masses.",
"pages": [
2,
3,
4
]
},
{
"title": "2.1.2. Does differential lensing introduce biases in the measurements?",
"content": "In Table 1, we give the magnification factors used to correct measured quantities for the lensing effect. The magnification factors are derived from optical imaging (typically from HST), and represent the average factor across the discs of these extended galaxies. However, in reality, different regions of the galaxies may be lensed differentially, with regions falling on or near caustics being more strongly magnified. Therefore, components of the galaxy having different spatial extents and compactness may not be magnified uniformly. This effect is known as differential lensing (e.g. Blandford & Narayan 1992). In practice, this could impact our analysis since both warm dust and high -J COtransitions are typically emitted in denser, more compact regions than cold dust or CO (1-0). Both the far-infrared SED and the CO spectral line energy distribution (SLED) might be biased, and standard calibrations to measure dust masses and temperatures, as well as excitation corrections to extrapolate total gas masses from a (3-2) line flux, may not apply (Downes et al. 1995; Serjeant 2012). For the specific case of galaxy lenses selected in large area, flux-limited submm/FIR surveys, there is a bias towards galaxies with very compact dusty star-forming regions, as these are most likely to benefit from the strongest magnifications (Hezaveh et al. 2012). With the exception of the Eyelash, our lenses are not submm/FIR-identified, but rather selected from cluster fields and SDSS imaging. In the discovery papers, SDSS galaxy lenses were typically identified visually as blue arcs. This includes the arc length, driven by the basic geometric configuration of lens and background object, as well as flux. They will therefore not suffer the specific biases discussed by Hezaveh et al. (2012), but the rest-UV emission used to derive the lensing model may still differ in extent or centroid from that at longer wavelengths. It is difficult to directly assess the impact of differential lensing, as the exercise requires detailed magnification maps, and high resolution imaging of the different components (optical/UV continuum, FIR continuum, CO line emission,...). The effect has therefore been so far quantified only based on simulations and very few, special galaxies. In particular, most of the literature on this concentrates on the special case of submm- or FIRselected galaxies such as the Eyelash (e.g. Hezaveh et al. 2012; Serjeant 2012; Fu et al. 2012; Wardlow et al. 2013). As can be seen in Figure 1, the Eyelash is a very special object, the only one in our sample to be submm-selected, and any conclusions drawn from such extreme dusty systems may not apply to all normal star-forming galaxies. We can however use these studies to get a sense of the amplitude of the issue in the most extreme cases. Using a submillimeter galaxy as a model, Serjeant (2012) estimate the impact of differential lensing on FIRselected galaxy samples. They find that even when comparing quantities measured from rest-frame optical/NIR wavelengths (e.g. stellar masses) and FIR observations (e.g. SFRs), the median differential magnification ratio is ∼ 0 . 8 with small dispersion. Therefore even in these cases, differential lensing does not affect the position of a galaxies in the SFR-mass plane significantly. Serjeant (2012) find a stronger effect of differential lensing on the CO SLED, but the effect manifests itself mostly for transitions with J upper > 4. Indeed, for the few galaxies in our sample where CO(1-0) measurements are also available, a typical (3-2)/(1-0) line ratio of ∼ 0 . 7 is retrieved (Riechers et al. 2010; Danielson et al. 2011), similar to what is measured in unlensed galaxies (e.g. Harris et al. 2010; Ivison et al. 2011; Bothwell et al. 2013). A common explanation for this ubiquitous value of the (32)/(1-0) line ratio is that the lines are emitted from the same moderately excited component of the ISM, but Harris et al. (2010) argue that it may instead reflect a generic feature of multi-component star-forming ISMs, in which the different lines are emitted by different but well mixed, optically thick and thermalized components with different but characteristic filling factors. Since our Herschel and IRAM observations are mostly unresolved, it is not possible to directly assess the impact of differential lensing on our measurements. Generally, this effect has not yet been extensively studied for normal star-forming high-redshift discs (as opposed to submm- or FIR-selected galaxies). However, based on the arguments above and the hypothesis that the FIR continuum and the CO line emission originate from similar physical regions (see however Fu et al. 2012), we can conclude that if occurring, differential lensing is un- likely to significantly affect the results of this study. It may be biasing our gas masses high if the CO SLEDs are affected, but by no more than 40%, comparing our adopted excitation correction R 13 = 2 . 0 and the value of 1.4 indicated by the observations of the Eye, Eyelash and cB58 (Riechers et al. 2010; Danielson et al. 2011, see also § 4.5). Similarly, differential lensing could be biasing the inferred dust properties. If the more compact, hotter regions of the galaxies are more strongly lensed than the diffuse component, the inferred dust temperatures could be too high (although Hezaveh et al. 2012, show that the effect can be reversed in systems with lower magnification factors of µ ∼ 10). This would result in under-estimated dust masses as well. However, the fact that we do not observe a strong effect on the CO SLED argues against a strong effect, but we cannot rule out the possibility of a small bias and include this caveat in the discussion in § 5.3.",
"pages": [
4,
5
]
},
{
"title": "2.2. Comparison samples",
"content": "We add to the z ∼ 2 lensed galaxy sample additional objects from the literature to serve as a comparison point. First, we use a compilation of 16 galaxies in the GOODS-North field for which CO line fluxes as well as deep Herschel photometry are available (Magnelli et al. 2012b). These galaxies are located mostly at 1 . 0 < z < 1 . 5 and sample well the SFR-stellar mass plane. We also use the compilation of submillimeter galaxies (SMGs) from Magnelli et al. (2012a), which expand the sample toward higher SFRs at fixed stellar mass and redshift. All the SMGs from the Magnelli et al. (2012a) compilation are used in the analysis when only dust measurements are involved, and the subset of these also found in Bothwell et al. (2013) whenever CO measurements are also needed. All the galaxies in the comparison sample have Herschel PACS and SPIRE photometry, which was processed identically as for the lensed galaxy sample in order to derive infrared luminosities, SFRs, dust temperatures and dust masses (see details in § 4.2). The lensed and comparison galaxy samples are shown in Figure 2 in the SFRM ∗ plane for two redshift intervals, 1 . 0 < z < 1 . 6 and 2 . 0 < z < 3 . 0, using the GOODS and EGS catalogs of Wuyts et al. (2011b) to provide a reference and define the main sequence (MS). These are the samples onto which our measurements of stellar mass and star formation rate are calibrated (sections 4.1 and 4.3), making them the best matched reference catalogs. The MS we derive from these is where a = [ -6 . 102 , -6 . 704 , -6 . 923] and b = [0 . 728 , 0 . 807 , 0 . 834] for the three redshift intervals of [1 . 0 -1 . 6], [2 . 0 -2 . 5] and [2 . 5 -3 . 0], respectively. When discussing the molecular gas properties of the lensed galaxies, we also use additional references. In particular, we make use of the data from the COLD GASS survey (Saintonge et al. 2011a), which includes CO(1-0) measurements for a representative sample of 365 SDSS-selected galaxies at 0 . 025 < z < 0 . 050 with M ∗ > 10 10 M /circledot . For high redshift galaxies, the PHIBSS sample (Tacconi et al. 2013) is by far the largest compilation of CO measurements for normal star-forming galax- s, and is the perfect reference to study e.g. the redshift evolution of the gas contents of galaxies (see section 5.3).",
"pages": [
5
]
},
{
"title": "3.1.1. PACS observations and data reduction",
"content": "We have obtained 70, 100 and 160 µ m 'miniscanmaps' of our targets using the PACS instrument (Poglitsch et al. 2010) on board the Herschel Space Observatory (Pilbratt et al. 2010). Total observing time per source was 1 hour at 70 and 100 µ m each, and 2 hours at 160 µ m wavelength, which is observed in parallel to both of the shorter wavelengths. The resulting PACS maps cover an area of ∼ 3 ' × 1 . 5 ' at more than half of the peak coverage, and have useful information over an area ∼ 6 ' × 2 . 5 ' . We processed the PACS data to maps using standard procedures similar to those described for the PEPproject in Lutz et al. (2011), in build 7.0.1786 of the Herschel HIPE software (Ott 2010). The Herschel blind pointing accuracy is ∼ 2 '' RMS (Pilbratt et al. 2010). To secure the astrometry, we therefore inspected the PACS maps to identify far-infrared sources clearly associated with counterparts having accurate astrometry (typically from SDSS). These reference positions were used to correct the astrometry of the PACS data to sub-arcsecond accuracy.",
"pages": [
5
]
},
{
"title": "3.1.2. SPIRE observations and data reduction",
"content": "We used the SPIRE instrument (Griffin et al. 2010) to simultaneously obtain 250, 350, and 500 µ m 'small maps' of our sources, using 14 repetitions and a total observing time of 35 minutes per source. Maps were produced with the standard reduction pipeline in HIPE (version 4.0.1349). Following the recommendation in the SPIRE Photometer Instrument Description, the maps are scaled with the appropriate flux correction factors of 1.02, 1.05, and 0.94 at 250 µ m, 350 µ m, and 500 µ m, respectively. The typical calibration accuracy of SPIRE maps is 15%. A preliminary source extraction at 250 µ m was performed, and the sources with counterparts in the PACS images (shifted as described above to the appropriate coordinate zero point) were used to correct the astrometry in all SPIRE bands.",
"pages": [
5
]
},
{
"title": "3.1.3. Far-infrared flux measurements",
"content": "We developed a measurement technique that combines aspects of blind source extraction, and guided extraction using prior information on the position of the sources. The procedure is aimed at measuring reliable fluxes for the lensed sources across the six Herschel bands, rather than produce complete catalogs in any given band. The first step is to perform a blind extraction with StarFinder (Diolaiti et al. 2000) on the 100 µ m, 160 µ m and 250 µ m maps. The resulting catalogs are used as prior information on the position of sources brighter than 3 σ , which is used to extract accurate fluxes. The main advantages of this PSF-fitting technique are that it accounts for most of the blending which could be an issue in these typically crowded fields at the longer wavelengths, and allows for photometry on any specific object that is reliable across the different bands (Magnelli et al. 2009, 2011; Lutz et al. 2011). For the PACS images, we use the merged blind 100 µ m and 160 µ m catalogs as a prior, while the 250 µ m blind catalog is used as the prior for the SPIRE bands. In cases where the lensed galaxy of interest is detected at less than 3 σ , the position of this galaxy is added in the prior catalogs based on its coordinates at shorter wavelengths. In most cases, the closest object with a 3 σ detection in the priors catalog, be it the lens or another object, is located 20-40 '' from the lensed galaxy and therefore easily separable at both 160 and 250 µ m. Only in the cases of cB58, the Cosmic Eye and J1133 is the closest neighboring object closer, at a distance of 10-15 '' . Even in these cases the objects are well separated in the PACS imaging, which we use as a guide for the SPIRE priors. The Herschel fluxes obtained from the PSF-fitting are finally aperture-corrected using factors of (0.883, 0.866, 0.811, 0.835, 0.848, 0.898), derived specifically for the PSFs used at (70, 100, 160, 250, 350, 500) µ m. Unless the lens is detected in the PACS/SPIRE images with > 3 σ , it is assumed that its FIR emission is negligible and does not contaminate the measurement for the lensed galaxy of interest. The final Herschel fluxes for the 17 lensed galaxies are given in Table 2. Values < 2 σ should be interpreted as upper limits, which occurs for 5 galaxies at both PACS 160 µ m and SPIRE 250 µ m.",
"pages": [
5,
6
]
},
{
"title": "3.2. IRAM-MAMBO photometry",
"content": "Photometric observations in the 1.2mm continuum were obtained for part of our sample during the pool observing sessions at the IRAM 30m telescope in the winters 2006/2007 and 2007/2008. We used the 117 element version of the Max Planck Millimeter Bolometer (MAMBO) array (Kreysa et al. 1998). On-off observa- tions were typically obtained in blocks of 6 scans of 20 subscans each, and repeated in later observing nights unless a detection was already reached. The data were reduced with standard procedures in the MOPSIC package developed by R. Zylka, using the default calibration files for the applicable pool periods. Table 2 lists the measured 1.2mm fluxes and their statistical uncertainties. We add the MAMBO detection which we already obtained for cB58 (Baker et al. 2001) and the SMA flux obtained by Swinbank et al. (2010) for the Eyelash. For J0901, we obtained a clear MAMBO detection at the target position centered on the southern bright lensed component but both the PACS maps and IRAM-PdB CO maps clearly indicate that the 11 '' MAMBO beam is missing flux. Based on the CO and PACS 100 µ m maps, the ratio between the total flux of J0901 and the flux in the southern component is independently measured to be 2.4 and 2.5, respectively. We hence scale the MAMBO flux from the southern component by a factor of 2.4 to infer the total flux before further use.",
"pages": [
6,
7
]
},
{
"title": "3.3. IRAM-PdBI CO mapping",
"content": "Molecular gas mass measurements through observations of the CO(3-2) line have been performed for 10 of the 17 galaxies in the Herschel sample, and we report here on both the previously published measurements and those coming from new PdBI observations. In sections 5 and 6, the full sample of 17 galaxies is used for the elements of the analysis that do not invoke molecular gas, and this subset of 10 galaxies for the rest of the analysis.",
"pages": [
7
]
},
{
"title": "3.3.1. Previous CO observations",
"content": "The CO line has been previously observed in seven galaxies from our Herschel sample. We briefly review the specifics of the observations of each object below, while a summary of the CO measurements is given in Table 6. Eye -The Cosmic Eye was observed in the CO(3-2) line with the IRAM PdBI (Coppin et al. 2007). The line width is 190 ± 24 km s -1 and the total line flux integrated over the line is 0 . 50 ± 0 . 07 Jy km s -1 . The CO emission appears to be spatially associated with component B 1 of the system, and Coppin et al. (2007) therefore suggest that the appropriate magnification correction factor for the CO line flux is 8 rather than the value of 30 found for the entire system (Dye et al. 2007). However, Riechers et al. (2010) report a detection of the CO(1-0) with the VLA that is spatially consistent with the bulk of the rest-frame UV emission, leading them to conclude that the total magnification value of the system should be used. This is the approach we adopt here. Eyelash -Multiple CO lines of both 12 CO and 13 CO were observed for this object (Danielson et al. 2011). Although a 12 CO(1-0) flux has been measured, for uniformity with the rest of the sample we adopt for the Eyelash the flux of 13 . 20 ± 0 . 10 Jy km s -1 measured in the (3-2) line. J0901 -Sharon (2013) reports on both EVLA CO(1-0) and PdBI CO(3-2) observations. For consistency with the rest of the dataset, we adopt the CO(3-2) flux of 19 . 8 ± 2 . 0 Jy km s -1 , which is obtained after primary beam correction given the large angular size of the source. cB58 -We adopt for this object the CO(3-2) flux of 0.37 ± 0.08 Jy km s -1 measured by Baker et al. (2004), which itself is consistent with an upper limit previously set by Frayer et al. (1997) and with the VLA CO(1-0) detection by Riechers et al. (2010) for the value of R 13 we adopt for the bulk of our sample (see § 4.5). 8:00arc -The 'eight o'clock arc' (thereafter the 8:00arc) was observed in 2007 May with the IRAM PdBI in compact configuration, using the 3mm SIS receivers to target the redshifted CO(3-2) line (Baker et al. in prep.). Following a standard reduction process, a 0.45 mJy continuum source associated with the lens was subtracted (see also Volino et al. 2010), and a final line flux of 0.85 ± 0.24 Jy km s -1 is measured. Clone, Horseshoe -Baker et al. (in prep.) report on IRAM PdBI compact configuration observations of these two sources, combining datasets taken in July to October 2007 and April 2009. Using standard reductions and integrating over the spatially extended CO emission, total line fluxes for the redshifted CO(3-2) line of 0 . 48 ± 0 . 15 (Clone) and 0 . 44 ± 0 . 18 (Horseshoe) are derived.",
"pages": [
7
]
},
{
"title": "3.3.2. New PdBI observations: J0900, J1137 and J1226",
"content": "In July-October 2011 we obtained CO(3-2) maps (rest frequency of 345.998 GHz) for three additional UV-bright lensed objects with the IRAM PdBI (Guilloteau et al. 1992). The three targets, J0900, J1137 and J1226, were chosen to extend the parameter space of objects with molecular gas measurements. Specifically, J0900 has the lowest metallicity of the sources with reliable Herschel flux measurements, and J1137 and J1226 are at the very low and very high ends of the redshift range of the sample, respectively. The lower redshift source, J1137, was observed in the 2mm band, while for the other two objects the CO(3-2) line was visible in the 3mm band. Observations were carried out under average to mediocre summer conditions with five of the six 15-m antennae in operation and in compact configuration. A total of 6-10 tracks of various duration per object were necessary to reach the required line sensitivities. Data were recorded with the dual polarization, large bandwidth WideX correlators, providing spectral resolution of 1.95 MHz over a total bandwidth of 3.6 GHz. The data were calibrated using the CLIC package and maps produced with MAPPING, within the IRAM GILDAS 7 software environment. A standard passband calibration scheme was first applied, followed by phase and amplitude calibration. Due to the poor observing conditions during some of the runs (e.g. high precipitable water vapour, strong winds, low elevation), particular care was taken to flag data with high phase noise. The absolute flux calibration was done using observation of reference bright quasars, and is typically accurate to better than 20% (Tacconi et al. 2010). The data cubes were examined for sources at the expected spatial and spectral positions. The CO(3-2) line is clearly detected in J1137, but not in J0900 and J1226. Assuming 200 km s -1 line widths, the 3 σ upper limit on the line flux is 0.16 Jy km s -1 for J1226 and 0.44 Jy km s -1 for J0900. The measured CO(3-2) integrated line flux of J1137 is 1 . 16 ± 0 . 12 Jy km s -1 , and the line FWHM is 137 km s -1 as determined by a gaussian fit. The integrated CO(3-2) line map and spectrum of J1137 are shown in Figure 3.",
"pages": [
7,
8
]
},
{
"title": "4.1. Stellar masses",
"content": "Stellar masses found in the literature can vary significantly for the same object depending on the measurement technique, and specific assumptions made regarding star formation histories, metallicities, and stellar population ages. For example there is an order of magnitude difference for cB58 between Siana et al. (2008) and Wuyts et al. (2012a), for the Eye between Richard et al. (2011) and Sommariva et al. (2012), and for the 8:00arc between Finkelstein et al. (2009) and Dessauges-Zavadsky et al. (2011) or Richard et al. (2011). Since homogeneity is paramount for the analysis we conduct here, we derive new stellar masses consistently for all the lensed galaxies in our sample. This is done with Spitzer /IRAC imaging, which is available for all the lensed galaxies with the exception of J1441. The calibrated images at 3.6 and 4.5 µ m were retrieved from the Spitzer archive, and fluxes measured using a custom-made pipeline. Since most of the lensed galaxies appear as resolved arcs in the IRAC images, and since they are also often situated in the wings of bright sources (generally, the lensing galaxy), standard photometric tools are not adequate. The details of the extraction technique are given in Appendix B, and the measured 3.6 and 4.5 µ m fluxes (un-corrected for lensing) are given in Table 3. To compute stellar masses from these IRAC fluxes that will be consistent with the masses of the comparison sample, we use the catalog of Wuyts et al. (2011b) for the GOODS fields as a calibration set. For each of our lensed galaxies, we extract from the calibration set all galaxies within ∆ z = ± 0 . 2 that are star forming based on their location in the SFRM ∗ plane. This subset of galaxies is used to determine the relation between observed 3.6 and 4.5 µ m fluxes and stellar mass. The scatter in these empirical relations varies from 0.13 dex for the lensed galaxies at z ∼ 1 . 5 to 0.19 dex at the highest redshifts. These uncertainties are comparable to the typical variations in stellar masses derived though SED modeling under varying assumptions (e.g. Forster Schreiber et al. 2004; Shapley et al. 2005; Maraston et al. 2010). Stellar masses for the lensed galaxies are then obtained by taking the observed IRAC fluxes (Table 3), correcting them for the lensing magnification, and then applying the empirical calibration determined for each object. The masses derived from the 3.6 and 4.5 µ m images are consistent within the errors, and we adopt as our stellar masses the mean between the two values. These values are summarized in Table 3 with the errors quoted obtained by propagating the uncertainties on the parameters of the fit to the calibration data set, the measurement errors on the IRAC fluxes, and the uncertainty on the magnification factor. The measurement and calibration uncertainties account for ∼ 10 -30% of the error budget, the rest coming from the uncertainty on the magnification. When available, we compare in Figure 4 previously published stellar masses (typically from SED-fitting) to our IRAC-derived masses. For cB58, the Cosmic Eye and the 8:00arc, two previously published values differing by ∼ one order of magnitude are shown. The IRAC-derived masses are always consistent with the higher of the two values. Adopting these higher estimates for these three galaxies, the scatter between the IRAC- and SED-derived stellar masses is 0.23 dex. This scatter and the outlier points are caused by the uncertainty on the calibration of our IRAC-based stellar masses, and by different assumptions about star formation histories in the SED modeling.",
"pages": [
8
]
},
{
"title": "4.2. Dust masses, dust temperatures and IR luminosities",
"content": "The far-infrared SEDs from the PACS, SPIRE and MAMBOphotometry are shown in Figure 5. They represent some of the highest fidelity individual Herschel SEDs of star-forming galaxies at z > 1 . 5. These SEDs are used to derive dust temperatures, dust masses, and total infrared luminosities. The procedure is identical to the one applied on the comparison samples. We summarize the key elements here, with the full details given in Magnelli et al. (2012b). Dust masses are calculated using the models of Draine & Li (2007) (DL07). A grid of models is created to sample the expected values of PAH abundances, radiation field intensities and dust fractions in the diffuse ISM. At each grid point, the model SED is compared with the Herschel photometry, with the dust mass given by the normalization of the SED minimizing the χ 2 . For each galaxy, the final dust mass assigned is the mean value over the grid points where χ 2 < χ 2 min + 1. To derive dust temperatures, a single modified black-body model with dust emissivity β = 1 . 5 is then fit to all the SED points with λ rest > 50 µ m. Dust temperatures and masses, along with their measured uncertainties, are given in Table 4. While infrared luminosities can be derived from the DL07 modeling at the same time as the dust masses, we adopt here the values that are derived from the 160 µ m fluxes (although for completeness we also give in Tab. 4 the values of L IR obtained by integrating in the wavelength range 8-1000 µ m the best fitting DL07 model SED). Nordon et al. (2012) have demonstrated that for our z ∼ 2 galaxies this method is robust, as the uncertainties on the 160 µ m-toL IR conversion factors provided by the Chary & Elbaz (2001) template library are /lessorsimilar 0 . 1 dex. Even in the cases where the lensed galaxies are not detected in the SPIRE bands, making the fitting of DL07 model templates difficult, there is always a /greaterorsimilar 3 σ detection at 160 µ m. For the galaxies with good detections in all PACS and SPIRE bands, there is an excellent agreement between the infrared luminosities derived by the DL07 modeling and by the extrapolation from 160 µ m flux (see also Elbaz et al. 2010).",
"pages": [
8
]
},
{
"title": "4.3. Star formation rates",
"content": "Dust-obscured SFRs can be obtained simply from the infrared luminosities as SFR IR = 10 -10 L IR , with L IR in units of solar luminosity and SFR IR in M /circledot yr -1 , assuming a Chabrier IMF. At high redshifts and at high total SFR (SFR tot ), SFR IR is the dominant contribution (e.g. Pannella et al. 2009; Reddy et al. 2010; Note . - Alternative published values of stellar masses (in log M /circledot units) are 10 . 02 ± 0 . 36 for the 8:00arc (Richard et al. 2011), 9 . 55 ± 0 . 14 for the Cosmic Eye (Sommariva et al. 2012), and 8 . 94 ± 0 . 15 for cB58 (Siana et al. 2008). Wuyts et al. 2011b; Whitaker et al. 2012; Nordon et al. 2013; Heinis et al. 2013) and it is commonly assumed that SFR tot ∼ SFR IR , neglecting the un-obscured component (SFR UV ). Since the lensed galaxies in the sample were mostly selected for being bright blue arcs in optical images (i.e. bright in rest frame UV), and since due to their large magnification factors their total intrinsic SFRs are modest, it is important in this case to estimate the contribution of SFR UV to SFR tot . Following Kennicutt (1998), we measure this for a Chabrier IMF as SFR UV = 8 . 2 × 10 -29 L ν, 1600 . The rest-frame 1600 ˚ A luminosity (in erg/s/Hz) is taken to be: where D L is the luminosity distance in cm, µ is the magnification factor from Table 1, and m 1600 the apparent AB-magnitude at a rest wavelength 1600 ˚ A. The most accurate approach to obtain m 1600 would be to interpolate between all available bands as was done for example by Nordon et al. (2013), but its main drawback for the lensed galaxy sample is in the lack of homogenous, multiwavelength photometry. While some galaxies have published photometry in several HST bands, some have been observed only with a single HST filter, while some others only have ground based photometry available. Given the nature of the available data and the fact that the SEDs of galaxies are typically relatively flat at these wavelengths, we directly adopt for m 1600 the observed magnitude in the available optical band the closest to rest-frame 1600 ˚ A. Given the typical UV slopes β in the range from -2.0 to -1.0 (see Fig. 1), the conversion factor from AB magnitude to SFR UV has only a very weak wavelength dependence, and therefore this approxima- Note . -SFR IR = L IR, 160 µm × 10 -10 , with L IR, 160 µm the infrared luminosity derived from the PACS 160 µ m flux. tion of adopting the closest band leads to systematic uncertainties of no more than 10%. We therefore adopt for the lensed galaxies a total SFR given by SFR UV + SFR IR . For the GOODSN comparison sample, we retrieve the ACS photometry from the multi-wavelength catalog of Berta et al. (2011) 8 and compute SFR UV from Eq. 3 using either B- or V-band magnitudes, depending on the redshift of each galaxy. For the SMG sample, we assume that SFR tot = SFR IR , based on their high IR luminosities and location in the SFRM ∗ plane, indicating a very high attenuation of the UV light and therefore a negligible contribution of SFR UV (Buat et al. 2005; Chapman et al. 2005; Wuyts et al. 2011a; Nordon et al. 2013; Casey et al. 2013).",
"pages": [
8,
9,
10
]
},
{
"title": "4.4. Metallicities",
"content": "In order to obtain a homogenous set of metallicities for all galaxies in our Herschel/IRAM sample, we have compiled from the literature [NII] and H α line fluxes (Table 5). Such fluxes were available for 11 of the 17 lensed galaxies. Using the observed [NII] /Hα ratio, we compute the nebular abundance using the calibration from Denicol'o et al. (2002). This is the calibration that produces the least amount of scatter in the mass-metallicity relation (Kewley & Ellison 2008) and that was used by Genzel et al. (2012) to derive a prescription for the metallicity-dependence of the CO-to-H 2 conversion factor. In the absence of a measured [NII] /Hα ratio, metallicities measured from the strong line indicator R 23 are adopted for the Eye, J1226 (see Appendix C.2) and J1441. These individual measurements are converted to the Denicol'o et al. (2002) scale using the appropriate relation given in Kewley & Ellison (2008). For the remaining three galaxies (Eyelash, J1133 and J1137), we use the mass-metallicity (MZ) relation as given in Genzel et al. (2012) to infer a metallicity. For the special case of J0901, we adopt the metallicity value derived from the MZ relation even in the presence of a measured [NII]/H α ratio as this value is likely affected by the presence of an AGN (details in § C.1). For galaxies in the comparison sample, the latter method is used as explained in Magnelli et al. (2012b). All the metallicities, as well as the H α luminosities and [NII]/H α line ratios when available, are summarized in Table 5.",
"pages": [
10
]
},
{
"title": "4.5. Molecular gas masses",
"content": "Starting from I CO (3 -2) , the integrated CO(3-2) line fluxes in Jy km s -1 presented in Table 6, we calculate the CO luminosity of the lensed galaxies, L ' CO , in units of (K km s -1 pc 2 ) following Solomon et al. (1997): where µ is the magnification factor as given in Table 1, D L is the luminosity distance in Mpc and ν obs the observed frequency in GHz. The factor R 13 ≡ I CO (1 -0) /I CO (3 -2) is the excitation correction to extrapolate the CO(1-0) line flux from our CO(3-2) observations. As in Tacconi et al. (2013), we adopt a value of R 13 = 2 based on recent studies of the CO spectral line energy distribution (e.g. Weiss et al. 2007; Dannerbauer et al. 2009; Harris et al. 2010; Ivison et al. 2011; Bauermeister et al. 2013; Bothwell et al. 2013). These studies have targeted both normal highz starforming galaxies and SMGs, with similar results pointing to a characteristic value of R 13 ∼ 2 that may be due to low excitation or to the typical filling factors of the two lines. We therefore apply this value of R 13 uniformly across our sample, and assume a conservative 20% uncertainty on this value compared to the typical errors presented in these individual studies. Total molecular gas masses are inferred from L ' CO using the CO-to-H 2 conversion factor, α CO ( M H 2 = α CO L ' CO ). The specific value of α CO to be applied for each galaxy must be determined with care, as most have sub-solar metallicities ((12 + log O / H) /circledot = 8 . 69 ± 0 . 05; Asplund et al. 2009). Under such low metallicity conditions, both observations and models suggest that the Note . - The metallicities in this table are either from the N2 indicator using the values of [NII]/H α from Column (3), the R 23 indicator, or derived from the mass-metallicity relation (MZ). In all cases, the metallicities are converted to the scale of the Denicol'o et al. (2002) calibration. value of α CO increases, as the CO molecule becomes a poor proxy for H 2 (e.g. Israel 1997; Dame et al. 2001; Rosolowsky et al. 2003; Blitz et al. 2007; Leroy et al. 2011; Glover & Mac Low 2011; Shetty et al. 2011; Feldmann et al. 2012; Genzel et al. 2012). Here, different methods to estimate α CO are investigated. The first is the 'inverse Kennicutt-Schmidt relation' method. Under the assumption that a tight relation exists between the surface density of molecular gas and the SFR surface density, the value of α CO can be estimated knowing Σ SFR and the CO luminosity. Using a compilation of local and high redshift galaxies, Genzel et al. (2012) calibrated a relation between α CO and metallicity using this approach. Since most recent studies suggest that the KS relation is near-linear (e.g. Leroy et al. 2008; Blanc et al. 2009; Genzel et al. 2010; Daddi et al. 2010b; Bigiel et al. 2011; Rahman et al. 2012; Saintonge et al. 2012; Shetty et al. 2013; Feldmann 2013), the problem is further simplified and α CO can be estimated as: where L ' CO is the CO(1-0) line luminosity in (K km s -1 pc 2 ), and α CO the CO-to-H 2 conversion factor in M /circledot (K km s -1 pc 2 ) -1 . In applying eq. 5, we adopt a redshift- endent depletion time, t dep (H 2 )= 1 . 5(1 + z ) -1 , as suggested by Tacconi et al. (2013) and further supported here to z = 2 -3 in § 5.3. In Figure 6, left panel, the value of α CO,KS is plotted against metallicity, for the lensed galaxies. The values of α CO scatter around the relation derived by Genzel et al. (2012) using this method and a sample of both local and high redshift galaxies, The second approach, the 'gas-to-dust ratio method', relies on a measurement of the dust mass and a motivated choice of a gas-to-dust ratio. This method has been used successfully in the local Universe (e.g. Israel 1997; Gratier et al. 2010; Leroy et al. 2011; Bolatto et al. 2011; Sandstrom et al. 2012), and shown to be applicable also at high redshifts (Magdis et al. 2011, 2012b; Magnelli et al. 2012b). In the case of high redshift galaxies, the conversion factor can be estimated simply as: where M dust is in solar masses, and we adopt a metallicity-dependent gas-to-dust ratio, δ GDR ( Z ), from Leroy et al. (2011). All the assumptions required to apply equation 7 to high redshift galaxies are extensively described in § 5.1 of Magnelli et al. (2012b). In particular, it needs to be assumed that the CO lines and the FIR continuum are emitted from the same physical regions of the galaxies given that Herschel does not resolve them, and that at high redshift M H 2 /greatermuch M HI and therefore that the atomic component of the ISM can be neglected in Eq. 7 (see § 5.3.1 for a justification of this assumption). A significant additional uncertainty lies in the assumption that the δ GDR ( Z ) relation of Leroy et al. (2011), calibrated on a handful of very nearby galaxies, applies directly at z ∼ 1 -3. Furthermore, other studies of nearby galaxies indicate that the scatter in the δ GDR -Z relation may be larger than suggested by this specific z = 0 sample, especially at low metallicities (Draine et al. 2007; Galametz et al. 2011). In Figure 6, right panel, α CO,dust measured from eq. 7 is shown as a function of metallicity. The z > 2 galaxies follow the inverse relation between α CO and metallicity seen in the z = 0 data and in the empirical relation of Genzel et al. (2012), although with a small systematic offset to lower values of α CO at fixed metallicity, the implications of which are discussed in § 5.2. Although affected by different sets of uncertainties and assumptions, the two methods of estimating α CO produce consistent results, which in the mean and within their errors reproduce the metallicity-dependence of α CO recently calibrated at z > 1 by Genzel et al. (2012), and previously also observed locally (e.g. Wilson 1995; Israel 1997; Boselli et al. 2002; Bolatto et al. 2011; Leroy et al. 2011). Throughout the rest of this paper, we therefore adopt as a consensus between α CO,KS and α CO,dust the value of α CO obtained from the prescription of Genzel et al. (2012), as given in eq. 6, using our best estimates of metallicities given in Table 5. These values of α CO , as well as the CO luminosities and derived molecular gas masses are presented in Table 6. Having derived accurate and homogeneous measures of M ∗ , SFR, T dust , M dust , M H 2 and 12+logO / Hacross the lensed galaxies and comparison samples, we now investigate the gas and dust properties of galaxies as a function of their position in the SFRM ∗ -z parameter space.",
"pages": [
11,
12,
13
]
},
{
"title": "5.1. High dust temperatures in z > 2 lensed galaxies",
"content": "In Figure 7, we plot the dust temperature against the offset from the star formation main sequence and the total infrared luminosity, both for the lensed and comparison galaxy samples. While we recover with the comparison sample the known trend between T dust and these two quantities (Dale et al. 2001; Chapman et al. 2003; Hwang et al. 2010; Magnelli et al. 2012b; Symeonidis et al. 2013), the z > 2 lensed galaxies occupy a significantly different region of the plot. Dust temperatures in these galaxies are very high ( T dust ∼ 50K), even though they are located on the main sequence and have modest infrared luminosities ( L IR ∼ 10 11 -10 12 L /circledot ) given their redshifts and stellar masses. Plotting dust temperatures against specific star formation rate rather than main sequence offset produces qualitatively equivalent results. There is evidence that the tightness of the T dust -L IR relation observed in classical samples of SMGs (e.g. Chapman et al. 2005, see also Fig. 7b) is due to selection biases (Casey et al. 2009; Magnelli et al. 2010, 2012a). In particular, the sub-millimeter selection technique favors colder objects, especially at low L IR . When also considering samples of dusty high-redshift star-forming galaxies selected through other techniques (e.g. the optically-faint radio galaxies of Chapman et al. 2004), dust temperatures can be significantly higher at fixed L IR . The lensed galaxies shown in Fig. 7b extend these results to galaxies with even lower IR luminosities and higher dust temperatures. To confirm the high T dust among the lensed galaxies, we also show in Figure 7c-d similar relations using the infrared color, defined as the ratio between rest-frame 60 µ m and 100 µ m fluxes ( S 60 /S 100 , where S 60 and S 100 are determined by linear interpolation of the Herschel photometry). This quantity, commonly used in IRASbased studies, is a proxy for T dust but is independent of any model assumptions. The same behavior is observed for the z > 2 lensed galaxies, with these objects having unusually high S 60 /S 100 ratios given their modest infrared luminosities. Among the z > 2 lensed galaxies, there are however two exceptions, J0901 and the Eyelash. With T dust ∼ 36K and S 60 /S 100 ∼ 0 . 5, these two galaxies follow the general trend between T dust and MS offset (or L IR ) traced by the various comparison samples (Magnelli et al. 2012a; Roseboom et al. 2012). These two galaxies also differ from the rest of the z > 2 lensed galaxy sample in other regards. For example, they have the largest dust masses ( M dust ∼ 10 9 M /circledot as compared to ∼ 1 -3 × 10 7 M /circledot ), and although no direct metallicity measurement is available for the Eyelash, J0901 has the highest metallicity of all the galaxies in the sample (but see Appendix C.1). The high dust temperatures in the rest of the z > 2 lensed galaxies therefore seem to be linked to their low dust contents compared to their SFRs. As we could not find evidence in 2.1.1 that the z > 2 lensed galaxies form a biased subsample of the underlying galaxy population, this observations suggests that ∼ 10 10 M /circledot galaxies at z = 2 -3 have higher dust temperatures and lower dust masses than similar galaxies at lower redshifts. Using deep Herschel PACS and SPIRE blind fields and a stacking technique, the mean T dust of normal star-forming galaxies with M ∗ ∼ 10 10 M /circledot at z > 2 is just barely measurable. Such studies indeed suggest a rise in temperature and intensity of the radiation field on the main sequence in the mass/redshift regime that we have now probed directly with the lensed galaxies (Reddy et al. 2012; Magdis et al. 2012a; Magnelli et al. 2013).",
"pages": [
13,
14
]
},
{
"title": "5.2. Gas-to-dust ratio",
"content": "In the local Universe, the typical gas-to-dust ratio, δ GDR , for star-forming galaxies with solar metallicities is of order 100 (Draine et al. 2007). This ratio has been shown to increase in low metallicity environments (e.g. Hunt et al. 2005; Engelbracht et al. 2008; Leroy et al. 2011), as predicted by dust formation models (e.g. Edmunds 2001). The value of δ GDR is also expected to vary in high density environments, such as the nuclear regions of starbursts, but observations suggest only a mild decrease of no more than a factor of 2 (Wilson et al. 2008; Clements et al. 2010; Santini et al. 2010). In the previous section, we inferred that the high dust temperatures of the z > 2 lensed galaxies were due to their low dust masses and metallicities. To understand whether the dust content of these galaxies is abnormally low for their other properties we plot in Figure 8 the measured value of δ GDR as a function of metallicity for the galaxies within 0.5 dex of the star formation main sequence. The gas-to-dust ratio is measured as δ GDR = M gas /M dust , where M gas is the molecular gas mass from Table 6 calculated from the CO luminosity using the metallicity-dependent conversion factor of Genzel et al. (2012), and M dust is derived from the Herschel photometry using the DL07 models (see § 4.2). Figure 8 reveals that the high redshift galaxies have a gas-to-dust ratio that scales inversely with metallicity like in the local Universe. However, fitting the δ GDR -Z relation of the z > 2 galaxies while keeping the slope fixed to the z = 0 reference relation reveals an increase in the mean δ GDR at fixed metallicity by a factor of 1.7. Since measuring CO in normal star-forming galaxies at high redshifts is challenging, there has been significant interest recently in using dust masses as a proxy for the total molecular gas contents (e.g. Magdis et al. 2011, 2012a; Scoville 2012). The method consists in measuring M dust using far-infrared and/or sub-mm photometry, and then applying the estimated gas-to-dust ratio ( δ GDR ( Z ), see § 4.5) to arrive at the gas mass. The systematic offset between the z > 2 normal star-forming galaxies and the reference relation of δ GDR ( Z ) therefore needs to be taken into consideration. There are at least three possible explanations: Are dust properties different at z > 2 ? -As detailed in § 4.2, dust masses for the lensed galaxies are computed using the model of DL07. In this model, dust is considered to be a combination of carbonaceous and amorphous silicate grains with a specific size distribution set to reproduce the Milky Way extinction curve. The model also assumes that a large fraction of the dust is found in the diffuse ISM, the rest in photodissociation regions, with the two components exposed to different radiation fields, the parameters of which are set to reproduce Milky Way conditions or left as free parameters. As argued by Bolatto et al. (2013), not only are dust grain properties poorly understood even in the local universe, the lack of a clear understanding of how dust is exactly produced and destroyed leads to significant uncertainties in applying the DL07 model to the denser, hotter interstellar medium of high redshift galaxies. Is α CO for the lensed galaxies smaller than expected? -If we instead assume that the dust masses are accurately measured in the lensed galaxies, the offset in δ GDR ( Z ) compared to z = 0 could be explained by a conversion factor α CO smaller on average by a factor of ∼ 1 . 7. The lensed galaxies are located on the main-sequence, which is interpreted as being the locus of galaxies where star formation takes place in virialized GMCs rather than in a centrallyconcentrated, dense starburst mode. Under such conditions, α CO should scale as ρ 0 . 5 /T (Tacconi et al. 2008; Bolatto et al. 2013). Assuming that gas and dust are thermalized, the large measured values of T dust may indicate a reduction of α CO , unless the gas density ρ also increases in proportion. The two z > 2 lensed galaxies with normal temperatures of ∼ 35K (J0901 and the Eyelash, see Fig. 7) also have gas-to-dust ratios that are elevated compared to the z = 0 relation, arguing that the ISM in the rest of the sample is not only warm but also denser such that α CO is not affected by the temperature variations. Is the gas-to-dust ratio really higher at z > 2 ? -As a last step, we can assume that none of these concerns apply and that both M gas and M dust are accurately measured in the lensed galaxies, leaving us with the conclusion that the gas-to-dust ratio is higher at fixed metallicity than in the local universe. This could happen for example if a smaller fraction of the metals are locked up in dust grains under the specific conditions prevailing in the ISM of the high redshift galaxies, or if the far-infrared emission used to compute M dust and the rest-frame optical line emission used to compute the metallicity are not emitted from the same physical regions. However, irrespectively of which one of these possible explanations is valid, we can conclude that when measuring dust and gas masses though standard techniques, an offset of 0.23 dex is obtained between the measured gas-to-dust ratio and the standard z = 0 prescription of Leroy et al. (2011). Care must therefore be taken in applying the gas-to-dust ratio method to estimate molecular gas masses of high redshift galaxies, as small but systematic differences with CO-based measurements may otherwise occur.",
"pages": [
14,
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},
{
"title": "5.3. Star formation efficiencies and gas fractions in z > 2 normal star-forming galaxies",
"content": "In § 5.1 above, we suggested that the high T dust of the lensed galaxies is caused by their low dust contents. To further illustrate this, we repeat the analy- is done in Figure 7, but this time normalizing L IR by the dust or the gas mass. We first show in Figure 9 how both T dust and the S 60 /S 100 ratio depend on the star formation efficiency (SFE ≡ SFR/M H 2 ). A correlation between these quantities is expected as T dust increases as a function of ∆ log( sSFR ) and L IR (Fig. 7), and there is also a positive correlation between SFE and ∆log( sSFR ) (Saintonge et al. 2012; Magdis et al. 2012a; Sargent et al. 2013). Figure 9 reveals that when the gas mass in each galaxy is taken into account, most of the offset between the z > 2 lensed galaxies and the comparison samples disappears, although the z > 2 normal star-forming galaxies still appear to have dust temperatures higher by ∼ 5 -10K at fixed SFE compared to the reference sample (gray symbols). The offset is strongest for those galaxies with the lowest metallicities. While metallicity may account for part of the remaining offset in T dust at fixed SFE, there could also be a contribution from differential lensing (see § 2.1.2), which if present at all, could manifest itself in an overestimation of the dust temperatures. In Figure 10 we then show a similar relation, but using the dust rather than the gas mass as the normalization factor, allowing us to include a larger number of galaxies for which dust masses are measured, but gas masses are not available. In this case again, some of the lensed galaxies appear to be systematically offset to higher T dust at fixed value of SFR /M dust . The z > 2 lensed galaxies having typically low metallicities, each dust grain will be exposed to more UV photons for a fixed SFR per unit gas mass, and therefore will have a hotter temperature. There is evidence for this in the fact that the galaxies with ∼ solar metallicities follow more closely the relation traced by the reference sample, in particular when using the S 60 /S 100 ratio (see Fig. 10). The combination of Figures 8-10 thus argues that in these galaxies, the balance between M dust , M gas and Z follows expected relations (modulo the possible change of δ GDR with redshift), and that the high dust temperatures are the result of the high SFRs per unit gas and dust mass, i.e. the high star formation efficiencies. This assertion that the z > 2 lensed galaxies have high star formation efficiencies (or put differently, short molecular gas depletion timescales, t dep (H 2 )) needs to be investigated in more detail. In the local Universe, normal starforming galaxies have t dep (H 2 ) ∼ 1 . 5 Gyr (Leroy et al. 2008; Bigiel et al. 2011; Saintonge et al. 2011b, 2012). In the PHIBSS survey, Tacconi et al. (2013) show that at z = 1 -1 . 5 the depletion time for main-sequence galaxies is reduced to ∼ 700 Myr (with 0.24 dex scatter in the log t dep (H 2 ) distribution, and assuming a Galactic conversion factor α CO ). Based on these observations, Tacconi et al. (2013) infer a redshift-dependence of the form t dep (H 2 )= 1 . 5(1 + z ) -1 . This redshift evolution is only slightly slower than the dependence of (1 + z ) -1 . 5 expected if t dep (H 2 ) is proportional to the dynamical timescale (Dav'e et al. 2012). The lensed and reference galaxies at z > 2 can be used to track the behavior of the depletion time at even higher redshifts. In Figure 11, left panel, the redshift evolution of t dep (H 2 ) is shown. The gray band shows the expected trend based on the empirical (1 + z ) -1 dependence (upper envelope), and the analytical expectation of a (1 + z ) -1 . 5 dependence (lower envelope). The mean values in four different redshift intervals are shown. For the highest redshift interval (2 < z < 3), we combine galaxies from the lensed and comparison samples as well as from PHIBSS. While the mean depletion time is ∼ 700 Myr at z ∼ 1 . 2, it decreases further to ∼ 450 Myr at z ∼ 2 . 2, consistently with the predicted redshift evolution. We caution however that the sample at z > 2 is by no means complete or representative of all normal star forming galaxies, but in terms of stellar masses and main sequence offset it is at least comparable to the z ∼ 1 -1 . 5 PHIBSS sample and the z = 0 reference sample from COLD GASS. This measurement at z > 2 suggests that the redshift-dependence of the molecular gas depletion time for main sequence galaxies, as predicted analytically (e.g. Dav'e et al. 2011, 2012) and recently observed up to z ∼ 1 . 5 (Tacconi et al. 2013), extends to z ∼ 3. Using the lensed galaxies and the various comparison samples, we can also trace the redshift evolution of the molecular gas mass fraction, In what follows, we restrict the samples only to galaxies found within 0.5 dex of the star formation main sequence at the appropriate redshift, i.e. we study the gas contents of normal, star-forming discs as a function of redshift. Several studies have now reported a rapid increase of f gas with redshift (Tacconi et al. 2010; Daddi et al. 2010a; Geach et al. 2011; Magdis et al. 2012a). The most robust analysis so far was performed by Tacconi et al. (2013). In that work, the PHIBSS data at z ∼ 1 and z ∼ 2 were corrected for incompleteness and compared to a matched local control sample extracted from the COLD GASS catalog, revealing an increase of f gas from 8% at z = 0 to 33% at z ∼ 1 and 47% at z ∼ 2. These three secure measurements are reproduced in Figure 11 (right panel). There are very few galaxies at z ∼ 0 . 5 with published CO measurements, but in Fig. 11 the few systems found in Geach et al. (2011) and Bauermeister et al. (2013) are compiled (CO measurements for several galaxies with 0 . 6 < z < 1 . 0 are published also in Combes et al. (2012), but we do not include them here as they are above-main sequence objects). As the PHIBSS sample extends to z = 2 . 4, we combine all the main-sequence galaxies in our lensed and comparison samples above that redshift to derive a mean gas fraction of 40 ± 15% at < z > = 2 . 8. We then apply the methodology of Tacconi et al. (2013) to correct for sample incompleteness. As the sample of lensed galaxies with z > 2 . 4 is richer in on- and below-main sequence galaxies, accounting for this bias rises the mean gas fraction to 44%. Our observations therefore suggest that the trend for increasing gas fraction with redshift does not extend beyond z ∼ 2, and may even be reversing. Can this flattening of the relation between gas fractions and redshift at z > 2 be expected under the equilibrium model? The definition of the gas fraction (eq. 9) can be re-expressed as: and the best predictions available for the redshift evolution of t dep and sSFR used to compute the expected behavior of f gas ( z ). As explained in § 5.3, it is estimated that with α measured to be -1.0 by Tacconi et al. (2013) and predicted to be -1.5 in the analytic model of Dav'e et al. (2012). The relation is normalized to the typical depletion time of 1.5 Gyr observed in local galaxies (Leroy et al. 2008; Bigiel et al. 2011; Saintonge et al. 2011b, 2012). Based on studies of the slope and redshift evolution of the star formation main sequence, the typical sSFR (in Gyr -1 ) of a star-forming galaxy of mass M ∗ at redshift z is The above equation is presentd by Lilly et al. (2013) based on results from a number of recent highredshift imaging surveys (Noeske et al. 2007; Elbaz et al. 2007; Daddi et al. 2007; Pannella et al. 2009; Stark et al. 2012). The expected redshift evolution of the gas fraction for galaxies of a given stellar mass can then be obtained by combining equations 9-11. For galaxies in the mass range 10 10 -5 × 10 11 and for α = -1 . 0 in eq. 10, the expected trend is shown in Figure 11 as the light gray band. At z > 2, f gas flattens out because of the shallower evolution of sSFR with redshift mostly canceling out the (1 + z ) -1 term from the t dep (H 2 ) relation (eq. 10). This model predicts a very modest evolution of the mean gas fraction from 47% at z = 2 . 2 to 49% at z = 2 . 8, consistently with our measurement. A different behavior for f gas ( z ) can be obtained by modifying the redshift-dependence of either t dep or sSFR (eq. 9). For example, if a value of α = -1 . 5 is used to set the evolution of t dep , the mean gas fraction expected at any redshift is lower than in the first model. However, since there is no observational evidence for a redshift evolution of t dep different from what has been assumed so far, we turn instead our attention to sSFR. Equation 11 assumes that at fixed stellar mass, the specific star formation rate increases as (1+ z ) 3 out to z = 2, and that this evolution then slows down to (1 + z ) 5 / 3 . If instead we took sSFR to keep increasing steadily with redshift, for example as sSFR ∝ M ∗ -0 . 3 (1 + z ) 2 . 8 (Tacconi et al. 2013), the predicted mean gas fraction at z = 2 . 8 would become 57%, in tension with our measured value of 40 ± 15%. Alternatively, it has been reported that the characteristic sSFR of main-sequence galaxies reaches a plateau at z ∼ 2 (Stark et al. 2009; Gonz'alez et al. 2010; Rodighiero et al. 2010; Weinmann et al. 2011; Reddy et al. 2012). If we assume that sSFR increases up to z = 2 according to eq. 11 and then remains constant at z > 2, the mean gas fraction is predicted to drop to 41% at z = 2 . 8. While this behavior is consistent with the observations presented here, the existence of such a plateau in sSFR at z ∼ 2 -7 has recently been challenged. After accounting for contamination by strong line emission, different authors suggest an increase of the mean sSFR by a factor of 2-5 between z = 2 and z = 7 (e.g. Bouwens et al. 2012; Stark et al. 2012; Gonzalez et al. 2012). This decrease in f gas at z > 2 should therefore be considered as a lower limit, with the relation based on eq. 11 providing the most realistic prediction of the redshift evolution of the gas fraction, as supported by our observations.",
"pages": [
15,
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},
{
"title": "5.3.1. Missing cold gas at z > 2 ?",
"content": "Another factor that could contribute to the redshift evolution of the gas fraction is the relative contribution of atomic hydrogen to the cold gas budget of the galaxies (M HI +M H 2 ) at the different epochs. For star forming galaxies with 10 . 5 < log M ∗ /M /circledot < 11 . 5 at z = 1 -2 such as those in the PHIBSS sample, it is generally assumed that M H 2 >> M HI , at least within the parts of the discs where star formation is actively taking place. This assumption is based on the observed high surface densities, above the characteristic threshold for the atomicto-molecular conversion. On the other hand, for nearby galaxies with log M ∗ /M /circledot > 10 . 0, it is observed that M HI ∼ 3 M H 2 , albeit with large galaxy-galaxy variations (Saintonge et al. 2011a). An important fraction of this atomic gas is located in the outer regions of galaxies, outside of the actively star-forming disks, but even within the central regions a significant fraction of the cold gas is in atomic form as the HI-to-H 2 transition typically occurs at a radius of ∼ 0 . 4 r 25 (Bigiel et al. 2008; Leroy et al. 2008). We can therefore estimate that on average within r 25 , M HI ∼ M H 2 , raising the cold gas mass fraction from 8% when only the molecular phase was considered to ∼ 15%. We cannot directly quantify the fraction of atomic gas in our high redshift lensed galaxies, but it could be significant given the low stellar masses. Not only do we know locally that the HI fractions increase as stellar masses decrease (Catinella et al. 2010, 2012; Cortese et al. 2011; Huang et al. 2012), it can also be expected that the gas surface densities are lower in the z > 2, log M ∗ /M /circledot ∼ 10 lensed galaxies than in others at the high-mass end of the star-formation main sequence. Therefore, an additional interpretation for the low mean value of f H 2 at < z > = 2 . 8 is that a higher fraction of the cold gas mass is in neutral form than in the sample at < z > = 1 . 2 and 2.2. However, were we missing a significant cold gas component, we would have measured lower than expected gas-to-dust ratios, as any dust should be mixed with both the molecular and atomic phases of the ISM. Figure 8 shows the opposite behavior, suggesting that the lower gas fraction measured at z = 2 . 8 is not the result of neglecting the atomic component.",
"pages": [
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},
{
"title": "6. SUMMARY OF OBSERVATIONAL RESULTS AND CONCLUSIONS",
"content": "The main observational results presented in this paper can be summarized as: The high values of T dust are a consequence of the fact that the lensed galaxies have low metallicities and short gas depletion times, as expected for their high redshifts and low stellar masses. Most of these results could be explained by assuming that our lensed galaxies sample, mostly selected in SDSS imaging based on high luminosities at rest-UV wavelengths, is heavily biased toward dust-poor objects. This could serve to explain for example the high dust temperatures, the low dust and gas masses, and the short depletion times. However, we have conducted extensive tests (see § 2.1.1), and could not find evidence of any such bias. Instead, the sample appears to be representative of the overall population of main sequence galaxies at < z > = 2 . 5 with M /circledot ∼ 10 10 M /circledot . While it should be kept in mind that sample biases may be present, in the absence of evidence to the contrary we have proceeded with the analysis under the assumption that our results apply to the bulk of the high redshift galaxy population. The combined Herschel and IRAM observations presented in this paper then suggest that main sequence galaxies with modest stellar masses (9 . 5 < log M ∗ /M /circledot < 10 . 5) at < z > = 2 . 5 have high star formation efficiencies and a molecular gas mass fraction no larger than measured at z = 1 -2, consistently with a simple model where the redshift evolution of the characteristic sSFR of mainsequence galaxies can be explained by a slowly varying gas depletion time and the measured gas fractions. The short depletion times and the possible redshift evolution of the gas-to-dust ratio imply that these high redshift galaxies have less dust at fixed SFR, producing the high T dust values we measure as each dust grain is exposed to more radiation (Fig. 7). Before concluding, we wish to point out that the equilibrium model on which this analysis relies requires that the gas depletion timescale is shorter than the accretion timescale. This balance is reached at a redshift z eq . At z > z eq , the star formation process cannot keep up with the accretion of new gas, and this is thus the epoch where the gas reservoirs of the galaxies are filling up with gas fractions expected to be high. The exact value of z eq is however still debated. For example, Dav'e et al. (2012) found that z eq ∼ 2 in the absence of outflows, but that even modest outflows raised the threshold to z eq ∼ 7, the latter value being in agreement with the analytical work of Bouch'e et al. (2010). On the other hand, Krumholz & Dekel (2012) suggest that z eq ∼ 2, and Papovich et al. (2011) estimate that z eq ∼ 4, using a sample of Lyman break galaxies and an indirect measurement of gas fractions. In this study, we have used direct measurements of CO line fluxes and the power of gravitational lensing to push to higher redshifts the study of the redshift evolution of gas fractions. After correcting for sample incompleteness, Tacconi et al. (2013) measure a mean gas fraction of 47% at z ∼ 2 . 2. In this study, we directly measure a mean gas fraction of 40% at < z > = 2 . 8, which is then corrected upward to 44% after accounting for sample incompleteness. This observation suggests that f gas does not increase significantly between z = 2 and 3. A similar conclusion was reached by Magdis et al. (2012b) based on CO observations of two z ∼ 3 LBGs. Since gas fractions are expected to be high during the gas accretion phase, these results may indicate that we have not yet reached observationally the regime where galaxies are out of equilibrium, and therefore that z eq > 3. The improved determination of stellar mass and star formation rates in large samples of galaxies beyond z = 2, as well as the direct measurement of gas masses in normal starforming galaxies at z > 3 with ALMA and NOEMA will be essential to refine this picture. We thank C. Feruglio for assistance with the reduction of the IRAM data and J. Richard for sharing the magnification factor for J1133 ahead of publication. We also thank T. Jones and C. Sharon for useful input, as well as the anonymous referee for constructive comments. PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAFIFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA).",
"pages": [
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},
{
"title": "REFERENCES",
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"pages": [
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},
{
"title": "SDSS J1137+4936 LENS MODEL",
"content": "J1137 is a galaxy-scale gravitational lens system, where a luminous red galaxy (i.e. the lens galaxy) at z = 0 . 45 is forming a bright blue arc of the background source (i.e. the source galaxy). Follow-up spectroscopy of the bright blue arc by Kubo et al. (2009) confirms a primary source galaxy redshift of z = 1 . 41 and shows evidence of a secondary source galaxy at z = 1 . 38. As shown in Figure 2 of Kubo et al. (2009), SDSS imaging of the blue arc resembles two split knots and does not distinguish between the two background sources that are very close in redshift, forming nearly overlapping multiple images. However, multiwavelength HST -WFPC2 imaging of SDSS J1137 indicates that the bright blue arc is comprised of several images, as shown in Figure 12( left ). Figure 12( right ) demonstrates the subtraction of the lens galaxy light profile, showing the lensed features more clearly: Therefore, an accurate lens model must account for all images except D . We use LENSFIT (Peng et al. 2006), an extension of the galaxy decomposition software GALFIT (Peng et al. 2002, 2010) for strong gravitational lens analysis, to derive the lens model. We refer the reader to Peng et al. (2006) and Bandara et al. (2013, ApJ submitted), for a detailed overview of LENSFIT and its application for the analysis of galaxy-scale gravitational lenses discovered in the Sloan Lens ACS (SLACS) Survey, and summarize our methodology below. To describe the mass distribution of the lens galaxy, we assume a singular isothermal ellipsoid mass model (SIE, Kormann et al. 1994) with an external shear field to model the tidal effects by nearby objects. The projected mass density of the SIE model is, where b is the mass scale and q is the axis ratio of the mass model. The mass scale parameter ( b ) is approximately the Einstein radius of the lens (denoted as b SIE ); however, this relation is only exact at the q = 1 limit (Kochanek et al. 2001; Peng et al. 2006). At this limit, b SIE is related to the physical quantities of the mass model by, where σ SIE is the velocity dispersion of the mass model and D LS and D S are angular diameter distances from the lens to the source galaxy and from the observer to the source galaxy respectively (Kochanek et al. 2001). The SIE mass model of LENSFIT is characterized by the following parameters: mass model centroid ( x SIE , y SIE ), Einstein radius ( b SIE ), axis ratio of the mass model ( q SIE ), position angle of the major axis measured E from N ( PA SIE ), external shear ( γ SIE ) and the position angle of the external shear component measured E from N ( PA γ ). Furthermore, LENSFIT is a parametric lensing code that describes the unlensed source galaxy light profile through a set of parametric functions. To model the source galaxies of J1137, we use S'ersic profiles (Sersic 1968), where Σ( r ) is the surface brightness at a given radius r , r hl is the half-light radius (i.e. also referred to as the effective radius, r e ), Σ e is the pixel surface brightness at the effective radius and n is the concentration parameter. The elliptically symmetric S'ersic profile is characterized by the following parameters: position of the S'ersic component ( x s , y s ), half-light radius ( r hl ), apparent magnitude ( m ), the S'ersic index ( n ), axis ratio of the elliptical profile ( q ) and the position angle of the major axis measured E from N ( PA SIE ). We perform the lens modeling of J1137 in the HST -WFPC2 filters F814W, F606W and F450W. Since the gravitational lensing phenomenon is achromatic, the mass-model should be identical across multiple filters within the systematic uncertainties. Therefore, when modeling a gravitational lens in multiple filters, the mass-model parameters are typically fixed to those determined by the highest signal-to-noise filter (in this case, the F814W filter). However, we allow the SIE mass model parameters to vary freely for the F606W filter (which also has high signal-to-noise), such that we can test whether the lens modeling process is sufficiently robust to converge to the same mass model parameters for different filters. We find that the SIE mass model parameters from F814W and F606W imaging are virtually indistinguishable, thus confirming that the lens model solution is robust. Since F450W imaging has a lower signal-to-noise, we initially fix the SIE mass model parameters to those constrained by F814W and F606W imaging. During the final iteration of the F450W lens model, we allow the mass model to vary freely and find that the fractional difference between F450W parameters and those obtained from F814W/F606W imaging is less than ∼ 2%. The morphology of the z = 1 . 41 unlensed source galaxy (which forms the lensed arc E on the image plane) is best described by three S'ersic components, in the F814W and F606W filters, and two S'ersic components in the F450W filter. In addition, the morphology of the z = 1 . 38 unlensed source galaxy (which forms features A through C on the image plane) is best described by a single S'ersic component in all three filters. Figure 13 shows the results of lens modeling of SDSS J1137 in the F814W filter. As indicated by the 'double residual' image of SDSS J1137, shown in the fourth panel of Figure 13, the complete lens model predicts images A through C and E but not D (which was unmasked during the final steps of the lensing analysis). LENSFIT output parameters include the unlensed flux of each S'ersic component of the source galaxy light profile. Due to the circular feedback mechanism between the mass model and the source galaxy light profile, the use of multiple S'ersic components to describe the unlensed source galaxy yields the best-fit mass model. In other words, it is important to minimize the residuals of the source galaxy light profile to fully constrain the mass model, using multiple S'ersic components if necessary. However, from the standpoint of computing the overall intrinsic properties of the source galaxy (e.g. for comparison with non-lensed galaxy samples), we require a simplified representation that mimics the analysis techniques of high-redshift studies which typically use a single S'ersic component in galaxy fitting. Therefore, we compute the global properties of the unlensed z = 1 . 41 source galaxy (i.e. the flux of a single S'ersic component that best minimizes the overall residuals) by performing an additional LENSFIT iteration using single S'ersic components. The SIE mass model, which was fully constrained by the use of multiple S'ersic components to define the z = 1 . 41 source galaxy, and the z = 1 . 38 source galaxy light profile parameters are fixed to those implied by the best-fit lens model. For this study, the quantity of interest is the magnification factor of the source galaxy at z = 1 . 41 resulting in lensed image E , defined as We use the SExtractor photometry package (Bertin & Arnouts 1996) to deblend and measure the flux within the lensed image E in all three filters. We then combine the total flux of the lensed image, obtained from the SExtractor photometry, and the unlensed flux of the global source galaxy profile to derive the magnification factor, according to equation A4. The total magnification factor of SDSS J1137 is the mean value of F814W, F606W and F450W filters and corresponds to ∼ 17 × . To estimate the uncertainty on this measurement, we propagate the errors coming from the determination of both the unlensed and lensed galaxy fluxes, of which the magnification factor is the ratio. The former includes the error on the slope of the mass model (Marshall et al. 2007), the error caused by the lens galaxy subtraction, and the error associated with the PSF model, while the lensed galaxy flux uncertainty comes only from the SExtractor photometry. Including all these sources of uncertainty, we arrive at a total error of 18% for the magnification factor of J1137. This also justifies our choice of a blanket 20% uncertainty on the magnification factors taken from the literature when published without an uncertainty value (see Table 1).",
"pages": [
23,
24,
25
]
},
{
"title": "IRAC PHOTOMETRY OF THE LENSED GALAXIES",
"content": "The Spitzer archive was queried for observations of the lensed galaxies in the 3.6 and 4.5 µ mbands, and the calibrated images retrieved. The only lensed galaxy in our sample without IRAC coverage is J1441. The 3.6 µ m images of the other 16 galaxies are shown in Figure 14. Most of the lensed galaxies appear as extended arcs in the IRAC images, and they are generally located within the wings of the foreground lenses, which are typically bright at these wavelengths. For these two main reasons, standard photometry packages are not optimal and instead a custom IDL script was written to measure accurate fluxes. We define an annulus, typically centered on the foreground lens, that encompasses fully the arc. The section of this annulus containing the arc is used to integrate the flux of the object, and the remaining section is used to estimate the background, as indicated n Figure 14. The advantages of this approach are twofold: (1) the shape of the aperture naturally matches well that of the lensed arcs, and (2) the background is estimated over a region with equivalent noise properties and contamination from the bright lens. The exact parameters defining the aperture (inner and outer radii, position angle and opening angle) are fixed using curve of growth arguments. In Figure 14, the curves of growth obtained by changing the outer radius of the apertures are shown. The background level within the aperture is estimated by measuring the mode of the sky pixels in the 'background aperture' and multiplying it by the area of the aperture. The mode is calculated with the MMM task in IDL (based of the DAOPHOT task of the same name), allowing an accurate measure of the sky properties even in the presence of bright point sources within the 'background aperture'. The error on the total flux of the lensed galaxy is obtained by adding in quadrature the uncertainty on the flux in the aperture from the IRAC error map, the scatter in the sky values, and the uncertainty in the mean sky brightness. Some of the lensed galaxies are however best modeled by a simple circular aperture than with an arc aperture. This is the case of the Eye, the Eyelash, cB58, J0744, J1149 and J1226. In those cases, a simple circular aperture is used, with a radius determined from the curve of growth (see Fig. 14). The background is determined from an annulus around this aperture, using the same technique described above. For the lensed galaxy J0712, neither technique produces reliable results, since the faint blue arc is heavily blended with the brighter lens, as shown in Fig. 14. For that galaxy, we adopt the stellar mass of log M ∗ /M /circledot = 10 . 23 ± 0 . 48 derived by Richard et al. (2011) from SED fitting to the HST photometry. The final, background subtracted fluxes at 3.6 and 4.5 µ m are summarized in Table 3, and our methodology to derive stellar masses from these fluxes is described in § 4.1.",
"pages": [
25,
26
]
},
{
"title": "AGN contamination in J0901?",
"content": "The lensed galaxy J0901 has been reported by Hainline et al. (2009) to harbor an AGN, based on a high [NII]/H α ratio, above the range expected from star forming galaxies, suggesting excitation of the lines through AGN and/or shocks (e.g. Kewley et al. 2001; Levesque et al. 2010). Care must therefore be taken that the AGN emission does not affect our estimates of stellar mass and star formation rate. Since our stellar masses are derived from the IRAC photometry at 3.6 and 4.5 µ m, we compare in Figure 15 the shape of the observed near-infrared spectral energy distribution to templates from the SWIRE library (Polletta et al. 2007). The emission observed in the IRAC bands is fully consistent with the starburst templates, suggesting that the 3.6 and 4.5 µ m fluxes (and therefore the stellar mass derived from them) are not affected by the presence of an AGN. Fadely et al. (2010) also concluded from Spitzer /IRS spectroscopy that the contribution of the AGN to the mid-infrared emission is insignificant and that the system is instead starburst-driven. In the absence of an AGN signature at near- and mid-infrared wavelengths, it can be safely concluded that there is no contamination at even longer wavelengths and therefore that the SFR and dust mass/temperature derived from the Herschel photometry are secure (see Rosario et al. 2012, for a discussion of AGN contamination in the far-infrared). However, given the un-physically high [NII]/H α ratio for star-forming regions, we adopt for this galaxy the metallicity given by the MZ relation, 12 + log O / H = 8 . 91. This value is consistent with the lower limit of 1.3 Z /circledot reported by Fadely et al. (2010) based on Argon line fluxes from Spitzer mid-infrared spectroscopy.",
"pages": [
26
]
},
{
"title": "Metallicity measurement for J1226",
"content": "In the absence of a [NII]/H α ratio measurement, we computed the metallicity for J1226 using the R 23 and O 23 indicators, using H β , [O III ] and [O II ] line fluxes from the spectrum of Wuyts et al. (2012a), further corrected for extinction (J.R. Rigby, private communication). The values derived are log R 23 = 1 . 00 ± 0 . 13 and log O 32 = 0 . 09 ± 0 . 18. To compute a metallicity from these line ratios, we use the calibration of McGaugh (1991) as parametrized by Kuzio de Naray et al. (2004). The measured values of R 23 and O 32 put this galaxy at the transition between the two branches of the relation, with 12+ logO/H=8.46 and 8.45 for the lower and upper branches, respectively. We can therefore obtain a reliable metallicity estimate, even without [NII]/H α to discriminate between the two branches. Finally, we use the relation of Kewley & Ellison (2008) to convert this metallicity on the Denicol'o et al. (2002) scale, for consistency with the rest of the sample (Section 4.4). The final value we adopt for J1226 is therefore 12+ logO/H= 8 . 27 ± 0 . 19, where the uncertainty is the sum in quadrature of the propagated measurement errors, and the average scatter of 0.15 in the McGaugh (1991) calibration.",
"pages": [
26,
28
]
}
]
2013ApJ...778...57G
https://arxiv.org/pdf/1308.3706.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_85><loc_88><loc_87></location>QUANTIFYING MASS SEGREGATION AND NEW CORE RADII FOR 54 MILKY WAY GLOBULAR CLUSTERS</section_header_level_1>
<text><location><page_1><loc_30><loc_81><loc_69><loc_84></location>Ryan Goldsbury 1 , Jeremy Heyl 1 , Harvey Richer 1 Draft version June 30, 2021</text>
<section_header_level_1><location><page_1><loc_45><loc_79><loc_55><loc_80></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_62><loc_86><loc_78></location>We present core radii for 54 Milky Way globular clusters determined by fitting King-Michie models to cumulative projected star count distributions. We find that fitting star counts rather than surface brightness profiles produces results that differ significantly due to the presence of mass segregation. The sample in each cluster is further broken down into various mass groups, each of which is fit independently, allowing us to determine how the concentration of each cluster varies with mass. The majority of the clusters in our sample show general agreement with the standard picture that more massive stars will be more centrally concentrated. We find that core radius vs. stellar mass can be fit with a two parameter power-law. The slope of this power-law is a value that describes the amount of mass segregation present in the cluster, and is measured independently of our distance from the cluster. This value correlates strongly with the core relaxation time and physical size of each cluster. Supplementary figures are also included showing the best fits and likelihood contours of fit parameters for all 54 clusters.</text>
<section_header_level_1><location><page_1><loc_22><loc_59><loc_35><loc_60></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_20><loc_48><loc_58></location>Comprehensive catalogues of Milky Way globular cluster parameters have been compiled by many different groups over the last two decades. The first large scale effort was put together by Trager et al. (1995), in which they presented surface brightness profiles for 125 Galactic globular clusters, and included parameters determined from model fitting for 63 of them. This work was expanded upon by McLaughlin & van der Marel (2005) who used data from the previous paper as well as new data, and fit multiple classes of models to the surface brightness profiles. The results from both of these papers, as well as many others, have been compiled in Harris (1996) (2010 edition). Our approach differs from these previous studies in that we do not assume a single mass-to-light ratio for the cluster. We instead work directly from star counts and consider bins 0.8 magnitudes wide to break each cluster down further, fitting the distribution of stars in each bin independently. Using stellar evolution models from Dotter et al. (2008) we can assign masses to each bin and analyze how the stellar concentration changes with mass. This approach avoids assuming a constant mass-to-light ratio for a cluster. Given the presence of any amount of mass segregation the assumption of constant mass-to-light ratio will not be true since mass-to-light ratio is a function of mass, and the average mass of stars changes as a function of distance from the cluster core. Because of this, this surface brightness profile does not necessarily reflect the underlying stellar density in a cluster.</text>
<section_header_level_1><location><page_1><loc_25><loc_18><loc_31><loc_19></location>2. DATA</section_header_level_1>
<text><location><page_1><loc_8><loc_12><loc_48><loc_17></location>All of the data used in this study are from the ACS Survey of Galactic Globular Clusters (Sarajedini et al. 2007). A thorough discussion of the reduction can be found in Anderson et al. (2008). Reduced cata-</text>
<text><location><page_1><loc_10><loc_7><loc_48><loc_10></location>1 Department of Physics & Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1; rgoldsb@phas.ubc.ca, heyl@phas.ubc.ca, richer@astro.ubc.ca</text>
<text><location><page_1><loc_52><loc_56><loc_92><loc_60></location>logues of both real and artificial stars can be found at: http://www.astro.ufl.edu/ ~ ata/public_hstgc/ databases.html .</text>
<section_header_level_1><location><page_1><loc_67><loc_54><loc_77><loc_55></location>3. METHODS</section_header_level_1>
<text><location><page_1><loc_52><loc_47><loc_92><loc_53></location>There is some ambiguity in the literature regarding a few parameters commonly used to characterize the concentration of star clusters. In an attempt to avoid this, we will use the following symbols and definitions for the remainder of the paper.</text>
<unordered_list>
<list_item><location><page_1><loc_54><loc_43><loc_67><loc_45></location>· core radius: R c</list_item>
<list_item><location><page_1><loc_54><loc_41><loc_67><loc_43></location>· King radius: r 0</list_item>
<list_item><location><page_1><loc_54><loc_38><loc_67><loc_40></location>· tidal radius: r t</list_item>
<list_item><location><page_1><loc_54><loc_36><loc_67><loc_38></location>· concentration: c</list_item>
</unordered_list>
<text><location><page_1><loc_53><loc_34><loc_88><loc_35></location>The latter three quanitites are defined as follows:</text>
<formula><location><page_1><loc_66><loc_30><loc_92><loc_33></location>Σ( R c ) = 1 2 Σ(0) (1)</formula>
<formula><location><page_1><loc_67><loc_25><loc_92><loc_28></location>r 0 = √ 9 σ 2 4 πGρ 0 (2)</formula>
<formula><location><page_1><loc_67><loc_21><loc_92><loc_24></location>c = log 10 ( r t r 0 ) (3)</formula>
<text><location><page_1><loc_52><loc_7><loc_92><loc_20></location>The quantity Σ here refers to the projected density distribution of the cluster. The core radius ( R c ) is the projected radial distance from the center of the cluster at which the projected density of the cluster drops to half of the central value. The quantity σ is the velocitydispersion parameter, and ρ 0 is the central density of the King model. The King radius ( r 0 ) can be calculated from the previous two quanities and describes the scale of the model similar to the core radius. The concentration ( c ) is defined following the convention in Binney & Tremaine</text>
<text><location><page_2><loc_8><loc_79><loc_48><loc_92></location>(1987) (hereafter BT87). This disagrees with the definition used by Harris (1996), in which c = log 10 ( r t /R c ). For a given r 0 and r t a King model has only one possible value of R c , but it is not equivalent to r 0 . It is also important to note here that we use the capital R to denote a projected radius, while the lower-case r refers to a three-dimensional radius. Equation 1 is intentionally ambiguous as it does not indicate whether the surface density is in units of luminosity, mass, or number of stars. This will be expanded upon in Section 3.2.</text>
<section_header_level_1><location><page_2><loc_10><loc_74><loc_46><loc_77></location>3.1. Solving for Projected Density Distributions of King-Michie Models</section_header_level_1>
<text><location><page_2><loc_8><loc_64><loc_48><loc_73></location>Our method involves generating a number of KingMichie (King 1966; Michie 1963) models over varying Ψ(0) /σ 2 , which are defined in Equations 4 and 5. Ψ(0) /σ 2 is referred to as the central dimensionless potential and is sometimes written Ψ 0 . We follow the prescription in Section 4.4 of BT87, ending with a series of models giving normalized surface density (Σ / ( ρ 0 r 0 )) as</text>
<text><location><page_2><loc_52><loc_90><loc_71><loc_92></location>a function of radius ( r/r 0 ).</text>
<text><location><page_2><loc_52><loc_77><loc_92><loc_90></location>To calculate these models, we begin with Equation 4132 from BT87, reproduced in Equation 4. We have two boundary conditions. The first is always the same: d Ψ /dr = 0 when r = 0. The second is the value of Ψ(0), which determines the concentration of the resulting density model for a given σ . Since we are only concerned with Ψ(0) /σ 2 , which will control the shape of the density distribution, and not Ψ(0) or σ independently, which determine the absolute scale of the system, we only consider Ψ(0) /σ 2 as a single combined parameter.</text>
<text><location><page_2><loc_52><loc_65><loc_92><loc_77></location>We use a fourth order Runge-Kutta method (RK4) to solve this equation numerically. The step size varies from 0 . 01 r 0 for the most concentrated to 10 -4 r 0 for the most extended models in our grid. The truncation error is less than 0 . 1% for all models calculated. A solution to this equation for a given set of boundary conditions gives Ψ( r ). This can then be fed into Equation 4-131 from BT87 (shown as Equation 5 in this paper) to go from ρ (Ψ) to ρ ( r ).</text>
<text><location><page_2><loc_53><loc_64><loc_57><loc_65></location>break</text>
<formula><location><page_2><loc_26><loc_57><loc_92><loc_62></location>d dr ( r 2 d Ψ dr ) = -4 πGρ 1 r 2 [ e Ψ /σ 2 erf ( √ Ψ σ ) -√ 4Ψ πσ 2 ( 1 + 2Ψ 3 σ 2 ) ] (4)</formula>
<formula><location><page_2><loc_32><loc_53><loc_92><loc_57></location>ρ (Ψ) = ρ 1 [ e Ψ /σ 2 erf ( √ Ψ σ ) -√ 4Ψ πσ 2 ( 1 + 2Ψ 3 σ 2 ) ] (5)</formula>
<text><location><page_2><loc_8><loc_45><loc_48><loc_49></location>Finally, to transform from a three dimensional density distribution to a projected density distribution, one must perform an Abel Transformation:</text>
<formula><location><page_2><loc_19><loc_40><loc_48><loc_44></location>Σ( R ) = ∫ ∞ R ρ ( r ) rdr √ r 2 -R 2 . (6)</formula>
<figure>
<location><page_2><loc_8><loc_18><loc_48><loc_39></location>
<caption>Figure 1. We have recalculated Figure 4-9b from BT87, showing the projected density distribution for various boundary conditions. These distributions correspond to (from left to right) Ψ(0) /σ 2 = 12 , 9 , 6 , 3 or c = 2 . 74 , 2 . 12 , 1 . 26 , 0 . 67 since there is a one-to-one relation between Ψ(0) /σ 2 and c .</caption>
</figure>
<text><location><page_2><loc_8><loc_7><loc_48><loc_11></location>A range of the resulting projected density distributions is shown in Figure 1. The brief summary in this section is not intended to be comprehensive, only to allow the</text>
<text><location><page_2><loc_52><loc_44><loc_92><loc_49></location>reader to replicate our procedure for calculating these models. A more thorough description, including motivation for the distribution function from which these models are derived, can be found in King (1966) or BT87.</text>
<section_header_level_1><location><page_2><loc_60><loc_42><loc_83><loc_43></location>3.2. Luminosity vs. Star Counts</section_header_level_1>
<text><location><page_2><loc_52><loc_11><loc_92><loc_41></location>In the interest of presenting a straightforward comparison to previously determined values of R c we first fit the distribution of all stars in a given cluster field together. While our grid is parametrized by r 0 , it is easy to determine the value of R c for a given model by finding the radius at which the surface density drops to half. This value is more commonly referenced in the literature. Our fitting method will be discussed in Section 3.4. A comparison between the values currently in the literature and our resulting best fit values and their uncertainties is shown in the left hand panel of Figure 2. Our uncertainties are estimated with an iterative fitting procedure that considers uncertainty in the location of the cluster center as well as uncertainty due to sample size through bootstrapping. We find that by fitting to the cumulative star count distribution and assuming a conservative 20% uncertainty in Harris' values the mean difference between our fit values and Harris' values is inconsistent with zero at a bit more than 4 σ , where the error used is the standard error. We tend to measure statistically significant larger core radii. This is not surprising, given that R c values reported in Harris (1996) are determined from fitting the luminosity profiles of clusters.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_11></location>If we repeat our fitting procedure, but now weight each star's contribution to the cumulative distribution by its luminosity, we then produce the result shown in the cen-</text>
<figure>
<location><page_3><loc_9><loc_73><loc_38><loc_92></location>
</figure>
<figure>
<location><page_3><loc_66><loc_73><loc_91><loc_92></location>
</figure>
<figure>
<location><page_3><loc_38><loc_73><loc_65><loc_92></location>
<caption>Figure 2. The left hand panel shows core radius values taken from Harris (1996) plotted against values determined using our approach of fitting the cumulative star count distribution. The center panel shows values determined by fitting the cumulative distribution in which objects are weighted by their luminosities. The right panel shows fits to the mass weighted cumulative distributions. The red line highlights a non-biased relation in each case. The horizontal error bars represent 1 σ and the Harris values have no error plotted.</caption>
</figure>
<text><location><page_3><loc_8><loc_60><loc_48><loc_64></location>ter panel of Figure 2. Here the bias described above is removed and the mean difference between our values and those of Harris is within 1 . 2 σ of zero.</text>
<text><location><page_3><loc_8><loc_43><loc_48><loc_60></location>The explanation for this discrepancy is as follows. The mass-to-light ratio is a function of radius in most globular clusters due to mass segregation; more massive (and therefore more luminous) stars will be more centrally concentrated than less massive (less luminous) stars. This implies that the mass-to-light ratio decreases toward the center of the star cluster and so converting between a distribution measured from surface brightness to the underlying distribution of stars is not a constant factor. In fact, how this mass-to-light ratio changes with radius also differs between clusters. So, even if we consider mass-to-light ratio as a function of R , there is no universal correction to be applied.</text>
<text><location><page_3><loc_8><loc_21><loc_48><loc_43></location>If we wish to determine the mass distribution of the cluster, then fitting the star count distribution still has an inherent bias for the same reason discussed above. We make a rough approximation of the core radius of the true underlying mass distribution by fitting cumulative distributions in which stars are weighted by mass. The masses are assigned by fitting isochrone models from Dotter et al. (2008) to the cluster CMDs. The results are shown in the right panel of Figure 2. These values fall somewhere in between the previous two, but the key point is that they are inconsistent with those determined from fitting the surface brightness profile. This indicates that the surface brightness profile is not an appropriate proxy for the mass surface density of a cluster. For the remainder of the paper we will attempt to quantify these effects by analyzing how subgroups of varying mass are distributed in each cluster.</text>
<section_header_level_1><location><page_3><loc_11><loc_18><loc_45><loc_19></location>3.3. Selecting Groups Along the Main-Sequence</section_header_level_1>
<text><location><page_3><loc_8><loc_7><loc_48><loc_17></location>For each cluster, we create ten independent groups along the main sequence. We begin by fitting a fiducial sequence to each cluster color-magnitude diagram (CMD) in F 606 W vs. ( F 606 W -F 814 W ). The fiducial is created using an iterative algorithm. A manual starting point is selected at the base of the CMD. The algorithm then chooses a direction to move by using the AMOEBA algorithm from Press et al. (2007) to mini-</text>
<text><location><page_3><loc_52><loc_33><loc_92><loc_64></location>mize the median of the absolute separation of the nearby points perpendicular to each proposed step. There is also an additional cost added based on the angle between sequential steps to prevent the sequence from bending back on itself. The result is a fiducial that traces out the center of the sequence. Once this fiducial is established for each cluster, the turn-off is then defined as the magnitude that produces the most negative derivative ( dcol/dmag ) of the fiducial. Ten magnitude bins are created starting from 0.5 magnitudes below this turn-off and covering eight magnitudes in total. The widths of the bins are determined by calculating the standard deviation of the CMD in the color direction from the fiducial. Only objects that fall within ± 3 σ are included. While this approach does require an initial point to be selected manually on the CMD, it was found that various starting points would converge to the same sequence within approximately 5 iterations. These initial burn-in points were removed and the fiducial was linearly interpolated backward from the later points. This effectively removes all human input, leading to a bin selection method that is entirely automated and consistent across all of the clusters used in this study. Figure 3 illustrates the location of these bins on a CMD.</text>
<figure>
<location><page_3><loc_52><loc_11><loc_71><loc_31></location>
</figure>
<figure>
<location><page_3><loc_72><loc_11><loc_92><loc_31></location>
<caption>Figure 3. The algorithmically determined fiducial sequence and the regions for the ten selected bins are shown for the cluster 47 Tucanae (NGC 104).</caption>
</figure>
<section_header_level_1><location><page_4><loc_17><loc_91><loc_40><loc_92></location>3.4. Fitting Models to Subgroups</section_header_level_1>
<text><location><page_4><loc_8><loc_74><loc_48><loc_90></location>For each cluster, we fit our grid of King-Michie models to the cumulative radial distribution of each of our 10 sub-groups. We fit only out to projected radii of 100 arc-seconds, due to the size limitations of a single ACS field. Out to 100 arc-seconds from the center ( R max in Equation 7), each annulus is fully within the field. Beyond this, more and more of each successive annulus will fall off of the field. It is possible to take this effect into account when fitting the models, but including these regions does not significantly improve our ability to constrain the best fitting model parameters, and so they are ignored in the interest of simplicity.</text>
<text><location><page_4><loc_8><loc_61><loc_48><loc_74></location>To generate a model to fit to our data we need to specify two of the three parameters: r 0 , r t , and c , since the third can always be determined from the other two. The tidal radius of the cluster in all cases will be many times the maximum radius that can be observed with our field, and we expect this to be a constant for all subgroups within the cluster. Therefore, instead of fitting for two parameters, we use the values of c and R c from Harris (1996) to calculate r t for each cluster and hold this value fixed.</text>
<text><location><page_4><loc_8><loc_49><loc_48><loc_61></location>Before fitting the models to the data, we also need to take into account the completeness of the stars in each bin. To do this, we use the artificial star tests mentioned in Section 2 to calculate the completeness as a function of radius for each sub-group in each cluster. The incompleteness ( I ) is the probability that a star of a given magnitude would be found at a given position in the field. These corrections are then applied to the model density distributions before fitting to the data.</text>
<text><location><page_4><loc_8><loc_45><loc_48><loc_49></location>Given the projected density distribution Σ( R ) and data incompleteness I ( R ), we calculate the cumulative distribution as:</text>
<formula><location><page_4><loc_16><loc_37><loc_48><loc_44></location>C ( R ) = ∫ R 0 Σ( R ' ) I ( R ' ) R ' dR ' ∫ R max 0 Σ( R ' ) I ( R ' ) R ' dR ' . (7)</formula>
<text><location><page_4><loc_8><loc_31><loc_48><loc_39></location>The angular component of this integral is not shown since our models are spherically symmetric and therefore azimuthally symmetric in projection, and the result simplifies to the equation shown above. Here I is only dependant on R since we are considering a group of objects that fall within a narrow magnitude range.</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_31></location>In determining the best fit to the cumulative distribution of each bin, there are two major sources of uncertainty. The first is the location of the center of the cluster. We use the center coordinates and uncertainties from Goldsbury et al. (2010), which were determined using this same data set. The second source of uncertainty is the sample size in each bin. Both of these are taken into account by using a bootstrap fitting method as follows. In each iteration, a sample of the same size as the full bin sample is chosen with replacement. The location of the center to use is drawn from a two dimensional Gaussian distribution centered at the value from Goldsbury et al. (2010) with the appropriate σ . The cumulative radial distribution is calculated and the best fitting model is determined by minimizing the maximum separation between the real distribution and the model, which is the Kolmogorov-Smirnov statistic D . In each iteration, we record a single value for the best fitting r 0 . After many it-</text>
<text><location><page_4><loc_52><loc_77><loc_92><loc_92></location>rations we can then generate a histogram of fit r 0 values. This distribution is roughly Gaussian and is broadened by both sources of error discussed above. We take the mean and standard deviation of these values as our best fitting r 0 and 1 σ uncertainty for each sub-group in each cluster. The dominant source of error in most cases is the √ N random uncertainty. However, in very well populated bins, such as the low mass groups in a very massive cluster like NGC 5139, the systematic uncertainty from the location of the cluster center contributes as much to the total error as the Poisson counting uncertainty.</text>
<section_header_level_1><location><page_4><loc_52><loc_75><loc_91><loc_76></location>3.5. Assigning Masses to Bins and Fitting Power Laws</section_header_level_1>
<text><location><page_4><loc_52><loc_58><loc_92><loc_74></location>After iterating the fitting procedure described in Section 3.4 for all sub-groups in all clusters, we can show how the concentration of stars along the main-sequence changes with magnitude for each cluster. However, this is not particularly useful for making comparisons among clusters, since the apparent magnitude of an object is a result of a number of factors: distance, extinction, mass, metallicity, and cluster age. In terms of dynamics, we would like to look at how the concentration changes with object mass. So, to assign a mass to each magnitude bin, we need to consider the effects of the other four parameters.</text>
<text><location><page_4><loc_52><loc_37><loc_92><loc_58></location>To do this, we again use an iterative approach. In each iteration we draw values of the four parameters listed above from Gaussian distributions. The parameters of these distributions are drawn from a number of sources. All of the distances, metallicities, and extinctions are taken from Harris (1996). The vast majority of these have no reported uncertainties in the original references, and so we assume conservative estimates: for distance we assume a standard deviation of 0.2 magnitudes, for extinction and metallicity we assume 20% uncertainties. The ages and uncertainties are taken from Dotter et al. (2010) with the exception of five clusters that were not listed in that paper. These are: NGC 1851 and NGC 2808 (Koleva et al. 2008), NGC 5139 (Forbes & Bridges 2010), NGC 6388 (Catelan et al. 2006), and NGC 6715 (Geisler et al. 2007).</text>
<text><location><page_4><loc_52><loc_10><loc_92><loc_37></location>These parameters are then used to define a stellar isochrone model from Dotter et al. (2008). For each magnitude bin, we assign masses to all of the objects using the model and then take the mean mass of all objects in each bin. So, for each iteration, we record one mass value for each bin. We then use these mass values along with our best fitting r 0 value and uncertainties to fit a two parameter power-law model to the relation between King radius and mass using a maximum likelihood approach. The isochrone model grid does not agree well with the structure of most CMDs at the very low mass end, so we ignore all bins for which the mean mass is less than 0.2 M /circledot . We also ignore points for which the best fitting r 0 falls outside of our field. In these cases, r 0 is not well constrained by our data. For each iteration, we save the likelihood results over the two power-law parameters shown in Equation 8. This power-law model is not physically but rather empirically motivated, as it fits a wide range of clusters well with a small number of parameters:</text>
<formula><location><page_4><loc_66><loc_5><loc_92><loc_9></location>r 0 = A ( M M /circledot ) B . (8)</formula>
<text><location><page_5><loc_8><loc_80><loc_48><loc_92></location>After many iterations as described above, we can average this stack of two dimensional likelihood surfaces. This amounts to a Monte Carlo integration over the likelihood of the four other parameters (distance, extinction, metallicity, and cluster age) with prior distributions on each parameter. This leaves the likelihood of only the two power-law parameters, but with the uncertainties in the other four propagated through. An example fit for 47 Tuc is shown in Figure 4.</text>
<figure>
<location><page_5><loc_8><loc_55><loc_27><loc_79></location>
</figure>
<figure>
<location><page_5><loc_31><loc_55><loc_48><loc_78></location>
<caption>Figure 4. The best fitting power-law model and likelihood of fit parameters for 47 Tuc. Similar figures can be found in the supplement showing power-laws fit to both r 0 and R c for all 54 clusters in our sample.</caption>
</figure>
<section_header_level_1><location><page_5><loc_24><loc_47><loc_32><loc_48></location>4. RESULTS</section_header_level_1>
<text><location><page_5><loc_8><loc_9><loc_48><loc_47></location>The parameter A is a normalization that corresponds to the King radius of cluster members at 1 solar mass. This value is given in projected coordinates which are related to the true three dimensional King radius by the distance to the cluster. The parameter B describes how much the concentration of stars changes with their mass, and is independent of the total scale of the cluster. Although we choose to fit our models for r 0 , we could just as easily parametrize them by R c and fit power-laws to this parameter in the same way as Equation 4. For all 54 of the clusters included in this analysis, these two powerlaw parameters as well as their uncertainties and the χ 2 values for the best fit (the value given is the total χ 2 , not per degree of freedom) are included in Table 2 for both r 0 and R c as a function of mass. For any of these clusters, this empirical relation can then be used to estimate the distribution of stars of a given mass. The values of A in this table are in units of arc-seconds, and the parameters correspond to a power-law function in the form of Equation 8, but in the right side the parameters are for fits to R c ( M ) rather than r 0 ( M ). Although r 0 is determined from the three dimensional density distribution, our values come from fitting the projected density distribution and so we fit in units of angle rather than distance. Since we are fitting in projected space, we leave our results in these projected units. These results can be converted to physical units by using the distance given in Harris (1996) or another source.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_9></location>Parameter A has units of angle and our measurement of it depends on our distance from the cluster. Parameter</text>
<text><location><page_5><loc_52><loc_75><loc_92><loc_92></location>B , on the other hand, is unitless and would be measured the same regardless of our distance from the cluster. This parameter is a quantitative descriptor of the mass segregation present in a cluster. We can use the apparent distance modulus and extinction from Harris (1996) to convert our fit values for A in arc-seconds (or pixels) to parsecs, which gives us a parameter to describe the intrinsic size of the cluster. After doing this, we find that A and B correlate strongly with each other such that larger clusters tend to have less mass segregation than small clusters. Both of these quantities also correlate with the core relaxation time from Harris (1996). Figure 5 shows these correlations.</text>
<text><location><page_5><loc_52><loc_59><loc_92><loc_75></location>The distribution inset in each diagram indicates the Pearson correlation coefficient ( R ) distribution determined through bootstrapping. A principal component analysis of this three-dimensional space indicates only a single significant component that links all three dimensions. This can be understood as follows: large clusters have longer crossing times and therefore longer relaxation times. The amount of mass segregation present in the cluster should approach some limit near equipartition as many relaxation times pass. Therefore, these larger and slower to relax clusters will exhibit less mass segregation than those that are small and quick to relax.</text>
<text><location><page_5><loc_53><loc_57><loc_92><loc_59></location>Lines of the following forms are fit to each correlation.</text>
<formula><location><page_5><loc_66><loc_53><loc_92><loc_55></location>log 10 ( A ) = b 1 + S 1 ∗ B (9)</formula>
<formula><location><page_5><loc_61><loc_50><loc_92><loc_52></location>log 10 ( t c ) = b 3 + S 3 ∗ log 10 ( A ) (11)</formula>
<formula><location><page_5><loc_66><loc_51><loc_92><loc_54></location>log 10 ( t c ) = b 2 + S 2 ∗ B (10)</formula>
<text><location><page_5><loc_52><loc_11><loc_92><loc_50></location>These fits are shown as blue dashed lines in Figure 5, and the values of the best fit parameters as well as their variances and covariances are listed in Table 1. As with the calculation of the correlation coefficients, core collapsed clusters are excluded from these fits. Classification of core collapsed clusters is taken from Harris (1996). These are derived from luminosity profiles and so this classification potentially suffers from the biases introduced by mass segregation that are discussed above. Of the eight clusters that are classified as core collapsed, we find that the projected density distribution measured from star counts is well modeled by just a powerlaw for six of them. Two of the clusters (NGC 6541 and NGC 6723) show clear evidence of leveling off toward the center and do not appear to have collapsed cores when the density is measured through star counts rather than luminosity. Similar discrepancies have been found by Miocchi et al. (2013), where clusters are found to show a central cusp in their surface brightness profile but no such feature in their stellar density profile. The authors attribute this to the presence of a few anomalous bright stars, but we believe this discrepancy could also be caused by the systematic effects of mass segregation. Despite the fact that clusters which appear to be core collapsed in surface brightness may not actually be so, we have still excluded all of these clusters from the correlation analysis shown in Figure 5 since this classification still carries a potential bias through the calculation of the relaxation time.</text>
<section_header_level_1><location><page_5><loc_66><loc_7><loc_78><loc_8></location>5. CONCLUSIONS</section_header_level_1>
<figure>
<location><page_6><loc_8><loc_73><loc_35><loc_92></location>
</figure>
<figure>
<location><page_6><loc_65><loc_73><loc_92><loc_92></location>
</figure>
<figure>
<location><page_6><loc_37><loc_73><loc_63><loc_92></location>
<caption>Figure 5. Correlations between our fit parameters and the core relaxation time of the cluster. Here A has been scaled by the distance to each cluster so that it is in units of parsecs. Red points indicate core collapsed clusters not used in calculating the correlation statistic. The insets show the histogram of measured Pearson correlation coefficients after bootstrapping and taking into account uncertainties in the fit parameters.</caption>
</figure>
<table>
<location><page_6><loc_8><loc_59><loc_51><loc_64></location>
<caption>Table 1 Fit Correlations</caption>
</table>
<text><location><page_6><loc_8><loc_38><loc_48><loc_57></location>We have shown that measuring the core radius of a cluster through the surface brightness introduces a bias that makes clusters appear more concentrated than the actual distribution of their main sequence stars or total mass on the main sequence. This bias is explained by the presence of mass segregation within clusters. Mass segregation creates a mass-to-light ratio that changes with radius, and so there is no constant factor that can be used to convert between the density distribution of a cluster as measured through the surface brightness profile and the true underlying stellar density. Cluster model parameters measured in this way will be biased to the most massive stars in the cluster, and will not accurately describe the distribution of lower mass stars. Such a discrepancy was also recently</text>
<text><location><page_6><loc_8><loc_16><loc_48><loc_38></location>We have attemped to quantify the amount of mass segregation in 54 clusters using public catalogs (Sarajedini et al. 2007). This was done by fitting KingMichie models to a series of narrow mass bins in each cluster, allowing us to measure how the King radius varies with mass. We found that a simple two parameter power-law fit these data well and we have presented the parameters of these fits for all of the clusters in Table 1. We also present the best fitting power-law and the likelihood contours of fit parameters for each cluster in the supplementary material. We have shown that a single concentration parameter does not accurately reflect the distribution of most stars within a cluster, and the models we have presented could be used to estimate the distribution of stars as a function of mass within the clusters.</text>
<text><location><page_6><loc_8><loc_12><loc_48><loc_16></location>We find that the exponent of our power-law model correlates strongly with both the core relaxation time and the absolute size of the cluster. This suggests that these</text>
<text><location><page_6><loc_52><loc_53><loc_92><loc_67></location>models could be used as a proxy for the underyling dynamical state of the cluster. However, the scatter in these correlations is larger than can be attributed to the errors alone. In the case of the second and third correlations, this is partly attributable to relaxation time values having no associated uncertainty. However, the scatter still appears to be larger than expected in the first correlation, for which uncertainties have been properly considered. This potentially indicates the effects of other factors, not considered in this work, that can influence the dynamical evolution of clusters.</text>
<section_header_level_1><location><page_6><loc_67><loc_46><loc_77><loc_47></location>REFERENCES</section_header_level_1>
<unordered_list>
<list_item><location><page_6><loc_52><loc_43><loc_90><loc_45></location>Anderson, J., Sarajedini, A., Bedin, L. R., et al. 2008, AJ, 135, 2055</list_item>
<list_item><location><page_6><loc_52><loc_40><loc_90><loc_42></location>Binney, J., & Tremaine, S. 1987, Galactic Dynamics (Princeton University Press)</list_item>
<list_item><location><page_6><loc_52><loc_38><loc_90><loc_40></location>Catelan, M., Stetson, P. B., Pritzl, B. J., et al. 2006, ApJ, 651, L133</list_item>
<list_item><location><page_6><loc_52><loc_36><loc_91><loc_38></location>Dotter, A., Chaboyer, B., Jevremovi'c, D., et al. 2008, ApJS, 178, 89</list_item>
<list_item><location><page_6><loc_52><loc_34><loc_92><loc_36></location>Dotter, A., Sarajedini, A., Anderson, J., et al. 2010, ApJ, 708, 698 Forbes, D. A., & Bridges, T. 2010, MNRAS, 404, 1203</list_item>
<list_item><location><page_6><loc_52><loc_33><loc_89><loc_34></location>Geisler, D., Wallerstein, G., Smith, V. V., & Casetti-Dinescu,</list_item>
<list_item><location><page_6><loc_53><loc_32><loc_70><loc_33></location>D. I. 2007, PASP, 119, 939</list_item>
<list_item><location><page_6><loc_52><loc_29><loc_91><loc_31></location>Goldsbury, R., Richer, H. B., Anderson, J., et al. 2010, AJ, 140, 1830</list_item>
<list_item><location><page_6><loc_52><loc_28><loc_72><loc_29></location>Harris, W. E. 1996, AJ, 112, 1487</list_item>
</unordered_list>
<text><location><page_6><loc_52><loc_27><loc_69><loc_28></location>King, I. R. 1966, AJ, 71, 64</text>
<unordered_list>
<list_item><location><page_6><loc_52><loc_26><loc_91><loc_27></location>Koleva, M., Prugniel, P., Ocvirk, P., Le Borgne, D., & Soubiran,</list_item>
<list_item><location><page_6><loc_53><loc_25><loc_70><loc_26></location>C. 2008, MNRAS, 385, 1998</list_item>
<list_item><location><page_6><loc_52><loc_23><loc_91><loc_25></location>McLaughlin, D. E., & van der Marel, R. P. 2005, ApJS, 161, 304 Michie, R. W. 1963, MNRAS, 126, 499</list_item>
<list_item><location><page_6><loc_52><loc_21><loc_88><loc_23></location>Miocchi, P., Lanzoni, B., Ferraro, F. R., et al. 2013, ArXiv e-prints</list_item>
<list_item><location><page_6><loc_52><loc_19><loc_90><loc_20></location>Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery,</list_item>
<list_item><location><page_6><loc_53><loc_18><loc_85><loc_19></location>B. P. 2007, Numerical Recipes: The Art of Scientific</list_item>
<list_item><location><page_6><loc_53><loc_17><loc_78><loc_18></location>Computing (Cambridge University Press)</list_item>
<list_item><location><page_6><loc_52><loc_15><loc_90><loc_17></location>Sarajedini, A., Bedin, L. R., Chaboyer, B., et al. 2007, AJ, 133, 1658</list_item>
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<text><location><page_6><loc_52><loc_14><loc_90><loc_15></location>Trager, S. C., King, I. R., & Djorgovski, S. 1995, AJ, 109, 218</text>
<text><location><page_7><loc_42><loc_89><loc_58><loc_90></location>Fit Powerlaw Parameters</text>
<table>
<location><page_7><loc_24><loc_20><loc_75><loc_89></location>
<caption>Table 2</caption>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "We present core radii for 54 Milky Way globular clusters determined by fitting King-Michie models to cumulative projected star count distributions. We find that fitting star counts rather than surface brightness profiles produces results that differ significantly due to the presence of mass segregation. The sample in each cluster is further broken down into various mass groups, each of which is fit independently, allowing us to determine how the concentration of each cluster varies with mass. The majority of the clusters in our sample show general agreement with the standard picture that more massive stars will be more centrally concentrated. We find that core radius vs. stellar mass can be fit with a two parameter power-law. The slope of this power-law is a value that describes the amount of mass segregation present in the cluster, and is measured independently of our distance from the cluster. This value correlates strongly with the core relaxation time and physical size of each cluster. Supplementary figures are also included showing the best fits and likelihood contours of fit parameters for all 54 clusters.",
"pages": [
1
]
},
{
"title": "QUANTIFYING MASS SEGREGATION AND NEW CORE RADII FOR 54 MILKY WAY GLOBULAR CLUSTERS",
"content": "Ryan Goldsbury 1 , Jeremy Heyl 1 , Harvey Richer 1 Draft version June 30, 2021",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Comprehensive catalogues of Milky Way globular cluster parameters have been compiled by many different groups over the last two decades. The first large scale effort was put together by Trager et al. (1995), in which they presented surface brightness profiles for 125 Galactic globular clusters, and included parameters determined from model fitting for 63 of them. This work was expanded upon by McLaughlin & van der Marel (2005) who used data from the previous paper as well as new data, and fit multiple classes of models to the surface brightness profiles. The results from both of these papers, as well as many others, have been compiled in Harris (1996) (2010 edition). Our approach differs from these previous studies in that we do not assume a single mass-to-light ratio for the cluster. We instead work directly from star counts and consider bins 0.8 magnitudes wide to break each cluster down further, fitting the distribution of stars in each bin independently. Using stellar evolution models from Dotter et al. (2008) we can assign masses to each bin and analyze how the stellar concentration changes with mass. This approach avoids assuming a constant mass-to-light ratio for a cluster. Given the presence of any amount of mass segregation the assumption of constant mass-to-light ratio will not be true since mass-to-light ratio is a function of mass, and the average mass of stars changes as a function of distance from the cluster core. Because of this, this surface brightness profile does not necessarily reflect the underlying stellar density in a cluster.",
"pages": [
1
]
},
{
"title": "2. DATA",
"content": "All of the data used in this study are from the ACS Survey of Galactic Globular Clusters (Sarajedini et al. 2007). A thorough discussion of the reduction can be found in Anderson et al. (2008). Reduced cata- 1 Department of Physics & Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1; rgoldsb@phas.ubc.ca, heyl@phas.ubc.ca, richer@astro.ubc.ca logues of both real and artificial stars can be found at: http://www.astro.ufl.edu/ ~ ata/public_hstgc/ databases.html .",
"pages": [
1
]
},
{
"title": "3. METHODS",
"content": "There is some ambiguity in the literature regarding a few parameters commonly used to characterize the concentration of star clusters. In an attempt to avoid this, we will use the following symbols and definitions for the remainder of the paper. The latter three quanitites are defined as follows: The quantity Σ here refers to the projected density distribution of the cluster. The core radius ( R c ) is the projected radial distance from the center of the cluster at which the projected density of the cluster drops to half of the central value. The quantity σ is the velocitydispersion parameter, and ρ 0 is the central density of the King model. The King radius ( r 0 ) can be calculated from the previous two quanities and describes the scale of the model similar to the core radius. The concentration ( c ) is defined following the convention in Binney & Tremaine (1987) (hereafter BT87). This disagrees with the definition used by Harris (1996), in which c = log 10 ( r t /R c ). For a given r 0 and r t a King model has only one possible value of R c , but it is not equivalent to r 0 . It is also important to note here that we use the capital R to denote a projected radius, while the lower-case r refers to a three-dimensional radius. Equation 1 is intentionally ambiguous as it does not indicate whether the surface density is in units of luminosity, mass, or number of stars. This will be expanded upon in Section 3.2.",
"pages": [
1,
2
]
},
{
"title": "3.1. Solving for Projected Density Distributions of King-Michie Models",
"content": "Our method involves generating a number of KingMichie (King 1966; Michie 1963) models over varying Ψ(0) /σ 2 , which are defined in Equations 4 and 5. Ψ(0) /σ 2 is referred to as the central dimensionless potential and is sometimes written Ψ 0 . We follow the prescription in Section 4.4 of BT87, ending with a series of models giving normalized surface density (Σ / ( ρ 0 r 0 )) as a function of radius ( r/r 0 ). To calculate these models, we begin with Equation 4132 from BT87, reproduced in Equation 4. We have two boundary conditions. The first is always the same: d Ψ /dr = 0 when r = 0. The second is the value of Ψ(0), which determines the concentration of the resulting density model for a given σ . Since we are only concerned with Ψ(0) /σ 2 , which will control the shape of the density distribution, and not Ψ(0) or σ independently, which determine the absolute scale of the system, we only consider Ψ(0) /σ 2 as a single combined parameter. We use a fourth order Runge-Kutta method (RK4) to solve this equation numerically. The step size varies from 0 . 01 r 0 for the most concentrated to 10 -4 r 0 for the most extended models in our grid. The truncation error is less than 0 . 1% for all models calculated. A solution to this equation for a given set of boundary conditions gives Ψ( r ). This can then be fed into Equation 4-131 from BT87 (shown as Equation 5 in this paper) to go from ρ (Ψ) to ρ ( r ). break Finally, to transform from a three dimensional density distribution to a projected density distribution, one must perform an Abel Transformation: A range of the resulting projected density distributions is shown in Figure 1. The brief summary in this section is not intended to be comprehensive, only to allow the reader to replicate our procedure for calculating these models. A more thorough description, including motivation for the distribution function from which these models are derived, can be found in King (1966) or BT87.",
"pages": [
2
]
},
{
"title": "3.2. Luminosity vs. Star Counts",
"content": "In the interest of presenting a straightforward comparison to previously determined values of R c we first fit the distribution of all stars in a given cluster field together. While our grid is parametrized by r 0 , it is easy to determine the value of R c for a given model by finding the radius at which the surface density drops to half. This value is more commonly referenced in the literature. Our fitting method will be discussed in Section 3.4. A comparison between the values currently in the literature and our resulting best fit values and their uncertainties is shown in the left hand panel of Figure 2. Our uncertainties are estimated with an iterative fitting procedure that considers uncertainty in the location of the cluster center as well as uncertainty due to sample size through bootstrapping. We find that by fitting to the cumulative star count distribution and assuming a conservative 20% uncertainty in Harris' values the mean difference between our fit values and Harris' values is inconsistent with zero at a bit more than 4 σ , where the error used is the standard error. We tend to measure statistically significant larger core radii. This is not surprising, given that R c values reported in Harris (1996) are determined from fitting the luminosity profiles of clusters. If we repeat our fitting procedure, but now weight each star's contribution to the cumulative distribution by its luminosity, we then produce the result shown in the cen- ter panel of Figure 2. Here the bias described above is removed and the mean difference between our values and those of Harris is within 1 . 2 σ of zero. The explanation for this discrepancy is as follows. The mass-to-light ratio is a function of radius in most globular clusters due to mass segregation; more massive (and therefore more luminous) stars will be more centrally concentrated than less massive (less luminous) stars. This implies that the mass-to-light ratio decreases toward the center of the star cluster and so converting between a distribution measured from surface brightness to the underlying distribution of stars is not a constant factor. In fact, how this mass-to-light ratio changes with radius also differs between clusters. So, even if we consider mass-to-light ratio as a function of R , there is no universal correction to be applied. If we wish to determine the mass distribution of the cluster, then fitting the star count distribution still has an inherent bias for the same reason discussed above. We make a rough approximation of the core radius of the true underlying mass distribution by fitting cumulative distributions in which stars are weighted by mass. The masses are assigned by fitting isochrone models from Dotter et al. (2008) to the cluster CMDs. The results are shown in the right panel of Figure 2. These values fall somewhere in between the previous two, but the key point is that they are inconsistent with those determined from fitting the surface brightness profile. This indicates that the surface brightness profile is not an appropriate proxy for the mass surface density of a cluster. For the remainder of the paper we will attempt to quantify these effects by analyzing how subgroups of varying mass are distributed in each cluster.",
"pages": [
2,
3
]
},
{
"title": "3.3. Selecting Groups Along the Main-Sequence",
"content": "For each cluster, we create ten independent groups along the main sequence. We begin by fitting a fiducial sequence to each cluster color-magnitude diagram (CMD) in F 606 W vs. ( F 606 W -F 814 W ). The fiducial is created using an iterative algorithm. A manual starting point is selected at the base of the CMD. The algorithm then chooses a direction to move by using the AMOEBA algorithm from Press et al. (2007) to mini- mize the median of the absolute separation of the nearby points perpendicular to each proposed step. There is also an additional cost added based on the angle between sequential steps to prevent the sequence from bending back on itself. The result is a fiducial that traces out the center of the sequence. Once this fiducial is established for each cluster, the turn-off is then defined as the magnitude that produces the most negative derivative ( dcol/dmag ) of the fiducial. Ten magnitude bins are created starting from 0.5 magnitudes below this turn-off and covering eight magnitudes in total. The widths of the bins are determined by calculating the standard deviation of the CMD in the color direction from the fiducial. Only objects that fall within ± 3 σ are included. While this approach does require an initial point to be selected manually on the CMD, it was found that various starting points would converge to the same sequence within approximately 5 iterations. These initial burn-in points were removed and the fiducial was linearly interpolated backward from the later points. This effectively removes all human input, leading to a bin selection method that is entirely automated and consistent across all of the clusters used in this study. Figure 3 illustrates the location of these bins on a CMD.",
"pages": [
3
]
},
{
"title": "3.4. Fitting Models to Subgroups",
"content": "For each cluster, we fit our grid of King-Michie models to the cumulative radial distribution of each of our 10 sub-groups. We fit only out to projected radii of 100 arc-seconds, due to the size limitations of a single ACS field. Out to 100 arc-seconds from the center ( R max in Equation 7), each annulus is fully within the field. Beyond this, more and more of each successive annulus will fall off of the field. It is possible to take this effect into account when fitting the models, but including these regions does not significantly improve our ability to constrain the best fitting model parameters, and so they are ignored in the interest of simplicity. To generate a model to fit to our data we need to specify two of the three parameters: r 0 , r t , and c , since the third can always be determined from the other two. The tidal radius of the cluster in all cases will be many times the maximum radius that can be observed with our field, and we expect this to be a constant for all subgroups within the cluster. Therefore, instead of fitting for two parameters, we use the values of c and R c from Harris (1996) to calculate r t for each cluster and hold this value fixed. Before fitting the models to the data, we also need to take into account the completeness of the stars in each bin. To do this, we use the artificial star tests mentioned in Section 2 to calculate the completeness as a function of radius for each sub-group in each cluster. The incompleteness ( I ) is the probability that a star of a given magnitude would be found at a given position in the field. These corrections are then applied to the model density distributions before fitting to the data. Given the projected density distribution Σ( R ) and data incompleteness I ( R ), we calculate the cumulative distribution as: The angular component of this integral is not shown since our models are spherically symmetric and therefore azimuthally symmetric in projection, and the result simplifies to the equation shown above. Here I is only dependant on R since we are considering a group of objects that fall within a narrow magnitude range. In determining the best fit to the cumulative distribution of each bin, there are two major sources of uncertainty. The first is the location of the center of the cluster. We use the center coordinates and uncertainties from Goldsbury et al. (2010), which were determined using this same data set. The second source of uncertainty is the sample size in each bin. Both of these are taken into account by using a bootstrap fitting method as follows. In each iteration, a sample of the same size as the full bin sample is chosen with replacement. The location of the center to use is drawn from a two dimensional Gaussian distribution centered at the value from Goldsbury et al. (2010) with the appropriate σ . The cumulative radial distribution is calculated and the best fitting model is determined by minimizing the maximum separation between the real distribution and the model, which is the Kolmogorov-Smirnov statistic D . In each iteration, we record a single value for the best fitting r 0 . After many it- rations we can then generate a histogram of fit r 0 values. This distribution is roughly Gaussian and is broadened by both sources of error discussed above. We take the mean and standard deviation of these values as our best fitting r 0 and 1 σ uncertainty for each sub-group in each cluster. The dominant source of error in most cases is the √ N random uncertainty. However, in very well populated bins, such as the low mass groups in a very massive cluster like NGC 5139, the systematic uncertainty from the location of the cluster center contributes as much to the total error as the Poisson counting uncertainty.",
"pages": [
4
]
},
{
"title": "3.5. Assigning Masses to Bins and Fitting Power Laws",
"content": "After iterating the fitting procedure described in Section 3.4 for all sub-groups in all clusters, we can show how the concentration of stars along the main-sequence changes with magnitude for each cluster. However, this is not particularly useful for making comparisons among clusters, since the apparent magnitude of an object is a result of a number of factors: distance, extinction, mass, metallicity, and cluster age. In terms of dynamics, we would like to look at how the concentration changes with object mass. So, to assign a mass to each magnitude bin, we need to consider the effects of the other four parameters. To do this, we again use an iterative approach. In each iteration we draw values of the four parameters listed above from Gaussian distributions. The parameters of these distributions are drawn from a number of sources. All of the distances, metallicities, and extinctions are taken from Harris (1996). The vast majority of these have no reported uncertainties in the original references, and so we assume conservative estimates: for distance we assume a standard deviation of 0.2 magnitudes, for extinction and metallicity we assume 20% uncertainties. The ages and uncertainties are taken from Dotter et al. (2010) with the exception of five clusters that were not listed in that paper. These are: NGC 1851 and NGC 2808 (Koleva et al. 2008), NGC 5139 (Forbes & Bridges 2010), NGC 6388 (Catelan et al. 2006), and NGC 6715 (Geisler et al. 2007). These parameters are then used to define a stellar isochrone model from Dotter et al. (2008). For each magnitude bin, we assign masses to all of the objects using the model and then take the mean mass of all objects in each bin. So, for each iteration, we record one mass value for each bin. We then use these mass values along with our best fitting r 0 value and uncertainties to fit a two parameter power-law model to the relation between King radius and mass using a maximum likelihood approach. The isochrone model grid does not agree well with the structure of most CMDs at the very low mass end, so we ignore all bins for which the mean mass is less than 0.2 M /circledot . We also ignore points for which the best fitting r 0 falls outside of our field. In these cases, r 0 is not well constrained by our data. For each iteration, we save the likelihood results over the two power-law parameters shown in Equation 8. This power-law model is not physically but rather empirically motivated, as it fits a wide range of clusters well with a small number of parameters: After many iterations as described above, we can average this stack of two dimensional likelihood surfaces. This amounts to a Monte Carlo integration over the likelihood of the four other parameters (distance, extinction, metallicity, and cluster age) with prior distributions on each parameter. This leaves the likelihood of only the two power-law parameters, but with the uncertainties in the other four propagated through. An example fit for 47 Tuc is shown in Figure 4.",
"pages": [
4,
5
]
},
{
"title": "4. RESULTS",
"content": "The parameter A is a normalization that corresponds to the King radius of cluster members at 1 solar mass. This value is given in projected coordinates which are related to the true three dimensional King radius by the distance to the cluster. The parameter B describes how much the concentration of stars changes with their mass, and is independent of the total scale of the cluster. Although we choose to fit our models for r 0 , we could just as easily parametrize them by R c and fit power-laws to this parameter in the same way as Equation 4. For all 54 of the clusters included in this analysis, these two powerlaw parameters as well as their uncertainties and the χ 2 values for the best fit (the value given is the total χ 2 , not per degree of freedom) are included in Table 2 for both r 0 and R c as a function of mass. For any of these clusters, this empirical relation can then be used to estimate the distribution of stars of a given mass. The values of A in this table are in units of arc-seconds, and the parameters correspond to a power-law function in the form of Equation 8, but in the right side the parameters are for fits to R c ( M ) rather than r 0 ( M ). Although r 0 is determined from the three dimensional density distribution, our values come from fitting the projected density distribution and so we fit in units of angle rather than distance. Since we are fitting in projected space, we leave our results in these projected units. These results can be converted to physical units by using the distance given in Harris (1996) or another source. Parameter A has units of angle and our measurement of it depends on our distance from the cluster. Parameter B , on the other hand, is unitless and would be measured the same regardless of our distance from the cluster. This parameter is a quantitative descriptor of the mass segregation present in a cluster. We can use the apparent distance modulus and extinction from Harris (1996) to convert our fit values for A in arc-seconds (or pixels) to parsecs, which gives us a parameter to describe the intrinsic size of the cluster. After doing this, we find that A and B correlate strongly with each other such that larger clusters tend to have less mass segregation than small clusters. Both of these quantities also correlate with the core relaxation time from Harris (1996). Figure 5 shows these correlations. The distribution inset in each diagram indicates the Pearson correlation coefficient ( R ) distribution determined through bootstrapping. A principal component analysis of this three-dimensional space indicates only a single significant component that links all three dimensions. This can be understood as follows: large clusters have longer crossing times and therefore longer relaxation times. The amount of mass segregation present in the cluster should approach some limit near equipartition as many relaxation times pass. Therefore, these larger and slower to relax clusters will exhibit less mass segregation than those that are small and quick to relax. Lines of the following forms are fit to each correlation. These fits are shown as blue dashed lines in Figure 5, and the values of the best fit parameters as well as their variances and covariances are listed in Table 1. As with the calculation of the correlation coefficients, core collapsed clusters are excluded from these fits. Classification of core collapsed clusters is taken from Harris (1996). These are derived from luminosity profiles and so this classification potentially suffers from the biases introduced by mass segregation that are discussed above. Of the eight clusters that are classified as core collapsed, we find that the projected density distribution measured from star counts is well modeled by just a powerlaw for six of them. Two of the clusters (NGC 6541 and NGC 6723) show clear evidence of leveling off toward the center and do not appear to have collapsed cores when the density is measured through star counts rather than luminosity. Similar discrepancies have been found by Miocchi et al. (2013), where clusters are found to show a central cusp in their surface brightness profile but no such feature in their stellar density profile. The authors attribute this to the presence of a few anomalous bright stars, but we believe this discrepancy could also be caused by the systematic effects of mass segregation. Despite the fact that clusters which appear to be core collapsed in surface brightness may not actually be so, we have still excluded all of these clusters from the correlation analysis shown in Figure 5 since this classification still carries a potential bias through the calculation of the relaxation time.",
"pages": [
5
]
},
{
"title": "5. CONCLUSIONS",
"content": "We have shown that measuring the core radius of a cluster through the surface brightness introduces a bias that makes clusters appear more concentrated than the actual distribution of their main sequence stars or total mass on the main sequence. This bias is explained by the presence of mass segregation within clusters. Mass segregation creates a mass-to-light ratio that changes with radius, and so there is no constant factor that can be used to convert between the density distribution of a cluster as measured through the surface brightness profile and the true underlying stellar density. Cluster model parameters measured in this way will be biased to the most massive stars in the cluster, and will not accurately describe the distribution of lower mass stars. Such a discrepancy was also recently We have attemped to quantify the amount of mass segregation in 54 clusters using public catalogs (Sarajedini et al. 2007). This was done by fitting KingMichie models to a series of narrow mass bins in each cluster, allowing us to measure how the King radius varies with mass. We found that a simple two parameter power-law fit these data well and we have presented the parameters of these fits for all of the clusters in Table 1. We also present the best fitting power-law and the likelihood contours of fit parameters for each cluster in the supplementary material. We have shown that a single concentration parameter does not accurately reflect the distribution of most stars within a cluster, and the models we have presented could be used to estimate the distribution of stars as a function of mass within the clusters. We find that the exponent of our power-law model correlates strongly with both the core relaxation time and the absolute size of the cluster. This suggests that these models could be used as a proxy for the underyling dynamical state of the cluster. However, the scatter in these correlations is larger than can be attributed to the errors alone. In the case of the second and third correlations, this is partly attributable to relaxation time values having no associated uncertainty. However, the scatter still appears to be larger than expected in the first correlation, for which uncertainties have been properly considered. This potentially indicates the effects of other factors, not considered in this work, that can influence the dynamical evolution of clusters.",
"pages": [
6
]
},
{
"title": "REFERENCES",
"content": "King, I. R. 1966, AJ, 71, 64 Trager, S. C., King, I. R., & Djorgovski, S. 1995, AJ, 109, 218 Fit Powerlaw Parameters",
"pages": [
6,
7
]
}
]
2013ApJ...778...61T
https://arxiv.org/pdf/1309.4096.pdf
<document>
<section_header_level_1><location><page_1><loc_11><loc_81><loc_89><loc_83></location>MERGERS IN GALAXY GROUPS - I. STRUCTURE AND PROPERTIES OF ELLIPTICAL REMNANTS</section_header_level_1>
<text><location><page_1><loc_33><loc_80><loc_68><loc_81></location>Dan S. Taranu, John Dubinski, and H.K.C. Yee</text>
<text><location><page_1><loc_12><loc_77><loc_89><loc_79></location>Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, Canada, M5S 3H4 Draft version July 21, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_75><loc_55><loc_76></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_49><loc_86><loc_74></location>We present collisionless simulations of dry mergers in groups of three to twenty-five galaxies to test the hypothesis that elliptical galaxies form at the centers of such groups. Mock observations of the central remnants confirm their similarity to ellipticals, despite having no dissipational component. We vary the profile of the original spiral's bulge and find that ellipticals formed from spirals with exponential bulges have too low Sersic indices. Mergers of spirals with de Vaucouleurs (classical) bulges produce remnants with larger Sersic indices correlated with luminosity, as with SDSS ellipticals. Exponential bulge mergers are better fits to faint ellipticals, whereas classical bulge mergers better match luminous ellipticals. Similarly, luminous ellipticals are better reproduced by remnants undergoing many ( > 5) mergers, and fainter ellipticals by those with fewer mergers. The remnants follow tight size-luminosity and velocity dispersion-luminosity (Faber-Jackson) relations ( < 0.12 dex scatter), demonstrating that stochastic merging can produce tight scaling relations if the merging galaxies also follow tight scaling relations. The slopes of the size-luminosity and Faber-Jackson relations are close to observations but slightly shallower in the former case. Both relations' intercepts are offset - remnants are too large but have too low dispersions at fixed luminosity. Some remnants show substantial (v/ σ > 0.1) rotational support, although most are slow rotators and few are very fast rotators (v/ σ > 0.5). These findings contrast with previous studies concluding that dissipation necessary to produce ellipticals from binary mergers of spirals. Multiple, mostly minor and dry mergers can produce bright ellipticals, whereas significant dissipation could be required to produce faint, rapidly-rotating ellipticals.</text>
<text><location><page_1><loc_14><loc_47><loc_85><loc_49></location>Subject headings: galaxies: elliptical - galaxies: evolution - galaxies: formation - galaxies:structure</text>
<section_header_level_1><location><page_1><loc_21><loc_44><loc_36><loc_45></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_27><loc_48><loc_43></location>Merging of spiral galaxies is a promising mechanism for producing elliptical galaxies. Although it was perhaps not until Toomre (1977) that interacting spirals became widely accepted as elliptical progenitors, simulations of interacting spirals date back at least to Toomre & Toomre (1972) and arguably as far as Holmberg (1941). Much of this work has focused on major mergers (mass ratio < 3:1) of pairs of spiral galaxies on parabolic orbits. While such binary major mergers certainly are observed in the local universe - and will likely be the ultimate fate of the Milky Way and M31 - they may not be as common as minor hierarchical mergers.</text>
<text><location><page_1><loc_8><loc_7><loc_48><loc_27></location>Observational evidence and numerical simulations suggest that most L* galaxies are found in groups (McGee et al. 2009), where the central galaxy is likely to have experienced multiple mergers and have several surviving satellites. Furthermore, late-type galaxies in groups follow a Schechter luminosity function (Schechter 1976) similar to those in other environments (Croton et al. 2005; Robotham et al. 2010). Thus, if high-redshift groups are composed primarily of spiral galaxies, they are likely dominated by several bright spirals with a larger number of fainter satellites, as in our Local Group. Our hypothesis is that groups of three or more spiral galaxies with luminosity distributions following a Schechter function will naturally merge to produce a central elliptical galaxy, and possibly fainter satellites.</text>
<text><location><page_1><loc_8><loc_3><loc_48><loc_7></location>We aim to test this hypothesis with numerical experiments. More specifically, we test whether the properties of the central galaxies formed through collisionless merg-</text>
<text><location><page_1><loc_52><loc_29><loc_92><loc_45></location>ers in groups of spirals are consistent with observations of local ellipticals. This first paper in a series outlines our methodology and demonstrates that the results are both qualitatively and quantitatively different from both the more prevalent studies on binary mergers and also the less abundant literature on galaxy group mergers. We present results on morphological and kinematical measures as well as two dimensional scaling relations. Paper II (Taranu et al. 2013) will focus on the three dimensional fundamental plane scaling relation. To further motivate this endeavour, we will outline some of the key results of the past several decades of work in this field.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_29></location>Carnevali et al. (1981) and Ishizawa et al. (1983) were amongst the first to produce simulations of mergers in groups of galaxies (10-20 each), using 20 and 100 particles per galaxy, respectively. Barnes (1985) introduced separate stellar and dark matter profiles, with 30 and 270 particles for each component, respectively. Barnes (1989) added stellar disks and bulges to the galaxy models, taking advantage of the newly invented N-body tree code (Barnes & Hut 1986) to increase resolution to 4096 luminous and dark particles each. While the arrangement of orbits was somewhat artificial (a pair of triple systems each consisting of a binary orbiting a more massive single galaxy), the study showed that mergers of realistic galaxies in compact groups can be rapid and produce central remnants with de Vaucouleurs profiles and shell structures, similar to local ellipticals.</text>
<text><location><page_1><loc_52><loc_3><loc_92><loc_8></location>A problem with the collisionless merger scenario was identified by Carlberg (1986). Collisionless mergers cannot increase central phase space density, which is necessary if disks are to merge and produce ellipticals with</text>
<text><location><page_2><loc_8><loc_79><loc_48><loc_87></location>higher central densities than their progenitors, as is often the case. Hernquist et al. (1993) directly addressed this issue with simulations of binary mergers of spirals with bulges (Hernquist 1993), which, unlike mergers of bulgeless spirals (Hernquist 1992), are capable of producing sufficiently centrally dense remnants.</text>
<text><location><page_2><loc_8><loc_45><loc_48><loc_79></location>Weil & Hernquist (1996) extended this methodology to mergers of groups of six equal-mass spirals. The resolution of these simulations was a factor of 18 higher than Barnes (1989), with almost 150,000 particles per galaxy, and included a comparison sample of pair mergers. Group mergers were shown to produce remnants with some rotation, in contrast to the non-rotating remnants typical of dry binary mergers (Cox et al. 2006). Both varieties of mergers produced early-type galaxies well fit by de Vaucouleurs profiles. However, group mergers with bulges did not maintain centrally concentrated profiles, exhibiting the same low central phase space density as found in bulgeless pair mergers. This may be seen as once again disfavoring the group merging scenario; however, it should be noted these mergers were of equal-mass galaxies and hence of very large mass ratios, which would tend to maximize this problem. Furthermore, Kormendy et al. (2009) argues that ellipticals have lower central densities ('cores') than expected from inward extrapolation of their outer surface brightness profiles, which is consistent with the group merging scenario and essentially the opposite of the central phase space density problem, although Kormendy et al. (2009) attribute these cores to scouring of inner stars by inspiraling supermassive black holes rather than purely stellar dynamical processes.</text>
<text><location><page_2><loc_8><loc_2><loc_48><loc_45></location>Since Weil & Hernquist (1996), the relatively rapid pace of advancements in the field of simulations of merging in galaxy groups has slowed, with the focus shifting to studies of hydrodynamical processes in group environments. This can be partially attributed to the findings of Robertson et al. (2006) that collisionless binary mergers are unable to produce ellipticals following a tilted fundamental plane relation, although Aceves & Vel'azquez (2005) found an appreciable tilt by merging spirals sampled from an appropriate Schechter luminosity function. Robertson et al. (2006) also found that collisionless binary mergers could only produce very slow-rotating remnants and that dissipation was required to produce significantly rotationally-supported ellipticals. However, between Weil & Hernquist (1996) and Robertson et al. (2006), very few studies have tested whether these results apply to multiple collisionless mergers as well. Galaxy clusters (Dubinski 1998) and starbursts in groups (Bekki 2001) have been considered. Ciotti et al. (2007) simulated consecutive mergers of spheroidal galaxies, roughly approximating hierarchical group merging. While this approach has provided useful estimates of the growth of stellar mass and size, the use of purely spheroidal progenitors is questionable given the prevalence of disks at high redshift. Hopkins et al. (2009) combined the results of binary merger simulations with cosmological merger trees, using empirical halo occupation models and spiral scaling relations to predict the evolution of early-type scaling relations. However, the models only incorporated multiple mergers by allowing binary merger remnants sufficient time to relax dynamically, whereas Moster et al. (2012) find that halos undergoing multiple mergers are</text>
<text><location><page_2><loc_52><loc_82><loc_92><loc_87></location>likely to have two mergers in quick succession. Galaxies in groups are thuse likely to undergo multiple mergers within a relatively short periods rather than a steady stream of isolated mergers.</text>
<text><location><page_2><loc_52><loc_53><loc_92><loc_82></location>More recently, fully cosmological simulations of mergers in a group or several groups of galaxies have been performed by, for example, Khalatyan et al. (2008) and Feldmann et al. (2011). Such simulations naturally incorporate hierarchical merging, typically by using the 'zoom-in' method of re-simulating a small sub-volume of a large dark matter-only cosmological simulation at higher resolution. Naab et al. (2009) and Oser et al. (2012) demonstrated that ellipticals can form in groups, with minor mergers being an important mechanism in controlling the evolution of the central galaxy size and mass. Feldmann et al. (2011) showed success in producing not only a central elliptical but early-type satellites by simulating a single group using this method. However, such ab initio simulations encounter difficulties achieving sufficient spatial resolution to produce realistic spiral galaxies. Typical softening lengths in such simulations are between 500 to 1000 pc, which significantly alters the inner profile of ellipticals, especially at low masses. Increasing the resolution can mitigate this problem but also greatly increases computational cost and limits the possible sample size to a few groups.</text>
<text><location><page_2><loc_52><loc_30><loc_92><loc_53></location>It is clear that there is a gap in the literature on multiple mergers in groups, even though multiple mergers likely create brightest cluster galaxies (Dubinski 1998). By contrast, observations of elliptical galaxies have advanced tremendously in recent years, providing public catalogs of morphologies of thousands of nearby ellipticals and spirals alike (e.g. Blanton et al. 2005; Hyde & Bernardi 2009a; Nair & Abraham 2010; Simard et al. 2011), mainly based on Sloan Digital Sky Survey (SDSS) images. The SAURON project (de Zeeuw et al. 2002) and its volume-limited successor survey Atlas3D (Cappellari et al. 2011) have provided integral-field kinematics of hundreds of early type galaxies at a comparable resolution to SDSS. It is increasingly necessary to match the large samples of observations with simulations and explore the vast parameter space of conditions in galaxy groups.</text>
<text><location><page_2><loc_52><loc_5><loc_92><loc_30></location>To meet the requirement of a large simulation sample, it is currently necessary to focus on dry merging and gravitational dynamics alone. Hydrodynamical simulations are more computationally expensive and add numerous parameters to initial conditions: disk gas fractions, gas disk scale heights and lengths relative to the stellar disk, the presence of a gaseous halo, etc. More importantly, existing literature has yet to establish the effects of collisionless gravitational dynamics in group mergers on central remnant structure. There is observational evidence suggesting that dry merging contributes significantly to the growth of massive galaxies, particularly ellipticals (e.g. van Dokkum et al. 2010). Even if exclusively dry merging is not the most common mechanism for forming ellipticals, many ellipticals will have experienced at least one dry merger in their lifetimes and it is instructive to ask what collisionless dynamics alone would predict before moving on to hydrodynamical processes.</text>
<text><location><page_2><loc_52><loc_2><loc_92><loc_5></location>The remainder of the paper is structured as follows: § 2 motivates and details the methods used in creating the</text>
<text><location><page_3><loc_8><loc_75><loc_48><loc_87></location>simulations, while § 3 details the analysis methodology and pipeline, with additional tests presented in App. A. A more detailed examination of numerical convergence can be found in App. B. Key results on morphology, scaling relations and kinematics of central remnants are presented in § 4. The implications of these results on theories of elliptical galaxy formation are detailed in § 5, with reference to prior studies on the subject. The conclusions are summarized in § 6.</text>
<section_header_level_1><location><page_3><loc_21><loc_73><loc_35><loc_74></location>2. SIMULATIONS</section_header_level_1>
<text><location><page_3><loc_8><loc_63><loc_48><loc_72></location>The simulations are designed to extend the methodology of binary galaxy merger simulations to groups of galaxies. This section details the parameters of the group sample ( § 2.1), as well as the two key ingredients in the initial conditions: group configuration ( § 2.2) and galaxy models ( § 2.3). Finally, the code and parameters used for the simulations are described in § 2.4.</text>
<text><location><page_3><loc_8><loc_47><loc_48><loc_63></location>Our choice of initial condition parameterization is designed to evenly sample the parameter space of groups which are likely to produce a central elliptical remnants, rather than be a unbiased sampling of real, nearby galaxy groups. This approach is similar to that used in binary mergers simulations, in which the orbits are typically nearly parabolic, with some cosmologicallymotivated distribution of pericentric distance and disk alignment. In our case, we model group-sized halos at the turnaround radius at z=1-2, such that the subhalos are likely to contain spiral galaxies which will eventually merge to form one central elliptical.</text>
<text><location><page_3><loc_8><loc_30><loc_48><loc_47></location>We use two galaxy models designed to reproduce the surface brightness profile and rotation curve of M31 and scale these models according to the Tully-Fisher relation (Tully & Fisher 1977). The only parameter we vary between the models is the profile of the bulge, which has a substantial impact on the structure of the merger remnant ( § 4.2). While this approach does not reproduce the variety of spiral galaxies found in the local universe, let alone at high redshift, it maintains the simplicity of the initial conditions. We do not vary the bulge fraction in the progenitors, as pre-formed bulges are required to produce sufficient central densities in the merger remnant. We discuss these choices further in § 5.</text>
<text><location><page_3><loc_8><loc_18><loc_48><loc_29></location>Although the simulations are nominally scale-free, as with any system of units having G=1, our simulations assume units of length in kpc, velocity in 100 kms -1 , time in 9.73 million years and mass in 2 . 325 × 10 9 M glyph[circledot] . The luminosity function sampling and initial group radius impose a unique, preferred scaling to each simulation, such that mergers of groups with the same number of galaxies but different luminosities are not simply re-scaled versions of each other.</text>
<section_header_level_1><location><page_3><loc_22><loc_15><loc_35><loc_16></location>2.1. Group Sample</section_header_level_1>
<text><location><page_3><loc_8><loc_2><loc_48><loc_14></location>We create groups with total luminosities from 0.1-10L* and masses between 2 × 10 11 -2 × 10 13 M glyph[circledot] . We incorporate several basic assumptions consistent with observations and cosmological simulation predictions. More massive groups contain more galaxies on average, with galaxies preferentially located closer to the center of the group. The group as a whole is initially collapsing, with galaxies located within R max = 2 × R 200 ,z =2 but having insufficient orbital energy to prevent collapse (i.e. the</text>
<table>
<location><page_3><loc_52><loc_74><loc_89><loc_84></location>
<caption>Table 1 Range of Numbers of Galaxies Initially in Each Group</caption>
</table>
<text><location><page_3><loc_52><loc_68><loc_92><loc_73></location>Note . - Each simulation sample is divided into those groups with relatively Few (F) or Many (M) galaxies for their mass, with three groups in either category per mass bin. The minimum and maximum number of galaxies in a group is listed, as well as the maximum for the F and the minimum for the M subsamples.</text>
<text><location><page_3><loc_52><loc_61><loc_92><loc_67></location>groups are sub-virial). We simulate each group configuration twice, with each simulation containing either spiral galaxies with exponential bulges or classical bulges (but not both), referring to the former sample as B.n s =1 and the latter as B.n s =4 for short.</text>
<text><location><page_3><loc_52><loc_46><loc_92><loc_61></location>There are 3 sets of simulations, each with different random seeds for the initial conditions. Each set has 7 target luminosity (or mass) bins, ranging from 1/8 to 8 L* and increasing by factors of 2. Each bin contains 8 groups, for a total of 3 × 7 × 8 = 168 simulations, of which 3 × 7 × 2 = 42 are mergers of spirals with equal masses, while the remaining 126 are sampled from a realistic luminosity function. Since each simulation is run twice (with different spiral bulge profiles), there are nominally 336 simulations, but only 168 different sets of galaxy masses and orbits.</text>
<text><location><page_3><loc_52><loc_34><loc_92><loc_46></location>Each group has a number of galaxies between N min = 3 and N max = 2+(5 / 6) × 10 · ( L/L ∗ ) 1 / 2 . Within each group luminosity bin, the number of galaxies in each simulation varies linearly from the minimum (3) to the maximum, so that L* groups have between 3 and 10 galaxies and the largest groups have 25 galaxies. This range of galaxy numbers roughly covers the number of bright galaxies one would expect in poor groups. The mass range covered by the groups is 2 . 0 × 10 11 M glyph[circledot] to 3 . 0 × 10 13 M glyph[circledot] .</text>
<text><location><page_3><loc_52><loc_12><loc_92><loc_34></location>We further subdivide the sample into groups with relatively many mergers (the Many-merger or 'M' subsample) or relatively few (Few-merger or 'F'). The groups in each mass bin with the three lowest initial galaxy counts are part of the F subsample, while the groups with the three largest initial galaxy counts qualify for the M subsample. Because the maximum number of galaxies changes in each mass bin, the dividing line between the Many-merger and Few-merger subsamples depends on mass and is not a fixed number of galaxies or mergers. Each mass bin also contains two groups with equal-mass galaxies ('Eq'), one with three galaxies ('F-Eq') and the other with the same number of galaxies - N min ,M - as the fourth group in the LF-sampled simulations (i.e., the group in the 'M' subsample with the fewest galaxies). The number of galaxies in a representative number of groups is listed in Tab. 1.</text>
<section_header_level_1><location><page_3><loc_63><loc_9><loc_81><loc_10></location>2.2. Group Configuration</section_header_level_1>
<text><location><page_3><loc_52><loc_5><loc_92><loc_9></location>Once the target luminosity and number of galaxies are selected, each group is initialized through the following steps:</text>
<unordered_list>
<list_item><location><page_3><loc_54><loc_2><loc_92><loc_4></location>1. Randomly select luminosities for all of the galaxies</list_item>
</unordered_list>
<text><location><page_4><loc_12><loc_85><loc_48><loc_87></location>from a restricted range of the spiral galaxy luminosity function.</text>
<unordered_list>
<list_item><location><page_4><loc_10><loc_81><loc_48><loc_84></location>2. Set the maximum radius within which to spawn galaxies, R max = 2 R 200 ,z =2 .</list_item>
<list_item><location><page_4><loc_10><loc_79><loc_45><loc_80></location>3. Place the most luminous galaxy in the center.</list_item>
<list_item><location><page_4><loc_10><loc_75><loc_48><loc_77></location>4. Place all other galaxies in order of decreasing luminosity.</list_item>
<list_item><location><page_4><loc_10><loc_71><loc_48><loc_74></location>5. Compute the group's gravitational potential energy.</list_item>
<list_item><location><page_4><loc_10><loc_66><loc_48><loc_70></location>6. Assign random velocities to the satellite galaxies, applying an inward and radial bias and re-selecting any velocities with v > v esc .</list_item>
</unordered_list>
<text><location><page_4><loc_8><loc_33><loc_48><loc_65></location>Galaxy luminosities are randomly sampled from the inclination- and extinction-corrected spiral luminosity (Schechter) function of Shao et al. (2007). For the r-band, the faint-end slope α = -1 . 26 and M ∗ r = -20 . 99 + 5 log( h = 0 . 71), or M ∗ r = -21 . 73, which is nearly identical to our standard M31 model's absolute magnitude of M r = -21 . 69. We set a minimum luminosity of 0 . 01 L ∗ for the spirals, as we do not expect the luminosity function to continue to arbitrarily faint magnitudes. Luminosities are drawn from a restricted range of the luminosity function with a width equal to ( N gals +2) / 10 dex, such that the integral under the curve is equal to the target group luminosity. This limited range produces groups with smaller magnitude differences between the brightest central galaxy and the next brightest satellite, making major mergers more likely especially in groups with few galaxies. We avoid simulating groups with a single luminous spiral and several much fainter satellites, since these groups would only produce relatively minor mergers and would be unlikely to produce ellipticals. The 42 groups with equal-mass spirals ('Eq') are not sampled from an LF and instead have exactly the target luminosity, split evenly between three or a larger number of galaxies.</text>
<text><location><page_4><loc_8><loc_16><loc_48><loc_33></location>Once a luminosity is determined for each group galaxy, the galaxies are randomly assigned locations within the group in order of decreasing luminosity. The most luminous galaxy is placed at the center of the halo, while subsequent galaxies are given a random radius with a likelihood inversely proportional to radius, i.e., ρ ∝ r -1 , and with r < R max , where R max = 2 R 200 ,z =2 . R 200 ,z =2 is the radius at which a volume enclosing the total group mass has a mean density of 200 times the critical density at z=2. Next, a random polar and azimuthal angle is given. The minimum distance between galaxies is set by 0 . 5 × R max ∗ ( M galaxy /M group ) 1 / 3 , which allows galaxy halos to be in contact but not overlap significantly.</text>
<text><location><page_4><loc_8><loc_2><loc_48><loc_15></location>All of the galaxies are given preferentially inward and radial orbits. The group itself is sub-virial to ensure collapse and no satellite is given a speed | v | > v escape . This is accomplished by giving each group a target virial ratio Q target = -2 T/W = 0 . 5, where W is the gravitational potential energy of the group. This is equivalent to a zero-energy parabolic orbit in a galaxy pair, where T=-W. The group velocity dispersion is determined by Q target : σ = ( -Q × ( W/a ) /M ) 0 . 5 , where a = 3 s -2 β and β is an orbital anisotropy parameter. Each galaxy's</text>
<text><location><page_4><loc_52><loc_79><loc_92><loc_87></location>radial velocity is then σ ∗ s , where s is a number randomly selected from a unit Gaussian centered on s=0.5. On average more than 70% of galaxies will have inward radial velocities. The azimuthal and polar velocities are given by σ ( s × (1 -β ) 0 . 5 ), where β = 0 . 5 directs most of the velocity of the galaxy radially.</text>
<text><location><page_4><loc_52><loc_63><loc_92><loc_79></location>In two of the three sets of simulations, the initial conditions in groups of similar mass (1/8 to 1/2, 1 to 4, and 8) are correlated, in the sense that galaxy positions are seeded in the same order (but not individual galaxy masses or orbits). This is intended to test the effect of adding additional galaxies to otherwise similar initial conditions. In the third set, all of the initial conditions are completely randomized. We note no statistically significant differences between the partially and completely random initial conditions for the relations presented in this paper; however, we will note some differences in the fundamental plane parameters in Paper II.</text>
<text><location><page_4><loc_52><loc_53><loc_92><loc_63></location>Finally, each galaxy has its own massive, extended dark halo. In practice, the these individual halos overlap in their outer regions, leaving little or no 'empty' space between galaxies. We have also experimented with including a separate dark matter halo for the group, not associated with any particular galaxy, but find that group galaxies then (unrealistically) merge with this invisible dark halo rather than with each other.</text>
<section_header_level_1><location><page_4><loc_65><loc_51><loc_79><loc_52></location>2.3. Galaxy Models</section_header_level_1>
<text><location><page_4><loc_52><loc_35><loc_92><loc_50></location>Initial spiral galaxy models are created using the GalactICS galaxy initial condition code (Widrow & Dubinski 2005). This code generates equilibrium models of galaxies with a bulge, disk and halo through spherical harmonic expansions of analytic potentials. Although the models begin in equilibrium and do not require additional time to settle, they have been tested in isolation. All models remain (statistically) unchanged for at least a Hubble time, even at the lowest resolution (55,000 particles per galaxy), which the vast majority of galaxies exceed.</text>
<text><location><page_4><loc_52><loc_18><loc_92><loc_35></location>The models are similar to the 'M31c' model of Widrow & Dubinski (2005), with some parameters adjusted following the approach of Widrow et al. (2008) to better reproduce the surface brightness profile and rotation curve of M31. M31 was chosen as a well-studied, nearby spiral having a sufficiently massive bulge to produce concentrated merger remnants. The models are bar-stable and contain a massive, non-rotating bulge, as well as a dark matter halo. The first variant uses a nearly exponential bulge with n s = 0 . 93, which will be referred to as exponential for convenience. A second variant uses an n s = 4 de Vaucouleurs or classical bulge but otherwise identical parameters.</text>
<text><location><page_4><loc_52><loc_5><loc_92><loc_18></location>The halo density profile is designed to match an NFW (Navarro et al. 1997) profile at large radii and smoothly drop to zero at large radii. The halo has a 6.07 kpc scale radius, ρ ∝ r -1 inner cusp and r -2 . 3 outer slope, an outer radius of 300 kpc and a total mass of 1185 units, or 2 . 75 × 10 12 M glyph[circledot] . This profile produces a 30:1 ratio in dark:baryonic (stellar) mass, which is a factor of two larger than estimates for M31 and the Milky Way (e.g. Watkins et al. 2010), but smaller than global estimate for the universal dark:stellar mass ratio.</text>
<text><location><page_4><loc_52><loc_2><loc_92><loc_5></location>The disk has a 5.8 kpc scale radius and a 750 pc sech 2 scale height, equivalent to a 375 pc exponential</text>
<figure>
<location><page_5><loc_8><loc_57><loc_48><loc_86></location>
<caption>Figure 1. The rotation curve of the M31 models used in these simulations. The rotation curve is dominated by the bulge inside in the inner 5 kpc and by the halo thereafter. The more concentrated n s = 4 bulge also produces a sharper, inner peak in the rotation curve at 1 kpc.</caption>
</figure>
<text><location><page_5><loc_8><loc_42><loc_48><loc_47></location>scale height. The disk is cut off past 6 scale radii, or 35 kpc, for a total mass of 25 simulation units, or 5 . 8 × 10 10 M glyph[circledot] . We adopt a disk stellar mass-to-light ratio of ( M/L r ) D = 3 . 4.</text>
<text><location><page_5><loc_8><loc_25><loc_48><loc_42></location>Each bulge has a 1.5 kpc effective radius. The exponential bulge and de Vaucouleurs bulge have masses of 14.75 units (3 . 4 × 10 10 M glyph[circledot] ) and 15 units (3 . 5 × 10 10 M glyph[circledot] ), respectively. The bulge-to-total mass ratio B/T M is about 33% in both models, larger than the 20-30% estimate for the R-band B/T ratio B/T R in M31 (Courteau et al. 2011). The models could compensate by using a lower ( M/L r ) B ; however, the bulge kinematics favor a lower value of 1.9 (Widrow & Dubinski 2005), which we adopt here. While ( M/L r ) B does not affect the simulations, the resulting bulge-to-total light ratio B/T r is 50%, and so mock images are more strongly weighted to bulge stars than disk stars.</text>
<text><location><page_5><loc_8><loc_12><loc_48><loc_25></location>The rotation curve for both models is shown in Fig. 1. The bulge dominates within the inner 4-5 kpc and the halo thereafter, with the disk contribution typically half that of the halo. The non-maximal disk is both consistent with recent observations of spiral galaxies (see van der Kruit & Freeman (2011) for a review) and also promotes bar stability. Although the exponential bulge can be torqued into a bar, the intrinsic bar stability means that any remnant properties such as rotation are a result of the merging process and not secular instabilities.</text>
<text><location><page_5><loc_8><loc_2><loc_48><loc_12></location>To scale our model to different masses, we multiply all masses by a factor m while retaining the same M/L R . Velocities are scaled by m 0 . 29 , assuring that the galaxies follow a Tully-Fisher relation V ∝ L 0 . 29 (Courteau et al. 2007). To maintain virial equilibrium ( R ∝ M/σ 2 , or, log( R ) ∝ log( M ) -2log( σ )), particle distances from the center of the galaxy are scaled by</text>
<figure>
<location><page_5><loc_52><loc_56><loc_90><loc_87></location>
<caption>Figure 2. Logarithmic surface density maps of the initial conditions for four of the simulated groups. The groups have a nominal total luminosity of 0.125, 1, and 8 L* in each row respectively. The number of galaxies in each group varies from three (leftmost column) to a mass-dependent maximum N max (rightmost column). The maximum radius within which galaxies are spawned (equivalent to 2 R 200 ,z =2 ) is shown in gray. Images are 1 Mpc across.</caption>
</figure>
<text><location><page_5><loc_52><loc_25><loc_92><loc_45></location>a factor of R ∝ M/σ 2 ∝ m 1 -2 × 0 . 29 , or, R ∝ m 0 . 42 . As a result, surface brightness scales weakly with mass -L/R 2 ∝ m 0 . 16 - consistent with the Tully-Fisher relation's assumption of nearly constant effective and/or central surface brightnesses. We do not incorporate scatter into the input galaxy scaling relations, so that scatter in the merger remnant scaling relations is both a lower limit and dependent on the formation process (merging) and bulge profile, rather than an additional input parameter like the Tully-Fisher relation's scatter. Similarly, we use the same bulge fraction for all galaxies. We deliberately avoid using bulgeless disks, as existing literature (e.g., Hernquist (1993)) shows that bulgeless disk mergers do not produce sufficiently high central densities. We will further discuss the implications of these choices in § 5.</text>
<text><location><page_5><loc_52><loc_5><loc_92><loc_25></location>The lowest resolution model has 5,000 bulge, 10,000 disk and 40,000 halo particles, for a 1:2:8 bulge:disk:halo ratio, and 15,000 stellar particles. More massive galaxies have larger particle counts by factors of two, up to a maximum of 480,000 disk particles. Most groups have at least three galaxies with 60,000 stellar particles and only a few tens of galaxies have fewer than 30,000 stellar particles. By scaling resolution this way, stellar particles all have the same mass within a factor of three, while dark particles are not more than 10 times more massive than star particles, limiting spurious numerical artifacts. App. B discusses the effects of numerical resolution in greater detail; in summary, this resolution is more than sufficient for the more massive galaxies and adequate for the least massive satellites.</text>
<paragraph><location><page_6><loc_8><loc_54><loc_48><loc_59></location>Figure 3. Logarithmic surface density maps of nine of the simulated groups after the full simulation time of 52,000 steps (about 10 Gyr). The groups shown are the same as in Fig. 2. The effective radius of the central galaxy in each group is shown in white. Images are 1 Mpc across, as in Fig. 2.</paragraph>
<text><location><page_6><loc_8><loc_34><loc_48><loc_53></location>Each group is simulated for 10 Gyr with the parallel N-body tree code PARTREE (Dubinski 1996). Figures 2 and 3 show a typical evolution for one such group. The simulations use 52,000 fixed timesteps of 0.02 units - about 195,000 years - and a softening length (spatial resolution) of 100 pc. We use an opening angle of θ = 0 . 9 with forces computed to quadrupole order. While this opening angle is somewhat larger than typical values of 0.7 to 0.8, PARTREE calculates forces between nearby particles in different trees directly, eliminating the source of the largest force errors. For θ = 1 . 0, PARTREE has been shown to produce median force errors under 0 . 2%, with 90% of force errors under 0 . 5% (Dubinski 1996); force errors with θ = 0 . 9 are considerably smaller.</text>
<section_header_level_1><location><page_6><loc_23><loc_31><loc_33><loc_33></location>3. ANALYSIS</section_header_level_1>
<text><location><page_6><loc_8><loc_6><loc_48><loc_31></location>The simulations are analyzed at three different epochs after 5.0, 7.7 and 10.3 Gyr, which correspond to formation redshifts of about 0.5, 1, and 2, respectively, if one assumes that the group formed at t=0 Gyr. Since the only redshift-dependent parameter in the initial conditions is the maximum radius of the group, analysis of the same group at different epochs is equivalent to assuming a different age for the group. Also, since galaxies are given an initial separation sufficient to prevent their halos from overlapping significantly, it typically takes 12 Gyr for the first mergers to occur. Groups with fewer galaxies complete the merger process after another 2-3 Gyr and so are not sensitive to the choice of formation time, while groups with more galaxies continue slowly accreting lower-mass satellites and growing even after 10 Gyr. Although we do not introduce additional galaxies into the simulation to mimic cosmological accretion, we note that the long merging time for less massive galaxies still allows for late-time mergers in richer groups.</text>
<figure>
<location><page_6><loc_52><loc_60><loc_92><loc_88></location>
<caption>Figure 4. Example mock image of the major axis projection of the central galaxy from a typical L* group after 10 Gyr of simulation. The image shows SDSS-equivalent r-band photometry down to the mean sky level, overlaying contours from the image itself (dashed, black) and the best fit GALFIT Sersic model (solid, black). The gray ellipse shows the effective radius but with no boxiness parameter altering its shape. The image is 29 kpc or 150 SDSS pixels across.</caption>
</figure>
<text><location><page_6><loc_52><loc_24><loc_92><loc_49></location>Once the simulations are complete, we create mock rband photometry and kinematics of each group at the three different epochs, placing the group at a mock redshift of 0.025 (about 100 Mpc away). In brief, we create SDSS-like photometry of the central galaxy out to 8 effective radii, including a sky background and appropriate signal-to-noise ratio. We use GALFIT (Peng et al. 2002, 2010) to fit a single Sersic profile to each galaxy. We also use GALMORPH (Hyde & Bernardi 2009a) to fit a de Vaucouleurs profile to the sky- and satellite-subtracted image, both for comparison to general Sersic fits and to the de Vaucouleurs fits of Hyde & Bernardi (2009b). Finally, we create spatially resolved kinematics at the same scale, and use these maps to measure kinematical quantities within the central region and the effective radius of the central galaxy. Although our simulations do not resolve faint satellites particular well, our pipeline is able to recover the properties of the central ellipticals with precision comparable to SDSS observations.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_23></location>Simulations are processed with our own imaging pipeline, which is intended to create images of the central galaxy in each group equivalent to those produced by the SDSS. We convert mass to luminosity to create nominal r-band images, using fixed stellar mass-to-light ratios for the bulge and disk components. We then extract a one-dimensional profile of the central galaxy in circular bins, masking out the central regions of satellite galaxies. A single Sersic profile is then fit to produce a rough estimate of the effective radius of the central galaxy ( R eff,est ).</text>
<text><location><page_6><loc_52><loc_2><loc_92><loc_9></location>Next, we create a FITS image out to 8 R eff,est around the central galaxy. The image is smoothed by a point spread function (PSF) with a full-width at halfmaximum (FWHM) of 1.43 arcseconds, typical for SDSS r-band observations (Stoughton et al. 2002; Abazajian</text>
<text><location><page_7><loc_8><loc_67><loc_48><loc_87></location>et al. 2009). Galaxies are imaged at a mock redshift of z obs = 0 . 025, typical for the SDSS spectroscopic sample used in (e.g. Hyde & Bernardi 2009a; Nair & Abraham 2010). Fig. 4 shows an example image of a typical galaxy. The pixel scale is identical to that used by SDSS, 0.396 arcsec/pixel. Most importantly, we add a sky background with both a mean surface brightness and variations comparable to SDSS observations. In the r band, the mean sky value is 20.86 and variations are Gaussian distributed with a standard deviation of 2 . 65%, equivalent to the SDSS asinh zero-flux magnitude of 24.80 (which itself was chosen to be approximately 1-sigma of typical sky noise). We also create maps of the projected dark matter distribution using the same pixel scale (but no PSF).</text>
<text><location><page_7><loc_8><loc_49><loc_48><loc_67></location>In addition to photometry, we create kinematical maps of the first four moments of the luminosity-weighted velocities of particles in each pixel (velocity, r.m.s. velocity dispersion σ , and v3 and v4). Although we do smooth these maps by the same PSF and use the same pixel scale as the photometry, we do not add a sky background or any instrument-specific noise. We do not perform any fitting to the kinematic quantities, choosing r.m.s. velocity dispersions rather than fitting any profiles, and so the kinematical maps remain largely instrument-agnostic beyond the choice of pixel scale and PSF. The maps can then be used both to measure central velocity dispersions and spatially resolved kinematics, comparing to SDSS and Atlas3D respectively.</text>
<text><location><page_7><loc_8><loc_26><loc_48><loc_49></location>Finally, we create an error map for the photometry, which will be used to perform χ 2 minimization in fitting profiles to the galaxies. The error is the square root of the luminosity in each pixel multiplied by some constant factor, which scales the signal-to-noise ratio across the image. The constant itself is simply related to the image exposure time, given a certain zero-point equivalent to 1 count per second (for SDSS r-band, this is about 26.7) and mean sky variation. This scheme contrasts with, e.g., Feldmann et al. (2011), and other simulations which use the square root of the number of particles in each pixel as the error. The per-pixel errors do not scale directly with the resolution of the simulation but should instead converge with increasing resolution. Similarly, setting a non-zero floor to the error map ensures that pixels with no signal are not ignored in the fit, which is necessary since the absence of a signal is meaningful.</text>
<text><location><page_7><loc_8><loc_2><loc_48><loc_26></location>For each galaxy, we create images in 10 randomly oriented but evenly spaced projections. These are the ten projections passing through opposite faces of a regular icosahedron, but arbitrarily rotated with respect to the central galaxy. We also use the three projections corresponding to estimates of the principal axes of the central galaxy. We fit every galaxy in the image with a single Sersic profile using GALFIT. Sufficiently large galaxies (including the central galaxy) fit a boxiness parameter (C0) as well, which allows for elliptical isophotes to vary from diamond-shaped (C0 < 0) to rectangular- or boxshaped (C0 > 0). For highly inclined disks with a bulge, this can also provide a better fit than an unmodified ellipse. The GALFIT fits also include a fixed sky background equal to the mean sky brightness. We do not allow for the sky value to vary, as doing so would result in over-fitting the sky, a common problem in observations. Since different surveys and even different data</text>
<text><location><page_7><loc_52><loc_79><loc_92><loc_87></location>releases of the SDSS have employed various methods for fitting sky backgrounds, we opt to avoid the difficulty of reproducing each methodology and simply fit with the known mean sky value. This does not, however, remove the pixel-to-pixel variation in sky brightness, which sets the effective limiting surface brightness in the image.</text>
<text><location><page_7><loc_52><loc_65><loc_92><loc_79></location>We use the GALFIT fits to create a sky- and satellitesubtracted image of the central galaxy in each frame. This image is used to measure various quantities, including alternative non-parametric half-light radii. We also use GALMORPH to fit a single de Vaucouleurs profile to this sky-substracted image. This provides a direct comparison to the methodology used in Hyde & Bernardi (2009b), with the caveat that our use of GALFIT to fit the profiles of satellite galaxies may not match the exact methodology employed in masking nearby sources in SDSS or other surveys.</text>
<section_header_level_1><location><page_7><loc_57><loc_62><loc_87><loc_63></location>3.2. Photometric and Kinematic Measures</section_header_level_1>
<text><location><page_7><loc_52><loc_46><loc_92><loc_61></location>Sizes and luminosities of the central remnants are measured several different ways. The preferred luminosity measure is the total luminosity within the deconvolved model image of the central galaxy, roughly equivalent to model magnitudes in SDSS and other surveys. For comparison, we also measure several other sizes and luminosities, including non-parametric Petrosian radii (Petrosian 1976) (see Abazajian et al. (2004) for the SDSS implementation and Graham et al. (2005) for analysis thereof). A thorough analysis of the suitability of these measures is presented in App. A.</text>
<text><location><page_7><loc_52><loc_23><loc_92><loc_46></location>Kinematical maps are used to measure the velocity distributions - mean velocities (for rotation measures), dispersions, and higher order moments. Generally, we use central dispersions within 1/8 R eff and rotation measures within R eff . Velocity dispersions in the central remnants do vary radially, generally dropping from peak central values. Integral field surveys such as Atlas3D can measure dispersions out to 0.5 to 1 R eff , whereas fiber dispersions from SDSS are measured within fixed angular diameters, and hence variable fractions of R eff . Aperture corrections are often applied to fiber dispersions to convert them to a fixed fraction of R eff , with 1/8 R eff a typical choice for SDSS observations (Hyde & Bernardi 2009a). However, we find that central dispersions are nearly identical to effective dispersions (within 1 R eff ), with most galaxies lying on a linear relation and only a handful of outliers, so aperture corrections are not necessary for the simulations.</text>
<text><location><page_7><loc_52><loc_13><loc_92><loc_22></location>The central velocity dispersions in simulations can be artificially depressed by softening of the gravitational potential. We mask out the central 300 pc to compensate, and measure central dispersions within 1/8 R eff where possible. For the few galaxies where 1/8 R eff is smaller than 300 pc, we enlarge the aperture by factors of 1/8 R eff until a reliable estimate is obtained.</text>
<text><location><page_7><loc_52><loc_3><loc_92><loc_13></location>We have also considered the kinetic energy measure S = √ σ 2 + v 2 , or equivalently S = σ × √ 1 + ( v/σ ) 2 . This is a more accurate measure of the stellar kinetic energy for galaxies with significant rotation. However, most simulated galaxies do not have sufficient rotational support for this correction to be significant, and there are not yet any large samples of galaxies with published dispersions and v/σ to compare to.</text>
<figure>
<location><page_8><loc_8><loc_73><loc_90><loc_88></location>
<caption>Figure 5. Example images of a typical galaxy. The leftmost image is as Fig. 4 but now sky subtracted. The remaining panels show, from left to right, the GALFIT Sersic model residuals ((image - model)/image), and velocity and velocity dispersion maps in units of kms -1 . All images are 29 kpc or 150 SDSS pixels across, as in Fig. 4.</caption>
</figure>
<section_header_level_1><location><page_8><loc_23><loc_67><loc_33><loc_68></location>4. RESULTS</section_header_level_1>
<text><location><page_8><loc_8><loc_56><loc_48><loc_67></location>The main results presented in this paper are the morphologies and kinematics of central group galaxies. Although we do fit satellites as well, this is mainly to exclude their contribution from the profile of the central galaxies. Few satellite galaxies are sufficiently well resolved to recover sizes and Sersic indices accurately, but we only require their total luminosities to be recovered and subtracted from the central galaxy's profile.</text>
<text><location><page_8><loc_8><loc_52><loc_48><loc_56></location>Unless otherwise noted, all radii measured with elliptical annuli are √ ( a × b ), where a and b are the major and minor axis lengths.</text>
<section_header_level_1><location><page_8><loc_17><loc_49><loc_40><loc_51></location>4.1. Observational Comparisons</section_header_level_1>
<text><location><page_8><loc_8><loc_33><loc_48><loc_49></location>Our results are compared to three published data sets for nearby galaxies. The Atlas3D survey (Cappellari et al. 2011)(hereafter A3D) is a volume-limited integral field unit survey of the kinematics of 260 nearby earlytype galaxies. A3D provides kinematical maps (Emsellem et al. 2011) with a pixel size about twice as large as that of SDSS. This is mitigated by the larger aperture and very low redshifts (z < 0.01) of the sample as compared to our nominal mock sample redshift (0.025). Sersic profile fits are also available from Krajnovi'c et al. (2012), with photometry from a variety of sources but typically comparable to or better than SDSS.</text>
<text><location><page_8><loc_8><loc_13><loc_48><loc_33></location>Simard et al. (2011)(hereafter S+11) published three different profile fits for over a million SDSS galaxies. We use the single Sersic decompositions for direct comparison and the free Sersic (bulge) plus exponential (disk) decompositions for diagnostic purposes. Although these fits were performed with a different code - GIM2D (Simard et al. 2002) - the fitting procedure is similar to our GALFIT fits. We select galaxies with spectroscopic redshifts 0 . 01 < z < 0 . 3 to ensure availability of a reliable V max volume correction term. We use the logarithmic median velocity dispersion between two sources - the SDSS DR7 and Princeton measurements (both included in DR7, Abazajian et al. (2009)). Stellar masses are based on the MPA-JHU DR7 catalog 1 , using fits to the multi-band photometry.</text>
<text><location><page_8><loc_8><loc_5><loc_48><loc_13></location>Detailed visual classification of of nearly 6,000 earlytype galaxies from SDSS with z < 0.1 is provided by Nair &Abraham (2010)(hereafter N+10). Volume corrections are applied with the standard 1/V max weighting scheme (Schmidt 1968). Profile fits from this catalog include Petrosian sizes from the SDSS pipeline (Stoughton et al.</text>
<table>
<location><page_8><loc_52><loc_57><loc_88><loc_65></location>
<caption>Table 2 Morphological types of cuts used for S+11 sample</caption>
</table>
<text><location><page_8><loc_52><loc_51><loc_92><loc_57></location>Note . - Each row lists the breakdown of visual morphological classifications from N+10 of each of the subsamples from S+11, which are based on empirical cuts on various parameters rather than visual classification. Empirical cuts generally produce complete but impure samples of ellipticals and spirals, with substantial contamination by S0s.</text>
<text><location><page_8><loc_52><loc_42><loc_92><loc_49></location>2002) and Sersic fits from S+11. Although the original catalog of N+10 contained over 14,000 galaxies, eliminating bad fits and unmatched/misclassified objects provides just over 11,000 galaxies, of which nearly 5,000 are early-types.</text>
<text><location><page_8><loc_52><loc_4><loc_92><loc_42></location>We exclude all SDSS galaxies with extreme velocity dispersions ( σ < 20 kms -1 or σ > 400 kms -1 ) or effective radii smaller than 0.3 kpc. Where visual classifications are available (A3D, N+10) we select galaxies with Hubble T-types (de Vaucouleurs 1959) less than 0 as earlytypes. T-types less than -3 are included in the elliptical sample while the remainder are classified as S0s. The majority of the S+11 sample does not have visual classifications, other than the small subset classified by N+10. We adopt a series of empirical cuts similar to those of Dutton et al. (2011) to identify early-type galaxies, testing these against the N+10 subset. The early-type sample contains galaxies with n s > 1, and - from the disk plus free n s fits - r-band bulge to total luminosity ratio B/T r > 0 . 4, disk inclination less than 63 degrees, and bulge r eff > 0 . 5kpc. Early-types must also have a spectroscopic eclass value less than -0.1 (see Yip et al. (2004), but note that the sign convention in SDSS is opposite), which selects galaxies with spectra consistent with a passive population. This early-type sample is subdivided into an elliptical subset, which imposes further cuts based on the single Sersic fits: g-band image smoothness S2 < 0 . 08, or g-band image smoothness S2 < 0 . 12 and B / T r < 0 . 6. These cuts are similar to those suggested by Simard et al. (2009) to select early-type galaxies from morphology alone, but also serve to decrease contamination from S0s and early-type spirals in the elliptical sample. All galaxies not classified as early-type but meeting the dispersion and R eff cut are identified as spirals.</text>
<text><location><page_8><loc_53><loc_2><loc_92><loc_4></location>The samples obtained by applying these cuts to the</text>
<text><location><page_9><loc_8><loc_69><loc_48><loc_87></location>N+10 catalog are listed in Tab. 2. The elliptical sample is 86% complete. While it is only 61% pure, the contamination mainly comes from S0s and not spirals. No cuts appear to be able to reliably classify S0s, which contaminate both elliptical and spiral samples. In principle, we could instead use the S+11 cuts on the N+10 sample rather than relying on visual classifications at all; however, visual classifications are repeatable and fairly robust (see Nair & Abraham (2010) for comparisons to previous classifications), and as seen in A3D, there are significant differences in rotational support between the elliptical and S0 population (Krajnovi'c et al. 2012), even if no automated morphological classification can separate them.</text>
<text><location><page_9><loc_8><loc_47><loc_48><loc_69></location>Additional weightings are necessary to compare these catalogs to our own simulations, which probe a range of about 5 in absolute magnitude and have a nearly flat luminosity function. We produce r-band luminosity functions for each sample, then weight by the ratio of the simulated luminosity function to the observed one. Elliptical and S0 subsamples use all simulated galaxies versus E/S0 classifications from observed catalogs - i.e., we do not morphologically classify simulated merger remnants. This weighting procedure turns each observational sample into a catalog with equal numbers of galaxies at each luminosity, directly comparable to our simulations. Although the weightings are not vital for tight scaling relations like the fundamental plane, they are necessary for fair comparisons of weaker correlations and histograms marginalizing over luminosity.</text>
<section_header_level_1><location><page_9><loc_23><loc_45><loc_34><loc_47></location>4.2. Morphology</section_header_level_1>
<text><location><page_9><loc_8><loc_29><loc_48><loc_45></location>As detailed in § 3.1, the central galaxies are fit with a single Sersic profile. Each profile has six free parameters (in addition to the two coordinates for the centre of the galaxy): the Sersic index n s , an effective half-light radius r eff , a surface brightness at this radius, an ellipticity glyph[epsilon1] , a position angle, and a boxiness parameter C0 modifying the shape of the ellipse from diamond (negative C0) to box-shaped (positive C0). Ellipticals have long been known to be best fit by larger Sersic indices than disks, to have small ellipticities and several correlations between size, luminosity and Sersic index. Any satellite galaxies in the image are also fit with a single Sersic profile.</text>
<section_header_level_1><location><page_9><loc_22><loc_27><loc_36><loc_28></location>4.2.1. Sersic Indices</section_header_level_1>
<text><location><page_9><loc_8><loc_2><loc_48><loc_26></location>Figure 6 shows various histograms of the Sersic index distribution for the B.n s =1, B.n s =4 and B.n s =all samples. Each individual bulge type produces a narrow distribution of Sersic indices. The B.n s =4 sample's distribution is narrower and peaked at a larger value of n s = 5 than the observational distributions. The B.n s =1 sample's peak at n s = 3 is significantly lower than those of N+10 and A3D ellipticals, and the distribution is narrower still than that of B.n s =4. The combined B.n s =all sample's n s distribution is nearly bimodal due to this separation and approximately twice as broad as B.n s =1 alone. By contrast, most observed distributions are unimodal, although the S+11 distributions show a larger peak of high n s galaxies, which is only reproduced in the B.n s =4 sample. There is also a hint of bimodality in the S0 distributions, which we have diminished by setting a lower limit of n s = 1. The peak of the S0 distribution is best reproduced by the B.n s =1, but, as will be</text>
<text><location><page_9><loc_52><loc_82><loc_92><loc_87></location>demonstrated in § 4.4, the remnants' rotational support is far lower than that of typical S0s. None of the simulation samples can reproduce the width of the observed S0 distributions.</text>
<text><location><page_9><loc_52><loc_66><loc_92><loc_82></location>Although each of the B.n s =1 and B.n s =4 samples are individually a poor fit to the elliptical data - particularly being too narrow of a distribution - the naive linear combination of the two (B.N s =all) provides a better match. The B.n s =all sample is also a better match to the elliptical distributions than the S0, the latter of which tend to smaller Sersic indices. While it is not a particularly realistic distribution - assuming that half of the groups in the universe contain galaxies with only exponential bulges while the other half contain de Vaucouleurs bulges - we will elaborate on the implications for more realistic bulge profile distributions in § 5.</text>
<text><location><page_9><loc_52><loc_51><loc_92><loc_66></location>The difference in Sersic index between the Many- and Few-merger subsamples is small in the L.F.-sampled case but is maximized at about 0.5 for equal-mass mergers. Furthermore, the distributions of the Few-, equal-mass merger remnants in the different bulge samples are sufficiently narrow that the combined B.n s =all, Few-merger subsample is distinctly bimodal. Thus, it appears that multiple mergers are sufficient to broaden the distributions of remnant Sersic indices, but sampling progenitors from a realistic luminosity function can accomplish the same purpose, even with relatively few mergers.</text>
<text><location><page_9><loc_52><loc_37><loc_92><loc_51></location>Major axis projections of central remnants have systematically larger Sersic indices than the medial or minor axis projections (bottom left and middle panels of Fig. 6), with the peak of the distribution shifted by about 1. Medial and minor axes have nearly identical distributions, even though their ellipticities and semi-major axes are not necessarily the same. As Fig. 7 shows, the variation in Sersic index for a single galaxy over different viewing angles is not usually much larger than one (and often smaller), so projections aligned near the major axis appear to produce the largest n s profiles.</text>
<text><location><page_9><loc_52><loc_16><loc_92><loc_37></location>Only B.n s =4 mergers produce a correlation between luminosity and n s , as shown in Fig. 7. This is partly a result of more massive ellipticals being produced by more mergers. In both B.n s samples, Many-merger remnants tend to have larger n s at fixed luminosity. However, in the B.n s =4 sample, even the Few-merger subsample shows a small positive slope in Sersic index, whereas the trend is flat or even slightly negative for B.n s =1. The overall trend is dependent both on the initial bulge profile and the number of mergers. A positive dependence of merger rate on halo mass is a prediction of ΛCDM (e.g. Hopkins et al. (2010)). Exponential bulges, however, are simply not concentrated enough to create merger remnants with n s > 4, even with repeated merging. Thus, luminous ellipticals are unlikely to be the product of only exponential bulge mergers.</text>
<text><location><page_9><loc_52><loc_4><loc_92><loc_15></location>The degree of agreement between simulations and observations is difficult to judge, since the observational samples do not completely agree. The N+10 n s -L relation appears to flatten above 10 10 L glyph[circledot] . This could be due to the larger redshift range of the S+11 sample; however, we find that GALFIT-derived Sersic fits to mock images at higher redshift tend to fit lower Sersic indices, so this systematic trend would have to be reversed in observed ellipticals to explain the shift.</text>
<figure>
<location><page_10><loc_8><loc_43><loc_92><loc_88></location>
<caption>Figure 6. Sersic indices of central ellipticals. The top row shows comparisons to observed elliptical galaxies. The bottom right panel shows distributions for equal mass mergers as well, as compared to observed S0 galaxies. The bottom left and middle panels show principal axis projections only. Histograms are offset slightly on the x-axis to prevent overlap. Simulation data are shown with several maximum n s - 8 (dashed line) for comparison with N+10 and S+11, and no limit (solid line) for the simulations themselves. B.n s =4 mergers match observed ellipticals best, but a range of bulge types appears to be required to reproduce the observed distributions.</caption>
</figure>
<section_header_level_1><location><page_10><loc_22><loc_35><loc_35><loc_36></location>4.2.2. Ellipticities</section_header_level_1>
<text><location><page_10><loc_8><loc_12><loc_48><loc_33></location>Ellipticities of the remnants are on the average slightly larger than observed elliptical samples but lower and more sharply peaked than S0s. Fig. 8 shows that there is only a small difference between the Many- and Fewmerger subsamples, while there is about a 0.05 shift towards rounder remnants from the B.n s =4 to B.n s =1 samples. On the whole the distributions are not unreasonable, lying closer to observations of ellipticals than of S0s, while lacking the tail of highly elliptical shapes found in S0s. Although the Many-merger remnants are slightly rounder on average than the Few-merger, the difference is not large even in equal-mass mergers. This is somewhat surprising, considering that the progenitor galaxy orbits are nearly isotropic and should tend to produce spheroidal remnants as the number of mergers increases. We will elaborate on this point further in § 5.</text>
<text><location><page_10><loc_8><loc_2><loc_48><loc_12></location>The intrinsic ellipticities of the remnants along the principal axis projections are also shown in the bottom left and middle panel of Fig. 8. The distributions are consistent with the remnants being triaxial, with the median value in each projection being both different than the others and greater than zero. The smallest axis ratios are found for the minor axis projection, which would be</text>
<text><location><page_10><loc_52><loc_29><loc_92><loc_36></location>the case for ellipsoids closer to prolate than oblate. Most galaxies have a medial axis ellipticity of around 0.4, with few being rounder than 0.2, indicating that almost all galaxies have a significantly shorter minor axis than the major axis.</text>
<text><location><page_10><loc_52><loc_12><loc_92><loc_29></location>In addition to having larger Sersic indices, brighter galaxies trend toward smaller ellipticities and rounder shapes (Fig. 9). This trend might be expected for more luminous galaxies with many mergers. If the orbits of the merging galaxies are isotropically distributed, the resulting remnant should be close to spherical. Such a trend is present in the simulations, although it appears stronger for the B.n s =4 sample. Much of the scatter in the relation appears to be due to projection effects of the inherently triaxial simulated galaxies, although median ellipticities show significant scatter as well. The B.n s =1 sample also appears to have few very round ( glyph[epsilon1] < 0 . 1) remnants, especially at low luminosities.</text>
<section_header_level_1><location><page_10><loc_64><loc_10><loc_80><loc_11></location>4.3. Scaling Relations</section_header_level_1>
<section_header_level_1><location><page_10><loc_55><loc_7><loc_89><loc_9></location>4.3.1. Size-Luminosity/Stellar Mass and Kormendy Relations</section_header_level_1>
<text><location><page_10><loc_52><loc_2><loc_92><loc_6></location>Fig. 10 shows the Sersic model size-luminosity relation for principal axis projection of simulated galaxies after 10.3 Gyr, connecting otherwise identical groups with dif-</text>
<figure>
<location><page_11><loc_8><loc_53><loc_49><loc_87></location>
<caption>Figure 7. Sersic indices of central ellipticals by galaxy luminosity. Classical-bulge mergers (right) have larger n s for the same initial conditions and show a strong dependence of n s on galaxy luminosity, as with observed ellipticals from S+11 but unlike exponential bulge mergers (left) and N+10 ellipticals. Different projections of the same group are shown as lines of best fit for clarity, with a single point marking the median projection. The length of the line shows the range of values from 10 random projections. Perpendicular lines cross at the 25th and 75th percentiles, with a length equivalent to the r.m.s. dispersion of points perpendicular to the line of best fit.</caption>
</figure>
<figure>
<location><page_11><loc_52><loc_53><loc_90><loc_87></location>
</figure>
<text><location><page_11><loc_50><loc_73><loc_51><loc_74></location>n</text>
<text><location><page_11><loc_50><loc_72><loc_51><loc_72></location>c</text>
<text><location><page_11><loc_50><loc_71><loc_51><loc_72></location>i</text>
<text><location><page_11><loc_50><loc_71><loc_51><loc_71></location>s</text>
<text><location><page_11><loc_50><loc_70><loc_51><loc_71></location>r</text>
<text><location><page_11><loc_50><loc_69><loc_51><loc_70></location>e</text>
<text><location><page_11><loc_50><loc_69><loc_51><loc_69></location>S</text>
<text><location><page_11><loc_8><loc_36><loc_48><loc_45></location>ferent spiral bulges. All relations have very small scatter. Part of the scatter is caused by the B.n s =1 sample having smaller sizes (a real effect) and lower luminosities (partly a real effect, but largely systematic, as will be shown in App. A). Regardless, both projection effects and different progenitor bulge profiles contribute to the scatter in the relation.</text>
<text><location><page_11><loc_8><loc_10><loc_48><loc_36></location>Table 3 lists best-fit Sersic model size-luminosity relations for simulations and observations alike, obtained by least-squares minimization of the orthogonal scatter. In all of the simulation samples, the scatter is relatively small at about 0.1 dex. The scatter does not appear to be mainly due to projection effects or combining progenitors. Fits to major axis projections have similar scatter to the ten equidistant but randomly aligned projections. Similarly, though some groups show projectiondependent sizes and luminosities, these variations are smaller than the scatter in median values, and are likely a result of the mild correlation between Sersic index and luminosity of projections of the same galaxy (evidenced in Fig. 7). If sizes and luminosities are generally accurate to within 10-20% or 0.04-0.08 dex, as suggested by our testing, then some of the scatter could be intrinsic. The scatter in the unweighted simulation data is comparable to that in observed ellipticals (slightly larger than N+10), while the slope is considerably shallower and the intercepts larger.</text>
<text><location><page_11><loc_8><loc_3><loc_48><loc_9></location>Separate fits to the Many- and Few-merger subsamples show a large difference of 0.05 to 0.1 in slope. Also, as Fig. 11 demonstrates, the Many-merger subsample is larger at fixed luminosity than the few merger sample. Thus, the slope of the predicted relation can be</text>
<text><location><page_11><loc_52><loc_33><loc_92><loc_45></location>maximized by giving a larger weight to luminous, Manymerger remnants (and a smaller weight to faint galaxies), while applying the opposite weighting to groups of relatively few galaxies, such that their weights are largest at low luminosities. We apply such a weighting in Tab. 3 and find that it can steepen the slope of the sizeluminosity relation further than even the Many-merger subsample alone, bringing it close to observed values for N+10 but still short of S+11 and A3D.</text>
<text><location><page_11><loc_52><loc_21><loc_92><loc_33></location>Table 3 also lists values for observational data, with both 1/V max corrections and optional weighting to match the luminosity function of the simulations. This weighting scheme only makes a significant difference in the S+11 sample - otherwise, most scaling relations are insensitive to weighting method, as one would expect if they are truly linear with uniform scatter. Some curvature may exist at the low- or high-luminosity extremes, but it is unclear whether it is real or systematic.</text>
<text><location><page_11><loc_52><loc_6><loc_92><loc_21></location>The Petrosian R 50 size-luminosity relation (Tab. 4) shows smaller scatter than Sersic sizes, despite the fact that uncorrected Petrosian half-light radii systematically underestimate the luminosities of pure Sersic profiles and simulated galaxies alike. This is especially true for the B.n s =4 sample, which has slightly lower scatter than B.n s =1 mergers, despite having greater systematic errors on R eff due to its larger mean n s . The slopes are still shallower than those observed in N+10, but the difference can shrink to less than 0.05 if considering weightings for both simulations and observations. The implications of these results will be discussed further in § 5.</text>
<text><location><page_11><loc_52><loc_3><loc_92><loc_5></location>The best-fit relations between size and stellar mass for the S+11 and N+10 catalogs are listed in Tab. 5.</text>
<figure>
<location><page_12><loc_8><loc_43><loc_92><loc_87></location>
<caption>Figure 8. Ellipticities of central ellipticals. Top panels show randomly oriented, evenly spaced projections of various models compared with observed ellipticals. Bottom left and middle panels show principal axis projections only, while the bottom right panel shows equal mass mergers and compares with observed S0s. Ellipticities of central galaxies are largely consistent with the observed distributions, though slightly more flattened on average. Simulated galaxies are intrinsically triaxial, with the minor axis projection being the roundest on average. The remnants are rounder than observed S0s and lack a tail of highly flattened ( glyph[epsilon1] > 0.5) objects.</caption>
</figure>
<text><location><page_12><loc_23><loc_43><loc_24><loc_44></location>/epsilon1</text>
<text><location><page_12><loc_51><loc_43><loc_52><loc_44></location>/epsilon1</text>
<text><location><page_12><loc_79><loc_43><loc_80><loc_44></location>/epsilon1</text>
<text><location><page_12><loc_8><loc_22><loc_48><loc_36></location>The slopes are slightly shallower than those for the sizeluminosity relations and closer to (but not quite matching) those predicted by the simulations, which do not have significant variations in the stellar mass-to-light ratio. Thus some of the tension between the slopes of the simulated and observed size-luminosity relations can be resolved by accounting for the variable stellar mass-tolight ratio of observed galaxies, which increases in more luminous observed ellipticals but is nearly constant by construction in the simulated remnants.</text>
<text><location><page_12><loc_8><loc_3><loc_48><loc_22></location>The Petrosian size-stellar mass relation shows slightly shallower slope, as with Sersic models. In fact, the slope and scatter of the weighted simulations (0.52 and 0.09) are within the quite small bootstrap errors (0.01) of the weighted observations (0.51 and 0.09), while the intercept is higher (-4.81 versus -5.08) but still also within the more generous error bars. Thus, it is entirely possible to match the slopes, and, to a lesser extent, the intercepts of the size-mass relation, depending on the fitting technique and sample weights. However, this alone does not justify either weighting scheme. The observational scheme is reasonable, since matching luminosity functions is necessary in order to make a fair comparison. The simulation scheme is not as well justified, since the number of mergers per group is somewhat arbitrary.</text>
<text><location><page_12><loc_52><loc_8><loc_92><loc_36></location>The Kormendy relation (Kormendy 1977), shown in Fig. 12, has large scatter and shallow slope, especially for the B.n s =1 relation, which is nearly flat. None of the observed relations are quite linear. While the kink at small sizes is likely a systematic artifact, the curvature near 5-6 kpc appears more robust and also more significant than the equivalent curvature in the size-luminosity relation. As in the size-luminosity relation, it appears as if the simulated galaxies are either too faint for their size or too large to be so faint. Interestingly, the relation for large ellipticals appears to asymptote towards the slope of constant luminosity (d log( µ e ) / dlog(R eff ) = 5), which suggests that bright ellipticals can grow significantly in size without adding a large amount of stellar mass. In fact, many BCGs have exceptionally large effective radii and faint mean surface brightnesses. However, most of the similar simulated remnants in Fig. 10 are mergers of many equal-mass spirals, rather than luminosity function-sampled remnants - without this M-Eq subsample, the simulated Kormendy relation is rather weak, especially for the B.n s =1 sample.</text>
<section_header_level_1><location><page_12><loc_62><loc_5><loc_82><loc_7></location>4.3.2. Faber-Jackson Relation</section_header_level_1>
<text><location><page_12><loc_52><loc_2><loc_92><loc_5></location>The Faber-Jackson (velocity dispersion-luminosity, Faber & Jackson (1976)) relation is shown in Fig. 13,</text>
<figure>
<location><page_13><loc_8><loc_53><loc_90><loc_87></location>
<caption>Figure 9. Ellipticities of central ellipticals by galaxy luminosity. Median ellipticities tend to decrease slightly with luminosity, both in simulations and observations, while B.n s =4 mergers are slightly rounder on average. Projection effects for single galaxies are quite large, with spreads of 0.2-0.3 in ellipticity being common. Line types are as in Fig. 7.</caption>
</figure>
<figure>
<location><page_13><loc_8><loc_16><loc_49><loc_48></location>
<caption>Figure 10. The size-luminosity relation of merger remnants after 10 Gyr. Each point shows one of the principal axis projections. Light (green) lines connect different projections of the same galaxy. Darker (red and black) lines connect the same projection for groups with different progenitor bulge profiles but otherwise identical initial conditions. The light (green) lines can be viewed as contributions to scatter in the relation from projection effects, while the darker lines show differences from progenitor galaxies.</caption>
</figure>
<text><location><page_13><loc_52><loc_22><loc_92><loc_47></location>with best fits tabulated in Tab. 7. The simulated relations have slopes fairly close to the observations, though the intercepts are significantly lower. The turnover or curvature at low velocity dispersions ( < 100 kms -1 ) is likely not entirely real, since such low dispersions are near the spectrograph's resolution limit and unlikely to be reliable (Aihara et al. 2011). The luminosity function weightings make a significant difference in slope for the N+10 sample, which is likely due to this same curvature. The scatter appears to be mostly due to projection effects at the low-luminosity end but increases at high luminosities, where the Many- and Few-merger samples appear to diverge. The most robust conclusions from the data are that the slope for the S0 sample is significantly steeper than that for ellipticals, which in turn is slightly steeper than the canonical slope of 0.25, depending on the weighting scheme used. The scatter in the simulated relations is also significantly lower than in the observations, even when both bulge samples are combined.</text>
<text><location><page_13><loc_52><loc_12><loc_92><loc_22></location>Unlike the size-mass/luminosity relations, the velocity dispersion-stellar mass relation (Tab. 8) is hardly changed from the velocity dispersion-luminosity relation, although the scatter shrinks slightly. The velocity dispersion-stellar mass relation also deviates from the canonical Faber-Jackson relation slope of 0.25, showing a scaling closer to σ ∝ M 0 . 3 ∗ .</text>
<section_header_level_1><location><page_13><loc_64><loc_11><loc_81><loc_12></location>4.4. Rotational Support</section_header_level_1>
<text><location><page_13><loc_52><loc_2><loc_92><loc_10></location>We measure v/σ as the luminosity-weighted average within R eff , as used in IFU observations like Atlas3D (Cappellari et al. 2011). Most simulated ellipticals are slow rotators (Fig. 14). However, some projections show v/σ as large as 0.35 and can be classified as fast rotators despite having been formed from dry mergers. Cox et al.</text>
<figure>
<location><page_14><loc_8><loc_51><loc_92><loc_86></location>
<caption>Figure 11. Sersic model size-luminosity relation of central ellipticals after 10 Gyr. The simulated relations have small scatter and a similar slope to observations but are offset slightly, being too large at fixed luminosity. Many merger galaxies appear to be a better fit for luminous ellipticals, whereas few mergers match low luminosity ellipticals better. Line types are as in Fig. 7.</caption>
</figure>
<figure>
<location><page_14><loc_8><loc_8><loc_92><loc_43></location>
<caption>Figure 12. The Kormendy relation of central ellipticals. Only B.n s =4 simulations show a distinct Kormendy relation, but they are also too faint at a fixed size. The observed relation can be better reproduced if few mergers and B.n s =1 are the preferred source of small galaxies and many/B.n s =4 produced large ellipticals. Line types are as in Fig. 7.</caption>
</figure>
<figure>
<location><page_15><loc_8><loc_30><loc_90><loc_64></location>
<caption>Figure 13. The Faber-Jackson relation. Both simulation samples follow a similar slope to observations but are offset to lower dispersions, regardless of which subsample is used. The scatter in simulations is tighter than observations, since the observed relations show 25th, 50th and 75th percentiles. Line types are as in Fig. 7.</caption>
</figure>
<paragraph><location><page_16><loc_17><loc_85><loc_40><loc_87></location>Table 3 Sersic model size-luminosity relations</paragraph>
<text><location><page_16><loc_8><loc_83><loc_47><loc_84></location>Simulations: Ten equally-spaced projections, randomly oriented</text>
<table>
<location><page_16><loc_8><loc_67><loc_47><loc_83></location>
<caption>Table 8</caption>
</table>
<section_header_level_1><location><page_16><loc_8><loc_66><loc_31><loc_66></location>Principal axis projections, unweighted</section_header_level_1>
<table>
<location><page_16><loc_8><loc_59><loc_44><loc_66></location>
</table>
<section_header_level_1><location><page_16><loc_8><loc_58><loc_16><loc_59></location>Observations</section_header_level_1>
<table>
<location><page_16><loc_8><loc_45><loc_48><loc_58></location>
</table>
<text><location><page_16><loc_8><loc_35><loc_48><loc_44></location>Note . - Slopes are given in log space, i.e., for log( R eff ) as a function of log( L ). Simulation data are from analyses after 10.3 Gyr, including various subsamples of randomly oriented (but equally spaced) projections, as detailed in the text, as well as principal axis projections. Observational data for each catalog (Cat.) are 1/V max corrected, with fits optionally weighted (Weight) or not by the difference between the simulated and observed luminosity functions. R.M.S. lists the r.m.s. orthogonal scatter of all points from the best-fit relation.</text>
<paragraph><location><page_16><loc_18><loc_30><loc_39><loc_32></location>Table 4 Petrosian size-luminosity relations</paragraph>
<text><location><page_16><loc_8><loc_28><loc_47><loc_29></location>Simulations: Ten equally-spaced projections, randomly oriented</text>
<table>
<location><page_16><loc_8><loc_11><loc_47><loc_28></location>
</table>
<section_header_level_1><location><page_16><loc_8><loc_11><loc_16><loc_11></location>Observations</section_header_level_1>
<table>
<location><page_16><loc_8><loc_4><loc_47><loc_11></location>
</table>
<text><location><page_16><loc_10><loc_3><loc_38><loc_4></location>Note . - Column definitions are as in Tab. 3.</text>
<table>
<location><page_16><loc_52><loc_75><loc_91><loc_84></location>
<caption>Table 5 Sersic size-stellar mass relations</caption>
</table>
<text><location><page_16><loc_53><loc_74><loc_81><loc_75></location>Note . - Column definitions are as in Tab. 3.</text>
<paragraph><location><page_16><loc_69><loc_70><loc_74><loc_71></location>Table 6</paragraph>
<section_header_level_1><location><page_16><loc_59><loc_69><loc_84><loc_70></location>Petrosian model size-stellar mass relation</section_header_level_1>
<table>
<location><page_16><loc_52><loc_62><loc_92><loc_68></location>
</table>
<text><location><page_16><loc_53><loc_60><loc_81><loc_61></location>Note . - Column definitions are as in Tab. 3.</text>
<paragraph><location><page_16><loc_69><loc_56><loc_74><loc_57></location>Table 7</paragraph>
<text><location><page_16><loc_61><loc_55><loc_83><loc_56></location>Sersic model Faber-Jackson relations</text>
<text><location><page_16><loc_52><loc_54><loc_90><loc_54></location>Simulations: Ten equally-spaced projections, randomly oriented</text>
<table>
<location><page_16><loc_52><loc_37><loc_89><loc_54></location>
</table>
<section_header_level_1><location><page_16><loc_52><loc_36><loc_75><loc_37></location>Principal axis projections, unweighted</section_header_level_1>
<table>
<location><page_16><loc_52><loc_30><loc_88><loc_36></location>
</table>
<section_header_level_1><location><page_16><loc_52><loc_29><loc_60><loc_30></location>Observations</section_header_level_1>
<table>
<location><page_16><loc_52><loc_20><loc_91><loc_29></location>
</table>
<text><location><page_16><loc_53><loc_19><loc_81><loc_20></location>Note . - Column definitions are as in Tab. 3.</text>
<text><location><page_16><loc_60><loc_14><loc_84><loc_15></location>Velocity dispersion-stellar mass relations</text>
<table>
<location><page_16><loc_52><loc_4><loc_91><loc_13></location>
</table>
<text><location><page_16><loc_53><loc_3><loc_81><loc_4></location>Note . - Column definitions are as in Tab. 3.</text>
<figure>
<location><page_17><loc_8><loc_54><loc_50><loc_87></location>
<caption>Figure 16 shows rotational support for the principal axis projections. The minor axis projection shows minimal rotation, which is expected if there are no stable orbits about the major axis. In general, B.n s =4 mergers are rounder despite having faster rotation for the same set of initial conditions. As with random projections, there appear to be two distinct tracks for galaxies, which is more readily apparent in the B.n s =4 mergers. Most galaxies have a range of ellipticities in their major axis projections but only show modest increases in rotational support from the minor to medial axis projections. These appear as horizontal lines with a shallow slope near the bottom of the figure. A smaller subset of galaxies are nearly round in the minor axis projections, with very modest rotation ( λ < 0 . 1), but are significantly flattened ( glyph[epsilon1] > 0 . 2 and rotationally supported ( λ > 0 . 2) in major and medial axes alike. In fact, for most of these</caption>
</figure>
<text><location><page_17><loc_31><loc_54><loc_32><loc_55></location>/epsilon1</text>
<paragraph><location><page_17><loc_8><loc_48><loc_48><loc_53></location>Figure 14. Rotational support of simulated galaxies by classical v/σ measure. Atlas3D ellipticals are shown with areas of points roughly corresponding to their relative weights on a logarithmic scale. Most simulated ellipticals are slow rotators, but some have modest rotational support.</paragraph>
<text><location><page_17><loc_8><loc_38><loc_48><loc_46></location>(2006) found that dry binary mergers only form slow rotators ( v/σ < 0.1), whereas some group mergers are clearly capable of producing fast rotators. Nonetheless, the scarcity of remnants with v/σ > 0 . 3 strongly suggests that dissipation is necessary to form fast rotators, as will be elaborated further in § 5.</text>
<text><location><page_17><loc_8><loc_25><loc_48><loc_38></location>We also measure rotation in Fig. 15 by the more physically motivated measure λ (Cappellari et al. 2011) - essentially a radially-weighted v/σ tracing net projected angular momentum. While the distribution of rotational support is not wildly different from Atlas3D, there is a significant excess of slow rotators (especially flattened ones) and a complete absence of simulated galaxies with λ > 0 . 4. B.n s =1 mergers and Many-merger remnants tend to be slightly slower rotators, but the differences in both cases are not large.</text>
<text><location><page_17><loc_52><loc_82><loc_92><loc_87></location>galaxies it appears as if the minor and medial axes are nearly identical, and so these galaxies are probably prolate spheroids. In this case, the distinction between major and medial axis projection is not very meaningful.</text>
<text><location><page_17><loc_52><loc_70><loc_92><loc_82></location>Rotational support decreases with increasing luminosity in A3D ellipticals but not in simulations, as shown in Fig. 17. This is largely due to the inability of dry mergers to produce fast rotating, faint ellipticals. Furthermore, even if the morphological properties of some remnants (particularly Sersic indices of B.n s =1 remnants) are more consistent with S0s than ellipticals, observed S0s have far more rotational support than the vast majority of the simulated galaxies.</text>
<text><location><page_17><loc_52><loc_55><loc_92><loc_70></location>There does not appear to be any strong correlation between rotational support and number of mergers in Fig. 14, or with total group mass or central galaxy luminosity. One might expect rotational support to at least correlate with net group specific angular momentum, assuming most of the halos merge and this angular momentum is conserved - however, Fig. 18 does not show any such correlation. It appears that repeated, mostly isotropic mergers cannot produce very fast rotators, even if the group itself has some net orbital angular momentum in one or more satellite galaxies.</text>
<section_header_level_1><location><page_17><loc_66><loc_53><loc_78><loc_54></location>5. DISCUSSION</section_header_level_1>
<text><location><page_17><loc_52><loc_41><loc_92><loc_52></location>The main results of § 4 are that collisionless mergers in groups can produce central remnants with properties very similar to nearby elliptical galaxies. However, we do note several key differences between the simulation predictions and observed elliptical galaxies, not all of which are easily reconciled with dissipationless merging. We will also highlight how and why these results differ from previously published simulations.</text>
<section_header_level_1><location><page_17><loc_66><loc_39><loc_78><loc_40></location>5.1. Morphology</section_header_level_1>
<text><location><page_17><loc_52><loc_8><loc_92><loc_38></location>In App. A, it is shown that at the resolutions used in this study, luminosities, sizes and Sersic indices of spherical Sersic model galaxies can be recovered within about 5%, usually underestimating the true values. For group merger remnants, Sersic fits typically recover luminosities and sizes to within 10%, although luminosities tend to be more precisely recovered. By contrast, Petrosian radii systematically underestimate galaxy sizes and luminosities, negating the advantage of a non-parametric fit unless corrected for. Thus we conclude that single Sersic model fits are suitable for the simulated galaxies and can be compared directly to S+11 catalog fits, with the caveat that Sersic indices are the least robust parameter at low resolutions and are likely systematically underestimated. However, it is also true that in practice, Sersic fits can produce larger scatter on size than on luminosity, whereas Petrosian half-light radii appear to limit scatter in sizes - likely because they systematically underestimate the total luminosity of galaxies with large Sersic indices. Given these issues, our solution is to compare sizes between simulations and observations as fairly as possible, so that any systematic errors are likely to be shared between simulated and observed galaxies.</text>
<text><location><page_17><loc_52><loc_2><loc_92><loc_8></location>We have compared Sersic index and ellipticity distribution to single Sersic profile fits of local (A3D) and SDSS (N+10, S+11) galaxies. Neither B.n s sample is a good fit to observed ellipticals alone, but the naive linear com-</text>
<section_header_level_1><location><page_18><loc_37><loc_85><loc_63><loc_87></location>All bulge models</section_header_level_1>
<figure>
<location><page_18><loc_8><loc_23><loc_92><loc_85></location>
<caption>Figure 15. Rotational support of simulated galaxies by dimensionless angular momentum measure λ . Observed data are shown with point sizes proportional to the logarithm of the relative weights to match the luminosity function of the simulated galaxies. Despite this weighting scheme, too many simulated galaxies have low rotational support, and none have very high support ( λ > 0.4).</caption>
</figure>
<text><location><page_18><loc_8><loc_3><loc_48><loc_17></location>bination of the two is a better fit while remaining inconsistent with S0s. However, such a naive combination still produces a near-bimodal distribution, in contrast to the single peak typical of observed ellipticals. A more natural choice of progenitors would likely smooth out this bimodality. For example, groups with half of the spirals having exponential bulges and the other half de Vaucouleurs would likely produce remnants with intermediate properties, filling in the gap between the two peaks of single-progenitor distributions. A smooth, realistic distribution of bulge profiles and bulge fractions</text>
<text><location><page_18><loc_52><loc_15><loc_92><loc_17></location>would likely flatten the peaks and further broaden the distribution of remnant Sersic indices.</text>
<text><location><page_18><loc_52><loc_3><loc_92><loc_15></location>Sersic indices of observed ellipticals are generally larger for more luminous galaxies, a trend reproduced by the simulated galaxies in Fig. 7. Hopkins et al. (2009) predicted that the dissipationless component in mergers (including both binary spiral mergers and some re-mergers of the resulting remnants) should show only a weak increase with luminosity and have low median values of about n s =3, with scatter of about 1. We find similar results for the B.n s =1 sample, for which n s is nearly con-</text>
<figure>
<location><page_19><loc_8><loc_56><loc_92><loc_87></location>
<caption>Figure 16. Rotational support of principal axis projections of simulated galaxies by dimensionless angular momentum measure λ . Different projections of the same galaxy are connected by lines, with dashed lines if the second point is lower on the y-axis than the first. Minor axis projections show very little rotation, while medial and major axis projections have similar amounts of rotational support, with medial axis projections being slightly more flattened.</caption>
</figure>
<figure>
<location><page_19><loc_8><loc_19><loc_49><loc_50></location>
<caption>Figure 17. Rotational support of elliptical galaxies by dimensionless angular momentum measure λ as a function of luminosity. Observational data show a trend of lower rotational support at higher luminosity, whereas the simulated trend is nearly flat. Almost all S0s are faster rotators than simulated remnants.</caption>
</figure>
<text><location><page_19><loc_8><loc_2><loc_48><loc_12></location>stant with luminosity at a mean of 3 and with a range from 2 to 4. By contrast, the B.n s =4 sample not only has larger mean n s at about 5, but the median n s increases with luminosity by ∼ 0.5-1 per dex. This slope is close to that observed for N+10 and shallower than that in S+11, the discrepancy between these two samples having no obvious cause beyond probable contamination by</text>
<text><location><page_19><loc_52><loc_36><loc_92><loc_50></location>S0s in the S+11 elliptical sample. Since the simulations have the same initial conditions other than their bulge profiles, this demonstrates that sufficiently concentrated progenitors can produce remnants with large n s through dissipationless merging. Furthermore, bulge n s =4 mergers appear to be a better fit for luminous ellipticals, while bulge n s =1 remnants match the less luminous ellipticals. If progenitor bulge profiles scale with luminosity (i.e., luminous spirals have larger bulge n s and merge to form luminous ellipticals), the scaling of elliptical Sersic index with luminosity can be matched more closely.</text>
<text><location><page_19><loc_52><loc_13><loc_92><loc_35></location>Both simulations and observations show a slight tendency for more luminous ellipticals to be rounder, especially above 10 11 L glyph[circledot] (Fig. 9). Again, the S+11 sample differs from N+10, in this case being more flattened on average - likely due to S0 contamination. Nonetheless, we also find similar luminosity-dependent behaviour as for Sersic indices, as the B.n s =4 sample is a better match for bright, rounder ellipticals. The B.n s =1 sample has a shallower slope and appears to be too flattened on average. The observed distribution does not obviously require a combination of both simulation samples in the same way as the Sersic index-luminosity relation does, but such a combination does not disagree with the S+11 relation either. The N+10 ellipticity-luminosity relation is considerably flatter and could be reproduced by the B.n s =4 sample alone, with the scatter due to the substantial projection effects.</text>
<section_header_level_1><location><page_19><loc_64><loc_10><loc_80><loc_11></location>5.2. Scaling Relations</section_header_level_1>
<section_header_level_1><location><page_19><loc_61><loc_8><loc_83><loc_10></location>5.2.1. Size-Luminosity Relation</section_header_level_1>
<text><location><page_19><loc_52><loc_2><loc_92><loc_8></location>Despite having randomized initial conditions, the simulated galaxies typically produce tight size-luminosity relations, with slight dependence on which size measure is used and whether different B.n s samples are com-</text>
<figure>
<location><page_20><loc_8><loc_54><loc_90><loc_88></location>
<caption>Figure 18. Rotational support of simulated galaxies as a function of initial group orbital angular momentum per unit mass. For reference, M31's estimated orbit about the Milky Way with a tangential velocity of 30 kms -1 and a distance of 700 kpc would be amongst the smaller values in the sample.</caption>
</figure>
<text><location><page_20><loc_8><loc_31><loc_48><loc_48></location>bined. The Sersic model relations are somewhat tighter ( ∼ 0 . 1 -0 . 12 dex scatter) than those reported by Nair et al. (2011) ( ∼ 0 . 12 -0 . 15 dex scatter). This is partly systematic, since Nair et al. (2011) used circular Sersic fits provided by Blanton et al. (2005). Using the elliptical Sersic model fits of S+11 - which are more directly comparable to our own methodology and overlap with the N+10 sample - yields smaller scatter in the N+10 sizeluminosity relation of ∼ 0 . 09 dex. We also find slightly tighter scatter in the remnant Petrosian size-luminosity relation ( ∼ 0 . 09 dex), whereas the scatter for N+10 ellipticals remains largely unchanged whether Sersic or Petrosian model sizes are used.</text>
<text><location><page_20><loc_8><loc_3><loc_48><loc_31></location>The small scatter in the size-luminosity relation should allay concerns that stochastic merging processes cannot produce tight scaling relations. Nipoti et al. (2009a) used simulations with multiple mergers of spheroidal galaxies to conclude that 'a remarkable degree of fine tuning is required to reproduce the tightness of the local scaling relations with dry mergers'. Instead, we find that mergers of many galaxies typically produce slightly tighter correlations than those with fewer galaxies, and the relations are tight regardless of which formation time is assumed for the groups (App. C). No fine tuning in galaxy orbits, number of mergers or any other parameters are required to produce tight scaling relations. Moreover, the Faber-Jackson relation has even tighter scatter than the size-luminosity relation. Rather than scattering galaxies away from existing scaling relations, multiple mergers appear to converge remnants towards a common relation, a behavior somewhat like the central limit theorem. However, it is still true that dry mergers of spirals in groups produce remnants with larger sizes and smaller velocity dispersions at fixed mass or luminosity, a problem shared</text>
<text><location><page_20><loc_52><loc_39><loc_92><loc_48></location>with mergers of spheroids (e.g. Nipoti et al. 2003; BoylanKolchin et al. 2006; Nipoti et al. 2009b). Also, the scatter does appear to increase slightly with luminosity. This could simply be a reflection of the wide range of galaxy and merger counts for the luminous groups, which may not match the true range of cosmological merger histories for galaxy groups.</text>
<text><location><page_20><loc_52><loc_19><loc_92><loc_39></location>We have tested mergers of spirals following a zeroscatter Tully-Fisher relation. The estimates for the scatter of merger remnant scaling relations can be considered lower limits, as they would likely have been higher had progenitors followed Tully-Fisher relations with intrinsic scatter and/or evolving slope and scatter. The observed Tully-Fisher relation does have significant scatter, even at low redshift (about 0.12 dex, from Courteau et al. (2007)), but the intrinsic scatter could be much lower. Hopkins et al. (2008) estimate that a scatter of 0.1 dex in the Tully-Fisher relation contributes about 0.04 dex scatter in the Fundamental Plane scaling relation; comparable scatter added to the existing size-luminosity relation scatter of 0.10-0.12 dex would make little difference if added in quadrature.</text>
<text><location><page_20><loc_52><loc_3><loc_92><loc_19></location>While limiting scatter does not appear to be a challenge, in almost all cases the slope of the size-luminosity relation is shallower than observed and the intercept larger, so most galaxies are too large for their luminosities. The slopes of the remnant size-luminosity relations (typically R ∝ L 0 . 5 -0 . 6 ) are steeper than the progenitor spiral scaling relation ( R ∝ L 0 . 42 ) and the group scaling relation ( ρ =constant, R ∝ L 1 / 3 ). However, the remnants slopes are still shallower than those for the observations, which range from R ∝ L 0 . 6 to R ∝ L 0 . 8 depending on the observational sample and size measure. Encouragingly, the best matches are found between simulated remnants</text>
<text><location><page_21><loc_8><loc_85><loc_48><loc_87></location>and N+10 ellipticals ( R ∝ L 0 . 66 ), the largest sample for which visual classifications are available.</text>
<text><location><page_21><loc_8><loc_70><loc_48><loc_85></location>The steeper slope of the S+11 elliptical Sersic sizeluminosity relation (0.75 to 0.78) is of some concern. However, the elliptical classification for S+11 is based on empirical cuts on various parameters and results in significant ( ∼ 30%) contamination by S0s (Tab. 2). The much smaller A3D elliptical sample also has a slightly larger slope than N+10, and the luminosity function weighting does not change the slope. Since A3D used a slightly different fitting methodology with a much smaller volume sample, it is not clear whether this discrepancy is significant.</text>
<text><location><page_21><loc_8><loc_57><loc_48><loc_70></location>The size-luminosity relation slopes for the simulated remnants are also steeper than that of R ∝ L ∼ 0 . 3 predicted for binary mergers remnants by Hopkins et al. (2009). However, those simulations began with a spiral scaling relation of similar slope (0.3), and so the merging process did not steepen the size-mass relation. By contrast, we have shown that group mergers are capable of steepening the slope of the size-luminosity relation by ∼ 0 . 1 -0 . 2 from progenitors to merger remnants without dissipation.</text>
<text><location><page_21><loc_8><loc_32><loc_48><loc_57></location>Our models predict virtually no dependence of stellar mass-to-light ratio on luminosity - while the bulge and disk stellar mass-to-light ratios have different values, the fraction of disk stars within the effective radius varies little. However, luminous ellipticals do tend to have larger stellar mass-to-light ratios, so comparing to observed size-stellar mass relations lessens the discrepancies in the slopes by about 0.05 dex, depending on the sample and size measure. Such a dependence could be produced by more massive progenitor spirals having larger mean stellar mass-to-light ratios. We also did not include any scatter in the progenitor spiral Tully-Fisher relation or any scatter or luminosity dependence in bulge fractions. Extra scatter in either of these input galaxy properties would likely result in increased scatter in the remnant scaling relations. Any realistic luminosity dependence in the large M31 model bulge fraction would likely flatten the slope still further, since faint ellipticals would be produced by faint spirals with weak bulges.</text>
<text><location><page_21><loc_8><loc_18><loc_48><loc_31></location>Dissipation is a tempting solution to the shallow sizeluminosity relation slope problem. Dissipation should decrease sizes at fixed luminosities and preferentially shrink faint ellipticals if their progenitors had larger gas fractions, resulting in a remnant with a larger fraction of stars formed in a central starburst. Luminositydependent gas fractions have been proposed by Robertson et al. (2006); Hopkins et al. (2008) as the source of the tilt in the fundamental plane scaling relation, a hypothesis which will be addressed in Paper II.</text>
<text><location><page_21><loc_8><loc_2><loc_48><loc_18></location>Another possible remedy to the shallower slopes of the simulated size-luminosity relations is to weight the contributions from various simulation subsamples differently. Applying a simple linear weighting scheme of favoring B.n s =1 groups at low luminosity and B.n s =4 at high luminosities yields a steeper slope than a uniform weighting and a closer match with observations. Such a weighting also produces steeper slopes than either the Few- or Many-merger relations alone and can be justified if more massive halos undergo more mergers. While average halo merger rates are not strongly mass dependent (Stewart et al. 2008; Fakhouri et al. 2010), the groups</text>
<text><location><page_21><loc_52><loc_85><loc_92><loc_87></location>we have simulated here would likely be those with higher than average merger rates.</text>
<text><location><page_21><loc_52><loc_63><loc_92><loc_85></location>Although these schemes could resolve the mismatch in slopes, none save dissipation are viable solutions to the problem that simulated remnants are generally too large at fixed luminosity (Fig. 11). Our estimated stellar mass-to-light ratios are already quite low, so making small galaxies brighter appears to be out of the question. Numerical resolution effects are not large (App. B). Barring a strong redshift dependence in the sizes of observed ellipticals, this discrepancy is real. As a result, the Kormendy relation (Fig. 12) is poorly reproduced. Remnants are too faint at fixed sizes, so their effective surface brightnesses are also too faint by about a magnitude for small galaxies. The shallower slopes of the size-mass relation also translate into a weak simulated Kormendy relation (nearly non-existent in the case of the B.n s =1 sample).</text>
<section_header_level_1><location><page_21><loc_62><loc_61><loc_82><loc_63></location>5.2.2. Faber-Jackson Relation</section_header_level_1>
<text><location><page_21><loc_52><loc_41><loc_92><loc_61></location>The Faber-Jackson relation of simulated galaxies shows even smaller scatter (0.04 dex) than their size-luminosity relation or any observed Faber-Jackson relation (typically 0.08 dex, as in Fig. 13 and Tab. 7). The simulated remnants also have slightly shallower slopes ( σ ∝ L 0 . 28 ) than the observations ( σ ∝ L 0 . 27 -0 . 37 ), again depending on sample and weighting scheme. Curiously, the slope of the Faber-Jackson relation is nearly identical to that of the progenitor spiral Tully-Fisher relation ( V ∝ L 0 . 29 ), so multiple mergers appear to preserve the scaling of orbital velocity with mass while converting ordered rotation into random motions. This is despite the fact that the virial ratio in each group varies significantly, and so galaxy orbits within each group are not scaled uniformly the same way that stellar orbits within galaxies are.</text>
<text><location><page_21><loc_52><loc_29><loc_92><loc_41></location>In virtually all cases, the slope of the remnant FaberJackson relation is steeper than the canonical value of 0.25 (or L ∝ σ 4 ). However, the observed relations show similar deviations and there is no compelling reason why ellipticals should follow this canonical relation. Indeed, the simulations of Boylan-Kolchin et al. (2006) predict scalings as steep as M ∝ σ 12 for major mergers with very small pericentric distances, so such deviations from the canonical relation are not unexpected.</text>
<text><location><page_21><loc_52><loc_2><loc_92><loc_29></location>In most samples, the simulations have smaller dispersions than observed galaxies of the same luminosity. No weighting scheme can resolve this mismatch in the intercept of the Faber-Jackson relation, which is of similar magnitude (but opposite sign) as the offset in the intercept of the size-luminosity relation. Increasing the stellar mass-to-light ratios of the simulations would make galaxies of the same dispersion fainter but would worsen the match to the size-luminosity relation by making small remnants even fainter. Dissipation appears to be necessary here - central starbursts have been shown to increase velocity dispersions and shrink effective radii compared to purely dissipationless mergers (Hopkins et al. 2009). However, it is not clear whether a mass-dependent gas fraction would preserve the slope or flatten it. The mild curvature in the observational relations may be a systematic effect at low dispersions, although we have attempted to minimize such systematics by including two independent dispersion measurements. On the other hand, the simulated relations are insensitive to the choice</text>
<text><location><page_22><loc_10><loc_86><loc_10><loc_87></location>0</text>
<text><location><page_22><loc_10><loc_86><loc_11><loc_87></location>.</text>
<text><location><page_22><loc_11><loc_86><loc_12><loc_87></location>00</text>
<text><location><page_22><loc_13><loc_86><loc_14><loc_87></location>≤</text>
<text><location><page_22><loc_14><loc_86><loc_15><loc_87></location>B</text>
<text><location><page_22><loc_15><loc_86><loc_16><loc_87></location>/</text>
<text><location><page_22><loc_16><loc_86><loc_17><loc_87></location>Tr</text>
<text><location><page_22><loc_18><loc_86><loc_19><loc_87></location>></text>
<text><location><page_22><loc_19><loc_86><loc_20><loc_87></location>0</text>
<text><location><page_22><loc_20><loc_86><loc_20><loc_87></location>.</text>
<text><location><page_22><loc_20><loc_86><loc_28><loc_87></location>15 ,N=2.7e4</text>
<figure>
<location><page_22><loc_8><loc_77><loc_27><loc_86></location>
<caption>0 . 30 ≤ B / Tr > 0 . 45 ,N=4.8e4</caption>
</figure>
<figure>
<location><page_22><loc_8><loc_64><loc_27><loc_75></location>
</figure>
<text><location><page_22><loc_30><loc_86><loc_48><loc_87></location>0 . 15 ≤ B / Tr > 0 . 30 ,N=4.8e4</text>
<figure>
<location><page_22><loc_30><loc_77><loc_48><loc_86></location>
<caption>0 . 45 ≤ B / Tr > 1 . 00 ,N=4.6e4</caption>
</figure>
<figure>
<location><page_22><loc_30><loc_64><loc_48><loc_75></location>
<caption>Figure 19. Relations between total luminosity and best-fit bulge Sersic index as a function of r-band bulge-to-total luminosity ratio in S+11 spiral galaxies. Plots show probability density on a logarithmic scale, with dark red the highest and blue the lowest densities. Only galaxies with a distinct bulge component are included. All galaxies have an F-test probability that a bulge component is not required for a good fit of less than 0 . 32. Each galaxy is weighted by 1/V max to correct for incompleteness.</caption>
</figure>
<text><location><page_22><loc_8><loc_49><loc_48><loc_53></location>of velocity dispersion measure (central or effective; including rotational support or not) and most observed relations are insensitive to various weighting schemes.</text>
<section_header_level_1><location><page_22><loc_16><loc_47><loc_41><loc_48></location>5.2.3. Time or Redshift Dependence</section_header_level_1>
<text><location><page_22><loc_8><loc_27><loc_48><loc_46></location>All of the results presented above apply to simulations analyzed after 10.3 Gyr, assuming an initial formation redshift of z=2.0 - a redshift at which pure dry mergers of disks are not likely to be common. However, the first merger in the group typically only occurs after another 1-2 Gyr. The scaling relations of remnants after 5 and 7.7 Gyr are similar to those at 10.3 Gyr, as shown in App. C, and so similar conclusions would be reached by assuming that the first merger occurred at z=0.5, when mergers were more likely to be dry or gas-poor. At face value, this also implies that the evolution in scaling relations is minimal; however, we caution that all of the groups are effectively the same age, so this prediction does not include any evolution from varied ages and assembly histories of real group galaxies.</text>
<section_header_level_1><location><page_22><loc_14><loc_24><loc_43><loc_25></location>5.3. Spiral Progenitors and Their Bulges</section_header_level_1>
<text><location><page_22><loc_8><loc_14><loc_48><loc_23></location>In the case of Sersic index distributions and scaling relations, it is tempting to consider whether a combination of progenitor bulge types (and possibly bulge fractions) could resolve the tensions with observations. To examine this further it is useful to ask what the distributions of bulge Sersic index and bulge fraction are for spirals as a function of luminosity.</text>
<text><location><page_22><loc_8><loc_2><loc_48><loc_14></location>Not all S+11 spirals have a distinct bulge, nor are most images of sufficient quality to accurately measure bulge properties, so we consider the subset for which a bulge plus disk fit is required - that is, those with an F-test probability that a de Vaucouleurs bulge is not required is less than 0.32. This is about half of the spiral sample. The proportion for which a free Sersic bulge is required over a de Vaucouleurs bulge is much smaller, so we do not limit the sample any further. Fig. 19 shows the probabil-</text>
<text><location><page_22><loc_53><loc_86><loc_93><loc_87></location>0 . 00 ≤ B / Tr > 0 . 15 ,N=1.0e3 0 . 15 ≤ B / Tr > 0 . 30 ,N=1.5e3</text>
<figure>
<location><page_22><loc_52><loc_77><loc_71><loc_86></location>
</figure>
<figure>
<location><page_22><loc_74><loc_77><loc_91><loc_86></location>
<caption>0 . 30 ≤ B / Tr > 0 . 45 ,N=1.1e3 0 . 45 ≤ B / Tr > 1 . 00 ,N=1.3e3</caption>
</figure>
<figure>
<location><page_22><loc_52><loc_64><loc_71><loc_75></location>
</figure>
<figure>
<location><page_22><loc_74><loc_64><loc_91><loc_74></location>
<caption>Figure 20. Spiral galaxy bulge properties as in Fig. 19, but now for visually classified N+10 spirals. Although the statistics are barely sufficient, there does appear to be a weak correlation between luminosity and bulge Sersic index for spirals with large bulge fractions, more so than in Fig. 19.</caption>
</figure>
<text><location><page_22><loc_52><loc_44><loc_92><loc_56></location>ity densities of bulge Sersic indices as a function of galaxy luminosity, split into different bulge fraction bins. In all bins, classical (n s =4) bulges are at least a local maximum, although extreme bulge Sersic indices (n s =0.5 and n s =8, which are the lower and upper limits for S+11) are often the most common. The dependence on luminosity is not very strong, but in most bulge fraction bins, fainter spirals are slightly more likely to have low Sersic index bulges than high.</text>
<text><location><page_22><loc_52><loc_28><loc_92><loc_44></location>The S+11 spiral sample is known to be contaminated by S0s. In Fig. 20, we instead use the much smaller but visually classified sample of spirals from N+10. This smaller sample does slight evidence for correlation between luminosity and bulge Sersic index, at least for more bulge-dominated spirals. Also, the large fraction of bulge Sersic indices below 1 is greatly diminished, suggesting that those could be primarily S0 contaminants in the S+11 sample, or possibly more poorly resolved, higherredshift spirals which appear in S+11 but not N+10. In either case, both samples contain substantial fractions of spirals with large bulge fractions.</text>
<text><location><page_22><loc_52><loc_8><loc_92><loc_28></location>The M31 model used in our simulations has a large bulge mass fraction (0.33) and luminosity fraction (0.5). Such fractions are not uncommon, even at low luminosities. de Vaucouleurs bulges are also quite common, whereas exponential bulges are at least not exceptionally rare, especially for bulge-dominated spirals. Even if groups of spirals have broad distributions of bulge profiles, as in Fig. 19, their median values could also lie close to the limiting cases of exponential or de Vaucouleurs in our simulations. Also, a wide distribution of bulge profiles is indeed a realistic solution to the problem of single-progenitor mergers producing remnants with narrow Sersic index distributions. Real mergers in groups would likely produce wider, less bimodal distributions of Sersic indices than the single-progenitor simulations.</text>
<section_header_level_1><location><page_22><loc_64><loc_5><loc_81><loc_7></location>5.4. Rotational Support</section_header_level_1>
<text><location><page_22><loc_52><loc_2><loc_92><loc_5></location>The abundance of fast-rotating, faint ellipticals is at odds with the simulation predictions. However, as Figs.</text>
<text><location><page_23><loc_8><loc_57><loc_48><loc_87></location>14 and 15 show, multiple mergers can product remnants with moderate rotational support. This contrasts with the results of Cox et al. (2006) that dissipationless binary mergers only produce slow-rotating remnants with v /σ < 0 . 15 - all the more so because Cox et al. (2006) measured major axis rotation curves, whereas our simulations (and A3D) average over R eff . Nonetheless, our simulations are unable to produce any remnants with λ > 0 . 35. The simulated remnants show little or no change in rotational support as a function of luminosity (Fig. 17), unlike observations, and do not produce any of the fast-rotating, moderately luminous S0s found in A3D. There is also an abundance of flattened remnants with minimal rotation, unlike in Atlas3D. Cox et al. (2006); Bois et al. (2011) and others have shown that significant rotation can be easily produced in gasrich mergers. Dissipation is likely necessary to produce some ellipticals, particularly faint ones, and most likely a large fraction of S0s - if S0s are formed through mergers. However, it should emphasized that many of the simulated galaxies are consistent with the properties of some A3D galaxies, particularly bright ellipticals, so dissipation may not be necessary in all cases.</text>
<section_header_level_1><location><page_23><loc_21><loc_54><loc_35><loc_56></location>6. CONCLUSIONS</section_header_level_1>
<text><location><page_23><loc_8><loc_46><loc_48><loc_54></location>We have investigated the hypothesis that elliptical galaxies can form through collisionless mergers of spiral galaxies by creating a sample of numerical simulations of such mergers and comparing the results directly with observations of local ellipticals. We draw the following key conclusions:</text>
<unordered_list>
<list_item><location><page_23><loc_10><loc_33><loc_48><loc_45></location>1. For a given fixed bulge type, central remnants have narrow distributions of Sersic indices, with mergers of spirals with exponential bulges producing less concentrated remnants ( ∼ n s =3) than classical-bulge merger remnants ( ∼ n s =5). Although classical-bulge mergers alone are a better fit than exponential-bulge mergers, a combination of progenitor bulge profiles is required to reproduce observed Sersic index distributions.</list_item>
<list_item><location><page_23><loc_10><loc_24><loc_48><loc_32></location>2. Classical-bulge mergers produce a correlation between luminosity and Sersic index, whereas exponential bulge mergers do not. The observed correlation is best reproduced if exponential bulge mergers preferentially produce faint ellipticals and classical bulge mergers produce bright ellipticals.</list_item>
<list_item><location><page_23><loc_10><loc_6><loc_48><loc_23></location>3. Every simulation sample produces tight scaling relations, with approximately 0.1 dex scatter for the size-mass relation and 0.04 dex scatter in the Faber-Jackson relation. Thus, even multiple dry mergers can produce ellipticals with exceptionally tight scaling relations. However, the scatter estimates represent a lower limit, because the progenitor spirals in our simulations follow a fixed, zero-scatter Tully-Fisher relations. The scatter in the remnants scaling relations would likely increase (though not necessarily significantly) if the progenitor scaling relations had larger intrinsic scatter or evolved with redshift.</list_item>
<list_item><location><page_23><loc_10><loc_2><loc_48><loc_5></location>4. The remnant size-luminosity relation typically has a shallower slope ( R ∝ L 0 . 5 -0 . 6 ) than observed re-</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_23><loc_56><loc_81><loc_92><loc_87></location>tions ( R ∝ L 0 . 6 -0 . 8 ), depending on the sample and weighting scheme used. The simulated slope is also steeper than that of the progenitor spiral sizeluminosity relation ( R ∝ L 0 . 42 ), suggesting that mergers can steepen the size-luminosity relation.</list_item>
<list_item><location><page_23><loc_54><loc_73><loc_92><loc_80></location>5. As a consequence of the shallower slopes and larger intercepts of the simulated size-luminosity relation, the simulated Kormendy relation is shallower than observed - nearly flat for exponential bulge mergers - and has larger scatter.</list_item>
<list_item><location><page_23><loc_54><loc_66><loc_92><loc_72></location>6. The remnant Faber-Jackson relation has a slightly shallower slope ( σ ∝ L 0 . 28 ) than most of the observed relations ( σ ∝ L 0 . 27 -0 . 37 ), but is virtually unchanged from the progenitor spiral Tully-Fisher relation, V ∝ L 0 . 29 .</list_item>
<list_item><location><page_23><loc_54><loc_59><loc_92><loc_65></location>7. The slopes of the scaling relations can be better reproduced if massive ellipticals are produced by many mergers and less massive by fewer mergers, or if stellar mass is compared instead of luminosity.</list_item>
<list_item><location><page_23><loc_54><loc_48><loc_92><loc_58></location>8. The intercepts of the size-mass and Faber-Jackson relations can be individually matched by adjusting the stellar mass-to-light ratios of the galaxies; however, each relation requires adjustment in the opposite sense (remnants of a fixed luminosity being too large and having too low of a velocity dispersion), so it is not possible to match both intercepts simultaneously.</list_item>
<list_item><location><page_23><loc_54><loc_40><loc_92><loc_47></location>9. Multiple mergers can produce remnants with modest rotational support (v/ σ > 0.1); however, most remnants are slow rotators, and there is no correlation between luminosity and v/ σ , whereas such a correlation is found in Atlas3D ellipticals.</list_item>
</unordered_list>
<text><location><page_23><loc_52><loc_21><loc_92><loc_39></location>These results demonstrate that many of the properties of elliptical galaxies are consistent with their emergence through multiple dry mergers of spiral galaxies. Perhaps most importantly, these properties also differ significantly from those of remnants formed through binary dry mergers of spirals, as reported in previous studies. This not only adds to an increasing body of evidence supporting the case for multiple mergers (e.g. Bournaud et al. 2007; Naab et al. 2009; Trujillo et al. 2011; Hilz et al. 2013) but also demonstrates that such mergers can produce tight scaling relations - in some cases tighter than observed ellipticals - as long as the progenitor spirals are drawn from a realistic luminosity function and scaled appropriately.</text>
<text><location><page_23><loc_52><loc_2><loc_92><loc_21></location>Several major concerns remain for a purely dissipationless formation scenario for elliptical galaxies. The first is the limited amount of rotational support in the merger remnants and the absence of any correlation between rotation and luminosity. The second is the large sizes (and low velocity dispersions) of faint ellipticals, which result in a shallow size-luminosity relation and poorly reproduced Kormendy relation. While this second point could be resolved without dissipation (e.g. by merging more compact disks at high redshift), dissipation does appear to be necessary to produce fast-rotating ellipticals. Dissipation could also solve the second problem, as central starbursts would produce more compact remnants with higher dispersions.</text>
<text><location><page_24><loc_8><loc_78><loc_48><loc_87></location>Perhaps the greatest challenge for dry mergers lies in matching the tilt of the fundamental plane with respect to the virial relation. Previous work has suggested that dissipational processes are the cause of this tilt and that dry mergers cannot produce any tilt (Robertson et al. 2006; Hopkins et al. 2008). This point will be addressed in Paper II of this series (Taranu et al. 2013).</text>
<section_header_level_1><location><page_24><loc_18><loc_75><loc_38><loc_77></location>7. ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_24><loc_8><loc_60><loc_48><loc_75></location>D.T. would like to thank B. Abraham, L. Bai, D. Krajnovic, T. Mendel and P. Nair for fruitful discussions and for providing data used herein, as well as the anonymous referee for helpful suggestions. D.T. acknowledges the support of Ontario Graduate Scholarships for this work. Simulations and analyses were performed on the Canadian Institute for Theoretical Astrophysics' Sunnyvale cluster and the University of Toronto's SciNet cluster. H.Y. acknowledges support from grants from the National Science Engineering Research Council of Canada and the Canada Research Chair program.</text>
<section_header_level_1><location><page_24><loc_24><loc_57><loc_33><loc_59></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_24><loc_14><loc_55><loc_43><loc_56></location>A. ANALYSIS PIPELINE TESTING</section_header_level_1>
<text><location><page_24><loc_8><loc_35><loc_48><loc_54></location>Several aspects of the simulation analysis pipeline merit further testing. First, we would like to determine if the pipeline can recover known or measurable quantities such as the total mass/luminosity and half-light radii in single galaxies. This is accomplished by analysing a sample of spherical, pure Sersic profile plus dark matter halo galaxies generated with GalactICS. This allows us to simultaneously test whether GalactICS can generate equilibrium Sersic profile models (which is how the bulges of progenitor spirals are initialized) and whether the analysis pipeline can successfully recover input parameters at arbitrary resolutions. We analyse these models before simulating them in any way. In App. B, we examine the results of simulating these simple models with PARTREE to test numerical convergence.</text>
<text><location><page_24><loc_8><loc_19><loc_48><loc_35></location>We can also use the group simulations themselves to test the analysis pipeline. Although we do not know the structural parameters of merger remnants a priori - indeed, they do not necessarily follow a single Sersic profile at all - the total luminosity is known in groups which have merged to a single remnant. Similarly, we can directly measure a half-light radius from mock images with no PSF or sky background in these cases and compare to observational estimates from the SDSS-equivalent mock images. This procedure allows us to determine whether single Sersic profile fits can simultaneously recover the total luminosity of a galaxy and its half-light radius.</text>
<text><location><page_24><loc_8><loc_12><loc_48><loc_19></location>In addition to Sersic fits from GALFIT, we fit de Vaucouleurs profiles with GALMORPH and measure nonparametric Petrosian radii to determine if these size measures can consistently recover the true half-light radius of a galaxy.</text>
<section_header_level_1><location><page_24><loc_18><loc_9><loc_39><loc_11></location>A.1. Sersic plus Halo Models</section_header_level_1>
<text><location><page_24><loc_8><loc_2><loc_48><loc_9></location>Our reference Sersic plus halo models consist of a single Sersic profile bulge and a dark halo with the same baryonic mass ratio as our fiducial M31 models. We produce n s = 2 and n s = 4 models to cover most of the range of typical elliptical surface brightness profiles. We</text>
<text><location><page_24><loc_52><loc_69><loc_92><loc_87></location>create models with R eff of 2, 4, 8 and 16 kpc, again covering ranges of typical elliptical galaxies and massive spiral bulges. The 2-kpc model is slightly larger than the 1.5-kpc bulge in our fiducial M31 model. All models are in virial equilibrium and follow a size-luminosity relation log( R eff ) = 0 . 7 log( L r ), with one model exactly on this relation and an extra model either over- or underluminous for its size. Each model is imaged at mock redshifts of 0.01, 0.025 and 0.1. These models will be used in the future to test recovery of scaling relations. However, for now we are mainly interested in whether the pipeline can recover the known values of n s , R eff and L for each model and whether the systematics depend on any of those parameters.</text>
<text><location><page_24><loc_52><loc_50><loc_92><loc_69></location>In addition to varying the galaxy luminosity as a function of size, each set of Sersic plus halo models is simulated at three resolutions. The lowest resolution has 15,000 star and 40,000 dark matter particles, identical to the lowest resolution model used in the simulations. The resolution increases by a factor of 8 each step such that the highest resolution model has 7,680,000 star and 20,480,000 dark particles, or at least a factor of two more than the total particle counts of the most massive group simulations. In principle these models should be rescaled versions of each other; however, the nominal SDSS PSF and signal-to-noise ratio set a physical scale for mock images, while our fixed softening length sets another physical scale for the simulation.</text>
<section_header_level_1><location><page_24><loc_64><loc_48><loc_80><loc_49></location>A.2. Sersic Quantities</section_header_level_1>
<text><location><page_24><loc_52><loc_28><loc_92><loc_47></location>For n s = 2 models imaged at z=0.025, GALFIT Sersic fits show excellent agreement with expectations, even at low numerical resolution. With just 15,000 star particles, sizes are recovered to within 1 ± 0 . 5% for 2 kpc radius, although larger galaxies have underestimated sizes to the level of 3 ± 1% at R eff = 16 kpc. However, n s is underestimated by 10% for R eff = 2 kpc galaxies, which improves to 4 ± 1% at R eff = 16 kpc. Luminosities, in turn, are underestimated at fairly constant levels of 3%, with standard deviations increasing with size from 0 . 1 to 1%. Similar trends are found at medium resolution but with smaller amplitudes - the largest errors on n s are just 1 . 4 ± 0 . 3% at 16 kpc, while errors on R eff are at most 3 . 5 ± 0 . 5% at 2 kpc and shrink to half of that value at 2 kpc.</text>
<text><location><page_24><loc_52><loc_14><loc_92><loc_27></location>Errors on parameters are reduced by about a factor of two by imaging at nearby redshifts (z=0.01) and increase by about the same factor by imaging at z=0.1. These errors are not eliminated by increasing the image size (and shrinking the PSF relative to R eff ) but shrink dramatically at the highest numerical resolution, to well under 1% in n s and L and about 1% in R eff . This suggests that these parameters are in principle completely recoverable with SDSS-equivalent imaging and good sky subtraction.</text>
<text><location><page_24><loc_52><loc_2><loc_92><loc_14></location>We have also fit n s = 4 de Vaucouleurs profiles using GALMORPH, as in Hyde & Bernardi (2009a). GALMORPH fits to the n s = 2 models show expectedly poor results. Sizes are overestimated by factors from 1.6 (at 2kpc) to 2.4 (at 16kpc) while luminosities are overestimated by 30 -60%. These results are not entirely unexpected - n s = 4 models have shallower outer profiles and hence more light at large radii compared to profiles with lower n s = 2. However, they do demonstrate that other</text>
<text><location><page_25><loc_8><loc_79><loc_48><loc_87></location>free parameters such as the effective radius and mean surface brightness cannot adjust to compensate for an incorrect profile choice, and so pure de Vaucouleurs profiles are not a good choice to fit ellipticals if their underlying surface brightness profiles are truly Sersic profile with n s significantly lower than 4.</text>
<text><location><page_25><loc_8><loc_65><loc_48><loc_79></location>For n s = 4 models, GALFIT free-n s fits show curiously constant fractional errors on sizes, consistently underestimating R eff by 8 -9 ± 1%. Underestimates of n s vary from a substantial 14 ± 1% at 2 kpc to 4 ± 0 . 6% at 16 kpc. Luminosity underestimates shrink from 5 ± 0 . 6% to 1 . 5 ± 0 . 3%. These errors are not improved by imaging at lower redshift and so are unrelated to the relative size of the PSF. Instead, they are reduced substantially by increasing numerical resolution. At the highest resolution, size estimates shrink to 5 -6 ± 1% at all sizes, while n s underestimates now scale from 8 -2%.</text>
<text><location><page_25><loc_8><loc_57><loc_48><loc_65></location>Since typical resolutions for central group galaxies are at the medium level or slightly higher, we expect that n s is underestimated by 7% on average for a pure de Vaucouleurs fit, with better performance at smaller n s . Size estimates are a minimum of 2% lower for large galaxies and up to 10% off for R eff = 2kpc.</text>
<text><location><page_25><loc_8><loc_43><loc_48><loc_57></location>By contrast, the GALMORPH fits with fixed n s = 4 accurately recover sizes and luminosities to better than 2% even at the smallest sizes and at medium resolution. We conclude that even in ideal situations, free-n s fits will systematically underestimate sizes and luminosities at the 5% level, whereas fixed-n s fits only perform better if the exact value of n s is known. In App. B we detail how these results change after 2 Gyr of simulation with PARTREE using a fixed 100 pc softening length, as in the simulations.</text>
<section_header_level_1><location><page_25><loc_17><loc_41><loc_40><loc_43></location>A.3. Group Simulation Results</section_header_level_1>
<text><location><page_25><loc_8><loc_16><loc_48><loc_41></location>As Fig. 21 shows, Sersic models generally do an acceptable job recovering central galaxy luminosities. For the n s = 1 sample, model luminosities are typically 85-90% of the total in groups with no satellites with relatively small scatter. The Sersic luminosities of n s = 4 central galaxies appear to have little or no systematic deviations from the true luminosities, although the scatter appears somewhat larger than in the n s = 1 case. The largest discrepancies are found for groups with many mergers, particularly equal mass mergers, in which case the models can overestimate the central galaxy's luminosity by at least 20-30%, largely due to runaway growth of the effective radius and Sersic index. However, in most cases Sersic profiles appear to be appropriate fits to the galaxies. The underestimation of n s = 1 merger luminosities appears to be a systematic effect. The underestimation of luminosities in groups with satellite galaxies is difficult to quantify, as the total luminosity in satellites is not easily separable from that of the central galaxy.</text>
<text><location><page_25><loc_8><loc_2><loc_48><loc_15></location>Testing whether half-light radii are recovered is also complicated by the presence of satellite galaxies. Nonetheless, we attempt to measure how closely R eff matches the 'true' half-light radius R 50 of the central galaxy in Fig. 22. We estimate R 50 as the radius enclosing half of the group luminosity in a given sky- and satellite-subtracted image, using the same best-fit ellipse as the Sersic model. The ratio should be unity if there are no satellite galaxies in the group and less than unity if there are. As Fig. 22 shows, half-light radii are more diffi-</text>
<text><location><page_25><loc_52><loc_81><loc_92><loc_87></location>cult to measure than total galaxy luminosities - or rather, errors on half-light radii from Sersic fits are considerably larger than for luminosities, which likely contributes to the significant scatter in the Sersic size-luminosity relation compared to the Faber-Jackson relation.</text>
<text><location><page_25><loc_52><loc_67><loc_92><loc_81></location>At first glance, the large scatter in the ratio of Sersic model to 'true' half-light radius might suggest that much of the error in the Sersic size-luminosity relation is due to systematics rather than any intrinsic scatter. However, the size-luminosity relation using total group luminosity and 'true' half-light radius still shows significant scatter (0.08 dex) even when limited to galaxies with no satellites. A much larger sample of higher-resolution simulations would be required to determine if this scatter is due to numerical effects or genuinely intrinsic.</text>
<section_header_level_1><location><page_25><loc_65><loc_64><loc_80><loc_66></location>A.4. Petrosian Radii</section_header_level_1>
<text><location><page_25><loc_52><loc_41><loc_92><loc_64></location>As in SDSS, the Petrosian radius R P is given by the radius at which the mean surface brightness in the ring bounded by 0 . 8 R P < r < 1 . 25 R P is 0.2 times the mean surface brightness within R P . As a non-parametric size measure, it requires no fitting to measure, unlike the Sersic R eff . Since the Sersic profile is an analytical solution, one can compute R P uniquely for any given n s . For n s = 3 to 6, R P /R eff ranges from 1.5 to 2. The Petrosian magnitude of a galaxy is often estimated as the flux contained within a radius of a factor N P larger than this Petrosian radius; SDSS uses N P = 2. Petrosian magnitudes effectively measure half-light radii within 3 -4 R eff rather than the nominal 8 R eff bounding box for the FITS images used to derive SDSS-equivalent magnitudes. We measure Petrosian radii using both circular apertures and elliptical apertures, using the best-fit ellipse from Sersic model fits in the latter case.</text>
<text><location><page_25><loc_52><loc_16><loc_92><loc_41></location>Unfortunately, as shown in Fig. 23, Petrosian luminosities appear to underestimate the true galaxy luminosity by a similar amount to the analytical relation for purely circular profiles (see Graham & Driver (2005) for a reference to various Sersic quantities). Sizes are also underestimated to a similar degree as predicted for a pure circular Sersic profile, which suggests that most galaxies do not deviate greatly from a pure Sersic profile. The slight excess could be due to a number of factors, including the Sersic models underestimating the true half-light radii and/or Sersic indices, radial variations in the ellipticity or shape of the isophotes, or deviations of the underlying profile from a pure Sersic model, all of which are plausible. In principle, one can correct for this 'missing' flux using fitting formulae valid for a wide range of Sersic or other profiles (Graham et al. 2005), but this seems unnecessary given that the Sersic fits appear sufficient and are available for all of the simulations and observational catalogs alike.</text>
<section_header_level_1><location><page_25><loc_58><loc_14><loc_86><loc_15></location>B. NUMERICAL CONVERGENCE</section_header_level_1>
<text><location><page_25><loc_52><loc_2><loc_92><loc_13></location>We test the numerical convergence of the spherical Sersic plus halo models by simulating every galaxy for 2 Gyr at 3 different resolutions (differing in particle number by a factor of 8 in each step). We also test a subset of the group simulations at similar resolutions. All measurements are made using the same analysis pipeline as the results above; the images also have the same nominal redshift of z = 0 . 025.</text>
<figure>
<location><page_26><loc_8><loc_51><loc_92><loc_85></location>
<caption>Figure 21. Ratio of Sersic model luminosities to total group luminosity. Groups with multiple galaxies are highlighted, since the fraction contained in the satellites is not well-constrained. Mergers of equal-mass spirals ('Eq') tend to show the largest deviations from unity.</caption>
</figure>
<figure>
<location><page_26><loc_8><loc_9><loc_92><loc_43></location>
<caption>Figure 22. Ratio of Sersic model effective radii to estimated half-light radii. Half-light radii are estimated by measuring the radius at which the enclosed light in a sky- and satellite-subtracted image equals half of the total group luminosity, and so are strictly larger than a true half-light radius if there are satellite galaxies. Regardless, Sersic half-light radii show considerably larger scatter relative to the 'true' half-light radius than do Sersic model luminosities to the total group luminosity (Fig. 21).</caption>
</figure>
<figure>
<location><page_27><loc_8><loc_52><loc_50><loc_87></location>
<caption>Figure 23. Ratio of Petrosian model to Sersic model luminosities and sizes. Petrosian luminosities are derived from the elliptical Petrosian half-light radii measured within twice the Petrosian radius. Petrosian sizes and luminosities generally follow the analytical relation for a pure Sersic profile, underestimating sizes and luminosities by larger fractions for large Sersic indices.</caption>
</figure>
<figure>
<location><page_27><loc_8><loc_19><loc_50><loc_52></location>
</figure>
<text><location><page_27><loc_28><loc_19><loc_34><loc_20></location>Sersic n</text>
<text><location><page_27><loc_8><loc_2><loc_48><loc_10></location>Convergence is generally quite good. With a 0.2 Myr timestep, total energy is conserved to better than one part in 10 5 . With the initial conditions re-centered to the barycenter, linear momentum remains small. The net angular momentum vector is the least well conserved quantity in Sersic plus halo models; each orthogonal com-</text>
<text><location><page_27><loc_52><loc_77><loc_92><loc_87></location>ponent can vary by up to 5% of the net rotation. However, the total angular momentum is usually dominated by a small number of dark matter halo particles at large distances from the galaxy center. Angular momentum conservation for baryons in isolated galaxies is considerably better, and deviations of 1 to 2% are typical for groups where the bulk of the angular momentum is initially in galaxy orbits.</text>
<text><location><page_27><loc_52><loc_61><loc_92><loc_77></location>Having tested input parameter (Sersic index and effective radius) recovery with the analysis pipeline, we now turn to examining how these same parameters evolve in a 100 pc softened potential with a fixed, 0.2 Myr timestep, as in the group simulations. While idealized, these simulations are comparable to both the central ellipticals (which are slowly rotating and close to Sersic profiles, albeit somewhat flattened) and the bulges of the input spirals (which are smaller than the Sersic models and also slightly flattened by the presence of the disk) and will give estimates for how galaxy structure is affected by numerical resolution.</text>
<section_header_level_1><location><page_27><loc_57><loc_59><loc_87><loc_60></location>B.1. Sersic plus Halo Model Convergence</section_header_level_1>
<text><location><page_27><loc_52><loc_29><loc_92><loc_58></location>For a typical model ( R eff =8 kpc) at very high resolution (7.68 million star particle), convergence of all parameters is achieved at the 1 to 2% level, with sizes, Sersic indices and dispersions shrinking slightly over 2 Gyr. Convergence is considerably worse for the n s = 4 model and is strongly resolution dependent. A factor eight drop to high resolution (0.96 million star particles) approximately double errors in all parameters to 2-4%. For medium resolution (120,000 star particle), n s = 4 models, parameters can shrink by over 10% - typical values being 5 to 15% for n s (4 to 3.4), 15% for sizes (8 kpc to 6.8 kpc) and 5% for dispersions. Thus, for larger ellipticals to be suitably resolved, a million or more stellar particles are required, especially if the profiles are as or more centrally concentrated than an n s = 4 model. Less centrally concentrated models such as n s = 2 are much less sensitive to numerical resolution and can be resolved by 100,000 stellar particles with at most 3 to 4% level drops in sizes and Sersic index. Only 4 simulations in the sample have fewer than 720,000 stellar particles, so central remnants are largely unaffected by numerical relaxation after formation regardless of their central concentration.</text>
<text><location><page_27><loc_52><loc_13><loc_92><loc_29></location>Unfortunately, the results are not as encouraging for smaller models. For the smallest R eff =2 kpc model at low (15,000 star particle) resolution, Sersic indices shrink up to 50% (from 2 to 1.5, or 4 to 2.3). Sizes typically drop by less than 10%, but dispersions also shrink up to 20%. At high resolution, Sersic indices converge at the 5 to 15% level (from 2 to 1.9 and 4 to 3.4). Sizes remain constant for n s = 2 and drop at most 5% for n s = 4, with dispersions also shrinking by 3 to 5%. Typical remnants are resolved at close to this high resolution, so the greatest effect would be on the Sersic indices of small, high n s ellipticals.</text>
<text><location><page_27><loc_52><loc_2><loc_92><loc_13></location>The greater concern with these results is the relaxation that occurs in the bulges of progenitor spirals. The effective radius of the M31 model is 1.5 kpc, but most galaxies are scaled to smaller sizes than this, with 0.5 to 1 kpc bulge R eff . Moreover, in groups with larger numbers of galaxies, total particle counts are larger, but individual spirals can have as few as 60,000 stellar particles, of which only 20,000 are in the bulge. The bulge is partially</text>
<text><location><page_28><loc_8><loc_79><loc_48><loc_87></location>stabilized (and flattened) by the disk, but the disk forms a core near the center of the galaxy, and so one might expect the behaviour of these compact, marginally resolved bulges to be similar to the Sersic plus halo models. We will now test this hypothesis with convergence studies of group mergers.</text>
<section_header_level_1><location><page_28><loc_16><loc_77><loc_42><loc_78></location>B.2. Group Simulation Convergence</section_header_level_1>
<text><location><page_28><loc_8><loc_63><loc_48><loc_76></location>We test numerical convergence in the groups by running a selected sample with a factor of eight higher and lower resolution and comparing parameters after the usual elapsed times (5.0, 7.7 and 10.3 Gyr). As all of the groups are resolved with an average of over a million stellar particles, numerical convergence is expected to be good once groups have merged. However, as detailed above, the least massive spirals in more massive groups are not as well resolved, so not all groups are expected to be converged at our standard resolution.</text>
<section_header_level_1><location><page_28><loc_19><loc_61><loc_38><loc_62></location>B.2.1. Parameter Recovery</section_header_level_1>
<text><location><page_28><loc_8><loc_48><loc_48><loc_60></location>Fig. 24 shows convergence for several identical groups on the size-luminosity and sizeσ relations after 10.3 Gyr. Central remnant luminosities are fairly constant across all resolutions, but low resolutions can have slightly lower values. Sizes and dispersions are larger at low resolutions. Both trends continue from fiducial/medium to high resolution, although it is not as extreme - sizes are usually not more than 10 percent smaller between medium and high resolution.</text>
<text><location><page_28><loc_8><loc_31><loc_48><loc_48></location>Sersic indices are systematically lower at low resolution by a factor of 1 to 2 (Fig. 25). The trend persists at high resolution, although n s typically increases by a smaller factor of 0.2 to 0.3 between mid to high resolutions. Of the four parameters tested, then, luminosity appears to be the most robust, while the Sersic index is most sensitive to resolution effects. The effects on sizes are too small to fully reconcile the mismatch between sizes of faint simulated galaxies compared to observed ellipticals (Fig. 11). Dispersions generally decrease with increasing resolution, and so numerical effects also cannot explain the lower intercept of the simulated Faber-Jackson relation compared to that of observed ellipticals (Fig. 13).</text>
<text><location><page_28><loc_8><loc_13><loc_48><loc_31></location>In general, increasing resolution by a factor of eight produces similar trends in the group simulations as in isolated Sersic plus halo models - Sersic indices increase, while sizes and dispersions decrease. The effects are not very large going from our standard (medium) to high resolution but are considerable when stepping down to low resolution. We recommend that a minimum of a million stellar particles be used to adequately resolve spheroidal galaxies. While luminosities and masses remain converged at low resolution, sizes and dispersions are overestimated. Sersic indices are especially untrustworthy, being systematically offset lower by one or two from higher resolutions.</text>
<section_header_level_1><location><page_28><loc_10><loc_10><loc_46><loc_12></location>C. SCALING RELATIONS AT DIFFERENT TIMES</section_header_level_1>
<text><location><page_28><loc_8><loc_2><loc_48><loc_9></location>The scaling relations presented in § 4.3 are nominally for a zero-redshift galaxy population, assuming evolution from z=2. We can instead consider scaling relations at younger ages, assuming a fixed formation time for all groups. This is equivalent to assuming evolution from</text>
<paragraph><location><page_28><loc_69><loc_86><loc_74><loc_87></location>Table 9</paragraph>
<text><location><page_28><loc_55><loc_85><loc_89><loc_85></location>Sersic model size-luminosity relations at different times</text>
<table>
<location><page_28><loc_52><loc_70><loc_90><loc_83></location>
<caption>Simulations, Sersic model L and r eff , Unweighted</caption>
</table>
<text><location><page_28><loc_52><loc_66><loc_92><loc_70></location>Note . - Sersic model size-luminosity relations of simulations after different times have elapsed (in Gyr) or, equivalently, assuming different formation redshifts (0.5, 1.0 and 2.0). Data are for ten equally-spaced, randomly oriented projections of each galaxy.</text>
<paragraph><location><page_28><loc_69><loc_62><loc_75><loc_63></location>Table 10</paragraph>
<text><location><page_28><loc_52><loc_60><loc_92><loc_62></location>Sersic model Faber-Jackson relations of simulations after different times</text>
<table>
<location><page_28><loc_52><loc_46><loc_91><loc_59></location>
<caption>Simulations, Sersic model L, Unweighted</caption>
</table>
<text><location><page_28><loc_52><loc_41><loc_92><loc_45></location>Note . - Sersic model Faber-Jackson relations of simulations after different times have elapsed. Format as in Tab. 9. The slopes generally flatten slightly while intercepts increase and scatter remains constant at 0.04 dex.</text>
<text><location><page_28><loc_52><loc_30><loc_92><loc_39></location>z=1 or z=0.5, since the only initial redshift-dependent parameter in the initial conditions is the group size. One might also consider combining groups from different snapshots into a single sample to simulate a sample with galaxies of different ages; however, this is best left to purely cosmological initial conditions with known merger trees and formation times.</text>
<text><location><page_28><loc_52><loc_5><loc_92><loc_30></location>With these caveats in mind, we now present predictions for the evolution of the slope and scatter of selected scaling relations assuming a fixed formation time for all groups. The best-fit relations measured in Tab. 9 show slight evolution with time in the slopes (increasing) and intercepts (decreasing) and limited evolution in scatter. The steepening of the slope and lowering of the intercept would seem to suggest that brighter ellipticals grow off the relation at later times while fainter ellipticals grow slowly, if it all - in our case largely by construction, since the Few-merger sample does not have any late-time mergers. This interpretation is complicated by the fact that some of the largest groups do not have a relaxed, early-type central remnant formed in the earlier time steps and so are not included in the sample at earlier times but are included later on. Thus, as in most observational catalogs, not all of the descendants can necessarily be clearly identified with a previous early-type ancestor.</text>
<text><location><page_28><loc_52><loc_2><loc_92><loc_5></location>The best-fit Faber-Jackson relations measured in Tab. 9 also show slight evolution of the slope, but in</text>
<figure>
<location><page_29><loc_8><loc_18><loc_91><loc_87></location>
<caption>Figure 24. Numerical convergence about size-luminosity and sizeσ relations for principal axis projections of selected groups after 10.3 Gyr. At low resolutions, galaxies generally have larger sizes and dispersions than at higher resolution.</caption>
</figure>
<text><location><page_29><loc_36><loc_18><loc_37><loc_19></location>/circledot</text>
<text><location><page_29><loc_77><loc_18><loc_78><loc_19></location>/circledot</text>
<text><location><page_29><loc_8><loc_11><loc_48><loc_14></location>the opposite sense (decreasing/flattening), with a corresponding increase in the intercept. However, the scatter remains largely unchanged at 0.04 dex.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "We present collisionless simulations of dry mergers in groups of three to twenty-five galaxies to test the hypothesis that elliptical galaxies form at the centers of such groups. Mock observations of the central remnants confirm their similarity to ellipticals, despite having no dissipational component. We vary the profile of the original spiral's bulge and find that ellipticals formed from spirals with exponential bulges have too low Sersic indices. Mergers of spirals with de Vaucouleurs (classical) bulges produce remnants with larger Sersic indices correlated with luminosity, as with SDSS ellipticals. Exponential bulge mergers are better fits to faint ellipticals, whereas classical bulge mergers better match luminous ellipticals. Similarly, luminous ellipticals are better reproduced by remnants undergoing many ( > 5) mergers, and fainter ellipticals by those with fewer mergers. The remnants follow tight size-luminosity and velocity dispersion-luminosity (Faber-Jackson) relations ( < 0.12 dex scatter), demonstrating that stochastic merging can produce tight scaling relations if the merging galaxies also follow tight scaling relations. The slopes of the size-luminosity and Faber-Jackson relations are close to observations but slightly shallower in the former case. Both relations' intercepts are offset - remnants are too large but have too low dispersions at fixed luminosity. Some remnants show substantial (v/ σ > 0.1) rotational support, although most are slow rotators and few are very fast rotators (v/ σ > 0.5). These findings contrast with previous studies concluding that dissipation necessary to produce ellipticals from binary mergers of spirals. Multiple, mostly minor and dry mergers can produce bright ellipticals, whereas significant dissipation could be required to produce faint, rapidly-rotating ellipticals. Subject headings: galaxies: elliptical - galaxies: evolution - galaxies: formation - galaxies:structure",
"pages": [
1
]
},
{
"title": "MERGERS IN GALAXY GROUPS - I. STRUCTURE AND PROPERTIES OF ELLIPTICAL REMNANTS",
"content": "Dan S. Taranu, John Dubinski, and H.K.C. Yee Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, Canada, M5S 3H4 Draft version July 21, 2018",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Merging of spiral galaxies is a promising mechanism for producing elliptical galaxies. Although it was perhaps not until Toomre (1977) that interacting spirals became widely accepted as elliptical progenitors, simulations of interacting spirals date back at least to Toomre & Toomre (1972) and arguably as far as Holmberg (1941). Much of this work has focused on major mergers (mass ratio < 3:1) of pairs of spiral galaxies on parabolic orbits. While such binary major mergers certainly are observed in the local universe - and will likely be the ultimate fate of the Milky Way and M31 - they may not be as common as minor hierarchical mergers. Observational evidence and numerical simulations suggest that most L* galaxies are found in groups (McGee et al. 2009), where the central galaxy is likely to have experienced multiple mergers and have several surviving satellites. Furthermore, late-type galaxies in groups follow a Schechter luminosity function (Schechter 1976) similar to those in other environments (Croton et al. 2005; Robotham et al. 2010). Thus, if high-redshift groups are composed primarily of spiral galaxies, they are likely dominated by several bright spirals with a larger number of fainter satellites, as in our Local Group. Our hypothesis is that groups of three or more spiral galaxies with luminosity distributions following a Schechter function will naturally merge to produce a central elliptical galaxy, and possibly fainter satellites. We aim to test this hypothesis with numerical experiments. More specifically, we test whether the properties of the central galaxies formed through collisionless merg- ers in groups of spirals are consistent with observations of local ellipticals. This first paper in a series outlines our methodology and demonstrates that the results are both qualitatively and quantitatively different from both the more prevalent studies on binary mergers and also the less abundant literature on galaxy group mergers. We present results on morphological and kinematical measures as well as two dimensional scaling relations. Paper II (Taranu et al. 2013) will focus on the three dimensional fundamental plane scaling relation. To further motivate this endeavour, we will outline some of the key results of the past several decades of work in this field. Carnevali et al. (1981) and Ishizawa et al. (1983) were amongst the first to produce simulations of mergers in groups of galaxies (10-20 each), using 20 and 100 particles per galaxy, respectively. Barnes (1985) introduced separate stellar and dark matter profiles, with 30 and 270 particles for each component, respectively. Barnes (1989) added stellar disks and bulges to the galaxy models, taking advantage of the newly invented N-body tree code (Barnes & Hut 1986) to increase resolution to 4096 luminous and dark particles each. While the arrangement of orbits was somewhat artificial (a pair of triple systems each consisting of a binary orbiting a more massive single galaxy), the study showed that mergers of realistic galaxies in compact groups can be rapid and produce central remnants with de Vaucouleurs profiles and shell structures, similar to local ellipticals. A problem with the collisionless merger scenario was identified by Carlberg (1986). Collisionless mergers cannot increase central phase space density, which is necessary if disks are to merge and produce ellipticals with higher central densities than their progenitors, as is often the case. Hernquist et al. (1993) directly addressed this issue with simulations of binary mergers of spirals with bulges (Hernquist 1993), which, unlike mergers of bulgeless spirals (Hernquist 1992), are capable of producing sufficiently centrally dense remnants. Weil & Hernquist (1996) extended this methodology to mergers of groups of six equal-mass spirals. The resolution of these simulations was a factor of 18 higher than Barnes (1989), with almost 150,000 particles per galaxy, and included a comparison sample of pair mergers. Group mergers were shown to produce remnants with some rotation, in contrast to the non-rotating remnants typical of dry binary mergers (Cox et al. 2006). Both varieties of mergers produced early-type galaxies well fit by de Vaucouleurs profiles. However, group mergers with bulges did not maintain centrally concentrated profiles, exhibiting the same low central phase space density as found in bulgeless pair mergers. This may be seen as once again disfavoring the group merging scenario; however, it should be noted these mergers were of equal-mass galaxies and hence of very large mass ratios, which would tend to maximize this problem. Furthermore, Kormendy et al. (2009) argues that ellipticals have lower central densities ('cores') than expected from inward extrapolation of their outer surface brightness profiles, which is consistent with the group merging scenario and essentially the opposite of the central phase space density problem, although Kormendy et al. (2009) attribute these cores to scouring of inner stars by inspiraling supermassive black holes rather than purely stellar dynamical processes. Since Weil & Hernquist (1996), the relatively rapid pace of advancements in the field of simulations of merging in galaxy groups has slowed, with the focus shifting to studies of hydrodynamical processes in group environments. This can be partially attributed to the findings of Robertson et al. (2006) that collisionless binary mergers are unable to produce ellipticals following a tilted fundamental plane relation, although Aceves & Vel'azquez (2005) found an appreciable tilt by merging spirals sampled from an appropriate Schechter luminosity function. Robertson et al. (2006) also found that collisionless binary mergers could only produce very slow-rotating remnants and that dissipation was required to produce significantly rotationally-supported ellipticals. However, between Weil & Hernquist (1996) and Robertson et al. (2006), very few studies have tested whether these results apply to multiple collisionless mergers as well. Galaxy clusters (Dubinski 1998) and starbursts in groups (Bekki 2001) have been considered. Ciotti et al. (2007) simulated consecutive mergers of spheroidal galaxies, roughly approximating hierarchical group merging. While this approach has provided useful estimates of the growth of stellar mass and size, the use of purely spheroidal progenitors is questionable given the prevalence of disks at high redshift. Hopkins et al. (2009) combined the results of binary merger simulations with cosmological merger trees, using empirical halo occupation models and spiral scaling relations to predict the evolution of early-type scaling relations. However, the models only incorporated multiple mergers by allowing binary merger remnants sufficient time to relax dynamically, whereas Moster et al. (2012) find that halos undergoing multiple mergers are likely to have two mergers in quick succession. Galaxies in groups are thuse likely to undergo multiple mergers within a relatively short periods rather than a steady stream of isolated mergers. More recently, fully cosmological simulations of mergers in a group or several groups of galaxies have been performed by, for example, Khalatyan et al. (2008) and Feldmann et al. (2011). Such simulations naturally incorporate hierarchical merging, typically by using the 'zoom-in' method of re-simulating a small sub-volume of a large dark matter-only cosmological simulation at higher resolution. Naab et al. (2009) and Oser et al. (2012) demonstrated that ellipticals can form in groups, with minor mergers being an important mechanism in controlling the evolution of the central galaxy size and mass. Feldmann et al. (2011) showed success in producing not only a central elliptical but early-type satellites by simulating a single group using this method. However, such ab initio simulations encounter difficulties achieving sufficient spatial resolution to produce realistic spiral galaxies. Typical softening lengths in such simulations are between 500 to 1000 pc, which significantly alters the inner profile of ellipticals, especially at low masses. Increasing the resolution can mitigate this problem but also greatly increases computational cost and limits the possible sample size to a few groups. It is clear that there is a gap in the literature on multiple mergers in groups, even though multiple mergers likely create brightest cluster galaxies (Dubinski 1998). By contrast, observations of elliptical galaxies have advanced tremendously in recent years, providing public catalogs of morphologies of thousands of nearby ellipticals and spirals alike (e.g. Blanton et al. 2005; Hyde & Bernardi 2009a; Nair & Abraham 2010; Simard et al. 2011), mainly based on Sloan Digital Sky Survey (SDSS) images. The SAURON project (de Zeeuw et al. 2002) and its volume-limited successor survey Atlas3D (Cappellari et al. 2011) have provided integral-field kinematics of hundreds of early type galaxies at a comparable resolution to SDSS. It is increasingly necessary to match the large samples of observations with simulations and explore the vast parameter space of conditions in galaxy groups. To meet the requirement of a large simulation sample, it is currently necessary to focus on dry merging and gravitational dynamics alone. Hydrodynamical simulations are more computationally expensive and add numerous parameters to initial conditions: disk gas fractions, gas disk scale heights and lengths relative to the stellar disk, the presence of a gaseous halo, etc. More importantly, existing literature has yet to establish the effects of collisionless gravitational dynamics in group mergers on central remnant structure. There is observational evidence suggesting that dry merging contributes significantly to the growth of massive galaxies, particularly ellipticals (e.g. van Dokkum et al. 2010). Even if exclusively dry merging is not the most common mechanism for forming ellipticals, many ellipticals will have experienced at least one dry merger in their lifetimes and it is instructive to ask what collisionless dynamics alone would predict before moving on to hydrodynamical processes. The remainder of the paper is structured as follows: § 2 motivates and details the methods used in creating the simulations, while § 3 details the analysis methodology and pipeline, with additional tests presented in App. A. A more detailed examination of numerical convergence can be found in App. B. Key results on morphology, scaling relations and kinematics of central remnants are presented in § 4. The implications of these results on theories of elliptical galaxy formation are detailed in § 5, with reference to prior studies on the subject. The conclusions are summarized in § 6.",
"pages": [
1,
2,
3
]
},
{
"title": "2. SIMULATIONS",
"content": "The simulations are designed to extend the methodology of binary galaxy merger simulations to groups of galaxies. This section details the parameters of the group sample ( § 2.1), as well as the two key ingredients in the initial conditions: group configuration ( § 2.2) and galaxy models ( § 2.3). Finally, the code and parameters used for the simulations are described in § 2.4. Our choice of initial condition parameterization is designed to evenly sample the parameter space of groups which are likely to produce a central elliptical remnants, rather than be a unbiased sampling of real, nearby galaxy groups. This approach is similar to that used in binary mergers simulations, in which the orbits are typically nearly parabolic, with some cosmologicallymotivated distribution of pericentric distance and disk alignment. In our case, we model group-sized halos at the turnaround radius at z=1-2, such that the subhalos are likely to contain spiral galaxies which will eventually merge to form one central elliptical. We use two galaxy models designed to reproduce the surface brightness profile and rotation curve of M31 and scale these models according to the Tully-Fisher relation (Tully & Fisher 1977). The only parameter we vary between the models is the profile of the bulge, which has a substantial impact on the structure of the merger remnant ( § 4.2). While this approach does not reproduce the variety of spiral galaxies found in the local universe, let alone at high redshift, it maintains the simplicity of the initial conditions. We do not vary the bulge fraction in the progenitors, as pre-formed bulges are required to produce sufficient central densities in the merger remnant. We discuss these choices further in § 5. Although the simulations are nominally scale-free, as with any system of units having G=1, our simulations assume units of length in kpc, velocity in 100 kms -1 , time in 9.73 million years and mass in 2 . 325 × 10 9 M glyph[circledot] . The luminosity function sampling and initial group radius impose a unique, preferred scaling to each simulation, such that mergers of groups with the same number of galaxies but different luminosities are not simply re-scaled versions of each other.",
"pages": [
3
]
},
{
"title": "2.1. Group Sample",
"content": "We create groups with total luminosities from 0.1-10L* and masses between 2 × 10 11 -2 × 10 13 M glyph[circledot] . We incorporate several basic assumptions consistent with observations and cosmological simulation predictions. More massive groups contain more galaxies on average, with galaxies preferentially located closer to the center of the group. The group as a whole is initially collapsing, with galaxies located within R max = 2 × R 200 ,z =2 but having insufficient orbital energy to prevent collapse (i.e. the Note . - Each simulation sample is divided into those groups with relatively Few (F) or Many (M) galaxies for their mass, with three groups in either category per mass bin. The minimum and maximum number of galaxies in a group is listed, as well as the maximum for the F and the minimum for the M subsamples. groups are sub-virial). We simulate each group configuration twice, with each simulation containing either spiral galaxies with exponential bulges or classical bulges (but not both), referring to the former sample as B.n s =1 and the latter as B.n s =4 for short. There are 3 sets of simulations, each with different random seeds for the initial conditions. Each set has 7 target luminosity (or mass) bins, ranging from 1/8 to 8 L* and increasing by factors of 2. Each bin contains 8 groups, for a total of 3 × 7 × 8 = 168 simulations, of which 3 × 7 × 2 = 42 are mergers of spirals with equal masses, while the remaining 126 are sampled from a realistic luminosity function. Since each simulation is run twice (with different spiral bulge profiles), there are nominally 336 simulations, but only 168 different sets of galaxy masses and orbits. Each group has a number of galaxies between N min = 3 and N max = 2+(5 / 6) × 10 · ( L/L ∗ ) 1 / 2 . Within each group luminosity bin, the number of galaxies in each simulation varies linearly from the minimum (3) to the maximum, so that L* groups have between 3 and 10 galaxies and the largest groups have 25 galaxies. This range of galaxy numbers roughly covers the number of bright galaxies one would expect in poor groups. The mass range covered by the groups is 2 . 0 × 10 11 M glyph[circledot] to 3 . 0 × 10 13 M glyph[circledot] . We further subdivide the sample into groups with relatively many mergers (the Many-merger or 'M' subsample) or relatively few (Few-merger or 'F'). The groups in each mass bin with the three lowest initial galaxy counts are part of the F subsample, while the groups with the three largest initial galaxy counts qualify for the M subsample. Because the maximum number of galaxies changes in each mass bin, the dividing line between the Many-merger and Few-merger subsamples depends on mass and is not a fixed number of galaxies or mergers. Each mass bin also contains two groups with equal-mass galaxies ('Eq'), one with three galaxies ('F-Eq') and the other with the same number of galaxies - N min ,M - as the fourth group in the LF-sampled simulations (i.e., the group in the 'M' subsample with the fewest galaxies). The number of galaxies in a representative number of groups is listed in Tab. 1.",
"pages": [
3
]
},
{
"title": "2.2. Group Configuration",
"content": "Once the target luminosity and number of galaxies are selected, each group is initialized through the following steps: from a restricted range of the spiral galaxy luminosity function. Galaxy luminosities are randomly sampled from the inclination- and extinction-corrected spiral luminosity (Schechter) function of Shao et al. (2007). For the r-band, the faint-end slope α = -1 . 26 and M ∗ r = -20 . 99 + 5 log( h = 0 . 71), or M ∗ r = -21 . 73, which is nearly identical to our standard M31 model's absolute magnitude of M r = -21 . 69. We set a minimum luminosity of 0 . 01 L ∗ for the spirals, as we do not expect the luminosity function to continue to arbitrarily faint magnitudes. Luminosities are drawn from a restricted range of the luminosity function with a width equal to ( N gals +2) / 10 dex, such that the integral under the curve is equal to the target group luminosity. This limited range produces groups with smaller magnitude differences between the brightest central galaxy and the next brightest satellite, making major mergers more likely especially in groups with few galaxies. We avoid simulating groups with a single luminous spiral and several much fainter satellites, since these groups would only produce relatively minor mergers and would be unlikely to produce ellipticals. The 42 groups with equal-mass spirals ('Eq') are not sampled from an LF and instead have exactly the target luminosity, split evenly between three or a larger number of galaxies. Once a luminosity is determined for each group galaxy, the galaxies are randomly assigned locations within the group in order of decreasing luminosity. The most luminous galaxy is placed at the center of the halo, while subsequent galaxies are given a random radius with a likelihood inversely proportional to radius, i.e., ρ ∝ r -1 , and with r < R max , where R max = 2 R 200 ,z =2 . R 200 ,z =2 is the radius at which a volume enclosing the total group mass has a mean density of 200 times the critical density at z=2. Next, a random polar and azimuthal angle is given. The minimum distance between galaxies is set by 0 . 5 × R max ∗ ( M galaxy /M group ) 1 / 3 , which allows galaxy halos to be in contact but not overlap significantly. All of the galaxies are given preferentially inward and radial orbits. The group itself is sub-virial to ensure collapse and no satellite is given a speed | v | > v escape . This is accomplished by giving each group a target virial ratio Q target = -2 T/W = 0 . 5, where W is the gravitational potential energy of the group. This is equivalent to a zero-energy parabolic orbit in a galaxy pair, where T=-W. The group velocity dispersion is determined by Q target : σ = ( -Q × ( W/a ) /M ) 0 . 5 , where a = 3 s -2 β and β is an orbital anisotropy parameter. Each galaxy's radial velocity is then σ ∗ s , where s is a number randomly selected from a unit Gaussian centered on s=0.5. On average more than 70% of galaxies will have inward radial velocities. The azimuthal and polar velocities are given by σ ( s × (1 -β ) 0 . 5 ), where β = 0 . 5 directs most of the velocity of the galaxy radially. In two of the three sets of simulations, the initial conditions in groups of similar mass (1/8 to 1/2, 1 to 4, and 8) are correlated, in the sense that galaxy positions are seeded in the same order (but not individual galaxy masses or orbits). This is intended to test the effect of adding additional galaxies to otherwise similar initial conditions. In the third set, all of the initial conditions are completely randomized. We note no statistically significant differences between the partially and completely random initial conditions for the relations presented in this paper; however, we will note some differences in the fundamental plane parameters in Paper II. Finally, each galaxy has its own massive, extended dark halo. In practice, the these individual halos overlap in their outer regions, leaving little or no 'empty' space between galaxies. We have also experimented with including a separate dark matter halo for the group, not associated with any particular galaxy, but find that group galaxies then (unrealistically) merge with this invisible dark halo rather than with each other.",
"pages": [
3,
4
]
},
{
"title": "2.3. Galaxy Models",
"content": "Initial spiral galaxy models are created using the GalactICS galaxy initial condition code (Widrow & Dubinski 2005). This code generates equilibrium models of galaxies with a bulge, disk and halo through spherical harmonic expansions of analytic potentials. Although the models begin in equilibrium and do not require additional time to settle, they have been tested in isolation. All models remain (statistically) unchanged for at least a Hubble time, even at the lowest resolution (55,000 particles per galaxy), which the vast majority of galaxies exceed. The models are similar to the 'M31c' model of Widrow & Dubinski (2005), with some parameters adjusted following the approach of Widrow et al. (2008) to better reproduce the surface brightness profile and rotation curve of M31. M31 was chosen as a well-studied, nearby spiral having a sufficiently massive bulge to produce concentrated merger remnants. The models are bar-stable and contain a massive, non-rotating bulge, as well as a dark matter halo. The first variant uses a nearly exponential bulge with n s = 0 . 93, which will be referred to as exponential for convenience. A second variant uses an n s = 4 de Vaucouleurs or classical bulge but otherwise identical parameters. The halo density profile is designed to match an NFW (Navarro et al. 1997) profile at large radii and smoothly drop to zero at large radii. The halo has a 6.07 kpc scale radius, ρ ∝ r -1 inner cusp and r -2 . 3 outer slope, an outer radius of 300 kpc and a total mass of 1185 units, or 2 . 75 × 10 12 M glyph[circledot] . This profile produces a 30:1 ratio in dark:baryonic (stellar) mass, which is a factor of two larger than estimates for M31 and the Milky Way (e.g. Watkins et al. 2010), but smaller than global estimate for the universal dark:stellar mass ratio. The disk has a 5.8 kpc scale radius and a 750 pc sech 2 scale height, equivalent to a 375 pc exponential scale height. The disk is cut off past 6 scale radii, or 35 kpc, for a total mass of 25 simulation units, or 5 . 8 × 10 10 M glyph[circledot] . We adopt a disk stellar mass-to-light ratio of ( M/L r ) D = 3 . 4. Each bulge has a 1.5 kpc effective radius. The exponential bulge and de Vaucouleurs bulge have masses of 14.75 units (3 . 4 × 10 10 M glyph[circledot] ) and 15 units (3 . 5 × 10 10 M glyph[circledot] ), respectively. The bulge-to-total mass ratio B/T M is about 33% in both models, larger than the 20-30% estimate for the R-band B/T ratio B/T R in M31 (Courteau et al. 2011). The models could compensate by using a lower ( M/L r ) B ; however, the bulge kinematics favor a lower value of 1.9 (Widrow & Dubinski 2005), which we adopt here. While ( M/L r ) B does not affect the simulations, the resulting bulge-to-total light ratio B/T r is 50%, and so mock images are more strongly weighted to bulge stars than disk stars. The rotation curve for both models is shown in Fig. 1. The bulge dominates within the inner 4-5 kpc and the halo thereafter, with the disk contribution typically half that of the halo. The non-maximal disk is both consistent with recent observations of spiral galaxies (see van der Kruit & Freeman (2011) for a review) and also promotes bar stability. Although the exponential bulge can be torqued into a bar, the intrinsic bar stability means that any remnant properties such as rotation are a result of the merging process and not secular instabilities. To scale our model to different masses, we multiply all masses by a factor m while retaining the same M/L R . Velocities are scaled by m 0 . 29 , assuring that the galaxies follow a Tully-Fisher relation V ∝ L 0 . 29 (Courteau et al. 2007). To maintain virial equilibrium ( R ∝ M/σ 2 , or, log( R ) ∝ log( M ) -2log( σ )), particle distances from the center of the galaxy are scaled by a factor of R ∝ M/σ 2 ∝ m 1 -2 × 0 . 29 , or, R ∝ m 0 . 42 . As a result, surface brightness scales weakly with mass -L/R 2 ∝ m 0 . 16 - consistent with the Tully-Fisher relation's assumption of nearly constant effective and/or central surface brightnesses. We do not incorporate scatter into the input galaxy scaling relations, so that scatter in the merger remnant scaling relations is both a lower limit and dependent on the formation process (merging) and bulge profile, rather than an additional input parameter like the Tully-Fisher relation's scatter. Similarly, we use the same bulge fraction for all galaxies. We deliberately avoid using bulgeless disks, as existing literature (e.g., Hernquist (1993)) shows that bulgeless disk mergers do not produce sufficiently high central densities. We will further discuss the implications of these choices in § 5. The lowest resolution model has 5,000 bulge, 10,000 disk and 40,000 halo particles, for a 1:2:8 bulge:disk:halo ratio, and 15,000 stellar particles. More massive galaxies have larger particle counts by factors of two, up to a maximum of 480,000 disk particles. Most groups have at least three galaxies with 60,000 stellar particles and only a few tens of galaxies have fewer than 30,000 stellar particles. By scaling resolution this way, stellar particles all have the same mass within a factor of three, while dark particles are not more than 10 times more massive than star particles, limiting spurious numerical artifacts. App. B discusses the effects of numerical resolution in greater detail; in summary, this resolution is more than sufficient for the more massive galaxies and adequate for the least massive satellites. Each group is simulated for 10 Gyr with the parallel N-body tree code PARTREE (Dubinski 1996). Figures 2 and 3 show a typical evolution for one such group. The simulations use 52,000 fixed timesteps of 0.02 units - about 195,000 years - and a softening length (spatial resolution) of 100 pc. We use an opening angle of θ = 0 . 9 with forces computed to quadrupole order. While this opening angle is somewhat larger than typical values of 0.7 to 0.8, PARTREE calculates forces between nearby particles in different trees directly, eliminating the source of the largest force errors. For θ = 1 . 0, PARTREE has been shown to produce median force errors under 0 . 2%, with 90% of force errors under 0 . 5% (Dubinski 1996); force errors with θ = 0 . 9 are considerably smaller.",
"pages": [
4,
5,
6
]
},
{
"title": "3. ANALYSIS",
"content": "The simulations are analyzed at three different epochs after 5.0, 7.7 and 10.3 Gyr, which correspond to formation redshifts of about 0.5, 1, and 2, respectively, if one assumes that the group formed at t=0 Gyr. Since the only redshift-dependent parameter in the initial conditions is the maximum radius of the group, analysis of the same group at different epochs is equivalent to assuming a different age for the group. Also, since galaxies are given an initial separation sufficient to prevent their halos from overlapping significantly, it typically takes 12 Gyr for the first mergers to occur. Groups with fewer galaxies complete the merger process after another 2-3 Gyr and so are not sensitive to the choice of formation time, while groups with more galaxies continue slowly accreting lower-mass satellites and growing even after 10 Gyr. Although we do not introduce additional galaxies into the simulation to mimic cosmological accretion, we note that the long merging time for less massive galaxies still allows for late-time mergers in richer groups. Once the simulations are complete, we create mock rband photometry and kinematics of each group at the three different epochs, placing the group at a mock redshift of 0.025 (about 100 Mpc away). In brief, we create SDSS-like photometry of the central galaxy out to 8 effective radii, including a sky background and appropriate signal-to-noise ratio. We use GALFIT (Peng et al. 2002, 2010) to fit a single Sersic profile to each galaxy. We also use GALMORPH (Hyde & Bernardi 2009a) to fit a de Vaucouleurs profile to the sky- and satellite-subtracted image, both for comparison to general Sersic fits and to the de Vaucouleurs fits of Hyde & Bernardi (2009b). Finally, we create spatially resolved kinematics at the same scale, and use these maps to measure kinematical quantities within the central region and the effective radius of the central galaxy. Although our simulations do not resolve faint satellites particular well, our pipeline is able to recover the properties of the central ellipticals with precision comparable to SDSS observations. Simulations are processed with our own imaging pipeline, which is intended to create images of the central galaxy in each group equivalent to those produced by the SDSS. We convert mass to luminosity to create nominal r-band images, using fixed stellar mass-to-light ratios for the bulge and disk components. We then extract a one-dimensional profile of the central galaxy in circular bins, masking out the central regions of satellite galaxies. A single Sersic profile is then fit to produce a rough estimate of the effective radius of the central galaxy ( R eff,est ). Next, we create a FITS image out to 8 R eff,est around the central galaxy. The image is smoothed by a point spread function (PSF) with a full-width at halfmaximum (FWHM) of 1.43 arcseconds, typical for SDSS r-band observations (Stoughton et al. 2002; Abazajian et al. 2009). Galaxies are imaged at a mock redshift of z obs = 0 . 025, typical for the SDSS spectroscopic sample used in (e.g. Hyde & Bernardi 2009a; Nair & Abraham 2010). Fig. 4 shows an example image of a typical galaxy. The pixel scale is identical to that used by SDSS, 0.396 arcsec/pixel. Most importantly, we add a sky background with both a mean surface brightness and variations comparable to SDSS observations. In the r band, the mean sky value is 20.86 and variations are Gaussian distributed with a standard deviation of 2 . 65%, equivalent to the SDSS asinh zero-flux magnitude of 24.80 (which itself was chosen to be approximately 1-sigma of typical sky noise). We also create maps of the projected dark matter distribution using the same pixel scale (but no PSF). In addition to photometry, we create kinematical maps of the first four moments of the luminosity-weighted velocities of particles in each pixel (velocity, r.m.s. velocity dispersion σ , and v3 and v4). Although we do smooth these maps by the same PSF and use the same pixel scale as the photometry, we do not add a sky background or any instrument-specific noise. We do not perform any fitting to the kinematic quantities, choosing r.m.s. velocity dispersions rather than fitting any profiles, and so the kinematical maps remain largely instrument-agnostic beyond the choice of pixel scale and PSF. The maps can then be used both to measure central velocity dispersions and spatially resolved kinematics, comparing to SDSS and Atlas3D respectively. Finally, we create an error map for the photometry, which will be used to perform χ 2 minimization in fitting profiles to the galaxies. The error is the square root of the luminosity in each pixel multiplied by some constant factor, which scales the signal-to-noise ratio across the image. The constant itself is simply related to the image exposure time, given a certain zero-point equivalent to 1 count per second (for SDSS r-band, this is about 26.7) and mean sky variation. This scheme contrasts with, e.g., Feldmann et al. (2011), and other simulations which use the square root of the number of particles in each pixel as the error. The per-pixel errors do not scale directly with the resolution of the simulation but should instead converge with increasing resolution. Similarly, setting a non-zero floor to the error map ensures that pixels with no signal are not ignored in the fit, which is necessary since the absence of a signal is meaningful. For each galaxy, we create images in 10 randomly oriented but evenly spaced projections. These are the ten projections passing through opposite faces of a regular icosahedron, but arbitrarily rotated with respect to the central galaxy. We also use the three projections corresponding to estimates of the principal axes of the central galaxy. We fit every galaxy in the image with a single Sersic profile using GALFIT. Sufficiently large galaxies (including the central galaxy) fit a boxiness parameter (C0) as well, which allows for elliptical isophotes to vary from diamond-shaped (C0 < 0) to rectangular- or boxshaped (C0 > 0). For highly inclined disks with a bulge, this can also provide a better fit than an unmodified ellipse. The GALFIT fits also include a fixed sky background equal to the mean sky brightness. We do not allow for the sky value to vary, as doing so would result in over-fitting the sky, a common problem in observations. Since different surveys and even different data releases of the SDSS have employed various methods for fitting sky backgrounds, we opt to avoid the difficulty of reproducing each methodology and simply fit with the known mean sky value. This does not, however, remove the pixel-to-pixel variation in sky brightness, which sets the effective limiting surface brightness in the image. We use the GALFIT fits to create a sky- and satellitesubtracted image of the central galaxy in each frame. This image is used to measure various quantities, including alternative non-parametric half-light radii. We also use GALMORPH to fit a single de Vaucouleurs profile to this sky-substracted image. This provides a direct comparison to the methodology used in Hyde & Bernardi (2009b), with the caveat that our use of GALFIT to fit the profiles of satellite galaxies may not match the exact methodology employed in masking nearby sources in SDSS or other surveys.",
"pages": [
6,
7
]
},
{
"title": "3.2. Photometric and Kinematic Measures",
"content": "Sizes and luminosities of the central remnants are measured several different ways. The preferred luminosity measure is the total luminosity within the deconvolved model image of the central galaxy, roughly equivalent to model magnitudes in SDSS and other surveys. For comparison, we also measure several other sizes and luminosities, including non-parametric Petrosian radii (Petrosian 1976) (see Abazajian et al. (2004) for the SDSS implementation and Graham et al. (2005) for analysis thereof). A thorough analysis of the suitability of these measures is presented in App. A. Kinematical maps are used to measure the velocity distributions - mean velocities (for rotation measures), dispersions, and higher order moments. Generally, we use central dispersions within 1/8 R eff and rotation measures within R eff . Velocity dispersions in the central remnants do vary radially, generally dropping from peak central values. Integral field surveys such as Atlas3D can measure dispersions out to 0.5 to 1 R eff , whereas fiber dispersions from SDSS are measured within fixed angular diameters, and hence variable fractions of R eff . Aperture corrections are often applied to fiber dispersions to convert them to a fixed fraction of R eff , with 1/8 R eff a typical choice for SDSS observations (Hyde & Bernardi 2009a). However, we find that central dispersions are nearly identical to effective dispersions (within 1 R eff ), with most galaxies lying on a linear relation and only a handful of outliers, so aperture corrections are not necessary for the simulations. The central velocity dispersions in simulations can be artificially depressed by softening of the gravitational potential. We mask out the central 300 pc to compensate, and measure central dispersions within 1/8 R eff where possible. For the few galaxies where 1/8 R eff is smaller than 300 pc, we enlarge the aperture by factors of 1/8 R eff until a reliable estimate is obtained. We have also considered the kinetic energy measure S = √ σ 2 + v 2 , or equivalently S = σ × √ 1 + ( v/σ ) 2 . This is a more accurate measure of the stellar kinetic energy for galaxies with significant rotation. However, most simulated galaxies do not have sufficient rotational support for this correction to be significant, and there are not yet any large samples of galaxies with published dispersions and v/σ to compare to.",
"pages": [
7
]
},
{
"title": "4. RESULTS",
"content": "The main results presented in this paper are the morphologies and kinematics of central group galaxies. Although we do fit satellites as well, this is mainly to exclude their contribution from the profile of the central galaxies. Few satellite galaxies are sufficiently well resolved to recover sizes and Sersic indices accurately, but we only require their total luminosities to be recovered and subtracted from the central galaxy's profile. Unless otherwise noted, all radii measured with elliptical annuli are √ ( a × b ), where a and b are the major and minor axis lengths.",
"pages": [
8
]
},
{
"title": "4.1. Observational Comparisons",
"content": "Our results are compared to three published data sets for nearby galaxies. The Atlas3D survey (Cappellari et al. 2011)(hereafter A3D) is a volume-limited integral field unit survey of the kinematics of 260 nearby earlytype galaxies. A3D provides kinematical maps (Emsellem et al. 2011) with a pixel size about twice as large as that of SDSS. This is mitigated by the larger aperture and very low redshifts (z < 0.01) of the sample as compared to our nominal mock sample redshift (0.025). Sersic profile fits are also available from Krajnovi'c et al. (2012), with photometry from a variety of sources but typically comparable to or better than SDSS. Simard et al. (2011)(hereafter S+11) published three different profile fits for over a million SDSS galaxies. We use the single Sersic decompositions for direct comparison and the free Sersic (bulge) plus exponential (disk) decompositions for diagnostic purposes. Although these fits were performed with a different code - GIM2D (Simard et al. 2002) - the fitting procedure is similar to our GALFIT fits. We select galaxies with spectroscopic redshifts 0 . 01 < z < 0 . 3 to ensure availability of a reliable V max volume correction term. We use the logarithmic median velocity dispersion between two sources - the SDSS DR7 and Princeton measurements (both included in DR7, Abazajian et al. (2009)). Stellar masses are based on the MPA-JHU DR7 catalog 1 , using fits to the multi-band photometry. Detailed visual classification of of nearly 6,000 earlytype galaxies from SDSS with z < 0.1 is provided by Nair &Abraham (2010)(hereafter N+10). Volume corrections are applied with the standard 1/V max weighting scheme (Schmidt 1968). Profile fits from this catalog include Petrosian sizes from the SDSS pipeline (Stoughton et al. Note . - Each row lists the breakdown of visual morphological classifications from N+10 of each of the subsamples from S+11, which are based on empirical cuts on various parameters rather than visual classification. Empirical cuts generally produce complete but impure samples of ellipticals and spirals, with substantial contamination by S0s. 2002) and Sersic fits from S+11. Although the original catalog of N+10 contained over 14,000 galaxies, eliminating bad fits and unmatched/misclassified objects provides just over 11,000 galaxies, of which nearly 5,000 are early-types. We exclude all SDSS galaxies with extreme velocity dispersions ( σ < 20 kms -1 or σ > 400 kms -1 ) or effective radii smaller than 0.3 kpc. Where visual classifications are available (A3D, N+10) we select galaxies with Hubble T-types (de Vaucouleurs 1959) less than 0 as earlytypes. T-types less than -3 are included in the elliptical sample while the remainder are classified as S0s. The majority of the S+11 sample does not have visual classifications, other than the small subset classified by N+10. We adopt a series of empirical cuts similar to those of Dutton et al. (2011) to identify early-type galaxies, testing these against the N+10 subset. The early-type sample contains galaxies with n s > 1, and - from the disk plus free n s fits - r-band bulge to total luminosity ratio B/T r > 0 . 4, disk inclination less than 63 degrees, and bulge r eff > 0 . 5kpc. Early-types must also have a spectroscopic eclass value less than -0.1 (see Yip et al. (2004), but note that the sign convention in SDSS is opposite), which selects galaxies with spectra consistent with a passive population. This early-type sample is subdivided into an elliptical subset, which imposes further cuts based on the single Sersic fits: g-band image smoothness S2 < 0 . 08, or g-band image smoothness S2 < 0 . 12 and B / T r < 0 . 6. These cuts are similar to those suggested by Simard et al. (2009) to select early-type galaxies from morphology alone, but also serve to decrease contamination from S0s and early-type spirals in the elliptical sample. All galaxies not classified as early-type but meeting the dispersion and R eff cut are identified as spirals. The samples obtained by applying these cuts to the N+10 catalog are listed in Tab. 2. The elliptical sample is 86% complete. While it is only 61% pure, the contamination mainly comes from S0s and not spirals. No cuts appear to be able to reliably classify S0s, which contaminate both elliptical and spiral samples. In principle, we could instead use the S+11 cuts on the N+10 sample rather than relying on visual classifications at all; however, visual classifications are repeatable and fairly robust (see Nair & Abraham (2010) for comparisons to previous classifications), and as seen in A3D, there are significant differences in rotational support between the elliptical and S0 population (Krajnovi'c et al. 2012), even if no automated morphological classification can separate them. Additional weightings are necessary to compare these catalogs to our own simulations, which probe a range of about 5 in absolute magnitude and have a nearly flat luminosity function. We produce r-band luminosity functions for each sample, then weight by the ratio of the simulated luminosity function to the observed one. Elliptical and S0 subsamples use all simulated galaxies versus E/S0 classifications from observed catalogs - i.e., we do not morphologically classify simulated merger remnants. This weighting procedure turns each observational sample into a catalog with equal numbers of galaxies at each luminosity, directly comparable to our simulations. Although the weightings are not vital for tight scaling relations like the fundamental plane, they are necessary for fair comparisons of weaker correlations and histograms marginalizing over luminosity.",
"pages": [
8,
9
]
},
{
"title": "4.2. Morphology",
"content": "As detailed in § 3.1, the central galaxies are fit with a single Sersic profile. Each profile has six free parameters (in addition to the two coordinates for the centre of the galaxy): the Sersic index n s , an effective half-light radius r eff , a surface brightness at this radius, an ellipticity glyph[epsilon1] , a position angle, and a boxiness parameter C0 modifying the shape of the ellipse from diamond (negative C0) to box-shaped (positive C0). Ellipticals have long been known to be best fit by larger Sersic indices than disks, to have small ellipticities and several correlations between size, luminosity and Sersic index. Any satellite galaxies in the image are also fit with a single Sersic profile.",
"pages": [
9
]
},
{
"title": "4.2.1. Sersic Indices",
"content": "Figure 6 shows various histograms of the Sersic index distribution for the B.n s =1, B.n s =4 and B.n s =all samples. Each individual bulge type produces a narrow distribution of Sersic indices. The B.n s =4 sample's distribution is narrower and peaked at a larger value of n s = 5 than the observational distributions. The B.n s =1 sample's peak at n s = 3 is significantly lower than those of N+10 and A3D ellipticals, and the distribution is narrower still than that of B.n s =4. The combined B.n s =all sample's n s distribution is nearly bimodal due to this separation and approximately twice as broad as B.n s =1 alone. By contrast, most observed distributions are unimodal, although the S+11 distributions show a larger peak of high n s galaxies, which is only reproduced in the B.n s =4 sample. There is also a hint of bimodality in the S0 distributions, which we have diminished by setting a lower limit of n s = 1. The peak of the S0 distribution is best reproduced by the B.n s =1, but, as will be demonstrated in § 4.4, the remnants' rotational support is far lower than that of typical S0s. None of the simulation samples can reproduce the width of the observed S0 distributions. Although each of the B.n s =1 and B.n s =4 samples are individually a poor fit to the elliptical data - particularly being too narrow of a distribution - the naive linear combination of the two (B.N s =all) provides a better match. The B.n s =all sample is also a better match to the elliptical distributions than the S0, the latter of which tend to smaller Sersic indices. While it is not a particularly realistic distribution - assuming that half of the groups in the universe contain galaxies with only exponential bulges while the other half contain de Vaucouleurs bulges - we will elaborate on the implications for more realistic bulge profile distributions in § 5. The difference in Sersic index between the Many- and Few-merger subsamples is small in the L.F.-sampled case but is maximized at about 0.5 for equal-mass mergers. Furthermore, the distributions of the Few-, equal-mass merger remnants in the different bulge samples are sufficiently narrow that the combined B.n s =all, Few-merger subsample is distinctly bimodal. Thus, it appears that multiple mergers are sufficient to broaden the distributions of remnant Sersic indices, but sampling progenitors from a realistic luminosity function can accomplish the same purpose, even with relatively few mergers. Major axis projections of central remnants have systematically larger Sersic indices than the medial or minor axis projections (bottom left and middle panels of Fig. 6), with the peak of the distribution shifted by about 1. Medial and minor axes have nearly identical distributions, even though their ellipticities and semi-major axes are not necessarily the same. As Fig. 7 shows, the variation in Sersic index for a single galaxy over different viewing angles is not usually much larger than one (and often smaller), so projections aligned near the major axis appear to produce the largest n s profiles. Only B.n s =4 mergers produce a correlation between luminosity and n s , as shown in Fig. 7. This is partly a result of more massive ellipticals being produced by more mergers. In both B.n s samples, Many-merger remnants tend to have larger n s at fixed luminosity. However, in the B.n s =4 sample, even the Few-merger subsample shows a small positive slope in Sersic index, whereas the trend is flat or even slightly negative for B.n s =1. The overall trend is dependent both on the initial bulge profile and the number of mergers. A positive dependence of merger rate on halo mass is a prediction of ΛCDM (e.g. Hopkins et al. (2010)). Exponential bulges, however, are simply not concentrated enough to create merger remnants with n s > 4, even with repeated merging. Thus, luminous ellipticals are unlikely to be the product of only exponential bulge mergers. The degree of agreement between simulations and observations is difficult to judge, since the observational samples do not completely agree. The N+10 n s -L relation appears to flatten above 10 10 L glyph[circledot] . This could be due to the larger redshift range of the S+11 sample; however, we find that GALFIT-derived Sersic fits to mock images at higher redshift tend to fit lower Sersic indices, so this systematic trend would have to be reversed in observed ellipticals to explain the shift.",
"pages": [
9
]
},
{
"title": "4.2.2. Ellipticities",
"content": "Ellipticities of the remnants are on the average slightly larger than observed elliptical samples but lower and more sharply peaked than S0s. Fig. 8 shows that there is only a small difference between the Many- and Fewmerger subsamples, while there is about a 0.05 shift towards rounder remnants from the B.n s =4 to B.n s =1 samples. On the whole the distributions are not unreasonable, lying closer to observations of ellipticals than of S0s, while lacking the tail of highly elliptical shapes found in S0s. Although the Many-merger remnants are slightly rounder on average than the Few-merger, the difference is not large even in equal-mass mergers. This is somewhat surprising, considering that the progenitor galaxy orbits are nearly isotropic and should tend to produce spheroidal remnants as the number of mergers increases. We will elaborate on this point further in § 5. The intrinsic ellipticities of the remnants along the principal axis projections are also shown in the bottom left and middle panel of Fig. 8. The distributions are consistent with the remnants being triaxial, with the median value in each projection being both different than the others and greater than zero. The smallest axis ratios are found for the minor axis projection, which would be the case for ellipsoids closer to prolate than oblate. Most galaxies have a medial axis ellipticity of around 0.4, with few being rounder than 0.2, indicating that almost all galaxies have a significantly shorter minor axis than the major axis. In addition to having larger Sersic indices, brighter galaxies trend toward smaller ellipticities and rounder shapes (Fig. 9). This trend might be expected for more luminous galaxies with many mergers. If the orbits of the merging galaxies are isotropically distributed, the resulting remnant should be close to spherical. Such a trend is present in the simulations, although it appears stronger for the B.n s =4 sample. Much of the scatter in the relation appears to be due to projection effects of the inherently triaxial simulated galaxies, although median ellipticities show significant scatter as well. The B.n s =1 sample also appears to have few very round ( glyph[epsilon1] < 0 . 1) remnants, especially at low luminosities.",
"pages": [
10
]
},
{
"title": "4.3.1. Size-Luminosity/Stellar Mass and Kormendy Relations",
"content": "Fig. 10 shows the Sersic model size-luminosity relation for principal axis projection of simulated galaxies after 10.3 Gyr, connecting otherwise identical groups with dif- n c i s r e S ferent spiral bulges. All relations have very small scatter. Part of the scatter is caused by the B.n s =1 sample having smaller sizes (a real effect) and lower luminosities (partly a real effect, but largely systematic, as will be shown in App. A). Regardless, both projection effects and different progenitor bulge profiles contribute to the scatter in the relation. Table 3 lists best-fit Sersic model size-luminosity relations for simulations and observations alike, obtained by least-squares minimization of the orthogonal scatter. In all of the simulation samples, the scatter is relatively small at about 0.1 dex. The scatter does not appear to be mainly due to projection effects or combining progenitors. Fits to major axis projections have similar scatter to the ten equidistant but randomly aligned projections. Similarly, though some groups show projectiondependent sizes and luminosities, these variations are smaller than the scatter in median values, and are likely a result of the mild correlation between Sersic index and luminosity of projections of the same galaxy (evidenced in Fig. 7). If sizes and luminosities are generally accurate to within 10-20% or 0.04-0.08 dex, as suggested by our testing, then some of the scatter could be intrinsic. The scatter in the unweighted simulation data is comparable to that in observed ellipticals (slightly larger than N+10), while the slope is considerably shallower and the intercepts larger. Separate fits to the Many- and Few-merger subsamples show a large difference of 0.05 to 0.1 in slope. Also, as Fig. 11 demonstrates, the Many-merger subsample is larger at fixed luminosity than the few merger sample. Thus, the slope of the predicted relation can be maximized by giving a larger weight to luminous, Manymerger remnants (and a smaller weight to faint galaxies), while applying the opposite weighting to groups of relatively few galaxies, such that their weights are largest at low luminosities. We apply such a weighting in Tab. 3 and find that it can steepen the slope of the sizeluminosity relation further than even the Many-merger subsample alone, bringing it close to observed values for N+10 but still short of S+11 and A3D. Table 3 also lists values for observational data, with both 1/V max corrections and optional weighting to match the luminosity function of the simulations. This weighting scheme only makes a significant difference in the S+11 sample - otherwise, most scaling relations are insensitive to weighting method, as one would expect if they are truly linear with uniform scatter. Some curvature may exist at the low- or high-luminosity extremes, but it is unclear whether it is real or systematic. The Petrosian R 50 size-luminosity relation (Tab. 4) shows smaller scatter than Sersic sizes, despite the fact that uncorrected Petrosian half-light radii systematically underestimate the luminosities of pure Sersic profiles and simulated galaxies alike. This is especially true for the B.n s =4 sample, which has slightly lower scatter than B.n s =1 mergers, despite having greater systematic errors on R eff due to its larger mean n s . The slopes are still shallower than those observed in N+10, but the difference can shrink to less than 0.05 if considering weightings for both simulations and observations. The implications of these results will be discussed further in § 5. The best-fit relations between size and stellar mass for the S+11 and N+10 catalogs are listed in Tab. 5. /epsilon1 /epsilon1 /epsilon1 The slopes are slightly shallower than those for the sizeluminosity relations and closer to (but not quite matching) those predicted by the simulations, which do not have significant variations in the stellar mass-to-light ratio. Thus some of the tension between the slopes of the simulated and observed size-luminosity relations can be resolved by accounting for the variable stellar mass-tolight ratio of observed galaxies, which increases in more luminous observed ellipticals but is nearly constant by construction in the simulated remnants. The Petrosian size-stellar mass relation shows slightly shallower slope, as with Sersic models. In fact, the slope and scatter of the weighted simulations (0.52 and 0.09) are within the quite small bootstrap errors (0.01) of the weighted observations (0.51 and 0.09), while the intercept is higher (-4.81 versus -5.08) but still also within the more generous error bars. Thus, it is entirely possible to match the slopes, and, to a lesser extent, the intercepts of the size-mass relation, depending on the fitting technique and sample weights. However, this alone does not justify either weighting scheme. The observational scheme is reasonable, since matching luminosity functions is necessary in order to make a fair comparison. The simulation scheme is not as well justified, since the number of mergers per group is somewhat arbitrary. The Kormendy relation (Kormendy 1977), shown in Fig. 12, has large scatter and shallow slope, especially for the B.n s =1 relation, which is nearly flat. None of the observed relations are quite linear. While the kink at small sizes is likely a systematic artifact, the curvature near 5-6 kpc appears more robust and also more significant than the equivalent curvature in the size-luminosity relation. As in the size-luminosity relation, it appears as if the simulated galaxies are either too faint for their size or too large to be so faint. Interestingly, the relation for large ellipticals appears to asymptote towards the slope of constant luminosity (d log( µ e ) / dlog(R eff ) = 5), which suggests that bright ellipticals can grow significantly in size without adding a large amount of stellar mass. In fact, many BCGs have exceptionally large effective radii and faint mean surface brightnesses. However, most of the similar simulated remnants in Fig. 10 are mergers of many equal-mass spirals, rather than luminosity function-sampled remnants - without this M-Eq subsample, the simulated Kormendy relation is rather weak, especially for the B.n s =1 sample.",
"pages": [
10,
11,
12
]
},
{
"title": "4.3.2. Faber-Jackson Relation",
"content": "The Faber-Jackson (velocity dispersion-luminosity, Faber & Jackson (1976)) relation is shown in Fig. 13, with best fits tabulated in Tab. 7. The simulated relations have slopes fairly close to the observations, though the intercepts are significantly lower. The turnover or curvature at low velocity dispersions ( < 100 kms -1 ) is likely not entirely real, since such low dispersions are near the spectrograph's resolution limit and unlikely to be reliable (Aihara et al. 2011). The luminosity function weightings make a significant difference in slope for the N+10 sample, which is likely due to this same curvature. The scatter appears to be mostly due to projection effects at the low-luminosity end but increases at high luminosities, where the Many- and Few-merger samples appear to diverge. The most robust conclusions from the data are that the slope for the S0 sample is significantly steeper than that for ellipticals, which in turn is slightly steeper than the canonical slope of 0.25, depending on the weighting scheme used. The scatter in the simulated relations is also significantly lower than in the observations, even when both bulge samples are combined. Unlike the size-mass/luminosity relations, the velocity dispersion-stellar mass relation (Tab. 8) is hardly changed from the velocity dispersion-luminosity relation, although the scatter shrinks slightly. The velocity dispersion-stellar mass relation also deviates from the canonical Faber-Jackson relation slope of 0.25, showing a scaling closer to σ ∝ M 0 . 3 ∗ .",
"pages": [
12,
13
]
},
{
"title": "4.4. Rotational Support",
"content": "We measure v/σ as the luminosity-weighted average within R eff , as used in IFU observations like Atlas3D (Cappellari et al. 2011). Most simulated ellipticals are slow rotators (Fig. 14). However, some projections show v/σ as large as 0.35 and can be classified as fast rotators despite having been formed from dry mergers. Cox et al. Simulations: Ten equally-spaced projections, randomly oriented",
"pages": [
13,
16
]
},
{
"title": "Observations",
"content": "Note . - Column definitions are as in Tab. 3. Velocity dispersion-stellar mass relations Note . - Column definitions are as in Tab. 3. /epsilon1 (2006) found that dry binary mergers only form slow rotators ( v/σ < 0.1), whereas some group mergers are clearly capable of producing fast rotators. Nonetheless, the scarcity of remnants with v/σ > 0 . 3 strongly suggests that dissipation is necessary to form fast rotators, as will be elaborated further in § 5. We also measure rotation in Fig. 15 by the more physically motivated measure λ (Cappellari et al. 2011) - essentially a radially-weighted v/σ tracing net projected angular momentum. While the distribution of rotational support is not wildly different from Atlas3D, there is a significant excess of slow rotators (especially flattened ones) and a complete absence of simulated galaxies with λ > 0 . 4. B.n s =1 mergers and Many-merger remnants tend to be slightly slower rotators, but the differences in both cases are not large. galaxies it appears as if the minor and medial axes are nearly identical, and so these galaxies are probably prolate spheroids. In this case, the distinction between major and medial axis projection is not very meaningful. Rotational support decreases with increasing luminosity in A3D ellipticals but not in simulations, as shown in Fig. 17. This is largely due to the inability of dry mergers to produce fast rotating, faint ellipticals. Furthermore, even if the morphological properties of some remnants (particularly Sersic indices of B.n s =1 remnants) are more consistent with S0s than ellipticals, observed S0s have far more rotational support than the vast majority of the simulated galaxies. There does not appear to be any strong correlation between rotational support and number of mergers in Fig. 14, or with total group mass or central galaxy luminosity. One might expect rotational support to at least correlate with net group specific angular momentum, assuming most of the halos merge and this angular momentum is conserved - however, Fig. 18 does not show any such correlation. It appears that repeated, mostly isotropic mergers cannot produce very fast rotators, even if the group itself has some net orbital angular momentum in one or more satellite galaxies.",
"pages": [
16,
17
]
},
{
"title": "Petrosian model size-stellar mass relation",
"content": "Note . - Column definitions are as in Tab. 3. Sersic model Faber-Jackson relations Simulations: Ten equally-spaced projections, randomly oriented",
"pages": [
16
]
},
{
"title": "5. DISCUSSION",
"content": "The main results of § 4 are that collisionless mergers in groups can produce central remnants with properties very similar to nearby elliptical galaxies. However, we do note several key differences between the simulation predictions and observed elliptical galaxies, not all of which are easily reconciled with dissipationless merging. We will also highlight how and why these results differ from previously published simulations.",
"pages": [
17
]
},
{
"title": "5.1. Morphology",
"content": "In App. A, it is shown that at the resolutions used in this study, luminosities, sizes and Sersic indices of spherical Sersic model galaxies can be recovered within about 5%, usually underestimating the true values. For group merger remnants, Sersic fits typically recover luminosities and sizes to within 10%, although luminosities tend to be more precisely recovered. By contrast, Petrosian radii systematically underestimate galaxy sizes and luminosities, negating the advantage of a non-parametric fit unless corrected for. Thus we conclude that single Sersic model fits are suitable for the simulated galaxies and can be compared directly to S+11 catalog fits, with the caveat that Sersic indices are the least robust parameter at low resolutions and are likely systematically underestimated. However, it is also true that in practice, Sersic fits can produce larger scatter on size than on luminosity, whereas Petrosian half-light radii appear to limit scatter in sizes - likely because they systematically underestimate the total luminosity of galaxies with large Sersic indices. Given these issues, our solution is to compare sizes between simulations and observations as fairly as possible, so that any systematic errors are likely to be shared between simulated and observed galaxies. We have compared Sersic index and ellipticity distribution to single Sersic profile fits of local (A3D) and SDSS (N+10, S+11) galaxies. Neither B.n s sample is a good fit to observed ellipticals alone, but the naive linear com-",
"pages": [
17
]
},
{
"title": "All bulge models",
"content": "bination of the two is a better fit while remaining inconsistent with S0s. However, such a naive combination still produces a near-bimodal distribution, in contrast to the single peak typical of observed ellipticals. A more natural choice of progenitors would likely smooth out this bimodality. For example, groups with half of the spirals having exponential bulges and the other half de Vaucouleurs would likely produce remnants with intermediate properties, filling in the gap between the two peaks of single-progenitor distributions. A smooth, realistic distribution of bulge profiles and bulge fractions would likely flatten the peaks and further broaden the distribution of remnant Sersic indices. Sersic indices of observed ellipticals are generally larger for more luminous galaxies, a trend reproduced by the simulated galaxies in Fig. 7. Hopkins et al. (2009) predicted that the dissipationless component in mergers (including both binary spiral mergers and some re-mergers of the resulting remnants) should show only a weak increase with luminosity and have low median values of about n s =3, with scatter of about 1. We find similar results for the B.n s =1 sample, for which n s is nearly con- stant with luminosity at a mean of 3 and with a range from 2 to 4. By contrast, the B.n s =4 sample not only has larger mean n s at about 5, but the median n s increases with luminosity by ∼ 0.5-1 per dex. This slope is close to that observed for N+10 and shallower than that in S+11, the discrepancy between these two samples having no obvious cause beyond probable contamination by S0s in the S+11 elliptical sample. Since the simulations have the same initial conditions other than their bulge profiles, this demonstrates that sufficiently concentrated progenitors can produce remnants with large n s through dissipationless merging. Furthermore, bulge n s =4 mergers appear to be a better fit for luminous ellipticals, while bulge n s =1 remnants match the less luminous ellipticals. If progenitor bulge profiles scale with luminosity (i.e., luminous spirals have larger bulge n s and merge to form luminous ellipticals), the scaling of elliptical Sersic index with luminosity can be matched more closely. Both simulations and observations show a slight tendency for more luminous ellipticals to be rounder, especially above 10 11 L glyph[circledot] (Fig. 9). Again, the S+11 sample differs from N+10, in this case being more flattened on average - likely due to S0 contamination. Nonetheless, we also find similar luminosity-dependent behaviour as for Sersic indices, as the B.n s =4 sample is a better match for bright, rounder ellipticals. The B.n s =1 sample has a shallower slope and appears to be too flattened on average. The observed distribution does not obviously require a combination of both simulation samples in the same way as the Sersic index-luminosity relation does, but such a combination does not disagree with the S+11 relation either. The N+10 ellipticity-luminosity relation is considerably flatter and could be reproduced by the B.n s =4 sample alone, with the scatter due to the substantial projection effects.",
"pages": [
18,
19
]
},
{
"title": "5.2.1. Size-Luminosity Relation",
"content": "Despite having randomized initial conditions, the simulated galaxies typically produce tight size-luminosity relations, with slight dependence on which size measure is used and whether different B.n s samples are com- bined. The Sersic model relations are somewhat tighter ( ∼ 0 . 1 -0 . 12 dex scatter) than those reported by Nair et al. (2011) ( ∼ 0 . 12 -0 . 15 dex scatter). This is partly systematic, since Nair et al. (2011) used circular Sersic fits provided by Blanton et al. (2005). Using the elliptical Sersic model fits of S+11 - which are more directly comparable to our own methodology and overlap with the N+10 sample - yields smaller scatter in the N+10 sizeluminosity relation of ∼ 0 . 09 dex. We also find slightly tighter scatter in the remnant Petrosian size-luminosity relation ( ∼ 0 . 09 dex), whereas the scatter for N+10 ellipticals remains largely unchanged whether Sersic or Petrosian model sizes are used. The small scatter in the size-luminosity relation should allay concerns that stochastic merging processes cannot produce tight scaling relations. Nipoti et al. (2009a) used simulations with multiple mergers of spheroidal galaxies to conclude that 'a remarkable degree of fine tuning is required to reproduce the tightness of the local scaling relations with dry mergers'. Instead, we find that mergers of many galaxies typically produce slightly tighter correlations than those with fewer galaxies, and the relations are tight regardless of which formation time is assumed for the groups (App. C). No fine tuning in galaxy orbits, number of mergers or any other parameters are required to produce tight scaling relations. Moreover, the Faber-Jackson relation has even tighter scatter than the size-luminosity relation. Rather than scattering galaxies away from existing scaling relations, multiple mergers appear to converge remnants towards a common relation, a behavior somewhat like the central limit theorem. However, it is still true that dry mergers of spirals in groups produce remnants with larger sizes and smaller velocity dispersions at fixed mass or luminosity, a problem shared with mergers of spheroids (e.g. Nipoti et al. 2003; BoylanKolchin et al. 2006; Nipoti et al. 2009b). Also, the scatter does appear to increase slightly with luminosity. This could simply be a reflection of the wide range of galaxy and merger counts for the luminous groups, which may not match the true range of cosmological merger histories for galaxy groups. We have tested mergers of spirals following a zeroscatter Tully-Fisher relation. The estimates for the scatter of merger remnant scaling relations can be considered lower limits, as they would likely have been higher had progenitors followed Tully-Fisher relations with intrinsic scatter and/or evolving slope and scatter. The observed Tully-Fisher relation does have significant scatter, even at low redshift (about 0.12 dex, from Courteau et al. (2007)), but the intrinsic scatter could be much lower. Hopkins et al. (2008) estimate that a scatter of 0.1 dex in the Tully-Fisher relation contributes about 0.04 dex scatter in the Fundamental Plane scaling relation; comparable scatter added to the existing size-luminosity relation scatter of 0.10-0.12 dex would make little difference if added in quadrature. While limiting scatter does not appear to be a challenge, in almost all cases the slope of the size-luminosity relation is shallower than observed and the intercept larger, so most galaxies are too large for their luminosities. The slopes of the remnant size-luminosity relations (typically R ∝ L 0 . 5 -0 . 6 ) are steeper than the progenitor spiral scaling relation ( R ∝ L 0 . 42 ) and the group scaling relation ( ρ =constant, R ∝ L 1 / 3 ). However, the remnants slopes are still shallower than those for the observations, which range from R ∝ L 0 . 6 to R ∝ L 0 . 8 depending on the observational sample and size measure. Encouragingly, the best matches are found between simulated remnants and N+10 ellipticals ( R ∝ L 0 . 66 ), the largest sample for which visual classifications are available. The steeper slope of the S+11 elliptical Sersic sizeluminosity relation (0.75 to 0.78) is of some concern. However, the elliptical classification for S+11 is based on empirical cuts on various parameters and results in significant ( ∼ 30%) contamination by S0s (Tab. 2). The much smaller A3D elliptical sample also has a slightly larger slope than N+10, and the luminosity function weighting does not change the slope. Since A3D used a slightly different fitting methodology with a much smaller volume sample, it is not clear whether this discrepancy is significant. The size-luminosity relation slopes for the simulated remnants are also steeper than that of R ∝ L ∼ 0 . 3 predicted for binary mergers remnants by Hopkins et al. (2009). However, those simulations began with a spiral scaling relation of similar slope (0.3), and so the merging process did not steepen the size-mass relation. By contrast, we have shown that group mergers are capable of steepening the slope of the size-luminosity relation by ∼ 0 . 1 -0 . 2 from progenitors to merger remnants without dissipation. Our models predict virtually no dependence of stellar mass-to-light ratio on luminosity - while the bulge and disk stellar mass-to-light ratios have different values, the fraction of disk stars within the effective radius varies little. However, luminous ellipticals do tend to have larger stellar mass-to-light ratios, so comparing to observed size-stellar mass relations lessens the discrepancies in the slopes by about 0.05 dex, depending on the sample and size measure. Such a dependence could be produced by more massive progenitor spirals having larger mean stellar mass-to-light ratios. We also did not include any scatter in the progenitor spiral Tully-Fisher relation or any scatter or luminosity dependence in bulge fractions. Extra scatter in either of these input galaxy properties would likely result in increased scatter in the remnant scaling relations. Any realistic luminosity dependence in the large M31 model bulge fraction would likely flatten the slope still further, since faint ellipticals would be produced by faint spirals with weak bulges. Dissipation is a tempting solution to the shallow sizeluminosity relation slope problem. Dissipation should decrease sizes at fixed luminosities and preferentially shrink faint ellipticals if their progenitors had larger gas fractions, resulting in a remnant with a larger fraction of stars formed in a central starburst. Luminositydependent gas fractions have been proposed by Robertson et al. (2006); Hopkins et al. (2008) as the source of the tilt in the fundamental plane scaling relation, a hypothesis which will be addressed in Paper II. Another possible remedy to the shallower slopes of the simulated size-luminosity relations is to weight the contributions from various simulation subsamples differently. Applying a simple linear weighting scheme of favoring B.n s =1 groups at low luminosity and B.n s =4 at high luminosities yields a steeper slope than a uniform weighting and a closer match with observations. Such a weighting also produces steeper slopes than either the Few- or Many-merger relations alone and can be justified if more massive halos undergo more mergers. While average halo merger rates are not strongly mass dependent (Stewart et al. 2008; Fakhouri et al. 2010), the groups we have simulated here would likely be those with higher than average merger rates. Although these schemes could resolve the mismatch in slopes, none save dissipation are viable solutions to the problem that simulated remnants are generally too large at fixed luminosity (Fig. 11). Our estimated stellar mass-to-light ratios are already quite low, so making small galaxies brighter appears to be out of the question. Numerical resolution effects are not large (App. B). Barring a strong redshift dependence in the sizes of observed ellipticals, this discrepancy is real. As a result, the Kormendy relation (Fig. 12) is poorly reproduced. Remnants are too faint at fixed sizes, so their effective surface brightnesses are also too faint by about a magnitude for small galaxies. The shallower slopes of the size-mass relation also translate into a weak simulated Kormendy relation (nearly non-existent in the case of the B.n s =1 sample).",
"pages": [
19,
20,
21
]
},
{
"title": "5.2.2. Faber-Jackson Relation",
"content": "The Faber-Jackson relation of simulated galaxies shows even smaller scatter (0.04 dex) than their size-luminosity relation or any observed Faber-Jackson relation (typically 0.08 dex, as in Fig. 13 and Tab. 7). The simulated remnants also have slightly shallower slopes ( σ ∝ L 0 . 28 ) than the observations ( σ ∝ L 0 . 27 -0 . 37 ), again depending on sample and weighting scheme. Curiously, the slope of the Faber-Jackson relation is nearly identical to that of the progenitor spiral Tully-Fisher relation ( V ∝ L 0 . 29 ), so multiple mergers appear to preserve the scaling of orbital velocity with mass while converting ordered rotation into random motions. This is despite the fact that the virial ratio in each group varies significantly, and so galaxy orbits within each group are not scaled uniformly the same way that stellar orbits within galaxies are. In virtually all cases, the slope of the remnant FaberJackson relation is steeper than the canonical value of 0.25 (or L ∝ σ 4 ). However, the observed relations show similar deviations and there is no compelling reason why ellipticals should follow this canonical relation. Indeed, the simulations of Boylan-Kolchin et al. (2006) predict scalings as steep as M ∝ σ 12 for major mergers with very small pericentric distances, so such deviations from the canonical relation are not unexpected. In most samples, the simulations have smaller dispersions than observed galaxies of the same luminosity. No weighting scheme can resolve this mismatch in the intercept of the Faber-Jackson relation, which is of similar magnitude (but opposite sign) as the offset in the intercept of the size-luminosity relation. Increasing the stellar mass-to-light ratios of the simulations would make galaxies of the same dispersion fainter but would worsen the match to the size-luminosity relation by making small remnants even fainter. Dissipation appears to be necessary here - central starbursts have been shown to increase velocity dispersions and shrink effective radii compared to purely dissipationless mergers (Hopkins et al. 2009). However, it is not clear whether a mass-dependent gas fraction would preserve the slope or flatten it. The mild curvature in the observational relations may be a systematic effect at low dispersions, although we have attempted to minimize such systematics by including two independent dispersion measurements. On the other hand, the simulated relations are insensitive to the choice 0 . 00 ≤ B / Tr > 0 . 15 ,N=2.7e4 0 . 15 ≤ B / Tr > 0 . 30 ,N=4.8e4 of velocity dispersion measure (central or effective; including rotational support or not) and most observed relations are insensitive to various weighting schemes.",
"pages": [
21,
22
]
},
{
"title": "5.2.3. Time or Redshift Dependence",
"content": "All of the results presented above apply to simulations analyzed after 10.3 Gyr, assuming an initial formation redshift of z=2.0 - a redshift at which pure dry mergers of disks are not likely to be common. However, the first merger in the group typically only occurs after another 1-2 Gyr. The scaling relations of remnants after 5 and 7.7 Gyr are similar to those at 10.3 Gyr, as shown in App. C, and so similar conclusions would be reached by assuming that the first merger occurred at z=0.5, when mergers were more likely to be dry or gas-poor. At face value, this also implies that the evolution in scaling relations is minimal; however, we caution that all of the groups are effectively the same age, so this prediction does not include any evolution from varied ages and assembly histories of real group galaxies.",
"pages": [
22
]
},
{
"title": "5.3. Spiral Progenitors and Their Bulges",
"content": "In the case of Sersic index distributions and scaling relations, it is tempting to consider whether a combination of progenitor bulge types (and possibly bulge fractions) could resolve the tensions with observations. To examine this further it is useful to ask what the distributions of bulge Sersic index and bulge fraction are for spirals as a function of luminosity. Not all S+11 spirals have a distinct bulge, nor are most images of sufficient quality to accurately measure bulge properties, so we consider the subset for which a bulge plus disk fit is required - that is, those with an F-test probability that a de Vaucouleurs bulge is not required is less than 0.32. This is about half of the spiral sample. The proportion for which a free Sersic bulge is required over a de Vaucouleurs bulge is much smaller, so we do not limit the sample any further. Fig. 19 shows the probabil- 0 . 00 ≤ B / Tr > 0 . 15 ,N=1.0e3 0 . 15 ≤ B / Tr > 0 . 30 ,N=1.5e3 ity densities of bulge Sersic indices as a function of galaxy luminosity, split into different bulge fraction bins. In all bins, classical (n s =4) bulges are at least a local maximum, although extreme bulge Sersic indices (n s =0.5 and n s =8, which are the lower and upper limits for S+11) are often the most common. The dependence on luminosity is not very strong, but in most bulge fraction bins, fainter spirals are slightly more likely to have low Sersic index bulges than high. The S+11 spiral sample is known to be contaminated by S0s. In Fig. 20, we instead use the much smaller but visually classified sample of spirals from N+10. This smaller sample does slight evidence for correlation between luminosity and bulge Sersic index, at least for more bulge-dominated spirals. Also, the large fraction of bulge Sersic indices below 1 is greatly diminished, suggesting that those could be primarily S0 contaminants in the S+11 sample, or possibly more poorly resolved, higherredshift spirals which appear in S+11 but not N+10. In either case, both samples contain substantial fractions of spirals with large bulge fractions. The M31 model used in our simulations has a large bulge mass fraction (0.33) and luminosity fraction (0.5). Such fractions are not uncommon, even at low luminosities. de Vaucouleurs bulges are also quite common, whereas exponential bulges are at least not exceptionally rare, especially for bulge-dominated spirals. Even if groups of spirals have broad distributions of bulge profiles, as in Fig. 19, their median values could also lie close to the limiting cases of exponential or de Vaucouleurs in our simulations. Also, a wide distribution of bulge profiles is indeed a realistic solution to the problem of single-progenitor mergers producing remnants with narrow Sersic index distributions. Real mergers in groups would likely produce wider, less bimodal distributions of Sersic indices than the single-progenitor simulations.",
"pages": [
22
]
},
{
"title": "5.4. Rotational Support",
"content": "The abundance of fast-rotating, faint ellipticals is at odds with the simulation predictions. However, as Figs. 14 and 15 show, multiple mergers can product remnants with moderate rotational support. This contrasts with the results of Cox et al. (2006) that dissipationless binary mergers only produce slow-rotating remnants with v /σ < 0 . 15 - all the more so because Cox et al. (2006) measured major axis rotation curves, whereas our simulations (and A3D) average over R eff . Nonetheless, our simulations are unable to produce any remnants with λ > 0 . 35. The simulated remnants show little or no change in rotational support as a function of luminosity (Fig. 17), unlike observations, and do not produce any of the fast-rotating, moderately luminous S0s found in A3D. There is also an abundance of flattened remnants with minimal rotation, unlike in Atlas3D. Cox et al. (2006); Bois et al. (2011) and others have shown that significant rotation can be easily produced in gasrich mergers. Dissipation is likely necessary to produce some ellipticals, particularly faint ones, and most likely a large fraction of S0s - if S0s are formed through mergers. However, it should emphasized that many of the simulated galaxies are consistent with the properties of some A3D galaxies, particularly bright ellipticals, so dissipation may not be necessary in all cases.",
"pages": [
22,
23
]
},
{
"title": "6. CONCLUSIONS",
"content": "We have investigated the hypothesis that elliptical galaxies can form through collisionless mergers of spiral galaxies by creating a sample of numerical simulations of such mergers and comparing the results directly with observations of local ellipticals. We draw the following key conclusions: These results demonstrate that many of the properties of elliptical galaxies are consistent with their emergence through multiple dry mergers of spiral galaxies. Perhaps most importantly, these properties also differ significantly from those of remnants formed through binary dry mergers of spirals, as reported in previous studies. This not only adds to an increasing body of evidence supporting the case for multiple mergers (e.g. Bournaud et al. 2007; Naab et al. 2009; Trujillo et al. 2011; Hilz et al. 2013) but also demonstrates that such mergers can produce tight scaling relations - in some cases tighter than observed ellipticals - as long as the progenitor spirals are drawn from a realistic luminosity function and scaled appropriately. Several major concerns remain for a purely dissipationless formation scenario for elliptical galaxies. The first is the limited amount of rotational support in the merger remnants and the absence of any correlation between rotation and luminosity. The second is the large sizes (and low velocity dispersions) of faint ellipticals, which result in a shallow size-luminosity relation and poorly reproduced Kormendy relation. While this second point could be resolved without dissipation (e.g. by merging more compact disks at high redshift), dissipation does appear to be necessary to produce fast-rotating ellipticals. Dissipation could also solve the second problem, as central starbursts would produce more compact remnants with higher dispersions. Perhaps the greatest challenge for dry mergers lies in matching the tilt of the fundamental plane with respect to the virial relation. Previous work has suggested that dissipational processes are the cause of this tilt and that dry mergers cannot produce any tilt (Robertson et al. 2006; Hopkins et al. 2008). This point will be addressed in Paper II of this series (Taranu et al. 2013).",
"pages": [
23,
24
]
},
{
"title": "7. ACKNOWLEDGMENTS",
"content": "D.T. would like to thank B. Abraham, L. Bai, D. Krajnovic, T. Mendel and P. Nair for fruitful discussions and for providing data used herein, as well as the anonymous referee for helpful suggestions. D.T. acknowledges the support of Ontario Graduate Scholarships for this work. Simulations and analyses were performed on the Canadian Institute for Theoretical Astrophysics' Sunnyvale cluster and the University of Toronto's SciNet cluster. H.Y. acknowledges support from grants from the National Science Engineering Research Council of Canada and the Canada Research Chair program.",
"pages": [
24
]
},
{
"title": "A. ANALYSIS PIPELINE TESTING",
"content": "Several aspects of the simulation analysis pipeline merit further testing. First, we would like to determine if the pipeline can recover known or measurable quantities such as the total mass/luminosity and half-light radii in single galaxies. This is accomplished by analysing a sample of spherical, pure Sersic profile plus dark matter halo galaxies generated with GalactICS. This allows us to simultaneously test whether GalactICS can generate equilibrium Sersic profile models (which is how the bulges of progenitor spirals are initialized) and whether the analysis pipeline can successfully recover input parameters at arbitrary resolutions. We analyse these models before simulating them in any way. In App. B, we examine the results of simulating these simple models with PARTREE to test numerical convergence. We can also use the group simulations themselves to test the analysis pipeline. Although we do not know the structural parameters of merger remnants a priori - indeed, they do not necessarily follow a single Sersic profile at all - the total luminosity is known in groups which have merged to a single remnant. Similarly, we can directly measure a half-light radius from mock images with no PSF or sky background in these cases and compare to observational estimates from the SDSS-equivalent mock images. This procedure allows us to determine whether single Sersic profile fits can simultaneously recover the total luminosity of a galaxy and its half-light radius. In addition to Sersic fits from GALFIT, we fit de Vaucouleurs profiles with GALMORPH and measure nonparametric Petrosian radii to determine if these size measures can consistently recover the true half-light radius of a galaxy.",
"pages": [
24
]
},
{
"title": "A.1. Sersic plus Halo Models",
"content": "Our reference Sersic plus halo models consist of a single Sersic profile bulge and a dark halo with the same baryonic mass ratio as our fiducial M31 models. We produce n s = 2 and n s = 4 models to cover most of the range of typical elliptical surface brightness profiles. We create models with R eff of 2, 4, 8 and 16 kpc, again covering ranges of typical elliptical galaxies and massive spiral bulges. The 2-kpc model is slightly larger than the 1.5-kpc bulge in our fiducial M31 model. All models are in virial equilibrium and follow a size-luminosity relation log( R eff ) = 0 . 7 log( L r ), with one model exactly on this relation and an extra model either over- or underluminous for its size. Each model is imaged at mock redshifts of 0.01, 0.025 and 0.1. These models will be used in the future to test recovery of scaling relations. However, for now we are mainly interested in whether the pipeline can recover the known values of n s , R eff and L for each model and whether the systematics depend on any of those parameters. In addition to varying the galaxy luminosity as a function of size, each set of Sersic plus halo models is simulated at three resolutions. The lowest resolution has 15,000 star and 40,000 dark matter particles, identical to the lowest resolution model used in the simulations. The resolution increases by a factor of 8 each step such that the highest resolution model has 7,680,000 star and 20,480,000 dark particles, or at least a factor of two more than the total particle counts of the most massive group simulations. In principle these models should be rescaled versions of each other; however, the nominal SDSS PSF and signal-to-noise ratio set a physical scale for mock images, while our fixed softening length sets another physical scale for the simulation.",
"pages": [
24
]
},
{
"title": "A.2. Sersic Quantities",
"content": "For n s = 2 models imaged at z=0.025, GALFIT Sersic fits show excellent agreement with expectations, even at low numerical resolution. With just 15,000 star particles, sizes are recovered to within 1 ± 0 . 5% for 2 kpc radius, although larger galaxies have underestimated sizes to the level of 3 ± 1% at R eff = 16 kpc. However, n s is underestimated by 10% for R eff = 2 kpc galaxies, which improves to 4 ± 1% at R eff = 16 kpc. Luminosities, in turn, are underestimated at fairly constant levels of 3%, with standard deviations increasing with size from 0 . 1 to 1%. Similar trends are found at medium resolution but with smaller amplitudes - the largest errors on n s are just 1 . 4 ± 0 . 3% at 16 kpc, while errors on R eff are at most 3 . 5 ± 0 . 5% at 2 kpc and shrink to half of that value at 2 kpc. Errors on parameters are reduced by about a factor of two by imaging at nearby redshifts (z=0.01) and increase by about the same factor by imaging at z=0.1. These errors are not eliminated by increasing the image size (and shrinking the PSF relative to R eff ) but shrink dramatically at the highest numerical resolution, to well under 1% in n s and L and about 1% in R eff . This suggests that these parameters are in principle completely recoverable with SDSS-equivalent imaging and good sky subtraction. We have also fit n s = 4 de Vaucouleurs profiles using GALMORPH, as in Hyde & Bernardi (2009a). GALMORPH fits to the n s = 2 models show expectedly poor results. Sizes are overestimated by factors from 1.6 (at 2kpc) to 2.4 (at 16kpc) while luminosities are overestimated by 30 -60%. These results are not entirely unexpected - n s = 4 models have shallower outer profiles and hence more light at large radii compared to profiles with lower n s = 2. However, they do demonstrate that other free parameters such as the effective radius and mean surface brightness cannot adjust to compensate for an incorrect profile choice, and so pure de Vaucouleurs profiles are not a good choice to fit ellipticals if their underlying surface brightness profiles are truly Sersic profile with n s significantly lower than 4. For n s = 4 models, GALFIT free-n s fits show curiously constant fractional errors on sizes, consistently underestimating R eff by 8 -9 ± 1%. Underestimates of n s vary from a substantial 14 ± 1% at 2 kpc to 4 ± 0 . 6% at 16 kpc. Luminosity underestimates shrink from 5 ± 0 . 6% to 1 . 5 ± 0 . 3%. These errors are not improved by imaging at lower redshift and so are unrelated to the relative size of the PSF. Instead, they are reduced substantially by increasing numerical resolution. At the highest resolution, size estimates shrink to 5 -6 ± 1% at all sizes, while n s underestimates now scale from 8 -2%. Since typical resolutions for central group galaxies are at the medium level or slightly higher, we expect that n s is underestimated by 7% on average for a pure de Vaucouleurs fit, with better performance at smaller n s . Size estimates are a minimum of 2% lower for large galaxies and up to 10% off for R eff = 2kpc. By contrast, the GALMORPH fits with fixed n s = 4 accurately recover sizes and luminosities to better than 2% even at the smallest sizes and at medium resolution. We conclude that even in ideal situations, free-n s fits will systematically underestimate sizes and luminosities at the 5% level, whereas fixed-n s fits only perform better if the exact value of n s is known. In App. B we detail how these results change after 2 Gyr of simulation with PARTREE using a fixed 100 pc softening length, as in the simulations.",
"pages": [
24,
25
]
},
{
"title": "A.3. Group Simulation Results",
"content": "As Fig. 21 shows, Sersic models generally do an acceptable job recovering central galaxy luminosities. For the n s = 1 sample, model luminosities are typically 85-90% of the total in groups with no satellites with relatively small scatter. The Sersic luminosities of n s = 4 central galaxies appear to have little or no systematic deviations from the true luminosities, although the scatter appears somewhat larger than in the n s = 1 case. The largest discrepancies are found for groups with many mergers, particularly equal mass mergers, in which case the models can overestimate the central galaxy's luminosity by at least 20-30%, largely due to runaway growth of the effective radius and Sersic index. However, in most cases Sersic profiles appear to be appropriate fits to the galaxies. The underestimation of n s = 1 merger luminosities appears to be a systematic effect. The underestimation of luminosities in groups with satellite galaxies is difficult to quantify, as the total luminosity in satellites is not easily separable from that of the central galaxy. Testing whether half-light radii are recovered is also complicated by the presence of satellite galaxies. Nonetheless, we attempt to measure how closely R eff matches the 'true' half-light radius R 50 of the central galaxy in Fig. 22. We estimate R 50 as the radius enclosing half of the group luminosity in a given sky- and satellite-subtracted image, using the same best-fit ellipse as the Sersic model. The ratio should be unity if there are no satellite galaxies in the group and less than unity if there are. As Fig. 22 shows, half-light radii are more diffi- cult to measure than total galaxy luminosities - or rather, errors on half-light radii from Sersic fits are considerably larger than for luminosities, which likely contributes to the significant scatter in the Sersic size-luminosity relation compared to the Faber-Jackson relation. At first glance, the large scatter in the ratio of Sersic model to 'true' half-light radius might suggest that much of the error in the Sersic size-luminosity relation is due to systematics rather than any intrinsic scatter. However, the size-luminosity relation using total group luminosity and 'true' half-light radius still shows significant scatter (0.08 dex) even when limited to galaxies with no satellites. A much larger sample of higher-resolution simulations would be required to determine if this scatter is due to numerical effects or genuinely intrinsic.",
"pages": [
25
]
},
{
"title": "A.4. Petrosian Radii",
"content": "As in SDSS, the Petrosian radius R P is given by the radius at which the mean surface brightness in the ring bounded by 0 . 8 R P < r < 1 . 25 R P is 0.2 times the mean surface brightness within R P . As a non-parametric size measure, it requires no fitting to measure, unlike the Sersic R eff . Since the Sersic profile is an analytical solution, one can compute R P uniquely for any given n s . For n s = 3 to 6, R P /R eff ranges from 1.5 to 2. The Petrosian magnitude of a galaxy is often estimated as the flux contained within a radius of a factor N P larger than this Petrosian radius; SDSS uses N P = 2. Petrosian magnitudes effectively measure half-light radii within 3 -4 R eff rather than the nominal 8 R eff bounding box for the FITS images used to derive SDSS-equivalent magnitudes. We measure Petrosian radii using both circular apertures and elliptical apertures, using the best-fit ellipse from Sersic model fits in the latter case. Unfortunately, as shown in Fig. 23, Petrosian luminosities appear to underestimate the true galaxy luminosity by a similar amount to the analytical relation for purely circular profiles (see Graham & Driver (2005) for a reference to various Sersic quantities). Sizes are also underestimated to a similar degree as predicted for a pure circular Sersic profile, which suggests that most galaxies do not deviate greatly from a pure Sersic profile. The slight excess could be due to a number of factors, including the Sersic models underestimating the true half-light radii and/or Sersic indices, radial variations in the ellipticity or shape of the isophotes, or deviations of the underlying profile from a pure Sersic model, all of which are plausible. In principle, one can correct for this 'missing' flux using fitting formulae valid for a wide range of Sersic or other profiles (Graham et al. 2005), but this seems unnecessary given that the Sersic fits appear sufficient and are available for all of the simulations and observational catalogs alike.",
"pages": [
25
]
},
{
"title": "B. NUMERICAL CONVERGENCE",
"content": "We test the numerical convergence of the spherical Sersic plus halo models by simulating every galaxy for 2 Gyr at 3 different resolutions (differing in particle number by a factor of 8 in each step). We also test a subset of the group simulations at similar resolutions. All measurements are made using the same analysis pipeline as the results above; the images also have the same nominal redshift of z = 0 . 025. Sersic n Convergence is generally quite good. With a 0.2 Myr timestep, total energy is conserved to better than one part in 10 5 . With the initial conditions re-centered to the barycenter, linear momentum remains small. The net angular momentum vector is the least well conserved quantity in Sersic plus halo models; each orthogonal com- ponent can vary by up to 5% of the net rotation. However, the total angular momentum is usually dominated by a small number of dark matter halo particles at large distances from the galaxy center. Angular momentum conservation for baryons in isolated galaxies is considerably better, and deviations of 1 to 2% are typical for groups where the bulk of the angular momentum is initially in galaxy orbits. Having tested input parameter (Sersic index and effective radius) recovery with the analysis pipeline, we now turn to examining how these same parameters evolve in a 100 pc softened potential with a fixed, 0.2 Myr timestep, as in the group simulations. While idealized, these simulations are comparable to both the central ellipticals (which are slowly rotating and close to Sersic profiles, albeit somewhat flattened) and the bulges of the input spirals (which are smaller than the Sersic models and also slightly flattened by the presence of the disk) and will give estimates for how galaxy structure is affected by numerical resolution.",
"pages": [
25,
27
]
},
{
"title": "B.1. Sersic plus Halo Model Convergence",
"content": "For a typical model ( R eff =8 kpc) at very high resolution (7.68 million star particle), convergence of all parameters is achieved at the 1 to 2% level, with sizes, Sersic indices and dispersions shrinking slightly over 2 Gyr. Convergence is considerably worse for the n s = 4 model and is strongly resolution dependent. A factor eight drop to high resolution (0.96 million star particles) approximately double errors in all parameters to 2-4%. For medium resolution (120,000 star particle), n s = 4 models, parameters can shrink by over 10% - typical values being 5 to 15% for n s (4 to 3.4), 15% for sizes (8 kpc to 6.8 kpc) and 5% for dispersions. Thus, for larger ellipticals to be suitably resolved, a million or more stellar particles are required, especially if the profiles are as or more centrally concentrated than an n s = 4 model. Less centrally concentrated models such as n s = 2 are much less sensitive to numerical resolution and can be resolved by 100,000 stellar particles with at most 3 to 4% level drops in sizes and Sersic index. Only 4 simulations in the sample have fewer than 720,000 stellar particles, so central remnants are largely unaffected by numerical relaxation after formation regardless of their central concentration. Unfortunately, the results are not as encouraging for smaller models. For the smallest R eff =2 kpc model at low (15,000 star particle) resolution, Sersic indices shrink up to 50% (from 2 to 1.5, or 4 to 2.3). Sizes typically drop by less than 10%, but dispersions also shrink up to 20%. At high resolution, Sersic indices converge at the 5 to 15% level (from 2 to 1.9 and 4 to 3.4). Sizes remain constant for n s = 2 and drop at most 5% for n s = 4, with dispersions also shrinking by 3 to 5%. Typical remnants are resolved at close to this high resolution, so the greatest effect would be on the Sersic indices of small, high n s ellipticals. The greater concern with these results is the relaxation that occurs in the bulges of progenitor spirals. The effective radius of the M31 model is 1.5 kpc, but most galaxies are scaled to smaller sizes than this, with 0.5 to 1 kpc bulge R eff . Moreover, in groups with larger numbers of galaxies, total particle counts are larger, but individual spirals can have as few as 60,000 stellar particles, of which only 20,000 are in the bulge. The bulge is partially stabilized (and flattened) by the disk, but the disk forms a core near the center of the galaxy, and so one might expect the behaviour of these compact, marginally resolved bulges to be similar to the Sersic plus halo models. We will now test this hypothesis with convergence studies of group mergers.",
"pages": [
27,
28
]
},
{
"title": "B.2. Group Simulation Convergence",
"content": "We test numerical convergence in the groups by running a selected sample with a factor of eight higher and lower resolution and comparing parameters after the usual elapsed times (5.0, 7.7 and 10.3 Gyr). As all of the groups are resolved with an average of over a million stellar particles, numerical convergence is expected to be good once groups have merged. However, as detailed above, the least massive spirals in more massive groups are not as well resolved, so not all groups are expected to be converged at our standard resolution.",
"pages": [
28
]
},
{
"title": "B.2.1. Parameter Recovery",
"content": "Fig. 24 shows convergence for several identical groups on the size-luminosity and sizeσ relations after 10.3 Gyr. Central remnant luminosities are fairly constant across all resolutions, but low resolutions can have slightly lower values. Sizes and dispersions are larger at low resolutions. Both trends continue from fiducial/medium to high resolution, although it is not as extreme - sizes are usually not more than 10 percent smaller between medium and high resolution. Sersic indices are systematically lower at low resolution by a factor of 1 to 2 (Fig. 25). The trend persists at high resolution, although n s typically increases by a smaller factor of 0.2 to 0.3 between mid to high resolutions. Of the four parameters tested, then, luminosity appears to be the most robust, while the Sersic index is most sensitive to resolution effects. The effects on sizes are too small to fully reconcile the mismatch between sizes of faint simulated galaxies compared to observed ellipticals (Fig. 11). Dispersions generally decrease with increasing resolution, and so numerical effects also cannot explain the lower intercept of the simulated Faber-Jackson relation compared to that of observed ellipticals (Fig. 13). In general, increasing resolution by a factor of eight produces similar trends in the group simulations as in isolated Sersic plus halo models - Sersic indices increase, while sizes and dispersions decrease. The effects are not very large going from our standard (medium) to high resolution but are considerable when stepping down to low resolution. We recommend that a minimum of a million stellar particles be used to adequately resolve spheroidal galaxies. While luminosities and masses remain converged at low resolution, sizes and dispersions are overestimated. Sersic indices are especially untrustworthy, being systematically offset lower by one or two from higher resolutions.",
"pages": [
28
]
},
{
"title": "C. SCALING RELATIONS AT DIFFERENT TIMES",
"content": "The scaling relations presented in § 4.3 are nominally for a zero-redshift galaxy population, assuming evolution from z=2. We can instead consider scaling relations at younger ages, assuming a fixed formation time for all groups. This is equivalent to assuming evolution from Sersic model size-luminosity relations at different times Note . - Sersic model size-luminosity relations of simulations after different times have elapsed (in Gyr) or, equivalently, assuming different formation redshifts (0.5, 1.0 and 2.0). Data are for ten equally-spaced, randomly oriented projections of each galaxy. Sersic model Faber-Jackson relations of simulations after different times Note . - Sersic model Faber-Jackson relations of simulations after different times have elapsed. Format as in Tab. 9. The slopes generally flatten slightly while intercepts increase and scatter remains constant at 0.04 dex. z=1 or z=0.5, since the only initial redshift-dependent parameter in the initial conditions is the group size. One might also consider combining groups from different snapshots into a single sample to simulate a sample with galaxies of different ages; however, this is best left to purely cosmological initial conditions with known merger trees and formation times. With these caveats in mind, we now present predictions for the evolution of the slope and scatter of selected scaling relations assuming a fixed formation time for all groups. The best-fit relations measured in Tab. 9 show slight evolution with time in the slopes (increasing) and intercepts (decreasing) and limited evolution in scatter. The steepening of the slope and lowering of the intercept would seem to suggest that brighter ellipticals grow off the relation at later times while fainter ellipticals grow slowly, if it all - in our case largely by construction, since the Few-merger sample does not have any late-time mergers. This interpretation is complicated by the fact that some of the largest groups do not have a relaxed, early-type central remnant formed in the earlier time steps and so are not included in the sample at earlier times but are included later on. Thus, as in most observational catalogs, not all of the descendants can necessarily be clearly identified with a previous early-type ancestor. The best-fit Faber-Jackson relations measured in Tab. 9 also show slight evolution of the slope, but in /circledot /circledot the opposite sense (decreasing/flattening), with a corresponding increase in the intercept. However, the scatter remains largely unchanged at 0.04 dex.",
"pages": [
28,
29
]
},
{
"title": "REFERENCES",
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"pages": [
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}
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2013ApJ...778...71G
https://arxiv.org/pdf/1309.5827.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_85><loc_87><loc_86></location>The diagnostic potential of Fe lines applied to protostellar jets 1</section_header_level_1>
<text><location><page_1><loc_14><loc_77><loc_86><loc_82></location>T. Giannini 1 , B. Nisini 1 , S. Antoniucci 1 , J. M. Alcal´a 2 , F. Bacciotti 3 , R. Bonito 4 , 5 , L. Podio 6 , B. Stelzer 4 , E.T. Whelan 7</text>
<text><location><page_1><loc_20><loc_72><loc_27><loc_74></location>Received</text>
<text><location><page_1><loc_48><loc_72><loc_49><loc_74></location>;</text>
<text><location><page_1><loc_52><loc_72><loc_59><loc_74></location>accepted</text>
<text><location><page_1><loc_42><loc_66><loc_58><loc_67></location>To appear in Ap. J.</text>
<text><location><page_1><loc_12><loc_41><loc_88><loc_46></location>1 INAF-Osservatorio Astronomico di Roma, via Frascati 33, I-00040 Monte Porzio Catone, Italy</text>
<text><location><page_1><loc_12><loc_36><loc_88><loc_40></location>2 INAF-Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131 Napoli, Italy</text>
<text><location><page_1><loc_14><loc_33><loc_84><loc_34></location>3 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy</text>
<text><location><page_1><loc_14><loc_30><loc_88><loc_32></location>4 INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, I-90134 Palermo,</text>
<text><location><page_1><loc_12><loc_27><loc_16><loc_29></location>Italy</text>
<text><location><page_1><loc_12><loc_21><loc_88><loc_26></location>5 Dipartimento di Fisica e Chimica, Universit´a di Palermo, Piazza del Parlamento 1, I- 90134 Palermo, Italy</text>
<text><location><page_1><loc_12><loc_16><loc_88><loc_20></location>6 UJF-Grenoble 1 / CNRS-INSU, Institut de Planetologie et d'Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble, F-38041, France</text>
<text><location><page_1><loc_14><loc_13><loc_88><loc_14></location>7 Institut f¨ur Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics,</text>
<text><location><page_1><loc_12><loc_10><loc_58><loc_11></location>Eberhard Karls Universit¨at, 72076 T¨ubingen, Germany</text>
<section_header_level_1><location><page_2><loc_44><loc_85><loc_56><loc_86></location>ABSTRACT</section_header_level_1>
<text><location><page_2><loc_17><loc_10><loc_83><loc_81></location>We investigate the diagnostic capabilities of the iron lines for tracing the physical conditions of the shock-excited gas in jets driven by pre-main sequence stars. We have analyzed the 3 000-25 000 ˚ A, X-shooter spectra of two jets driven by the pre-main sequence stars ESO-H α 574 and Par-Lup 3-4. Both spectra are very rich in [Fe II ] lines over the whole spectral range; in addition, lines from [Fe III ] are detected in the ESO-H α 574 spectrum. NLTE codes solving the equations of the statistical equilibrium along with codes for the ionization equilibrium are used to derive the gas excitation conditions of electron temperature and density, and fractional ionization. An estimate of the iron gas-phase abundance is provided by comparing the iron lines emissivity with that of neutral oxygen at 6300 ˚ A. The [Fe II ] line analysis indicates that the jet driven by ESO-H α 574 is, on average, colder ( T e ∼ 9 000 K), less dense ( n e ∼ 2 10 4 cm -3 ) and more ionized ( x e ∼ 0.7) than the Par-Lup 3-4 jet ( T e ∼ 13 000 K, n e ∼ 6 10 4 cm -3 , x e < 0.4), even if the existence of a higher density component ( n e ∼ 2 10 5 cm -3 ) is probed by the [Fe III ] and [Fe II ] ultra-violet lines. The physical conditions derived from the iron lines are compared with shock models suggesting that the shock at work in ESO-H α 574 is faster and likely more energetic than the Par-Lup 3-4 shock. This latter feature is confirmed by the high percentage of gas-phase iron measured in ESO-H α 574 (50-60% of its solar abundance in comparison with less than 30% in Par-Lup 3-4), which testifies that the ESO-H α 574 shock is powerful enough to partially destroy the dust present inside the jet. This work demonstrates that a multiline Fe analysis can be effectively used to probe the excitation and ionization conditions of the gas in a jet without any assumption on ionic abundances. The main limitation on the diagnostics resides in the large uncertainties of the atomic data, which, however, can be overcome</text>
<text><location><page_3><loc_17><loc_85><loc_60><loc_86></location>through a statistical approach involving many lines.</text>
<text><location><page_3><loc_17><loc_77><loc_82><loc_81></location>Subject headings: ISM:jets and outflows - stars:pre-main sequence - ISM: lines and bands - ISM: individual objects: ESO-H α 574, Par-Lup 3-4</text>
<section_header_level_1><location><page_4><loc_42><loc_85><loc_58><loc_86></location>1. Introduction</section_header_level_1>
<text><location><page_4><loc_12><loc_36><loc_88><loc_81></location>Jets from young stars play a key role in the dynamics of star formation and disk evolution. They regulate the process of stellar accretion, by both removing the angular momentum generated by accreting material in the disk, and modifying the inner disk physics, thus influencing the evolution of proto-planetary systems. The specific role of jets in the dynamics and evolution of the accreting system strongly depends on the parameters that characterize their structure and excitation, which are in turn related to their formation and heating mechanisms. From an observational point of view, information on the jet physics and dynamics can be retrieved through the analysis of the forbidden lines emitted by the jet plasma when it gets excited in shocks: to this aim, strong optical lines, such as [O I ], [S II ] and [N II ] lines, are widely used and specific diagnostic tools, able to retrieve a complete set of parameters (namely electron density, n e , temperature, T e and ionization fraction, x e ) have been developed (e.g. Bacciotti & Eisloffel 1999). The knowledge of these parameters is fundamental to an understanding of jet acceleration mechanisms (e.g. MHD disk-winds or X-winds, e.g., Shu et al. 1994; Ferreira 1997) and for measuring the mass flux rate ( ˙ M jet ). ˙ M jet is the quantity regulating the efficiency of the jet and is directly related to the disk mass accretion rate ( ˙ M acc ).</text>
<text><location><page_4><loc_12><loc_21><loc_88><loc_34></location>Although widely exploited and constantly refined, the diagnostic tools based on bright optical lines suffer from several intrinsic limitations. Firstly, optical lines trace specific excitation conditions and hence force the assumption that the gas in the jet has a constant temperature and density. This assumption is in contrast with combined optical/near infrared line analysis, which has shown that gradients in temperature and density up to</text>
<text><location><page_5><loc_12><loc_41><loc_88><loc_86></location>orders of magnitudes usually occur in the cooling region behind the shock front (Nisini et al. 2005, Podio et al. 2006). Secondly, diagnostic tools based on ratios between lines of different atomic species require one to assume a set of elemental abundances, which in turn imply an uncertainty on the parameters (temperature and density) more than 40% (Podio et al. 2006). Finally, the use of optical lines requires an a priori knowledge of the visual extinction, a circumstance that often makes the optical diagnostic applicable only to jets of the more evolved sources, where the reddening is negligible. All the above limitations can be circumvented by using different lines of the same species, covering a wide range of wavelengths particularly sensitive to extinction variations. In this respect diagnostic of iron (in different ionization stages) represents a very well suited tool. Indeed, since iron line spectrum covers all the wavelengths between the Ultraviolet (UV) and the Near-Infrared (NIR), it is sensitive to a large range of excitation conditions, allowing therefore to derive a complete view of the post-shock cooling region. The aim of the present paper is to probe the potential of the iron lines in probing the jet physical parameters. Our test-cases are two jets we have observed with the X-shooter spectrograph in the wavelength range ∼ 3 000 -25 000 ˚ A, namely the jets excited by the sources ESO-H α 574 and Par-Lup 3-4.</text>
<text><location><page_5><loc_12><loc_13><loc_88><loc_36></location>ESO-H α 574 ( α J 2000 . 0 = 11 h 16 m 03 s .7, δ J 2000 . 0 = -76 · 24 ' 53 '' ), spectral type K8, is a low-luminosity source in the Chamaleon I star-forming region located at a distance d =160 ± 17 pc (Wichmann et al. 1998). The low luminosity of 3.4 10 -3 L /circledot (Luhman 2007), which is a factor ∼ 150 lower than the luminosity of the typical T Tauri stars of the same spectral type, is interpreted as due to a disk seen edge-on. The source powers a bipolar jet (HH 872) of total projected length of 0.015 pc (3140 AU). It was discovered by Comer'on & Reipurth (2006) in a [S II ] image at 6728 ˚ A as a chain of knots, of which knots A1, A, B, C, D form the blue-shifted jet and knot E forms the red-shifted jet.</text>
<text><location><page_6><loc_12><loc_67><loc_88><loc_86></location>Par-Lup 3-4 ( α J 2000 . 0 = 16 h 08 m 51 s .44, δ J 2000 . 0 = -39 · 05 ' 30 '' ), spectral type M5, is located in the Lupus III dark cloud at d =200 ± 40 pc (Comer'on et al. 2003). This object also appears to be under-luminous, being about 25 times fainter than typical M5 pre-main sequence objects (L=3 10 -3 L /circledot , Mer'ın et al. 2008). As in the case of ESO-H α 574, its low luminosity is likely due to the obscuration of the star by an edge-on viewing disk (Hu'elamo et al. 2010). The jet was discovered by Fern'andez & Comer'on (2005), with emission extending in opposite directions with respect to the star for a total length of ∼ 1240 AU.</text>
<text><location><page_6><loc_16><loc_63><loc_87><loc_65></location>The X-shooter spectra of the two objects have been already investigated by Bacciotti</text>
<text><location><page_6><loc_12><loc_28><loc_88><loc_62></location>et al. (2011, hereinafter BWA11) and by Whelan et al. (2013, hereinafter WBA13). Both these papers determine the mass ejection to mass accretion ratio ˙ M jet / ˙ M acc . While in Par-Lup 3-4 this ratio is at the upper end of the range predicted by jet models, the value found in ESO-H α 574 of ∼ 90 can be partially reconciled with the predictions of magneto-centrifugal jet acceleration mechanisms only if the edge-on disk severely reduces the luminosity of the accretion tracers. The numerous spectral lines detected in the two jets, along with the kinematical properties derived in the line profiles are presented in WBA13. In the present paper we concentrate our analysis on the many iron lines detected in both spectra. The outline is the following: in Sect. 2 we briefly summarize the details of the observations and present the spectroscopic data; in Sect. 3 we describe the iron excitation and ionization models and derive the jet physical parameters. In Sect. 4 we discuss the results, which are summarized in Sect. 5.</text>
<section_header_level_1><location><page_6><loc_36><loc_21><loc_64><loc_22></location>2. Observations and Results</section_header_level_1>
<text><location><page_6><loc_12><loc_11><loc_87><loc_18></location>The present work is part of a coherent series of papers that deal with our X-shooter survey of Pre-Main Sequence (PMS) objects. The overall aspects, such as scopes, data reduction procedures, calibrations and results are thoroughly discussed in Alcal'a et al.</text>
<text><location><page_7><loc_12><loc_85><loc_87><loc_86></location>2011 and Alcal'a et al. 2013. Here we just recall the information which is essential for the</text>
<text><location><page_7><loc_12><loc_38><loc_88><loc_83></location>presented subject. The X-shooter spectra of ESO-H α 574 and Par-Lup 3-4 were acquired on April 7 2010, with an integration time of ∼ 1 hr per object. The slit, aligned with the jet axis, was set to achieve a resolving power of 5 100, 8 800 and 5 600 for the UVB (3 000 5 900 ˚ A), VIS (5 450 - 10 200 ˚ A) and NIR arm (9900 - 24 700 ˚ A), respectively (slit widths: 1 . '' 0, 0 . '' 9, 0 . '' 9). The pixel scale is 0 . '' 16 for the UVB and VIS arms and 0 . '' 21 for the NIR arm. The data reduction was performed independently for each arm using the X-shooter pipeline version 1.1., which provides 2-dimensional spectra, background-subtracted and calibrated in wavelength. Post-pipeline procedures were then applied by using routines within the IRAF and MIDAS packages to subtract sky lines and obtain 1-dimensional spectra. These are then divided by a telluric spectrum to remove the atmospheric features, and to do the flux-calibration. The complete spectrum was obtained by comparing the flux densities in the overlapping portions of the spectra of adjacent arms. While UVB and VIS spectra are perfectly aligned, the NIR spectrum of ESO-H α 574 appears lower of a factor ∼ 1.26. Flux losses in the NIR arm are not uncommon in the X-Shooter spectra and are caused by a misalignment between the NIR with respect to the VIS and UVB arms (Alcal'a et al. 2013). No correction was need to re-align the three arms spectra of Par-Lup 3-4.</text>
<text><location><page_7><loc_12><loc_20><loc_88><loc_36></location>As far as the ESO-H α 574 jet is concerned (hereinafter ESO-H α 574), we concentrate here on the iron lines detected in the brightest knot A1. This is also the closest to the exciting source, extending from the source itself up to 2 '' away (320 AU, see Figure 2 of BWA11, upper panel). The Par-Lup 3-4 jet (hereinafter Par-Lup 3-4) was integrated up to a distance of 1 '' from the source continuum, which corresponds to 200 AU (see Figure 2 of BWA11, lower panel).</text>
<text><location><page_7><loc_12><loc_12><loc_88><loc_19></location>Figures 1 and 2 show the portions of the spectra of the two objects where iron lines are detected, while Figure 3 shows the Grotrian diagram of Fe + levels from which the detected lines originate. The maximum energy level is at more than 30 000 cm -1 above the ground</text>
<text><location><page_8><loc_12><loc_73><loc_88><loc_86></location>state, and the line wavelengths cover the whole investigated range (in blue, green, and red we indicate ultra-violet, optical, and near-infrared lines, respectively). Similarly, Figure 4 gives the diagram of Fe ++ levels. Note that, due to the level structure, all the emitted lines lie only in the ultra-violet range, although the covered energy range is comparable to that of Fe + .</text>
<text><location><page_8><loc_12><loc_38><loc_88><loc_71></location>The line fluxes of all the detected lines are listed in Table A.1 of WBA13. Here we give in Table 1 the observed line ratios R ESO -Hα 574 and R Par -Lup 3 -4 of the [Fe II ] lines detected in the two objects with respect to the bright line at 4277 ˚ A. Lines originating from the same multiplet are grouped together and listed in order of decreasing energy of the upper level. In the last column, the ratio R ESO -Hα 574 /R Par -Lup 3 -4 is reported. Since the differential extinction between the two objects is negligible (see Sect. 3.1.1), this ratio gives a qualitative indication on whether or not the excitation conditions are similar in the two objects. Indeed, lines with excitation energy /greaterorsimilar 20 000 cm -1 (ultra-violet and optical lines) and those with excitation energy /lessorsimilar 20 000 cm -1 (near-infrared lines) have < R ESO -Hα 574 /R Par -Lup 3 -4 > ≈ 1.15 and < R ESO -Hα 574 /R Par -Lup 3 -4 > ≈ 2.6, respectively. This in practice suggests that in Par-Lup 3-4 the most excited lines are brighter (in comparison to the 4277 ˚ A) than in ESO-H α</text>
<text><location><page_8><loc_12><loc_35><loc_88><loc_39></location>574, a circumstance that could reflect a higher gas temperature.</text>
<text><location><page_8><loc_12><loc_23><loc_87><loc_33></location>Notably, [Fe III ] lines are detected only in ESO-H α 574 (see Table 2). This result cannot be explained with a different sensitivity in the X-shooter spectra of the two jets, which are similarly bright and were integrated for a comparable amount of time. Therefore, this feature also points to different excitation conditions in the two objects.</text>
<section_header_level_1><location><page_9><loc_39><loc_85><loc_61><loc_86></location>3. Line fitting model</section_header_level_1>
<text><location><page_9><loc_12><loc_74><loc_88><loc_81></location>In this section we describe the excitation and ionization models which provide the main physical parameters of the two investigated jets. The results of the comparison between observations and models are summarized in Table 4.</text>
<section_header_level_1><location><page_9><loc_37><loc_67><loc_63><loc_68></location>3.1. The excitation model</section_header_level_1>
<text><location><page_9><loc_12><loc_16><loc_88><loc_64></location>The observed ratios between lines from the same ionic species (e.g. Fe + or Fe ++ ) can be compared with the predictions by an excitation model to derive the physical conditions of the gas. To this aim we adopted a Non Local Thermal Equilibrium (NLTE) approximation for line excitation. One of the main issues of such line modeling regards the choice of the atomic dataset. The complexity of the iron atomic system, which involves hundreds of energy levels (with multiple metastable levels), makes it very difficult to get accurate atomic data sets (both radiative and collisional). For example, seven different computations of the Einstein coefficients for the spontaneous radiative decay (A-values) have been implemented for Fe + , which may differ from each other by more than 50%. Bautista et al. (2013) have evaluated the uncertainties in the line emissivities due to the combinations of the uncertainties on A-values, collisional coefficients and propagation of these two on the level populations. For typical shock-excitation conditions, namely T e ∼ 10 000 K and density between 10 2 -10 8 cm -3 , they find a very wide range of uncertainties, which vary from less than 10% (e.g. lines at 1.256 µ m and 8616 ˚ A) to more than 60% (e.g. lines at 5.330 µ m and at 5527 ˚ A). As shown by the same authors, the most effective way to circumvent the problem is to apply a statistical approach by including in the analysis a large number of lines.</text>
<text><location><page_9><loc_12><loc_13><loc_88><loc_14></location>In our model we use the up-to-date atomic database of the XSTAR compilation (Bautista</text>
<text><location><page_10><loc_12><loc_76><loc_88><loc_86></location>& Kallman 2001 2 ), which gives energy levels, A-values and rates for collisions with electrons (these latter for temperatures between 2 000 K and 20 000 K) for the first 159 and 34 fine-structure levels of Fe + and Fe ++ , respectively. The implications on the results when adopting different data sets will be commented in Sect.3.1.1.</text>
<text><location><page_10><loc_12><loc_16><loc_88><loc_73></location>The NLTE model assumes electronic collisional excitation/de-excitation and spontaneous radiative decay. Possible contributions on line emissivities due to radiative processes are discarded at this step of the analysis but will be considered in Sect. 3.3. The free parameters of the excitation model are the electron temperature T e and the density n e , which can be derived from the observed flux ratios once these latter are corrected for the visual extinction ( A V ) along the line of sight. This latter parameter is usually derived from the flux ratio of lines emitted from the same upper level, this being independent from the level population and therefore a function only of the line frequencies and the A-coefficients. As stated above, however, the large uncertainties associated with these latter values, are reflected in a poor estimate of A V , especially if one considers only two or three lines, as is often done with the NIR [Fe II ] lines (see also Giannini et al. 2008). Therefore, we have taken the extinction as a further free parameter of the excitation model. To derive the differential extinction at each line wavelength, we adopt the extinction curve by Draine (2003). To minimize the uncertainties, we included in the fit only the un-blended lines detected with a signal-to-noise (snr) ratio larger than 5 (i.e. 35 lines for ESO-H α 574 and 20 lines for Par-Lup 3-4) and checked the compatibility of the fit with line fluxes at lower snr ratio a posteriori . First, we constructed a grid of model solutions in the parameter space 2 000 K < T e < 30 000 K (in steps of δT e =1000 K); 10 2 cm -3 < n e < 10 7 cm -3 (in steps of log 10 ( δn e /cm -3 ) = 0.1) and A V ≤ 2 mag (in steps of δA V =0.5 mag). Then, following the method for line fitting proposed by Hartigan & Morse (2007), we have iteratively changed</text>
<text><location><page_11><loc_12><loc_79><loc_88><loc_86></location>the line used for the normalization, hence considering all the possible sets of line ratios. Each of them was then compared with the grid of theoretical values to find the model with the lowest value of χ 2 .</text>
<section_header_level_1><location><page_11><loc_41><loc_72><loc_59><loc_73></location>3.1.1. [Fe II ] lines fit</section_header_level_1>
<text><location><page_11><loc_12><loc_29><loc_88><loc_68></location>The result of the excitation model considering the complete set of [Fe II ] lines detected in ESO-H α 574 is depicted in Figure 5. The minimum χ 2 -value is found if the line at 4277 ˚ A is taken as a reference and the corresponding line ratios are reported in Table 1. The best-fit of the ESO-H α 574 [Fe II ] lines gives the following parameters : A V = 0 mag, T e = 9000 K, and n e = 2.0 10 4 cm -3 . A gas component at a single pair ( T e , n e ) fits reasonably well all the lines but systematically underestimates those coming from some doublets and sextets levels (b 2 H, a 6 S, and a 2 G), shown with different colors in Figure 5 and reported in Table 3. In particular, ratios involving lines from a 6 S and a 2 G levels (8 lines) are underestimated by a factor of two, while those from level b 2 H (2 lines) are underestimated by a factor of four. This systematic behavior, which can be reasonably ascribed to the poor knowledge of the atomic parameters, has been already evidenced by Bautista & Pradhan (1998) for the a 6 S level. Notably, however, the same model is selected as best-fit irrespective from including or not the doublets and sextets in the fit, although with a higher minimum reducedχ 2 (hereinafter χ 2 ) in the latter case.</text>
<text><location><page_11><loc_12><loc_12><loc_88><loc_28></location>The sensitivity of the line ratios to the fitted parameters is probed in Figure 6, where we plot the χ 2 -contours in the density-temperature plane for A V =0 mag (minimum χ 2 = 0.9). Higher A V values return fits with substantially higher χ 2 and are therefore discarded (for example minimum χ 2 = 1.9 for A V =0.5 mag); this indicates that extinction decreases slightly from the ESO-H α 574 central source (where A V ∼ 1.5 mag, WBA13) to the jet. The plotted contours refer to increasing χ 2 values of 30%, 60%, and 90% with respect to the</text>
<text><location><page_12><loc_12><loc_79><loc_88><loc_86></location>minimum χ 2 value. From this plot we derive that temperature and density do not exceed (inside a confidence of 3σ ) the ranges 8 000 K /lessorsimilar T e /lessorsimilar 11 000 K and 6 10 3 cm -3 /lessorsimilar n e /lessorsimilar 6 10 4 cm -3 , respectively.</text>
<text><location><page_12><loc_12><loc_40><loc_88><loc_76></location>To check the reliability of the results, we have also attempted two different approaches: i ) to fit the data with a different set of collisional coefficients (Bautista & Pradhan 1998), which returns the same physical parameters but with a higher minimum χ 2 , and ii ) to fit the ultra-violet component and the infrared components separately, with the aim to test the possibility of the presence of different gas components. The χ 2 -contours of the ultra-violet lines fit (Figure 7) shows a best-fit value not significantly different from that obtained by the all-lines fit. Analogously, the temperature range does not significantly differ from that found in the all-lines fit. More interestingly, the density range traced by the ultra-violet lines points to higher densities (i.e. up to 10 5 . 8 cm -3 within a 3σ confidence level). This suggests that while temperature is fairly constant in the probed region (or that its variations occur over spatial scales much smaller than the angular resolution), density may be subjected to stronger gradients. Finally, the fit of the infrared lines (not shown here) gives results in good agreement with the all-lines fit.</text>
<text><location><page_12><loc_12><loc_13><loc_88><loc_37></location>In Figure 8 we show the best-fit model for the [Fe II ] lines observed in Par-Lup 3-4. The minimum χ 2 is found, as in the case of ESO-H α 574, by taking as a reference the 4277 ˚ A line. To better compare the line emission observed in this object with that of ESO-H α 574, we plot, together with the line ratios of the detected lines, also the 2σ upper limits at the wavelength of the lines detected only in ESO-H α 574. As anticipated in Sec. 2, in Par-Lup 3-4 the ratios between ultra-violet and optical/near-infrared lines are substantially higher. This circumstance is a consequence of the higher temperature probed ( T e =13000 K). The inferred electron density and extinction are n e = 6.0 10 4 cm -3 and A V =0 mag, respectively. As for ESO-H α 574, we find that the predictions of sextet and doublet levels</text>
<text><location><page_13><loc_12><loc_73><loc_88><loc_86></location>are systematically underestimating the observed ratios of a factor between two and three. Within a confidence level of 90%, the χ 2 -contour plot gives 11 000 K /lessorsimilar T e /lessorsimilar 20 000 K and 1.8 10 4 cm -3 /lessorsimilar n e /lessorsimilar 1.8 10 5 cm -3 (see Figure 9). Finally, if the collisional coefficients by Bautista & Pradhan (1998) are adopted, the best-fit gives T e =16000 K, n e = 8.0 10 4 cm -3 , A V = 0 mag.</text>
<section_header_level_1><location><page_13><loc_40><loc_66><loc_60><loc_67></location>3.1.2. [Fe III ] lines fit</section_header_level_1>
<text><location><page_13><loc_12><loc_32><loc_88><loc_63></location>The fit of [Fe III ] lines detected in ESO-H α 574 is presented in Figure 10. The best-fit model is obtained by taking as a reference the line at 4930 ˚ A (see also Table 2). This gives the following parameters: T e = 19000 K, n e = 2.0 10 5 cm -3 , A V = 0 mag. At variance with Fe + lines, lines of Fe ++ lie all in the ultra-violet range and come all from levels with similar upper energy. Consequently, we expect that Fe ++ lines are poorly sensitive to the temperature. This is clear in the χ 2 -contour plot of Figure 11, where all temperatures in the grid of NLTE solutions above 8 000 K are compatible with the observations (within a confidence level of 90%). Conversely, the electron density is better constrained within the range 1 10 5 cm -3 /lessorsimilar n e /lessorsimilar 6 10 5 cm -3 . This result confirms that indeed a density gradient exists along the jet of ESO-H α 574, and that [Fe III ] and [Fe II ] ultra-violet lines likely probe the same, high-density gas component.</text>
<section_header_level_1><location><page_13><loc_37><loc_25><loc_63><loc_26></location>3.2. The ionization model</section_header_level_1>
<text><location><page_13><loc_12><loc_12><loc_87><loc_22></location>To consistently interpret the [Fe II ] and [Fe III ] emission in ESO-H α 574 and to derive the fractional abundance Fe + /Fe ++ , we applied a ionization equilibrium code that involves the first 4 ionization stages of iron. The following processes have been taken into account: direct ionization, radiative and dielectronic recombination (data from Arnaud</text>
<text><location><page_14><loc_12><loc_64><loc_88><loc_86></location>& Raymond 1992), and direct and inverse charge-exchange with hydrogen (data from Kingdon & Ferland 1996). Notably, while the first three processes are a function only of the electron temperature, direct and inverse charge-exchange rates also depend on the fractional ionization x e = n e / n H , where n H = n H 0 + n H + . Moreover, since the electron transfer is more efficient when the involved ions (e.g. H 0 and Fe + ) have similar ionization potentials (IP) 3 , the charge-exchange rate is relevant only for the process Fe + + H + /arrowparrrightleft Fe ++ + H 0 . Therefore, it returns relevant results for the Fe + /Fe ++ abundance ratio, while it is negligible for both the Fe 0 /Fe + and Fe ++ /Fe +3 abundance ratios.</text>
<text><location><page_14><loc_12><loc_41><loc_88><loc_63></location>For T e = 8000 K, namely the lowest temperature derived from the χ 2 -contours of Figures 6 and 7, our model predicts a substantial fraction of iron in neutral form even if the gas is almost fully ionized (e.g. we get 30% of Fe 0 , 52% of Fe + , and 18% of Fe ++ for x e = 0.9). This strongly contrasts with the simultaneous lack of any Fe 0 line in the ESO-H α 574 spectrum together with the presence of Fe ++ lines. However, just a slight increase of the electron temperature at 9 000 K makes the neutral Fe 0 percentage drop to less than 10%, and that of Fe ++ to increase to more than 20%, in agreement with the observations. The expected percentage of Fe +3 is negligible for the whole range of temperature considered in</text>
<section_header_level_1><location><page_14><loc_12><loc_38><loc_25><loc_39></location>Figures 6 and 7.</section_header_level_1>
<text><location><page_14><loc_12><loc_20><loc_88><loc_36></location>To derive x e we solved the ionization equilibrium equations (together with the excitation equilibrium for each of the two species) to predict a number of [Fe II ]/[Fe III ] line ratios. We constructed a grid of model solutions in the range 0 ≤ x e ≤ 1 (in steps of δx e =0.05) and 9 000 K ≤ T e ≤ 14 000 K, being the upper value that derived from the χ 2 -contours of Figure 7. To estimate x e we consider the [Fe II ] ultra-violet lines and the [Fe III ] lines, assuming that they come from the same portion of the post-shock gas (see Sect. 3.1.2). We</text>
<text><location><page_15><loc_12><loc_70><loc_88><loc_86></location>consider 14 line ratios involving 7 [Fe II ] lines with two bright [Fe III ] lines at 4701.59 ˚ A and 5270.53 ˚ A. As an example, we show in Figure 12, upper panel, the [Fe II ]4244/[Fe III ]5270 ratio as a function of x e for the considered range of temperature. The observations are in agreement with 0.65 /lessorsimilar x e /lessorsimilar 0.85, where the lower (upper) value refers to the highest (lowest) temperature assumed. This value of x e is the same found (within the error range) if all the 14 ratios are considered.</text>
<text><location><page_15><loc_12><loc_58><loc_86><loc_69></location>For Par-Lup 3-4 we can derive an upper limit on x e by considering the upper limits on the [Fe III ] lines. Taking a grid in the range 11 000 < T e < 20 000 K (see Sect. 3.1.1 and Figure 9), we get x e /lessorsimilar 0.4. As an example, the derivation of x e from the ratio [Fe II ]4244/[Fe III ]5270) is shown in Figure 12, lower panel.</text>
<text><location><page_15><loc_12><loc_44><loc_87><loc_57></location>Typical x e values in protostellar jets range from 0.03 to 0.6 (Ray et al. 2007, Nisini et al. 2005, Podio et al. 2009), although x e = 0.8 is found in the High Velocity Component (HVC) of the DG Tau B jet Podio et al (2011). Therefore, while the fractional ionization of Par-Lup 3-4 is in the range of the most common values, that of ESO-H α 574 appears remarkably high.</text>
<section_header_level_1><location><page_15><loc_33><loc_37><loc_67><loc_38></location>3.3. Photoexcitation contribution</section_header_level_1>
<text><location><page_15><loc_12><loc_14><loc_88><loc_33></location>In Section 3.1.1 the observed line ratios have been interpreted in the light of collisional excitation. In this Section we explore whether an additional contribution from fluorescence excitation can be relevant. In ESO-H α 574 this possibility is supported by the detection of bright [Ni II] lines at 7377.8 ˚ A and 7411.6 ˚ A(WBA13), whose intensity is easily enhanced because of the pumping of an ultra-violet field (Lucy 1995), though the observed intensity ratio of around 10, is compatible only with collisional excitation (see Figure 2 of Bautista et al. 1996). In Par-Lup 3-4, only the 7377.8 ˚ A line is detected.</text>
<text><location><page_15><loc_12><loc_12><loc_85><loc_13></location>To better investigate the role of photo-excitation in ESO-H α 574, we have included in</text>
<text><location><page_16><loc_12><loc_61><loc_88><loc_86></location>the excitation model a radiation field, which can be produced either from the stellar photosphere or by a hot spot on the stellar surface produced by the accretion shock of the infalling matter. Both these fields have been approximated as W × B ν (T eff ) , where B ν is the black-body function at the stellar (or hot spot) temperature and W= 1/4 (R/r) 2 is the dilution factor, having adopted the stellar radius R= 3 R /circledot and the distance of the knot A1 from the star, r, equal to 100 AU (i.e. 0.2 '' , see BWA11). We take T eff = 4000 K for the stellar temperature and 6 000 K ≤ T eff ≤ 12 000 K for the hot spot temperature, following the model of Calvet & Gullbring (1998). The hot spot area has been taken between 10-30% of the stellar surface.</text>
<text><location><page_16><loc_12><loc_14><loc_88><loc_60></location>As shown by Lucy (1995), a powerful way to evaluate the relevance of photo-excitation, is to compute the so-called excitation parameter (U ex ), which is defined as the ratio between all the radiative and collisional excitation rates involving two given levels. From U ex , the 'second critical electron density' can be also derived, n ∗ e = U ex n e , such that for n ∗ e /greatermuch n e , fluorescent excitation is predominant with respect to collisional excitation. Assuming a stellar field and for n e = 210 4 cm -3 (Sect. 3.1.1), we get n ∗ e /lessorsimilar 10 2 cm -3 (or U ex < 5 10 -3 ) for all the levels, indicating that fluorescence excitation is negligible in this case. The importance of the hot-spot field was tested by varying both T eff and W in the ranges given above, obtaining n ∗ e up to 10 5 cm -3 . Thus, in principle, the presence of a hot-spot could have a role in fluorescence excitation. However, the comparison of the predicted intensity ratios with those observed in the ESO-H α 574 spectrum, indicates a marginal compatibility only for the lowest values of T eff and W (i.e. T eff ≤ 8 000 K and hot-spot area not exceeding 10% of the stellar surface). Hence, even if a hot-spot may exist, certainly it is not the main cause of the observed emission. As a note, and with reference to Sect. 3.1.1 and Table 3, we also report that none of the line ratios systematically underestimated by the collisional model can be reproduced even if fluorescence excitation is considered.</text>
<text><location><page_16><loc_12><loc_12><loc_85><loc_13></location>Finally, in the Par-Lup 3-4 case the distance between the central source and the jet is</text>
<text><location><page_17><loc_12><loc_79><loc_88><loc_86></location>not well defined as in the ESO-H α 574 case. Taking different values of W, we estimate that photo-excitation contribution, and in particular that due to the hot spot field, can be relevant for distances closer than 5-10 AU from the central source.</text>
<section_header_level_1><location><page_17><loc_43><loc_72><loc_57><loc_73></location>4. Discussion</section_header_level_1>
<section_header_level_1><location><page_17><loc_32><loc_67><loc_68><loc_68></location>4.1. Comparison with shock models</section_header_level_1>
<text><location><page_17><loc_12><loc_22><loc_88><loc_64></location>Once derived the physical conditions, the origin of the iron emission in the two jets was investigated in the framework of shock models. Figure 13, adapted from Figure 1 of Hartigan, Raymond, & Morse (1994), shows the variation of the ionization fraction, electron density, and temperature with the distance behind the shock front for a low velocity (35 km s -1 ) and an intermediate velocity (70 km s -1 ) shock, in the approximation of a slab geometry and for assumed values of the pre-shock density and magnetic field. For each combination of these parameters, we computed the intensity of the most prominent iron lines, then deriving their expected intensity variation along the overall post-shock region. In particular we show, in the left panels, the peak-normalized intensity profiles of ultra-violet, optical, near-infrared [Fe II ] lines (those coming from levels a 4 G, a 4 P, and a 4 D), and in the middle panels the profiles of [Fe III ] lines coming from level a 3 F. Notably, lines at different wavelengths peak at different distance from the shock front, in the dimensional scale of ∼ 10 13 - 10 14 cm. At the distance of our objects these scales correspond to hundredths of arcsec, which are not resolved at our spatial resolution, and therefore the excitation model of Fe + gives only average quantities.</text>
<text><location><page_17><loc_12><loc_10><loc_88><loc_20></location>It is also important to notice that the physical parameters derived in ESO-H α 574 and Par-Lup 3-4 cannot be directly compared with those depicted in Figure 13, which strongly depend on the assumed conditions of pre-shock density of the gas, magnetic field strength and shock velocity. Nevertheless, a trend between post- and pre- shock parameters can</text>
<text><location><page_18><loc_12><loc_41><loc_88><loc_86></location>be evidenced. We computed (see Table 5) the average < T e > , < x e > , < n e > and the compression factor C = n post -shock /n pre -shock , weighted by the intensity profiles of the various (groups of) lines depicted in Figure 13. By examining the data of Table 5, a number of conclusions can be drawn: 1) for a given shock velocity, lines at decreasing wavelengths trace progressively higher temperatures. Ionization fraction and electron density slightly increase with decreasing wavelength in the model with v shock =70 km s -1 , while they remain fairly constant and significantly lower if v shock =35 km s -1 ; 2) the average parameters probed by the mean of all [Fe II ] lines (fourth line of Table 5) indicate that increasing shock velocities correspond to decreasing temperatures and to increasing ionization fraction, electron density and compression factor. This points toward a higher shock-velocity in ESO-H α 574, where temperature is lower and electron density and ionization fraction are higher than in Par Lup 3-4 (see Table 4). Moreover, in the intermediate-velocity shock model, the [Fe III ] lines trace more specifically the portion of the post-shock region extending up to ∼ 10 13 cm behind the shock front, where the electron density reaches its maximum value. This region should therefore correspond to that traced by the observed [Fe III ] line ratios.</text>
<text><location><page_18><loc_12><loc_26><loc_88><loc_39></location>We also note that the above scenario is also consistent with the abundance ratios of the Fe 0 , Fe + , and Fe ++ depicted in the right panels of Figure 13. Indeed, while for a low-velocity shock the bulk of iron is singly ionized, for an intermediate velocity shock the ratio Fe + /Fe ++ ∼ 8 (at distances of the order of 10 13 cm), again consistent with the detection of Fe ++ only in ESO-H α 574.</text>
<text><location><page_18><loc_12><loc_12><loc_88><loc_25></location>Finally, we again remark that although the above analysis allows us to interpret the observations in a consistent framework of shocked origin, the pre-shock parameters of the two models taken as a reference are not consistent with the derived post-shock parameters. For example, for the measured < n e > and the compression factors of Table 5, the pre-shock density would be < n 0 > ∼ 7 10 3 cm -3 and ∼ 6 10 4 cm -3 for ESO-H α 574 and Par Lup</text>
<text><location><page_19><loc_12><loc_82><loc_88><loc_86></location>3-4, respectively, which are higher than the n 0 values at which the two models of Hartigan et al. (1994) are computed.</text>
<section_header_level_1><location><page_19><loc_35><loc_75><loc_65><loc_76></location>4.2. Gas-phase Fe abundance</section_header_level_1>
<text><location><page_19><loc_12><loc_17><loc_88><loc_71></location>The gas-phase Fe abundance x (Fe) is an indirect measure of the presence of dust inside the jet. In general jet launching models predict that the jet is dust-free as dust is completely destroyed in the launching region by the stellar radiation. Conversely, if the jet originates from a disk region extending beyond the dust evaporation radius, it could eventually transport some dust. This, in turn, could be then partially destroyed by the shock because of vaporisation and sputtering of energetic particles (e.g. Seab 1987; Jones 1999: Guillet et al. 2009). The degree of iron depletion is therefore also a function of the shock efficiency. Previous studies of x (Fe) in shock environments have given sparse results, from values close to solar abundance (e.g. Beck-Winchatz et al. 1996), up to intermediate (Nisini et al. 2002, Podio et al. 2006, 2009) and very high depletion factors (Mouri & Taniguchi 2000; Nisini et al. 2005). A powerful way to estimate the percentage of gas-phase iron ( δ Fe ), relies on intensity ratios involving lines of non-refractory species emitted in similar excitation conditions, as for example the [Fe II ]1.25 µ m/[P II ]1.18 µ m, as suggested by Oliva et al. (2001). Since phosphorous lines are not detected in our spectra, we investigate the possibility of using ratios involving [O I ] lines. To this aim, we solved the equations of ionization equilibrium for the first three ionic stages of oxygen, together with the statistical equilibrium for the first five levels of O 0 . The radiative coefficients are taken from the NIST database 4 while the rates for collisions with electrons are from Bhatia & Kastner (1995). As a result, we get the percentage of neutral oxygen and the</text>
<text><location><page_20><loc_12><loc_58><loc_88><loc_86></location>peak-normalized intensity profile along the post-shock region. In particular, that of [O I ] 6300 ˚ A shown in the middle panels of Figure 13, well resembles that of [Fe II ] ultra-violet lines. Therefore, we conclude that [Fe II ] ultra-violet lines and [O I ] 6300 ˚ A trace the same shock region and are therefore suited to measure δ Fe inside the shock. This is also roughly confirmed by the average parameters traced by the [O I] optical lines reported in Table 5 and taken from Bacciotti & Eisloffel (1999). Note also that other tracers commonly used to derive δ Fe , such as [S II ] 6740 ˚ A, are not as powerful as [O I ] 6300 ˚ A since their shock profile does not resemble that of any iron line (see e.g. Figure 3 of Bacciotti & Eisloffel, 1999). The same problem arises if the [O I ] 6300 ˚ A is compared with [Fe II ] near-infrared lines (see Figure 13).</text>
<text><location><page_20><loc_12><loc_20><loc_88><loc_57></location>To derive δ Fe , we thus selected several ratios [O I ] 6300 ˚ A 5 over bright ultra-violet [Fe II ] lines, whose observed values are compared with those expected for the < T e > , < n e > and < x e > determinations derived from the iron analysis. By assuming the solar iron and oxygen abundances with respect to hydrogen of 3.16 10 -5 and 6.76 10 -4 (Grevesse & Sauval 1998), we estimate δ Fe = 0.55 ± 0.05 and δ Fe = 0.30 ± 0.03 for ESO-H α 574 and Par-Lup 3-4, respectively. This result is in agreement with the shock interpretation given in the previous section. The higher efficiency in destroying the dust in the shock in ESO-H α 574 is due to its higher velocity, as expected from models of dissociative shocks (Guillet et al. 2009). In this respect, further observational evidence is provided by the detection in ESO-H α 574 of bright lines from other refractory species, such as Ca and Ni, which, on the contrary, are barely detected in Par-Lup 3-4 (BWA11, WBA13). Finally, we note that the derived values of δ Fe belong to the group of 'intermediate' depletion values, where the shock has not a sufficient strength to completely destroy dust. The presence of dust inside</text>
<text><location><page_21><loc_12><loc_82><loc_85><loc_86></location>the shock is in turn an indication that the jet launching region is larger than the dust sublimation zone.</text>
<section_header_level_1><location><page_21><loc_20><loc_75><loc_80><loc_76></location>4.3. Comparison with the diagnostics of other atomic species</section_header_level_1>
<text><location><page_21><loc_12><loc_12><loc_88><loc_71></location>Together with iron lines, the spectra of ESO-H α 574 and Par-Lup 3-4 are rich in other atomic emission lines (BWA11, WBA13), some of which commonly used to diagnose the physical conditions of the emitting gas. In this section we intend to compare the parameters derived from iron lines with those traced by ratios of lines of oxygen, nitrogen and sulphur. To derive the theoretical values of such ratios we have implemented simple NLTE codes for the lowest 5 fine structure levels of each species. The radiative coefficients are taken from the NIST database, while the electronic collision coefficients are taken from Pradhan (1976, [O II ]), Pequignot, & Aldovrandi (1976, [N I ]), Mendoza (1983, [N II ]), Hollenbach, & McKee (1989, [S II ]). The main results of this analysis, which are summarized in Table 6 are the following: i ) on average the temperature probed in ESO-H α 574 is in agreement with that probed with iron lines. In Par-Lup 3-4 the derived temperatures give sparse results, with T e ([O I ]) lower than T e ([Fe II ]) and with T e ([S II ]) not consistent with T e ([N II ]); ii ) ratios of different species probe different electron densities, with n e ([O II ]) > n e ([N I ]) > n e ([S II ]). This result can be explained by comparing the fitted values with the critical densities of the involved lines, which, at T e = 10000 K are of ∼ 10 8 cm -3 , ∼ 10 6 cm -3 , and ∼ 10 4 cm -3 for [O II ], [N I ] and [S II ], lines, respectively. While the densities traced with the [S II ] ratio are close to the critical value, and therefore not completely reliable, this is not the case for the density indicated by the [O II ] flux ratio. In ESO-H α 574 this density is the same as that inferred from the [Fe III ] and [Fe II ] ultra-violet lines, thus again supporting the result of a density gradient inside the jet. Notably, the [O II ] line ratio indicates that in Par-Lup 3-4 the density is higher than in ESO-H α 574, in agreement with what found with</text>
<text><location><page_22><loc_12><loc_85><loc_37><loc_86></location>the [Fe II ] VIS and NIR lines.</text>
<text><location><page_22><loc_12><loc_70><loc_88><loc_83></location>In conclusion, care should be taken to compare physical conditions derived from different atomic species and lines, due to the their different sensitivity to variations of physical parameters behind the shock front. In this respect, the rich iron spectrum from UV to NIR, with lines sensitive to a large range of excitation conditions, is particularly suited to obtain a more complete view of the post-shock cooling region.</text>
<section_header_level_1><location><page_22><loc_43><loc_63><loc_57><loc_64></location>5. Summary</section_header_level_1>
<text><location><page_22><loc_12><loc_49><loc_88><loc_60></location>We have analyzed the 3 000-25 000 ˚ A, X-shooter spectra, of two jets driven by low-luminosity pre-main sequence stars, ESO-H α 574 and Par-Lup 3-4, with the aim of investigating the diagnostic capabilities of the iron lines. Our analysis and main results can be summarized as follows:</text>
<unordered_list>
<list_item><location><page_22><loc_15><loc_21><loc_88><loc_46></location>- The spectra of the two objects are both rich in iron emission. More than 70 lines are detected in ESO-H α 574, (knot A1, up to 2 '' from the source), while around 35 lines are detected in the Par-Lup 3-4 jet (integrated up to 1 '' from the source). The spectra show substantially different features. While in the Par-Lup 3-4 jet only [Fe II ] lines are detected, the spectrum of ESO-H α 574 shows both [Fe II ] and [Fe III ] emission. The [Fe II ] lines are detected over the whole spectral range, coming from levels with energy up to more than 30 000 cm -1 . While in ESO-H α 574 the low-excitation, near-infrared lines are stronger than the high-excitation, ultra-violet lines, the opposite occurs in Par-Lup 3-4.</list_item>
<list_item><location><page_22><loc_15><loc_11><loc_88><loc_18></location>- Both [Fe II ] and [Fe III ] line ratios are interpreted through NLTE models. These allow us to derive both the gas parameters (electron density and temperature) along with the visual extinction. The [Fe II ] line fit indicates that the jet driven by ESO-H α</list_item>
</unordered_list>
<text><location><page_23><loc_17><loc_70><loc_88><loc_86></location>574 is, on average, colder ( T e ∼ 9 000 K) and less dense ( n e ∼ 2 10 4 cm -3 ) than the Par-Lup 3-4 jet ( T e ∼ 13 000 K, n e ∼ 6 10 4 cm -3 ). A more compact component ( n e ∼ 2 10 5 cm -3 ) inside the jet is revealed in ESO-H α 574 if the ultra-violet lines are fitted separately from the optical and near-infrared lines. This component, whose temperature is not well constrained, is likely the same responsible for the [Fe III ] line emission. The extinction appears to be negligible in both jets.</text>
<unordered_list>
<list_item><location><page_23><loc_15><loc_60><loc_88><loc_67></location>- The contribution of fluorescence excitation due to photons emitted from the central star was investigated. In ESO-H α 574 this effect is negligible, while it can have a role in Par-Lup 3-4 up to distances less than 10 AU from the central star.</list_item>
<list_item><location><page_23><loc_15><loc_47><loc_87><loc_57></location>- A ionization equilibrium code was applied to derive the fractional ionization ( x e ) inside the two jets. We get x e ∼ 0.7 in ESO-H α 574 and x e /lessorsimilar 0.4 in Par-Lup 3-4. In particular the value detected in ESO-H α 574 is remarkably high, as expected in high-velocity shocks.</list_item>
<list_item><location><page_23><loc_15><loc_23><loc_88><loc_44></location>- The observational differences evidenced in the iron spectra of the two jets have been qualitatively interpreted in the framework of shock models. The physical parameters derived from the excitation analysis are consistent with shocks with different velocities, with the shock of ESO-H α 574 being significantly faster than that of Par-Lup 3-4. Plots of post-shock [Fe II ] line intensities vs. distance from the shock front indicate that lines at different wavelengths trace different post-shock regions. In particular [Fe II ] ultra-violet and [Fe III ] lines are emitted only close to the shock front (within a distance of ∼ 10 13 cm), where the post-shock density reaches its maximum value.</list_item>
<list_item><location><page_23><loc_15><loc_10><loc_88><loc_20></location>- The shock strength of the jets is probed by measuring the gas-phase iron abundance ( δ Fe ). This was derived from the ratios of fluxes of ultra-violet [Fe II ] lines with that of [O I ] 6 300 ˚ A. Under the assumption of solar Fe and O abundances, we derive δ Fe ∼ 0.55 and 0.30 in ESO-H α 574 and Par-Lup, respectively. This evidence is in</list_item>
</unordered_list>
<text><location><page_24><loc_17><loc_82><loc_88><loc_86></location>agreement with the higher shock-velocity of ESO-H α 574, which in turn corresponds in a higher kinetic energy able to partially destroy the dust particles.</text>
<unordered_list>
<list_item><location><page_24><loc_15><loc_54><loc_88><loc_79></location>- The gas diagnostic derived from iron lines was compared with that obtained from bright lines of other atomic species detected in the X-shooter spectra. Although the average trend of temperature and density is the same (with ESO-H α 574 colder than Par-Lup 3-4), the derived values are in general not consistent with each-other. We ascribe this behavior to the low number of the used lines, able to cover a limited parameter range that depends on the specific line excitation energies and critical densities. Conversely, thanks both to the very rich spectrum of iron and to the wide spectral range covered with X-shooter, the analysis of iron lines allows us to get a very comprehensive and consistent view of the gas physics in the post-shock region.</list_item>
</unordered_list>
<section_header_level_1><location><page_24><loc_39><loc_47><loc_61><loc_48></location>6. Acknowledgments</section_header_level_1>
<text><location><page_24><loc_12><loc_37><loc_88><loc_44></location>We are grateful to Manuel Bautista and to an anonymous referee for their suggestions and constructive discussions. TG and JMA thank also G. Attusino. The ESO staff is acknowledged for support with the observations and the X-shooter pipeline.</text>
<table>
<location><page_25><loc_15><loc_14><loc_85><loc_81></location>
<caption>Table 1. [Fe II ] lines</caption>
</table>
<table>
<location><page_26><loc_15><loc_15><loc_85><loc_81></location>
<caption>Table 1-Continued</caption>
</table>
<table>
<location><page_27><loc_16><loc_41><loc_84><loc_79></location>
<caption>Table 1-Continued</caption>
</table>
<unordered_list>
<list_item><location><page_27><loc_17><loc_29><loc_54><loc_31></location>∗∗ Line with signal-to-noise ratio between 2 and 3.</list_item>
<list_item><location><page_27><loc_17><loc_26><loc_54><loc_28></location>a Blended with [Cr II] c 2 F 7 / 2 -a 4 G 9 / 2 at 5164.45 ˚ A.</list_item>
<list_item><location><page_27><loc_17><loc_23><loc_50><loc_24></location>b Blended with [Fe III] 3 P 2 -5 D 1 at 5412.08 ˚ A.</list_item>
<list_item><location><page_27><loc_17><loc_19><loc_54><loc_21></location>c Blended with [Fe II] b 2 P 1 / 2 -a 4 D 1 / 2 at 5527.60 ˚ A.</list_item>
<list_item><location><page_27><loc_17><loc_16><loc_50><loc_18></location>d In Par-Lup 3-4 blended with He I 3 P 0 -3 D 0 .</list_item>
</unordered_list>
<table>
<location><page_28><loc_34><loc_35><loc_66><loc_73></location>
<caption>Table 2. [Fe III ] lines in ESO-H α 574- knot A1.</caption>
</table>
<unordered_list>
<list_item><location><page_28><loc_34><loc_21><loc_66><loc_25></location>a Blended with [Fe II] a 2 D 3 / 2 -a 4 F 5 / 2 at 5412.65 ˚ A.</list_item>
</unordered_list>
<paragraph><location><page_29><loc_27><loc_82><loc_86><loc_84></location>] lines coming from doublets and sextets levels underestimated by the</paragraph>
<table>
<location><page_29><loc_39><loc_47><loc_60><loc_78></location>
<caption>Table 3. [Fe II NLTE model.Table 4. Fitted physical parameters.</caption>
</table>
<table>
<location><page_29><loc_28><loc_20><loc_72><loc_36></location>
</table>
<table>
<location><page_30><loc_16><loc_52><loc_84><loc_79></location>
<caption>Table 5. Intensity-weighted parameters in the shock cooling region (computed from the models of Fig. 13).</caption>
</table>
<unordered_list>
<list_item><location><page_30><loc_18><loc_49><loc_49><loc_51></location>a Taken from Bacciotti & Eisloffel, (1999).</list_item>
</unordered_list>
<table>
<location><page_30><loc_19><loc_13><loc_81><loc_40></location>
<caption>Table 6. Diagnostics of other atomic lines</caption>
</table>
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<figure>
<location><page_34><loc_15><loc_27><loc_77><loc_80></location>
<caption>Fig. 1.- UVB spectrum of ESO-H α 574 (black) and Par-Lup 3.4 (red) where iron lines are detected. Green labels : [Fe II ] lines; blue labels : [Fe III ] lines: magenta labels: blends. For clarity, the spectrum of Par-Lup 3-4 was augmented by a factor of 5 (in the reported units).</caption>
</figure>
<figure>
<location><page_35><loc_16><loc_24><loc_77><loc_76></location>
<caption>Fig. 2.- As in Figure 1 for the VIS and NIR spectra.</caption>
</figure>
<figure>
<location><page_36><loc_14><loc_33><loc_71><loc_73></location>
<caption>Fig. 3.- Grotrian diagram of Fe + levels associated with the observed lines. Groups of lines detected in different X-shooter arms are depicted with different colors : blue: ultra-violet lines, green: optical lines, red: near-infrared lines.</caption>
</figure>
<figure>
<location><page_37><loc_12><loc_30><loc_71><loc_70></location>
<caption>Fig. 4.- As in Figure 3 for the Fe ++ levels.</caption>
</figure>
<figure>
<location><page_38><loc_12><loc_34><loc_77><loc_80></location>
<caption>Fig. 5.- NLTE best-fit model of the [Fe II ] lines detected in ESO-H α 574. In the fitting procedure we have included the lines detected with snr ≥ 5, represented as filled circles (black: data, red: model). Lines detected with 3 ≤ snr < 5 and 2 ≤ snr < 3 are reported with open circles and open triangles, respectively. Down arrows are the fluxes of blended lines. Blue filled circles and magenta filled circles indicate the observed data and model predictions of lines coming from levels b 2 H, a 6 S, and a 2 G, which are not included in the fitting procedure (see text). The best-fit parameters are reported as well.</caption>
</figure>
<figure>
<location><page_39><loc_13><loc_58><loc_56><loc_83></location>
<caption>Fig. 6.χ 2 -contours of the fit through the [Fe II ] lines detected in ESO-H α 574. The curves refer to increasing values of χ 2 of 30 %, 60 %, 90 %. The minimum χ 2 value is given, as well.</caption>
</figure>
<figure>
<location><page_39><loc_13><loc_17><loc_55><loc_43></location>
<caption>Fig. 7.- As in Figure 6 for the fit of the [Fe II ] ultra-violet lines detected in ESO-H α 574.</caption>
</figure>
<figure>
<location><page_40><loc_13><loc_35><loc_77><loc_81></location>
<caption>Fig. 8.- NLTE best-fit model of the [Fe II ] lines detected in Par-Lup 3-4. In the fitting procedure the lines detected with snr ≥ 5, represented as filled circles (black: data, red: model) have been included. Lines detected with 3 ≤ snr < 5 and 2 ≤ snr < 3 are reported with open circles and open triangles, respectively. Down arrows are the blended lines or the 2σ upper limits of lines not detected in Par-Lup 3-4 but detected in ESO-H α 574. With blue and magenta symbols we indicate the observed data and model predictions of lines coming from levels b 2 H, a 6 S, and a 2 G, which are not included in the fitting procedure. The best-fit parameters are reported as well.</caption>
</figure>
<figure>
<location><page_41><loc_13><loc_38><loc_55><loc_63></location>
<caption>Fig. 9.- As in Figure 6 for the fit of the [Fe II ] lines detected in Par-Lup 3-4.</caption>
</figure>
<figure>
<location><page_42><loc_13><loc_27><loc_78><loc_57></location>
<caption>Fig. 10.- NLTE best-fit model of the [Fe III ] lines detected in ESO-H α 574. The symbols have the same meaning as in Figure 5.</caption>
</figure>
<figure>
<location><page_43><loc_13><loc_38><loc_55><loc_63></location>
<caption>Fig. 11.- As in Figure 6 for the fit of the [Fe III ] lines detected in ESO-H α 574.</caption>
</figure>
<figure>
<location><page_44><loc_16><loc_34><loc_60><loc_74></location>
<caption>Fig. 12.- [Fe II ]4244 ˚ A/[Fe III ]5270 ˚ A line ratio as a function of the fractional ionization for different values of the electron temperature. The ratio measured in ESO-H α 574 (upper panel) and Par-Lup 3-4 (lower panel) is depicted with an horizontal line.</caption>
</figure>
<figure>
<location><page_45><loc_12><loc_41><loc_72><loc_84></location>
<caption>Fig. 13.- Post-shock intensities relative to peak values vs. distance from the shock front, adapted from Figure 1 of Hartigan, Morse, & Raymond (1994). [Fe II ] and [Fe III ] lines (left and middle panels, respectively) are shown for two shock velocities (35 km s -1 , upper panel, and 70 km s -1 , lower panel). For [Fe II ] lines, the peak-normalized intensity profile of ultra-violet (blue), optical (green) and near-infrared (red) lines is shown. Temperature (in K, divided by 10 5 for v shock = 35 km s -1 , and 2 10 5 for v shock = 70 km s -1 ), electron density (in cm -3 , divided by 10 4 ), fractional ionization (multiplied by 10 for v shock = 35 km s -1 ), and compression factor ( C = n post -shock /n pre -shock , divided by 100 for v shock = 70 km s -1 and by 10 for v shock = 35 km s -1 ) are plotted with dotted, short-dashed, long-dashed, and dot-short-dashed curves, respectively. The assumed pre-shock gas conditions in terms of density and magnetic field strength are reported, as well. In the middle panels is also shown the peak-normalized intensity profile of the [O I ] 6300 ˚ A line. The right panels give the relative fraction of Fe 0 , Fe + , and Fe ++ with respect to the total Fe abundance along the</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "We investigate the diagnostic capabilities of the iron lines for tracing the physical conditions of the shock-excited gas in jets driven by pre-main sequence stars. We have analyzed the 3 000-25 000 ˚ A, X-shooter spectra of two jets driven by the pre-main sequence stars ESO-H α 574 and Par-Lup 3-4. Both spectra are very rich in [Fe II ] lines over the whole spectral range; in addition, lines from [Fe III ] are detected in the ESO-H α 574 spectrum. NLTE codes solving the equations of the statistical equilibrium along with codes for the ionization equilibrium are used to derive the gas excitation conditions of electron temperature and density, and fractional ionization. An estimate of the iron gas-phase abundance is provided by comparing the iron lines emissivity with that of neutral oxygen at 6300 ˚ A. The [Fe II ] line analysis indicates that the jet driven by ESO-H α 574 is, on average, colder ( T e ∼ 9 000 K), less dense ( n e ∼ 2 10 4 cm -3 ) and more ionized ( x e ∼ 0.7) than the Par-Lup 3-4 jet ( T e ∼ 13 000 K, n e ∼ 6 10 4 cm -3 , x e < 0.4), even if the existence of a higher density component ( n e ∼ 2 10 5 cm -3 ) is probed by the [Fe III ] and [Fe II ] ultra-violet lines. The physical conditions derived from the iron lines are compared with shock models suggesting that the shock at work in ESO-H α 574 is faster and likely more energetic than the Par-Lup 3-4 shock. This latter feature is confirmed by the high percentage of gas-phase iron measured in ESO-H α 574 (50-60% of its solar abundance in comparison with less than 30% in Par-Lup 3-4), which testifies that the ESO-H α 574 shock is powerful enough to partially destroy the dust present inside the jet. This work demonstrates that a multiline Fe analysis can be effectively used to probe the excitation and ionization conditions of the gas in a jet without any assumption on ionic abundances. The main limitation on the diagnostics resides in the large uncertainties of the atomic data, which, however, can be overcome through a statistical approach involving many lines. Subject headings: ISM:jets and outflows - stars:pre-main sequence - ISM: lines and bands - ISM: individual objects: ESO-H α 574, Par-Lup 3-4",
"pages": [
2,
3
]
},
{
"title": "The diagnostic potential of Fe lines applied to protostellar jets 1",
"content": "T. Giannini 1 , B. Nisini 1 , S. Antoniucci 1 , J. M. Alcal´a 2 , F. Bacciotti 3 , R. Bonito 4 , 5 , L. Podio 6 , B. Stelzer 4 , E.T. Whelan 7 Received ; accepted To appear in Ap. J. 1 INAF-Osservatorio Astronomico di Roma, via Frascati 33, I-00040 Monte Porzio Catone, Italy 2 INAF-Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131 Napoli, Italy 3 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 4 INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, I-90134 Palermo, Italy 5 Dipartimento di Fisica e Chimica, Universit´a di Palermo, Piazza del Parlamento 1, I- 90134 Palermo, Italy 6 UJF-Grenoble 1 / CNRS-INSU, Institut de Planetologie et d'Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble, F-38041, France 7 Institut f¨ur Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics, Eberhard Karls Universit¨at, 72076 T¨ubingen, Germany",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Jets from young stars play a key role in the dynamics of star formation and disk evolution. They regulate the process of stellar accretion, by both removing the angular momentum generated by accreting material in the disk, and modifying the inner disk physics, thus influencing the evolution of proto-planetary systems. The specific role of jets in the dynamics and evolution of the accreting system strongly depends on the parameters that characterize their structure and excitation, which are in turn related to their formation and heating mechanisms. From an observational point of view, information on the jet physics and dynamics can be retrieved through the analysis of the forbidden lines emitted by the jet plasma when it gets excited in shocks: to this aim, strong optical lines, such as [O I ], [S II ] and [N II ] lines, are widely used and specific diagnostic tools, able to retrieve a complete set of parameters (namely electron density, n e , temperature, T e and ionization fraction, x e ) have been developed (e.g. Bacciotti & Eisloffel 1999). The knowledge of these parameters is fundamental to an understanding of jet acceleration mechanisms (e.g. MHD disk-winds or X-winds, e.g., Shu et al. 1994; Ferreira 1997) and for measuring the mass flux rate ( ˙ M jet ). ˙ M jet is the quantity regulating the efficiency of the jet and is directly related to the disk mass accretion rate ( ˙ M acc ). Although widely exploited and constantly refined, the diagnostic tools based on bright optical lines suffer from several intrinsic limitations. Firstly, optical lines trace specific excitation conditions and hence force the assumption that the gas in the jet has a constant temperature and density. This assumption is in contrast with combined optical/near infrared line analysis, which has shown that gradients in temperature and density up to orders of magnitudes usually occur in the cooling region behind the shock front (Nisini et al. 2005, Podio et al. 2006). Secondly, diagnostic tools based on ratios between lines of different atomic species require one to assume a set of elemental abundances, which in turn imply an uncertainty on the parameters (temperature and density) more than 40% (Podio et al. 2006). Finally, the use of optical lines requires an a priori knowledge of the visual extinction, a circumstance that often makes the optical diagnostic applicable only to jets of the more evolved sources, where the reddening is negligible. All the above limitations can be circumvented by using different lines of the same species, covering a wide range of wavelengths particularly sensitive to extinction variations. In this respect diagnostic of iron (in different ionization stages) represents a very well suited tool. Indeed, since iron line spectrum covers all the wavelengths between the Ultraviolet (UV) and the Near-Infrared (NIR), it is sensitive to a large range of excitation conditions, allowing therefore to derive a complete view of the post-shock cooling region. The aim of the present paper is to probe the potential of the iron lines in probing the jet physical parameters. Our test-cases are two jets we have observed with the X-shooter spectrograph in the wavelength range ∼ 3 000 -25 000 ˚ A, namely the jets excited by the sources ESO-H α 574 and Par-Lup 3-4. ESO-H α 574 ( α J 2000 . 0 = 11 h 16 m 03 s .7, δ J 2000 . 0 = -76 · 24 ' 53 '' ), spectral type K8, is a low-luminosity source in the Chamaleon I star-forming region located at a distance d =160 ± 17 pc (Wichmann et al. 1998). The low luminosity of 3.4 10 -3 L /circledot (Luhman 2007), which is a factor ∼ 150 lower than the luminosity of the typical T Tauri stars of the same spectral type, is interpreted as due to a disk seen edge-on. The source powers a bipolar jet (HH 872) of total projected length of 0.015 pc (3140 AU). It was discovered by Comer'on & Reipurth (2006) in a [S II ] image at 6728 ˚ A as a chain of knots, of which knots A1, A, B, C, D form the blue-shifted jet and knot E forms the red-shifted jet. Par-Lup 3-4 ( α J 2000 . 0 = 16 h 08 m 51 s .44, δ J 2000 . 0 = -39 · 05 ' 30 '' ), spectral type M5, is located in the Lupus III dark cloud at d =200 ± 40 pc (Comer'on et al. 2003). This object also appears to be under-luminous, being about 25 times fainter than typical M5 pre-main sequence objects (L=3 10 -3 L /circledot , Mer'ın et al. 2008). As in the case of ESO-H α 574, its low luminosity is likely due to the obscuration of the star by an edge-on viewing disk (Hu'elamo et al. 2010). The jet was discovered by Fern'andez & Comer'on (2005), with emission extending in opposite directions with respect to the star for a total length of ∼ 1240 AU. The X-shooter spectra of the two objects have been already investigated by Bacciotti et al. (2011, hereinafter BWA11) and by Whelan et al. (2013, hereinafter WBA13). Both these papers determine the mass ejection to mass accretion ratio ˙ M jet / ˙ M acc . While in Par-Lup 3-4 this ratio is at the upper end of the range predicted by jet models, the value found in ESO-H α 574 of ∼ 90 can be partially reconciled with the predictions of magneto-centrifugal jet acceleration mechanisms only if the edge-on disk severely reduces the luminosity of the accretion tracers. The numerous spectral lines detected in the two jets, along with the kinematical properties derived in the line profiles are presented in WBA13. In the present paper we concentrate our analysis on the many iron lines detected in both spectra. The outline is the following: in Sect. 2 we briefly summarize the details of the observations and present the spectroscopic data; in Sect. 3 we describe the iron excitation and ionization models and derive the jet physical parameters. In Sect. 4 we discuss the results, which are summarized in Sect. 5.",
"pages": [
4,
5,
6
]
},
{
"title": "2. Observations and Results",
"content": "The present work is part of a coherent series of papers that deal with our X-shooter survey of Pre-Main Sequence (PMS) objects. The overall aspects, such as scopes, data reduction procedures, calibrations and results are thoroughly discussed in Alcal'a et al. 2011 and Alcal'a et al. 2013. Here we just recall the information which is essential for the presented subject. The X-shooter spectra of ESO-H α 574 and Par-Lup 3-4 were acquired on April 7 2010, with an integration time of ∼ 1 hr per object. The slit, aligned with the jet axis, was set to achieve a resolving power of 5 100, 8 800 and 5 600 for the UVB (3 000 5 900 ˚ A), VIS (5 450 - 10 200 ˚ A) and NIR arm (9900 - 24 700 ˚ A), respectively (slit widths: 1 . '' 0, 0 . '' 9, 0 . '' 9). The pixel scale is 0 . '' 16 for the UVB and VIS arms and 0 . '' 21 for the NIR arm. The data reduction was performed independently for each arm using the X-shooter pipeline version 1.1., which provides 2-dimensional spectra, background-subtracted and calibrated in wavelength. Post-pipeline procedures were then applied by using routines within the IRAF and MIDAS packages to subtract sky lines and obtain 1-dimensional spectra. These are then divided by a telluric spectrum to remove the atmospheric features, and to do the flux-calibration. The complete spectrum was obtained by comparing the flux densities in the overlapping portions of the spectra of adjacent arms. While UVB and VIS spectra are perfectly aligned, the NIR spectrum of ESO-H α 574 appears lower of a factor ∼ 1.26. Flux losses in the NIR arm are not uncommon in the X-Shooter spectra and are caused by a misalignment between the NIR with respect to the VIS and UVB arms (Alcal'a et al. 2013). No correction was need to re-align the three arms spectra of Par-Lup 3-4. As far as the ESO-H α 574 jet is concerned (hereinafter ESO-H α 574), we concentrate here on the iron lines detected in the brightest knot A1. This is also the closest to the exciting source, extending from the source itself up to 2 '' away (320 AU, see Figure 2 of BWA11, upper panel). The Par-Lup 3-4 jet (hereinafter Par-Lup 3-4) was integrated up to a distance of 1 '' from the source continuum, which corresponds to 200 AU (see Figure 2 of BWA11, lower panel). Figures 1 and 2 show the portions of the spectra of the two objects where iron lines are detected, while Figure 3 shows the Grotrian diagram of Fe + levels from which the detected lines originate. The maximum energy level is at more than 30 000 cm -1 above the ground state, and the line wavelengths cover the whole investigated range (in blue, green, and red we indicate ultra-violet, optical, and near-infrared lines, respectively). Similarly, Figure 4 gives the diagram of Fe ++ levels. Note that, due to the level structure, all the emitted lines lie only in the ultra-violet range, although the covered energy range is comparable to that of Fe + . The line fluxes of all the detected lines are listed in Table A.1 of WBA13. Here we give in Table 1 the observed line ratios R ESO -Hα 574 and R Par -Lup 3 -4 of the [Fe II ] lines detected in the two objects with respect to the bright line at 4277 ˚ A. Lines originating from the same multiplet are grouped together and listed in order of decreasing energy of the upper level. In the last column, the ratio R ESO -Hα 574 /R Par -Lup 3 -4 is reported. Since the differential extinction between the two objects is negligible (see Sect. 3.1.1), this ratio gives a qualitative indication on whether or not the excitation conditions are similar in the two objects. Indeed, lines with excitation energy /greaterorsimilar 20 000 cm -1 (ultra-violet and optical lines) and those with excitation energy /lessorsimilar 20 000 cm -1 (near-infrared lines) have < R ESO -Hα 574 /R Par -Lup 3 -4 > ≈ 1.15 and < R ESO -Hα 574 /R Par -Lup 3 -4 > ≈ 2.6, respectively. This in practice suggests that in Par-Lup 3-4 the most excited lines are brighter (in comparison to the 4277 ˚ A) than in ESO-H α 574, a circumstance that could reflect a higher gas temperature. Notably, [Fe III ] lines are detected only in ESO-H α 574 (see Table 2). This result cannot be explained with a different sensitivity in the X-shooter spectra of the two jets, which are similarly bright and were integrated for a comparable amount of time. Therefore, this feature also points to different excitation conditions in the two objects.",
"pages": [
6,
7,
8
]
},
{
"title": "3. Line fitting model",
"content": "In this section we describe the excitation and ionization models which provide the main physical parameters of the two investigated jets. The results of the comparison between observations and models are summarized in Table 4.",
"pages": [
9
]
},
{
"title": "3.1. The excitation model",
"content": "The observed ratios between lines from the same ionic species (e.g. Fe + or Fe ++ ) can be compared with the predictions by an excitation model to derive the physical conditions of the gas. To this aim we adopted a Non Local Thermal Equilibrium (NLTE) approximation for line excitation. One of the main issues of such line modeling regards the choice of the atomic dataset. The complexity of the iron atomic system, which involves hundreds of energy levels (with multiple metastable levels), makes it very difficult to get accurate atomic data sets (both radiative and collisional). For example, seven different computations of the Einstein coefficients for the spontaneous radiative decay (A-values) have been implemented for Fe + , which may differ from each other by more than 50%. Bautista et al. (2013) have evaluated the uncertainties in the line emissivities due to the combinations of the uncertainties on A-values, collisional coefficients and propagation of these two on the level populations. For typical shock-excitation conditions, namely T e ∼ 10 000 K and density between 10 2 -10 8 cm -3 , they find a very wide range of uncertainties, which vary from less than 10% (e.g. lines at 1.256 µ m and 8616 ˚ A) to more than 60% (e.g. lines at 5.330 µ m and at 5527 ˚ A). As shown by the same authors, the most effective way to circumvent the problem is to apply a statistical approach by including in the analysis a large number of lines. In our model we use the up-to-date atomic database of the XSTAR compilation (Bautista & Kallman 2001 2 ), which gives energy levels, A-values and rates for collisions with electrons (these latter for temperatures between 2 000 K and 20 000 K) for the first 159 and 34 fine-structure levels of Fe + and Fe ++ , respectively. The implications on the results when adopting different data sets will be commented in Sect.3.1.1. The NLTE model assumes electronic collisional excitation/de-excitation and spontaneous radiative decay. Possible contributions on line emissivities due to radiative processes are discarded at this step of the analysis but will be considered in Sect. 3.3. The free parameters of the excitation model are the electron temperature T e and the density n e , which can be derived from the observed flux ratios once these latter are corrected for the visual extinction ( A V ) along the line of sight. This latter parameter is usually derived from the flux ratio of lines emitted from the same upper level, this being independent from the level population and therefore a function only of the line frequencies and the A-coefficients. As stated above, however, the large uncertainties associated with these latter values, are reflected in a poor estimate of A V , especially if one considers only two or three lines, as is often done with the NIR [Fe II ] lines (see also Giannini et al. 2008). Therefore, we have taken the extinction as a further free parameter of the excitation model. To derive the differential extinction at each line wavelength, we adopt the extinction curve by Draine (2003). To minimize the uncertainties, we included in the fit only the un-blended lines detected with a signal-to-noise (snr) ratio larger than 5 (i.e. 35 lines for ESO-H α 574 and 20 lines for Par-Lup 3-4) and checked the compatibility of the fit with line fluxes at lower snr ratio a posteriori . First, we constructed a grid of model solutions in the parameter space 2 000 K < T e < 30 000 K (in steps of δT e =1000 K); 10 2 cm -3 < n e < 10 7 cm -3 (in steps of log 10 ( δn e /cm -3 ) = 0.1) and A V ≤ 2 mag (in steps of δA V =0.5 mag). Then, following the method for line fitting proposed by Hartigan & Morse (2007), we have iteratively changed the line used for the normalization, hence considering all the possible sets of line ratios. Each of them was then compared with the grid of theoretical values to find the model with the lowest value of χ 2 .",
"pages": [
9,
10,
11
]
},
{
"title": "3.1.1. [Fe II ] lines fit",
"content": "The result of the excitation model considering the complete set of [Fe II ] lines detected in ESO-H α 574 is depicted in Figure 5. The minimum χ 2 -value is found if the line at 4277 ˚ A is taken as a reference and the corresponding line ratios are reported in Table 1. The best-fit of the ESO-H α 574 [Fe II ] lines gives the following parameters : A V = 0 mag, T e = 9000 K, and n e = 2.0 10 4 cm -3 . A gas component at a single pair ( T e , n e ) fits reasonably well all the lines but systematically underestimates those coming from some doublets and sextets levels (b 2 H, a 6 S, and a 2 G), shown with different colors in Figure 5 and reported in Table 3. In particular, ratios involving lines from a 6 S and a 2 G levels (8 lines) are underestimated by a factor of two, while those from level b 2 H (2 lines) are underestimated by a factor of four. This systematic behavior, which can be reasonably ascribed to the poor knowledge of the atomic parameters, has been already evidenced by Bautista & Pradhan (1998) for the a 6 S level. Notably, however, the same model is selected as best-fit irrespective from including or not the doublets and sextets in the fit, although with a higher minimum reducedχ 2 (hereinafter χ 2 ) in the latter case. The sensitivity of the line ratios to the fitted parameters is probed in Figure 6, where we plot the χ 2 -contours in the density-temperature plane for A V =0 mag (minimum χ 2 = 0.9). Higher A V values return fits with substantially higher χ 2 and are therefore discarded (for example minimum χ 2 = 1.9 for A V =0.5 mag); this indicates that extinction decreases slightly from the ESO-H α 574 central source (where A V ∼ 1.5 mag, WBA13) to the jet. The plotted contours refer to increasing χ 2 values of 30%, 60%, and 90% with respect to the minimum χ 2 value. From this plot we derive that temperature and density do not exceed (inside a confidence of 3σ ) the ranges 8 000 K /lessorsimilar T e /lessorsimilar 11 000 K and 6 10 3 cm -3 /lessorsimilar n e /lessorsimilar 6 10 4 cm -3 , respectively. To check the reliability of the results, we have also attempted two different approaches: i ) to fit the data with a different set of collisional coefficients (Bautista & Pradhan 1998), which returns the same physical parameters but with a higher minimum χ 2 , and ii ) to fit the ultra-violet component and the infrared components separately, with the aim to test the possibility of the presence of different gas components. The χ 2 -contours of the ultra-violet lines fit (Figure 7) shows a best-fit value not significantly different from that obtained by the all-lines fit. Analogously, the temperature range does not significantly differ from that found in the all-lines fit. More interestingly, the density range traced by the ultra-violet lines points to higher densities (i.e. up to 10 5 . 8 cm -3 within a 3σ confidence level). This suggests that while temperature is fairly constant in the probed region (or that its variations occur over spatial scales much smaller than the angular resolution), density may be subjected to stronger gradients. Finally, the fit of the infrared lines (not shown here) gives results in good agreement with the all-lines fit. In Figure 8 we show the best-fit model for the [Fe II ] lines observed in Par-Lup 3-4. The minimum χ 2 is found, as in the case of ESO-H α 574, by taking as a reference the 4277 ˚ A line. To better compare the line emission observed in this object with that of ESO-H α 574, we plot, together with the line ratios of the detected lines, also the 2σ upper limits at the wavelength of the lines detected only in ESO-H α 574. As anticipated in Sec. 2, in Par-Lup 3-4 the ratios between ultra-violet and optical/near-infrared lines are substantially higher. This circumstance is a consequence of the higher temperature probed ( T e =13000 K). The inferred electron density and extinction are n e = 6.0 10 4 cm -3 and A V =0 mag, respectively. As for ESO-H α 574, we find that the predictions of sextet and doublet levels are systematically underestimating the observed ratios of a factor between two and three. Within a confidence level of 90%, the χ 2 -contour plot gives 11 000 K /lessorsimilar T e /lessorsimilar 20 000 K and 1.8 10 4 cm -3 /lessorsimilar n e /lessorsimilar 1.8 10 5 cm -3 (see Figure 9). Finally, if the collisional coefficients by Bautista & Pradhan (1998) are adopted, the best-fit gives T e =16000 K, n e = 8.0 10 4 cm -3 , A V = 0 mag.",
"pages": [
11,
12,
13
]
},
{
"title": "3.1.2. [Fe III ] lines fit",
"content": "The fit of [Fe III ] lines detected in ESO-H α 574 is presented in Figure 10. The best-fit model is obtained by taking as a reference the line at 4930 ˚ A (see also Table 2). This gives the following parameters: T e = 19000 K, n e = 2.0 10 5 cm -3 , A V = 0 mag. At variance with Fe + lines, lines of Fe ++ lie all in the ultra-violet range and come all from levels with similar upper energy. Consequently, we expect that Fe ++ lines are poorly sensitive to the temperature. This is clear in the χ 2 -contour plot of Figure 11, where all temperatures in the grid of NLTE solutions above 8 000 K are compatible with the observations (within a confidence level of 90%). Conversely, the electron density is better constrained within the range 1 10 5 cm -3 /lessorsimilar n e /lessorsimilar 6 10 5 cm -3 . This result confirms that indeed a density gradient exists along the jet of ESO-H α 574, and that [Fe III ] and [Fe II ] ultra-violet lines likely probe the same, high-density gas component.",
"pages": [
13
]
},
{
"title": "3.2. The ionization model",
"content": "To consistently interpret the [Fe II ] and [Fe III ] emission in ESO-H α 574 and to derive the fractional abundance Fe + /Fe ++ , we applied a ionization equilibrium code that involves the first 4 ionization stages of iron. The following processes have been taken into account: direct ionization, radiative and dielectronic recombination (data from Arnaud & Raymond 1992), and direct and inverse charge-exchange with hydrogen (data from Kingdon & Ferland 1996). Notably, while the first three processes are a function only of the electron temperature, direct and inverse charge-exchange rates also depend on the fractional ionization x e = n e / n H , where n H = n H 0 + n H + . Moreover, since the electron transfer is more efficient when the involved ions (e.g. H 0 and Fe + ) have similar ionization potentials (IP) 3 , the charge-exchange rate is relevant only for the process Fe + + H + /arrowparrrightleft Fe ++ + H 0 . Therefore, it returns relevant results for the Fe + /Fe ++ abundance ratio, while it is negligible for both the Fe 0 /Fe + and Fe ++ /Fe +3 abundance ratios. For T e = 8000 K, namely the lowest temperature derived from the χ 2 -contours of Figures 6 and 7, our model predicts a substantial fraction of iron in neutral form even if the gas is almost fully ionized (e.g. we get 30% of Fe 0 , 52% of Fe + , and 18% of Fe ++ for x e = 0.9). This strongly contrasts with the simultaneous lack of any Fe 0 line in the ESO-H α 574 spectrum together with the presence of Fe ++ lines. However, just a slight increase of the electron temperature at 9 000 K makes the neutral Fe 0 percentage drop to less than 10%, and that of Fe ++ to increase to more than 20%, in agreement with the observations. The expected percentage of Fe +3 is negligible for the whole range of temperature considered in",
"pages": [
13,
14
]
},
{
"title": "Figures 6 and 7.",
"content": "To derive x e we solved the ionization equilibrium equations (together with the excitation equilibrium for each of the two species) to predict a number of [Fe II ]/[Fe III ] line ratios. We constructed a grid of model solutions in the range 0 ≤ x e ≤ 1 (in steps of δx e =0.05) and 9 000 K ≤ T e ≤ 14 000 K, being the upper value that derived from the χ 2 -contours of Figure 7. To estimate x e we consider the [Fe II ] ultra-violet lines and the [Fe III ] lines, assuming that they come from the same portion of the post-shock gas (see Sect. 3.1.2). We consider 14 line ratios involving 7 [Fe II ] lines with two bright [Fe III ] lines at 4701.59 ˚ A and 5270.53 ˚ A. As an example, we show in Figure 12, upper panel, the [Fe II ]4244/[Fe III ]5270 ratio as a function of x e for the considered range of temperature. The observations are in agreement with 0.65 /lessorsimilar x e /lessorsimilar 0.85, where the lower (upper) value refers to the highest (lowest) temperature assumed. This value of x e is the same found (within the error range) if all the 14 ratios are considered. For Par-Lup 3-4 we can derive an upper limit on x e by considering the upper limits on the [Fe III ] lines. Taking a grid in the range 11 000 < T e < 20 000 K (see Sect. 3.1.1 and Figure 9), we get x e /lessorsimilar 0.4. As an example, the derivation of x e from the ratio [Fe II ]4244/[Fe III ]5270) is shown in Figure 12, lower panel. Typical x e values in protostellar jets range from 0.03 to 0.6 (Ray et al. 2007, Nisini et al. 2005, Podio et al. 2009), although x e = 0.8 is found in the High Velocity Component (HVC) of the DG Tau B jet Podio et al (2011). Therefore, while the fractional ionization of Par-Lup 3-4 is in the range of the most common values, that of ESO-H α 574 appears remarkably high.",
"pages": [
14,
15
]
},
{
"title": "3.3. Photoexcitation contribution",
"content": "In Section 3.1.1 the observed line ratios have been interpreted in the light of collisional excitation. In this Section we explore whether an additional contribution from fluorescence excitation can be relevant. In ESO-H α 574 this possibility is supported by the detection of bright [Ni II] lines at 7377.8 ˚ A and 7411.6 ˚ A(WBA13), whose intensity is easily enhanced because of the pumping of an ultra-violet field (Lucy 1995), though the observed intensity ratio of around 10, is compatible only with collisional excitation (see Figure 2 of Bautista et al. 1996). In Par-Lup 3-4, only the 7377.8 ˚ A line is detected. To better investigate the role of photo-excitation in ESO-H α 574, we have included in the excitation model a radiation field, which can be produced either from the stellar photosphere or by a hot spot on the stellar surface produced by the accretion shock of the infalling matter. Both these fields have been approximated as W × B ν (T eff ) , where B ν is the black-body function at the stellar (or hot spot) temperature and W= 1/4 (R/r) 2 is the dilution factor, having adopted the stellar radius R= 3 R /circledot and the distance of the knot A1 from the star, r, equal to 100 AU (i.e. 0.2 '' , see BWA11). We take T eff = 4000 K for the stellar temperature and 6 000 K ≤ T eff ≤ 12 000 K for the hot spot temperature, following the model of Calvet & Gullbring (1998). The hot spot area has been taken between 10-30% of the stellar surface. As shown by Lucy (1995), a powerful way to evaluate the relevance of photo-excitation, is to compute the so-called excitation parameter (U ex ), which is defined as the ratio between all the radiative and collisional excitation rates involving two given levels. From U ex , the 'second critical electron density' can be also derived, n ∗ e = U ex n e , such that for n ∗ e /greatermuch n e , fluorescent excitation is predominant with respect to collisional excitation. Assuming a stellar field and for n e = 210 4 cm -3 (Sect. 3.1.1), we get n ∗ e /lessorsimilar 10 2 cm -3 (or U ex < 5 10 -3 ) for all the levels, indicating that fluorescence excitation is negligible in this case. The importance of the hot-spot field was tested by varying both T eff and W in the ranges given above, obtaining n ∗ e up to 10 5 cm -3 . Thus, in principle, the presence of a hot-spot could have a role in fluorescence excitation. However, the comparison of the predicted intensity ratios with those observed in the ESO-H α 574 spectrum, indicates a marginal compatibility only for the lowest values of T eff and W (i.e. T eff ≤ 8 000 K and hot-spot area not exceeding 10% of the stellar surface). Hence, even if a hot-spot may exist, certainly it is not the main cause of the observed emission. As a note, and with reference to Sect. 3.1.1 and Table 3, we also report that none of the line ratios systematically underestimated by the collisional model can be reproduced even if fluorescence excitation is considered. Finally, in the Par-Lup 3-4 case the distance between the central source and the jet is not well defined as in the ESO-H α 574 case. Taking different values of W, we estimate that photo-excitation contribution, and in particular that due to the hot spot field, can be relevant for distances closer than 5-10 AU from the central source.",
"pages": [
15,
16,
17
]
},
{
"title": "4.1. Comparison with shock models",
"content": "Once derived the physical conditions, the origin of the iron emission in the two jets was investigated in the framework of shock models. Figure 13, adapted from Figure 1 of Hartigan, Raymond, & Morse (1994), shows the variation of the ionization fraction, electron density, and temperature with the distance behind the shock front for a low velocity (35 km s -1 ) and an intermediate velocity (70 km s -1 ) shock, in the approximation of a slab geometry and for assumed values of the pre-shock density and magnetic field. For each combination of these parameters, we computed the intensity of the most prominent iron lines, then deriving their expected intensity variation along the overall post-shock region. In particular we show, in the left panels, the peak-normalized intensity profiles of ultra-violet, optical, near-infrared [Fe II ] lines (those coming from levels a 4 G, a 4 P, and a 4 D), and in the middle panels the profiles of [Fe III ] lines coming from level a 3 F. Notably, lines at different wavelengths peak at different distance from the shock front, in the dimensional scale of ∼ 10 13 - 10 14 cm. At the distance of our objects these scales correspond to hundredths of arcsec, which are not resolved at our spatial resolution, and therefore the excitation model of Fe + gives only average quantities. It is also important to notice that the physical parameters derived in ESO-H α 574 and Par-Lup 3-4 cannot be directly compared with those depicted in Figure 13, which strongly depend on the assumed conditions of pre-shock density of the gas, magnetic field strength and shock velocity. Nevertheless, a trend between post- and pre- shock parameters can be evidenced. We computed (see Table 5) the average < T e > , < x e > , < n e > and the compression factor C = n post -shock /n pre -shock , weighted by the intensity profiles of the various (groups of) lines depicted in Figure 13. By examining the data of Table 5, a number of conclusions can be drawn: 1) for a given shock velocity, lines at decreasing wavelengths trace progressively higher temperatures. Ionization fraction and electron density slightly increase with decreasing wavelength in the model with v shock =70 km s -1 , while they remain fairly constant and significantly lower if v shock =35 km s -1 ; 2) the average parameters probed by the mean of all [Fe II ] lines (fourth line of Table 5) indicate that increasing shock velocities correspond to decreasing temperatures and to increasing ionization fraction, electron density and compression factor. This points toward a higher shock-velocity in ESO-H α 574, where temperature is lower and electron density and ionization fraction are higher than in Par Lup 3-4 (see Table 4). Moreover, in the intermediate-velocity shock model, the [Fe III ] lines trace more specifically the portion of the post-shock region extending up to ∼ 10 13 cm behind the shock front, where the electron density reaches its maximum value. This region should therefore correspond to that traced by the observed [Fe III ] line ratios. We also note that the above scenario is also consistent with the abundance ratios of the Fe 0 , Fe + , and Fe ++ depicted in the right panels of Figure 13. Indeed, while for a low-velocity shock the bulk of iron is singly ionized, for an intermediate velocity shock the ratio Fe + /Fe ++ ∼ 8 (at distances of the order of 10 13 cm), again consistent with the detection of Fe ++ only in ESO-H α 574. Finally, we again remark that although the above analysis allows us to interpret the observations in a consistent framework of shocked origin, the pre-shock parameters of the two models taken as a reference are not consistent with the derived post-shock parameters. For example, for the measured < n e > and the compression factors of Table 5, the pre-shock density would be < n 0 > ∼ 7 10 3 cm -3 and ∼ 6 10 4 cm -3 for ESO-H α 574 and Par Lup 3-4, respectively, which are higher than the n 0 values at which the two models of Hartigan et al. (1994) are computed.",
"pages": [
17,
18,
19
]
},
{
"title": "4.2. Gas-phase Fe abundance",
"content": "The gas-phase Fe abundance x (Fe) is an indirect measure of the presence of dust inside the jet. In general jet launching models predict that the jet is dust-free as dust is completely destroyed in the launching region by the stellar radiation. Conversely, if the jet originates from a disk region extending beyond the dust evaporation radius, it could eventually transport some dust. This, in turn, could be then partially destroyed by the shock because of vaporisation and sputtering of energetic particles (e.g. Seab 1987; Jones 1999: Guillet et al. 2009). The degree of iron depletion is therefore also a function of the shock efficiency. Previous studies of x (Fe) in shock environments have given sparse results, from values close to solar abundance (e.g. Beck-Winchatz et al. 1996), up to intermediate (Nisini et al. 2002, Podio et al. 2006, 2009) and very high depletion factors (Mouri & Taniguchi 2000; Nisini et al. 2005). A powerful way to estimate the percentage of gas-phase iron ( δ Fe ), relies on intensity ratios involving lines of non-refractory species emitted in similar excitation conditions, as for example the [Fe II ]1.25 µ m/[P II ]1.18 µ m, as suggested by Oliva et al. (2001). Since phosphorous lines are not detected in our spectra, we investigate the possibility of using ratios involving [O I ] lines. To this aim, we solved the equations of ionization equilibrium for the first three ionic stages of oxygen, together with the statistical equilibrium for the first five levels of O 0 . The radiative coefficients are taken from the NIST database 4 while the rates for collisions with electrons are from Bhatia & Kastner (1995). As a result, we get the percentage of neutral oxygen and the peak-normalized intensity profile along the post-shock region. In particular, that of [O I ] 6300 ˚ A shown in the middle panels of Figure 13, well resembles that of [Fe II ] ultra-violet lines. Therefore, we conclude that [Fe II ] ultra-violet lines and [O I ] 6300 ˚ A trace the same shock region and are therefore suited to measure δ Fe inside the shock. This is also roughly confirmed by the average parameters traced by the [O I] optical lines reported in Table 5 and taken from Bacciotti & Eisloffel (1999). Note also that other tracers commonly used to derive δ Fe , such as [S II ] 6740 ˚ A, are not as powerful as [O I ] 6300 ˚ A since their shock profile does not resemble that of any iron line (see e.g. Figure 3 of Bacciotti & Eisloffel, 1999). The same problem arises if the [O I ] 6300 ˚ A is compared with [Fe II ] near-infrared lines (see Figure 13). To derive δ Fe , we thus selected several ratios [O I ] 6300 ˚ A 5 over bright ultra-violet [Fe II ] lines, whose observed values are compared with those expected for the < T e > , < n e > and < x e > determinations derived from the iron analysis. By assuming the solar iron and oxygen abundances with respect to hydrogen of 3.16 10 -5 and 6.76 10 -4 (Grevesse & Sauval 1998), we estimate δ Fe = 0.55 ± 0.05 and δ Fe = 0.30 ± 0.03 for ESO-H α 574 and Par-Lup 3-4, respectively. This result is in agreement with the shock interpretation given in the previous section. The higher efficiency in destroying the dust in the shock in ESO-H α 574 is due to its higher velocity, as expected from models of dissociative shocks (Guillet et al. 2009). In this respect, further observational evidence is provided by the detection in ESO-H α 574 of bright lines from other refractory species, such as Ca and Ni, which, on the contrary, are barely detected in Par-Lup 3-4 (BWA11, WBA13). Finally, we note that the derived values of δ Fe belong to the group of 'intermediate' depletion values, where the shock has not a sufficient strength to completely destroy dust. The presence of dust inside the shock is in turn an indication that the jet launching region is larger than the dust sublimation zone.",
"pages": [
19,
20,
21
]
},
{
"title": "4.3. Comparison with the diagnostics of other atomic species",
"content": "Together with iron lines, the spectra of ESO-H α 574 and Par-Lup 3-4 are rich in other atomic emission lines (BWA11, WBA13), some of which commonly used to diagnose the physical conditions of the emitting gas. In this section we intend to compare the parameters derived from iron lines with those traced by ratios of lines of oxygen, nitrogen and sulphur. To derive the theoretical values of such ratios we have implemented simple NLTE codes for the lowest 5 fine structure levels of each species. The radiative coefficients are taken from the NIST database, while the electronic collision coefficients are taken from Pradhan (1976, [O II ]), Pequignot, & Aldovrandi (1976, [N I ]), Mendoza (1983, [N II ]), Hollenbach, & McKee (1989, [S II ]). The main results of this analysis, which are summarized in Table 6 are the following: i ) on average the temperature probed in ESO-H α 574 is in agreement with that probed with iron lines. In Par-Lup 3-4 the derived temperatures give sparse results, with T e ([O I ]) lower than T e ([Fe II ]) and with T e ([S II ]) not consistent with T e ([N II ]); ii ) ratios of different species probe different electron densities, with n e ([O II ]) > n e ([N I ]) > n e ([S II ]). This result can be explained by comparing the fitted values with the critical densities of the involved lines, which, at T e = 10000 K are of ∼ 10 8 cm -3 , ∼ 10 6 cm -3 , and ∼ 10 4 cm -3 for [O II ], [N I ] and [S II ], lines, respectively. While the densities traced with the [S II ] ratio are close to the critical value, and therefore not completely reliable, this is not the case for the density indicated by the [O II ] flux ratio. In ESO-H α 574 this density is the same as that inferred from the [Fe III ] and [Fe II ] ultra-violet lines, thus again supporting the result of a density gradient inside the jet. Notably, the [O II ] line ratio indicates that in Par-Lup 3-4 the density is higher than in ESO-H α 574, in agreement with what found with the [Fe II ] VIS and NIR lines. In conclusion, care should be taken to compare physical conditions derived from different atomic species and lines, due to the their different sensitivity to variations of physical parameters behind the shock front. In this respect, the rich iron spectrum from UV to NIR, with lines sensitive to a large range of excitation conditions, is particularly suited to obtain a more complete view of the post-shock cooling region.",
"pages": [
21,
22
]
},
{
"title": "5. Summary",
"content": "We have analyzed the 3 000-25 000 ˚ A, X-shooter spectra, of two jets driven by low-luminosity pre-main sequence stars, ESO-H α 574 and Par-Lup 3-4, with the aim of investigating the diagnostic capabilities of the iron lines. Our analysis and main results can be summarized as follows: 574 is, on average, colder ( T e ∼ 9 000 K) and less dense ( n e ∼ 2 10 4 cm -3 ) than the Par-Lup 3-4 jet ( T e ∼ 13 000 K, n e ∼ 6 10 4 cm -3 ). A more compact component ( n e ∼ 2 10 5 cm -3 ) inside the jet is revealed in ESO-H α 574 if the ultra-violet lines are fitted separately from the optical and near-infrared lines. This component, whose temperature is not well constrained, is likely the same responsible for the [Fe III ] line emission. The extinction appears to be negligible in both jets. agreement with the higher shock-velocity of ESO-H α 574, which in turn corresponds in a higher kinetic energy able to partially destroy the dust particles.",
"pages": [
22,
23,
24
]
},
{
"title": "6. Acknowledgments",
"content": "We are grateful to Manuel Bautista and to an anonymous referee for their suggestions and constructive discussions. TG and JMA thank also G. Attusino. The ESO staff is acknowledged for support with the observations and the X-shooter pipeline.",
"pages": [
24
]
},
{
"title": "REFERENCES",
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"pages": [
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}
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2013ApJ...778...72F
https://arxiv.org/pdf/1310.1011.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_86><loc_88><loc_87></location>MULTIPLE MONOPOLAR OUTFLOWS DRIVEN BY MASSIVE PROTOSTARS IN IRAS 18162-2048</section_header_level_1>
<text><location><page_1><loc_16><loc_83><loc_83><loc_85></location>M. Fern'andez-L'opez 1 , J. M. Girart 2 , S. Curiel 3 , L. A. Zapata 4 , J. P. Fonfr'ıa 3 , and K. Qiu 5 Draft version May 21, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_80><loc_55><loc_81></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_65><loc_86><loc_80></location>In this paper we present Combined Array for Research in Millimeter-wave Astronomy (CARMA) 3.5 mm observations and SubMillimeter Array (SMA) 870 µ m observations toward the high-mass star-forming region IRAS 18162-2048, the core of the HH 80/81/80N system. Molecular emission from HCN, HCO + and SiO is tracing two molecular outflows (the so-called Northeast and Northwest outflows). These outflows have their origin in a region close to the position of MM2, a millimeter source known to harbor a couple protostars. We estimate for the first time the physical characteristics of these molecular outflows, which are similar to those of 10 3 -5 × 10 3 L /circledot protostars, suggesting that MM2 harbors high-mass protostars. High-angular resolution CO observations show an additional outflow due southeast. We identify for the first time its driving source, MM2(E), and see evidence of precession. All three outflows have a monopolar appearance, but we link the NW and SE lobes, explaining their asymmetric shape as a consequence of possible deflection.</text>
<text><location><page_1><loc_14><loc_62><loc_86><loc_65></location>Subject headings: circumstellar matter - ISM: individual (GGD27, HH 80-81, IRAS 18162-2048) stars: formation - stars: early type - submillimeter: ISM</text>
<section_header_level_1><location><page_1><loc_22><loc_59><loc_35><loc_60></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_22><loc_48><loc_56></location>High-mass stars exert a huge influence on the interstellar medium, ejecting powerful winds and large amounts of ionizing photons or exploding as supernova. Despite their importance injecting energy and momentum into the gas of galaxies, how they form is still not well understood. High-mass protostars are typically more than 1 kpc from Earth (Cesaroni et al. 2005). They are deeply embedded inside dense molecular clouds and often accompanied by close members of clusters. Hence, well detailed studies have only been possible in a few cases. From a handful of studies based on high-angular resolution observations, we know that early B-type protostars posses accretion disks (e.g., Shepherd et al. 2001; Cesaroni et al. 2005; Patel et al. 2005; Galv'an-Madrid et al. 2010; Kraus et al. 2010; Fern'andez-L'opez et al. 2011a), while for O-type protostars there is no unambiguous evidence for accretion disks yet, but vast toroids of dust and molecular gas have been seen rotating around them, showing signs of infalling gas (e.g., Sollins & Ho 2005; Sollins et al. 2005; Beltr'an et al. 2006; Zapata et al. 2009; Qiu et al. 2012; Jim'enez-Serra et al. 2012; Palau et al. 2013). On the other hand, molecular outflow studies seem to display diverse scenarios in the few cases where both, highangular resolution and relatively nearby targets were available. For example, the two nearest to the Earth</text>
<text><location><page_1><loc_8><loc_17><loc_48><loc_21></location>1 Department of Astronomy, University of Illinois at UrbanaChampaign, 1002 West Green Street, Urbana, IL 61801, USA; manferna@illinois.edu</text>
<unordered_list>
<list_item><location><page_1><loc_8><loc_14><loc_48><loc_18></location>2 Institut de Ciencies de l'Espai, (CSIC-IEEC),Campus UAB, Facultat de Ciencies, Torre C5-parell 2, 08193 Bellaterra, Catalunya, Spain; girart@ieec.cat</list_item>
<list_item><location><page_1><loc_8><loc_11><loc_48><loc_14></location>3 Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico (UNAM), Apartado Postal 70-264, 04510 M'exico, DF, Mexico</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_1><loc_8><loc_7><loc_48><loc_9></location>5 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_33><loc_92><loc_60></location>massive protostars (Orion BN/KL and Cep A HW2) have molecular outflows that are very difficult to interpret. It has been suggested that Orion BN/KL shows an explosion-like isotropic ejection of molecular material (Zapata et al. 2009), while Cep A HW2 drives a very fast jet (Curiel et al. 2006), which is apparently pulsing and precessing due to the gravitational interaction of a small cluster of protostars (Cunningham et al. 2009; Zapata et al. 2013). These two examples provide how analyzing the outflow activity of these kind of regions can provide important insight on the nature of each region. They are also showing us that although accretion disks could be ubiquitously found around all kind of protostars, a complete understanding on the real nature of the massive star-formation processes should almost inevitably include the interaction between close-by protostars. This introduces much more complexity to the observations, not only because of the possible interaction between multiple outflows, but also the difficulty to resolve the multiple systems with most telescopes.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_33></location>IRAS 18162-2048, also known as the GGD27 nebula, is associated with the unique HH 80/81/80N radio jet. It is among the nearest high-mass protostars (1.7 kpc). The large radio jet ends at HH 80/81 in the south (Mart'ı et al. 1993) and at HH 80N in the north (Girart et al. 2004), wiggling in a precessing motion across a projected distance of 7.5 pc (Heathcote et al. 1998). Recently, Masqu'e et al. (2012) have proposed that this radio jet could be larger, extending to the north, with a total length of 18.4 pc. Linear polarization has been detected for the first time in radio emission from this jet indicating the presence of magnetic fields in a protostellar jet (Carrasco-Gonz'alez et al. 2010). The jet is apparently launched close to the position of a young and massive protostar, surrounded by a very massive disk of dust and gas rotating around it (Fern'andez-L'opez et al. 2011a,b; Carrasco-Gonz'alez et al. 2012). In spite of this apparently easy-to-interpret scenario, the region contains other massive protostars (Qiu & Zhang 2009;</text>
<table>
<location><page_2><loc_16><loc_70><loc_83><loc_88></location>
<caption>Table 1 IRAS 18162-2048 observations</caption>
</table>
<unordered_list>
<list_item><location><page_2><loc_16><loc_68><loc_35><loc_70></location>(a) Number of tracks observed.</list_item>
<list_item><location><page_2><loc_16><loc_67><loc_71><loc_69></location>(b) Data extracted from Cologne Database for Molecular Spectroscopy (CDMS) catalogue.</list_item>
<list_item><location><page_2><loc_16><loc_66><loc_26><loc_67></location>(c) Per channel.</list_item>
</unordered_list>
<figure>
<location><page_2><loc_11><loc_41><loc_46><loc_66></location>
<caption>Figure 1. The color scale image shows the sensitivity response of the CARMA mosaic pattern taken toward IRAS 18162-2048. The white area is that with rms noise = 2 mJy beam -1 , and each shaded zone has an rms noise of 1.1, 1.2, 1.3, 1.4 and 1.5 × 2 mJy beam -1 . The red crosses mark the center of each mosaic pointing and the red circle has a radius of 70 '' . Inside this red circle, the Nyquist-sampling of the mosaic insures an almost uniform rms noise. The synthesized beam is plotted in the bottom right corner.</caption>
</figure>
<text><location><page_2><loc_8><loc_7><loc_48><loc_29></location>Fern'andez-L'opez et al. 2011a) which are driving other high velocity outflows. (Sub)mm observations of the central part of the radio jet revealed two main sources, MM1 and MM2, separated about 7 '' (G'omez et al. 2003), that are probably in different evolutionary stages (Fern'andez-L'opez et al. 2011a). MM1 is at the origin of the thermal radio jet, while MM2 is spatially coincident with a water maser and has been resolved into a possible massive Class 0 protostar and a second even younger source (Fern'andez-L'opez et al. 2011b). MM2 is also associated with a young monopolar southeast CO outflow (Qiu & Zhang 2009). There is evidence of a possible third source, MC, which is observed as a compact molecular core, detected only through several millimeter molecular lines. At present, MC is interpreted as a hot core, but the observed molecular line emission could also be explained by shocks originated by the interaction</text>
<text><location><page_2><loc_52><loc_63><loc_92><loc_65></location>between the outflowing gas from MM2 and a molecular core or clump (Fern'andez-L'opez et al. 2011b).</text>
<text><location><page_2><loc_52><loc_44><loc_92><loc_63></location>In this paper we focus our attention on the CO(32), SiO(2-1), HCO + (1-0) and HCN(1-0) emission toward IRAS 18162-2048, aiming at detecting and characterizing outflows from the massive protostars inside the IRAS 18162-2048 core and their interaction with the molecular cloud. Millimeter continuum and line observations of IRAS 18162-2048 were made with CARMA, while submillimeter continuum and line observations were made with the SMA. In Section 2, we describe the observations undertook in this study. In Section 3 we present the main results and in section 4 we carry out some analysis of the data. Section 5 is dedicated to discuss the data, and in Section 6 we give the main conclusions of this study.</text>
<section_header_level_1><location><page_2><loc_65><loc_42><loc_78><loc_43></location>2. OBSERVATIONS</section_header_level_1>
<section_header_level_1><location><page_2><loc_67><loc_40><loc_77><loc_41></location>2.1. CARMA</section_header_level_1>
<text><location><page_2><loc_52><loc_32><loc_92><loc_40></location>IRAS 18162-2048 observations used the CARMA 23element mode: six 10.4 m antennas, nine 6.1 m antennas, and eight 3.5 m antennas. This CARMA23 observing mode included up to 253 baselines that provide extra short spacing baselines, thus minimizing missing flux when observing extended objects.</text>
<text><location><page_2><loc_52><loc_15><loc_92><loc_32></location>The 3.5 mm (84.8 GHz) observations were obtained in 2011 October 9, 11 and 2011 November 11. The weather was good for 3 mm observations in all three nights, with τ 230 GHz fluctuating between 0.2 and 0.6. The system temperature varied between 200 and 400 K. During those epochs, CARMA was in its D configuration and the baselines ranged from 5 to 148 m. The present observations are thus sensitive to structures between 6 '' and 70 '' , approximately 6 . We made a hexagonal nineteen pointing Nyquist-sampled mosaic with the central tile pointing at R.A.(J2000.0)=18 h 19 m 12 . s 430 and DEC(J2000.0)= -20 · 47 ' 23 . '' 80 (see Fig. 1). This kind of mosaic pattern is used to reach a uniform rms noise in</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_13></location>6 Note that while the observations are often said to be sensitive to structures on scales Θ = λ/B min /similarequal 206 '' [ λ ( mm ) /B min ( m )], the interferometer flux filtering could reduce the actual largest scale detection roughly in a factor 2, so that Θ /similarequal 100 '' [ λ ( mm ) /B min ( m )] (see appendix in Wilner & Welch 1994). Here we report a value estimated using the last expression.</text>
<text><location><page_3><loc_8><loc_88><loc_48><loc_92></location>the whole area covered. The field of view of the mosaic has thus, a radius of about 70 '' . Beyond this radius the rms noise increases over 20%.</text>
<text><location><page_3><loc_8><loc_73><loc_48><loc_88></location>At the time of the observations, the correlator of the CARMA 23-element array provided 4 separate spectral bands of variable width. One spectral band was set with a bandwidth of 244 MHz, used for the 3.5 mm continuum emission. The other three bands were set with a bandwidth of 125 MHz, providing 80 channels with a spectral resolution of 1.56 MHz ( ∼ 5 . 2 km s -1 ). One of these three bands was always tuned at the SiO (2-1) frequency. The other two bands were tuned at two of the following lines: HCO + (1-0), and HCN (1-0) and HC 3 N (10-9).</text>
<text><location><page_3><loc_8><loc_59><loc_48><loc_73></location>The calibration scheme was the same during all the nights, with an on-source time of about 2 hours each. The gain calibrator was J1733-130 (with a measured flux density of 4.1 ± 0.4 Jy), which we also used as the bandpass calibrator. Observations of MWC349, with an adopted flux of 1.245 Jy provided the absolute scale for the flux calibration. The uncertainty in the flux scale was 10% between the three observations, while the absolute flux error for CARMA is estimated to be 15-20%. In the remainder of the paper we will only consider statistical uncertainties.</text>
<text><location><page_3><loc_8><loc_43><loc_48><loc_59></location>The data were edited, calibrated, imaged and analyzed using the MIRIAD package (Sault et al. 1995) in a standard way. GILDAS 7 was also used for imaging. Continuum emission was built in the uv-plane from the line-free channels and was imaged using a uniform weighting to improve the angular resolution of the data. For the line emission, the continuum was removed from the uv-plane. We then imaged the continuum-free emission using a natural weighting to improve the signal-to-noise ratio. The continuum and the line images have rms noises of about 2 mJy beam -1 and 20-35 mJy beam -1 per channel, respectively.</text>
<section_header_level_1><location><page_3><loc_25><loc_40><loc_32><loc_41></location>2.2. SMA</section_header_level_1>
<text><location><page_3><loc_8><loc_29><loc_48><loc_39></location>The CO(3-2) maps at subarcsecond angular resolution presented in this paper were obtained from SMA observations taken on 2011 July 18 and October 3 in the extended and very extended configurations, respectively. The calibration, reduction and imaging procedures used in these observations are described in Girart et al. (2013, in preparation). Table 1 shows the imaging parameters for this data set.</text>
<text><location><page_3><loc_8><loc_19><loc_48><loc_29></location>The CO(3-2) maps at an angular resolution of /similarequal 2 '' presented in this paper were obtained from the public SMA data archive. The observations were done in the compact configuration on 2010 April 9. We downloaded the calibrated data and used the spectral windows around the CO(3-2) to produce integrated intensity maps.</text>
<section_header_level_1><location><page_3><loc_24><loc_17><loc_32><loc_18></location>3. RESULTS</section_header_level_1>
<section_header_level_1><location><page_3><loc_15><loc_15><loc_41><loc_16></location>3.1. Continuum emission at 3.5 mm</section_header_level_1>
<text><location><page_3><loc_8><loc_10><loc_48><loc_14></location>Using CARMA we detected a resolved 3.5 mm source composed of two components (Fig. 2). The spatial distribution of the continuum emission resembles</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_9></location>7 The GILDAS package is available at http://www.iram.fr/IRAMFR/GILDAS</text>
<text><location><page_3><loc_52><loc_65><loc_92><loc_92></location>very well that of the continuum at 1.4 mm shown by Fern'andez-L'opez et al. 2011a. We applied a twocomponent Gaussian model to fit the continuum data, which left no residuals over a 3σ level. From this fit, the main southwestern component is spatially coincident with the exciting source of the thermal jet, MM1, and has an integrated flux of 94 mJy beam -1 . The component to the northeast coincides with the position of MM2 and has an integrated flux of 36 mJy beam -1 . This is the first time that a measurement of MM2 can be obtained at 3 mm, and it is in good agreement with the flux density expected at this wavelength from the estimated spectral index for MM2, using previous millimeter and submillimeter observations (Fern'andez-L'opez et al. 2011a). Although no continuum emission from the molecular core (MC) detected by Qiu & Zhang (2009) was needed to explain the whole continuum emission, the angular resolution of the present CARMA observations does not allow us to rule out the possibility that the molecular core contributes to the observed dust emission.</text>
<text><location><page_3><loc_52><loc_56><loc_92><loc_65></location>The total flux measured on the field of view is 145 ± 15 mJy beam -1 , which is also consistent with the total flux reported by G'omez et al. 2003. However, they proposed that all the emission is associated with MM1, while we find that a large fraction of the emission is associated with MM1 and the rest of the emission comes from MM2.</text>
<section_header_level_1><location><page_3><loc_63><loc_53><loc_66><loc_54></location>3.2.</section_header_level_1>
<section_header_level_1><location><page_3><loc_67><loc_53><loc_80><loc_54></location>Molecular emission</section_header_level_1>
<section_header_level_1><location><page_3><loc_63><loc_51><loc_81><loc_52></location>3.2.1. Detection of outflows</section_header_level_1>
<text><location><page_3><loc_52><loc_29><loc_92><loc_51></location>CARMA observations of the classical outflow tracers SiO(2-1), HCN(1-0) and HCO + (1-0) were aimed to studying the emission from the outflow associated with MM1 and its powerful radio jet (Ridge & Moore 2001; Benedettini et al. 2004). We do not detect any emission associated with this radio jet in the velocity range (211,+189) km s -1 covered by the observations, although part of the HCN and HCO + low-velocity emission could be associated with material encasing the collimated radio jet. In what follows we adopt v lsr = 11 . 8 km s -1 as the cloud velocity (Fern'andez-L'opez et al. 2011b). The CARMA observations do not show molecular emission from the southeast monopolar outflow associated with MM2 and previously reported by Qiu & Zhang (2009), but they show two other monopolar outflows, possibly originated from MM2 and/or MC (Fig. 3).</text>
<text><location><page_3><loc_52><loc_17><loc_92><loc_29></location>The SiO(2-1) arises only at redshifted velocities (from +4 to +38 km s -1 with respect to the cloud velocity 8 ) and from two different spots: a well collimated lobe running in the northeast direction, east of MM2, and a low collimated lobe running in the northwest direction, north of MM2 and apparently coinciding with the infrared reflection nebula (e.g., Aspin & Geballe 1992) seen in grey scale in Fig 5 (which only shows the channels with the main emission from -12 to +26 km s -1 ).</text>
<text><location><page_3><loc_52><loc_10><loc_92><loc_17></location>Fig. 6 shows the HCN(1-0) velocity cube. The spectral resolution of these observations does not resolve the hyperfine structure of this line, since the two strongest transitions are within 5 km s -1 . The redshifted HCN(10) emission mainly coincides with the same two outflows</text>
<figure>
<location><page_4><loc_23><loc_70><loc_75><loc_90></location>
<caption>Figure 2. Left: CARMA image of the 3.5 mm continuum emission (black contours and color scale) toward the central region of IRAS 18162-2048. Contours are 4, 6, 9, 12, 15, 18 and 21 × 2 mJy beam -1 , the rms noise of the image. The star mark the position of the putative molecular core (Qiu & Zhang 2009) and the circles mark the position of the millimeter sources reported in Fern'andez-L'opez et al. (2011a). The synthesized beam is shown in the bottom left corner. Right: 3.5 mm continuum emission (color scale) overlapped with the two-Gaussian fit made to the image (black contours as in left panel, see § 3.1) and the residuals left after the fit (red contours at -4, -3, -1.5, -0.75, 0.75, 1.5, 3 and 4 × 2 mJy beam -1 ).</caption>
</figure>
<figure>
<location><page_4><loc_24><loc_19><loc_77><loc_57></location>
<caption>Figure 3. Black contours represent the velocity-integrated flux maps of SiO(2-1) (top left panel), obtained by integrating the emission at about -1 km s -1 and +31 km s -1 ; HCN(1-0) (top right panel), obtained by integrating the emission at about -7 km s -1 and +47 km s -1 ; HCO + (1-0) (bottom left panel) obtained by integrating the emission at about -12 km s -1 and +10 km s -1 ; HC 3 N(10-9) (bottom right panel), obtained by integrating the emission at about -1 km s -1 and +4 km s -1 . The contours are -35, -25, -15, 15, 25, 35, 45, 55, 65, 75, 85 and 95% of the peak flux in all the maps. The color scale image represents the 2MASS K-band emission. The magenta circles and the star mark the position of the millimeter cores and the molecular core, respectively. Two red arrows show the redshifted NE and NW outflow directions, while the blue arrow show the blueshifted SE outflow path (see § 3.2.1). The region inside the dashed circle has an almost uniform rms noise level. The synthesized beam is shown in the bottom right corner of each panel.</caption>
</figure>
<figure>
<location><page_5><loc_11><loc_40><loc_46><loc_92></location>
<caption>Figure 4. SMA redshifted (red contours) and blueshifted (blue contours) emission of the CO(3-2) line. In the two upper panels the magenta circles mark the positions of MM1 and MM2. The dashed black circles represent the SMA primary beam and the synthesized beam is shown in the bottom left corner of each panel. Top panel: LV component obtained by integrating the emission at about -15 km s -1 and +20 km s -1 (blue: from -22 to -10 km s -1 ; red: from +14 to +26 km s -1 ). Contour levels are 5, 13, 25, 40, 60, 80 and 100 × 60 mJy beam -1 . The synthesized beam is 3 . '' 0 × 2 . '' 6, P.A.= -6 · . Middle panel: MVcomponent obtained by integrating the emission at about -50 km s -1 and +50 km s -1 (blue: from -72 to -2 km s -1 ; red: from +24 to +74 km s -1 ). Contour levels are 5, 13, 25, 40, 60, 80 and 100 × 10 mJy beam -1 . The synthesized beam is 3 . '' 0 × 2 . '' 6, P.A.= -6 · . Bottom panel: Zoom to the MM2 surroundings of the HV component obtained by integrating the emission at about -85 km s -1 and +115 km s -1 (blue: from -118 to -74 km s -1 ; red: from +106 to +130 km s -1 ). CO(3-2) contour levels are 3, 5, 7, 10, 14, 18, 22, 26 and 30 × 3 . 2 mJy beam -1 . The synthesized beam is 0 . '' 45 × 0 . '' 34, P.A.= 31 · . The grey scale shows the SMA 870 µ m continuum emission from the easternmost MM2(E) and the westernmost MM2(W).</caption>
</figure>
<text><location><page_5><loc_8><loc_7><loc_48><loc_15></location>seen in SiO(2-1), but with lower velocities (near the cloud velocity), and the emission is more spread out, nearly matching the spatial distribution of the infrared reflection nebula and following the radio jet path. At these velocities, the HCN(1-0) also shows elongated emission due southwest (see also Fig. 3).</text>
<text><location><page_5><loc_52><loc_85><loc_92><loc_92></location>HCO + (1-0) redshifted emission is weaker, but also coinciding with the two SiO(2-1) outflows (Fig. 7). The low velocity emission from this molecular line also appears extended and mostly associated with the infrared reflection nebula and possibly the radio jet path.</text>
<text><location><page_5><loc_52><loc_57><loc_92><loc_85></location>In addition to CARMA observations, we present here new high-angular resolution SMA observations showing CO(3-2) emission from the southeast outflow. Fig. 4 shows the CO(3-2) SMA images in three panels showing three different velocity regimes: low (LV), medium (MV) and high velocity (HV). The southeast outflow appears in all of them, comprised of blue and redshifted emission at LV and MV, and mostly blueshifted emission at HV. The HV panel, the map with highest angular resolution, clearly shows that the origin of the southeast outflow is MM2(E). Inspecting that image, it is evident that this outflow is wiggling, being ejected due east at the origin, and then turning southeast (about ∼ 30 · change in position angle). Such behavior can be produced by precessing motion of the source (e.g., Raga et al. 2009). The middle panel shows a possible additional east turn in the blueshifted emission far away from MM2(E). Finally, the LV and MV maps also show the redshifted northwest lobe and hints of the northeast lobe, but the angular resolution of these maps do not allow identification of their origin.</text>
<text><location><page_5><loc_52><loc_52><loc_92><loc_57></location>From now on we will call the two high velocity SiO lobes, NE (Northeast, P.A.= 71 · ) and NW (Northwest, P.A.= -22 · ) outflows. The southeast outflow (P.A.= 126 · ) will be designated as the SE outflow.</text>
<section_header_level_1><location><page_5><loc_59><loc_48><loc_85><loc_49></location>3.2.2. Gas tracing the reflection nebula</section_header_level_1>
<text><location><page_5><loc_52><loc_34><loc_92><loc_47></location>Figures 3, 5, 6 and 7 allow comparisons between the observed molecular emission and the 2MASS K-band emission from the infrared reflection nebula and the 6 cm VLA radio continuum emission from the radio jet launched from MM1. The HCN(1-0) and HCO + (1-0) lines are tracing the gas from the molecular outflows, but also other kind of structures. We have also detected emission from the HC 3 N(10-9) transition, which is weaker than the other detected lines and at velocities close to systemic (Fig. 8).</text>
<text><location><page_5><loc_52><loc_7><loc_92><loc_34></location>The 2MASS K-band image shows the well-known bipolar reflection nebula (Aspin et al. 1991; Aspin & Geballe 1992). This nebula wraps the radio jet path. Toward the position of MM1, the nebula becomes narrower and splits into two U-shaped lobes. The north lobe, with an intricate structure, matches quite well most of the HCN(1-0), HCO + (1-0) and HC 3 N(10-9) emission between -12 and +4 km s -1 . The blueshifted molecular line emission (12 to -7 km s -1 ) traces the basis of the reflexion nebula and spreads out due north, covering the whole northern lobe at velocities in the range -1 km s -1 and +4 km s -1 . At these velocities, part of the HCN(1-0) and HCO + (10) emission follows the radio jet trajectory towards the north of the MC position. The southern lobe has fainter K-band emission than the northern lobe and it has no molecular emission associated with it. The lack of strong molecular line emission in that area of the nebula is probably due to the lack of dense molecular gas, as shown in the C 17 O(2-1) emission map by (Fern'andez-L'opez et al. 2011b).</text>
<figure>
<location><page_6><loc_15><loc_63><loc_86><loc_92></location>
<caption>Figure 5. SiO(2-1) emission velocity map (black contours) overlapped with a VLA HH 80/81/80N radio jet image (green contours) at 6 cm and a 2MASS K-band infrared image (grey scale). SiO(2-1) contours are -3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14 and 16 × 20 mJy beam -1 , the rms noise of the image. Symbols are the same that those of Fig. 3.</caption>
</figure>
<figure>
<location><page_6><loc_15><loc_28><loc_86><loc_57></location>
<caption>Figure 6. HCN(1-0) emission velocity map (black contours) overlapped with a VLA HH 80/81/80N radio jet image (green contours) at 6 cm and a 2MASS K-band infrared image (grey scale). HCN(1-0) contours are -3, 3, 5, 8, 12, 17, 23, 30, 38, 47 and 57 × 30 mJy beam -1 , the rms noise of the image. Symbols are the same that those of Fig. 3.</caption>
</figure>
<section_header_level_1><location><page_6><loc_8><loc_20><loc_48><loc_22></location>4. ANALYSIS OF THE OUTFLOW PROPERTIES FROM SIO EMISSION</section_header_level_1>
<text><location><page_6><loc_8><loc_8><loc_48><loc_20></location>SiO is a good tracer of outflows (e.g. Jim'enez-Serra et al. 2010; L'opez-Sepulcre et al. 2011 and references therein). Its abundance is dramatically increased in shocks (e.g., Schilke et al. 1997; Pineau des Forets & Flower 1996; Gusdorf et al. 2008a,b; Guillet et al. 2009) and has been used as a tool to map the innermost part of outflows (Martin-Pintado et al. 1992; Santiago-Garc'ıa et al. 2009). In addition, it does suffer minimal contamination</text>
<text><location><page_6><loc_52><loc_16><loc_92><loc_23></location>from quiescent and infalling envelopes and can not be masked by blended hyperfine components as in the case of the HCN(1-0) line. Thus, we choose this molecular outflow tracer to derive the characteristics of the NE and NW outflows.</text>
<text><location><page_6><loc_52><loc_8><loc_92><loc_16></location>SiO(2-1) has been previously detected toward IRAS 18162-2048 with the IRAM 30 m by Acord (1997). They reported an integrated flux of 1 . 9 ± 0 . 4 K km s -1 , while here we have measured 2 . 5 ± 0 . 05 K km s -1 . Applying a 27 '' beam dilution (the angular size of the IRAM 30 m beam), the CARMA measurement becomes</text>
<figure>
<location><page_7><loc_15><loc_63><loc_86><loc_92></location>
<caption>Figure 7. HCO + (1-0) emission velocity map (black contours) overlapped with a VLA HH 80/81/80N radio jet image (green contours) at 6 cm and a 2MASS K-band infrared image (grey scale). HCO + (1-0) contours are -3, 3, 4, 6, 9, 11, 16, 22 and 29 × 30 mJy beam -1 , the rms noise of the image. Symbols are the same that those of Fig. 3.</caption>
</figure>
<figure>
<location><page_7><loc_15><loc_27><loc_86><loc_57></location>
<caption>Figure 8. HC 3 N(10-9) emission velocity map (black contours) overlapped with a VLA HH 80/81/80N radio jet image (green contours) at 6 cm and a 2MASS K-band infrared image (grey scale). HC 3 N(10-9) contours are -3, 3, 4, 6, 9, 11 and 16 × 35 mJy beam -1 , the rms noise of the image. Symbols are the same that those of Fig. 3.</caption>
</figure>
<text><location><page_7><loc_8><loc_17><loc_48><loc_23></location>1.9 K km s -1 . This implies that CARMA is recovering the same flux as the IRAM 30 m for this transition and hence, we can use it to estimate some physical parameters of the outflow.</text>
<text><location><page_7><loc_8><loc_12><loc_48><loc_17></location>In the first place, we estimate the column density using the following equation (derived from the expressions contained in the appendix of Frau et al. 2010), written in convenient units:</text>
<formula><location><page_7><loc_9><loc_6><loc_48><loc_9></location>N H 2 = 2 . 04 × 10 20 χ ( SiO ) Q ( T ex ) e E u /T ex Ω s ν 3 Sµ 2 ∫ S ν dv ,</formula>
<text><location><page_7><loc_52><loc_8><loc_92><loc_23></location>where χ ( SiO ) is the abundance of SiO relative to H 2 , Q ( T ex ) is the partition function, E u and T ex are the upper energy level and the excitation temperature, both in K, Ω is the angular size of the outflow in square arcseconds, ν is the rest frequency of the transition in GHz, Sµ 2 is the product of the intrinsic line strength in erg cm 3 D -2 and the squared dipole momentum in D 2 . Following the NH 3 observations of G'omez et al. (2003), we adopted an excitation temperature of 30 K. E u , Q ( T ex ) and Sµ 2 = 19 . 2 erg cm 3 were extracted from the CDMS catalogue (Muller et al. 2005). The</text>
<table>
<location><page_8><loc_8><loc_70><loc_96><loc_88></location>
<caption>Table 2 Outflow parameters</caption>
</table>
<text><location><page_8><loc_8><loc_66><loc_95><loc_69></location>Note . - Outflow parameters from top to bottom: SiO column density (N(SiO)), position angle, length ( λ ), maximum velocity (v max ), dynamical time (t dyn ), mass (M), mass injection rate ( ˙ M ), momentum (P), momentum rate ( ˙ P ), energy (E), and mechanical luminosity (L mech ). The uncertainties are just statistical and the inclination with respect to the plane of the sky has not been taken into account.</text>
<unordered_list>
<list_item><location><page_8><loc_8><loc_65><loc_34><loc_66></location>(a) Data extracted from Qiu & Zhang (2009).</list_item>
<list_item><location><page_8><loc_8><loc_64><loc_61><loc_65></location>(b) Values were extracted from low-mass Class 0 sources observed in several works, see text.</list_item>
<list_item><location><page_8><loc_8><loc_62><loc_91><loc_64></location>(c) Data relative to outflows from high-mass protostars with luminosities below 10 3 L /circledot were averaged from observations by Zhang et al. (2005).</list_item>
<list_item><location><page_8><loc_8><loc_60><loc_95><loc_62></location>(d) Data relative to outflows from high-mass protostars with luminosities between 10 3 L /circledot and 5 × 10 3 L /circledot were averaged from observations by Zhang et al. (2005).</list_item>
</unordered_list>
<text><location><page_8><loc_8><loc_56><loc_48><loc_59></location>term ∫ S ν dv is the measured integrated line emission in Jy beam -1 km s -1 .</text>
<text><location><page_8><loc_8><loc_47><loc_48><loc_56></location>We analyzed the two SiO outflows separately (NE and NW outflows), finding average column densities of 1 . 1 ± 0 . 2 × 10 13 cm -2 and 1 . 2 ± 0 . 2 × 10 13 cm -2 respectively. We used these values to derive the mass ( M = N H 2 µ g m H 2 Ω s ) and other properties of the outflow, assuming a mean gas atomic weight µ g = 1 . 36. The results are shown in Table 2.</text>
<text><location><page_8><loc_8><loc_13><loc_48><loc_47></location>We used an SiO abundance of χ ( SiO ) = 5 × 10 -9 . This value is in good agreement with the χ ( SiO ) found in outflows driven by other intermediate/high-mass protostars. For instance, in a recent paper, S'anchez-Monge et al. (2013) found the χ ( SiO ) in 14 high-mass protostars outflows range between 10 -9 and 10 -8 (similar values were also encountered in Hatchell et al. 2001; Qiu et al. 2007; Codella et al. 2013). It is important to mention that the uncertainty in the SiO abundance is probably one of the main sources of error for our estimates (e.g., Qiu et al. 2009). One order of magnitude in χ ( SiO ) translates into one order of magnitude in most of the outflow properties given in Table 2 (M, ˙ M , P, ˙ P , E and L mech ). Most of the outflow properties in this table were derived following the approach of Palau et al. (2007). In order to derive the mass and momentum rates, together with the mechanical luminosity, we need to know the dynamical timescale of the outflows, which can be determined as t dyn = λ/v max , where λ is the outflow length. No correction for the inclination was included. Hence, we estimate the dynamical time of the NE and NW outflows as 18000 yr and 5000 yr, and their masses as 1.60 and 0.74 M /circledot , respectively (Table 2). In addition, it is worth to note that most of the mass (about 80%) of the NE outflow is concentrated in its brightest condensation (see Fig. 3).</text>
<section_header_level_1><location><page_8><loc_23><loc_9><loc_34><loc_10></location>5. DISCUSSION</section_header_level_1>
<text><location><page_8><loc_10><loc_8><loc_46><loc_9></location>5.1. Asymmetric outflows and their possible origin</text>
<text><location><page_8><loc_52><loc_41><loc_92><loc_59></location>Recent work has provided evidence supporting MM1 being a 11-15 M /circledot high-mass protostar, probably still accreting material from a 4 M /circledot rotating disk, and MM2 being a massive dusty core containing at least one highmass protostar, MM2(E), in a less evolved stage than MM1 (Fern'andez-L'opez et al. 2011b). MM2 also contains another core, MM2(W), thought to be in a still earlier evolutionary stage. In addition to these well known protostars, at present it is controversial whether MC harbors another protostar, since neither a thermal radio continuum nor a dust continuum emission has been detected in this molecular core. Hence in the area MM2-MC we can count two or three protostars that could be associated with the molecular outflows reported in this work.</text>
<text><location><page_8><loc_52><loc_10><loc_92><loc_41></location>IRAS 18162-2048 has protostars surrounded by accretion disks and/or envelopes, and these protostars are associated with molecular outflows and jets, resembling low-mass star-forming systems. The most apparent case would be that of MM1, with one of the largest bipolar radio jets associated with a protostar. Being a scaled up version of the low-mass star formation scenario implies more energetic outflows (as we see in MM1 and MM2), which means larger accretion rates, but it also implies large amounts of momentum impinged on the surrounding environment, affecting occasionally the nearby protostellar neighbors and their outflows. In this case, IRAS 18162-2048 is a region with multiple outflows, mostly seen monopolar. Our observations show no well opposed counterlobes for the NW, NE and SE outflows. That could be explained if the counterlobes are passing through the cavity excavated by the radio jet or through regions of low molecular abundance. Another explanation for the asymmetry of outflows, perhaps more feasible in a high-mass star-forming scenario with a number of protostars and powerful outflows, is deflection after hitting a dense clump of gas and dust (e.g., Raga et al. 2002).</text>
<text><location><page_8><loc_52><loc_7><loc_92><loc_10></location>Adding a little bit more to the complexity of the region, the SE outflow is comprised of two very different</text>
<text><location><page_9><loc_8><loc_44><loc_48><loc_92></location>kinematical components. Fig. 4 shows high velocity emission at two well separated velocities laying almost at the same spatial projected path. A blueshifted and redshifted spatial overlap has been usually interpreted as an outflow laying close to the plane of the sky. However, the SE outflow has large radial blueshifted and redshifted velocities ( ∼ ± 50 km s -1 ). One possibility is that the high velocity emission of the SE outflow comes from two molecular outflows, one redshifted and the other one blueshifted, both originated around the MM2 position. A second possibility is that the SE outflow is precessing with an angle α = 15 · (see Fig. 2 in Raga et al. 2009). We derived this angle by taking half the observed wiggling angle of the outflow axis in projection (from P.A. /similarequal 95 · at the origin of the outflow to P.A. /similarequal 126 · away from it). Hence, the absolute velocity of the ejecta should be ∼ 200 km s -1 , which seems to be a reasonable value (see e.g., Bally 2009). A smaller α angle would imply a higher outflow velocity. From the central panel of Fig. 4, we roughly estimate the period of the wiggles of the SE outflow as λ /similarequal 16 '' (27000 AU). We have assumed that the SE outflow completes one precessing period between the position of MM2(E) and the end of the CO(3-2) blueshifted emission in our map. Thus, the precession period is τ p = λ/ ( v j cos α ) /similarequal 660 yr . If the precession is caused by the tidal interaction between the disk of a protostar in MM2(E) and a companion protostar in a non-coplanar orbit (e.g. Terquem et al. 1999b; Montgomery 2009) then it is possible to obtain some information about the binary system. We use an equivalent form of equation (37) from Montgomery (2009) for circular precessing Keplerian disks. This equation relates the angular velocity at the disk edge ( ω d ), the Keplerian orbital angular velocity of the companion around the primary protostar ( ω o ) and the retrograde precession rate of the disk and the outflow ( ω p ):</text>
<formula><location><page_9><loc_21><loc_39><loc_36><loc_42></location>ω p = -15 32 ω 2 o ω d cos α ,</formula>
<text><location><page_9><loc_8><loc_9><loc_48><loc_38></location>where α is the inclination of the orbit of the companion with respect to the plane of the disk (or obliquity angle), and this angle is the same as the angle of the outflow precession (i.e. the angle between the outflow axis and the line of maximum deviation of the outflow from this axis; Terquem et al. 1999a). Using this expression and adopting reasonable values for the mass of the primary protostar and the radius of its disk we can constrain the orbital period and the radius of the companion protostar. G'omez et al. (1995) assigned a B4 ZAMS spectral type to MM2(E) based on its flux at 3.5 cm. A B4 spectral type protostar has about 6-7 M /circledot (Table 5 in Molinari et al. 1998). On the other side, the dust emission of MM2(E) has a radius no larger than 300 AU (Fern'andez-L'opez et al. 2011a) and we consider 50 AU as a reasonable lower limit for the disk radius. With all of this we derived an orbital period between 200 yr and 800 yr and an orbital radius between 35 M 1 / 3 2 AU and 86 M 1 / 3 2 AU for the putative MM2(E) binary system, being M 2 the mass of the secondary protostar expressed in solar masses.</text>
<text><location><page_9><loc_8><loc_7><loc_48><loc_9></location>The HV panel of Fig. 4 clearly shows for the first time the precise origin of the SE-blueshifted outflow:</text>
<text><location><page_9><loc_52><loc_69><loc_92><loc_92></location>MM2(E). Its corresponding redshifted counterlobe appears truncated between MM2(E) and MM2(W). However, the CO(3-2) LV and MV panels show that the redshifted NW outflow reaches the position of MM2 (see also HCN(1-0) emission in Fig. 6). Then, it is possible that the SE outflow counterpart is the NW outflow. This resolves the monopolar nature of both the NW and SE outflows and explains the prominent redshifted wing spectral line profiles of H 2 CO and SO transitions previously observed by Fern'andez-L'opez et al. (2011b) at the position of MC, as being produced by a receding outflow from MM2(E) colliding with a dense cloudlet at the position of MC. The change in the SE-NW outflow direction could be explained by a deflection due north of the NW outflow. The cause of the deflection could be a direct impact against MM2(W), or the action of the powerful HH 80/81/80N wind over the NW lobe.</text>
<text><location><page_9><loc_52><loc_59><loc_92><loc_69></location>The NE outflow has a monopolar structure at first sight too (Figs. 5, 3 and 6). Its origin cannot be well determined due to angular resolution constraints, but it is also in the MM2-MC area. Figs. 3, 6 and 7 (channels at 1 and +4 km s -1 ) show some signs that a low-velocity counterlobe may exist with a position angle of -112 · , almost opposite to the NE outflow. It would spread out 0.20-0.25 pc from the MM2-MC position.</text>
<section_header_level_1><location><page_9><loc_54><loc_55><loc_90><loc_57></location>5.2. Evolutionary stage of outflows and protostars</section_header_level_1>
<text><location><page_9><loc_52><loc_16><loc_92><loc_55></location>As stated in several works, SiO is a commonly used molecular tracer of shocked gas in outflows from lowmass protostars (e.g., Hirano et al. 2006; Lee et al. 2008; Santiago-Garc'ıa et al. 2009), but SiO is also found in outflows from high-mass protostars (e.g., Cesaroni et al. 1999; Hatchell et al. 2001; Qiu et al. 2009; Beuther et al. 2004; Zhang et al. 2007a,b; L'opez-Sepulcre et al. 2011; Zapata et al. 2012; Leurini et al. 2013). However, with the present CARMA observations we have not detected SiO(2-1) emission associated with the HH 80/81/80N radio jet, nor with the SE outflow along the ∼ 400 km s -1 of the CARMA SiO(2-1) window bandwidth. The HH 80/81/80N and the SE outflows have not been observed either in the other molecular transitions of this study, HCO + (1-0) and HCN(1-0), also known to be good outflow tracers. If anything, some HCO + and HCN emission may come from gas pushed away by the collimated and high-velocity jets or may be due to lowvelocity winds. Both outflows have been observed in CO lines at the velocities sampled by the CARMA observations, though. Then, what is producing the different chemistry in the outflows of the region? Why is the SiO(2-1) not detected in the SE outflow nor the radio jet, while the other two outflows NE and NW are? Furthermore, why is only one lobe of each the NE and NW outflows detected? There are other cases in the literature where a similar behavior is observed in CO and SiO (e.g., Zhang et al. 2007a,b; Reid & Matthews 2008; Zapata et al. 2012; Codella et al. 2013).</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_16></location>It has been proposed that the SiO abundance can decrease with the age of the outflow (Codella et al. 1999; Miettinen et al. 2006; Sakai et al. 2010; L'opez-Sepulcre et al. 2011), which could explain the differences of SiO emission from the outflows of the same region. This hypothesis implies that during the early stages, the gas surrounding the protostar is denser</text>
<text><location><page_10><loc_8><loc_50><loc_48><loc_92></location>and rich in grains, producing stronger shocks between the outflow and the ambient material, and thus producing an abundant release of SiO molecules. After that, in more evolved stages, the outflow digs a large cavity close to the protostar and thus the shocks are weaker, and grains are rarer. The hypothesis has further support in the shorter SiO depletion timescale (before it freezes out onto the dust grains) with respect to the typical outflow's timescale (some 10 4 yr), together with its disappearance from the gas phase favoring the creation of SiO 2 (Pineau des Forets & Flower 1996; Gibb et al. 2004) and could describe well the HH 80/81/80N jet case, since it produced a cavity probably devoid of dust grains. Now, we can compare the timescale of the outflows in the central region of IRAS 18162-2048. The radio jet HH 80/81/80N is 10 6 yr (Benedettini et al. 2004), the SE outflow has an age of about 2 × 10 3 years (Qiu & Zhang 2009) and the NE and NW outflows have 2 × 10 4 and 5 × 10 3 years. Therefore, except for the SE outflow, the outflow timescales would be in good agreement with SiO decreasing its abundance with time. Actually, as indicated before, the case for the SE outflow is more complex. It has high velocity gas and it is apparently precessing. That can therefore indicate a larger outflow path. In addition CO(3-2) observations are constrained by the SMA primary beam, implying that the outflow could be larger than observed and thereby older. In any case, if the SE outflow is the counterlobe of the NW one (see § 5.1), then a different explanation must be found to account for the chemical differences between these two outflow lobes.</text>
<text><location><page_10><loc_8><loc_8><loc_48><loc_50></location>We can also compare the characteristics of the NE and NW outflows with those of outflows ejected by (i) highmass protostars and (ii) low-mass Class 0 protostars, in order to put additional constrains on the ejecting sources. Zhang et al. (2005) made an outflow survey toward highmass star-forming regions using CO single-dish observations. After inspecting this work we summarize the properties belonging to outflows from L < 10 3 L /circledot protostars and L ∈ (10 3 , 5 × 10 3 ) L /circledot protostars in Table 2. This Table shows in addition, information on outflows from lowmass Class 0 protostars as well, gathered up from several sources (Arce & Sargent 2004, 2005; Kwon et al. 2006; Davidson et al. 2011). The characteristics of the NE and NWoutflows in IRAS 18162-2048 (as well as those of the SE outflow) are similar to outflows from high-mass protostars, being the NE outflow more energetic and with higher momentum than the NW and SE outflows. On the contrary, the properties of outflows from low-mass Class 0 protostars, although similar in length and dynamical time, have in general lower mass, and overall kinetic energy, about four orders of magnitude lower. All of this indicates that the NE, NW and SE outflows in IRAS 181622048 could be associated with intermediate or high-mass protostars in a very early evolutive stage (massive class 0 protostars). Therefore, given the powerful outflowing activity from MM2, the protostars would be undergoing a powerful accretion process in which the gas from the dusty envelope (about 11 M /circledot ) is probably falling directly onto the protostars. This kind of objects is very rare. Maybe the closest case is that of Cepheus E (Smith et al. 2003). The outflow from this intermediate-mass protostar, which is surrounded by a massive ∼ 25 M /circledot enve-</text>
<text><location><page_10><loc_52><loc_88><loc_92><loc_92></location>is very young (t dyn ∼ 1 × 10 3 yr), with a mass and an energy (M ∼ 0 . 3 M /circledot , E ∼ 5 × 10 45 ergs) resembling those obtained for the NE, NW and SE outflows.</text>
<section_header_level_1><location><page_10><loc_66><loc_85><loc_78><loc_86></location>6. CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_52><loc_70><loc_92><loc_85></location>We have carried out CARMA low-angular resolution observations at 3.5 mm and SMA high-angular resolution observations at 870 µ m toward the massive star-forming region IRAS 18162-2048. We have also included the analysis of SMA low-angular resolution archive data analysis of the CO(3-2) line. The analysis of several molecular lines, all of which are good outflow tracers, resulted in the physical characterization of two previously not well detected outflows (NE and NW outflows) and the clear identification of the driving source of a third outflow (SE outflow). The main results of this work are as follows:</text>
<unordered_list>
<list_item><location><page_10><loc_54><loc_54><loc_92><loc_68></location>· We observed three apparently monopolar or asymmetric outflows in IRAS 18162-2048. The NE and NWoutflows were detected in most of the observed molecular lines (SiO, HCN, HCO + and CO), while the SE outflow was only clearly detected in CO. The outflow associated with HH 80/81/80N was undetected. At most, it could explain some HCN and HCO + low-velocity emission associated with the infrared reflection nebula, which could be produced by dragged gas or a wide open angle lowvelocity wind from MM1.</list_item>
<list_item><location><page_10><loc_54><loc_50><loc_92><loc_52></location>· The NE and NW outflows have their origins close to MM2.</list_item>
<list_item><location><page_10><loc_54><loc_41><loc_92><loc_48></location>· We have estimated the physical properties of the NE and NW outflows from their SiO emission. They have similar characteristics to those found in molecular outflows from massive protostars, being the NE outflow more massive and energetic than the NW and SE outflows.</list_item>
<list_item><location><page_10><loc_54><loc_25><loc_92><loc_39></location>· SMA high-angular resolution CO(3-2) observations have identified the driving source of the SE outflow: MM2(E). These observations provide evidence of precession along this outflow, which show a change of about 30 · in the position angle and a period of 660 yr. If the precession of the SE outflow is caused by the misalignment between the plane of the disk and the orbit of a binary companion, then the orbital period of the binary system is 200-800 yr and the orbital radius is 35-86 M 1 / 3 2 AU.</list_item>
<list_item><location><page_10><loc_54><loc_14><loc_92><loc_24></location>· We discuss the monopolar or asymmetric appearance of all three outflows. We provide evidence that the SE and NW outflows are linked and that precession and a possible deflection are the causes of the asymmetry of the outflow. In addition, the NE outflow could have a smaller and slower southwest counterlobe, maybe associated with elongated HCN and HCO + emission.</list_item>
<list_item><location><page_10><loc_54><loc_7><loc_92><loc_12></location>· Finally, we discuss that the SiO outflow content in IRAS 18162-2048 could be related to outflow age. This would explain the SiO non-detection of the radio jet HH 80/81/80N.</list_item>
</unordered_list>
<text><location><page_11><loc_8><loc_88><loc_48><loc_92></location>The authors want to bring a special reminder of our good fellow Yolanda G'omez, who helped in the very beginning of this work with her contagious optimism.</text>
<text><location><page_11><loc_8><loc_72><loc_48><loc_88></location>We thank all members of the CARMA and SMA staff that made these observations possible. We thank Pau Frau for helping with the SMA observations. MFL acknowledges financial support from University of Illinois and thanks John Carpenter and Melvin Wright for their patience with CARMA explanations. MFL also thanks the hospitality of the Instituto de Astronom'ıa (UNAM), M'exico D.F., and of the CRyA, Morelia. JMG are supported by the Spanish MICINN AYA2011-30228-C03-02 and the Catalan AGAUR 2009SGR1172 grants. SC acknowledges support from CONACyT grants 60581 and 168251. LAZ acknowledges support from CONACyT.</text>
<text><location><page_11><loc_8><loc_59><loc_48><loc_72></location>Support for CARMA construction was derived from the Gordon and Betty Moore Foundation, the Kenneth T. and Eileen L. Norris Foundation, the James S. McDonnell Foundation, the Associates of the California Institute of Technology, the University of Chicago, the states of Illinois, California, and Maryland, and the National Science Foundation. Ongoing CARMA development and operations are supported by the National Science Foundation under a cooperative agreement, and by the CARMA partner universities.</text>
<text><location><page_11><loc_8><loc_52><loc_48><loc_59></location>The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.</text>
<text><location><page_11><loc_10><loc_51><loc_28><loc_52></location>Facilities: CARMA, SMA</text>
<section_header_level_1><location><page_11><loc_24><loc_48><loc_33><loc_49></location>REFERENCES</section_header_level_1>
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<list_item><location><page_12><loc_8><loc_87><loc_48><loc_92></location>Terquem, C., Papaloizou, J. C. B., & Nelson, R. P. 1999b, in Astronomical Society of the Pacific Conference Series, Vol. 160, Astrophysical Discs - an EC Summer School, ed. J. A. Sellwood & J. Goodman, 71</list_item>
<list_item><location><page_12><loc_8><loc_82><loc_48><loc_87></location>Wilner, D. J., & Welch, W. J. 1994, ApJ, 427, 898 Zapata, L. A., Fern'andez-L'opez, M., Curiel, S., Patel, N., & Rodriguez, L. F. 2013, ArXiv e-prints, arXiv:1305.4084 [astro-ph.GA] Zapata, L. A., Ho, P. T. P., Schilke, P., et al. 2009, ApJ, 698, 1422</list_item>
<list_item><location><page_12><loc_52><loc_88><loc_90><loc_92></location>Zapata, L. A., Loinard, L., Su, Y.-N., et al. 2012, ApJ, 744, 86 Zhang, Q., Hunter, T. R., Beuther, H., et al. 2007a, ApJ, 658, 1152</list_item>
<list_item><location><page_12><loc_52><loc_85><loc_90><loc_88></location>Zhang, Q., Hunter, T. R., Brand, J., et al. 2005, ApJ, 625, 864 Zhang, Q., Sridharan, T. K., Hunter, T. R., et al. 2007b, A&A, 470, 269</list_item>
</document>
[
{
"title": "ABSTRACT",
"content": "In this paper we present Combined Array for Research in Millimeter-wave Astronomy (CARMA) 3.5 mm observations and SubMillimeter Array (SMA) 870 µ m observations toward the high-mass star-forming region IRAS 18162-2048, the core of the HH 80/81/80N system. Molecular emission from HCN, HCO + and SiO is tracing two molecular outflows (the so-called Northeast and Northwest outflows). These outflows have their origin in a region close to the position of MM2, a millimeter source known to harbor a couple protostars. We estimate for the first time the physical characteristics of these molecular outflows, which are similar to those of 10 3 -5 × 10 3 L /circledot protostars, suggesting that MM2 harbors high-mass protostars. High-angular resolution CO observations show an additional outflow due southeast. We identify for the first time its driving source, MM2(E), and see evidence of precession. All three outflows have a monopolar appearance, but we link the NW and SE lobes, explaining their asymmetric shape as a consequence of possible deflection. Subject headings: circumstellar matter - ISM: individual (GGD27, HH 80-81, IRAS 18162-2048) stars: formation - stars: early type - submillimeter: ISM",
"pages": [
1
]
},
{
"title": "MULTIPLE MONOPOLAR OUTFLOWS DRIVEN BY MASSIVE PROTOSTARS IN IRAS 18162-2048",
"content": "M. Fern'andez-L'opez 1 , J. M. Girart 2 , S. Curiel 3 , L. A. Zapata 4 , J. P. Fonfr'ıa 3 , and K. Qiu 5 Draft version May 21, 2018",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "High-mass stars exert a huge influence on the interstellar medium, ejecting powerful winds and large amounts of ionizing photons or exploding as supernova. Despite their importance injecting energy and momentum into the gas of galaxies, how they form is still not well understood. High-mass protostars are typically more than 1 kpc from Earth (Cesaroni et al. 2005). They are deeply embedded inside dense molecular clouds and often accompanied by close members of clusters. Hence, well detailed studies have only been possible in a few cases. From a handful of studies based on high-angular resolution observations, we know that early B-type protostars posses accretion disks (e.g., Shepherd et al. 2001; Cesaroni et al. 2005; Patel et al. 2005; Galv'an-Madrid et al. 2010; Kraus et al. 2010; Fern'andez-L'opez et al. 2011a), while for O-type protostars there is no unambiguous evidence for accretion disks yet, but vast toroids of dust and molecular gas have been seen rotating around them, showing signs of infalling gas (e.g., Sollins & Ho 2005; Sollins et al. 2005; Beltr'an et al. 2006; Zapata et al. 2009; Qiu et al. 2012; Jim'enez-Serra et al. 2012; Palau et al. 2013). On the other hand, molecular outflow studies seem to display diverse scenarios in the few cases where both, highangular resolution and relatively nearby targets were available. For example, the two nearest to the Earth 1 Department of Astronomy, University of Illinois at UrbanaChampaign, 1002 West Green Street, Urbana, IL 61801, USA; manferna@illinois.edu massive protostars (Orion BN/KL and Cep A HW2) have molecular outflows that are very difficult to interpret. It has been suggested that Orion BN/KL shows an explosion-like isotropic ejection of molecular material (Zapata et al. 2009), while Cep A HW2 drives a very fast jet (Curiel et al. 2006), which is apparently pulsing and precessing due to the gravitational interaction of a small cluster of protostars (Cunningham et al. 2009; Zapata et al. 2013). These two examples provide how analyzing the outflow activity of these kind of regions can provide important insight on the nature of each region. They are also showing us that although accretion disks could be ubiquitously found around all kind of protostars, a complete understanding on the real nature of the massive star-formation processes should almost inevitably include the interaction between close-by protostars. This introduces much more complexity to the observations, not only because of the possible interaction between multiple outflows, but also the difficulty to resolve the multiple systems with most telescopes. IRAS 18162-2048, also known as the GGD27 nebula, is associated with the unique HH 80/81/80N radio jet. It is among the nearest high-mass protostars (1.7 kpc). The large radio jet ends at HH 80/81 in the south (Mart'ı et al. 1993) and at HH 80N in the north (Girart et al. 2004), wiggling in a precessing motion across a projected distance of 7.5 pc (Heathcote et al. 1998). Recently, Masqu'e et al. (2012) have proposed that this radio jet could be larger, extending to the north, with a total length of 18.4 pc. Linear polarization has been detected for the first time in radio emission from this jet indicating the presence of magnetic fields in a protostellar jet (Carrasco-Gonz'alez et al. 2010). The jet is apparently launched close to the position of a young and massive protostar, surrounded by a very massive disk of dust and gas rotating around it (Fern'andez-L'opez et al. 2011a,b; Carrasco-Gonz'alez et al. 2012). In spite of this apparently easy-to-interpret scenario, the region contains other massive protostars (Qiu & Zhang 2009; Fern'andez-L'opez et al. 2011a) which are driving other high velocity outflows. (Sub)mm observations of the central part of the radio jet revealed two main sources, MM1 and MM2, separated about 7 '' (G'omez et al. 2003), that are probably in different evolutionary stages (Fern'andez-L'opez et al. 2011a). MM1 is at the origin of the thermal radio jet, while MM2 is spatially coincident with a water maser and has been resolved into a possible massive Class 0 protostar and a second even younger source (Fern'andez-L'opez et al. 2011b). MM2 is also associated with a young monopolar southeast CO outflow (Qiu & Zhang 2009). There is evidence of a possible third source, MC, which is observed as a compact molecular core, detected only through several millimeter molecular lines. At present, MC is interpreted as a hot core, but the observed molecular line emission could also be explained by shocks originated by the interaction between the outflowing gas from MM2 and a molecular core or clump (Fern'andez-L'opez et al. 2011b). In this paper we focus our attention on the CO(32), SiO(2-1), HCO + (1-0) and HCN(1-0) emission toward IRAS 18162-2048, aiming at detecting and characterizing outflows from the massive protostars inside the IRAS 18162-2048 core and their interaction with the molecular cloud. Millimeter continuum and line observations of IRAS 18162-2048 were made with CARMA, while submillimeter continuum and line observations were made with the SMA. In Section 2, we describe the observations undertook in this study. In Section 3 we present the main results and in section 4 we carry out some analysis of the data. Section 5 is dedicated to discuss the data, and in Section 6 we give the main conclusions of this study.",
"pages": [
1,
2
]
},
{
"title": "2.1. CARMA",
"content": "IRAS 18162-2048 observations used the CARMA 23element mode: six 10.4 m antennas, nine 6.1 m antennas, and eight 3.5 m antennas. This CARMA23 observing mode included up to 253 baselines that provide extra short spacing baselines, thus minimizing missing flux when observing extended objects. The 3.5 mm (84.8 GHz) observations were obtained in 2011 October 9, 11 and 2011 November 11. The weather was good for 3 mm observations in all three nights, with τ 230 GHz fluctuating between 0.2 and 0.6. The system temperature varied between 200 and 400 K. During those epochs, CARMA was in its D configuration and the baselines ranged from 5 to 148 m. The present observations are thus sensitive to structures between 6 '' and 70 '' , approximately 6 . We made a hexagonal nineteen pointing Nyquist-sampled mosaic with the central tile pointing at R.A.(J2000.0)=18 h 19 m 12 . s 430 and DEC(J2000.0)= -20 · 47 ' 23 . '' 80 (see Fig. 1). This kind of mosaic pattern is used to reach a uniform rms noise in 6 Note that while the observations are often said to be sensitive to structures on scales Θ = λ/B min /similarequal 206 '' [ λ ( mm ) /B min ( m )], the interferometer flux filtering could reduce the actual largest scale detection roughly in a factor 2, so that Θ /similarequal 100 '' [ λ ( mm ) /B min ( m )] (see appendix in Wilner & Welch 1994). Here we report a value estimated using the last expression. the whole area covered. The field of view of the mosaic has thus, a radius of about 70 '' . Beyond this radius the rms noise increases over 20%. At the time of the observations, the correlator of the CARMA 23-element array provided 4 separate spectral bands of variable width. One spectral band was set with a bandwidth of 244 MHz, used for the 3.5 mm continuum emission. The other three bands were set with a bandwidth of 125 MHz, providing 80 channels with a spectral resolution of 1.56 MHz ( ∼ 5 . 2 km s -1 ). One of these three bands was always tuned at the SiO (2-1) frequency. The other two bands were tuned at two of the following lines: HCO + (1-0), and HCN (1-0) and HC 3 N (10-9). The calibration scheme was the same during all the nights, with an on-source time of about 2 hours each. The gain calibrator was J1733-130 (with a measured flux density of 4.1 ± 0.4 Jy), which we also used as the bandpass calibrator. Observations of MWC349, with an adopted flux of 1.245 Jy provided the absolute scale for the flux calibration. The uncertainty in the flux scale was 10% between the three observations, while the absolute flux error for CARMA is estimated to be 15-20%. In the remainder of the paper we will only consider statistical uncertainties. The data were edited, calibrated, imaged and analyzed using the MIRIAD package (Sault et al. 1995) in a standard way. GILDAS 7 was also used for imaging. Continuum emission was built in the uv-plane from the line-free channels and was imaged using a uniform weighting to improve the angular resolution of the data. For the line emission, the continuum was removed from the uv-plane. We then imaged the continuum-free emission using a natural weighting to improve the signal-to-noise ratio. The continuum and the line images have rms noises of about 2 mJy beam -1 and 20-35 mJy beam -1 per channel, respectively.",
"pages": [
2,
3
]
},
{
"title": "2.2. SMA",
"content": "The CO(3-2) maps at subarcsecond angular resolution presented in this paper were obtained from SMA observations taken on 2011 July 18 and October 3 in the extended and very extended configurations, respectively. The calibration, reduction and imaging procedures used in these observations are described in Girart et al. (2013, in preparation). Table 1 shows the imaging parameters for this data set. The CO(3-2) maps at an angular resolution of /similarequal 2 '' presented in this paper were obtained from the public SMA data archive. The observations were done in the compact configuration on 2010 April 9. We downloaded the calibrated data and used the spectral windows around the CO(3-2) to produce integrated intensity maps.",
"pages": [
3
]
},
{
"title": "3.1. Continuum emission at 3.5 mm",
"content": "Using CARMA we detected a resolved 3.5 mm source composed of two components (Fig. 2). The spatial distribution of the continuum emission resembles 7 The GILDAS package is available at http://www.iram.fr/IRAMFR/GILDAS very well that of the continuum at 1.4 mm shown by Fern'andez-L'opez et al. 2011a. We applied a twocomponent Gaussian model to fit the continuum data, which left no residuals over a 3σ level. From this fit, the main southwestern component is spatially coincident with the exciting source of the thermal jet, MM1, and has an integrated flux of 94 mJy beam -1 . The component to the northeast coincides with the position of MM2 and has an integrated flux of 36 mJy beam -1 . This is the first time that a measurement of MM2 can be obtained at 3 mm, and it is in good agreement with the flux density expected at this wavelength from the estimated spectral index for MM2, using previous millimeter and submillimeter observations (Fern'andez-L'opez et al. 2011a). Although no continuum emission from the molecular core (MC) detected by Qiu & Zhang (2009) was needed to explain the whole continuum emission, the angular resolution of the present CARMA observations does not allow us to rule out the possibility that the molecular core contributes to the observed dust emission. The total flux measured on the field of view is 145 ± 15 mJy beam -1 , which is also consistent with the total flux reported by G'omez et al. 2003. However, they proposed that all the emission is associated with MM1, while we find that a large fraction of the emission is associated with MM1 and the rest of the emission comes from MM2.",
"pages": [
3
]
},
{
"title": "3.2.1. Detection of outflows",
"content": "CARMA observations of the classical outflow tracers SiO(2-1), HCN(1-0) and HCO + (1-0) were aimed to studying the emission from the outflow associated with MM1 and its powerful radio jet (Ridge & Moore 2001; Benedettini et al. 2004). We do not detect any emission associated with this radio jet in the velocity range (211,+189) km s -1 covered by the observations, although part of the HCN and HCO + low-velocity emission could be associated with material encasing the collimated radio jet. In what follows we adopt v lsr = 11 . 8 km s -1 as the cloud velocity (Fern'andez-L'opez et al. 2011b). The CARMA observations do not show molecular emission from the southeast monopolar outflow associated with MM2 and previously reported by Qiu & Zhang (2009), but they show two other monopolar outflows, possibly originated from MM2 and/or MC (Fig. 3). The SiO(2-1) arises only at redshifted velocities (from +4 to +38 km s -1 with respect to the cloud velocity 8 ) and from two different spots: a well collimated lobe running in the northeast direction, east of MM2, and a low collimated lobe running in the northwest direction, north of MM2 and apparently coinciding with the infrared reflection nebula (e.g., Aspin & Geballe 1992) seen in grey scale in Fig 5 (which only shows the channels with the main emission from -12 to +26 km s -1 ). Fig. 6 shows the HCN(1-0) velocity cube. The spectral resolution of these observations does not resolve the hyperfine structure of this line, since the two strongest transitions are within 5 km s -1 . The redshifted HCN(10) emission mainly coincides with the same two outflows seen in SiO(2-1), but with lower velocities (near the cloud velocity), and the emission is more spread out, nearly matching the spatial distribution of the infrared reflection nebula and following the radio jet path. At these velocities, the HCN(1-0) also shows elongated emission due southwest (see also Fig. 3). HCO + (1-0) redshifted emission is weaker, but also coinciding with the two SiO(2-1) outflows (Fig. 7). The low velocity emission from this molecular line also appears extended and mostly associated with the infrared reflection nebula and possibly the radio jet path. In addition to CARMA observations, we present here new high-angular resolution SMA observations showing CO(3-2) emission from the southeast outflow. Fig. 4 shows the CO(3-2) SMA images in three panels showing three different velocity regimes: low (LV), medium (MV) and high velocity (HV). The southeast outflow appears in all of them, comprised of blue and redshifted emission at LV and MV, and mostly blueshifted emission at HV. The HV panel, the map with highest angular resolution, clearly shows that the origin of the southeast outflow is MM2(E). Inspecting that image, it is evident that this outflow is wiggling, being ejected due east at the origin, and then turning southeast (about ∼ 30 · change in position angle). Such behavior can be produced by precessing motion of the source (e.g., Raga et al. 2009). The middle panel shows a possible additional east turn in the blueshifted emission far away from MM2(E). Finally, the LV and MV maps also show the redshifted northwest lobe and hints of the northeast lobe, but the angular resolution of these maps do not allow identification of their origin. From now on we will call the two high velocity SiO lobes, NE (Northeast, P.A.= 71 · ) and NW (Northwest, P.A.= -22 · ) outflows. The southeast outflow (P.A.= 126 · ) will be designated as the SE outflow.",
"pages": [
3,
5
]
},
{
"title": "3.2.2. Gas tracing the reflection nebula",
"content": "Figures 3, 5, 6 and 7 allow comparisons between the observed molecular emission and the 2MASS K-band emission from the infrared reflection nebula and the 6 cm VLA radio continuum emission from the radio jet launched from MM1. The HCN(1-0) and HCO + (1-0) lines are tracing the gas from the molecular outflows, but also other kind of structures. We have also detected emission from the HC 3 N(10-9) transition, which is weaker than the other detected lines and at velocities close to systemic (Fig. 8). The 2MASS K-band image shows the well-known bipolar reflection nebula (Aspin et al. 1991; Aspin & Geballe 1992). This nebula wraps the radio jet path. Toward the position of MM1, the nebula becomes narrower and splits into two U-shaped lobes. The north lobe, with an intricate structure, matches quite well most of the HCN(1-0), HCO + (1-0) and HC 3 N(10-9) emission between -12 and +4 km s -1 . The blueshifted molecular line emission (12 to -7 km s -1 ) traces the basis of the reflexion nebula and spreads out due north, covering the whole northern lobe at velocities in the range -1 km s -1 and +4 km s -1 . At these velocities, part of the HCN(1-0) and HCO + (10) emission follows the radio jet trajectory towards the north of the MC position. The southern lobe has fainter K-band emission than the northern lobe and it has no molecular emission associated with it. The lack of strong molecular line emission in that area of the nebula is probably due to the lack of dense molecular gas, as shown in the C 17 O(2-1) emission map by (Fern'andez-L'opez et al. 2011b).",
"pages": [
5
]
},
{
"title": "4. ANALYSIS OF THE OUTFLOW PROPERTIES FROM SIO EMISSION",
"content": "SiO is a good tracer of outflows (e.g. Jim'enez-Serra et al. 2010; L'opez-Sepulcre et al. 2011 and references therein). Its abundance is dramatically increased in shocks (e.g., Schilke et al. 1997; Pineau des Forets & Flower 1996; Gusdorf et al. 2008a,b; Guillet et al. 2009) and has been used as a tool to map the innermost part of outflows (Martin-Pintado et al. 1992; Santiago-Garc'ıa et al. 2009). In addition, it does suffer minimal contamination from quiescent and infalling envelopes and can not be masked by blended hyperfine components as in the case of the HCN(1-0) line. Thus, we choose this molecular outflow tracer to derive the characteristics of the NE and NW outflows. SiO(2-1) has been previously detected toward IRAS 18162-2048 with the IRAM 30 m by Acord (1997). They reported an integrated flux of 1 . 9 ± 0 . 4 K km s -1 , while here we have measured 2 . 5 ± 0 . 05 K km s -1 . Applying a 27 '' beam dilution (the angular size of the IRAM 30 m beam), the CARMA measurement becomes 1.9 K km s -1 . This implies that CARMA is recovering the same flux as the IRAM 30 m for this transition and hence, we can use it to estimate some physical parameters of the outflow. In the first place, we estimate the column density using the following equation (derived from the expressions contained in the appendix of Frau et al. 2010), written in convenient units: where χ ( SiO ) is the abundance of SiO relative to H 2 , Q ( T ex ) is the partition function, E u and T ex are the upper energy level and the excitation temperature, both in K, Ω is the angular size of the outflow in square arcseconds, ν is the rest frequency of the transition in GHz, Sµ 2 is the product of the intrinsic line strength in erg cm 3 D -2 and the squared dipole momentum in D 2 . Following the NH 3 observations of G'omez et al. (2003), we adopted an excitation temperature of 30 K. E u , Q ( T ex ) and Sµ 2 = 19 . 2 erg cm 3 were extracted from the CDMS catalogue (Muller et al. 2005). The Note . - Outflow parameters from top to bottom: SiO column density (N(SiO)), position angle, length ( λ ), maximum velocity (v max ), dynamical time (t dyn ), mass (M), mass injection rate ( ˙ M ), momentum (P), momentum rate ( ˙ P ), energy (E), and mechanical luminosity (L mech ). The uncertainties are just statistical and the inclination with respect to the plane of the sky has not been taken into account. term ∫ S ν dv is the measured integrated line emission in Jy beam -1 km s -1 . We analyzed the two SiO outflows separately (NE and NW outflows), finding average column densities of 1 . 1 ± 0 . 2 × 10 13 cm -2 and 1 . 2 ± 0 . 2 × 10 13 cm -2 respectively. We used these values to derive the mass ( M = N H 2 µ g m H 2 Ω s ) and other properties of the outflow, assuming a mean gas atomic weight µ g = 1 . 36. The results are shown in Table 2. We used an SiO abundance of χ ( SiO ) = 5 × 10 -9 . This value is in good agreement with the χ ( SiO ) found in outflows driven by other intermediate/high-mass protostars. For instance, in a recent paper, S'anchez-Monge et al. (2013) found the χ ( SiO ) in 14 high-mass protostars outflows range between 10 -9 and 10 -8 (similar values were also encountered in Hatchell et al. 2001; Qiu et al. 2007; Codella et al. 2013). It is important to mention that the uncertainty in the SiO abundance is probably one of the main sources of error for our estimates (e.g., Qiu et al. 2009). One order of magnitude in χ ( SiO ) translates into one order of magnitude in most of the outflow properties given in Table 2 (M, ˙ M , P, ˙ P , E and L mech ). Most of the outflow properties in this table were derived following the approach of Palau et al. (2007). In order to derive the mass and momentum rates, together with the mechanical luminosity, we need to know the dynamical timescale of the outflows, which can be determined as t dyn = λ/v max , where λ is the outflow length. No correction for the inclination was included. Hence, we estimate the dynamical time of the NE and NW outflows as 18000 yr and 5000 yr, and their masses as 1.60 and 0.74 M /circledot , respectively (Table 2). In addition, it is worth to note that most of the mass (about 80%) of the NE outflow is concentrated in its brightest condensation (see Fig. 3).",
"pages": [
6,
7,
8
]
},
{
"title": "5. DISCUSSION",
"content": "5.1. Asymmetric outflows and their possible origin Recent work has provided evidence supporting MM1 being a 11-15 M /circledot high-mass protostar, probably still accreting material from a 4 M /circledot rotating disk, and MM2 being a massive dusty core containing at least one highmass protostar, MM2(E), in a less evolved stage than MM1 (Fern'andez-L'opez et al. 2011b). MM2 also contains another core, MM2(W), thought to be in a still earlier evolutionary stage. In addition to these well known protostars, at present it is controversial whether MC harbors another protostar, since neither a thermal radio continuum nor a dust continuum emission has been detected in this molecular core. Hence in the area MM2-MC we can count two or three protostars that could be associated with the molecular outflows reported in this work. IRAS 18162-2048 has protostars surrounded by accretion disks and/or envelopes, and these protostars are associated with molecular outflows and jets, resembling low-mass star-forming systems. The most apparent case would be that of MM1, with one of the largest bipolar radio jets associated with a protostar. Being a scaled up version of the low-mass star formation scenario implies more energetic outflows (as we see in MM1 and MM2), which means larger accretion rates, but it also implies large amounts of momentum impinged on the surrounding environment, affecting occasionally the nearby protostellar neighbors and their outflows. In this case, IRAS 18162-2048 is a region with multiple outflows, mostly seen monopolar. Our observations show no well opposed counterlobes for the NW, NE and SE outflows. That could be explained if the counterlobes are passing through the cavity excavated by the radio jet or through regions of low molecular abundance. Another explanation for the asymmetry of outflows, perhaps more feasible in a high-mass star-forming scenario with a number of protostars and powerful outflows, is deflection after hitting a dense clump of gas and dust (e.g., Raga et al. 2002). Adding a little bit more to the complexity of the region, the SE outflow is comprised of two very different kinematical components. Fig. 4 shows high velocity emission at two well separated velocities laying almost at the same spatial projected path. A blueshifted and redshifted spatial overlap has been usually interpreted as an outflow laying close to the plane of the sky. However, the SE outflow has large radial blueshifted and redshifted velocities ( ∼ ± 50 km s -1 ). One possibility is that the high velocity emission of the SE outflow comes from two molecular outflows, one redshifted and the other one blueshifted, both originated around the MM2 position. A second possibility is that the SE outflow is precessing with an angle α = 15 · (see Fig. 2 in Raga et al. 2009). We derived this angle by taking half the observed wiggling angle of the outflow axis in projection (from P.A. /similarequal 95 · at the origin of the outflow to P.A. /similarequal 126 · away from it). Hence, the absolute velocity of the ejecta should be ∼ 200 km s -1 , which seems to be a reasonable value (see e.g., Bally 2009). A smaller α angle would imply a higher outflow velocity. From the central panel of Fig. 4, we roughly estimate the period of the wiggles of the SE outflow as λ /similarequal 16 '' (27000 AU). We have assumed that the SE outflow completes one precessing period between the position of MM2(E) and the end of the CO(3-2) blueshifted emission in our map. Thus, the precession period is τ p = λ/ ( v j cos α ) /similarequal 660 yr . If the precession is caused by the tidal interaction between the disk of a protostar in MM2(E) and a companion protostar in a non-coplanar orbit (e.g. Terquem et al. 1999b; Montgomery 2009) then it is possible to obtain some information about the binary system. We use an equivalent form of equation (37) from Montgomery (2009) for circular precessing Keplerian disks. This equation relates the angular velocity at the disk edge ( ω d ), the Keplerian orbital angular velocity of the companion around the primary protostar ( ω o ) and the retrograde precession rate of the disk and the outflow ( ω p ): where α is the inclination of the orbit of the companion with respect to the plane of the disk (or obliquity angle), and this angle is the same as the angle of the outflow precession (i.e. the angle between the outflow axis and the line of maximum deviation of the outflow from this axis; Terquem et al. 1999a). Using this expression and adopting reasonable values for the mass of the primary protostar and the radius of its disk we can constrain the orbital period and the radius of the companion protostar. G'omez et al. (1995) assigned a B4 ZAMS spectral type to MM2(E) based on its flux at 3.5 cm. A B4 spectral type protostar has about 6-7 M /circledot (Table 5 in Molinari et al. 1998). On the other side, the dust emission of MM2(E) has a radius no larger than 300 AU (Fern'andez-L'opez et al. 2011a) and we consider 50 AU as a reasonable lower limit for the disk radius. With all of this we derived an orbital period between 200 yr and 800 yr and an orbital radius between 35 M 1 / 3 2 AU and 86 M 1 / 3 2 AU for the putative MM2(E) binary system, being M 2 the mass of the secondary protostar expressed in solar masses. The HV panel of Fig. 4 clearly shows for the first time the precise origin of the SE-blueshifted outflow: MM2(E). Its corresponding redshifted counterlobe appears truncated between MM2(E) and MM2(W). However, the CO(3-2) LV and MV panels show that the redshifted NW outflow reaches the position of MM2 (see also HCN(1-0) emission in Fig. 6). Then, it is possible that the SE outflow counterpart is the NW outflow. This resolves the monopolar nature of both the NW and SE outflows and explains the prominent redshifted wing spectral line profiles of H 2 CO and SO transitions previously observed by Fern'andez-L'opez et al. (2011b) at the position of MC, as being produced by a receding outflow from MM2(E) colliding with a dense cloudlet at the position of MC. The change in the SE-NW outflow direction could be explained by a deflection due north of the NW outflow. The cause of the deflection could be a direct impact against MM2(W), or the action of the powerful HH 80/81/80N wind over the NW lobe. The NE outflow has a monopolar structure at first sight too (Figs. 5, 3 and 6). Its origin cannot be well determined due to angular resolution constraints, but it is also in the MM2-MC area. Figs. 3, 6 and 7 (channels at 1 and +4 km s -1 ) show some signs that a low-velocity counterlobe may exist with a position angle of -112 · , almost opposite to the NE outflow. It would spread out 0.20-0.25 pc from the MM2-MC position.",
"pages": [
8,
9
]
},
{
"title": "5.2. Evolutionary stage of outflows and protostars",
"content": "As stated in several works, SiO is a commonly used molecular tracer of shocked gas in outflows from lowmass protostars (e.g., Hirano et al. 2006; Lee et al. 2008; Santiago-Garc'ıa et al. 2009), but SiO is also found in outflows from high-mass protostars (e.g., Cesaroni et al. 1999; Hatchell et al. 2001; Qiu et al. 2009; Beuther et al. 2004; Zhang et al. 2007a,b; L'opez-Sepulcre et al. 2011; Zapata et al. 2012; Leurini et al. 2013). However, with the present CARMA observations we have not detected SiO(2-1) emission associated with the HH 80/81/80N radio jet, nor with the SE outflow along the ∼ 400 km s -1 of the CARMA SiO(2-1) window bandwidth. The HH 80/81/80N and the SE outflows have not been observed either in the other molecular transitions of this study, HCO + (1-0) and HCN(1-0), also known to be good outflow tracers. If anything, some HCO + and HCN emission may come from gas pushed away by the collimated and high-velocity jets or may be due to lowvelocity winds. Both outflows have been observed in CO lines at the velocities sampled by the CARMA observations, though. Then, what is producing the different chemistry in the outflows of the region? Why is the SiO(2-1) not detected in the SE outflow nor the radio jet, while the other two outflows NE and NW are? Furthermore, why is only one lobe of each the NE and NW outflows detected? There are other cases in the literature where a similar behavior is observed in CO and SiO (e.g., Zhang et al. 2007a,b; Reid & Matthews 2008; Zapata et al. 2012; Codella et al. 2013). It has been proposed that the SiO abundance can decrease with the age of the outflow (Codella et al. 1999; Miettinen et al. 2006; Sakai et al. 2010; L'opez-Sepulcre et al. 2011), which could explain the differences of SiO emission from the outflows of the same region. This hypothesis implies that during the early stages, the gas surrounding the protostar is denser and rich in grains, producing stronger shocks between the outflow and the ambient material, and thus producing an abundant release of SiO molecules. After that, in more evolved stages, the outflow digs a large cavity close to the protostar and thus the shocks are weaker, and grains are rarer. The hypothesis has further support in the shorter SiO depletion timescale (before it freezes out onto the dust grains) with respect to the typical outflow's timescale (some 10 4 yr), together with its disappearance from the gas phase favoring the creation of SiO 2 (Pineau des Forets & Flower 1996; Gibb et al. 2004) and could describe well the HH 80/81/80N jet case, since it produced a cavity probably devoid of dust grains. Now, we can compare the timescale of the outflows in the central region of IRAS 18162-2048. The radio jet HH 80/81/80N is 10 6 yr (Benedettini et al. 2004), the SE outflow has an age of about 2 × 10 3 years (Qiu & Zhang 2009) and the NE and NW outflows have 2 × 10 4 and 5 × 10 3 years. Therefore, except for the SE outflow, the outflow timescales would be in good agreement with SiO decreasing its abundance with time. Actually, as indicated before, the case for the SE outflow is more complex. It has high velocity gas and it is apparently precessing. That can therefore indicate a larger outflow path. In addition CO(3-2) observations are constrained by the SMA primary beam, implying that the outflow could be larger than observed and thereby older. In any case, if the SE outflow is the counterlobe of the NW one (see § 5.1), then a different explanation must be found to account for the chemical differences between these two outflow lobes. We can also compare the characteristics of the NE and NW outflows with those of outflows ejected by (i) highmass protostars and (ii) low-mass Class 0 protostars, in order to put additional constrains on the ejecting sources. Zhang et al. (2005) made an outflow survey toward highmass star-forming regions using CO single-dish observations. After inspecting this work we summarize the properties belonging to outflows from L < 10 3 L /circledot protostars and L ∈ (10 3 , 5 × 10 3 ) L /circledot protostars in Table 2. This Table shows in addition, information on outflows from lowmass Class 0 protostars as well, gathered up from several sources (Arce & Sargent 2004, 2005; Kwon et al. 2006; Davidson et al. 2011). The characteristics of the NE and NWoutflows in IRAS 18162-2048 (as well as those of the SE outflow) are similar to outflows from high-mass protostars, being the NE outflow more energetic and with higher momentum than the NW and SE outflows. On the contrary, the properties of outflows from low-mass Class 0 protostars, although similar in length and dynamical time, have in general lower mass, and overall kinetic energy, about four orders of magnitude lower. All of this indicates that the NE, NW and SE outflows in IRAS 181622048 could be associated with intermediate or high-mass protostars in a very early evolutive stage (massive class 0 protostars). Therefore, given the powerful outflowing activity from MM2, the protostars would be undergoing a powerful accretion process in which the gas from the dusty envelope (about 11 M /circledot ) is probably falling directly onto the protostars. This kind of objects is very rare. Maybe the closest case is that of Cepheus E (Smith et al. 2003). The outflow from this intermediate-mass protostar, which is surrounded by a massive ∼ 25 M /circledot enve- is very young (t dyn ∼ 1 × 10 3 yr), with a mass and an energy (M ∼ 0 . 3 M /circledot , E ∼ 5 × 10 45 ergs) resembling those obtained for the NE, NW and SE outflows.",
"pages": [
9,
10
]
},
{
"title": "6. CONCLUSIONS",
"content": "We have carried out CARMA low-angular resolution observations at 3.5 mm and SMA high-angular resolution observations at 870 µ m toward the massive star-forming region IRAS 18162-2048. We have also included the analysis of SMA low-angular resolution archive data analysis of the CO(3-2) line. The analysis of several molecular lines, all of which are good outflow tracers, resulted in the physical characterization of two previously not well detected outflows (NE and NW outflows) and the clear identification of the driving source of a third outflow (SE outflow). The main results of this work are as follows: The authors want to bring a special reminder of our good fellow Yolanda G'omez, who helped in the very beginning of this work with her contagious optimism. We thank all members of the CARMA and SMA staff that made these observations possible. We thank Pau Frau for helping with the SMA observations. MFL acknowledges financial support from University of Illinois and thanks John Carpenter and Melvin Wright for their patience with CARMA explanations. MFL also thanks the hospitality of the Instituto de Astronom'ıa (UNAM), M'exico D.F., and of the CRyA, Morelia. JMG are supported by the Spanish MICINN AYA2011-30228-C03-02 and the Catalan AGAUR 2009SGR1172 grants. SC acknowledges support from CONACyT grants 60581 and 168251. LAZ acknowledges support from CONACyT. Support for CARMA construction was derived from the Gordon and Betty Moore Foundation, the Kenneth T. and Eileen L. Norris Foundation, the James S. McDonnell Foundation, the Associates of the California Institute of Technology, the University of Chicago, the states of Illinois, California, and Maryland, and the National Science Foundation. Ongoing CARMA development and operations are supported by the National Science Foundation under a cooperative agreement, and by the CARMA partner universities. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. Facilities: CARMA, SMA",
"pages": [
10,
11
]
}
]
2013ApJ...778..121L
https://arxiv.org/pdf/1310.3884.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_85><loc_89><loc_87></location>PREDICTING THE AMOUNT OF HYDROGEN STRIPPED BY THE SUPERNOVA EXPLOSION FOR SN 2002CX-LIKE SNE IA</section_header_level_1>
<text><location><page_1><loc_24><loc_81><loc_76><loc_84></location>Zheng-Wei. Liu 1,2,3,4 , M. Kromer 4 , M. Fink 6 , R. Pakmor 5 , F. K. Ropke 6 , X. F. Chen 1,2 , B. Wang 1,2 and Z. W. Han 1,2</text>
<text><location><page_1><loc_42><loc_80><loc_58><loc_81></location>Draft version May 28, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_77><loc_55><loc_79></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_66><loc_86><loc_77></location>The most favored progenitor scenarios for Type Ia supernovae (SNe Ia) involve the single-degenerate (SD) scenario and the double-degenerate scenario. The absence of stripped hydrogen (H) in the nebular spectra of SNe Ia challenges the SD progenitor models. Recently, it was shown that pure deflagration explosion models of Chandrasekhar-mass white dwarfs ignited off-center reproduce the characteristic observational features of 2002cx-like SNe Ia very well. In this work we predict, for the first time, the amount of stripped H for the off-center pure deflagration explosions. We find that their low kinetic energies lead to inefficient H mass stripping ( /lessorsimilar 0 . 01 M /circledot ), indicating that the stripped H may be hidden in (observed) late-time spectra of SN 2002cx-like SNe Ia.</text>
<text><location><page_1><loc_14><loc_65><loc_71><loc_66></location>Subject headings: supernovae: general - binaries: close - methods: numerical</text>
<section_header_level_1><location><page_1><loc_21><loc_61><loc_36><loc_62></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_37><loc_48><loc_61></location>Type Ia supernovae (SNe Ia) are instrumental as distance indicators on a cosmic scale to determine the expansion history of the Universe (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999). They are widely believed to be caused by thermonuclear explosions of carbon/oxygen white dwarfs (C/O WDs) in binary systems. The two favored classes of SN Ia progenitors are the single-degenerate (SD) scenario and double-degenerate (DD) scenario. In the DDS, two C/O WDs merge due to gravitational wave radiation, leading to a SN Ia thermonuclear explosion (DD scenario, e.g., Iben & Tutukov 1984). In the SD scenario, WDs accrete H/He-rich matters from companions that could be main-sequence (MS) stars, sub-giants, red giants (RGs) or He stars. They ignite SN Ia explosions when approaching the Chandrasekhar-mass ( M Ch ) limit (e.g., Whelan & Iben 1973; Hachisu et al. 1996; Han & Podsiadlowski 2004).</text>
<text><location><page_1><loc_8><loc_22><loc_48><loc_37></location>Recently, some observational and hydrodynamical studies (see, e.g., Li et al. 2011; Nugent et al. 2011; Chomiuk et al. 2012; Horesh et al. 2012; Bloom et al. 2012; Schaefer & Pagnotta 2012; Pakmor et al. 2010, 2011, 2012) support the viability of DD scenario. There are some observational indications (see, e.g., Patat et al. 2007; Sternberg et al. 2011; Foley et al. 2012; Dilday et al. 2012), suggesting that the progenitors of some SNe Ia may come from the SD scenario. However, the exact nature of SN Ia progenitors remains uncertain (see Hillebrandt & Niemeyer 2000;</text>
<section_header_level_1><location><page_1><loc_10><loc_20><loc_25><loc_21></location>Email: zwliu@ynao.ac.cn</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_10><loc_18><loc_48><loc_20></location>1 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China</list_item>
<list_item><location><page_1><loc_10><loc_15><loc_48><loc_18></location>2 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China 3 University of Chinese Academy of Sciences, Beijing 100049,</list_item>
<list_item><location><page_1><loc_10><loc_14><loc_13><loc_15></location>China</list_item>
<list_item><location><page_1><loc_10><loc_11><loc_48><loc_14></location>4 Max-Planck-Institut fur Astrophysik, Karl-SchwarzschildStr. 1, 85741 Garching, Germany</list_item>
<list_item><location><page_1><loc_10><loc_9><loc_48><loc_12></location>5 Heidelberger Institut fur Theoretische Studien, SchlossWolfsbrunnenweg 35, 69118 Heidelberg, Germany</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_9></location>6 Institut fur Theoretische Physik und Astrophysik, Universitat Wurzburg, Am Hubland, 97074 Wurzburg, Germany</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_61><loc_77><loc_62></location>Hillebrandt et al. 2013 for reviews).</text>
<text><location><page_1><loc_52><loc_19><loc_92><loc_61></location>The spectra of normal SNe Ia are characterized by the absence of H and He lines and a strong silicon absorption feature. To date, no direct observation shows the signature of H lines in late-time, nebular spectra of SNe Ia (Mattila et al. 2005; Leonard 2007; Shappee et al. 2013). One of the signatures of the SD scenario is that the SN Ia explosion is expected to remove H/Herich material from its non-degenerate companion star (Wheeler et al. 1975). Hydrodynamical simulations with a classical SN Ia explosion model (i.e., the W7 model, see Nomoto et al. 1984) showed that about 0 . 1 M /circledot Hrich material are expected to be stripped off from a MS companion star by the impact of the SN Ia ejecta (see, e.g., Marietta et al. 2000; Pakmor et al. 2008; Liu et al. 2012, 2013; Pan et al. 2012). Almost the whole envelope of a RG companion ( ∼ 0 . 5 M /circledot ) is removed (see, e.g., Marietta et al. 2000; Pan et al. 2012). The amount of stripped H is significantly above the most stringent upper limits on non-detection of H ( ∼ 0.01-0.03 M /circledot , see Leonard 2007; Lundqvist et al. 2013) which were derived from observations of normal SNe Ia. Moreover, Shappee et al. (2013) obtained a lower limit of ∼ 0 . 001 M /circledot on detection of stripped H for SN 2011fe which is the nearest SN Ia in the last 25 years and has been observed in unprecedented detail. Therefore, the absence of H α in late time nebular spectra of SNe Ia poses some problems for the SD scenario and favors other progenitor channel such as WD merger (see Pakmor et al. 2010, 2011, 2012). However, all previous hydrodynamical simulations were performed with the classical W7 explosion model which is suitable for normal SNe Ia in nickel production and kinetic energy.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_19></location>SN 2002cx-like SNe are spectroscopically peculiar and faint objects compared to other SNe Ia. Their spectra are characterized by very low expansion velocities and show strong mixing of the explosion ejecta (Jha et al. 2006; Phillips et al. 2007). Moreover, SN 2002cx-like SNe are proposed to originate from Chandrasekhar mass deflagrations, i.e., SD H-accreting progenitors. Very recently, Kromer et al. (2013) performed hydrodynamics</text>
<figure>
<location><page_2><loc_9><loc_61><loc_48><loc_92></location>
<caption>Fig. 1.Mass vs. radius profiles for four main-sequence companion models used in the impact simulations.</caption>
</figure>
<text><location><page_2><loc_8><loc_33><loc_48><loc_55></location>(see also Jordan et al. 2012) and radiative-transfer calculations for a three-dimensional (3D) full-star pure deflagration model (i.e., the N5def model, see Fink et al. 2013) which is able to reproduce the characteristic observational features of SN 2005hk (a prototypical 2002cxlike SN Ia). In the N5def model, only a part of the M Ch WD, ∼ 0 . 37 M /circledot , is ejected with a much lower kinetic energy ( ∼ 1 . 34 × 10 50 erg) than models for normal SNe Ia. The thermonuclear explosion fails to completely unbind the WD and leaves behind a bound remnant of ∼ 1 . 03 M /circledot which consists mainly of unburned C/O (see Jordan et al. 2012; Kromer et al. 2013; Fink et al. 2013). The small amount of kinetic energy released in this pure deflagration model might significantly decrease the stripped companion mass, potentially avoiding a signature of H lines in late-time spectra of SN 2002cx-like SNe Ia.</text>
<text><location><page_2><loc_8><loc_17><loc_48><loc_32></location>Here we calculate the amount of stripped H for the N5def model (which is the best current model for SN 2002cx-like SNe Ia) in the SD scenario using 3D hydrodynamical simulations of the impact of the SN ejecta on MS companion stars. The paper is organized as follows. In Section 2, we describe the methods and codes used in this work. Section 3 presents the results from hydrodynamical simulations. The distribution of the unbound mass from population synthesis calculations is shown in Section 4. Some discusions based on results of impact simulations are presented in Section 5. Finally, we summarize the basic results of simulations in Section 6.</text>
<section_header_level_1><location><page_2><loc_13><loc_14><loc_44><loc_15></location>2. NUMERICAL METHOD AND MODEL</section_header_level_1>
<text><location><page_2><loc_8><loc_7><loc_48><loc_13></location>In order to construct a detailed companion structure at the moment of the SN Ia explosion, we used the same method as described in Liu et al. (2012) to trace binary evolution in which a WD accretes H-rich material from a MS companion star (i.e., WD+MS M Ch explosion sce-</text>
<figure>
<location><page_2><loc_52><loc_69><loc_92><loc_92></location>
<caption>Fig. 2.Radial profiles of the density for a 1D companion model (i.e., Model A) and its corresponding 3D SPH model generated by the healpix method (see Pakmor et al. 2012).</caption>
</figure>
<text><location><page_2><loc_52><loc_36><loc_92><loc_61></location>nario). We think the WD would explode as a SN Ia when its mass increases to the M Ch limit. Here, we adopted the Eggleton's stellar evolution code, Roche-lobe overflow and the optically thick wind model of Hachisu et al. (1996) were included into the code to treat mass transfer in the binary. With a series of consistent binary evolution calculations, we selected four companion star models as input models of hydrodynamical simulations. These four companion stars were constructed with different initial WD masses, companion masses and orbital periods (see Table 1), which leads to companion models different in mass, orbital period and detailed structure at the moment of the SN explosion. Four companion models created in 1D binary evolution calculations are summarized in Table 1, their radial mass profiles are shown in Figure 1. We then performed 3D hydrodynamical simulations of the impact of SN Ia ejecta on the companion star employing the SPH code Stellar GADGET (Pakmor et al. 2012; Springel 2005).</text>
<text><location><page_2><loc_52><loc_20><loc_92><loc_36></location>In this work, all initial conditions and basic setup for the impact simulations are the same as those in Liu et al. (2012). We use the healpix method described in Pakmor et al. (2012) to map the 1D profiles of density and internal energy of a 1D companion star model to a particle distribution suitable for the SPH code. Before we start the actual impact simulations, the SPH model of each companion star is relaxed for several dynamical timescales to reduce numerical noise introduced by the mapping. A comparison of density profiles between the 1D stellar model and its consistent SPH model for Model A are shown in Figure 2.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_20></location>The SN Ia explosion was represented by the pure deflagration model of Kromer et al. (2013) (i.e., the N5def model). This model has been shown to reproduce the characteristic observational features of 2002cx-like SNe Ia well (Kromer et al. 2013; Jordan et al. 2012). In this simulation, only the 0 . 37 M /circledot of ejected material with a total kinetic energy of 1 . 34 × 10 50 erg were used to represent the SN Ia explosion, we did not include the 1 . 03 M /circledot bound remnant of the M Ch WD into the simulations. Based on the angle averaged 1D ejecta structure of the</text>
<table>
<location><page_3><loc_23><loc_79><loc_76><loc_89></location>
<caption>TABLE 1 Initial models and results from hydrodynamical simulations.</caption>
</table>
<text><location><page_3><loc_8><loc_74><loc_92><loc_78></location>Note . -M WD and M 2 , i present the WD and companion mass at the beginning of mass transfer. M 2 , f , R 2 , f and a f demonstrate the companion mass, companion radius and binary separation at the time of the explosion. ∆ M Def and ∆ M W7 show the unbound companion masses in the impact simulations for the pure deflagration model and the W7 model. v W7 kick and v Def kick correspond to the companion kick velocities.</text>
<text><location><page_3><loc_8><loc_54><loc_48><loc_71></location>N5def model, SPH particles were placed randomly in shells to reproduce the mass (density) profile and gain the radial velocities they should have at their positions. The composition of a particle was then set to the values of the initial 1D model at a radius equal to the radial coordinate of the particle. Here, the effect a mild degree of asymmetry caused by off-center pure deflagration explosion of a M Ch WD was ignored. However, only in the direction opposites to the one-sided ignition region the velocities are somewhat lower (Fink et al. 2013). The orientation of an asymmetry of SN ejecta plays an inefficient role in mass stripping by the time of interaction with the companion. 1</text>
<text><location><page_3><loc_8><loc_38><loc_48><loc_54></location>We used 6 × 10 6 million SPH particles to represent the He companion stars in all simulations of this work. 2 Because all SPH particle was set up with the same mass, the number of particles representing the supernova explosion is then fixed. The supernova was placed at a distance to the companion star given by the separation at the moment of SN Ia explosion in our 1D binary-evolution calculations. The impact of the SN Ia ejecta on their binary companions was then simulated for 5000 s, at which point the mass stripped off from the companion star and its kick velocity due to the impact have reached constant values.</text>
<section_header_level_1><location><page_3><loc_15><loc_35><loc_42><loc_37></location>3. HYDRODYNAMICAL RESULTS</section_header_level_1>
<text><location><page_3><loc_8><loc_16><loc_48><loc_35></location>Figure 3 shows the typical evolution of the density distribution in the impact simulations for the companion star Model A. Compared to our previous hydrodynamical simulations for the same MS companion star model (but with a different explosion model, see Liu et al. 2012), the basic impact processes are quite similar. The SN explodes at the right side of the companion star. After the SN explosion, the SN ejecta expand freely for a while and hit the MS companion star, removing solar-metallicity companion material and forming a bow shock. Subsequently, the bow shock propagates through the companion star, causing an additional loss of H-rich companion material from the far side of the star. Finally, the final unbound H-rich material of the companion star</text>
<text><location><page_3><loc_52><loc_66><loc_92><loc_71></location>caused by the SN impact is largely embedded in lowvelocity SN debris behind the companion star, and the strongly impacted companion starts to relax to become almost spherical again.</text>
<text><location><page_3><loc_52><loc_49><loc_92><loc_66></location>After the SN explosion, the non-degenerate companion star is significantly hit by the SN ejecta. The SN impact remove H-rich material from outer layers of the companion star through the ablation (SN heating) and the stripping (momentum transfer) mechanism. The late-time spectra of SN Ia probably show a signature of H lines if a large amount of H-rich material can be stripped from the companion star during the interaction with the SN ejecta. To calculate the amount of final unbound companion mass due to the SN impact, we sum the masses of all unbound SPH particles (ablated+stripped particles) that originally belongs to the companion star at the end of the simulations.</text>
<text><location><page_3><loc_52><loc_39><loc_92><loc_49></location>For four different MS companion star models, our impact simulations show that the companion star received a kick velocity of ∼ 15 (Model D)-25 km s -1 (Model A) at the end of the simulations. Moreover, only a small amount of 0 . 013 (Model C)-0 . 016 M /circledot (Model D) of Hrich material is removed (ablation+stripping) from the companion stars due to the SN impact (see Table 1).</text>
<text><location><page_3><loc_52><loc_22><loc_92><loc_39></location>For a comparison, numerical results for a classical explosion model of a M Ch WD for normal SNe Ia, the W7 model, are also shown in Table 1. Because the N5def model does not burn the complete WD but leaves behind a ∼ 1 . 0 M /circledot bound remnant, it produces a much lower kinetic energy (1 . 34 × 10 50 erg) than the W7 model (1 . 23 × 10 51 erg). Therefore, a much smaller amount of H-rich material of only ∼ 0 . 015 M /circledot is removed from the companion stars by the impact of the pure deflagration explosion of a M Ch WD. In contrast, the amount of final unbound companion mass is more than ten times larger ( > 0 . 1 M /circledot ) for the W7 explosion model with the same companion star models at the same separations.</text>
<text><location><page_3><loc_52><loc_7><loc_92><loc_22></location>A large amount of unbound companion mass in the impact simulations for the W7 explosion model seems to indicate that normal SNe Ia are not likely produced from the WD+MS M Ch explosion scenario (see also Liu et al. 2012). Here, it is shown that the mass stripping is inefficient for 2002cx-like SNe Ia due to the low kinetic energies of off-center pure deflagration explosions of a M Ch WD, leading to that a amount of unbound companion material caused by the SN impact is quite small. Therefore, the H lines probably be hidden in late-time spectra of SN 2002cx-like SNe Ia. However, whether or</text>
<figure>
<location><page_4><loc_11><loc_60><loc_90><loc_92></location>
<caption>Fig. 3.Density distributions of all gas material as a function of the explosion time in the impact simulations for Model A. The color scale shows the logarithm of the density in g cm -3 . The plots are made with the freely available SPLASH tool (Price 2007).</caption>
</figure>
<table>
<location><page_4><loc_15><loc_41><loc_42><loc_51></location>
<caption>TABLE 2 Fitting parameters for equation (1) and (2)</caption>
</table>
<text><location><page_4><loc_8><loc_31><loc_48><loc_38></location>not such small amount of stripped H mass would expect to show a signature of H lines in late-time spectra of SN 2002cx-like SNe Ia, which needs to analyze late-time spectra of 2002cx-like SNe to obtain the lower limit of stripped mass for detecting the H lines. 3</text>
<section_header_level_1><location><page_4><loc_13><loc_29><loc_44><loc_30></location>4. POPULATION SYNTHESIS RESULTS</section_header_level_1>
<section_header_level_1><location><page_4><loc_21><loc_27><loc_36><loc_28></location>4.1. Power-law fitting</section_header_level_1>
<text><location><page_4><loc_8><loc_12><loc_48><loc_26></location>Different WD+MS binary systems evolve to different evolutionary stages and have different binary parameters when the WD explodes as an SN Ia. Consequently, the companion radius and binary separation of binary systems differ significantly from the MS companion models used in our present simulations. We therefore investigated the dependence of the numerical results on the ratio of binary separation to the companion radius ( a f /R 2 , f ) at the time of the explosion. Here, we used the same method as in Liu et al. (2012) to artificially adjust the binary separations for a fixed companion star model</text>
<text><location><page_4><loc_52><loc_52><loc_92><loc_54></location>(which means that all parameters but the orbital separation are kept constant).</text>
<text><location><page_4><loc_52><loc_42><loc_92><loc_52></location>Figure 4 presents the amount of final unbound companion mass and kick velocity as a function of the parameter of a f /R 2 , f for four companion star models. It shows that the unbound mass and kick velocity significantly decrease when increasing the orbital separation of the binary. Generally, these relations can be fitted with power law functions in good approximation (see Figure 4):</text>
<formula><location><page_4><loc_60><loc_38><loc_92><loc_41></location>M unbound = C 1 ( a f R 2 , f ) -α M /circledot , (1)</formula>
<formula><location><page_4><loc_61><loc_33><loc_92><loc_37></location>v kick = C 2 ( a f R 2 , f ) -β kms -1 , (2)</formula>
<text><location><page_4><loc_52><loc_27><loc_92><loc_33></location>where a f is the binary separation, R 2 , f is the radius of the MS companion star at the onset of the SN explosion. C 1 and C 2 are two constant, α and β are the power-law indices (see Table 2).</text>
<section_header_level_1><location><page_4><loc_58><loc_25><loc_87><loc_26></location>4.2. Unbound masses and kick velocities</section_header_level_1>
<text><location><page_4><loc_52><loc_7><loc_92><loc_24></location>Wang et al. (2010)(hereafter WLH10) performed comprehensive binary population synthesis (BPS) calculations obtaining a large sample of WD+MS SN Ia progenitor models. They predicted many properties of the companion stars and binary systems at the moment of the SN explosion (e.g. the companion masses, the companion radii, the orbital periods, etc). The distributions of the ratio of binary separations to the companion radii ( a f /R 2 , f ) in the WLH10 sample (their model with α CE × λ = 0 . 5 , 1 . 5) are presented in Figure 5. It shows that most systems are concentrated at a f /R 2 , f ∼ 2 . 9 and 3 . 4. Based on the distribution of a f /R 2 , f , we calculated the final unbound companion masses (and kick velocities)</text>
<figure>
<location><page_5><loc_10><loc_67><loc_47><loc_92></location>
<caption>Figure 4 shows that the fitting parameters of four different companion models are different, which indicates that the companion structure results from the characteristics of the original binary system and the details of the mass transfer also can affect the final unbound mass. In reality, the companion structure is not independent on the binary separation in the binary evolutions. Therefore, the companion structure would be different with different a f /R 2 , f from BPS calculations.</caption>
</figure>
<figure>
<location><page_5><loc_10><loc_41><loc_46><loc_66></location>
<caption>Fig. 4.Dependence of unbound companion mass ( panel a ) or kick velocity ( panel b ) on the binary separation in the impact simulations of four different companion models. The corresponding fitting parameters are shown in Table 2.</caption>
</figure>
<text><location><page_5><loc_8><loc_21><loc_48><loc_33></location>due to the SN impact by adopting the power law relations of four companion star models in Figure 4, which are displayed in Figure 6. As it is shown, the impact of off-center pure deflagration explosions of M Ch WDs lead to a small amount of mass loss of the companion star ( /lessorsimilar 0 . 015 M /circledot ) in almost all WD+MS binary systems from BPS calculations. Moreover, the SN impact delivers a small kick velocity of /lessorsimilar 25 kms -1 to the companion star.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_21></location>In Section 2, initial MS companion star models were set up with a H abundance of X = 0 . 7, a He abundance of Y = 0 . 28 and a metallicity of Z = 0 . 02 when we constructed the MS companion star model at the moment of the SN explosion. Therefore, pure deflagrations of the M Ch WDs strip off a small amount of pure H of /lessorsimilar 0 . 01 M /circledot from the companion stars in almost all WD+MS models. These small stripped H masses are consistent with the lower mass limit for detecting H α emission lines in nebular spectra of normal SNe Ia (0 . 01 M /circledot , see Leonard 2007). The inefficient mass strip-</text>
<figure>
<location><page_5><loc_53><loc_64><loc_93><loc_92></location>
<caption>Fig. 5.Distribution of the ratio of binary separation to companion radius ( a f /R 2 , f ) in the population synthesis calculations for WD+MS progenitor models. The solid and dash-dotted line show results of the models with α CE λ = 0 . 5 and α CE λ = 1 . 5 in WLH10.</caption>
</figure>
<text><location><page_5><loc_52><loc_38><loc_92><loc_56></location>ing seems to imply the stripped H may be hidden in (observed) late-time spectra of most of SN 2002cx-like SNe Ia. However, only normal SNe Ia were looked at when Leonard (2007) obtained the upper limit for nondetection of stripped H α of 0 . 01 M /circledot , the observational limits for SN 2002cx-like SNe may be smaller/larger than the value for normal SNe Ia. Most of stripped H-rich material in this simulations ends up at velocities below 10 3 kms -1 so that it is confined to the innermost part of explosion ejecta. Whether or not H α emission will be detectable, can only be answered by performing sophisticated radiative transfer simulations on the abundance structure of our explosion models.</text>
<section_header_level_1><location><page_5><loc_66><loc_36><loc_78><loc_37></location>5. DISCUSSION</section_header_level_1>
<section_header_level_1><location><page_5><loc_55><loc_34><loc_89><loc_35></location>5.1. The influence of the companion structures</section_header_level_1>
<text><location><page_5><loc_52><loc_25><loc_92><loc_33></location>To test influence of the ratio of binary separation to the companion radius of a f /R 2 , f , we artificially adjusted the binary separation for a fixed companion star model. The final unbound mass due to the SN impact decreases by a factor of 10 as the parameter of a f /R 2 , f increases by a factor of 2.</text>
<text><location><page_5><loc_52><loc_7><loc_92><loc_13></location>In Section 4.2, we only used the same power-law relation obtained from a fixed companion star model (for example, the power-law relation in Model A) to calculate the total mass loss of the companion star caused by the SN impact for all WD+MS models, which ignores the</text>
<text><location><page_6><loc_8><loc_77><loc_48><loc_92></location>influence of details of the companion structure. Figure 6 show that a comparison of the results that calculated by using four different power-law relations between the final unbound mass (and the kick velocities) and the parameter of a f /R 2 , f in Model A, Model B, Model C, and Model D. Some differences (but not big differences) in the distribution of final unbound masses are seen in Figure 6a,c, which implies that the parameter of a f /R 2 , f is not the only factor to determine the final stripped companion mass, the companion structure can also affect the results.</text>
<section_header_level_1><location><page_6><loc_21><loc_75><loc_36><loc_76></location>5.2. Explosion energy</section_header_level_1>
<text><location><page_6><loc_8><loc_50><loc_48><loc_74></location>For a comparison, we performed the impact simulations for the same companion star by adopting both the N5def model and the W7 explosion model to represent the SN Ia explosion. A factor of 10 lower explosion energy in the N5def model leads to the stripped material reduced by a factor of 10 (see Table 1). Moreover, Pakmor et al. (2008) investigated the influence of the SN explosion energy on the interaction with the companion star. They also found that explosion energy range covers a factor of 2 therefore leads to the unbound companion mass varies by a factor of 2. These results indicate that the SN explosion energy have only a small effect on the total mass loss of the companion star as compared to the effect of the parameter of a f /R 2 , f discussed above. Therefore, the ratio of the binary separation to the radius of companion star ( a f /R 2 , f ) is the most important parameter to determine the final unbound companion mass (see also Liu et al. 2012, 2013).</text>
<section_header_level_1><location><page_6><loc_15><loc_48><loc_42><loc_49></location>5.3. The class of SN 2002cx-like SNe</section_header_level_1>
<text><location><page_6><loc_8><loc_19><loc_48><loc_47></location>The SN 2002cx was discovered as a new class of peculiar SNe Ia by Li et al. (2003). From a volumelimited sample of the Lick Observatory Supernova Search (LOSS), Li et al. (2003) estimate that SN 2002cs-like SNe Ia contribute at about 5 per cent to the total SN Ia rate. Very recently, Foley et al. (2013) concluded that 'SNe Iax' (the prototype of which is SN 2002cx SNe) are the most common peculiar class of SNe, they estimated that in a given volume SNe Iax could contribute ∼ 1 / 3 of total SNe Ia. Nonetheless, to date, only 25 SNe Iax are confirmed that they are observationally similar to its prototypical member, SN 2002cx (see Foley et al. 2013). This sample consists of 25 members is a very small fraction of total SNe Ia. In this work, the results obtained from the impact simulations only apply to the subclass of peculiar 2002cx-like SNe but not the bulk of SNe Ia. Therefore, even the off-center pure deflagration explosion of a M Ch WD removes H-rich material during the interaction with the MS companion star, the H stripped from the companion star may not be observed in the ejecta of such relatively rare events.</text>
<section_header_level_1><location><page_6><loc_15><loc_17><loc_42><loc_18></location>5.4. Post-explosion fate of the binary</section_header_level_1>
<text><location><page_6><loc_8><loc_7><loc_48><loc_16></location>In this work, the N5def pure deflagration model was used to represent the SN Ia explosion in our impact simulations. The hydrodynamics calculations of Jordan et al. (2012) and Kromer et al. (2013) showed that the N5def model does not burn the complete WD but leaves behind a ∼ 1 . 0 M /circledot bound remnant. Unfortunately, this bound remnant cannot be properly spatially resolved until late</text>
<text><location><page_6><loc_52><loc_89><loc_92><loc_92></location>time in hydrodynamical simulations due to the strong expansion of the SN ejecta.</text>
<text><location><page_6><loc_52><loc_60><loc_92><loc_89></location>Our simulations show that the companion star receives a small kick velocity ( < 30 kms -1 ) during the interaction with the SN ejecta. Therefore, whether the WD+MS binary system would be destoried after the SN explosion, which depends primarily on the kick velocity of the bound remnant. If the bound remnant receives a large kick velocity that can overcome its gravitational force, the existence of abundance-enriched MS-like stars and WDs with peculiar spatial velocities are indicators (Jordan et al. 2012) of this studied progenitor scenario. Otherwise, the new binary system would survive the SN explosion. 4 In this study, it is found that the post-impact radii of companions at 5000 s after the explosion are larger than the initial binary separations used for the impact simulations due to their extreme expansions caused by the SN heating. 5 This indicates that the surviving binary system may evolve and merge into a single object with a rapid rotation velocity, or experience a common envelope phase. The details of this post-explosion evolution should be addressed in future work. However, fully resolving the detailed structure of the bound remnants is a prerequisite for this investigation.</text>
<section_header_level_1><location><page_6><loc_58><loc_57><loc_86><loc_58></location>6. SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_6><loc_52><loc_46><loc_92><loc_57></location>We presented 3D hydrodynamical simulations of the impact of SN Ia explosions on their companion stars for the WD+MS scenario for the pure deflagration model presented in Kromer et al. (2013). For four different companion star models, we find that the much lower kinetic energy of the pure deflagration model compared to models for normal SNe Ia leads to a much lower stripped H-rich mass of only 0 . 013 M /circledot -0 . 016 M /circledot .</text>
<text><location><page_6><loc_52><loc_29><loc_92><loc_46></location>Moreover, using the distribution of a f /R 2 , f from BPS calculations, we discussed the distribution of the amount unbound H-rich material of the companion star using a power law relation between the total unbound mass and the ratio of binary separation to the companion radius ( a f /R 2 , f ). We find that the off-center pure deflagration explosions strip off a small amount of H ( /lessorsimilar 0 . 01 M /circledot ) from MS companion stars in most WD+MS progenitor models of SN 2002cx-like SNe Ia. The inefficient mass stripping may lead to the stripped H is fully hidden in their late-time spectra. Therefore, it will be very interesting to analyze late-time spectra of 2002cx-like SNe Ia for the presence of hydrogen emission.</text>
<section_header_level_1><location><page_6><loc_63><loc_26><loc_81><loc_27></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_6><loc_52><loc_16><loc_92><loc_26></location>Z.W.L acknowledges the financial support from the MPG-CAS Joint Doctoral Promotion Program (DPP) and from Max Planck Institute for Astrophysics (MPA). This work is supported by the National Basic Research Program of China (Grant No. 2009CB824800), the National Natural Science Foundation of China (Grant Nos. 11033008 and 11103072) and the Chinese Academy of</text>
<unordered_list>
<list_item><location><page_6><loc_52><loc_9><loc_92><loc_15></location>4 After the SN explosion, it was found that the bound remnant receives small kick velocities of ∼ 36 km s -1 (see Kromer et al. 2013) or large kick velocities up to 520 km s -1 (see Jordan et al. 2012). The difference of kick velocity of bound remnant may originate from the different gravity solvers used.</list_item>
<list_item><location><page_6><loc_52><loc_7><loc_92><loc_9></location>5 However, the effect of the 1 . 03 M /circledot bound remnant of the M ch WD was not considered in our impact simulations.</list_item>
</unordered_list>
<text><location><page_7><loc_8><loc_87><loc_48><loc_92></location>Sciences (Grant N0. KJCX2-YW-T24). The work of F.K.R was supported by Deutsche Forschungsgemeinschaft via the Emmy Noether Program (RO 3676/1-1) and by the ARCHES prize of the German Federal Min-</text>
<text><location><page_7><loc_52><loc_88><loc_92><loc_92></location>istry of Education and Research (BMBF). The simulations were carried out at the Computing Center of the Max Plank Society, Garching, Germany.</text>
<section_header_level_1><location><page_7><loc_45><loc_83><loc_55><loc_84></location>REFERENCES</section_header_level_1>
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<figure>
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<caption>Fig. 6.Distributions of the final unbound companion mass (first cloumn) and the kick velocity (second cloumn) due to the SN impact in the simulations. Different color show the results that are calculated by using the relation obtained from the power-law fitting (see Figure 4) for Model A (bule lines), Modle B (red lines), Model C (green lines), and Model D (yellow lines). The solid (top row) and dash-dotted lines (bottom row) show results of the models with α CE λ = 0 . 5 and α CE λ = 1 . 5 in WLH10.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "The most favored progenitor scenarios for Type Ia supernovae (SNe Ia) involve the single-degenerate (SD) scenario and the double-degenerate scenario. The absence of stripped hydrogen (H) in the nebular spectra of SNe Ia challenges the SD progenitor models. Recently, it was shown that pure deflagration explosion models of Chandrasekhar-mass white dwarfs ignited off-center reproduce the characteristic observational features of 2002cx-like SNe Ia very well. In this work we predict, for the first time, the amount of stripped H for the off-center pure deflagration explosions. We find that their low kinetic energies lead to inefficient H mass stripping ( /lessorsimilar 0 . 01 M /circledot ), indicating that the stripped H may be hidden in (observed) late-time spectra of SN 2002cx-like SNe Ia. Subject headings: supernovae: general - binaries: close - methods: numerical",
"pages": [
1
]
},
{
"title": "PREDICTING THE AMOUNT OF HYDROGEN STRIPPED BY THE SUPERNOVA EXPLOSION FOR SN 2002CX-LIKE SNE IA",
"content": "Zheng-Wei. Liu 1,2,3,4 , M. Kromer 4 , M. Fink 6 , R. Pakmor 5 , F. K. Ropke 6 , X. F. Chen 1,2 , B. Wang 1,2 and Z. W. Han 1,2 Draft version May 28, 2018",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Type Ia supernovae (SNe Ia) are instrumental as distance indicators on a cosmic scale to determine the expansion history of the Universe (Riess et al. 1998; Schmidt et al. 1998; Perlmutter et al. 1999). They are widely believed to be caused by thermonuclear explosions of carbon/oxygen white dwarfs (C/O WDs) in binary systems. The two favored classes of SN Ia progenitors are the single-degenerate (SD) scenario and double-degenerate (DD) scenario. In the DDS, two C/O WDs merge due to gravitational wave radiation, leading to a SN Ia thermonuclear explosion (DD scenario, e.g., Iben & Tutukov 1984). In the SD scenario, WDs accrete H/He-rich matters from companions that could be main-sequence (MS) stars, sub-giants, red giants (RGs) or He stars. They ignite SN Ia explosions when approaching the Chandrasekhar-mass ( M Ch ) limit (e.g., Whelan & Iben 1973; Hachisu et al. 1996; Han & Podsiadlowski 2004). Recently, some observational and hydrodynamical studies (see, e.g., Li et al. 2011; Nugent et al. 2011; Chomiuk et al. 2012; Horesh et al. 2012; Bloom et al. 2012; Schaefer & Pagnotta 2012; Pakmor et al. 2010, 2011, 2012) support the viability of DD scenario. There are some observational indications (see, e.g., Patat et al. 2007; Sternberg et al. 2011; Foley et al. 2012; Dilday et al. 2012), suggesting that the progenitors of some SNe Ia may come from the SD scenario. However, the exact nature of SN Ia progenitors remains uncertain (see Hillebrandt & Niemeyer 2000;",
"pages": [
1
]
},
{
"title": "Email: zwliu@ynao.ac.cn",
"content": "Hillebrandt et al. 2013 for reviews). The spectra of normal SNe Ia are characterized by the absence of H and He lines and a strong silicon absorption feature. To date, no direct observation shows the signature of H lines in late-time, nebular spectra of SNe Ia (Mattila et al. 2005; Leonard 2007; Shappee et al. 2013). One of the signatures of the SD scenario is that the SN Ia explosion is expected to remove H/Herich material from its non-degenerate companion star (Wheeler et al. 1975). Hydrodynamical simulations with a classical SN Ia explosion model (i.e., the W7 model, see Nomoto et al. 1984) showed that about 0 . 1 M /circledot Hrich material are expected to be stripped off from a MS companion star by the impact of the SN Ia ejecta (see, e.g., Marietta et al. 2000; Pakmor et al. 2008; Liu et al. 2012, 2013; Pan et al. 2012). Almost the whole envelope of a RG companion ( ∼ 0 . 5 M /circledot ) is removed (see, e.g., Marietta et al. 2000; Pan et al. 2012). The amount of stripped H is significantly above the most stringent upper limits on non-detection of H ( ∼ 0.01-0.03 M /circledot , see Leonard 2007; Lundqvist et al. 2013) which were derived from observations of normal SNe Ia. Moreover, Shappee et al. (2013) obtained a lower limit of ∼ 0 . 001 M /circledot on detection of stripped H for SN 2011fe which is the nearest SN Ia in the last 25 years and has been observed in unprecedented detail. Therefore, the absence of H α in late time nebular spectra of SNe Ia poses some problems for the SD scenario and favors other progenitor channel such as WD merger (see Pakmor et al. 2010, 2011, 2012). However, all previous hydrodynamical simulations were performed with the classical W7 explosion model which is suitable for normal SNe Ia in nickel production and kinetic energy. SN 2002cx-like SNe are spectroscopically peculiar and faint objects compared to other SNe Ia. Their spectra are characterized by very low expansion velocities and show strong mixing of the explosion ejecta (Jha et al. 2006; Phillips et al. 2007). Moreover, SN 2002cx-like SNe are proposed to originate from Chandrasekhar mass deflagrations, i.e., SD H-accreting progenitors. Very recently, Kromer et al. (2013) performed hydrodynamics (see also Jordan et al. 2012) and radiative-transfer calculations for a three-dimensional (3D) full-star pure deflagration model (i.e., the N5def model, see Fink et al. 2013) which is able to reproduce the characteristic observational features of SN 2005hk (a prototypical 2002cxlike SN Ia). In the N5def model, only a part of the M Ch WD, ∼ 0 . 37 M /circledot , is ejected with a much lower kinetic energy ( ∼ 1 . 34 × 10 50 erg) than models for normal SNe Ia. The thermonuclear explosion fails to completely unbind the WD and leaves behind a bound remnant of ∼ 1 . 03 M /circledot which consists mainly of unburned C/O (see Jordan et al. 2012; Kromer et al. 2013; Fink et al. 2013). The small amount of kinetic energy released in this pure deflagration model might significantly decrease the stripped companion mass, potentially avoiding a signature of H lines in late-time spectra of SN 2002cx-like SNe Ia. Here we calculate the amount of stripped H for the N5def model (which is the best current model for SN 2002cx-like SNe Ia) in the SD scenario using 3D hydrodynamical simulations of the impact of the SN ejecta on MS companion stars. The paper is organized as follows. In Section 2, we describe the methods and codes used in this work. Section 3 presents the results from hydrodynamical simulations. The distribution of the unbound mass from population synthesis calculations is shown in Section 4. Some discusions based on results of impact simulations are presented in Section 5. Finally, we summarize the basic results of simulations in Section 6.",
"pages": [
1,
2
]
},
{
"title": "2. NUMERICAL METHOD AND MODEL",
"content": "In order to construct a detailed companion structure at the moment of the SN Ia explosion, we used the same method as described in Liu et al. (2012) to trace binary evolution in which a WD accretes H-rich material from a MS companion star (i.e., WD+MS M Ch explosion sce- nario). We think the WD would explode as a SN Ia when its mass increases to the M Ch limit. Here, we adopted the Eggleton's stellar evolution code, Roche-lobe overflow and the optically thick wind model of Hachisu et al. (1996) were included into the code to treat mass transfer in the binary. With a series of consistent binary evolution calculations, we selected four companion star models as input models of hydrodynamical simulations. These four companion stars were constructed with different initial WD masses, companion masses and orbital periods (see Table 1), which leads to companion models different in mass, orbital period and detailed structure at the moment of the SN explosion. Four companion models created in 1D binary evolution calculations are summarized in Table 1, their radial mass profiles are shown in Figure 1. We then performed 3D hydrodynamical simulations of the impact of SN Ia ejecta on the companion star employing the SPH code Stellar GADGET (Pakmor et al. 2012; Springel 2005). In this work, all initial conditions and basic setup for the impact simulations are the same as those in Liu et al. (2012). We use the healpix method described in Pakmor et al. (2012) to map the 1D profiles of density and internal energy of a 1D companion star model to a particle distribution suitable for the SPH code. Before we start the actual impact simulations, the SPH model of each companion star is relaxed for several dynamical timescales to reduce numerical noise introduced by the mapping. A comparison of density profiles between the 1D stellar model and its consistent SPH model for Model A are shown in Figure 2. The SN Ia explosion was represented by the pure deflagration model of Kromer et al. (2013) (i.e., the N5def model). This model has been shown to reproduce the characteristic observational features of 2002cx-like SNe Ia well (Kromer et al. 2013; Jordan et al. 2012). In this simulation, only the 0 . 37 M /circledot of ejected material with a total kinetic energy of 1 . 34 × 10 50 erg were used to represent the SN Ia explosion, we did not include the 1 . 03 M /circledot bound remnant of the M Ch WD into the simulations. Based on the angle averaged 1D ejecta structure of the Note . -M WD and M 2 , i present the WD and companion mass at the beginning of mass transfer. M 2 , f , R 2 , f and a f demonstrate the companion mass, companion radius and binary separation at the time of the explosion. ∆ M Def and ∆ M W7 show the unbound companion masses in the impact simulations for the pure deflagration model and the W7 model. v W7 kick and v Def kick correspond to the companion kick velocities. N5def model, SPH particles were placed randomly in shells to reproduce the mass (density) profile and gain the radial velocities they should have at their positions. The composition of a particle was then set to the values of the initial 1D model at a radius equal to the radial coordinate of the particle. Here, the effect a mild degree of asymmetry caused by off-center pure deflagration explosion of a M Ch WD was ignored. However, only in the direction opposites to the one-sided ignition region the velocities are somewhat lower (Fink et al. 2013). The orientation of an asymmetry of SN ejecta plays an inefficient role in mass stripping by the time of interaction with the companion. 1 We used 6 × 10 6 million SPH particles to represent the He companion stars in all simulations of this work. 2 Because all SPH particle was set up with the same mass, the number of particles representing the supernova explosion is then fixed. The supernova was placed at a distance to the companion star given by the separation at the moment of SN Ia explosion in our 1D binary-evolution calculations. The impact of the SN Ia ejecta on their binary companions was then simulated for 5000 s, at which point the mass stripped off from the companion star and its kick velocity due to the impact have reached constant values.",
"pages": [
2,
3
]
},
{
"title": "3. HYDRODYNAMICAL RESULTS",
"content": "Figure 3 shows the typical evolution of the density distribution in the impact simulations for the companion star Model A. Compared to our previous hydrodynamical simulations for the same MS companion star model (but with a different explosion model, see Liu et al. 2012), the basic impact processes are quite similar. The SN explodes at the right side of the companion star. After the SN explosion, the SN ejecta expand freely for a while and hit the MS companion star, removing solar-metallicity companion material and forming a bow shock. Subsequently, the bow shock propagates through the companion star, causing an additional loss of H-rich companion material from the far side of the star. Finally, the final unbound H-rich material of the companion star caused by the SN impact is largely embedded in lowvelocity SN debris behind the companion star, and the strongly impacted companion starts to relax to become almost spherical again. After the SN explosion, the non-degenerate companion star is significantly hit by the SN ejecta. The SN impact remove H-rich material from outer layers of the companion star through the ablation (SN heating) and the stripping (momentum transfer) mechanism. The late-time spectra of SN Ia probably show a signature of H lines if a large amount of H-rich material can be stripped from the companion star during the interaction with the SN ejecta. To calculate the amount of final unbound companion mass due to the SN impact, we sum the masses of all unbound SPH particles (ablated+stripped particles) that originally belongs to the companion star at the end of the simulations. For four different MS companion star models, our impact simulations show that the companion star received a kick velocity of ∼ 15 (Model D)-25 km s -1 (Model A) at the end of the simulations. Moreover, only a small amount of 0 . 013 (Model C)-0 . 016 M /circledot (Model D) of Hrich material is removed (ablation+stripping) from the companion stars due to the SN impact (see Table 1). For a comparison, numerical results for a classical explosion model of a M Ch WD for normal SNe Ia, the W7 model, are also shown in Table 1. Because the N5def model does not burn the complete WD but leaves behind a ∼ 1 . 0 M /circledot bound remnant, it produces a much lower kinetic energy (1 . 34 × 10 50 erg) than the W7 model (1 . 23 × 10 51 erg). Therefore, a much smaller amount of H-rich material of only ∼ 0 . 015 M /circledot is removed from the companion stars by the impact of the pure deflagration explosion of a M Ch WD. In contrast, the amount of final unbound companion mass is more than ten times larger ( > 0 . 1 M /circledot ) for the W7 explosion model with the same companion star models at the same separations. A large amount of unbound companion mass in the impact simulations for the W7 explosion model seems to indicate that normal SNe Ia are not likely produced from the WD+MS M Ch explosion scenario (see also Liu et al. 2012). Here, it is shown that the mass stripping is inefficient for 2002cx-like SNe Ia due to the low kinetic energies of off-center pure deflagration explosions of a M Ch WD, leading to that a amount of unbound companion material caused by the SN impact is quite small. Therefore, the H lines probably be hidden in late-time spectra of SN 2002cx-like SNe Ia. However, whether or not such small amount of stripped H mass would expect to show a signature of H lines in late-time spectra of SN 2002cx-like SNe Ia, which needs to analyze late-time spectra of 2002cx-like SNe to obtain the lower limit of stripped mass for detecting the H lines. 3",
"pages": [
3,
4
]
},
{
"title": "4.1. Power-law fitting",
"content": "Different WD+MS binary systems evolve to different evolutionary stages and have different binary parameters when the WD explodes as an SN Ia. Consequently, the companion radius and binary separation of binary systems differ significantly from the MS companion models used in our present simulations. We therefore investigated the dependence of the numerical results on the ratio of binary separation to the companion radius ( a f /R 2 , f ) at the time of the explosion. Here, we used the same method as in Liu et al. (2012) to artificially adjust the binary separations for a fixed companion star model (which means that all parameters but the orbital separation are kept constant). Figure 4 presents the amount of final unbound companion mass and kick velocity as a function of the parameter of a f /R 2 , f for four companion star models. It shows that the unbound mass and kick velocity significantly decrease when increasing the orbital separation of the binary. Generally, these relations can be fitted with power law functions in good approximation (see Figure 4): where a f is the binary separation, R 2 , f is the radius of the MS companion star at the onset of the SN explosion. C 1 and C 2 are two constant, α and β are the power-law indices (see Table 2).",
"pages": [
4
]
},
{
"title": "4.2. Unbound masses and kick velocities",
"content": "Wang et al. (2010)(hereafter WLH10) performed comprehensive binary population synthesis (BPS) calculations obtaining a large sample of WD+MS SN Ia progenitor models. They predicted many properties of the companion stars and binary systems at the moment of the SN explosion (e.g. the companion masses, the companion radii, the orbital periods, etc). The distributions of the ratio of binary separations to the companion radii ( a f /R 2 , f ) in the WLH10 sample (their model with α CE × λ = 0 . 5 , 1 . 5) are presented in Figure 5. It shows that most systems are concentrated at a f /R 2 , f ∼ 2 . 9 and 3 . 4. Based on the distribution of a f /R 2 , f , we calculated the final unbound companion masses (and kick velocities) due to the SN impact by adopting the power law relations of four companion star models in Figure 4, which are displayed in Figure 6. As it is shown, the impact of off-center pure deflagration explosions of M Ch WDs lead to a small amount of mass loss of the companion star ( /lessorsimilar 0 . 015 M /circledot ) in almost all WD+MS binary systems from BPS calculations. Moreover, the SN impact delivers a small kick velocity of /lessorsimilar 25 kms -1 to the companion star. In Section 2, initial MS companion star models were set up with a H abundance of X = 0 . 7, a He abundance of Y = 0 . 28 and a metallicity of Z = 0 . 02 when we constructed the MS companion star model at the moment of the SN explosion. Therefore, pure deflagrations of the M Ch WDs strip off a small amount of pure H of /lessorsimilar 0 . 01 M /circledot from the companion stars in almost all WD+MS models. These small stripped H masses are consistent with the lower mass limit for detecting H α emission lines in nebular spectra of normal SNe Ia (0 . 01 M /circledot , see Leonard 2007). The inefficient mass strip- ing seems to imply the stripped H may be hidden in (observed) late-time spectra of most of SN 2002cx-like SNe Ia. However, only normal SNe Ia were looked at when Leonard (2007) obtained the upper limit for nondetection of stripped H α of 0 . 01 M /circledot , the observational limits for SN 2002cx-like SNe may be smaller/larger than the value for normal SNe Ia. Most of stripped H-rich material in this simulations ends up at velocities below 10 3 kms -1 so that it is confined to the innermost part of explosion ejecta. Whether or not H α emission will be detectable, can only be answered by performing sophisticated radiative transfer simulations on the abundance structure of our explosion models.",
"pages": [
4,
5
]
},
{
"title": "5.1. The influence of the companion structures",
"content": "To test influence of the ratio of binary separation to the companion radius of a f /R 2 , f , we artificially adjusted the binary separation for a fixed companion star model. The final unbound mass due to the SN impact decreases by a factor of 10 as the parameter of a f /R 2 , f increases by a factor of 2. In Section 4.2, we only used the same power-law relation obtained from a fixed companion star model (for example, the power-law relation in Model A) to calculate the total mass loss of the companion star caused by the SN impact for all WD+MS models, which ignores the influence of details of the companion structure. Figure 6 show that a comparison of the results that calculated by using four different power-law relations between the final unbound mass (and the kick velocities) and the parameter of a f /R 2 , f in Model A, Model B, Model C, and Model D. Some differences (but not big differences) in the distribution of final unbound masses are seen in Figure 6a,c, which implies that the parameter of a f /R 2 , f is not the only factor to determine the final stripped companion mass, the companion structure can also affect the results.",
"pages": [
5,
6
]
},
{
"title": "5.2. Explosion energy",
"content": "For a comparison, we performed the impact simulations for the same companion star by adopting both the N5def model and the W7 explosion model to represent the SN Ia explosion. A factor of 10 lower explosion energy in the N5def model leads to the stripped material reduced by a factor of 10 (see Table 1). Moreover, Pakmor et al. (2008) investigated the influence of the SN explosion energy on the interaction with the companion star. They also found that explosion energy range covers a factor of 2 therefore leads to the unbound companion mass varies by a factor of 2. These results indicate that the SN explosion energy have only a small effect on the total mass loss of the companion star as compared to the effect of the parameter of a f /R 2 , f discussed above. Therefore, the ratio of the binary separation to the radius of companion star ( a f /R 2 , f ) is the most important parameter to determine the final unbound companion mass (see also Liu et al. 2012, 2013).",
"pages": [
6
]
},
{
"title": "5.3. The class of SN 2002cx-like SNe",
"content": "The SN 2002cx was discovered as a new class of peculiar SNe Ia by Li et al. (2003). From a volumelimited sample of the Lick Observatory Supernova Search (LOSS), Li et al. (2003) estimate that SN 2002cs-like SNe Ia contribute at about 5 per cent to the total SN Ia rate. Very recently, Foley et al. (2013) concluded that 'SNe Iax' (the prototype of which is SN 2002cx SNe) are the most common peculiar class of SNe, they estimated that in a given volume SNe Iax could contribute ∼ 1 / 3 of total SNe Ia. Nonetheless, to date, only 25 SNe Iax are confirmed that they are observationally similar to its prototypical member, SN 2002cx (see Foley et al. 2013). This sample consists of 25 members is a very small fraction of total SNe Ia. In this work, the results obtained from the impact simulations only apply to the subclass of peculiar 2002cx-like SNe but not the bulk of SNe Ia. Therefore, even the off-center pure deflagration explosion of a M Ch WD removes H-rich material during the interaction with the MS companion star, the H stripped from the companion star may not be observed in the ejecta of such relatively rare events.",
"pages": [
6
]
},
{
"title": "5.4. Post-explosion fate of the binary",
"content": "In this work, the N5def pure deflagration model was used to represent the SN Ia explosion in our impact simulations. The hydrodynamics calculations of Jordan et al. (2012) and Kromer et al. (2013) showed that the N5def model does not burn the complete WD but leaves behind a ∼ 1 . 0 M /circledot bound remnant. Unfortunately, this bound remnant cannot be properly spatially resolved until late time in hydrodynamical simulations due to the strong expansion of the SN ejecta. Our simulations show that the companion star receives a small kick velocity ( < 30 kms -1 ) during the interaction with the SN ejecta. Therefore, whether the WD+MS binary system would be destoried after the SN explosion, which depends primarily on the kick velocity of the bound remnant. If the bound remnant receives a large kick velocity that can overcome its gravitational force, the existence of abundance-enriched MS-like stars and WDs with peculiar spatial velocities are indicators (Jordan et al. 2012) of this studied progenitor scenario. Otherwise, the new binary system would survive the SN explosion. 4 In this study, it is found that the post-impact radii of companions at 5000 s after the explosion are larger than the initial binary separations used for the impact simulations due to their extreme expansions caused by the SN heating. 5 This indicates that the surviving binary system may evolve and merge into a single object with a rapid rotation velocity, or experience a common envelope phase. The details of this post-explosion evolution should be addressed in future work. However, fully resolving the detailed structure of the bound remnants is a prerequisite for this investigation.",
"pages": [
6
]
},
{
"title": "6. SUMMARY AND CONCLUSIONS",
"content": "We presented 3D hydrodynamical simulations of the impact of SN Ia explosions on their companion stars for the WD+MS scenario for the pure deflagration model presented in Kromer et al. (2013). For four different companion star models, we find that the much lower kinetic energy of the pure deflagration model compared to models for normal SNe Ia leads to a much lower stripped H-rich mass of only 0 . 013 M /circledot -0 . 016 M /circledot . Moreover, using the distribution of a f /R 2 , f from BPS calculations, we discussed the distribution of the amount unbound H-rich material of the companion star using a power law relation between the total unbound mass and the ratio of binary separation to the companion radius ( a f /R 2 , f ). We find that the off-center pure deflagration explosions strip off a small amount of H ( /lessorsimilar 0 . 01 M /circledot ) from MS companion stars in most WD+MS progenitor models of SN 2002cx-like SNe Ia. The inefficient mass stripping may lead to the stripped H is fully hidden in their late-time spectra. Therefore, it will be very interesting to analyze late-time spectra of 2002cx-like SNe Ia for the presence of hydrogen emission.",
"pages": [
6
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "Z.W.L acknowledges the financial support from the MPG-CAS Joint Doctoral Promotion Program (DPP) and from Max Planck Institute for Astrophysics (MPA). This work is supported by the National Basic Research Program of China (Grant No. 2009CB824800), the National Natural Science Foundation of China (Grant Nos. 11033008 and 11103072) and the Chinese Academy of Sciences (Grant N0. KJCX2-YW-T24). The work of F.K.R was supported by Deutsche Forschungsgemeinschaft via the Emmy Noether Program (RO 3676/1-1) and by the ARCHES prize of the German Federal Min- istry of Education and Research (BMBF). The simulations were carried out at the Computing Center of the Max Plank Society, Garching, Germany.",
"pages": [
6,
7
]
},
{
"title": "REFERENCES",
"content": "Frontiers of Physics, 8, 116 Hillebrandt, W., & Niemeyer, J. C. 2000, ARA&A, 38, 191 Horesh, A., Kulkarni, S. R., Fox, D. B., et al. 2012, ApJ, 746, 21 Iben, I., Jr., & Tutukov, A. V. 1984, ApJS, 54, 335 Jha, S., Branch, D., Chornock, R., et al. 2006, AJ, 132, 189 Jordan, G. C., IV, Perets, H. B., Fisher, R. T., & van Rossum, ⊙ ⊙",
"pages": [
7,
8
]
}
]
2013ApJ...778..137P
https://arxiv.org/pdf/1307.5255.pdf
<document>
<section_header_level_1><location><page_1><loc_25><loc_85><loc_75><loc_86></location>A Primeval Magellanic Stream and Others</section_header_level_1>
<text><location><page_1><loc_43><loc_81><loc_57><loc_82></location>P. J. E. Peebles</text>
<text><location><page_1><loc_20><loc_78><loc_80><loc_79></location>Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544</text>
<text><location><page_1><loc_44><loc_75><loc_56><loc_76></location>R. Brent Tully</text>
<text><location><page_1><loc_12><loc_71><loc_88><loc_73></location>Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, H I 96822</text>
<section_header_level_1><location><page_1><loc_44><loc_67><loc_56><loc_68></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_32><loc_83><loc_64></location>The Magellanic Stream might have grown out of tidal interactions at high redshift, when the young galaxies were close together, rather than from later interactions among the Magellanic Clouds and Milky Way. This is illustrated in solutions for the orbits of Local Group galaxies under the cosmological condition of growing peculiar velocities at high redshift. Massless test particles initially near and moving with the Large Magellanic Cloud in these solutions end up with distributions in angular position and redshift similar to the Magellanic Stream, though with the usual overly prominent leading component that the Milky Way corona might have suppressed. Another possible example of the effect of conditions at high redshift is a model primeval stream around the Local Group galaxy NGC6822. Depending on the solution for Local Group dynamics this primeval stream can end up with position angle similar to the H I around this galaxy, and a redshift gradient in the observed direction. The gradient is much smaller than observed, but might have been increased by dissipative contraction. Presented also is an even more speculative illustration of the possible effect of initial conditions, primeval stellar streams around M 31.</text>
<text><location><page_1><loc_17><loc_26><loc_83><loc_31></location>Key words: galaxies: NGC 6822, Large Magellanic Cloud - galaxies: kinematics and dynamics - galaxies: interactions - Local Group - large-scale structure of universe</text>
<section_header_level_1><location><page_1><loc_42><loc_19><loc_58><loc_21></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_10><loc_88><loc_17></location>The examples presented here of how intergalactic tidal streams could have been triggered by interactions among the young galaxies at high redshift, when they were all close together, are based on solutions to a dynamical model for the Local Group (LG) under the condition of growing peculiar velocities at high redshift, in analogy to the growing mode of departure from</text>
<text><location><page_2><loc_12><loc_77><loc_88><loc_86></location>a homogeneous expanding universe in perturbation theory. In these solutions a reasonable approximation to the Magellanic Stream (MS) grew largely from the interaction of the young Large Magellanic Cloud (LMC) with its nearest massive neighbor at high redshift, the Milky Way (MW). That is, we arrive back at the picture introduced by Fujimoto & Sofue (1976) and Lin & Lynden-Bell (1977), but applied at high redshift under cosmological initial conditions.</text>
<text><location><page_2><loc_12><loc_41><loc_88><loc_75></location>The MS certainly is expected to have been affected by subsequent tidal interactions, perhaps between the LMC and Small Magellanic Cloud (SMC), as noted by Fujimoto & Sofue (1976) and Lin & Lynden-Bell (1977), and by interaction with the MW mass and corona (Meurer, Bicknell, & Gingold 1985; Moore & Davis 1994). Analyses of these effects (Gardiner & Noguchi 1996; Mastropietro et al. 2005; Connors, Kawata, & Gibson 2006; Diaz & Bekki 2011, 2012; Besla, Kallivayalil, Hernquist, et al. 2010, 2012; and references therein) show that tidal and hydrodynamical interactions at modest redshifts can produce plausible approximations to MS without reference to conditions at high redshift. Exploration of a primeval origin nevertheless is called for. Gravitational interactions among the young galaxies certainly are real, as exemplified by the cosmic web (Bond, Kofman, & Pogosyan 1996). Exploration of the consequences in observations of the galaxies near us requires a prediction of how the young galaxies were positioned at high redshift. We seem to have that now for the LMC in a dynamical LG model constrained by initial conditions from cosmology and by the now considerable number of measurements of nearby galaxy redshifts, distances and proper motions (Peebles 2010; Peebles & Tully 2013, PT). This invites exploration of the effect of primeval tidal interactions on the distribution of matter around the young LMC. The result is a credible first approximation to MS.</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_40></location>The LG dynamical model and methods of its solution are outlined in Section 2 and discussed in more detail in PT and references therein. The dynamical actors and observational constraints are the same as in PT except that the actors that are meant to represent the effect of external mass are allowed the freedom to adjust angular positions as well as distances to aid the fit to the LG parameter constraints. The primeval streams presented in Section 3 show purely gravitational motion of massless test particles, in the tradition of Toomre & Toomre (1972). This simplifies the computation of streams in this preliminary exploration that might motivate more complete analyses that take account of hydrodynamics and self-gravitation. Section 3.1 shows the evolution of the model for a primeval Magellanic Stream. It produces a reasonable-looking fit to the MS H I angular and redshift distributions without special parameter adjustments. This result motivates the exploration in Section 3.2 of primeval streams around NGC 6822. The observed H I envelope around this LG dwarf almost certainly is gravitationally bound to the galaxy, a very different situation from MS. The results suggest that the H I envelope might have grown by dissipative contraction of a primeval H I stream, though substantiating that idea would require a considerable parameter</text>
<text><location><page_3><loc_12><loc_71><loc_88><loc_86></location>search. A crude estimate of the situation is offered in Section 4. Section 3.3 shows an even more speculative example, the development of streams around M 31 from its interactions with M33, NGC185, and NGC147 at high redshift. In the LG model solutions none of these galaxies passed close to M 31 at modest redshifts, but streams form. This certainly cannot make the case for a primeval origin of streams around M 31, because there are many other neighbors that could have produced streams at more modest redshifts, but it offers the possibility of a primeval component. We summarize our assessment of the results from the primeval stream models in Section 4.</text>
<section_header_level_1><location><page_3><loc_33><loc_64><loc_67><loc_66></location>2. Dynamical Model and Solutions</section_header_level_1>
<text><location><page_3><loc_12><loc_31><loc_88><loc_62></location>The starting assumption for the dynamical LG model is that the mass now concentrated around a galaxy was at high redshift in a patch whose motion may be traced by the position of its effective center of mass. This of course allows the galaxy to grow by accretion, provided it is accretion within the patch traced by the effective center. The initial condition is that the peculiar velocities of the mass patches are small and growing at high redshift. The condition that the galaxies end up where they are observed - or else how they are observed to be moving - presents mixed boundary conditions that are fitted by relaxation of the orbits to a stationary point of the action (in the NAM method introduced in Peebles 1989 and made more efficient in Peebles, Tully & Shaya 2011). In NAM solutions the equivalent of the decaying mode in linear perturbation theory is suppressed but not eliminated, as illustrated in Figure 1 in PT. It shows that, for the model parameters used in PT and here, peculiar velocities in the solutions are growing at redshift z < ∼ 20 in a reasonable approximation to the wanted growing mode, while earlier than that the decaying mode that inevitably appears in a numerical solution dominates and diverges as a ( t ) → 0. The advantage over a numerical integration back in time from given present positions and velocities is that NAM shifts domination of the decaying mode to high redshift where it seems likely to be harmless.</text>
<text><location><page_3><loc_12><loc_26><loc_88><loc_29></location>The NAM solutions are based on the ΛCDM cosmology with Hubble and matter density parameters</text>
<formula><location><page_3><loc_33><loc_23><loc_88><loc_25></location>H o = 70 km s -1 Mpc -1 , Ω m = 0 . 27 , (1)</formula>
<text><location><page_3><loc_12><loc_11><loc_88><loc_22></location>where Ω m represents the sum of masses in baryons and dark matter, the mass in radiation is neglected, space sections are flat, and Einstein's Λ is constant. The numerical solutions trace back in time by expansion factor 1 + z = 10 (to redshift z = 9) in 500 time steps uniformly spaced in the expansion parameter a(t). Numerical accuracy is checked by numerical integration forward in time in 5000 steps uniformly spaced in a(t) from positions and velocities at 1 + z = 10 from the action solution. The present positions and velocities</text>
<text><location><page_4><loc_12><loc_81><loc_88><loc_86></location>from this forward integration generally agree with the action solution to better than 0.1 kpc and 0.3 km s -1 , apart from some solutions for Leo 1, whose close passage of MW produces differences as large as 3 kpc and 5 km s -1 .</text>
<text><location><page_4><loc_12><loc_66><loc_88><loc_79></location>Solutions are found starting from random trial orbits with random initial assignments of distances, redshifts and masses within the nominal uncertainties, the orbits relaxed to a stationary point of the action, and the parameters then iteratively adjusted and relaxed to a stationary point to improve the fit to the measurements of LG redshifts, distances, luminosities, and peculiar velocities. The mixed boundary conditions allow many discretely different solutions; we choose the more plausible ones by comparison to the data. More details are in PT.</text>
<text><location><page_4><loc_12><loc_20><loc_88><loc_64></location>Table 1 names the LG galaxies in the dynamical model. The adopted LG parameter values and their measured or estimated uncertainties are entered under the headers 'catalog' (or 'cat'). Entered under the headers 'solution' are the parameter values in three numerical solutions to the dynamical model, ordered by the goodness of fit to the data. The catalog distances and their uncertainties, redshifts, and luminosities are from the Local Universe (LU) catalog maintained and provided on-line by Tully. 1 The adopted nominal uncertainty in each redshift, 10 km s -1 , is meant to allow for possible motion of the galaxy of stars relative to its dark matter halo. The nominal catalog masses (baryonic plus dark matter) are computed from the K-band luminosities using mass-to-light ratio M/L K = 50 M glyph[circledot] /L glyph[circledot] , meaning the nominal value of M/L K in Table 1 is 50. The nominal uncertainties in the LG galaxy masses are placed on a logarithmic scale, with a factor of 1.5 at one standard deviation. The exception is MW, whose nominal mass ratio to M 31 is unity with a factor of 1.1 at one standard deviation (PT eqs. [5] and [6]). The nominal rms galaxy peculiar velocity at 1 + z = 10 is taken to be v i = 50 km s -1 for LG and external actors. This is roughly what might be expected from the growth to rms peculiar velocities several times that at the present epoch. The velocity of the Sun relative to the local standard of rest is from Schonrich, Binney & Dehnen (2010), with no allowance for uncertainty in this relatively small term. The circular velocity of the local standard of rest has catalog value v c = 230 ± 10 km s -1 . The mass distribution in each actor is rigid and spherical with density run ρ ∝ r -2 cut off at the radius that produces the model mass for given v c . Assigned circular velocities without uncertainties are v c = 250 km s -1 in M31 and v c = 100 km s -1 in all the other actors except MW.</text>
<section_header_level_1><location><page_5><loc_12><loc_22><loc_14><loc_74></location>T able 1: Lo cal Group Mo del Distances, Redshifts, Masses, and Initial V elo cities</section_header_level_1>
<text><location><page_5><loc_65><loc_11><loc_67><loc_30></location>units: Mp c, km s -1 , M glyph[circledot] /L glyph[circledot]</text>
<table>
<location><page_5><loc_13><loc_11><loc_65><loc_86></location>
</table>
<section_header_level_1><location><page_6><loc_12><loc_40><loc_14><loc_56></location>T able 2: Prop er Motions</section_header_level_1>
<table>
<location><page_6><loc_15><loc_11><loc_38><loc_86></location>
</table>
<text><location><page_6><loc_38><loc_25><loc_39><loc_26></location>1</text>
<text><location><page_6><loc_38><loc_24><loc_39><loc_25></location>-</text>
<text><location><page_6><loc_39><loc_24><loc_40><loc_24></location>y</text>
<text><location><page_6><loc_39><loc_21><loc_40><loc_23></location>sec</text>
<text><location><page_6><loc_39><loc_19><loc_40><loc_21></location>arc</text>
<text><location><page_6><loc_39><loc_15><loc_40><loc_18></location>milli</text>
<text><location><page_6><loc_39><loc_12><loc_40><loc_15></location>unit:</text>
<section_header_level_1><location><page_6><loc_44><loc_41><loc_46><loc_57></location>T able 3: External Actors</section_header_level_1>
<figure>
<location><page_6><loc_47><loc_9><loc_69><loc_91></location>
</figure>
<table>
<location><page_6><loc_47><loc_9><loc_69><loc_91></location>
</table>
<text><location><page_6><loc_69><loc_9><loc_71><loc_28></location>units: Mp c, km s -1 , 10 11 M glyph[circledot]</text>
<text><location><page_7><loc_12><loc_77><loc_88><loc_86></location>Table 2 lists proper motions, where µ α is the motion in the direction of increasing right ascension and µ δ is the motion in the direction of increasing declination. The measurement and uncertainty for M31 is from Sohn, Anderson & van der Marel (2012), for LMC from Kallivayalil et al. (2013), for M33 from Brunthaler et al. (2005), for IC 10 from Brunthaler et al. (2007), and for LeoI from Sohn, Besla, van der Marel, et al. (2012).</text>
<text><location><page_7><loc_12><loc_48><loc_88><loc_75></location>The external actors named in Table 3 are meant to give a phenomenological description of the effect of the external mass distribution on LG by allowing their present positions and masses to float to aid the model fit to the catalog LG parameters. In a departure from PT, the angular positions as well as distances of these actors are allowed to float. The nominal angular positions (columns 2 and 3 in Table 3) are luminosity-weighted means for the galaxies concentrated around the Sculptor group, the Maffei-IC 342 system, the M 81 group, and the Centaurus-M94 system. The positions are given in supergalactic coordinates, because the nearby galaxies outside LG are concentrated near the supergalactic plane. The distances δD between the three-dimensional positions in the catalog and the model solutions have nominal allowed rms value 0.5 Mpc. The nominal redshifts are luminosity-weighted means, with adopted uncertainties 50 km s -1 . The catalog masses are computed from the sums of Kband luminosities with M/L K = 50 M glyph[circledot] /L glyph[circledot] , and the mass uncertainties are on a logarithmic scale with one standard deviation at a factor of two difference between catalog and model solution.</text>
<text><location><page_7><loc_12><loc_10><loc_88><loc_46></location>The measured or adopted uncertainties in the catalog parameters are treated as standard deviations in a χ 2 sum of squares of differences between model and catalog values divided by standard deviations. There are 69 LG parameters: 14 distances, 14 redshifts, 15 masses, 10 components of proper motion, the MW circular velocity, and 15 primeval velocities. (The last are more properly counted as 45 primeval velocity components, each with a Gaussian velocity distribution, less three components because the center of mass is at rest, but this is too fine for the present purpose). There are 24 external actor parameters: the redshift, mass, primeval velocity, and three components of present position for each of the four actors. Solutions 1 through 3 have χ 2 = 92, 98, and 114, close to the total of 93 parameters in χ 2 . This is not very meaningful, however, for two reasons. First, many of the nominal standard deviations are at best only informed guesses. Second, the multiple solutions allowed by the mixed boundary conditions allow multiple choices among which we choose those with the smallest χ 2 . That is, if model, measurements and standard deviations were accurate enough for a meaningful value of χ 2 we would expect it to be less than 93. The sums over LG parameters alone are χ 2 LG = 70, 76, and 89. These are not much larger than the 69 LG parameters, but again one would have expected smaller because the solutions were chosen for their fit to the catalog parameters, and because the external parameters were adjusted to reduce χ 2 LG . If it were supposed that the external actor parameters are in effect free, because</text>
<text><location><page_8><loc_12><loc_80><loc_88><loc_86></location>their constraints are quite loose, one might expect χ 2 LG = 69 -16 = 53 (discounting only the masses and present positions, which most matter for LG orbits ), which would put the reduced χ 2 LG values at about 1.5, well above statistical expectation.</text>
<text><location><page_8><loc_12><loc_56><loc_88><loc_79></location>The initial peculiar velocities of the external actors are less than about 25 km s -1 , which seems acceptably small. The numerical solutions put the mass of the M81 actor at or above its nominal value and the other masses below nominal, and M81 is placed at heliocentric distance ∼ 2 Mpc, well short of the LU distance to the galaxy M 81, 3 . 65 ± 0 . 18 Mpc, while the other three actors are placed close to their catalog positions, at median δD ∼ 0 . 3 Mpc. Parameter adjustments are allowed to move the external actors away from the plane, but the model positions now and at 1 + z = 10 are close to the plane, supergalactic latitude SGB close to zero (columns 4 to 9 in Table 3). A possibly significant exception is that the solutions prefer the M81 actor below the plane. This is in the direction that would help compensate for the striking scarcity of galaxies - and likely mass - in the Local Void immediately above the plane. We hope to investigate this and other aspects of the influence on LG of the external mass distribution in due course.</text>
<text><location><page_8><loc_12><loc_30><loc_88><loc_54></location>The parameters in the numerical solutions in Tables 1 to 3 that differ from catalog by more than two nominal standard deviations are entered in italics. (There are no 3σ differences in LG parameters.) The model prefers a long LMC distance, at 1 . 8 σ in Solution 1, 2 . 5 σ in Solution 3. Since measurements of this distance have been thoroughly examined we expect the discrepancy indicates a systematic error in the model, perhaps in the simple approximation to the mass distribution in MW. The other 2σ discrepancy in Table 1 is the short distance to NGC 185 in Solution 3. None of the model solutions fit both catalog proper motion components of IC 10 to better than two standard deviations, and only Solution 1 fits the proper motion of M 33, but it fails the proper motion of Leo I at two standard deviations. Here again the problem certainly may be with the model, but since these proper motion measurements have not been so thoroughly reconsidered one may imagine some of the errors assigned to these difficult measurements are underestimates.</text>
<text><location><page_8><loc_12><loc_11><loc_88><loc_29></location>In Solutions 1 to 3 respectively the MW circular velocity is v c = 229, 238, and 246 km s -1 . The increase with decreasing quality of fit to the measurements may be accidental. The preference for a value larger than the standard 220 km s -1 is in the direction of but smaller than that found by Reid, Menten, Zheng, Brunthaler, et al. (2009). The preference for greater mass in MW than M31 is present also in the larger number of solutions in PT with generally poorer fits to the constraints. In models 1 to 3 the MW masses in units of 10 11 M glyph[circledot] are 15.2, 16.3, and 18.3, respectively, and the M 31 masses are 12.9, 13.6 and 15.8. If the relative motion of MW and M31 were not affected by other actors the sum of the derived masses would decrease with increasing v c because the derived galactocentric redshift of M 31 de-</text>
<text><location><page_9><loc_12><loc_81><loc_88><loc_86></location>creases with increasing v c . The trend the other way in the three solutions is not inconsistent with this argument because the other actors significantly affect the relative motion of MW and M31, curving the orbit.</text>
<text><location><page_9><loc_12><loc_68><loc_88><loc_79></location>In future work aimed at improving the LG dynamical model we expect to use more realistic descriptions of the mass distributions within MW and M 31. It may help also to let the characteristic radius of the mass distribution within M 31 be an adjustable parameter (in analogy to the adjustment of v c in a truncated limiting isothermal sphere). And, perhaps most important, the treatment of the effect of the external mass distribution on LG should more closely refer to the observed external galaxy distribution and peculiar motions.</text>
<text><location><page_9><loc_12><loc_53><loc_88><loc_66></location>The numerical solutions are not formally statistically consistent with the full catalog of parameters within their uncertainties. That is not surprising, because the dynamical model is crude and the catalog likely to contain errors. It is encouraging that the solutions match a considerable number and variety of constraints with relatively few discrepancies beyond two nominal standard deviations and, in LG, none beyond 3 σ . This degree of fit to the constraints argues that we have a reasonably secure basis for exploration of the effect of initial conditions on the development of streams within the Local Group.</text>
<section_header_level_1><location><page_9><loc_40><loc_46><loc_60><loc_48></location>3. Primeval Streams</section_header_level_1>
<text><location><page_9><loc_12><loc_13><loc_88><loc_44></location>Figure 1 shows model solutions for the motions of galaxies relative to MW. The righthand coordinate system is galactic, with the z -axis at b = 90 · and the x -axis at b = l = 0. The lengths are physical. Positions are plotted relative to MW at the plus sign. We consider possible examples of remnant primeval streams around LMC, plotted in black, and NGC6822, plotted in red. The green curves show the relative position of the other massive LG actor, M 31. Present positions are at the crosses, and the other ends of the orbits show the young galaxies moving apart at redshift z = 9. The solid lines show Solution 1 in Tables 1 to 3, the dashed lines Solution 2, and dotted, 3. These solutions are at different stationary points of the action and local minima of χ 2 . The three orbits of LMC relative to MW are quite similar, and they are similar too in the greater number of solutions in PT. The apparently well-determined initial situation of LMC invites our exploration of the effect of the initial conditions on a cloud of test particles that might approximate the behavior of an HI envelope. The motion of NGC 6822 relative to MW is similar in Solutions 1 and 2, quite different in 3. This illustrates the multiple solutions allowed by the mixed boundary conditions. For this galaxy an assessment of the situation has to guide selection of the more likely solution.</text>
<figure>
<location><page_10><loc_34><loc_41><loc_66><loc_86></location>
<caption>Fig. 1.Orbits around MW, at the origin, for LG galaxies LMC (black), NGC 6822 (red), and M 31 (green). Solution 1 is plotted as the solid lines, 2 as long dashes, and 3 as dots.</caption>
</figure>
<text><location><page_10><loc_12><loc_24><loc_88><loc_32></location>The test particles in a model stream around a chosen LG actor move in the given gravitational field of the solution for the actors with mass, making it easy to accumulate a dense sample of test particle paths. The test particles are placed at z = 9 uniformly at random within the gravitational radius x g of the actor, where</text>
<formula><location><page_10><loc_41><loc_21><loc_88><loc_23></location>x g = (2 GM/ Ω H 2 o ) 1 / 3 . (2)</formula>
<text><location><page_10><loc_12><loc_10><loc_88><loc_19></location>The comoving length x g is normalized to the physical radius at the present epoch. The galaxy mass, M , is the same as the mass within x g in a homogeneous mass distribution at the cosmic mean density. This means that a test particle closer than x g tends to be gravitationally attracted to the actor, in comoving coordinates, and at x > x g a test particle tends to be pulled away. The radii x g also are the characteristic separations of the actors at</text>
<text><location><page_11><loc_12><loc_83><loc_88><loc_86></location>high redshift, where the peculiar accelerations of the actors are bounded by the condition that the orbits approximate the expansion of a near-homogeneous mass distribution.</text>
<text><location><page_11><loc_12><loc_64><loc_88><loc_81></location>In a model stream the test particles are initially at rest in comoving coordinates relative to the chosen actor, meaning the particles initially are streaming away from the actor with the general expansion of the universe. The condition for capture of a test particle by the chosen actor varies with direction as well as distance relative to x g , of course. Trials that take account of this by rejecting test particles with initial relative peculiar accelerations directed away from the chosen actor, and trials with initial velocities set to what is derived from the initial peculiar gravitational acceleration in linear perturbation theory, do not to produce very different streams, so for simplicity these refinements are not used in the results presented here.</text>
<section_header_level_1><location><page_11><loc_36><loc_57><loc_64><loc_59></location>3.1. The Magellanic Stream</section_header_level_1>
<text><location><page_11><loc_12><loc_38><loc_88><loc_55></location>Figure 2 shows present positions and redshifts of the test particles that are uniformly distributed within r g around LMC at z = 9 in Solution 1. The plots derived from the other two solutions look quite similar (and the gravitational radii defined in eq. [2] are similar; the physical values at 1 + z = 10 are r g = 82, 90, and 87 kpc in Solutions 1 to 3). The sharp cutoff in the initial distribution of test particles can produce features in the present distribution of the particles plotted in black that are at initial physical distance 0 . 75 r g to r g from LMC, but the effect is not prominent here. There is not much indication of orbit mixing; the appearance rather is that the initial distribution has been smeared by a smooth velocity field.</text>
<text><location><page_11><loc_12><loc_10><loc_88><loc_36></location>Figure 2 is plotted in the MS coordinates defined by Nidever, Majewski, & Burton (2008), where LMC is at the origin, the stream is centered near B MS = 0, and the trailing stream is at L MS < 0. Figure 2 can be compared to Figure 8 in Nidever et al. (2010), which shows in these coordinates the measured H I angular distribution and the heliocentric redshift as a function of L MS . The orientation of the model stream and the variation of the heliocentric redshift with L MS are close to what is observed, though the model lacks the fine structure in MS, including the SMC, and the leading stream is much too prominent. Figure 2 also is similar to other MS models, including Figures 6 and 7 in Gardiner & Noguchi (1996); Figures 9 and 12 in Mastropietro et al. (2005); Figures 7 and 9 in Connors, Kawata, & Gibson (2006); Figure 7 in Diaz & Bekki (2012); and Figures 7 and 9 in Besla et al. (2012). In Figure 2 the distance to MS is nearly constant from -50 · < ∼ L MS < ∼ 50 · at about 60 kpc. The variation of distance with L MS is similar in Connors, Kawata, & Gibson, and in Diaz & Bekki, while Besla et al. show a more rapid decrease of distance with increasing L MS . An</text>
<figure>
<location><page_12><loc_19><loc_41><loc_81><loc_86></location>
<caption>Fig. 2.The Magellanic Stream in Solution 1. Particles initially less than 41 kpc from LMC are plotted in yellow, those initially at 41 < r < 62 kpc in red, and those at 62 < r < 82 kpc in black.</caption>
</figure>
<text><location><page_12><loc_12><loc_28><loc_88><loc_31></location>observational check does not seem likely, however. The notable overall similarity of results in a variety of models is discussed in Section 4.</text>
<text><location><page_12><loc_12><loc_11><loc_88><loc_26></location>Figure 3 illustrates the evolution of the cloud of test particles around LMC in Solution 1. Solutions 2 and 3 look similar. The effect of the sharp cutoff in the distribution of black particles is more apparent here than in Figure 2. At z = 9 the physical distance between MW and LMC is 250 kpc, comparable to the MW gravitational radius 220 kpc (eq. [2]) at z = 9, and M31 is about 500 kpc from LMC. At z = 3, in the left-hand panel, MW is at the blue circle near the bottom edge. Its distance from LMC has about doubled, growing slightly less than the factor 2.5 general expansion. M 31 is to the left, outside the boundary of the figure. The physical width of the distribution of the initially innermost particles plotted in</text>
<figure>
<location><page_13><loc_12><loc_64><loc_88><loc_86></location>
<caption>Fig. 3.Evolution of the Magellanic Stream model in Figure 2, with the same color scheme. MW is at the blue circle. Coordinates are galactic, lengths physical.</caption>
</figure>
<text><location><page_13><loc_12><loc_47><loc_88><loc_55></location>yellow is about the same at z = 3 as at z = 9, while the outer envelope marked by the black particles has expanded by about as much as the general expansion. The elongated distribution of test particles at z = 3 points to MW, even among the initially innermost particles particles shown in yellow.</text>
<text><location><page_13><loc_12><loc_30><loc_88><loc_46></location>In the central panel in Figure 3, at z = 1, MW is close to its maximum separation from LMC, 700 kpc. It is below the panel and positioned about in line with the prominent red stream and the long axis of the slightly eccentric distribution of yellow particles. MW reappears in the right-hand panel, at z = 0 . 3. By this time some of the test particles are concentrated around MW. The band of black particles in the upper left part of the righthand panel is suggestive of folding in singe-particle phase space. The yellow particles that initially were less than 41 kpc from LMC are still concentrated around LMC at about this radius at z = 0 . 3. They end up smeared into the yellow stream in Figure 2.</text>
<text><location><page_13><loc_12><loc_23><loc_88><loc_29></location>The point of Figure 3 is that the cloud of test particles around LMC carries some memory of its interaction with MW when they were close, at high redshift. This is seen in features in the three panels of Figure 3 and, at z = 0, in the stream in Figure 2.</text>
<section_header_level_1><location><page_13><loc_34><loc_17><loc_66><loc_18></location>3.2. A Stream around NGC6822</section_header_level_1>
<text><location><page_13><loc_12><loc_11><loc_88><loc_15></location>The atomic hydrogen around the LG galaxy NGC 6822 extends to angular radius ∼ 30 ' , projected separation r ∼ 4 kpc. The HI redshifts of the outer parts differ from the center</text>
<figure>
<location><page_14><loc_26><loc_42><loc_74><loc_86></location>
<caption>Fig. 4.Model streams around NGC6822 in equatorial coordinates for the three solutions in the color scheme of Figure 2. The nominal redshift cz is the component of the heliocentric velocity in the direction to NGC6822. Solutions 1 and 2 are labeled; 3 is in the bottom panels.</caption>
</figure>
<text><location><page_14><loc_12><loc_11><loc_88><loc_31></location>by v ∼ ± 50 km s -1 . These quantities define a characteristic mass, v 2 r/G ∼ 3 × 10 9 M glyph[circledot] . The catalog mass of this galaxy, 6 × 10 9 M glyph[circledot] , is not well tested by the dynamical model because it is too small to have much effect on the other actors, but its similarity to v 2 r/G does suggest that the H I could be gravitationally bound to the galaxy. And, if the H I were not bound, the relative velocity of 50 km s -1 would soon carry the hydrogen far beyond its projected separation from the galaxy, unless the relative motion of galaxy and gas were directed almost exactly along the line of sight, which seems unlikely. Thus we ought to study the formation of a gravitationally bound H I cloud. This cannot be simulated by the simple gravitational motions of test particles, but we can consider initial conditions for dissipative contraction that might produce the H I envelope of NGC 6822.</text>
<text><location><page_15><loc_12><loc_69><loc_88><loc_86></location>Figure 4 shows angular distributions and radial velocities of the primeval streams of test particles initially uniformly distributed around NGC 6822 in the three solutions. The particles initially less than r g / 2 = 17 kpc from NGC 6822 are plotted in yellow, those initially at 17 to 25 kpc in red, and those at 25 to 34 kpc in black. Here again the cutoff at r g in the initial test particle distribution produces the sharp edges in the figure. The range of angles plotted in Figure 4 is large enough that the heliocentric motion of the galaxy causes a significant variation of the line-of-sight velocity across the figure. This effect is removed by plotting an effective redshift cz defined as the component of the heliocentric velocity of each test particle along the heliocentric direction to the center of NGC 6822.</text>
<text><location><page_15><loc_12><loc_48><loc_88><loc_67></location>The orbital histories of NGC 6822 in Solutions 1 and 2 are similar (Fig. 1), as are the present test particle distributions in Figure 4, though there are systematic differences. The stream position angles in these two solutions are about 140 · (measured from the direction of increasing declination toward the direction of increasing right ascension), similar to the orientation of the H I stream around this galaxy (Roberts 1972; de Blok & Walter 2000). In these two model streams the redshift increases with decreasing declination and increasing right ascension, in the direction of the observations, though the gradient is much smaller than observed. This is in line with the idea that the primeval stream may model the precursor to dissipative contraction. Solution 3 has a different history (Fig. 1) and different present distributions of positions and redshifts that seem less promising.</text>
<text><location><page_15><loc_12><loc_32><loc_88><loc_46></location>The lower panel in Figure 5 shows the evolution in Solution 1 of the numbers of particles within physical distances 15 and 30 kpc from NGC 6822, plotted as the ratio to the number initially within 30 kpc. The concentrations initially decrease because the cloud is expanding with the general expansion of the universe. The first minima, at z ∼ 4 for r = 15 kpc and z ∼ 3 for r = 30 kpc, are artificially deep because the initial velocities relative to NGC 6822 are artificially radial. At z < 1 the concentration within 15 kpc is nearly constant at about the value at z = 9, and the concentration within 30 kpc is about one fifth its initial value.</text>
<text><location><page_15><loc_12><loc_11><loc_88><loc_31></location>The black curves in the upper panel of Figure 5 show the evolution of the components (in galactic coordinates) of the mean (specific) angular momentum per particle relative to the position and motion of NGC 6822 for the particles that are at physical distance r < 30 kpc from NGC6822. The mean angular momentum at r < 15 kpc is smaller but the components evolve in a similar way. The angular momentum evolves in part because particles are streaming past the 30 kpc limiting distance, and in part because of the torques from other actors. These effects are separated by identifying the particles that at (1 + z ) -1 = 0 . 75 are within 30 kpc from NGC6822. The components of mean angular momentum of this fixed set of particles are plotted at lower redshift as the red curves in Figure 5. The red and black curves differ because of the motions of particles through r = 30 kpc. The similarities</text>
<figure>
<location><page_16><loc_29><loc_50><loc_71><loc_86></location>
<caption>Fig. 5.Evolution of the cloud of test particles around NGC 6822 in Solution 1. The lower panel shows test particle concentrations. Black curves in the upper panel are components of mean angular momentum of the particles within 30 kpc. Red curves are components for the fixed set of particles that are within 30 kpc at redshift z = 1 / 3.</caption>
</figure>
<text><location><page_16><loc_12><loc_24><loc_88><loc_37></location>show that gravitational torques substantially affect the mean angular momentum per particle near NGC6822 as it lingers near its maximum distance from MW and M 31 approaches. The angular momentum may be compared to the maximum to be expected from the tidal torque by MW integrated over a Hubble time, L ∼ GMr 2 / ( H o R 3 ) ∼ 500 kpc km s -1 for MW mass M ∼ 10 12 M glyph[circledot] , NGC6822 distance R ∼ 500 kpc, and moment arm r ∼ 30 kpc. This is an order of magnitude larger than the mean angular momentum within r = 30 kpc at z = 0 in the model, L ∼ 50 kpc km s -1 .</text>
<text><location><page_16><loc_12><loc_11><loc_88><loc_23></location>At z = 0 and r = 30 kpc the angular momentum vector has position angle ∼ 220 · and inclination i ∼ 55 · (where i is the angle between the angular momentum vector and the direction from NGC 6822 to MW, meaning the angular momentum is tilted from the plane of the sky toward us by about 35 · ). Over the range of limiting radius r = 10 kpc to 100 kpc the direction of the model angular momentum vector does not change much, but the magnitude increases with increasing r from about 20 kpc km s -1 at r = 10 kpc to about 200</text>
<text><location><page_17><loc_12><loc_75><loc_88><loc_86></location>kpc km s -1 at r = 100 kpc. The observable component of angular momentum per unit mass in the H I around NGC 6822 is about 100 kpc km s -1 . The position angle is similar to the model stream, perhaps a significant coincidence. It will be noted, however, that the model stream acquired its angular momentum at redshift z < ∼ 0 . 3, so it could be a progenitor of the HI cloud around NGC6822 only if the accretion of this H I were a recent development. This special condition is discussed in Section 4.</text>
<figure>
<location><page_17><loc_34><loc_29><loc_66><loc_73></location>
<caption>Fig. 6.Orbits around M 31, at the origin, for M 33 (blue), NGC 185 (black), NGC 147 (red), and MW (green). Solution 1 is plotted as the solid line, 2 as long dashes, and 3 as dots.</caption>
</figure>
<section_header_level_1><location><page_17><loc_37><loc_15><loc_63><loc_17></location>3.3. Streams around M31</section_header_level_1>
<text><location><page_17><loc_12><loc_10><loc_88><loc_13></location>The streams around M 31 are dominated by stars (Richardson, Irwin, McConnachie, et al. 2011, and references therein), a different situation from the loose stream of H I around</text>
<text><location><page_18><loc_12><loc_67><loc_88><loc_86></location>the Magellanic Clouds or the gravitationally bound H I around NGC 6822. The prominent optical stream between M 31 and M 33 invites the idea that these two galaxies suffered a close passage, which did not happen in the LG solutions used here. This is illustrated in Figure 6, which shows in galactic coordinates the paths of galaxies relative to the position of M31 at the plus sign. The green curves show MW approaching M 31 from the right, in the mirror image of the approach of M 31 to MW in Figure 1. The orbits of NGC 185 (black) and NGC147 (red) are similar in Solutions 1 and 2 and rather different in Solution (3). The situation may be compared to NGC 6822 in Figure 1. In all three solutions M 33 (blue) has been well away from M 31 and MW. But we can offer an illustration of how M 33 and M 31 might be connected by a primeval stream.</text>
<figure>
<location><page_18><loc_22><loc_24><loc_78><loc_65></location>
<caption>Fig. 7.Present distributions of the test particles in Solution 1 that at z = 9 were close to M 33, now at the blue square to the lower left in the left panel, initially close to NGC 185, now at the left of the top two squares in the middle panel, and NGC 147, at the right uppermost square in the right-hand panel. M 31 is at the center square.</caption>
</figure>
<paragraph><location><page_18><loc_16><loc_11><loc_88><loc_12></location>Figure 7 shows streams of test particles around M 33 in the left panel, NGC 185 in the</paragraph>
<text><location><page_19><loc_12><loc_67><loc_88><loc_86></location>central panel, and NGC 147 in the right-hand panel. The test particles plotted as yellow were initially closer to the galaxy than r g / 2, and red initially at r g / 2 to 3 r g / 4, where r g ∼ 100 kpc in M 33 and ∼ 40 kpc in NGC 185 and NGC 147. Test particles initially further out add more diffuse streaks. To reduce clutter we refrain from including IC 10 and the stars that might have been pulled out of M 31. If stars formed around the young M 33 and were drawn away by tidal fields of MW, which was 230 kpc away at z = 9, and M 31, which was 250 kpc from M33, then the left-hand panel in Figure 7 suggests the stars might end up in an observable stream that passes across M 31, at the central square in the figure. A primeval stream of stars drawn from the outskirts of NGC 185 could appear somewhat tighter, and a stream from NGC147 could be tighter still, though both avoid M 33.</text>
<section_header_level_1><location><page_19><loc_43><loc_60><loc_57><loc_62></location>4. Discussion</section_header_level_1>
<text><location><page_19><loc_12><loc_23><loc_88><loc_58></location>The plausible approximation to MS in Figure 2 is based on a dynamical model that fits the considerable number of measurements in Tables 1 and 2 about as well as could be expected. This dynamical model offers a reasonably unambiguous prediction of the positions of the galaxies near LMC at high redshift. The primeval stream model assumes the young LMChad an envelope of H I or cool plasma whose response to the presence of the neighboring galaxies at high redshift may be approximated by a cloud of massless test particles initially moving with LMC. It also assumes memory of the conditions at high redshift is preserved in the present state of the H I. Memory in a cloud of test particles is illustrated in Figure 3. Besla et al. (2010) summarize arguments that MS is 'a young feature (1 - 2 Gyr)'. In the primeval stream model LMC would have entered an MW corona that extends to 300 kpc at redshift z ∼ 0 . 1, or about 1 Gyr ago, which may be recent enough that the trailing component largely survived moving through the plasma while losing much of the leading component. Prior to that LMC would have been well separated from large galaxies and its proto-MS might have survived in the same manner as the H I at similar surface densities around other isolated gas-rich dwarfs. These are not many assumptions, and they are applied in a straightforward way, which lends support for the result. A test by hydrodynamic simulation could provide a stronger argument, but the evidence we have now is that gravitational interactions among the young galaxies can have produced MS.</text>
<text><location><page_19><loc_12><loc_12><loc_88><loc_21></location>We must consider that plausible MS models have been obtained without reference to initial conditions, and from a variety of ways to model the orbits and interactions among LMC, SMC and MW (Gardiner & Noguchi 1996; Mastropietro et al. 2005; Connors, Kawata, & Gibson 2006; Diaz & Bekki 2011, 2012; Besla, Kallivayalil, Hernquist, et al. 2010, 2012). Though these approaches generally require close attention to parameter choices, it appears</text>
<text><location><page_20><loc_12><loc_75><loc_88><loc_86></location>that MS has properties of an attractor, capable at arriving at a good approximation to its present state - and, we must expect, of being destroyed - under a variety of interactions along the way. The case that MS originated as a primeval stream rests on the demonstration that such a stream would have formed if cool baryons in the young galaxy were in a position to be tidally disturbed and then not seriously disturbed thereafter. We must be cautious, however, for MS seems to have a generic tendency to end up looking like Figure 2.</text>
<text><location><page_20><loc_12><loc_39><loc_88><loc_73></location>In contrast to MS, the case for formation of the H I envelope of NGC 6822 out of a primeval stream requires a very special situation. However, the situation may be indicated by the demonstration by Demers, Battinelli, & Kunkel (2006) that NGC 6822 is a polar ring galaxy. The stellar halo traced by RGB stars extends about as far from the galaxy as the H I envelope, but the long axis of the stellar distribution has position angle 60 · , while the HI has position angle is 130 · (de Blok & Walter 2000). The redshift gradients of the halo stars and the H I envelope both point along the long axis of their angular distribution, and both gradients are close to constant at 15 km s -1 kpc -1 . The contributions to the angular momenta of stars and H I by the observed redshift gradients have position angles differing by ∼ 70 · . This could signify a strong departure from axial symmetry, as in bars. (Hodge 1977 notes that the stars in the inner ∼ 0 . 5 kpc, with PA ∼ 10 · , may be a bar.) The alternative is that the angular momenta of stellar halo and H I envelope have quite different directions. Similar tilts of the long axis of the stellar distribution from the H I redshift gradient are observed in isolated dwarfs (Stanonik, Platen, Arag'on-Calvo, van Gorkom, et al. 2009; Kreckel, Peebles, van Gorkom, van de Weygaert, & van der Hulst 2011). This might indicate that the H I envelopes dissipatively settled onto these galaxies without adding many stars.</text>
<text><location><page_20><loc_12><loc_10><loc_88><loc_38></location>The orientation and the direction of the redshift gradient in the primeval stream around NGC6822 in Solutions 1 and 2 (Fig. [4]) agree with the H I envelope of this galaxy. The model redshift gradient is much smaller than observed, but that could be because of the dissipative contraction of the H I. This picture requires that the H I settled after redshift z ∼ 0 . 3, when the primeval stream acquired its angular momentum (Fig. [5]), a very special condition that may be crudely modeled as follows. Suppose that at z ∼ 0 . 3 the stars in NGC6822 were centered on a sheet of diffuse baryons with number density n b ∼ 10 -4 cm -3 and thickness h ∼ 30 kpc, or angular width ∼ 3 · at the distance of this galaxy, which is on the scale of what is plotted in Figure 4. The characteristic baryon surface density is ∼ n b h ∼ 10 19 cm -2 , or baryon surface mass density Σ b ∼ 10 5 M glyph[circledot] kpc -2 . With total surface density Σ m ∼ 6Σ b , to take account of dark matter, pressure support requires plasma temperature kT ∼ 2 πG Σ m m p h ∼ 3 eV. That makes the plasma cooling time ( τ ∼ 10 11 . 4 √ T/n b in cgs units, ignoring line emission) about 10 10 y, roughly what is wanted for late accretion of an HI envelope around NGC6822. If an h by h (30 kpc by 30 kpc) piece of the slab collapsed</text>
<text><location><page_21><loc_12><loc_75><loc_88><loc_86></location>by a factor of three in each direction it would gather baryon mass Σ b h 2 ∼ 10 8 M glyph[circledot] in a region of width ∼ 10 kpc ∼ 1 · at NGC6822, roughly what is detected in 21-cm radiation (de Blok & Walter 2000). Conservation of angular momentum would bring the redshift gradient in Figure 4, 0.4 km s -1 kpc -1 , up by a factor of 9 to ∼ 4 km s -1 kpc -1 , approaching what de Blok and Walter observe. The point of this crude set of estimates is that late collapse might happen around NGC6822 as well as other polar ring galaxies.</text>
<text><location><page_21><loc_12><loc_29><loc_88><loc_73></location>Our numerical method of finding the orbits of LG actors is not efficient at arriving at solutions in which actors have orbited each other several times. This is not a problem for the motion of LMC around MW, because the present conditions seem to be well enough known to exclude multiple orbit passages (Besla, Kallivayalil, Hernquist, et al. 2007). It does not seem to be a problem for NGC 6822, either, because its present slow motion away from MW with standard estimates of the MW mass make it likely that NGC 6822 has not completed more than one orbit. The limitation of the numerical method is more serious for the smaller galaxies now near M 31. In the Local Group solutions used here the pair of dwarf spheroidal galaxies NGC 185 and NGC 147 has mass ∼ 10 10 M glyph[circledot] , above van den Bergh's (1998) estimate of what is required if these galaxies are a bound system, and indeed our solutions show the two galaxies completing about one orbit of relative motion. A solution with completion of several orbits, perhaps leaving trails of stars, could have been missed, however. The dynamical solutions show M 31 and M 33 approaching each other for the first time after separating at high redshift (Fig. [6]). That does not agree with the idea that the optical stream between M 31 and M 33 is a remnant of a close passage of the two galaxies. Perhaps we have not found the right orbit for M 33. Perhaps the stream between M 33 and M31 formed by close passage of one of the other galaxies near M 31, as discussed by Sadoun, Mohayaee, & Colin (2013). Or perhaps this is a primeval stream that happened to have been loaded with stars. We must add that the primeval stream picture requires the special postulate that stars are pulled out of the young galaxies in and around M 31, while H I would have been pulled out the Magellanic Clouds to make the Magellanic Stream. But Figure 7 does show that primeval streams can run across M 31, which may merit closer consideration.</text>
<text><location><page_21><loc_12><loc_19><loc_88><loc_26></location>We have benefited from discussions with Ed Shaya and support from the NASA Astrophysics Data Analysis Program award NNX12AE70G and from a series of awards from the Space Telescope Science Institute, most recently associated with programs AR-11285, GO-11584, and GO-12546.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "The Magellanic Stream might have grown out of tidal interactions at high redshift, when the young galaxies were close together, rather than from later interactions among the Magellanic Clouds and Milky Way. This is illustrated in solutions for the orbits of Local Group galaxies under the cosmological condition of growing peculiar velocities at high redshift. Massless test particles initially near and moving with the Large Magellanic Cloud in these solutions end up with distributions in angular position and redshift similar to the Magellanic Stream, though with the usual overly prominent leading component that the Milky Way corona might have suppressed. Another possible example of the effect of conditions at high redshift is a model primeval stream around the Local Group galaxy NGC6822. Depending on the solution for Local Group dynamics this primeval stream can end up with position angle similar to the H I around this galaxy, and a redshift gradient in the observed direction. The gradient is much smaller than observed, but might have been increased by dissipative contraction. Presented also is an even more speculative illustration of the possible effect of initial conditions, primeval stellar streams around M 31. Key words: galaxies: NGC 6822, Large Magellanic Cloud - galaxies: kinematics and dynamics - galaxies: interactions - Local Group - large-scale structure of universe",
"pages": [
1
]
},
{
"title": "A Primeval Magellanic Stream and Others",
"content": "P. J. E. Peebles Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 R. Brent Tully Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, H I 96822",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The examples presented here of how intergalactic tidal streams could have been triggered by interactions among the young galaxies at high redshift, when they were all close together, are based on solutions to a dynamical model for the Local Group (LG) under the condition of growing peculiar velocities at high redshift, in analogy to the growing mode of departure from a homogeneous expanding universe in perturbation theory. In these solutions a reasonable approximation to the Magellanic Stream (MS) grew largely from the interaction of the young Large Magellanic Cloud (LMC) with its nearest massive neighbor at high redshift, the Milky Way (MW). That is, we arrive back at the picture introduced by Fujimoto & Sofue (1976) and Lin & Lynden-Bell (1977), but applied at high redshift under cosmological initial conditions. The MS certainly is expected to have been affected by subsequent tidal interactions, perhaps between the LMC and Small Magellanic Cloud (SMC), as noted by Fujimoto & Sofue (1976) and Lin & Lynden-Bell (1977), and by interaction with the MW mass and corona (Meurer, Bicknell, & Gingold 1985; Moore & Davis 1994). Analyses of these effects (Gardiner & Noguchi 1996; Mastropietro et al. 2005; Connors, Kawata, & Gibson 2006; Diaz & Bekki 2011, 2012; Besla, Kallivayalil, Hernquist, et al. 2010, 2012; and references therein) show that tidal and hydrodynamical interactions at modest redshifts can produce plausible approximations to MS without reference to conditions at high redshift. Exploration of a primeval origin nevertheless is called for. Gravitational interactions among the young galaxies certainly are real, as exemplified by the cosmic web (Bond, Kofman, & Pogosyan 1996). Exploration of the consequences in observations of the galaxies near us requires a prediction of how the young galaxies were positioned at high redshift. We seem to have that now for the LMC in a dynamical LG model constrained by initial conditions from cosmology and by the now considerable number of measurements of nearby galaxy redshifts, distances and proper motions (Peebles 2010; Peebles & Tully 2013, PT). This invites exploration of the effect of primeval tidal interactions on the distribution of matter around the young LMC. The result is a credible first approximation to MS. The LG dynamical model and methods of its solution are outlined in Section 2 and discussed in more detail in PT and references therein. The dynamical actors and observational constraints are the same as in PT except that the actors that are meant to represent the effect of external mass are allowed the freedom to adjust angular positions as well as distances to aid the fit to the LG parameter constraints. The primeval streams presented in Section 3 show purely gravitational motion of massless test particles, in the tradition of Toomre & Toomre (1972). This simplifies the computation of streams in this preliminary exploration that might motivate more complete analyses that take account of hydrodynamics and self-gravitation. Section 3.1 shows the evolution of the model for a primeval Magellanic Stream. It produces a reasonable-looking fit to the MS H I angular and redshift distributions without special parameter adjustments. This result motivates the exploration in Section 3.2 of primeval streams around NGC 6822. The observed H I envelope around this LG dwarf almost certainly is gravitationally bound to the galaxy, a very different situation from MS. The results suggest that the H I envelope might have grown by dissipative contraction of a primeval H I stream, though substantiating that idea would require a considerable parameter search. A crude estimate of the situation is offered in Section 4. Section 3.3 shows an even more speculative example, the development of streams around M 31 from its interactions with M33, NGC185, and NGC147 at high redshift. In the LG model solutions none of these galaxies passed close to M 31 at modest redshifts, but streams form. This certainly cannot make the case for a primeval origin of streams around M 31, because there are many other neighbors that could have produced streams at more modest redshifts, but it offers the possibility of a primeval component. We summarize our assessment of the results from the primeval stream models in Section 4.",
"pages": [
1,
2,
3
]
},
{
"title": "2. Dynamical Model and Solutions",
"content": "The starting assumption for the dynamical LG model is that the mass now concentrated around a galaxy was at high redshift in a patch whose motion may be traced by the position of its effective center of mass. This of course allows the galaxy to grow by accretion, provided it is accretion within the patch traced by the effective center. The initial condition is that the peculiar velocities of the mass patches are small and growing at high redshift. The condition that the galaxies end up where they are observed - or else how they are observed to be moving - presents mixed boundary conditions that are fitted by relaxation of the orbits to a stationary point of the action (in the NAM method introduced in Peebles 1989 and made more efficient in Peebles, Tully & Shaya 2011). In NAM solutions the equivalent of the decaying mode in linear perturbation theory is suppressed but not eliminated, as illustrated in Figure 1 in PT. It shows that, for the model parameters used in PT and here, peculiar velocities in the solutions are growing at redshift z < ∼ 20 in a reasonable approximation to the wanted growing mode, while earlier than that the decaying mode that inevitably appears in a numerical solution dominates and diverges as a ( t ) → 0. The advantage over a numerical integration back in time from given present positions and velocities is that NAM shifts domination of the decaying mode to high redshift where it seems likely to be harmless. The NAM solutions are based on the ΛCDM cosmology with Hubble and matter density parameters where Ω m represents the sum of masses in baryons and dark matter, the mass in radiation is neglected, space sections are flat, and Einstein's Λ is constant. The numerical solutions trace back in time by expansion factor 1 + z = 10 (to redshift z = 9) in 500 time steps uniformly spaced in the expansion parameter a(t). Numerical accuracy is checked by numerical integration forward in time in 5000 steps uniformly spaced in a(t) from positions and velocities at 1 + z = 10 from the action solution. The present positions and velocities from this forward integration generally agree with the action solution to better than 0.1 kpc and 0.3 km s -1 , apart from some solutions for Leo 1, whose close passage of MW produces differences as large as 3 kpc and 5 km s -1 . Solutions are found starting from random trial orbits with random initial assignments of distances, redshifts and masses within the nominal uncertainties, the orbits relaxed to a stationary point of the action, and the parameters then iteratively adjusted and relaxed to a stationary point to improve the fit to the measurements of LG redshifts, distances, luminosities, and peculiar velocities. The mixed boundary conditions allow many discretely different solutions; we choose the more plausible ones by comparison to the data. More details are in PT. Table 1 names the LG galaxies in the dynamical model. The adopted LG parameter values and their measured or estimated uncertainties are entered under the headers 'catalog' (or 'cat'). Entered under the headers 'solution' are the parameter values in three numerical solutions to the dynamical model, ordered by the goodness of fit to the data. The catalog distances and their uncertainties, redshifts, and luminosities are from the Local Universe (LU) catalog maintained and provided on-line by Tully. 1 The adopted nominal uncertainty in each redshift, 10 km s -1 , is meant to allow for possible motion of the galaxy of stars relative to its dark matter halo. The nominal catalog masses (baryonic plus dark matter) are computed from the K-band luminosities using mass-to-light ratio M/L K = 50 M glyph[circledot] /L glyph[circledot] , meaning the nominal value of M/L K in Table 1 is 50. The nominal uncertainties in the LG galaxy masses are placed on a logarithmic scale, with a factor of 1.5 at one standard deviation. The exception is MW, whose nominal mass ratio to M 31 is unity with a factor of 1.1 at one standard deviation (PT eqs. [5] and [6]). The nominal rms galaxy peculiar velocity at 1 + z = 10 is taken to be v i = 50 km s -1 for LG and external actors. This is roughly what might be expected from the growth to rms peculiar velocities several times that at the present epoch. The velocity of the Sun relative to the local standard of rest is from Schonrich, Binney & Dehnen (2010), with no allowance for uncertainty in this relatively small term. The circular velocity of the local standard of rest has catalog value v c = 230 ± 10 km s -1 . The mass distribution in each actor is rigid and spherical with density run ρ ∝ r -2 cut off at the radius that produces the model mass for given v c . Assigned circular velocities without uncertainties are v c = 250 km s -1 in M31 and v c = 100 km s -1 in all the other actors except MW.",
"pages": [
3,
4
]
},
{
"title": "T able 1: Lo cal Group Mo del Distances, Redshifts, Masses, and Initial V elo cities",
"content": "units: Mp c, km s -1 , M glyph[circledot] /L glyph[circledot]",
"pages": [
5
]
},
{
"title": "T able 2: Prop er Motions",
"content": "1 - y sec arc milli unit:",
"pages": [
6
]
},
{
"title": "T able 3: External Actors",
"content": "units: Mp c, km s -1 , 10 11 M glyph[circledot] Table 2 lists proper motions, where µ α is the motion in the direction of increasing right ascension and µ δ is the motion in the direction of increasing declination. The measurement and uncertainty for M31 is from Sohn, Anderson & van der Marel (2012), for LMC from Kallivayalil et al. (2013), for M33 from Brunthaler et al. (2005), for IC 10 from Brunthaler et al. (2007), and for LeoI from Sohn, Besla, van der Marel, et al. (2012). The external actors named in Table 3 are meant to give a phenomenological description of the effect of the external mass distribution on LG by allowing their present positions and masses to float to aid the model fit to the catalog LG parameters. In a departure from PT, the angular positions as well as distances of these actors are allowed to float. The nominal angular positions (columns 2 and 3 in Table 3) are luminosity-weighted means for the galaxies concentrated around the Sculptor group, the Maffei-IC 342 system, the M 81 group, and the Centaurus-M94 system. The positions are given in supergalactic coordinates, because the nearby galaxies outside LG are concentrated near the supergalactic plane. The distances δD between the three-dimensional positions in the catalog and the model solutions have nominal allowed rms value 0.5 Mpc. The nominal redshifts are luminosity-weighted means, with adopted uncertainties 50 km s -1 . The catalog masses are computed from the sums of Kband luminosities with M/L K = 50 M glyph[circledot] /L glyph[circledot] , and the mass uncertainties are on a logarithmic scale with one standard deviation at a factor of two difference between catalog and model solution. The measured or adopted uncertainties in the catalog parameters are treated as standard deviations in a χ 2 sum of squares of differences between model and catalog values divided by standard deviations. There are 69 LG parameters: 14 distances, 14 redshifts, 15 masses, 10 components of proper motion, the MW circular velocity, and 15 primeval velocities. (The last are more properly counted as 45 primeval velocity components, each with a Gaussian velocity distribution, less three components because the center of mass is at rest, but this is too fine for the present purpose). There are 24 external actor parameters: the redshift, mass, primeval velocity, and three components of present position for each of the four actors. Solutions 1 through 3 have χ 2 = 92, 98, and 114, close to the total of 93 parameters in χ 2 . This is not very meaningful, however, for two reasons. First, many of the nominal standard deviations are at best only informed guesses. Second, the multiple solutions allowed by the mixed boundary conditions allow multiple choices among which we choose those with the smallest χ 2 . That is, if model, measurements and standard deviations were accurate enough for a meaningful value of χ 2 we would expect it to be less than 93. The sums over LG parameters alone are χ 2 LG = 70, 76, and 89. These are not much larger than the 69 LG parameters, but again one would have expected smaller because the solutions were chosen for their fit to the catalog parameters, and because the external parameters were adjusted to reduce χ 2 LG . If it were supposed that the external actor parameters are in effect free, because their constraints are quite loose, one might expect χ 2 LG = 69 -16 = 53 (discounting only the masses and present positions, which most matter for LG orbits ), which would put the reduced χ 2 LG values at about 1.5, well above statistical expectation. The initial peculiar velocities of the external actors are less than about 25 km s -1 , which seems acceptably small. The numerical solutions put the mass of the M81 actor at or above its nominal value and the other masses below nominal, and M81 is placed at heliocentric distance ∼ 2 Mpc, well short of the LU distance to the galaxy M 81, 3 . 65 ± 0 . 18 Mpc, while the other three actors are placed close to their catalog positions, at median δD ∼ 0 . 3 Mpc. Parameter adjustments are allowed to move the external actors away from the plane, but the model positions now and at 1 + z = 10 are close to the plane, supergalactic latitude SGB close to zero (columns 4 to 9 in Table 3). A possibly significant exception is that the solutions prefer the M81 actor below the plane. This is in the direction that would help compensate for the striking scarcity of galaxies - and likely mass - in the Local Void immediately above the plane. We hope to investigate this and other aspects of the influence on LG of the external mass distribution in due course. The parameters in the numerical solutions in Tables 1 to 3 that differ from catalog by more than two nominal standard deviations are entered in italics. (There are no 3σ differences in LG parameters.) The model prefers a long LMC distance, at 1 . 8 σ in Solution 1, 2 . 5 σ in Solution 3. Since measurements of this distance have been thoroughly examined we expect the discrepancy indicates a systematic error in the model, perhaps in the simple approximation to the mass distribution in MW. The other 2σ discrepancy in Table 1 is the short distance to NGC 185 in Solution 3. None of the model solutions fit both catalog proper motion components of IC 10 to better than two standard deviations, and only Solution 1 fits the proper motion of M 33, but it fails the proper motion of Leo I at two standard deviations. Here again the problem certainly may be with the model, but since these proper motion measurements have not been so thoroughly reconsidered one may imagine some of the errors assigned to these difficult measurements are underestimates. In Solutions 1 to 3 respectively the MW circular velocity is v c = 229, 238, and 246 km s -1 . The increase with decreasing quality of fit to the measurements may be accidental. The preference for a value larger than the standard 220 km s -1 is in the direction of but smaller than that found by Reid, Menten, Zheng, Brunthaler, et al. (2009). The preference for greater mass in MW than M31 is present also in the larger number of solutions in PT with generally poorer fits to the constraints. In models 1 to 3 the MW masses in units of 10 11 M glyph[circledot] are 15.2, 16.3, and 18.3, respectively, and the M 31 masses are 12.9, 13.6 and 15.8. If the relative motion of MW and M31 were not affected by other actors the sum of the derived masses would decrease with increasing v c because the derived galactocentric redshift of M 31 de- creases with increasing v c . The trend the other way in the three solutions is not inconsistent with this argument because the other actors significantly affect the relative motion of MW and M31, curving the orbit. In future work aimed at improving the LG dynamical model we expect to use more realistic descriptions of the mass distributions within MW and M 31. It may help also to let the characteristic radius of the mass distribution within M 31 be an adjustable parameter (in analogy to the adjustment of v c in a truncated limiting isothermal sphere). And, perhaps most important, the treatment of the effect of the external mass distribution on LG should more closely refer to the observed external galaxy distribution and peculiar motions. The numerical solutions are not formally statistically consistent with the full catalog of parameters within their uncertainties. That is not surprising, because the dynamical model is crude and the catalog likely to contain errors. It is encouraging that the solutions match a considerable number and variety of constraints with relatively few discrepancies beyond two nominal standard deviations and, in LG, none beyond 3 σ . This degree of fit to the constraints argues that we have a reasonably secure basis for exploration of the effect of initial conditions on the development of streams within the Local Group.",
"pages": [
6,
7,
8,
9
]
},
{
"title": "3. Primeval Streams",
"content": "Figure 1 shows model solutions for the motions of galaxies relative to MW. The righthand coordinate system is galactic, with the z -axis at b = 90 · and the x -axis at b = l = 0. The lengths are physical. Positions are plotted relative to MW at the plus sign. We consider possible examples of remnant primeval streams around LMC, plotted in black, and NGC6822, plotted in red. The green curves show the relative position of the other massive LG actor, M 31. Present positions are at the crosses, and the other ends of the orbits show the young galaxies moving apart at redshift z = 9. The solid lines show Solution 1 in Tables 1 to 3, the dashed lines Solution 2, and dotted, 3. These solutions are at different stationary points of the action and local minima of χ 2 . The three orbits of LMC relative to MW are quite similar, and they are similar too in the greater number of solutions in PT. The apparently well-determined initial situation of LMC invites our exploration of the effect of the initial conditions on a cloud of test particles that might approximate the behavior of an HI envelope. The motion of NGC 6822 relative to MW is similar in Solutions 1 and 2, quite different in 3. This illustrates the multiple solutions allowed by the mixed boundary conditions. For this galaxy an assessment of the situation has to guide selection of the more likely solution. The test particles in a model stream around a chosen LG actor move in the given gravitational field of the solution for the actors with mass, making it easy to accumulate a dense sample of test particle paths. The test particles are placed at z = 9 uniformly at random within the gravitational radius x g of the actor, where The comoving length x g is normalized to the physical radius at the present epoch. The galaxy mass, M , is the same as the mass within x g in a homogeneous mass distribution at the cosmic mean density. This means that a test particle closer than x g tends to be gravitationally attracted to the actor, in comoving coordinates, and at x > x g a test particle tends to be pulled away. The radii x g also are the characteristic separations of the actors at high redshift, where the peculiar accelerations of the actors are bounded by the condition that the orbits approximate the expansion of a near-homogeneous mass distribution. In a model stream the test particles are initially at rest in comoving coordinates relative to the chosen actor, meaning the particles initially are streaming away from the actor with the general expansion of the universe. The condition for capture of a test particle by the chosen actor varies with direction as well as distance relative to x g , of course. Trials that take account of this by rejecting test particles with initial relative peculiar accelerations directed away from the chosen actor, and trials with initial velocities set to what is derived from the initial peculiar gravitational acceleration in linear perturbation theory, do not to produce very different streams, so for simplicity these refinements are not used in the results presented here.",
"pages": [
9,
10,
11
]
},
{
"title": "3.1. The Magellanic Stream",
"content": "Figure 2 shows present positions and redshifts of the test particles that are uniformly distributed within r g around LMC at z = 9 in Solution 1. The plots derived from the other two solutions look quite similar (and the gravitational radii defined in eq. [2] are similar; the physical values at 1 + z = 10 are r g = 82, 90, and 87 kpc in Solutions 1 to 3). The sharp cutoff in the initial distribution of test particles can produce features in the present distribution of the particles plotted in black that are at initial physical distance 0 . 75 r g to r g from LMC, but the effect is not prominent here. There is not much indication of orbit mixing; the appearance rather is that the initial distribution has been smeared by a smooth velocity field. Figure 2 is plotted in the MS coordinates defined by Nidever, Majewski, & Burton (2008), where LMC is at the origin, the stream is centered near B MS = 0, and the trailing stream is at L MS < 0. Figure 2 can be compared to Figure 8 in Nidever et al. (2010), which shows in these coordinates the measured H I angular distribution and the heliocentric redshift as a function of L MS . The orientation of the model stream and the variation of the heliocentric redshift with L MS are close to what is observed, though the model lacks the fine structure in MS, including the SMC, and the leading stream is much too prominent. Figure 2 also is similar to other MS models, including Figures 6 and 7 in Gardiner & Noguchi (1996); Figures 9 and 12 in Mastropietro et al. (2005); Figures 7 and 9 in Connors, Kawata, & Gibson (2006); Figure 7 in Diaz & Bekki (2012); and Figures 7 and 9 in Besla et al. (2012). In Figure 2 the distance to MS is nearly constant from -50 · < ∼ L MS < ∼ 50 · at about 60 kpc. The variation of distance with L MS is similar in Connors, Kawata, & Gibson, and in Diaz & Bekki, while Besla et al. show a more rapid decrease of distance with increasing L MS . An observational check does not seem likely, however. The notable overall similarity of results in a variety of models is discussed in Section 4. Figure 3 illustrates the evolution of the cloud of test particles around LMC in Solution 1. Solutions 2 and 3 look similar. The effect of the sharp cutoff in the distribution of black particles is more apparent here than in Figure 2. At z = 9 the physical distance between MW and LMC is 250 kpc, comparable to the MW gravitational radius 220 kpc (eq. [2]) at z = 9, and M31 is about 500 kpc from LMC. At z = 3, in the left-hand panel, MW is at the blue circle near the bottom edge. Its distance from LMC has about doubled, growing slightly less than the factor 2.5 general expansion. M 31 is to the left, outside the boundary of the figure. The physical width of the distribution of the initially innermost particles plotted in yellow is about the same at z = 3 as at z = 9, while the outer envelope marked by the black particles has expanded by about as much as the general expansion. The elongated distribution of test particles at z = 3 points to MW, even among the initially innermost particles particles shown in yellow. In the central panel in Figure 3, at z = 1, MW is close to its maximum separation from LMC, 700 kpc. It is below the panel and positioned about in line with the prominent red stream and the long axis of the slightly eccentric distribution of yellow particles. MW reappears in the right-hand panel, at z = 0 . 3. By this time some of the test particles are concentrated around MW. The band of black particles in the upper left part of the righthand panel is suggestive of folding in singe-particle phase space. The yellow particles that initially were less than 41 kpc from LMC are still concentrated around LMC at about this radius at z = 0 . 3. They end up smeared into the yellow stream in Figure 2. The point of Figure 3 is that the cloud of test particles around LMC carries some memory of its interaction with MW when they were close, at high redshift. This is seen in features in the three panels of Figure 3 and, at z = 0, in the stream in Figure 2.",
"pages": [
11,
12,
13
]
},
{
"title": "3.2. A Stream around NGC6822",
"content": "The atomic hydrogen around the LG galaxy NGC 6822 extends to angular radius ∼ 30 ' , projected separation r ∼ 4 kpc. The HI redshifts of the outer parts differ from the center by v ∼ ± 50 km s -1 . These quantities define a characteristic mass, v 2 r/G ∼ 3 × 10 9 M glyph[circledot] . The catalog mass of this galaxy, 6 × 10 9 M glyph[circledot] , is not well tested by the dynamical model because it is too small to have much effect on the other actors, but its similarity to v 2 r/G does suggest that the H I could be gravitationally bound to the galaxy. And, if the H I were not bound, the relative velocity of 50 km s -1 would soon carry the hydrogen far beyond its projected separation from the galaxy, unless the relative motion of galaxy and gas were directed almost exactly along the line of sight, which seems unlikely. Thus we ought to study the formation of a gravitationally bound H I cloud. This cannot be simulated by the simple gravitational motions of test particles, but we can consider initial conditions for dissipative contraction that might produce the H I envelope of NGC 6822. Figure 4 shows angular distributions and radial velocities of the primeval streams of test particles initially uniformly distributed around NGC 6822 in the three solutions. The particles initially less than r g / 2 = 17 kpc from NGC 6822 are plotted in yellow, those initially at 17 to 25 kpc in red, and those at 25 to 34 kpc in black. Here again the cutoff at r g in the initial test particle distribution produces the sharp edges in the figure. The range of angles plotted in Figure 4 is large enough that the heliocentric motion of the galaxy causes a significant variation of the line-of-sight velocity across the figure. This effect is removed by plotting an effective redshift cz defined as the component of the heliocentric velocity of each test particle along the heliocentric direction to the center of NGC 6822. The orbital histories of NGC 6822 in Solutions 1 and 2 are similar (Fig. 1), as are the present test particle distributions in Figure 4, though there are systematic differences. The stream position angles in these two solutions are about 140 · (measured from the direction of increasing declination toward the direction of increasing right ascension), similar to the orientation of the H I stream around this galaxy (Roberts 1972; de Blok & Walter 2000). In these two model streams the redshift increases with decreasing declination and increasing right ascension, in the direction of the observations, though the gradient is much smaller than observed. This is in line with the idea that the primeval stream may model the precursor to dissipative contraction. Solution 3 has a different history (Fig. 1) and different present distributions of positions and redshifts that seem less promising. The lower panel in Figure 5 shows the evolution in Solution 1 of the numbers of particles within physical distances 15 and 30 kpc from NGC 6822, plotted as the ratio to the number initially within 30 kpc. The concentrations initially decrease because the cloud is expanding with the general expansion of the universe. The first minima, at z ∼ 4 for r = 15 kpc and z ∼ 3 for r = 30 kpc, are artificially deep because the initial velocities relative to NGC 6822 are artificially radial. At z < 1 the concentration within 15 kpc is nearly constant at about the value at z = 9, and the concentration within 30 kpc is about one fifth its initial value. The black curves in the upper panel of Figure 5 show the evolution of the components (in galactic coordinates) of the mean (specific) angular momentum per particle relative to the position and motion of NGC 6822 for the particles that are at physical distance r < 30 kpc from NGC6822. The mean angular momentum at r < 15 kpc is smaller but the components evolve in a similar way. The angular momentum evolves in part because particles are streaming past the 30 kpc limiting distance, and in part because of the torques from other actors. These effects are separated by identifying the particles that at (1 + z ) -1 = 0 . 75 are within 30 kpc from NGC6822. The components of mean angular momentum of this fixed set of particles are plotted at lower redshift as the red curves in Figure 5. The red and black curves differ because of the motions of particles through r = 30 kpc. The similarities show that gravitational torques substantially affect the mean angular momentum per particle near NGC6822 as it lingers near its maximum distance from MW and M 31 approaches. The angular momentum may be compared to the maximum to be expected from the tidal torque by MW integrated over a Hubble time, L ∼ GMr 2 / ( H o R 3 ) ∼ 500 kpc km s -1 for MW mass M ∼ 10 12 M glyph[circledot] , NGC6822 distance R ∼ 500 kpc, and moment arm r ∼ 30 kpc. This is an order of magnitude larger than the mean angular momentum within r = 30 kpc at z = 0 in the model, L ∼ 50 kpc km s -1 . At z = 0 and r = 30 kpc the angular momentum vector has position angle ∼ 220 · and inclination i ∼ 55 · (where i is the angle between the angular momentum vector and the direction from NGC 6822 to MW, meaning the angular momentum is tilted from the plane of the sky toward us by about 35 · ). Over the range of limiting radius r = 10 kpc to 100 kpc the direction of the model angular momentum vector does not change much, but the magnitude increases with increasing r from about 20 kpc km s -1 at r = 10 kpc to about 200 kpc km s -1 at r = 100 kpc. The observable component of angular momentum per unit mass in the H I around NGC 6822 is about 100 kpc km s -1 . The position angle is similar to the model stream, perhaps a significant coincidence. It will be noted, however, that the model stream acquired its angular momentum at redshift z < ∼ 0 . 3, so it could be a progenitor of the HI cloud around NGC6822 only if the accretion of this H I were a recent development. This special condition is discussed in Section 4.",
"pages": [
13,
14,
15,
16,
17
]
},
{
"title": "3.3. Streams around M31",
"content": "The streams around M 31 are dominated by stars (Richardson, Irwin, McConnachie, et al. 2011, and references therein), a different situation from the loose stream of H I around the Magellanic Clouds or the gravitationally bound H I around NGC 6822. The prominent optical stream between M 31 and M 33 invites the idea that these two galaxies suffered a close passage, which did not happen in the LG solutions used here. This is illustrated in Figure 6, which shows in galactic coordinates the paths of galaxies relative to the position of M31 at the plus sign. The green curves show MW approaching M 31 from the right, in the mirror image of the approach of M 31 to MW in Figure 1. The orbits of NGC 185 (black) and NGC147 (red) are similar in Solutions 1 and 2 and rather different in Solution (3). The situation may be compared to NGC 6822 in Figure 1. In all three solutions M 33 (blue) has been well away from M 31 and MW. But we can offer an illustration of how M 33 and M 31 might be connected by a primeval stream. central panel, and NGC 147 in the right-hand panel. The test particles plotted as yellow were initially closer to the galaxy than r g / 2, and red initially at r g / 2 to 3 r g / 4, where r g ∼ 100 kpc in M 33 and ∼ 40 kpc in NGC 185 and NGC 147. Test particles initially further out add more diffuse streaks. To reduce clutter we refrain from including IC 10 and the stars that might have been pulled out of M 31. If stars formed around the young M 33 and were drawn away by tidal fields of MW, which was 230 kpc away at z = 9, and M 31, which was 250 kpc from M33, then the left-hand panel in Figure 7 suggests the stars might end up in an observable stream that passes across M 31, at the central square in the figure. A primeval stream of stars drawn from the outskirts of NGC 185 could appear somewhat tighter, and a stream from NGC147 could be tighter still, though both avoid M 33.",
"pages": [
17,
18,
19
]
},
{
"title": "4. Discussion",
"content": "The plausible approximation to MS in Figure 2 is based on a dynamical model that fits the considerable number of measurements in Tables 1 and 2 about as well as could be expected. This dynamical model offers a reasonably unambiguous prediction of the positions of the galaxies near LMC at high redshift. The primeval stream model assumes the young LMChad an envelope of H I or cool plasma whose response to the presence of the neighboring galaxies at high redshift may be approximated by a cloud of massless test particles initially moving with LMC. It also assumes memory of the conditions at high redshift is preserved in the present state of the H I. Memory in a cloud of test particles is illustrated in Figure 3. Besla et al. (2010) summarize arguments that MS is 'a young feature (1 - 2 Gyr)'. In the primeval stream model LMC would have entered an MW corona that extends to 300 kpc at redshift z ∼ 0 . 1, or about 1 Gyr ago, which may be recent enough that the trailing component largely survived moving through the plasma while losing much of the leading component. Prior to that LMC would have been well separated from large galaxies and its proto-MS might have survived in the same manner as the H I at similar surface densities around other isolated gas-rich dwarfs. These are not many assumptions, and they are applied in a straightforward way, which lends support for the result. A test by hydrodynamic simulation could provide a stronger argument, but the evidence we have now is that gravitational interactions among the young galaxies can have produced MS. We must consider that plausible MS models have been obtained without reference to initial conditions, and from a variety of ways to model the orbits and interactions among LMC, SMC and MW (Gardiner & Noguchi 1996; Mastropietro et al. 2005; Connors, Kawata, & Gibson 2006; Diaz & Bekki 2011, 2012; Besla, Kallivayalil, Hernquist, et al. 2010, 2012). Though these approaches generally require close attention to parameter choices, it appears that MS has properties of an attractor, capable at arriving at a good approximation to its present state - and, we must expect, of being destroyed - under a variety of interactions along the way. The case that MS originated as a primeval stream rests on the demonstration that such a stream would have formed if cool baryons in the young galaxy were in a position to be tidally disturbed and then not seriously disturbed thereafter. We must be cautious, however, for MS seems to have a generic tendency to end up looking like Figure 2. In contrast to MS, the case for formation of the H I envelope of NGC 6822 out of a primeval stream requires a very special situation. However, the situation may be indicated by the demonstration by Demers, Battinelli, & Kunkel (2006) that NGC 6822 is a polar ring galaxy. The stellar halo traced by RGB stars extends about as far from the galaxy as the H I envelope, but the long axis of the stellar distribution has position angle 60 · , while the HI has position angle is 130 · (de Blok & Walter 2000). The redshift gradients of the halo stars and the H I envelope both point along the long axis of their angular distribution, and both gradients are close to constant at 15 km s -1 kpc -1 . The contributions to the angular momenta of stars and H I by the observed redshift gradients have position angles differing by ∼ 70 · . This could signify a strong departure from axial symmetry, as in bars. (Hodge 1977 notes that the stars in the inner ∼ 0 . 5 kpc, with PA ∼ 10 · , may be a bar.) The alternative is that the angular momenta of stellar halo and H I envelope have quite different directions. Similar tilts of the long axis of the stellar distribution from the H I redshift gradient are observed in isolated dwarfs (Stanonik, Platen, Arag'on-Calvo, van Gorkom, et al. 2009; Kreckel, Peebles, van Gorkom, van de Weygaert, & van der Hulst 2011). This might indicate that the H I envelopes dissipatively settled onto these galaxies without adding many stars. The orientation and the direction of the redshift gradient in the primeval stream around NGC6822 in Solutions 1 and 2 (Fig. [4]) agree with the H I envelope of this galaxy. The model redshift gradient is much smaller than observed, but that could be because of the dissipative contraction of the H I. This picture requires that the H I settled after redshift z ∼ 0 . 3, when the primeval stream acquired its angular momentum (Fig. [5]), a very special condition that may be crudely modeled as follows. Suppose that at z ∼ 0 . 3 the stars in NGC6822 were centered on a sheet of diffuse baryons with number density n b ∼ 10 -4 cm -3 and thickness h ∼ 30 kpc, or angular width ∼ 3 · at the distance of this galaxy, which is on the scale of what is plotted in Figure 4. The characteristic baryon surface density is ∼ n b h ∼ 10 19 cm -2 , or baryon surface mass density Σ b ∼ 10 5 M glyph[circledot] kpc -2 . With total surface density Σ m ∼ 6Σ b , to take account of dark matter, pressure support requires plasma temperature kT ∼ 2 πG Σ m m p h ∼ 3 eV. That makes the plasma cooling time ( τ ∼ 10 11 . 4 √ T/n b in cgs units, ignoring line emission) about 10 10 y, roughly what is wanted for late accretion of an HI envelope around NGC6822. If an h by h (30 kpc by 30 kpc) piece of the slab collapsed by a factor of three in each direction it would gather baryon mass Σ b h 2 ∼ 10 8 M glyph[circledot] in a region of width ∼ 10 kpc ∼ 1 · at NGC6822, roughly what is detected in 21-cm radiation (de Blok & Walter 2000). Conservation of angular momentum would bring the redshift gradient in Figure 4, 0.4 km s -1 kpc -1 , up by a factor of 9 to ∼ 4 km s -1 kpc -1 , approaching what de Blok and Walter observe. The point of this crude set of estimates is that late collapse might happen around NGC6822 as well as other polar ring galaxies. Our numerical method of finding the orbits of LG actors is not efficient at arriving at solutions in which actors have orbited each other several times. This is not a problem for the motion of LMC around MW, because the present conditions seem to be well enough known to exclude multiple orbit passages (Besla, Kallivayalil, Hernquist, et al. 2007). It does not seem to be a problem for NGC 6822, either, because its present slow motion away from MW with standard estimates of the MW mass make it likely that NGC 6822 has not completed more than one orbit. The limitation of the numerical method is more serious for the smaller galaxies now near M 31. In the Local Group solutions used here the pair of dwarf spheroidal galaxies NGC 185 and NGC 147 has mass ∼ 10 10 M glyph[circledot] , above van den Bergh's (1998) estimate of what is required if these galaxies are a bound system, and indeed our solutions show the two galaxies completing about one orbit of relative motion. A solution with completion of several orbits, perhaps leaving trails of stars, could have been missed, however. The dynamical solutions show M 31 and M 33 approaching each other for the first time after separating at high redshift (Fig. [6]). That does not agree with the idea that the optical stream between M 31 and M 33 is a remnant of a close passage of the two galaxies. Perhaps we have not found the right orbit for M 33. Perhaps the stream between M 33 and M31 formed by close passage of one of the other galaxies near M 31, as discussed by Sadoun, Mohayaee, & Colin (2013). Or perhaps this is a primeval stream that happened to have been loaded with stars. We must add that the primeval stream picture requires the special postulate that stars are pulled out of the young galaxies in and around M 31, while H I would have been pulled out the Magellanic Clouds to make the Magellanic Stream. But Figure 7 does show that primeval streams can run across M 31, which may merit closer consideration. We have benefited from discussions with Ed Shaya and support from the NASA Astrophysics Data Analysis Program award NNX12AE70G and from a series of awards from the Space Telescope Science Institute, most recently associated with programs AR-11285, GO-11584, and GO-12546.",
"pages": [
19,
20,
21
]
},
{
"title": "REFERENCES",
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"pages": [
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}
]
2013ApJ...778..152S
https://arxiv.org/pdf/1306.5748.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_86><loc_85><loc_87></location>COSMOLOGICAL SIMULATIONS OF ISOTROPIC CONDUCTION IN GALAXY CLUSTERS</section_header_level_1>
<text><location><page_1><loc_25><loc_84><loc_76><loc_85></location>Britton Smith, Brian W. O'Shea 1 , G. Mark Voit, David Ventimiglia</text>
<text><location><page_1><loc_23><loc_83><loc_78><loc_84></location>Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824</text>
<text><location><page_1><loc_49><loc_81><loc_51><loc_82></location>and</text>
<text><location><page_1><loc_42><loc_80><loc_58><loc_81></location>Samuel W. Skillman 2</text>
<text><location><page_1><loc_10><loc_77><loc_92><loc_79></location>Center for Astrophysics and Space Astronomy, Department of Astrophysical & Planetary Science, University of Colorado, Boulder, CO 80309, USA</text>
<text><location><page_1><loc_41><loc_76><loc_60><loc_77></location>Draft version October 20, 2021</text>
<section_header_level_1><location><page_1><loc_45><loc_73><loc_55><loc_75></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_49><loc_86><loc_73></location>Simulations of galaxy clusters have a difficult time reproducing the radial gas-property gradients and red central galaxies observed to exist in the cores of galaxy clusters. Thermal conduction has been suggested as a mechanism that can help bring simulations of cluster cores into better alignment with observations by stabilizing the feedback processes that regulate gas cooling, but this idea has not yet been well tested with cosmological numerical simulations. Here we present cosmological simulations of ten galaxy clusters performed with five different levels of isotropic Spitzer conduction, which alters both the cores and outskirts of clusters, but not dramatically. In the cores, conduction flattens central temperature gradients, making them nearly isothermal and slightly lowering the central density but failing to prevent a cooling catastrophe there. Conduction has little effect on temperature gradients outside of cluster cores because outward conductive heat flow tends to inflate the outer parts of the intracluster medium (ICM) instead of raising its temperature. In general, conduction tends reduce temperature inhomogeneity in the ICM, but our simulations indicate that those homogenizing effects would be extremely difficult to observe in ∼ 5 keV clusters. Outside the virial radius, our conduction implementation lowers the gas densities and temperatures because it reduces the Mach numbers of accretion shocks. We conclude that despite the numerous small ways in which conduction alters the structure of galaxy clusters, none of these effects are significant enough to make the efficiency of conduction easily measurable unless its effects are more pronounced in clusters hotter than those we have simulated.</text>
<text><location><page_1><loc_14><loc_48><loc_34><loc_49></location>Subject headings: cosmology</text>
<section_header_level_1><location><page_1><loc_22><loc_44><loc_35><loc_45></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_22><loc_48><loc_43></location>Numerical simulations of galaxy clusters have not yet succeeded in producing objects with properties identical to those of observed galaxy clusters. The most serious discrepancies with observations are in the cores (e.g, Nagai et al. 2007; Borgani & Kravtsov 2011; Skory et al. 2013). Within the central ∼ 100 kpc of clusters whose central cooling time is less than a Hubble time, radiative cooling of intracluster gas in the simulations tends to be too efficient, leading to overproduction of young stars and excessive entropy levels in the core gas, as higherentropy material flows inward to replace the gas that has condensed. The resulting cluster cores in such simulations consequently have temperature profiles that decline from ∼ 10 kpc outward, in stark disagreement with the observed temperature profiles of cool-core clusters, which rise in the ∼ 10-100 kpc range.</text>
<text><location><page_1><loc_8><loc_13><loc_48><loc_22></location>Feedback from a central active galactic nucleus helps solve this problem because it slows the process of cooling. However, the core structures of clusters produced in cosmological simulations depend sensitively on the details of the implementation of feedback, the radiative cooling scheme, and numerical resolution. For example, Dubois et al. (2011) simulated the same cosmological galaxy clus-</text>
<text><location><page_1><loc_10><loc_10><loc_21><loc_11></location>smit1685@msu.edu</text>
<text><location><page_1><loc_52><loc_35><loc_92><loc_45></location>ter with two different mechanisms for feedback and two different cooling algorithms, one with metal-line cooling and one without. The core structures of the resulting clusters were quite diverse, with kinetic jet feedback and metal-free cooling producing the most realistic-looking cores. But discouragingly, allowing cooling via metal lines had the effect of supercharging AGN feedback and producing a cluster core with excessively high entropy.</text>
<text><location><page_1><loc_52><loc_14><loc_92><loc_35></location>Another oft-proposed solution to the core-structure problem in galaxy clusters is thermal conduction. It has been invoked many times to mitigate the prodigious mass flows predicted for cool-core clusters lacking a central heat source (e.g., Tucker & Rosner 1983; Bertschinger & Meiksin 1986; Narayan & Medvedev 2001) but cannot be in stable balance with cooling (e.g., Bregman & David 1988; Soker 2003; Guo et al. 2008) and does not completely compensate for radiative cooling in all cluster cores (Voigt et al. 2002; Zakamska & Narayan 2003). Nevertheless, the temperature and density profiles of many cool-core clusters are tantalizingly close to being in conductive balance, suggesting that AGN feedback is triggered only when inward thermal conduction, perhaps assisted by turbulent heat transport, fails to compensate for radiative losses from the core (e.g., Voit 2011).</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_13></location>The nebular line emission observed in many cool-core clusters also suggests that thermal conduction may be important. Ultraviolet light from young stars accounts for some of the line emission but cannot explain all of it, implying that another heat source is present (e.g., John-</text>
<text><location><page_2><loc_8><loc_76><loc_48><loc_92></location>stone et al. 1987; Voit & Donahue 1997; Werner et al. 2013). Observations of the optical and infrared line ratios are consistent with heat input by a population of suprathermal electrons, possibly entering the nebular gas from the ambient hot medium (Ferland et al. 2009). The emission-line studies are not conclusive on this point, but recent observations of ultraviolet emission from a filament in the Virgo cluster have bolstered the case for conduction (Sparks et al. 2009, 2011). The C IV 1550 ˚ A and He II 1640 ˚ A emission lines from this filament are consistent with a model in which the filament is surrounded by a conductive sheath that channels heat into the filament.</text>
<text><location><page_2><loc_8><loc_45><loc_48><loc_75></location>If conduction is indeed important in determining the structure of cluster cores and mediating the heat flow into cool gas clouds, then it ought to be included in cosmological simulations of galaxy clusters. Dolag et al. (2004) presented the first simulations of conductive intracluster media in cosmological galaxy clusters, using the Jubelgas et al. (2004) implementation of isotropic conduction in the GADGET smooth-particle hydrodynamics code. These simulations were notable in that they vividly demonstrated the inability of conduction to prevent a cooling catastrophe. Even at 1/3 of the full Spitzer rate, thermal conduction could not prevent a large cooling flow from developing in the cluster core. Conduction delayed the cooling catastrophe but did not diminish the magnitude of the eventual mass inflow. An updated set of cluster simulations using the same conduction algorithm was recently performed by Fabjan et al. (2011). Isotropic conductivity was set to 1/3 of the full Spitzer value for simulations including cooling and star formation but without AGN feedback. As with the previous simulation set, there was not much difference between the global properties of the final states of clusters simulated with and without thermal conduction.</text>
<text><location><page_2><loc_8><loc_24><loc_48><loc_45></location>Conduction is also expected to smooth out temperature inhomogeneities in hot clusters (e.g., Markevitch et al. 2003). One of the clusters simulated by Dolag et al. (2004) was quite hot, with a peak gas temperature of ∼ 12 keV, and conduction in that cluster produced a much more homogeneous ICM, with far less azimuthal temperature variation than its conduction-free counterpart. Producing simulated clusters with smallscale temperature variations similar to those of observed clusters continues to be somewhat tricky (see, for example, Ventimiglia et al. 2012). In particular, some recent simulations of sloshing, magnetized cluster cores with anisotropic conduction are suggesting that conduction needs to be somewhat suppressed along the magnetic field lines into order to explain observations of cluster cores with cold fronts (ZuHone et al. 2013)</text>
<text><location><page_2><loc_8><loc_8><loc_48><loc_24></location>Implementations of anisotropic conduction directed along magnetic field lines are also of interest because the MHD instabilities that result may have important implications for the structure of cluster cores and the generation of turbulence in cluster outskirts (e.g., Parrish et al. 2009, 2012). Ruszkowski et al. (2011) recently simulated a cosmological cluster with anisotropic conduction using the FLASH adaptive mesh-refinement (AMR) code. Runs were performed with radiative cooling on and off, but no AGN feedback was included. In this case, anisotropic conduction significantly reduced mass accretion into the central galaxy, even though the magnetic field geometry</text>
<text><location><page_2><loc_52><loc_88><loc_92><loc_92></location>significantly suppressed radial heat conduction. However, the ∼ 31 kpc/ h effective resolution was insufficient to adequately resolve core structure.</text>
<text><location><page_2><loc_52><loc_71><loc_92><loc_88></location>In this paper we present a new set of cosmological cluster simulations performed with the AMR code Enzo , including isotropic Spitzer conduction. We simulate 10 clusters spanning a mass range of 2 -8 × 10 14 M glyph[circledot] . We also model a range of suppression factors in order to evaluate the effects of thermal conductivity on core structure. As with previous calculations, we do not include AGN feedback, and a cooling catastrophe therefore results, once again confirming that conduction alone cannot prevent strong cooling flows from developing in galaxy clusters. However, our focus here is on how conduction affects the radial profiles and homogeneity of gas density and temperature outside the central ∼ 20 kpc.</text>
<text><location><page_2><loc_52><loc_57><loc_92><loc_70></location>Our presentation of these conductive cluster simulations proceeds as follows. In § 2 we describe the primary features of the Enzo code employed in this work and describe our recent implementation of isotropic Spitzer conduction, with the simulation setup given in § 3. Section 4 discusses the qualitative morphological properties of the simulated clusters, while § 5 compares their radial profiles. Section 6 focuses on conduction affects the thermal homogeneity of the ICM, § 7 looks at the star formation histories of the clusters, and § 8 summarizes our results.</text>
<section_header_level_1><location><page_2><loc_63><loc_55><loc_81><loc_56></location>2. NUMERICAL METHODS</section_header_level_1>
<text><location><page_2><loc_52><loc_42><loc_92><loc_54></location>In this work, we use the open-source cosmological, adaptive mesh-refinement (AMR) hydrodynamic + Nbody code Enzo 3 (Bryan & Norman 1997a,b; Norman & Bryan 1999; O'Shea et al. 2004, 2005). Below, we describe the primary features of Enzo used in this work, including newly-added machinery for solving thermal conduction. All analysis is performed with the open-source simulation analysis toolkit, yt 4 (Turk et al. 2011) using the Parallel-HOP halo finder (Skory et al. 2010).</text>
<section_header_level_1><location><page_2><loc_65><loc_39><loc_79><loc_41></location>2.1. The Enzo code</section_header_level_1>
<text><location><page_2><loc_52><loc_11><loc_92><loc_39></location>The Enzo code couples an N-body particle-mesh (PM) solver (Efstathiou et al. 1985; Hockney & Eastwood 1988) used to follow the evolution of a collisionless dark matter component with an Eulerian AMR method for ideal gas dynamics by Berger & Colella (1989), which allows high dynamic range in gravitational physics and hydrodynamics in an expanding universe. This AMR method (referred to as structured AMR) utilizes an adaptive hierarchy of grid patches at varying levels of resolution. Each rectangular grid patch (referred to as a 'grid') covers some region of space in its parent grid which requires higher resolution, and can itself become the parent grid to an even more highly resolved child grid . Enzo 's implementation of structured AMR places no fundamental restrictions on the number of grids at a given level of refinement, or on the number of levels of refinement. However, owing to limited computational resources it is practical to institute a maximum level of refinement, glyph[lscript] max . Additionally, the Enzo AMR implementation allows arbitrary integer ratios of parent and child grid resolution, though in general for cosmological</text>
<text><location><page_3><loc_8><loc_89><loc_48><loc_92></location>simulations (including the work described in this paper) a refinement ratio of 2 is used.</text>
<text><location><page_3><loc_8><loc_68><loc_48><loc_89></location>Multiple hydrodynamic methods are implemented in Enzo . For this work, we use the method from the ZEUS code (Stone & Norman 1992a,b) for its robustness in conditions with steep internal energy gradients, which are common in cosmological structure formation when radiative cooling is used. We use the metallicity-dependent radiative cooling method described in Smith et al. (2008) and Smith et al. (2011). This method solves the nonequilibrium chemistry and cooling for atomic H and He (Abel et al. 1997; Anninos et al. 1997) and calculates the cooling and heating from metals by interpolating from multi-dimensional tables created with the photoionization software, Cloudy 5 (Ferland et al. 1998). Both the primordial and metal cooling also take into account heating from a time-dependent, spatially uniform metagalactic UV background (Haardt & Madau 2001).</text>
<text><location><page_3><loc_8><loc_51><loc_48><loc_68></location>To model the effects of star formation and feedback, we use a modified version of the algorithm presented by Cen & Ostriker (1992). A grid cell is capable of forming stars if the following criteria are met: the baryon overdensity is above some threshold (here 1000), the velocity divergence is negative (i.e., the flow is converging), the gas mass in the cell is greater than the Jeans mass, and the cooling time is less than the self-gravitational dynamical time. Because conduction can potentially balance radiative cooling, instead of the classical definition of the cooling time, e/ ˙ e , we use a modified cooling time that includes the change in energy from conduction, defined as</text>
<formula><location><page_3><loc_21><loc_48><loc_48><loc_51></location>t cool = e ˙ e rad + ˙ e cond , (1)</formula>
<text><location><page_3><loc_8><loc_41><loc_48><loc_47></location>where ˙ e rad is the cooling rate and ˙ e cond is the conduction rate. In the original implementation, if all of the above criteria are satisfied, then a star particle representing a large, coeval group of stars with the following mass is formed:</text>
<formula><location><page_3><loc_21><loc_38><loc_48><loc_41></location>m ∗ = f ∗ m cell ∆ t t dyn , (2)</formula>
<text><location><page_3><loc_8><loc_24><loc_48><loc_37></location>where f ∗ is an efficiency parameter, m cell is the baryon mass in the cell, t dyn is the dynamical time of the combined baryon and dark matter fluid, and ∆ t is the timestep taken by the grid containing the cell in question. The factor of t dyn is added to provide a connection between the physical conditions of the gas and the timescale over which star formation and feedback occur. When a star particle is created, feedback in the form of thermal energy, gas, and metals is returned to the grid at a rate given by</text>
<formula><location><page_3><loc_14><loc_20><loc_48><loc_23></location>∆ m sf = m ∗ ∆ t t dyn ( t -t ∗ ) t dyn e -( t -t ∗ ) /t dyn , (3)</formula>
<text><location><page_3><loc_8><loc_10><loc_48><loc_19></location>where t ∗ is the creation time for the particle, so that ∆ m sf rises linearly over one dynamical time, then falls off exponentially after that. This feature ensures that the feedback response unfolds over a dynamical time, even if star formation in the simulation is formally instantaneous. We also use a distributed feedback model, in which feedback material is distributed evenly over a</text>
<text><location><page_3><loc_52><loc_81><loc_92><loc_92></location>3 × 3 × 3 cube of cells centered on the star particle's location. This was shown by Smith et al. (2011) to more effectively transport hot, metal-enriched gas out of galaxies and to minimize overcooling issues that are common to simulations of structure formation. This method was also shown by Skory et al. (2013) to provide a better match to observed cluster properties than injecting all feedback into a single grid cell.</text>
<text><location><page_3><loc_52><loc_41><loc_92><loc_81></location>In practice, Enzo simulations often implement an additional restriction, allowing creation of a star particle only if m ∗ is above some minimum mass. This prevents the formation of a large number of low-mass star particles whose presence can significantly slow down a simulation. However, the use of this algorithm in concert with conduction must be done with care. As we discuss in § 2.2, explicit modeling of thermal conduction can require timesteps that are significantly shorter than the hydrodynamic Courant condition, which has the effect of considerably reducing the value of m ∗ . If the value of m ∗ is only slightly less than the minimum particle size, this condition will likely be satisfied in a relatively short time as condensation proceeds, causing the gas density to increase and its dynamical time to decrease. When the conduction algorithm is active, the resulting time delay can allow cold, dense gas to persist while in thermal contact with the hot ICM. Since radiative cooling is proportional to the square of the density, heat conducted into high-density gas can be easily radiated away. This artificial time delay therefore produces a spurious heat sink in the ICM that can potentially boost the rates of cooling and star formation. In order to avoid this unphysical behavior, we remove the factor of ∆ t/t dyn from Equation 2 and adopt a constant value of 10 Myr for t dyn for use in Equation 3. We have tested the effects of these changes by comparing two simulations run without conduction using both the original star-formation implementation and this modification and find the stellar masses at any given time differ by less than one percent.</text>
<section_header_level_1><location><page_3><loc_62><loc_39><loc_81><loc_40></location>2.2. Conduction Algorithm</section_header_level_1>
<text><location><page_3><loc_52><loc_33><loc_92><loc_38></location>We implement the equations of isotropic heat conduction in a manner similar to that of Jubelgas et al. (2004), where the heat flux, j , for a temperature field, T , is given by</text>
<formula><location><page_3><loc_68><loc_31><loc_92><loc_33></location>j = -κ ∇ T, (4)</formula>
<text><location><page_3><loc_52><loc_24><loc_92><loc_30></location>where κ is the conductivity coefficient. In an ionized plasma, heat transport is mediated by Coulomb interactions between electrons. In this scenario, referred to as Spitzer conduction (Spitzer 1962), the conductivity coefficient is given by</text>
<formula><location><page_3><loc_61><loc_19><loc_92><loc_23></location>κ sp = 1 . 31 n e λ e k ( k B T e m e ) 1 / 2 , (5)</formula>
<text><location><page_3><loc_52><loc_14><loc_92><loc_18></location>where m e , n e , λ e , and T e are the electron mass, number density, mean free path, and temperature, and k B is the Boltzmann constant. The electron mean free path is</text>
<formula><location><page_3><loc_64><loc_10><loc_92><loc_13></location>λ e = 3 3 / 2 ( k B T ) 2 4 π 1 / 2 e 4 n e ln Λ , (6)</formula>
<text><location><page_3><loc_52><loc_7><loc_92><loc_9></location>where e is the electron charge and ln Λ is the Coulomb logarithm, defined to be the ratio of the maximum and</text>
<text><location><page_4><loc_8><loc_87><loc_48><loc_92></location>minimum impact parameters over which Coulomb collisions yield significant momentum change in the interacting particles, which are electrons in this case. The Coulomb logarithm for electron-electron collisions is</text>
<formula><location><page_4><loc_9><loc_80><loc_49><loc_85></location>ln Λ = 23 . 5 -ln ( n 1 / 2 e T -5 / 4 e ) -( 10 -5 +(ln ( T e ) -2) 2 16 ) 1 / 2 (7)</formula>
<text><location><page_4><loc_8><loc_69><loc_48><loc_80></location>(Huba 2011, pg. 34). Because of its relative insensitivity to n e and T e , we follow Jubelgas et al. (2004) and Sarazin (1988) and adopt a constant value of ln Λ = 37 . 8, corresponding to n e = 1 cm -3 and T e = 10 6 K. For values more appropriate to the ICM ( n e = 10 -3 cm -3 , T e = 10 7 K), ln Λ = 43 . 6. At a constant density, ln Λ varies by ∼ 10-20% over the range of temperatures relevant here. The Spitzer conductivity then reduces to</text>
<formula><location><page_4><loc_13><loc_67><loc_48><loc_68></location>κ sp = 4 . 9 × 10 -7 T 5 / 2 erg s -1 cm -1 K -1 . (8)</formula>
<text><location><page_4><loc_8><loc_54><loc_48><loc_66></location>Since ln Λ increases with T and κ sp ∝ 1 / ln Λ, including a direct calculation of ln Λ would serve to somewhat soften dependence of κ sp on T . In low density plasmas with large temperature gradients, the characteristic length scale of the temperature gradient, glyph[lscript] T ≡ T/ |∇ T | , can be smaller than the electron mean free path, at which point Equation 5 no longer applies. Instead, the maximum allowable heat flux is described by a saturation term (Cowie & McKee 1977), given as</text>
<formula><location><page_4><loc_18><loc_50><loc_48><loc_53></location>j sat glyph[similarequal] 0 . 4 n e k B T ( 2 kT πm e ) 1 / 2 . (9)</formula>
<text><location><page_4><loc_8><loc_45><loc_48><loc_49></location>To smoothly connect the saturated and unsaturated regimes, we use an effective conductivity (Sarazin 1988) given by</text>
<formula><location><page_4><loc_21><loc_42><loc_48><loc_45></location>κ eff = κ sp 1 + 4 . 2 λ e /glyph[lscript] T . (10)</formula>
<text><location><page_4><loc_8><loc_39><loc_48><loc_42></location>The rate of change of the internal energy, u , due to conduction is then</text>
<formula><location><page_4><loc_23><loc_37><loc_48><loc_39></location>du dt = -1 ρ ∇· j. (11)</formula>
<text><location><page_4><loc_8><loc_23><loc_48><loc_36></location>Our numerical implementation closely follows that of Parrish & Stone (2005). We use an explicit, first-order, forward time, centered space algorithm, which is the most straightforward to implement in an AMR code. For each cell, Equation 11 is solved by summing the heat fluxes from all grid cell faces, and calculating the electron density and temperature on the grid cell face as the arithmetic mean of the cell and its neighbor sharing that face. The time step stability criterion for an explicit solution of the conduction equation is</text>
<formula><location><page_4><loc_24><loc_19><loc_48><loc_22></location>dt < 0 . 5 ∆ x 2 α (12)</formula>
<text><location><page_4><loc_8><loc_16><loc_48><loc_18></location>where ∆ x is the grid cell size and α is the thermal diffusivity, defined as</text>
<formula><location><page_4><loc_24><loc_13><loc_48><loc_16></location>α = κ n e k B . (13)</formula>
<text><location><page_4><loc_8><loc_7><loc_48><loc_12></location>Equation (12) has the potentially to be considerably more constraining than the hydrodyamical Courant condition, which is proportional to T -1 / 2 ∆ x , and so the conduction timestep in conditions typical of the ICM is often</text>
<text><location><page_4><loc_52><loc_64><loc_92><loc_92></location>much shorter than the hydrodynamic timestep. The conduction timestep is calculated on a per-level basis and is taken to be the minimum of all such values on a given level. Enzo pads each AMR grid patch with three rows of ghost zones from neighboring grids. This allows the conduction routine to take three sub-cycled time steps for every hydrodynamic step, and thus allows the minimum grid timestep to be a factor of three larger than Equation 12. After three steps, temperature information from the outermost ghost zone has propagated to the edge of the active region of the grid, so performing additional conduction cycles would yield inconsistent solutions with neighboring grids. By default, Enzo rebuilds the adaptive mesh hierarchy for a given refinement level after each timestep, which is computationally expensive. However, because conduction does not explicitly change the density field (the value of which is the only quantity that determines mesh refinement), we modify this behavior such that the hierarchy is only rebuilt after an amount of time has passed equivalent to what the minimum timestep would be if conduction were not enabled.</text>
<text><location><page_4><loc_52><loc_25><loc_92><loc_64></location>This implementation models the case of isotropic Spitzer conduction, where heat flows unimpeded along temperature gradients. If magnetic fields are present, and strong enough that the electron gyroradius is small compared to the physical scales of interest (which is likely to be true in the intracluster medium), then heat is restricted to flow primarily along magnetic field lines, so the heat flux becomes the dot product of the temperature gradient with the magnetic field direction (i.e., j = -κ bb · ∇ T , where b is the unit vector pointing in the direction of the magnetic field). Diffusion perpendicular to magnetic field lines is generally considered to be negligible. We do not consider magnetic fields in this work, but instead approximate the suppression of conduction by magnetic fields by adding a suppression factor, f sp , varying from 0 to 1, to the heat flux calculation. In reality, the strength and orientation of magnetic fields in galaxy clusters is poorly understood. Hence, the degree to which conduction is suppressed from its maximum efficiency is not known. The extreme limit of tangled magnetic fields is equivalent to isotropic conduction with f sp = 1 / 3. A number of works have shown that the presence of magnetic fields alongside conduction can create a variety of instabilities that greatly influence the effective level of isotropic heat transport (e.g., Parrish & Stone 2005, 2007; Ruszkowski & Oh 2010; McCourt et al. 2011, 2012). For this reason, we simulate a large range of values of f sp , from strongly suppressed ( f sp = 0 . 01) to fully unsuppressed ( f sp = 1).</text>
<section_header_level_1><location><page_4><loc_64><loc_24><loc_80><loc_25></location>3. SIMULATION SETUP</section_header_level_1>
<text><location><page_4><loc_52><loc_7><loc_92><loc_23></location>The simulations described in this work are are initialized at z = 99 assuming the WMAP Year 7 'best fit' cosmological model (Larson et al. 2011; Komatsu et al. 2011): Ω m = 0 . 268, Ω b = 0 . 0441, Ω CDM = 0 . 2239, Ω Λ = 0 . 732, h = 0 . 704 (in units of 100 km/s/Mpc), σ 8 = 0 . 809, and using an Eisenstein & Hu power spectrum (Eisenstein & Hu 1999) with a spectral index of n = 0 . 96. We use a single cosmological realization with a box size of 128 Mpc/ h (comoving) and a simulation resolution of 256 3 root grid cells and dark matter particles. From this realization, we select the 10 most massive clusters and resimulate each individually, allowing</text>
<table>
<location><page_5><loc_16><loc_73><loc_39><loc_88></location>
<caption>Table 1 Galaxy Cluster Sample</caption>
</table>
<text><location><page_5><loc_16><loc_67><loc_39><loc_73></location>Note . - Masses and equivalent temperatures (as given by Equation 15) for the galaxy clusters in the sample. Note, the clusters are ordered by the masses computed by a halo finder in an original exploratory simulation.</text>
<text><location><page_5><loc_8><loc_43><loc_48><loc_65></location>adaptive mesh refinement only in a region that minimally encloses the initial positions of all particles that end up in the halo in question at z = 0. We allow a maximum of 5 levels of refinement, each by a factor of 2, refining on baryon and dark matter overdensities of 8. This gives us a maximum comoving spatial resolution of 15.6 kpc/ h . We set the Enzo parameter MinimumMassForRefinementLevelExponent to -0 . 2 for both dark matter and baryon-based refinement, resulting in slightly super-Lagrangian refinement behavior. For each halo in our sample, we perform a control simulation with f sp = 0 and conductive simulations with f sp = 0 . 01, 0.1, 0.3, and 1, evolving each to z = 0. All of the simulations employ the star formation/feedback and radiative cooling methods described above in § 2.1. Table 1 lists the masses and equivalent temperatures for all 10 clusters in the sample.</text>
<section_header_level_1><location><page_5><loc_17><loc_41><loc_40><loc_42></location>4. QUALITATIVE MORPHOLOGY</section_header_level_1>
<text><location><page_5><loc_8><loc_14><loc_48><loc_40></location>Perhaps the most surprising finding to emerge from our simulations is the insensitivity of the qualitative morphological features of the ICM to thermal conduction, even in clusters with typical gas temperatures ∼ 5 keV. Figures 1, 2, and 3 show gas density, temperature, and X-ray surface brightness projections for three clusters at the minimum and maximum conduction efficiency. We choose these clusters in order to capture a wide range of potential influence from conduction. Halo 0 is the most massive cluster in the sample (with a mass of 8 × 10 14 M glyph[circledot] ), Halo 2 is a highly unrelaxed cluster, and Halo 5 is relatively relaxed and is close to the average mass for the sample (M 200 glyph[similarequal] 4 . 9 × 10 14 M glyph[circledot] ) In order to highlight differences between values of f sp , we limit the projected line of sight to 0.5 Mpc. The morphological similarity between these conductive and non-conductive clusters starkly contrasts with the obvious differences found by Dolag et al. (2004) for a ∼ 12 keV cluster. In their work, it is immediately obvious which simulation included conduction.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_13></location>There are, however, some subtle but noticeable morphological differences between our conductive and nonconductive clusters. The overall density structure is slightly smoother for f sp = 1 in all three clusters, and the temperature field is visibly smoother in Halo 5. In all</text>
<figure>
<location><page_5><loc_54><loc_57><loc_90><loc_92></location>
<caption>Figure 1. Projections of a 500 kpc thick slab centered on the most massive cluster at z = 0 ( M 200 = 8 . 0 × 10 14 M glyph[circledot] , T 200 = 5 . 6 keV) for the simulations with f sp = 0 (left) and 1. (right) The top and middle rows show average density and temperature, weighted by X-ray emissivity in the energy range of 0.5 to 2 keV. The bottom row shows the X-ray surface brightness in the same energy range.</caption>
</figure>
<figure>
<location><page_5><loc_54><loc_12><loc_90><loc_47></location>
<caption>Figure 2. Projections of a 500 kpc thick slab centered on the third most massive cluster at z = 0 (M 200 = 5 . 6 × 10 14 M glyph[circledot] , T 200 = 4 . 4 keV) for the simulations with f sp = 0 (left) and 1 (right).</caption>
</figure>
<figure>
<location><page_6><loc_10><loc_57><loc_46><loc_92></location>
<caption>Figure 3. Projections of a 500 kpc thick slab centered on the sixth most massive halo at z = 0 (M 200 = 4 . 6 × 10 14 M glyph[circledot] , T 200 = 3 . 9 keV) for the simulations with f sp = 0 (left) and 1 (right).</caption>
</figure>
<text><location><page_6><loc_8><loc_36><loc_48><loc_51></location>three clusters shown, the core appears to be marginally less dense with conduction on and is also markedly hotter in Halos 2 and 5. Nevertheless, while it is for the most part apparent that conduction is present in our cluster sample with f sp = 1, its effects are not nearly as striking as in Dolag et al. (2004). Quite possibly, this difference is due to the strong temperature dependence of Spitzer conductivity, suggesting that it may be possible to estimate the typical conductivity of the ICM by analyzing the dependence of azimuthal temperature homogeneity on cluster temperature (or lack thereof) in a large sample of galaxy clusters.</text>
<section_header_level_1><location><page_6><loc_21><loc_33><loc_36><loc_34></location>5. RADIAL PROFILES</section_header_level_1>
<text><location><page_6><loc_8><loc_10><loc_48><loc_32></location>Systematic conductivity-dependent differences among our simulated clusters are hard to see in individual cases but are more apparent in comparisons of average properties of sample sets with differing conductivity. Therefore, in order to understand the systematic effects of increasing the conduction efficiency, we aggregate the radial profiles from all the clusters in our simulated sample for each value of f sp , as shown in Figure 4. To minimize artifacts that result from the rebinning of profile data, we perform the initial profiling in physical distance units to calculate the virial radius for each cluster (which we take for convenience to be r 200 with respect to the critical density). We then create a second set of profiles in units of r/r 200 that are combined to make the aggregate profiles presented here. We refer to these as 'averaged profiles' and employ this method for Figures 4 and 5 and all subsequent figures showing aggregate properties.</text>
<section_header_level_1><location><page_6><loc_21><loc_7><loc_36><loc_9></location>5.1. Cluster Interiors</section_header_level_1>
<text><location><page_6><loc_52><loc_51><loc_92><loc_92></location>The presence of conduction leads to higher gas temperatures both inside and outside of the conduction-free cluster's characteristic temperature peak at r/r 200 ∼ 0 . 07. Within r/r 200 ∼ 0 . 07, gas temperatures in conductive clusters are greater than those in the non-conductive control simulation, with values of f sp glyph[greaterorsimilar] 0 . 1 leading to nearly isothermal cores and central temperatures ∼ 40% greater than in non-conductive cluster simulations. The temperature profiles of our conductive clusters are in reasonably good agreement with those of Ruszkowski et al. (2011) for their cluster with isotropic conduction and f sp = 1 / 3, although the core of their cluster is less isothermal than ours, with cooler temperatures near the center. This could be because the simulations of Ruszkowski et al. (2011) did not include a prescription for star formation and feedback in cluster galaxies, which can increase the core temperature both by consuming the cold gas and through thermal and mechanical feedback. Agreement with the temperature profiles of the simulated clusters from Dolag et al. (2004) is not nearly as good. Those clusters show a significant reduction in both the peak and central temperatures with conduction present, which we do not observe. The theoretical temperature profiles of McCourt et al. (2013) for the 10 14 . 5 M glyph[circledot] clusters show an elevation in the core temperatures in conductive clusters in rough agreement with our results. We also see a marginal inward movement of the location of the temperature peak as they predict, although not nearly to the same degree. The temperature peak for our clusters is also at a smaller radius to begin with.</text>
<text><location><page_6><loc_52><loc_20><loc_92><loc_50></location>Despite its ability to maintain approximate isothermality in the cluster cores, conduction at even maximal efficiency is unable to avert a cooling catastrophe at the very center of the halo. This cooling catastrophe and the resulting condensation and star-formation activity produce sharp peaks at r/r 200 glyph[lessorsimilar] 0 . 03 in all the gas density profiles in Figure 4, as well as a slight temperature increase at the same location in the more highly conductive clusters. The elevated central temperatures at the centers of conductive clusters reduce the central gas density relative to the control clusters by ∼ 20-30%. Material that would have fallen into the center is displaced to larger radii, as can be seen by the elevated density at 0 . 04 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 6 in the more conductive runs. Gas density is enhanced primarily in the range 0 . 04 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 2, but the enhancement appears to saturate at ∼ 20% for f sp = 0 . 33. For f sp = 1, the density is actually lower than for f sp = 0 . 33 in this range, but is then higher for 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 6. It is unclear what causes the outward transport (or prevention of inflow) to stall just inside 0.2 r 200 , but it seems that it can be overcome for some value of f sp above 0.33, likely closer to 1.</text>
<text><location><page_6><loc_52><loc_8><loc_92><loc_20></location>Further evidence of conduction-driven inflation of the core can be seen in Figure 5, where we plot averaged profiles of M/r , including the individual contributions from dark matter, stars, and gas. Despite the lack of perfect monotonicity in the density and temperature profiles, the gaseous component of the potential decreases monotonically with increasing f sp , indicating that the clusters are indeed responding to the presence of conduction by puffing up their cores and redistributing gas to larger radii.</text>
<text><location><page_7><loc_8><loc_85><loc_48><loc_92></location>As is the case for most simulated galaxy clusters, the stellar component is extremely centrally concentrated, dominating the gravitational potential inside 0.04 r 200 . The stellar and dark matter components of the potential are largely unaffected by conduction.</text>
<text><location><page_7><loc_8><loc_81><loc_48><loc_85></location>In the right panel of Figure 4, we plot normalized entropy profiles, where the normalization term, K 200 (Voit 2005; Voit et al. 2005), is given by</text>
<formula><location><page_7><loc_20><loc_79><loc_48><loc_80></location>K 200 = k B T 200 ¯ n e -2 / 3 , (14)</formula>
<text><location><page_7><loc_8><loc_77><loc_18><loc_78></location>where T 200 is</text>
<formula><location><page_7><loc_20><loc_73><loc_48><loc_76></location>k B T 200 = GM 200 µm p 2 r 200 , (15)</formula>
<text><location><page_7><loc_8><loc_51><loc_48><loc_72></location>and ¯ n e is the average electron number density within r 200 , assuming a fully ionized plasma of primordial composition. The decrease in density and increase in temperature in the cluster cores with conduction yield entropy profiles that are enhanced by 40-70%, but still show the steep decline in the very center that is typical of simulated clusters. This enhancement decreases out to r/r 200 ∼ 0 . 07, where the mean entropy values are approximately equivalent for all values of f sp . The entropy profiles for the clusters with conduction then dip below the control sample by roughly 10% out to r/r 200 ∼ 0 . 2. In the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 6, the combined temperature and density enhancements appear to be in perfect balance, producing nearly identical entropy profiles across the entire cluster sample with a very small variance.</text>
<text><location><page_7><loc_8><loc_46><loc_48><loc_51></location>Voit et al. (2005) find that for a sample of nonradiative, non-conducting clusters simulated with Enzo , the entropy profiles in the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 1 are best fit by the power-law</text>
<formula><location><page_7><loc_17><loc_43><loc_48><loc_45></location>K ( r ) /K 200 = 1 . 51 ( r/r 200 ) 1 . 24 , (16)</formula>
<text><location><page_7><loc_8><loc_37><loc_48><loc_42></location>which we overplot on Figure 4 with a black dashed line. This power law matches our sample well in normalization, but has a slightly steeper slope. We find that our sample is best matched by</text>
<formula><location><page_7><loc_17><loc_35><loc_48><loc_36></location>K ( r ) /K 200 = 1 . 50 ( r/r 200 ) 1 . 09 . (17)</formula>
<text><location><page_7><loc_8><loc_18><loc_48><loc_34></location>For comparison, we also plot the predicted entropy profile for a pure cooling model (Voit & Bryan 2001; Voit et al. 2002) of a 5 keV cluster. This model assumes gas and dark matter density distributions that follow an NFW profile (Navarro et al. 1997) with concentration c = 6. The gas is initially in hydrostatic equilibrium and allowed to cool for a Hubble time. The pure cooling model agrees quite well with our clusters in the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 1, and especially well in the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 5. Outside of 0.5 r 200 , the slope of the pure-cooling model is slightly too steep and is closer in slope to the Voit et al. (2005) fit.</text>
<section_header_level_1><location><page_7><loc_20><loc_15><loc_36><loc_16></location>5.2. Cluster Exteriors</section_header_level_1>
<text><location><page_7><loc_8><loc_7><loc_48><loc_15></location>The outskirts of galaxy clusters, while having lower temperatures and hence lower conductivities, are not subject to intermittent heat injection from stellar feedback, and thus offer an intriguing laboratory for studying the effects of conduction. We identify two distinct regions in the cluster outskirts where conduction appears to have</text>
<text><location><page_7><loc_52><loc_76><loc_92><loc_92></location>influence, at ∼ r 200 and at ∼ 3 r 200 . At r/r 200 glyph[greaterorsimilar] 0 . 6, the density excess seen in the cluster interiors turns into a deficit for all values of conduction simultaneously, as can be see in Figure 4. From this point out to well past the virial radius, there exists a perfectly monotonic trend of lower gas densities for higher values of f sp . For the maximum value of f sp , the average gas density at the virial radius is 10% lower than that of the clusters simulated without conduction. In fact, the density is measurably lower for all values of f sp ≥ 0 . 1 out to a few virial radii. The temperature just inside r 200 is marginally higher, while the temperature just outside r 200 is reduced.</text>
<text><location><page_7><loc_52><loc_47><loc_92><loc_76></location>Conduction transports heat outward in these regions because the gravitational potential there produces a declining gas temperature gradient. This heat transfer causes the entropy of the outer gas to increase. However, because gas in the cluster outskirts is not pressureconfined like the gas in the core, it is free to expand outward and decrease in both density and temperature, while its temperature gradient remains determined by the gravitational potential. This happens because the timescale for conduction in the outer regions is substantially greater than the sound-crossing time. Consequently, conduction of heat outward causes the outer gas to expand without much change in the temperature gradient. Beyond r 200 , however, we see a slight steepening in the temperature profile as predicted for conductive clusters by McCourt et al. (2013), presumably because inflation of the ICM due to conduction pushes gas near the virial radius farther from the cluster center without adding much thermal energy. The fact that the entropy profiles of conductive clusters are lower than those of non-conductive clusters beyond the virial radius supports this interpretation.</text>
<text><location><page_7><loc_52><loc_37><loc_92><loc_47></location>Further out, at ∼ 3 r 200 , there is another systematic decrease in both density and temperature. Interestingly, as was pointed out by Skillman et al. (2008), this is the typical location for a galaxy cluster's accretion shocks, which are responsible for heating gas up to the virial temperature. This raises a critical question: How can conduction affect a large halo's accretion shocks?</text>
<text><location><page_7><loc_52><loc_8><loc_92><loc_37></location>To test this question in a more controlled environment than a cosmological simulation, we performed a pair of one-dimensional simulations designed to mimic the conditions of an accretion shock around a galaxy cluster. Following the shock properties characterized by Skillman et al. (2008), we initiated a Mach 100 standing shockwave with preshock conditions of n = 10 -4 cm -3 and T = 10 4 K, comparable to gas that is falling directly onto a galaxy cluster (i.e., through spherical accretion of the surrounding intergalactic medium, rather than being accreted via filaments). We ran simulations with f sp = 0 and 1, with profiles shown in Figure 6. Within tens of Myr, a conduction front in the f sp = 1 simulation races ahead of the original shock. The conduction front continues to advance with ever-decreasing speed and settles into a nearsteady state by t = 200 Myr, the time shown in the figure. Conduction of heat ahead of the main shock front therefore results in a separation of the density and temperature jumps, effectively creating two shocks, one produced by conductive preheating and a second nearly isothermal shock front some distance downstream. The lower-right panel of Figure 6 shows the Mach numbers determined</text>
<figure>
<location><page_8><loc_9><loc_64><loc_91><loc_92></location>
<caption>Figure 4. Averaged radial profiles for all ten halos in the sample, showing volume-weighted density, mass-weighted temperature, and entropy constructed from volume-weighted electron density and mass-weighted temperature profiles. The colors denote the value of f sp and the shaded regions indicate the variance within the cluster sample. The lower panels show profiles of the enhancement or decrement in the fields profiled above with respect to the clusters with simulated without conduction. Note, the ratios shown in the lower panels were computed for each cluster, then aggregated, and as such are not strictly the ratio of the values plotted above. The right panel also includes the power-law fit to non-radiative Enzo simulations of Voit et al. (2005) (Equation 16, black, dashed line), a power-law fit to the sample presented here (Equation 17, dashed, lime-green line), and a pure cooling model for a 5 keV cluster (Voit & Bryan 2001; Voit et al. 2002, dashed, pink line).</caption>
</figure>
<figure>
<location><page_8><loc_10><loc_26><loc_46><loc_54></location>
<caption>Figure 5. Averaged profiles of the radial mass distribution in terms of M/r , including individual contributions from dark matter, stars, and gas for all clusters. The process by which the averaged profiles are created is described in § 5 and in the caption of Figure 4.</caption>
</figure>
<text><location><page_8><loc_8><loc_7><loc_48><loc_19></location>using the shock-finding algorithm described in Skillman et al. (2008). Bifurcation of the original Mach 100 shock has produced a Mach ∼ 70 conductive-precursor shock followed by a Mach ∼ 1 . 5 shock where the main density jump occurs. This splitting is also evident in the plot showing the local Mach ratio, v/c s (lower-middle panel of Figure 6) on either side of the temperature jump. The heat transfer upstream also causes a slight drop in pressure at the original shock front, shifting the primary den-</text>
<text><location><page_8><loc_52><loc_53><loc_80><loc_54></location>jump downstream by about 5 kpc.</text>
<text><location><page_8><loc_52><loc_28><loc_92><loc_53></location>These results are independent of resolution. The simulations shown in Figure 6 are for a grid 128 cells across, and we observe nearly identical behavior down to a resolution of 16 cells. For this configuration, the results are strongly dependent on f sp . At f sp = 0 . 67, the distance between the separated shocks is approximately half of that at f sp = 1 and the Mach number of the primary shock is only reduced to just under 90. For f sp glyph[lessorsimilar] 0 . 4, the results are indistinguishable from those without conduction. However, we find that it is possible to produce significant shockwave alteration for lower values of f sp simply by lowering the initial Mach number and increasing the preshock temperature (resulting in gas in a thermodynamic regime comparable to gas that is being accreted from cosmological filaments). As long as the postshock gas is able to reach temperatures in the range of 10 7 K, where the Spitzer conductivity becomes considerable, conduction is capable of bifurcating the shock front.</text>
<text><location><page_8><loc_52><loc_13><loc_92><loc_27></location>Figure 7 shows the average Mach number as a function of gas density in our simulations of the most massive halo for all shocks in the subvolume in which refinement is allowed. Conduction reduces the Mach numbers of the strongest shocks, which occur preferentially in the lowest density gas, by roughly 10% from f sp = 0 to f sp = 1. We therefore conclude that the density, temperature, and entropy deficits observed beyond the virial radius in our conductive clusters are indeed due to shock bifurcations similar to those seen in our idealized one-dimensional simulations of conductive shock fronts.</text>
<text><location><page_8><loc_52><loc_8><loc_92><loc_13></location>Yet, the physics of the actual accretion shocks around real galaxy clusters is undoubtedly more complex. In particular, it is important to note that the electron mean free path in that gas is several times greater than the</text>
<figure>
<location><page_9><loc_9><loc_60><loc_91><loc_92></location>
<caption>Figure 6. Profiles of a one-dimensional, Mach 100 standing shockwave after 200 Myr for f sp = 0 and 1 (black and blue lines, respectively). Note, the curves for f sp = 0 also denote the initial configuration of the shock, which is stationary in the case where conduction is negligible.</caption>
</figure>
<figure>
<location><page_9><loc_10><loc_28><loc_47><loc_56></location>
<caption>Figure 7. Profiles of the average Mach number as a function of gas density for the entire region where refinement is allowed in the simulations of the most massive galaxy cluster. The average Mach number is calculated as a weighted mean, where the weight is the rate of kinetic energy processed through the shock, given by 1 2 ρv 3 A , where ρ , v , and A are the pre-shock density, shock velocity, and shock surface area.</caption>
</figure>
<text><location><page_9><loc_8><loc_7><loc_48><loc_17></location>intershock distance in our one-dimensional simulations, and furthermore that real accretion shocks are likely to be collisionless and magnetically-mediated. Nonetheless, the general qualitative point these simulations illustrate is interesting: Heating of preshock gas by a hot electron precursor has the potential to alter the expected relationships between the sizes and locations of the density and temperature jumps in accretion and merger shocks.</text>
<text><location><page_9><loc_52><loc_26><loc_92><loc_56></location>Progress in understanding the effects of conduction on accretion shocks will require modeling the gas as a fully ionized plasma (Zel'dovich 1957; Shafranov 1957), which is beyond the scope of this work. The high Mach number of an accretion shock combined with a post-shock temperature high enough for significant heat flux create conditions similar to a supercritical radiative shock, as described by Lowrie & Rauenzahn (2007). When simulated with a two-fluid approach, the combination of preheating of the preshock medium via conduction in the electrons, electron-ion coupling, and compression of the ion fluid in the postshock region can produce a small region, known as a Zel'dovich spike, where the ion temperature is slightly higher than the equilibrium postshock temperature (Lowrie & Rauenzahn 2007; Lowrie & Edwards 2008; Masser et al. 2011). Our single-fluid simulations, despite their limitations, produce shocks quite similar to the non-equilibrium results of Lowrie & Edwards (2008), where a diffusion term proportional to T 5 / 2 (like Spitzer conductivity) is used. Nevertheless, a more detailed study of the characteristics of accretion shocks employing a two-fluid MHD treatment along with physically motivated conduction and cooling rates seems warranted.</text>
<section_header_level_1><location><page_9><loc_60><loc_23><loc_84><loc_24></location>6. TEMPERATURE HOMOGENEITY</section_header_level_1>
<text><location><page_9><loc_52><loc_7><loc_92><loc_23></location>Conduction strong enough to alter the temperature gradients in cluster cores should also be effective at smoothing out small-scale thermal variations in the ICM. However, those effects turn out to be rather subtle, as shown in Figure 8, which plots the normalized variance of the temperature field as a function of radius, averaged over all clusters in the sample that have the same level of conductivity. It reveals an extremely weak trend of greater temperature homogeneity with increasing f sp , but at all radii the difference in homogeneity among cluster sets with different values of f sp is less than the clusterto-cluster variation. Furthermore, this trend reverses be-</text>
<figure>
<location><page_10><loc_10><loc_64><loc_47><loc_92></location>
<caption>Figure 8. Averaged radial profile of the mass-weighted temperature variance normalized to mean temperature within each radial bin. The process by which the averaged profiles are created is described in § 5 and in the caption of Figure 4.</caption>
</figure>
<text><location><page_10><loc_31><loc_63><loc_32><loc_64></location>200</text>
<text><location><page_10><loc_8><loc_54><loc_48><loc_56></location>yond the virial radius, at 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 3, albeit at a nearly marginal level.</text>
<text><location><page_10><loc_8><loc_29><loc_48><loc_54></location>The homogenizing effects of conduction are even harder to see in projection, which perhaps is not surprising given the scarcity of obvious conduction-dependent morphological differences in Figures 1, 2, and 3. Those differences are further diluted when projected over a 4 Mpc line of sight through each cluster, as in Figures 911. These latter figures show mean temperature weighted by the X-ray emission in the 0.5-2 keV energy band, and X-ray emission is calculated by interpolating from density, temperature, and metallicity-dependent emissivity tables computed with the Cloudy code. Since Xray emission is proportional to n 2 e T 1 / 2 e , this weighting should highlight clumpiness and differences in temperature. However, as stated previously, these results contrast considerably with the cluster map comparison of Dolag et al. (2004), which shows a significantly hotter cluster in which conduction should be much more efficient.</text>
<text><location><page_10><loc_8><loc_7><loc_48><loc_29></location>The change in temperature homogeneity due to conduction is quite small in our simulated cluster sample, but important insights into the physics of the ICM could be gained if the effective value of f sp could be measured observationally. To evaluate this possibility, we created X-ray-weighted temperature maps of the central 300 kpc for each of the clusters in the sample, masking out the pixels within 40 kpc of the cluster centers to remove features that would be considered part of the central galaxy. We then quantified the amount of azimuthal temperature structure by dividing each temperature map into azimuthal bins and calculating both the mean temperature in each bin and the temperature variance among all azimuthal bins in the map. We performed this calculation multiple times for each map while rotating the azimuthal bins, and took the maximum variance calculated as the value for that map. Finally, we averaged</text>
<figure>
<location><page_10><loc_54><loc_57><loc_90><loc_92></location>
<caption>Figure 9. Similar projections to Figure 1 for Halo 0, but with a projected depth of 4 Mpc.</caption>
</figure>
<figure>
<location><page_10><loc_54><loc_13><loc_90><loc_48></location>
<caption>Figure 10. Similar projections to Figure 2 for Halo 2, but with a projected depth of 4 Mpc.</caption>
</figure>
<figure>
<location><page_11><loc_10><loc_57><loc_46><loc_92></location>
<caption>Figure 11. Similar projections to Figure 3 for Halo 5, but with a projected depth of 4 Mpc.</caption>
</figure>
<text><location><page_11><loc_8><loc_34><loc_48><loc_53></location>the values together for all clusters in the sample and performed the entire exercise over a range in the total number of azimuthal bins, from 2 to 9. Figure 12 plots the maximum variance as a function of the number of bins for each value of f sp . We find that, in general, the maximum variance is lower for the simulations with conduction, but only by approximately 10%. There does not appear to be any sort of monotonic trend with increasing f sp . We repeated this experiment, varying the inner and outer radius for the temperature maps, but were unable to find conditions that produce a better trend than can be seen in Figure 12. Thus, we conclude that the efficiency of conduction is difficult to determine solely from the observable temperature homogeneity of the ICM, at least for clusters of temperature glyph[lessorsimilar] 6 keV.</text>
<section_header_level_1><location><page_11><loc_21><loc_31><loc_36><loc_32></location>7. STAR FORMATION</section_header_level_1>
<text><location><page_11><loc_8><loc_7><loc_48><loc_31></location>Figure 13 shows the average difference in stellar mass between the clusters with conduction and those without as a function of time. The difference in stellar mass at z = 0 for all levels of conduction is less than 5%. Surprisingly, the clusters with higher levels of conduction form more stars in our calculations (see also Dolag et al. 2004). However, the shaded regions in Figure 13 give an indication of just how much variation there is between clusters. As we have shown, even the highest level of conduction is unable to prevent a cooling catastrophe in the center of a cluster. Therefore, one should not expect a large difference in the amount of star formation. Given the coarseness with which the star forming regions are resolved, it is possible that the enhancement in star formation with increasing f sp is numerical and not physical. We propose two potential numerical explanations. First, the short timesteps required for the stability of the conduction algorithm may not provide enough time for</text>
<figure>
<location><page_11><loc_54><loc_64><loc_91><loc_92></location>
<caption>Figure 12. The maximum variance in X-ray-weighted temperature projections within a region 40 kpc ≤ r ≤ 150 kpc as a function of the number of azimuthal bins.</caption>
</figure>
<text><location><page_11><loc_52><loc_51><loc_92><loc_59></location>a single star particle to sufficiently heat up a grid cell and quench star formation in a given cell in the following timestep. Second, conduction may transport thermal energy too quickly out of regions heated by recent star formation. This would allow a star-forming region to recool and form additional stars too rapidly.</text>
<text><location><page_11><loc_52><loc_38><loc_92><loc_51></location>Because conduction is able to create a nearly isothermal core for f sp ≥ 0 . 1, yet the final stellar mass from the clusters with f sp = 0 . 1 is consistent with no change, we find it reasonable to conclude that the isothermality of the core in a conductive cluster has no influence on the star formation rate Enzo calculates for the central galaxy. However, because we do not resolve the interface between the ICM and the interstellar medium (ISM), we cannot definitively state the effect of conduction on star-forming gas in a cosmological simulation.</text>
<section_header_level_1><location><page_11><loc_66><loc_35><loc_78><loc_36></location>8. CONCLUSIONS</section_header_level_1>
<text><location><page_11><loc_52><loc_11><loc_92><loc_35></location>We have performed cosmological simulations of 10 galaxy clusters using isotropic thermal conduction with five values of the conductive suppression factor in order to study the effects of conduction on galaxy cluster cores and the intracluster medium. By studying the aggregate properties of the clusters in our sample, we find that the presence of conduction even at its maximum possible efficiency induces changes to the density and temperature structure on the order of only 20-30%. For f sp ≥ 0 . 1, the cluster cores become roughly isothermal. However, conduction at any level is incapable of stopping the cooling catastrophe at the very centers of our clusters, where the density profile is always very sharply peaked. To some extent, this is due to our limited spatial resolution, since the temperature gradients on which heat conduction depends are limited by the scale of the smallest grid cell, which at ∼ 15 kpc/ h is still quite large compared to the scales of galaxies and the Field length.</text>
<text><location><page_11><loc_52><loc_7><loc_92><loc_11></location>However, the extremely well-resolved study of Li & Bryan (2012) also finds that conduction can at best slightly delay the cooling catastrophe. While conduction</text>
<figure>
<location><page_12><loc_9><loc_63><loc_47><loc_92></location>
<caption>Figure 13. The ratio of total stellar mass in the simulations with f sp > 0 to the control run as a function of cosmic time for all clusters in the sample, with the shaded regions denoting the cluster to cluster variance. The process by which the averaged quantities are calculated is described in § 5 and in the caption of Figure 4.</caption>
</figure>
<text><location><page_12><loc_8><loc_40><loc_48><loc_56></location>is unable to prevent the cooling catastrophe, the elevation of gas entropy in a conductive, isothermal core displaces some of the core gas, moving it out to larger radii. For values of f sp up to 0.33, this material is redistributed mostly within ∼ 0 . 2 r 200 . For higher values of f sp , it is transported out even further, up to ∼ 0 . 6 r 200 . A similar phenomenon occurs around r 200 , where the negative temperature gradient allows outward heat conduction to inflate the outer parts of the cluster. However, because this material is not deep in the potential well, it is free to expand and cool, leading to slightly lower temperatures just outside the virial radius.</text>
<text><location><page_12><loc_8><loc_7><loc_48><loc_40></location>More surprisingly, we observe a systematic decrease in both the density and temperature with increasing f sp at large radii, out to ∼ 3 r 200 . We hypothesize that this is due to alteration of the accretion shocks by conduction. To test this, we perform one-dimensional 'shock tube' simulations with conditions characterizing an accretion shock around a galaxy cluster, with the level of conduction treated as the sole free parameter. As long as the post-shock temperature is high enough for the Spitzer conductivity to be efficient ( T glyph[greaterorsimilar] 10 7 K), conduction moves the temperature jump upstream and the density jump downstream of the original shock face. This creates two distinct shocks, both with Mach numbers less than the original shock. We conclude that conduction is responsible for the systematic decrease in density and temperature in the outskirts of our simulated clusters, because it acts to weaken the shocks. We acknowledge that our modeling of this problem is not totally accurate, and instead requires a two-fluid MHD approach, which is beyond the current capabilities of our simulation tool. However, more rigorous two-fluid simulations of shockwaves in fully ionized plasmas show qualitatively similar behavior, save a tiny feature in the ion temperature that cannot be achieved in a single-fluid approach. Unfortunately, because the effect on clusters is only at the 10%</text>
<text><location><page_12><loc_52><loc_87><loc_92><loc_92></location>level and at very large radii, where the X-ray surface brightness of the plasma is extremely low, observing the effects of conduction on accretion shocks may never be possible.</text>
<text><location><page_12><loc_52><loc_64><loc_92><loc_86></location>We also find that in addition to altering temperature gradients, conduction is able to make the intracluster medium more thermally uniform. This effect, while measurable in spherically-averaged radial profiles, is almost totally lost in projection. Our results contrast with the temperature maps of Dolag et al. (2004), wherein the effect of conduction is instantly recognizable. The cluster shown in Dolag et al. (2004) is significantly more massive than our most massive cluster, so it is possible that a hotter ICM, with a higher thermal conductivity, is made more homogeneous, suggesting that the temperature dependence of temperature inhomogeneity in a large cluster sample could help reveal the typical conductivity of the ICM. We attempted to find a means of distinguishing the level of conduction observationally by measuring the variance in our projected temperature maps, but without success.</text>
<text><location><page_12><loc_52><loc_37><loc_92><loc_64></location>Finally, conduction appears to have very little influence on the star formation rate within our simulated clusters. When determining whether a grid cell should form a star particle, we include the energy change from conduction in the calculation of the cooling time, but this seems to have very little influence. This is likely because star-forming grid cells are surrounded mostly by cells that are also quite cool. Somewhat surprisingly, we observe a marginal increase in the total stellar mass with increasing conduction, such that the sample with f sp = 1 shows an enhancement in star formation rate of ∼ 5%. The fact that conduction cannot suppress star formation is directly related to its inability to prevent the cooling catastrophe in the very center of the cluster. However, the reasons for the slight increase in star formation may be more numerical than physical. Further progress on understanding the effects of thermal conduction on star formation in cluster cores will require properly resolving the interface between ISM and the ICM, which at present is impractical in cosmological galaxy cluster simulations.</text>
<text><location><page_12><loc_52><loc_13><loc_92><loc_35></location>This work was supported by NASA through grant NNX09AD80G and NNX12AC98G, and by the NSF through AST grant 0908819. The simulations presented here were performed and analyzed on the NICS Kraken and Nautilus supercomputing resources under XSEDE allocations TG-AST090040 and TG-AST120009. We thank Greg Bryan, Gus Evrard, Eric Hallman, Andrey Kravtsov, Jack Burns, Matthew Turk, and Stephen Skory for helpful discussions during the course of preparing this manuscript. SWS has been supported by a DOE Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308. BWO was supported in part by the MSU Institute for Cyber-Enabled Research. Enzo and yt are developed by a large number of independent research from numerous institutions around the world. Their committment to open science has helped make this work possible.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "Simulations of galaxy clusters have a difficult time reproducing the radial gas-property gradients and red central galaxies observed to exist in the cores of galaxy clusters. Thermal conduction has been suggested as a mechanism that can help bring simulations of cluster cores into better alignment with observations by stabilizing the feedback processes that regulate gas cooling, but this idea has not yet been well tested with cosmological numerical simulations. Here we present cosmological simulations of ten galaxy clusters performed with five different levels of isotropic Spitzer conduction, which alters both the cores and outskirts of clusters, but not dramatically. In the cores, conduction flattens central temperature gradients, making them nearly isothermal and slightly lowering the central density but failing to prevent a cooling catastrophe there. Conduction has little effect on temperature gradients outside of cluster cores because outward conductive heat flow tends to inflate the outer parts of the intracluster medium (ICM) instead of raising its temperature. In general, conduction tends reduce temperature inhomogeneity in the ICM, but our simulations indicate that those homogenizing effects would be extremely difficult to observe in ∼ 5 keV clusters. Outside the virial radius, our conduction implementation lowers the gas densities and temperatures because it reduces the Mach numbers of accretion shocks. We conclude that despite the numerous small ways in which conduction alters the structure of galaxy clusters, none of these effects are significant enough to make the efficiency of conduction easily measurable unless its effects are more pronounced in clusters hotter than those we have simulated. Subject headings: cosmology",
"pages": [
1
]
},
{
"title": "COSMOLOGICAL SIMULATIONS OF ISOTROPIC CONDUCTION IN GALAXY CLUSTERS",
"content": "Britton Smith, Brian W. O'Shea 1 , G. Mark Voit, David Ventimiglia Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824 and Samuel W. Skillman 2 Center for Astrophysics and Space Astronomy, Department of Astrophysical & Planetary Science, University of Colorado, Boulder, CO 80309, USA Draft version October 20, 2021",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Numerical simulations of galaxy clusters have not yet succeeded in producing objects with properties identical to those of observed galaxy clusters. The most serious discrepancies with observations are in the cores (e.g, Nagai et al. 2007; Borgani & Kravtsov 2011; Skory et al. 2013). Within the central ∼ 100 kpc of clusters whose central cooling time is less than a Hubble time, radiative cooling of intracluster gas in the simulations tends to be too efficient, leading to overproduction of young stars and excessive entropy levels in the core gas, as higherentropy material flows inward to replace the gas that has condensed. The resulting cluster cores in such simulations consequently have temperature profiles that decline from ∼ 10 kpc outward, in stark disagreement with the observed temperature profiles of cool-core clusters, which rise in the ∼ 10-100 kpc range. Feedback from a central active galactic nucleus helps solve this problem because it slows the process of cooling. However, the core structures of clusters produced in cosmological simulations depend sensitively on the details of the implementation of feedback, the radiative cooling scheme, and numerical resolution. For example, Dubois et al. (2011) simulated the same cosmological galaxy clus- smit1685@msu.edu ter with two different mechanisms for feedback and two different cooling algorithms, one with metal-line cooling and one without. The core structures of the resulting clusters were quite diverse, with kinetic jet feedback and metal-free cooling producing the most realistic-looking cores. But discouragingly, allowing cooling via metal lines had the effect of supercharging AGN feedback and producing a cluster core with excessively high entropy. Another oft-proposed solution to the core-structure problem in galaxy clusters is thermal conduction. It has been invoked many times to mitigate the prodigious mass flows predicted for cool-core clusters lacking a central heat source (e.g., Tucker & Rosner 1983; Bertschinger & Meiksin 1986; Narayan & Medvedev 2001) but cannot be in stable balance with cooling (e.g., Bregman & David 1988; Soker 2003; Guo et al. 2008) and does not completely compensate for radiative cooling in all cluster cores (Voigt et al. 2002; Zakamska & Narayan 2003). Nevertheless, the temperature and density profiles of many cool-core clusters are tantalizingly close to being in conductive balance, suggesting that AGN feedback is triggered only when inward thermal conduction, perhaps assisted by turbulent heat transport, fails to compensate for radiative losses from the core (e.g., Voit 2011). The nebular line emission observed in many cool-core clusters also suggests that thermal conduction may be important. Ultraviolet light from young stars accounts for some of the line emission but cannot explain all of it, implying that another heat source is present (e.g., John- stone et al. 1987; Voit & Donahue 1997; Werner et al. 2013). Observations of the optical and infrared line ratios are consistent with heat input by a population of suprathermal electrons, possibly entering the nebular gas from the ambient hot medium (Ferland et al. 2009). The emission-line studies are not conclusive on this point, but recent observations of ultraviolet emission from a filament in the Virgo cluster have bolstered the case for conduction (Sparks et al. 2009, 2011). The C IV 1550 ˚ A and He II 1640 ˚ A emission lines from this filament are consistent with a model in which the filament is surrounded by a conductive sheath that channels heat into the filament. If conduction is indeed important in determining the structure of cluster cores and mediating the heat flow into cool gas clouds, then it ought to be included in cosmological simulations of galaxy clusters. Dolag et al. (2004) presented the first simulations of conductive intracluster media in cosmological galaxy clusters, using the Jubelgas et al. (2004) implementation of isotropic conduction in the GADGET smooth-particle hydrodynamics code. These simulations were notable in that they vividly demonstrated the inability of conduction to prevent a cooling catastrophe. Even at 1/3 of the full Spitzer rate, thermal conduction could not prevent a large cooling flow from developing in the cluster core. Conduction delayed the cooling catastrophe but did not diminish the magnitude of the eventual mass inflow. An updated set of cluster simulations using the same conduction algorithm was recently performed by Fabjan et al. (2011). Isotropic conductivity was set to 1/3 of the full Spitzer value for simulations including cooling and star formation but without AGN feedback. As with the previous simulation set, there was not much difference between the global properties of the final states of clusters simulated with and without thermal conduction. Conduction is also expected to smooth out temperature inhomogeneities in hot clusters (e.g., Markevitch et al. 2003). One of the clusters simulated by Dolag et al. (2004) was quite hot, with a peak gas temperature of ∼ 12 keV, and conduction in that cluster produced a much more homogeneous ICM, with far less azimuthal temperature variation than its conduction-free counterpart. Producing simulated clusters with smallscale temperature variations similar to those of observed clusters continues to be somewhat tricky (see, for example, Ventimiglia et al. 2012). In particular, some recent simulations of sloshing, magnetized cluster cores with anisotropic conduction are suggesting that conduction needs to be somewhat suppressed along the magnetic field lines into order to explain observations of cluster cores with cold fronts (ZuHone et al. 2013) Implementations of anisotropic conduction directed along magnetic field lines are also of interest because the MHD instabilities that result may have important implications for the structure of cluster cores and the generation of turbulence in cluster outskirts (e.g., Parrish et al. 2009, 2012). Ruszkowski et al. (2011) recently simulated a cosmological cluster with anisotropic conduction using the FLASH adaptive mesh-refinement (AMR) code. Runs were performed with radiative cooling on and off, but no AGN feedback was included. In this case, anisotropic conduction significantly reduced mass accretion into the central galaxy, even though the magnetic field geometry significantly suppressed radial heat conduction. However, the ∼ 31 kpc/ h effective resolution was insufficient to adequately resolve core structure. In this paper we present a new set of cosmological cluster simulations performed with the AMR code Enzo , including isotropic Spitzer conduction. We simulate 10 clusters spanning a mass range of 2 -8 × 10 14 M glyph[circledot] . We also model a range of suppression factors in order to evaluate the effects of thermal conductivity on core structure. As with previous calculations, we do not include AGN feedback, and a cooling catastrophe therefore results, once again confirming that conduction alone cannot prevent strong cooling flows from developing in galaxy clusters. However, our focus here is on how conduction affects the radial profiles and homogeneity of gas density and temperature outside the central ∼ 20 kpc. Our presentation of these conductive cluster simulations proceeds as follows. In § 2 we describe the primary features of the Enzo code employed in this work and describe our recent implementation of isotropic Spitzer conduction, with the simulation setup given in § 3. Section 4 discusses the qualitative morphological properties of the simulated clusters, while § 5 compares their radial profiles. Section 6 focuses on conduction affects the thermal homogeneity of the ICM, § 7 looks at the star formation histories of the clusters, and § 8 summarizes our results.",
"pages": [
1,
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},
{
"title": "2. NUMERICAL METHODS",
"content": "In this work, we use the open-source cosmological, adaptive mesh-refinement (AMR) hydrodynamic + Nbody code Enzo 3 (Bryan & Norman 1997a,b; Norman & Bryan 1999; O'Shea et al. 2004, 2005). Below, we describe the primary features of Enzo used in this work, including newly-added machinery for solving thermal conduction. All analysis is performed with the open-source simulation analysis toolkit, yt 4 (Turk et al. 2011) using the Parallel-HOP halo finder (Skory et al. 2010).",
"pages": [
2
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},
{
"title": "2.1. The Enzo code",
"content": "The Enzo code couples an N-body particle-mesh (PM) solver (Efstathiou et al. 1985; Hockney & Eastwood 1988) used to follow the evolution of a collisionless dark matter component with an Eulerian AMR method for ideal gas dynamics by Berger & Colella (1989), which allows high dynamic range in gravitational physics and hydrodynamics in an expanding universe. This AMR method (referred to as structured AMR) utilizes an adaptive hierarchy of grid patches at varying levels of resolution. Each rectangular grid patch (referred to as a 'grid') covers some region of space in its parent grid which requires higher resolution, and can itself become the parent grid to an even more highly resolved child grid . Enzo 's implementation of structured AMR places no fundamental restrictions on the number of grids at a given level of refinement, or on the number of levels of refinement. However, owing to limited computational resources it is practical to institute a maximum level of refinement, glyph[lscript] max . Additionally, the Enzo AMR implementation allows arbitrary integer ratios of parent and child grid resolution, though in general for cosmological simulations (including the work described in this paper) a refinement ratio of 2 is used. Multiple hydrodynamic methods are implemented in Enzo . For this work, we use the method from the ZEUS code (Stone & Norman 1992a,b) for its robustness in conditions with steep internal energy gradients, which are common in cosmological structure formation when radiative cooling is used. We use the metallicity-dependent radiative cooling method described in Smith et al. (2008) and Smith et al. (2011). This method solves the nonequilibrium chemistry and cooling for atomic H and He (Abel et al. 1997; Anninos et al. 1997) and calculates the cooling and heating from metals by interpolating from multi-dimensional tables created with the photoionization software, Cloudy 5 (Ferland et al. 1998). Both the primordial and metal cooling also take into account heating from a time-dependent, spatially uniform metagalactic UV background (Haardt & Madau 2001). To model the effects of star formation and feedback, we use a modified version of the algorithm presented by Cen & Ostriker (1992). A grid cell is capable of forming stars if the following criteria are met: the baryon overdensity is above some threshold (here 1000), the velocity divergence is negative (i.e., the flow is converging), the gas mass in the cell is greater than the Jeans mass, and the cooling time is less than the self-gravitational dynamical time. Because conduction can potentially balance radiative cooling, instead of the classical definition of the cooling time, e/ ˙ e , we use a modified cooling time that includes the change in energy from conduction, defined as where ˙ e rad is the cooling rate and ˙ e cond is the conduction rate. In the original implementation, if all of the above criteria are satisfied, then a star particle representing a large, coeval group of stars with the following mass is formed: where f ∗ is an efficiency parameter, m cell is the baryon mass in the cell, t dyn is the dynamical time of the combined baryon and dark matter fluid, and ∆ t is the timestep taken by the grid containing the cell in question. The factor of t dyn is added to provide a connection between the physical conditions of the gas and the timescale over which star formation and feedback occur. When a star particle is created, feedback in the form of thermal energy, gas, and metals is returned to the grid at a rate given by where t ∗ is the creation time for the particle, so that ∆ m sf rises linearly over one dynamical time, then falls off exponentially after that. This feature ensures that the feedback response unfolds over a dynamical time, even if star formation in the simulation is formally instantaneous. We also use a distributed feedback model, in which feedback material is distributed evenly over a 3 × 3 × 3 cube of cells centered on the star particle's location. This was shown by Smith et al. (2011) to more effectively transport hot, metal-enriched gas out of galaxies and to minimize overcooling issues that are common to simulations of structure formation. This method was also shown by Skory et al. (2013) to provide a better match to observed cluster properties than injecting all feedback into a single grid cell. In practice, Enzo simulations often implement an additional restriction, allowing creation of a star particle only if m ∗ is above some minimum mass. This prevents the formation of a large number of low-mass star particles whose presence can significantly slow down a simulation. However, the use of this algorithm in concert with conduction must be done with care. As we discuss in § 2.2, explicit modeling of thermal conduction can require timesteps that are significantly shorter than the hydrodynamic Courant condition, which has the effect of considerably reducing the value of m ∗ . If the value of m ∗ is only slightly less than the minimum particle size, this condition will likely be satisfied in a relatively short time as condensation proceeds, causing the gas density to increase and its dynamical time to decrease. When the conduction algorithm is active, the resulting time delay can allow cold, dense gas to persist while in thermal contact with the hot ICM. Since radiative cooling is proportional to the square of the density, heat conducted into high-density gas can be easily radiated away. This artificial time delay therefore produces a spurious heat sink in the ICM that can potentially boost the rates of cooling and star formation. In order to avoid this unphysical behavior, we remove the factor of ∆ t/t dyn from Equation 2 and adopt a constant value of 10 Myr for t dyn for use in Equation 3. We have tested the effects of these changes by comparing two simulations run without conduction using both the original star-formation implementation and this modification and find the stellar masses at any given time differ by less than one percent.",
"pages": [
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},
{
"title": "2.2. Conduction Algorithm",
"content": "We implement the equations of isotropic heat conduction in a manner similar to that of Jubelgas et al. (2004), where the heat flux, j , for a temperature field, T , is given by where κ is the conductivity coefficient. In an ionized plasma, heat transport is mediated by Coulomb interactions between electrons. In this scenario, referred to as Spitzer conduction (Spitzer 1962), the conductivity coefficient is given by where m e , n e , λ e , and T e are the electron mass, number density, mean free path, and temperature, and k B is the Boltzmann constant. The electron mean free path is where e is the electron charge and ln Λ is the Coulomb logarithm, defined to be the ratio of the maximum and minimum impact parameters over which Coulomb collisions yield significant momentum change in the interacting particles, which are electrons in this case. The Coulomb logarithm for electron-electron collisions is (Huba 2011, pg. 34). Because of its relative insensitivity to n e and T e , we follow Jubelgas et al. (2004) and Sarazin (1988) and adopt a constant value of ln Λ = 37 . 8, corresponding to n e = 1 cm -3 and T e = 10 6 K. For values more appropriate to the ICM ( n e = 10 -3 cm -3 , T e = 10 7 K), ln Λ = 43 . 6. At a constant density, ln Λ varies by ∼ 10-20% over the range of temperatures relevant here. The Spitzer conductivity then reduces to Since ln Λ increases with T and κ sp ∝ 1 / ln Λ, including a direct calculation of ln Λ would serve to somewhat soften dependence of κ sp on T . In low density plasmas with large temperature gradients, the characteristic length scale of the temperature gradient, glyph[lscript] T ≡ T/ |∇ T | , can be smaller than the electron mean free path, at which point Equation 5 no longer applies. Instead, the maximum allowable heat flux is described by a saturation term (Cowie & McKee 1977), given as To smoothly connect the saturated and unsaturated regimes, we use an effective conductivity (Sarazin 1988) given by The rate of change of the internal energy, u , due to conduction is then Our numerical implementation closely follows that of Parrish & Stone (2005). We use an explicit, first-order, forward time, centered space algorithm, which is the most straightforward to implement in an AMR code. For each cell, Equation 11 is solved by summing the heat fluxes from all grid cell faces, and calculating the electron density and temperature on the grid cell face as the arithmetic mean of the cell and its neighbor sharing that face. The time step stability criterion for an explicit solution of the conduction equation is where ∆ x is the grid cell size and α is the thermal diffusivity, defined as Equation (12) has the potentially to be considerably more constraining than the hydrodyamical Courant condition, which is proportional to T -1 / 2 ∆ x , and so the conduction timestep in conditions typical of the ICM is often much shorter than the hydrodynamic timestep. The conduction timestep is calculated on a per-level basis and is taken to be the minimum of all such values on a given level. Enzo pads each AMR grid patch with three rows of ghost zones from neighboring grids. This allows the conduction routine to take three sub-cycled time steps for every hydrodynamic step, and thus allows the minimum grid timestep to be a factor of three larger than Equation 12. After three steps, temperature information from the outermost ghost zone has propagated to the edge of the active region of the grid, so performing additional conduction cycles would yield inconsistent solutions with neighboring grids. By default, Enzo rebuilds the adaptive mesh hierarchy for a given refinement level after each timestep, which is computationally expensive. However, because conduction does not explicitly change the density field (the value of which is the only quantity that determines mesh refinement), we modify this behavior such that the hierarchy is only rebuilt after an amount of time has passed equivalent to what the minimum timestep would be if conduction were not enabled. This implementation models the case of isotropic Spitzer conduction, where heat flows unimpeded along temperature gradients. If magnetic fields are present, and strong enough that the electron gyroradius is small compared to the physical scales of interest (which is likely to be true in the intracluster medium), then heat is restricted to flow primarily along magnetic field lines, so the heat flux becomes the dot product of the temperature gradient with the magnetic field direction (i.e., j = -κ bb · ∇ T , where b is the unit vector pointing in the direction of the magnetic field). Diffusion perpendicular to magnetic field lines is generally considered to be negligible. We do not consider magnetic fields in this work, but instead approximate the suppression of conduction by magnetic fields by adding a suppression factor, f sp , varying from 0 to 1, to the heat flux calculation. In reality, the strength and orientation of magnetic fields in galaxy clusters is poorly understood. Hence, the degree to which conduction is suppressed from its maximum efficiency is not known. The extreme limit of tangled magnetic fields is equivalent to isotropic conduction with f sp = 1 / 3. A number of works have shown that the presence of magnetic fields alongside conduction can create a variety of instabilities that greatly influence the effective level of isotropic heat transport (e.g., Parrish & Stone 2005, 2007; Ruszkowski & Oh 2010; McCourt et al. 2011, 2012). For this reason, we simulate a large range of values of f sp , from strongly suppressed ( f sp = 0 . 01) to fully unsuppressed ( f sp = 1).",
"pages": [
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},
{
"title": "3. SIMULATION SETUP",
"content": "The simulations described in this work are are initialized at z = 99 assuming the WMAP Year 7 'best fit' cosmological model (Larson et al. 2011; Komatsu et al. 2011): Ω m = 0 . 268, Ω b = 0 . 0441, Ω CDM = 0 . 2239, Ω Λ = 0 . 732, h = 0 . 704 (in units of 100 km/s/Mpc), σ 8 = 0 . 809, and using an Eisenstein & Hu power spectrum (Eisenstein & Hu 1999) with a spectral index of n = 0 . 96. We use a single cosmological realization with a box size of 128 Mpc/ h (comoving) and a simulation resolution of 256 3 root grid cells and dark matter particles. From this realization, we select the 10 most massive clusters and resimulate each individually, allowing Note . - Masses and equivalent temperatures (as given by Equation 15) for the galaxy clusters in the sample. Note, the clusters are ordered by the masses computed by a halo finder in an original exploratory simulation. adaptive mesh refinement only in a region that minimally encloses the initial positions of all particles that end up in the halo in question at z = 0. We allow a maximum of 5 levels of refinement, each by a factor of 2, refining on baryon and dark matter overdensities of 8. This gives us a maximum comoving spatial resolution of 15.6 kpc/ h . We set the Enzo parameter MinimumMassForRefinementLevelExponent to -0 . 2 for both dark matter and baryon-based refinement, resulting in slightly super-Lagrangian refinement behavior. For each halo in our sample, we perform a control simulation with f sp = 0 and conductive simulations with f sp = 0 . 01, 0.1, 0.3, and 1, evolving each to z = 0. All of the simulations employ the star formation/feedback and radiative cooling methods described above in § 2.1. Table 1 lists the masses and equivalent temperatures for all 10 clusters in the sample.",
"pages": [
4,
5
]
},
{
"title": "4. QUALITATIVE MORPHOLOGY",
"content": "Perhaps the most surprising finding to emerge from our simulations is the insensitivity of the qualitative morphological features of the ICM to thermal conduction, even in clusters with typical gas temperatures ∼ 5 keV. Figures 1, 2, and 3 show gas density, temperature, and X-ray surface brightness projections for three clusters at the minimum and maximum conduction efficiency. We choose these clusters in order to capture a wide range of potential influence from conduction. Halo 0 is the most massive cluster in the sample (with a mass of 8 × 10 14 M glyph[circledot] ), Halo 2 is a highly unrelaxed cluster, and Halo 5 is relatively relaxed and is close to the average mass for the sample (M 200 glyph[similarequal] 4 . 9 × 10 14 M glyph[circledot] ) In order to highlight differences between values of f sp , we limit the projected line of sight to 0.5 Mpc. The morphological similarity between these conductive and non-conductive clusters starkly contrasts with the obvious differences found by Dolag et al. (2004) for a ∼ 12 keV cluster. In their work, it is immediately obvious which simulation included conduction. There are, however, some subtle but noticeable morphological differences between our conductive and nonconductive clusters. The overall density structure is slightly smoother for f sp = 1 in all three clusters, and the temperature field is visibly smoother in Halo 5. In all three clusters shown, the core appears to be marginally less dense with conduction on and is also markedly hotter in Halos 2 and 5. Nevertheless, while it is for the most part apparent that conduction is present in our cluster sample with f sp = 1, its effects are not nearly as striking as in Dolag et al. (2004). Quite possibly, this difference is due to the strong temperature dependence of Spitzer conductivity, suggesting that it may be possible to estimate the typical conductivity of the ICM by analyzing the dependence of azimuthal temperature homogeneity on cluster temperature (or lack thereof) in a large sample of galaxy clusters.",
"pages": [
5,
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},
{
"title": "5. RADIAL PROFILES",
"content": "Systematic conductivity-dependent differences among our simulated clusters are hard to see in individual cases but are more apparent in comparisons of average properties of sample sets with differing conductivity. Therefore, in order to understand the systematic effects of increasing the conduction efficiency, we aggregate the radial profiles from all the clusters in our simulated sample for each value of f sp , as shown in Figure 4. To minimize artifacts that result from the rebinning of profile data, we perform the initial profiling in physical distance units to calculate the virial radius for each cluster (which we take for convenience to be r 200 with respect to the critical density). We then create a second set of profiles in units of r/r 200 that are combined to make the aggregate profiles presented here. We refer to these as 'averaged profiles' and employ this method for Figures 4 and 5 and all subsequent figures showing aggregate properties.",
"pages": [
6
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},
{
"title": "5.1. Cluster Interiors",
"content": "The presence of conduction leads to higher gas temperatures both inside and outside of the conduction-free cluster's characteristic temperature peak at r/r 200 ∼ 0 . 07. Within r/r 200 ∼ 0 . 07, gas temperatures in conductive clusters are greater than those in the non-conductive control simulation, with values of f sp glyph[greaterorsimilar] 0 . 1 leading to nearly isothermal cores and central temperatures ∼ 40% greater than in non-conductive cluster simulations. The temperature profiles of our conductive clusters are in reasonably good agreement with those of Ruszkowski et al. (2011) for their cluster with isotropic conduction and f sp = 1 / 3, although the core of their cluster is less isothermal than ours, with cooler temperatures near the center. This could be because the simulations of Ruszkowski et al. (2011) did not include a prescription for star formation and feedback in cluster galaxies, which can increase the core temperature both by consuming the cold gas and through thermal and mechanical feedback. Agreement with the temperature profiles of the simulated clusters from Dolag et al. (2004) is not nearly as good. Those clusters show a significant reduction in both the peak and central temperatures with conduction present, which we do not observe. The theoretical temperature profiles of McCourt et al. (2013) for the 10 14 . 5 M glyph[circledot] clusters show an elevation in the core temperatures in conductive clusters in rough agreement with our results. We also see a marginal inward movement of the location of the temperature peak as they predict, although not nearly to the same degree. The temperature peak for our clusters is also at a smaller radius to begin with. Despite its ability to maintain approximate isothermality in the cluster cores, conduction at even maximal efficiency is unable to avert a cooling catastrophe at the very center of the halo. This cooling catastrophe and the resulting condensation and star-formation activity produce sharp peaks at r/r 200 glyph[lessorsimilar] 0 . 03 in all the gas density profiles in Figure 4, as well as a slight temperature increase at the same location in the more highly conductive clusters. The elevated central temperatures at the centers of conductive clusters reduce the central gas density relative to the control clusters by ∼ 20-30%. Material that would have fallen into the center is displaced to larger radii, as can be seen by the elevated density at 0 . 04 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 6 in the more conductive runs. Gas density is enhanced primarily in the range 0 . 04 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 2, but the enhancement appears to saturate at ∼ 20% for f sp = 0 . 33. For f sp = 1, the density is actually lower than for f sp = 0 . 33 in this range, but is then higher for 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 6. It is unclear what causes the outward transport (or prevention of inflow) to stall just inside 0.2 r 200 , but it seems that it can be overcome for some value of f sp above 0.33, likely closer to 1. Further evidence of conduction-driven inflation of the core can be seen in Figure 5, where we plot averaged profiles of M/r , including the individual contributions from dark matter, stars, and gas. Despite the lack of perfect monotonicity in the density and temperature profiles, the gaseous component of the potential decreases monotonically with increasing f sp , indicating that the clusters are indeed responding to the presence of conduction by puffing up their cores and redistributing gas to larger radii. As is the case for most simulated galaxy clusters, the stellar component is extremely centrally concentrated, dominating the gravitational potential inside 0.04 r 200 . The stellar and dark matter components of the potential are largely unaffected by conduction. In the right panel of Figure 4, we plot normalized entropy profiles, where the normalization term, K 200 (Voit 2005; Voit et al. 2005), is given by where T 200 is and ¯ n e is the average electron number density within r 200 , assuming a fully ionized plasma of primordial composition. The decrease in density and increase in temperature in the cluster cores with conduction yield entropy profiles that are enhanced by 40-70%, but still show the steep decline in the very center that is typical of simulated clusters. This enhancement decreases out to r/r 200 ∼ 0 . 07, where the mean entropy values are approximately equivalent for all values of f sp . The entropy profiles for the clusters with conduction then dip below the control sample by roughly 10% out to r/r 200 ∼ 0 . 2. In the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 6, the combined temperature and density enhancements appear to be in perfect balance, producing nearly identical entropy profiles across the entire cluster sample with a very small variance. Voit et al. (2005) find that for a sample of nonradiative, non-conducting clusters simulated with Enzo , the entropy profiles in the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 1 are best fit by the power-law which we overplot on Figure 4 with a black dashed line. This power law matches our sample well in normalization, but has a slightly steeper slope. We find that our sample is best matched by For comparison, we also plot the predicted entropy profile for a pure cooling model (Voit & Bryan 2001; Voit et al. 2002) of a 5 keV cluster. This model assumes gas and dark matter density distributions that follow an NFW profile (Navarro et al. 1997) with concentration c = 6. The gas is initially in hydrostatic equilibrium and allowed to cool for a Hubble time. The pure cooling model agrees quite well with our clusters in the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 1, and especially well in the range 0 . 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 0 . 5. Outside of 0.5 r 200 , the slope of the pure-cooling model is slightly too steep and is closer in slope to the Voit et al. (2005) fit.",
"pages": [
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},
{
"title": "5.2. Cluster Exteriors",
"content": "The outskirts of galaxy clusters, while having lower temperatures and hence lower conductivities, are not subject to intermittent heat injection from stellar feedback, and thus offer an intriguing laboratory for studying the effects of conduction. We identify two distinct regions in the cluster outskirts where conduction appears to have influence, at ∼ r 200 and at ∼ 3 r 200 . At r/r 200 glyph[greaterorsimilar] 0 . 6, the density excess seen in the cluster interiors turns into a deficit for all values of conduction simultaneously, as can be see in Figure 4. From this point out to well past the virial radius, there exists a perfectly monotonic trend of lower gas densities for higher values of f sp . For the maximum value of f sp , the average gas density at the virial radius is 10% lower than that of the clusters simulated without conduction. In fact, the density is measurably lower for all values of f sp ≥ 0 . 1 out to a few virial radii. The temperature just inside r 200 is marginally higher, while the temperature just outside r 200 is reduced. Conduction transports heat outward in these regions because the gravitational potential there produces a declining gas temperature gradient. This heat transfer causes the entropy of the outer gas to increase. However, because gas in the cluster outskirts is not pressureconfined like the gas in the core, it is free to expand outward and decrease in both density and temperature, while its temperature gradient remains determined by the gravitational potential. This happens because the timescale for conduction in the outer regions is substantially greater than the sound-crossing time. Consequently, conduction of heat outward causes the outer gas to expand without much change in the temperature gradient. Beyond r 200 , however, we see a slight steepening in the temperature profile as predicted for conductive clusters by McCourt et al. (2013), presumably because inflation of the ICM due to conduction pushes gas near the virial radius farther from the cluster center without adding much thermal energy. The fact that the entropy profiles of conductive clusters are lower than those of non-conductive clusters beyond the virial radius supports this interpretation. Further out, at ∼ 3 r 200 , there is another systematic decrease in both density and temperature. Interestingly, as was pointed out by Skillman et al. (2008), this is the typical location for a galaxy cluster's accretion shocks, which are responsible for heating gas up to the virial temperature. This raises a critical question: How can conduction affect a large halo's accretion shocks? To test this question in a more controlled environment than a cosmological simulation, we performed a pair of one-dimensional simulations designed to mimic the conditions of an accretion shock around a galaxy cluster. Following the shock properties characterized by Skillman et al. (2008), we initiated a Mach 100 standing shockwave with preshock conditions of n = 10 -4 cm -3 and T = 10 4 K, comparable to gas that is falling directly onto a galaxy cluster (i.e., through spherical accretion of the surrounding intergalactic medium, rather than being accreted via filaments). We ran simulations with f sp = 0 and 1, with profiles shown in Figure 6. Within tens of Myr, a conduction front in the f sp = 1 simulation races ahead of the original shock. The conduction front continues to advance with ever-decreasing speed and settles into a nearsteady state by t = 200 Myr, the time shown in the figure. Conduction of heat ahead of the main shock front therefore results in a separation of the density and temperature jumps, effectively creating two shocks, one produced by conductive preheating and a second nearly isothermal shock front some distance downstream. The lower-right panel of Figure 6 shows the Mach numbers determined using the shock-finding algorithm described in Skillman et al. (2008). Bifurcation of the original Mach 100 shock has produced a Mach ∼ 70 conductive-precursor shock followed by a Mach ∼ 1 . 5 shock where the main density jump occurs. This splitting is also evident in the plot showing the local Mach ratio, v/c s (lower-middle panel of Figure 6) on either side of the temperature jump. The heat transfer upstream also causes a slight drop in pressure at the original shock front, shifting the primary den- jump downstream by about 5 kpc. These results are independent of resolution. The simulations shown in Figure 6 are for a grid 128 cells across, and we observe nearly identical behavior down to a resolution of 16 cells. For this configuration, the results are strongly dependent on f sp . At f sp = 0 . 67, the distance between the separated shocks is approximately half of that at f sp = 1 and the Mach number of the primary shock is only reduced to just under 90. For f sp glyph[lessorsimilar] 0 . 4, the results are indistinguishable from those without conduction. However, we find that it is possible to produce significant shockwave alteration for lower values of f sp simply by lowering the initial Mach number and increasing the preshock temperature (resulting in gas in a thermodynamic regime comparable to gas that is being accreted from cosmological filaments). As long as the postshock gas is able to reach temperatures in the range of 10 7 K, where the Spitzer conductivity becomes considerable, conduction is capable of bifurcating the shock front. Figure 7 shows the average Mach number as a function of gas density in our simulations of the most massive halo for all shocks in the subvolume in which refinement is allowed. Conduction reduces the Mach numbers of the strongest shocks, which occur preferentially in the lowest density gas, by roughly 10% from f sp = 0 to f sp = 1. We therefore conclude that the density, temperature, and entropy deficits observed beyond the virial radius in our conductive clusters are indeed due to shock bifurcations similar to those seen in our idealized one-dimensional simulations of conductive shock fronts. Yet, the physics of the actual accretion shocks around real galaxy clusters is undoubtedly more complex. In particular, it is important to note that the electron mean free path in that gas is several times greater than the intershock distance in our one-dimensional simulations, and furthermore that real accretion shocks are likely to be collisionless and magnetically-mediated. Nonetheless, the general qualitative point these simulations illustrate is interesting: Heating of preshock gas by a hot electron precursor has the potential to alter the expected relationships between the sizes and locations of the density and temperature jumps in accretion and merger shocks. Progress in understanding the effects of conduction on accretion shocks will require modeling the gas as a fully ionized plasma (Zel'dovich 1957; Shafranov 1957), which is beyond the scope of this work. The high Mach number of an accretion shock combined with a post-shock temperature high enough for significant heat flux create conditions similar to a supercritical radiative shock, as described by Lowrie & Rauenzahn (2007). When simulated with a two-fluid approach, the combination of preheating of the preshock medium via conduction in the electrons, electron-ion coupling, and compression of the ion fluid in the postshock region can produce a small region, known as a Zel'dovich spike, where the ion temperature is slightly higher than the equilibrium postshock temperature (Lowrie & Rauenzahn 2007; Lowrie & Edwards 2008; Masser et al. 2011). Our single-fluid simulations, despite their limitations, produce shocks quite similar to the non-equilibrium results of Lowrie & Edwards (2008), where a diffusion term proportional to T 5 / 2 (like Spitzer conductivity) is used. Nevertheless, a more detailed study of the characteristics of accretion shocks employing a two-fluid MHD treatment along with physically motivated conduction and cooling rates seems warranted.",
"pages": [
7,
8,
9
]
},
{
"title": "6. TEMPERATURE HOMOGENEITY",
"content": "Conduction strong enough to alter the temperature gradients in cluster cores should also be effective at smoothing out small-scale thermal variations in the ICM. However, those effects turn out to be rather subtle, as shown in Figure 8, which plots the normalized variance of the temperature field as a function of radius, averaged over all clusters in the sample that have the same level of conductivity. It reveals an extremely weak trend of greater temperature homogeneity with increasing f sp , but at all radii the difference in homogeneity among cluster sets with different values of f sp is less than the clusterto-cluster variation. Furthermore, this trend reverses be- 200 yond the virial radius, at 2 glyph[lessorsimilar] r/r 200 glyph[lessorsimilar] 3, albeit at a nearly marginal level. The homogenizing effects of conduction are even harder to see in projection, which perhaps is not surprising given the scarcity of obvious conduction-dependent morphological differences in Figures 1, 2, and 3. Those differences are further diluted when projected over a 4 Mpc line of sight through each cluster, as in Figures 911. These latter figures show mean temperature weighted by the X-ray emission in the 0.5-2 keV energy band, and X-ray emission is calculated by interpolating from density, temperature, and metallicity-dependent emissivity tables computed with the Cloudy code. Since Xray emission is proportional to n 2 e T 1 / 2 e , this weighting should highlight clumpiness and differences in temperature. However, as stated previously, these results contrast considerably with the cluster map comparison of Dolag et al. (2004), which shows a significantly hotter cluster in which conduction should be much more efficient. The change in temperature homogeneity due to conduction is quite small in our simulated cluster sample, but important insights into the physics of the ICM could be gained if the effective value of f sp could be measured observationally. To evaluate this possibility, we created X-ray-weighted temperature maps of the central 300 kpc for each of the clusters in the sample, masking out the pixels within 40 kpc of the cluster centers to remove features that would be considered part of the central galaxy. We then quantified the amount of azimuthal temperature structure by dividing each temperature map into azimuthal bins and calculating both the mean temperature in each bin and the temperature variance among all azimuthal bins in the map. We performed this calculation multiple times for each map while rotating the azimuthal bins, and took the maximum variance calculated as the value for that map. Finally, we averaged the values together for all clusters in the sample and performed the entire exercise over a range in the total number of azimuthal bins, from 2 to 9. Figure 12 plots the maximum variance as a function of the number of bins for each value of f sp . We find that, in general, the maximum variance is lower for the simulations with conduction, but only by approximately 10%. There does not appear to be any sort of monotonic trend with increasing f sp . We repeated this experiment, varying the inner and outer radius for the temperature maps, but were unable to find conditions that produce a better trend than can be seen in Figure 12. Thus, we conclude that the efficiency of conduction is difficult to determine solely from the observable temperature homogeneity of the ICM, at least for clusters of temperature glyph[lessorsimilar] 6 keV.",
"pages": [
9,
10,
11
]
},
{
"title": "7. STAR FORMATION",
"content": "Figure 13 shows the average difference in stellar mass between the clusters with conduction and those without as a function of time. The difference in stellar mass at z = 0 for all levels of conduction is less than 5%. Surprisingly, the clusters with higher levels of conduction form more stars in our calculations (see also Dolag et al. 2004). However, the shaded regions in Figure 13 give an indication of just how much variation there is between clusters. As we have shown, even the highest level of conduction is unable to prevent a cooling catastrophe in the center of a cluster. Therefore, one should not expect a large difference in the amount of star formation. Given the coarseness with which the star forming regions are resolved, it is possible that the enhancement in star formation with increasing f sp is numerical and not physical. We propose two potential numerical explanations. First, the short timesteps required for the stability of the conduction algorithm may not provide enough time for a single star particle to sufficiently heat up a grid cell and quench star formation in a given cell in the following timestep. Second, conduction may transport thermal energy too quickly out of regions heated by recent star formation. This would allow a star-forming region to recool and form additional stars too rapidly. Because conduction is able to create a nearly isothermal core for f sp ≥ 0 . 1, yet the final stellar mass from the clusters with f sp = 0 . 1 is consistent with no change, we find it reasonable to conclude that the isothermality of the core in a conductive cluster has no influence on the star formation rate Enzo calculates for the central galaxy. However, because we do not resolve the interface between the ICM and the interstellar medium (ISM), we cannot definitively state the effect of conduction on star-forming gas in a cosmological simulation.",
"pages": [
11
]
},
{
"title": "8. CONCLUSIONS",
"content": "We have performed cosmological simulations of 10 galaxy clusters using isotropic thermal conduction with five values of the conductive suppression factor in order to study the effects of conduction on galaxy cluster cores and the intracluster medium. By studying the aggregate properties of the clusters in our sample, we find that the presence of conduction even at its maximum possible efficiency induces changes to the density and temperature structure on the order of only 20-30%. For f sp ≥ 0 . 1, the cluster cores become roughly isothermal. However, conduction at any level is incapable of stopping the cooling catastrophe at the very centers of our clusters, where the density profile is always very sharply peaked. To some extent, this is due to our limited spatial resolution, since the temperature gradients on which heat conduction depends are limited by the scale of the smallest grid cell, which at ∼ 15 kpc/ h is still quite large compared to the scales of galaxies and the Field length. However, the extremely well-resolved study of Li & Bryan (2012) also finds that conduction can at best slightly delay the cooling catastrophe. While conduction is unable to prevent the cooling catastrophe, the elevation of gas entropy in a conductive, isothermal core displaces some of the core gas, moving it out to larger radii. For values of f sp up to 0.33, this material is redistributed mostly within ∼ 0 . 2 r 200 . For higher values of f sp , it is transported out even further, up to ∼ 0 . 6 r 200 . A similar phenomenon occurs around r 200 , where the negative temperature gradient allows outward heat conduction to inflate the outer parts of the cluster. However, because this material is not deep in the potential well, it is free to expand and cool, leading to slightly lower temperatures just outside the virial radius. More surprisingly, we observe a systematic decrease in both the density and temperature with increasing f sp at large radii, out to ∼ 3 r 200 . We hypothesize that this is due to alteration of the accretion shocks by conduction. To test this, we perform one-dimensional 'shock tube' simulations with conditions characterizing an accretion shock around a galaxy cluster, with the level of conduction treated as the sole free parameter. As long as the post-shock temperature is high enough for the Spitzer conductivity to be efficient ( T glyph[greaterorsimilar] 10 7 K), conduction moves the temperature jump upstream and the density jump downstream of the original shock face. This creates two distinct shocks, both with Mach numbers less than the original shock. We conclude that conduction is responsible for the systematic decrease in density and temperature in the outskirts of our simulated clusters, because it acts to weaken the shocks. We acknowledge that our modeling of this problem is not totally accurate, and instead requires a two-fluid MHD approach, which is beyond the current capabilities of our simulation tool. However, more rigorous two-fluid simulations of shockwaves in fully ionized plasmas show qualitatively similar behavior, save a tiny feature in the ion temperature that cannot be achieved in a single-fluid approach. Unfortunately, because the effect on clusters is only at the 10% level and at very large radii, where the X-ray surface brightness of the plasma is extremely low, observing the effects of conduction on accretion shocks may never be possible. We also find that in addition to altering temperature gradients, conduction is able to make the intracluster medium more thermally uniform. This effect, while measurable in spherically-averaged radial profiles, is almost totally lost in projection. Our results contrast with the temperature maps of Dolag et al. (2004), wherein the effect of conduction is instantly recognizable. The cluster shown in Dolag et al. (2004) is significantly more massive than our most massive cluster, so it is possible that a hotter ICM, with a higher thermal conductivity, is made more homogeneous, suggesting that the temperature dependence of temperature inhomogeneity in a large cluster sample could help reveal the typical conductivity of the ICM. We attempted to find a means of distinguishing the level of conduction observationally by measuring the variance in our projected temperature maps, but without success. Finally, conduction appears to have very little influence on the star formation rate within our simulated clusters. When determining whether a grid cell should form a star particle, we include the energy change from conduction in the calculation of the cooling time, but this seems to have very little influence. This is likely because star-forming grid cells are surrounded mostly by cells that are also quite cool. Somewhat surprisingly, we observe a marginal increase in the total stellar mass with increasing conduction, such that the sample with f sp = 1 shows an enhancement in star formation rate of ∼ 5%. The fact that conduction cannot suppress star formation is directly related to its inability to prevent the cooling catastrophe in the very center of the cluster. However, the reasons for the slight increase in star formation may be more numerical than physical. Further progress on understanding the effects of thermal conduction on star formation in cluster cores will require properly resolving the interface between ISM and the ICM, which at present is impractical in cosmological galaxy cluster simulations. This work was supported by NASA through grant NNX09AD80G and NNX12AC98G, and by the NSF through AST grant 0908819. The simulations presented here were performed and analyzed on the NICS Kraken and Nautilus supercomputing resources under XSEDE allocations TG-AST090040 and TG-AST120009. We thank Greg Bryan, Gus Evrard, Eric Hallman, Andrey Kravtsov, Jack Burns, Matthew Turk, and Stephen Skory for helpful discussions during the course of preparing this manuscript. SWS has been supported by a DOE Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308. BWO was supported in part by the MSU Institute for Cyber-Enabled Research. Enzo and yt are developed by a large number of independent research from numerous institutions around the world. Their committment to open science has helped make this work possible.",
"pages": [
11,
12
]
},
{
"title": "REFERENCES",
"content": "Abel, T., Anninos, P., Zhang, Y., & Norman, M. L. 1997, New Astronomy, 2, 181 Bregman, J. N. & David, L. P. 1988, ApJ, 326, 639 Bryan, G. & Norman, M. 1997a, 12th Kingston Meeting on Theoretical Astrophysics, proceedings of meeting held in Halifax; Nova Scotia; Canada October 17-19; 1996 (ASP Conference Series # 123), ed. D. Clarke. & M. Fall -. 1997b, Workshop on Structured Adaptive Mesh Refinement Hockney, R. W. & Eastwood, J. W. 1988, Computer Simulation Using Particles (Institute of Physics Publishing) Huba, J. 2011, NRL Plasma Formulary, ed. Huba, J.D. Johnstone, R. M., Fabian, A. C., & Nulsen, P. E. J. 1987, MNRAS, 224, 75 Parrish, I. J., Quataert, E., & Sharma, P. 2009, ApJ, 703, 96 Parrish, I. J. & Stone, J. M. 2005, ApJ, 633, 334 -. 2007, ApJ, 664, 135 Zel'dovich, Y. B. 1957, Sov. Phys. JETP, 5, 919",
"pages": [
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]
}
]
2013ApJ...778..166O
https://arxiv.org/pdf/1401.0932.pdf
<document>
<section_header_level_1><location><page_1><loc_25><loc_85><loc_75><loc_86></location>Spin-down dynamics of magnetized solar-type stars</section_header_level_1>
<text><location><page_1><loc_34><loc_82><loc_66><loc_83></location>R. L. F. Oglethorpe 1 & P. Garaud 2</text>
<text><location><page_1><loc_12><loc_73><loc_88><loc_80></location>1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 2 Department of Applied Mathematics and Statistics, Baskin School of Engineering, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA</text>
<section_header_level_1><location><page_1><loc_44><loc_68><loc_56><loc_70></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_16><loc_32><loc_84><loc_65></location>It has long been known that solar-type stars undergo significant spin-down, via magnetic braking, during their Main-Sequence lifetimes. However, magnetic braking only operates on the surface layers; it is not yet completely understood how angular momentum is transported within the star, and how rapidly the spin-down information is communicated to the deep interior. In this work, we use insight from recent progress in understanding internal solar dynamics to model the interior of other solar-type stars. We assume, following Gough and McIntyre (1998), that the bulk of the radiation zone of these stars is held in uniform rotation by the presence of an embedded large-scale primordial field, confined below a stably-stratified, magnetic-free tachocline by large-scale meridional flows downwelling from the convection zone. We derive simple equations to describe the response of this model interior to spin-down of the surface layers, that are identical to the two-zone model of MacGregor and Brenner (1991), with a coupling timescale proportional to the local Eddington-Sweet timescale across the tachocline. This timescale depends both on the rotation rate of the star and on the thickness of the tachocline, and can vary from a few hundred thousand years to a few Gyr, depending on stellar properties. Qualitative predictions of the model appear to be consistent with observations, although depend sensitively on the assumed functional dependence of the tachocline thickness on the stellar rotation rate.</text>
<text><location><page_1><loc_16><loc_28><loc_80><loc_30></location>Subject headings: MHD - Sun: interior - Sun: magnetic fields - Sun: rotation</text>
<section_header_level_1><location><page_1><loc_40><loc_23><loc_60><loc_24></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_10><loc_88><loc_20></location>It has long been known that there exists a remarkable dichotomy between stars of masses M glyph[star] > 1 . 5 M glyph[circledot] , which typically always remain rapid rotators, and stars of masses M glyph[star] < 1 . 5 M glyph[circledot] , which undergo significant spin-down during their Main-Sequence lifetimes. This dichotomy was first resolved by Schatzman (1962) (see also Schatzman 1959) who noted that the transition coincides with the disappearance of the outer convection zone with increasing stellar mass, and deduced that the latter must play an important role in the spin-down process. He argued, following Parker (1955),</text>
<text><location><page_2><loc_12><loc_73><loc_88><loc_86></location>that convection is necessary for dynamo action, and that magnetic activity drives the ejection of mass from the surface of the star into the interstellar medium. He then estimated the amount of angular momentum lost by the star assuming that the ejecta is forced to corotate with the surface magnetic field until a point where the field strength is no longer strong enough to act on the plasma. He found that for reasonable field strengths this point was sufficiently far out from the star that even a tiny amount of mass-loss could lead to very significant angular-momentum loss. The theory of stellar spin-down via magnetic braking was born.</text>
<text><location><page_2><loc_12><loc_61><loc_88><loc_72></location>Schatzman's original calculation has been refined further over the past five decades, following two parallel lines of investigation. The majority of the effort has been dedicated to improving stellar wind models, and the manner in which they act on the stellar surface (Weber & Davis 1967; Mestel 1968; Li 1999). Questions of interest include how the spin-down process is affected by the strength and geometry of the stellar wind and how the latter depend on the star's mass and rotation rate (Aib'eo et al. 2007; Matt et al. 2012; Reiners & Mohanty 2012).</text>
<text><location><page_2><loc_12><loc_49><loc_88><loc_60></location>However, magnetic braking as currently understood only operates on the surface layers of the star. A second line of investigation has therefore focused on modeling angular-momentum transport within the star, to see how the spin-down information is communicated to the deep interior. Unfortunately, and despite decades of research, only little is known about the subject. Part of the problem resides in the fact that the internal rotation profile of Main Sequence solar-type stars (the Sun excepted) remains extremely difficult to observe.</text>
<text><location><page_2><loc_12><loc_33><loc_88><loc_48></location>Nevertheless, useful information can still be learned about the process by comparing simple idealized models with observations. Since turbulent convection can redistribute angular momentum across an entire convection zone within a matter of weeks to a few years (depending on the stellar type), the outer convective region is usually assumed to rotate more-or-less uniformly. Within the radiation zone, one can either assume that each layer individually conserves angular momentum (Kawaler 1988), or that the entire region is in a state of uniform rotation. In the latter scenario, an additional assumption must be made to determine how angular momentum is transported from the radiation zone to the convection zone.</text>
<text><location><page_2><loc_12><loc_23><loc_88><loc_32></location>Following this idea, MacGregor & Brenner (1991) proposed a 'two-zone' model, now commonly used in statistical comparisons of models with observations. The model assumes that both radiation and convection zones are rotating uniformly with rotation rates Ω core and Ω cz respectively, and that the angular-momentum redistribution between the two regions occurs on a coupling timescale τ c . The system then evolves according to the following coupled ordinary differential equations:</text>
<formula><location><page_2><loc_34><loc_12><loc_88><loc_22></location>d J core d t = -∆ J τ c , d J cz d t + d J core d t = -˙ J w , where ∆ J = I cz I core I core + I cz (Ω core -Ω cz ) , (1)</formula>
<text><location><page_2><loc_12><loc_10><loc_88><loc_11></location>where J represents the angular momentum and I the moment of inertia (such that J = I Ω) of</text>
<text><location><page_3><loc_12><loc_79><loc_88><loc_86></location>each region considered and ˙ J w is the rate of angular-momentum extraction from the entire star by the stellar wind. The model allows for analytical solutions in certain limits, and is very easy to integrate numerically in conjunction with stellar evolution. The real difficulty, if one wishes to use it for quantitative purposes, is to express τ c and ˙ J w as functions of known stellar properties.</text>
<text><location><page_3><loc_12><loc_58><loc_88><loc_78></location>Qualitatively speaking, however, the model provides a simple way of studying the difference in rotational evolution between stars in solid-body rotation and stars which can retain significant differential rotation between the core and the envelope, by suitably choosing τ c . With strong coupling (small τ c ), the region effectively being spun down by the wind encompasses the entire star. By contrast, with weak coupling the region being spun down is at first limited to the convection zone, while the radiation zone only feels the effect of spin-down later, when ∆ J has grown to be sufficiently large. Hence, for the same angular-momentum extraction rate, the apparent spin-down rate of the surface layers is at first much slower for solid-body rotators than for differential rotators. This model then predicts dramatically different rotational histories in the two cases, a prediction that can and has been tested against observations to estimate τ c (Allain 1998; Irwin et al. 2007; Denissenkov et al. 2010; Spada et al. 2011; Gallet & Bouvier 2013).</text>
<text><location><page_3><loc_12><loc_43><loc_88><loc_56></location>Before we proceed to discuss these results, first note that stellar rotation rates in a given very young cluster and a given mass bin, usually exhibit a significant spread, with 'rapid rotators' rotating up to a few tens of times the speed of the slower rotators (Herbst et al. 2001; Lamm et al. 2005; Irwin et al. 2008). This spread is usually attributed to a spread in the initial pre-stellar core conditions, and the length of the initial pre-Main Sequence disk-locking phase (Bouvier et al. 1997). The spread of rotational velocities in these 'initial conditions' then propagates to later ages, with rapid rotators and slow rotators in each mass bin having distinct evolutionary paths.</text>
<text><location><page_3><loc_12><loc_22><loc_88><loc_42></location>By comparing observations to the predictions of a two-zone model, Irwin et al. (2007) found that the rotational periods of rapid rotators are well-explained by assuming solid-body rotation at all time, for any mass bin. By contrast, the solid-body rotation assumption is not consistent with observations for slow-rotators in the mass range 0 . 7 M glyph[circledot] -1 . 1 M glyph[circledot] . A coupling time τ c as large as a Gyr appears to fit the data much better (see also Allain 1998). This suggests that τ c must depend on multiple factors, such as the mass and rotation rate of the star. Their conclusion was confirmed by subsequent work by Denissenkov et al. (2010), Spada et al. (2011) and Gallet & Bouvier (2013). Note that the rotation rates of low-mass stars ( M glyph[star] < 0 . 7 M glyph[circledot] ), by contrast, are fairly well-explained by solid-body rotation. This is not surprising since the latter are fully convective for M glyph[star] < 0 . 35 M glyph[circledot] and nearly fully convective for 0 . 35 M glyph[circledot] < M glyph[star] < 0 . 7 M glyph[circledot] (see also Reiners & Mohanty 2012, for further work on the rotational histories of very low-mass stars in older clusters).</text>
<text><location><page_3><loc_12><loc_15><loc_88><loc_20></location>Going beyond the two-zone model, however, and understanding from a more physical point of view what the source of the dynamical coupling between the radiative and convective regions is, and whether the radiation zone is indeed in uniform rotation, requires peering below the surface.</text>
<text><location><page_4><loc_12><loc_75><loc_88><loc_86></location>Today, this can only be done for a single Main-Sequence star, the Sun 1 . Our goal in this work is to use insight from recent progress in understanding internal solar dynamics to model the transport of angular momentum in the interior of other stars. This, of course, implicitly assumes that the Sun is representative of all solar-type stars, and that its internal workings are not fundamentally different from theirs. Comparing the predictions of any model based on this assumption with observations may, to some extent, help us establish whether it is valid or not.</text>
<text><location><page_4><loc_12><loc_50><loc_88><loc_74></location>Thanks to helioseismology, we now have a good view of the large-scale internal dynamics of the Sun. Its outer convective region spans roughly a third of the solar radius and 2.5% of the solar mass. It is rotating differentially, with an equatorial rotation rate Ω eq glyph[similarequal] 2 . 9 × 10 -6 s -1 , and a polar rotation rate about 25% slower (Thompson et al. 1996; Schou et al. 1998). Meanwhile, the radiative interior is rotating uniformly, with an angular velocity similar to that of the surface at mid-latitudes. The shear layer between the two regions is called the tachocline. It is remarkably thin, with a thickness estimated at 2-4% of the solar radius (Kosovichev et al. 1997; Charbonneau et al. 1999; Elliott & Gough 1999). Further to the question of how the spin-down of the surface layers is communicated to the interior, these observations raise new puzzles: why is the radiation zone rotating uniformly despite the latitudinal shear imposed by the overlying convection zone; and why is the tachocline so thin? Answering these questions in conjunction with the spin-down problem prompts a closer inspection of the various angular-momentum transport processes thought to take place in the Sun (and by proxy, in all solar-type stars).</text>
<text><location><page_4><loc_12><loc_23><loc_88><loc_48></location>Angular-momentum transporters in the solar radiation zone can be split into two categories: hydrodynamic processes and magnetohydrodynamic (MHD) processes. The former include stratified turbulence, large-scale meridional flows, and gravity waves. Purely hydrodynamic models of the solar interior based on all three types of processes, in isolation or in combination, have been studied at length (Spiegel & Zahn 1992; Elliott 1997; Talon et al. 2002; Charbonnel & Talon 2005; Rogers & Glatzmaier 2006; Brun et al. 2011; Wood & Brummell 2012). As discussed by McIntyre (1994), Gough & McIntyre (1998), Gough (2007) and Zahn (2007), however, these models remain problematic. On the one hand, they tend to have difficulties explaining how to maintain uniform rotation, since they often rely on some level of differential rotation to be effective. This is particularly true of turbulent transport, and transport by large-scale flows. On the other hand, they also have difficulty explaining how the gradual extraction of angular momentum from the radiation zone during spin-down can proceed without concurrently inducing significant compositional mixing throughout the interior, which would be inconsistent with helioseismic inversions (Gough & Kosovichev 1990).</text>
<text><location><page_4><loc_12><loc_18><loc_88><loc_21></location>MHDprocesses, which do not suffer from the same limitations, have recently gained popularity as a means to explain helioseismic and related observations of the dynamics of the solar interior</text>
<text><location><page_5><loc_12><loc_70><loc_88><loc_86></location>(see the review by Garaud 2007). Mestel & Weiss (1987) first discussed how the presence of a large-scale primordial field confined to the radiation zone would rapidly suppress any differential rotation within that region. Indeed, the magnetic diffusivity is low in conditions relevant for stellar interiors. If, in addition, meridional flows are weak (which is true in strongly stratified regions), then Ferraro's isorotation law (Ferraro 1937) applies, which states that angular velocity must be constant on magnetic field lines. Although Ferraro's law does not preclude differential rotation entirely since different field lines could, in principle, rotate at different speeds, Mestel & Weiss (1987) argued that interactions between Alfv'en waves along neighboring field lines, through a process called phase mixing, should rapidly suppress any remaining shear.</text>
<text><location><page_5><loc_12><loc_50><loc_88><loc_68></location>Charbonneau & MacGregor (1993) later used this idea to study the effect of spin-down in the solar interior. They considered a model similar to the one of Mestel & Weiss (1987), in which the Sun is threaded by a fixed poloidal field, confined beneath a fixed radius r f . They assumed that the convection zone is gradually spun down by magnetic braking, while remaining at all time in a state of uniform rotation. They then studied the role of the magnetic field in promoting angularmomentum transport, both between the two zones and throughout the radiative interior, by solving simultaneously the axisymmetric azimuthal components of the momentum and magnetic induction equations. In their model, which ignores phase mixing and any meridional flows, (turbulent) viscosity had to be invoked to damp any remaining differential rotation across field lines, and to promote angular-momentum transport in regions where there is no field.</text>
<text><location><page_5><loc_12><loc_26><loc_88><loc_48></location>They found that if the poloidal field has significant overlap with the convection zone (i.e. if r f > r cz , where r cz is the radius of the radiative-convective interface), then the spin-down is very rapidly communicated to the interior. For large enough viscosity, the entire star rotates more-orless as a solid body at all times 2 . On the other hand, if the poloidal field is confined strictly below the convection zone (i.e. if r f < r cz ), then viscosity is also needed to communicate the spin-down from the convection zone to the magnetically-dominated part of the radiation zone. Charbonneau & MacGregor (1993) showed that the system eventually evolves toward a quasi-steady state where both the convection zone and the deep interior are rotating uniformly, but at distinct angular velocities with Ω core > Ω cz . A shear layer separates the two zones, with a thickness that depends on r cz -r f . In this quasi-steady state, the total viscous angular-momentum flux across this shear layer is equal to that extracted by the wind from the surface, and fixes the relative core-envelope lag (Ω core -Ω cz ) / Ω cz .</text>
<text><location><page_5><loc_12><loc_20><loc_88><loc_25></location>These ideas have recently been revisited by Denissenkov (2010), who, in addition to the question of spin-down, also attempted to address the issue of the solar tachocline and of the light-element abundances at the same time. Starting from the model proposed by Charbonneau & MacGregor</text>
<text><location><page_6><loc_12><loc_66><loc_88><loc_86></location>(1993) but ultimately allowing for a differentially rotating convection zone, he first argued that the high value of the viscosity needed to explain the uniform rotation of the radiation zone (to ensure that all field lines are rotating at the same rate) must be of turbulent origin. He then assumed that this turbulence would also transport chemical species at the same rate, and showed that this leads to inconsistent predictions for the surface Li and Be abundance in the Sun. To resolve the problem, he then invoked the work of Spiegel & Zahn (1992) and Zahn (1992) to argue that turbulence in the radiation zone must be highly anisotropic, redistributing angular momentum rapidly in the horizontal direction but very slowly in the radial direction - and similarly for chemical species. Under this assumption, and using a related prescription for the turbulent transport coefficients, he was able to explain simultaneously both the uniform rotation of the solar interior, the thin solar tachocline and the surface light-element abundances.</text>
<text><location><page_6><loc_12><loc_33><loc_88><loc_64></location>However, while compelling in its ability to reproduce observations, this model suffers from a number of inconsistencies. First, note that as long as the field is confined beneath the convection zone, phase mixing is likely to drive the radiation zone towards uniform rotation without the need to invoke turbulence, and thus without causing significant concurrent compositional mixing. The added anisotropic turbulent transport central to the work of Denissenkov (2010) should thus be viewed a 'rapid fix' for a problem that does not necessarily exist. Second, the fix is itself inconsistent, since it uses a prescription for turbulent transport that was created for a purely hydrodynamic system, in an environment dominated by magnetic fields. As shown by Tobias et al. (2007), the turbulent viscosity prescription of Spiegel & Zahn (1992) is unlikely to apply in any circumstance. Indeed, in a purely hydrodynamic setting, strongly stratified turbulence drives a system away from, rather than towards uniform rotation. In the presence of a weak magnetic field, Tobias et al. (2007) showed that the fluid motions rapidly evolve into a state where Reynolds stresses and Maxwell stresses cancel out, and where the efficacy of turbulent angular-momentum transport is strongly quenched. Finally, note that all of these models (Mestel & Weiss 1987; Charbonneau & MacGregor 1993; Denissenkov 2010) assume the existence of a confined poloidal field, but do not explain how confinement is maintained. As it turns out, it is only by answering the fundamental question of magnetic confinement that new light can be shed on the problem.</text>
<text><location><page_6><loc_12><loc_11><loc_88><loc_32></location>Gough & McIntyre (1998) (GM98 hereafter) were the first to put forward a global and selfconsistent theory of the large-scale rotational dynamics of the solar interior, albeit in a steady state (that is, without spin-down). They first addressed the magnetic confinement question. They argued that large-scale meridional flows must be driven by gyroscopic pumping from the differentiallyrotating convection zone down into the radiation zone, and that by pushing on the magnetic field lines they can confine the primordial field strictly below the base of the convection zone. The solar tachocline thus emerges as a 'magnetic-free' region (to be precise, a region where the Lorentz force is insignificant), lying between the base of the convection zone and the top of the magneticallydominated, uniformly rotating radiative interior. GM98 then studied the tachocline dynamics more quantitatively, and estimated that these flows could transport angular momentum and chemical species across the tachocline on a local Eddington-Sweet timescale, which is a few Myr in the</text>
<text><location><page_7><loc_12><loc_79><loc_88><loc_86></location>present-day Sun. The flows, however, do not penetrate below the base of the tachocline, hence satisfying observations of the light-element abundances. Angular-momentum extraction by the tachocline from the deeper regions is done by magnetic stresses, through a very thin magnetic boundary layer, now known as the tachopause (see Figure 1 for detail).</text>
<text><location><page_7><loc_12><loc_69><loc_88><loc_78></location>Gough & McIntyre's vision of the solar interior was recently confirmed by semi-analytical calculations (Wood & McIntyre 2011; Wood et al. 2011) and by numerical simulations (Acevedo-Arreguin et al. 2013). Given its success in explaining (at least qualitatively so far) existing observations for the Sun, we take the natural next step by assuming a similar structure for all solar-type stars, and studying its response to spin-down.</text>
<text><location><page_7><loc_12><loc_55><loc_88><loc_67></location>Our complete model setup is presented in Section 2. Our goal is to derive an analytical or semi-analytical description of the spin-down problem, and study how the latter varies with stellar parameters. To do so, we are forced to abandon the spherical geometry and model the star as a cylinder. This approximation is introduced and discussed in Section 2 and then tested on a simplified system for which analytical solutions exist for both cylindrical and spherical geometry, in Section 3. The derivations presented in that Section also serve the pedagogical role of introducing our methodology.</text>
<text><location><page_7><loc_12><loc_37><loc_88><loc_53></location>In Section 4, we first consider the spin-down of a non-magnetic solar-type star. Our results recover many aspects of Spiegel & Zahn (1992). Including the effects of a primordial magnetic field following the GM98 model requires modeling the transport of angular momentum out of the deep interior, across the tachopause and the tachocline. This is done in Section 5. Finally, in Section 6, we re-interpret our findings in view of their application to stellar evolution, casting our mathematical results in a more astrophysical light, and discussing their limitations and possible implications in explaining observations. This final Section is written in such a way that it may be read independently of Sections 3-5, for readers who, in a first pass, are primarily interested in the results rather than their derivation.</text>
<section_header_level_1><location><page_7><loc_42><loc_31><loc_58><loc_32></location>2. THE MODEL</section_header_level_1>
<text><location><page_7><loc_12><loc_17><loc_88><loc_29></location>The principal difficulty involved in extending the GM98 model to study its behavior under spindown is the nonlinear nature of the equations of motion and of the magnetic induction equation. In fact, following GM98, we do not even attempt to address it: the full nonlinear problem is so complex that it cannot be treated exactly analytically (although see Wood & McIntyre 2011, for a first nonlinear solution of a simplified version of the GM98 model). Instead, we shall make a number of assumptions and order-of-magnitude estimates to model the effects of the nonlinearites. These will be introduced and discussed in detail as they arise.</text>
<text><location><page_7><loc_12><loc_10><loc_88><loc_15></location>A second difficulty lies in the geometry of the system. Since gravity (the vertical axis) is not parallel to the rotation axis in a star (except at the poles), even the restricted hydrodynamic linearized version of the governing equations does not usually have a simple analytical solution.</text>
<figure>
<location><page_8><loc_11><loc_64><loc_88><loc_86></location>
<caption>Fig. 1.- Figure contrasting two geometries: complete with convection zone and magnetic field.</caption>
</figure>
<text><location><page_8><loc_12><loc_53><loc_88><loc_58></location>Indeed, the misalignment between these two axes implies that eigen-solutions of the problem are not separable in the vertical and horizontal directions, which makes the problem much more complicated analytically.</text>
<text><location><page_8><loc_12><loc_37><loc_88><loc_52></location>By contrast with the issue of nonlinearities, this second problem can be dealt with, at least approximately. To do so, we further simplify the model by considering a cylinder instead of a sphere, where gravity is by construction parallel to the rotation axis. This cylinder can, for instance, be viewed as the polar regions of the star (see Figure 1). This simplification adds an order-unity geometrical error to all of our results, but on the other hand allows for fully analytical solutions of the linearized equations. We show in Section 3 that in some simple limit it is possible to find analytical solutions of the linearized system in both spherical and cylindrical geometries, and the solutions agree up to a geometrical factor, hence justifying our procedure.</text>
<section_header_level_1><location><page_8><loc_37><loc_31><loc_63><loc_33></location>2.1. The 'cylindrical star'</section_header_level_1>
<text><location><page_8><loc_12><loc_21><loc_88><loc_29></location>Our cylindrical model is presented in Figure 1. As in GM98, we consider that the star is divided into four dynamically distinct regions, as shown in Figure 1a: from the surface downward, the convection zone (yellow), the tachocline (green), the tachopause (blue), and the uniformlyrotating part of the radiation zone (purple), threaded by a primordial magnetic field (red). Figure 1b shows the equivalent cylindrical model setup.</text>
<text><location><page_8><loc_12><loc_10><loc_88><loc_19></location>The 'cylindrical star' has radius R and total height H . The region with z ∈ [ z cz , H ] (yellow) represents the convection zone. The latter is assumed to rotate with a uniform angular velocity Ω cz ( t ) (as expressed in an inertial frame) which decreases over time via magnetic braking. Henceforth, we work in a frame that is rotating with angular velocity Ω cz ( t ), so that by construction, the convection zone is at rest in the rotating frame. Note that we neglect here the possibility of</text>
<text><location><page_9><loc_12><loc_77><loc_88><loc_86></location>any differential rotation in the convection zone. This is done in order to simplify our calculations, enabling us to neglect any meridional flows driven by differential rotation (via gyroscopic pumping) in favor of those driven by the spin-down torque. We expect this assumption to hold for young stars, which undergo rapid spin-down, but not necessarily for older ones like the Sun (see Section 6.3 for further discussion of this point).</text>
<text><location><page_9><loc_12><loc_65><loc_88><loc_76></location>We use cylindrical coordinates ( s, φ, z ) aligned with the rotation axis (or equivalently, the vertical axis e z ), and assume axial symmetry. In these coordinates, u = ( u, v, w ) is the velocity field relative to the rotating frame. The equations describing the dynamics of the system are the momentum equation, the mass conservation equation, the thermal energy equation, the induction equation and the solenoidal condition. Before studying them in more detail, we first proceed to describe each of the four regions listed above, and the assumptions made in each of them.</text>
<section_header_level_1><location><page_9><loc_38><loc_59><loc_62><loc_61></location>2.2. The convection zone</section_header_level_1>
<text><location><page_9><loc_12><loc_30><loc_88><loc_57></location>The dynamics of stellar convective zones are quite complex, and result from the nonlinear interplay between convection, rotation, large-scale flows and magnetic fields. We shall not attempt to model them in any detail here. Our main goal is merely to account for the rapid transport of angular momentum between the surface and the top of the radiation zone. To do this, we model the convection zone as in Bretherton & Spiegel (1968), who studied stellar spin-down by treating the effect of the convection on large-scale flows (and on momentum transport in general) through a Darcy friction term (i.e. a linear damping term). A similar method was used by Garaud & Acevedo-Arreguin (2009), Wood et al. (2011) and Acevedo-Arreguin et al. (2013) in their models of the solar interior. We therefore replace the Reynolds stress term -u · ∇ u in the momentum equation (where u is the velocity field expressed in the rotating frame) with the term -u /τ , where τ is a damping timescale. Dimensionally speaking, one can assume that τ is of the same order as the convective turnover timescale. For simplicity, we assume that τ is constant in the convective region, and that 1 /τ is zero in the radiative region. Finally, we also assume that the convection zone is adiabatically stratified, and transports heat very efficiently compared with all other timescales in the system.</text>
<section_header_level_1><location><page_9><loc_41><loc_24><loc_59><loc_25></location>2.3. The tachocline</section_header_level_1>
<text><location><page_9><loc_12><loc_11><loc_88><loc_22></location>The region z ∈ [ z tc + δ, z cz ] (green) represents the tachocline. Its thickness, ∆ = z cz -( z tc + δ ), depends on a number of factors including intrinsic properties of the star's structure, as well as its rotation rate and the strength of the primordial field (see GM98 or Wood et al. 2011, for estimates of ∆ in the steady-state solar case, and Appendix E for further discussion of ∆ in the spin-down case). It is stably stratified, with a mean buoyancy frequency ¯ N tc ≥ 0 and mean background density ¯ ρ tc . In this region we make several assumptions to simplify the equations of motion (as in,</text>
<text><location><page_10><loc_12><loc_70><loc_88><loc_86></location>for example, Spiegel & Zahn 1992; Gough & McIntyre 1998; Wood & McIntyre 2011; Wood et al. 2011). First, we assume that the thickness of the tachocline is small compared with a pressure and density scaleheight, and use the Boussinesq approximation (Spiegel & Veronis 1960) to study its dynamics. We also assume that, in the rotating frame, tachocline flows are slow enough to neglect the inertial term in the momentum equation, and also neglect the effect of viscous forces. These assumptions then imply that the tachocline is in hydrostatic and geostrophic equilibrium. We also assume that the flows are sufficiently steady and slow for the system to be in thermal equilibrium, with heat diffusion being balanced by the advection of the background entropy. Finally, as suggested by GM98, we assume that the Lorentz force is negligible in the tachocline.</text>
<section_header_level_1><location><page_10><loc_31><loc_64><loc_69><loc_65></location>2.4. The tachopause and the deep interior</section_header_level_1>
<text><location><page_10><loc_12><loc_44><loc_88><loc_62></location>The region z ∈ [ z tc , z tc + δ ] (blue) represents the tachopause. As discussed in Section 1, the tachopause is a magnetized boundary layer through which the spin-down torque is ultimately transmitted to the interior. As in the case of the tachocline, its thickness δ depends on the local thermodynamical properties of the star, as well as its rotation rate and the strength of the primordial field (see GM98 or Wood et al. 2011). All of the assumptions concerning the dynamics of the tachocline made above apply here as well, with the exception of course of the one concerning the magnetic field - the tachopause is by definition a layer in which the Lorentz force plays a fundamental role. In addition, the thickness of the tachopause is assumed to be much smaller than the thickness of the tachocline ( δ glyph[lessmuch] ∆) to allow for a boundary layer analysis of various balances within.</text>
<text><location><page_10><loc_12><loc_31><loc_88><loc_42></location>Finally, the region 0 < z < z tc (pink) represents the magnetized radiative core, below the tachopause. We do not solve any equations to model that region. Instead, we merely assume that temperature perturbations and large-scale flows vanish for z < z tc , and that the entire region is held in uniform rotation with angular velocity Ω c ( t ) (expressed in the rotating frame, so Ω c = Ω core -Ω cz ). The latter is controlled by the spin-down rate of the regions above, through the torques acting within the tachopause.</text>
<text><location><page_10><loc_12><loc_21><loc_88><loc_30></location>We now begin our theoretical investigation, which ultimately results in deriving an equation for the evolution of the total angular momentum of the core under this model that is identical to the two-zone model of MacGregor & Brenner (1991), but with a coupling timescale that is related to the local Eddington-Sweet timescale across the tachocline. The reader interested principally in a discussion of this result rather than its derivation may, as a first pass, skip directly to Section 6.</text>
<section_header_level_1><location><page_10><loc_26><loc_15><loc_74><loc_17></location>3. CYLINDRICAL VS. SPHERICAL GEOMETRY</section_header_level_1>
<text><location><page_10><loc_12><loc_10><loc_88><loc_13></location>We begin our investigation by showing that results obtained using a cylindrical geometry are, within a geometrical factor of order unity, consistent with those obtained using a more realistic</text>
<text><location><page_11><loc_12><loc_85><loc_27><loc_86></location>spherical geometry.</text>
<text><location><page_11><loc_12><loc_73><loc_88><loc_83></location>To do so, we consider the simplest possible problem, of a non-magnetic star with an 'unstratified' interior of constant density. While this case does not have any astrophysical relevance, it can easily be solved analytically in both cylindrical and spherical geometry (Bretherton & Spiegel 1968). As such, it can serve as an illustration of the validity of the 'cylindrical star' assumption, as well as a pedagogical tool to introduce the method of solution of the governing equations we shall use throughout.</text>
<text><location><page_11><loc_12><loc_60><loc_88><loc_71></location>We also assume in this Section that there is no magnetic field. In this case, the tachopause does not exist, and the 'tachocline' fills the entire radiation zone, extending all the way to the center of the star. In this model, the only difference between the 'convective' and 'radiative' regions is the presence of the Darcy forcing term modeling angular-momentum transport by convective motions. Under the assumptions discussed in Section 2, regardless of geometry, the equations of motion are the momentum equation and incompressibility (the thermal energy equation is not needed):</text>
<formula><location><page_11><loc_30><loc_56><loc_88><loc_59></location>∂ u ∂t +2 Ω cz × u + ˙ Ω cz × r + u τ = -1 ¯ ρ ∇ p, ∇ . u = 0 , (2)</formula>
<text><location><page_11><loc_12><loc_47><loc_88><loc_54></location>where r is the position vector, ¯ ρ is the constant density of the fluid, and p is the pressure perturbation away from hydrostatic equilibrium. The second term in Equation (2) is the Coriolis force, the third is Euler's force (which is due to the deceleration of the frame) and the fourth term is the Darcy friction term (see Section 2), which is assumed to be zero in the 'radiation zone'.</text>
<section_header_level_1><location><page_11><loc_39><loc_42><loc_61><loc_43></location>3.1. Spherical geometry</section_header_level_1>
<text><location><page_11><loc_12><loc_31><loc_88><loc_39></location>Bretherton & Spiegel (1968) were the first to study the spin-down of such an unstratified star using the model described above. They found an analytical solution of the problem in a spherical geometry, assuming that the angular velocity of the convection zone Ω cz ( t ) (expressed in an inertial frame) decays exponentially over time. We now repeat their calculation and consider any functional form for Ω cz ( t ) for more generality.</text>
<text><location><page_11><loc_12><loc_22><loc_88><loc_29></location>Bretherton & Spiegel (1968) first assumed that the spin-down rate is low, so that | ˙ Ω cz / Ω cz | glyph[lessmuch] Ω cz . They then looked for a quasi-steady solution in the rotating frame, requiring the term ∂ u /∂t in the momentum equation to be negligible. The resulting quasi-steady equations of motion in the 'radiative interior' (0 < r < r cz ) are</text>
<formula><location><page_11><loc_34><loc_18><loc_88><loc_21></location>2 Ω cz × u + ˙ Ω cz × r = -1 ¯ ρ ∇ p, ∇ . u = 0 . (3)</formula>
<text><location><page_11><loc_12><loc_15><loc_54><loc_16></location>In the 'convection zone' ( r cz < r < R ), they reduce to</text>
<formula><location><page_11><loc_40><loc_10><loc_88><loc_13></location>1 τ u = -1 ¯ ρ ∇ p, ∇ . u = 0 , (4)</formula>
<text><location><page_12><loc_12><loc_84><loc_84><loc_86></location>assuming that the Darcy timescale τ is significantly smaller than the rotation timescale Ω -1 cz .</text>
<text><location><page_12><loc_12><loc_74><loc_88><loc_83></location>Solving these equations, then matching p and the normal velocity at the radiative-convective interface (at r = r cz ), yields the shape of the streamlines (see Figure 3), as well the angularvelocity perturbation Ω( r, θ, t ) everywhere in the star (see their original work for the details of the calculation). The angular velocity in the radiative interior turns out to be uniform, with value Ω c such that</text>
<formula><location><page_12><loc_40><loc_71><loc_88><loc_75></location>Ω c Ω cz = -˙ Ω cz 4Ω 3 cz τ 3 r 5 cz +2 R 5 R 5 -r 5 cz , (5)</formula>
<text><location><page_12><loc_12><loc_61><loc_88><loc_70></location>as expressed in the rotating frame, where R is the radius of the star and r cz that of the base of the convection zone. Since ˙ Ω cz < 0, we have Ω c > 0, implying that the interior is always rotating faster than the convection zone, or in other words, lagging behind in the context of the spin-down process. The relative lag is measured by Ω c / Ω cz ; whether it increases or decreases with time depends on the behavior of ˙ Ω cz / Ω 3 cz . The implications of Equation (5) are discussed in Section 3.3.</text>
<section_header_level_1><location><page_12><loc_38><loc_56><loc_62><loc_57></location>3.2. Cylindrical geometry</section_header_level_1>
<text><location><page_12><loc_12><loc_39><loc_88><loc_54></location>We now solve the same equations under the same assumptions, but this time in the cylindrical geometry presented in Section 2. Inspection of the expression for the meridional flows given by Bretherton & Spiegel (1968) (see their Equations 7 and 8) reveals that they have equatorial symmetry. To obtain solutions with the same symmetry in cylindrical geometry, we set the vertical velocity at z = 0 to be zero. In addition, the vertical velocity at the surface of the star is also zero. Following Bretherton & Spiegel (1968), we solve the equations for z > z cz and z < z cz separately, and match p and the vertical velocity w at the radiative-convective interface. Our vertical boundary conditions are 3 therefore:</text>
<formula><location><page_12><loc_21><loc_36><loc_88><loc_37></location>w = 0 at z = 0 , z = H, p ( z = z -cz ) = p ( z = z + cz ) , w ( z = z -cz ) = w ( z = z + cz ) . (6)</formula>
<text><location><page_12><loc_12><loc_30><loc_88><loc_33></location>Boundary conditions at the side wall ( s = R ) are more difficult to choose, as we want them to have as little influence as possible on the dynamics inside the cylinder. For simplicity, we select</text>
<formula><location><page_12><loc_44><loc_27><loc_88><loc_28></location>p = 0 at s = R. (7)</formula>
<text><location><page_12><loc_12><loc_18><loc_88><loc_25></location>This boundary condition allows a radial flow across the side wall, as required by analogy with the spherical solution (see Figure 3). By conservation of mass the fluid must somehow return to the convection zone, a process which necessarily occurs outside of the cylinder considered, and that we cannot model explicitly. In what follows, we assume that this return flow does not play any</text>
<text><location><page_13><loc_12><loc_83><loc_88><loc_86></location>fundamental role in the spin-down process (in the sense that it does not change the solution beyond a factor of order unity). This assumption is verified a posteriori to be correct (see Section 3.3).</text>
<text><location><page_13><loc_12><loc_78><loc_88><loc_81></location>We first consider the radiative region ( z < z cz ). The various components of Equation (2), when expressed in a cylindrical geometry, become</text>
<formula><location><page_13><loc_30><loc_74><loc_88><loc_77></location>∂p ∂z = 0 , ∂v ∂t +2Ω cz u + ˙ Ω cz s = 0 , 2Ω cz v = 1 ¯ ρ ∂p ∂s . (8)</formula>
<text><location><page_13><loc_12><loc_71><loc_52><loc_72></location>where u = ( u, v, w ). We also have incompressibility:</text>
<formula><location><page_13><loc_42><loc_66><loc_88><loc_69></location>1 s ∂ ∂s ( su ) + ∂w ∂z = 0 . (9)</formula>
<text><location><page_13><loc_12><loc_64><loc_83><loc_65></location>Following Bretherton & Spiegel (1968), we take ∂v/∂t = 0, and combine (8) and (9) to find</text>
<formula><location><page_13><loc_35><loc_58><loc_88><loc_62></location>u = -˙ Ω cz 2Ω cz s, w = ˙ Ω cz Ω cz z for z < z cz . (10)</formula>
<text><location><page_13><loc_15><loc_55><loc_52><loc_56></location>In the convection zone, expanding (4) we have</text>
<formula><location><page_13><loc_36><loc_50><loc_88><loc_53></location>1 ¯ ρ ∂p ∂s = -u τ , v = 0 , 1 ¯ ρ ∂p ∂z = -w τ . (11)</formula>
<text><location><page_13><loc_12><loc_48><loc_54><loc_49></location>Combining these with incompressibility ∇ . u = 0 gives</text>
<formula><location><page_13><loc_42><loc_44><loc_88><loc_46></location>∇ 2 p = 0 , ∇ 2 w = 0 . (12)</formula>
<text><location><page_13><loc_12><loc_41><loc_65><loc_42></location>Using the boundary conditions (6) and (7), we can write w and p as</text>
<formula><location><page_13><loc_28><loc_36><loc_88><loc_39></location>w = ∑ n B n sinh ( λ n z -H R ) J 0 ( λ n s R ) , (13)</formula>
<formula><location><page_13><loc_27><loc_31><loc_88><loc_35></location>1 ¯ ρ p = -∑ n B n τ R λ n J 0 ( λ n s R ) cosh ( λ n z -H R ) , for z > z cz , (14)</formula>
<text><location><page_13><loc_12><loc_25><loc_88><loc_30></location>where J 0 is the zeroth-order regular Bessel function, the { λ n } are its zeros, and the coefficients { B n } are integration constants. The latter are found by matching w in Equations (10) and (13) at z = z cz , so that</text>
<formula><location><page_13><loc_36><loc_21><loc_88><loc_25></location>B n = ˙ Ω cz Ω cz 2 z cz λ n J 1 ( λ n ) sinh ( λ n z cz -H R ) . (15)</formula>
<text><location><page_13><loc_12><loc_17><loc_88><loc_20></location>where J 1 is the first order Bessel function. Equation (8) shows that, for z < z cz , p is constant with height, so that matching p at z = z cz and using (15) gives</text>
<formula><location><page_13><loc_24><loc_11><loc_88><loc_15></location>1 ¯ ρ p = ˙ Ω cz Ω cz τ ∑ n J 0 ( λ n s R ) [ 2 z cz R λ 2 n J 1 ( λ n ) tanh ( λ n H -z cz R ) ] , for z < z cz , (16)</formula>
<figure>
<location><page_14><loc_28><loc_60><loc_69><loc_85></location>
<caption>Fig. 2.- Comparison of the two geometrical factors multiplying -˙ Ω cz /τ Ω 3 cz in Equations (5) and (17). The lengths are scaled such that z cz /H = z cz /R = 0 . 7 for the cylindrical geometry, and r cz /R = 0 . 7 for the spherical geometry.</caption>
</figure>
<text><location><page_14><loc_12><loc_45><loc_88><loc_48></location>and, using (8), we find that the angular velocity of the radiative region, expressed in the rotating frame, becomes</text>
<formula><location><page_14><loc_28><loc_41><loc_88><loc_45></location>Ω( s, z ) Ω cz = v ( s ) s Ω cz = -˙ Ω cz τ Ω 3 cz ∑ n J 1 ( λ n s R ) sλ n J 1 ( λ n ) z cz tanh ( λ n H -z cz R ) . (17)</formula>
<text><location><page_14><loc_12><loc_37><loc_88><loc_40></location>The latter is always positive, and is a function of s only, as expected from the Taylor-Proudman constraint.</text>
<section_header_level_1><location><page_14><loc_43><loc_31><loc_57><loc_33></location>3.3. Discussion</section_header_level_1>
<text><location><page_14><loc_12><loc_13><loc_88><loc_29></location>The solutions in spherical and cylindrical geometries bear strong similarities. In both cases, we find that the azimuthal velocity is constant along the rotation axis, as expected from the TaylorProudman constraint since the fluid is unstratified. Furthermore, the radiation zone is always rotating more rapidly than the convection zone. The relative lag between the two regions, in both cases, is equal to the prefactor -˙ Ω cz /τ Ω 3 cz , times a non-dimensional term that depends only on the geometry of the system. This term is shown as a function of s in Figure 2 for both cylindrical and spherical geometries. It is constant in the spherical case, and increases with s in the cylindrical case. However, the two are consistent (within a factor of order unity). This shows that the difference in the results obtained in the two geometries is not dramatic.</text>
<text><location><page_14><loc_15><loc_10><loc_88><loc_11></location>The structure of the meridional circulation is also very similar in both cases, as shown in Figure</text>
<figure>
<location><page_15><loc_15><loc_61><loc_47><loc_85></location>
</figure>
<figure>
<location><page_15><loc_55><loc_60><loc_87><loc_84></location>
<caption>Fig. 3.- Comparison of the streamlines in the spherical and cylindrical solutions. The cylindrical case (right) can be viewed as a distorted version of the spherical case (left), as long as the solution of the latter is truncated at a cylindrical radius roughly equal to s = 0 . 6 R glyph[star] .</caption>
</figure>
<text><location><page_15><loc_12><loc_44><loc_88><loc_49></location>3. In the cylindrical case, one could imagine the streamlines closing back on themselves outside of the domain, as they do in the spherical case. This changes the global angular momentum balance somewhat, but as discussed above, does not affect the outcome by more than an order unity factor.</text>
<text><location><page_15><loc_12><loc_35><loc_88><loc_41></location>While Ω cz ( t ) should in principle be calculated self-consistently from a stellar wind model, it is informative to look at specific 'plausible' spin-down laws. For the purpose of the following discussion, we either assume that Ω cz ( t ) decays exponentially, with</text>
<formula><location><page_15><loc_39><loc_32><loc_88><loc_34></location>Ω cz ( t ) = Ω 0 exp( -k ( t -t 0 )) , (18)</formula>
<text><location><page_15><loc_12><loc_29><loc_30><loc_30></location>or as a power law, with</text>
<formula><location><page_15><loc_42><loc_27><loc_88><loc_29></location>Ω cz ( t ) = Ω 0 ( t/t 0 ) -α , (19)</formula>
<text><location><page_15><loc_12><loc_21><loc_88><loc_26></location>for some α > 0. In these laws, Ω 0 = Ω cz ( t 0 ), where t 0 is the initial timescale considered (e.g. the end of the disk-locking phase, for instance, or the Zero Age Main Sequence). The parameters α and k are unspecified here, but can be fitted to any desired spin-down model.</text>
<text><location><page_15><loc_12><loc_14><loc_88><loc_20></location>The -˙ Ω cz / Ω 3 cz prefactor in Equations (5) and (17) shows that the relative lag between the radiative and convective regions always increases exponentially with time for an exponential spindown law. This suggests a break-down of the quasi-steady approximation for this case 4 . Using a</text>
<text><location><page_16><loc_12><loc_75><loc_88><loc_86></location>power-law to model spin-down reveals that Ω( s ) / Ω cz increases with time if α > 1 / 2, is constant if α = 1 / 2, and decreases with time if α < 1 / 2. This, again, suggests a break-down of the quasisteady approximation if spin-down occurs faster than the Skumanich power-law, which has α = 1 / 2 (Skumanich 1972). Although interesting, we believe that the correspondence between the critical power-index α = 1 / 2 and the Skumanich law is a coincidence, given the simplistic nature of this particular unstratified model.</text>
<text><location><page_16><loc_12><loc_65><loc_88><loc_74></location>To conclude this Section, since the solutions in (5) and (17) are the same up to a purely geometrical factor, and since that geometrical factor is typically of order unity, we now make the assumption that results from our cylindrical model may carry across to the spherical geometry, up to a geometrical error, for more complex systems such as ones including stratification or a magnetic field. Hence from now on, all our analysis will be restricted to cylindrical geometry.</text>
<section_header_level_1><location><page_16><loc_27><loc_59><loc_73><loc_61></location>4. SPIN-DOWN IN A NON-MAGNETIC STAR</section_header_level_1>
<text><location><page_16><loc_12><loc_39><loc_88><loc_57></location>We now consider the spin-down of a non-magnetic but more realistically stratified star. As in Section 3, we study only solar-type stars, with an outer convection zone ( z cz < z < H ) and an inner radiation zone (0 < z < z cz ). In preparation for the implementation of the GM98 model, which is our ultimate goal, we assume that the radiation zone is sub-divided into two regions, a uniformly rotating core (for z < z tc ) and a tachocline (for z tc < z < z cz ). Here, we do not specify the mechanism by which the core is held in solid-body rotation. Furthermore, we assume that this core is not dynamically connected to the tachocline, but instead, merely acts as a passive boundary whose only role is to be impenetrable to the fluid. Within that approximation, note the core cannot be spun-down, as no angular momentum can be extracted from it. Finally, since the tachopause does not exist, we take δ = 0.</text>
<text><location><page_16><loc_60><loc_25><loc_60><loc_26></location>glyph[negationslash]</text>
<text><location><page_16><loc_12><loc_19><loc_88><loc_37></location>While this setup may seem odd at first, note that one could use it to represent a normal nonmagnetic star (that is, with no rigidly rotating core) simply by taking the regular limit z tc → 0; the impenetrability of the boundary at z = z tc simply becomes an equatorial symmetry condition (see Section 3). However, it is important to remember that this limit is generally inconsistent with the Boussinesq approximation, which requires ∆ = z cz -z tc to be smaller than a pressure scaleheight. We therefore advise the reader against indiscriminately using our results in this fashion. Instead, we note that the solutions derived in this Section with z tc = 0 will be applicable (with a few modifications) to model the complete stellar spin-down problem in Section 5. In this sense, our work in this section should be viewed once again as a pedagogical step toward understanding the final result.</text>
<section_header_level_1><location><page_17><loc_40><loc_85><loc_60><loc_86></location>4.1. Model equations</section_header_level_1>
<text><location><page_17><loc_12><loc_72><loc_88><loc_83></location>The convection zone is assumed to be very nearly adiabatic with a buoyancy frequency ¯ N = 0, and entropy perturbations are assumed to be negligible. For more realism, and in this Section only, we allow the background density ¯ ρ to vary with height in the convection zone, and use the anelastic approximation. We shall show that the results are identical to those derived in the Boussinesq case. The equations governing the dynamics within the convection zone thus become, under similar assumptions to the ones discussed in Section 2:</text>
<formula><location><page_17><loc_35><loc_68><loc_88><loc_71></location>1 τ u = -1 ¯ ρ ∇ p, ∇· (¯ ρ u ) = 0 , T = 0 , (20)</formula>
<text><location><page_17><loc_12><loc_65><loc_44><loc_66></location>where T is the temperature perturbation.</text>
<text><location><page_17><loc_12><loc_55><loc_88><loc_64></location>Since the tachocline is thought to be thin, we use the Boussinesq approximation (Spiegel & Veronis 1960) to model it, and assume that the buoyancy frequency, gravity, density and temperature do not depart significantly from their mean tachocline values ¯ N tc , ¯ g tc , ¯ ρ tc and ¯ T tc . In this approximation, density and temperature perturbations are formally related through a linearized equation of state, in which pressure perturbations are negligible. Hence</text>
<formula><location><page_17><loc_45><loc_50><loc_88><loc_53></location>ρ ¯ ρ tc = -T ¯ T tc . (21)</formula>
<text><location><page_17><loc_12><loc_48><loc_52><loc_49></location>The momentum equation in the tachocline becomes</text>
<formula><location><page_17><loc_32><loc_43><loc_88><loc_47></location>∂ u ∂t +2 Ω cz × u + ˙ Ω cz × r = -1 ¯ ρ tc ∇ p + ¯ g tc ¯ T tc T ˆ e z . (22)</formula>
<text><location><page_17><loc_12><loc_39><loc_88><loc_42></location>Combining the radial and vertical components of (22), with the assumptions discussed in Section 2 and above, yields the well-known thermal-wind equation</text>
<formula><location><page_17><loc_43><loc_34><loc_88><loc_38></location>2Ω cz ∂v ∂z = ¯ g tc ¯ T tc ∂T ∂s . (23)</formula>
<text><location><page_17><loc_12><loc_32><loc_61><loc_33></location>The tachocline is also assumed to be in thermal equilibrium, so</text>
<formula><location><page_17><loc_43><loc_27><loc_88><loc_31></location>¯ N 2 tc ¯ T tc ¯ g tc w = κ tc ∇ 2 T, (24)</formula>
<text><location><page_17><loc_12><loc_25><loc_47><loc_26></location>where κ tc is its (constant) thermal diffusivity.</text>
<text><location><page_17><loc_12><loc_16><loc_88><loc_23></location>The boundary conditions for velocity and pressure are similar to those of the previous Section, but with the lower boundary raised to z = z tc . We require impermeability ( w = 0) at z = z tc and z = H , p = 0 at s = R , and that p and w must be continuous at z = z cz . The temperature perturbations are assumed to vanish at z = z tc , and at z = z cz .</text>
<text><location><page_17><loc_12><loc_10><loc_88><loc_15></location>We now first study this system of equations in the same 'quasi-steady' state discussed in the previous Section, and then solve for the complete time-dependence of the system to determine under which conditions this quasi-steady state is valid.</text>
<section_header_level_1><location><page_18><loc_38><loc_85><loc_62><loc_86></location>4.2. Quasi-steady solution</section_header_level_1>
<text><location><page_18><loc_12><loc_76><loc_88><loc_83></location>As in Section 3, we define the 'quasi-steady' state as the solution of the governing equations in which ∂ u /∂t is neglected. We find solutions separately for z > z cz and z < z cz , and match w and p at the radiative-convective interface. The full calculation is given in Appendix A. We find that the azimuthal velocity in the tachocline is</text>
<formula><location><page_18><loc_25><loc_71><loc_88><loc_75></location>v ( s, z, t ) = -˙ Ω cz Ω 2 cz ∑ n J 1 ( λ n s R ) λ n J 1 ( λ n ) [ ∆ τ tanh ( λ n H -z cz R ) + ¯ N 2 tc κ tc G n ( z ) ] , (25)</formula>
<text><location><page_18><loc_12><loc_67><loc_88><loc_70></location>where ∆ = z cz -z tc is the thickness of the tachocline (since δ = 0), and where G n is the geometrical factor</text>
<formula><location><page_18><loc_15><loc_62><loc_88><loc_66></location>G n ( z ) = R λ n { ∆ R λ n sinh ( λ n ∆ R ) [ cosh ( λ n z -z tc R ) -cosh ( λ n ∆ R )] -( z -z tc ) 2 -∆ 2 2 } . (26)</formula>
<text><location><page_18><loc_12><loc_49><loc_88><loc_61></location>Equation (25) reduces to (17) when the tachocline is unstratified ( ¯ N tc = 0), regardless of the position of the lower boundary z tc . This is not surprising: the Taylor-Proudman constraint requires v to be independent of z in that limit. It is also shown in Appendix A that the density variation in the convection zone has no effect on the angular velocity within the tachocline, as long as the density is continuous across z = z cz . In what follows, we can therefore equivalently use ∇· u = 0 in the convection zone for mathematical simplicity even though the latter does not actually satisfy the Boussinesq approximation.</text>
<figure>
<location><page_18><loc_32><loc_25><loc_68><loc_46></location>
<caption>Fig. 4.- Contour plot of the quasi-steady solution for Ω( s, z ) = v ( s, z ) /s given in (25), with z tc = 0 . 5 z cz , where Ω cz = Ω 0 ( t/t 0 ) -1 / 2 , and where R , H , τ , ¯ N tc , κ tc , ¯ ρ tc , t 0 and Ω 0 are otherwise given in Table 1. Note that Ω is strictly positive everywhere, and increases with depth and cylindrical radius. The solid black lines show flow streamlines, with the flow direction being downward and outward.</caption>
</figure>
<text><location><page_18><loc_65><loc_12><loc_65><loc_13></location>glyph[negationslash]</text>
<text><location><page_18><loc_12><loc_10><loc_88><loc_13></location>The angular velocity and meridional circulation profile for ¯ N tc = 0 is shown in Figure 4. To understand its properties, first note that since G ( z cz ) = 0, the angular velocity just below the base</text>
<text><location><page_19><loc_12><loc_73><loc_88><loc_86></location>of the convection zone is the same in the stratified and unstratified cases. However, v is no longer independent of z , but instead increases with depth. This effect is due to the added buoyancy force in the momentum equation, which relaxes the Taylor-Proudman constraint. The latter is replaced by the thermal-wind constraint given in Equation (23), which relates any variation of angular velocity along the rotation axis to gradients of temperature perpendicular to it. The relative lag between the base of the tachocline and the convection zone is thus controlled simultaneously by thermal-wind balance and by thermal equilibrium within the tachocline.</text>
<text><location><page_19><loc_12><loc_65><loc_88><loc_72></location>The combination of these two constraints yields a simple estimate for the relative angularvelocity shear across the tachocline as a function of input stellar parameters. Indeed, first note that the downwelling flow velocity across the tachocline is primarily controlled by the tachocline thickness, and by the spin-down rate through:</text>
<formula><location><page_19><loc_43><loc_60><loc_88><loc_64></location>w = ˙ Ω cz Ω cz ( z -z tc ) , (27)</formula>
<text><location><page_19><loc_12><loc_58><loc_82><loc_59></location>(see Appendix A for detail). An order-of-magnitude approximation of this equation yields</text>
<formula><location><page_19><loc_44><loc_53><loc_88><loc_57></location>| w | ∼ -∆ ˙ Ω cz Ω cz . (28)</formula>
<text><location><page_19><loc_12><loc_47><loc_88><loc_52></location>In thermal equilibrium (see Equation 24), however, the advection of the background entropy stratification by these flows must balance the diffusion of the induced temperature perturbations T . This sets the typical amplitude of T in the tachocline to be</text>
<formula><location><page_19><loc_43><loc_42><loc_88><loc_46></location>T ∼ ∆ 2 κ tc ¯ N 2 tc ¯ T tc ¯ g tc | w | . (29)</formula>
<text><location><page_19><loc_12><loc_36><loc_88><loc_41></location>Finally, by thermal-wind balance, latitudinal variations in T control the allowable shear across the tachocline and therefore the total angular-velocity difference (Ω b -0) between the base of the tachocline (where Ω ∼ Ω b ) and the radiative-convective interface (where Ω = 0):</text>
<formula><location><page_19><loc_43><loc_32><loc_88><loc_35></location>2Ω cz R Ω b ∆ ∼ ¯ g tc ¯ T tc T R . (30)</formula>
<text><location><page_19><loc_12><loc_30><loc_40><loc_31></location>Combining all these estimates yields</text>
<formula><location><page_19><loc_31><loc_25><loc_88><loc_28></location>t ES (Ω b ) ≡ ¯ N 2 tc 2Ω cz Ω b ( ∆ R ) 2 ∆ 2 κ tc ∼ -Ω cz ˙ Ω cz ≡ t sd (Ω cz ) , (31)</formula>
<text><location><page_19><loc_12><loc_20><loc_89><loc_24></location>which states that the system adjusts itself (by selecting Ω b ) such that the Eddington-Sweet timescale 5 based on that angular-velocity lag and on the thickness of the tachocline (expressed in the left-</text>
<text><location><page_20><loc_12><loc_83><loc_88><loc_86></location>nd-side of this equation) is equal to the spin-down timescale of the star (expressed in the righthand-side). We then have</text>
<formula><location><page_20><loc_21><loc_78><loc_88><loc_81></location>Ω b Ω cz ∼ -˙ Ω cz Ω 3 cz ( ∆ R ) 2 ¯ N 2 tc ∆ 2 2 κ tc = t ES (Ω cz ) t sd (Ω cz ) , where t ES (Ω cz ) = ¯ N 2 tc 2Ω 2 cz ( ∆ R ) 2 ∆ 2 κ tc . (32)</formula>
<text><location><page_20><loc_12><loc_66><loc_88><loc_76></location>This shows that the relative angular-velocity shear between the top and the bottom of the tachocline is equal to the ratio of the local Eddington-Sweet timescale (based, this time, on the rotation rate of the convection zone) to the spin-down timescale. We then expect the shear across the tachocline to be larger (i) if the background stratification is larger (ii) if the local thermal diffusivity is smaller, (iii) if the spin-down rate is larger or (iv) if the tachocline is thicker. This is indeed what the exact Equation (25) and its order of magnitude approximation (32) both show.</text>
<text><location><page_20><loc_12><loc_54><loc_88><loc_65></location>Finally, note that the relative lag is also proportional to -˙ Ω cz / Ω 3 cz , as in the unstratified case. This is not entirely surprising, since the unstratified case is a regular limit of this stratified problem as ¯ N tc → 0. As such, as long as ∆ is constant, Ω b / Ω cz diverges with time for an exponential spin-down law, or for any power-law with α > 1 / 2, as discussed in Section 3.3. However, since ∆ likely depends on time as well through its dependence on Ω cz (see Appendix E), other criteria apply (see Sections 5 and 6 for detail).</text>
<section_header_level_1><location><page_20><loc_39><loc_48><loc_61><loc_49></location>4.3. Transient solution</section_header_level_1>
<text><location><page_20><loc_12><loc_39><loc_88><loc_46></location>Having found a quasi-steady solution to the spin-down problem, we now revisit the original time-dependent equations to determine when that solution is valid, and how rapidly the system relaxes to it. Guided by the steady-state solution, we expand the azimuthal velocity v on the same basis of Bessel functions, namely</text>
<formula><location><page_20><loc_37><loc_34><loc_88><loc_38></location>v ( s, z, t ) = ∑ n dJ 0 ( λ n s R ) ds v n ( z, t ) . (33)</formula>
<text><location><page_20><loc_12><loc_29><loc_88><loc_32></location>Combining (22), (23) and incompressibility with this ansatz, and retaining the time-derivative in the azimuthal component of the momentum equation, gives</text>
<formula><location><page_20><loc_22><loc_24><loc_88><loc_28></location>∂v n ∂t + ( 2Ω cz ( t ) R ¯ N tc λ n ) 2 κ tc ∂ 4 v n ∂z 4 -( 2Ω cz ( t ) ¯ N tc ) 2 κ tc ∂ 2 v n ∂z 2 = ˙ Ω cz ( t ) 4 R 2 λ 3 n J 1 ( λ n ) , (34)</formula>
<text><location><page_20><loc_12><loc_10><loc_88><loc_23></location>where we have explicitly written in the time-dependence of Ω cz and ˙ Ω cz to remember that it must be taken into account. This equation is quite similar to the one derived by Spiegel & Zahn (1992) in the context of the evolution of the differential rotation profile within the solar tachocline. This is not surprising, as our underlying assumptions (thermal-wind balance, thermal equilibrium) are essentially the same. The first and second terms on the left-hand-side of (34) are the same as theirs (see their Equation 4.10). The hyperdiffusion term arises from the advection of angular momentum by local Eddington-Sweet flows, a transport process that Spiegel & Zahn (1992) called</text>
<text><location><page_21><loc_12><loc_70><loc_88><loc_86></location>'thermal spreading'. The third term on the left-hand side is also part of the thermal spreading process, but was neglected by Spiegel & Zahn (1992) on the grounds that it is quite small when ∆ glyph[lessmuch] R . For consistency with the Boussinesq approximation, which requires ∆ to be smaller than a pressure scaleheight, we also neglect it from here on. Finally, the right-hand side contains the global forcing term arising from Euler's force. Since viscous and turbulent transport are neglected here, the evolution of the angular momentum in the tachocline has two contributions only: transport by meridional flows, and global extraction by Euler's force. As such, we expect the system to behave in rather different ways if the spin-down timescale is much larger or much smaller than local Eddington-Sweet mixing timescale.</text>
<text><location><page_21><loc_12><loc_65><loc_88><loc_68></location>In what follows, we introduce the new variable x = ( z -z tc ) / ∆. Together with the simplification discussed above, Equation (34) becomes</text>
<formula><location><page_21><loc_32><loc_60><loc_88><loc_64></location>∂v n ∂t + 4Ω 2 cz ( t ) R 2 ¯ N 2 tc λ 2 n κ tc ∆ 4 ( t ) ∂ 4 v n ∂x 4 = ˙ Ω cz ( t ) 4 R 2 λ 3 n J 1 ( λ n ) , (35)</formula>
<text><location><page_21><loc_12><loc_55><loc_88><loc_58></location>where we have also explicitly written in the time-dependence of ∆. To solve Equation (35), we note that it is further separable in x and t , and write</text>
<formula><location><page_21><loc_39><loc_51><loc_88><loc_54></location>v n ( x, t ) = ∑ m V nm ( t ) Z nm ( x ) , (36)</formula>
<text><location><page_21><loc_12><loc_48><loc_47><loc_49></location>where the vertical eigenmodes Z nm ( x ) satisfy</text>
<formula><location><page_21><loc_38><loc_43><loc_88><loc_46></location>L ( Z nm ) ≡ d 4 Z nm dx 4 = µ 4 nm Z nm , (37)</formula>
<text><location><page_21><loc_12><loc_39><loc_88><loc_42></location>for some constants µ nm . It can be shown that the Z nm functions form an orthogonal set, so that projecting (35) onto each of them individually gives, for each n and m ,</text>
<formula><location><page_21><loc_27><loc_33><loc_88><loc_37></location>dV nm dt + V nm τ ES nm ( t ) = ˙ Ω cz ( t ) 4 R 2 λ 3 n J 1 ( λ n ) ∫ 1 0 Z nm ( x ) dx ∫ 1 0 Z 2 nm ( x ) dx ≡ F nm ( t ) , (38)</formula>
<text><location><page_21><loc_12><loc_30><loc_80><loc_31></location>where additional information on Z nm and µ nm , are given in Appendix B. The quantity</text>
<formula><location><page_21><loc_31><loc_25><loc_88><loc_29></location>τ ES nm ( t ) = ¯ N 2 tc ∆ 4 ( t ) 4Ω 2 cz ( t ) R 2 κ tc λ 2 n µ 4 nm = λ 2 n 2 µ 4 nm t ES (Ω cz ( t )) , (39)</formula>
<text><location><page_21><loc_12><loc_19><loc_88><loc_24></location>where t ES (Ω cz ) was defined in Equation (32), naturally emerges from this calculation, and can be interpreted as a local Eddington-Sweet timescale based on the typical geometry of the spatial eigenmode considered.</text>
<text><location><page_21><loc_15><loc_16><loc_64><loc_17></location>Equation (38) can easily be solved using an integrating factor,</text>
<formula><location><page_21><loc_38><loc_11><loc_88><loc_14></location>µ ( t ) = exp (∫ t t 0 1 τ ES nm ( t ' ) dt ' ) , (40)</formula>
<text><location><page_22><loc_12><loc_85><loc_21><loc_86></location>which yields</text>
<formula><location><page_22><loc_17><loc_76><loc_88><loc_84></location>V nm ( t ) = 1 µ ( t ) [ V nm ( t 0 ) + ∫ t t 0 µ ( t ' ) F nm ( t ' ) dt ' ] = exp ( -∫ t t 0 1 τ ES nm ( t ' ) dt ' ) V nm ( t 0 ) + ∫ t t 0 exp ( ∫ t ' t 1 τ ES nm ( t '' ) dt '' ) F nm ( t ' ) dt ' . (41)</formula>
<text><location><page_22><loc_12><loc_59><loc_88><loc_73></location>As expected, we see that V nm ( t ) contains two terms, one that depends on the initial conditions (the first term on the right-hand side of Equation 41) and one that depends on the forcing applied to the system (the second term on the right-hand side of Equation 41). For V nm ( t ) to tend to the quasi-steady solution discussed in Section 4.2 as t → + ∞ (see Equation 25), the effect of the initial conditions must decay, since (25) is independent of the initial differential rotation profile of the star. Furthermore, the terms containing the forcing in Equation (41), when recombined as in Equation (36) must eventually recover (25). Whether this occurs or not clearly depends on the behavior of the integrating factor µ ( t ). We now study the latter in more detail.</text>
<text><location><page_22><loc_15><loc_56><loc_46><loc_57></location>We first re-write the integral in µ ( t ) as</text>
<formula><location><page_22><loc_20><loc_51><loc_88><loc_55></location>∫ t t 0 1 τ ES nm ( t ' ) dt ' = -2 µ 4 nm λ 2 n ∫ t t 0 t sd ( t ' ) t ES ( t ' ) ˙ Ω cz ( t ' ) Ω cz ( t ' ) dt ' = -2 µ 4 nm λ 2 n ∫ Ω cz ( t ) Ω 0 t sd (Ω cz ) t ES (Ω cz ) d Ω cz Ω cz , (42)</formula>
<text><location><page_22><loc_12><loc_47><loc_88><loc_50></location>where the spin-down timescale t sd was defined in Equation (31). Writing it in this form enables us to study the long-term behavior of this integral for a fairly broad class of problems.</text>
<text><location><page_22><loc_12><loc_42><loc_88><loc_45></location>First, note that for an exponential spin-down law (see Equation 18), t sd is constant (and equal to 1 /k ) while for a power-law spin-down rate (see Equation 19), then</text>
<formula><location><page_22><loc_31><loc_37><loc_88><loc_41></location>t sd (Ω cz ) = t 0 α ( Ω cz Ω 0 ) -1 α = t sd (Ω 0 ) ( Ω cz Ω 0 ) -α -1 . (43)</formula>
<text><location><page_22><loc_12><loc_33><loc_88><loc_36></location>The second form of t sd written above can actually be used to describe both exponential and powerlaw spin-down models if one views the exponential case as having α -1 = 0.</text>
<text><location><page_22><loc_12><loc_22><loc_88><loc_31></location>Next, note that both GM98 and Wood et al. (2011) found ∆ to be a power-law function of the mean stellar rotation rate. While their findings do not directly apply here, since they were derived assuming that the tachocline circulation is driven by the latitudinal shear in the convection zone rather than by spin-down, we may nevertheless safely assume a similar functional dependence 6 , taking</text>
<formula><location><page_22><loc_43><loc_19><loc_88><loc_22></location>∆ = ∆ 0 ( Ω cz Ω 0 ) β , (44)</formula>
<text><location><page_23><loc_12><loc_85><loc_75><loc_86></location>(where we anticipate that β ≥ 0, and ∆ 0 is by construction ∆ at t = t 0 ), so that</text>
<formula><location><page_23><loc_37><loc_80><loc_88><loc_83></location>t ES (Ω cz ) = t ES (Ω 0 ) ( Ω cz Ω 0 ) 4 β -2 . (45)</formula>
<text><location><page_23><loc_12><loc_77><loc_58><loc_78></location>Combining Equations (45) and (43) with (42), we find that</text>
<text><location><page_23><loc_70><loc_69><loc_70><loc_70></location>glyph[negationslash]</text>
<formula><location><page_23><loc_27><loc_65><loc_88><loc_76></location>∫ t t 0 1 τ ES nm ( t ' ) dt ' = -t sd (Ω 0 ) τ ES nm (Ω 0 ) ∫ Ω cz ( t ) Ω 0 ( Ω cz Ω 0 ) 1 -α -1 -4 β d Ω cz Ω 0 = -1 q t sd (Ω 0 ) τ ES nm (Ω 0 ) [( Ω cz Ω 0 ) q -1 ] if q = 0 , -t sd (Ω 0 ) τ ES nm (Ω 0 ) ln ( Ω cz Ω 0 ) if q = 0 , (46)</formula>
<text><location><page_23><loc_12><loc_60><loc_88><loc_63></location>where q = 2 -α -1 -4 β . We therefore see that the behavior of the integrating factor µ ( t ) depends only on the sign of q .</text>
<text><location><page_23><loc_12><loc_50><loc_88><loc_59></location>If q > 0, then the right-hand-side of Equation (46) tends to a constant as the star spins down. In that case, µ ( t ) also tends to a constant as t → + ∞ , which implies that the contribution of the initial conditions to V nm ( t ) does not disappear in Equation (41). In other words, the star cannot relax to the state described by the quasi-steady solution discussed in Section 4.2, and the latter becomes irrelevant to the spin-down problem.</text>
<text><location><page_23><loc_12><loc_36><loc_88><loc_48></location>On the other hand, if q ≤ 0 then the right-hand-side of Equation (46) is positive, and increases towards + ∞ as the star spins down. In that case, the integrating factor µ ( t ) also increases with time (super-exponentially when q < 0, and as Ω -1 cz ( t ) when q = 0), which implies that the contribution of the initial conditions to V nm ( t ) rapidly disappears. For q < 0, the latter decays exponentially roughly on the local Eddington-Sweet timescale across the tachocline. This timescale decreases with time and rapidly becomes much smaller than the age of the star except if the tachocline is very thick (which we explicitly assumed was not the case).</text>
<text><location><page_23><loc_12><loc_25><loc_88><loc_34></location>Furthermore, it can be shown with additional algebra that the complete transient solution given by Equation (36) actually tends to the quasi-steady solution (25) when t → + ∞ , when q ≤ 0 (see Appendix B2). In other words, the quasi-steady solution derived and discussed in Section 4.2 is a meaningful description of stellar spin-down, after a transient phase which is short compared with the age of the star, during which all knowledge of the initial rotation profile disappears.</text>
<text><location><page_23><loc_12><loc_21><loc_88><loc_24></location>The physical interpretation of q as a critical value of this problem is quite straightforward given that q is effectively defined so that</text>
<formula><location><page_23><loc_42><loc_16><loc_88><loc_20></location>t sd (Ω cz ) t ES (Ω cz ) ∝ ( Ω cz Ω 0 ) q . (47)</formula>
<text><location><page_23><loc_12><loc_10><loc_88><loc_15></location>If q > 0, then the ratio of the spin-down timescale to the local angular-momentum transport timescale across the tachocline decreases as the star spins down. This implies that the meridional flows are less and less efficient at extracting angular momentum from the tachocline relative to</text>
<text><location><page_24><loc_12><loc_75><loc_88><loc_86></location>the rate at which it is removed from the envelope. The lag between the envelope and the bottom of the tachocline then increases with time, until such a point where new dynamics not taken into account here, such as turbulent transport, must come into play. In other words, the quasi-steady solution derived in the previous Section is only of limited validity. When q ≤ 0 on the other hand, the converse is true: the system rapidly tends to the quasi-steady solution regardless of the initial conditions.</text>
<section_header_level_1><location><page_24><loc_43><loc_70><loc_57><loc_71></location>4.4. Discussion</section_header_level_1>
<text><location><page_24><loc_12><loc_49><loc_88><loc_67></location>Our findings regarding the dynamics of both transient and quasi-steady spin-down solutions can easily be summarized as follows. If, for a given spin-down law Ω cz ( t ) and a given tachocline structure (characterized by its local thermodynamic properties and its thickness), the system is such that the ratio of the spin-down timescale to the local Eddington-Sweet timescale t sd (Ω cz ) /t ES (Ω cz ) (where t sd is given in Equation 43 and t ES is given in Equation 45) monotonically decreases as the star spins down, then angular-momentum transport by large-scale meridional flows across the tachocline is not sufficient to maintain dynamical coupling with the envelope. After some time, the shear across the tachocline is likely to become large enough to be unstable to shearing instabilities. Angular-momentum transport will then be dominated by turbulent motions, and must be described using an entirely different formalism (not discussed here).</text>
<text><location><page_24><loc_12><loc_35><loc_88><loc_48></location>If, on the other hand, the system is such that t sd (Ω cz ) /t ES (Ω cz ) increases or remains constant as the star spins down, then the tachocline remains coupled to the envelope, and the shear simply adjusts itself geostrophically so that the angular momentum flux transported by the meridional flows out of the tachocline is, at all times, equal to angular-momentum flux removed from the star by the wind. The relative lag Ω b / Ω cz between the base and the top of the tachocline, in this case, is correctly given by the quasi-steady solution, and is roughly equal to t ES (Ω cz ) /t sd (Ω cz ) (which is either constant, or decreases with time), see Equation (32).</text>
<text><location><page_24><loc_12><loc_25><loc_88><loc_34></location>We therefore find that the quasi-steady solution is conveniently valid whenever it makes sense, that is, whenever Ω b / Ω cz ∝ t ES (Ω cz ) /t sd (Ω cz ) decreases (or at least remains constant and much smaller than one) as the star spins down. In other words, we can actually avoid the calculation of the transient solution entirely, since the quasi-steady solution itself provides all the information needed as to the limits of its own validity.</text>
<section_header_level_1><location><page_24><loc_29><loc_19><loc_71><loc_20></location>5. SPIN-DOWN OF A MAGNETIZED STAR</section_header_level_1>
<text><location><page_24><loc_12><loc_10><loc_88><loc_17></location>We now finally return to the originally-posed problem and investigate the manner in which the tachocline spin-down is finally communicated to the deep radiative interior. In the GM98 model, this process is mediated by magnetic torques within the tachopause, a thin boundary layer that separates the tachocline from the magnetically-dominated, uniformly-rotating region below</text>
<text><location><page_25><loc_12><loc_73><loc_88><loc_86></location>(see Figure 1a). These torques are generated as the primordial magnetic field, confined below the tachocline by downwelling meridional flows, is wound up into a significant toroidal field by the rotational shear. In order to model angular-momentum transport across the tachopause exactly, one should therefore solve the magnetic induction equation in addition to the previously discussed equations describing the tachocline dynamics (i.e. Equations 22 and 24), and include the Lorentz force in the momentum balance. The nonlinear nature of the added terms, unfortunately, makes it impossible to derive exact analytical solutions of the problem without further assumptions.</text>
<text><location><page_25><loc_12><loc_58><loc_88><loc_72></location>Wood et al. (2011), however, were able to derive analytical solutions for a simplified version of the GM98 model. They found that the mathematical equations describing tachopause are, in many ways, analogous to those describing a viscous Ekman layer, with the viscous drag force replaced by a magnetic one. In other words, they showed that one can develop physical insight into the problem and obtain quantitatively meaningful results by considering a thought experiment in which the uniformly rotating part of the radiative interior is simply a massive, impenetrable solid sphere (or, in our case, a cylinder), whose rotation rate is gradually spun-down by the fluid lying above through friction.</text>
<text><location><page_25><loc_12><loc_47><loc_88><loc_56></location>Our final model is thus constructed as follows. We consider the same setup as the one described and studied in Section 4, but the base of the tachocline is now no longer passive. Instead, it hosts a thin tachopause of thickness δ , which communicates the tachocline spin-down to the rigidlyrotating, impermeable interior via magnetic torques. The latter will be modeled using a boundary layer jump condition. We thus recover the picture first presented in Figure 1b.</text>
<section_header_level_1><location><page_25><loc_32><loc_41><loc_68><loc_43></location>5.1. Global angular-momentum balance</section_header_level_1>
<text><location><page_25><loc_12><loc_29><loc_88><loc_39></location>Let Ω c be the angular velocity of the rigidly rotating 'core' region of the radiation zone, expressed in the rotating frame. Rigid-body rotation throughout the entire star implies that Ω c glyph[similarequal] 0, while Ω c > 0 expresses a lag between the core and the convection zone. In our cylindrical model, the core spans the region z < z tc , s < R (see Figure 1). Recall that the tachopause spans the interval [ z tc , z tc + δ ], and is assumed to be thinner than the tachocline, which lies above (with z ∈ [ z tc + δ, z cz ]).</text>
<text><location><page_25><loc_12><loc_20><loc_88><loc_27></location>Although negligible in the tachocline, magnetic stresses are significant within the tachopause, and must be included when studying the global angular-momentum balance. To find an evolution equation for Ω c ( t ), we thus begin by writing the complete angular-momentum conservation equation as</text>
<formula><location><page_25><loc_31><loc_17><loc_88><loc_20></location>∂ ∂t (¯ ρsv + ¯ ρs 2 Ω cz ) + ∇· ( ¯ ρ u s 2 Ω cz -sB φ B 4 π ) = 0 , (48)</formula>
<text><location><page_25><loc_12><loc_11><loc_88><loc_16></location>where B = ( B s , B φ , B z ) is the magnetic field, and where we have ignored viscous stresses on the grounds that they are most likely negligible. Note that we assume that the system is laminar, and ignore the contribution of turbulent transport - this assumption is discussed in Sections 5.2 and 6.</text>
<text><location><page_26><loc_12><loc_83><loc_88><loc_86></location>Integrating Equation (48) over the volume V of the core up to the top of the tachopause, and using the divergence theorem then yields</text>
<formula><location><page_26><loc_19><loc_74><loc_88><loc_81></location>2 π ∫ z tc + δ 0 ∫ R 0 ∂ ∂t (¯ ρsv + ¯ ρs 2 Ω cz ) sdsdz +2 πR ∫ z tc + δ z tc ( ¯ ρus 2 Ω cz -sB φ B s 4 π ) s = R dz +2 π ∫ R 0 ( ¯ ρws 2 Ω cz -sB φ B z 4 π ) z = z tc + δ sds = 0 , (49)</formula>
<text><location><page_26><loc_12><loc_64><loc_88><loc_72></location>where the second integral is a surface integral through the side of the tachopause, and the third integral is a surface integral through the top of the tachopause. To derive this equation, we have used the fact that the angular-momentum flux through the bottom and side boundaries of the core is zero. Indeed, u and B φ disappear since the core is assumed to be rigidly rotating and impermeable.</text>
<text><location><page_26><loc_12><loc_48><loc_90><loc_62></location>By definition of the tachocline, the magnetic torque becomes negligible just above the tachopause, and thus disappears from the surface integral at z = z tc + δ . We assume that it also disappears from the integral on the side-boundary. This assumption is somewhat difficult to justify a priori, and will require verification when full numerical solutions of the problem, in a spherical geometry, are available. However, it is consistent with the assumption that the dynamics occurring beyond the sides of the cylinders do not directly affect spin-down. We argue that it is at least plausible as long as the magnitude of the toroidal field B φ on the side-wall of the tachopause remains small, which requires in turn that the radial angular-velocity gradient ∂ Ω /∂s at the same location be small.</text>
<text><location><page_26><loc_15><loc_45><loc_61><loc_46></location>The remaining terms in Equation (49) can be expressed as</text>
<formula><location><page_26><loc_13><loc_40><loc_88><loc_43></location>d dt [ I core (Ω c +Ω cz )] + 2 π ¯ ρ tc Ω cz R 3 ∫ z tc + δ z tc u ( R,z, t ) dz +2 π ¯ ρ tc Ω cz ∫ R 0 w ( s, z tc + δ, t ) s 3 ds ≈ 0 , (50)</formula>
<text><location><page_26><loc_12><loc_35><loc_88><loc_38></location>where, as in the previous Section, ¯ ρ tc is defined as the mean density of the tachocline and tachopause region, and where I core is the moment of inertia of the core and tachopause combined, defined as</text>
<formula><location><page_26><loc_29><loc_30><loc_88><loc_34></location>I core = ∫ V ¯ ρ ( z ) s 2 dV = π 2 R 4 ∫ z tc + δ 0 ¯ ρ ( z ) dz = M core 2 R 2 , (51)</formula>
<text><location><page_26><loc_12><loc_24><loc_88><loc_29></location>where M core is the mass of the cylinder included in the volume V . Note that in order to derive (50), we have used the fact that v = s Ω c within the core, and assumed that v glyph[similarequal] s Ω c in the tachopause as well. Since the tachopause is very thin, the error made has negligible impact on the result.</text>
<text><location><page_26><loc_12><loc_10><loc_88><loc_22></location>Within the scope of these assumptions, angular momentum is extracted from the core through a series of channels, which unfold as follows. The surface layers are spun-down by the magnetized wind torque, and then communicate the spin-down information to the rest of the convection zone by turbulent stresses. The spin-down torque also drives large-scale meridional flows in the convection zone, which extract angular momentum from the tachocline and the top of the tachopause (by flowing downward from the convection zone and then outward in the tachopause), and at the same time confine the internal magnetic field. The tachopause then finally spins the core down via</text>
<text><location><page_27><loc_12><loc_75><loc_88><loc_86></location>magnetic stresses, generated as the local core-tachopause shear winds the internal poloidal field into a toroidal one. Equation (50) describes only the hydrodynamic processes in the tachocline, but the other channels are implied in the assumptions that (1) the convection zone rotates at the velocity Ω cz , (2) the tachopause rotates at a velocity close to Ω c , and (3) the vertical flow at the base of the tachocline is given by the magnetic jump condition that describes the tachopause dynamics. We study the latter in Section 5.2.</text>
<text><location><page_27><loc_12><loc_69><loc_88><loc_74></location>To estimate the second term in Equation (50), note that the mass flux entering the tachopause through the surface z = z tc + δ must be the same as that leaving through the side wall, since the core is impermeable. Hence:</text>
<formula><location><page_27><loc_29><loc_64><loc_88><loc_67></location>2 π ∫ R 0 w ( s, z tc + δ, t ) sds = -2 πR ∫ z tc + δ z tc u ( R,z, t ) dz. (52)</formula>
<text><location><page_27><loc_12><loc_61><loc_36><loc_62></location>Combining this with (50) gives</text>
<formula><location><page_27><loc_32><loc_56><loc_88><loc_59></location>dJ core dt +2 π ¯ ρ tc Ω cz ∫ R 0 sw 0 ( s, t )( R 2 -s 2 ) ds glyph[similarequal] 0 , (53)</formula>
<text><location><page_27><loc_12><loc_53><loc_16><loc_54></location>where</text>
<formula><location><page_27><loc_41><loc_51><loc_88><loc_53></location>J core = I core (Ω cz +Ω c ) (54)</formula>
<text><location><page_27><loc_12><loc_43><loc_88><loc_50></location>is the total angular momentum of the core 7 in the inertial frame, and where we have introduced w 0 ( s, t ) = w ( s, z tc + δ, t ) for simplicity. Equation (53) thus shows that the rate of angular-momentum transport between the surface layers and the core is fully determined once ¯ ρ tc w 0 ( s, t ), the vertical mass flux downwelling from the tachocline into tachopause, is known.</text>
<text><location><page_27><loc_12><loc_27><loc_88><loc_41></location>To find w 0 ( s, t ), we must once more solve the equations describing the dynamics of the tachocline and of the convection zone, and match them to one another at the radiative-convective interface. The main difference with the work presented in Section 4.2 lies in the treatment of the lower boundary of the tachocline, which is no longer passive nor strictly impermeable, but must instead be modified to take the presence of the tachopause into account. This is done by replacing the impermeability condition used in Section 4.2 by a 'jump condition', that relates w 0 to v 0 ( s, t ) = v ( s, z tc + δ, t ) at the bottom of the tachocline, and depends on the magnetohydrodynamics of the tachopause. This jump condition is now derived.</text>
<section_header_level_1><location><page_27><loc_35><loc_21><loc_65><loc_23></location>5.2. Tachopause jump condition</section_header_level_1>
<text><location><page_27><loc_12><loc_14><loc_88><loc_19></location>Guided by the analogy between the tachopause and an Ekman layer suggested by the work of Wood et al. (2011), we begin by deriving a tachopause jump condition assuming that its dynamics are dominated by viscous torques only. This assumption greatly facilitates our derivation, and the</text>
<text><location><page_28><loc_12><loc_83><loc_88><loc_86></location>result can then be used without any further algebra to deduce the equivalent jump condition for a magnetized tachopause.</text>
<text><location><page_28><loc_12><loc_72><loc_88><loc_81></location>While well-known in the context of a steadily-rotating frame, the derivation of the Ekman jump condition has not yet, to our knowledge, been done in a frame that is spinning down. The steps of the calculation are essentially identical, however, and are presented in Appendix C. We find that the quasi-steady vertical and azimuthal velocity profiles at the base of the tachocline, w 0 ( s ) and v 0 ( s ), are related via:</text>
<formula><location><page_28><loc_32><loc_68><loc_88><loc_71></location>w 0 = ˙ Ω cz Ω cz δ E ( 1 -1 √ 2 ) -δ E √ 2 s ∂ ∂s ( s 2 Ω c -sv 0 ) , (55)</formula>
<text><location><page_28><loc_12><loc_59><loc_88><loc_67></location>where δ E = √ ν tc / 2Ω cz is the thickness of the Ekman layer that mimics the tachopause, and is based on the local viscosity ν tc of the star at z = z tc . The second term on the right-hand-side is the standard jump condition (expressed in a cylindrical coordinate system), while the first term is the correction arising from Euler's force.</text>
<text><location><page_28><loc_12><loc_55><loc_88><loc_58></location>In stars, however, the tachopause transmits the spin-down torque via magnetic rather than viscous friction. Wood et al. (2011) found that its thickness 8 is given by</text>
<formula><location><page_28><loc_42><loc_49><loc_88><loc_53></location>δ = √ 2 π ¯ ρ tc η tc Ω cz R 2 B 2 0 , (56)</formula>
<text><location><page_28><loc_12><loc_39><loc_88><loc_48></location>where B 0 is the strength of the confined magnetic field (just below z tc ), and η tc is the local magnetic diffusivity of the star near z = z tc . They also found that the jump condition relating the vertical and azimuthal velocities in the tachocline is exactly of the same mathematical form as that of an Ekman layer, albeit with the numerical constant 1 / √ 2 replaced by π/ 4, and δ E replaced by δ . The magnetic jump condition thus becomes</text>
<formula><location><page_28><loc_33><loc_35><loc_88><loc_38></location>w 0 = ˙ Ω cz Ω cz δ ( 1 -π 4 ) -π 4 δ s ∂ ∂s ( s 2 Ω c -sv 0 ) . (57)</formula>
<text><location><page_28><loc_12><loc_24><loc_88><loc_33></location>In both cases discussed above, the tachopause is assumed to be laminar. The angularmomentum transport budget within this region involves only large-scale flows, and either magnetic or viscous stresses. The reason behind the similarity of the two jump conditions comes from the fact that viscous and magnetic stresses are directly proportional to the local angular-velocity shear 9 . As a result, one may conjecture that any boundary layer in which the angular-momentum</text>
<text><location><page_29><loc_12><loc_79><loc_88><loc_86></location>balance relies on large-scale meridional flows and some form of stress that is proportional to the angular-velocity shear will also result in the same type of jump condition, even if that boundary layer is not laminar. From this argument, we propose that the general form of the jump condition should be</text>
<formula><location><page_29><loc_34><loc_76><loc_88><loc_79></location>w 0 = ˙ Ω cz Ω cz δ (1 -C ) -C δ s ∂ ∂s ( s 2 Ω c -sv 0 ) , (58)</formula>
<text><location><page_29><loc_12><loc_70><loc_88><loc_75></location>where C is a constant of order unity which depends on the type of stresses acting in the tachopause, and δ is its thickness, which is no longer necessarily related to B 0 via (56) specifically, but likely depends on B 0 in some form or another.</text>
<text><location><page_29><loc_12><loc_56><loc_88><loc_68></location>In any case, as we demonstrate below, the long-term behavior of the spin-down problem is ultimately controlled only by the slowest timescale in the sequence of processes responsible for angular-momentum transport from the core to the surface. In most cases, this turns out to be the Eddington-Sweet timescale across the tachocline, rather than any timescale intrinsic to the tachopause. In this sense, the global spin-down timescale is usually independent of the exact nature and structure of the tachopause (at least, in an explicit sense, see Section 6.3 and Appendix E), unless the latter is very thick - which we have previously assumed is not the case.</text>
<section_header_level_1><location><page_29><loc_29><loc_50><loc_71><loc_51></location>5.3. Evolution of the core angular momentum</section_header_level_1>
<text><location><page_29><loc_12><loc_30><loc_88><loc_48></location>We can now solve the system of equations governing the convection zone and the tachocline as in Section 4, using this time the jump condition (58) as a lower boundary condition for the tachocline flows. At this junction we have two possibilities: to solve the full time-dependent tachocline dynamics as in Section 4.3, or to study them using a quasi-steady approximation as in Section 4.2. Having proved in Section 4.3 that the time-dependent solution very rapidly tends to the quasi-steady solution (when it is well-behaved), and given that the latter is much more easily derived, we assume here that the tachocline dynamics are in a quasi-steady state 10 . The derivation of this solution is presented in Appendix D, and eventually yields the mass flux into the tachopause, ¯ ρ tc w 0 ( s ). Using the expression obtained into Equation (53), we then find that for δ glyph[lessmuch] ∆ (i.e. when the tachopause is much thinner than the tachocline), then</text>
<formula><location><page_29><loc_36><loc_25><loc_88><loc_28></location>dJ core dt = -˙ Ω cz I tc + Ω cz ∆ I tc ∑ n 32 λ 4 n a n b n , (59)</formula>
<text><location><page_29><loc_12><loc_22><loc_16><loc_23></location>where</text>
<formula><location><page_29><loc_44><loc_19><loc_88><loc_22></location>I tc = ¯ ρ tc π ∆ R 4 2 (60)</formula>
<text><location><page_30><loc_12><loc_85><loc_52><loc_86></location>is the moment of inertia of the tachocline, and with</text>
<formula><location><page_30><loc_21><loc_79><loc_88><loc_83></location>a n = ˙ Ω cz Ω cz ∆ -2 Cδ Ω c -Cδ ¯ N 2 tc ∆ 2Ω cz κ tc ˙ Ω cz Ω cz ( R λ n cosh ( λ n ∆ R ) -1 sinh ( λ n ∆ R ) -∆ 2 ) , (61)</formula>
<formula><location><page_30><loc_21><loc_75><loc_88><loc_79></location>b n = 1 + Cδλ n 2Ω cz R [ 2 ¯ N 2 tc R 2 κ tc λ 2 n 1 -cosh ( λ n ∆ R ) sinh ( λ n ∆ R ) + ¯ N 2 tc ∆ R κ tc λ n + 1 τ tanh ( λ n H -z cz R ) ] , (62)</formula>
<text><location><page_30><loc_12><loc_64><loc_88><loc_73></location>where ∆ glyph[similarequal] z cz -z tc is the thickness of the tachocline. Note that for most stars whose outer convection zone is not too thin, and where τ is not too small, the last term in the square brackets of Equation (62) is negligible compared with the first two. In what follows, we neglect it, but bear in mind that it may be important for solar-type stars whose mass approach the critical mass above which the outer convection zone disappears.</text>
<text><location><page_30><loc_15><loc_62><loc_84><loc_63></location>While somewhat obscure at first, Equation (59) has a simple limit. Indeed, when δ = 0,</text>
<formula><location><page_30><loc_41><loc_57><loc_88><loc_60></location>a n = ˙ Ω cz Ω cz ∆ and b n = 1 . (63)</formula>
<text><location><page_30><loc_12><loc_54><loc_79><loc_55></location>Using properties of Bessel Functions, it can be shown that ∑ n 32 /λ 4 n = 1. As a result,</text>
<formula><location><page_30><loc_46><loc_49><loc_88><loc_52></location>dJ core dt = 0 , (64)</formula>
<text><location><page_30><loc_12><loc_41><loc_88><loc_48></location>which implies that, when viewed in an inertial frame, the core retains its initial angular momentum. This result is as expected, since δ = 0 means that the tachopause is absent, and without it the tachocline cannot exert any torque on the core. In other words, the convective envelope and the tachocline both spin down exactly as described in Section 4.2, but the core does not.</text>
<text><location><page_30><loc_12><loc_16><loc_88><loc_40></location>When δ > 0, by contrast, the angular velocity of the core evolves with time in response to the spin-down torque communicated by the tachopause. This is illustrated in Figure 5, which shows a plot of the relative core-envelope lag Ω c / Ω cz , as a function of time, in two idealized test cases. Note that the moment of inertia of the core is assumed to be constant to simplify the interpretation of the results. At time t = t 0 we also assume for simplicity that Ω c ( t 0 ) = 0 (or in other words, that the star is uniformly rotating). In Figure 5a, we show Ω c / Ω cz for a 'reference' star whose parameters are summarized in Table 1, using a Skumanich spin-down law with Ω cz ( t ) = Ω 0 ( t/t 0 ) -1 / 2 (i.e. α = 1 / 2), and with a tachocline of constant thickness (setting β = 0 in Equation 44). In Figure 5b, we evolve the same star but allow for a tachocline whose thickness varies with Ω cz , taking β = 2 / 3 in Equation (44) and choosing ∆ 0 such that the values of ∆( t ) in both Figure 5a and Figure 5b agree at t = 10 3 t 0 (which more-or-less represents the 'present day' time for the Sun). In both figures the relative core-envelope lag first increases rapidly, then eventually converges to a global quasi-steady state 11 whose time-dependence can be predicted analytically (see Section 5.4). The</text>
<text><location><page_31><loc_12><loc_81><loc_88><loc_86></location>time taken to reach this global quasi-steady state, however, depends sensitively on the thickness of the tachocline (see Section 5.5), and is much larger in Figure 5b, which has a much larger initial tachocline thickness, than in Figure 5a.</text>
<figure>
<location><page_31><loc_12><loc_59><loc_52><loc_79></location>
</figure>
<figure>
<location><page_31><loc_54><loc_60><loc_85><loc_78></location>
<caption>Fig. 5.- (a) Evolution of Ω c / Ω cz for a star with parameters shown in Table 1, undergoing spindown following Skumanich's law. The solid (red) line is the exact solution of Equation (59) using 96 terms in the sum. The dot-dashed (blue) line is the exact solution of Equation (65), which keeps only one term in the sum. The latter is a fairly good approximation to the 'true' solution. The dashed and dotted lines are the quasi-steady approximations to the full solutions, obtained by dropping the time-derivative of Ω c in Equation (59) and solving for Ω c algebraically. The 1-term quasi-steady solution is given by Equation (71). They are a good approximation to the full solution after a transient period whose duration depends on the thickness of the tachocline. (b) Same figure, but for a tachocline whose thickness varies as (Ω cz / Ω 0 ) 2 / 3 (see main text for detail). Because the initial tachocline thickness in this model is much thicker, the true solution takes much longer to approach the quasi-steady solution.</caption>
</figure>
<text><location><page_31><loc_12><loc_31><loc_88><loc_34></location>To understand the results, first note that a good approximation to Equation (59) can be obtained by keeping a single term in the sum over all spatial eigenmodes, that is, by using</text>
<formula><location><page_31><loc_39><loc_26><loc_88><loc_30></location>dJ core dt = -˙ Ω cz I tc + Ω cz ∆ I tc a 1 b 1 , (65)</formula>
<text><location><page_31><loc_12><loc_18><loc_88><loc_25></location>instead, where we have replaced 32 /λ 4 1 in the second term on the right-hand-side by 1, to ensure that the core is not spun-down by the tachocline when δ = 0 (see discussion above). Since λ 1 glyph[similarequal] 2 . 4, this is a fairly good approximation anyway. In addition, as long as the tachocline and tachopause are thin, a 1 and b 1 further simplify by Taylor expansion to</text>
<formula><location><page_31><loc_34><loc_10><loc_88><loc_17></location>a 1 = ˙ Ω cz Ω cz ∆ -2 Cδ Ω c + Cλ 2 1 24 ˙ Ω cz t ES (Ω cz ) δ, b 1 = 1 + Cλ 2 1 12 Ω cz t ES (Ω cz ) δ ∆ . (66)</formula>
<table>
<location><page_32><loc_18><loc_49><loc_77><loc_65></location>
<caption>Table 1: Parameters for reference model.</caption>
</table>
<text><location><page_33><loc_12><loc_83><loc_88><loc_86></location>Substituting these terms into Equation (65), reducing the right-hand side to the same denominator and simplifying, yields</text>
<formula><location><page_33><loc_20><loc_77><loc_88><loc_82></location>dJ core dt = -Cδ ∆ Ω cz I tc ( 2Ω c + λ 2 1 24 ˙ Ω cz t ES (Ω cz ) 1 + Cλ 2 1 12 Ω cz t ES (Ω cz ) δ ∆ ) glyph[similarequal] -I tc ( 24Ω c λ 2 1 t ES (Ω cz ) + ˙ Ω cz 2 ) . (67)</formula>
<text><location><page_33><loc_12><loc_62><loc_88><loc_76></location>In writing the second expression, we have simplified the denominator further by noting that the second term in b 1 is usually larger than 1 by many orders of magnitude for any physically meaningful values of δ/ ∆ (except of course in the strict limit δ → 0 discussed earlier, which we do not consider here). The resulting expression for the rate of change of J core is now completely independent of Cδ , or in other words, independent of the detailed nature and structure of the tachopause. This property of the solution is discussed in more detail in Section 6, but essentially stems from the fact that the tachopause can propagate the spin-down torque near-instantaneously to the core when it is thin, and thus does not introduce any new timescale in the problem.</text>
<text><location><page_33><loc_12><loc_55><loc_88><loc_60></location>Finally, note that since dJ core /dt = d ( I core Ω c ) /dt + d ( I core Ω cz ) /dt , and since the assumption of a thin tachocline fundamental to this work implies that I tc glyph[lessmuch] I core , we can neglect I tc ˙ Ω cz in the right-hand side of Equation (67) in comparison with the I core ˙ Ω cz term on its left-hand side, so that:</text>
<formula><location><page_33><loc_32><loc_51><loc_88><loc_54></location>dJ core dt glyph[similarequal] -24 λ 2 1 Ω c I tc t ES (Ω cz ) = -K (Ω core -Ω cz ) I tc t ES (Ω cz ) , (68)</formula>
<text><location><page_33><loc_12><loc_43><loc_88><loc_50></location>where K is a constant of order unity, and Ω core is the angular velocity of the core in an inertial frame. Written in this final form, our model bears some obvious similarities with the two-zone model of MacGregor & Brenner (1991). In fact, it can be cast exactly as in Equation (1) provided we define the coupling timescale between the core and the envelope to be</text>
<formula><location><page_33><loc_38><loc_39><loc_88><loc_42></location>τ c = t ES (Ω cz ) K I core I cz I tc ( I core + I cz ) , (69)</formula>
<text><location><page_33><loc_12><loc_35><loc_88><loc_38></location>where I cz is the moment of inertia of the convection zone. This result is discussed in more detail in Section 6.1.</text>
<text><location><page_33><loc_12><loc_21><loc_88><loc_33></location>The comparison between the solution of the exact Equation (59), and that the much simpler Equation (68) is shown in Figure 5. The two are within ten percent of one another at all times. Given that our cylindrical geometry solutions approximate a real star to within a geometrical factor of order unity at best anyway, the error made in using (68) instead of (59) is of a similar nature, and can be incorporated in the former. In what follows, we therefore advocate the use of (68) as a much simpler and more physically meaningful, and yet equivalent description of the spin-down problem.</text>
<section_header_level_1><location><page_33><loc_30><loc_15><loc_70><loc_16></location>5.4. Properties of the quasi-steady solution</section_header_level_1>
<text><location><page_33><loc_12><loc_10><loc_88><loc_13></location>We now use this simpler expression to derive a global quasi-steady approximation to the solution, which gives insight into the long-term behavior of the system. By analogy with Section</text>
<text><location><page_34><loc_12><loc_83><loc_88><loc_86></location>4.2, we derive it by neglecting the acceleration, but keeping Euler's force in the momentum equation. This yields the following algebraic equation instead,</text>
<formula><location><page_34><loc_40><loc_78><loc_88><loc_82></location>I core ˙ Ω cz = -24 λ 2 1 Ω c I tc t ES (Ω cz ) , (70)</formula>
<text><location><page_34><loc_12><loc_70><loc_88><loc_77></location>which can then be solved for Ω c , and thus yields its quasi-steady approximation Ω qs c . Note that we have again assumed here that I core varies sufficiently slowly with time that its derivative can be neglected. This is done for simplicity of interpretation of the results but is not necessarily valid during all evolutionary stages of the star. We then find that:</text>
<formula><location><page_34><loc_27><loc_65><loc_88><loc_69></location>Ω qs c Ω cz = λ 2 1 24 t ES (Ω cz ) t sd (Ω cz ) I core I tc = λ 2 1 24 t ES (Ω 0 ) t sd (Ω 0 ) ∆ I core ∆ 0 I tc ( Ω cz Ω 0 ) -( q + β ) , (71)</formula>
<text><location><page_34><loc_12><loc_62><loc_34><loc_64></location>where q + β = 2 -α -1 -3 β .</text>
<text><location><page_34><loc_12><loc_44><loc_88><loc_61></location>The scaling shown in Equation (71) is similar to that discussed in Section 4.2 (see Equation 32), and can therefore also be understood using order-of-magnitude arguments based on thermal-wind balance, thermal equilibrium and mass conservation. However, it contains new factor which is the ratio I core /I tc . The presence of this factor can be understood physically by noting that this time, the core is spun-down as well as the tachocline. A much larger torque is needed to spin down a more massive core, and as a consequence, if the same spin-down torque is applied, then the core-envelope lag is proportionally larger for a more massive core. Furthermore, since I core /I tc is inversely proportional to the thickness of the tachocline, Ω qs c / Ω cz now scales with the third instead of the fourth power of ∆, and varies as (Ω cz / Ω 0 ) -( q + β ) instead of (Ω cz / Ω 0 ) -q .</text>
<text><location><page_34><loc_12><loc_28><loc_88><loc_43></location>We can now apply the same reasoning as in Section 4.3 as to the limits of validity of our laminar solution. If q + β > 0 then Ω qs c / Ω cz increases as the star spins down. In this case, the quasi-steady solution is not a good approximation to the actual time-dependent problem, and the laminar solution probably eventually breaks down to a turbulent one with different scalings instead. If q + β ≤ 0 on the other hand then Ω qs c / Ω cz remains constant or decreases, and the laminar solution is likely always valid. When q + β < 0, the system always tends to solid-body rotation as the star spins down. By contrast, if q + β = 0 then Ω qs c / Ω cz tends to a constant, which implies that the system eventually maintains a non-zero core-envelope lag as t →∞ .</text>
<text><location><page_34><loc_12><loc_20><loc_88><loc_27></location>This behavior is illustrated in Figure 6, which shows the long-term evolution of the 'reference' star, for α = 1 / 2 and β = 0, β = 1 / 3 and β = 2 / 3 respectively. We see that, for large times, the exact solution for Ω c / Ω cz does indeed tend to the quasi-steady solution as expected. The latter is either constant or decreases with time, depending on the value of q + β .</text>
<text><location><page_34><loc_12><loc_10><loc_88><loc_19></location>We now briefly examine under which conditions q + β = 2 -α -1 -3 β ≤ 0. Recall that α is the spin-down law index (with the convention that α -1 = 0 for an exponential spin-down law, see Equation 43) and β is the index of the power-law describing the variation of the tachocline thickness with Ω cz (see Equation 44). The condition 2 -α -1 -3 β ≤ 0 is then automatically satisfied whenever α ≤ 1 / 2, as long as β > 0, or whenever β ≥ 2 / 3, regardless of α .</text>
<text><location><page_35><loc_12><loc_77><loc_88><loc_86></location>The fact that the observationally-favored Skumanich law (Skumanich 1972), which has α = 1 / 2, lies in the region of parameter space for which our quasi-steady laminar solutions are valid regardless of β is a very nice - if somewhat unexpected - feature of our model, and speaks to its relevance for observations. The fact that it is also a critical parameter value, on the other hand, is probably just a coincidence.</text>
<text><location><page_35><loc_12><loc_63><loc_88><loc_76></location>In general, however, α is not known a priori - it is an outcome of the complete spin-down problem (see Reiners & Mohanty 2012, for instance, who found both exponential and power-law spin-down solutions depending on their assumed wind model). Our theoretical results suggest that if β ≥ 2 / 3, then again a laminar quasi-steady solution always exists regardless of the spin-down law. If β < 2 / 3, on the other hand, whether the laminar solution holds or not also depends on the actual spin-down rate of the convection zone, and must therefore be determined 'on the fly' while the solution of (68) is being computed.</text>
<text><location><page_35><loc_12><loc_45><loc_88><loc_62></location>Unfortunately, it is difficult to constrain β from theory alone (see Appendix E). Indeed, while GM98 and Wood et al. (2011) both found that β > 2 / 3 in a related but distinctly different model system, their results cannot be used here. In this particular instance, help in constraining β comes from observations instead. The requirement that fast rotators should be in solid-body rotation at all times immediately constrains β to be strictly smaller than 2 / 3 (see Section 6 for detail), which does imply that our model could break down if the spin-down rate is too rapid. For further constraints, we note that surface abundances of light elements and other tracers can potentially be used to estimate the depth of the chemically mixed tachocline, and study its variation with stellar rotation rate (see Section 6.3).</text>
<section_header_level_1><location><page_35><loc_29><loc_40><loc_71><loc_41></location>5.5. Properties of the initial transient solution</section_header_level_1>
<text><location><page_35><loc_12><loc_29><loc_88><loc_37></location>The initial, nearly linear increase in the core-envelope lag seen in Figures 5 and 6 can be understood by noting that the angular-momentum transport rate across the tachopause depends on the local torques, which in turn depend on the local angular-velocity shear. At first, the latter is small, so the torques are not strong enough to spin the core down. When viewed in an inertial frame, the latter continues to spin at its original rate, so that</text>
<formula><location><page_35><loc_36><loc_25><loc_88><loc_27></location>Ω c ( t ) = -Ω cz ( t ) glyph[similarequal] -˙ Ω cz ( t 0 )( t -t 0 ) , (72)</formula>
<text><location><page_35><loc_12><loc_22><loc_49><loc_23></location>for t -t 0 smaller than the spin-down timescale.</text>
<text><location><page_35><loc_12><loc_12><loc_88><loc_21></location>As the core-envelope lag increases, so does the shear, until a point where the torque exerted is just strong enough to communicate the surface spin-down to the interior. When this happens, the core locks on to the tachocline, and the angular-momentum flux becomes independent of radius. The system reaches a quasi-steady state in the spinning-down frame, in which both convective zone, tachocline and core concurrently spin-down at more-or-less the same rate. Equating the early-time</text>
<figure>
<location><page_36><loc_28><loc_44><loc_70><loc_69></location>
<caption>Fig. 6.- Evolution of Ω c / Ω cz for the reference star described in Table 1, undergoing a Skumanich law spin-down. The tachocline thickness varies with Ω cz as in Equation (44), with β = 0, 1 / 3 and 2 / 3 respectively. In all cases ∆ 0 is chosen such that ∆ = 1 . 5 × 10 9 cm at t = 10 3 t 0 (see Table 1), so ∆ 0 = 4 . 7 × 10 9 cm for β = 1 / 3, and ∆ 0 = 1 . 5 × 10 10 cm for β = 2 / 3. The exact solution to Equation (65) is shown by the solid, dashed and dot-dashed lines, and the quasi-steady state approximation to that solution (given by Equation (71)) is shown by the dotted lines. For large t , Ω c / Ω cz ∝ t α ( q + β ) (see Section 5.4).</caption>
</figure>
<text><location><page_37><loc_12><loc_85><loc_87><loc_86></location>solution (72) with the quasi-steady solution (71), we find that this happens (very roughly) when</text>
<formula><location><page_37><loc_36><loc_80><loc_88><loc_84></location>t -t 0 = λ 2 1 24 t ES (Ω cz ( t )) I core I tc ( t ) ˙ Ω cz ( t ) ˙ Ω cz ( t 0 ) . (73)</formula>
<text><location><page_37><loc_12><loc_70><loc_88><loc_79></location>Although implicit for t , and therefore difficult to solve algebraically, this expression readily shows that the duration of the initial transient phase is proportional to the local Eddington-Sweet timescale across the tachocline, times I core /I tc , a quantity that is overall proportional to ∆ 3 / Ω 2 cz . In other words, it is much longer if the initial tachocline thickness is larger, which explains the results of Figure 6.</text>
<text><location><page_37><loc_12><loc_63><loc_88><loc_68></location>In what follows, we now summarize our results, importing them into a spherical geometry and casting them in a more astrophysically relevant terminology, and discuss their implications for stellar spin-down and related observations.</text>
<section_header_level_1><location><page_37><loc_33><loc_58><loc_67><loc_59></location>6. SUMMARY AND DISCUSSION</section_header_level_1>
<section_header_level_1><location><page_37><loc_37><loc_54><loc_63><loc_55></location>6.1. Summary of the results</section_header_level_1>
<text><location><page_37><loc_12><loc_43><loc_88><loc_52></location>In this work, we have studied the impact of spin-down on solar-type stars whose internal dynamics are assumed to be analogous to that of the Sun as first introduced by GM98. More specifically, we considered stars with a radiation zone held in uniform rotation by the presence of a large-scale primordial magnetic field, that is confined strictly below the base of the convection zone by large-scale meridional flows. The geometry of such stars was shown in Figure 1.</text>
<text><location><page_37><loc_12><loc_31><loc_88><loc_42></location>Separating the convection zone and the bulk of the radiation zone (the 'core', hereafter), which are rotating with angular velocities Ω cz and Ω core respectively (as expressed in an inertial frame), lie two thin nested shear layers (GM98; Wood & McIntyre 2011; Wood et al. 2011; Acevedo-Arreguin et al. 2013): the tachocline, which resides just beneath the radiative-convective interface, and is dynamically speaking at least - magnetic free, and the tachopause, which lies below the tachocline, and connects the latter magnetically to the core.</text>
<text><location><page_37><loc_12><loc_19><loc_88><loc_30></location>The extraction of angular momentum from the star by the stellar wind in our model takes a rather different form across each of these regions. Magnetic braking exerts a torque on the surface layers, which is nearly instantaneously communicated down to the base of the convection zone by the turbulence. Angular-momentum transport (and chemical mixing) across the underlying tachocline, on the other hand, is mediated principally by large-scale meridional flows and roughly takes place on a local Eddington-Sweet timescale:</text>
<formula><location><page_37><loc_39><loc_14><loc_88><loc_18></location>t ES (Ω cz ) = ¯ N 2 tc 2Ω 2 cz ( ∆ r cz ) 2 ∆ 2 κ tc , (74)</formula>
<text><location><page_37><loc_12><loc_10><loc_88><loc_13></location>where ¯ N tc , κ tc are the buoyancy frequency and thermal diffusivity of the fluid within the tachocline, ∆ is its thickness, and r cz is the radius of the base of the convection zone. The spin-down torque is</text>
<text><location><page_38><loc_12><loc_77><loc_88><loc_86></location>then finally communicated through the tachopause down to the deep interior primarily by magnetic torques. The mostly-dipolar primordial field is wound-up by the radial shear, which generates a significant toroidal field. The resulting Lorentz force reacts against the shear, and thus extracts angular momentum from the core. This happens on an Alfv'enic timescale which is more or less instantaneous compared with the spin-down timescale or the tachocline mixing timescale.</text>
<text><location><page_38><loc_12><loc_67><loc_88><loc_76></location>The tachocline is clearly the 'bottleneck' of this angular-momentum extraction sequence, and therefore controls the overall rotational evolution of the star. As a result, the timescale t ES (Ω cz ) introduced above plays a role that is similar (but not identical, see below) to the coupling timescale between the core and the envelope in the two-zone model of MacGregor & Brenner (1991) (see τ c in Equation 1).</text>
<text><location><page_38><loc_12><loc_60><loc_88><loc_65></location>Thanks to the help of an idealized model for which exact solutions exist, we have formally shown that the concurrent evolution of the rigidly-rotating core and the convective envelope can typically (i.e. under reasonable assumptions usually valid in most solar-type stars) be modeled as:</text>
<formula><location><page_38><loc_40><loc_56><loc_88><loc_59></location>dJ core dt = -∆ J τ c , (75)</formula>
<formula><location><page_38><loc_34><loc_52><loc_88><loc_55></location>dJ cz dt + dJ core dt = -˙ J w , where (76)</formula>
<formula><location><page_38><loc_42><loc_49><loc_88><loc_52></location>∆ J =(Ω core -Ω cz ) I core I cz I core + I cz , (77)</formula>
<text><location><page_38><loc_12><loc_44><loc_88><loc_47></location>that is, exactly as in the two-zone model of MacGregor & Brenner (1991) (see Equation 1), with a coupling timescale τ c given by</text>
<formula><location><page_38><loc_38><loc_39><loc_88><loc_43></location>τ c = t ES (Ω cz ) K I core I cz I tc ( I core + I cz ) , (78)</formula>
<text><location><page_38><loc_12><loc_33><loc_88><loc_38></location>where K is a positive geometrical constant of order unity, and where I tc is the moment of inertia of the tachocline (which, to a good approximation, is I tc = 4 π ¯ ρ tc r 4 cz ∆, where ¯ ρ tc is the local density within the tachocline). Equation (75) was derived in Section 5.3.</text>
<text><location><page_38><loc_12><loc_21><loc_88><loc_31></location>Inspection of Equations (75) - (78) shows that they do not explicitly depend on the magnetohydrodynamics of the tachopause 12 , a result inherently tied to the assumption that the tachopause is much thinner than the tachocline, and that it is at all time in complete dynamical and thermal equilibrium. When this is the case, the tachopause responds near-instantaneously to any perturbation and thus cannot introduce any additional timescale in the system (see Section 5.3 for detail). This property of our model turns out to be quite convenient: as discussed by Acevedo-Arreguin et al.</text>
<text><location><page_39><loc_12><loc_79><loc_88><loc_86></location>(2013), the specific nature and dynamical properties of the tachopause are arguably the 'weakest link' of the GM98 model, being the most sensitive to any dynamics that were purposefully neglected (turbulence, gravity waves, etc.). But as Equations (75) - (78) show, this model-dependence does not have any direct impact on the long-term evolution of the angular velocity of the core.</text>
<text><location><page_39><loc_12><loc_54><loc_88><loc_78></location>On the other hand, Equation (75) is very sensitive to the properties of the tachocline, and in particular to its thickness ∆. The latter presumably depends on the star's mean rotation rate, on its spin-down rate, on the strength of the internal primordial field, and on the position and local thermodynamical properties of the base of the convection zone. Both GM98 and Wood et al. (2011) propose scalings for ∆ as a function of these quantities in the solar case, where the largescale meridional flows are driven by the latitudinal shear within the convection zone rather than by spin-down. Unfortunately, as discussed in Appendix E, these scalings do not directly apply here. Furthermore, any attempt to estimate ∆ from first principles necessarily yields a result that depends sensitively on the structure of the tachopause, which we have just argued is both poorly constrained and strongly dependent on the model considered. In this sense, while the dynamics of the tachopause do not explicitly participate in Equations (75) - (78), they nevertheless indirectly influence the rotational evolution of the star by controlling ∆ (which appears in t ES (Ω cz ) and therefore in τ c ).</text>
<text><location><page_39><loc_12><loc_47><loc_88><loc_52></location>For these reasons, instead of proposing a carefully derived, mathematically correct but highly model-dependent formula for the tachocline thickness ∆, we suggest the following simple parametric prescription:</text>
<formula><location><page_39><loc_37><loc_43><loc_88><loc_47></location>∆ = ∆ 0 ( B 0 ) ( Ω cz Ω 0 ) β ( r cz r cz ( t 0 ) ) γ . (79)</formula>
<text><location><page_39><loc_12><loc_30><loc_88><loc_42></location>In this model, we have hidden all information about the unknown (and non-observable) internal field strength B 0 into ∆ 0 ( B 0 ). Any information about the time-dependence induced by spin-down is contained in the second term, and any information about the local properties of the tachocline is contained in the third term 13 . While β and γ are difficult to estimate from theory alone (see Appendix E for detail), we hope that they can, in the future, be constrained observationally by studying simultaneously the rotational histories of solar-type stars in young clusters and their light-element surface abundances (see Section 6.3).</text>
<text><location><page_39><loc_12><loc_18><loc_88><loc_28></location>In general, Equations (75)-(79) have to be solved - and should be solved - numerically, in conjunction with the equations for stellar evolution which yield I core , I cz and τ c at each point in time. However, good insight into the long-term behavior of the solutions can be obtained by considering a 'quasi-steady' approximation, in which (1) we assume that the spin-down rate ˙ Ω cz is known, (2) I core does not vary too rapidly with time and (3) Ω core -Ω cz is not too large. Using all three approximations implies that dJ core /dt glyph[similarequal] I core ˙ Ω cz is known, and one can then simply</text>
<text><location><page_40><loc_12><loc_83><loc_88><loc_86></location>solve Equation (75) analytically for Ω core . The relative core-envelope lag, in this quasi-steady approximation, is given by</text>
<formula><location><page_40><loc_30><loc_78><loc_88><loc_81></location>Ω core -Ω cz Ω cz = K t ES (Ω cz ) t sd (Ω cz ) I core I tc = -K ˙ Ω cz Ω 3 cz ¯ N 2 tc ∆ 4 r 2 cz κ tc I core I tc (80)</formula>
<text><location><page_40><loc_12><loc_69><loc_88><loc_76></location>where t sd = | Ω cz / ˙ Ω cz | is the spin-down timescale of the convective envelope. Physically speaking, this formula is equivalent to stating that the star adjusts itself in such a way that the integrated angular-momentum flux out of each spherical shell is constant with radius, and equal to that extracted from the star by the stellar wind.</text>
<text><location><page_40><loc_12><loc_53><loc_88><loc_68></location>We have shown in Sections 4.3 and 5.4 that our model is only technically valid when this solution is bounded (in the sense that the quasi-steady relative core-envelope lag remains constant or decreases with time). This happens when 2 -α -1 -3 β ≤ 0, where β is defined in (79), and where α is defined such that t sd ∝ Ω -α -1 cz . When the model applies, then the quasi-steady solution is also an attracting solution of the governing equations (which means that the system relaxes to this state regardless of its initial conditions). All stars satisfying this condition are therefore expected to have a core-envelope lag given by (80), after a transient period whose duration is of the order of t ES (Ω cz ) evaluated at t = t 0 (see Section 5.5 for detail).</text>
<section_header_level_1><location><page_40><loc_38><loc_47><loc_62><loc_49></location>6.2. Caveats of the model</section_header_level_1>
<text><location><page_40><loc_12><loc_40><loc_88><loc_45></location>Before we proceed to discuss the observational implications of our model, let us briefly address its caveats and limitations. In many ways, they are the same as those of the GM98 model, listed and discussed at length by GM98 and by Acevedo-Arreguin et al. (2013).</text>
<text><location><page_40><loc_12><loc_30><loc_88><loc_39></location>Central to our calculation is the assumption that the star has a dynamical structure similar to the Sun, with an outer convection zone and a uniformly rotating magnetized core both in solidbody rotation, separated by a thin magnetic-free tachocline, and an even thinner tachopause which, by contrast, is essentially magnetic in nature. As discussed by Acevedo-Arreguin et al. (2013), a necessary condition for such a layered model to exist is</text>
<formula><location><page_40><loc_43><loc_25><loc_88><loc_29></location>¯ N tc Ω cz √ ν tc κ tc glyph[lessmuch] r cz ∆ , (81)</formula>
<text><location><page_40><loc_12><loc_15><loc_88><loc_24></location>where ν tc is the viscosity in the tachocline region. If this condition is not satisfied, then the meridional flows downwelling from the convection zone are unable to confine the magnetic field, and a different model must be used (see Acevedo-Arreguin et al. 2013, for details). However, since the Sun satisfies this property, we expect that most young and thus more rapidly rotating stars are likely to satisfy it as well.</text>
<text><location><page_40><loc_12><loc_10><loc_88><loc_13></location>Even if (81) is satisfied, the existence of such a layered structure is not yet guaranteed. While it has now been revealed in fully nonlinear, full-sphere, steady-state simulations of the solar interior</text>
<text><location><page_41><loc_12><loc_70><loc_88><loc_86></location>for the first time (Acevedo-Arreguin et al. 2013), one should still verify that it can also be achieved in a spin-down problem. Indeed, as mentioned in Section 6.1, the mechanisms driving the largescale tachocline flows, which are responsible for confining the internal magnetic field within and below the tachopause, are subtly different in the solar steady-state case and in the spin-down case. We defer the task of running full-sphere numerical simulations of the spin-down problem to a future publication. Beyond the question of existence of a tachocline and tachopause, such a calculation could furthermore yield a first estimate of the possible relationships between their thicknesses and other stellar parameters, a result that cannot be robustly obtained from linear theory alone here (see Appendix E).</text>
<text><location><page_41><loc_12><loc_50><loc_88><loc_68></location>The next major assumption we need to verify is whether the effects of turbulence can indeed be neglected while modeling the tachocline. As discussed by Acevedo-Arreguin et al. (2013), thermalwind balance and thermal equilibrium - two key balances in the system - are both still likely to hold even in the presence of turbulence (given reasonable assumptions as to its source). On the other hand, angular-momentum balance is much more sensitive to any added effects, and must be studied carefully. For our model to hold, radial angular-momentum transport across the tachocline must be dominated by advection by large-scale flows, rather than by turbulence. Naturally, and as discussed throughout this work, this assumption can only hold as long as the shear across the tachocline is 'weak enough' not to cause significant turbulent transport - the question being what 'weak enough' means in this context.</text>
<text><location><page_41><loc_12><loc_36><loc_88><loc_48></location>One may first ask under which conditions the tachocline is linearly unstable to shear instabilities. As studied by Ligni'eres et al. (1999), this could depend both on the Richardson number Ri glyph[similarequal] ¯ N 2 tc ∆ 2 /r 2 cz (Ω cz -Ω core ) 2 , and on the P'eclet number Pe = r cz (Ω cz -Ω core )∆ /κ tc . In this particular problem, however, the P'eclet number is typically so large that the relevant criterion for global, tachocline-scale shear instabilities is the standard Ri < O (1) rather than RiPe < O (1) advocated by Zahn (1974) (which is only applicable to the small Pe limit). Using solar values as guidance (see Table 1), we find that our model is expected to break down completely only when</text>
<formula><location><page_41><loc_28><loc_31><loc_88><loc_34></location>2 , 300 ( ¯ N tc 8 × 10 -4 ) 2 ( ∆ /r cz 0 . 03 ) 2 ( 5 × 10 -7 Ω core -Ω cz ) 2 < O (1) . (82)</formula>
<text><location><page_41><loc_12><loc_20><loc_88><loc_29></location>For this inequality to hold, we therefore see that a substantial shear is required. Although unlikely in older stars, this could happen during the early stages of the spin-down process where Ω cz is much larger, and where rapid core-contraction can result in significant core-envelope shear. One should therefore monitor Ri carefully in the process of time-stepping Equations (75)-(79), and use a turbulent coupling timescale instead should Ri drop below 1.</text>
<text><location><page_41><loc_12><loc_14><loc_88><loc_19></location>Alternatively, ignoring linear stability considerations, one could simply assume that the system becomes unstable to finite amplitude perturbations for much weaker shearing rates. In that state, Prat & Ligni'eres (2013) (see also Zahn 1974, 1992) suggest that turbulent transport can be described</text>
<text><location><page_42><loc_12><loc_85><loc_50><loc_86></location>using the vertical turbulent diffusion coefficient 14</text>
<formula><location><page_42><loc_43><loc_81><loc_88><loc_83></location>D t glyph[similarequal] 0 . 05 κ tc Ri -1 . (83)</formula>
<text><location><page_42><loc_12><loc_72><loc_88><loc_79></location>In the presence of this kind of stratified shear turbulence, our model holds provided the timescale for advection of angular momentum across the tachocline by large-scale flows (given more-or-less by t ES (Ω cz )) is shorter than the timescale for the turbulent diffusion of angular momentum (given more-or-less by ∆ 2 /D t ). This implies that our model is expected to apply whenever</text>
<formula><location><page_42><loc_30><loc_67><loc_88><loc_71></location>¯ N 2 tc ∆ 4 Ω 2 cz r 2 cz κ tc D t ∆ 2 < O (1) ⇔ 0 . 05 (Ω core -Ω cz ) 2 Ω 2 cz < O (1) . (84)</formula>
<text><location><page_42><loc_12><loc_57><loc_88><loc_66></location>This criterion, rather interestingly and perhaps surprisingly, depends only on the relative coreenvelope lag iself. We then find that the use of our model is also always justified unless (Ω core -Ω cz ) is unrealistically large (that is, much larger than Ω cz itself). It thus appears that neglecting turbulent angular-momentum transport in the tachocline could well be justified, except perhaps very early on in the spin-down process if the Richardson number Ri ever drops below 1.</text>
<section_header_level_1><location><page_42><loc_35><loc_52><loc_65><loc_53></location>6.3. Observational implications</section_header_level_1>
<text><location><page_42><loc_12><loc_41><loc_88><loc_49></location>While a complete discussion of the observational implications of our model will have to be done by applying it in conjunction with stellar evolution, and statistically comparing its predictions against observations (as in Allain 1998; Irwin et al. 2007; Denissenkov et al. 2010; Spada et al. 2011; Reiners & Mohanty 2012; Gallet & Bouvier 2013, for instance), we can nevertheless already discuss its prospects in the light of previous work.</text>
<text><location><page_42><loc_12><loc_17><loc_88><loc_39></location>As reported in Section 1, these previous studies found that stars that begin their lives as rapid rotators can, at all times, be modeled assuming solid-body rotation, while the rotation rates of stars in the mass-range 0 . 7 M glyph[circledot] -1 . 1 M glyph[circledot] that are initially slow rotators are best modeled with the two-zone model of MacGregor & Brenner (1991) assuming a rather long core-envelope coupling timescale (of the order of hundreds of Myr up to a Gyr). It is very difficult to explain such long timescales using a magnetic model, unless rather dramatic assumptions are made concerning the degree of confinement of the field (which must then also be explained). It is also difficult to explain observations with a purely turbulent model, since the latter does not easily explain why the core should be mostly in solid-body rotation. By contrast, our model naturally results in a system that behaves like the two-zone model, with a coupling timescale that depends both on the stellar structure and on the rotation rate of the star (see Equation 78), and that can be very substantial for slower rotators with fairly thick tachoclines.</text>
<text><location><page_43><loc_12><loc_70><loc_88><loc_86></location>Specifically, we find that the coupling timescale is proportional to ∆ 3 /r 2 cz Ω 2 cz . As such, for similar tachocline thicknesses, it is naturally much shorter for fast rotators than for slow rotators, and for higher-mass stars (which have a larger radiation zone) than for lower-mass stars. Both results are qualitatively consistent with the aforementioned observations. Of course, the unknown dependence of ∆ on Ω cz and r cz makes it difficult at this point to give strict estimates of how strong this effect may be. Nevertheless, we can already infer from the data (whereby fast rotators should also be solid-body rotators) that ∆ 3 /r 2 cz Ω 2 cz must be a decreasing function of Ω cz . This constrains β (see Equation 79) to be strictly smaller than 2 / 3, and quite possibly substantially smaller than that.</text>
<text><location><page_43><loc_12><loc_54><loc_88><loc_68></location>Additional information on the tachocline thickness ∆ may be obtained by studying the relative differences in surface chemical abundances of stars within the same cluster. Light-element such as lithium and beryllium undergo significant Main-Sequence depletion, that can only be explained by extra mixing below the base of the convection zone (see for instance the review by Pinsonneault 1997). Within the scope of our model, we expect their respective depletion rates to depend sensitively on ∆( t ), so that present-day surface abundances of a given star provide an integrated view of the variation of the tachocline depth with time (and therefore with rotation rate). Concurrently fitting rotational histories with Li and Be abundances may thus help constrain both β and γ .</text>
<text><location><page_43><loc_12><loc_32><loc_88><loc_52></location>For stars in older clusters ( > few hundred Myr), we generally expect Equation (80) to hold. Indeed, as long as the condition 2 -α -1 -3 β ≤ 0 is satisfied (see 5.4 and 6.1 for detail), stars should have relaxed to their quasi-steady state by that age. In that state, aside from the surface abundances which depend on the rotational history of the star (as discussed above), all dynamical information about the star's initial conditions is lost, and the core-envelope lag only depends on present-day parameters. We then see from (80) that everything else being equal , their core-envelope lag should be much larger (1) if the spin-down rate is larger or if the star is rotating more slowly; (2) for lower-mass solar-type stars; (3) for stars with thicker tachoclines. While asteroseismology has not yet been able to detect any core-envelope lag in solar-type stars other than the Sun, one can only hope that such detection may be possible at some point in the future, and will independently help constrain our model.</text>
<text><location><page_43><loc_12><loc_11><loc_88><loc_31></location>Finally, note that our model predicts that stars in this quasi-steady state, with 2 -α -1 -3 β < 0, always eventually reach solid-body rotation (at least in a radial sense) and that ∆ → 0 as Ω cz → 0 and ˙ Ω cz → 0. However, substantial latitudinal differential rotation is likely to persist in their convection zones, as it does in the Sun. This latitudinal shear by itself also drives large-scale meridional flows by gyroscopic pumping. As studied by GM98, these flows transport angular momentum and interact with the embedded primordial field, leading to a finite tachocline thickness (see Appendix E) even when ˙ Ω cz = 0. Although we have ignored this effect here for simplicity, and on the grounds that these shear-induced flows are presumably weaker than those driven by the spin-down torque for young stars, it can no longer be ignored for much older stars. In future work, we shall attempt to model simultaneously the effects of spin-down and of latitudinal shear in the convection zone in driving the tachocline flows, so as to present an integrated model of</text>
<text><location><page_44><loc_12><loc_83><loc_88><loc_86></location>the rotational evolution of solar-type stars that can be used all the way from the Zero-Age Main Sequence to the present-day Sun.</text>
<section_header_level_1><location><page_44><loc_41><loc_77><loc_59><loc_78></location>Acknowledgements</section_header_level_1>
<text><location><page_44><loc_12><loc_66><loc_88><loc_75></location>This work originated from R. L. F. Oglethorpe's summer project at the Woods Hole GFD Summer Program in 2012. We thank the NSF and the ONR for supporting this excellent program. R. L. F. Oglethorpe acknowledges funding from an EPSRC studentship. P. Garaud acknowledges funding from the NSF (CAREER-0847477). We thank Nic Brummell, Douglas Gough, Subhanjoy Mohanty, Nigel Weiss and Toby Wood for fruitful discussions.</text>
<section_header_level_1><location><page_44><loc_20><loc_59><loc_80><loc_62></location>A. QUASI-STEADY SOLUTION FOR THE SPIN-DOWN OF A NON-MAGNETIC STAR</section_header_level_1>
<text><location><page_44><loc_12><loc_51><loc_88><loc_56></location>In this Appendix, we derive the result presented in Equation (25). Assuming a quasi-steady state, we drop the time-derivative in the momentum equation (22). Its azimuthal component then reduces to</text>
<formula><location><page_44><loc_44><loc_48><loc_88><loc_52></location>u ( s ) = -˙ Ω cz 2Ω cz s, (A1)</formula>
<text><location><page_44><loc_12><loc_44><loc_88><loc_47></location>implying (using mass conservation) that w must be a linear function of z , exactly as in Section 3. To satisfy impermeability at z = z tc , we must therefore have</text>
<formula><location><page_44><loc_42><loc_40><loc_88><loc_43></location>w ( z ) = ˙ Ω cz Ω cz ( z -z tc ) . (A2)</formula>
<text><location><page_44><loc_12><loc_37><loc_87><loc_39></location>The boundary condition (7), combined with the vertical component of the momentum equation,</text>
<formula><location><page_44><loc_44><loc_33><loc_88><loc_36></location>1 ¯ ρ tc ∂p ∂z = ¯ g tc ¯ T tc T, (A3)</formula>
<text><location><page_44><loc_12><loc_29><loc_88><loc_32></location>implies that T = 0 on s = R . Solving the thermal energy equation (24) with this boundary condition, along with T = 0 at z = z tc then gives</text>
<formula><location><page_44><loc_25><loc_24><loc_88><loc_28></location>T ( s, z ) = ∑ n J 0 ( λ n s R ) [ α n sinh ( λ n z -z tc R ) -C n ( z -z tc ) R 2 λ 2 n ] , (A4)</formula>
<text><location><page_44><loc_12><loc_21><loc_66><loc_23></location>where the constants { λ n } are the zeros of the Bessel function J 0 , and</text>
<formula><location><page_44><loc_40><loc_17><loc_88><loc_20></location>C n = ¯ N 2 tc ¯ T tc ¯ g tc κ tc ˙ Ω cz Ω cz 2 λ n J 1 ( λ n ) . (A5)</formula>
<text><location><page_44><loc_12><loc_14><loc_70><loc_16></location>Using the fact that T = 0 at z = z cz determines the { α n } coefficients to be</text>
<formula><location><page_44><loc_42><loc_9><loc_88><loc_13></location>α n = R 2 λ 2 n C n ∆ sinh ( λ n ∆ R ) , (A6)</formula>
<text><location><page_45><loc_12><loc_85><loc_54><loc_86></location>where ∆ = z cz -z tc is the thickness of the tachocline.</text>
<text><location><page_45><loc_12><loc_80><loc_88><loc_83></location>Equations (A3) and (23) can then be used to derive p and v , up to the unknown set of integration constants { p n } (which are the same in both equations):</text>
<formula><location><page_45><loc_15><loc_75><loc_88><loc_79></location>p ( s, z ) = ¯ ρ tc ¯ g tc ¯ T tc ∑ n J 0 ( λ n s R ) [ α n R λ n cosh ( λ n z -z tc R ) -C n ( z -z tc ) 2 R 2 2 λ 2 n + p n ] , (A7)</formula>
<formula><location><page_45><loc_15><loc_71><loc_88><loc_75></location>v ( s, z ) = ¯ g tc 2Ω cz ¯ T tc ∑ n d ds J 0 ( λ n s R ) [ α n R λ n cosh ( λ n z -z tc R ) -C n ( z -z tc ) 2 R 2 2 λ 2 n + p n ] . (A8)</formula>
<text><location><page_45><loc_12><loc_64><loc_88><loc_70></location>To find { p n } , we need to solve for the dynamics of the convection zone, and match them onto the tachocline solution. Using the same method as the one outlined in Section 3, but this time with the anelastic mass conservation equation (see Equation 20), we find that</text>
<formula><location><page_45><loc_24><loc_59><loc_88><loc_63></location>¯ ρ ( z ) w ( s, z ) = ∑ n ˜ B n sinh ( λ n z -H R ) J 0 ( λ n s R ) , (A9)</formula>
<formula><location><page_45><loc_27><loc_55><loc_88><loc_59></location>p ( s, z ) = -∑ n J 0 ( λ n s R ) ˜ B n τ R λ n cosh ( λ n z -H R ) , for z > z cz , (A10)</formula>
<text><location><page_45><loc_12><loc_52><loc_52><loc_54></location>with { ˜ B n } given again by matching w at z cz so that</text>
<formula><location><page_45><loc_34><loc_47><loc_88><loc_51></location>˜ B n = ¯ ρ ( z cz ) ˙ Ω cz Ω cz 2∆ λ n J 1 ( λ n ) sinh ( λ n z cz -H R ) . (A11)</formula>
<text><location><page_45><loc_12><loc_43><loc_88><loc_46></location>Matching p at z = z cz , and using the fact that ¯ ρ tc in the tachocline is, to a first approximation, equal to ¯ ρ ( z cz ) gives</text>
<formula><location><page_45><loc_22><loc_38><loc_88><loc_42></location>p n = -¯ T tc ¯ ρ tc ¯ g tc ˜ B n R τλ n cosh ( λ n H -z cz R ) -α n R λ n cosh ( λ n ∆ R ) + C n ∆ 2 R 2 2 λ 2 n , (A12)</formula>
<text><location><page_45><loc_12><loc_36><loc_58><loc_37></location>so that the azimuthal velocity (A8) leads to Equation (25).</text>
<section_header_level_1><location><page_45><loc_17><loc_30><loc_87><loc_31></location>TRANSIENT SOLUTION FOR THE SPIN-DOWN OF A NON-MAGNETIC</section_header_level_1>
<section_header_level_1><location><page_45><loc_13><loc_28><loc_55><loc_31></location>B. STAR</section_header_level_1>
<section_header_level_1><location><page_45><loc_38><loc_25><loc_62><loc_26></location>B.1. Vertical eigenmodes</section_header_level_1>
<text><location><page_45><loc_12><loc_18><loc_88><loc_23></location>In this Section we derive solutions of Equation (37) in the form of the eigenfunctions Z nm ( x ) and their associated eigenvalues µ nm , where x = ( z -z tc ) / ∆. To do this, we first need to specify the various boundary conditions on Z nm . The boundary conditions at z = z tc ( x = 0) are</text>
<formula><location><page_45><loc_37><loc_13><loc_88><loc_16></location>T = 0 ⇒ ∂v n ∂z = 0 ⇒ dZ nm dx = 0 , (B1)</formula>
<formula><location><page_45><loc_37><loc_10><loc_88><loc_13></location>w = 0 ⇒ ∂ 3 v n ∂z 3 = 0 ⇒ d 3 Z nm dx 3 = 0 . (B2)</formula>
<text><location><page_46><loc_12><loc_77><loc_88><loc_86></location>In the convection zone, by contrast, we assume that the dynamics always relax to the steady state on a very rapid timescale. Furthermore, we have shown in Section 4.2 that, within the context of a Darcy friction model, one can interchangeably use the Boussinesq approximation or the more realistic anelastic approximation. Here we adopt the former for the convection zone, hence (13) and (14) hold. The boundary conditions at z = z cz ( x = 1) are then</text>
<formula><location><page_46><loc_22><loc_73><loc_88><loc_76></location>T continuous ⇒ T = 0 ⇒ ∂v n ∂z = 0 ⇒ dZ nm dx = 0 , (B3)</formula>
<formula><location><page_46><loc_23><loc_69><loc_88><loc_72></location>p continuous ⇒ ∂p ∂s continuous ⇒ 2Ω cz v n = -B n R τλ n cosh ( λ n z cz -H R ) , (B4)</formula>
<formula><location><page_46><loc_22><loc_65><loc_88><loc_69></location>w continuous ⇒ 2Ω cz κ tc ¯ N 2 tc ∂ 3 v n ∂z 3 = B n sinh ( λ n z cz -H R ) , (B5)</formula>
<text><location><page_46><loc_12><loc_60><loc_88><loc_64></location>using (B3), where { B n } remain to be determined. Equations (B4) and (B5) can finally be combined to give</text>
<formula><location><page_46><loc_24><loc_56><loc_88><loc_60></location>Z nm = R τλ n 1 tanh( λ n H -z cz R ) κ tc ¯ N 2 tc ∆ 3 d 3 Z nm dx 3 ≡ K n d 3 Z nm dx 3 at x = 1 , (B6)</formula>
<text><location><page_46><loc_12><loc_49><loc_88><loc_56></location>which defines the constants { K n } , and shows them to be positive. We then see that the eigenvalue problem defined by Equation (37) and associated boundary conditions listed above is homogeneous. It can easily be shown that the operator L on the left-hand-side of Equation (37) is self-adjoint with these boundary conditions, which implies that the vertical eigenmodes are orthogonal, with</text>
<formula><location><page_46><loc_33><loc_44><loc_88><loc_47></location>∫ 1 0 Z nm ( x ) Z nm ' ( x ) dx = δ mm ' ∫ 1 0 Z 2 nm ( x ) dx. (B7)</formula>
<text><location><page_46><loc_12><loc_35><loc_88><loc_42></location>It can also be shown by considering the integral ∫ 1 0 Z nm L ( Z nm ) dx , suitably integrating it by parts, and applying the boundary conditions, that the eigenvalues associated with the operator L and our boundary conditions must be strictly positive, hence our choice of writing them as µ 4 nm in Equation (37).</text>
<text><location><page_46><loc_12><loc_27><loc_88><loc_34></location>Since (37) is an equation with constant coefficients, we seek solutions of the form e σ nm z and find four solutions for σ nm : ± µ nm and ± iµ nm . Using this information, solutions of (37) that satisfy Z ' nm (0) = 0 and Z ''' nm (0) = 0 can be written as a linear combination of cosh( µ nm x ) and cos( µ nm x ) Applying the boundary condition (B3) we then have</text>
<formula><location><page_46><loc_32><loc_22><loc_88><loc_25></location>Z nm ( x ) = sin( µ nm ) sinh( µ nm ) cosh( µ nm x ) + cos( µ nm x ) , (B8)</formula>
<text><location><page_46><loc_12><loc_19><loc_76><loc_20></location>while the µ nm coefficients can be found by applying (B6). They are the solution of</text>
<formula><location><page_46><loc_36><loc_14><loc_88><loc_17></location>1 tanh( µ nm ) + 1 tan( µ nm ) = 2 K n µ 3 nm . (B9)</formula>
<section_header_level_1><location><page_47><loc_31><loc_85><loc_69><loc_86></location>B.2. Limit of the time-dependent solution</section_header_level_1>
<text><location><page_47><loc_12><loc_80><loc_88><loc_83></location>We now seek to show that the solution to the time-dependent problem given in Equation (41) tends to the quasi-steady solution derived in Section 4.2 in the limit of large time for q ≤ 0.</text>
<text><location><page_47><loc_15><loc_77><loc_87><loc_78></location>Since the contribution of the initial conditions disappear as t → + ∞ for q ≤ 0, we find that</text>
<formula><location><page_47><loc_37><loc_72><loc_88><loc_76></location>V nm ( t ) → 1 µ ( t ) ∫ t t 0 µ ( t ' ) F nm ( t ' ) dt ' . (B10)</formula>
<text><location><page_47><loc_12><loc_70><loc_35><loc_71></location>For ease of notation, we write</text>
<formula><location><page_47><loc_22><loc_64><loc_88><loc_68></location>F nm ( t ) = f n ˙ Ω cz ( t ) = -f n Ω cz t sd (Ω cz ) , where f n = 4 R 2 λ 3 n J 1 ( λ n ) ∫ 1 0 Z nm ( x ) dx ∫ 1 0 Z 2 nm ( x ) dx , (B11)</formula>
<text><location><page_47><loc_12><loc_62><loc_17><loc_63></location>so that</text>
<formula><location><page_47><loc_35><loc_58><loc_88><loc_62></location>V nm ( t ) → f n µ (Ω cz ( t )) ∫ Ω cz ( t ) Ω 0 µ (Ω cz ) d Ω cz . (B12)</formula>
<text><location><page_47><loc_15><loc_55><loc_49><loc_57></location>In the case that q < 0, Equation (46) gives</text>
<formula><location><page_47><loc_24><loc_50><loc_88><loc_54></location>V nm ( t ) → f n µ (Ω cz ( t )) ∫ Ω cz ( t ) Ω 0 exp [ -1 q t sd (Ω 0 ) τ ES nm (Ω 0 ) [( Ω cz Ω 0 ) q -1 ]] d Ω cz . (B13)</formula>
<text><location><page_47><loc_12><loc_46><loc_88><loc_49></location>Using the method of steepest descent (as in Riley et al. 2006, for instance), noting that Ω cz < Ω 0 and that q < 0, we find that</text>
<formula><location><page_47><loc_21><loc_41><loc_88><loc_45></location>V nm ( t ) →-f n µ (Ω cz ( t )) exp [ -1 q t sd (Ω 0 ) τ ES nm (Ω 0 ) [( Ω cz ( t ) Ω 0 ) q -1 ]] Ω q 0 Ω 1 -q cz τ ES nm (Ω 0 ) t sd (Ω 0 ) , (B14)</formula>
<text><location><page_47><loc_12><loc_39><loc_60><loc_40></location>which, using Equations (39), (45), (43) and (B11), reduces to</text>
<formula><location><page_47><loc_37><loc_35><loc_88><loc_37></location>V nm ( t ) → F nm ( t ) τ ES nm ( t ) = V qs nm ( t ) , (B15)</formula>
<text><location><page_47><loc_12><loc_29><loc_88><loc_34></location>where V qs nm ( t ) is defined as the solution to Equation (38) without the dV nm /dt term, and so by definition is the projection of the quasi-steady solution (Equation 25) onto the horizontal and vertical eigenmodes.</text>
<text><location><page_47><loc_15><loc_26><loc_70><loc_27></location>In the case that q = 0, in the limit of large t , Equation (B12) becomes</text>
<formula><location><page_47><loc_21><loc_10><loc_88><loc_25></location>V nm ( t ) → f n µ (Ω cz ( t )) ∫ Ω cz ( t ) Ω 0 ( Ω cz Ω 0 ) -t sd (Ω 0 ) /τ ES nm (Ω 0 ) d Ω cz = f n Ω 0 µ ( t ) 1 1 -t sd (Ω 0 ) τ ES nm (Ω 0 ) ( ( Ω cz ( t ) Ω 0 ) 1 -t sd (Ω 0 ) /τ ES nm (Ω 0 ) -1 ) = f n Ω 0 1 1 -t sd (Ω 0 ) τ ES nm (Ω 0 ) ( ( Ω cz ( t ) Ω 0 ) -( Ω cz ( t ) Ω 0 ) t sd (Ω 0 ) /τ ES nm (Ω 0 ) ) . (B16)</formula>
<text><location><page_48><loc_12><loc_81><loc_88><loc_86></location>Since q = 0, the ratio of t sd to τ ES nm is a constant, for each n , m (see Equation 47), and whether the transient solution tends to the quasi-steady solution depends on this ratio. If t sd glyph[greatermuch] τ ES nm (or equivalently t sd glyph[greatermuch] t ES ), then</text>
<formula><location><page_48><loc_30><loc_76><loc_88><loc_79></location>V nm ( t ) → f n Ω 0 τ ES nm (Ω 0 ) t sd (Ω 0 ) = F nm ( t ) τ ES nm ( t ) = V qs nm ( t ) . (B17)</formula>
<text><location><page_48><loc_12><loc_73><loc_41><loc_75></location>If, on the other hand, t sd glyph[lessmuch] τ ES nm , then</text>
<text><location><page_48><loc_78><loc_69><loc_78><loc_70></location>glyph[negationslash]</text>
<formula><location><page_48><loc_14><loc_66><loc_88><loc_72></location>V nm ( t ) →-f n Ω 0 ( Ω cz ( t ) Ω 0 ) t sd (Ω 0 ) /τ ES nm (Ω 0 ) = F nm ( t ) t sd ( t ) ( Ω cz ( t ) Ω 0 ) t sd (Ω 0 ) /τ ES nm (Ω 0 ) -1 = V qs nm ( t ) . (B18)</formula>
<text><location><page_48><loc_12><loc_64><loc_88><loc_66></location>Hence, the system only relaxes to the quasi-steady solution when q = 0 provided t ES ( t 0 ) glyph[lessmuch] t sd ( t 0 ).</text>
<section_header_level_1><location><page_48><loc_20><loc_57><loc_80><loc_60></location>C. DERIVATION OF THE EKMAN JUMP CONDITION IN A SPINNING-DOWN FRAME</section_header_level_1>
<text><location><page_48><loc_12><loc_50><loc_88><loc_55></location>In this Section we derive the viscous jump condition across the Ekman layer reported in Equation (55). Assuming that the Ekman layer is sufficiently thin to always be in balance, we apply the quasi-steady approximation to the momentum equation, which now reads</text>
<formula><location><page_48><loc_34><loc_45><loc_88><loc_48></location>2 Ω cz × u + ˙ Ω cz × r = -1 ¯ ρ tc ∇ p + ν tc ∂ 2 u ∂z 2 , (C1)</formula>
<text><location><page_48><loc_12><loc_38><loc_88><loc_43></location>where ν tc is the local viscosity, and where we have approximated the Laplacian in the viscous term by keeping only the vertical derivatives. Combining this momentum equation with conservation of mass gives</text>
<formula><location><page_48><loc_43><loc_35><loc_88><loc_38></location>-∂v ∂z = ν 2 tc 4Ω 2 cz ∂ 5 v ∂z 5 . (C2)</formula>
<text><location><page_48><loc_12><loc_30><loc_88><loc_34></location>To solve this equation, we first define δ E = √ ν tc / 2Ω cz and introduce the boundary-layer variable ζ = ( z -z tc ) /δ E . Hence</text>
<formula><location><page_48><loc_45><loc_27><loc_88><loc_30></location>-∂v ∂ζ = ∂ 5 v ∂ζ 5 . (C3)</formula>
<text><location><page_48><loc_12><loc_17><loc_88><loc_26></location>By construction, the variable ζ remains of order unity within the tachopause, and rapidly tends to infinity above it, or in other words, as z enters the tachocline. We therefore have v → v 0 ( s ) as ζ → + ∞ , where v 0 ( s ) is the azimuthal velocity profile near the base of the tachocline. We also have v ( s, ζ ) = s Ω c at ζ = 0 assuming a no-slip boundary condition with the core. Applying these two conditions yields</text>
<formula><location><page_48><loc_23><loc_13><loc_88><loc_16></location>v ( s, ζ ) = v 0 ( s ) + e -ζ/ √ 2 [ ( s Ω c -v 0 ( s )) cos ( ζ √ 2 ) + c ( s ) sin ( ζ √ 2 )] , (C4)</formula>
<text><location><page_48><loc_12><loc_10><loc_65><loc_11></location>where c ( s ) is an integrating function that remains to be determined.</text>
<text><location><page_49><loc_12><loc_83><loc_88><loc_86></location>A no-slip boundary condition also applies to the radial velocity, so u ( s, ζ ) = 0 at ζ = 0. Using the azimuthal component of the momentum equation to find u ( s, ζ ), we have</text>
<formula><location><page_49><loc_22><loc_78><loc_88><loc_81></location>u ( s, ζ ) = -˙ Ω cz 2Ω cz s + e -ζ/ √ 2 [ ( s Ω c -v 0 ( s )) sin ( ζ √ 2 ) -c ( s ) cos ( ζ √ 2 )] , (C5)</formula>
<text><location><page_49><loc_12><loc_76><loc_13><loc_77></location>so</text>
<formula><location><page_49><loc_44><loc_72><loc_88><loc_76></location>c ( s ) = -˙ Ω cz 2Ω cz s. (C6)</formula>
<text><location><page_49><loc_12><loc_68><loc_88><loc_71></location>Finally, we require that the core be impermeable. To do so, we apply mass conservation to find ∂w ( s, ζ ) /∂ζ :</text>
<formula><location><page_49><loc_20><loc_62><loc_88><loc_66></location>∂w ∂ζ = δ E ˙ Ω cz Ω cz -δ E e -ζ/ √ 2 [ 1 s ∂ ∂s ( s 2 Ω c -sv 0 ( s ) ) sin ( ζ √ 2 ) + ˙ Ω cz Ω cz cos ( ζ √ 2 ) ] . (C7)</formula>
<text><location><page_49><loc_15><loc_59><loc_84><loc_60></location>Integrating Equation (C7) from ζ = 0 upward, and requiring w ( s, ζ ) = 0 at ζ = 0 yields</text>
<formula><location><page_49><loc_21><loc_45><loc_88><loc_57></location>w ( s, ζ ) = δ E ˙ Ω cz Ω cz ζ + δ E s ∂ ∂s ( s 2 Ω c -sv 0 ( s ) ) e -ζ/ √ 2 sin ( ζ √ 2 ) +cos ( ζ √ 2 ) √ 2 ζ 0 -δ E ˙ Ω cz Ω cz e -ζ/ √ 2 sin ( ζ √ 2 ) -cos ( ζ √ 2 ) √ 2 ζ 0 . (C8)</formula>
<text><location><page_49><loc_12><loc_43><loc_30><loc_44></location>We see that, as ζ →∞ ,</text>
<formula><location><page_49><loc_29><loc_38><loc_88><loc_42></location>w ( s, ζ ) → ˙ Ω cz Ω cz δ E ( ζ -1 √ 2 ) -δ E √ 2 s ∂ ∂s ( s 2 Ω c -sv 0 ( s ) ) . (C9)</formula>
<text><location><page_49><loc_12><loc_36><loc_84><loc_37></location>Matching this onto the tachocline solution (see Equation D1) yields the jump condition (55)</text>
<formula><location><page_49><loc_29><loc_31><loc_88><loc_34></location>w 0 ( s ) = ˙ Ω cz Ω cz δ E ( 1 -1 √ 2 ) -δ E √ 2 s ∂ ∂s ( s 2 Ω c -sv 0 ( s ) ) . (C10)</formula>
<section_header_level_1><location><page_49><loc_19><loc_25><loc_81><loc_27></location>D. DERIVATION OF THE EVOLUTION EQUATION FOR J core ( t )</section_header_level_1>
<text><location><page_49><loc_12><loc_14><loc_88><loc_23></location>In this Appendix, we derive the evolution equation for the angular momentum of the rigidly rotating core, reported in Equation (59). As in Appendix A, we solve the set of governing equations separately in the convection zone and in the tachocline, and match these solutions to the boundary conditions (at the top and side-walls of the domain), to the jump condition (at the base of the tachocline at z = z tc + δ ), and to each other (at the radiative-convective interface at z = z cz ).</text>
<text><location><page_49><loc_12><loc_10><loc_88><loc_13></location>In the convection zone ( z cz < z < H ), assuming a Boussinesq system (which was proved to yield the same results as in the anelastic case in Appendix A), w and p are given by Equations</text>
<text><location><page_50><loc_12><loc_83><loc_88><loc_86></location>(13) and (14), where { B n } are integration constants that need to be determined by matching these solutions to the tachocline.</text>
<text><location><page_50><loc_12><loc_74><loc_88><loc_82></location>In the tachocline ( z tc + δ < z < z cz ), we still have ∂w/∂z = ˙ Ω cz / Ω cz (see Equation (A1) and using mass conservation). However, we can no longer directly apply the impermeability condition at z = z tc , since w must first be matched onto the tachopause solution. Hence, we write instead that</text>
<formula><location><page_50><loc_38><loc_71><loc_88><loc_75></location>w = w 0 ( s ) + ˙ Ω cz Ω cz ( z -( z tc + δ )) , (D1)</formula>
<text><location><page_50><loc_12><loc_69><loc_67><loc_70></location>where w 0 ( s ) is an integration function, that remains to be determined.</text>
<text><location><page_50><loc_15><loc_66><loc_76><loc_67></location>Requiring continuity of w at the radiative-convective interface ( z = z cz ) yields</text>
<formula><location><page_50><loc_29><loc_61><loc_88><loc_65></location>w 0 ( s ) = ∑ n J 0 ( λ n s R ) B n sinh ( λ n z cz -H R ) -˙ Ω cz Ω cz ∆ . (D2)</formula>
<text><location><page_50><loc_12><loc_58><loc_41><loc_59></location>Substituting this into (53) then gives</text>
<formula><location><page_50><loc_26><loc_53><loc_88><loc_56></location>dJ core dt = -˙ Ω cz I tc + 16Ω cz I tc ∆ ∑ n J 1 ( λ n ) λ 3 n B n sinh ( λ n z cz -H R ) , (D3)</formula>
<text><location><page_50><loc_12><loc_48><loc_88><loc_51></location>where I tc is the moment of inertia of the tachocline defined in Equation (60) and where the only remaining unknowns are the { B n } . The following calculations show how to derive them.</text>
<text><location><page_50><loc_12><loc_43><loc_88><loc_46></location>Solving the thermal equilibrium equation (24) for T in the tachocline, with w given by (D1) yields</text>
<formula><location><page_50><loc_21><loc_34><loc_88><loc_42></location>T = ∑ n J 0 ( λ n s R ) [ α n sinh ( λ n z -( z tc + δ ) R ) + β n cosh ( λ n z -( z tc + δ ) R ) -¯ N 2 tc R 2 ¯ T κ tc λ 2 n ¯ g tc ( B n sinh ( λ n z cz -H R ) + 2( z -z cz ) λ n J 1 ( λ n ) ˙ Ω cz Ω cz )] , (D4)</formula>
<text><location><page_50><loc_12><loc_25><loc_88><loc_32></location>where α n and β n are found by applying the following boundary conditions: T = 0 at z = z cz and at z = z tc + δ . The second of these two boundary conditions can be justified only when the tachopause is much thinner than the tachocline, and much thinner than a thermal diffusion length. This is usually the case so</text>
<formula><location><page_50><loc_23><loc_20><loc_88><loc_24></location>α n sinh ( λ n ∆ R ) = ¯ N 2 tc R 2 ¯ T tc κ tc λ 2 n ¯ g tc B n sinh ( λ n z cz -H R ) -β n cosh ( λ n ∆ R ) (D5)</formula>
<formula><location><page_50><loc_32><loc_16><loc_88><loc_20></location>β n = ¯ N 2 tc R 2 ¯ T tc κ tc λ 2 n ¯ g tc ( B n sinh ( λ n z cz -H R ) -˙ Ω cz Ω cz 2∆ λ n J 1 ( λ n ) ) , (D6)</formula>
<text><location><page_50><loc_12><loc_13><loc_32><loc_14></location>where ∆ = z cz -( z tc + δ ).</text>
<text><location><page_51><loc_12><loc_83><loc_88><loc_86></location>The vertical component of the momentum equation expressed in (A3) can then be used to calculate p in the tachocline, so that</text>
<formula><location><page_51><loc_16><loc_73><loc_88><loc_81></location>1 ¯ ρ tc p = ¯ g tc ¯ T tc ∑ n J 0 ( λ n s R ) [ α n R λ n cosh ( λ n z -( z tc + δ ) R ) + β n R λ n sinh ( λ n z -( z tc + δ ) R ) -¯ N 2 tc R 2 ¯ T tc κ tc λ 2 n ¯ g tc ( ( z -( z tc + δ )) B n sinh ( λ n z cz -H R ) + ˙ Ω cz Ω cz ( z -z cz ) 2 λ n J 1 ( λ n ) ) + P n ] , (D7)</formula>
<text><location><page_51><loc_12><loc_68><loc_88><loc_72></location>where the { P n } are found from matching this solution with that of the convection zone (see Equation 14) at z = z cz . After some algebra, we find that</text>
<formula><location><page_51><loc_25><loc_59><loc_88><loc_67></location>P n = B n ¯ T tc ¯ g tc sinh ( λ n z cz -H R ) [ ¯ N 2 tc R 2 ∆ κ tc λ 2 n -R τλ n tanh ( λ n z cz -H R ) ] -α n R λ n cosh ( λ n ∆ R ) -β n R λ n sinh ( λ n ∆ R ) . (D8)</formula>
<text><location><page_51><loc_15><loc_55><loc_72><loc_57></location>Finally, using the radial component of the momentum equation yields v ,</text>
<formula><location><page_51><loc_44><loc_51><loc_88><loc_54></location>v = 1 2¯ ρ tc Ω cz ∂p ∂s , (D9)</formula>
<text><location><page_51><loc_12><loc_48><loc_59><loc_49></location>which can be used to calculate v 0 ( s ) ≡ v ( s, z tc + δ ). We find</text>
<formula><location><page_51><loc_23><loc_42><loc_88><loc_47></location>v 0 ( s ) = ¯ g tc 2Ω cz ¯ T tc ∑ n dJ 0 ( λ n s R ) ds [ α n R λ n -¯ N 2 tc R 2 ¯ T tc κ tc λ 2 n ¯ g tc ˙ Ω cz Ω cz ∆ 2 λ n J 1 ( λ n ) + P n ] . (D10)</formula>
<text><location><page_51><loc_12><loc_38><loc_88><loc_41></location>Substituting (D10) and (D2) into the jump condition (58) finally provides an equation for the { B n } , which, after significant algebra, can be cast in the form</text>
<formula><location><page_51><loc_35><loc_33><loc_88><loc_36></location>B n sinh ( λ n z cz -H R ) = 2 λ n J 1 ( λ n ) a n b n , (D11)</formula>
<text><location><page_51><loc_12><loc_28><loc_88><loc_32></location>where a n and b n recover the formulae given by Equations (61) and (62) in the limit δ glyph[lessmuch] ∆. Using (D11) in (D3) then leads to (59).</text>
<text><location><page_51><loc_15><loc_26><loc_82><loc_27></location>Finally, if we want to calculate w 0 ( s ), we simply substitute B n back into (D2) to get:</text>
<formula><location><page_51><loc_32><loc_20><loc_88><loc_24></location>w 0 ( s ) = ∑ n J 0 ( λ n s R ) [ 2 λ n J 1 ( λ n ) a n b n ] -˙ Ω cz Ω cz ∆ . (D12)</formula>
<section_header_level_1><location><page_51><loc_32><loc_15><loc_68><loc_16></location>E. THE TACHOCLINE THICKNESS</section_header_level_1>
<text><location><page_51><loc_12><loc_10><loc_88><loc_13></location>In GM98 and Wood et al. (2011), the radial mass flux downwelling into the tachocline is caused by the gyroscopic pumping associated with the turbulent torques that permanently drive</text>
<text><location><page_52><loc_12><loc_77><loc_88><loc_86></location>the observed latitudinal shear in the convection zone, rather than by the spin-down torque. For this reason, their respective estimates of the tachocline thickness ∆ as a function of other stellar parameters do not directly apply here. Nevertheless, we can apply a similar method to the one they use to infer ∆ in the spin-down case. We now proceed to describe this method and its limitations, first applied to the steady-state solar case, and then applied to our own spin-down problem.</text>
<text><location><page_52><loc_12><loc_69><loc_88><loc_76></location>In the solar case studied by GM98 and Wood et al. (2011), the thickness of the tachocline ∆ is obtained by matching the vertical mass flux through the tachocline to the vertical mass flux through the tachopause. The latter is estimated by assuming advection-diffusion balance of the flux of horizontal magnetic field across the tachopause (whose thickness is δ ):</text>
<formula><location><page_52><loc_45><loc_65><loc_88><loc_68></location>w tc glyph[similarequal] η tc δ , (E1)</formula>
<text><location><page_52><loc_12><loc_60><loc_88><loc_64></location>while the former is obtained, as in Section 4.2, by considering thermal-wind balance and thermal equilibrium across the tachocline, which yields</text>
<formula><location><page_52><loc_42><loc_55><loc_88><loc_59></location>w tc glyph[similarequal] 2 χ Ω 2 cz ¯ N 2 tc r 2 cz κ tc ∆ 3 , (E2)</formula>
<text><location><page_52><loc_12><loc_51><loc_88><loc_54></location>where χ Ω cz is an estimate of the amplitude of the latitudinal differential rotation in the convection zone. In the Sun, χ ∼ 0 . 1. Combining (E1) and (E2) yields a relationship between δ and ∆:</text>
<formula><location><page_52><loc_42><loc_46><loc_88><loc_50></location>∆ 3 glyph[similarequal] 2 χ Ω 2 cz ¯ N 2 tc κ tc η tc r 2 cz δ . (E3)</formula>
<text><location><page_52><loc_12><loc_42><loc_88><loc_45></location>This equation is quite robust, since it relies on basic balances that are not easily upset by additional dynamics, and has been verified against numerical simulations (Acevedo-Arreguin et al. 2013).</text>
<text><location><page_52><loc_12><loc_31><loc_88><loc_40></location>However, in order to obtain ∆ as a function of known stellar parameters and independently of δ , one must make further assumptions concerning the nature and structure of the tachopause. This final step, unfortunately, is quite model-dependent. GM98 and Wood et al. (2011) propose different scalings for δ as a function of B 0 and Ω cz for instance. Both assume that the tachopause is laminar, but disagree on its thermal properties, leading to</text>
<formula><location><page_52><loc_32><loc_26><loc_88><loc_30></location>∆ r cz ∝ ( | B 0 | √ ¯ ρ tc r cz √ κ tc η tc ) -1 / 9 ( κ tc η tc ) 1 / 3 ( Ω cz ¯ N tc ) 7 / 9 (E4)</formula>
<text><location><page_52><loc_12><loc_23><loc_58><loc_25></location>for GM98 (using their Equation 6 as a definition of δ ), and</text>
<formula><location><page_52><loc_33><loc_18><loc_88><loc_22></location>∆ r cz ∝ ( Ω cz ¯ N tc ) 2 / 3 ( κ tc η tc ) 1 / 3 ( ¯ ρ tc η tc Ω cz B 2 0 ) 1 / 6 , (E5)</formula>
<text><location><page_52><loc_12><loc_10><loc_88><loc_17></location>for Wood et al. (2011) (using Equation 56 as a definition of δ ). Moreover, neither of these scalings apply if turbulence also plays a role in the angular-momentum transport balance across the tachopause (which cannot a priori be ruled out). In short, while Equation (E3) robustly relates ∆ to δ in the solar steady-state model, it is not sufficient on its own to derive a reliable estimate</text>
<text><location><page_53><loc_12><loc_79><loc_88><loc_86></location>of how ∆ varies with stellar parameters without further constraints on the tachopause structure. The latter can only be obtained in direct numerical simulations of the system, which are not yet available at this point. Not surprisingly, we find that the same problem affects the determination of ∆ in a spin-down model.</text>
<text><location><page_53><loc_12><loc_71><loc_88><loc_78></location>In Section 5, we found that w in the tachocline is neither constant with distance from the rotation axis nor with depth, so that a direct application of Equation (E1) is not possible. Nevertheless, one can require advection-diffusion balance on average in the tachopause by setting (within the context of the cylindrical model used throughout this work)</text>
<formula><location><page_53><loc_40><loc_66><loc_88><loc_69></location>∣ ∣ ∣ ∣ 2 π πR 2 ∫ R 0 w 0 ( s ) sds ∣ ∣ ∣ ∣ = η tc δ . (E6)</formula>
<text><location><page_53><loc_12><loc_63><loc_80><loc_64></location>Substituting w 0 ( s ) given in (D12) into this equation and evaluating the integral yields:</text>
<formula><location><page_53><loc_28><loc_58><loc_88><loc_62></location>∣ ∣ ∣ ∣ ∣ ∑ n 4 λ 2 n a n b n -˙ Ω cz Ω cz ∆ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∑ n 4 λ 2 n ( a n b n -8 λ 2 n ˙ Ω cz Ω cz ∆ )∣ ∣ ∣ ∣ ∣ = η tc δ , (E7)</formula>
<text><location><page_53><loc_12><loc_53><loc_88><loc_56></location>where we have once again used the property ∑ n (32 /λ 4 n ) = 1. Dropping all but the first term in this sum (as in Section 5.3), and substituting a 1 and b 1 given in (66) yields</text>
<formula><location><page_53><loc_28><loc_47><loc_88><loc_52></location>4 λ 2 1 ∣ ∣ ∣ ∣ ∣ ∣ ˙ Ω cz Ω cz ∆ -2 Cδ Ω c + Cλ 2 1 24 ˙ Ω cz t ES (Ω cz ) δ 1 + Cλ 2 1 12 Ω cz t ES (Ω cz ) δ ∆ -8 λ 2 1 ˙ Ω cz Ω cz ∆ ∣ ∣ ∣ ∣ ∣ ∣ = η tc δ . (E8)</formula>
<text><location><page_53><loc_12><loc_41><loc_88><loc_44></location>While somewhat obscure, this expression is indeed the equivalent of (E3) in the spin-down case. To see this, note that if ˙ Ω cz = 0, then</text>
<formula><location><page_53><loc_36><loc_36><loc_88><loc_40></location>4 λ 2 1 [ 2 Cδ Ω c 1 + Cλ 2 1 12 Ω cz t ES (Ω cz ) δ ∆ ] = η tc δ , (E9)</formula>
<text><location><page_53><loc_12><loc_27><loc_88><loc_34></location>which recovers the same scalings as those of GM98 and Wood et al. (2011) (see Equation E3), as long as Ω c is re-interpreted as the latitudinal shear driving the large-scale flows χ Ω cz , and the second term in the denominator is much larger than 1 (which is usually the case unless δ/ ∆ is unrealistically small).</text>
<text><location><page_53><loc_12><loc_17><loc_88><loc_26></location>For young solar-type stars, however, spin-down dominates the dynamics of the system. Equation (E8) then takes a different form during the initial transient and the later quasi-steady state phases. During the transient, Ω c = Ω core -Ω cz glyph[similarequal] -Ω cz , as discussed in Section 5.5. In that case, and using the fact that t ES (Ω cz ) glyph[lessmuch] t sd (Ω cz ) for our model to apply anyway, Equation (E8) can be approximated as</text>
<formula><location><page_53><loc_43><loc_14><loc_88><loc_17></location>-c 1 ˙ Ω cz Ω cz ∆ glyph[similarequal] η tc δ , (E10)</formula>
<text><location><page_53><loc_12><loc_10><loc_88><loc_13></location>where c 1 is a constant of order unity. Using this equation in conjunction with a given tachopause model (as in Equation 56 or as in Equation 6 of GM98) does yield an estimate for ∆ as a function</text>
<text><location><page_54><loc_12><loc_83><loc_88><loc_86></location>of stellar parameters. However, that estimate is very sensitive to any model uncertainty on the nature and structure of the tachopause, as discussed above in the context of the Sun.</text>
<text><location><page_54><loc_12><loc_78><loc_88><loc_81></location>In the quasi-steady phase, the problem is even worse. Since the core-envelope lag Ω c = Ω core -Ω cz is now given by Equation (80), Equation (E8) becomes</text>
<formula><location><page_54><loc_41><loc_74><loc_88><loc_77></location>-c 2 ˙ Ω cz Ω cz ∆ I core I tc glyph[similarequal] η tc δ , (E11)</formula>
<text><location><page_54><loc_12><loc_56><loc_88><loc_73></location>where c 2 is also a constant of order unity. We then see that the dependence on ∆ on the left-hand side vanishes altogether (since I tc ∝ ∆), which implies that this method cannot be used to constrain ∆ directly. Instead, Equation (E11) provides a second constrain on δ - the first one being given by various balances within the tachopause, leading for instance to Equation (56), or Equation (6) of GM98 - and therefore defines the position of the tachopause to be the radius where the amplitude of the primordial field B 0 is such that the two definitions of δ coincide. Since the tachopause lies by construction at the bottom of the tachocline, one could in principle use this method to determine ∆ if the radial variation of B 0 is known. However, any estimate of ∆ based on this method will, once again, be uncomfortably model-dependent.</text>
<text><location><page_54><loc_12><loc_40><loc_88><loc_55></location>In summary, we conclude that theory alone cannot robustly predict how the thickness of the tachocline varies with stellar parameters. Any estimate of ∆ made by applying mass continuity across the interface between the tachocline and the tachopause, as in GM98, relies sensitively on the assumed structure of the tachopause, which is itself sensitively dependent on the nature and balance of forces, thermal energy transport and angular-momentum transport within. For this reason, until such a time where the tachopause is better understood (through direct numerical simulations for instance), we advocate the use of a more general tachocline law, as in Equation (79) for instance.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "It has long been known that solar-type stars undergo significant spin-down, via magnetic braking, during their Main-Sequence lifetimes. However, magnetic braking only operates on the surface layers; it is not yet completely understood how angular momentum is transported within the star, and how rapidly the spin-down information is communicated to the deep interior. In this work, we use insight from recent progress in understanding internal solar dynamics to model the interior of other solar-type stars. We assume, following Gough and McIntyre (1998), that the bulk of the radiation zone of these stars is held in uniform rotation by the presence of an embedded large-scale primordial field, confined below a stably-stratified, magnetic-free tachocline by large-scale meridional flows downwelling from the convection zone. We derive simple equations to describe the response of this model interior to spin-down of the surface layers, that are identical to the two-zone model of MacGregor and Brenner (1991), with a coupling timescale proportional to the local Eddington-Sweet timescale across the tachocline. This timescale depends both on the rotation rate of the star and on the thickness of the tachocline, and can vary from a few hundred thousand years to a few Gyr, depending on stellar properties. Qualitative predictions of the model appear to be consistent with observations, although depend sensitively on the assumed functional dependence of the tachocline thickness on the stellar rotation rate. Subject headings: MHD - Sun: interior - Sun: magnetic fields - Sun: rotation",
"pages": [
1
]
},
{
"title": "Spin-down dynamics of magnetized solar-type stars",
"content": "R. L. F. Oglethorpe 1 & P. Garaud 2 1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK 2 Department of Applied Mathematics and Statistics, Baskin School of Engineering, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "It has long been known that there exists a remarkable dichotomy between stars of masses M glyph[star] > 1 . 5 M glyph[circledot] , which typically always remain rapid rotators, and stars of masses M glyph[star] < 1 . 5 M glyph[circledot] , which undergo significant spin-down during their Main-Sequence lifetimes. This dichotomy was first resolved by Schatzman (1962) (see also Schatzman 1959) who noted that the transition coincides with the disappearance of the outer convection zone with increasing stellar mass, and deduced that the latter must play an important role in the spin-down process. He argued, following Parker (1955), that convection is necessary for dynamo action, and that magnetic activity drives the ejection of mass from the surface of the star into the interstellar medium. He then estimated the amount of angular momentum lost by the star assuming that the ejecta is forced to corotate with the surface magnetic field until a point where the field strength is no longer strong enough to act on the plasma. He found that for reasonable field strengths this point was sufficiently far out from the star that even a tiny amount of mass-loss could lead to very significant angular-momentum loss. The theory of stellar spin-down via magnetic braking was born. Schatzman's original calculation has been refined further over the past five decades, following two parallel lines of investigation. The majority of the effort has been dedicated to improving stellar wind models, and the manner in which they act on the stellar surface (Weber & Davis 1967; Mestel 1968; Li 1999). Questions of interest include how the spin-down process is affected by the strength and geometry of the stellar wind and how the latter depend on the star's mass and rotation rate (Aib'eo et al. 2007; Matt et al. 2012; Reiners & Mohanty 2012). However, magnetic braking as currently understood only operates on the surface layers of the star. A second line of investigation has therefore focused on modeling angular-momentum transport within the star, to see how the spin-down information is communicated to the deep interior. Unfortunately, and despite decades of research, only little is known about the subject. Part of the problem resides in the fact that the internal rotation profile of Main Sequence solar-type stars (the Sun excepted) remains extremely difficult to observe. Nevertheless, useful information can still be learned about the process by comparing simple idealized models with observations. Since turbulent convection can redistribute angular momentum across an entire convection zone within a matter of weeks to a few years (depending on the stellar type), the outer convective region is usually assumed to rotate more-or-less uniformly. Within the radiation zone, one can either assume that each layer individually conserves angular momentum (Kawaler 1988), or that the entire region is in a state of uniform rotation. In the latter scenario, an additional assumption must be made to determine how angular momentum is transported from the radiation zone to the convection zone. Following this idea, MacGregor & Brenner (1991) proposed a 'two-zone' model, now commonly used in statistical comparisons of models with observations. The model assumes that both radiation and convection zones are rotating uniformly with rotation rates Ω core and Ω cz respectively, and that the angular-momentum redistribution between the two regions occurs on a coupling timescale τ c . The system then evolves according to the following coupled ordinary differential equations: where J represents the angular momentum and I the moment of inertia (such that J = I Ω) of each region considered and ˙ J w is the rate of angular-momentum extraction from the entire star by the stellar wind. The model allows for analytical solutions in certain limits, and is very easy to integrate numerically in conjunction with stellar evolution. The real difficulty, if one wishes to use it for quantitative purposes, is to express τ c and ˙ J w as functions of known stellar properties. Qualitatively speaking, however, the model provides a simple way of studying the difference in rotational evolution between stars in solid-body rotation and stars which can retain significant differential rotation between the core and the envelope, by suitably choosing τ c . With strong coupling (small τ c ), the region effectively being spun down by the wind encompasses the entire star. By contrast, with weak coupling the region being spun down is at first limited to the convection zone, while the radiation zone only feels the effect of spin-down later, when ∆ J has grown to be sufficiently large. Hence, for the same angular-momentum extraction rate, the apparent spin-down rate of the surface layers is at first much slower for solid-body rotators than for differential rotators. This model then predicts dramatically different rotational histories in the two cases, a prediction that can and has been tested against observations to estimate τ c (Allain 1998; Irwin et al. 2007; Denissenkov et al. 2010; Spada et al. 2011; Gallet & Bouvier 2013). Before we proceed to discuss these results, first note that stellar rotation rates in a given very young cluster and a given mass bin, usually exhibit a significant spread, with 'rapid rotators' rotating up to a few tens of times the speed of the slower rotators (Herbst et al. 2001; Lamm et al. 2005; Irwin et al. 2008). This spread is usually attributed to a spread in the initial pre-stellar core conditions, and the length of the initial pre-Main Sequence disk-locking phase (Bouvier et al. 1997). The spread of rotational velocities in these 'initial conditions' then propagates to later ages, with rapid rotators and slow rotators in each mass bin having distinct evolutionary paths. By comparing observations to the predictions of a two-zone model, Irwin et al. (2007) found that the rotational periods of rapid rotators are well-explained by assuming solid-body rotation at all time, for any mass bin. By contrast, the solid-body rotation assumption is not consistent with observations for slow-rotators in the mass range 0 . 7 M glyph[circledot] -1 . 1 M glyph[circledot] . A coupling time τ c as large as a Gyr appears to fit the data much better (see also Allain 1998). This suggests that τ c must depend on multiple factors, such as the mass and rotation rate of the star. Their conclusion was confirmed by subsequent work by Denissenkov et al. (2010), Spada et al. (2011) and Gallet & Bouvier (2013). Note that the rotation rates of low-mass stars ( M glyph[star] < 0 . 7 M glyph[circledot] ), by contrast, are fairly well-explained by solid-body rotation. This is not surprising since the latter are fully convective for M glyph[star] < 0 . 35 M glyph[circledot] and nearly fully convective for 0 . 35 M glyph[circledot] < M glyph[star] < 0 . 7 M glyph[circledot] (see also Reiners & Mohanty 2012, for further work on the rotational histories of very low-mass stars in older clusters). Going beyond the two-zone model, however, and understanding from a more physical point of view what the source of the dynamical coupling between the radiative and convective regions is, and whether the radiation zone is indeed in uniform rotation, requires peering below the surface. Today, this can only be done for a single Main-Sequence star, the Sun 1 . Our goal in this work is to use insight from recent progress in understanding internal solar dynamics to model the transport of angular momentum in the interior of other stars. This, of course, implicitly assumes that the Sun is representative of all solar-type stars, and that its internal workings are not fundamentally different from theirs. Comparing the predictions of any model based on this assumption with observations may, to some extent, help us establish whether it is valid or not. Thanks to helioseismology, we now have a good view of the large-scale internal dynamics of the Sun. Its outer convective region spans roughly a third of the solar radius and 2.5% of the solar mass. It is rotating differentially, with an equatorial rotation rate Ω eq glyph[similarequal] 2 . 9 × 10 -6 s -1 , and a polar rotation rate about 25% slower (Thompson et al. 1996; Schou et al. 1998). Meanwhile, the radiative interior is rotating uniformly, with an angular velocity similar to that of the surface at mid-latitudes. The shear layer between the two regions is called the tachocline. It is remarkably thin, with a thickness estimated at 2-4% of the solar radius (Kosovichev et al. 1997; Charbonneau et al. 1999; Elliott & Gough 1999). Further to the question of how the spin-down of the surface layers is communicated to the interior, these observations raise new puzzles: why is the radiation zone rotating uniformly despite the latitudinal shear imposed by the overlying convection zone; and why is the tachocline so thin? Answering these questions in conjunction with the spin-down problem prompts a closer inspection of the various angular-momentum transport processes thought to take place in the Sun (and by proxy, in all solar-type stars). Angular-momentum transporters in the solar radiation zone can be split into two categories: hydrodynamic processes and magnetohydrodynamic (MHD) processes. The former include stratified turbulence, large-scale meridional flows, and gravity waves. Purely hydrodynamic models of the solar interior based on all three types of processes, in isolation or in combination, have been studied at length (Spiegel & Zahn 1992; Elliott 1997; Talon et al. 2002; Charbonnel & Talon 2005; Rogers & Glatzmaier 2006; Brun et al. 2011; Wood & Brummell 2012). As discussed by McIntyre (1994), Gough & McIntyre (1998), Gough (2007) and Zahn (2007), however, these models remain problematic. On the one hand, they tend to have difficulties explaining how to maintain uniform rotation, since they often rely on some level of differential rotation to be effective. This is particularly true of turbulent transport, and transport by large-scale flows. On the other hand, they also have difficulty explaining how the gradual extraction of angular momentum from the radiation zone during spin-down can proceed without concurrently inducing significant compositional mixing throughout the interior, which would be inconsistent with helioseismic inversions (Gough & Kosovichev 1990). MHDprocesses, which do not suffer from the same limitations, have recently gained popularity as a means to explain helioseismic and related observations of the dynamics of the solar interior (see the review by Garaud 2007). Mestel & Weiss (1987) first discussed how the presence of a large-scale primordial field confined to the radiation zone would rapidly suppress any differential rotation within that region. Indeed, the magnetic diffusivity is low in conditions relevant for stellar interiors. If, in addition, meridional flows are weak (which is true in strongly stratified regions), then Ferraro's isorotation law (Ferraro 1937) applies, which states that angular velocity must be constant on magnetic field lines. Although Ferraro's law does not preclude differential rotation entirely since different field lines could, in principle, rotate at different speeds, Mestel & Weiss (1987) argued that interactions between Alfv'en waves along neighboring field lines, through a process called phase mixing, should rapidly suppress any remaining shear. Charbonneau & MacGregor (1993) later used this idea to study the effect of spin-down in the solar interior. They considered a model similar to the one of Mestel & Weiss (1987), in which the Sun is threaded by a fixed poloidal field, confined beneath a fixed radius r f . They assumed that the convection zone is gradually spun down by magnetic braking, while remaining at all time in a state of uniform rotation. They then studied the role of the magnetic field in promoting angularmomentum transport, both between the two zones and throughout the radiative interior, by solving simultaneously the axisymmetric azimuthal components of the momentum and magnetic induction equations. In their model, which ignores phase mixing and any meridional flows, (turbulent) viscosity had to be invoked to damp any remaining differential rotation across field lines, and to promote angular-momentum transport in regions where there is no field. They found that if the poloidal field has significant overlap with the convection zone (i.e. if r f > r cz , where r cz is the radius of the radiative-convective interface), then the spin-down is very rapidly communicated to the interior. For large enough viscosity, the entire star rotates more-orless as a solid body at all times 2 . On the other hand, if the poloidal field is confined strictly below the convection zone (i.e. if r f < r cz ), then viscosity is also needed to communicate the spin-down from the convection zone to the magnetically-dominated part of the radiation zone. Charbonneau & MacGregor (1993) showed that the system eventually evolves toward a quasi-steady state where both the convection zone and the deep interior are rotating uniformly, but at distinct angular velocities with Ω core > Ω cz . A shear layer separates the two zones, with a thickness that depends on r cz -r f . In this quasi-steady state, the total viscous angular-momentum flux across this shear layer is equal to that extracted by the wind from the surface, and fixes the relative core-envelope lag (Ω core -Ω cz ) / Ω cz . These ideas have recently been revisited by Denissenkov (2010), who, in addition to the question of spin-down, also attempted to address the issue of the solar tachocline and of the light-element abundances at the same time. Starting from the model proposed by Charbonneau & MacGregor (1993) but ultimately allowing for a differentially rotating convection zone, he first argued that the high value of the viscosity needed to explain the uniform rotation of the radiation zone (to ensure that all field lines are rotating at the same rate) must be of turbulent origin. He then assumed that this turbulence would also transport chemical species at the same rate, and showed that this leads to inconsistent predictions for the surface Li and Be abundance in the Sun. To resolve the problem, he then invoked the work of Spiegel & Zahn (1992) and Zahn (1992) to argue that turbulence in the radiation zone must be highly anisotropic, redistributing angular momentum rapidly in the horizontal direction but very slowly in the radial direction - and similarly for chemical species. Under this assumption, and using a related prescription for the turbulent transport coefficients, he was able to explain simultaneously both the uniform rotation of the solar interior, the thin solar tachocline and the surface light-element abundances. However, while compelling in its ability to reproduce observations, this model suffers from a number of inconsistencies. First, note that as long as the field is confined beneath the convection zone, phase mixing is likely to drive the radiation zone towards uniform rotation without the need to invoke turbulence, and thus without causing significant concurrent compositional mixing. The added anisotropic turbulent transport central to the work of Denissenkov (2010) should thus be viewed a 'rapid fix' for a problem that does not necessarily exist. Second, the fix is itself inconsistent, since it uses a prescription for turbulent transport that was created for a purely hydrodynamic system, in an environment dominated by magnetic fields. As shown by Tobias et al. (2007), the turbulent viscosity prescription of Spiegel & Zahn (1992) is unlikely to apply in any circumstance. Indeed, in a purely hydrodynamic setting, strongly stratified turbulence drives a system away from, rather than towards uniform rotation. In the presence of a weak magnetic field, Tobias et al. (2007) showed that the fluid motions rapidly evolve into a state where Reynolds stresses and Maxwell stresses cancel out, and where the efficacy of turbulent angular-momentum transport is strongly quenched. Finally, note that all of these models (Mestel & Weiss 1987; Charbonneau & MacGregor 1993; Denissenkov 2010) assume the existence of a confined poloidal field, but do not explain how confinement is maintained. As it turns out, it is only by answering the fundamental question of magnetic confinement that new light can be shed on the problem. Gough & McIntyre (1998) (GM98 hereafter) were the first to put forward a global and selfconsistent theory of the large-scale rotational dynamics of the solar interior, albeit in a steady state (that is, without spin-down). They first addressed the magnetic confinement question. They argued that large-scale meridional flows must be driven by gyroscopic pumping from the differentiallyrotating convection zone down into the radiation zone, and that by pushing on the magnetic field lines they can confine the primordial field strictly below the base of the convection zone. The solar tachocline thus emerges as a 'magnetic-free' region (to be precise, a region where the Lorentz force is insignificant), lying between the base of the convection zone and the top of the magneticallydominated, uniformly rotating radiative interior. GM98 then studied the tachocline dynamics more quantitatively, and estimated that these flows could transport angular momentum and chemical species across the tachocline on a local Eddington-Sweet timescale, which is a few Myr in the present-day Sun. The flows, however, do not penetrate below the base of the tachocline, hence satisfying observations of the light-element abundances. Angular-momentum extraction by the tachocline from the deeper regions is done by magnetic stresses, through a very thin magnetic boundary layer, now known as the tachopause (see Figure 1 for detail). Gough & McIntyre's vision of the solar interior was recently confirmed by semi-analytical calculations (Wood & McIntyre 2011; Wood et al. 2011) and by numerical simulations (Acevedo-Arreguin et al. 2013). Given its success in explaining (at least qualitatively so far) existing observations for the Sun, we take the natural next step by assuming a similar structure for all solar-type stars, and studying its response to spin-down. Our complete model setup is presented in Section 2. Our goal is to derive an analytical or semi-analytical description of the spin-down problem, and study how the latter varies with stellar parameters. To do so, we are forced to abandon the spherical geometry and model the star as a cylinder. This approximation is introduced and discussed in Section 2 and then tested on a simplified system for which analytical solutions exist for both cylindrical and spherical geometry, in Section 3. The derivations presented in that Section also serve the pedagogical role of introducing our methodology. In Section 4, we first consider the spin-down of a non-magnetic solar-type star. Our results recover many aspects of Spiegel & Zahn (1992). Including the effects of a primordial magnetic field following the GM98 model requires modeling the transport of angular momentum out of the deep interior, across the tachopause and the tachocline. This is done in Section 5. Finally, in Section 6, we re-interpret our findings in view of their application to stellar evolution, casting our mathematical results in a more astrophysical light, and discussing their limitations and possible implications in explaining observations. This final Section is written in such a way that it may be read independently of Sections 3-5, for readers who, in a first pass, are primarily interested in the results rather than their derivation.",
"pages": [
1,
2,
3,
4,
5,
6,
7
]
},
{
"title": "2. THE MODEL",
"content": "The principal difficulty involved in extending the GM98 model to study its behavior under spindown is the nonlinear nature of the equations of motion and of the magnetic induction equation. In fact, following GM98, we do not even attempt to address it: the full nonlinear problem is so complex that it cannot be treated exactly analytically (although see Wood & McIntyre 2011, for a first nonlinear solution of a simplified version of the GM98 model). Instead, we shall make a number of assumptions and order-of-magnitude estimates to model the effects of the nonlinearites. These will be introduced and discussed in detail as they arise. A second difficulty lies in the geometry of the system. Since gravity (the vertical axis) is not parallel to the rotation axis in a star (except at the poles), even the restricted hydrodynamic linearized version of the governing equations does not usually have a simple analytical solution. Indeed, the misalignment between these two axes implies that eigen-solutions of the problem are not separable in the vertical and horizontal directions, which makes the problem much more complicated analytically. By contrast with the issue of nonlinearities, this second problem can be dealt with, at least approximately. To do so, we further simplify the model by considering a cylinder instead of a sphere, where gravity is by construction parallel to the rotation axis. This cylinder can, for instance, be viewed as the polar regions of the star (see Figure 1). This simplification adds an order-unity geometrical error to all of our results, but on the other hand allows for fully analytical solutions of the linearized equations. We show in Section 3 that in some simple limit it is possible to find analytical solutions of the linearized system in both spherical and cylindrical geometries, and the solutions agree up to a geometrical factor, hence justifying our procedure.",
"pages": [
7,
8
]
},
{
"title": "2.1. The 'cylindrical star'",
"content": "Our cylindrical model is presented in Figure 1. As in GM98, we consider that the star is divided into four dynamically distinct regions, as shown in Figure 1a: from the surface downward, the convection zone (yellow), the tachocline (green), the tachopause (blue), and the uniformlyrotating part of the radiation zone (purple), threaded by a primordial magnetic field (red). Figure 1b shows the equivalent cylindrical model setup. The 'cylindrical star' has radius R and total height H . The region with z ∈ [ z cz , H ] (yellow) represents the convection zone. The latter is assumed to rotate with a uniform angular velocity Ω cz ( t ) (as expressed in an inertial frame) which decreases over time via magnetic braking. Henceforth, we work in a frame that is rotating with angular velocity Ω cz ( t ), so that by construction, the convection zone is at rest in the rotating frame. Note that we neglect here the possibility of any differential rotation in the convection zone. This is done in order to simplify our calculations, enabling us to neglect any meridional flows driven by differential rotation (via gyroscopic pumping) in favor of those driven by the spin-down torque. We expect this assumption to hold for young stars, which undergo rapid spin-down, but not necessarily for older ones like the Sun (see Section 6.3 for further discussion of this point). We use cylindrical coordinates ( s, φ, z ) aligned with the rotation axis (or equivalently, the vertical axis e z ), and assume axial symmetry. In these coordinates, u = ( u, v, w ) is the velocity field relative to the rotating frame. The equations describing the dynamics of the system are the momentum equation, the mass conservation equation, the thermal energy equation, the induction equation and the solenoidal condition. Before studying them in more detail, we first proceed to describe each of the four regions listed above, and the assumptions made in each of them.",
"pages": [
8,
9
]
},
{
"title": "2.2. The convection zone",
"content": "The dynamics of stellar convective zones are quite complex, and result from the nonlinear interplay between convection, rotation, large-scale flows and magnetic fields. We shall not attempt to model them in any detail here. Our main goal is merely to account for the rapid transport of angular momentum between the surface and the top of the radiation zone. To do this, we model the convection zone as in Bretherton & Spiegel (1968), who studied stellar spin-down by treating the effect of the convection on large-scale flows (and on momentum transport in general) through a Darcy friction term (i.e. a linear damping term). A similar method was used by Garaud & Acevedo-Arreguin (2009), Wood et al. (2011) and Acevedo-Arreguin et al. (2013) in their models of the solar interior. We therefore replace the Reynolds stress term -u · ∇ u in the momentum equation (where u is the velocity field expressed in the rotating frame) with the term -u /τ , where τ is a damping timescale. Dimensionally speaking, one can assume that τ is of the same order as the convective turnover timescale. For simplicity, we assume that τ is constant in the convective region, and that 1 /τ is zero in the radiative region. Finally, we also assume that the convection zone is adiabatically stratified, and transports heat very efficiently compared with all other timescales in the system.",
"pages": [
9
]
},
{
"title": "2.3. The tachocline",
"content": "The region z ∈ [ z tc + δ, z cz ] (green) represents the tachocline. Its thickness, ∆ = z cz -( z tc + δ ), depends on a number of factors including intrinsic properties of the star's structure, as well as its rotation rate and the strength of the primordial field (see GM98 or Wood et al. 2011, for estimates of ∆ in the steady-state solar case, and Appendix E for further discussion of ∆ in the spin-down case). It is stably stratified, with a mean buoyancy frequency ¯ N tc ≥ 0 and mean background density ¯ ρ tc . In this region we make several assumptions to simplify the equations of motion (as in, for example, Spiegel & Zahn 1992; Gough & McIntyre 1998; Wood & McIntyre 2011; Wood et al. 2011). First, we assume that the thickness of the tachocline is small compared with a pressure and density scaleheight, and use the Boussinesq approximation (Spiegel & Veronis 1960) to study its dynamics. We also assume that, in the rotating frame, tachocline flows are slow enough to neglect the inertial term in the momentum equation, and also neglect the effect of viscous forces. These assumptions then imply that the tachocline is in hydrostatic and geostrophic equilibrium. We also assume that the flows are sufficiently steady and slow for the system to be in thermal equilibrium, with heat diffusion being balanced by the advection of the background entropy. Finally, as suggested by GM98, we assume that the Lorentz force is negligible in the tachocline.",
"pages": [
9,
10
]
},
{
"title": "2.4. The tachopause and the deep interior",
"content": "The region z ∈ [ z tc , z tc + δ ] (blue) represents the tachopause. As discussed in Section 1, the tachopause is a magnetized boundary layer through which the spin-down torque is ultimately transmitted to the interior. As in the case of the tachocline, its thickness δ depends on the local thermodynamical properties of the star, as well as its rotation rate and the strength of the primordial field (see GM98 or Wood et al. 2011). All of the assumptions concerning the dynamics of the tachocline made above apply here as well, with the exception of course of the one concerning the magnetic field - the tachopause is by definition a layer in which the Lorentz force plays a fundamental role. In addition, the thickness of the tachopause is assumed to be much smaller than the thickness of the tachocline ( δ glyph[lessmuch] ∆) to allow for a boundary layer analysis of various balances within. Finally, the region 0 < z < z tc (pink) represents the magnetized radiative core, below the tachopause. We do not solve any equations to model that region. Instead, we merely assume that temperature perturbations and large-scale flows vanish for z < z tc , and that the entire region is held in uniform rotation with angular velocity Ω c ( t ) (expressed in the rotating frame, so Ω c = Ω core -Ω cz ). The latter is controlled by the spin-down rate of the regions above, through the torques acting within the tachopause. We now begin our theoretical investigation, which ultimately results in deriving an equation for the evolution of the total angular momentum of the core under this model that is identical to the two-zone model of MacGregor & Brenner (1991), but with a coupling timescale that is related to the local Eddington-Sweet timescale across the tachocline. The reader interested principally in a discussion of this result rather than its derivation may, as a first pass, skip directly to Section 6.",
"pages": [
10
]
},
{
"title": "3. CYLINDRICAL VS. SPHERICAL GEOMETRY",
"content": "We begin our investigation by showing that results obtained using a cylindrical geometry are, within a geometrical factor of order unity, consistent with those obtained using a more realistic spherical geometry. To do so, we consider the simplest possible problem, of a non-magnetic star with an 'unstratified' interior of constant density. While this case does not have any astrophysical relevance, it can easily be solved analytically in both cylindrical and spherical geometry (Bretherton & Spiegel 1968). As such, it can serve as an illustration of the validity of the 'cylindrical star' assumption, as well as a pedagogical tool to introduce the method of solution of the governing equations we shall use throughout. We also assume in this Section that there is no magnetic field. In this case, the tachopause does not exist, and the 'tachocline' fills the entire radiation zone, extending all the way to the center of the star. In this model, the only difference between the 'convective' and 'radiative' regions is the presence of the Darcy forcing term modeling angular-momentum transport by convective motions. Under the assumptions discussed in Section 2, regardless of geometry, the equations of motion are the momentum equation and incompressibility (the thermal energy equation is not needed): where r is the position vector, ¯ ρ is the constant density of the fluid, and p is the pressure perturbation away from hydrostatic equilibrium. The second term in Equation (2) is the Coriolis force, the third is Euler's force (which is due to the deceleration of the frame) and the fourth term is the Darcy friction term (see Section 2), which is assumed to be zero in the 'radiation zone'.",
"pages": [
10,
11
]
},
{
"title": "3.1. Spherical geometry",
"content": "Bretherton & Spiegel (1968) were the first to study the spin-down of such an unstratified star using the model described above. They found an analytical solution of the problem in a spherical geometry, assuming that the angular velocity of the convection zone Ω cz ( t ) (expressed in an inertial frame) decays exponentially over time. We now repeat their calculation and consider any functional form for Ω cz ( t ) for more generality. Bretherton & Spiegel (1968) first assumed that the spin-down rate is low, so that | ˙ Ω cz / Ω cz | glyph[lessmuch] Ω cz . They then looked for a quasi-steady solution in the rotating frame, requiring the term ∂ u /∂t in the momentum equation to be negligible. The resulting quasi-steady equations of motion in the 'radiative interior' (0 < r < r cz ) are In the 'convection zone' ( r cz < r < R ), they reduce to assuming that the Darcy timescale τ is significantly smaller than the rotation timescale Ω -1 cz . Solving these equations, then matching p and the normal velocity at the radiative-convective interface (at r = r cz ), yields the shape of the streamlines (see Figure 3), as well the angularvelocity perturbation Ω( r, θ, t ) everywhere in the star (see their original work for the details of the calculation). The angular velocity in the radiative interior turns out to be uniform, with value Ω c such that as expressed in the rotating frame, where R is the radius of the star and r cz that of the base of the convection zone. Since ˙ Ω cz < 0, we have Ω c > 0, implying that the interior is always rotating faster than the convection zone, or in other words, lagging behind in the context of the spin-down process. The relative lag is measured by Ω c / Ω cz ; whether it increases or decreases with time depends on the behavior of ˙ Ω cz / Ω 3 cz . The implications of Equation (5) are discussed in Section 3.3.",
"pages": [
11,
12
]
},
{
"title": "3.2. Cylindrical geometry",
"content": "We now solve the same equations under the same assumptions, but this time in the cylindrical geometry presented in Section 2. Inspection of the expression for the meridional flows given by Bretherton & Spiegel (1968) (see their Equations 7 and 8) reveals that they have equatorial symmetry. To obtain solutions with the same symmetry in cylindrical geometry, we set the vertical velocity at z = 0 to be zero. In addition, the vertical velocity at the surface of the star is also zero. Following Bretherton & Spiegel (1968), we solve the equations for z > z cz and z < z cz separately, and match p and the vertical velocity w at the radiative-convective interface. Our vertical boundary conditions are 3 therefore: Boundary conditions at the side wall ( s = R ) are more difficult to choose, as we want them to have as little influence as possible on the dynamics inside the cylinder. For simplicity, we select This boundary condition allows a radial flow across the side wall, as required by analogy with the spherical solution (see Figure 3). By conservation of mass the fluid must somehow return to the convection zone, a process which necessarily occurs outside of the cylinder considered, and that we cannot model explicitly. In what follows, we assume that this return flow does not play any fundamental role in the spin-down process (in the sense that it does not change the solution beyond a factor of order unity). This assumption is verified a posteriori to be correct (see Section 3.3). We first consider the radiative region ( z < z cz ). The various components of Equation (2), when expressed in a cylindrical geometry, become where u = ( u, v, w ). We also have incompressibility: Following Bretherton & Spiegel (1968), we take ∂v/∂t = 0, and combine (8) and (9) to find In the convection zone, expanding (4) we have Combining these with incompressibility ∇ . u = 0 gives Using the boundary conditions (6) and (7), we can write w and p as where J 0 is the zeroth-order regular Bessel function, the { λ n } are its zeros, and the coefficients { B n } are integration constants. The latter are found by matching w in Equations (10) and (13) at z = z cz , so that where J 1 is the first order Bessel function. Equation (8) shows that, for z < z cz , p is constant with height, so that matching p at z = z cz and using (15) gives and, using (8), we find that the angular velocity of the radiative region, expressed in the rotating frame, becomes The latter is always positive, and is a function of s only, as expected from the Taylor-Proudman constraint.",
"pages": [
12,
13,
14
]
},
{
"title": "3.3. Discussion",
"content": "The solutions in spherical and cylindrical geometries bear strong similarities. In both cases, we find that the azimuthal velocity is constant along the rotation axis, as expected from the TaylorProudman constraint since the fluid is unstratified. Furthermore, the radiation zone is always rotating more rapidly than the convection zone. The relative lag between the two regions, in both cases, is equal to the prefactor -˙ Ω cz /τ Ω 3 cz , times a non-dimensional term that depends only on the geometry of the system. This term is shown as a function of s in Figure 2 for both cylindrical and spherical geometries. It is constant in the spherical case, and increases with s in the cylindrical case. However, the two are consistent (within a factor of order unity). This shows that the difference in the results obtained in the two geometries is not dramatic. The structure of the meridional circulation is also very similar in both cases, as shown in Figure 3. In the cylindrical case, one could imagine the streamlines closing back on themselves outside of the domain, as they do in the spherical case. This changes the global angular momentum balance somewhat, but as discussed above, does not affect the outcome by more than an order unity factor. While Ω cz ( t ) should in principle be calculated self-consistently from a stellar wind model, it is informative to look at specific 'plausible' spin-down laws. For the purpose of the following discussion, we either assume that Ω cz ( t ) decays exponentially, with or as a power law, with for some α > 0. In these laws, Ω 0 = Ω cz ( t 0 ), where t 0 is the initial timescale considered (e.g. the end of the disk-locking phase, for instance, or the Zero Age Main Sequence). The parameters α and k are unspecified here, but can be fitted to any desired spin-down model. The -˙ Ω cz / Ω 3 cz prefactor in Equations (5) and (17) shows that the relative lag between the radiative and convective regions always increases exponentially with time for an exponential spindown law. This suggests a break-down of the quasi-steady approximation for this case 4 . Using a power-law to model spin-down reveals that Ω( s ) / Ω cz increases with time if α > 1 / 2, is constant if α = 1 / 2, and decreases with time if α < 1 / 2. This, again, suggests a break-down of the quasisteady approximation if spin-down occurs faster than the Skumanich power-law, which has α = 1 / 2 (Skumanich 1972). Although interesting, we believe that the correspondence between the critical power-index α = 1 / 2 and the Skumanich law is a coincidence, given the simplistic nature of this particular unstratified model. To conclude this Section, since the solutions in (5) and (17) are the same up to a purely geometrical factor, and since that geometrical factor is typically of order unity, we now make the assumption that results from our cylindrical model may carry across to the spherical geometry, up to a geometrical error, for more complex systems such as ones including stratification or a magnetic field. Hence from now on, all our analysis will be restricted to cylindrical geometry.",
"pages": [
14,
15,
16
]
},
{
"title": "4. SPIN-DOWN IN A NON-MAGNETIC STAR",
"content": "We now consider the spin-down of a non-magnetic but more realistically stratified star. As in Section 3, we study only solar-type stars, with an outer convection zone ( z cz < z < H ) and an inner radiation zone (0 < z < z cz ). In preparation for the implementation of the GM98 model, which is our ultimate goal, we assume that the radiation zone is sub-divided into two regions, a uniformly rotating core (for z < z tc ) and a tachocline (for z tc < z < z cz ). Here, we do not specify the mechanism by which the core is held in solid-body rotation. Furthermore, we assume that this core is not dynamically connected to the tachocline, but instead, merely acts as a passive boundary whose only role is to be impenetrable to the fluid. Within that approximation, note the core cannot be spun-down, as no angular momentum can be extracted from it. Finally, since the tachopause does not exist, we take δ = 0. glyph[negationslash] While this setup may seem odd at first, note that one could use it to represent a normal nonmagnetic star (that is, with no rigidly rotating core) simply by taking the regular limit z tc → 0; the impenetrability of the boundary at z = z tc simply becomes an equatorial symmetry condition (see Section 3). However, it is important to remember that this limit is generally inconsistent with the Boussinesq approximation, which requires ∆ = z cz -z tc to be smaller than a pressure scaleheight. We therefore advise the reader against indiscriminately using our results in this fashion. Instead, we note that the solutions derived in this Section with z tc = 0 will be applicable (with a few modifications) to model the complete stellar spin-down problem in Section 5. In this sense, our work in this section should be viewed once again as a pedagogical step toward understanding the final result.",
"pages": [
16
]
},
{
"title": "4.1. Model equations",
"content": "The convection zone is assumed to be very nearly adiabatic with a buoyancy frequency ¯ N = 0, and entropy perturbations are assumed to be negligible. For more realism, and in this Section only, we allow the background density ¯ ρ to vary with height in the convection zone, and use the anelastic approximation. We shall show that the results are identical to those derived in the Boussinesq case. The equations governing the dynamics within the convection zone thus become, under similar assumptions to the ones discussed in Section 2: where T is the temperature perturbation. Since the tachocline is thought to be thin, we use the Boussinesq approximation (Spiegel & Veronis 1960) to model it, and assume that the buoyancy frequency, gravity, density and temperature do not depart significantly from their mean tachocline values ¯ N tc , ¯ g tc , ¯ ρ tc and ¯ T tc . In this approximation, density and temperature perturbations are formally related through a linearized equation of state, in which pressure perturbations are negligible. Hence The momentum equation in the tachocline becomes Combining the radial and vertical components of (22), with the assumptions discussed in Section 2 and above, yields the well-known thermal-wind equation The tachocline is also assumed to be in thermal equilibrium, so where κ tc is its (constant) thermal diffusivity. The boundary conditions for velocity and pressure are similar to those of the previous Section, but with the lower boundary raised to z = z tc . We require impermeability ( w = 0) at z = z tc and z = H , p = 0 at s = R , and that p and w must be continuous at z = z cz . The temperature perturbations are assumed to vanish at z = z tc , and at z = z cz . We now first study this system of equations in the same 'quasi-steady' state discussed in the previous Section, and then solve for the complete time-dependence of the system to determine under which conditions this quasi-steady state is valid.",
"pages": [
17
]
},
{
"title": "4.2. Quasi-steady solution",
"content": "As in Section 3, we define the 'quasi-steady' state as the solution of the governing equations in which ∂ u /∂t is neglected. We find solutions separately for z > z cz and z < z cz , and match w and p at the radiative-convective interface. The full calculation is given in Appendix A. We find that the azimuthal velocity in the tachocline is where ∆ = z cz -z tc is the thickness of the tachocline (since δ = 0), and where G n is the geometrical factor Equation (25) reduces to (17) when the tachocline is unstratified ( ¯ N tc = 0), regardless of the position of the lower boundary z tc . This is not surprising: the Taylor-Proudman constraint requires v to be independent of z in that limit. It is also shown in Appendix A that the density variation in the convection zone has no effect on the angular velocity within the tachocline, as long as the density is continuous across z = z cz . In what follows, we can therefore equivalently use ∇· u = 0 in the convection zone for mathematical simplicity even though the latter does not actually satisfy the Boussinesq approximation. glyph[negationslash] The angular velocity and meridional circulation profile for ¯ N tc = 0 is shown in Figure 4. To understand its properties, first note that since G ( z cz ) = 0, the angular velocity just below the base of the convection zone is the same in the stratified and unstratified cases. However, v is no longer independent of z , but instead increases with depth. This effect is due to the added buoyancy force in the momentum equation, which relaxes the Taylor-Proudman constraint. The latter is replaced by the thermal-wind constraint given in Equation (23), which relates any variation of angular velocity along the rotation axis to gradients of temperature perpendicular to it. The relative lag between the base of the tachocline and the convection zone is thus controlled simultaneously by thermal-wind balance and by thermal equilibrium within the tachocline. The combination of these two constraints yields a simple estimate for the relative angularvelocity shear across the tachocline as a function of input stellar parameters. Indeed, first note that the downwelling flow velocity across the tachocline is primarily controlled by the tachocline thickness, and by the spin-down rate through: (see Appendix A for detail). An order-of-magnitude approximation of this equation yields In thermal equilibrium (see Equation 24), however, the advection of the background entropy stratification by these flows must balance the diffusion of the induced temperature perturbations T . This sets the typical amplitude of T in the tachocline to be Finally, by thermal-wind balance, latitudinal variations in T control the allowable shear across the tachocline and therefore the total angular-velocity difference (Ω b -0) between the base of the tachocline (where Ω ∼ Ω b ) and the radiative-convective interface (where Ω = 0): Combining all these estimates yields which states that the system adjusts itself (by selecting Ω b ) such that the Eddington-Sweet timescale 5 based on that angular-velocity lag and on the thickness of the tachocline (expressed in the left- nd-side of this equation) is equal to the spin-down timescale of the star (expressed in the righthand-side). We then have This shows that the relative angular-velocity shear between the top and the bottom of the tachocline is equal to the ratio of the local Eddington-Sweet timescale (based, this time, on the rotation rate of the convection zone) to the spin-down timescale. We then expect the shear across the tachocline to be larger (i) if the background stratification is larger (ii) if the local thermal diffusivity is smaller, (iii) if the spin-down rate is larger or (iv) if the tachocline is thicker. This is indeed what the exact Equation (25) and its order of magnitude approximation (32) both show. Finally, note that the relative lag is also proportional to -˙ Ω cz / Ω 3 cz , as in the unstratified case. This is not entirely surprising, since the unstratified case is a regular limit of this stratified problem as ¯ N tc → 0. As such, as long as ∆ is constant, Ω b / Ω cz diverges with time for an exponential spin-down law, or for any power-law with α > 1 / 2, as discussed in Section 3.3. However, since ∆ likely depends on time as well through its dependence on Ω cz (see Appendix E), other criteria apply (see Sections 5 and 6 for detail).",
"pages": [
18,
19,
20
]
},
{
"title": "4.3. Transient solution",
"content": "Having found a quasi-steady solution to the spin-down problem, we now revisit the original time-dependent equations to determine when that solution is valid, and how rapidly the system relaxes to it. Guided by the steady-state solution, we expand the azimuthal velocity v on the same basis of Bessel functions, namely Combining (22), (23) and incompressibility with this ansatz, and retaining the time-derivative in the azimuthal component of the momentum equation, gives where we have explicitly written in the time-dependence of Ω cz and ˙ Ω cz to remember that it must be taken into account. This equation is quite similar to the one derived by Spiegel & Zahn (1992) in the context of the evolution of the differential rotation profile within the solar tachocline. This is not surprising, as our underlying assumptions (thermal-wind balance, thermal equilibrium) are essentially the same. The first and second terms on the left-hand-side of (34) are the same as theirs (see their Equation 4.10). The hyperdiffusion term arises from the advection of angular momentum by local Eddington-Sweet flows, a transport process that Spiegel & Zahn (1992) called 'thermal spreading'. The third term on the left-hand side is also part of the thermal spreading process, but was neglected by Spiegel & Zahn (1992) on the grounds that it is quite small when ∆ glyph[lessmuch] R . For consistency with the Boussinesq approximation, which requires ∆ to be smaller than a pressure scaleheight, we also neglect it from here on. Finally, the right-hand side contains the global forcing term arising from Euler's force. Since viscous and turbulent transport are neglected here, the evolution of the angular momentum in the tachocline has two contributions only: transport by meridional flows, and global extraction by Euler's force. As such, we expect the system to behave in rather different ways if the spin-down timescale is much larger or much smaller than local Eddington-Sweet mixing timescale. In what follows, we introduce the new variable x = ( z -z tc ) / ∆. Together with the simplification discussed above, Equation (34) becomes where we have also explicitly written in the time-dependence of ∆. To solve Equation (35), we note that it is further separable in x and t , and write where the vertical eigenmodes Z nm ( x ) satisfy for some constants µ nm . It can be shown that the Z nm functions form an orthogonal set, so that projecting (35) onto each of them individually gives, for each n and m , where additional information on Z nm and µ nm , are given in Appendix B. The quantity where t ES (Ω cz ) was defined in Equation (32), naturally emerges from this calculation, and can be interpreted as a local Eddington-Sweet timescale based on the typical geometry of the spatial eigenmode considered. Equation (38) can easily be solved using an integrating factor, which yields As expected, we see that V nm ( t ) contains two terms, one that depends on the initial conditions (the first term on the right-hand side of Equation 41) and one that depends on the forcing applied to the system (the second term on the right-hand side of Equation 41). For V nm ( t ) to tend to the quasi-steady solution discussed in Section 4.2 as t → + ∞ (see Equation 25), the effect of the initial conditions must decay, since (25) is independent of the initial differential rotation profile of the star. Furthermore, the terms containing the forcing in Equation (41), when recombined as in Equation (36) must eventually recover (25). Whether this occurs or not clearly depends on the behavior of the integrating factor µ ( t ). We now study the latter in more detail. We first re-write the integral in µ ( t ) as where the spin-down timescale t sd was defined in Equation (31). Writing it in this form enables us to study the long-term behavior of this integral for a fairly broad class of problems. First, note that for an exponential spin-down law (see Equation 18), t sd is constant (and equal to 1 /k ) while for a power-law spin-down rate (see Equation 19), then The second form of t sd written above can actually be used to describe both exponential and powerlaw spin-down models if one views the exponential case as having α -1 = 0. Next, note that both GM98 and Wood et al. (2011) found ∆ to be a power-law function of the mean stellar rotation rate. While their findings do not directly apply here, since they were derived assuming that the tachocline circulation is driven by the latitudinal shear in the convection zone rather than by spin-down, we may nevertheless safely assume a similar functional dependence 6 , taking (where we anticipate that β ≥ 0, and ∆ 0 is by construction ∆ at t = t 0 ), so that Combining Equations (45) and (43) with (42), we find that glyph[negationslash] where q = 2 -α -1 -4 β . We therefore see that the behavior of the integrating factor µ ( t ) depends only on the sign of q . If q > 0, then the right-hand-side of Equation (46) tends to a constant as the star spins down. In that case, µ ( t ) also tends to a constant as t → + ∞ , which implies that the contribution of the initial conditions to V nm ( t ) does not disappear in Equation (41). In other words, the star cannot relax to the state described by the quasi-steady solution discussed in Section 4.2, and the latter becomes irrelevant to the spin-down problem. On the other hand, if q ≤ 0 then the right-hand-side of Equation (46) is positive, and increases towards + ∞ as the star spins down. In that case, the integrating factor µ ( t ) also increases with time (super-exponentially when q < 0, and as Ω -1 cz ( t ) when q = 0), which implies that the contribution of the initial conditions to V nm ( t ) rapidly disappears. For q < 0, the latter decays exponentially roughly on the local Eddington-Sweet timescale across the tachocline. This timescale decreases with time and rapidly becomes much smaller than the age of the star except if the tachocline is very thick (which we explicitly assumed was not the case). Furthermore, it can be shown with additional algebra that the complete transient solution given by Equation (36) actually tends to the quasi-steady solution (25) when t → + ∞ , when q ≤ 0 (see Appendix B2). In other words, the quasi-steady solution derived and discussed in Section 4.2 is a meaningful description of stellar spin-down, after a transient phase which is short compared with the age of the star, during which all knowledge of the initial rotation profile disappears. The physical interpretation of q as a critical value of this problem is quite straightforward given that q is effectively defined so that If q > 0, then the ratio of the spin-down timescale to the local angular-momentum transport timescale across the tachocline decreases as the star spins down. This implies that the meridional flows are less and less efficient at extracting angular momentum from the tachocline relative to the rate at which it is removed from the envelope. The lag between the envelope and the bottom of the tachocline then increases with time, until such a point where new dynamics not taken into account here, such as turbulent transport, must come into play. In other words, the quasi-steady solution derived in the previous Section is only of limited validity. When q ≤ 0 on the other hand, the converse is true: the system rapidly tends to the quasi-steady solution regardless of the initial conditions.",
"pages": [
20,
21,
22,
23,
24
]
},
{
"title": "4.4. Discussion",
"content": "Our findings regarding the dynamics of both transient and quasi-steady spin-down solutions can easily be summarized as follows. If, for a given spin-down law Ω cz ( t ) and a given tachocline structure (characterized by its local thermodynamic properties and its thickness), the system is such that the ratio of the spin-down timescale to the local Eddington-Sweet timescale t sd (Ω cz ) /t ES (Ω cz ) (where t sd is given in Equation 43 and t ES is given in Equation 45) monotonically decreases as the star spins down, then angular-momentum transport by large-scale meridional flows across the tachocline is not sufficient to maintain dynamical coupling with the envelope. After some time, the shear across the tachocline is likely to become large enough to be unstable to shearing instabilities. Angular-momentum transport will then be dominated by turbulent motions, and must be described using an entirely different formalism (not discussed here). If, on the other hand, the system is such that t sd (Ω cz ) /t ES (Ω cz ) increases or remains constant as the star spins down, then the tachocline remains coupled to the envelope, and the shear simply adjusts itself geostrophically so that the angular momentum flux transported by the meridional flows out of the tachocline is, at all times, equal to angular-momentum flux removed from the star by the wind. The relative lag Ω b / Ω cz between the base and the top of the tachocline, in this case, is correctly given by the quasi-steady solution, and is roughly equal to t ES (Ω cz ) /t sd (Ω cz ) (which is either constant, or decreases with time), see Equation (32). We therefore find that the quasi-steady solution is conveniently valid whenever it makes sense, that is, whenever Ω b / Ω cz ∝ t ES (Ω cz ) /t sd (Ω cz ) decreases (or at least remains constant and much smaller than one) as the star spins down. In other words, we can actually avoid the calculation of the transient solution entirely, since the quasi-steady solution itself provides all the information needed as to the limits of its own validity.",
"pages": [
24
]
},
{
"title": "5. SPIN-DOWN OF A MAGNETIZED STAR",
"content": "We now finally return to the originally-posed problem and investigate the manner in which the tachocline spin-down is finally communicated to the deep radiative interior. In the GM98 model, this process is mediated by magnetic torques within the tachopause, a thin boundary layer that separates the tachocline from the magnetically-dominated, uniformly-rotating region below (see Figure 1a). These torques are generated as the primordial magnetic field, confined below the tachocline by downwelling meridional flows, is wound up into a significant toroidal field by the rotational shear. In order to model angular-momentum transport across the tachopause exactly, one should therefore solve the magnetic induction equation in addition to the previously discussed equations describing the tachocline dynamics (i.e. Equations 22 and 24), and include the Lorentz force in the momentum balance. The nonlinear nature of the added terms, unfortunately, makes it impossible to derive exact analytical solutions of the problem without further assumptions. Wood et al. (2011), however, were able to derive analytical solutions for a simplified version of the GM98 model. They found that the mathematical equations describing tachopause are, in many ways, analogous to those describing a viscous Ekman layer, with the viscous drag force replaced by a magnetic one. In other words, they showed that one can develop physical insight into the problem and obtain quantitatively meaningful results by considering a thought experiment in which the uniformly rotating part of the radiative interior is simply a massive, impenetrable solid sphere (or, in our case, a cylinder), whose rotation rate is gradually spun-down by the fluid lying above through friction. Our final model is thus constructed as follows. We consider the same setup as the one described and studied in Section 4, but the base of the tachocline is now no longer passive. Instead, it hosts a thin tachopause of thickness δ , which communicates the tachocline spin-down to the rigidlyrotating, impermeable interior via magnetic torques. The latter will be modeled using a boundary layer jump condition. We thus recover the picture first presented in Figure 1b.",
"pages": [
24,
25
]
},
{
"title": "5.1. Global angular-momentum balance",
"content": "Let Ω c be the angular velocity of the rigidly rotating 'core' region of the radiation zone, expressed in the rotating frame. Rigid-body rotation throughout the entire star implies that Ω c glyph[similarequal] 0, while Ω c > 0 expresses a lag between the core and the convection zone. In our cylindrical model, the core spans the region z < z tc , s < R (see Figure 1). Recall that the tachopause spans the interval [ z tc , z tc + δ ], and is assumed to be thinner than the tachocline, which lies above (with z ∈ [ z tc + δ, z cz ]). Although negligible in the tachocline, magnetic stresses are significant within the tachopause, and must be included when studying the global angular-momentum balance. To find an evolution equation for Ω c ( t ), we thus begin by writing the complete angular-momentum conservation equation as where B = ( B s , B φ , B z ) is the magnetic field, and where we have ignored viscous stresses on the grounds that they are most likely negligible. Note that we assume that the system is laminar, and ignore the contribution of turbulent transport - this assumption is discussed in Sections 5.2 and 6. Integrating Equation (48) over the volume V of the core up to the top of the tachopause, and using the divergence theorem then yields where the second integral is a surface integral through the side of the tachopause, and the third integral is a surface integral through the top of the tachopause. To derive this equation, we have used the fact that the angular-momentum flux through the bottom and side boundaries of the core is zero. Indeed, u and B φ disappear since the core is assumed to be rigidly rotating and impermeable. By definition of the tachocline, the magnetic torque becomes negligible just above the tachopause, and thus disappears from the surface integral at z = z tc + δ . We assume that it also disappears from the integral on the side-boundary. This assumption is somewhat difficult to justify a priori, and will require verification when full numerical solutions of the problem, in a spherical geometry, are available. However, it is consistent with the assumption that the dynamics occurring beyond the sides of the cylinders do not directly affect spin-down. We argue that it is at least plausible as long as the magnitude of the toroidal field B φ on the side-wall of the tachopause remains small, which requires in turn that the radial angular-velocity gradient ∂ Ω /∂s at the same location be small. The remaining terms in Equation (49) can be expressed as where, as in the previous Section, ¯ ρ tc is defined as the mean density of the tachocline and tachopause region, and where I core is the moment of inertia of the core and tachopause combined, defined as where M core is the mass of the cylinder included in the volume V . Note that in order to derive (50), we have used the fact that v = s Ω c within the core, and assumed that v glyph[similarequal] s Ω c in the tachopause as well. Since the tachopause is very thin, the error made has negligible impact on the result. Within the scope of these assumptions, angular momentum is extracted from the core through a series of channels, which unfold as follows. The surface layers are spun-down by the magnetized wind torque, and then communicate the spin-down information to the rest of the convection zone by turbulent stresses. The spin-down torque also drives large-scale meridional flows in the convection zone, which extract angular momentum from the tachocline and the top of the tachopause (by flowing downward from the convection zone and then outward in the tachopause), and at the same time confine the internal magnetic field. The tachopause then finally spins the core down via magnetic stresses, generated as the local core-tachopause shear winds the internal poloidal field into a toroidal one. Equation (50) describes only the hydrodynamic processes in the tachocline, but the other channels are implied in the assumptions that (1) the convection zone rotates at the velocity Ω cz , (2) the tachopause rotates at a velocity close to Ω c , and (3) the vertical flow at the base of the tachocline is given by the magnetic jump condition that describes the tachopause dynamics. We study the latter in Section 5.2. To estimate the second term in Equation (50), note that the mass flux entering the tachopause through the surface z = z tc + δ must be the same as that leaving through the side wall, since the core is impermeable. Hence: Combining this with (50) gives where is the total angular momentum of the core 7 in the inertial frame, and where we have introduced w 0 ( s, t ) = w ( s, z tc + δ, t ) for simplicity. Equation (53) thus shows that the rate of angular-momentum transport between the surface layers and the core is fully determined once ¯ ρ tc w 0 ( s, t ), the vertical mass flux downwelling from the tachocline into tachopause, is known. To find w 0 ( s, t ), we must once more solve the equations describing the dynamics of the tachocline and of the convection zone, and match them to one another at the radiative-convective interface. The main difference with the work presented in Section 4.2 lies in the treatment of the lower boundary of the tachocline, which is no longer passive nor strictly impermeable, but must instead be modified to take the presence of the tachopause into account. This is done by replacing the impermeability condition used in Section 4.2 by a 'jump condition', that relates w 0 to v 0 ( s, t ) = v ( s, z tc + δ, t ) at the bottom of the tachocline, and depends on the magnetohydrodynamics of the tachopause. This jump condition is now derived.",
"pages": [
25,
26,
27
]
},
{
"title": "5.2. Tachopause jump condition",
"content": "Guided by the analogy between the tachopause and an Ekman layer suggested by the work of Wood et al. (2011), we begin by deriving a tachopause jump condition assuming that its dynamics are dominated by viscous torques only. This assumption greatly facilitates our derivation, and the result can then be used without any further algebra to deduce the equivalent jump condition for a magnetized tachopause. While well-known in the context of a steadily-rotating frame, the derivation of the Ekman jump condition has not yet, to our knowledge, been done in a frame that is spinning down. The steps of the calculation are essentially identical, however, and are presented in Appendix C. We find that the quasi-steady vertical and azimuthal velocity profiles at the base of the tachocline, w 0 ( s ) and v 0 ( s ), are related via: where δ E = √ ν tc / 2Ω cz is the thickness of the Ekman layer that mimics the tachopause, and is based on the local viscosity ν tc of the star at z = z tc . The second term on the right-hand-side is the standard jump condition (expressed in a cylindrical coordinate system), while the first term is the correction arising from Euler's force. In stars, however, the tachopause transmits the spin-down torque via magnetic rather than viscous friction. Wood et al. (2011) found that its thickness 8 is given by where B 0 is the strength of the confined magnetic field (just below z tc ), and η tc is the local magnetic diffusivity of the star near z = z tc . They also found that the jump condition relating the vertical and azimuthal velocities in the tachocline is exactly of the same mathematical form as that of an Ekman layer, albeit with the numerical constant 1 / √ 2 replaced by π/ 4, and δ E replaced by δ . The magnetic jump condition thus becomes In both cases discussed above, the tachopause is assumed to be laminar. The angularmomentum transport budget within this region involves only large-scale flows, and either magnetic or viscous stresses. The reason behind the similarity of the two jump conditions comes from the fact that viscous and magnetic stresses are directly proportional to the local angular-velocity shear 9 . As a result, one may conjecture that any boundary layer in which the angular-momentum balance relies on large-scale meridional flows and some form of stress that is proportional to the angular-velocity shear will also result in the same type of jump condition, even if that boundary layer is not laminar. From this argument, we propose that the general form of the jump condition should be where C is a constant of order unity which depends on the type of stresses acting in the tachopause, and δ is its thickness, which is no longer necessarily related to B 0 via (56) specifically, but likely depends on B 0 in some form or another. In any case, as we demonstrate below, the long-term behavior of the spin-down problem is ultimately controlled only by the slowest timescale in the sequence of processes responsible for angular-momentum transport from the core to the surface. In most cases, this turns out to be the Eddington-Sweet timescale across the tachocline, rather than any timescale intrinsic to the tachopause. In this sense, the global spin-down timescale is usually independent of the exact nature and structure of the tachopause (at least, in an explicit sense, see Section 6.3 and Appendix E), unless the latter is very thick - which we have previously assumed is not the case.",
"pages": [
27,
28,
29
]
},
{
"title": "5.3. Evolution of the core angular momentum",
"content": "We can now solve the system of equations governing the convection zone and the tachocline as in Section 4, using this time the jump condition (58) as a lower boundary condition for the tachocline flows. At this junction we have two possibilities: to solve the full time-dependent tachocline dynamics as in Section 4.3, or to study them using a quasi-steady approximation as in Section 4.2. Having proved in Section 4.3 that the time-dependent solution very rapidly tends to the quasi-steady solution (when it is well-behaved), and given that the latter is much more easily derived, we assume here that the tachocline dynamics are in a quasi-steady state 10 . The derivation of this solution is presented in Appendix D, and eventually yields the mass flux into the tachopause, ¯ ρ tc w 0 ( s ). Using the expression obtained into Equation (53), we then find that for δ glyph[lessmuch] ∆ (i.e. when the tachopause is much thinner than the tachocline), then where is the moment of inertia of the tachocline, and with where ∆ glyph[similarequal] z cz -z tc is the thickness of the tachocline. Note that for most stars whose outer convection zone is not too thin, and where τ is not too small, the last term in the square brackets of Equation (62) is negligible compared with the first two. In what follows, we neglect it, but bear in mind that it may be important for solar-type stars whose mass approach the critical mass above which the outer convection zone disappears. While somewhat obscure at first, Equation (59) has a simple limit. Indeed, when δ = 0, Using properties of Bessel Functions, it can be shown that ∑ n 32 /λ 4 n = 1. As a result, which implies that, when viewed in an inertial frame, the core retains its initial angular momentum. This result is as expected, since δ = 0 means that the tachopause is absent, and without it the tachocline cannot exert any torque on the core. In other words, the convective envelope and the tachocline both spin down exactly as described in Section 4.2, but the core does not. When δ > 0, by contrast, the angular velocity of the core evolves with time in response to the spin-down torque communicated by the tachopause. This is illustrated in Figure 5, which shows a plot of the relative core-envelope lag Ω c / Ω cz , as a function of time, in two idealized test cases. Note that the moment of inertia of the core is assumed to be constant to simplify the interpretation of the results. At time t = t 0 we also assume for simplicity that Ω c ( t 0 ) = 0 (or in other words, that the star is uniformly rotating). In Figure 5a, we show Ω c / Ω cz for a 'reference' star whose parameters are summarized in Table 1, using a Skumanich spin-down law with Ω cz ( t ) = Ω 0 ( t/t 0 ) -1 / 2 (i.e. α = 1 / 2), and with a tachocline of constant thickness (setting β = 0 in Equation 44). In Figure 5b, we evolve the same star but allow for a tachocline whose thickness varies with Ω cz , taking β = 2 / 3 in Equation (44) and choosing ∆ 0 such that the values of ∆( t ) in both Figure 5a and Figure 5b agree at t = 10 3 t 0 (which more-or-less represents the 'present day' time for the Sun). In both figures the relative core-envelope lag first increases rapidly, then eventually converges to a global quasi-steady state 11 whose time-dependence can be predicted analytically (see Section 5.4). The time taken to reach this global quasi-steady state, however, depends sensitively on the thickness of the tachocline (see Section 5.5), and is much larger in Figure 5b, which has a much larger initial tachocline thickness, than in Figure 5a. To understand the results, first note that a good approximation to Equation (59) can be obtained by keeping a single term in the sum over all spatial eigenmodes, that is, by using instead, where we have replaced 32 /λ 4 1 in the second term on the right-hand-side by 1, to ensure that the core is not spun-down by the tachocline when δ = 0 (see discussion above). Since λ 1 glyph[similarequal] 2 . 4, this is a fairly good approximation anyway. In addition, as long as the tachocline and tachopause are thin, a 1 and b 1 further simplify by Taylor expansion to Substituting these terms into Equation (65), reducing the right-hand side to the same denominator and simplifying, yields In writing the second expression, we have simplified the denominator further by noting that the second term in b 1 is usually larger than 1 by many orders of magnitude for any physically meaningful values of δ/ ∆ (except of course in the strict limit δ → 0 discussed earlier, which we do not consider here). The resulting expression for the rate of change of J core is now completely independent of Cδ , or in other words, independent of the detailed nature and structure of the tachopause. This property of the solution is discussed in more detail in Section 6, but essentially stems from the fact that the tachopause can propagate the spin-down torque near-instantaneously to the core when it is thin, and thus does not introduce any new timescale in the problem. Finally, note that since dJ core /dt = d ( I core Ω c ) /dt + d ( I core Ω cz ) /dt , and since the assumption of a thin tachocline fundamental to this work implies that I tc glyph[lessmuch] I core , we can neglect I tc ˙ Ω cz in the right-hand side of Equation (67) in comparison with the I core ˙ Ω cz term on its left-hand side, so that: where K is a constant of order unity, and Ω core is the angular velocity of the core in an inertial frame. Written in this final form, our model bears some obvious similarities with the two-zone model of MacGregor & Brenner (1991). In fact, it can be cast exactly as in Equation (1) provided we define the coupling timescale between the core and the envelope to be where I cz is the moment of inertia of the convection zone. This result is discussed in more detail in Section 6.1. The comparison between the solution of the exact Equation (59), and that the much simpler Equation (68) is shown in Figure 5. The two are within ten percent of one another at all times. Given that our cylindrical geometry solutions approximate a real star to within a geometrical factor of order unity at best anyway, the error made in using (68) instead of (59) is of a similar nature, and can be incorporated in the former. In what follows, we therefore advocate the use of (68) as a much simpler and more physically meaningful, and yet equivalent description of the spin-down problem.",
"pages": [
29,
30,
31,
33
]
},
{
"title": "5.4. Properties of the quasi-steady solution",
"content": "We now use this simpler expression to derive a global quasi-steady approximation to the solution, which gives insight into the long-term behavior of the system. By analogy with Section 4.2, we derive it by neglecting the acceleration, but keeping Euler's force in the momentum equation. This yields the following algebraic equation instead, which can then be solved for Ω c , and thus yields its quasi-steady approximation Ω qs c . Note that we have again assumed here that I core varies sufficiently slowly with time that its derivative can be neglected. This is done for simplicity of interpretation of the results but is not necessarily valid during all evolutionary stages of the star. We then find that: where q + β = 2 -α -1 -3 β . The scaling shown in Equation (71) is similar to that discussed in Section 4.2 (see Equation 32), and can therefore also be understood using order-of-magnitude arguments based on thermal-wind balance, thermal equilibrium and mass conservation. However, it contains new factor which is the ratio I core /I tc . The presence of this factor can be understood physically by noting that this time, the core is spun-down as well as the tachocline. A much larger torque is needed to spin down a more massive core, and as a consequence, if the same spin-down torque is applied, then the core-envelope lag is proportionally larger for a more massive core. Furthermore, since I core /I tc is inversely proportional to the thickness of the tachocline, Ω qs c / Ω cz now scales with the third instead of the fourth power of ∆, and varies as (Ω cz / Ω 0 ) -( q + β ) instead of (Ω cz / Ω 0 ) -q . We can now apply the same reasoning as in Section 4.3 as to the limits of validity of our laminar solution. If q + β > 0 then Ω qs c / Ω cz increases as the star spins down. In this case, the quasi-steady solution is not a good approximation to the actual time-dependent problem, and the laminar solution probably eventually breaks down to a turbulent one with different scalings instead. If q + β ≤ 0 on the other hand then Ω qs c / Ω cz remains constant or decreases, and the laminar solution is likely always valid. When q + β < 0, the system always tends to solid-body rotation as the star spins down. By contrast, if q + β = 0 then Ω qs c / Ω cz tends to a constant, which implies that the system eventually maintains a non-zero core-envelope lag as t →∞ . This behavior is illustrated in Figure 6, which shows the long-term evolution of the 'reference' star, for α = 1 / 2 and β = 0, β = 1 / 3 and β = 2 / 3 respectively. We see that, for large times, the exact solution for Ω c / Ω cz does indeed tend to the quasi-steady solution as expected. The latter is either constant or decreases with time, depending on the value of q + β . We now briefly examine under which conditions q + β = 2 -α -1 -3 β ≤ 0. Recall that α is the spin-down law index (with the convention that α -1 = 0 for an exponential spin-down law, see Equation 43) and β is the index of the power-law describing the variation of the tachocline thickness with Ω cz (see Equation 44). The condition 2 -α -1 -3 β ≤ 0 is then automatically satisfied whenever α ≤ 1 / 2, as long as β > 0, or whenever β ≥ 2 / 3, regardless of α . The fact that the observationally-favored Skumanich law (Skumanich 1972), which has α = 1 / 2, lies in the region of parameter space for which our quasi-steady laminar solutions are valid regardless of β is a very nice - if somewhat unexpected - feature of our model, and speaks to its relevance for observations. The fact that it is also a critical parameter value, on the other hand, is probably just a coincidence. In general, however, α is not known a priori - it is an outcome of the complete spin-down problem (see Reiners & Mohanty 2012, for instance, who found both exponential and power-law spin-down solutions depending on their assumed wind model). Our theoretical results suggest that if β ≥ 2 / 3, then again a laminar quasi-steady solution always exists regardless of the spin-down law. If β < 2 / 3, on the other hand, whether the laminar solution holds or not also depends on the actual spin-down rate of the convection zone, and must therefore be determined 'on the fly' while the solution of (68) is being computed. Unfortunately, it is difficult to constrain β from theory alone (see Appendix E). Indeed, while GM98 and Wood et al. (2011) both found that β > 2 / 3 in a related but distinctly different model system, their results cannot be used here. In this particular instance, help in constraining β comes from observations instead. The requirement that fast rotators should be in solid-body rotation at all times immediately constrains β to be strictly smaller than 2 / 3 (see Section 6 for detail), which does imply that our model could break down if the spin-down rate is too rapid. For further constraints, we note that surface abundances of light elements and other tracers can potentially be used to estimate the depth of the chemically mixed tachocline, and study its variation with stellar rotation rate (see Section 6.3).",
"pages": [
33,
34,
35
]
},
{
"title": "5.5. Properties of the initial transient solution",
"content": "The initial, nearly linear increase in the core-envelope lag seen in Figures 5 and 6 can be understood by noting that the angular-momentum transport rate across the tachopause depends on the local torques, which in turn depend on the local angular-velocity shear. At first, the latter is small, so the torques are not strong enough to spin the core down. When viewed in an inertial frame, the latter continues to spin at its original rate, so that for t -t 0 smaller than the spin-down timescale. As the core-envelope lag increases, so does the shear, until a point where the torque exerted is just strong enough to communicate the surface spin-down to the interior. When this happens, the core locks on to the tachocline, and the angular-momentum flux becomes independent of radius. The system reaches a quasi-steady state in the spinning-down frame, in which both convective zone, tachocline and core concurrently spin-down at more-or-less the same rate. Equating the early-time solution (72) with the quasi-steady solution (71), we find that this happens (very roughly) when Although implicit for t , and therefore difficult to solve algebraically, this expression readily shows that the duration of the initial transient phase is proportional to the local Eddington-Sweet timescale across the tachocline, times I core /I tc , a quantity that is overall proportional to ∆ 3 / Ω 2 cz . In other words, it is much longer if the initial tachocline thickness is larger, which explains the results of Figure 6. In what follows, we now summarize our results, importing them into a spherical geometry and casting them in a more astrophysically relevant terminology, and discuss their implications for stellar spin-down and related observations.",
"pages": [
35,
37
]
},
{
"title": "6.1. Summary of the results",
"content": "In this work, we have studied the impact of spin-down on solar-type stars whose internal dynamics are assumed to be analogous to that of the Sun as first introduced by GM98. More specifically, we considered stars with a radiation zone held in uniform rotation by the presence of a large-scale primordial magnetic field, that is confined strictly below the base of the convection zone by large-scale meridional flows. The geometry of such stars was shown in Figure 1. Separating the convection zone and the bulk of the radiation zone (the 'core', hereafter), which are rotating with angular velocities Ω cz and Ω core respectively (as expressed in an inertial frame), lie two thin nested shear layers (GM98; Wood & McIntyre 2011; Wood et al. 2011; Acevedo-Arreguin et al. 2013): the tachocline, which resides just beneath the radiative-convective interface, and is dynamically speaking at least - magnetic free, and the tachopause, which lies below the tachocline, and connects the latter magnetically to the core. The extraction of angular momentum from the star by the stellar wind in our model takes a rather different form across each of these regions. Magnetic braking exerts a torque on the surface layers, which is nearly instantaneously communicated down to the base of the convection zone by the turbulence. Angular-momentum transport (and chemical mixing) across the underlying tachocline, on the other hand, is mediated principally by large-scale meridional flows and roughly takes place on a local Eddington-Sweet timescale: where ¯ N tc , κ tc are the buoyancy frequency and thermal diffusivity of the fluid within the tachocline, ∆ is its thickness, and r cz is the radius of the base of the convection zone. The spin-down torque is then finally communicated through the tachopause down to the deep interior primarily by magnetic torques. The mostly-dipolar primordial field is wound-up by the radial shear, which generates a significant toroidal field. The resulting Lorentz force reacts against the shear, and thus extracts angular momentum from the core. This happens on an Alfv'enic timescale which is more or less instantaneous compared with the spin-down timescale or the tachocline mixing timescale. The tachocline is clearly the 'bottleneck' of this angular-momentum extraction sequence, and therefore controls the overall rotational evolution of the star. As a result, the timescale t ES (Ω cz ) introduced above plays a role that is similar (but not identical, see below) to the coupling timescale between the core and the envelope in the two-zone model of MacGregor & Brenner (1991) (see τ c in Equation 1). Thanks to the help of an idealized model for which exact solutions exist, we have formally shown that the concurrent evolution of the rigidly-rotating core and the convective envelope can typically (i.e. under reasonable assumptions usually valid in most solar-type stars) be modeled as: that is, exactly as in the two-zone model of MacGregor & Brenner (1991) (see Equation 1), with a coupling timescale τ c given by where K is a positive geometrical constant of order unity, and where I tc is the moment of inertia of the tachocline (which, to a good approximation, is I tc = 4 π ¯ ρ tc r 4 cz ∆, where ¯ ρ tc is the local density within the tachocline). Equation (75) was derived in Section 5.3. Inspection of Equations (75) - (78) shows that they do not explicitly depend on the magnetohydrodynamics of the tachopause 12 , a result inherently tied to the assumption that the tachopause is much thinner than the tachocline, and that it is at all time in complete dynamical and thermal equilibrium. When this is the case, the tachopause responds near-instantaneously to any perturbation and thus cannot introduce any additional timescale in the system (see Section 5.3 for detail). This property of our model turns out to be quite convenient: as discussed by Acevedo-Arreguin et al. (2013), the specific nature and dynamical properties of the tachopause are arguably the 'weakest link' of the GM98 model, being the most sensitive to any dynamics that were purposefully neglected (turbulence, gravity waves, etc.). But as Equations (75) - (78) show, this model-dependence does not have any direct impact on the long-term evolution of the angular velocity of the core. On the other hand, Equation (75) is very sensitive to the properties of the tachocline, and in particular to its thickness ∆. The latter presumably depends on the star's mean rotation rate, on its spin-down rate, on the strength of the internal primordial field, and on the position and local thermodynamical properties of the base of the convection zone. Both GM98 and Wood et al. (2011) propose scalings for ∆ as a function of these quantities in the solar case, where the largescale meridional flows are driven by the latitudinal shear within the convection zone rather than by spin-down. Unfortunately, as discussed in Appendix E, these scalings do not directly apply here. Furthermore, any attempt to estimate ∆ from first principles necessarily yields a result that depends sensitively on the structure of the tachopause, which we have just argued is both poorly constrained and strongly dependent on the model considered. In this sense, while the dynamics of the tachopause do not explicitly participate in Equations (75) - (78), they nevertheless indirectly influence the rotational evolution of the star by controlling ∆ (which appears in t ES (Ω cz ) and therefore in τ c ). For these reasons, instead of proposing a carefully derived, mathematically correct but highly model-dependent formula for the tachocline thickness ∆, we suggest the following simple parametric prescription: In this model, we have hidden all information about the unknown (and non-observable) internal field strength B 0 into ∆ 0 ( B 0 ). Any information about the time-dependence induced by spin-down is contained in the second term, and any information about the local properties of the tachocline is contained in the third term 13 . While β and γ are difficult to estimate from theory alone (see Appendix E for detail), we hope that they can, in the future, be constrained observationally by studying simultaneously the rotational histories of solar-type stars in young clusters and their light-element surface abundances (see Section 6.3). In general, Equations (75)-(79) have to be solved - and should be solved - numerically, in conjunction with the equations for stellar evolution which yield I core , I cz and τ c at each point in time. However, good insight into the long-term behavior of the solutions can be obtained by considering a 'quasi-steady' approximation, in which (1) we assume that the spin-down rate ˙ Ω cz is known, (2) I core does not vary too rapidly with time and (3) Ω core -Ω cz is not too large. Using all three approximations implies that dJ core /dt glyph[similarequal] I core ˙ Ω cz is known, and one can then simply solve Equation (75) analytically for Ω core . The relative core-envelope lag, in this quasi-steady approximation, is given by where t sd = | Ω cz / ˙ Ω cz | is the spin-down timescale of the convective envelope. Physically speaking, this formula is equivalent to stating that the star adjusts itself in such a way that the integrated angular-momentum flux out of each spherical shell is constant with radius, and equal to that extracted from the star by the stellar wind. We have shown in Sections 4.3 and 5.4 that our model is only technically valid when this solution is bounded (in the sense that the quasi-steady relative core-envelope lag remains constant or decreases with time). This happens when 2 -α -1 -3 β ≤ 0, where β is defined in (79), and where α is defined such that t sd ∝ Ω -α -1 cz . When the model applies, then the quasi-steady solution is also an attracting solution of the governing equations (which means that the system relaxes to this state regardless of its initial conditions). All stars satisfying this condition are therefore expected to have a core-envelope lag given by (80), after a transient period whose duration is of the order of t ES (Ω cz ) evaluated at t = t 0 (see Section 5.5 for detail).",
"pages": [
37,
38,
39,
40
]
},
{
"title": "6.2. Caveats of the model",
"content": "Before we proceed to discuss the observational implications of our model, let us briefly address its caveats and limitations. In many ways, they are the same as those of the GM98 model, listed and discussed at length by GM98 and by Acevedo-Arreguin et al. (2013). Central to our calculation is the assumption that the star has a dynamical structure similar to the Sun, with an outer convection zone and a uniformly rotating magnetized core both in solidbody rotation, separated by a thin magnetic-free tachocline, and an even thinner tachopause which, by contrast, is essentially magnetic in nature. As discussed by Acevedo-Arreguin et al. (2013), a necessary condition for such a layered model to exist is where ν tc is the viscosity in the tachocline region. If this condition is not satisfied, then the meridional flows downwelling from the convection zone are unable to confine the magnetic field, and a different model must be used (see Acevedo-Arreguin et al. 2013, for details). However, since the Sun satisfies this property, we expect that most young and thus more rapidly rotating stars are likely to satisfy it as well. Even if (81) is satisfied, the existence of such a layered structure is not yet guaranteed. While it has now been revealed in fully nonlinear, full-sphere, steady-state simulations of the solar interior for the first time (Acevedo-Arreguin et al. 2013), one should still verify that it can also be achieved in a spin-down problem. Indeed, as mentioned in Section 6.1, the mechanisms driving the largescale tachocline flows, which are responsible for confining the internal magnetic field within and below the tachopause, are subtly different in the solar steady-state case and in the spin-down case. We defer the task of running full-sphere numerical simulations of the spin-down problem to a future publication. Beyond the question of existence of a tachocline and tachopause, such a calculation could furthermore yield a first estimate of the possible relationships between their thicknesses and other stellar parameters, a result that cannot be robustly obtained from linear theory alone here (see Appendix E). The next major assumption we need to verify is whether the effects of turbulence can indeed be neglected while modeling the tachocline. As discussed by Acevedo-Arreguin et al. (2013), thermalwind balance and thermal equilibrium - two key balances in the system - are both still likely to hold even in the presence of turbulence (given reasonable assumptions as to its source). On the other hand, angular-momentum balance is much more sensitive to any added effects, and must be studied carefully. For our model to hold, radial angular-momentum transport across the tachocline must be dominated by advection by large-scale flows, rather than by turbulence. Naturally, and as discussed throughout this work, this assumption can only hold as long as the shear across the tachocline is 'weak enough' not to cause significant turbulent transport - the question being what 'weak enough' means in this context. One may first ask under which conditions the tachocline is linearly unstable to shear instabilities. As studied by Ligni'eres et al. (1999), this could depend both on the Richardson number Ri glyph[similarequal] ¯ N 2 tc ∆ 2 /r 2 cz (Ω cz -Ω core ) 2 , and on the P'eclet number Pe = r cz (Ω cz -Ω core )∆ /κ tc . In this particular problem, however, the P'eclet number is typically so large that the relevant criterion for global, tachocline-scale shear instabilities is the standard Ri < O (1) rather than RiPe < O (1) advocated by Zahn (1974) (which is only applicable to the small Pe limit). Using solar values as guidance (see Table 1), we find that our model is expected to break down completely only when For this inequality to hold, we therefore see that a substantial shear is required. Although unlikely in older stars, this could happen during the early stages of the spin-down process where Ω cz is much larger, and where rapid core-contraction can result in significant core-envelope shear. One should therefore monitor Ri carefully in the process of time-stepping Equations (75)-(79), and use a turbulent coupling timescale instead should Ri drop below 1. Alternatively, ignoring linear stability considerations, one could simply assume that the system becomes unstable to finite amplitude perturbations for much weaker shearing rates. In that state, Prat & Ligni'eres (2013) (see also Zahn 1974, 1992) suggest that turbulent transport can be described using the vertical turbulent diffusion coefficient 14 In the presence of this kind of stratified shear turbulence, our model holds provided the timescale for advection of angular momentum across the tachocline by large-scale flows (given more-or-less by t ES (Ω cz )) is shorter than the timescale for the turbulent diffusion of angular momentum (given more-or-less by ∆ 2 /D t ). This implies that our model is expected to apply whenever This criterion, rather interestingly and perhaps surprisingly, depends only on the relative coreenvelope lag iself. We then find that the use of our model is also always justified unless (Ω core -Ω cz ) is unrealistically large (that is, much larger than Ω cz itself). It thus appears that neglecting turbulent angular-momentum transport in the tachocline could well be justified, except perhaps very early on in the spin-down process if the Richardson number Ri ever drops below 1.",
"pages": [
40,
41,
42
]
},
{
"title": "6.3. Observational implications",
"content": "While a complete discussion of the observational implications of our model will have to be done by applying it in conjunction with stellar evolution, and statistically comparing its predictions against observations (as in Allain 1998; Irwin et al. 2007; Denissenkov et al. 2010; Spada et al. 2011; Reiners & Mohanty 2012; Gallet & Bouvier 2013, for instance), we can nevertheless already discuss its prospects in the light of previous work. As reported in Section 1, these previous studies found that stars that begin their lives as rapid rotators can, at all times, be modeled assuming solid-body rotation, while the rotation rates of stars in the mass-range 0 . 7 M glyph[circledot] -1 . 1 M glyph[circledot] that are initially slow rotators are best modeled with the two-zone model of MacGregor & Brenner (1991) assuming a rather long core-envelope coupling timescale (of the order of hundreds of Myr up to a Gyr). It is very difficult to explain such long timescales using a magnetic model, unless rather dramatic assumptions are made concerning the degree of confinement of the field (which must then also be explained). It is also difficult to explain observations with a purely turbulent model, since the latter does not easily explain why the core should be mostly in solid-body rotation. By contrast, our model naturally results in a system that behaves like the two-zone model, with a coupling timescale that depends both on the stellar structure and on the rotation rate of the star (see Equation 78), and that can be very substantial for slower rotators with fairly thick tachoclines. Specifically, we find that the coupling timescale is proportional to ∆ 3 /r 2 cz Ω 2 cz . As such, for similar tachocline thicknesses, it is naturally much shorter for fast rotators than for slow rotators, and for higher-mass stars (which have a larger radiation zone) than for lower-mass stars. Both results are qualitatively consistent with the aforementioned observations. Of course, the unknown dependence of ∆ on Ω cz and r cz makes it difficult at this point to give strict estimates of how strong this effect may be. Nevertheless, we can already infer from the data (whereby fast rotators should also be solid-body rotators) that ∆ 3 /r 2 cz Ω 2 cz must be a decreasing function of Ω cz . This constrains β (see Equation 79) to be strictly smaller than 2 / 3, and quite possibly substantially smaller than that. Additional information on the tachocline thickness ∆ may be obtained by studying the relative differences in surface chemical abundances of stars within the same cluster. Light-element such as lithium and beryllium undergo significant Main-Sequence depletion, that can only be explained by extra mixing below the base of the convection zone (see for instance the review by Pinsonneault 1997). Within the scope of our model, we expect their respective depletion rates to depend sensitively on ∆( t ), so that present-day surface abundances of a given star provide an integrated view of the variation of the tachocline depth with time (and therefore with rotation rate). Concurrently fitting rotational histories with Li and Be abundances may thus help constrain both β and γ . For stars in older clusters ( > few hundred Myr), we generally expect Equation (80) to hold. Indeed, as long as the condition 2 -α -1 -3 β ≤ 0 is satisfied (see 5.4 and 6.1 for detail), stars should have relaxed to their quasi-steady state by that age. In that state, aside from the surface abundances which depend on the rotational history of the star (as discussed above), all dynamical information about the star's initial conditions is lost, and the core-envelope lag only depends on present-day parameters. We then see from (80) that everything else being equal , their core-envelope lag should be much larger (1) if the spin-down rate is larger or if the star is rotating more slowly; (2) for lower-mass solar-type stars; (3) for stars with thicker tachoclines. While asteroseismology has not yet been able to detect any core-envelope lag in solar-type stars other than the Sun, one can only hope that such detection may be possible at some point in the future, and will independently help constrain our model. Finally, note that our model predicts that stars in this quasi-steady state, with 2 -α -1 -3 β < 0, always eventually reach solid-body rotation (at least in a radial sense) and that ∆ → 0 as Ω cz → 0 and ˙ Ω cz → 0. However, substantial latitudinal differential rotation is likely to persist in their convection zones, as it does in the Sun. This latitudinal shear by itself also drives large-scale meridional flows by gyroscopic pumping. As studied by GM98, these flows transport angular momentum and interact with the embedded primordial field, leading to a finite tachocline thickness (see Appendix E) even when ˙ Ω cz = 0. Although we have ignored this effect here for simplicity, and on the grounds that these shear-induced flows are presumably weaker than those driven by the spin-down torque for young stars, it can no longer be ignored for much older stars. In future work, we shall attempt to model simultaneously the effects of spin-down and of latitudinal shear in the convection zone in driving the tachocline flows, so as to present an integrated model of the rotational evolution of solar-type stars that can be used all the way from the Zero-Age Main Sequence to the present-day Sun.",
"pages": [
42,
43,
44
]
},
{
"title": "Acknowledgements",
"content": "This work originated from R. L. F. Oglethorpe's summer project at the Woods Hole GFD Summer Program in 2012. We thank the NSF and the ONR for supporting this excellent program. R. L. F. Oglethorpe acknowledges funding from an EPSRC studentship. P. Garaud acknowledges funding from the NSF (CAREER-0847477). We thank Nic Brummell, Douglas Gough, Subhanjoy Mohanty, Nigel Weiss and Toby Wood for fruitful discussions.",
"pages": [
44
]
},
{
"title": "A. QUASI-STEADY SOLUTION FOR THE SPIN-DOWN OF A NON-MAGNETIC STAR",
"content": "In this Appendix, we derive the result presented in Equation (25). Assuming a quasi-steady state, we drop the time-derivative in the momentum equation (22). Its azimuthal component then reduces to implying (using mass conservation) that w must be a linear function of z , exactly as in Section 3. To satisfy impermeability at z = z tc , we must therefore have The boundary condition (7), combined with the vertical component of the momentum equation, implies that T = 0 on s = R . Solving the thermal energy equation (24) with this boundary condition, along with T = 0 at z = z tc then gives where the constants { λ n } are the zeros of the Bessel function J 0 , and Using the fact that T = 0 at z = z cz determines the { α n } coefficients to be where ∆ = z cz -z tc is the thickness of the tachocline. Equations (A3) and (23) can then be used to derive p and v , up to the unknown set of integration constants { p n } (which are the same in both equations): To find { p n } , we need to solve for the dynamics of the convection zone, and match them onto the tachocline solution. Using the same method as the one outlined in Section 3, but this time with the anelastic mass conservation equation (see Equation 20), we find that with { ˜ B n } given again by matching w at z cz so that Matching p at z = z cz , and using the fact that ¯ ρ tc in the tachocline is, to a first approximation, equal to ¯ ρ ( z cz ) gives so that the azimuthal velocity (A8) leads to Equation (25).",
"pages": [
44,
45
]
},
{
"title": "B.1. Vertical eigenmodes",
"content": "In this Section we derive solutions of Equation (37) in the form of the eigenfunctions Z nm ( x ) and their associated eigenvalues µ nm , where x = ( z -z tc ) / ∆. To do this, we first need to specify the various boundary conditions on Z nm . The boundary conditions at z = z tc ( x = 0) are In the convection zone, by contrast, we assume that the dynamics always relax to the steady state on a very rapid timescale. Furthermore, we have shown in Section 4.2 that, within the context of a Darcy friction model, one can interchangeably use the Boussinesq approximation or the more realistic anelastic approximation. Here we adopt the former for the convection zone, hence (13) and (14) hold. The boundary conditions at z = z cz ( x = 1) are then using (B3), where { B n } remain to be determined. Equations (B4) and (B5) can finally be combined to give which defines the constants { K n } , and shows them to be positive. We then see that the eigenvalue problem defined by Equation (37) and associated boundary conditions listed above is homogeneous. It can easily be shown that the operator L on the left-hand-side of Equation (37) is self-adjoint with these boundary conditions, which implies that the vertical eigenmodes are orthogonal, with It can also be shown by considering the integral ∫ 1 0 Z nm L ( Z nm ) dx , suitably integrating it by parts, and applying the boundary conditions, that the eigenvalues associated with the operator L and our boundary conditions must be strictly positive, hence our choice of writing them as µ 4 nm in Equation (37). Since (37) is an equation with constant coefficients, we seek solutions of the form e σ nm z and find four solutions for σ nm : ± µ nm and ± iµ nm . Using this information, solutions of (37) that satisfy Z ' nm (0) = 0 and Z ''' nm (0) = 0 can be written as a linear combination of cosh( µ nm x ) and cos( µ nm x ) Applying the boundary condition (B3) we then have while the µ nm coefficients can be found by applying (B6). They are the solution of",
"pages": [
45,
46
]
},
{
"title": "B.2. Limit of the time-dependent solution",
"content": "We now seek to show that the solution to the time-dependent problem given in Equation (41) tends to the quasi-steady solution derived in Section 4.2 in the limit of large time for q ≤ 0. Since the contribution of the initial conditions disappear as t → + ∞ for q ≤ 0, we find that For ease of notation, we write so that In the case that q < 0, Equation (46) gives Using the method of steepest descent (as in Riley et al. 2006, for instance), noting that Ω cz < Ω 0 and that q < 0, we find that which, using Equations (39), (45), (43) and (B11), reduces to where V qs nm ( t ) is defined as the solution to Equation (38) without the dV nm /dt term, and so by definition is the projection of the quasi-steady solution (Equation 25) onto the horizontal and vertical eigenmodes. In the case that q = 0, in the limit of large t , Equation (B12) becomes Since q = 0, the ratio of t sd to τ ES nm is a constant, for each n , m (see Equation 47), and whether the transient solution tends to the quasi-steady solution depends on this ratio. If t sd glyph[greatermuch] τ ES nm (or equivalently t sd glyph[greatermuch] t ES ), then If, on the other hand, t sd glyph[lessmuch] τ ES nm , then glyph[negationslash] Hence, the system only relaxes to the quasi-steady solution when q = 0 provided t ES ( t 0 ) glyph[lessmuch] t sd ( t 0 ).",
"pages": [
47,
48
]
},
{
"title": "C. DERIVATION OF THE EKMAN JUMP CONDITION IN A SPINNING-DOWN FRAME",
"content": "In this Section we derive the viscous jump condition across the Ekman layer reported in Equation (55). Assuming that the Ekman layer is sufficiently thin to always be in balance, we apply the quasi-steady approximation to the momentum equation, which now reads where ν tc is the local viscosity, and where we have approximated the Laplacian in the viscous term by keeping only the vertical derivatives. Combining this momentum equation with conservation of mass gives To solve this equation, we first define δ E = √ ν tc / 2Ω cz and introduce the boundary-layer variable ζ = ( z -z tc ) /δ E . Hence By construction, the variable ζ remains of order unity within the tachopause, and rapidly tends to infinity above it, or in other words, as z enters the tachocline. We therefore have v → v 0 ( s ) as ζ → + ∞ , where v 0 ( s ) is the azimuthal velocity profile near the base of the tachocline. We also have v ( s, ζ ) = s Ω c at ζ = 0 assuming a no-slip boundary condition with the core. Applying these two conditions yields where c ( s ) is an integrating function that remains to be determined. A no-slip boundary condition also applies to the radial velocity, so u ( s, ζ ) = 0 at ζ = 0. Using the azimuthal component of the momentum equation to find u ( s, ζ ), we have so Finally, we require that the core be impermeable. To do so, we apply mass conservation to find ∂w ( s, ζ ) /∂ζ : Integrating Equation (C7) from ζ = 0 upward, and requiring w ( s, ζ ) = 0 at ζ = 0 yields We see that, as ζ →∞ , Matching this onto the tachocline solution (see Equation D1) yields the jump condition (55)",
"pages": [
48,
49
]
},
{
"title": "D. DERIVATION OF THE EVOLUTION EQUATION FOR J core ( t )",
"content": "In this Appendix, we derive the evolution equation for the angular momentum of the rigidly rotating core, reported in Equation (59). As in Appendix A, we solve the set of governing equations separately in the convection zone and in the tachocline, and match these solutions to the boundary conditions (at the top and side-walls of the domain), to the jump condition (at the base of the tachocline at z = z tc + δ ), and to each other (at the radiative-convective interface at z = z cz ). In the convection zone ( z cz < z < H ), assuming a Boussinesq system (which was proved to yield the same results as in the anelastic case in Appendix A), w and p are given by Equations (13) and (14), where { B n } are integration constants that need to be determined by matching these solutions to the tachocline. In the tachocline ( z tc + δ < z < z cz ), we still have ∂w/∂z = ˙ Ω cz / Ω cz (see Equation (A1) and using mass conservation). However, we can no longer directly apply the impermeability condition at z = z tc , since w must first be matched onto the tachopause solution. Hence, we write instead that where w 0 ( s ) is an integration function, that remains to be determined. Requiring continuity of w at the radiative-convective interface ( z = z cz ) yields Substituting this into (53) then gives where I tc is the moment of inertia of the tachocline defined in Equation (60) and where the only remaining unknowns are the { B n } . The following calculations show how to derive them. Solving the thermal equilibrium equation (24) for T in the tachocline, with w given by (D1) yields where α n and β n are found by applying the following boundary conditions: T = 0 at z = z cz and at z = z tc + δ . The second of these two boundary conditions can be justified only when the tachopause is much thinner than the tachocline, and much thinner than a thermal diffusion length. This is usually the case so where ∆ = z cz -( z tc + δ ). The vertical component of the momentum equation expressed in (A3) can then be used to calculate p in the tachocline, so that where the { P n } are found from matching this solution with that of the convection zone (see Equation 14) at z = z cz . After some algebra, we find that Finally, using the radial component of the momentum equation yields v , which can be used to calculate v 0 ( s ) ≡ v ( s, z tc + δ ). We find Substituting (D10) and (D2) into the jump condition (58) finally provides an equation for the { B n } , which, after significant algebra, can be cast in the form where a n and b n recover the formulae given by Equations (61) and (62) in the limit δ glyph[lessmuch] ∆. Using (D11) in (D3) then leads to (59). Finally, if we want to calculate w 0 ( s ), we simply substitute B n back into (D2) to get:",
"pages": [
49,
50,
51
]
},
{
"title": "E. THE TACHOCLINE THICKNESS",
"content": "In GM98 and Wood et al. (2011), the radial mass flux downwelling into the tachocline is caused by the gyroscopic pumping associated with the turbulent torques that permanently drive the observed latitudinal shear in the convection zone, rather than by the spin-down torque. For this reason, their respective estimates of the tachocline thickness ∆ as a function of other stellar parameters do not directly apply here. Nevertheless, we can apply a similar method to the one they use to infer ∆ in the spin-down case. We now proceed to describe this method and its limitations, first applied to the steady-state solar case, and then applied to our own spin-down problem. In the solar case studied by GM98 and Wood et al. (2011), the thickness of the tachocline ∆ is obtained by matching the vertical mass flux through the tachocline to the vertical mass flux through the tachopause. The latter is estimated by assuming advection-diffusion balance of the flux of horizontal magnetic field across the tachopause (whose thickness is δ ): while the former is obtained, as in Section 4.2, by considering thermal-wind balance and thermal equilibrium across the tachocline, which yields where χ Ω cz is an estimate of the amplitude of the latitudinal differential rotation in the convection zone. In the Sun, χ ∼ 0 . 1. Combining (E1) and (E2) yields a relationship between δ and ∆: This equation is quite robust, since it relies on basic balances that are not easily upset by additional dynamics, and has been verified against numerical simulations (Acevedo-Arreguin et al. 2013). However, in order to obtain ∆ as a function of known stellar parameters and independently of δ , one must make further assumptions concerning the nature and structure of the tachopause. This final step, unfortunately, is quite model-dependent. GM98 and Wood et al. (2011) propose different scalings for δ as a function of B 0 and Ω cz for instance. Both assume that the tachopause is laminar, but disagree on its thermal properties, leading to for GM98 (using their Equation 6 as a definition of δ ), and for Wood et al. (2011) (using Equation 56 as a definition of δ ). Moreover, neither of these scalings apply if turbulence also plays a role in the angular-momentum transport balance across the tachopause (which cannot a priori be ruled out). In short, while Equation (E3) robustly relates ∆ to δ in the solar steady-state model, it is not sufficient on its own to derive a reliable estimate of how ∆ varies with stellar parameters without further constraints on the tachopause structure. The latter can only be obtained in direct numerical simulations of the system, which are not yet available at this point. Not surprisingly, we find that the same problem affects the determination of ∆ in a spin-down model. In Section 5, we found that w in the tachocline is neither constant with distance from the rotation axis nor with depth, so that a direct application of Equation (E1) is not possible. Nevertheless, one can require advection-diffusion balance on average in the tachopause by setting (within the context of the cylindrical model used throughout this work) Substituting w 0 ( s ) given in (D12) into this equation and evaluating the integral yields: where we have once again used the property ∑ n (32 /λ 4 n ) = 1. Dropping all but the first term in this sum (as in Section 5.3), and substituting a 1 and b 1 given in (66) yields While somewhat obscure, this expression is indeed the equivalent of (E3) in the spin-down case. To see this, note that if ˙ Ω cz = 0, then which recovers the same scalings as those of GM98 and Wood et al. (2011) (see Equation E3), as long as Ω c is re-interpreted as the latitudinal shear driving the large-scale flows χ Ω cz , and the second term in the denominator is much larger than 1 (which is usually the case unless δ/ ∆ is unrealistically small). For young solar-type stars, however, spin-down dominates the dynamics of the system. Equation (E8) then takes a different form during the initial transient and the later quasi-steady state phases. During the transient, Ω c = Ω core -Ω cz glyph[similarequal] -Ω cz , as discussed in Section 5.5. In that case, and using the fact that t ES (Ω cz ) glyph[lessmuch] t sd (Ω cz ) for our model to apply anyway, Equation (E8) can be approximated as where c 1 is a constant of order unity. Using this equation in conjunction with a given tachopause model (as in Equation 56 or as in Equation 6 of GM98) does yield an estimate for ∆ as a function of stellar parameters. However, that estimate is very sensitive to any model uncertainty on the nature and structure of the tachopause, as discussed above in the context of the Sun. In the quasi-steady phase, the problem is even worse. Since the core-envelope lag Ω c = Ω core -Ω cz is now given by Equation (80), Equation (E8) becomes where c 2 is also a constant of order unity. We then see that the dependence on ∆ on the left-hand side vanishes altogether (since I tc ∝ ∆), which implies that this method cannot be used to constrain ∆ directly. Instead, Equation (E11) provides a second constrain on δ - the first one being given by various balances within the tachopause, leading for instance to Equation (56), or Equation (6) of GM98 - and therefore defines the position of the tachopause to be the radius where the amplitude of the primordial field B 0 is such that the two definitions of δ coincide. Since the tachopause lies by construction at the bottom of the tachocline, one could in principle use this method to determine ∆ if the radial variation of B 0 is known. However, any estimate of ∆ based on this method will, once again, be uncomfortably model-dependent. In summary, we conclude that theory alone cannot robustly predict how the thickness of the tachocline varies with stellar parameters. Any estimate of ∆ made by applying mass continuity across the interface between the tachocline and the tachopause, as in GM98, relies sensitively on the assumed structure of the tachopause, which is itself sensitively dependent on the nature and balance of forces, thermal energy transport and angular-momentum transport within. For this reason, until such a time where the tachopause is better understood (through direct numerical simulations for instance), we advocate the use of a more general tachocline law, as in Equation (79) for instance.",
"pages": [
51,
52,
53,
54
]
},
{
"title": "REFERENCES",
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"pages": [
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2013ApJ...779...18B
https://arxiv.org/pdf/1307.5324.pdf
<document>
<text><location><page_1><loc_11><loc_85><loc_89><loc_87></location>A SEARCH FOR FAST OPTICAL TRANSIENTS IN THE PAN-STARRS1 MEDIUM-DEEP SURVEY: M DWARF FLARES, ASTEROIDS, LIMITS ON EXTRAGALACTIC RATES, AND IMPLICATIONS FOR LSST</text>
<text><location><page_1><loc_10><loc_81><loc_90><loc_84></location>E. BERGER 1 , C. N. LEIBLER 1,2 , R. CHORNOCK 1 , A. REST 3 , R. J. FOLEY 1 , A. M. SODERBERG 1 , P. A. PRICE 4 , W. S. BURGETT 5 , K. C. CHAMBERS 5 , H. FLEWELLING 5 , M. E. HUBER 5 , E. A. MAGNIER 5 , N. METCALFE 6 , C. W. STUBBS 7 , & J. L. TONRY 5</text>
<text><location><page_1><loc_42><loc_80><loc_59><loc_81></location>Draft version September 1, 2021</text>
<section_header_level_1><location><page_1><loc_46><loc_77><loc_54><loc_79></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_49><loc_86><loc_77></location>We present a search for fast optical transients ( τ ∼ 0 . 5hr -1d) using repeated observations of the PanSTARRS1 Medium-Deep Survey (PS1/MDS) fields. Our search takes advantage of the consecutive g P1 r P1 observations (16 . 5 min in each filter), by requiring detections in both bands, with non-detections on preceding and subsequent nights. We identify 19 transients brighter than 22 . 5 AB mag (S / N /greaterorsimilar 10). Of these, 11 events exhibit quiescent counterparts in the deep PS1/MDS templates that we identify as M4-M9 dwarfs at d ≈ 0 . 2 -1 . 2 kpc. The remaining 8 transients lack quiescent counterparts, exhibit mild but significant astrometric shifts between the g P1 and r P1 images, colors of ( g -r )P1 ≈ 0 . 5 -0 . 8mag, non-varyinglight curves, and locations near the ecliptic plane with solar elongations of about 130 deg, which are all indicative of main-belt asteroids near the stationary point of their orbits. With identifications for all 19 transients, we place an upper limit of R FOT( τ ∼ 0 . 5hr) /lessorsimilar 0 . 12 deg -2 d -1 (95% confidence level) on the sky-projected rate of extragalactic fast transients at /lessorsimilar 22 . 5 mag, a factor of 30 -50 times lower than previous limits; the limit for a timescale of ∼ day is R FOT /lessorsimilar 2 . 4 × 10 -3 deg -2 d -1 . To convert these sky-projected rates to volumetric rates, we explore the expected peak luminosities of fast optical transients powered by various mechanisms, and find that nonrelativistic events are limited to M ≈ -10 mag ( M ≈ -14 mag) for a timescale of ∼ 0 . 5 hr ( ∼ day), while relativistic sources (e.g., gamma-ray bursts, magnetar-powered transients) can reach much larger luminosities. The resulting volumetric rates are /lessorsimilar 13 Mpc -3 yr -1 ( M ≈ -10 mag), /lessorsimilar 0 . 05 Mpc -3 yr -1 ( M ≈ -14 mag) and /lessorsimilar 10 -6 Mpc -3 yr -1 ( M ≈ -24 mag), significantly above the nova, supernova, and GRB rates, respectively, indicating that much larger surveys are required to provide meaningful constraints. Motivated by the results of our search we discuss strategies for identifying fast optical transients in the LSST main survey, and reach the optimistic conclusion that the veil of foreground contaminants can be lifted with the survey data, without the need for expensive follow-up observations.</text>
<text><location><page_1><loc_14><loc_47><loc_70><loc_48></location>Subject headings: stars: flare, asteroids: general, supernovae: general, novae, surveys</text>
<section_header_level_1><location><page_1><loc_21><loc_43><loc_35><loc_44></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_22><loc_48><loc_43></location>For nearly a century, optical observations aimed at the discovery and study of astrophysical transients have largely focused on events with durations of days to months. This is mainly due to a fortuitous match with the timescales of the most common extragalactic events (novae and supernovae), whose typical luminosities and intrinsic rates require coverage of large numbers of galaxies and/or blank sky areas, leading to a natural search cadence of several days. Thus, novae have a much higher intrinsic rate than supernovae ( ∼ 2 . 2 yr -1 per 10 10 LK , /circledot versus ∼ 2 × 10 -3 yr -1 per 10 10 LK , /circledot , respectively; e.g., Williams & Shafter 2004; Li et al. 2011), but supernovae are significantly more luminous than novae ( ∼ -18 mag versus ∼ -8 mag, respectively; Gallagher & Starrfield 1978; Filippenko 1997). As a result, for a given survey limiting magnitude tens of novae can be discovered per year by</text>
<text><location><page_1><loc_52><loc_32><loc_92><loc_44></location>targeting a few nearby galaxies with a cadence of few days, while discovering a similar number of supernovae requires monitoring of ∼ 10 4 galaxies (or hundreds of deg 2 ), thereby necessitating a similar cadence of several days; for the purpose of sheer discovery rate, a faster cadence is not profitable for nova and supernova searches. Over the past few decades such surveys have been highly successful at discovering about one hundred novae and supernovae per year (e.g., Williams & Shafter 2004; Leaman et al. 2011).</text>
<text><location><page_1><loc_52><loc_5><loc_92><loc_32></location>The advent of large format cameras on dedicated wide-field telescopes, coupled with serendipitous discoveries of transients outside of the traditional nova and supernova luminosity and timescale ranges, has opened up a new discovery space for astrophysical transients. By repeatedly targeting the same fields, such surveys are in principle capable of exploring a wide range of timescales, from the duration of single exposures (i.e., minutes) to years. In practice, most surveys are still primarily focused on supernovae (driven to a large extent by Type Ia supernova cosmology at increasingly larger redshifts), and therefore cover wider fields to greater depth at the expense of a faster temporal cadence to maximize the supernova discovery rate while preserving adequate light curve coverage. Still, some surveys have been utilized to perform initial searches for fast optical transients (FOTs) on timescales as short as ∼ 0 . 5 hr. Clearly, the effective areal exposure of such searches (i.e., the product of survey area and exposure time) becomes progressively smaller at faster cadence as sky coverage has to be sacrificed for repeated short-cadence observations.</text>
<text><location><page_2><loc_8><loc_57><loc_48><loc_92></location>In this context, the Deep Lens Survey (DLS) was utilized to search for FOTs on a timescale of about 1300 s to a depth of B ≈ 23 . 8 mag (with a total exposure of 1.1 deg 2 d) and led to an upper limit on the extragalactic sky-projected rate of R FOT /lessorsimilar 6 . 5 deg -2 d -1 (95% confidence level; Becker et al. 2004). The DLS search uncovered three fast transients, which were shown to be flares from Galactic M dwarf stars (Becker et al. 2004; Kulkarni & Rau 2006). A search for transients with a timescale of /greaterorsimilar 0 . 5 hr to a depth of about 17.5 mag (with a total exposure of 635 deg 2 d) using the Robotic Optical Transient Search Experiment-III (ROTSE-III) yielded a limit on the extragalactic rate of R FOT /lessorsimilar 5 × 10 -3 deg -2 d -1 , and uncovered a single M dwarf flare (Rykoff et al. 2005); a similar search with the MASTER telescope system yielded comparable limits (Lipunov et al. 2007), and two uncharacterized candidate fast transients (Gorbovskoy et al. 2013). Similarly, a targeted search for transients on a timescale of about 0.5 hr and to a depth of B ≈ 21 . 3 mag in the Fornax galaxy cluster (with a total exposure of 1.9 deg 2 d) placed a limit on the extragalactic rate of R FOT /lessorsimilar 3 . 3 deg -2 d -1 (Rau et al. 2008). Two fast transients were detected in this search, both shown to be M dwarf flares (Rau et al. 2008). At the bright end, the 'Pi of the Sky' project placed a limit on transients brighter than 11 mag with a duration of /greaterorsimilar 10 s of R FOT /lessorsimilar 5 × 10 -5 deg -2 d -1 (Sokołowski et al. 2010).</text>
<text><location><page_2><loc_8><loc_36><loc_48><loc_58></location>Fast optical transients have also been found serendipitously by other surveys, but they have generally been shown to be Galactic in origin 8 . A notable exception is the transient PTF11agg (Cenko et al. 2013), which faded by about 1.2 mag in 5.3 hours, and 3.9 mag in 2.2 d, and was accompanied by radio emission that may be indicative of relativistic expansion (although the distance of this transient is not known, thereby complicating its interpretation). We note that such an event, while fast compared to the general nova and supernova population, is still of longer duration than the timescales probed by the DLS and Fornax searches, as well as the Pan-STARRS1 search we describe here. Thus, the existing searches for extragalactic FOTs have mainly raised the awareness that the foreground of M dwarf flares is large, with an estimated allsky rate of ∼ 10 8 yr -1 at a limiting magnitude of ∼ 24 mag (Becker et al. 2004; Kulkarni & Rau 2006).</text>
<text><location><page_2><loc_8><loc_12><loc_48><loc_36></location>Here, we present a search for fast optical transients with an effective timescale of about 0.5 hr to 1 d and to a depth of about 22.5 mag in the first 1 . 5 years of data from the PanSTARRS1 Medium-Deep Survey (PS1/MDS). This search uncovered a substantial sample of 19 fast transients, both with and without quiescent counterparts. We describe the survey strategy and selection criteria in §2. In §3 we summarize the properties of the 19 detected transients and classify them using a combination of color information, astrometry, sky location, and the properties of quiescent counterparts (when detected). With a unique identification of all 19 transients as Solar system or Galactic in origin, we place a limit on the rate of extragalactic fast transients that is 30 -50 times better than the limits from previous searches (§4). We further investigate for the first time the limits on the volumetric rates of FOTs from our survey and previous searches using the survey limiting magnitudes and fiducial transients luminosities. In §5 we expand on this point and discuss the expected peak luminosities</text>
<text><location><page_2><loc_52><loc_85><loc_92><loc_92></location>of FOTs for a range of physically motivated models. Finally, since our search is the first one to utilize observations that are similar to the Large Synoptic Survey Telescope (LSST) main survey strategy, we conclude by drawing implications for fast optical transient searches in the LSST data §6.</text>
<section_header_level_1><location><page_2><loc_65><loc_83><loc_79><loc_84></location>2. OBSERVATIONS</section_header_level_1>
<section_header_level_1><location><page_2><loc_63><loc_81><loc_81><loc_82></location>2.1. PS1 Survey Summary</section_header_level_1>
<text><location><page_2><loc_52><loc_57><loc_92><loc_81></location>The PS1 telescope, located on Mount Haleakala, is a highetendue wide-field survey instrument with a 1.8-m diameter primary mirror and a 3 . 3 · diameter field-of-view imaged by an array of sixty 4800 × 4800 pixel detectors, with a pixel scale of 0 . 258 '' (Kaiser et al. 2010; Tonry & Onaka 2009). The observations are obtained through five broad-band filters ( g P1 r P1 i P1 z P1 y P1), with some differences relative to the Sloan Digital Sky Survey (SDSS); the g P1 filter extends 200 Å redward of g SDSS to achieve greater sensitivity and lower systematics for photometric redshifts, while the z P1 filter terminates at 9300 Å, unlike z SDSS which is defined by the detector response (Tonry et al. 2012). PS1 photometry is in the 'natural' system, m = -2 . 5log( F ν ) + m ' , with a single zero-point adjustment ( m ' ) in each band to conform to the AB magnitude scale. Magnitudes are interpreted as being at the top of the atmosphere, with 1.2 airmasses of atmospheric attenuation included in the system response function (Tonry et al. 2012).</text>
<text><location><page_2><loc_52><loc_40><loc_92><loc_57></location>The PS1 Medium-Deep Survey (MDS) consists of 10 fields (each with a single PS1 imager footprint) observed on a nearly nightly basis by cycling through the five filters in 3 -4 nights to a typical 5 σ depth of ∼ 23 . 3 mag in g P1 r P1 i P1 z P1, and ∼ 21 . 7 mag in y P1. The MDS images are processed through the Image Processing Pipeline (IPP; Magnier 2006), which includes flat-fielding ('de-trending'), a flux-conserving warping to a sky-based image plane, masking and artifact removal, and object detection and photometry. For the fast transient search described here we produced difference images from the stacked nightly images using the photpipe pipeline (Rest et al. 2005) running on the Odyssey computer cluster at Harvard University.</text>
<section_header_level_1><location><page_2><loc_59><loc_38><loc_86><loc_39></location>2.2. A Search for Fast Optical Transients</section_header_level_1>
<text><location><page_2><loc_52><loc_12><loc_92><loc_37></location>For the purpose of detecting fast optical transients we take advantage of the consecutive MDS g P1 r P1 observations, with eight 113 s exposures in each filter providing a total timespan of about 33 min for a full sequence. We carry out the search using the stacked g P1 and r P1 images from each visit through image subtraction relative to deep multi-epoch templates, and subsequently utilize the individual exposures to construct light curves; representative discovery and template images are shown in Figure 1 and light curves are shown in Figure 2. We limit the timescale of the transients to /lessorsimilar 1 d by further requiring no additional detections in the g P1 r P1 i P1 z P1 filters on preceding and subsequent nights (extending to ± 5 nights). To ensure that this constraint is met we only perform our search on the subset of MDS data for which consecutive nights of observations are available. In the first 1 . 5 years of data we searched a total of 277 nights of g P1 r P1 observations across the 10 MDS fields, leading to a total areal exposure of 40.4 deg 2 d for a timescale of 0.5 hr and 1940 deg 2 d for a timescale of 1 d.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_11></location>To select transients in the g P1 r P1 difference images we utilize a signal-to-noise ratio threshold 9 of S / N = 10, with resulting limiting magnitudes of g P1 ≈ 22 . 7 mag and r P1 ≈ 22 . 4</text>
<figure>
<location><page_3><loc_8><loc_51><loc_92><loc_92></location>
<caption>FIG. 1.- PS1/MDS images of a representative M dwarf flare (left; PSO J164.3814 + 58.3011) and an asteroid (right; PSO J352.5968 -0.4471) found in our fast transients search. In each case we show the template images ( g P1 r P1 i P1 z P1) in the left column, the fast transient discovery images in the middle column, and the difference images in the right column. Each panel is 20 '' × 20 '' oriented with north up and east to the left. The red optical colors of the M dwarf counterpart are evident, as is the lack of a counterpart in the case of the asteroid. The light curves resulting from these detections are shown in Figure 2.</caption>
</figure>
<text><location><page_3><loc_8><loc_32><loc_48><loc_44></location>mag. We additionally require sources in the g P1 r P1 difference images to astrometrically match within 0 . 35 '' , corresponding to a 1.2 pixel radius around each detection (this is about 6 times the typical astrometric error; see §3.1). Using these cuts we find a total of 227 candidates, which were visually inspected by one of us (C.L.) leading to a final list of 19 sources that were further validated by two of us (E.B. and R.C.); the remaining 208 events were predominantly spurious detections near saturated stars.</text>
<section_header_level_1><location><page_3><loc_8><loc_28><loc_48><loc_30></location>3. THE PROPERTIES OF PAN-STARRS1 FAST OPTICAL TRANSIENTS</section_header_level_1>
<text><location><page_3><loc_8><loc_11><loc_48><loc_27></location>Using the procedure described in the previous section we found 19 genuine fast transients, spanning a brightness range of g P1 ≈ 18 . 5 -22 . 7 mag and r P1 ≈ 18 . 8 -22 . 4 mag. In Figure 3 we show the g P1 versus ( g -r )P1 color-magnitude diagram for all 19 sources; the photometry is summarized in Tables 1 and 2. The faint end of the distribution is determined by our requirement of S / N /greaterorsimilar 10 in the subtractions of the individual g P1 r P1 nightly stacks (16.5 min in each filter) from the deep templates. On the other hand, the bright end of the observed distribution is about 2 mag fainter than the saturation limit of our images, indicating a genuine dearth of apparently bright fast transients in our search area. The ( g -r )P1</text>
<text><location><page_3><loc_8><loc_5><loc_48><loc_9></location>tections by measuring the flux and uncertainty at random positions in the difference images in the same manner as the transient flux, and then determining a correction factor which leads to a distribution with a reduced χ 2 of unity.</text>
<figure>
<location><page_3><loc_52><loc_20><loc_92><loc_44></location>
<caption>FIG. 2.- Light curves in the g P1 (blue) and r P1 (red) filters for a representative M dwarf flare (squares) and an asteroid (circles) detected in our search for fast optical transients (see Figure 1). In each case the lines mark the best linear fit. The M dwarf flare exhibits a 3.5 mag decline during the time-span of our observation, while the asteroid exhibits constant brightness. Missing light curve points are due to individual exposures in which the source was located in a chip gap.</caption>
</figure>
<text><location><page_3><loc_52><loc_5><loc_92><loc_9></location>colors span a wide range of about -2 . 0 to + 0 . 9 mag, but we stress that for transients that rapidly vary in brightness within the time-span of each observation (e.g., Figure 2), the non-</text>
<figure>
<location><page_4><loc_9><loc_68><loc_48><loc_92></location>
<caption>FIG. 3.- Color-magnitude diagram for the fast transients discovered in our PS1/MDS search. The sources are divided into those with detected quiescent counterparts in the deep template images (blue squares), and those lacking counterparts to limits of g P1 r P1 i P1 z P1 /greaterorsimilar 24 . 5 -25 mag (red circles). The dashed line marks the r P1 limit of our search (the region below the line is inaccessible to the survey). The hatched region marks the expected ( g -r )P1 color range for asteroids. All of the fast transients lacking quiescent counterparts reside in this color range.</caption>
</figure>
<text><location><page_4><loc_8><loc_54><loc_48><loc_57></location>simultaneous g P1 and r P1 measurements do not reflect the true instantaneous colors.</text>
<section_header_level_1><location><page_4><loc_11><loc_52><loc_46><loc_53></location>3.1. Fast Transients Lacking Quiescent Counterparts</section_header_level_1>
<text><location><page_4><loc_8><loc_9><loc_48><loc_51></location>Of the 19 fast transients discovered in our search, 8 events lack quiescent counterparts in any of the deep template images ( g P1 r P1 i P1 z P1) to typical limits of /greaterorsimilar 24 . 5 -25 mag. These sources are in principle a promising population of distant extragalactic transients with undetected host galaxies. We utilize a combination of color information, the 33 min time-span of the g P1 r P1 observations, and the sky locations to investigate the nature of these sources. We first note that all 8 sources span a narrow color range with ( g -r )P1 ≈ 0 . 55 -0 . 8 mag (Figure 3 and Table 2), which is typical of main-belt asteroids (e.g., Ivezi'c et al. 2001). This interpretation naturally explains the lack of quiescent counterparts in the template images. We further test this scenario by comparing the astrometric positions of each transient in the g P1 and r P1 images. In Figure 4 we show the distribution of positional shifts for the 8 sources compared to unresolved field sources in the same images. The median offset for field sources is about 53 mas (with a standard deviation of about 31 mas), indicative of the astrometric alignment precision of the MDS images. On the other hand, the 8 transients exhibit shifts of 11 -340 mas (bounded by our initial cut of /lessorsimilar 0 . 35 '' shift; §2.2), with a median value of about 230 mas. This is well in excess of the point source population, and a Kolmogorov-Smirnov(K-S) test gives a p -value of only 1 . 1 × 10 -4 for the null hypothesis that the positional offsets of the 8 transients and the field sources are drawn from the same underlying distribution. This clearly indicates that the 8 transients lacking quiescent counterparts exhibit larger than average astrometric shifts, supporting their identification as asteroids. We further inspect the g P1 and r P1 light curves of the 8 transients and find that none exhibit variability larger than the photometric uncertainties (e.g., Figure 2).</text>
<text><location><page_4><loc_8><loc_4><loc_48><loc_9></location>Finally, we note that all 8 sources are located in the three MDS fields (MD04, MD09, and MD10) that are positioned within ± 10 · of the ecliptic plane (Table 2). In particular, the</text>
<figure>
<location><page_4><loc_52><loc_67><loc_92><loc_92></location>
<caption>FIG. 4.- Astrometric shift between the g P1 and r P1 centroids for the fast transients lacking quiescent counterparts (red arrows) in comparison to the shifts for field stars from the same images (gray hatched histogram). The median positional shift for the field stars, which indicates the typical astrometric uncertainty of the MDS images, is about 53 mas. The fast transients lacking quiescent counterparts exhibit generally larger shifts of up to ≈ 340 mas (the limit allowed by our search), with a median of 230 mas. A K-S test indicates a p -value of only 1 . 1 × 10 -4 for the null hypothesis that the two populations are drawn from the same underlying distribution.</caption>
</figure>
<text><location><page_4><loc_52><loc_45><loc_92><loc_55></location>asteroids were discovered in these fields on dates concentrated at solar elongation values of about 130 deg, at which mainbelt asteroids go through a stationary point with negligible apparent motion. We therefore conclude based on their colors, astrometric motions, light curve behavior, and ecliptic coordinates that the 8 fast transients lacking quiescent counterparts are simply main-belt asteroids near the stationary point of their orbit.</text>
<section_header_level_1><location><page_4><loc_53><loc_42><loc_91><loc_43></location>3.2. Fast Transients with Detected Quiescent Counterparts</section_header_level_1>
<text><location><page_4><loc_52><loc_14><loc_92><loc_41></location>We now turn to the 11 fast transients that exhibit quiescent counterparts in some or all of the deep MDS template images. In all cases we find that the counterparts are unresolved (with a typical seeing of about 1 '' ) and have red colors that are indicative of M dwarf stars. Photometry of the quiescent counterparts from the PS1/MDS templates, and from SDSS when available, is summarized in Table 1. Using these magnitudes we determine the spectral type of each counterpart by comparing to the SDSS colors of M dwarfs (West et al. 2011); from sources with both PS1 and SDSS photometry we infer color transformations of ( g -r )P1 ≈ 0 . 94 × ( g -r )SDSS and ( i -z )P1 ≈ 0 . 93 × ( i -z )SDSS to account for the difference between the g P1and z P1 filters compared to the g SDSS and z SDSS filters (§2.1). The results are shown in Figure 5 indicating that 7 counterparts have spectral types of about M4-M5, while the remaining 4 counterparts have spectral types of M7-M9 (see Table 1 for the inferred spectral types). We further infer the distances to these M dwarfs using their associated absolute magnitudes (Bochanski et al. 2011) and find d ≈ 0 . 2 -1 . 2 kpc (Table 1).</text>
<text><location><page_4><loc_52><loc_8><loc_92><loc_14></location>Thus, as in previous fast transient searches, all 11 fast transients with quiescent counterparts in our survey are M dwarf flares. Using the flare magnitudes and inferred spectral types, we compare the resulting flare and bolometric luminosities in Figure 6. We find that the flares span a luminosity 10 range</text>
<figure>
<location><page_5><loc_9><loc_68><loc_48><loc_92></location>
<caption>FIG. 5.- Color-color phase-space for the quiescent counterparts of all 11 fast transients with a detected counterpart (stars; arrows indicate limits). Also shown are the color ranges for M0-M9 dwarf stars (West et al. 2011). All of the detected counterparts track the M dwarf sequence, with 7 sources exhibiting the colors of M4-M5 dwarfs and 4 sources exhibiting colors typical of M7-M9 dwarfs.</caption>
</figure>
<text><location><page_5><loc_29><loc_68><loc_30><loc_69></location>P1</text>
<text><location><page_5><loc_8><loc_48><loc_48><loc_59></location>of Lf , g ≈ (6 -150) × 10 28 erg s -1 and Lf , r ≈ (4 -80) × 10 28 erg s -1 , with no apparent dependence on spectral type. However, since the bolometric luminosity declines from about 3 . 3 × 10 31 erg s -1 at spectral type M4 to about 1 . 3 × 10 30 erg s -1 at spectral type M9, the relative flare luminosities increase with later spectral type. We find relative flare luminosities of ≈ 0 . 006 -0 . 07 Lbol at ∼ M5, and larger values of ≈ 0 . 03 -1 Lbol at M8-M9.</text>
<text><location><page_5><loc_8><loc_25><loc_48><loc_48></location>It is instructive to compare the properties of the flares and M dwarfs uncovered in our blind fast transients search to those from targeted M dwarf variability studies. In particular, Kowalski et al. (2009) searched for flares from 50,130 pre-selected M0-M6 dwarfs in the SDSS Stripe 82 and found 271 flares, with an apparent increase in the flare rate with later spectral type. This may explain the lack of M0-M4 dwarfs in our relatively small sample. For the M4-M6 dwarfs Kowalski et al. (2009) find a mean flare amplitude of ∆ u ≈ 1 . 5 mag, comparable to our mean value of ∆ g ≈ 1 . 1 mag, when taking into account that M dwarfs flares are generally brighter in u -band than in g -band due a typical temperature of ∼ 10 4 K. The flare luminosities for the Stripe 82 M4-M6 dwarfs are Lf , u ≈ (2 -100) × 10 28 erg s -1 , again comparable to the g P1-band luminosities of the ∼ M5 dwarfs in our sample. We note that there are no M7-M9 dwarfs in the Stripe 82 sample.</text>
<text><location><page_5><loc_8><loc_9><loc_48><loc_25></location>Kowalski et al. (2009) also found that for M4-M6 dwarfs there is a strong dependence of the flare rate on vertical distance from the Galactic plane; namely, the fraction of time in which a star flares decreases by about an order of magnitude over a vertical distance range of about 50 -150 pc. The M4-M6 dwarfs in our sample are all located at larger vertical distances of ≈ 190 -560 pc, with a mean of about 390 pc, suggesting that the decline in flaring activity with vertical distance from the Galactic plane may not be as steep as previously inferred. Moreover, the Stripe 82 data exhibit a trend of steeper decline in the flare rate as a function of vertical distance with increasing spectral type (Kowalski et al. 2009),</text>
<text><location><page_5><loc_8><loc_5><loc_48><loc_8></location>over the widths of the g P1 and r P1 filters, with δν ≈ 1 . 287 × 10 14 Hz and ≈ 7 . 721 × 10 13 Hz, respectively.</text>
<figure>
<location><page_5><loc_52><loc_68><loc_92><loc_92></location>
<caption>FIG. 6.- Luminosities of the fast transients (flares) associated with M dwarfs counterparts (vertical bars) as a function of source spectral type. The luminosity range for each source is defined by the g P1 and r P1 detections. The solid red line marks the bolometric luminosity as a function of spectral type, while dashed lines mark fractions of the bolometric luminosity as indicated in the figure. We find that the flares from the ∼ M5 sources have a range of ≈ 0 . 006 -0 . 07 Lbol, while the flares from the M7-M9 dwarfs are relatively larger, with ≈ 0 . 03 -1 Lbol.</caption>
</figure>
<text><location><page_5><loc_52><loc_48><loc_92><loc_57></location>while here we find that the M7-M9 dwarfs have a similar mean vertical distance from the Galactic plane to the ∼ M5 dwarfs. Clearly, a more systematic search for M dwarf flares in the PS1/MDS is required to study these trends. In particular, it is likely that we have missed some flares due to the requirement of no additional variability within a ± 5 night window around each detection (§2.2).</text>
<section_header_level_1><location><page_5><loc_52><loc_44><loc_91><loc_46></location>4. LIMITS ON THE RATE OF EXTRAGALACTIC FAST OPTICAL TRANSIENTS</section_header_level_1>
<text><location><page_5><loc_52><loc_23><loc_92><loc_43></location>The 19 fast transients uncovered by our search cleanly divide into two categories: (i) main-belt asteroids near the stationary point of their orbits (§3.1); and (ii) flares from M dwarf stars (§3.2). Neither category is unexpected given that our search is based on consecutive g P1 r P1 detections with a time-span of about 0.5 hr. M dwarf flares typically exhibit blue colors indicative of T ∼ 10 4 K, with timescales of minutes to hours, and are thus ubiquitous in searches that utilize rapid observations in the ultraviolet (Welsh et al. 2005, 2006) or blue optical bands (Becker et al. 2004; Kulkarni & Rau 2006; Rau et al. 2008). Similarly, our requirement of two consecutive detections within ∼ 0 . 5 hr, with non-detections on preceding or subsequent nights is effective at capturing asteroids near the stationary point of their orbits (i.e., at solar elongations of about 130 deg for main-belt asteroids).</text>
<text><location><page_5><loc_52><loc_5><loc_92><loc_23></location>Since we account for all 19 fast transients as Solar system or Galactic in origin, we can place a robust upper limit on the rate of extragalactic fast optical transients. While we could in principle detect a sufficiently bright transient with a timescale as short as about 4 min, corresponding to a detection in only the final exposure in one filter and the first exposure in the second filter, a more reasonable timescale probed by our search is about 0.5 hr, the time-span of a full g P1 r P1 exposure sequence (e.g., Figure 2). Similarly, our selection criteria could in principle accommodate transients with durations as long as ∼ 2 d, but for the bulk of the search the maximum timescale is /lessorsimilar 1 d. Thus, we consider our search to place limits on fast transients spanning about 0.5 hr to 1 d. In the discussion below</text>
<figure>
<location><page_6><loc_12><loc_48><loc_88><loc_92></location>
<caption>FIG. 7.- Limits on the sky-projected rate of extragalactic fast optical transients as a function of timescale and survey limiting magnitude. Shown are the limits from our survey (black) and from the literature (red; Becker et al. 2004; Rykoff et al. 2005; Lipunov et al. 2007; Rau et al. 2008). The much larger effective areal exposure of our survey provides constraints that are about 30 -50 times deeper than previous surveys with similar limiting magnitudes. Also shown are the expected limits from the LSST main survey (blue) for 1 night (top limit) and 1 year (bottom limit) of observations.</caption>
</figure>
<text><location><page_6><loc_8><loc_39><loc_48><loc_41></location>we provide upper limits for the upper and lower bounds of the timescale distribution.</text>
<text><location><page_6><loc_8><loc_32><loc_48><loc_38></location>The total areal exposure of our survey for a timescale of 0.5 hr is EA ≈ 40 . 4 deg 2 d, while for a timescale of ∼ day it is correspondingly longer, EA ≈ 1940 deg 2 d. Thus, we can place a 95% confidence limit ( /lessorsimilar 3 events) on the skyprojected rate of:</text>
<formula><location><page_6><loc_23><loc_30><loc_48><loc_32></location>R FOT = N //epsilon1 2 EA , (1)</formula>
<text><location><page_6><loc_8><loc_23><loc_48><loc_29></location>where we estimate the detection efficiency per filter at /epsilon1 ≈ 0 . 8 based on the overall search for transients in the MDS fields. Thus, for a timescale of 0.5 hr we place a limit of R FOT /lessorsimilar 0 . 12 deg -2 d -1 , while for a timescale of ∼ day it is R FOT /lessorsimilar 2 . 4 × 10 -3 deg -2 d -1 (see Table 3).</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_23></location>Our limit on fast transients with a 0.5 hr timescale improves on existing searches with similar limiting magnitudes by a factor of 30 -50 (Figure 7) thanks to the much larger effective areal exposure. DLS ( EA ≈ 1 . 1 deg d) placed an upper limit of R FOT /lessorsimilar 6 . 5 events deg -2 d -1 for transients with a timescale of about 0 . 36 hr (Becker et al. 2004), while the Fornax galaxy cluster search ( EA ≈ 1 . 9 deg d) placed a limit of R FOT /lessorsimilar 3 . 3 deg -2 d -1 for transients with a timescale of about 0 . 55 hr (Rau et al. 2008). The ROTSE-III search ( EA ≈ 635 deg d) placed a limit of R FOT /lessorsimilar 5 × 10 -3 deg -2 d -1 for transients with a timescale of about 0 . 5 hr, but was significantly shallower (Rykoff et al. 2005).</text>
<text><location><page_6><loc_10><loc_5><loc_48><loc_6></location>The sky-projected rates do not take into account the differ-</text>
<text><location><page_6><loc_52><loc_31><loc_92><loc_41></location>ence in limiting magnitudes between various searches. Our survey limiting magnitude is about 1.3 mag shallower than the DLS search, but about 1.2 mag deeper than the Fornax cluster search, and about 5 mag deeper than the ROTSE-III search. This information is summarized in Table 3, and shown in the relevant three-dimensional phase-space of survey limiting magnitude, timescale, and sky-projected rate limit in Figure 7.</text>
<text><location><page_6><loc_52><loc_14><loc_92><loc_30></location>The survey depth impacts the inferred limits on volumetric rates. For example, for a population of fast transients with a fiducial absolute magnitude of -10 (comparable to the most luminous known novae; see §5), the limits on the volumetric rates 11 on a timescale of 0.5 hr are /lessorsimilar 13 Mpc -3 yr -1 for our survey, /lessorsimilar 1 . 3 × 10 2 Mpc -3 yr -1 for DLS, and /lessorsimilar 2 . 0 × 10 3 Mpc -3 yr -1 for Fornax. The small volume probed by the ROTSE-III search for transients with -10 mag ( d ≈ 3 Mpc) provides no real insight on the extragalactic population. For a fiducial absolute magnitude of -14 (comparable to the least luminous supernovae), the limits are /lessorsimilar 0 . 05 (PS1/MDS), /lessorsimilar 0 . 5 (DLS), and /lessorsimilar 7 . 8 (Fornax) Mpc -3 yr -1 ; at this peak</text>
<text><location><page_6><loc_52><loc_5><loc_92><loc_13></location>11 We assume a uniform underlying distribution of galaxies within the volume probed by each search, which is not strictly the case for the maximum distances associated with a peak magnitude of -10: 32 Mpc for our search, 56 Mpc for DLS, and 18 Mpc for Fornax. With the exception of the Fornax search, which was centered on the Fornax galaxy cluster at d ≈ 16 Mpc, this indicates that the actual limits from our survey and from DLS are subject to the underlying non-uniform galaxy distribution.</text>
<text><location><page_7><loc_8><loc_81><loc_48><loc_92></location>magnitude the maximal detection distances are large enough to uniformly sample the galaxy distribution (about 200 Mpc for our search; 360 Mpc for DLS, and 115 Mpc for Fornax). Finally, for a fiducial absolute magnitude of -24 (intermediate between long and short gamma-ray burst afterglows; see §5), the volumetric rate limits are /lessorsimilar 10 -6 (PS1/MDS), /lessorsimilar 6 × 10 -5 (DLS), and /lessorsimilar 3 × 10 -5 (Fornax) Mpc -3 yr -1 . The various limits are summarized in Table 3.</text>
<text><location><page_7><loc_8><loc_68><loc_48><loc_81></location>We note that the inferred limits at -10 and -14 mag are substantially higher than the volumetric rate of supernovae, ≈ 10 -4 Mpc -3 yr -1 (Li et al. 2011), and novae 12 , ≈ 0 . 1 Mpc -3 yr -1 ; similarly, the limits at -24 mag are at least two orders of magnitude larger than the on-axis GRB rate. This indicates that any source population of extragalactic fast optical transients with a timescale of 0.5 hr would have to be much more abundant than nova, supernova, or GRB progenitors, or produce multiple ( ∼ 10 2 -10 3 ) events per progenitor system to be detected with current surveys.</text>
<text><location><page_7><loc_8><loc_55><loc_48><loc_67></location>Utilizing the B -band luminosity density in the local universe ( ≈ 1 . 4 × 10 8 LB , /circledot Mpc -3 ) the upper limits on the volumetric rates at -10 mag can be recast as /lessorsimilar 9 . 2 × 10 2 (PS1/MDS) and /lessorsimilar 9 . 3 × 10 3 (DLS) yr -1 per 10 10 LB , /circledot ; for the Fornax survey, which targeted a galaxy cluster environment, the limit is /lessorsimilar 350 yr -1 per 10 10 LB , /circledot (Rau et al. 2008). For a fiducial brightness of -14 mag, the limits are /lessorsimilar 3 . 6 (PS1/MDS), /lessorsimilar 36 (DLS), and /lessorsimilar 75 yr -1 per 10 10 LB , /circledot ; the latter value is from Rau et al. (2008).</text>
<text><location><page_7><loc_8><loc_45><loc_48><loc_55></location>For the fiducial timescale of about 1 d (the upper bound of our survey), the volumetric rate limits are /lessorsimilar 0 . 3 ( -10 mag), /lessorsimilar 10 -3 ( -14 mag), and /lessorsimilar 4 × 10 -8 ( -24 mag) Mpc -3 yr -1 . The rates at -14 and -24 mag are of interest since they are only an order of magnitude larger than the supernova and GRB rates, indicating that an expansion of our search might yield interesting limits on the rate of fast transients with ∼ 1 d timescale.</text>
<section_header_level_1><location><page_7><loc_12><loc_42><loc_46><loc_44></location>5. THEORETICAL EXPECTATIONS FOR THE LUMINOSITY OF FAST OPTICAL TRANSIENTS</section_header_level_1>
<text><location><page_7><loc_8><loc_19><loc_48><loc_41></location>In the discussion above we used fiducial peak absolute magnitudes of -10, -14, and -24 mag to infer limits on the volumetric rates of extragalactic fast optical transients. These particular values were chosen to match the most luminous classical novae, the least luminous supernovae, and typical long and short GRB afterglows, respectively. However, it is instructive to explore what peak luminosities are expected for sources with a fiducial timescale of ∼ 0 . 5 hr to ∼ day for a range of potential mechanisms. We defer an exhaustive investigation of this question to future work and focus on the basic arguments here. At the most basic level, we note that explosive sources with a characteristic velocity of ∼ 10 4 kms -1 that emit thermally with a peak in the optical ( T ∼ 10 4 K) will be limited to an absolute magnitude of /greaterorsimilar -14; only sources undergoing relativistic expansion (e.g., GRBs) may significantly exceed this limits. We expand on this general theme below.</text>
<text><location><page_7><loc_8><loc_10><loc_48><loc_19></location>We first investigate the possibility of unusually fast and luminous classical novae. In general, novae follow the maximum magnitude versus rate of decline (MMRD) relation (e.g., della Valle & Livio 1995), which indicates that faster novae are also more luminous. However, the MMRD relation flattens to a maximal observed value of about -10 mag for timescales shorter than a few days (della Valle & Livio 1995).</text>
<text><location><page_7><loc_52><loc_76><loc_92><loc_92></location>Indeed, the shortest observed timescales (quantified as t 2 or t 3, the timescales to decline by 2 or 3 magnitudes, respectively) are ≈ 3 -5 d (Czekala et al. 2013). Theoretical models point to maximal magnitudes of about -10 and timescales as short as t 2 ≈ 1 d (Yaron et al. 2005). Moreover, the fraction of novae that achieve such high peak luminosity and rapid fading is /lessorsimilar 1% (e.g., Shafter et al. 2009), leading to a rate of /lessorsimilar 10 -3 Mpc -3 yr -1 , orders of magnitude below the upper bounds for -10 mag derived from the various searches (e.g., 0.3 Mpc -3 yr -1 for our search; Table 3). Thus, novae are not expected to contribute a population of luminous and fast extragalactic transients for current surveys.</text>
<text><location><page_7><loc_52><loc_68><loc_92><loc_75></location>We next explore the generic case of an explosive event with ejecta that cool adiabatically due to free expansion ( R = v ej t ); i.e., a supernova-like explosion, but lacking internal heating. In this case, with the expansion driven by radiation pressure and using the standard diffusion approximation (Arnett 1982), the luminosity is given by:</text>
<formula><location><page_7><loc_61><loc_63><loc_92><loc_67></location>L ( t ) = L 0 exp [ -t τ d , 0 -t 2 √ 2 τ d , 0 τ h ] , (2)</formula>
<text><location><page_7><loc_52><loc_38><loc_92><loc_63></location>where τ h = R 0 / v ej is the initial hydrodynamic timescale, τ d , 0 = B κ M ej / cR 0 is the initial diffusion timescale, κ is the opacity (for simplicity we use κ = κ es = 0 . 4 cm 2 g -1 ), M ej is the ejecta mass, B ≈ 0 . 07 is a geometric factor, and L 0 = E 0 /τ d , 0 is the initial luminosity. A fiducial timescale of 0.5 hr requires R 0 /lessorsimilar 0 . 5hr × v ej ≈ 2 × 10 12 ( v ej / 10 4 kms -1 ) cm; i.e., more extended sources are unlikely to produce transients with a duration as short as ∼ 0 . 5 hr. Using the appropriate size limit we find that for a source with L (0 . 5hr) / L 0 ∼ 0 . 1 the resulting peak bolometric magnitude ranges from about -8 mag (for a white dwarf with R 0 ≈ 10 9 cm) to about -15 mag (for a star with R 0 ≈ 10 R /circledot ). The resulting characteristic effective temperature ranges from T eff = ( L 0 / 4 π R 2 σ ) 1 / 4 ≈ 2 . 2 × 10 4 K (for R 0 ≈ 10 9 cm) to ≈ 1 . 1 × 10 5 K(for R 0 ≈ 10 R /circledot ), or a peak in the ultraviolet. This means that the peak absolute magnitude in gr will be a few magnitudes fainter, corresponding to the faint end of the magnitude range considered in the previous section (i.e., /greaterorsimilar -14 mag).</text>
<text><location><page_7><loc_52><loc_5><loc_92><loc_38></location>The luminosity can be enhanced by appealing to an internal energy source, for example radioactive heating as in the case of Type I supernovae. In this scenario the luminosity is maximized if the radioactive decay timescale is wellmatched to the diffusion timescale, τ r ≈ τ d ≈ B κ M ej / cR ≈ 0 . 5 hr. The fast timescale investigated here requires different radioactive material than 56 Ni, which powers the optical light curves of Type I supernovae, since the latter has τ r ≈ 8 . 8 d. Instead, a better match may be provided by the radioactive decay of r -process elements, for which there is a broad range of timescales (e.g., Li & Paczy'nski 1998; Metzger et al. 2010). Using a timescale of 0.5 hr we find that the required ejecta mass is low, M ej ≈ 2 × 10 -6 ( v ej / 10 4 km / s) 2 M /circledot . For radioactive heating the total energy generation rate is /epsilon1 ( t ) = ( f M ej c 2 /τ r )e -t / τ r , where f /lessmuch 1 is an efficiency factor, likely in the range of 10 -6 -10 -5 (Metzger et al. 2010). For the ejecta mass inferred above we find /epsilon1 ( t ) ≈ 10 45 f erg s -1 , and hence L ∼ 10 40 erg s -1 (for f ∼ 10 -5 ), or an absolute magnitude of about -12 mag. As in the discussion above, the spectrum will peak in the ultraviolet, and the optical emission ( g -band) will be dimmer by about 2 mag (i.e., to a peak of about -10 mag). For a fiducial timescale of ∼ day, the peak absolute magnitude is larger, reaching ≈ -14 mag in the optical. This is essentially the 'kilonova' model invoked for compact</text>
<figure>
<location><page_8><loc_12><loc_47><loc_88><loc_92></location>
<caption>FIG. 8.- Limits on the volumetric rate of extragalactic fast optical transients as a function of timescale from our PS1/MDS survey and from the literature (Becker et al. 2004; Rykoff et al. 2005; Lipunov et al. 2007; Rau et al. 2008). For each survey we provide limits for a range of fiducial absolute magnitudes (top to bottom) of -10 (most luminous novae), -14 (least luminous SNe), -20 (on-axis short GRB), and -27 (on-axis long GRB). Also marked are the actual volumetric rates of luminous novae, supernovae, binary neutron star mergers, and on-axis GRBs. The inferred limits from all existing surveys are orders of magnitude larger than the known volumetric rates, indicating that large new populations of currently-unknown astrophysical fast transients are required for detectability, or that much larger surveys are essential to produce meaningful limits. The single night and full year limits for the LSST main survey indicate that this survey should detect some fast optical transients (the LSST estimates have been shifted along the time axis for clarity).</caption>
</figure>
<text><location><page_8><loc_8><loc_18><loc_48><loc_36></location>object binary mergers (Li & Paczy'nski 1998; Metzger et al. 2010; Berger et al. 2013), with a predicted volumetric rate of ∼ 10 -8 -10 -5 Mpc -3 yr -1 (e.g., Abadie et al. 2010). We note, however, that recent opacity calculations for r -process material indicate that the actual peak brightness in the optical bands will be an order of magnitude fainter than suggested by the above calculation (Barnes & Kasen 2013). A similar scenario is the thermonuclear .Ia supernova model with heating by radioactive α -chain elements (Ca, Ti), leading to a peak optical brightness of ∼ -15 mag, but with a longer timescale of ∼ week even for low helium shell masses (Shen et al. 2010). Thus, models with radioactive heating are limited to ∼ -10 mag ( ∼ 0 . 5 hr) or ∼ -14 mag ( ∼ day).</text>
<text><location><page_8><loc_8><loc_5><loc_48><loc_19></location>An alternative internal energy source is the spin-down of a newly-born millisecond magnetar (Kasen & Bildsten 2010; Woosley 2010). In this scenario the maximal luminosity is achieved when the magnetar spin-down timescale, τ sd ≈ 4 × 10 3 ( B / 10 14 G) -2 ( P / 1ms) 2 s, is comparable to the diffusion timescale, τ d ≈ 0 . 5 hr; here B is the magnetic field strength and P is the initial rotation period. The available rotational energy is large, E rot ≈ 2 × 10 52 ( P / 1ms) -2 erg, and therefore a magnetar engine can in principle produce extremely luminous fast transients, with a peak brightness of</text>
<text><location><page_8><loc_52><loc_15><loc_92><loc_37></location>L ≈ E rot t sd / t 2 d ≈ 5 × 10 48 erg s -1 . However, since τ d ≈ 0 . 5 hr requires a low ejecta mass of ∼ 10 -6 M /circledot , this scenario necessitates magnetar birth with negligible ejected mass, and results in a relativistic outflow; this is essentially the scenario invoked in magnetar models of gamma-ray bursts. It is not obvious in this scenario whether the large energy reservoir will be emitted in the optical band, or primarily at X-ray/ γ -ray energies. Thus, the optical signature may still be rather weak, and perhaps dominated instead by interaction with the ambient medium (i.e., an afterglow). Overall, the estimated magnetar birth rate is ∼ 10% of the core-collapse supernova rate (Kouveliotou et al. 1998), while the GRB rate is /lessorsimilar 1% of the supernova rate (Wanderman & Piran 2010). This indicates that any magnetar-powered fast optical transients will have a rate orders of magnitude below the upper limits inferred by the existing searches.</text>
<text><location><page_8><loc_52><loc_11><loc_92><loc_15></location>Finally, luminous rapid optical transients can be produced by circumstellar interaction of a relativistic outflow, such as an on-axis 13 GRB afterglow. The typical absolute magnitude</text>
<text><location><page_9><loc_8><loc_65><loc_48><loc_92></location>of a short-duration GRB afterglow at δ t ≈ 0 . 5 hr is about -20 mag (based on a typical apparent brightness of about 22 mag and a typical redshift of z ≈ 0 . 5; Berger 2010). At this fiducial luminosity the inferred volumetric upper limits from the fast optical transient searches are /lessorsimilar 3 × 10 -5 (PS1/MDS), /lessorsimilar 3 × 10 -4 (DLS), /lessorsimilar 2 × 10 -3 (Fornax), and /lessorsimilar 7 × 10 -4 (ROTSE-III) Mpc -3 yr -1 . The actual volumetric rate of on-axis short GRBs is about 10 -8 Mpc -3 yr -1 (Nakar et al. 2006), at least 3 × 10 3 times lower than the limits reached by the surveys. For longduration GRBs the typical absolute magnitude at δ t ≈ 0 . 5 hr is much larger, about -27 mag (based on a typical apparent brightness of about 17 mag and a typical redshift of z ≈ 2; Kann et al. 2010). At this luminosity the volumetric upper limits for the various searches are 14 /lessorsimilar 10 -6 (PS1/MDS), /lessorsimilar 2 × 10 -5 (DLS), /lessorsimilar 1 × 10 -5 (Fornax), and /lessorsimilar 3 × 10 -7 (ROTSE-III) Mpc -3 yr -1 . The inferred on-axis volumetric rate (at z ∼ 2) is ∼ 10 -8 Mpc -3 yr -1 (Wanderman & Piran 2010). Thus, unless there is a substantial population of relativistic explosions that do not produce γ -ray emission, the existing limits are much too shallow in the context of on-axis GRB rates.</text>
<text><location><page_9><loc_8><loc_62><loc_48><loc_65></location>We therefore conclude based on the comparison to various potential models and populations that:</text>
<unordered_list>
<list_item><location><page_9><loc_11><loc_45><loc_48><loc_61></location>· non-relativistic fast optical transients with a timescale of ∼ 0 . 5 hr ( ∼ day) are generically limited to a peak optical brightness of ∼ -10 ( ∼ -14) mag even if they are powered by radioactive heating; revised opacities for r -process matter may reduce these values by about an order of magnitude. Similarly, unusually fast classical novae are not likely to reach peak magnitudes larger than about -10 mag. Overall, the limits achieved by existing searches at these luminosities (Table 3) are orders of magnitude higher than the known and expected event rates for such transients (e.g., /lessorsimilar 10 -3 Mpc -3 yr -1 for luminous novae).</list_item>
<list_item><location><page_9><loc_11><loc_30><loc_48><loc_43></location>· relativistic sources such as on-axis GRBs or transients powered by the spin-down of millisecond magnetars can produce much larger luminosities on a timescale of ∼ 0 . 5 hr, but the volumetric rate limits from the existing searches are still orders of magnitude larger than the very low known or anticipated rates of relativistic explosions. A possible exception is a fast transient recently discovered by Cenko et al. (2013), although with a timescale of ∼ few d.</list_item>
</unordered_list>
<text><location><page_9><loc_8><loc_11><loc_48><loc_30></location>In Figure 8 we summarize this information by comparing the volumetric rate limits from the various surveys at fiducial peak absolute magnitudes of -10, -14, -20 (short GRB), and -27 (long GRB) to the rates of known transients (e.g., novae, supernovae, neutron star binary mergers, on-axis short and long GRBs). The results demonstrate that the existing survey limits are orders of magnitude above the anticipated event rates based on the known classes. This indicates that much larger surveys are essential to robustly explore the transient sky on the ∼ hour -day timescale. In addition, we stress that given the low expected luminosity for non-relativistic fast transients ( /greaterorsimilar -14), a profitable strategy is to systematically target nearby galaxies (as was done for example in the Fornax Cluster search; Rau et al. 2008) rather than image wide blank</text>
<text><location><page_9><loc_52><loc_83><loc_92><loc_92></location>fields. On the other hand, searches for the rare but highly luminous relativistic fast transients are best focused on covering wide fields (preferably ∼ all -sky) at the expense of survey depth since the projected rate is at most ∼ few per sky per day, while the limiting factor for distance coverage is the use of optical filters (limited to z ∼ 6) rather than depth.</text>
<section_header_level_1><location><page_9><loc_61><loc_81><loc_82><loc_82></location>6. IMPLICATIONS FOR LSST</section_header_level_1>
<text><location><page_9><loc_52><loc_55><loc_92><loc_80></location>Our search for fast optical transients proved highly effective at identifying foreground events without the need for expensive follow-up observations . This is unlike previous searches such as DLS and Fornax. This has been accomplished thanks to the use of dual-filter observations spanning a timescale of about 0.5 hr, with each observation composed of multiple exposures that help to eliminate individual spurious detections. In addition, the availability of color information ( g -r in this case), coupled with the time baseline allowed us to effectively identify main-belt asteroids near the stationary point. Similarly, deep multi-band templates allowed us to unambiguously identify M dwarf counterparts for all detected flares to a distance of at least ∼ 1 . 2 kpc and with spectral types extending to about M9. With this information we were able to account for all the fast optical transients found in the survey, and to place the deepest limit to date on the rate of extragalactic fast transients. Our results are reassuring given that follow-up spectroscopy in the era of on-going and future large surveys is in limited supply.</text>
<text><location><page_9><loc_52><loc_22><loc_92><loc_55></location>With this in mind, it is instructive to consider our results in the context of the anticipated LSST survey strategy (Ivezic et al. 2008). The LSST main survey is intended to cover a total of ≈ 18 , 000 deg 2 , accounting for about 90% of the observing time. Each pointing will consist of two visits per field with a separation of about 15 -60 min, with each visit consisting of a pair of 15 s exposures with a 3 σ depth of ∼ 24 . 5 mag (or about 23.5 mag at 10 σ ). It is anticipated that a total of about 3000 deg 2 will be imaged on any given night, potentially with more than one filter, leading to an areal exposure of about 62 deg 2 d in a single night, or about 15 1 . 6 × 10 4 deg 2 d per year, for the fiducial timescale of 0.5 hr. This survey will therefore achieve a sky projected rate limit (95% confidence level) of 0 . 07 deg -2 d -1 in a single night and 3 × 10 -4 deg -2 d -1 in a year. The latter value represents a factor of 400 times improvement relative to our PS1/MDS search. In terms of a volumetric rate, the greater depth achieved by LSST will probe rates of ≈ 9 × 10 -3 Mpc -3 yr -1 for -10 mag and ≈ 3 × 10 -5 Mpc -3 yr -1 for -14 mag for 1 year of operations. At a typical brightness of short and long GRBs, the rate will be about 2 × 10 -9 Mpc -3 yr -1 (limited to z /lessorsimilar 6 by the zy filters), a few times deeper than the actual onaxis GRB rate, indicating that LSST is likely to detect such events.</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_22></location>The LSST data set can form the basis for a search similar to the one described here, with the main difference that contemporaneous color information may not be available. However, with a S / N ≈ 10 cut as imposed here, the observations will reach about 1 mag deeper than our search, indicating that they will uncover M dwarf flares to distances that are only about 50% larger than those found here (i.e., generally /lessorsimilar 2 kpc). At the same time, the LSST templates will reach a comparable depth to the PS1/MDS templates after only ∼ 10 visits per field per filter, i.e., within a year of when the survey</text>
<text><location><page_10><loc_8><loc_84><loc_48><loc_92></location>commences. Thus, we conclude that it will be simple to identify M dwarf counterparts in the template images for essentially all M dwarf flares, negating the need for follow-up spectroscopy. This approach will effectively eliminate the largest known contaminating foreground for ∼ hour timescales.</text>
<text><location><page_10><loc_8><loc_69><loc_48><loc_85></location>Similarly, the time baseline of 15 -60 min per field will allow for astrometric rejection of most asteroids even near the stationary point, although as we found in our PS1/MDS search some asteroids have negligible motions that are indistinguishable from the astrometric scatter of field sources (2 / 8 in our search). Such asteroids can in principle be recognized using color information, but this is not likely to be available with LSST. On the other hand, a constant brightness level between visits 15 -60 min apart, combined with a location near the ecliptic plane and in particular with solar elongation of ∼ 130 deg, will be indicative of an asteroid origin. Such fields can simply be avoided in searches for fast transients.</text>
<text><location><page_10><loc_8><loc_57><loc_48><loc_69></location>Thus, the two primary contaminants for extragalactic fast optical transient searches (M dwarf flares and asteroids near the stationary point) will be identifiable with the LSST baseline survey strategy. Furthermore, we anticipate based on the comparison to known transient classes that with a year of operations LSST is unlikely to reveal large numbers of extragalactic fast transients unless they result from a source population that far exceeds classical novae, supernova progenitors, or compact object binaries (Figure 8).</text>
<section_header_level_1><location><page_10><loc_22><loc_54><loc_35><loc_55></location>7. CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_8><loc_36><loc_48><loc_53></location>Wepresent a search for fast optical transients on a timescale of about 0.5 hr to 1 d in consecutive g P1 r P1 observations of the PS1/MDS fields, by requiring a detection in both filters with no additional detections on preceding or subsequent nights. The search yielded 19 astrophysical transients, of which 8 events that lack quiescent counterparts are identified as mainbelt asteroids near the stationary point of their orbits, while the remaining 11 transients are identified as flares from M5M9 dwarf stars at distances of about 0 . 2 -1 . 2 kpc. The flare properties are generally similar to those from M dwarfs studied in the SDSS Stripe 82, although our sample extends to later spectral types and to much greater vertical distances from the Galactic plane.</text>
<text><location><page_10><loc_8><loc_20><loc_48><loc_36></location>A key result of our search is a limit on the sky-projected rate of extragalactic fast transients of R FOT /lessorsimilar 0 . 12 deg -2 d -1 ( ∼ 0 . 5 hr) that is about 30 -50 time deeper than previous limits; the limit for a timescale of ∼ day is /lessorsimilar 2 . 4 × 10 -3 deg -2 d -1 . The upper bounds on the volumetric rates at fiducial absolute magnitudes of -10, -14, and -24 mag are likewise an order of magnitude deeper than from previous searches (Table 3). With an additional 3 years of PS1/MDS data in hand we can improve these estimates by a factor of a few. We also anticipate additional M dwarf flare detections that will allow us to better characterize the distribution of flare properties, as well as the properties of flaring M dwarfs in general.</text>
<text><location><page_10><loc_8><loc_9><loc_48><loc_20></location>To guide the conversion from our sky-projected rate to volumetric rates, and to motivate future searches for fast transients, we also explore the expected luminosities of such transients for a range of physically motivated models. We find that non-relativistic fast transients are generally limited to about -10 mag for a timescale of ∼ 0 . 5 hr and -14 mag for a timescale of ∼ day even if powered by radioactive decay.</text>
<text><location><page_10><loc_52><loc_68><loc_92><loc_92></location>This is simply a reflection of the low ejecta mass required to achieve a rapid diffusion timescale of /lessorsimilar day. Such low luminosity events are best explored through targeted searches of galaxies in the local universe, with anticipated event rates of /lessorsimilar 10 -4 Mpc -3 yr -1 (Figure 8). A separate class of relativistic fast transients (on-axis GRBs, magnetar engines) can exceed -25 mag on a timescale of ∼ 0 . 5 hr, but they are exceedingly rare, with anticipated volumetric rates of ∼ 10 -8 Mpc -3 yr -1 (or sky-projected rates of a few per sky per day). Such event rates are orders of magnitude below the level probed in current searches, but will be reached by the LSST main survey in a full year. Another strategy to find such events is a shallower search of a wider sky area than is planned for LSST. We note that a broad exploration of the various mechanisms that can power optical transients is essential to guide and motivate future searches for fast extragalactic transients. The initial investigation performed here already places clear bounds on the luminosities and rates for a wide range of mechanisms.</text>
<text><location><page_10><loc_52><loc_38><loc_92><loc_67></location>Finally, since the PS1/MDS survey is currently the only close analogue to the main LSST survey in terms of depth, cadence, and choice of filters, we use the results of our search to investigate the efficacy of fast transient searches with LSST. We demonstrate that the main contaminants recognized here should be identifiable with the LSST survey strategy, without the need for expensive follow-up spectroscopy. Namely, asteroids near the stationary point can be recognized through a larger than average astrometric shift, a constant brightness in visits separated by 15 -60 min, and an expected location near the ecliptic plane with solar elongation of 130 deg. LSST may not provide near-simultaneous color information, which can serve as an additional discriminant for asteroids. For M dwarf flares we show that LSST's increased sensitivity will probe a larger volume of the Galaxy, but the correspondingly deeper templates will still allow for the identification of quiescent counterparts in essentially all cases. The PS1/MDS survey demonstrates that with multi-band photometry it is possible to identify M dwarf flares without the need for follow-up spectroscopy (as in previous searches; e.g., Kulkarni & Rau 2006). This result indicates that extragalactic fast optical transients should be able to pierce the veil of foreground flares.</text>
<text><location><page_10><loc_52><loc_12><loc_92><loc_34></location>We thank Mario Juric for helpful information on the planned LSST survey strategy. E. B. acknowledges support for this work from the National Science Foundation through Grant AST-1008361. PS1 has been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS1 Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, and the Las Cumbres Observatory Global Telescope Network, Incorporated, the National Central University of Taiwan, and the National Aeronautics and Space Administration under Grant NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate.</text>
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<table>
<location><page_12><loc_9><loc_35><loc_91><loc_88></location>
<caption>TABLE 1 FAST OPTICAL TRANSIENTS WITH QUIESCENT COUNTERPARTS</caption>
</table>
<text><location><page_12><loc_9><loc_33><loc_90><loc_35></location>NOTE. - Properties of the fast transients that exhibit quiescent counterparts, as well as multi-band photometry of the counterparts. We find that all 11 transients arise from M dwarfs, and provide the inferred spectral types and distances of these sources.</text>
<table>
<location><page_12><loc_15><loc_13><loc_86><loc_26></location>
<caption>TABLE 2 FAST OPTICAL TRANSIENTS LACKING QUIESCENT COUNTERPARTS</caption>
</table>
<text><location><page_12><loc_15><loc_11><loc_85><loc_13></location>NOTE. - Properties of the fast transients that lack quiescent counterparts. We find that all 8 transients exhibit the colors and expected ecliptic coordinates of main-belt asteroids near the stationary point of their orbits.</text>
<table>
<location><page_13><loc_11><loc_76><loc_90><loc_88></location>
<caption>TABLE 3 SURVEYS FOR FAST OPTICAL TRANSIENTS WITH A TIMESCALE OF ∼ 0 . 5 HR</caption>
</table>
<text><location><page_13><loc_11><loc_73><loc_89><loc_75></location>NOTE. - Survey parameters and resulting limits on the projected sky rates and volumetric rates of extragalactic fast optical transients with a timescale of ∼ 0 . 5 hr.</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We present a search for fast optical transients ( τ ∼ 0 . 5hr -1d) using repeated observations of the PanSTARRS1 Medium-Deep Survey (PS1/MDS) fields. Our search takes advantage of the consecutive g P1 r P1 observations (16 . 5 min in each filter), by requiring detections in both bands, with non-detections on preceding and subsequent nights. We identify 19 transients brighter than 22 . 5 AB mag (S / N /greaterorsimilar 10). Of these, 11 events exhibit quiescent counterparts in the deep PS1/MDS templates that we identify as M4-M9 dwarfs at d ≈ 0 . 2 -1 . 2 kpc. The remaining 8 transients lack quiescent counterparts, exhibit mild but significant astrometric shifts between the g P1 and r P1 images, colors of ( g -r )P1 ≈ 0 . 5 -0 . 8mag, non-varyinglight curves, and locations near the ecliptic plane with solar elongations of about 130 deg, which are all indicative of main-belt asteroids near the stationary point of their orbits. With identifications for all 19 transients, we place an upper limit of R FOT( τ ∼ 0 . 5hr) /lessorsimilar 0 . 12 deg -2 d -1 (95% confidence level) on the sky-projected rate of extragalactic fast transients at /lessorsimilar 22 . 5 mag, a factor of 30 -50 times lower than previous limits; the limit for a timescale of ∼ day is R FOT /lessorsimilar 2 . 4 × 10 -3 deg -2 d -1 . To convert these sky-projected rates to volumetric rates, we explore the expected peak luminosities of fast optical transients powered by various mechanisms, and find that nonrelativistic events are limited to M ≈ -10 mag ( M ≈ -14 mag) for a timescale of ∼ 0 . 5 hr ( ∼ day), while relativistic sources (e.g., gamma-ray bursts, magnetar-powered transients) can reach much larger luminosities. The resulting volumetric rates are /lessorsimilar 13 Mpc -3 yr -1 ( M ≈ -10 mag), /lessorsimilar 0 . 05 Mpc -3 yr -1 ( M ≈ -14 mag) and /lessorsimilar 10 -6 Mpc -3 yr -1 ( M ≈ -24 mag), significantly above the nova, supernova, and GRB rates, respectively, indicating that much larger surveys are required to provide meaningful constraints. Motivated by the results of our search we discuss strategies for identifying fast optical transients in the LSST main survey, and reach the optimistic conclusion that the veil of foreground contaminants can be lifted with the survey data, without the need for expensive follow-up observations. Subject headings: stars: flare, asteroids: general, supernovae: general, novae, surveys",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "For nearly a century, optical observations aimed at the discovery and study of astrophysical transients have largely focused on events with durations of days to months. This is mainly due to a fortuitous match with the timescales of the most common extragalactic events (novae and supernovae), whose typical luminosities and intrinsic rates require coverage of large numbers of galaxies and/or blank sky areas, leading to a natural search cadence of several days. Thus, novae have a much higher intrinsic rate than supernovae ( ∼ 2 . 2 yr -1 per 10 10 LK , /circledot versus ∼ 2 × 10 -3 yr -1 per 10 10 LK , /circledot , respectively; e.g., Williams & Shafter 2004; Li et al. 2011), but supernovae are significantly more luminous than novae ( ∼ -18 mag versus ∼ -8 mag, respectively; Gallagher & Starrfield 1978; Filippenko 1997). As a result, for a given survey limiting magnitude tens of novae can be discovered per year by targeting a few nearby galaxies with a cadence of few days, while discovering a similar number of supernovae requires monitoring of ∼ 10 4 galaxies (or hundreds of deg 2 ), thereby necessitating a similar cadence of several days; for the purpose of sheer discovery rate, a faster cadence is not profitable for nova and supernova searches. Over the past few decades such surveys have been highly successful at discovering about one hundred novae and supernovae per year (e.g., Williams & Shafter 2004; Leaman et al. 2011). The advent of large format cameras on dedicated wide-field telescopes, coupled with serendipitous discoveries of transients outside of the traditional nova and supernova luminosity and timescale ranges, has opened up a new discovery space for astrophysical transients. By repeatedly targeting the same fields, such surveys are in principle capable of exploring a wide range of timescales, from the duration of single exposures (i.e., minutes) to years. In practice, most surveys are still primarily focused on supernovae (driven to a large extent by Type Ia supernova cosmology at increasingly larger redshifts), and therefore cover wider fields to greater depth at the expense of a faster temporal cadence to maximize the supernova discovery rate while preserving adequate light curve coverage. Still, some surveys have been utilized to perform initial searches for fast optical transients (FOTs) on timescales as short as ∼ 0 . 5 hr. Clearly, the effective areal exposure of such searches (i.e., the product of survey area and exposure time) becomes progressively smaller at faster cadence as sky coverage has to be sacrificed for repeated short-cadence observations. In this context, the Deep Lens Survey (DLS) was utilized to search for FOTs on a timescale of about 1300 s to a depth of B ≈ 23 . 8 mag (with a total exposure of 1.1 deg 2 d) and led to an upper limit on the extragalactic sky-projected rate of R FOT /lessorsimilar 6 . 5 deg -2 d -1 (95% confidence level; Becker et al. 2004). The DLS search uncovered three fast transients, which were shown to be flares from Galactic M dwarf stars (Becker et al. 2004; Kulkarni & Rau 2006). A search for transients with a timescale of /greaterorsimilar 0 . 5 hr to a depth of about 17.5 mag (with a total exposure of 635 deg 2 d) using the Robotic Optical Transient Search Experiment-III (ROTSE-III) yielded a limit on the extragalactic rate of R FOT /lessorsimilar 5 × 10 -3 deg -2 d -1 , and uncovered a single M dwarf flare (Rykoff et al. 2005); a similar search with the MASTER telescope system yielded comparable limits (Lipunov et al. 2007), and two uncharacterized candidate fast transients (Gorbovskoy et al. 2013). Similarly, a targeted search for transients on a timescale of about 0.5 hr and to a depth of B ≈ 21 . 3 mag in the Fornax galaxy cluster (with a total exposure of 1.9 deg 2 d) placed a limit on the extragalactic rate of R FOT /lessorsimilar 3 . 3 deg -2 d -1 (Rau et al. 2008). Two fast transients were detected in this search, both shown to be M dwarf flares (Rau et al. 2008). At the bright end, the 'Pi of the Sky' project placed a limit on transients brighter than 11 mag with a duration of /greaterorsimilar 10 s of R FOT /lessorsimilar 5 × 10 -5 deg -2 d -1 (Sokołowski et al. 2010). Fast optical transients have also been found serendipitously by other surveys, but they have generally been shown to be Galactic in origin 8 . A notable exception is the transient PTF11agg (Cenko et al. 2013), which faded by about 1.2 mag in 5.3 hours, and 3.9 mag in 2.2 d, and was accompanied by radio emission that may be indicative of relativistic expansion (although the distance of this transient is not known, thereby complicating its interpretation). We note that such an event, while fast compared to the general nova and supernova population, is still of longer duration than the timescales probed by the DLS and Fornax searches, as well as the Pan-STARRS1 search we describe here. Thus, the existing searches for extragalactic FOTs have mainly raised the awareness that the foreground of M dwarf flares is large, with an estimated allsky rate of ∼ 10 8 yr -1 at a limiting magnitude of ∼ 24 mag (Becker et al. 2004; Kulkarni & Rau 2006). Here, we present a search for fast optical transients with an effective timescale of about 0.5 hr to 1 d and to a depth of about 22.5 mag in the first 1 . 5 years of data from the PanSTARRS1 Medium-Deep Survey (PS1/MDS). This search uncovered a substantial sample of 19 fast transients, both with and without quiescent counterparts. We describe the survey strategy and selection criteria in §2. In §3 we summarize the properties of the 19 detected transients and classify them using a combination of color information, astrometry, sky location, and the properties of quiescent counterparts (when detected). With a unique identification of all 19 transients as Solar system or Galactic in origin, we place a limit on the rate of extragalactic fast transients that is 30 -50 times better than the limits from previous searches (§4). We further investigate for the first time the limits on the volumetric rates of FOTs from our survey and previous searches using the survey limiting magnitudes and fiducial transients luminosities. In §5 we expand on this point and discuss the expected peak luminosities of FOTs for a range of physically motivated models. Finally, since our search is the first one to utilize observations that are similar to the Large Synoptic Survey Telescope (LSST) main survey strategy, we conclude by drawing implications for fast optical transient searches in the LSST data §6.",
"pages": [
1,
2
]
},
{
"title": "2.1. PS1 Survey Summary",
"content": "The PS1 telescope, located on Mount Haleakala, is a highetendue wide-field survey instrument with a 1.8-m diameter primary mirror and a 3 . 3 · diameter field-of-view imaged by an array of sixty 4800 × 4800 pixel detectors, with a pixel scale of 0 . 258 '' (Kaiser et al. 2010; Tonry & Onaka 2009). The observations are obtained through five broad-band filters ( g P1 r P1 i P1 z P1 y P1), with some differences relative to the Sloan Digital Sky Survey (SDSS); the g P1 filter extends 200 Å redward of g SDSS to achieve greater sensitivity and lower systematics for photometric redshifts, while the z P1 filter terminates at 9300 Å, unlike z SDSS which is defined by the detector response (Tonry et al. 2012). PS1 photometry is in the 'natural' system, m = -2 . 5log( F ν ) + m ' , with a single zero-point adjustment ( m ' ) in each band to conform to the AB magnitude scale. Magnitudes are interpreted as being at the top of the atmosphere, with 1.2 airmasses of atmospheric attenuation included in the system response function (Tonry et al. 2012). The PS1 Medium-Deep Survey (MDS) consists of 10 fields (each with a single PS1 imager footprint) observed on a nearly nightly basis by cycling through the five filters in 3 -4 nights to a typical 5 σ depth of ∼ 23 . 3 mag in g P1 r P1 i P1 z P1, and ∼ 21 . 7 mag in y P1. The MDS images are processed through the Image Processing Pipeline (IPP; Magnier 2006), which includes flat-fielding ('de-trending'), a flux-conserving warping to a sky-based image plane, masking and artifact removal, and object detection and photometry. For the fast transient search described here we produced difference images from the stacked nightly images using the photpipe pipeline (Rest et al. 2005) running on the Odyssey computer cluster at Harvard University.",
"pages": [
2
]
},
{
"title": "2.2. A Search for Fast Optical Transients",
"content": "For the purpose of detecting fast optical transients we take advantage of the consecutive MDS g P1 r P1 observations, with eight 113 s exposures in each filter providing a total timespan of about 33 min for a full sequence. We carry out the search using the stacked g P1 and r P1 images from each visit through image subtraction relative to deep multi-epoch templates, and subsequently utilize the individual exposures to construct light curves; representative discovery and template images are shown in Figure 1 and light curves are shown in Figure 2. We limit the timescale of the transients to /lessorsimilar 1 d by further requiring no additional detections in the g P1 r P1 i P1 z P1 filters on preceding and subsequent nights (extending to ± 5 nights). To ensure that this constraint is met we only perform our search on the subset of MDS data for which consecutive nights of observations are available. In the first 1 . 5 years of data we searched a total of 277 nights of g P1 r P1 observations across the 10 MDS fields, leading to a total areal exposure of 40.4 deg 2 d for a timescale of 0.5 hr and 1940 deg 2 d for a timescale of 1 d. To select transients in the g P1 r P1 difference images we utilize a signal-to-noise ratio threshold 9 of S / N = 10, with resulting limiting magnitudes of g P1 ≈ 22 . 7 mag and r P1 ≈ 22 . 4 mag. We additionally require sources in the g P1 r P1 difference images to astrometrically match within 0 . 35 '' , corresponding to a 1.2 pixel radius around each detection (this is about 6 times the typical astrometric error; see §3.1). Using these cuts we find a total of 227 candidates, which were visually inspected by one of us (C.L.) leading to a final list of 19 sources that were further validated by two of us (E.B. and R.C.); the remaining 208 events were predominantly spurious detections near saturated stars.",
"pages": [
2,
3
]
},
{
"title": "3. THE PROPERTIES OF PAN-STARRS1 FAST OPTICAL TRANSIENTS",
"content": "Using the procedure described in the previous section we found 19 genuine fast transients, spanning a brightness range of g P1 ≈ 18 . 5 -22 . 7 mag and r P1 ≈ 18 . 8 -22 . 4 mag. In Figure 3 we show the g P1 versus ( g -r )P1 color-magnitude diagram for all 19 sources; the photometry is summarized in Tables 1 and 2. The faint end of the distribution is determined by our requirement of S / N /greaterorsimilar 10 in the subtractions of the individual g P1 r P1 nightly stacks (16.5 min in each filter) from the deep templates. On the other hand, the bright end of the observed distribution is about 2 mag fainter than the saturation limit of our images, indicating a genuine dearth of apparently bright fast transients in our search area. The ( g -r )P1 tections by measuring the flux and uncertainty at random positions in the difference images in the same manner as the transient flux, and then determining a correction factor which leads to a distribution with a reduced χ 2 of unity. colors span a wide range of about -2 . 0 to + 0 . 9 mag, but we stress that for transients that rapidly vary in brightness within the time-span of each observation (e.g., Figure 2), the non- simultaneous g P1 and r P1 measurements do not reflect the true instantaneous colors.",
"pages": [
3,
4
]
},
{
"title": "3.1. Fast Transients Lacking Quiescent Counterparts",
"content": "Of the 19 fast transients discovered in our search, 8 events lack quiescent counterparts in any of the deep template images ( g P1 r P1 i P1 z P1) to typical limits of /greaterorsimilar 24 . 5 -25 mag. These sources are in principle a promising population of distant extragalactic transients with undetected host galaxies. We utilize a combination of color information, the 33 min time-span of the g P1 r P1 observations, and the sky locations to investigate the nature of these sources. We first note that all 8 sources span a narrow color range with ( g -r )P1 ≈ 0 . 55 -0 . 8 mag (Figure 3 and Table 2), which is typical of main-belt asteroids (e.g., Ivezi'c et al. 2001). This interpretation naturally explains the lack of quiescent counterparts in the template images. We further test this scenario by comparing the astrometric positions of each transient in the g P1 and r P1 images. In Figure 4 we show the distribution of positional shifts for the 8 sources compared to unresolved field sources in the same images. The median offset for field sources is about 53 mas (with a standard deviation of about 31 mas), indicative of the astrometric alignment precision of the MDS images. On the other hand, the 8 transients exhibit shifts of 11 -340 mas (bounded by our initial cut of /lessorsimilar 0 . 35 '' shift; §2.2), with a median value of about 230 mas. This is well in excess of the point source population, and a Kolmogorov-Smirnov(K-S) test gives a p -value of only 1 . 1 × 10 -4 for the null hypothesis that the positional offsets of the 8 transients and the field sources are drawn from the same underlying distribution. This clearly indicates that the 8 transients lacking quiescent counterparts exhibit larger than average astrometric shifts, supporting their identification as asteroids. We further inspect the g P1 and r P1 light curves of the 8 transients and find that none exhibit variability larger than the photometric uncertainties (e.g., Figure 2). Finally, we note that all 8 sources are located in the three MDS fields (MD04, MD09, and MD10) that are positioned within ± 10 · of the ecliptic plane (Table 2). In particular, the asteroids were discovered in these fields on dates concentrated at solar elongation values of about 130 deg, at which mainbelt asteroids go through a stationary point with negligible apparent motion. We therefore conclude based on their colors, astrometric motions, light curve behavior, and ecliptic coordinates that the 8 fast transients lacking quiescent counterparts are simply main-belt asteroids near the stationary point of their orbit.",
"pages": [
4
]
},
{
"title": "3.2. Fast Transients with Detected Quiescent Counterparts",
"content": "We now turn to the 11 fast transients that exhibit quiescent counterparts in some or all of the deep MDS template images. In all cases we find that the counterparts are unresolved (with a typical seeing of about 1 '' ) and have red colors that are indicative of M dwarf stars. Photometry of the quiescent counterparts from the PS1/MDS templates, and from SDSS when available, is summarized in Table 1. Using these magnitudes we determine the spectral type of each counterpart by comparing to the SDSS colors of M dwarfs (West et al. 2011); from sources with both PS1 and SDSS photometry we infer color transformations of ( g -r )P1 ≈ 0 . 94 × ( g -r )SDSS and ( i -z )P1 ≈ 0 . 93 × ( i -z )SDSS to account for the difference between the g P1and z P1 filters compared to the g SDSS and z SDSS filters (§2.1). The results are shown in Figure 5 indicating that 7 counterparts have spectral types of about M4-M5, while the remaining 4 counterparts have spectral types of M7-M9 (see Table 1 for the inferred spectral types). We further infer the distances to these M dwarfs using their associated absolute magnitudes (Bochanski et al. 2011) and find d ≈ 0 . 2 -1 . 2 kpc (Table 1). Thus, as in previous fast transient searches, all 11 fast transients with quiescent counterparts in our survey are M dwarf flares. Using the flare magnitudes and inferred spectral types, we compare the resulting flare and bolometric luminosities in Figure 6. We find that the flares span a luminosity 10 range P1 of Lf , g ≈ (6 -150) × 10 28 erg s -1 and Lf , r ≈ (4 -80) × 10 28 erg s -1 , with no apparent dependence on spectral type. However, since the bolometric luminosity declines from about 3 . 3 × 10 31 erg s -1 at spectral type M4 to about 1 . 3 × 10 30 erg s -1 at spectral type M9, the relative flare luminosities increase with later spectral type. We find relative flare luminosities of ≈ 0 . 006 -0 . 07 Lbol at ∼ M5, and larger values of ≈ 0 . 03 -1 Lbol at M8-M9. It is instructive to compare the properties of the flares and M dwarfs uncovered in our blind fast transients search to those from targeted M dwarf variability studies. In particular, Kowalski et al. (2009) searched for flares from 50,130 pre-selected M0-M6 dwarfs in the SDSS Stripe 82 and found 271 flares, with an apparent increase in the flare rate with later spectral type. This may explain the lack of M0-M4 dwarfs in our relatively small sample. For the M4-M6 dwarfs Kowalski et al. (2009) find a mean flare amplitude of ∆ u ≈ 1 . 5 mag, comparable to our mean value of ∆ g ≈ 1 . 1 mag, when taking into account that M dwarfs flares are generally brighter in u -band than in g -band due a typical temperature of ∼ 10 4 K. The flare luminosities for the Stripe 82 M4-M6 dwarfs are Lf , u ≈ (2 -100) × 10 28 erg s -1 , again comparable to the g P1-band luminosities of the ∼ M5 dwarfs in our sample. We note that there are no M7-M9 dwarfs in the Stripe 82 sample. Kowalski et al. (2009) also found that for M4-M6 dwarfs there is a strong dependence of the flare rate on vertical distance from the Galactic plane; namely, the fraction of time in which a star flares decreases by about an order of magnitude over a vertical distance range of about 50 -150 pc. The M4-M6 dwarfs in our sample are all located at larger vertical distances of ≈ 190 -560 pc, with a mean of about 390 pc, suggesting that the decline in flaring activity with vertical distance from the Galactic plane may not be as steep as previously inferred. Moreover, the Stripe 82 data exhibit a trend of steeper decline in the flare rate as a function of vertical distance with increasing spectral type (Kowalski et al. 2009), over the widths of the g P1 and r P1 filters, with δν ≈ 1 . 287 × 10 14 Hz and ≈ 7 . 721 × 10 13 Hz, respectively. while here we find that the M7-M9 dwarfs have a similar mean vertical distance from the Galactic plane to the ∼ M5 dwarfs. Clearly, a more systematic search for M dwarf flares in the PS1/MDS is required to study these trends. In particular, it is likely that we have missed some flares due to the requirement of no additional variability within a ± 5 night window around each detection (§2.2).",
"pages": [
4,
5
]
},
{
"title": "4. LIMITS ON THE RATE OF EXTRAGALACTIC FAST OPTICAL TRANSIENTS",
"content": "The 19 fast transients uncovered by our search cleanly divide into two categories: (i) main-belt asteroids near the stationary point of their orbits (§3.1); and (ii) flares from M dwarf stars (§3.2). Neither category is unexpected given that our search is based on consecutive g P1 r P1 detections with a time-span of about 0.5 hr. M dwarf flares typically exhibit blue colors indicative of T ∼ 10 4 K, with timescales of minutes to hours, and are thus ubiquitous in searches that utilize rapid observations in the ultraviolet (Welsh et al. 2005, 2006) or blue optical bands (Becker et al. 2004; Kulkarni & Rau 2006; Rau et al. 2008). Similarly, our requirement of two consecutive detections within ∼ 0 . 5 hr, with non-detections on preceding or subsequent nights is effective at capturing asteroids near the stationary point of their orbits (i.e., at solar elongations of about 130 deg for main-belt asteroids). Since we account for all 19 fast transients as Solar system or Galactic in origin, we can place a robust upper limit on the rate of extragalactic fast optical transients. While we could in principle detect a sufficiently bright transient with a timescale as short as about 4 min, corresponding to a detection in only the final exposure in one filter and the first exposure in the second filter, a more reasonable timescale probed by our search is about 0.5 hr, the time-span of a full g P1 r P1 exposure sequence (e.g., Figure 2). Similarly, our selection criteria could in principle accommodate transients with durations as long as ∼ 2 d, but for the bulk of the search the maximum timescale is /lessorsimilar 1 d. Thus, we consider our search to place limits on fast transients spanning about 0.5 hr to 1 d. In the discussion below we provide upper limits for the upper and lower bounds of the timescale distribution. The total areal exposure of our survey for a timescale of 0.5 hr is EA ≈ 40 . 4 deg 2 d, while for a timescale of ∼ day it is correspondingly longer, EA ≈ 1940 deg 2 d. Thus, we can place a 95% confidence limit ( /lessorsimilar 3 events) on the skyprojected rate of: where we estimate the detection efficiency per filter at /epsilon1 ≈ 0 . 8 based on the overall search for transients in the MDS fields. Thus, for a timescale of 0.5 hr we place a limit of R FOT /lessorsimilar 0 . 12 deg -2 d -1 , while for a timescale of ∼ day it is R FOT /lessorsimilar 2 . 4 × 10 -3 deg -2 d -1 (see Table 3). Our limit on fast transients with a 0.5 hr timescale improves on existing searches with similar limiting magnitudes by a factor of 30 -50 (Figure 7) thanks to the much larger effective areal exposure. DLS ( EA ≈ 1 . 1 deg d) placed an upper limit of R FOT /lessorsimilar 6 . 5 events deg -2 d -1 for transients with a timescale of about 0 . 36 hr (Becker et al. 2004), while the Fornax galaxy cluster search ( EA ≈ 1 . 9 deg d) placed a limit of R FOT /lessorsimilar 3 . 3 deg -2 d -1 for transients with a timescale of about 0 . 55 hr (Rau et al. 2008). The ROTSE-III search ( EA ≈ 635 deg d) placed a limit of R FOT /lessorsimilar 5 × 10 -3 deg -2 d -1 for transients with a timescale of about 0 . 5 hr, but was significantly shallower (Rykoff et al. 2005). The sky-projected rates do not take into account the differ- ence in limiting magnitudes between various searches. Our survey limiting magnitude is about 1.3 mag shallower than the DLS search, but about 1.2 mag deeper than the Fornax cluster search, and about 5 mag deeper than the ROTSE-III search. This information is summarized in Table 3, and shown in the relevant three-dimensional phase-space of survey limiting magnitude, timescale, and sky-projected rate limit in Figure 7. The survey depth impacts the inferred limits on volumetric rates. For example, for a population of fast transients with a fiducial absolute magnitude of -10 (comparable to the most luminous known novae; see §5), the limits on the volumetric rates 11 on a timescale of 0.5 hr are /lessorsimilar 13 Mpc -3 yr -1 for our survey, /lessorsimilar 1 . 3 × 10 2 Mpc -3 yr -1 for DLS, and /lessorsimilar 2 . 0 × 10 3 Mpc -3 yr -1 for Fornax. The small volume probed by the ROTSE-III search for transients with -10 mag ( d ≈ 3 Mpc) provides no real insight on the extragalactic population. For a fiducial absolute magnitude of -14 (comparable to the least luminous supernovae), the limits are /lessorsimilar 0 . 05 (PS1/MDS), /lessorsimilar 0 . 5 (DLS), and /lessorsimilar 7 . 8 (Fornax) Mpc -3 yr -1 ; at this peak 11 We assume a uniform underlying distribution of galaxies within the volume probed by each search, which is not strictly the case for the maximum distances associated with a peak magnitude of -10: 32 Mpc for our search, 56 Mpc for DLS, and 18 Mpc for Fornax. With the exception of the Fornax search, which was centered on the Fornax galaxy cluster at d ≈ 16 Mpc, this indicates that the actual limits from our survey and from DLS are subject to the underlying non-uniform galaxy distribution. magnitude the maximal detection distances are large enough to uniformly sample the galaxy distribution (about 200 Mpc for our search; 360 Mpc for DLS, and 115 Mpc for Fornax). Finally, for a fiducial absolute magnitude of -24 (intermediate between long and short gamma-ray burst afterglows; see §5), the volumetric rate limits are /lessorsimilar 10 -6 (PS1/MDS), /lessorsimilar 6 × 10 -5 (DLS), and /lessorsimilar 3 × 10 -5 (Fornax) Mpc -3 yr -1 . The various limits are summarized in Table 3. We note that the inferred limits at -10 and -14 mag are substantially higher than the volumetric rate of supernovae, ≈ 10 -4 Mpc -3 yr -1 (Li et al. 2011), and novae 12 , ≈ 0 . 1 Mpc -3 yr -1 ; similarly, the limits at -24 mag are at least two orders of magnitude larger than the on-axis GRB rate. This indicates that any source population of extragalactic fast optical transients with a timescale of 0.5 hr would have to be much more abundant than nova, supernova, or GRB progenitors, or produce multiple ( ∼ 10 2 -10 3 ) events per progenitor system to be detected with current surveys. Utilizing the B -band luminosity density in the local universe ( ≈ 1 . 4 × 10 8 LB , /circledot Mpc -3 ) the upper limits on the volumetric rates at -10 mag can be recast as /lessorsimilar 9 . 2 × 10 2 (PS1/MDS) and /lessorsimilar 9 . 3 × 10 3 (DLS) yr -1 per 10 10 LB , /circledot ; for the Fornax survey, which targeted a galaxy cluster environment, the limit is /lessorsimilar 350 yr -1 per 10 10 LB , /circledot (Rau et al. 2008). For a fiducial brightness of -14 mag, the limits are /lessorsimilar 3 . 6 (PS1/MDS), /lessorsimilar 36 (DLS), and /lessorsimilar 75 yr -1 per 10 10 LB , /circledot ; the latter value is from Rau et al. (2008). For the fiducial timescale of about 1 d (the upper bound of our survey), the volumetric rate limits are /lessorsimilar 0 . 3 ( -10 mag), /lessorsimilar 10 -3 ( -14 mag), and /lessorsimilar 4 × 10 -8 ( -24 mag) Mpc -3 yr -1 . The rates at -14 and -24 mag are of interest since they are only an order of magnitude larger than the supernova and GRB rates, indicating that an expansion of our search might yield interesting limits on the rate of fast transients with ∼ 1 d timescale.",
"pages": [
5,
6,
7
]
},
{
"title": "5. THEORETICAL EXPECTATIONS FOR THE LUMINOSITY OF FAST OPTICAL TRANSIENTS",
"content": "In the discussion above we used fiducial peak absolute magnitudes of -10, -14, and -24 mag to infer limits on the volumetric rates of extragalactic fast optical transients. These particular values were chosen to match the most luminous classical novae, the least luminous supernovae, and typical long and short GRB afterglows, respectively. However, it is instructive to explore what peak luminosities are expected for sources with a fiducial timescale of ∼ 0 . 5 hr to ∼ day for a range of potential mechanisms. We defer an exhaustive investigation of this question to future work and focus on the basic arguments here. At the most basic level, we note that explosive sources with a characteristic velocity of ∼ 10 4 kms -1 that emit thermally with a peak in the optical ( T ∼ 10 4 K) will be limited to an absolute magnitude of /greaterorsimilar -14; only sources undergoing relativistic expansion (e.g., GRBs) may significantly exceed this limits. We expand on this general theme below. We first investigate the possibility of unusually fast and luminous classical novae. In general, novae follow the maximum magnitude versus rate of decline (MMRD) relation (e.g., della Valle & Livio 1995), which indicates that faster novae are also more luminous. However, the MMRD relation flattens to a maximal observed value of about -10 mag for timescales shorter than a few days (della Valle & Livio 1995). Indeed, the shortest observed timescales (quantified as t 2 or t 3, the timescales to decline by 2 or 3 magnitudes, respectively) are ≈ 3 -5 d (Czekala et al. 2013). Theoretical models point to maximal magnitudes of about -10 and timescales as short as t 2 ≈ 1 d (Yaron et al. 2005). Moreover, the fraction of novae that achieve such high peak luminosity and rapid fading is /lessorsimilar 1% (e.g., Shafter et al. 2009), leading to a rate of /lessorsimilar 10 -3 Mpc -3 yr -1 , orders of magnitude below the upper bounds for -10 mag derived from the various searches (e.g., 0.3 Mpc -3 yr -1 for our search; Table 3). Thus, novae are not expected to contribute a population of luminous and fast extragalactic transients for current surveys. We next explore the generic case of an explosive event with ejecta that cool adiabatically due to free expansion ( R = v ej t ); i.e., a supernova-like explosion, but lacking internal heating. In this case, with the expansion driven by radiation pressure and using the standard diffusion approximation (Arnett 1982), the luminosity is given by: where τ h = R 0 / v ej is the initial hydrodynamic timescale, τ d , 0 = B κ M ej / cR 0 is the initial diffusion timescale, κ is the opacity (for simplicity we use κ = κ es = 0 . 4 cm 2 g -1 ), M ej is the ejecta mass, B ≈ 0 . 07 is a geometric factor, and L 0 = E 0 /τ d , 0 is the initial luminosity. A fiducial timescale of 0.5 hr requires R 0 /lessorsimilar 0 . 5hr × v ej ≈ 2 × 10 12 ( v ej / 10 4 kms -1 ) cm; i.e., more extended sources are unlikely to produce transients with a duration as short as ∼ 0 . 5 hr. Using the appropriate size limit we find that for a source with L (0 . 5hr) / L 0 ∼ 0 . 1 the resulting peak bolometric magnitude ranges from about -8 mag (for a white dwarf with R 0 ≈ 10 9 cm) to about -15 mag (for a star with R 0 ≈ 10 R /circledot ). The resulting characteristic effective temperature ranges from T eff = ( L 0 / 4 π R 2 σ ) 1 / 4 ≈ 2 . 2 × 10 4 K (for R 0 ≈ 10 9 cm) to ≈ 1 . 1 × 10 5 K(for R 0 ≈ 10 R /circledot ), or a peak in the ultraviolet. This means that the peak absolute magnitude in gr will be a few magnitudes fainter, corresponding to the faint end of the magnitude range considered in the previous section (i.e., /greaterorsimilar -14 mag). The luminosity can be enhanced by appealing to an internal energy source, for example radioactive heating as in the case of Type I supernovae. In this scenario the luminosity is maximized if the radioactive decay timescale is wellmatched to the diffusion timescale, τ r ≈ τ d ≈ B κ M ej / cR ≈ 0 . 5 hr. The fast timescale investigated here requires different radioactive material than 56 Ni, which powers the optical light curves of Type I supernovae, since the latter has τ r ≈ 8 . 8 d. Instead, a better match may be provided by the radioactive decay of r -process elements, for which there is a broad range of timescales (e.g., Li & Paczy'nski 1998; Metzger et al. 2010). Using a timescale of 0.5 hr we find that the required ejecta mass is low, M ej ≈ 2 × 10 -6 ( v ej / 10 4 km / s) 2 M /circledot . For radioactive heating the total energy generation rate is /epsilon1 ( t ) = ( f M ej c 2 /τ r )e -t / τ r , where f /lessmuch 1 is an efficiency factor, likely in the range of 10 -6 -10 -5 (Metzger et al. 2010). For the ejecta mass inferred above we find /epsilon1 ( t ) ≈ 10 45 f erg s -1 , and hence L ∼ 10 40 erg s -1 (for f ∼ 10 -5 ), or an absolute magnitude of about -12 mag. As in the discussion above, the spectrum will peak in the ultraviolet, and the optical emission ( g -band) will be dimmer by about 2 mag (i.e., to a peak of about -10 mag). For a fiducial timescale of ∼ day, the peak absolute magnitude is larger, reaching ≈ -14 mag in the optical. This is essentially the 'kilonova' model invoked for compact object binary mergers (Li & Paczy'nski 1998; Metzger et al. 2010; Berger et al. 2013), with a predicted volumetric rate of ∼ 10 -8 -10 -5 Mpc -3 yr -1 (e.g., Abadie et al. 2010). We note, however, that recent opacity calculations for r -process material indicate that the actual peak brightness in the optical bands will be an order of magnitude fainter than suggested by the above calculation (Barnes & Kasen 2013). A similar scenario is the thermonuclear .Ia supernova model with heating by radioactive α -chain elements (Ca, Ti), leading to a peak optical brightness of ∼ -15 mag, but with a longer timescale of ∼ week even for low helium shell masses (Shen et al. 2010). Thus, models with radioactive heating are limited to ∼ -10 mag ( ∼ 0 . 5 hr) or ∼ -14 mag ( ∼ day). An alternative internal energy source is the spin-down of a newly-born millisecond magnetar (Kasen & Bildsten 2010; Woosley 2010). In this scenario the maximal luminosity is achieved when the magnetar spin-down timescale, τ sd ≈ 4 × 10 3 ( B / 10 14 G) -2 ( P / 1ms) 2 s, is comparable to the diffusion timescale, τ d ≈ 0 . 5 hr; here B is the magnetic field strength and P is the initial rotation period. The available rotational energy is large, E rot ≈ 2 × 10 52 ( P / 1ms) -2 erg, and therefore a magnetar engine can in principle produce extremely luminous fast transients, with a peak brightness of L ≈ E rot t sd / t 2 d ≈ 5 × 10 48 erg s -1 . However, since τ d ≈ 0 . 5 hr requires a low ejecta mass of ∼ 10 -6 M /circledot , this scenario necessitates magnetar birth with negligible ejected mass, and results in a relativistic outflow; this is essentially the scenario invoked in magnetar models of gamma-ray bursts. It is not obvious in this scenario whether the large energy reservoir will be emitted in the optical band, or primarily at X-ray/ γ -ray energies. Thus, the optical signature may still be rather weak, and perhaps dominated instead by interaction with the ambient medium (i.e., an afterglow). Overall, the estimated magnetar birth rate is ∼ 10% of the core-collapse supernova rate (Kouveliotou et al. 1998), while the GRB rate is /lessorsimilar 1% of the supernova rate (Wanderman & Piran 2010). This indicates that any magnetar-powered fast optical transients will have a rate orders of magnitude below the upper limits inferred by the existing searches. Finally, luminous rapid optical transients can be produced by circumstellar interaction of a relativistic outflow, such as an on-axis 13 GRB afterglow. The typical absolute magnitude of a short-duration GRB afterglow at δ t ≈ 0 . 5 hr is about -20 mag (based on a typical apparent brightness of about 22 mag and a typical redshift of z ≈ 0 . 5; Berger 2010). At this fiducial luminosity the inferred volumetric upper limits from the fast optical transient searches are /lessorsimilar 3 × 10 -5 (PS1/MDS), /lessorsimilar 3 × 10 -4 (DLS), /lessorsimilar 2 × 10 -3 (Fornax), and /lessorsimilar 7 × 10 -4 (ROTSE-III) Mpc -3 yr -1 . The actual volumetric rate of on-axis short GRBs is about 10 -8 Mpc -3 yr -1 (Nakar et al. 2006), at least 3 × 10 3 times lower than the limits reached by the surveys. For longduration GRBs the typical absolute magnitude at δ t ≈ 0 . 5 hr is much larger, about -27 mag (based on a typical apparent brightness of about 17 mag and a typical redshift of z ≈ 2; Kann et al. 2010). At this luminosity the volumetric upper limits for the various searches are 14 /lessorsimilar 10 -6 (PS1/MDS), /lessorsimilar 2 × 10 -5 (DLS), /lessorsimilar 1 × 10 -5 (Fornax), and /lessorsimilar 3 × 10 -7 (ROTSE-III) Mpc -3 yr -1 . The inferred on-axis volumetric rate (at z ∼ 2) is ∼ 10 -8 Mpc -3 yr -1 (Wanderman & Piran 2010). Thus, unless there is a substantial population of relativistic explosions that do not produce γ -ray emission, the existing limits are much too shallow in the context of on-axis GRB rates. We therefore conclude based on the comparison to various potential models and populations that: In Figure 8 we summarize this information by comparing the volumetric rate limits from the various surveys at fiducial peak absolute magnitudes of -10, -14, -20 (short GRB), and -27 (long GRB) to the rates of known transients (e.g., novae, supernovae, neutron star binary mergers, on-axis short and long GRBs). The results demonstrate that the existing survey limits are orders of magnitude above the anticipated event rates based on the known classes. This indicates that much larger surveys are essential to robustly explore the transient sky on the ∼ hour -day timescale. In addition, we stress that given the low expected luminosity for non-relativistic fast transients ( /greaterorsimilar -14), a profitable strategy is to systematically target nearby galaxies (as was done for example in the Fornax Cluster search; Rau et al. 2008) rather than image wide blank fields. On the other hand, searches for the rare but highly luminous relativistic fast transients are best focused on covering wide fields (preferably ∼ all -sky) at the expense of survey depth since the projected rate is at most ∼ few per sky per day, while the limiting factor for distance coverage is the use of optical filters (limited to z ∼ 6) rather than depth.",
"pages": [
7,
8,
9
]
},
{
"title": "6. IMPLICATIONS FOR LSST",
"content": "Our search for fast optical transients proved highly effective at identifying foreground events without the need for expensive follow-up observations . This is unlike previous searches such as DLS and Fornax. This has been accomplished thanks to the use of dual-filter observations spanning a timescale of about 0.5 hr, with each observation composed of multiple exposures that help to eliminate individual spurious detections. In addition, the availability of color information ( g -r in this case), coupled with the time baseline allowed us to effectively identify main-belt asteroids near the stationary point. Similarly, deep multi-band templates allowed us to unambiguously identify M dwarf counterparts for all detected flares to a distance of at least ∼ 1 . 2 kpc and with spectral types extending to about M9. With this information we were able to account for all the fast optical transients found in the survey, and to place the deepest limit to date on the rate of extragalactic fast transients. Our results are reassuring given that follow-up spectroscopy in the era of on-going and future large surveys is in limited supply. With this in mind, it is instructive to consider our results in the context of the anticipated LSST survey strategy (Ivezic et al. 2008). The LSST main survey is intended to cover a total of ≈ 18 , 000 deg 2 , accounting for about 90% of the observing time. Each pointing will consist of two visits per field with a separation of about 15 -60 min, with each visit consisting of a pair of 15 s exposures with a 3 σ depth of ∼ 24 . 5 mag (or about 23.5 mag at 10 σ ). It is anticipated that a total of about 3000 deg 2 will be imaged on any given night, potentially with more than one filter, leading to an areal exposure of about 62 deg 2 d in a single night, or about 15 1 . 6 × 10 4 deg 2 d per year, for the fiducial timescale of 0.5 hr. This survey will therefore achieve a sky projected rate limit (95% confidence level) of 0 . 07 deg -2 d -1 in a single night and 3 × 10 -4 deg -2 d -1 in a year. The latter value represents a factor of 400 times improvement relative to our PS1/MDS search. In terms of a volumetric rate, the greater depth achieved by LSST will probe rates of ≈ 9 × 10 -3 Mpc -3 yr -1 for -10 mag and ≈ 3 × 10 -5 Mpc -3 yr -1 for -14 mag for 1 year of operations. At a typical brightness of short and long GRBs, the rate will be about 2 × 10 -9 Mpc -3 yr -1 (limited to z /lessorsimilar 6 by the zy filters), a few times deeper than the actual onaxis GRB rate, indicating that LSST is likely to detect such events. The LSST data set can form the basis for a search similar to the one described here, with the main difference that contemporaneous color information may not be available. However, with a S / N ≈ 10 cut as imposed here, the observations will reach about 1 mag deeper than our search, indicating that they will uncover M dwarf flares to distances that are only about 50% larger than those found here (i.e., generally /lessorsimilar 2 kpc). At the same time, the LSST templates will reach a comparable depth to the PS1/MDS templates after only ∼ 10 visits per field per filter, i.e., within a year of when the survey commences. Thus, we conclude that it will be simple to identify M dwarf counterparts in the template images for essentially all M dwarf flares, negating the need for follow-up spectroscopy. This approach will effectively eliminate the largest known contaminating foreground for ∼ hour timescales. Similarly, the time baseline of 15 -60 min per field will allow for astrometric rejection of most asteroids even near the stationary point, although as we found in our PS1/MDS search some asteroids have negligible motions that are indistinguishable from the astrometric scatter of field sources (2 / 8 in our search). Such asteroids can in principle be recognized using color information, but this is not likely to be available with LSST. On the other hand, a constant brightness level between visits 15 -60 min apart, combined with a location near the ecliptic plane and in particular with solar elongation of ∼ 130 deg, will be indicative of an asteroid origin. Such fields can simply be avoided in searches for fast transients. Thus, the two primary contaminants for extragalactic fast optical transient searches (M dwarf flares and asteroids near the stationary point) will be identifiable with the LSST baseline survey strategy. Furthermore, we anticipate based on the comparison to known transient classes that with a year of operations LSST is unlikely to reveal large numbers of extragalactic fast transients unless they result from a source population that far exceeds classical novae, supernova progenitors, or compact object binaries (Figure 8).",
"pages": [
9,
10
]
},
{
"title": "7. CONCLUSIONS",
"content": "Wepresent a search for fast optical transients on a timescale of about 0.5 hr to 1 d in consecutive g P1 r P1 observations of the PS1/MDS fields, by requiring a detection in both filters with no additional detections on preceding or subsequent nights. The search yielded 19 astrophysical transients, of which 8 events that lack quiescent counterparts are identified as mainbelt asteroids near the stationary point of their orbits, while the remaining 11 transients are identified as flares from M5M9 dwarf stars at distances of about 0 . 2 -1 . 2 kpc. The flare properties are generally similar to those from M dwarfs studied in the SDSS Stripe 82, although our sample extends to later spectral types and to much greater vertical distances from the Galactic plane. A key result of our search is a limit on the sky-projected rate of extragalactic fast transients of R FOT /lessorsimilar 0 . 12 deg -2 d -1 ( ∼ 0 . 5 hr) that is about 30 -50 time deeper than previous limits; the limit for a timescale of ∼ day is /lessorsimilar 2 . 4 × 10 -3 deg -2 d -1 . The upper bounds on the volumetric rates at fiducial absolute magnitudes of -10, -14, and -24 mag are likewise an order of magnitude deeper than from previous searches (Table 3). With an additional 3 years of PS1/MDS data in hand we can improve these estimates by a factor of a few. We also anticipate additional M dwarf flare detections that will allow us to better characterize the distribution of flare properties, as well as the properties of flaring M dwarfs in general. To guide the conversion from our sky-projected rate to volumetric rates, and to motivate future searches for fast transients, we also explore the expected luminosities of such transients for a range of physically motivated models. We find that non-relativistic fast transients are generally limited to about -10 mag for a timescale of ∼ 0 . 5 hr and -14 mag for a timescale of ∼ day even if powered by radioactive decay. This is simply a reflection of the low ejecta mass required to achieve a rapid diffusion timescale of /lessorsimilar day. Such low luminosity events are best explored through targeted searches of galaxies in the local universe, with anticipated event rates of /lessorsimilar 10 -4 Mpc -3 yr -1 (Figure 8). A separate class of relativistic fast transients (on-axis GRBs, magnetar engines) can exceed -25 mag on a timescale of ∼ 0 . 5 hr, but they are exceedingly rare, with anticipated volumetric rates of ∼ 10 -8 Mpc -3 yr -1 (or sky-projected rates of a few per sky per day). Such event rates are orders of magnitude below the level probed in current searches, but will be reached by the LSST main survey in a full year. Another strategy to find such events is a shallower search of a wider sky area than is planned for LSST. We note that a broad exploration of the various mechanisms that can power optical transients is essential to guide and motivate future searches for fast extragalactic transients. The initial investigation performed here already places clear bounds on the luminosities and rates for a wide range of mechanisms. Finally, since the PS1/MDS survey is currently the only close analogue to the main LSST survey in terms of depth, cadence, and choice of filters, we use the results of our search to investigate the efficacy of fast transient searches with LSST. We demonstrate that the main contaminants recognized here should be identifiable with the LSST survey strategy, without the need for expensive follow-up spectroscopy. Namely, asteroids near the stationary point can be recognized through a larger than average astrometric shift, a constant brightness in visits separated by 15 -60 min, and an expected location near the ecliptic plane with solar elongation of 130 deg. LSST may not provide near-simultaneous color information, which can serve as an additional discriminant for asteroids. For M dwarf flares we show that LSST's increased sensitivity will probe a larger volume of the Galaxy, but the correspondingly deeper templates will still allow for the identification of quiescent counterparts in essentially all cases. The PS1/MDS survey demonstrates that with multi-band photometry it is possible to identify M dwarf flares without the need for follow-up spectroscopy (as in previous searches; e.g., Kulkarni & Rau 2006). This result indicates that extragalactic fast optical transients should be able to pierce the veil of foreground flares. We thank Mario Juric for helpful information on the planned LSST survey strategy. E. B. acknowledges support for this work from the National Science Foundation through Grant AST-1008361. PS1 has been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS1 Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, and the Las Cumbres Observatory Global Telescope Network, Incorporated, the National Central University of Taiwan, and the National Aeronautics and Space Administration under Grant NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate. Facilities: Pan-STARRS1 Barnes, J., & Kasen, D. 2013, arXiv:1303.5787 Becker, A. C., et al. 2004, ApJ, 611, 418 Berger, E. 2010, ApJ, 722, 1946 Bochanski, J. J., Hawley, S. L., & West, A. A. 2011, AJ, 141, 98 Cole, S., et al. 2001, MNRAS, 326, 255 Czekala, I., et al. 2013, ApJ, 765, 57 della Valle, M., & Livio, M. 1995, ApJ, 452, 704 Filippenko, A. V. 1997, ARA&A, 35, 309 Gallagher, J. S., & Starrfield, S. 1978, ARA&A, 16, 171 Gorbovskoy, E. S., et al. 2013, Astronomy Reports, 57, 233 Ivezi'c, Ž., et al. 2001, AJ, 122, 2749 Ivezic, Z., et al. 2008, arXiv:0805.2366 Kann, D. A., et al. 2010, ApJ, 720, 1513 Yaron, O., Prialnik, D., Shara, M. M., & Kovetz, A. 2005, ApJ, 623, 398 NOTE. - Properties of the fast transients that exhibit quiescent counterparts, as well as multi-band photometry of the counterparts. We find that all 11 transients arise from M dwarfs, and provide the inferred spectral types and distances of these sources. NOTE. - Properties of the fast transients that lack quiescent counterparts. We find that all 8 transients exhibit the colors and expected ecliptic coordinates of main-belt asteroids near the stationary point of their orbits. NOTE. - Survey parameters and resulting limits on the projected sky rates and volumetric rates of extragalactic fast optical transients with a timescale of ∼ 0 . 5 hr.",
"pages": [
10,
11,
12,
13
]
}
]
2013ApJ...779...99W
https://arxiv.org/pdf/1310.3535.pdf
<document>
<section_header_level_1><location><page_1><loc_23><loc_86><loc_78><loc_87></location>CYCLIC SPECTROSCOPY OF THE MILLISECOND PULSAR, B1937+21</section_header_level_1>
<section_header_level_1><location><page_1><loc_44><loc_84><loc_56><loc_85></location>Mark A. Walker</section_header_level_1>
<text><location><page_1><loc_32><loc_83><loc_69><loc_84></location>Manly Astrophysics, 3/22 Cliff Street, Manly 2095, Australia</text>
<section_header_level_1><location><page_1><loc_43><loc_80><loc_57><loc_80></location>Paul B. Demorest</section_header_level_1>
<text><location><page_1><loc_28><loc_78><loc_72><loc_79></location>National Radio Astronomy Observatory, Charlottesville, VA 22903, USA</text>
<section_header_level_1><location><page_1><loc_42><loc_75><loc_58><loc_75></location>Willem van Straten</section_header_level_1>
<text><location><page_1><loc_20><loc_72><loc_80><loc_74></location>Swinburne University of Technology, Astrophysics and Supercomputing, Hawthorn 3122, Australia Submitted to ApJ 16th January 2013</text>
<section_header_level_1><location><page_1><loc_45><loc_70><loc_55><loc_71></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_48><loc_86><loc_69></location>Cyclic spectroscopy is a signal processing technique that was originally developed for engineering applications and has recently been introduced into the field of pulsar astronomy. It is a powerful technique with many attractive features, not least of which is the explicit rendering of information about the relative phases in any filtering imposed on the signal, thus making holography a more straightforward proposition. Here we present methods for determining optimum estimates of both the filter itself and the statistics of the unfiltered signal, starting from a measured cyclic spectrum. In the context of radio pulsars these quantities tell us the impulse response of the interstellar medium and the intrinsic pulse profile. We demonstrate our techniques by application to 428 MHz Arecibo data on the millisecond pulsar B1937+21, obtaining the pulse profile free from the effects of interstellar scattering. As expected, the intrinsic profile exhibits main- and inter-pulse components that are narrower than they appear in the scattered profile; it also manifests some weak, but sharp features that are revealed for the first time at low frequency. We determine the structure of the received electric-field envelope as a function of delay and Doppler-shift. Our delay-Doppler image has a high dynamic-range and displays some pronounced, low-level power concentrations at large delays. These concentrations imply strong clumpiness in the ionized interstellar medium, on AU size-scales, which must adversely affect the timing of B1937+21.</text>
<text><location><page_1><loc_14><loc_45><loc_86><loc_47></location>Subject headings: methods: data analysis pulsars: general pulsars: individual (PSR B1937+21) ISM: general scattering</text>
<section_header_level_1><location><page_1><loc_22><loc_41><loc_35><loc_42></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_26><loc_48><loc_41></location>Although radio pulsars emit intrinsically broad-band radiation, spectroscopy of these sources often reveals a great deal of narrow-band structure (e.g. Rickett 1990). This structure arises during propagation of the signal through the interstellar medium (ISM), where it is scattered by inhomogeneities in the ionized gas - it is interference between the various scattered waves which creates the observed fringes. Consequently high resolution spectroscopy of pulsars has proved to be a powerful tool for investigating the ISM (Roberts and Ables 1982; Cordes and Wolszczan 1985; Stinebring et al 2001).</text>
<text><location><page_1><loc_8><loc_12><loc_48><loc_26></location>Traditionally pulsar spectroscopy is undertaken by forming the power-spectrum of the signal in a pulsephase window where the flux is high (i.e. 'on-pulse'), and subtracting the power-spectrum from a window where the flux is low ('off-pulse'), so as to remove the steady, background power level. Recently Demorest (2011) has drawn attention to an alternative approach, known as cyclic spectroscopy, in which one measures the modulation of the spectrum across the entire pulse profile. Cyclic spectroscopy was developed in engineering disciplines for studying signals whose statistics are pe-</text>
<text><location><page_1><loc_10><loc_7><loc_32><loc_10></location>Mark.Walker@manlyastrophysics.org pdemores@nrao.edu willem@swin.edu.au</text>
<text><location><page_1><loc_52><loc_35><loc_92><loc_43></location>riodically modulated (Gardner 1992; Antoni 2007). Signals of this type are common and are referred to as cyclostationary. The electric field received from a radio pulsar can be thought of as periodically-amplitude-modulated noise (Rickett 1975), so radio pulsars provide an example of a signal which is cyclo-stationary.</text>
<text><location><page_1><loc_52><loc_13><loc_92><loc_35></location>As described by Demorest (2011), cyclic spectroscopy has several advantages over the simpler method of differencing on-pulse and off-pulse power spectra. Periodic amplitude modulation of the pulsar's radio-frequency noise, introduced by rotation of the pulsar beam, splits the received signal into upper- and lower-sidebands. By construction, the cyclic spectrum is the product of the lower sideband with the complex conjugate of the upper sideband. It is thus a complex quantity and as such it explicitly manifests information about the phase of any filtering which has occured prior to reception. For radio pulsars observed at low frequencies the dominant filtering is due to the ISM - specifically, to dispersion and scattering of the waves. Thus a time-domain representation of the filter is, to a good approximation, just the impulse-response of the ISM.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_13></location>In the present paper we show how to determine the filter given a measured cyclic spectrum. We also show how to determine the intrinsic cyclic spectrum of the signal in other words the (Fourier Transform of the) pulse pro-</text>
<text><location><page_2><loc_8><loc_80><loc_48><loc_92></location>file which would have been observed in the absence of any scattering or dispersion. These determinations are both made in the narrow-band approximation, appropriate to our data, where there is assumed to be no variation of the intrinsic cyclic spectrum across the observed radiofrequency band. Our main dataset is a 4 MHz bandwidth voltage recording, centered on 428 MHz, of the original millisecond pulsar, B1937+21 (Backer et al 1982), made with the Arecibo radio telescope. 1</text>
<text><location><page_2><loc_8><loc_48><loc_48><loc_80></location>As far as we are aware the methods presented in this paper are the first attempts to determine both the filter and the intrinsic cyclic spectrum for any astronomical signal. It is possible that our techniques may be useful in fields other than pulsar astronomy, but we do not attempt to identify appropriate fields. Rather we encourage readers to consider applications in other contexts. To aid that process we note here the requirements for validity of our approach: first, the signal must be cyclostationary - i.e. stationary at each phase in its cycle in order for the cyclic spectrum to be well-defined. Secondly, our least-squares fitting assumes that the intrinsic cyclic spectrum is just white-noise that is periodically amplitude-modulated, so non-pulsar applications of our techniques are limited to signals which can be described in this or similar fashion. And, finally, the filter must not change significantly within the averaging time over which each cyclic spectrum is constructed. In addition to these requirements, the stopping criterion we employ for our optimizations is based on the assumption of Gaussian noise; but it would be straightforward to modify that criterion. We note that source code is freely available for all the software used herein (see § 5), so readers are free to adapt our code to their purpose.</text>
<text><location><page_2><loc_8><loc_20><loc_48><loc_48></location>This paper is organised as follows. In the next section we give some background to the particular problems tackled in this paper. Then in § 3 we show how to determine the filter function and the intrinsic cyclic spectrum by direct construction. In § 4 we consider the issue of optimization - i.e. obtaining representations of these quantities which best fit the measured cyclic spectrum. In doing so we see that our direct estimate of the intrinsic profile, given in § 3, is in fact the optimum estimate in a least-squares sense. But § 4 does highlight deficiencies in our direct approach to the filter function; so for this quantity we utilize a large-scale optimization of the filter coefficients. Our implementation of this optimization is coded in the language 'C' and is freely available; it is described in § 5. In § 6 we present results obtained by applying our methods to low-frequency data on PSR B1937+21; both filter functions and intrinsic pulse profiles are presented. Discussion ( § 7) and Conclusions ( § 8) round out the paper. Two Appendices detail (i) the results of various tests we used to evaluate the code, and (ii) an analysis of the uncertainties in best-fit parameters.</text>
<section_header_level_1><location><page_2><loc_10><loc_18><loc_47><loc_19></location>2. BACKGROUND AND GENERAL CONSIDERATIONS</section_header_level_1>
<text><location><page_2><loc_8><loc_12><loc_48><loc_18></location>Procedures for constructing the cyclic spectrum itself, from a set of recorded voltages, are given by Demorest (2011). We begin our development by quoting the relationship between a signal, x ( t ), a function of time, with</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_11></location>1 The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. M'endez-Universidad Metropolitana, and the Universities Space Research Association.</text>
<text><location><page_2><loc_52><loc_88><loc_92><loc_92></location>Fourier Transform X ( ν ), and the cyclic spectrum of that signal, S x . At modulation frequency α we have (Gardner 1992; Antoni 2007; Demorest 2011)</text>
<formula><location><page_2><loc_58><loc_86><loc_92><loc_87></location>S x ( α, ν ) ≡ 〈 X ( ν + α/ 2) X ∗ ( ν -α/ 2) 〉 , (1)</formula>
<text><location><page_2><loc_52><loc_79><loc_92><loc_85></location>where the time-average is taken over integer multiples of the period of the system. Thus if we apply a filter, H ( ν ), such that the filtered signal is Z ( ν ) = H ( ν ) X ( ν ) then the cyclic spectrum of the filtered signal is</text>
<formula><location><page_2><loc_55><loc_77><loc_92><loc_79></location>S z ( α, ν ) = H ( ν + α/ 2) H ∗ ( ν -α/ 2) S x ( α, ν ) . (2)</formula>
<text><location><page_2><loc_52><loc_64><loc_92><loc_76></location>In the case of a radio pulsar the signals X,Z are just electric fields, and the frequency ν is the radio frequency. Filtering of the signal occurs as a result of propagation, notably dispersion and scattering in the ionized ISM, and in the process of reception, e.g. the bandpass filter. The filter resulting from interstellar propagation evolves on some time-scale, and the average in equation (1) must be restricted to times which are short compared to that evolution time-scale.</text>
<text><location><page_2><loc_52><loc_59><loc_92><loc_64></location>Throughout this paper we confine attention to the case of small fractional radio bandwidths, for which we expect the intrinsic cyclic spectrum to be approximately independent of ν :</text>
<formula><location><page_2><loc_65><loc_57><loc_92><loc_59></location>S x ( α, ν ) → S x ( α ) . (3)</formula>
<text><location><page_2><loc_52><loc_46><loc_92><loc_56></location>The quantity S x ( α ) is already familiar to astronomers from conventional analysis of radio pulsar signals: it is just the Fourier Transform of the pulse profile. But we emphasise that it is the transform of the intrinsic pulse profile, rather than the transform of the measured pulse profile - the difference being that the latter includes the influence of scattering and other contributions to the filter H .</text>
<text><location><page_2><loc_52><loc_38><loc_92><loc_46></location>In general both S z and S x are complex quantities, but in the particular case α = 0 we obtain the zeromodulation-frequency components of the filtered and unfiltered signals, respectively. As these are just the time-averaged power-spectra of the signals they are nonnegative real numbers.</text>
<section_header_level_1><location><page_2><loc_66><loc_35><loc_78><loc_36></location>2.1. Degeneracies</section_header_level_1>
<text><location><page_2><loc_52><loc_29><loc_92><loc_35></location>Before extracting estimates from our data it is necessary to identify and eliminate any degeneracies in the model. Equation (2) shows that there are degeneracies which are multiplicative in form. Writing</text>
<formula><location><page_2><loc_65><loc_27><loc_92><loc_28></location>H ( ν ) → H ( ν ) Q ( ν ) , (4)</formula>
<text><location><page_2><loc_52><loc_25><loc_76><loc_26></location>we see that S z → S z if and only if</text>
<formula><location><page_2><loc_59><loc_21><loc_92><loc_24></location>S x ( α, ν ) → S x ( α, ν ) Q ( ν + α/ 2) Q ∗ ( ν -α/ 2) . (5)</formula>
<text><location><page_2><loc_52><loc_14><loc_92><loc_20></location>Thus if S x and H are completely unconstrained then there may be a great deal of degeneracy between these quantities in our model of S z : features seen in the data might be attributed to the intrinsic spectrum or to the effects of an imposed filter.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_13></location>In circumstances where the intrinsic cyclic spectrum is independent of radio-frequency (equation 3), the scope of the degeneracy is limited to functions Q ( ν ) such that Q ( ν + α/ 2) Q ∗ ( ν -α/ 2) is independent of ν . This condition should hold for all α . In the case of small α , by</text>
<text><location><page_3><loc_8><loc_89><loc_48><loc_92></location>expanding to first order in α , we see that the form of Q is restricted to those functions satisfying</text>
<formula><location><page_3><loc_23><loc_87><loc_48><loc_88></location>| Q ( ν ) | = const ., (6)</formula>
<text><location><page_3><loc_8><loc_85><loc_11><loc_86></location>and</text>
<text><location><page_3><loc_52><loc_87><loc_92><loc_92></location>completely independent. Because of these limitations, the actual number of constraints provided by the data may be smaller than the number of grid points in the cyclic spectrum.</text>
<section_header_level_1><location><page_3><loc_65><loc_84><loc_79><loc_85></location>2.3. Noise and bias</section_header_level_1>
<text><location><page_3><loc_52><loc_78><loc_92><loc_83></location>The computed cyclic spectrum includes measurement noise which we can characterize in the following way. Suppose that the recorded voltage is Z ( ν ) + N ( ν ), then we expect the measured cyclic spectrum to be</text>
<formula><location><page_3><loc_58><loc_76><loc_92><loc_77></location>〈 D ( α, ν ) 〉 = S z ( α, ν ) + 〈| N ( ν ) | 2 〉 δ ( α ) , (10)</formula>
<text><location><page_3><loc_52><loc_71><loc_92><loc_75></location>where the delta-function appears because the measurement noise is stationary. Thus our measured cyclic spectrum is free of noise bias except at α = 0.</text>
<text><location><page_3><loc_52><loc_55><loc_92><loc_71></location>Because modulation is the fundamental characteriztic of pulsar radiation which allows it to be distinguished from measurement noise, estimating the unmodulated part of the cyclic spectrum, S z (0 , ν ), from D (0 , ν ) is ambiguous. In this paper we therefore make no attempt to quantify S z (0 , ν ), nor do we make direct use of D (0 , ν ) in our estimates of the signal properties S x ( α ) and H ( ν ). In turn this means that we are giving up any possibility of determining S x (0), the zero-frequency term in the Fourier representation of the intrinsic pulse profile. We therefore adopt the convention S x (0) = 0 in our models throughout the rest of this paper.</text>
<text><location><page_3><loc_52><loc_49><loc_92><loc_55></location>The actual data which we record, D ( α, ν ), will differ from 〈 D 〉 because of measurement noise and because the signal itself is stochastic in nature. If there is no averaging (see discussion following equation 14) the variance of the measured cyclic spectrum is given by (Antoni 2007)</text>
<formula><location><page_3><loc_53><loc_47><loc_92><loc_48></location>var { D ( α, ν ) } = 〈 D (0 , ν -α/ 2) 〉 〈 D (0 , ν + α/ 2) 〉 . (11)</formula>
<text><location><page_3><loc_52><loc_41><loc_92><loc_46></location>At zero modulation frequency, we recover from equation 11 the familiar result for stationary signals that the variance of the unaveraged power is just the square of the mean power.</text>
<text><location><page_3><loc_52><loc_37><loc_92><loc_40></location>For observations of radio pulsars with current instrumentation, noise power is usually the dominant contribution to D (0 , ν ) and in this case we have</text>
<formula><location><page_3><loc_53><loc_34><loc_92><loc_36></location>var { D ( α, ν ) } glyph[similarequal] 〈| N ( ν -α/ 2) | 2 〉 〈| N ( ν + α/ 2) | 2 〉 . (12)</formula>
<text><location><page_3><loc_52><loc_31><loc_92><loc_33></location>If the measurement noise is white, as is often the case in practice, then equation 12 yields a uniform variance,</text>
<formula><location><page_3><loc_62><loc_29><loc_92><loc_30></location>var { D } = 〈| N ( ν ) | 2 〉 2 = σ 2 , (13)</formula>
<text><location><page_3><loc_52><loc_23><loc_92><loc_28></location>over the entire cyclic spectrum. It is straightforward to estimate σ , because at zero modulation frequency the cyclic spectrum is just a power spectrum. Thus the noise level is just</text>
<formula><location><page_3><loc_67><loc_19><loc_92><loc_23></location>σ = S sys √ ∆ t ∆ ν , (14)</formula>
<text><location><page_3><loc_52><loc_12><loc_92><loc_19></location>where S sys is the system-equivalent-flux-density, ∆ t is the integration time and ∆ ν the channel width. (Here we consider only a single polarization state, but clearly the results can be generalised to different combinations of polarization states.)</text>
<text><location><page_3><loc_52><loc_7><loc_92><loc_12></location>Equation 14 clarifies what is meant by the 'no averaging' requirement immediately preceding equation 11. For cyclic spectroscopy of pulsars the natural choice of spectral resolution is ∆ ν = ∆ α , and we always have</text>
<formula><location><page_3><loc_19><loc_82><loc_48><loc_85></location>d d ν Im { log Q ( ν ) } = const . (7)</formula>
<text><location><page_3><loc_8><loc_78><loc_48><loc_82></location>Hence if we do not know the actual form of S x ( α ), then the filter function can only be determined up to an arbitrary multiplicative factor of</text>
<formula><location><page_3><loc_19><loc_76><loc_48><loc_77></location>Q ( ν ) = exp[ i ( τν + φ ) + ρ ] , (8)</formula>
<text><location><page_3><loc_8><loc_70><loc_48><loc_75></location>where τ , φ and ρ are real constants. In other words the overall normalization of H , its phase and its phase gradient are all arbitrary, because the simultaneous transformation</text>
<formula><location><page_3><loc_17><loc_68><loc_48><loc_69></location>S x ( α ) → S x ( α ) exp[ -iτα -2 ρ ] , (9)</formula>
<text><location><page_3><loc_8><loc_65><loc_23><loc_67></location>leaves S z unchanged.</text>
<text><location><page_3><loc_8><loc_59><loc_48><loc_65></location>If, however, S x ( α ) is already known, from previous observations, then the only remaining degeneracy is the overall phase of H . This phase is always arbitrary, as can be seen by noting that φ does not appear in equation (9).</text>
<section_header_level_1><location><page_3><loc_23><loc_57><loc_33><loc_58></location>2.2. Sampling</section_header_level_1>
<text><location><page_3><loc_8><loc_36><loc_48><loc_56></location>For a periodic modulation with period P = 1 / Ω, as is the case with signals from a radio pulsar, the cyclic spectrum is expected to be zero everywhere except at α = m Ω, where m is an integer, so those are the only modulation frequencies which we sample. In practice the data are also sampled discretely in the radio-frequency dimension, so we have measurements on a grid, with spacing ∆ α = Ω, and ∆ ν which we are at liberty to choose. In choosing ∆ ν the primary consideration relates to structure in the filter function: if we wish to capture signal components which are delayed by times up to τ then we need to have a resolution ∆ ν ≤ 1 / 2 τ . One could choose the resolution to be glyph[lessmuch] 1 / 2 τ but that would entail a greater computational load in constructing the cyclic spectrum.</text>
<text><location><page_3><loc_8><loc_27><loc_48><loc_36></location>There is a natural limit to the fineness of the spectral resolution set by ∆ ν = ∆ α = Ω, corresponding to delays τ = ± P/ 2, where P is the pulse period. If there are signal components at delays greater than half the pulse period then the cyclic spectrum is intrinsically undersampled in α , because the modulation imposed by the filter function changes significantly on scales δα < Ω.</text>
<text><location><page_3><loc_8><loc_23><loc_48><loc_27></location>On the other hand there is no difficulty in setting ∆ ν glyph[greatermuch] Ω, providing that there are no significant signal components at delays greater than 1 / ∆ ν .</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_23></location>Although the cyclic spectrum is normally computed on a rectangular grid, values at large | α | and | ν | may not contain any information. If the voltage signal has a bandwidth B , sampled at the Nyquist rate, then the resulting cyclic spectrum is only valid within a diamond-shaped region around the origin, with | α/ 2 | + | ν | < B/ 2 (Demorest 2011). We also note that there cannot be more information in the cyclic spectrum than was present in the sampled voltage signal from which it was derived. Thus if the cyclic spectrum includes pulse harmonic numbers m> N p (the number of pulses averaged-over), then the pixels in the cyclic spectrum may not be statistically</text>
<text><location><page_4><loc_8><loc_84><loc_48><loc_92></location>∆ α = 1 /P , where P is the pulse period. Thus for ∆ t = P we have a time-bandwidth product of unity - a single sample of the signal - and σ = S sys . Equation 11 is then appropriate to a single pulse, and if the cyclic spectrum is averaged over N p pulses the variance is smaller by a factor 1 / N p .</text>
<section_header_level_1><location><page_4><loc_9><loc_82><loc_47><loc_83></location>3. DIRECT CONSTRUCTION OF FILTER AND PROFILE</section_header_level_1>
<text><location><page_4><loc_8><loc_75><loc_48><loc_81></location>We now turn to the task of estimating the filter function (ISM impulse response) and the intrinsic (unscattered) pulse profile starting from a measured cyclic spectrum. We can approach both of these tasks by iteration, as we now describe.</text>
<section_header_level_1><location><page_4><loc_16><loc_73><loc_41><loc_74></location>3.1. Determining the filter function</section_header_level_1>
<text><location><page_4><loc_8><loc_65><loc_48><loc_72></location>Suppose we have a model for S x , but we have incomplete knowledge of H . If we know the value of H at a single frequency, ν 1 , we can determine its value at nearby frequencies using the measured cyclic spectrum in the vicinity of ν 1 , thus:</text>
<formula><location><page_4><loc_16><loc_62><loc_48><loc_65></location>H ( ν 1 + α ) glyph[similarequal] D ( α, ν 1 + α/ 2) H ∗ ( ν 1 ) S x ( α ) . (15)</formula>
<text><location><page_4><loc_8><loc_55><loc_48><loc_61></location>We can make a better estimate of H at a given frequency if we know several nearby values of H . Making the replacement ν → ν -α/ 2 in eq. (2), multiplying by H ( ν -α ) S ∗ x ( α ) and summing yields</text>
<text><location><page_4><loc_18><loc_53><loc_18><loc_53></location>glyph[negationslash]</text>
<formula><location><page_4><loc_11><loc_50><loc_48><loc_55></location>H ( ν ) = ∑ α =0 D ( α, ν -α/ 2) H ( ν -α ) S ∗ x ( α ) ∑ α =0 | H ( ν -α ) | 2 | S x ( α ) | 2 , (16)</formula>
<text><location><page_4><loc_21><loc_50><loc_21><loc_51></location>glyph[negationslash]</text>
<text><location><page_4><loc_8><loc_40><loc_48><loc_49></location>where we have used the data, D , as our estimate of S z . This equation allows us to construct H , in regions where it is unknown, from nearby regions where it has already been determined, providing only that we have formed an estimate of S x . We note that equation (16) includes equation (15) as a special case where H is known only at a single frequency.</text>
<text><location><page_4><loc_8><loc_30><loc_48><loc_39></location>Although the development in this section has focused on the idea of obtaining an estimate of H at frequencies where it is not known, it is clear that one could employ equation (16) even if we already have an estimate of H ( ν ) for all frequencies, so it can also be viewed as a procedure for updating an existing model of H . We will return to this idea in § 3.3 and § 4.</text>
<section_header_level_1><location><page_4><loc_14><loc_28><loc_42><loc_29></location>3.2. Determining the intrinsic spectrum</section_header_level_1>
<text><location><page_4><loc_8><loc_24><loc_48><loc_28></location>Now suppose that we have a model for H , what then do the data tell us about S x ? Multiplying equation (2) by H ∗ ( ν + α/ 2) H ( ν -α/ 2) and summing over ν gives</text>
<formula><location><page_4><loc_10><loc_18><loc_48><loc_23></location>S x ( α ) = ∑ ν D ( α, ν ) H ( ν -α/ 2) H ∗ ( ν + α/ 2) ∑ ν | H ( ν -α/ 2) | 2 | H ( ν + α/ 2) | 2 , (17)</formula>
<text><location><page_4><loc_8><loc_7><loc_48><loc_17></location>where, again, we have used the data, D , as our estimate for S z . Thus, given data and a model for the filter function, we can obtain an estimate of the intrinsic pulse profile implied by the observed cyclic spectrum. Notice that this formula implies a unique estimate of S x associated with any given pair D,H . We shall see in § 4 that equation (17) is actually the optimum estimate of S x , in a least-squares sense, given the data D and the filter H .</text>
<section_header_level_1><location><page_4><loc_67><loc_91><loc_77><loc_92></location>3.3. Bootstrap</section_header_level_1>
<text><location><page_4><loc_52><loc_75><loc_92><loc_90></location>From the foregoing we can see that it is straightforward to form an estimate of H given S x , and vice versa. But initially we might not know either. In this situation it is natural to proceeed iteratively, starting with crude estimates and then using equations (16) and (17) repeatedly to improve those estimates. One way of starting the process is to initialize the intrinsic cyclic spectrum to S x ( α ) ← 〈 D ( α, ν ) 〉 ν , i.e. the observed (scattered) pulse profile. This corresponds to the model H ( ν ) = 1 and we could commence iteration of equations (16) and (17) using this approximation.</text>
<text><location><page_4><loc_52><loc_57><loc_92><loc_75></location>Alternatively, having specified our initial estimate of S x we can build up our estimate of H gradually, using equation (16), starting from an estimate of its value at a single frequency, H ( ν 1 ). Because the overall phase of H is arbitrary ( § 2.1) we are free to choose the phase of H ( ν 1 ), e.g. phase zero, so only | H ( ν 1 ) | need be specified in order to start the iteration. One possible initialization is thus H ( ν 1 ) ← √ | D (Ω , ν 1 ) /S x (Ω) | , and from there we can gradually build H over the full range of radio frequencies, with information flowing outward from ν 1 towards the edges of the band. In this approach one simply initializes H to zero for frequencies where no estimate has previously been made, so that those frequencies make no contribution to the estimator in equation (16).</text>
<text><location><page_4><loc_52><loc_47><loc_92><loc_56></location>Once this is done we can improve our estimate of the intrinsic cyclic spectrum, S x , by application of equation (17), then we can get a better estimate of H by applying equation (16), and these iterations can be repeated. Thus, if we know neither S x nor H , we can build bootstrap estimates for both of these quantities, given a measured cyclic spectrum.</text>
<text><location><page_4><loc_52><loc_35><loc_92><loc_47></location>The procedure just described is the method which we initially used to solve for H and S x , from the first measured cyclic spectra of a radio pulsar (i.e. the data used in § 5). Broadly speaking the method works: we found that it provided a good representation of much of the structure in the cyclic spectra, and the intrinsic profile was significantly narrower than the scattered profile (see figure 3 in Demorest, 2011). But it did also exhibit some deficiencies, as we describe below.</text>
<section_header_level_1><location><page_4><loc_59><loc_33><loc_85><loc_34></location>3.4. Deficiencies of the direct method</section_header_level_1>
<text><location><page_4><loc_52><loc_7><loc_92><loc_32></location>One problem which we anticipated is the difficulty of constructing H in regions where | H | is small. In these regions the solution for H is sensitive to noise in the data. In particular it is susceptible to phase jumps at points where | H | → 0: the solutions on either side of the zero can be mutually inconsistent. There are two reasons why this problem arises. One is fundamental: a zero in | H | is an absence of phase information at that particular point, and this cannot be overcome by using different methods of solution. The other reason is specific to the solution method we have presented: the summation in equation (16) includes information coming from both sides of the zero, so each side tries to rotate the phases of the other in order to bring about consistency, but neither side succeeds. In other words, phase discontinuities at zeros of | H | constitute traps for this method of solution. It is not necessary for | H | to be precisely zero in order for a trap to form; it suffices for the signal-to-noise ratio to be low ( glyph[lessorsimilar] 1 on a per-channel basis). Trapping was indeed</text>
<text><location><page_5><loc_8><loc_88><loc_48><loc_92></location>observed in the results we obtained using the approach described above, with significant residuals commonly occuring in the vicinity of points where | H | is small.</text>
<text><location><page_5><loc_8><loc_69><loc_48><loc_88></location>It is clearly possible to modify the solution method so as to be less susceptible to these traps. Most obviously, one can restrict the summations in equation (16) to values of α with a single sign - so that we are only using the information from frequencies > ν (or < ν ) in our estimate for H ( ν ). In this scheme information flows in only one direction across the zeros, so one side dictates phase to the other. In practice we observed that this modification did decrease the prevalence of trapping. However, in using only one sign of α we are ignoring half of the information available to constrain H at any given value of ν , so the resulting solution for H cannot be optimum. In the next section we present methods for obtaining the best fit solutions for H and S x .</text>
<section_header_level_1><location><page_5><loc_10><loc_67><loc_46><loc_68></location>4. OPTIMUM ESTIMATES OF FILTER AND PROFILE</section_header_level_1>
<text><location><page_5><loc_8><loc_61><loc_48><loc_66></location>In estimating H and S x what we really want are the models which best fit the data, so we have an optimization problem. We introduce the residual between model and data:</text>
<formula><location><page_5><loc_18><loc_59><loc_48><loc_60></location>R ( α, ν ) ≡ S z ( α, ν ) -D ( α, ν ) , (18)</formula>
<text><location><page_5><loc_8><loc_56><loc_48><loc_57></location>and we seek to minimize the magnitude of these residuals.</text>
<text><location><page_5><loc_8><loc_40><loc_48><loc_56></location>Suppose our data, D , have N ν radio-frequency channels, and N α modulation-frequency bins. In this case we are modelling a filter with N ν complex unknowns, and an intrinsic cyclic spectrum with N α / 2 complex unknowns. (The pulse profile is a real quantity, so the spectrum at negative modulation frequencies is simply the complex conjugate of that at positive frequencies.) Thus there are N ν +N α / 2 complex unknowns and ∼ N ν × N α / 2 complex constraints provided by the data, so for N ν , N α glyph[greatermuch] 1 the model is over-determined. In this situation we cannot make the residuals zero everywhere and we simply aim to make them small.</text>
<text><location><page_5><loc_8><loc_38><loc_48><loc_40></location>Here we follow the usual practice of minimizing the sum-of-squares of the residuals</text>
<formula><location><page_5><loc_22><loc_34><loc_48><loc_37></location>M ≡ ∑ ν,α =0 R ∗ R, (19)</formula>
<text><location><page_5><loc_28><loc_34><loc_28><loc_35></location>glyph[negationslash]</text>
<text><location><page_5><loc_8><loc_30><loc_48><loc_32></location>with respect to all of the model parameters. We then have</text>
<formula><location><page_5><loc_21><loc_26><loc_48><loc_30></location>∂M ∂q = 2 ∑ ν,α =0 R ∂S ∗ z ∂q , (20)</formula>
<text><location><page_5><loc_29><loc_26><loc_29><loc_27></location>glyph[negationslash]</text>
<text><location><page_5><loc_8><loc_23><loc_48><loc_25></location>where q represents any of the model parameters which define H and S x , and minimization of M implies</text>
<formula><location><page_5><loc_25><loc_19><loc_48><loc_22></location>∂M ∂q = 0 (21)</formula>
<text><location><page_5><loc_8><loc_17><loc_24><loc_18></location>for every parameter q .</text>
<text><location><page_5><loc_8><loc_10><loc_48><loc_16></location>We compute the derivative for each parameter in turn. Each value of H and S x is complex and thus involves two distinct real parameters. We take these to be the real and imaginary parts of the coefficients. For S rm := Re { S x ( α m ) } , S im := Im { S x ( α m ) } , we have</text>
<formula><location><page_5><loc_17><loc_6><loc_48><loc_9></location>∂M ∂S rm + i ∂M ∂S im = ∇ S M ∣ ∣ α = α m , (22)</formula>
<text><location><page_5><loc_52><loc_91><loc_56><loc_92></location>where</text>
<formula><location><page_5><loc_54><loc_87><loc_92><loc_90></location>∇ S M := 4 ∑ ν R ( α, ν ) H ( ν -α/ 2) H ∗ ( ν + α/ 2) . (23)</formula>
<text><location><page_5><loc_52><loc_85><loc_92><loc_86></location>And for H rk := Re { H ( ν k ) } , H ik := Im { H ( ν k ) } , we have</text>
<formula><location><page_5><loc_61><loc_81><loc_92><loc_84></location>∂M ∂H rk + i ∂M ∂H ik = ∇ H M ∣ ∣ ν = ν k , (24)</formula>
<text><location><page_5><loc_52><loc_79><loc_56><loc_80></location>where</text>
<formula><location><page_5><loc_54><loc_76><loc_92><loc_79></location>∇ H M := 4 ∑ α =0 R ( α, ν -α/ 2) H ( ν -α ) S ∗ x ( α ) . (25)</formula>
<text><location><page_5><loc_63><loc_76><loc_63><loc_77></location>glyph[negationslash]</text>
<text><location><page_5><loc_52><loc_63><loc_92><loc_75></location>Having determined a demerit function, M , and the gradient of M with respect to each of the parameters of interest, we are in a position to employ one of various standard methods (e.g. Nocedal and Wright 1999) to the problem of optimizing our solutions. Before turning to the choice of method, and the details of its application, it is helpful to establish the relationship between our 'direct' solutions of § 3 and the optimum estimates which we are seeking.</text>
<section_header_level_1><location><page_5><loc_54><loc_61><loc_90><loc_62></location>4.1. Relationship of direct solution to least-squares</section_header_level_1>
<text><location><page_5><loc_52><loc_53><loc_92><loc_60></location>We have already noted ( § 3) that our 'direct' procedure for constructing H - i.e. equation (16) - could also be regarded as an algorithm for updating H , given an existing estimate. Explicitly, the update is H → H +∆ H , where</text>
<text><location><page_5><loc_63><loc_51><loc_63><loc_51></location>glyph[negationslash]</text>
<formula><location><page_5><loc_53><loc_48><loc_92><loc_53></location>∆ H ( ν ) := -∑ α =0 R ( α, ν -α/ 2) H ( ν -α ) S ∗ x ( α ) ∑ α =0 | H ( ν -α ) | 2 | S x ( α ) | 2 . (26)</formula>
<text><location><page_5><loc_67><loc_48><loc_67><loc_49></location>glyph[negationslash]</text>
<text><location><page_5><loc_52><loc_44><loc_92><loc_47></location>We can also rewrite equation (17) as an update for the intrinsic spectrum, S x → S x +∆ S x , with</text>
<formula><location><page_5><loc_54><loc_37><loc_92><loc_44></location>∆ S x ( α ) := -∑ ν R ( α, ν ) H ( ν -α/ 2) H ∗ ( ν + α/ 2) ∑ ν | H ( ν -α/ 2) | 2 | H ( ν + α/ 2) | 2 . (27)</formula>
<text><location><page_5><loc_52><loc_27><loc_92><loc_37></location>In both cases we recognise the numerator to be (up to a constant factor) just the gradient of -M with respect to the corresponding parameters. This is comforting because it suggests that our 'direct' method is moving the estimates in a direction which will improve the model. To be confident that this is the case we need to gauge the step-size, not just its direction, and to achieve that it is helpful to evaluate the second derivatives of M .</text>
<text><location><page_5><loc_52><loc_23><loc_92><loc_27></location>The curvature of M with respect to our various parameters is given by differentiating equations 23 and 25. The results are</text>
<formula><location><page_5><loc_55><loc_15><loc_92><loc_22></location>∂ 2 M ∂S 2 rm = 4 ∑ ν | H ( ν + α m / 2) | 2 | H ( ν -α m / 2) | 2 , = ∂ 2 M ∂S 2 im , (28)</formula>
<text><location><page_5><loc_52><loc_13><loc_55><loc_15></location>and</text>
<text><location><page_5><loc_67><loc_9><loc_67><loc_10></location>glyph[negationslash]</text>
<formula><location><page_5><loc_59><loc_6><loc_92><loc_13></location>∂ 2 M ∂H 2 rk = 4 ∑ α =0 | H ( ν k -α ) | 2 | S x ( α ) | 2 , = ∂ 2 M ∂H 2 ik . (29)</formula>
<text><location><page_6><loc_8><loc_88><loc_48><loc_92></location>We can now see that for each of our real parameters, q , the 'direct' estimate in § 3 is an iteration with updates (equations 26, 27) ∆ q :</text>
<formula><location><page_6><loc_20><loc_84><loc_48><loc_87></location>∆ q = -[ ∂ 2 M ∂q 2 ] -1 ∂M ∂q . (30)</formula>
<text><location><page_6><loc_8><loc_78><loc_48><loc_83></location>This form is just Newton's method applied to each parameter separately. Equivalently: it is a simultaneous multi-parameter quasi-Newton method in which the off diagonal elements of the Hessian are neglected.</text>
<text><location><page_6><loc_8><loc_58><loc_48><loc_78></location>We can check whether or not this is a good approximation to the actual Hessian by explicitly computing the off-diagonal terms. In the case where both q i and q j relate to S x these off-diagonal elements are all zero. Furthermore, because the diagonal terms (equation 28) are independent of S x , all of the higher derivatives of M with respect to S x are zero - the hypersurface of M is quadratic in S x when H is fixed. This is no surprise because the residual (equation 18) is linear in S x , and M is quadratic in the residual. It follows that Newton's method yields an exact solution for S x in a single step. Thus we see that our direct estimate of S x , given in equation (17), is also the least-squares solution appropriate to the filter H and the data D ; no additional optimization steps are necessary.</text>
<text><location><page_6><loc_8><loc_49><loc_48><loc_58></location>Unfortunately this is not true of the filter function, H : neither the off-diagonal elements of the Hessian nor the higher order derivatives are zero in this case. The fact that the off-diagonal terms of the Hessian are nonzero means that we should not expect the filter update (equations 16, 26) to yield a good model. We now turn to the problem of optimizing our model filter function.</text>
<section_header_level_1><location><page_6><loc_18><loc_47><loc_39><loc_48></location>4.2. Optimisation of the filter</section_header_level_1>
<text><location><page_6><loc_8><loc_35><loc_48><loc_46></location>To optimize our model filter we can employ one of the established quasi-Newton methods, in which an approximate (inverse-) Hessian is constructed at each iteration, based on the local properties of the hypersurface M revealed in previous iterations (see, e.g., Nocedal and Wright 1999). A popular choice is the BroydenFletcher-Goldfarb-Shanno (BFGS) update, and that is the method we employ in § 5 and subsequently.</text>
<text><location><page_6><loc_8><loc_21><loc_48><loc_35></location>By utilizing a BFGS update to our estimate of H ( ν ) we expect to do significantly better than the update in equation 16. This is a general expectation, but it also applies to the particular problems noted in § 3.4: by including the non-zero off-diagonal curvatures of M we provide some of the information needed for the algorithm to escape the traps introduced by zeros in | H ( ν ) | . However, we ought to be able to do better still if we do not actually seek to construct H in frequency-space, where the traps are localised, but in the Fourier-space conjugate to frequency, i.e. lag-space.</text>
<text><location><page_6><loc_8><loc_16><loc_48><loc_21></location>We introduce the lag-space description of the filter, h ( τ ), which is related to the frequency-space description of the filter, H ( ν ), by the usual Fourier relationships for discretely sampled functions:</text>
<formula><location><page_6><loc_16><loc_12><loc_48><loc_15></location>h j ≡ h ( τ j ) = ∑ k H k exp[2 πiτ j ν k ] , (31)</formula>
<text><location><page_6><loc_8><loc_10><loc_11><loc_11></location>and</text>
<text><location><page_6><loc_52><loc_85><loc_92><loc_92></location>To optimize our model filter in lag-space we need to know the gradient of M with respect to the lag-space filter coefficients, h rj := Re { h ( τ j ) } and h ij := Im { h ( τ j ) } . Noting that { h j } and { H k } are different representations of the same information we can write</text>
<formula><location><page_6><loc_56><loc_81><loc_92><loc_85></location>∂M ∂h rj = ∑ k { ∂H rk ∂h rj ∂M ∂H rk + ∂H ik ∂h rj ∂M ∂H ik } , (33)</formula>
<text><location><page_6><loc_52><loc_78><loc_92><loc_80></location>and similarly for the derivative with respect to h ij . In this way we find</text>
<formula><location><page_6><loc_62><loc_74><loc_92><loc_77></location>∂M ∂h rj + i ∂M ∂h ij = ∇ h M ∣ ∣ τ = τ j , (34)</formula>
<text><location><page_6><loc_52><loc_72><loc_56><loc_73></location>where</text>
<formula><location><page_6><loc_60><loc_69><loc_92><loc_72></location>∇ h M := 1 N ν ∑ ν ∇ H M exp[2 πiτν ] . (35)</formula>
<text><location><page_6><loc_53><loc_67><loc_73><loc_68></location>Similarly one can show that</text>
<formula><location><page_6><loc_61><loc_63><loc_92><loc_66></location>∇ H M = ∑ τ ∇ h M exp[ -2 πiτν ] . (36)</formula>
<text><location><page_6><loc_52><loc_52><loc_92><loc_62></location>So as an alternative to computing the frequency-space derivatives and determining the lag-space derivatives from them, we can compute the lag-space derivatives first and then determine the frequency-space derivatives. Formally the two different paths to either frequency-space or lag-space derivatives are equivalent. In practice we computed the lag-space derivatives as our primary quantities, using</text>
<formula><location><page_6><loc_52><loc_48><loc_92><loc_51></location>∇ h M = 4 N ν ∑ ν,α =0 R ( α, ν ) S ∗ x ( α ) H ( ν -α/ 2) exp[ πiτ j (2 ν + α )] ,</formula>
<text><location><page_6><loc_61><loc_48><loc_61><loc_48></location>glyph[negationslash]</text>
<text><location><page_6><loc_89><loc_46><loc_92><loc_47></location>(37)</text>
<text><location><page_6><loc_52><loc_41><loc_92><loc_46></location>and determined the frequency-space derivatives, if required, using equation 36. There appears to be no significant difference in computation time between the two approaches.</text>
<section_header_level_1><location><page_6><loc_58><loc_38><loc_86><loc_39></location>4.3. Uncertainties in best-fit parameters</section_header_level_1>
<text><location><page_6><loc_52><loc_32><loc_92><loc_38></location>Suppose that we have obtained our best-fit model, the question then arises 'how accurate is that model?' To address this issue we need a description of the behaviour of the demerit, M , in the vicinity of the best fit.</text>
<text><location><page_6><loc_52><loc_26><loc_92><loc_32></location>At the best-fit point in parameter space, which we denote by { q jo } , ∇ S M = 0, and either ∇ H M = 0 or ∇ h M = 0. If M = M o at the best-fit point, then in the immediate neighbourhood of this point the variation of M can be approximated by</text>
<formula><location><page_6><loc_54><loc_21><loc_92><loc_25></location>M glyph[similarequal] M o + ∑ j,m 1 2 ∂ 2 M ∂q m ∂q j ( q m -q mo )( q j -q jo ) . (38)</formula>
<text><location><page_6><loc_52><loc_11><loc_92><loc_20></location>For Gaussian noise the normalized demerit, M/σ 2 , is distributed like χ 2 with N dof glyph[similarequal] (N ν -1)(N α -2) degrees of freedom, and we expect M o glyph[similarequal] N dof σ 2 . The fit becomes significantly worse if we move away from the optimum point to any other point such that M -M o = σ 2 (Avni 1976), and this contour of M delineates the range of uncertainties in our fit.</text>
<text><location><page_6><loc_52><loc_7><loc_92><loc_11></location>Uncertainties in the individual fit parameters can be readily determined if the Hessian, ∂ 2 M/∂q m ∂q j , is diagonal so that the parameters are all independent of each</text>
<formula><location><page_6><loc_12><loc_6><loc_48><loc_9></location>H k ≡ H ( ν k ) = 1 N ν ∑ j h j exp[ -2 πiτ j ν k ] . (32)</formula>
<text><location><page_7><loc_8><loc_89><loc_48><loc_92></location>other. In this case the standard deviation, δq j , is given by</text>
<formula><location><page_7><loc_19><loc_85><loc_48><loc_89></location>( δq j ) 2 = 2 σ 2 ( ∂ 2 M ∂q 2 j ) -1 . (39)</formula>
<text><location><page_7><loc_8><loc_67><loc_48><loc_84></location>If the Hessian is not diagonal then the parameters are covariant and it is a much more difficult task to describe the uncertainties in the fit. Because we know how M depends on each of the various parameters, we can evaluate the elements of the Hessian explicitly. Doing so we find that the Hessian is indeed diagonal with respect to the set of parameters describing S x , so equation 39 correctly describes the constraints which our model places on those parameters. However, the Hessian is not diagonal with respect to either { H k } or { h j } . The standard errors as given by equation 39 are evaluated in an Appendix, while in the next section we discuss parameter covariance.</text>
<section_header_level_1><location><page_7><loc_19><loc_64><loc_37><loc_65></location>4.3.1. Covariances of { h j }</section_header_level_1>
<text><location><page_7><loc_8><loc_51><loc_48><loc_63></location>Unfortunately the curvatures given in equation A8 are not the whole story when it comes to describing the uncertainties in the impulse-response function, because there are non-zero off-diagonal elements of the Hessian in respect of these parameters. It is beyond the scope of this paper to give a detailed description of the effect of these mixed curvature terms; here we only draw attention to their significance, deferring a thorough treatment to a later paper.</text>
<text><location><page_7><loc_8><loc_26><loc_48><loc_51></location>To illustrate the importance of the off-diagonal elements of the Hessian we employ the simplest filter model, H ( ν ) = 1. In this case we find by direct calculation that in addition to the leading diagonal (described by equation A8), there is a single reverse-diagonal on which the curvatures are non-zero. This reverse-diagonal cuts the leading diagonal at τ m = τ j = 0, and for | τ m -τ j | glyph[lessmuch] w , the pulse-width, the mixed curvatures are comparable in size to the diagonal elements. The upshot of this is that the combination of complex coefficients h ( τ j )+ h ∗ ( -τ j ) is tightly constrained, whereas the combination h ( τ j ) -h ∗ ( -τ j ) is poorly constrained. The former combination can be thought of as a pure-amplitude modification of the filter H ( ν ), whereas the latter is a purephase modification. And the fact that these particular combinations of parameters are well-constrained/poorlyconstrained for | τ m -τ j | glyph[lessmuch] w is directly attributable to the (in)sensitivity of H ( ν + α/ 2) H ∗ ( ν -α/ 2) to these types of modification.</text>
<section_header_level_1><location><page_7><loc_11><loc_23><loc_45><loc_24></location>5. IMPLEMENTATION OF FILTER OPTIMIZATION</section_header_level_1>
<text><location><page_7><loc_8><loc_8><loc_48><loc_23></location>Having already established that the simple quasiNewton method of § 3 works tolerably well for our optimization problem, even though all the off-diagonal elements of the Hessian are neglected, our next step is to implement a more sophisticated quasi-Newton method, the BFGS algorithm, to optimize our filter coefficients. More precisely, because of the large number of parameters ( ∼ 10 4 ) needed to describe the filter coefficients, we utilize a 'limited memory' algorithm, which we call L-BFGS, in which the full inverse-Hessian is not constructed (Nocedal 1980; Liu and Nocedal 1989).</text>
<text><location><page_7><loc_10><loc_7><loc_48><loc_8></location>We employed the L-BFGS algorithm coded in the</text>
<text><location><page_7><loc_52><loc_77><loc_92><loc_92></location>NLopt library 2 (Steven G. Johnson, 'The NLopt nonlinear-optimization package' ). The NLopt package was chosen because it is free, portable and offers a wide variety of optimization algorithms (see § 5.3.1). In addition we utilized the FFTW Fourier Transform package 3 from the same group (Frigo and Johnson 2005). Our code is written in C and is freely available. 4 It makes use of the PSRCHIVE library 5 (Hotan et al 2004; van Straten et al 2012) for file input and therefore can accept data in a variety of formats, including the standard PSRFITS pulsar data format (Hotan et al 2004).</text>
<text><location><page_7><loc_52><loc_60><loc_92><loc_77></location>Perhaps the first point to make, here, is that we have chosen to optimize the filter coefficients separately from the parameters which describe our model of the intrinsic cyclic spectrum, S x ( α ). There are several reasons for this choice. The strongest motivation is that it allows us to enforce a common timing reference on all our filter solutions, by using the same intrinsic cyclic spectrum throughout. A common timing reference is of paramount importance for all astrophysical studies which rely on pulse arrival-time measurements. Furthermore, by using a common timing (pulse-phase) reference, we can obtain a high signal-to-noise ratio measurement of the intrinsic spectrum by averaging over all our data.</text>
<text><location><page_7><loc_52><loc_46><loc_92><loc_60></location>The degeneracies discussed in § 2.1 provide further, minor motivations for separate optimization of filter and intrinsic cyclic spectrum models, as these degeneracies must be eliminated in order for any algorithm to identify the best fit solution. For the overall normalization and phase of the filter that is fairly straightforward, but controlling the degeneracy in phase-gradient is not so easy if both H and S x are simultaneously adjusted. By contrast, there is no degeneracy in phase-gradient if S x is fixed.</text>
<text><location><page_7><loc_52><loc_29><loc_92><loc_46></location>We noted in § 2.1 that the overall phase of H is always arbitrary, and this degeneracy must be eliminated before we can determine the model filter which best fits the data. We remove this freedom by forcing the imaginary part of h ( τ ) (or H ( ν )) to be zero at the point where | h ( τ ) | (or | H ( ν ) | ) attains its largest value. Because this choice is arbitrary, once an optimized filter is obtained we are free to rotate its overall phase to any preferred value. If we have a temporal sequence of filters (see § 7.5), the appropriate choice of phase for a given filter is the one which yields the closest match between the current and the previous (or subsequent) filter, leaving only a single, arbitrary phase for the whole temporal sequence.</text>
<section_header_level_1><location><page_7><loc_66><loc_27><loc_78><loc_28></location>5.1. Initialisation</section_header_level_1>
<text><location><page_7><loc_52><loc_13><loc_92><loc_26></location>We make use of two different initializations, which we refer to as 'Unit' and 'Proximate'. In the case of Unit initialization we begin with | H ( ν ) | = 1, for all radio frequencies, and a constant phase gradient in H ( ν ), chosen to match the mean phases seen in the data at α = Ω. For lag-space optimization this initialization corresponds to a delta-function model for h ( τ ). Naturally, Unit initialization is only sensible if the overall normalization of our model S x ( α ) is consistent with that of the data, D ( α, ν ), and we therefore also normalize S x ( α ) appropriately.</text>
<text><location><page_8><loc_8><loc_73><loc_48><loc_92></location>Unit initialization is appropriate if we have no prior information on the actual structure which is present in the filter function at the time the cyclic spectrum was recorded. Usually there are many cyclic spectra recorded during a single epoch of observation - e.g. in § 6 we present data from three separate epochs of observation, totalling several hundred cyclic spectra. In such cases the averaging time for each spectrum is chosen to be small enough that the changes in the filter function between adjacent cyclic spectra are small. Consequently, if we have already optimized the filter appropriate to one cyclic spectrum then that model provides us with a good starting point for modelling the next filter function: that scheme is what we refer to as Proximate initialization.</text>
<section_header_level_1><location><page_8><loc_20><loc_71><loc_36><loc_72></location>5.2. Stopping criterion</section_header_level_1>
<text><location><page_8><loc_8><loc_58><loc_48><loc_70></location>At what point should we stop the optimization? The NLopt algorithms include various criteria which may be used to recognise that the optimization is complete. Our aim is to find the minimum of M , but we do not know ahead of time the precise value of that minimum, so a natural choice of stopping criterion is that M should change by less than a certain, small fractional value during a single iteration of the algorithm. We can determine what that fractional tolerance should be as follows.</text>
<text><location><page_8><loc_8><loc_49><loc_48><loc_58></location>In § 2.3 we gave expressions for the variance of D ( α, ν ). In particular we noted that var { D } = σ 2 , a constant, is usually a good approximation in practice. Furthermore, at large modulation frequencies, α glyph[greatermuch] Ω, the noise is usually much larger than the signal we're interested in, so it is straightforward to get an estimate of σ 2 directly from the data.</text>
<text><location><page_8><loc_8><loc_33><loc_48><loc_49></location>For Gaussian noise, which is appropriate to the thermal noise component, we expect the best-fit value of M to conform to a χ 2 distribution, with N dof glyph[similarequal] (N ν -1)(N α -2) degrees of freedom. In this case the minimum demerit is expected to be M min glyph[similarequal] N dof σ 2 , and σ 2 is a significant change in M (Avni 1976), so it is appropriate to stop the optimization once the changes in M are small compared to σ 2 . This translates directly into the requirement that fractional changes in M should be small compared to 1 / N dof . Therefore in this paper the usual stopping criterion is that the fractional change in M should be less than 0 . 1 / N dof .</text>
<text><location><page_8><loc_8><loc_19><loc_48><loc_33></location>If the noise is not uniform - e.g. at the edges of the band, where the instrumental response rolls off, or because of strong Radio Frequency Interference - one can determine the variance at each point in the cyclic spectrum using equation 11. In this case the residuals (equation 19) should be normalized by the variance at each point ( α, ν ) prior to summation. The resulting figure of merit will then be distributed like χ 2 . It is straightforward to measure the noise variation across the band, as per § 6.1 (see the top panel in figure 1).</text>
<section_header_level_1><location><page_8><loc_15><loc_17><loc_41><loc_18></location>5.3. Choice of optimization approach</section_header_level_1>
<text><location><page_8><loc_8><loc_10><loc_48><loc_16></location>The various tests described in the Appendix demonstrate that, of the various optimization approaches we tried, the best method for this problem is L-BFGS in lag-space from Proximate initial conditions; we therefore utilize that method.</text>
<text><location><page_8><loc_52><loc_61><loc_92><loc_92></location>All of the data utilized in this paper are Arecibo observations of PSR B1937+21 (Backer et al 1982), at radio frequencies close to 428 MHz. The ATNF Pulsar Catalogue 6 (Manchester et al 2005) reports the following characteriztics for this pulsar: a period of 1.558 ms, a dispersion measure of 71 pc cm -3 (Cognard et al 1995), and a mean 400 MHz flux of 240 mJy (Foster, Fairhead and Backer 1991). Most of the data we use come from a single 4 MHz band centered on 428 MHz, with the exception being an additional 4 MHz chunk, centered on 432 MHz, that we use exclusively in § 6.3 (intrinsic pulse profile determination). We observed at three different epochs: MJD53791, MJD53847 and MJD53873. Dualpolarization voltages were recorded for intervals of order an hour at each epoch, using the Arecibo Signal Processor baseband recorder (ASP; Demorest 2007), with digitisation at 8-bits per sample. This high dynamic range sampling proved valuable in mitigating the effects of Radio Frequency Interference ( § 6.4). We did not attempt any polarization calibration for our data; all the results reported here are based on summing the two polarizations (i.e. the orthogonal feeds of the telescope), as an approximation to Stokes-I.</text>
<text><location><page_8><loc_52><loc_39><loc_92><loc_61></location>Individual cyclic spectra were generated from the recorded voltages, using the method described by Demorest (2011). In our first processing of the data we constructed cyclic spectra, averaged over 15 seconds, with 6230 radio-frequency channels and 511 pulse-phase bins. These values were chosen so as to make ∆ ν as nearly equal to ∆ α as possible, because our first attempts at modelling H and S x (using the method described in § 3), avoided interpolations. However, the improved fitting method described in §§ 4,5 employs precise interpolation, so it is no longer necessary to match the resolutions in this way. Nor is it preferred, as array sizes which are integer powers of two are better matched to the Fast Fourier Transform algorithm, which we utilize. All the tests of our optimization software, reported in an Appendix, were conducted on the cyclic spectra obtained in our first processing of the data.</text>
<text><location><page_8><loc_52><loc_25><loc_92><loc_39></location>Analysis of the cyclic spectra from our first processing revealed some leakage at the edges of the bandpass filter. This is undesirable, particularly because any out-of-band signal is aliased by ± 4 MHz, and will thus appear delayed by approximately ± 30 ms due to incorrect dedispersion. In turn this leaked signal may introduce low-level contamination into our profile estimates or our filter models, or both. We therefore decided to completely reprocess our data, to deal with the leakage and to correct some other minor defects which we were aware of.</text>
<text><location><page_8><loc_52><loc_11><loc_92><loc_25></location>In the second processing we produced cyclic spectra averaged over 15 seconds, with 4608 channels and 1024 pulse-phase bins. This reprocessing utilized the cyclic spectrum implementation now freely available as part of the DSPSR software package 7 (van Straten and Bailes 2011). To eliminate the aliased (leakage) signals we then trimmed the spectral array down to 4096 channels, so the final bandwidth was approximately 3 . 56 MHz. With the exception of § 6.1 and § 6.2, all of the results presented in this section were obtained using the trimmed cyclic spectra from the second processing of our data.</text>
<figure>
<location><page_9><loc_9><loc_53><loc_48><loc_92></location>
<caption>Figure 1. Two estimates of the amplitude of the instrumental bandpass filter for the ASP baseband recorder. The upper plot shows a traditional estimate for the bandpass, formed from the square-root of the total power √ 〈 D (0 , ν ) 〉 , averaged over the data taken on MJD53791. The lower plot shows the result of averaging | H ( ν ) | over all three epochs of observation. The vertical, dashed lines in the lower plot delimit the regions which we trimmed off, to eliminate aliased signals leaking in at the edges of the band.</caption>
</figure>
<section_header_level_1><location><page_9><loc_21><loc_41><loc_36><loc_42></location>6.1. Bandpass Filter</section_header_level_1>
<text><location><page_9><loc_8><loc_31><loc_48><loc_40></location>If we want to know the profile of the bandpass filter of our instrument there are two methods available to us: we can measure the average total power as a function of radio-frequency, or we can make use of the filter functions, H , obtained from our fitting. (One can also inject artificial pulsed power, of known spectral shape, into the signal chain, but we did not record such data.)</text>
<text><location><page_9><loc_8><loc_10><loc_48><loc_31></location>Our estimates of H incorporate all of the filtering imposed on the signal. We expect there to be contributions from the ISM, the Solar wind, the Earth's ionosphere, and from our instrument (telescope, front-end and backend). Of these various contributions, only the receiver system is expected to be stable over long time-scales. As H is a complex quantity, averaging it will yield zero, but we can instead form 〈| H ( ν ) |〉 , which we take as an estimate of the bandpass filter, | H rec ( ν ) | . Averaging over all filter solutions for all three epochs we obtain the result shown in figure 1 for | H rec ( ν ) | . Also shown in figure 1 is the result of estimating the bandpass in a more traditional way, using the square-root of the average total power: √ 〈 D (0 , ν ) 〉 . (The square-root appears here because the power is a quadratic function of the filter response.)</text>
<text><location><page_9><loc_8><loc_7><loc_48><loc_9></location>Although H ISM fluctuates quite rapidly, the amplitude of those fluctuations is large, so a long total observation</text>
<text><location><page_9><loc_52><loc_75><loc_92><loc_92></location>time is required in order to form an accurate estimate of | H rec ( ν ) | . With our three epochs combined we have approximately 4.5 hours of data, and the scintillation time-scale is of order a minute so we expect our estimate of the filter response to be accurate to ∼ 6%. That is approximately the level of fluctuation seen in our estimate of | H rec ( ν ) | across most of the band. Thus the only clearly significant structure we find in | H rec ( ν ) | is the roll-off of the filter at the band edges. A cause for concern is the abrupt rise in the estimated filter response at both extremes of the frequency range. These upturns indicate that that there is some leakage of signal from outside the nominal band of the filter.</text>
<text><location><page_9><loc_56><loc_55><loc_56><loc_56></location>glyph[negationslash]</text>
<text><location><page_9><loc_52><loc_52><loc_92><loc_75></location>By contrast with | H rec ( ν ) | , the estimate √ 〈 D (0 , ν ) 〉 shows evidence of an upturn only at one end of the band. The reason for this difference is unclear. The other points of distinction between the two results are (i) that the noise on the traditional estimate is much smaller, even though only a third as much data was used, and (ii) Radio Frequency Interference (RFI) is manifest in the traditional estimate. To some extent the effect of the RFI could be mitigated by averaging using the median estimator, rather than the mean, but this would not help for steady interference. The reason for the lower noiselevel on the traditional bandpass estimate can be seen from equation (10). Our solutions for H ( ν ) - whence the | H rec ( ν ) | estimate - are based on the pulsed power, i.e. α = 0, whereas the zero-modulation-frequency data, D (0 , ν ), are dominated by the system noise, N ( ν ), which is both large and unpulsed.</text>
<text><location><page_9><loc_52><loc_41><loc_92><loc_52></location>As mentioned at the start of § 6, the leakage at the band edges, most evident in the lower panel of figure 1, can introduce low-level artifacts into our filter or pulse-profile estimates. Consequently we decided to fully reprocess our data, trimming off the edges of the band as we did so. The results described in § 6.3 and later sections of this paper were obtained from the second processing in which the spectral band was trimmed.</text>
<section_header_level_1><location><page_9><loc_55><loc_39><loc_88><loc_40></location>6.2. Bootstrap approach to the intrinsic profile</section_header_level_1>
<text><location><page_9><loc_52><loc_24><loc_92><loc_39></location>Lacking prior knowledge of the intrinsic pulse profile we are obliged, as in § 3.3, to commence our modelling using the observed, scattered pulse profile as an approximation to the intrinsic profile. We then obtain our first model of the filter function, for each sample cyclic spectrum, by fitting to the data in the way described in §§ 4,5. The filters obtained in this way are then used to obtain a better estimate of the intrinsic pulse profile, and the whole process is iterated, obtaining better approximations to S x , and the various H , on each pass through the data.</text>
<text><location><page_9><loc_52><loc_15><loc_92><loc_24></location>Once an accurate model of the intrinsic profile is obtained, other data-sets for the same pulsar taken with the same instrumental configuration can use that profile to obtain model filters in a single pass through the data. But new instrumental configurations - e.g. different observing frequencies - may force a return to the bootstrap approach.</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_15></location>Because it requires multiple passes through the data, a bootstrap can be slow. We can, however, speed things up to some degree because at the second and subsequent profile-iterations we already have available a set of model filters appropriate to each recorded cyclic spectrum. These filters can be used to initialize subsequent</text>
<text><location><page_10><loc_8><loc_84><loc_48><loc_92></location>models prior to optimization. As our model of the intrinsic pulse profile approaches the true intrinsic profile we expect the model filters to change very little between successive iterations, so this procedure should accelerate the optimization substantially. This expectation was borne out in practice, as we now describe.</text>
<text><location><page_10><loc_8><loc_59><loc_48><loc_84></location>To enable a rapid approach to the intrinsic profile we initially used a subset of the data (roughly 20 minutes of observation) from one epoch (MJD53873), iterating several times on this subset, and then adding in the rest of the data from this epoch in order to improve the signalto-noise ratio of our profile estimate. For the first set of filter solutions, using Proximate initialization, we found that on average 289 NLopt steps were required to fit each cyclic spectrum in the data subset. Subsequently, using the filter models obtained at the previous iteration as our starting point, the number of NLopt steps declined to 222, 17 and 14 for the second, third and fourth iterations, respectively. 8 The small decrease in the required number of steps between the third and fourth iterations, contrasting with the large decrease between the second and third iterations, suggested that we had reached the noise floor of the data subset, so for subsequent iterations we utilized all of the data from MJD53873 - a total of approximately 2 hours.</text>
<text><location><page_10><loc_8><loc_47><loc_48><loc_58></location>For iteration five we needed to obtain the first filter solutions for the bulk of the data from this epoch, using Proximate initialization, which required on average 241 NLopt steps per cyclic spectrum. But for all subsequent iterations we were able to initialize our models using the previous set of filter solutions. We found that iterations six and seven required only 12 and 3, respectively, 8 NLopt steps for each cyclic spectrum, indicating very rapid convergence of our estimate of the intrinsic pulse profile.</text>
<text><location><page_10><loc_8><loc_36><loc_48><loc_47></location>Separately we have observed, when using an existing set of optimized filter models as our starting point, that our code requires a minimum of 3 NLopt steps to return an optimized solution, even when the same reference profile is used for both solutions. We therefore conclude that our intrinsic profile estimates for B1937+21 do not differ significantly between iterations six and seven, and further iterations are unwarranted.</text>
<text><location><page_10><loc_8><loc_25><loc_48><loc_36></location>Use of the previous set of filter solutions to initialize our models clearly leads to a substantial saving in computation time. Using Proximate initialization we expect that the bootstrap would have required a total of 10 days of CPU time, whereas the sequence just described used only a third of that time. In fact our procedure needed only one quarter more time than a single pass through the same data using a given reference pulse profile.</text>
<section_header_level_1><location><page_10><loc_15><loc_20><loc_41><loc_21></location>6.3. Intrinsic versus scattered profile</section_header_level_1>
<text><location><page_10><loc_8><loc_10><loc_48><loc_19></location>In figure 2 we show our estimate of the intrinsic profile, together with the scattered profile, using all the data from MJD53873. This epoch was chosen because we obtained significantly more data on that date than on either of the other epochs. As expected, the intrinsic modulation profile of the signal is much sharper than the apparent modulation, because of the contribution of the</text>
<figure>
<location><page_10><loc_53><loc_51><loc_92><loc_92></location>
<caption>Figure 2. Intrinsic (red) and scattered (black) pulse profiles for B1937+21, at 428 MHz, observed on MJD53873. Two complete rotations are shown. The zero-point of the profile amplitude is arbitrarily chosen, whereas phase-zero corresponds to the peak of the (intrinsic) main-pulse. The top panel shows the full range of the pulse, while the lower panel shows a close-up of the lowest 3% of the profile.</caption>
</figure>
<figure>
<location><page_10><loc_52><loc_17><loc_91><loc_37></location>
<caption>Figure 3. Intrinsic (right-hand-side: positive harmonics) and scattered (left-hand-side: negative harmonics) pulsed-power vs. harmonic number for B1937+21, at 428 MHz, observed on MJD53873. The pulse profile is real, so the power-spectrum is an even-function of the harmonic number. Odd-numbered harmonics (i.e. α = (2 m + 1)Ω, with m an integer) are shown in black; even-numbered harmonics are shown in red.</caption>
</figure>
<figure>
<location><page_11><loc_9><loc_71><loc_48><loc_92></location>
<caption>Figure 4. Comparison of intrinsic pulse profiles derived independently for MJD53873 and MJD53791. The mean of these profiles is shown in the upper curve, while the difference is shown in the lower curve. The scaling of this plot is as for figure 2, so the full scale of the mean profile has a range of 100. For clarity of presentation, arbitrary vertical offsets have been applied to both the mean and the difference profiles. Two complete rotations are shown.</caption>
</figure>
<figure>
<location><page_11><loc_9><loc_40><loc_48><loc_60></location>
<caption>Figure 5. Comparison of intrinsic pulse profiles derived independently at 428 MHz (upper, black line) and 432 MHz (lower, black line) for MJD53791. The scaling of this plot is as for figure 1; arbitrary vertical offsets have been applied, for clarity, and two complete rotations are plotted. The red/blue curves show calculated profiles appropriate to leakage signals at the lower/upper edge of the 4 MHz band of the 428 MHz (upper curves) and 432 MHz (lower curves) data. These signals are delayed/advanced by roughly 30 ms, as a result of aliasing and the associated incorrect dedispersion. Normalisation of the red/blue curves is arbitrarily chosen. We find no indication of residual contamination by signal leakage in our results (see text, § 6.3).</caption>
</figure>
<text><location><page_11><loc_8><loc_22><loc_48><loc_25></location>scattered (delayed) waves to the apparent profile. The 'scattered tail' of the pulse is absent from our estimate of the intrinsic profile.</text>
<text><location><page_11><loc_8><loc_8><loc_48><loc_21></location>Figure 2 (lower panel) also reveals the presence of several low-level (a fraction of 1% of the peak height), but sharp features in the 'baseline' of the intrinsic pulse. These features are difficult to recognise in the scattered profile for two reasons. First, interstellar scattering broadens them, while decreasing the peak amplitude of each. Secondly, the features that are present immediately after the main-pulse or the inter-pulse are swamped by the delayed signal from those two, very strong components of the pulse profile.</text>
<text><location><page_11><loc_10><loc_7><loc_48><loc_8></location>An equivalent description of the pulse modulation is</text>
<text><location><page_11><loc_52><loc_79><loc_92><loc_92></location>available by Fourier-transforming the scattered and intrinsic profiles. The resulting harmonic powers are shown in figure 3, demonstrating that the high harmonics of the intrinsic profile contain a great deal more power than the scattered profile. This is just as expected. The scattered profile is a convolution of the intrinsic profile with the impulse response function, so in the Fourier domain the relationship is multiplicative, and the multiplier declines from near unity at low harmonic numbers to very small values at high harmonic numbers.</text>
<text><location><page_11><loc_52><loc_63><loc_92><loc_78></location>Because the low-level features evident in figure 2 are seen here for the first time at these radio frequencies, and the signal-processing we have used to reveal these structures is itself novel, we would like to have some confirmation of their reality. We have therefore undertaken a completely independent bootstrap estimate of the intrinsic profile for another epoch, MJD53791. In this comparison we are not interested in any timing (pulse-phase) offset between the two epochs, so in comparing the two intrinsic profiles we have applied a pulse-phase shift and a scaling, chosen so as to minimize the difference between the profiles.</text>
<text><location><page_11><loc_52><loc_47><loc_92><loc_62></location>The result of our two independent bootstrap solutions can be seen in figure 4, where we show the mean of the intrinsic profiles and their difference. The latter curve appears noise-like, without any clearly significant differences between the two, independently derived intrinsic profiles. In particular we note that the largest differences occur underneath the main-pulse and inter-pulse components, where the signal is very strong and the noise is therefore greater than at other pulse-phases. There is no apparent systematic difference between the two profiles at those pulse-phases where the weak, low-level features are seen.</text>
<text><location><page_11><loc_52><loc_32><loc_92><loc_47></location>As a final check on the reality of the features revealed in figures 2, 4, we have also compared the intrinsic pulse profiles obtained from independent bootstrap estimates at two different frequencies, 428 MHz and 432 MHz, for the epoch MJD53791 - this comparison is shown in figure 5. Although the 432 MHz data exhibit more system noise than the 428 MHz profile, because the integration time for the latter is larger by a factor of 1.5, the two profiles appear otherwise very similar in respect of the low-level features which are revealed by construction of the intrinsic profile.</text>
<text><location><page_11><loc_52><loc_7><loc_92><loc_32></location>An important aspect of the inter-band comparison in figure 5 is that it excludes signal leakage ( § 6.1) as a possible origin for the low-level structures which we see in the intrinsic profile. Even though we have trimmed the band edges, which should eliminate the bulk of that problem, it is possible that some traces of leakage remain. This concern is heightened by the fact that the sharp feature at a pulse-phase of 500 µ s lies close to the expected location of the aliased main pulse component, for signals leaking across the low-frequency edge of the 428 MHz band (upper red curve in figure 5). The inter-band comparison makes it plain that this is not a viable explanation for that feature, because at 432 MHz the corresponding alias should lie at 1,200 µ s, where no profile feature is seen - yet the observed 500 µ s peak appears very similar in the two bands. We also note that interpreting the 500 µ s feature as an alias of the main-pulse implies that there should be a counterpart feature from the interpulse, roughly half a turn later, whereas no such feature</text>
<text><location><page_12><loc_8><loc_87><loc_48><loc_92></location>is observed in either band. Overall, the aliased signals from the band edges do not correspond with the low-level features we see in the pulse profiles in either band, and we conclude that they are not due to out-of-band signals.</text>
<text><location><page_12><loc_8><loc_65><loc_48><loc_86></location>In fact residual out-of-band signals are expected to appear as broad structures in the time-domain, because the dispersive delay is a strong function of frequency. The sharpness of the features shown in the red and blue curves in figure 5 is due to the fact that only frequencies immediately adjacent to the band edges have been considered. The red and blue curves are simply calculated as delayed (and scaled) versions of the mean pulse profile, with the delay/advance equal to the difference in dispersive delay between the upper and lower edges of the band. At MJD53791 the dispersion measure of PSR B1937+21 was 71.023 pccm -3 , and the period was 1.5577 ms, so the aliased signals appear at ± 30 . 068 ms (428 MHz band) and ± 29 . 240 ms (432 MHz band). Modulo the pulse period these become, respectively, ± 0 . 470 and ± 1 . 201 milliseconds.</text>
<text><location><page_12><loc_8><loc_39><loc_48><loc_65></location>Some of the 'new' structure that we see in the intrinsic pulse profile corresponds well with features of B1937+21 which have been found by others, as follows. The distinct peaks seen immediately after the main- and inter-pulse have previously been observed by a number of authors at higher radio-frequencies, where the delayed, scattered signal is much weaker - see, particularly, figure 1 of Thorsett and Stinebring (1990). Here we are presumably seeing the emission regions which are responsible for the giant pulses of B1937+21 (Cognard et al 1996), and the consequently high modulation index at these pulsephases (Jenet, Anderson and Prince 2001). The sharp feature we see at a pulse-phase of 500 µ s (0 . 3 turns) has a counterpart which was noted in L-band observations by Yan et al (2011). Residual dispersion smearing in the Yan et al (2011) data is significant, so it is not surprising that their feature appears broader than the one we observe. Finally, the gradual rise we see in the 0 . 2 turns immediately preceding the main-pulse is also manifest in the Yan et al (2011) data.</text>
<text><location><page_12><loc_8><loc_30><loc_48><loc_38></location>The consistency of our intrinsic profile estimates across different epochs and spectral sub-bands, and the connections we can make between individual features and previous observations of B1937+21 at other frequencies, give confidence that the statistically-significant features we see in our intrinsic profile are indeed real.</text>
<section_header_level_1><location><page_12><loc_21><loc_27><loc_36><loc_28></location>6.4. Dynamic Spectra</section_header_level_1>
<text><location><page_12><loc_8><loc_10><loc_48><loc_27></location>A measured cyclic spectrum quantifies the power spectrum of the signal as the zero-modulation-frequency array D (0 , ν ) (see § 2). We compute our cyclic spectra with a cadence of 15 seconds, and thus we can trivially obtain a dynamic spectrum from the temporal sequence of D (0 , ν ). This dynamic spectrum is a simple timeaverage, not a difference of on-pulse and off-pulse power, so it includes all power contributions: noise from the receiver and the sky, the pulsar signal, and any terrestrial signals reaching the receiver, i.e. RFI. Because RFI can cause severe problems for some types of radio astronomical investigations, it is useful to examine the dynamic spectrum in order to gauge its impact.</text>
<figure>
<location><page_12><loc_55><loc_70><loc_91><loc_92></location>
<caption>Figure 6 shows the dynamic spectrum for a 512-channel spectral segment recorded on MJD53791; RFI is mani-</caption>
</figure>
<figure>
<location><page_12><loc_55><loc_46><loc_91><loc_68></location>
<caption>Figure 6. Inverted grey-scale image of the dynamic spectrum, D (0 , ν, t ) (lower panel), recorded on MJD53791, together with the corresponding dynamic filter power, | H ( ν, t ) | 2 (top panel). Only a fraction ( glyph[similarequal] 0 . 44 MHz) of the recorded bandwidth is shown; the temporal extent is approximately 98 mins. Two short gaps in the temporal coverage are visible as discontinuities, running horizontally in both images. Radio-Frequency Interference is evident in the dynamic spectrum as thin, black, vertical lines; but it is almost completely absent from the dynamic filter power.</caption>
</figure>
<text><location><page_12><loc_52><loc_19><loc_92><loc_33></location>fest in this segment as narrow spectral lines. None of these lines is so strong that the voltage signal exceeds the dynamic range of the sampler, nor is any impulsive RFI evident in figure 6. These aspects of the data reassure us that the observations were taken under relatively benign RFI conditions, and in this circumstance we can reasonably expect a high level of immunity from RFI in our models of S x and H . In particular, if the RFI is both accurately captured and not modulated at the frequency Ω = 1 /P , or its harmonics, then cyclic spectra will be free of RFI contamination.</text>
<text><location><page_12><loc_52><loc_7><loc_92><loc_19></location>To demonstrate that the observed RFI does not propagate into our model filters we also show in figure 6 the squared-modulus of the dynamic filter, i.e. | H ( ν, t ) | 2 . This quantity is our estimate of the contribution of the pulsar to the dynamic spectrum; the spectral structure | H ( ν, t ) | 2 can also be seen in the total power signal. It is evident that the RFI present in the total power signal is absent from the dynamic filter. We emphasise that the specific, small fraction of the spectrum shown in figure</text>
<figure>
<location><page_13><loc_9><loc_56><loc_48><loc_92></location>
<caption>Figure 7. The impulse response function, h ( τ ), for B1937+21 observed on MJD53873. The top panel shows the real part of h (linear scale, arbitrary normalization) for the first cyclic spectrum recorded at that epoch, while the lower panel shows the average 〈| h ( τ ) | 2 〉 (normalized by the maximum of 〈| h ( τ ) | 2 〉 ) over all the data taken at that epoch. The data in both plots covers a total range of 1,152 µ s in delay. The impulse response function itself is characterized by 4,096 complex coefficients, evenly spaced in lag.</caption>
</figure>
<text><location><page_13><loc_8><loc_43><loc_48><loc_46></location>6 was chosen at random: it was not selected because it displays good immunity from RFI.</text>
<section_header_level_1><location><page_13><loc_21><loc_41><loc_35><loc_42></location>6.5. Dynamic fields</section_header_level_1>
<text><location><page_13><loc_8><loc_24><loc_48><loc_40></location>Whereas the dynamic spectrum is a quantity which pulsar astronomers routinely measure, it has been much more difficult to get at the dynamic electric field because the latter requires information on the phases, and that information is usually not explicit in the measured intensities. The requisite phases can sometimes be retrieved - e.g. if the field is sparse in some representation - but to date this has been successfully demonstrated for only one dynamic spectrum (Walker et al 2008). By contrast, cyclic spectroscopy provides us with access to the electric field envelope, including the phase information; as such it is an intrinsically holographic method.</text>
<text><location><page_13><loc_8><loc_17><loc_48><loc_24></location>There are various possible representations of the dynamic fields because they may be described in terms of frequency-space (filter) or lag-space (impulse-response) coefficients, and the dynamic nature of the field can be represented either as a temporal sequence or in terms of the conjugate Fourier variable, i.e. a frequency.</text>
<section_header_level_1><location><page_13><loc_17><loc_14><loc_39><loc_15></location>6.5.1. Impulse-response functions</section_header_level_1>
<text><location><page_13><loc_8><loc_7><loc_48><loc_13></location>Figure 7 (top panel) shows one possible representation of the field: the real part of the impulse-response function, h ( τ ), determined from the first cyclic spectrum we observed on MJD53873. This function spans a lag range of 1 , 152 µ s, and we see that the amplitude of the re-</text>
<figure>
<location><page_13><loc_53><loc_63><loc_92><loc_92></location>
<caption>Figure 8. The squared-modulus of the delay-Doppler field image, | h ( τ, ω ) | 2 , for B1937+21 observed on MJD53873. The field intensity is represented as an inverse, logarithmic grey-scale, over a range of 50 dB (almost the full dynamic range of the image, which is 51 dB). In this image the vertical dimension is delay, spanning the range | τ | ≤ 576 µ s, and the horizontal dimension is Dopplershift, with | ω | ≤ 100 / 3 mHz.</caption>
</figure>
<text><location><page_13><loc_52><loc_42><loc_92><loc_54></location>onse falls off on lag-scale glyph[lessorsimilar] 50 µ s. There is, however, a low-level tail to the response, extending to lags that are a substantial fraction of the pulse period. To bring out these low-level features we took the modulus of the impulse response, and then averaged it over all the data at this epoch of observation. The lower panel of figure 7 presents the resulting 〈| h ( τ ) | 2 〉 , which demonstrates that the low-level tail of h continues out to delays of at least 400 µ s relative to the peak of the response.</text>
<text><location><page_13><loc_52><loc_34><loc_92><loc_42></location>At extreme negative lags there is an obvious rise in | h | . The origin of this feature is not completely clear; however, a preliminary analysis suggests that parameter covariances in { h j } (see Appendix) may give rise to increased noise near the lag limits of the cyclic spectra, and we therefore consider this to be an artifact.</text>
<text><location><page_13><loc_52><loc_29><loc_92><loc_34></location>On the other hand the features seen in the vicinity of τ ∼ +300 µ s appear to be bona fide structure in h . The delay-Doppler image, which we present in the next section, gives more information on these features.</text>
<section_header_level_1><location><page_13><loc_61><loc_26><loc_83><loc_27></location>6.5.2. Delay-Doppler field images</section_header_level_1>
<text><location><page_13><loc_52><loc_12><loc_92><loc_25></location>Finally we present our results in the Fourier domain conjugate to ( ν, t ). The conjugate variables ( τ, ω ) have immediate physical meaning as the delay and Dopplershift, respectively, that accumulate during propagation of the wave (Harmon and Coles 1983; Cordes et al 2006). The Fourier Transform, h ( τ, ω ), of the dynamic electric field, H ( ν, t ), is therefore a quantity of particular interest we call this the 'delay-Doppler image'. Figure 8 shows the squared-amplitude of the delay-Doppler image for our data taken on MJD53873.</text>
<text><location><page_13><loc_52><loc_7><loc_92><loc_12></location>The lower-half of figure 8 is largely free of signal, as expected for negative lags (which are acausal). The only signals that can be recognised at negative lags are the band of scattered power running horizontally across the</text>
<figure>
<location><page_14><loc_9><loc_63><loc_48><loc_92></location>
<caption>Figure 9. The power-spectrum of the dynamic-spectrum (often called the 'secondary spectrum'), |F{| H ( ν, t ) | 2 }| 2 , for B1937+21 observed on MJD53873. Power is represented as an inverse, logarithmic grey-scale, over a range of 50 dB. The full dynamic range of the secondary spectrum - i.e. the ratio of the peak value to the noise-floor - is 80 dB.</caption>
</figure>
<text><location><page_14><loc_8><loc_40><loc_48><loc_55></location>figure (discussed later), and a handul of thin, faint, vertical streaks in the region | ω | glyph[lessorsimilar] 4 mHz, 0 > τ glyph[greaterorsimilar] -100 µ s. We are uncertain as to the cause of these streaks, but we suspect that they may be sidelobes caused by the sharp truncation of the spectrum which we introduced by trimming the band ( § 6.1). These streaks were not seen in the delay-Doppler image that we obtained in our first processing of the data. The enhanced noise at extreme negative lags, plainly seen in the average signal in figure 7, is also present in figure 8 but is difficult to discern without averaging.</text>
<text><location><page_14><loc_8><loc_14><loc_48><loc_40></location>By contrast, in the upper half-plane of figure 8 there is an abundance of structure. Most of the power is concentrated in a broad distribution centered on zero-Dopplershift. And the overall distribution appears to have an approximately parabolic envelope, as is now familiar for many pulsars (Stinebring 2001; Cordes et al 2006). But there are also some discrete concentrations of power. Most evident of these are the concentrations in the range 200 glyph[lessorsimilar] τ ( µ s) glyph[lessorsimilar] 400 on the right-hand-side of the figure. These concentrations indicate that there are particular regions, within a few milli-arseconds of the direct lineof-sight to B1937+21, which are strongly diffracting, or refracting signals from this pulsar into our telescope. Apparently similar features were discovered by Hill et al (2005), in a multi-epoch study of PSR B0834+06, who found that their features appear to move through the delay-Doppler plane at constant velocity, consistent with the observed proper-motion of the pulsar. At present we don't know whether that property also holds for the features seen in figure 8.</text>
<text><location><page_14><loc_8><loc_7><loc_48><loc_13></location>In addition to the real structure just discussed, a strong artifact is plain in figure 8: around zero delay there is a broad, horizontal stripe in the image. The nature of this feature is clear: it is 'scattered power' caused by discontinuities between successive values of H ( ν ) (or h ( τ ))</text>
<text><location><page_14><loc_52><loc_81><loc_92><loc_92></location>in our temporal sequence. These discontinuities might arise in several ways, for example: inadequate sampling of the evolving H ( ν ); amplitude fluctuations in the pulsar; arbitrary phase rotations between successive filter solutions (per the degeneracy in overall phase, § 2.1); or gaps in the data record. We have considered each of the above possibilities, but none provides a satisfactory explanation, as we now detail.</text>
<text><location><page_14><loc_52><loc_59><loc_92><loc_81></location>First, the evolution of the filter H ( ν ) is well sampled by our 15-second cadence, as can be seen from the upper panel of figure 6. Secondly, there are ∼ 10 4 pulses within each of our cyclic spectra, so the variations in average intensity between samples will be small, glyph[lessorsimilar] 1%. In fact even this variation is irrelevant to figure 8 as we have normalized each filter solution such that it has a rootmean-square value of unity. Thirdly, the arbitrary phase of each filter (see § 2.1) has been chosen so that each solution H ( ν, t n ) matches the previous solution H ( ν, t n -1 ) as closely as possible, in a least-squares sense. Finally, although there is indeed a gap of 30 seconds in our temporal coverage (caused by a change of hard-disk during observing), we have interpolated across this gap before constructing figure 8. For these reasons we do not expect any of these effects to be responsible for the high levels of scattered power seen in figure 8.</text>
<text><location><page_14><loc_52><loc_28><loc_92><loc_58></location>Aclue to the origin of the scattered power can be found in figure 9, which shows the 'secondary spectrum' - i.e. the power-spectrum of the dynamic spectrum, | H ( ν, t ) | 2 - for our data. By contrast with figure 8, this quantity shows quite low levels of power scattered to large Doppler-shifts. In forming the spectrum, | H ( ν ) | 2 , we are erasing all information on the phase of the filter H ( ν ), so the difference in scattered power levels between figures 8 and 9 indicates that the source of the scattered power in figure 8 is phase-discontinuities between adjacent filters, H ( ν ). As noted above, we have matched the phases of adjacent filters, to the extent that this can be done with a uniform phase rotation of H . Therefore our filter solutions contain non-uniform phase structure that is discontinuous between adjacent samples. In § 4.3.1 we noted that our filter solutions may exhibit covariance between the lag coefficients h ( τ m ) and h ( τ j ), for lag separations small compared to the pulse-width ( | τ m -τ j | glyph[lessmuch] w ), and that the poorly constrained combination ( h ( τ j ) -h ∗ ( -τ j )) modifies only the phase of H . We therefore attribute the scattered power evident in figure 8 to these parameter covariances. We defer a detailed treatment of these issues to a later paper.</text>
<text><location><page_14><loc_52><loc_7><loc_92><loc_28></location>As figures 8 and 9 both display the response of the interstellar medium in the delay-Doppler coordinate frame, it is worth clarifying the relationship between them. Recall that h ( τ, ω ) is just the Fourier Transform of the sequence H ( ν, t ). Thus the Fourier transform of the dynamic spectrum, which is the Fourier transform of the product H ( ν, t ) H ∗ ( ν, t ), is just the convolution of h ( τ, ω ) with h ∗ ( τ, ω ). Consequently the arc that appears around the origin in figure 8 is echoed in a series of inverted arclets in figure 9; each of these arclets is centered on one of the power concentrations visible in figure 8 - cf. figure 5 of Walker et al (2004). Because the 'secondary spectrum' (figure 9) is equivalent to a self-convolution of the delay-Doppler image (figure 8), the latter is more fundamental and will typically be the more useful quantity for two reasons. First because the delay-Doppler</text>
<text><location><page_15><loc_8><loc_84><loc_48><loc_92></location>image exhibits the scattered field with greater clarity: in the secondary spectrum the scattered field is tangled up with itself. Secondly, convolution is a smoothing operation, so faint power concentrations are more easily seen in the delay-Doppler image. These points are well demonstrated by comparing figures 8 and 9.</text>
<text><location><page_15><loc_8><loc_68><loc_48><loc_84></location>Despite the fact that figure 9 is derived from the dynamic spectrum, it could not have been obtained by conventional spectroscopic methods, in which the on-pulse power-spectrum is determined within a window of width comparable to the width of the main-pulse (or interpulse) component. The reason is simply that windowing restricts the lag-range of the resulting secondary spectrum to the width of that window. Refering to figure 2 we see that the main-pulse would be completely contained within a window of width ∼ 100 µ s, so the resulting lag range would be -50 ≤ τ ( µ s) ≤ 50 - a tiny fraction of the actual lag range of figure 9.</text>
<section_header_level_1><location><page_15><loc_13><loc_66><loc_43><loc_67></location>7. DISCUSSION AND FUTURE DIRECTIONS</section_header_level_1>
<text><location><page_15><loc_8><loc_59><loc_48><loc_65></location>Because cyclic spectroscopy has not previously been applied to radio pulsar signals, there are many related issues that deserve consideration. Here we confine ourselves to a brief discussion of three aspects that the present study calls attention to.</text>
<section_header_level_1><location><page_15><loc_14><loc_57><loc_43><loc_58></location>7.1. Precision timing of PSR B1937+21</section_header_level_1>
<text><location><page_15><loc_8><loc_29><loc_48><loc_56></location>It is well known that the small-scale structure of the ISM can have a significant effect on the measured arrival times of radio pulses, in consequence of the delays (geometric and wave-speed) associated with signal propagation (e.g. Foster and Cordes 1990). These effects are of particular importance if they are epoch dependent, which is the case if the scattering properties of the medium are not statistically uniform transverse to the line-of-sight. It is plain from figure 8 that some of the scattering material towards B1937+21 is indeed very clumpy, with several flux concentrations appearing far from the origin, albeit at low power levels. Previous studies of the dispersion and scattering on this line-of-sight (Cordes et al 1990; Ramachandran et al 2006) preferred a near-Kolmogorov model of the structure, but the clumpiness we see is quite different from the expectations of a uniform Kolmogorov model (Cordes et al 2006; Walker et al 2004). As B1937+21 is routinely used for precision timing experiments (e.g Verbiest et al 2009), a better understanding of this scattering material is desirable.</text>
<text><location><page_15><loc_8><loc_7><loc_48><loc_29></location>It has previously been reported (Cognard et al 1993; Lestrade, Rickett and Cognard 1998) that B1937+21 occasionally exhibits timing fluctuations, correlated with flux variations, whose properties are suggestive of 'Extreme Scattering Events' - that is, plasma-lensing events (Fiedler et al 1987, 1994; Romani, Blandford and Cordes 1987). Such events require close alignment between the observer, plasma-lens and pulsar, and these events are consequently rare. If the alignment is not so close then the lens will cause smaller flux changes, but may still have a significant effect on the pulse arrival time because the extra path-length traversed by the faint images may be large. Furthermore these poorly aligned lens configurations should be relatively common. It is possible that plasma-lensing is responsible for the discrete flux concentrations that we see in the vicinity of τ ∼ 300 µ s (figure 7 and 8), with each concentration</text>
<text><location><page_15><loc_52><loc_77><loc_92><loc_92></location>being due to one or more additional faint images. We note that at this epoch (MJD53873) the features appear at such large delays that the scattered pulse has little overlap with the unscattered signal, so the pulse arrival time estimate should not be greatly affected. But at later epochs, when the scattering structures are closer to the line-of-sight to the pulsar, the scattered signals may appear at delays τ ∼ 100 µ s where they can exert a substantial influence on the measured arrival time. We defer a quantitative examination of pulse arrival time variations to a later paper.</text>
<text><location><page_15><loc_52><loc_60><loc_92><loc_77></location>Depending on the electron column-density structure, and the pulsar-lens-observer configuration, several additional images may arise from one plasma lens, so it is possible that all of the flux concentrations we see near τ ∼ 300 µ s in figures 7 and 8 are due to a single lens. Under that hypothesis, the observed range of delays (200 glyph[lessorsimilar] τ ( µ s) glyph[lessorsimilar] 400) tells us something about the size of the lens. Assuming that the pulsar is at a distance ∼ 5 kpc, and that the lens is near the midpoint, one finds that the lens diameter is ∼ 4 AU. This is comparable to the dimensions that have previously been inferred for the lenses responsible for Extreme Scattering Events (e.g. Romani, Blandford and Cordes 1987).</text>
<text><location><page_15><loc_52><loc_43><loc_92><loc_60></location>Unfortunately, with the techniques currently available to us, it is not possible to distinguish between lenslike, refractive behaviour and diffractive scattering as the cause of the observed power concentrations around τ ∼ 300 µ s. The clearest way to distinguish between these possibilities would be to undertake rigorous, quantitative physical modelling of the particular wave-propagation paths for this line-of-sight at the epoch(s) of observation. Such modelling would also tell us the relationship between the pulse arrival times actually observed, and those that would have been observed in the absence of the scattering medium. Physical modelling is, however, beyond the scope of this paper.</text>
<section_header_level_1><location><page_15><loc_61><loc_41><loc_82><loc_42></location>7.2. Cyclic spectropolarimetry</section_header_level_1>
<text><location><page_15><loc_52><loc_8><loc_92><loc_40></location>We have seen how cyclic spectroscopy gives access to the intrinsic modulation (pulse) profile of the signal, and that this can reveal new structure (figure 2) which is otherwise masked by the effects of scattering. It is the sharp features of the profile - those which include a large fraction of high-modulation-frequency Fourier components which are most affected by the scattering. All the results shown in this paper are based on a signal combination which approximates Stokes-I (recall that our data have not been polarization calibrated). But many pulsars exhibit highly polarized radio emission, and the polarized pulse profiles may be quite complex (van Straten 2006; Johnston et al 2008). For example, there are pulsars where the profile shows rapid transitions between orthogonal, elliptically-polarized states - usually referred to as 'orthogonal mode jumps'. Such transitions will be strongly affected by any filtering (temporal smearing) of the signal (Karastergiou 2009). More generally, it is clear that interstellar scattering can have a profound effect on the apparent polarization properties of pulsars at low frequencies (Li and Han 2003; Kramer and Johnston 2008), and we therefore expect the fidelity of polarization profiles to improve substantially when intrinsic profiles, rather than scattered profiles are used.</text>
<text><location><page_15><loc_53><loc_7><loc_92><loc_8></location>Furthermore, it has been emphasised by van Straten</text>
<text><location><page_16><loc_8><loc_87><loc_48><loc_92></location>(2006) that the most accurate pulse-timing requires accurate polarimetry. These are strong motivations to further develop the methods of this paper to encompass cyclic spectropolarimetry.</text>
<section_header_level_1><location><page_16><loc_16><loc_84><loc_41><loc_85></location>7.3. Covariance of filter coefficients</section_header_level_1>
<text><location><page_16><loc_8><loc_67><loc_48><loc_83></location>In § 4.3.1 we drew attention to the issue of covariance amongst the parameters describing the filter coefficients (or, equivalently, the impulse-response coefficients). The effects of these covariances are not easy to quantify because (i) the total number of parameters needed to describe the filter is very large ( ∼ 10 4 in the present case), and (ii) the covariances depend on the properties of both the filter and the pulse profile - neither of which is known a priori. What is clear, though, is the qualitative point that the actual uncertainty in the filter coefficients can be much larger than the standard deviation for a single parameter taken in isolation.</text>
<text><location><page_16><loc_8><loc_54><loc_48><loc_67></location>We have argued that there are two aspects of the impulse-response functions, seen in figures 7 and 8, that are probably due to parameter covariances. And one of these - the power near zero delay, scattered to large Doppler-shifts - is a very strong feature indeed, being evidently well above the noise floor and potentially masking real features of h ( τ, ω ). In other respects cyclic spectroscopy seems to be a near-ideal tool for studying the propagation of radio-pulsar signals, and the issue of parameter covariance consequently deserves further study.</text>
<text><location><page_16><loc_8><loc_35><loc_48><loc_54></location>We can identify two aspects that merit particular attention. The first is a thorough understanding of the origin of parameter covariance, and thus how it manifests itself in different representations of the data. Our preliminary analysis ( § 4.3.1) suggests that strong covariance can be traced to pure-phase modifications of the filter. That analysis was only carried through for the simplest possible filter model ( H ( ν ) = 1), and needs to be revisited using more general models. In cases where the data cannot constrain pure-phase modifications of the filter to be small compared to 1 radian, the problem is akin to one of phase-retrieval. Such problems are notoriously difficult, and the difficulty is associated with non-convexity of the target set (Bauschke, Combettes and Luke 2002).</text>
<text><location><page_16><loc_8><loc_25><loc_48><loc_35></location>With an understanding of the origin of the covariances one would be in a good position to tackle the key question of how to mitigate their effects on the filter models. For example, in § 4.3.1 we noted that the welldetermined/poorly-determined parameter combinations are sum/difference terms of h ( τ j ) and h ∗ ( -τ j ), so one might think of enforcing causality in the solutions, such that h ( τ j ) = 0 for all τ j < 0.</text>
<section_header_level_1><location><page_16><loc_16><loc_22><loc_40><loc_23></location>8. SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_16><loc_8><loc_7><loc_48><loc_21></location>Cyclic spectroscopy of PSR B1937+21 was undertaken with a 15 second cadence over a 4 MHz band at 428 MHz, starting from voltages recorded with the Arecibo radio telescope. By least-squares fitting we determined the impulse response function of the ISM for each cyclic spectrum separately, and the intrinsic pulse-profile averaged over the whole observation. In this way we obtained the 428 MHz pulse-profile of B1937+21 free of the influence of interstellar scattering, revealing some weak, but sharp features that had not previously been seen at low radiofrequencies.</text>
<text><location><page_16><loc_52><loc_68><loc_92><loc_92></location>From our temporal sequence of impulse response functions we derive the delay-Doppler field image. This image exhibits a noise floor at -51 dB relative to the peak power, and we are thus able to see faint features in the angular structure of the received field. Several power concentrations are visible in the delay range 200 -400 µ s. These concentrations can plausibly be attributed to a single plasma-lens, a few AU in diameter, but alternative interpretations are possible. Regardless of their physical origin, the scattered power concentrations are expected to have a deleterious effect on the pulse-timing experiments that are utilizing this pulsar. To accurately describe and remove these effects it is necessary to have a physical model of the various propagation paths by which the signal reaches the telescope. We did not attempt any physical modelling, but we have shown that cyclic spectroscopy provides us with a large quantity of information on these paths, and thus faciltates that process.</text>
<text><location><page_16><loc_52><loc_59><loc_92><loc_68></location>We caution that our fitting procedure is adversely affected by covariance amongst some combinations of the ∼ 10 4 fit parameters. These covariances were identified as the origin of the scattered power artifact in our delayDoppler image. Parameter covariance appears to be the main challenge currently facing widespread application of cyclic spectroscopy.</text>
<text><location><page_16><loc_52><loc_53><loc_92><loc_57></location>We thank Dan Stinebring for helpful discussions that prompted our examination of parameter covariances. This paper is dedicated to the memory of Don Backer.</text>
<section_header_level_1><location><page_16><loc_67><loc_50><loc_77><loc_51></location>REFERENCES</section_header_level_1>
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<table>
<location><page_17><loc_8><loc_66><loc_92><loc_88></location>
<caption>Table 1 Results of modelling ten sample cyclic spectra.</caption>
</table>
<text><location><page_17><loc_8><loc_63><loc_92><loc_65></location>Note . - This table summarizes the results of the tests described in this Appendix. Each line represents the outcomes from least-squares modelling of H ( ν ), or h ( τ ), for 10 sample cyclic spectra of B1937+21. In each case there are approximately 3 × 10 6 degrees of freedom and the total number of parameters in the model is roughly 13,000.</text>
<text><location><page_17><loc_8><loc_43><loc_48><loc_61></location>Kraft D., 1994, ACM Trans. Math. Software, 20 (3), 262 Kramer M., Johnston S., 2008, MNRAS, 390, 87 Lestrade J.-F., Rickett B.J., Cognard I., 1998, A&A, 334, 1068 Li X.H.., Han J.L., 2003, A&A, 410, 253 Liu D.C., Nocedal J., 1989, Math. Programming, 45, 503 Manchester R.N., Hobbs G.B., Teoh A., Hobbs M., 2005, AJ, 129, 1993 Nocedal J., 1980, Math. Comput., 35, 773 Nocedal J., Wright S.J. 1999 'Numerical Optimization' (Springer: New York) Ramachandran R., Demorest P.B., Backer D.C., Cognard I., Lommen A., 2006, ApJ, 645, 303 Rickett B.J., 1975, ApJ, 197, 185 Rickett B.J., 1990, ARA&A, 28, 561 Roberts J.A., Ables J.G., 1982, MNRAS, 201, 1119 Romani R.W., Blandford R.D., Cordes J.M., 1987, Nature, 328,</text>
<text><location><page_17><loc_10><loc_42><loc_12><loc_43></location>324</text>
<section_header_level_1><location><page_17><loc_46><loc_39><loc_54><loc_40></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_17><loc_34><loc_37><loc_67><loc_38></location>TESTS OF THE FILTER OPTIMIZATION CODE</section_header_level_1>
<text><location><page_17><loc_8><loc_31><loc_92><loc_36></location>Here we describe tests which we have undertaken to evaluate the performance of our software. Three different aspects of the optimization were compared: L-BFGS versus other algorithms; lag-space versus frequency-space optimization; and Unit versus Proximate initializations. All of these comparisons were made using cyclic spectrum samples #2-11 of PSR B1937+21 recorded at Arecibo on MJD53873 (first processing of the data: see § 6).</text>
<text><location><page_17><loc_8><loc_25><loc_92><loc_31></location>We do not expect that our conclusions regarding the relative merits of the different optimization paths are machine dependent. But for reference: the machine used for these tests was a MacBook Pro with a dual core 2.7 GHz Intel processor and 8 GB RAM installed. With this machine almost all algorithms required approximately 2 seconds to complete a single iteration, so run-times for the various approaches can be compared directly from the number of steps required to complete the optimization.</text>
<text><location><page_17><loc_8><loc_14><loc_92><loc_24></location>Table 1 sets out the results of our tests. The first three columns show the NLopt algorithm used, the space in which the filter was optimized, and the initialization conditions. Column four shows the average number of steps (rounded to the nearest integer) required to find the best-fit model for the ten sample cyclic spectra. Column five shows the number of sample cyclic spectra in which a particular configuration yielded the best result (i.e. lowest value of M min ) out of all of the configurations tested. And the final column shows the average value of M min , relative to the best-performing configuration, in units of σ 2 (rounded to the nearest integer). The ordering of the outcomes in the table was dictated by the results given in the last column, because a high-quality fit is our main objective. In the following sections we consider the outcomes presented in table 1, and their implications for the choice of optimization approach.</text>
<section_header_level_1><location><page_17><loc_41><loc_11><loc_60><loc_12></location>L-BFGS vs other algorithms</section_header_level_1>
<text><location><page_17><loc_8><loc_7><loc_92><loc_11></location>By design the NLopt package makes it possible to switch easily between a variety of different optimization algorithms, and thus to select the best one for the task at hand: to change algorithms is simply a matter of altering one line of code. The algorithms available within NLopt include both global and local methods. Global methods are not practical</text>
<text><location><page_17><loc_52><loc_60><loc_92><loc_61></location>Stinebring, D.R., McLaughlin, M.A., Cordes, J.M., Becker, K.M.,</text>
<text><location><page_17><loc_53><loc_59><loc_91><loc_60></location>Espinoza-Goodman, J., Kramer, M.A., Sheckard, J.L., Smith,</text>
<text><location><page_17><loc_52><loc_44><loc_91><loc_58></location>C.T., 2001, ApJ, 549, L97 Svanberg K., 2002, SIAM J. Optim. 12 (2), 555 Thorsett S.E., Stinebring D.R., 1990, ApJ361, 644 van Straten W., 2006, ApJ, 642, 1004 van Straten W., Bailes M., 2011, PASA, 28, 1 van Straten W., Demorest P.B., Oslowski S., 2012, Astron. Res. Tech., 9, 237 Verbiest J.P.W., et al 2009, MNRAS, 400, 951 Vlˇcek J., Lukˇsan L., 2006, J. Comp. Appl. Math. 186, 365 Walker M.A., Melrose D.B., Stinebring D.R., Zhang C.M., 2004, MNRAS, 354, 43 Walker M.A., Koopmans L.V.E., Stinebring D.R., van Straten W., 2008, MNRAS, 388, 1214</text>
<text><location><page_17><loc_52><loc_43><loc_78><loc_44></location>Yan W.M., et al, 2011, MNRAS, 414, 2087</text>
<text><location><page_18><loc_8><loc_89><loc_92><loc_92></location>for our problem because of the large-scale nature of the optimization: it would be necessary to thoroughly search a space of ∼ 10 4 dimensions in order to find the global minimum.</text>
<text><location><page_18><loc_8><loc_80><loc_92><loc_89></location>Of the local methods, there are algorithms which require derivatives of M to be supplied, and those which do not. As we are able to supply derivatives, and this is a major advantage in exploring the hypersurface of M , we restrict ourselves to those algorithms which make use of the gradient of M ; there are five such algorithms available in NLopt. One of these, SLSQP ('Sequential Least Squares Quadratic Programming'; Kraft 1994), had not completed a single step after more than an hour of run-time, at which point we terminated the optimization by force. The failure of SLSQP on our optimization problem is not surprising: it uses dense-matrix methods which, for our problem, requires ∼ 10 4 times more storage space and run-time than a limited-memory algorithm.</text>
<text><location><page_18><loc_8><loc_71><loc_92><loc_80></location>Results for the remaining four algorithms are given in table 1. We can see a clear division between these four: the Method of Moving Asymptotes (MMA; Svanberg 2002) and the Truncated Newton method (TNewtonPR; Dembo and Steihaug 1982) both performed poorly on our optimization task, in terms of the quality of fit and run-time, when compared to the Variable Metric (in either rank 1 or rank 2 forms: VarMetric1,2; Vlˇcek and Lukˇsan 2006) and L-BFGS algorithms. We note the failure of TNewtonPR to complete the optimization task from Proximate initialization, or from Unit initialization in frequency-space, hence the omission of those results. It is clear that MMA and TNewtonPR are uncompetitive for our optimization task and we do not consider them further.</text>
<text><location><page_18><loc_8><loc_65><loc_92><loc_70></location>It is not surprising that the VarMetric and L-BFGS algorithms yield similar results as they are similar algorithms. Nevertheless, our tests do show a clear preference for L-BFGS over either of the variable metric methods, with L-BFGS providing the three top-performing configurations, as gauged by δM min , and 9/10 of the best individual fits (column 5 of table 1).</text>
<section_header_level_1><location><page_18><loc_41><loc_63><loc_60><loc_64></location>Lag-space vs. frequency-space</section_header_level_1>
<text><location><page_18><loc_8><loc_56><loc_92><loc_63></location>We have already noted ( § 4.2) that lag-space optimization is expected to be superior to a frequency-space approach, because of the traps present in the latter space. This expectation is borne out in practice, with lag-space optimization yielding better fits than the corresponding frequency-space optimization in almost every case in table 1. However, the difference is not very great. We interpret this as meaning that L-BFGS and the VarMetric algorithms obtain enough information on the hyper-surface of M to allow them to avoid most of the traps.</text>
<text><location><page_18><loc_8><loc_52><loc_92><loc_56></location>One potential problem which we noticed during our tests is that L-BFGS, when used in frequency-space, would sometimes oscillate as it progressed towards the minimum. This phenomenon was most noticeable with Unit initialization; it appears to be responsible for the 30% extra steps required for L-BFGS-Freq-Unit relative to L-BFGS-Lag-Unit.</text>
<text><location><page_18><loc_8><loc_48><loc_92><loc_52></location>We note that the cyclic spectra used for these tests (see § 6) have typical signal-to-noise ratio greater than unity, for low harmonic numbers, on individual channels. It remains to be seen whether frequency-space optimization remains competitive for cyclic spectra which exhibit low signal-to-noise ratio at all harmonic numbers.</text>
<section_header_level_1><location><page_18><loc_42><loc_46><loc_59><loc_47></location>Variation of initialization</section_header_level_1>
<text><location><page_18><loc_8><loc_38><loc_92><loc_45></location>The algorithms tested here are local methods. That is, they locate a minimum of M in the vicinity of the starting point, but this minimum is not guaranteed to be the global minimum of M . The local nature of our solutions is something that readers should be aware of. However, reliably finding the true, global minimum of M in a space with ∼ 10 4 dimensions is a difficult problem which does not seem tractable with the computational technologies currently available. Given the difficulty of finding the true minimum of M , it behoves us to examine the sensitivity of our results to the starting point from which the optimization of H proceeds.</text>
<text><location><page_18><loc_8><loc_32><loc_92><loc_37></location>Unsurprisingly, table 1 shows that optimization from a Proximate initialization is roughly a factor of two quicker than from Unit initialization. And Proximate initialization always yields a significantly better fit, for a given choice of algorithm and optimization-space. Bearing in mind the large-scale nature of the optimization, with ∼ 10 4 parameters, some sensitivity to the initialization conditions is not surprising.</text>
<text><location><page_18><loc_8><loc_18><loc_92><loc_32></location>The fact that there are significant differences between Unit and Proximate initializations suggests the specific question 'how far are our best results from the corresponding global minima?' As a partial answer to that question we can compare the results of different Proximate initializations, because each of the 10 sample cyclic spectra used in our tests has cyclic spectra taken immediately before and immediately after, and we can step through this sequence in either direction. Referring to the L-BFGS-Lag-Proximate results in table 1 as 'Forward' initialization, we find that the corresponding 'Backward' initialization typically gives worse results, with the average M min being larger by 7 σ 2 and needing 31 more steps per cyclic spectrum, on average, to complete. Forward initialization produced a better fit than Backward for eight of the ten spectra, 9 and the root-mean-square difference between the corresponding M min values is approximately 21 σ 2 . Clearly the variations of the L-BFGS-Lag-Proximate outcomes, relative to the true minimum for each spectrum, must therefore be at least as large as 21 σ 2 , indicating that there is room for some significant improvement.</text>
<text><location><page_18><loc_8><loc_11><loc_92><loc_18></location>This point was confirmed by the following: we ran the whole suite of optimization tests again, but with a tighter fractional tolerance on M of 0 . 01 / N dof for the stopping criterion. For each of the ten sample cyclic spectra, we took the lowest value of M min (regardless of the configuration which achieved that result) as a reference point. Compared to that reference point, we find that the best-performing configuration of the standard-precision tests (i.e. L-BFGS-Lag-Prox; table 1) is worse by δM min glyph[similarequal] 41 σ 2 , on average, for each cyclic spectrum.</text>
<text><location><page_18><loc_8><loc_8><loc_48><loc_10></location>9 This level of asymmetry between Forward and Backward initialization is slightly surprising, being expected only once in 18</text>
<text><location><page_18><loc_52><loc_7><loc_92><loc_9></location>trials, but we have no explanation other than as a random occurence.</text>
<text><location><page_19><loc_8><loc_54><loc_11><loc_56></location>and</text>
<text><location><page_19><loc_8><loc_81><loc_92><loc_92></location>In the high-precision suite of tests we observed that none of the consistent outcomes of table 1 - i.e. L-BFGS better than other algorithms, Prox better than Unit, Lag better than Freq - were reproduced. Not surprisingly, the differences in M min amongst the 12 tested configurations were considerably smaller than shown in table 1, with the worst-performing configuration being only 7 σ 2 above the best (cf. 39 σ 2 in table 1). These facts suggest that in the high-precision tests all configurations have penetrated well into the noise-limited region of the optimization. The penalty for doing so, of course, is that many more steps are required to achieve that outcome - 784 steps, on average, for L-BFGS-Lag-Prox, which is more than 3 times the number of steps required to satisfy our usual stopping criterion (see table 1).</text>
<section_header_level_1><location><page_19><loc_46><loc_79><loc_54><loc_79></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_19><loc_35><loc_77><loc_65><loc_77></location>ESTIMATION OF MODEL UNCERTAINTIES</section_header_level_1>
<text><location><page_19><loc_8><loc_72><loc_92><loc_76></location>We have already determined the curvature of M with respect to the coefficients describing S x and H (equations 28 and 29). For the parameters describing the lag-space representation of the filter, the curvatures can be obtained by taking the real and imaginary parts of the relations</text>
<formula><location><page_19><loc_36><loc_68><loc_92><loc_71></location>∂ 2 M ∂h rm ∂h rj + i ∂ 2 M ∂h rm ∂h ij = A mj + C mj , (A1)</formula>
<text><location><page_19><loc_8><loc_66><loc_11><loc_67></location>and</text>
<formula><location><page_19><loc_36><loc_63><loc_92><loc_66></location>∂ 2 M ∂h im ∂h ij -i ∂ 2 M ∂h im ∂h rj = A mj -C mj , (A2)</formula>
<text><location><page_19><loc_8><loc_61><loc_38><loc_62></location>where the matrices A and C are given by</text>
<formula><location><page_19><loc_31><loc_57><loc_92><loc_60></location>A mj = 4 N 3 ν ∑ n,α =0 | S x ( α ) | 2 cos [2 πα ( τ j -τ m )] h ∗ n h n + j -m , (A3)</formula>
<text><location><page_19><loc_40><loc_57><loc_40><loc_57></location>glyph[negationslash]</text>
<formula><location><page_19><loc_31><loc_51><loc_92><loc_55></location>C mj = 4 N 3 ν ∑ n,α =0 | S x ( α ) | 2 cos [2 πα ( τ j -τ n )] h n h m + j -n . (A4)</formula>
<text><location><page_19><loc_41><loc_51><loc_41><loc_52></location>glyph[negationslash]</text>
<text><location><page_19><loc_8><loc_48><loc_92><loc_50></location>Here we have used notation such that h n + j -m means h ( τ n + τ j -τ m ), for example; and we have neglected the contribution from a sum over the residuals, whose expectation is zero.</text>
<formula><location><page_19><loc_41><loc_45><loc_59><loc_46></location>Noise levels for H ( ν ) = 1</formula>
<text><location><page_19><loc_8><loc_41><loc_92><loc_44></location>It is clear that the uncertainties in our parameter estimates depend on the filter coefficients and intrinsic pulse profile. But for our purposes here it suffices to determine rough estimates of the parameter uncertainties. To proceed we therefore consider the particular case H ( ν ) = 1. For this circumstance we obtain</text>
<formula><location><page_19><loc_42><loc_36><loc_92><loc_40></location>∂ 2 M ∂S 2 rm = ∂ 2 M ∂S 2 im = 4N ν , (A5)</formula>
<text><location><page_19><loc_8><loc_34><loc_11><loc_36></location>and</text>
<formula><location><page_19><loc_42><loc_31><loc_92><loc_35></location>∂ 2 M ∂H 2 rk = ∂ 2 M ∂H 2 ik = 4 F 2 , (A6)</formula>
<text><location><page_19><loc_8><loc_29><loc_45><loc_30></location>where F is a measure of the total pulsed flux, with</text>
<formula><location><page_19><loc_43><loc_25><loc_92><loc_28></location>F 2 := ∑ α =0 | S x ( α ) | 2 . (A7)</formula>
<text><location><page_19><loc_48><loc_25><loc_48><loc_26></location>glyph[negationslash]</text>
<text><location><page_19><loc_8><loc_23><loc_41><loc_24></location>For the lag representation of the filter we find</text>
<text><location><page_19><loc_59><loc_20><loc_59><loc_21></location>glyph[negationslash]</text>
<formula><location><page_19><loc_38><loc_19><loc_92><loc_22></location>∂ 2 M ∂h 2 rj = ∂ 2 M ∂h 2 ij = 4 N ν F 2 , ( τ j = 0) , (A8)</formula>
<text><location><page_19><loc_8><loc_14><loc_92><loc_18></location>and for τ j = 0 the curvature with respect to the real part of the coefficient h j is twice this value, whereas there is no curvature with respect to the imaginary part. This last point, which implies a formally infinite uncertainty, should not cause concern because the overall phase of the filter is completely arbitrary.</text>
<text><location><page_19><loc_10><loc_13><loc_86><loc_14></location>Using equation 39 we can immediately translate these curvatures into standard deviations. The results are</text>
<formula><location><page_19><loc_45><loc_9><loc_92><loc_12></location>δS m = σ √ 2N ν , (A9)</formula>
<formula><location><page_19><loc_45><loc_6><loc_92><loc_9></location>δH k = σ F √ 2 , (A10)</formula>
<text><location><page_20><loc_8><loc_91><loc_11><loc_92></location>and</text>
<text><location><page_20><loc_56><loc_88><loc_56><loc_90></location>glyph[negationslash]</text>
<formula><location><page_20><loc_41><loc_87><loc_92><loc_91></location>δh j = σ F √ N ν 2 , ( τ j = 0) . (A11)</formula>
<text><location><page_20><loc_8><loc_83><loc_92><loc_87></location>In all these cases the coefficients are complex; the quoted uncertainty is the uncertainty in the real part of the coefficient, which is equal to the uncertainty in the imaginary part. With the exception of one coefficient of h , the standard deviation is uniform across each set of coefficients.</text>
<text><location><page_20><loc_8><loc_80><loc_92><loc_83></location>In practice the system noise, σ , is dependent on the total number of radio-frequency channels, N ν , because we have a fixed total bandwidth, B , for the instrument. Thus N ν ∆ ν = B , and equation 14 can be written</text>
<formula><location><page_20><loc_44><loc_76><loc_92><loc_79></location>σ = S sys √ N ν B ∆ t . (A12)</formula>
<text><location><page_20><loc_8><loc_72><loc_92><loc_75></location>A further simplification is appropriate. For cyclic spectroscopy of a pulsar with period P , the pulsar's rotation frequency Ω = 1 /P is necessarily equal to the spacing in modulation frequency, ∆ α , and in turn this is the natural choice for channelisation, ∆ ν . Thus the natural configuration is PB = N ν , and for this circumstance we obtain</text>
<formula><location><page_20><loc_44><loc_68><loc_92><loc_71></location>δS m = S sys √ 2 B ∆ t , (A13)</formula>
<formula><location><page_20><loc_43><loc_64><loc_92><loc_67></location>δH k = S sys F √ P 2∆ t , (A14)</formula>
<text><location><page_20><loc_58><loc_60><loc_58><loc_61></location>glyph[negationslash]</text>
<formula><location><page_20><loc_39><loc_59><loc_92><loc_62></location>δh j = S sys P F √ B 2∆ t , ( τ j = 0) . (A15)</formula>
<section_header_level_1><location><page_20><loc_38><loc_56><loc_63><loc_57></location>Noise levels for more general filters</section_header_level_1>
<text><location><page_20><loc_8><loc_52><loc_92><loc_56></location>The curvature of the demerit function with-respect-to the various model parameters depends on the structure in the filter functions, as manifest in equations 28, 29, A3, A4, but we have so far considered only the simplest filter, H ( ν ) = 1. We now consider how structure in the filter affects the noise level on various parameters.</text>
<text><location><page_20><loc_8><loc_46><loc_92><loc_52></location>It is, of course, possible to concoct bizarre examples of filters which imply correspondingly unusual noise properties. But we shall ignore such possibilities as our purpose here is to describe what one might normally expect to encounter in practice. To that end we will restrict our discussion to cases where 〈| H ( ν ) | 2 〉 ∼ 〈| H ( ν ) | 4 〉 ∼ 1, and we will characterize the impulse response function by a typical scattering time, τ s , corresponding to a filter decorrelation bandwidth ∼ 1 /τ s .</text>
<text><location><page_20><loc_8><loc_36><loc_92><loc_46></location>Consider first the noise level for the pulse harmonic coefficients. For low harmonics the summation in equation 28 is approximately N ν 〈| H ( ν ) | 4 〉 . But at higher harmonics, where | α m | τ s ∼ 1, there is some decorrelation between | H ( ν -α m / 2) | and | H ( ν + α m / 2) | and the sum declines. In the limit of complete decorrelation, | α m | τ s glyph[greatermuch] 1, the summation yields N ν 〈| H ( ν ) | 2 〉 2 . Providing that both second- and fourth-order expectation values are of order unity, this is not a big effect. For example, in the random-phasor picture for the electric field the intensity statistics are exponential, so 〈| H ( ν ) | 2 〉 = 1 and 〈| H ( ν ) | 4 〉 = 2, yielding a noise level for high harmonics which is √ 2 larger than for low harmonics. In this picture, the noise level for high harmonics coincides with the value quoted in equation A13, for the case H ( ν ) = 1.</text>
<text><location><page_20><loc_8><loc_23><loc_92><loc_35></location>Quite a different situation arises for the filter coefficients H k . It is evident that the curvatures given in equation 29 may be much less than 4 F 2 in regions where the filter function is small, with correspondingly large errors on those coefficients. As with the noise on the pulse harmonics, there are two different limiting cases relating to the value of the typical scattering time. Most of the pulsed flux, F , is contributed by harmonics up to | α m | ∼ 1 /w , where w is the temporal width of the pulse. If τ s glyph[lessmuch] w then the filter function H ( ν k -α ) is almost constant over the range of α which contributes most to F , so the curvature in equation 29 becomes 4 F 2 | H k | 2 . Clearly this curvature could be very large (small) in comparison with the estimate given in equation A6, leading to correspondingly small (large) errors in the H k estimates. In the opposite limit, where τ s glyph[greatermuch] w , the filter coefficient | H ( ν -α ) | changes rapidly with harmonic number and we obtain a curvature estimate ∼ 4 F 2 〈| H ( ν ) | 2 〉 ∼ 4 F 2 , comparable to that given in equation A6.</text>
<text><location><page_20><loc_8><loc_14><loc_92><loc_23></location>Finally we consider the effect of a structured filter on the errors associated with the lag-space filter coefficients, h j . The curvatures of the merit function with respect to real and imaginary parts are (equations A3 and A4) made up of two terms. The first term is the same in both cases and we expect it to be 4 F 2 〈| H ( ν ) | 2 〉 / N ν ∼ 4 F 2 / N ν . The second term differs in sign between the real and imaginary parts of the coefficients; it is the real part of a sum of complex numbers. In normal circumstances those complex numbers bear no particular phase relationship to each other, so the second term is typically small in comparison with the first. We therefore neglect it, and we conclude that in normal circumstances the curvatures given in equation A8 are appropriate to all lag-space filter coefficients.</text>
<text><location><page_20><loc_8><loc_62><loc_11><loc_63></location>and</text>
</document>
[
{
"title": "ABSTRACT",
"content": "Cyclic spectroscopy is a signal processing technique that was originally developed for engineering applications and has recently been introduced into the field of pulsar astronomy. It is a powerful technique with many attractive features, not least of which is the explicit rendering of information about the relative phases in any filtering imposed on the signal, thus making holography a more straightforward proposition. Here we present methods for determining optimum estimates of both the filter itself and the statistics of the unfiltered signal, starting from a measured cyclic spectrum. In the context of radio pulsars these quantities tell us the impulse response of the interstellar medium and the intrinsic pulse profile. We demonstrate our techniques by application to 428 MHz Arecibo data on the millisecond pulsar B1937+21, obtaining the pulse profile free from the effects of interstellar scattering. As expected, the intrinsic profile exhibits main- and inter-pulse components that are narrower than they appear in the scattered profile; it also manifests some weak, but sharp features that are revealed for the first time at low frequency. We determine the structure of the received electric-field envelope as a function of delay and Doppler-shift. Our delay-Doppler image has a high dynamic-range and displays some pronounced, low-level power concentrations at large delays. These concentrations imply strong clumpiness in the ionized interstellar medium, on AU size-scales, which must adversely affect the timing of B1937+21. Subject headings: methods: data analysis pulsars: general pulsars: individual (PSR B1937+21) ISM: general scattering",
"pages": [
1
]
},
{
"title": "Mark A. Walker",
"content": "Manly Astrophysics, 3/22 Cliff Street, Manly 2095, Australia",
"pages": [
1
]
},
{
"title": "Paul B. Demorest",
"content": "National Radio Astronomy Observatory, Charlottesville, VA 22903, USA",
"pages": [
1
]
},
{
"title": "Willem van Straten",
"content": "Swinburne University of Technology, Astrophysics and Supercomputing, Hawthorn 3122, Australia Submitted to ApJ 16th January 2013",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Although radio pulsars emit intrinsically broad-band radiation, spectroscopy of these sources often reveals a great deal of narrow-band structure (e.g. Rickett 1990). This structure arises during propagation of the signal through the interstellar medium (ISM), where it is scattered by inhomogeneities in the ionized gas - it is interference between the various scattered waves which creates the observed fringes. Consequently high resolution spectroscopy of pulsars has proved to be a powerful tool for investigating the ISM (Roberts and Ables 1982; Cordes and Wolszczan 1985; Stinebring et al 2001). Traditionally pulsar spectroscopy is undertaken by forming the power-spectrum of the signal in a pulsephase window where the flux is high (i.e. 'on-pulse'), and subtracting the power-spectrum from a window where the flux is low ('off-pulse'), so as to remove the steady, background power level. Recently Demorest (2011) has drawn attention to an alternative approach, known as cyclic spectroscopy, in which one measures the modulation of the spectrum across the entire pulse profile. Cyclic spectroscopy was developed in engineering disciplines for studying signals whose statistics are pe- Mark.Walker@manlyastrophysics.org pdemores@nrao.edu willem@swin.edu.au riodically modulated (Gardner 1992; Antoni 2007). Signals of this type are common and are referred to as cyclostationary. The electric field received from a radio pulsar can be thought of as periodically-amplitude-modulated noise (Rickett 1975), so radio pulsars provide an example of a signal which is cyclo-stationary. As described by Demorest (2011), cyclic spectroscopy has several advantages over the simpler method of differencing on-pulse and off-pulse power spectra. Periodic amplitude modulation of the pulsar's radio-frequency noise, introduced by rotation of the pulsar beam, splits the received signal into upper- and lower-sidebands. By construction, the cyclic spectrum is the product of the lower sideband with the complex conjugate of the upper sideband. It is thus a complex quantity and as such it explicitly manifests information about the phase of any filtering which has occured prior to reception. For radio pulsars observed at low frequencies the dominant filtering is due to the ISM - specifically, to dispersion and scattering of the waves. Thus a time-domain representation of the filter is, to a good approximation, just the impulse-response of the ISM. In the present paper we show how to determine the filter given a measured cyclic spectrum. We also show how to determine the intrinsic cyclic spectrum of the signal in other words the (Fourier Transform of the) pulse pro- file which would have been observed in the absence of any scattering or dispersion. These determinations are both made in the narrow-band approximation, appropriate to our data, where there is assumed to be no variation of the intrinsic cyclic spectrum across the observed radiofrequency band. Our main dataset is a 4 MHz bandwidth voltage recording, centered on 428 MHz, of the original millisecond pulsar, B1937+21 (Backer et al 1982), made with the Arecibo radio telescope. 1 As far as we are aware the methods presented in this paper are the first attempts to determine both the filter and the intrinsic cyclic spectrum for any astronomical signal. It is possible that our techniques may be useful in fields other than pulsar astronomy, but we do not attempt to identify appropriate fields. Rather we encourage readers to consider applications in other contexts. To aid that process we note here the requirements for validity of our approach: first, the signal must be cyclostationary - i.e. stationary at each phase in its cycle in order for the cyclic spectrum to be well-defined. Secondly, our least-squares fitting assumes that the intrinsic cyclic spectrum is just white-noise that is periodically amplitude-modulated, so non-pulsar applications of our techniques are limited to signals which can be described in this or similar fashion. And, finally, the filter must not change significantly within the averaging time over which each cyclic spectrum is constructed. In addition to these requirements, the stopping criterion we employ for our optimizations is based on the assumption of Gaussian noise; but it would be straightforward to modify that criterion. We note that source code is freely available for all the software used herein (see § 5), so readers are free to adapt our code to their purpose. This paper is organised as follows. In the next section we give some background to the particular problems tackled in this paper. Then in § 3 we show how to determine the filter function and the intrinsic cyclic spectrum by direct construction. In § 4 we consider the issue of optimization - i.e. obtaining representations of these quantities which best fit the measured cyclic spectrum. In doing so we see that our direct estimate of the intrinsic profile, given in § 3, is in fact the optimum estimate in a least-squares sense. But § 4 does highlight deficiencies in our direct approach to the filter function; so for this quantity we utilize a large-scale optimization of the filter coefficients. Our implementation of this optimization is coded in the language 'C' and is freely available; it is described in § 5. In § 6 we present results obtained by applying our methods to low-frequency data on PSR B1937+21; both filter functions and intrinsic pulse profiles are presented. Discussion ( § 7) and Conclusions ( § 8) round out the paper. Two Appendices detail (i) the results of various tests we used to evaluate the code, and (ii) an analysis of the uncertainties in best-fit parameters.",
"pages": [
1,
2
]
},
{
"title": "2. BACKGROUND AND GENERAL CONSIDERATIONS",
"content": "Procedures for constructing the cyclic spectrum itself, from a set of recorded voltages, are given by Demorest (2011). We begin our development by quoting the relationship between a signal, x ( t ), a function of time, with 1 The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. M'endez-Universidad Metropolitana, and the Universities Space Research Association. Fourier Transform X ( ν ), and the cyclic spectrum of that signal, S x . At modulation frequency α we have (Gardner 1992; Antoni 2007; Demorest 2011) where the time-average is taken over integer multiples of the period of the system. Thus if we apply a filter, H ( ν ), such that the filtered signal is Z ( ν ) = H ( ν ) X ( ν ) then the cyclic spectrum of the filtered signal is In the case of a radio pulsar the signals X,Z are just electric fields, and the frequency ν is the radio frequency. Filtering of the signal occurs as a result of propagation, notably dispersion and scattering in the ionized ISM, and in the process of reception, e.g. the bandpass filter. The filter resulting from interstellar propagation evolves on some time-scale, and the average in equation (1) must be restricted to times which are short compared to that evolution time-scale. Throughout this paper we confine attention to the case of small fractional radio bandwidths, for which we expect the intrinsic cyclic spectrum to be approximately independent of ν : The quantity S x ( α ) is already familiar to astronomers from conventional analysis of radio pulsar signals: it is just the Fourier Transform of the pulse profile. But we emphasise that it is the transform of the intrinsic pulse profile, rather than the transform of the measured pulse profile - the difference being that the latter includes the influence of scattering and other contributions to the filter H . In general both S z and S x are complex quantities, but in the particular case α = 0 we obtain the zeromodulation-frequency components of the filtered and unfiltered signals, respectively. As these are just the time-averaged power-spectra of the signals they are nonnegative real numbers.",
"pages": [
2
]
},
{
"title": "2.1. Degeneracies",
"content": "Before extracting estimates from our data it is necessary to identify and eliminate any degeneracies in the model. Equation (2) shows that there are degeneracies which are multiplicative in form. Writing we see that S z → S z if and only if Thus if S x and H are completely unconstrained then there may be a great deal of degeneracy between these quantities in our model of S z : features seen in the data might be attributed to the intrinsic spectrum or to the effects of an imposed filter. In circumstances where the intrinsic cyclic spectrum is independent of radio-frequency (equation 3), the scope of the degeneracy is limited to functions Q ( ν ) such that Q ( ν + α/ 2) Q ∗ ( ν -α/ 2) is independent of ν . This condition should hold for all α . In the case of small α , by expanding to first order in α , we see that the form of Q is restricted to those functions satisfying and completely independent. Because of these limitations, the actual number of constraints provided by the data may be smaller than the number of grid points in the cyclic spectrum.",
"pages": [
2,
3
]
},
{
"title": "2.3. Noise and bias",
"content": "The computed cyclic spectrum includes measurement noise which we can characterize in the following way. Suppose that the recorded voltage is Z ( ν ) + N ( ν ), then we expect the measured cyclic spectrum to be where the delta-function appears because the measurement noise is stationary. Thus our measured cyclic spectrum is free of noise bias except at α = 0. Because modulation is the fundamental characteriztic of pulsar radiation which allows it to be distinguished from measurement noise, estimating the unmodulated part of the cyclic spectrum, S z (0 , ν ), from D (0 , ν ) is ambiguous. In this paper we therefore make no attempt to quantify S z (0 , ν ), nor do we make direct use of D (0 , ν ) in our estimates of the signal properties S x ( α ) and H ( ν ). In turn this means that we are giving up any possibility of determining S x (0), the zero-frequency term in the Fourier representation of the intrinsic pulse profile. We therefore adopt the convention S x (0) = 0 in our models throughout the rest of this paper. The actual data which we record, D ( α, ν ), will differ from 〈 D 〉 because of measurement noise and because the signal itself is stochastic in nature. If there is no averaging (see discussion following equation 14) the variance of the measured cyclic spectrum is given by (Antoni 2007) At zero modulation frequency, we recover from equation 11 the familiar result for stationary signals that the variance of the unaveraged power is just the square of the mean power. For observations of radio pulsars with current instrumentation, noise power is usually the dominant contribution to D (0 , ν ) and in this case we have If the measurement noise is white, as is often the case in practice, then equation 12 yields a uniform variance, over the entire cyclic spectrum. It is straightforward to estimate σ , because at zero modulation frequency the cyclic spectrum is just a power spectrum. Thus the noise level is just where S sys is the system-equivalent-flux-density, ∆ t is the integration time and ∆ ν the channel width. (Here we consider only a single polarization state, but clearly the results can be generalised to different combinations of polarization states.) Equation 14 clarifies what is meant by the 'no averaging' requirement immediately preceding equation 11. For cyclic spectroscopy of pulsars the natural choice of spectral resolution is ∆ ν = ∆ α , and we always have Hence if we do not know the actual form of S x ( α ), then the filter function can only be determined up to an arbitrary multiplicative factor of where τ , φ and ρ are real constants. In other words the overall normalization of H , its phase and its phase gradient are all arbitrary, because the simultaneous transformation leaves S z unchanged. If, however, S x ( α ) is already known, from previous observations, then the only remaining degeneracy is the overall phase of H . This phase is always arbitrary, as can be seen by noting that φ does not appear in equation (9).",
"pages": [
3
]
},
{
"title": "2.2. Sampling",
"content": "For a periodic modulation with period P = 1 / Ω, as is the case with signals from a radio pulsar, the cyclic spectrum is expected to be zero everywhere except at α = m Ω, where m is an integer, so those are the only modulation frequencies which we sample. In practice the data are also sampled discretely in the radio-frequency dimension, so we have measurements on a grid, with spacing ∆ α = Ω, and ∆ ν which we are at liberty to choose. In choosing ∆ ν the primary consideration relates to structure in the filter function: if we wish to capture signal components which are delayed by times up to τ then we need to have a resolution ∆ ν ≤ 1 / 2 τ . One could choose the resolution to be glyph[lessmuch] 1 / 2 τ but that would entail a greater computational load in constructing the cyclic spectrum. There is a natural limit to the fineness of the spectral resolution set by ∆ ν = ∆ α = Ω, corresponding to delays τ = ± P/ 2, where P is the pulse period. If there are signal components at delays greater than half the pulse period then the cyclic spectrum is intrinsically undersampled in α , because the modulation imposed by the filter function changes significantly on scales δα < Ω. On the other hand there is no difficulty in setting ∆ ν glyph[greatermuch] Ω, providing that there are no significant signal components at delays greater than 1 / ∆ ν . Although the cyclic spectrum is normally computed on a rectangular grid, values at large | α | and | ν | may not contain any information. If the voltage signal has a bandwidth B , sampled at the Nyquist rate, then the resulting cyclic spectrum is only valid within a diamond-shaped region around the origin, with | α/ 2 | + | ν | < B/ 2 (Demorest 2011). We also note that there cannot be more information in the cyclic spectrum than was present in the sampled voltage signal from which it was derived. Thus if the cyclic spectrum includes pulse harmonic numbers m> N p (the number of pulses averaged-over), then the pixels in the cyclic spectrum may not be statistically ∆ α = 1 /P , where P is the pulse period. Thus for ∆ t = P we have a time-bandwidth product of unity - a single sample of the signal - and σ = S sys . Equation 11 is then appropriate to a single pulse, and if the cyclic spectrum is averaged over N p pulses the variance is smaller by a factor 1 / N p .",
"pages": [
3,
4
]
},
{
"title": "3. DIRECT CONSTRUCTION OF FILTER AND PROFILE",
"content": "We now turn to the task of estimating the filter function (ISM impulse response) and the intrinsic (unscattered) pulse profile starting from a measured cyclic spectrum. We can approach both of these tasks by iteration, as we now describe.",
"pages": [
4
]
},
{
"title": "3.1. Determining the filter function",
"content": "Suppose we have a model for S x , but we have incomplete knowledge of H . If we know the value of H at a single frequency, ν 1 , we can determine its value at nearby frequencies using the measured cyclic spectrum in the vicinity of ν 1 , thus: We can make a better estimate of H at a given frequency if we know several nearby values of H . Making the replacement ν → ν -α/ 2 in eq. (2), multiplying by H ( ν -α ) S ∗ x ( α ) and summing yields glyph[negationslash] glyph[negationslash] where we have used the data, D , as our estimate of S z . This equation allows us to construct H , in regions where it is unknown, from nearby regions where it has already been determined, providing only that we have formed an estimate of S x . We note that equation (16) includes equation (15) as a special case where H is known only at a single frequency. Although the development in this section has focused on the idea of obtaining an estimate of H at frequencies where it is not known, it is clear that one could employ equation (16) even if we already have an estimate of H ( ν ) for all frequencies, so it can also be viewed as a procedure for updating an existing model of H . We will return to this idea in § 3.3 and § 4.",
"pages": [
4
]
},
{
"title": "3.2. Determining the intrinsic spectrum",
"content": "Now suppose that we have a model for H , what then do the data tell us about S x ? Multiplying equation (2) by H ∗ ( ν + α/ 2) H ( ν -α/ 2) and summing over ν gives where, again, we have used the data, D , as our estimate for S z . Thus, given data and a model for the filter function, we can obtain an estimate of the intrinsic pulse profile implied by the observed cyclic spectrum. Notice that this formula implies a unique estimate of S x associated with any given pair D,H . We shall see in § 4 that equation (17) is actually the optimum estimate of S x , in a least-squares sense, given the data D and the filter H .",
"pages": [
4
]
},
{
"title": "3.3. Bootstrap",
"content": "From the foregoing we can see that it is straightforward to form an estimate of H given S x , and vice versa. But initially we might not know either. In this situation it is natural to proceeed iteratively, starting with crude estimates and then using equations (16) and (17) repeatedly to improve those estimates. One way of starting the process is to initialize the intrinsic cyclic spectrum to S x ( α ) ← 〈 D ( α, ν ) 〉 ν , i.e. the observed (scattered) pulse profile. This corresponds to the model H ( ν ) = 1 and we could commence iteration of equations (16) and (17) using this approximation. Alternatively, having specified our initial estimate of S x we can build up our estimate of H gradually, using equation (16), starting from an estimate of its value at a single frequency, H ( ν 1 ). Because the overall phase of H is arbitrary ( § 2.1) we are free to choose the phase of H ( ν 1 ), e.g. phase zero, so only | H ( ν 1 ) | need be specified in order to start the iteration. One possible initialization is thus H ( ν 1 ) ← √ | D (Ω , ν 1 ) /S x (Ω) | , and from there we can gradually build H over the full range of radio frequencies, with information flowing outward from ν 1 towards the edges of the band. In this approach one simply initializes H to zero for frequencies where no estimate has previously been made, so that those frequencies make no contribution to the estimator in equation (16). Once this is done we can improve our estimate of the intrinsic cyclic spectrum, S x , by application of equation (17), then we can get a better estimate of H by applying equation (16), and these iterations can be repeated. Thus, if we know neither S x nor H , we can build bootstrap estimates for both of these quantities, given a measured cyclic spectrum. The procedure just described is the method which we initially used to solve for H and S x , from the first measured cyclic spectra of a radio pulsar (i.e. the data used in § 5). Broadly speaking the method works: we found that it provided a good representation of much of the structure in the cyclic spectra, and the intrinsic profile was significantly narrower than the scattered profile (see figure 3 in Demorest, 2011). But it did also exhibit some deficiencies, as we describe below.",
"pages": [
4
]
},
{
"title": "3.4. Deficiencies of the direct method",
"content": "One problem which we anticipated is the difficulty of constructing H in regions where | H | is small. In these regions the solution for H is sensitive to noise in the data. In particular it is susceptible to phase jumps at points where | H | → 0: the solutions on either side of the zero can be mutually inconsistent. There are two reasons why this problem arises. One is fundamental: a zero in | H | is an absence of phase information at that particular point, and this cannot be overcome by using different methods of solution. The other reason is specific to the solution method we have presented: the summation in equation (16) includes information coming from both sides of the zero, so each side tries to rotate the phases of the other in order to bring about consistency, but neither side succeeds. In other words, phase discontinuities at zeros of | H | constitute traps for this method of solution. It is not necessary for | H | to be precisely zero in order for a trap to form; it suffices for the signal-to-noise ratio to be low ( glyph[lessorsimilar] 1 on a per-channel basis). Trapping was indeed observed in the results we obtained using the approach described above, with significant residuals commonly occuring in the vicinity of points where | H | is small. It is clearly possible to modify the solution method so as to be less susceptible to these traps. Most obviously, one can restrict the summations in equation (16) to values of α with a single sign - so that we are only using the information from frequencies > ν (or < ν ) in our estimate for H ( ν ). In this scheme information flows in only one direction across the zeros, so one side dictates phase to the other. In practice we observed that this modification did decrease the prevalence of trapping. However, in using only one sign of α we are ignoring half of the information available to constrain H at any given value of ν , so the resulting solution for H cannot be optimum. In the next section we present methods for obtaining the best fit solutions for H and S x .",
"pages": [
4,
5
]
},
{
"title": "4. OPTIMUM ESTIMATES OF FILTER AND PROFILE",
"content": "In estimating H and S x what we really want are the models which best fit the data, so we have an optimization problem. We introduce the residual between model and data: and we seek to minimize the magnitude of these residuals. Suppose our data, D , have N ν radio-frequency channels, and N α modulation-frequency bins. In this case we are modelling a filter with N ν complex unknowns, and an intrinsic cyclic spectrum with N α / 2 complex unknowns. (The pulse profile is a real quantity, so the spectrum at negative modulation frequencies is simply the complex conjugate of that at positive frequencies.) Thus there are N ν +N α / 2 complex unknowns and ∼ N ν × N α / 2 complex constraints provided by the data, so for N ν , N α glyph[greatermuch] 1 the model is over-determined. In this situation we cannot make the residuals zero everywhere and we simply aim to make them small. Here we follow the usual practice of minimizing the sum-of-squares of the residuals glyph[negationslash] with respect to all of the model parameters. We then have glyph[negationslash] where q represents any of the model parameters which define H and S x , and minimization of M implies for every parameter q . We compute the derivative for each parameter in turn. Each value of H and S x is complex and thus involves two distinct real parameters. We take these to be the real and imaginary parts of the coefficients. For S rm := Re { S x ( α m ) } , S im := Im { S x ( α m ) } , we have where And for H rk := Re { H ( ν k ) } , H ik := Im { H ( ν k ) } , we have where glyph[negationslash] Having determined a demerit function, M , and the gradient of M with respect to each of the parameters of interest, we are in a position to employ one of various standard methods (e.g. Nocedal and Wright 1999) to the problem of optimizing our solutions. Before turning to the choice of method, and the details of its application, it is helpful to establish the relationship between our 'direct' solutions of § 3 and the optimum estimates which we are seeking.",
"pages": [
5
]
},
{
"title": "4.1. Relationship of direct solution to least-squares",
"content": "We have already noted ( § 3) that our 'direct' procedure for constructing H - i.e. equation (16) - could also be regarded as an algorithm for updating H , given an existing estimate. Explicitly, the update is H → H +∆ H , where glyph[negationslash] glyph[negationslash] We can also rewrite equation (17) as an update for the intrinsic spectrum, S x → S x +∆ S x , with In both cases we recognise the numerator to be (up to a constant factor) just the gradient of -M with respect to the corresponding parameters. This is comforting because it suggests that our 'direct' method is moving the estimates in a direction which will improve the model. To be confident that this is the case we need to gauge the step-size, not just its direction, and to achieve that it is helpful to evaluate the second derivatives of M . The curvature of M with respect to our various parameters is given by differentiating equations 23 and 25. The results are and glyph[negationslash] We can now see that for each of our real parameters, q , the 'direct' estimate in § 3 is an iteration with updates (equations 26, 27) ∆ q : This form is just Newton's method applied to each parameter separately. Equivalently: it is a simultaneous multi-parameter quasi-Newton method in which the off diagonal elements of the Hessian are neglected. We can check whether or not this is a good approximation to the actual Hessian by explicitly computing the off-diagonal terms. In the case where both q i and q j relate to S x these off-diagonal elements are all zero. Furthermore, because the diagonal terms (equation 28) are independent of S x , all of the higher derivatives of M with respect to S x are zero - the hypersurface of M is quadratic in S x when H is fixed. This is no surprise because the residual (equation 18) is linear in S x , and M is quadratic in the residual. It follows that Newton's method yields an exact solution for S x in a single step. Thus we see that our direct estimate of S x , given in equation (17), is also the least-squares solution appropriate to the filter H and the data D ; no additional optimization steps are necessary. Unfortunately this is not true of the filter function, H : neither the off-diagonal elements of the Hessian nor the higher order derivatives are zero in this case. The fact that the off-diagonal terms of the Hessian are nonzero means that we should not expect the filter update (equations 16, 26) to yield a good model. We now turn to the problem of optimizing our model filter function.",
"pages": [
5,
6
]
},
{
"title": "4.2. Optimisation of the filter",
"content": "To optimize our model filter we can employ one of the established quasi-Newton methods, in which an approximate (inverse-) Hessian is constructed at each iteration, based on the local properties of the hypersurface M revealed in previous iterations (see, e.g., Nocedal and Wright 1999). A popular choice is the BroydenFletcher-Goldfarb-Shanno (BFGS) update, and that is the method we employ in § 5 and subsequently. By utilizing a BFGS update to our estimate of H ( ν ) we expect to do significantly better than the update in equation 16. This is a general expectation, but it also applies to the particular problems noted in § 3.4: by including the non-zero off-diagonal curvatures of M we provide some of the information needed for the algorithm to escape the traps introduced by zeros in | H ( ν ) | . However, we ought to be able to do better still if we do not actually seek to construct H in frequency-space, where the traps are localised, but in the Fourier-space conjugate to frequency, i.e. lag-space. We introduce the lag-space description of the filter, h ( τ ), which is related to the frequency-space description of the filter, H ( ν ), by the usual Fourier relationships for discretely sampled functions: and To optimize our model filter in lag-space we need to know the gradient of M with respect to the lag-space filter coefficients, h rj := Re { h ( τ j ) } and h ij := Im { h ( τ j ) } . Noting that { h j } and { H k } are different representations of the same information we can write and similarly for the derivative with respect to h ij . In this way we find where Similarly one can show that So as an alternative to computing the frequency-space derivatives and determining the lag-space derivatives from them, we can compute the lag-space derivatives first and then determine the frequency-space derivatives. Formally the two different paths to either frequency-space or lag-space derivatives are equivalent. In practice we computed the lag-space derivatives as our primary quantities, using glyph[negationslash] (37) and determined the frequency-space derivatives, if required, using equation 36. There appears to be no significant difference in computation time between the two approaches.",
"pages": [
6
]
},
{
"title": "4.3. Uncertainties in best-fit parameters",
"content": "Suppose that we have obtained our best-fit model, the question then arises 'how accurate is that model?' To address this issue we need a description of the behaviour of the demerit, M , in the vicinity of the best fit. At the best-fit point in parameter space, which we denote by { q jo } , ∇ S M = 0, and either ∇ H M = 0 or ∇ h M = 0. If M = M o at the best-fit point, then in the immediate neighbourhood of this point the variation of M can be approximated by For Gaussian noise the normalized demerit, M/σ 2 , is distributed like χ 2 with N dof glyph[similarequal] (N ν -1)(N α -2) degrees of freedom, and we expect M o glyph[similarequal] N dof σ 2 . The fit becomes significantly worse if we move away from the optimum point to any other point such that M -M o = σ 2 (Avni 1976), and this contour of M delineates the range of uncertainties in our fit. Uncertainties in the individual fit parameters can be readily determined if the Hessian, ∂ 2 M/∂q m ∂q j , is diagonal so that the parameters are all independent of each other. In this case the standard deviation, δq j , is given by If the Hessian is not diagonal then the parameters are covariant and it is a much more difficult task to describe the uncertainties in the fit. Because we know how M depends on each of the various parameters, we can evaluate the elements of the Hessian explicitly. Doing so we find that the Hessian is indeed diagonal with respect to the set of parameters describing S x , so equation 39 correctly describes the constraints which our model places on those parameters. However, the Hessian is not diagonal with respect to either { H k } or { h j } . The standard errors as given by equation 39 are evaluated in an Appendix, while in the next section we discuss parameter covariance.",
"pages": [
6,
7
]
},
{
"title": "4.3.1. Covariances of { h j }",
"content": "Unfortunately the curvatures given in equation A8 are not the whole story when it comes to describing the uncertainties in the impulse-response function, because there are non-zero off-diagonal elements of the Hessian in respect of these parameters. It is beyond the scope of this paper to give a detailed description of the effect of these mixed curvature terms; here we only draw attention to their significance, deferring a thorough treatment to a later paper. To illustrate the importance of the off-diagonal elements of the Hessian we employ the simplest filter model, H ( ν ) = 1. In this case we find by direct calculation that in addition to the leading diagonal (described by equation A8), there is a single reverse-diagonal on which the curvatures are non-zero. This reverse-diagonal cuts the leading diagonal at τ m = τ j = 0, and for | τ m -τ j | glyph[lessmuch] w , the pulse-width, the mixed curvatures are comparable in size to the diagonal elements. The upshot of this is that the combination of complex coefficients h ( τ j )+ h ∗ ( -τ j ) is tightly constrained, whereas the combination h ( τ j ) -h ∗ ( -τ j ) is poorly constrained. The former combination can be thought of as a pure-amplitude modification of the filter H ( ν ), whereas the latter is a purephase modification. And the fact that these particular combinations of parameters are well-constrained/poorlyconstrained for | τ m -τ j | glyph[lessmuch] w is directly attributable to the (in)sensitivity of H ( ν + α/ 2) H ∗ ( ν -α/ 2) to these types of modification.",
"pages": [
7
]
},
{
"title": "5. IMPLEMENTATION OF FILTER OPTIMIZATION",
"content": "Having already established that the simple quasiNewton method of § 3 works tolerably well for our optimization problem, even though all the off-diagonal elements of the Hessian are neglected, our next step is to implement a more sophisticated quasi-Newton method, the BFGS algorithm, to optimize our filter coefficients. More precisely, because of the large number of parameters ( ∼ 10 4 ) needed to describe the filter coefficients, we utilize a 'limited memory' algorithm, which we call L-BFGS, in which the full inverse-Hessian is not constructed (Nocedal 1980; Liu and Nocedal 1989). We employed the L-BFGS algorithm coded in the NLopt library 2 (Steven G. Johnson, 'The NLopt nonlinear-optimization package' ). The NLopt package was chosen because it is free, portable and offers a wide variety of optimization algorithms (see § 5.3.1). In addition we utilized the FFTW Fourier Transform package 3 from the same group (Frigo and Johnson 2005). Our code is written in C and is freely available. 4 It makes use of the PSRCHIVE library 5 (Hotan et al 2004; van Straten et al 2012) for file input and therefore can accept data in a variety of formats, including the standard PSRFITS pulsar data format (Hotan et al 2004). Perhaps the first point to make, here, is that we have chosen to optimize the filter coefficients separately from the parameters which describe our model of the intrinsic cyclic spectrum, S x ( α ). There are several reasons for this choice. The strongest motivation is that it allows us to enforce a common timing reference on all our filter solutions, by using the same intrinsic cyclic spectrum throughout. A common timing reference is of paramount importance for all astrophysical studies which rely on pulse arrival-time measurements. Furthermore, by using a common timing (pulse-phase) reference, we can obtain a high signal-to-noise ratio measurement of the intrinsic spectrum by averaging over all our data. The degeneracies discussed in § 2.1 provide further, minor motivations for separate optimization of filter and intrinsic cyclic spectrum models, as these degeneracies must be eliminated in order for any algorithm to identify the best fit solution. For the overall normalization and phase of the filter that is fairly straightforward, but controlling the degeneracy in phase-gradient is not so easy if both H and S x are simultaneously adjusted. By contrast, there is no degeneracy in phase-gradient if S x is fixed. We noted in § 2.1 that the overall phase of H is always arbitrary, and this degeneracy must be eliminated before we can determine the model filter which best fits the data. We remove this freedom by forcing the imaginary part of h ( τ ) (or H ( ν )) to be zero at the point where | h ( τ ) | (or | H ( ν ) | ) attains its largest value. Because this choice is arbitrary, once an optimized filter is obtained we are free to rotate its overall phase to any preferred value. If we have a temporal sequence of filters (see § 7.5), the appropriate choice of phase for a given filter is the one which yields the closest match between the current and the previous (or subsequent) filter, leaving only a single, arbitrary phase for the whole temporal sequence.",
"pages": [
7
]
},
{
"title": "5.1. Initialisation",
"content": "We make use of two different initializations, which we refer to as 'Unit' and 'Proximate'. In the case of Unit initialization we begin with | H ( ν ) | = 1, for all radio frequencies, and a constant phase gradient in H ( ν ), chosen to match the mean phases seen in the data at α = Ω. For lag-space optimization this initialization corresponds to a delta-function model for h ( τ ). Naturally, Unit initialization is only sensible if the overall normalization of our model S x ( α ) is consistent with that of the data, D ( α, ν ), and we therefore also normalize S x ( α ) appropriately. Unit initialization is appropriate if we have no prior information on the actual structure which is present in the filter function at the time the cyclic spectrum was recorded. Usually there are many cyclic spectra recorded during a single epoch of observation - e.g. in § 6 we present data from three separate epochs of observation, totalling several hundred cyclic spectra. In such cases the averaging time for each spectrum is chosen to be small enough that the changes in the filter function between adjacent cyclic spectra are small. Consequently, if we have already optimized the filter appropriate to one cyclic spectrum then that model provides us with a good starting point for modelling the next filter function: that scheme is what we refer to as Proximate initialization.",
"pages": [
7,
8
]
},
{
"title": "5.2. Stopping criterion",
"content": "At what point should we stop the optimization? The NLopt algorithms include various criteria which may be used to recognise that the optimization is complete. Our aim is to find the minimum of M , but we do not know ahead of time the precise value of that minimum, so a natural choice of stopping criterion is that M should change by less than a certain, small fractional value during a single iteration of the algorithm. We can determine what that fractional tolerance should be as follows. In § 2.3 we gave expressions for the variance of D ( α, ν ). In particular we noted that var { D } = σ 2 , a constant, is usually a good approximation in practice. Furthermore, at large modulation frequencies, α glyph[greatermuch] Ω, the noise is usually much larger than the signal we're interested in, so it is straightforward to get an estimate of σ 2 directly from the data. For Gaussian noise, which is appropriate to the thermal noise component, we expect the best-fit value of M to conform to a χ 2 distribution, with N dof glyph[similarequal] (N ν -1)(N α -2) degrees of freedom. In this case the minimum demerit is expected to be M min glyph[similarequal] N dof σ 2 , and σ 2 is a significant change in M (Avni 1976), so it is appropriate to stop the optimization once the changes in M are small compared to σ 2 . This translates directly into the requirement that fractional changes in M should be small compared to 1 / N dof . Therefore in this paper the usual stopping criterion is that the fractional change in M should be less than 0 . 1 / N dof . If the noise is not uniform - e.g. at the edges of the band, where the instrumental response rolls off, or because of strong Radio Frequency Interference - one can determine the variance at each point in the cyclic spectrum using equation 11. In this case the residuals (equation 19) should be normalized by the variance at each point ( α, ν ) prior to summation. The resulting figure of merit will then be distributed like χ 2 . It is straightforward to measure the noise variation across the band, as per § 6.1 (see the top panel in figure 1).",
"pages": [
8
]
},
{
"title": "5.3. Choice of optimization approach",
"content": "The various tests described in the Appendix demonstrate that, of the various optimization approaches we tried, the best method for this problem is L-BFGS in lag-space from Proximate initial conditions; we therefore utilize that method. All of the data utilized in this paper are Arecibo observations of PSR B1937+21 (Backer et al 1982), at radio frequencies close to 428 MHz. The ATNF Pulsar Catalogue 6 (Manchester et al 2005) reports the following characteriztics for this pulsar: a period of 1.558 ms, a dispersion measure of 71 pc cm -3 (Cognard et al 1995), and a mean 400 MHz flux of 240 mJy (Foster, Fairhead and Backer 1991). Most of the data we use come from a single 4 MHz band centered on 428 MHz, with the exception being an additional 4 MHz chunk, centered on 432 MHz, that we use exclusively in § 6.3 (intrinsic pulse profile determination). We observed at three different epochs: MJD53791, MJD53847 and MJD53873. Dualpolarization voltages were recorded for intervals of order an hour at each epoch, using the Arecibo Signal Processor baseband recorder (ASP; Demorest 2007), with digitisation at 8-bits per sample. This high dynamic range sampling proved valuable in mitigating the effects of Radio Frequency Interference ( § 6.4). We did not attempt any polarization calibration for our data; all the results reported here are based on summing the two polarizations (i.e. the orthogonal feeds of the telescope), as an approximation to Stokes-I. Individual cyclic spectra were generated from the recorded voltages, using the method described by Demorest (2011). In our first processing of the data we constructed cyclic spectra, averaged over 15 seconds, with 6230 radio-frequency channels and 511 pulse-phase bins. These values were chosen so as to make ∆ ν as nearly equal to ∆ α as possible, because our first attempts at modelling H and S x (using the method described in § 3), avoided interpolations. However, the improved fitting method described in §§ 4,5 employs precise interpolation, so it is no longer necessary to match the resolutions in this way. Nor is it preferred, as array sizes which are integer powers of two are better matched to the Fast Fourier Transform algorithm, which we utilize. All the tests of our optimization software, reported in an Appendix, were conducted on the cyclic spectra obtained in our first processing of the data. Analysis of the cyclic spectra from our first processing revealed some leakage at the edges of the bandpass filter. This is undesirable, particularly because any out-of-band signal is aliased by ± 4 MHz, and will thus appear delayed by approximately ± 30 ms due to incorrect dedispersion. In turn this leaked signal may introduce low-level contamination into our profile estimates or our filter models, or both. We therefore decided to completely reprocess our data, to deal with the leakage and to correct some other minor defects which we were aware of. In the second processing we produced cyclic spectra averaged over 15 seconds, with 4608 channels and 1024 pulse-phase bins. This reprocessing utilized the cyclic spectrum implementation now freely available as part of the DSPSR software package 7 (van Straten and Bailes 2011). To eliminate the aliased (leakage) signals we then trimmed the spectral array down to 4096 channels, so the final bandwidth was approximately 3 . 56 MHz. With the exception of § 6.1 and § 6.2, all of the results presented in this section were obtained using the trimmed cyclic spectra from the second processing of our data.",
"pages": [
8
]
},
{
"title": "6.1. Bandpass Filter",
"content": "If we want to know the profile of the bandpass filter of our instrument there are two methods available to us: we can measure the average total power as a function of radio-frequency, or we can make use of the filter functions, H , obtained from our fitting. (One can also inject artificial pulsed power, of known spectral shape, into the signal chain, but we did not record such data.) Our estimates of H incorporate all of the filtering imposed on the signal. We expect there to be contributions from the ISM, the Solar wind, the Earth's ionosphere, and from our instrument (telescope, front-end and backend). Of these various contributions, only the receiver system is expected to be stable over long time-scales. As H is a complex quantity, averaging it will yield zero, but we can instead form 〈| H ( ν ) |〉 , which we take as an estimate of the bandpass filter, | H rec ( ν ) | . Averaging over all filter solutions for all three epochs we obtain the result shown in figure 1 for | H rec ( ν ) | . Also shown in figure 1 is the result of estimating the bandpass in a more traditional way, using the square-root of the average total power: √ 〈 D (0 , ν ) 〉 . (The square-root appears here because the power is a quadratic function of the filter response.) Although H ISM fluctuates quite rapidly, the amplitude of those fluctuations is large, so a long total observation time is required in order to form an accurate estimate of | H rec ( ν ) | . With our three epochs combined we have approximately 4.5 hours of data, and the scintillation time-scale is of order a minute so we expect our estimate of the filter response to be accurate to ∼ 6%. That is approximately the level of fluctuation seen in our estimate of | H rec ( ν ) | across most of the band. Thus the only clearly significant structure we find in | H rec ( ν ) | is the roll-off of the filter at the band edges. A cause for concern is the abrupt rise in the estimated filter response at both extremes of the frequency range. These upturns indicate that that there is some leakage of signal from outside the nominal band of the filter. glyph[negationslash] By contrast with | H rec ( ν ) | , the estimate √ 〈 D (0 , ν ) 〉 shows evidence of an upturn only at one end of the band. The reason for this difference is unclear. The other points of distinction between the two results are (i) that the noise on the traditional estimate is much smaller, even though only a third as much data was used, and (ii) Radio Frequency Interference (RFI) is manifest in the traditional estimate. To some extent the effect of the RFI could be mitigated by averaging using the median estimator, rather than the mean, but this would not help for steady interference. The reason for the lower noiselevel on the traditional bandpass estimate can be seen from equation (10). Our solutions for H ( ν ) - whence the | H rec ( ν ) | estimate - are based on the pulsed power, i.e. α = 0, whereas the zero-modulation-frequency data, D (0 , ν ), are dominated by the system noise, N ( ν ), which is both large and unpulsed. As mentioned at the start of § 6, the leakage at the band edges, most evident in the lower panel of figure 1, can introduce low-level artifacts into our filter or pulse-profile estimates. Consequently we decided to fully reprocess our data, trimming off the edges of the band as we did so. The results described in § 6.3 and later sections of this paper were obtained from the second processing in which the spectral band was trimmed.",
"pages": [
9
]
},
{
"title": "6.2. Bootstrap approach to the intrinsic profile",
"content": "Lacking prior knowledge of the intrinsic pulse profile we are obliged, as in § 3.3, to commence our modelling using the observed, scattered pulse profile as an approximation to the intrinsic profile. We then obtain our first model of the filter function, for each sample cyclic spectrum, by fitting to the data in the way described in §§ 4,5. The filters obtained in this way are then used to obtain a better estimate of the intrinsic pulse profile, and the whole process is iterated, obtaining better approximations to S x , and the various H , on each pass through the data. Once an accurate model of the intrinsic profile is obtained, other data-sets for the same pulsar taken with the same instrumental configuration can use that profile to obtain model filters in a single pass through the data. But new instrumental configurations - e.g. different observing frequencies - may force a return to the bootstrap approach. Because it requires multiple passes through the data, a bootstrap can be slow. We can, however, speed things up to some degree because at the second and subsequent profile-iterations we already have available a set of model filters appropriate to each recorded cyclic spectrum. These filters can be used to initialize subsequent models prior to optimization. As our model of the intrinsic pulse profile approaches the true intrinsic profile we expect the model filters to change very little between successive iterations, so this procedure should accelerate the optimization substantially. This expectation was borne out in practice, as we now describe. To enable a rapid approach to the intrinsic profile we initially used a subset of the data (roughly 20 minutes of observation) from one epoch (MJD53873), iterating several times on this subset, and then adding in the rest of the data from this epoch in order to improve the signalto-noise ratio of our profile estimate. For the first set of filter solutions, using Proximate initialization, we found that on average 289 NLopt steps were required to fit each cyclic spectrum in the data subset. Subsequently, using the filter models obtained at the previous iteration as our starting point, the number of NLopt steps declined to 222, 17 and 14 for the second, third and fourth iterations, respectively. 8 The small decrease in the required number of steps between the third and fourth iterations, contrasting with the large decrease between the second and third iterations, suggested that we had reached the noise floor of the data subset, so for subsequent iterations we utilized all of the data from MJD53873 - a total of approximately 2 hours. For iteration five we needed to obtain the first filter solutions for the bulk of the data from this epoch, using Proximate initialization, which required on average 241 NLopt steps per cyclic spectrum. But for all subsequent iterations we were able to initialize our models using the previous set of filter solutions. We found that iterations six and seven required only 12 and 3, respectively, 8 NLopt steps for each cyclic spectrum, indicating very rapid convergence of our estimate of the intrinsic pulse profile. Separately we have observed, when using an existing set of optimized filter models as our starting point, that our code requires a minimum of 3 NLopt steps to return an optimized solution, even when the same reference profile is used for both solutions. We therefore conclude that our intrinsic profile estimates for B1937+21 do not differ significantly between iterations six and seven, and further iterations are unwarranted. Use of the previous set of filter solutions to initialize our models clearly leads to a substantial saving in computation time. Using Proximate initialization we expect that the bootstrap would have required a total of 10 days of CPU time, whereas the sequence just described used only a third of that time. In fact our procedure needed only one quarter more time than a single pass through the same data using a given reference pulse profile.",
"pages": [
9,
10
]
},
{
"title": "6.3. Intrinsic versus scattered profile",
"content": "In figure 2 we show our estimate of the intrinsic profile, together with the scattered profile, using all the data from MJD53873. This epoch was chosen because we obtained significantly more data on that date than on either of the other epochs. As expected, the intrinsic modulation profile of the signal is much sharper than the apparent modulation, because of the contribution of the scattered (delayed) waves to the apparent profile. The 'scattered tail' of the pulse is absent from our estimate of the intrinsic profile. Figure 2 (lower panel) also reveals the presence of several low-level (a fraction of 1% of the peak height), but sharp features in the 'baseline' of the intrinsic pulse. These features are difficult to recognise in the scattered profile for two reasons. First, interstellar scattering broadens them, while decreasing the peak amplitude of each. Secondly, the features that are present immediately after the main-pulse or the inter-pulse are swamped by the delayed signal from those two, very strong components of the pulse profile. An equivalent description of the pulse modulation is available by Fourier-transforming the scattered and intrinsic profiles. The resulting harmonic powers are shown in figure 3, demonstrating that the high harmonics of the intrinsic profile contain a great deal more power than the scattered profile. This is just as expected. The scattered profile is a convolution of the intrinsic profile with the impulse response function, so in the Fourier domain the relationship is multiplicative, and the multiplier declines from near unity at low harmonic numbers to very small values at high harmonic numbers. Because the low-level features evident in figure 2 are seen here for the first time at these radio frequencies, and the signal-processing we have used to reveal these structures is itself novel, we would like to have some confirmation of their reality. We have therefore undertaken a completely independent bootstrap estimate of the intrinsic profile for another epoch, MJD53791. In this comparison we are not interested in any timing (pulse-phase) offset between the two epochs, so in comparing the two intrinsic profiles we have applied a pulse-phase shift and a scaling, chosen so as to minimize the difference between the profiles. The result of our two independent bootstrap solutions can be seen in figure 4, where we show the mean of the intrinsic profiles and their difference. The latter curve appears noise-like, without any clearly significant differences between the two, independently derived intrinsic profiles. In particular we note that the largest differences occur underneath the main-pulse and inter-pulse components, where the signal is very strong and the noise is therefore greater than at other pulse-phases. There is no apparent systematic difference between the two profiles at those pulse-phases where the weak, low-level features are seen. As a final check on the reality of the features revealed in figures 2, 4, we have also compared the intrinsic pulse profiles obtained from independent bootstrap estimates at two different frequencies, 428 MHz and 432 MHz, for the epoch MJD53791 - this comparison is shown in figure 5. Although the 432 MHz data exhibit more system noise than the 428 MHz profile, because the integration time for the latter is larger by a factor of 1.5, the two profiles appear otherwise very similar in respect of the low-level features which are revealed by construction of the intrinsic profile. An important aspect of the inter-band comparison in figure 5 is that it excludes signal leakage ( § 6.1) as a possible origin for the low-level structures which we see in the intrinsic profile. Even though we have trimmed the band edges, which should eliminate the bulk of that problem, it is possible that some traces of leakage remain. This concern is heightened by the fact that the sharp feature at a pulse-phase of 500 µ s lies close to the expected location of the aliased main pulse component, for signals leaking across the low-frequency edge of the 428 MHz band (upper red curve in figure 5). The inter-band comparison makes it plain that this is not a viable explanation for that feature, because at 432 MHz the corresponding alias should lie at 1,200 µ s, where no profile feature is seen - yet the observed 500 µ s peak appears very similar in the two bands. We also note that interpreting the 500 µ s feature as an alias of the main-pulse implies that there should be a counterpart feature from the interpulse, roughly half a turn later, whereas no such feature is observed in either band. Overall, the aliased signals from the band edges do not correspond with the low-level features we see in the pulse profiles in either band, and we conclude that they are not due to out-of-band signals. In fact residual out-of-band signals are expected to appear as broad structures in the time-domain, because the dispersive delay is a strong function of frequency. The sharpness of the features shown in the red and blue curves in figure 5 is due to the fact that only frequencies immediately adjacent to the band edges have been considered. The red and blue curves are simply calculated as delayed (and scaled) versions of the mean pulse profile, with the delay/advance equal to the difference in dispersive delay between the upper and lower edges of the band. At MJD53791 the dispersion measure of PSR B1937+21 was 71.023 pccm -3 , and the period was 1.5577 ms, so the aliased signals appear at ± 30 . 068 ms (428 MHz band) and ± 29 . 240 ms (432 MHz band). Modulo the pulse period these become, respectively, ± 0 . 470 and ± 1 . 201 milliseconds. Some of the 'new' structure that we see in the intrinsic pulse profile corresponds well with features of B1937+21 which have been found by others, as follows. The distinct peaks seen immediately after the main- and inter-pulse have previously been observed by a number of authors at higher radio-frequencies, where the delayed, scattered signal is much weaker - see, particularly, figure 1 of Thorsett and Stinebring (1990). Here we are presumably seeing the emission regions which are responsible for the giant pulses of B1937+21 (Cognard et al 1996), and the consequently high modulation index at these pulsephases (Jenet, Anderson and Prince 2001). The sharp feature we see at a pulse-phase of 500 µ s (0 . 3 turns) has a counterpart which was noted in L-band observations by Yan et al (2011). Residual dispersion smearing in the Yan et al (2011) data is significant, so it is not surprising that their feature appears broader than the one we observe. Finally, the gradual rise we see in the 0 . 2 turns immediately preceding the main-pulse is also manifest in the Yan et al (2011) data. The consistency of our intrinsic profile estimates across different epochs and spectral sub-bands, and the connections we can make between individual features and previous observations of B1937+21 at other frequencies, give confidence that the statistically-significant features we see in our intrinsic profile are indeed real.",
"pages": [
10,
11,
12
]
},
{
"title": "6.4. Dynamic Spectra",
"content": "A measured cyclic spectrum quantifies the power spectrum of the signal as the zero-modulation-frequency array D (0 , ν ) (see § 2). We compute our cyclic spectra with a cadence of 15 seconds, and thus we can trivially obtain a dynamic spectrum from the temporal sequence of D (0 , ν ). This dynamic spectrum is a simple timeaverage, not a difference of on-pulse and off-pulse power, so it includes all power contributions: noise from the receiver and the sky, the pulsar signal, and any terrestrial signals reaching the receiver, i.e. RFI. Because RFI can cause severe problems for some types of radio astronomical investigations, it is useful to examine the dynamic spectrum in order to gauge its impact. fest in this segment as narrow spectral lines. None of these lines is so strong that the voltage signal exceeds the dynamic range of the sampler, nor is any impulsive RFI evident in figure 6. These aspects of the data reassure us that the observations were taken under relatively benign RFI conditions, and in this circumstance we can reasonably expect a high level of immunity from RFI in our models of S x and H . In particular, if the RFI is both accurately captured and not modulated at the frequency Ω = 1 /P , or its harmonics, then cyclic spectra will be free of RFI contamination. To demonstrate that the observed RFI does not propagate into our model filters we also show in figure 6 the squared-modulus of the dynamic filter, i.e. | H ( ν, t ) | 2 . This quantity is our estimate of the contribution of the pulsar to the dynamic spectrum; the spectral structure | H ( ν, t ) | 2 can also be seen in the total power signal. It is evident that the RFI present in the total power signal is absent from the dynamic filter. We emphasise that the specific, small fraction of the spectrum shown in figure 6 was chosen at random: it was not selected because it displays good immunity from RFI.",
"pages": [
12,
13
]
},
{
"title": "6.5. Dynamic fields",
"content": "Whereas the dynamic spectrum is a quantity which pulsar astronomers routinely measure, it has been much more difficult to get at the dynamic electric field because the latter requires information on the phases, and that information is usually not explicit in the measured intensities. The requisite phases can sometimes be retrieved - e.g. if the field is sparse in some representation - but to date this has been successfully demonstrated for only one dynamic spectrum (Walker et al 2008). By contrast, cyclic spectroscopy provides us with access to the electric field envelope, including the phase information; as such it is an intrinsically holographic method. There are various possible representations of the dynamic fields because they may be described in terms of frequency-space (filter) or lag-space (impulse-response) coefficients, and the dynamic nature of the field can be represented either as a temporal sequence or in terms of the conjugate Fourier variable, i.e. a frequency.",
"pages": [
13
]
},
{
"title": "6.5.1. Impulse-response functions",
"content": "Figure 7 (top panel) shows one possible representation of the field: the real part of the impulse-response function, h ( τ ), determined from the first cyclic spectrum we observed on MJD53873. This function spans a lag range of 1 , 152 µ s, and we see that the amplitude of the re- onse falls off on lag-scale glyph[lessorsimilar] 50 µ s. There is, however, a low-level tail to the response, extending to lags that are a substantial fraction of the pulse period. To bring out these low-level features we took the modulus of the impulse response, and then averaged it over all the data at this epoch of observation. The lower panel of figure 7 presents the resulting 〈| h ( τ ) | 2 〉 , which demonstrates that the low-level tail of h continues out to delays of at least 400 µ s relative to the peak of the response. At extreme negative lags there is an obvious rise in | h | . The origin of this feature is not completely clear; however, a preliminary analysis suggests that parameter covariances in { h j } (see Appendix) may give rise to increased noise near the lag limits of the cyclic spectra, and we therefore consider this to be an artifact. On the other hand the features seen in the vicinity of τ ∼ +300 µ s appear to be bona fide structure in h . The delay-Doppler image, which we present in the next section, gives more information on these features.",
"pages": [
13
]
},
{
"title": "6.5.2. Delay-Doppler field images",
"content": "Finally we present our results in the Fourier domain conjugate to ( ν, t ). The conjugate variables ( τ, ω ) have immediate physical meaning as the delay and Dopplershift, respectively, that accumulate during propagation of the wave (Harmon and Coles 1983; Cordes et al 2006). The Fourier Transform, h ( τ, ω ), of the dynamic electric field, H ( ν, t ), is therefore a quantity of particular interest we call this the 'delay-Doppler image'. Figure 8 shows the squared-amplitude of the delay-Doppler image for our data taken on MJD53873. The lower-half of figure 8 is largely free of signal, as expected for negative lags (which are acausal). The only signals that can be recognised at negative lags are the band of scattered power running horizontally across the figure (discussed later), and a handul of thin, faint, vertical streaks in the region | ω | glyph[lessorsimilar] 4 mHz, 0 > τ glyph[greaterorsimilar] -100 µ s. We are uncertain as to the cause of these streaks, but we suspect that they may be sidelobes caused by the sharp truncation of the spectrum which we introduced by trimming the band ( § 6.1). These streaks were not seen in the delay-Doppler image that we obtained in our first processing of the data. The enhanced noise at extreme negative lags, plainly seen in the average signal in figure 7, is also present in figure 8 but is difficult to discern without averaging. By contrast, in the upper half-plane of figure 8 there is an abundance of structure. Most of the power is concentrated in a broad distribution centered on zero-Dopplershift. And the overall distribution appears to have an approximately parabolic envelope, as is now familiar for many pulsars (Stinebring 2001; Cordes et al 2006). But there are also some discrete concentrations of power. Most evident of these are the concentrations in the range 200 glyph[lessorsimilar] τ ( µ s) glyph[lessorsimilar] 400 on the right-hand-side of the figure. These concentrations indicate that there are particular regions, within a few milli-arseconds of the direct lineof-sight to B1937+21, which are strongly diffracting, or refracting signals from this pulsar into our telescope. Apparently similar features were discovered by Hill et al (2005), in a multi-epoch study of PSR B0834+06, who found that their features appear to move through the delay-Doppler plane at constant velocity, consistent with the observed proper-motion of the pulsar. At present we don't know whether that property also holds for the features seen in figure 8. In addition to the real structure just discussed, a strong artifact is plain in figure 8: around zero delay there is a broad, horizontal stripe in the image. The nature of this feature is clear: it is 'scattered power' caused by discontinuities between successive values of H ( ν ) (or h ( τ )) in our temporal sequence. These discontinuities might arise in several ways, for example: inadequate sampling of the evolving H ( ν ); amplitude fluctuations in the pulsar; arbitrary phase rotations between successive filter solutions (per the degeneracy in overall phase, § 2.1); or gaps in the data record. We have considered each of the above possibilities, but none provides a satisfactory explanation, as we now detail. First, the evolution of the filter H ( ν ) is well sampled by our 15-second cadence, as can be seen from the upper panel of figure 6. Secondly, there are ∼ 10 4 pulses within each of our cyclic spectra, so the variations in average intensity between samples will be small, glyph[lessorsimilar] 1%. In fact even this variation is irrelevant to figure 8 as we have normalized each filter solution such that it has a rootmean-square value of unity. Thirdly, the arbitrary phase of each filter (see § 2.1) has been chosen so that each solution H ( ν, t n ) matches the previous solution H ( ν, t n -1 ) as closely as possible, in a least-squares sense. Finally, although there is indeed a gap of 30 seconds in our temporal coverage (caused by a change of hard-disk during observing), we have interpolated across this gap before constructing figure 8. For these reasons we do not expect any of these effects to be responsible for the high levels of scattered power seen in figure 8. Aclue to the origin of the scattered power can be found in figure 9, which shows the 'secondary spectrum' - i.e. the power-spectrum of the dynamic spectrum, | H ( ν, t ) | 2 - for our data. By contrast with figure 8, this quantity shows quite low levels of power scattered to large Doppler-shifts. In forming the spectrum, | H ( ν ) | 2 , we are erasing all information on the phase of the filter H ( ν ), so the difference in scattered power levels between figures 8 and 9 indicates that the source of the scattered power in figure 8 is phase-discontinuities between adjacent filters, H ( ν ). As noted above, we have matched the phases of adjacent filters, to the extent that this can be done with a uniform phase rotation of H . Therefore our filter solutions contain non-uniform phase structure that is discontinuous between adjacent samples. In § 4.3.1 we noted that our filter solutions may exhibit covariance between the lag coefficients h ( τ m ) and h ( τ j ), for lag separations small compared to the pulse-width ( | τ m -τ j | glyph[lessmuch] w ), and that the poorly constrained combination ( h ( τ j ) -h ∗ ( -τ j )) modifies only the phase of H . We therefore attribute the scattered power evident in figure 8 to these parameter covariances. We defer a detailed treatment of these issues to a later paper. As figures 8 and 9 both display the response of the interstellar medium in the delay-Doppler coordinate frame, it is worth clarifying the relationship between them. Recall that h ( τ, ω ) is just the Fourier Transform of the sequence H ( ν, t ). Thus the Fourier transform of the dynamic spectrum, which is the Fourier transform of the product H ( ν, t ) H ∗ ( ν, t ), is just the convolution of h ( τ, ω ) with h ∗ ( τ, ω ). Consequently the arc that appears around the origin in figure 8 is echoed in a series of inverted arclets in figure 9; each of these arclets is centered on one of the power concentrations visible in figure 8 - cf. figure 5 of Walker et al (2004). Because the 'secondary spectrum' (figure 9) is equivalent to a self-convolution of the delay-Doppler image (figure 8), the latter is more fundamental and will typically be the more useful quantity for two reasons. First because the delay-Doppler image exhibits the scattered field with greater clarity: in the secondary spectrum the scattered field is tangled up with itself. Secondly, convolution is a smoothing operation, so faint power concentrations are more easily seen in the delay-Doppler image. These points are well demonstrated by comparing figures 8 and 9. Despite the fact that figure 9 is derived from the dynamic spectrum, it could not have been obtained by conventional spectroscopic methods, in which the on-pulse power-spectrum is determined within a window of width comparable to the width of the main-pulse (or interpulse) component. The reason is simply that windowing restricts the lag-range of the resulting secondary spectrum to the width of that window. Refering to figure 2 we see that the main-pulse would be completely contained within a window of width ∼ 100 µ s, so the resulting lag range would be -50 ≤ τ ( µ s) ≤ 50 - a tiny fraction of the actual lag range of figure 9.",
"pages": [
13,
14,
15
]
},
{
"title": "7. DISCUSSION AND FUTURE DIRECTIONS",
"content": "Because cyclic spectroscopy has not previously been applied to radio pulsar signals, there are many related issues that deserve consideration. Here we confine ourselves to a brief discussion of three aspects that the present study calls attention to.",
"pages": [
15
]
},
{
"title": "7.1. Precision timing of PSR B1937+21",
"content": "It is well known that the small-scale structure of the ISM can have a significant effect on the measured arrival times of radio pulses, in consequence of the delays (geometric and wave-speed) associated with signal propagation (e.g. Foster and Cordes 1990). These effects are of particular importance if they are epoch dependent, which is the case if the scattering properties of the medium are not statistically uniform transverse to the line-of-sight. It is plain from figure 8 that some of the scattering material towards B1937+21 is indeed very clumpy, with several flux concentrations appearing far from the origin, albeit at low power levels. Previous studies of the dispersion and scattering on this line-of-sight (Cordes et al 1990; Ramachandran et al 2006) preferred a near-Kolmogorov model of the structure, but the clumpiness we see is quite different from the expectations of a uniform Kolmogorov model (Cordes et al 2006; Walker et al 2004). As B1937+21 is routinely used for precision timing experiments (e.g Verbiest et al 2009), a better understanding of this scattering material is desirable. It has previously been reported (Cognard et al 1993; Lestrade, Rickett and Cognard 1998) that B1937+21 occasionally exhibits timing fluctuations, correlated with flux variations, whose properties are suggestive of 'Extreme Scattering Events' - that is, plasma-lensing events (Fiedler et al 1987, 1994; Romani, Blandford and Cordes 1987). Such events require close alignment between the observer, plasma-lens and pulsar, and these events are consequently rare. If the alignment is not so close then the lens will cause smaller flux changes, but may still have a significant effect on the pulse arrival time because the extra path-length traversed by the faint images may be large. Furthermore these poorly aligned lens configurations should be relatively common. It is possible that plasma-lensing is responsible for the discrete flux concentrations that we see in the vicinity of τ ∼ 300 µ s (figure 7 and 8), with each concentration being due to one or more additional faint images. We note that at this epoch (MJD53873) the features appear at such large delays that the scattered pulse has little overlap with the unscattered signal, so the pulse arrival time estimate should not be greatly affected. But at later epochs, when the scattering structures are closer to the line-of-sight to the pulsar, the scattered signals may appear at delays τ ∼ 100 µ s where they can exert a substantial influence on the measured arrival time. We defer a quantitative examination of pulse arrival time variations to a later paper. Depending on the electron column-density structure, and the pulsar-lens-observer configuration, several additional images may arise from one plasma lens, so it is possible that all of the flux concentrations we see near τ ∼ 300 µ s in figures 7 and 8 are due to a single lens. Under that hypothesis, the observed range of delays (200 glyph[lessorsimilar] τ ( µ s) glyph[lessorsimilar] 400) tells us something about the size of the lens. Assuming that the pulsar is at a distance ∼ 5 kpc, and that the lens is near the midpoint, one finds that the lens diameter is ∼ 4 AU. This is comparable to the dimensions that have previously been inferred for the lenses responsible for Extreme Scattering Events (e.g. Romani, Blandford and Cordes 1987). Unfortunately, with the techniques currently available to us, it is not possible to distinguish between lenslike, refractive behaviour and diffractive scattering as the cause of the observed power concentrations around τ ∼ 300 µ s. The clearest way to distinguish between these possibilities would be to undertake rigorous, quantitative physical modelling of the particular wave-propagation paths for this line-of-sight at the epoch(s) of observation. Such modelling would also tell us the relationship between the pulse arrival times actually observed, and those that would have been observed in the absence of the scattering medium. Physical modelling is, however, beyond the scope of this paper.",
"pages": [
15
]
},
{
"title": "7.2. Cyclic spectropolarimetry",
"content": "We have seen how cyclic spectroscopy gives access to the intrinsic modulation (pulse) profile of the signal, and that this can reveal new structure (figure 2) which is otherwise masked by the effects of scattering. It is the sharp features of the profile - those which include a large fraction of high-modulation-frequency Fourier components which are most affected by the scattering. All the results shown in this paper are based on a signal combination which approximates Stokes-I (recall that our data have not been polarization calibrated). But many pulsars exhibit highly polarized radio emission, and the polarized pulse profiles may be quite complex (van Straten 2006; Johnston et al 2008). For example, there are pulsars where the profile shows rapid transitions between orthogonal, elliptically-polarized states - usually referred to as 'orthogonal mode jumps'. Such transitions will be strongly affected by any filtering (temporal smearing) of the signal (Karastergiou 2009). More generally, it is clear that interstellar scattering can have a profound effect on the apparent polarization properties of pulsars at low frequencies (Li and Han 2003; Kramer and Johnston 2008), and we therefore expect the fidelity of polarization profiles to improve substantially when intrinsic profiles, rather than scattered profiles are used. Furthermore, it has been emphasised by van Straten (2006) that the most accurate pulse-timing requires accurate polarimetry. These are strong motivations to further develop the methods of this paper to encompass cyclic spectropolarimetry.",
"pages": [
15,
16
]
},
{
"title": "7.3. Covariance of filter coefficients",
"content": "In § 4.3.1 we drew attention to the issue of covariance amongst the parameters describing the filter coefficients (or, equivalently, the impulse-response coefficients). The effects of these covariances are not easy to quantify because (i) the total number of parameters needed to describe the filter is very large ( ∼ 10 4 in the present case), and (ii) the covariances depend on the properties of both the filter and the pulse profile - neither of which is known a priori. What is clear, though, is the qualitative point that the actual uncertainty in the filter coefficients can be much larger than the standard deviation for a single parameter taken in isolation. We have argued that there are two aspects of the impulse-response functions, seen in figures 7 and 8, that are probably due to parameter covariances. And one of these - the power near zero delay, scattered to large Doppler-shifts - is a very strong feature indeed, being evidently well above the noise floor and potentially masking real features of h ( τ, ω ). In other respects cyclic spectroscopy seems to be a near-ideal tool for studying the propagation of radio-pulsar signals, and the issue of parameter covariance consequently deserves further study. We can identify two aspects that merit particular attention. The first is a thorough understanding of the origin of parameter covariance, and thus how it manifests itself in different representations of the data. Our preliminary analysis ( § 4.3.1) suggests that strong covariance can be traced to pure-phase modifications of the filter. That analysis was only carried through for the simplest possible filter model ( H ( ν ) = 1), and needs to be revisited using more general models. In cases where the data cannot constrain pure-phase modifications of the filter to be small compared to 1 radian, the problem is akin to one of phase-retrieval. Such problems are notoriously difficult, and the difficulty is associated with non-convexity of the target set (Bauschke, Combettes and Luke 2002). With an understanding of the origin of the covariances one would be in a good position to tackle the key question of how to mitigate their effects on the filter models. For example, in § 4.3.1 we noted that the welldetermined/poorly-determined parameter combinations are sum/difference terms of h ( τ j ) and h ∗ ( -τ j ), so one might think of enforcing causality in the solutions, such that h ( τ j ) = 0 for all τ j < 0.",
"pages": [
16
]
},
{
"title": "8. SUMMARY AND CONCLUSIONS",
"content": "Cyclic spectroscopy of PSR B1937+21 was undertaken with a 15 second cadence over a 4 MHz band at 428 MHz, starting from voltages recorded with the Arecibo radio telescope. By least-squares fitting we determined the impulse response function of the ISM for each cyclic spectrum separately, and the intrinsic pulse-profile averaged over the whole observation. In this way we obtained the 428 MHz pulse-profile of B1937+21 free of the influence of interstellar scattering, revealing some weak, but sharp features that had not previously been seen at low radiofrequencies. From our temporal sequence of impulse response functions we derive the delay-Doppler field image. This image exhibits a noise floor at -51 dB relative to the peak power, and we are thus able to see faint features in the angular structure of the received field. Several power concentrations are visible in the delay range 200 -400 µ s. These concentrations can plausibly be attributed to a single plasma-lens, a few AU in diameter, but alternative interpretations are possible. Regardless of their physical origin, the scattered power concentrations are expected to have a deleterious effect on the pulse-timing experiments that are utilizing this pulsar. To accurately describe and remove these effects it is necessary to have a physical model of the various propagation paths by which the signal reaches the telescope. We did not attempt any physical modelling, but we have shown that cyclic spectroscopy provides us with a large quantity of information on these paths, and thus faciltates that process. We caution that our fitting procedure is adversely affected by covariance amongst some combinations of the ∼ 10 4 fit parameters. These covariances were identified as the origin of the scattered power artifact in our delayDoppler image. Parameter covariance appears to be the main challenge currently facing widespread application of cyclic spectroscopy. We thank Dan Stinebring for helpful discussions that prompted our examination of parameter covariances. This paper is dedicated to the memory of Don Backer.",
"pages": [
16
]
},
{
"title": "REFERENCES",
"content": "Antoni J., 2007, Mechanical Systems and Signal Processing, 21, 597 Non-probabilistic Theory MNRAS, 388, 261 Karastergiou A., 2009, MNRAS, 392, L60 Note . - This table summarizes the results of the tests described in this Appendix. Each line represents the outcomes from least-squares modelling of H ( ν ), or h ( τ ), for 10 sample cyclic spectra of B1937+21. In each case there are approximately 3 × 10 6 degrees of freedom and the total number of parameters in the model is roughly 13,000. Kraft D., 1994, ACM Trans. Math. Software, 20 (3), 262 Kramer M., Johnston S., 2008, MNRAS, 390, 87 Lestrade J.-F., Rickett B.J., Cognard I., 1998, A&A, 334, 1068 Li X.H.., Han J.L., 2003, A&A, 410, 253 Liu D.C., Nocedal J., 1989, Math. Programming, 45, 503 Manchester R.N., Hobbs G.B., Teoh A., Hobbs M., 2005, AJ, 129, 1993 Nocedal J., 1980, Math. Comput., 35, 773 Nocedal J., Wright S.J. 1999 'Numerical Optimization' (Springer: New York) Ramachandran R., Demorest P.B., Backer D.C., Cognard I., Lommen A., 2006, ApJ, 645, 303 Rickett B.J., 1975, ApJ, 197, 185 Rickett B.J., 1990, ARA&A, 28, 561 Roberts J.A., Ables J.G., 1982, MNRAS, 201, 1119 Romani R.W., Blandford R.D., Cordes J.M., 1987, Nature, 328, 324",
"pages": [
16,
17
]
},
{
"title": "TESTS OF THE FILTER OPTIMIZATION CODE",
"content": "Here we describe tests which we have undertaken to evaluate the performance of our software. Three different aspects of the optimization were compared: L-BFGS versus other algorithms; lag-space versus frequency-space optimization; and Unit versus Proximate initializations. All of these comparisons were made using cyclic spectrum samples #2-11 of PSR B1937+21 recorded at Arecibo on MJD53873 (first processing of the data: see § 6). We do not expect that our conclusions regarding the relative merits of the different optimization paths are machine dependent. But for reference: the machine used for these tests was a MacBook Pro with a dual core 2.7 GHz Intel processor and 8 GB RAM installed. With this machine almost all algorithms required approximately 2 seconds to complete a single iteration, so run-times for the various approaches can be compared directly from the number of steps required to complete the optimization. Table 1 sets out the results of our tests. The first three columns show the NLopt algorithm used, the space in which the filter was optimized, and the initialization conditions. Column four shows the average number of steps (rounded to the nearest integer) required to find the best-fit model for the ten sample cyclic spectra. Column five shows the number of sample cyclic spectra in which a particular configuration yielded the best result (i.e. lowest value of M min ) out of all of the configurations tested. And the final column shows the average value of M min , relative to the best-performing configuration, in units of σ 2 (rounded to the nearest integer). The ordering of the outcomes in the table was dictated by the results given in the last column, because a high-quality fit is our main objective. In the following sections we consider the outcomes presented in table 1, and their implications for the choice of optimization approach.",
"pages": [
17
]
},
{
"title": "L-BFGS vs other algorithms",
"content": "By design the NLopt package makes it possible to switch easily between a variety of different optimization algorithms, and thus to select the best one for the task at hand: to change algorithms is simply a matter of altering one line of code. The algorithms available within NLopt include both global and local methods. Global methods are not practical Stinebring, D.R., McLaughlin, M.A., Cordes, J.M., Becker, K.M., Espinoza-Goodman, J., Kramer, M.A., Sheckard, J.L., Smith, C.T., 2001, ApJ, 549, L97 Svanberg K., 2002, SIAM J. Optim. 12 (2), 555 Thorsett S.E., Stinebring D.R., 1990, ApJ361, 644 van Straten W., 2006, ApJ, 642, 1004 van Straten W., Bailes M., 2011, PASA, 28, 1 van Straten W., Demorest P.B., Oslowski S., 2012, Astron. Res. Tech., 9, 237 Verbiest J.P.W., et al 2009, MNRAS, 400, 951 Vlˇcek J., Lukˇsan L., 2006, J. Comp. Appl. Math. 186, 365 Walker M.A., Melrose D.B., Stinebring D.R., Zhang C.M., 2004, MNRAS, 354, 43 Walker M.A., Koopmans L.V.E., Stinebring D.R., van Straten W., 2008, MNRAS, 388, 1214 Yan W.M., et al, 2011, MNRAS, 414, 2087 for our problem because of the large-scale nature of the optimization: it would be necessary to thoroughly search a space of ∼ 10 4 dimensions in order to find the global minimum. Of the local methods, there are algorithms which require derivatives of M to be supplied, and those which do not. As we are able to supply derivatives, and this is a major advantage in exploring the hypersurface of M , we restrict ourselves to those algorithms which make use of the gradient of M ; there are five such algorithms available in NLopt. One of these, SLSQP ('Sequential Least Squares Quadratic Programming'; Kraft 1994), had not completed a single step after more than an hour of run-time, at which point we terminated the optimization by force. The failure of SLSQP on our optimization problem is not surprising: it uses dense-matrix methods which, for our problem, requires ∼ 10 4 times more storage space and run-time than a limited-memory algorithm. Results for the remaining four algorithms are given in table 1. We can see a clear division between these four: the Method of Moving Asymptotes (MMA; Svanberg 2002) and the Truncated Newton method (TNewtonPR; Dembo and Steihaug 1982) both performed poorly on our optimization task, in terms of the quality of fit and run-time, when compared to the Variable Metric (in either rank 1 or rank 2 forms: VarMetric1,2; Vlˇcek and Lukˇsan 2006) and L-BFGS algorithms. We note the failure of TNewtonPR to complete the optimization task from Proximate initialization, or from Unit initialization in frequency-space, hence the omission of those results. It is clear that MMA and TNewtonPR are uncompetitive for our optimization task and we do not consider them further. It is not surprising that the VarMetric and L-BFGS algorithms yield similar results as they are similar algorithms. Nevertheless, our tests do show a clear preference for L-BFGS over either of the variable metric methods, with L-BFGS providing the three top-performing configurations, as gauged by δM min , and 9/10 of the best individual fits (column 5 of table 1).",
"pages": [
17,
18
]
},
{
"title": "Lag-space vs. frequency-space",
"content": "We have already noted ( § 4.2) that lag-space optimization is expected to be superior to a frequency-space approach, because of the traps present in the latter space. This expectation is borne out in practice, with lag-space optimization yielding better fits than the corresponding frequency-space optimization in almost every case in table 1. However, the difference is not very great. We interpret this as meaning that L-BFGS and the VarMetric algorithms obtain enough information on the hyper-surface of M to allow them to avoid most of the traps. One potential problem which we noticed during our tests is that L-BFGS, when used in frequency-space, would sometimes oscillate as it progressed towards the minimum. This phenomenon was most noticeable with Unit initialization; it appears to be responsible for the 30% extra steps required for L-BFGS-Freq-Unit relative to L-BFGS-Lag-Unit. We note that the cyclic spectra used for these tests (see § 6) have typical signal-to-noise ratio greater than unity, for low harmonic numbers, on individual channels. It remains to be seen whether frequency-space optimization remains competitive for cyclic spectra which exhibit low signal-to-noise ratio at all harmonic numbers.",
"pages": [
18
]
},
{
"title": "Variation of initialization",
"content": "The algorithms tested here are local methods. That is, they locate a minimum of M in the vicinity of the starting point, but this minimum is not guaranteed to be the global minimum of M . The local nature of our solutions is something that readers should be aware of. However, reliably finding the true, global minimum of M in a space with ∼ 10 4 dimensions is a difficult problem which does not seem tractable with the computational technologies currently available. Given the difficulty of finding the true minimum of M , it behoves us to examine the sensitivity of our results to the starting point from which the optimization of H proceeds. Unsurprisingly, table 1 shows that optimization from a Proximate initialization is roughly a factor of two quicker than from Unit initialization. And Proximate initialization always yields a significantly better fit, for a given choice of algorithm and optimization-space. Bearing in mind the large-scale nature of the optimization, with ∼ 10 4 parameters, some sensitivity to the initialization conditions is not surprising. The fact that there are significant differences between Unit and Proximate initializations suggests the specific question 'how far are our best results from the corresponding global minima?' As a partial answer to that question we can compare the results of different Proximate initializations, because each of the 10 sample cyclic spectra used in our tests has cyclic spectra taken immediately before and immediately after, and we can step through this sequence in either direction. Referring to the L-BFGS-Lag-Proximate results in table 1 as 'Forward' initialization, we find that the corresponding 'Backward' initialization typically gives worse results, with the average M min being larger by 7 σ 2 and needing 31 more steps per cyclic spectrum, on average, to complete. Forward initialization produced a better fit than Backward for eight of the ten spectra, 9 and the root-mean-square difference between the corresponding M min values is approximately 21 σ 2 . Clearly the variations of the L-BFGS-Lag-Proximate outcomes, relative to the true minimum for each spectrum, must therefore be at least as large as 21 σ 2 , indicating that there is room for some significant improvement. This point was confirmed by the following: we ran the whole suite of optimization tests again, but with a tighter fractional tolerance on M of 0 . 01 / N dof for the stopping criterion. For each of the ten sample cyclic spectra, we took the lowest value of M min (regardless of the configuration which achieved that result) as a reference point. Compared to that reference point, we find that the best-performing configuration of the standard-precision tests (i.e. L-BFGS-Lag-Prox; table 1) is worse by δM min glyph[similarequal] 41 σ 2 , on average, for each cyclic spectrum. 9 This level of asymmetry between Forward and Backward initialization is slightly surprising, being expected only once in 18 trials, but we have no explanation other than as a random occurence. and In the high-precision suite of tests we observed that none of the consistent outcomes of table 1 - i.e. L-BFGS better than other algorithms, Prox better than Unit, Lag better than Freq - were reproduced. Not surprisingly, the differences in M min amongst the 12 tested configurations were considerably smaller than shown in table 1, with the worst-performing configuration being only 7 σ 2 above the best (cf. 39 σ 2 in table 1). These facts suggest that in the high-precision tests all configurations have penetrated well into the noise-limited region of the optimization. The penalty for doing so, of course, is that many more steps are required to achieve that outcome - 784 steps, on average, for L-BFGS-Lag-Prox, which is more than 3 times the number of steps required to satisfy our usual stopping criterion (see table 1).",
"pages": [
18,
19
]
},
{
"title": "ESTIMATION OF MODEL UNCERTAINTIES",
"content": "We have already determined the curvature of M with respect to the coefficients describing S x and H (equations 28 and 29). For the parameters describing the lag-space representation of the filter, the curvatures can be obtained by taking the real and imaginary parts of the relations and where the matrices A and C are given by glyph[negationslash] glyph[negationslash] Here we have used notation such that h n + j -m means h ( τ n + τ j -τ m ), for example; and we have neglected the contribution from a sum over the residuals, whose expectation is zero. It is clear that the uncertainties in our parameter estimates depend on the filter coefficients and intrinsic pulse profile. But for our purposes here it suffices to determine rough estimates of the parameter uncertainties. To proceed we therefore consider the particular case H ( ν ) = 1. For this circumstance we obtain and where F is a measure of the total pulsed flux, with glyph[negationslash] For the lag representation of the filter we find glyph[negationslash] and for τ j = 0 the curvature with respect to the real part of the coefficient h j is twice this value, whereas there is no curvature with respect to the imaginary part. This last point, which implies a formally infinite uncertainty, should not cause concern because the overall phase of the filter is completely arbitrary. Using equation 39 we can immediately translate these curvatures into standard deviations. The results are and glyph[negationslash] In all these cases the coefficients are complex; the quoted uncertainty is the uncertainty in the real part of the coefficient, which is equal to the uncertainty in the imaginary part. With the exception of one coefficient of h , the standard deviation is uniform across each set of coefficients. In practice the system noise, σ , is dependent on the total number of radio-frequency channels, N ν , because we have a fixed total bandwidth, B , for the instrument. Thus N ν ∆ ν = B , and equation 14 can be written A further simplification is appropriate. For cyclic spectroscopy of a pulsar with period P , the pulsar's rotation frequency Ω = 1 /P is necessarily equal to the spacing in modulation frequency, ∆ α , and in turn this is the natural choice for channelisation, ∆ ν . Thus the natural configuration is PB = N ν , and for this circumstance we obtain glyph[negationslash]",
"pages": [
19,
20
]
},
{
"title": "Noise levels for more general filters",
"content": "The curvature of the demerit function with-respect-to the various model parameters depends on the structure in the filter functions, as manifest in equations 28, 29, A3, A4, but we have so far considered only the simplest filter, H ( ν ) = 1. We now consider how structure in the filter affects the noise level on various parameters. It is, of course, possible to concoct bizarre examples of filters which imply correspondingly unusual noise properties. But we shall ignore such possibilities as our purpose here is to describe what one might normally expect to encounter in practice. To that end we will restrict our discussion to cases where 〈| H ( ν ) | 2 〉 ∼ 〈| H ( ν ) | 4 〉 ∼ 1, and we will characterize the impulse response function by a typical scattering time, τ s , corresponding to a filter decorrelation bandwidth ∼ 1 /τ s . Consider first the noise level for the pulse harmonic coefficients. For low harmonics the summation in equation 28 is approximately N ν 〈| H ( ν ) | 4 〉 . But at higher harmonics, where | α m | τ s ∼ 1, there is some decorrelation between | H ( ν -α m / 2) | and | H ( ν + α m / 2) | and the sum declines. In the limit of complete decorrelation, | α m | τ s glyph[greatermuch] 1, the summation yields N ν 〈| H ( ν ) | 2 〉 2 . Providing that both second- and fourth-order expectation values are of order unity, this is not a big effect. For example, in the random-phasor picture for the electric field the intensity statistics are exponential, so 〈| H ( ν ) | 2 〉 = 1 and 〈| H ( ν ) | 4 〉 = 2, yielding a noise level for high harmonics which is √ 2 larger than for low harmonics. In this picture, the noise level for high harmonics coincides with the value quoted in equation A13, for the case H ( ν ) = 1. Quite a different situation arises for the filter coefficients H k . It is evident that the curvatures given in equation 29 may be much less than 4 F 2 in regions where the filter function is small, with correspondingly large errors on those coefficients. As with the noise on the pulse harmonics, there are two different limiting cases relating to the value of the typical scattering time. Most of the pulsed flux, F , is contributed by harmonics up to | α m | ∼ 1 /w , where w is the temporal width of the pulse. If τ s glyph[lessmuch] w then the filter function H ( ν k -α ) is almost constant over the range of α which contributes most to F , so the curvature in equation 29 becomes 4 F 2 | H k | 2 . Clearly this curvature could be very large (small) in comparison with the estimate given in equation A6, leading to correspondingly small (large) errors in the H k estimates. In the opposite limit, where τ s glyph[greatermuch] w , the filter coefficient | H ( ν -α ) | changes rapidly with harmonic number and we obtain a curvature estimate ∼ 4 F 2 〈| H ( ν ) | 2 〉 ∼ 4 F 2 , comparable to that given in equation A6. Finally we consider the effect of a structured filter on the errors associated with the lag-space filter coefficients, h j . The curvatures of the merit function with respect to real and imaginary parts are (equations A3 and A4) made up of two terms. The first term is the same in both cases and we expect it to be 4 F 2 〈| H ( ν ) | 2 〉 / N ν ∼ 4 F 2 / N ν . The second term differs in sign between the real and imaginary parts of the coefficients; it is the real part of a sum of complex numbers. In normal circumstances those complex numbers bear no particular phase relationship to each other, so the second term is typically small in comparison with the first. We therefore neglect it, and we conclude that in normal circumstances the curvatures given in equation A8 are appropriate to all lag-space filter coefficients. and",
"pages": [
20
]
}
]
2013ApJ...779..131H
https://arxiv.org/pdf/1310.1913.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_86><loc_85><loc_87></location>DISCOVERY OF GEV EMISSION FROM THE CIRCINUS GALAXY WITH THE FERMI -LAT</section_header_level_1>
<text><location><page_1><loc_13><loc_81><loc_86><loc_85></location>Masaaki Hayashida 1,2, 3 , /suppressLukasz Stawarz 4, 5 , Chi C. Cheung 6 , Keith Bechtol 7 , Greg M. Madejski 2 , Marco Ajello 8 , Francesco Massaro 2 , Igor V. Moskalenko 2 , Andrew Strong 9 , and Luigi Tibaldo 2 Accepted to the Astrophysical Journal</text>
<section_header_level_1><location><page_1><loc_45><loc_79><loc_55><loc_80></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_64><loc_86><loc_78></location>We report the discovery of γ -ray emission from the Circinus galaxy using the Large Area Telescope (LAT) onboard the Fermi Gamma-ray Space Telescope . Circinus is a nearby ( ∼ 4 Mpc) starburst with a heavily obscured Seyfert-type active nucleus, bipolar radio lobes perpendicular to the spiral disk, and kpc-scale jet-like structures. Our analysis of 0.1-100 GeV events collected during 4 years of LAT observations reveals a significant ( /similarequal 7 . 3 σ ) excess above the background. We find no indications of variability or spatial extension beyond the LAT point-spread function. A power-law model used to describe the 0 . 1 -100GeV γ -ray spectrum yields a flux of (18 . 8 ± 5 . 8) × 10 -9 phcm -2 s -1 and photon index 2 . 19 ± 0 . 12, corresponding to an isotropic γ -ray luminosity of 3 × 10 40 erg s -1 . This observed γ -ray luminosity exceeds the luminosity expected from cosmic-ray interactions in the interstellar medium and inverse Compton radiation from the radio lobes. Thus the origin of the GeV excess requires further investigation.</text>
<text><location><page_1><loc_14><loc_61><loc_86><loc_63></location>Subject headings: radiation mechanisms: non-thermal - galaxies: active - galaxies: individual (Circinus) - galaxies: jets - galaxies: Seyfert - gamma rays: galaxies</text>
<section_header_level_1><location><page_1><loc_22><loc_57><loc_35><loc_58></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_34><loc_48><loc_57></location>Active galactic nuclei (AGN) often possess fast nuclear outflows and collimated radio-emitting jets powered by accretion onto supermassive black holes (e.g., Krolik 1999). Relativistic jets produced in radio-loud AGN (such as blazars and radio galaxies), which are typically hosted by early-type galaxies, are well-established sources of Doppler-boosted γ -ray emission, dominating the extragalactic source population in the GeV range (Nolan et al. 2012). The bulk of the observed γ -ray jet emission in such systems is believed to originate within a few parsecs of the central black hole (e.g., Abdo et al. 2010a). However, Fermi -LAT observations of nearby radio galaxies indicate that large-scale structures - hereafter 'lobes' or 'bubbles' - formed by jets outside their hosts due to interactions with the intergalactic medium, can also be bright GeV emitters (Abdo et al. 2010b; see also Katsuta et al. 2013).</text>
<text><location><page_1><loc_8><loc_27><loc_48><loc_34></location>Radio-quiet AGN, the most luminous of which are classified as Seyferts by their optical spectra, are typically hosted by late-type galaxies. The 'radio-quiet' label does not necessarily mean these sources are completely radio-silent', and the nuclear jets found in these sys-</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_24><loc_48><loc_26></location>1 Institute for Cosmic Ray Research, University of Tokyo, 51-5 Kashiwanoha, Kashiwa, Chiba, 277-8582, Japan</list_item>
<list_item><location><page_1><loc_10><loc_21><loc_48><loc_24></location>2 Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Stanford University, 2575 Sand Hill Road M/S 29, Menlo Park, CA 94025, USA</list_item>
<list_item><location><page_1><loc_11><loc_20><loc_32><loc_21></location>3 email: mahaya@icrr.u-tokyo.ac.jp</list_item>
<list_item><location><page_1><loc_11><loc_19><loc_48><loc_20></location>4 Institute of Space and Astronautical Science, JAXA, 3-1-1</list_item>
<list_item><location><page_1><loc_10><loc_18><loc_48><loc_19></location>Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan</list_item>
<list_item><location><page_1><loc_10><loc_16><loc_48><loc_18></location>5 Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Krak'ow, Poland</list_item>
<list_item><location><page_1><loc_10><loc_14><loc_48><loc_16></location>6 Space Science Division, Naval Research Laboratory, Washington, DC 20375-5352, USA</list_item>
<list_item><location><page_1><loc_10><loc_11><loc_48><loc_14></location>7 Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637</list_item>
<list_item><location><page_1><loc_10><loc_9><loc_48><loc_12></location>8 Space Sciences Laboratory, 7 Gauss Way, University of California, Berkeley, CA 94720-7450, USA</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_9></location>9 Max-Planck Institut fur extraterrestrische Physik, 85748 Garching, Germany</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_50><loc_92><loc_58></location>tems are non-relativistic and comparatively low-powered (e.g., Lal et al. 2011), so that AGN and starburst activities may both be equally important factors in forming of kpc-scale outflows and lobes (e.g., Gallimore et al. 2006). Seyfert galaxies seem γ -ray quiet as a class (Teng et al. 2011; Ackermann et al. 2012a).</text>
<text><location><page_1><loc_52><loc_33><loc_92><loc_50></location>Besides radio-loud AGN, another class of established extragalactic γ -ray sources consists of galaxies lacking any pronounced nuclear activity, but experiencing a burst of vigorous star formation. The observed GeV emission from these sources is most readily explained by interactions of galactic cosmic rays (CRs) with ambient matter and radiation fields of the interstellar medium (ISM; Ackermann et al. 2012b, and references therein). Starburst-driven outflows routinely found in such systems may lead to the formation of large-scale bipolar structures extending perpendicular to the galactic disks, and somewhat resembling jet-driven AGN lobes (Veilleux et al. 2005).</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_33></location>Circinus is one of the nearest (distance D = 4 . 2 ± 0 . 7Mpc; Tully et al. 2009) and most extensively studied composite starburst/AGN systems, but its location behind the intense foreground of the Milky Way disk, ( l , b ) = (311 . 3 · , -3 . 8 · ), resulted in exclusion from several previous population studies using LAT data (Ackermann et al. 2012a,b). Circinus exhibits a polarized broad H α line with FWHM ∼ 3000kms -1 (Oliva et al. 1998), evidencing a heavily obscured Seyfert 2 nucleus which ranks as the third brightest Comptonthick AGN (Yang et al. 2009). Radio observations have mapped bipolar bubbles extending orthogonal to the spiral host, as well as kpc-scale jet-like structures likely responsible for the formation of the lobes (Elmouttie et al. 1998). The extended morphology of the system including the lobes has been recently resolved and studied in X-rays (Mingo et al. 2012).</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_10></location>Here, we report the discovery of a significant γ -ray excess positionally coinciding with the Circinus galaxy</text>
<figure>
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<caption>Fig. 1.The left color map represents the 1 · × 1 · (0 · . 02 pixels) spatial TS variation of the > 100 MeV γ -ray signal excess centered on Circinus. The green lines denote positional errors of the γ -ray excess at 68 % and 95 % confidence levels for inner and outer lines, respectively. The magenta cross indicates the position of the galaxy core while the cyan ellipse corresponds to the positional error at 95% confidence of 2FGL J1415.7 -6520. The black square corresponds to the area of the right panel, which shows the Australia Telescope Compact Array (ATCA) 1.4 GHz radio contours at /similarequal 20 '' resolution (Elmouttie et al. 1998) superposed with the 2MASS H -band color image (Jarrett et al. 2003); here the galactic disk extends in the NE-SW direction, while the radio lobes in the SE-NW direction.</caption>
</figure>
<text><location><page_2><loc_8><loc_60><loc_48><loc_63></location>using 4 years of Fermi -LAT data and discuss possible origins for this emission.</text>
<section_header_level_1><location><page_2><loc_15><loc_58><loc_42><loc_59></location>2. LAT DATA ANALYSIS AND RESULTS</section_header_level_1>
<text><location><page_2><loc_8><loc_28><loc_48><loc_57></location>The LAT is a pair-production telescope onboard the Fermi satellite with large effective area (6500 cm 2 on axis for > 1 GeV photons) and large field of view (2.4 sr), sensitive from 20 MeV to > 300GeV (Atwood et al. 2009). Here, we analyzed LAT data for the Circinus region collected between 2008 August 5 and 2012 August 5, following the standard procedure 10 , using the LAT analysis software ScienceTools v9r29v0 with the P7SOURCE V6 instrument response functions. Events in the energy range 0.1-100GeV were extracted within a 17 · × 17 · region of interest (RoI) centered on the galaxy core position (RA= 213 . · 2913, Dec = -65 . · 3390: J2000). γ -ray fluxes and spectra were determined by a maximum likelihood fit with gtlike for events divided into 0 . · 1-sized pixels and 30 uniformly-spaced log-energy bins. The background model included all known γ -ray sources within the RoI from the 2nd LAT catalog (2FGL: Nolan et al. 2012), except for 2FGL J1415.7 -6520, located 0 . · 266 away from Circinus. Additionally, the model included the isotropic and Galactic diffuse emission components 11 ; flux normalizations for the diffuse and background sources were left free in the fitting procedure.</text>
<text><location><page_2><loc_8><loc_11><loc_48><loc_28></location>Our analysis yields a test statistic (TS) of 58 for the maximum likelihood fit when placing a new candidate source at the core position of Circinus, corresponding to a formal detection significance of /similarequal 7 . 3 σ . Figure 1 shows the spatial variation in the TS value for the candidate source when evaluated over a grid of positions around the galaxy (TS map). Circinus is located within the 68% confidence region for the direction of the γ -ray excess. The distribution of the γ -ray excess is consistent with a pointlike feature, but we note that the LAT does not have sufficient angular resolution (see e.g., Ackermann et al. 2013b) to spatially resolve the core, the radio lobes, or the galactic disk (all shown in the right panel of Figure 1)</text>
<figure>
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<caption>Fig. 2.GeV spectrum of Circinus as measured by LAT. The black lines represent the best-fit PL model (1 σ confidence band). The vertical bars denote 1 σ statistical errors for the flux estimates within energy bins given by horizontal bars. The arrow denotes a 95% confidence level upper limit.</caption>
</figure>
<text><location><page_2><loc_52><loc_29><loc_62><loc_31></location>of the system.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_29></location>Assuming 2FGLJ1415.7 -6520 is an additional γ -ray source distinct from the Circinus galaxy, we repeat the analysis including 2FGL J1415.7 -6520 in the background model. The γ -ray excess at the position of Circinus yields in such a case a TS of 52, while no significant excess can be seen at the formal position of 2FGL J1415.7 -6520 (TS /similarequal 5.6). Our analysis confirms therefore a single point-like γ -ray excess positionally coinciding with Circinus, which may be identified with 2FGLJ1415.7 -6520 after the localization of the 2FGL source is refined using the 4 yearaccumulation of LAT data. The 2FGL analysis flagged 2FGL J1415.7 -6520 as a source with a low signal-tobackground ratio (2FGL flag was set to '4), meaning its position can be relatively strongly affected by systematic uncertainties in the Galactic diffuse emission model (see also Ackermann et al. 2012c), and the overall positional</text>
<figure>
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<caption>Fig. 3.The color map represents the 1 · × 1 · (0 · . 02 pixels) spatial TS variation of the > 1 GeV γ -ray signal excess centered on Circinus. The green lines denote positional errors of the γ -ray excess at 68 % and 95 % confidence levels. The magenta cross indicates the position of the galaxy core while the cyan ellipse corresponds to the positional error at 95% confidence of 2FGLJ1415.7 -6520.</caption>
</figure>
<figure>
<location><page_3><loc_9><loc_41><loc_45><loc_62></location>
<caption>Fig. 4.Observed counts profile of > 1 GeV γ rays for the Circinus source with events within ± 0 . 5 degree in Declination (northsouth) projected along the R.A. (east-west) direction. The red curve represents the best-fit emission model with the point-like source and diffuse emission. The observed profile is consistent with a point-like source, indicating no significant spatial extension in the γ -ray excess associated with Circinus.</caption>
</figure>
<text><location><page_3><loc_8><loc_24><loc_48><loc_31></location>error of 2FGL J1415.7 -6520 could be larger than indicated. Our analysis here of 4 years of accumulated LAT data for the source allowed it to be detected at higher energies, thus decreasing the systematic uncertainty in the location.</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_24></location>A simple power-law model (PL) adequately describes the γ -ray spectrum of the source, yielding a γ -ray flux F > 0 . 1 GeV = (18 . 8 ± 5 . 8) × 10 -9 phcm -2 s -1 and photon index Γ = 2 . 19 ± 0 . 12, corresponding to an isotropic 0 . 1 -100GeV luminosity of L γ /similarequal (2 . 9 ± 0 . 5) × 10 40 erg s -1 . The best-fit model together with flux points of the spectral energy distribution (SED) are plotted in Figure 2. We also perform spectral fits using a broken powerlaw model and a log-parabola model, but no significant improvement ( < 1 . 2 σ ) can be seen in the likelihood values when compared with the PL parametrization. The results of the spectral fits for Circinus are summarized in Table 1. For comparison, we also derived</text>
<figure>
<location><page_3><loc_52><loc_72><loc_92><loc_92></location>
<caption>Fig. 5.The top panel (a) shows the light curve of Circinus above 100 MeV, binned in half-year intervals. For bins with TS < 4, a 95 % confidence level upper limit is plotted. The bottom panel (b) presents the cumulative TS of the γ -ray excess at the source position.</caption>
</figure>
<text><location><page_3><loc_52><loc_59><loc_92><loc_64></location>γ -ray fluxes and spectra for four LAT-detected starburst galaxies (Ackermann et al. 2012b) with 4-yr LAT data by performing the same analysis procedure described above, and summarize the results in Table 1.</text>
<text><location><page_3><loc_52><loc_45><loc_92><loc_59></location>Since Circinus is located near the Galactic plane, it is important to consider systematic errors caused by uncertainties in the Galactic diffuse emission model. We test eight alternative diffuse models (following Ackermann et al. 2013a; de Palma et al. 2013), and find that the derived source flux above 0.1 GeV shifts by no more than 3%, which is much smaller than the statistical uncertainty. The detection significance at the source position remains stable for the different alternative diffuse models (TS /similarequal 58 -60).</text>
<text><location><page_3><loc_52><loc_17><loc_92><loc_45></location>We also analyze a limited event sample with an energy range 1 -100GeV within a 10 · × 10 · RoI. In this energy range, reduced systematic effects related to diffuse emission modeling are expected, owing in part to the improved spatial resolution of the LAT. The powerlaw character of the γ -ray continuum is confirmed with Γ = 2 . 19 ± 0 . 41, and we find that much of the statistical power of our search comes from photons above 1 GeV (TS=42). The extrapolated flux above 0.1 GeV from the higher energy analysis corresponds to (16 . 4 ± 5 . 6) × 10 -9 phcm -2 s -1 , in good agreement with the result obtained using events from 0.1 GeV. Figure 3 shows a TS map created using events with this energy band, and we have confirmed that Circinus is located at the edge of the 68% confidence contour of the γ -ray source. We also produced an observed counts profile of Circinus and compared it with the best-fit point-like source model as shown in Figure 4. The observed counts profile is consistent with the model shape of a point-like source, indicating no significant spatial extension of the γ -ray excess.</text>
<text><location><page_3><loc_52><loc_7><loc_92><loc_17></location>In order to search for variability in the γ -ray flux, we divided the 4-year observation window into halfyear intervals and derived γ -ray fluxes above 0.1 GeV for each time interval using a PL model with photon index fixed at the best-fit value obtained for the 4-year dataset. No significant variability is found in the γ -ray flux ( χ 2 /d.o.f = 5 . 2 / 7). We also checked the cumulative significance of the γ -ray excess at the source position</text>
<table>
<location><page_4><loc_14><loc_79><loc_86><loc_87></location>
<caption>TABLE 1 Starburst galaxies detected with Fermi -LAT.</caption>
</table>
<figure>
<location><page_4><loc_13><loc_50><loc_48><loc_75></location>
</figure>
<figure>
<location><page_4><loc_52><loc_50><loc_88><loc_75></location>
<caption>Fig. 6.Comparisons between (a:left) the γ -ray (0.1-100 GeV) and total IR (8 -1000 µ m), (b:right) the γ -ray and 1.4 GHz radio luminosities of star-forming galaxies and Seyferts. Circinus is denoted by the red filled circle. Eight γ -ray detected star-forming galaxies are plotted by magenta points. γ -ray upper limits for non-detected starburst galaxies selected from the HCN survey are denoted by blue arrows, and for hard X-ray selected Seyfert galaxies by black arrows. The orange line represents the best-fit power-law relation between L γ and L IR (or νL radio ) for star-forming galaxies, and the orange bands includes the fit uncertainty and intrinsic dispersion around the fitted relation. The dotted red line corresponds to the calorimetric limit assuming an average CR luminosity per supernova of ηE SN = 10 50 erg.</caption>
</figure>
<text><location><page_4><loc_8><loc_36><loc_48><loc_41></location>during the 4-year observation; the TS showed a gradual increase, as expected for a steady source. Both the source light curve and the cumulative TS are presented in Figure 5.</text>
<section_header_level_1><location><page_4><loc_23><loc_34><loc_34><loc_35></location>3. DISCUSSION</section_header_level_1>
<text><location><page_4><loc_8><loc_13><loc_48><loc_33></location>In Figure 6, we plot the γ -ray luminosity of Circinus, L γ , normalized by the total IR (8 -1000 µ m) luminosity, L IR , as a function of L IR . For comparison, the plot includes γ -ray upper limits (95% confidence level) derived for a large sample of starbursts selected from the HCN survey, together with eight γ -ray detected star-forming galaxies (the four Local Group galaxies and four starbursts). These all follow Ackermann et al. (2012b), with the exception of the four starbursts where the results are from our updated 4-yr LAT analysis (Table 1). In addition, we also show the γ -ray upper limits for the sample of hard X-ray selected Seyferts (Ackermann et al. 2012a). The IR luminosities are derived from the fluxes measured by the Infrared Astronomical Satellite (IRAS) in four bands, following Sanders & Mirabel (1996).</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_13></location>As shown in Figure 6, and discussed quantitatively in Ackermann et al. (2012b), the γ -ray emission related to CRs in the ISM scales with the total IR luminosity, which serves as a proxy for the injected CR power within the supernova remnant paradigm for galactic CR origin. Circi-</text>
<text><location><page_4><loc_52><loc_16><loc_92><loc_41></location>us is more than five times over-luminous in the γ -ray band when compared with this scaling relation (see also Table 1). In fact, the L γ /L IR luminosity ratio for Circinus exceeds the so-called 'calorimetric limit' expected to hold when CRs interact faster than they can escape the galactic disk, and as much as η /similarequal 10 % energy per supernova explosion goes into CR acceleration, ηE SN /similarequal 10 50 erg (see, e.g., Lacki et al. 2011; Ackermann et al. 2012b). Circinus also does not follow the correlation found for star-forming galaxies between L γ and total monochromatic 1.4 GHz radio luminosity νL R , by a factor of six (see Table 1 and the right panel of Figure 6). Following the approach of Ackermann et al. (2012b), and assuming that star-forming galaxies do follow a simple power-law scaling between γ -ray and total-IR luminosity, we find that adjusting either the normalization of this relation or the intrinsic scatter to accommodate Circinus would tend to overpredict the total number of galaxies which have been actually detected by LAT.</text>
<text><location><page_4><loc_52><loc_8><loc_92><loc_16></location>Let us consider the possibility that the total IR and radio luminosities listed in Table 1 underestimate the star formation rate (SFR). Whereas standard scaling relations using IRAS fluxes yield SFR /similarequal 2 M /circledot yr -1 , the most recent mid-IR studies of Circinus with Spitzer presented by For et al. (2012) suggest SFR /similarequal (3 -8) M /circledot yr -1 (de-</text>
<text><location><page_5><loc_8><loc_65><loc_48><loc_92></location>pending on the particular calibration method). The differences between SFRs derived from mid-IR and total-IR (IRAS) measurements might be explained if Circinus is a relatively dust-poor system for which the far-IR luminosity is not a reliable indicator of the SFR. The SFR converted from the observed total 1.4 GHz fluxes (Yun et al. 2001) is only /similarequal 1 . 5 M /circledot yr -1 , but the radio luminosity could also under-represent the actual CR injection power if the interstellar magnetic field strength were below average. On the other hand, the total flux density ratio log ( F 70 µ m /F 1 . 4 GHz ) = 2 . 40 demonstrates that Circinus obeys the 'far-IR/radio' correlation established for local star-forming and starburst systems (e.g., Seymour et al. 2009). The luminosities of NGC 1068 and NGC4945, which are similar starburst/Seyfert composite galaxies, are consistent with the γ -ray-to-total-IR and γ -ray-toradio correlations (Ackermann et al. 2012b, see also Figure 6). These complexities in the Circinus system make a firm conclusion regarding the attribution of γ rays to CR interactions difficult.</text>
<text><location><page_5><loc_8><loc_55><loc_48><loc_65></location>Other than CR interactions, the disk coronae or accretion disks of Seyfert galaxies can be considered as possible γ -ray emission sites (e.g., Nied'zwiecki et al. 2013). However, LAT studies of a large sample of hard X-rayselected Seyferts devoid of prominent jets revealed that such sources are γ -ray quiet as a class, down to the level of 1 -10% of the hard X-ray fluxes (Ackermann et al. 2012a).</text>
<text><location><page_5><loc_8><loc_32><loc_48><loc_54></location>One of the striking characteristics of the Circinus system is the presence of well-defined radio lobes and kpc-scale jet-like features, the 'plumes' (Elmouttie et al. 1998), which are also resolved at X-ray frequencies with Chandra (Mingo et al. 2012). Both of these might be relevant γ -ray emission sites. To investigate this idea quantitatively, we apply standard leptonic synchrotron and inverse-Compton (IC) modeling to the radio spectra of the lobes, investigating whether the extrapolation of the high-energy emission continuum may account for the GeV flux from the system. In the IC calculations we consider seed photons provided by the cosmic background radiation, the observed IR-to-optical emission of the galactic disk, as well as the UV-to-hard X-ray photon field due to the active nucleus, corrected for obscuration (assuming the intrinsic UV emission of the accretion disk /similarequal 3 × 10 43 erg s -1 ; Prieto et al. 2010).</text>
<text><location><page_5><loc_8><loc_8><loc_48><loc_32></location>Figure 7 presents the broad-band, multi-component SED of Circinus, including new LAT measurements. Integrated far-IR-to-optical measurements represent the dominant starlight and the dust emission of the galaxy, with a negligible contribution from a heavily obscured AGN. In the X-ray regime, the hard X-ray fluxes are dominated by the heavily absorbed emission of the accretion disk and disk coronae. As demonstrated in Elmouttie et al. (1998), the total radio fluxes of the source in the 0 . 4 -8 . 6GHz range are due to a superposition of various emission components characterized by different spectral properties. Using their published ATCA 1.4, 2.4, 4.8, and 8.6 GHz maps, we measured fluxes for each distinct component separately, namely for the nucleus (circle with radius, r = 1 '' /similarequal 20pc), galaxy core including the central starburst region ( r = 35 '' circle), the outer parts of the galaxy disk, NW lobe ( r = 44 . 6 '' circle), NW plume (50 '' × 25 '' box centered 68 . 15 '' from</text>
<text><location><page_5><loc_52><loc_81><loc_92><loc_92></location>the nucleus), SE lobe (ellipse with radii of 70 '' and 55 '' ), and SE plume (62 . 5 '' × 25 '' box centered 89 . 30 '' from the nucleus). The total radio emission of the Circinus system at > GHz frequencies is dominated by the central starburst region, and at lower frequencies by the outer parts of the galaxy disk; lobes and plumes contribute to the observed emission at the level of 10%; radio emission of the unresolved nucleus is negligible.</text>
<text><location><page_5><loc_52><loc_52><loc_92><loc_81></location>We fit the radio spectra of the lobes and plumes, assuming energy equipartition between the radiating electrons and the magnetic field (energy density ratio u e /u B ≡ 1), and a standard form of the electron energy distribution consisting of a power-law dN e /dγ ∝ γ -2 between electron energies γ min ≡ 1 and γ br , breaking to dN e /dγ ∝ γ -3 between γ br and γ max ≡ 10 6 . We adjust both the electron normalization and break Lorentz factors γ br to obtain satisfactory fits to the radio data for each region separately, and then evaluate the expected IC emission. The results of the modeling are given as blue curves in Figure 7, where we show in addition the modeled emission of the accretion disk and the disk coronae using the MyTORUS model (red curves) (Murphy & Yaqoob 2009). For comparison, the figure presents also the interstellar radiation field/ GALPROP models (Porter et al. 2008; Strong et al. 2010) for the Milky Way placed at the distance of Circinus (gray curves). Note that the 'clump' structure seen around 10 23 Hz in the γ -ray continuum evaluated with GALPROP is due to the pion decay component dominating over the leptonic (IC and bremsstrahlung) ISM emission.</text>
<text><location><page_5><loc_52><loc_13><loc_92><loc_52></location>The evaluated IC emission of the lobes severely underestimates the measured GeV emission of Circinus. In the model, magnetic field intensities read as B /similarequal 5 -10 µ G, in agreement with the values claimed by Elmouttie et al. (1998). The total energy stored in the entire structure is E tot /similarequal 10 54 erg. This is about an order of magnitude lower than the total energy of the lobes inferred by Mingo et al. (2012) based on X-ray observations, indicating either a significant departure from the energy equipartition condition or dominant pressure support within the lobes provided by hot thermal plasma or relativistic protons. The departures from the minimum energy condition u e /u B /greatermuch 1, which may enhance the expected IC radiation for a given synchrotron (radio) flux, are often claimed for lobes in radio galaxies and quasars (see Takeuchi et al. 2012, and references therein). In the particular case of Circinus, however, the effect would have to be extreme in order to account for the flux detected with the LAT, u e /u B /similarequal 10 4 . Although we cannot exclude this possibility, we consider it rather unlikely. Deep, high-resolution X-ray observations could in principle be used in the near future to validate the u e /u B /greatermuch 1 hypothesis for the Circinus lobes, but at this moment the very limited photon statistics of the available Chandra maps precludes any robust detection of a non-thermal lobe-related emission component at keV photon energies. We note however that the IC model curve calculated for u e /u B = 1 and shown in Figure 7 is below the corresponding upper limits.</text>
<text><location><page_5><loc_52><loc_8><loc_92><loc_13></location>Yet another possibility is the presence of an energetically significant population of CR protons associated with the radio lobes. Relativistic protons injected by the kpc-scale jets/plumes, or accelerated at the bow-shocks</text>
<figure>
<location><page_6><loc_8><loc_69><loc_48><loc_92></location>
<caption>Fig. 7.Broad-band, multi-component SED of Circinus. Black open squares represent the total 'lobes + plumes' radio fluxes, with the assumed 10% uncertainties. Black filled circles denote the total (integrated) fluxes of the system from the Parkes Catalog (0 . 4 -8 . 6 GHz; Wright & Otrupcek 1990), ISO (170 -52 µ m; Brauher et al. 2008), IRAS (100 -25 µ m), Spitzer and 2MASS (70 -1 µ m; For et al. 2012), RC3.9 catalog (extinction-corrected V , B , and U -bands; de Vaucouleurs et al. 1991), ROSAT (0 . 1 -2 . 4 keV; Brinkmann et al. 1994), Suzaku (2 -10 keV; Yang et al. 2009), Swift -BAT (14 -195 keV; Baumgartner et al. 2010) and LAT (0 . 1 -100 GeV; this paper). Gray curves correspond to the interstellar radiation field/ GALPROP models for the Milky Way (Porter et al. 2008; Strong et al. 2010) placed at the distance of Circinus. Red curves denote the modeled emission of the accretion disk and the disk coronae using the MyTORUS model. Blue curves represent the modeled synchrotron and IC emission of the lobes and plumes. The thick gray dotted line represents the γ -ray spectrum of a starforming system (assumed Γ = 2 . 2) corresponding to the IR luminosity of Circinus and IRγ correlation found in Ackermann et al. (2012b).</caption>
</figure>
<text><location><page_6><loc_8><loc_34><loc_48><loc_44></location>of the expanding lobes (see Mingo et al. 2012), may generate a non-negligible γ -ray emission due to the decay of pions produced during CR interactions with surrounding matter. However, due to the sparseness of thermal gas within the galactic halo (gas number density /lessorsimilar 10 -3 cm -3 ), implying a low efficiency for this process, the resulting GeV fluxes are likely below the γ -ray output of the galactic disk.</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_34></location>It is interesting to comment in this context on the similarities between the Circinus lobes and the Fermi -LAT discovered giant 'Fermi Bubbles' in our Galaxy (Dobler et al. 2010; Su et al. 2010). Even though the origin of both structures is still under the debate, the jet activity of their central supermassive black holes which are of comparable masses, namely M BH /similarequal (1 . 7 ± 0 . 3) × 10 6 M /circledot for Circinus (Greenhill et al. 2003) and (4 . 5 ± 0 . 4) × 10 6 M /circledot for the Milky Way (e.g., Ghez et al. 2008) - is the widely considered scenario. In the case of Circinus, the jet scenario is evidenced directly by the presence of collimated outflows supplying the lobes with energetic magnetized plasma (see the discussion in Elmouttie et al. 1998; Mingo et al. 2012), while in the case of our Galaxy, it is supported by general energetic arguments and numerical simulations reproducing well the observed properties of bubbles (e.g., Guo & Mathews 2012; Yang et al. 2012). In both systems, the lobes extend to kpc scales across the galactic disks and are characterized by magnetic field strengths of the order of</text>
<text><location><page_6><loc_52><loc_83><loc_92><loc_92></location>∼ 10 µ G (see Su et al. 2010; Mertsch & Sarkar 2011, for Fermi Bubbles). Thus it seems that structures analogous to the Fermi Bubbles and Circinus lobes may not be uncommon in late-type galaxies undergoing episodic outbursts of AGN (jet) activity, but their contributions to the total γ -ray outputs of the systems do not exceed ∼ 10%.</text>
<section_header_level_1><location><page_6><loc_66><loc_80><loc_78><loc_81></location>4. CONCLUSIONS</section_header_level_1>
<text><location><page_6><loc_52><loc_51><loc_92><loc_80></location>Here we report the detection of a steady and spatially unresolved γ -ray source at the position of Circinus, consistent with 2FGL J1415.7 -6520 based on a refined analysis using 4 years of LAT data. Although the observed power-law spectrum (Γ = 2 . 19 ± 0 . 12) is similar to that of other LAT-detected starburst systems, Circinus is γ -ray over-luminous by a factor of 5-6 relative to what is expected for emission from the ISM, based on multiwavelength correlations observed for nearby star-forming galaxies. However, the range of SFRs estimated from radio, mid-IR, and far-IR luminosities span a factor of /similarequal 4, indicating large uncertainties in the expected CR power delivered to the ISM. We presented several alternative possibilities for the origin of the GeV excess, including emission from the extended radio lobes, but found no conclusive answer to fully account for the GeV emission. This issue may be resolved by future studies, as Circinus is a compelling target for observations in the very high energy γ -ray regime with ground-based Cherenkov telescopes; the TeV-detected starburst systems NGC 253 and M82 (Acero et al. 2009; Acciari et al. 2009) are actually fainter in the GeV range.</text>
<section_header_level_1><location><page_6><loc_64><loc_48><loc_80><loc_49></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_6><loc_52><loc_44><loc_92><loc_48></location>/suppress L. S. was supported by Polish NSC grant DEC2012/04/A/ST9/00083. Work by C.C.C. at NRL is supported in part by NASA DPR S-15633-Y.</text>
<text><location><page_6><loc_52><loc_21><loc_92><loc_44></location>The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat 'a l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucl'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.</text>
<text><location><page_6><loc_52><loc_16><loc_92><loc_21></location>Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d' ' Etudes Spatiales in France.</text>
<text><location><page_6><loc_52><loc_13><loc_92><loc_16></location>We thank M. Elmouttie for providing the ATCA images.</text>
<section_header_level_1><location><page_7><loc_45><loc_91><loc_55><loc_92></location>REFERENCES</section_header_level_1>
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</document>
[
{
"title": "ABSTRACT",
"content": "We report the discovery of γ -ray emission from the Circinus galaxy using the Large Area Telescope (LAT) onboard the Fermi Gamma-ray Space Telescope . Circinus is a nearby ( ∼ 4 Mpc) starburst with a heavily obscured Seyfert-type active nucleus, bipolar radio lobes perpendicular to the spiral disk, and kpc-scale jet-like structures. Our analysis of 0.1-100 GeV events collected during 4 years of LAT observations reveals a significant ( /similarequal 7 . 3 σ ) excess above the background. We find no indications of variability or spatial extension beyond the LAT point-spread function. A power-law model used to describe the 0 . 1 -100GeV γ -ray spectrum yields a flux of (18 . 8 ± 5 . 8) × 10 -9 phcm -2 s -1 and photon index 2 . 19 ± 0 . 12, corresponding to an isotropic γ -ray luminosity of 3 × 10 40 erg s -1 . This observed γ -ray luminosity exceeds the luminosity expected from cosmic-ray interactions in the interstellar medium and inverse Compton radiation from the radio lobes. Thus the origin of the GeV excess requires further investigation. Subject headings: radiation mechanisms: non-thermal - galaxies: active - galaxies: individual (Circinus) - galaxies: jets - galaxies: Seyfert - gamma rays: galaxies",
"pages": [
1
]
},
{
"title": "DISCOVERY OF GEV EMISSION FROM THE CIRCINUS GALAXY WITH THE FERMI -LAT",
"content": "Masaaki Hayashida 1,2, 3 , /suppressLukasz Stawarz 4, 5 , Chi C. Cheung 6 , Keith Bechtol 7 , Greg M. Madejski 2 , Marco Ajello 8 , Francesco Massaro 2 , Igor V. Moskalenko 2 , Andrew Strong 9 , and Luigi Tibaldo 2 Accepted to the Astrophysical Journal",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Active galactic nuclei (AGN) often possess fast nuclear outflows and collimated radio-emitting jets powered by accretion onto supermassive black holes (e.g., Krolik 1999). Relativistic jets produced in radio-loud AGN (such as blazars and radio galaxies), which are typically hosted by early-type galaxies, are well-established sources of Doppler-boosted γ -ray emission, dominating the extragalactic source population in the GeV range (Nolan et al. 2012). The bulk of the observed γ -ray jet emission in such systems is believed to originate within a few parsecs of the central black hole (e.g., Abdo et al. 2010a). However, Fermi -LAT observations of nearby radio galaxies indicate that large-scale structures - hereafter 'lobes' or 'bubbles' - formed by jets outside their hosts due to interactions with the intergalactic medium, can also be bright GeV emitters (Abdo et al. 2010b; see also Katsuta et al. 2013). Radio-quiet AGN, the most luminous of which are classified as Seyferts by their optical spectra, are typically hosted by late-type galaxies. The 'radio-quiet' label does not necessarily mean these sources are completely radio-silent', and the nuclear jets found in these sys- tems are non-relativistic and comparatively low-powered (e.g., Lal et al. 2011), so that AGN and starburst activities may both be equally important factors in forming of kpc-scale outflows and lobes (e.g., Gallimore et al. 2006). Seyfert galaxies seem γ -ray quiet as a class (Teng et al. 2011; Ackermann et al. 2012a). Besides radio-loud AGN, another class of established extragalactic γ -ray sources consists of galaxies lacking any pronounced nuclear activity, but experiencing a burst of vigorous star formation. The observed GeV emission from these sources is most readily explained by interactions of galactic cosmic rays (CRs) with ambient matter and radiation fields of the interstellar medium (ISM; Ackermann et al. 2012b, and references therein). Starburst-driven outflows routinely found in such systems may lead to the formation of large-scale bipolar structures extending perpendicular to the galactic disks, and somewhat resembling jet-driven AGN lobes (Veilleux et al. 2005). Circinus is one of the nearest (distance D = 4 . 2 ± 0 . 7Mpc; Tully et al. 2009) and most extensively studied composite starburst/AGN systems, but its location behind the intense foreground of the Milky Way disk, ( l , b ) = (311 . 3 · , -3 . 8 · ), resulted in exclusion from several previous population studies using LAT data (Ackermann et al. 2012a,b). Circinus exhibits a polarized broad H α line with FWHM ∼ 3000kms -1 (Oliva et al. 1998), evidencing a heavily obscured Seyfert 2 nucleus which ranks as the third brightest Comptonthick AGN (Yang et al. 2009). Radio observations have mapped bipolar bubbles extending orthogonal to the spiral host, as well as kpc-scale jet-like structures likely responsible for the formation of the lobes (Elmouttie et al. 1998). The extended morphology of the system including the lobes has been recently resolved and studied in X-rays (Mingo et al. 2012). Here, we report the discovery of a significant γ -ray excess positionally coinciding with the Circinus galaxy using 4 years of Fermi -LAT data and discuss possible origins for this emission.",
"pages": [
1,
2
]
},
{
"title": "2. LAT DATA ANALYSIS AND RESULTS",
"content": "The LAT is a pair-production telescope onboard the Fermi satellite with large effective area (6500 cm 2 on axis for > 1 GeV photons) and large field of view (2.4 sr), sensitive from 20 MeV to > 300GeV (Atwood et al. 2009). Here, we analyzed LAT data for the Circinus region collected between 2008 August 5 and 2012 August 5, following the standard procedure 10 , using the LAT analysis software ScienceTools v9r29v0 with the P7SOURCE V6 instrument response functions. Events in the energy range 0.1-100GeV were extracted within a 17 · × 17 · region of interest (RoI) centered on the galaxy core position (RA= 213 . · 2913, Dec = -65 . · 3390: J2000). γ -ray fluxes and spectra were determined by a maximum likelihood fit with gtlike for events divided into 0 . · 1-sized pixels and 30 uniformly-spaced log-energy bins. The background model included all known γ -ray sources within the RoI from the 2nd LAT catalog (2FGL: Nolan et al. 2012), except for 2FGL J1415.7 -6520, located 0 . · 266 away from Circinus. Additionally, the model included the isotropic and Galactic diffuse emission components 11 ; flux normalizations for the diffuse and background sources were left free in the fitting procedure. Our analysis yields a test statistic (TS) of 58 for the maximum likelihood fit when placing a new candidate source at the core position of Circinus, corresponding to a formal detection significance of /similarequal 7 . 3 σ . Figure 1 shows the spatial variation in the TS value for the candidate source when evaluated over a grid of positions around the galaxy (TS map). Circinus is located within the 68% confidence region for the direction of the γ -ray excess. The distribution of the γ -ray excess is consistent with a pointlike feature, but we note that the LAT does not have sufficient angular resolution (see e.g., Ackermann et al. 2013b) to spatially resolve the core, the radio lobes, or the galactic disk (all shown in the right panel of Figure 1) of the system. Assuming 2FGLJ1415.7 -6520 is an additional γ -ray source distinct from the Circinus galaxy, we repeat the analysis including 2FGL J1415.7 -6520 in the background model. The γ -ray excess at the position of Circinus yields in such a case a TS of 52, while no significant excess can be seen at the formal position of 2FGL J1415.7 -6520 (TS /similarequal 5.6). Our analysis confirms therefore a single point-like γ -ray excess positionally coinciding with Circinus, which may be identified with 2FGLJ1415.7 -6520 after the localization of the 2FGL source is refined using the 4 yearaccumulation of LAT data. The 2FGL analysis flagged 2FGL J1415.7 -6520 as a source with a low signal-tobackground ratio (2FGL flag was set to '4), meaning its position can be relatively strongly affected by systematic uncertainties in the Galactic diffuse emission model (see also Ackermann et al. 2012c), and the overall positional error of 2FGL J1415.7 -6520 could be larger than indicated. Our analysis here of 4 years of accumulated LAT data for the source allowed it to be detected at higher energies, thus decreasing the systematic uncertainty in the location. A simple power-law model (PL) adequately describes the γ -ray spectrum of the source, yielding a γ -ray flux F > 0 . 1 GeV = (18 . 8 ± 5 . 8) × 10 -9 phcm -2 s -1 and photon index Γ = 2 . 19 ± 0 . 12, corresponding to an isotropic 0 . 1 -100GeV luminosity of L γ /similarequal (2 . 9 ± 0 . 5) × 10 40 erg s -1 . The best-fit model together with flux points of the spectral energy distribution (SED) are plotted in Figure 2. We also perform spectral fits using a broken powerlaw model and a log-parabola model, but no significant improvement ( < 1 . 2 σ ) can be seen in the likelihood values when compared with the PL parametrization. The results of the spectral fits for Circinus are summarized in Table 1. For comparison, we also derived γ -ray fluxes and spectra for four LAT-detected starburst galaxies (Ackermann et al. 2012b) with 4-yr LAT data by performing the same analysis procedure described above, and summarize the results in Table 1. Since Circinus is located near the Galactic plane, it is important to consider systematic errors caused by uncertainties in the Galactic diffuse emission model. We test eight alternative diffuse models (following Ackermann et al. 2013a; de Palma et al. 2013), and find that the derived source flux above 0.1 GeV shifts by no more than 3%, which is much smaller than the statistical uncertainty. The detection significance at the source position remains stable for the different alternative diffuse models (TS /similarequal 58 -60). We also analyze a limited event sample with an energy range 1 -100GeV within a 10 · × 10 · RoI. In this energy range, reduced systematic effects related to diffuse emission modeling are expected, owing in part to the improved spatial resolution of the LAT. The powerlaw character of the γ -ray continuum is confirmed with Γ = 2 . 19 ± 0 . 41, and we find that much of the statistical power of our search comes from photons above 1 GeV (TS=42). The extrapolated flux above 0.1 GeV from the higher energy analysis corresponds to (16 . 4 ± 5 . 6) × 10 -9 phcm -2 s -1 , in good agreement with the result obtained using events from 0.1 GeV. Figure 3 shows a TS map created using events with this energy band, and we have confirmed that Circinus is located at the edge of the 68% confidence contour of the γ -ray source. We also produced an observed counts profile of Circinus and compared it with the best-fit point-like source model as shown in Figure 4. The observed counts profile is consistent with the model shape of a point-like source, indicating no significant spatial extension of the γ -ray excess. In order to search for variability in the γ -ray flux, we divided the 4-year observation window into halfyear intervals and derived γ -ray fluxes above 0.1 GeV for each time interval using a PL model with photon index fixed at the best-fit value obtained for the 4-year dataset. No significant variability is found in the γ -ray flux ( χ 2 /d.o.f = 5 . 2 / 7). We also checked the cumulative significance of the γ -ray excess at the source position during the 4-year observation; the TS showed a gradual increase, as expected for a steady source. Both the source light curve and the cumulative TS are presented in Figure 5.",
"pages": [
2,
3,
4
]
},
{
"title": "3. DISCUSSION",
"content": "In Figure 6, we plot the γ -ray luminosity of Circinus, L γ , normalized by the total IR (8 -1000 µ m) luminosity, L IR , as a function of L IR . For comparison, the plot includes γ -ray upper limits (95% confidence level) derived for a large sample of starbursts selected from the HCN survey, together with eight γ -ray detected star-forming galaxies (the four Local Group galaxies and four starbursts). These all follow Ackermann et al. (2012b), with the exception of the four starbursts where the results are from our updated 4-yr LAT analysis (Table 1). In addition, we also show the γ -ray upper limits for the sample of hard X-ray selected Seyferts (Ackermann et al. 2012a). The IR luminosities are derived from the fluxes measured by the Infrared Astronomical Satellite (IRAS) in four bands, following Sanders & Mirabel (1996). As shown in Figure 6, and discussed quantitatively in Ackermann et al. (2012b), the γ -ray emission related to CRs in the ISM scales with the total IR luminosity, which serves as a proxy for the injected CR power within the supernova remnant paradigm for galactic CR origin. Circi- us is more than five times over-luminous in the γ -ray band when compared with this scaling relation (see also Table 1). In fact, the L γ /L IR luminosity ratio for Circinus exceeds the so-called 'calorimetric limit' expected to hold when CRs interact faster than they can escape the galactic disk, and as much as η /similarequal 10 % energy per supernova explosion goes into CR acceleration, ηE SN /similarequal 10 50 erg (see, e.g., Lacki et al. 2011; Ackermann et al. 2012b). Circinus also does not follow the correlation found for star-forming galaxies between L γ and total monochromatic 1.4 GHz radio luminosity νL R , by a factor of six (see Table 1 and the right panel of Figure 6). Following the approach of Ackermann et al. (2012b), and assuming that star-forming galaxies do follow a simple power-law scaling between γ -ray and total-IR luminosity, we find that adjusting either the normalization of this relation or the intrinsic scatter to accommodate Circinus would tend to overpredict the total number of galaxies which have been actually detected by LAT. Let us consider the possibility that the total IR and radio luminosities listed in Table 1 underestimate the star formation rate (SFR). Whereas standard scaling relations using IRAS fluxes yield SFR /similarequal 2 M /circledot yr -1 , the most recent mid-IR studies of Circinus with Spitzer presented by For et al. (2012) suggest SFR /similarequal (3 -8) M /circledot yr -1 (de- pending on the particular calibration method). The differences between SFRs derived from mid-IR and total-IR (IRAS) measurements might be explained if Circinus is a relatively dust-poor system for which the far-IR luminosity is not a reliable indicator of the SFR. The SFR converted from the observed total 1.4 GHz fluxes (Yun et al. 2001) is only /similarequal 1 . 5 M /circledot yr -1 , but the radio luminosity could also under-represent the actual CR injection power if the interstellar magnetic field strength were below average. On the other hand, the total flux density ratio log ( F 70 µ m /F 1 . 4 GHz ) = 2 . 40 demonstrates that Circinus obeys the 'far-IR/radio' correlation established for local star-forming and starburst systems (e.g., Seymour et al. 2009). The luminosities of NGC 1068 and NGC4945, which are similar starburst/Seyfert composite galaxies, are consistent with the γ -ray-to-total-IR and γ -ray-toradio correlations (Ackermann et al. 2012b, see also Figure 6). These complexities in the Circinus system make a firm conclusion regarding the attribution of γ rays to CR interactions difficult. Other than CR interactions, the disk coronae or accretion disks of Seyfert galaxies can be considered as possible γ -ray emission sites (e.g., Nied'zwiecki et al. 2013). However, LAT studies of a large sample of hard X-rayselected Seyferts devoid of prominent jets revealed that such sources are γ -ray quiet as a class, down to the level of 1 -10% of the hard X-ray fluxes (Ackermann et al. 2012a). One of the striking characteristics of the Circinus system is the presence of well-defined radio lobes and kpc-scale jet-like features, the 'plumes' (Elmouttie et al. 1998), which are also resolved at X-ray frequencies with Chandra (Mingo et al. 2012). Both of these might be relevant γ -ray emission sites. To investigate this idea quantitatively, we apply standard leptonic synchrotron and inverse-Compton (IC) modeling to the radio spectra of the lobes, investigating whether the extrapolation of the high-energy emission continuum may account for the GeV flux from the system. In the IC calculations we consider seed photons provided by the cosmic background radiation, the observed IR-to-optical emission of the galactic disk, as well as the UV-to-hard X-ray photon field due to the active nucleus, corrected for obscuration (assuming the intrinsic UV emission of the accretion disk /similarequal 3 × 10 43 erg s -1 ; Prieto et al. 2010). Figure 7 presents the broad-band, multi-component SED of Circinus, including new LAT measurements. Integrated far-IR-to-optical measurements represent the dominant starlight and the dust emission of the galaxy, with a negligible contribution from a heavily obscured AGN. In the X-ray regime, the hard X-ray fluxes are dominated by the heavily absorbed emission of the accretion disk and disk coronae. As demonstrated in Elmouttie et al. (1998), the total radio fluxes of the source in the 0 . 4 -8 . 6GHz range are due to a superposition of various emission components characterized by different spectral properties. Using their published ATCA 1.4, 2.4, 4.8, and 8.6 GHz maps, we measured fluxes for each distinct component separately, namely for the nucleus (circle with radius, r = 1 '' /similarequal 20pc), galaxy core including the central starburst region ( r = 35 '' circle), the outer parts of the galaxy disk, NW lobe ( r = 44 . 6 '' circle), NW plume (50 '' × 25 '' box centered 68 . 15 '' from the nucleus), SE lobe (ellipse with radii of 70 '' and 55 '' ), and SE plume (62 . 5 '' × 25 '' box centered 89 . 30 '' from the nucleus). The total radio emission of the Circinus system at > GHz frequencies is dominated by the central starburst region, and at lower frequencies by the outer parts of the galaxy disk; lobes and plumes contribute to the observed emission at the level of 10%; radio emission of the unresolved nucleus is negligible. We fit the radio spectra of the lobes and plumes, assuming energy equipartition between the radiating electrons and the magnetic field (energy density ratio u e /u B ≡ 1), and a standard form of the electron energy distribution consisting of a power-law dN e /dγ ∝ γ -2 between electron energies γ min ≡ 1 and γ br , breaking to dN e /dγ ∝ γ -3 between γ br and γ max ≡ 10 6 . We adjust both the electron normalization and break Lorentz factors γ br to obtain satisfactory fits to the radio data for each region separately, and then evaluate the expected IC emission. The results of the modeling are given as blue curves in Figure 7, where we show in addition the modeled emission of the accretion disk and the disk coronae using the MyTORUS model (red curves) (Murphy & Yaqoob 2009). For comparison, the figure presents also the interstellar radiation field/ GALPROP models (Porter et al. 2008; Strong et al. 2010) for the Milky Way placed at the distance of Circinus (gray curves). Note that the 'clump' structure seen around 10 23 Hz in the γ -ray continuum evaluated with GALPROP is due to the pion decay component dominating over the leptonic (IC and bremsstrahlung) ISM emission. The evaluated IC emission of the lobes severely underestimates the measured GeV emission of Circinus. In the model, magnetic field intensities read as B /similarequal 5 -10 µ G, in agreement with the values claimed by Elmouttie et al. (1998). The total energy stored in the entire structure is E tot /similarequal 10 54 erg. This is about an order of magnitude lower than the total energy of the lobes inferred by Mingo et al. (2012) based on X-ray observations, indicating either a significant departure from the energy equipartition condition or dominant pressure support within the lobes provided by hot thermal plasma or relativistic protons. The departures from the minimum energy condition u e /u B /greatermuch 1, which may enhance the expected IC radiation for a given synchrotron (radio) flux, are often claimed for lobes in radio galaxies and quasars (see Takeuchi et al. 2012, and references therein). In the particular case of Circinus, however, the effect would have to be extreme in order to account for the flux detected with the LAT, u e /u B /similarequal 10 4 . Although we cannot exclude this possibility, we consider it rather unlikely. Deep, high-resolution X-ray observations could in principle be used in the near future to validate the u e /u B /greatermuch 1 hypothesis for the Circinus lobes, but at this moment the very limited photon statistics of the available Chandra maps precludes any robust detection of a non-thermal lobe-related emission component at keV photon energies. We note however that the IC model curve calculated for u e /u B = 1 and shown in Figure 7 is below the corresponding upper limits. Yet another possibility is the presence of an energetically significant population of CR protons associated with the radio lobes. Relativistic protons injected by the kpc-scale jets/plumes, or accelerated at the bow-shocks of the expanding lobes (see Mingo et al. 2012), may generate a non-negligible γ -ray emission due to the decay of pions produced during CR interactions with surrounding matter. However, due to the sparseness of thermal gas within the galactic halo (gas number density /lessorsimilar 10 -3 cm -3 ), implying a low efficiency for this process, the resulting GeV fluxes are likely below the γ -ray output of the galactic disk. It is interesting to comment in this context on the similarities between the Circinus lobes and the Fermi -LAT discovered giant 'Fermi Bubbles' in our Galaxy (Dobler et al. 2010; Su et al. 2010). Even though the origin of both structures is still under the debate, the jet activity of their central supermassive black holes which are of comparable masses, namely M BH /similarequal (1 . 7 ± 0 . 3) × 10 6 M /circledot for Circinus (Greenhill et al. 2003) and (4 . 5 ± 0 . 4) × 10 6 M /circledot for the Milky Way (e.g., Ghez et al. 2008) - is the widely considered scenario. In the case of Circinus, the jet scenario is evidenced directly by the presence of collimated outflows supplying the lobes with energetic magnetized plasma (see the discussion in Elmouttie et al. 1998; Mingo et al. 2012), while in the case of our Galaxy, it is supported by general energetic arguments and numerical simulations reproducing well the observed properties of bubbles (e.g., Guo & Mathews 2012; Yang et al. 2012). In both systems, the lobes extend to kpc scales across the galactic disks and are characterized by magnetic field strengths of the order of ∼ 10 µ G (see Su et al. 2010; Mertsch & Sarkar 2011, for Fermi Bubbles). Thus it seems that structures analogous to the Fermi Bubbles and Circinus lobes may not be uncommon in late-type galaxies undergoing episodic outbursts of AGN (jet) activity, but their contributions to the total γ -ray outputs of the systems do not exceed ∼ 10%.",
"pages": [
4,
5,
6
]
},
{
"title": "4. CONCLUSIONS",
"content": "Here we report the detection of a steady and spatially unresolved γ -ray source at the position of Circinus, consistent with 2FGL J1415.7 -6520 based on a refined analysis using 4 years of LAT data. Although the observed power-law spectrum (Γ = 2 . 19 ± 0 . 12) is similar to that of other LAT-detected starburst systems, Circinus is γ -ray over-luminous by a factor of 5-6 relative to what is expected for emission from the ISM, based on multiwavelength correlations observed for nearby star-forming galaxies. However, the range of SFRs estimated from radio, mid-IR, and far-IR luminosities span a factor of /similarequal 4, indicating large uncertainties in the expected CR power delivered to the ISM. We presented several alternative possibilities for the origin of the GeV excess, including emission from the extended radio lobes, but found no conclusive answer to fully account for the GeV emission. This issue may be resolved by future studies, as Circinus is a compelling target for observations in the very high energy γ -ray regime with ground-based Cherenkov telescopes; the TeV-detected starburst systems NGC 253 and M82 (Acero et al. 2009; Acciari et al. 2009) are actually fainter in the GeV range.",
"pages": [
6
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "/suppress L. S. was supported by Polish NSC grant DEC2012/04/A/ST9/00083. Work by C.C.C. at NRL is supported in part by NASA DPR S-15633-Y. The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat 'a l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucl'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d' ' Etudes Spatiales in France. We thank M. Elmouttie for providing the ATCA images.",
"pages": [
6
]
},
{
"title": "REFERENCES",
"content": "Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010a, Nature, 463, 919 Baumgartner, W. H., Tueller, J., Markwardt, C., & Skinner, G. 2010, Bulletin of the American Astronomical Society, 42, 675 Brauher, J. R., Dale, D. A., & Helou, G. 2008, ApJS, 178, 280 Brinkmann, W., Siebert, J., & Boller, T. 1994, A&A, 281, 355 de Palma, F., Brandt, T. J., Johannesson, G., & Tibaldo, L., for Krolik, J. H. 1999, Active galactic nuclei: from the central black hole to the galactic environment (Princeton: Princeton University Press)",
"pages": [
7
]
}
]
2013ApJ...779..182Q
https://arxiv.org/pdf/1311.0566.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_81><loc_87><loc_86></location>From Poloidal to Toroidal: Detection of Well-ordered Magnetic Field in High-mass Proto-cluster G35.2 -0.74 N</section_header_level_1>
<text><location><page_1><loc_16><loc_79><loc_84><loc_80></location>Keping Qiu 1 , 2 , Qizhou Zhang 3 , Karl M. Menten 4 , Hauyu B. Liu 5 , Ya-Wen Tang 5</text>
<text><location><page_1><loc_42><loc_75><loc_58><loc_77></location>kpqiu@nju.edu.cn</text>
<section_header_level_1><location><page_1><loc_44><loc_71><loc_56><loc_72></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_26><loc_83><loc_68></location>We report on detection of an ordered magnetic field (B field) threading a massive star-forming clump in the molecular cloud G35.2 -0.74, using Submillimeter Array observations of polarized dust emission. Thanks to the sensitive and high-angular-resolution observations, we are able to resolve the morphology of the B field in the plane of sky and detect a great turn of 90 · in the B field direction: Over the northern part of the clump, where a velocity gradient is evident, the B field is largely aligned with the long axis of the clump, whereas in the southern part, where the velocity field appears relatively uniform, the B field is slightly pinched with its mean direction perpendicular to the clump elongation. We suggest that the clump forms as its parent cloud collapses more along the large scale B field. In this process, the northern part carries over most of the angular momentum, forming a fast rotating system, and pulls the B field into a toroidal configuration. In contrast, the southern part is not significantly rotating and the B field remains in a poloidal configuration. A statistical analysis of the observed polarization dispersion yields a B field strength of ∼ 1 mG, a turbulent-to-magnetic energy ratio of order unity, and a mass-to-magnetic flux ratio of ∼ 2-3 in units of the critical value. Detailed calculations support our hypothesis that the B field in the northern part is being rotationally distorted. Our observations, in conjunction with early single-dish data, suggest that the B field may play a critical role in the formation of the dense clump, whereas rotation and turbulence could also be important in further dynamical evolution</text>
<text><location><page_2><loc_17><loc_82><loc_83><loc_86></location>of the clump. The observations also provide evidence for a wide-angle outflow driven from a strongly rotating region whose B field is largely toroidal.</text>
<text><location><page_2><loc_17><loc_77><loc_83><loc_80></location>Subject headings: ISM: magnetic fields - stars: formation - stars: early-type - techniques: polarimetric - techniques: interferometric</text>
<section_header_level_1><location><page_2><loc_42><loc_71><loc_58><loc_72></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_39><loc_88><loc_68></location>The role of magnetic fields in the evolution of molecular clouds and the formation of stars has long been subject to great debate (see Crutcher 2012, for a review). In a classic model of low-mass star formation, molecular clouds are supported by magnetic fields, and stay in subcritical states, evolving quasi-statically; ambipolar diffusion induces the formation of supercritical cores which dynamically collapse to form stars (Shu et al. 1987; Basu & Mouschovias 1994; Mouschovias et al. 2006). The other view is that molecular clouds are short lived, dynamically evolving, and producing stars rapidly (Elmegreen 2000; Hartmann et al. 2012), with magnetic field being implicitly weak, or not really appreciable in cloud evolution and star formation; Mac Low and Klessen (2004) further strongly argue that supersonic turbulence, instead of magnetic field, supports molecular clouds and regulates star formation. On the other hand, our understanding of high-mass star formation is far less clear, but the competing views on the role of magnetic field/turbulence are equally, if not more strongly debated. Recent magnetohydrodynamic (MHD) simulations suggest that magnetic fields are dynamically important in high-mass star formation, in particular at suppressing complete fragmentation and creating bipolar outflows (Banerjee & Pudritz 2007;</text>
<text><location><page_2><loc_12><loc_37><loc_99><loc_38></location>Peters et al. 2011; Hennebelle et al. 2011; Commer¸con et al. 2011; Seifried et al. 2012; Myers et al. 2013</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_35></location>The 'magnetic support' model predicts a well-ordered or, in the extreme case, uniform magnetic field permeating a molecular cloud (Ostriker et al. 2001). Since there is an increased support against gravity in the direction perpendicular to the magnetic field compared to the direction parallel to it, the cloud contracts more along the field, forming flattened cores orthogonal to the mean direction of the field (Matsumoto & Tomisaka 2004; Tassis et al. 2009). In contrast, in the 'weak field' or 'turbulent support' model, magnetic field is expected to show an irregular and even chaotic morphology due to overwhelming turbulent twisting (Ostriker et al. 2001; Padoan et al. 2001). Therefore, mapping the morphology of magnetic field provides a straightforward method to distinguish between the two competing paradigms, or to provide insights into the possibility of a scenario where both magnetic field and turbulence are important. Focusing on massive clumps or cores with a typical size scale of 0.1 pc and a distance of a few kpc, high-angular-resolution observations of polarized dust emission are needed to spatially resolve the magnetic field morphology.</text>
<text><location><page_3><loc_12><loc_74><loc_88><loc_86></location>Submillimeter Array (SMA) observations have been playing a major role in recent studies of this kind, though the observations are still limited to a small number of case studies (e.g., Girart et al. 2009; Tang et al. 2013; Liu et al. 2013; Girart et al. 2013). Here we present an SMA 6 study of a massive cluster-forming clump which shows an elongated morphology projected in the plane of sky, and thus defines an axis ready to be compared to the direction of the magnetic field.</text>
<text><location><page_3><loc_12><loc_39><loc_88><loc_73></location>The targeted clump (hereafter G35.2N) lies in the northern part of G35.2 -0.74, a molecular cloud first discovered by Brown et al. (1982) and located at a distance of 2.19 kpc (Zhang et al. 2009a). It is associated with the IRAS source 18566+0136, which has a total luminosity of 3 × 10 4 L /circledot (Dent et al. 1989; S'anchez-Monge et al. 2013). Early molecular line observations revealed a velocity gradient along the long axis of the clump as well as a large scale bipolar outflow approximately orthogonal to the clump elongation (Dent et al. 1985; Little et al. 1985; Brebner et al. 1987). G35.2N was thus interpreted as a rotating interstellar disk or toroid. More detailed studies of the molecular outflow were presented by Gibbs et al. (2003) and Birks et al. (2006). Most recently, Zhang et al. (2013) presented SOFIA-FORCAST mid-infrared observations of G35.2N and modeled the object as a single high-mass protostar forming by an ordered and symmetric collapse of a massive core with a radius of 0.1 pc. However, both Atacama Large Millimeter/submillimeter Array (ALMA) cycle 0 observations (S'anchez-Monge et al. 2013) and the SMA data presented here reveal an apparently filamentary and highly fragmented structure for the dense gas on a 0.15 pc scale. We further infer a well-ordered B field from sensitive observations of the polarized dust emission and find that the B field morphology correlates with the dense gas kinematics.</text>
<section_header_level_1><location><page_3><loc_32><loc_33><loc_68><loc_35></location>2. Observations and Data Reduction</section_header_level_1>
<text><location><page_3><loc_12><loc_20><loc_88><loc_31></location>The observations were carried out with the SMA from 2010 to 2012, using three array configuration under excellent weather conditions. Detailed information on the observations, including the observing dates, array configurations, number of available antennas, atmospheric opacities at 225 GHz, and various calibrators, is presented in Table 1. For the Subcompact and Extended observations, the 345 GHz receiver was tuned to cover roughly 332-336 GHz in the lower sideband and 344-348 GHz in the upper sideband. For the Com-</text>
<text><location><page_4><loc_12><loc_79><loc_88><loc_86></location>pact observations, the frequency setup covers about 333.5-337.5 GHz in the lower sideband and 345.5-349.5 GHz in the upper sideband. For all the observations, the correlator was configured to have a uniform spectral resolution of 812.5 kHz ( ∼ 0.7 kms -1 ).</text>
<text><location><page_4><loc_12><loc_55><loc_88><loc_79></location>We performed basic data calibration, including bandpass, time dependent gain, and flux calibration, with the IDL MIR package, and output the data into MIRIAD for further processing. The intrinsic instrumental polarization (i.e., leakage) was removed to a 0.1% accuracy (Marrone & Rao 2008) with the MIRIAD task GPCAL. For each sideband, a pseudo-continuum data set was created from spectral line-free channels using the MIRIAD task UVLIN. The calibrated visibilities were jointly imaged to make Stokes I , Q , and U maps. We performed self-calibration with the continuum data in Stokes I , and applied the solutions to both continuum and spectral line data in Stokes I , Q , U . Finally, the Stokes I , Q , and U maps were combined to produce maps of the polarized emission intensity, the fractional polarization, and the polarization angle, using the MIRIAD task IMPOL. No significant polarization is detected in molecular line emission, so only Stokes I maps are presented for spectral lines.</text>
<section_header_level_1><location><page_4><loc_45><loc_49><loc_55><loc_51></location>3. Results</section_header_level_1>
<section_header_level_1><location><page_4><loc_31><loc_46><loc_69><loc_47></location>3.1. Dust Emission and Magnetic Field</section_header_level_1>
<text><location><page_4><loc_12><loc_18><loc_88><loc_43></location>Figure 1 shows the total dust emission on a variety of size scales. In the James Cleak Maxwell Telescope 7 (JCMT) SCUBA 850 µ m map with a 15 '' resolution (Figure 1a), the emission unveils a ∼ 0.5 pc, slightly elongated clump, which shows hierarchical fragmentation in our SMA observations. In the SMA 880 µ m map made by combining the Subcompact and Compact observations (Figure 1b), the inner clump splits into three cores, MM1-3, each with a diameter of order 0.05 pc. Convolving the SMA map to the SCUBA beam and comparing its peak flux ( ∼ 6 Jy) to that of the SCUBA map ( ∼ 8.5 Jy), the SMA observations recover approximately 70% of the total flux. Figure 1c shows the SMA map made from all the observations spread over five tracks (Table 1). With a uniform weighting of the data, we obtain an angular resolution of 1 . '' 0 × 0 . '' 6, and resolve each of the three cores into at least two condensations with diameters of 0.01-0.02 pc. These condensations are distributed approximately along the major axis of the clump, but in MM2, there are minor emission peaks to the west. MM1b approximately coincides with an unresolved radio source first</text>
<text><location><page_5><loc_12><loc_80><loc_88><loc_86></location>detected at 8.5 GHz (Gibb et al. 2003). S'anchez-Monge et al. (2013) presented the ALMA 860 µ m continuum map at a 0 . '' 4 resolution. Our SMA map shown in Figure 1c is in general consistent with the ALMA observations.</text>
<text><location><page_5><loc_12><loc_45><loc_88><loc_79></location>To achieve the highest possible sensitivity, which is desirable for a polarization study, we jointly imaged all the SMA observations under a natural weighting, which results in an r.m.s. noise level, σ , of 1 . 5 mJybeam -1 . Strong polarization of the dust emission (signalto-noise ratios greater than 3) is detected toward MM1, MM3, and part of MM2 (Figure 2a), and thus the inferred B field in the plane of sky is well resolved (Figure 2b). We detect a clearly ordered B field threading the clump. Over the northern two cores MM1 and MM3, the B field direction is overwhelmingly aligned with the long axis of the clump, whereas approaching the southern core MM2, the B field direction drastically changes by about 90 · so that it is perpendicular to the clump elongation. Interestingly, early single-dish polarimetric observations with a 14 '' beam revealed a mean direction at a position angle of 56 · for the B field permeating the parent cloud of the clump (Vall'ee & Bastien 2000, also see Figure 2b). Therefore, over MM1 and MM3, the B field in the dense clump is approximately perpendicular to the mean direction of the large scale B field, while toward MM2, the B field in the clump roughly follows that in the cloud. In addition, to the southwest of MM1 and to the east of MM2, there are B field segments inclined to the direction of the cloud B field. Even for the northern part of the clump, the direction of the B field on the edges of the clump tends to deviate more from the clump major axis.</text>
<section_header_level_1><location><page_5><loc_35><loc_39><loc_65><loc_41></location>3.2. Molecular Line Emission</section_header_level_1>
<text><location><page_5><loc_12><loc_32><loc_88><loc_37></location>Our SMA observations cover a large number of spectral lines tracing molecular gas under a variety of physical conditions. No significant polarization is detected in these lines, but they allow a detailed investigation of gas kinematics.</text>
<section_header_level_1><location><page_5><loc_23><loc_25><loc_77><loc_27></location>3.2.1. Velocity gradient in high-density tracing molecular lines</section_header_level_1>
<text><location><page_5><loc_12><loc_18><loc_88><loc_23></location>We investigate the kinematics of the dense gas by examining the first moment (intensity weighted velocity) maps (Figures 3a-c) and position-velocity ( PV ) diagrams (Figures 3d-f) of various high-density tracing molecular lines.</text>
<text><location><page_5><loc_12><loc_11><loc_88><loc_16></location>In Figure 3, the H 13 CO + (4-3) emission traces the bulk dense gas of the clump seen in dust continuum. The most remarkable feature in the first moment map of the emission is a velocity shear lying between MM1 and MM2. From the PV diagram cut along the</text>
<text><location><page_6><loc_12><loc_66><loc_88><loc_86></location>major axis of the clump (position angle -35 · ), MM2, which has a mean velocity close to the systemic velocity of the ambient cloud G35.2 -0.74 (Roman-Duval et al. 2009), can be readily distinguished from the rest of the clump, MM1 and MM3, and a velocity gradient is clearly seen between the latter two cores' locations. The CH 3 OH (7 1 , 7 -6 1 , 6 ) A emission preferentially traces the inner regions of the clump, and reveals an overall velocity field consistent with that seen in H 13 CO + (4-3). The HC 3 N (38-37) line has an upper level energy of 324 K above the ground, and probes the two condensations, MM1a and MM1b, embedded within MM1. There is a clear velocity gradient across the two condensations in the first moment map and PV diagram of the HC 3 N (38-37) emission. We also investigated our SMA 230 GHz observations, and found a similar velocity structure along the clump.</text>
<text><location><page_6><loc_12><loc_53><loc_88><loc_65></location>The dense gas kinematics and the B field morphology both suggest that the clump can be understood as a two-component structure: the northern part, which consists of MM1 and MM3, exhibits a velocity gradient of order 50 km s -1 pc -1 , and the mean direction of the B field is aligned with the long axis of the clump; the southern part, i.e., MM2, does not show a clear velocity gradient, and the B field direction is perpendicular to the clump axis, such that it is aligned with the B field in the cloud.</text>
<section_header_level_1><location><page_6><loc_28><loc_47><loc_72><loc_49></location>3.2.2. Hot molecular cores and a wide-angle outflow</section_header_level_1>
<text><location><page_6><loc_12><loc_30><loc_88><loc_45></location>The two dust condensations embedded within MM1 (i.e., MM1a,b) are the brightest in the clump, and show the richest molecular lines. MM1b coincides with (within 0 . '' 5) a faint and spatially unresolved radio source detected at 8.5 GHz (Gibb et al. 2003). Following Qiu & Zhang (2009) and Qiu et al. (2011b), we perform a local thermodynamical equilibrium (LTE) fitting to the K -ladder of the CH 3 CN (19-18) emission that is detected toward these two condensations. In Figure 4, the best-fit model agrees well with the observation, and yields a temperature of 135 +35 -15 K, indicating that the two sources are hot molecular cores (HMCs) and confirming that the clump is actively forming high-mass stars.</text>
<text><location><page_6><loc_12><loc_11><loc_88><loc_28></location>The CO (3-2) line is covered in our SMA observations. However, due to the nearly equatorial position of the source and a limited number of antennas available in the interferometer, the CO maps are heavily affected by sidelobe and missing flux issues, and do not allow a reliable investigation of the outflow structure. We instead show our Atacama Pathfinder Experiment (APEX) CO (7-6) observations, which reveal a wide-angle bipolar outflow centered on the two HMCs (Figure 5). The outflow morphology is in general agreement with previous lowJ CO observations (Gibb et al. 2003; Birks et al. 2006). A more detailed study of the outflow nature in G35.2N, using multi-frequency observations, will be presented in a forthcoming paper. Here we stress that the outflow axis is approximately per-</text>
<text><location><page_7><loc_12><loc_82><loc_88><loc_86></location>pendicular to the clump elongation (thus also perpendicular to the B field in the northern part of the clump).</text>
<section_header_level_1><location><page_7><loc_43><loc_76><loc_57><loc_78></location>4. Discussion</section_header_level_1>
<section_header_level_1><location><page_7><loc_14><loc_71><loc_86><loc_74></location>4.1. The B field strength and its significance compared to turbulence and gravity</section_header_level_1>
<text><location><page_7><loc_12><loc_51><loc_88><loc_69></location>We detect a structured B field threading the clump (Figure 2). The observed P.A. dispersion is attributed to both ordered and random perturbations to the B field, and the latter is probably dominated by turbulence. But dynamical processes related to cluster formation could perturb the B field in a manner rendering some disordered to random variations. We perform a statistical analysis of the observed P.A. dispersion to quantify the random perturbations (Hildebrand et al. 2009; Houde et al. 2009; Koch et al. 2010). Assuming the observed B field to be composed of a large-scale, ordered component, B 0 , and a turbulent component, B t , Houde et al. (2009) show that the disperse function 1 -〈 cos[∆Φ( l )] 〉 , which is measurable with a polarization map, can be approximately expressed as</text>
<formula><location><page_7><loc_27><loc_45><loc_73><loc_50></location>1 -〈 cos[∆Φ( l )] 〉 /similarequal 1 N 〈 B 2 t 〉 〈 B 2 0 〉 × [1 -e -l 2 / 2( δ 2 +2 W 2 ) ] + a ' 2 l 2 ,</formula>
<text><location><page_7><loc_12><loc_37><loc_88><loc_44></location>where ∆Φ( l ) is the difference in P.A. measured at two positions separated by a distance l and 〈· · ·〉 denotes an average, δ is the turbulent correlation length, W is the beam radius (i.e., the FWHM beam divided by √ 8ln2), and a ' 2 is the slope of the second-order term. N is the number of independent turbulent cells probed by observations, and is defined by</text>
<formula><location><page_7><loc_42><loc_31><loc_58><loc_35></location>N = ( δ 2 +2 W 2 )∆ ' √ 2 πδ 3 ,</formula>
<text><location><page_7><loc_12><loc_15><loc_88><loc_30></location>where ∆ ' is the effective depth of the cloud along the line of sight. Here turbulence is implicitly adopted to be the only source of the random perturbations. Figure 6(a) shows the measured function taking into account polarization detections with signal-to-noise ratios > 4 in the northern part of the G35.2N clump 8 . In our observations, W = 0 . '' 63, and ∆ ' ∼ 8-22 '' , which are approximately the dimensions of the clump projected on the plane of sky (Figure 2). We then fit the measured function with the above relation to solve for 〈 B 2 t 〉 / 〈 B 2 0 〉 , δ , and a ' 2 , of which the former two are of our interest. Since the relation is valid when l is less than a few times W , we fit the data points at l /lessorsimilar 6 '' . The solid curve in Figure 6a shows the fitting</text>
<text><location><page_8><loc_12><loc_80><loc_88><loc_86></location>results, and the dashed curve visualizes the sum of the integrated turbulent contribution, [ 〈 B 2 t 〉 / 〈 B 2 0 〉 ] /N , and the large-scale contribution, a ' 2 l 2 . The correlated turbulent component of the dispersion function,</text>
<formula><location><page_8><loc_41><loc_76><loc_59><loc_81></location>1 N 〈 B 2 t 〉 〈 B 2 0 〉 e -l 2 / 2( δ 2 +2 W 2 ) ,</formula>
<text><location><page_8><loc_12><loc_70><loc_88><loc_76></location>is shown in Figure 6b. With the fitting we obtain δ = 15 . 4 mpc (1 . '' 45), which is approximately equal to the value derived in OMC-1 (Houde et al. 2009)), and 〈 B 2 t 〉 / 〈 B 2 0 〉 = 0 . 230.64 for N = 3 . 0-8.4, which depends on ∆ ' . According to Chandrasekhar & Fermi (1953),</text>
<formula><location><page_8><loc_42><loc_64><loc_58><loc_69></location>[ 〈 B 2 t 〉 〈 B 2 0 〉 ] 1 / 2 ∼ δV los V A ,</formula>
<text><location><page_8><loc_12><loc_51><loc_88><loc_64></location>where δV los is the one-dimensional velocity dispersion and V A = B 0 / √ 4 πρ is the Alfv'en velocity at mass density ρ . Consequently we can estimate the energy ratio of the turbulence to the ordered B field, β turb ∼ 3 〈 B 2 t 〉 / 〈 B 2 0 〉 , to be 0.7-1.9. The ratio is somewhat overestimated considering that we are probing the B field in the plane of sky and B t is not necessarily all ascribed to turbulence. Nevertheless, the statistical analysis suggests that the ordered B field and turbulence are energetically comparable.</text>
<text><location><page_8><loc_16><loc_48><loc_66><loc_50></location>The strength of the ordered B field can be derived following</text>
<formula><location><page_8><loc_37><loc_42><loc_63><loc_47></location>B 0 = √ 4 πρ · δV los · [ 〈 B 2 t 〉 〈 B 2 0 〉 ] -1 / 2 .</formula>
<text><location><page_8><loc_12><loc_10><loc_88><loc_41></location>The H 13 CO + (4-3) emission traces a structure similar to that is seen in the dust emission, and we estimate δV los ∼ 1 kms -1 from the line-of-sight velocity dispersion of the H 13 CO + line. To estimate ρ , we first compute the mass of the clump from its dust emission, which requires the information on the dust temperature and opacity. In our SMA observations, the clump consists of three cores, MM1-3, each of which harbors a couple of dust condensations. Toward MM2 and MM3, early NH 3 (1,1) and (2,2) observations deduced a rotational temperature of 20-25 K, corresponding to a kinetic temperature of 30-40 K (Little et al. 1985). Toward MM1, our LTE model of the CH 3 CN (19-18) K -ladder gives a temperature of 135 K for the embedded HMCs, while recent high-angular-resolution NH 3 observations suggest a temperature > 50 K (Codella et al. 2010). Thus we adopt a gas temperature of 50-135 K for MM1. Assuming thermal equilibrium between gas and dust, the derived gas temperature approximates the dust temperature. The dust opacity index, β , is derived by comparing the dust emission fluxes at 880 µ m and 1.3 mm (the latter is obtained from our new SMA observations). The dust opacity at 880 µ m is then extrapolated from 10[ ν/ (1 . 2 THz)] β , where ν is frequency, following Hildebrand (1983). Assuming a canonical gas-to-dust mass ratio of 100, the masses of the three cores are then computed from their dust emission fluxes (Table 2).</text>
<text><location><page_9><loc_12><loc_72><loc_88><loc_86></location>The total mass of the clump amounts to ∼ 150 M /circledot , and results in an averaged column density of ∼ 3 . 7 × 10 23 cm -2 over a measured area, s , of 200 square arc sec (0.015 pc 2 ) where the emission is detected with signal-to-noise ratios above 5. If we adopt a volume of 4 / 3( s 3 /π ) 1 / 2 , the averaged volume density is ∼ 1 . 0 × 10 6 cm -3 . We then obtain B 0 ∼ 1 . 4-0.9 mG, again depending on ∆ ' . Consequently, the mass-to-magnetic flux ratio, M/φ B , is 1.9-3.2 in units of a critical value, 1 / (2 π √ G ), where G is the gravitational constant (Nakano & Nakamura 1978), indicating that the clump is unstable against gravitational collapse and fragmentation</text>
<text><location><page_9><loc_12><loc_11><loc_88><loc_71></location>It has been shown that a star-forming cloud generally collapses to a flattened or filamentary configuration to allow efficient fragmentation to follow (Larson 1985; Pon et al. 2011). The observations of the G35.2N clump is consistent with this theoretical prediction. To further understand the fragmentation property of the clump, it is instructive to compute the thermal Jeans mass, which is about 1.5 M /circledot at a density of 1 . 0 × 10 6 cm -3 and a temperature of 30 K. The analyses of the gravitational instability of molecular gas sheets and filaments by Larson (1985) are probably of more relevance to G35.2N. Larson (1985) argued that rotation and magnetic field do not fundamentally change the characteristic mass, M c ( M /circledot ) = 2 . 4 T 2 /µ, where T is the gas temperature and µ is the surface density in M /circledot pc -2 , of the fragments. For G35.2N, M c /similarequal 0 . 22 M /circledot . Even without taking into account the mass of the forming stars and of the gas already dispersed by the wide-angle outflow, the clump mass is orders of magnitude greater than the Jeans mass or M c , and is expected to fragment into hundreds of density enhancements if the dynamics is solely controlled by the gravity and thermal pressure. In Figure 1c, there are only six prominent condensations detected at a spatial resolution (0.01 pc) finer than the Jeans length (0.04 pc). We also perform a multi-component Gaussian fitting to the dust emission using the MIRIAD task IMFIT, and identify three new possible condensations, MM1c, MM2c, and MM2d (see Figure 7). Nevertheless, the total number of the condensations is far less than what is expected in thermal fragmentation. Such observational results have been seen in some other high-mass star-forming regions (Zhang et al. 2009b ? ), and strongly suggest that other mechanisms, e.g., turbulence, magnetic field, and rotation, are playing important roles in the dynamical evolution of the clump. A highly turbulent core with a radius of 0.1 pc and a mass of 240 M /circledot , as described by Zhang et al. (2013) in applying the model of McKee & Tan (2003) to G35.2N, is obviously inconsistent with the high-angular-resolution observations obtained here with the SMA and with the ALMA (S'anchez-Monge et al. 2013). Recent numerical simulations of the collapse and fragmentation of magnetized, massive star-forming cores show that cores with M/φ B ∼ 2 form a single star rather than fragmenting to a small cluster (e.g., Commer¸con et al. 2011; Myers et al. 2013). This is inconsistent with our observations either; the measured M/φ B in G35.2N is ∼ 2-3, and the clump is highly fragmented. We expect that both the B field and turbulence help to increase the effective Jeans mass or</text>
<text><location><page_10><loc_12><loc_82><loc_88><loc_86></location>the characteristic mass of the fragments, but neither (or their combination) is sufficient to completely suppress fragmentation on /lessorsimilar 0 . 1 pc scales.</text>
<section_header_level_1><location><page_10><loc_18><loc_76><loc_82><loc_78></location>4.2. Winding in the deep - rotationally distorted magnetic field?</section_header_level_1>
<section_header_level_1><location><page_10><loc_38><loc_73><loc_62><loc_74></location>4.2.1. A schematic picture</section_header_level_1>
<text><location><page_10><loc_12><loc_43><loc_88><loc_71></location>The sub-parsec clump in G35.2N has been interpreted as a prototype of a massive interstellar disk or toroid, with the 'disk' being nearly edge-on, having LSR velocities increasing from northwest to southeast (Dent et al. 1985; Little et al. 1985; Brebner et al. 1987; Little et al. 1998; L'opez-Sepulcre et al. 2009). Our observations provide significantly new insights into the dynamics of the clump. Going from the northern to the southern part of the clump, the B field direction takes a 90 · turn and is correlated with the velocity field revealed by the high-density tracing molecular lines (see Section 3.2.1). We also find that the B field in the southern part appears to be slightly pinched and can be fitted with a set of parabola (Figure 8), which provides marginal evidence for a magnetically regulated collapse of MM2 (e.g., Girart et al. 2006; Gon¸calves et al. 2008). In Figure 7, the dust condensations in the northern part are distributed approximately along the major axis of the clump, while those in the southern part have a Trapezium-like distribution. All this suggests that the clump is composed of two dynamically different components rather than coherently rotating as a whole.</text>
<text><location><page_10><loc_12><loc_10><loc_88><loc_41></location>One interpretation of the observed velocity gradient in the northern part of the clump is that it arises from the rotation of the dense gas. Considering the orientation of the B field in the cloud revealed by the single-dish observations (see Figure 2b), we speculate that the dense clump forms as the cloud collapses more along the direction of the large scale B field. In this dynamical process, the dense gas breaks into two parts. The northern part carries over most of the angular momentum that is not dissipated and forms a fast rotating system. The system could be a rotating toroid embedded with multiple fragments. Or it is a hierarchical structure consisting of two cores (MM1 and MM3) in an orbital motion, and each core further fragments and collapses into a binary or multiple star. In the latter case, the projected separation of ∼ 0 . 05 pc and the line-of-sight velocity difference of ∼ 2 . 5 kms -1 between MM1 and MM3 yield a dynamical mass of /greaterorsimilar 73 M /circledot , which is compatible with the combined mass of the two cores (Table 2). The southern part, however, does not participate in the rotating motion and is expected to have a mean velocity similar to that of the parent cloud. This hypothesis is indeed supported by the large scale kinematics seen in 13 CO (1-0) observations obtained from the Galactic Ring Survey, from which an LSR velocity of 35.2 km s -1 is derived for the ambient cloud of G35.2N (Jackson et al. 2006; Roman-Duval et al. 2009, also see Figure 3).</text>
<text><location><page_11><loc_12><loc_70><loc_88><loc_86></location>Also, one may expect that the angular momentum in the northern part has a significant influence on the fragmentation, causing the resulted fragments to be distributed within the rotational plane. The observed morphology of the B field in the clump can be naturally incorporated in the above scenario. Figure 9 provides an overall picture of our interpretations of the B field structure: in the northern part, the B field is pulled by the rotation into a toroidal configuration, whereas in the southern part, the B field is only gently squeezed by the collapse and remains to be in a poloidal configuration (aligned with the B field in the cloud).</text>
<section_header_level_1><location><page_11><loc_36><loc_64><loc_64><loc_66></location>4.2.2. Quantitative comparisons</section_header_level_1>
<text><location><page_11><loc_12><loc_43><loc_88><loc_62></location>We quantitatively investigate the feasibility of our interpretation of a rotationally distorted B field in the northern part of the clump. According to Matsumoto et al. (2006), a 90 · misalignment between the axis of an outflow and the mean direction of an ordered B field may occur if the rotational energy is at least comparable to the magnetic energy (also see Rao et al. 2009). Assuming the measured velocity gradient, ∆ v/L , in the northern part of G35.2N is arising from a disk-like structure with a diameter L , the rotational energy amounts to 1 / 2( ML 2 / 8)(∆ v/L ) 2 , where M is the mass of the rotating structure. The averaged ratio of the rotational energy to the magnetic energy, β rot = (∆ v ) 2 / (8 V 2 A ), is found to be 0.5-1.3 for an end-to-end velocity difference of ∼ 4 kms -1 and B 0 ∼ 1 . 4-0.9 mG. Hence the rotational energy does appear to be comparable to the magnetic energy.</text>
<text><location><page_11><loc_12><loc_11><loc_88><loc_41></location>Alternatively, if a B field is wound by rotation into an overwhelmingly toroidal configuration, the centrifugal force is presumably overcoming the magnetic force. More quantitatively, Machida et al. (2005) found that whether the centrifugal force or the magnetic force regulates the evolution of a collapsing core can be evaluated by the ratio of the angular velocity to the magnetic field strength, ω/B . If this ratio exceeds a critical value, 0 . 39 √ G/c s , where c s is the sound speed, the centrifugal force dominates the dynamics. For the G35.2N clump, ω ∼ 5 . 1 × 10 -5 year -1 and c s ∼ 0 . 33 kms -1 at 30 K, leading to ω/B /similarequal (3 . 6-5 . 7) × 10 -8 yr -1 µ G -1 , which is on the same order but lower than the critical value of 9 . 7 × 10 -8 yr -1 µ G -1 . We stress that both ω and c s are averaged over the entire clump. Toward the inner part of the clump, ω increases since the cores appear to rotate faster, and c s also increases since the gas temperature is higher. Thus the measured ω/B would be greater than the critical value. For example, considering the dense gas in MM1, ω measures ∼ 1 . 5 × 10 -4 year -1 (Figure 3f) and c s reaches 0.71 km s -1 ; the measured ω/B is more than 2 times greater than the critical value, confirming that the centrifugal force is dominating over the magnetic force.</text>
<text><location><page_12><loc_12><loc_70><loc_88><loc_86></location>The above calculations show that energy-wise, the B field is likely to be significantly distorted by the rotation. But, is there sufficient time for a toroidal B field to develop? The rotation period, 2 πL/ ∆ v , is estimated to be 1 . 2 × 10 5 yr. The dense clump with an averaged density of 1 . 0 × 10 6 cm -3 presumably form from a cloud with a lower density ( /lessorsimilar 10 4 cm -3 ), which has a free-fall time > 10 5 yr. In addition, with the detection of HMCs and a radio source in MM1, it is feasible to expect that the clump has a dynamical age of order 10 5 yr (e.g., Charnley 1997). Therefore, it is likely that the rotation has proceeded for a few periods, and significantly distorted the B field into a predominantly toroidal configuration.</text>
<text><location><page_12><loc_12><loc_57><loc_88><loc_69></location>Finally, another piece of evidence supporting for the presence of a toroidal B field in the clump comes from polarization observations (0 . '' 2 resolution) of OH maser emission; Hutawarakorn & Cohen (1999) detected a few Zeeman pairs within 0 . '' 5 of MM1b and found that the direction of the B field along the line of sight reverses from the southeast to the northwest of MM1b, and the direction reversal was interpreted as being due to a toroidal B field.</text>
<section_header_level_1><location><page_12><loc_31><loc_51><loc_69><loc_53></location>4.2.3. Connecting the B field to the outflow</section_header_level_1>
<text><location><page_12><loc_12><loc_24><loc_95><loc_49></location>Both well-collimated and wide-angle outflows have been seen in high-mass star-forming regions (e.g., Qiu et al. 2008; Qiu & Zhang 2009; Qiu et al. 2009; Qiu et al. 2011a; Qiu et al. 2012). Here in G35.2N, the CO (7-6) observations reveal a parsec-sized wide-angle outflow (Figure 5, also see Figure 9). The outflow appears to originate from one of the two HMCs (MM1a or MM1b, see Section 3.2.2), which must have been undergoing collapse and spun up to form a rotating disk at the center. Indeed, there is evidence for the presence of a rotating disk in MM1b from the ALMA observations (S'anchez-Monge et al. 2013), though it is yet to be confirmed whether the outflow is driven from MM1b. In line with our above interpretations, the ambient B field of the inner ∼ 0 . 1 pc part of the outflow is dominated by a toroidal component, and that component could be further traced down to a 0.01 pc scale taking into account previous OH maser observations (Hutawarakorn & Cohen 1999). Thus the wide-angle outflow is most likely associated with a strong toroidal B field. How is this outflow driven?</text>
<text><location><page_12><loc_12><loc_11><loc_88><loc_22></location>Bipolar outflows in protostars or young stars are believed to be magnetically driven. In the most quoted magneto-centrifugal wind theory, the outflowing gas is centrifugally ejected, either from the inner edge of an accretion disk or over a wide range of disk radii, along open poloidal field lines (e.g., Shu et al. 1994; Shang et al. 2006; Pudritz et al. 2007; Fendt 2009). On the other hand, the magnetic tower model suggests that an outflow is accelerated by the magnetic pressure gradient of a wound-up volume of a toroidal B field</text>
<text><location><page_13><loc_12><loc_36><loc_97><loc_86></location>(Uchida & Shibata 1985; Lynden-Bell 1996; Lynden-Bell 2003). Many MHD simulations of collapsing magnetized cores have shown that a bipolar outflow is launched from a strongly rotating region when a predominantly toroidal B field has been built up by the collapse-spin up process (e.g., Tomisaka 1998; Matsumoto & Tomisaka 2004; Banerjee & Pudritz 2006; Banerjee & Pudritz 2007; Seifried et al. 2012). The observations of G35.2N are apparently consistent with such a scenario. Furthermore, numerical simulations often found two-component outflows: a larger, low-velocity, and less collimated outflow emerging earlier than an inner, faster, and jet-like outflow (e.g., Tomisaka 2002; Banerjee & Pudritz 2006; Machida et al. 2008; Seifried et al. 2012; Tomida et al. 2013). While some claim that the former is a magnetic tower flow and the latter is launched by the magneto-centrifugal force (Banerjee & Pudritz 2006; Banerjee & Pudritz 2007), others argue for an opposite view (Machida et al. 2008; Tomida et al. 2013). The situation can be even more complicated considering possible dependence of the outflow morphology on the B field strength (e.g., Hennebelle & Fromang 2008; Seifried et al. 2012; Bate et al. 2013). In addition, radiation pressure could be important in widening massive outflows (Vaidya et al. 2011). Thus we are not able to infer the driving mechanism of the G35.2N outflow solely based on its morphology. Regarding the B field morphology, existing observations suggest a strong toroidal component in the inner part of the outflow, but a poloidal component certainly exists. To determine whether a magnetic tower or the centrifugal force dominates the acceleration of the outflowing gas, one may need a complete knowledge of three-dimensional B field and velocity structures (Seifried et al. 2012). Since we are only probing the B field projected in the plane of sky, and the estimate of the B field strength is indirect, it is not possible to compare the toroidal to poloidal components in detail. Hence whether or to what extent a magnetic tower plays a role in accelerating the outflow remains open. But if the growing tower is important, the central collapse-spin up process significantly helps to create at least part of the observed toroidal B field.</text>
<section_header_level_1><location><page_13><loc_30><loc_30><loc_70><loc_32></location>4.3. Infall along a magnetized cylinder?</section_header_level_1>
<text><location><page_13><loc_12><loc_16><loc_88><loc_28></location>Provided the remarkable filamentary morphology of the clump seen in Figure 1c, it is likely that we are observing a dense gas cylinder. A gas cylinder may fragment along its major axis due to the 'sausage' or 'varicose' instability, and the resulted fragments will have a characteristic spacing (e.g., Jackson et al. 2010). In Fgiure 7a, the dust condensations in the northern part are nearly equally spaced, in qualitative agreement with the fragmentation of a cylinder. Since a cylinder/filament is not expected to be rotating too fast about its minor</text>
<text><location><page_14><loc_12><loc_71><loc_88><loc_86></location>axis 9 , the observed velocity gradient ( ∼ 50 kms -1 pc -1 ) is most likely arising from an infall motion along the cylinder axis rather than from a rotation. From Figure 3d, the infall velocity is ∼ 2/sin i kms -1 , where i is the inclination angle of the cylinder with respective to the plane of sky. For a cylinder with a mass of 100 M /circledot (roughly the gas mass of the northern clump plus the mass of the forming stars) and a length of 0.1 pc, the line-of-sight component of the free-fall velocity reaches ∼ 4(cos i ) 1 / 2 sin i kms -1 , which is not incompatible (e.g., for i /similarequal 54 · ) with the observed velocity gradient.</text>
<text><location><page_14><loc_12><loc_37><loc_89><loc_71></location>However, in the SCUBA map (Figure 1a), the clump is much less elongated, and on a larger scale, we do not detect any filamentary structure where the cylinder is embedded. Furthermore, self-gravitating cylinders have a critical linear mass density, 2 v 2 /G , where v is the sound speed c s in case of thermal support (Stod'olkiewicz 1963; Ostriker 1964) or the turbulent velocity dispersion δV los in case of turbulent support (Fiege & Pudritz 2000). Above the critical value, the cylinder would radially collapse into a line. The B field can somewhat increase the critical linear mass density, but probably by small factors ∼ 1 (Fiege & Pudritz 2000). For G35.2N, the measured linear mass density is of order 1000 M /circledot pc -(without taking into account the mass of the forming stars and of the gas already dispersed by the outflow), which is about two times greater than the critical value of 470 M /circledot pc -1 . Unless the cylinder is largely inclined to the line of sight (i.e., i > 60 · ), the linear mass density seems to be too high to allow a stable cylinder to exist. Also, the southern part of the clump, which appears very different from the northern part in the B field, kinematics, and fragmentation properties, cannot be easily incorporated in a unified picture of a cylinder. Considering all these difficulties in understanding the observed clump as a cylinder, the interpretation of a rotating system (Section 4.2) for the northern part of the clump appears more robust than that of a collapsing cylinder.</text>
<section_header_level_1><location><page_14><loc_42><loc_31><loc_58><loc_33></location>5. Conclusions</section_header_level_1>
<text><location><page_14><loc_12><loc_18><loc_88><loc_29></location>With the sensitive and high-angular-resolution observations made with the SMA, we detect a well-ordered B field threading G35.2N, which is an apparently filamentary, massive, and cluster-forming clump. Based on a statistical analysis of the observed polarization dispersion, we derive a B field strength of ∼ 1 mG, a turbulent-to-magnetic energy ratio of order unity, and a mass-to-magnetic flux ratio of ∼ 2-3 times the critical value.</text>
<text><location><page_14><loc_16><loc_17><loc_88><loc_18></location>The B field morphology is found to vary depending on the kinematics revealed by high-</text>
<text><location><page_14><loc_89><loc_56><loc_90><loc_57></location>1</text>
<text><location><page_15><loc_12><loc_58><loc_88><loc_86></location>density tracing molecular lines. In the northern part of the clump, which exhibits a velocity gradient, the B field is largely aligned with the long axis of the clump (hence follows the velocity gradient). Approaching the southern part, which has a relatively uniform velocity field, the B field takes a great turn of 90 · to a direction perpendicular to the clump elongation. Although we cannot rule out the possibility that the velocity gradient is due to an infall motion along a gas cylinder, a synthesis of all the available observations prefers a rotating motion to be the origin of the velocity gradient. Consequently the B field in the northern part is pulled by the centrifugal force into an overwhelmingly toroidal configuration, whereas the B field in the southern part remains in a poloidal configuration, that is, aligned with a large scale B field which guides the collapse of the cloud from which the clump forms. Therefore we are likely witnessing a transition from a poloidal to toroidal configuration for the B field in this region. Detailed calculations adopting the B field strength derived from the statistical analysis supports such a scenario. Our observations also provide evidence for a wide-angle outflow driven from a strongly rotating region whose B field is largely toroidal.</text>
<text><location><page_15><loc_12><loc_47><loc_88><loc_57></location>Last, but not least, the supercritical clump is observed to be significantly fragmenting, but to a degree far from expected for a purely thermal fragmentation. Our observations and calculations suggest that both the B field and turbulence, as well as the potential rotation, all play a role in interplaying with the gravity and shaping the dynamical evolution of the clump.</text>
<text><location><page_15><loc_12><loc_26><loc_88><loc_44></location>Part of the data were obtained in the context of the SMA legacy project: 'Filaments, Magnetic Fields, and Massive Star Formation' (PI: Qizhou Zhang). We acknowledge all the members of the SMA staff who made these observations possible. The JCMT SCUBA data were obtained from the JCMT archive (Program ID: M97BH07). This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. Part of this research was undertaken when KQ was a postdoctoral fellow at Max-Planck-Institut fuer Radioastronomie. KQ is supported by the 985 project of Nanjing University. QZ is partially supported by the NSFC grant 11328301.</text>
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<text><location><page_20><loc_65><loc_26><loc_65><loc_27></location>s</text>
<figure>
<location><page_20><loc_35><loc_24><loc_65><loc_87></location>
<caption>Fig. 1.- The total dust continuum emission, shown in both inverse gray scale and contours, at 850 µ m observed with the JCMT SCUBA and at 880 µ m with the SMA. (a) The SCUBA map, with the starting and spacing contour levels of 1 Jy beam -1 . (b) The SMA map made from the Subcompact and Compact data, with the contour levels of 0 . 04 × (1 , 2 , 3 , ... ) 1 . 5 Jybeam -1 . (c) The SMA map made from all the data available, with the contour levels of 0 . 015 × (1 , 2 , 3 , ... ) 1 . 4 Jybeam -1 . A plus sign marks the position of a weak radio continuum source (Gibb et al. 2003). Negative emissions are invisible since their absolute levels are all below the lowest contour levels shown here. Hereafter, a filled ellipse in the lower right or left of a panel depicts the beam size at FWHM.</caption>
</figure>
<figure>
<location><page_21><loc_29><loc_30><loc_71><loc_87></location>
<caption>Fig. 2.- (a): The total dust emission (Stokes I ) shown in contours (solid for positive and dotted for negative emission) and the linearly polarized emission ( √ Q 2 + U 2 -σ 2 ) detected with signal-to-noise ratios ≥ 3 are shown in gray scale; the contour levels are ± 0 . 03 × (1 , 2 , 3 , ... ) 1 . 5 Jybeam -1 . Yellow segments indicate the directions of the linear polarization, with their lengths proportional to the fractional degree of the polarization; a scale bar in the lower right corresponds to a polarization degree of 5%. Star symbols mark the peak positions of the six dust condensations. (b): Contours and star symbols are the same as in (a). Blue segments with an arbitrary length show the B field directions, deduced by rotating the polarization directions by 90 · . A large green bar indicates the direction of the B field obtained from early JCMT observations of the polarized dust emission at 760 µ m (Vall'ee & Bastien 2000); a dashed circle marks the 14 '' beam of the observations. Brown curves are drawn following a method proposed by Li et al. (2010), and outline the continuous variation in the direction of the B field at representative positions.</caption>
</figure>
<text><location><page_22><loc_18><loc_69><loc_19><loc_70></location>01</text>
<text><location><page_22><loc_18><loc_48><loc_19><loc_49></location>01</text>
<text><location><page_22><loc_21><loc_83><loc_23><loc_84></location>45''</text>
<text><location><page_22><loc_21><loc_76><loc_23><loc_77></location>35''</text>
<text><location><page_22><loc_19><loc_69><loc_20><loc_70></location>o</text>
<text><location><page_22><loc_20><loc_69><loc_23><loc_70></location>40'25''</text>
<text><location><page_22><loc_21><loc_62><loc_23><loc_63></location>45''</text>
<text><location><page_22><loc_21><loc_55><loc_23><loc_56></location>35''</text>
<text><location><page_22><loc_19><loc_48><loc_20><loc_49></location>o</text>
<text><location><page_22><loc_20><loc_48><loc_23><loc_49></location>40'25''</text>
<text><location><page_22><loc_27><loc_85><loc_29><loc_86></location>13</text>
<text><location><page_22><loc_26><loc_84><loc_27><loc_85></location>H</text>
<text><location><page_22><loc_25><loc_77><loc_26><loc_80></location>0.1pc</text>
<text><location><page_22><loc_25><loc_68><loc_26><loc_69></location>h</text>
<text><location><page_22><loc_26><loc_67><loc_27><loc_68></location>58</text>
<text><location><page_22><loc_27><loc_68><loc_28><loc_69></location>m</text>
<text><location><page_22><loc_26><loc_63><loc_28><loc_64></location>CH</text>
<text><location><page_22><loc_29><loc_84><loc_31><loc_85></location>CO</text>
<text><location><page_22><loc_31><loc_85><loc_32><loc_86></location>+</text>
<text><location><page_22><loc_31><loc_68><loc_31><loc_69></location>s</text>
<text><location><page_22><loc_28><loc_67><loc_31><loc_68></location>13.5</text>
<text><location><page_22><loc_29><loc_63><loc_33><loc_64></location>OH (7</text>
<text><location><page_22><loc_31><loc_46><loc_31><loc_47></location>s</text>
<text><location><page_22><loc_28><loc_63><loc_29><loc_64></location>3</text>
<text><location><page_22><loc_25><loc_56><loc_26><loc_59></location>0.1pc</text>
<text><location><page_22><loc_25><loc_46><loc_26><loc_47></location>h</text>
<text><location><page_22><loc_24><loc_46><loc_25><loc_47></location>18</text>
<text><location><page_22><loc_26><loc_46><loc_27><loc_47></location>58</text>
<text><location><page_22><loc_27><loc_46><loc_28><loc_47></location>m</text>
<text><location><page_22><loc_28><loc_46><loc_31><loc_47></location>13.5</text>
<text><location><page_22><loc_42><loc_84><loc_45><loc_86></location>(a)</text>
<text><location><page_22><loc_43><loc_68><loc_43><loc_69></location>s</text>
<text><location><page_22><loc_40><loc_67><loc_43><loc_68></location>12.5</text>
<text><location><page_22><loc_42><loc_63><loc_45><loc_65></location>(b)</text>
<text><location><page_22><loc_43><loc_46><loc_43><loc_47></location>s</text>
<text><location><page_22><loc_40><loc_46><loc_43><loc_47></location>12.5</text>
<figure>
<location><page_22><loc_17><loc_23><loc_52><loc_44></location>
<caption>Fig. 3.- (a-c) Color images show the first moment (intensity weighted velocity) maps of the H 13 CO (4-3), CH 3 OH (7 1 , 7 -6 1 , 6 ) A , and HC 3 N (38-37) lines, respectively. A dashdotted arrow delineates the PV cut at a position angle of -35 · and points to the direction of increasing offsets. A dashed line divides the clump into two parts which are discussed in the text (Sections 3.2.1 and 4.2). Other symbols (stars and curves) are the same as in Figure 2b. (d-f) PV diagrams in the three spectral lines shown in both gray scale and contours (solid for positive and dotted for negative emission). The contour levels are ± 0 . 15 × (1 , 2 , 3 , ... ) 1 . 5 Jybeam -1 . A dashed line marks the position of the dashed line in panels (a-c). A vertical line indicates the systemic velocity of G35.2-0.74 (Roman-Duval et al. 2009). The positions of the six condensations are denoted in each panel.</caption>
</figure>
<text><location><page_22><loc_58><loc_83><loc_59><loc_84></location>10</text>
<text><location><page_22><loc_58><loc_79><loc_59><loc_80></location>5</text>
<text><location><page_22><loc_58><loc_76><loc_59><loc_77></location>0</text>
<text><location><page_22><loc_58><loc_73><loc_59><loc_74></location>-5</text>
<text><location><page_22><loc_57><loc_70><loc_59><loc_71></location>-10</text>
<text><location><page_22><loc_58><loc_62><loc_59><loc_63></location>10</text>
<text><location><page_22><loc_58><loc_57><loc_59><loc_59></location>5</text>
<text><location><page_22><loc_58><loc_53><loc_59><loc_54></location>0</text>
<text><location><page_22><loc_58><loc_48><loc_59><loc_49></location>-5</text>
<text><location><page_22><loc_62><loc_84><loc_63><loc_85></location>H</text>
<text><location><page_22><loc_63><loc_85><loc_64><loc_86></location>13</text>
<text><location><page_22><loc_60><loc_81><loc_63><loc_82></location>MM2a</text>
<text><location><page_22><loc_60><loc_80><loc_63><loc_80></location>MM2b</text>
<text><location><page_22><loc_60><loc_78><loc_63><loc_78></location>MM1b</text>
<text><location><page_22><loc_60><loc_77><loc_63><loc_77></location>MM1a</text>
<text><location><page_22><loc_60><loc_75><loc_63><loc_75></location>MM3a</text>
<text><location><page_22><loc_60><loc_74><loc_63><loc_74></location>MM3b</text>
<text><location><page_22><loc_62><loc_67><loc_63><loc_68></location>25</text>
<text><location><page_22><loc_68><loc_67><loc_70><loc_68></location>30</text>
<text><location><page_22><loc_75><loc_67><loc_77><loc_68></location>35</text>
<text><location><page_22><loc_61><loc_63><loc_63><loc_64></location>CH</text>
<text><location><page_22><loc_63><loc_63><loc_64><loc_63></location>3</text>
<text><location><page_22><loc_60><loc_60><loc_63><loc_61></location>MM2a</text>
<text><location><page_22><loc_60><loc_58><loc_63><loc_58></location>MM2b</text>
<text><location><page_22><loc_60><loc_55><loc_63><loc_55></location>MM1b</text>
<text><location><page_22><loc_60><loc_53><loc_63><loc_54></location>MM1a</text>
<text><location><page_22><loc_60><loc_50><loc_63><loc_51></location>MM3a</text>
<text><location><page_22><loc_60><loc_49><loc_63><loc_50></location>MM3b</text>
<text><location><page_22><loc_59><loc_46><loc_60><loc_47></location>20</text>
<text><location><page_22><loc_65><loc_46><loc_66><loc_47></location>25</text>
<text><location><page_22><loc_70><loc_46><loc_72><loc_47></location>30</text>
<text><location><page_22><loc_76><loc_46><loc_78><loc_47></location>35</text>
<text><location><page_22><loc_58><loc_43><loc_59><loc_44></location>8</text>
<text><location><page_22><loc_58><loc_40><loc_59><loc_41></location>6</text>
<text><location><page_22><loc_58><loc_38><loc_59><loc_39></location>4</text>
<text><location><page_22><loc_58><loc_35><loc_59><loc_36></location>2</text>
<text><location><page_22><loc_58><loc_32><loc_59><loc_33></location>0</text>
<text><location><page_22><loc_58><loc_30><loc_59><loc_31></location>-2</text>
<text><location><page_22><loc_58><loc_27><loc_59><loc_28></location>-4</text>
<text><location><page_22><loc_60><loc_42><loc_63><loc_43></location>MM2a</text>
<text><location><page_22><loc_60><loc_39><loc_63><loc_40></location>MM2b</text>
<text><location><page_22><loc_60><loc_35><loc_63><loc_36></location>MM1b</text>
<text><location><page_22><loc_60><loc_33><loc_63><loc_33></location>MM1a</text>
<text><location><page_22><loc_60><loc_28><loc_63><loc_29></location>MM3a</text>
<text><location><page_22><loc_60><loc_27><loc_63><loc_27></location>MM3b</text>
<text><location><page_22><loc_59><loc_25><loc_60><loc_26></location>20</text>
<text><location><page_22><loc_65><loc_25><loc_66><loc_26></location>25</text>
<text><location><page_22><loc_70><loc_25><loc_72><loc_26></location>30</text>
<text><location><page_22><loc_76><loc_25><loc_78><loc_26></location>35</text>
<text><location><page_22><loc_65><loc_23><loc_67><loc_25></location>V</text>
<text><location><page_22><loc_68><loc_63><loc_70><loc_63></location>1,7</text>
<text><location><page_22><loc_67><loc_42><loc_70><loc_43></location>HC</text>
<text><location><page_22><loc_67><loc_23><loc_69><loc_24></location>LSR</text>
<text><location><page_22><loc_69><loc_23><loc_75><loc_25></location>(km s</text>
<text><location><page_22><loc_76><loc_23><loc_77><loc_25></location>)</text>
<text><location><page_22><loc_70><loc_42><loc_70><loc_42></location>3</text>
<text><location><page_22><loc_70><loc_42><loc_77><loc_43></location>N (38-37)</text>
<text><location><page_22><loc_75><loc_24><loc_76><loc_25></location>-1</text>
<text><location><page_22><loc_64><loc_63><loc_68><loc_64></location>OH (7</text>
<text><location><page_22><loc_24><loc_67><loc_25><loc_68></location>18</text>
<text><location><page_22><loc_32><loc_84><loc_36><loc_85></location>(4-3)</text>
<text><location><page_22><loc_35><loc_63><loc_36><loc_64></location>-6</text>
<text><location><page_22><loc_33><loc_63><loc_35><loc_64></location>1,7</text>
<text><location><page_22><loc_38><loc_63><loc_38><loc_64></location>)</text>
<text><location><page_22><loc_38><loc_63><loc_40><loc_64></location>A</text>
<text><location><page_22><loc_36><loc_63><loc_38><loc_64></location>1,6</text>
<text><location><page_22><loc_47><loc_86><loc_49><loc_86></location>36.6</text>
<text><location><page_22><loc_47><loc_82><loc_49><loc_83></location>35.0</text>
<text><location><page_22><loc_47><loc_79><loc_49><loc_80></location>33.4</text>
<text><location><page_22><loc_47><loc_75><loc_49><loc_76></location>31.8</text>
<text><location><page_22><loc_47><loc_72><loc_49><loc_72></location>30.2</text>
<text><location><page_22><loc_47><loc_68><loc_49><loc_69></location>28.6</text>
<text><location><page_22><loc_47><loc_64><loc_49><loc_65></location>35.8</text>
<text><location><page_22><loc_47><loc_60><loc_49><loc_61></location>34.0</text>
<text><location><page_22><loc_47><loc_56><loc_49><loc_57></location>32.2</text>
<text><location><page_22><loc_47><loc_51><loc_49><loc_52></location>30.4</text>
<text><location><page_22><loc_47><loc_47><loc_49><loc_48></location>28.6</text>
<text><location><page_22><loc_55><loc_30><loc_57><loc_41></location>Offset (arcsec)</text>
<text><location><page_22><loc_67><loc_85><loc_67><loc_86></location>+</text>
<text><location><page_22><loc_64><loc_84><loc_67><loc_85></location>CO</text>
<text><location><page_22><loc_67><loc_84><loc_71><loc_85></location>(4-3)</text>
<text><location><page_22><loc_70><loc_63><loc_71><loc_64></location>-6</text>
<text><location><page_22><loc_73><loc_63><loc_74><loc_64></location>)</text>
<text><location><page_22><loc_74><loc_63><loc_75><loc_64></location>A</text>
<text><location><page_22><loc_71><loc_63><loc_73><loc_63></location>1,6</text>
<text><location><page_22><loc_79><loc_84><loc_82><loc_86></location>(d)</text>
<text><location><page_22><loc_79><loc_63><loc_82><loc_65></location>(e)</text>
<text><location><page_22><loc_79><loc_42><loc_81><loc_44></location>(f)</text>
<figure>
<location><page_23><loc_12><loc_33><loc_88><loc_74></location>
<caption>Fig. 4.- Solid histograms show the observed spectra of the CH 3 CN (19-18) K -ladder, overlaid with the best-fit LTE model, which yields a gas temperature of 135 K, shown in a dashed curve. A very strong spectral line with a (sky) frequency of ∼ 349 . 07 GHz is CH 3 OH (14 1 , 13 -14 0 , 14 ) A , and is irrelevant to the fitting to the CH 3 CN lines.</caption>
</figure>
<figure>
<location><page_24><loc_12><loc_28><loc_88><loc_78></location>
<caption>Fig. 5.- The velocity integrated CO (7-6) emission observed with the APEX telescope. The blue background and solid contours show the blueshifted emission, and the red background and dotted contours show the redshifted emission. Two arrows approximately mark the outflow axis. Other symbols are the same as in Figure 2b.</caption>
</figure>
<figure>
<location><page_25><loc_19><loc_26><loc_81><loc_86></location>
<caption>Fig. 6.- (a) Plus signs show the measured dispersion function, 1 -〈 cos[∆Φ( l )] 〉 . For most data points, the error bars are too small to recognize (barely visible at l > 9 '' ). A solid curve shows the fitted function, 1 N 〈 B 2 t 〉 〈 B 2 0 〉 × [1 -e -l 2 / 2( δ 2 +2 W 2 ) ] + a ' 2 l 2 , for l /lessorsimilar 6 '' ; a dashed curve visualizes the sum of the integrated turbulent contribution and the large scale contribution, i.e., 1 N 〈 B 2 t 〉 〈 B 2 0 〉 + a ' 2 l 2 . (b) The correlated component of the dispersion function: data points (shown in plus signs) are derived by subtracting the measured dispersion function from 1 N 〈 B 2 t 〉 〈 B 2 0 〉 + a ' 2 l 2 ; the fitted 1 N 〈 B 2 t 〉 〈 B 2 0 〉 e -l 2 / 2( δ 2 +2 W 2 ) is shown in a solid curve. A dashed curve visualizes the correlation solely due to the beam of the observation.</caption>
</figure>
<figure>
<location><page_26><loc_12><loc_40><loc_88><loc_70></location>
<caption>Fig. 7.- The Left panel shows a model of the total dust emission derived from a multicomponent Gaussian fitting to Figure 1c; the starting and spacing contour levels are both 0.018 Jy beam -1 . Star symbols are the same as those in Figure 2, and squares denote the three newly identified condensations. The Right panel shows the residual derived by subtracting the model from the observation; solid and dotted contours show the positive and negative emissions, respectively, with the contour levels of ± 0 . 018 , 0 . 036 Jy beam -1 .</caption>
</figure>
<figure>
<location><page_27><loc_12><loc_30><loc_88><loc_85></location>
<caption>Fig. 8.- Contours are the same as those in Figure 2; symbols of stars and squares are the same as those in Figure 7. Red segments show the measured directions of the B field in the southern part of the clump. We fit this part of the B field with a set of parabola, y -y 0 = g i + g i C ( x -x 0 ) 2 , following Girart et al. (2006) and Rao et al. (2009), and the bestfit model yields C = 0 . 053, the center of symmetry, ( x 0 , y 0 ), located at (R . A ., Decl . ) J2000 = (18 h 58 m 13 . s 139 , +01 d 40 m 31 . s 62) (denoted as a plus sign), and a position angle of -51 · for the y -axis of the parabola. Dashed curves show a representative set of the fitted parabola. Blue segments depict the tangential directions of the fitted parabola at the positions where the B field directions are measured.</caption>
</figure>
<figure>
<location><page_28><loc_11><loc_30><loc_88><loc_87></location>
<caption>Fig. 9.- A schematic view of our interpretation of the kinematics and magnetic fields in G35.2N. A filamentary structure in dark gray depicts the clump observed with the SMA, and six black dots indicate the deeply embedded dust condensations (see Figure 1c). The northern part of the clump is suggested to be rotating, as denoted by curved arrows, whose colors represent the sense of the rotation. A large ellipse filled with light gray shows the parent cloud observable with the JCMT/SCUBA. Two parabolic structures in light blue and red illustrate the blueshifted and redshifted lobes of a bipolar outflow. Solid and dotted lines visualize the magnetic field inside the clump (observed with the SMA) and that outside the clump (inferred from early JCMT observations), respectively. The cartoon is schematic only, and not to be scaled with the observed sizes of the various structures.</caption>
</figure>
<table>
<location><page_29><loc_12><loc_57><loc_88><loc_81></location>
<caption>Table 1. List of Observational Parameters</caption>
</table>
<table>
<location><page_29><loc_13><loc_28><loc_87><loc_42></location>
<caption>Table 2. Measured and computed parameters of the three dust cores</caption>
</table>
<text><location><page_29><loc_13><loc_22><loc_87><loc_26></location>a The upper limit is from our LTE fitting to the CH 3 CN (19-18) emission, and the lower limit is from recent NH 3 observations (Codella et al. 2010).</text>
<text><location><page_29><loc_15><loc_19><loc_67><loc_21></location>b From early NH 3 (1,1), (2,2) observations (Little et al. 1985).</text>
<text><location><page_29><loc_13><loc_15><loc_87><loc_18></location>c Derived by comparing the 340 GHz and 225 GHz continuum maps constructed with the same ( u, v ) range.</text>
<table>
<location><page_30><loc_23><loc_46><loc_77><loc_55></location>
<caption>Table 3. Results from the statistical analysis of the P.A. dispersion a</caption>
</table>
<text><location><page_30><loc_23><loc_39><loc_77><loc_44></location>a Except δ , all the derived parameters depend on the effective depth, ∆ ' ∼ 8-22 '' .</text>
</document>
[
{
"title": "ABSTRACT",
"content": "We report on detection of an ordered magnetic field (B field) threading a massive star-forming clump in the molecular cloud G35.2 -0.74, using Submillimeter Array observations of polarized dust emission. Thanks to the sensitive and high-angular-resolution observations, we are able to resolve the morphology of the B field in the plane of sky and detect a great turn of 90 · in the B field direction: Over the northern part of the clump, where a velocity gradient is evident, the B field is largely aligned with the long axis of the clump, whereas in the southern part, where the velocity field appears relatively uniform, the B field is slightly pinched with its mean direction perpendicular to the clump elongation. We suggest that the clump forms as its parent cloud collapses more along the large scale B field. In this process, the northern part carries over most of the angular momentum, forming a fast rotating system, and pulls the B field into a toroidal configuration. In contrast, the southern part is not significantly rotating and the B field remains in a poloidal configuration. A statistical analysis of the observed polarization dispersion yields a B field strength of ∼ 1 mG, a turbulent-to-magnetic energy ratio of order unity, and a mass-to-magnetic flux ratio of ∼ 2-3 in units of the critical value. Detailed calculations support our hypothesis that the B field in the northern part is being rotationally distorted. Our observations, in conjunction with early single-dish data, suggest that the B field may play a critical role in the formation of the dense clump, whereas rotation and turbulence could also be important in further dynamical evolution of the clump. The observations also provide evidence for a wide-angle outflow driven from a strongly rotating region whose B field is largely toroidal. Subject headings: ISM: magnetic fields - stars: formation - stars: early-type - techniques: polarimetric - techniques: interferometric",
"pages": [
1,
2
]
},
{
"title": "From Poloidal to Toroidal: Detection of Well-ordered Magnetic Field in High-mass Proto-cluster G35.2 -0.74 N",
"content": "Keping Qiu 1 , 2 , Qizhou Zhang 3 , Karl M. Menten 4 , Hauyu B. Liu 5 , Ya-Wen Tang 5 kpqiu@nju.edu.cn",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The role of magnetic fields in the evolution of molecular clouds and the formation of stars has long been subject to great debate (see Crutcher 2012, for a review). In a classic model of low-mass star formation, molecular clouds are supported by magnetic fields, and stay in subcritical states, evolving quasi-statically; ambipolar diffusion induces the formation of supercritical cores which dynamically collapse to form stars (Shu et al. 1987; Basu & Mouschovias 1994; Mouschovias et al. 2006). The other view is that molecular clouds are short lived, dynamically evolving, and producing stars rapidly (Elmegreen 2000; Hartmann et al. 2012), with magnetic field being implicitly weak, or not really appreciable in cloud evolution and star formation; Mac Low and Klessen (2004) further strongly argue that supersonic turbulence, instead of magnetic field, supports molecular clouds and regulates star formation. On the other hand, our understanding of high-mass star formation is far less clear, but the competing views on the role of magnetic field/turbulence are equally, if not more strongly debated. Recent magnetohydrodynamic (MHD) simulations suggest that magnetic fields are dynamically important in high-mass star formation, in particular at suppressing complete fragmentation and creating bipolar outflows (Banerjee & Pudritz 2007; Peters et al. 2011; Hennebelle et al. 2011; Commer¸con et al. 2011; Seifried et al. 2012; Myers et al. 2013 The 'magnetic support' model predicts a well-ordered or, in the extreme case, uniform magnetic field permeating a molecular cloud (Ostriker et al. 2001). Since there is an increased support against gravity in the direction perpendicular to the magnetic field compared to the direction parallel to it, the cloud contracts more along the field, forming flattened cores orthogonal to the mean direction of the field (Matsumoto & Tomisaka 2004; Tassis et al. 2009). In contrast, in the 'weak field' or 'turbulent support' model, magnetic field is expected to show an irregular and even chaotic morphology due to overwhelming turbulent twisting (Ostriker et al. 2001; Padoan et al. 2001). Therefore, mapping the morphology of magnetic field provides a straightforward method to distinguish between the two competing paradigms, or to provide insights into the possibility of a scenario where both magnetic field and turbulence are important. Focusing on massive clumps or cores with a typical size scale of 0.1 pc and a distance of a few kpc, high-angular-resolution observations of polarized dust emission are needed to spatially resolve the magnetic field morphology. Submillimeter Array (SMA) observations have been playing a major role in recent studies of this kind, though the observations are still limited to a small number of case studies (e.g., Girart et al. 2009; Tang et al. 2013; Liu et al. 2013; Girart et al. 2013). Here we present an SMA 6 study of a massive cluster-forming clump which shows an elongated morphology projected in the plane of sky, and thus defines an axis ready to be compared to the direction of the magnetic field. The targeted clump (hereafter G35.2N) lies in the northern part of G35.2 -0.74, a molecular cloud first discovered by Brown et al. (1982) and located at a distance of 2.19 kpc (Zhang et al. 2009a). It is associated with the IRAS source 18566+0136, which has a total luminosity of 3 × 10 4 L /circledot (Dent et al. 1989; S'anchez-Monge et al. 2013). Early molecular line observations revealed a velocity gradient along the long axis of the clump as well as a large scale bipolar outflow approximately orthogonal to the clump elongation (Dent et al. 1985; Little et al. 1985; Brebner et al. 1987). G35.2N was thus interpreted as a rotating interstellar disk or toroid. More detailed studies of the molecular outflow were presented by Gibbs et al. (2003) and Birks et al. (2006). Most recently, Zhang et al. (2013) presented SOFIA-FORCAST mid-infrared observations of G35.2N and modeled the object as a single high-mass protostar forming by an ordered and symmetric collapse of a massive core with a radius of 0.1 pc. However, both Atacama Large Millimeter/submillimeter Array (ALMA) cycle 0 observations (S'anchez-Monge et al. 2013) and the SMA data presented here reveal an apparently filamentary and highly fragmented structure for the dense gas on a 0.15 pc scale. We further infer a well-ordered B field from sensitive observations of the polarized dust emission and find that the B field morphology correlates with the dense gas kinematics.",
"pages": [
2,
3
]
},
{
"title": "2. Observations and Data Reduction",
"content": "The observations were carried out with the SMA from 2010 to 2012, using three array configuration under excellent weather conditions. Detailed information on the observations, including the observing dates, array configurations, number of available antennas, atmospheric opacities at 225 GHz, and various calibrators, is presented in Table 1. For the Subcompact and Extended observations, the 345 GHz receiver was tuned to cover roughly 332-336 GHz in the lower sideband and 344-348 GHz in the upper sideband. For the Com- pact observations, the frequency setup covers about 333.5-337.5 GHz in the lower sideband and 345.5-349.5 GHz in the upper sideband. For all the observations, the correlator was configured to have a uniform spectral resolution of 812.5 kHz ( ∼ 0.7 kms -1 ). We performed basic data calibration, including bandpass, time dependent gain, and flux calibration, with the IDL MIR package, and output the data into MIRIAD for further processing. The intrinsic instrumental polarization (i.e., leakage) was removed to a 0.1% accuracy (Marrone & Rao 2008) with the MIRIAD task GPCAL. For each sideband, a pseudo-continuum data set was created from spectral line-free channels using the MIRIAD task UVLIN. The calibrated visibilities were jointly imaged to make Stokes I , Q , and U maps. We performed self-calibration with the continuum data in Stokes I , and applied the solutions to both continuum and spectral line data in Stokes I , Q , U . Finally, the Stokes I , Q , and U maps were combined to produce maps of the polarized emission intensity, the fractional polarization, and the polarization angle, using the MIRIAD task IMPOL. No significant polarization is detected in molecular line emission, so only Stokes I maps are presented for spectral lines.",
"pages": [
3,
4
]
},
{
"title": "3.1. Dust Emission and Magnetic Field",
"content": "Figure 1 shows the total dust emission on a variety of size scales. In the James Cleak Maxwell Telescope 7 (JCMT) SCUBA 850 µ m map with a 15 '' resolution (Figure 1a), the emission unveils a ∼ 0.5 pc, slightly elongated clump, which shows hierarchical fragmentation in our SMA observations. In the SMA 880 µ m map made by combining the Subcompact and Compact observations (Figure 1b), the inner clump splits into three cores, MM1-3, each with a diameter of order 0.05 pc. Convolving the SMA map to the SCUBA beam and comparing its peak flux ( ∼ 6 Jy) to that of the SCUBA map ( ∼ 8.5 Jy), the SMA observations recover approximately 70% of the total flux. Figure 1c shows the SMA map made from all the observations spread over five tracks (Table 1). With a uniform weighting of the data, we obtain an angular resolution of 1 . '' 0 × 0 . '' 6, and resolve each of the three cores into at least two condensations with diameters of 0.01-0.02 pc. These condensations are distributed approximately along the major axis of the clump, but in MM2, there are minor emission peaks to the west. MM1b approximately coincides with an unresolved radio source first detected at 8.5 GHz (Gibb et al. 2003). S'anchez-Monge et al. (2013) presented the ALMA 860 µ m continuum map at a 0 . '' 4 resolution. Our SMA map shown in Figure 1c is in general consistent with the ALMA observations. To achieve the highest possible sensitivity, which is desirable for a polarization study, we jointly imaged all the SMA observations under a natural weighting, which results in an r.m.s. noise level, σ , of 1 . 5 mJybeam -1 . Strong polarization of the dust emission (signalto-noise ratios greater than 3) is detected toward MM1, MM3, and part of MM2 (Figure 2a), and thus the inferred B field in the plane of sky is well resolved (Figure 2b). We detect a clearly ordered B field threading the clump. Over the northern two cores MM1 and MM3, the B field direction is overwhelmingly aligned with the long axis of the clump, whereas approaching the southern core MM2, the B field direction drastically changes by about 90 · so that it is perpendicular to the clump elongation. Interestingly, early single-dish polarimetric observations with a 14 '' beam revealed a mean direction at a position angle of 56 · for the B field permeating the parent cloud of the clump (Vall'ee & Bastien 2000, also see Figure 2b). Therefore, over MM1 and MM3, the B field in the dense clump is approximately perpendicular to the mean direction of the large scale B field, while toward MM2, the B field in the clump roughly follows that in the cloud. In addition, to the southwest of MM1 and to the east of MM2, there are B field segments inclined to the direction of the cloud B field. Even for the northern part of the clump, the direction of the B field on the edges of the clump tends to deviate more from the clump major axis.",
"pages": [
4,
5
]
},
{
"title": "3.2. Molecular Line Emission",
"content": "Our SMA observations cover a large number of spectral lines tracing molecular gas under a variety of physical conditions. No significant polarization is detected in these lines, but they allow a detailed investigation of gas kinematics.",
"pages": [
5
]
},
{
"title": "3.2.1. Velocity gradient in high-density tracing molecular lines",
"content": "We investigate the kinematics of the dense gas by examining the first moment (intensity weighted velocity) maps (Figures 3a-c) and position-velocity ( PV ) diagrams (Figures 3d-f) of various high-density tracing molecular lines. In Figure 3, the H 13 CO + (4-3) emission traces the bulk dense gas of the clump seen in dust continuum. The most remarkable feature in the first moment map of the emission is a velocity shear lying between MM1 and MM2. From the PV diagram cut along the major axis of the clump (position angle -35 · ), MM2, which has a mean velocity close to the systemic velocity of the ambient cloud G35.2 -0.74 (Roman-Duval et al. 2009), can be readily distinguished from the rest of the clump, MM1 and MM3, and a velocity gradient is clearly seen between the latter two cores' locations. The CH 3 OH (7 1 , 7 -6 1 , 6 ) A emission preferentially traces the inner regions of the clump, and reveals an overall velocity field consistent with that seen in H 13 CO + (4-3). The HC 3 N (38-37) line has an upper level energy of 324 K above the ground, and probes the two condensations, MM1a and MM1b, embedded within MM1. There is a clear velocity gradient across the two condensations in the first moment map and PV diagram of the HC 3 N (38-37) emission. We also investigated our SMA 230 GHz observations, and found a similar velocity structure along the clump. The dense gas kinematics and the B field morphology both suggest that the clump can be understood as a two-component structure: the northern part, which consists of MM1 and MM3, exhibits a velocity gradient of order 50 km s -1 pc -1 , and the mean direction of the B field is aligned with the long axis of the clump; the southern part, i.e., MM2, does not show a clear velocity gradient, and the B field direction is perpendicular to the clump axis, such that it is aligned with the B field in the cloud.",
"pages": [
5,
6
]
},
{
"title": "3.2.2. Hot molecular cores and a wide-angle outflow",
"content": "The two dust condensations embedded within MM1 (i.e., MM1a,b) are the brightest in the clump, and show the richest molecular lines. MM1b coincides with (within 0 . '' 5) a faint and spatially unresolved radio source detected at 8.5 GHz (Gibb et al. 2003). Following Qiu & Zhang (2009) and Qiu et al. (2011b), we perform a local thermodynamical equilibrium (LTE) fitting to the K -ladder of the CH 3 CN (19-18) emission that is detected toward these two condensations. In Figure 4, the best-fit model agrees well with the observation, and yields a temperature of 135 +35 -15 K, indicating that the two sources are hot molecular cores (HMCs) and confirming that the clump is actively forming high-mass stars. The CO (3-2) line is covered in our SMA observations. However, due to the nearly equatorial position of the source and a limited number of antennas available in the interferometer, the CO maps are heavily affected by sidelobe and missing flux issues, and do not allow a reliable investigation of the outflow structure. We instead show our Atacama Pathfinder Experiment (APEX) CO (7-6) observations, which reveal a wide-angle bipolar outflow centered on the two HMCs (Figure 5). The outflow morphology is in general agreement with previous lowJ CO observations (Gibb et al. 2003; Birks et al. 2006). A more detailed study of the outflow nature in G35.2N, using multi-frequency observations, will be presented in a forthcoming paper. Here we stress that the outflow axis is approximately per- pendicular to the clump elongation (thus also perpendicular to the B field in the northern part of the clump).",
"pages": [
6,
7
]
},
{
"title": "4.1. The B field strength and its significance compared to turbulence and gravity",
"content": "We detect a structured B field threading the clump (Figure 2). The observed P.A. dispersion is attributed to both ordered and random perturbations to the B field, and the latter is probably dominated by turbulence. But dynamical processes related to cluster formation could perturb the B field in a manner rendering some disordered to random variations. We perform a statistical analysis of the observed P.A. dispersion to quantify the random perturbations (Hildebrand et al. 2009; Houde et al. 2009; Koch et al. 2010). Assuming the observed B field to be composed of a large-scale, ordered component, B 0 , and a turbulent component, B t , Houde et al. (2009) show that the disperse function 1 -〈 cos[∆Φ( l )] 〉 , which is measurable with a polarization map, can be approximately expressed as where ∆Φ( l ) is the difference in P.A. measured at two positions separated by a distance l and 〈· · ·〉 denotes an average, δ is the turbulent correlation length, W is the beam radius (i.e., the FWHM beam divided by √ 8ln2), and a ' 2 is the slope of the second-order term. N is the number of independent turbulent cells probed by observations, and is defined by where ∆ ' is the effective depth of the cloud along the line of sight. Here turbulence is implicitly adopted to be the only source of the random perturbations. Figure 6(a) shows the measured function taking into account polarization detections with signal-to-noise ratios > 4 in the northern part of the G35.2N clump 8 . In our observations, W = 0 . '' 63, and ∆ ' ∼ 8-22 '' , which are approximately the dimensions of the clump projected on the plane of sky (Figure 2). We then fit the measured function with the above relation to solve for 〈 B 2 t 〉 / 〈 B 2 0 〉 , δ , and a ' 2 , of which the former two are of our interest. Since the relation is valid when l is less than a few times W , we fit the data points at l /lessorsimilar 6 '' . The solid curve in Figure 6a shows the fitting results, and the dashed curve visualizes the sum of the integrated turbulent contribution, [ 〈 B 2 t 〉 / 〈 B 2 0 〉 ] /N , and the large-scale contribution, a ' 2 l 2 . The correlated turbulent component of the dispersion function, is shown in Figure 6b. With the fitting we obtain δ = 15 . 4 mpc (1 . '' 45), which is approximately equal to the value derived in OMC-1 (Houde et al. 2009)), and 〈 B 2 t 〉 / 〈 B 2 0 〉 = 0 . 230.64 for N = 3 . 0-8.4, which depends on ∆ ' . According to Chandrasekhar & Fermi (1953), where δV los is the one-dimensional velocity dispersion and V A = B 0 / √ 4 πρ is the Alfv'en velocity at mass density ρ . Consequently we can estimate the energy ratio of the turbulence to the ordered B field, β turb ∼ 3 〈 B 2 t 〉 / 〈 B 2 0 〉 , to be 0.7-1.9. The ratio is somewhat overestimated considering that we are probing the B field in the plane of sky and B t is not necessarily all ascribed to turbulence. Nevertheless, the statistical analysis suggests that the ordered B field and turbulence are energetically comparable. The strength of the ordered B field can be derived following The H 13 CO + (4-3) emission traces a structure similar to that is seen in the dust emission, and we estimate δV los ∼ 1 kms -1 from the line-of-sight velocity dispersion of the H 13 CO + line. To estimate ρ , we first compute the mass of the clump from its dust emission, which requires the information on the dust temperature and opacity. In our SMA observations, the clump consists of three cores, MM1-3, each of which harbors a couple of dust condensations. Toward MM2 and MM3, early NH 3 (1,1) and (2,2) observations deduced a rotational temperature of 20-25 K, corresponding to a kinetic temperature of 30-40 K (Little et al. 1985). Toward MM1, our LTE model of the CH 3 CN (19-18) K -ladder gives a temperature of 135 K for the embedded HMCs, while recent high-angular-resolution NH 3 observations suggest a temperature > 50 K (Codella et al. 2010). Thus we adopt a gas temperature of 50-135 K for MM1. Assuming thermal equilibrium between gas and dust, the derived gas temperature approximates the dust temperature. The dust opacity index, β , is derived by comparing the dust emission fluxes at 880 µ m and 1.3 mm (the latter is obtained from our new SMA observations). The dust opacity at 880 µ m is then extrapolated from 10[ ν/ (1 . 2 THz)] β , where ν is frequency, following Hildebrand (1983). Assuming a canonical gas-to-dust mass ratio of 100, the masses of the three cores are then computed from their dust emission fluxes (Table 2). The total mass of the clump amounts to ∼ 150 M /circledot , and results in an averaged column density of ∼ 3 . 7 × 10 23 cm -2 over a measured area, s , of 200 square arc sec (0.015 pc 2 ) where the emission is detected with signal-to-noise ratios above 5. If we adopt a volume of 4 / 3( s 3 /π ) 1 / 2 , the averaged volume density is ∼ 1 . 0 × 10 6 cm -3 . We then obtain B 0 ∼ 1 . 4-0.9 mG, again depending on ∆ ' . Consequently, the mass-to-magnetic flux ratio, M/φ B , is 1.9-3.2 in units of a critical value, 1 / (2 π √ G ), where G is the gravitational constant (Nakano & Nakamura 1978), indicating that the clump is unstable against gravitational collapse and fragmentation It has been shown that a star-forming cloud generally collapses to a flattened or filamentary configuration to allow efficient fragmentation to follow (Larson 1985; Pon et al. 2011). The observations of the G35.2N clump is consistent with this theoretical prediction. To further understand the fragmentation property of the clump, it is instructive to compute the thermal Jeans mass, which is about 1.5 M /circledot at a density of 1 . 0 × 10 6 cm -3 and a temperature of 30 K. The analyses of the gravitational instability of molecular gas sheets and filaments by Larson (1985) are probably of more relevance to G35.2N. Larson (1985) argued that rotation and magnetic field do not fundamentally change the characteristic mass, M c ( M /circledot ) = 2 . 4 T 2 /µ, where T is the gas temperature and µ is the surface density in M /circledot pc -2 , of the fragments. For G35.2N, M c /similarequal 0 . 22 M /circledot . Even without taking into account the mass of the forming stars and of the gas already dispersed by the wide-angle outflow, the clump mass is orders of magnitude greater than the Jeans mass or M c , and is expected to fragment into hundreds of density enhancements if the dynamics is solely controlled by the gravity and thermal pressure. In Figure 1c, there are only six prominent condensations detected at a spatial resolution (0.01 pc) finer than the Jeans length (0.04 pc). We also perform a multi-component Gaussian fitting to the dust emission using the MIRIAD task IMFIT, and identify three new possible condensations, MM1c, MM2c, and MM2d (see Figure 7). Nevertheless, the total number of the condensations is far less than what is expected in thermal fragmentation. Such observational results have been seen in some other high-mass star-forming regions (Zhang et al. 2009b ? ), and strongly suggest that other mechanisms, e.g., turbulence, magnetic field, and rotation, are playing important roles in the dynamical evolution of the clump. A highly turbulent core with a radius of 0.1 pc and a mass of 240 M /circledot , as described by Zhang et al. (2013) in applying the model of McKee & Tan (2003) to G35.2N, is obviously inconsistent with the high-angular-resolution observations obtained here with the SMA and with the ALMA (S'anchez-Monge et al. 2013). Recent numerical simulations of the collapse and fragmentation of magnetized, massive star-forming cores show that cores with M/φ B ∼ 2 form a single star rather than fragmenting to a small cluster (e.g., Commer¸con et al. 2011; Myers et al. 2013). This is inconsistent with our observations either; the measured M/φ B in G35.2N is ∼ 2-3, and the clump is highly fragmented. We expect that both the B field and turbulence help to increase the effective Jeans mass or the characteristic mass of the fragments, but neither (or their combination) is sufficient to completely suppress fragmentation on /lessorsimilar 0 . 1 pc scales.",
"pages": [
7,
8,
9,
10
]
},
{
"title": "4.2.1. A schematic picture",
"content": "The sub-parsec clump in G35.2N has been interpreted as a prototype of a massive interstellar disk or toroid, with the 'disk' being nearly edge-on, having LSR velocities increasing from northwest to southeast (Dent et al. 1985; Little et al. 1985; Brebner et al. 1987; Little et al. 1998; L'opez-Sepulcre et al. 2009). Our observations provide significantly new insights into the dynamics of the clump. Going from the northern to the southern part of the clump, the B field direction takes a 90 · turn and is correlated with the velocity field revealed by the high-density tracing molecular lines (see Section 3.2.1). We also find that the B field in the southern part appears to be slightly pinched and can be fitted with a set of parabola (Figure 8), which provides marginal evidence for a magnetically regulated collapse of MM2 (e.g., Girart et al. 2006; Gon¸calves et al. 2008). In Figure 7, the dust condensations in the northern part are distributed approximately along the major axis of the clump, while those in the southern part have a Trapezium-like distribution. All this suggests that the clump is composed of two dynamically different components rather than coherently rotating as a whole. One interpretation of the observed velocity gradient in the northern part of the clump is that it arises from the rotation of the dense gas. Considering the orientation of the B field in the cloud revealed by the single-dish observations (see Figure 2b), we speculate that the dense clump forms as the cloud collapses more along the direction of the large scale B field. In this dynamical process, the dense gas breaks into two parts. The northern part carries over most of the angular momentum that is not dissipated and forms a fast rotating system. The system could be a rotating toroid embedded with multiple fragments. Or it is a hierarchical structure consisting of two cores (MM1 and MM3) in an orbital motion, and each core further fragments and collapses into a binary or multiple star. In the latter case, the projected separation of ∼ 0 . 05 pc and the line-of-sight velocity difference of ∼ 2 . 5 kms -1 between MM1 and MM3 yield a dynamical mass of /greaterorsimilar 73 M /circledot , which is compatible with the combined mass of the two cores (Table 2). The southern part, however, does not participate in the rotating motion and is expected to have a mean velocity similar to that of the parent cloud. This hypothesis is indeed supported by the large scale kinematics seen in 13 CO (1-0) observations obtained from the Galactic Ring Survey, from which an LSR velocity of 35.2 km s -1 is derived for the ambient cloud of G35.2N (Jackson et al. 2006; Roman-Duval et al. 2009, also see Figure 3). Also, one may expect that the angular momentum in the northern part has a significant influence on the fragmentation, causing the resulted fragments to be distributed within the rotational plane. The observed morphology of the B field in the clump can be naturally incorporated in the above scenario. Figure 9 provides an overall picture of our interpretations of the B field structure: in the northern part, the B field is pulled by the rotation into a toroidal configuration, whereas in the southern part, the B field is only gently squeezed by the collapse and remains to be in a poloidal configuration (aligned with the B field in the cloud).",
"pages": [
10,
11
]
},
{
"title": "4.2.2. Quantitative comparisons",
"content": "We quantitatively investigate the feasibility of our interpretation of a rotationally distorted B field in the northern part of the clump. According to Matsumoto et al. (2006), a 90 · misalignment between the axis of an outflow and the mean direction of an ordered B field may occur if the rotational energy is at least comparable to the magnetic energy (also see Rao et al. 2009). Assuming the measured velocity gradient, ∆ v/L , in the northern part of G35.2N is arising from a disk-like structure with a diameter L , the rotational energy amounts to 1 / 2( ML 2 / 8)(∆ v/L ) 2 , where M is the mass of the rotating structure. The averaged ratio of the rotational energy to the magnetic energy, β rot = (∆ v ) 2 / (8 V 2 A ), is found to be 0.5-1.3 for an end-to-end velocity difference of ∼ 4 kms -1 and B 0 ∼ 1 . 4-0.9 mG. Hence the rotational energy does appear to be comparable to the magnetic energy. Alternatively, if a B field is wound by rotation into an overwhelmingly toroidal configuration, the centrifugal force is presumably overcoming the magnetic force. More quantitatively, Machida et al. (2005) found that whether the centrifugal force or the magnetic force regulates the evolution of a collapsing core can be evaluated by the ratio of the angular velocity to the magnetic field strength, ω/B . If this ratio exceeds a critical value, 0 . 39 √ G/c s , where c s is the sound speed, the centrifugal force dominates the dynamics. For the G35.2N clump, ω ∼ 5 . 1 × 10 -5 year -1 and c s ∼ 0 . 33 kms -1 at 30 K, leading to ω/B /similarequal (3 . 6-5 . 7) × 10 -8 yr -1 µ G -1 , which is on the same order but lower than the critical value of 9 . 7 × 10 -8 yr -1 µ G -1 . We stress that both ω and c s are averaged over the entire clump. Toward the inner part of the clump, ω increases since the cores appear to rotate faster, and c s also increases since the gas temperature is higher. Thus the measured ω/B would be greater than the critical value. For example, considering the dense gas in MM1, ω measures ∼ 1 . 5 × 10 -4 year -1 (Figure 3f) and c s reaches 0.71 km s -1 ; the measured ω/B is more than 2 times greater than the critical value, confirming that the centrifugal force is dominating over the magnetic force. The above calculations show that energy-wise, the B field is likely to be significantly distorted by the rotation. But, is there sufficient time for a toroidal B field to develop? The rotation period, 2 πL/ ∆ v , is estimated to be 1 . 2 × 10 5 yr. The dense clump with an averaged density of 1 . 0 × 10 6 cm -3 presumably form from a cloud with a lower density ( /lessorsimilar 10 4 cm -3 ), which has a free-fall time > 10 5 yr. In addition, with the detection of HMCs and a radio source in MM1, it is feasible to expect that the clump has a dynamical age of order 10 5 yr (e.g., Charnley 1997). Therefore, it is likely that the rotation has proceeded for a few periods, and significantly distorted the B field into a predominantly toroidal configuration. Finally, another piece of evidence supporting for the presence of a toroidal B field in the clump comes from polarization observations (0 . '' 2 resolution) of OH maser emission; Hutawarakorn & Cohen (1999) detected a few Zeeman pairs within 0 . '' 5 of MM1b and found that the direction of the B field along the line of sight reverses from the southeast to the northwest of MM1b, and the direction reversal was interpreted as being due to a toroidal B field.",
"pages": [
11,
12
]
},
{
"title": "4.2.3. Connecting the B field to the outflow",
"content": "Both well-collimated and wide-angle outflows have been seen in high-mass star-forming regions (e.g., Qiu et al. 2008; Qiu & Zhang 2009; Qiu et al. 2009; Qiu et al. 2011a; Qiu et al. 2012). Here in G35.2N, the CO (7-6) observations reveal a parsec-sized wide-angle outflow (Figure 5, also see Figure 9). The outflow appears to originate from one of the two HMCs (MM1a or MM1b, see Section 3.2.2), which must have been undergoing collapse and spun up to form a rotating disk at the center. Indeed, there is evidence for the presence of a rotating disk in MM1b from the ALMA observations (S'anchez-Monge et al. 2013), though it is yet to be confirmed whether the outflow is driven from MM1b. In line with our above interpretations, the ambient B field of the inner ∼ 0 . 1 pc part of the outflow is dominated by a toroidal component, and that component could be further traced down to a 0.01 pc scale taking into account previous OH maser observations (Hutawarakorn & Cohen 1999). Thus the wide-angle outflow is most likely associated with a strong toroidal B field. How is this outflow driven? Bipolar outflows in protostars or young stars are believed to be magnetically driven. In the most quoted magneto-centrifugal wind theory, the outflowing gas is centrifugally ejected, either from the inner edge of an accretion disk or over a wide range of disk radii, along open poloidal field lines (e.g., Shu et al. 1994; Shang et al. 2006; Pudritz et al. 2007; Fendt 2009). On the other hand, the magnetic tower model suggests that an outflow is accelerated by the magnetic pressure gradient of a wound-up volume of a toroidal B field (Uchida & Shibata 1985; Lynden-Bell 1996; Lynden-Bell 2003). Many MHD simulations of collapsing magnetized cores have shown that a bipolar outflow is launched from a strongly rotating region when a predominantly toroidal B field has been built up by the collapse-spin up process (e.g., Tomisaka 1998; Matsumoto & Tomisaka 2004; Banerjee & Pudritz 2006; Banerjee & Pudritz 2007; Seifried et al. 2012). The observations of G35.2N are apparently consistent with such a scenario. Furthermore, numerical simulations often found two-component outflows: a larger, low-velocity, and less collimated outflow emerging earlier than an inner, faster, and jet-like outflow (e.g., Tomisaka 2002; Banerjee & Pudritz 2006; Machida et al. 2008; Seifried et al. 2012; Tomida et al. 2013). While some claim that the former is a magnetic tower flow and the latter is launched by the magneto-centrifugal force (Banerjee & Pudritz 2006; Banerjee & Pudritz 2007), others argue for an opposite view (Machida et al. 2008; Tomida et al. 2013). The situation can be even more complicated considering possible dependence of the outflow morphology on the B field strength (e.g., Hennebelle & Fromang 2008; Seifried et al. 2012; Bate et al. 2013). In addition, radiation pressure could be important in widening massive outflows (Vaidya et al. 2011). Thus we are not able to infer the driving mechanism of the G35.2N outflow solely based on its morphology. Regarding the B field morphology, existing observations suggest a strong toroidal component in the inner part of the outflow, but a poloidal component certainly exists. To determine whether a magnetic tower or the centrifugal force dominates the acceleration of the outflowing gas, one may need a complete knowledge of three-dimensional B field and velocity structures (Seifried et al. 2012). Since we are only probing the B field projected in the plane of sky, and the estimate of the B field strength is indirect, it is not possible to compare the toroidal to poloidal components in detail. Hence whether or to what extent a magnetic tower plays a role in accelerating the outflow remains open. But if the growing tower is important, the central collapse-spin up process significantly helps to create at least part of the observed toroidal B field.",
"pages": [
12,
13
]
},
{
"title": "4.3. Infall along a magnetized cylinder?",
"content": "Provided the remarkable filamentary morphology of the clump seen in Figure 1c, it is likely that we are observing a dense gas cylinder. A gas cylinder may fragment along its major axis due to the 'sausage' or 'varicose' instability, and the resulted fragments will have a characteristic spacing (e.g., Jackson et al. 2010). In Fgiure 7a, the dust condensations in the northern part are nearly equally spaced, in qualitative agreement with the fragmentation of a cylinder. Since a cylinder/filament is not expected to be rotating too fast about its minor axis 9 , the observed velocity gradient ( ∼ 50 kms -1 pc -1 ) is most likely arising from an infall motion along the cylinder axis rather than from a rotation. From Figure 3d, the infall velocity is ∼ 2/sin i kms -1 , where i is the inclination angle of the cylinder with respective to the plane of sky. For a cylinder with a mass of 100 M /circledot (roughly the gas mass of the northern clump plus the mass of the forming stars) and a length of 0.1 pc, the line-of-sight component of the free-fall velocity reaches ∼ 4(cos i ) 1 / 2 sin i kms -1 , which is not incompatible (e.g., for i /similarequal 54 · ) with the observed velocity gradient. However, in the SCUBA map (Figure 1a), the clump is much less elongated, and on a larger scale, we do not detect any filamentary structure where the cylinder is embedded. Furthermore, self-gravitating cylinders have a critical linear mass density, 2 v 2 /G , where v is the sound speed c s in case of thermal support (Stod'olkiewicz 1963; Ostriker 1964) or the turbulent velocity dispersion δV los in case of turbulent support (Fiege & Pudritz 2000). Above the critical value, the cylinder would radially collapse into a line. The B field can somewhat increase the critical linear mass density, but probably by small factors ∼ 1 (Fiege & Pudritz 2000). For G35.2N, the measured linear mass density is of order 1000 M /circledot pc -(without taking into account the mass of the forming stars and of the gas already dispersed by the outflow), which is about two times greater than the critical value of 470 M /circledot pc -1 . Unless the cylinder is largely inclined to the line of sight (i.e., i > 60 · ), the linear mass density seems to be too high to allow a stable cylinder to exist. Also, the southern part of the clump, which appears very different from the northern part in the B field, kinematics, and fragmentation properties, cannot be easily incorporated in a unified picture of a cylinder. Considering all these difficulties in understanding the observed clump as a cylinder, the interpretation of a rotating system (Section 4.2) for the northern part of the clump appears more robust than that of a collapsing cylinder.",
"pages": [
13,
14
]
},
{
"title": "5. Conclusions",
"content": "With the sensitive and high-angular-resolution observations made with the SMA, we detect a well-ordered B field threading G35.2N, which is an apparently filamentary, massive, and cluster-forming clump. Based on a statistical analysis of the observed polarization dispersion, we derive a B field strength of ∼ 1 mG, a turbulent-to-magnetic energy ratio of order unity, and a mass-to-magnetic flux ratio of ∼ 2-3 times the critical value. The B field morphology is found to vary depending on the kinematics revealed by high- 1 density tracing molecular lines. In the northern part of the clump, which exhibits a velocity gradient, the B field is largely aligned with the long axis of the clump (hence follows the velocity gradient). Approaching the southern part, which has a relatively uniform velocity field, the B field takes a great turn of 90 · to a direction perpendicular to the clump elongation. Although we cannot rule out the possibility that the velocity gradient is due to an infall motion along a gas cylinder, a synthesis of all the available observations prefers a rotating motion to be the origin of the velocity gradient. Consequently the B field in the northern part is pulled by the centrifugal force into an overwhelmingly toroidal configuration, whereas the B field in the southern part remains in a poloidal configuration, that is, aligned with a large scale B field which guides the collapse of the cloud from which the clump forms. Therefore we are likely witnessing a transition from a poloidal to toroidal configuration for the B field in this region. Detailed calculations adopting the B field strength derived from the statistical analysis supports such a scenario. Our observations also provide evidence for a wide-angle outflow driven from a strongly rotating region whose B field is largely toroidal. Last, but not least, the supercritical clump is observed to be significantly fragmenting, but to a degree far from expected for a purely thermal fragmentation. Our observations and calculations suggest that both the B field and turbulence, as well as the potential rotation, all play a role in interplaying with the gravity and shaping the dynamical evolution of the clump. Part of the data were obtained in the context of the SMA legacy project: 'Filaments, Magnetic Fields, and Massive Star Formation' (PI: Qizhou Zhang). We acknowledge all the members of the SMA staff who made these observations possible. The JCMT SCUBA data were obtained from the JCMT archive (Program ID: M97BH07). This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. Part of this research was undertaken when KQ was a postdoctoral fellow at Max-Planck-Institut fuer Radioastronomie. KQ is supported by the 985 project of Nanjing University. QZ is partially supported by the NSFC grant 11328301.",
"pages": [
14,
15
]
},
{
"title": "REFERENCES",
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"pages": [
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}
]
2013ApJS..206...23H
https://arxiv.org/pdf/1205.5791.pdf
<document>
<section_header_level_1><location><page_1><loc_22><loc_86><loc_78><loc_87></location>RECONSTRUCTING THE SHAPE OF THE CORRELATION FUNCTION</section_header_level_1>
<text><location><page_1><loc_28><loc_83><loc_73><loc_84></location>Department of Physics, University of Miami, Coral Gables, Florida 33146</text>
<text><location><page_1><loc_33><loc_82><loc_67><loc_85></location>K. M. Huffenberger, M. Galeazzi, E. Ursino Draft version September 1, 2021</text>
<section_header_level_1><location><page_1><loc_45><loc_79><loc_55><loc_80></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_70><loc_86><loc_79></location>We develop an estimator for the correlation function which, in the ensemble average, returns the shape of the correlation function, even for signals that have significant correlations on the scale of the survey region. Our estimator is general and works in any number of dimensions. We develop versions of the estimator for both diffuse and discrete signals. As an application, we examine Monte Carlo simulations of X-ray background measurements. These include a realistic, spatially-inhomogeneous population of spurious detector events. We discuss applying the estimator to the averaging of correlation functions evaluated on several small fields, and to other cosmological applications.</text>
<text><location><page_1><loc_14><loc_67><loc_86><loc_69></location>Subject headings: cosmology: theory-methods: numerical-methods: data analysis-methods: statistical-X-rays: diffuse background-galaxies: clustering</text>
<section_header_level_1><location><page_1><loc_43><loc_62><loc_57><loc_63></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_58><loc_92><loc_62></location>Two-point statistics encode valuable information about the fields that they describe, such as the cosmological matter density traced by galaxies or the intensity of radiation in backgrounds like the Cosmic Microwave Background (CMB), the Cosmic Infrared Background (CIB), or the Diffuse X-ray Background (DXB).</text>
<text><location><page_1><loc_8><loc_55><loc_92><loc_58></location>For discrete objects, the two-point, dimensionless correlation function can be defined in terms of the probability of finding a pair of objects in two small cells, with sizes δ Ω 1 and δ Ω 2 , separated by θ 12 (Peebles 1980, § 31, 45):</text>
<formula><location><page_1><loc_39><loc_53><loc_92><loc_54></location>δP 12 = N 2 δ Ω 1 δ Ω 2 [1 + w ( θ 12 )] (1)</formula>
<text><location><page_1><loc_8><loc_51><loc_92><loc_52></location>where N is the mean density of sources. For diffuse fields, the equivalent definition for a signal s with mean 〈 s 〉 = µ is</text>
<formula><location><page_1><loc_41><loc_48><loc_92><loc_50></location>〈 s 1 s 2 〉 = µ 2 [1 + w ( θ 12 )] , (2)</formula>
<text><location><page_1><loc_8><loc_41><loc_92><loc_48></location>where here and throughout 〈 . . . 〉 denotes the ensemble average. 1 We denote the covariance of s as C ( θ ) = µ 2 w ( θ ), which we also refer to as the (dimensionful) correlation function. This work mostly deals with the dimensionful correlation function and addresses the bias in its estimation. With similar expressions, we can define correlation functions in any number of dimensions, replacing the angular separation θ by a linear separation or time interval or whatever is appropriate.</text>
<text><location><page_1><loc_8><loc_29><loc_92><loc_41></location>The estimation of the correlation function has been studied extensively in the literature. For galaxy clustering, Hewett (1982), Davis & Peebles (1983), and Hamilton (1993) suggest different Monte Carlo estimators, but the most common estimator now in use was advocated by Landy & Szalay (1993), which employs the data in concert with a synthetic, random catalog. Their estimator combines counts of objects pairs within and between the data and random catalogs. This estimator is biased, but for surveys where the correlation length of the objects is much smaller than the survey area, the bias is small (Bernstein 1994). Such is the case for modern galaxy surveys like 2dF and SDSS (Percival et al. 2001; York et al. 2000). However, the bias can become significant when structures approach the size of the survey (Kerscher 1999). This bias can be corrected (e.g. Scranton et al. 2002), but the correction depends on same correlation function that is being estimated.</text>
<text><location><page_1><loc_8><loc_26><loc_92><loc_29></location>For diffuse signals like the CMB, where using the dimensionful correlation function is more common, a typical estimator looks like (Hinshaw et al. 1996; Copi et al. 2007):</text>
<formula><location><page_1><loc_38><loc_22><loc_92><loc_25></location>˜ C 0 ( θ ) = ∑ ij α i α j ( s i -˜ µ )( s j -˜ µ ) ∑ ij α i α j (3)</formula>
<text><location><page_1><loc_8><loc_17><loc_92><loc_21></location>where α i are the weights applied to the pixels or cells (for the purpose of downweighting noisy regions), ˜ µ is an estimate of the mean, and the sum over ij refers to pixels separated by θ . These estimators suffer the same biases on small fields.</text>
<text><location><page_1><loc_8><loc_11><loc_92><loc_17></location>In this paper we introduce a new method to address the biases in these above estimators. Our estimator is also biased, but biased in a particularly convenient way: regardless of the survey geometry or weighting, the shape of the correlation function is preserved on average, and only information about a constant offset is lost. This permits the straightforward averaging of correlation functions from several small patches across the sky. Building upon the estimator in eqn. (3), we develop classes of estimators for both diffuse signals and discrete objects.</text>
<text><location><page_2><loc_8><loc_83><loc_92><loc_92></location>This work was prompted by our group's efforts to compute correlation function from observations of the diffuse X-ray background. The signal in that case comes from a diffuse, gaseous source, but arrives and is recorded as individual, discrete X-ray photons, and so can be analyzed with either scheme above. Indeed, for simulations of diffuse X-ray emission from the WHIM, Ursino et al. (2011) found that the Landy & Szalay (1993) estimator gave roughly equivalent results to an estimator of the type in eqn. (3). We focused on the correlation function biases because the angular correlation scale of this gas (several arcminutes) is substantial compared to the field-of-view ( ∼ 8 arcminutes) for single-field observations with the Chandra X-ray Observatory.</text>
<text><location><page_2><loc_8><loc_76><loc_92><loc_82></location>The paper is organized as follows. In section 2 we find the bias for the naive estimator (eqn. 3), verifying our result with Monte Carlo simulations, and introduce a method for correcting it up to a constant offset. In section 3 we extend this estimate to Poisson-distributed counts, allowing for the possibility of a spatially-varying set of spurious detector events. Finally, we summarize our conclusions in section 4. An appendix contains the detailed derivations of the bias terms.</text>
<section_header_level_1><location><page_2><loc_33><loc_74><loc_67><loc_75></location>2. CORRELATION FUNCTION ESTIMATOR BIAS</section_header_level_1>
<text><location><page_2><loc_8><loc_71><loc_92><loc_73></location>We begin by defining our signals. Let s i represent a pixelized, diffuse signal that is statistically homogeneous and isotropic. Let it be described by a mean and covariance as follows:</text>
<formula><location><page_2><loc_36><loc_67><loc_92><loc_70></location>〈 s i 〉 = µ (4) 〈 ( s i -µ )( s j -µ ) 〉 = 〈 s i s j 〉 -µ 2 = C ( θ ij )</formula>
<text><location><page_2><loc_8><loc_58><loc_92><loc_66></location>where θ ij represents the separation between cells i and j . In our derivations we use C ( θ ) rather that w ( θ ) because the examination of biases is convenient; C ( θ ) also makes sense for diffuse fields where µ = 0. No other special properties of s are required, except that the covariance matrix is positive semi-definite: 0 ≤ | C ( θ ) | ≤ C (0). In particular, the signal need not be a Gaussian random field: we could define higher-order moments without disrupting our following arguments. Note that by this definition, the correlation function C ( θ ) is a property of the probability distribution for our signal s , and it is not a descriptive statistic.</text>
<text><location><page_2><loc_10><loc_57><loc_80><loc_58></location>With a set of weights on the pixels, α i , we can compute a weighted average to estimate the mean,</text>
<formula><location><page_2><loc_46><loc_53><loc_54><loc_56></location>˜ µ = ∑ i α i s i ∑ i α i</formula>
<formula><location><page_2><loc_90><loc_54><loc_92><loc_55></location>(5)</formula>
<text><location><page_2><loc_8><loc_48><loc_92><loc_52></location>where the sum is over all pixels. These weights could be chosen to be uniform or to suppress noisy or polluted portions of the measurement. Throughout we mark estimated quantities with tildes. This mean estimate is unbiased, 〈 ˜ µ 〉 = µ . Additionally we define the deviation between the true mean and the estimated mean by</text>
<formula><location><page_2><loc_46><loc_46><loc_92><loc_48></location>δ ˜ µ = ˜ µ -µ (6)</formula>
<text><location><page_2><loc_8><loc_44><loc_18><loc_46></location>with 〈 δ ˜ µ 〉 = 0.</text>
<section_header_level_1><location><page_2><loc_35><loc_42><loc_65><loc_43></location>2.1. Naive correlation function estimator</section_header_level_1>
<text><location><page_2><loc_8><loc_39><loc_92><loc_42></location>Based on the estimated mean, we make an initial estimate of the correlation function in a bin labeled by θ p , which we call the naive estimator:</text>
<formula><location><page_2><loc_35><loc_36><loc_92><loc_39></location>˜ C 0 ( θ p ) = ∑ ij d ij ( θ p ) α i α j ( s i -˜ µ )( s j -˜ µ ) ∑ ij d ij ( θ p ) α i α j (7)</formula>
<text><location><page_2><loc_8><loc_34><loc_53><loc_35></location>This is just a more explicit rewriting of eqn. (3). The function</text>
<formula><location><page_2><loc_31><loc_30><loc_92><loc_33></location>d ij ( θ p ) = { 1 , if i and j are separated by θ p ± δθ /2 0 , otherwise (8)</formula>
<text><location><page_2><loc_8><loc_20><loc_92><loc_29></location>chooses the separation bin to which the pixel sum contributes. 2 Evaluation of the estimator costs O ( N 2 ) operations over N pixels. If the true mean µ replaces the estimated mean ˜ µ in eqn. (7), then this correlation function estimate is unbiased, 3 and we find 〈 ˜ C 0 ( θ p ) 〉 = C ( θ p ). However, since we do not know the true mean, our estimate will be biased, because we are forced to use the same (correlated) set of pixels to compute the mean and the correlation function. The smaller the survey compared to the correlation length of the signal, the worse this bias-the 'integral constraint'-becomes. (See Hamilton (1993) for further discussion of bias due to the mean error and other approaches to avoid it.)</text>
<text><location><page_2><loc_8><loc_17><loc_92><loc_19></location>In the appendix, we compute the bias explicitly. We further show that the ensemble average of the naive, biased estimator may be cast as a linear operation applied to the true correlation function:</text>
<formula><location><page_2><loc_41><loc_13><loc_92><loc_16></location>〈 ˜ C 0 ( θ p ) 〉 = ∑ q M pq C ( θ q ) , (9)</formula>
<figure>
<location><page_3><loc_25><loc_62><loc_75><loc_92></location>
<caption>Fig. 1.The input correlation function (black) was used to create a set of N MC = 1000 Monte Carlo realizations of a simulated map (without shot noise). At each angular separation, 95 percent of naive estimates ˜ C 0 ( θ ) for the correlation function fall within the pink region. The average of the Monte Carlo ensemble of naive estimates is solid blue, and has fluctuations reduced by a factor √ N MC ∼ 30. The sum of the bias terms computed from the input C ( θ ) is shown as the dashed red line. The ensemble average minus the bias terms is shown with the dash-dot blue line, and closely matches the input.</caption>
</figure>
<text><location><page_3><loc_8><loc_52><loc_25><loc_53></location>or as a matrix equation,</text>
<formula><location><page_3><loc_45><loc_50><loc_92><loc_52></location>〈 ˜ C 0 〉 = MC . (10)</formula>
<text><location><page_3><loc_8><loc_47><loc_92><loc_49></location>Writing it this way is somewhat analogous to the MASTER technique (Hivon et al. 2002) for CMB power spectrum estimation on the partial sky.</text>
<text><location><page_3><loc_10><loc_46><loc_40><loc_47></location>In the appendix we find that the matrix is</text>
<formula><location><page_3><loc_35><loc_41><loc_92><loc_45></location>M pq = δ pq -2 ∑ ij d ij ( θ p ) α i α j D (1) iq ∑ ij d ij ( θ p ) α i α j + D (2) q (A10)</formula>
<text><location><page_3><loc_8><loc_39><loc_30><loc_40></location>where the auxiliary operations</text>
<formula><location><page_3><loc_31><loc_34><loc_92><loc_38></location>D (1) iq = ∑ k α k d ik ( θ q ) ∑ k α k D (2) q = ∑ kl α l α k d kl ( θ q ) ( ∑ k α k ) 2 (A7, A9)</formula>
<text><location><page_3><loc_8><loc_30><loc_92><loc_34></location>are functions of the pixel weights. This matrix is composed of three terms. The first term is the identity matrix and the following two terms are responsible for the bias. The matrix costs O ( N 2 ) operations to compute, the same as the naive correlation function estimator.</text>
<section_header_level_1><location><page_3><loc_40><loc_27><loc_60><loc_28></location>2.2. Monte Carlo simulation</section_header_level_1>
<text><location><page_3><loc_8><loc_19><loc_92><loc_27></location>To test our expression for the bias terms, we performed Monte Carlo simulations of continuous, diffuse fields; later we will include shot noise. The survey size, roughly 7 ' × 8 ' , mimics an actual observation with Chandra. For the correlation function C ( θ ) in the simulation, we use a Gaussian function with correlation length (i.e. standard deviation) of 3 . 9 ' , significant compared to the size of the field. For weights we use the inverse of the exposure for a real set of observations. These downweight the edges of the observations compared to the center (and correspond to inverse-variance pixel weights in the Poisson-noise-dominated limit.)</text>
<text><location><page_3><loc_8><loc_12><loc_92><loc_19></location>Figure 1 shows the input correlation, and the ensemble average (and dispersion) of the naive estimates, which are biased. Compared to the input correlation, the ensemble average is offset and the shape differs. The bias terms capture this difference, but the bias terms depend on the input correlation function, and so when working with data are not directly available. We address this shortcoming in the next section. The matrix M for our example is depicted in Figure 2.</text>
<text><location><page_3><loc_8><loc_7><loc_92><loc_12></location>In the simulations shown, we generated the diffuse signal s as a Gaussian random field, but obtain the same results with a log-normal random field (constructed with the recipe from Carron & Neyrinck (2012) to keep the same mean and correlation function). The ensemble average and bias terms are the same in the Gaussian and non-Gaussian cases, however the non-Gaussianities substantially increase the dispersion of the naive estimates.</text>
<text><location><page_4><loc_51><loc_91><loc_52><loc_92></location>10</text>
<figure>
<location><page_4><loc_9><loc_65><loc_91><loc_92></location>
<caption>Fig. 2.Left: the matrix M which relates the true correlation function to the ensemble average of the naive estimate. The columns represent the input scale and the rows the output scale. The matrix is dimensionless. Right: Without the identity matrix, we have the biasing terms only.</caption>
</figure>
<section_header_level_1><location><page_4><loc_38><loc_56><loc_63><loc_58></location>2.3. Correcting the naive estimator</section_header_level_1>
<text><location><page_4><loc_10><loc_54><loc_90><loc_56></location>Once we have M , we can define a reconstructed correlation function ˜ C ( θ q ) as the solution to the linear equation</text>
<formula><location><page_4><loc_42><loc_50><loc_92><loc_53></location>˜ C 0 ( θ p ) = ∑ q M pq ˜ C ( θ q ) , (11)</formula>
<text><location><page_4><loc_8><loc_43><loc_92><loc_49></location>where the left-hand-side is the naive estimate we already obtained and the right-hand-side contains our reconstruction. Unfortunately this equation does not have a unique solution. Explicit computation in the appendix shows that M maps any constant offset to zero. Thus constant offsets to the correlation function are in the null space of the matrix. In particular this implies that M is not invertible, ruling out a straightforward solution to the linear equation. However, we can recover the true C ( θ ) in the ensemble average up to an unknown constant function.</text>
<text><location><page_4><loc_8><loc_40><loc_92><loc_43></location>Since we know this matrix has a non-empty null space, we analyze it by singular value decomposition, factoring it as</text>
<formula><location><page_4><loc_46><loc_39><loc_92><loc_40></location>M = UsV T (12)</formula>
<text><location><page_4><loc_8><loc_34><loc_92><loc_38></location>where U and V are orthogonal and s is diagonal and contains the singular values. The matrix has one singular value near zero, and the column of V that corresponds to the singular mode contains the constant function we identified previously as being in the null space.</text>
<text><location><page_4><loc_10><loc_33><loc_88><loc_34></location>The upshot of this discussion is that although M does not have an inverse, we can construct a pseudo-inverse</text>
<formula><location><page_4><loc_45><loc_30><loc_92><loc_32></location>M + = Vs + U T (13)</formula>
<text><location><page_4><loc_8><loc_26><loc_92><loc_29></location>where s + is a diagonal matrix constructed from the reciprocal of the diagonal of s except at the singular value where it is set to zero. Then the reconstructed correlation function</text>
<formula><location><page_4><loc_42><loc_23><loc_92><loc_26></location>˜ C ( θ p ) = ∑ q M + pq ˜ C 0 ( θ q ) (14)</formula>
<text><location><page_4><loc_8><loc_18><loc_92><loc_22></location>solves equation (11). This solution is not unique, however, since adding any constant function also yields a solution. This procedure chooses the solution which minimizes the squared norm of the reconstructed correlation function (e.g. Press et al. 1992)</text>
<formula><location><page_4><loc_46><loc_15><loc_92><loc_17></location>∑ p | ˜ C ( θ p ) | 2 . (15)</formula>
<text><location><page_4><loc_8><loc_10><loc_92><loc_13></location>Therefore, in the ensemble average, we can reconstruct the correlation matrix up to a constant offset factor, as shown in Fig. 3 for our Monte Carlo simulation. This shows how the incorrect shape of the ensemble average has been repaired in the reconstruction, except for residual fluctuations in the ensemble average.</text>
<text><location><page_4><loc_10><loc_8><loc_19><loc_9></location>Thus we have</text>
<formula><location><page_4><loc_41><loc_7><loc_92><loc_8></location>〈 ˜ C ( θ p ) 〉 = C ( θ p ) + const. (16)</formula>
<figure>
<location><page_5><loc_25><loc_62><loc_75><loc_92></location>
<caption>Fig. 3.The ensemble of 1000 realizations made with the input correlation shown in black yields the average naive correlation function shown in blue. Multiplying the ensemble average by M + , the pseudo-inverse of the biasing matrix, gives the reconstructed correlation function (in green), which has the same shape as the input spectrum, but has lost the information about the constant offset. It resembles the input spectrum after the input is offset to minimize the square norm.</caption>
</figure>
<text><location><page_5><loc_8><loc_53><loc_92><loc_55></location>where the constant is unknown. Our estimator is therefore biased. Note however, that the shape is not biased, as we can see from a comparison of the reconstructed correlation function at two separations:</text>
<formula><location><page_5><loc_26><loc_51><loc_92><loc_52></location>〈 ˜ C ( θ p ) -˜ C ( θ q ) 〉 = C ( θ p ) + const. -C ( θ q ) -const. = C ( θ p ) -C ( θ q ) (17)</formula>
<text><location><page_5><loc_8><loc_46><loc_92><loc_50></location>for any scales θ p and θ q accessible by the survey. Thus we can say that the shape information is preserved in an unbiased way. If we further have theoretical expectations or other constraints, these can help fix the offset for the correlation function.</text>
<section_header_level_1><location><page_5><loc_41><loc_44><loc_59><loc_45></location>3. POISSON SHOT NOISE</section_header_level_1>
<text><location><page_5><loc_8><loc_33><loc_92><loc_43></location>If the observations have significant shot noise from measuring discrete photons or objects, additional bias terms appear. We use a Poisson model (Peebles 1980, § 33) for our computations. Let N i be the count of events in pixel or cell i . This quantity is Poisson-distributed with a mean parameter λ i that is proportional to our diffuse signal. In our X-ray example, λ i = s i t i A , where s i is our diffuse signal from before, representing a photon rate per area, time t i is the duration of the pixel's exposure, and A is the pixel's collecting area. 4 Note λ i is a mean number of counts, and so is dimensionless. The Chandra observations we have studied have a large fraction of counts ( ∼ 85 percent) that are spurious events unrelated to the cosmic signal. We first derive the bias and corrections for the naive estimator neglecting these spurious counts, and then including them.</text>
<section_header_level_1><location><page_5><loc_39><loc_30><loc_61><loc_31></location>3.1. No spurious contamination</section_header_level_1>
<text><location><page_5><loc_10><loc_28><loc_81><loc_29></location>If all the counts are genuinely related to the cosmic signal, the observed rate ( R ) of signal events is</text>
<formula><location><page_5><loc_46><loc_26><loc_92><loc_27></location>R i = N i /t i A (18)</formula>
<text><location><page_5><loc_8><loc_24><loc_53><loc_25></location>which has the same units as s i . The ensemble average of R i is</text>
<formula><location><page_5><loc_40><loc_20><loc_92><loc_23></location>〈 R i 〉 = 〈 N i 〉 t i A = 〈 s i 〉 t i A t i A = µ. (19)</formula>
<text><location><page_5><loc_8><loc_18><loc_39><loc_19></location>We can estimate the mean of our rate map</text>
<formula><location><page_5><loc_45><loc_15><loc_92><loc_18></location>¯ R = ∑ i α i R i ∑ i α i (20)</formula>
<text><location><page_5><loc_8><loc_13><loc_68><loc_14></location>which is an unbiased estimate: 〈 ¯ R 〉 = µ . The fluctuation in the map's mean we call</text>
<formula><location><page_5><loc_46><loc_11><loc_92><loc_12></location>δ ¯ R = ¯ R -µ (21)</formula>
<text><location><page_6><loc_8><loc_91><loc_54><loc_92></location>which has 〈 δ ¯ R 〉 = 0. The covariance of the observed rate map is</text>
<formula><location><page_6><loc_39><loc_87><loc_92><loc_90></location>Cov( R i , R j ) = µ t i A δ ij + C ( θ ij ) . (22)</formula>
<text><location><page_6><loc_8><loc_82><loc_92><loc_86></location>This has an additional shot noise component compared to the covariance of the diffuse signal. The shot noise term can be avoided if the sums over pixel pairs exclude common pixels, at the cost of slightly more complicated pixel accounting. Here we include it in our computations for completeness.</text>
<text><location><page_6><loc_8><loc_77><loc_92><loc_82></location>Note that since C ( θ ) is a property of the diffuse field's probability distribution, in the discrete case it is not subject to any particular new constraints compared to the continuous case. The total number of counts (or objects) summed over all pixels is a random variable, and is not fixed (Peebles 1980, cf. § 31, 33 vs. § 32), and there is no specific constraint on the integral of C ( θ ).</text>
<text><location><page_6><loc_8><loc_69><loc_92><loc_77></location>The field s , representing a rate of counts or objects, must be non-negative, which implies that its statistics are nonGaussian. For the derivation of the estimator biases, this matters little because, as before, the higher-order moments do not appear in our argument. On the other hand, it may matter more when constructing simulations. A Gaussian random field can be a suitable approximation for s , but only if the particular realizations do not contain negative pixels, which would lead to negative (and thus ill-defined) expected counts. Otherwise, a log-normal random field, which is positive-definite and which we employ below, provides another useful candidate.</text>
<text><location><page_6><loc_10><loc_68><loc_54><loc_69></location>As before we make a naive estimate of the correlation function</text>
<formula><location><page_6><loc_35><loc_63><loc_92><loc_67></location>˜ C R 0 ( θ ) = ∑ ij d ij ( θ ) α i α j ( R i -¯ R )( R j -¯ R ) ∑ ij d ij ( θ ) α i α j . (23)</formula>
<text><location><page_6><loc_8><loc_60><loc_92><loc_62></location>In the appendix, we show that the ensemble average of the naive estimator for the discrete field can be written as a linear function of both the true mean and the true correlation function.</text>
<formula><location><page_6><loc_39><loc_56><loc_92><loc_59></location>〈 ˜ C R 0 ( θ p ) 〉 = v R p µ + ∑ q M pq C ( θ q ) (B7)</formula>
<text><location><page_6><loc_8><loc_54><loc_12><loc_55></location>where</text>
<formula><location><page_6><loc_33><loc_45><loc_92><loc_53></location>v R p = ∑ ij d ij ( θ p ) α i α j [(1 /t i A ) δ ij -2 E (1) i + E (2) ] ∑ ij d ij ( θ p ) α i α j E (1) i = α i /t i A ∑ α k E (2) = ∑ k α 2 k /t k A ( ∑ α k ) 2 (B3, B5, B6)</formula>
<formula><location><page_6><loc_42><loc_45><loc_61><loc_46></location>k k</formula>
<text><location><page_6><loc_8><loc_43><loc_34><loc_44></location>and M is the same matrix as before.</text>
<text><location><page_6><loc_10><loc_42><loc_46><loc_43></location>We can express this relationship in matrix form as</text>
<formula><location><page_6><loc_37><loc_38><loc_92><loc_41></location>( 〈 ¯ R 〉 〈 ˜ C R 0 〉 ) = ( 1 (0 . . . 0) v R M )( µ C ) (24)</formula>
<text><location><page_6><loc_8><loc_35><loc_46><loc_37></location>where we used that ¯ R is an unbiased estimator for µ .</text>
<text><location><page_6><loc_8><loc_33><loc_92><loc_35></location>Like M before, this larger square matrix is amenable to the construction of a pseudo-inverse by singular value decomposition. Analogous to equation (11), we can solve the linear equation</text>
<formula><location><page_6><loc_38><loc_28><loc_92><loc_32></location>( ¯ R ˜ C R 0 ) = ( 1 (0 . . . 0) v R M )( ˜ µ ˜ C ) (25)</formula>
<text><location><page_6><loc_8><loc_22><loc_92><loc_27></location>to reconstruct estimates (on the right-hand side) for the mean (this estimate is unbiased because it just takes the already unbiased ¯ R directly) and correlation function, with the same limitation as before: a constant function added to the correlation function is unconstrained. As before, the shape of the reconstructed correlation function in the ensemble average matches the true correlation function.</text>
<section_header_level_1><location><page_6><loc_38><loc_19><loc_62><loc_20></location>3.2. With spurious contamination</section_header_level_1>
<text><location><page_6><loc_8><loc_13><loc_92><loc_19></location>In the presence of an uncorrelated, but spatially varying, set of spurious counts, the analysis changes slightly, with the spurious counts contributing additional shot noise terms. In the case of Chandra data, these spurious counts are well-characterized in the sense that their mean rate is well-understood. However, counts cannot be classified as signal or spurious on an individual basis.</text>
<text><location><page_6><loc_8><loc_9><loc_92><loc_14></location>Now our counts include events from both the signal and the spurious set: N i = N s i + N sp i . Then the ensemble average photon count is 〈 N i 〉 = µt i A + λ sp i , where λ sp i is the known spurious mean count for each pixel. We redefine the signal rate map as</text>
<formula><location><page_6><loc_45><loc_6><loc_92><loc_9></location>R i = N i -λ sp i t i A (26)</formula>
<figure>
<location><page_7><loc_25><loc_62><loc_75><loc_92></location>
<caption>Fig. 4.Similar to figure 1, except including shot noise from signal photons and background events, based on 5000 log-normal random fields. The Poisson bias terms (dashed green and cyan) are very small except in the first bin, which contains common pixel pairs. Accounting for all bias terms, the average closely matches the input, including at the first bin. so that 〈 R i 〉 = µ . Defining the map mean as before yields 〈 ¯ R 〉 = µ and the fluctuation from the mean has average 〈 δ ¯ R 〉 = 0. From here the analysis proceeds much as before. Noting that</caption>
</figure>
<formula><location><page_7><loc_32><loc_51><loc_92><loc_53></location>〈 ( N i -λ sp i )( N j -λ sp j ) 〉 = 〈 N s i N s j 〉 +Cov( N sp i , N sp j ) (27)</formula>
<text><location><page_7><loc_8><loc_49><loc_21><loc_50></location>we can show that</text>
<formula><location><page_7><loc_35><loc_46><loc_92><loc_49></location>Cov( R i , R j ) = ( µ t i A + λ sp i t 2 i A 2 ) δ ij + C ( θ ij ) . (28)</formula>
<text><location><page_7><loc_8><loc_44><loc_65><loc_45></location>which includes an additional shot noise term compared to the similar eqn. (22).</text>
<text><location><page_7><loc_8><loc_40><loc_92><loc_44></location>This allows the ensemble average of the naive estimate to be written as the sum of the spurious-event-free naive estimate and additional shot-noise terms which depend on the known mean spurious rate, λ sp i . In the appendix we show that this is:</text>
<formula><location><page_7><loc_14><loc_36><loc_92><loc_40></location>〈 ˜ C R,sp 0 ( θ ) 〉 = 〈 ˜ C R 0 ( θ ) 〉 + ∑ ij d ij ( θ ) α i α j ( λ sp i /t 2 i A 2 ) δ ij -2 ( α i λ sp i / ( t 2 i A 2 ∑ k α k ) ) ∑ ij d ij ( θ ) α i α j + ∑ k α 2 k λ sp k /t 2 k A 2 ( ∑ k α k ) 2 (B10)</formula>
<text><location><page_7><loc_8><loc_33><loc_92><loc_35></location>Subtracting away these spurious terms, we can proceed to reconstruct the correlation function as described at the end of section 3.1.</text>
<section_header_level_1><location><page_7><loc_37><loc_30><loc_63><loc_31></location>3.3. Poisson Monte Carlo simulation</section_header_level_1>
<text><location><page_7><loc_8><loc_15><loc_92><loc_29></location>For a set of 5000 Monte Carlo realizations that include shot noise, we show in Fig. 4 the ensemble average and dispersion for the naive estimate, and also the analytic computation of the bias terms. The mean rate of photons, µ = 4 . 3 × 10 -9 counts/s/pixel, was chosen based on a real Chandra observation, and is low enough that a Gaussian random field with this correlation function will have negative pixels. For this reason we used a log-normal random field in this case, which accounts for much of the increase in the dispersion compared to Fig. 1. The shot-noise bias terms are large in the first bin of the correlation function, which contains the same-pixel pairs. Elsewhere, they are small because in this application, we have enough photons to make the shot noise contribution to δ ¯ R sub-dominant. The bias terms we computed account for the shot noise well. The dispersion due to shot noise is extreme at > 9 ' separations for two reasons: only the periphery of the map provides these separations, so there are few pixel pairs, and the effective exposure for pixels at the edge of the map is less, so there are many fewer photons than at the center of the field.</text>
<text><location><page_7><loc_8><loc_12><loc_92><loc_15></location>In Fig. 5, we demonstrate that the reconstruction of the correlation function by the singular value decomposition method works well to correct the shape distortion in the ensemble average.</text>
<section_header_level_1><location><page_7><loc_44><loc_10><loc_56><loc_11></location>4. CONCLUSIONS</section_header_level_1>
<text><location><page_7><loc_8><loc_7><loc_92><loc_9></location>We have developed an estimator for the correlation function which allows the shape, but not the overall offset, of the correlation function to be estimated properly in the ensemble average. If there are significant signal correlations</text>
<figure>
<location><page_8><loc_25><loc_62><loc_75><loc_92></location>
<caption>Fig. 5.Similar to figure 3, reconstructing the correlation function, except including shot noise from signal photons and background events.</caption>
</figure>
<text><location><page_8><loc_8><loc_51><loc_92><loc_57></location>on the largest scales that the survey region can probe, as with X-ray observations and some other astronomical data sets, the large sample variance will limit the utility of the correlation function shape measurement. However, when ˜ C ( θ )'s from multiple fields are averaged, we beat down the noise on the shape, while the average of unknown offsets simply yields a new unknown offset. Put another way, averaging improves our knowledge of the shape but does not worsen our lack of knowledge about the offset.</text>
<text><location><page_8><loc_8><loc_45><loc_92><loc_50></location>The estimators written here, although motivated by observations of the diffuse X-ray background, easily generalize to galaxy counts-in-cells (setting λ i = N i ∆Ω in section 3). The estimator can be trivially adapted for cross-correlations between fields, or extended from angular correlations in two dimensions to linear or time-series correlations in one dimension or spatial correlations in three dimensions.</text>
<text><location><page_8><loc_8><loc_37><loc_92><loc_45></location>These estimators may be usefully applied to any situation with correlations on the scale of the observed region. One example is the CMB, which in the ΛCDM model has significant correlations even between points on the sky separated by 180 · . However, estimates from the COBE and WMAP data (Hinshaw et al. 1996; Spergel et al. 2003; Copi et al. 2007, 2009) show surprisingly little correlations at scales larger than 60 · . These authors have used the biased, naive estimator (equation 7), but our preliminary tests on WMAP maps and the ΛCDM CMB correlation function indicate that the bias terms we have computed here are too small to account for this difference.</text>
<text><location><page_8><loc_8><loc_27><loc_92><loc_37></location>We have computed the variance of our estimates in Monte Carlo simulations, but not analytically, nor have we tried to find optimal weights to minimize the variance. When sample variance dominates the covariance for the correlation function, it is unlikely that the optimal weighting can be done on a pixel-by-pixel basis, and instead pixel pairs will need to be jointly weighted by the inverse covariance for that pair, accounting for the signal covariance and the signal and spurious shot noise. Compared to the real-space estimators we examine here, Efstathiou (2004) and Efstathiou et al. (2010) argue that a correlation function estimate built from a maximum likelihood estimate of the harmonic space power spectrum will have lower variance, because it effectively gives pixel pairs closer-to-optimal weights in this way. This task we leave for future work.</text>
<section_header_level_1><location><page_8><loc_42><loc_24><loc_58><loc_25></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_8><loc_8><loc_19><loc_92><loc_24></location>We thank Enzo Branchini and the anonymous referee for useful comments on earlier versions of this work. We thank Gabriela Degwitz for help in the preparation of this manuscript. This work was supported by NASA through the Smithsonian Astrophysical Observatory (SAO), award G0112177X, and NASA award NNX11AF80G. KMH also receives support from NASA-JPL subcontract 1363745.</text>
<section_header_level_1><location><page_8><loc_45><loc_16><loc_55><loc_17></location>REFERENCES</section_header_level_1>
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<text><location><page_8><loc_52><loc_14><loc_92><loc_15></location>Efstathiou, G., Ma, Y.-Z., & Hanson, D. 2010, MNRAS, 407, 2530</text>
<text><location><page_8><loc_52><loc_10><loc_90><loc_14></location>Hamilton, A. J. S. 1993, ApJ, 417, 19 Hewett, P. C. 1982, MNRAS, 201, 867 Hinshaw, G., Branday, A. J., Bennett, C. L., Gorski, K. M., Kogut, A., Lineweaver, C. H., Smoot, G. F., & Wright, E. L.</text>
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<text><location><page_9><loc_8><loc_91><loc_47><loc_92></location>Hivon, E., G'orski, K. M., Netterfield, C. B., Crill, B. P., Prunet,</text>
<text><location><page_9><loc_8><loc_82><loc_47><loc_90></location>S., & Hansen, F. 2002, ApJ, 567, 2 Kerscher, M. 1999, A&A, 343, 333 Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64 Peebles, P. J. E. 1980, The large-scale structure of the universe Percival, W. J., et al. 2001, MNRAS, 327, 1297 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific computing</text>
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<section_header_level_1><location><page_9><loc_46><loc_80><loc_54><loc_80></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_9><loc_38><loc_78><loc_62><loc_79></location>BIAS TERMS: CONTINUOUS CASE</section_header_level_1>
<text><location><page_9><loc_8><loc_74><loc_92><loc_77></location>In this appendix we compute the bias terms for the continuous signal. Rewriting ˜ µ = µ + δ ˜ µ , the ensemble average of the numerator of the naive estimator (7) is</text>
<formula><location><page_9><loc_33><loc_71><loc_92><loc_73></location>∑ ij d ij ( θ p ) α i α j [ C ( θ ij ) -〈 s i δ ˜ µ 〉 - 〈 s j δ ˜ µ 〉 + 〈 δ ˜ µ 2 〉 ] (A1)</formula>
<text><location><page_9><loc_8><loc_68><loc_87><loc_70></location>where we have used 〈 δ ˜ µ 〉 = 0. Further we can use the sum's symmetry between i and j to show that it equals</text>
<formula><location><page_9><loc_35><loc_64><loc_92><loc_67></location>∑ ij d ij ( θ p ) α i α j [ C ( θ ij ) -2 〈 s i δ ˜ µ 〉 + 〈 δ ˜ µ 2 〉 ] . (A2)</formula>
<text><location><page_9><loc_8><loc_59><loc_92><loc_63></location>If we had used the true mean, only the C ( θ ij ) term would be present, and we could pull it out of the sum as C ( θ p ). The sum over weights would cancel the denominator, and we would indeed find that 〈 ˜ C 0 ( θ p ) 〉 = C ( θ p ). This is not the case here because of the middle and last terms in the brackets, which are responsible for the bias.</text>
<text><location><page_9><loc_8><loc_56><loc_92><loc_59></location>We can compute both bias terms from the field's correlation function. We call the first bias term B (1) i because it is first order in the mean estimation error δ ˜ µ , and compute it as</text>
<formula><location><page_9><loc_38><loc_46><loc_62><loc_55></location>B (1) i = 〈 s i δ ˜ µ 〉 = 〈 s i (˜ µ -µ ) 〉 = ∑ k α k 〈 s i s k 〉 ∑ k α k -µ 2 = ∑ k α k C ( θ ik ) ∑ k α k .</formula>
<text><location><page_9><loc_8><loc_44><loc_86><loc_46></location>The second bias term, B (2) , which is second order in the mean's error, has no dependence on the pixel index.</text>
<formula><location><page_9><loc_35><loc_31><loc_65><loc_43></location>B (2) = 〈 δ ˜ µ 2 〉 = 〈 (˜ µ -µ ) 2 〉 = 〈∑ k α k s k ∑ k α k ∑ l α l s l ∑ l α l 〉 -µ 2 = ∑ kl α l α k 〈 s k s l 〉 ( ∑ k α k ) 2 -µ 2 = ∑ kl α l α k C ( θ kl ) ( ∑ k α k ) 2</formula>
<text><location><page_9><loc_8><loc_27><loc_92><loc_30></location>Because B (2) does not depend on the pixel index, this term too can slip outside the sum over pixel pairs in eqn. (A2). Therefore, finally, we have</text>
<formula><location><page_9><loc_33><loc_23><loc_92><loc_27></location>〈 ˜ C 0 ( θ p ) 〉 = C ( θ p ) -2 ∑ ij d ij ( θ p ) α i α j B (1) i ∑ ij d ij ( θ p ) α i α j + B (2) (A3)</formula>
<text><location><page_9><loc_8><loc_20><loc_92><loc_23></location>which states the bias in our estimate explicitly. Each bias term costs O ( N 2 ) operations to compute, the same as the correlation function.</text>
<text><location><page_9><loc_8><loc_10><loc_92><loc_20></location>Note that our naive estimator has a peculiar reaction to correlation functions such as C ( θ ) = c for all separations sampled by our survey. 5 In this case 〈 ˜ C 0 ( θ ) 〉 = 0, which we show by examining the bias terms. If C ( θ ) = c , then the constant can be set outside the sums, which cancel the denominators. Therefore bias factors B (1) i = c and B (2) = c , and the middle term of eqn. (A3) is -2 c . Therefore 〈 ˜ C 0 ( θ ) 〉 = c -2 c + c = 0. Thus, if the naive estimator is viewed as a linear operator on the input correlation function, constant functions are in the null space of the operator, since any constant maps to zero. Moreover, the naive estimator loses the information about any constant baseline in the correlation function, although the information about the shape is preserved.</text>
<text><location><page_10><loc_8><loc_88><loc_92><loc_92></location>The bias terms depend on C ( θ ) only on scales accessible by the survey region, and not on any larger scales. This permits an (imperfect) reconstruction of the correlation function. To proceed, we can rewrite eqn. (A3) as a matrix multiplication:</text>
<formula><location><page_10><loc_41><loc_85><loc_92><loc_88></location>〈 ˜ C 0 ( θ p ) 〉 = ∑ q M pq C ( θ q ) (A4)</formula>
<text><location><page_10><loc_66><loc_83><loc_67><loc_84></location>.</text>
<text><location><page_10><loc_8><loc_81><loc_66><loc_84></location>where the sum is over the angular bins. Then we set about finding the matrix M To write down M , we make use of the relationship</text>
<formula><location><page_10><loc_41><loc_77><loc_92><loc_80></location>C ( θ ik ) = ∑ q d ik ( θ q ) C ( θ q ) . (A5)</formula>
<text><location><page_10><loc_8><loc_74><loc_92><loc_76></location>Note that this sum is over angular bin, not pixel. We rewrite the bias terms more explicitly as linear operations on the vector C ( θ q ). The first bias term is</text>
<formula><location><page_10><loc_34><loc_69><loc_92><loc_73></location>B (1) i = ∑ kq α k d ik ( θ q ) C ( θ q ) ∑ k α k = ∑ q D (1) iq C ( θ q ) , (A6)</formula>
<text><location><page_10><loc_8><loc_67><loc_20><loc_68></location>where we define</text>
<formula><location><page_10><loc_42><loc_64><loc_92><loc_67></location>D (1) iq = ∑ k α k d ik ( θ q ) ∑ k α k . (A7)</formula>
<text><location><page_10><loc_8><loc_62><loc_70><loc_63></location>Note that the first index refers to pixel and the second to bin. The second bias term is</text>
<formula><location><page_10><loc_33><loc_58><loc_92><loc_61></location>B (2) = ∑ kl α l α k d kl ( θ q ) C ( θ q ) ( ∑ k α k ) 2 = ∑ q D (2) q C ( θ q ) , (A8)</formula>
<text><location><page_10><loc_8><loc_56><loc_20><loc_57></location>where we define</text>
<formula><location><page_10><loc_41><loc_52><loc_92><loc_56></location>D (2) q = ∑ kl α l α k d kl ( θ q ) ( ∑ k α k ) 2 . (A9)</formula>
<text><location><page_10><loc_8><loc_50><loc_39><loc_51></location>Since C ( θ p ) = ∑ q δ pq C ( θ q ), we finally have</text>
<formula><location><page_10><loc_35><loc_45><loc_92><loc_49></location>M pq = δ pq -2 ∑ ij d ij ( θ p ) α i α j D (1) iq ∑ ij d ij ( θ p ) α i α j + D (2) q (A10)</formula>
<text><location><page_10><loc_8><loc_39><loc_92><loc_44></location>To sum up, in this appendix we have: (1) computed the bias of the naive correlation function estimator; (2) shown that the ensemble average of the naive estimate is a linear operation acting upon the true correlation; (3) computed that linear operator in terms of the pixel weights; and (4) shown that constant offsets are in the null space of that operator. The method to estimate the shape of the correlation function in section 2.3 depends on these results.</text>
<section_header_level_1><location><page_10><loc_39><loc_37><loc_61><loc_38></location>BIAS TERMS: DISCRETE CASE</section_header_level_1>
<text><location><page_10><loc_41><loc_35><loc_60><loc_36></location>No spurious contamination</text>
<text><location><page_10><loc_8><loc_32><loc_92><loc_34></location>To compute the bias for the discrete case, we write the numerator of the naive estimator (23) in terms of the fluctuation of the mean δ ¯ R and take the ensemble average:</text>
<formula><location><page_10><loc_28><loc_19><loc_92><loc_31></location>〈 ∑ ij d ij ( θ ) α i α j ( R i -µ -δ ¯ R )( R j -µ -δ ¯ R ) 〉 = 〈 ∑ ij d ij ( θ ) α i α j [ ( R i -µ )( R j -µ ) -2( R i -µ ) δ ¯ R +( δ ¯ R ) 2 ] 〉 = ∑ ij d ij ( θ ) α i α j [ ( µ/t i A ) δ ij + C ( θ ij ) -2 〈 R i δ ¯ R 〉 + 〈 ( δ ¯ R ) 2 〉 ] (B1)</formula>
<text><location><page_10><loc_8><loc_17><loc_82><loc_18></location>Now we examine the last two terms, which are analogous to the bias terms for the diffuse signal. First,</text>
<formula><location><page_10><loc_34><loc_7><loc_92><loc_16></location>B R (1) i = 〈 R i δ ¯ R 〉 = ∑ k α k 〈 R i R k 〉 ∑ k α k -µ 2 = ∑ k α k [( µ/t i A ) δ ik + C ( θ ik )] ∑ k α k = E (1) i µ + B (1) i (B2)</formula>
<text><location><page_11><loc_8><loc_91><loc_20><loc_92></location>where we define</text>
<text><location><page_11><loc_8><loc_72><loc_20><loc_73></location>where we define</text>
<formula><location><page_11><loc_45><loc_87><loc_92><loc_91></location>E (1) i = α i /t i A ∑ k α k . (B3)</formula>
<text><location><page_11><loc_8><loc_84><loc_92><loc_87></location>This shows that for a signal of discrete photons, this bias term can be written as a sum of a new shot noise term and the old B (1) bias term from the diffuse case.</text>
<text><location><page_11><loc_10><loc_83><loc_22><loc_84></location>The final term is</text>
<formula><location><page_11><loc_33><loc_74><loc_92><loc_82></location>B R (2) = 〈 ( δ ¯ R ) 2 〉 = ∑ kl α k 〈 R k R l 〉 ∑ kl α k α l -µ 2 = ∑ kl α k α l [( µ/t i A ) δ kl + C ( θ kl )] ( ∑ k α k ) 2 = E (2) µ + B (2) (B4)</formula>
<formula><location><page_11><loc_43><loc_68><loc_92><loc_72></location>E (2) = ∑ k α 2 k /t k A ( ∑ k α k ) 2 . (B5)</formula>
<text><location><page_11><loc_8><loc_65><loc_92><loc_68></location>Again this bias term has a new, shot-noise component added to the old bias term from the diffuse signal. These shot noise bias terms cannot be avoided by excluding i = j from the naive estimator's pixel sums.</text>
<text><location><page_11><loc_8><loc_62><loc_92><loc_65></location>Each of the new shot noise terms is proportional to µ . We can gather those terms together and notice that the remaining terms are just those which appear on the right side of eqn. (A3), so that:</text>
<formula><location><page_11><loc_25><loc_58><loc_92><loc_61></location>〈 ˜ C R 0 ( θ p ) 〉 = [ ∑ ij d ij ( θ p ) α i α j [(1 /t i A ) δ ij -2 E (1) i + E (2) ] ∑ ij d ij ( θ p ) α i α j ] µ + 〈 ˜ C 0 ( θ p ) 〉 (B6)</formula>
<text><location><page_11><loc_8><loc_53><loc_92><loc_57></location>Therefore the ensemble average of the naive estimate for the discrete signal equals the ensemble average of the naive estimate for the diffuse signal plus an additional shot noise bias term which is proportional to the mean of the diffuse field.</text>
<text><location><page_11><loc_8><loc_50><loc_92><loc_53></location>Thus the ensemble average of the naive estimator for the discrete field can be written as a linear function of the true mean and correlation function.</text>
<formula><location><page_11><loc_38><loc_48><loc_92><loc_50></location>〈 ˜ C R 0 ( θ p ) 〉 = v R p µ + ∑ q M pq C ( θ q ) . (B7)</formula>
<text><location><page_11><loc_8><loc_45><loc_83><loc_46></location>This formulation leads to the reconstruction method for the correlation function discussed in section 3.1.</text>
<section_header_level_1><location><page_11><loc_38><loc_43><loc_62><loc_44></location>Including spurious contamination</section_header_level_1>
<text><location><page_11><loc_8><loc_40><loc_92><loc_42></location>Starting from equations (26) and (28), we find that the two bias terms also have additional shot noise components due to the spurious signal. Instead of eqn. (B2) we have</text>
<formula><location><page_11><loc_33><loc_35><loc_92><loc_39></location>B R (1) i = 〈 R i δ ¯ R 〉 = E (1) i µ + B (1) i + α i λ sp i t 2 i A 2 ∑ k α k , (B8)</formula>
<text><location><page_11><loc_8><loc_33><loc_32><loc_35></location>and instead of eqn. (B4) we have</text>
<formula><location><page_11><loc_33><loc_29><loc_92><loc_33></location>B R (2) = 〈 ( δ ¯ R ) 2 〉 = E (2) µ + B (2) + ∑ k α 2 k λ sp k /t 2 k A 2 ( ∑ k α k ) 2 . (B9)</formula>
<text><location><page_11><loc_8><loc_26><loc_92><loc_28></location>Thus there are additional terms which can be subtracted away to yield the naive estimator in the contamination-free case.</text>
<formula><location><page_11><loc_14><loc_21><loc_92><loc_25></location>〈 ˜ C R,sp 0 ( θ ) 〉 = 〈 ˜ C R 0 ( θ ) 〉 + ∑ ij d ij ( θ ) α i α j ( λ sp i /t 2 i A 2 ) δ ij -2 ( α i λ sp i / ( t 2 i A 2 ∑ k α k ) ) ∑ ij d ij ( θ ) α i α j + ∑ k α 2 k λ sp k /t 2 k A 2 ( ∑ k α k ) 2 (B10)</formula>
</document>
[
{
"title": "ABSTRACT",
"content": "We develop an estimator for the correlation function which, in the ensemble average, returns the shape of the correlation function, even for signals that have significant correlations on the scale of the survey region. Our estimator is general and works in any number of dimensions. We develop versions of the estimator for both diffuse and discrete signals. As an application, we examine Monte Carlo simulations of X-ray background measurements. These include a realistic, spatially-inhomogeneous population of spurious detector events. We discuss applying the estimator to the averaging of correlation functions evaluated on several small fields, and to other cosmological applications. Subject headings: cosmology: theory-methods: numerical-methods: data analysis-methods: statistical-X-rays: diffuse background-galaxies: clustering",
"pages": [
1
]
},
{
"title": "RECONSTRUCTING THE SHAPE OF THE CORRELATION FUNCTION",
"content": "Department of Physics, University of Miami, Coral Gables, Florida 33146 K. M. Huffenberger, M. Galeazzi, E. Ursino Draft version September 1, 2021",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Two-point statistics encode valuable information about the fields that they describe, such as the cosmological matter density traced by galaxies or the intensity of radiation in backgrounds like the Cosmic Microwave Background (CMB), the Cosmic Infrared Background (CIB), or the Diffuse X-ray Background (DXB). For discrete objects, the two-point, dimensionless correlation function can be defined in terms of the probability of finding a pair of objects in two small cells, with sizes δ Ω 1 and δ Ω 2 , separated by θ 12 (Peebles 1980, § 31, 45): where N is the mean density of sources. For diffuse fields, the equivalent definition for a signal s with mean 〈 s 〉 = µ is where here and throughout 〈 . . . 〉 denotes the ensemble average. 1 We denote the covariance of s as C ( θ ) = µ 2 w ( θ ), which we also refer to as the (dimensionful) correlation function. This work mostly deals with the dimensionful correlation function and addresses the bias in its estimation. With similar expressions, we can define correlation functions in any number of dimensions, replacing the angular separation θ by a linear separation or time interval or whatever is appropriate. The estimation of the correlation function has been studied extensively in the literature. For galaxy clustering, Hewett (1982), Davis & Peebles (1983), and Hamilton (1993) suggest different Monte Carlo estimators, but the most common estimator now in use was advocated by Landy & Szalay (1993), which employs the data in concert with a synthetic, random catalog. Their estimator combines counts of objects pairs within and between the data and random catalogs. This estimator is biased, but for surveys where the correlation length of the objects is much smaller than the survey area, the bias is small (Bernstein 1994). Such is the case for modern galaxy surveys like 2dF and SDSS (Percival et al. 2001; York et al. 2000). However, the bias can become significant when structures approach the size of the survey (Kerscher 1999). This bias can be corrected (e.g. Scranton et al. 2002), but the correction depends on same correlation function that is being estimated. For diffuse signals like the CMB, where using the dimensionful correlation function is more common, a typical estimator looks like (Hinshaw et al. 1996; Copi et al. 2007): where α i are the weights applied to the pixels or cells (for the purpose of downweighting noisy regions), ˜ µ is an estimate of the mean, and the sum over ij refers to pixels separated by θ . These estimators suffer the same biases on small fields. In this paper we introduce a new method to address the biases in these above estimators. Our estimator is also biased, but biased in a particularly convenient way: regardless of the survey geometry or weighting, the shape of the correlation function is preserved on average, and only information about a constant offset is lost. This permits the straightforward averaging of correlation functions from several small patches across the sky. Building upon the estimator in eqn. (3), we develop classes of estimators for both diffuse signals and discrete objects. This work was prompted by our group's efforts to compute correlation function from observations of the diffuse X-ray background. The signal in that case comes from a diffuse, gaseous source, but arrives and is recorded as individual, discrete X-ray photons, and so can be analyzed with either scheme above. Indeed, for simulations of diffuse X-ray emission from the WHIM, Ursino et al. (2011) found that the Landy & Szalay (1993) estimator gave roughly equivalent results to an estimator of the type in eqn. (3). We focused on the correlation function biases because the angular correlation scale of this gas (several arcminutes) is substantial compared to the field-of-view ( ∼ 8 arcminutes) for single-field observations with the Chandra X-ray Observatory. The paper is organized as follows. In section 2 we find the bias for the naive estimator (eqn. 3), verifying our result with Monte Carlo simulations, and introduce a method for correcting it up to a constant offset. In section 3 we extend this estimate to Poisson-distributed counts, allowing for the possibility of a spatially-varying set of spurious detector events. Finally, we summarize our conclusions in section 4. An appendix contains the detailed derivations of the bias terms.",
"pages": [
1,
2
]
},
{
"title": "2. CORRELATION FUNCTION ESTIMATOR BIAS",
"content": "We begin by defining our signals. Let s i represent a pixelized, diffuse signal that is statistically homogeneous and isotropic. Let it be described by a mean and covariance as follows: where θ ij represents the separation between cells i and j . In our derivations we use C ( θ ) rather that w ( θ ) because the examination of biases is convenient; C ( θ ) also makes sense for diffuse fields where µ = 0. No other special properties of s are required, except that the covariance matrix is positive semi-definite: 0 ≤ | C ( θ ) | ≤ C (0). In particular, the signal need not be a Gaussian random field: we could define higher-order moments without disrupting our following arguments. Note that by this definition, the correlation function C ( θ ) is a property of the probability distribution for our signal s , and it is not a descriptive statistic. With a set of weights on the pixels, α i , we can compute a weighted average to estimate the mean, where the sum is over all pixels. These weights could be chosen to be uniform or to suppress noisy or polluted portions of the measurement. Throughout we mark estimated quantities with tildes. This mean estimate is unbiased, 〈 ˜ µ 〉 = µ . Additionally we define the deviation between the true mean and the estimated mean by with 〈 δ ˜ µ 〉 = 0.",
"pages": [
2
]
},
{
"title": "2.1. Naive correlation function estimator",
"content": "Based on the estimated mean, we make an initial estimate of the correlation function in a bin labeled by θ p , which we call the naive estimator: This is just a more explicit rewriting of eqn. (3). The function chooses the separation bin to which the pixel sum contributes. 2 Evaluation of the estimator costs O ( N 2 ) operations over N pixels. If the true mean µ replaces the estimated mean ˜ µ in eqn. (7), then this correlation function estimate is unbiased, 3 and we find 〈 ˜ C 0 ( θ p ) 〉 = C ( θ p ). However, since we do not know the true mean, our estimate will be biased, because we are forced to use the same (correlated) set of pixels to compute the mean and the correlation function. The smaller the survey compared to the correlation length of the signal, the worse this bias-the 'integral constraint'-becomes. (See Hamilton (1993) for further discussion of bias due to the mean error and other approaches to avoid it.) In the appendix, we compute the bias explicitly. We further show that the ensemble average of the naive, biased estimator may be cast as a linear operation applied to the true correlation function: or as a matrix equation, Writing it this way is somewhat analogous to the MASTER technique (Hivon et al. 2002) for CMB power spectrum estimation on the partial sky. In the appendix we find that the matrix is where the auxiliary operations are functions of the pixel weights. This matrix is composed of three terms. The first term is the identity matrix and the following two terms are responsible for the bias. The matrix costs O ( N 2 ) operations to compute, the same as the naive correlation function estimator.",
"pages": [
2,
3
]
},
{
"title": "2.2. Monte Carlo simulation",
"content": "To test our expression for the bias terms, we performed Monte Carlo simulations of continuous, diffuse fields; later we will include shot noise. The survey size, roughly 7 ' × 8 ' , mimics an actual observation with Chandra. For the correlation function C ( θ ) in the simulation, we use a Gaussian function with correlation length (i.e. standard deviation) of 3 . 9 ' , significant compared to the size of the field. For weights we use the inverse of the exposure for a real set of observations. These downweight the edges of the observations compared to the center (and correspond to inverse-variance pixel weights in the Poisson-noise-dominated limit.) Figure 1 shows the input correlation, and the ensemble average (and dispersion) of the naive estimates, which are biased. Compared to the input correlation, the ensemble average is offset and the shape differs. The bias terms capture this difference, but the bias terms depend on the input correlation function, and so when working with data are not directly available. We address this shortcoming in the next section. The matrix M for our example is depicted in Figure 2. In the simulations shown, we generated the diffuse signal s as a Gaussian random field, but obtain the same results with a log-normal random field (constructed with the recipe from Carron & Neyrinck (2012) to keep the same mean and correlation function). The ensemble average and bias terms are the same in the Gaussian and non-Gaussian cases, however the non-Gaussianities substantially increase the dispersion of the naive estimates. 10",
"pages": [
3,
4
]
},
{
"title": "2.3. Correcting the naive estimator",
"content": "Once we have M , we can define a reconstructed correlation function ˜ C ( θ q ) as the solution to the linear equation where the left-hand-side is the naive estimate we already obtained and the right-hand-side contains our reconstruction. Unfortunately this equation does not have a unique solution. Explicit computation in the appendix shows that M maps any constant offset to zero. Thus constant offsets to the correlation function are in the null space of the matrix. In particular this implies that M is not invertible, ruling out a straightforward solution to the linear equation. However, we can recover the true C ( θ ) in the ensemble average up to an unknown constant function. Since we know this matrix has a non-empty null space, we analyze it by singular value decomposition, factoring it as where U and V are orthogonal and s is diagonal and contains the singular values. The matrix has one singular value near zero, and the column of V that corresponds to the singular mode contains the constant function we identified previously as being in the null space. The upshot of this discussion is that although M does not have an inverse, we can construct a pseudo-inverse where s + is a diagonal matrix constructed from the reciprocal of the diagonal of s except at the singular value where it is set to zero. Then the reconstructed correlation function solves equation (11). This solution is not unique, however, since adding any constant function also yields a solution. This procedure chooses the solution which minimizes the squared norm of the reconstructed correlation function (e.g. Press et al. 1992) Therefore, in the ensemble average, we can reconstruct the correlation matrix up to a constant offset factor, as shown in Fig. 3 for our Monte Carlo simulation. This shows how the incorrect shape of the ensemble average has been repaired in the reconstruction, except for residual fluctuations in the ensemble average. Thus we have where the constant is unknown. Our estimator is therefore biased. Note however, that the shape is not biased, as we can see from a comparison of the reconstructed correlation function at two separations: for any scales θ p and θ q accessible by the survey. Thus we can say that the shape information is preserved in an unbiased way. If we further have theoretical expectations or other constraints, these can help fix the offset for the correlation function.",
"pages": [
4,
5
]
},
{
"title": "3. POISSON SHOT NOISE",
"content": "If the observations have significant shot noise from measuring discrete photons or objects, additional bias terms appear. We use a Poisson model (Peebles 1980, § 33) for our computations. Let N i be the count of events in pixel or cell i . This quantity is Poisson-distributed with a mean parameter λ i that is proportional to our diffuse signal. In our X-ray example, λ i = s i t i A , where s i is our diffuse signal from before, representing a photon rate per area, time t i is the duration of the pixel's exposure, and A is the pixel's collecting area. 4 Note λ i is a mean number of counts, and so is dimensionless. The Chandra observations we have studied have a large fraction of counts ( ∼ 85 percent) that are spurious events unrelated to the cosmic signal. We first derive the bias and corrections for the naive estimator neglecting these spurious counts, and then including them.",
"pages": [
5
]
},
{
"title": "3.1. No spurious contamination",
"content": "If all the counts are genuinely related to the cosmic signal, the observed rate ( R ) of signal events is which has the same units as s i . The ensemble average of R i is We can estimate the mean of our rate map which is an unbiased estimate: 〈 ¯ R 〉 = µ . The fluctuation in the map's mean we call which has 〈 δ ¯ R 〉 = 0. The covariance of the observed rate map is This has an additional shot noise component compared to the covariance of the diffuse signal. The shot noise term can be avoided if the sums over pixel pairs exclude common pixels, at the cost of slightly more complicated pixel accounting. Here we include it in our computations for completeness. Note that since C ( θ ) is a property of the diffuse field's probability distribution, in the discrete case it is not subject to any particular new constraints compared to the continuous case. The total number of counts (or objects) summed over all pixels is a random variable, and is not fixed (Peebles 1980, cf. § 31, 33 vs. § 32), and there is no specific constraint on the integral of C ( θ ). The field s , representing a rate of counts or objects, must be non-negative, which implies that its statistics are nonGaussian. For the derivation of the estimator biases, this matters little because, as before, the higher-order moments do not appear in our argument. On the other hand, it may matter more when constructing simulations. A Gaussian random field can be a suitable approximation for s , but only if the particular realizations do not contain negative pixels, which would lead to negative (and thus ill-defined) expected counts. Otherwise, a log-normal random field, which is positive-definite and which we employ below, provides another useful candidate. As before we make a naive estimate of the correlation function In the appendix, we show that the ensemble average of the naive estimator for the discrete field can be written as a linear function of both the true mean and the true correlation function. where and M is the same matrix as before. We can express this relationship in matrix form as where we used that ¯ R is an unbiased estimator for µ . Like M before, this larger square matrix is amenable to the construction of a pseudo-inverse by singular value decomposition. Analogous to equation (11), we can solve the linear equation to reconstruct estimates (on the right-hand side) for the mean (this estimate is unbiased because it just takes the already unbiased ¯ R directly) and correlation function, with the same limitation as before: a constant function added to the correlation function is unconstrained. As before, the shape of the reconstructed correlation function in the ensemble average matches the true correlation function.",
"pages": [
5,
6
]
},
{
"title": "3.2. With spurious contamination",
"content": "In the presence of an uncorrelated, but spatially varying, set of spurious counts, the analysis changes slightly, with the spurious counts contributing additional shot noise terms. In the case of Chandra data, these spurious counts are well-characterized in the sense that their mean rate is well-understood. However, counts cannot be classified as signal or spurious on an individual basis. Now our counts include events from both the signal and the spurious set: N i = N s i + N sp i . Then the ensemble average photon count is 〈 N i 〉 = µt i A + λ sp i , where λ sp i is the known spurious mean count for each pixel. We redefine the signal rate map as we can show that which includes an additional shot noise term compared to the similar eqn. (22). This allows the ensemble average of the naive estimate to be written as the sum of the spurious-event-free naive estimate and additional shot-noise terms which depend on the known mean spurious rate, λ sp i . In the appendix we show that this is: Subtracting away these spurious terms, we can proceed to reconstruct the correlation function as described at the end of section 3.1.",
"pages": [
6,
7
]
},
{
"title": "3.3. Poisson Monte Carlo simulation",
"content": "For a set of 5000 Monte Carlo realizations that include shot noise, we show in Fig. 4 the ensemble average and dispersion for the naive estimate, and also the analytic computation of the bias terms. The mean rate of photons, µ = 4 . 3 × 10 -9 counts/s/pixel, was chosen based on a real Chandra observation, and is low enough that a Gaussian random field with this correlation function will have negative pixels. For this reason we used a log-normal random field in this case, which accounts for much of the increase in the dispersion compared to Fig. 1. The shot-noise bias terms are large in the first bin of the correlation function, which contains the same-pixel pairs. Elsewhere, they are small because in this application, we have enough photons to make the shot noise contribution to δ ¯ R sub-dominant. The bias terms we computed account for the shot noise well. The dispersion due to shot noise is extreme at > 9 ' separations for two reasons: only the periphery of the map provides these separations, so there are few pixel pairs, and the effective exposure for pixels at the edge of the map is less, so there are many fewer photons than at the center of the field. In Fig. 5, we demonstrate that the reconstruction of the correlation function by the singular value decomposition method works well to correct the shape distortion in the ensemble average.",
"pages": [
7
]
},
{
"title": "4. CONCLUSIONS",
"content": "We have developed an estimator for the correlation function which allows the shape, but not the overall offset, of the correlation function to be estimated properly in the ensemble average. If there are significant signal correlations on the largest scales that the survey region can probe, as with X-ray observations and some other astronomical data sets, the large sample variance will limit the utility of the correlation function shape measurement. However, when ˜ C ( θ )'s from multiple fields are averaged, we beat down the noise on the shape, while the average of unknown offsets simply yields a new unknown offset. Put another way, averaging improves our knowledge of the shape but does not worsen our lack of knowledge about the offset. The estimators written here, although motivated by observations of the diffuse X-ray background, easily generalize to galaxy counts-in-cells (setting λ i = N i ∆Ω in section 3). The estimator can be trivially adapted for cross-correlations between fields, or extended from angular correlations in two dimensions to linear or time-series correlations in one dimension or spatial correlations in three dimensions. These estimators may be usefully applied to any situation with correlations on the scale of the observed region. One example is the CMB, which in the ΛCDM model has significant correlations even between points on the sky separated by 180 · . However, estimates from the COBE and WMAP data (Hinshaw et al. 1996; Spergel et al. 2003; Copi et al. 2007, 2009) show surprisingly little correlations at scales larger than 60 · . These authors have used the biased, naive estimator (equation 7), but our preliminary tests on WMAP maps and the ΛCDM CMB correlation function indicate that the bias terms we have computed here are too small to account for this difference. We have computed the variance of our estimates in Monte Carlo simulations, but not analytically, nor have we tried to find optimal weights to minimize the variance. When sample variance dominates the covariance for the correlation function, it is unlikely that the optimal weighting can be done on a pixel-by-pixel basis, and instead pixel pairs will need to be jointly weighted by the inverse covariance for that pair, accounting for the signal covariance and the signal and spurious shot noise. Compared to the real-space estimators we examine here, Efstathiou (2004) and Efstathiou et al. (2010) argue that a correlation function estimate built from a maximum likelihood estimate of the harmonic space power spectrum will have lower variance, because it effectively gives pixel pairs closer-to-optimal weights in this way. This task we leave for future work.",
"pages": [
7,
8
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "We thank Enzo Branchini and the anonymous referee for useful comments on earlier versions of this work. We thank Gabriela Degwitz for help in the preparation of this manuscript. This work was supported by NASA through the Smithsonian Astrophysical Observatory (SAO), award G0112177X, and NASA award NNX11AF80G. KMH also receives support from NASA-JPL subcontract 1363745.",
"pages": [
8
]
},
{
"title": "REFERENCES",
"content": "Bernstein, G. M. 1994, ApJ, 424, 569 Carron, J., & Neyrinck, M. C. 2012, ApJ, 750, 28 Copi, C. J., Huterer, D., Schwarz, D. J., & Starkman, G. D. 2007, Phys. Rev. D, 75, 023507 -. 2009, MNRAS, 399, 295 Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465 Efstathiou, G. 2004, MNRAS, 348, 885 Efstathiou, G., Ma, Y.-Z., & Hanson, D. 2010, MNRAS, 407, 2530 Hamilton, A. J. S. 1993, ApJ, 417, 19 Hewett, P. C. 1982, MNRAS, 201, 867 Hinshaw, G., Branday, A. J., Bennett, C. L., Gorski, K. M., Kogut, A., Lineweaver, C. H., Smoot, G. F., & Wright, E. L. 1996, ApJ, 464, L25 Hivon, E., G'orski, K. M., Netterfield, C. B., Crill, B. P., Prunet, S., & Hansen, F. 2002, ApJ, 567, 2 Kerscher, M. 1999, A&A, 343, 333 Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64 Peebles, P. J. E. 1980, The large-scale structure of the universe Percival, W. J., et al. 2001, MNRAS, 327, 1297 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific computing Scranton, R., et al. 2002, ApJ, 579, 48 Spergel, D. N., et al. 2003, ApJS, 148, 175 Ursino, E., Branchini, E., Galeazzi, M., Marulli, F., Moscardini, L., Piro, L., Roncarelli, M., & Takei, Y. 2011, MNRAS, 414, 2970 York, D. G., et al. 2000, AJ, 120, 1579",
"pages": [
8,
9
]
},
{
"title": "BIAS TERMS: CONTINUOUS CASE",
"content": "In this appendix we compute the bias terms for the continuous signal. Rewriting ˜ µ = µ + δ ˜ µ , the ensemble average of the numerator of the naive estimator (7) is where we have used 〈 δ ˜ µ 〉 = 0. Further we can use the sum's symmetry between i and j to show that it equals If we had used the true mean, only the C ( θ ij ) term would be present, and we could pull it out of the sum as C ( θ p ). The sum over weights would cancel the denominator, and we would indeed find that 〈 ˜ C 0 ( θ p ) 〉 = C ( θ p ). This is not the case here because of the middle and last terms in the brackets, which are responsible for the bias. We can compute both bias terms from the field's correlation function. We call the first bias term B (1) i because it is first order in the mean estimation error δ ˜ µ , and compute it as The second bias term, B (2) , which is second order in the mean's error, has no dependence on the pixel index. Because B (2) does not depend on the pixel index, this term too can slip outside the sum over pixel pairs in eqn. (A2). Therefore, finally, we have which states the bias in our estimate explicitly. Each bias term costs O ( N 2 ) operations to compute, the same as the correlation function. Note that our naive estimator has a peculiar reaction to correlation functions such as C ( θ ) = c for all separations sampled by our survey. 5 In this case 〈 ˜ C 0 ( θ ) 〉 = 0, which we show by examining the bias terms. If C ( θ ) = c , then the constant can be set outside the sums, which cancel the denominators. Therefore bias factors B (1) i = c and B (2) = c , and the middle term of eqn. (A3) is -2 c . Therefore 〈 ˜ C 0 ( θ ) 〉 = c -2 c + c = 0. Thus, if the naive estimator is viewed as a linear operator on the input correlation function, constant functions are in the null space of the operator, since any constant maps to zero. Moreover, the naive estimator loses the information about any constant baseline in the correlation function, although the information about the shape is preserved. The bias terms depend on C ( θ ) only on scales accessible by the survey region, and not on any larger scales. This permits an (imperfect) reconstruction of the correlation function. To proceed, we can rewrite eqn. (A3) as a matrix multiplication: . where the sum is over the angular bins. Then we set about finding the matrix M To write down M , we make use of the relationship Note that this sum is over angular bin, not pixel. We rewrite the bias terms more explicitly as linear operations on the vector C ( θ q ). The first bias term is where we define Note that the first index refers to pixel and the second to bin. The second bias term is where we define Since C ( θ p ) = ∑ q δ pq C ( θ q ), we finally have To sum up, in this appendix we have: (1) computed the bias of the naive correlation function estimator; (2) shown that the ensemble average of the naive estimate is a linear operation acting upon the true correlation; (3) computed that linear operator in terms of the pixel weights; and (4) shown that constant offsets are in the null space of that operator. The method to estimate the shape of the correlation function in section 2.3 depends on these results.",
"pages": [
9,
10
]
},
{
"title": "BIAS TERMS: DISCRETE CASE",
"content": "No spurious contamination To compute the bias for the discrete case, we write the numerator of the naive estimator (23) in terms of the fluctuation of the mean δ ¯ R and take the ensemble average: Now we examine the last two terms, which are analogous to the bias terms for the diffuse signal. First, where we define where we define This shows that for a signal of discrete photons, this bias term can be written as a sum of a new shot noise term and the old B (1) bias term from the diffuse case. The final term is Again this bias term has a new, shot-noise component added to the old bias term from the diffuse signal. These shot noise bias terms cannot be avoided by excluding i = j from the naive estimator's pixel sums. Each of the new shot noise terms is proportional to µ . We can gather those terms together and notice that the remaining terms are just those which appear on the right side of eqn. (A3), so that: Therefore the ensemble average of the naive estimate for the discrete signal equals the ensemble average of the naive estimate for the diffuse signal plus an additional shot noise bias term which is proportional to the mean of the diffuse field. Thus the ensemble average of the naive estimator for the discrete field can be written as a linear function of the true mean and correlation function. This formulation leads to the reconstruction method for the correlation function discussed in section 3.1.",
"pages": [
10,
11
]
},
{
"title": "Including spurious contamination",
"content": "Starting from equations (26) and (28), we find that the two bias terms also have additional shot noise components due to the spurious signal. Instead of eqn. (B2) we have and instead of eqn. (B4) we have Thus there are additional terms which can be subtracted away to yield the naive estimator in the contamination-free case.",
"pages": [
11
]
}
]
2013ApJS..207....1A
https://arxiv.org/pdf/1304.3746.pdf
<document>
<section_header_level_1><location><page_1><loc_36><loc_86><loc_64><loc_87></location>HOT GAS LINES IN T TAURI STARS</section_header_level_1>
<text><location><page_1><loc_9><loc_79><loc_92><loc_85></location>David R. Ardila 1 , Gregory J. Herczeg 2 , Scott G. Gregory 3,4 , Laura Ingleby 5 , Kevin France 6 , Alexander Brown 6 , Suzan Edwards 7 , Christopher Johns-Krull 8 , Jeffrey L. Linsky 9 , Hao Yang 10 , Jeff A. Valenti 11 , Herv'e Abgrall 12 , Richard D. Alexander 13 , Edwin Bergin 5 , Thomas Bethell 5 , Joanna M. Brown 14 , Nuria Calvet 5 , Catherine Espaillat 14 , Lynne A. Hillenbrand 3 , Gaitee Hussain 15 , Evelyne Roueff 12 , Rebecca Schindhelm 16 , Frederick M. Walter 17</text>
<text><location><page_1><loc_40><loc_78><loc_60><loc_79></location>Draft version September 13, 2021</text>
<section_header_level_1><location><page_1><loc_45><loc_75><loc_55><loc_76></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_41><loc_86><loc_75></location>For Classical T Tauri Stars (CTTSs), the resonance doublets of N v , Si iv , and C iv , as well as the He ii 1640 ˚ A line, trace hot gas flows and act as diagnostics of the accretion process. In this paper we assemble a large high-resolution, high-sensitivity dataset of these lines in CTTSs and Weak T Tauri Stars (WTTSs). The sample comprises 35 stars: one Herbig Ae star, 28 CTTSs, and 6 WTTSs. We find that the C iv , Si iv , and N v lines in CTTSs all have similar shapes. We decompose the C iv and He ii lines into broad and narrow Gaussian components (BC & NC). The most common (50 %) C iv line morphology in CTTSs is that of a low-velocity NC together with a redshifted BC. For CTTSs, a strong BC is the result of the accretion process. The contribution fraction of the NC to the C iv line flux in CTTSs increases with accretion rate, from ∼ 20% to up to ∼ 80%. The velocity centroids of the BCs and NCs are such that V BC /greaterorsimilar 4 V NC , consistent with the predictions of the accretion shock model, in at most 12 out of 22 CTTSs. We do not find evidence of the post-shock becoming buried in the stellar photosphere due to the pressure of the accretion flow. The He ii CTTSs lines are generally symmetric and narrow, with FWHM and redshifts comparable to those of WTTSs. They are less redshifted than the CTTSs C iv lines, by ∼ 10 km s -1 . The amount of flux in the BC of the He ii line is small compared to that of the C iv line, and we show that this is consistent with models of the pre-shock column emission. Overall, the observations are consistent with the presence of multiple accretion columns with different densities or with accretion models that predict a slow-moving, lowdensity region in the periphery of the accretion column. For HN Tau A and RW Aur A, most of the C iv line is blueshifted suggesting that the C iv emission is produced by shocks within outflow jets. In our sample, the Herbig Ae star DX Cha is the only object for which we find a P-Cygni profile in the C iv line, which argues for the presence of a hot (10 5 K) wind. For the overall sample, the Si iv and N v line luminosities are correlated with the C iv line luminosities, although the relationship between Si iv and C iv shows large scatter about a linear relationship and suggests that TW Hya, V4046 Sgr, AA Tau, DF Tau, GM Aur, and V1190 Sco are silicon-poor, while CV Cha, DX Cha, RU Lup, and RW Aur may be silicon-rich.</text>
<text><location><page_1><loc_14><loc_38><loc_86><loc_41></location>Keywords: Surveys - Protoplanetary disks - Stars: pre-main sequence - Stars: variables: T Tauri, Herbig Ae/Be - Ultraviolet: stars</text>
<section_header_level_1><location><page_1><loc_65><loc_35><loc_79><loc_36></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_31><loc_48><loc_35></location>1 ardila@ipac.caltech.edu; NASA Herschel Science Center, California Institute of Technology, MC 100-22, Pasadena, CA 91125, USA</text>
<unordered_list>
<list_item><location><page_1><loc_8><loc_29><loc_48><loc_31></location>2 The Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China</list_item>
<list_item><location><page_1><loc_8><loc_27><loc_48><loc_29></location>3 Cahill Center for Astronomy and Astrophysics, California Institute of Technology, MC 249-17, Pasadena, CA 91125, USA</list_item>
<list_item><location><page_1><loc_8><loc_25><loc_48><loc_27></location>4 School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK</list_item>
<list_item><location><page_1><loc_8><loc_23><loc_48><loc_25></location>5 Department of Astronomy, University of Michigan, 830 Dennison Building, 500 Church Street, Ann Arbor, MI 48109</list_item>
<list_item><location><page_1><loc_8><loc_21><loc_48><loc_23></location>6 Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309-0389, USA</list_item>
<list_item><location><page_1><loc_8><loc_19><loc_48><loc_21></location>7 Department of Astronomy, Smith College, Northampton, MA 01063, USA</list_item>
<list_item><location><page_1><loc_8><loc_17><loc_48><loc_19></location>8 Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA</list_item>
<list_item><location><page_1><loc_8><loc_15><loc_48><loc_17></location>9 JILA, University of Colorado and NIST, 440 UCB Boulder, CO 80309-0440, USA</list_item>
<list_item><location><page_1><loc_8><loc_12><loc_48><loc_15></location>10 Institute for Astrophysics, Central China Normal University, Wuhan, China 430079</list_item>
<list_item><location><page_1><loc_8><loc_10><loc_48><loc_13></location>11 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA</list_item>
<list_item><location><page_1><loc_8><loc_8><loc_48><loc_10></location>12 LUTH and UMR 8102 du CNRS, Observatoire de Paris, Section de Meudon, Place J. Janssen, F-92195 Meudon, France</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_8></location>13 Department of Physics and Astronomy, University of Leices-</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_22><loc_92><loc_34></location>Classical T Tauri stars (CTTSs) are low-mass, young stellar objects surrounded by an accretion disk. They provide us with a laboratory to study the interaction between stars, magnetic fields, and accretion disks. In addition to optical and longer wavelength excesses, this interaction is responsible for a strong ultraviolet (UV) excess, Lyα emission and soft X-ray excesses, all of which have a significant impact on the disk evolution, the rate of planet formation, and the circumstellar environment.</text>
<text><location><page_1><loc_52><loc_19><loc_92><loc_22></location>The observed long rotational periods, the large widths of the Balmer lines, and the presence of optical and UV</text>
<unordered_list>
<list_item><location><page_1><loc_52><loc_17><loc_80><loc_18></location>ter, University Road, Leicester LE1 7RH, UK</list_item>
<list_item><location><page_1><loc_52><loc_15><loc_92><loc_17></location>14 Harvard-Smithsonian Center for Astrophysics, 60 Garden St. MS 78, Cambridge, MA 02138, USA</list_item>
<list_item><location><page_1><loc_52><loc_12><loc_92><loc_15></location>15 ESO, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei M ' 'unchen, Germany</list_item>
<list_item><location><page_1><loc_52><loc_10><loc_92><loc_12></location>16 Southwest Research Institute, Department of Space Studies, Boulder, CO 80303, USA</list_item>
<list_item><location><page_1><loc_52><loc_8><loc_92><loc_10></location>17 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA</list_item>
</unordered_list>
<text><location><page_2><loc_8><loc_68><loc_48><loc_92></location>excesses of CTTSs are naturally explained by the magnetospheric accretion paradigm (e.g. Uchida & Shibata 1984, Koenigl 1991, Shu et al. 1994). According to this paradigm, the gas disk is truncated some distance from the star ( ∼ 5 R ∗ , Meyer et al. 1997) by the pressure of the stellar magnetosphere. Gas from the disk slides down the stellar gravitational potential along the magnetic field lines, reaching speeds comparable to the free-fall velocity ( ∼ 300 km s -1 ). These velocities are much larger than the local ∼ 20 km s -1 sound speed (but less than the Alfven speed of ∼ 500 km s -1 implied by a 2 kG magnetic field Johns-Krull 2007). The density of the accretion stream depends on the accretion rate and the filling factor but it is typically of the order of ∼ 10 12 cm -3 (pre-shock, see Calvet & Gullbring 1998). The supersonic flow, confined by the magnetic field, produces a strong shock upon reaching the star, which converts most of the kinetic energy of the gas into thermal energy (e.g. Lamzin 1995).</text>
<text><location><page_2><loc_8><loc_52><loc_48><loc_68></location>The gas reaches temperatures of the order of a million degrees at the shock surface and cools radiatively until it merges with the stellar photosphere. Part of the cooling radiation will heat the stellar surface, resulting in a hot spot, observed spectroscopically as an excess continuum (the 'veiling'). Cooling radiation emitted away from the star will illuminate gas before the shock surface, producing a radiative precursor of warm (T ∼ 10 4 K), ionized gas (Calvet & Gullbring 1998). In this paper we will use the term 'accretion shock region' as shorthand for the region that includes the pre-shock, the shock surface, the post-shock column, and the heated photosphere.</text>
<text><location><page_2><loc_8><loc_32><loc_48><loc_52></location>Because the accretion column should be in pressure equilibrium with the stellar photosphere, Drake (2005) suggested that for typical accretion rates ( ∼ 10 -8 M /circledot /yr) the post-shock region would be buried, in the sense that the shortest escape paths for post-shock photons would go through a significant column of photospheric gas. Sacco et al. (2010) have argued that the burying effects appear at accretion rates as small as a few times ∼ 10 -10 M /circledot /yr and that the absorption of the X-rays by the stellar photosphere may explain the one to two orders of magnitude discrepancy between the accretion rates calculated from X-ray line emission and those calculated from optical veiling or near-UV excesses, and the lack of dense (n e /greaterorsimilar 10 11 cm -3 ) X-ray emitting plasma in objects such as T Tau (Gudel et al. 2007).</text>
<text><location><page_2><loc_8><loc_20><loc_48><loc_32></location>Time-dependent models of the accretion column by Sacco et al. (2008) predict that emission from a single, homogeneous, magnetically dominated post-shock column should be quasi-periodic, on timescales of ∼ 400 sec, because plasma instabilities can collapse the column. Such periodicity has not been observed (Drake et al. 2009; Gunther et al. 2010), perhaps suggesting that accretion streams are inhomogeneous, or that there are multiple, uncorrelated accretion columns.</text>
<text><location><page_2><loc_8><loc_8><loc_48><loc_20></location>Observations during the last two decades have resulted in spectroscopic and photometric evidence for the presence of accretion hot spots or rings with filling factors of up to a few percent (see review by Bouvier et al. 2007). The surface topology of the magnetic field (e.g. Gregory et al. 2006; Mohanty & Shu 2008) and/or the misalignment between the rotational and magnetic axes result in a rotationally modulated surface flux (Johns & Basri 1995; Argiroffi et al. 2011,</text>
<text><location><page_2><loc_52><loc_80><loc_92><loc_92></location>2012a), with a small filling factor (Calvet & Gullbring 1998; Valenti & Johns-Krull 2004). The spots appear and disappear over timescales of days (Rucinski et al. 2008) to years (Bouvier et al. 1993). Analysis of the possible magnetic field configurations indicates that although CTTSs have very complex surface magnetic fields, the portion of the field that carries gas from the inner disk to the star, is well ordered globally (Johns-Krull et al. 1999a; Adams & Gregory 2012).</text>
<text><location><page_2><loc_52><loc_69><loc_92><loc_80></location>In this paper we use the strong emission lines of ionized metals in order to probe the characteristics of the accretion shock region. In particular, we are interested in understanding the role that accretion has in shaping these lines, where the lines originate, and what the lines reveal about the geometry of the accretion process. Our long-term goal is to clarify the UV evolution of young stars and its impact on the surrounding accretion disk.</text>
<text><location><page_2><loc_52><loc_54><loc_92><loc_69></location>We analyze the resonance doublets of N v ( λλ 1238.82, 1242.80 ˚ A), Si iv ( λλ 1393.76, 1402.77 ˚ A), and C iv ( λλ 1548.19, 1550.77 ˚ A), as well as the He ii ( λ 1640.47 ˚ A) line. If they are produced by collisional excitation in a low-density medium, their presence suggests a high temperature ( ∼ 10 5 K, assuming collisional ionization equilibrium) or a photoionized environment. In solartype main sequence stars, these 'hot' lines are formed in the transition region, the narrow region between the chromosphere and the corona, and they are sometimes called transition-region lines.</text>
<text><location><page_2><loc_52><loc_31><loc_92><loc_54></location>The C iv resonance doublet lines are among the strongest lines in the UV spectra of CTTSs (Ardila et al. 2002). Using International Ultraviolet Explorer (IUE) data, Johns-Krull et al. (2000) showed that the surface flux in the C iv resonant lines can be as much as an order of magnitude larger than the largest flux observed in Weak T Tauri stars (WTTSs), main sequence dwarfs, or RS CVn stars. They also showed that the high surface flux in the C iv lines of CTTSs is uncorrelated with measures of stellar activity but it is strongly correlated with accretion rate, for accretion rates from 10 -8 M /circledot /yr to 10 -5 M /circledot /yr. The strong correlations among accretion rate, C iv flux and Far Ultraviolet (FUV) luminosity have been confirmed by Ingleby et al. (2011) and Yang et al. (2012) using ACS/SBC and STIS data. Those results further suggest a causal relationship between the accretion process and the hot line flux.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_31></location>Previous surveys of the UV emission lines in lowresolution spectra of T Tauri stars include the analysis of IUE spectra (Valenti et al. 2000; Johns-Krull et al. 2000) and GHRS, ACS, and STIS spectra (Yang et al. 2012). Prior analysis of high resolution observations of the hot gas lines in T Tauri stars had been published for some objects (BP Tau, DF Tau, DG Tau, DR Tau, EG Cha, EP Cha, GM Aur, RU Lup, RW Aur A, RY Tau, T Tau, TW Hya, TWA 5, V4046 Sgr, and HBC 388, see Lamzin 2000a,b; Errico et al. 2000, 2001; Ardila et al. 2002; Herczeg et al. 2002; Lamzin et al. 2004; Herczeg et al. 2005, 2006; Gunther & Schmitt 2008; Ingleby et al. 2011). In addition, analysis of the C iv lines for the brown dwarf 2M1207 has been published by France et al. (2010). The observations show that the C iv lines in CTTSs have asymmetric shapes and wings that often extend to ± 400 km s -1 from the line rest velocity. The emission lines are mostly centered</text>
<text><location><page_3><loc_8><loc_68><loc_48><loc_92></location>or redshifted, although some stars occasionally present strong blueshifted emission (e.g. DG Tau, DR Tau, RY Tau). Doublet ratios are not always 2:1 (e.g. DG Tau, DR Tau, RU Lup, and RW Aur A). Overall, the UV spectra of CTTSs also show large numbers of narrow H 2 emission lines, and CO bands in absorption and emission (France et al. 2011; Schindhelm et al. 2012), produced primarily by Lyα fluorescence. In most observations published so far, the Si iv line is strongly contaminated by H 2 lines, and the N v 1243 ˚ A line is absorbed by circumstellar or interstellar N I. High spectral resolution observations are crucial to fully exploit the diagnostic power of the UV observations as the H 2 emission lines can be kinematically separated from the hot lines only when the resolution is high enough. In addition, we will show here that the hot gas lines have multiple kinematic components that can only be analyzed in high resolution spectra.</text>
<text><location><page_3><loc_8><loc_45><loc_48><loc_68></location>Both the correlation between accretion rate and C iv surface flux and the presence of redshifted hot gas line profiles in some stars, are consistent with formation in a high-latitude accretion flow. However, according to Lamzin (2003a,b) C iv line formation in the accretion shock region should result in double-peaked line profiles, which are generally not observed. In this context Gunther & Schmitt (2008) explored the shape of the hot gas lines (primarily O vi and C iv ) in a sample of 7 stars. They considered formation in the accretion shock, in an outflow, in the surface of the disk, in an equatorial boundary layer, and in the stellar transition region, and concluded that no single explanation or region can be responsible for all the observed line characteristics. In particular, they concluded that the shape of the redshifted lines was incompatible with models of magnetospheric accretion.</text>
<text><location><page_3><loc_8><loc_20><loc_48><loc_45></location>With the primary goal of providing a unified description of the hot gas lines and understanding their origin, we have obtained Far (FUV) and Near Ultraviolet (NUV) spectra of a large sample of CTTSs and WTTSs, using the Cosmic Origins Spectrograph ( COS ) and the Space Telescope Imaging Spectrograph ( STIS ). Most of the data for this paper comes from the Cycle 17 Hubble Space Telescope ( HST ) proposal 'The Disks, Accretion, and Outflows (DAO) of T Tau stars' (PI G. Herczeg, Prop. ID HST-GO-11616). The DAO program is the largest and most sensitive high resolution spectroscopic survey of young stars in the UV ever undertaken and as such it provides a rich source of information for these objects. The program is described in more detail in Herczeg et al. (2013). We have complemented the DAO data with GTO data from HST programs 11533 and 12036 (PI J. Green - BP Tau, DF Tau, RU Lup, V4046 Sgr) as well as UV spectra from the literature and from the Mikulski Archive at STScI ( MAST ).</text>
<text><location><page_3><loc_8><loc_8><loc_48><loc_20></location>As shown below, there is a wide diversity of profiles in all lines for the stars in our sample. The spectra are rich and a single paper cannot do justice to their variety nor to all the physical mechanisms that likely contribute to their formation. Here we take a broad view in an attempt to obtain general statements about accretion in CTTSs. Details about the sample and the data reduction are presented in Section 2. We then analyze the C iv , Si iv , He ii and N v lines. The analysis of the C iv</text>
<text><location><page_3><loc_52><loc_76><loc_92><loc_92></location>lines takes up most of the paper, as this is the strongest line in the set, and the least affected by absorptions or emissions by other species (Section 5). We examine the line shapes, the relationship with accretion rate, and correlations among quantities associated with the lines and other CTTSs parameters. We also obtain from the literature multi-epoch information on line variability (Section 5.4). The other lines play a supporting role in this analysis and are examined in Sections 6 and 7. Section 8 contains a summary of the observational conclusions and a discussion of their implications. The conclusions are in Section 9.</text>
<section_header_level_1><location><page_3><loc_65><loc_73><loc_78><loc_74></location>2. OBSERVATIONS</section_header_level_1>
<text><location><page_3><loc_52><loc_65><loc_92><loc_73></location>Tables 1, 2, and 3 list the 35 stars we will be analyzing in this paper and the references for all the ancillary data we consider. Table 4 indicates the origin of the data (DAO or some other project), the datasets and slit sizes used for the observations. Details about exposure times will appear in Herczeg et al. (2013).</text>
<text><location><page_3><loc_52><loc_49><loc_92><loc_65></location>The data considered here encompass most of the published high resolution HST observations of the C iv doublet lines for CTTSs. Ardila et al. (2002) provides references to additional high-resolution C iv data for CTTSs obtained with the Goddard High Resolution Spectrograph ( GHRS ). We do not re-analyze those spectra here, but they provide additional context to our paper. Non-DAO STIS data were downloaded from the HST STIS Echelle Spectral Catalog of Stars (StarCAT, Ayres 2010). NonDAO COS data were downloaded from the Mikulski Archive for Space Telescopes (MAST) and reduced as described below.</text>
<text><location><page_3><loc_52><loc_24><loc_92><loc_49></location>The sample of stars considered here includes objects with spectral types ranging from A7 (the Herbig Ae star DX Cha 18 ) to M2, although most objects have mid-K spectral types. We assume stellar ages and distances to be ∼ 2 Myr and 140 pc, respectively, for Taurus-Aurigae (see Loinard et al. 2007 and references therein), ∼ 2 Myr and 150 pc for Lupus I (Comer'on et al. 2009), ∼ 5 Myr and 145 pc for Upper-Scorpius (see Alencar et al. 2003 and references therein), ∼ 5 Myr and 114 pc for the /epsilon1 Chamaeleontis cluster (see Lyo et al. 2008 and references therein), ∼ 5 Myr and 160 pc for Chamaeleon I (see Hussain et al. 2009 and references therein), ∼ 8 Myr and 97 pc for the η Chamaeleontis cluster (Mamajek et al. 1999), ∼ 10 Myr and 55 pc for the TW Hydrae association (Zuckerman & Song 2004), ∼ 12 Myr and 72 pc for V4046Sgr in the β Pictoris moving group (Torres et al. 2006). The sample includes MP Mus ( ∼ 7 Myr and 100 pc, Kastner et al. 2010 and references therein), not known to be associated with any young region.</text>
<text><location><page_3><loc_52><loc_10><loc_92><loc_24></location>The sample includes stars with transition disks (TD) (Espaillat et al. 2011) CS Cha, DM Tau, GM Aur, IP Tau, TW Hya, UX Tau A, and V1079 Tau (LkCa 15). Here we take the term 'transition disk' to mean a disk showing infrared evidence of a hole or a gap. As a group, transition disks may have lower accretion rates than other CTTS disks (Espaillat et al. 2012). However, for the targets included here the difference between the accretion rates of TDs and those of the the rest of the accreting stars is not significant. The sample also includes</text>
<text><location><page_4><loc_8><loc_88><loc_48><loc_92></location>six Weak T Tauri Stars (WTTSs): EG Cha (RECX 1), V396 Aur (LkCa 19), V1068 Tau (LkCa 4), TWA 7, V397 Aur, and V410 Tau.</text>
<text><location><page_4><loc_8><loc_77><loc_48><loc_88></location>Of the 35 stars considered here, 12 are known to be part of binary systems. Dynamical interactions among the binary components may re-arrange the circumstellar disk or preclude its existence altogether. The presence of an unaccounted companion may result in larger-thanexpected accretion diagnostic lines. In addition, large instantaneous radial velocities may be observed in the targeted lines at certain points of the binary orbit.</text>
<text><location><page_4><loc_8><loc_64><loc_48><loc_77></location>AK Sco, DX Cha, and V4046 Sgr are spectroscopic binaries with circumbinary and/or circumstellar disks. CS Cha is a candidate long-period spectroscopic binary, although the characteristics of the companion are unknown (Guenther et al. 2007). The accretion rates listed in Table 3 are obtained from optical veiling and NUV excesses and represent total accretion for the overall system. At the end of this paper we will conclude that binarity may be affecting the line centroid determination only in AK Sco.</text>
<text><location><page_4><loc_8><loc_55><loc_48><loc_64></location>For non-spectroscopic binaries, the effect of the binarity may be relevant only if both companions are within the COS or STIS apertures. This is the case for DF Tau, RW Aur A, and the WTTSs EG Cha, V397 Aur, and V410 Tau. For DF Tau and RW Aur A, the primary component dominates the FUV emission (see Herczeg et al. 2006; Alencar et al. 2005).</text>
<text><location><page_4><loc_8><loc_36><loc_48><loc_55></location>Table 3 lists the accretion rate, obtained from literature sources. For some stars, the DAO dataset contains simultaneous NUV and FUV observations, obtained during the same HST visit. Ingleby et al. (2013) describe those NUV observations in more detail and calculate accretion rates based on them. In turn, we use those accretion rate determinations here. The uncertainty in the accretion rate is dominated by systematic factors such as the adopted extinction correction and the color of the underlying photosphere. These may result in errors as large as a factor of 10 in the accretion rate. We do not list v sin i measurements for the sample but typical values for young stars are ∼ 10-20 km s -1 (Basri & Batalha 1990).</text>
<section_header_level_1><location><page_4><loc_21><loc_34><loc_35><loc_35></location>2.1. Data Reduction</section_header_level_1>
<text><location><page_4><loc_8><loc_31><loc_48><loc_33></location>The COS and STIS DAO data were all taken in timetagged mode.</text>
<unordered_list>
<list_item><location><page_4><loc_11><loc_20><loc_48><loc_30></location>· COS observations: The FUV (1150-1790 ˚ A) spectra were recorded using multiple exposures with the G130M grating (1291, 1327 ˚ A settings) and the G160M grating (1577, 1600, 1623 ˚ A settings). These provide a velocity resolution of ∆ v ∼ 17 km s -1 (R ∼ 17,000) with seven pixels per resolution element (Osterman et al. 2011; Green et al. 2012).</list_item>
</unordered_list>
<text><location><page_4><loc_12><loc_10><loc_48><loc_19></location>The maximum exposure time was accumulated in the region of the Si iv resonance lines and the exposures ranged from 3 ksec (for most stars with 2 orbit visits) to a maximum of 8 ksec for a few stars with 4 orbit visits. Use of multiple grating positions ensured full wavelength coverage and reduced the effects of fixed pattern noise.</text>
<text><location><page_4><loc_12><loc_7><loc_48><loc_9></location>All observations were taken using the primary science aperture (PSA), which is a 2.5' diameter</text>
<text><location><page_4><loc_56><loc_80><loc_92><loc_92></location>circular aperture. The acquisition observations used the ACQ/SEARCH algorithm followed by ACQ/IMAGE (Dixon 2011). The absolute wavelength scale accuracy is ∼ 15 km s -1 (1 σ ), where the error is dominated by pointing errors. We obtained one dimensional, co-added spectra using the COS calibration pipeline (CALCOS) with alignment and co-addition obtained using the IDL routines described by Danforth et al. (2010).</text>
<unordered_list>
<list_item><location><page_4><loc_54><loc_69><loc_92><loc_79></location>· STIS observations: Those targets that are too bright to be observed by COS were observed using the STIS E140M echelle grating, which has a spectral resolution of ∆ v ∼ 7 km s -1 (R ∼ 45,000), over the region 1150 - 1700 ˚ A . All the observations were obtained using the 0.2' × 0.2' 'photometric' aperture, during two HST orbits.</list_item>
<list_item><location><page_4><loc_56><loc_60><loc_92><loc_69></location>The STIS FUV spectra were calibrated and the echelle orders co-added to provide a single spectrum using the IDL software package developed for the StarCAT project (Ayres 2010). This reduction procedure provides a wavelength-scale accuracy of 1 σ = 3 km s -1 . Non-DAO STIS data were taken directly from StarCAT.</list_item>
</unordered_list>
<text><location><page_4><loc_52><loc_40><loc_92><loc_59></location>For a given pointing, errors in the positioning of the target within the aperture result in an offset in the wavelength scale. As indicated above, these are supposed to be 15 km s -1 for COS and 3 km s -1 for STIS , according to the instrument observing manuals (Dixon 2011; Ely 2011). In addition, the geometric correction necessary to account for the curvature of the COS FUV detector may make longer wavelength features appear redder than they really are. The offset depends on the exact position of the target on the detector. In most cases this intra-spectrum wavelength uncertainty is < 10 km s -1 , but in a few extreme cases it will introduce a ∼ 15 km s -1 shift from the reddest to the bluest wavelengths in a single COS grating mode (See Figure 4 from Linsky et al. 2012).</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_40></location>To determine how important the errors due to pointing and calibration are in the COS data, we focus on the H 2 lines. We use the line measurements from K. France (personal communication, see also France et al. 2012) for P(2)(0-5) at 1398.95 ˚ A, R(11)(2-8) at 1555.89 ˚ A, R(6)(1-8) at 1556.87 ˚ A, and P(5)(4-11) at 1613.72 ˚ A. France et al. (2012) show that in the case of DK Tau, ET Cha (RECX 15), HN Tau, IP Tau, RU Lup, RW Aur, and V1079 Tau (LkCa 15), some H 2 lines show a redshifted peak and a blueshifted low-level emission, which makes them asymmetric. Ignoring these stars, the four H 2 lines are centered at the stellar rest velocity, with a standard deviations of 7.1, 6.5, 6.1, and 12.1 km s -1 , respectively. The large scatter in the P(5)(4-11) line measurement is partly the result of low signal-to-noise in this region of the spectrum. Unlike the results reported by Linsky et al. 2012), we do not observe a systematic increase in the line center as a function of wavelength, neither star by star nor in the average of all stars for a given line. This is likely the result of the different acquisition procedure followed here. For the STIS data, the average standard deviation in all the H 2 wavelengths is 4.4 km s -1 .</text>
<text><location><page_4><loc_52><loc_7><loc_92><loc_9></location>This shows that the errors in the COS wavelength scale are smaller than the 15 km s -1 reported in the man-</text>
<text><location><page_5><loc_8><loc_81><loc_48><loc_92></location>uals and that no systematic shifts with wavelength are present. For the rest of the paper we will assume that the pointing errors in the COS data result in a velocity uncertainty of 7 km s -1 (the average of the first three H 2 lines considered) while the STIS errors are 5 km s -1 . We do not correct the spectra for the H 2 velocities, as it is not clear how to implement this correction in the case of stars with asymmetric H 2 lines.</text>
<section_header_level_1><location><page_5><loc_11><loc_79><loc_46><loc_80></location>3. INTRODUCTION TO THE SHAPE OF THE LINES</section_header_level_1>
<text><location><page_5><loc_8><loc_63><loc_48><loc_78></location>Figures A.1 to A.5 in the Appendix show the lines that we are discussing in this paper. Each line is plotted in velocity space (km s -1 ) centered on the stellar photospheric rest frame. The ordinate gives flux density in units 10 -14 erg sec -1 cm -2 ˚ A -1 . In the case of doublets, the nominal wavelength of the strongest member of the line is set to zero velocity, and the positions of the strongest H 2 lines are marked with dashed lines. Dotted lines mark the positions of other features in the spectra. The plotted spectra have been smoothed by a 5-point median.</text>
<text><location><page_5><loc_8><loc_50><loc_48><loc_63></location>The C iv , Si iv , and N v lines are resonance doublets with slightly offset upper levels. For each doublet, both lines should have the same shape when emitted. This redundancy allow us to identify extra spectral features and to distinguish real features from noise. If the lines are emitted from an optically thin or effectively thin plasma, their flux ratio should be 2:1. If the lines are emitted from a medium that has a thermalization depth smaller than its optical depth (i.e. the medium is optically thick, but not effectively thin) the flux ratio will tend to 1.</text>
<text><location><page_5><loc_8><loc_25><loc_48><loc_50></location>For a plasma at rest in coronal ionization equilibrium, the peak ion abundance occurs at temperatures of log(T)=5.3 (K) for N v , log(T)=5.0 (K) for C iv , log(T)=4.9 (K) for Si iv , and log(T)=4.7 (K) for He ii (Mazzotta et al. 1998). In the context of the magnetospheric accretion paradigm, the velocities of the postshock gas are high enough that the collisional timescales are longer than the dynamical timescale (Ardila 2007). This implies that the post-shock gas lines may trace lower temperature, higher density plasma than in the coronal ionization equilibrium case. The pre-shock plasma is radiatively ionized to temperatures of ∼ 10 4 K. If the preand post-shock regions both contribute to the emission, conclusions derived from the usual differential emission analyses are not valid (e.g. Brooks et al. 2001). In the case of non-accreting stars, the resonance N v , C iv , and Si iv lines are collisionally excited while the He ii line is populated by radiative recombination in the X-ray ionized plasma (Zirin 1975).</text>
<text><location><page_5><loc_8><loc_21><loc_48><loc_25></location>Figure 1 compares the hot gas lines of BP Tau and the WTTS V396 Aur. These stars provide examples of the main observational points we will make in this paper:</text>
<unordered_list>
<list_item><location><page_5><loc_10><loc_14><loc_48><loc_19></location>1. For N v and C iv the lines of the CTTSs have broad wings not present in the WTTSs. This is generally also the case for Si iv , although for BP Tau the lines are weak compared to H 2 .</list_item>
<list_item><location><page_5><loc_10><loc_7><loc_48><loc_13></location>2. The N v 1243 ˚ Aline of BP Tau (the redder member of the multiplet), and of CTTSs in general, looks truncated when compared to the 1239 ˚ A line (the blue member), in the sense of not having a sharp</list_item>
</unordered_list>
<text><location><page_5><loc_56><loc_89><loc_92><loc_92></location>emission component. This is the result of N i circumstellar absorption.</text>
<unordered_list>
<list_item><location><page_5><loc_54><loc_78><loc_92><loc_88></location>3. The Si iv lines in CTTSs are strongly affected by H 2 lines and weakly affected by CO A-X absorption bands (France et al. 2011; Schindhelm et al. 2012) and O iv . At the wavelengths of the Si iv lines, the dominant emission seen in BP Tau and other CTTSs is primarily due to fluorescent H 2 lines. The WTTS do not show H 2 lines or CO bands within ± 400 km s -1 of the gas lines we study here.</list_item>
<list_item><location><page_5><loc_54><loc_70><loc_92><loc_77></location>4. The C iv BP Tau lines have similar shapes to each other, to the 1239 N v line, and to the Si iv lines, when observed. Generally, the CTTS C iv lines are asymmetric to the red and slightly redshifted with respect to the WTTS line.</list_item>
<list_item><location><page_5><loc_54><loc_65><loc_92><loc_69></location>5. The He ii lines are similar (in shape, width, and velocity centroid) in WTTSs and CTTSs, and similar to the narrow component of C iv .</list_item>
</unordered_list>
<text><location><page_5><loc_52><loc_52><loc_92><loc_64></location>The general statements above belie the remarkable diversity of line profiles in this sample. Some of this diversity is showcased in Figure 2. For analysis and interpretation, we focus on the C iv doublet lines, as they are the brightest and 'cleanest' of the set, with the others playing a supporting role. In order to test the predictions of the magnetospheric accretion model, in section 5.3 we perform a Gaussian decomposition of the C iv profiles, and representative results are shown in Figure 2.</text>
<text><location><page_5><loc_52><loc_32><loc_92><loc_52></location>TW Hya has a profile and decomposition similar to those of BP Tau: strong narrow component (NC) plus a redshifted, lower-peak broad component (BC). ∼ 50% (12/22) of the stars for which a Gaussian decomposition is possible show this kind of profile and ∼ 70% (21/29) of CTTSs in our sample have redshifted C iv peaks. HN Tau also has a NC plus a redshifted BC, but the former is blueshifted with respect to the stellar rest velocity by 80 km s -1 . For IP Tau and V1190 Sco, the BC is blueshifted with respect to the NC. The magnetospheric accretion model may explain some of objects with morphologies analogous to TW Hya but blueshifted emission requires extensions to the model or the contributions of other emitting regions besides the accretion funnel (Gunther & Schmitt 2008).</text>
<text><location><page_5><loc_52><loc_11><loc_92><loc_32></location>Figure 3 shows the C iv and He ii lines in all the WTTSs, scaled to the blue wings of the lines. For all except for V1068 Tau (Lk Ca 4) and V410 Tau, the C iv and He ii lines appear very similar to each other in shape, width, and shift. They are all fairly symmetric, with velocity maxima within ∼ 20 km s -1 of zero, and FWHM from 60 to 100 km s -1 . V1068 Tau and V410 Tau have the largest FWHM in both C iv and He ii . The lines of these two stars appear either truncated or broadened with respect to the rest of the WTTSs. In the case of V410 Tau, v sin i = 73 km s -1 (G/suppressl¸ebocki & Gnaci'nski 2005), which means that rotational broadening is responsible for a significant fraction of the width. However, V1068 Tau has v sin i = 26 km s -1 . For the rest of the WTTSs, v sin i ranges from 4 to 20 km s -1 (G/suppressl¸ebocki & Gnaci'nski 2005; Torres et al. 2006).</text>
<text><location><page_5><loc_52><loc_7><loc_92><loc_11></location>To characterize the line shapes of He ii and C iv we have defined non-parametric and parametric shape measurements. The non-parametric measurements (Table 5)</text>
<text><location><page_6><loc_8><loc_79><loc_48><loc_92></location>do not make strong assumptions about the line shapes and they provide a intuitive summary description of the line. These are the velocity at maximum flux, the full width at half maximum (FWHM), and the skewness, defined in Section 5.2. For C iv we have also measured the ratio of the 1548 ˚ A to the 1550 ˚ A line, by scaling the line wings to match each other. This provides a measure of the line optical depth. The parametric measurements (Tables 6 and 7) assume that each C iv and He ii line is a combination of two Gaussians.</text>
<text><location><page_6><loc_8><loc_71><loc_48><loc_78></location>Table 8 lists the flux measurements that we will be considering in the following sections. The fluxes are obtained by direct integration of the spectra over the spectral range listed in the table, after subtracting the continuum and interpolating over known blending features (see Section 5).</text>
<section_header_level_1><location><page_6><loc_10><loc_67><loc_47><loc_69></location>4. THE RELATIONSHIP BETWEEN ACCRETION RATE AND C IV LUMINOSITY</section_header_level_1>
<text><location><page_6><loc_8><loc_45><loc_48><loc_66></location>As mentioned in the introduction, Johns-Krull et al. (2000) showed that the accretion rate is correlated with excess C iv luminosity. The excess C iv luminosity is obtained by subtracting the stellar atmosphere contribution to the observed line. They estimated the stellar contribution to be 6 × 10 -5 L /circledot for a 2R /circledot object. They also showed that the correlation of C iv excess luminosity with accretion rate is very sensitive to extinction estimates. More recently, Yang et al. (2012) obtain a linear correlation between the C iv luminosity (from STIS and ACS/SBC low-resolution spectra) and the accretion luminosity (from literature values) for 91 CTTSs. Here we show that our data are consistent with a correlation between accretion rate C iv luminosity and explore the role that the lack of simultaneous observations or different extinction estimates play in this relationship.</text>
<text><location><page_6><loc_8><loc_34><loc_48><loc_45></location>Figure 4 compares accretion rates (references given in Table 1) with the C iv line luminosities. Blue diamonds correspond to objects with simultaneous determinations of ˙ M and L CIV when available (Ingleby et al. 2013) and green diamonds correspond to ˙ M determinations for the rest of the objects. Note that the accretion rate estimates from Ingleby et al. (2013) are derived using the extinction values from Furlan et al. (2009, 2011).</text>
<text><location><page_6><loc_8><loc_20><loc_48><loc_34></location>Using only the simultaneous values (blue diamonds), we obtain a Pearson product-moment correlation coefficient r=0 . 73 (p-value=0.3% 19 ) while with all values we obtain r=0 . 61 (p-value= < 0.05%). The difference between using all of the data or only the simultaneous data is not significant for the purposes of the correlation. Including all data, we obtain log ˙ M = ( -5 . 4 ± 0 . 2)+ (0 . 8 ± 0 . 1) log L CIV /L /circledot , where ˙ M is given in M /circledot /yr and the errors indicate 1 σ values obtained by the bootstrap method. The correlation is plotted in Figure 4.</text>
<text><location><page_6><loc_8><loc_13><loc_48><loc_20></location>As argued by Johns-Krull et al. (2000), the observed relationship between log L CIV /L /circledot and accretion rate is very sensitive to extinction estimates. This is shown in Figure 4, for which the C iv luminosities indicated by the black triangles were calculated using the extinction esti-</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_12></location>19 The p-value is the probability of obtaining a value of the test statistic at least as extreme as the observed one, assuming the null hypothesis is true. In this case, it is the probability that the Pearson's r is as large as measured or larger, if the two quantities are uncorrelated. We reject the null hypothesis if p-value ≤ 0.05.</text>
<text><location><page_6><loc_52><loc_85><loc_92><loc_92></location>mates from Furlan et al. (2009, 2011), for all the targets we have in common with that work. For log L CIV /L /circledot and log ˙ M the correlation is weakened when using the Furlan et al. (2009, 2011) extinction values: the value of the Pearson's r is 0 . 44, with p-value=5%.</text>
<text><location><page_6><loc_52><loc_61><loc_92><loc_85></location>The extinction values we adopt in this paper come from a variety of sources, but a significant fraction come from Gullbring et al. (1998, 2000). They argue that the colors of the WTTSs underlying the CTTSs are anomalous for their spectral types, which biases the near-IR extinction estimates. They obtain the extinctions reported here by deriving models of the UV excess in CTTSs. For other stars, we have adopted extinction estimates based on spectroscopic observations of the accretion veiling, when possible. The Furlan et al. (2009, 2011) extinction values are obtained by de-reddening the observed near-infrared colors until they match the colors for the target's spectral type. They are significantly larger than the values we adopt in this paper, resulting in larger estimates of the C iv luminosity. Differences in the extinction estimates can have a substantial impact in the adopted flux, as a 10% increase in the value of A V results in a 30% increase in the de-reddened C iv line flux.</text>
<text><location><page_6><loc_52><loc_46><loc_92><loc_61></location>Figure 4 also shows the relationship between accretion rate and C iv luminosity derived by Johns-Krull et al. (2000) (their equation 2) assuming that all stars have a radius of 2R /circledot . They obtained their relationship based on the accretion rates and extinctions from Hartigan et al. (1995). Both of those quantities are higher, on average, than the ones we adopt here, and so their correlation predicts larger accretion rates. Note that the relationship from Johns-Krull et al. (2000) is not defined for stars with excess C iv surface fluxes smaller than 10 6 erg sec -1 cm -2 .</text>
<text><location><page_6><loc_52><loc_32><loc_92><loc_46></location>Overall, there is enough evidence to confirm that for most CTTSs the C iv line luminosity is powered primarily by accretion, and we will adopt this hypothesis here. However, the exact relationship between accretion rate and C iv luminosity remains uncertain. This is not surprising considering the complexity of the processes that contribute to the line flux, as we show in this work. The dominant uncertainty is the exact value of the extinction, which depends on the assumed stellar colors and on the shape of the extinction law in the UV (Johns-Krull et al. 2000; Calvet et al. 2004a).</text>
<text><location><page_6><loc_52><loc_21><loc_92><loc_32></location>We do not detect a monotonic decrease in the CTTSs C iv luminosity as a function of age, for the range of ages considered here (2 - 10 Myrs). We do not observe a significant difference in the C iv luminosities of the TDs as compared with the rest of the sample, consistent with the results from Ingleby et al. (2011) who found no correlations between FUV luminosities and tracers of dust evolution in the disk.</text>
<section_header_level_1><location><page_6><loc_63><loc_19><loc_81><loc_20></location>5. THE C IV LINE SHAPE</section_header_level_1>
<text><location><page_6><loc_52><loc_12><loc_92><loc_19></location>The C iv lines of CTTSs are generally redshifted, broad (with emission within ∼ 400 km s -1 of the stellar rest velocity), and asymmetric to the red (positive skewness). We will show that none of these characteristics is correlated with the line luminosity or with accretion rate.</text>
<section_header_level_1><location><page_6><loc_59><loc_10><loc_84><loc_11></location>5.1. Comparing the two C IV lines</section_header_level_1>
<text><location><page_6><loc_52><loc_7><loc_92><loc_9></location>If optically thin, both C iv lines form should have the same shape. Differences between the components can</text>
<text><location><page_7><loc_8><loc_77><loc_48><loc_92></location>help us discover the presence of extra sources of absorption or emission. To exploit this redundancy, in Figure A.6 we plot both members of the C iv doublet, with the 1550 ˚ A line scaled to match the 1548 ˚ A one. The scaling is done by matching the line peaks or the line wings from ∼ 0 to ∼ 150 km s -1 . We expect this scaling factor to be 2 for optically thin or effectively thin emission. However, as we discuss in Section 5.2.4, the opacity characteristics of the broad and narrow line components are different, and the overall scaling factor may not be the best predictor of opacity.</text>
<text><location><page_7><loc_8><loc_56><loc_48><loc_77></location>Figure A.6 shows that the line wings tend to follow each other closely, at least until about +200 to +300 km s -1 , when the 1548 ˚ A line start to bump into the 1550 ˚ A one. The 1548 ˚ A line is usually contaminated by the H 2 line R(3)1- 8 (1547.3 ˚ A), at -167 km s -1 with respect to its rest velocity (e.g. DK Tau). Figure A.6 also reveals examples of extra emission at ∼ -100 km s -1 which do not appear in the 1550 ˚ A line (AK Sco, DE Tau, DK Tau, DR Tau, HN Tau A, RU Lup, and UX Tau A). We tentatively identify emission from Fe ii (1547.807, -73.6 km s -1 from the rest velocity of the 1548 C iv line), C ii (1547.465; -139.8 km s -1 ), and Si i (1547.452 ˚ A, -142.3 km s -1 ; 1547.731 ˚ A, -88.3 km s -1 ), for these stars, as responsible for the emission to the blue of the C iv 1548 ˚ A line.</text>
<text><location><page_7><loc_8><loc_41><loc_48><loc_56></location>To measure the C iv line flux listed in Table 8 we subtracted the continuum and interpolated over the H 2 R(3)1-8 line and the Si i , C ii , Fe ii -complex, when present. The resulting spectrum is then integrated from -400 km s -1 to 900 km s -1 of the 1548 C iv line. We also detect the CO A-X (0-0) absorption band at 1544.4 ˚ A (-730 km s -1 from the 1548 ˚ A line, see France et al. 2011; McJunkin et al. 2013) in a significant fraction of the sample. The wing of the CO absorption may extend to the blue edge of the 1548 C iv line. However its impact in the overall C iv flux is negligible.</text>
<text><location><page_7><loc_8><loc_36><loc_48><loc_41></location>Note that the red wings of each C iv line for DX Cha, RW Aur A, DF Tau, and RU Lup do not follow each other, and AK Sco and CS Cha have extra emission features near the 1550 ˚ A.</text>
<text><location><page_7><loc_8><loc_32><loc_48><loc_35></location>For DX Cha we will argue in section 8 that the peculiar shape of the C iv lines can be explained by the existence of a hot wind.</text>
<text><location><page_7><loc_8><loc_7><loc_48><loc_31></location>The strange appearance of C iv lines of RW Aur A is due to a bipolar outflow (See Figure 5). France et al. (2012) show that the H 2 lines from RW Aur A are redshifted by ∼ 100 km s -1 in at least two progressions ([v',J']=[1,4] and [1,7]), as they originate in the receding part of the outflow. In the observations we present here, the C iv doublet lines are blueshifted by ∼ -100 km s -1 , as can be seen from the position of the 1550 C iv line. This points to an origin in the approaching part of the outflow. The net result of these two outflows is that the blueshifted 1548 C iv line is buried under the redshifted H 2 emission. RW Aur A is the only unequivocal example of this coincidence in the current dataset. For HN Tau A the H 2 R(3) 1-8 line is also redshifted, although the redshift (+30 km s -1 ) is within the 2 σ of the error introduced by the pointing uncertainty (see also France et al. 2012). Therefore, HN Tau A may be another object for which we are observing two sides of the outflow, although</text>
<text><location><page_7><loc_52><loc_89><loc_92><loc_92></location>the velocities of the blueshifted and redshifted sides do not match as well as in RW Aur.</text>
<text><location><page_7><loc_52><loc_74><loc_92><loc_89></location>For DF Tau, the apparent red wing in the 1550 ˚ A line is an artifact of the line scaling, because for this star the ratio between the two C iv lines is almost 3, indicating either extra absorption or emission in one of the components. Unlike the case in RW Aur A, redshifted H 2 emission is not responsible for the extra emission, as the R(3) 1-8 line is observed at -167 km s -1 . Other observations show that the ratio between the two lines remains high among different epochs (Section 5.4). The origin of this extra emission in the 1548 ˚ A line or absorption in the 1550 ˚ A one remains unexplained in this work.</text>
<text><location><page_7><loc_52><loc_63><loc_92><loc_74></location>For RU Lup, the extra 'bump' 200 km s -1 to the red of the 1548 ˚ A C iv line is also present in other epoch observations of the system and remains unexplained in this work. AK Sco and CS Cha also present anomalous profiles, but of a different kind. AK Sco shows a broad emission -250 km s -1 from the 1550 ˚ A line, not present in the 1548 ˚ A line. CS Cha also shows and extra emission in the 1550 ˚ A line at ∼ -100 km s -1 and ∼ +200 km s -1 .</text>
<section_header_level_1><location><page_7><loc_60><loc_61><loc_83><loc_62></location>5.2. Non-parametric description</section_header_level_1>
<section_header_level_1><location><page_7><loc_53><loc_58><loc_90><loc_60></location>5.2.1. The velocity of the peak emission as a function of luminosity and accretion rate</section_header_level_1>
<text><location><page_7><loc_52><loc_48><loc_92><loc_57></location>The velocity of peak emission for each line is defined as the mean velocity of the top 5% of the flux between -100 and +100 km s -1 (for the 1548 C iv line) or between +400 and +600 km s -1 (for the 1550 ˚ A line). This is calculated from spectra that have been smoothed by 5point median. The velocity of peak listed in Table 5 is the average of both lines.</text>
<text><location><page_7><loc_52><loc_22><loc_92><loc_48></location>The C iv line is centered or redshifted for 21 out of 29 CTTSs, uncorrelated with line luminosity or accretion rate (Figure 6 - Top). The most significant exceptions are: DK Tau which shows two emission peaks, one blueshifted and the other redshifted, present in both lines; HN Tau A, for which both C iv lines have an emission peak at ∼ -80 km s -1 from their rest velocity 20 ; RW Aur A, for which the peak of the 1550 ˚ A line is at -86 km s -1 . In the case of AK Sco there is extra emission in the 1548 ˚ A line that may be due to Si i but there is also extra emission ∼ -250 km s -1 from the 1550 ˚ A line. It is unclear therefore, whether the C iv lines are centered or blueshifted, although both the Si iv and the N v lines (which follow the C iv lines) are well centered. Those four stars are among the higher accretors in the sample and have jets seen in the high-velocity components of forbidden lines (Hartigan et al. 1995). Four other CTTSs have slightly blueshifted peaks, but with velocities > -4 km s -1 .</text>
<text><location><page_7><loc_52><loc_17><loc_92><loc_22></location>The average velocity at maximum flux for the C iv CTTSs lines is V Max = 18 ± 4 km s -1 (Table 9. In this paper the uncertainties refer to uncertainties in the calculated mean values), not including the four stars with</text>
<text><location><page_7><loc_52><loc_7><loc_92><loc_16></location>20 We note that for HN Tau A, the radial velocity (4.6 ± 0.6 km s -1 ), derived by Nguyen et al. (2012), which we use here, deviates significantly from the velocity of the surrounding molecular cloud (21 km s -1 ; Kenyon & Hartmann 1995). Using the cloud velocity would shift the C iv lines to the blue even more. France et al. 2012 finds that the H 2 lines have maxima at ∼ 20 km s -1 from the rest velocity, although the line asymmetry suggest components at lower velocity.</text>
<text><location><page_8><loc_8><loc_77><loc_48><loc_92></location>strong blueshifts (DK Tau, HN Tau A, and RW Aur A for which outflowing material is contributing to the profile, nor AK Sco, for which the two C iv lines are different from each other). We have argued that the velocity uncertainties for this COS dataset are ∼ 7 km s -1 . A onesample Kolmogorov - Smirnov (KS) test confirms that the probability of the observed distribution being normal centered at zero, with 1 σ = 7 km s -1 is negligibly small. The conclusion is the same if we take 1 σ = 15 km s -1 , the nominal wavelength error for COS . In other words, the overall redshift of the CTTSs C iv lines is significant.</text>
<text><location><page_8><loc_8><loc_60><loc_48><loc_77></location>The WTTSs are also redshifted, with V Max = 11 ± 4 km s -1 . A two-sided KS test comparing the CTTSs and WTTSs V Max distribution gives p-value=0.6, which indicates that they are consistent with the null hypothesis. In other words, although the mean velocity of the WTTSs is ∼ 2 σ less that the mean CTTS velocity, the observations are consistent with the two quantities having the same distribution. Linsky et al. (2012) have shown that for rotation periods such as those observed in T Tauri stars, the Si iv and C iv lines in low-mass dwarfs present redshifts of ∼ 7 km s -1 , the result of gas flows produced by magnetic heating. Given the scatter in their sample, our redshifts in WTTSs are consistent with theirs.</text>
<section_header_level_1><location><page_8><loc_11><loc_56><loc_46><loc_59></location>5.2.2. The line width as a function of luminosity and accretion rate</section_header_level_1>
<text><location><page_8><loc_8><loc_49><loc_48><loc_56></location>Overall, most of the CTTSs C iv lines show detectable emission within ± 400 km s -1 of the nominal velocity. The FWHM is measured using the 1550 ˚ A line of the smoothed C iv spectrum. The smoothing is the same used to measure the velocity of maximum flux.</text>
<text><location><page_8><loc_8><loc_42><loc_48><loc_49></location>Figure 6 (Second row) shows that, as a group, the C iv WTTSs lines are narrower ( FWHM = 90 ± 10 km s -1 ) than the CTTSs ones ( FWHM = 210 ± 20 km s -1 ). The p-value of the KS test comparing the two samples is 0.02, implying that the difference is significant.</text>
<text><location><page_8><loc_8><loc_30><loc_48><loc_42></location>The WTTS also have a smaller FWHM range (from 60 to 145 km s -1 ) than the CTTSs (from 45 to 387 km s -1 ). Note that the FWHM of the WTTSs is comparable to that of the CTTSs in some stars, the result of the strong narrow component dominating the CTTS lines (see for example the C iv lines for DS Tau in Figure A.2). The FWHMclearly fails to capture most relevant information regarding the line as it does not take into account the multi-component nature of the lines.</text>
<text><location><page_8><loc_8><loc_23><loc_48><loc_30></location>The observed FWHM is uncorrelated with line luminosity or accretion rate. The plots in Figure 6 (Second row) suggest that the FWHM scatter increases with accretion rate, but this apparent increase is not statistically significant.</text>
<section_header_level_1><location><page_8><loc_11><loc_20><loc_46><loc_22></location>5.2.3. The skewness as a function of luminosity and accretion rate</section_header_level_1>
<text><location><page_8><loc_8><loc_7><loc_48><loc_19></location>In Figure 6 (Third row) we plot skewness versus C iv luminosity. The skewness compares the velocity of the peak to the mean velocity of the profile. It is defined as ( V Max -V ) / ∆ V , where V Max is the velocity at maximum flux and V is the flux-weighted mean velocity over an interval ∆ V centered on the maximum velocity. To calculate the skewness of the C iv line, we subtracted the continuum and interpolated over the H 2 lines present in the C iv intervals. For the 1548 and 1550 ˚ A lines, V is</text>
<text><location><page_8><loc_52><loc_86><loc_92><loc_92></location>measured within ± 250 km s -1 and ± 150 km s -1 from the maximum, respectively. To make the values comparable, we normalize to ∆ V = 250 km s -1 in each line. The 1548 ˚ A and 1550 ˚ A values are then averaged.</text>
<text><location><page_8><loc_52><loc_78><loc_92><loc_86></location>Qualitatively, values of skewness within ∼ ± 0 . 02 indicate a symmetric line. Positive values indicate a line extending to the red. The absolute value of the skewness serves as a quantitative measure of asymmetry. 70% (20/29) CTTS have skewness greater or equal to zero, with 52% (15/29) having positive ( > 0 . 02) skewness.</text>
<text><location><page_8><loc_52><loc_73><loc_92><loc_78></location>All the WTTS have skewness values consistent with symmetric lines. A KS test comparing both populations shows that the difference in asymmetry is significant (pvalue=0.002).</text>
<text><location><page_8><loc_52><loc_70><loc_92><loc_73></location>As with FWHM, the skewness is uncorrelated with line luminosity or accretion rate.</text>
<section_header_level_1><location><page_8><loc_54><loc_67><loc_90><loc_69></location>5.2.4. The line scaling as a function of luminosity and accretion rate</section_header_level_1>
<text><location><page_8><loc_52><loc_56><loc_92><loc_66></location>The fourth row panels of Figure 6 compare the ratios of the 1548 ˚ A to the 1550 ˚ A C iv lines as a function of L CIV (left) and ˙ M (right). These are not flux ratios, but the scaling factors used to match both line profiles in Figure A.6. As mentioned before (Section 5.1), the scaling factors between the C iv lines are indicative of the line's optical depth compared to the thermalization depth (Table 8).</text>
<text><location><page_8><loc_52><loc_46><loc_92><loc_55></location>All WTTSs and over half of the CTTSs (19/29 - 66%; 6/7 - 86% of TD, 12/22 - 55% of the non-TD) have ratios > 1.65, consistent within the errors with thin or effectively thin emission. The difference in line opacity between stars with TDs and those without is not significant. For the rest of CTTSs, 10/29 (34%) have ratios which are consistent with small absorption mean free paths.</text>
<text><location><page_8><loc_52><loc_38><loc_92><loc_46></location>We do not observe a correlation between this measure of opacity and either line luminosity or accretion rate. The apparent increase in scatter at high accretion rates is not significant, according to a two-sided KS test (pvalue=0.6). Furthermore, objects with C iv ratios < 1 . 7 are found for all luminosities and accretion rates.</text>
<text><location><page_8><loc_52><loc_33><loc_92><loc_38></location>As indicated in Section 5.1, DF Tau presents a particular case for which the scaling factor is significantly > 2, an impossible value unless there is extra emission in the 1548 ˚ A line or extra absorption in the 1550 ˚ A one.</text>
<section_header_level_1><location><page_8><loc_52><loc_29><loc_92><loc_32></location>5.2.5. The role of the inclination in the measured luminosity and accretion rate</section_header_level_1>
<text><location><page_8><loc_52><loc_19><loc_92><loc_29></location>Geometric explanations are often invoked in the literature to explain the shapes of these emission lines. For example, Lamzin et al. (2004) suggest that the observed accretion in TW Hya occurs at low stellar latitudes, and that the lack of separate line components coming from the pre- and post-shock in other stars may be due to equatorial layer accretion.</text>
<text><location><page_8><loc_52><loc_7><loc_92><loc_19></location>Figure 6 (Bottom) shows the relationship between inclination and line luminosity or accretion rate. We do not observe particularly larger or smaller fluxes at high or low inclinations. According to the KS test, the distributions of L CIV and ˙ M are the same between objects with i > 45 · and those with i ≤ 45 · . Also, the distribution of inclinations for objects with high ˙ M or high L CIV is statistically the same that for objects with low ˙ M or low L CIV . In particular, the apparently empty region</text>
<text><location><page_9><loc_8><loc_84><loc_48><loc_92></location>in the panel at high inclinations and low accretion rates is not significant and does not provide evidence that the disk or the accretion flow are obscuring the accretion diagnostics. Furthermore, we do not find any significant correlations between the line ratio, V Max , FWHM, or skewness, and inclination.</text>
<text><location><page_9><loc_8><loc_60><loc_48><loc_84></location>If the C iv lines originate in a localized accretion stream one would expect to detect more stars at low inclinations than at high inclinations: for face-on systems the accretion stream will always be visible, while for systems almost edge-on this is not the case, if the region below the disk is blocked from view. For a random distribution of accretion spot positions in the stellar hemisphere, we expect to see an accretion spot in 58% of stars with 0 · < i < 45 · and in 42% of stars with 45 · < i < 90 · (ignoring disk flaring). In our sample, we have 37% of targets with inclinations larger than 45 · , and the standard deviation of this count is 10%. Therefore, the observed difference in the number of stars with high and low inclinations is not significant. Over a hundred CTTSs with known inclinations are needed before we can distinguish 42% from 58% at the 3 σ level. The current inclination dataset is not complete enough to reveal geometric information about the C iv lines.</text>
<text><location><page_9><loc_8><loc_49><loc_48><loc_60></location>On the other hand, one could assume that the C iv UV lines are not emitted from a particular place but covers the whole star. For a given star, the C iv luminosity should then decrease linearly, by a factor of two, as the inclination increases from 0 · to 90 · . We do not observe this effect either, indicating, at least, that the observed scatter is dominated by intrinsic differences in the objects and not by geometry.</text>
<section_header_level_1><location><page_9><loc_11><loc_46><loc_46><loc_47></location>5.2.6. Conclusions from the non-parametric analysis</section_header_level_1>
<text><location><page_9><loc_8><loc_23><loc_48><loc_46></location>In conclusion, WTTS and CTTS C iv lines have comparable velocities at maximum flux, but the CTTS lines are generally broader and more asymmetric. In the case of CTTSs, neither the velocity of maximum flux, the FWHM, the skewness of the line, nor the ratio between the two C iv lines are correlated with C iv luminosity or accretion rate. The right column panels from Figure 6 do show increased scatter in these quantities as the accretion rate increases, suggesting that objects with large accretion rates have more diverse line shapes. However, the differences in the distributions (of the velocity, FWHM, skewness) with high and low accretion rates are not significant. More observations of stars with accretion rates ≤ 4 × 10 -9 M /circledot /yr are needed. Among the three pairs of shape quantities ( V Max vs. FWHM, FWHM vs. Skewness, V Max vs. Skewness) there are no significant correlations.</text>
<text><location><page_9><loc_8><loc_14><loc_48><loc_23></location>The scaling of the 1548 ˚ A line to the 1550 ˚ A one should be 2 if the lines are emitted from a thin or effectively thin medium. This is the case in all the WTTSs and in about 70% of CTTSs. This measure of the opacity is not correlated with accretion rate or line luminosity. However, in Section 5.3.1 we show that the NC of the line is correlated with accretion rate.</text>
<text><location><page_9><loc_8><loc_7><loc_48><loc_14></location>There are no correlations between line shape parameters and inclination or between inclination and accretion rate or line luminosity, but we conclude that the data are not complete enough for inclination to be a strong descriptor in the sample.</text>
<section_header_level_1><location><page_9><loc_60><loc_91><loc_84><loc_92></location>5.3. C IV Gaussian decomposition</section_header_level_1>
<text><location><page_9><loc_52><loc_74><loc_92><loc_90></location>Different regions of the T Tauri system may be contributing to the C iv lines. In the context of the magnetospheric accretion paradigm, the pre- and post-shock regions should be the dominant sources of the observed line emission. Shocks in an outflow, hot winds, and the stellar atmosphere may also contribute to the emission. To examine the line kinematics of the different regions we decompose each C iv line in one or two Gaussian functions. We are not asserting that the mechanism giving origin to the lines produces Gaussian shapes, although turbulent flows will do so (Gunther & Schmitt 2008). Representative decompositions are shown in Figure 2.</text>
<text><location><page_9><loc_52><loc_59><loc_92><loc_74></location>The primary goal of the Gaussian decomposition is to obtain widths and centroids for the main line components. According to the magnetospheric accretion model, post-shock emission lines should have small velocity centroid distribution about the stellar rest frame velocity, and if no turbulence is present, the lines should be narrow. Pre-shock emission lines, which likely originate in a larger volume upstream from the accretion flow (Calvet & Gullbring 1998), should have larger velocity centroid distribution in the stellar rest frame and broader lines.</text>
<text><location><page_9><loc_52><loc_37><loc_92><loc_59></location>Table 6 presents the results of fitting one or two Gaussians to each of the C iv lines, after subtracting the continuum and interpolating over the H 2 line R(3)1-8, and the Si i , C ii , Fe ii -complex, when present. For each target, we assume that the C iv lines are always separated by 500.96 km s -1 , and that they have the same shape. This results in a 4-parameter fit when fitting one Gaussian to each line: Height for the 1548 ˚ A line; height for the 1550 ˚ A line; width σ s; centroid velocity for the 1548 ˚ A line. When fitting two Gaussians to each line, we have an 8-parameter fit: For the 1548 ˚ A line, the heights of the broad (A BC ) and narrow (A NC ) components; analogous parameters for the 1550 ˚ A line; σ for the broad component; σ for the narrow component; centroid velocities for the broad (v BC ) and narrow (v NC ) components of the 1548 ˚ A line.</text>
<text><location><page_9><loc_52><loc_28><loc_92><loc_37></location>Table 6 also indicates whether the data were taken with COS (possible systematic wavelength error assumed to be 7 km s -1 ) or STIS (possible systematic wavelength error of 3 km s -1 ). We also list the parameters derived for the multi-epoch observations of BP Tau, DF Tau, DR Tau, RU Lup, and T Tau N that we will consider in Section 5.4.</text>
<text><location><page_9><loc_52><loc_12><loc_92><loc_28></location>Wood et al. (1997) analyzed the C iv lines of 12 stars with spectral types F5 to M0. They found that the observed line profiles could be better fit with both a narrow and a broad Gaussian component, than with a single Gaussian component. For the type of stars included in their sample (dwarfs, giants, spectroscopic binaries, and the Sun), the C iv line is a transition-region line, as it is in the WTTSs. Because of this, we fitted both NCs and BCs to all the WTTSs, except V410 Tau. For V410 Tau, each C iv line can be well fitted with only one NC, although this may just be the result of the low S/N in the spectrum.</text>
<text><location><page_9><loc_52><loc_8><loc_92><loc_12></location>When comparing the two C iv lines in section 5.1, we mentioned that AK Sco, CS Cha, DX Cha, DF Tau, RU Lup, and RW Aur A were objects for which the line dou-</text>
<text><location><page_10><loc_8><loc_81><loc_48><loc_92></location>blet members had different shapes and/or extra unidentified emission in one of the doublet members. We do not perform the Gaussian decomposition for RW Aur A or DX Cha. For AK Sco, we list the parameters derived from the Gaussian fits (Table 6), but we do not use these results when exploring correlations. For the rest, we interpolate the profiles over the apparent the extra emission.</text>
<text><location><page_10><loc_8><loc_58><loc_48><loc_81></location>For 4 objects (CY Tau, DM Tau, ET Cha, and UX Tau A) we decompose the C iv lines in only one broad Gaussian component. For the rest of the CTTSs, the decomposition requires both a narrow and a broad Gaussian components, and Figure 7 shows the distributions of velocity centroids and FWHM. Average velocity and FWHM values are given in Table 9. Typical full widths at half maxima of CTTS NCs range from 50 to 240 km s -1 , with an average of 130 km s -1 , while BCs widths range from 140 to a 470 km s -1 , with an average of 350 km s -1 . The velocity centroids range from -100 km s -1 to 200 km s -1 . The BC velocity is larger than the NC velocity in 70% of the CTTS sample, and the distribution of BC velocities tend to be more positive ( V BC ∼ 40 km s -1 ) than that of the NC velocities ( V NC ∼ 30 km s -1 ), giving some of the profiles the characteristic 'skewed to the red' shape.</text>
<section_header_level_1><location><page_10><loc_12><loc_56><loc_45><loc_57></location>5.3.1. The optical depth as a function of accretion</section_header_level_1>
<text><location><page_10><loc_8><loc_28><loc_48><loc_55></location>Figure 8 (Top) shows the ratio of the height of the NC in the 1548 ˚ A line to the NC in the 1550 ˚ A C iv line as a function of accretion rate, as well as the ratio of the heights of the BCs. Most observations are grouped around 2 although there is a lot of scatter, particularly at high accretion rates. The plot reveals that the NC of DF Tau is anomalous ( > 2), while the BC has an optically thin ratio. Individual Gaussian components of other objects (CS Cha, DR Tau) are also anomalous. Furthermore, there is a population of objects with NC ratios close to 1: AA Tau, DE Tau, DR Tau, RU Lup,, SU Aur, and T Tau. However, the difference in the distribution of BC ratios and NC ratios is marginal according to the KS test (p-value=6%), and the correlation of the ratio of NC luminosities with accretion rate is not significant (Pearson's r=-0.37, p-value=10%). We conclude that for most objects, both the NC and BC ratios are close to 2, although there is considerable scatter, and that there are some peculiar objects at accretion rates > 4 × 10 -9 M /circledot /yr.</text>
<text><location><page_10><loc_8><loc_7><loc_48><loc_28></location>The bottom panel of Figure 8 shows that the contribution of the NC to the overall profile increases with accretion rate. For low accretion rates ( < 4 × 10 -9 M /circledot /yr), the average NC contribution to the luminosity is ∼ 20% while for high accretion rates it is ∼ 40% on average. While the Pearson's r=0.4 (p-value=6%) suggest that this correlation is not significant, this statistical test assumes that the quantities being compared are sampled from a bivariate Gaussian distribution. This is likely not the correct assumption for the heterogeneous sample of CTTSs being considered here. A better correlation test uses the Kendall rank correlation statistic, which considers only (non-parametric) rank orderings between the data (Feigelson & Jogesh Babu 2012). The Kendall's τ =0.30 (p-value=0.05) is at the threshold of what we consider significant.</text>
<text><location><page_10><loc_52><loc_80><loc_92><loc_92></location>If the increase in accretion rate is due to larger accreting area or larger density, and both the narrow and broad components are emitted from regions that are optically thin or effectively thin, no correlation should be observed. This is because the rate of collisional excitation will change linearly with density and both lines will increase at the same rate. The observed correlation implies that the region responsible for the BC may be becoming optically thick at high accretion rates.</text>
<text><location><page_10><loc_52><loc_71><loc_92><loc_80></location>As in main-sequence stars, the WTTS line shapes (blue labels in Figure 8) are characterized by a strong NC and a weak BC (Wood et al. 1997; Linsky et al. 2012) and tend to have stronger NC contributions to the total flux than low accretion rate CTTSs. The luminosity in NCs and BCs, in both WTTSs and CTTSs increases with total C iv luminosity (not shown).</text>
<text><location><page_10><loc_52><loc_48><loc_92><loc_71></location>The bottom panel of Figure 8 raises the issue of the transition from WTTS to CTTS as a function of accretion rate. The WTTSs generally have strong NC, while low accretion rate objects have very weak NC, compared to BC. Does the accretion process suppress the NC present in the WTTSs or does it enhance the BC? We believe the latter to be true. The lowest accretion rate object shown in Figure 8 (Bottom) is EP Cha (RECX 11). For this star the total C iv luminosity is ∼ 6 times larger than for most of the WTTSs, with the exception of V1068 Tau. However, the NC luminosity of EP Cha is 4 × 10 -6 L /circledot , similar to that of a low-luminosity WTTSs. In other words, most of the extra C iv luminosity that distinguishes this CTTS from the WTTSs is due to the generation of the BC. This suggest that the accretion process generates first a BC, with the NC becoming increasingly important at larger accretion rates.</text>
<text><location><page_10><loc_52><loc_17><loc_92><loc_48></location>From this it follows that all accreting stars should show a BC. In average, the flux in the BC grows at a slower rate than the flux in the NC as the accretion rate increases, but it is not clear why some stars develop a strong NC (like DS Tau) while others do not (like GM Aur). These statements may not be valid outside the mass or accretion rate range of the sample considered here. For example, France et al. (2010) present UV spectra of the brown dwarf 2M1207, from which they derive an accretion rate of 10 -10 M /circledot /yr, comparable to that of EP Cha. This value is derived using the calibration between C iv flux and accretion rate from Johns-Krull et al. (2000). On the other hand Herczeg et al. (2009) derives an accretion rate of 10 -12 M /circledot /yr based on the Balmer excess emission. At any rate, each C iv line can be fit with only one, very narrow, Gaussian component with FWHM=36.3 ± 2.3 km s -1 . 21 This is then a case of a low-mass accreting young object without a BC. The very narrow C iv lines may be the result of the smaller gas infall velocity in 2M1207 ( ∼ 200 km s -1 , using the stellar parameters from Riaz & Gizis 2007) compared to the sample presented here ( ∼ 300 km s -1 ). The smaller gas infall velocity will result in lower turbulence broadening.</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_16></location>21 In their published analysis, France et al. (2010) fit the C iv lines of 2M1207 with two Gaussian components each. However, that analysis is based on an early reduction of the COS data. A new re-processing and new Gaussian decomposition shows that the C iv line can be fit with one Gaussian component (K. France, personal communication).</text>
<text><location><page_11><loc_10><loc_89><loc_46><loc_92></location>5.3.2. The kinematic predictions of the magnetospheric accretion model</text>
<text><location><page_11><loc_8><loc_79><loc_48><loc_88></location>For the magnetospheric accretion model, the gas speed in the accretion flow before the shock should reach velocities ∼ 300 km s -1 for typical stellar parameters (Calvet & Gullbring 1998), although the interplay between line-of-sight and the complex magnetospheric structure may result in line-of-sight velocities smaller than this.</text>
<text><location><page_11><loc_8><loc_61><loc_48><loc_78></location>If it originates primarily in the accretion shock region, the C iv line emission (as well as the Si iv , N v , and He ii lines) comes from plasma very close to the stellar surface and the gas in this region should be moving away from the observer. Therefore, we expect that the velocity of the flow should be positive (v BC > 0, v NC > 0). If the dominant emission in the broad and narrow components comes only from the pre- and post-shock regions, respectively, then v BC > v NC . The velocity of the post-shock gas decreases after the shock surface, and depending on the origin of our observational diagnostic, we may observe velocities ranging from 4 × less than the pre-shock gas velocity to zero (Lamzin 1995, 2003a,b).</text>
<text><location><page_11><loc_8><loc_44><loc_48><loc_61></location>Observationally, v BC and v NC are uncorrelated with accretion rate or line luminosity. Figure 9 compares the velocity centroids of the broad and narrow components, for 22 CTTSs and 5 WTTSs. Black labels indicate CTTSs observed with COS , while orange labels are CTTSs observed with STIS . The plot also shows, in red, the velocity centroids of the WTTSs, all observed with COS . The errors in these velocities are dominated by the systematic wavelength scale error, illustrated by the boxes on the upper-left corner of Figure 9. We have argued in Section 5.2 that the errors in the COS dataset are ∼ 7 km s -1 . Errors in the wavelength scale cause the data points to move parallel to the dotted line.</text>
<text><location><page_11><loc_8><loc_27><loc_48><loc_44></location>We focus first on the upper right quadrant of Figure 9, those objects with both v BC > 0 and v NC > 0. The hatched region of the plot corresponds to v BC ≥ 4 v NC . As we can see, only TW Hya and CY Tau reside fully within this region. In addition, errors in the wavelength calibration may explain why some objects (AA Tau, BP Tau, CS Cha, DE Tau, DN Tau, DS Tau, EP Cha, MP Mus, V1079 Tau, and V4046 Sgr) reside away from the hatched region. Note that while the rotation of the star may contribute to v BC and v NC , typical values of v sin i for CTTSs are 10 to 20 km s -1 , and so the velocity pair would only move to higher or lower velocities by up to this amount, parallel to the dotted line.</text>
<text><location><page_11><loc_8><loc_13><loc_48><loc_27></location>The other 10 objects are 'anomalous,' either because one or both of the velocities is negative and/or v BC /lessorsimilar v NC , beyond what could be explained by pointing errors. Stellar rotation alone will not bring these objects to the hatched region. As a group, they all have accretion rates larger than 4 × 10 -9 M /circledot /yr, and they make up half of the objects with accretion rates this large or larger. From the point of view of the magnetospheric accretion paradigm, they present a considerable explanatory challenge.</text>
<text><location><page_11><loc_8><loc_7><loc_48><loc_13></location>In turn, these anomalous objects come in two groups: those for which both components are positive or close to zero but v BC < v NC (DF Tau, perhaps DR Tau, GM Aur, RU Lup, SU Aur), and those objects for which one or both of the components are negative (DK Tau, HN Tau,</text>
<text><location><page_11><loc_52><loc_89><loc_92><loc_92></location>IP Tau, T Tau N, V1190 Sco). Emission from the latter group may have contributions from outflows or winds.</text>
<section_header_level_1><location><page_11><loc_55><loc_85><loc_89><loc_87></location>5.4. The accretion process in time: multi-epoch information in C IV</section_header_level_1>
<text><location><page_11><loc_52><loc_79><loc_92><loc_84></location>Line flux variability in CTTSs occurs on all time scales, from minutes to years. It is therefore relevant to ask how does the line shape change in time and what is the impact of these changes on the general statements we have made.</text>
<text><location><page_11><loc_52><loc_59><loc_92><loc_79></location>High-resolution multi-epoch observations of the C iv lines are available for a subset of our objects, plotted in Figure 10. The spectra were obtained from Ardila et al. (2002) (GHRS observations: BP Tau, DF Tau, DR Tau, RU Lup, T Tau, and RW Aur), Herczeg et al. (2005) ( STIS : RU Lup), Herczeg et al. (2006) ( STIS : DF Tau) and the MAST archive (HST-GO 8206; STIS : DR Tau). The epoch of the observations is indicated in the figure. Note that the apertures of the three spectrographs are different. The GHRS observations were performed with the Large Science Aperture (2' × 2' before 1994, 1.74' × 1.74' after 1994), the Primary Science Aperture for COS is a circle 2.5' in diameter, and the STIS observations where obtained with the apertures 0.2' × 0.06' (for T Tau) or 0.2' × 0.2' (for the other objects).</text>
<text><location><page_11><loc_52><loc_24><loc_92><loc_59></location>For DR Tau the DAO observations show double-peaked C iv emission as well as strong emission in the blue wing of the 1548 ˚ A line. We have identified the additional blue wing emission as a mixture of H 2 , Si i , C ii , and Fe ii . The parameters that we have listed for DR Tau in Table 6 provide a good fit to the overall profile, but not to the double-peaked C iv emission. In the GHRS observation (red, from 1995) the extra Si i , C ii , and Fe ii emissions are not present, leaving only a low S/N H 2 line. The low-velocity peak of the C iv line observed in the DAO data is not present in the GHRS data. In addition to this 1995 observation, Ardila et al. (2002) describes a 1993 observation (not shown in Figure 10), which shows blueshifted emission at -250 km s -1 present in both line components. The STIS observations from HST program GO 8206 (PI Calvet) show a strong H 2 line. The centroids of the C iv lines are either redshifted by ∼ 200 km s -1 or the centers (within ± 150 km s -1 of the rest velocity) are being absorbed. If redshifted, this is the largest redshift observed in the sample, although a comparable shift is seen in v BC for CS Cha (Figure A.3). We may be observing extended C iv emission that is not seen in the narrow-slit STIS observations (Schneider et al. 2013), C iv absorption from a turbulent saturated wind or the disappearing of the accretion spot behind the stellar limb as the star rotates.</text>
<text><location><page_11><loc_52><loc_16><loc_92><loc_24></location>T Tau shows a change in the H 2 emission and a small decrease in the strength on the NC. Walter et al. (2003); Saucedo et al. (2003) show that the H 2 emission around T Tau is extended over angular scales comparable to the GHRS aperture, and the smaller STIS flux is due to the smaller aperture.</text>
<text><location><page_11><loc_52><loc_8><loc_92><loc_16></location>For DF Tau, the ratio between the two line members remain anomalously high, ∼ 3 in all epochs. For BP Tau, a decrease in the flux is accompanied by a decrease in both components, although the NC decreases more strongly. Changes over time in the velocity centroids are not significant.</text>
<text><location><page_11><loc_53><loc_7><loc_92><loc_8></location>For RW Aur A the difference between GHRS and STIS</text>
<text><location><page_12><loc_8><loc_69><loc_48><loc_92></location>DAO observations is also dramatic. As we have shown, the DAO observations can be explained by assuming that we are observing two sides of a bipolar outflow. The GHRS flux is larger, and dominated by three peaks, the bluemost of which is likely the R(3)1-8 H 2 line. The other two do not match each other in velocity and so they cannot both be C iv . Errico et al. (2000) have suggested that the early GHRS observations may be affected by Fe ii absorption. We do not find evidence for Fe ii absorption within the C iv lines for any other star, nor for the DAO observations of RW Aur, and so we discard this possibility. Based on the spectroscopic and photometric variability, Gahm et al. (1999) have suggested that a brown dwarf secondary is present in the system, although its role in the dynamics of the primary is uncertain. The COS aperture is larger, suggesting that the changes are due to true varaibility.</text>
<text><location><page_12><loc_8><loc_57><loc_48><loc_69></location>Dramatic changes are also seen in RU Lup, as noted by Herczeg et al. (2006) and France et al. (2012). The C iv line, which is almost absent in the GHRS observations is 4 × stronger in the STIS observations. The DAO observations presented here are similar to the latter, although the strength of the extra Si i , C ii , and Fe ii emissions is also variable. The excess emission in the blue wing of the 1548 ˚ A line described in Section 5.1 is also present in the STIS observations.</text>
<text><location><page_12><loc_8><loc_49><loc_48><loc_57></location>For all of the multi-epoch CTTSs observations except RW Aur A, Figure 6 (Fourth row) shows the path that the line ratio (the optical depth indicator) follows as a function of line luminosity. The changes in the value of the line ratio are not correlated with the changes in the C iv line luminosity, as we observed before.</text>
<section_header_level_1><location><page_12><loc_12><loc_46><loc_45><loc_47></location>5.5. Conclusions from the parametric analysis</section_header_level_1>
<text><location><page_12><loc_8><loc_32><loc_48><loc_46></location>Most CTTSs C iv line profiles can be decomposed into narrow ( FWHM ∼ 130 km s -1 ) and broad ( FWHM ∼ 350 km s -1 ) components, with the BC redshifted with respect to the NC in 70% of the CTTSs sample. The fractional contribution to the flux in the NC increases (from ∼ 20% to 40% on average) and may become more optically thick with increasing accretion rate. Strong narrow components will be present in objects of high accretion rate, but high accretion rate by itself does not guarantee that the C iv lines will have strong NCs.</text>
<text><location><page_12><loc_8><loc_11><loc_48><loc_32></location>The component velocities in about 12 out of 23 CTTSs are roughly consistent with predictions of the magnetospheric accretion model, in the sense that v BC > 0, v NC > 0 and v BC /greaterorsimilar 4 v NC . For most of the 12 the NC velocity seems too large or the BC velocity too small, compared to predictions, although this may be the result of errors in the wavelength calibration. For 11 of the CTTSs the kinematic characteristics of the C iv line cannot be explained by the magnetospheric accretion model: these objects (AK Sco, DF Tau, DK Tau, DR Tau, GM Aur, HN Tau A, IP Tau, RU Lup, SU Aur, T Tau N, V1190 Sco) have BC velocities smaller than their NC velocities, or one of the velocity components is negative. An examination of the systematic pointing errors which produce offsets in the wavelength scale lead us to conclude that these do not impact the conclusions significantly.</text>
<text><location><page_12><loc_8><loc_7><loc_48><loc_11></location>Multi-epoch observations reveal significant changes in morphology in all the lines, from one epoch to the next. The line velocity centroids remain relatively constant in</text>
<text><location><page_12><loc_52><loc_84><loc_92><loc_92></location>low accretion rate objects such as DF Tau and BP Tau, but the overall line appearance change significantly for high accretion rate CTTSs, like DR Tau. In the case of RU Lup, the observed variability of the line may be consistent with the accretion spot in C iv coming in and out of view.</text>
<section_header_level_1><location><page_12><loc_65><loc_82><loc_78><loc_83></location>6. THE HE II LINE</section_header_level_1>
<text><location><page_12><loc_52><loc_63><loc_92><loc_82></location>The 1640 ˚ A He ii line is the 'Helium α ' line, analogous to neutral hydrogen line H α , corresponding to a hydrogenic de-excitation from level 3 to level 2 (Brown et al. 1984). In principle, the line is a blend of 1640.33 ˚ A, 1640.34 ˚ A, 1640.37 ˚ A, 1640.39 ˚ A, 1640.47 ˚ A, 1640.49 ˚ A, and 1640.53 ˚ A. Of these, 1640.47 ˚ A ( Log ( gf ) = 0 . 39884) should dominate the emission, followed by 1640.33 ˚ A ( Log ( gf ) = 0 . 14359), as the recombination coefficient is largest to 3d 1 D (Osterbrock 1989). He ii is strongly correlated with C iv , implying, as with Si iv , that they are powered by the same process (Johns-Krull et al. 2000; Ingleby et al. 2011; G'omez de Castro & Marcos-Arenal 2012; Yang et al. 2012).</text>
<text><location><page_12><loc_52><loc_48><loc_92><loc_63></location>For a sample of 31 CTTSs Beristain et al. (2001) modeled the He ii 4686 ˚ Aline with a single Gaussian function. Those lines are narrow, with an average FWHM of 52 km s -1 and somewhat redshifted, with an average centroid of 10 km s -1 . Based on the high excitation energy of the line (40.8 eV), and the redshift in the velocity centroid, Beristain et al. (2001) argued that the He ii optical emission originated from the post-shock gas. Below we find that the centroid shifts in the He ii UV line are comparable with the average values reported by Beristain et al. (2001) for the He ii optical line.</text>
<text><location><page_12><loc_52><loc_39><loc_92><loc_48></location>If the He ii 1640 ˚ A line originates in the accretion postshock we expect it to be redshifted, similar in shape to the NC of the C iv line, although with lower velocity centroids as it will be emitted from a cooler, slower region of the post shock. Below we show that, indeed, the shape is similar to the NC C iv line, and the velocity centroid is smaller.</text>
<text><location><page_12><loc_52><loc_23><loc_92><loc_39></location>In Figure A.7 we compare the 1550 ˚ A C iv line to the He ii line, scaled to the same maximum flux. The He ii lines are similar to the NCs of the C iv lines, if present, although the latter appears slightly redshifted with respect to He ii . Most He ii lines can be described as having a strong narrow core and a low-level broad component. Significant emission is present within ± 200 km s -1 of the nominal wavelength. The exceptions to this description are HN Tau A and RW Aur A, which are blueshifted and present a strong BC, and DX Cha, for which no He ii is observed in the background of a strong continuum.</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_23></location>G'omez de Castro & Marcos-Arenal (2012) state that He ii is observed only in a subset of the stars that show C iv emission. In their sample of ACS and IUE lowresolution spectra, 15 stars show He ii out of the 20 stars that present C iv emission. We do not confirm this statement, as we observe the He ii line in all of the stars in our sample. The difference between detection rates is likely due to the differing spectral resolutions and sensitivities between our sample (R > 10000) and theirs (R ∼ 40 at 1640 ˚ A).</text>
<text><location><page_12><loc_52><loc_7><loc_92><loc_9></location>It is well-known that the C iv and He ii luminosities are correlated to each other (e.g. Ingleby et al. 2011;</text>
<text><location><page_13><loc_8><loc_80><loc_48><loc_92></location>Yang et al. 2012). The C iv -to-He ii luminosity ratio measured here is significantly larger for CTTSs (3 . 5 ± 0 . 4) than for WTTSs (1 . 3 ± 0 . 2). Ingleby et al. (2011) noted that the C iv -to-He ii luminosity ratio is close to one in field stars, but larger in CTTSs. The fact that we measure a ratio close to unity also in WTTSs supports Ingleby et al. (2011)'s assertion that the C iv -to-He ii luminosity ratio is controlled by accretion, and not by the underlying stellar chromosphere (Alexander et al. 2005).</text>
<text><location><page_13><loc_8><loc_73><loc_48><loc_80></location>For the purposes of a deeper analysis we first use the non-parametric measurements (the velocity of maximum flux, the skewness and the FWHM, Table 5) to describe the line. We also fit one or two Gaussians to the lines (Table 7).</text>
<section_header_level_1><location><page_13><loc_13><loc_69><loc_43><loc_72></location>6.1. The He II line shape: non-parametric measurements</section_header_level_1>
<text><location><page_13><loc_8><loc_54><loc_48><loc_69></location>We find that, except for HN Tau A and RW Aur A, the CTTSs He ii lines are well-centered or slightly redshifted: the average V Max ignoring these two stars is 7 ± 3 km s -1 (Figure 11). As with C iv , we conclude that the error on the COS wavelength scale for He ii should be smaller than the nominal value. This is because only 7 out of 23 objects (again ignoring HN Tau A and RW Aur A) have V Max < 0, a situation expected to occur with negligible probability if the wavelength errors are normally distributed around zero km s -1 . The difference in redshift between CTTSs and WTTSs is not significant.</text>
<text><location><page_13><loc_8><loc_50><loc_48><loc_54></location>The C iv lines are redder than the He ii lines. For the CTTSs V Max CIV -V Max HeII = 11 ± 4 km s -1 . For WTTSs, the difference is 7 ± 6 km s -1 (Table 9).</text>
<text><location><page_13><loc_8><loc_42><loc_48><loc_50></location>The FWHM for He ii CTTSs range range from ∼ 50 to 400 km s -1 , with an average of 96 ± 9 km s -1 , which is significantly narrower than in C iv . On the other hand, the FWHM of He ii lines of CTTSs and WTTSs are consistent with being drawn from the same sample, according to the KS test.</text>
<text><location><page_13><loc_8><loc_34><loc_48><loc_42></location>The average skewness for the CTTS He ii sample is 0 . 01 ± 0 . 01, whereas the skewness of the WTTS sample is -0 . 01 ± 0 . 01: the difference between the two populations is not significant. The He ii lines are significantly more symmetric (as measured by the absolute skewness) than the C iv lines.</text>
<text><location><page_13><loc_8><loc_30><loc_48><loc_34></location>As with C iv , neither the velocity shift, FWHM, or skewness are correlated with line luminosity, accretion rate or inclination.</text>
<section_header_level_1><location><page_13><loc_10><loc_27><loc_46><loc_29></location>6.2. The He II line shape: Gaussian decomposition</section_header_level_1>
<text><location><page_13><loc_8><loc_14><loc_48><loc_27></location>Most He ii lines require a narrow and a broad Gaussian component to fit the core and the wings of the line, respectively. The average of the ratio of the He ii NC line luminosity to the total He ii line luminosity is 0.6 and uncorrelated with accretion rate, and it is the same in CTTSs and WTTSs. This large contribution of the NC gives the lines their sharp, peaked appearance. As shown in Figure 7 the BC of the He ii lines span a much broader range than those of C iv , and their distributions are significantly different.</text>
<text><location><page_13><loc_8><loc_7><loc_48><loc_14></location>Overall, this Gaussian decomposition confirms the conclusions from the non-parametric analysis. The average value of v NC for He ii is the same for the CTTS and WTTS (V CTTS NC HeII - V WTTS NC HeII =2 ± 3 km s -1 ), the C iv CTTS line is redshifted with respect to</text>
<text><location><page_13><loc_52><loc_86><loc_92><loc_92></location>the He ii line (V CTTS NC CIV - V CTTS NC HeII =20 ± 6 km s -1 ), and for WTTSs the velocities of He ii and C iv are the same (V WTTS NC CIV - V WTTS NC HeII = -2 ± 5 km s -1 ).</text>
<text><location><page_13><loc_52><loc_77><loc_92><loc_86></location>Based both on the non-parametric analysis and the Gaussian decomposition we conclude that the He ii line is comparable in terms of redshift and FWHM in WTTSs and in CTTSs. The line is blueshifted with respect to C iv CTTSs but has the same velocity shift as a C iv WTTS line, and as the NC of the CTTSs in C iv . We discuss this further in Section 8.</text>
<section_header_level_1><location><page_13><loc_56><loc_75><loc_88><loc_76></location>7. SI IV AND N V: ANOMALOUS ABUNDANCES</section_header_level_1>
<text><location><page_13><loc_52><loc_66><loc_92><loc_74></location>Analyses as detailed as those performed before are not possible for Si iv and N v : the lines are weaker and the extra emissions and absorptions due to other species make a detailed study of the line shape unreliable. Here we provide a high-level description of the lines and compare their fluxes to that of C iv .</text>
<section_header_level_1><location><page_13><loc_64><loc_64><loc_80><loc_65></location>7.1. Si IV description</section_header_level_1>
<text><location><page_13><loc_52><loc_34><loc_92><loc_63></location>The two Si iv lines are separated by 1938 km s -1 (see Figures A.1 to A.5). The wavelength of the Si iv doublet members coincides with the bright H 2 lines R(0) 0-5 (1393.7 ˚ A, -9 km s -1 from the 1394 ˚ A line), R(1) 0-5 (43 km s -1 from the 1394 ˚ A line), and P(3) 0-5 (1402.6 ˚ A, -26 km s -1 from the 1403 ˚ A line). The strong narrow line between the two Si iv lines is H 2 P(2) 0-5 (1399.0 ˚ A, 1117 km s -1 away from the 1394 ˚ A Si iv line). Additional H 2 lines (R(2) 0-5, 1395.3 ˚ A; P(1) 0-5, 1396.3 ˚ A; R(11) 2-5, 1399.3 ˚ A) are observed between the doublet members of DF Tau and TW Hya. For this paper, the H 2 lines are considered contaminants in the spectra and we will ignore them. A detailed study of their characteristics has appeared in France et al. (2012). The panels in the appendix also show the O iv line at 1401.16 ˚ A, sometimes observed as a narrow emission line blueward of the 1403 ˚ A Si iv line (see for example DE Tau). We have also indicated the position of the CO 5-0 bandhead (France et al. 2011). In the case of WTTSs, no H 2 lines are observed in the Si iv region and narrow well-centered Si iv lines characterize the emission.</text>
<text><location><page_13><loc_52><loc_15><loc_92><loc_34></location>To the extent that the Si iv lines can be seen under the H 2 emission, their shape is similar to that of the C iv lines, as was suggested in Ardila et al. (2002) (see for example IP Tau in Fig. A.3). A notable exception is the WTTS EG Cha, for which the Si iv lines have broad wings that extend beyond ± 500 km s -1 , not present in C iv . Because the observations were taken in TIMETAG mode, we have time-resolved spectra of the Si iv region that shows sharp increase in the count rate ( ∼ 10 × in 300 secs) followed by a slow return to quiescence over 1000 secs. These observations indicate that the WTTS EG Cha was caught during a stellar flare. The broad line wings are observed only during the flare. A detailed study of this flare event is in preparation.</text>
<section_header_level_1><location><page_13><loc_64><loc_13><loc_79><loc_14></location>7.2. N V description</section_header_level_1>
<text><location><page_13><loc_52><loc_7><loc_92><loc_13></location>The two N v lines are separated by 964 km s -1 . Figures A.1 to A.5 also indicate the positions of the H 2 lines R(11) 2-2 (1237.54 ˚ A), P(8) 1-2 (1237.88 ˚ A), and P(11) 1-5 (1240.87 ˚ A). In addition to the H 2 lines, we note</text>
<text><location><page_14><loc_8><loc_87><loc_48><loc_92></location>the presence of N i absorptions (Herczeg et al. 2005) at 1243.18 ˚ A and 1243.31 ˚ A (1055 km s -1 and 1085 km s -1 from the 1239 ˚ A line).</text>
<text><location><page_14><loc_8><loc_77><loc_48><loc_87></location>The two N i lines are clearly seen in the 1243 N v member (the red line) of EP Cha (Figure A.4). N i absorptions are observed in most CTTSs (Exceptions: AA Tau - N i in emission, DN Tau, EG Cha; Uncertain: AK Sco, CV Cha, CY Tau). Some objects (e.g. RU Lup) show a clear wind signature in N i , with a wide blueshifted absorption which in this object absorbs most of the N v line.</text>
<text><location><page_14><loc_8><loc_56><loc_48><loc_77></location>The 1239 ˚ A line of N v (the blue line, not affected by extra absorption) and the 1550 ˚ A line of C iv (the red line, not affected by H 2 ) have comparable wing extensions, velocity centroids, and overall shapes. As for C iv and Si iv , the doublet lines of N v should be in a 2:1 ratio if effectively thin although the presence of the N i line makes estimating the ratio impossible. For CTTSs there are other unidentified absorption sources in vicinity of the 1243 ˚ A N v line. These can be seen most easily in the spectra of DE Tau ( -32 km s -1 ), TW Hya (-87 km s -1 , -32 km s -1 ), GM Aur, and V4046 Sgr (50 km s -1 ) with respect to the rest velocity of the 1243 N v line. It is unclear if these are N i absorption features in a highly structured gas flow or absorptions from a different species.</text>
<text><location><page_14><loc_8><loc_40><loc_48><loc_56></location>For the WTTSs both N v lines are copies of each other, and copies of the WTTS Si iv , C iv , and He ii lines. Therefore, the extra absorption in the CTTSs are due to the 'classical' T Tauri Star phenomena: they may represent absorption in a velocity-structured wind, or in a disk atmosphere. In particular, the N i feature is likley due to absorption in the CTTSs outflow or disk. We note here the similar excitation N i absorption lines are observed at 1492.62 ˚ A and 1494.67 ˚ A for all stars that show N i absorption in the N v region. A detailed study of the wind signatures in the sample will appear in a future paper.</text>
<section_header_level_1><location><page_14><loc_20><loc_38><loc_37><loc_39></location>7.3. Flux measurements</section_header_level_1>
<text><location><page_14><loc_8><loc_35><loc_48><loc_37></location>Figure 12 shows the relationship between the C iv luminosity and the Si iv and N v luminosities.</text>
<text><location><page_14><loc_8><loc_17><loc_48><loc_35></location>To measure the flux in the Si iv lines (Table 8), we have integrated each of them between -400 and 400 km s -1 , interpolating over the contaminating H 2 lines. The Si iv lines are much broader than the H 2 ones, and at these resolutions can be separated from them, at least in the cases in which the Si iv lines are actually observed. Assuming the H 2 lines are optically thin, the emission in R(0) 0-5 should be 2 × weaker than P(2) 0-5 and R(1) 0-5 should be 1.4 × weaker than P(3) 0-5, as explained in Ardila et al. (2002). In 11 objects no Si iv line is seen under the H 2 emission. For these stars, we give the 3 σ upper limit to the flux, over the same velocity range we used to measure the C iv line in the same star.</text>
<text><location><page_14><loc_8><loc_12><loc_48><loc_17></location>Table 8 also lists the flux in the N v lines. Note that the flux in the 1243 ˚ A line is the observed flux, and it has not been corrected for N i absorption or by any of the other absorptions mentioned above.</text>
<text><location><page_14><loc_8><loc_7><loc_48><loc_12></location>Figure 12 shows that the C iv and Si iv luminosities of CTTSs are correlated (log L SiIV /L /circledot = (0 . 9 ± 0 . 6) + (1 . 4 ± 0 . 2) log L CIV /L /circledot , ignoring the non-detections and the WTTSs). To understand the nature of this correla-</text>
<text><location><page_14><loc_52><loc_79><loc_92><loc_92></location>on, we perform the following calculation. We assume that each line has a contribution from two regions, a preand a post-shock, and that the ratio of the contributions between the two regions is the same in Si iv and in C iv . In other words, L SiIV pre L SiIV post = L CIVpre L CIVpost . This is likely appropriate if all the emission regions contributing to the line are optically thin. If this is the case, the total luminosity in each line can be shown to be proportional to the post-shock luminosity, and we have:</text>
<formula><location><page_14><loc_54><loc_74><loc_89><loc_77></location>L SiIV L CIV = L SiIV post L CIVpost /similarequal N SiIV N CIV Ab SiIV Ab CIV C SiIV C CIV λ CIV λ SiIV</formula>
<text><location><page_14><loc_69><loc_72><loc_75><loc_73></location>= 0 . 111</text>
<text><location><page_14><loc_52><loc_57><loc_92><loc_71></location>further assuming that the post-shock gas is an optically thin plasma in collisional equilibrium and the emission measure (EM) is the same for both lines over the emitting region. N x is the fraction of the species in that ionization state, at that temperature, Ab x is the number abundance of the element and C x is the collisional excitation rate. We assume a Si/C number abundance ratio of 0.13 for the present day Sun (Grevesse et al. 2007). To calculate the collision rate we use the analytical approximation (Burgess & Tully 1992; Dere et al. 1997):</text>
<formula><location><page_14><loc_64><loc_53><loc_80><loc_56></location>C x ∝ 1 T 0 . 5 Υexp( -hν kT )</formula>
<text><location><page_14><loc_52><loc_46><loc_92><loc_52></location>where Υ is the thermally averaged collision strength. We use Chianti V. 7.0 (Dere et al. 1997; Landi et al. 2012) to calculate this quantity and assume that the Si iv and the C iv emission come from a plasma at Log ( T ) = 4 . 9 (K) and Log ( T ) = 5 . 0 (K), respectively.</text>
<text><location><page_14><loc_52><loc_38><loc_92><loc_46></location>The relationship L SiIV = 0.111 L CIV is indicated with a solid line in Figure 12 (Top). A model that does not assume the same EM for Si iv and C iv will move the solid line up or down (modestly: a 30% larger emission measure in Si iv than in C iv moves the solid line up by 0.1 dex), but will not change the slope.</text>
<text><location><page_14><loc_52><loc_17><loc_92><loc_38></location>To perform the same comparison between C iv and N v requires a correction from the observed values in the latter, because we only fully observe the 1239 ˚ A doublet member, as the 1243 ˚ A doublet member is generally absorbed by N i . Therefore, we assume that the ratio of the flux in the 1239 ˚ A to the 1243 ˚ A N v lines is the same as in the C iv lines and calculate the total flux that would be observed in both N v doublet lines in the absence of N i . This is N v flux plotted against C iv in Figure 12 (Bottom). For N v the observed relationship is log L NV /L /circledot = ( -1 . 5 ± 0 . 4)+(0 . 8 ± 0 . 1) log L CIV /L /circledot , ignoring the WTTSs. The expected relationship between the luminosities of both lines is L NV = 0 . 183 L CIV , using N/C abundance of 0.25, and maximum ionization temperature of Log ( T ) = 5 . 3 (K).</text>
<text><location><page_14><loc_52><loc_7><loc_92><loc_17></location>In the case of Si iv the observed linear fit (dashed line in Figure 12) is not consistent with our simple model (solid line, Figure 12). For the current assumptions, V4046 Sgr and TW Hya show a flux deficit in Si iv (or an excess in C iv ), while objects such as CV Cha, CY Tau, DX Cha, RU Lup, and RW Aur present a flux excess in Si iv (or a deficit in C iv ). In average, the WTTSs are above the line predicted by the model. For N v vs. C</text>
<text><location><page_15><loc_8><loc_89><loc_48><loc_92></location>iv this simple model succeeds in explaining the observed linear correlation.</text>
<text><location><page_15><loc_8><loc_76><loc_48><loc_89></location>Based on the absence of Si iii lines as well as the weak Si ii lines in TW Hya, Herczeg et al. (2002) proposed that for this star the silicon has been locked in grains in the disk and does not participate in the accretion. Consistent with this idea, Kastner et al. (2002) and Stelzer & Schmitt (2004) found depletion of Fe and O in TW Hya. For V4046 Sgr, Gunther et al. (2006) found a high Ne/O ratio, similar to that which was found in TW Hya and suggesting that a similar process may be at work.</text>
<text><location><page_15><loc_8><loc_63><loc_48><loc_76></location>More generally, Drake et al. (2005) argues that the abundance of refractory species reflects the evolutionary status of the circumstellar disk. We see no evidence of this in our sample. The distribution of accretion rates for stars above and below the model line for Si iv /C iv is not significantly different. We find that 80% ± 10% of the non-TD CTTSs are above the Si/C model line, compared to 60% ± 20% of the TD stars. This difference is not significant either. It is, however, noteworthy that our analysis suggests that most CTTSs are Si iv -rich.</text>
<text><location><page_15><loc_8><loc_37><loc_48><loc_63></location>Four out of the 5 WTTSs for which we measure the Si iv luminosity lie above the model line. This is reminiscent of the First Ionization Potential (FIP) effect seen in the Sun (e.g. Mohan et al. 2000) in which elements in the upper solar atmosphere (the transition region and the corona) show anomalous abundances when compared to the lower atmosphere. Upper atmosphere elements with low FIPs (FIP < 10eV, like silicon) show abundance excesses by 4 × , on average, while high FIP elements (FIP > 10 eV, like carbon and nitrogen) show the same abundances between the photosphere and the corona. Possible explanations for the effect include the gravitational settling of neutrals in the chromospheric plateau (Vauclair & Meyer 1985) and/or diffusion of neutrals driven by electromagnetic forces (H'enoux 1998; Laming 2004). The fact that the luminosity in N v for WTTSs follow the expected relationship with C iv , is consistent with the Si iv excess flux in WTTSs being due to the FIP effect (Wood & Linsky 2011).</text>
<text><location><page_15><loc_8><loc_19><loc_48><loc_37></location>However, abundance studies of active, non-accreting, low-mass stars based on X-ray observations, find a mass-dependent inverse FIP-effect abundance pattern in Fe/Ne (see Gudel 2007; Testa 2010 for reviews). If this applies to WTTSs, we would expect to see a silicon deficit in them, as compared to carbon. If we were to move the model (solid) line from the top panel of Figure 12 upwards, in order to make the WTTSs siliconpoor, most CTTSs would become silicon-poor, suggesting that the disk grain evolution observed in TW Hya starts at younger ages. This is speculative and requires further study. We note here that Telleschi et al. (2007) find a mass-dependent inverse FIP-effect abundance pattern also in CTTSs.</text>
<text><location><page_15><loc_8><loc_8><loc_48><loc_19></location>These statements are only intended to identify candidates for further study. What we can tell from our observations is that there is considerable scatter of CTTSs to both sides of the linear relation between L SiIV and L CIV . EM analyses for each star are necessary before significant conclusions can be drawn from this sample regarding abundances. These are possible with the DAO dataset, but they are beyond the scope of this paper.</text>
<section_header_level_1><location><page_15><loc_67><loc_91><loc_77><loc_92></location>8. DISCUSSION</section_header_level_1>
<text><location><page_15><loc_52><loc_77><loc_92><loc_90></location>In this paper we are primarily interested in what the hot gas lines are telling us about the region they are emitted from. In particular, we want to know whether the observed profiles are consistent with emission in an accretion shock, a stellar transition region, or other volumes within the system, such as shocks in the stellar outflow or hot winds. We conclude that extensions of the accretion shock model involving inhomogeneous or multiple columns, or emission far from the accretion spot are necessary to account for the observations.</text>
<text><location><page_15><loc_52><loc_71><loc_92><loc_77></location>The observational description reveals a remarkable diversity of line shapes, and interpretations about them are necessarily qualified with exceptions. However, there are a few general statements that describe the systems:</text>
<unordered_list>
<list_item><location><page_15><loc_54><loc_59><loc_92><loc_70></location>1. The C iv , N v , and Si iv lines generally have the same shape as each other, while the He ii line tends to be narrower and symmetric. The 1548 ˚ A and 1550 ˚ A C iv lines are generally similar to each other, except for a scale factor (Exceptions: Different red wings: DF Tau, DX Cha, RU Lup, RW Aur A; Extra emission close to the 1550 ˚ A line: AK Sco, CS Cha)</list_item>
<list_item><location><page_15><loc_54><loc_50><loc_92><loc_58></location>2. On average, for CTTSs the C iv lines are redshifted by ∼ 20 km s -1 from the stellar rest velocity. The He ii lines for CTTSs are redshifted by ∼ 10 km s -1 . The C iv lines in WTTS may be redshifted from the stellar rest velocity (by ∼ 11 km s -1 at the 2 σ level).</list_item>
<list_item><location><page_15><loc_54><loc_39><loc_92><loc_49></location>3. For CTTSs, the C iv lines are broader (FWHM ∼ 200 km s -1 ) than for WTTSs (FWHM ∼ 90 km s -1 ). On average the He ii lines have the same widths in CTTSs and WTTSs (FWHM ∼ 100 km s -1 ). The CTTSs C iv lines are skewed to the red more than the He ii lines, and more than C iv lines in WTTSs.</list_item>
<list_item><location><page_15><loc_54><loc_26><loc_92><loc_38></location>4. For C iv the NC contributes about 20% of the total line flux at low accretion rates ( < 4 × 10 -9 M /circledot /yr). At high accretion rates the NC contribution to the flux becomes comparable to and larger than the BC contribution for some stars. The multi-epoch data also suggest that increases in C iv luminosity are accompanied by an increase in the strength of the NC. For He ii , the NC contributes about 60% of the total line flux at all accretion rates.</list_item>
<list_item><location><page_15><loc_54><loc_20><loc_92><loc_25></location>5. The C iv line luminosity is stronger than the He ii line luminosity by a factor of 3 to 4 in CTTSs, while the luminosities are comparable to each other in WTTSs.</list_item>
<list_item><location><page_15><loc_54><loc_7><loc_92><loc_19></location>6. For CTTSs that show BC and NC in C iv , both components are redshifted (exceptions: DF Tau, DK Tau, HN Tau A, IP Tau, MP Mus, T Tau N, V1190 Sco) and the BC is redshifted with respect to the NC (exceptions: DF Tau, GM Aur, IP Tau, RU Lup, SU Aur, T Tau, V1190 Sco). For 12 out of 22 objects (AA Tau, BP Tau, CS Cha, CY Tau, DE Tau, DN Tau, DS Tau, EP Cha, MP Mus, TW Hya, V1079 Tau, V4046 Sgr), the velocities of the</list_item>
</unordered_list>
<text><location><page_16><loc_12><loc_84><loc_48><loc_92></location>narrow and broad components are consistent with a magnetospheric origin, within the instrumental velocity errors. For the rest (DF Tau, DK Tau, DR Tau, GM Aur, HN Tau A, IP Tau, RU Lup, SU Aur, T Tau, V1190 Sco) other explanations are necessary.</text>
<unordered_list>
<list_item><location><page_16><loc_10><loc_76><loc_48><loc_83></location>7. There are no significant correlations between the C iv luminosity or accretion rate and the velocity at peak flux, FWHM, skewness, inclination, 1548-to1550 line ratio, or velocity of the Gaussian components.</list_item>
</unordered_list>
<section_header_level_1><location><page_16><loc_13><loc_74><loc_43><loc_75></location>8.1. Emission from the accretion column?</section_header_level_1>
<text><location><page_16><loc_8><loc_60><loc_48><loc_73></location>If the lines originate primarily in an accretion shock, one can make at least four predictions: (1) the C iv and He ii line profiles should be redshifted, with the latter having smaller velocities; (2) the line emission should be well localized on the stellar surface; (3) if increases in the accretion rate are due to density increases, the postshock gas emission will be quenched, due to burying of the post-shock column; (4) the velocity of the narrow and broad components of the C iv lines should be related as V BC /greaterorsimilar 4 V NC .</text>
<text><location><page_16><loc_8><loc_45><loc_48><loc_60></location>Indeed, we observe C iv and He ii redshifted, with the former redshifted more than the latter. This, however, is not a very constraining prediction. Linsky et al. (2012) have shown that C iv lines in dwarfs are redshifted by an amount correlated with their rotation period and Ayres et al. (1983) have shown that the C iv lines in late-type giants are redshifted with respect to the He ii lines by variable amounts of up to ∼ 20 km s -1 . So, for non-CTTSs gas flows in the upper stellar atmosphere may result in line shifts comparable to those we observe in our sample.</text>
<text><location><page_16><loc_8><loc_27><loc_48><loc_45></location>The predictions regarding the localization of the emitting region in the stellar surface are difficult to test with this dataset, because, as we have argued in section 5.2.5, the sample is not large enough for the inclination to serve as a discriminator of the line origin. On the other hand, there is a large body of evidence (see the references in the introduction) suggesting that the accretion continuum and the (optical) line emission are well localized on the stellar surface. No unequivocal rotational modulation of the lines studied here has been reported in the literature, but this may be because the available dataset does not provide enough rotational coverage. The change in line flux observed among the different RU Lup epochs (Figure 10) may result from rotational modulation.</text>
<text><location><page_16><loc_8><loc_8><loc_48><loc_27></location>If we assume that the NC of C iv is due to post-shock emission, we would naively expect its importance to diminish for high accretion rates as the post-shock is buried in the photosphere (Drake 2005). However, we observe the opposite, with the NC flux increasing with respect to the BC flux as the accretion rate increases. One possible explanation is that increased accretion produces larger accretion areas on the star rather than much higher densities in the accretion column. This has been shown for BP Tau by Ardila & Basri (2000). Larger areas provide more escape paths for the photons emitted from a buried column. Area coverages as small as 0.1% of the stellar surface of a 2R /circledot star have a radius of ∼ 10 5 km, which is larger than the deepest likely burials (Sacco et al. 2010).</text>
<text><location><page_16><loc_10><loc_7><loc_48><loc_8></location>A complementary insight comes from X-ray observa-</text>
<text><location><page_16><loc_52><loc_68><loc_92><loc_92></location>tions. Models that assume a uniform density accretion column also predict quenching of the X-ray flux, due to absorption of X-rays in the stellar layers, for accretion rates as small as a few times 10 -10 M /circledot /yr (Sacco et al. 2010). However, models by Romanova et al. (2004) indicate that the accretion column is likely non-uniform in density and the denser core may be surrounded by a slower-moving, lower-density region. Sacco et al. (2010) argue that most of the observed X-ray post-shock flux should come from this low-density region. In addition, post-shock columns are believed to be unstable to density perturbations, and should collapse on timescales of minutes (Sacco et al. 2008). The lack of observed periodicities in the X-ray fluxes indicate that multiple incoherent columns should be present. Orlando et al. (2010) also conclude that the presence of multiple columns with different densities is necessary to explain the low accretion rates derived from X-ray observations.</text>
<text><location><page_16><loc_52><loc_63><loc_92><loc_68></location>In addition, models of low-resolution CTTSs spectra from the near-UV to the infrared indicate that the observed accretion continuum is consistent with the presence of multiple accretion spots (Ingleby et al. 2013).</text>
<text><location><page_16><loc_52><loc_49><loc_92><loc_63></location>In summary, we observe that the C iv flux does not decrease with accretion, which suggests that if burying is occurring, it does not affect the observed flux substantially. This may be because the aspect ratio of the accretion spots is such that the post-shock radiation can escape without interacting with the photosphere, and/or that the flux we observe is emitted from unburied lowdensity edges of the accretion column, and/or that multiple columns with different densities and buried by different amounts are the source of the emission.</text>
<section_header_level_1><location><page_16><loc_58><loc_47><loc_86><loc_48></location>8.1.1. He ii emission in the pre-shock gas</section_header_level_1>
<text><location><page_16><loc_52><loc_33><loc_92><loc_46></location>As we have shown, the BC of He ii is weak compared to its NC, in contrast with the BC for C iv . For a BC produced in the pre-shock gas, this implies that the preshock emits more strongly in C iv than in He ii . Observationally, for CTTSs the ratio between luminosity in the BC of the C iv to the He ii line ranges from 2.5 to 10.4, with a median of 5.6. The ranges are comparable if instead of using the whole sample we use only those CTTSs with redshifted profiles. Are these values consistent with a pre-shock origin?</text>
<text><location><page_16><loc_52><loc_19><loc_92><loc_33></location>The pre-shock region is heated and ionized by radiation from the post-shock gas. The model described next confirms that the He ii pre-shock gas contribution to the line is produced by recombination of He iii to He ii , while the pre-shock gas contribution to C iv 1550 ˚ A line is the result of collisions from the ground state. The He ii 1640 ˚ A line is emitted from a smaller region in the pre-shock, closer to the star, than the C iv 1550 ˚ A lines, because the energy required to ionize He ii to He iii is 54.4 eV, while the energy to ionize C iii to C iv is 47.9 eV.</text>
<text><location><page_16><loc_52><loc_7><loc_92><loc_19></location>Weuse Cloudy version 07.02.02 (Ferland et al. 1998) to simulate the pre-shock structure and calculate the ratio between the C iv lines at 1550 ˚ Aand the He ii line at 1640 ˚ A. We illuminate the pre-shock gas with a 4000 K stellar photosphere, a shock continuum with the same energy as contained in half of the incoming flux, and half of the cooling energy from the post-shock gas. For the purposes of this model the post-shock cooling radiation is calculated by solving the mass, momentum, and energy con-</text>
<text><location><page_17><loc_8><loc_77><loc_48><loc_92></location>rvation equations as described in Calvet & Gullbring (1998), to derive a temperature and density structure and using Chianti V. 7.0 (Dere et al. 1997; Landi et al. 2012) to calculate the emissivity of the plasma at each point in the post-shock. This procedure assumes that the post-shock gas is an optically thin plasma in collisional equilibrium. These models are parametrized by incoming gas velocities and densities. A typical density for an incoming accretion flow with ˙ M = 10 -8 M /circledot /yr, covering 1% of the stellar surface area, is 5 × 10 12 cm -3 (Calvet & Gullbring 1998).</text>
<text><location><page_17><loc_8><loc_59><loc_48><loc_77></location>We find that the C iv pre-shock gas emission is always larger than He ii emission by factors ranging from 2 (at 10 14 cm -3 ) to 6 (at 10 10 cm -3 ), for incoming pre-shock gas velocities of 300 km s -1 . This increase with density implies that the pre-shock emission is beginning to become optically thick at high densities, consistent with Figure 8 (bottom). Higher velocities result in larger C iv emission with respect to He ii : for incoming velocities of 400 km s -1 and densitites of 10 10 cm -3 the pre-shock emits 10 times more flux in C iv 1550 ˚ A, than in He ii 1640 ˚ A. In these simple models, the C iv post-shock emission as a fraction of the total emission varies from 0.7 (at 10 10 cm -3 ) to ∼ 1 (at 10 14 cm -3 ).</text>
<text><location><page_17><loc_8><loc_48><loc_48><loc_59></location>These are only illustrative models but they suggest that the observed values are within the range of what is produced by the accretion shock region. It is noteworthy that low pre-shock gas densities (10 10 cm -3 ) are required to explain the median C iv /He ii ratio. This again suggests that the accretion spots are very large, or that the observed emission comes from low-density regions in the accretion columns.</text>
<section_header_level_1><location><page_17><loc_9><loc_46><loc_47><loc_47></location>8.1.2. Kinematic predictions of the accretion shock model</section_header_level_1>
<text><location><page_17><loc_8><loc_20><loc_48><loc_45></location>A crucial prediction of the accretion model is that V BC /greaterorsimilar 4 V NC . Strictly, the observations are consistent with this prediction for only two objects: TW Hya and CY Tau. However, given the size of the COS and STIS pointing errors, half of the CTTSs sample may comply with the accretion model predictions (see Figure 9). The rest of the objects are anomalous, as the standard magnetospheric accretion model has trouble explaining stars for which the BC velocity is significantly smaller than the NC velocity, or those stars in which the velocity of one or both of the components is significantly negative. Possible explanations for these anomalous objects include the target having a significant extra radial velocity due to the presence of a close companion, multiple columns being responsible for the emission, regions far from the accretion shock surface or parts of an an outflow contributing to the emission, and winds or outflows dominating the emission. We examine the first three possibilities in this section. We discuss outflows in Section 8.2.</text>
<text><location><page_17><loc_8><loc_7><loc_48><loc_20></location>For binaries, the relative velocity between close binary components will result in shifts in the velocity centroid of the C iv line, especially if one component dominates the accretion or if a circumbinary disk is present (Artymowicz & Lubow 1996). For example, AK Sco is a well know spectroscopic binary in which the radial velocity of the stellar components can reach 100 km s -1 with respect to the system's center-of-mass (Alencar et al. 2003). If the characteristics of the accretion stream to each component were different (different accretion</text>
<text><location><page_17><loc_52><loc_52><loc_92><loc_92></location>rates, different pre- and post-shock emission contributions among the components, etc), this would result in shifted velocity centroids. We have not considered AK Sco in the general description because the Gaussian decomposition is problematic. Regarding the other spectroscopic binaries in the sample, V4046 Sgr and CS Cha have velocity contrasts between the BC and NC close to what is expected from the accretion paradigm. Orbital modulation of X-rays has been detected in V4046 Sgr (Argiroffi et al. 2012b), which may be responsible for the large NC velocity, compared to the predictions of the accretion shock model. The C iv profiles of DX Cha are slightly redshfited, but they cannot be decomposed into velocity components and this object may belong to a different class altogether, as the only Herbig Ae star in the sample. The other systems for which multiplicity may be relevant are DF Tau and RW Aur A. DF Tau has v BC -v NC ∼ -27 km s -1 , but the component separation of 12 AU is too large to induce these velocity shifts. In the case of RW Aur A, we do not decompose the C iv lines into Gaussian components, and argue that the strong blueshift is the result of outflow emission. Gahm et al. (1999) have suggested that the system is accompanied by a brown dwarf companion. According to their observations, the companion produces radial velocity variations smaller than 10 km s -1 . In summary, with the possible exceptions of AK Sco and V4046 Sgr, binarity does not play a significant role in altering the values of V BC and V NC for any of the CTTSs we have termed anomalous.</text>
<text><location><page_17><loc_52><loc_31><loc_92><loc_52></location>Models of the hot gas line shapes have been performed by Lamzin (2003a,b) and Lamzin et al. (2004), among others. For a broad range of geometries, those models predict redshifted double-peaked line profiles. Lamzin (2003a) modeled the C iv emission assuming planeparallel geometry with the pre- and post-shock emission lines thermally broadened, while Lamzin (2003b) consider emission from an accretion ring at high stellar latitudes, with gas flow falling perpendicular to the stellar surface. Both sets of models recover two emission kinematic components, one from the pre-shock and one from the post-shock. The peak separation between the preand post-shock contributions depends on the velocity of the incoming flow. With the possible exceptions of DK Tau and DR Tau, these double-peaked profiles are not observed.</text>
<text><location><page_17><loc_52><loc_7><loc_92><loc_31></location>The failure of those models in predicting observed line profiles, led Lamzin (2003b) to argue that the incoming flow cannot be perpendicular to the stellar surface, and that a substantial tangential component must be present in the gas velocity. In models of the UV spectrum of TW Hya, Lamzin et al. (2004) argue that to explain the C iv line shape, the accretion flow must fall at a very low stellar latitude but in a direction almost parallel to the stellar surface, in such a way that we are able to observe the accretion streams from both sides of the disk through the inner disk hole. However, Donati et al. (2011) showed that the magnetic topology in TW Hya is such that the accretion streams have to be located at high stellar latitudes. We have shown here that the C iv line shape in TW Hya is the most common one ( ∼ 50%) in our sample and therefore whatever process is responsible for it must be fairly general. A more general explanation that the one from Lamzin et al. (2004) is required to understand</text>
<text><location><page_18><loc_8><loc_91><loc_19><loc_92></location>the line shapes.</text>
<text><location><page_18><loc_8><loc_71><loc_48><loc_90></location>Gunther & Schmitt (2008) considered hot gas observations of 7 CTTSs (RU Lup, T Tau, DF Tau, V4046 Sgr, TWA 5, GM Aur, and TW Hya). Their model of the post-shock contribution to the O vi profiles of TW Hya, including only thermal broadening, results in a line skewed to the red, but very narrow compared to the observations. A turbulent velocity of 150 km s -1 is necessary to obtain widths comparable to those observed. However, this turbulence results in a very symmetric line, with significant emission to the blue of the line, which is not observed. Based on these analyses, Gunther & Schmitt (2008) concluded that the O vi emission in those stars that show redshifted profiles (DF Tau, V4046 Sgr, TWA 5, GM Aur, and TW Hya) is incompatible with current models of magnetospheric accretion.</text>
<text><location><page_18><loc_8><loc_51><loc_48><loc_71></location>The larger sample that we present here provides some insights into these issues. Focusing only on those objects for which the velocities are such that they could in principle be produced in an accretion shock (the wedge with 12 CTTSs in the upper right quadrant of Figure 9, below the dotted line, including MP Mus but excluding DR Tau), we conclude that the lack of observed double peaked profiles is due to the small difference between velocity components and to the fact that both components are very broad (perhaps as a result of turbulence in the flow). Notwithstanding the conclusions from Lamzin et al. (2004) and Gunther & Schmitt (2008), the ratio between the velocity components for the C iv profile from TW Hya is perfectly consistent with magnetospheric accretion.</text>
<text><location><page_18><loc_8><loc_32><loc_48><loc_51></location>The small difference between velocity components is due to either too large V NC values or too small V BC values, compared with expected infall speeds. For TW Hya, for example, the models by Gunther et al. (2007) predict an infall velocity of 525 km s -1 , and we observe V BC = 116 km s -1 . This difference requires an angle between the line of sight and the accretion column of ∼ 77 degrees. For most of the rest of the stars likely to come from an accretion shock, similarly large angles are implied: AA Tau: 76 · , BP Tau: 82 · , CY Tau: 68 · , CS Cha: 49 · , DE Tau: 80 · , DN Tau: 75 · , DS Tau: 83 · , EP Cha: 85 · , MP Mus: 89 · , V1079 Tau: 83 · , V4046 Sgr: 76 · (assuming literature values for stellar masses and radii).</text>
<text><location><page_18><loc_8><loc_11><loc_48><loc_32></location>These values are calculated using the observed BC velocity and the predicted free-fall velocity. Observationally, they represent a flux-weighted average of the velocities along the line of sight. It is surprising that most are close to 90 degrees, indicating that the average column is seen sideways and that no emission is observed from the top of the accretion column. However, observations of red-wing absorption in the He i 1.1 µ mline (Fischer et al. 2008) indicate that the accretion flow is slower than the free-fall velolocities by ∼ 50%. In addition, in the models by Romanova et al. (2004, 2011) mentioned before, the periphery of the accretion column is moving more slowly, by factors of ∼ 2, than the column core. If the BC emission is not coming from the fast free-falling core of the pre-shock gas flow but from the slower edges, the calculated angle will be smaller.</text>
<text><location><page_18><loc_8><loc_8><loc_48><loc_11></location>So far, this exploration of the expected relationship between the broad and narrow velocity components as-</text>
<text><location><page_18><loc_52><loc_61><loc_92><loc_92></location>sumes that the pre- and post-shock flows share the same line-of-sight angle. There are two situations in which this may not be the case. If some of the post-shock ionizing radiation reaches regions of the accretion flow in which the line-of-sight to the observer is different than for the accretion spot, we may end up with BC velocities that are unrelated to the NC velocities. After all, even close to the stellar surface the magnetic field twists and curves (Gregory et al. 2008; Mohanty & Shu 2008) and photoionized regions may be produced in flow moving in different directions. The Cloudy models we develop in section 8.1.1 create fully ionized post-shock columns with sizes ranging from 10000 km for densities ∼ 10 12 cm -3 to ∼ R /circledot , for densities ∼ 10 10 cm -3 (Calvet & Gullbring 1998). In other words, for low densities the ionizing photons may reach far from the stellar surface. Without more detailed models including at least some notional information regarding the configuration of the magnetosphere, it is not possible to say if this concept is relevant, but it may offer an explanation for objects with either small or negative BC velocities, but positive NC velocities (upper half of Figure 9), such as DF Tau, DR Tau, GM Aur, IP Tau, RU Lup, SU Aur, and V1190 Sco.</text>
<text><location><page_18><loc_52><loc_44><loc_92><loc_61></location>It is also possible that we are observing multiple columns for which the ratios of pre- to post-shock emission are not the same in all columns, resulting in a situation in which we observe the pre-shock of one column but the post-shock of another with a different orientation. This may occur, for example, if the post-shock is occulted by the stellar limb, or buried, or if the columns have different optical depths. In this case, we will see pre- and post-shock velocities that are essentially unrelated to each other. This effect may explain objects for which both velocities are positive, but the BC is small compared to the NC (e.g., DF Tau, DR Tau, GM Aur, maybe IP Tau, RU Lup, SU Aur).</text>
<section_header_level_1><location><page_18><loc_53><loc_42><loc_90><loc_43></location>8.1.3. A contribution from the stellar transition region?</section_header_level_1>
<text><location><page_18><loc_52><loc_23><loc_92><loc_41></location>An alternative hypothesis to the line origin in an accretion shock is that some of the observed flux originates in the stellar transition region outside of the accretion spot. Models by Cranmer (2008, 2009) indicate that accretion energy may contribute to the powering of the corona. Furthermore, based on iron and helium line observations in the optical of five CTTSs, Petrov et al. (2011) suggest that an area of enhanced chromospheric emission, more extended than the hot accretion spot, is produced by the accretion process. In addition, observations by Brickhouse et al. (2010) suggest the existence of a larger region than the accretion spot as the source of a third Xray component (after the corona and the accretion spot itself).</text>
<text><location><page_18><loc_52><loc_7><loc_92><loc_23></location>The idea of an atmospheric contribution to the observed lines in CTTSs is not new (Herbig 1970), although its limitations were the inspiration for the magnetospheric accretion paradigm. Cram (1979) and Calvet et al. (1984) showed that a dense chromosphere cannot reproduce the strength of the observed H α line in CTTSs, and Batalha & Basri (1993) showed that chromospheric-based models are unable to reproduce the veiling or the size of the Balmer jump in CTTSs. On the other hand, magnetospheric accretion models are able to reproduce the hydrogen-line fluxes and shapes (e.g. Muzerolle et al. 1998, Kurosawa et al. 2006).</text>
<text><location><page_19><loc_8><loc_76><loc_48><loc_92></location>If the lines are primarily emitted from the transition region, then a model would predict that: (1) the line profiles should have comparable redshifts in WTTSs and CTTSs; (2) the emission should not be localized to a small area of the stellar surface; and (3) the He ii and C iv lines should have similar shapes. Our observations do not confirm any of the predictions of a transition region origin for the lines: the C iv line redshifts are different in CTTS and WTTSs, the He ii and C iv lines have different shapes, and other observations show that the accretion indicators are localized in a small area of the stellar surface.</text>
<text><location><page_19><loc_8><loc_53><loc_48><loc_76></location>However, we observe that the NCs of the He ii lines have comparable widths in CTTSs and in WTTSs, and the same widths for the NC of C iv . Within the picture of an accretion shock column, it is surprising that the WTTSs line widths, formed in the upper stellar atmosphere, should be the same as the CTTSs NCs line widths, formed in the turbulent post-shock gas. The velocity differential between high and low density regions in the accretion column will result in large amounts of turbulence, which tends to produce broad lines (Gunther & Schmitt 2008). It may be that at least some of the flux in the NCs in CTTSs comes from the stellar transition region, while the BCs come from the pre-shock gas. This would only apply to objects in which both Gaussian components have positive velocities as there is no evidence of blueshifted C iv or He ii profiles in the atmospheres of young active stars.</text>
<text><location><page_19><loc_8><loc_9><loc_48><loc_53></location>Could the stellar transition region respond to accretion by producing enough C iv or He ii emission to contribute to the observed lines? For He ii a detailed model of the heating process would have to show that the X-ray emission from the accretion spot in CTTSs is enough to increase the He ii line luminosity in the stellar transition region by approximately one order of magnitude from the WTTS values (Yang et al. 2012 and this work). In the Sun, between 30% (quiet regions) and 60% (active regions) of the 1640 ˚ A He ii flux comes from ionization by soft X-rays followed by radiative recombination. The rest is due to collisional or radiative excitation of ground-level He ii (Hartmann et al. 1979; Kohl 1977). In CTTSs, the accretion spot acts as a source of soft X-rays (Kastner et al. 2002; Stelzer & Schmitt 2004; Gunther et al. 2007), although the overall X-ray emission is dominated by hot plasma produced by enhanced magnetic activity. Even the corona may increase its X-ray emission as a response to accretion events (Dupree et al. 2012). Typical observed values of L HeII ∼ 10 30 erg/sec are comparable to X-ray luminosities between 0.2 and 10 keV (Ingleby et al. 2011; G'omez de Castro & Marcos-Arenal 2012). However, based on solar models, Hartmann et al. (1979) concluded L X =50 L HeII in the 0.25 keV band. This suggests that the observed amount of X-ray flux in young stars is small compared to what would be required to produce the observed He ii line, and perhaps other mechanisms besides radiative recombination may be at play. On the other hand, the models of the coronal heating by Cranmer (2008) show that it is plausible to assume that the accretion energy is sufficient to drive CTTS stellar winds and coronal X-ray emission.</text>
<text><location><page_19><loc_10><loc_8><loc_48><loc_9></location>The C iv resonance doublet observed in stellar atmo-</text>
<text><location><page_19><loc_52><loc_75><loc_92><loc_92></location>spheres is the result of collisional excitation from C iv followed by radiative de-excitation (Golub & Pasachoff 2009), and an increase ranging from one to two orders of magnitude from WTTSs values would be required to match the surface flux (Johns-Krull et al. 2000) or luminosity (Figure 6) observed in CTTSs. This would require a proportional increase in atmospheric density. High densities of hot plasma gas are indeed observed in CTTSs, but at temperatures consistent with an origin in an accretion shock (Sacco et al. 2008; Argiroffi et al. 2009). As is the case for He ii , the increased X-ray flux due to accretion may result in a larger C iv population, and larger observed doublet flux.</text>
<text><location><page_19><loc_52><loc_68><loc_92><loc_75></location>In summary, the relative contribution of the stellar transition region to the total He ii or C iv remains uncertain. While unlikely, we cannot rule out with these observations that at least some fraction of the NC in He ii or C iv originates in the stellar atmosphere of CTTSs.</text>
<section_header_level_1><location><page_19><loc_59><loc_65><loc_85><loc_66></location>8.2. Blueshifted profiles and outflows</section_header_level_1>
<text><location><page_19><loc_52><loc_51><loc_92><loc_64></location>We have argued that if the post-shock radiation ionizes material far away from the accretion spot, we may end up with broad components having small or even negative velocities. On the other hand, the objects for which we observe a negative velocity in the NC of C iv , or in the overall profile, present considerable challenges. These are AK Sco, DK Tau, HN Tau A, T Tau N, and RW Aur A. The He ii line matches the C iv line for RW Aur A and HN Tau A (Figure A.7) but is centered at velocities closer to zero than C iv for the other stars.</text>
<text><location><page_19><loc_52><loc_38><loc_92><loc_51></location>If we observe a CTTS for which the accretion stream is moving towards us, the velocity components would be negative. This would be the case, for example, if we observe the stream below the disk through the inner truncation hole of the accretion disk. The only case in which this is a possibility is T Tau N as this is the only target for which v NC < 0, v BC < 0, and | v NC | < | v BC | . This requires an inclination close to face-on, which the system has, and perhaps a large difference between the stellar rotation axis and the magnetic field axis.</text>
<text><location><page_19><loc_52><loc_8><loc_92><loc_38></location>Outflows present a more likely explanation for the blueshifted profiles. Outflow phenomena are common in CTTSs and high speed shocks between the jet material and the ISM may result in C iv emission. The C iv blueshifted emission in DG Tau (Ardila et al. 2002) is clearly related to the beautiful outflow imaged with HST/NICMOS (Padgett et al. 1999) and the C iv emission is likely produced by shocks in the jet (Schneider et al. 2013). In order to generate high temperature ( > 10 5 K) gas via an outflow shock, velocities larger than 100 km s -1 in the strong shock limit are necessary (Gunther & Schmitt 2008), resulting in postshock (observed) C iv velocities > 25 km s -1 . The five objects we are considering have absolute NC velocities or velocities at maximum flux larger than this value. Therefore, at least energetically, it is possible for the emission to be produced by a shock in the outflow. Both HN Tau A and RW Aur A are known to have outflows (Hirth et al. 1994; Hartigan et al. 1995), and velocities in the approaching and receding jets that are comparable to the ones we observe in C iv and H 2 (Melnikov et al. 2009; Coffey et al. 2012).</text>
<text><location><page_19><loc_53><loc_7><loc_92><loc_8></location>For objects such as AK Sco and DK Tau the accretion</text>
<text><location><page_20><loc_8><loc_75><loc_48><loc_92></location>shock and the outflow regions may both be contributing to the emission. For HN Tau A and RW Aur, if we accept that the observed profiles originate primarily in an outflow, the lack of accretion shock emission becomes puzzling. These are high-accretion rate objects, and perhaps in these conditions the low-density region in the periphery of the accretion column is not present, and so the shock is truly buried. Or maybe we are observing the objects in a rotational phase such that the accretion spot is away from us. The GHRS observations of RW Aur (red trace, Figure 10) show the presence of additional emission components to the red of the nominal C iv lines, which may be due to the accretion spot.</text>
<text><location><page_20><loc_8><loc_50><loc_48><loc_75></location>Overall, the relationship between H 2 asymmetries and hot line shapes remains to be fully explored. The analysis of DAO data by France et al. (2012) shows that some H 2 emission lines in DK Tau, ET Cha (RECX 15), HN Tau, IP Tau, RU Lup, RW Aur, and V1079 Tau (LkCa 15) are asymmetric, presenting redshifted peaks and lowlevel emission to the blue of the profiles. In general, the systematic errors in the COS wavelength scale make it difficult to determine whether the line peaks are truly shifted in velocity. For HN Tau A and RW Aur, the peak of the H 2 line R(6) (1-8), at 1556.87 ˚ A is shifted +19 km s -1 and +88 km s -1 respectively, from the stellar rest frame, larger than would be expected from pointing errors alone and suggests that outflows that are fast enough in C iv are accompanied by H 2 flows away from the observer. For RW Aur A, we observe that the (redshifted) H 2 emission covers the (blueshifted) C iv emission (Figure 5).</text>
<text><location><page_20><loc_8><loc_33><loc_48><loc_50></location>If the gas heating occurs very close to the star, or if the wind is launched hot, one would expect to observe P-Cygni-like profiles in the hot gas lines. Dupree et al. (2005) argued that the asymmetric shape of the O vi profiles of TW Hya is the result of a hot wind in the star. Because the two C iv lines are close to each other, blueshifted wind absorption in the 1550 ˚ A line will decrease emission in the red wing of the 1548 ˚ A line. Johns-Krull & Herczeg (2007) compared both C iv lines and concluded that their similarity, as well as the absence of absorption below the local continuum (as seen in many neutral and singly ionized lines), suggest that a hot wind is not present in the case of TW Hya.</text>
<text><location><page_20><loc_8><loc_11><loc_48><loc_33></location>Within our sample, DX Cha is the only object that may have a high temperature wind, as it shows a deficit in the red wing of the 1548 ˚ A line for C iv , compared to the red wing of the 1550 ˚ Aline and a very sharp blue cutoff in the C iv and the Si iv lines. A blueshifted absorption is seen in the 1548 ˚ AC iv line, suggesting outflow speeds of up to 400 km s -1 . A blueshifted absorption is also seen in the 1403 ˚ A line of Si iv (the red doublet member). However, note that there is an O iv emission line at -340 km s -1 of the line, and its presence may give the illusion of a wind absorption. The narrow H 2 lines observed in the Si iv profiles suggest that the putative wind is collimated or inhomogeneous, as the H 2 lines are not absorbed by the wind. The difference between both members of N v is not due to a wind but to absorption of the 1243 ˚ A line by circumstellar absorbers.</text>
<text><location><page_20><loc_8><loc_7><loc_48><loc_11></location>DX Cha also shows absorption features in the region near He ii , although it is not clear that they are related to the outflows. Lower temperature outflows, like those</text>
<text><location><page_20><loc_52><loc_84><loc_92><loc_92></location>observed in other CTTSs (Herczeg et al. 2005) are also observed in the DAO spectra of DX Cha: the Si ii λ 1526.71 ˚ A and 1533.43 ˚ A (not shown here) present very clear P-Cygni profile with wind absorption up to 600 km s -1 from the star. Further examination of the outflows will appear in a future paper.</text>
<section_header_level_1><location><page_20><loc_60><loc_81><loc_84><loc_82></location>8.3. Accretion in Herbig Ae stars</section_header_level_1>
<text><location><page_20><loc_52><loc_74><loc_92><loc_80></location>In this analysis, we have considered DX Cha as one more member of the overall sample, in order to contrast the characteristics of Herbig Ae stars with those of CTTSs. It has the largest mass ( ∼ 2.2 M /circledot , Bohm et al. 2004), and earliest spectral type (A7.5) in the sample.</text>
<text><location><page_20><loc_52><loc_54><loc_92><loc_74></location>DX Cha is a spectroscopic system with a K3 secondary and an average separation of ∼ 0.15 AU between components (Bohm et al. 2004). Even at these small separations, small circumstellar disks may be present, in addition to the circumbinary disk (de Val-Borro et al. 2011). To add to the complexity of the system, observations by Tatulli et al. (2007) are consistent with the presence of a wind launched in the 0.5 AU region of the disk. Testa et al. (2008) show that two different temperature plasmas are responsible for the X-ray emission. The overall flux is emitted from a relatively high-density region and dominated by the primary star. They argue that the hot component is created in the companion's corona, while the low-temperature component originates in an accretion shock.</text>
<text><location><page_20><loc_52><loc_39><loc_92><loc_54></location>In the observations presented here, DX Cha does appear peculiar when compared to the CTTSs. The system has the largest Si iv luminosity of the sample, and the second largest C iv luminosity. The 1550 ˚ A C iv line has the second largest FWHM of the sample, and as we have noted, the C iv lines are not alike in shape, suggesting extra emission or absorption in one of the members. The Si iv lines are unlike any other Si iv or C iv lines in shape, in that they show a very sharp, blue cutoff. As indicated above, DX Cha is also the only clear candidate in the sample for the presence of a hot wind.</text>
<text><location><page_20><loc_52><loc_22><loc_92><loc_39></location>For DX Cha we observe a strong continuum in the He ii region and no clear emission line. If in Figure A.1 we identify the depression at +60 km s -1 in the He ii panel of DX Cha as He ii in absorption, this would be the largest redshift of any He ii line in our sample, and larger than the 11.5 km s -1 observed for the peak of C iv . The absence of the He ii line in emission is put in context by Calvet et al. (2004a), who present low-resolution UV spectra of accreting objects with masses comparable to DX Cha, all of which show He ii in emission. For all of them L HeII /L /circledot ∼ 10 -5 to 10 -4 . 5 , comparable to the luminosities of other stars observed here. This makes the absence of an He ii line a mystery.</text>
<text><location><page_20><loc_52><loc_13><loc_92><loc_22></location>Are these characteristics the continuation of the standard accretion rate phenomena to larger masses, the consequence of the close companion and/or an outflow, or new phenomena related to the weak magnetic fields associated with Herbig Ae objects? A larger sample of highresolution UV spectra of higher mass objects is required to answer these questions.</text>
<section_header_level_1><location><page_20><loc_66><loc_10><loc_78><loc_11></location>9. CONCLUSIONS</section_header_level_1>
<text><location><page_20><loc_52><loc_7><loc_92><loc_9></location>The goal of this paper is to describe the hot gas lines of CTTSs and to provide measurements that will con-</text>
<text><location><page_21><loc_8><loc_81><loc_48><loc_92></location>tribute to understand their origin. We describe the resonance doublets of N v ( λλ 1238.82, 1242.80 ˚ A), Si iv ( λλ 1393.76, 1402.77 ˚ A), and C iv ( λλ 1548.19, 1550.77 ˚ A), as well as the He ii ( λ 1640.47 ˚ A) line. If produced by collisional excitation in a low-density medium, these UV lines suggest the presence of a plasma with temperatures ∼ 10 5 K. We focus primarily on the C iv doublet lines, with the other emission lines playing a supporting role.</text>
<text><location><page_21><loc_8><loc_69><loc_48><loc_81></location>We combine high resolution COS and STIS data from the Cycle 17 Hubble Space Telescope ( HST ) proposal 'The Disks, Accretion, and Outflows (DAO) of T Tau stars' (PI G. Herczeg) with archive and literature data for 35 stars: one Herbig Ae star, 28 CTTSs, and 6 WTTSs. The sample includes 7 stars with transition disks. This is the largest single study of the UV hot gas lines in CTTSs and WTTSs, with high resolution and high sensitivity.</text>
<text><location><page_21><loc_8><loc_63><loc_48><loc_69></location>We use the centroids of the H 2 lines to argue that the systematic wavelength errors in these COS observations are ∼ 7 km s -1 . We do not perform any systematic velocity correction to the spectra to account for these errors.</text>
<text><location><page_21><loc_8><loc_55><loc_48><loc_63></location>The WTTSs establish the baseline characteristics of the hot line emission, and help to separate the effect of accretion from purely atmospheric effects. In particular, they provide the line luminosities and shapes that would be emitted by the young stars in the absence of the accretion process.</text>
<text><location><page_21><loc_8><loc_46><loc_48><loc_55></location>The observations were analyzed using non-parametric shape measurements such as the integrated flux, the velocity at maximum flux (V Max ), the FWHM, and the line skewness. We also decomposed each He ii and C iv line into narrow and broad Gaussian components. We obtained accretion rate measurements from the literature (see Table 3).</text>
<section_header_level_1><location><page_21><loc_19><loc_43><loc_38><loc_45></location>9.1. The shape of the lines</section_header_level_1>
<unordered_list>
<list_item><location><page_21><loc_11><loc_24><loc_48><loc_43></location>· The most common (50 %) C iv line morphology is that of a strong, narrow emission component together with a weaker, redshifted, broad component (see for example BP Tau. Figure 1). In general, the C iv CTTSs lines are broad, with significant emission within ± 400 km s -1 of the lines. When compared to the WTTSs lines, the C iv lines in CTTSs are skewed to the red, broader (FWHM ∼ 200 km s -1 for CTTSs, FWHM ∼ 100 km s -1 for WTTSs), and more redshifted (20 km s -1 for CTTSs, 10 km s -1 for WTTSs). See Table 9. The 1548 ˚ A member of the C iv doublet is sometimes contaminated by the H 2 R(3)1-8 line and by Si i , C ii , and Fe ii emission lines.</list_item>
<list_item><location><page_21><loc_11><loc_13><loc_48><loc_23></location>· Overall, the C iv , Si iv , and N v lines in CTTSs all have similar shapes. The 1243 ˚ A N v line is strongly absorbed by circumstellar N i and both lines of the Si iv doublet are strongly affected by H 2 emission (Figures A.1 to A.5). We do no detect H 2 emission within ± 400 km s -1 of the hot gas lines for WTTSs.</list_item>
<list_item><location><page_21><loc_11><loc_7><loc_48><loc_12></location>· In general, the He ii CTTSs lines are symmetric and narrow, with FWHM ∼ 100 km s -1 . The FWHM and redshifts are comparable to the same values in WTTSs. The He ii lines also have the</list_item>
</unordered_list>
<text><location><page_21><loc_56><loc_84><loc_92><loc_92></location>same FWHM as the narrow component of C iv in CTTSs. They are less redshifted than the CTTSs C iv lines, by ∼ 10 km s -1 , but have the same redshift as the WTTSs. A comparison of the Gaussian parameters for C iv and He ii is shown in Figure 7.</text>
<section_header_level_1><location><page_21><loc_66><loc_81><loc_78><loc_82></location>9.2. Correlations</section_header_level_1>
<unordered_list>
<list_item><location><page_21><loc_54><loc_67><loc_92><loc_80></location>· We confirm that the C iv line luminosities are correlated with accretion luminosities and accretion rates, although the exact correlation depends on the set of extinctions that is adopted (Figure 4). C iv and He ii luminosities are also correlated with each other, and the C iv /He ii luminosity ratio is a factor of three to four times larger in CTTSs than in WTTS. This confirms that the ratio depends crucially on accretion phenomena and it is not an intrinsic property of the stellar atmosphere.</list_item>
<list_item><location><page_21><loc_54><loc_52><loc_92><loc_65></location>· We do not find any significant correlation between inclination and C iv V Max , FWHM, or skewness, or between inclination and any of the Gaussian parameters for the C iv decomposition (Figure 6). We conclude that the dataset is not large enough for inclination to be a strong descriptor in the sample. The variability observed in the C iv lines of RU Lup and DR Tau (Figure 10) may be consistent with the C iv accretion spot coming in and out of view.</list_item>
<list_item><location><page_21><loc_54><loc_44><loc_92><loc_51></location>· We do not find correlations among V Max , FWHM, or skewness, with each other or with accretion rate or line luminosity. On the same token, we do not find correlations between the width of the Gaussian components and accretion rate or line luminosity.</list_item>
<list_item><location><page_21><loc_54><loc_30><loc_92><loc_43></location>· The ratio between the flux in the 1548 ˚ A C iv line to the flux in the 1550 ˚ A C iv line is a measure of the opacity of the emitting region. For CTTSs, 70% (19/29) of the stars have blue-to-red C iv line ratios consistent with optically thin or effectively thin emitting regions. DF Tau is anomalous, with a blue-to-red C iv line ratio close to 3, indicating either emission in the 1548 ˚ A line or absorption on the 1550 ˚ A one.</list_item>
<list_item><location><page_21><loc_54><loc_18><loc_92><loc_29></location>· We find that the contribution fraction of the NC to the C iv line flux in CTTSs increases with accretion rate, from ∼ 20% up to ∼ 80%, for the range of accretion rates considered here (Section 5.2.4, Figure 8). This suggests that for some stars the region responsible for the BC becomes optically thick with accretion rate. As a response to the accretion process, the C iv lines develop a BC first.</list_item>
<list_item><location><page_21><loc_54><loc_7><loc_92><loc_16></location>· We show resolved multi-epoch observations in C iv for six CTTSs: BP Tau, DF Tau, DR Tau, RU Lup, RW Aur A, and T Tau N. The kind of variability observed is different for each star, but we confirm that the NC of BP Tau, DF Tau, and RU Lup increase with an increase in the line flux (Section 5.4, Figure 10).</list_item>
</unordered_list>
<section_header_level_1><location><page_22><loc_18><loc_91><loc_38><loc_92></location>9.3. Abundance Anomalies?</section_header_level_1>
<unordered_list>
<list_item><location><page_22><loc_11><loc_69><loc_48><loc_90></location>· The Si iv and N v line luminosities are correlated with the C iv line luminosities (Figure 12). The relationship between Si iv and C iv shows large scatter with respect to a linear relationship. If we model the emitting region as an optically thin plasma in coronal ionization equilibrium, we conclude that TW Hya and V4046 Sgr show evidence of silicon depletion with respect to carbon, as has already been noted in previous papers (Herczeg et al. 2002; Kastner et al. 2002; Stelzer & Schmitt 2004; Gunther et al. 2006). Other stars (AA Tau, DF Tau, GM Aur, and V1190 Sco) may also be siliconpoor, while CV Cha, DX Cha, RU Lup, and RW Aur may be silicon-rich. The relationship between N v and C iv shows significantly less scatter.</list_item>
<list_item><location><page_22><loc_11><loc_51><loc_48><loc_68></location>· For WTTSs, we observe silicon to be generally more abundant than expected within our simple model. This is reminiscent of the first ionization potential (FIP) effect. However, an inverse FIP has been observed in the Fe/Ne abundance ratios of active, low-mass, non-accreting stars, which should behave similarly to WTTSs (Gudel 2007). It is unclear whether the discrepancy is due to the simplicity of the assumed model or if the Si/C ratios in WTTSs behave differently than the Fe/Ne ratios in active stars. This points to the need for a more sophisticated model of the relationship between Si iv , C iv , and N v both in WTTSs and CTTSs.</list_item>
</unordered_list>
<section_header_level_1><location><page_22><loc_23><loc_48><loc_34><loc_50></location>9.4. Age effects</section_header_level_1>
<unordered_list>
<list_item><location><page_22><loc_11><loc_37><loc_48><loc_48></location>· Our sample covers a range of ages from ∼ 2 Myr to ∼ 10 Myr, and a wide range in disk evolutionary state. We do no detect changes in any of the measured quantities (line luminosities, FWHM, velocity centroids, etc) with age. We do not find systematic differences in any quantity considered here between the CTTS subsample with transition disks and the whole CTTS sample.</list_item>
<list_item><location><page_22><loc_11><loc_32><loc_48><loc_36></location>· We do not find any significant difference in the accretion rates of the Si iv -rich objects compared to those Si iv -poor, nor in their disk characteristics.</list_item>
</unordered_list>
<section_header_level_1><location><page_22><loc_16><loc_30><loc_41><loc_31></location>9.5. The origin of the hot gas lines</section_header_level_1>
<unordered_list>
<list_item><location><page_22><loc_11><loc_7><loc_48><loc_29></location>· We find no evidence for a decrease in any of the Gaussian components of C iv that could be interpreted as burying of the accretion column in the stellar photosphere due to the ram pressure of the accretion flow (Drake 2005; Sacco et al. 2010). In particular, we do not observe a decrease in the strength of the NC as a function of accretion, but exactly the opposite. In the context of the magnetospheric accretion model, we interpret this as an argument in favor of a line origin in either (a) an accretion spot with an aspect ratio such that the post-shock photons can escape the columns (f > 0.001), and/or (b) the low-density, slow-moving periphery of an inhomogeneous accretion column (Romanova et al. 2004), and/or (c) multiple, uncorrelated accretion columns of different densities (Sacco et al. 2010; Orlando et al.</list_item>
<list_item><location><page_22><loc_56><loc_89><loc_92><loc_92></location>2010; Ingleby et al. 2013), which appear as the result of increasing accretion rate.</list_item>
<list_item><location><page_22><loc_54><loc_70><loc_92><loc_88></location>· The accretion shock model predicts that the velocity of the post-shock emission should be 4 times smaller than the velocity of the pre-shock gas emission. If we identify the broad Gaussian component with the pre-shock emission and the NC with the post-shock emission, we find that V BC /greaterorsimilar 4 V NC is the case in only 2 out of 22 CTTSs. Accounting for possible pointing errors, 10 more objects could be within the accretion shock predictions (see Figure 9). For six systems the Gaussian decomposition is impossible (DX Cha, RW Aur) or requires only one Gaussian component (CV Cha, ET Cha, DM Tau, UX Tau A).</list_item>
<list_item><location><page_22><loc_54><loc_40><loc_92><loc_69></location>· When compared to the predicted infall velocities, the measured V BC values of these objects imply large ( ∼ 90 · ) inclinations of the flow with respect to the line of sight. Because this is unlikely in general, we suggest that the accretion flow responsible for the emission may not be at the free-fall velocity (for example if the emission comes from the slowregion at the edge of the column). Another alternative is that we are observing emission from multiple columns with different lines of sight, in which case the ratio of pre- to post-shock emission is not the same. Finally, radiation from the post-shock gas may be ionizing regions far from it, which, because of the shape of the magnetic field channeling the flow, may have line-of-sight velocities unrelated to the post-shock ones. Our models of the pre-shock gas show that pre-shock sizes can range from a few tens of kilometers to ∼ 1 R /circledot for typical parameters. These concepts may also explain cases for which the NC velocity is positive, but the BC velocity is small or negative (DF Tau, DR Tau, GM Aur, IP Tau, RU Lup, SU Aur, and V1190 Sco).</list_item>
<list_item><location><page_22><loc_54><loc_26><loc_92><loc_38></location>· The accretion shock model predicts that, because the formation temperature of He ii is lower than that of C iv , the post-shock line emission from the former should have a lower velocity than the latter. We confirm this, by identifying the NC emission with the post-shock gas emission. However, we note that gas flows in the stellar transition region may result in similar velocity offsets between C iv and He ii .</list_item>
<list_item><location><page_22><loc_54><loc_14><loc_92><loc_24></location>· Observationally, the amount of flux in the BC of the He ii line is small compared to the C iv line. We model the pre-shock column and conclude that for typical parameters, it can produce line ratios between C iv and He ii within the observed range. In other words, the weakness of the BC component in He ii is consistent with its origin in the pre-shock gas.</list_item>
<list_item><location><page_22><loc_54><loc_7><loc_92><loc_12></location>· Overall, we favor the origin of the emission lines in an accretion shock, but we cannot rule out that the true transition region is contributing part of the NC flux.</list_item>
</unordered_list>
<section_header_level_1><location><page_23><loc_16><loc_91><loc_40><loc_92></location>9.6. Outflow shocks and hot winds</section_header_level_1>
<text><location><page_23><loc_8><loc_87><loc_48><loc_90></location>We find three different types of profiles that show evidence of outflows (Section 8.2):</text>
<unordered_list>
<list_item><location><page_23><loc_11><loc_77><loc_48><loc_86></location>· For HN Tau A and RW Aur A, most of the C iv and He ii line fluxes are blueshifted and the peaks of the H 2 lines are redshifted (France et al. 2012). The C iv and He ii emission in this case should be produced by shocks within outflow jets. For these stars we may be observing the opposite sides of an outflow simultaneously (Section 5.1, Figure 5).</list_item>
<list_item><location><page_23><loc_11><loc_66><loc_48><loc_76></location>· For the CTTSs AK Sco, DK Tau, T Tau N, we find that the NC velocity is negative. These represent less extreme examples of objects such as HN Tau A and RW Aur, although the observed velocities are high enough that internal shocks in the outflow can result in hot gas emission. For these stars we do not observe any redshifted NC emission that would correspond to the postshock.</list_item>
<list_item><location><page_23><loc_11><loc_55><loc_48><loc_65></location>· Within our sample, DX Cha is the only candidate for a high temperature wind, as it shows a deficit in the red wing of the 1548 ˚ A line for C iv , compared to the red wing of the 1550 ˚ A line, and blue absorption in 1548 ˚ A C iv line (Figure 13). Lower temperature outflows in Si ii are also observed in DX Cha.</list_item>
</unordered_list>
<section_header_level_1><location><page_23><loc_21><loc_53><loc_36><loc_54></location>9.7. Peculiar Objects</section_header_level_1>
<text><location><page_23><loc_8><loc_48><loc_48><loc_52></location>Certain well-known objects are tagged as peculiar in this work. In addition to DX Cha, HN Tau, and RW Aur, mentioned before, the most peculiar are:</text>
<unordered_list>
<list_item><location><page_23><loc_11><loc_42><loc_48><loc_47></location>· AK Sco: Even when accounting for extra emission lines, the C iv lines are different form each other, perhaps as a result of it being a spectroscopic binary. It may be Si-rich.</list_item>
<list_item><location><page_23><loc_11><loc_37><loc_48><loc_41></location>· DF Tau: The 1548-to-1550 ratio is too large. The BC velocity is small compared to expectations of the accretion model.</list_item>
<list_item><location><page_23><loc_11><loc_33><loc_48><loc_36></location>· DK Tau: The BC velocity is negative, the C iv profile is double-peaked, may be Si-rich.</list_item>
<list_item><location><page_23><loc_11><loc_22><loc_48><loc_32></location>· DR Tau: For C iv , the BC velocity is small compared to expectations, the thermalization depth for the NC is small, but the BC flux ratio is anomalously high. The C iv lines are double-peaked. Multi-epoch observations show either absorption within 150 km s -1 of the line center, or evidence for rotational modulation in the profile. It may be Si-rich.</list_item>
<list_item><location><page_23><loc_11><loc_12><loc_48><loc_21></location>· RU Lup: The red wings of the C iv lines are different from each other, the BC velocity, the thermalization depth for the NC is small, and it may be Si-rich. In addition, some of the H 2 lines present a low level blushifted emission. As in DR Tau, we may be observing rotational modulation of the C iv profile in multi-epoch observations.</list_item>
</unordered_list>
<text><location><page_23><loc_8><loc_7><loc_48><loc_11></location>High-spectral resolution UV observations provide a crucial piece of the puzzle posed by young stellar evolution. They sample a much hotter plasma than optical</text>
<text><location><page_23><loc_52><loc_79><loc_92><loc_92></location>observations and complement and enhance X-ray data. However the data considered here has two important limitations. One is the lack of time-domain information, which makes it difficult to understand what is the average behavior of a given target. The other is the small range of accretion rates well-covered by objects. Future observational work should focus on resolving these limitations. Even with these limitations, it is clear that with this work we have only scratched the surface of this magnificent dataset.</text>
<figure>
<location><page_24><loc_9><loc_72><loc_48><loc_90></location>
<caption>Figure 1. Comparison between the CTTS BP Tau (black line) and the WTTS V396 Aur (blue line) for the four emission lines we are analyzing. The vertical scale is arbitrary. The abscissas are velocities (km s -1 ) in the stellar rest frame. Solid lines : Nominal line positions; Dashed lines : Nominal locations of the strongest H 2 lines. N V, dotted lines : N i (1243.18 ˚ A, +1055 km s -1 ; 1243.31 ˚ A, +1085 km s -1 ). Si IV , dotted lines : CO A-X (5-0) bandhead (1392.5 ˚ A, -271 km s -1 ), O iv (1401.17 ˚ A, +1633 km s -1 ). He II, dotted line : Location of the secondary He ii line.</caption>
</figure>
<figure>
<location><page_24><loc_9><loc_38><loc_48><loc_57></location>
</figure>
<text><location><page_24><loc_28><loc_38><loc_31><loc_38></location>km sec</text>
<figure>
<location><page_24><loc_54><loc_72><loc_92><loc_92></location>
<caption>Figure 2. The diversity of C iv CTTSs profiles. Dashed lines : Narrow and broad gaussian components; Smooth solid lines : Total model fit. The most common morphology is that of TW Hya, with a lower peak BC redshifted with respect to the NC. Stars like IP Tau have a BC blueshifted with respect to the NC. For V1190 Sco, both components have similar widths. For HN Tau the NC profile is blueshifted.</caption>
</figure>
<text><location><page_24><loc_70><loc_72><loc_74><loc_72></location>km sec</text>
<figure>
<location><page_24><loc_53><loc_25><loc_91><loc_44></location>
<caption>Figure 3. The WTTSs spectra in C iv and He ii . All of the spectra have been smoothed by a 5-point median. All the spectra have been scaled to have the same mean value in the wing between 150 and -50 km s -1 . The solid vertical lines mark the rest-velocity positions of the C iv and He ii lines. Top : C iv . The dashed vertical line indicates the location at which the H 2 line R(3)1- 8 would be, if present. The large value of the TWA 7 line is an artifact of the scaling procedure, due to the line redshift (25.3 km s -1 ). Bottom : He ii . Except for V1068 Tau and V410 Tau, the WTTSs have similar characteristics (shape, shifts, and FWHM) in He ii and C iv . V1068 Tau and V410 Tau appear to be truncated or broadened in both lines.Figure 4. Accretion rate vs. C iv luminosity. The blue diamonds correspond to stars with simultaneous determinations of accretion rate and line luminosity. The red diamonds are objects with nonsimultaneous determinations of accretion rate. The solid line is the correlation obtained by using all of the diamonds. Black triangles use the extinctions from Furlan et al. (2009, 2011) to calculate L CIV . Errors in L CIV /L /circledot are ∼ 5-10%. The dashed line is Equation 2 from Johns-Krull et al. (2000), assuming R ∗ = 2R /circledot for all stars. The lowest luminosity blue diamond corresponds to ET Cha (RECX 15) which has R ∗ = 0.9 R /circledot (Siess et al. 2000).</caption>
</figure>
<figure>
<location><page_25><loc_11><loc_72><loc_48><loc_91></location>
<caption>Figure 5. H 2 contamination of the C iv line in RW Aur. The plot shows that the C iv and H 2 emissions are blueshifted and redshifted, respectively, by almost 100 km s -1 . The result is that most of the C iv 1548 ˚ A line emission is covered by the redshifted H 2 . Vertical solid lines : nominal positions of the C iv doublet lines; Vertical dashed line : nominal R(3) 1-8 H 2 line position. Black trace : observed spectrum in the C iv region; Red trace : the C iv spectrum blueshifted by 500.96 km s -1 . With this shift, the nominal position of the 1550 ˚ A line should match the nominal position of the 1548 ˚ A line. Blue trace : The R(3)1-7 H 2 line (at 1489.57 ˚ A), but redshifted to the nominal position of the R(3)1-8 H 2 line (at 1547.34 ˚ A).</caption>
</figure>
<figure>
<location><page_26><loc_9><loc_38><loc_79><loc_92></location>
<caption>Figure 6. Measurements of the line shape. An abbreviated name is used for each star. WTTSs are plotted in red. To avoid crowding the plots, error bars are not shown. First row : Velocity at the maximum flux (measurement errors: ∼ 2 km s -1 ; systematic error: ∼ 3 km s -1 for STIS data, ∼ 15 km s -1 for COS ). Second row : FWHM (errors: ∼ 5 km s -1 ). Third row : Skewness (errors: ∼ 0.005). For the skewness, the dashed lines indicate ± 0 . 02, the region within which we consider the profiles to be symmetric. Fourth row: Scale factor from the 1550 ˚ A to the 1548 ˚ A C iv line. Errors are ∼ 0.1 - 0.3. We also indicate the variability path followed by BP Tau, DF Tau, DR Tau, T Tau N, and RU Lup.; Fifth row: Inclination, with zero meaning 'face-on.' When the inclination angle errors are given in the literature, they are ∼ 10 deg.</caption>
</figure>
<figure>
<location><page_27><loc_9><loc_72><loc_48><loc_91></location>
<caption>Figure 7. Histogram of line components for CTTSs. Left Column : Full widths at half maxima for C iv and He ii . Right Column : Velocity at maximum flux for C iv and He ii . Clear histogram : BC; Diagonal hatch histogram : NC</caption>
</figure>
<figure>
<location><page_27><loc_9><loc_44><loc_48><loc_63></location>
</figure>
<text><location><page_27><loc_32><loc_44><loc_33><loc_44></location>Sun</text>
<figure>
<location><page_27><loc_53><loc_72><loc_90><loc_91></location>
<caption>Figure 8. The C iv line shape as a function of accretion rate. Typical errors in the ordinate axes are ∼ 10%. Top : The ratio of the luminosity 1548 ˚ A to 1550 ˚ A. Red labels indicate the ratio of the luminosities of the BCs. Black labels are the ratio of the luminosities of the NCs. Bottom : The fraction of the line luminosity in the NC. Black labels are for CTTSs, while red are for WTTSs. For CTTSs, the NC contributes about 20% of the luminosity at low accretion rates, and up to 80% at high accretion rates.Figure 9. Narrow vs. the broad velocity components for the Gaussian decomposition of the C iv lines. The plot is divided in quadrants (solid horizontal and vertical lines). blue labels indicate targets observed with STIS, while black labels are for targets observed with COS. Red labels are for WTTSs (all observed with COS). The errors in the velocities are dominated by systematic errors, shown as boxes in the upper left of the diagram. Errors in the wavelength scale will move the points parallel to the dotted line. If the line emission is dominated by the magnetospheric shock, the data points should reside within the hatched region (v NC < v BC / 4). More generally, when taking into account the wavelength errors, all points below the dotted line in the upper right-hand quadrant may reside within the hatched region</caption>
</figure>
<figure>
<location><page_28><loc_8><loc_56><loc_78><loc_91></location>
<caption>Figure 10. CTTSs with multi-epoch, high-resolution observations of C iv . Nominal line centroids are indicated by a vertical solid line and the position of the R(3)1- 8 H 2 line is indicated by a dashed line. Black : Observations from this paper. Red : GHRS observations from Ardila et al. (2002). The systematic error in the GHRS wavelength scale is 20 km s -1 . For DR Tau there are two GHRS observations available in the literature, one from 1993 and the other from 1995. To avoid crowding the figure we plot here only the one from 1995. Blue : For DR Tau, this observation belongs to HST program GO 8206 (PI Calvet). For DF Tau and RU Lup, the spectra are from Herczeg et al. (2005, 2006).</caption>
</figure>
<text><location><page_29><loc_9><loc_88><loc_10><loc_89></location>-1</text>
<text><location><page_29><loc_29><loc_46><loc_30><loc_46></location>CIV</text>
<text><location><page_29><loc_31><loc_46><loc_32><loc_46></location>Sun</text>
<figure>
<location><page_29><loc_10><loc_46><loc_48><loc_91></location>
<caption>Figure 11. Shape characteristics for He ii compared with C iv . Diamonds indicate the C iv values, blue triangles the CTTSs He ii values, red symbols the WTTSs. Values for the same star are joined by a segment.</caption>
</figure>
<figure>
<location><page_30><loc_10><loc_72><loc_48><loc_92></location>
<caption>Figure 12. Si iv luminosity and N v luminosity vs. C iv luminosity. Errors are ∼ 10-30% for the Si iv and N v measurements and ∼ 1-5% for C iv , as indicated in Table 8. Red labels are for WTTS. Downward arrows indicate upper limits. In both plots, the dashed line is the linear fit to the data, ignoring the non-detections and the WTTSs and the solid line corresponds to L SiIV = 0 . 111 L CIV ( Top ) or L NV = 0 . 183 L CIV ( Bottom ).</caption>
</figure>
<figure>
<location><page_30><loc_8><loc_40><loc_46><loc_59></location>
<caption>Figure 13. The N v , Si iv , and C iv doublets for DX Cha. The red line is the redder member of the doublet, scaled to match the blue member. The red member of the N v lines shows strong N i absorption. The red member of Si iv show O iv emission at -341 km s -1 . Notice the very sharp blue cutoff in both Si iv lines. In the case of C iv , we believe that red wing of the bluer line (shown in black) is being absorbed by a hot wind. Blueshifted absorption is seen in the blue wing of the blue member.</caption>
</figure>
<text><location><page_31><loc_46><loc_88><loc_54><loc_89></location>APPENDIX</text>
<text><location><page_31><loc_42><loc_86><loc_59><loc_87></location>MULTI-PANEL FIGURES</text>
<figure>
<location><page_32><loc_15><loc_21><loc_85><loc_91></location>
<caption>Figure A.1. Hot lines for CTTSs ordered by decreasing ˙ M . Left to right: N v , Si iv , C iv , He ii . The plots show flux density (10 -14 erg sec -1 cm -2 ˚ A -1 ) versus velocities (km s -1 ) in the stellar rest frame. The spectra are not extinction-corrected nor continuum-subtracted. Transition disks are indicated with 'TD' after the name. Solid lines : Nominal line positions; Dashed lines : Nominal locations of the strongest H 2 lines. N v, dotted lines : N i (1243.18 ˚ A, +1055 km s -1 ; 1243.31 ˚ A, +1085 km s -1 ). Si iv, dotted lines : CO A-X (5-0) bandhead (1392.5 ˚ A, -271 km s -1 ), O iv (1401.17 ˚ A, +1633 km s -1 ). He ii, dotted line : Location of the secondary He ii line (Section 6).</caption>
</figure>
<figure>
<location><page_33><loc_15><loc_21><loc_85><loc_91></location>
<caption>Figure A.2. Same as Figure A.1</caption>
</figure>
<figure>
<location><page_34><loc_15><loc_21><loc_85><loc_91></location>
<caption>Figure A.3. Same as Figure A.1</caption>
</figure>
<figure>
<location><page_35><loc_16><loc_48><loc_85><loc_91></location>
<caption>Figure A.4. Same as Figure A.1</caption>
</figure>
<figure>
<location><page_36><loc_15><loc_39><loc_85><loc_91></location>
<caption>Figure A.5. The 'hot' lines for WTTS. The stars are listed in alphabetical order. Axes and markings are the same as in Figure A.1</caption>
</figure>
<figure>
<location><page_37><loc_15><loc_28><loc_85><loc_91></location>
<caption>Figure A.6. Both members of the C iv doublet, scaled and overplotted. The 1550 ˚ A line (the red member of the doublet) is shown in red, and it has been scaled to match the red wing of the 1548 ˚ A line. Axes and units are the same as in figure A.1.</caption>
</figure>
<figure>
<location><page_38><loc_15><loc_28><loc_85><loc_91></location>
<caption>Figure A.7. The black trace shows the He ii line overploted to the 1550 ˚ A C iv and scaled to the same maximum value. Axes and units are the same as in figure A.1.</caption>
</figure>
<text><location><page_38><loc_15><loc_89><loc_16><loc_90></location>2.5</text>
<text><location><page_38><loc_15><loc_88><loc_16><loc_89></location>2.0</text>
<text><location><page_38><loc_15><loc_87><loc_16><loc_88></location>1.5</text>
<text><location><page_38><loc_15><loc_86><loc_16><loc_86></location>1.0</text>
<text><location><page_38><loc_15><loc_85><loc_16><loc_85></location>0.5</text>
<text><location><page_38><loc_15><loc_84><loc_16><loc_84></location>0.0</text>
<text><location><page_38><loc_15><loc_71><loc_16><loc_72></location>1.2</text>
<text><location><page_38><loc_15><loc_70><loc_16><loc_71></location>1.0</text>
<text><location><page_38><loc_15><loc_69><loc_16><loc_70></location>0.8</text>
<text><location><page_38><loc_15><loc_68><loc_16><loc_69></location>0.6</text>
<text><location><page_38><loc_15><loc_67><loc_16><loc_68></location>0.4</text>
<text><location><page_38><loc_15><loc_66><loc_16><loc_67></location>0.2</text>
<text><location><page_38><loc_15><loc_65><loc_16><loc_66></location>0.0</text>
<text><location><page_38><loc_15><loc_53><loc_16><loc_54></location>2.5</text>
<text><location><page_38><loc_15><loc_52><loc_16><loc_52></location>2.0</text>
<text><location><page_38><loc_15><loc_51><loc_16><loc_51></location>1.5</text>
<text><location><page_38><loc_15><loc_49><loc_16><loc_50></location>1.0</text>
<text><location><page_38><loc_15><loc_48><loc_16><loc_49></location>0.5</text>
<text><location><page_38><loc_15><loc_47><loc_16><loc_48></location>0.0</text>
<text><location><page_39><loc_8><loc_84><loc_92><loc_92></location>Based on observations made with the NASA/ESA Hubble Space Telescope. Support for this paper was provided by NASA through grant numbers HST-GO-11616.10 and HST-GO-12161.01 from the Space Telescope Science Institute (STScI), which is operated by Association of Universities for Research in Astronomy, Inc (AURA) under NASA contract NAS 5-26555. SGG acknowledges support from the Science & Technology Facilities Council (STFC) via an Ernest Rutherford Fellowship [ST/J003255/1]. RDA acknowledges support from the UK's Science & Technology Facilities Council (STFC) through an Advanced Fellowship (ST/G00711X/1).</text>
<text><location><page_39><loc_8><loc_80><loc_92><loc_84></location>This research has made use of NASA's Astrophysics Data System Bibliographic Services and CHIANTI, a collaborative project involving George Mason University, the University of Michigan (USA) and the University of Cambridge (UK).</text>
<text><location><page_39><loc_8><loc_77><loc_92><loc_80></location>We thank the team from HST GTO programs 11533 and 12036 (PI J. Green) for allowing us early access to their data.</text>
<section_header_level_1><location><page_39><loc_10><loc_75><loc_34><loc_77></location>Facilities: HST ( COS , STIS ). REFERENCES</section_header_level_1>
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<text><location><page_42><loc_9><loc_93><loc_10><loc_94></location>42</text>
<text><location><page_42><loc_45><loc_93><loc_56><loc_94></location>Ardila et al.</text>
<text><location><page_42><loc_10><loc_56><loc_11><loc_56></location>r</text>
<text><location><page_42><loc_10><loc_55><loc_11><loc_56></location>e</text>
<text><location><page_42><loc_10><loc_55><loc_11><loc_55></location>p</text>
<text><location><page_42><loc_10><loc_54><loc_11><loc_55></location>a</text>
<text><location><page_42><loc_10><loc_54><loc_11><loc_54></location>p</text>
<text><location><page_42><loc_10><loc_53><loc_11><loc_53></location>s</text>
<text><location><page_42><loc_10><loc_52><loc_11><loc_53></location>i</text>
<text><location><page_42><loc_10><loc_52><loc_11><loc_52></location>h</text>
<text><location><page_42><loc_10><loc_51><loc_11><loc_52></location>t</text>
<text><location><page_42><loc_10><loc_51><loc_11><loc_51></location>n</text>
<text><location><page_42><loc_10><loc_50><loc_11><loc_51></location>i</text>
<text><location><page_42><loc_10><loc_49><loc_11><loc_50></location>d</text>
<text><location><page_42><loc_10><loc_49><loc_11><loc_49></location>e</text>
<text><location><page_42><loc_10><loc_48><loc_11><loc_49></location>z</text>
<text><location><page_42><loc_10><loc_48><loc_11><loc_48></location>y</text>
<text><location><page_42><loc_10><loc_47><loc_11><loc_48></location>l</text>
<text><location><page_42><loc_10><loc_47><loc_11><loc_47></location>a</text>
<text><location><page_42><loc_10><loc_46><loc_11><loc_47></location>n</text>
<text><location><page_42><loc_10><loc_46><loc_11><loc_46></location>a</text>
<text><location><page_42><loc_10><loc_45><loc_11><loc_45></location>s</text>
<text><location><page_42><loc_10><loc_45><loc_11><loc_45></location>r</text>
<text><location><page_42><loc_10><loc_44><loc_11><loc_45></location>a</text>
<text><location><page_42><loc_10><loc_44><loc_11><loc_44></location>t</text>
<text><location><page_42><loc_10><loc_43><loc_11><loc_44></location>S</text>
<table>
<location><page_42><loc_12><loc_22><loc_87><loc_88></location>
</table>
<text><location><page_42><loc_13><loc_22><loc_14><loc_22></location>l</text>
<text><location><page_42><loc_13><loc_21><loc_14><loc_22></location>A</text>
<text><location><page_42><loc_13><loc_17><loc_14><loc_18></location>)</text>
<text><location><page_42><loc_13><loc_17><loc_14><loc_17></location>a</text>
<text><location><page_42><loc_13><loc_17><loc_14><loc_17></location>(</text>
<text><location><page_42><loc_13><loc_16><loc_14><loc_16></location>e</text>
<text><location><page_42><loc_13><loc_15><loc_14><loc_16></location>m</text>
<text><location><page_42><loc_13><loc_14><loc_14><loc_15></location>a</text>
<text><location><page_42><loc_13><loc_13><loc_14><loc_14></location>N</text>
<text><location><page_42><loc_48><loc_21><loc_49><loc_22></location>B</text>
<text><location><page_42><loc_48><loc_20><loc_49><loc_21></location>H</text>
<text><location><page_42><loc_63><loc_21><loc_64><loc_22></location>C</text>
<text><location><page_42><loc_63><loc_21><loc_64><loc_21></location>k</text>
<text><location><page_42><loc_63><loc_20><loc_64><loc_21></location>L</text>
<text><location><page_42><loc_40><loc_17><loc_64><loc_18></location>A A o r A u</text>
<text><location><page_42><loc_41><loc_17><loc_42><loc_17></location>s</text>
<text><location><page_42><loc_41><loc_16><loc_42><loc_17></location>u</text>
<text><location><page_42><loc_41><loc_15><loc_42><loc_16></location>M</text>
<text><location><page_42><loc_41><loc_14><loc_42><loc_15></location>P</text>
<text><location><page_42><loc_41><loc_13><loc_42><loc_14></location>M</text>
<text><location><page_42><loc_44><loc_17><loc_45><loc_17></location>r</text>
<text><location><page_42><loc_44><loc_16><loc_45><loc_17></location>u</text>
<text><location><page_42><loc_44><loc_15><loc_45><loc_16></location>A</text>
<text><location><page_42><loc_44><loc_14><loc_45><loc_15></location>W</text>
<text><location><page_42><loc_44><loc_13><loc_45><loc_14></location>R</text>
<text><location><page_42><loc_47><loc_16><loc_48><loc_17></location>N</text>
<text><location><page_42><loc_47><loc_15><loc_48><loc_16></location>u</text>
<text><location><page_42><loc_47><loc_15><loc_48><loc_15></location>a</text>
<text><location><page_42><loc_47><loc_14><loc_48><loc_15></location>T</text>
<text><location><page_42><loc_47><loc_13><loc_48><loc_14></location>T</text>
<text><location><page_42><loc_48><loc_17><loc_49><loc_17></location>c</text>
<text><location><page_42><loc_48><loc_16><loc_49><loc_17></location>S</text>
<text><location><page_42><loc_48><loc_16><loc_49><loc_16></location>0</text>
<text><location><page_42><loc_48><loc_15><loc_49><loc_16></location>9</text>
<text><location><page_42><loc_48><loc_14><loc_49><loc_15></location>1</text>
<text><location><page_42><loc_48><loc_14><loc_49><loc_14></location>1</text>
<text><location><page_42><loc_48><loc_13><loc_49><loc_14></location>V</text>
<text><location><page_42><loc_49><loc_17><loc_51><loc_18></location>g</text>
<text><location><page_42><loc_49><loc_16><loc_51><loc_17></location>S</text>
<text><location><page_42><loc_49><loc_16><loc_51><loc_16></location>6</text>
<text><location><page_42><loc_49><loc_15><loc_51><loc_16></location>4</text>
<text><location><page_42><loc_49><loc_14><loc_51><loc_15></location>0</text>
<text><location><page_42><loc_49><loc_14><loc_51><loc_14></location>4</text>
<text><location><page_42><loc_49><loc_13><loc_51><loc_14></location>V</text>
<text><location><page_42><loc_56><loc_16><loc_57><loc_17></location>u</text>
<text><location><page_42><loc_56><loc_16><loc_57><loc_16></location>a</text>
<text><location><page_42><loc_56><loc_15><loc_57><loc_16></location>T</text>
<text><location><page_42><loc_56><loc_14><loc_57><loc_15></location>M</text>
<text><location><page_42><loc_56><loc_13><loc_57><loc_14></location>D</text>
<text><location><page_42><loc_57><loc_17><loc_58><loc_17></location>r</text>
<text><location><page_42><loc_57><loc_16><loc_58><loc_17></location>u</text>
<text><location><page_42><loc_57><loc_15><loc_58><loc_16></location>A</text>
<text><location><page_42><loc_57><loc_14><loc_58><loc_15></location>M</text>
<text><location><page_42><loc_57><loc_13><loc_58><loc_14></location>G</text>
<text><location><page_42><loc_60><loc_17><loc_61><loc_17></location>a</text>
<text><location><page_42><loc_60><loc_16><loc_61><loc_17></location>y</text>
<text><location><page_42><loc_60><loc_15><loc_61><loc_16></location>H</text>
<text><location><page_42><loc_60><loc_14><loc_61><loc_15></location>W</text>
<text><location><page_42><loc_60><loc_13><loc_61><loc_14></location>T</text>
<text><location><page_42><loc_63><loc_17><loc_64><loc_18></location>a</text>
<text><location><page_42><loc_63><loc_16><loc_64><loc_17></location>T</text>
<text><location><page_42><loc_63><loc_16><loc_64><loc_16></location>9</text>
<text><location><page_42><loc_63><loc_15><loc_64><loc_16></location>7</text>
<text><location><page_42><loc_63><loc_14><loc_64><loc_15></location>0</text>
<text><location><page_42><loc_63><loc_14><loc_64><loc_14></location>1</text>
<text><location><page_42><loc_63><loc_13><loc_64><loc_14></location>V</text>
<text><location><page_42><loc_72><loc_21><loc_73><loc_22></location>a</text>
<text><location><page_42><loc_72><loc_21><loc_73><loc_21></location>C</text>
<text><location><page_42><loc_72><loc_20><loc_73><loc_21></location>k</text>
<text><location><page_42><loc_72><loc_19><loc_73><loc_20></location>L</text>
<text><location><page_42><loc_72><loc_17><loc_73><loc_18></location>r</text>
<text><location><page_42><loc_72><loc_17><loc_73><loc_17></location>u</text>
<text><location><page_42><loc_72><loc_16><loc_73><loc_17></location>A</text>
<text><location><page_42><loc_72><loc_15><loc_73><loc_16></location>6</text>
<text><location><page_42><loc_72><loc_14><loc_73><loc_15></location>9</text>
<text><location><page_42><loc_72><loc_14><loc_73><loc_14></location>3</text>
<text><location><page_42><loc_72><loc_13><loc_73><loc_14></location>V</text>
<text><location><page_42><loc_80><loc_21><loc_81><loc_22></location>a</text>
<text><location><page_42><loc_80><loc_21><loc_81><loc_21></location>t</text>
<text><location><page_42><loc_80><loc_21><loc_81><loc_21></location>s</text>
<text><location><page_42><loc_80><loc_20><loc_81><loc_20></location>e</text>
<text><location><page_42><loc_80><loc_20><loc_81><loc_20></location>h</text>
<text><location><page_42><loc_80><loc_19><loc_81><loc_20></location>t</text>
<text><location><page_42><loc_80><loc_19><loc_81><loc_19></location>o</text>
<text><location><page_42><loc_80><loc_18><loc_81><loc_19></location>t</text>
<text><location><page_42><loc_80><loc_17><loc_81><loc_18></location>r</text>
<text><location><page_42><loc_80><loc_17><loc_81><loc_18></location>e</text>
<text><location><page_42><loc_80><loc_17><loc_81><loc_17></location>f</text>
<text><location><page_42><loc_80><loc_16><loc_81><loc_17></location>e</text>
<text><location><page_42><loc_80><loc_16><loc_81><loc_16></location>r</text>
<text><location><page_42><loc_80><loc_15><loc_81><loc_16></location>e</text>
<text><location><page_42><loc_80><loc_15><loc_81><loc_15></location>w</text>
<text><location><page_42><loc_80><loc_14><loc_81><loc_15></location>r</text>
<text><location><page_42><loc_80><loc_14><loc_81><loc_14></location>e</text>
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<text><location><page_42><loc_80><loc_13><loc_81><loc_13></location>a</text>
<text><location><page_42><loc_80><loc_12><loc_81><loc_13></location>p</text>
<text><location><page_42><loc_80><loc_12><loc_81><loc_12></location>s</text>
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<text><location><page_42><loc_80><loc_11><loc_81><loc_11></location>t</text>
<text><location><page_42><loc_80><loc_10><loc_81><loc_10></location>n</text>
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<text><location><page_42><loc_81><loc_20><loc_82><loc_21></location>l</text>
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<text><location><page_42><loc_81><loc_19><loc_82><loc_20></location>t</text>
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<text><location><page_42><loc_81><loc_16><loc_82><loc_17></location>t</text>
<text><location><page_42><loc_81><loc_16><loc_82><loc_16></location>,</text>
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<text><location><page_42><loc_81><loc_9><loc_82><loc_9></location>)</text>
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<text><location><page_42><loc_31><loc_14><loc_33><loc_15></location>N</text>
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<text><location><page_42><loc_33><loc_16><loc_34><loc_17></location>u</text>
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<text><location><page_42><loc_33><loc_14><loc_34><loc_15></location>R</text>
<text><location><page_42><loc_33><loc_13><loc_34><loc_14></location>D</text>
<text><location><page_42><loc_34><loc_16><loc_35><loc_17></location>u</text>
<text><location><page_42><loc_34><loc_16><loc_35><loc_16></location>a</text>
<text><location><page_42><loc_34><loc_15><loc_35><loc_16></location>T</text>
<text><location><page_42><loc_34><loc_14><loc_35><loc_15></location>S</text>
<text><location><page_42><loc_34><loc_13><loc_35><loc_14></location>D</text>
<text><location><page_42><loc_36><loc_16><loc_37><loc_17></location>a</text>
<text><location><page_42><loc_36><loc_16><loc_37><loc_16></location>h</text>
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<text><location><page_42><loc_36><loc_14><loc_37><loc_15></location>X</text>
<text><location><page_42><loc_36><loc_13><loc_37><loc_14></location>D</text>
<text><location><page_42><loc_37><loc_16><loc_38><loc_17></location>a</text>
<text><location><page_42><loc_37><loc_16><loc_38><loc_16></location>h</text>
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<text><location><page_42><loc_37><loc_14><loc_38><loc_15></location>P</text>
<text><location><page_42><loc_37><loc_13><loc_38><loc_14></location>E</text>
<text><location><page_42><loc_38><loc_16><loc_40><loc_17></location>a</text>
<text><location><page_42><loc_38><loc_16><loc_40><loc_16></location>h</text>
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<text><location><page_42><loc_38><loc_14><loc_40><loc_15></location>T</text>
<text><location><page_42><loc_38><loc_13><loc_40><loc_14></location>E</text>
<text><location><page_42><loc_40><loc_16><loc_41><loc_17></location>u</text>
<text><location><page_42><loc_40><loc_16><loc_41><loc_16></location>a</text>
<text><location><page_42><loc_40><loc_15><loc_41><loc_16></location>T</text>
<text><location><page_42><loc_40><loc_14><loc_41><loc_15></location>N</text>
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<text><location><page_42><loc_42><loc_16><loc_44><loc_16></location>u</text>
<text><location><page_42><loc_42><loc_15><loc_44><loc_16></location>L</text>
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<text><location><page_42><loc_42><loc_13><loc_44><loc_14></location>R</text>
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<text><location><page_42><loc_45><loc_16><loc_47><loc_16></location>u</text>
<text><location><page_42><loc_45><loc_15><loc_47><loc_16></location>A</text>
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<text><location><page_42><loc_54><loc_13><loc_56><loc_14></location>C</text>
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<text><location><page_42><loc_58><loc_15><loc_60><loc_16></location>a</text>
<text><location><page_42><loc_58><loc_15><loc_60><loc_15></location>T</text>
<text><location><page_42><loc_58><loc_13><loc_60><loc_14></location>P</text>
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<text><location><page_42><loc_61><loc_16><loc_63><loc_17></location>u</text>
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<text><location><page_42><loc_61><loc_15><loc_63><loc_16></location>T</text>
<text><location><page_42><loc_61><loc_14><loc_63><loc_15></location>X</text>
<text><location><page_42><loc_61><loc_13><loc_63><loc_14></location>U</text>
<text><location><page_42><loc_68><loc_16><loc_69><loc_17></location>a</text>
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<text><location><page_42><loc_68><loc_13><loc_69><loc_14></location>E</text>
<text><location><page_42><loc_69><loc_21><loc_70><loc_22></location>A</text>
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<text><location><page_42><loc_70><loc_21><loc_72><loc_22></location>C</text>
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<text><location><page_42><loc_70><loc_18><loc_72><loc_18></location>u</text>
<text><location><page_42><loc_70><loc_17><loc_72><loc_18></location>a</text>
<text><location><page_42><loc_70><loc_16><loc_72><loc_17></location>T</text>
<text><location><page_42><loc_70><loc_16><loc_72><loc_16></location>8</text>
<text><location><page_42><loc_70><loc_15><loc_72><loc_16></location>6</text>
<text><location><page_42><loc_70><loc_14><loc_72><loc_15></location>0</text>
<text><location><page_42><loc_70><loc_14><loc_72><loc_14></location>1</text>
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<text><location><page_42><loc_73><loc_17><loc_74><loc_18></location>r</text>
<text><location><page_42><loc_73><loc_17><loc_74><loc_17></location>u</text>
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<text><location><page_42><loc_73><loc_15><loc_74><loc_16></location>7</text>
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<text><location><page_42><loc_73><loc_14><loc_74><loc_14></location>3</text>
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<text><location><page_42><loc_74><loc_17><loc_76><loc_18></location>u</text>
<text><location><page_42><loc_74><loc_17><loc_76><loc_17></location>a</text>
<text><location><page_42><loc_74><loc_16><loc_76><loc_17></location>T</text>
<text><location><page_42><loc_74><loc_15><loc_76><loc_16></location>0</text>
<text><location><page_42><loc_74><loc_14><loc_76><loc_15></location>1</text>
<text><location><page_42><loc_74><loc_14><loc_76><loc_14></location>4</text>
<text><location><page_42><loc_74><loc_13><loc_76><loc_14></location>V</text>
<text><location><page_42><loc_79><loc_12><loc_80><loc_13></location>-</text>
<text><location><page_42><loc_79><loc_11><loc_80><loc_11></location>.</text>
<text><location><page_42><loc_79><loc_11><loc_80><loc_11></location>e</text>
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<text><location><page_42><loc_79><loc_10><loc_80><loc_10></location>o</text>
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<text><location><page_43><loc_20><loc_90><loc_22><loc_91></location>)</text>
<text><location><page_43><loc_20><loc_90><loc_22><loc_90></location>7</text>
<text><location><page_43><loc_20><loc_89><loc_22><loc_90></location>0</text>
<text><location><page_43><loc_20><loc_89><loc_22><loc_89></location>0</text>
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<text><location><page_43><loc_20><loc_88><loc_22><loc_88></location>(</text>
<text><location><page_43><loc_20><loc_10><loc_22><loc_11></location>a</text>
<text><location><page_43><loc_20><loc_10><loc_22><loc_10></location>T</text>
<text><location><page_43><loc_20><loc_9><loc_22><loc_9></location>A</text>
<text><location><page_43><loc_20><loc_8><loc_22><loc_9></location>A</text>
<text><location><page_43><loc_23><loc_91><loc_24><loc_91></location>)</text>
<text><location><page_43><loc_23><loc_90><loc_24><loc_91></location>3</text>
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<text><location><page_43><loc_23><loc_88><loc_24><loc_89></location>1</text>
<text><location><page_43><loc_23><loc_88><loc_24><loc_88></location>(</text>
<text><location><page_43><loc_26><loc_90><loc_27><loc_91></location>)</text>
<text><location><page_43><loc_26><loc_90><loc_27><loc_90></location>7</text>
<text><location><page_43><loc_26><loc_89><loc_27><loc_90></location>0</text>
<text><location><page_43><loc_26><loc_89><loc_27><loc_89></location>0</text>
<text><location><page_43><loc_26><loc_88><loc_27><loc_89></location>2</text>
<text><location><page_43><loc_26><loc_88><loc_27><loc_88></location>(</text>
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<text><location><page_43><loc_26><loc_10><loc_27><loc_10></location>T</text>
<text><location><page_43><loc_26><loc_9><loc_27><loc_9></location>Y</text>
<text><location><page_43><loc_26><loc_8><loc_27><loc_9></location>C</text>
<text><location><page_43><loc_27><loc_90><loc_29><loc_91></location>)</text>
<text><location><page_43><loc_27><loc_90><loc_29><loc_90></location>7</text>
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<text><location><page_43><loc_27><loc_89><loc_29><loc_89></location>0</text>
<text><location><page_43><loc_27><loc_88><loc_29><loc_89></location>2</text>
<text><location><page_43><loc_27><loc_88><loc_29><loc_88></location>(</text>
<text><location><page_43><loc_29><loc_90><loc_30><loc_90></location>)</text>
<text><location><page_43><loc_29><loc_89><loc_30><loc_90></location>3</text>
<text><location><page_43><loc_29><loc_89><loc_30><loc_89></location>9</text>
<text><location><page_43><loc_29><loc_88><loc_30><loc_89></location>9</text>
<text><location><page_43><loc_29><loc_88><loc_30><loc_88></location>1</text>
<text><location><page_43><loc_31><loc_90><loc_33><loc_91></location>)</text>
<text><location><page_43><loc_31><loc_90><loc_33><loc_90></location>7</text>
<text><location><page_43><loc_31><loc_89><loc_33><loc_90></location>0</text>
<text><location><page_43><loc_31><loc_89><loc_33><loc_89></location>0</text>
<text><location><page_43><loc_31><loc_88><loc_33><loc_89></location>2</text>
<text><location><page_43><loc_31><loc_88><loc_33><loc_88></location>(</text>
<text><location><page_43><loc_31><loc_10><loc_33><loc_11></location>a</text>
<text><location><page_43><loc_33><loc_91><loc_34><loc_91></location>)</text>
<text><location><page_43><loc_33><loc_90><loc_34><loc_91></location>3</text>
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<text><location><page_43><loc_33><loc_89><loc_34><loc_90></location>9</text>
<text><location><page_43><loc_33><loc_88><loc_34><loc_89></location>1</text>
<text><location><page_43><loc_33><loc_88><loc_34><loc_88></location>(</text>
<text><location><page_43><loc_34><loc_90><loc_36><loc_91></location>)</text>
<text><location><page_43><loc_34><loc_90><loc_36><loc_90></location>7</text>
<text><location><page_43><loc_34><loc_89><loc_36><loc_90></location>0</text>
<text><location><page_43><loc_34><loc_89><loc_36><loc_89></location>0</text>
<text><location><page_43><loc_34><loc_88><loc_36><loc_89></location>2</text>
<text><location><page_43><loc_34><loc_88><loc_36><loc_88></location>(</text>
<text><location><page_43><loc_38><loc_90><loc_40><loc_90></location>)</text>
<text><location><page_43><loc_38><loc_89><loc_40><loc_90></location>4</text>
<text><location><page_43><loc_38><loc_89><loc_40><loc_90></location>0</text>
<text><location><page_43><loc_38><loc_88><loc_40><loc_89></location>0</text>
<text><location><page_43><loc_38><loc_88><loc_40><loc_88></location>2</text>
<text><location><page_43><loc_38><loc_93><loc_61><loc_94></location>Hot lines in T Tauri stars</text>
<text><location><page_43><loc_90><loc_93><loc_92><loc_94></location>43</text>
<text><location><page_43><loc_40><loc_92><loc_41><loc_92></location>)</text>
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<text><location><page_43><loc_40><loc_90><loc_41><loc_91></location>0</text>
<text><location><page_43><loc_40><loc_90><loc_41><loc_90></location>2</text>
<text><location><page_43><loc_40><loc_90><loc_41><loc_90></location>(</text>
<text><location><page_43><loc_40><loc_89><loc_41><loc_89></location>.</text>
<text><location><page_43><loc_40><loc_89><loc_41><loc_89></location>l</text>
<text><location><page_43><loc_40><loc_88><loc_41><loc_89></location>a</text>
<text><location><page_43><loc_41><loc_91><loc_42><loc_91></location>)</text>
<text><location><page_43><loc_41><loc_90><loc_42><loc_91></location>1</text>
<text><location><page_43><loc_41><loc_89><loc_42><loc_90></location>1</text>
<text><location><page_43><loc_41><loc_89><loc_42><loc_89></location>0</text>
<text><location><page_43><loc_41><loc_88><loc_42><loc_89></location>2</text>
<text><location><page_43><loc_41><loc_88><loc_42><loc_88></location>(</text>
<text><location><page_43><loc_44><loc_91><loc_45><loc_91></location>)</text>
<text><location><page_43><loc_44><loc_90><loc_45><loc_91></location>1</text>
<text><location><page_43><loc_44><loc_90><loc_45><loc_90></location>1</text>
<text><location><page_43><loc_44><loc_89><loc_45><loc_90></location>0</text>
<text><location><page_43><loc_44><loc_88><loc_45><loc_89></location>2</text>
<text><location><page_43><loc_44><loc_88><loc_45><loc_88></location>(</text>
<text><location><page_43><loc_45><loc_88><loc_47><loc_88></location>)</text>
<text><location><page_43><loc_47><loc_91><loc_48><loc_91></location>;</text>
<text><location><page_43><loc_47><loc_90><loc_48><loc_91></location>)</text>
<text><location><page_43><loc_47><loc_90><loc_48><loc_90></location>9</text>
<text><location><page_43><loc_47><loc_89><loc_48><loc_90></location>9</text>
<text><location><page_43><loc_47><loc_89><loc_48><loc_89></location>9</text>
<text><location><page_43><loc_47><loc_88><loc_48><loc_89></location>1</text>
<section_header_level_1><location><page_43><loc_48><loc_89><loc_55><loc_90></location>0 2 0</section_header_level_1>
<paragraph><location><page_43><loc_48><loc_89><loc_55><loc_91></location>0 6 ) 0 1 2 ) ) 0 )</paragraph>
<text><location><page_43><loc_48><loc_89><loc_49><loc_89></location>2</text>
<text><location><page_43><loc_48><loc_88><loc_49><loc_89></location>(</text>
<text><location><page_43><loc_49><loc_89><loc_51><loc_89></location>2</text>
<text><location><page_43><loc_49><loc_88><loc_51><loc_89></location>(</text>
<text><location><page_43><loc_49><loc_10><loc_51><loc_11></location>u</text>
<text><location><page_43><loc_49><loc_10><loc_51><loc_10></location>A</text>
<text><location><page_43><loc_49><loc_8><loc_51><loc_9></location>U</text>
<text><location><page_43><loc_49><loc_8><loc_51><loc_8></location>S</text>
<text><location><page_43><loc_51><loc_89><loc_52><loc_89></location>8</text>
<text><location><page_43><loc_51><loc_88><loc_52><loc_89></location>9</text>
<text><location><page_43><loc_51><loc_88><loc_52><loc_88></location>1</text>
<text><location><page_43><loc_54><loc_88><loc_55><loc_89></location>0</text>
<text><location><page_43><loc_54><loc_88><loc_55><loc_88></location>2</text>
<text><location><page_43><loc_59><loc_92><loc_60><loc_92></location>)</text>
<text><location><page_43><loc_59><loc_91><loc_60><loc_92></location>7</text>
<text><location><page_43><loc_59><loc_91><loc_60><loc_91></location>0</text>
<text><location><page_43><loc_59><loc_90><loc_60><loc_91></location>0</text>
<text><location><page_43><loc_59><loc_89><loc_60><loc_90></location>2</text>
<text><location><page_43><loc_59><loc_89><loc_60><loc_90></location>(</text>
<text><location><page_43><loc_59><loc_88><loc_60><loc_89></location>.</text>
<text><location><page_43><loc_59><loc_88><loc_60><loc_88></location>l</text>
<text><location><page_43><loc_59><loc_88><loc_60><loc_88></location>a</text>
<text><location><page_43><loc_60><loc_90><loc_61><loc_91></location>)</text>
<text><location><page_43><loc_60><loc_90><loc_61><loc_90></location>7</text>
<text><location><page_43><loc_60><loc_89><loc_61><loc_90></location>0</text>
<text><location><page_43><loc_60><loc_89><loc_61><loc_89></location>0</text>
<text><location><page_43><loc_60><loc_88><loc_61><loc_89></location>2</text>
<text><location><page_43><loc_60><loc_88><loc_61><loc_88></location>(</text>
<text><location><page_43><loc_60><loc_11><loc_61><loc_11></location>a</text>
<text><location><page_43><loc_61><loc_90><loc_63><loc_91></location>)</text>
<text><location><page_43><loc_61><loc_90><loc_63><loc_90></location>7</text>
<text><location><page_43><loc_61><loc_89><loc_63><loc_90></location>0</text>
<text><location><page_43><loc_61><loc_89><loc_63><loc_89></location>0</text>
<text><location><page_43><loc_61><loc_88><loc_63><loc_89></location>2</text>
<text><location><page_43><loc_61><loc_88><loc_63><loc_88></location>(</text>
<text><location><page_43><loc_63><loc_90><loc_64><loc_91></location>)</text>
<text><location><page_43><loc_63><loc_90><loc_64><loc_90></location>7</text>
<text><location><page_43><loc_63><loc_89><loc_64><loc_90></location>0</text>
<text><location><page_43><loc_63><loc_89><loc_64><loc_89></location>0</text>
<text><location><page_43><loc_63><loc_88><loc_64><loc_89></location>2</text>
<text><location><page_43><loc_63><loc_88><loc_64><loc_88></location>(</text>
<text><location><page_43><loc_63><loc_10><loc_64><loc_11></location>u</text>
<text><location><page_43><loc_63><loc_10><loc_64><loc_10></location>a</text>
<text><location><page_43><loc_63><loc_9><loc_64><loc_10></location>T</text>
<text><location><page_43><loc_63><loc_8><loc_64><loc_9></location>P</text>
<text><location><page_43><loc_63><loc_8><loc_64><loc_8></location>I</text>
<text><location><page_43><loc_67><loc_91><loc_68><loc_91></location>)</text>
<text><location><page_43><loc_67><loc_90><loc_68><loc_91></location>1</text>
<text><location><page_43><loc_67><loc_90><loc_68><loc_90></location>1</text>
<text><location><page_43><loc_67><loc_89><loc_68><loc_90></location>0</text>
<text><location><page_43><loc_67><loc_88><loc_68><loc_89></location>2</text>
<text><location><page_43><loc_67><loc_88><loc_68><loc_88></location>(</text>
<text><location><page_43><loc_70><loc_90><loc_71><loc_91></location>)</text>
<text><location><page_43><loc_70><loc_90><loc_71><loc_90></location>7</text>
<text><location><page_43><loc_70><loc_89><loc_71><loc_90></location>0</text>
<text><location><page_43><loc_70><loc_89><loc_71><loc_89></location>0</text>
<text><location><page_43><loc_70><loc_88><loc_71><loc_89></location>2</text>
<text><location><page_43><loc_70><loc_88><loc_71><loc_88></location>(</text>
<text><location><page_43><loc_70><loc_10><loc_71><loc_11></location>a</text>
<text><location><page_43><loc_70><loc_10><loc_71><loc_10></location>T</text>
<text><location><page_43><loc_70><loc_9><loc_71><loc_9></location>X</text>
<text><location><page_43><loc_70><loc_8><loc_71><loc_9></location>U</text>
<text><location><page_43><loc_71><loc_90><loc_72><loc_91></location>)</text>
<text><location><page_43><loc_71><loc_90><loc_72><loc_90></location>7</text>
<text><location><page_43><loc_71><loc_89><loc_72><loc_90></location>0</text>
<text><location><page_43><loc_71><loc_89><loc_72><loc_89></location>0</text>
<text><location><page_43><loc_71><loc_88><loc_72><loc_89></location>2</text>
<text><location><page_43><loc_71><loc_88><loc_72><loc_88></location>(</text>
<text><location><page_43><loc_76><loc_95><loc_77><loc_96></location>)</text>
<text><location><page_43><loc_76><loc_95><loc_77><loc_95></location>2</text>
<text><location><page_43><loc_76><loc_94><loc_77><loc_95></location>0</text>
<text><location><page_43><loc_76><loc_94><loc_77><loc_94></location>0</text>
<text><location><page_43><loc_76><loc_93><loc_77><loc_94></location>2</text>
<text><location><page_43><loc_76><loc_93><loc_77><loc_93></location>(</text>
<text><location><page_43><loc_76><loc_92><loc_77><loc_92></location>s</text>
<text><location><page_43><loc_76><loc_91><loc_77><loc_92></location>n</text>
<text><location><page_43><loc_76><loc_91><loc_77><loc_91></location>e</text>
<text><location><page_43><loc_76><loc_90><loc_77><loc_91></location>z</text>
<text><location><page_43><loc_76><loc_90><loc_77><loc_90></location>t</text>
<text><location><page_43><loc_76><loc_90><loc_77><loc_90></location>o</text>
<text><location><page_43><loc_76><loc_89><loc_77><loc_90></location>G</text>
<text><location><page_43><loc_76><loc_88><loc_77><loc_89></location>-</text>
<text><location><page_43><loc_76><loc_88><loc_77><loc_88></location>r</text>
<text><location><page_43><loc_80><loc_90><loc_81><loc_91></location>)</text>
<text><location><page_43><loc_80><loc_90><loc_81><loc_90></location>7</text>
<text><location><page_43><loc_80><loc_89><loc_81><loc_90></location>0</text>
<text><location><page_43><loc_80><loc_89><loc_81><loc_89></location>0</text>
<text><location><page_43><loc_80><loc_88><loc_81><loc_89></location>2</text>
<text><location><page_43><loc_80><loc_88><loc_81><loc_88></location>(</text>
<text><location><page_43><loc_81><loc_90><loc_83><loc_91></location>)</text>
<text><location><page_43><loc_81><loc_90><loc_83><loc_90></location>7</text>
<text><location><page_43><loc_81><loc_89><loc_83><loc_90></location>0</text>
<text><location><page_43><loc_81><loc_89><loc_83><loc_89></location>0</text>
<text><location><page_43><loc_81><loc_88><loc_83><loc_89></location>2</text>
<text><location><page_43><loc_81><loc_88><loc_83><loc_88></location>(</text>
<text><location><page_43><loc_81><loc_11><loc_83><loc_11></location>A</text>
<text><location><page_43><loc_81><loc_10><loc_83><loc_10></location>6</text>
<text><location><page_43><loc_81><loc_9><loc_83><loc_10></location>9</text>
<text><location><page_43><loc_81><loc_9><loc_83><loc_9></location>3</text>
<text><location><page_43><loc_81><loc_8><loc_83><loc_9></location>V</text>
<text><location><page_43><loc_83><loc_90><loc_84><loc_91></location>)</text>
<text><location><page_43><loc_83><loc_90><loc_84><loc_90></location>1</text>
<text><location><page_43><loc_83><loc_89><loc_84><loc_90></location>0</text>
<text><location><page_43><loc_83><loc_89><loc_84><loc_89></location>0</text>
<text><location><page_43><loc_83><loc_88><loc_84><loc_89></location>2</text>
<text><location><page_43><loc_83><loc_88><loc_84><loc_88></location>(</text>
<table>
<location><page_43><loc_7><loc_11><loc_96><loc_88></location>
</table>
<text><location><page_43><loc_22><loc_10><loc_23><loc_11></location>c</text>
<text><location><page_43><loc_22><loc_10><loc_23><loc_10></location>S</text>
<text><location><page_43><loc_22><loc_9><loc_23><loc_9></location>K</text>
<text><location><page_43><loc_22><loc_8><loc_23><loc_9></location>A</text>
<text><location><page_43><loc_23><loc_10><loc_24><loc_11></location>a</text>
<text><location><page_43><loc_23><loc_10><loc_24><loc_10></location>T</text>
<text><location><page_43><loc_23><loc_9><loc_24><loc_9></location>P</text>
<text><location><page_43><loc_23><loc_8><loc_24><loc_9></location>B</text>
<text><location><page_43><loc_27><loc_10><loc_29><loc_11></location>a</text>
<text><location><page_43><loc_27><loc_10><loc_29><loc_10></location>T</text>
<text><location><page_43><loc_27><loc_9><loc_29><loc_9></location>E</text>
<text><location><page_43><loc_27><loc_8><loc_29><loc_9></location>D</text>
<text><location><page_43><loc_29><loc_10><loc_30><loc_11></location>a</text>
<text><location><page_43><loc_29><loc_10><loc_30><loc_10></location>T</text>
<text><location><page_43><loc_29><loc_9><loc_30><loc_9></location>F</text>
<text><location><page_43><loc_29><loc_8><loc_30><loc_9></location>D</text>
<text><location><page_43><loc_30><loc_91><loc_31><loc_91></location>)</text>
<text><location><page_43><loc_30><loc_90><loc_31><loc_91></location>6</text>
<text><location><page_43><loc_30><loc_90><loc_31><loc_90></location>0</text>
<text><location><page_43><loc_30><loc_89><loc_31><loc_90></location>0</text>
<text><location><page_43><loc_30><loc_89><loc_31><loc_89></location>2</text>
<text><location><page_43><loc_30><loc_88><loc_31><loc_89></location>(</text>
<text><location><page_43><loc_30><loc_10><loc_31><loc_11></location>a</text>
<text><location><page_43><loc_30><loc_10><loc_31><loc_11></location>T</text>
<text><location><page_43><loc_30><loc_9><loc_31><loc_9></location>K</text>
<text><location><page_43><loc_30><loc_8><loc_31><loc_9></location>D</text>
<text><location><page_43><loc_31><loc_10><loc_33><loc_11></location>T</text>
<text><location><page_43><loc_31><loc_9><loc_33><loc_9></location>N</text>
<text><location><page_43><loc_31><loc_8><loc_33><loc_9></location>D</text>
<text><location><page_43><loc_34><loc_10><loc_36><loc_11></location>a</text>
<text><location><page_43><loc_34><loc_10><loc_36><loc_10></location>T</text>
<text><location><page_43><loc_34><loc_9><loc_36><loc_9></location>S</text>
<text><location><page_43><loc_34><loc_8><loc_36><loc_9></location>D</text>
<text><location><page_43><loc_44><loc_10><loc_45><loc_11></location>M</text>
<text><location><page_43><loc_44><loc_9><loc_45><loc_9></location>P</text>
<text><location><page_43><loc_44><loc_8><loc_45><loc_9></location>M</text>
<text><location><page_43><loc_47><loc_10><loc_48><loc_11></location>A</text>
<text><location><page_43><loc_47><loc_8><loc_48><loc_10></location>W</text>
<text><location><page_43><loc_47><loc_8><loc_48><loc_9></location>R</text>
<text><location><page_43><loc_51><loc_10><loc_52><loc_11></location>u</text>
<text><location><page_43><loc_51><loc_10><loc_52><loc_10></location>a</text>
<text><location><page_43><loc_51><loc_9><loc_52><loc_10></location>T</text>
<text><location><page_43><loc_51><loc_8><loc_52><loc_9></location>T</text>
<text><location><page_43><loc_52><loc_10><loc_54><loc_11></location>0</text>
<text><location><page_43><loc_52><loc_10><loc_54><loc_10></location>9</text>
<text><location><page_43><loc_52><loc_9><loc_54><loc_10></location>1</text>
<text><location><page_43><loc_52><loc_9><loc_54><loc_9></location>1</text>
<text><location><page_43><loc_52><loc_8><loc_54><loc_9></location>V</text>
<text><location><page_43><loc_54><loc_10><loc_55><loc_11></location>6</text>
<text><location><page_43><loc_54><loc_10><loc_55><loc_10></location>4</text>
<text><location><page_43><loc_54><loc_9><loc_55><loc_10></location>0</text>
<text><location><page_43><loc_54><loc_9><loc_55><loc_9></location>4</text>
<text><location><page_43><loc_54><loc_8><loc_55><loc_9></location>V</text>
<text><location><page_43><loc_59><loc_10><loc_60><loc_11></location>h</text>
<text><location><page_43><loc_59><loc_9><loc_60><loc_10></location>C</text>
<text><location><page_43><loc_59><loc_9><loc_60><loc_9></location>S</text>
<text><location><page_43><loc_59><loc_8><loc_60><loc_9></location>C</text>
<text><location><page_43><loc_60><loc_10><loc_61><loc_11></location>T</text>
<text><location><page_43><loc_60><loc_9><loc_61><loc_10></location>M</text>
<text><location><page_43><loc_60><loc_8><loc_61><loc_9></location>D</text>
<text><location><page_43><loc_61><loc_10><loc_63><loc_11></location>A</text>
<text><location><page_43><loc_61><loc_9><loc_63><loc_10></location>M</text>
<text><location><page_43><loc_61><loc_8><loc_63><loc_9></location>G</text>
<text><location><page_43><loc_67><loc_10><loc_68><loc_11></location>H</text>
<text><location><page_43><loc_67><loc_9><loc_68><loc_10></location>W</text>
<text><location><page_43><loc_67><loc_8><loc_68><loc_9></location>T</text>
<text><location><page_43><loc_71><loc_10><loc_72><loc_11></location>9</text>
<text><location><page_43><loc_71><loc_10><loc_72><loc_10></location>7</text>
<text><location><page_43><loc_71><loc_9><loc_72><loc_10></location>0</text>
<text><location><page_43><loc_71><loc_9><loc_72><loc_9></location>1</text>
<text><location><page_43><loc_71><loc_8><loc_72><loc_9></location>V</text>
<text><location><page_43><loc_80><loc_10><loc_81><loc_11></location>8</text>
<text><location><page_43><loc_80><loc_10><loc_81><loc_10></location>6</text>
<text><location><page_43><loc_80><loc_9><loc_81><loc_10></location>0</text>
<text><location><page_43><loc_80><loc_9><loc_81><loc_9></location>1</text>
<text><location><page_43><loc_80><loc_8><loc_81><loc_9></location>V</text>
<text><location><page_43><loc_83><loc_11><loc_84><loc_11></location>A</text>
<text><location><page_43><loc_83><loc_10><loc_84><loc_10></location>7</text>
<text><location><page_43><loc_83><loc_9><loc_84><loc_10></location>9</text>
<text><location><page_43><loc_83><loc_9><loc_84><loc_9></location>3</text>
<text><location><page_43><loc_83><loc_8><loc_84><loc_9></location>V</text>
<text><location><page_43><loc_76><loc_10><loc_77><loc_11></location>h</text>
<text><location><page_43><loc_76><loc_10><loc_77><loc_10></location>C</text>
<text><location><page_43><loc_76><loc_8><loc_77><loc_9></location>G</text>
<text><location><page_43><loc_76><loc_8><loc_77><loc_8></location>E</text>
<text><location><page_43><loc_45><loc_10><loc_47><loc_11></location>u</text>
<text><location><page_43><loc_45><loc_10><loc_47><loc_10></location>L</text>
<text><location><page_43><loc_45><loc_9><loc_47><loc_9></location>U</text>
<text><location><page_43><loc_45><loc_8><loc_47><loc_9></location>R</text>
<text><location><page_43><loc_41><loc_10><loc_42><loc_11></location>h</text>
<text><location><page_43><loc_41><loc_10><loc_42><loc_10></location>C</text>
<text><location><page_43><loc_41><loc_8><loc_42><loc_9></location>T</text>
<text><location><page_43><loc_41><loc_8><loc_42><loc_8></location>E</text>
<text><location><page_43><loc_40><loc_10><loc_41><loc_11></location>h</text>
<text><location><page_43><loc_40><loc_10><loc_41><loc_10></location>C</text>
<text><location><page_43><loc_40><loc_8><loc_41><loc_9></location>P</text>
<text><location><page_43><loc_40><loc_8><loc_41><loc_8></location>E</text>
<text><location><page_43><loc_38><loc_10><loc_40><loc_11></location>h</text>
<text><location><page_43><loc_38><loc_10><loc_40><loc_10></location>C</text>
<text><location><page_43><loc_38><loc_9><loc_40><loc_9></location>X</text>
<text><location><page_43><loc_38><loc_8><loc_40><loc_9></location>D</text>
<text><location><page_43><loc_33><loc_10><loc_34><loc_11></location>a</text>
<text><location><page_43><loc_33><loc_10><loc_34><loc_10></location>T</text>
<text><location><page_43><loc_33><loc_9><loc_34><loc_9></location>R</text>
<text><location><page_43><loc_33><loc_8><loc_34><loc_9></location>D</text>
<text><location><page_43><loc_22><loc_91><loc_23><loc_91></location>)</text>
<text><location><page_43><loc_22><loc_90><loc_23><loc_91></location>3</text>
<text><location><page_43><loc_22><loc_90><loc_23><loc_90></location>0</text>
<text><location><page_43><loc_22><loc_89><loc_23><loc_90></location>0</text>
<text><location><page_43><loc_22><loc_89><loc_23><loc_89></location>2</text>
<text><location><page_43><loc_22><loc_88><loc_23><loc_89></location>(</text>
<text><location><page_43><loc_25><loc_10><loc_26><loc_11></location>h</text>
<text><location><page_43><loc_25><loc_10><loc_26><loc_10></location>C</text>
<text><location><page_43><loc_25><loc_9><loc_26><loc_9></location>V</text>
<text><location><page_43><loc_25><loc_8><loc_26><loc_9></location>C</text>
<text><location><page_43><loc_43><loc_91><loc_44><loc_91></location>)</text>
<text><location><page_43><loc_43><loc_90><loc_44><loc_91></location>6</text>
<text><location><page_43><loc_43><loc_90><loc_44><loc_90></location>0</text>
<text><location><page_43><loc_43><loc_89><loc_44><loc_90></location>0</text>
<text><location><page_43><loc_43><loc_89><loc_44><loc_89></location>2</text>
<text><location><page_43><loc_43><loc_88><loc_44><loc_89></location>(</text>
<text><location><page_43><loc_43><loc_10><loc_44><loc_11></location>a</text>
<text><location><page_43><loc_43><loc_10><loc_44><loc_10></location>T</text>
<text><location><page_43><loc_43><loc_9><loc_44><loc_9></location>N</text>
<text><location><page_43><loc_43><loc_8><loc_44><loc_9></location>H</text>
<text><location><page_43><loc_79><loc_92><loc_80><loc_92></location>)</text>
<text><location><page_43><loc_79><loc_91><loc_80><loc_92></location>5</text>
<text><location><page_43><loc_79><loc_91><loc_80><loc_91></location>0</text>
<text><location><page_43><loc_79><loc_90><loc_80><loc_91></location>0</text>
<text><location><page_43><loc_79><loc_90><loc_80><loc_90></location>2</text>
<text><location><page_43><loc_79><loc_89><loc_80><loc_90></location>(</text>
<text><location><page_43><loc_79><loc_89><loc_80><loc_89></location>.</text>
<text><location><page_43><loc_79><loc_88><loc_80><loc_89></location>l</text>
<text><location><page_43><loc_79><loc_88><loc_80><loc_88></location>a</text>
<text><location><page_43><loc_79><loc_9><loc_80><loc_10></location>A</text>
<text><location><page_43><loc_79><loc_9><loc_80><loc_10></location>W</text>
<text><location><page_43><loc_79><loc_8><loc_80><loc_9></location>T</text>
<text><location><page_43><loc_84><loc_91><loc_86><loc_91></location>)</text>
<text><location><page_43><loc_84><loc_90><loc_86><loc_91></location>6</text>
<text><location><page_43><loc_84><loc_90><loc_86><loc_90></location>0</text>
<text><location><page_43><loc_84><loc_89><loc_86><loc_90></location>0</text>
<text><location><page_43><loc_84><loc_89><loc_86><loc_89></location>2</text>
<text><location><page_43><loc_84><loc_88><loc_86><loc_89></location>(</text>
<text><location><page_43><loc_84><loc_11><loc_86><loc_11></location>T</text>
<text><location><page_43><loc_84><loc_10><loc_86><loc_10></location>0</text>
<text><location><page_43><loc_84><loc_9><loc_86><loc_10></location>1</text>
<text><location><page_43><loc_84><loc_9><loc_86><loc_9></location>4</text>
<text><location><page_43><loc_84><loc_8><loc_86><loc_9></location>V</text>
<text><location><page_44><loc_9><loc_93><loc_10><loc_94></location>44</text>
<text><location><page_44><loc_45><loc_93><loc_56><loc_94></location>Ardila et al.</text>
<table>
<location><page_44><loc_7><loc_85><loc_97><loc_89></location>
</table>
<text><location><page_44><loc_11><loc_85><loc_13><loc_86></location>n</text>
<text><location><page_44><loc_11><loc_85><loc_13><loc_85></location>i</text>
<text><location><page_44><loc_11><loc_84><loc_13><loc_85></location>B</text>
<text><location><page_44><loc_11><loc_83><loc_13><loc_84></location>.</text>
<text><location><page_44><loc_11><loc_83><loc_13><loc_83></location>f</text>
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<text><location><page_44><loc_11><loc_75><loc_13><loc_75></location>)</text>
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<text><location><page_44><loc_11><loc_73><loc_13><loc_74></location>y</text>
<text><location><page_44><loc_11><loc_73><loc_13><loc_73></location>t</text>
<text><location><page_44><loc_11><loc_72><loc_13><loc_73></location>i</text>
<text><location><page_44><loc_11><loc_72><loc_13><loc_72></location>r</text>
<text><location><page_44><loc_11><loc_71><loc_13><loc_72></location>a</text>
<text><location><page_44><loc_11><loc_71><loc_13><loc_71></location>n</text>
<text><location><page_44><loc_11><loc_71><loc_13><loc_71></location>i</text>
<text><location><page_44><loc_11><loc_70><loc_13><loc_71></location>B</text>
<text><location><page_44><loc_11><loc_54><loc_13><loc_54></location>.</text>
<text><location><page_44><loc_11><loc_53><loc_13><loc_54></location>c</text>
<text><location><page_44><loc_11><loc_53><loc_13><loc_53></location>n</text>
<text><location><page_44><loc_11><loc_52><loc_13><loc_53></location>I</text>
<text><location><page_44><loc_11><loc_51><loc_13><loc_52></location>.</text>
<text><location><page_44><loc_11><loc_51><loc_13><loc_51></location>f</text>
<text><location><page_44><loc_11><loc_51><loc_13><loc_51></location>e</text>
<text><location><page_44><loc_11><loc_50><loc_13><loc_51></location>R</text>
<text><location><page_44><loc_11><loc_48><loc_13><loc_49></location>)</text>
<text><location><page_44><loc_11><loc_48><loc_13><loc_48></location>a</text>
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<text><location><page_44><loc_11><loc_46><loc_13><loc_47></location>.</text>
<text><location><page_44><loc_11><loc_46><loc_13><loc_46></location>c</text>
<text><location><page_44><loc_11><loc_45><loc_13><loc_46></location>n</text>
<text><location><page_44><loc_11><loc_45><loc_13><loc_45></location>I</text>
<text><location><page_44><loc_11><loc_34><loc_13><loc_35></location>.</text>
<text><location><page_44><loc_11><loc_34><loc_13><loc_34></location>l</text>
<text><location><page_44><loc_11><loc_33><loc_13><loc_34></location>e</text>
<text><location><page_44><loc_11><loc_33><loc_13><loc_34></location>V</text>
<text><location><page_44><loc_11><loc_32><loc_13><loc_32></location>.</text>
<text><location><page_44><loc_11><loc_31><loc_13><loc_32></location>d</text>
<text><location><page_44><loc_11><loc_31><loc_13><loc_31></location>a</text>
<text><location><page_44><loc_11><loc_30><loc_13><loc_31></location>R</text>
<text><location><page_44><loc_11><loc_29><loc_13><loc_30></location>.</text>
<text><location><page_44><loc_11><loc_29><loc_13><loc_29></location>f</text>
<text><location><page_44><loc_11><loc_28><loc_13><loc_29></location>e</text>
<text><location><page_44><loc_11><loc_28><loc_13><loc_28></location>R</text>
<text><location><page_44><loc_11><loc_26><loc_13><loc_26></location>y</text>
<text><location><page_44><loc_11><loc_25><loc_13><loc_26></location>t</text>
<text><location><page_44><loc_11><loc_25><loc_13><loc_25></location>i</text>
<text><location><page_44><loc_11><loc_25><loc_13><loc_25></location>c</text>
<text><location><page_44><loc_11><loc_24><loc_13><loc_25></location>o</text>
<text><location><page_44><loc_11><loc_24><loc_13><loc_24></location>l</text>
<text><location><page_44><loc_11><loc_23><loc_13><loc_24></location>e</text>
<text><location><page_44><loc_11><loc_23><loc_13><loc_23></location>V</text>
<text><location><page_44><loc_11><loc_22><loc_13><loc_22></location>l</text>
<text><location><page_44><loc_11><loc_22><loc_13><loc_22></location>a</text>
<text><location><page_44><loc_11><loc_21><loc_13><loc_21></location>i</text>
<text><location><page_44><loc_11><loc_21><loc_13><loc_21></location>d</text>
<text><location><page_44><loc_11><loc_20><loc_13><loc_21></location>a</text>
<text><location><page_44><loc_11><loc_19><loc_13><loc_20></location>R</text>
<text><location><page_44><loc_11><loc_10><loc_13><loc_10></location>e</text>
<text><location><page_44><loc_11><loc_9><loc_13><loc_10></location>m</text>
<text><location><page_44><loc_11><loc_9><loc_13><loc_9></location>a</text>
<text><location><page_44><loc_11><loc_8><loc_13><loc_9></location>N</text>
<text><location><page_44><loc_8><loc_57><loc_10><loc_57></location>d</text>
<text><location><page_44><loc_8><loc_56><loc_10><loc_57></location>e</text>
<text><location><page_44><loc_8><loc_56><loc_10><loc_56></location>u</text>
<text><location><page_44><loc_8><loc_55><loc_10><loc_56></location>n</text>
<text><location><page_44><loc_8><loc_55><loc_10><loc_55></location>i</text>
<text><location><page_44><loc_8><loc_54><loc_10><loc_55></location>t</text>
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<text><location><page_44><loc_8><loc_52><loc_10><loc_53></location>C</text>
<text><location><page_44><loc_8><loc_51><loc_10><loc_52></location>-</text>
<text><location><page_44><loc_8><loc_50><loc_9><loc_50></location>2</text>
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<text><location><page_44><loc_8><loc_48><loc_9><loc_49></location>l</text>
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<text><location><page_44><loc_13><loc_47><loc_14><loc_48></location>g</text>
<text><location><page_44><loc_13><loc_46><loc_14><loc_47></location>e</text>
<text><location><page_44><loc_13><loc_46><loc_14><loc_47></location>d</text>
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<text><location><page_44><loc_13><loc_25><loc_14><loc_25></location>)</text>
<text><location><page_44><loc_13><loc_24><loc_13><loc_25></location>1</text>
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<text><location><page_44><loc_13><loc_23><loc_14><loc_23></location>s</text>
<text><location><page_44><loc_13><loc_22><loc_14><loc_23></location>m</text>
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<text><location><page_44><loc_17><loc_12><loc_17><loc_13></location>-</text>
<text><location><page_44><loc_17><loc_11><loc_17><loc_11></location>.</text>
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<text><location><page_44><loc_17><loc_10><loc_17><loc_10></location>o</text>
<text><location><page_44><loc_17><loc_9><loc_17><loc_10></location>N</text>
<text><location><page_44><loc_18><loc_70><loc_18><loc_70></location>.</text>
<text><location><page_44><loc_18><loc_69><loc_18><loc_70></location>s</text>
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<text><location><page_44><loc_18><loc_68><loc_18><loc_68></location>d</text>
<text><location><page_44><loc_18><loc_68><loc_18><loc_68></location>a</text>
<text><location><page_44><loc_18><loc_67><loc_18><loc_68></location>r</text>
<text><location><page_44><loc_18><loc_67><loc_18><loc_67></location>r</text>
<text><location><page_44><loc_18><loc_66><loc_18><loc_67></location>a</text>
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<text><location><page_44><loc_18><loc_61><loc_18><loc_62></location>a</text>
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<text><location><page_44><loc_20><loc_8><loc_20><loc_9></location>(</text>
<text><location><page_45><loc_11><loc_37><loc_13><loc_37></location>Note</text>
<text><location><page_45><loc_13><loc_36><loc_14><loc_37></location>.</text>
<text><location><page_45><loc_14><loc_36><loc_15><loc_37></location>-</text>
<text><location><page_45><loc_10><loc_34><loc_92><loc_36></location>(a) Simultaneous (within ∼ 10 hrs) near and far ultraviolet observations are available for some of the DAO targets (Herczeg et al. 2013). The last column indicates whether simultaneous NUV observations were used to calculate the accretion rate, as described in Ingleby et al. (2013). (b) We assume ˙ M =1 × 10 -8 M /circledot /yr for AK Sco (G'omez de Castro 2009).</text>
<section_header_level_1><location><page_45><loc_38><loc_93><loc_61><loc_94></location>Hot lines in T Tauri stars</section_header_level_1>
<table>
<location><page_45><loc_22><loc_38><loc_78><loc_89></location>
<caption>Table 3 Ancillary Data (cont.)</caption>
</table>
<text><location><page_46><loc_46><loc_90><loc_54><loc_91></location>Data Sources</text>
<table>
<location><page_46><loc_24><loc_40><loc_76><loc_90></location>
<caption>Table 4</caption>
</table>
<text><location><page_46><loc_11><loc_39><loc_62><loc_39></location>Note . - (a) In addition to the DAO data; we have made use of data from the following HST proposals:</text>
<text><location><page_46><loc_10><loc_33><loc_72><loc_38></location>GO 8157: Molecular Hydrogen in the Circumstellar Environments of T Tauri Stars; PI Walter GO 8206: The Structure of the Accretion Flow on pre-main-sequence stars; PI: Calvet GTO 11533: Accretion Flows and Winds of Pre-Main Sequence Stars; PI: Green GO 11608: How Far Does H2 Go: Constraining FUV Variability in the Gaseous Inner Holes of Protoplanetary Disks; PI: Calvet GTO 12036: Accretion Flows and Winds of Pre-Main Sequence Stars Part 2; PI: Green (b ) The data set columns indicate the suffix or the full name (if only one) of the HST dataset used. (c) PSA: Primary Science Aperture (for COS). 2.5' diameter. For the STIS data the slit size used is indicated.</text>
<text><location><page_47><loc_8><loc_51><loc_9><loc_51></location>5</text>
<text><location><page_47><loc_8><loc_50><loc_9><loc_50></location>e</text>
<text><location><page_47><loc_8><loc_50><loc_9><loc_50></location>l</text>
<text><location><page_47><loc_8><loc_49><loc_9><loc_50></location>b</text>
<text><location><page_47><loc_8><loc_48><loc_9><loc_49></location>a</text>
<text><location><page_47><loc_8><loc_47><loc_9><loc_48></location>T</text>
<text><location><page_47><loc_10><loc_57><loc_11><loc_58></location>s</text>
<text><location><page_47><loc_10><loc_57><loc_11><loc_57></location>t</text>
<text><location><page_47><loc_10><loc_56><loc_11><loc_57></location>n</text>
<text><location><page_47><loc_10><loc_56><loc_11><loc_56></location>e</text>
<text><location><page_47><loc_10><loc_55><loc_11><loc_56></location>m</text>
<text><location><page_47><loc_10><loc_54><loc_11><loc_55></location>e</text>
<text><location><page_47><loc_10><loc_54><loc_11><loc_54></location>r</text>
<text><location><page_47><loc_10><loc_53><loc_11><loc_54></location>u</text>
<text><location><page_47><loc_10><loc_53><loc_11><loc_53></location>s</text>
<text><location><page_47><loc_10><loc_52><loc_11><loc_53></location>a</text>
<text><location><page_47><loc_10><loc_52><loc_11><loc_52></location>e</text>
<text><location><page_47><loc_10><loc_51><loc_11><loc_52></location>m</text>
<text><location><page_47><loc_10><loc_50><loc_11><loc_51></location>e</text>
<text><location><page_47><loc_10><loc_50><loc_11><loc_50></location>n</text>
<text><location><page_47><loc_10><loc_49><loc_11><loc_50></location>i</text>
<text><location><page_47><loc_10><loc_49><loc_11><loc_49></location>l</text>
<text><location><page_47><loc_10><loc_48><loc_11><loc_49></location>c</text>
<text><location><page_47><loc_10><loc_48><loc_11><loc_48></location>i</text>
<text><location><page_47><loc_10><loc_47><loc_11><loc_48></location>r</text>
<text><location><page_47><loc_10><loc_47><loc_11><loc_47></location>t</text>
<text><location><page_47><loc_10><loc_47><loc_11><loc_47></location>e</text>
<text><location><page_47><loc_10><loc_46><loc_11><loc_47></location>m</text>
<text><location><page_47><loc_10><loc_45><loc_11><loc_46></location>a</text>
<text><location><page_47><loc_10><loc_45><loc_11><loc_45></location>r</text>
<text><location><page_47><loc_10><loc_44><loc_11><loc_45></location>a</text>
<text><location><page_47><loc_10><loc_44><loc_11><loc_44></location>p</text>
<text><location><page_47><loc_10><loc_43><loc_11><loc_44></location>-</text>
<text><location><page_47><loc_10><loc_43><loc_11><loc_43></location>n</text>
<text><location><page_47><loc_10><loc_42><loc_11><loc_43></location>o</text>
<text><location><page_47><loc_10><loc_41><loc_11><loc_42></location>N</text>
<text><location><page_47><loc_13><loc_72><loc_14><loc_72></location>i</text>
<text><location><page_47><loc_13><loc_72><loc_14><loc_72></location>i</text>
<text><location><page_47><loc_13><loc_71><loc_14><loc_71></location>e</text>
<text><location><page_47><loc_13><loc_70><loc_14><loc_71></location>H</text>
<text><location><page_47><loc_13><loc_41><loc_14><loc_42></location>v</text>
<text><location><page_47><loc_13><loc_41><loc_14><loc_41></location>i</text>
<text><location><page_47><loc_13><loc_40><loc_14><loc_40></location>C</text>
<text><location><page_47><loc_16><loc_82><loc_17><loc_82></location>)</text>
<text><location><page_47><loc_16><loc_81><loc_17><loc_81></location>f</text>
<text><location><page_47><loc_16><loc_81><loc_17><loc_81></location>(</text>
<text><location><page_47><loc_16><loc_80><loc_17><loc_80></location>s</text>
<text><location><page_47><loc_16><loc_80><loc_17><loc_80></location>s</text>
<text><location><page_47><loc_16><loc_79><loc_17><loc_80></location>e</text>
<text><location><page_47><loc_16><loc_78><loc_17><loc_79></location>n</text>
<text><location><page_47><loc_16><loc_78><loc_17><loc_78></location>w</text>
<text><location><page_47><loc_16><loc_77><loc_17><loc_78></location>e</text>
<text><location><page_47><loc_16><loc_77><loc_17><loc_77></location>k</text>
<text><location><page_47><loc_16><loc_76><loc_17><loc_77></location>S</text>
<text><location><page_47><loc_16><loc_74><loc_17><loc_75></location>x</text>
<text><location><page_47><loc_16><loc_74><loc_17><loc_74></location>u</text>
<text><location><page_47><loc_16><loc_73><loc_17><loc_74></location>fl</text>
<text><location><page_47><loc_16><loc_72><loc_17><loc_73></location>.</text>
<text><location><page_47><loc_16><loc_72><loc_17><loc_72></location>x</text>
<text><location><page_47><loc_16><loc_71><loc_17><loc_72></location>a</text>
<text><location><page_47><loc_16><loc_70><loc_17><loc_71></location>m</text>
<text><location><page_47><loc_16><loc_70><loc_17><loc_70></location>t</text>
<text><location><page_47><loc_16><loc_69><loc_17><loc_70></location>a</text>
<text><location><page_47><loc_16><loc_68><loc_17><loc_69></location>.</text>
<text><location><page_47><loc_16><loc_68><loc_17><loc_68></location>l</text>
<text><location><page_47><loc_16><loc_67><loc_17><loc_68></location>e</text>
<text><location><page_47><loc_16><loc_67><loc_17><loc_68></location>V</text>
<text><location><page_47><loc_16><loc_65><loc_17><loc_66></location>)</text>
<text><location><page_47><loc_16><loc_65><loc_17><loc_65></location>e</text>
<text><location><page_47><loc_16><loc_64><loc_17><loc_65></location>(</text>
<text><location><page_47><loc_16><loc_63><loc_17><loc_64></location>M</text>
<text><location><page_47><loc_16><loc_62><loc_17><loc_63></location>H</text>
<text><location><page_47><loc_16><loc_61><loc_17><loc_62></location>W</text>
<text><location><page_47><loc_16><loc_60><loc_17><loc_61></location>F</text>
<text><location><page_47><loc_16><loc_57><loc_17><loc_58></location>)</text>
<text><location><page_47><loc_16><loc_57><loc_17><loc_57></location>d</text>
<text><location><page_47><loc_16><loc_56><loc_17><loc_57></location>(</text>
<text><location><page_47><loc_16><loc_56><loc_17><loc_56></location>s</text>
<text><location><page_47><loc_16><loc_55><loc_17><loc_56></location>s</text>
<text><location><page_47><loc_16><loc_55><loc_17><loc_55></location>e</text>
<text><location><page_47><loc_16><loc_54><loc_17><loc_55></location>n</text>
<text><location><page_47><loc_16><loc_53><loc_17><loc_54></location>w</text>
<text><location><page_47><loc_16><loc_53><loc_17><loc_53></location>e</text>
<text><location><page_47><loc_16><loc_52><loc_17><loc_53></location>k</text>
<text><location><page_47><loc_16><loc_52><loc_17><loc_52></location>S</text>
<text><location><page_47><loc_16><loc_50><loc_17><loc_50></location>)</text>
<text><location><page_47><loc_16><loc_50><loc_17><loc_50></location>c</text>
<text><location><page_47><loc_16><loc_49><loc_17><loc_50></location>(</text>
<text><location><page_47><loc_16><loc_48><loc_17><loc_49></location>x</text>
<text><location><page_47><loc_16><loc_48><loc_17><loc_48></location>u</text>
<text><location><page_47><loc_16><loc_47><loc_17><loc_48></location>fl</text>
<text><location><page_47><loc_16><loc_46><loc_17><loc_47></location>.</text>
<text><location><page_47><loc_16><loc_46><loc_17><loc_46></location>x</text>
<text><location><page_47><loc_16><loc_45><loc_17><loc_46></location>a</text>
<text><location><page_47><loc_16><loc_44><loc_17><loc_45></location>m</text>
<text><location><page_47><loc_16><loc_43><loc_17><loc_44></location>t</text>
<text><location><page_47><loc_16><loc_43><loc_17><loc_43></location>a</text>
<text><location><page_47><loc_16><loc_42><loc_17><loc_42></location>.</text>
<text><location><page_47><loc_16><loc_42><loc_17><loc_42></location>l</text>
<text><location><page_47><loc_16><loc_41><loc_17><loc_42></location>e</text>
<text><location><page_47><loc_16><loc_41><loc_17><loc_41></location>V</text>
<text><location><page_47><loc_16><loc_39><loc_17><loc_39></location>)</text>
<text><location><page_47><loc_16><loc_38><loc_17><loc_39></location>b</text>
<text><location><page_47><loc_16><loc_38><loc_17><loc_38></location>(</text>
<text><location><page_47><loc_16><loc_37><loc_17><loc_38></location>M</text>
<text><location><page_47><loc_16><loc_36><loc_17><loc_37></location>H</text>
<text><location><page_47><loc_16><loc_35><loc_17><loc_36></location>W</text>
<text><location><page_47><loc_16><loc_34><loc_17><loc_35></location>F</text>
<text><location><page_47><loc_16><loc_32><loc_17><loc_33></location>)</text>
<text><location><page_47><loc_16><loc_32><loc_17><loc_32></location>a</text>
<text><location><page_47><loc_16><loc_31><loc_17><loc_32></location>(</text>
<text><location><page_47><loc_16><loc_30><loc_17><loc_31></location>d</text>
<text><location><page_47><loc_16><loc_30><loc_17><loc_30></location>e</text>
<text><location><page_47><loc_16><loc_29><loc_17><loc_30></location>R</text>
<text><location><page_47><loc_16><loc_29><loc_17><loc_29></location>/</text>
<text><location><page_47><loc_16><loc_28><loc_17><loc_29></location>e</text>
<text><location><page_47><loc_16><loc_28><loc_17><loc_28></location>u</text>
<text><location><page_47><loc_16><loc_27><loc_17><loc_28></location>l</text>
<text><location><page_47><loc_16><loc_27><loc_17><loc_27></location>B</text>
<text><location><page_47><loc_16><loc_26><loc_17><loc_26></location>o</text>
<text><location><page_47><loc_16><loc_25><loc_17><loc_26></location>i</text>
<text><location><page_47><loc_16><loc_25><loc_17><loc_25></location>t</text>
<text><location><page_47><loc_16><loc_24><loc_17><loc_25></location>a</text>
<text><location><page_47><loc_16><loc_24><loc_17><loc_24></location>R</text>
<text><location><page_47><loc_16><loc_21><loc_17><loc_21></location>e</text>
<text><location><page_47><loc_16><loc_20><loc_17><loc_21></location>m</text>
<text><location><page_47><loc_38><loc_93><loc_61><loc_94></location>Hot lines in T Tauri stars</text>
<text><location><page_47><loc_90><loc_93><loc_92><loc_94></location>47</text>
<table>
<location><page_47><loc_16><loc_20><loc_83><loc_88></location>
</table>
<text><location><page_47><loc_16><loc_19><loc_17><loc_20></location>a</text>
<text><location><page_47><loc_23><loc_19><loc_24><loc_20></location>T</text>
<text><location><page_47><loc_24><loc_20><loc_26><loc_20></location>c</text>
<text><location><page_47><loc_24><loc_19><loc_26><loc_20></location>S</text>
<text><location><page_47><loc_27><loc_19><loc_28><loc_20></location>C</text>
<text><location><page_47><loc_29><loc_19><loc_30><loc_20></location>T</text>
<text><location><page_47><loc_30><loc_19><loc_31><loc_20></location>T</text>
<text><location><page_47><loc_31><loc_19><loc_33><loc_20></location>T</text>
<text><location><page_47><loc_33><loc_19><loc_34><loc_20></location>T</text>
<unordered_list>
<list_item><location><page_47><loc_16><loc_18><loc_34><loc_19></location>N A K P V Y E F K</list_item>
</unordered_list>
<text><location><page_47><loc_26><loc_20><loc_27><loc_20></location>a</text>
<text><location><page_47><loc_26><loc_19><loc_27><loc_20></location>T</text>
<text><location><page_47><loc_34><loc_19><loc_35><loc_20></location>T</text>
<text><location><page_47><loc_34><loc_18><loc_35><loc_19></location>N</text>
<text><location><page_47><loc_35><loc_19><loc_37><loc_20></location>T</text>
<text><location><page_47><loc_35><loc_18><loc_37><loc_19></location>R</text>
<text><location><page_47><loc_37><loc_20><loc_38><loc_20></location>a</text>
<text><location><page_47><loc_37><loc_19><loc_38><loc_20></location>T</text>
<text><location><page_47><loc_37><loc_18><loc_38><loc_19></location>S</text>
<text><location><page_47><loc_38><loc_19><loc_39><loc_20></location>C</text>
<text><location><page_47><loc_38><loc_18><loc_39><loc_19></location>X</text>
<text><location><page_47><loc_39><loc_19><loc_41><loc_20></location>C</text>
<text><location><page_47><loc_39><loc_18><loc_41><loc_19></location>P</text>
<text><location><page_47><loc_41><loc_19><loc_42><loc_20></location>C</text>
<text><location><page_47><loc_41><loc_18><loc_42><loc_19></location>T</text>
<text><location><page_47><loc_42><loc_19><loc_43><loc_20></location>T</text>
<text><location><page_47><loc_42><loc_18><loc_43><loc_19></location>N</text>
<text><location><page_47><loc_45><loc_19><loc_46><loc_20></location>L</text>
<text><location><page_47><loc_45><loc_18><loc_46><loc_19></location>U</text>
<text><location><page_47><loc_48><loc_19><loc_49><loc_20></location>A</text>
<text><location><page_47><loc_48><loc_18><loc_49><loc_19></location>U</text>
<text><location><page_47><loc_61><loc_19><loc_62><loc_20></location>a</text>
<text><location><page_47><loc_61><loc_19><loc_62><loc_20></location>T</text>
<text><location><page_47><loc_44><loc_19><loc_45><loc_20></location>M</text>
<text><location><page_47><loc_44><loc_18><loc_45><loc_19></location>P</text>
<text><location><page_47><loc_49><loc_20><loc_50><loc_20></location>u</text>
<text><location><page_47><loc_49><loc_19><loc_50><loc_20></location>a</text>
<text><location><page_47><loc_49><loc_18><loc_50><loc_19></location>T</text>
<text><location><page_47><loc_50><loc_20><loc_52><loc_20></location>0</text>
<text><location><page_47><loc_50><loc_19><loc_52><loc_20></location>9</text>
<text><location><page_47><loc_50><loc_19><loc_52><loc_19></location>1</text>
<text><location><page_47><loc_50><loc_18><loc_52><loc_19></location>1</text>
<text><location><page_47><loc_52><loc_20><loc_53><loc_20></location>6</text>
<text><location><page_47><loc_52><loc_19><loc_53><loc_20></location>4</text>
<text><location><page_47><loc_52><loc_19><loc_53><loc_19></location>0</text>
<text><location><page_47><loc_52><loc_18><loc_53><loc_19></location>4</text>
<text><location><page_47><loc_65><loc_20><loc_66><loc_20></location>9</text>
<text><location><page_47><loc_65><loc_19><loc_66><loc_20></location>7</text>
<text><location><page_47><loc_65><loc_19><loc_66><loc_19></location>0</text>
<text><location><page_47><loc_65><loc_18><loc_66><loc_19></location>1</text>
<text><location><page_47><loc_72><loc_20><loc_74><loc_20></location>8</text>
<text><location><page_47><loc_72><loc_19><loc_74><loc_20></location>6</text>
<text><location><page_47><loc_72><loc_19><loc_74><loc_19></location>0</text>
<text><location><page_47><loc_72><loc_18><loc_74><loc_19></location>1</text>
<text><location><page_47><loc_74><loc_19><loc_75><loc_20></location>6</text>
<text><location><page_47><loc_74><loc_19><loc_75><loc_19></location>9</text>
<text><location><page_47><loc_74><loc_18><loc_75><loc_19></location>3</text>
<text><location><page_47><loc_75><loc_19><loc_76><loc_20></location>7</text>
<text><location><page_47><loc_75><loc_19><loc_76><loc_19></location>9</text>
<text><location><page_47><loc_75><loc_18><loc_76><loc_19></location>3</text>
<text><location><page_47><loc_76><loc_19><loc_78><loc_20></location>0</text>
<text><location><page_47><loc_76><loc_19><loc_78><loc_19></location>1</text>
<text><location><page_47><loc_76><loc_18><loc_78><loc_19></location>4</text>
<unordered_list>
<list_item><location><page_47><loc_23><loc_17><loc_78><loc_18></location>A A B C C D D D D D D D E E H M R R S T V V C D G I P T U V E T V V V V</list_item>
</unordered_list>
<text><location><page_47><loc_63><loc_19><loc_65><loc_20></location>T</text>
<text><location><page_47><loc_63><loc_18><loc_65><loc_19></location>X</text>
<text><location><page_47><loc_70><loc_19><loc_71><loc_20></location>C</text>
<text><location><page_47><loc_70><loc_18><loc_71><loc_19></location>G</text>
<text><location><page_47><loc_71><loc_19><loc_72><loc_20></location>A</text>
<text><location><page_47><loc_71><loc_18><loc_72><loc_19></location>W</text>
<text><location><page_47><loc_81><loc_20><loc_82><loc_20></location>c</text>
<text><location><page_47><loc_81><loc_19><loc_82><loc_19></location>e</text>
<text><location><page_47><loc_81><loc_19><loc_82><loc_19></location>r</text>
<text><location><page_47><loc_81><loc_18><loc_82><loc_19></location>a</text>
<text><location><page_47><loc_81><loc_17><loc_82><loc_18></location>s</text>
<text><location><page_47><loc_81><loc_17><loc_82><loc_17></location>e</text>
<text><location><page_47><loc_81><loc_17><loc_82><loc_17></location>i</text>
<text><location><page_47><loc_81><loc_17><loc_82><loc_17></location>t</text>
<text><location><page_47><loc_81><loc_16><loc_82><loc_16></location>i</text>
<text><location><page_47><loc_81><loc_16><loc_82><loc_16></location>c</text>
<text><location><page_47><loc_81><loc_15><loc_82><loc_16></location>o</text>
<text><location><page_47><loc_81><loc_15><loc_82><loc_15></location>l</text>
<text><location><page_47><loc_81><loc_15><loc_82><loc_15></location>e</text>
<text><location><page_47><loc_82><loc_78><loc_83><loc_79></location>.</text>
<text><location><page_47><loc_82><loc_78><loc_83><loc_78></location>y</text>
<text><location><page_47><loc_82><loc_78><loc_83><loc_78></location>l</text>
<text><location><page_47><loc_82><loc_77><loc_83><loc_78></location>n</text>
<text><location><page_47><loc_82><loc_77><loc_83><loc_77></location>o</text>
<text><location><page_47><loc_82><loc_75><loc_83><loc_75></location>s</text>
<text><location><page_47><loc_82><loc_74><loc_83><loc_75></location>m</text>
<text><location><page_47><loc_82><loc_74><loc_83><loc_74></location>k</text>
<text><location><page_47><loc_82><loc_73><loc_83><loc_73></location>0</text>
<text><location><page_47><loc_82><loc_72><loc_83><loc_73></location>0</text>
<text><location><page_47><loc_82><loc_72><loc_83><loc_72></location>1</text>
<text><location><page_47><loc_82><loc_71><loc_83><loc_72></location>o</text>
<text><location><page_47><loc_82><loc_71><loc_83><loc_71></location>t</text>
<text><location><page_47><loc_82><loc_70><loc_83><loc_71></location>0</text>
<text><location><page_47><loc_82><loc_69><loc_83><loc_70></location>m</text>
<text><location><page_47><loc_82><loc_69><loc_83><loc_69></location>o</text>
<text><location><page_47><loc_82><loc_68><loc_83><loc_69></location>r</text>
<text><location><page_47><loc_82><loc_68><loc_83><loc_68></location>f</text>
<text><location><page_47><loc_82><loc_67><loc_83><loc_68></location>d</text>
<text><location><page_47><loc_82><loc_67><loc_83><loc_67></location>e</text>
<text><location><page_47><loc_82><loc_67><loc_83><loc_67></location>t</text>
<text><location><page_47><loc_82><loc_66><loc_83><loc_67></location>a</text>
<text><location><page_47><loc_82><loc_66><loc_83><loc_66></location>l</text>
<text><location><page_47><loc_82><loc_65><loc_83><loc_66></location>u</text>
<text><location><page_47><loc_82><loc_65><loc_83><loc_65></location>c</text>
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<text><location><page_47><loc_87><loc_48><loc_88><loc_48></location>m</text>
<text><location><page_47><loc_87><loc_47><loc_88><loc_48></location>r</text>
<text><location><page_47><loc_87><loc_47><loc_88><loc_47></location>o</text>
<text><location><page_47><loc_87><loc_47><loc_88><loc_47></location>n</text>
<text><location><page_47><loc_87><loc_46><loc_88><loc_46></location>e</text>
<text><location><page_47><loc_87><loc_45><loc_88><loc_46></location>w</text>
<text><location><page_47><loc_87><loc_45><loc_88><loc_45></location>,</text>
<text><location><page_47><loc_87><loc_44><loc_88><loc_45></location>e</text>
<text><location><page_47><loc_87><loc_44><loc_88><loc_44></location>g</text>
<text><location><page_47><loc_87><loc_43><loc_88><loc_44></location>a</text>
<text><location><page_47><loc_87><loc_43><loc_88><loc_43></location>r</text>
<text><location><page_47><loc_87><loc_43><loc_88><loc_43></location>e</text>
<text><location><page_47><loc_87><loc_42><loc_88><loc_43></location>v</text>
<text><location><page_47><loc_87><loc_42><loc_88><loc_42></location>a</text>
<text><location><page_47><loc_87><loc_41><loc_88><loc_42></location>e</text>
<text><location><page_47><loc_87><loc_41><loc_88><loc_41></location>h</text>
<text><location><page_47><loc_87><loc_40><loc_88><loc_41></location>t</text>
<text><location><page_47><loc_87><loc_40><loc_88><loc_40></location>e</text>
<text><location><page_47><loc_87><loc_39><loc_88><loc_40></location>t</text>
<text><location><page_47><loc_87><loc_39><loc_88><loc_39></location>a</text>
<text><location><page_47><loc_87><loc_39><loc_88><loc_39></location>l</text>
<text><location><page_47><loc_87><loc_38><loc_88><loc_39></location>u</text>
<text><location><page_47><loc_87><loc_38><loc_88><loc_38></location>c</text>
<text><location><page_47><loc_87><loc_38><loc_88><loc_38></location>l</text>
<text><location><page_47><loc_87><loc_37><loc_88><loc_38></location>a</text>
<text><location><page_47><loc_87><loc_37><loc_88><loc_37></location>c</text>
<text><location><page_47><loc_87><loc_36><loc_88><loc_36></location>o</text>
<text><location><page_47><loc_87><loc_35><loc_88><loc_36></location>T</text>
<text><location><page_47><loc_87><loc_35><loc_88><loc_35></location>.</text>
<text><location><page_47><loc_87><loc_34><loc_88><loc_35></location>y</text>
<text><location><page_47><loc_87><loc_34><loc_88><loc_34></location>l</text>
<text><location><page_47><loc_87><loc_34><loc_88><loc_34></location>e</text>
<text><location><page_47><loc_87><loc_33><loc_88><loc_34></location>v</text>
<text><location><page_47><loc_87><loc_33><loc_88><loc_33></location>i</text>
<text><location><page_47><loc_87><loc_33><loc_88><loc_33></location>t</text>
<text><location><page_47><loc_87><loc_32><loc_88><loc_33></location>c</text>
<text><location><page_47><loc_87><loc_32><loc_88><loc_32></location>e</text>
<text><location><page_47><loc_87><loc_31><loc_88><loc_32></location>p</text>
<text><location><page_47><loc_87><loc_31><loc_88><loc_31></location>s</text>
<text><location><page_47><loc_87><loc_31><loc_88><loc_31></location>e</text>
<text><location><page_47><loc_87><loc_30><loc_88><loc_31></location>r</text>
<text><location><page_47><loc_87><loc_30><loc_88><loc_30></location>,</text>
<text><location><page_47><loc_87><loc_29><loc_88><loc_30></location>m</text>
<text><location><page_47><loc_87><loc_29><loc_88><loc_29></location>u</text>
<text><location><page_47><loc_87><loc_28><loc_88><loc_29></location>m</text>
<text><location><page_47><loc_87><loc_28><loc_88><loc_28></location>i</text>
<text><location><page_47><loc_87><loc_27><loc_88><loc_28></location>x</text>
<text><location><page_47><loc_87><loc_27><loc_88><loc_27></location>a</text>
<text><location><page_47><loc_87><loc_26><loc_88><loc_27></location>m</text>
<text><location><page_47><loc_87><loc_25><loc_88><loc_26></location>e</text>
<text><location><page_47><loc_87><loc_25><loc_88><loc_25></location>h</text>
<text><location><page_47><loc_87><loc_25><loc_88><loc_25></location>t</text>
<text><location><page_47><loc_87><loc_24><loc_88><loc_24></location>m</text>
<text><location><page_47><loc_87><loc_23><loc_88><loc_24></location>o</text>
<text><location><page_47><loc_87><loc_23><loc_88><loc_23></location>r</text>
<text><location><page_47><loc_87><loc_23><loc_88><loc_23></location>f</text>
<text><location><page_47><loc_86><loc_22><loc_87><loc_22></location>1</text>
<text><location><page_47><loc_86><loc_21><loc_87><loc_22></location>-</text>
<text><location><page_47><loc_87><loc_21><loc_88><loc_21></location>s</text>
<text><location><page_47><loc_87><loc_20><loc_88><loc_20></location>m</text>
<text><location><page_47><loc_87><loc_19><loc_88><loc_20></location>k</text>
<text><location><page_47><loc_87><loc_19><loc_88><loc_19></location>0</text>
<text><location><page_47><loc_87><loc_18><loc_88><loc_19></location>5</text>
<text><location><page_47><loc_87><loc_18><loc_88><loc_18></location>1</text>
<text><location><page_47><loc_87><loc_17><loc_88><loc_18></location>±</text>
<text><location><page_47><loc_87><loc_16><loc_88><loc_17></location>d</text>
<text><location><page_47><loc_87><loc_16><loc_88><loc_16></location>n</text>
<text><location><page_47><loc_87><loc_15><loc_88><loc_16></location>a</text>
<text><location><page_47><loc_89><loc_28><loc_90><loc_29></location>.</text>
<text><location><page_47><loc_89><loc_28><loc_90><loc_28></location>1</text>
<text><location><page_47><loc_89><loc_27><loc_89><loc_28></location>-</text>
<text><location><page_47><loc_89><loc_27><loc_90><loc_27></location>s</text>
<text><location><page_47><loc_89><loc_26><loc_90><loc_26></location>m</text>
<text><location><page_47><loc_89><loc_25><loc_90><loc_26></location>k</text>
<text><location><page_47><loc_89><loc_25><loc_90><loc_25></location>5</text>
<text><location><page_47><loc_89><loc_24><loc_90><loc_25></location>±</text>
<text><location><page_47><loc_89><loc_23><loc_90><loc_24></location>s</text>
<text><location><page_47><loc_89><loc_23><loc_90><loc_23></location>i</text>
<text><location><page_47><loc_89><loc_22><loc_90><loc_23></location>r</text>
<text><location><page_47><loc_89><loc_22><loc_90><loc_22></location>o</text>
<text><location><page_47><loc_89><loc_22><loc_90><loc_22></location>r</text>
<text><location><page_47><loc_89><loc_21><loc_90><loc_22></location>r</text>
<text><location><page_47><loc_89><loc_21><loc_90><loc_21></location>e</text>
<text><location><page_47><loc_89><loc_20><loc_90><loc_21></location>e</text>
<text><location><page_47><loc_89><loc_20><loc_90><loc_20></location>h</text>
<text><location><page_47><loc_89><loc_19><loc_90><loc_20></location>T</text>
<text><location><page_47><loc_89><loc_19><loc_90><loc_19></location>.</text>
<text><location><page_47><loc_89><loc_18><loc_90><loc_19></location>e</text>
<text><location><page_47><loc_89><loc_18><loc_90><loc_18></location>n</text>
<text><location><page_47><loc_89><loc_17><loc_90><loc_18></location>i</text>
<text><location><page_47><loc_89><loc_17><loc_90><loc_17></location>l</text>
<text><location><page_47><loc_89><loc_17><loc_90><loc_17></location>i</text>
<text><location><page_47><loc_89><loc_16><loc_90><loc_17></location>i</text>
<text><location><page_47><loc_89><loc_16><loc_90><loc_16></location>e</text>
<text><location><page_47><loc_89><loc_15><loc_90><loc_16></location>H</text>
<text><location><page_48><loc_9><loc_93><loc_10><loc_94></location>48</text>
<text><location><page_48><loc_45><loc_93><loc_56><loc_94></location>Ardila et al.</text>
<paragraph><location><page_48><loc_42><loc_90><loc_57><loc_92></location>0 . 1 0 . 1 0 . 1 . . . 0 . 1 . . . 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1</paragraph>
<text><location><page_48><loc_42><loc_88><loc_43><loc_88></location>6</text>
<text><location><page_48><loc_42><loc_9><loc_43><loc_9></location>N</text>
<text><location><page_48><loc_42><loc_8><loc_43><loc_9></location>D</text>
<text><location><page_48><loc_44><loc_88><loc_45><loc_88></location>5</text>
<text><location><page_48><loc_44><loc_9><loc_45><loc_9></location>R</text>
<text><location><page_48><loc_44><loc_8><loc_45><loc_9></location>D</text>
<text><location><page_48><loc_45><loc_88><loc_46><loc_88></location>0</text>
<text><location><page_48><loc_45><loc_9><loc_46><loc_9></location>R</text>
<text><location><page_48><loc_45><loc_8><loc_46><loc_9></location>D</text>
<text><location><page_48><loc_46><loc_88><loc_48><loc_88></location>2</text>
<text><location><page_48><loc_55><loc_88><loc_56><loc_88></location>6</text>
<text><location><page_48><loc_63><loc_91><loc_64><loc_92></location>.</text>
<text><location><page_48><loc_63><loc_91><loc_64><loc_91></location>.</text>
<text><location><page_48><loc_63><loc_90><loc_64><loc_91></location>.</text>
<text><location><page_48><loc_63><loc_88><loc_64><loc_88></location>1</text>
<text><location><page_48><loc_78><loc_91><loc_79><loc_92></location>6</text>
<text><location><page_48><loc_78><loc_91><loc_79><loc_91></location>.</text>
<text><location><page_48><loc_78><loc_90><loc_79><loc_91></location>0</text>
<text><location><page_48><loc_78><loc_88><loc_79><loc_88></location>4</text>
<text><location><page_48><loc_88><loc_91><loc_89><loc_92></location>5</text>
<text><location><page_48><loc_88><loc_91><loc_89><loc_91></location>0</text>
<text><location><page_48><loc_88><loc_91><loc_89><loc_91></location>.</text>
<text><location><page_48><loc_88><loc_90><loc_89><loc_91></location>0</text>
<text><location><page_48><loc_88><loc_88><loc_89><loc_88></location>1</text>
<table>
<location><page_48><loc_7><loc_9><loc_93><loc_88></location>
</table>
<text><location><page_48><loc_18><loc_9><loc_19><loc_9></location>a</text>
<text><location><page_48><loc_18><loc_8><loc_19><loc_9></location>N</text>
<text><location><page_48><loc_48><loc_88><loc_49><loc_88></location>2</text>
<text><location><page_48><loc_48><loc_9><loc_49><loc_9></location>S</text>
<text><location><page_48><loc_48><loc_8><loc_49><loc_9></location>D</text>
<text><location><page_48><loc_62><loc_91><loc_63><loc_92></location>3</text>
<text><location><page_48><loc_62><loc_91><loc_63><loc_91></location>.</text>
<text><location><page_48><loc_62><loc_90><loc_63><loc_91></location>0</text>
<text><location><page_48><loc_62><loc_88><loc_63><loc_88></location>2</text>
<text><location><page_48><loc_62><loc_8><loc_63><loc_9></location>U</text>
<text><location><page_48><loc_62><loc_8><loc_63><loc_8></location>S</text>
<text><location><page_48><loc_66><loc_91><loc_67><loc_92></location>1</text>
<text><location><page_48><loc_66><loc_91><loc_67><loc_91></location>.</text>
<text><location><page_48><loc_66><loc_90><loc_67><loc_91></location>0</text>
<text><location><page_48><loc_66><loc_88><loc_67><loc_88></location>0</text>
<text><location><page_48><loc_66><loc_9><loc_67><loc_9></location>1</text>
<text><location><page_48><loc_66><loc_8><loc_67><loc_9></location>V</text>
<text><location><page_48><loc_67><loc_91><loc_68><loc_92></location>4</text>
<text><location><page_48><loc_67><loc_91><loc_68><loc_91></location>.</text>
<text><location><page_48><loc_67><loc_90><loc_68><loc_91></location>0</text>
<text><location><page_48><loc_67><loc_88><loc_68><loc_88></location>7</text>
<text><location><page_48><loc_67><loc_9><loc_68><loc_9></location>4</text>
<text><location><page_48><loc_67><loc_8><loc_68><loc_9></location>V</text>
<text><location><page_48><loc_72><loc_91><loc_73><loc_92></location>2</text>
<text><location><page_48><loc_72><loc_91><loc_73><loc_91></location>.</text>
<text><location><page_48><loc_72><loc_90><loc_73><loc_91></location>0</text>
<text><location><page_48><loc_72><loc_88><loc_73><loc_88></location>2</text>
<text><location><page_48><loc_72><loc_9><loc_73><loc_9></location>S</text>
<text><location><page_48><loc_72><loc_8><loc_73><loc_9></location>C</text>
<text><location><page_48><loc_76><loc_91><loc_78><loc_92></location>1</text>
<text><location><page_48><loc_76><loc_91><loc_78><loc_91></location>.</text>
<text><location><page_48><loc_76><loc_90><loc_78><loc_91></location>0</text>
<text><location><page_48><loc_76><loc_88><loc_78><loc_88></location>3</text>
<text><location><page_48><loc_76><loc_8><loc_78><loc_9></location>P</text>
<text><location><page_48><loc_76><loc_8><loc_78><loc_8></location>I</text>
<text><location><page_48><loc_80><loc_91><loc_82><loc_92></location>1</text>
<text><location><page_48><loc_80><loc_91><loc_82><loc_91></location>.</text>
<text><location><page_48><loc_80><loc_90><loc_82><loc_91></location>0</text>
<text><location><page_48><loc_80><loc_88><loc_82><loc_88></location>4</text>
<text><location><page_48><loc_80><loc_9><loc_82><loc_9></location>1</text>
<text><location><page_48><loc_80><loc_8><loc_82><loc_9></location>V</text>
<text><location><page_48><loc_88><loc_9><loc_89><loc_9></location>1</text>
<text><location><page_48><loc_88><loc_8><loc_89><loc_9></location>V</text>
<text><location><page_48><loc_89><loc_91><loc_91><loc_92></location>1</text>
<text><location><page_48><loc_89><loc_91><loc_91><loc_91></location>.</text>
<text><location><page_48><loc_89><loc_90><loc_91><loc_91></location>0</text>
<text><location><page_48><loc_89><loc_88><loc_91><loc_88></location>4</text>
<text><location><page_48><loc_89><loc_9><loc_91><loc_9></location>3</text>
<text><location><page_48><loc_89><loc_8><loc_91><loc_9></location>V</text>
<text><location><page_48><loc_78><loc_9><loc_79><loc_10></location>W</text>
<text><location><page_48><loc_78><loc_8><loc_79><loc_9></location>T</text>
<text><location><page_48><loc_55><loc_9><loc_56><loc_9></location>P</text>
<text><location><page_48><loc_55><loc_8><loc_56><loc_9></location>M</text>
<text><location><page_48><loc_46><loc_9><loc_48><loc_9></location>R</text>
<text><location><page_48><loc_46><loc_8><loc_48><loc_9></location>D</text>
<text><location><page_48><loc_50><loc_88><loc_52><loc_88></location>4</text>
<text><location><page_48><loc_50><loc_8><loc_52><loc_9></location>P</text>
<text><location><page_48><loc_50><loc_8><loc_52><loc_8></location>E</text>
<text><location><page_48><loc_52><loc_88><loc_53><loc_88></location>8</text>
<text><location><page_48><loc_52><loc_8><loc_53><loc_9></location>T</text>
<text><location><page_48><loc_52><loc_8><loc_53><loc_8></location>E</text>
<text><location><page_48><loc_53><loc_88><loc_55><loc_88></location>8</text>
<text><location><page_48><loc_53><loc_9><loc_55><loc_9></location>N</text>
<text><location><page_48><loc_53><loc_8><loc_55><loc_9></location>H</text>
<text><location><page_48><loc_56><loc_88><loc_57><loc_88></location>5</text>
<text><location><page_48><loc_56><loc_9><loc_57><loc_9></location>U</text>
<text><location><page_48><loc_56><loc_8><loc_57><loc_9></location>R</text>
<text><location><page_48><loc_15><loc_92><loc_17><loc_92></location>.</text>
<text><location><page_48><loc_15><loc_91><loc_17><loc_92></location>p</text>
<text><location><page_48><loc_15><loc_91><loc_17><loc_91></location>m</text>
<text><location><page_48><loc_15><loc_90><loc_17><loc_91></location>o</text>
<text><location><page_48><loc_15><loc_89><loc_17><loc_90></location>C</text>
<text><location><page_48><loc_15><loc_88><loc_17><loc_89></location>d</text>
<text><location><page_48><loc_15><loc_88><loc_17><loc_88></location>a</text>
<text><location><page_48><loc_18><loc_92><loc_19><loc_92></location>3</text>
<text><location><page_48><loc_18><loc_91><loc_19><loc_92></location>A</text>
<text><location><page_48><loc_18><loc_90><loc_19><loc_91></location>∆</text>
<text><location><page_48><loc_20><loc_91><loc_21><loc_92></location>x</text>
<text><location><page_48><loc_20><loc_91><loc_21><loc_91></location>u</text>
<text><location><page_48><loc_20><loc_90><loc_21><loc_91></location>fl</text>
<text><location><page_48><loc_20><loc_88><loc_21><loc_88></location>x</text>
<text><location><page_48><loc_27><loc_91><loc_28><loc_92></location>1</text>
<text><location><page_48><loc_27><loc_91><loc_28><loc_91></location>.</text>
<text><location><page_48><loc_27><loc_90><loc_28><loc_91></location>0</text>
<text><location><page_48><loc_27><loc_88><loc_28><loc_88></location>7</text>
<text><location><page_48><loc_27><loc_9><loc_28><loc_9></location>A</text>
<text><location><page_48><loc_27><loc_8><loc_28><loc_9></location>A</text>
<text><location><page_48><loc_28><loc_91><loc_30><loc_92></location>1</text>
<text><location><page_48><loc_28><loc_91><loc_30><loc_91></location>.</text>
<text><location><page_48><loc_28><loc_90><loc_30><loc_91></location>0</text>
<text><location><page_48><loc_28><loc_88><loc_30><loc_88></location>6</text>
<text><location><page_48><loc_28><loc_9><loc_30><loc_9></location>K</text>
<text><location><page_48><loc_28><loc_8><loc_30><loc_9></location>A</text>
<text><location><page_48><loc_30><loc_91><loc_31><loc_92></location>1</text>
<text><location><page_48><loc_30><loc_91><loc_31><loc_91></location>.</text>
<text><location><page_48><loc_30><loc_90><loc_31><loc_91></location>0</text>
<text><location><page_48><loc_30><loc_88><loc_31><loc_88></location>9</text>
<text><location><page_48><loc_30><loc_9><loc_31><loc_9></location>P</text>
<text><location><page_48><loc_30><loc_8><loc_31><loc_9></location>B</text>
<text><location><page_48><loc_31><loc_91><loc_32><loc_92></location>2</text>
<text><location><page_48><loc_31><loc_91><loc_32><loc_91></location>.</text>
<text><location><page_48><loc_31><loc_90><loc_32><loc_91></location>0</text>
<text><location><page_48><loc_31><loc_88><loc_32><loc_88></location>4</text>
<text><location><page_48><loc_31><loc_9><loc_32><loc_9></location>P</text>
<text><location><page_48><loc_31><loc_8><loc_32><loc_9></location>B</text>
<text><location><page_48><loc_32><loc_91><loc_34><loc_92></location>1</text>
<text><location><page_48><loc_32><loc_91><loc_34><loc_91></location>.</text>
<text><location><page_48><loc_32><loc_90><loc_34><loc_91></location>0</text>
<text><location><page_48><loc_32><loc_88><loc_34><loc_88></location>6</text>
<text><location><page_48><loc_32><loc_9><loc_34><loc_9></location>V</text>
<text><location><page_48><loc_32><loc_8><loc_34><loc_9></location>C</text>
<text><location><page_48><loc_34><loc_91><loc_35><loc_92></location>1</text>
<text><location><page_48><loc_34><loc_91><loc_35><loc_91></location>.</text>
<text><location><page_48><loc_34><loc_90><loc_35><loc_91></location>0</text>
<text><location><page_48><loc_34><loc_88><loc_35><loc_88></location>0</text>
<text><location><page_48><loc_34><loc_9><loc_35><loc_9></location>Y</text>
<text><location><page_48><loc_34><loc_8><loc_35><loc_9></location>C</text>
<text><location><page_48><loc_35><loc_91><loc_37><loc_92></location>2</text>
<text><location><page_48><loc_35><loc_91><loc_37><loc_91></location>.</text>
<text><location><page_48><loc_35><loc_90><loc_37><loc_91></location>0</text>
<text><location><page_48><loc_35><loc_88><loc_37><loc_88></location>6</text>
<text><location><page_48><loc_35><loc_9><loc_37><loc_9></location>E</text>
<text><location><page_48><loc_35><loc_8><loc_37><loc_9></location>D</text>
<text><location><page_48><loc_37><loc_91><loc_38><loc_92></location>1</text>
<text><location><page_48><loc_37><loc_91><loc_38><loc_91></location>.</text>
<text><location><page_48><loc_37><loc_90><loc_38><loc_91></location>0</text>
<text><location><page_48><loc_37><loc_88><loc_38><loc_88></location>8</text>
<text><location><page_48><loc_37><loc_9><loc_38><loc_9></location>F</text>
<text><location><page_48><loc_37><loc_8><loc_38><loc_9></location>D</text>
<text><location><page_48><loc_38><loc_91><loc_39><loc_92></location>1</text>
<text><location><page_48><loc_38><loc_91><loc_39><loc_91></location>.</text>
<text><location><page_48><loc_38><loc_90><loc_39><loc_91></location>0</text>
<text><location><page_48><loc_38><loc_88><loc_39><loc_88></location>6</text>
<text><location><page_48><loc_38><loc_9><loc_39><loc_9></location>F</text>
<text><location><page_48><loc_38><loc_8><loc_39><loc_9></location>D</text>
<text><location><page_48><loc_39><loc_91><loc_41><loc_92></location>2</text>
<text><location><page_48><loc_39><loc_91><loc_41><loc_91></location>.</text>
<text><location><page_48><loc_39><loc_90><loc_41><loc_91></location>0</text>
<text><location><page_48><loc_39><loc_88><loc_41><loc_88></location>2</text>
<text><location><page_48><loc_39><loc_9><loc_41><loc_9></location>F</text>
<text><location><page_48><loc_39><loc_8><loc_41><loc_9></location>D</text>
<text><location><page_48><loc_41><loc_91><loc_42><loc_92></location>1</text>
<text><location><page_48><loc_41><loc_91><loc_42><loc_91></location>.</text>
<text><location><page_48><loc_41><loc_90><loc_42><loc_91></location>0</text>
<text><location><page_48><loc_41><loc_88><loc_42><loc_88></location>9</text>
<text><location><page_48><loc_41><loc_9><loc_42><loc_9></location>K</text>
<text><location><page_48><loc_41><loc_8><loc_42><loc_9></location>D</text>
<text><location><page_48><loc_49><loc_9><loc_50><loc_9></location>X</text>
<text><location><page_48><loc_49><loc_8><loc_50><loc_9></location>D</text>
<text><location><page_48><loc_57><loc_91><loc_59><loc_92></location>1</text>
<text><location><page_48><loc_57><loc_91><loc_59><loc_91></location>.</text>
<text><location><page_48><loc_57><loc_90><loc_59><loc_91></location>0</text>
<text><location><page_48><loc_57><loc_88><loc_59><loc_88></location>6</text>
<text><location><page_48><loc_57><loc_9><loc_59><loc_9></location>U</text>
<text><location><page_48><loc_57><loc_8><loc_59><loc_9></location>R</text>
<text><location><page_48><loc_59><loc_91><loc_60><loc_91></location>1</text>
<text><location><page_48><loc_59><loc_9><loc_60><loc_9></location>U</text>
<text><location><page_48><loc_59><loc_8><loc_60><loc_9></location>R</text>
<text><location><page_48><loc_60><loc_91><loc_61><loc_92></location>.</text>
<text><location><page_48><loc_60><loc_91><loc_61><loc_91></location>.</text>
<text><location><page_48><loc_60><loc_90><loc_61><loc_91></location>.</text>
<text><location><page_48><loc_60><loc_8><loc_61><loc_10></location>W</text>
<text><location><page_48><loc_60><loc_8><loc_61><loc_9></location>R</text>
<text><location><page_48><loc_63><loc_8><loc_64><loc_9></location>T</text>
<text><location><page_48><loc_64><loc_91><loc_66><loc_92></location>1</text>
<text><location><page_48><loc_64><loc_91><loc_66><loc_91></location>.</text>
<text><location><page_48><loc_64><loc_90><loc_66><loc_91></location>0</text>
<text><location><page_48><loc_64><loc_88><loc_66><loc_88></location>5</text>
<text><location><page_48><loc_64><loc_8><loc_66><loc_9></location>T</text>
<text><location><page_48><loc_73><loc_91><loc_75><loc_92></location>1</text>
<text><location><page_48><loc_73><loc_91><loc_75><loc_91></location>.</text>
<text><location><page_48><loc_73><loc_90><loc_75><loc_91></location>0</text>
<text><location><page_48><loc_73><loc_88><loc_75><loc_88></location>8</text>
<text><location><page_48><loc_73><loc_9><loc_75><loc_10></location>M</text>
<text><location><page_48><loc_73><loc_8><loc_75><loc_9></location>D</text>
<text><location><page_48><loc_75><loc_91><loc_76><loc_92></location>1</text>
<text><location><page_48><loc_75><loc_91><loc_76><loc_91></location>.</text>
<text><location><page_48><loc_75><loc_90><loc_76><loc_91></location>0</text>
<text><location><page_48><loc_75><loc_88><loc_76><loc_88></location>4</text>
<text><location><page_48><loc_75><loc_9><loc_76><loc_10></location>M</text>
<text><location><page_48><loc_75><loc_8><loc_76><loc_9></location>G</text>
<text><location><page_48><loc_79><loc_91><loc_80><loc_92></location>1</text>
<text><location><page_48><loc_79><loc_91><loc_80><loc_91></location>.</text>
<text><location><page_48><loc_79><loc_90><loc_80><loc_91></location>0</text>
<text><location><page_48><loc_79><loc_88><loc_80><loc_88></location>2</text>
<text><location><page_48><loc_79><loc_9><loc_80><loc_9></location>X</text>
<text><location><page_48><loc_79><loc_8><loc_80><loc_9></location>U</text>
<text><location><page_48><loc_85><loc_91><loc_87><loc_92></location>1</text>
<text><location><page_48><loc_85><loc_91><loc_87><loc_91></location>.</text>
<text><location><page_48><loc_85><loc_90><loc_87><loc_91></location>0</text>
<text><location><page_48><loc_85><loc_88><loc_87><loc_88></location>3</text>
<text><location><page_48><loc_85><loc_8><loc_87><loc_9></location>G</text>
<text><location><page_48><loc_85><loc_8><loc_87><loc_8></location>E</text>
<text><location><page_48><loc_87><loc_91><loc_88><loc_92></location>1</text>
<text><location><page_48><loc_87><loc_91><loc_88><loc_91></location>.</text>
<text><location><page_48><loc_87><loc_90><loc_88><loc_91></location>0</text>
<text><location><page_48><loc_87><loc_88><loc_88><loc_88></location>8</text>
<text><location><page_48><loc_87><loc_9><loc_88><loc_10></location>W</text>
<text><location><page_48><loc_87><loc_8><loc_88><loc_9></location>T</text>
<text><location><page_49><loc_8><loc_55><loc_10><loc_55></location>d</text>
<text><location><page_49><loc_8><loc_54><loc_10><loc_55></location>e</text>
<text><location><page_49><loc_8><loc_54><loc_10><loc_54></location>u</text>
<text><location><page_49><loc_8><loc_53><loc_10><loc_54></location>n</text>
<text><location><page_49><loc_8><loc_53><loc_10><loc_53></location>i</text>
<text><location><page_49><loc_8><loc_53><loc_10><loc_53></location>t</text>
<text><location><page_49><loc_8><loc_52><loc_10><loc_53></location>n</text>
<text><location><page_49><loc_8><loc_51><loc_10><loc_52></location>o</text>
<text><location><page_49><loc_8><loc_51><loc_10><loc_51></location>C</text>
<text><location><page_49><loc_8><loc_49><loc_10><loc_50></location>-</text>
<text><location><page_49><loc_8><loc_48><loc_9><loc_49></location>6</text>
<text><location><page_49><loc_8><loc_47><loc_9><loc_47></location>e</text>
<text><location><page_49><loc_8><loc_47><loc_9><loc_47></location>l</text>
<text><location><page_49><loc_8><loc_46><loc_9><loc_47></location>b</text>
<text><location><page_49><loc_8><loc_45><loc_9><loc_46></location>a</text>
<text><location><page_49><loc_8><loc_45><loc_9><loc_45></location>T</text>
<text><location><page_49><loc_11><loc_49><loc_13><loc_50></location>e</text>
<text><location><page_49><loc_11><loc_49><loc_13><loc_49></location>n</text>
<text><location><page_49><loc_11><loc_48><loc_13><loc_49></location>i</text>
<text><location><page_49><loc_11><loc_48><loc_13><loc_48></location>L</text>
<text><location><page_49><loc_11><loc_47><loc_13><loc_47></location>e</text>
<text><location><page_49><loc_11><loc_46><loc_13><loc_47></location>u</text>
<text><location><page_49><loc_11><loc_46><loc_13><loc_46></location>l</text>
<text><location><page_49><loc_11><loc_45><loc_13><loc_46></location>B</text>
<text><location><page_49><loc_14><loc_92><loc_15><loc_92></location>.</text>
<text><location><page_49><loc_14><loc_91><loc_15><loc_92></location>p</text>
<text><location><page_49><loc_14><loc_91><loc_15><loc_91></location>m</text>
<text><location><page_49><loc_14><loc_90><loc_15><loc_91></location>o</text>
<text><location><page_49><loc_14><loc_89><loc_15><loc_90></location>C</text>
<text><location><page_49><loc_14><loc_64><loc_15><loc_65></location>.</text>
<text><location><page_49><loc_14><loc_64><loc_15><loc_64></location>p</text>
<text><location><page_49><loc_14><loc_63><loc_15><loc_64></location>m</text>
<text><location><page_49><loc_14><loc_62><loc_15><loc_63></location>o</text>
<text><location><page_49><loc_14><loc_61><loc_15><loc_62></location>C</text>
<text><location><page_49><loc_14><loc_61><loc_15><loc_61></location>d</text>
<text><location><page_49><loc_14><loc_60><loc_15><loc_61></location>a</text>
<text><location><page_49><loc_14><loc_59><loc_15><loc_60></location>o</text>
<text><location><page_49><loc_14><loc_59><loc_15><loc_59></location>r</text>
<text><location><page_49><loc_14><loc_58><loc_15><loc_59></location>B</text>
<text><location><page_49><loc_14><loc_37><loc_15><loc_37></location>.</text>
<text><location><page_49><loc_14><loc_36><loc_15><loc_37></location>p</text>
<text><location><page_49><loc_14><loc_35><loc_15><loc_36></location>m</text>
<text><location><page_49><loc_14><loc_35><loc_15><loc_35></location>o</text>
<text><location><page_49><loc_14><loc_34><loc_15><loc_35></location>C</text>
<text><location><page_49><loc_14><loc_33><loc_15><loc_33></location>w</text>
<text><location><page_49><loc_14><loc_32><loc_15><loc_33></location>o</text>
<text><location><page_49><loc_14><loc_32><loc_15><loc_32></location>r</text>
<text><location><page_49><loc_14><loc_31><loc_15><loc_32></location>r</text>
<text><location><page_49><loc_14><loc_31><loc_15><loc_31></location>a</text>
<text><location><page_49><loc_14><loc_30><loc_15><loc_31></location>N</text>
<text><location><page_49><loc_17><loc_92><loc_18><loc_92></location>3</text>
<text><location><page_49><loc_17><loc_91><loc_18><loc_92></location>A</text>
<text><location><page_49><loc_17><loc_90><loc_18><loc_91></location>∆</text>
<text><location><page_49><loc_17><loc_72><loc_18><loc_73></location>σ</text>
<text><location><page_49><loc_29><loc_92><loc_29><loc_92></location>-</text>
<text><location><page_49><loc_29><loc_91><loc_29><loc_92></location>v</text>
<text><location><page_49><loc_29><loc_91><loc_29><loc_91></location>(</text>
<text><location><page_49><loc_29><loc_90><loc_29><loc_91></location>-</text>
<text><location><page_49><loc_29><loc_90><loc_29><loc_90></location>(</text>
<text><location><page_49><loc_29><loc_89><loc_29><loc_90></location>p</text>
<text><location><page_49><loc_29><loc_73><loc_29><loc_73></location>n</text>
<text><location><page_49><loc_29><loc_72><loc_29><loc_73></location>a</text>
<text><location><page_49><loc_29><loc_72><loc_29><loc_72></location>i</text>
<table>
<location><page_49><loc_7><loc_73><loc_93><loc_90></location>
</table>
<text><location><page_49><loc_22><loc_73><loc_23><loc_73></location>1</text>
<text><location><page_49><loc_22><loc_72><loc_23><loc_73></location>.</text>
<text><location><page_49><loc_23><loc_73><loc_25><loc_73></location>.</text>
<text><location><page_49><loc_23><loc_72><loc_25><loc_72></location>.</text>
<text><location><page_49><loc_30><loc_73><loc_31><loc_73></location>.</text>
<text><location><page_49><loc_30><loc_73><loc_31><loc_73></location>)</text>
<text><location><page_49><loc_30><loc_72><loc_31><loc_72></location>23</text>
<text><location><page_49><loc_29><loc_66><loc_29><loc_66></location>c</text>
<text><location><page_49><loc_29><loc_65><loc_29><loc_65></location>e</text>
<text><location><page_49><loc_29><loc_64><loc_29><loc_65></location>h</text>
<text><location><page_49><loc_29><loc_64><loc_29><loc_64></location>t</text>
<text><location><page_49><loc_29><loc_63><loc_29><loc_64></location>n</text>
<text><location><page_49><loc_29><loc_63><loc_29><loc_63></location>I</text>
<text><location><page_49><loc_29><loc_62><loc_29><loc_63></location>.</text>
<text><location><page_49><loc_29><loc_62><loc_29><loc_62></location>s</text>
<text><location><page_49><loc_29><loc_61><loc_29><loc_62></location>n</text>
<text><location><page_49><loc_29><loc_61><loc_29><loc_61></location>a</text>
<text><location><page_49><loc_29><loc_61><loc_29><loc_61></location>i</text>
<text><location><page_49><loc_29><loc_60><loc_29><loc_61></location>s</text>
<text><location><page_49><loc_29><loc_60><loc_29><loc_60></location>s</text>
<text><location><page_49><loc_29><loc_60><loc_29><loc_60></location>u</text>
<text><location><page_49><loc_29><loc_59><loc_29><loc_60></location>a</text>
<text><location><page_49><loc_29><loc_59><loc_29><loc_59></location>g</text>
<text><location><page_49><loc_29><loc_58><loc_29><loc_58></location>r</text>
<text><location><page_49><loc_29><loc_58><loc_29><loc_58></location>u</text>
<text><location><page_49><loc_29><loc_57><loc_29><loc_58></location>o</text>
<text><location><page_49><loc_29><loc_57><loc_29><loc_57></location>f</text>
<text><location><page_49><loc_29><loc_56><loc_29><loc_57></location>r</text>
<text><location><page_49><loc_29><loc_56><loc_29><loc_56></location>o</text>
<text><location><page_49><loc_29><loc_55><loc_29><loc_56></location>o</text>
<text><location><page_49><loc_29><loc_55><loc_29><loc_55></location>w</text>
<text><location><page_49><loc_29><loc_54><loc_29><loc_55></location>t</text>
<text><location><page_49><loc_29><loc_54><loc_29><loc_54></location>r</text>
<text><location><page_49><loc_29><loc_53><loc_29><loc_54></location>e</text>
<text><location><page_49><loc_29><loc_53><loc_29><loc_53></location>h</text>
<text><location><page_49><loc_29><loc_52><loc_29><loc_53></location>t</text>
<text><location><page_49><loc_29><loc_52><loc_29><loc_52></location>i</text>
<text><location><page_49><loc_29><loc_52><loc_29><loc_52></location>e</text>
<text><location><page_49><loc_29><loc_51><loc_29><loc_51></location>t</text>
<text><location><page_49><loc_29><loc_51><loc_29><loc_51></location>i</text>
<text><location><page_49><loc_29><loc_51><loc_29><loc_51></location>f</text>
<text><location><page_49><loc_29><loc_50><loc_29><loc_50></location>e</text>
<text><location><page_49><loc_29><loc_49><loc_29><loc_50></location>w</text>
<text><location><page_49><loc_29><loc_49><loc_29><loc_49></location>s</text>
<text><location><page_49><loc_29><loc_48><loc_29><loc_49></location>e</text>
<text><location><page_49><loc_29><loc_48><loc_29><loc_48></location>n</text>
<text><location><page_49><loc_29><loc_48><loc_29><loc_48></location>i</text>
<text><location><page_49><loc_29><loc_47><loc_29><loc_48></location>l</text>
<text><location><page_49><loc_29><loc_47><loc_29><loc_47></location>v</text>
<text><location><page_49><loc_29><loc_46><loc_29><loc_47></location>i</text>
<text><location><page_49><loc_29><loc_46><loc_29><loc_46></location>C</text>
<text><location><page_49><loc_29><loc_45><loc_29><loc_45></location>o</text>
<text><location><page_49><loc_29><loc_44><loc_29><loc_45></location>w</text>
<text><location><page_49><loc_29><loc_44><loc_29><loc_44></location>t</text>
<text><location><page_49><loc_29><loc_43><loc_29><loc_44></location>f</text>
<text><location><page_49><loc_29><loc_43><loc_29><loc_43></location>o</text>
<text><location><page_49><loc_29><loc_42><loc_29><loc_43></location>x</text>
<text><location><page_49><loc_29><loc_42><loc_29><loc_42></location>e</text>
<text><location><page_49><loc_29><loc_41><loc_29><loc_42></location>l</text>
<text><location><page_49><loc_29><loc_41><loc_29><loc_42></location>p</text>
<text><location><page_49><loc_29><loc_40><loc_29><loc_41></location>m</text>
<text><location><page_49><loc_29><loc_40><loc_29><loc_40></location>o</text>
<text><location><page_49><loc_29><loc_40><loc_29><loc_40></location>c</text>
<text><location><page_49><loc_29><loc_39><loc_29><loc_39></location>h</text>
<text><location><page_49><loc_29><loc_38><loc_29><loc_39></location>c</text>
<text><location><page_49><loc_29><loc_38><loc_29><loc_38></location>a</text>
<text><location><page_49><loc_29><loc_38><loc_29><loc_38></location>e</text>
<text><location><page_49><loc_29><loc_37><loc_29><loc_37></location>r</text>
<text><location><page_49><loc_29><loc_37><loc_29><loc_37></location>o</text>
<text><location><page_49><loc_29><loc_36><loc_29><loc_37></location>F</text>
<text><location><page_49><loc_29><loc_35><loc_29><loc_36></location>.</text>
<text><location><page_49><loc_28><loc_35><loc_29><loc_35></location>2</text>
<text><location><page_49><loc_28><loc_34><loc_29><loc_35></location>-</text>
<text><location><page_49><loc_29><loc_33><loc_29><loc_34></location>m</text>
<text><location><page_49><loc_29><loc_33><loc_29><loc_33></location>c</text>
<text><location><page_49><loc_28><loc_32><loc_29><loc_33></location>1</text>
<text><location><page_49><loc_28><loc_32><loc_29><loc_32></location>-</text>
<text><location><page_49><loc_29><loc_31><loc_29><loc_32></location>c</text>
<text><location><page_49><loc_29><loc_31><loc_29><loc_31></location>e</text>
<text><location><page_49><loc_29><loc_30><loc_29><loc_31></location>s</text>
<text><location><page_49><loc_29><loc_30><loc_29><loc_30></location>s</text>
<text><location><page_49><loc_29><loc_29><loc_29><loc_30></location>g</text>
<text><location><page_49><loc_29><loc_29><loc_29><loc_29></location>r</text>
<text><location><page_49><loc_29><loc_28><loc_29><loc_29></location>e</text>
<text><location><page_49><loc_28><loc_28><loc_29><loc_28></location>4</text>
<text><location><page_49><loc_28><loc_27><loc_29><loc_28></location>1</text>
<text><location><page_49><loc_28><loc_27><loc_29><loc_27></location>-</text>
<text><location><page_49><loc_29><loc_26><loc_29><loc_27></location>0</text>
<text><location><page_49><loc_29><loc_26><loc_29><loc_26></location>1</text>
<text><location><page_49><loc_29><loc_25><loc_29><loc_25></location>f</text>
<text><location><page_49><loc_29><loc_25><loc_29><loc_25></location>o</text>
<text><location><page_49><loc_29><loc_24><loc_29><loc_24></location>s</text>
<text><location><page_49><loc_29><loc_24><loc_29><loc_24></location>t</text>
<text><location><page_49><loc_29><loc_24><loc_29><loc_24></location>i</text>
<text><location><page_49><loc_29><loc_23><loc_29><loc_24></location>n</text>
<text><location><page_49><loc_29><loc_23><loc_29><loc_23></location>u</text>
<text><location><page_49><loc_29><loc_22><loc_29><loc_22></location>n</text>
<text><location><page_49><loc_29><loc_22><loc_29><loc_22></location>i</text>
<text><location><page_49><loc_29><loc_21><loc_29><loc_21></location>t</text>
<text><location><page_49><loc_29><loc_21><loc_29><loc_21></location>i</text>
<text><location><page_49><loc_29><loc_20><loc_29><loc_21></location>f</text>
<text><location><page_49><loc_29><loc_20><loc_29><loc_20></location>n</text>
<text><location><page_49><loc_29><loc_19><loc_29><loc_20></location>e</text>
<text><location><page_49><loc_29><loc_19><loc_29><loc_19></location>e</text>
<text><location><page_49><loc_29><loc_18><loc_29><loc_19></location>b</text>
<text><location><page_49><loc_29><loc_18><loc_29><loc_18></location>s</text>
<text><location><page_49><loc_29><loc_17><loc_29><loc_18></location>a</text>
<text><location><page_49><loc_29><loc_17><loc_29><loc_17></location>h</text>
<text><location><page_49><loc_29><loc_16><loc_29><loc_16></location>x</text>
<text><location><page_49><loc_29><loc_16><loc_29><loc_16></location>u</text>
<text><location><page_49><loc_29><loc_15><loc_29><loc_16></location>l</text>
<text><location><page_49><loc_29><loc_15><loc_29><loc_15></location>f</text>
<text><location><page_49><loc_29><loc_14><loc_29><loc_15></location>e</text>
<text><location><page_49><loc_29><loc_14><loc_29><loc_14></location>h</text>
<text><location><page_49><loc_29><loc_13><loc_29><loc_14></location>T</text>
<text><location><page_49><loc_29><loc_12><loc_29><loc_13></location>-</text>
<text><location><page_49><loc_29><loc_11><loc_29><loc_11></location>.</text>
<text><location><page_49><loc_29><loc_11><loc_29><loc_11></location>e</text>
<text><location><page_49><loc_29><loc_10><loc_29><loc_11></location>t</text>
<text><location><page_49><loc_29><loc_10><loc_29><loc_10></location>o</text>
<text><location><page_49><loc_29><loc_9><loc_29><loc_10></location>N</text>
<text><location><page_49><loc_30><loc_66><loc_31><loc_66></location>v</text>
<text><location><page_49><loc_30><loc_65><loc_31><loc_66></location>(</text>
<text><location><page_49><loc_30><loc_65><loc_31><loc_65></location>-</text>
<text><location><page_49><loc_30><loc_64><loc_31><loc_65></location>(</text>
<text><location><page_49><loc_30><loc_64><loc_31><loc_64></location>p</text>
<text><location><page_49><loc_30><loc_63><loc_31><loc_64></location>x</text>
<text><location><page_49><loc_30><loc_63><loc_31><loc_63></location>e</text>
<text><location><page_49><loc_30><loc_62><loc_31><loc_63></location>3</text>
<text><location><page_49><loc_30><loc_62><loc_31><loc_62></location>A</text>
<text><location><page_49><loc_30><loc_61><loc_31><loc_61></location>+</text>
<text><location><page_49><loc_30><loc_60><loc_31><loc_61></location>)</text>
<text><location><page_49><loc_30><loc_60><loc_31><loc_60></location>20</text>
<text><location><page_49><loc_30><loc_59><loc_31><loc_60></location>σ</text>
<text><location><page_49><loc_30><loc_59><loc_31><loc_59></location>2</text>
<text><location><page_49><loc_30><loc_58><loc_31><loc_59></location>/</text>
<text><location><page_49><loc_30><loc_58><loc_30><loc_58></location>2</text>
<text><location><page_49><loc_30><loc_57><loc_31><loc_58></location>)</text>
<text><location><page_49><loc_30><loc_57><loc_31><loc_57></location>6</text>
<text><location><page_49><loc_30><loc_57><loc_31><loc_57></location>9</text>
<text><location><page_49><loc_30><loc_56><loc_31><loc_57></location>.</text>
<text><location><page_49><loc_30><loc_56><loc_31><loc_56></location>0</text>
<text><location><page_49><loc_30><loc_55><loc_31><loc_56></location>0</text>
<text><location><page_49><loc_30><loc_55><loc_31><loc_55></location>5</text>
<text><location><page_49><loc_30><loc_54><loc_31><loc_55></location>-</text>
<text><location><page_49><loc_30><loc_53><loc_31><loc_54></location>v</text>
<text><location><page_49><loc_30><loc_53><loc_31><loc_53></location>(</text>
<text><location><page_49><loc_30><loc_52><loc_31><loc_53></location>-</text>
<text><location><page_49><loc_30><loc_52><loc_31><loc_52></location>(</text>
<text><location><page_49><loc_30><loc_52><loc_31><loc_52></location>p</text>
<text><location><page_49><loc_30><loc_51><loc_31><loc_52></location>x</text>
<text><location><page_49><loc_30><loc_51><loc_31><loc_51></location>e</text>
<text><location><page_49><loc_30><loc_50><loc_31><loc_50></location>2</text>
<text><location><page_49><loc_30><loc_49><loc_31><loc_50></location>A</text>
<text><location><page_49><loc_30><loc_49><loc_31><loc_49></location>+</text>
<text><location><page_49><loc_30><loc_48><loc_31><loc_48></location>)</text>
<text><location><page_49><loc_30><loc_47><loc_31><loc_48></location>21</text>
<text><location><page_49><loc_30><loc_47><loc_31><loc_47></location>σ</text>
<text><location><page_49><loc_30><loc_47><loc_31><loc_47></location>2</text>
<text><location><page_49><loc_30><loc_46><loc_31><loc_47></location>/</text>
<text><location><page_49><loc_30><loc_46><loc_30><loc_46></location>2</text>
<text><location><page_49><loc_30><loc_45><loc_31><loc_46></location>)</text>
<text><location><page_49><loc_30><loc_45><loc_31><loc_45></location>1</text>
<text><location><page_49><loc_30><loc_44><loc_31><loc_45></location>µ</text>
<text><location><page_49><loc_30><loc_43><loc_31><loc_44></location>-</text>
<text><location><page_49><loc_30><loc_43><loc_31><loc_43></location>v</text>
<text><location><page_49><loc_30><loc_42><loc_31><loc_43></location>(</text>
<text><location><page_49><loc_30><loc_42><loc_31><loc_42></location>-</text>
<text><location><page_49><loc_30><loc_41><loc_31><loc_42></location>(</text>
<text><location><page_49><loc_30><loc_41><loc_31><loc_41></location>p</text>
<text><location><page_49><loc_30><loc_40><loc_31><loc_41></location>x</text>
<text><location><page_49><loc_30><loc_40><loc_31><loc_40></location>e</text>
<text><location><page_49><loc_30><loc_39><loc_31><loc_40></location>1</text>
<text><location><page_49><loc_30><loc_38><loc_31><loc_39></location>A</text>
<text><location><page_49><loc_30><loc_38><loc_31><loc_38></location>+</text>
<text><location><page_49><loc_30><loc_37><loc_31><loc_37></location>)</text>
<text><location><page_49><loc_30><loc_37><loc_31><loc_37></location>20</text>
<text><location><page_49><loc_30><loc_36><loc_31><loc_37></location>σ</text>
<text><location><page_49><loc_30><loc_36><loc_31><loc_36></location>2</text>
<text><location><page_49><loc_30><loc_35><loc_31><loc_36></location>/</text>
<text><location><page_49><loc_30><loc_35><loc_30><loc_35></location>2</text>
<text><location><page_49><loc_30><loc_34><loc_31><loc_35></location>)</text>
<text><location><page_49><loc_31><loc_35><loc_32><loc_35></location>.</text>
<text><location><page_49><loc_31><loc_35><loc_32><loc_35></location>s</text>
<text><location><page_49><loc_31><loc_34><loc_32><loc_35></location>i</text>
<text><location><page_49><loc_31><loc_34><loc_32><loc_34></location>s</text>
<text><location><page_49><loc_30><loc_34><loc_31><loc_34></location>0</text>
<text><location><page_49><loc_30><loc_33><loc_31><loc_34></location>µ</text>
<text><location><page_49><loc_30><loc_32><loc_31><loc_33></location>-</text>
<text><location><page_49><loc_30><loc_32><loc_31><loc_32></location>v</text>
<text><location><page_49><loc_30><loc_31><loc_31><loc_32></location>(</text>
<text><location><page_49><loc_30><loc_31><loc_31><loc_31></location>-</text>
<text><location><page_49><loc_30><loc_30><loc_31><loc_31></location>(</text>
<text><location><page_49><loc_30><loc_30><loc_31><loc_30></location>p</text>
<text><location><page_49><loc_30><loc_29><loc_31><loc_30></location>x</text>
<text><location><page_49><loc_30><loc_29><loc_31><loc_29></location>e</text>
<text><location><page_49><loc_30><loc_28><loc_31><loc_29></location>0</text>
<text><location><page_49><loc_30><loc_28><loc_31><loc_28></location>A</text>
<text><location><page_49><loc_30><loc_27><loc_31><loc_27></location>=</text>
<text><location><page_49><loc_30><loc_26><loc_31><loc_26></location>F</text>
<text><location><page_49><loc_30><loc_25><loc_31><loc_25></location>:</text>
<text><location><page_49><loc_30><loc_25><loc_31><loc_25></location>s</text>
<text><location><page_49><loc_30><loc_24><loc_31><loc_25></location>n</text>
<text><location><page_49><loc_30><loc_24><loc_31><loc_24></location>a</text>
<text><location><page_49><loc_30><loc_24><loc_31><loc_24></location>i</text>
<text><location><page_49><loc_30><loc_23><loc_31><loc_24></location>s</text>
<text><location><page_49><loc_30><loc_23><loc_31><loc_23></location>s</text>
<text><location><page_49><loc_30><loc_22><loc_31><loc_23></location>u</text>
<text><location><page_49><loc_30><loc_22><loc_31><loc_23></location>a</text>
<text><location><page_49><loc_30><loc_22><loc_31><loc_22></location>g</text>
<text><location><page_49><loc_30><loc_21><loc_31><loc_21></location>r</text>
<text><location><page_49><loc_30><loc_21><loc_31><loc_21></location>u</text>
<text><location><page_49><loc_30><loc_20><loc_31><loc_21></location>o</text>
<text><location><page_49><loc_30><loc_20><loc_31><loc_20></location>f</text>
<text><location><page_49><loc_30><loc_19><loc_31><loc_20></location>f</text>
<text><location><page_49><loc_30><loc_19><loc_31><loc_19></location>o</text>
<text><location><page_49><loc_30><loc_18><loc_31><loc_19></location>e</text>
<text><location><page_49><loc_30><loc_18><loc_31><loc_18></location>s</text>
<text><location><page_49><loc_30><loc_17><loc_31><loc_18></location>a</text>
<text><location><page_49><loc_30><loc_17><loc_31><loc_17></location>c</text>
<text><location><page_49><loc_30><loc_16><loc_31><loc_17></location>e</text>
<text><location><page_49><loc_30><loc_16><loc_31><loc_16></location>h</text>
<text><location><page_49><loc_30><loc_16><loc_31><loc_16></location>t</text>
<text><location><page_49><loc_30><loc_15><loc_31><loc_15></location>n</text>
<text><location><page_49><loc_30><loc_14><loc_31><loc_15></location>I</text>
<text><location><page_49><loc_30><loc_14><loc_31><loc_14></location>.</text>
<text><location><page_49><loc_30><loc_13><loc_31><loc_14></location>)</text>
<text><location><page_49><loc_30><loc_13><loc_31><loc_13></location>20</text>
<text><location><page_49><loc_30><loc_12><loc_31><loc_13></location>σ</text>
<text><location><page_49><loc_30><loc_12><loc_31><loc_12></location>2</text>
<text><location><page_49><loc_30><loc_12><loc_31><loc_12></location>/</text>
<text><location><page_49><loc_30><loc_11><loc_30><loc_11></location>2</text>
<text><location><page_49><loc_30><loc_11><loc_31><loc_11></location>)</text>
<text><location><page_49><loc_30><loc_10><loc_31><loc_11></location>6</text>
<text><location><page_49><loc_30><loc_10><loc_31><loc_10></location>9</text>
<text><location><page_49><loc_30><loc_10><loc_31><loc_10></location>.</text>
<text><location><page_49><loc_30><loc_9><loc_31><loc_10></location>0</text>
<text><location><page_49><loc_30><loc_9><loc_31><loc_9></location>0</text>
<text><location><page_49><loc_30><loc_8><loc_31><loc_9></location>5</text>
<text><location><page_49><loc_31><loc_34><loc_32><loc_34></location>e</text>
<text><location><page_49><loc_31><loc_33><loc_32><loc_34></location>h</text>
<text><location><page_49><loc_31><loc_33><loc_32><loc_33></location>t</text>
<text><location><page_49><loc_31><loc_32><loc_32><loc_33></location>n</text>
<text><location><page_49><loc_31><loc_32><loc_32><loc_32></location>e</text>
<text><location><page_49><loc_31><loc_32><loc_32><loc_32></location>r</text>
<text><location><page_49><loc_31><loc_31><loc_32><loc_32></location>a</text>
<text><location><page_49><loc_31><loc_31><loc_32><loc_31></location>p</text>
<text><location><page_49><loc_31><loc_30><loc_32><loc_30></location>n</text>
<text><location><page_49><loc_31><loc_30><loc_32><loc_30></location>i</text>
<text><location><page_49><loc_31><loc_29><loc_32><loc_29></location>d</text>
<text><location><page_49><loc_31><loc_28><loc_32><loc_29></location>e</text>
<text><location><page_49><loc_31><loc_28><loc_32><loc_28></location>t</text>
<text><location><page_49><loc_31><loc_28><loc_32><loc_28></location>a</text>
<text><location><page_49><loc_31><loc_27><loc_32><loc_28></location>c</text>
<text><location><page_49><loc_31><loc_27><loc_32><loc_27></location>i</text>
<text><location><page_49><loc_31><loc_27><loc_32><loc_27></location>d</text>
<text><location><page_49><loc_31><loc_26><loc_32><loc_27></location>n</text>
<text><location><page_49><loc_31><loc_26><loc_32><loc_26></location>i</text>
<text><location><page_49><loc_31><loc_25><loc_32><loc_26></location>s</text>
<text><location><page_49><loc_31><loc_25><loc_32><loc_25></location>i</text>
<text><location><page_49><loc_31><loc_24><loc_32><loc_25></location>a</text>
<text><location><page_49><loc_31><loc_24><loc_32><loc_24></location>t</text>
<text><location><page_49><loc_31><loc_24><loc_32><loc_24></location>a</text>
<text><location><page_49><loc_31><loc_23><loc_32><loc_24></location>d</text>
<text><location><page_49><loc_31><loc_22><loc_32><loc_23></location>e</text>
<text><location><page_49><loc_31><loc_22><loc_32><loc_22></location>h</text>
<text><location><page_49><loc_31><loc_22><loc_32><loc_22></location>t</text>
<text><location><page_49><loc_31><loc_21><loc_32><loc_21></location>n</text>
<text><location><page_49><loc_31><loc_20><loc_32><loc_21></location>i</text>
<text><location><page_49><loc_31><loc_20><loc_32><loc_21></location>a</text>
<text><location><page_49><loc_31><loc_20><loc_32><loc_20></location>t</text>
<text><location><page_49><loc_31><loc_19><loc_32><loc_20></location>b</text>
<text><location><page_49><loc_31><loc_19><loc_32><loc_19></location>o</text>
<text><location><page_49><loc_31><loc_18><loc_32><loc_19></location>o</text>
<text><location><page_49><loc_31><loc_18><loc_32><loc_18></location>t</text>
<text><location><page_49><loc_31><loc_17><loc_32><loc_17></location>d</text>
<text><location><page_49><loc_31><loc_17><loc_32><loc_17></location>e</text>
<text><location><page_49><loc_31><loc_16><loc_32><loc_17></location>s</text>
<text><location><page_49><loc_31><loc_16><loc_32><loc_16></location>u</text>
<text><location><page_49><loc_31><loc_15><loc_32><loc_16></location>t</text>
<text><location><page_49><loc_31><loc_15><loc_32><loc_15></location>n</text>
<text><location><page_49><loc_31><loc_14><loc_32><loc_15></location>e</text>
<text><location><page_49><loc_31><loc_14><loc_32><loc_14></location>m</text>
<text><location><page_49><loc_31><loc_13><loc_32><loc_14></location>u</text>
<text><location><page_49><loc_31><loc_13><loc_32><loc_13></location>r</text>
<text><location><page_49><loc_31><loc_13><loc_32><loc_13></location>t</text>
<text><location><page_49><loc_31><loc_12><loc_32><loc_13></location>s</text>
<text><location><page_49><loc_31><loc_12><loc_32><loc_12></location>n</text>
<text><location><page_49><loc_31><loc_11><loc_32><loc_12></location>i</text>
<text><location><page_49><loc_31><loc_11><loc_32><loc_11></location>e</text>
<text><location><page_49><loc_31><loc_10><loc_32><loc_11></location>h</text>
<text><location><page_49><loc_31><loc_10><loc_32><loc_10></location>T</text>
<text><location><page_49><loc_31><loc_9><loc_32><loc_9></location>)</text>
<text><location><page_49><loc_31><loc_9><loc_32><loc_9></location>a</text>
<text><location><page_49><loc_31><loc_8><loc_32><loc_9></location>(</text>
<text><location><page_49><loc_17><loc_63><loc_18><loc_64></location>1</text>
<text><location><page_49><loc_17><loc_63><loc_18><loc_63></location>V</text>
<text><location><page_49><loc_17><loc_62><loc_18><loc_63></location>∆</text>
<text><location><page_49><loc_17><loc_58><loc_18><loc_59></location>1</text>
<text><location><page_49><loc_17><loc_57><loc_18><loc_58></location>V</text>
<text><location><page_49><loc_17><loc_55><loc_18><loc_55></location>1</text>
<text><location><page_49><loc_17><loc_54><loc_18><loc_55></location>A</text>
<text><location><page_49><loc_17><loc_53><loc_18><loc_54></location>∆</text>
<text><location><page_49><loc_17><loc_50><loc_18><loc_51></location>1</text>
<text><location><page_49><loc_17><loc_50><loc_18><loc_50></location>A</text>
<text><location><page_49><loc_17><loc_45><loc_18><loc_45></location>0</text>
<text><location><page_49><loc_17><loc_44><loc_18><loc_45></location>σ</text>
<text><location><page_49><loc_17><loc_43><loc_18><loc_44></location>∆</text>
<text><location><page_49><loc_17><loc_40><loc_18><loc_40></location>0</text>
<text><location><page_49><loc_17><loc_39><loc_18><loc_40></location>σ</text>
<text><location><page_49><loc_17><loc_36><loc_18><loc_36></location>0</text>
<text><location><page_49><loc_17><loc_35><loc_18><loc_36></location>V</text>
<text><location><page_49><loc_17><loc_34><loc_18><loc_35></location>∆</text>
<text><location><page_49><loc_17><loc_30><loc_18><loc_31></location>0</text>
<text><location><page_49><loc_17><loc_30><loc_18><loc_30></location>V</text>
<text><location><page_49><loc_17><loc_27><loc_18><loc_27></location>0</text>
<text><location><page_49><loc_17><loc_26><loc_18><loc_27></location>A</text>
<text><location><page_49><loc_17><loc_25><loc_18><loc_26></location>∆</text>
<text><location><page_49><loc_17><loc_22><loc_18><loc_23></location>0</text>
<text><location><page_49><loc_17><loc_22><loc_18><loc_22></location>A</text>
<text><location><page_49><loc_17><loc_12><loc_18><loc_12></location>)</text>
<text><location><page_49><loc_17><loc_11><loc_18><loc_12></location>a</text>
<text><location><page_49><loc_17><loc_11><loc_18><loc_11></location>(</text>
<text><location><page_49><loc_17><loc_10><loc_18><loc_10></location>e</text>
<text><location><page_49><loc_17><loc_9><loc_18><loc_10></location>m</text>
<text><location><page_49><loc_17><loc_9><loc_18><loc_9></location>a</text>
<text><location><page_49><loc_17><loc_8><loc_18><loc_9></location>N</text>
<text><location><page_49><loc_18><loc_91><loc_20><loc_92></location>x</text>
<text><location><page_49><loc_18><loc_91><loc_20><loc_91></location>u</text>
<text><location><page_49><loc_18><loc_90><loc_20><loc_91></location>fl</text>
<text><location><page_49><loc_18><loc_72><loc_20><loc_73></location>s</text>
<text><location><page_49><loc_18><loc_64><loc_19><loc_65></location>1</text>
<text><location><page_49><loc_18><loc_63><loc_19><loc_64></location>-</text>
<text><location><page_49><loc_18><loc_63><loc_20><loc_63></location>s</text>
<text><location><page_49><loc_18><loc_62><loc_20><loc_63></location>m</text>
<text><location><page_49><loc_18><loc_61><loc_20><loc_62></location>k</text>
<text><location><page_49><loc_18><loc_59><loc_19><loc_60></location>1</text>
<text><location><page_49><loc_18><loc_59><loc_19><loc_59></location>-</text>
<text><location><page_49><loc_18><loc_58><loc_20><loc_59></location>s</text>
<text><location><page_49><loc_18><loc_57><loc_20><loc_58></location>m</text>
<text><location><page_49><loc_18><loc_56><loc_20><loc_57></location>k</text>
<text><location><page_49><loc_18><loc_54><loc_20><loc_55></location>x</text>
<text><location><page_49><loc_18><loc_54><loc_20><loc_54></location>u</text>
<text><location><page_49><loc_18><loc_53><loc_20><loc_54></location>fl</text>
<text><location><page_49><loc_18><loc_50><loc_20><loc_51></location>x</text>
<text><location><page_49><loc_18><loc_50><loc_20><loc_50></location>u</text>
<text><location><page_49><loc_18><loc_49><loc_20><loc_50></location>fl</text>
<text><location><page_49><loc_18><loc_46><loc_19><loc_46></location>1</text>
<text><location><page_49><loc_18><loc_45><loc_19><loc_46></location>-</text>
<text><location><page_49><loc_18><loc_44><loc_20><loc_45></location>s</text>
<text><location><page_49><loc_18><loc_43><loc_20><loc_44></location>m</text>
<text><location><page_49><loc_18><loc_43><loc_20><loc_43></location>k</text>
<text><location><page_49><loc_18><loc_41><loc_19><loc_41></location>1</text>
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<text><location><page_49><loc_33><loc_54><loc_34><loc_54></location>u</text>
<text><location><page_49><loc_33><loc_53><loc_34><loc_54></location>o</text>
<text><location><page_49><loc_33><loc_53><loc_34><loc_53></location>f</text>
<text><location><page_49><loc_33><loc_52><loc_34><loc_53></location>a</text>
<text><location><page_49><loc_33><loc_52><loc_34><loc_52></location>r</text>
<text><location><page_49><loc_33><loc_51><loc_34><loc_52></location>e</text>
<text><location><page_49><loc_33><loc_51><loc_34><loc_51></location>h</text>
<text><location><page_49><loc_33><loc_50><loc_34><loc_51></location>t</text>
<text><location><page_49><loc_33><loc_50><loc_34><loc_50></location>i</text>
<text><location><page_49><loc_33><loc_50><loc_34><loc_50></location>e</text>
<text><location><page_49><loc_33><loc_49><loc_34><loc_50></location>n</text>
<text><location><page_49><loc_33><loc_49><loc_34><loc_49></location>d</text>
<text><location><page_49><loc_33><loc_48><loc_34><loc_49></location>n</text>
<text><location><page_49><loc_33><loc_48><loc_34><loc_48></location>a</text>
<text><location><page_49><loc_33><loc_47><loc_34><loc_47></location>e</text>
<text><location><page_49><loc_33><loc_47><loc_34><loc_47></location>n</text>
<text><location><page_49><loc_33><loc_46><loc_34><loc_47></location>i</text>
<text><location><page_49><loc_33><loc_46><loc_34><loc_46></location>l</text>
<text><location><page_49><loc_33><loc_45><loc_34><loc_46></location>e</text>
<text><location><page_49><loc_33><loc_45><loc_34><loc_45></location>u</text>
<text><location><page_49><loc_33><loc_45><loc_34><loc_45></location>l</text>
<text><location><page_49><loc_33><loc_44><loc_34><loc_45></location>b</text>
<text><location><page_49><loc_33><loc_43><loc_34><loc_44></location>e</text>
<text><location><page_49><loc_33><loc_43><loc_34><loc_43></location>h</text>
<text><location><page_49><loc_33><loc_43><loc_34><loc_43></location>t</text>
<text><location><page_49><loc_33><loc_42><loc_34><loc_42></location>f</text>
<text><location><page_49><loc_33><loc_42><loc_34><loc_42></location>o</text>
<text><location><page_49><loc_33><loc_41><loc_34><loc_41></location>g</text>
<text><location><page_49><loc_33><loc_40><loc_34><loc_41></location>n</text>
<text><location><page_49><loc_33><loc_40><loc_34><loc_40></location>i</text>
<text><location><page_49><loc_33><loc_40><loc_34><loc_40></location>w</text>
<text><location><page_49><loc_33><loc_39><loc_34><loc_39></location>d</text>
<text><location><page_49><loc_33><loc_38><loc_34><loc_39></location>e</text>
<text><location><page_49><loc_33><loc_38><loc_34><loc_38></location>r</text>
<text><location><page_49><loc_33><loc_37><loc_34><loc_38></location>e</text>
<text><location><page_49><loc_33><loc_37><loc_34><loc_37></location>h</text>
<text><location><page_49><loc_33><loc_37><loc_34><loc_37></location>t</text>
<text><location><page_49><loc_33><loc_36><loc_34><loc_36></location>n</text>
<text><location><page_49><loc_33><loc_35><loc_34><loc_36></location>a</text>
<text><location><page_49><loc_33><loc_35><loc_34><loc_35></location>h</text>
<text><location><page_49><loc_33><loc_35><loc_34><loc_35></location>t</text>
<text><location><page_49><loc_33><loc_34><loc_34><loc_34></location>t</text>
<text><location><page_49><loc_33><loc_34><loc_34><loc_34></location>n</text>
<text><location><page_49><loc_33><loc_33><loc_34><loc_34></location>e</text>
<text><location><page_49><loc_33><loc_33><loc_34><loc_33></location>r</text>
<text><location><page_49><loc_33><loc_32><loc_34><loc_33></location>e</text>
<text><location><page_49><loc_33><loc_32><loc_34><loc_32></location>f</text>
<text><location><page_49><loc_33><loc_32><loc_34><loc_32></location>f</text>
<text><location><page_49><loc_33><loc_32><loc_34><loc_32></location>i</text>
<text><location><page_49><loc_33><loc_31><loc_34><loc_32></location>d</text>
<text><location><page_49><loc_33><loc_31><loc_34><loc_31></location>s</text>
<text><location><page_49><loc_33><loc_30><loc_34><loc_30></location>i</text>
<text><location><page_49><loc_33><loc_30><loc_34><loc_30></location>e</text>
<text><location><page_49><loc_33><loc_29><loc_34><loc_30></location>n</text>
<text><location><page_49><loc_33><loc_29><loc_34><loc_29></location>i</text>
<text><location><page_49><loc_33><loc_29><loc_34><loc_29></location>l</text>
<text><location><page_49><loc_33><loc_28><loc_34><loc_28></location>d</text>
<text><location><page_49><loc_33><loc_27><loc_34><loc_28></location>e</text>
<text><location><page_49><loc_33><loc_27><loc_34><loc_27></location>r</text>
<text><location><page_49><loc_33><loc_26><loc_34><loc_27></location>e</text>
<text><location><page_49><loc_33><loc_26><loc_34><loc_26></location>h</text>
<text><location><page_49><loc_33><loc_26><loc_34><loc_26></location>t</text>
<text><location><page_49><loc_33><loc_25><loc_34><loc_25></location>f</text>
<text><location><page_49><loc_33><loc_25><loc_34><loc_25></location>o</text>
<text><location><page_49><loc_33><loc_24><loc_34><loc_24></location>g</text>
<text><location><page_49><loc_33><loc_23><loc_34><loc_24></location>n</text>
<text><location><page_49><loc_33><loc_23><loc_34><loc_23></location>i</text>
<text><location><page_49><loc_33><loc_23><loc_34><loc_23></location>w</text>
<text><location><page_49><loc_33><loc_22><loc_34><loc_22></location>d</text>
<text><location><page_49><loc_33><loc_21><loc_34><loc_22></location>e</text>
<text><location><page_49><loc_33><loc_21><loc_34><loc_21></location>r</text>
<text><location><page_49><loc_33><loc_20><loc_34><loc_21></location>e</text>
<text><location><page_49><loc_33><loc_20><loc_34><loc_20></location>h</text>
<text><location><page_49><loc_33><loc_20><loc_34><loc_20></location>t</text>
<text><location><page_49><loc_33><loc_19><loc_34><loc_19></location>r</text>
<text><location><page_49><loc_33><loc_18><loc_34><loc_19></location>u</text>
<text><location><page_49><loc_33><loc_18><loc_34><loc_18></location>A</text>
<text><location><page_49><loc_33><loc_17><loc_34><loc_18></location>W</text>
<text><location><page_49><loc_33><loc_16><loc_34><loc_17></location>R</text>
<text><location><page_49><loc_33><loc_15><loc_34><loc_16></location>d</text>
<text><location><page_49><loc_33><loc_15><loc_34><loc_15></location>n</text>
<text><location><page_49><loc_33><loc_14><loc_34><loc_15></location>a</text>
<text><location><page_49><loc_33><loc_14><loc_34><loc_14></location>a</text>
<text><location><page_49><loc_33><loc_13><loc_34><loc_14></location>h</text>
<text><location><page_49><loc_33><loc_13><loc_34><loc_13></location>C</text>
<text><location><page_49><loc_33><loc_12><loc_34><loc_12></location>X</text>
<text><location><page_49><loc_33><loc_11><loc_34><loc_12></location>D</text>
<text><location><page_49><loc_33><loc_11><loc_34><loc_11></location>r</text>
<text><location><page_49><loc_33><loc_10><loc_34><loc_11></location>o</text>
<text><location><page_49><loc_33><loc_10><loc_34><loc_10></location>F</text>
<text><location><page_49><loc_33><loc_9><loc_34><loc_9></location>)</text>
<text><location><page_49><loc_33><loc_9><loc_34><loc_9></location>c</text>
<text><location><page_49><loc_33><loc_8><loc_34><loc_9></location>(</text>
<text><location><page_49><loc_38><loc_93><loc_61><loc_94></location>Hot lines in T Tauri stars</text>
<text><location><page_49><loc_90><loc_93><loc_92><loc_94></location>49</text>
<text><location><page_50><loc_9><loc_93><loc_10><loc_94></location>50</text>
<text><location><page_50><loc_45><loc_93><loc_56><loc_94></location>Ardila et al.</text>
<text><location><page_50><loc_10><loc_54><loc_11><loc_54></location>s</text>
<text><location><page_50><loc_10><loc_53><loc_11><loc_54></location>t</text>
<text><location><page_50><loc_10><loc_53><loc_11><loc_53></location>fi</text>
<text><location><page_50><loc_10><loc_52><loc_11><loc_52></location>n</text>
<text><location><page_50><loc_10><loc_51><loc_11><loc_52></location>a</text>
<text><location><page_50><loc_10><loc_51><loc_11><loc_51></location>i</text>
<text><location><page_50><loc_10><loc_50><loc_11><loc_51></location>s</text>
<text><location><page_50><loc_10><loc_50><loc_11><loc_50></location>s</text>
<text><location><page_50><loc_10><loc_49><loc_11><loc_50></location>u</text>
<text><location><page_50><loc_10><loc_49><loc_11><loc_49></location>a</text>
<text><location><page_50><loc_10><loc_48><loc_11><loc_49></location>G</text>
<text><location><page_50><loc_10><loc_47><loc_11><loc_48></location>i</text>
<text><location><page_50><loc_10><loc_47><loc_11><loc_47></location>i</text>
<text><location><page_50><loc_10><loc_46><loc_11><loc_47></location>e</text>
<text><location><page_50><loc_10><loc_45><loc_11><loc_46></location>H</text>
<table>
<location><page_50><loc_19><loc_22><loc_80><loc_88></location>
</table>
<text><location><page_50><loc_46><loc_22><loc_48><loc_22></location>a</text>
<text><location><page_50><loc_20><loc_22><loc_21><loc_23></location>T</text>
<text><location><page_50><loc_22><loc_22><loc_23><loc_22></location>S</text>
<text><location><page_50><loc_23><loc_22><loc_24><loc_22></location>T</text>
<text><location><page_50><loc_24><loc_22><loc_26><loc_23></location>C</text>
<text><location><page_50><loc_26><loc_22><loc_27><loc_23></location>T</text>
<text><location><page_50><loc_27><loc_22><loc_28><loc_22></location>T</text>
<text><location><page_50><loc_28><loc_22><loc_30><loc_22></location>T</text>
<text><location><page_50><loc_30><loc_22><loc_31><loc_23></location>T</text>
<text><location><page_50><loc_31><loc_22><loc_33><loc_23></location>T</text>
<text><location><page_50><loc_33><loc_22><loc_34><loc_23></location>T</text>
<text><location><page_50><loc_34><loc_22><loc_35><loc_22></location>T</text>
<text><location><page_50><loc_35><loc_22><loc_37><loc_23></location>C</text>
<text><location><page_50><loc_37><loc_22><loc_38><loc_22></location>C</text>
<text><location><page_50><loc_38><loc_22><loc_39><loc_22></location>C</text>
<text><location><page_50><loc_40><loc_22><loc_41><loc_23></location>T</text>
<text><location><page_50><loc_41><loc_22><loc_42><loc_23></location>M</text>
<text><location><page_50><loc_42><loc_22><loc_44><loc_22></location>L</text>
<text><location><page_50><loc_45><loc_22><loc_46><loc_22></location>A</text>
<text><location><page_50><loc_48><loc_22><loc_49><loc_22></location>9</text>
<text><location><page_50><loc_49><loc_22><loc_51><loc_22></location>4</text>
<text><location><page_50><loc_58><loc_22><loc_60><loc_23></location>a</text>
<text><location><page_50><loc_58><loc_21><loc_60><loc_22></location>T</text>
<text><location><page_50><loc_61><loc_22><loc_62><loc_23></location>T</text>
<text><location><page_50><loc_62><loc_22><loc_64><loc_22></location>7</text>
<text><location><page_50><loc_62><loc_21><loc_64><loc_22></location>0</text>
<text><location><page_50><loc_20><loc_20><loc_64><loc_22></location>A A A K B P C V C Y D E D F D K D N D R D S D X E P E T H N M P R U R W S U T T V 1 1 V 4 0 C S D M G M I P T W U X V 1</text>
<text><location><page_50><loc_70><loc_22><loc_71><loc_22></location>6</text>
<text><location><page_50><loc_70><loc_21><loc_71><loc_22></location>0</text>
<text><location><page_50><loc_70><loc_21><loc_71><loc_21></location>1</text>
<text><location><page_50><loc_70><loc_20><loc_71><loc_21></location>V</text>
<text><location><page_50><loc_72><loc_22><loc_73><loc_22></location>6</text>
<text><location><page_50><loc_72><loc_21><loc_73><loc_22></location>9</text>
<text><location><page_50><loc_72><loc_21><loc_73><loc_21></location>3</text>
<text><location><page_50><loc_72><loc_20><loc_73><loc_21></location>V</text>
<text><location><page_50><loc_73><loc_22><loc_74><loc_22></location>7</text>
<text><location><page_50><loc_73><loc_21><loc_74><loc_22></location>9</text>
<text><location><page_50><loc_73><loc_21><loc_74><loc_21></location>3</text>
<text><location><page_50><loc_73><loc_20><loc_74><loc_21></location>V</text>
<text><location><page_50><loc_74><loc_22><loc_76><loc_22></location>0</text>
<text><location><page_50><loc_74><loc_21><loc_76><loc_22></location>1</text>
<text><location><page_50><loc_74><loc_21><loc_76><loc_21></location>4</text>
<text><location><page_50><loc_74><loc_20><loc_76><loc_21></location>V</text>
<text><location><page_50><loc_80><loc_54><loc_81><loc_55></location>.</text>
<text><location><page_50><loc_80><loc_54><loc_81><loc_54></location>s</text>
<text><location><page_50><loc_80><loc_54><loc_81><loc_54></location>i</text>
<text><location><page_50><loc_80><loc_53><loc_81><loc_54></location>s</text>
<text><location><page_50><loc_80><loc_53><loc_81><loc_53></location>e</text>
<text><location><page_50><loc_80><loc_53><loc_81><loc_53></location>h</text>
<text><location><page_50><loc_80><loc_52><loc_81><loc_53></location>t</text>
<text><location><page_50><loc_80><loc_52><loc_81><loc_52></location>n</text>
<text><location><page_50><loc_80><loc_51><loc_81><loc_52></location>e</text>
<text><location><page_50><loc_80><loc_51><loc_81><loc_51></location>r</text>
<text><location><page_50><loc_80><loc_51><loc_81><loc_51></location>a</text>
<text><location><page_50><loc_80><loc_50><loc_81><loc_51></location>p</text>
<text><location><page_50><loc_80><loc_49><loc_81><loc_50></location>n</text>
<text><location><page_50><loc_80><loc_49><loc_81><loc_49></location>i</text>
<text><location><page_50><loc_80><loc_48><loc_81><loc_49></location>d</text>
<text><location><page_50><loc_80><loc_48><loc_81><loc_48></location>e</text>
<text><location><page_50><loc_80><loc_48><loc_81><loc_48></location>t</text>
<text><location><page_50><loc_80><loc_47><loc_81><loc_48></location>a</text>
<text><location><page_50><loc_80><loc_47><loc_81><loc_47></location>c</text>
<text><location><page_50><loc_80><loc_47><loc_81><loc_47></location>i</text>
<text><location><page_50><loc_80><loc_46><loc_81><loc_47></location>d</text>
<text><location><page_50><loc_80><loc_46><loc_81><loc_46></location>n</text>
<text><location><page_50><loc_80><loc_45><loc_81><loc_46></location>i</text>
<text><location><page_50><loc_80><loc_45><loc_81><loc_45></location>s</text>
<text><location><page_50><loc_80><loc_45><loc_81><loc_45></location>i</text>
<text><location><page_50><loc_80><loc_44><loc_81><loc_44></location>a</text>
<text><location><page_50><loc_80><loc_43><loc_81><loc_44></location>t</text>
<text><location><page_50><loc_80><loc_43><loc_81><loc_43></location>a</text>
<text><location><page_50><loc_80><loc_43><loc_81><loc_43></location>d</text>
<text><location><page_50><loc_80><loc_42><loc_81><loc_42></location>e</text>
<text><location><page_50><loc_80><loc_41><loc_81><loc_42></location>h</text>
<text><location><page_50><loc_80><loc_41><loc_81><loc_41></location>t</text>
<text><location><page_50><loc_80><loc_40><loc_81><loc_41></location>n</text>
<text><location><page_50><loc_80><loc_40><loc_81><loc_40></location>i</text>
<text><location><page_50><loc_80><loc_40><loc_81><loc_40></location>a</text>
<text><location><page_50><loc_80><loc_39><loc_81><loc_40></location>t</text>
<text><location><page_50><loc_80><loc_39><loc_81><loc_39></location>b</text>
<text><location><page_50><loc_80><loc_38><loc_81><loc_39></location>o</text>
<text><location><page_50><loc_80><loc_38><loc_81><loc_38></location>o</text>
<text><location><page_50><loc_80><loc_37><loc_81><loc_38></location>t</text>
<text><location><page_50><loc_80><loc_37><loc_81><loc_37></location>d</text>
<text><location><page_50><loc_80><loc_36><loc_81><loc_37></location>e</text>
<text><location><page_50><loc_80><loc_36><loc_81><loc_36></location>s</text>
<text><location><page_50><loc_80><loc_35><loc_81><loc_36></location>u</text>
<text><location><page_50><loc_80><loc_35><loc_81><loc_35></location>t</text>
<text><location><page_50><loc_80><loc_34><loc_81><loc_35></location>n</text>
<text><location><page_50><loc_80><loc_34><loc_81><loc_34></location>e</text>
<text><location><page_50><loc_80><loc_33><loc_81><loc_34></location>m</text>
<text><location><page_50><loc_80><loc_33><loc_81><loc_33></location>u</text>
<text><location><page_50><loc_80><loc_32><loc_81><loc_33></location>r</text>
<text><location><page_50><loc_80><loc_32><loc_81><loc_32></location>t</text>
<text><location><page_50><loc_80><loc_32><loc_81><loc_32></location>s</text>
<text><location><page_50><loc_80><loc_31><loc_81><loc_32></location>n</text>
<text><location><page_50><loc_80><loc_31><loc_81><loc_31></location>i</text>
<text><location><page_50><loc_80><loc_30><loc_81><loc_31></location>e</text>
<text><location><page_50><loc_80><loc_30><loc_81><loc_30></location>h</text>
<text><location><page_50><loc_80><loc_29><loc_81><loc_30></location>T</text>
<text><location><page_50><loc_80><loc_29><loc_81><loc_29></location>;</text>
<text><location><page_50><loc_80><loc_28><loc_81><loc_29></location>e</text>
<text><location><page_50><loc_80><loc_28><loc_81><loc_28></location>m</text>
<text><location><page_50><loc_80><loc_27><loc_81><loc_28></location>a</text>
<text><location><page_50><loc_80><loc_27><loc_81><loc_27></location>n</text>
<text><location><page_50><loc_80><loc_26><loc_81><loc_26></location>r</text>
<text><location><page_50><loc_80><loc_26><loc_81><loc_26></location>i</text>
<text><location><page_50><loc_80><loc_25><loc_81><loc_26></location>e</text>
<text><location><page_50><loc_80><loc_25><loc_81><loc_25></location>h</text>
<text><location><page_50><loc_80><loc_25><loc_81><loc_25></location>t</text>
<text><location><page_50><loc_80><loc_24><loc_81><loc_24></location>r</text>
<text><location><page_50><loc_80><loc_24><loc_81><loc_24></location>e</text>
<text><location><page_50><loc_80><loc_23><loc_81><loc_24></location>t</text>
<text><location><page_50><loc_80><loc_23><loc_81><loc_23></location>f</text>
<text><location><page_50><loc_80><loc_23><loc_81><loc_23></location>a</text>
<text><location><page_50><loc_80><loc_22><loc_81><loc_22></location>)</text>
<text><location><page_50><loc_80><loc_21><loc_81><loc_22></location>W</text>
<text><location><page_50><loc_80><loc_21><loc_81><loc_21></location>(</text>
<text><location><page_50><loc_80><loc_20><loc_81><loc_20></location>a</text>
<text><location><page_50><loc_80><loc_19><loc_81><loc_20></location>h</text>
<text><location><page_50><loc_80><loc_19><loc_81><loc_19></location>t</text>
<text><location><page_50><loc_80><loc_19><loc_81><loc_19></location>i</text>
<text><location><page_50><loc_80><loc_18><loc_81><loc_19></location>w</text>
<text><location><page_50><loc_80><loc_17><loc_81><loc_18></location>d</text>
<text><location><page_50><loc_80><loc_17><loc_81><loc_17></location>e</text>
<text><location><page_50><loc_80><loc_17><loc_81><loc_17></location>t</text>
<text><location><page_50><loc_80><loc_16><loc_81><loc_17></location>a</text>
<text><location><page_50><loc_80><loc_16><loc_81><loc_16></location>c</text>
<text><location><page_50><loc_80><loc_15><loc_81><loc_16></location>i</text>
<text><location><page_50><loc_80><loc_15><loc_81><loc_15></location>d</text>
<text><location><page_50><loc_80><loc_15><loc_81><loc_15></location>n</text>
<text><location><page_50><loc_80><loc_14><loc_81><loc_15></location>i</text>
<text><location><page_50><loc_80><loc_14><loc_81><loc_14></location>e</text>
<text><location><page_50><loc_80><loc_13><loc_81><loc_14></location>r</text>
<text><location><page_50><loc_80><loc_13><loc_81><loc_13></location>a</text>
<text><location><page_50><loc_80><loc_12><loc_81><loc_13></location>s</text>
<text><location><page_50><loc_80><loc_12><loc_81><loc_12></location>S</text>
<text><location><page_50><loc_80><loc_11><loc_81><loc_12></location>T</text>
<text><location><page_50><loc_80><loc_11><loc_81><loc_11></location>T</text>
<text><location><page_50><loc_80><loc_10><loc_81><loc_11></location>W</text>
<text><location><page_50><loc_80><loc_9><loc_81><loc_9></location>)</text>
<text><location><page_50><loc_80><loc_9><loc_81><loc_9></location>a</text>
<text><location><page_50><loc_80><loc_8><loc_81><loc_9></location>(</text>
<text><location><page_50><loc_54><loc_22><loc_55><loc_22></location>C</text>
<text><location><page_50><loc_8><loc_51><loc_9><loc_51></location>7</text>
<text><location><page_50><loc_8><loc_50><loc_9><loc_50></location>e</text>
<text><location><page_50><loc_8><loc_50><loc_9><loc_50></location>l</text>
<text><location><page_50><loc_8><loc_49><loc_9><loc_50></location>b</text>
<text><location><page_50><loc_8><loc_48><loc_9><loc_49></location>a</text>
<text><location><page_50><loc_8><loc_47><loc_9><loc_48></location>T</text>
<text><location><page_50><loc_13><loc_77><loc_14><loc_77></location>1</text>
<text><location><page_50><loc_13><loc_76><loc_14><loc_77></location>σ</text>
<text><location><page_50><loc_13><loc_71><loc_14><loc_71></location>1</text>
<text><location><page_50><loc_13><loc_70><loc_14><loc_71></location>V</text>
<text><location><page_50><loc_13><loc_62><loc_14><loc_62></location>1</text>
<text><location><page_50><loc_13><loc_61><loc_14><loc_62></location>A</text>
<text><location><page_50><loc_13><loc_53><loc_14><loc_53></location>0</text>
<text><location><page_50><loc_13><loc_52><loc_14><loc_53></location>σ</text>
<text><location><page_50><loc_13><loc_47><loc_14><loc_47></location>0</text>
<text><location><page_50><loc_13><loc_46><loc_14><loc_47></location>V</text>
<text><location><page_50><loc_13><loc_37><loc_14><loc_38></location>0</text>
<text><location><page_50><loc_13><loc_37><loc_14><loc_37></location>A</text>
<text><location><page_50><loc_13><loc_27><loc_14><loc_27></location>)</text>
<text><location><page_50><loc_13><loc_26><loc_14><loc_27></location>a</text>
<text><location><page_50><loc_13><loc_26><loc_14><loc_26></location>(</text>
<text><location><page_50><loc_13><loc_25><loc_14><loc_26></location>e</text>
<text><location><page_50><loc_13><loc_24><loc_14><loc_25></location>m</text>
<text><location><page_50><loc_13><loc_24><loc_14><loc_24></location>a</text>
<text><location><page_50><loc_13><loc_23><loc_14><loc_24></location>N</text>
<text><location><page_50><loc_14><loc_78><loc_15><loc_79></location>1</text>
<text><location><page_50><loc_14><loc_77><loc_15><loc_78></location>-</text>
<text><location><page_50><loc_14><loc_77><loc_16><loc_77></location>s</text>
<text><location><page_50><loc_14><loc_76><loc_16><loc_77></location>m</text>
<text><location><page_50><loc_14><loc_75><loc_16><loc_76></location>k</text>
<text><location><page_50><loc_14><loc_72><loc_15><loc_72></location>1</text>
<text><location><page_50><loc_14><loc_71><loc_15><loc_72></location>-</text>
<text><location><page_50><loc_14><loc_71><loc_16><loc_71></location>s</text>
<text><location><page_50><loc_14><loc_70><loc_16><loc_70></location>m</text>
<text><location><page_50><loc_14><loc_69><loc_16><loc_70></location>k</text>
<text><location><page_50><loc_14><loc_67><loc_15><loc_67></location>2</text>
<text><location><page_50><loc_14><loc_66><loc_15><loc_67></location>-</text>
<text><location><page_50><loc_14><loc_65><loc_16><loc_66></location>m</text>
<text><location><page_50><loc_14><loc_65><loc_16><loc_65></location>c</text>
<text><location><page_50><loc_14><loc_64><loc_15><loc_64></location>1</text>
<text><location><page_50><loc_14><loc_63><loc_15><loc_64></location>-</text>
<text><location><page_50><loc_14><loc_62><loc_16><loc_63></location>c</text>
<text><location><page_50><loc_14><loc_62><loc_16><loc_62></location>e</text>
<text><location><page_50><loc_14><loc_62><loc_16><loc_62></location>s</text>
<text><location><page_50><loc_14><loc_61><loc_16><loc_61></location>s</text>
<text><location><page_50><loc_14><loc_60><loc_16><loc_61></location>g</text>
<text><location><page_50><loc_14><loc_60><loc_16><loc_60></location>r</text>
<text><location><page_50><loc_14><loc_59><loc_16><loc_60></location>e</text>
<text><location><page_50><loc_14><loc_58><loc_15><loc_59></location>4</text>
<text><location><page_50><loc_14><loc_58><loc_15><loc_58></location>1</text>
<text><location><page_50><loc_14><loc_57><loc_15><loc_58></location>-</text>
<text><location><page_50><loc_14><loc_57><loc_16><loc_57></location>0</text>
<text><location><page_50><loc_14><loc_56><loc_16><loc_57></location>1</text>
<text><location><page_50><loc_14><loc_54><loc_15><loc_54></location>1</text>
<text><location><page_50><loc_14><loc_53><loc_15><loc_54></location>-</text>
<text><location><page_50><loc_14><loc_53><loc_16><loc_53></location>s</text>
<text><location><page_50><loc_14><loc_51><loc_16><loc_52></location>m</text>
<text><location><page_50><loc_14><loc_51><loc_16><loc_51></location>k</text>
<text><location><page_50><loc_14><loc_48><loc_15><loc_48></location>1</text>
<text><location><page_50><loc_14><loc_47><loc_15><loc_48></location>-</text>
<text><location><page_50><loc_14><loc_47><loc_16><loc_47></location>s</text>
<text><location><page_50><loc_14><loc_45><loc_16><loc_46></location>m</text>
<text><location><page_50><loc_14><loc_45><loc_16><loc_45></location>k</text>
<text><location><page_50><loc_14><loc_42><loc_15><loc_43></location>2</text>
<text><location><page_50><loc_14><loc_42><loc_15><loc_42></location>-</text>
<text><location><page_50><loc_14><loc_41><loc_16><loc_42></location>m</text>
<text><location><page_50><loc_14><loc_40><loc_16><loc_41></location>c</text>
<text><location><page_50><loc_14><loc_39><loc_15><loc_40></location>1</text>
<text><location><page_50><loc_14><loc_39><loc_15><loc_39></location>-</text>
<text><location><page_50><loc_14><loc_38><loc_16><loc_39></location>c</text>
<text><location><page_50><loc_14><loc_38><loc_16><loc_38></location>e</text>
<text><location><page_50><loc_14><loc_37><loc_16><loc_38></location>s</text>
<text><location><page_50><loc_14><loc_36><loc_16><loc_37></location>s</text>
<text><location><page_50><loc_14><loc_36><loc_16><loc_36></location>g</text>
<text><location><page_50><loc_14><loc_35><loc_16><loc_36></location>r</text>
<text><location><page_50><loc_14><loc_35><loc_16><loc_35></location>e</text>
<text><location><page_50><loc_14><loc_34><loc_15><loc_35></location>4</text>
<text><location><page_50><loc_14><loc_34><loc_15><loc_34></location>1</text>
<text><location><page_50><loc_14><loc_33><loc_15><loc_34></location>-</text>
<text><location><page_50><loc_14><loc_32><loc_16><loc_33></location>0</text>
<text><location><page_50><loc_14><loc_32><loc_16><loc_32></location>1</text>
<text><location><page_50><loc_18><loc_55><loc_19><loc_55></location>s</text>
<text><location><page_50><loc_18><loc_54><loc_19><loc_55></location>r</text>
<text><location><page_50><loc_18><loc_54><loc_19><loc_54></location>a</text>
<text><location><page_50><loc_18><loc_53><loc_19><loc_54></location>t</text>
<text><location><page_50><loc_18><loc_53><loc_19><loc_53></location>S</text>
<text><location><page_50><loc_18><loc_52><loc_19><loc_52></location>i</text>
<text><location><page_50><loc_18><loc_52><loc_19><loc_52></location>r</text>
<text><location><page_50><loc_18><loc_51><loc_19><loc_52></location>u</text>
<text><location><page_50><loc_18><loc_50><loc_19><loc_51></location>a</text>
<text><location><page_50><loc_18><loc_50><loc_19><loc_51></location>T</text>
<text><location><page_50><loc_18><loc_49><loc_19><loc_49></location>T</text>
<text><location><page_50><loc_18><loc_48><loc_19><loc_48></location>l</text>
<text><location><page_50><loc_18><loc_47><loc_19><loc_48></location>a</text>
<text><location><page_50><loc_18><loc_47><loc_19><loc_47></location>c</text>
<text><location><page_50><loc_18><loc_47><loc_19><loc_47></location>i</text>
<text><location><page_50><loc_18><loc_46><loc_19><loc_47></location>s</text>
<text><location><page_50><loc_18><loc_46><loc_19><loc_46></location>s</text>
<text><location><page_50><loc_18><loc_45><loc_19><loc_46></location>a</text>
<text><location><page_50><loc_18><loc_45><loc_19><loc_45></location>l</text>
<text><location><page_50><loc_18><loc_44><loc_19><loc_45></location>C</text>
<text><location><page_50><loc_67><loc_22><loc_69><loc_23></location>C</text>
<text><location><page_50><loc_67><loc_21><loc_69><loc_21></location>G</text>
<text><location><page_50><loc_67><loc_20><loc_69><loc_21></location>E</text>
<text><location><page_50><loc_69><loc_22><loc_70><loc_22></location>A</text>
<text><location><page_50><loc_69><loc_21><loc_70><loc_22></location>W</text>
<text><location><page_50><loc_69><loc_20><loc_70><loc_21></location>T</text>
<text><location><page_50><loc_79><loc_22><loc_80><loc_22></location>S</text>
<text><location><page_50><loc_79><loc_21><loc_80><loc_22></location>K</text>
<text><location><page_50><loc_79><loc_20><loc_80><loc_21></location>A</text>
<text><location><page_50><loc_79><loc_20><loc_80><loc_20></location>f</text>
<text><location><page_50><loc_79><loc_19><loc_80><loc_20></location>o</text>
<text><location><page_50><loc_79><loc_19><loc_80><loc_19></location>n</text>
<text><location><page_50><loc_79><loc_18><loc_80><loc_19></location>o</text>
<text><location><page_50><loc_79><loc_18><loc_80><loc_18></location>i</text>
<text><location><page_50><loc_79><loc_17><loc_80><loc_18></location>g</text>
<text><location><page_50><loc_79><loc_17><loc_80><loc_17></location>e</text>
<text><location><page_50><loc_79><loc_17><loc_80><loc_17></location>r</text>
<text><location><page_50><loc_79><loc_16><loc_80><loc_16></location>i</text>
<text><location><page_50><loc_79><loc_16><loc_80><loc_16></location>i</text>
<text><location><page_50><loc_79><loc_15><loc_80><loc_16></location>e</text>
<text><location><page_50><loc_79><loc_15><loc_80><loc_15></location>H</text>
<text><location><page_50><loc_79><loc_14><loc_80><loc_14></location>e</text>
<text><location><page_50><loc_79><loc_14><loc_80><loc_14></location>h</text>
<text><location><page_50><loc_79><loc_13><loc_80><loc_14></location>T</text>
<text><location><page_50><loc_79><loc_12><loc_80><loc_13></location>-</text>
<text><location><page_50><loc_79><loc_11><loc_80><loc_11></location>.</text>
<text><location><page_50><loc_79><loc_11><loc_80><loc_11></location>e</text>
<text><location><page_50><loc_79><loc_10><loc_80><loc_11></location>t</text>
<text><location><page_50><loc_79><loc_10><loc_80><loc_10></location>o</text>
<text><location><page_50><loc_79><loc_9><loc_80><loc_10></location>N</text>
<text><location><page_51><loc_8><loc_51><loc_9><loc_51></location>8</text>
<text><location><page_51><loc_8><loc_50><loc_9><loc_50></location>e</text>
<text><location><page_51><loc_8><loc_50><loc_9><loc_50></location>l</text>
<text><location><page_51><loc_8><loc_49><loc_9><loc_50></location>b</text>
<text><location><page_51><loc_8><loc_48><loc_9><loc_49></location>a</text>
<text><location><page_51><loc_8><loc_47><loc_9><loc_48></location>T</text>
<text><location><page_51><loc_13><loc_14><loc_14><loc_14></location>e</text>
<text><location><page_51><loc_13><loc_13><loc_14><loc_14></location>m</text>
<text><location><page_51><loc_13><loc_12><loc_14><loc_13></location>a</text>
<text><location><page_51><loc_13><loc_11><loc_14><loc_12></location>N</text>
<text><location><page_51><loc_14><loc_88><loc_15><loc_89></location>2</text>
<text><location><page_51><loc_15><loc_88><loc_15><loc_88></location>-</text>
<text><location><page_51><loc_80><loc_92><loc_81><loc_92></location>o</text>
<text><location><page_51><loc_80><loc_92><loc_81><loc_92></location>t</text>
<text><location><page_51><loc_80><loc_91><loc_81><loc_91></location>0</text>
<text><location><page_51><loc_80><loc_91><loc_81><loc_91></location>0</text>
<text><location><page_51><loc_80><loc_90><loc_81><loc_91></location>8</text>
<text><location><page_51><loc_80><loc_90><loc_81><loc_90></location>-</text>
<text><location><page_51><loc_80><loc_89><loc_81><loc_89></location>m</text>
<text><location><page_51><loc_80><loc_88><loc_81><loc_89></location>o</text>
<text><location><page_51><loc_80><loc_88><loc_81><loc_88></location>r</text>
<text><location><page_51><loc_80><loc_88><loc_81><loc_88></location>f</text>
<table>
<location><page_51><loc_10><loc_22><loc_90><loc_88></location>
</table>
<text><location><page_51><loc_13><loc_22><loc_14><loc_22></location>2</text>
<text><location><page_51><loc_13><loc_21><loc_14><loc_22></location>1</text>
<text><location><page_51><loc_13><loc_21><loc_14><loc_21></location>-</text>
<text><location><page_51><loc_15><loc_22><loc_16><loc_22></location>e</text>
<text><location><page_51><loc_15><loc_22><loc_16><loc_22></location>s</text>
<text><location><page_51><loc_15><loc_21><loc_16><loc_21></location>g</text>
<text><location><page_51><loc_21><loc_22><loc_22><loc_23></location>±</text>
<text><location><page_51><loc_21><loc_21><loc_22><loc_22></location>1</text>
<text><location><page_51><loc_21><loc_21><loc_22><loc_21></location>2</text>
<text><location><page_51><loc_22><loc_22><loc_23><loc_23></location>±</text>
<text><location><page_51><loc_22><loc_21><loc_23><loc_22></location>3</text>
<text><location><page_51><loc_22><loc_21><loc_23><loc_21></location>4</text>
<text><location><page_51><loc_22><loc_20><loc_23><loc_21></location>.</text>
<text><location><page_51><loc_22><loc_13><loc_23><loc_14></location>o</text>
<text><location><page_51><loc_22><loc_13><loc_23><loc_13></location>c</text>
<text><location><page_51><loc_22><loc_12><loc_23><loc_13></location>S</text>
<text><location><page_51><loc_22><loc_11><loc_23><loc_12></location>K</text>
<text><location><page_51><loc_22><loc_10><loc_23><loc_11></location>A</text>
<text><location><page_51><loc_23><loc_22><loc_24><loc_23></location>±</text>
<text><location><page_51><loc_23><loc_21><loc_25><loc_22></location>6</text>
<text><location><page_51><loc_23><loc_21><loc_25><loc_21></location>8</text>
<text><location><page_51><loc_25><loc_22><loc_26><loc_23></location>±</text>
<text><location><page_51><loc_25><loc_21><loc_26><loc_22></location>3</text>
<text><location><page_51><loc_25><loc_21><loc_26><loc_21></location>6</text>
<text><location><page_51><loc_25><loc_20><loc_26><loc_21></location>.</text>
<text><location><page_51><loc_25><loc_14><loc_26><loc_14></location>a</text>
<text><location><page_51><loc_25><loc_13><loc_26><loc_14></location>h</text>
<text><location><page_51><loc_25><loc_12><loc_26><loc_13></location>C</text>
<text><location><page_51><loc_25><loc_11><loc_26><loc_12></location>V</text>
<text><location><page_51><loc_25><loc_10><loc_26><loc_11></location>C</text>
<text><location><page_51><loc_26><loc_22><loc_27><loc_23></location>±</text>
<text><location><page_51><loc_26><loc_21><loc_27><loc_22></location>0</text>
<text><location><page_51><loc_26><loc_21><loc_27><loc_21></location>2</text>
<text><location><page_51><loc_26><loc_20><loc_27><loc_21></location>.</text>
<text><location><page_51><loc_26><loc_13><loc_27><loc_14></location>u</text>
<text><location><page_51><loc_26><loc_13><loc_27><loc_13></location>a</text>
<text><location><page_51><loc_26><loc_12><loc_27><loc_13></location>T</text>
<text><location><page_51><loc_26><loc_11><loc_27><loc_12></location>Y</text>
<text><location><page_51><loc_26><loc_10><loc_27><loc_11></location>C</text>
<text><location><page_51><loc_27><loc_22><loc_29><loc_23></location>±</text>
<text><location><page_51><loc_27><loc_21><loc_29><loc_22></location>5</text>
<text><location><page_51><loc_27><loc_21><loc_29><loc_21></location>7</text>
<text><location><page_51><loc_29><loc_22><loc_30><loc_23></location>±</text>
<text><location><page_51><loc_29><loc_21><loc_30><loc_22></location>0</text>
<text><location><page_51><loc_29><loc_21><loc_30><loc_21></location>8</text>
<text><location><page_51><loc_30><loc_22><loc_31><loc_23></location>±</text>
<text><location><page_51><loc_30><loc_21><loc_31><loc_22></location>5</text>
<text><location><page_51><loc_30><loc_21><loc_31><loc_21></location>4</text>
<text><location><page_51><loc_31><loc_22><loc_33><loc_23></location>±</text>
<text><location><page_51><loc_31><loc_21><loc_33><loc_22></location>1</text>
<text><location><page_51><loc_31><loc_21><loc_33><loc_21></location>0</text>
<text><location><page_51><loc_33><loc_22><loc_34><loc_23></location>±</text>
<text><location><page_51><loc_33><loc_21><loc_34><loc_22></location>7</text>
<text><location><page_51><loc_33><loc_21><loc_34><loc_21></location>8</text>
<text><location><page_51><loc_34><loc_22><loc_35><loc_23></location>±</text>
<text><location><page_51><loc_34><loc_21><loc_35><loc_22></location>4</text>
<text><location><page_51><loc_34><loc_21><loc_35><loc_21></location>8</text>
<text><location><page_51><loc_34><loc_20><loc_35><loc_21></location>.</text>
<text><location><page_51><loc_34><loc_13><loc_35><loc_14></location>u</text>
<text><location><page_51><loc_34><loc_13><loc_35><loc_13></location>a</text>
<text><location><page_51><loc_34><loc_12><loc_35><loc_13></location>T</text>
<text><location><page_51><loc_34><loc_11><loc_35><loc_12></location>S</text>
<text><location><page_51><loc_34><loc_10><loc_35><loc_11></location>D</text>
<text><location><page_51><loc_36><loc_22><loc_37><loc_23></location>±</text>
<text><location><page_51><loc_36><loc_21><loc_37><loc_22></location>2</text>
<text><location><page_51><loc_36><loc_21><loc_37><loc_22></location>9</text>
<text><location><page_51><loc_36><loc_21><loc_37><loc_21></location>.</text>
<text><location><page_51><loc_37><loc_22><loc_38><loc_23></location>±</text>
<text><location><page_51><loc_37><loc_21><loc_38><loc_22></location>3</text>
<text><location><page_51><loc_37><loc_21><loc_38><loc_21></location>8</text>
<text><location><page_51><loc_38><loc_93><loc_61><loc_94></location>Hot lines in T Tauri stars</text>
<text><location><page_51><loc_90><loc_93><loc_92><loc_94></location>51</text>
<text><location><page_51><loc_38><loc_22><loc_39><loc_23></location>±</text>
<text><location><page_51><loc_38><loc_21><loc_40><loc_22></location>1</text>
<text><location><page_51><loc_38><loc_21><loc_40><loc_21></location>1</text>
<text><location><page_51><loc_40><loc_22><loc_41><loc_23></location>±</text>
<text><location><page_51><loc_40><loc_21><loc_41><loc_22></location>7</text>
<text><location><page_51><loc_40><loc_21><loc_41><loc_21></location>3</text>
<text><location><page_51><loc_41><loc_22><loc_42><loc_23></location>±</text>
<text><location><page_51><loc_41><loc_21><loc_42><loc_22></location>8</text>
<text><location><page_51><loc_41><loc_21><loc_42><loc_21></location>6</text>
<text><location><page_51><loc_41><loc_20><loc_42><loc_21></location>.</text>
<text><location><page_51><loc_41><loc_14><loc_42><loc_14></location>s</text>
<text><location><page_51><loc_41><loc_13><loc_42><loc_14></location>u</text>
<text><location><page_51><loc_41><loc_12><loc_42><loc_13></location>M</text>
<text><location><page_51><loc_41><loc_11><loc_42><loc_12></location>P</text>
<text><location><page_51><loc_41><loc_10><loc_42><loc_11></location>M</text>
<text><location><page_51><loc_42><loc_22><loc_44><loc_23></location>±</text>
<text><location><page_51><loc_42><loc_21><loc_44><loc_22></location>4</text>
<text><location><page_51><loc_42><loc_21><loc_44><loc_21></location>7</text>
<text><location><page_51><loc_44><loc_22><loc_45><loc_23></location>±</text>
<text><location><page_51><loc_44><loc_21><loc_45><loc_22></location>2</text>
<text><location><page_51><loc_44><loc_21><loc_45><loc_21></location>1</text>
<text><location><page_51><loc_45><loc_22><loc_46><loc_23></location>±</text>
<text><location><page_51><loc_45><loc_21><loc_46><loc_22></location>3</text>
<text><location><page_51><loc_45><loc_21><loc_46><loc_21></location>6</text>
<text><location><page_51><loc_47><loc_22><loc_48><loc_23></location>±</text>
<text><location><page_51><loc_46><loc_21><loc_48><loc_22></location>4</text>
<text><location><page_51><loc_46><loc_21><loc_48><loc_21></location>1</text>
<text><location><page_51><loc_46><loc_20><loc_48><loc_21></location>.</text>
<text><location><page_51><loc_46><loc_14><loc_48><loc_14></location>N</text>
<text><location><page_51><loc_46><loc_13><loc_48><loc_13></location>u</text>
<text><location><page_51><loc_46><loc_12><loc_48><loc_13></location>a</text>
<text><location><page_51><loc_46><loc_11><loc_48><loc_12></location>T</text>
<text><location><page_51><loc_46><loc_10><loc_48><loc_11></location>T</text>
<text><location><page_51><loc_48><loc_22><loc_49><loc_23></location>±</text>
<text><location><page_51><loc_48><loc_21><loc_49><loc_22></location>0</text>
<text><location><page_51><loc_48><loc_21><loc_49><loc_21></location>0</text>
<text><location><page_51><loc_49><loc_22><loc_50><loc_23></location>±</text>
<text><location><page_51><loc_49><loc_22><loc_50><loc_22></location>6</text>
<text><location><page_51><loc_49><loc_21><loc_50><loc_22></location>6</text>
<text><location><page_51><loc_49><loc_21><loc_50><loc_21></location>.</text>
<text><location><page_51><loc_54><loc_22><loc_55><loc_23></location>±</text>
<text><location><page_51><loc_54><loc_21><loc_55><loc_22></location>1</text>
<text><location><page_51><loc_54><loc_21><loc_55><loc_21></location>9</text>
<text><location><page_51><loc_55><loc_22><loc_57><loc_23></location>±</text>
<text><location><page_51><loc_55><loc_21><loc_57><loc_22></location>3</text>
<text><location><page_51><loc_55><loc_21><loc_57><loc_21></location>5</text>
<text><location><page_51><loc_57><loc_22><loc_58><loc_23></location>±</text>
<text><location><page_51><loc_57><loc_21><loc_58><loc_22></location>3</text>
<text><location><page_51><loc_57><loc_21><loc_58><loc_21></location>8</text>
<text><location><page_51><loc_58><loc_22><loc_59><loc_23></location>±</text>
<text><location><page_51><loc_58><loc_21><loc_59><loc_22></location>1</text>
<text><location><page_51><loc_58><loc_21><loc_59><loc_21></location>1</text>
<text><location><page_51><loc_60><loc_22><loc_61><loc_23></location>±</text>
<text><location><page_51><loc_59><loc_22><loc_61><loc_22></location>0</text>
<text><location><page_51><loc_59><loc_21><loc_61><loc_22></location>3</text>
<text><location><page_51><loc_59><loc_21><loc_61><loc_21></location>.</text>
<text><location><page_51><loc_61><loc_22><loc_62><loc_23></location>±</text>
<text><location><page_51><loc_61><loc_21><loc_62><loc_22></location>9</text>
<text><location><page_51><loc_61><loc_21><loc_62><loc_21></location>4</text>
<text><location><page_51><loc_62><loc_22><loc_63><loc_23></location>±</text>
<text><location><page_51><loc_62><loc_21><loc_64><loc_22></location>5</text>
<text><location><page_51><loc_62><loc_21><loc_64><loc_21></location>8</text>
<text><location><page_51><loc_67><loc_22><loc_68><loc_23></location>±</text>
<text><location><page_51><loc_67><loc_21><loc_68><loc_22></location>8</text>
<text><location><page_51><loc_67><loc_21><loc_68><loc_21></location>4</text>
<text><location><page_51><loc_69><loc_22><loc_70><loc_23></location>±</text>
<text><location><page_51><loc_68><loc_21><loc_70><loc_22></location>9</text>
<text><location><page_51><loc_68><loc_21><loc_70><loc_21></location>5</text>
<text><location><page_51><loc_70><loc_22><loc_71><loc_23></location>±</text>
<text><location><page_51><loc_70><loc_21><loc_71><loc_22></location>3</text>
<text><location><page_51><loc_70><loc_21><loc_71><loc_21></location>1</text>
<text><location><page_51><loc_71><loc_22><loc_72><loc_23></location>±</text>
<text><location><page_51><loc_71><loc_21><loc_73><loc_22></location>5</text>
<text><location><page_51><loc_71><loc_21><loc_73><loc_21></location>3</text>
<text><location><page_51><loc_73><loc_22><loc_74><loc_23></location>±</text>
<text><location><page_51><loc_73><loc_21><loc_74><loc_22></location>5</text>
<text><location><page_51><loc_73><loc_21><loc_74><loc_21></location>9</text>
<text><location><page_51><loc_74><loc_22><loc_75><loc_23></location>±</text>
<text><location><page_51><loc_74><loc_21><loc_75><loc_22></location>0</text>
<text><location><page_51><loc_74><loc_21><loc_75><loc_21></location>3</text>
<text><location><page_51><loc_74><loc_20><loc_75><loc_21></location>.</text>
<text><location><page_51><loc_74><loc_14><loc_75><loc_15></location>u</text>
<text><location><page_51><loc_74><loc_14><loc_75><loc_14></location>a</text>
<text><location><page_51><loc_74><loc_13><loc_75><loc_14></location>T</text>
<text><location><page_51><loc_74><loc_12><loc_75><loc_13></location>0</text>
<text><location><page_51><loc_74><loc_12><loc_75><loc_12></location>1</text>
<text><location><page_51><loc_74><loc_11><loc_75><loc_12></location>4</text>
<text><location><page_51><loc_74><loc_10><loc_75><loc_11></location>V</text>
<text><location><page_51><loc_83><loc_22><loc_84><loc_23></location>w</text>
<text><location><page_51><loc_83><loc_21><loc_84><loc_22></location>A</text>
<text><location><page_51><loc_83><loc_21><loc_84><loc_21></location>r</text>
<text><location><page_51><loc_80><loc_22><loc_81><loc_22></location>r</text>
<text><location><page_51><loc_80><loc_22><loc_81><loc_22></location>o</text>
<text><location><page_51><loc_80><loc_21><loc_81><loc_22></location>f</text>
<text><location><page_51><loc_80><loc_20><loc_81><loc_21></location>d</text>
<text><location><page_51><loc_80><loc_18><loc_81><loc_18></location>c</text>
<text><location><page_51><loc_80><loc_17><loc_81><loc_17></location>t</text>
<text><location><page_51><loc_80><loc_16><loc_81><loc_17></location>o</text>
<text><location><page_51><loc_80><loc_16><loc_81><loc_16></location>n</text>
<text><location><page_51><loc_80><loc_15><loc_81><loc_16></location>s</text>
<text><location><page_51><loc_80><loc_15><loc_81><loc_15></location>e</text>
<text><location><page_51><loc_80><loc_15><loc_81><loc_15></location>x</text>
<text><location><page_51><loc_80><loc_14><loc_81><loc_15></location>u</text>
<text><location><page_51><loc_80><loc_14><loc_81><loc_14></location>l</text>
<text><location><page_51><loc_80><loc_14><loc_81><loc_14></location>f</text>
<text><location><page_51><loc_80><loc_13><loc_81><loc_13></location>d</text>
<text><location><page_51><loc_80><loc_12><loc_81><loc_13></location>e</text>
<text><location><page_51><loc_80><loc_12><loc_81><loc_12></location>r</text>
<text><location><page_51><loc_80><loc_12><loc_81><loc_12></location>u</text>
<text><location><page_51><loc_80><loc_11><loc_81><loc_12></location>s</text>
<text><location><page_51><loc_80><loc_11><loc_81><loc_11></location>a</text>
<text><location><page_51><loc_80><loc_10><loc_81><loc_11></location>e</text>
<text><location><page_51><loc_80><loc_10><loc_81><loc_10></location>M</text>
<text><location><page_51><loc_80><loc_9><loc_81><loc_9></location>)</text>
<text><location><page_51><loc_80><loc_9><loc_81><loc_9></location>a</text>
<text><location><page_51><loc_80><loc_8><loc_81><loc_9></location>(</text>
<text><location><page_51><loc_82><loc_22><loc_83><loc_22></location>o</text>
<text><location><page_51><loc_82><loc_22><loc_83><loc_22></location>f</text>
<text><location><page_51><loc_82><loc_21><loc_83><loc_21></location>d</text>
<text><location><page_51><loc_82><loc_20><loc_83><loc_21></location>e</text>
<text><location><page_51><loc_82><loc_17><loc_83><loc_18></location>t</text>
<text><location><page_51><loc_82><loc_17><loc_83><loc_17></location>o</text>
<text><location><page_51><loc_82><loc_16><loc_83><loc_17></location>n</text>
<text><location><page_51><loc_82><loc_16><loc_83><loc_16></location>s</text>
<text><location><page_51><loc_82><loc_15><loc_83><loc_16></location>e</text>
<text><location><page_51><loc_82><loc_15><loc_83><loc_15></location>x</text>
<text><location><page_51><loc_82><loc_14><loc_83><loc_15></location>u</text>
<text><location><page_51><loc_82><loc_14><loc_83><loc_14></location>l</text>
<text><location><page_51><loc_82><loc_14><loc_83><loc_14></location>f</text>
<text><location><page_51><loc_82><loc_13><loc_83><loc_14></location>d</text>
<text><location><page_51><loc_82><loc_13><loc_83><loc_13></location>e</text>
<text><location><page_51><loc_82><loc_12><loc_83><loc_13></location>r</text>
<text><location><page_51><loc_82><loc_12><loc_83><loc_12></location>u</text>
<text><location><page_51><loc_82><loc_12><loc_83><loc_12></location>s</text>
<text><location><page_51><loc_82><loc_11><loc_83><loc_12></location>a</text>
<text><location><page_51><loc_82><loc_11><loc_83><loc_11></location>e</text>
<text><location><page_51><loc_82><loc_10><loc_83><loc_11></location>M</text>
<text><location><page_51><loc_82><loc_9><loc_83><loc_10></location>)</text>
<text><location><page_51><loc_82><loc_9><loc_83><loc_9></location>b</text>
<text><location><page_51><loc_82><loc_8><loc_83><loc_9></location>(</text>
<text><location><page_51><loc_21><loc_13><loc_22><loc_14></location>u</text>
<text><location><page_51><loc_21><loc_13><loc_22><loc_13></location>a</text>
<text><location><page_51><loc_21><loc_12><loc_22><loc_13></location>T</text>
<text><location><page_51><loc_21><loc_11><loc_22><loc_12></location>A</text>
<text><location><page_51><loc_21><loc_10><loc_22><loc_11></location>A</text>
<text><location><page_51><loc_23><loc_13><loc_25><loc_14></location>u</text>
<text><location><page_51><loc_23><loc_13><loc_25><loc_13></location>a</text>
<text><location><page_51><loc_23><loc_12><loc_25><loc_13></location>T</text>
<text><location><page_51><loc_23><loc_11><loc_25><loc_12></location>P</text>
<text><location><page_51><loc_23><loc_10><loc_25><loc_11></location>B</text>
<text><location><page_51><loc_27><loc_13><loc_29><loc_14></location>u</text>
<text><location><page_51><loc_27><loc_13><loc_29><loc_13></location>a</text>
<text><location><page_51><loc_27><loc_12><loc_29><loc_13></location>T</text>
<text><location><page_51><loc_27><loc_11><loc_29><loc_12></location>E</text>
<text><location><page_51><loc_27><loc_10><loc_29><loc_11></location>D</text>
<text><location><page_51><loc_29><loc_13><loc_30><loc_14></location>u</text>
<text><location><page_51><loc_29><loc_13><loc_30><loc_13></location>a</text>
<text><location><page_51><loc_29><loc_12><loc_30><loc_13></location>T</text>
<text><location><page_51><loc_29><loc_11><loc_30><loc_12></location>F</text>
<text><location><page_51><loc_29><loc_10><loc_30><loc_11></location>D</text>
<text><location><page_51><loc_30><loc_14><loc_31><loc_14></location>u</text>
<text><location><page_51><loc_30><loc_13><loc_31><loc_14></location>a</text>
<text><location><page_51><loc_30><loc_12><loc_31><loc_13></location>T</text>
<text><location><page_51><loc_30><loc_11><loc_31><loc_12></location>K</text>
<text><location><page_51><loc_30><loc_10><loc_31><loc_11></location>D</text>
<text><location><page_51><loc_31><loc_13><loc_33><loc_14></location>u</text>
<text><location><page_51><loc_31><loc_13><loc_33><loc_13></location>a</text>
<text><location><page_51><loc_31><loc_12><loc_33><loc_13></location>T</text>
<text><location><page_51><loc_31><loc_11><loc_33><loc_12></location>N</text>
<text><location><page_51><loc_31><loc_10><loc_33><loc_11></location>D</text>
<text><location><page_51><loc_33><loc_13><loc_34><loc_14></location>u</text>
<text><location><page_51><loc_33><loc_13><loc_34><loc_13></location>a</text>
<text><location><page_51><loc_33><loc_12><loc_34><loc_13></location>T</text>
<text><location><page_51><loc_33><loc_11><loc_34><loc_12></location>R</text>
<text><location><page_51><loc_33><loc_10><loc_34><loc_11></location>D</text>
<text><location><page_51><loc_36><loc_14><loc_37><loc_14></location>a</text>
<text><location><page_51><loc_36><loc_13><loc_37><loc_14></location>h</text>
<text><location><page_51><loc_36><loc_12><loc_37><loc_13></location>C</text>
<text><location><page_51><loc_36><loc_11><loc_37><loc_12></location>X</text>
<text><location><page_51><loc_36><loc_10><loc_37><loc_11></location>D</text>
<text><location><page_51><loc_37><loc_13><loc_38><loc_14></location>a</text>
<text><location><page_51><loc_37><loc_13><loc_38><loc_13></location>h</text>
<text><location><page_51><loc_37><loc_12><loc_38><loc_13></location>C</text>
<text><location><page_51><loc_37><loc_11><loc_38><loc_12></location>P</text>
<text><location><page_51><loc_37><loc_10><loc_38><loc_11></location>E</text>
<text><location><page_51><loc_38><loc_14><loc_40><loc_14></location>a</text>
<text><location><page_51><loc_38><loc_13><loc_40><loc_14></location>h</text>
<text><location><page_51><loc_38><loc_12><loc_40><loc_13></location>C</text>
<text><location><page_51><loc_38><loc_11><loc_40><loc_12></location>T</text>
<text><location><page_51><loc_38><loc_10><loc_40><loc_11></location>E</text>
<text><location><page_51><loc_40><loc_14><loc_41><loc_15></location>A</text>
<text><location><page_51><loc_40><loc_13><loc_41><loc_14></location>u</text>
<text><location><page_51><loc_40><loc_13><loc_41><loc_13></location>a</text>
<text><location><page_51><loc_40><loc_12><loc_41><loc_13></location>T</text>
<text><location><page_51><loc_40><loc_11><loc_41><loc_12></location>N</text>
<text><location><page_51><loc_40><loc_10><loc_41><loc_11></location>H</text>
<text><location><page_51><loc_42><loc_13><loc_44><loc_14></location>p</text>
<text><location><page_51><loc_42><loc_13><loc_44><loc_13></location>u</text>
<text><location><page_51><loc_42><loc_12><loc_44><loc_13></location>L</text>
<text><location><page_51><loc_42><loc_11><loc_44><loc_12></location>U</text>
<text><location><page_51><loc_42><loc_10><loc_44><loc_11></location>R</text>
<text><location><page_51><loc_44><loc_15><loc_45><loc_15></location>A</text>
<text><location><page_51><loc_44><loc_14><loc_45><loc_14></location>r</text>
<text><location><page_51><loc_44><loc_13><loc_45><loc_14></location>u</text>
<text><location><page_51><loc_44><loc_12><loc_45><loc_13></location>A</text>
<text><location><page_51><loc_44><loc_11><loc_45><loc_12></location>W</text>
<text><location><page_51><loc_44><loc_10><loc_45><loc_11></location>R</text>
<text><location><page_51><loc_45><loc_13><loc_46><loc_14></location>r</text>
<text><location><page_51><loc_45><loc_13><loc_46><loc_13></location>u</text>
<text><location><page_51><loc_45><loc_12><loc_46><loc_13></location>A</text>
<text><location><page_51><loc_45><loc_11><loc_46><loc_12></location>U</text>
<text><location><page_51><loc_45><loc_10><loc_46><loc_11></location>S</text>
<text><location><page_51><loc_48><loc_15><loc_49><loc_15></location>o</text>
<text><location><page_51><loc_48><loc_14><loc_49><loc_15></location>c</text>
<text><location><page_51><loc_48><loc_14><loc_49><loc_14></location>S</text>
<text><location><page_51><loc_48><loc_13><loc_49><loc_13></location>0</text>
<text><location><page_51><loc_48><loc_12><loc_49><loc_13></location>9</text>
<text><location><page_51><loc_48><loc_12><loc_49><loc_12></location>1</text>
<text><location><page_51><loc_48><loc_11><loc_49><loc_12></location>1</text>
<text><location><page_51><loc_48><loc_10><loc_49><loc_11></location>V</text>
<text><location><page_51><loc_49><loc_15><loc_50><loc_15></location>r</text>
<text><location><page_51><loc_49><loc_14><loc_50><loc_15></location>g</text>
<text><location><page_51><loc_49><loc_14><loc_50><loc_14></location>S</text>
<text><location><page_51><loc_49><loc_13><loc_50><loc_13></location>6</text>
<text><location><page_51><loc_49><loc_12><loc_50><loc_13></location>4</text>
<text><location><page_51><loc_49><loc_12><loc_50><loc_12></location>0</text>
<text><location><page_51><loc_49><loc_11><loc_50><loc_12></location>4</text>
<text><location><page_51><loc_49><loc_10><loc_50><loc_11></location>V</text>
<text><location><page_51><loc_54><loc_13><loc_55><loc_14></location>a</text>
<text><location><page_51><loc_54><loc_13><loc_55><loc_13></location>h</text>
<text><location><page_51><loc_54><loc_12><loc_55><loc_13></location>C</text>
<text><location><page_51><loc_54><loc_11><loc_55><loc_12></location>S</text>
<text><location><page_51><loc_54><loc_10><loc_55><loc_11></location>C</text>
<text><location><page_51><loc_55><loc_14><loc_57><loc_14></location>u</text>
<text><location><page_51><loc_55><loc_13><loc_57><loc_14></location>a</text>
<text><location><page_51><loc_55><loc_12><loc_57><loc_13></location>T</text>
<text><location><page_51><loc_55><loc_11><loc_57><loc_12></location>M</text>
<text><location><page_51><loc_55><loc_10><loc_57><loc_11></location>D</text>
<text><location><page_51><loc_57><loc_14><loc_58><loc_14></location>r</text>
<text><location><page_51><loc_57><loc_13><loc_58><loc_14></location>u</text>
<text><location><page_51><loc_57><loc_12><loc_58><loc_13></location>A</text>
<text><location><page_51><loc_57><loc_11><loc_58><loc_12></location>M</text>
<text><location><page_51><loc_57><loc_10><loc_58><loc_11></location>G</text>
<text><location><page_51><loc_58><loc_13><loc_59><loc_14></location>u</text>
<text><location><page_51><loc_58><loc_12><loc_59><loc_13></location>a</text>
<text><location><page_51><loc_58><loc_12><loc_59><loc_13></location>T</text>
<text><location><page_51><loc_58><loc_11><loc_59><loc_11></location>P</text>
<text><location><page_51><loc_58><loc_10><loc_59><loc_11></location>I</text>
<text><location><page_51><loc_59><loc_14><loc_61><loc_14></location>a</text>
<text><location><page_51><loc_59><loc_13><loc_61><loc_14></location>y</text>
<text><location><page_51><loc_59><loc_13><loc_61><loc_13></location>H</text>
<text><location><page_51><loc_59><loc_11><loc_61><loc_12></location>W</text>
<text><location><page_51><loc_59><loc_10><loc_61><loc_11></location>T</text>
<text><location><page_51><loc_61><loc_14><loc_62><loc_15></location>A</text>
<text><location><page_51><loc_61><loc_13><loc_62><loc_14></location>u</text>
<text><location><page_51><loc_61><loc_13><loc_62><loc_13></location>a</text>
<text><location><page_51><loc_61><loc_12><loc_62><loc_13></location>T</text>
<text><location><page_51><loc_61><loc_11><loc_62><loc_12></location>X</text>
<text><location><page_51><loc_61><loc_10><loc_62><loc_11></location>U</text>
<text><location><page_51><loc_62><loc_15><loc_64><loc_15></location>u</text>
<text><location><page_51><loc_62><loc_14><loc_64><loc_15></location>a</text>
<text><location><page_51><loc_62><loc_14><loc_64><loc_14></location>T</text>
<text><location><page_51><loc_62><loc_13><loc_64><loc_13></location>9</text>
<text><location><page_51><loc_62><loc_12><loc_64><loc_13></location>7</text>
<text><location><page_51><loc_62><loc_12><loc_64><loc_12></location>0</text>
<text><location><page_51><loc_62><loc_11><loc_64><loc_12></location>1</text>
<text><location><page_51><loc_62><loc_10><loc_64><loc_11></location>V</text>
<text><location><page_51><loc_67><loc_14><loc_68><loc_14></location>a</text>
<text><location><page_51><loc_67><loc_13><loc_68><loc_14></location>h</text>
<text><location><page_51><loc_67><loc_12><loc_68><loc_13></location>C</text>
<text><location><page_51><loc_67><loc_11><loc_68><loc_12></location>G</text>
<text><location><page_51><loc_67><loc_10><loc_68><loc_11></location>E</text>
<text><location><page_51><loc_68><loc_13><loc_70><loc_14></location>7</text>
<text><location><page_51><loc_68><loc_12><loc_70><loc_13></location>A</text>
<text><location><page_51><loc_68><loc_11><loc_70><loc_12></location>W</text>
<text><location><page_51><loc_68><loc_10><loc_70><loc_11></location>T</text>
<text><location><page_51><loc_70><loc_15><loc_71><loc_15></location>u</text>
<text><location><page_51><loc_70><loc_14><loc_71><loc_15></location>a</text>
<text><location><page_51><loc_70><loc_14><loc_71><loc_14></location>T</text>
<text><location><page_51><loc_70><loc_13><loc_71><loc_13></location>8</text>
<text><location><page_51><loc_70><loc_12><loc_71><loc_13></location>6</text>
<text><location><page_51><loc_70><loc_12><loc_71><loc_12></location>0</text>
<text><location><page_51><loc_70><loc_11><loc_71><loc_12></location>1</text>
<text><location><page_51><loc_70><loc_10><loc_71><loc_11></location>V</text>
<text><location><page_51><loc_71><loc_14><loc_73><loc_15></location>r</text>
<text><location><page_51><loc_71><loc_14><loc_73><loc_14></location>u</text>
<text><location><page_51><loc_71><loc_13><loc_73><loc_14></location>A</text>
<text><location><page_51><loc_71><loc_12><loc_73><loc_13></location>6</text>
<text><location><page_51><loc_71><loc_12><loc_73><loc_12></location>9</text>
<text><location><page_51><loc_71><loc_11><loc_73><loc_12></location>3</text>
<text><location><page_51><loc_71><loc_10><loc_73><loc_11></location>V</text>
<text><location><page_51><loc_73><loc_14><loc_74><loc_15></location>r</text>
<text><location><page_51><loc_73><loc_14><loc_74><loc_14></location>u</text>
<text><location><page_51><loc_73><loc_13><loc_74><loc_14></location>A</text>
<text><location><page_51><loc_73><loc_12><loc_74><loc_13></location>7</text>
<text><location><page_51><loc_73><loc_12><loc_74><loc_12></location>9</text>
<text><location><page_51><loc_73><loc_11><loc_74><loc_12></location>3</text>
<text><location><page_51><loc_73><loc_10><loc_74><loc_11></location>V</text>
<text><location><page_51><loc_79><loc_12><loc_80><loc_13></location>-</text>
<text><location><page_51><loc_79><loc_11><loc_80><loc_11></location>.</text>
<text><location><page_51><loc_79><loc_11><loc_79><loc_11></location>e</text>
<text><location><page_51><loc_79><loc_10><loc_79><loc_11></location>t</text>
<text><location><page_51><loc_79><loc_10><loc_79><loc_10></location>o</text>
<text><location><page_51><loc_79><loc_9><loc_79><loc_10></location>N</text>
<text><location><page_51><loc_81><loc_13><loc_82><loc_13></location>.</text>
<text><location><page_51><loc_81><loc_12><loc_81><loc_13></location>1</text>
<text><location><page_51><loc_81><loc_12><loc_81><loc_12></location>-</text>
<text><location><page_51><loc_81><loc_11><loc_82><loc_12></location>s</text>
<text><location><page_51><loc_81><loc_10><loc_82><loc_11></location>m</text>
<text><location><page_51><loc_81><loc_10><loc_82><loc_10></location>k</text>
<text><location><page_51><loc_81><loc_9><loc_82><loc_10></location>0</text>
<text><location><page_51><loc_81><loc_9><loc_82><loc_9></location>0</text>
<text><location><page_51><loc_81><loc_8><loc_82><loc_9></location>6</text>
<text><location><page_51><loc_83><loc_17><loc_84><loc_18></location>r</text>
<text><location><page_51><loc_83><loc_17><loc_84><loc_17></location>o</text>
<text><location><page_51><loc_83><loc_17><loc_84><loc_17></location>f</text>
<text><location><page_51><loc_83><loc_16><loc_84><loc_16></location>x</text>
<text><location><page_51><loc_83><loc_15><loc_84><loc_16></location>u</text>
<text><location><page_51><loc_83><loc_15><loc_84><loc_15></location>l</text>
<text><location><page_51><loc_83><loc_15><loc_84><loc_15></location>f</text>
<text><location><page_51><loc_83><loc_14><loc_84><loc_15></location>e</text>
<text><location><page_51><loc_83><loc_14><loc_84><loc_14></location>n</text>
<text><location><page_51><loc_83><loc_13><loc_84><loc_14></location>i</text>
<text><location><page_51><loc_83><loc_13><loc_84><loc_13></location>l</text>
<text><location><page_51><loc_83><loc_12><loc_84><loc_13></location>v</text>
<text><location><page_51><loc_83><loc_12><loc_84><loc_12></location>i</text>
<text><location><page_51><loc_83><loc_11><loc_84><loc_12></location>C</text>
<text><location><page_51><loc_83><loc_11><loc_84><loc_11></location>e</text>
<text><location><page_51><loc_83><loc_10><loc_84><loc_11></location>h</text>
<text><location><page_51><loc_83><loc_10><loc_84><loc_10></location>T</text>
<text><location><page_51><loc_83><loc_9><loc_84><loc_9></location>)</text>
<text><location><page_51><loc_83><loc_9><loc_84><loc_9></location>c</text>
<text><location><page_51><loc_83><loc_8><loc_84><loc_9></location>(</text>
<text><location><page_52><loc_39><loc_90><loc_61><loc_91></location>Average Line Kinematic Parameters</text>
<table>
<location><page_52><loc_22><loc_61><loc_78><loc_89></location>
<caption>Table 9</caption>
</table>
<text><location><page_52><loc_11><loc_59><loc_47><loc_60></location>Note . - All calculations include the H 2 velocity correction (Section 2.1).</text>
<unordered_list>
<list_item><location><page_52><loc_10><loc_58><loc_33><loc_59></location>(a) Does not include HN Tau, RW Aur, AK Sco.</list_item>
<list_item><location><page_52><loc_10><loc_57><loc_72><loc_58></location>(b) Velocity differences measured in the same spectrum (for example V CIV -V HeII ) are not subject to errors due to pointing.</list_item>
</document>
[
{
"title": "ABSTRACT",
"content": "For Classical T Tauri Stars (CTTSs), the resonance doublets of N v , Si iv , and C iv , as well as the He ii 1640 ˚ A line, trace hot gas flows and act as diagnostics of the accretion process. In this paper we assemble a large high-resolution, high-sensitivity dataset of these lines in CTTSs and Weak T Tauri Stars (WTTSs). The sample comprises 35 stars: one Herbig Ae star, 28 CTTSs, and 6 WTTSs. We find that the C iv , Si iv , and N v lines in CTTSs all have similar shapes. We decompose the C iv and He ii lines into broad and narrow Gaussian components (BC & NC). The most common (50 %) C iv line morphology in CTTSs is that of a low-velocity NC together with a redshifted BC. For CTTSs, a strong BC is the result of the accretion process. The contribution fraction of the NC to the C iv line flux in CTTSs increases with accretion rate, from ∼ 20% to up to ∼ 80%. The velocity centroids of the BCs and NCs are such that V BC /greaterorsimilar 4 V NC , consistent with the predictions of the accretion shock model, in at most 12 out of 22 CTTSs. We do not find evidence of the post-shock becoming buried in the stellar photosphere due to the pressure of the accretion flow. The He ii CTTSs lines are generally symmetric and narrow, with FWHM and redshifts comparable to those of WTTSs. They are less redshifted than the CTTSs C iv lines, by ∼ 10 km s -1 . The amount of flux in the BC of the He ii line is small compared to that of the C iv line, and we show that this is consistent with models of the pre-shock column emission. Overall, the observations are consistent with the presence of multiple accretion columns with different densities or with accretion models that predict a slow-moving, lowdensity region in the periphery of the accretion column. For HN Tau A and RW Aur A, most of the C iv line is blueshifted suggesting that the C iv emission is produced by shocks within outflow jets. In our sample, the Herbig Ae star DX Cha is the only object for which we find a P-Cygni profile in the C iv line, which argues for the presence of a hot (10 5 K) wind. For the overall sample, the Si iv and N v line luminosities are correlated with the C iv line luminosities, although the relationship between Si iv and C iv shows large scatter about a linear relationship and suggests that TW Hya, V4046 Sgr, AA Tau, DF Tau, GM Aur, and V1190 Sco are silicon-poor, while CV Cha, DX Cha, RU Lup, and RW Aur may be silicon-rich. Keywords: Surveys - Protoplanetary disks - Stars: pre-main sequence - Stars: variables: T Tauri, Herbig Ae/Be - Ultraviolet: stars",
"pages": [
1
]
},
{
"title": "HOT GAS LINES IN T TAURI STARS",
"content": "David R. Ardila 1 , Gregory J. Herczeg 2 , Scott G. Gregory 3,4 , Laura Ingleby 5 , Kevin France 6 , Alexander Brown 6 , Suzan Edwards 7 , Christopher Johns-Krull 8 , Jeffrey L. Linsky 9 , Hao Yang 10 , Jeff A. Valenti 11 , Herv'e Abgrall 12 , Richard D. Alexander 13 , Edwin Bergin 5 , Thomas Bethell 5 , Joanna M. Brown 14 , Nuria Calvet 5 , Catherine Espaillat 14 , Lynne A. Hillenbrand 3 , Gaitee Hussain 15 , Evelyne Roueff 12 , Rebecca Schindhelm 16 , Frederick M. Walter 17 Draft version September 13, 2021",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "1 ardila@ipac.caltech.edu; NASA Herschel Science Center, California Institute of Technology, MC 100-22, Pasadena, CA 91125, USA Classical T Tauri stars (CTTSs) are low-mass, young stellar objects surrounded by an accretion disk. They provide us with a laboratory to study the interaction between stars, magnetic fields, and accretion disks. In addition to optical and longer wavelength excesses, this interaction is responsible for a strong ultraviolet (UV) excess, Lyα emission and soft X-ray excesses, all of which have a significant impact on the disk evolution, the rate of planet formation, and the circumstellar environment. The observed long rotational periods, the large widths of the Balmer lines, and the presence of optical and UV excesses of CTTSs are naturally explained by the magnetospheric accretion paradigm (e.g. Uchida & Shibata 1984, Koenigl 1991, Shu et al. 1994). According to this paradigm, the gas disk is truncated some distance from the star ( ∼ 5 R ∗ , Meyer et al. 1997) by the pressure of the stellar magnetosphere. Gas from the disk slides down the stellar gravitational potential along the magnetic field lines, reaching speeds comparable to the free-fall velocity ( ∼ 300 km s -1 ). These velocities are much larger than the local ∼ 20 km s -1 sound speed (but less than the Alfven speed of ∼ 500 km s -1 implied by a 2 kG magnetic field Johns-Krull 2007). The density of the accretion stream depends on the accretion rate and the filling factor but it is typically of the order of ∼ 10 12 cm -3 (pre-shock, see Calvet & Gullbring 1998). The supersonic flow, confined by the magnetic field, produces a strong shock upon reaching the star, which converts most of the kinetic energy of the gas into thermal energy (e.g. Lamzin 1995). The gas reaches temperatures of the order of a million degrees at the shock surface and cools radiatively until it merges with the stellar photosphere. Part of the cooling radiation will heat the stellar surface, resulting in a hot spot, observed spectroscopically as an excess continuum (the 'veiling'). Cooling radiation emitted away from the star will illuminate gas before the shock surface, producing a radiative precursor of warm (T ∼ 10 4 K), ionized gas (Calvet & Gullbring 1998). In this paper we will use the term 'accretion shock region' as shorthand for the region that includes the pre-shock, the shock surface, the post-shock column, and the heated photosphere. Because the accretion column should be in pressure equilibrium with the stellar photosphere, Drake (2005) suggested that for typical accretion rates ( ∼ 10 -8 M /circledot /yr) the post-shock region would be buried, in the sense that the shortest escape paths for post-shock photons would go through a significant column of photospheric gas. Sacco et al. (2010) have argued that the burying effects appear at accretion rates as small as a few times ∼ 10 -10 M /circledot /yr and that the absorption of the X-rays by the stellar photosphere may explain the one to two orders of magnitude discrepancy between the accretion rates calculated from X-ray line emission and those calculated from optical veiling or near-UV excesses, and the lack of dense (n e /greaterorsimilar 10 11 cm -3 ) X-ray emitting plasma in objects such as T Tau (Gudel et al. 2007). Time-dependent models of the accretion column by Sacco et al. (2008) predict that emission from a single, homogeneous, magnetically dominated post-shock column should be quasi-periodic, on timescales of ∼ 400 sec, because plasma instabilities can collapse the column. Such periodicity has not been observed (Drake et al. 2009; Gunther et al. 2010), perhaps suggesting that accretion streams are inhomogeneous, or that there are multiple, uncorrelated accretion columns. Observations during the last two decades have resulted in spectroscopic and photometric evidence for the presence of accretion hot spots or rings with filling factors of up to a few percent (see review by Bouvier et al. 2007). The surface topology of the magnetic field (e.g. Gregory et al. 2006; Mohanty & Shu 2008) and/or the misalignment between the rotational and magnetic axes result in a rotationally modulated surface flux (Johns & Basri 1995; Argiroffi et al. 2011, 2012a), with a small filling factor (Calvet & Gullbring 1998; Valenti & Johns-Krull 2004). The spots appear and disappear over timescales of days (Rucinski et al. 2008) to years (Bouvier et al. 1993). Analysis of the possible magnetic field configurations indicates that although CTTSs have very complex surface magnetic fields, the portion of the field that carries gas from the inner disk to the star, is well ordered globally (Johns-Krull et al. 1999a; Adams & Gregory 2012). In this paper we use the strong emission lines of ionized metals in order to probe the characteristics of the accretion shock region. In particular, we are interested in understanding the role that accretion has in shaping these lines, where the lines originate, and what the lines reveal about the geometry of the accretion process. Our long-term goal is to clarify the UV evolution of young stars and its impact on the surrounding accretion disk. We analyze the resonance doublets of N v ( λλ 1238.82, 1242.80 ˚ A), Si iv ( λλ 1393.76, 1402.77 ˚ A), and C iv ( λλ 1548.19, 1550.77 ˚ A), as well as the He ii ( λ 1640.47 ˚ A) line. If they are produced by collisional excitation in a low-density medium, their presence suggests a high temperature ( ∼ 10 5 K, assuming collisional ionization equilibrium) or a photoionized environment. In solartype main sequence stars, these 'hot' lines are formed in the transition region, the narrow region between the chromosphere and the corona, and they are sometimes called transition-region lines. The C iv resonance doublet lines are among the strongest lines in the UV spectra of CTTSs (Ardila et al. 2002). Using International Ultraviolet Explorer (IUE) data, Johns-Krull et al. (2000) showed that the surface flux in the C iv resonant lines can be as much as an order of magnitude larger than the largest flux observed in Weak T Tauri stars (WTTSs), main sequence dwarfs, or RS CVn stars. They also showed that the high surface flux in the C iv lines of CTTSs is uncorrelated with measures of stellar activity but it is strongly correlated with accretion rate, for accretion rates from 10 -8 M /circledot /yr to 10 -5 M /circledot /yr. The strong correlations among accretion rate, C iv flux and Far Ultraviolet (FUV) luminosity have been confirmed by Ingleby et al. (2011) and Yang et al. (2012) using ACS/SBC and STIS data. Those results further suggest a causal relationship between the accretion process and the hot line flux. Previous surveys of the UV emission lines in lowresolution spectra of T Tauri stars include the analysis of IUE spectra (Valenti et al. 2000; Johns-Krull et al. 2000) and GHRS, ACS, and STIS spectra (Yang et al. 2012). Prior analysis of high resolution observations of the hot gas lines in T Tauri stars had been published for some objects (BP Tau, DF Tau, DG Tau, DR Tau, EG Cha, EP Cha, GM Aur, RU Lup, RW Aur A, RY Tau, T Tau, TW Hya, TWA 5, V4046 Sgr, and HBC 388, see Lamzin 2000a,b; Errico et al. 2000, 2001; Ardila et al. 2002; Herczeg et al. 2002; Lamzin et al. 2004; Herczeg et al. 2005, 2006; Gunther & Schmitt 2008; Ingleby et al. 2011). In addition, analysis of the C iv lines for the brown dwarf 2M1207 has been published by France et al. (2010). The observations show that the C iv lines in CTTSs have asymmetric shapes and wings that often extend to ± 400 km s -1 from the line rest velocity. The emission lines are mostly centered or redshifted, although some stars occasionally present strong blueshifted emission (e.g. DG Tau, DR Tau, RY Tau). Doublet ratios are not always 2:1 (e.g. DG Tau, DR Tau, RU Lup, and RW Aur A). Overall, the UV spectra of CTTSs also show large numbers of narrow H 2 emission lines, and CO bands in absorption and emission (France et al. 2011; Schindhelm et al. 2012), produced primarily by Lyα fluorescence. In most observations published so far, the Si iv line is strongly contaminated by H 2 lines, and the N v 1243 ˚ A line is absorbed by circumstellar or interstellar N I. High spectral resolution observations are crucial to fully exploit the diagnostic power of the UV observations as the H 2 emission lines can be kinematically separated from the hot lines only when the resolution is high enough. In addition, we will show here that the hot gas lines have multiple kinematic components that can only be analyzed in high resolution spectra. Both the correlation between accretion rate and C iv surface flux and the presence of redshifted hot gas line profiles in some stars, are consistent with formation in a high-latitude accretion flow. However, according to Lamzin (2003a,b) C iv line formation in the accretion shock region should result in double-peaked line profiles, which are generally not observed. In this context Gunther & Schmitt (2008) explored the shape of the hot gas lines (primarily O vi and C iv ) in a sample of 7 stars. They considered formation in the accretion shock, in an outflow, in the surface of the disk, in an equatorial boundary layer, and in the stellar transition region, and concluded that no single explanation or region can be responsible for all the observed line characteristics. In particular, they concluded that the shape of the redshifted lines was incompatible with models of magnetospheric accretion. With the primary goal of providing a unified description of the hot gas lines and understanding their origin, we have obtained Far (FUV) and Near Ultraviolet (NUV) spectra of a large sample of CTTSs and WTTSs, using the Cosmic Origins Spectrograph ( COS ) and the Space Telescope Imaging Spectrograph ( STIS ). Most of the data for this paper comes from the Cycle 17 Hubble Space Telescope ( HST ) proposal 'The Disks, Accretion, and Outflows (DAO) of T Tau stars' (PI G. Herczeg, Prop. ID HST-GO-11616). The DAO program is the largest and most sensitive high resolution spectroscopic survey of young stars in the UV ever undertaken and as such it provides a rich source of information for these objects. The program is described in more detail in Herczeg et al. (2013). We have complemented the DAO data with GTO data from HST programs 11533 and 12036 (PI J. Green - BP Tau, DF Tau, RU Lup, V4046 Sgr) as well as UV spectra from the literature and from the Mikulski Archive at STScI ( MAST ). As shown below, there is a wide diversity of profiles in all lines for the stars in our sample. The spectra are rich and a single paper cannot do justice to their variety nor to all the physical mechanisms that likely contribute to their formation. Here we take a broad view in an attempt to obtain general statements about accretion in CTTSs. Details about the sample and the data reduction are presented in Section 2. We then analyze the C iv , Si iv , He ii and N v lines. The analysis of the C iv lines takes up most of the paper, as this is the strongest line in the set, and the least affected by absorptions or emissions by other species (Section 5). We examine the line shapes, the relationship with accretion rate, and correlations among quantities associated with the lines and other CTTSs parameters. We also obtain from the literature multi-epoch information on line variability (Section 5.4). The other lines play a supporting role in this analysis and are examined in Sections 6 and 7. Section 8 contains a summary of the observational conclusions and a discussion of their implications. The conclusions are in Section 9.",
"pages": [
1,
2,
3
]
},
{
"title": "2. OBSERVATIONS",
"content": "Tables 1, 2, and 3 list the 35 stars we will be analyzing in this paper and the references for all the ancillary data we consider. Table 4 indicates the origin of the data (DAO or some other project), the datasets and slit sizes used for the observations. Details about exposure times will appear in Herczeg et al. (2013). The data considered here encompass most of the published high resolution HST observations of the C iv doublet lines for CTTSs. Ardila et al. (2002) provides references to additional high-resolution C iv data for CTTSs obtained with the Goddard High Resolution Spectrograph ( GHRS ). We do not re-analyze those spectra here, but they provide additional context to our paper. Non-DAO STIS data were downloaded from the HST STIS Echelle Spectral Catalog of Stars (StarCAT, Ayres 2010). NonDAO COS data were downloaded from the Mikulski Archive for Space Telescopes (MAST) and reduced as described below. The sample of stars considered here includes objects with spectral types ranging from A7 (the Herbig Ae star DX Cha 18 ) to M2, although most objects have mid-K spectral types. We assume stellar ages and distances to be ∼ 2 Myr and 140 pc, respectively, for Taurus-Aurigae (see Loinard et al. 2007 and references therein), ∼ 2 Myr and 150 pc for Lupus I (Comer'on et al. 2009), ∼ 5 Myr and 145 pc for Upper-Scorpius (see Alencar et al. 2003 and references therein), ∼ 5 Myr and 114 pc for the /epsilon1 Chamaeleontis cluster (see Lyo et al. 2008 and references therein), ∼ 5 Myr and 160 pc for Chamaeleon I (see Hussain et al. 2009 and references therein), ∼ 8 Myr and 97 pc for the η Chamaeleontis cluster (Mamajek et al. 1999), ∼ 10 Myr and 55 pc for the TW Hydrae association (Zuckerman & Song 2004), ∼ 12 Myr and 72 pc for V4046Sgr in the β Pictoris moving group (Torres et al. 2006). The sample includes MP Mus ( ∼ 7 Myr and 100 pc, Kastner et al. 2010 and references therein), not known to be associated with any young region. The sample includes stars with transition disks (TD) (Espaillat et al. 2011) CS Cha, DM Tau, GM Aur, IP Tau, TW Hya, UX Tau A, and V1079 Tau (LkCa 15). Here we take the term 'transition disk' to mean a disk showing infrared evidence of a hole or a gap. As a group, transition disks may have lower accretion rates than other CTTS disks (Espaillat et al. 2012). However, for the targets included here the difference between the accretion rates of TDs and those of the the rest of the accreting stars is not significant. The sample also includes six Weak T Tauri Stars (WTTSs): EG Cha (RECX 1), V396 Aur (LkCa 19), V1068 Tau (LkCa 4), TWA 7, V397 Aur, and V410 Tau. Of the 35 stars considered here, 12 are known to be part of binary systems. Dynamical interactions among the binary components may re-arrange the circumstellar disk or preclude its existence altogether. The presence of an unaccounted companion may result in larger-thanexpected accretion diagnostic lines. In addition, large instantaneous radial velocities may be observed in the targeted lines at certain points of the binary orbit. AK Sco, DX Cha, and V4046 Sgr are spectroscopic binaries with circumbinary and/or circumstellar disks. CS Cha is a candidate long-period spectroscopic binary, although the characteristics of the companion are unknown (Guenther et al. 2007). The accretion rates listed in Table 3 are obtained from optical veiling and NUV excesses and represent total accretion for the overall system. At the end of this paper we will conclude that binarity may be affecting the line centroid determination only in AK Sco. For non-spectroscopic binaries, the effect of the binarity may be relevant only if both companions are within the COS or STIS apertures. This is the case for DF Tau, RW Aur A, and the WTTSs EG Cha, V397 Aur, and V410 Tau. For DF Tau and RW Aur A, the primary component dominates the FUV emission (see Herczeg et al. 2006; Alencar et al. 2005). Table 3 lists the accretion rate, obtained from literature sources. For some stars, the DAO dataset contains simultaneous NUV and FUV observations, obtained during the same HST visit. Ingleby et al. (2013) describe those NUV observations in more detail and calculate accretion rates based on them. In turn, we use those accretion rate determinations here. The uncertainty in the accretion rate is dominated by systematic factors such as the adopted extinction correction and the color of the underlying photosphere. These may result in errors as large as a factor of 10 in the accretion rate. We do not list v sin i measurements for the sample but typical values for young stars are ∼ 10-20 km s -1 (Basri & Batalha 1990).",
"pages": [
3,
4
]
},
{
"title": "2.1. Data Reduction",
"content": "The COS and STIS DAO data were all taken in timetagged mode. The maximum exposure time was accumulated in the region of the Si iv resonance lines and the exposures ranged from 3 ksec (for most stars with 2 orbit visits) to a maximum of 8 ksec for a few stars with 4 orbit visits. Use of multiple grating positions ensured full wavelength coverage and reduced the effects of fixed pattern noise. All observations were taken using the primary science aperture (PSA), which is a 2.5' diameter circular aperture. The acquisition observations used the ACQ/SEARCH algorithm followed by ACQ/IMAGE (Dixon 2011). The absolute wavelength scale accuracy is ∼ 15 km s -1 (1 σ ), where the error is dominated by pointing errors. We obtained one dimensional, co-added spectra using the COS calibration pipeline (CALCOS) with alignment and co-addition obtained using the IDL routines described by Danforth et al. (2010). For a given pointing, errors in the positioning of the target within the aperture result in an offset in the wavelength scale. As indicated above, these are supposed to be 15 km s -1 for COS and 3 km s -1 for STIS , according to the instrument observing manuals (Dixon 2011; Ely 2011). In addition, the geometric correction necessary to account for the curvature of the COS FUV detector may make longer wavelength features appear redder than they really are. The offset depends on the exact position of the target on the detector. In most cases this intra-spectrum wavelength uncertainty is < 10 km s -1 , but in a few extreme cases it will introduce a ∼ 15 km s -1 shift from the reddest to the bluest wavelengths in a single COS grating mode (See Figure 4 from Linsky et al. 2012). To determine how important the errors due to pointing and calibration are in the COS data, we focus on the H 2 lines. We use the line measurements from K. France (personal communication, see also France et al. 2012) for P(2)(0-5) at 1398.95 ˚ A, R(11)(2-8) at 1555.89 ˚ A, R(6)(1-8) at 1556.87 ˚ A, and P(5)(4-11) at 1613.72 ˚ A. France et al. (2012) show that in the case of DK Tau, ET Cha (RECX 15), HN Tau, IP Tau, RU Lup, RW Aur, and V1079 Tau (LkCa 15), some H 2 lines show a redshifted peak and a blueshifted low-level emission, which makes them asymmetric. Ignoring these stars, the four H 2 lines are centered at the stellar rest velocity, with a standard deviations of 7.1, 6.5, 6.1, and 12.1 km s -1 , respectively. The large scatter in the P(5)(4-11) line measurement is partly the result of low signal-to-noise in this region of the spectrum. Unlike the results reported by Linsky et al. 2012), we do not observe a systematic increase in the line center as a function of wavelength, neither star by star nor in the average of all stars for a given line. This is likely the result of the different acquisition procedure followed here. For the STIS data, the average standard deviation in all the H 2 wavelengths is 4.4 km s -1 . This shows that the errors in the COS wavelength scale are smaller than the 15 km s -1 reported in the man- uals and that no systematic shifts with wavelength are present. For the rest of the paper we will assume that the pointing errors in the COS data result in a velocity uncertainty of 7 km s -1 (the average of the first three H 2 lines considered) while the STIS errors are 5 km s -1 . We do not correct the spectra for the H 2 velocities, as it is not clear how to implement this correction in the case of stars with asymmetric H 2 lines.",
"pages": [
4,
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]
},
{
"title": "3. INTRODUCTION TO THE SHAPE OF THE LINES",
"content": "Figures A.1 to A.5 in the Appendix show the lines that we are discussing in this paper. Each line is plotted in velocity space (km s -1 ) centered on the stellar photospheric rest frame. The ordinate gives flux density in units 10 -14 erg sec -1 cm -2 ˚ A -1 . In the case of doublets, the nominal wavelength of the strongest member of the line is set to zero velocity, and the positions of the strongest H 2 lines are marked with dashed lines. Dotted lines mark the positions of other features in the spectra. The plotted spectra have been smoothed by a 5-point median. The C iv , Si iv , and N v lines are resonance doublets with slightly offset upper levels. For each doublet, both lines should have the same shape when emitted. This redundancy allow us to identify extra spectral features and to distinguish real features from noise. If the lines are emitted from an optically thin or effectively thin plasma, their flux ratio should be 2:1. If the lines are emitted from a medium that has a thermalization depth smaller than its optical depth (i.e. the medium is optically thick, but not effectively thin) the flux ratio will tend to 1. For a plasma at rest in coronal ionization equilibrium, the peak ion abundance occurs at temperatures of log(T)=5.3 (K) for N v , log(T)=5.0 (K) for C iv , log(T)=4.9 (K) for Si iv , and log(T)=4.7 (K) for He ii (Mazzotta et al. 1998). In the context of the magnetospheric accretion paradigm, the velocities of the postshock gas are high enough that the collisional timescales are longer than the dynamical timescale (Ardila 2007). This implies that the post-shock gas lines may trace lower temperature, higher density plasma than in the coronal ionization equilibrium case. The pre-shock plasma is radiatively ionized to temperatures of ∼ 10 4 K. If the preand post-shock regions both contribute to the emission, conclusions derived from the usual differential emission analyses are not valid (e.g. Brooks et al. 2001). In the case of non-accreting stars, the resonance N v , C iv , and Si iv lines are collisionally excited while the He ii line is populated by radiative recombination in the X-ray ionized plasma (Zirin 1975). Figure 1 compares the hot gas lines of BP Tau and the WTTS V396 Aur. These stars provide examples of the main observational points we will make in this paper: emission component. This is the result of N i circumstellar absorption. The general statements above belie the remarkable diversity of line profiles in this sample. Some of this diversity is showcased in Figure 2. For analysis and interpretation, we focus on the C iv doublet lines, as they are the brightest and 'cleanest' of the set, with the others playing a supporting role. In order to test the predictions of the magnetospheric accretion model, in section 5.3 we perform a Gaussian decomposition of the C iv profiles, and representative results are shown in Figure 2. TW Hya has a profile and decomposition similar to those of BP Tau: strong narrow component (NC) plus a redshifted, lower-peak broad component (BC). ∼ 50% (12/22) of the stars for which a Gaussian decomposition is possible show this kind of profile and ∼ 70% (21/29) of CTTSs in our sample have redshifted C iv peaks. HN Tau also has a NC plus a redshifted BC, but the former is blueshifted with respect to the stellar rest velocity by 80 km s -1 . For IP Tau and V1190 Sco, the BC is blueshifted with respect to the NC. The magnetospheric accretion model may explain some of objects with morphologies analogous to TW Hya but blueshifted emission requires extensions to the model or the contributions of other emitting regions besides the accretion funnel (Gunther & Schmitt 2008). Figure 3 shows the C iv and He ii lines in all the WTTSs, scaled to the blue wings of the lines. For all except for V1068 Tau (Lk Ca 4) and V410 Tau, the C iv and He ii lines appear very similar to each other in shape, width, and shift. They are all fairly symmetric, with velocity maxima within ∼ 20 km s -1 of zero, and FWHM from 60 to 100 km s -1 . V1068 Tau and V410 Tau have the largest FWHM in both C iv and He ii . The lines of these two stars appear either truncated or broadened with respect to the rest of the WTTSs. In the case of V410 Tau, v sin i = 73 km s -1 (G/suppressl¸ebocki & Gnaci'nski 2005), which means that rotational broadening is responsible for a significant fraction of the width. However, V1068 Tau has v sin i = 26 km s -1 . For the rest of the WTTSs, v sin i ranges from 4 to 20 km s -1 (G/suppressl¸ebocki & Gnaci'nski 2005; Torres et al. 2006). To characterize the line shapes of He ii and C iv we have defined non-parametric and parametric shape measurements. The non-parametric measurements (Table 5) do not make strong assumptions about the line shapes and they provide a intuitive summary description of the line. These are the velocity at maximum flux, the full width at half maximum (FWHM), and the skewness, defined in Section 5.2. For C iv we have also measured the ratio of the 1548 ˚ A to the 1550 ˚ A line, by scaling the line wings to match each other. This provides a measure of the line optical depth. The parametric measurements (Tables 6 and 7) assume that each C iv and He ii line is a combination of two Gaussians. Table 8 lists the flux measurements that we will be considering in the following sections. The fluxes are obtained by direct integration of the spectra over the spectral range listed in the table, after subtracting the continuum and interpolating over known blending features (see Section 5).",
"pages": [
5,
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},
{
"title": "4. THE RELATIONSHIP BETWEEN ACCRETION RATE AND C IV LUMINOSITY",
"content": "As mentioned in the introduction, Johns-Krull et al. (2000) showed that the accretion rate is correlated with excess C iv luminosity. The excess C iv luminosity is obtained by subtracting the stellar atmosphere contribution to the observed line. They estimated the stellar contribution to be 6 × 10 -5 L /circledot for a 2R /circledot object. They also showed that the correlation of C iv excess luminosity with accretion rate is very sensitive to extinction estimates. More recently, Yang et al. (2012) obtain a linear correlation between the C iv luminosity (from STIS and ACS/SBC low-resolution spectra) and the accretion luminosity (from literature values) for 91 CTTSs. Here we show that our data are consistent with a correlation between accretion rate C iv luminosity and explore the role that the lack of simultaneous observations or different extinction estimates play in this relationship. Figure 4 compares accretion rates (references given in Table 1) with the C iv line luminosities. Blue diamonds correspond to objects with simultaneous determinations of ˙ M and L CIV when available (Ingleby et al. 2013) and green diamonds correspond to ˙ M determinations for the rest of the objects. Note that the accretion rate estimates from Ingleby et al. (2013) are derived using the extinction values from Furlan et al. (2009, 2011). Using only the simultaneous values (blue diamonds), we obtain a Pearson product-moment correlation coefficient r=0 . 73 (p-value=0.3% 19 ) while with all values we obtain r=0 . 61 (p-value= < 0.05%). The difference between using all of the data or only the simultaneous data is not significant for the purposes of the correlation. Including all data, we obtain log ˙ M = ( -5 . 4 ± 0 . 2)+ (0 . 8 ± 0 . 1) log L CIV /L /circledot , where ˙ M is given in M /circledot /yr and the errors indicate 1 σ values obtained by the bootstrap method. The correlation is plotted in Figure 4. As argued by Johns-Krull et al. (2000), the observed relationship between log L CIV /L /circledot and accretion rate is very sensitive to extinction estimates. This is shown in Figure 4, for which the C iv luminosities indicated by the black triangles were calculated using the extinction esti- 19 The p-value is the probability of obtaining a value of the test statistic at least as extreme as the observed one, assuming the null hypothesis is true. In this case, it is the probability that the Pearson's r is as large as measured or larger, if the two quantities are uncorrelated. We reject the null hypothesis if p-value ≤ 0.05. mates from Furlan et al. (2009, 2011), for all the targets we have in common with that work. For log L CIV /L /circledot and log ˙ M the correlation is weakened when using the Furlan et al. (2009, 2011) extinction values: the value of the Pearson's r is 0 . 44, with p-value=5%. The extinction values we adopt in this paper come from a variety of sources, but a significant fraction come from Gullbring et al. (1998, 2000). They argue that the colors of the WTTSs underlying the CTTSs are anomalous for their spectral types, which biases the near-IR extinction estimates. They obtain the extinctions reported here by deriving models of the UV excess in CTTSs. For other stars, we have adopted extinction estimates based on spectroscopic observations of the accretion veiling, when possible. The Furlan et al. (2009, 2011) extinction values are obtained by de-reddening the observed near-infrared colors until they match the colors for the target's spectral type. They are significantly larger than the values we adopt in this paper, resulting in larger estimates of the C iv luminosity. Differences in the extinction estimates can have a substantial impact in the adopted flux, as a 10% increase in the value of A V results in a 30% increase in the de-reddened C iv line flux. Figure 4 also shows the relationship between accretion rate and C iv luminosity derived by Johns-Krull et al. (2000) (their equation 2) assuming that all stars have a radius of 2R /circledot . They obtained their relationship based on the accretion rates and extinctions from Hartigan et al. (1995). Both of those quantities are higher, on average, than the ones we adopt here, and so their correlation predicts larger accretion rates. Note that the relationship from Johns-Krull et al. (2000) is not defined for stars with excess C iv surface fluxes smaller than 10 6 erg sec -1 cm -2 . Overall, there is enough evidence to confirm that for most CTTSs the C iv line luminosity is powered primarily by accretion, and we will adopt this hypothesis here. However, the exact relationship between accretion rate and C iv luminosity remains uncertain. This is not surprising considering the complexity of the processes that contribute to the line flux, as we show in this work. The dominant uncertainty is the exact value of the extinction, which depends on the assumed stellar colors and on the shape of the extinction law in the UV (Johns-Krull et al. 2000; Calvet et al. 2004a). We do not detect a monotonic decrease in the CTTSs C iv luminosity as a function of age, for the range of ages considered here (2 - 10 Myrs). We do not observe a significant difference in the C iv luminosities of the TDs as compared with the rest of the sample, consistent with the results from Ingleby et al. (2011) who found no correlations between FUV luminosities and tracers of dust evolution in the disk.",
"pages": [
6
]
},
{
"title": "5. THE C IV LINE SHAPE",
"content": "The C iv lines of CTTSs are generally redshifted, broad (with emission within ∼ 400 km s -1 of the stellar rest velocity), and asymmetric to the red (positive skewness). We will show that none of these characteristics is correlated with the line luminosity or with accretion rate.",
"pages": [
6
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},
{
"title": "5.1. Comparing the two C IV lines",
"content": "If optically thin, both C iv lines form should have the same shape. Differences between the components can help us discover the presence of extra sources of absorption or emission. To exploit this redundancy, in Figure A.6 we plot both members of the C iv doublet, with the 1550 ˚ A line scaled to match the 1548 ˚ A one. The scaling is done by matching the line peaks or the line wings from ∼ 0 to ∼ 150 km s -1 . We expect this scaling factor to be 2 for optically thin or effectively thin emission. However, as we discuss in Section 5.2.4, the opacity characteristics of the broad and narrow line components are different, and the overall scaling factor may not be the best predictor of opacity. Figure A.6 shows that the line wings tend to follow each other closely, at least until about +200 to +300 km s -1 , when the 1548 ˚ A line start to bump into the 1550 ˚ A one. The 1548 ˚ A line is usually contaminated by the H 2 line R(3)1- 8 (1547.3 ˚ A), at -167 km s -1 with respect to its rest velocity (e.g. DK Tau). Figure A.6 also reveals examples of extra emission at ∼ -100 km s -1 which do not appear in the 1550 ˚ A line (AK Sco, DE Tau, DK Tau, DR Tau, HN Tau A, RU Lup, and UX Tau A). We tentatively identify emission from Fe ii (1547.807, -73.6 km s -1 from the rest velocity of the 1548 C iv line), C ii (1547.465; -139.8 km s -1 ), and Si i (1547.452 ˚ A, -142.3 km s -1 ; 1547.731 ˚ A, -88.3 km s -1 ), for these stars, as responsible for the emission to the blue of the C iv 1548 ˚ A line. To measure the C iv line flux listed in Table 8 we subtracted the continuum and interpolated over the H 2 R(3)1-8 line and the Si i , C ii , Fe ii -complex, when present. The resulting spectrum is then integrated from -400 km s -1 to 900 km s -1 of the 1548 C iv line. We also detect the CO A-X (0-0) absorption band at 1544.4 ˚ A (-730 km s -1 from the 1548 ˚ A line, see France et al. 2011; McJunkin et al. 2013) in a significant fraction of the sample. The wing of the CO absorption may extend to the blue edge of the 1548 C iv line. However its impact in the overall C iv flux is negligible. Note that the red wings of each C iv line for DX Cha, RW Aur A, DF Tau, and RU Lup do not follow each other, and AK Sco and CS Cha have extra emission features near the 1550 ˚ A. For DX Cha we will argue in section 8 that the peculiar shape of the C iv lines can be explained by the existence of a hot wind. The strange appearance of C iv lines of RW Aur A is due to a bipolar outflow (See Figure 5). France et al. (2012) show that the H 2 lines from RW Aur A are redshifted by ∼ 100 km s -1 in at least two progressions ([v',J']=[1,4] and [1,7]), as they originate in the receding part of the outflow. In the observations we present here, the C iv doublet lines are blueshifted by ∼ -100 km s -1 , as can be seen from the position of the 1550 C iv line. This points to an origin in the approaching part of the outflow. The net result of these two outflows is that the blueshifted 1548 C iv line is buried under the redshifted H 2 emission. RW Aur A is the only unequivocal example of this coincidence in the current dataset. For HN Tau A the H 2 R(3) 1-8 line is also redshifted, although the redshift (+30 km s -1 ) is within the 2 σ of the error introduced by the pointing uncertainty (see also France et al. 2012). Therefore, HN Tau A may be another object for which we are observing two sides of the outflow, although the velocities of the blueshifted and redshifted sides do not match as well as in RW Aur. For DF Tau, the apparent red wing in the 1550 ˚ A line is an artifact of the line scaling, because for this star the ratio between the two C iv lines is almost 3, indicating either extra absorption or emission in one of the components. Unlike the case in RW Aur A, redshifted H 2 emission is not responsible for the extra emission, as the R(3) 1-8 line is observed at -167 km s -1 . Other observations show that the ratio between the two lines remains high among different epochs (Section 5.4). The origin of this extra emission in the 1548 ˚ A line or absorption in the 1550 ˚ A one remains unexplained in this work. For RU Lup, the extra 'bump' 200 km s -1 to the red of the 1548 ˚ A C iv line is also present in other epoch observations of the system and remains unexplained in this work. AK Sco and CS Cha also present anomalous profiles, but of a different kind. AK Sco shows a broad emission -250 km s -1 from the 1550 ˚ A line, not present in the 1548 ˚ A line. CS Cha also shows and extra emission in the 1550 ˚ A line at ∼ -100 km s -1 and ∼ +200 km s -1 .",
"pages": [
6,
7
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},
{
"title": "5.2.1. The velocity of the peak emission as a function of luminosity and accretion rate",
"content": "The velocity of peak emission for each line is defined as the mean velocity of the top 5% of the flux between -100 and +100 km s -1 (for the 1548 C iv line) or between +400 and +600 km s -1 (for the 1550 ˚ A line). This is calculated from spectra that have been smoothed by 5point median. The velocity of peak listed in Table 5 is the average of both lines. The C iv line is centered or redshifted for 21 out of 29 CTTSs, uncorrelated with line luminosity or accretion rate (Figure 6 - Top). The most significant exceptions are: DK Tau which shows two emission peaks, one blueshifted and the other redshifted, present in both lines; HN Tau A, for which both C iv lines have an emission peak at ∼ -80 km s -1 from their rest velocity 20 ; RW Aur A, for which the peak of the 1550 ˚ A line is at -86 km s -1 . In the case of AK Sco there is extra emission in the 1548 ˚ A line that may be due to Si i but there is also extra emission ∼ -250 km s -1 from the 1550 ˚ A line. It is unclear therefore, whether the C iv lines are centered or blueshifted, although both the Si iv and the N v lines (which follow the C iv lines) are well centered. Those four stars are among the higher accretors in the sample and have jets seen in the high-velocity components of forbidden lines (Hartigan et al. 1995). Four other CTTSs have slightly blueshifted peaks, but with velocities > -4 km s -1 . The average velocity at maximum flux for the C iv CTTSs lines is V Max = 18 ± 4 km s -1 (Table 9. In this paper the uncertainties refer to uncertainties in the calculated mean values), not including the four stars with 20 We note that for HN Tau A, the radial velocity (4.6 ± 0.6 km s -1 ), derived by Nguyen et al. (2012), which we use here, deviates significantly from the velocity of the surrounding molecular cloud (21 km s -1 ; Kenyon & Hartmann 1995). Using the cloud velocity would shift the C iv lines to the blue even more. France et al. 2012 finds that the H 2 lines have maxima at ∼ 20 km s -1 from the rest velocity, although the line asymmetry suggest components at lower velocity. strong blueshifts (DK Tau, HN Tau A, and RW Aur A for which outflowing material is contributing to the profile, nor AK Sco, for which the two C iv lines are different from each other). We have argued that the velocity uncertainties for this COS dataset are ∼ 7 km s -1 . A onesample Kolmogorov - Smirnov (KS) test confirms that the probability of the observed distribution being normal centered at zero, with 1 σ = 7 km s -1 is negligibly small. The conclusion is the same if we take 1 σ = 15 km s -1 , the nominal wavelength error for COS . In other words, the overall redshift of the CTTSs C iv lines is significant. The WTTSs are also redshifted, with V Max = 11 ± 4 km s -1 . A two-sided KS test comparing the CTTSs and WTTSs V Max distribution gives p-value=0.6, which indicates that they are consistent with the null hypothesis. In other words, although the mean velocity of the WTTSs is ∼ 2 σ less that the mean CTTS velocity, the observations are consistent with the two quantities having the same distribution. Linsky et al. (2012) have shown that for rotation periods such as those observed in T Tauri stars, the Si iv and C iv lines in low-mass dwarfs present redshifts of ∼ 7 km s -1 , the result of gas flows produced by magnetic heating. Given the scatter in their sample, our redshifts in WTTSs are consistent with theirs.",
"pages": [
7,
8
]
},
{
"title": "5.2.2. The line width as a function of luminosity and accretion rate",
"content": "Overall, most of the CTTSs C iv lines show detectable emission within ± 400 km s -1 of the nominal velocity. The FWHM is measured using the 1550 ˚ A line of the smoothed C iv spectrum. The smoothing is the same used to measure the velocity of maximum flux. Figure 6 (Second row) shows that, as a group, the C iv WTTSs lines are narrower ( FWHM = 90 ± 10 km s -1 ) than the CTTSs ones ( FWHM = 210 ± 20 km s -1 ). The p-value of the KS test comparing the two samples is 0.02, implying that the difference is significant. The WTTS also have a smaller FWHM range (from 60 to 145 km s -1 ) than the CTTSs (from 45 to 387 km s -1 ). Note that the FWHM of the WTTSs is comparable to that of the CTTSs in some stars, the result of the strong narrow component dominating the CTTS lines (see for example the C iv lines for DS Tau in Figure A.2). The FWHMclearly fails to capture most relevant information regarding the line as it does not take into account the multi-component nature of the lines. The observed FWHM is uncorrelated with line luminosity or accretion rate. The plots in Figure 6 (Second row) suggest that the FWHM scatter increases with accretion rate, but this apparent increase is not statistically significant.",
"pages": [
8
]
},
{
"title": "5.2.3. The skewness as a function of luminosity and accretion rate",
"content": "In Figure 6 (Third row) we plot skewness versus C iv luminosity. The skewness compares the velocity of the peak to the mean velocity of the profile. It is defined as ( V Max -V ) / ∆ V , where V Max is the velocity at maximum flux and V is the flux-weighted mean velocity over an interval ∆ V centered on the maximum velocity. To calculate the skewness of the C iv line, we subtracted the continuum and interpolated over the H 2 lines present in the C iv intervals. For the 1548 and 1550 ˚ A lines, V is measured within ± 250 km s -1 and ± 150 km s -1 from the maximum, respectively. To make the values comparable, we normalize to ∆ V = 250 km s -1 in each line. The 1548 ˚ A and 1550 ˚ A values are then averaged. Qualitatively, values of skewness within ∼ ± 0 . 02 indicate a symmetric line. Positive values indicate a line extending to the red. The absolute value of the skewness serves as a quantitative measure of asymmetry. 70% (20/29) CTTS have skewness greater or equal to zero, with 52% (15/29) having positive ( > 0 . 02) skewness. All the WTTS have skewness values consistent with symmetric lines. A KS test comparing both populations shows that the difference in asymmetry is significant (pvalue=0.002). As with FWHM, the skewness is uncorrelated with line luminosity or accretion rate.",
"pages": [
8
]
},
{
"title": "5.2.4. The line scaling as a function of luminosity and accretion rate",
"content": "The fourth row panels of Figure 6 compare the ratios of the 1548 ˚ A to the 1550 ˚ A C iv lines as a function of L CIV (left) and ˙ M (right). These are not flux ratios, but the scaling factors used to match both line profiles in Figure A.6. As mentioned before (Section 5.1), the scaling factors between the C iv lines are indicative of the line's optical depth compared to the thermalization depth (Table 8). All WTTSs and over half of the CTTSs (19/29 - 66%; 6/7 - 86% of TD, 12/22 - 55% of the non-TD) have ratios > 1.65, consistent within the errors with thin or effectively thin emission. The difference in line opacity between stars with TDs and those without is not significant. For the rest of CTTSs, 10/29 (34%) have ratios which are consistent with small absorption mean free paths. We do not observe a correlation between this measure of opacity and either line luminosity or accretion rate. The apparent increase in scatter at high accretion rates is not significant, according to a two-sided KS test (pvalue=0.6). Furthermore, objects with C iv ratios < 1 . 7 are found for all luminosities and accretion rates. As indicated in Section 5.1, DF Tau presents a particular case for which the scaling factor is significantly > 2, an impossible value unless there is extra emission in the 1548 ˚ A line or extra absorption in the 1550 ˚ A one.",
"pages": [
8
]
},
{
"title": "5.2.5. The role of the inclination in the measured luminosity and accretion rate",
"content": "Geometric explanations are often invoked in the literature to explain the shapes of these emission lines. For example, Lamzin et al. (2004) suggest that the observed accretion in TW Hya occurs at low stellar latitudes, and that the lack of separate line components coming from the pre- and post-shock in other stars may be due to equatorial layer accretion. Figure 6 (Bottom) shows the relationship between inclination and line luminosity or accretion rate. We do not observe particularly larger or smaller fluxes at high or low inclinations. According to the KS test, the distributions of L CIV and ˙ M are the same between objects with i > 45 · and those with i ≤ 45 · . Also, the distribution of inclinations for objects with high ˙ M or high L CIV is statistically the same that for objects with low ˙ M or low L CIV . In particular, the apparently empty region in the panel at high inclinations and low accretion rates is not significant and does not provide evidence that the disk or the accretion flow are obscuring the accretion diagnostics. Furthermore, we do not find any significant correlations between the line ratio, V Max , FWHM, or skewness, and inclination. If the C iv lines originate in a localized accretion stream one would expect to detect more stars at low inclinations than at high inclinations: for face-on systems the accretion stream will always be visible, while for systems almost edge-on this is not the case, if the region below the disk is blocked from view. For a random distribution of accretion spot positions in the stellar hemisphere, we expect to see an accretion spot in 58% of stars with 0 · < i < 45 · and in 42% of stars with 45 · < i < 90 · (ignoring disk flaring). In our sample, we have 37% of targets with inclinations larger than 45 · , and the standard deviation of this count is 10%. Therefore, the observed difference in the number of stars with high and low inclinations is not significant. Over a hundred CTTSs with known inclinations are needed before we can distinguish 42% from 58% at the 3 σ level. The current inclination dataset is not complete enough to reveal geometric information about the C iv lines. On the other hand, one could assume that the C iv UV lines are not emitted from a particular place but covers the whole star. For a given star, the C iv luminosity should then decrease linearly, by a factor of two, as the inclination increases from 0 · to 90 · . We do not observe this effect either, indicating, at least, that the observed scatter is dominated by intrinsic differences in the objects and not by geometry.",
"pages": [
8,
9
]
},
{
"title": "5.2.6. Conclusions from the non-parametric analysis",
"content": "In conclusion, WTTS and CTTS C iv lines have comparable velocities at maximum flux, but the CTTS lines are generally broader and more asymmetric. In the case of CTTSs, neither the velocity of maximum flux, the FWHM, the skewness of the line, nor the ratio between the two C iv lines are correlated with C iv luminosity or accretion rate. The right column panels from Figure 6 do show increased scatter in these quantities as the accretion rate increases, suggesting that objects with large accretion rates have more diverse line shapes. However, the differences in the distributions (of the velocity, FWHM, skewness) with high and low accretion rates are not significant. More observations of stars with accretion rates ≤ 4 × 10 -9 M /circledot /yr are needed. Among the three pairs of shape quantities ( V Max vs. FWHM, FWHM vs. Skewness, V Max vs. Skewness) there are no significant correlations. The scaling of the 1548 ˚ A line to the 1550 ˚ A one should be 2 if the lines are emitted from a thin or effectively thin medium. This is the case in all the WTTSs and in about 70% of CTTSs. This measure of the opacity is not correlated with accretion rate or line luminosity. However, in Section 5.3.1 we show that the NC of the line is correlated with accretion rate. There are no correlations between line shape parameters and inclination or between inclination and accretion rate or line luminosity, but we conclude that the data are not complete enough for inclination to be a strong descriptor in the sample.",
"pages": [
9
]
},
{
"title": "5.3. C IV Gaussian decomposition",
"content": "Different regions of the T Tauri system may be contributing to the C iv lines. In the context of the magnetospheric accretion paradigm, the pre- and post-shock regions should be the dominant sources of the observed line emission. Shocks in an outflow, hot winds, and the stellar atmosphere may also contribute to the emission. To examine the line kinematics of the different regions we decompose each C iv line in one or two Gaussian functions. We are not asserting that the mechanism giving origin to the lines produces Gaussian shapes, although turbulent flows will do so (Gunther & Schmitt 2008). Representative decompositions are shown in Figure 2. The primary goal of the Gaussian decomposition is to obtain widths and centroids for the main line components. According to the magnetospheric accretion model, post-shock emission lines should have small velocity centroid distribution about the stellar rest frame velocity, and if no turbulence is present, the lines should be narrow. Pre-shock emission lines, which likely originate in a larger volume upstream from the accretion flow (Calvet & Gullbring 1998), should have larger velocity centroid distribution in the stellar rest frame and broader lines. Table 6 presents the results of fitting one or two Gaussians to each of the C iv lines, after subtracting the continuum and interpolating over the H 2 line R(3)1-8, and the Si i , C ii , Fe ii -complex, when present. For each target, we assume that the C iv lines are always separated by 500.96 km s -1 , and that they have the same shape. This results in a 4-parameter fit when fitting one Gaussian to each line: Height for the 1548 ˚ A line; height for the 1550 ˚ A line; width σ s; centroid velocity for the 1548 ˚ A line. When fitting two Gaussians to each line, we have an 8-parameter fit: For the 1548 ˚ A line, the heights of the broad (A BC ) and narrow (A NC ) components; analogous parameters for the 1550 ˚ A line; σ for the broad component; σ for the narrow component; centroid velocities for the broad (v BC ) and narrow (v NC ) components of the 1548 ˚ A line. Table 6 also indicates whether the data were taken with COS (possible systematic wavelength error assumed to be 7 km s -1 ) or STIS (possible systematic wavelength error of 3 km s -1 ). We also list the parameters derived for the multi-epoch observations of BP Tau, DF Tau, DR Tau, RU Lup, and T Tau N that we will consider in Section 5.4. Wood et al. (1997) analyzed the C iv lines of 12 stars with spectral types F5 to M0. They found that the observed line profiles could be better fit with both a narrow and a broad Gaussian component, than with a single Gaussian component. For the type of stars included in their sample (dwarfs, giants, spectroscopic binaries, and the Sun), the C iv line is a transition-region line, as it is in the WTTSs. Because of this, we fitted both NCs and BCs to all the WTTSs, except V410 Tau. For V410 Tau, each C iv line can be well fitted with only one NC, although this may just be the result of the low S/N in the spectrum. When comparing the two C iv lines in section 5.1, we mentioned that AK Sco, CS Cha, DX Cha, DF Tau, RU Lup, and RW Aur A were objects for which the line dou- blet members had different shapes and/or extra unidentified emission in one of the doublet members. We do not perform the Gaussian decomposition for RW Aur A or DX Cha. For AK Sco, we list the parameters derived from the Gaussian fits (Table 6), but we do not use these results when exploring correlations. For the rest, we interpolate the profiles over the apparent the extra emission. For 4 objects (CY Tau, DM Tau, ET Cha, and UX Tau A) we decompose the C iv lines in only one broad Gaussian component. For the rest of the CTTSs, the decomposition requires both a narrow and a broad Gaussian components, and Figure 7 shows the distributions of velocity centroids and FWHM. Average velocity and FWHM values are given in Table 9. Typical full widths at half maxima of CTTS NCs range from 50 to 240 km s -1 , with an average of 130 km s -1 , while BCs widths range from 140 to a 470 km s -1 , with an average of 350 km s -1 . The velocity centroids range from -100 km s -1 to 200 km s -1 . The BC velocity is larger than the NC velocity in 70% of the CTTS sample, and the distribution of BC velocities tend to be more positive ( V BC ∼ 40 km s -1 ) than that of the NC velocities ( V NC ∼ 30 km s -1 ), giving some of the profiles the characteristic 'skewed to the red' shape.",
"pages": [
9,
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]
},
{
"title": "5.3.1. The optical depth as a function of accretion",
"content": "Figure 8 (Top) shows the ratio of the height of the NC in the 1548 ˚ A line to the NC in the 1550 ˚ A C iv line as a function of accretion rate, as well as the ratio of the heights of the BCs. Most observations are grouped around 2 although there is a lot of scatter, particularly at high accretion rates. The plot reveals that the NC of DF Tau is anomalous ( > 2), while the BC has an optically thin ratio. Individual Gaussian components of other objects (CS Cha, DR Tau) are also anomalous. Furthermore, there is a population of objects with NC ratios close to 1: AA Tau, DE Tau, DR Tau, RU Lup,, SU Aur, and T Tau. However, the difference in the distribution of BC ratios and NC ratios is marginal according to the KS test (p-value=6%), and the correlation of the ratio of NC luminosities with accretion rate is not significant (Pearson's r=-0.37, p-value=10%). We conclude that for most objects, both the NC and BC ratios are close to 2, although there is considerable scatter, and that there are some peculiar objects at accretion rates > 4 × 10 -9 M /circledot /yr. The bottom panel of Figure 8 shows that the contribution of the NC to the overall profile increases with accretion rate. For low accretion rates ( < 4 × 10 -9 M /circledot /yr), the average NC contribution to the luminosity is ∼ 20% while for high accretion rates it is ∼ 40% on average. While the Pearson's r=0.4 (p-value=6%) suggest that this correlation is not significant, this statistical test assumes that the quantities being compared are sampled from a bivariate Gaussian distribution. This is likely not the correct assumption for the heterogeneous sample of CTTSs being considered here. A better correlation test uses the Kendall rank correlation statistic, which considers only (non-parametric) rank orderings between the data (Feigelson & Jogesh Babu 2012). The Kendall's τ =0.30 (p-value=0.05) is at the threshold of what we consider significant. If the increase in accretion rate is due to larger accreting area or larger density, and both the narrow and broad components are emitted from regions that are optically thin or effectively thin, no correlation should be observed. This is because the rate of collisional excitation will change linearly with density and both lines will increase at the same rate. The observed correlation implies that the region responsible for the BC may be becoming optically thick at high accretion rates. As in main-sequence stars, the WTTS line shapes (blue labels in Figure 8) are characterized by a strong NC and a weak BC (Wood et al. 1997; Linsky et al. 2012) and tend to have stronger NC contributions to the total flux than low accretion rate CTTSs. The luminosity in NCs and BCs, in both WTTSs and CTTSs increases with total C iv luminosity (not shown). The bottom panel of Figure 8 raises the issue of the transition from WTTS to CTTS as a function of accretion rate. The WTTSs generally have strong NC, while low accretion rate objects have very weak NC, compared to BC. Does the accretion process suppress the NC present in the WTTSs or does it enhance the BC? We believe the latter to be true. The lowest accretion rate object shown in Figure 8 (Bottom) is EP Cha (RECX 11). For this star the total C iv luminosity is ∼ 6 times larger than for most of the WTTSs, with the exception of V1068 Tau. However, the NC luminosity of EP Cha is 4 × 10 -6 L /circledot , similar to that of a low-luminosity WTTSs. In other words, most of the extra C iv luminosity that distinguishes this CTTS from the WTTSs is due to the generation of the BC. This suggest that the accretion process generates first a BC, with the NC becoming increasingly important at larger accretion rates. From this it follows that all accreting stars should show a BC. In average, the flux in the BC grows at a slower rate than the flux in the NC as the accretion rate increases, but it is not clear why some stars develop a strong NC (like DS Tau) while others do not (like GM Aur). These statements may not be valid outside the mass or accretion rate range of the sample considered here. For example, France et al. (2010) present UV spectra of the brown dwarf 2M1207, from which they derive an accretion rate of 10 -10 M /circledot /yr, comparable to that of EP Cha. This value is derived using the calibration between C iv flux and accretion rate from Johns-Krull et al. (2000). On the other hand Herczeg et al. (2009) derives an accretion rate of 10 -12 M /circledot /yr based on the Balmer excess emission. At any rate, each C iv line can be fit with only one, very narrow, Gaussian component with FWHM=36.3 ± 2.3 km s -1 . 21 This is then a case of a low-mass accreting young object without a BC. The very narrow C iv lines may be the result of the smaller gas infall velocity in 2M1207 ( ∼ 200 km s -1 , using the stellar parameters from Riaz & Gizis 2007) compared to the sample presented here ( ∼ 300 km s -1 ). The smaller gas infall velocity will result in lower turbulence broadening. 21 In their published analysis, France et al. (2010) fit the C iv lines of 2M1207 with two Gaussian components each. However, that analysis is based on an early reduction of the COS data. A new re-processing and new Gaussian decomposition shows that the C iv line can be fit with one Gaussian component (K. France, personal communication). 5.3.2. The kinematic predictions of the magnetospheric accretion model For the magnetospheric accretion model, the gas speed in the accretion flow before the shock should reach velocities ∼ 300 km s -1 for typical stellar parameters (Calvet & Gullbring 1998), although the interplay between line-of-sight and the complex magnetospheric structure may result in line-of-sight velocities smaller than this. If it originates primarily in the accretion shock region, the C iv line emission (as well as the Si iv , N v , and He ii lines) comes from plasma very close to the stellar surface and the gas in this region should be moving away from the observer. Therefore, we expect that the velocity of the flow should be positive (v BC > 0, v NC > 0). If the dominant emission in the broad and narrow components comes only from the pre- and post-shock regions, respectively, then v BC > v NC . The velocity of the post-shock gas decreases after the shock surface, and depending on the origin of our observational diagnostic, we may observe velocities ranging from 4 × less than the pre-shock gas velocity to zero (Lamzin 1995, 2003a,b). Observationally, v BC and v NC are uncorrelated with accretion rate or line luminosity. Figure 9 compares the velocity centroids of the broad and narrow components, for 22 CTTSs and 5 WTTSs. Black labels indicate CTTSs observed with COS , while orange labels are CTTSs observed with STIS . The plot also shows, in red, the velocity centroids of the WTTSs, all observed with COS . The errors in these velocities are dominated by the systematic wavelength scale error, illustrated by the boxes on the upper-left corner of Figure 9. We have argued in Section 5.2 that the errors in the COS dataset are ∼ 7 km s -1 . Errors in the wavelength scale cause the data points to move parallel to the dotted line. We focus first on the upper right quadrant of Figure 9, those objects with both v BC > 0 and v NC > 0. The hatched region of the plot corresponds to v BC ≥ 4 v NC . As we can see, only TW Hya and CY Tau reside fully within this region. In addition, errors in the wavelength calibration may explain why some objects (AA Tau, BP Tau, CS Cha, DE Tau, DN Tau, DS Tau, EP Cha, MP Mus, V1079 Tau, and V4046 Sgr) reside away from the hatched region. Note that while the rotation of the star may contribute to v BC and v NC , typical values of v sin i for CTTSs are 10 to 20 km s -1 , and so the velocity pair would only move to higher or lower velocities by up to this amount, parallel to the dotted line. The other 10 objects are 'anomalous,' either because one or both of the velocities is negative and/or v BC /lessorsimilar v NC , beyond what could be explained by pointing errors. Stellar rotation alone will not bring these objects to the hatched region. As a group, they all have accretion rates larger than 4 × 10 -9 M /circledot /yr, and they make up half of the objects with accretion rates this large or larger. From the point of view of the magnetospheric accretion paradigm, they present a considerable explanatory challenge. In turn, these anomalous objects come in two groups: those for which both components are positive or close to zero but v BC < v NC (DF Tau, perhaps DR Tau, GM Aur, RU Lup, SU Aur), and those objects for which one or both of the components are negative (DK Tau, HN Tau, IP Tau, T Tau N, V1190 Sco). Emission from the latter group may have contributions from outflows or winds.",
"pages": [
10,
11
]
},
{
"title": "5.4. The accretion process in time: multi-epoch information in C IV",
"content": "Line flux variability in CTTSs occurs on all time scales, from minutes to years. It is therefore relevant to ask how does the line shape change in time and what is the impact of these changes on the general statements we have made. High-resolution multi-epoch observations of the C iv lines are available for a subset of our objects, plotted in Figure 10. The spectra were obtained from Ardila et al. (2002) (GHRS observations: BP Tau, DF Tau, DR Tau, RU Lup, T Tau, and RW Aur), Herczeg et al. (2005) ( STIS : RU Lup), Herczeg et al. (2006) ( STIS : DF Tau) and the MAST archive (HST-GO 8206; STIS : DR Tau). The epoch of the observations is indicated in the figure. Note that the apertures of the three spectrographs are different. The GHRS observations were performed with the Large Science Aperture (2' × 2' before 1994, 1.74' × 1.74' after 1994), the Primary Science Aperture for COS is a circle 2.5' in diameter, and the STIS observations where obtained with the apertures 0.2' × 0.06' (for T Tau) or 0.2' × 0.2' (for the other objects). For DR Tau the DAO observations show double-peaked C iv emission as well as strong emission in the blue wing of the 1548 ˚ A line. We have identified the additional blue wing emission as a mixture of H 2 , Si i , C ii , and Fe ii . The parameters that we have listed for DR Tau in Table 6 provide a good fit to the overall profile, but not to the double-peaked C iv emission. In the GHRS observation (red, from 1995) the extra Si i , C ii , and Fe ii emissions are not present, leaving only a low S/N H 2 line. The low-velocity peak of the C iv line observed in the DAO data is not present in the GHRS data. In addition to this 1995 observation, Ardila et al. (2002) describes a 1993 observation (not shown in Figure 10), which shows blueshifted emission at -250 km s -1 present in both line components. The STIS observations from HST program GO 8206 (PI Calvet) show a strong H 2 line. The centroids of the C iv lines are either redshifted by ∼ 200 km s -1 or the centers (within ± 150 km s -1 of the rest velocity) are being absorbed. If redshifted, this is the largest redshift observed in the sample, although a comparable shift is seen in v BC for CS Cha (Figure A.3). We may be observing extended C iv emission that is not seen in the narrow-slit STIS observations (Schneider et al. 2013), C iv absorption from a turbulent saturated wind or the disappearing of the accretion spot behind the stellar limb as the star rotates. T Tau shows a change in the H 2 emission and a small decrease in the strength on the NC. Walter et al. (2003); Saucedo et al. (2003) show that the H 2 emission around T Tau is extended over angular scales comparable to the GHRS aperture, and the smaller STIS flux is due to the smaller aperture. For DF Tau, the ratio between the two line members remain anomalously high, ∼ 3 in all epochs. For BP Tau, a decrease in the flux is accompanied by a decrease in both components, although the NC decreases more strongly. Changes over time in the velocity centroids are not significant. For RW Aur A the difference between GHRS and STIS DAO observations is also dramatic. As we have shown, the DAO observations can be explained by assuming that we are observing two sides of a bipolar outflow. The GHRS flux is larger, and dominated by three peaks, the bluemost of which is likely the R(3)1-8 H 2 line. The other two do not match each other in velocity and so they cannot both be C iv . Errico et al. (2000) have suggested that the early GHRS observations may be affected by Fe ii absorption. We do not find evidence for Fe ii absorption within the C iv lines for any other star, nor for the DAO observations of RW Aur, and so we discard this possibility. Based on the spectroscopic and photometric variability, Gahm et al. (1999) have suggested that a brown dwarf secondary is present in the system, although its role in the dynamics of the primary is uncertain. The COS aperture is larger, suggesting that the changes are due to true varaibility. Dramatic changes are also seen in RU Lup, as noted by Herczeg et al. (2006) and France et al. (2012). The C iv line, which is almost absent in the GHRS observations is 4 × stronger in the STIS observations. The DAO observations presented here are similar to the latter, although the strength of the extra Si i , C ii , and Fe ii emissions is also variable. The excess emission in the blue wing of the 1548 ˚ A line described in Section 5.1 is also present in the STIS observations. For all of the multi-epoch CTTSs observations except RW Aur A, Figure 6 (Fourth row) shows the path that the line ratio (the optical depth indicator) follows as a function of line luminosity. The changes in the value of the line ratio are not correlated with the changes in the C iv line luminosity, as we observed before.",
"pages": [
11,
12
]
},
{
"title": "5.5. Conclusions from the parametric analysis",
"content": "Most CTTSs C iv line profiles can be decomposed into narrow ( FWHM ∼ 130 km s -1 ) and broad ( FWHM ∼ 350 km s -1 ) components, with the BC redshifted with respect to the NC in 70% of the CTTSs sample. The fractional contribution to the flux in the NC increases (from ∼ 20% to 40% on average) and may become more optically thick with increasing accretion rate. Strong narrow components will be present in objects of high accretion rate, but high accretion rate by itself does not guarantee that the C iv lines will have strong NCs. The component velocities in about 12 out of 23 CTTSs are roughly consistent with predictions of the magnetospheric accretion model, in the sense that v BC > 0, v NC > 0 and v BC /greaterorsimilar 4 v NC . For most of the 12 the NC velocity seems too large or the BC velocity too small, compared to predictions, although this may be the result of errors in the wavelength calibration. For 11 of the CTTSs the kinematic characteristics of the C iv line cannot be explained by the magnetospheric accretion model: these objects (AK Sco, DF Tau, DK Tau, DR Tau, GM Aur, HN Tau A, IP Tau, RU Lup, SU Aur, T Tau N, V1190 Sco) have BC velocities smaller than their NC velocities, or one of the velocity components is negative. An examination of the systematic pointing errors which produce offsets in the wavelength scale lead us to conclude that these do not impact the conclusions significantly. Multi-epoch observations reveal significant changes in morphology in all the lines, from one epoch to the next. The line velocity centroids remain relatively constant in low accretion rate objects such as DF Tau and BP Tau, but the overall line appearance change significantly for high accretion rate CTTSs, like DR Tau. In the case of RU Lup, the observed variability of the line may be consistent with the accretion spot in C iv coming in and out of view.",
"pages": [
12
]
},
{
"title": "6. THE HE II LINE",
"content": "The 1640 ˚ A He ii line is the 'Helium α ' line, analogous to neutral hydrogen line H α , corresponding to a hydrogenic de-excitation from level 3 to level 2 (Brown et al. 1984). In principle, the line is a blend of 1640.33 ˚ A, 1640.34 ˚ A, 1640.37 ˚ A, 1640.39 ˚ A, 1640.47 ˚ A, 1640.49 ˚ A, and 1640.53 ˚ A. Of these, 1640.47 ˚ A ( Log ( gf ) = 0 . 39884) should dominate the emission, followed by 1640.33 ˚ A ( Log ( gf ) = 0 . 14359), as the recombination coefficient is largest to 3d 1 D (Osterbrock 1989). He ii is strongly correlated with C iv , implying, as with Si iv , that they are powered by the same process (Johns-Krull et al. 2000; Ingleby et al. 2011; G'omez de Castro & Marcos-Arenal 2012; Yang et al. 2012). For a sample of 31 CTTSs Beristain et al. (2001) modeled the He ii 4686 ˚ Aline with a single Gaussian function. Those lines are narrow, with an average FWHM of 52 km s -1 and somewhat redshifted, with an average centroid of 10 km s -1 . Based on the high excitation energy of the line (40.8 eV), and the redshift in the velocity centroid, Beristain et al. (2001) argued that the He ii optical emission originated from the post-shock gas. Below we find that the centroid shifts in the He ii UV line are comparable with the average values reported by Beristain et al. (2001) for the He ii optical line. If the He ii 1640 ˚ A line originates in the accretion postshock we expect it to be redshifted, similar in shape to the NC of the C iv line, although with lower velocity centroids as it will be emitted from a cooler, slower region of the post shock. Below we show that, indeed, the shape is similar to the NC C iv line, and the velocity centroid is smaller. In Figure A.7 we compare the 1550 ˚ A C iv line to the He ii line, scaled to the same maximum flux. The He ii lines are similar to the NCs of the C iv lines, if present, although the latter appears slightly redshifted with respect to He ii . Most He ii lines can be described as having a strong narrow core and a low-level broad component. Significant emission is present within ± 200 km s -1 of the nominal wavelength. The exceptions to this description are HN Tau A and RW Aur A, which are blueshifted and present a strong BC, and DX Cha, for which no He ii is observed in the background of a strong continuum. G'omez de Castro & Marcos-Arenal (2012) state that He ii is observed only in a subset of the stars that show C iv emission. In their sample of ACS and IUE lowresolution spectra, 15 stars show He ii out of the 20 stars that present C iv emission. We do not confirm this statement, as we observe the He ii line in all of the stars in our sample. The difference between detection rates is likely due to the differing spectral resolutions and sensitivities between our sample (R > 10000) and theirs (R ∼ 40 at 1640 ˚ A). It is well-known that the C iv and He ii luminosities are correlated to each other (e.g. Ingleby et al. 2011; Yang et al. 2012). The C iv -to-He ii luminosity ratio measured here is significantly larger for CTTSs (3 . 5 ± 0 . 4) than for WTTSs (1 . 3 ± 0 . 2). Ingleby et al. (2011) noted that the C iv -to-He ii luminosity ratio is close to one in field stars, but larger in CTTSs. The fact that we measure a ratio close to unity also in WTTSs supports Ingleby et al. (2011)'s assertion that the C iv -to-He ii luminosity ratio is controlled by accretion, and not by the underlying stellar chromosphere (Alexander et al. 2005). For the purposes of a deeper analysis we first use the non-parametric measurements (the velocity of maximum flux, the skewness and the FWHM, Table 5) to describe the line. We also fit one or two Gaussians to the lines (Table 7).",
"pages": [
12,
13
]
},
{
"title": "6.1. The He II line shape: non-parametric measurements",
"content": "We find that, except for HN Tau A and RW Aur A, the CTTSs He ii lines are well-centered or slightly redshifted: the average V Max ignoring these two stars is 7 ± 3 km s -1 (Figure 11). As with C iv , we conclude that the error on the COS wavelength scale for He ii should be smaller than the nominal value. This is because only 7 out of 23 objects (again ignoring HN Tau A and RW Aur A) have V Max < 0, a situation expected to occur with negligible probability if the wavelength errors are normally distributed around zero km s -1 . The difference in redshift between CTTSs and WTTSs is not significant. The C iv lines are redder than the He ii lines. For the CTTSs V Max CIV -V Max HeII = 11 ± 4 km s -1 . For WTTSs, the difference is 7 ± 6 km s -1 (Table 9). The FWHM for He ii CTTSs range range from ∼ 50 to 400 km s -1 , with an average of 96 ± 9 km s -1 , which is significantly narrower than in C iv . On the other hand, the FWHM of He ii lines of CTTSs and WTTSs are consistent with being drawn from the same sample, according to the KS test. The average skewness for the CTTS He ii sample is 0 . 01 ± 0 . 01, whereas the skewness of the WTTS sample is -0 . 01 ± 0 . 01: the difference between the two populations is not significant. The He ii lines are significantly more symmetric (as measured by the absolute skewness) than the C iv lines. As with C iv , neither the velocity shift, FWHM, or skewness are correlated with line luminosity, accretion rate or inclination.",
"pages": [
13
]
},
{
"title": "6.2. The He II line shape: Gaussian decomposition",
"content": "Most He ii lines require a narrow and a broad Gaussian component to fit the core and the wings of the line, respectively. The average of the ratio of the He ii NC line luminosity to the total He ii line luminosity is 0.6 and uncorrelated with accretion rate, and it is the same in CTTSs and WTTSs. This large contribution of the NC gives the lines their sharp, peaked appearance. As shown in Figure 7 the BC of the He ii lines span a much broader range than those of C iv , and their distributions are significantly different. Overall, this Gaussian decomposition confirms the conclusions from the non-parametric analysis. The average value of v NC for He ii is the same for the CTTS and WTTS (V CTTS NC HeII - V WTTS NC HeII =2 ± 3 km s -1 ), the C iv CTTS line is redshifted with respect to the He ii line (V CTTS NC CIV - V CTTS NC HeII =20 ± 6 km s -1 ), and for WTTSs the velocities of He ii and C iv are the same (V WTTS NC CIV - V WTTS NC HeII = -2 ± 5 km s -1 ). Based both on the non-parametric analysis and the Gaussian decomposition we conclude that the He ii line is comparable in terms of redshift and FWHM in WTTSs and in CTTSs. The line is blueshifted with respect to C iv CTTSs but has the same velocity shift as a C iv WTTS line, and as the NC of the CTTSs in C iv . We discuss this further in Section 8.",
"pages": [
13
]
},
{
"title": "7. SI IV AND N V: ANOMALOUS ABUNDANCES",
"content": "Analyses as detailed as those performed before are not possible for Si iv and N v : the lines are weaker and the extra emissions and absorptions due to other species make a detailed study of the line shape unreliable. Here we provide a high-level description of the lines and compare their fluxes to that of C iv .",
"pages": [
13
]
},
{
"title": "7.1. Si IV description",
"content": "The two Si iv lines are separated by 1938 km s -1 (see Figures A.1 to A.5). The wavelength of the Si iv doublet members coincides with the bright H 2 lines R(0) 0-5 (1393.7 ˚ A, -9 km s -1 from the 1394 ˚ A line), R(1) 0-5 (43 km s -1 from the 1394 ˚ A line), and P(3) 0-5 (1402.6 ˚ A, -26 km s -1 from the 1403 ˚ A line). The strong narrow line between the two Si iv lines is H 2 P(2) 0-5 (1399.0 ˚ A, 1117 km s -1 away from the 1394 ˚ A Si iv line). Additional H 2 lines (R(2) 0-5, 1395.3 ˚ A; P(1) 0-5, 1396.3 ˚ A; R(11) 2-5, 1399.3 ˚ A) are observed between the doublet members of DF Tau and TW Hya. For this paper, the H 2 lines are considered contaminants in the spectra and we will ignore them. A detailed study of their characteristics has appeared in France et al. (2012). The panels in the appendix also show the O iv line at 1401.16 ˚ A, sometimes observed as a narrow emission line blueward of the 1403 ˚ A Si iv line (see for example DE Tau). We have also indicated the position of the CO 5-0 bandhead (France et al. 2011). In the case of WTTSs, no H 2 lines are observed in the Si iv region and narrow well-centered Si iv lines characterize the emission. To the extent that the Si iv lines can be seen under the H 2 emission, their shape is similar to that of the C iv lines, as was suggested in Ardila et al. (2002) (see for example IP Tau in Fig. A.3). A notable exception is the WTTS EG Cha, for which the Si iv lines have broad wings that extend beyond ± 500 km s -1 , not present in C iv . Because the observations were taken in TIMETAG mode, we have time-resolved spectra of the Si iv region that shows sharp increase in the count rate ( ∼ 10 × in 300 secs) followed by a slow return to quiescence over 1000 secs. These observations indicate that the WTTS EG Cha was caught during a stellar flare. The broad line wings are observed only during the flare. A detailed study of this flare event is in preparation.",
"pages": [
13
]
},
{
"title": "7.2. N V description",
"content": "The two N v lines are separated by 964 km s -1 . Figures A.1 to A.5 also indicate the positions of the H 2 lines R(11) 2-2 (1237.54 ˚ A), P(8) 1-2 (1237.88 ˚ A), and P(11) 1-5 (1240.87 ˚ A). In addition to the H 2 lines, we note the presence of N i absorptions (Herczeg et al. 2005) at 1243.18 ˚ A and 1243.31 ˚ A (1055 km s -1 and 1085 km s -1 from the 1239 ˚ A line). The two N i lines are clearly seen in the 1243 N v member (the red line) of EP Cha (Figure A.4). N i absorptions are observed in most CTTSs (Exceptions: AA Tau - N i in emission, DN Tau, EG Cha; Uncertain: AK Sco, CV Cha, CY Tau). Some objects (e.g. RU Lup) show a clear wind signature in N i , with a wide blueshifted absorption which in this object absorbs most of the N v line. The 1239 ˚ A line of N v (the blue line, not affected by extra absorption) and the 1550 ˚ A line of C iv (the red line, not affected by H 2 ) have comparable wing extensions, velocity centroids, and overall shapes. As for C iv and Si iv , the doublet lines of N v should be in a 2:1 ratio if effectively thin although the presence of the N i line makes estimating the ratio impossible. For CTTSs there are other unidentified absorption sources in vicinity of the 1243 ˚ A N v line. These can be seen most easily in the spectra of DE Tau ( -32 km s -1 ), TW Hya (-87 km s -1 , -32 km s -1 ), GM Aur, and V4046 Sgr (50 km s -1 ) with respect to the rest velocity of the 1243 N v line. It is unclear if these are N i absorption features in a highly structured gas flow or absorptions from a different species. For the WTTSs both N v lines are copies of each other, and copies of the WTTS Si iv , C iv , and He ii lines. Therefore, the extra absorption in the CTTSs are due to the 'classical' T Tauri Star phenomena: they may represent absorption in a velocity-structured wind, or in a disk atmosphere. In particular, the N i feature is likley due to absorption in the CTTSs outflow or disk. We note here the similar excitation N i absorption lines are observed at 1492.62 ˚ A and 1494.67 ˚ A for all stars that show N i absorption in the N v region. A detailed study of the wind signatures in the sample will appear in a future paper.",
"pages": [
13,
14
]
},
{
"title": "7.3. Flux measurements",
"content": "Figure 12 shows the relationship between the C iv luminosity and the Si iv and N v luminosities. To measure the flux in the Si iv lines (Table 8), we have integrated each of them between -400 and 400 km s -1 , interpolating over the contaminating H 2 lines. The Si iv lines are much broader than the H 2 ones, and at these resolutions can be separated from them, at least in the cases in which the Si iv lines are actually observed. Assuming the H 2 lines are optically thin, the emission in R(0) 0-5 should be 2 × weaker than P(2) 0-5 and R(1) 0-5 should be 1.4 × weaker than P(3) 0-5, as explained in Ardila et al. (2002). In 11 objects no Si iv line is seen under the H 2 emission. For these stars, we give the 3 σ upper limit to the flux, over the same velocity range we used to measure the C iv line in the same star. Table 8 also lists the flux in the N v lines. Note that the flux in the 1243 ˚ A line is the observed flux, and it has not been corrected for N i absorption or by any of the other absorptions mentioned above. Figure 12 shows that the C iv and Si iv luminosities of CTTSs are correlated (log L SiIV /L /circledot = (0 . 9 ± 0 . 6) + (1 . 4 ± 0 . 2) log L CIV /L /circledot , ignoring the non-detections and the WTTSs). To understand the nature of this correla- on, we perform the following calculation. We assume that each line has a contribution from two regions, a preand a post-shock, and that the ratio of the contributions between the two regions is the same in Si iv and in C iv . In other words, L SiIV pre L SiIV post = L CIVpre L CIVpost . This is likely appropriate if all the emission regions contributing to the line are optically thin. If this is the case, the total luminosity in each line can be shown to be proportional to the post-shock luminosity, and we have: = 0 . 111 further assuming that the post-shock gas is an optically thin plasma in collisional equilibrium and the emission measure (EM) is the same for both lines over the emitting region. N x is the fraction of the species in that ionization state, at that temperature, Ab x is the number abundance of the element and C x is the collisional excitation rate. We assume a Si/C number abundance ratio of 0.13 for the present day Sun (Grevesse et al. 2007). To calculate the collision rate we use the analytical approximation (Burgess & Tully 1992; Dere et al. 1997): where Υ is the thermally averaged collision strength. We use Chianti V. 7.0 (Dere et al. 1997; Landi et al. 2012) to calculate this quantity and assume that the Si iv and the C iv emission come from a plasma at Log ( T ) = 4 . 9 (K) and Log ( T ) = 5 . 0 (K), respectively. The relationship L SiIV = 0.111 L CIV is indicated with a solid line in Figure 12 (Top). A model that does not assume the same EM for Si iv and C iv will move the solid line up or down (modestly: a 30% larger emission measure in Si iv than in C iv moves the solid line up by 0.1 dex), but will not change the slope. To perform the same comparison between C iv and N v requires a correction from the observed values in the latter, because we only fully observe the 1239 ˚ A doublet member, as the 1243 ˚ A doublet member is generally absorbed by N i . Therefore, we assume that the ratio of the flux in the 1239 ˚ A to the 1243 ˚ A N v lines is the same as in the C iv lines and calculate the total flux that would be observed in both N v doublet lines in the absence of N i . This is N v flux plotted against C iv in Figure 12 (Bottom). For N v the observed relationship is log L NV /L /circledot = ( -1 . 5 ± 0 . 4)+(0 . 8 ± 0 . 1) log L CIV /L /circledot , ignoring the WTTSs. The expected relationship between the luminosities of both lines is L NV = 0 . 183 L CIV , using N/C abundance of 0.25, and maximum ionization temperature of Log ( T ) = 5 . 3 (K). In the case of Si iv the observed linear fit (dashed line in Figure 12) is not consistent with our simple model (solid line, Figure 12). For the current assumptions, V4046 Sgr and TW Hya show a flux deficit in Si iv (or an excess in C iv ), while objects such as CV Cha, CY Tau, DX Cha, RU Lup, and RW Aur present a flux excess in Si iv (or a deficit in C iv ). In average, the WTTSs are above the line predicted by the model. For N v vs. C iv this simple model succeeds in explaining the observed linear correlation. Based on the absence of Si iii lines as well as the weak Si ii lines in TW Hya, Herczeg et al. (2002) proposed that for this star the silicon has been locked in grains in the disk and does not participate in the accretion. Consistent with this idea, Kastner et al. (2002) and Stelzer & Schmitt (2004) found depletion of Fe and O in TW Hya. For V4046 Sgr, Gunther et al. (2006) found a high Ne/O ratio, similar to that which was found in TW Hya and suggesting that a similar process may be at work. More generally, Drake et al. (2005) argues that the abundance of refractory species reflects the evolutionary status of the circumstellar disk. We see no evidence of this in our sample. The distribution of accretion rates for stars above and below the model line for Si iv /C iv is not significantly different. We find that 80% ± 10% of the non-TD CTTSs are above the Si/C model line, compared to 60% ± 20% of the TD stars. This difference is not significant either. It is, however, noteworthy that our analysis suggests that most CTTSs are Si iv -rich. Four out of the 5 WTTSs for which we measure the Si iv luminosity lie above the model line. This is reminiscent of the First Ionization Potential (FIP) effect seen in the Sun (e.g. Mohan et al. 2000) in which elements in the upper solar atmosphere (the transition region and the corona) show anomalous abundances when compared to the lower atmosphere. Upper atmosphere elements with low FIPs (FIP < 10eV, like silicon) show abundance excesses by 4 × , on average, while high FIP elements (FIP > 10 eV, like carbon and nitrogen) show the same abundances between the photosphere and the corona. Possible explanations for the effect include the gravitational settling of neutrals in the chromospheric plateau (Vauclair & Meyer 1985) and/or diffusion of neutrals driven by electromagnetic forces (H'enoux 1998; Laming 2004). The fact that the luminosity in N v for WTTSs follow the expected relationship with C iv , is consistent with the Si iv excess flux in WTTSs being due to the FIP effect (Wood & Linsky 2011). However, abundance studies of active, non-accreting, low-mass stars based on X-ray observations, find a mass-dependent inverse FIP-effect abundance pattern in Fe/Ne (see Gudel 2007; Testa 2010 for reviews). If this applies to WTTSs, we would expect to see a silicon deficit in them, as compared to carbon. If we were to move the model (solid) line from the top panel of Figure 12 upwards, in order to make the WTTSs siliconpoor, most CTTSs would become silicon-poor, suggesting that the disk grain evolution observed in TW Hya starts at younger ages. This is speculative and requires further study. We note here that Telleschi et al. (2007) find a mass-dependent inverse FIP-effect abundance pattern also in CTTSs. These statements are only intended to identify candidates for further study. What we can tell from our observations is that there is considerable scatter of CTTSs to both sides of the linear relation between L SiIV and L CIV . EM analyses for each star are necessary before significant conclusions can be drawn from this sample regarding abundances. These are possible with the DAO dataset, but they are beyond the scope of this paper.",
"pages": [
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},
{
"title": "8. DISCUSSION",
"content": "In this paper we are primarily interested in what the hot gas lines are telling us about the region they are emitted from. In particular, we want to know whether the observed profiles are consistent with emission in an accretion shock, a stellar transition region, or other volumes within the system, such as shocks in the stellar outflow or hot winds. We conclude that extensions of the accretion shock model involving inhomogeneous or multiple columns, or emission far from the accretion spot are necessary to account for the observations. The observational description reveals a remarkable diversity of line shapes, and interpretations about them are necessarily qualified with exceptions. However, there are a few general statements that describe the systems: narrow and broad components are consistent with a magnetospheric origin, within the instrumental velocity errors. For the rest (DF Tau, DK Tau, DR Tau, GM Aur, HN Tau A, IP Tau, RU Lup, SU Aur, T Tau, V1190 Sco) other explanations are necessary.",
"pages": [
15,
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]
},
{
"title": "8.1. Emission from the accretion column?",
"content": "If the lines originate primarily in an accretion shock, one can make at least four predictions: (1) the C iv and He ii line profiles should be redshifted, with the latter having smaller velocities; (2) the line emission should be well localized on the stellar surface; (3) if increases in the accretion rate are due to density increases, the postshock gas emission will be quenched, due to burying of the post-shock column; (4) the velocity of the narrow and broad components of the C iv lines should be related as V BC /greaterorsimilar 4 V NC . Indeed, we observe C iv and He ii redshifted, with the former redshifted more than the latter. This, however, is not a very constraining prediction. Linsky et al. (2012) have shown that C iv lines in dwarfs are redshifted by an amount correlated with their rotation period and Ayres et al. (1983) have shown that the C iv lines in late-type giants are redshifted with respect to the He ii lines by variable amounts of up to ∼ 20 km s -1 . So, for non-CTTSs gas flows in the upper stellar atmosphere may result in line shifts comparable to those we observe in our sample. The predictions regarding the localization of the emitting region in the stellar surface are difficult to test with this dataset, because, as we have argued in section 5.2.5, the sample is not large enough for the inclination to serve as a discriminator of the line origin. On the other hand, there is a large body of evidence (see the references in the introduction) suggesting that the accretion continuum and the (optical) line emission are well localized on the stellar surface. No unequivocal rotational modulation of the lines studied here has been reported in the literature, but this may be because the available dataset does not provide enough rotational coverage. The change in line flux observed among the different RU Lup epochs (Figure 10) may result from rotational modulation. If we assume that the NC of C iv is due to post-shock emission, we would naively expect its importance to diminish for high accretion rates as the post-shock is buried in the photosphere (Drake 2005). However, we observe the opposite, with the NC flux increasing with respect to the BC flux as the accretion rate increases. One possible explanation is that increased accretion produces larger accretion areas on the star rather than much higher densities in the accretion column. This has been shown for BP Tau by Ardila & Basri (2000). Larger areas provide more escape paths for the photons emitted from a buried column. Area coverages as small as 0.1% of the stellar surface of a 2R /circledot star have a radius of ∼ 10 5 km, which is larger than the deepest likely burials (Sacco et al. 2010). A complementary insight comes from X-ray observa- tions. Models that assume a uniform density accretion column also predict quenching of the X-ray flux, due to absorption of X-rays in the stellar layers, for accretion rates as small as a few times 10 -10 M /circledot /yr (Sacco et al. 2010). However, models by Romanova et al. (2004) indicate that the accretion column is likely non-uniform in density and the denser core may be surrounded by a slower-moving, lower-density region. Sacco et al. (2010) argue that most of the observed X-ray post-shock flux should come from this low-density region. In addition, post-shock columns are believed to be unstable to density perturbations, and should collapse on timescales of minutes (Sacco et al. 2008). The lack of observed periodicities in the X-ray fluxes indicate that multiple incoherent columns should be present. Orlando et al. (2010) also conclude that the presence of multiple columns with different densities is necessary to explain the low accretion rates derived from X-ray observations. In addition, models of low-resolution CTTSs spectra from the near-UV to the infrared indicate that the observed accretion continuum is consistent with the presence of multiple accretion spots (Ingleby et al. 2013). In summary, we observe that the C iv flux does not decrease with accretion, which suggests that if burying is occurring, it does not affect the observed flux substantially. This may be because the aspect ratio of the accretion spots is such that the post-shock radiation can escape without interacting with the photosphere, and/or that the flux we observe is emitted from unburied lowdensity edges of the accretion column, and/or that multiple columns with different densities and buried by different amounts are the source of the emission.",
"pages": [
16
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},
{
"title": "8.1.1. He ii emission in the pre-shock gas",
"content": "As we have shown, the BC of He ii is weak compared to its NC, in contrast with the BC for C iv . For a BC produced in the pre-shock gas, this implies that the preshock emits more strongly in C iv than in He ii . Observationally, for CTTSs the ratio between luminosity in the BC of the C iv to the He ii line ranges from 2.5 to 10.4, with a median of 5.6. The ranges are comparable if instead of using the whole sample we use only those CTTSs with redshifted profiles. Are these values consistent with a pre-shock origin? The pre-shock region is heated and ionized by radiation from the post-shock gas. The model described next confirms that the He ii pre-shock gas contribution to the line is produced by recombination of He iii to He ii , while the pre-shock gas contribution to C iv 1550 ˚ A line is the result of collisions from the ground state. The He ii 1640 ˚ A line is emitted from a smaller region in the pre-shock, closer to the star, than the C iv 1550 ˚ A lines, because the energy required to ionize He ii to He iii is 54.4 eV, while the energy to ionize C iii to C iv is 47.9 eV. Weuse Cloudy version 07.02.02 (Ferland et al. 1998) to simulate the pre-shock structure and calculate the ratio between the C iv lines at 1550 ˚ Aand the He ii line at 1640 ˚ A. We illuminate the pre-shock gas with a 4000 K stellar photosphere, a shock continuum with the same energy as contained in half of the incoming flux, and half of the cooling energy from the post-shock gas. For the purposes of this model the post-shock cooling radiation is calculated by solving the mass, momentum, and energy con- rvation equations as described in Calvet & Gullbring (1998), to derive a temperature and density structure and using Chianti V. 7.0 (Dere et al. 1997; Landi et al. 2012) to calculate the emissivity of the plasma at each point in the post-shock. This procedure assumes that the post-shock gas is an optically thin plasma in collisional equilibrium. These models are parametrized by incoming gas velocities and densities. A typical density for an incoming accretion flow with ˙ M = 10 -8 M /circledot /yr, covering 1% of the stellar surface area, is 5 × 10 12 cm -3 (Calvet & Gullbring 1998). We find that the C iv pre-shock gas emission is always larger than He ii emission by factors ranging from 2 (at 10 14 cm -3 ) to 6 (at 10 10 cm -3 ), for incoming pre-shock gas velocities of 300 km s -1 . This increase with density implies that the pre-shock emission is beginning to become optically thick at high densities, consistent with Figure 8 (bottom). Higher velocities result in larger C iv emission with respect to He ii : for incoming velocities of 400 km s -1 and densitites of 10 10 cm -3 the pre-shock emits 10 times more flux in C iv 1550 ˚ A, than in He ii 1640 ˚ A. In these simple models, the C iv post-shock emission as a fraction of the total emission varies from 0.7 (at 10 10 cm -3 ) to ∼ 1 (at 10 14 cm -3 ). These are only illustrative models but they suggest that the observed values are within the range of what is produced by the accretion shock region. It is noteworthy that low pre-shock gas densities (10 10 cm -3 ) are required to explain the median C iv /He ii ratio. This again suggests that the accretion spots are very large, or that the observed emission comes from low-density regions in the accretion columns.",
"pages": [
16,
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},
{
"title": "8.1.2. Kinematic predictions of the accretion shock model",
"content": "A crucial prediction of the accretion model is that V BC /greaterorsimilar 4 V NC . Strictly, the observations are consistent with this prediction for only two objects: TW Hya and CY Tau. However, given the size of the COS and STIS pointing errors, half of the CTTSs sample may comply with the accretion model predictions (see Figure 9). The rest of the objects are anomalous, as the standard magnetospheric accretion model has trouble explaining stars for which the BC velocity is significantly smaller than the NC velocity, or those stars in which the velocity of one or both of the components is significantly negative. Possible explanations for these anomalous objects include the target having a significant extra radial velocity due to the presence of a close companion, multiple columns being responsible for the emission, regions far from the accretion shock surface or parts of an an outflow contributing to the emission, and winds or outflows dominating the emission. We examine the first three possibilities in this section. We discuss outflows in Section 8.2. For binaries, the relative velocity between close binary components will result in shifts in the velocity centroid of the C iv line, especially if one component dominates the accretion or if a circumbinary disk is present (Artymowicz & Lubow 1996). For example, AK Sco is a well know spectroscopic binary in which the radial velocity of the stellar components can reach 100 km s -1 with respect to the system's center-of-mass (Alencar et al. 2003). If the characteristics of the accretion stream to each component were different (different accretion rates, different pre- and post-shock emission contributions among the components, etc), this would result in shifted velocity centroids. We have not considered AK Sco in the general description because the Gaussian decomposition is problematic. Regarding the other spectroscopic binaries in the sample, V4046 Sgr and CS Cha have velocity contrasts between the BC and NC close to what is expected from the accretion paradigm. Orbital modulation of X-rays has been detected in V4046 Sgr (Argiroffi et al. 2012b), which may be responsible for the large NC velocity, compared to the predictions of the accretion shock model. The C iv profiles of DX Cha are slightly redshfited, but they cannot be decomposed into velocity components and this object may belong to a different class altogether, as the only Herbig Ae star in the sample. The other systems for which multiplicity may be relevant are DF Tau and RW Aur A. DF Tau has v BC -v NC ∼ -27 km s -1 , but the component separation of 12 AU is too large to induce these velocity shifts. In the case of RW Aur A, we do not decompose the C iv lines into Gaussian components, and argue that the strong blueshift is the result of outflow emission. Gahm et al. (1999) have suggested that the system is accompanied by a brown dwarf companion. According to their observations, the companion produces radial velocity variations smaller than 10 km s -1 . In summary, with the possible exceptions of AK Sco and V4046 Sgr, binarity does not play a significant role in altering the values of V BC and V NC for any of the CTTSs we have termed anomalous. Models of the hot gas line shapes have been performed by Lamzin (2003a,b) and Lamzin et al. (2004), among others. For a broad range of geometries, those models predict redshifted double-peaked line profiles. Lamzin (2003a) modeled the C iv emission assuming planeparallel geometry with the pre- and post-shock emission lines thermally broadened, while Lamzin (2003b) consider emission from an accretion ring at high stellar latitudes, with gas flow falling perpendicular to the stellar surface. Both sets of models recover two emission kinematic components, one from the pre-shock and one from the post-shock. The peak separation between the preand post-shock contributions depends on the velocity of the incoming flow. With the possible exceptions of DK Tau and DR Tau, these double-peaked profiles are not observed. The failure of those models in predicting observed line profiles, led Lamzin (2003b) to argue that the incoming flow cannot be perpendicular to the stellar surface, and that a substantial tangential component must be present in the gas velocity. In models of the UV spectrum of TW Hya, Lamzin et al. (2004) argue that to explain the C iv line shape, the accretion flow must fall at a very low stellar latitude but in a direction almost parallel to the stellar surface, in such a way that we are able to observe the accretion streams from both sides of the disk through the inner disk hole. However, Donati et al. (2011) showed that the magnetic topology in TW Hya is such that the accretion streams have to be located at high stellar latitudes. We have shown here that the C iv line shape in TW Hya is the most common one ( ∼ 50%) in our sample and therefore whatever process is responsible for it must be fairly general. A more general explanation that the one from Lamzin et al. (2004) is required to understand the line shapes. Gunther & Schmitt (2008) considered hot gas observations of 7 CTTSs (RU Lup, T Tau, DF Tau, V4046 Sgr, TWA 5, GM Aur, and TW Hya). Their model of the post-shock contribution to the O vi profiles of TW Hya, including only thermal broadening, results in a line skewed to the red, but very narrow compared to the observations. A turbulent velocity of 150 km s -1 is necessary to obtain widths comparable to those observed. However, this turbulence results in a very symmetric line, with significant emission to the blue of the line, which is not observed. Based on these analyses, Gunther & Schmitt (2008) concluded that the O vi emission in those stars that show redshifted profiles (DF Tau, V4046 Sgr, TWA 5, GM Aur, and TW Hya) is incompatible with current models of magnetospheric accretion. The larger sample that we present here provides some insights into these issues. Focusing only on those objects for which the velocities are such that they could in principle be produced in an accretion shock (the wedge with 12 CTTSs in the upper right quadrant of Figure 9, below the dotted line, including MP Mus but excluding DR Tau), we conclude that the lack of observed double peaked profiles is due to the small difference between velocity components and to the fact that both components are very broad (perhaps as a result of turbulence in the flow). Notwithstanding the conclusions from Lamzin et al. (2004) and Gunther & Schmitt (2008), the ratio between the velocity components for the C iv profile from TW Hya is perfectly consistent with magnetospheric accretion. The small difference between velocity components is due to either too large V NC values or too small V BC values, compared with expected infall speeds. For TW Hya, for example, the models by Gunther et al. (2007) predict an infall velocity of 525 km s -1 , and we observe V BC = 116 km s -1 . This difference requires an angle between the line of sight and the accretion column of ∼ 77 degrees. For most of the rest of the stars likely to come from an accretion shock, similarly large angles are implied: AA Tau: 76 · , BP Tau: 82 · , CY Tau: 68 · , CS Cha: 49 · , DE Tau: 80 · , DN Tau: 75 · , DS Tau: 83 · , EP Cha: 85 · , MP Mus: 89 · , V1079 Tau: 83 · , V4046 Sgr: 76 · (assuming literature values for stellar masses and radii). These values are calculated using the observed BC velocity and the predicted free-fall velocity. Observationally, they represent a flux-weighted average of the velocities along the line of sight. It is surprising that most are close to 90 degrees, indicating that the average column is seen sideways and that no emission is observed from the top of the accretion column. However, observations of red-wing absorption in the He i 1.1 µ mline (Fischer et al. 2008) indicate that the accretion flow is slower than the free-fall velolocities by ∼ 50%. In addition, in the models by Romanova et al. (2004, 2011) mentioned before, the periphery of the accretion column is moving more slowly, by factors of ∼ 2, than the column core. If the BC emission is not coming from the fast free-falling core of the pre-shock gas flow but from the slower edges, the calculated angle will be smaller. So far, this exploration of the expected relationship between the broad and narrow velocity components as- sumes that the pre- and post-shock flows share the same line-of-sight angle. There are two situations in which this may not be the case. If some of the post-shock ionizing radiation reaches regions of the accretion flow in which the line-of-sight to the observer is different than for the accretion spot, we may end up with BC velocities that are unrelated to the NC velocities. After all, even close to the stellar surface the magnetic field twists and curves (Gregory et al. 2008; Mohanty & Shu 2008) and photoionized regions may be produced in flow moving in different directions. The Cloudy models we develop in section 8.1.1 create fully ionized post-shock columns with sizes ranging from 10000 km for densities ∼ 10 12 cm -3 to ∼ R /circledot , for densities ∼ 10 10 cm -3 (Calvet & Gullbring 1998). In other words, for low densities the ionizing photons may reach far from the stellar surface. Without more detailed models including at least some notional information regarding the configuration of the magnetosphere, it is not possible to say if this concept is relevant, but it may offer an explanation for objects with either small or negative BC velocities, but positive NC velocities (upper half of Figure 9), such as DF Tau, DR Tau, GM Aur, IP Tau, RU Lup, SU Aur, and V1190 Sco. It is also possible that we are observing multiple columns for which the ratios of pre- to post-shock emission are not the same in all columns, resulting in a situation in which we observe the pre-shock of one column but the post-shock of another with a different orientation. This may occur, for example, if the post-shock is occulted by the stellar limb, or buried, or if the columns have different optical depths. In this case, we will see pre- and post-shock velocities that are essentially unrelated to each other. This effect may explain objects for which both velocities are positive, but the BC is small compared to the NC (e.g., DF Tau, DR Tau, GM Aur, maybe IP Tau, RU Lup, SU Aur).",
"pages": [
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},
{
"title": "8.1.3. A contribution from the stellar transition region?",
"content": "An alternative hypothesis to the line origin in an accretion shock is that some of the observed flux originates in the stellar transition region outside of the accretion spot. Models by Cranmer (2008, 2009) indicate that accretion energy may contribute to the powering of the corona. Furthermore, based on iron and helium line observations in the optical of five CTTSs, Petrov et al. (2011) suggest that an area of enhanced chromospheric emission, more extended than the hot accretion spot, is produced by the accretion process. In addition, observations by Brickhouse et al. (2010) suggest the existence of a larger region than the accretion spot as the source of a third Xray component (after the corona and the accretion spot itself). The idea of an atmospheric contribution to the observed lines in CTTSs is not new (Herbig 1970), although its limitations were the inspiration for the magnetospheric accretion paradigm. Cram (1979) and Calvet et al. (1984) showed that a dense chromosphere cannot reproduce the strength of the observed H α line in CTTSs, and Batalha & Basri (1993) showed that chromospheric-based models are unable to reproduce the veiling or the size of the Balmer jump in CTTSs. On the other hand, magnetospheric accretion models are able to reproduce the hydrogen-line fluxes and shapes (e.g. Muzerolle et al. 1998, Kurosawa et al. 2006). If the lines are primarily emitted from the transition region, then a model would predict that: (1) the line profiles should have comparable redshifts in WTTSs and CTTSs; (2) the emission should not be localized to a small area of the stellar surface; and (3) the He ii and C iv lines should have similar shapes. Our observations do not confirm any of the predictions of a transition region origin for the lines: the C iv line redshifts are different in CTTS and WTTSs, the He ii and C iv lines have different shapes, and other observations show that the accretion indicators are localized in a small area of the stellar surface. However, we observe that the NCs of the He ii lines have comparable widths in CTTSs and in WTTSs, and the same widths for the NC of C iv . Within the picture of an accretion shock column, it is surprising that the WTTSs line widths, formed in the upper stellar atmosphere, should be the same as the CTTSs NCs line widths, formed in the turbulent post-shock gas. The velocity differential between high and low density regions in the accretion column will result in large amounts of turbulence, which tends to produce broad lines (Gunther & Schmitt 2008). It may be that at least some of the flux in the NCs in CTTSs comes from the stellar transition region, while the BCs come from the pre-shock gas. This would only apply to objects in which both Gaussian components have positive velocities as there is no evidence of blueshifted C iv or He ii profiles in the atmospheres of young active stars. Could the stellar transition region respond to accretion by producing enough C iv or He ii emission to contribute to the observed lines? For He ii a detailed model of the heating process would have to show that the X-ray emission from the accretion spot in CTTSs is enough to increase the He ii line luminosity in the stellar transition region by approximately one order of magnitude from the WTTS values (Yang et al. 2012 and this work). In the Sun, between 30% (quiet regions) and 60% (active regions) of the 1640 ˚ A He ii flux comes from ionization by soft X-rays followed by radiative recombination. The rest is due to collisional or radiative excitation of ground-level He ii (Hartmann et al. 1979; Kohl 1977). In CTTSs, the accretion spot acts as a source of soft X-rays (Kastner et al. 2002; Stelzer & Schmitt 2004; Gunther et al. 2007), although the overall X-ray emission is dominated by hot plasma produced by enhanced magnetic activity. Even the corona may increase its X-ray emission as a response to accretion events (Dupree et al. 2012). Typical observed values of L HeII ∼ 10 30 erg/sec are comparable to X-ray luminosities between 0.2 and 10 keV (Ingleby et al. 2011; G'omez de Castro & Marcos-Arenal 2012). However, based on solar models, Hartmann et al. (1979) concluded L X =50 L HeII in the 0.25 keV band. This suggests that the observed amount of X-ray flux in young stars is small compared to what would be required to produce the observed He ii line, and perhaps other mechanisms besides radiative recombination may be at play. On the other hand, the models of the coronal heating by Cranmer (2008) show that it is plausible to assume that the accretion energy is sufficient to drive CTTS stellar winds and coronal X-ray emission. The C iv resonance doublet observed in stellar atmo- spheres is the result of collisional excitation from C iv followed by radiative de-excitation (Golub & Pasachoff 2009), and an increase ranging from one to two orders of magnitude from WTTSs values would be required to match the surface flux (Johns-Krull et al. 2000) or luminosity (Figure 6) observed in CTTSs. This would require a proportional increase in atmospheric density. High densities of hot plasma gas are indeed observed in CTTSs, but at temperatures consistent with an origin in an accretion shock (Sacco et al. 2008; Argiroffi et al. 2009). As is the case for He ii , the increased X-ray flux due to accretion may result in a larger C iv population, and larger observed doublet flux. In summary, the relative contribution of the stellar transition region to the total He ii or C iv remains uncertain. While unlikely, we cannot rule out with these observations that at least some fraction of the NC in He ii or C iv originates in the stellar atmosphere of CTTSs.",
"pages": [
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},
{
"title": "8.2. Blueshifted profiles and outflows",
"content": "We have argued that if the post-shock radiation ionizes material far away from the accretion spot, we may end up with broad components having small or even negative velocities. On the other hand, the objects for which we observe a negative velocity in the NC of C iv , or in the overall profile, present considerable challenges. These are AK Sco, DK Tau, HN Tau A, T Tau N, and RW Aur A. The He ii line matches the C iv line for RW Aur A and HN Tau A (Figure A.7) but is centered at velocities closer to zero than C iv for the other stars. If we observe a CTTS for which the accretion stream is moving towards us, the velocity components would be negative. This would be the case, for example, if we observe the stream below the disk through the inner truncation hole of the accretion disk. The only case in which this is a possibility is T Tau N as this is the only target for which v NC < 0, v BC < 0, and | v NC | < | v BC | . This requires an inclination close to face-on, which the system has, and perhaps a large difference between the stellar rotation axis and the magnetic field axis. Outflows present a more likely explanation for the blueshifted profiles. Outflow phenomena are common in CTTSs and high speed shocks between the jet material and the ISM may result in C iv emission. The C iv blueshifted emission in DG Tau (Ardila et al. 2002) is clearly related to the beautiful outflow imaged with HST/NICMOS (Padgett et al. 1999) and the C iv emission is likely produced by shocks in the jet (Schneider et al. 2013). In order to generate high temperature ( > 10 5 K) gas via an outflow shock, velocities larger than 100 km s -1 in the strong shock limit are necessary (Gunther & Schmitt 2008), resulting in postshock (observed) C iv velocities > 25 km s -1 . The five objects we are considering have absolute NC velocities or velocities at maximum flux larger than this value. Therefore, at least energetically, it is possible for the emission to be produced by a shock in the outflow. Both HN Tau A and RW Aur A are known to have outflows (Hirth et al. 1994; Hartigan et al. 1995), and velocities in the approaching and receding jets that are comparable to the ones we observe in C iv and H 2 (Melnikov et al. 2009; Coffey et al. 2012). For objects such as AK Sco and DK Tau the accretion shock and the outflow regions may both be contributing to the emission. For HN Tau A and RW Aur, if we accept that the observed profiles originate primarily in an outflow, the lack of accretion shock emission becomes puzzling. These are high-accretion rate objects, and perhaps in these conditions the low-density region in the periphery of the accretion column is not present, and so the shock is truly buried. Or maybe we are observing the objects in a rotational phase such that the accretion spot is away from us. The GHRS observations of RW Aur (red trace, Figure 10) show the presence of additional emission components to the red of the nominal C iv lines, which may be due to the accretion spot. Overall, the relationship between H 2 asymmetries and hot line shapes remains to be fully explored. The analysis of DAO data by France et al. (2012) shows that some H 2 emission lines in DK Tau, ET Cha (RECX 15), HN Tau, IP Tau, RU Lup, RW Aur, and V1079 Tau (LkCa 15) are asymmetric, presenting redshifted peaks and lowlevel emission to the blue of the profiles. In general, the systematic errors in the COS wavelength scale make it difficult to determine whether the line peaks are truly shifted in velocity. For HN Tau A and RW Aur, the peak of the H 2 line R(6) (1-8), at 1556.87 ˚ A is shifted +19 km s -1 and +88 km s -1 respectively, from the stellar rest frame, larger than would be expected from pointing errors alone and suggests that outflows that are fast enough in C iv are accompanied by H 2 flows away from the observer. For RW Aur A, we observe that the (redshifted) H 2 emission covers the (blueshifted) C iv emission (Figure 5). If the gas heating occurs very close to the star, or if the wind is launched hot, one would expect to observe P-Cygni-like profiles in the hot gas lines. Dupree et al. (2005) argued that the asymmetric shape of the O vi profiles of TW Hya is the result of a hot wind in the star. Because the two C iv lines are close to each other, blueshifted wind absorption in the 1550 ˚ A line will decrease emission in the red wing of the 1548 ˚ A line. Johns-Krull & Herczeg (2007) compared both C iv lines and concluded that their similarity, as well as the absence of absorption below the local continuum (as seen in many neutral and singly ionized lines), suggest that a hot wind is not present in the case of TW Hya. Within our sample, DX Cha is the only object that may have a high temperature wind, as it shows a deficit in the red wing of the 1548 ˚ A line for C iv , compared to the red wing of the 1550 ˚ Aline and a very sharp blue cutoff in the C iv and the Si iv lines. A blueshifted absorption is seen in the 1548 ˚ AC iv line, suggesting outflow speeds of up to 400 km s -1 . A blueshifted absorption is also seen in the 1403 ˚ A line of Si iv (the red doublet member). However, note that there is an O iv emission line at -340 km s -1 of the line, and its presence may give the illusion of a wind absorption. The narrow H 2 lines observed in the Si iv profiles suggest that the putative wind is collimated or inhomogeneous, as the H 2 lines are not absorbed by the wind. The difference between both members of N v is not due to a wind but to absorption of the 1243 ˚ A line by circumstellar absorbers. DX Cha also shows absorption features in the region near He ii , although it is not clear that they are related to the outflows. Lower temperature outflows, like those observed in other CTTSs (Herczeg et al. 2005) are also observed in the DAO spectra of DX Cha: the Si ii λ 1526.71 ˚ A and 1533.43 ˚ A (not shown here) present very clear P-Cygni profile with wind absorption up to 600 km s -1 from the star. Further examination of the outflows will appear in a future paper.",
"pages": [
19,
20
]
},
{
"title": "8.3. Accretion in Herbig Ae stars",
"content": "In this analysis, we have considered DX Cha as one more member of the overall sample, in order to contrast the characteristics of Herbig Ae stars with those of CTTSs. It has the largest mass ( ∼ 2.2 M /circledot , Bohm et al. 2004), and earliest spectral type (A7.5) in the sample. DX Cha is a spectroscopic system with a K3 secondary and an average separation of ∼ 0.15 AU between components (Bohm et al. 2004). Even at these small separations, small circumstellar disks may be present, in addition to the circumbinary disk (de Val-Borro et al. 2011). To add to the complexity of the system, observations by Tatulli et al. (2007) are consistent with the presence of a wind launched in the 0.5 AU region of the disk. Testa et al. (2008) show that two different temperature plasmas are responsible for the X-ray emission. The overall flux is emitted from a relatively high-density region and dominated by the primary star. They argue that the hot component is created in the companion's corona, while the low-temperature component originates in an accretion shock. In the observations presented here, DX Cha does appear peculiar when compared to the CTTSs. The system has the largest Si iv luminosity of the sample, and the second largest C iv luminosity. The 1550 ˚ A C iv line has the second largest FWHM of the sample, and as we have noted, the C iv lines are not alike in shape, suggesting extra emission or absorption in one of the members. The Si iv lines are unlike any other Si iv or C iv lines in shape, in that they show a very sharp, blue cutoff. As indicated above, DX Cha is also the only clear candidate in the sample for the presence of a hot wind. For DX Cha we observe a strong continuum in the He ii region and no clear emission line. If in Figure A.1 we identify the depression at +60 km s -1 in the He ii panel of DX Cha as He ii in absorption, this would be the largest redshift of any He ii line in our sample, and larger than the 11.5 km s -1 observed for the peak of C iv . The absence of the He ii line in emission is put in context by Calvet et al. (2004a), who present low-resolution UV spectra of accreting objects with masses comparable to DX Cha, all of which show He ii in emission. For all of them L HeII /L /circledot ∼ 10 -5 to 10 -4 . 5 , comparable to the luminosities of other stars observed here. This makes the absence of an He ii line a mystery. Are these characteristics the continuation of the standard accretion rate phenomena to larger masses, the consequence of the close companion and/or an outflow, or new phenomena related to the weak magnetic fields associated with Herbig Ae objects? A larger sample of highresolution UV spectra of higher mass objects is required to answer these questions.",
"pages": [
20
]
},
{
"title": "9. CONCLUSIONS",
"content": "The goal of this paper is to describe the hot gas lines of CTTSs and to provide measurements that will con- tribute to understand their origin. We describe the resonance doublets of N v ( λλ 1238.82, 1242.80 ˚ A), Si iv ( λλ 1393.76, 1402.77 ˚ A), and C iv ( λλ 1548.19, 1550.77 ˚ A), as well as the He ii ( λ 1640.47 ˚ A) line. If produced by collisional excitation in a low-density medium, these UV lines suggest the presence of a plasma with temperatures ∼ 10 5 K. We focus primarily on the C iv doublet lines, with the other emission lines playing a supporting role. We combine high resolution COS and STIS data from the Cycle 17 Hubble Space Telescope ( HST ) proposal 'The Disks, Accretion, and Outflows (DAO) of T Tau stars' (PI G. Herczeg) with archive and literature data for 35 stars: one Herbig Ae star, 28 CTTSs, and 6 WTTSs. The sample includes 7 stars with transition disks. This is the largest single study of the UV hot gas lines in CTTSs and WTTSs, with high resolution and high sensitivity. We use the centroids of the H 2 lines to argue that the systematic wavelength errors in these COS observations are ∼ 7 km s -1 . We do not perform any systematic velocity correction to the spectra to account for these errors. The WTTSs establish the baseline characteristics of the hot line emission, and help to separate the effect of accretion from purely atmospheric effects. In particular, they provide the line luminosities and shapes that would be emitted by the young stars in the absence of the accretion process. The observations were analyzed using non-parametric shape measurements such as the integrated flux, the velocity at maximum flux (V Max ), the FWHM, and the line skewness. We also decomposed each He ii and C iv line into narrow and broad Gaussian components. We obtained accretion rate measurements from the literature (see Table 3).",
"pages": [
20,
21
]
},
{
"title": "9.1. The shape of the lines",
"content": "same FWHM as the narrow component of C iv in CTTSs. They are less redshifted than the CTTSs C iv lines, by ∼ 10 km s -1 , but have the same redshift as the WTTSs. A comparison of the Gaussian parameters for C iv and He ii is shown in Figure 7.",
"pages": [
21
]
},
{
"title": "9.6. Outflow shocks and hot winds",
"content": "We find three different types of profiles that show evidence of outflows (Section 8.2):",
"pages": [
23
]
},
{
"title": "9.7. Peculiar Objects",
"content": "Certain well-known objects are tagged as peculiar in this work. In addition to DX Cha, HN Tau, and RW Aur, mentioned before, the most peculiar are: High-spectral resolution UV observations provide a crucial piece of the puzzle posed by young stellar evolution. They sample a much hotter plasma than optical observations and complement and enhance X-ray data. However the data considered here has two important limitations. One is the lack of time-domain information, which makes it difficult to understand what is the average behavior of a given target. The other is the small range of accretion rates well-covered by objects. Future observational work should focus on resolving these limitations. Even with these limitations, it is clear that with this work we have only scratched the surface of this magnificent dataset. km sec km sec Sun -1 CIV Sun APPENDIX MULTI-PANEL FIGURES 2.5 2.0 1.5 1.0 0.5 0.0 1.2 1.0 0.8 0.6 0.4 0.2 0.0 2.5 2.0 1.5 1.0 0.5 0.0 Based on observations made with the NASA/ESA Hubble Space Telescope. Support for this paper was provided by NASA through grant numbers HST-GO-11616.10 and HST-GO-12161.01 from the Space Telescope Science Institute (STScI), which is operated by Association of Universities for Research in Astronomy, Inc (AURA) under NASA contract NAS 5-26555. SGG acknowledges support from the Science & Technology Facilities Council (STFC) via an Ernest Rutherford Fellowship [ST/J003255/1]. RDA acknowledges support from the UK's Science & Technology Facilities Council (STFC) through an Advanced Fellowship (ST/G00711X/1). This research has made use of NASA's Astrophysics Data System Bibliographic Services and CHIANTI, a collaborative project involving George Mason University, the University of Michigan (USA) and the University of Cambridge (UK). We thank the team from HST GTO programs 11533 and 12036 (PI J. Green) for allowing us early access to their data.",
"pages": [
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27,
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31,
38,
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},
{
"title": "Facilities: HST ( COS , STIS ). REFERENCES",
"content": "Bertout, C. 2007, in IAU Symposium, Vol. 243, IAU Symposium, ed. J. Bouvier & I. Appenzeller, 1-12 L., Saucedo, J. L., & Gordon, K. D. 2004a, AJ, 128, 1294 -. 2004b, AJ, 128, 1294 Comer'on, F., Spezzi, L., & L'opez Mart'ı, B. 2009, A&A, 500, 1045 Correia, S., Zinnecker, H., Ratzka, T., & Sterzik, M. F. 2006, A&A, 459, 909 Espaillat, C., Furlan, E., D'Alessio, P., et al. 2011, ApJ, 728, 49 Kenyon, S. J. & Hartmann, L. 1995, ApJS, 101, 117 Reipurth, B. & Cernicharo, J. 1995, in Revista Mexicana de Astronomia y Astrofisica, vol. 27, Vol. 1, Revista Mexicana de Astronomia y Astrofisica Conference Series, ed. S. Lizano & J. M. Torrelles, 43 42 Ardila et al. r e p a p s i h t n i d e z y l a n a s r a t S l A ) a ( e m a N B H C k L A A o r A u s u M P M r u A W R N u a T T c S 0 9 1 1 V g S 6 4 0 4 V u a T M D r u A M G a y H W T a T 9 7 0 1 V a C k L r u A 6 9 3 V a t s e h t o t r e f e r e w r e p a p s i h t n I ) a ( y t l a r t c e p s e h t , s r a t s y r a n i b r o F ) b ( 1 e l b a T u a T A A o c S K A u a T P B a h C V C u a T Y C u a T E D u a T F D u a T K D u a T N D u a T R D u a T S D a h C X D a h C P E a h C T E u a T N H p u L U R r u A U S a h C S C u a T P I u a T X U a h C G E A M 2 7 A W T C k L u a T 8 6 0 1 V r u A 7 9 3 V u a T 0 1 4 V - . e t o N e m a N ) 7 0 0 2 ( a T A A ) 3 9 9 1 ( ) 7 0 0 2 ( a T Y C ) 7 0 0 2 ( ) 3 9 9 1 ) 7 0 0 2 ( a ) 3 9 9 1 ( ) 7 0 0 2 ( ) 4 0 0 2 Hot lines in T Tauri stars 43 ) 6 0 0 2 ( . l a ) 1 1 0 2 ( ) 1 1 0 2 ( ) ; ) 9 9 9 1",
"pages": [
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]
},
{
"title": "0 2 0",
"content": "2 ( 2 ( u A U S 8 9 1 0 2 ) 7 0 0 2 ( . l a ) 7 0 0 2 ( a ) 7 0 0 2 ( ) 7 0 0 2 ( u a T P I ) 1 1 0 2 ( ) 7 0 0 2 ( a T X U ) 7 0 0 2 ( ) 2 0 0 2 ( s n e z t o G - r ) 7 0 0 2 ( ) 7 0 0 2 ( A 6 9 3 V ) 1 0 0 2 ( c S K A a T P B a T E D a T F D ) 6 0 0 2 ( a T K D T N D a T S D M P M A W R u a T T 0 9 1 1 V 6 4 0 4 V h C S C T M D A M G H W T 9 7 0 1 V 8 6 0 1 V A 7 9 3 V h C G E u L U R h C T E h C P E h C X D a T R D ) 3 0 0 2 ( h C V C ) 6 0 0 2 ( a T N H ) 5 0 0 2 ( . l a A W T ) 6 0 0 2 ( T 0 1 4 V 44 Ardila et al. n i B . f ) b ( y t i r a n i B . c n I . f e R ) a ( . c n I . l e V . d a R . f e R y t i c o l e V l a i d a R e m a N d e u n i t n o C - 2 e l b a T ) g e d ( ) 1 - s m k ( - . e t o N . s u i d a r r a l l e t s e h t d n a , d o i r e p e h t , i n i s ∗ v d e t r o p e r e h t m o r f t i d e v i r e d e v a h e w ' m o r f d e v i r e D ' s a d e t s i l s i n o i t a n i l c n i n e h W . ) · 0 = ( n o - e c a f m o r f n o i t a n i l c n I ) a ( . s t n e n o p m o c e h t f o n o i t a r a p e s e h t s e t a c i d n i s i s e h t n e r a p n i r e b m u n e h T . y r a n i B c i p o c s o r t c e p S = B S ; e l p i t l u M = M ; e l g n i S = S : y t i r a n i B ) b ( . ) 7 0 0 2 . l a t e r e h t n e u G ( s c i t s i r e t c a r a h c n o i n a p m o c n w o n k n U . y r a n i b ) s y a d 2 8 4 2 > ( d o i r e p - g n o l c i p o c s o r t c e p s e l b i s s o P ) c ( Note . - (a) Simultaneous (within ∼ 10 hrs) near and far ultraviolet observations are available for some of the DAO targets (Herczeg et al. 2013). The last column indicates whether simultaneous NUV observations were used to calculate the accretion rate, as described in Ingleby et al. (2013). (b) We assume ˙ M =1 × 10 -8 M /circledot /yr for AK Sco (G'omez de Castro 2009).",
"pages": [
43,
44,
45
]
},
{
"title": "Hot lines in T Tauri stars",
"content": "Data Sources Note . - (a) In addition to the DAO data; we have made use of data from the following HST proposals: GO 8157: Molecular Hydrogen in the Circumstellar Environments of T Tauri Stars; PI Walter GO 8206: The Structure of the Accretion Flow on pre-main-sequence stars; PI: Calvet GTO 11533: Accretion Flows and Winds of Pre-Main Sequence Stars; PI: Green GO 11608: How Far Does H2 Go: Constraining FUV Variability in the Gaseous Inner Holes of Protoplanetary Disks; PI: Calvet GTO 12036: Accretion Flows and Winds of Pre-Main Sequence Stars Part 2; PI: Green (b ) The data set columns indicate the suffix or the full name (if only one) of the HST dataset used. (c) PSA: Primary Science Aperture (for COS). 2.5' diameter. For the STIS data the slit size used is indicated. 5 e l b a T s t n e m e r u s a e m e n i l c i r t e m a r a p - n o N i i e H v i C ) f ( s s e n w e k S x u fl . x a m t a . l e V ) e ( M H W F ) d ( s s e n w e k S ) c ( x u fl . x a m t a . l e V ) b ( M H W F ) a ( d e R / e u l B o i t a R e m Hot lines in T Tauri stars 47 a T c S C T T T T a T T N T R a T S C X C P C T T N L U A U a T M P u a T 0 9 1 1 6 4 0 4 9 7 0 1 8 6 0 1 6 9 3 7 9 3 0 1 4 T X C G A W c e r a s e i t i c o l e . y l n o s m k 0 0 1 o t 0 m o r f d e t a l u c l a c s i g n i l a c s e h t , A r u A W R r o F . s m k 0 5 1 d n a 0 n e e w t e b s e n i l h t o b f o s g n i w d e r e h t g n i h c t a m y b d e t a l u c l a c s i s e n i l v i C o w t e h . e n i l c i r t e m m y s a e t a c i d n i ] 2 0 . 0 , 2 0 . 0 - [ ∼ n i h t i w s s e n w e k s e h t f o s e u l a V . s e n i l 2 H e h t r e v o d e t a l o p r e t n i d e t o N e . 1 - s m k 5 ± s i r o r r e e h T . e n i l v i C d e r e h t f o M H W . e n i l d e r e h t f o y t i c o l e v e h t y l n o s i t i A r u A W R r o F . s e n i l t e l b u o d o w t e h t f o e g a r e v a e h t s i s i h t , A r u A W R r o f t p e c x E . e l i f o r p v i C e h t r o f x u l f m u m i x a s l a v r e t n i e h t , s e n i l d e r d n a e u l b e h t r o F . V ∆ l a v r e t n i n a r e v o y t i c o l e v n a e m d e t h g i e w - x u l f e h t s i V e r e h w , V ∆ / ) V - x a M V ( s a d e n i f e d s i s s e n w e k s , e n i l h c a e r o F . s e n i l o w t e h t r e v o d e g a r e v a , e l i f o r p v i C e h . l a v r e t n i v i C e h t o t d e z i l a m r o n t u b , 1 - s m k 0 0 1 ± r e v o d e r u s a e M . e l i f o r p i i e H e h h T ) a ( i h T ) b ( l e V ) c ( 2 ± e r a e k S ) f ( e k S ) d ( n o c e h t W F ) e ( A W h C S T M A M a H W d e t c a r t b u s e v a h e w , s s e n w e k s e h t e t a l u c l a c o T . 0 5 1 / 0 5 2 y b g n i d i v i d y b , e n o e u l b e h t o t s s e n w e k s v i C d e r e h t e z i l a m r o n e w , e g a r e v a e h t e t a l u c l a c o T . y l e v i t c e p s e r , m u m i x a m e h t m o r f 1 - s m k 0 5 1 ± d n a . 1 - s m k 5 ± s i r o r r e e h T . e n i l i i e H 48 Ardila et al. 6 N D 5 R D 0 R D 2 6 . . . 1 6 . 0 4 5 0 . 0 1 a N 2 S D 3 . 0 2 U S 1 . 0 0 1 V 4 . 0 7 4 V 2 . 0 2 S C 1 . 0 3 P I 1 . 0 4 1 V 1 V 1 . 0 4 3 V W T P M R D 4 P E 8 T E 8 N H 5 U R . p m o C d a 3 A ∆ x u fl x 1 . 0 7 A A 1 . 0 6 K A 1 . 0 9 P B 2 . 0 4 P B 1 . 0 6 V C 1 . 0 0 Y C 2 . 0 6 E D 1 . 0 8 F D 1 . 0 6 F D 2 . 0 2 F D 1 . 0 9 K D X D 1 . 0 6 U R 1 U R . . . W R T 1 . 0 5 T 1 . 0 8 M D 1 . 0 4 M G 1 . 0 2 X U 1 . 0 3 G E 1 . 0 8 W T d e u n i t n o C - 6 e l b a T e n i L e u l B . p m o C . p m o C d a o r B . p m o C w o r r a N 3 A ∆ σ - v ( - ( p n a i 1 . . . . ) 23 c e h t n I . s n a i s s u a g r u o f r o o w t r e h t i e t i f e w s e n i l v i C o w t f o x e l p m o c h c a e r o F . 2 - m c 1 - c e s s g r e 4 1 - 0 1 f o s t i n u n i t i f n e e b s a h x u l f e h T - . e t o N v ( - ( p x e 3 A + ) 20 σ 2 / 2 ) 6 9 . 0 0 5 - v ( - ( p x e 2 A + ) 21 σ 2 / 2 ) 1 µ - v ( - ( p x e 1 A + ) 20 σ 2 / 2 ) . s i s 0 µ - v ( - ( p x e 0 A = F : s n a i s s u a g r u o f f o e s a c e h t n I . ) 20 σ 2 / 2 ) 6 9 . 0 0 5 e h t n e r a p n i d e t a c i d n i s i a t a d e h t n i a t b o o t d e s u t n e m u r t s n i e h T ) a ( 1 V ∆ 1 V 1 A ∆ 1 A 0 σ ∆ 0 σ 0 V ∆ 0 V 0 A ∆ 0 A ) a ( e m a N x u fl s 1 - s m k 1 - s m k x u fl x u fl 1 - s m k 1 - s m k 1 - s m k 1 - s m k x u fl x u fl 1 . 0 1 . 0 7 0 . 3 - 1 . 0 4 4 . 0 1 . 0 8 2 . 3 2 1 . 0 1 4 . 1 - 1 . 0 4 5 . 0 ) S O C ( r u A 7 9 3 V . . . . . . . . . . . . . . . 7 . 2 1 5 . 6 6 0 . 1 4 0 . 0 - 1 . 0 2 4 . 1 ) S I T S ( u a T 0 1 4 V . % 5 s i o i t a r e h t n i r o r r e e h T . C B e h t f o y t i c o l e v e h t f o n o i t c a r f a s a C N e h t d n a C B e h t n e e w t e b e c n e r e f f i d y t i c o l e V ) b ( s i t i f r e t e m a r a p t h g i e n a r o n r u o f a r e h t i e n d n a e n i l e u l b e h t f o g n i w d e r e h t n a h t t n e r e f f i d s i e n i l d e r e h t f o g n i w d e r e h t r u A W R d n a a h C X D r o F ) c ( Hot lines in T Tauri stars 49 50 Ardila et al. s t fi n a i s s u a G i i e H a T S T C T T T T T T T C C C T M L A 9 4 a T T 7 0 A A A K B P C V C Y D E D F D K D N D R D S D X E P E T H N M P R U R W S U T T V 1 1 V 4 0 C S D M G M I P T W U X V 1 6 0 1 V 6 9 3 V 7 9 3 V 0 1 4 V . s i s e h t n e r a p n i d e t a c i d n i s i a t a d e h t n i a t b o o t d e s u t n e m u r t s n i e h T ; e m a n r i e h t r e t f a ) W ( a h t i w d e t a c i d n i e r a s S T T W ) a ( C 7 e l b a T 1 σ 1 V 1 A 0 σ 0 V 0 A ) a ( e m a N 1 - s m k 1 - s m k 2 - m c 1 - c e s s g r e 4 1 - 0 1 1 - s m k 1 - s m k 2 - m c 1 - c e s s g r e 4 1 - 0 1 s r a t S i r u a T T l a c i s s a l C C G E A W T S K A f o n o i g e r i i e H e h T - . e t o N 8 e l b a T e m a N 2 - o t 0 0 8 - m o r f 2 1 - e s g ± 1 2 ± 3 4 . o c S K A ± 6 8 ± 3 6 . a h C V C ± 0 2 . u a T Y C ± 5 7 ± 0 8 ± 5 4 ± 1 0 ± 7 8 ± 4 8 . u a T S D ± 2 9 . ± 3 8 Hot lines in T Tauri stars 51 ± 1 1 ± 7 3 ± 8 6 . s u M P M ± 4 7 ± 2 1 ± 3 6 ± 4 1 . N u a T T ± 0 0 ± 6 6 . ± 1 9 ± 3 5 ± 3 8 ± 1 1 ± 0 3 . ± 9 4 ± 5 8 ± 8 4 ± 9 5 ± 3 1 ± 5 3 ± 5 9 ± 0 3 . u a T 0 1 4 V w A r r o f d c t o n s e x u l f d e r u s a e M ) a ( o f d e t o n s e x u l f d e r u s a e M ) b ( u a T A A u a T P B u a T E D u a T F D u a T K D u a T N D u a T R D a h C X D a h C P E a h C T E A u a T N H p u L U R A r u A W R r u A U S o c S 0 9 1 1 V r g S 6 4 0 4 V a h C S C u a T M D r u A M G u a T P I a y H W T A u a T X U u a T 9 7 0 1 V a h C G E 7 A W T u a T 8 6 0 1 V r u A 6 9 3 V r u A 7 9 3 V - . e t o N . 1 - s m k 0 0 6 r o f x u l f e n i l v i C e h T ) c ( Average Line Kinematic Parameters Note . - All calculations include the H 2 velocity correction (Section 2.1).",
"pages": [
46,
47,
48,
49,
50,
51,
52
]
}
]
2013ApJS..209...16S
https://arxiv.org/pdf/1309.7204.pdf
<document>
<section_header_level_1><location><page_1><loc_19><loc_86><loc_82><loc_87></location>MAGNETOACOUSTIC WAVES IN A PARTIALLY IONIZED TWO-FLUID PLASMA</section_header_level_1>
<text><location><page_1><loc_28><loc_84><loc_72><loc_85></location>Roberto Soler 1 , Marc Carbonell 2 , & Jose Luis Ballester 1</text>
<text><location><page_1><loc_16><loc_80><loc_84><loc_84></location>1 Departament de F'ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain and 2 Departament de Matem'atiques i Inform'atica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Draft version June 18, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_78><loc_55><loc_79></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_57><loc_86><loc_77></location>Compressible disturbances propagate in a plasma in the form of magnetoacoustic waves driven by both gas pressure and magnetic forces. In partially ionized plasmas the dynamics of ionized and neutral species are coupled due to ion-neutral collisions. As a consequence, magnetoacoustic waves propagating through a partially ionized medium are affected by the ion-neutral coupling. The degree to which the behavior of the classic waves is modified depends on the physical properties of the various species and on the relative value of the wave frequency compared to the ion-neutral collision frequency. Here, we perform a comprehensive theoretical investigation of magnetoacoustic wave propagation in a partially ionized plasma using the two-fluid formalism. We consider an extensive range of values for the collision frequency, ionization ratio, and plasma β , so that the results are applicable to a wide variety of astrophysical plasmas. We determine the modification of the wave frequencies and study the frictional damping due to ion-neutral collisions. Approximate analytic expressions to the frequencies are given in the limit case of strongly coupled ions and neutrals, while numerically obtained dispersion diagrams are provided for arbitrary collision frequencies. In addition, we discuss the presence of cutoffs in the dispersion diagrams that constrain wave propagation for certain combinations of parameters. A specific application to propagation of compressible waves in the solar chromosphere is given.</text>
<text><location><page_1><loc_14><loc_55><loc_86><loc_57></location>Subject headings: magnetic fields - magnetohydrodynamics (MHD) - plasmas - Sun: atmosphere Sun: oscillations - waves</text>
<section_header_level_1><location><page_1><loc_21><loc_51><loc_36><loc_52></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_31><loc_48><loc_51></location>The theoretical study of magnetohydrodynamic (MHD) waves in partially ionized plasmas has received increasing attention in the recent years. Energy dissipation due to ion-neutral collisions may be important in many astrophysical plasmas, e.g., in the low solar atmosphere, because of its connection to plasma heating (e.g., Khomenko & Collados 2012; Mart'ınez-Sykora et al. 2012; Russell & Fletcher 2013). Some recent examples of theoretical works that focused on the investigation of MHD waves in partially ionized plasmas are, e.g., P'ecseli & Engvold (2000); De Pontieu et al. (2001); Kumar & Roberts (2003); Khodachenko et al. (2004, 2006); Leake et al. (2005); Zaqarashvili et al. (2011, 2013); Soler et al. (2012) among many others.</text>
<text><location><page_1><loc_8><loc_10><loc_48><loc_31></location>In the present paper, we continue the research started in Soler et al. (2013a), hereafter Paper I, about wave propagation in a partially ionized two-fluid plasma. The two-fluid formalism used here assumes that ions and electrons form together an ion-electron fluid, while neutrals form a separate fluid. The ion-electron fluid and the neutral fluid interact by means of ion-neutral collisions. Paper I was devoted to the study of Alfv'en waves. Here we tackle the investigation of magnetoacoustic waves. The dynamics of magnetoacoustic waves is more involved than that of Alfv'en waves even in the case of a fully ionized plasma. The reason is that Alfv'en waves are incompressible and are only driven by the magnetic tension force. On the contrary, magnetoacoustic waves produce compression of the plasma and are driven by both magnetic and gas pressure forces (see, e.g., Goossens 2003).</text>
<text><location><page_1><loc_10><loc_7><loc_51><loc_8></location>roberto.soler@uib.es, marc.carbonell@uib.es, joseluis.ballester@uib.es</text>
<text><location><page_1><loc_52><loc_35><loc_92><loc_52></location>As a consequence, the behavior of magnetoacoustic waves is determined by the relative values of the sound and Alfv'en velocities in the plasma. When the medium is partially ionized, the situation is even more complex because, on the one hand, magnetoacoustic waves also produce compression in the neutral fluid due to collisions with ions and, on the other hand, the neutral fluid is able to support its own acoustic waves that may couple with the plasma magnetoacoustic waves. Thus, the study of propagation of magnetoacoustic disturbances in a partially ionized medium is interesting from the physical point of view and challenging from the theoretical point of view.</text>
<text><location><page_1><loc_52><loc_13><loc_92><loc_35></location>This work can be related to the previous papers by Zaqarashvili et al. (2011) and Mouschovias et al. (2011). Zaqarashvili et al. (2011) derived the dispersion relation for magnetoacoustic waves in a partially ionized twofluid plasma. They focused their study on the derivation of the two-fluid equations from the more general threefluid equations and in comparing the two-fluid results with those in the single-fluid approximation. They concluded that the single-fluid approximation breaks down when the value of the ion-neutral collision frequency is on the same order of magnitude or lower than the wave frequency. Here we proceed as in Zaqarashvili et al. (2011) and use the more general two-fluid theory instead of the single-fluid approximation. Mouschovias et al. (2011) also considered the two-fluid theory and performed a very detailed investigation of wave propagation and instabilities in interstellar molecular clouds.</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_13></location>To realistically represent the plasma in molecular clouds, Mouschovias et al. (2011) included in addition to ion-neutral collisions other effects as, e.g., ionization and recombination and self-gravity. The main purpose</text>
<text><location><page_2><loc_8><loc_55><loc_48><loc_92></location>of the present work is to obtain physical insight on the effect of ion-neutral collisions on waves without taking into account other effects that could hinder the specific role of collisions. For this reason, our approach is closer to Zaqarashvili et al. (2011) than to Mouschovias et al. (2011). As stated before, Zaqarashvili et al. (2011) put their emphasis in the derivation of the equations and the dispersion relation. As examples, Zaqarashvili et al. (2011) computed some solutions, but they restricted themselves to a specific set of parameters and did not perform an in-depth parameter study. We follow the spirit of Paper I and do not focus on a specific astrophysical plasma. Instead, we take the relevant physical quantities as, e.g., the collision frequency, the ionization fraction, the Alfv'en and sound velocities, etc., as free parameters whose values can be conveniently chosen. An inconvenience of the present approach is that the space of parameters to be explored is very big. The advantage is that this approach makes possible to apply the results of this work to a wide variety of situations. Compared to Zaqarashvili et al. (2011), we cover a wider parameter range. In particular, we consider lower ionization levels and weaker magnetic fields than those used by Zaqarashvili et al. (2011). The output of this investigation will be also useful to understand the recent results by Soler et al. (2013b) concerning the behavior of slow waves propagating along a partially ionized magnetic flux tube.</text>
<text><location><page_2><loc_8><loc_39><loc_48><loc_55></location>This paper is organized as follows. Section 2 contains the description of the equilibrium state and the basic equations. In Section 3 we explore the limit cases of uncoupled fluids and strongly coupled fluids and derive approximate expressions to the wave frequencies in those cases. Later, we investigate the general case in Section 4 by plotting dispersion diagrams when the collision frequency is varied between the uncoupled and strongly coupled regimes. We perform a specific application to magnetoacoustic waves propagating in the low solar atmosphere in Section 5. Finally, Section 6 contains some concluding remarks.</text>
<section_header_level_1><location><page_2><loc_11><loc_37><loc_45><loc_38></location>2. EQUILIBRIUM AND BASIC EQUATIONS</section_header_level_1>
<text><location><page_2><loc_8><loc_17><loc_48><loc_36></location>The equilibrium configuration considered here is a uniform and unbounded partially ionized hydrogen plasma composed of ions, electrons, and neutrals. As in Paper I, the plasma dynamics is studied using the two-fluid formalism (see also, e.g., Zaqarashvili et al. 2011). Ions and electrons are considered together as forming a single fluid, while neutrals are treated as a separate fluid. The ion-electron fluid and the neutral fluid interact by means of particle collisions, which transfer momentum between the species. It is implicitly assumed in the present version of the two-fluid theory that ion-electron and ion-ion collisions occur always more frequently than ion-neutral collisions, so that ions and electrons display fluid behavior regardless of the presence of neutrals.</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_17></location>The medium is permeated by a straight and constant magnetic field, B . We use Cartesian coordinates and conveniently choose the reference frame so that the equilibrium magnetic field is orientated along the z -direction, namely B = B ˆ z , with B constant. We ignore the effect of gravity. The omission of gravity is approximately valid as long as we consider wavelengths much shorter than the gravitational stratification scale height. We also assume</text>
<text><location><page_2><loc_52><loc_87><loc_92><loc_92></location>that there are no flows in the equilibrium. We study linear adiabatic perturbations superimposed on the equilibrium state. The equations governing the behavior of the perturbations are the same as in Paper I, namely</text>
<formula><location><page_2><loc_55><loc_83><loc_92><loc_86></location>ρ i ∂ v i ∂t = -∇ p i + 1 µ ( ∇× b ) × B -α in ( v i -v n ) , (1)</formula>
<formula><location><page_2><loc_54><loc_79><loc_92><loc_82></location>ρ n ∂ v n ∂t = -∇ p n -α in ( v n -v i ) , (2)</formula>
<formula><location><page_2><loc_57><loc_76><loc_92><loc_79></location>∂ b ∂t = ∇× ( v i × B ) , (3)</formula>
<formula><location><page_2><loc_57><loc_73><loc_92><loc_76></location>∂p i ∂t = -γP i ∇· v i , (4)</formula>
<formula><location><page_2><loc_56><loc_70><loc_92><loc_73></location>∂p n ∂t = -γP n ∇· v n , (5)</formula>
<text><location><page_2><loc_52><loc_60><loc_92><loc_70></location>where v i , p i , P i , and ρ i are the velocity perturbation, gas pressure perturbation, equilibrium gas pressure, and equilibrium density of the ion-electron fluid, v n , p n , P n , and ρ n are the respective quantities for the neutral fluid, b is the magnetic field perturbation, µ is the magnetic permeability, γ is the adiabatic index, and α in is the ionneutral friction coefficient.</text>
<text><location><page_2><loc_52><loc_42><loc_92><loc_60></location>The friction coefficient, α in , plays a very important role. It determines the strength of the ion-neutral friction force and, therefore, the importance of ion-neutral coupling. Braginskii (1965) gives the expression of α in in a hydrogen plasma. As in Paper I, we take α in as a free parameter and do not use a specific expression of α in . We do so to conveniently control the strength of the ion-neutral friction force. Our aim is to keep the investigation as general as possible for the results of the present article to be applicable in a wide range of astrophysical situations. Instead of using α in , in the remaining of this article we write the equations using the ion-neutral, ν in , and neutral-ion, ν ni collision frequencies, which have a more obvious physical meaning. They are defined as</text>
<formula><location><page_2><loc_63><loc_38><loc_92><loc_41></location>ν in = α in ρ i , ν ni = α in ρ n . (6)</formula>
<text><location><page_2><loc_52><loc_36><loc_89><loc_38></location>Hence, both frequencies are related by ρ i ν in = ρ n ν ni .</text>
<section_header_level_1><location><page_2><loc_62><loc_34><loc_82><loc_35></location>2.1. Normal mode analysis</section_header_level_1>
<text><location><page_2><loc_52><loc_23><loc_92><loc_33></location>Here we perform a normal mode analysis. The temporal dependence of the perturbations is put proportional to exp ( -iωt ), where ω is the angular frequency. In addition, we perform a Fourier analysis of the perturbations in space. The spatial dependence of perturbations is put proportional to exp ( ik x x + ik y y + ik z z ), where k x , k y , and k z are the components of the wavevector in the x -, y -, and z -directions, respectively.</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_23></location>Alfv'en waves and magnetoacoustic waves are two distinct classes of MHD waves in a uniform plasma (see, e.g., Cramer 2001; Goossens 2003). Alfv'en waves are incompressible and propagate vorticity perturbations. Magnetoacoustic waves are compressible and do not produce vorticity perturbations. Alfv'en waves were investigated in Paper I. Here we focus on magnetoacoustic waves. Compressibility is an appropriate quantity to describe magnetoacoustic waves (see, e.g., Lighthill 1960). By working with compressibility perturbations we are able to decouple magnetoacoustic waves from Alfv'en waves without imposing any restriction on the direction of the</text>
<text><location><page_3><loc_8><loc_88><loc_48><loc_92></location>wavevector. We define ∆ i and ∆ n as the compressibility perturbations of the ion-electron fluid and the neutral fluid, respectively,</text>
<formula><location><page_3><loc_14><loc_85><loc_48><loc_87></location>∆ i ≡∇· v i = ik x v x, i + ik y v y, i + ik z v z, i , (7)</formula>
<formula><location><page_3><loc_14><loc_83><loc_48><loc_85></location>∆ n ≡∇· v n = ik x v x, n + ik y v y, n + ik z v z, n (8)</formula>
<text><location><page_3><loc_8><loc_79><loc_48><loc_83></location>We combine Equations (1)-(5) and after some algebraic manipulations we obtain the two following coupled equations involving ∆ i and ∆ n only, namely</text>
<formula><location><page_3><loc_14><loc_67><loc_48><loc_79></location>( ω 4 -ω 2 k 2 ( c 2 i + c 2 A ) + k 2 k 2 z c 2 A c 2 i ) ∆ i = -iν in ω 3 (∆ i -∆ n ) + iν in ω + i ( ν ni + ν in ) k 2 k 2 z c 2 A ( c 2 i ∆ i -c 2 n ∆ n ) , (9) ( ω 2 -k 2 c 2 n ) ∆ n = -iν ni ω (∆ n -∆ i ) , (10)</formula>
<text><location><page_3><loc_8><loc_65><loc_48><loc_69></location>where k 2 = k 2 x + k 2 y + k 2 z and c A , c i , and c n are the Alfv'en velocity, the ion-electron sound velocity, and the neutral sound velocity defined as</text>
<formula><location><page_3><loc_14><loc_61><loc_48><loc_64></location>c 2 A = B 2 µρ i , c 2 i = γP i ρ i , c 2 n = γP n ρ n . (11)</formula>
<text><location><page_3><loc_8><loc_53><loc_48><loc_60></location>Note that the Alfv'en velocity is here defined using the density of the ionized fluid only. Equations (9) and (10) are the governing equations for linear compressibility perturbations. Therefore, they govern the behavior of magnetoacoustic waves.</text>
<text><location><page_3><loc_8><loc_48><loc_48><loc_53></location>The magnetoacoustic character of the waves is determined by the relative values of the Alfv'en and sound velocities. In this work we use the following definition of the ion-electron β i ,</text>
<formula><location><page_3><loc_22><loc_44><loc_48><loc_47></location>β i = c 2 i c 2 A = γP i B 2 /µ . (12)</formula>
<text><location><page_3><loc_8><loc_27><loc_48><loc_43></location>Thus, using the present definition of β i we can quantify the relative importance of the Alfv'en velocity, c A , and ion-electron sound velocity, c i . No restriction on the value of the neutral sound velocity, c n , has been imposed so far. However, to simplify matters and decrease the number of parameters involved, we proceed as in Zaqarashvili et al. (2011) and Soler et al. (2013b) and assume that there is a strong thermal coupling between the ion-electron fluid and the neutral fluid. As a consequence, the equilibrium temperature, T , is the same for both fluids. We use the ideal gas law to relate the equilibrium gas pressure, density, and temperature as</text>
<formula><location><page_3><loc_18><loc_25><loc_48><loc_27></location>P i = 2 ρ i RT, P n = ρ n RT, (13)</formula>
<text><location><page_3><loc_8><loc_14><loc_48><loc_24></location>where R is the ideal gas constant. The factor 2 in the expression for ion-electrons is present because the partial pressures of both electrons and ions have to be summed up. In a hydrogen plasma, the partial pressures of electrons and ions are the same. From Equation (13), we deduce that the requirement that all the species have the same temperature results in imposing a relation between the sound velocities as,</text>
<formula><location><page_3><loc_25><loc_11><loc_48><loc_13></location>c 2 i = 2 c 2 n . (14)</formula>
<text><location><page_3><loc_8><loc_7><loc_48><loc_11></location>Hence, using Equations (12) and (14) we see that fixing the value of β i is enough to establish the relation between the Alfv'en velocity and the two sound velocities.</text>
<text><location><page_3><loc_52><loc_76><loc_92><loc_92></location>For the subsequent analysis, we decompose the wavevector, k , in its components parallel and perpendicular to the equilibrium magnetic field as k = k z ˆ z + k ⊥ 1 ⊥ , where k ⊥ = √ k 2 x + k 2 y and 1 ⊥ is the unity vector in the perpendicular direction to B . We define θ as the angle that forms the wavevector, k , with the equilibrium magnetic field, B . Then, we can write k z = k cos θ and k ⊥ = k sin θ . For propagation perpendicular to the magnetic field, θ = π/ 2, whereas for propagation parallel to the magnetic field, θ = 0. Intermediate values of θ represent oblique propagation.</text>
<section_header_level_1><location><page_3><loc_53><loc_72><loc_90><loc_75></location>3. APPROXIMATIONS FOR LIMIT VALUES OF THE COLLISION FREQUENCIES</section_header_level_1>
<text><location><page_3><loc_52><loc_64><loc_92><loc_72></location>Before tackling the general case for arbitrary values of the collision frequencies, it is instructive to consider two paradigmatic limit cases, namely the uncoupled case in which the two fluids do not interact, and the strongly coupled case in which the two fluids behave as a single fluid.</text>
<section_header_level_1><location><page_3><loc_65><loc_61><loc_79><loc_62></location>3.1. Uncoupled case</section_header_level_1>
<text><location><page_3><loc_52><loc_56><loc_92><loc_60></location>Here, we remove the effect of ion-neutral collisions. We represent this situation by setting ν in = ν ni = 0. Equations (9) and (10) become</text>
<formula><location><page_3><loc_56><loc_49><loc_92><loc_55></location>( ω 4 -ω 2 k 2 ( c 2 i + c 2 A ) + k 4 c 2 A c 2 i cos 2 θ ) ∆ i =0 , (15) ( ω 2 -k 2 c 2 n ) ∆ n =0 . (16)</formula>
<text><location><page_3><loc_52><loc_47><loc_92><loc_51></location>Now, the equations governing compressibility perturbations in the two fluids are decoupled. The waves in the two fluids can be studied separately.</text>
<text><location><page_3><loc_52><loc_43><loc_92><loc_47></location>We start by considering waves in the ion-electron fluid. For nonzero ∆ i , the solutions to Equation (15) must satisfy</text>
<formula><location><page_3><loc_56><loc_38><loc_92><loc_42></location>ω 4 -ω 2 k 2 ( c 2 i + c 2 A ) + k 4 c 2 A c 2 i cos 2 θ = 0 . (17)</formula>
<text><location><page_3><loc_52><loc_35><loc_92><loc_39></location>Equation (17) is the well-known dispersion relation of magnetoacoustic waves in a fully ionized plasma (see, e.g., Lighthill 1960). The solutions to Equation (17) are</text>
<formula><location><page_3><loc_53><loc_29><loc_92><loc_34></location>ω 2 = k 2 c 2 A + c 2 i 2 ± k 2 c 2 A + c 2 i 2 [ 1 -4 c 2 A c 2 i cos 2 θ ( c 2 A + c 2 i ) 2 ] 1 / 2 , (18)</formula>
<text><location><page_3><loc_52><loc_24><loc_92><loc_29></location>where the + sign is for the fast wave and the -sign is for the slow wave. The magnetoacoustic character of these two distinct wave modes depends on the relative values of c A and c i (see, e.g., Goossens 2003).</text>
<text><location><page_3><loc_52><loc_21><loc_92><loc_24></location>In the lowβ i case, c 2 A /greatermuch c 2 i and the solutions simplify to</text>
<text><location><page_3><loc_52><loc_17><loc_67><loc_18></location>for the fast wave and</text>
<formula><location><page_3><loc_62><loc_17><loc_92><loc_21></location>ω 2 ≈ k 2 ( c 2 A + c 2 i ) ≈ k 2 c 2 A , (19)</formula>
<formula><location><page_3><loc_59><loc_13><loc_92><loc_16></location>ω 2 ≈ k 2 c 2 A c 2 i c 2 A + c 2 i cos 2 θ ≈ k 2 c 2 i cos 2 θ, (20)</formula>
<text><location><page_3><loc_52><loc_7><loc_92><loc_12></location>for the slow wave. When c 2 A /greatermuch c 2 i the fast wave is a magnetic wave that propagates isotropically at the Alfv'en velocity, c A , while the slow wave is essentially a sound wave that is guided by the magnetic field and travels at the</text>
<text><location><page_4><loc_8><loc_88><loc_48><loc_92></location>ion-electron sound velocity, c i . Conversely, in the highβ i case c 2 i /greatermuch c 2 A and the solutions simplify to</text>
<text><location><page_4><loc_8><loc_85><loc_23><loc_86></location>for the fast wave and</text>
<formula><location><page_4><loc_19><loc_85><loc_48><loc_88></location>ω 2 ≈ k 2 ( c 2 A + c 2 i ) ≈ k 2 c 2 i , (21)</formula>
<formula><location><page_4><loc_15><loc_81><loc_48><loc_84></location>ω 2 ≈ k 2 c 2 A c 2 i c 2 A + c 2 i cos 2 θ ≈ k 2 c 2 A cos 2 θ, (22)</formula>
<text><location><page_4><loc_8><loc_77><loc_48><loc_80></location>for the slow wave. Now the fast wave behaves as an isotropic sound wave, while the slow wave is a magnetic wave guided by the magnetic field.</text>
<text><location><page_4><loc_8><loc_74><loc_48><loc_76></location>We turn to the waves in the neutral fluid. The solutions to Equation (16) for nonzero ∆ n are</text>
<formula><location><page_4><loc_24><loc_71><loc_48><loc_73></location>ω 2 = k 2 c 2 n . (23)</formula>
<text><location><page_4><loc_8><loc_66><loc_48><loc_71></location>Equation (23) represent acoustic (or sound) waves in a gas. These waves propagate isotropically at the neutral sound velocity, c n , and are unaffected by the magnetic field.</text>
<text><location><page_4><loc_8><loc_51><loc_48><loc_66></location>As expected, in the absence of ion-neutral collisions we consistently recover the classic magnetoacoustic waves in the ion-electron fluid and the classic acoustic waves in the neutral fluid. Therefore, three distinct waves are present in the uncoupled, collisionless case. These waves do not interact and are undamped in the absence of collisions. The waves would be damped if nonideal processes as, e.g., viscosity, resistivity, or thermal conduction, are taken into account. In the remaining of this article, we use the adjective 'classic' to refer to the waves found in the uncoupled case.</text>
<section_header_level_1><location><page_4><loc_19><loc_49><loc_38><loc_50></location>3.2. Strongly coupled case</section_header_level_1>
<text><location><page_4><loc_31><loc_41><loc_31><loc_43></location>/negationslash</text>
<text><location><page_4><loc_8><loc_36><loc_48><loc_48></location>Conversely to the uncoupled case, the strongly coupled limit represents the situation in which ion-electrons and neutrals behave as a single fluid. To study this case, we take the limits ν in → ∞ and ν ni → ∞ in Equations (9) and (10). We realize that, if ω = 0, it is necessary that ∆ i = ∆ n for the equations to remain finite. This is equivalent to assume that the two fluids move as a whole. Then, when ν in → ∞ and ν ni → ∞ , Equations (9) and (10) reduce to a single equation, namely</text>
<formula><location><page_4><loc_16><loc_28><loc_48><loc_36></location>( ω 4 -ω 2 k 2 c 2 A + c 2 i + χc 2 n 1 + χ + k 4 c 2 A ( c 2 i + χc 2 n ) (1 + χ ) 2 cos 2 θ ) ∆ i , n = 0 , (24)</formula>
<text><location><page_4><loc_8><loc_25><loc_48><loc_27></location>where we use ∆ i , n to represent either ∆ i or ∆ n and the ionization fraction of the plasma, χ , is defined as</text>
<formula><location><page_4><loc_25><loc_21><loc_48><loc_24></location>χ = ρ n ρ i . (25)</formula>
<text><location><page_4><loc_8><loc_18><loc_48><loc_21></location>For nonzero ∆ i , n , the solutions to Equation (24) must satisfy</text>
<formula><location><page_4><loc_9><loc_12><loc_48><loc_17></location>ω 4 -ω 2 k 2 c 2 A + c 2 i + χc 2 n 1 + χ + k 4 c 2 A ( c 2 i + χc 2 n ) (1 + χ ) 2 cos 2 θ = 0 . (26)</formula>
<text><location><page_4><loc_8><loc_10><loc_48><loc_12></location>This is the wave dispersion relation in the strongly coupled case. Its solutions are</text>
<formula><location><page_4><loc_10><loc_6><loc_25><loc_9></location>ω 2 = k 2 c 2 A + c 2 i + χc 2 n 2(1 + χ )</formula>
<formula><location><page_4><loc_55><loc_88><loc_92><loc_92></location>± k 2 c 2 A + c 2 i + χc 2 n 2(1 + χ ) [ 1 -4 c 2 A ( c 2 i + χc 2 n ) cos 2 θ ( c 2 A + c 2 i + χc 2 n ) 2 ] 1 / 2 , (27)</formula>
<text><location><page_4><loc_52><loc_81><loc_92><loc_87></location>where the + sign is for the modified fast wave and the -sign is for the modified slow wave. We use the adjective 'modified' to stress that these waves are the counterparts of the classic fast and slow modes but modified by the presence of the neutral fluid.</text>
<text><location><page_4><loc_52><loc_63><loc_92><loc_81></location>The first important difference between the uncoupled and strongly coupled cases is in the number of solutions. In the uncoupled case there are three distinct waves, namely the classic slow and fast magnetoacoustic waves and the neutral acoustic wave, but in the strongly coupled limit we only find the modified version of the slow and fast magnetoacoustic modes. The modified counterpart of the classic neutral acoustic mode is apparently absent. The question then arises, what happened to the neutral acoustic mode? To answer this question, we study in the following paragraphs the character of the modified fast and slow modes depending on the relative values of the Alfv'en and sound velocities.</text>
<text><location><page_4><loc_52><loc_47><loc_92><loc_63></location>A remark should be made before tackling the approximate study of the solutions in Equation (27). We recall that the neutral and ion-electron sound velocities are related by c 2 i = 2 c 2 n . This condition means that c 2 i and c 2 n are of the same order of magnitude. However, this is not the end of the story. We notice that the value of the ionization fraction, χ , also plays a role in determining the solutions to Equation (27). Specifically, χ appears multiplying c 2 n , which means that χ can increase or decrease the effective value of the neutral sound velocity. Thus, we need to compare χc 2 n with c 2 A and c 2 i in order to determine the nature of the solutions.</text>
<text><location><page_4><loc_52><loc_44><loc_92><loc_47></location>The approximate study of the solutions given in Equation (27) is done considering eight typical cases:</text>
<unordered_list>
<list_item><location><page_4><loc_54><loc_39><loc_92><loc_43></location>1. Case c 2 A /greatermuch c 2 i /greatermuch χc 2 n . This case corresponds to an almost fully ionized ( χ /lessmuch 1) lowβ i plasma. The solutions given in Equation (27) simplify to</list_item>
</unordered_list>
<formula><location><page_4><loc_68><loc_36><loc_92><loc_38></location>ω 2 ≈ k 2 c 2 A , (28)</formula>
<text><location><page_4><loc_56><loc_35><loc_78><loc_36></location>for the modified fast wave and</text>
<formula><location><page_4><loc_66><loc_32><loc_92><loc_34></location>ω 2 ≈ k 2 c 2 i cos 2 θ, (29)</formula>
<text><location><page_4><loc_56><loc_25><loc_92><loc_32></location>for the modified slow wave. We recover the fast and slow waves of the uncoupled case (Equations (19) and (20)). No trace of the neutral acoustic mode remains in this limit because the amount of neutrals is negligible.</text>
<unordered_list>
<list_item><location><page_4><loc_54><loc_20><loc_92><loc_24></location>2. Case c 2 A /greatermuch c 2 i ∼ χc 2 n . This case corresponds to a lowβ i plasma with χ ∼ 1. The solutions are approximated as</list_item>
</unordered_list>
<formula><location><page_4><loc_67><loc_16><loc_92><loc_19></location>ω 2 ≈ k 2 c 2 A 1 + χ , (30)</formula>
<text><location><page_4><loc_56><loc_14><loc_78><loc_15></location>for the modified fast wave and</text>
<formula><location><page_4><loc_63><loc_10><loc_92><loc_13></location>ω 2 ≈ k 2 c 2 i + χc 2 n 1 + χ cos 2 θ, (31)</formula>
<text><location><page_4><loc_56><loc_7><loc_92><loc_9></location>for the modified slow wave. By comparing Equations (19) and (30), we see that the square of</text>
<text><location><page_5><loc_12><loc_69><loc_48><loc_92></location>the fast mode frequency is reduced by the factor ( χ +1) -1 compared to the value in the uncoupled case. As a consequence, the effective Alfv'en velocity in the strongly coupled limit is c A / √ 1 + χ . This is equivalent to replace the ion density, ρ i , by the sum of the ion and neutral densities, ρ i + ρ n , in the definition of the Alfv'en velocity. The same result was obtained by Kumar & Roberts (2003) and Soler et al. (2013b) for fast magnetoacoustic waves in the case c 2 n = 0. This is also the same result obtained in Paper I and Soler et al. (2012) for Alfv'en waves. On the other hand, the modification of the slow mode frequency is more complicated (compare Equations (20) and (31)). The expression of the slow mode frequency involves both c i and c n . Consequently, an effective sound velocity, c eff , can be defined as</text>
<formula><location><page_5><loc_17><loc_65><loc_48><loc_68></location>c 2 eff = c 2 i + χc 2 n 1 + χ = ρ i c 2 i + ρ n c 2 n ρ i + ρ n . (32)</formula>
<text><location><page_5><loc_12><loc_51><loc_48><loc_64></location>The effective sound velocity is the weighted average of the sound velocities of ion-electrons and neutrals, and was first obtained by Soler et al. (2013b) for slow magnetoacoustic waves propagating along magnetic flux tubes. The fact that the expression of the effective sound velocity (Equation (32)) involves both neutral and ion-electron sound velocities suggests that, in this case, the modified slow mode is the descendant of both the classic slow mode and the neutral acoustic mode.</text>
<unordered_list>
<list_item><location><page_5><loc_10><loc_43><loc_48><loc_50></location>3. Case c 2 A /greatermuch χc 2 n /greatermuch c 2 i . This case corresponds to a weakly ionized ( χ /greatermuch 1) lowβ i plasma, but the magnetic field is strong enough for c 2 A to be much larger than χc 2 n . The solutions given in Equation (27) become</list_item>
</unordered_list>
<formula><location><page_5><loc_24><loc_38><loc_48><loc_42></location>ω 2 ≈ k 2 c 2 A χ , (33)</formula>
<text><location><page_5><loc_12><loc_36><loc_34><loc_38></location>for the modified fast wave and</text>
<formula><location><page_5><loc_22><loc_33><loc_48><loc_36></location>ω 2 ≈ k 2 c 2 n cos 2 θ, (34)</formula>
<text><location><page_5><loc_12><loc_20><loc_48><loc_33></location>for the modified slow wave. The effective Alfv'en velocity is now c A / √ χ . This is equivalent to replace the ion density, ρ i , by the neutral density, ρ n , in the definition of the Alfv'en velocity. On the other hand, the slow wave is the descendant of the neutral acoustic wave that becomes guided by the magnetic field. Therefore, the frequencies of both fast and slow waves are determined by the density of the neutral fluid alone, which is indirectly affected by the magnetic field.</text>
<unordered_list>
<list_item><location><page_5><loc_10><loc_14><loc_48><loc_19></location>4. Case χc 2 n /greatermuch c 2 A /greatermuch c 2 i . In this case we consider a very weakly ionized ( χ →∞ ) lowβ i plasma. The approximate frequencies turn to be</list_item>
</unordered_list>
<formula><location><page_5><loc_24><loc_11><loc_48><loc_14></location>ω 2 ≈ k 2 c 2 n , (35)</formula>
<text><location><page_5><loc_12><loc_10><loc_34><loc_11></location>for the modified fast wave and</text>
<formula><location><page_5><loc_22><loc_6><loc_48><loc_9></location>ω 2 ≈ k 2 c 2 A χ cos 2 θ, (36)</formula>
<text><location><page_5><loc_56><loc_77><loc_92><loc_92></location>for the modified slow wave. The modified fast wave can straightforwardly be related to the isotropic acoustic mode of the neutral fluid (Equation (23)), while the modified slow wave is a guided magnetic wave with the effective Alfv'en velocity depending on the neutral density alone. As in case (3), neither the ion-electron sound velocity nor the ion density play a role. The fact that the plasma is weakly ionized changes completely the physical nature of the solutions compared to the classic magnetoacoustic waves in a lowβ i fully ionized plasma.</text>
<unordered_list>
<list_item><location><page_5><loc_54><loc_72><loc_92><loc_76></location>5. Case c 2 i /greatermuch c 2 A /greatermuch χc 2 n . This corresponds to an almost fully ionized ( χ /lessmuch 1) highβ i plasma. The solutions given in Equation (27) simplify to</list_item>
</unordered_list>
<formula><location><page_5><loc_68><loc_69><loc_92><loc_72></location>ω 2 ≈ k 2 c 2 i , (37)</formula>
<text><location><page_5><loc_56><loc_68><loc_78><loc_69></location>for the modified fast wave and</text>
<formula><location><page_5><loc_66><loc_65><loc_92><loc_67></location>ω 2 ≈ k 2 c 2 A cos 2 θ, (38)</formula>
<text><location><page_5><loc_56><loc_60><loc_92><loc_65></location>for the modified slow wave. We recover Equations (21) and (22) obtained in the highβ i collisionless case. No trace of the neutral acoustic mode remains in this limit.</text>
<unordered_list>
<list_item><location><page_5><loc_54><loc_54><loc_92><loc_59></location>6. Case c 2 i /greatermuch χc 2 n /greatermuch c 2 A . This corresponds again to an almost fully ionized ( χ /lessmuch 1) highβ i plasma, but now the magnetic field is so weak that χc 2 n /greatermuch c 2 A . The solutions given in Equation (27) simplify to</list_item>
</unordered_list>
<formula><location><page_5><loc_68><loc_50><loc_92><loc_53></location>ω 2 ≈ k 2 c 2 i , (39)</formula>
<text><location><page_5><loc_56><loc_49><loc_78><loc_50></location>for the modified fast wave and</text>
<formula><location><page_5><loc_66><loc_46><loc_92><loc_49></location>ω 2 ≈ k 2 c 2 A cos 2 θ, (40)</formula>
<text><location><page_5><loc_56><loc_34><loc_92><loc_46></location>for the modified slow wave. As in the previous case (5), we revert to Equations (21) and (22) obtained in the uncoupled case. Cases (5) and (6) point out that the relative values of c 2 A and χc 2 n are not important as long as c 2 i remains much larger than both of them. The same approximations to the frequencies are found in cases (5) and (6). The properties of the neutral fluid are not important in both cases (5) and (6).</text>
<unordered_list>
<list_item><location><page_5><loc_54><loc_29><loc_92><loc_33></location>7. Case χc 2 n ∼ c 2 i /greatermuch c 2 A . This is a highβ i plasma with χ ∼ 1. The solutions given in Equation (27) simplify to</list_item>
</unordered_list>
<formula><location><page_5><loc_63><loc_25><loc_92><loc_28></location>ω 2 ≈ k 2 c 2 i + χc 2 n 1 + χ = k 2 c 2 eff , (41)</formula>
<text><location><page_5><loc_56><loc_23><loc_78><loc_25></location>for the modified fast wave and</text>
<formula><location><page_5><loc_64><loc_20><loc_92><loc_23></location>ω 2 ≈ k 2 c 2 A 1 + χ cos 2 θ, (42)</formula>
<text><location><page_5><loc_56><loc_7><loc_92><loc_19></location>for the modified slow wave. Equations (41) and (42) are the modified version of Equations (21) and (22) obtained in the highβ i uncoupled case, where c eff replaces c i and c A / √ 1 + χ replaces c A . Here we find that the modified fast mode is the descendant of both the classic fast mode and the neutral acoustic mode, while the modified slow mode is a guided magnetic wave whose frequency depends on both ion and neutral densities.</text>
<unordered_list>
<list_item><location><page_6><loc_10><loc_88><loc_48><loc_92></location>8. Case χc 2 n /greatermuch c 2 i /greatermuch c 2 A . This corresponds to a weakly ionized ( χ /greatermuch 1) highβ i plasma. The solutions given in Equation (27) simplify to</list_item>
</unordered_list>
<formula><location><page_6><loc_24><loc_85><loc_48><loc_87></location>ω 2 ≈ k 2 c 2 n , (43)</formula>
<text><location><page_6><loc_12><loc_83><loc_34><loc_85></location>for the modified fast wave and</text>
<formula><location><page_6><loc_22><loc_80><loc_48><loc_83></location>ω 2 ≈ k 2 c 2 A χ cos 2 θ, (44)</formula>
<text><location><page_6><loc_12><loc_72><loc_48><loc_79></location>for the modified slow wave. The approximation found here are the same as in case (4). The properties of the neutral fluid completely determine the behavior of the waves. Neutrals feel the magnetic field indirectly through the collisions with ions.</text>
<text><location><page_6><loc_8><loc_53><loc_48><loc_71></location>The results of the above approximate study are summarized in Table 1. Based on this study, we can now answer the previous question about the apparent absence of the neutral acoustic mode in a strongly coupled plasma. We conclude that the classic neutral acoustic mode and the classic ion-electron magnetoacoustic modes heavily interact when ion-neutral collisions are at work. The two resulting modes in the strongly coupled regime (modified fast mode and modified slow mode) have, in general, mixed properties and are affected by the physical conditions in the two fluids. The degree to which the properties of the classic waves are present in the resulting waves depends on the relative values of the Alfv'en and sound velocities and on the ionization fraction of the plasma.</text>
<text><location><page_6><loc_8><loc_36><loc_48><loc_53></location>In addition, the wave frequencies in the strongly coupled limit are real as in the uncoupled case. This means that the waves are undamped in the limit of high collision frequencies as well. The damping of magnetoacoustic waves due to ion-neutral collisions takes place for intermediate collision frequencies. As stated before, the waves would be damped if additional damping mechanisms are considered, but this falls beyond the aim of the present article. Both the ion-neutral damping and the coupling between modes are investigated in Section 4 by studying the modification of the wave frequencies when a progressive variation of the collision frequencies between the uncoupled limit and strongly coupled limit is considered.</text>
<section_header_level_1><location><page_6><loc_12><loc_32><loc_45><loc_34></location>4. NUMERICAL STUDY FOR ARBITRARY COLLISION FREQUENCIES</section_header_level_1>
<text><location><page_6><loc_8><loc_26><loc_48><loc_31></location>From here on, we consider arbitrary values of ν in and ν ni . With no approximation, we combine the coupled Equations (9) and (10) to obtain separate equations for ∆ i and ∆ n as</text>
<formula><location><page_6><loc_26><loc_23><loc_48><loc_25></location>D ( ω ) ∆ i =0 , (45)</formula>
<formula><location><page_6><loc_21><loc_20><loc_48><loc_23></location>iν ni ω D ( ω ) D n ( ω ) ∆ n =0 . (46)</formula>
<formula><location><page_6><loc_17><loc_16><loc_48><loc_18></location>D ( ω ) = D i ( ω ) D n ( ω ) + D 2 c ( ω ) , (47)</formula>
<text><location><page_6><loc_8><loc_18><loc_11><loc_19></location>with</text>
<text><location><page_6><loc_8><loc_14><loc_42><loc_16></location>where D i ( ω ), D n ( ω ), and D 2 c ( ω ) are defined as</text>
<formula><location><page_6><loc_12><loc_11><loc_39><loc_14></location>D i ( ω ) = ω 3 ( ω + iν in ) -ω 2 k 2 c 2 A + c 2 i</formula>
<formula><location><page_6><loc_18><loc_9><loc_48><loc_14></location>( ) + ω + iν ni ω + i ( ν in + ν ni ) k 4 c 2 A c 2 i cos 2 θ, (48)</formula>
<formula><location><page_6><loc_12><loc_6><loc_48><loc_8></location>D n ( ω ) = ω ( ω + iν ni ) -k 2 c 2 n , (49)</formula>
<formula><location><page_6><loc_55><loc_87><loc_92><loc_93></location>D 2 c ( ω ) = ων in ν ni ω + i ( ν in + ν ni ) [ ω 3 ( ω + i ( ν in + ν ni )) -k 4 c 2 A c 2 n cos 2 θ . (50)</formula>
<text><location><page_6><loc_52><loc_83><loc_92><loc_89></location>] The waves in the ion-electron fluid and in the neutral fluid are described by the same dispersion relation. The dispersion relation is</text>
<formula><location><page_6><loc_68><loc_80><loc_92><loc_82></location>D ( ω ) = 0 , (51)</formula>
<text><location><page_6><loc_52><loc_67><loc_92><loc_80></location>and is equivalent to the dispersion relation given by Zaqarashvili et al. (2011) in their Equation (57), although here we use a different notation. Expressed in polynomial form, the dispersion relation is a 7th-order polynomial in ω , so that there are seven solutions (see Zaqarashvili et al. 2011). The frequencies are complex, namely ω = ω R + iω I , where ω R and ω I are the real and imaginary parts. The imaginary part of the frequency is negative and accounts for the exponential damping rate of the perturbations due to ion-neutral collisions.</text>
<text><location><page_6><loc_82><loc_29><loc_82><loc_31></location>/negationslash</text>
<text><location><page_6><loc_52><loc_12><loc_92><loc_67></location>Before exploring the solutions to the dispersion relation, it is necessary to make a comment on the nature of the solutions. We distinguish between two different kind of solutions depending of whether ω R is zero or nonzero. (1) The first kind of solutions are those with ω R = 0, i.e., solutions that represent purely decaying disturbances. These solutions are not waves in the strict sense because they do not propagate. The solutions with ω R = 0 were called 'vortex modes' by Zaqarashvili et al. (2011) but, in our view, the name 'vortex modes' does not reflect the true nature of these solutions. The reason is that vortex modes in a partially ionized plasma are intrinsically related to Alfv'en waves (see Paper I), not to magnetoacoustic waves. Magnetoacoustic modes do not produce vorticity perturbations (see, e.g., Goossens 2003), hence the purely imaginary solutions of the magnetoacoustic dispersion relation cannot represent fluid vorticity. Instead, we propose to physically interpret the solutions with ω R = 0 as 'entropy modes' (see, e.g., Goedbloed & Poedts 2004; Murawski et al. 2011). Entropy modes represent perturbations of density and, therefore, compression of the plasma. Compression is an intrinsic property of magnetoacoustic modes. Entropy modes have zero frequency in the absence of collisions. Due to ion-neutral collisions, perturbations in density are damped and the entropy mode frequency acquires an imaginary part, while its real part remains zero. (2) The second kind of solutions are those with ω R = 0. These solutions represent propagating waves and appear in pairs, namely ω 1 = ω R + iω I and ω 2 = -ω R + iω I . Both ω 1 and ω 2 represent the same magnetoacoustic wave, but the real parts of their frequencies have opposite signs (see Carbonell et al. 2004). The different signs of the real parts account for forward ( ω R > 0) and backward ( ω R < 0) propagation with respect to the direction of the wavevector, k . Since the equilibrium is static, the two directions of propagation are physically equivalent. As shown later, the number of entropy modes and propagating waves is not constant. Depending on the physical parameters considered, there can be conversion between these two kind of solutions.</text>
<text><location><page_6><loc_52><loc_7><loc_92><loc_12></location>Unlike the limit cases studied in Section 3, no simple analytic solutions of the dispersion relation can be obtained when ν in and ν ni are arbitrary. Instead, we solve the dispersion relation numerically. The numerically ob-</text>
<text><location><page_7><loc_8><loc_89><loc_48><loc_92></location>ined dispersion diagrams are plotted as functions of the averaged collision frequency, ¯ ν , defined as</text>
<formula><location><page_7><loc_17><loc_85><loc_48><loc_88></location>¯ ν = ρ i ν in + ρ n ν ni ρ i + ρ n = 2 ρ i + ρ n α in . (52)</formula>
<text><location><page_7><loc_8><loc_65><loc_48><loc_84></location>Because of the complex behavior displayed by the solutions and the large number of parameters involved, we organize the presentation of the results and their discussion as follows. First, we investigate the specific situation in which the wave propagation is strictly perpendicular to the magnetic field direction. This paradigmatic case is helpful to get physical insight of the behavior of the various solutions and to illustrate some general results that repeatedly appear in this investigation. Slow modes are absent from the discussion for perpendicular propagation because they are unable to propagate across the magnetic field. Subsequently, we study the other limit case, i.e., wave propagation parallel to the magnetic field, so that slow modes are added to the discussion. Finally, the general case of oblique propagation is taken into account.</text>
<section_header_level_1><location><page_7><loc_17><loc_62><loc_40><loc_63></location>4.1. Perpendicular propagation</section_header_level_1>
<text><location><page_7><loc_8><loc_52><loc_48><loc_61></location>We start by studying the case of perpendicular propagation to the magnetic field. Hence we set θ = π/ 2. The dispersion relation (Equation (51)) reduces to a 5th-order polynomial because the slow magnetoacoustic wave is absent. Conversely, the fast magnetoacoustic mode and the neutral acoustic mode can propagate across the field and so they remain when θ = π/ 2.</text>
<text><location><page_7><loc_8><loc_31><loc_48><loc_52></location>We compute the real and imaginary parts of the frequency of the various solutions of the dispersion relation as functions of ¯ ν . Frequencies are expressed in units of kc i so that the results can be applied to the astrophysical plasma of interest by providing appropriate numeric values of k and c i . The averaged collision frequency is varied four orders of magnitude between ¯ ν/kc i = 0 . 01 and ¯ ν/kc i = 100. These two values are chosen to represent the uncoupled and strongly coupled limits, respectively. Beyond these two values of ¯ ν/kc i the behavior of the solutions remains essentially unaltered. Figures 1 and 2 show the results for β i = 0 . 04 (lowβ i case) and β i = 25 (highβ i case), respectively. Three values of χ are considered in each case, namely χ = 0 . 2, 2, and 20, which cover the cases of largely ionized, moderately ionized, and weakly ionized plasmas, respectively.</text>
<text><location><page_7><loc_8><loc_7><loc_48><loc_31></location>First, we analyze the real part of the frequency. The comparison of the numerically obtained ω R with the analytic approximations of Section 3 helps us identifying the various propagating waves in the limit values of ¯ ν . The modes are labeled and plotted with different line styles in Figures 1 and 2. When ¯ ν/kc i = 0 . 01 we find the classic fast magnetoacoustic (solid black line) and neutral acoustic (dashed blue line) modes, while for ¯ ν/kc i = 100 only the modified fast wave is present as a propagating solution, in agreement with Section 3. As expected, the modes do not display the properties of the classic waves when the averaged collision frequency takes intermediate values. To understand how the solutions change due to the effect of collisions we continuously follow the various solutions when ¯ ν/kc i increases from ¯ ν/kc i = 0 . 01 to ¯ ν/kc i = 100. We find that one of the two solutions present at the low collision frequency limit has a cutoff when ¯ ν/kc i reaches a certain value. At the cutoff, the</text>
<text><location><page_7><loc_52><loc_85><loc_92><loc_92></location>forward ( ω R > 0) and backward ( ω R < 0) waves of this specific solution merge and their ω R becomes zero. On the contrary, the other solution does not have a cutoff and eventually becomes the modified fast wave at the high collision frequency limit.</text>
<text><location><page_7><loc_52><loc_41><loc_92><loc_85></location>The location of the cutoff is important because it determines the number of distinct waves that are able to propagate for a certain combination of parameters. The effect of the various parameters on the location of the cutoff can be studied as in Paper I by taking advantage of the fact that the dispersion relation is a polynomial in ω . So, we can compute the polynomial discriminant of the dispersion relation. By definition, the discriminant is zero when the dispersion relation has a multiple solution. Since the cutoff takes place when a forward and a backward propagating wave merge, the dispersion relation has necessarily a double root at the cutoff. The discriminant is a function of χ , ¯ ν/kc i , and β i . Hence, the zeros of the discriminant inform us about the relation between these parameters at the cutoff. We omit here the expression of the discriminant because it is rather cumbersome and can be straightforwardly obtained from the dispersion relation using standard algebraic methods. The number of propagating solutions for a given set of parameters is represented in Figure 3 in the χ -¯ ν/kc i plane for three different values of β i . The zero of the discriminant is plotted with a line. Below this line two propagating waves are possible, namely the classic fast and acoustic modes, while above the line only the modified fast wave is present as a propagating wave. Importantly, we find that a third region, where propagating waves are not possible, appears when β i /lessmuch 1 (see the shaded area in Figure 3 when β i = 10 -3 ). This forbidden region takes place for large ionization ratio (weakly ionized plasmas), very low β i (strong magnetic fields) and relatively high collision frequency (strongly coupled plasmas). This could be the case of the plasma in intense magnetic tubes of the low solar atmosphere.</text>
<text><location><page_7><loc_52><loc_19><loc_92><loc_41></location>The physical reason for the existence of wave cutoffs is qualitatively the same as discussed in Paper I for the case of Alfv'en waves (see also, e.g., Kulsrud & Pearce 1969; Pudritz 1990; Kamaya & Nishi 1998; Mouschovias et al. 2011). There is competition between the ion-neutral friction force and the wave restoring forces. Friction becomes the dominant force and overcomes the remaining forces at the cutoff. Hence, waves are unable to propagate when the strength of friction greatly exceeds those of the restoring forces. The cutoff found here corresponds to the situation in which ions and neutrals are sufficiently coupled for the neutral fluid to be significantly affected by the magnetic field. As a consequence, the neutral fluid is not able to support its own pure acoustic modes, which start to behave as slow-like magnetoacoustic waves. For θ = π/ 2 slow modes cannot propagate and so these solutions are cut off.</text>
<text><location><page_7><loc_52><loc_8><loc_92><loc_19></location>We turn to the imaginary part of the frequency. For presentation purposes, we plot the absolute value of ω I in logarithmic scale. The line styles used in the graphics of ω I are the same as in the plots of ω R so that the various solutions can be easily identified. The inspection of ω I in Figures 1 and 2 informs us of the existence of a third purely imaginary mode (red dotted line). This mode is the descendant of the undamped entropy mode, which</text>
<text><location><page_8><loc_8><loc_68><loc_48><loc_92></location>becomes damped by collisions when ¯ ν/kc i increases. The damping rate of this nonpropagating mode monotonically increases with ¯ ν/kc i . Regarding the propagating waves, we see that the solution that has the cutoff in ω R displays a bifurcation in ω I at the same location. Two purely imaginary branches emerge at the bifurcation. We identify these new branches as entropy modes. On the one hand, the upper branch is heavily damped and follows the behavior of the original entropy mode plotted with a red dotted line. On the other hand, the lower branch is less damped and becomes an undamped entropy mode when ¯ ν/kc i →∞ . Finally, the remaining solution whose ω I does not have a bifurcation corresponds to the propagating wave that becomes the modified fast wave when ¯ ν/kc i /greatermuch 1. The damping rate of this wave is nonmonotonic. Its | ω I | is maximal, i.e., the damping is most efficient, when ¯ ν ≈ ω R . This is the same result as for Alfv'en waves (see Paper I).</text>
<section_header_level_1><location><page_8><loc_20><loc_65><loc_37><loc_67></location>4.2. Parallel propagation</section_header_level_1>
<text><location><page_8><loc_8><loc_50><loc_48><loc_65></location>We move to the case of parallel propagation to the magnetic field. We set θ = 0 and perform the same computations as in the previous section. These results are displayed in Figures 4 and 5, where we have considered the same parameters as before. As expected, the slow magnetoacoustic wave is now present (dash-dotted green line). As in the perpendicular propagation case, we analyze Figures 4 and 5 by observing how the modes present at the low collision frequency limit evolve as we increase the collision frequency toward the high collision frequency limit.</text>
<text><location><page_8><loc_8><loc_16><loc_48><loc_50></location>In the lowβ i case (Figure 4), the slow magnetoacoustic wave and the neutral acoustic wave strongly interact. There is a change of character between these two waves depending on the value of χ . When χ /lessmuch 1 , i.e., largely ionized plasma, the neutral acoustic mode has a cutoff and the classic slow mode becomes the modified slow mode in the limit of large ¯ ν/kc i . On the contrary, the situation is reversed when χ /greaterorsimilar 1, so that the solution that is cut off is the classic slow mode while the neutral acoustic mode is the solution that becomes the modified slow mode. These results are consistent with the approximate study of Section 3.2. On the other hand, the behavior of the fast magnetoacoustic wave in the lowβ i case is very similar to that of the Alfv'en waves studied in Paper I. It is well-known that in a lowβ i plasma fast waves behave as Alfv'en waves for parallel propagation to the magnetic field (see, e.g., Goossens 2003). As in the case of Alfv'en waves, fast magnetoacoustic waves are nonpropagating in an interval of ¯ ν/kc i when χ > 8 (see extensive details in Paper I for the case of Alfv'en waves). This nonpropagating interval can be seen in Figure 4 in the results for χ = 20. In this interval the fast wave becomes a purely imaginary solution. The boundaries of the nonpropagating interval can be obtained from Equation (20) of Paper I. Using the present notation, the boundaries of the nonpropagating interval are</text>
<formula><location><page_8><loc_8><loc_9><loc_49><loc_15></location>¯ ν kc i = 2 χ β 1 / 2 i ( χ +1) [ χ 2 +20 χ -8 8 ( χ +1) 3 ± χ 1 / 2 ( χ -8) 3 / 2 8 ( χ +1) 3 ] 1 / 2 , (53)</formula>
<text><location><page_8><loc_8><loc_7><loc_48><loc_9></location>where the -and + signs correspond to the left and right boundaries of the nonpropagating range.</text>
<text><location><page_8><loc_52><loc_75><loc_92><loc_92></location>In the highβ i case (Figure 5), the slow magnetoacoustic wave displays little interaction with the other two solutions. In this case, the behavior of the fast magnetoacoustic wave and the neutral acoustic wave is similar to that found when θ = π/ 2, while the slow wave behaves as an Alfv'en wave (Paper I). Now, it is the slow wave and not the fast wave that has a nonpropagating interval of ¯ ν/kc i . Equation (53) holds for the slow wave in the highβ i case too. Since Equation (53) depends upon β -1 / 2 i , the nonpropagating interval is now shifted towards smaller ¯ ν/kc i than in the lowβ i case (compare the results for χ = 20 in Figure 4 and 5).</text>
<text><location><page_8><loc_52><loc_50><loc_92><loc_75></location>We study in more detail the presence of cutoffs and nonpropagating intervals by using again the discriminant of the dispersion relation. This is shown in Figure 6 for β i = 0 . 04 and β i = 25. As in the case of perpendicular propagation (see Figure 3) we find two zones in the χ -¯ ν/kc i plane separated by a cutoff (black solid line). These separate zones correspond to the region where the classic slow, fast, and acoustic waves live and the region where the modified slow and fast modes exist. In addition, we find the presence of the nonpropagating interval previously discussed (area between the red lines). In the lowβ i case (Figure 6a) the nonpropagating interval is located above the boundary between the two regions. Within the nonpropagating interval, the modified slow wave is the only propagating solution. On the contrary, in the highβ i case (Figure 6b) the nonpropagating interval intersects the boundary between the two regions. Hence the number of possible scenarios increases, as schematically indicated in Figure 6b.</text>
<text><location><page_8><loc_52><loc_34><loc_92><loc_50></location>It is appropriate to compare our results with those plotted in Figure 3 of Zaqarashvili et al. (2011) in order to check whether the present work is consistent with that previous study. In our notation, the parameters used by Zaqarashvili et al. (2011) are χ = 1, β i = 0 . 25, and θ = 0. These parameters are close to those used in Figure 4(c)-(d). To perform a proper comparison we should use the same notation as Zaqarashvili et al. (2011). They used two dimensionless quantities, w and a , that correspond to the wave frequency and to the inverse of the collision frequency, respectively. These two quantities are expressed in our notation as</text>
<formula><location><page_8><loc_64><loc_30><loc_92><loc_33></location>w = √ β i (1 + χ ) cos θ ω kc i , (54)</formula>
<text><location><page_8><loc_52><loc_10><loc_92><loc_28></location>√ We solve the dispersion relation using the above parameters and reproduce in Figure 7 the results shown in Figure 3 of Zaqarashvili et al. (2011). A perfect agreement between our results and those of Zaqarashvili et al. (2011) is obtained. With the help of the discriminant of the dispersion relation, we compute that the cutoff of the neutral acoustic wave takes place at ¯ ν/kc i ≈ 0 . 84, which corresponds to a ≈ 3 . 4 in the notation of Zaqarashvili et al. (2011). Since χ < 8, the fast mode does not have a nonpropagating interval. Thus, our results confirm and fully agree with the specific example studied by Zaqarashvili et al. (2011).</text>
<formula><location><page_8><loc_65><loc_27><loc_92><loc_30></location>a = 2 cos θ β i (1 + χ ) kc i ¯ ν . (55)</formula>
<section_header_level_1><location><page_8><loc_63><loc_8><loc_81><loc_9></location>4.3. Oblique propagation</section_header_level_1>
<text><location><page_9><loc_8><loc_69><loc_48><loc_92></location>Finally, here we consider the case in which the direction of wave propagation is in between the limit cases studied in the previous two Subsections. Hence, we use θ = π/ 4. The results are displayed in Figures 8 and 9. The behavior of the various solutions is more complex than in the previous limits and can be understood as a combination of the results for perpendicular and parallel propagation. Compared to the case with θ = 0, the acoustic and fast modes display similar behavior, while the slow mode is the solution whose behavior is most altered, specially in the lowβ i case (compare Figures 4 and 8). Both slow and fast modes display nonpropagating intervals for oblique propagation, whereas only the fast mode have a nonpropagating interval for parallel propagation. The fast mode nonpropagating interval is also approximately described by Equation (53), which points out that the location of this interval does not depend on θ .</text>
<text><location><page_9><loc_8><loc_53><loc_48><loc_69></location>We proceed as before and show in Figure 10 the various possible scenarios for wave propagation in the χ -¯ ν/kc i plane of parameters. The highβ i case (Figure 10b) is very similar to the equivalent result for θ = 0 (Figure 6b), so that no additional comments are needed. However, in the lowβ i case, the presence of nonpropagating intervals for both slow and fast modes and their intersections cause the presence of six possible scenarios (Figure 10a). The result represented in Figure 10a clearly points out the high complexity of the interactions between the various waves in a partially ionized plasma, which do not occur in fully ionized plasmas (Mouschovias et al. 2011).</text>
<text><location><page_9><loc_8><loc_33><loc_48><loc_53></location>The existence of two nonpropagating intervals in the lowβ i case is consistent with the results by Soler et al. (2013b) of slow waves propagating along a partially ionized magnetic flux tube. Soler et al. (2013b) also found the presence of two nonpropagating intervals, see their Figure 5. Here we see that this double nonpropagating interval is a result also present for magnetoacoustic waves in a homogeneous, partially ionized medium. Thus, we can now conclude that this result is not caused by the geometry of the waveguide, but it is an intrinsic property of magnetoacoustic waves propagating obliquely to the magnetic field in a partially ionized plasma. The slow magnetoacoustic waves in a magnetic flux tube studied by Soler et al. (2013b) behave as the waves studied here for oblique propagation.</text>
<section_header_level_1><location><page_9><loc_15><loc_28><loc_42><loc_30></location>5. APPLICATION TO THE SOLAR CHROMOSPHERE</section_header_level_1>
<text><location><page_9><loc_8><loc_11><loc_48><loc_27></location>Here we perform a specific application to magnetoacoustic waves propagating in the partially ionized solar chromosphere. This application is motivated by the recent observations of ubiquitous compressive waves in the low solar atmosphere (Morton et al. 2011, 2012). To represent the chromospheric plasma we use the same simplified model considered in Paper I. We use the quiet sun model C of Vernazza et al. (1981), hereafter VALC model, to account for the variation of physical parameters with height from the photosphere to the base of the corona. The expression of α in is taken after Braginskii (1965), namely</text>
<formula><location><page_9><loc_19><loc_5><loc_48><loc_9></location>α in = 1 2 ρ i ρ n m i √ 16 k B T πm i σ in , (56)</formula>
<text><location><page_9><loc_52><loc_65><loc_92><loc_92></location>where m i is the ion (proton) mass, k B is Boltzmann's constant, and σ in is the collision cross section. Equation (56) considers only hydrogen and ignores the influence of heavier species. In Equation (56) it is implicitly assumed that the ion and neutral masses are approximately equal. For the collision cross section of protons with neutral hydrogen atoms we consider the typical value of σ in ≈ 5 × 10 -19 m 2 (used in, e.g., Khodachenko et al. 2004; Leake et al. 2005; Arber et al. 2007; Soler et al. 2009; Khomenko & Collados 2012, among others). Here we must note that, in the application done in Paper I for the case of Alfv'en waves, we took σ in ≈ 10 -20 m 2 based on the hard sphere collision model (see, e.g., Braginskii 1965; Zaqarashvili et al. 2013). In a recent paper, Vranjes & Krstic (2013) claim that the realistic value of σ in in the solar chromosphere is about two orders of magnitude higher than that estimated in the hard sphere model. The value σ in ≈ 5 × 10 -19 m 2 used here is closer to the value proposed by Vranjes & Krstic (2013) than to the hard sphere value.</text>
<text><location><page_9><loc_52><loc_60><loc_92><loc_65></location>We adopt the magnetic field strength model by Leake & Arber (2006), which aims to represent the field strength in a chromospheric vertical magnetic flux tube expanding with height, namely</text>
<formula><location><page_9><loc_65><loc_55><loc_92><loc_59></location>B = B ph ( ρ ρ ph ) 0 . 3 , (57)</formula>
<text><location><page_9><loc_52><loc_38><loc_92><loc_55></location>where ρ = ρ i + ρ n is the total density, and B ph and ρ ph = 2 . 74 × 10 -4 kg m -3 are the magnetic field strength and the total density, respectively, at the photospheric level. The variation of ρ with height is taken from the VALC model. Accordingly, the magnetic field strength decreases with height. Regarding the value of B ph , we consider two possible scenarios: an active region and the quiet Sun. For the active region case, we set B ph = 1 . 5 kG, so that B ≈ 100 G at 1,000 km and B ≈ 20 G at 2,000 km above the photosphere. For the quiet Sun case, we set B ph = 100 G, so that B ≈ 7 G at 1,000 km and B ≈ 1 G at 2,000 km above the photosphere.</text>
<text><location><page_9><loc_52><loc_20><loc_92><loc_39></location>As explained in Paper I, the physical parameters in this simplified model of the low solar atmosphere depend on the vertical direction, while in the previous theoretical analysis all the parameters are taken constant. Here we perform a local analysis and use the dispersion relation derived for a homogeneous plasma. We use the physical parameters at a given height to locally solve the dispersion relation at that height. A limitation of the present approach is that it ignores the possible presence of cutoff frequencies due to gravitational stratification (see, e.g., Roberts 2006). The condition for this method to be approximately valid is that the wavelength, λ = 2 π/k , is much shorter than the stratification scale height. This is fulfilled by the wavelengths used in this analysis.</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_20></location>Figure 11 displays the variation with height above the photosphere of the three physical quantities relevant for the behavior of the waves: the ionization fraction, χ , the ionized fluid β i , and the averaged ion-neutral collision frequency, ¯ ν . Only β i is affected by the magnetic field scenario considered. First of all, we see that χ ranges several orders of magnitude from very weakly ionized plasma at the low levels of the chromosphere to fully ionized plasma when the transition region to the solar corona is reached. Full ionization of hydrogen takes place</text>
<text><location><page_10><loc_8><loc_79><loc_48><loc_92></location>at 2,100 km above the photospheric level, approximately. The value of β i in the active region case is much smaller than unity throughout the chromosphere, which informs us that we are dealing with a lowβ i situation. Conversely, in the quiet Sun case the value of β i ranges from β i /lessmuch 1 at low levels to β i /greatermuch 1 at high levels in the chromosphere. Moreover, ¯ ν also ranges several orders of magnitude, specially in the low chromosphere up to 700 km, approximately. Then, ¯ ν takes values in between 10 2 Hz and 10 3 Hz until the full ionization level is reached.</text>
<text><location><page_10><loc_8><loc_33><loc_48><loc_78></location>The high values of ¯ ν displayed in Figure 11(c) indicate that only waves with short wavelengths, i.e., high frequencies, would be affected by two-fluid effects. To investigate this, we use the discriminant of the dispersion relation to determine the nature of the propagating solutions as function of height for a given value of the wavelength. This is shown in Figure 12 for three values of θ , namely θ = 0, π/ 4, and π/ 2. In the active region case, we see that two-fluid effects are important for propagation of magnetoacoustic waves when λ /lessorsimilar 1 km. For longer wavelengths, ions and neutrals behave as a single fluid and, consequently, the propagating waves are the slow and fast waves modified due to the presence of neutrals (see Section 3.2). For short wavelengths, however, there appear a number of nonpropagating intervals that constrain the propagation of the various modes. Also, due to the strong ion-neutral coupling, the presence of neutral acoustic waves in the active region chromosphere is only possible for very short wavelengths. In the case of the quiet Sun, the nonpropagating intervals are shifted towards values of λ about an order or magnitude shorter than in the active region case. This result points out that two-fluid effects are of less relevance in those regions of the chromosphere where the magnetic field is weak. For representation purposes, in Figure 12 we have varied λ in between 10 -7 km and 10 2 km, although we must warn the reader that the fluid approximation for the various species may not hold for wavelengths approaching the lower boundary of this range. Therefore, the results shown in Figure 12 for wavelengths near the lower limit should be taken with extreme caution. Also, the assumption that gravitational effects are negligible breaks down for very long wavelengths, although this probably happens for wavelengths longer than those in Figure 12.</text>
<text><location><page_10><loc_8><loc_6><loc_48><loc_33></location>Next, we define the damping rate as δ = -ω I /ω R . This quantity informs us about the efficiency of damping due to ion-neutral collisions. Indirectly, δ is also an indicator of the ability of ion-neutral collisions to heat the plasma by dissipating magnetoacoustic waves. For the active region case, Figure 13 shows the damping rate of the modified slow and fast waves as function of height for λ = 10 km and λ = 1 km. When λ = 10 km (Figure 13(a)-(b)), the waves are unaffected by two-fluid effects and do not have nonpropagating intervals. The only effect of ion-neutral collisions is to produce the damping of the waves. The fast wave damping rate is independent of the propagation angle, θ , and is maximal at 1,900 km, approximately. This suggests that the main contribution of fast waves to plasma heating might take place at the higher levels of the active region chromosphere. Conversely, the slow wave damping rate strongly depends on θ . The damping of the slow wave is in general very weak. We have to consider propagation almost perpendicular to the magnetic field, i.e., θ → π/ 2, to obtain a significant</text>
<text><location><page_10><loc_52><loc_72><loc_92><loc_92></location>slow wave damping. Two-fluid effects play a role when the wavelength is decreased to λ = 1 km (Figure 13(c)(d)). Consistent with Figure 12, when λ = 1 km the fast wave has a nonpropagating zone around 1,500 km above the photosphere. Again, the results for the fast wave are independent of θ . Near the location of the nonpropagating zone, the fast wave damping rate boosts dramatically since δ → ∞ in the nonpropagating interval. Conversely, the slow wave only has nonpropagating regions when propagation is nearly perpendicular to the magnetic field. In this case, the slow wave has two nonpropagating intervals at two different heights (Soler et al. 2013b). These forbidden intervals for the nearly perpendicular slow wave cover a significant part of the chromosphere.</text>
<text><location><page_10><loc_52><loc_56><loc_92><loc_72></location>Figure 14 shows the same results as Figure 13 with λ = 1 km but for the quiet Sun case. Now, the solutions do not display nonpropagating intervals. As indicated by Figure 12, in the quiet Sun case we need to consider shorter wavelengths than in the active region case for nonpropagating intervals to be present. In addition, Figure 14 shows that both fast and slow waves damping rates depend upon the propagation angle. The reason for this result is that in the quiet Sun case β i is higher than in the active region case and, in fact, β i > 1 at the upper levels. This causes the fast wave behavior to be dependent on the propagation angle too.</text>
<text><location><page_10><loc_52><loc_32><loc_92><loc_56></location>In summary, in this application we conclude that magnetoacoustic waves propagating in regions of the solar chromosphere with strong magnetic fields are affected by two-fluid effects when λ /lessorsimilar 1 km. The presence of nonpropagating intervals heavily constrains the propagation of magnetoacoustic waves of short wavelengths. In addition, damping due to ion-neutral collisions is very efficient in the vicinity of the nonpropagating regions, which points out that significant wave energy dissipation may take place at those heights in the chromosphere. The fast wave is the propagating solution that may contribute the most to heat the plasma. For λ /greaterorsimilar 1 km, however, twofluid effects are of less relevance because ions and neutrals behave as a single fluid. Then, wave damping is appropriately described by the single-fluid approximation (e.g., Khodachenko et al. 2004). In quiet Sun regions, we need to consider much shorter wavelengths for two-fluid effects to be relevant.</text>
<text><location><page_10><loc_52><loc_15><loc_92><loc_32></location>At present, we lack of appropriate observations to compare with the theoretical predictions discussed above. Unfortunately, current instruments do not have enough spatial and temporal resolutions to observe the range of wavelengths where two-fluid effects would be important. It is possible that new and future instruments as, e.g., the Atacama Large Millimeter/submillimeter Array (ALMA), may reach sufficiently high resolutions to observe wavelengths of 1 km and shorter. Angular resolution of 0.01 arcseconds or smaller is needed to observe wavelengths on the order of kilometres in the chromosphere. The ALMA telescope may reach the required resolution.</text>
<section_header_level_1><location><page_10><loc_61><loc_13><loc_83><loc_14></location>6. CONCLUDING REMARKS</section_header_level_1>
<text><location><page_10><loc_52><loc_7><loc_92><loc_12></location>In this paper we continued the work started in Soler et al. (2013a) about the effect of ion-neutral collisions on MHD wave propagation in a two-fluid plasma. In the previous paper, we focused on incompressible</text>
<text><location><page_11><loc_8><loc_71><loc_48><loc_92></location>Alfv'en waves. Here, we investigated compressible magnetoacoustic waves. The present work is also related to recent investigation of wave propagation in a partially ionized magnetic flux tube (Soler et al. 2013b). The study presented in this paper reveals that ion-neutral coupling strongly affects the well-known behavior and properties of classic magnetoacoustic waves. There is a large number of possible scenarios for wave propagation depending on the plasma physical properties. As pointed out by Mouschovias et al. (2011), the various waves supported by a partially ionized plasma display complex interactions and couplings that are not present in the fully ionized case. Among these complex interactions, the presence of cutoffs and forbidden intervals has an strong impact on waves since the allowed wavelengths of the propagating modes get constrained.</text>
<text><location><page_11><loc_8><loc_44><loc_48><loc_71></location>After performing a general study, we considered the particular case of the solar chromosphere and showed that magnetoacoustic waves with λ /lessorsimilar 1 km are affected by two-fluid effects in regions with intense magnetic fields, while much shorter wavelengths have to be considered for these effects to be relevant in quiet Sun conditions. In addition, we discussed the possible role of these waves in heating the chromospheric plasma due to dissipation by ion-neutral collisions. Here, we must refer to the comment by Khodachenko et al. (2004) about that the correct description of MHD wave damping in the solar atmosphere requires the consideration of all energy dissipation mechanisms. We have focused on the effect of ion-neutral collisions alone and have not taken into account the roles of, e.g., thermal conduction, viscosity, resistivity, etc., that may have an important impact on the dissipation of wave energy. The consideration of these mechanisms using the two-fluid description of a partially ionized plasma is an interesting task to be done in future works.</text>
<text><location><page_11><loc_8><loc_29><loc_48><loc_44></location>Finally, a natural extension of the investigation done in Soler et al. (2013a,b) and in the present paper is to abandon the normal mode approach and to study the behavior of impulsively excited disturbances. This requires the solution of the initial-value problem by means of time-dependent numerical simulations. The combination of results from both the initial-value problem and the normal mode analysis would give us a complete picture of wave excitation and propagation in a partially ionized plasma. The investigation of the initial-value problem will be tackled in future studies.</text>
<text><location><page_11><loc_52><loc_87><loc_92><loc_92></location>We acknowledge the support from MINECO and FEDER Funds through grant AYA2011-22846 and from CAIB through the 'grups competitius' scheme and FEDER Funds.</text>
<section_header_level_1><location><page_11><loc_67><loc_83><loc_77><loc_84></location>REFERENCES</section_header_level_1>
<text><location><page_11><loc_52><loc_75><loc_91><loc_80></location>Arber, T. D., Haynes, M., & Leake, J. E. 2007, ApJ, 666, 541 Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205 Carbonell, M., Oliver, R., & Ballester, J. L. 2004, A&A, 415, 739 Cramer, N. F. 2001, The Physics of Alfv'en Waves (Wiley-VCH) De Pontieu, B., Martens, P. C. H., & Hudson, H. S. 2001, ApJ,</text>
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<list_item><location><page_11><loc_52><loc_33><loc_92><loc_75></location>558, 859 Goedbloed, J. P., & Poedts, S. 2004, Principles of magnetohydrodynamics (Cambridge University Press) Goossens, M. 2003, An introduction to plasma astrophysics and magnetohydrodynamics (Kluwer Academic Publishers) Kamaya, H., & Nishi, R. 1998, ApJ, 500, 257 Khodachenko, M. L., Arber, T. D., Rucker, H. O., & Hanslmeier, A. 2004, A&A, 422, 1073 Khodachenko, M. L., Rucker, H. O., Oliver, R., Arber, T. D., & Hanslmeier, A. 2006, Advances in Space Research, 37, 447 Khomenko, E., & Collados, M. 2012, ApJ, 747, 87 Kulsrud, R., & Pearce, W. P. 1969, ApJ, 156, 445 Kumar, N., & Roberts, B. 2003, Sol. Phys., 214, 241 Leake, J. E., & Arber, T. D. 2006, A&A, 450, 805 Leake, J. E., Arber, T. D., & Khodachenko, M. L. 2005, A&A, 442, 1091 Lighthill, M. J. 1960, Royal Society of London Philosophical Transactions Series A, 252, 397 Mart'ınez-Sykora, J., De Pontieu, B., & Hansteen, V. 2012, ApJ, 753, 161 Morton, R. J., Erd'elyi, R., Jess, D. B., & Mathioudakis, M. 2011, ApJ, 729, L18 Morton, R. J., Verth, G., Jess, D. B., et al. 2012, Nature Communications, 3 Mouschovias, T. C., Ciolek, G. E., & Morton, S. A. 2011, MNRAS, 415, 1751 Murawski, K., Zaqarashvili, T. V., & Nakariakov, V. M. 2011, A&A, 533, A18 P'ecseli, H., & Engvold, O. 2000, Sol. Phys., 194, 73 Pudritz, R. E. 1990, ApJ, 350, 195 Roberts, B. 2006, Royal Society of London Philosophical Transactions Series A, 364, 447 Russell, A. J. B., & Fletcher, L. 2013, ApJ, 765, 81 Soler, R., Andries, J., & Goossens, M. 2012, A&A, 537, A84 Soler, R., Carbonell, M., Ballester, J. L., & Terradas, J. 2013a, ApJ, 767, 171 (Paper I) Soler, R., D'ıaz, A. J., Ballester, J. L., & Goossens, M. 2013b, A&A, 551, A86 Soler, R., Oliver, R., & Ballester, J. L. 2009, ApJ, 699, 1553 Vernazza, J. E., Avrett, E. H., & Loeser, R. 1981, ApJS, 45, 635 Vranjes, J., & Krstic, P. S. 2013, A&A, 554, A22 Zaqarashvili, T. V., Khodachenko, M. L., & Rucker, H. O. 2011, A&A, 529, A82 Zaqarashvili, T. V., Khodachenko, M. L., & Soler, R. 2013, A&A,</list_item>
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<caption>Figure 1. Real (left) and imaginary (right) parts of the frequency of the various waves versus the averaged collision frequency (in logarithmic scale) for propagation perpendicular to the magnetic field, i.e., θ = π/ 2, with β i = 0 . 04. Panels (a)-(b) are for χ = 0 . 2, panels (c)-(b) for χ = 2, and panels (c)-(b) for χ = 20. All frequencies are expressed in units of kc i . Note that the absolute value of ω I is plotted.</caption>
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<caption>Figure 2. Same as Figure 1 but for β i = 25.</caption>
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<caption>Figure 3. Number and nature of the propagating solutions of the dispersion relation in the χ -¯ ν/kc i plane for θ = π/ 2 and β i = 10 -3 , 0.04, and 25. Each line denotes the boundary between the various regions for a given value of β i . The shaded area denotes a region in the case β i = 10 -3 where propagation is forbidden. This forbidden region is also distinguished as a tiny feature in the line for β i = 0 . 04.</caption>
</figure>
<text><location><page_14><loc_18><loc_35><loc_81><loc_37></location>Behavior of the modified fast and slow waves and their approximate frequencies in the highly collisional limit depending on the ordering of the characteristic velocities.</text>
<table>
<location><page_14><loc_18><loc_17><loc_82><loc_34></location>
<caption>Table 1</caption>
</table>
<text><location><page_14><loc_18><loc_15><loc_81><loc_17></location>Note . - IMW, IAW, GMW, and GAW denote isotropic magnetic wave, isotropic acoustic wave, guided magnetic wave, and guided acoustic wave, respectively.</text>
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<caption>Figure 4. Same as Figure 1 but for propagation parallel to the magnetic field, i.e, θ = 0. We use β i = 0 . 04.</caption>
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<caption>Figure 5. Same as Figure 4 but for β i = 25.</caption>
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<caption>Figure 6. Number and nature of the propagating solutions of the dispersion relation in the χ -¯ ν/kc i plane for θ = 0 and (a) β i = 0 . 04 and (b) β i = 25. The red lines correspond to the interval given in Equation (53).</caption>
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<caption>Figure 7. Reproduction of Figure 3 of Zaqarashvili et al. (2011) with the results of the present work. (a) Real and (b) imaginary parts of w as functions of a . The line colors have the same meaning as in Figure 3 of Zaqarashvili et al. (2011). The vertical dotted line denote the location of the neutral acoustic wave cutoff. Purely imaginary solutions are omitted in these plots.</caption>
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<caption>Figure 8. Same as Figure 1 but for oblique propagation with θ = π/ 4. We use β i = 0 . 04.</caption>
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<caption>Figure 9. Same as Figure 8 but for β i = 25.</caption>
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<caption>Figure 10. Number and nature of the propagating solutions of the dispersion relation in the χ -¯ ν/kc i plane for θ = π/ 4 and (a) β i = 0 . 04 and (b) β i = 25. The red and blue lines correspond to the nonpropagating intervals for slow and fast waves.</caption>
</figure>
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<caption>Figure 11. Variation of (a) χ , (b) β i , and (c) ¯ ν with height in the solar chromosphere model used in Section 5.</caption>
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<caption>Figure 12. Number and nature of the propagating waves with a given wavelength, λ , as function of height in the solar chromosphere for the active region case (left) and the quiet Sun case (right) with θ = 0 (top), θ = π/ 4 (middle), and θ = π/ 2 (bottom). The vertical dashed line denotes the height at which hydrogen becomes fully ionized.</caption>
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<caption>Figure 13. Active region case. Damping rate, δ = -ω I /ω R , of (a) fast waves and (b) slow waves as function of height in the solar chromosphere for λ = 10 km. The various lines are for θ = 0 (solid), π/ 4 (dotted), and π/ 2 . 1 (dashed). Panels (c) and (d) are the same as panels (a) and (b) but for λ = 1 km.</caption>
</figure>
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<caption>Figure 14. Same as Figure 13 but for the quiet Sun case with λ = 1 km.</caption>
</figure>
</document>
[
{
"title": "ABSTRACT",
"content": "Compressible disturbances propagate in a plasma in the form of magnetoacoustic waves driven by both gas pressure and magnetic forces. In partially ionized plasmas the dynamics of ionized and neutral species are coupled due to ion-neutral collisions. As a consequence, magnetoacoustic waves propagating through a partially ionized medium are affected by the ion-neutral coupling. The degree to which the behavior of the classic waves is modified depends on the physical properties of the various species and on the relative value of the wave frequency compared to the ion-neutral collision frequency. Here, we perform a comprehensive theoretical investigation of magnetoacoustic wave propagation in a partially ionized plasma using the two-fluid formalism. We consider an extensive range of values for the collision frequency, ionization ratio, and plasma β , so that the results are applicable to a wide variety of astrophysical plasmas. We determine the modification of the wave frequencies and study the frictional damping due to ion-neutral collisions. Approximate analytic expressions to the frequencies are given in the limit case of strongly coupled ions and neutrals, while numerically obtained dispersion diagrams are provided for arbitrary collision frequencies. In addition, we discuss the presence of cutoffs in the dispersion diagrams that constrain wave propagation for certain combinations of parameters. A specific application to propagation of compressible waves in the solar chromosphere is given. Subject headings: magnetic fields - magnetohydrodynamics (MHD) - plasmas - Sun: atmosphere Sun: oscillations - waves",
"pages": [
1
]
},
{
"title": "MAGNETOACOUSTIC WAVES IN A PARTIALLY IONIZED TWO-FLUID PLASMA",
"content": "Roberto Soler 1 , Marc Carbonell 2 , & Jose Luis Ballester 1 1 Departament de F'ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain and 2 Departament de Matem'atiques i Inform'atica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Draft version June 18, 2018",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "The theoretical study of magnetohydrodynamic (MHD) waves in partially ionized plasmas has received increasing attention in the recent years. Energy dissipation due to ion-neutral collisions may be important in many astrophysical plasmas, e.g., in the low solar atmosphere, because of its connection to plasma heating (e.g., Khomenko & Collados 2012; Mart'ınez-Sykora et al. 2012; Russell & Fletcher 2013). Some recent examples of theoretical works that focused on the investigation of MHD waves in partially ionized plasmas are, e.g., P'ecseli & Engvold (2000); De Pontieu et al. (2001); Kumar & Roberts (2003); Khodachenko et al. (2004, 2006); Leake et al. (2005); Zaqarashvili et al. (2011, 2013); Soler et al. (2012) among many others. In the present paper, we continue the research started in Soler et al. (2013a), hereafter Paper I, about wave propagation in a partially ionized two-fluid plasma. The two-fluid formalism used here assumes that ions and electrons form together an ion-electron fluid, while neutrals form a separate fluid. The ion-electron fluid and the neutral fluid interact by means of ion-neutral collisions. Paper I was devoted to the study of Alfv'en waves. Here we tackle the investigation of magnetoacoustic waves. The dynamics of magnetoacoustic waves is more involved than that of Alfv'en waves even in the case of a fully ionized plasma. The reason is that Alfv'en waves are incompressible and are only driven by the magnetic tension force. On the contrary, magnetoacoustic waves produce compression of the plasma and are driven by both magnetic and gas pressure forces (see, e.g., Goossens 2003). roberto.soler@uib.es, marc.carbonell@uib.es, joseluis.ballester@uib.es As a consequence, the behavior of magnetoacoustic waves is determined by the relative values of the sound and Alfv'en velocities in the plasma. When the medium is partially ionized, the situation is even more complex because, on the one hand, magnetoacoustic waves also produce compression in the neutral fluid due to collisions with ions and, on the other hand, the neutral fluid is able to support its own acoustic waves that may couple with the plasma magnetoacoustic waves. Thus, the study of propagation of magnetoacoustic disturbances in a partially ionized medium is interesting from the physical point of view and challenging from the theoretical point of view. This work can be related to the previous papers by Zaqarashvili et al. (2011) and Mouschovias et al. (2011). Zaqarashvili et al. (2011) derived the dispersion relation for magnetoacoustic waves in a partially ionized twofluid plasma. They focused their study on the derivation of the two-fluid equations from the more general threefluid equations and in comparing the two-fluid results with those in the single-fluid approximation. They concluded that the single-fluid approximation breaks down when the value of the ion-neutral collision frequency is on the same order of magnitude or lower than the wave frequency. Here we proceed as in Zaqarashvili et al. (2011) and use the more general two-fluid theory instead of the single-fluid approximation. Mouschovias et al. (2011) also considered the two-fluid theory and performed a very detailed investigation of wave propagation and instabilities in interstellar molecular clouds. To realistically represent the plasma in molecular clouds, Mouschovias et al. (2011) included in addition to ion-neutral collisions other effects as, e.g., ionization and recombination and self-gravity. The main purpose of the present work is to obtain physical insight on the effect of ion-neutral collisions on waves without taking into account other effects that could hinder the specific role of collisions. For this reason, our approach is closer to Zaqarashvili et al. (2011) than to Mouschovias et al. (2011). As stated before, Zaqarashvili et al. (2011) put their emphasis in the derivation of the equations and the dispersion relation. As examples, Zaqarashvili et al. (2011) computed some solutions, but they restricted themselves to a specific set of parameters and did not perform an in-depth parameter study. We follow the spirit of Paper I and do not focus on a specific astrophysical plasma. Instead, we take the relevant physical quantities as, e.g., the collision frequency, the ionization fraction, the Alfv'en and sound velocities, etc., as free parameters whose values can be conveniently chosen. An inconvenience of the present approach is that the space of parameters to be explored is very big. The advantage is that this approach makes possible to apply the results of this work to a wide variety of situations. Compared to Zaqarashvili et al. (2011), we cover a wider parameter range. In particular, we consider lower ionization levels and weaker magnetic fields than those used by Zaqarashvili et al. (2011). The output of this investigation will be also useful to understand the recent results by Soler et al. (2013b) concerning the behavior of slow waves propagating along a partially ionized magnetic flux tube. This paper is organized as follows. Section 2 contains the description of the equilibrium state and the basic equations. In Section 3 we explore the limit cases of uncoupled fluids and strongly coupled fluids and derive approximate expressions to the wave frequencies in those cases. Later, we investigate the general case in Section 4 by plotting dispersion diagrams when the collision frequency is varied between the uncoupled and strongly coupled regimes. We perform a specific application to magnetoacoustic waves propagating in the low solar atmosphere in Section 5. Finally, Section 6 contains some concluding remarks.",
"pages": [
1,
2
]
},
{
"title": "2. EQUILIBRIUM AND BASIC EQUATIONS",
"content": "The equilibrium configuration considered here is a uniform and unbounded partially ionized hydrogen plasma composed of ions, electrons, and neutrals. As in Paper I, the plasma dynamics is studied using the two-fluid formalism (see also, e.g., Zaqarashvili et al. 2011). Ions and electrons are considered together as forming a single fluid, while neutrals are treated as a separate fluid. The ion-electron fluid and the neutral fluid interact by means of particle collisions, which transfer momentum between the species. It is implicitly assumed in the present version of the two-fluid theory that ion-electron and ion-ion collisions occur always more frequently than ion-neutral collisions, so that ions and electrons display fluid behavior regardless of the presence of neutrals. The medium is permeated by a straight and constant magnetic field, B . We use Cartesian coordinates and conveniently choose the reference frame so that the equilibrium magnetic field is orientated along the z -direction, namely B = B ˆ z , with B constant. We ignore the effect of gravity. The omission of gravity is approximately valid as long as we consider wavelengths much shorter than the gravitational stratification scale height. We also assume that there are no flows in the equilibrium. We study linear adiabatic perturbations superimposed on the equilibrium state. The equations governing the behavior of the perturbations are the same as in Paper I, namely where v i , p i , P i , and ρ i are the velocity perturbation, gas pressure perturbation, equilibrium gas pressure, and equilibrium density of the ion-electron fluid, v n , p n , P n , and ρ n are the respective quantities for the neutral fluid, b is the magnetic field perturbation, µ is the magnetic permeability, γ is the adiabatic index, and α in is the ionneutral friction coefficient. The friction coefficient, α in , plays a very important role. It determines the strength of the ion-neutral friction force and, therefore, the importance of ion-neutral coupling. Braginskii (1965) gives the expression of α in in a hydrogen plasma. As in Paper I, we take α in as a free parameter and do not use a specific expression of α in . We do so to conveniently control the strength of the ion-neutral friction force. Our aim is to keep the investigation as general as possible for the results of the present article to be applicable in a wide range of astrophysical situations. Instead of using α in , in the remaining of this article we write the equations using the ion-neutral, ν in , and neutral-ion, ν ni collision frequencies, which have a more obvious physical meaning. They are defined as Hence, both frequencies are related by ρ i ν in = ρ n ν ni .",
"pages": [
2
]
},
{
"title": "2.1. Normal mode analysis",
"content": "Here we perform a normal mode analysis. The temporal dependence of the perturbations is put proportional to exp ( -iωt ), where ω is the angular frequency. In addition, we perform a Fourier analysis of the perturbations in space. The spatial dependence of perturbations is put proportional to exp ( ik x x + ik y y + ik z z ), where k x , k y , and k z are the components of the wavevector in the x -, y -, and z -directions, respectively. Alfv'en waves and magnetoacoustic waves are two distinct classes of MHD waves in a uniform plasma (see, e.g., Cramer 2001; Goossens 2003). Alfv'en waves are incompressible and propagate vorticity perturbations. Magnetoacoustic waves are compressible and do not produce vorticity perturbations. Alfv'en waves were investigated in Paper I. Here we focus on magnetoacoustic waves. Compressibility is an appropriate quantity to describe magnetoacoustic waves (see, e.g., Lighthill 1960). By working with compressibility perturbations we are able to decouple magnetoacoustic waves from Alfv'en waves without imposing any restriction on the direction of the wavevector. We define ∆ i and ∆ n as the compressibility perturbations of the ion-electron fluid and the neutral fluid, respectively, We combine Equations (1)-(5) and after some algebraic manipulations we obtain the two following coupled equations involving ∆ i and ∆ n only, namely where k 2 = k 2 x + k 2 y + k 2 z and c A , c i , and c n are the Alfv'en velocity, the ion-electron sound velocity, and the neutral sound velocity defined as Note that the Alfv'en velocity is here defined using the density of the ionized fluid only. Equations (9) and (10) are the governing equations for linear compressibility perturbations. Therefore, they govern the behavior of magnetoacoustic waves. The magnetoacoustic character of the waves is determined by the relative values of the Alfv'en and sound velocities. In this work we use the following definition of the ion-electron β i , Thus, using the present definition of β i we can quantify the relative importance of the Alfv'en velocity, c A , and ion-electron sound velocity, c i . No restriction on the value of the neutral sound velocity, c n , has been imposed so far. However, to simplify matters and decrease the number of parameters involved, we proceed as in Zaqarashvili et al. (2011) and Soler et al. (2013b) and assume that there is a strong thermal coupling between the ion-electron fluid and the neutral fluid. As a consequence, the equilibrium temperature, T , is the same for both fluids. We use the ideal gas law to relate the equilibrium gas pressure, density, and temperature as where R is the ideal gas constant. The factor 2 in the expression for ion-electrons is present because the partial pressures of both electrons and ions have to be summed up. In a hydrogen plasma, the partial pressures of electrons and ions are the same. From Equation (13), we deduce that the requirement that all the species have the same temperature results in imposing a relation between the sound velocities as, Hence, using Equations (12) and (14) we see that fixing the value of β i is enough to establish the relation between the Alfv'en velocity and the two sound velocities. For the subsequent analysis, we decompose the wavevector, k , in its components parallel and perpendicular to the equilibrium magnetic field as k = k z ˆ z + k ⊥ 1 ⊥ , where k ⊥ = √ k 2 x + k 2 y and 1 ⊥ is the unity vector in the perpendicular direction to B . We define θ as the angle that forms the wavevector, k , with the equilibrium magnetic field, B . Then, we can write k z = k cos θ and k ⊥ = k sin θ . For propagation perpendicular to the magnetic field, θ = π/ 2, whereas for propagation parallel to the magnetic field, θ = 0. Intermediate values of θ represent oblique propagation.",
"pages": [
2,
3
]
},
{
"title": "3. APPROXIMATIONS FOR LIMIT VALUES OF THE COLLISION FREQUENCIES",
"content": "Before tackling the general case for arbitrary values of the collision frequencies, it is instructive to consider two paradigmatic limit cases, namely the uncoupled case in which the two fluids do not interact, and the strongly coupled case in which the two fluids behave as a single fluid.",
"pages": [
3
]
},
{
"title": "3.1. Uncoupled case",
"content": "Here, we remove the effect of ion-neutral collisions. We represent this situation by setting ν in = ν ni = 0. Equations (9) and (10) become Now, the equations governing compressibility perturbations in the two fluids are decoupled. The waves in the two fluids can be studied separately. We start by considering waves in the ion-electron fluid. For nonzero ∆ i , the solutions to Equation (15) must satisfy Equation (17) is the well-known dispersion relation of magnetoacoustic waves in a fully ionized plasma (see, e.g., Lighthill 1960). The solutions to Equation (17) are where the + sign is for the fast wave and the -sign is for the slow wave. The magnetoacoustic character of these two distinct wave modes depends on the relative values of c A and c i (see, e.g., Goossens 2003). In the lowβ i case, c 2 A /greatermuch c 2 i and the solutions simplify to for the fast wave and for the slow wave. When c 2 A /greatermuch c 2 i the fast wave is a magnetic wave that propagates isotropically at the Alfv'en velocity, c A , while the slow wave is essentially a sound wave that is guided by the magnetic field and travels at the ion-electron sound velocity, c i . Conversely, in the highβ i case c 2 i /greatermuch c 2 A and the solutions simplify to for the fast wave and for the slow wave. Now the fast wave behaves as an isotropic sound wave, while the slow wave is a magnetic wave guided by the magnetic field. We turn to the waves in the neutral fluid. The solutions to Equation (16) for nonzero ∆ n are Equation (23) represent acoustic (or sound) waves in a gas. These waves propagate isotropically at the neutral sound velocity, c n , and are unaffected by the magnetic field. As expected, in the absence of ion-neutral collisions we consistently recover the classic magnetoacoustic waves in the ion-electron fluid and the classic acoustic waves in the neutral fluid. Therefore, three distinct waves are present in the uncoupled, collisionless case. These waves do not interact and are undamped in the absence of collisions. The waves would be damped if nonideal processes as, e.g., viscosity, resistivity, or thermal conduction, are taken into account. In the remaining of this article, we use the adjective 'classic' to refer to the waves found in the uncoupled case.",
"pages": [
3,
4
]
},
{
"title": "3.2. Strongly coupled case",
"content": "/negationslash Conversely to the uncoupled case, the strongly coupled limit represents the situation in which ion-electrons and neutrals behave as a single fluid. To study this case, we take the limits ν in → ∞ and ν ni → ∞ in Equations (9) and (10). We realize that, if ω = 0, it is necessary that ∆ i = ∆ n for the equations to remain finite. This is equivalent to assume that the two fluids move as a whole. Then, when ν in → ∞ and ν ni → ∞ , Equations (9) and (10) reduce to a single equation, namely where we use ∆ i , n to represent either ∆ i or ∆ n and the ionization fraction of the plasma, χ , is defined as For nonzero ∆ i , n , the solutions to Equation (24) must satisfy This is the wave dispersion relation in the strongly coupled case. Its solutions are where the + sign is for the modified fast wave and the -sign is for the modified slow wave. We use the adjective 'modified' to stress that these waves are the counterparts of the classic fast and slow modes but modified by the presence of the neutral fluid. The first important difference between the uncoupled and strongly coupled cases is in the number of solutions. In the uncoupled case there are three distinct waves, namely the classic slow and fast magnetoacoustic waves and the neutral acoustic wave, but in the strongly coupled limit we only find the modified version of the slow and fast magnetoacoustic modes. The modified counterpart of the classic neutral acoustic mode is apparently absent. The question then arises, what happened to the neutral acoustic mode? To answer this question, we study in the following paragraphs the character of the modified fast and slow modes depending on the relative values of the Alfv'en and sound velocities. A remark should be made before tackling the approximate study of the solutions in Equation (27). We recall that the neutral and ion-electron sound velocities are related by c 2 i = 2 c 2 n . This condition means that c 2 i and c 2 n are of the same order of magnitude. However, this is not the end of the story. We notice that the value of the ionization fraction, χ , also plays a role in determining the solutions to Equation (27). Specifically, χ appears multiplying c 2 n , which means that χ can increase or decrease the effective value of the neutral sound velocity. Thus, we need to compare χc 2 n with c 2 A and c 2 i in order to determine the nature of the solutions. The approximate study of the solutions given in Equation (27) is done considering eight typical cases: for the modified fast wave and for the modified slow wave. We recover the fast and slow waves of the uncoupled case (Equations (19) and (20)). No trace of the neutral acoustic mode remains in this limit because the amount of neutrals is negligible. for the modified fast wave and for the modified slow wave. By comparing Equations (19) and (30), we see that the square of the fast mode frequency is reduced by the factor ( χ +1) -1 compared to the value in the uncoupled case. As a consequence, the effective Alfv'en velocity in the strongly coupled limit is c A / √ 1 + χ . This is equivalent to replace the ion density, ρ i , by the sum of the ion and neutral densities, ρ i + ρ n , in the definition of the Alfv'en velocity. The same result was obtained by Kumar & Roberts (2003) and Soler et al. (2013b) for fast magnetoacoustic waves in the case c 2 n = 0. This is also the same result obtained in Paper I and Soler et al. (2012) for Alfv'en waves. On the other hand, the modification of the slow mode frequency is more complicated (compare Equations (20) and (31)). The expression of the slow mode frequency involves both c i and c n . Consequently, an effective sound velocity, c eff , can be defined as The effective sound velocity is the weighted average of the sound velocities of ion-electrons and neutrals, and was first obtained by Soler et al. (2013b) for slow magnetoacoustic waves propagating along magnetic flux tubes. The fact that the expression of the effective sound velocity (Equation (32)) involves both neutral and ion-electron sound velocities suggests that, in this case, the modified slow mode is the descendant of both the classic slow mode and the neutral acoustic mode. for the modified fast wave and for the modified slow wave. The effective Alfv'en velocity is now c A / √ χ . This is equivalent to replace the ion density, ρ i , by the neutral density, ρ n , in the definition of the Alfv'en velocity. On the other hand, the slow wave is the descendant of the neutral acoustic wave that becomes guided by the magnetic field. Therefore, the frequencies of both fast and slow waves are determined by the density of the neutral fluid alone, which is indirectly affected by the magnetic field. for the modified fast wave and for the modified slow wave. The modified fast wave can straightforwardly be related to the isotropic acoustic mode of the neutral fluid (Equation (23)), while the modified slow wave is a guided magnetic wave with the effective Alfv'en velocity depending on the neutral density alone. As in case (3), neither the ion-electron sound velocity nor the ion density play a role. The fact that the plasma is weakly ionized changes completely the physical nature of the solutions compared to the classic magnetoacoustic waves in a lowβ i fully ionized plasma. for the modified fast wave and for the modified slow wave. We recover Equations (21) and (22) obtained in the highβ i collisionless case. No trace of the neutral acoustic mode remains in this limit. for the modified fast wave and for the modified slow wave. As in the previous case (5), we revert to Equations (21) and (22) obtained in the uncoupled case. Cases (5) and (6) point out that the relative values of c 2 A and χc 2 n are not important as long as c 2 i remains much larger than both of them. The same approximations to the frequencies are found in cases (5) and (6). The properties of the neutral fluid are not important in both cases (5) and (6). for the modified fast wave and for the modified slow wave. Equations (41) and (42) are the modified version of Equations (21) and (22) obtained in the highβ i uncoupled case, where c eff replaces c i and c A / √ 1 + χ replaces c A . Here we find that the modified fast mode is the descendant of both the classic fast mode and the neutral acoustic mode, while the modified slow mode is a guided magnetic wave whose frequency depends on both ion and neutral densities. for the modified fast wave and for the modified slow wave. The approximation found here are the same as in case (4). The properties of the neutral fluid completely determine the behavior of the waves. Neutrals feel the magnetic field indirectly through the collisions with ions. The results of the above approximate study are summarized in Table 1. Based on this study, we can now answer the previous question about the apparent absence of the neutral acoustic mode in a strongly coupled plasma. We conclude that the classic neutral acoustic mode and the classic ion-electron magnetoacoustic modes heavily interact when ion-neutral collisions are at work. The two resulting modes in the strongly coupled regime (modified fast mode and modified slow mode) have, in general, mixed properties and are affected by the physical conditions in the two fluids. The degree to which the properties of the classic waves are present in the resulting waves depends on the relative values of the Alfv'en and sound velocities and on the ionization fraction of the plasma. In addition, the wave frequencies in the strongly coupled limit are real as in the uncoupled case. This means that the waves are undamped in the limit of high collision frequencies as well. The damping of magnetoacoustic waves due to ion-neutral collisions takes place for intermediate collision frequencies. As stated before, the waves would be damped if additional damping mechanisms are considered, but this falls beyond the aim of the present article. Both the ion-neutral damping and the coupling between modes are investigated in Section 4 by studying the modification of the wave frequencies when a progressive variation of the collision frequencies between the uncoupled limit and strongly coupled limit is considered.",
"pages": [
4,
5,
6
]
},
{
"title": "4. NUMERICAL STUDY FOR ARBITRARY COLLISION FREQUENCIES",
"content": "From here on, we consider arbitrary values of ν in and ν ni . With no approximation, we combine the coupled Equations (9) and (10) to obtain separate equations for ∆ i and ∆ n as with where D i ( ω ), D n ( ω ), and D 2 c ( ω ) are defined as ] The waves in the ion-electron fluid and in the neutral fluid are described by the same dispersion relation. The dispersion relation is and is equivalent to the dispersion relation given by Zaqarashvili et al. (2011) in their Equation (57), although here we use a different notation. Expressed in polynomial form, the dispersion relation is a 7th-order polynomial in ω , so that there are seven solutions (see Zaqarashvili et al. 2011). The frequencies are complex, namely ω = ω R + iω I , where ω R and ω I are the real and imaginary parts. The imaginary part of the frequency is negative and accounts for the exponential damping rate of the perturbations due to ion-neutral collisions. /negationslash Before exploring the solutions to the dispersion relation, it is necessary to make a comment on the nature of the solutions. We distinguish between two different kind of solutions depending of whether ω R is zero or nonzero. (1) The first kind of solutions are those with ω R = 0, i.e., solutions that represent purely decaying disturbances. These solutions are not waves in the strict sense because they do not propagate. The solutions with ω R = 0 were called 'vortex modes' by Zaqarashvili et al. (2011) but, in our view, the name 'vortex modes' does not reflect the true nature of these solutions. The reason is that vortex modes in a partially ionized plasma are intrinsically related to Alfv'en waves (see Paper I), not to magnetoacoustic waves. Magnetoacoustic modes do not produce vorticity perturbations (see, e.g., Goossens 2003), hence the purely imaginary solutions of the magnetoacoustic dispersion relation cannot represent fluid vorticity. Instead, we propose to physically interpret the solutions with ω R = 0 as 'entropy modes' (see, e.g., Goedbloed & Poedts 2004; Murawski et al. 2011). Entropy modes represent perturbations of density and, therefore, compression of the plasma. Compression is an intrinsic property of magnetoacoustic modes. Entropy modes have zero frequency in the absence of collisions. Due to ion-neutral collisions, perturbations in density are damped and the entropy mode frequency acquires an imaginary part, while its real part remains zero. (2) The second kind of solutions are those with ω R = 0. These solutions represent propagating waves and appear in pairs, namely ω 1 = ω R + iω I and ω 2 = -ω R + iω I . Both ω 1 and ω 2 represent the same magnetoacoustic wave, but the real parts of their frequencies have opposite signs (see Carbonell et al. 2004). The different signs of the real parts account for forward ( ω R > 0) and backward ( ω R < 0) propagation with respect to the direction of the wavevector, k . Since the equilibrium is static, the two directions of propagation are physically equivalent. As shown later, the number of entropy modes and propagating waves is not constant. Depending on the physical parameters considered, there can be conversion between these two kind of solutions. Unlike the limit cases studied in Section 3, no simple analytic solutions of the dispersion relation can be obtained when ν in and ν ni are arbitrary. Instead, we solve the dispersion relation numerically. The numerically ob- ined dispersion diagrams are plotted as functions of the averaged collision frequency, ¯ ν , defined as Because of the complex behavior displayed by the solutions and the large number of parameters involved, we organize the presentation of the results and their discussion as follows. First, we investigate the specific situation in which the wave propagation is strictly perpendicular to the magnetic field direction. This paradigmatic case is helpful to get physical insight of the behavior of the various solutions and to illustrate some general results that repeatedly appear in this investigation. Slow modes are absent from the discussion for perpendicular propagation because they are unable to propagate across the magnetic field. Subsequently, we study the other limit case, i.e., wave propagation parallel to the magnetic field, so that slow modes are added to the discussion. Finally, the general case of oblique propagation is taken into account.",
"pages": [
6,
7
]
},
{
"title": "4.1. Perpendicular propagation",
"content": "We start by studying the case of perpendicular propagation to the magnetic field. Hence we set θ = π/ 2. The dispersion relation (Equation (51)) reduces to a 5th-order polynomial because the slow magnetoacoustic wave is absent. Conversely, the fast magnetoacoustic mode and the neutral acoustic mode can propagate across the field and so they remain when θ = π/ 2. We compute the real and imaginary parts of the frequency of the various solutions of the dispersion relation as functions of ¯ ν . Frequencies are expressed in units of kc i so that the results can be applied to the astrophysical plasma of interest by providing appropriate numeric values of k and c i . The averaged collision frequency is varied four orders of magnitude between ¯ ν/kc i = 0 . 01 and ¯ ν/kc i = 100. These two values are chosen to represent the uncoupled and strongly coupled limits, respectively. Beyond these two values of ¯ ν/kc i the behavior of the solutions remains essentially unaltered. Figures 1 and 2 show the results for β i = 0 . 04 (lowβ i case) and β i = 25 (highβ i case), respectively. Three values of χ are considered in each case, namely χ = 0 . 2, 2, and 20, which cover the cases of largely ionized, moderately ionized, and weakly ionized plasmas, respectively. First, we analyze the real part of the frequency. The comparison of the numerically obtained ω R with the analytic approximations of Section 3 helps us identifying the various propagating waves in the limit values of ¯ ν . The modes are labeled and plotted with different line styles in Figures 1 and 2. When ¯ ν/kc i = 0 . 01 we find the classic fast magnetoacoustic (solid black line) and neutral acoustic (dashed blue line) modes, while for ¯ ν/kc i = 100 only the modified fast wave is present as a propagating solution, in agreement with Section 3. As expected, the modes do not display the properties of the classic waves when the averaged collision frequency takes intermediate values. To understand how the solutions change due to the effect of collisions we continuously follow the various solutions when ¯ ν/kc i increases from ¯ ν/kc i = 0 . 01 to ¯ ν/kc i = 100. We find that one of the two solutions present at the low collision frequency limit has a cutoff when ¯ ν/kc i reaches a certain value. At the cutoff, the forward ( ω R > 0) and backward ( ω R < 0) waves of this specific solution merge and their ω R becomes zero. On the contrary, the other solution does not have a cutoff and eventually becomes the modified fast wave at the high collision frequency limit. The location of the cutoff is important because it determines the number of distinct waves that are able to propagate for a certain combination of parameters. The effect of the various parameters on the location of the cutoff can be studied as in Paper I by taking advantage of the fact that the dispersion relation is a polynomial in ω . So, we can compute the polynomial discriminant of the dispersion relation. By definition, the discriminant is zero when the dispersion relation has a multiple solution. Since the cutoff takes place when a forward and a backward propagating wave merge, the dispersion relation has necessarily a double root at the cutoff. The discriminant is a function of χ , ¯ ν/kc i , and β i . Hence, the zeros of the discriminant inform us about the relation between these parameters at the cutoff. We omit here the expression of the discriminant because it is rather cumbersome and can be straightforwardly obtained from the dispersion relation using standard algebraic methods. The number of propagating solutions for a given set of parameters is represented in Figure 3 in the χ -¯ ν/kc i plane for three different values of β i . The zero of the discriminant is plotted with a line. Below this line two propagating waves are possible, namely the classic fast and acoustic modes, while above the line only the modified fast wave is present as a propagating wave. Importantly, we find that a third region, where propagating waves are not possible, appears when β i /lessmuch 1 (see the shaded area in Figure 3 when β i = 10 -3 ). This forbidden region takes place for large ionization ratio (weakly ionized plasmas), very low β i (strong magnetic fields) and relatively high collision frequency (strongly coupled plasmas). This could be the case of the plasma in intense magnetic tubes of the low solar atmosphere. The physical reason for the existence of wave cutoffs is qualitatively the same as discussed in Paper I for the case of Alfv'en waves (see also, e.g., Kulsrud & Pearce 1969; Pudritz 1990; Kamaya & Nishi 1998; Mouschovias et al. 2011). There is competition between the ion-neutral friction force and the wave restoring forces. Friction becomes the dominant force and overcomes the remaining forces at the cutoff. Hence, waves are unable to propagate when the strength of friction greatly exceeds those of the restoring forces. The cutoff found here corresponds to the situation in which ions and neutrals are sufficiently coupled for the neutral fluid to be significantly affected by the magnetic field. As a consequence, the neutral fluid is not able to support its own pure acoustic modes, which start to behave as slow-like magnetoacoustic waves. For θ = π/ 2 slow modes cannot propagate and so these solutions are cut off. We turn to the imaginary part of the frequency. For presentation purposes, we plot the absolute value of ω I in logarithmic scale. The line styles used in the graphics of ω I are the same as in the plots of ω R so that the various solutions can be easily identified. The inspection of ω I in Figures 1 and 2 informs us of the existence of a third purely imaginary mode (red dotted line). This mode is the descendant of the undamped entropy mode, which becomes damped by collisions when ¯ ν/kc i increases. The damping rate of this nonpropagating mode monotonically increases with ¯ ν/kc i . Regarding the propagating waves, we see that the solution that has the cutoff in ω R displays a bifurcation in ω I at the same location. Two purely imaginary branches emerge at the bifurcation. We identify these new branches as entropy modes. On the one hand, the upper branch is heavily damped and follows the behavior of the original entropy mode plotted with a red dotted line. On the other hand, the lower branch is less damped and becomes an undamped entropy mode when ¯ ν/kc i →∞ . Finally, the remaining solution whose ω I does not have a bifurcation corresponds to the propagating wave that becomes the modified fast wave when ¯ ν/kc i /greatermuch 1. The damping rate of this wave is nonmonotonic. Its | ω I | is maximal, i.e., the damping is most efficient, when ¯ ν ≈ ω R . This is the same result as for Alfv'en waves (see Paper I).",
"pages": [
7,
8
]
},
{
"title": "4.2. Parallel propagation",
"content": "We move to the case of parallel propagation to the magnetic field. We set θ = 0 and perform the same computations as in the previous section. These results are displayed in Figures 4 and 5, where we have considered the same parameters as before. As expected, the slow magnetoacoustic wave is now present (dash-dotted green line). As in the perpendicular propagation case, we analyze Figures 4 and 5 by observing how the modes present at the low collision frequency limit evolve as we increase the collision frequency toward the high collision frequency limit. In the lowβ i case (Figure 4), the slow magnetoacoustic wave and the neutral acoustic wave strongly interact. There is a change of character between these two waves depending on the value of χ . When χ /lessmuch 1 , i.e., largely ionized plasma, the neutral acoustic mode has a cutoff and the classic slow mode becomes the modified slow mode in the limit of large ¯ ν/kc i . On the contrary, the situation is reversed when χ /greaterorsimilar 1, so that the solution that is cut off is the classic slow mode while the neutral acoustic mode is the solution that becomes the modified slow mode. These results are consistent with the approximate study of Section 3.2. On the other hand, the behavior of the fast magnetoacoustic wave in the lowβ i case is very similar to that of the Alfv'en waves studied in Paper I. It is well-known that in a lowβ i plasma fast waves behave as Alfv'en waves for parallel propagation to the magnetic field (see, e.g., Goossens 2003). As in the case of Alfv'en waves, fast magnetoacoustic waves are nonpropagating in an interval of ¯ ν/kc i when χ > 8 (see extensive details in Paper I for the case of Alfv'en waves). This nonpropagating interval can be seen in Figure 4 in the results for χ = 20. In this interval the fast wave becomes a purely imaginary solution. The boundaries of the nonpropagating interval can be obtained from Equation (20) of Paper I. Using the present notation, the boundaries of the nonpropagating interval are where the -and + signs correspond to the left and right boundaries of the nonpropagating range. In the highβ i case (Figure 5), the slow magnetoacoustic wave displays little interaction with the other two solutions. In this case, the behavior of the fast magnetoacoustic wave and the neutral acoustic wave is similar to that found when θ = π/ 2, while the slow wave behaves as an Alfv'en wave (Paper I). Now, it is the slow wave and not the fast wave that has a nonpropagating interval of ¯ ν/kc i . Equation (53) holds for the slow wave in the highβ i case too. Since Equation (53) depends upon β -1 / 2 i , the nonpropagating interval is now shifted towards smaller ¯ ν/kc i than in the lowβ i case (compare the results for χ = 20 in Figure 4 and 5). We study in more detail the presence of cutoffs and nonpropagating intervals by using again the discriminant of the dispersion relation. This is shown in Figure 6 for β i = 0 . 04 and β i = 25. As in the case of perpendicular propagation (see Figure 3) we find two zones in the χ -¯ ν/kc i plane separated by a cutoff (black solid line). These separate zones correspond to the region where the classic slow, fast, and acoustic waves live and the region where the modified slow and fast modes exist. In addition, we find the presence of the nonpropagating interval previously discussed (area between the red lines). In the lowβ i case (Figure 6a) the nonpropagating interval is located above the boundary between the two regions. Within the nonpropagating interval, the modified slow wave is the only propagating solution. On the contrary, in the highβ i case (Figure 6b) the nonpropagating interval intersects the boundary between the two regions. Hence the number of possible scenarios increases, as schematically indicated in Figure 6b. It is appropriate to compare our results with those plotted in Figure 3 of Zaqarashvili et al. (2011) in order to check whether the present work is consistent with that previous study. In our notation, the parameters used by Zaqarashvili et al. (2011) are χ = 1, β i = 0 . 25, and θ = 0. These parameters are close to those used in Figure 4(c)-(d). To perform a proper comparison we should use the same notation as Zaqarashvili et al. (2011). They used two dimensionless quantities, w and a , that correspond to the wave frequency and to the inverse of the collision frequency, respectively. These two quantities are expressed in our notation as √ We solve the dispersion relation using the above parameters and reproduce in Figure 7 the results shown in Figure 3 of Zaqarashvili et al. (2011). A perfect agreement between our results and those of Zaqarashvili et al. (2011) is obtained. With the help of the discriminant of the dispersion relation, we compute that the cutoff of the neutral acoustic wave takes place at ¯ ν/kc i ≈ 0 . 84, which corresponds to a ≈ 3 . 4 in the notation of Zaqarashvili et al. (2011). Since χ < 8, the fast mode does not have a nonpropagating interval. Thus, our results confirm and fully agree with the specific example studied by Zaqarashvili et al. (2011).",
"pages": [
8
]
},
{
"title": "4.3. Oblique propagation",
"content": "Finally, here we consider the case in which the direction of wave propagation is in between the limit cases studied in the previous two Subsections. Hence, we use θ = π/ 4. The results are displayed in Figures 8 and 9. The behavior of the various solutions is more complex than in the previous limits and can be understood as a combination of the results for perpendicular and parallel propagation. Compared to the case with θ = 0, the acoustic and fast modes display similar behavior, while the slow mode is the solution whose behavior is most altered, specially in the lowβ i case (compare Figures 4 and 8). Both slow and fast modes display nonpropagating intervals for oblique propagation, whereas only the fast mode have a nonpropagating interval for parallel propagation. The fast mode nonpropagating interval is also approximately described by Equation (53), which points out that the location of this interval does not depend on θ . We proceed as before and show in Figure 10 the various possible scenarios for wave propagation in the χ -¯ ν/kc i plane of parameters. The highβ i case (Figure 10b) is very similar to the equivalent result for θ = 0 (Figure 6b), so that no additional comments are needed. However, in the lowβ i case, the presence of nonpropagating intervals for both slow and fast modes and their intersections cause the presence of six possible scenarios (Figure 10a). The result represented in Figure 10a clearly points out the high complexity of the interactions between the various waves in a partially ionized plasma, which do not occur in fully ionized plasmas (Mouschovias et al. 2011). The existence of two nonpropagating intervals in the lowβ i case is consistent with the results by Soler et al. (2013b) of slow waves propagating along a partially ionized magnetic flux tube. Soler et al. (2013b) also found the presence of two nonpropagating intervals, see their Figure 5. Here we see that this double nonpropagating interval is a result also present for magnetoacoustic waves in a homogeneous, partially ionized medium. Thus, we can now conclude that this result is not caused by the geometry of the waveguide, but it is an intrinsic property of magnetoacoustic waves propagating obliquely to the magnetic field in a partially ionized plasma. The slow magnetoacoustic waves in a magnetic flux tube studied by Soler et al. (2013b) behave as the waves studied here for oblique propagation.",
"pages": [
9
]
},
{
"title": "5. APPLICATION TO THE SOLAR CHROMOSPHERE",
"content": "Here we perform a specific application to magnetoacoustic waves propagating in the partially ionized solar chromosphere. This application is motivated by the recent observations of ubiquitous compressive waves in the low solar atmosphere (Morton et al. 2011, 2012). To represent the chromospheric plasma we use the same simplified model considered in Paper I. We use the quiet sun model C of Vernazza et al. (1981), hereafter VALC model, to account for the variation of physical parameters with height from the photosphere to the base of the corona. The expression of α in is taken after Braginskii (1965), namely where m i is the ion (proton) mass, k B is Boltzmann's constant, and σ in is the collision cross section. Equation (56) considers only hydrogen and ignores the influence of heavier species. In Equation (56) it is implicitly assumed that the ion and neutral masses are approximately equal. For the collision cross section of protons with neutral hydrogen atoms we consider the typical value of σ in ≈ 5 × 10 -19 m 2 (used in, e.g., Khodachenko et al. 2004; Leake et al. 2005; Arber et al. 2007; Soler et al. 2009; Khomenko & Collados 2012, among others). Here we must note that, in the application done in Paper I for the case of Alfv'en waves, we took σ in ≈ 10 -20 m 2 based on the hard sphere collision model (see, e.g., Braginskii 1965; Zaqarashvili et al. 2013). In a recent paper, Vranjes & Krstic (2013) claim that the realistic value of σ in in the solar chromosphere is about two orders of magnitude higher than that estimated in the hard sphere model. The value σ in ≈ 5 × 10 -19 m 2 used here is closer to the value proposed by Vranjes & Krstic (2013) than to the hard sphere value. We adopt the magnetic field strength model by Leake & Arber (2006), which aims to represent the field strength in a chromospheric vertical magnetic flux tube expanding with height, namely where ρ = ρ i + ρ n is the total density, and B ph and ρ ph = 2 . 74 × 10 -4 kg m -3 are the magnetic field strength and the total density, respectively, at the photospheric level. The variation of ρ with height is taken from the VALC model. Accordingly, the magnetic field strength decreases with height. Regarding the value of B ph , we consider two possible scenarios: an active region and the quiet Sun. For the active region case, we set B ph = 1 . 5 kG, so that B ≈ 100 G at 1,000 km and B ≈ 20 G at 2,000 km above the photosphere. For the quiet Sun case, we set B ph = 100 G, so that B ≈ 7 G at 1,000 km and B ≈ 1 G at 2,000 km above the photosphere. As explained in Paper I, the physical parameters in this simplified model of the low solar atmosphere depend on the vertical direction, while in the previous theoretical analysis all the parameters are taken constant. Here we perform a local analysis and use the dispersion relation derived for a homogeneous plasma. We use the physical parameters at a given height to locally solve the dispersion relation at that height. A limitation of the present approach is that it ignores the possible presence of cutoff frequencies due to gravitational stratification (see, e.g., Roberts 2006). The condition for this method to be approximately valid is that the wavelength, λ = 2 π/k , is much shorter than the stratification scale height. This is fulfilled by the wavelengths used in this analysis. Figure 11 displays the variation with height above the photosphere of the three physical quantities relevant for the behavior of the waves: the ionization fraction, χ , the ionized fluid β i , and the averaged ion-neutral collision frequency, ¯ ν . Only β i is affected by the magnetic field scenario considered. First of all, we see that χ ranges several orders of magnitude from very weakly ionized plasma at the low levels of the chromosphere to fully ionized plasma when the transition region to the solar corona is reached. Full ionization of hydrogen takes place at 2,100 km above the photospheric level, approximately. The value of β i in the active region case is much smaller than unity throughout the chromosphere, which informs us that we are dealing with a lowβ i situation. Conversely, in the quiet Sun case the value of β i ranges from β i /lessmuch 1 at low levels to β i /greatermuch 1 at high levels in the chromosphere. Moreover, ¯ ν also ranges several orders of magnitude, specially in the low chromosphere up to 700 km, approximately. Then, ¯ ν takes values in between 10 2 Hz and 10 3 Hz until the full ionization level is reached. The high values of ¯ ν displayed in Figure 11(c) indicate that only waves with short wavelengths, i.e., high frequencies, would be affected by two-fluid effects. To investigate this, we use the discriminant of the dispersion relation to determine the nature of the propagating solutions as function of height for a given value of the wavelength. This is shown in Figure 12 for three values of θ , namely θ = 0, π/ 4, and π/ 2. In the active region case, we see that two-fluid effects are important for propagation of magnetoacoustic waves when λ /lessorsimilar 1 km. For longer wavelengths, ions and neutrals behave as a single fluid and, consequently, the propagating waves are the slow and fast waves modified due to the presence of neutrals (see Section 3.2). For short wavelengths, however, there appear a number of nonpropagating intervals that constrain the propagation of the various modes. Also, due to the strong ion-neutral coupling, the presence of neutral acoustic waves in the active region chromosphere is only possible for very short wavelengths. In the case of the quiet Sun, the nonpropagating intervals are shifted towards values of λ about an order or magnitude shorter than in the active region case. This result points out that two-fluid effects are of less relevance in those regions of the chromosphere where the magnetic field is weak. For representation purposes, in Figure 12 we have varied λ in between 10 -7 km and 10 2 km, although we must warn the reader that the fluid approximation for the various species may not hold for wavelengths approaching the lower boundary of this range. Therefore, the results shown in Figure 12 for wavelengths near the lower limit should be taken with extreme caution. Also, the assumption that gravitational effects are negligible breaks down for very long wavelengths, although this probably happens for wavelengths longer than those in Figure 12. Next, we define the damping rate as δ = -ω I /ω R . This quantity informs us about the efficiency of damping due to ion-neutral collisions. Indirectly, δ is also an indicator of the ability of ion-neutral collisions to heat the plasma by dissipating magnetoacoustic waves. For the active region case, Figure 13 shows the damping rate of the modified slow and fast waves as function of height for λ = 10 km and λ = 1 km. When λ = 10 km (Figure 13(a)-(b)), the waves are unaffected by two-fluid effects and do not have nonpropagating intervals. The only effect of ion-neutral collisions is to produce the damping of the waves. The fast wave damping rate is independent of the propagation angle, θ , and is maximal at 1,900 km, approximately. This suggests that the main contribution of fast waves to plasma heating might take place at the higher levels of the active region chromosphere. Conversely, the slow wave damping rate strongly depends on θ . The damping of the slow wave is in general very weak. We have to consider propagation almost perpendicular to the magnetic field, i.e., θ → π/ 2, to obtain a significant slow wave damping. Two-fluid effects play a role when the wavelength is decreased to λ = 1 km (Figure 13(c)(d)). Consistent with Figure 12, when λ = 1 km the fast wave has a nonpropagating zone around 1,500 km above the photosphere. Again, the results for the fast wave are independent of θ . Near the location of the nonpropagating zone, the fast wave damping rate boosts dramatically since δ → ∞ in the nonpropagating interval. Conversely, the slow wave only has nonpropagating regions when propagation is nearly perpendicular to the magnetic field. In this case, the slow wave has two nonpropagating intervals at two different heights (Soler et al. 2013b). These forbidden intervals for the nearly perpendicular slow wave cover a significant part of the chromosphere. Figure 14 shows the same results as Figure 13 with λ = 1 km but for the quiet Sun case. Now, the solutions do not display nonpropagating intervals. As indicated by Figure 12, in the quiet Sun case we need to consider shorter wavelengths than in the active region case for nonpropagating intervals to be present. In addition, Figure 14 shows that both fast and slow waves damping rates depend upon the propagation angle. The reason for this result is that in the quiet Sun case β i is higher than in the active region case and, in fact, β i > 1 at the upper levels. This causes the fast wave behavior to be dependent on the propagation angle too. In summary, in this application we conclude that magnetoacoustic waves propagating in regions of the solar chromosphere with strong magnetic fields are affected by two-fluid effects when λ /lessorsimilar 1 km. The presence of nonpropagating intervals heavily constrains the propagation of magnetoacoustic waves of short wavelengths. In addition, damping due to ion-neutral collisions is very efficient in the vicinity of the nonpropagating regions, which points out that significant wave energy dissipation may take place at those heights in the chromosphere. The fast wave is the propagating solution that may contribute the most to heat the plasma. For λ /greaterorsimilar 1 km, however, twofluid effects are of less relevance because ions and neutrals behave as a single fluid. Then, wave damping is appropriately described by the single-fluid approximation (e.g., Khodachenko et al. 2004). In quiet Sun regions, we need to consider much shorter wavelengths for two-fluid effects to be relevant. At present, we lack of appropriate observations to compare with the theoretical predictions discussed above. Unfortunately, current instruments do not have enough spatial and temporal resolutions to observe the range of wavelengths where two-fluid effects would be important. It is possible that new and future instruments as, e.g., the Atacama Large Millimeter/submillimeter Array (ALMA), may reach sufficiently high resolutions to observe wavelengths of 1 km and shorter. Angular resolution of 0.01 arcseconds or smaller is needed to observe wavelengths on the order of kilometres in the chromosphere. The ALMA telescope may reach the required resolution.",
"pages": [
9,
10
]
},
{
"title": "6. CONCLUDING REMARKS",
"content": "In this paper we continued the work started in Soler et al. (2013a) about the effect of ion-neutral collisions on MHD wave propagation in a two-fluid plasma. In the previous paper, we focused on incompressible Alfv'en waves. Here, we investigated compressible magnetoacoustic waves. The present work is also related to recent investigation of wave propagation in a partially ionized magnetic flux tube (Soler et al. 2013b). The study presented in this paper reveals that ion-neutral coupling strongly affects the well-known behavior and properties of classic magnetoacoustic waves. There is a large number of possible scenarios for wave propagation depending on the plasma physical properties. As pointed out by Mouschovias et al. (2011), the various waves supported by a partially ionized plasma display complex interactions and couplings that are not present in the fully ionized case. Among these complex interactions, the presence of cutoffs and forbidden intervals has an strong impact on waves since the allowed wavelengths of the propagating modes get constrained. After performing a general study, we considered the particular case of the solar chromosphere and showed that magnetoacoustic waves with λ /lessorsimilar 1 km are affected by two-fluid effects in regions with intense magnetic fields, while much shorter wavelengths have to be considered for these effects to be relevant in quiet Sun conditions. In addition, we discussed the possible role of these waves in heating the chromospheric plasma due to dissipation by ion-neutral collisions. Here, we must refer to the comment by Khodachenko et al. (2004) about that the correct description of MHD wave damping in the solar atmosphere requires the consideration of all energy dissipation mechanisms. We have focused on the effect of ion-neutral collisions alone and have not taken into account the roles of, e.g., thermal conduction, viscosity, resistivity, etc., that may have an important impact on the dissipation of wave energy. The consideration of these mechanisms using the two-fluid description of a partially ionized plasma is an interesting task to be done in future works. Finally, a natural extension of the investigation done in Soler et al. (2013a,b) and in the present paper is to abandon the normal mode approach and to study the behavior of impulsively excited disturbances. This requires the solution of the initial-value problem by means of time-dependent numerical simulations. The combination of results from both the initial-value problem and the normal mode analysis would give us a complete picture of wave excitation and propagation in a partially ionized plasma. The investigation of the initial-value problem will be tackled in future studies. We acknowledge the support from MINECO and FEDER Funds through grant AYA2011-22846 and from CAIB through the 'grups competitius' scheme and FEDER Funds.",
"pages": [
10,
11
]
},
{
"title": "REFERENCES",
"content": "Arber, T. D., Haynes, M., & Leake, J. E. 2007, ApJ, 666, 541 Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205 Carbonell, M., Oliver, R., & Ballester, J. L. 2004, A&A, 415, 739 Cramer, N. F. 2001, The Physics of Alfv'en Waves (Wiley-VCH) De Pontieu, B., Martens, P. C. H., & Hudson, H. S. 2001, ApJ, Behavior of the modified fast and slow waves and their approximate frequencies in the highly collisional limit depending on the ordering of the characteristic velocities. Note . - IMW, IAW, GMW, and GAW denote isotropic magnetic wave, isotropic acoustic wave, guided magnetic wave, and guided acoustic wave, respectively.",
"pages": [
11,
14
]
}
]
2013ApPhL.102a1123K
https://arxiv.org/pdf/1208.5776.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_87><loc_86><loc_91></location>Hot electron bolometer heterodyne receiver with a 4.7-THz quantum cascade laser as a local oscillator</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_18><loc_78><loc_85><loc_82></location>J. L. Kloosterman, 1, a) D. J. Hayton, 2 Y. Ren, 3, 4 T. Y. Kao, 5 J. N. Hovenier, 3 J. R. Gao, 2, 3, b) T. M. Klapwijk, 3 Q. Hu, 5 C. K. Walker, 6 and J. L. Reno 7</list_item>
</unordered_list>
<text><location><page_1><loc_18><loc_70><loc_85><loc_74></location>1) Department of Electrical and Computer Engineering, 1230 E. Speedway Blvd., University of Arizona, Tucson, AZ 85721 USA</text>
<unordered_list>
<list_item><location><page_1><loc_18><loc_62><loc_77><loc_66></location>2) SRON Netherlands Institute for Space Research, Groningen/Utrecht, Netherlands</list_item>
</unordered_list>
<text><location><page_1><loc_18><loc_54><loc_84><loc_58></location>3) Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands</text>
<text><location><page_1><loc_18><loc_40><loc_83><loc_50></location>4) Purple Mountain Observatory (PMO), Chinese Academy of Sciences, 2 West Beijing Road, Nanjing, JiangSu 210008, China, and Graduate School, Chinese Academy of Sciences, 19A Yu Quan Road, Beijing 100049, China</text>
<text><location><page_1><loc_18><loc_29><loc_85><loc_36></location>5) Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts 02139, USA</text>
<text><location><page_1><loc_18><loc_21><loc_83><loc_26></location>6) Steward Observatory, 933 N Cherry Ave., Rm N204, University of Arizona, Tucson, AZ 85721 USA</text>
<text><location><page_1><loc_18><loc_13><loc_71><loc_17></location>7) Sandia National Laboratories, Albuquerque, NM 87185-0601, USA</text>
<text><location><page_1><loc_18><loc_7><loc_38><loc_8></location>(Dated: 4 November 2021)</text>
<text><location><page_2><loc_18><loc_71><loc_88><loc_91></location>We report on a heterodyne receiver designed to observe the astrophysically important neutral atomic oxygen [OI] line at 4.7448 THz. The local oscillator is a third-order distributed feedback Quantum Cascade Laser operating in continuous wave mode at 4.741 THz. A quasi-optical, superconducting NbN hot electron bolometer is used as the mixer. We recorded a double sideband receiver noise temperature (T DSB rec ) of 815 K, which is ∼ 7 times the quantum noise limit ( h ν 2k B ) and an Allan variance time of 15 s at an effective noise fluctuation bandwidth of 18 MHz. Heterodyne performance was confirmed by measuring a methanol line spectrum.</text>
<text><location><page_3><loc_12><loc_68><loc_88><loc_91></location>Astronomers have long been interested in the fine structure line of [OI] at 4.7448 THz. [OI] probes the star formation process and is the most important cooling line in the interstellar medium (ISM) 1 for gas clouds with densities of n > 10 4 cm -3 . Large scale surveys with extremely high spectral resolution and sensitivity are required to disentangle large scale kinematics and energetics within these clouds. Such high spectral resolution observations require the development of super-THz ( > 3 THz) heterodyne receivers. Due to strong water absorption in the atmosphere, it is not possible to observe the [OI] line from ground-based telescopes. Therefore, an astronomical [OI] receiver requires a compact local oscillator (LO) that can be readily integrated into space-based or suborbital observatories.</text>
<text><location><page_3><loc_12><loc_39><loc_88><loc_67></location>There are several candidate THz technologies for use in LO systems. These technologies include Schottky diode based multiplier chains, gas lasers, and quantum cascade lasers (QCLs). QCLs are currently the only technological approach that leads to devices small and powerful enough 2 to be used in a variety of space-based, super-THz applications. Furthermore, THz QCLs operating in CW mode have yielded line widths of ∼ 100 Hz 3 , excellent power stability 4 , and output powers over 100 mW 5 , making them well-suited for high resolution spectroscopy. Much progress has been made toward overcoming the challenges associated with using a QCL as an LO. Frequency stabilization without the need of another THz source has been achieved using an absorption line within a methanol gas cell 6,7 . Diverging far-field beam patterns and mode selectivity have been improved by using a 3 rd -order distributed feedback (DFB) grating 8,9 .</text>
<text><location><page_3><loc_12><loc_12><loc_88><loc_38></location>In this letter we report on a full demonstration of a heterodyne receiver using a THz quantum cascade laser as local oscillator. In contrast to previous publications 4,10 , significant progress has been made on DFB QCLs by changing the tapered corrugations. At 4.7 THz, this QCL is the highest frequency ever reported using the 3 rd -order DFB structure. Furthermore, by introducing an array of 21 DFB lasers with a linear frequency coverage and a 7.5 GHz frequency spacing, we can target a specific LO frequency. An unprecedented high sensitivity for a heterodyne receiver was measured at 4.741 THz along with a 15 s Allan variance time, the first time such stability has been reported with this combination. Lastly, a theoretical model for methanol molecular lines has been verified at 4.7 THz, which was not possible until now.</text>
<text><location><page_3><loc_12><loc_7><loc_88><loc_11></location>The THz QCL active region is based on a four-well resonant-phonon depopulation design in a metal-metal waveguide. Cavity structure with a lateral corrugated third-order DFB</text>
<figure>
<location><page_4><loc_29><loc_66><loc_67><loc_91></location>
<caption>FIG. 1. (Color online) CW spectra of a 4.7 THz QCL (at 77 K) measured at different bias voltages. (a) Frequencies of an array with 11 devices in pulse mode (at 10 K) demonstrating the frequency selectivity of a 21-element third-order DFB array. (b) Beam pattern of the QCL. (c) Scanning Electron Microscope (SEM) image for a taper-horn third-order DFB laser. The contact pad connects to the side of the last period of the DFB grating.</caption>
</figure>
<text><location><page_4><loc_12><loc_16><loc_88><loc_46></location>grating similar to those demonstrated in Amanti et al. 8 were used to provide frequencyselectivity and also to the improve far-field beam pattern. We improve upon this design by changing the shape of the corrugated gratings from a traditional square tooth to a tapered shape as shown in Fig. 1c. According to an electromagnetic finite-element (FEM) simulation, the taper-horn shape increases the radiation loss from the unwanted upper bandedge mode while marginally reducing the radiation loss for the desired third-order DFB mode, hence improving mode selectivity in order to ensure a robust single-mode operation. Effectively, this approach leverages a trade-off between the output power efficiency and mode discrimination. With this improved frequency selectivity, we realize a linear frequency coverage of 440 GHz, from 4.61 to 5.05 THz as shown in Figure 1a with robust single-mode operation on the same gain medium. These grating periods range from 28.5 to 25 µ m, which cover ∼ 80% of the gain spectrum.</text>
<text><location><page_4><loc_12><loc_7><loc_88><loc_14></location>The third-order DFB QCL arrays were fabricated using standard metal-metal waveguide fabrication techniques, contact lithography, and inductively-coupled-reactive ion etching (ICP-RIE) to define the laser mesas with the Ti/Au top contact acting as the self-aligned</text>
<text><location><page_5><loc_12><loc_79><loc_88><loc_91></location>etch mask. A 300 nm SiO 2 electrical insulation layer was used for the isolation of the contact pads. Each array consists of 21 DFB lasers arranged in a similar manner as in Lee et al. 11 with a ∼ 7.5 GHz frequency spacing. The position where the contact pads connect to the DFB laser was chosen to minimize unwanted perturbation to the grating boundary condition.</text>
<text><location><page_5><loc_12><loc_63><loc_88><loc_77></location>The device used in the heterodyne experiment has a width of 17 µ m and 27 grating periods with an overall device length of ∼ 0.76 mm. The measured CW output power is 0.25 mWwith ∼ 0.7 W DC of power dissipation at 10 K and a main beam divergence of ∼ 12 · , as shown in Figure 1b. CW lasing at 4.7471 THz is realized at 77 K with a 12.4 V bias voltage, which is within 3 GHz of the target [OI] line (see Fig. 1a). For the heterodyne measurement described below, the device is operated at ∼ 10 K.</text>
<text><location><page_5><loc_12><loc_39><loc_88><loc_61></location>HEBs are the preferred mixer for frequencies above 1.5 THz and have been used up to 1.9 THz in the Herschel Space Telescope 12 and the Stratospheric THz Observatory 13 , and up to 2.5 THz in the Stratospheric Observatory For Infrared Astronomy 14,15 . We use a NbN HEB mixer, which was developed by SRON and TU Delft. Nb contact pads connect a 2 × 0.2 µ m 2 superconducting bridge to a tight winding spiral antenna, which is suitable for super-THz frequencies. With the application of both electrical bias and optical pumping from an LO source, a temperature distribution of hot electrons is maintained producing a resistive state in the center of the bridge. Incoming signals modulate temperature causing a modulation in the resistance to create heterodyne mixing 16 .</text>
<text><location><page_5><loc_12><loc_7><loc_88><loc_38></location>The test setup for measuring T DSB rec is shown in Fig. 2. The QCL was mounted in a liquid helium cryostat and operated at ∼ 10 K. The beam was focused by an ultra-high molecular weight polyethylene (UHMW-PE) lens, which is ∼ 80% transmissive at 4.7-THz. A voice coil attenuator together with a proportional - integral - derivative (PID) feedback loop is used to stabilize the power output of the QCL during T DSB rec and Allan variance measurements 17 (where noted). In this case the HEB DC current is used as a power reference signal. The beam entered a blackbody hot/cold vacuum setup attached to the HEB cryostat via a second UHMW-PE window and then was reflected by a 3 µ m mylar beam splitter. This cryostat was cooled to 4.2 K. The HEB was mounted to the back side of a 10 mm Si lens with an anti-reflection coating designed for 4.25 THz. The first stage low noise amplifier (LNA) was attached to the cold plate and operated at 4.2 K. The LNA noise temperature was 3 K with a gain of 42 dB measured at 15 K. Outside the dewar, room temperature amplifiers and an</text>
<figure>
<location><page_6><loc_31><loc_62><loc_68><loc_89></location>
<caption>FIG. 2. (Color online) Laboratory setup for heterodyne QCL-HEB measurements.</caption>
</figure>
<text><location><page_6><loc_12><loc_47><loc_88><loc_54></location>80 MHz wide band pass filter (BPF) centered at 1.5 GHz were used to further condition the intermediate frequency (IF) signal before the total power was read using an Agilent E4418B power meter.</text>
<text><location><page_6><loc_12><loc_26><loc_88><loc_46></location>The total optical losses in the setup are ∼ 20 dB or about 99% of the QCL emission, including the mylar beam splitter efficiency and atmospheric absorption in the optical path from the LO to the HEB mixer 18 . Based on the QCL output power of 250 µ W, a maximum of 2.5 µ Wcan couple into the detector. By using the isothermal method based on IV curves 19 , the maximum LO power recorded by the detector is ∼ 290 nW. Thus, with a lens, 10% to 15% of the available power was coupled into the HEB. Because of the beam pattern of the third-order DFB structure, this is considerably improved over the 1.4% coupling efficiency of previous generation QCLs 4 .</text>
<text><location><page_6><loc_12><loc_18><loc_88><loc_25></location>Receiver sensitivity was measured using the Y-factor method. Eq. 1 is used to convert a Y-factor to a T DSB rec . The Callen-Welton temperatures at 4.7 THz are T eff , hot = 309 K and T eff , cold = 126 K 20 .</text>
<formula><location><page_6><loc_39><loc_12><loc_88><loc_16></location>T N , rec = T eff , hot -YT eff , cold Y -1 (1)</formula>
<text><location><page_6><loc_12><loc_7><loc_88><loc_11></location>The Y-factor was measured using three different methods. In Fig. 3a the measured IF power was swept as a function of bias voltage with a fixed LO power. We corrected for direct</text>
<text><location><page_7><loc_32><loc_89><loc_32><loc_90></location>/s32</text>
<figure>
<location><page_7><loc_12><loc_61><loc_91><loc_89></location>
<caption>FIG. 3. (Color Online) (a) IF power measurements as functions of bias voltage with the calculated T DSB rec plotted on the right hand side. (b) IF power measurements as functions of stabilized bias current with the calculated T DSB rec plotted on the right hand side. (c) T DSB rec for 4.25, 4.74, and 5.25 THz using a 3 µ m beam splitter. A gas laser was used as an LO at 4.25 ad 5.25 THz and a QCL was used as an LO at 4.74 THz. Ten times the quantum noise limit is also shown with the dashed line.</caption>
</figure>
<text><location><page_7><loc_30><loc_61><loc_30><loc_61></location>/s98/s105/s97/s115/s32</text>
<text><location><page_7><loc_12><loc_12><loc_88><loc_37></location>detection by adjusting the LO power so that the IV curves were on top of one another. The best T DSB rec was found to be 825 K at a bias of 0.7 mV and 30 µ A. Recently it has become possible to accurately sweep LO power by attenuating the LO signal with a stabilized voice coil attenuator 17 and plot the resulting HEB bias current as a function of output power (see Fig. 3b). This method reduces direct detection and results in an average T DSB rec of 810 K around a bias of 0.65 mV and 29 µ A. This current corresponds to 220 nW of LO power. The third method, not shown, chops between hot and cold loads with a stabilized current. It also produced a T DSB rec of 810 K at the same operating point. Thus we obtain a T DSB rec of 815 K by averaging the three methods. This T DSB rec is ∼ 7 times the quantum noise limit ( hν 2 k B ) .</text>
<text><location><page_7><loc_12><loc_7><loc_88><loc_11></location>To demonstrate the QCL adds no additional noise to the receiver system, T DSB rec measurements were taken with a gas laser at 4.25 and 5.25 THz. We recorded 750 K at 4.25</text>
<text><location><page_8><loc_51><loc_90><loc_52><loc_90></location>/s32</text>
<figure>
<location><page_8><loc_25><loc_61><loc_73><loc_89></location>
<caption>FIG. 4. (Color online) Measurements in air for (a) non-stabilized spectroscopic and (b) stabilized spectroscopic Allan variance curves. In the inset is an Allan variance curve for a non-stabilized spectroscopic measurement taken with the air purged by nitrogen gas.</caption>
</figure>
<text><location><page_8><loc_12><loc_39><loc_88><loc_48></location>THz and 950 K at 5.25 THz with the same HEB receiver. Fig. 3c shows that all three T DSB rec scale linearly with frequency. This suggests that the QCL is a clean LO source. These measurements improve upon the previously published T DSB rec of 860 K at 4.25 THz and 1150 K at 5.25 THz 21 . We attribute most of this ( > ∼ 12%) improvement to a new IF mixer circuit.</text>
<text><location><page_8><loc_12><loc_15><loc_88><loc_38></location>HEB receivers have been plagued by stability issues, which can now largely be attributed to instability in the received LO power at the detector 17 . Allan variance measurements are important in determining the optimum integration time on a source between instrument calibrations. For this purpose the noise temperature setup of Fig. 2 was modified to include a two-way power splitter at the end of the IF chain. Each of the output channels was then sent through a band pass filter, one centered at 1.25 GHz and the other at 1.75 GHz. This enabled measurements of the Allan variance in the spectroscopic configuration (the spectral difference between the two channels), which yields greater Allan variance times because it effectively filters out longer period baseline variations.</text>
<text><location><page_8><loc_12><loc_7><loc_88><loc_14></location>The results from our receiver are shown in Fig. 4. We found that the non-stabilized Allan variance time was ∼ 1 s and the stabilized Allan variance time was ∼ 15 s with an 18 MHz noise fluctuation bandwidth. The resulting Allan variance time from a shorted IF chain was</text>
<text><location><page_8><loc_74><loc_76><loc_75><loc_76></location>/s32</text>
<text><location><page_9><loc_45><loc_90><loc_45><loc_90></location>/s32</text>
<figure>
<location><page_9><loc_26><loc_63><loc_74><loc_90></location>
<caption>FIG. 5. (Color online) A DSB spectrum of methanol with an LO frequency of 4.74093 THz compared with the predicted spectrum from the JPL catalog.</caption>
</figure>
<text><location><page_9><loc_12><loc_39><loc_88><loc_51></location>sufficiently long enough to eliminate the IF chain as a source of instability. Next, we purged the air between the QCL window and vacuum setup with nitrogen gas. This improves the non-stabilized, spectroscopic Allan time to ∼ 7 s as shown in the inset of Fig 4, suggesting that atmospheric turbulence at 4.7 THz may be a large contributor to the instability in the system.</text>
<text><location><page_9><loc_12><loc_10><loc_88><loc_38></location>In order to demonstrate the functionality of the receiver for heterodyne spectroscopy, the receiver was used to measure a spectrum of methanol gas (CH 3 OH). A methanol gas cell was attached to an external input port on the hot/cold vacuum setup so that there was no air in the signal path. The QCL was operated at a bias voltage of 11.8 V. The results, averaged over 18 s of integration time, are shown in Fig. 5 along with a simulation 22 at 0.25 mbar that predicts line widths based on the frequencies and line strengths from the JPL spectral catalog 23,24 . The lines from 1500-1700 MHz are attenuated because the FFTS upper band (1500-3000 MHz) high pass filter has a cut-off frequency of 1700 MHz. The best-fit frequency for the QCL is 4.740493 THz, which is close to the HEB bandwidth ( ∼ 4 GHz) for the [OI] line. The verification of the JPL spectral catalog is also important for the frequency locking of the QCL 7 .</text>
<text><location><page_9><loc_14><loc_7><loc_88><loc_8></location>In conclusion, we have demonstrated a 4.7-THz HEB-QCL receiver with a measured</text>
<text><location><page_10><loc_12><loc_76><loc_88><loc_91></location>sensitivity of 815 K and spectroscopic Allan time of 15 s. This T DSB rec is 85 times lower than a previous Schottky receiver 25 . Heterodyne performance was verified by observing a methanol spectrum. The performance of this receiver indicates THz receiver technology has reached a level of maturity that will permit large-scale [OI] surveys of the interstellar medium to take place, such as those planned by the Gal/Xgal Ultra-Long Duration Spectroscopic Stratospheric Terahertz Observatory (GUSSTO).</text>
<text><location><page_10><loc_12><loc_49><loc_88><loc_75></location>We acknowledge G. Goltsmans group at MSPU for the provision of NbN films. We would like to thank John C. Pearson for his help in understanding methanol lines in the JPL catalog near 4.7 THz. The work of the University of Arizona was supported by NASA grant NN612PK37C. The work in the Netherlands is supported by NWO, KNAW, and NATO SFP. The work at MIT is supported by NASA and NSF. The work at Sandia was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy National Nuclear Security Administration under contract DE-AC04-94AL85000.</text>
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[
{
"title": "Hot electron bolometer heterodyne receiver with a 4.7-THz quantum cascade laser as a local oscillator",
"content": "1) Department of Electrical and Computer Engineering, 1230 E. Speedway Blvd., University of Arizona, Tucson, AZ 85721 USA 3) Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 4) Purple Mountain Observatory (PMO), Chinese Academy of Sciences, 2 West Beijing Road, Nanjing, JiangSu 210008, China, and Graduate School, Chinese Academy of Sciences, 19A Yu Quan Road, Beijing 100049, China 5) Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts 02139, USA 6) Steward Observatory, 933 N Cherry Ave., Rm N204, University of Arizona, Tucson, AZ 85721 USA 7) Sandia National Laboratories, Albuquerque, NM 87185-0601, USA (Dated: 4 November 2021) We report on a heterodyne receiver designed to observe the astrophysically important neutral atomic oxygen [OI] line at 4.7448 THz. The local oscillator is a third-order distributed feedback Quantum Cascade Laser operating in continuous wave mode at 4.741 THz. A quasi-optical, superconducting NbN hot electron bolometer is used as the mixer. We recorded a double sideband receiver noise temperature (T DSB rec ) of 815 K, which is ∼ 7 times the quantum noise limit ( h ν 2k B ) and an Allan variance time of 15 s at an effective noise fluctuation bandwidth of 18 MHz. Heterodyne performance was confirmed by measuring a methanol line spectrum. Astronomers have long been interested in the fine structure line of [OI] at 4.7448 THz. [OI] probes the star formation process and is the most important cooling line in the interstellar medium (ISM) 1 for gas clouds with densities of n > 10 4 cm -3 . Large scale surveys with extremely high spectral resolution and sensitivity are required to disentangle large scale kinematics and energetics within these clouds. Such high spectral resolution observations require the development of super-THz ( > 3 THz) heterodyne receivers. Due to strong water absorption in the atmosphere, it is not possible to observe the [OI] line from ground-based telescopes. Therefore, an astronomical [OI] receiver requires a compact local oscillator (LO) that can be readily integrated into space-based or suborbital observatories. There are several candidate THz technologies for use in LO systems. These technologies include Schottky diode based multiplier chains, gas lasers, and quantum cascade lasers (QCLs). QCLs are currently the only technological approach that leads to devices small and powerful enough 2 to be used in a variety of space-based, super-THz applications. Furthermore, THz QCLs operating in CW mode have yielded line widths of ∼ 100 Hz 3 , excellent power stability 4 , and output powers over 100 mW 5 , making them well-suited for high resolution spectroscopy. Much progress has been made toward overcoming the challenges associated with using a QCL as an LO. Frequency stabilization without the need of another THz source has been achieved using an absorption line within a methanol gas cell 6,7 . Diverging far-field beam patterns and mode selectivity have been improved by using a 3 rd -order distributed feedback (DFB) grating 8,9 . In this letter we report on a full demonstration of a heterodyne receiver using a THz quantum cascade laser as local oscillator. In contrast to previous publications 4,10 , significant progress has been made on DFB QCLs by changing the tapered corrugations. At 4.7 THz, this QCL is the highest frequency ever reported using the 3 rd -order DFB structure. Furthermore, by introducing an array of 21 DFB lasers with a linear frequency coverage and a 7.5 GHz frequency spacing, we can target a specific LO frequency. An unprecedented high sensitivity for a heterodyne receiver was measured at 4.741 THz along with a 15 s Allan variance time, the first time such stability has been reported with this combination. Lastly, a theoretical model for methanol molecular lines has been verified at 4.7 THz, which was not possible until now. The THz QCL active region is based on a four-well resonant-phonon depopulation design in a metal-metal waveguide. Cavity structure with a lateral corrugated third-order DFB grating similar to those demonstrated in Amanti et al. 8 were used to provide frequencyselectivity and also to the improve far-field beam pattern. We improve upon this design by changing the shape of the corrugated gratings from a traditional square tooth to a tapered shape as shown in Fig. 1c. According to an electromagnetic finite-element (FEM) simulation, the taper-horn shape increases the radiation loss from the unwanted upper bandedge mode while marginally reducing the radiation loss for the desired third-order DFB mode, hence improving mode selectivity in order to ensure a robust single-mode operation. Effectively, this approach leverages a trade-off between the output power efficiency and mode discrimination. With this improved frequency selectivity, we realize a linear frequency coverage of 440 GHz, from 4.61 to 5.05 THz as shown in Figure 1a with robust single-mode operation on the same gain medium. These grating periods range from 28.5 to 25 µ m, which cover ∼ 80% of the gain spectrum. The third-order DFB QCL arrays were fabricated using standard metal-metal waveguide fabrication techniques, contact lithography, and inductively-coupled-reactive ion etching (ICP-RIE) to define the laser mesas with the Ti/Au top contact acting as the self-aligned etch mask. A 300 nm SiO 2 electrical insulation layer was used for the isolation of the contact pads. Each array consists of 21 DFB lasers arranged in a similar manner as in Lee et al. 11 with a ∼ 7.5 GHz frequency spacing. The position where the contact pads connect to the DFB laser was chosen to minimize unwanted perturbation to the grating boundary condition. The device used in the heterodyne experiment has a width of 17 µ m and 27 grating periods with an overall device length of ∼ 0.76 mm. The measured CW output power is 0.25 mWwith ∼ 0.7 W DC of power dissipation at 10 K and a main beam divergence of ∼ 12 · , as shown in Figure 1b. CW lasing at 4.7471 THz is realized at 77 K with a 12.4 V bias voltage, which is within 3 GHz of the target [OI] line (see Fig. 1a). For the heterodyne measurement described below, the device is operated at ∼ 10 K. HEBs are the preferred mixer for frequencies above 1.5 THz and have been used up to 1.9 THz in the Herschel Space Telescope 12 and the Stratospheric THz Observatory 13 , and up to 2.5 THz in the Stratospheric Observatory For Infrared Astronomy 14,15 . We use a NbN HEB mixer, which was developed by SRON and TU Delft. Nb contact pads connect a 2 × 0.2 µ m 2 superconducting bridge to a tight winding spiral antenna, which is suitable for super-THz frequencies. With the application of both electrical bias and optical pumping from an LO source, a temperature distribution of hot electrons is maintained producing a resistive state in the center of the bridge. Incoming signals modulate temperature causing a modulation in the resistance to create heterodyne mixing 16 . The test setup for measuring T DSB rec is shown in Fig. 2. The QCL was mounted in a liquid helium cryostat and operated at ∼ 10 K. The beam was focused by an ultra-high molecular weight polyethylene (UHMW-PE) lens, which is ∼ 80% transmissive at 4.7-THz. A voice coil attenuator together with a proportional - integral - derivative (PID) feedback loop is used to stabilize the power output of the QCL during T DSB rec and Allan variance measurements 17 (where noted). In this case the HEB DC current is used as a power reference signal. The beam entered a blackbody hot/cold vacuum setup attached to the HEB cryostat via a second UHMW-PE window and then was reflected by a 3 µ m mylar beam splitter. This cryostat was cooled to 4.2 K. The HEB was mounted to the back side of a 10 mm Si lens with an anti-reflection coating designed for 4.25 THz. The first stage low noise amplifier (LNA) was attached to the cold plate and operated at 4.2 K. The LNA noise temperature was 3 K with a gain of 42 dB measured at 15 K. Outside the dewar, room temperature amplifiers and an 80 MHz wide band pass filter (BPF) centered at 1.5 GHz were used to further condition the intermediate frequency (IF) signal before the total power was read using an Agilent E4418B power meter. The total optical losses in the setup are ∼ 20 dB or about 99% of the QCL emission, including the mylar beam splitter efficiency and atmospheric absorption in the optical path from the LO to the HEB mixer 18 . Based on the QCL output power of 250 µ W, a maximum of 2.5 µ Wcan couple into the detector. By using the isothermal method based on IV curves 19 , the maximum LO power recorded by the detector is ∼ 290 nW. Thus, with a lens, 10% to 15% of the available power was coupled into the HEB. Because of the beam pattern of the third-order DFB structure, this is considerably improved over the 1.4% coupling efficiency of previous generation QCLs 4 . Receiver sensitivity was measured using the Y-factor method. Eq. 1 is used to convert a Y-factor to a T DSB rec . The Callen-Welton temperatures at 4.7 THz are T eff , hot = 309 K and T eff , cold = 126 K 20 . The Y-factor was measured using three different methods. In Fig. 3a the measured IF power was swept as a function of bias voltage with a fixed LO power. We corrected for direct /s32 /s98/s105/s97/s115/s32 detection by adjusting the LO power so that the IV curves were on top of one another. The best T DSB rec was found to be 825 K at a bias of 0.7 mV and 30 µ A. Recently it has become possible to accurately sweep LO power by attenuating the LO signal with a stabilized voice coil attenuator 17 and plot the resulting HEB bias current as a function of output power (see Fig. 3b). This method reduces direct detection and results in an average T DSB rec of 810 K around a bias of 0.65 mV and 29 µ A. This current corresponds to 220 nW of LO power. The third method, not shown, chops between hot and cold loads with a stabilized current. It also produced a T DSB rec of 810 K at the same operating point. Thus we obtain a T DSB rec of 815 K by averaging the three methods. This T DSB rec is ∼ 7 times the quantum noise limit ( hν 2 k B ) . To demonstrate the QCL adds no additional noise to the receiver system, T DSB rec measurements were taken with a gas laser at 4.25 and 5.25 THz. We recorded 750 K at 4.25 /s32 THz and 950 K at 5.25 THz with the same HEB receiver. Fig. 3c shows that all three T DSB rec scale linearly with frequency. This suggests that the QCL is a clean LO source. These measurements improve upon the previously published T DSB rec of 860 K at 4.25 THz and 1150 K at 5.25 THz 21 . We attribute most of this ( > ∼ 12%) improvement to a new IF mixer circuit. HEB receivers have been plagued by stability issues, which can now largely be attributed to instability in the received LO power at the detector 17 . Allan variance measurements are important in determining the optimum integration time on a source between instrument calibrations. For this purpose the noise temperature setup of Fig. 2 was modified to include a two-way power splitter at the end of the IF chain. Each of the output channels was then sent through a band pass filter, one centered at 1.25 GHz and the other at 1.75 GHz. This enabled measurements of the Allan variance in the spectroscopic configuration (the spectral difference between the two channels), which yields greater Allan variance times because it effectively filters out longer period baseline variations. The results from our receiver are shown in Fig. 4. We found that the non-stabilized Allan variance time was ∼ 1 s and the stabilized Allan variance time was ∼ 15 s with an 18 MHz noise fluctuation bandwidth. The resulting Allan variance time from a shorted IF chain was /s32 /s32 sufficiently long enough to eliminate the IF chain as a source of instability. Next, we purged the air between the QCL window and vacuum setup with nitrogen gas. This improves the non-stabilized, spectroscopic Allan time to ∼ 7 s as shown in the inset of Fig 4, suggesting that atmospheric turbulence at 4.7 THz may be a large contributor to the instability in the system. In order to demonstrate the functionality of the receiver for heterodyne spectroscopy, the receiver was used to measure a spectrum of methanol gas (CH 3 OH). A methanol gas cell was attached to an external input port on the hot/cold vacuum setup so that there was no air in the signal path. The QCL was operated at a bias voltage of 11.8 V. The results, averaged over 18 s of integration time, are shown in Fig. 5 along with a simulation 22 at 0.25 mbar that predicts line widths based on the frequencies and line strengths from the JPL spectral catalog 23,24 . The lines from 1500-1700 MHz are attenuated because the FFTS upper band (1500-3000 MHz) high pass filter has a cut-off frequency of 1700 MHz. The best-fit frequency for the QCL is 4.740493 THz, which is close to the HEB bandwidth ( ∼ 4 GHz) for the [OI] line. The verification of the JPL spectral catalog is also important for the frequency locking of the QCL 7 . In conclusion, we have demonstrated a 4.7-THz HEB-QCL receiver with a measured sensitivity of 815 K and spectroscopic Allan time of 15 s. This T DSB rec is 85 times lower than a previous Schottky receiver 25 . Heterodyne performance was verified by observing a methanol spectrum. The performance of this receiver indicates THz receiver technology has reached a level of maturity that will permit large-scale [OI] surveys of the interstellar medium to take place, such as those planned by the Gal/Xgal Ultra-Long Duration Spectroscopic Stratospheric Terahertz Observatory (GUSSTO). We acknowledge G. Goltsmans group at MSPU for the provision of NbN films. We would like to thank John C. Pearson for his help in understanding methanol lines in the JPL catalog near 4.7 THz. The work of the University of Arizona was supported by NASA grant NN612PK37C. The work in the Netherlands is supported by NWO, KNAW, and NATO SFP. The work at MIT is supported by NASA and NSF. The work at Sandia was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy National Nuclear Security Administration under contract DE-AC04-94AL85000.",
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2013ApPhL.103t2602N
https://arxiv.org/pdf/1310.7287.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_85><loc_93></location>High-resolution gamma-ray spectroscopy with a microwave-multiplexed transition-edge sensor array</section_header_level_1>
<text><location><page_1><loc_15><loc_85><loc_92><loc_89></location>Omid Noroozian, 1, 2, a) John A. B. Mates, 1 Douglas A. Bennett, 1 Justus A. Brevik, 1 Joseph W. Fowler, 1 Jiansong Gao, 1 Gene C. Hilton, 1 Robert D. Horansky, 1 Kent D. Irwin, 1 Zhao Kang, 3 Daniel R. Schmidt, 1 Leila R. Vale, 1 and Joel N. Ullom 1</text>
<unordered_list>
<list_item><location><page_1><loc_15><loc_83><loc_65><loc_85></location>1) National Institute of Standards and Technology, Boulder, CO 80305, USA</list_item>
<list_item><location><page_1><loc_15><loc_82><loc_78><loc_83></location>2) Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309,</list_item>
</unordered_list>
<text><location><page_1><loc_15><loc_81><loc_18><loc_82></location>USA</text>
<text><location><page_1><loc_15><loc_80><loc_64><loc_81></location>3) Department of Physics, University of Colorado, Boulder, CO 80309, USA</text>
<text><location><page_1><loc_15><loc_78><loc_32><loc_79></location>(Dated: 29 October 2013)</text>
<text><location><page_1><loc_15><loc_65><loc_92><loc_77></location>We demonstrate very high resolution photon spectroscopy with a microwave-multiplexed two-pixel transitionedge sensor (TES) array. We measured a 153 Gd photon source and achieved an energy resolution of 63 eV full-width-at-half-maximum at 97 keV and an equivalent readout system noise of 86 pA/ √ Hz at the TES. The readout circuit consists of superconducting microwave resonators coupled to radio-frequency superconductingquantum-interference-devices (SQUID) and transduces changes in input current to changes in phase of a microwave signal. We use flux-ramp modulation to linearize the response and evade low-frequency noise. This demonstration establishes one path for the readout of cryogenic X-ray and gamma-ray sensor arrays with more than 10 3 elements and spectral resolving powers R = λ/ ∆ λ > 10 3 .</text>
<text><location><page_1><loc_9><loc_38><loc_49><loc_62></location>Multiplexed readout of sub-Kelvin cryogenic detectors is an essential requirement for large focal plane arrays. Next-generation instruments for the detection of electromagnetic radiation from gamma-ray to far-infrared wavelengths will have pixel counts in the 10 3 -10 6 range and require readout techniques that do not compromise their sensitivity. To date, many instruments have used time-, frequency-, or code-domain SQUID multiplexing schemes 1-3 . One such instrument, the TES bolometer camera SCUBA2, has achieved background-limited sensitivity in 10 4 pixels using time-domain multiplexing (TDM) 4 . Similarly, calorimetric gamma-ray/X-ray spectrometers that use TDM have reached excellent energy resolutions of δE ≈ 50 eV at 100 keV in a 256-pixel array 5 . However, the scalability of these readout approaches is limited by the finite measurement bandwidth ( ∼ 10 MHz) achievable in a flux-locked loop.</text>
<text><location><page_1><loc_9><loc_13><loc_49><loc_38></location>Kinetic Inductance Detectors (KIDs) 6,7 , on the other hand, provide a possible path to higher multiplexing factors. These devices are naturally frequency-multiplexed and the ultimate limit on the available bandwidth is many gigahertz, which is set by the readout cryogenic amplifier. Present limits in room-temperature electronics impose a 550 MHz bandwidth limit 8 , but this figure will improve steadily. However, the sensing element is part of a thin-film superconducting resonator, so readout and signal generation can be difficult to simultaneously optimize. This challenge is particularly severe for spectroscopic X-ray and gamma-ray detectors, which must stop high-energy photons and where spatial variation in the device response must be smaller than 0.1%. X-ray and gamma-ray spectroscopy results achieved to date with KIDs are not yet compellingly better than conventional semiconducting detectors 9,10 .</text>
<text><location><page_1><loc_52><loc_31><loc_92><loc_63></location>Microwave SQUID multiplexing 11,12 ( µ Mux) is a readout technique that potentially combines the proven sensitivity of TESs and the scalable multiplexing power found in KIDs. Microwave SQUID multiplexing uses radiofrequency (rf) SQUIDs coupled to high quality-factor ( Q ) microwave resonators and has sufficiently low noise to read out the most sensitive cryogenic detectors. Additionally, it allows independent optimization of the detector and the multiplexer, and provides signal modulation that evades low-frequency resonator noise. Previously, we demonstrated device-noise limited µ Mux readout of two 150-GHz TES polarimeter bolometers 13 . Here we demonstrate the use of µ Mux with TES gamma-ray detectors that respond to single incident photons. Specifically, we demonstrate spectroscopy with resolving powers R glyph[greaterorsimilar] 1500 using µ Mux and a two-pixel TES array. Our achieved resolving power is close to an order of magnitude higher than state-of-the-art high-purity germanium (HPGe) detectors. Furthermore, the number of pixels can easily be scaled to values > 10 3 in future instruments to provide useful system-level count rates and collecting areas.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_30></location>The main goal of this experiment was to demonstrate the readout of two gamma-ray TESs using microwave SQUID multiplexing with little degradation in energy resolution and in a scalable fashion that establishes a path for instruments with much larger detector count. A photograph of the device used is shown in Fig. 1. The sensor chip consists of 21 TES microcalorimeters similar in design to devices reported by Bennett et al. 5 , but modified for higher energy resolution and smaller dynamic range. The TESs are made from a Mo-Cu bilayer with T c = 107 mK and heat capacity C 1 ≈ 2 . 3 pJ/K, and are placed on a 1 . 38 × 1 . 38 mm 2 Si 3 N 4 membrane 1 µ m thick that provides a thermal conductance G 1 ≈ 2 . 2 nW/K. Bulk absorbers of polycrystaline Sn with dimensions of 1 . 1 × 1 . 1 mm 2 × 250 µ m are glued using Stycast 1266 to</text>
<figure>
<location><page_2><loc_9><loc_75><loc_50><loc_94></location>
<caption>FIG. 1. Photograph of the microwave SQUID multiplexer and gamma-ray TESs with bulk Sn absorbers. TESs numbered 1 and 2 are wirebonded to SQUID input coils on the multiplexer chip through a central interface (IF) chip. Up to 35 resonators on the µ Mux chip are read out using the single microwave feedline at bottom. Inset shows the full device box with two microwave SMA connectors.</caption>
</figure>
<text><location><page_2><loc_9><loc_54><loc_49><loc_61></location>SU8 epoxy posts on the SiN membrane. The thermal conductance between absorber and TES was approximately G 2 ≈ 31 nW/K. The absorber has a heat capacity of C 2 ≈ 6 pJ/K and provides ∼ 27 % absorption efficiency for a 100 keV photon.</text>
<text><location><page_2><loc_9><loc_41><loc_49><loc_53></location>An interface (IF) chip was used to provide a bias shunt resistance of R sh = 0 . 33 mΩ in parallel with each TES and a wirebond-selectable Nyquist inductor, L N , in series to increase the pulse rise-times. The value for L N can be selected from values of 0, 270, and 690 nH. Although the IF chip was not specifically designed for this experiment it provided reasonable values of resistance and inductance. We used L N = 690 nH for TES 1 and L N = 0 nH for TES 2.</text>
<text><location><page_2><loc_9><loc_10><loc_49><loc_40></location>The multiplexer chip consists of 35 quarter-wave coplanar waveguide (CPW) microwave resonators made from a 200 nm thick Nb film deposited on high-resistivity silicon with ρ > 10 kΩ cm. The resonator CPWs have a 10 µ m center strip and 6 µ m gap widths. Adjacent resonances are separated by ∼ 6 MHz and are centered around 5.5 GHz with coupling quality factors 11 , 500 glyph[lessorsimilar] Q c glyph[lessorsimilar] 34 , 000 and internal quality factors 48 , 000 glyph[lessorsimilar] Q i glyph[lessorsimilar] 120 , 000. We chose two resonances at 5.503 GHz and 5.566 GHz with Q c 's of 12,000 and 32,000 and Q i 's of 80,000 and 110,000 to read out TES 1 and 2 respectively. The short circuit end of each resonator inductively couples to an rf SQUID (with geometric inductance L s ≈ 20 pH and critical current I c ≈ 5 µ A) that acts as a flux-dependent nonlinear inductor. The rf SQUID transduces a change in input flux into a change of resonance frequency. In turn, the TES current couples flux into the SQUID through an input coil with mutual inductance M ≈ 88 pH. A common flux line is inductively coupled to all the SQUIDs to provide flux-ramp modulation 14 ability (see below). The details of the chip can be found in Mates's PhD thesis 13 .</text>
<text><location><page_2><loc_10><loc_9><loc_49><loc_10></location>The chips were mounted in a gold-plated copper sam-</text>
<text><location><page_2><loc_52><loc_76><loc_92><loc_93></location>ple box. A G-10 circuit board provides DC connectivity for the chips. Two Duroid circuit boards with microstrip to CPW transitions connect the microwave input and output lines to the CPW feedline on the resonator chip and to SMA connectors on the box. In this unoptimized setup long aluminum wirebonds ( glyph[lessorsimilar] 1 cm) connect the SQUID input coils to the IF chip and the TESs, and bring in the DC bias; these long free-space connections are a likely source of 1/f noise. Gold wirebonds were used for heat-sinking the TES chip. A small hole in the copper box lid (not shown here) above the TES chip increases the gamma-ray flux reaching the absorbers.</text>
<text><location><page_2><loc_52><loc_37><loc_92><loc_75></location>The sample box was mounted inside a cryostat and was connected to a pulse-tube backed adiabatic demagnetization refrigerator (ADR), and its temperature was regulated at 85 mK using the ADR magnet. During the initial cooldown from 300 K a magnetic shield made from mu-metal was placed around the cryostat to avoid trapping earth's field inside the resonators, SQUIDs, and Sn absorbers. This shield was removed after reaching base temperature to increase the gamma-ray count rate. Once this shield was removed, there was no magnetic shielding for the experiment. The IV curves for some of the TESs showed distortions from magnetic field trapped in the nearby Sn absorber, which resulted in lower pulse heights. This flux trapping likely occurred when the ADR was cycled to reach 85 mK. However, TESs 1 and 2 showed good IV characteristics with no evidence of flux trapping. Successful unshielded operation at 85 mK (after removing the mu-metal) in the presence of earth's field and the ADR field bodes well for future robustness. The final gamma-ray path contained a 0.8 mm thick carbon fiber window in the cryostat vacuum shell and three access windows cut into the 60 K and 3 K radiation shields and the Cu box lid. All three access windows were covered with 0.1 mm thick aluminum tape. We positioned a weak 153 Gd radioisotope source outside the carbon fiber window approximately 10 cm from the detectors to provide a photon count rate of ∼ 0 . 75 Hz per detector.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_36></location>The circuit diagram for simultaneous readout of two TESs is shown in Fig. 2. The number of sensors in our demonstration was set by the availability of roomtemperature microwave electronics but, as shown in Fig. 2, the circuit architecture is compatible with a much larger number of sensors that share the same microwave feedline, and can easily be scaled by using softwaredefined radio electronics 8,16 . In our experiment two microwave signal generators tuned close to the frequencies of the resonators inject two tones into the feedline. Each tone's frequency and power is adjusted to optimize the signal-to-noise 13 . The final readout powers were P µ w = -69 dBm and -73 dBm at the µ Mux chip feedline for the resonators connected to TESs 1 and 2, respectively. The TESs are voltage biased in their resistive transition at ∼ 20 % of their normal-state resistance R n by use of a single DC bias signal. A gamma-ray photon event in a TES increases the temperature and therefore the resistance of the TES, which in turn reduces the cur-</text>
<text><location><page_3><loc_47><loc_90><loc_48><loc_95></location>I</text>
<text><location><page_3><loc_47><loc_65><loc_48><loc_70></location>Q</text>
<figure>
<location><page_3><loc_9><loc_67><loc_50><loc_93></location>
<caption>FIG. 2. Circuit schematic for multiplexed gamma-ray spectroscopy. Resonator circuitry is shown in black, rf-SQUID components in purple, TES components in orange, and flux ramp components in green. Dashed blue line contains circuit components at the cold stage of the ADR. Microwave attenuators at various stages have been omitted for clarity.</caption>
</figure>
<text><location><page_3><loc_48><loc_66><loc_48><loc_69></location>2</text>
<text><location><page_3><loc_9><loc_22><loc_49><loc_54></location>rent I TES passing through the TES. This time-dependent current applies flux Φ in the SQUID loop as Φ = MI TES . The value of M sets the transduction gain. The SQUID inductance is periodic with flux, as L ≈ L s + L J / cos φ where L J = Φ 0 / (2 πI c ) is the Josephson inductance, φ = 2 π Φ / Φ 0 is the phase shift across the junction, and Φ 0 is the magnetic flux quantum. Therefore, a large input flux signal can cause an excursion of several flux periods in the SQUID inductance and consequently in the resonance frequency. Changes in the resonance frequencies of the two resonators change the complex microwave transmission S 21 across the feedline. These changes are amplified with a cryogenic high-electron-mobility-transistor (HEMT) amplifier and further amplified at room temperature. Two IQ mixers then downconvert the signals in each channel using copies of the original microwave tones. The in-phase ( V I ) and quadrature-phase ( V Q ) signals are then digitized at a sample rate of 2 MHz in a computer. We used only the V Q signal components, rotating the resonance IQ planes with phase shifters such that they coincide with the electronics IQ plane in each channel.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_21></location>In order to linearize the signal response we implemented flux-ramp modulation 14 by applying a f s = 40 kHz sawtooth flux-ramp signal V fr (see Fig. 3) to all of the SQUIDs. The amplitude of the ramp is tuned such that it provides ∼ 3Φ 0 of flux per ramp period and modulates V Q 1 and V Q 2 at a carrier frequency of f c ≈ 120 kHz. Since f c is significantly larger than the frequency content of a gamma-ray pulse, the phase shift φ of V Q during each ramp period is effectively constant and pro-</text>
<figure>
<location><page_3><loc_52><loc_67><loc_93><loc_93></location>
<caption>FIG. 3. The two top figures show the measured quadraturephase component V Q 2 (blue circles) of the microwave signal, the flux-ramp modulation sawtooth signal V fr (green triangles), and a pure sinusoidal waveform (orange dashed) over a 32 µ s window sometime before (1) and during (2) a 97 keV gamma-ray pulse in TES 2. The bottom panel shows the corresponding TES current (left axis) and resonance phase shift (right axis) pulse after demodulation. The inset is a snapshot from an oscilloscope simultaneously measuring V Q 1 for TES 1 (top yellow) and V Q 2 for TES 2 (bottom purple) just before digitization (see Fig. 2). The faint phase-shifted yellow waveform represents a gamma-ray event. A real-time movie showing gamma-ray events in both TESs is available online 15 .</caption>
</figure>
<text><location><page_3><loc_52><loc_11><loc_92><loc_44></location>nal to the current signal as I TES = Φ 0 φ/ (2 πM ). The Nyquist signal sampling rate is therefore 40 kHz. Flux-ramp modulation has the added benefit that the signal is upconverted to frequencies above the low-frequency two-level system (TLS) noise that is intrinsic to the resonator 7,17-19 . The two top panels in Fig. 3 show the measured V Q ( t ) component for the TES 2 channel during a 32 µ s window before and after a gamma-ray pulse (blue circles). In order to demodulate the data stream we apply on-the-fly Fourier analysis to extract the phase shift φ of the fundamental frequency ( f c ) component in V Q for each 25 µ s ramp window as φ = arctan ( ∑ V Q ( t ) sin 2 πf c t ∑ V Q ( t ) cos 2 πf c t ) . The sine and cosine are defined over an integer number of periods (2 in this case) ending at the ramp reset (see orange dashed line in Fig. 3 top panels). The remaining ∼ 1 period of V Q at the start of the ramp is ignored to prevent the unwanted transient behavior caused by the ramp reset from contaminating the data. f c is measured from a sinusoidal fit to the flux-ramp response when the TES is superconducting (i.e., zero input flux signal). The phase shift and corresponding TES current for a measured 97 keV pulse after demodulation are shown in the bottom panel in Fig. 3.</text>
<text><location><page_3><loc_53><loc_9><loc_92><loc_10></location>After simultaneously collecting data for TES 1 and 2</text>
<figure>
<location><page_4><loc_9><loc_68><loc_50><loc_93></location>
<caption>FIG. 4. Energy spectrum of a 153 Gd source measured using our two-pixel TES array and the microwave SQUID multiplexer. The spectrum from the TES 1 pixel has been horizontally and vertically shifted for clarity. The top right four insets show a zoom-in of the two 97.4 keV and 103.2 keV photopeaks and corresponding Gaussian fits for TES 1 (top green) and TES 2 (bottom red). The full-width-at-half-maximum (FWHM) energy resolution δE for each peak is indicated. The top left inset shows a zoom-in from 40-49 keV where europium K α and K β complexes can be seen.</caption>
</figure>
<text><location><page_4><loc_9><loc_10><loc_49><loc_50></location>over a seven-hour period we excluded pulse records contaminated by pile-up and other nonidealities. An optimal filter was then applied determined by the power spectral density (PSD) of measured noise (below; see Fig. 5) and average pulse shape 20 . A correction was then made due to drift in the peak energies over the time of the measurement. Finally, we calibrated the energy scale using four known spectral features. The resulting spectra for two simultaneously measured TESs are shown in Fig. 4 where the two most prominent 153 Gd gamma-ray photopeaks at 97.4 keV and 103.2 keV are shown in the upper right insets. Weighted Gaussian fits to these lines give FWHM energy resolutions of δE = 63 . 0 ± 2 . 2 eV and 63 . 8 ± 2 . 9 eV for TES 2, and δE = 87 . 3 ± 2 . 6 eV and 78 . 1 ± 3 . 6 eV for TES 1, for the 97.4 and 103.2 keV peaks, respectively. These resolutions are close to the expected resolution of 55 eV for TES 2 and 66 eV for TES 1, which were obtained from the noise PSD and average 97 keV pulse shapes 21 . The difference can be attributed to several factors, including residual pulse-tube noise at lower frequencies, uncorrected gain drift, and positiondependence of the TES response, all of which degrade the performance. More specifically, the pulse-tube noise was non-stationary such that consecutive gamma-ray pulses did not always experience the same noise level. The upper left inset shows the Eu K α and K β complexes. The remaining lines include Sn X-ray escape peaks and fluorescence from the gold plating of the sample box.</text>
<text><location><page_4><loc_10><loc_9><loc_49><loc_10></location>In order to evaluate the noise performance of our read-</text>
<figure>
<location><page_4><loc_52><loc_76><loc_93><loc_93></location>
<caption>FIG. 5. Measured noise from the microwave multiplexer and TES device for channel 2. The plot is given in terms of the equivalent current noise referred to the TES. The blue line shows the total system noise when the TES is biased in the transition at 20 % of R n . The green line shows the total system noise when the TES is superconducting. Far above the L/R roll-off the ∼ 86 pA/ √ Hz remaining noise is from the µ Mux readout system. Spikes due to noise pickup have been removed from data for clarity.</caption>
</figure>
<text><location><page_4><loc_52><loc_9><loc_92><loc_58></location>out circuit we made noise measurements with TES 2 biased in the transition at 20 % of R n and also unbiased (superconducting), as shown in Fig. 5. When biased, the TES noise contribution rolls off at ∼ 20 kHz. When the TES is superconducting the shunt resistance contribution to the total noise rolls off at ∼ 3 kHz, above which the remaining ∼ 86 pA/ √ Hz noise is the contribution from the readout circuit. The electrical roll-offs are consistent with the shunt and TES resistance and inductance values (including approximate wirebond inductance) in the circuit. It can be inferred that the difference in noise power between the blue curve and the readout noise level obtained by subtracting in quadrature is the TES contribution to the noise. To confirm this, we compared the inferred low-frequency TES noise level to an independent measurement of TES noise in a similar device performed using a TDM readout. The TDM data give a TES noise level of 150 pA/ √ Hz, which closely matches the inferred TES noise level of 140 pA/ √ Hz. From this we conclude that the µ Mux readout noise is a factor of 1.6 below the signal-band TES noise. We also conducted a set of TDM measurements where the readout noise was effectively negligible and where, consequently, we expected slightly better energy resolution. Indeed, measurements using TDM with 12 TESs from the same fabrication batch achieved resolutions of ∼ 50 ± 8 eV. The slight resolution advantage of the TDM results is consistent with the quadrature noise penalty from the current implementation of µ Mux readout. Although our achieved resolution of 63 eV is already sufficient for most spectroscopic applications, we expect that a number of simple modifications to the readout circuit should provide even better performance. Lower readout noise comfortably below the TES noise can be achieved with a larger input</text>
<text><location><page_5><loc_9><loc_88><loc_49><loc_93></location>coil mutual inductance M that boosts the TES signal before the rf-SQUIDs, and faster sensors can be read out with lowerQ resonators that allow for higher sampling rates above 40 kHz.</text>
<text><location><page_5><loc_9><loc_47><loc_49><loc_87></location>In summary, we have shown that microwave SQUID multiplexing readout is an excellent candidate for future large focal-plane arrays of spectroscopic sensors. In contrast to previous microwave measurements of cryogenic X-ray/gamma-ray sensors 9,10 , we have demonstrated energy resolutions substantially better than conventional semiconducting detectors and that approach state-of-theart results using traditional low-frequency readout. Dramatic increases in pixel count per readout channel will be straightforward to achieve by coupling additional resonators to the same feedline in order to use more of the 10 GHz of available HEMT bandwidth. Potential electromagnetic crosstalk between resonators in large arrays can be avoided by properly designing the resonators and embedding circuitry 22,23 . In the near term, the multiplexing factor will be constrained by the availability of multichannel microwave electronics to synthesize and demodulate large numbers of readout tones. This type of signal processing has recently been demonstrated for 256 sensors using high-performance but low-cost commercial electronics 8 and further improvements are certain. Our approach is compatible with TES sensors designed for other applications such as low-energy X-ray spectroscopy and the detection of single optical photons 24,25 as well as with magnetic calorimeters 26 . Even larger increases in the multiplexing factor will be achievable by embedding code-division multiplexed sensor columns in each microwave resonator 27 .</text>
<text><location><page_5><loc_9><loc_37><loc_49><loc_47></location>This work was supported by DHS under grant 2011DN-077-ARI051, by the US Department of Energy through the Office of Nonproliferation Research and Development and the Office of Nuclear Energy, and by NASA, under contract NNH11AR83I. The authors thank A. Betz and C. Bockstiegel for useful discussions and help with the experiment.</text>
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</document>
[
{
"title": "High-resolution gamma-ray spectroscopy with a microwave-multiplexed transition-edge sensor array",
"content": "Omid Noroozian, 1, 2, a) John A. B. Mates, 1 Douglas A. Bennett, 1 Justus A. Brevik, 1 Joseph W. Fowler, 1 Jiansong Gao, 1 Gene C. Hilton, 1 Robert D. Horansky, 1 Kent D. Irwin, 1 Zhao Kang, 3 Daniel R. Schmidt, 1 Leila R. Vale, 1 and Joel N. Ullom 1 USA 3) Department of Physics, University of Colorado, Boulder, CO 80309, USA (Dated: 29 October 2013) We demonstrate very high resolution photon spectroscopy with a microwave-multiplexed two-pixel transitionedge sensor (TES) array. We measured a 153 Gd photon source and achieved an energy resolution of 63 eV full-width-at-half-maximum at 97 keV and an equivalent readout system noise of 86 pA/ √ Hz at the TES. The readout circuit consists of superconducting microwave resonators coupled to radio-frequency superconductingquantum-interference-devices (SQUID) and transduces changes in input current to changes in phase of a microwave signal. We use flux-ramp modulation to linearize the response and evade low-frequency noise. This demonstration establishes one path for the readout of cryogenic X-ray and gamma-ray sensor arrays with more than 10 3 elements and spectral resolving powers R = λ/ ∆ λ > 10 3 . Multiplexed readout of sub-Kelvin cryogenic detectors is an essential requirement for large focal plane arrays. Next-generation instruments for the detection of electromagnetic radiation from gamma-ray to far-infrared wavelengths will have pixel counts in the 10 3 -10 6 range and require readout techniques that do not compromise their sensitivity. To date, many instruments have used time-, frequency-, or code-domain SQUID multiplexing schemes 1-3 . One such instrument, the TES bolometer camera SCUBA2, has achieved background-limited sensitivity in 10 4 pixels using time-domain multiplexing (TDM) 4 . Similarly, calorimetric gamma-ray/X-ray spectrometers that use TDM have reached excellent energy resolutions of δE ≈ 50 eV at 100 keV in a 256-pixel array 5 . However, the scalability of these readout approaches is limited by the finite measurement bandwidth ( ∼ 10 MHz) achievable in a flux-locked loop. Kinetic Inductance Detectors (KIDs) 6,7 , on the other hand, provide a possible path to higher multiplexing factors. These devices are naturally frequency-multiplexed and the ultimate limit on the available bandwidth is many gigahertz, which is set by the readout cryogenic amplifier. Present limits in room-temperature electronics impose a 550 MHz bandwidth limit 8 , but this figure will improve steadily. However, the sensing element is part of a thin-film superconducting resonator, so readout and signal generation can be difficult to simultaneously optimize. This challenge is particularly severe for spectroscopic X-ray and gamma-ray detectors, which must stop high-energy photons and where spatial variation in the device response must be smaller than 0.1%. X-ray and gamma-ray spectroscopy results achieved to date with KIDs are not yet compellingly better than conventional semiconducting detectors 9,10 . Microwave SQUID multiplexing 11,12 ( µ Mux) is a readout technique that potentially combines the proven sensitivity of TESs and the scalable multiplexing power found in KIDs. Microwave SQUID multiplexing uses radiofrequency (rf) SQUIDs coupled to high quality-factor ( Q ) microwave resonators and has sufficiently low noise to read out the most sensitive cryogenic detectors. Additionally, it allows independent optimization of the detector and the multiplexer, and provides signal modulation that evades low-frequency resonator noise. Previously, we demonstrated device-noise limited µ Mux readout of two 150-GHz TES polarimeter bolometers 13 . Here we demonstrate the use of µ Mux with TES gamma-ray detectors that respond to single incident photons. Specifically, we demonstrate spectroscopy with resolving powers R glyph[greaterorsimilar] 1500 using µ Mux and a two-pixel TES array. Our achieved resolving power is close to an order of magnitude higher than state-of-the-art high-purity germanium (HPGe) detectors. Furthermore, the number of pixels can easily be scaled to values > 10 3 in future instruments to provide useful system-level count rates and collecting areas. The main goal of this experiment was to demonstrate the readout of two gamma-ray TESs using microwave SQUID multiplexing with little degradation in energy resolution and in a scalable fashion that establishes a path for instruments with much larger detector count. A photograph of the device used is shown in Fig. 1. The sensor chip consists of 21 TES microcalorimeters similar in design to devices reported by Bennett et al. 5 , but modified for higher energy resolution and smaller dynamic range. The TESs are made from a Mo-Cu bilayer with T c = 107 mK and heat capacity C 1 ≈ 2 . 3 pJ/K, and are placed on a 1 . 38 × 1 . 38 mm 2 Si 3 N 4 membrane 1 µ m thick that provides a thermal conductance G 1 ≈ 2 . 2 nW/K. Bulk absorbers of polycrystaline Sn with dimensions of 1 . 1 × 1 . 1 mm 2 × 250 µ m are glued using Stycast 1266 to SU8 epoxy posts on the SiN membrane. The thermal conductance between absorber and TES was approximately G 2 ≈ 31 nW/K. The absorber has a heat capacity of C 2 ≈ 6 pJ/K and provides ∼ 27 % absorption efficiency for a 100 keV photon. An interface (IF) chip was used to provide a bias shunt resistance of R sh = 0 . 33 mΩ in parallel with each TES and a wirebond-selectable Nyquist inductor, L N , in series to increase the pulse rise-times. The value for L N can be selected from values of 0, 270, and 690 nH. Although the IF chip was not specifically designed for this experiment it provided reasonable values of resistance and inductance. We used L N = 690 nH for TES 1 and L N = 0 nH for TES 2. The multiplexer chip consists of 35 quarter-wave coplanar waveguide (CPW) microwave resonators made from a 200 nm thick Nb film deposited on high-resistivity silicon with ρ > 10 kΩ cm. The resonator CPWs have a 10 µ m center strip and 6 µ m gap widths. Adjacent resonances are separated by ∼ 6 MHz and are centered around 5.5 GHz with coupling quality factors 11 , 500 glyph[lessorsimilar] Q c glyph[lessorsimilar] 34 , 000 and internal quality factors 48 , 000 glyph[lessorsimilar] Q i glyph[lessorsimilar] 120 , 000. We chose two resonances at 5.503 GHz and 5.566 GHz with Q c 's of 12,000 and 32,000 and Q i 's of 80,000 and 110,000 to read out TES 1 and 2 respectively. The short circuit end of each resonator inductively couples to an rf SQUID (with geometric inductance L s ≈ 20 pH and critical current I c ≈ 5 µ A) that acts as a flux-dependent nonlinear inductor. The rf SQUID transduces a change in input flux into a change of resonance frequency. In turn, the TES current couples flux into the SQUID through an input coil with mutual inductance M ≈ 88 pH. A common flux line is inductively coupled to all the SQUIDs to provide flux-ramp modulation 14 ability (see below). The details of the chip can be found in Mates's PhD thesis 13 . The chips were mounted in a gold-plated copper sam- ple box. A G-10 circuit board provides DC connectivity for the chips. Two Duroid circuit boards with microstrip to CPW transitions connect the microwave input and output lines to the CPW feedline on the resonator chip and to SMA connectors on the box. In this unoptimized setup long aluminum wirebonds ( glyph[lessorsimilar] 1 cm) connect the SQUID input coils to the IF chip and the TESs, and bring in the DC bias; these long free-space connections are a likely source of 1/f noise. Gold wirebonds were used for heat-sinking the TES chip. A small hole in the copper box lid (not shown here) above the TES chip increases the gamma-ray flux reaching the absorbers. The sample box was mounted inside a cryostat and was connected to a pulse-tube backed adiabatic demagnetization refrigerator (ADR), and its temperature was regulated at 85 mK using the ADR magnet. During the initial cooldown from 300 K a magnetic shield made from mu-metal was placed around the cryostat to avoid trapping earth's field inside the resonators, SQUIDs, and Sn absorbers. This shield was removed after reaching base temperature to increase the gamma-ray count rate. Once this shield was removed, there was no magnetic shielding for the experiment. The IV curves for some of the TESs showed distortions from magnetic field trapped in the nearby Sn absorber, which resulted in lower pulse heights. This flux trapping likely occurred when the ADR was cycled to reach 85 mK. However, TESs 1 and 2 showed good IV characteristics with no evidence of flux trapping. Successful unshielded operation at 85 mK (after removing the mu-metal) in the presence of earth's field and the ADR field bodes well for future robustness. The final gamma-ray path contained a 0.8 mm thick carbon fiber window in the cryostat vacuum shell and three access windows cut into the 60 K and 3 K radiation shields and the Cu box lid. All three access windows were covered with 0.1 mm thick aluminum tape. We positioned a weak 153 Gd radioisotope source outside the carbon fiber window approximately 10 cm from the detectors to provide a photon count rate of ∼ 0 . 75 Hz per detector. The circuit diagram for simultaneous readout of two TESs is shown in Fig. 2. The number of sensors in our demonstration was set by the availability of roomtemperature microwave electronics but, as shown in Fig. 2, the circuit architecture is compatible with a much larger number of sensors that share the same microwave feedline, and can easily be scaled by using softwaredefined radio electronics 8,16 . In our experiment two microwave signal generators tuned close to the frequencies of the resonators inject two tones into the feedline. Each tone's frequency and power is adjusted to optimize the signal-to-noise 13 . The final readout powers were P µ w = -69 dBm and -73 dBm at the µ Mux chip feedline for the resonators connected to TESs 1 and 2, respectively. The TESs are voltage biased in their resistive transition at ∼ 20 % of their normal-state resistance R n by use of a single DC bias signal. A gamma-ray photon event in a TES increases the temperature and therefore the resistance of the TES, which in turn reduces the cur- I Q 2 rent I TES passing through the TES. This time-dependent current applies flux Φ in the SQUID loop as Φ = MI TES . The value of M sets the transduction gain. The SQUID inductance is periodic with flux, as L ≈ L s + L J / cos φ where L J = Φ 0 / (2 πI c ) is the Josephson inductance, φ = 2 π Φ / Φ 0 is the phase shift across the junction, and Φ 0 is the magnetic flux quantum. Therefore, a large input flux signal can cause an excursion of several flux periods in the SQUID inductance and consequently in the resonance frequency. Changes in the resonance frequencies of the two resonators change the complex microwave transmission S 21 across the feedline. These changes are amplified with a cryogenic high-electron-mobility-transistor (HEMT) amplifier and further amplified at room temperature. Two IQ mixers then downconvert the signals in each channel using copies of the original microwave tones. The in-phase ( V I ) and quadrature-phase ( V Q ) signals are then digitized at a sample rate of 2 MHz in a computer. We used only the V Q signal components, rotating the resonance IQ planes with phase shifters such that they coincide with the electronics IQ plane in each channel. In order to linearize the signal response we implemented flux-ramp modulation 14 by applying a f s = 40 kHz sawtooth flux-ramp signal V fr (see Fig. 3) to all of the SQUIDs. The amplitude of the ramp is tuned such that it provides ∼ 3Φ 0 of flux per ramp period and modulates V Q 1 and V Q 2 at a carrier frequency of f c ≈ 120 kHz. Since f c is significantly larger than the frequency content of a gamma-ray pulse, the phase shift φ of V Q during each ramp period is effectively constant and pro- nal to the current signal as I TES = Φ 0 φ/ (2 πM ). The Nyquist signal sampling rate is therefore 40 kHz. Flux-ramp modulation has the added benefit that the signal is upconverted to frequencies above the low-frequency two-level system (TLS) noise that is intrinsic to the resonator 7,17-19 . The two top panels in Fig. 3 show the measured V Q ( t ) component for the TES 2 channel during a 32 µ s window before and after a gamma-ray pulse (blue circles). In order to demodulate the data stream we apply on-the-fly Fourier analysis to extract the phase shift φ of the fundamental frequency ( f c ) component in V Q for each 25 µ s ramp window as φ = arctan ( ∑ V Q ( t ) sin 2 πf c t ∑ V Q ( t ) cos 2 πf c t ) . The sine and cosine are defined over an integer number of periods (2 in this case) ending at the ramp reset (see orange dashed line in Fig. 3 top panels). The remaining ∼ 1 period of V Q at the start of the ramp is ignored to prevent the unwanted transient behavior caused by the ramp reset from contaminating the data. f c is measured from a sinusoidal fit to the flux-ramp response when the TES is superconducting (i.e., zero input flux signal). The phase shift and corresponding TES current for a measured 97 keV pulse after demodulation are shown in the bottom panel in Fig. 3. After simultaneously collecting data for TES 1 and 2 over a seven-hour period we excluded pulse records contaminated by pile-up and other nonidealities. An optimal filter was then applied determined by the power spectral density (PSD) of measured noise (below; see Fig. 5) and average pulse shape 20 . A correction was then made due to drift in the peak energies over the time of the measurement. Finally, we calibrated the energy scale using four known spectral features. The resulting spectra for two simultaneously measured TESs are shown in Fig. 4 where the two most prominent 153 Gd gamma-ray photopeaks at 97.4 keV and 103.2 keV are shown in the upper right insets. Weighted Gaussian fits to these lines give FWHM energy resolutions of δE = 63 . 0 ± 2 . 2 eV and 63 . 8 ± 2 . 9 eV for TES 2, and δE = 87 . 3 ± 2 . 6 eV and 78 . 1 ± 3 . 6 eV for TES 1, for the 97.4 and 103.2 keV peaks, respectively. These resolutions are close to the expected resolution of 55 eV for TES 2 and 66 eV for TES 1, which were obtained from the noise PSD and average 97 keV pulse shapes 21 . The difference can be attributed to several factors, including residual pulse-tube noise at lower frequencies, uncorrected gain drift, and positiondependence of the TES response, all of which degrade the performance. More specifically, the pulse-tube noise was non-stationary such that consecutive gamma-ray pulses did not always experience the same noise level. The upper left inset shows the Eu K α and K β complexes. The remaining lines include Sn X-ray escape peaks and fluorescence from the gold plating of the sample box. In order to evaluate the noise performance of our read- out circuit we made noise measurements with TES 2 biased in the transition at 20 % of R n and also unbiased (superconducting), as shown in Fig. 5. When biased, the TES noise contribution rolls off at ∼ 20 kHz. When the TES is superconducting the shunt resistance contribution to the total noise rolls off at ∼ 3 kHz, above which the remaining ∼ 86 pA/ √ Hz noise is the contribution from the readout circuit. The electrical roll-offs are consistent with the shunt and TES resistance and inductance values (including approximate wirebond inductance) in the circuit. It can be inferred that the difference in noise power between the blue curve and the readout noise level obtained by subtracting in quadrature is the TES contribution to the noise. To confirm this, we compared the inferred low-frequency TES noise level to an independent measurement of TES noise in a similar device performed using a TDM readout. The TDM data give a TES noise level of 150 pA/ √ Hz, which closely matches the inferred TES noise level of 140 pA/ √ Hz. From this we conclude that the µ Mux readout noise is a factor of 1.6 below the signal-band TES noise. We also conducted a set of TDM measurements where the readout noise was effectively negligible and where, consequently, we expected slightly better energy resolution. Indeed, measurements using TDM with 12 TESs from the same fabrication batch achieved resolutions of ∼ 50 ± 8 eV. The slight resolution advantage of the TDM results is consistent with the quadrature noise penalty from the current implementation of µ Mux readout. Although our achieved resolution of 63 eV is already sufficient for most spectroscopic applications, we expect that a number of simple modifications to the readout circuit should provide even better performance. Lower readout noise comfortably below the TES noise can be achieved with a larger input coil mutual inductance M that boosts the TES signal before the rf-SQUIDs, and faster sensors can be read out with lowerQ resonators that allow for higher sampling rates above 40 kHz. In summary, we have shown that microwave SQUID multiplexing readout is an excellent candidate for future large focal-plane arrays of spectroscopic sensors. In contrast to previous microwave measurements of cryogenic X-ray/gamma-ray sensors 9,10 , we have demonstrated energy resolutions substantially better than conventional semiconducting detectors and that approach state-of-theart results using traditional low-frequency readout. Dramatic increases in pixel count per readout channel will be straightforward to achieve by coupling additional resonators to the same feedline in order to use more of the 10 GHz of available HEMT bandwidth. Potential electromagnetic crosstalk between resonators in large arrays can be avoided by properly designing the resonators and embedding circuitry 22,23 . In the near term, the multiplexing factor will be constrained by the availability of multichannel microwave electronics to synthesize and demodulate large numbers of readout tones. This type of signal processing has recently been demonstrated for 256 sensors using high-performance but low-cost commercial electronics 8 and further improvements are certain. Our approach is compatible with TES sensors designed for other applications such as low-energy X-ray spectroscopy and the detection of single optical photons 24,25 as well as with magnetic calorimeters 26 . Even larger increases in the multiplexing factor will be achievable by embedding code-division multiplexed sensor columns in each microwave resonator 27 . This work was supported by DHS under grant 2011DN-077-ARI051, by the US Department of Energy through the Office of Nonproliferation Research and Development and the Office of Nuclear Energy, and by NASA, under contract NNH11AR83I. The authors thank A. Betz and C. Bockstiegel for useful discussions and help with the experiment. son, B. Burger, T. Chuter, I. M. Coulson, W. B. Doriese, C. Dunare, F. Economou, M. D. Niemack, H. A. L. Parsons, C. D. Reintsema, B. Sibthorpe, I. Smail, R. Sudiwala, and H. S. Thomas, Monthly Notices of the Royal Astronomical Society 430 , 2513-2533 (2013).",
"pages": [
1,
2,
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2013AsBio..13..415G
https://arxiv.org/pdf/1303.6804.pdf
<document>
<text><location><page_1><loc_18><loc_78><loc_83><loc_80></location>Potential Biosignatures in Super-Earth Atmospheres II. Photochemical Responses</text>
<text><location><page_1><loc_25><loc_66><loc_75><loc_70></location>J. L. Grenfell 1 , S. Gebauer 1 , M. Godolt 1 , K. Palczynski 1,* , H. Rauer 1,2 , J. Stock 2 , P. v. Paris 2,# , R. Lehmann 3 , and F. Selsis 4</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_55><loc_89><loc_61></location>1 Zentrum für Astronomie und Astrophysik, Technische Universität Berlin (TUB), Hardenbergstr. 36, 10623 Berlin, Germany, email: lee.grenfell@dlr.de -to whom correspondance should be addressed</list_item>
</unordered_list>
<text><location><page_1><loc_9><loc_50><loc_90><loc_54></location>2 Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Rutherford Str. 2, 12489 Berlin, Germany</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_45><loc_85><loc_49></location>3 Alfred-Wegener-Institut für Polar- und Meeresforschung, Telegrafenberg A43, 14473 Potsdam, Germany</list_item>
<list_item><location><page_1><loc_9><loc_41><loc_24><loc_44></location>4,# Present address</list_item>
<list_item><location><page_1><loc_9><loc_38><loc_53><loc_40></location>(ii) CNRS, LAB, UMR 5804, F-33270, Floirac, France</list_item>
<list_item><location><page_1><loc_9><loc_40><loc_60><loc_42></location>(i) Univ. Bordeaux, LAB, UMR 5804, F-33270, Floirac, France</list_item>
</unordered_list>
<text><location><page_1><loc_9><loc_34><loc_23><loc_37></location>* Present address:</text>
<text><location><page_1><loc_9><loc_31><loc_89><loc_35></location>Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1, 14109 Berlin, Germany</text>
<text><location><page_1><loc_9><loc_17><loc_53><loc_19></location>Running title: Photochemistry Earth-like biosignatures</text>
<text><location><page_2><loc_9><loc_17><loc_90><loc_71></location>Abstract: Spectral characterization of Super-Earth atmospheres for planets orbiting in the Habitable Zone of M-dwarf stars is a key focus in exoplanet science. A central challenge is to understand and predict the expected spectral signals of atmospheric biosignatures (species associated with life). Our work applies a global-mean radiative-convective-photochemical column model assuming a planet with an Earth-like biomass and planetary development. We investigated planets with gravities of 1g and 3g and a surface pressure of one bar around central stars with spectral classes from M0 to M7. The spectral signals of the calculated planetary scenarios have been presented by Rauer et al. (2011). The main motivation of the present work is to perform a deeper analysis of the chemical processes in the planetary atmospheres. We apply a diagnostic tool, the Pathway Analysis Program, to shed light on the photochemical pathways that form and destroy biosignature species. Ozone is a potential biosignature for complex- life. An important result of our analysis is a shift in the ozone photochemistry from mainly Chapman production (which dominates in Earth's stratosphere) to smog-dominated ozone production for planets in the Habitable Zone of cooler (M5-M7)-class dwarf stars. This result is associated with a lower energy flux in the UVB wavelength range from the central star, hence slower planetary atmospheric photolysis of molecular oxygen, which slows the Chapman ozone production. This is important for future atmospheric characterziation missions because it provides an indication of</text>
<text><location><page_2><loc_9><loc_10><loc_91><loc_15></location>different chemical environments that can lead to very different responses of ozone, for example, cosmic rays. Nitrous oxide, a biosignature for simple bacterial life is favored for low stratospheric UV</text>
<text><location><page_3><loc_9><loc_83><loc_89><loc_92></location>conditions, that is, on planets orbiting cooler stars. Transport of this species from its surface source to the stratosphere where it is destroyed can also be a key process. Comparing 1g with 3g scenarios, our analysis suggests it is important to include the effects of interactive chemistry.</text>
<text><location><page_3><loc_9><loc_41><loc_69><loc_43></location>Key words: Exoplanets, Earth-like, M-dwarf, photochemistry, biosignatures</text>
<section_header_level_1><location><page_3><loc_9><loc_38><loc_22><loc_40></location>1. Introduction</section_header_level_1>
<text><location><page_3><loc_9><loc_10><loc_91><loc_33></location>Understanding the photochemical responses of Super-Earth (SE) atmospheres in the Habitable Zone (HZ) of M-dwarf stars is a central goal of exoplanet science, since it is feasible that such environments may present the first opportunities to search for biosignature spectral signals. Gliese 581d (Mayor et al. 2009; Udry et al., 2007) is the first SE to be found that may orbit in the HZ of its M-dwarf star. Recently, initial constraints on the composition of hot transiting SEs such as CoRoT-7b (e.g., Guenther et al., 2011) and GJ1214b (e.g., Bean et al., 2011; Croll et al., 2011) have been discussed. Kepler 22b (Borucki et al., 2012) is the first transiting object found to occur in the HZ of a solar-type</text>
<text><location><page_4><loc_9><loc_87><loc_90><loc_92></location>star; several Earth-sized objects have been found orbiting a cool M-dwarf (e.g., Muirhead et al., 2012) and detection of further SEs in the HZ is just beginning (e.g., Bonfils et al., 2013).</text>
<text><location><page_4><loc_9><loc_66><loc_91><loc_85></location>There exist a large number of possible parameters that could influence the abundances of possible biosignature species in hypothetical Earth-like atmospheres. Our motivation here is to take two parameters that are relatively well-known, namely, stellar class and planetary gravity, and perform a sensitivity study assuming an Earth-like biomass and development in order to determine their effect upon the photochemistry and climate, and hence the potential biosignatures. Other works (e.g., Segura et al., 2005; Grenfell et al., 2007) have also adopted this approach.</text>
<text><location><page_4><loc_9><loc_10><loc_91><loc_64></location>In this work, we analyzed the photochemical responses of key species from the same scenarios as the earlier work of Rauer et al. (2011) (hereafter Paper I), who analyzed spectral signals for Earthlike planets with gravities of 1g and 3g orbiting in the HZ of M-dwarf stars with classes from M0 to M7. In an earlier study, Segura et al. (2005) also discussed photochemical responses of (1g) Earth-like planets orbiting in the HZ of M-dwarf stars. They calculated enhanced abundances of methane (CH4) (by about x100) and nitrous oxide (N2O) (by about x5) compared with those of Earth related to the weaker UV emissions of M-dwarf stars. Their results also suggest a reduction in the ozone (O3) column by up to about a factor of 7 compared with that of Earth, associated with weakened UV leading to a slowing in the O3 photochemical source. This result was already broadly anticipated in the early 1990s (see Segura et al. 2005 and references therein). In the present study, we aimed to examine the nature of these photochemical responses in more depth. We applied a diagnostic tool termed the Pathway Analysis Program (PAP) written by Lehmann, (2004) to investigate the photochemical responses. PAP delivers unique information on chemical pathways of key species and has identified new chemical atmospheric pathways on Earth (Grenfell et al., 2006) and on Mars (Stock et al. 2012 a,b ) . PAP is a key tool for understanding atmospheric sources and sinks of the biosignatures and related compounds. The usual mechanisms that operate in Earth's atmosphere (e.g., O3 catalytic cycles etc.) are complex and</text>
<text><location><page_5><loc_9><loc_87><loc_91><loc_92></location>may be very different for Earth-like planets orbiting M-dwarf scenarios, which is a good motivation for applying such a tool.</text>
<text><location><page_5><loc_15><loc_83><loc_84><loc_85></location>The primary driver of the photochemistry is the Top-of-Atmosphere (TOA) stellar flux,</text>
<text><location><page_5><loc_9><loc_59><loc_91><loc_82></location>especially in the UVB and UVC regions, which weaken with decreasing effective stellar temperature. Therefore, we first analyzed the Ultra-Violet (UV) fluxes in our planetary atmospheres. Then, we focused on their influence on atmospheric ozone (O3) since this is not only an important biosignature but also a key UVB absorber governing the abundances of other chemical species. We then investigated the biomarker N2O, which is sensitive to UVB. Finally, we analyzed the photochemistry of CH4 and water (H2O) since these key greenhouse gases can influence surface habitability. We now present a brief overview of the photochemistry of the above four species.</text>
<section_header_level_1><location><page_5><loc_9><loc_52><loc_30><loc_54></location>1.1 Photochemistry of O3</section_header_level_1>
<text><location><page_5><loc_9><loc_31><loc_91><loc_50></location>O3 on Earth is a potential biosignature associated mainly with molecular oxygen (O2), which arises mostly via photosynthesis. In Earth's atmosphere, about 90% (10%) of O3 resides in the stratosphere (troposphere). Production of O3 in the Earth's stratosphere occurs mainly via the Chapman mechanism (Chapman, 1930) via O2 photolysis. Production of O3 in the troposphere occurs mostly via the smog mechanism (Haagen-Smit, 1952), which requires volatile organic compounds (VOCs), nitrogen oxides, and Ultraviolet (UV).</text>
<text><location><page_5><loc_9><loc_10><loc_91><loc_29></location>Destruction of O3 in the stratosphere proceeds mainly via catalytic cycles involving hydrogen-, nitrogen, or chlorine-oxides (e.g., Crutzen, 1970) (designated HOx, NOx, and ClOx respectively). These molecules can be stored in so-called reservoir species, the atmospheric distributions of which are reasonably well-defined for Earth (e.g., World Meteorological Organization (WMO) Report, 1995). Changes in, for example, temperature and/or UV can lead to the reservoirs releasing their HOx-NOxClOx, associated with rapid stratospheric O3 removal in sunlight. Destruction of O3 in the troposphere</text>
<text><location><page_6><loc_9><loc_87><loc_88><loc_92></location>occurs, for example, via wet and dry deposition and/or gas-phase removal via fast removal with, for example, NOx.</text>
<text><location><page_6><loc_9><loc_73><loc_91><loc_85></location>O3 can be formed abiotically in CO2 atmospheres (e.g., Segura et al., 2007). O3 layers (albeit very weak compared to that on Earth) have been documented on Mars (Fast et al., 2009) and on Venus (Montmessin et al., 2011), so caution is warranted when interpreting O3 signals as indicative of biology or not (e.g., Selsis et al. 2002).</text>
<section_header_level_1><location><page_6><loc_9><loc_66><loc_32><loc_68></location>1.2 Photochemistry of N2O</section_header_level_1>
<text><location><page_6><loc_9><loc_48><loc_91><loc_64></location>N2O is a biosignature produced almost exclusively on Earth from microbes in the soil as part of the nitrogen cycle (International Panel on Climate Change (IPCC, 2001)). Minor inorganic sources include, for example, the reaction of molecular nitrogen with electronically excited atomic oxygen: N2+O( 1 D)+M N2O+M (e.g. Estupiñan et al. 2002). Destruction of N2O occurs in the stratosphere mainly via photolysis or via removal with excited oxygen atoms.</text>
<section_header_level_1><location><page_6><loc_9><loc_41><loc_56><loc_43></location>1.3 Photochemistry of CH4 and Methyl choride (CH3Cl)</section_header_level_1>
<text><location><page_6><loc_9><loc_24><loc_90><loc_40></location>CH4 is a strong greenhouse gas affecting climate and hence habitability. It is destroyed in the troposphere up to the mid-stratosphere mainly by oxidative degradation pathways with hydroxyl (OH) and in the upper stratosphere via photolysis. CH4 is a possible indicator of life (bioindicator) but not a definite proof since this species (on Earth) has, in addition to biogenic sources, also some geological origins (IPCC, 2001).</text>
<text><location><page_6><loc_9><loc_10><loc_90><loc_22></location>CH3Cl on Earth has important biogenic sources associated with vegetation, although its sourcesink budget and net anthropogenic ccontribution is not well known (Keppler et al. 2005). Like CH4, its removal is controlled by reaction with OH, although the chlorine atom leads to increased reactivity (with an enhanced rate constant of about a factor 6 for this reaction) compared with CH4.</text>
<section_header_level_1><location><page_7><loc_9><loc_87><loc_32><loc_89></location>1.4 Photochemistry of H2O</section_header_level_1>
<text><location><page_7><loc_9><loc_69><loc_89><loc_85></location>Although not a biosignature, H2O is essential for life as we know it. Like CH4, H2O is an efficient greenhouse gas. Production of H2O in Earth's stratosphere proceeds via CH4 oxidation, whereas destruction of H2O occurs in the upper stratosphere via photolysis (World Meteorological Organisation (WMO), 1994). In the troposphere, H2O is subject to the hydrological cycle, including evaporation and condensation.</text>
<section_header_level_1><location><page_7><loc_9><loc_62><loc_26><loc_64></location>1.5 Key Questions</section_header_level_1>
<text><location><page_7><loc_9><loc_45><loc_90><loc_61></location>O3 is formed on Earth in different ways, that is, via the smog mechanism (~10% on Earth) and the Chapman mechanism (~90%). How and why these values may change for different exoplanet scenarios is not well investigated, yet this is important information for predicting and interpreting spectra. A flaring M-dwarf star, for example, will induce a photochemical response creating NOx, which destroys 'Chapman'-produced O3 but could actually enhance a 'smog' O3 signal.</text>
<text><location><page_7><loc_9><loc_10><loc_90><loc_43></location>N2O is destroyed via photolysis in the stratosphere by UVB radiation in the stratosphere, but its supply upwards from the surface is controlled by atmospheric transport and mixing. Models with fast upwards transport will ultimately lead to reduced N2O abundances since in the case of faster transport, the N2O molecules reach the altitudes of efficient destruction earlier, that is, the lifetime of N2O molecules is reduced, which (at a constant emission rate) leads to smaller N2O concentrations. To improve knowledge of potential N2O spectral signals in exoplanet environments, it is important to understand which processes (photochemistry or transport) dominate the abundance of N2O in different environments. For example, N2O on Earth is affected by both stratospheric UVB (which depends on, e.g., the solar spectra, radiative transfer, atmospheric photochemistry, etc.) as well as on troposphericto-stratospheric transport processes.</text>
<text><location><page_8><loc_9><loc_83><loc_91><loc_92></location>To begin to address such questions, we apply a new chemical diagnostic tool, the Pathway Analysis Program (PAP), which sheds unique light into the chemical pathways that control biosignature abundances.</text>
<section_header_level_1><location><page_8><loc_9><loc_76><loc_30><loc_78></location>2. Models and Scenarios</section_header_level_1>
<section_header_level_1><location><page_8><loc_9><loc_69><loc_19><loc_71></location>2.1 Models</section_header_level_1>
<text><location><page_8><loc_9><loc_10><loc_91><loc_64></location>The model details for the atmospheric coupled climate-chemistry column model and the theoretical spectral model have been described in Paper 1. Recent model updates include, for example, a new offline binning routine for calculating the input stellar spectra and a variable vertical atmospheric height in the model; more details were given by Rauer et al. (2011). The radiative-convective module is based on the work of Toon et al. (1989) for the shortwave region and RRTM (Rapid Radiative Transfer Module) for the thermal radiation. Since a main focus in this work is on photochemical effects, we will now provide a detailed description of the photochemical module. The model simulates 1D globalaverage, cloud-free conditions, although the effects of clouds were considered in a straightforward way by adjusting the surface albedo until the mean surface temperature of Earth (288 K) was attained for the Earth control run, as in earlier studies (Paper 1, Segura et al., 2003). The scheme solved the central chemical continuity equations by applying an implicit Euler solver that used the LU (Lower Upper) triangular matrix decomposition method with variable iterative stepping such that the stepsize was halved whenever the abundance of a long-lived species changed by more than 30% over a single step. The version used here employs chemical kinetic data from the Jet Propulsion Laboratory (JPL) Evaluation 14 (2003) report. The scheme includes the key inorganic gas-phase and photolytic chemical reactions commonly applied in Earth's atmosphere, that is, with hydrogen-, nitrogen, and chlorine-</text>
<text><location><page_9><loc_9><loc_83><loc_91><loc_92></location>oxide reactions and their reservoirs. The scheme was considered to be converged when the relative change in concentration for any species in any layer changes by less than 10 -4 over a chemical iteration that exceeded 10 5 s.</text>
<text><location><page_9><loc_9><loc_62><loc_91><loc_82></location>From a total of 55 chemical species, 34 were 'long-lived,' that is, the transport timescales are long compared with those of the photochemistry. Their concentrations were calculated by solving the full Jacobian matrix; 3 species, namely, CO2, N2, and O2 were set to constant isoprofile values based on modern Earth, and the remainder of the species were 'short-lived,' that is, assumed to be in steadystate, and therefore calculated from the long-lived species. The steady-state assumption simplifies the numerical solution.</text>
<text><location><page_9><loc_9><loc_45><loc_91><loc_61></location>Surface biogenic and source gas fluxes for CH4, (=531Tg/yr) N2O (=8.6 Tg N contained in N2O /yr) , CO (=1796Tg/yr) and CH3Cl (=3.4Tg/yr) were set such that for the Earth control run, Earth's modern-day concentrations were achieved at the surface - this procedure was commonly used in earlier approaches for Earth-like exoplanets (e.g., Paper 1, Segura et al., 2003). H2 at the surface was removed with a constant deposition velocity of 7.7x10 -4</text>
<text><location><page_9><loc_9><loc_10><loc_91><loc_44></location>cm s -1 . Dry and wet deposition removal fluxes for other key species were included via molecular velocities and Henry's Law coefficients respectively. Volcanic fluxes of SO2 and H2S were based on modern Earth. Tropospheric lightning sources of NO were based on the Earth lightning model of Chameides et al. (1977), assuming chemical equilibrium between N2, O2, and NO, a freeze-out temperature of 3500K and equilibrium constants taken from the Chemical Rubber Company (CRC) 1976 handbook. Modern Earth's atmosphere has ~44 lightning flashes s -1 global mean (with flashes mainly generated over land in the tropics), which produces ~5Tg N in the form of NOx globally per year (Schumann and Huntreiser, 2007). Clearly, these values depend, for example, on atmospheric transport, convective activity, and the land-sea distribution, etc. for Earth-like exoplanets, which are not well-constrained parameters. At the model upper boundary, a constant, downwards (effusion) flux of</text>
<text><location><page_10><loc_9><loc_87><loc_88><loc_92></location>CO and O is set, which represents the photolysis products of CO2 that are formed above the model's upper lid.</text>
<text><location><page_10><loc_9><loc_76><loc_90><loc_85></location>Atmospheric mixing between the 64 vertical chemical layers was calculated via eddy diffusion constants (K in cm 2 s -1 ), where log(K) varied from ~5.0 at the surface, decreased to a minimum value of ~3.6 at ~16km, and then increased to ~5.7 at the model upper boundary.</text>
<text><location><page_10><loc_9><loc_38><loc_91><loc_75></location>Photolysis rates included the major absorbers, including important (E)UV absorbers such as O2, CO2, H2O, O3, NO, CH4, and SO2. The O2 photolysis absorbtion coefficients were calculated with the mean exponential sums method. The O3 coefficients included the Hartley-Huggins T-dependence based on data measured at 203K and 273K (and linearly interpolated between). Species that photolyze in the UVB that are relevant for O3 destruction were also included, for example, nitric acid (HNO3) photolysis was included - this is important for NOx release. Finally, weakly bound species that photolyze in the UVA/visible region, for example, NO3, N2O5 were included. Photolysis rates were calculated based on insolation fluxes from the delta two-stream module (Toon et al. 1989). One hundred eight wavelength intervals were included from (175.4-855) nm in the UV and visible, nine intervals in the EUV from (130-175) nm, and one Lyman-alpha interval at 121.6 nm. Rayleigh scattering for N2, O2, and CO2 was included.</text>
<text><location><page_10><loc_9><loc_10><loc_91><loc_36></location>The Pathway Analysis Program (PAP) was developed by Lehmann (2004) and applied by Grenfell et al. (2006) to Earth's stratosphere and by Stock et al. (2012 a,b ) to the martian atmosphere. In the present work, it is applied to Super-Earth planetary atmospheres. The PAP algorithm identifies and quantifies chemical pathways in chemical systems. Starting with individual reactions as pathways, PAP constructs longer pathways step-by-step. To achieve this, short pathways already found are connected at so-called 'branching point' species, whereby each pathway that forms a particular species is connected with each pathway that destroys it. Branching point species are chosen based on increasing lifetime with respect to the pathways constructed so far. In this work, all species with a chemical</text>
<text><location><page_11><loc_9><loc_48><loc_91><loc_92></location>lifetime shorter than the chemical lifetime of the species being studied (i.e., the biosignatures O3, N2O, and the greenhouse gas CH4) are treated as branching point species. Since in general the chemical lifetime of all species varies with altitude, the choice of branching point species adapts to the local chemical and physical conditions. A detailed description of the PAP algorithm is given by Lehmann (2004). To avoid a prohibitively long computational time, pathways with a rate smaller than a userdefined threshold (in the present study, fmin=10 -8 parts per billion by volume per second (ppbv/s)) are deleted. The chosen f_min = 10 -8 ppbv/s is sufficient for finding the 5 dominant pathways (e.g., of N2O, CH4 loss) as shown in the main table (Appendix 1). Stock et al. (2012 a ) discussed the effect of varying this parameter. PAP calculates the chemical pathways by taking as input (i) a list of chemical species, (ii) chemical reactions, (iii) time-averaged concentrations and reaction rates, and (iv) concentration changes arising only from the gas-phase chemical reactions only (i.e., not including changes in abundance from, e.g., mixing, deposition, etc). PAP calculates as output the identified chemical pathways with their associated rates. Information from PAP is used to interpret chemical responses.</text>
<section_header_level_1><location><page_11><loc_9><loc_41><loc_21><loc_43></location>2.2 Scenarios</section_header_level_1>
<text><location><page_11><loc_9><loc_10><loc_91><loc_36></location>Here, we analyze the model scenarios described in Paper I. We considered planets with masses corresponding to 1g and 3g with Earth-like (i.e., N2-O2) atmospheres with Earth's source gas emissions and initial p,T, and abundance profiles as for modern Earth. There are currently no observational constraints for the surface pressure of SE planets. On the one hand, theoretical studies, for example, that of Elkins-Tanton and Seager (2008), have suggested a wide-range of possible atmospheric masses resulting from outgassing on SE planets, whereas on the other hand, for example, Stamenković et al. (2012), who included a pressure-depenence of viscosity in the mantle, suggested rather weak SE outgassing rates. Given the current uncertainties, we therefore assume 1 bar surface pressure to be</text>
<text><location><page_12><loc_9><loc_52><loc_91><loc_92></location>comparable with Paper 1 and earlier studies and to compare with our 1g scenarios. Our modeled p, T, and chemical output profiles are calculated self-consistently for planets around different central Mdwarf stars in the HZ (with the Sun-Earth case for comparison). We explore an extensive parameter range, considering planets orbiting M-dwarf stellar classes from M0 to M7. This is neccessary because atmospheric chemistry-climate coupling is strongly non-linear and, hence, general results from one set of stellar classes (e.g., M0 to M4) cannot be simply extrapolated to other stellar classes (e.g., M5-M7) instead each scenario has to be calculated separately. Mixing ratios for radiative species are fed back into the climate module, which calculates a new T, p profile, and this is again fed back into the chemistry module. This iterative process continues until T, p, and concentrations all converge. The planets are placed at an orbital distance from their star such that the total energy input at the TOA equals the modern Solar constant of 1366 Wm -2 (see Paper 1 for the stellar input spectra used). In total, the following eleven scenarios were investigated:</text>
<table>
<location><page_12><loc_15><loc_10><loc_84><loc_46></location>
</table>
<unordered_list>
<list_item><location><page_13><loc_9><loc_83><loc_88><loc_89></location>* Segura et al. (2005) and Rauer et al. (2011) adopted a spectral class of 4.5 based on the SIMBAD database, whereas Hawley and Pettersen (1991) used a value of 3.5.</list_item>
</unordered_list>
<section_header_level_1><location><page_13><loc_9><loc_76><loc_41><loc_78></location>2.3 Planetary Radiation Environment</section_header_level_1>
<text><location><page_13><loc_9><loc_38><loc_91><loc_71></location>Incoming Stellar fluxes (F*) - These are the primary driver of planetary atmospheric photochemistry, especially in the UVB and UVC range, and are also central to habitability for life as we know it on Earth. A significant proportion of cooler M-dwarfs like those considered in our work may be active emitters of UV from their chromospheres or/and transition regions (see e.g., Walkowicz et al., 2008, France et al., 2012 in press). This could have a considerable impact upon the planetary photochemistry, climate, and associated biosignatures. How efficiently the UV is absorbed throughout the atmospheric column is closely linked with the photochemical responses and, hence, determines the final abundances of the biosignature. We therefore start our analysis by investigating the planetary radiation environment. We discuss UV radiation at the TOA and at the planetary surface, and present a validation of surface UV based on Earth observations</text>
<text><location><page_13><loc_9><loc_20><loc_91><loc_33></location>Planetary TOA Radiation Analysis - We analyzed the planetary TOA F* in the UVA, UVB, and UVC wavelength range for the different stellar scenarios in the top model layer. UVA corresponds to the model wavelength intervals from ( 315-400) nm; UVB corresponds to (280-315) nm; UVC corresponds to ( 175.4-280) nm.</text>
<text><location><page_13><loc_9><loc_10><loc_91><loc_19></location>To be comparable with Paper 1, we approximated the TOA stellar spectra for the M0 to M7 Mdwarf stars as Planck functions (other than for the Sun, which is for solar mean conditions based on the work of Gueymard et al. (2004), and for AD-Leo, for which measured UV-spectra are available, see</text>
<text><location><page_14><loc_9><loc_48><loc_91><loc_92></location>Paper I). The approach used in Paper 1 and, therefore, in this study as well was to employ Planck curve spectra that correspond to quiet M-dwarf stars with little emitted UV fluxes. Recent results (Reiners et al., 2012) suggest that >90% of hotter (M0 to M2) M-dwarf stars sampled are quiet , whereas >50% of the cooler stars (M4 and cooler) are active . Clearly, we are well-aware that smooth Planck functions do not include, for example, enhanced Lyman-alpha and UVC features, etc. characteristic of cool M-dwarf stars that may have very active chromospheric and coronal regions. However, direct observations of stellar spectra for the cooler M-dwarf stars (M5-M7) in the critical wavelength range ( λ <UVA) in our photolysis scheme are presently not available, and hence we prefer to adopt such a Planck -spectrum approach. Future work will study the effect of varying (E)UV characteristic emissions in the input spectra. Further, by comparing results from scenarios in which Planck curve spectra are used with those for active stars, we can isolate the photochemical effects in the planetary atmosphere of varying stellar activity. Firstly, to get an overview, Table 1 compares ratios of UV emission for our considered Mdwarf scenarios with the Sun.</text>
<table>
<location><page_14><loc_17><loc_15><loc_83><loc_26></location>
<caption>Table 1: Ratios of UV radiation for our M-dwarf-star (M7) scenario compared with the Sun (upper row) and for ADL.Table 1 (row 1) suggests that our cool (M7) M-dwarf would emit less than 1% of the UVA,</caption>
</table>
<text><location><page_15><loc_9><loc_83><loc_90><loc_92></location>UVB, and UVC radiation compared with the Sun. Comparing row 2) the active AD Leo M-dwarf star with the Sun suggests that UVA, UVB, and UVC for the flaring star amount to only (1-7)% of the total Solar radiation</text>
<text><location><page_15><loc_9><loc_66><loc_91><loc_82></location>Figures (1a-1c) show the TOA UVA, UVB, and UVC net flux (W m -2 ). Figure 1 shows an increase with increasing stellar effective temperature as expected. The active AD-Leo flaring case is an especially strong emitter of UV due to its extremely active chromosphere. Modeled TOA UVB flux for Earth (~18.3 Wm -2 ) compare reasonably well with available observations (e.g. 16 ± 3 Wm -2 ; Benestad, 2006).</text>
<text><location><page_15><loc_9><loc_41><loc_91><loc_61></location>Planetary Surface Radiation - In the chemistry module, the UVA and UVB net fluxes required for the photolysis scheme are calculated from the top layer downward via the twostream module with Rayleigh scattering. Figures (2a-2b) show UVA and UVB net flux (Wm -2 ) at the planetary surface as calculated in the chemistry module of this work. UVC is essentially zero at the surface so is not shown in Figure 2, and similarly for Figure 3. Generally, Figures 2a and 2b show an increase in planetary surface UV radiation with higher stellar temperatures, as for the TOA cases shown in Figure 1.</text>
<text><location><page_15><loc_9><loc_13><loc_91><loc_36></location>Comparison with Earth Surface UV Radiation - Global satellite observations from 1992-1994 (Wang et al. 2000, their Figure 6b) suggest observed UVB surface radiation for Earth of ~1.4Wm -2 for cloud-free conditions. By comparison, Figure 2b suggests that our model over-estimates this value, calculating 2.3 Wm -2 UVB for the Earth control run. Uncertainties include, for example, our straightforward treatment of clouds whereby we adjust the surface albedo (see above) as well as the challenge of representing, for example, time-dependent and, for example, latitude-varying O3 photochemistry and UV absorption in a global-averaged 1D model.</text>
<text><location><page_16><loc_9><loc_80><loc_91><loc_92></location>Ratio of Surface to TOA UV Flux - This ratio (R) is shown for the 1g and 3g cases in Figures 3a and 3b for UVA and UVB respectively. R is an inverse measure of the UV shielding of an atmosphere. Figure 3a suggests that UVA passes efficiently through the atmospheres considered, as expected, since most values of Rnet,UVA are >0.7. The UVA ratio is not greatly dependent on the stellar temperature.</text>
<text><location><page_16><loc_9><loc_62><loc_91><loc_78></location>Figure 3b shows as expected a much stronger atmospheric extinction of UVB than for the UVA wavelengths, and there is now a clear dependency on stellar temperature. Weaker overhead O3 columns in the cool M-dwarf cases lead to a strong rise in the ratio in Figure 3b. For the 3g scenarios (circles), a lowering in the atmospheric column by a factor of three resulted in less UV shielding and a rise in the surface UV.</text>
<section_header_level_1><location><page_16><loc_9><loc_55><loc_27><loc_57></location>3. Chemical Analysis</section_header_level_1>
<text><location><page_16><loc_9><loc_41><loc_91><loc_50></location>Here, we first compare briefly previous results (Segura et al., 2005) reported in the literature. Then, we discuss the general trends in column abundances of the biosignatures and related key species. Finally, we discuss the chemical responses for the vertical profiles that were also shown in Paper I.</text>
<section_header_level_1><location><page_16><loc_9><loc_34><loc_40><loc_36></location>3.1 Column Biomarkers (1g planets)</section_header_level_1>
<text><location><page_16><loc_9><loc_17><loc_91><loc_29></location>Column O3 in Figure 4a (blue diamonds) mostly decreased with increasing star class (i.e., decreasing Teff of the star) related to less UVB, therefore there was a slowing in the photolysis of molecular O2 and hence a slowing in the Chapman cycle, a major source of O3. The O3 profile responses are discussed in more detail in section 3.6. The column values are shown in appendix 1.</text>
<text><location><page_16><loc_9><loc_10><loc_91><loc_12></location>Column N2O in Figure 4a (red squares) generally increased with increasing star class. The cooler stars</text>
<text><location><page_17><loc_9><loc_87><loc_91><loc_92></location>emit less UVB, which suggests a slowing in the photolytic loss of N2O in the planetary atmosphere and hence an increase in its abundance.</text>
<text><location><page_17><loc_9><loc_69><loc_91><loc_82></location>Column CH3Cl in Figure 4a (green triangles) generally increased with increasing star class due to less OH, its major sink (see OH analysis, Table 2). The response is comparable to CH4 (discussed in next section), which has a similar photochemistry. Spectral features of CH3Cl, however, were too weak to be evident in the calculations of Paper 1 despite the enhanced column amounts for the cooler stars.</text>
<section_header_level_1><location><page_17><loc_9><loc_62><loc_41><loc_64></location>3.2 Column Biosignatures (3g planets)</section_header_level_1>
<text><location><page_17><loc_9><loc_41><loc_90><loc_57></location>For the 3g planets, we assumed a constant surface pressure of 1 bar, which led to the total atmospheric column being reduced by a factor of three, as already mentioned (Figure 4b). The general trends for O3 and N2O remain for the 3g scenarios, that is, mostly similar to the corresponding 1g scenarios already discussed, although the reduced total column resulted in a cooling of the lower atmosphere due to a weaker greenhouse effect, as we will show (see Paper I also).</text>
<text><location><page_17><loc_9><loc_20><loc_89><loc_40></location>The N2O 3g response is linked with enhanced UVB penetrating the reduced atmospheric column compared with 1g, which leads to more photolytic loss of N2O. A transport effect also took place. For the 3g case (with its lower model lid due to less atmospheric mass and higher gravity), the upward tropospheric diffusion of N2O was faster, for example, by about 50% in the mid to upper troposphere than the 1g case. This meant that N2O for the 3g case could reach the stratosphere faster, where it would be rapidly photolyzed.</text>
<section_header_level_1><location><page_17><loc_9><loc_13><loc_46><loc_15></location>3.3 Column Greenhouse Gases (1g planets)</section_header_level_1>
<text><location><page_18><loc_9><loc_83><loc_91><loc_92></location>In this section, we discuss the planetary atmospheric column abundances of CH4 and H2O since they have a major impact on temperature via the greenhouse effect. Vertical profiles will be discussed later and can also be found in Paper 1.</text>
<text><location><page_18><loc_9><loc_69><loc_90><loc_78></location>CH4 Column Response - Since the only source of CH4 in the model is fixed biomass surface emission, the CH4 response for the various runs is controlled by the main atmospheric CH4 sink, that is, removal via the hydroxyl (OH) radical. OH is affected by three main processes:</text>
<text><location><page_18><loc_9><loc_59><loc_91><loc_64></location>OH Source(s) : for example, H2O+O( 1 D) 2OH (where O( 1 D) comes mainly from O3 photolysis in the UV).</text>
<text><location><page_18><loc_9><loc_52><loc_90><loc_57></location>OH Recyling reactions in which NOx species can interconvert HOx (defined here as OH+HO2) family members via, for example, NO+HO2 NO2+OH.</text>
<text><location><page_18><loc_9><loc_48><loc_87><loc_50></location>OH Sinks, for example, reaction with CH4 and CO (see e.g. Grenfell et al., 1999 for an overview).</text>
<text><location><page_18><loc_9><loc_13><loc_90><loc_43></location>Figure 4c suggests a strong CH4 (green diamonds) increase with decreasing effective stellar temperature. Cooler stars are weak UV emitters, which favors a slowing in the OH source reaction above. Note also that greenhouse warming by the enhanced CH4 favors a damp troposphere (more evaporation) and, hence, all else being equal would favor actually more OH (via more H2O, see source OH reaction above). This is an opposing process which our results suggest is not the dominant effect. So, for a given model, calculating accurately the net effect will depend, for example, on a good treatment of, for example, the hydrological cycle, which is challenging for a global column model. To aid in understanding the CH4 response, which is controlled by OH, Table 2 summarizes the OH sources, sinks, recycling budget, and associated quantities.</text>
<table>
<location><page_19><loc_8><loc_25><loc_92><loc_78></location>
<caption>Table 2: Modeled (lowest atmospheric layer) and observed (surface) global-mean key species abundances (molecules cm -3 ) and reaction rates (molecules cm -3 s -1 ) affecting CH4 (and H2O) for various 1g scenarios. *From Lelieveld et al. (2002).</caption>
</table>
<text><location><page_19><loc_9><loc_13><loc_88><loc_22></location>OH Abundances - Control run (1g Sun) OH abundances in Table 2 are within ~20% of global-mean observed OH proxies for Earth. Table 2 suggests a strong decrease in OH from left to right (i.e., for decreasing stellar effective temperature) especially for the M7 case.</text>
<text><location><page_20><loc_9><loc_66><loc_91><loc_92></location>OH Source Reaction Rates - The source reaction rate (Sun) in Table 2, that is, O( 1 D)+H2O 2OH, is about 12 times weaker than indicated by the Whalley et al. (2010) study, which investigates (Earth) clean-air, tropical northern-hemisphere daytime OH. The factor 12 difference reflects a lowering due to day-night averaging in our global mean model (which accounts for ~factor 2 of the difference in OH) and the fact that the Whalley study considered tropical conditions. Concentrations of the trace specie O( 1 D) in the control run (=6x10 -8 ppbv at 30km) compared reasonably well with Earth observations (~3x10 -8 ppbv, Brasseur and Solomon, 2005). Table 2 suggests that the source reaction rate decreases from left to right, which is consistent with the decrease in OH.</text>
<text><location><page_20><loc_9><loc_34><loc_91><loc_61></location>OH Recycling Reaction Rates - Our (Sun case) recycling reaction was comparable with that of the Whalley et al. (2010) study to within about 50%. Earlier (Earth) modeling studies, for example, that of Savage et al. (2001), suggest that the OH recycling reaction dominates the source reaction even in quite clean air-masses (NOx ~250pptv and below), which is somewhat in contrast to this and the Whalley study. In Table 2, the recycling reaction rates (like the source reaction) also decreases from left to right, which favors the decrease in OH, although the change in the source reaction is the stronger effect. For cooler stars, the recycling reaction becomes increasingly important compared with the source reaction, and it dominates for the ADL and M7 cases.</text>
<text><location><page_20><loc_9><loc_10><loc_90><loc_29></location>HOx and NOx Ratios - These ratios are sensitive markers of changes in HOx and NOx chemistry and hence affect, for example, O3 cycles and CH4. The ratios (HO2/OH) and (NO2/NO) in Table 2 increase strongly for the cooler stars. These ratios are strongly affected by the concentration of O3, whose production via the Chapman mechanism (discussed in 3.5) weakens for the cooler stars. The ratios for the cooler stars are far from their 'Earth' values, so the interactions between HOx and NOx are much perturbed. This is a hint that the usual mechanisms that operate on Earth (e.g., O3 catalytic cycles etc.)</text>
<text><location><page_21><loc_9><loc_87><loc_87><loc_92></location>may be very different for the cooler star scenarios - a good motivation for applying PAP as already mentioned.</text>
<text><location><page_21><loc_9><loc_66><loc_89><loc_82></location>Atmospheric response for AD Leo - Although the 1g ADL scenario featured lower OH (Table 2) than for M5, ADL featured lower CH4 (Paper I) than M5. The upper layers (>60km) of the 1g ADL run showed very rapid destruction of CH4 via OH - about five times faster than for M5. This was consistent with the high Lymanα output of ADL leading to faster HOx enhancement via H2O photolysis.</text>
<text><location><page_21><loc_9><loc_34><loc_90><loc_61></location>Water Column Response - Figure 4c suggests that the increased CH4 columns (green diamonds), with decreasing stellar effective temperature generally (except for M7), lead to higher H2O columns (green squares). Generally, for the cooler star scenarios, (up to and including M5), more CH4 greenhouse heating leads to more water evaporation in the troposphere, and in the stratosphere, faster CH4 oxidation leads to faster H2O production. However, for the M7 case (Figure 4c), although CH4 increased, surface temperature did not, which suggests a saturation in the CH4 greenhouse from M5 to M7, where the lower atmosphere becomes optically thick at very high CH4 abundances. Surface cooling from M5 to M7 is also seen in the temperature profiles in Paper 1 (their Figure 3).</text>
<section_header_level_1><location><page_21><loc_9><loc_27><loc_46><loc_29></location>3.4 Column Greenhouse Gases (3g planets)</section_header_level_1>
<text><location><page_21><loc_9><loc_10><loc_91><loc_22></location>CH4 and H2O - Figure 4d has a similar format to Figure 4c but instead shows results for the 3g (instead of 1g in 4c)scenarios. The basic response to decreasing the effective stellar temperature at 3g is similar to the 1g case, that is, results suggest a column rise in CH4 and in H2O but with a drop-off in the latter for the cooler stars. To gain more insight into the effect of changing gravity, upon CH4, Table 3</text>
<text><location><page_22><loc_9><loc_90><loc_85><loc_92></location>shows the ratio (1g/3g) of the CH4 column and for the near-surface atmospheric OH abundance:</text>
<table>
<location><page_22><loc_8><loc_64><loc_92><loc_75></location>
<caption>Table 3: Ratio (1g/3g) for the CH4 atmospheric column and for near-surface OH (midpoint of lowermost gridbox) for the Sun compared with M-dwarf star scenarios.</caption>
</table>
<text><location><page_22><loc_9><loc_17><loc_91><loc_61></location>Without calculating interactive photochemistry, a passive tracer would undergo a column reduction by a factor of three from 1g to 3g, because at constant surface pressure, increasing gravity by a factor of three leads to column collapse and a reduction in the overhead column by the same factor as the increase in gravity. In Table 3, therefore, a hypothetical, passive tracer (with no chemistry) would have a value of exactly three. The actual (with chemistry) CH4 column ratios (row 1), however, are all less than three. The reduction is consistent with faster chemical loss at 1g than at 3g. To investigate this further, OH ratios are shown in Table 3 (row 2). They mostly (except ADL) increase for the cooler stars, suggesting a lowering in the 3g OH abundances compared with the corresponding 1g cases for the cooler stars. This is consistent with faster chemical loss at 1g. The reduction in OH for the 3g scenarios implies that, for example, the increase in UVB due to weaker shielding of some 3g atmospheres ( favouring OH production) is out-weighed by the (opposing) feedback where reduced greenhouse warming at 3g led to a drier troposphere (disfavoring OH which is produced via O( 1 D)+H2O 2OH).</text>
<text><location><page_22><loc_9><loc_10><loc_89><loc_15></location>This is confirmed by the water column (open circles in Figure 4d), which suggests that the 3g compared with 1g (Figure 4c) scenarios led to a weakening in the greenhouse effect and hence</text>
<text><location><page_23><loc_9><loc_80><loc_91><loc_92></location>tropospheric cooling (as seen in Figure 2 of Paper 1) and a general lowering in the H2O column (due to more condensation) by around a factor of ten (Figure 4d) compared with the 1g case (Figure 4c). In general, however, note that responses in chemical abundances do not scale directly with the column reduction at 3g compared with 1g since the effects of, for example, photochemistry are important.</text>
<text><location><page_23><loc_9><loc_48><loc_90><loc_78></location>Figures (4e, 4f) show the ratios (1g column/3g column) for biosignature and greenhouse gases respectively. The main point is that the values can lie far from a value of three (which would be expected for a passive tracer). This shows that it is important to include the effects of interactive chemistry. For the biosignature O3 there is some indication of an increase in the ratio shown in Figure 4e for the cooler stars, which will be the subject of future study. For CH3Cl (Figure 4e) and CH4 (Figure 4f) (which both have similar OH removal chemistry), the trend is downward for the cooler stars. The H2O (Figure 4f) scenarios are relatively more damp (with values >3) than for a purely passive tracer. This suggests more efficient production of H2O from CH4 for the cooler stars at 3g than at 1g, for example, due to more UV in the thinner, 3g atmospheres.</text>
<section_header_level_1><location><page_23><loc_9><loc_41><loc_64><loc_43></location>3.5 Column-Integrated Pathway Analysis Program (PAP) Results</section_header_level_1>
<text><location><page_23><loc_9><loc_17><loc_90><loc_36></location>Figure 5 shows output of O3 cycles from the PAP. The cycles (divided into production and loss cycles) found have been quantified according to the rate of O3 production or loss through each particular cycle expressed as a percentage of the total rate of production or loss found by PAP (see also description of Appendix 1 below). Values are integrated over the model vertical domain. PAP analyses were performed for each of the 64 vertical column model chemistry levels, and the column-integrated values are shown in Figure 5. The full cycles referred to in Figure 5 can be found in the Appendix.</text>
<text><location><page_23><loc_9><loc_10><loc_87><loc_12></location>Sun PAP Analysis Figure 5 confirms the expected result for O3 production, that is, the Chapman</text>
<text><location><page_24><loc_9><loc_73><loc_91><loc_92></location>mechanism dominates over the smog mechanism. For O3 destruction, the column model suggests strong NOx contributions in the lower stratosphere, although an Earth GCM study (Grenfell et al., 2006) suggests a strong HOx contribution there. This result could reflect the challenge of 1D models of capturing 3D variations in photochemistry. Also, the column model does not include industrial emissions unlike the Earth 3D model. The result should be explored in future comparisons between the column model and 3D runs.</text>
<text><location><page_24><loc_9><loc_52><loc_91><loc_68></location>Column-Integrated O3 (1g) Production Figure 5a suggests a change from a mainly Chapman-based O3 production for the 1g Sun and the warmer 1g M-dwarf stars, switching to a slower, mainly smogbased O3 production for the cooler stars (1g M5 and 1g M7). This was related to the decrease in UVB for the cooler star scenarios, since UVB is required to initiate the Chapman mechanism via photolysis of O2.</text>
<text><location><page_24><loc_9><loc_34><loc_91><loc_47></location>Column integrated O3 (1g) Destruction - Figure 5a also suggests that the classical NOx and HOx cycles (see also Figures 6 and 7) that operate mainly in the stratosphere were the most dominant O3 loss pathways for the Sun and warmer M-dwarf scenarios. For the cooler stars scenarios, the enhanced CO concentrations led to a CO-oxidation cycle gaining in importance.</text>
<text><location><page_24><loc_9><loc_17><loc_91><loc_29></location>Column O3 (3g) - Behavior at 3g (Figure 5b) was broadly similar to 1g, except at 3g both Chapman and smog were important O3 producers for the M5 case (i.e., not just smog as in the 1g case). Weaker atmospheric UVB absorption led to more penetration of UVB and hence an increased role for Chapman in the layers below.</text>
<text><location><page_24><loc_9><loc_10><loc_88><loc_12></location>Column-Integrated Results Table for O3, N2O and CH4 - Appendix (1a-1c) shows the integrated</text>
<text><location><page_25><loc_9><loc_69><loc_90><loc_92></location>column mean PAP output for O3, N2O, and CH4 respectively. Shown are (i) the column integrated rates (CIR) (in molecules cm -2 s -1 ) for all pathways found by PAP ('Found_PAP'), (ii) the CIR for only the pathways shown in the Appendix ('Shown_PAP') (shown are either the 5 dominant pathways or the first pathways that together account for >90% of the total formation or loss of found_PAP, whichever condition is fulfilled first), and (iii) the CIR as calculated in the chemistry scheme of the atmospheric column model ('total_chem'). Percent values for a particular cycle show its individual rate as a percentage of Found_PAP.</text>
<text><location><page_25><loc_9><loc_38><loc_91><loc_68></location>Comparing these three CIR values, it can be seen that for the O3 production, which is relatively straightforward, the pathways found by PAP can account very well for the rate calculated in the column model chemistry module. For the O3 loss pathways, which are rather more complex than the production, PAP can still account for generally more than ~90% of the rate from the chemistry module. For the sometimes very complex CH4 pathways, with the value of fmin chosen for this study, PAP can account for only up to about 50% of the rate from the chemistry module. Further tests suggested that decreasing the PAP input parameter fmin (the minimum considered flux, currently set to 10 -8 ppbv s -1 for all runs) leads to improvement, but the resulting complex CH4 cycles are beyond the scope of this paper (see also 3.6.3). We now discuss the individual cycles for each scenario.</text>
<section_header_level_1><location><page_25><loc_9><loc_31><loc_37><loc_33></location>O3 Column-Integrated Pathways</section_header_level_1>
<text><location><page_25><loc_9><loc_10><loc_91><loc_26></location>Chemical pathways for the 1g Sun scenario in Appendix 1a mostly compare well with established results for Earth as discussed above. Appendix 1a suggests that for the 1g M0 scenario - due to less stellar UVB emission compared with the Sun - the Chapman mechanism for producing O3 is somewhat suppressed (89.2%) and a new CO sink ('CO oxidation 1', 7.4%) appears, since CO is abundant. For the 3g M0 scenario, results suggest that Chapman features more strongly (96.7%) in the</text>
<text><location><page_26><loc_9><loc_27><loc_91><loc_92></location>thinner 3g atmosphere compared with the corresponding 1g case. HOx and NOx remain important chemical sinks for both the 3g and 1g cases. The active star (1g ADL) features a stronger Chapman contribution (97.2%) compared with 1g M0 since ADL is especially active in the UV, which is important for Chapman-initiation (via molecular oxygen) with only modest changes for the 3g ADL case. For cooler non-active stars (1g M5), large changes are apparent compared with the warmer star cases. Less UVB emission from the cool M5 star leads to a switch to smog-type O3 production ('smog 1', 57.8%). As discussed, the atmosphere is abundant in CH4 and CO. Thus, the 'CO-oxidation-1' cycle is an important O3 loss pathway (36.8%). For the (3g M5) case, the thinner total column at 3g compared with 1g leads to a rise in UV, which is consistent with more Chapman O3 (47.8%) production than the 1g case (7.5%). For O3 loss, a complex CH4 oxidation pathway involving CH3OOH becomes important (46.8%), which is not evident at 1g. The changed UV environment leads to a modest rise in HOx in the upper troposphere at 3g. Finally, for the coolest M-dwarf case (1g M7), O3 production occurs via numerous types of smog mechanisms involving the oxidation of different VOCs, for example, CO, HCHO, and CH3OOH. CO smog cycles become a key means of producing O3 especially for the cooler stars. Like CH4, an important sink for CO is the reaction with OH. As discussed, weakening UV emissions for the cooler stars leads to less OH and therefore an enhanced abundance of CO. Near the surface, CO mixing ratios correspond to: 0.09 (Sun), 9.0 (M4), 64 (ADL), and 426 (M7) parts per million (ppm). O3 loss also involves NOx cycles but also a smog mechanism ('smog 7') where O3 is the net oxidant, which is consumed to oxidize CH4 and a CO oxidation cycle.</text>
<text><location><page_26><loc_9><loc_10><loc_90><loc_26></location>Smog cycles have larger rates for the M5 and M7 scenarios than for the Sun and M0 scenarios. This is because the important smog 1 cycle (producing O3) is in competition with the CO-oxidation 1 pathway (destroying O3). At high O3 concentrations (for the Sun and M0 scenarios), (i) the reaction NO+O3 NO2+O2 shifts the NOx family to favor NO2. The reduction in NO leads to a slowing in the key reaction NO + HO2 NO2+OH and hence slows the smog 1 cycle. Also at high O3 concentrations,</text>
<text><location><page_27><loc_9><loc_76><loc_90><loc_92></location>(ii) the reaction HO2+O3 OH + 2O2 favors the CO-oxidation 1 pathway. These two effects together, favor large smog rates for the M5 and M7 scenarios. In summary, total vertically integrated O3 production and loss rates for the 1g Sun (=1.9x10 13 molecules cm -2 s -1 ) are 68 times larger than for the 1g M7 case (=2.8x10 11 molecules cm -2 s -1 ), which illustrates the change in the dominance from the rather fast Chapman chemistry to the slower smog mechanism.</text>
<section_header_level_1><location><page_27><loc_9><loc_69><loc_39><loc_71></location>N2O Column-Integrated Pathways</section_header_level_1>
<text><location><page_27><loc_9><loc_24><loc_91><loc_64></location>The main result of the PAP is that loss pathways from the N2O 'viewpoint' are non-catalytic for all scenarios. In other words, loss occurs mainly directly via photolysis, which can be calculated from the photolysis rate without performing a PAP analysis for N2O. We therefore only show (Appendix 1b) one scenario as an illustration, that is, the Sun scenario, which confirms results measured for Earth, that is, ~95% loss via photolysis (i.e., the sum of the 4 cycles involving N2O photolysis in Appendix 1b), and ~5% loss via catalytic reaction with O( 1 D) is similar to observed values quoted for Earth (e.g., 9095% photolytic loss, 5-10% via reaction with O( 1 D), IPCC Third Assessment Report, see discussion to Table 4.4). The PAP finds no formation pathways of N2O via inorganic reactions, as expected since these are insignificant compared with surface biogenic input. For the M-dwarf scenarios, photolysis similarly remained the main removal mechanism, and the overall column integrated rate of removal decreased by about a factor of two for the M7 compared with the Sun case since the cooler stars emit less UV.</text>
<section_header_level_1><location><page_27><loc_9><loc_17><loc_39><loc_19></location>CH4 Column-Integrated Pathways</section_header_level_1>
<text><location><page_27><loc_15><loc_10><loc_85><loc_12></location>Appendix 1c shows the PAP output for CH4. Results suggest a large number of complex</text>
<text><location><page_28><loc_9><loc_69><loc_91><loc_92></location>removal pathways that oxidize CH4. PAP found no in-situ production pathways, since there are no inorganic reactions in our model that produce CH4 in the atmosphere. The net removal can involve either complete oxidation of CH4 to its stable combustion products: H2O and CO2 (as in the 'oxidation 2O2-a' pathway for the 1g Sun scenario) but can also involve only partial oxidation, for example, to intermediate organic species such as formaldehyde (HCHO), for example, as in the 'Oxidation O2' pathway (1g M0). Clearly, more complete oxidation is favored in oxidizing environments, for example, damp atmospheres with strong UV where OH is abundant.</text>
<text><location><page_28><loc_9><loc_41><loc_91><loc_68></location>The choice of oxidant in the net reaction will depend on the central star's particular UVB radiation output and its ability to release, for example, HOx, Ox, or NOx from their reservoirs in the planetary atmosphere. Importantly for O3 photochemistry, there are CH4 cycles in which O3 itself is the oxidant in the net reaction (see e.g. net reaction for several cycles from the 3g Sun case). This is an example where CH4-oxidation does not lead to the more familiar O3 (smog) production, but to the reverse effect where O3 is consumed. Many of the CH4 pathways are NOx-catalyzed, as on Earth, although this is not the case for all scenarios (e.g., pathway 'CH3OOH-d' (3g M5) does not include NOx).</text>
<section_header_level_1><location><page_28><loc_9><loc_34><loc_40><loc_36></location>3.6 Altitude-Dependent PAP Results</section_header_level_1>
<text><location><page_28><loc_9><loc_24><loc_88><loc_29></location>In this section we will present PAP results from the same scenarios as the previous section. However, here we will discuss the contribution of the PAP cycles as profiles varying in the vertical.</text>
<section_header_level_1><location><page_28><loc_9><loc_17><loc_65><loc_19></location>3.6.1 Vertical Changes in Ozone (O3) Production and Loss Cycles</section_header_level_1>
<text><location><page_28><loc_15><loc_10><loc_91><loc_12></location>Figure 6 shows the altitude-dependent PAP results, comparing production and loss pathways for</text>
<text><location><page_29><loc_9><loc_62><loc_90><loc_92></location>the Earth case (Figures 6a, 6b) with the M7 case (Figures 6c, 6d). Similarly, Figures 7a, 7b compare ADL (1g) with M5 (1g) (Figures 7c, d). In Figures 6 and 7, the logarithmic x-axis shows the rate of change of O3 associated with a particular cycle found by PAP, in molecules cm -3 s -1 . The black and white text labels on these Figures indicate the names of the O3 pathways, which can be found in Appendix 1a. Note that the logarithmic x-axis where results are plotted cumulatively (meaning to estimate the contribution of a pathway at a particular height one must subtract its left-hand side x-axis boundary from its right-hand side x-axis boundary) in Figures 6 and 7 means that the pathways shown on the right-hand side of the Figure can make up a strong overall contribution to the net rate of change despite having only a thin section (relatively small area).</text>
<text><location><page_29><loc_9><loc_10><loc_91><loc_61></location>For the Earth results (Figures 6a, 6b), the O3 production and loss rates output by PAP compare well with middle atmosphere O3 budgets derived for Earth, see for example the work of Jucks et al. (1996), their Figure 4. The Earth results (Figure 6a) in the top model layer show an uppermost region of O3 production (thin, blue stripe), which arose due to the single reaction: O2+O( 3 P)+M O3+M. This is linked with the model's upper boundary condition, where a downward flux of CO and O( 3 P) is imposed. This is done to parameterize the effects of CO2 photolysis (forming CO and O( 3 P)), which takes place above the model's lid, for example, above the mid mesosphere. The resulting enhanced O( 3 P) in the uppermost model layer favors the direct O3 formation pathway found by PAP. The enhanced O3 source was balanced by an increase in the photolysis rate of O3, and therefore the abundance decreased smoothly with altitude as expected. The effect of varying the upper boundary will be the subject of future work. NOx loss cycles dominate (>60%) the Earth lower stratosphere; HOx cycles are more important in the upper stratosphere. For the 3g case (3g Sun), the O3 production pathways are similar to those of Earth, but HOx destruction is stronger (~70%) in the lower stratosphere, which is consistent with more UV penetration (releasing HOx from its reservoirs) for the thinner (3g) atmospheric column compared with the 1g case. The enhanced tropospheric HOx, which</text>
<text><location><page_30><loc_9><loc_90><loc_82><loc_92></location>also stimulated the 'CO oxidation 1' cycle, accounted for (30-50%) of tropospheric O3 loss.</text>
<text><location><page_30><loc_9><loc_52><loc_91><loc_89></location>For the warm M-dwarf star scenarios (e.g., 1g M0) - here, like the control (1g Sun), smog 1 dominates 50-60% of the O3 production in the troposphere (with 10-20% arising from CH4 smog cycles). The influence of the smog mechanism extends to high altitudes (up to about 20km) compared with the Earth control (which extends up to about 16km). 'Chapman 1' (Appendix 1a) dominated the stratosphere. O3 loss was dominated by the 'CO-oxidation 1' pathway (60-80%) in the troposphere, NOx loss pathways in the mid-stratosphere, and HOx loss pathways in the upper stratosphere. For the 3g case (3g M0), the 'smog 1' pathway contributes ~70% of O3 production in the troposphere with the ~(10-15%) remainder in the troposphere coming from CH4 smog pathways. 'Chapman 1' is dominant in the stratosphere, and 'Chapman 2' is dominant in the uppermost layers (see discussion above for Earth run 1). O3 loss, like the 1g case, was dominated by 'CO-oxidation 1' pathway in the troposphere (~90%) with different HOx cycles important for loss in the upper levels.</text>
<text><location><page_30><loc_9><loc_10><loc_91><loc_50></location>In Figure 7, ADL O3 photochemistry production (Figure 7a) is rather similar, for example, to the Earth control (1g Sun) case (Figure 6a) in that Chapman production dominates the stratosphere and smog in the troposphere. However, for the 1g M5 run, results are very different from what occurs on Earth, since O3 production is now dominated by the smog mechanism through much of the atmosphere. For ADL, O3 production occurred mostly via 'Smog 1' (70-80%) in the troposphere, with various CH4 smog pathways making up between 10-20% in this region. 'Chapman 1' dominated the stratosphere. O3 loss was again dominated by 'CO-oxidation 1' in the troposphere (70-90%) with a variety of HOx cycles important for loss in the upper levels. Intense Lymanα radiation favored some enhancement of H2O photolysis (hence more O3 loss via HOx) in the 1g ADL scenario compared to, for example, the Earth control (run 1), but the effect was quickly damped (in the uppermost ~2 model layers) and the overall change in O3 was small. For the corresponding 3g case (3g ADL), O3 production pathways did not change greatly with altitude compared with the 1g case. O3 loss pathways were also rather similar</text>
<text><location><page_31><loc_9><loc_90><loc_91><loc_92></location>to the 1g ADL case, with the 'CO-oxidation 1' pathway for 3g ADL dominating the lower atmosphere.</text>
<text><location><page_31><loc_9><loc_48><loc_91><loc_89></location>The cooler stars (M5, M7) show significant changes in the O3 photochemistry compared with the other M-dwarf scenarios. The rather weak UV radiation of these cooler stars means that Chapman chemistry (requiring UV to break the strong O2 molecule) is now only significant (up to ~50% O3 production) (1g M5) in the uppermost (>60km) altitudes. The 'CO-smog 1' pathway, however, is now significant over all altitudes, accounting for 60% of O3 production in the troposphere and about 30% in the upper atmosphere. A variety of CH4 smog pathways make up most of the remaining O3 production (1g M5). For O3 loss, the 'CO-oxidation 1' pathway is again significant (50-70%) in the lower half of the model domain, whereas a variety of NOx cycles are important in the upper regions. For the coolest star considered (1g M7), the O3 abundance is determined by mainly CO and CH4 oxidation. First, 'classical' smog production - with OH as the oxidant (mainly CO smog 1 and various CH4 oxidation pathways)--produce O3 but, on the other hand, O3 in the M7 scenario can also act as an oxidant in pathways that oxidise, for example, CH4 and CO.</text>
<text><location><page_31><loc_9><loc_10><loc_91><loc_47></location>The M7 case (Figures 6c-6d) shows that the CO smog mechanism dominates the O3 production, whereas the CO oxidation cycle and the classical NOx cycle dominate the O3 loss. Near the surface, some direct removal of O3 occurred via the reaction: NO+O3 O2+NO2 (Figure 6d). On Earth, more NOx usually leads to more O3 production via the smog mechanism; the direct removal reaction is, however, sometimes important at high NOx abundances, for example, in city centers. In our M7 scenario, which does not have industrial NOx emissions, an important source of lower atmosphere NOx is from lightning. For the cool M-dwarf 3g case (3g M5, not shown), the 'CO-smog 1' and 'Chapman 1' pathways make almost equal contributions to the O3 production budget in the middle atmosphere. 'Chapman 1' contributes up to ~80% of local production in the upper levels (where UV is abundant), whereas the smog mechanism contributes up to ~70% in the lower layers. The smog contribution has a minimum of ~20% local production near the cold trap, which is consistent with low temperatures and a</text>
<text><location><page_32><loc_9><loc_87><loc_88><loc_92></location>rather low OH abundance. For the O3 loss pathways, results suggest an increase in complex CO and CH4 smog pathways that consume O3.</text>
<text><location><page_32><loc_9><loc_73><loc_90><loc_82></location>3.6.2 N2O - For all scenarios, non-catalytic photolytic removal (>90%) is the main loss mechanism in the stratosphere. Catalytic removal involving reaction with O( 1 D) makes up the remainder (occurring mostly in the mid to upper stratosphere) of the N2O loss.</text>
<text><location><page_32><loc_9><loc_24><loc_90><loc_68></location>3.6.3 CH4 - Results suggest that a large number of loss pathways occur near the cold trap. For example, at 16km (1g Sun), the CH4 pathways found by PAP with the value of fmin chosen in this study could account for only about 20% of the total CH4 change calculated in the column model. Low OH abundances and cold temperatures in this region are consistent with rather slow oxidation and a resulting complex mix of only weakly oxidized organic species with individual pathway contributions lying below the PAP threshold criteria chosen for the present study, but whose net effect is important. For this study, the PAP detection threshold was set to fmin = 10 -8 ppbv/s. OH-initiated oxidation of CH4 is more favored on the lower layers but with relatively more O( 1 D)-initiated oxidation on the upper levels, where this species is more abundant. A test run (not shown) where the fmin value is decreased to 10 -9 ppbv/s was found to address the above problem, that is, PAP was then able to account, for example, three times more CH4 net change (for the Earth run), though with a notable increase in the overall number of pathways, each with small contributions to net the overall chemical change, beyond the scope of our work.</text>
<text><location><page_32><loc_9><loc_10><loc_91><loc_19></location>3.7 Comparison with previous studies - Compared with the results of Segura et al. (2005), our results are similar for N2O and CH4 within 10-20% for the inactive (e.g., M4) and active (ADL) cases. For O3, our atmopsheric column amounts are ~40% thicker (=270DU) compared with the Segura et al. (2005)</text>
<text><location><page_33><loc_9><loc_80><loc_90><loc_92></location>(=164DU) value for the ADL case. This results from changes in our photochemical scheme, including, for example, the parameterization of the lower boundary flux of H2, as discussed in Paper I. Also our stellar insolation corresponds to 1366Wm-2 at the TOA, whereas Segura et al. (2005) scaled their incoming spectrum to obtain a surface temperature of 288K.</text>
<section_header_level_1><location><page_33><loc_9><loc_73><loc_42><loc_75></location>4. Spectral Detectability of Biomarkers</section_header_level_1>
<text><location><page_33><loc_9><loc_48><loc_90><loc_68></location>O3 - Paper I shows that the detection of O3 is challenging especially for M7. To better understand O3 detectability, improved stellar spectra for the cooler stars in the (E)UV are desirable especially in the UVB and UVC, where O3 responds sensitively. M7 stars are statistically older and burn more slowly compared with lower spectral class stars, which means more developed convection zones and possibly larger differences in UV between flaring and quiet states for M7 than considered in our work (see Reiners et al. 2012).</text>
<text><location><page_33><loc_9><loc_31><loc_91><loc_43></location>N2O - Clearly, the most favorable (planet/star) contrast ratios are associated with cool stars such as M7. However, Paper I shows that some spectral absorption features can be weakened, partly due to the large CH4 abundance, which warms the stratosphere. The N2O spectral features were weak for the scenarios analyzed.</text>
<section_header_level_1><location><page_33><loc_9><loc_24><loc_22><loc_26></location>5. Conclusions</section_header_level_1>
<unordered_list>
<list_item><location><page_33><loc_12><loc_10><loc_88><loc_19></location>· The potential complex-life biosignature O3 has a very different photochemistry for planets orbiting in the HZ of cool M-dwarf stars compared to that of Earth since the key mechanism switches from Chapman production to slower, smog production. Expected responses of O3</list_item>
</unordered_list>
<text><location><page_34><loc_15><loc_80><loc_87><loc_92></location>produced by the smog cycle (which could be favored by increases in HOx and NOx, e.g., by cosmic rays) could be very different than Chapman-produced O3 (where HOx and NOx catalytically destroy O3). This is important to consider when predicting and interpreting O3 spectral features.</text>
<unordered_list>
<list_item><location><page_34><loc_12><loc_59><loc_89><loc_75></location>· The simple microbial-life biosignature N2O increases for the cooler stars, mostly related to weaker photolytic loss of N2O via weaker UVB in the middle atmosphere, as found too by earlier studies. In some cases, however, variations in transport become important. The amount of N2O in the middle atmosphere depends on the UV and on the rate at which this species can be transported upwards from the troposphere into the stratosphere where it is photolyzed.</list_item>
<list_item><location><page_34><loc_12><loc_38><loc_89><loc_50></location>· The greenhouse gas CH4 responses and its removal pathways become complex especially for the cooler stars. CH4 abundances generally increase for the cooler stars, a result also found in earlier studies, due to a lowering in OH, its main sink, which is associated mainly with a weakening in the main OH source reaction that requires UVB.</list_item>
<list_item><location><page_34><loc_12><loc_20><loc_90><loc_33></location>· The potential vegetation biosignature CH3Cl is enhanced in abundance by more than three orders of magnitude compared with the Earth run especially for cool M-star scenarios associated with low OH since reaction with this species is the main sink (see also CH4 above). Like earlier studies, our results suggest that its spectral features are nevertheless very weak.</list_item>
<list_item><location><page_34><loc_12><loc_10><loc_89><loc_15></location>· Comparison of the 1g and 3g scenarios suggests that it is important to include interactive photochemistry when calculating biosignatures and greenhouse gas abundances. Reducing the</list_item>
</unordered_list>
<text><location><page_35><loc_15><loc_83><loc_91><loc_92></location>mass of the atmosphere by, for example, a factor of three does not always lead to a reduction in, for example, biosignatures and greenhouse gases by a factor of three, due to interactive climatephotochemical effects.</text>
<section_header_level_1><location><page_35><loc_9><loc_77><loc_25><loc_79></location>Acknowledgements</section_header_level_1>
<text><location><page_35><loc_9><loc_74><loc_91><loc_77></location>This research has been partly supported by the Helmholtz Gemeinschaft (HGF) through the HGF research alliance "Planetary Evolution and Life." F. Selsis and P. von Paris acknowledge support from the European Research Council (Starting Grant 209622: E3ARTHs).</text>
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<text><location><page_36><loc_9><loc_53><loc_84><loc_56></location>Segura, A., Meadows, V., Kasting, J. F., Crisp, D., and Cohen, M., (2007) Abiotic formation of O2 and O3 in high-CO2 terrestrial atmospheres, Astrobiology , 472, 665-679.</text>
<text><location><page_36><loc_9><loc_50><loc_88><loc_52></location>Segura, A., Walkowicz, L. M., Meadows, V., Kasting, J., and Hawley, S. (2010) The effect of a strong stellar flare on the atmospheric chemistry of an Earth-like planet orbiting an M-dwarf, Astrobiology , 10, 7, 751-771, doi: 10.1089/ast.2009.0376.</text>
<text><location><page_36><loc_9><loc_48><loc_83><loc_49></location>Selsis, F., Despois, D., and Parisot, J.-P. (2002) Signature of life on exoplanets - can Darwin produce false detections? A&A , 388, 985-1003.</text>
<text><location><page_36><loc_9><loc_45><loc_90><loc_47></location>St amenković, V., Noack, L., Breuer, D., and Spohn, T., The influence of pressure -dependent viscosity on the thermal evolution of Super-Earths, ApJ , 748, 41, 2012.</text>
<text><location><page_36><loc_9><loc_41><loc_87><loc_44></location>Stock a , J., Grenfell, J. L., Lehmann, R., Patzer, A. B. C., and Rauer, H. (2012) Chemical pathway analysis of the lower Martian atmosphere: the CO2 stability problem, Plan. Spa. Sci ., 68, 18-24.</text>
<text><location><page_36><loc_9><loc_38><loc_86><loc_41></location>Stock b , J., Boxe, C. S., Lehmann, R., Grenfell, J. L., Patzer, A. B. C., Rauer, H., and Yung, Y. L. (2012) Chemical pathway analysis of the Martian atmosphere: CO2-formation pathways, Icarus , 219, 13-24.</text>
<text><location><page_36><loc_9><loc_34><loc_85><loc_37></location>Toon, O. B., McKay, C. P., Ackerman, T. P. and Santhanam, K. (1989) Rapid calculation of radiative heating rates and photodissociation rates in inhomogeneous multiple scattering atmospheres, J. Geophys. Res ., 94, 16,287-16,301.</text>
<text><location><page_36><loc_9><loc_32><loc_68><loc_34></location>Udry, S., Bonfils, X., Delfosse X., et al. (2007) Super-Earths (5 and 8 ME) in a 3-planet system, A&A, 469, L43.</text>
<text><location><page_36><loc_9><loc_30><loc_78><loc_31></location>Walkowicz, L. M., Jojhns-Krull, C. M., and Hawley, S. L., Characterising the Near-UV environment of (2008) ApJ, 677, 593-606.</text>
<text><location><page_36><loc_9><loc_26><loc_87><loc_29></location>Wang, P., Li, Z., Cihlar, J., Wardle, D. I., and Kerr, J. (2000) Validations of an UV inversion algorithm using satellite and surface measurements, J. Geophys. Res ., 105, 5037-5058.</text>
<text><location><page_36><loc_9><loc_23><loc_90><loc_25></location>Whalley, L. K., Furneaux, K. L., Goddard, A., Lee, J. D., Mahajan, A., et al. (2010) The chemistry of OH and HO2 radicals in the boundary layer over the tropical Atlantic ocean, Atmos. Chem. Phys ., 10, 1555-1576.</text>
<text><location><page_36><loc_9><loc_19><loc_90><loc_22></location>World Meteorological Organisation, (WMO) Scientific Assessment of Ozone Depletion 1994 (1995) Global ozone research and monitoring project report No.37, Geneva.</text>
<section_header_level_1><location><page_37><loc_9><loc_63><loc_21><loc_64></location>Figure Captions</section_header_level_1>
<text><location><page_37><loc_9><loc_57><loc_89><loc_60></location>Figure 1: Planetary Global Mean Top-of-Atmosphere Incoming Radiation (W m -2 ) for UVA (Figure 1a), UVB (Figure 1b) UVC (Figure 1c) for Earth's gravity.</text>
<text><location><page_37><loc_9><loc_53><loc_86><loc_56></location>Figure 2: As for Figure 1 but at the planetary surface for UVA (Figure 2a), UVB (Figure 2b) and UVC (Figure 2c) for Earth's gravity (1g).</text>
<text><location><page_37><loc_9><loc_48><loc_89><loc_51></location>Figure 3: As for Figure 1 but showing the Ratio (Surface/TOA) (at 1g and 3g) for UVA (Figure 3a), UVB (Figure 3b) and UVC (Figure 3c) Radiation (1g).</text>
<text><location><page_37><loc_9><loc_44><loc_88><loc_47></location>Figure 4a: Atmospheric columns (Dobson Units, DU) (1g) of biosignatures, ozone (O3), nitrous oxide (N2O) and methyl chloride (CH3Cl).</text>
<text><location><page_37><loc_9><loc_41><loc_65><loc_43></location>Figure 4c: As for Figure 4a but for column CH4 (Dobson Units, DU) and H2O (DU).</text>
<unordered_list>
<list_item><location><page_37><loc_9><loc_43><loc_68><loc_44></location>Figure 4b: As for Figure 4a but for (3g) scenarios (same surface pressure (=1 bar) as 1g).</list_item>
<list_item><location><page_37><loc_9><loc_40><loc_75><loc_41></location>Figure 4d: As for Figure 4c but for (3g) scenarios assume same surface pressure (1 bar) as 1g cases.</list_item>
<list_item><location><page_37><loc_9><loc_37><loc_66><loc_38></location>Figure 4f: Column (1g/3g) ratio for the same greenhouse gases as shown in Figure 4c.</list_item>
<list_item><location><page_37><loc_9><loc_38><loc_63><loc_40></location>Figure 4e: Column (1g/3g) ratio for the same biosignatures as shown in Figure 4a.</list_item>
</unordered_list>
<text><location><page_37><loc_9><loc_31><loc_90><loc_35></location>Figure 5: Pathway Analysis results for Global Mean Ozone Sources (Figure 5a) and Sinks (Figure 5b) for the Sun and for the M-dwarf star scenarios (1g) calculated by the Pathway Analysis Program. The pathways are shown in the PAP Tables in the Appendix.</text>
<text><location><page_37><loc_9><loc_21><loc_91><loc_30></location>Figure 6: Pathway Analysis results showing cumulative contribution of altitude-dependent O3 production and loss pathways for the 1g Sun (Figures 6a, 6b) and for the 1g M7 scenarios (Figures 6c, 6d) plotted in the vertical and shown in (molecules cm -3 s -1 ). Black and white labels on the Figure correspond to the names of the individual cycles as shown in Appendix 1. Logarithmic x-axis tick labels correspond to factors of x2 x5 and x8 respectively. Note that the model vertical grid is variable depending on, e.g., greenhouse gas heating, which leads to an expansion in the vertical for cooler effective stellar temperatures.</text>
<text><location><page_37><loc_9><loc_18><loc_80><loc_20></location>Figure 7: As for Figure 6 but for the 1g ADL (Figures 7a, 7b) and for the 1g M5 scenarios (Figures 7c, 7d).</text>
<text><location><page_38><loc_23><loc_88><loc_27><loc_89></location>1000</text>
<text><location><page_38><loc_24><loc_83><loc_27><loc_85></location>100</text>
<text><location><page_38><loc_25><loc_79><loc_27><loc_80></location>10</text>
<text><location><page_38><loc_25><loc_70><loc_27><loc_71></location>0.1</text>
<text><location><page_38><loc_22><loc_76><loc_24><loc_79></location>š</text>
<text><location><page_38><loc_24><loc_16><loc_27><loc_17></location>0.01</text>
<text><location><page_38><loc_32><loc_17><loc_36><loc_19></location>M5</text>
<text><location><page_38><loc_31><loc_13><loc_33><loc_16></location>;</text>
<text><location><page_38><loc_42><loc_26><loc_45><loc_29></location>M0</text>
<text><location><page_38><loc_38><loc_21><loc_42><loc_23></location>ADL</text>
<text><location><page_38><loc_36><loc_18><loc_39><loc_20></location>M4</text>
<text><location><page_38><loc_39><loc_13><loc_41><loc_16></location>;</text>
<text><location><page_38><loc_38><loc_89><loc_59><loc_91></location>TOA Net Radiation (Wm-?)</text>
<text><location><page_38><loc_55><loc_82><loc_59><loc_84></location>Sun</text>
<text><location><page_38><loc_35><loc_79><loc_39><loc_81></location>M4</text>
<text><location><page_38><loc_33><loc_74><loc_36><loc_76></location>M5</text>
<text><location><page_38><loc_31><loc_72><loc_34><loc_74></location>M7</text>
<text><location><page_38><loc_41><loc_79><loc_44><loc_81></location>M0</text>
<text><location><page_38><loc_38><loc_74><loc_42><loc_76></location>ADL</text>
<text><location><page_38><loc_39><loc_67><loc_40><loc_70></location>;</text>
<text><location><page_38><loc_47><loc_67><loc_48><loc_70></location>;</text>
<text><location><page_38><loc_50><loc_67><loc_52><loc_70></location>;</text>
<text><location><page_38><loc_54><loc_67><loc_56><loc_70></location>;</text>
<text><location><page_38><loc_32><loc_65><loc_55><loc_67></location>Stellar Effective Temperature (K)</text>
<figure>
<location><page_38><loc_22><loc_38><loc_61><loc_63></location>
<caption>Figure Ic: UVC TOA Net Radiation (Wm-2)</caption>
</figure>
<text><location><page_38><loc_25><loc_32><loc_27><loc_33></location>10</text>
<text><location><page_38><loc_57><loc_28><loc_61><loc_31></location>Sun</text>
<text><location><page_38><loc_56><loc_13><loc_58><loc_16></location>;</text>
<text><location><page_38><loc_33><loc_11><loc_56><loc_13></location>Stellar Effective Temperature (K)</text>
<text><location><page_38><loc_60><loc_13><loc_62><loc_16></location>;</text>
<figure>
<location><page_39><loc_27><loc_55><loc_66><loc_80></location>
<caption>Figure 2b: 1g UVB Surface Net Radiation (Wm-?)</caption>
</figure>
<figure>
<location><page_39><loc_28><loc_22><loc_66><loc_46></location>
</figure>
<figure>
<location><page_40><loc_21><loc_56><loc_63><loc_81></location>
<caption>Figure 3b: 1g and 3g (UVB SurfacelUVB TOA)</caption>
</figure>
<figure>
<location><page_40><loc_22><loc_25><loc_64><loc_46></location>
</figure>
<figure>
<location><page_41><loc_13><loc_54><loc_87><loc_91></location>
<caption>Figure 4b: Column (DU) Biomarkers (3g)</caption>
</figure>
<figure>
<location><page_41><loc_13><loc_11><loc_86><loc_46></location>
</figure>
<figure>
<location><page_42><loc_13><loc_54><loc_86><loc_91></location>
</figure>
<figure>
<location><page_42><loc_15><loc_13><loc_85><loc_46></location>
</figure>
<figure>
<location><page_43><loc_14><loc_54><loc_86><loc_88></location>
<caption>Figure 4f: (1g/3g) column methane and water</caption>
</figure>
<figure>
<location><page_43><loc_14><loc_11><loc_86><loc_46></location>
</figure>
<figure>
<location><page_44><loc_12><loc_50><loc_87><loc_90></location>
<caption>Figure 5b: Ozone Atmospheric Production and Loss (3g)</caption>
</figure>
<figure>
<location><page_44><loc_12><loc_8><loc_87><loc_45></location>
</figure>
<figure>
<location><page_45><loc_10><loc_52><loc_90><loc_82></location>
<caption>Figure 6a: 1g Sun O3 Production Figure 6b: 1g Sun O3 Destruction</caption>
</figure>
<text><location><page_45><loc_9><loc_50><loc_26><loc_53></location>Chapmanhi</text>
<figure>
<location><page_45><loc_10><loc_14><loc_90><loc_43></location>
<caption>Figure 6c: 1g M7 O3 Production Figure 6d: 1g M7 O3 Destruction</caption>
</figure>
<text><location><page_45><loc_15><loc_11><loc_35><loc_13></location>rate (molecule cm</text>
<text><location><page_45><loc_35><loc_12><loc_36><loc_13></location>-3</text>
<text><location><page_45><loc_38><loc_12><loc_40><loc_13></location>-1</text>
<text><location><page_45><loc_36><loc_11><loc_38><loc_13></location>s</text>
<text><location><page_45><loc_40><loc_11><loc_41><loc_13></location>)</text>
<text><location><page_45><loc_56><loc_11><loc_75><loc_13></location>rate (molecule cm</text>
<text><location><page_45><loc_75><loc_12><loc_76><loc_13></location>-3</text>
<text><location><page_45><loc_79><loc_12><loc_80><loc_13></location>-1</text>
<text><location><page_45><loc_76><loc_11><loc_79><loc_13></location>s</text>
<text><location><page_45><loc_80><loc_11><loc_82><loc_13></location>)</text>
<figure>
<location><page_46><loc_10><loc_50><loc_90><loc_82></location>
<caption>Figure 7a: 1g ADL O3 Production Figure 7b: 1g ADL O3 DestructionFigure 7c: 1g M5 O3 Production Figure 7d: 1g M5 O3 Destruction</caption>
</figure>
<figure>
<location><page_46><loc_11><loc_12><loc_92><loc_42></location>
</figure>
<text><location><page_46><loc_15><loc_10><loc_36><loc_12></location>rate (molecule cm</text>
<text><location><page_46><loc_36><loc_11><loc_38><loc_12></location>-3</text>
<text><location><page_46><loc_40><loc_11><loc_42><loc_12></location>-1</text>
<text><location><page_46><loc_38><loc_10><loc_40><loc_12></location>s</text>
<text><location><page_46><loc_42><loc_10><loc_45><loc_12></location>)</text>
<text><location><page_46><loc_56><loc_10><loc_75><loc_12></location>rate (molecule cm</text>
<text><location><page_46><loc_75><loc_11><loc_76><loc_12></location>-3</text>
<text><location><page_46><loc_79><loc_11><loc_80><loc_12></location>-1</text>
<text><location><page_46><loc_76><loc_10><loc_79><loc_12></location>s</text>
<text><location><page_46><loc_80><loc_10><loc_82><loc_12></location>)</text>
<section_header_level_1><location><page_47><loc_9><loc_90><loc_35><loc_92></location>Appendix 1: (a) Ozone Pathways</section_header_level_1>
<unordered_list>
<list_item><location><page_47><loc_9><loc_79><loc_90><loc_89></location>' Found_PAP ' denotes the Column Integrated Rates (CIR) (in molecules cm -2 s -1 ) of change shown for production and loss for all atmospheric pathways found by PAP. ' Shown_PAP ' denotes the CIR only for the pathways shown in this Appendix. Shown are either the 5 dominant pathways or the first pathways which together account for >90% of Found_PAP, whichever criterion is fulfilled first. ' Total _chem ' denotes the CIR as calculated in the chemistry scheme of the atmospheric column model. Percent values for a particular cycle show its individual rate as a percentage of Found_PAP.</list_item>
</unordered_list>
<table>
<location><page_47><loc_19><loc_66><loc_78><loc_75></location>
<caption>1g Sun Ozone Production: Found_PAP =1.88x10 13 , Shown_PAP=1.87 x10 13 , Total_chem =1.88x10 13</caption>
</table>
<text><location><page_47><loc_9><loc_63><loc_79><loc_65></location>1g Sun Ozone Loss: Found_PAP =1.81x10 13 , Shown_PAP=1.06 x10 13 , Total_chem=1.88x10 13</text>
<table>
<location><page_47><loc_9><loc_53><loc_93><loc_63></location>
</table>
<text><location><page_47><loc_12><loc_50><loc_89><loc_52></location>3g Sun Ozone Production: Found_PAP = 1.28x10 13 , Shown_PAP=1.27 x10 13 , Total_chem = 1.28x10 13</text>
<table>
<location><page_47><loc_36><loc_46><loc_61><loc_49></location>
</table>
<text><location><page_47><loc_9><loc_44><loc_80><loc_46></location>3g Sun Ozone Loss: Found_PAP =1.20x10 13 , Shown_PAP=6.06x10 12 , Total_chem = 1.26x10 13</text>
<table>
<location><page_47><loc_14><loc_37><loc_85><loc_43></location>
<caption>1g M0 Ozone Production : Found_PAP =1.65x10 12 , Shown_PAP=1.63x10 12 , Total_chem = 1.70x10 12</caption>
</table>
<table>
<location><page_47><loc_9><loc_19><loc_88><loc_32></location>
<caption>1g M0 Ozone Loss : Found_PAP =1.52x10 12 , Shown_PAP=8.92x10 11 , Total_chem = 1.64x10 12</caption>
</table>
<table>
<location><page_47><loc_9><loc_7><loc_91><loc_15></location>
</table>
<text><location><page_48><loc_9><loc_90><loc_84><loc_92></location>3g M0 Ozone Production : Found_PAP =1.31x10 12 , Shown_PAP=1.30x10 12 , Total_chem = 1.32x10 12</text>
<table>
<location><page_48><loc_30><loc_86><loc_67><loc_90></location>
</table>
<text><location><page_48><loc_9><loc_83><loc_85><loc_85></location>3g M0 Ozone Loss : Total = Found_PAP =9.75x10 11 , Shown_PAP=5.27x10 11 , Total_chem = 1.20x10 12</text>
<table>
<location><page_48><loc_6><loc_78><loc_97><loc_83></location>
</table>
<section_header_level_1><location><page_48><loc_9><loc_75><loc_88><loc_77></location>1g AD-Leo Ozone Production : Found_PAP =1.72x10 12 , Shown_PAP=1.71x10 12 , Total_chem = 1.72x10 12</section_header_level_1>
<table>
<location><page_48><loc_15><loc_63><loc_85><loc_75></location>
</table>
<text><location><page_48><loc_9><loc_60><loc_83><loc_62></location>1g AD-Leo Ozone Loss : Found_PAP =1.43x10 12 , Shown_PAP=6.71x10 11 , Total_chem = 1.69x10 12</text>
<table>
<location><page_48><loc_5><loc_53><loc_96><loc_59></location>
</table>
<text><location><page_48><loc_9><loc_51><loc_35><loc_53></location>3g AD-Leo Ozone Production:</text>
<text><location><page_48><loc_35><loc_51><loc_51><loc_53></location>Found_PAP =1.56x10</text>
<text><location><page_48><loc_39><loc_48><loc_49><loc_50></location>Chapman 1</text>
<text><location><page_48><loc_38><loc_47><loc_44><loc_48></location>(99.5%)</text>
<text><location><page_48><loc_52><loc_51><loc_69><loc_53></location>, Shown_PAP=1.56x10</text>
<text><location><page_48><loc_52><loc_48><loc_58><loc_50></location>Smog 1</text>
<text><location><page_48><loc_51><loc_52><loc_52><loc_53></location>12</text>
<text><location><page_48><loc_50><loc_47><loc_57><loc_48></location>(0.4%)</text>
<table>
<location><page_48><loc_9><loc_34><loc_94><loc_44></location>
<caption>3g AD-Leo Ozone Loss: Found_PAP =1.13x10 12 , Shown_PAP=4.78x10 11 , Total_chem = 1.34x10 12</caption>
</table>
<section_header_level_1><location><page_48><loc_9><loc_31><loc_85><loc_33></location>1g M5 Ozone Production : Found_PAP =2.92x10 11 , Shown_PAP=2.72x10 11 , Total_chem = 3.24x10 11</section_header_level_1>
<table>
<location><page_48><loc_2><loc_17><loc_98><loc_30></location>
</table>
<section_header_level_1><location><page_48><loc_9><loc_15><loc_79><loc_17></location>1g M5 Ozone Loss : Found_PAP =2.57x10 11 , Shown_PAP=2.19x10 11 , Total_chem = 2.91x10 11</section_header_level_1>
<table>
<location><page_48><loc_2><loc_0><loc_98><loc_14></location>
</table>
<text><location><page_48><loc_69><loc_52><loc_70><loc_53></location>12</text>
<text><location><page_48><loc_87><loc_52><loc_88><loc_53></location>12</text>
<text><location><page_48><loc_70><loc_51><loc_87><loc_53></location>, Total_chem = 1.56x10</text>
<text><location><page_49><loc_9><loc_90><loc_84><loc_92></location>3g M5 Ozone Production : Found_PAP =1.35x10 11 , Shown_PAP=1.32x10 11 , Total_chem = 1.42x10 11</text>
<text><location><page_49><loc_33><loc_86><loc_40><loc_88></location>Chapman 1 (47.8%)</text>
<text><location><page_49><loc_47><loc_86><loc_51><loc_88></location>Smog 1 (47.6%)</text>
<text><location><page_49><loc_58><loc_86><loc_65><loc_88></location>Chapman 2 (2.4%)</text>
<text><location><page_49><loc_9><loc_81><loc_79><loc_83></location>3g M5 Ozone Loss : Found_PAP =1.19x10 11 , Shown_PAP=1.08x10 11 , Total_chem = 1.28x10 11</text>
<table>
<location><page_49><loc_12><loc_65><loc_89><loc_79></location>
<caption>1g M7 Ozone Production : Found_PAP =2.79x10 11 , Shown_PAP=2.59x10 11 , Total_chem = 2.92x10 11</caption>
</table>
<table>
<location><page_49><loc_13><loc_50><loc_89><loc_60></location>
<caption>1g M7 Ozone Loss : Found_PAP =2.43x10 11 , Shown_PAP=2.01x10 11 , Total_chem = 2.74x10 11</caption>
</table>
<table>
<location><page_49><loc_4><loc_34><loc_95><loc_46></location>
</table>
<section_header_level_1><location><page_49><loc_9><loc_30><loc_46><loc_32></location>1b. Nitrous Oxide Pathways (Sun only)</section_header_level_1>
<text><location><page_49><loc_9><loc_26><loc_84><loc_28></location>1g Sun Nitrous Oxide Loss : Found_PAP =9.63x10 8 , Shown_PAP=8.78x10 8 , Total_chem = 1.15x10 9</text>
<text><location><page_49><loc_9><loc_16><loc_24><loc_23></location>N2O-NOx (69.5%): 2(N2O +h ν → N 2 + O( 3 P)) O( 3 P) + O2 + M → O 3 + M NO2 + O( 3 P) → NO + O 2 NO + O3 → NO 2 + O2 net: 2N2 O → O 2 + 2N2</text>
<text><location><page_49><loc_26><loc_17><loc_41><loc_23></location>N2O-Ox (9.6%): 2(N2O +h ν → N 2 + O( 3 P)) O( 3 P) + O2 + M → O 3 + M O( 3 P) + O3 → O 2 + O2 net: 2N2 O → O 2 + 2N2</text>
<text><location><page_49><loc_60><loc_16><loc_75><loc_23></location>N2O-HOx-1 (3.4%): 2(N2O +h ν → N 2 + O( 3 P)) 2(O( 3 P)+O2 +M → O 2+M) OH + O3 → HO 2 + O2 HO2+O3 → OH+O 2+O2 net: 2N2 O → O 2 + 2N2</text>
<text><location><page_49><loc_43><loc_16><loc_58><loc_23></location>N2-O( 1 D) (5.4%): 2( N2O +O( 1 D) → N 2 + O2) 2(O( 3 P)+O2 +M → O 3+M) 2(O3 + h ν → O 2 + O( 1 D)) O2 + h ν → O( 3 P) + O( 3 P) net: 2N2 O → O 2 + 2N2</text>
<text><location><page_49><loc_77><loc_16><loc_92><loc_23></location>N2O-HOx-2 (3.3%): 2(N2O + h ν → N 2 + O( 3 P) O( 3 P) + O2 + M → O 3 + M HO2 + O( 3 P) → OH + O 2 OH + O3 → HO 2 + O2 net: 2N2 O → O 2 + 2N2</text>
<section_header_level_1><location><page_50><loc_9><loc_88><loc_30><loc_90></location>1c. Methane Pathways</section_header_level_1>
<text><location><page_50><loc_9><loc_84><loc_81><loc_86></location>1g Sun Methane Loss : Found_PAP =1.15x10 10 , Shown_PAP=7.68x10 9 , Total_chem = 1.24x10 11</text>
<text><location><page_50><loc_12><loc_65><loc_26><loc_82></location>Oxidation 3O2 -a (29.6%): 2(CH4+OH → CH3+H 2 O) 2(CH3+O2+M → CH3O2+M) 2 (CH3O2+NO → H3CO+NO2) 2 (H3CO+O2 → H2CO+HO2) 2 (H2CO + h ν → H2 + CO) NO + HO2 → NO2 + OH 3 (NO2 + h ν → NO + O( 3 P)) 3 (O( 3 P) + O2 + M → O3 + M) 3 (HO2+O3 → OH+O2+O2) 2 (CO + OH → CO2 + H) 2 (H + O2 + M → HO2 +M) net:2CH4+3O2 → 2H2O+ 2H2+2CO2</text>
<text><location><page_50><loc_29><loc_66><loc_43><loc_81></location>Oxidation 6O2 (23.0%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + h ν → H2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 2 (NO + HO2 → NO2 + OH) 3 (NO2 + h ν → NO + O( 3 P)) 3 (O( 3 P) + O2 + M → O3 + M) net: CH4 + 6O2 → H2O + 3O3 + H2 + CO2</text>
<text><location><page_50><loc_63><loc_66><loc_77><loc_82></location>Oxidation 8O2 (6.0%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO HCO + O2 → HO2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 3(NO + HO2 → NO2 + OH) 4(NO2 + h ν → NO + O( 3 P)) 4(O( 3 P) + O2 + M → O3 + M) net: CH4 + 8O2 → 2H2O + 4O3 + CO2</text>
<text><location><page_50><loc_46><loc_66><loc_60><loc_82></location>Oxidation 2O2-a (6.2%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO NO + HO2 → NO2 + OH 2(NO2 + h ν → NO + O( 3 P) ) 2(O( 3 P) + O2 + M → O3 + M) HCO + O2 → HO2 + CO 2(HO2+O3 → OH+O2+O2) CO + OH → CO2 + H H + O2 + M → HO2 +M net:CH4 +2O2 → 2H2O + CO2</text>
<text><location><page_50><loc_80><loc_67><loc_94><loc_82></location>Oxidation 3O2-b (2.2%) 2(CH4 + OH → CH3 + H2O) 2(CH3+O2+M → CH3O2+M) 2(CH3O2+NO → H3CO+NO2) 2(H3CO+O2 → H2CO+HO2) 2(H2CO + hv → H2 + CO) 2(CO + OH → CO2 + H) 2(H + O2 + M → HO2 +M) 4(NO + HO2 → NO2 + OH) 3(NO2 + h ν → NO + O( 3 P)) 3(NO2 + O( 3 P) → NO + O2) net: 2CH4 + 3O2 → 2H2O + 2H2 + 2CO2</text>
<section_header_level_1><location><page_50><loc_9><loc_62><loc_81><loc_64></location>3g Sun Methane Loss : Found_PAP =3.21x10 10 , Shown_PAP=1.63x10 10 , Total_chem = 1.24x10 11</section_header_level_1>
<text><location><page_50><loc_9><loc_47><loc_23><loc_58></location>Oxidation CH3OOH-a (26.5%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 H2O + O( 1 D) → OH + OH CO + OH → CO2 + H H + O2 + M → HO2 +M O3 + h ν → O2 + O( 1 D) net: CH4 + O3 + CO → CH3OOH + CO2</text>
<text><location><page_50><loc_26><loc_45><loc_40><loc_59></location>Oxidation 2O2-b (7.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+OH → H3CO+HO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO HCO + O2 → HO2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 4(HO2 + O( 3 P) → OH + O2) 2(O2 + h ν → O( 3 P) + O( 3 P)) net:CH4 +2O2 → 2H2O+CO2</text>
<text><location><page_50><loc_60><loc_45><loc_74><loc_61></location>Oxidation 2O2-c (5.4%) CH4 + O( 1 D) → CH 3 + OH CH3 + O2 + M → CH 3O2 + M CO + OH → CO 2 + H H + O2 + M → HO 2 +M CH3O2 +OH →H 3CO+HO2 H3CO + O2 → H 2CO + HO2 3(HO2 + O( 3 P) → OH + O 2) OH + HO2 → H 2O + O2 H2 CO + OH → H 2O + HCO HCO + O2 → HO 2 + CO O( 3 P) + O2 + M → O 3 + M O3 + h ν → O 2 + O( 1 D) 2(O2 + h ν → O( 3 P) + O( 3 P)) net:CH4 +2O2 →2H 2O+CO2</text>
<text><location><page_50><loc_43><loc_45><loc_57><loc_59></location>Oxidation O3-a (5.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M H2CO + h ν → H2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 2(HO2+O3 → OH+O2+O2) net: CH4 +O3 → H2O+H2+CO2</text>
<text><location><page_50><loc_77><loc_46><loc_91><loc_60></location>Oxidation O3-b (5.3%) CH4 + OH → CH 3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 CH3OOH+h ν → H 3CO+OH H3CO+O2 → H 2CO+HO2 H2CO + h ν → H 2 + CO CO + OH → CO 2 + H H + O2 + M → HO2 +M HO2 + O3 → OH + O2 + O2 net: CH4 +O3 → H2O+ H2+CO2</text>
<section_header_level_1><location><page_50><loc_9><loc_42><loc_81><loc_44></location>1g M0 Methane Loss: Found_PAP =3.25x10 10 , Shown_PAP=1.93x10 10 , Total_chem = 1.24x10 11</section_header_level_1>
<text><location><page_50><loc_13><loc_33><loc_25><loc_34></location>Oxidation 3O2-a (24.4%)</text>
<text><location><page_50><loc_29><loc_25><loc_43><loc_40></location>Oxidation CH3OOH-b (12.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 H2O + O( 1 D) → OH + OH 2(H + O2 + M → HO2 +M) NO + HO2 → NO2 + OH 2(CO + OH → CO2 + H) NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + 2CO + 2O2 → CH3OOH + 2CO2</text>
<text><location><page_50><loc_65><loc_33><loc_76><loc_34></location>Oxidation 6O2 (6.1%)</text>
<text><location><page_50><loc_46><loc_28><loc_60><loc_38></location>Oxidation O2 (10.4%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + O3 → OH + O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M net: CH4 + O2 → H2CO+H2O</text>
<text><location><page_50><loc_80><loc_25><loc_94><loc_39></location>Oxidation CH3OOH-c (5.7%) 2(CH4 + OH → CH3 + H2O) 2(CH3+O2+M → CH3O2+M) CH3O2+OH → CH3OOH+O2 CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) H2O + O( 1 D) → OH + OH net: 2CH4 + 2O2 → H2CO + H2O + CH3OOH</text>
<table>
<location><page_50><loc_8><loc_12><loc_93><loc_18></location>
<caption>3g M0 Methane Loss : Found_PAP =5.67x10 10 , Shown_PAP=2.16x10 10 , Total_chem = 1.24x10 11</caption>
</table>
<table>
<location><page_51><loc_9><loc_72><loc_94><loc_87></location>
<caption>1g ADL Methane Loss : Found_PAP =6.14x10 10 , Shown_PAP=3.95x10 10 , Total_chem = 1.24x10 113g ADL Methane Loss : Found_PAP =7.06x10 10 , Shown_PAP=3.31x10 10 , Total_chem = 1.24x10 11</caption>
</table>
<table>
<location><page_51><loc_10><loc_49><loc_95><loc_68></location>
<caption>1g M5 Methane Loss: Found_PAP =5.65x10 10 , Shown_PAP=4.04x10 10 , Total_chem = 1.24x10 11</caption>
</table>
<table>
<location><page_51><loc_9><loc_29><loc_95><loc_45></location>
<caption>3g M5 Methane Loss : Found_PAP =7.26x10 10 , Shown_PAP=4.57x10 10 , Total_chem = 1.24x10 11</caption>
</table>
<text><location><page_51><loc_9><loc_9><loc_24><loc_22></location>Oxidation CH3OOH-d (24.5%) 2(CH4 + O( 1 D) → CH3 + OH) 2(CH3+O2+M → CH3O2+M) 2(CO + OH → CO2 + H) 2(H + O2 + M → HO2 +M) 2(CH3O2+HO2>CH3OOH+O2) 2(O( 3 P) + O2 + M → O3 + M) 2(O2 + h ν → O2 + O( 1 D)) O2 + h ν → O( 3 P) + O( 3 P) net: 2CH4 + 2CO + 3O2 → 2CH3OOH + 2CO2</text>
<text><location><page_51><loc_26><loc_8><loc_42><loc_22></location>Oxidation CH3OOH-e (23.7%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 2(H + O2 + M → HO2 +M ) NO + HO2 → NO2 + OH 2( CO + OH → CO2 + H) NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + 2CO + 2O2 → CH3OOH + 2CO2</text>
<text><location><page_51><loc_62><loc_7><loc_77><loc_22></location>Oxidation H2O2-b (4.5%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H H + O2 + M → HO2 +M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + HO2 → H2O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + CO + 2O2 → H2CO + H2O2 + CO2</text>
<text><location><page_51><loc_44><loc_7><loc_59><loc_22></location>Oxidation H2O2-a (5.8%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H H + O2 + M → HO2 +M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + HO2 → H2O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) H2CO + h ν → H2 + CO net:CH4 +2O2 →H 2O2+H2+CO2</text>
<text><location><page_51><loc_79><loc_7><loc_94><loc_23></location>Oxidation H2O2-c (4.4%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + h ν → HCO + H HCO + O2 → HO2 + CO NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) 2(H + O2 + M → HO2 +M) 2(HO2 + HO2 → H2O2 + O2) net: CH4 +3O2 →2H 2O2+CO2</text>
<table>
<location><page_52><loc_8><loc_68><loc_93><loc_85></location>
<caption>1g M7 Methane Loss : Found_PAP =5.48x10 10 , Shown_PAP=3.40x10 10 , Total_chem = 1.24x10 11</caption>
</table>
<section_header_level_1><location><page_53><loc_9><loc_87><loc_19><loc_90></location>Appendix 1</section_header_level_1>
<text><location><page_53><loc_9><loc_80><loc_88><loc_84></location>Ozone column in Dobson Units (DU) for the 1g, 3g scenarios corresponding to the values plotted in Figures 4a, 4b as a function of stellar effective temperature (Teff) (K).</text>
<table>
<location><page_53><loc_8><loc_55><loc_92><loc_74></location>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "Potential Biosignatures in Super-Earth Atmospheres II. Photochemical Responses J. L. Grenfell 1 , S. Gebauer 1 , M. Godolt 1 , K. Palczynski 1,* , H. Rauer 1,2 , J. Stock 2 , P. v. Paris 2,# , R. Lehmann 3 , and F. Selsis 4 2 Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Rutherford Str. 2, 12489 Berlin, Germany * Present address: Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Hahn-Meitner-Platz 1, 14109 Berlin, Germany Running title: Photochemistry Earth-like biosignatures Abstract: Spectral characterization of Super-Earth atmospheres for planets orbiting in the Habitable Zone of M-dwarf stars is a key focus in exoplanet science. A central challenge is to understand and predict the expected spectral signals of atmospheric biosignatures (species associated with life). Our work applies a global-mean radiative-convective-photochemical column model assuming a planet with an Earth-like biomass and planetary development. We investigated planets with gravities of 1g and 3g and a surface pressure of one bar around central stars with spectral classes from M0 to M7. The spectral signals of the calculated planetary scenarios have been presented by Rauer et al. (2011). The main motivation of the present work is to perform a deeper analysis of the chemical processes in the planetary atmospheres. We apply a diagnostic tool, the Pathway Analysis Program, to shed light on the photochemical pathways that form and destroy biosignature species. Ozone is a potential biosignature for complex- life. An important result of our analysis is a shift in the ozone photochemistry from mainly Chapman production (which dominates in Earth's stratosphere) to smog-dominated ozone production for planets in the Habitable Zone of cooler (M5-M7)-class dwarf stars. This result is associated with a lower energy flux in the UVB wavelength range from the central star, hence slower planetary atmospheric photolysis of molecular oxygen, which slows the Chapman ozone production. This is important for future atmospheric characterziation missions because it provides an indication of different chemical environments that can lead to very different responses of ozone, for example, cosmic rays. Nitrous oxide, a biosignature for simple bacterial life is favored for low stratospheric UV conditions, that is, on planets orbiting cooler stars. Transport of this species from its surface source to the stratosphere where it is destroyed can also be a key process. Comparing 1g with 3g scenarios, our analysis suggests it is important to include the effects of interactive chemistry. Key words: Exoplanets, Earth-like, M-dwarf, photochemistry, biosignatures",
"pages": [
1,
2,
3
]
},
{
"title": "1. Introduction",
"content": "Understanding the photochemical responses of Super-Earth (SE) atmospheres in the Habitable Zone (HZ) of M-dwarf stars is a central goal of exoplanet science, since it is feasible that such environments may present the first opportunities to search for biosignature spectral signals. Gliese 581d (Mayor et al. 2009; Udry et al., 2007) is the first SE to be found that may orbit in the HZ of its M-dwarf star. Recently, initial constraints on the composition of hot transiting SEs such as CoRoT-7b (e.g., Guenther et al., 2011) and GJ1214b (e.g., Bean et al., 2011; Croll et al., 2011) have been discussed. Kepler 22b (Borucki et al., 2012) is the first transiting object found to occur in the HZ of a solar-type star; several Earth-sized objects have been found orbiting a cool M-dwarf (e.g., Muirhead et al., 2012) and detection of further SEs in the HZ is just beginning (e.g., Bonfils et al., 2013). There exist a large number of possible parameters that could influence the abundances of possible biosignature species in hypothetical Earth-like atmospheres. Our motivation here is to take two parameters that are relatively well-known, namely, stellar class and planetary gravity, and perform a sensitivity study assuming an Earth-like biomass and development in order to determine their effect upon the photochemistry and climate, and hence the potential biosignatures. Other works (e.g., Segura et al., 2005; Grenfell et al., 2007) have also adopted this approach. In this work, we analyzed the photochemical responses of key species from the same scenarios as the earlier work of Rauer et al. (2011) (hereafter Paper I), who analyzed spectral signals for Earthlike planets with gravities of 1g and 3g orbiting in the HZ of M-dwarf stars with classes from M0 to M7. In an earlier study, Segura et al. (2005) also discussed photochemical responses of (1g) Earth-like planets orbiting in the HZ of M-dwarf stars. They calculated enhanced abundances of methane (CH4) (by about x100) and nitrous oxide (N2O) (by about x5) compared with those of Earth related to the weaker UV emissions of M-dwarf stars. Their results also suggest a reduction in the ozone (O3) column by up to about a factor of 7 compared with that of Earth, associated with weakened UV leading to a slowing in the O3 photochemical source. This result was already broadly anticipated in the early 1990s (see Segura et al. 2005 and references therein). In the present study, we aimed to examine the nature of these photochemical responses in more depth. We applied a diagnostic tool termed the Pathway Analysis Program (PAP) written by Lehmann, (2004) to investigate the photochemical responses. PAP delivers unique information on chemical pathways of key species and has identified new chemical atmospheric pathways on Earth (Grenfell et al., 2006) and on Mars (Stock et al. 2012 a,b ) . PAP is a key tool for understanding atmospheric sources and sinks of the biosignatures and related compounds. The usual mechanisms that operate in Earth's atmosphere (e.g., O3 catalytic cycles etc.) are complex and may be very different for Earth-like planets orbiting M-dwarf scenarios, which is a good motivation for applying such a tool. The primary driver of the photochemistry is the Top-of-Atmosphere (TOA) stellar flux, especially in the UVB and UVC regions, which weaken with decreasing effective stellar temperature. Therefore, we first analyzed the Ultra-Violet (UV) fluxes in our planetary atmospheres. Then, we focused on their influence on atmospheric ozone (O3) since this is not only an important biosignature but also a key UVB absorber governing the abundances of other chemical species. We then investigated the biomarker N2O, which is sensitive to UVB. Finally, we analyzed the photochemistry of CH4 and water (H2O) since these key greenhouse gases can influence surface habitability. We now present a brief overview of the photochemistry of the above four species.",
"pages": [
3,
4,
5
]
},
{
"title": "1.1 Photochemistry of O3",
"content": "O3 on Earth is a potential biosignature associated mainly with molecular oxygen (O2), which arises mostly via photosynthesis. In Earth's atmosphere, about 90% (10%) of O3 resides in the stratosphere (troposphere). Production of O3 in the Earth's stratosphere occurs mainly via the Chapman mechanism (Chapman, 1930) via O2 photolysis. Production of O3 in the troposphere occurs mostly via the smog mechanism (Haagen-Smit, 1952), which requires volatile organic compounds (VOCs), nitrogen oxides, and Ultraviolet (UV). Destruction of O3 in the stratosphere proceeds mainly via catalytic cycles involving hydrogen-, nitrogen, or chlorine-oxides (e.g., Crutzen, 1970) (designated HOx, NOx, and ClOx respectively). These molecules can be stored in so-called reservoir species, the atmospheric distributions of which are reasonably well-defined for Earth (e.g., World Meteorological Organization (WMO) Report, 1995). Changes in, for example, temperature and/or UV can lead to the reservoirs releasing their HOx-NOxClOx, associated with rapid stratospheric O3 removal in sunlight. Destruction of O3 in the troposphere occurs, for example, via wet and dry deposition and/or gas-phase removal via fast removal with, for example, NOx. O3 can be formed abiotically in CO2 atmospheres (e.g., Segura et al., 2007). O3 layers (albeit very weak compared to that on Earth) have been documented on Mars (Fast et al., 2009) and on Venus (Montmessin et al., 2011), so caution is warranted when interpreting O3 signals as indicative of biology or not (e.g., Selsis et al. 2002).",
"pages": [
5,
6
]
},
{
"title": "1.2 Photochemistry of N2O",
"content": "N2O is a biosignature produced almost exclusively on Earth from microbes in the soil as part of the nitrogen cycle (International Panel on Climate Change (IPCC, 2001)). Minor inorganic sources include, for example, the reaction of molecular nitrogen with electronically excited atomic oxygen: N2+O( 1 D)+M N2O+M (e.g. Estupiñan et al. 2002). Destruction of N2O occurs in the stratosphere mainly via photolysis or via removal with excited oxygen atoms.",
"pages": [
6
]
},
{
"title": "1.3 Photochemistry of CH4 and Methyl choride (CH3Cl)",
"content": "CH4 is a strong greenhouse gas affecting climate and hence habitability. It is destroyed in the troposphere up to the mid-stratosphere mainly by oxidative degradation pathways with hydroxyl (OH) and in the upper stratosphere via photolysis. CH4 is a possible indicator of life (bioindicator) but not a definite proof since this species (on Earth) has, in addition to biogenic sources, also some geological origins (IPCC, 2001). CH3Cl on Earth has important biogenic sources associated with vegetation, although its sourcesink budget and net anthropogenic ccontribution is not well known (Keppler et al. 2005). Like CH4, its removal is controlled by reaction with OH, although the chlorine atom leads to increased reactivity (with an enhanced rate constant of about a factor 6 for this reaction) compared with CH4.",
"pages": [
6
]
},
{
"title": "1.4 Photochemistry of H2O",
"content": "Although not a biosignature, H2O is essential for life as we know it. Like CH4, H2O is an efficient greenhouse gas. Production of H2O in Earth's stratosphere proceeds via CH4 oxidation, whereas destruction of H2O occurs in the upper stratosphere via photolysis (World Meteorological Organisation (WMO), 1994). In the troposphere, H2O is subject to the hydrological cycle, including evaporation and condensation.",
"pages": [
7
]
},
{
"title": "1.5 Key Questions",
"content": "O3 is formed on Earth in different ways, that is, via the smog mechanism (~10% on Earth) and the Chapman mechanism (~90%). How and why these values may change for different exoplanet scenarios is not well investigated, yet this is important information for predicting and interpreting spectra. A flaring M-dwarf star, for example, will induce a photochemical response creating NOx, which destroys 'Chapman'-produced O3 but could actually enhance a 'smog' O3 signal. N2O is destroyed via photolysis in the stratosphere by UVB radiation in the stratosphere, but its supply upwards from the surface is controlled by atmospheric transport and mixing. Models with fast upwards transport will ultimately lead to reduced N2O abundances since in the case of faster transport, the N2O molecules reach the altitudes of efficient destruction earlier, that is, the lifetime of N2O molecules is reduced, which (at a constant emission rate) leads to smaller N2O concentrations. To improve knowledge of potential N2O spectral signals in exoplanet environments, it is important to understand which processes (photochemistry or transport) dominate the abundance of N2O in different environments. For example, N2O on Earth is affected by both stratospheric UVB (which depends on, e.g., the solar spectra, radiative transfer, atmospheric photochemistry, etc.) as well as on troposphericto-stratospheric transport processes. To begin to address such questions, we apply a new chemical diagnostic tool, the Pathway Analysis Program (PAP), which sheds unique light into the chemical pathways that control biosignature abundances.",
"pages": [
7,
8
]
},
{
"title": "2.1 Models",
"content": "The model details for the atmospheric coupled climate-chemistry column model and the theoretical spectral model have been described in Paper 1. Recent model updates include, for example, a new offline binning routine for calculating the input stellar spectra and a variable vertical atmospheric height in the model; more details were given by Rauer et al. (2011). The radiative-convective module is based on the work of Toon et al. (1989) for the shortwave region and RRTM (Rapid Radiative Transfer Module) for the thermal radiation. Since a main focus in this work is on photochemical effects, we will now provide a detailed description of the photochemical module. The model simulates 1D globalaverage, cloud-free conditions, although the effects of clouds were considered in a straightforward way by adjusting the surface albedo until the mean surface temperature of Earth (288 K) was attained for the Earth control run, as in earlier studies (Paper 1, Segura et al., 2003). The scheme solved the central chemical continuity equations by applying an implicit Euler solver that used the LU (Lower Upper) triangular matrix decomposition method with variable iterative stepping such that the stepsize was halved whenever the abundance of a long-lived species changed by more than 30% over a single step. The version used here employs chemical kinetic data from the Jet Propulsion Laboratory (JPL) Evaluation 14 (2003) report. The scheme includes the key inorganic gas-phase and photolytic chemical reactions commonly applied in Earth's atmosphere, that is, with hydrogen-, nitrogen, and chlorine- oxide reactions and their reservoirs. The scheme was considered to be converged when the relative change in concentration for any species in any layer changes by less than 10 -4 over a chemical iteration that exceeded 10 5 s. From a total of 55 chemical species, 34 were 'long-lived,' that is, the transport timescales are long compared with those of the photochemistry. Their concentrations were calculated by solving the full Jacobian matrix; 3 species, namely, CO2, N2, and O2 were set to constant isoprofile values based on modern Earth, and the remainder of the species were 'short-lived,' that is, assumed to be in steadystate, and therefore calculated from the long-lived species. The steady-state assumption simplifies the numerical solution. Surface biogenic and source gas fluxes for CH4, (=531Tg/yr) N2O (=8.6 Tg N contained in N2O /yr) , CO (=1796Tg/yr) and CH3Cl (=3.4Tg/yr) were set such that for the Earth control run, Earth's modern-day concentrations were achieved at the surface - this procedure was commonly used in earlier approaches for Earth-like exoplanets (e.g., Paper 1, Segura et al., 2003). H2 at the surface was removed with a constant deposition velocity of 7.7x10 -4 cm s -1 . Dry and wet deposition removal fluxes for other key species were included via molecular velocities and Henry's Law coefficients respectively. Volcanic fluxes of SO2 and H2S were based on modern Earth. Tropospheric lightning sources of NO were based on the Earth lightning model of Chameides et al. (1977), assuming chemical equilibrium between N2, O2, and NO, a freeze-out temperature of 3500K and equilibrium constants taken from the Chemical Rubber Company (CRC) 1976 handbook. Modern Earth's atmosphere has ~44 lightning flashes s -1 global mean (with flashes mainly generated over land in the tropics), which produces ~5Tg N in the form of NOx globally per year (Schumann and Huntreiser, 2007). Clearly, these values depend, for example, on atmospheric transport, convective activity, and the land-sea distribution, etc. for Earth-like exoplanets, which are not well-constrained parameters. At the model upper boundary, a constant, downwards (effusion) flux of CO and O is set, which represents the photolysis products of CO2 that are formed above the model's upper lid. Atmospheric mixing between the 64 vertical chemical layers was calculated via eddy diffusion constants (K in cm 2 s -1 ), where log(K) varied from ~5.0 at the surface, decreased to a minimum value of ~3.6 at ~16km, and then increased to ~5.7 at the model upper boundary. Photolysis rates included the major absorbers, including important (E)UV absorbers such as O2, CO2, H2O, O3, NO, CH4, and SO2. The O2 photolysis absorbtion coefficients were calculated with the mean exponential sums method. The O3 coefficients included the Hartley-Huggins T-dependence based on data measured at 203K and 273K (and linearly interpolated between). Species that photolyze in the UVB that are relevant for O3 destruction were also included, for example, nitric acid (HNO3) photolysis was included - this is important for NOx release. Finally, weakly bound species that photolyze in the UVA/visible region, for example, NO3, N2O5 were included. Photolysis rates were calculated based on insolation fluxes from the delta two-stream module (Toon et al. 1989). One hundred eight wavelength intervals were included from (175.4-855) nm in the UV and visible, nine intervals in the EUV from (130-175) nm, and one Lyman-alpha interval at 121.6 nm. Rayleigh scattering for N2, O2, and CO2 was included. The Pathway Analysis Program (PAP) was developed by Lehmann (2004) and applied by Grenfell et al. (2006) to Earth's stratosphere and by Stock et al. (2012 a,b ) to the martian atmosphere. In the present work, it is applied to Super-Earth planetary atmospheres. The PAP algorithm identifies and quantifies chemical pathways in chemical systems. Starting with individual reactions as pathways, PAP constructs longer pathways step-by-step. To achieve this, short pathways already found are connected at so-called 'branching point' species, whereby each pathway that forms a particular species is connected with each pathway that destroys it. Branching point species are chosen based on increasing lifetime with respect to the pathways constructed so far. In this work, all species with a chemical lifetime shorter than the chemical lifetime of the species being studied (i.e., the biosignatures O3, N2O, and the greenhouse gas CH4) are treated as branching point species. Since in general the chemical lifetime of all species varies with altitude, the choice of branching point species adapts to the local chemical and physical conditions. A detailed description of the PAP algorithm is given by Lehmann (2004). To avoid a prohibitively long computational time, pathways with a rate smaller than a userdefined threshold (in the present study, fmin=10 -8 parts per billion by volume per second (ppbv/s)) are deleted. The chosen f_min = 10 -8 ppbv/s is sufficient for finding the 5 dominant pathways (e.g., of N2O, CH4 loss) as shown in the main table (Appendix 1). Stock et al. (2012 a ) discussed the effect of varying this parameter. PAP calculates the chemical pathways by taking as input (i) a list of chemical species, (ii) chemical reactions, (iii) time-averaged concentrations and reaction rates, and (iv) concentration changes arising only from the gas-phase chemical reactions only (i.e., not including changes in abundance from, e.g., mixing, deposition, etc). PAP calculates as output the identified chemical pathways with their associated rates. Information from PAP is used to interpret chemical responses.",
"pages": [
8,
9,
10,
11
]
},
{
"title": "2.2 Scenarios",
"content": "Here, we analyze the model scenarios described in Paper I. We considered planets with masses corresponding to 1g and 3g with Earth-like (i.e., N2-O2) atmospheres with Earth's source gas emissions and initial p,T, and abundance profiles as for modern Earth. There are currently no observational constraints for the surface pressure of SE planets. On the one hand, theoretical studies, for example, that of Elkins-Tanton and Seager (2008), have suggested a wide-range of possible atmospheric masses resulting from outgassing on SE planets, whereas on the other hand, for example, Stamenković et al. (2012), who included a pressure-depenence of viscosity in the mantle, suggested rather weak SE outgassing rates. Given the current uncertainties, we therefore assume 1 bar surface pressure to be comparable with Paper 1 and earlier studies and to compare with our 1g scenarios. Our modeled p, T, and chemical output profiles are calculated self-consistently for planets around different central Mdwarf stars in the HZ (with the Sun-Earth case for comparison). We explore an extensive parameter range, considering planets orbiting M-dwarf stellar classes from M0 to M7. This is neccessary because atmospheric chemistry-climate coupling is strongly non-linear and, hence, general results from one set of stellar classes (e.g., M0 to M4) cannot be simply extrapolated to other stellar classes (e.g., M5-M7) instead each scenario has to be calculated separately. Mixing ratios for radiative species are fed back into the climate module, which calculates a new T, p profile, and this is again fed back into the chemistry module. This iterative process continues until T, p, and concentrations all converge. The planets are placed at an orbital distance from their star such that the total energy input at the TOA equals the modern Solar constant of 1366 Wm -2 (see Paper 1 for the stellar input spectra used). In total, the following eleven scenarios were investigated:",
"pages": [
11,
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]
},
{
"title": "2.3 Planetary Radiation Environment",
"content": "Incoming Stellar fluxes (F*) - These are the primary driver of planetary atmospheric photochemistry, especially in the UVB and UVC range, and are also central to habitability for life as we know it on Earth. A significant proportion of cooler M-dwarfs like those considered in our work may be active emitters of UV from their chromospheres or/and transition regions (see e.g., Walkowicz et al., 2008, France et al., 2012 in press). This could have a considerable impact upon the planetary photochemistry, climate, and associated biosignatures. How efficiently the UV is absorbed throughout the atmospheric column is closely linked with the photochemical responses and, hence, determines the final abundances of the biosignature. We therefore start our analysis by investigating the planetary radiation environment. We discuss UV radiation at the TOA and at the planetary surface, and present a validation of surface UV based on Earth observations Planetary TOA Radiation Analysis - We analyzed the planetary TOA F* in the UVA, UVB, and UVC wavelength range for the different stellar scenarios in the top model layer. UVA corresponds to the model wavelength intervals from ( 315-400) nm; UVB corresponds to (280-315) nm; UVC corresponds to ( 175.4-280) nm. To be comparable with Paper 1, we approximated the TOA stellar spectra for the M0 to M7 Mdwarf stars as Planck functions (other than for the Sun, which is for solar mean conditions based on the work of Gueymard et al. (2004), and for AD-Leo, for which measured UV-spectra are available, see Paper I). The approach used in Paper 1 and, therefore, in this study as well was to employ Planck curve spectra that correspond to quiet M-dwarf stars with little emitted UV fluxes. Recent results (Reiners et al., 2012) suggest that >90% of hotter (M0 to M2) M-dwarf stars sampled are quiet , whereas >50% of the cooler stars (M4 and cooler) are active . Clearly, we are well-aware that smooth Planck functions do not include, for example, enhanced Lyman-alpha and UVC features, etc. characteristic of cool M-dwarf stars that may have very active chromospheric and coronal regions. However, direct observations of stellar spectra for the cooler M-dwarf stars (M5-M7) in the critical wavelength range ( λ Table 1: Ratios of UV radiation for our M-dwarf-star (M7) scenario compared with the Sun (upper row) and for ADL.Table 1 (row 1) suggests that our cool (M7) M-dwarf would emit less than 1% of the UVA, Table 1: Ratios of UV radiation for our M-dwarf-star (M7) scenario compared with the Sun (upper row) and for ADL.Table 1 (row 1) suggests that our cool (M7) M-dwarf would emit less than 1% of the UVA, UVB, and UVC radiation compared with the Sun. Comparing row 2) the active AD Leo M-dwarf star with the Sun suggests that UVA, UVB, and UVC for the flaring star amount to only (1-7)% of the total Solar radiation Figures (1a-1c) show the TOA UVA, UVB, and UVC net flux (W m -2 ). Figure 1 shows an increase with increasing stellar effective temperature as expected. The active AD-Leo flaring case is an especially strong emitter of UV due to its extremely active chromosphere. Modeled TOA UVB flux for Earth (~18.3 Wm -2 ) compare reasonably well with available observations (e.g. 16 ± 3 Wm -2 ; Benestad, 2006). Planetary Surface Radiation - In the chemistry module, the UVA and UVB net fluxes required for the photolysis scheme are calculated from the top layer downward via the twostream module with Rayleigh scattering. Figures (2a-2b) show UVA and UVB net flux (Wm -2 ) at the planetary surface as calculated in the chemistry module of this work. UVC is essentially zero at the surface so is not shown in Figure 2, and similarly for Figure 3. Generally, Figures 2a and 2b show an increase in planetary surface UV radiation with higher stellar temperatures, as for the TOA cases shown in Figure 1. Comparison with Earth Surface UV Radiation - Global satellite observations from 1992-1994 (Wang et al. 2000, their Figure 6b) suggest observed UVB surface radiation for Earth of ~1.4Wm -2 for cloud-free conditions. By comparison, Figure 2b suggests that our model over-estimates this value, calculating 2.3 Wm -2 UVB for the Earth control run. Uncertainties include, for example, our straightforward treatment of clouds whereby we adjust the surface albedo (see above) as well as the challenge of representing, for example, time-dependent and, for example, latitude-varying O3 photochemistry and UV absorption in a global-averaged 1D model. Ratio of Surface to TOA UV Flux - This ratio (R) is shown for the 1g and 3g cases in Figures 3a and 3b for UVA and UVB respectively. R is an inverse measure of the UV shielding of an atmosphere. Figure 3a suggests that UVA passes efficiently through the atmospheres considered, as expected, since most values of Rnet,UVA are >0.7. The UVA ratio is not greatly dependent on the stellar temperature. Figure 3b shows as expected a much stronger atmospheric extinction of UVB than for the UVA wavelengths, and there is now a clear dependency on stellar temperature. Weaker overhead O3 columns in the cool M-dwarf cases lead to a strong rise in the ratio in Figure 3b. For the 3g scenarios (circles), a lowering in the atmospheric column by a factor of three resulted in less UV shielding and a rise in the surface UV. 3. Chemical Analysis Here, we first compare briefly previous results (Segura et al., 2005) reported in the literature. Then, we discuss the general trends in column abundances of the biosignatures and related key species. Finally, we discuss the chemical responses for the vertical profiles that were also shown in Paper I. 3.1 Column Biomarkers (1g planets) Column O3 in Figure 4a (blue diamonds) mostly decreased with increasing star class (i.e., decreasing Teff of the star) related to less UVB, therefore there was a slowing in the photolysis of molecular O2 and hence a slowing in the Chapman cycle, a major source of O3. The O3 profile responses are discussed in more detail in section 3.6. The column values are shown in appendix 1. Column N2O in Figure 4a (red squares) generally increased with increasing star class. The cooler stars emit less UVB, which suggests a slowing in the photolytic loss of N2O in the planetary atmosphere and hence an increase in its abundance. Column CH3Cl in Figure 4a (green triangles) generally increased with increasing star class due to less OH, its major sink (see OH analysis, Table 2). The response is comparable to CH4 (discussed in next section), which has a similar photochemistry. Spectral features of CH3Cl, however, were too weak to be evident in the calculations of Paper 1 despite the enhanced column amounts for the cooler stars. 3.2 Column Biosignatures (3g planets) For the 3g planets, we assumed a constant surface pressure of 1 bar, which led to the total atmospheric column being reduced by a factor of three, as already mentioned (Figure 4b). The general trends for O3 and N2O remain for the 3g scenarios, that is, mostly similar to the corresponding 1g scenarios already discussed, although the reduced total column resulted in a cooling of the lower atmosphere due to a weaker greenhouse effect, as we will show (see Paper I also). The N2O 3g response is linked with enhanced UVB penetrating the reduced atmospheric column compared with 1g, which leads to more photolytic loss of N2O. A transport effect also took place. For the 3g case (with its lower model lid due to less atmospheric mass and higher gravity), the upward tropospheric diffusion of N2O was faster, for example, by about 50% in the mid to upper troposphere than the 1g case. This meant that N2O for the 3g case could reach the stratosphere faster, where it would be rapidly photolyzed. 3.3 Column Greenhouse Gases (1g planets) In this section, we discuss the planetary atmospheric column abundances of CH4 and H2O since they have a major impact on temperature via the greenhouse effect. Vertical profiles will be discussed later and can also be found in Paper 1. CH4 Column Response - Since the only source of CH4 in the model is fixed biomass surface emission, the CH4 response for the various runs is controlled by the main atmospheric CH4 sink, that is, removal via the hydroxyl (OH) radical. OH is affected by three main processes: OH Source(s) : for example, H2O+O( 1 D) 2OH (where O( 1 D) comes mainly from O3 photolysis in the UV). OH Recyling reactions in which NOx species can interconvert HOx (defined here as OH+HO2) family members via, for example, NO+HO2 NO2+OH. OH Sinks, for example, reaction with CH4 and CO (see e.g. Grenfell et al., 1999 for an overview). Figure 4c suggests a strong CH4 (green diamonds) increase with decreasing effective stellar temperature. Cooler stars are weak UV emitters, which favors a slowing in the OH source reaction above. Note also that greenhouse warming by the enhanced CH4 favors a damp troposphere (more evaporation) and, hence, all else being equal would favor actually more OH (via more H2O, see source OH reaction above). This is an opposing process which our results suggest is not the dominant effect. So, for a given model, calculating accurately the net effect will depend, for example, on a good treatment of, for example, the hydrological cycle, which is challenging for a global column model. To aid in understanding the CH4 response, which is controlled by OH, Table 2 summarizes the OH sources, sinks, recycling budget, and associated quantities. Table 2: Modeled (lowest atmospheric layer) and observed (surface) global-mean key species abundances (molecules cm -3 ) and reaction rates (molecules cm -3 s -1 ) affecting CH4 (and H2O) for various 1g scenarios. *From Lelieveld et al. (2002). Table 2: Modeled (lowest atmospheric layer) and observed (surface) global-mean key species abundances (molecules cm -3 ) and reaction rates (molecules cm -3 s -1 ) affecting CH4 (and H2O) for various 1g scenarios. *From Lelieveld et al. (2002). OH Abundances - Control run (1g Sun) OH abundances in Table 2 are within ~20% of global-mean observed OH proxies for Earth. Table 2 suggests a strong decrease in OH from left to right (i.e., for decreasing stellar effective temperature) especially for the M7 case. OH Source Reaction Rates - The source reaction rate (Sun) in Table 2, that is, O( 1 D)+H2O 2OH, is about 12 times weaker than indicated by the Whalley et al. (2010) study, which investigates (Earth) clean-air, tropical northern-hemisphere daytime OH. The factor 12 difference reflects a lowering due to day-night averaging in our global mean model (which accounts for ~factor 2 of the difference in OH) and the fact that the Whalley study considered tropical conditions. Concentrations of the trace specie O( 1 D) in the control run (=6x10 -8 ppbv at 30km) compared reasonably well with Earth observations (~3x10 -8 ppbv, Brasseur and Solomon, 2005). Table 2 suggests that the source reaction rate decreases from left to right, which is consistent with the decrease in OH. OH Recycling Reaction Rates - Our (Sun case) recycling reaction was comparable with that of the Whalley et al. (2010) study to within about 50%. Earlier (Earth) modeling studies, for example, that of Savage et al. (2001), suggest that the OH recycling reaction dominates the source reaction even in quite clean air-masses (NOx ~250pptv and below), which is somewhat in contrast to this and the Whalley study. In Table 2, the recycling reaction rates (like the source reaction) also decreases from left to right, which favors the decrease in OH, although the change in the source reaction is the stronger effect. For cooler stars, the recycling reaction becomes increasingly important compared with the source reaction, and it dominates for the ADL and M7 cases. HOx and NOx Ratios - These ratios are sensitive markers of changes in HOx and NOx chemistry and hence affect, for example, O3 cycles and CH4. The ratios (HO2/OH) and (NO2/NO) in Table 2 increase strongly for the cooler stars. These ratios are strongly affected by the concentration of O3, whose production via the Chapman mechanism (discussed in 3.5) weakens for the cooler stars. The ratios for the cooler stars are far from their 'Earth' values, so the interactions between HOx and NOx are much perturbed. This is a hint that the usual mechanisms that operate on Earth (e.g., O3 catalytic cycles etc.) may be very different for the cooler star scenarios - a good motivation for applying PAP as already mentioned. Atmospheric response for AD Leo - Although the 1g ADL scenario featured lower OH (Table 2) than for M5, ADL featured lower CH4 (Paper I) than M5. The upper layers (>60km) of the 1g ADL run showed very rapid destruction of CH4 via OH - about five times faster than for M5. This was consistent with the high Lymanα output of ADL leading to faster HOx enhancement via H2O photolysis. Water Column Response - Figure 4c suggests that the increased CH4 columns (green diamonds), with decreasing stellar effective temperature generally (except for M7), lead to higher H2O columns (green squares). Generally, for the cooler star scenarios, (up to and including M5), more CH4 greenhouse heating leads to more water evaporation in the troposphere, and in the stratosphere, faster CH4 oxidation leads to faster H2O production. However, for the M7 case (Figure 4c), although CH4 increased, surface temperature did not, which suggests a saturation in the CH4 greenhouse from M5 to M7, where the lower atmosphere becomes optically thick at very high CH4 abundances. Surface cooling from M5 to M7 is also seen in the temperature profiles in Paper 1 (their Figure 3). 3.4 Column Greenhouse Gases (3g planets) CH4 and H2O - Figure 4d has a similar format to Figure 4c but instead shows results for the 3g (instead of 1g in 4c)scenarios. The basic response to decreasing the effective stellar temperature at 3g is similar to the 1g case, that is, results suggest a column rise in CH4 and in H2O but with a drop-off in the latter for the cooler stars. To gain more insight into the effect of changing gravity, upon CH4, Table 3 shows the ratio (1g/3g) of the CH4 column and for the near-surface atmospheric OH abundance: Table 3: Ratio (1g/3g) for the CH4 atmospheric column and for near-surface OH (midpoint of lowermost gridbox) for the Sun compared with M-dwarf star scenarios. Table 3: Ratio (1g/3g) for the CH4 atmospheric column and for near-surface OH (midpoint of lowermost gridbox) for the Sun compared with M-dwarf star scenarios. Without calculating interactive photochemistry, a passive tracer would undergo a column reduction by a factor of three from 1g to 3g, because at constant surface pressure, increasing gravity by a factor of three leads to column collapse and a reduction in the overhead column by the same factor as the increase in gravity. In Table 3, therefore, a hypothetical, passive tracer (with no chemistry) would have a value of exactly three. The actual (with chemistry) CH4 column ratios (row 1), however, are all less than three. The reduction is consistent with faster chemical loss at 1g than at 3g. To investigate this further, OH ratios are shown in Table 3 (row 2). They mostly (except ADL) increase for the cooler stars, suggesting a lowering in the 3g OH abundances compared with the corresponding 1g cases for the cooler stars. This is consistent with faster chemical loss at 1g. The reduction in OH for the 3g scenarios implies that, for example, the increase in UVB due to weaker shielding of some 3g atmospheres ( favouring OH production) is out-weighed by the (opposing) feedback where reduced greenhouse warming at 3g led to a drier troposphere (disfavoring OH which is produced via O( 1 D)+H2O 2OH). This is confirmed by the water column (open circles in Figure 4d), which suggests that the 3g compared with 1g (Figure 4c) scenarios led to a weakening in the greenhouse effect and hence tropospheric cooling (as seen in Figure 2 of Paper 1) and a general lowering in the H2O column (due to more condensation) by around a factor of ten (Figure 4d) compared with the 1g case (Figure 4c). In general, however, note that responses in chemical abundances do not scale directly with the column reduction at 3g compared with 1g since the effects of, for example, photochemistry are important. Figures (4e, 4f) show the ratios (1g column/3g column) for biosignature and greenhouse gases respectively. The main point is that the values can lie far from a value of three (which would be expected for a passive tracer). This shows that it is important to include the effects of interactive chemistry. For the biosignature O3 there is some indication of an increase in the ratio shown in Figure 4e for the cooler stars, which will be the subject of future study. For CH3Cl (Figure 4e) and CH4 (Figure 4f) (which both have similar OH removal chemistry), the trend is downward for the cooler stars. The H2O (Figure 4f) scenarios are relatively more damp (with values >3) than for a purely passive tracer. This suggests more efficient production of H2O from CH4 for the cooler stars at 3g than at 1g, for example, due to more UV in the thinner, 3g atmospheres. 3.5 Column-Integrated Pathway Analysis Program (PAP) Results Figure 5 shows output of O3 cycles from the PAP. The cycles (divided into production and loss cycles) found have been quantified according to the rate of O3 production or loss through each particular cycle expressed as a percentage of the total rate of production or loss found by PAP (see also description of Appendix 1 below). Values are integrated over the model vertical domain. PAP analyses were performed for each of the 64 vertical column model chemistry levels, and the column-integrated values are shown in Figure 5. The full cycles referred to in Figure 5 can be found in the Appendix. Sun PAP Analysis Figure 5 confirms the expected result for O3 production, that is, the Chapman mechanism dominates over the smog mechanism. For O3 destruction, the column model suggests strong NOx contributions in the lower stratosphere, although an Earth GCM study (Grenfell et al., 2006) suggests a strong HOx contribution there. This result could reflect the challenge of 1D models of capturing 3D variations in photochemistry. Also, the column model does not include industrial emissions unlike the Earth 3D model. The result should be explored in future comparisons between the column model and 3D runs. Column-Integrated O3 (1g) Production Figure 5a suggests a change from a mainly Chapman-based O3 production for the 1g Sun and the warmer 1g M-dwarf stars, switching to a slower, mainly smogbased O3 production for the cooler stars (1g M5 and 1g M7). This was related to the decrease in UVB for the cooler star scenarios, since UVB is required to initiate the Chapman mechanism via photolysis of O2. Column integrated O3 (1g) Destruction - Figure 5a also suggests that the classical NOx and HOx cycles (see also Figures 6 and 7) that operate mainly in the stratosphere were the most dominant O3 loss pathways for the Sun and warmer M-dwarf scenarios. For the cooler stars scenarios, the enhanced CO concentrations led to a CO-oxidation cycle gaining in importance. Column O3 (3g) - Behavior at 3g (Figure 5b) was broadly similar to 1g, except at 3g both Chapman and smog were important O3 producers for the M5 case (i.e., not just smog as in the 1g case). Weaker atmospheric UVB absorption led to more penetration of UVB and hence an increased role for Chapman in the layers below. Column-Integrated Results Table for O3, N2O and CH4 - Appendix (1a-1c) shows the integrated column mean PAP output for O3, N2O, and CH4 respectively. Shown are (i) the column integrated rates (CIR) (in molecules cm -2 s -1 ) for all pathways found by PAP ('Found_PAP'), (ii) the CIR for only the pathways shown in the Appendix ('Shown_PAP') (shown are either the 5 dominant pathways or the first pathways that together account for >90% of the total formation or loss of found_PAP, whichever condition is fulfilled first), and (iii) the CIR as calculated in the chemistry scheme of the atmospheric column model ('total_chem'). Percent values for a particular cycle show its individual rate as a percentage of Found_PAP. Comparing these three CIR values, it can be seen that for the O3 production, which is relatively straightforward, the pathways found by PAP can account very well for the rate calculated in the column model chemistry module. For the O3 loss pathways, which are rather more complex than the production, PAP can still account for generally more than ~90% of the rate from the chemistry module. For the sometimes very complex CH4 pathways, with the value of fmin chosen for this study, PAP can account for only up to about 50% of the rate from the chemistry module. Further tests suggested that decreasing the PAP input parameter fmin (the minimum considered flux, currently set to 10 -8 ppbv s -1 for all runs) leads to improvement, but the resulting complex CH4 cycles are beyond the scope of this paper (see also 3.6.3). We now discuss the individual cycles for each scenario. O3 Column-Integrated Pathways Chemical pathways for the 1g Sun scenario in Appendix 1a mostly compare well with established results for Earth as discussed above. Appendix 1a suggests that for the 1g M0 scenario - due to less stellar UVB emission compared with the Sun - the Chapman mechanism for producing O3 is somewhat suppressed (89.2%) and a new CO sink ('CO oxidation 1', 7.4%) appears, since CO is abundant. For the 3g M0 scenario, results suggest that Chapman features more strongly (96.7%) in the thinner 3g atmosphere compared with the corresponding 1g case. HOx and NOx remain important chemical sinks for both the 3g and 1g cases. The active star (1g ADL) features a stronger Chapman contribution (97.2%) compared with 1g M0 since ADL is especially active in the UV, which is important for Chapman-initiation (via molecular oxygen) with only modest changes for the 3g ADL case. For cooler non-active stars (1g M5), large changes are apparent compared with the warmer star cases. Less UVB emission from the cool M5 star leads to a switch to smog-type O3 production ('smog 1', 57.8%). As discussed, the atmosphere is abundant in CH4 and CO. Thus, the 'CO-oxidation-1' cycle is an important O3 loss pathway (36.8%). For the (3g M5) case, the thinner total column at 3g compared with 1g leads to a rise in UV, which is consistent with more Chapman O3 (47.8%) production than the 1g case (7.5%). For O3 loss, a complex CH4 oxidation pathway involving CH3OOH becomes important (46.8%), which is not evident at 1g. The changed UV environment leads to a modest rise in HOx in the upper troposphere at 3g. Finally, for the coolest M-dwarf case (1g M7), O3 production occurs via numerous types of smog mechanisms involving the oxidation of different VOCs, for example, CO, HCHO, and CH3OOH. CO smog cycles become a key means of producing O3 especially for the cooler stars. Like CH4, an important sink for CO is the reaction with OH. As discussed, weakening UV emissions for the cooler stars leads to less OH and therefore an enhanced abundance of CO. Near the surface, CO mixing ratios correspond to: 0.09 (Sun), 9.0 (M4), 64 (ADL), and 426 (M7) parts per million (ppm). O3 loss also involves NOx cycles but also a smog mechanism ('smog 7') where O3 is the net oxidant, which is consumed to oxidize CH4 and a CO oxidation cycle. Smog cycles have larger rates for the M5 and M7 scenarios than for the Sun and M0 scenarios. This is because the important smog 1 cycle (producing O3) is in competition with the CO-oxidation 1 pathway (destroying O3). At high O3 concentrations (for the Sun and M0 scenarios), (i) the reaction NO+O3 NO2+O2 shifts the NOx family to favor NO2. The reduction in NO leads to a slowing in the key reaction NO + HO2 NO2+OH and hence slows the smog 1 cycle. Also at high O3 concentrations, (ii) the reaction HO2+O3 OH + 2O2 favors the CO-oxidation 1 pathway. These two effects together, favor large smog rates for the M5 and M7 scenarios. In summary, total vertically integrated O3 production and loss rates for the 1g Sun (=1.9x10 13 molecules cm -2 s -1 ) are 68 times larger than for the 1g M7 case (=2.8x10 11 molecules cm -2 s -1 ), which illustrates the change in the dominance from the rather fast Chapman chemistry to the slower smog mechanism. N2O Column-Integrated Pathways The main result of the PAP is that loss pathways from the N2O 'viewpoint' are non-catalytic for all scenarios. In other words, loss occurs mainly directly via photolysis, which can be calculated from the photolysis rate without performing a PAP analysis for N2O. We therefore only show (Appendix 1b) one scenario as an illustration, that is, the Sun scenario, which confirms results measured for Earth, that is, ~95% loss via photolysis (i.e., the sum of the 4 cycles involving N2O photolysis in Appendix 1b), and ~5% loss via catalytic reaction with O( 1 D) is similar to observed values quoted for Earth (e.g., 9095% photolytic loss, 5-10% via reaction with O( 1 D), IPCC Third Assessment Report, see discussion to Table 4.4). The PAP finds no formation pathways of N2O via inorganic reactions, as expected since these are insignificant compared with surface biogenic input. For the M-dwarf scenarios, photolysis similarly remained the main removal mechanism, and the overall column integrated rate of removal decreased by about a factor of two for the M7 compared with the Sun case since the cooler stars emit less UV. CH4 Column-Integrated Pathways Appendix 1c shows the PAP output for CH4. Results suggest a large number of complex removal pathways that oxidize CH4. PAP found no in-situ production pathways, since there are no inorganic reactions in our model that produce CH4 in the atmosphere. The net removal can involve either complete oxidation of CH4 to its stable combustion products: H2O and CO2 (as in the 'oxidation 2O2-a' pathway for the 1g Sun scenario) but can also involve only partial oxidation, for example, to intermediate organic species such as formaldehyde (HCHO), for example, as in the 'Oxidation O2' pathway (1g M0). Clearly, more complete oxidation is favored in oxidizing environments, for example, damp atmospheres with strong UV where OH is abundant. The choice of oxidant in the net reaction will depend on the central star's particular UVB radiation output and its ability to release, for example, HOx, Ox, or NOx from their reservoirs in the planetary atmosphere. Importantly for O3 photochemistry, there are CH4 cycles in which O3 itself is the oxidant in the net reaction (see e.g. net reaction for several cycles from the 3g Sun case). This is an example where CH4-oxidation does not lead to the more familiar O3 (smog) production, but to the reverse effect where O3 is consumed. Many of the CH4 pathways are NOx-catalyzed, as on Earth, although this is not the case for all scenarios (e.g., pathway 'CH3OOH-d' (3g M5) does not include NOx). 3.6 Altitude-Dependent PAP Results In this section we will present PAP results from the same scenarios as the previous section. However, here we will discuss the contribution of the PAP cycles as profiles varying in the vertical. 3.6.1 Vertical Changes in Ozone (O3) Production and Loss Cycles Figure 6 shows the altitude-dependent PAP results, comparing production and loss pathways for the Earth case (Figures 6a, 6b) with the M7 case (Figures 6c, 6d). Similarly, Figures 7a, 7b compare ADL (1g) with M5 (1g) (Figures 7c, d). In Figures 6 and 7, the logarithmic x-axis shows the rate of change of O3 associated with a particular cycle found by PAP, in molecules cm -3 s -1 . The black and white text labels on these Figures indicate the names of the O3 pathways, which can be found in Appendix 1a. Note that the logarithmic x-axis where results are plotted cumulatively (meaning to estimate the contribution of a pathway at a particular height one must subtract its left-hand side x-axis boundary from its right-hand side x-axis boundary) in Figures 6 and 7 means that the pathways shown on the right-hand side of the Figure can make up a strong overall contribution to the net rate of change despite having only a thin section (relatively small area). For the Earth results (Figures 6a, 6b), the O3 production and loss rates output by PAP compare well with middle atmosphere O3 budgets derived for Earth, see for example the work of Jucks et al. (1996), their Figure 4. The Earth results (Figure 6a) in the top model layer show an uppermost region of O3 production (thin, blue stripe), which arose due to the single reaction: O2+O( 3 P)+M O3+M. This is linked with the model's upper boundary condition, where a downward flux of CO and O( 3 P) is imposed. This is done to parameterize the effects of CO2 photolysis (forming CO and O( 3 P)), which takes place above the model's lid, for example, above the mid mesosphere. The resulting enhanced O( 3 P) in the uppermost model layer favors the direct O3 formation pathway found by PAP. The enhanced O3 source was balanced by an increase in the photolysis rate of O3, and therefore the abundance decreased smoothly with altitude as expected. The effect of varying the upper boundary will be the subject of future work. NOx loss cycles dominate (>60%) the Earth lower stratosphere; HOx cycles are more important in the upper stratosphere. For the 3g case (3g Sun), the O3 production pathways are similar to those of Earth, but HOx destruction is stronger (~70%) in the lower stratosphere, which is consistent with more UV penetration (releasing HOx from its reservoirs) for the thinner (3g) atmospheric column compared with the 1g case. The enhanced tropospheric HOx, which also stimulated the 'CO oxidation 1' cycle, accounted for (30-50%) of tropospheric O3 loss. For the warm M-dwarf star scenarios (e.g., 1g M0) - here, like the control (1g Sun), smog 1 dominates 50-60% of the O3 production in the troposphere (with 10-20% arising from CH4 smog cycles). The influence of the smog mechanism extends to high altitudes (up to about 20km) compared with the Earth control (which extends up to about 16km). 'Chapman 1' (Appendix 1a) dominated the stratosphere. O3 loss was dominated by the 'CO-oxidation 1' pathway (60-80%) in the troposphere, NOx loss pathways in the mid-stratosphere, and HOx loss pathways in the upper stratosphere. For the 3g case (3g M0), the 'smog 1' pathway contributes ~70% of O3 production in the troposphere with the ~(10-15%) remainder in the troposphere coming from CH4 smog pathways. 'Chapman 1' is dominant in the stratosphere, and 'Chapman 2' is dominant in the uppermost layers (see discussion above for Earth run 1). O3 loss, like the 1g case, was dominated by 'CO-oxidation 1' pathway in the troposphere (~90%) with different HOx cycles important for loss in the upper levels. In Figure 7, ADL O3 photochemistry production (Figure 7a) is rather similar, for example, to the Earth control (1g Sun) case (Figure 6a) in that Chapman production dominates the stratosphere and smog in the troposphere. However, for the 1g M5 run, results are very different from what occurs on Earth, since O3 production is now dominated by the smog mechanism through much of the atmosphere. For ADL, O3 production occurred mostly via 'Smog 1' (70-80%) in the troposphere, with various CH4 smog pathways making up between 10-20% in this region. 'Chapman 1' dominated the stratosphere. O3 loss was again dominated by 'CO-oxidation 1' in the troposphere (70-90%) with a variety of HOx cycles important for loss in the upper levels. Intense Lymanα radiation favored some enhancement of H2O photolysis (hence more O3 loss via HOx) in the 1g ADL scenario compared to, for example, the Earth control (run 1), but the effect was quickly damped (in the uppermost ~2 model layers) and the overall change in O3 was small. For the corresponding 3g case (3g ADL), O3 production pathways did not change greatly with altitude compared with the 1g case. O3 loss pathways were also rather similar to the 1g ADL case, with the 'CO-oxidation 1' pathway for 3g ADL dominating the lower atmosphere. The cooler stars (M5, M7) show significant changes in the O3 photochemistry compared with the other M-dwarf scenarios. The rather weak UV radiation of these cooler stars means that Chapman chemistry (requiring UV to break the strong O2 molecule) is now only significant (up to ~50% O3 production) (1g M5) in the uppermost (>60km) altitudes. The 'CO-smog 1' pathway, however, is now significant over all altitudes, accounting for 60% of O3 production in the troposphere and about 30% in the upper atmosphere. A variety of CH4 smog pathways make up most of the remaining O3 production (1g M5). For O3 loss, the 'CO-oxidation 1' pathway is again significant (50-70%) in the lower half of the model domain, whereas a variety of NOx cycles are important in the upper regions. For the coolest star considered (1g M7), the O3 abundance is determined by mainly CO and CH4 oxidation. First, 'classical' smog production - with OH as the oxidant (mainly CO smog 1 and various CH4 oxidation pathways)--produce O3 but, on the other hand, O3 in the M7 scenario can also act as an oxidant in pathways that oxidise, for example, CH4 and CO. The M7 case (Figures 6c-6d) shows that the CO smog mechanism dominates the O3 production, whereas the CO oxidation cycle and the classical NOx cycle dominate the O3 loss. Near the surface, some direct removal of O3 occurred via the reaction: NO+O3 O2+NO2 (Figure 6d). On Earth, more NOx usually leads to more O3 production via the smog mechanism; the direct removal reaction is, however, sometimes important at high NOx abundances, for example, in city centers. In our M7 scenario, which does not have industrial NOx emissions, an important source of lower atmosphere NOx is from lightning. For the cool M-dwarf 3g case (3g M5, not shown), the 'CO-smog 1' and 'Chapman 1' pathways make almost equal contributions to the O3 production budget in the middle atmosphere. 'Chapman 1' contributes up to ~80% of local production in the upper levels (where UV is abundant), whereas the smog mechanism contributes up to ~70% in the lower layers. The smog contribution has a minimum of ~20% local production near the cold trap, which is consistent with low temperatures and a rather low OH abundance. For the O3 loss pathways, results suggest an increase in complex CO and CH4 smog pathways that consume O3. 3.6.2 N2O - For all scenarios, non-catalytic photolytic removal (>90%) is the main loss mechanism in the stratosphere. Catalytic removal involving reaction with O( 1 D) makes up the remainder (occurring mostly in the mid to upper stratosphere) of the N2O loss. 3.6.3 CH4 - Results suggest that a large number of loss pathways occur near the cold trap. For example, at 16km (1g Sun), the CH4 pathways found by PAP with the value of fmin chosen in this study could account for only about 20% of the total CH4 change calculated in the column model. Low OH abundances and cold temperatures in this region are consistent with rather slow oxidation and a resulting complex mix of only weakly oxidized organic species with individual pathway contributions lying below the PAP threshold criteria chosen for the present study, but whose net effect is important. For this study, the PAP detection threshold was set to fmin = 10 -8 ppbv/s. OH-initiated oxidation of CH4 is more favored on the lower layers but with relatively more O( 1 D)-initiated oxidation on the upper levels, where this species is more abundant. A test run (not shown) where the fmin value is decreased to 10 -9 ppbv/s was found to address the above problem, that is, PAP was then able to account, for example, three times more CH4 net change (for the Earth run), though with a notable increase in the overall number of pathways, each with small contributions to net the overall chemical change, beyond the scope of our work. 3.7 Comparison with previous studies - Compared with the results of Segura et al. (2005), our results are similar for N2O and CH4 within 10-20% for the inactive (e.g., M4) and active (ADL) cases. For O3, our atmopsheric column amounts are ~40% thicker (=270DU) compared with the Segura et al. (2005) (=164DU) value for the ADL case. This results from changes in our photochemical scheme, including, for example, the parameterization of the lower boundary flux of H2, as discussed in Paper I. Also our stellar insolation corresponds to 1366Wm-2 at the TOA, whereas Segura et al. (2005) scaled their incoming spectrum to obtain a surface temperature of 288K. 4. Spectral Detectability of Biomarkers O3 - Paper I shows that the detection of O3 is challenging especially for M7. To better understand O3 detectability, improved stellar spectra for the cooler stars in the (E)UV are desirable especially in the UVB and UVC, where O3 responds sensitively. M7 stars are statistically older and burn more slowly compared with lower spectral class stars, which means more developed convection zones and possibly larger differences in UV between flaring and quiet states for M7 than considered in our work (see Reiners et al. 2012). N2O - Clearly, the most favorable (planet/star) contrast ratios are associated with cool stars such as M7. However, Paper I shows that some spectral absorption features can be weakened, partly due to the large CH4 abundance, which warms the stratosphere. The N2O spectral features were weak for the scenarios analyzed. 5. Conclusions · The potential complex-life biosignature O3 has a very different photochemistry for planets orbiting in the HZ of cool M-dwarf stars compared to that of Earth since the key mechanism switches from Chapman production to slower, smog production. Expected responses of O3 produced by the smog cycle (which could be favored by increases in HOx and NOx, e.g., by cosmic rays) could be very different than Chapman-produced O3 (where HOx and NOx catalytically destroy O3). This is important to consider when predicting and interpreting O3 spectral features. · The simple microbial-life biosignature N2O increases for the cooler stars, mostly related to weaker photolytic loss of N2O via weaker UVB in the middle atmosphere, as found too by earlier studies. In some cases, however, variations in transport become important. The amount of N2O in the middle atmosphere depends on the UV and on the rate at which this species can be transported upwards from the troposphere into the stratosphere where it is photolyzed. · The greenhouse gas CH4 responses and its removal pathways become complex especially for the cooler stars. CH4 abundances generally increase for the cooler stars, a result also found in earlier studies, due to a lowering in OH, its main sink, which is associated mainly with a weakening in the main OH source reaction that requires UVB. · The potential vegetation biosignature CH3Cl is enhanced in abundance by more than three orders of magnitude compared with the Earth run especially for cool M-star scenarios associated with low OH since reaction with this species is the main sink (see also CH4 above). Like earlier studies, our results suggest that its spectral features are nevertheless very weak. · Comparison of the 1g and 3g scenarios suggests that it is important to include interactive photochemistry when calculating biosignatures and greenhouse gas abundances. Reducing the mass of the atmosphere by, for example, a factor of three does not always lead to a reduction in, for example, biosignatures and greenhouse gases by a factor of three, due to interactive climatephotochemical effects. Acknowledgements This research has been partly supported by the Helmholtz Gemeinschaft (HGF) through the HGF research alliance \"Planetary Evolution and Life.\" F. Selsis and P. von Paris acknowledge support from the European Research Council (Starting Grant 209622: E3ARTHs). References Bean, J. L., Desert, J.-M., Kabarg, P., Stalder, B., Seager, S., Miller-Ricki-Kempton, E., Berta, Z. K., Homeier, D., Walsh, S., and Seifahrt, A. 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World Meteorological Organisation, (WMO) Scientific Assessment of Ozone Depletion 1994 (1995) Global ozone research and monitoring project report No.37, Geneva. Figure Captions Figure 1: Planetary Global Mean Top-of-Atmosphere Incoming Radiation (W m -2 ) for UVA (Figure 1a), UVB (Figure 1b) UVC (Figure 1c) for Earth's gravity. Figure 2: As for Figure 1 but at the planetary surface for UVA (Figure 2a), UVB (Figure 2b) and UVC (Figure 2c) for Earth's gravity (1g). Figure 3: As for Figure 1 but showing the Ratio (Surface/TOA) (at 1g and 3g) for UVA (Figure 3a), UVB (Figure 3b) and UVC (Figure 3c) Radiation (1g). Figure 4a: Atmospheric columns (Dobson Units, DU) (1g) of biosignatures, ozone (O3), nitrous oxide (N2O) and methyl chloride (CH3Cl). Figure 4c: As for Figure 4a but for column CH4 (Dobson Units, DU) and H2O (DU). Figure 4b: As for Figure 4a but for (3g) scenarios (same surface pressure (=1 bar) as 1g). Figure 4d: As for Figure 4c but for (3g) scenarios assume same surface pressure (1 bar) as 1g cases. Figure 4f: Column (1g/3g) ratio for the same greenhouse gases as shown in Figure 4c. Figure 4e: Column (1g/3g) ratio for the same biosignatures as shown in Figure 4a. Figure 5: Pathway Analysis results for Global Mean Ozone Sources (Figure 5a) and Sinks (Figure 5b) for the Sun and for the M-dwarf star scenarios (1g) calculated by the Pathway Analysis Program. The pathways are shown in the PAP Tables in the Appendix. Figure 6: Pathway Analysis results showing cumulative contribution of altitude-dependent O3 production and loss pathways for the 1g Sun (Figures 6a, 6b) and for the 1g M7 scenarios (Figures 6c, 6d) plotted in the vertical and shown in (molecules cm -3 s -1 ). Black and white labels on the Figure correspond to the names of the individual cycles as shown in Appendix 1. Logarithmic x-axis tick labels correspond to factors of x2 x5 and x8 respectively. Note that the model vertical grid is variable depending on, e.g., greenhouse gas heating, which leads to an expansion in the vertical for cooler effective stellar temperatures. Figure 7: As for Figure 6 but for the 1g ADL (Figures 7a, 7b) and for the 1g M5 scenarios (Figures 7c, 7d). 1000 100 10 0.1 š 0.01 M5 ; M0 ADL M4 ; TOA Net Radiation (Wm-?) Sun M4 M5 M7 M0 ADL ; ; ; ; Stellar Effective Temperature (K) Figure Ic: UVC TOA Net Radiation (Wm-2) Figure Ic: UVC TOA Net Radiation (Wm-2) 10 Sun ; Stellar Effective Temperature (K) ; Figure 2b: 1g UVB Surface Net Radiation (Wm-?) Figure 2b: 1g UVB Surface Net Radiation (Wm-?) Figure 3b: 1g and 3g (UVB SurfacelUVB TOA) Figure 3b: 1g and 3g (UVB SurfacelUVB TOA) Figure 4b: Column (DU) Biomarkers (3g) Figure 4b: Column (DU) Biomarkers (3g) Figure 4f: (1g/3g) column methane and water Figure 4f: (1g/3g) column methane and water Figure 5b: Ozone Atmospheric Production and Loss (3g) Figure 5b: Ozone Atmospheric Production and Loss (3g) Figure 6a: 1g Sun O3 Production Figure 6b: 1g Sun O3 Destruction Figure 6a: 1g Sun O3 Production Figure 6b: 1g Sun O3 Destruction Chapmanhi Figure 6c: 1g M7 O3 Production Figure 6d: 1g M7 O3 Destruction Figure 6c: 1g M7 O3 Production Figure 6d: 1g M7 O3 Destruction rate (molecule cm -3 -1 s ) rate (molecule cm -3 -1 s ) Figure 7a: 1g ADL O3 Production Figure 7b: 1g ADL O3 DestructionFigure 7c: 1g M5 O3 Production Figure 7d: 1g M5 O3 Destruction Figure 7a: 1g ADL O3 Production Figure 7b: 1g ADL O3 DestructionFigure 7c: 1g M5 O3 Production Figure 7d: 1g M5 O3 Destruction rate (molecule cm -3 -1 s ) rate (molecule cm -3 -1 s ) Appendix 1: (a) Ozone Pathways ' Found_PAP ' denotes the Column Integrated Rates (CIR) (in molecules cm -2 s -1 ) of change shown for production and loss for all atmospheric pathways found by PAP. ' Shown_PAP ' denotes the CIR only for the pathways shown in this Appendix. Shown are either the 5 dominant pathways or the first pathways which together account for >90% of Found_PAP, whichever criterion is fulfilled first. ' Total _chem ' denotes the CIR as calculated in the chemistry scheme of the atmospheric column model. Percent values for a particular cycle show its individual rate as a percentage of Found_PAP. 1g Sun Ozone Production: Found_PAP =1.88x10 13 , Shown_PAP=1.87 x10 13 , Total_chem =1.88x10 13 1g Sun Ozone Production: Found_PAP =1.88x10 13 , Shown_PAP=1.87 x10 13 , Total_chem =1.88x10 13 1g Sun Ozone Loss: Found_PAP =1.81x10 13 , Shown_PAP=1.06 x10 13 , Total_chem=1.88x10 13 3g Sun Ozone Production: Found_PAP = 1.28x10 13 , Shown_PAP=1.27 x10 13 , Total_chem = 1.28x10 13 3g Sun Ozone Loss: Found_PAP =1.20x10 13 , Shown_PAP=6.06x10 12 , Total_chem = 1.26x10 13 1g M0 Ozone Production : Found_PAP =1.65x10 12 , Shown_PAP=1.63x10 12 , Total_chem = 1.70x10 12 1g M0 Ozone Production : Found_PAP =1.65x10 12 , Shown_PAP=1.63x10 12 , Total_chem = 1.70x10 12 1g M0 Ozone Loss : Found_PAP =1.52x10 12 , Shown_PAP=8.92x10 11 , Total_chem = 1.64x10 12 1g M0 Ozone Loss : Found_PAP =1.52x10 12 , Shown_PAP=8.92x10 11 , Total_chem = 1.64x10 12 3g M0 Ozone Production : Found_PAP =1.31x10 12 , Shown_PAP=1.30x10 12 , Total_chem = 1.32x10 12 3g M0 Ozone Loss : Total = Found_PAP =9.75x10 11 , Shown_PAP=5.27x10 11 , Total_chem = 1.20x10 12 1g AD-Leo Ozone Production : Found_PAP =1.72x10 12 , Shown_PAP=1.71x10 12 , Total_chem = 1.72x10 12 1g AD-Leo Ozone Loss : Found_PAP =1.43x10 12 , Shown_PAP=6.71x10 11 , Total_chem = 1.69x10 12 3g AD-Leo Ozone Production: Found_PAP =1.56x10 Chapman 1 (99.5%) , Shown_PAP=1.56x10 Smog 1 12 (0.4%) 3g AD-Leo Ozone Loss: Found_PAP =1.13x10 12 , Shown_PAP=4.78x10 11 , Total_chem = 1.34x10 12 3g AD-Leo Ozone Loss: Found_PAP =1.13x10 12 , Shown_PAP=4.78x10 11 , Total_chem = 1.34x10 12 1g M5 Ozone Production : Found_PAP =2.92x10 11 , Shown_PAP=2.72x10 11 , Total_chem = 3.24x10 11 1g M5 Ozone Loss : Found_PAP =2.57x10 11 , Shown_PAP=2.19x10 11 , Total_chem = 2.91x10 11 12 12 , Total_chem = 1.56x10 3g M5 Ozone Production : Found_PAP =1.35x10 11 , Shown_PAP=1.32x10 11 , Total_chem = 1.42x10 11 Chapman 1 (47.8%) Smog 1 (47.6%) Chapman 2 (2.4%) 3g M5 Ozone Loss : Found_PAP =1.19x10 11 , Shown_PAP=1.08x10 11 , Total_chem = 1.28x10 11 1g M7 Ozone Production : Found_PAP =2.79x10 11 , Shown_PAP=2.59x10 11 , Total_chem = 2.92x10 11 1g M7 Ozone Production : Found_PAP =2.79x10 11 , Shown_PAP=2.59x10 11 , Total_chem = 2.92x10 11 1g M7 Ozone Loss : Found_PAP =2.43x10 11 , Shown_PAP=2.01x10 11 , Total_chem = 2.74x10 11 1g M7 Ozone Loss : Found_PAP =2.43x10 11 , Shown_PAP=2.01x10 11 , Total_chem = 2.74x10 11 1b. Nitrous Oxide Pathways (Sun only) 1g Sun Nitrous Oxide Loss : Found_PAP =9.63x10 8 , Shown_PAP=8.78x10 8 , Total_chem = 1.15x10 9 N2O-NOx (69.5%): 2(N2O +h ν → N 2 + O( 3 P)) O( 3 P) + O2 + M → O 3 + M NO2 + O( 3 P) → NO + O 2 NO + O3 → NO 2 + O2 net: 2N2 O → O 2 + 2N2 N2O-Ox (9.6%): 2(N2O +h ν → N 2 + O( 3 P)) O( 3 P) + O2 + M → O 3 + M O( 3 P) + O3 → O 2 + O2 net: 2N2 O → O 2 + 2N2 N2O-HOx-1 (3.4%): 2(N2O +h ν → N 2 + O( 3 P)) 2(O( 3 P)+O2 +M → O 2+M) OH + O3 → HO 2 + O2 HO2+O3 → OH+O 2+O2 net: 2N2 O → O 2 + 2N2 N2-O( 1 D) (5.4%): 2( N2O +O( 1 D) → N 2 + O2) 2(O( 3 P)+O2 +M → O 3+M) 2(O3 + h ν → O 2 + O( 1 D)) O2 + h ν → O( 3 P) + O( 3 P) net: 2N2 O → O 2 + 2N2 N2O-HOx-2 (3.3%): 2(N2O + h ν → N 2 + O( 3 P) O( 3 P) + O2 + M → O 3 + M HO2 + O( 3 P) → OH + O 2 OH + O3 → HO 2 + O2 net: 2N2 O → O 2 + 2N2 1c. Methane Pathways 1g Sun Methane Loss : Found_PAP =1.15x10 10 , Shown_PAP=7.68x10 9 , Total_chem = 1.24x10 11 Oxidation 3O2 -a (29.6%): 2(CH4+OH → CH3+H 2 O) 2(CH3+O2+M → CH3O2+M) 2 (CH3O2+NO → H3CO+NO2) 2 (H3CO+O2 → H2CO+HO2) 2 (H2CO + h ν → H2 + CO) NO + HO2 → NO2 + OH 3 (NO2 + h ν → NO + O( 3 P)) 3 (O( 3 P) + O2 + M → O3 + M) 3 (HO2+O3 → OH+O2+O2) 2 (CO + OH → CO2 + H) 2 (H + O2 + M → HO2 +M) net:2CH4+3O2 → 2H2O+ 2H2+2CO2 Oxidation 6O2 (23.0%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + h ν → H2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 2 (NO + HO2 → NO2 + OH) 3 (NO2 + h ν → NO + O( 3 P)) 3 (O( 3 P) + O2 + M → O3 + M) net: CH4 + 6O2 → H2O + 3O3 + H2 + CO2 Oxidation 8O2 (6.0%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO HCO + O2 → HO2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 3(NO + HO2 → NO2 + OH) 4(NO2 + h ν → NO + O( 3 P)) 4(O( 3 P) + O2 + M → O3 + M) net: CH4 + 8O2 → 2H2O + 4O3 + CO2 Oxidation 2O2-a (6.2%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO NO + HO2 → NO2 + OH 2(NO2 + h ν → NO + O( 3 P) ) 2(O( 3 P) + O2 + M → O3 + M) HCO + O2 → HO2 + CO 2(HO2+O3 → OH+O2+O2) CO + OH → CO2 + H H + O2 + M → HO2 +M net:CH4 +2O2 → 2H2O + CO2 Oxidation 3O2-b (2.2%) 2(CH4 + OH → CH3 + H2O) 2(CH3+O2+M → CH3O2+M) 2(CH3O2+NO → H3CO+NO2) 2(H3CO+O2 → H2CO+HO2) 2(H2CO + hv → H2 + CO) 2(CO + OH → CO2 + H) 2(H + O2 + M → HO2 +M) 4(NO + HO2 → NO2 + OH) 3(NO2 + h ν → NO + O( 3 P)) 3(NO2 + O( 3 P) → NO + O2) net: 2CH4 + 3O2 → 2H2O + 2H2 + 2CO2 3g Sun Methane Loss : Found_PAP =3.21x10 10 , Shown_PAP=1.63x10 10 , Total_chem = 1.24x10 11 Oxidation CH3OOH-a (26.5%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 H2O + O( 1 D) → OH + OH CO + OH → CO2 + H H + O2 + M → HO2 +M O3 + h ν → O2 + O( 1 D) net: CH4 + O3 + CO → CH3OOH + CO2 Oxidation 2O2-b (7.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+OH → H3CO+HO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO HCO + O2 → HO2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 4(HO2 + O( 3 P) → OH + O2) 2(O2 + h ν → O( 3 P) + O( 3 P)) net:CH4 +2O2 → 2H2O+CO2 Oxidation 2O2-c (5.4%) CH4 + O( 1 D) → CH 3 + OH CH3 + O2 + M → CH 3O2 + M CO + OH → CO 2 + H H + O2 + M → HO 2 +M CH3O2 +OH →H 3CO+HO2 H3CO + O2 → H 2CO + HO2 3(HO2 + O( 3 P) → OH + O 2) OH + HO2 → H 2O + O2 H2 CO + OH → H 2O + HCO HCO + O2 → HO 2 + CO O( 3 P) + O2 + M → O 3 + M O3 + h ν → O 2 + O( 1 D) 2(O2 + h ν → O( 3 P) + O( 3 P)) net:CH4 +2O2 →2H 2O+CO2 Oxidation O3-a (5.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M H2CO + h ν → H2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 2(HO2+O3 → OH+O2+O2) net: CH4 +O3 → H2O+H2+CO2 Oxidation O3-b (5.3%) CH4 + OH → CH 3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 CH3OOH+h ν → H 3CO+OH H3CO+O2 → H 2CO+HO2 H2CO + h ν → H 2 + CO CO + OH → CO 2 + H H + O2 + M → HO2 +M HO2 + O3 → OH + O2 + O2 net: CH4 +O3 → H2O+ H2+CO2 1g M0 Methane Loss: Found_PAP =3.25x10 10 , Shown_PAP=1.93x10 10 , Total_chem = 1.24x10 11 Oxidation 3O2-a (24.4%) Oxidation CH3OOH-b (12.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 H2O + O( 1 D) → OH + OH 2(H + O2 + M → HO2 +M) NO + HO2 → NO2 + OH 2(CO + OH → CO2 + H) NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + 2CO + 2O2 → CH3OOH + 2CO2 Oxidation 6O2 (6.1%) Oxidation O2 (10.4%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + O3 → OH + O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M net: CH4 + O2 → H2CO+H2O Oxidation CH3OOH-c (5.7%) 2(CH4 + OH → CH3 + H2O) 2(CH3+O2+M → CH3O2+M) CH3O2+OH → CH3OOH+O2 CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) H2O + O( 1 D) → OH + OH net: 2CH4 + 2O2 → H2CO + H2O + CH3OOH 3g M0 Methane Loss : Found_PAP =5.67x10 10 , Shown_PAP=2.16x10 10 , Total_chem = 1.24x10 11 3g M0 Methane Loss : Found_PAP =5.67x10 10 , Shown_PAP=2.16x10 10 , Total_chem = 1.24x10 11 1g ADL Methane Loss : Found_PAP =6.14x10 10 , Shown_PAP=3.95x10 10 , Total_chem = 1.24x10 113g ADL Methane Loss : Found_PAP =7.06x10 10 , Shown_PAP=3.31x10 10 , Total_chem = 1.24x10 11 1g ADL Methane Loss : Found_PAP =6.14x10 10 , Shown_PAP=3.95x10 10 , Total_chem = 1.24x10 113g ADL Methane Loss : Found_PAP =7.06x10 10 , Shown_PAP=3.31x10 10 , Total_chem = 1.24x10 11 1g M5 Methane Loss: Found_PAP =5.65x10 10 , Shown_PAP=4.04x10 10 , Total_chem = 1.24x10 11 1g M5 Methane Loss: Found_PAP =5.65x10 10 , Shown_PAP=4.04x10 10 , Total_chem = 1.24x10 11 3g M5 Methane Loss : Found_PAP =7.26x10 10 , Shown_PAP=4.57x10 10 , Total_chem = 1.24x10 11 3g M5 Methane Loss : Found_PAP =7.26x10 10 , Shown_PAP=4.57x10 10 , Total_chem = 1.24x10 11 Oxidation CH3OOH-d (24.5%) 2(CH4 + O( 1 D) → CH3 + OH) 2(CH3+O2+M → CH3O2+M) 2(CO + OH → CO2 + H) 2(H + O2 + M → HO2 +M) 2(CH3O2+HO2>CH3OOH+O2) 2(O( 3 P) + O2 + M → O3 + M) 2(O2 + h ν → O2 + O( 1 D)) O2 + h ν → O( 3 P) + O( 3 P) net: 2CH4 + 2CO + 3O2 → 2CH3OOH + 2CO2 Oxidation CH3OOH-e (23.7%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 2(H + O2 + M → HO2 +M ) NO + HO2 → NO2 + OH 2( CO + OH → CO2 + H) NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + 2CO + 2O2 → CH3OOH + 2CO2 Oxidation H2O2-b (4.5%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H H + O2 + M → HO2 +M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + HO2 → H2O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + CO + 2O2 → H2CO + H2O2 + CO2 Oxidation H2O2-a (5.8%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H H + O2 + M → HO2 +M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + HO2 → H2O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) H2CO + h ν → H2 + CO net:CH4 +2O2 →H 2O2+H2+CO2 Oxidation H2O2-c (4.4%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + h ν → HCO + H HCO + O2 → HO2 + CO NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) 2(H + O2 + M → HO2 +M) 2(HO2 + HO2 → H2O2 + O2) net: CH4 +3O2 →2H 2O2+CO2 1g M7 Methane Loss : Found_PAP =5.48x10 10 , Shown_PAP=3.40x10 10 , Total_chem = 1.24x10 11 1g M7 Methane Loss : Found_PAP =5.48x10 10 , Shown_PAP=3.40x10 10 , Total_chem = 1.24x10 11 Appendix 1 Ozone column in Dobson Units (DU) for the 1g, 3g scenarios corresponding to the values plotted in Figures 4a, 4b as a function of stellar effective temperature (Teff) (K). UVB, and UVC radiation compared with the Sun. Comparing row 2) the active AD Leo M-dwarf star with the Sun suggests that UVA, UVB, and UVC for the flaring star amount to only (1-7)% of the total Solar radiation Figures (1a-1c) show the TOA UVA, UVB, and UVC net flux (W m -2 ). Figure 1 shows an increase with increasing stellar effective temperature as expected. The active AD-Leo flaring case is an especially strong emitter of UV due to its extremely active chromosphere. Modeled TOA UVB flux for Earth (~18.3 Wm -2 ) compare reasonably well with available observations (e.g. 16 ± 3 Wm -2 ; Benestad, 2006). Planetary Surface Radiation - In the chemistry module, the UVA and UVB net fluxes required for the photolysis scheme are calculated from the top layer downward via the twostream module with Rayleigh scattering. Figures (2a-2b) show UVA and UVB net flux (Wm -2 ) at the planetary surface as calculated in the chemistry module of this work. UVC is essentially zero at the surface so is not shown in Figure 2, and similarly for Figure 3. Generally, Figures 2a and 2b show an increase in planetary surface UV radiation with higher stellar temperatures, as for the TOA cases shown in Figure 1. Comparison with Earth Surface UV Radiation - Global satellite observations from 1992-1994 (Wang et al. 2000, their Figure 6b) suggest observed UVB surface radiation for Earth of ~1.4Wm -2 for cloud-free conditions. By comparison, Figure 2b suggests that our model over-estimates this value, calculating 2.3 Wm -2 UVB for the Earth control run. Uncertainties include, for example, our straightforward treatment of clouds whereby we adjust the surface albedo (see above) as well as the challenge of representing, for example, time-dependent and, for example, latitude-varying O3 photochemistry and UV absorption in a global-averaged 1D model. Ratio of Surface to TOA UV Flux - This ratio (R) is shown for the 1g and 3g cases in Figures 3a and 3b for UVA and UVB respectively. R is an inverse measure of the UV shielding of an atmosphere. Figure 3a suggests that UVA passes efficiently through the atmospheres considered, as expected, since most values of Rnet,UVA are >0.7. The UVA ratio is not greatly dependent on the stellar temperature. Figure 3b shows as expected a much stronger atmospheric extinction of UVB than for the UVA wavelengths, and there is now a clear dependency on stellar temperature. Weaker overhead O3 columns in the cool M-dwarf cases lead to a strong rise in the ratio in Figure 3b. For the 3g scenarios (circles), a lowering in the atmospheric column by a factor of three resulted in less UV shielding and a rise in the surface UV.",
"pages": [
13,
14,
15,
16
]
},
{
"title": "3. Chemical Analysis",
"content": "Here, we first compare briefly previous results (Segura et al., 2005) reported in the literature. Then, we discuss the general trends in column abundances of the biosignatures and related key species. Finally, we discuss the chemical responses for the vertical profiles that were also shown in Paper I.",
"pages": [
16
]
},
{
"title": "3.1 Column Biomarkers (1g planets)",
"content": "Column O3 in Figure 4a (blue diamonds) mostly decreased with increasing star class (i.e., decreasing Teff of the star) related to less UVB, therefore there was a slowing in the photolysis of molecular O2 and hence a slowing in the Chapman cycle, a major source of O3. The O3 profile responses are discussed in more detail in section 3.6. The column values are shown in appendix 1. Column N2O in Figure 4a (red squares) generally increased with increasing star class. The cooler stars emit less UVB, which suggests a slowing in the photolytic loss of N2O in the planetary atmosphere and hence an increase in its abundance. Column CH3Cl in Figure 4a (green triangles) generally increased with increasing star class due to less OH, its major sink (see OH analysis, Table 2). The response is comparable to CH4 (discussed in next section), which has a similar photochemistry. Spectral features of CH3Cl, however, were too weak to be evident in the calculations of Paper 1 despite the enhanced column amounts for the cooler stars.",
"pages": [
16,
17
]
},
{
"title": "3.2 Column Biosignatures (3g planets)",
"content": "For the 3g planets, we assumed a constant surface pressure of 1 bar, which led to the total atmospheric column being reduced by a factor of three, as already mentioned (Figure 4b). The general trends for O3 and N2O remain for the 3g scenarios, that is, mostly similar to the corresponding 1g scenarios already discussed, although the reduced total column resulted in a cooling of the lower atmosphere due to a weaker greenhouse effect, as we will show (see Paper I also). The N2O 3g response is linked with enhanced UVB penetrating the reduced atmospheric column compared with 1g, which leads to more photolytic loss of N2O. A transport effect also took place. For the 3g case (with its lower model lid due to less atmospheric mass and higher gravity), the upward tropospheric diffusion of N2O was faster, for example, by about 50% in the mid to upper troposphere than the 1g case. This meant that N2O for the 3g case could reach the stratosphere faster, where it would be rapidly photolyzed.",
"pages": [
17
]
},
{
"title": "3.3 Column Greenhouse Gases (1g planets)",
"content": "In this section, we discuss the planetary atmospheric column abundances of CH4 and H2O since they have a major impact on temperature via the greenhouse effect. Vertical profiles will be discussed later and can also be found in Paper 1. CH4 Column Response - Since the only source of CH4 in the model is fixed biomass surface emission, the CH4 response for the various runs is controlled by the main atmospheric CH4 sink, that is, removal via the hydroxyl (OH) radical. OH is affected by three main processes: OH Source(s) : for example, H2O+O( 1 D) 2OH (where O( 1 D) comes mainly from O3 photolysis in the UV). OH Recyling reactions in which NOx species can interconvert HOx (defined here as OH+HO2) family members via, for example, NO+HO2 NO2+OH. OH Sinks, for example, reaction with CH4 and CO (see e.g. Grenfell et al., 1999 for an overview). Figure 4c suggests a strong CH4 (green diamonds) increase with decreasing effective stellar temperature. Cooler stars are weak UV emitters, which favors a slowing in the OH source reaction above. Note also that greenhouse warming by the enhanced CH4 favors a damp troposphere (more evaporation) and, hence, all else being equal would favor actually more OH (via more H2O, see source OH reaction above). This is an opposing process which our results suggest is not the dominant effect. So, for a given model, calculating accurately the net effect will depend, for example, on a good treatment of, for example, the hydrological cycle, which is challenging for a global column model. To aid in understanding the CH4 response, which is controlled by OH, Table 2 summarizes the OH sources, sinks, recycling budget, and associated quantities. OH Abundances - Control run (1g Sun) OH abundances in Table 2 are within ~20% of global-mean observed OH proxies for Earth. Table 2 suggests a strong decrease in OH from left to right (i.e., for decreasing stellar effective temperature) especially for the M7 case. OH Source Reaction Rates - The source reaction rate (Sun) in Table 2, that is, O( 1 D)+H2O 2OH, is about 12 times weaker than indicated by the Whalley et al. (2010) study, which investigates (Earth) clean-air, tropical northern-hemisphere daytime OH. The factor 12 difference reflects a lowering due to day-night averaging in our global mean model (which accounts for ~factor 2 of the difference in OH) and the fact that the Whalley study considered tropical conditions. Concentrations of the trace specie O( 1 D) in the control run (=6x10 -8 ppbv at 30km) compared reasonably well with Earth observations (~3x10 -8 ppbv, Brasseur and Solomon, 2005). Table 2 suggests that the source reaction rate decreases from left to right, which is consistent with the decrease in OH. OH Recycling Reaction Rates - Our (Sun case) recycling reaction was comparable with that of the Whalley et al. (2010) study to within about 50%. Earlier (Earth) modeling studies, for example, that of Savage et al. (2001), suggest that the OH recycling reaction dominates the source reaction even in quite clean air-masses (NOx ~250pptv and below), which is somewhat in contrast to this and the Whalley study. In Table 2, the recycling reaction rates (like the source reaction) also decreases from left to right, which favors the decrease in OH, although the change in the source reaction is the stronger effect. For cooler stars, the recycling reaction becomes increasingly important compared with the source reaction, and it dominates for the ADL and M7 cases. HOx and NOx Ratios - These ratios are sensitive markers of changes in HOx and NOx chemistry and hence affect, for example, O3 cycles and CH4. The ratios (HO2/OH) and (NO2/NO) in Table 2 increase strongly for the cooler stars. These ratios are strongly affected by the concentration of O3, whose production via the Chapman mechanism (discussed in 3.5) weakens for the cooler stars. The ratios for the cooler stars are far from their 'Earth' values, so the interactions between HOx and NOx are much perturbed. This is a hint that the usual mechanisms that operate on Earth (e.g., O3 catalytic cycles etc.) may be very different for the cooler star scenarios - a good motivation for applying PAP as already mentioned. Atmospheric response for AD Leo - Although the 1g ADL scenario featured lower OH (Table 2) than for M5, ADL featured lower CH4 (Paper I) than M5. The upper layers (>60km) of the 1g ADL run showed very rapid destruction of CH4 via OH - about five times faster than for M5. This was consistent with the high Lymanα output of ADL leading to faster HOx enhancement via H2O photolysis. Water Column Response - Figure 4c suggests that the increased CH4 columns (green diamonds), with decreasing stellar effective temperature generally (except for M7), lead to higher H2O columns (green squares). Generally, for the cooler star scenarios, (up to and including M5), more CH4 greenhouse heating leads to more water evaporation in the troposphere, and in the stratosphere, faster CH4 oxidation leads to faster H2O production. However, for the M7 case (Figure 4c), although CH4 increased, surface temperature did not, which suggests a saturation in the CH4 greenhouse from M5 to M7, where the lower atmosphere becomes optically thick at very high CH4 abundances. Surface cooling from M5 to M7 is also seen in the temperature profiles in Paper 1 (their Figure 3).",
"pages": [
18,
19,
20,
21
]
},
{
"title": "3.4 Column Greenhouse Gases (3g planets)",
"content": "CH4 and H2O - Figure 4d has a similar format to Figure 4c but instead shows results for the 3g (instead of 1g in 4c)scenarios. The basic response to decreasing the effective stellar temperature at 3g is similar to the 1g case, that is, results suggest a column rise in CH4 and in H2O but with a drop-off in the latter for the cooler stars. To gain more insight into the effect of changing gravity, upon CH4, Table 3 shows the ratio (1g/3g) of the CH4 column and for the near-surface atmospheric OH abundance: Without calculating interactive photochemistry, a passive tracer would undergo a column reduction by a factor of three from 1g to 3g, because at constant surface pressure, increasing gravity by a factor of three leads to column collapse and a reduction in the overhead column by the same factor as the increase in gravity. In Table 3, therefore, a hypothetical, passive tracer (with no chemistry) would have a value of exactly three. The actual (with chemistry) CH4 column ratios (row 1), however, are all less than three. The reduction is consistent with faster chemical loss at 1g than at 3g. To investigate this further, OH ratios are shown in Table 3 (row 2). They mostly (except ADL) increase for the cooler stars, suggesting a lowering in the 3g OH abundances compared with the corresponding 1g cases for the cooler stars. This is consistent with faster chemical loss at 1g. The reduction in OH for the 3g scenarios implies that, for example, the increase in UVB due to weaker shielding of some 3g atmospheres ( favouring OH production) is out-weighed by the (opposing) feedback where reduced greenhouse warming at 3g led to a drier troposphere (disfavoring OH which is produced via O( 1 D)+H2O 2OH). This is confirmed by the water column (open circles in Figure 4d), which suggests that the 3g compared with 1g (Figure 4c) scenarios led to a weakening in the greenhouse effect and hence tropospheric cooling (as seen in Figure 2 of Paper 1) and a general lowering in the H2O column (due to more condensation) by around a factor of ten (Figure 4d) compared with the 1g case (Figure 4c). In general, however, note that responses in chemical abundances do not scale directly with the column reduction at 3g compared with 1g since the effects of, for example, photochemistry are important. Figures (4e, 4f) show the ratios (1g column/3g column) for biosignature and greenhouse gases respectively. The main point is that the values can lie far from a value of three (which would be expected for a passive tracer). This shows that it is important to include the effects of interactive chemistry. For the biosignature O3 there is some indication of an increase in the ratio shown in Figure 4e for the cooler stars, which will be the subject of future study. For CH3Cl (Figure 4e) and CH4 (Figure 4f) (which both have similar OH removal chemistry), the trend is downward for the cooler stars. The H2O (Figure 4f) scenarios are relatively more damp (with values >3) than for a purely passive tracer. This suggests more efficient production of H2O from CH4 for the cooler stars at 3g than at 1g, for example, due to more UV in the thinner, 3g atmospheres.",
"pages": [
21,
22,
23
]
},
{
"title": "3.5 Column-Integrated Pathway Analysis Program (PAP) Results",
"content": "Figure 5 shows output of O3 cycles from the PAP. The cycles (divided into production and loss cycles) found have been quantified according to the rate of O3 production or loss through each particular cycle expressed as a percentage of the total rate of production or loss found by PAP (see also description of Appendix 1 below). Values are integrated over the model vertical domain. PAP analyses were performed for each of the 64 vertical column model chemistry levels, and the column-integrated values are shown in Figure 5. The full cycles referred to in Figure 5 can be found in the Appendix. Sun PAP Analysis Figure 5 confirms the expected result for O3 production, that is, the Chapman mechanism dominates over the smog mechanism. For O3 destruction, the column model suggests strong NOx contributions in the lower stratosphere, although an Earth GCM study (Grenfell et al., 2006) suggests a strong HOx contribution there. This result could reflect the challenge of 1D models of capturing 3D variations in photochemistry. Also, the column model does not include industrial emissions unlike the Earth 3D model. The result should be explored in future comparisons between the column model and 3D runs. Column-Integrated O3 (1g) Production Figure 5a suggests a change from a mainly Chapman-based O3 production for the 1g Sun and the warmer 1g M-dwarf stars, switching to a slower, mainly smogbased O3 production for the cooler stars (1g M5 and 1g M7). This was related to the decrease in UVB for the cooler star scenarios, since UVB is required to initiate the Chapman mechanism via photolysis of O2. Column integrated O3 (1g) Destruction - Figure 5a also suggests that the classical NOx and HOx cycles (see also Figures 6 and 7) that operate mainly in the stratosphere were the most dominant O3 loss pathways for the Sun and warmer M-dwarf scenarios. For the cooler stars scenarios, the enhanced CO concentrations led to a CO-oxidation cycle gaining in importance. Column O3 (3g) - Behavior at 3g (Figure 5b) was broadly similar to 1g, except at 3g both Chapman and smog were important O3 producers for the M5 case (i.e., not just smog as in the 1g case). Weaker atmospheric UVB absorption led to more penetration of UVB and hence an increased role for Chapman in the layers below. Column-Integrated Results Table for O3, N2O and CH4 - Appendix (1a-1c) shows the integrated column mean PAP output for O3, N2O, and CH4 respectively. Shown are (i) the column integrated rates (CIR) (in molecules cm -2 s -1 ) for all pathways found by PAP ('Found_PAP'), (ii) the CIR for only the pathways shown in the Appendix ('Shown_PAP') (shown are either the 5 dominant pathways or the first pathways that together account for >90% of the total formation or loss of found_PAP, whichever condition is fulfilled first), and (iii) the CIR as calculated in the chemistry scheme of the atmospheric column model ('total_chem'). Percent values for a particular cycle show its individual rate as a percentage of Found_PAP. Comparing these three CIR values, it can be seen that for the O3 production, which is relatively straightforward, the pathways found by PAP can account very well for the rate calculated in the column model chemistry module. For the O3 loss pathways, which are rather more complex than the production, PAP can still account for generally more than ~90% of the rate from the chemistry module. For the sometimes very complex CH4 pathways, with the value of fmin chosen for this study, PAP can account for only up to about 50% of the rate from the chemistry module. Further tests suggested that decreasing the PAP input parameter fmin (the minimum considered flux, currently set to 10 -8 ppbv s -1 for all runs) leads to improvement, but the resulting complex CH4 cycles are beyond the scope of this paper (see also 3.6.3). We now discuss the individual cycles for each scenario.",
"pages": [
23,
24,
25
]
},
{
"title": "O3 Column-Integrated Pathways",
"content": "Chemical pathways for the 1g Sun scenario in Appendix 1a mostly compare well with established results for Earth as discussed above. Appendix 1a suggests that for the 1g M0 scenario - due to less stellar UVB emission compared with the Sun - the Chapman mechanism for producing O3 is somewhat suppressed (89.2%) and a new CO sink ('CO oxidation 1', 7.4%) appears, since CO is abundant. For the 3g M0 scenario, results suggest that Chapman features more strongly (96.7%) in the thinner 3g atmosphere compared with the corresponding 1g case. HOx and NOx remain important chemical sinks for both the 3g and 1g cases. The active star (1g ADL) features a stronger Chapman contribution (97.2%) compared with 1g M0 since ADL is especially active in the UV, which is important for Chapman-initiation (via molecular oxygen) with only modest changes for the 3g ADL case. For cooler non-active stars (1g M5), large changes are apparent compared with the warmer star cases. Less UVB emission from the cool M5 star leads to a switch to smog-type O3 production ('smog 1', 57.8%). As discussed, the atmosphere is abundant in CH4 and CO. Thus, the 'CO-oxidation-1' cycle is an important O3 loss pathway (36.8%). For the (3g M5) case, the thinner total column at 3g compared with 1g leads to a rise in UV, which is consistent with more Chapman O3 (47.8%) production than the 1g case (7.5%). For O3 loss, a complex CH4 oxidation pathway involving CH3OOH becomes important (46.8%), which is not evident at 1g. The changed UV environment leads to a modest rise in HOx in the upper troposphere at 3g. Finally, for the coolest M-dwarf case (1g M7), O3 production occurs via numerous types of smog mechanisms involving the oxidation of different VOCs, for example, CO, HCHO, and CH3OOH. CO smog cycles become a key means of producing O3 especially for the cooler stars. Like CH4, an important sink for CO is the reaction with OH. As discussed, weakening UV emissions for the cooler stars leads to less OH and therefore an enhanced abundance of CO. Near the surface, CO mixing ratios correspond to: 0.09 (Sun), 9.0 (M4), 64 (ADL), and 426 (M7) parts per million (ppm). O3 loss also involves NOx cycles but also a smog mechanism ('smog 7') where O3 is the net oxidant, which is consumed to oxidize CH4 and a CO oxidation cycle. Smog cycles have larger rates for the M5 and M7 scenarios than for the Sun and M0 scenarios. This is because the important smog 1 cycle (producing O3) is in competition with the CO-oxidation 1 pathway (destroying O3). At high O3 concentrations (for the Sun and M0 scenarios), (i) the reaction NO+O3 NO2+O2 shifts the NOx family to favor NO2. The reduction in NO leads to a slowing in the key reaction NO + HO2 NO2+OH and hence slows the smog 1 cycle. Also at high O3 concentrations, (ii) the reaction HO2+O3 OH + 2O2 favors the CO-oxidation 1 pathway. These two effects together, favor large smog rates for the M5 and M7 scenarios. In summary, total vertically integrated O3 production and loss rates for the 1g Sun (=1.9x10 13 molecules cm -2 s -1 ) are 68 times larger than for the 1g M7 case (=2.8x10 11 molecules cm -2 s -1 ), which illustrates the change in the dominance from the rather fast Chapman chemistry to the slower smog mechanism.",
"pages": [
25,
26,
27
]
},
{
"title": "N2O Column-Integrated Pathways",
"content": "The main result of the PAP is that loss pathways from the N2O 'viewpoint' are non-catalytic for all scenarios. In other words, loss occurs mainly directly via photolysis, which can be calculated from the photolysis rate without performing a PAP analysis for N2O. We therefore only show (Appendix 1b) one scenario as an illustration, that is, the Sun scenario, which confirms results measured for Earth, that is, ~95% loss via photolysis (i.e., the sum of the 4 cycles involving N2O photolysis in Appendix 1b), and ~5% loss via catalytic reaction with O( 1 D) is similar to observed values quoted for Earth (e.g., 9095% photolytic loss, 5-10% via reaction with O( 1 D), IPCC Third Assessment Report, see discussion to Table 4.4). The PAP finds no formation pathways of N2O via inorganic reactions, as expected since these are insignificant compared with surface biogenic input. For the M-dwarf scenarios, photolysis similarly remained the main removal mechanism, and the overall column integrated rate of removal decreased by about a factor of two for the M7 compared with the Sun case since the cooler stars emit less UV.",
"pages": [
27
]
},
{
"title": "CH4 Column-Integrated Pathways",
"content": "Appendix 1c shows the PAP output for CH4. Results suggest a large number of complex removal pathways that oxidize CH4. PAP found no in-situ production pathways, since there are no inorganic reactions in our model that produce CH4 in the atmosphere. The net removal can involve either complete oxidation of CH4 to its stable combustion products: H2O and CO2 (as in the 'oxidation 2O2-a' pathway for the 1g Sun scenario) but can also involve only partial oxidation, for example, to intermediate organic species such as formaldehyde (HCHO), for example, as in the 'Oxidation O2' pathway (1g M0). Clearly, more complete oxidation is favored in oxidizing environments, for example, damp atmospheres with strong UV where OH is abundant. The choice of oxidant in the net reaction will depend on the central star's particular UVB radiation output and its ability to release, for example, HOx, Ox, or NOx from their reservoirs in the planetary atmosphere. Importantly for O3 photochemistry, there are CH4 cycles in which O3 itself is the oxidant in the net reaction (see e.g. net reaction for several cycles from the 3g Sun case). This is an example where CH4-oxidation does not lead to the more familiar O3 (smog) production, but to the reverse effect where O3 is consumed. Many of the CH4 pathways are NOx-catalyzed, as on Earth, although this is not the case for all scenarios (e.g., pathway 'CH3OOH-d' (3g M5) does not include NOx).",
"pages": [
27,
28
]
},
{
"title": "3.6 Altitude-Dependent PAP Results",
"content": "In this section we will present PAP results from the same scenarios as the previous section. However, here we will discuss the contribution of the PAP cycles as profiles varying in the vertical.",
"pages": [
28
]
},
{
"title": "3.6.1 Vertical Changes in Ozone (O3) Production and Loss Cycles",
"content": "Figure 6 shows the altitude-dependent PAP results, comparing production and loss pathways for the Earth case (Figures 6a, 6b) with the M7 case (Figures 6c, 6d). Similarly, Figures 7a, 7b compare ADL (1g) with M5 (1g) (Figures 7c, d). In Figures 6 and 7, the logarithmic x-axis shows the rate of change of O3 associated with a particular cycle found by PAP, in molecules cm -3 s -1 . The black and white text labels on these Figures indicate the names of the O3 pathways, which can be found in Appendix 1a. Note that the logarithmic x-axis where results are plotted cumulatively (meaning to estimate the contribution of a pathway at a particular height one must subtract its left-hand side x-axis boundary from its right-hand side x-axis boundary) in Figures 6 and 7 means that the pathways shown on the right-hand side of the Figure can make up a strong overall contribution to the net rate of change despite having only a thin section (relatively small area). For the Earth results (Figures 6a, 6b), the O3 production and loss rates output by PAP compare well with middle atmosphere O3 budgets derived for Earth, see for example the work of Jucks et al. (1996), their Figure 4. The Earth results (Figure 6a) in the top model layer show an uppermost region of O3 production (thin, blue stripe), which arose due to the single reaction: O2+O( 3 P)+M O3+M. This is linked with the model's upper boundary condition, where a downward flux of CO and O( 3 P) is imposed. This is done to parameterize the effects of CO2 photolysis (forming CO and O( 3 P)), which takes place above the model's lid, for example, above the mid mesosphere. The resulting enhanced O( 3 P) in the uppermost model layer favors the direct O3 formation pathway found by PAP. The enhanced O3 source was balanced by an increase in the photolysis rate of O3, and therefore the abundance decreased smoothly with altitude as expected. The effect of varying the upper boundary will be the subject of future work. NOx loss cycles dominate (>60%) the Earth lower stratosphere; HOx cycles are more important in the upper stratosphere. For the 3g case (3g Sun), the O3 production pathways are similar to those of Earth, but HOx destruction is stronger (~70%) in the lower stratosphere, which is consistent with more UV penetration (releasing HOx from its reservoirs) for the thinner (3g) atmospheric column compared with the 1g case. The enhanced tropospheric HOx, which also stimulated the 'CO oxidation 1' cycle, accounted for (30-50%) of tropospheric O3 loss. For the warm M-dwarf star scenarios (e.g., 1g M0) - here, like the control (1g Sun), smog 1 dominates 50-60% of the O3 production in the troposphere (with 10-20% arising from CH4 smog cycles). The influence of the smog mechanism extends to high altitudes (up to about 20km) compared with the Earth control (which extends up to about 16km). 'Chapman 1' (Appendix 1a) dominated the stratosphere. O3 loss was dominated by the 'CO-oxidation 1' pathway (60-80%) in the troposphere, NOx loss pathways in the mid-stratosphere, and HOx loss pathways in the upper stratosphere. For the 3g case (3g M0), the 'smog 1' pathway contributes ~70% of O3 production in the troposphere with the ~(10-15%) remainder in the troposphere coming from CH4 smog pathways. 'Chapman 1' is dominant in the stratosphere, and 'Chapman 2' is dominant in the uppermost layers (see discussion above for Earth run 1). O3 loss, like the 1g case, was dominated by 'CO-oxidation 1' pathway in the troposphere (~90%) with different HOx cycles important for loss in the upper levels. In Figure 7, ADL O3 photochemistry production (Figure 7a) is rather similar, for example, to the Earth control (1g Sun) case (Figure 6a) in that Chapman production dominates the stratosphere and smog in the troposphere. However, for the 1g M5 run, results are very different from what occurs on Earth, since O3 production is now dominated by the smog mechanism through much of the atmosphere. For ADL, O3 production occurred mostly via 'Smog 1' (70-80%) in the troposphere, with various CH4 smog pathways making up between 10-20% in this region. 'Chapman 1' dominated the stratosphere. O3 loss was again dominated by 'CO-oxidation 1' in the troposphere (70-90%) with a variety of HOx cycles important for loss in the upper levels. Intense Lymanα radiation favored some enhancement of H2O photolysis (hence more O3 loss via HOx) in the 1g ADL scenario compared to, for example, the Earth control (run 1), but the effect was quickly damped (in the uppermost ~2 model layers) and the overall change in O3 was small. For the corresponding 3g case (3g ADL), O3 production pathways did not change greatly with altitude compared with the 1g case. O3 loss pathways were also rather similar to the 1g ADL case, with the 'CO-oxidation 1' pathway for 3g ADL dominating the lower atmosphere. The cooler stars (M5, M7) show significant changes in the O3 photochemistry compared with the other M-dwarf scenarios. The rather weak UV radiation of these cooler stars means that Chapman chemistry (requiring UV to break the strong O2 molecule) is now only significant (up to ~50% O3 production) (1g M5) in the uppermost (>60km) altitudes. The 'CO-smog 1' pathway, however, is now significant over all altitudes, accounting for 60% of O3 production in the troposphere and about 30% in the upper atmosphere. A variety of CH4 smog pathways make up most of the remaining O3 production (1g M5). For O3 loss, the 'CO-oxidation 1' pathway is again significant (50-70%) in the lower half of the model domain, whereas a variety of NOx cycles are important in the upper regions. For the coolest star considered (1g M7), the O3 abundance is determined by mainly CO and CH4 oxidation. First, 'classical' smog production - with OH as the oxidant (mainly CO smog 1 and various CH4 oxidation pathways)--produce O3 but, on the other hand, O3 in the M7 scenario can also act as an oxidant in pathways that oxidise, for example, CH4 and CO. The M7 case (Figures 6c-6d) shows that the CO smog mechanism dominates the O3 production, whereas the CO oxidation cycle and the classical NOx cycle dominate the O3 loss. Near the surface, some direct removal of O3 occurred via the reaction: NO+O3 O2+NO2 (Figure 6d). On Earth, more NOx usually leads to more O3 production via the smog mechanism; the direct removal reaction is, however, sometimes important at high NOx abundances, for example, in city centers. In our M7 scenario, which does not have industrial NOx emissions, an important source of lower atmosphere NOx is from lightning. For the cool M-dwarf 3g case (3g M5, not shown), the 'CO-smog 1' and 'Chapman 1' pathways make almost equal contributions to the O3 production budget in the middle atmosphere. 'Chapman 1' contributes up to ~80% of local production in the upper levels (where UV is abundant), whereas the smog mechanism contributes up to ~70% in the lower layers. The smog contribution has a minimum of ~20% local production near the cold trap, which is consistent with low temperatures and a rather low OH abundance. For the O3 loss pathways, results suggest an increase in complex CO and CH4 smog pathways that consume O3. 3.6.2 N2O - For all scenarios, non-catalytic photolytic removal (>90%) is the main loss mechanism in the stratosphere. Catalytic removal involving reaction with O( 1 D) makes up the remainder (occurring mostly in the mid to upper stratosphere) of the N2O loss. 3.6.3 CH4 - Results suggest that a large number of loss pathways occur near the cold trap. For example, at 16km (1g Sun), the CH4 pathways found by PAP with the value of fmin chosen in this study could account for only about 20% of the total CH4 change calculated in the column model. Low OH abundances and cold temperatures in this region are consistent with rather slow oxidation and a resulting complex mix of only weakly oxidized organic species with individual pathway contributions lying below the PAP threshold criteria chosen for the present study, but whose net effect is important. For this study, the PAP detection threshold was set to fmin = 10 -8 ppbv/s. OH-initiated oxidation of CH4 is more favored on the lower layers but with relatively more O( 1 D)-initiated oxidation on the upper levels, where this species is more abundant. A test run (not shown) where the fmin value is decreased to 10 -9 ppbv/s was found to address the above problem, that is, PAP was then able to account, for example, three times more CH4 net change (for the Earth run), though with a notable increase in the overall number of pathways, each with small contributions to net the overall chemical change, beyond the scope of our work. 3.7 Comparison with previous studies - Compared with the results of Segura et al. (2005), our results are similar for N2O and CH4 within 10-20% for the inactive (e.g., M4) and active (ADL) cases. For O3, our atmopsheric column amounts are ~40% thicker (=270DU) compared with the Segura et al. (2005) (=164DU) value for the ADL case. This results from changes in our photochemical scheme, including, for example, the parameterization of the lower boundary flux of H2, as discussed in Paper I. Also our stellar insolation corresponds to 1366Wm-2 at the TOA, whereas Segura et al. (2005) scaled their incoming spectrum to obtain a surface temperature of 288K.",
"pages": [
28,
29,
30,
31,
32,
33
]
},
{
"title": "4. Spectral Detectability of Biomarkers",
"content": "O3 - Paper I shows that the detection of O3 is challenging especially for M7. To better understand O3 detectability, improved stellar spectra for the cooler stars in the (E)UV are desirable especially in the UVB and UVC, where O3 responds sensitively. M7 stars are statistically older and burn more slowly compared with lower spectral class stars, which means more developed convection zones and possibly larger differences in UV between flaring and quiet states for M7 than considered in our work (see Reiners et al. 2012). N2O - Clearly, the most favorable (planet/star) contrast ratios are associated with cool stars such as M7. However, Paper I shows that some spectral absorption features can be weakened, partly due to the large CH4 abundance, which warms the stratosphere. The N2O spectral features were weak for the scenarios analyzed.",
"pages": [
33
]
},
{
"title": "5. Conclusions",
"content": "produced by the smog cycle (which could be favored by increases in HOx and NOx, e.g., by cosmic rays) could be very different than Chapman-produced O3 (where HOx and NOx catalytically destroy O3). This is important to consider when predicting and interpreting O3 spectral features. mass of the atmosphere by, for example, a factor of three does not always lead to a reduction in, for example, biosignatures and greenhouse gases by a factor of three, due to interactive climatephotochemical effects.",
"pages": [
34,
35
]
},
{
"title": "Acknowledgements",
"content": "This research has been partly supported by the Helmholtz Gemeinschaft (HGF) through the HGF research alliance \"Planetary Evolution and Life.\" F. Selsis and P. von Paris acknowledge support from the European Research Council (Starting Grant 209622: E3ARTHs).",
"pages": [
35
]
},
{
"title": "References",
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"pages": [
35,
36
]
},
{
"title": "Figure Captions",
"content": "Figure 1: Planetary Global Mean Top-of-Atmosphere Incoming Radiation (W m -2 ) for UVA (Figure 1a), UVB (Figure 1b) UVC (Figure 1c) for Earth's gravity. Figure 2: As for Figure 1 but at the planetary surface for UVA (Figure 2a), UVB (Figure 2b) and UVC (Figure 2c) for Earth's gravity (1g). Figure 3: As for Figure 1 but showing the Ratio (Surface/TOA) (at 1g and 3g) for UVA (Figure 3a), UVB (Figure 3b) and UVC (Figure 3c) Radiation (1g). Figure 4a: Atmospheric columns (Dobson Units, DU) (1g) of biosignatures, ozone (O3), nitrous oxide (N2O) and methyl chloride (CH3Cl). Figure 4c: As for Figure 4a but for column CH4 (Dobson Units, DU) and H2O (DU). Figure 5: Pathway Analysis results for Global Mean Ozone Sources (Figure 5a) and Sinks (Figure 5b) for the Sun and for the M-dwarf star scenarios (1g) calculated by the Pathway Analysis Program. The pathways are shown in the PAP Tables in the Appendix. Figure 6: Pathway Analysis results showing cumulative contribution of altitude-dependent O3 production and loss pathways for the 1g Sun (Figures 6a, 6b) and for the 1g M7 scenarios (Figures 6c, 6d) plotted in the vertical and shown in (molecules cm -3 s -1 ). Black and white labels on the Figure correspond to the names of the individual cycles as shown in Appendix 1. Logarithmic x-axis tick labels correspond to factors of x2 x5 and x8 respectively. Note that the model vertical grid is variable depending on, e.g., greenhouse gas heating, which leads to an expansion in the vertical for cooler effective stellar temperatures. Figure 7: As for Figure 6 but for the 1g ADL (Figures 7a, 7b) and for the 1g M5 scenarios (Figures 7c, 7d). 1000 100 10 0.1 š 0.01 M5 ; M0 ADL M4 ; TOA Net Radiation (Wm-?) Sun M4 M5 M7 M0 ADL ; ; ; ; Stellar Effective Temperature (K) 10 Sun ; Stellar Effective Temperature (K) ; Chapmanhi rate (molecule cm -3 -1 s ) rate (molecule cm -3 -1 s ) rate (molecule cm -3 -1 s ) rate (molecule cm -3 -1 s )",
"pages": [
37,
38,
45,
46
]
},
{
"title": "Appendix 1: (a) Ozone Pathways",
"content": "1g Sun Ozone Loss: Found_PAP =1.81x10 13 , Shown_PAP=1.06 x10 13 , Total_chem=1.88x10 13 3g Sun Ozone Production: Found_PAP = 1.28x10 13 , Shown_PAP=1.27 x10 13 , Total_chem = 1.28x10 13 3g Sun Ozone Loss: Found_PAP =1.20x10 13 , Shown_PAP=6.06x10 12 , Total_chem = 1.26x10 13 3g M0 Ozone Production : Found_PAP =1.31x10 12 , Shown_PAP=1.30x10 12 , Total_chem = 1.32x10 12 3g M0 Ozone Loss : Total = Found_PAP =9.75x10 11 , Shown_PAP=5.27x10 11 , Total_chem = 1.20x10 12",
"pages": [
47,
48
]
},
{
"title": "1g AD-Leo Ozone Production : Found_PAP =1.72x10 12 , Shown_PAP=1.71x10 12 , Total_chem = 1.72x10 12",
"content": "1g AD-Leo Ozone Loss : Found_PAP =1.43x10 12 , Shown_PAP=6.71x10 11 , Total_chem = 1.69x10 12 3g AD-Leo Ozone Production: Found_PAP =1.56x10 Chapman 1 (99.5%) , Shown_PAP=1.56x10 Smog 1 12 (0.4%)",
"pages": [
48
]
},
{
"title": "1g M5 Ozone Loss : Found_PAP =2.57x10 11 , Shown_PAP=2.19x10 11 , Total_chem = 2.91x10 11",
"content": "12 12 , Total_chem = 1.56x10 3g M5 Ozone Production : Found_PAP =1.35x10 11 , Shown_PAP=1.32x10 11 , Total_chem = 1.42x10 11 Chapman 1 (47.8%) Smog 1 (47.6%) Chapman 2 (2.4%) 3g M5 Ozone Loss : Found_PAP =1.19x10 11 , Shown_PAP=1.08x10 11 , Total_chem = 1.28x10 11",
"pages": [
48,
49
]
},
{
"title": "1b. Nitrous Oxide Pathways (Sun only)",
"content": "1g Sun Nitrous Oxide Loss : Found_PAP =9.63x10 8 , Shown_PAP=8.78x10 8 , Total_chem = 1.15x10 9 N2O-NOx (69.5%): 2(N2O +h ν → N 2 + O( 3 P)) O( 3 P) + O2 + M → O 3 + M NO2 + O( 3 P) → NO + O 2 NO + O3 → NO 2 + O2 net: 2N2 O → O 2 + 2N2 N2O-Ox (9.6%): 2(N2O +h ν → N 2 + O( 3 P)) O( 3 P) + O2 + M → O 3 + M O( 3 P) + O3 → O 2 + O2 net: 2N2 O → O 2 + 2N2 N2O-HOx-1 (3.4%): 2(N2O +h ν → N 2 + O( 3 P)) 2(O( 3 P)+O2 +M → O 2+M) OH + O3 → HO 2 + O2 HO2+O3 → OH+O 2+O2 net: 2N2 O → O 2 + 2N2 N2-O( 1 D) (5.4%): 2( N2O +O( 1 D) → N 2 + O2) 2(O( 3 P)+O2 +M → O 3+M) 2(O3 + h ν → O 2 + O( 1 D)) O2 + h ν → O( 3 P) + O( 3 P) net: 2N2 O → O 2 + 2N2 N2O-HOx-2 (3.3%): 2(N2O + h ν → N 2 + O( 3 P) O( 3 P) + O2 + M → O 3 + M HO2 + O( 3 P) → OH + O 2 OH + O3 → HO 2 + O2 net: 2N2 O → O 2 + 2N2",
"pages": [
49
]
},
{
"title": "1c. Methane Pathways",
"content": "1g Sun Methane Loss : Found_PAP =1.15x10 10 , Shown_PAP=7.68x10 9 , Total_chem = 1.24x10 11 Oxidation 3O2 -a (29.6%): 2(CH4+OH → CH3+H 2 O) 2(CH3+O2+M → CH3O2+M) 2 (CH3O2+NO → H3CO+NO2) 2 (H3CO+O2 → H2CO+HO2) 2 (H2CO + h ν → H2 + CO) NO + HO2 → NO2 + OH 3 (NO2 + h ν → NO + O( 3 P)) 3 (O( 3 P) + O2 + M → O3 + M) 3 (HO2+O3 → OH+O2+O2) 2 (CO + OH → CO2 + H) 2 (H + O2 + M → HO2 +M) net:2CH4+3O2 → 2H2O+ 2H2+2CO2 Oxidation 6O2 (23.0%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + h ν → H2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 2 (NO + HO2 → NO2 + OH) 3 (NO2 + h ν → NO + O( 3 P)) 3 (O( 3 P) + O2 + M → O3 + M) net: CH4 + 6O2 → H2O + 3O3 + H2 + CO2 Oxidation 8O2 (6.0%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO HCO + O2 → HO2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 3(NO + HO2 → NO2 + OH) 4(NO2 + h ν → NO + O( 3 P)) 4(O( 3 P) + O2 + M → O3 + M) net: CH4 + 8O2 → 2H2O + 4O3 + CO2 Oxidation 2O2-a (6.2%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO NO + HO2 → NO2 + OH 2(NO2 + h ν → NO + O( 3 P) ) 2(O( 3 P) + O2 + M → O3 + M) HCO + O2 → HO2 + CO 2(HO2+O3 → OH+O2+O2) CO + OH → CO2 + H H + O2 + M → HO2 +M net:CH4 +2O2 → 2H2O + CO2 Oxidation 3O2-b (2.2%) 2(CH4 + OH → CH3 + H2O) 2(CH3+O2+M → CH3O2+M) 2(CH3O2+NO → H3CO+NO2) 2(H3CO+O2 → H2CO+HO2) 2(H2CO + hv → H2 + CO) 2(CO + OH → CO2 + H) 2(H + O2 + M → HO2 +M) 4(NO + HO2 → NO2 + OH) 3(NO2 + h ν → NO + O( 3 P)) 3(NO2 + O( 3 P) → NO + O2) net: 2CH4 + 3O2 → 2H2O + 2H2 + 2CO2",
"pages": [
50
]
},
{
"title": "3g Sun Methane Loss : Found_PAP =3.21x10 10 , Shown_PAP=1.63x10 10 , Total_chem = 1.24x10 11",
"content": "Oxidation CH3OOH-a (26.5%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 H2O + O( 1 D) → OH + OH CO + OH → CO2 + H H + O2 + M → HO2 +M O3 + h ν → O2 + O( 1 D) net: CH4 + O3 + CO → CH3OOH + CO2 Oxidation 2O2-b (7.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+OH → H3CO+HO2 H3CO + O2 → H2CO + HO2 H2CO + OH → H2O + HCO HCO + O2 → HO2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 4(HO2 + O( 3 P) → OH + O2) 2(O2 + h ν → O( 3 P) + O( 3 P)) net:CH4 +2O2 → 2H2O+CO2 Oxidation 2O2-c (5.4%) CH4 + O( 1 D) → CH 3 + OH CH3 + O2 + M → CH 3O2 + M CO + OH → CO 2 + H H + O2 + M → HO 2 +M CH3O2 +OH →H 3CO+HO2 H3CO + O2 → H 2CO + HO2 3(HO2 + O( 3 P) → OH + O 2) OH + HO2 → H 2O + O2 H2 CO + OH → H 2O + HCO HCO + O2 → HO 2 + CO O( 3 P) + O2 + M → O 3 + M O3 + h ν → O 2 + O( 1 D) 2(O2 + h ν → O( 3 P) + O( 3 P)) net:CH4 +2O2 →2H 2O+CO2 Oxidation O3-a (5.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M H2CO + h ν → H2 + CO CO + OH → CO2 + H H + O2 + M → HO2 +M 2(HO2+O3 → OH+O2+O2) net: CH4 +O3 → H2O+H2+CO2 Oxidation O3-b (5.3%) CH4 + OH → CH 3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 CH3OOH+h ν → H 3CO+OH H3CO+O2 → H 2CO+HO2 H2CO + h ν → H 2 + CO CO + OH → CO 2 + H H + O2 + M → HO2 +M HO2 + O3 → OH + O2 + O2 net: CH4 +O3 → H2O+ H2+CO2",
"pages": [
50
]
},
{
"title": "1g M0 Methane Loss: Found_PAP =3.25x10 10 , Shown_PAP=1.93x10 10 , Total_chem = 1.24x10 11",
"content": "Oxidation 3O2-a (24.4%) Oxidation CH3OOH-b (12.8%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 H2O + O( 1 D) → OH + OH 2(H + O2 + M → HO2 +M) NO + HO2 → NO2 + OH 2(CO + OH → CO2 + H) NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + 2CO + 2O2 → CH3OOH + 2CO2 Oxidation 6O2 (6.1%) Oxidation O2 (10.4%) CH4 + OH → CH3 + H2O CH3 + O2 + M → CH3O2 + M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + O3 → OH + O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M net: CH4 + O2 → H2CO+H2O Oxidation CH3OOH-c (5.7%) 2(CH4 + OH → CH3 + H2O) 2(CH3+O2+M → CH3O2+M) CH3O2+OH → CH3OOH+O2 CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) H2O + O( 1 D) → OH + OH net: 2CH4 + 2O2 → H2CO + H2O + CH3OOH Oxidation CH3OOH-d (24.5%) 2(CH4 + O( 1 D) → CH3 + OH) 2(CH3+O2+M → CH3O2+M) 2(CO + OH → CO2 + H) 2(H + O2 + M → HO2 +M) 2(CH3O2+HO2>CH3OOH+O2) 2(O( 3 P) + O2 + M → O3 + M) 2(O2 + h ν → O2 + O( 1 D)) O2 + h ν → O( 3 P) + O( 3 P) net: 2CH4 + 2CO + 3O2 → 2CH3OOH + 2CO2 Oxidation CH3OOH-e (23.7%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CH3O2+HO2 → CH3OOH+O2 2(H + O2 + M → HO2 +M ) NO + HO2 → NO2 + OH 2( CO + OH → CO2 + H) NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + 2CO + 2O2 → CH3OOH + 2CO2 Oxidation H2O2-b (4.5%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H H + O2 + M → HO2 +M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + HO2 → H2O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) net: CH4 + CO + 2O2 → H2CO + H2O2 + CO2 Oxidation H2O2-a (5.8%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H H + O2 + M → HO2 +M CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 HO2 + HO2 → H2O2 + O2 NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) H2CO + h ν → H2 + CO net:CH4 +2O2 →H 2O2+H2+CO2 Oxidation H2O2-c (4.4%) CH4 + O( 1 D) → CH3 + OH CH3 + O2 + M → CH3O2 + M CO + OH → CO2 + H CH3O2 + NO → H3CO + NO2 H3CO + O2 → H2CO + HO2 H2CO + h ν → HCO + H HCO + O2 → HO2 + CO NO2 + h ν → NO + O( 3 P) O( 3 P) + O2 + M → O3 + M O3 + h ν → O2 + O( 1 D) 2(H + O2 + M → HO2 +M) 2(HO2 + HO2 → H2O2 + O2) net: CH4 +3O2 →2H 2O2+CO2",
"pages": [
50,
51
]
},
{
"title": "Appendix 1",
"content": "Ozone column in Dobson Units (DU) for the 1g, 3g scenarios corresponding to the values plotted in Figures 4a, 4b as a function of stellar effective temperature (Teff) (K).",
"pages": [
53
]
}
]
2013AstL...39..446G
https://arxiv.org/pdf/1306.2651.pdf
<document>
<section_header_level_1><location><page_1><loc_18><loc_80><loc_82><loc_84></location>Magnetically Active Stars in Taurus-Auriga: Activity and Rotation</section_header_level_1>
<section_header_level_1><location><page_1><loc_42><loc_75><loc_58><loc_76></location>K. N. Grankin</section_header_level_1>
<text><location><page_1><loc_20><loc_71><loc_80><loc_73></location>Crimean Astrophysical Observatory, Nauchny, Crimea, 98409 Ukraine</text>
<text><location><page_1><loc_37><loc_68><loc_63><loc_69></location>konstantin.grankin@rambler.ru</text>
<section_header_level_1><location><page_1><loc_45><loc_60><loc_55><loc_62></location>Abstract</section_header_level_1>
<text><location><page_1><loc_14><loc_39><loc_86><loc_59></location>A sample of 70 magnetically active stars toward the Taurus-Auriga star-forming region has been investigated. The positions of the sample stars on the Rossby diagram have been analyzed. All stars are shown to be in the regime of a saturated dynamo, where the X-ray luminosity reaches its maximum and does not depend on the Rossby number. A correlation has been found between the lithium line equivalent width and the age of a solar-mass (from 0.7 to 1.2 M /circledot ) pre-main-sequence star. The older the age, the smaller the Li line equivalent width. Analysis of the longterm photometric variability of these stars has shown that the photometric activity of the youngest stars is appreciably higher than that of the older objects from the sample. This result can be an indirect confirmation of the evolution of the magnetic field in pre-main-sequence stars from predominantly dipole and axisymmetric to multipole and nonaxisymmetric.</text>
<text><location><page_1><loc_12><loc_34><loc_85><loc_35></location>Key words: stars - physical properties, rotation, activity, pre-main-sequence stars.</text>
<section_header_level_1><location><page_1><loc_10><loc_30><loc_32><loc_31></location>INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_10><loc_8><loc_91><loc_29></location>Previously (Grankin 2013b), we investigated a sample of 74 magnetically active stars toward the Taurus-Auriga star-forming region (SFR). The sample contains 24 well-known young premain-sequence (PMS) stars and 60 candidates for PMS stars from Wichmann et al. (1996). All sample objects exhibit no optical and near-infrared excesses and, consequently, have no accretion disks. Based on accurate data on their main physical parameters (see Tables 3 and 4 in Grankin 2013a) and published data on their proper motions, X-ray luminosities, and equivalent widths of the H α and Li lines (see Table 1 in Grankin 2013b), we refined the evolutionary status of these objects. As a result, we identified a group of 70 objects with ages 1-40 Myr. We showed that 50 stars from this group belong to the Taurus-Auriga SFR with a high probability. Other 20 objects have a controversial evolutionary status and can belong to both the Taurus-Auriga SFR and the Gould Belt (see Table 3 in Grankin 2013b). For 50 PMS stars with known rotation periods, we analyzed the relationship between their rotation,</text>
<text><location><page_2><loc_10><loc_81><loc_91><loc_91></location>mass, and age. The rotation was shown to depend on both mass and age of young stars. We investigated the angular momentum evolution for the sample stars within the first 40 Myr. An active interaction between the sample stars and their protoplanetary disks was shown to have occurred on a time scale from 0.7 to 10 Myr. In this paper, we discuss various magnetic activity parameters for the sample PMS stars and investigate the relationship between their activity and rotation.</text>
<section_header_level_1><location><page_2><loc_10><loc_77><loc_46><loc_79></location>ACTIVITY AND ROTATION</section_header_level_1>
<text><location><page_2><loc_10><loc_69><loc_91><loc_76></location>All of the PMS stars from our sample exhibit an enhanced variable X-ray emission, which is evidence for the existence of a hot corona and, hence, magnetic activity. The key unsolved question regarding this X-ray emission is whether an analogy exists between the magnetic activity of PMS stars and that of the Sun.</text>
<text><location><page_2><loc_10><loc_32><loc_91><loc_69></location>On the Sun and on all of the stars whose internal structure consists of a radiative core and a convective envelope, magnetic activity is probably produced by the so-called α Ω dynamo mechanism. This mechanism operates in a thin shell lying at the interface between the radiative and convective zones and is generated by the interaction between differential rotation and convective motion (see Schrijver and Zwaan (2000) and references therein). This hypothesis is confirmed by the existence of magnetic activity, which manifests itself through cool photospheric spots, the chromospheric emission in the calcium H and K lines and the H α line, and the coronal X-ray emission. In the long run, this type of dynamo is governed by stellar rotation. As a result, a strong activity-rotation correlation is observed for solar-type stars with ages > 100 Myr. This correlation was first found by Skumanich (1972) and was subsequently confirmed by numerous studies (see, e.g., Patten and Simon 1996; Terndrup et al. 2000; Barnes 2001). The correlation usually manifests itself as a linear increase in activity indicators with increasing rotation rate accompanied by activity saturation at a high rotation rate. However, PMS stars are known to be fully convective and, thus, cannot provide a basis for the existence of a solar-type dynamo. As PMS stars evolve toward the main sequence (MS), their rotation rate changes greatly. This can lead to a change in the magnetic field generation mechanisms themselves and, as a consequence, to a change in magnetic activity properties and their relationship to stellar rotation. Thus, investigation of the relationship between magnetic activity and rotation for PMS stars can provide an understanding of the fundamental changes in the physics of stars that occur in the range of ages 1-100 Myr. In the next sections, we will investigate the relationship between the rotation of PMS stars in Taurus-Auriga and various magnetic activity indicators for these stars.</text>
<text><location><page_2><loc_10><loc_18><loc_91><loc_31></location>The nature of the relationship between magnetic activity and rotation can be complex, because it depends on the stellar age, mass, internal structure, and, possibly, interaction with the disks at early evolutionary stages. To separate the various processes, we will begin by analyzing the equivalent width of the H α line (EW(H α )) as a function of the temperature or spectral type. It should be noted that most of the objects from our sample were identified in X-ray surveys and the sample may be biased toward more active objects. Therefore, when discussing activity, we prefer to use criteria based on the upper limit of activity in our sample and not on the lower limit, because the latter can be shifted greatly.</text>
<section_header_level_1><location><page_2><loc_10><loc_14><loc_49><loc_15></location>CHROMOSPHERIC ACTIVITY</section_header_level_1>
<text><location><page_2><loc_10><loc_9><loc_91><loc_13></location>The H α line is commonly used as an indicator of chromospheric activity resulting from photoionization and collisions in a hot chromosphere. In Fig. 1, EW(H α ) is plotted against</text>
<text><location><page_3><loc_10><loc_61><loc_91><loc_91></location>the effective temperature represented by the spectral type in our case. The black and white symbols denote the objects with a reliable and unreliable evolutionary status, respectively (for details, see Grankin 2013b). The objects classified as weak-lined T Tauri stars (WTTS) with ages < 10 Myr and as post T Tauri stars (PTTS) with ages > 10 Myr are marked by the circles and squares, respectively. It can be seen from the figure that EW(H α ) is an obvious function of the spectral type. Whereas the H α line in G-type stars is in absorption (EW(H α ) ∼ 2 ˚ A), H α in K1-K4 stars exhibits a gradual transition to an emission state. For stars later than K5, H α is always in emission. The scatter of EW(H α ) increases and EW(H α ) for M-type stars lie within the range from 0 to -7 ˚ A. The dramatic change in EW(H α ) with spectral type reflects not only the change in chromospheric activity but also the additional effects related to a reduction in the continuum level with decreasing stellar luminosity and to a change in photospheric absorption in H α , which is zero for M dwarfs and increases toward earlier spectral types. The combined effects of a reduction in the continuum level and photospheric absorption in H α were estimated from spectroscopic observations of standard nonactive stars by Scholz et al. (2007). The linear approximation of EW(H α ) as a function of the spectral type is indicated in Fig. 1 by the dash-dotted line. This line coincides with the dependence of EW(H α ) on spectral type for stars in the Hyades and for field stars of various spectral types (for details, see Scholz et al. 2007). Thus, this line is an estimate for the purely photospheric contribution to EW(H α ).</text>
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<caption>Figure 1: EW(H α ) versus spectral type. The dash-dotted line indicates the upper limit of EW(H α ) for nonactive field stars. The dashed line corresponds to zero for EW(H α ). The black and white symbols denote the objects with a reliable and unreliable evolutionary status, respectively. The objects classified as WTTS (with ages < 10 Myr) and PTTS (with ages > 10 Myr) are marked by the circles and squares, respectively.</caption>
</figure>
<text><location><page_3><loc_10><loc_10><loc_91><loc_18></location>It can be seen from the figure that the dash-dotted line is the lower envelope for the stars of our sample: almost all of the PMS stars in Taurus-Auriga lie above this line, except for seven objects that exhibit no measurable chromospheric activity. Thus, most of the stars from our sample are chromospherically active objects. The maximum activity level increases rapidly for K7-M4 stars.</text>
<section_header_level_1><location><page_4><loc_10><loc_90><loc_33><loc_91></location>X-RAY ACTIVITY</section_header_level_1>
<text><location><page_4><loc_10><loc_70><loc_91><loc_89></location>First of all, we investigated the possible relationship between the rotation period ( P rot ) and various X-ray activity parameters of PMS stars: the X-ray luminosity ( L X ), the X-ray surface flux ( F X ), and the X-ray luminosity excess defined as the X-ray to bolometric luminosity ratio ( L X /L bol ). To calculate these X-ray activity parameters, we used data from our two previous papers (Grankin 2013a, 2013b). Figure 2 shows the corresponding graphs on a logarithmic scale. It can be seen from the figure that a very weak correlation is traceable between P rot and L X (see Fig. 2a). We failed to find any correlation between the rotation period and L X /L bol (see Fig. 2b). The most significant correlation is observed between P rot and F X (see Fig. 2c). Nevertheless, it would be unreasonable to assert that there exists an unambiguous correlation between these activity parameters of PMS stars and their rotation period because of the small correlation coefficients.</text>
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<caption>Figure 2: Relationship between rotation period and X-ray activity parameters: the X-ray luminosity L X (a), the ratio L X /L bol (b), and the X-ray surface flux F X (c). The position of the Sun is denoted by the corresponding symbol. The designations of the objects are the same as those in Fig. 1.</caption>
</figure>
<text><location><page_4><loc_10><loc_8><loc_91><loc_11></location>We attempted to investigate the possible correlation between the X-ray activity of PMS stars and their angular momentum, which is a no less informative characteristic of stellar rotation.</text>
<text><location><page_5><loc_10><loc_80><loc_91><loc_91></location>The angular momentum is J = Iω , where I is the moment of inertia and ω is the angular velocity of the star. The angular velocity of a star is easy to calculate if its rotation period is known: ω = 2 π/P . The moment of inertia of a star can be determined from the formula I = M ( kR ) 2 , where M is the stellar mass, R is the stellar radius, and k is the radius of inertia dependent on the rotation period and shape of the star. Thus, the angular momentum of a star depends on three parameters: its mass, radius, and rotation period. To reduce the number of unknown parameters, we can introduce some normalized angular momentum:</text>
<formula><location><page_5><loc_33><loc_75><loc_67><loc_78></location>j = J M = Iω M = 2 πM ( kR ) 2 MP = 2 πk 2 R 2 P .</formula>
<text><location><page_5><loc_10><loc_69><loc_91><loc_74></location>For a spherically symmetric star, k = √ 2 / 3 and the normalized angular momentum can be calculated from the formula j = 4 π 3 R 2 P . For the Sun, P = 25 days, R = 6 . 96 × 10 10 cm, and the angular momentum j /circledot = 9 . 39 × 10 15 cm 2 s -1 .</text>
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<caption>Figure 3: Relationship between angular momentum and various X-ray activity parameters. The sign of the Sun in the lower left corners marks its position. The designations are the same as those in Fig. 1.</caption>
</figure>
<text><location><page_5><loc_10><loc_8><loc_91><loc_11></location>We calculated the ratio j/j /circledot for all of the stars from our sample with known rotation periods. Based on these data, we plotted the normalized angular momentum( j/j /circledot ) against</text>
<text><location><page_6><loc_10><loc_86><loc_91><loc_91></location>various X-ray activity parameters of PMS stars: L X (Fig. 3a), L X /L bol (Fig. 3b), and F X (Fig. 3c). It can be seen from Fig. 3 that there is no significant correlation between the various X-ray activity parameters and angular momentum, as in the case with the rotation period.</text>
<section_header_level_1><location><page_6><loc_10><loc_82><loc_35><loc_84></location>ROSSBY DIAGRAM</section_header_level_1>
<text><location><page_6><loc_10><loc_52><loc_91><loc_81></location>The chromospheric and coronal activity of stars is known to be related to their rotation and to the depth of the convective zone or the convective turnover time ( τ c ). The existence of such a relationship is in good agreement with qualitative predictions of the α Ω dynamo theory explaining the generation of a magnetic field. The Rossby diagram is one of the best tools for demonstrating the existence of a relationship between magnetic field generation and stellar activity. As a rule, the Rossby diagram displays the relationship between some stellar activity indicator and the Rossby number ( R 0 ) calculated from the formula R 0 = P rot /τ c . In particular, previous studies have shown that the slowly rotating stars in the Hyades cluster and most of the dwarf field stars exhibit a decrease in log( L X /L bol ) with increasing Rossby number. In contrast, the rapidly rotating field stars and G and K dwarfs in the Pleiades and α Persei clusters exhibit no obvious relationship between log( L X /L bol ) and the Rossby number (Hempelmann et al. 1995; Patten and Simon 1996; Randich et al. 1996; Queloz et al. 1998). Subsequently, Pizzolato et al. (2003) showed that the relationship between the X-ray luminosity and rotation period of a star could be roughly described by a power law irrespective of its mass and spectral type. Thus, stellar rotation dominates over convection for slowly rotating solartype stars. At the same time, the X-ray luminosity of rapidly rotating stars depends only on L bol and, consequently, depends on stellar-structure characteristics.</text>
<text><location><page_6><loc_10><loc_40><loc_91><loc_52></location>Recent studies have shown that for slowly rotating stars there is a tendency for L X /L bol to grow with increasing rotation velocity up to /revsimilar 15 km s -1 , while stars with higher velocities have approximately the same L X /L bol near the saturation level ( L X /L bol /revsimilar 10 -3 ), with this saturation limit being observed for stars in a wide range of spectral types, from G to M. Thus, the most active stars exhibit a maximum X-ray luminosity at a level of L X /L bol /revsimilar 10 -3 , which does not depend on the rotation velocity. This phenomenon was called the saturated dynamo effect.</text>
<text><location><page_6><loc_10><loc_35><loc_91><loc_40></location>In Fig. 4, log( L X /L bol ) is plotted against the Rossby number for all of the stars from our sample with known rotation periods. We estimated the convective turnover time τ c from an empirical relation given in Noyes et al. (1984):</text>
<formula><location><page_6><loc_26><loc_30><loc_73><loc_34></location>log τ c = { 1 . 362 -0 . 166 x +0 . 025 x 2 -5 . 323 x 3 , x > 0 , 1 . 362 -0 . 14 x, x < 0 ,</formula>
<text><location><page_6><loc_10><loc_22><loc_91><loc_28></location>where x = 1 -( B -V ). We used the extinction-corrected color index as ( B -V ). To compare the activity of PMS stars with the activity of other solar-type stars, we showed the positions of the MS stars and the stars from the Hyades, Pleiades, IC 2391, and IC 2602 open clusters investigated by Pizzolato et al. (2003) in Fig. 4.</text>
<text><location><page_6><loc_10><loc_13><loc_91><loc_21></location>It can be seen from Fig. 4 that the X-ray luminosity excess ( L X /L bol ) for the sample of active stars from Pizzolato et al. (2003) increases with decreasing Rossby number ( R 0 ). However, the increase in L X /L bol ceases at a level of about log( L X /L bol ) = -3, when R 0 reaches /revsimilar 0 . 28 -0 . 1 (log R 0 = -0 . 56 --0 . 98). From this time on, the so-called saturation regime is observed, where the X-ray luminosity excess reaches its maximum values and ceases to depend on R 0 .</text>
<text><location><page_6><loc_10><loc_8><loc_91><loc_13></location>All PMS stars from our sample exhibit the same L X /L bol and R 0 as the stars from the IC 2602 and Pleiades open clusters with ages within the range 30-100 Myr. In other words, the X-ray activity of PMS stars in the Taurus-Auriga SFR closely coincides with that of the cluster</text>
<text><location><page_7><loc_10><loc_83><loc_91><loc_91></location>stars in the regime of saturated activity. It should be noted that there are slightly more stars with small log R 0 in the range from -1.7 to -2.1 in the Pleiades. Since the stars of our sample are located in the zone of a saturated dynamo, the fact that we failed to find an unequivocal correlation between the rotation period and various X-ray activity parameters (see the previous section) becomes explainable.</text>
<figure>
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<caption>Figure 4: L X /L bol versus Rossby number. The crosses are stars from the Pleiades, IC 2391, and IC 2602 open clusters; the asterisks are Hyades stars; the pluses are MS dwarfs from Pizzolato et al. (2003). The position of the Sun at the maximum of the activity cycle is also marked. The designations of PMS objects are the same as those in Fig. 1.</caption>
</figure>
<section_header_level_1><location><page_7><loc_10><loc_33><loc_46><loc_35></location>PHOTOSPHERIC ACTIVITY</section_header_level_1>
<text><location><page_7><loc_10><loc_8><loc_91><loc_32></location>In previous sections, we have pointed out that the magnetic activity of young solar-type stars manifests itself through the chromospheric emission in the calcium H and K lines and the H α line or through the coronal X-ray emission. In addition, the magnetic activity can also manifest itself through the maximum photometric variability amplitude in the optical spectral range (∆ V max ). Indeed, the maximum photometric variability amplitude depends primarily on the degree of nonuniformity in the distribution of spotted regions over the stellar surface and, consequently, on the total surface magnetic flux. Before investigating the possible relationship between ∆ V max and various rotation parameters, we analyzed the possible correlations of ∆ V max with such parameters as the spectral type of a star and its age. In Fig. 5a, the maximum variability amplitude is plotted against the spectral type. It can be seen from the figure that the maximum amplitude gradually increases from earlier spectral types to later ones and reaches its maximum near a spectral type K7-M1. This effect can be due to a change in the contrast of dark spots against the background of the photosphere. The results of modeling this effect are represented by the solid line. It can be clearly seen that the amplitude of the periodicity</text>
<text><location><page_8><loc_10><loc_81><loc_91><loc_91></location>increases from relatively early spectral types to later ones. It should be noted that our sample contains five most active WTTS whose maximum amplitudes are considerably larger than those of the remaining PMS stars: LkCa 4, LkCa 7, V827 Tau, V830 Tau, and TAP 41. These stars are separated from the main group by the horizontal dashed line at a level of 0 . m 35. We excluded these objects from the subsequent statistical analysis and will discuss their properties separately.</text>
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<caption>Figure 5: Maximum photometric variability amplitude versus spectral type (a), Li line equivalent width (b), age (c), and H α equivalent width (d). The designations of PMS objects are the same as those in Fig. 1.</caption>
</figure>
<text><location><page_8><loc_10><loc_23><loc_91><loc_37></location>In Fig. 5b, ∆ V max is plotted against EW(Li). We found a weak correlation between ∆ V max and EW(Li) with a correlation coefficient k = 0.39. The maximum photometric variability amplitude increases with increasing lithium line equivalent width. The existence of such a positive correlation between ∆ V max and EW(Li) may reflect the fact that younger PMS stars are simultaneously also more active objects. For example, it is easy to notice that the five most active WTTS lying above the dashed line have values of EW(Li) that are among the largest ones. This result is quite intriguing, because the presence of a lithium absorption line is considered primarily as a signature of youth and not as a signature of stellar activity.</text>
<text><location><page_8><loc_10><loc_13><loc_91><loc_23></location>In Fig. 5c, the maximum photometric variability amplitude is plotted against the age of PMS stars. The solid line is a linear approximation for all of the stars lying below the dashed line (∆ V max < 0 . m 35). A weak correlation between the maximum photometric variability amplitude and age with a correlation coefficient k = 0.45 is noticeable. The maximum amplitude decreases with increasing stellar age. The four stars exhibiting the largest variability amplitudes and lying above the horizontal dashed line have ages 2 . 5 -3 . 1 Myr.</text>
<text><location><page_8><loc_10><loc_8><loc_91><loc_13></location>In Fig. 5d, ∆ V max is plotted against EW(H α ). The solid line is a linear approximation for all of the stars lying below the horizontal dashed line (∆ V max < 0 . m 35). It can be seen from the figure that there is a clear correlation between ∆ V max and EW(H α ) with a correlation</text>
<text><location><page_9><loc_10><loc_83><loc_91><loc_91></location>coefficient k = 0.46. As the H α line passes from an absorption state to an emission one, the maximum photometric variability amplitude increases monotonically. This result confirms our assumption that the maximum photometric variability amplitude can be used as an indicator of photospheric activity, while the H α line is an indicator of chromospheric activity for PMS stars.</text>
<text><location><page_9><loc_10><loc_55><loc_91><loc_83></location>In the previous section, we showed that the PMS stars in Taurus-Auriga are in the regime of a saturated dynamo, where the X-ray flux reaches saturation and ceases to depend on the rotation rate. Since the X-ray flux is related to the number of active regions on the stellar surface, we can assume that the active regions should cover almost the entire stellar surface in the regime of a saturated dynamo. In that case, we may expect maximum photometric variability amplitudes for PMS stars, of course, only when the active regions are distributed over the stellar surface highly nonuniformly. Therefore, it is interesting to investigate the relationship between ∆ V max and such stellar rotation parameters as the rotation period, the Rossby number, the equatorial rotation velocity, and the angular momentum. For this purpose, we constructed the corresponding dependences but failed to find a clear correlation between ∆ V max and the rotation parameters listed above. This result confirms our conclusion that the PMS stars are in a state of saturated dynamo. Regarding the five most active stars that exhibit the largest variability amplitudes (∆ V max > 0 . m 35), it should be noted that they have moderate rotation velocities ( v eq = 10 -30 km s -1 ) and are in the region of the transition into the zone of a saturated dynamo that corresponds to Rossby numbers in the range log( P rot /τ c ) = -0 . 56 --0 . 98.</text>
<section_header_level_1><location><page_9><loc_10><loc_52><loc_65><loc_53></location>Li EVOLUTION DURING THE PMS STAGE</section_header_level_1>
<text><location><page_9><loc_10><loc_37><loc_91><loc_50></location>Lithium, just as other light elements, such as beryllium and boron, is burnt out in thermonuclear reactions at relatively low temperatures in the stellar interior ((2 . 5 -3 . 0) × 10 6 K). In the case of initial evolution of low-mass ( M < 1 . 2 M /circledot ) stars, efficient mixing can rapidly transport a lithium-depleted material from the central regions of a PMS star to its photosphere. For this reason, measurements of the photospheric Li abundance provide one of the few means for probing the stellar interior and are a sensitive test of evolution models for PMS stars. Understanding the Li depletion mechanisms at the stage of PMS stars also makes it possible to estimate the ages of young stars.</text>
<text><location><page_9><loc_10><loc_16><loc_91><loc_37></location>A large number of observational and theoretical works were devoted to understanding the initial abundance of Li and its subsequent PMS evolution (see the recent review by Jeffries 2006). According to classical models, the photospheric depletion of Li begins near 2 Myr for a star with a mass of 1 M /circledot and should end in an age of about 15 Myr. This window moves toward older ages for stars with lower masses. However, the degree of Li depletion depends very strongly on mass, convection efficiency, opacity, metallicity, and other model parameters. Thus, the amount of photospheric Li can serve as a characteristic of the youth of a PMS star. Nevertheless, numerous observations of the photospheric lithium abundance in hundreds of young stars in open clusters suggest that the degree of Li depletion in these stars is considerably smaller than what is predicted by standard models. In addition, the K-type stars in clusters with ages of less than 250 Myr are characterized by a significant dispersion in Li abundance whose cause is not yet completely understood (Jeffries 2006).</text>
<text><location><page_9><loc_10><loc_10><loc_91><loc_16></location>This and other puzzling peculiarities that are unexpected within the framework of standard models suggest that Li depletion is governed not only by convection and that there exist other, unknown processes that have not been included in the classical theory. In recent years, several nonstandard models explaining the physics of the possible processes leading to Li depletion</text>
<text><location><page_10><loc_10><loc_86><loc_91><loc_91></location>have been proposed. However, the mechanisms governing Li depletion still remain poorly studied. Additional observational constraints for present-day models are badly needed for further progress in understanding the evolution of Li during the PMS stage.</text>
<text><location><page_10><loc_10><loc_76><loc_91><loc_86></location>That is why we attempted to reveal any correlations or relationships between the Li equivalent width and other physical parameters of the PMS stars from our sample. Finding such correlations can shed light on the problem of Li depletion at the PMS evolutionary phase of young stars. In the previous section, we pointed out that there exists a weak positive correlation between EW(Li) and ∆ V max for PMS stars. At the same time, we failed to find any correlation between EW(Li) and the X-ray luminosity.</text>
<text><location><page_10><loc_10><loc_73><loc_90><loc_76></location>Below, we investigate the possible correlation between EW(Li) and such parameters of PMS stars as the theoretical age ( t ) and the rotation period ( P rot ).</text>
<text><location><page_10><loc_10><loc_61><loc_91><loc_72></location>If the entire sample of stars is considered as a single group, then no correlation is observed between EW(Li) and t . It should be noted that our sample includes PMS stars with quite different physical parameters. For example, the masses of the PMS stars from our sample lie within the range 0.26 - 2.2 M /circledot . Since the degree of Li depletion depends very strongly on mass, we attempted to find a possible correlation between EW(Li), age, and rotation period for stars with masses fairly close to the solar mass (in the range 0.7 - 1.2 M /circledot ). In Fig. 6, EW(Li) is plotted against t (a) and P rot (b).</text>
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</figure>
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<caption>Figure 6: EW(Li) versus age (a) and rotation period (b). The designations are the same as those in Fig. 1.</caption>
</figure>
<text><location><page_10><loc_10><loc_22><loc_91><loc_33></location>For solar-mass stars, there exists a statistically significant correlation between EW(Li) and t with a correlation coefficient of 0.68. The older the age, the smaller the equivalent width EW(Li). The stars with ages /revsimilar 2 -3 Myr have maximum values of EW(Li) ( /revsimilar 0 . 58 ˚ A). In contrast, the stars with ages older than 30 Myr exhibit minimum values of EW(Li) (about 0.20 ˚ A). This result is in excellent agreement with the predictions of the classical models explaining the evolution of the atmospheric Li abundance during the PMS evolutionary stage of solar-mass stars.</text>
<text><location><page_10><loc_10><loc_8><loc_91><loc_21></location>If the entire sample of stars is considered as a single group, then no correlation is observed between EW(Li) and P rot . It can also be noted that the stars with rotation periods longer than 5 days have relatively broad Li absorption lines (EW(Li) > 0.4 ˚ A). In contrast, the stars with periods shorter than 5 days have various values of EW(Li), from 0.2 to 0.7 ˚ A. If each subgroup of stars is considered separately, then the following can be identified: (1) the reliable WTTS (black circles) have EW(Li) /revsimilar 0 . 6 ˚ A for the entire range of rotation periods from 0.5 to 10 days; (2) the reliable PTTS (black squares) with ages older than 10 Myr have EW(Li) /revsimilar 0 . 4 ˚ A for their range of rotation periods from 0.5 to 5 days; (3) the PTTS with an unreliable evolutionary</text>
<text><location><page_11><loc_10><loc_86><loc_91><loc_91></location>status rotate more rapidly ( P rot in the range from 0.5 to 2 days) and have EW(Li) /revsimilar 0 . 2 ˚ A. In other words, the same dependence of the lithium line equivalent width on age, but not on rotation period, is observed.</text>
<section_header_level_1><location><page_11><loc_10><loc_82><loc_75><loc_84></location>PROPERTIES OF THE MOST ACTIVE PMS STARS</section_header_level_1>
<text><location><page_11><loc_10><loc_64><loc_91><loc_81></location>Several most active stars that exhibit the record maximum photometric variability amplitudes deserve particular attention: LkCa 4 (0 m . 81), LkCa 7 (0 m . 58), V827 Tau [TAP 42] (0 m . 51), V830 Tau (0 m . 45), and V1075 Tau [TAP 41] (0 m . 39). It should be noted that there are two more objects that exhibit large variability amplitudes: V410 Tau (0 m . 63) and V836 Tau (0 m . 62). However, we do not discuss their properties here, because we failed to determine reliable luminosities, radii, masses, and ages for them. Such large amplitudes of light variations can be due to the existence of very large and extended spotted regions in the photospheres of these stars; these extended spotted regions must be distributed over the surface highly nonuniformly, otherwise the very large amplitudes of periodic light variations reaching 0 m . 4 -0 m . 8 cannot be explained.</text>
<text><location><page_11><loc_10><loc_49><loc_91><loc_64></location>Aprevious analysis of long-term photometric observations for a sample of well-known WTTS from the Taurus-Auriga SFR showed that some of these objects exhibit stability of the phase of minimum light ( ϕ min ) over several years of observations (see, e.g., Grankin et al. (2008) and references therein). Only seven stars from the entire sample of known WTTS show stability of ϕ min in the interval from 5 to 19 years: LkCa 4, LkCa 7, V819 Tau, V827 Tau, V830 Tau, V836 Tau, and V410 Tau. Such long-term stability of ϕ min can be due to the existence of the so-called active longitudes at which short-lived groups of spots are located (Grankin et al. 1995). Similar long-lived active regions are known on the Sun and some of the RS CVn binary stars.</text>
<text><location><page_11><loc_10><loc_40><loc_91><loc_48></location>It should be noted that almost all of these stars enter into the list of seven most active objects that exhibit the largest photometric variability amplitudes (see above). Thus, the stability of the phase of minimum light for these objects is somehow related to the existence of very large variability amplitudes. Given the unusual photometric properties of these active stars, we decided to discuss their main physical parameters in more detail.</text>
<text><location><page_11><loc_10><loc_28><loc_91><loc_40></location>First, all of the most active stars with known parameters have very similar spectral types in the range K6-K7 and, hence, almost identical surface temperatures. Second, the rotation periods of these active stars lie within a fairly narrow range, from 2.4 to 5.7 days. Third, the radii of these stars are fairly close and lie within the range from 1.30 to 1.75 R /circledot . Fourth, analysis of their positions on the Hertzsprung-Russell diagram showed that their masses also have close values (from 0.74 to 0.92 M /circledot ), while their ages lie within the range from 2.5 to 8.2 Myr.</text>
<text><location><page_11><loc_10><loc_20><loc_91><loc_28></location>Apart from the similarity of the main parameters of these stars noted above, it should be noted that they exhibit the broadest lithium lines with EW(Li) > 0 . 57 ˚ A. This fact confirms our assumption that the lithium line equivalent width can be used as an activity and youth criterion for stars, because there exists a significant correlation between EW(Li), the mean photometric variability amplitude, and the age of PMS stars (see Figs. 5b and 6a).</text>
<text><location><page_11><loc_10><loc_13><loc_91><loc_19></location>We limited our sample of stars in mass and considered only those objects whose masses were close to the solar mass (within the range from 0.7 to 1.2 M /circledot ). In this case, all five most active stars exhibit the strongest H α emission lines and are among the youngest ones, with ages from 2.5 to 3 Myr, except TAP 41 whose age is estimated to be 8 Myr.</text>
<text><location><page_11><loc_10><loc_9><loc_91><loc_12></location>Thus, if the subgroup of solar-mass PMS stars is considered, then it can be asserted that the most active and youngest stars with ages of no more than 8 Myr have the largest photometric</text>
<text><location><page_12><loc_10><loc_86><loc_91><loc_91></location>variability amplitudes (reaching 0 m . 39 -0 m . 81), show H α emission in the range from -0.5 to -4.0 ˚ A, and exhibit the most stable phase light curves and the strongest lithium absorption line (EW(Li) > 0.57 ˚ A).</text>
<section_header_level_1><location><page_12><loc_10><loc_82><loc_69><loc_84></location>EVOLUTION OF THE PHASE LIGHT CURVES</section_header_level_1>
<text><location><page_12><loc_10><loc_68><loc_91><loc_81></location>The evolution of the phase light curves for the most active PMS stars was analyzed in detail by Grankin et al. (2008). Here, we will point out the most interesting results. Despite the stability of the phase light curves in the sense of stability of the phase of minimum light, all active stars exhibit significant changes in the amplitude and shape of the phase light curve from season to season. For example, the amplitude for LkCa 7 changes from 0 m . 33 to 0 m . 58. The most symmetric (relative to ϕ min ) phase light curve, as a rule, corresponds to the season with a maximum amplitude. Conversely, the most asymmetric phase light curves correspond to minimum amplitudes of light variations.</text>
<text><location><page_12><loc_10><loc_61><loc_91><loc_67></location>Whereas many of the stars show gradual changes in the amplitude and shape of the light curve from season to season (for example, LkCa 4 and LkCa 7), there are examples of a completely different behavior. The amplitude of periodic light variations can change noticeably by a few tenths of a magnitude during one season, as in the case of TAP 41.</text>
<text><location><page_12><loc_10><loc_45><loc_91><loc_60></location>Although the amplitude of the light curve can change noticeably, the mean brightness level is essentially constant. Simple simulations showed that the stability of the mean brightness level from season to season suggests that the total number of spots on the surfaces of active stars changes much less than their distribution over the stellar surface. In other words, the decrease in the amplitude of the periodicity is attributable not to a decrease in the total area of the spots (in this case, the mean brightness level should increase) but to a more uniform distribution of the spots over the stellar surface. However, it should be noted that some of the stars exhibit noticeable changes in the mean brightness level from season to season (V819 Tau, V827 Tau, V836 Tau, and VY Tau).</text>
<text><location><page_12><loc_10><loc_23><loc_91><loc_45></location>All the noted peculiarities of the evolution of the phase light curves concerned the most active and youngest stars of our sample. However, the overwhelming majority of sample stars show a slightly different photometric behavior that was not discussed in previous papers. In particular, many of the PMS stars exhibit modest variability amplitudes that do not exceed 0 m . 1 -0 m . 2. In addition, periodic light variations are observed much more rarely than in the most active PMS stars whose properties have been discussed above. The differences in photometric behavior between the most active PMS stars and the remaining sample stars are presented in Fig. 7. Figures 7a and 7b show the seasonal phase light curves, respectively, for V819 Tau, one of the most active and youngest WTTS with an age of 3.3 Myr, and for W62 (RXJ0452.5+1730), a reliable PTTS with an age of 22 Myr. It can be seen from the figure that clear periodic light variations in W62 were detected only in two of the six seasons: in 2001 and 2004. In contrast, periodic light variations in V819 Tau were observed during each observing season. The phase of minimum light remained stable over 6 years (from 1999 to 2004).</text>
<text><location><page_12><loc_10><loc_14><loc_91><loc_23></location>To quantitatively characterize the frequency of occurrence of a periodicity, we used a simple parameter, f = N p /N s , where N p is the number of observing seasons with periodic light variations and N s is the total number of observing seasons (see Grankin 2013a). For example, the most active PMS stars exhibit periodic light variations virtually during each observing season, i.e., the frequency of occurrence of a periodicity is f = 1.</text>
<text><location><page_12><loc_10><loc_9><loc_91><loc_14></location>The less active PMS stars show periodic light variations with a mean frequency /revsimilar 0.5, i.e., periodic light variations were detected in half of the observing seasons. Since most of the stars from our sample are not so young and active, we investigated the relationship between the</text>
<text><location><page_13><loc_10><loc_83><loc_91><loc_91></location>frequency of occurrence of a periodicity and such parameters of PMS stars as EW(Li) (Fig. 8a) and the age (Fig. 8b). It can be seen from Fig. 8a that the frequency of occurrence of a periodicity increases with increasing EW(Li). The dependence of f on age (Fig. 8b) shows that the frequency of occurrence of a periodicity is at a maximum for the youngest stars and gradually decreases for older objects.</text>
<figure>
<location><page_13><loc_15><loc_45><loc_77><loc_80></location>
<caption>Figure 7: Phase light curves over six years of observations (from 1999 to 2004) for V819 Tau with an age of 3.3 Myr (a) and for W62 (RXJ0452.5+1730) with an age of 22 Myr (b).</caption>
</figure>
<text><location><page_13><loc_10><loc_8><loc_91><loc_39></location>In our previous papers, we showed that a small periodicity amplitude suggests a more uniform distribution of spots over the stellar surface, while a large amplitude is typical of the case where the spots are concentrated in one or two high-latitude regions, i.e., they are distributed highly nonuniformly (see Grankin 1999; Grankin et al. 2008). These conclusions are confirmed by the Doppler mapping of the surfaces of selected PMS stars. In particular, cool long-lived high-latitude spots were shown to have dominated on the surface of V410 Tau in the period 1992-1993, when the photometric variability amplitude was at a maximum and reached 0 m . 5 -0 m . 6 (Rice and Strassmeier 1996). In contrast, in the period 2007-2009, when the photometric variability amplitude decreased to 0 m . 06 -0 m . 10 (Grankin and Artemenko 2009), Doppler mapping showed that quite a few low-latitude spots distributed in longitude almost uniformly were present on the stellar surface (see Fig. 4 in Skelly et al. 2010). Thus, it can be assumed that the above differences in photometric behavior between the youngest PMS stars and older sample objects are attributable to different patterns of distribution of the spots over the stellar surface. Since the positions of cool magnetic spots on the surfaces of stars with convective envelopes are related to the locations of local magnetic fields, it is obvious that the pattern of distribution of the spots over the surface will depend on the magnetic field structure. Since the photometric behavior of the most active and youngest objects suggests that long-lived spots are concentrated at high latitudes, it can be assumed that the magnetic field of these</text>
<text><location><page_14><loc_10><loc_76><loc_91><loc_91></location>stars has a simpler and fairly symmetric dipole structure. Owing to such a structure, the spots are concentrated predominantly near the magnetic poles and retain their positions during many rotation cycles. It is such a behavior that we observe in the case of the most active and youngest objects. In contrast, the relatively old stars can have a more complex magnetic field structure. Therefore, the spots are distributed over the surface more uniformly; the amplitude of the periodicity is much smaller or it is not observed at all. Such a photometric behavior is typical of the older objects from our sample. In other words, the existence of a correlation between the photometric behavior and age can be a consequence of the evolution of the magnetic field structure for PMS stars.</text>
<figure>
<location><page_14><loc_15><loc_56><loc_47><loc_73></location>
</figure>
<figure>
<location><page_14><loc_52><loc_56><loc_82><loc_73></location>
<caption>Figure 8: Frequency of occurrence of a periodicity f = N p /N s versus lithium line equivalent width (a) and age (b).</caption>
</figure>
<text><location><page_14><loc_10><loc_24><loc_91><loc_49></location>This assumption is in good agreement with the results of a recent study of the magnetic field topology for several PMS stars performed within the MaPP (Magnetic Protostars and Planets) Program (see, e.g., Donati et al. (2010, 2011) and references therein). In particular, these studies showed that the magnetic field structure evolves from predominantly dipole and axisymmetric (in the case of fully convective stars) to octupole and axisymmetric (when the radiative core is less than half the stellar radius) and then to multipole and nonaxisymmetric (when the convective zone occupies less than half the stellar radius). The fact that the five most active stars discussed above lie on the Hertzsprung-Russell diagram in the region where fully convective PMS stars with a predominantly dipole and axisymmetric magnetic field structure are located can serve as an additional argument for such evolution of the magnetic field. Our future cooperative studies of the magnetic field topology for 40 PMS stars planned as part of the MaTYSSE (Magnetic Topologies of Young Stars & the Survival of close-in massive Exoplanets) Program performed on TBL (NARVAL spectrograph) and CFHT (ESPaDOnS spectrograph) will show whether the changes in the amplitude of the light curve are accompanied by significant changes in the distribution of spots and/or magnetic field topology.</text>
<section_header_level_1><location><page_14><loc_10><loc_20><loc_29><loc_21></location>CONCLUSIONS</section_header_level_1>
<text><location><page_14><loc_10><loc_14><loc_91><loc_18></location>We analyzed a sample of 70 magnetically active stars toward the Taurus-Auriga SFR and investigated the relationship between magnetic activity and rotation for these objects. In particular, we obtained the following results.</text>
<text><location><page_14><loc_10><loc_8><loc_91><loc_13></location>We analyzed the relationship between various X-ray activity parameters and rotation for PMS stars in the Taurus-Auriga SFR. We showed that there is no significant correlation between various X-ray activity parameters ( L X , L X /L bol , and F X ) and rotation parameters, such</text>
<text><location><page_15><loc_10><loc_85><loc_91><loc_91></location>as the period and the angular momentum. We investigated the positions of PMS stars on the Rossby diagram. All sample stars exhibit the same L X /L bol and R 0 as the stars from the Pleiades and IC 2602 clusters with ages within the range 30 -100 Myr, i.e., they are in the regime of a saturated dynamo.</text>
<text><location><page_15><loc_10><loc_78><loc_91><loc_84></location>We analyzed the photospheric activity of PMS stars based on original long-term photometric observations. The maximum photometric variability amplitude was found, on average, to decrease with increasing age of the sample objects and to increase with increasing equivalent width of the H α emission line and the lithium absorption line.</text>
<text><location><page_15><loc_10><loc_69><loc_91><loc_77></location>We found a statistically significant correlation between the lithium line equivalent width and the age of solar-mass (in the range from 0.7 to 1.2 M /circledot ) PMS stars. The older the age, the smaller the Li line equivalent width. This result is in excellent agreement with the predictions of the classical models explaining the evolution of the atmospheric Li abundance during the PMS stage of evolution of solar-mass stars.</text>
<text><location><page_15><loc_10><loc_61><loc_91><loc_69></location>We identified a group of five most active PMS stars that exhibit maximum photometric variability amplitudes reaching 0 . m 4 -0 . m 8. All these stars have very similar physical parameters: spectral types (K6-K7), rotation periods (2.4-5.7 days), radii (1.3-1.75 R /circledot ), masses (0.74-0.92 M /circledot ), and ages (2.5-8.2 Myr). In addition, they show a prominent emission in H α (from -0.5 ˚ A to -4.0 ˚ A) and the strongest lithium absorption line (EW(Li) > 0.57 ˚ A).</text>
<text><location><page_15><loc_10><loc_45><loc_91><loc_60></location>The most interesting feature of the photometric behavior of these active stars is related to the stability of the phase light curve over several observing seasons. The long-termstability of the phase light curves manifests itself in the fact that the phase of minimum light ( ϕ min ) retains its value in the interval from 5 to 19 years. Such a feature of the photometric behavior may be attributable to peculiarities of the magnetic field configuration for these stars. The most active and youngest stars from our sample most likely have mainly large-scale magnetic fields with an axisymmetric poloidal configuration. In this case, extended spotted regions are concentrated near the locations of two magnetic poles. The long existence of extended spotted regions suggests that the structure of these dipole fields is fairly stable over several years.</text>
<text><location><page_15><loc_10><loc_26><loc_91><loc_45></location>The remaining sample stars exhibit small photometric variability amplitudes (no more than 0 m . 15), with a periodicity being observed not in each observing season. The frequency of occurrence of a periodicity was shown to be maximal for the youngest stars and to gradually decrease for older objects. It may well be that the existence of this relationship is an indirect confirmation of the evolution of the magnetic field structure for young stars from predominantly dipole and axisymmetric (in the case of fully convective stars) to octupole and axisymmetric (when the radiative core is less than half the stellar radius) and then to multipole and nonaxisymmetric (when the convective zone occupies less than half the stellar radius). The fact that the five most active stars lie on the Hertzsprung-Russell diagram in the region where fully convective PMS stars with a fairly simple dipolar magnetic field structure are located can serve as an additional argument for such evolution of the magnetic field.</text>
<section_header_level_1><location><page_15><loc_10><loc_22><loc_28><loc_24></location>REFERENCES</section_header_level_1>
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</document>
[
{
"title": "K. N. Grankin",
"content": "Crimean Astrophysical Observatory, Nauchny, Crimea, 98409 Ukraine konstantin.grankin@rambler.ru",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "A sample of 70 magnetically active stars toward the Taurus-Auriga star-forming region has been investigated. The positions of the sample stars on the Rossby diagram have been analyzed. All stars are shown to be in the regime of a saturated dynamo, where the X-ray luminosity reaches its maximum and does not depend on the Rossby number. A correlation has been found between the lithium line equivalent width and the age of a solar-mass (from 0.7 to 1.2 M /circledot ) pre-main-sequence star. The older the age, the smaller the Li line equivalent width. Analysis of the longterm photometric variability of these stars has shown that the photometric activity of the youngest stars is appreciably higher than that of the older objects from the sample. This result can be an indirect confirmation of the evolution of the magnetic field in pre-main-sequence stars from predominantly dipole and axisymmetric to multipole and nonaxisymmetric. Key words: stars - physical properties, rotation, activity, pre-main-sequence stars.",
"pages": [
1
]
},
{
"title": "INTRODUCTION",
"content": "Previously (Grankin 2013b), we investigated a sample of 74 magnetically active stars toward the Taurus-Auriga star-forming region (SFR). The sample contains 24 well-known young premain-sequence (PMS) stars and 60 candidates for PMS stars from Wichmann et al. (1996). All sample objects exhibit no optical and near-infrared excesses and, consequently, have no accretion disks. Based on accurate data on their main physical parameters (see Tables 3 and 4 in Grankin 2013a) and published data on their proper motions, X-ray luminosities, and equivalent widths of the H α and Li lines (see Table 1 in Grankin 2013b), we refined the evolutionary status of these objects. As a result, we identified a group of 70 objects with ages 1-40 Myr. We showed that 50 stars from this group belong to the Taurus-Auriga SFR with a high probability. Other 20 objects have a controversial evolutionary status and can belong to both the Taurus-Auriga SFR and the Gould Belt (see Table 3 in Grankin 2013b). For 50 PMS stars with known rotation periods, we analyzed the relationship between their rotation, mass, and age. The rotation was shown to depend on both mass and age of young stars. We investigated the angular momentum evolution for the sample stars within the first 40 Myr. An active interaction between the sample stars and their protoplanetary disks was shown to have occurred on a time scale from 0.7 to 10 Myr. In this paper, we discuss various magnetic activity parameters for the sample PMS stars and investigate the relationship between their activity and rotation.",
"pages": [
1,
2
]
},
{
"title": "ACTIVITY AND ROTATION",
"content": "All of the PMS stars from our sample exhibit an enhanced variable X-ray emission, which is evidence for the existence of a hot corona and, hence, magnetic activity. The key unsolved question regarding this X-ray emission is whether an analogy exists between the magnetic activity of PMS stars and that of the Sun. On the Sun and on all of the stars whose internal structure consists of a radiative core and a convective envelope, magnetic activity is probably produced by the so-called α Ω dynamo mechanism. This mechanism operates in a thin shell lying at the interface between the radiative and convective zones and is generated by the interaction between differential rotation and convective motion (see Schrijver and Zwaan (2000) and references therein). This hypothesis is confirmed by the existence of magnetic activity, which manifests itself through cool photospheric spots, the chromospheric emission in the calcium H and K lines and the H α line, and the coronal X-ray emission. In the long run, this type of dynamo is governed by stellar rotation. As a result, a strong activity-rotation correlation is observed for solar-type stars with ages > 100 Myr. This correlation was first found by Skumanich (1972) and was subsequently confirmed by numerous studies (see, e.g., Patten and Simon 1996; Terndrup et al. 2000; Barnes 2001). The correlation usually manifests itself as a linear increase in activity indicators with increasing rotation rate accompanied by activity saturation at a high rotation rate. However, PMS stars are known to be fully convective and, thus, cannot provide a basis for the existence of a solar-type dynamo. As PMS stars evolve toward the main sequence (MS), their rotation rate changes greatly. This can lead to a change in the magnetic field generation mechanisms themselves and, as a consequence, to a change in magnetic activity properties and their relationship to stellar rotation. Thus, investigation of the relationship between magnetic activity and rotation for PMS stars can provide an understanding of the fundamental changes in the physics of stars that occur in the range of ages 1-100 Myr. In the next sections, we will investigate the relationship between the rotation of PMS stars in Taurus-Auriga and various magnetic activity indicators for these stars. The nature of the relationship between magnetic activity and rotation can be complex, because it depends on the stellar age, mass, internal structure, and, possibly, interaction with the disks at early evolutionary stages. To separate the various processes, we will begin by analyzing the equivalent width of the H α line (EW(H α )) as a function of the temperature or spectral type. It should be noted that most of the objects from our sample were identified in X-ray surveys and the sample may be biased toward more active objects. Therefore, when discussing activity, we prefer to use criteria based on the upper limit of activity in our sample and not on the lower limit, because the latter can be shifted greatly.",
"pages": [
2
]
},
{
"title": "CHROMOSPHERIC ACTIVITY",
"content": "The H α line is commonly used as an indicator of chromospheric activity resulting from photoionization and collisions in a hot chromosphere. In Fig. 1, EW(H α ) is plotted against the effective temperature represented by the spectral type in our case. The black and white symbols denote the objects with a reliable and unreliable evolutionary status, respectively (for details, see Grankin 2013b). The objects classified as weak-lined T Tauri stars (WTTS) with ages < 10 Myr and as post T Tauri stars (PTTS) with ages > 10 Myr are marked by the circles and squares, respectively. It can be seen from the figure that EW(H α ) is an obvious function of the spectral type. Whereas the H α line in G-type stars is in absorption (EW(H α ) ∼ 2 ˚ A), H α in K1-K4 stars exhibits a gradual transition to an emission state. For stars later than K5, H α is always in emission. The scatter of EW(H α ) increases and EW(H α ) for M-type stars lie within the range from 0 to -7 ˚ A. The dramatic change in EW(H α ) with spectral type reflects not only the change in chromospheric activity but also the additional effects related to a reduction in the continuum level with decreasing stellar luminosity and to a change in photospheric absorption in H α , which is zero for M dwarfs and increases toward earlier spectral types. The combined effects of a reduction in the continuum level and photospheric absorption in H α were estimated from spectroscopic observations of standard nonactive stars by Scholz et al. (2007). The linear approximation of EW(H α ) as a function of the spectral type is indicated in Fig. 1 by the dash-dotted line. This line coincides with the dependence of EW(H α ) on spectral type for stars in the Hyades and for field stars of various spectral types (for details, see Scholz et al. 2007). Thus, this line is an estimate for the purely photospheric contribution to EW(H α ). It can be seen from the figure that the dash-dotted line is the lower envelope for the stars of our sample: almost all of the PMS stars in Taurus-Auriga lie above this line, except for seven objects that exhibit no measurable chromospheric activity. Thus, most of the stars from our sample are chromospherically active objects. The maximum activity level increases rapidly for K7-M4 stars.",
"pages": [
2,
3
]
},
{
"title": "X-RAY ACTIVITY",
"content": "First of all, we investigated the possible relationship between the rotation period ( P rot ) and various X-ray activity parameters of PMS stars: the X-ray luminosity ( L X ), the X-ray surface flux ( F X ), and the X-ray luminosity excess defined as the X-ray to bolometric luminosity ratio ( L X /L bol ). To calculate these X-ray activity parameters, we used data from our two previous papers (Grankin 2013a, 2013b). Figure 2 shows the corresponding graphs on a logarithmic scale. It can be seen from the figure that a very weak correlation is traceable between P rot and L X (see Fig. 2a). We failed to find any correlation between the rotation period and L X /L bol (see Fig. 2b). The most significant correlation is observed between P rot and F X (see Fig. 2c). Nevertheless, it would be unreasonable to assert that there exists an unambiguous correlation between these activity parameters of PMS stars and their rotation period because of the small correlation coefficients. We attempted to investigate the possible correlation between the X-ray activity of PMS stars and their angular momentum, which is a no less informative characteristic of stellar rotation. The angular momentum is J = Iω , where I is the moment of inertia and ω is the angular velocity of the star. The angular velocity of a star is easy to calculate if its rotation period is known: ω = 2 π/P . The moment of inertia of a star can be determined from the formula I = M ( kR ) 2 , where M is the stellar mass, R is the stellar radius, and k is the radius of inertia dependent on the rotation period and shape of the star. Thus, the angular momentum of a star depends on three parameters: its mass, radius, and rotation period. To reduce the number of unknown parameters, we can introduce some normalized angular momentum: For a spherically symmetric star, k = √ 2 / 3 and the normalized angular momentum can be calculated from the formula j = 4 π 3 R 2 P . For the Sun, P = 25 days, R = 6 . 96 × 10 10 cm, and the angular momentum j /circledot = 9 . 39 × 10 15 cm 2 s -1 . We calculated the ratio j/j /circledot for all of the stars from our sample with known rotation periods. Based on these data, we plotted the normalized angular momentum( j/j /circledot ) against various X-ray activity parameters of PMS stars: L X (Fig. 3a), L X /L bol (Fig. 3b), and F X (Fig. 3c). It can be seen from Fig. 3 that there is no significant correlation between the various X-ray activity parameters and angular momentum, as in the case with the rotation period.",
"pages": [
4,
5,
6
]
},
{
"title": "ROSSBY DIAGRAM",
"content": "The chromospheric and coronal activity of stars is known to be related to their rotation and to the depth of the convective zone or the convective turnover time ( τ c ). The existence of such a relationship is in good agreement with qualitative predictions of the α Ω dynamo theory explaining the generation of a magnetic field. The Rossby diagram is one of the best tools for demonstrating the existence of a relationship between magnetic field generation and stellar activity. As a rule, the Rossby diagram displays the relationship between some stellar activity indicator and the Rossby number ( R 0 ) calculated from the formula R 0 = P rot /τ c . In particular, previous studies have shown that the slowly rotating stars in the Hyades cluster and most of the dwarf field stars exhibit a decrease in log( L X /L bol ) with increasing Rossby number. In contrast, the rapidly rotating field stars and G and K dwarfs in the Pleiades and α Persei clusters exhibit no obvious relationship between log( L X /L bol ) and the Rossby number (Hempelmann et al. 1995; Patten and Simon 1996; Randich et al. 1996; Queloz et al. 1998). Subsequently, Pizzolato et al. (2003) showed that the relationship between the X-ray luminosity and rotation period of a star could be roughly described by a power law irrespective of its mass and spectral type. Thus, stellar rotation dominates over convection for slowly rotating solartype stars. At the same time, the X-ray luminosity of rapidly rotating stars depends only on L bol and, consequently, depends on stellar-structure characteristics. Recent studies have shown that for slowly rotating stars there is a tendency for L X /L bol to grow with increasing rotation velocity up to /revsimilar 15 km s -1 , while stars with higher velocities have approximately the same L X /L bol near the saturation level ( L X /L bol /revsimilar 10 -3 ), with this saturation limit being observed for stars in a wide range of spectral types, from G to M. Thus, the most active stars exhibit a maximum X-ray luminosity at a level of L X /L bol /revsimilar 10 -3 , which does not depend on the rotation velocity. This phenomenon was called the saturated dynamo effect. In Fig. 4, log( L X /L bol ) is plotted against the Rossby number for all of the stars from our sample with known rotation periods. We estimated the convective turnover time τ c from an empirical relation given in Noyes et al. (1984): where x = 1 -( B -V ). We used the extinction-corrected color index as ( B -V ). To compare the activity of PMS stars with the activity of other solar-type stars, we showed the positions of the MS stars and the stars from the Hyades, Pleiades, IC 2391, and IC 2602 open clusters investigated by Pizzolato et al. (2003) in Fig. 4. It can be seen from Fig. 4 that the X-ray luminosity excess ( L X /L bol ) for the sample of active stars from Pizzolato et al. (2003) increases with decreasing Rossby number ( R 0 ). However, the increase in L X /L bol ceases at a level of about log( L X /L bol ) = -3, when R 0 reaches /revsimilar 0 . 28 -0 . 1 (log R 0 = -0 . 56 --0 . 98). From this time on, the so-called saturation regime is observed, where the X-ray luminosity excess reaches its maximum values and ceases to depend on R 0 . All PMS stars from our sample exhibit the same L X /L bol and R 0 as the stars from the IC 2602 and Pleiades open clusters with ages within the range 30-100 Myr. In other words, the X-ray activity of PMS stars in the Taurus-Auriga SFR closely coincides with that of the cluster stars in the regime of saturated activity. It should be noted that there are slightly more stars with small log R 0 in the range from -1.7 to -2.1 in the Pleiades. Since the stars of our sample are located in the zone of a saturated dynamo, the fact that we failed to find an unequivocal correlation between the rotation period and various X-ray activity parameters (see the previous section) becomes explainable.",
"pages": [
6,
7
]
},
{
"title": "PHOTOSPHERIC ACTIVITY",
"content": "In previous sections, we have pointed out that the magnetic activity of young solar-type stars manifests itself through the chromospheric emission in the calcium H and K lines and the H α line or through the coronal X-ray emission. In addition, the magnetic activity can also manifest itself through the maximum photometric variability amplitude in the optical spectral range (∆ V max ). Indeed, the maximum photometric variability amplitude depends primarily on the degree of nonuniformity in the distribution of spotted regions over the stellar surface and, consequently, on the total surface magnetic flux. Before investigating the possible relationship between ∆ V max and various rotation parameters, we analyzed the possible correlations of ∆ V max with such parameters as the spectral type of a star and its age. In Fig. 5a, the maximum variability amplitude is plotted against the spectral type. It can be seen from the figure that the maximum amplitude gradually increases from earlier spectral types to later ones and reaches its maximum near a spectral type K7-M1. This effect can be due to a change in the contrast of dark spots against the background of the photosphere. The results of modeling this effect are represented by the solid line. It can be clearly seen that the amplitude of the periodicity increases from relatively early spectral types to later ones. It should be noted that our sample contains five most active WTTS whose maximum amplitudes are considerably larger than those of the remaining PMS stars: LkCa 4, LkCa 7, V827 Tau, V830 Tau, and TAP 41. These stars are separated from the main group by the horizontal dashed line at a level of 0 . m 35. We excluded these objects from the subsequent statistical analysis and will discuss their properties separately. In Fig. 5b, ∆ V max is plotted against EW(Li). We found a weak correlation between ∆ V max and EW(Li) with a correlation coefficient k = 0.39. The maximum photometric variability amplitude increases with increasing lithium line equivalent width. The existence of such a positive correlation between ∆ V max and EW(Li) may reflect the fact that younger PMS stars are simultaneously also more active objects. For example, it is easy to notice that the five most active WTTS lying above the dashed line have values of EW(Li) that are among the largest ones. This result is quite intriguing, because the presence of a lithium absorption line is considered primarily as a signature of youth and not as a signature of stellar activity. In Fig. 5c, the maximum photometric variability amplitude is plotted against the age of PMS stars. The solid line is a linear approximation for all of the stars lying below the dashed line (∆ V max < 0 . m 35). A weak correlation between the maximum photometric variability amplitude and age with a correlation coefficient k = 0.45 is noticeable. The maximum amplitude decreases with increasing stellar age. The four stars exhibiting the largest variability amplitudes and lying above the horizontal dashed line have ages 2 . 5 -3 . 1 Myr. In Fig. 5d, ∆ V max is plotted against EW(H α ). The solid line is a linear approximation for all of the stars lying below the horizontal dashed line (∆ V max < 0 . m 35). It can be seen from the figure that there is a clear correlation between ∆ V max and EW(H α ) with a correlation coefficient k = 0.46. As the H α line passes from an absorption state to an emission one, the maximum photometric variability amplitude increases monotonically. This result confirms our assumption that the maximum photometric variability amplitude can be used as an indicator of photospheric activity, while the H α line is an indicator of chromospheric activity for PMS stars. In the previous section, we showed that the PMS stars in Taurus-Auriga are in the regime of a saturated dynamo, where the X-ray flux reaches saturation and ceases to depend on the rotation rate. Since the X-ray flux is related to the number of active regions on the stellar surface, we can assume that the active regions should cover almost the entire stellar surface in the regime of a saturated dynamo. In that case, we may expect maximum photometric variability amplitudes for PMS stars, of course, only when the active regions are distributed over the stellar surface highly nonuniformly. Therefore, it is interesting to investigate the relationship between ∆ V max and such stellar rotation parameters as the rotation period, the Rossby number, the equatorial rotation velocity, and the angular momentum. For this purpose, we constructed the corresponding dependences but failed to find a clear correlation between ∆ V max and the rotation parameters listed above. This result confirms our conclusion that the PMS stars are in a state of saturated dynamo. Regarding the five most active stars that exhibit the largest variability amplitudes (∆ V max > 0 . m 35), it should be noted that they have moderate rotation velocities ( v eq = 10 -30 km s -1 ) and are in the region of the transition into the zone of a saturated dynamo that corresponds to Rossby numbers in the range log( P rot /τ c ) = -0 . 56 --0 . 98.",
"pages": [
7,
8,
9
]
},
{
"title": "Li EVOLUTION DURING THE PMS STAGE",
"content": "Lithium, just as other light elements, such as beryllium and boron, is burnt out in thermonuclear reactions at relatively low temperatures in the stellar interior ((2 . 5 -3 . 0) × 10 6 K). In the case of initial evolution of low-mass ( M < 1 . 2 M /circledot ) stars, efficient mixing can rapidly transport a lithium-depleted material from the central regions of a PMS star to its photosphere. For this reason, measurements of the photospheric Li abundance provide one of the few means for probing the stellar interior and are a sensitive test of evolution models for PMS stars. Understanding the Li depletion mechanisms at the stage of PMS stars also makes it possible to estimate the ages of young stars. A large number of observational and theoretical works were devoted to understanding the initial abundance of Li and its subsequent PMS evolution (see the recent review by Jeffries 2006). According to classical models, the photospheric depletion of Li begins near 2 Myr for a star with a mass of 1 M /circledot and should end in an age of about 15 Myr. This window moves toward older ages for stars with lower masses. However, the degree of Li depletion depends very strongly on mass, convection efficiency, opacity, metallicity, and other model parameters. Thus, the amount of photospheric Li can serve as a characteristic of the youth of a PMS star. Nevertheless, numerous observations of the photospheric lithium abundance in hundreds of young stars in open clusters suggest that the degree of Li depletion in these stars is considerably smaller than what is predicted by standard models. In addition, the K-type stars in clusters with ages of less than 250 Myr are characterized by a significant dispersion in Li abundance whose cause is not yet completely understood (Jeffries 2006). This and other puzzling peculiarities that are unexpected within the framework of standard models suggest that Li depletion is governed not only by convection and that there exist other, unknown processes that have not been included in the classical theory. In recent years, several nonstandard models explaining the physics of the possible processes leading to Li depletion have been proposed. However, the mechanisms governing Li depletion still remain poorly studied. Additional observational constraints for present-day models are badly needed for further progress in understanding the evolution of Li during the PMS stage. That is why we attempted to reveal any correlations or relationships between the Li equivalent width and other physical parameters of the PMS stars from our sample. Finding such correlations can shed light on the problem of Li depletion at the PMS evolutionary phase of young stars. In the previous section, we pointed out that there exists a weak positive correlation between EW(Li) and ∆ V max for PMS stars. At the same time, we failed to find any correlation between EW(Li) and the X-ray luminosity. Below, we investigate the possible correlation between EW(Li) and such parameters of PMS stars as the theoretical age ( t ) and the rotation period ( P rot ). If the entire sample of stars is considered as a single group, then no correlation is observed between EW(Li) and t . It should be noted that our sample includes PMS stars with quite different physical parameters. For example, the masses of the PMS stars from our sample lie within the range 0.26 - 2.2 M /circledot . Since the degree of Li depletion depends very strongly on mass, we attempted to find a possible correlation between EW(Li), age, and rotation period for stars with masses fairly close to the solar mass (in the range 0.7 - 1.2 M /circledot ). In Fig. 6, EW(Li) is plotted against t (a) and P rot (b). For solar-mass stars, there exists a statistically significant correlation between EW(Li) and t with a correlation coefficient of 0.68. The older the age, the smaller the equivalent width EW(Li). The stars with ages /revsimilar 2 -3 Myr have maximum values of EW(Li) ( /revsimilar 0 . 58 ˚ A). In contrast, the stars with ages older than 30 Myr exhibit minimum values of EW(Li) (about 0.20 ˚ A). This result is in excellent agreement with the predictions of the classical models explaining the evolution of the atmospheric Li abundance during the PMS evolutionary stage of solar-mass stars. If the entire sample of stars is considered as a single group, then no correlation is observed between EW(Li) and P rot . It can also be noted that the stars with rotation periods longer than 5 days have relatively broad Li absorption lines (EW(Li) > 0.4 ˚ A). In contrast, the stars with periods shorter than 5 days have various values of EW(Li), from 0.2 to 0.7 ˚ A. If each subgroup of stars is considered separately, then the following can be identified: (1) the reliable WTTS (black circles) have EW(Li) /revsimilar 0 . 6 ˚ A for the entire range of rotation periods from 0.5 to 10 days; (2) the reliable PTTS (black squares) with ages older than 10 Myr have EW(Li) /revsimilar 0 . 4 ˚ A for their range of rotation periods from 0.5 to 5 days; (3) the PTTS with an unreliable evolutionary status rotate more rapidly ( P rot in the range from 0.5 to 2 days) and have EW(Li) /revsimilar 0 . 2 ˚ A. In other words, the same dependence of the lithium line equivalent width on age, but not on rotation period, is observed.",
"pages": [
9,
10,
11
]
},
{
"title": "PROPERTIES OF THE MOST ACTIVE PMS STARS",
"content": "Several most active stars that exhibit the record maximum photometric variability amplitudes deserve particular attention: LkCa 4 (0 m . 81), LkCa 7 (0 m . 58), V827 Tau [TAP 42] (0 m . 51), V830 Tau (0 m . 45), and V1075 Tau [TAP 41] (0 m . 39). It should be noted that there are two more objects that exhibit large variability amplitudes: V410 Tau (0 m . 63) and V836 Tau (0 m . 62). However, we do not discuss their properties here, because we failed to determine reliable luminosities, radii, masses, and ages for them. Such large amplitudes of light variations can be due to the existence of very large and extended spotted regions in the photospheres of these stars; these extended spotted regions must be distributed over the surface highly nonuniformly, otherwise the very large amplitudes of periodic light variations reaching 0 m . 4 -0 m . 8 cannot be explained. Aprevious analysis of long-term photometric observations for a sample of well-known WTTS from the Taurus-Auriga SFR showed that some of these objects exhibit stability of the phase of minimum light ( ϕ min ) over several years of observations (see, e.g., Grankin et al. (2008) and references therein). Only seven stars from the entire sample of known WTTS show stability of ϕ min in the interval from 5 to 19 years: LkCa 4, LkCa 7, V819 Tau, V827 Tau, V830 Tau, V836 Tau, and V410 Tau. Such long-term stability of ϕ min can be due to the existence of the so-called active longitudes at which short-lived groups of spots are located (Grankin et al. 1995). Similar long-lived active regions are known on the Sun and some of the RS CVn binary stars. It should be noted that almost all of these stars enter into the list of seven most active objects that exhibit the largest photometric variability amplitudes (see above). Thus, the stability of the phase of minimum light for these objects is somehow related to the existence of very large variability amplitudes. Given the unusual photometric properties of these active stars, we decided to discuss their main physical parameters in more detail. First, all of the most active stars with known parameters have very similar spectral types in the range K6-K7 and, hence, almost identical surface temperatures. Second, the rotation periods of these active stars lie within a fairly narrow range, from 2.4 to 5.7 days. Third, the radii of these stars are fairly close and lie within the range from 1.30 to 1.75 R /circledot . Fourth, analysis of their positions on the Hertzsprung-Russell diagram showed that their masses also have close values (from 0.74 to 0.92 M /circledot ), while their ages lie within the range from 2.5 to 8.2 Myr. Apart from the similarity of the main parameters of these stars noted above, it should be noted that they exhibit the broadest lithium lines with EW(Li) > 0 . 57 ˚ A. This fact confirms our assumption that the lithium line equivalent width can be used as an activity and youth criterion for stars, because there exists a significant correlation between EW(Li), the mean photometric variability amplitude, and the age of PMS stars (see Figs. 5b and 6a). We limited our sample of stars in mass and considered only those objects whose masses were close to the solar mass (within the range from 0.7 to 1.2 M /circledot ). In this case, all five most active stars exhibit the strongest H α emission lines and are among the youngest ones, with ages from 2.5 to 3 Myr, except TAP 41 whose age is estimated to be 8 Myr. Thus, if the subgroup of solar-mass PMS stars is considered, then it can be asserted that the most active and youngest stars with ages of no more than 8 Myr have the largest photometric variability amplitudes (reaching 0 m . 39 -0 m . 81), show H α emission in the range from -0.5 to -4.0 ˚ A, and exhibit the most stable phase light curves and the strongest lithium absorption line (EW(Li) > 0.57 ˚ A).",
"pages": [
11,
12
]
},
{
"title": "EVOLUTION OF THE PHASE LIGHT CURVES",
"content": "The evolution of the phase light curves for the most active PMS stars was analyzed in detail by Grankin et al. (2008). Here, we will point out the most interesting results. Despite the stability of the phase light curves in the sense of stability of the phase of minimum light, all active stars exhibit significant changes in the amplitude and shape of the phase light curve from season to season. For example, the amplitude for LkCa 7 changes from 0 m . 33 to 0 m . 58. The most symmetric (relative to ϕ min ) phase light curve, as a rule, corresponds to the season with a maximum amplitude. Conversely, the most asymmetric phase light curves correspond to minimum amplitudes of light variations. Whereas many of the stars show gradual changes in the amplitude and shape of the light curve from season to season (for example, LkCa 4 and LkCa 7), there are examples of a completely different behavior. The amplitude of periodic light variations can change noticeably by a few tenths of a magnitude during one season, as in the case of TAP 41. Although the amplitude of the light curve can change noticeably, the mean brightness level is essentially constant. Simple simulations showed that the stability of the mean brightness level from season to season suggests that the total number of spots on the surfaces of active stars changes much less than their distribution over the stellar surface. In other words, the decrease in the amplitude of the periodicity is attributable not to a decrease in the total area of the spots (in this case, the mean brightness level should increase) but to a more uniform distribution of the spots over the stellar surface. However, it should be noted that some of the stars exhibit noticeable changes in the mean brightness level from season to season (V819 Tau, V827 Tau, V836 Tau, and VY Tau). All the noted peculiarities of the evolution of the phase light curves concerned the most active and youngest stars of our sample. However, the overwhelming majority of sample stars show a slightly different photometric behavior that was not discussed in previous papers. In particular, many of the PMS stars exhibit modest variability amplitudes that do not exceed 0 m . 1 -0 m . 2. In addition, periodic light variations are observed much more rarely than in the most active PMS stars whose properties have been discussed above. The differences in photometric behavior between the most active PMS stars and the remaining sample stars are presented in Fig. 7. Figures 7a and 7b show the seasonal phase light curves, respectively, for V819 Tau, one of the most active and youngest WTTS with an age of 3.3 Myr, and for W62 (RXJ0452.5+1730), a reliable PTTS with an age of 22 Myr. It can be seen from the figure that clear periodic light variations in W62 were detected only in two of the six seasons: in 2001 and 2004. In contrast, periodic light variations in V819 Tau were observed during each observing season. The phase of minimum light remained stable over 6 years (from 1999 to 2004). To quantitatively characterize the frequency of occurrence of a periodicity, we used a simple parameter, f = N p /N s , where N p is the number of observing seasons with periodic light variations and N s is the total number of observing seasons (see Grankin 2013a). For example, the most active PMS stars exhibit periodic light variations virtually during each observing season, i.e., the frequency of occurrence of a periodicity is f = 1. The less active PMS stars show periodic light variations with a mean frequency /revsimilar 0.5, i.e., periodic light variations were detected in half of the observing seasons. Since most of the stars from our sample are not so young and active, we investigated the relationship between the frequency of occurrence of a periodicity and such parameters of PMS stars as EW(Li) (Fig. 8a) and the age (Fig. 8b). It can be seen from Fig. 8a that the frequency of occurrence of a periodicity increases with increasing EW(Li). The dependence of f on age (Fig. 8b) shows that the frequency of occurrence of a periodicity is at a maximum for the youngest stars and gradually decreases for older objects. In our previous papers, we showed that a small periodicity amplitude suggests a more uniform distribution of spots over the stellar surface, while a large amplitude is typical of the case where the spots are concentrated in one or two high-latitude regions, i.e., they are distributed highly nonuniformly (see Grankin 1999; Grankin et al. 2008). These conclusions are confirmed by the Doppler mapping of the surfaces of selected PMS stars. In particular, cool long-lived high-latitude spots were shown to have dominated on the surface of V410 Tau in the period 1992-1993, when the photometric variability amplitude was at a maximum and reached 0 m . 5 -0 m . 6 (Rice and Strassmeier 1996). In contrast, in the period 2007-2009, when the photometric variability amplitude decreased to 0 m . 06 -0 m . 10 (Grankin and Artemenko 2009), Doppler mapping showed that quite a few low-latitude spots distributed in longitude almost uniformly were present on the stellar surface (see Fig. 4 in Skelly et al. 2010). Thus, it can be assumed that the above differences in photometric behavior between the youngest PMS stars and older sample objects are attributable to different patterns of distribution of the spots over the stellar surface. Since the positions of cool magnetic spots on the surfaces of stars with convective envelopes are related to the locations of local magnetic fields, it is obvious that the pattern of distribution of the spots over the surface will depend on the magnetic field structure. Since the photometric behavior of the most active and youngest objects suggests that long-lived spots are concentrated at high latitudes, it can be assumed that the magnetic field of these stars has a simpler and fairly symmetric dipole structure. Owing to such a structure, the spots are concentrated predominantly near the magnetic poles and retain their positions during many rotation cycles. It is such a behavior that we observe in the case of the most active and youngest objects. In contrast, the relatively old stars can have a more complex magnetic field structure. Therefore, the spots are distributed over the surface more uniformly; the amplitude of the periodicity is much smaller or it is not observed at all. Such a photometric behavior is typical of the older objects from our sample. In other words, the existence of a correlation between the photometric behavior and age can be a consequence of the evolution of the magnetic field structure for PMS stars. This assumption is in good agreement with the results of a recent study of the magnetic field topology for several PMS stars performed within the MaPP (Magnetic Protostars and Planets) Program (see, e.g., Donati et al. (2010, 2011) and references therein). In particular, these studies showed that the magnetic field structure evolves from predominantly dipole and axisymmetric (in the case of fully convective stars) to octupole and axisymmetric (when the radiative core is less than half the stellar radius) and then to multipole and nonaxisymmetric (when the convective zone occupies less than half the stellar radius). The fact that the five most active stars discussed above lie on the Hertzsprung-Russell diagram in the region where fully convective PMS stars with a predominantly dipole and axisymmetric magnetic field structure are located can serve as an additional argument for such evolution of the magnetic field. Our future cooperative studies of the magnetic field topology for 40 PMS stars planned as part of the MaTYSSE (Magnetic Topologies of Young Stars & the Survival of close-in massive Exoplanets) Program performed on TBL (NARVAL spectrograph) and CFHT (ESPaDOnS spectrograph) will show whether the changes in the amplitude of the light curve are accompanied by significant changes in the distribution of spots and/or magnetic field topology.",
"pages": [
12,
13,
14
]
},
{
"title": "CONCLUSIONS",
"content": "We analyzed a sample of 70 magnetically active stars toward the Taurus-Auriga SFR and investigated the relationship between magnetic activity and rotation for these objects. In particular, we obtained the following results. We analyzed the relationship between various X-ray activity parameters and rotation for PMS stars in the Taurus-Auriga SFR. We showed that there is no significant correlation between various X-ray activity parameters ( L X , L X /L bol , and F X ) and rotation parameters, such as the period and the angular momentum. We investigated the positions of PMS stars on the Rossby diagram. All sample stars exhibit the same L X /L bol and R 0 as the stars from the Pleiades and IC 2602 clusters with ages within the range 30 -100 Myr, i.e., they are in the regime of a saturated dynamo. We analyzed the photospheric activity of PMS stars based on original long-term photometric observations. The maximum photometric variability amplitude was found, on average, to decrease with increasing age of the sample objects and to increase with increasing equivalent width of the H α emission line and the lithium absorption line. We found a statistically significant correlation between the lithium line equivalent width and the age of solar-mass (in the range from 0.7 to 1.2 M /circledot ) PMS stars. The older the age, the smaller the Li line equivalent width. This result is in excellent agreement with the predictions of the classical models explaining the evolution of the atmospheric Li abundance during the PMS stage of evolution of solar-mass stars. We identified a group of five most active PMS stars that exhibit maximum photometric variability amplitudes reaching 0 . m 4 -0 . m 8. All these stars have very similar physical parameters: spectral types (K6-K7), rotation periods (2.4-5.7 days), radii (1.3-1.75 R /circledot ), masses (0.74-0.92 M /circledot ), and ages (2.5-8.2 Myr). In addition, they show a prominent emission in H α (from -0.5 ˚ A to -4.0 ˚ A) and the strongest lithium absorption line (EW(Li) > 0.57 ˚ A). The most interesting feature of the photometric behavior of these active stars is related to the stability of the phase light curve over several observing seasons. The long-termstability of the phase light curves manifests itself in the fact that the phase of minimum light ( ϕ min ) retains its value in the interval from 5 to 19 years. Such a feature of the photometric behavior may be attributable to peculiarities of the magnetic field configuration for these stars. The most active and youngest stars from our sample most likely have mainly large-scale magnetic fields with an axisymmetric poloidal configuration. In this case, extended spotted regions are concentrated near the locations of two magnetic poles. The long existence of extended spotted regions suggests that the structure of these dipole fields is fairly stable over several years. The remaining sample stars exhibit small photometric variability amplitudes (no more than 0 m . 15), with a periodicity being observed not in each observing season. The frequency of occurrence of a periodicity was shown to be maximal for the youngest stars and to gradually decrease for older objects. It may well be that the existence of this relationship is an indirect confirmation of the evolution of the magnetic field structure for young stars from predominantly dipole and axisymmetric (in the case of fully convective stars) to octupole and axisymmetric (when the radiative core is less than half the stellar radius) and then to multipole and nonaxisymmetric (when the convective zone occupies less than half the stellar radius). The fact that the five most active stars lie on the Hertzsprung-Russell diagram in the region where fully convective PMS stars with a fairly simple dipolar magnetic field structure are located can serve as an additional argument for such evolution of the magnetic field.",
"pages": [
14,
15
]
}
]
2013CEAB...37..319T
https://arxiv.org/pdf/1211.1563.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_74><loc_82><loc_77></location>GETTING TO KNOW THE CATACLYSMIC VARIABLE BENEATH THE NOVA ERUPTION</section_header_level_1>
<text><location><page_1><loc_20><loc_69><loc_74><loc_71></location>C. Tappert 1 , L. Schmidtobreick 2 , A. Ederoclite 3 and N. Vogt 1</text>
<text><location><page_1><loc_14><loc_60><loc_80><loc_68></location>1 Dpto. de Física y Astronomía, Universidad de Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile 2 European Southern Observatory, Alonso de Cordova 3106, Santiago, Chile 3 Centro de Estudios de Física del Cosmos de Aragón, Plaza San Juan 1, Planta 2, Teruel, E44001, Spain</text>
<text><location><page_1><loc_12><loc_41><loc_83><loc_58></location>Abstract. The eruption of a (classical) nova is widely accepted to be a recurrent event in the lifetime of a cataclysmic binary star. In-between eruptions the system should therefore behave as a 'normal' cataclysmic variable (CV), i.e. according to its characteristic properties like the mass-transfer rate or the strength of the magnetic field of the white dwarf. How important are these characteristics for the nova eruption itself, i.e. which type of systems preferably undergo a nova eruption? This question could in principle be addressed by comparing the post-nova systems with the general CV population. However, information on post-novae is scarce, even to the extent that the identification of the post-nova is ambiguous in most cases. In this paper we inform on the progress of a project that has been undertaken to significantly improve the number of confirmed post-novae, thus ultimately providing the means for a better understanding of these objects.</text>
<text><location><page_1><loc_12><loc_38><loc_80><loc_39></location>Key words: binaries: close - novae, cataclysmic variables - stars: variables: general</text>
<section_header_level_1><location><page_1><loc_38><loc_33><loc_56><loc_35></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_18><loc_83><loc_31></location>A nova eruption in a cataclysmic variable (CV) occurs as a thermonuclear explosion on the surface of the white dwarf primary star once it has accumulated a critical mass from its late-type, usually main-sequence, companion. In the process material is ejected into the interstellar medium, typically amounting to 10 -5 to 10 -4 M /circledot yr -1 (e.g., Yaron et al. , 2005). Since the system is not destroyed by the eruption, this is thought to be a recurrent event, with recurrence times > 10 3 yr (see Shara et al. , 2012a).</text>
<text><location><page_1><loc_12><loc_10><loc_83><loc_18></location>In-between nova eruptions the binary is supposed to appear as a 'normal' CV, i.e. its behaviour is dominated by its current mass-transfer rate and the magnetic field strength of the white dwarf (Vogt, 1989). Furthermore, the hibernation model predicts that most of the time between erup-</text>
<section_header_level_1><location><page_2><loc_35><loc_85><loc_71><loc_86></location>AUTHORS (CONTROLED BY \ def \ aut{})</section_header_level_1>
<text><location><page_2><loc_17><loc_63><loc_95><loc_82></location>tions the system passes as a detached binary (Shara et al. , 1986; Prialnik and Shara, 1986). While there is still no observational evidence for this scenario, i.e. the state of actual 'hibernation' (e.g., Naylor et al. , 1992), it is already well established that old novae are part of the CV community. For example, the system DQ Her (Nova Her 1934) is known as the prototype intermediate polar, while RR Pic (Nova Pic 1925) shows the characteristics of an SW Sex CV (Schmidtobreick et al. , 2003a). Especially important in this context has been the discovery of nova shells around the dwarf novae Z Cam and AT Cnc (Shara et al. , 2007, 2012b), since it proves that (at least some) original CVs are also old novae.</text>
<text><location><page_2><loc_17><loc_44><loc_88><loc_62></location>All in all, however, our knowledge on old novae is largely incomplete. There are about 200 reported nova eruptions that occurred before 1980 (Downes et al. , 2005), but for less than half of them a spectrum of the postnova has been obtained, and about 80 objects even still lack an unambiguous identification. A first attempt to remedy this situation was undertaken by Schmidtobreick et al. (2005) who, however, concentrated exclusively novae with large eruption amplitudes. Among others, they recovered the old nova V840 Oph, an apparently carbon-rich CV (Schmidtobreick et al. , 2003b) which raises the question if the abundance pattern of novae is different from other CVs.</text>
<text><location><page_2><loc_17><loc_24><loc_88><loc_43></location>Furthermore, only for 28 pre-1980 novae the orbital period is well established. Because the period distribution diagram is one of the most important tools in the study of the evolution of compact binaries, this scarcity of respective information presents a severe obstacle for any systematic study on old novae and their place within the general CV population. This concerns questions like the importance of magnetic fields for the nova eruption (what is the fraction of magnetic CVs among novae compared to the general CV population?), the mass-transfer rate averaged over long time-scales (is there a bias against intrinsically faint systems?), and the hibernation model (do post-novae eventually become detached?).</text>
<section_header_level_1><location><page_2><loc_37><loc_20><loc_68><loc_21></location>2. The search for old novae</section_header_level_1>
<text><location><page_2><loc_17><loc_10><loc_88><loc_18></location>In order to establish a sample of properly identified post-novae we have begun observations of the nova candidates listed in the Downes et al. (2005) catalogue. We limit our research to novae that were reported to have erupted at least 30 years ago (i.e. before 1980) because in most systems the con-</text>
<figure>
<location><page_3><loc_11><loc_25><loc_83><loc_78></location>
<caption>Figure 1 : The data on the old nova V728 Sco, taken in three runs using EFOSC2 (Eckert, Hofstadt and Melnick, 1989) on the ESO-NTT, La Silla, Chile. Top: Colourcolour diagram of the 4 . 5 ' × 4 . 5 ' field centred on coordinates taken from the Downes et al. (2005) catalogue. The nova is marked by a circle. Middle: Low-resolution spectrum that confirmed the nova candidate. Bottom: Photometric light curve folded on the orbital period of 3.32 h.</caption>
</figure>
<section_header_level_1><location><page_4><loc_35><loc_85><loc_71><loc_86></location>AUTHORS (CONTROLED BY \ def \ aut{})</section_header_level_1>
<figure>
<location><page_4><loc_17><loc_66><loc_89><loc_82></location>
<caption>Figure 2 : The period distribution of the pre-1980 novae. New systems are shown in black.</caption>
</figure>
<text><location><page_4><loc_17><loc_52><loc_88><loc_57></location>tribution of the nova shell in the optical range becomes negligible after about three decades. First results of this survey have been published in Tappert et al. (2012).</text>
<text><location><page_4><loc_17><loc_33><loc_88><loc_50></location>We use UBVR photometry to select candidates for the post-nova based on their position in the colour-colour diagram. The typical components of CVs (white dwarf + red dwarf + accretion disc or stream) place these systems away from the main-sequence, the actual position depending strongly on the relation between the individual contributions. The candidates are then examined with long-slit spectroscopy for CV characteristic features, like emission lines of the hydrogen and helium series. Finally for the brightest confirmed post-novae we attempt to derive the orbital period via timeseries spectroscopy or photometric light curves.</text>
<text><location><page_4><loc_17><loc_11><loc_88><loc_31></location>In Fig. 1 we present our study on the old nova V728 Sco as an example. The system erupted in October 1862 (Tebbutt, 1878), putting it among the five oldest novae in the southern hemisphere. Our UBVR diagram (top plot) shows the nova well separated from the main-sequence. The longslit spectrum presents for a nova unusually strong emission lines of the Balmer and He I series, indicating a comparatively low mass-transfer rate. The presence of He II λ 4686 emission, on the other hand, is evidence for a hot component somewhere in the system. The width especially of the lowexcitation lines indicates a high system inclination. The latter is confirmed by our time-series photometry (bottom plot) that reveals an eclipsing system with an orbital period of 3.32 h.</text>
<section_header_level_1><location><page_5><loc_36><loc_80><loc_58><loc_81></location>3. The story so far</section_header_level_1>
<text><location><page_5><loc_12><loc_53><loc_83><loc_78></location>The initial sample of reported pre-1980 nova eruptions in the southern hemisphere (DEC < +20 ) consisted of 28 novae with known orbital period, 9 confirmed novae with unknown period, 32 targets without a post-nova spectrum, and 86 objects with no proper identification. In the course of our project we have since then confirmed 13 novae spectroscopically and identified two variable stars (potentially Miras) whose variability had been mistaken for a nova eruption (Tappert et al. , 2012, give details on the majority of these results). We have furthermore determined the orbital period for eight novae, which already represents an increase of almost 30% with respect to the previously known periods. Among those eight there are two eclipsing novae: V728 Sco, with P orb = 3 . 32 h , and V909 Sgr, with P orb = 3 . 44 h . Note that the latter object had already been reported by Diaz and Bruch (1997) to be an eclipsing nova with a possible period of 3.36 h.</text>
<text><location><page_5><loc_12><loc_34><loc_83><loc_53></location>In Fig. 2 we show the period distribution of our sample. The eight additions of our present research further emphasise the apparent clustering of the orbital periods around 3-6 h, with 2/3 of the novae having periods in this range. This is the region where the CVs with the highest mass-transfer rates are situated (Townsley and Gänsicke, 2009). This clustering is therefore not unexpected since it appears to support the simple idea that the white dwarfs in CVs with high mass-transfer rates accumulate the critical mass for a nova eruption faster, leading to shorter eruption recurrence times than for low-mass-transfer CVs. However, the current sample size of 36 novae is much too small for definite conclusions.</text>
<text><location><page_5><loc_12><loc_26><loc_83><loc_34></location>Our project still very much represents work in progress. New observations are already underway, with more planned for the future. We therefore expect to significantly improve on the nova sample in the next years, so that it can be used for in-depth statistical analyses.</text>
<section_header_level_1><location><page_5><loc_37><loc_22><loc_57><loc_23></location>Acknowledgements</section_header_level_1>
<text><location><page_5><loc_12><loc_10><loc_83><loc_20></location>This research was supported by FONDECYT Regular grant 1120338 (CT and NV). The CEFCA is funded by the Fondo de Inversiones de Teruel, supported by both the Government of Spain (50%) and the regional Government of Aragón (50%). This work has been partially funded by the spanish Ministerio de Ciencia e Innovación through the PNAYA, under grants</text>
<text><location><page_6><loc_17><loc_80><loc_62><loc_82></location>AYA2006-14056 and through the ICTS 2009-14</text>
<text><location><page_6><loc_17><loc_74><loc_91><loc_80></location>We furthermore gratefully acknowledge ample use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System Bibliographic Services.</text>
<section_header_level_1><location><page_6><loc_47><loc_70><loc_59><loc_72></location>References</section_header_level_1>
<text><location><page_6><loc_19><loc_67><loc_77><loc_68></location>Diaz, M. P. and Bruch, A.: 1997, Astron. Astrophys. 322 , 807-816.</text>
<unordered_list>
<list_item><location><page_6><loc_19><loc_63><loc_88><loc_66></location>Downes, R. A., Webbink, R. F., Shara, M. M., Ritter, H., Kolb, U., and Duerbeck, H. W.: 2005, Journal of Astronomical Data 11 , 2-+.</list_item>
</unordered_list>
<text><location><page_6><loc_19><loc_62><loc_84><loc_63></location>Eckert, W., Hofstadt, D., and Melnick, J.: 1989, The Messenger 57 , 66-68.</text>
<text><location><page_6><loc_19><loc_58><loc_88><loc_61></location>Naylor, T., Charles, P. A., Mukai, K., and Evans, A.: 1992, Mon. Not. R. Astron. Soc. 258 , 449-456.</text>
<text><location><page_6><loc_19><loc_56><loc_76><loc_58></location>Prialnik, D. and Shara, M. M.: 1986, Astrophys. J. 311 , 172-182.</text>
<text><location><page_6><loc_19><loc_53><loc_88><loc_56></location>Schmidtobreick, L., Tappert, C., and Saviane, I.: 2003a, Mon. Not. R. Astron. Soc. 342 , 145-150.</text>
<text><location><page_6><loc_19><loc_49><loc_88><loc_52></location>Schmidtobreick, L., Tappert, C., Bianchini, A., and Mennickent, R. E.: 2003b, Astron. Astrophys. 410 , 943-949.</text>
<text><location><page_6><loc_19><loc_46><loc_88><loc_49></location>Schmidtobreick, L., Tappert, C., Bianchini, A., and Mennickent, R. E.: 2005, Astron. Astrophys. 432 , 199-205.</text>
<text><location><page_6><loc_19><loc_42><loc_88><loc_45></location>Shara, M. M., Livio, M., Moffat, A. F. J., and Orio, M.: 1986, Astrophys. J. 311 , 163-171.</text>
<text><location><page_6><loc_19><loc_35><loc_88><loc_42></location>Shara, M. M., Martin, C. D., Seibert, M., Rich, R. M., Salim, S., Reitzel, D., Schiminovich, D., Deliyannis, C. P., Sarrazine, A. R., Kulkarni, S. R., Ofek, E. O., Brosch, N., Lépine, S., Zurek, D., de Marco, O., and Jacoby, G.: 2007, Nature 446 , 159-162.</text>
<text><location><page_6><loc_19><loc_32><loc_88><loc_35></location>Shara, M. M., Mizusawa, T., Zurek, D., Martin, C. D., Neill, J. D., and Seibert, M.: 2012a, Astrophys. J. 756 , 107.</text>
<text><location><page_6><loc_19><loc_28><loc_88><loc_31></location>Shara, M. M., Mizusawa, T., Wehinger, P., Zurek, D., Martin, C. D., Neill, J. D., Forster, K., and Seibert, M.: 2012b, Astrophys. J. 758 , 121.</text>
<text><location><page_6><loc_19><loc_25><loc_88><loc_28></location>Tappert, C., Ederoclite, A., Mennickent, R. E., Schmidtobreick, L., and Vogt, N.: 2012, Mon. Not. R. Astron. Soc. 423 , 2476-2485.</text>
<text><location><page_6><loc_19><loc_23><loc_66><loc_24></location>Tebbutt, J.: 1878, Mon. Not. R. Astron. Soc. 38 , 330.</text>
<text><location><page_6><loc_19><loc_21><loc_84><loc_22></location>Townsley, D. M. and Gänsicke, B. T.: 2009, Astrophys. J. 693 , 1007-1021.</text>
<text><location><page_6><loc_19><loc_18><loc_88><loc_21></location>Vogt, N.: 1989, in M. F. Bode and A. Evans (eds.), Classical Novae , Wiley, pp. 225-247.</text>
<text><location><page_6><loc_19><loc_14><loc_88><loc_17></location>Yaron, O., Prialnik, D., Shara, M. M., and Kovetz, A.: 2005, Astrophys. J. 623 , 398-410.</text>
</document>
[
{
"title": "GETTING TO KNOW THE CATACLYSMIC VARIABLE BENEATH THE NOVA ERUPTION",
"content": "C. Tappert 1 , L. Schmidtobreick 2 , A. Ederoclite 3 and N. Vogt 1 1 Dpto. de Física y Astronomía, Universidad de Valparaíso, Avda. Gran Bretaña 1111, Valparaíso, Chile 2 European Southern Observatory, Alonso de Cordova 3106, Santiago, Chile 3 Centro de Estudios de Física del Cosmos de Aragón, Plaza San Juan 1, Planta 2, Teruel, E44001, Spain Abstract. The eruption of a (classical) nova is widely accepted to be a recurrent event in the lifetime of a cataclysmic binary star. In-between eruptions the system should therefore behave as a 'normal' cataclysmic variable (CV), i.e. according to its characteristic properties like the mass-transfer rate or the strength of the magnetic field of the white dwarf. How important are these characteristics for the nova eruption itself, i.e. which type of systems preferably undergo a nova eruption? This question could in principle be addressed by comparing the post-nova systems with the general CV population. However, information on post-novae is scarce, even to the extent that the identification of the post-nova is ambiguous in most cases. In this paper we inform on the progress of a project that has been undertaken to significantly improve the number of confirmed post-novae, thus ultimately providing the means for a better understanding of these objects. Key words: binaries: close - novae, cataclysmic variables - stars: variables: general",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "A nova eruption in a cataclysmic variable (CV) occurs as a thermonuclear explosion on the surface of the white dwarf primary star once it has accumulated a critical mass from its late-type, usually main-sequence, companion. In the process material is ejected into the interstellar medium, typically amounting to 10 -5 to 10 -4 M /circledot yr -1 (e.g., Yaron et al. , 2005). Since the system is not destroyed by the eruption, this is thought to be a recurrent event, with recurrence times > 10 3 yr (see Shara et al. , 2012a). In-between nova eruptions the binary is supposed to appear as a 'normal' CV, i.e. its behaviour is dominated by its current mass-transfer rate and the magnetic field strength of the white dwarf (Vogt, 1989). Furthermore, the hibernation model predicts that most of the time between erup-",
"pages": [
1
]
},
{
"title": "AUTHORS (CONTROLED BY \\ def \\ aut{})",
"content": "tribution of the nova shell in the optical range becomes negligible after about three decades. First results of this survey have been published in Tappert et al. (2012). We use UBVR photometry to select candidates for the post-nova based on their position in the colour-colour diagram. The typical components of CVs (white dwarf + red dwarf + accretion disc or stream) place these systems away from the main-sequence, the actual position depending strongly on the relation between the individual contributions. The candidates are then examined with long-slit spectroscopy for CV characteristic features, like emission lines of the hydrogen and helium series. Finally for the brightest confirmed post-novae we attempt to derive the orbital period via timeseries spectroscopy or photometric light curves. In Fig. 1 we present our study on the old nova V728 Sco as an example. The system erupted in October 1862 (Tebbutt, 1878), putting it among the five oldest novae in the southern hemisphere. Our UBVR diagram (top plot) shows the nova well separated from the main-sequence. The longslit spectrum presents for a nova unusually strong emission lines of the Balmer and He I series, indicating a comparatively low mass-transfer rate. The presence of He II λ 4686 emission, on the other hand, is evidence for a hot component somewhere in the system. The width especially of the lowexcitation lines indicates a high system inclination. The latter is confirmed by our time-series photometry (bottom plot) that reveals an eclipsing system with an orbital period of 3.32 h.",
"pages": [
4
]
},
{
"title": "2. The search for old novae",
"content": "In order to establish a sample of properly identified post-novae we have begun observations of the nova candidates listed in the Downes et al. (2005) catalogue. We limit our research to novae that were reported to have erupted at least 30 years ago (i.e. before 1980) because in most systems the con-",
"pages": [
2
]
},
{
"title": "3. The story so far",
"content": "The initial sample of reported pre-1980 nova eruptions in the southern hemisphere (DEC < +20 ) consisted of 28 novae with known orbital period, 9 confirmed novae with unknown period, 32 targets without a post-nova spectrum, and 86 objects with no proper identification. In the course of our project we have since then confirmed 13 novae spectroscopically and identified two variable stars (potentially Miras) whose variability had been mistaken for a nova eruption (Tappert et al. , 2012, give details on the majority of these results). We have furthermore determined the orbital period for eight novae, which already represents an increase of almost 30% with respect to the previously known periods. Among those eight there are two eclipsing novae: V728 Sco, with P orb = 3 . 32 h , and V909 Sgr, with P orb = 3 . 44 h . Note that the latter object had already been reported by Diaz and Bruch (1997) to be an eclipsing nova with a possible period of 3.36 h. In Fig. 2 we show the period distribution of our sample. The eight additions of our present research further emphasise the apparent clustering of the orbital periods around 3-6 h, with 2/3 of the novae having periods in this range. This is the region where the CVs with the highest mass-transfer rates are situated (Townsley and Gänsicke, 2009). This clustering is therefore not unexpected since it appears to support the simple idea that the white dwarfs in CVs with high mass-transfer rates accumulate the critical mass for a nova eruption faster, leading to shorter eruption recurrence times than for low-mass-transfer CVs. However, the current sample size of 36 novae is much too small for definite conclusions. Our project still very much represents work in progress. New observations are already underway, with more planned for the future. We therefore expect to significantly improve on the nova sample in the next years, so that it can be used for in-depth statistical analyses.",
"pages": [
5
]
},
{
"title": "Acknowledgements",
"content": "This research was supported by FONDECYT Regular grant 1120338 (CT and NV). The CEFCA is funded by the Fondo de Inversiones de Teruel, supported by both the Government of Spain (50%) and the regional Government of Aragón (50%). This work has been partially funded by the spanish Ministerio de Ciencia e Innovación through the PNAYA, under grants AYA2006-14056 and through the ICTS 2009-14 We furthermore gratefully acknowledge ample use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System Bibliographic Services.",
"pages": [
5,
6
]
},
{
"title": "References",
"content": "Diaz, M. P. and Bruch, A.: 1997, Astron. Astrophys. 322 , 807-816. Eckert, W., Hofstadt, D., and Melnick, J.: 1989, The Messenger 57 , 66-68. Naylor, T., Charles, P. A., Mukai, K., and Evans, A.: 1992, Mon. Not. R. Astron. Soc. 258 , 449-456. Prialnik, D. and Shara, M. M.: 1986, Astrophys. J. 311 , 172-182. Schmidtobreick, L., Tappert, C., and Saviane, I.: 2003a, Mon. Not. R. Astron. Soc. 342 , 145-150. Schmidtobreick, L., Tappert, C., Bianchini, A., and Mennickent, R. E.: 2003b, Astron. Astrophys. 410 , 943-949. Schmidtobreick, L., Tappert, C., Bianchini, A., and Mennickent, R. E.: 2005, Astron. Astrophys. 432 , 199-205. Shara, M. M., Livio, M., Moffat, A. F. J., and Orio, M.: 1986, Astrophys. J. 311 , 163-171. Shara, M. M., Martin, C. D., Seibert, M., Rich, R. M., Salim, S., Reitzel, D., Schiminovich, D., Deliyannis, C. P., Sarrazine, A. R., Kulkarni, S. R., Ofek, E. O., Brosch, N., Lépine, S., Zurek, D., de Marco, O., and Jacoby, G.: 2007, Nature 446 , 159-162. Shara, M. M., Mizusawa, T., Zurek, D., Martin, C. D., Neill, J. D., and Seibert, M.: 2012a, Astrophys. J. 756 , 107. Shara, M. M., Mizusawa, T., Wehinger, P., Zurek, D., Martin, C. D., Neill, J. D., Forster, K., and Seibert, M.: 2012b, Astrophys. J. 758 , 121. Tappert, C., Ederoclite, A., Mennickent, R. E., Schmidtobreick, L., and Vogt, N.: 2012, Mon. Not. R. Astron. Soc. 423 , 2476-2485. Tebbutt, J.: 1878, Mon. Not. R. Astron. Soc. 38 , 330. Townsley, D. M. and Gänsicke, B. T.: 2009, Astrophys. J. 693 , 1007-1021. Vogt, N.: 1989, in M. F. Bode and A. Evans (eds.), Classical Novae , Wiley, pp. 225-247. Yaron, O., Prialnik, D., Shara, M. M., and Kovetz, A.: 2005, Astrophys. J. 623 , 398-410.",
"pages": [
6
]
}
]
2013CEAB...37..417G
https://arxiv.org/pdf/1401.1739.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_74><loc_78><loc_77></location>VARIATIONS OF SOLAR NON-AXISYMMETRIC ACTIVITY</section_header_level_1>
<text><location><page_1><loc_29><loc_69><loc_65><loc_71></location>Gyenge, N., T. Baranyi and A. Ludmány</text>
<text><location><page_1><loc_13><loc_64><loc_82><loc_68></location>Heliophysical Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, Debrecen, P.O.Box 30, H-4010, H-4010 Debrecen, P.O.Box 30. Hungary</text>
<section_header_level_1><location><page_1><loc_12><loc_57><loc_20><loc_58></location>Abstract.</section_header_level_1>
<text><location><page_1><loc_12><loc_38><loc_83><loc_56></location>The temporal behaviour of solar active longitudes has been examined by using two sunspot catalogues, the Greenwich Photoheliographic Results (GPR) and the Debrecen Photoheliographic Data (DPD). The time-longitude diagrams of the activity distribution reveal the preferred longitudinal zones and their migration with respect to the Carrington frame. The migration paths outline a set of patterns in which the activity zone has alternating prograde/retrograde angular velocities with respect to the Carrington rotation rate. The time profiles of these variations can be described by a set of successive parabolae. Two similar migration paths have been selected from these datasets, one northern path during cycles 21 - 22 and one southern path during cycles 13 - 14, for closer examination and comparison of their dynamical behaviours. The rates of sunspot emergence exhibited in both migration paths similar periodicities, close to 1.3 years. This behaviour may imply that the active longitude is connected to the bottom of convection zone.</text>
<text><location><page_1><loc_12><loc_35><loc_44><loc_37></location>Key words: sunspots, active longitudes</text>
<section_header_level_1><location><page_1><loc_38><loc_31><loc_56><loc_33></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_10><loc_83><loc_29></location>The probability of magnetic field emergence is not uniform at all solar longitudes. Numerous works have been devoted to identify and follow the most active longitudinal belts but the different datasets, time intervals, approaches and preassumptions resulted in a broad variety of spatial patterns and temporal behaviours. In the history of this research field two larger groups can be separated. One of them disregards the spatially resolved active region data and focuses on the temporal variation of the activity by assuming that the active zones belong to a frame having different rotation rate from that of the Carrington frame. This approach is restricted by the preassumption that this different rotation rate is constant (Bai, 1987;</text>
<section_header_level_1><location><page_2><loc_43><loc_85><loc_63><loc_86></location>ACTIVE LONGITUDES</section_header_level_1>
<text><location><page_2><loc_17><loc_72><loc_88><loc_82></location>Balthasar, 2007; Bogart, 1982; Jetsu et al., 1997; Olemskoy and Kitchatinov, 2007). The other group uses the position data of active regions but also with restricting preassumptions allowing e.g. the impact of differential rotation which necessarily implies a cyclic behaviour (Berdyugina and Usoskin, 2003; Usoskin et al., 2005; Zhang et al., 2011).</text>
<text><location><page_2><loc_17><loc_63><loc_88><loc_72></location>Our work tries to combine these two approaches. At first we want to identify the active belts and its motion with respect to the Carrington frame without assuming that this motion is invariable or cycle dependent. In the next phase the study of the temporal variation can focus on the moving active longitudinal zone.</text>
<text><location><page_2><loc_17><loc_34><loc_88><loc_62></location>Our previous paper (Gyenge et al., 2012, hereinafter Paper I) presented a possible method to localise and follow the longitudinal zone of enhanced activity. The diagrams of that paper show a characteristic migration pattern in the time-longitude diagram. In contrast to the well recognizable Spörer pattern in the time-latitude diagram, the longitudinal migration of enhanced activity does not show connection with the cycle profiles. A parabola has been fitted to the migration path and along this curve the width of the active zone was about 30 degrees, the flip-flop phenomenon was well identifiable. The migration path comprised the decreasing phase of cycle 21, the entire cycle 22 and the beginning of cycle 23 so it is highly improbable that the phenomenon of active longitude is connected with the solar cycle or the differential rotation. That work was restricted to the time interval of the Debrecen Photoheliographic Data (DPD): 1979-2011. The aim of the recent work is to extend the study to earlier cycles and to scrutinize closer the dynamics of flux emergence within the activity belt.</text>
<section_header_level_1><location><page_2><loc_23><loc_30><loc_83><loc_31></location>2. Time-Longitude Analysis of Sunspot Distributions</section_header_level_1>
<text><location><page_2><loc_17><loc_17><loc_88><loc_28></location>The work is based on the two detailed sunspot catalogue, the DPD (Győri et al., 2011) and the GPR (Greenwich Photoheliographic Results, Royal Observatory Greenwich). The procedure was similar to that of Paper I. The 360 · longitudinal circumference of the Sun was divided into 10 · bins and the normalized weight of activity has been computed in each bin and each Carrington rotation by the formula:</text>
<formula><location><page_2><loc_44><loc_10><loc_88><loc_14></location>W i = A i ∑ 36 j =1 A j (1)</formula>
<figure>
<location><page_3><loc_22><loc_57><loc_72><loc_81></location>
<caption>Figure 1 : Upper panel: migration of the active longitudinal zone with respect to the Carrington frame in the northern hemisphere between Carrington rotations 1740 - 1935 (between years 1984 - 1996). On the vertical axis the solar circumference is plotted three times. The lowest part of the panel contains the simultaneous cycle profiles plotted by using the International Sunspot Number (SIDC-team). Lower panel: longitudinal distribution of activity as measured from the parabola.</caption>
</figure>
<figure>
<location><page_3><loc_28><loc_40><loc_66><loc_54></location>
</figure>
<text><location><page_3><loc_46><loc_39><loc_51><loc_40></location>Longitude</text>
<text><location><page_3><loc_12><loc_10><loc_83><loc_23></location>The W i quantity represents the fraction of the total activity emerging at a certain longitudinal bin so if its highest values are tracked through the rotations it may reveal the migration of the most active longitudinal zone with respect to the Carrington frame. The first step in the identification of the path was a search by visual pattern recognition in the time - longitude diagram, this was a forward and backward shift of the active zone between the Carrington rotations 1740 - 1935 (between years 1984 - 1996) in the north-</text>
<section_header_level_1><location><page_4><loc_43><loc_85><loc_63><loc_86></location>ACTIVE LONGITUDES</section_header_level_1>
<text><location><page_4><loc_17><loc_63><loc_88><loc_82></location>ern hemisphere. This subjective choice has been checked with more objective procedures: fitting of a parabola on the points of the highest activity, the determination of the width of active zone along the parabola path, the detection of a flip-flop phenomenon. In the present work this procedure has been repeated with a somewhat different approach: in each Carrington rotation all longitudinal bins were disregarded in which the fraction of entire activity in the given rotation was smaller than 0.28 which is twice as high as the mean standard deviation of the averaged W i in the rotations. The remaining bins show up the migration path of the active zone, the parabola fitted on them is practically the same as in Paper I:</text>
<formula><location><page_4><loc_37><loc_58><loc_88><loc_59></location>l = -0 . 081( r -1837) 2 +720 (2)</formula>
<text><location><page_4><loc_21><loc_54><loc_84><loc_56></location>Where l is the longitude, and r is the Carrington rotation number.</text>
<text><location><page_4><loc_17><loc_33><loc_88><loc_54></location>Figure 1 shows the path of the active zone, the selected points of the time - longitude diagram and the fitted parabola. The solar longitudinal circumference is plotted three times on the vertical axis in order to follow the migration. It is remarkable that the forward motion in the Carrington frame starts in the decreasing phase of cycle 21, it returns at the time of maximum of cycle 22 and it ends during the rising phase of cycle 23. The lower panel shows the longitudinal distribution of the activity in a moving reference frame in which the position of 60 · mark moves along the parabola of the migration path. The distributions in each Carrington rotation were averaged for the total length of the path. The activity distribution has a smaller secondary maximum at the opposite longitude of the main maximum.</text>
<text><location><page_4><loc_17><loc_18><loc_88><loc_33></location>The temporal behaviour of the active zone was investigated by using autocorrelation analysis. Figure 2 shows the temporal variation and the autocorrelogram of monthly total sunspot area in the longitudinal zone of ± 15 · width on both sides of the parabola curve. The highest peak of the curve is at the rotation 18 which corresponds nearly to 1.3 years. To check whether this period belongs really to the active zone the autocorrelation of the entire activity has also been computed, see the lower panel of Figure 1, it does not contain this peak or some other signatures of any periodicity.</text>
<text><location><page_4><loc_17><loc_10><loc_88><loc_18></location>The above presented investigations have been carried out by using the data of DPD. In order to extend the investigations the time-longitude diagrams of the GPR-period have been studied and another similar migration path has been selected in the southern hemisphere at the time of cycle 14</text>
<figure>
<location><page_5><loc_25><loc_47><loc_69><loc_81></location>
<caption>Figure 2 : Upper panel: temporal variation of monthly total sunspot area in the migrating active zone. Middle panel: the autocorrelogram of this data series. Lower panel: the autocorrelogram of the entire activity in the same time interval.</caption>
</figure>
<text><location><page_5><loc_12><loc_28><loc_83><loc_36></location>between Carrington rotations 420 - 620 (years 1885 - 1900). The procedure of curve fitting was the same as in the case of the path presented above. The obtained curve and the longitudinal activity distribution around it are plotted in Figure 3.</text>
<text><location><page_5><loc_12><loc_15><loc_83><loc_28></location>There are remarkable similarities between these diagrams and those of Figure 1, primarily the parabola shape of the migration, the return of the migration at the time of cycle 14 maximum, and the main and secondary maxima at opposite positions in the longitudinal distribution of the activity obtained along the path. The most important difference is that the lengths of the forward-backward migrations are not the same, at around the rotation No. 600 a new migration path starts in forward direction, not followed here.</text>
<text><location><page_5><loc_12><loc_10><loc_83><loc_14></location>The most intriguing property of the northern migration path between 1984-1996 is that the activity within its narrow belt exhibits a variation</text>
<section_header_level_1><location><page_6><loc_43><loc_85><loc_63><loc_86></location>ACTIVE LONGITUDES</section_header_level_1>
<figure>
<location><page_6><loc_28><loc_57><loc_78><loc_81></location>
<caption>Figure 3 : The same diagrams as in Figure 1 for the migration path between Carrington rotations 420 - 620 (years 1885 - 1900)</caption>
</figure>
<figure>
<location><page_6><loc_34><loc_40><loc_71><loc_54></location>
</figure>
<text><location><page_6><loc_52><loc_39><loc_57><loc_40></location>Longitude</text>
<text><location><page_6><loc_17><loc_11><loc_88><loc_26></location>with a period of about 1.3 years and this period cannot be detected in the entire activity of the same time interval. The same variation has also been studied in the migration path coinciding with cycles 13-14, the results are shown in Figure 4. In the autocorrelogram of Figure 4 the significant peak also exists at rotation 18 (1.33 years) but two smaller peaks are also present at rotations 22 and 24 (1.62 and 1.77 years). The check of the curve, the autocorrelogram of the entire activity is similar to that of the studied interval at cycles 21 - 23, it has no significant peaks at all.</text>
<figure>
<location><page_7><loc_25><loc_59><loc_69><loc_81></location>
<caption>Figure 4 : The same diagrams as in the two lower panels of Figure 2 for the path shown in Figure 3.</caption>
</figure>
<section_header_level_1><location><page_7><loc_39><loc_47><loc_55><loc_49></location>3. Discussion</section_header_level_1>
<text><location><page_7><loc_12><loc_15><loc_83><loc_45></location>The obtained results of the above case studies may have connections to several other findings. The shapes of the migration paths of the active longitudinal zones are fairly similar to those variations which have been found by Juckett (2006) with a quite different method. The variations of the flux emergences within the active zones exhibit a period of 1.3 years, the active zone during cycle 14 has also two further periods of smaller peaks. These periods are absent in the entire activity. This may imply that the magnetic fluxes of these active zones emerge from the bottom of the convective zone where the 1.3 year radial torsional oscillation has been detected (Howe et al, 2000). This interpretation is supported by the theoretical considerations of Bigazzi and Ruzmaikin (2004) that the active longitude can only be pertinent at the bottom of the convection zone because at higher layers the differential rotation would disarrange it. If the conjectured connection to the depth of the tachocline zone really exists it could support the "shallow layer" model of the active longitudes proposed by Dikpati and Gilman (2005).</text>
<text><location><page_7><loc_12><loc_10><loc_83><loc_14></location>The present work studied a northern and a southern migration path at an interval of about a century from each other and several similarities have</text>
<section_header_level_1><location><page_8><loc_43><loc_85><loc_63><loc_86></location>ACTIVE LONGITUDES</section_header_level_1>
<text><location><page_8><loc_17><loc_76><loc_88><loc_82></location>been found between them. A large statistical study is in preparation for the detailed search for further identifiable migrating active longitudes in the entire time interval covered by the DPD and GPR.</text>
<section_header_level_1><location><page_8><loc_42><loc_72><loc_63><loc_73></location>Acknowledgements</section_header_level_1>
<text><location><page_8><loc_17><loc_65><loc_88><loc_70></location>The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2012-2015) under grant agreement No. 284461.</text>
<section_header_level_1><location><page_8><loc_47><loc_61><loc_59><loc_62></location>References</section_header_level_1>
<text><location><page_8><loc_19><loc_57><loc_56><loc_59></location>Bai, T., 1987, Astrophys. J. , 314, 795-807.</text>
<text><location><page_8><loc_19><loc_55><loc_65><loc_57></location>Balthasar, H., 2007, Astron. Astrophys. 471, 281-287.</text>
<text><location><page_8><loc_19><loc_53><loc_86><loc_55></location>Berdyugina, S. V. & Usoskin, I. G., 2003, Astron. Astrophys. 405, 1121-1128.</text>
<text><location><page_8><loc_19><loc_51><loc_75><loc_53></location>Bigazzi, A. & Ruzmaikin, A., 2004, Astrophys. J. , 604, 944-959.</text>
<text><location><page_8><loc_19><loc_49><loc_57><loc_51></location>Bogart, R. S., 1982, Sol. Phys. , 76, 155-165.</text>
<text><location><page_8><loc_19><loc_47><loc_74><loc_49></location>Dikpati, M. & Gilman, P., 2005, Astrophys. J. , 635, L193-L196.</text>
<text><location><page_8><loc_19><loc_45><loc_88><loc_46></location>Gyenge, N., Baranyi, T. & Ludmány, A., 2012, CEAB , 36, No.1., 9-16. Paper I.</text>
<text><location><page_8><loc_19><loc_43><loc_72><loc_44></location>Győri, L., Baranyi, T., & Ludmány, A. 2011, IAUS 273, 403.</text>
<text><location><page_8><loc_23><loc_41><loc_69><loc_43></location>see: http://fenyi.solarobs.unideb.hu/DPD/index.html</text>
<text><location><page_8><loc_19><loc_38><loc_88><loc_41></location>Howe, R., Christensen-Dalsgaard, J., Hill, F., et al., 2000, Science , 287, 24562460.</text>
<text><location><page_8><loc_19><loc_34><loc_88><loc_37></location>Jetsu, L., Pohjolainen, S., Pelt, J., & Tuominen, I., 1997, Astron. Astrophys. , 318, 293-307.</text>
<text><location><page_8><loc_19><loc_32><loc_57><loc_33></location>Juckett, D. A., 2006, Sol. Phys. , 245, 37-53.</text>
<text><location><page_8><loc_19><loc_30><loc_84><loc_31></location>Olemskoy, S. V. & Kitchatinov, L. L., 2007, Geomagn.Aeron. , 49, 866-870.</text>
<text><location><page_8><loc_19><loc_26><loc_88><loc_29></location>Royal Observatory Greenwich, Greenwich Photoheliographic Results, 1874-1976, in 103 volumes, see: http://solarscience.msfc.nasa.gov/greenwch.shtml</text>
<text><location><page_8><loc_19><loc_21><loc_88><loc_25></location>SIDC-team, World Data Center for the Sunspot Index, Royal Observatory of Belgium, Monthly Report on the International Sunspot Number, online catalogue of the sunspot index: http://www.sidc.be/sunspot-data/</text>
<text><location><page_8><loc_19><loc_17><loc_88><loc_20></location>Usoskin, I. G., Berdyugina, S. V. & Poutanen, J., 2005, Astron. Astrophys. , 441, 347-352.</text>
<text><location><page_8><loc_19><loc_13><loc_88><loc_16></location>Zhang, L.Y., Mursula, K., Usoskin, I. G. & Wang, H.N., 2011, Astron. Astrophys. , 529, 23.</text>
</document>
[
{
"title": "VARIATIONS OF SOLAR NON-AXISYMMETRIC ACTIVITY",
"content": "Gyenge, N., T. Baranyi and A. Ludmány Heliophysical Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, Debrecen, P.O.Box 30, H-4010, H-4010 Debrecen, P.O.Box 30. Hungary",
"pages": [
1
]
},
{
"title": "Abstract.",
"content": "The temporal behaviour of solar active longitudes has been examined by using two sunspot catalogues, the Greenwich Photoheliographic Results (GPR) and the Debrecen Photoheliographic Data (DPD). The time-longitude diagrams of the activity distribution reveal the preferred longitudinal zones and their migration with respect to the Carrington frame. The migration paths outline a set of patterns in which the activity zone has alternating prograde/retrograde angular velocities with respect to the Carrington rotation rate. The time profiles of these variations can be described by a set of successive parabolae. Two similar migration paths have been selected from these datasets, one northern path during cycles 21 - 22 and one southern path during cycles 13 - 14, for closer examination and comparison of their dynamical behaviours. The rates of sunspot emergence exhibited in both migration paths similar periodicities, close to 1.3 years. This behaviour may imply that the active longitude is connected to the bottom of convection zone. Key words: sunspots, active longitudes",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The probability of magnetic field emergence is not uniform at all solar longitudes. Numerous works have been devoted to identify and follow the most active longitudinal belts but the different datasets, time intervals, approaches and preassumptions resulted in a broad variety of spatial patterns and temporal behaviours. In the history of this research field two larger groups can be separated. One of them disregards the spatially resolved active region data and focuses on the temporal variation of the activity by assuming that the active zones belong to a frame having different rotation rate from that of the Carrington frame. This approach is restricted by the preassumption that this different rotation rate is constant (Bai, 1987;",
"pages": [
1
]
},
{
"title": "ACTIVE LONGITUDES",
"content": "been found between them. A large statistical study is in preparation for the detailed search for further identifiable migrating active longitudes in the entire time interval covered by the DPD and GPR.",
"pages": [
8
]
},
{
"title": "2. Time-Longitude Analysis of Sunspot Distributions",
"content": "The work is based on the two detailed sunspot catalogue, the DPD (Győri et al., 2011) and the GPR (Greenwich Photoheliographic Results, Royal Observatory Greenwich). The procedure was similar to that of Paper I. The 360 · longitudinal circumference of the Sun was divided into 10 · bins and the normalized weight of activity has been computed in each bin and each Carrington rotation by the formula: Longitude The W i quantity represents the fraction of the total activity emerging at a certain longitudinal bin so if its highest values are tracked through the rotations it may reveal the migration of the most active longitudinal zone with respect to the Carrington frame. The first step in the identification of the path was a search by visual pattern recognition in the time - longitude diagram, this was a forward and backward shift of the active zone between the Carrington rotations 1740 - 1935 (between years 1984 - 1996) in the north-",
"pages": [
2,
3
]
},
{
"title": "3. Discussion",
"content": "The obtained results of the above case studies may have connections to several other findings. The shapes of the migration paths of the active longitudinal zones are fairly similar to those variations which have been found by Juckett (2006) with a quite different method. The variations of the flux emergences within the active zones exhibit a period of 1.3 years, the active zone during cycle 14 has also two further periods of smaller peaks. These periods are absent in the entire activity. This may imply that the magnetic fluxes of these active zones emerge from the bottom of the convective zone where the 1.3 year radial torsional oscillation has been detected (Howe et al, 2000). This interpretation is supported by the theoretical considerations of Bigazzi and Ruzmaikin (2004) that the active longitude can only be pertinent at the bottom of the convection zone because at higher layers the differential rotation would disarrange it. If the conjectured connection to the depth of the tachocline zone really exists it could support the \"shallow layer\" model of the active longitudes proposed by Dikpati and Gilman (2005). The present work studied a northern and a southern migration path at an interval of about a century from each other and several similarities have",
"pages": [
7
]
},
{
"title": "Acknowledgements",
"content": "The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2012-2015) under grant agreement No. 284461.",
"pages": [
8
]
},
{
"title": "References",
"content": "Bai, T., 1987, Astrophys. J. , 314, 795-807. Balthasar, H., 2007, Astron. Astrophys. 471, 281-287. Berdyugina, S. V. & Usoskin, I. G., 2003, Astron. Astrophys. 405, 1121-1128. Bigazzi, A. & Ruzmaikin, A., 2004, Astrophys. J. , 604, 944-959. Bogart, R. S., 1982, Sol. Phys. , 76, 155-165. Dikpati, M. & Gilman, P., 2005, Astrophys. J. , 635, L193-L196. Gyenge, N., Baranyi, T. & Ludmány, A., 2012, CEAB , 36, No.1., 9-16. Paper I. Győri, L., Baranyi, T., & Ludmány, A. 2011, IAUS 273, 403. see: http://fenyi.solarobs.unideb.hu/DPD/index.html Howe, R., Christensen-Dalsgaard, J., Hill, F., et al., 2000, Science , 287, 24562460. Jetsu, L., Pohjolainen, S., Pelt, J., & Tuominen, I., 1997, Astron. Astrophys. , 318, 293-307. Juckett, D. A., 2006, Sol. Phys. , 245, 37-53. Olemskoy, S. V. & Kitchatinov, L. L., 2007, Geomagn.Aeron. , 49, 866-870. Royal Observatory Greenwich, Greenwich Photoheliographic Results, 1874-1976, in 103 volumes, see: http://solarscience.msfc.nasa.gov/greenwch.shtml SIDC-team, World Data Center for the Sunspot Index, Royal Observatory of Belgium, Monthly Report on the International Sunspot Number, online catalogue of the sunspot index: http://www.sidc.be/sunspot-data/ Usoskin, I. G., Berdyugina, S. V. & Poutanen, J., 2005, Astron. Astrophys. , 441, 347-352. Zhang, L.Y., Mursula, K., Usoskin, I. G. & Wang, H.N., 2011, Astron. Astrophys. , 529, 23.",
"pages": [
8
]
}
]
2013CKA....10..189A
https://arxiv.org/pdf/1310.1967.pdf
<document>
<section_header_level_1><location><page_1><loc_19><loc_93><loc_71><loc_95></location>COMPARATIVE ANALYSIS OF NUMERICAL METHODS FOR PARAMETER DETERMINATION</section_header_level_1>
<text><location><page_1><loc_29><loc_89><loc_61><loc_91></location>Ivan L. Andronov, Maria G. Tkachenko</text>
<text><location><page_1><loc_22><loc_84><loc_67><loc_88></location>Department 'High and Applied Mathematics', Odessa National Maritime University, Odessa, Ukraine tt_ari @ ukr.net, masha.vodn @ yandex.ua</text>
<text><location><page_1><loc_24><loc_80><loc_76><loc_82></location>To be published in: Częstochowski Kalendarz Astronomiczny 2014</text>
<text><location><page_1><loc_14><loc_62><loc_76><loc_78></location>ABSTRACT. We made a comparative analysis of numerical methods for multidimensional optimization. The main parameter is a number of computations of the test function to reach necessary accuracy, as it is computationally "slow". For complex functions, analytic differentiation by many parameters can cause problems associated with a significant complication of the program and thus slowing its operation. For comparison, we used the methods: "brute force" (or minimization on a regular grid), Monte Carlo, steepest descent, conjugate gradients, Brent's method (golden section search), parabolic interpolation etc. The Monte-Carlo method was applied to the eclipsing binary system AM Leo.</text>
<text><location><page_1><loc_14><loc_33><loc_76><loc_60></location>Determination of statistically optimal parameters fitting the observations is a common task in science, particularly, in astronomy. In modeling eclipsing binary stars, there is an important problem of dependence of the optimal model parameters on each other, which may locally be described by high correlation of the deviations of these parameters near the point of optimal solution. And, extremely, by a presence of regions in multidimensional space, which correspond to nearly equal quality of the approximating light curves. There are some well-known programs based on the method of Wilson & Devinney (1971), which allow modeling of eclipsing binary stars, such as e.g. the program by Wilson (1993), Binary Maker (Bradstreet and Steelman, 2002), PHOEBE and EBAI (Prsa et al., 2012) and others. Zoła et al. (1997, 2010) presented a program for the parameter determination using the Monte-Carlo method only. Although the program works very effectively, it needs a lot of computational time, which theoretically could be decreased using other methods for faster convergence to a minimum of the test function describing "distance" between the observations and the theoretical curve.</text>
<text><location><page_1><loc_14><loc_30><loc_76><loc_32></location>And so we try to find the best method for the determination of the parameters of eclipsing binary stars.</text>
<text><location><page_2><loc_14><loc_89><loc_82><loc_94></location>For this purpose, we have used observations of one eclipsing binary system, which was analyzed by Zola et al. (2010). This star is AM Leonis, which was observed using 3 filters (B, V, R).</text>
<text><location><page_2><loc_14><loc_81><loc_76><loc_89></location>For the analysis, we used the computer code written by Professor Stanisław Zoła (Zoła et al. 1997, 2010). In the program, the method MonteCarlo is implemented. As a result: the parameters were determined and the corresponding light curves (assuming statistically optimal values of the parameters) are shown in Fig. 1.</text>
<figure>
<location><page_2><loc_14><loc_68><loc_76><loc_79></location>
<caption>Fig. 1. A test star: AM Leo (Observations presented by S.Zoła et al. (2010)), Phase light curve of AM Leo: observations (left) and best theoretical approximation (right).</caption>
</figure>
<figure>
<location><page_2><loc_15><loc_32><loc_77><loc_62></location>
<caption>Fig. 2.Parameter-parameter diagrams: best 1500 points after hundred thousands of evaluations.</caption>
</figure>
<text><location><page_3><loc_14><loc_74><loc_76><loc_92></location>Most important pairs are: inclination versus potential of the primary or secondary star; potentials of both stars. With different colors are shown results for different numbers of computations (from which only the best 1500 points are shown). One may see that initially the points are distributed nearly homogeneously. With an increasing number of evaluations, the points are being concentrated to smaller and smaller regions. And, finally, the 'cloud' should converge to a single point. Practically this process is very slow. This is why we try to find more effective algorithms. At the 'potential - potential' diagram, we see that the best solution corresponds to an 'over-contact' system, which makes an addition link 1= 2 and corresponding decrease of the number of unknown parameters.</text>
<text><location><page_3><loc_14><loc_61><loc_76><loc_72></location>Such a method needs a lot of computation time. We had made fitting using a hundred thousands sets of model parameters. The best 1500 (user defined) points are stored in the file and one may plot the 'parameter parameter diagrams'. Of course, the number of parameters is large, so one may choose many pairs of parameters. However, some parameters are suggested to be fixed, and thus a smaller number of parameters is to be determined.</text>
<table>
<location><page_3><loc_14><loc_29><loc_76><loc_58></location>
<caption>All the parameters are listed in the following table.</caption>
</table>
<text><location><page_4><loc_14><loc_84><loc_76><loc_94></location>Looking for the 'parameter-parameter' diagrams, we see that there are strong correlations between the parameters. E.g. the temperature in our computations is fixed for one star. If not, the temperature difference is only slightly dependent on temperature, thus both temperatures may not be determined accurately from modeling. So the best solution may not be unique; it may fill some sub-space in the space of parameters.</text>
<text><location><page_4><loc_14><loc_79><loc_76><loc_84></location>This is a common problem: the parameter estimates are dependent. Our tests were made on another function, which is similar in behavior to a test function used for modeling of eclipsing binaries.</text>
<text><location><page_4><loc_20><loc_78><loc_62><loc_79></location>We have used the following test function ( x 1 =x, x 2 =y )</text>
<formula><location><page_4><loc_18><loc_75><loc_75><loc_77></location>Z ( x 1 ,x 2) = ( x 1 2 -x 2) 2 + . ( x 2 -1.0201) 2 . (1)</formula>
<text><location><page_4><loc_14><loc_65><loc_76><loc_75></location>It is shown at Fig. 3. This function has an exact symmetric solution for a minimum for >0: x 1=±1.01, x 2=1.0201. However, if =0, there is an infinite number of solutions, located at the line x 1 2 = x 2. Thus this simple function fits the criteria of complexity - two local minima instead of one global; the 'ravine' is present, which links the minima; the minima are 'shallow' for small <<1 and completely vanishing for =0.</text>
<figure>
<location><page_4><loc_14><loc_49><loc_75><loc_65></location>
<caption>Fig. 3. 2D test function Z ( x 1 ,x 2) with 2 symmetric minima: for =0.01 (left) and =1 (right).</caption>
</figure>
<formula><location><page_4><loc_14><loc_38><loc_52><loc_44></location>The tangent surface is usually defined as Z . . +0.5</formula>
<formula><location><page_4><loc_20><loc_38><loc_75><loc_41></location>( x 1 ,x 2) = Z ( x 01 ,x 02)+ Z 1 ' ( x 1x 01) + Z2' ( x 2x 02)+ (2) . ( Z 11 ' . ( x 1x 01) 2 +2 . Z 12 ' . ( x 1x 01) . ( x 2x 02)+ Z 22 ' . ( x 2x 02) 2 )</formula>
<text><location><page_4><loc_14><loc_29><loc_76><loc_37></location>(e.g. Korn and Korn, 1968), where the derivatives Zj' ≡∂ Z/ ∂ xi , Zij' ≡∂ 2 Z/ (∂ xi ∂ xj ), i , j =1..2, are defined at the point of contact ( x 01 ,x 02). The shape of such surface is defined by a determinant of the Hessian matrix (combined from partial derivatives of second order). In a case of two variables, it may be defined as D = Z 11 . Z 22 -Z 12 2 .</text>
<text><location><page_5><loc_14><loc_91><loc_71><loc_94></location>For D > 0, D = 0 and D < 0, the surface is an elliptic paraboloid, parabolic cylinder and an elliptic hyperboloid, respectively.</text>
<text><location><page_5><loc_14><loc_84><loc_76><loc_91></location>Using Eq.(2), one may estimate a position of a stationary point (where all first derivatives vanish: Zj' =0). For D >0, this stationary point corresponds to an extremum (maximum for Z 11<0, else minimum), and this is a good estimate for a next point to iterate using the method of conjugate gradients.</text>
<text><location><page_5><loc_14><loc_78><loc_76><loc_84></location>For D <0, the point of convergence is a 'saddle' point, so it is not correct to use conjugate gradients at points with D <0. Instead, we suggest to use 'steepest descent' and then return to 'conjugate gradients', when next iteration point will correspond to D >0.</text>
<text><location><page_5><loc_14><loc_60><loc_76><loc_77></location>As expected theoretically, the fastest method should be 'Conjugate gradients'. However, it needs computation not only of the function, but also of derivatives of the second and first orders. At figure 5, one may see changes of the function Z ( x 1 ,x 2) with the parameter . At the right part, one may see 3 regions. Only in the red part, the method of the 'Conjugate gradients' is working good. In this case, the graph of the function may be locally represented as an elliptic paraboloid. In the blue zone, the approximating surface is a hyperbolic paraboloid. In this case, the destination point is not a minimum, but to a singular point of the type 'saddle'. So we recommended to use the method of 'Steepest Descent' in blue and green regions and to continue to use 'Conjugate Gradients' in the red zone.</text>
<text><location><page_5><loc_14><loc_53><loc_76><loc_59></location>The series of images of Fig. 5 shows subsequent sets of best hundred of computations. As in binary stars, we see a sequential shrinkage of points towards parabola marked as a green line. Best points change positions with larger and larger number of computations.</text>
<text><location><page_5><loc_14><loc_43><loc_76><loc_53></location>We have used the search for minimal value using the Monte-Carlo method. Let n be a number of computations Z [ n ]= Z ( x 1, x 2) made for randomly generated sets of parameters ( x 1, x 2), and ni are numbers of successive 'best solutions', i.e. Z [ ni ]< Z [ ni -1]<…< Z [ n 1]; ni > ni -1>…> n 1. These numbers ni are very rarely distributed among all possible numbers 1.. n . Also one may compute a distance between successive 'best points' i= ni -ni -1.</text>
<text><location><page_5><loc_14><loc_30><loc_76><loc_43></location>At this plot, we see the dependence of the accuracy on the number of trial computations. One may see that this dependence is nearly linear in a double logarithmic scale. After nearly 10 9 computations, the accuracy was close to ≈10 -10 . We also computed similar dependence for other number of parameters m and we see that a slope at a double logarithmic scale is nearly equal to = 2/ m . The upper line shows a dependence of a number of computations i between successes on a total number of computations n . The statistical dependence is linear for all values of the number of parameters m .</text>
<figure>
<location><page_6><loc_16><loc_33><loc_73><loc_94></location>
<caption>Fig. 4. Dependence of the test function Z ( x 1 ,x 2) for -2 ≤ x 1≤2 , -2≤ x 2≤2 on the parameter (0.01 (up), 0.5 (middle), 1 (bottom)). Left: levels of the function; right: zones of D >0, D 11>0 (red), D <0, D 11<0 (blue), D <0, D 11>0 (green).</caption>
</figure>
<figure>
<location><page_7><loc_14><loc_71><loc_75><loc_92></location>
<caption>Fig. 5. Diagrams for best (100 - blue and 10 - red) pairs of points (x1, x2) after different number of computations (100, 600, 1600, 30000, respectively). Green cross shows the best point. Green parabola corresponds to the 'ravine' x2= x1 2 .</caption>
</figure>
<figure>
<location><page_7><loc_14><loc_35><loc_74><loc_58></location>
<caption>Fig. 6. Dependence of the number of trials n to get deviation of signal from exact minimum (down) and dependence number between successful computations on a total number of computations. The graph is in a double logarithmic scale.</caption>
</figure>
<section_header_level_1><location><page_8><loc_20><loc_92><loc_49><loc_94></location>We have tried also other methods:</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_17><loc_91><loc_34><loc_92></location>· Steepest descent</list_item>
<list_item><location><page_8><loc_17><loc_89><loc_36><loc_91></location>· Conjugate gradients</list_item>
<list_item><location><page_8><loc_17><loc_88><loc_49><loc_89></location>· 'Brute Force' (checking on the grid)</list_item>
<list_item><location><page_8><loc_17><loc_86><loc_51><loc_87></location>· Minimization on subsequent directions</list_item>
<list_item><location><page_8><loc_17><loc_84><loc_67><loc_86></location>· Numerical estimates of derivatives of first and second order</list_item>
</unordered_list>
<section_header_level_1><location><page_8><loc_20><loc_82><loc_39><loc_83></location>For 1D minimization:</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_17><loc_80><loc_66><loc_82></location>· 'Brute Force' (checking on the grid with decreasing step)</list_item>
<list_item><location><page_8><loc_17><loc_79><loc_35><loc_80></location>· 'Golden Section'</list_item>
<list_item><location><page_8><loc_17><loc_77><loc_31><loc_78></location>· 'Dichotomy'</list_item>
</unordered_list>
<text><location><page_8><loc_17><loc_75><loc_63><loc_77></location>The corresponding programs had been written and tested.</text>
<section_header_level_1><location><page_8><loc_20><loc_73><loc_32><loc_74></location>Future plans:</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_17><loc_70><loc_71><loc_73></location>1. To apply elaborated programs to determine parameters of spotted eclipsing binary stars.</list_item>
<list_item><location><page_8><loc_17><loc_67><loc_74><loc_69></location>2. To improve existing programs to a case of spots, which migrate from season to season.</list_item>
</unordered_list>
<text><location><page_8><loc_14><loc_57><loc_76><loc_65></location>Acknowledgements. We thank to Dr. Bogdan Wszołek and the Institute of Physics of the Jan Długosz University for hospitality and Prof. Stanisław Zoła for allowing to use the program of modeling and sample data on AM Leo. This work is a part of the projects 'Inter-Longitude Astronomy' (Andronov et al., 2010) and 'Ukrainian Virtual Observatory' (Vavilova et al., 2012).</text>
<section_header_level_1><location><page_8><loc_20><loc_53><loc_30><loc_55></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_17><loc_51><loc_65><loc_52></location>1. Andronov I.L. et al., 2010, Odessa Astron. Publ., 23, 8</list_item>
<list_item><location><page_8><loc_17><loc_49><loc_65><loc_51></location>2. Bradstreet D.H., Steelman D.P., 2002AAS...201.7502B</list_item>
<list_item><location><page_8><loc_17><loc_46><loc_76><loc_49></location>3. Korn G.A., Korn Th.M.: Mathematical Handbook for Scientists and Engineers. - McGraw-Hill Book Company, N.Y. et al. (1968)</list_item>
<list_item><location><page_8><loc_17><loc_43><loc_76><loc_46></location>4. Prša A., Guinan E.F., Devinney E.J., Degroote P., Bloemen S., Matijevič G., 2012IAUS..282..271P</list_item>
<list_item><location><page_8><loc_17><loc_40><loc_76><loc_43></location>5. Wilson R.E.: 1993, Documentation of Eclipsing Binary Computer Model, University of Florida</list_item>
<list_item><location><page_8><loc_17><loc_36><loc_76><loc_39></location>6. Vavilova I. B. et al., 2012, Kinematics and Physics of Celestial Bodies, 28, 85.</list_item>
<list_item><location><page_8><loc_17><loc_35><loc_61><loc_36></location>7. Wilson R.E., Devinney, E.J.: 1971, ApJ, 166, 605</list_item>
<list_item><location><page_8><loc_17><loc_33><loc_68><loc_34></location>8. Zola S., Kolonko M., Szczech M.: 1997, A& A, 324, 1010</list_item>
<list_item><location><page_8><loc_17><loc_28><loc_76><loc_33></location>9. Zola S., Gazeas K., Kreiner J.M., Ogloza W., Siwak M., KozielWierzbowska D., Winiarski M.: 2010, Mon. Not. R. Astron. Soc., 408, 464</list_item>
</unordered_list>
</document>
[
{
"title": "COMPARATIVE ANALYSIS OF NUMERICAL METHODS FOR PARAMETER DETERMINATION",
"content": "Ivan L. Andronov, Maria G. Tkachenko Department 'High and Applied Mathematics', Odessa National Maritime University, Odessa, Ukraine tt_ari @ ukr.net, masha.vodn @ yandex.ua To be published in: Częstochowski Kalendarz Astronomiczny 2014 ABSTRACT. We made a comparative analysis of numerical methods for multidimensional optimization. The main parameter is a number of computations of the test function to reach necessary accuracy, as it is computationally \"slow\". For complex functions, analytic differentiation by many parameters can cause problems associated with a significant complication of the program and thus slowing its operation. For comparison, we used the methods: \"brute force\" (or minimization on a regular grid), Monte Carlo, steepest descent, conjugate gradients, Brent's method (golden section search), parabolic interpolation etc. The Monte-Carlo method was applied to the eclipsing binary system AM Leo. Determination of statistically optimal parameters fitting the observations is a common task in science, particularly, in astronomy. In modeling eclipsing binary stars, there is an important problem of dependence of the optimal model parameters on each other, which may locally be described by high correlation of the deviations of these parameters near the point of optimal solution. And, extremely, by a presence of regions in multidimensional space, which correspond to nearly equal quality of the approximating light curves. There are some well-known programs based on the method of Wilson & Devinney (1971), which allow modeling of eclipsing binary stars, such as e.g. the program by Wilson (1993), Binary Maker (Bradstreet and Steelman, 2002), PHOEBE and EBAI (Prsa et al., 2012) and others. Zoła et al. (1997, 2010) presented a program for the parameter determination using the Monte-Carlo method only. Although the program works very effectively, it needs a lot of computational time, which theoretically could be decreased using other methods for faster convergence to a minimum of the test function describing \"distance\" between the observations and the theoretical curve. And so we try to find the best method for the determination of the parameters of eclipsing binary stars. For this purpose, we have used observations of one eclipsing binary system, which was analyzed by Zola et al. (2010). This star is AM Leonis, which was observed using 3 filters (B, V, R). For the analysis, we used the computer code written by Professor Stanisław Zoła (Zoła et al. 1997, 2010). In the program, the method MonteCarlo is implemented. As a result: the parameters were determined and the corresponding light curves (assuming statistically optimal values of the parameters) are shown in Fig. 1. Most important pairs are: inclination versus potential of the primary or secondary star; potentials of both stars. With different colors are shown results for different numbers of computations (from which only the best 1500 points are shown). One may see that initially the points are distributed nearly homogeneously. With an increasing number of evaluations, the points are being concentrated to smaller and smaller regions. And, finally, the 'cloud' should converge to a single point. Practically this process is very slow. This is why we try to find more effective algorithms. At the 'potential - potential' diagram, we see that the best solution corresponds to an 'over-contact' system, which makes an addition link 1= 2 and corresponding decrease of the number of unknown parameters. Such a method needs a lot of computation time. We had made fitting using a hundred thousands sets of model parameters. The best 1500 (user defined) points are stored in the file and one may plot the 'parameter parameter diagrams'. Of course, the number of parameters is large, so one may choose many pairs of parameters. However, some parameters are suggested to be fixed, and thus a smaller number of parameters is to be determined. Looking for the 'parameter-parameter' diagrams, we see that there are strong correlations between the parameters. E.g. the temperature in our computations is fixed for one star. If not, the temperature difference is only slightly dependent on temperature, thus both temperatures may not be determined accurately from modeling. So the best solution may not be unique; it may fill some sub-space in the space of parameters. This is a common problem: the parameter estimates are dependent. Our tests were made on another function, which is similar in behavior to a test function used for modeling of eclipsing binaries. We have used the following test function ( x 1 =x, x 2 =y ) It is shown at Fig. 3. This function has an exact symmetric solution for a minimum for >0: x 1=±1.01, x 2=1.0201. However, if =0, there is an infinite number of solutions, located at the line x 1 2 = x 2. Thus this simple function fits the criteria of complexity - two local minima instead of one global; the 'ravine' is present, which links the minima; the minima are 'shallow' for small <<1 and completely vanishing for =0. (e.g. Korn and Korn, 1968), where the derivatives Zj' ≡∂ Z/ ∂ xi , Zij' ≡∂ 2 Z/ (∂ xi ∂ xj ), i , j =1..2, are defined at the point of contact ( x 01 ,x 02). The shape of such surface is defined by a determinant of the Hessian matrix (combined from partial derivatives of second order). In a case of two variables, it may be defined as D = Z 11 . Z 22 -Z 12 2 . For D > 0, D = 0 and D < 0, the surface is an elliptic paraboloid, parabolic cylinder and an elliptic hyperboloid, respectively. Using Eq.(2), one may estimate a position of a stationary point (where all first derivatives vanish: Zj' =0). For D >0, this stationary point corresponds to an extremum (maximum for Z 11<0, else minimum), and this is a good estimate for a next point to iterate using the method of conjugate gradients. For D <0, the point of convergence is a 'saddle' point, so it is not correct to use conjugate gradients at points with D <0. Instead, we suggest to use 'steepest descent' and then return to 'conjugate gradients', when next iteration point will correspond to D >0. As expected theoretically, the fastest method should be 'Conjugate gradients'. However, it needs computation not only of the function, but also of derivatives of the second and first orders. At figure 5, one may see changes of the function Z ( x 1 ,x 2) with the parameter . At the right part, one may see 3 regions. Only in the red part, the method of the 'Conjugate gradients' is working good. In this case, the graph of the function may be locally represented as an elliptic paraboloid. In the blue zone, the approximating surface is a hyperbolic paraboloid. In this case, the destination point is not a minimum, but to a singular point of the type 'saddle'. So we recommended to use the method of 'Steepest Descent' in blue and green regions and to continue to use 'Conjugate Gradients' in the red zone. The series of images of Fig. 5 shows subsequent sets of best hundred of computations. As in binary stars, we see a sequential shrinkage of points towards parabola marked as a green line. Best points change positions with larger and larger number of computations. We have used the search for minimal value using the Monte-Carlo method. Let n be a number of computations Z [ n ]= Z ( x 1, x 2) made for randomly generated sets of parameters ( x 1, x 2), and ni are numbers of successive 'best solutions', i.e. Z [ ni ]< Z [ ni -1]<…< Z [ n 1]; ni > ni -1>…> n 1. These numbers ni are very rarely distributed among all possible numbers 1.. n . Also one may compute a distance between successive 'best points' i= ni -ni -1. At this plot, we see the dependence of the accuracy on the number of trial computations. One may see that this dependence is nearly linear in a double logarithmic scale. After nearly 10 9 computations, the accuracy was close to ≈10 -10 . We also computed similar dependence for other number of parameters m and we see that a slope at a double logarithmic scale is nearly equal to = 2/ m . The upper line shows a dependence of a number of computations i between successes on a total number of computations n . The statistical dependence is linear for all values of the number of parameters m .",
"pages": [
1,
2,
3,
4,
5
]
},
{
"title": "For 1D minimization:",
"content": "The corresponding programs had been written and tested.",
"pages": [
8
]
},
{
"title": "Future plans:",
"content": "Acknowledgements. We thank to Dr. Bogdan Wszołek and the Institute of Physics of the Jan Długosz University for hospitality and Prof. Stanisław Zoła for allowing to use the program of modeling and sample data on AM Leo. This work is a part of the projects 'Inter-Longitude Astronomy' (Andronov et al., 2010) and 'Ukrainian Virtual Observatory' (Vavilova et al., 2012).",
"pages": [
8
]
}
]
2013CQGra..30g5011H
https://arxiv.org/pdf/1210.1449.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_77><loc_68><loc_81></location>Strong lensing, plane gravitational waves and transient flashes</section_header_level_1>
<section_header_level_1><location><page_1><loc_27><loc_74><loc_42><loc_75></location>Abraham I. Harte</section_header_level_1>
<text><location><page_1><loc_27><loc_71><loc_69><loc_73></location>Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut Am M¨uhlenberg 1, D-14476 Golm, Germany</text>
<text><location><page_1><loc_27><loc_69><loc_32><loc_70></location>E-mail:</text>
<text><location><page_1><loc_32><loc_69><loc_43><loc_70></location>harte@aei.mpg.de</text>
<text><location><page_1><loc_27><loc_46><loc_76><loc_64></location>Abstract. Plane-symmetric gravitational waves are considered as gravitational lenses. Numbers of images, frequency shifts, mutual angles, and image distortion parameters are computed exactly in essentially all non-singular plane wave spacetimes. For a fixed observation event in a particular plane wave spacetime, the number of images is found to be the same for almost every source. This number can be any positive integer, including infinity. Wavepackets of finite width are discussed in detail as well as waves which maintain a constant amplitude for all time. Short wavepackets are found to generically produce up to two images of each source which appear (separately) only some time after the wave has passed. They are initially infinitely bright, infinitely blueshifted images of the infinitely distant past. Later, these images become dim and acquire a rapidly-increasing redshift. For sufficiently weak wavepackets, one such 'flash' almost always exists. The appearance of a second flash requires that the Ricci tensor inside the wave exceed a certain threshold. This might occur if a gravitational plane wave is sourced by, e.g., a sufficiently strong electromagnetic plane wave.</text>
<section_header_level_1><location><page_1><loc_16><loc_40><loc_28><loc_41></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_16><loc_22><loc_76><loc_39></location>The theory of gravitational lensing has by now reached a considerable degree of sophistication [1, 2, 3, 4]. Theorems have been found predicting (or bounding) the number of images in very general systems [5, 6, 7]. Shapes of stable caustics have been exhaustively classified [2, 3, 8, 9], a non-perturbative notion of the lens map has been obtained [10], and various universal behaviors of images have been found for sources lying near caustics [11, 12, 13]. These general results have been complemented by a number of detailed calculations for specific types of lenses. The majority of such calculations have been performed within the quasi-Newtonian viewpoint of gravitational lensing commonly used in astrophysics [3]. While various flavors of this formalism exist, most require that bending angles be small and that all lenses be nearly-Newtonian mass distributions.</text>
<text><location><page_1><loc_16><loc_15><loc_76><loc_22></location>This is to be contrasted with the more fundamental picture of gravitational lensing where light rays are modelled as null geodesics in a Lorentzian spacetime. Within this context, specific lensing calculations have been performed in Kerr, ReissnerNordstrom, and a handful of other geometries [1]. While curvatures and bending angles may be large in these examples, they all involve (at least conformally) stationary</text>
<text><location><page_2><loc_16><loc_85><loc_76><loc_88></location>spacetimes. It is of interest to understand if qualitatively new effects appear in dynamical cases.</text>
<text><location><page_2><loc_16><loc_67><loc_76><loc_85></location>This paper considers the bending of light by (decidedly non-stationary) planesymmetric gravitational waves. Gravitational lensing by gravitational waves has previously been considered by a number of authors, although almost all of this work has been carried out within the weak-field regime [14, 15, 16, 17, 18]. One exception is [19], where redshifts were computed in an exact solution to the vacuum Einstein equation representing a plane gravitational wave. This work considered only very specific waveforms, and was confined to a coordinate patch too small to include caustics and other effects associated with the formation of multiple images. Separately, extensive work has been devoted to non-perturbatively understanding the geodesic structure of generic plane wave spacetimes [20, 21, 22, 23, 24, 25, 26]. This is clearly a subject closely related to gravitational lensing, although few explicit relations between the two subjects appear to have been made (see, however, remarks in [1]).</text>
<text><location><page_2><loc_16><loc_47><loc_76><loc_66></location>It is the purpose of this work to provide a comprehensive and non-perturbative discussion of lensing in plane wave spacetimes. These geometries are a well-known subclass of pp -waves; plane-fronted waves with parallel rays. Many plane wave spacetimes are exact solutions to the vacuum Einstein equation. Others may be interpreted as, e.g., exact solutions in Einstein-Maxwell theory. While rather idealized from the astrophysical perspective, plane wave spacetimes admit a wide variety of interesting phenomena. Depending on the waveform, any number of astigmatic and anastigmatic caustics may exist. Examples admitting any specified number of images even an infinite number - are easily constructed. In particular, even numbers of images can exist [1]. The image count can also change in time even when a source does not cross an observer's caustic. Despite all of these properties, plane wave spacetimes are geometrically very mild. They are topologically equivalent to R 4 and admit coordinate systems which cover the entire manifold.</text>
<text><location><page_2><loc_16><loc_26><loc_76><loc_47></location>Aside from their value as models of gravitational radiation, the plane wave spacetimes considered here have also found numerous applications via the Penrose limit. This limit provides a sense in which the metric near any null geodesic in any spacetime is equivalent to the metric of an appropriate plane wave spacetime [28, 29]. It allows problems in relatively complicated spacetimes to be reduced to equivalent problems in plane wave spacetimes (which are often much simpler). This has been particularly valuable within string theory and related fields [30, 31]. Penrose limits have also been applied to ordinary quantum field theory in order to investigate causality and effective indices of refraction for photons and gravitons propagating in curved spacetimes [24, 25, 32]. More recently, Penrose limits were used to deduce the effect of caustics on Green functions associated with the propagation of classical fields in curved spacetimes [23]. Given the content of the Penrose limit, lensing in plane wave spacetimes might be related to lensing in generic spacetimes as seen by ultrarelativistic observers. We make no attempt to justify this conjecture, however.</text>
<text><location><page_2><loc_16><loc_14><loc_76><loc_26></location>This paper starts by providing a self-contained review of plane wave spacetimes in Sect. 2. Although most of this material is not new [20, 21, 22, 23, 24, 25, 26, 27], it is not widely known. Sect. 3 then derives the number of images of a point source that may be viewed in plane wave spacetimes. Under generic conditions, this is found to depend only on the waveform and a certain time parameter associated with the observation event. The number of images does not depend on any properties of the source. Once this is established, Sect. 4 computes image positions, frequency shifts, angles, and image distortion parameters in general plane wave spacetimes. Sect. 5</text>
<text><location><page_3><loc_16><loc_67><loc_76><loc_88></location>applies these results to symmetric plane waves, which have constant waveforms. These geometries produce an infinite number of images of almost every source. Their lensing properties are found to change significantly if the Ricci tensor is increased beyond a certain threshold. Lastly, Sect. 6 discusses 'sandwich waves;' wavepackets with finite width. These spacetimes generically admit images which appear at discrete times and then persist indefinitely. Such images initially provide infinitely blueshifted, infinitely bright pictures into the infinitely distant past. Very quickly, however, such images become highly redshifted and effectively fade away. One of these 'transient flashes' is produced by almost every sufficiently weak vacuum (Ricci-flat) wave. For weak waves, a second flash appears only if the Ricci tensor of the wavepacket exceeds a certain threshold. Throughout this work, the spacetime is assumed to be everywhere transparent. The language used also assumes that the geometric optics approximation [3] holds even in situations where it would be severely strained (such as when light is emitted near an observer's caustic).</text>
<section_header_level_1><location><page_3><loc_16><loc_63><loc_22><loc_65></location>Notation</section_header_level_1>
<text><location><page_3><loc_16><loc_48><loc_76><loc_62></location>This paper restricts attention only to plane wave spacetimes in four spacetime dimensions. Our sign conventions follow those of Wald [33]. The signature is -+++. Latin letters a, b, . . . (and occasionally A,B,... ) from the beginning of the alphabet are used to denote abstract indices. Greek letters µ, ν, . . . are used to denote fourdimensional coordinate indices. The Latin letters i, j, . . . are instead coordinate indices associated with the two directions transverse to the direction of wave propagation. Objects involving the latter type of index are often written in boldface with all indices suppressed. They are then manipulated using the standard notation of linear algebra [e.g., A ki B kj = ( A ᵀ B ) ij and | x | = √ x i x i ]. Overall, notation related to plane wave spacetimes closely follows the conventions of [23].</text>
<section_header_level_1><location><page_3><loc_16><loc_44><loc_47><loc_45></location>2. Geometry of plane wave spacetimes</section_header_level_1>
<text><location><page_3><loc_16><loc_34><loc_76><loc_43></location>Plane wave geometries may be interpreted as simple models for gravitational waves emitted from distant sources. Alternatively, they arise as universal limits for the geometries near null geodesics in any spacetime [28, 29]. The typical definition of a plane wave spacetime ( M,g ab ) requires that M = R 4 and that there exist global coordinates ( u, v, x ) = ( u, v, x 1 , x 2 ) : M → R 4 such that the line element takes the form</text>
<formula><location><page_3><loc_27><loc_31><loc_76><loc_33></location>g µν d x µ d x ν = -2d u d v + H ij ( u ) x i x j d u 2 +(d x 1 ) 2 +(d x 2 ) 2 . (1)</formula>
<text><location><page_3><loc_16><loc_23><loc_76><loc_31></location>H ij = ( H ) ij is any symmetric 2 × 2 matrix. Its components describe the waveforms associated with a wave's three polarization states ‡ . The u coordinate is interpreted as a phase parameter for the wave, while v affinely parametrizes its rays. The remaining two coordinates x i span spacelike wavefronts transverse to the wave's direction of propagation.</text>
<text><location><page_4><loc_16><loc_82><loc_76><loc_88></location>Note that if H = 0 in some region, the spacetime is locally flat there. In terms of a Minkowski coordinate system ( t, x 1 , x 2 , x 3 ), u and v satisfy u = ( t + x 3 ) / √ 2 and v = ( t -x 3 ) / √ 2 in such a region. We consider only nontrivial plane waves, so H cannot vanish everywhere.</text>
<text><location><page_4><loc_16><loc_67><loc_76><loc_82></location>The physical interpretation of the so-called Brinkmann metric (1) as a planesymmetric gravitational wave follows from considering the integral curves of the vector field glyph[lscript] a = ( ∂/∂v ) a . These curves form a null geodesic congruence which may be interpreted as the rays of the gravitational wave. ∇ a glyph[lscript] b = 0, so these rays have vanishing expansion, shear, and twist. There is therefore a sense in which they are everywhere parallel to one another. All rays are also orthogonal to the family of spacelike 2-surfaces generated by the two commuting spacelike vector fields X a ( i ) = ( ∂/∂x i ) a . The induced metric on each such surface is flat: The wavefronts are 2-planes. The curvature is constant on these planes in the sense that the X a ( i ) are curvature collineations:</text>
<formula><location><page_4><loc_27><loc_64><loc_76><loc_66></location>L X ( i ) R abc d = 0 . (2)</formula>
<text><location><page_4><loc_16><loc_53><loc_76><loc_64></location>Despite this, the X a ( i ) are not everywhere Killing. There do, however, exist linear combinations of X a ( i ) and glyph[lscript] a which are Killing. glyph[lscript] a itself is also Killing, which may be interpreted as a statement that plane waves do not deform along their characteristics. Plane wave spacetimes admit a minimum of five linearly independent Killing fields. Note that in flat spacetime, five (out of the total of ten) Killing fields are symmetries of all electromagnetic plane waves [27]. Killing fields of plane wave spacetimes are discussed more fully in Sect. 2.3.</text>
<text><location><page_4><loc_16><loc_50><loc_76><loc_53></location>All non-vanishing coordinate components of the Riemann tensor may be determined from</text>
<formula><location><page_4><loc_27><loc_47><loc_76><loc_49></location>R uiuj = -H ij . (3)</formula>
<text><location><page_4><loc_16><loc_46><loc_46><loc_47></location>It follows from this that the Ricci tensor is</text>
<formula><location><page_4><loc_27><loc_43><loc_76><loc_45></location>R ab = -Tr H glyph[lscript] a glyph[lscript] b , (4)</formula>
<text><location><page_4><loc_16><loc_31><loc_76><loc_43></location>where Tr denotes the ordinary (Euclidean) trace of the 2 × 2 matrix H . The Ricci scalar always vanishes in plane wave spacetimes. More generally, there are no nonzero scalars formed by local contractions of the metric, the curvature, and its derivatives: R ab R ab = R abcd R abcd = R abcd glyph[epsilon1] abfh R fh cd = . . . = 0. This is analogous to the fact that plane electromagnetic waves satisfy, e.g., F ab F ab = glyph[epsilon1] abcd F ab F cd = 0. Note, however, that electromagnetic plane waves are not the only electromagnetic fields with vanishing field scalars. Similarly, plane wave spacetimes are not the only curved geometries with vanishing curvature scalars [35].</text>
<text><location><page_4><loc_16><loc_25><loc_76><loc_31></location>It follows from (4) that plane wave spacetimes satisfying the vacuum Einstein equation (and the vacuum equations of many alternative theories of gravity [36]) are characterized by the simple algebraic constraint Tr H = 0. For vacuum waves, there exist two scalar functions h + and h × such that</text>
<formula><location><page_4><loc_27><loc_21><loc_76><loc_24></location>H = ( -h + h × h × h + ) . (5)</formula>
<text><location><page_4><loc_16><loc_15><loc_76><loc_21></location>h + and h × describe the waveforms for the two polarization states of a gravitational plane wave propagating in vacuum. A plane wave is said to be linearly polarized if h + and h × are linearly dependent (in which case one of these functions can be eliminated by a suitable rotation of the transverse coordinates x ).</text>
<text><location><page_5><loc_16><loc_76><loc_76><loc_88></location>If h + and h × have compact support, the geometry is said to be a sandwich wave. This name evinces the image of a curved region of spacetime 'sandwiched' between null hyperplanes in a geometry that is otherwise Minkowski. Physically, it corresponds to a wavepacket of finite length. Note that the planar symmetry considered here is very special in the sense that passing waves do not necessarily leave any 'tail' behind them. After interacting with a sandwich wave, all observers enter a region of spacetime which is perfectly flat. There is a sense in which test fields propagating on plane wave spacetimes also have no tails [23, 37].</text>
<text><location><page_5><loc_16><loc_71><loc_76><loc_75></location>A general (not necessarily vacuum) wave profile H may be built by adding to (5) a term proportional to the identity matrix δ . There then exists a third polarization function h ‖ such that</text>
<formula><location><page_5><loc_27><loc_67><loc_76><loc_70></location>H = ( -h + -h ‖ h × h × h + -h ‖ ) . (6)</formula>
<text><location><page_5><loc_16><loc_61><loc_76><loc_67></location>If the Ricci tensor of such a wave is associated with a stress-energy tensor via Einstein's equation, that stress-energy tensor obeys the weak energy condition if and only if h ‖ ≥ 0. Assuming this, the stress-energy tensors associated with (6) are very simple. They could be generated by, e.g., electromagnetic plane waves with the form</text>
<formula><location><page_5><loc_27><loc_57><loc_76><loc_60></location>F ab = 2 h 1 2 ‖ ∇ [ a u ∇ b ] x 1 . (7)</formula>
<text><location><page_5><loc_16><loc_55><loc_76><loc_57></location>Alternatively, (6) could be associated with the stress-energy tensor of the massless Klein-Gordon plane wave</text>
<formula><location><page_5><loc_27><loc_51><loc_76><loc_54></location>φ = ∫ u h 1 2 ‖ ( w )d w. (8)</formula>
<text><location><page_5><loc_16><loc_43><loc_76><loc_50></location>Besides the vacuum case h ‖ = 0, another interesting class of wave profiles are those that are conformally flat. These satisfy h + = h × = 0, so H ∝ δ . As gravitational lenses, all caustics of conformally-flat plane waves are associated with 'perfect' anastigmatic focusing. For more general plane waves, caustics are typically (but not necessarily) associated with astigmatic focusing.</text>
<section_header_level_1><location><page_5><loc_16><loc_40><loc_35><loc_41></location>2.1. The matrices A and B</section_header_level_1>
<text><location><page_5><loc_16><loc_32><loc_76><loc_39></location>The geometry of plane wave spacetimes has been analyzed in detail by a number of authors [20, 21, 22, 23, 24, 25, 26, 27]. One essential conclusion of this work is that nearly all interesting properties of plane wave spacetimes may be deduced from the properties of 2 × 2 matrices E = E ( u ) satisfying the differential equation</text>
<formula><location><page_5><loc_27><loc_31><loc_76><loc_32></location>E = HE . (9)</formula>
<text><location><page_5><loc_16><loc_23><loc_76><loc_30></location>This is a 'generalized oscillator equation' with -H acting like a matrix of squared frequencies. Eq. (9) arises when solving for geodesics or Jacobi fields in plane wave spacetimes. Bitensors such as Synge's function and the parallel propagator may be written explicitly in terms of its solutions. The same is also true for a plane wave's Killing vectors.</text>
<text><location><page_5><loc_16><loc_19><loc_76><loc_22></location>It is convenient to write all possible matrices E in terms of two particular solutions. Fix any § u o ∈ R and define A ( · , u o ) and B ( · , u o ) to be solutions to (9)</text>
<text><location><page_6><loc_16><loc_85><loc_76><loc_88></location>(where derivatives are applied to the first arguments of A and B ) with the initial conditions</text>
<formula><location><page_6><loc_28><loc_82><loc_76><loc_84></location>lim u s → u o A ( u s , u o ) = lim u s → u o ∂ (1) B ( u s , u o ) = δ , (10)</formula>
<formula><location><page_6><loc_27><loc_80><loc_76><loc_81></location>lim u s → u o ∂ (1) A ( u s , u o ) = lim u s → u o B ( u s , u o ) = 0 . (11)</formula>
<text><location><page_6><loc_16><loc_73><loc_76><loc_79></location>Here, the notation ∂ (1) A indicates a partial derivative with respect to the first argument of A . We assume for simplicity that H is a matrix of piecewise-continuous functions and that A and B are at least C 1 (and piecewiseC 2 ) in both of their arguments. Example expressions for A and B are discussed in Sects. 5 and 6.</text>
<text><location><page_6><loc_16><loc_70><loc_76><loc_73></location>Given any two solutions E 1 and E 2 to (9), it is easily verified that their Wronskian is conserved:</text>
<formula><location><page_6><loc_27><loc_67><loc_76><loc_69></location>E ᵀ 1 ˙ E 2 -˙ E ᵀ 1 E 2 = constant . (12)</formula>
<text><location><page_6><loc_16><loc_64><loc_76><loc_67></location>Here, ᵀ denotes a matrix transpose. Applying this formula with E 1 → A and E 2 → B shows that</text>
<formula><location><page_6><loc_27><loc_61><loc_76><loc_64></location>A ᵀ ∂ (1) B -∂ (1) A ᵀ B = δ . (13)</formula>
<text><location><page_6><loc_16><loc_58><loc_76><loc_61></location>Neither A nor B are necessarily symmetric matrices. Nevertheless, (12) and (13) may be used to show that the products</text>
<formula><location><page_6><loc_24><loc_56><loc_76><loc_58></location>A ᵀ ∂ (1) A , ∂ (1) AA -1 B ᵀ ∂ (1) B , ∂ (1) BB -1 , BA ᵀ , B -1 A (14)</formula>
<text><location><page_6><loc_16><loc_53><loc_76><loc_55></location>are symmetric wherever they exist [23]. Letting E 1 → B ( · , u o ) and E 2 → B ( · , u s ) in (12) shows that</text>
<formula><location><page_6><loc_27><loc_50><loc_76><loc_52></location>B ( u s , u o ) = -B ᵀ ( u o , u s ) (15)</formula>
<text><location><page_6><loc_16><loc_47><loc_76><loc_50></location>for all u s , u o ∈ R . This is essentially Etherington's reciprocity law [1, 3, 38, 39]. A similar calculation may be used to show that</text>
<formula><location><page_6><loc_27><loc_44><loc_76><loc_46></location>∂ (1) A ( u s , u o ) = -∂ (1) A ᵀ ( u o , u s ) (16)</formula>
<text><location><page_6><loc_16><loc_43><loc_21><loc_44></location>as well.</text>
<text><location><page_6><loc_16><loc_38><loc_76><loc_42></location>It is sometimes useful to consider partial derivatives ∂ (2) with respect to the second arguments of A and B . The resulting matrices remain solutions to (9). Comparing initial conditions shows that</text>
<formula><location><page_6><loc_20><loc_35><loc_76><loc_37></location>∂ (2) A ( u s , u o ) = -B ( u s , u o ) H ( u o ) , ∂ (2) B ( u s , u o ) = -A ( u s , u o ) . (17)</formula>
<text><location><page_6><loc_16><loc_34><loc_75><loc_35></location>A may therefore be derived from B . The opposite is also true wherever det H = 0.</text>
<text><location><page_6><loc_16><loc_20><loc_76><loc_34></location>Geometrically, B corresponds to the transverse coordinate components of a Jacobi propagator describing the evolution of deviation vectors along geodesics passing between different pairs of points [23]. B is also related to image distortion. Up to an overall time dilation factor, it translates small differences in image position on an observer's sky to spatial deviations from a fiducial source point. In the language of [11], B is proportional to the Jacobi map. The symmetric matrix ∂ (1) BB -1 plays a similar role, but translates source separations to emission (rather than observation) angles. It is proportional to an object typically referred to as the optical deformation matrix. These statements are explained more fully in Sect. 4.3.</text>
<text><location><page_6><loc_72><loc_33><loc_72><loc_35></location>glyph[negationslash]</text>
<section_header_level_1><location><page_7><loc_16><loc_86><loc_30><loc_88></location>2.2. Conjugate pairs</section_header_level_1>
<text><location><page_7><loc_16><loc_82><loc_76><loc_85></location>It is often useful when working with plane wave spacetimes to consider hypersurfaces of 'constant phase.' Recalling the interpretation of the u coordinate as a phase, let</text>
<formula><location><page_7><loc_27><loc_80><loc_76><loc_82></location>S u o := { p ∈ M : u ( p ) = u o } (18)</formula>
<text><location><page_7><loc_72><loc_77><loc_72><loc_80></location>glyph[negationslash]</text>
<text><location><page_7><loc_16><loc_77><loc_76><loc_80></location>denote such a hypersurface. Given two points p s ∈ S u s and p o ∈ S u o (with u s = u o ), define the multiplicity or 'index' of these points to be [20]</text>
<formula><location><page_7><loc_27><loc_74><loc_76><loc_76></location>I ( p s , p o ) := 2 -rank B ( u s , u o ) . (19)</formula>
<text><location><page_7><loc_38><loc_69><loc_38><loc_71></location>glyph[negationslash]</text>
<text><location><page_7><loc_16><loc_64><loc_76><loc_74></location>We also set I ( p o , p s ) = 0 whenever u o = u s . It follows from (15) that I ( p s , p o ) = I ( p o , p s ). For most pairs of points, I = 0. Such pairs are said to be 'disconjugate.' Pairs p s , p o satisfying I ( p s , p o ) = 0 are instead said to be conjugate with multiplicity I ( p s , p o ). Similarly, we call the pairs S u s , S u o 'conjugate hyperplanes' and the pairs of real numbers u s , u o 'conjugate phases' when rank B ( u s , u o ) < 2. Despite the appearance of (19), the index map I : M × M →{ 0 , 1 , 2 } describes phenomena which do not depend on any choice of coordinate system.</text>
<text><location><page_7><loc_16><loc_49><loc_76><loc_63></location>Conjugate pairs as described here are closely related to the conjugate points commonly considered in differential geometry and optics. In general, distinct points p s and p o on a given geodesic are said to be conjugate if and only if there exist nontrivial deviation vectors along that geodesic which vanish at both p o and p s . In plane wave spacetimes, this condition reduces to I ( p s , p o ) > 0. Defining the multiplicity of a pair of conjugate points to be the number of linearly independent deviation vectors which vanish at those points, that multiplicity is equal to I ( p s , p o ). The concept of conjugacy associated with I does not, however, require the specification of any particular geodesic. It is uniquely defined even for pairs of points connected by multiple geodesics or by none. Indeed, these are the only cases where I = 0.</text>
<text><location><page_7><loc_16><loc_44><loc_76><loc_48></location>All strong lensing effects associated with plane wave spacetimes are related to the existence of conjugate hyperplanes. It follows from (19) that every pair of conjugate phases u s , u o satisfies</text>
<text><location><page_7><loc_61><loc_48><loc_61><loc_50></location>glyph[negationslash]</text>
<formula><location><page_7><loc_28><loc_42><loc_76><loc_43></location>det B ( u s , u o ) = det B ( u o , u s ) = 0 . (20)</formula>
<text><location><page_7><loc_16><loc_31><loc_76><loc_41></location>Finding conjugate pairs and their multiplicities may be viewed as a matter of direct computation once H is specified. Alternatively, various Sturm-type comparison theorems can be used to make general statements regarding the existence and separations of conjugate pairs for various classes of plane wave. See, e.g., Chapt. XI of [40] for results relating to mathematical problems of this type and [20] for an application to 'tame' plane wave spacetimes. More specific examples are discussed in Sects. 5 and 6 below.</text>
<text><location><page_7><loc_16><loc_23><loc_76><loc_31></location>The qualitative structure of geodesics in plane wave spacetimes is closely related to the index I . First note that every disconjugate pair of points p s , p o is connected by exactly one geodesic. If two points are conjugate, the number of connecting geodesics is either zero or infinity. Sizes of geodesically connected regions may be summarized by</text>
<formula><location><page_7><loc_20><loc_20><loc_76><loc_23></location>dim[(all points geodesically connected to p o ) ∩ S u s ] + I ( p s , p o ) = 3 . (21)</formula>
<text><location><page_7><loc_16><loc_19><loc_53><loc_20></location>A similar relation exists for null cones when u o = u s :</text>
<text><location><page_7><loc_50><loc_18><loc_50><loc_21></location>glyph[negationslash]</text>
<text><location><page_7><loc_20><loc_16><loc_76><loc_18></location>dim[(all points connected to p o via null geodesics) ∩ S u s ] + I ( p s , p o ) = 2 . (22)</text>
<text><location><page_7><loc_16><loc_13><loc_76><loc_16></location>The latter result has been referred to as an 'index theorem' in [20]. Eq. (21) implies that geodesics emanating from p o and intersecting a hyperplane S u s that</text>
<figure>
<location><page_8><loc_33><loc_68><loc_60><loc_87></location>
<caption>Figure 1: Given a preferred point p o or a preferred u = constant hyperplane S u o , plane wave spacetimes naturally divide into a number of open four-dimensional regions N n ( u o ). These regions are separated from each other by the hyperplanes S τ n ( u o ) conjugate to S u o .</caption>
</figure>
<text><location><page_8><loc_16><loc_43><loc_76><loc_58></location>is disconjugate to S u o form a 3-dimensional region. Indeed, these geodesics fill the entire hyperplane. More interestingly, geodesics intersecting a conjugate hyperplane with multiplicity 1 fill only a 2-dimensional region on that 3-dimensional surface. Geodesics intersecting a hyperplane with multiplicity 2 form a line. Similarly, the null cone of a point reduces to a 1-dimensional curve on every hyperplane with multiplicity 1. A null cone intersecting a hyperplane with multiplicity 2 is focused to a single point on that hyperplane. These two cases correspond to astigmatic and anastigmatic focusing, respectively. Anastigmatic focusing tends to be unstable in the sense that perturbations tend to split a single multiplicity 2 phase into two closely-spaced phases each with multiplicity 1.</text>
<text><location><page_8><loc_16><loc_23><loc_76><loc_43></location>In many applications, there exists a preferred point p o , or perhaps a preferred hyperplane S u o . p o may, for example, represent the position of an observer at a particular time. Fixing this point, the set of hyperplanes conjugate to S u o divides a plane wave spacetime into a (possibly infinite) number of open regions N n ( u o ). Let N 0 ( u o ) denote the largest connected region containing S u o and excluding any portion of a hyperplane conjugate to S u o . If there exists a smallest τ 1 ( u o ) > u o conjugate to u o , the surface S τ 1 ( u o ) is clearly contained in the boundary of N 0 ( u o ). N 1 ( u o ) may then be defined as the largest connected region which includes S τ 1 ( u o ) as a boundary and contains points p s satisfying u s > τ 1 ( u o ) and I ( p s , p o ) = 0. This continues the spacetime 'above' S τ 1 ( u o ) . Similar constructions may be used to define τ n ( u o ) and N n ( u o ) for values of n other than 1. See Fig. 1. Fixing a particular nonzero integer n and real number u o , it is not necessary that τ n ( u o ) exist at all. In general, the domain of τ n is an open subset of R . This domain can be empty for some n .</text>
<text><location><page_8><loc_61><loc_19><loc_61><loc_21></location>glyph[negationslash]</text>
<text><location><page_8><loc_16><loc_14><loc_76><loc_23></location>The geodesic uniqueness result described above can now be reduced to the statement that a point p o is connected to another point p s = p o by exactly one geodesic if and only if there exists some n such that p s ∈ N n ( u o ). Two-point tensors like Synge's world function, the parallel propagator, and the van Vleck determinant may be defined unambiguously throughout ' M × ( ∪ n N n ).' This excludes from M × M only a set of measure zero.</text>
<section_header_level_1><location><page_9><loc_16><loc_86><loc_26><loc_88></location>2.3. Geodesics</section_header_level_1>
<text><location><page_9><loc_16><loc_75><loc_76><loc_85></location>Beyond the qualitative geodesic structure of plane wave spacetimes discussed above, it is not difficult to obtain explicit coordinate expressions for all geodesics. Let Γ ⊂ M denote some geodesic and γ : R → M an affine parametrization of it. The vector field glyph[lscript] a = ( ∂/∂v ) a generating the characteristics of the gravitational wave is Killing, so ˙ γ a glyph[lscript] a must be constant on Γ. If this constant vanishes, Γ is confined to a hypersurface of constant phase. Such a geodesic has the form of a (Euclidean) straight line in the coordinates ( v, x ).</text>
<text><location><page_9><loc_38><loc_73><loc_38><loc_75></location>glyph[negationslash]</text>
<text><location><page_9><loc_16><loc_66><loc_76><loc_75></location>Geodesics satisfying ˙ γ a glyph[lscript] a = 0 are more interesting. In these cases, the affine parameter can always be rescaled such that ˙ γ a glyph[lscript] a = -1. It is then possible to identify that parameter with the phase coordinate u . Doing, so u ( γ ( u s )) = u s for all u s ∈ R . The spatial components γ := x ( γ ) of any geodesic are fixed everywhere once γ o := γ ( u o ) and ˙ γ o := ˙ γ ( u o ) have been specified at some fiducial phase u o . In terms of the matrices A and B defined in Sect. 2.1,</text>
<formula><location><page_9><loc_27><loc_64><loc_76><loc_65></location>γ ( u s ) = A ( u s , u o ) γ o + B ( u s , u o ) ˙ γ o . (23)</formula>
<text><location><page_9><loc_16><loc_60><loc_76><loc_63></location>The associated v coordinate of Γ may be efficiently derived using that fact that the vector field (2 v∂ v + x i ∂ i ) a is a homothety [23]. This implies that</text>
<formula><location><page_9><loc_21><loc_57><loc_76><loc_60></location>v ( γ ( u s )) = v ( γ ( u o )) + 1 2 [ κ s ( u s -u o ) + γ ( u s ) · ˙ γ ( u s ) -γ o · ˙ γ o ] , (24)</formula>
<text><location><page_9><loc_16><loc_55><loc_20><loc_56></location>where</text>
<text><location><page_9><loc_16><loc_51><loc_25><loc_52></location>is a constant.</text>
<text><location><page_9><loc_16><loc_48><loc_76><loc_51></location>κ s is closely related to the conserved quantity on Γ associated with the Killing field glyph[lscript] a . The (unit) 4-velocity U a s tangent to Γ is related to ˙ γ a via</text>
<formula><location><page_9><loc_27><loc_44><loc_76><loc_48></location>U a s = ˙ γ a √ κ s . (26)</formula>
<text><location><page_9><loc_16><loc_43><loc_26><loc_44></location>It follows that</text>
<formula><location><page_9><loc_27><loc_52><loc_76><loc_55></location>κ s := -˙ γ a ˙ γ a (25)</formula>
<formula><location><page_9><loc_28><loc_39><loc_76><loc_42></location>-glyph[lscript] a U a s = 1 √ κ s . (27)</formula>
<text><location><page_9><loc_16><loc_30><loc_76><loc_39></location>If the spacetime is nearly flat (so H ≈ 0), κ s reduces to a particle's specific energy minus its specific momentum in the direction of the gravitational wave. The limit -glyph[lscript] a U a → 0 (or κ s → ∞ ) may therefore be interpreted as ultrarelativistic motion in the direction of the gravitational wave. By contrast, the limit -glyph[lscript] a U a → ∞ (or κ s → 0) corresponds to ultrarelativistic motion against the background wave.</text>
<text><location><page_9><loc_16><loc_28><loc_76><loc_31></location>Other Killing fields present in essentially all plane wave spacetimes may be written as</text>
<formula><location><page_9><loc_27><loc_26><loc_76><loc_28></location>( x i ˙ Ξ i ) glyph[lscript] a +Ξ i X a ( i ) , (28)</formula>
<text><location><page_9><loc_16><loc_24><loc_47><loc_25></location>where Ξ i = Ξ i is any 2-vector with the form</text>
<formula><location><page_9><loc_27><loc_21><loc_76><loc_23></location>Ξ = A ( · , u o ) Ξ ( u o ) + B ( · , u o ) ˙ Ξ ( u o ) . (29)</formula>
<text><location><page_9><loc_16><loc_17><loc_76><loc_21></location>For each choice of u o , there exists a four-parameter family of such vector fields. Each of these is associated with a conservation law. Two such conserved quantities may be summarized by</text>
<formula><location><page_9><loc_27><loc_13><loc_76><loc_16></location>P ( u o ) := 1 √ κ s [ A ᵀ ( u s , u o ) ˙ γ ( u s ) -∂ (1) A ᵀ ( u s , u o ) γ ( u s ) ] . (30)</formula>
<text><location><page_10><loc_16><loc_85><loc_76><loc_88></location>Fixing any u o , this 2-vector is conserved in the sense that it is independent of u s . Another conserved 2-vector may be defined by</text>
<formula><location><page_10><loc_27><loc_81><loc_76><loc_84></location>C ( u o ) := 1 √ κ s [ B ᵀ ( u s , u o ) ˙ γ ( u s ) -∂ (1) B ᵀ ( u s , u o ) γ ( u s ) ] . (31)</formula>
<text><location><page_10><loc_16><loc_72><loc_76><loc_81></location>In the weak-field limit, P corresponds to the specific momentum transverse to the gravitational wave. In this same context, C may be interpreted as the conserved quantity associated with boosts transverse to the gravitational wave. It constrains transverse displacements. Note that both P and C depend on a choice of u o . This is analogous to the choice of origin necessary to define angular momentum in elementary mechanics.</text>
<text><location><page_10><loc_16><loc_65><loc_76><loc_72></location>In stationary spacetimes, it is common to discuss various quantities related to gravitational lensing in terms of stationary observers (and often stationary sources). While plane wave spacetimes are not stationary, there does exist sufficient symmetry to define similarly preferred sources and observers. Two geodesics Γ and Γ ' can be said to be 'instantaneously comoving' at u = u o when</text>
<formula><location><page_10><loc_27><loc_62><loc_76><loc_64></location>κ s = κ ' s , P ( u o ) = P ' ( u o ) . (32)</formula>
<text><location><page_10><loc_16><loc_56><loc_76><loc_62></location>These conditions imply that the 4-velocities of both geodesics are parallel-transported versions of each other on the constant-phase hyperplane S u o . Note, however, that geodesics which are comoving at one phase are not necessarily comoving at any other phase.</text>
<text><location><page_10><loc_16><loc_50><loc_76><loc_56></location>As implied by (21), bundles of geodesics are strongly focused on conjugate hyperplanes. Consider such a hyperplane associated with a phase τ n ( u o ) conjugate to u o with multiplicity 1. It is clear from (19) that ˆ B n ( u o ) := B ( τ n ( u o ) , u o ) is a matrix with rank 1. There therefore exists a unit 2-vector ˆ q n ( u o ) such that</text>
<formula><location><page_10><loc_27><loc_48><loc_76><loc_50></location>ˆ q ᵀ n ( u o ) ˆ B n ( u o ) = 0 . (33)</formula>
<text><location><page_10><loc_16><loc_44><loc_76><loc_47></location>ˆ q n ( u o ) is unique up to sign. Choosing any ˆ p n ( u o ) orthogonal to ˆ q n ( u o ), the transverse spatial coordinates of all geodesics starting at a given point p o focus to the line</text>
<formula><location><page_10><loc_27><loc_43><loc_76><loc_44></location>γ ( τ n ) = ˆ A n γ o + w ˆ p n (34)</formula>
<text><location><page_10><loc_16><loc_39><loc_76><loc_42></location>as they pass through S τ n ( u 0 ) . Here, w is any real number. All v coordinates may be reached on S τ n ( u 0 ) by appropriate geodesics.</text>
<text><location><page_10><loc_16><loc_36><loc_76><loc_39></location>Geodesics starting at p o and intersecting a hyperplane S τ n ( u o ) with multiplicity 2 all focus to the single transverse position</text>
<formula><location><page_10><loc_27><loc_34><loc_76><loc_36></location>γ ( τ n ) = ˆ A n γ o (35)</formula>
<text><location><page_10><loc_16><loc_25><loc_76><loc_34></location>as they pass through S τ n ( u o ) . As in the multiplicity 1 case, all values of v may be reached by appropriate geodesics. Eqs. (34) and (35) illustrate explicitly how conjugate hyperplanes with multiplicities 1 and 2 are associated with astigmatic and anastigmatic focusing, respectively. The former case involves focusing in only one transverse direction, while the latter case involves simultaneous focusing in both directions transverse to the gravitational wave.</text>
<section_header_level_1><location><page_10><loc_16><loc_22><loc_26><loc_23></location>2.4. Distances</section_header_level_1>
<text><location><page_10><loc_16><loc_16><loc_76><loc_21></location>As noted above, all pairs of points not lying on conjugate hyperplanes are connected by exactly one geodesic. There is therefore no ambiguity in ascribing geodesic distances to these pairs. In particular, Synge's world function</text>
<formula><location><page_10><loc_27><loc_13><loc_76><loc_16></location>σ ( p s , p o ) := 1 2 (squared geodesic distance between p s and p o ) (36)</formula>
<text><location><page_11><loc_16><loc_83><loc_76><loc_88></location>is well-defined whenever its arguments do not lie on conjugate hyperplanes. Plane wave spacetimes constitute one of the few examples where σ is known essentially in closed form:</text>
<formula><location><page_11><loc_20><loc_78><loc_76><loc_83></location>σ ( p s , p o ) = 1 2 ( u s -u o ) [ -2( v s -v o ) + x ᵀ s ∂ (1) B ( u s , u o ) B -1 ( u s , u o ) x s + x ᵀ o B -1 ( u s , u o ) A ( u s , u o ) x o -2 x ᵀ o B -1 ( u s , u o ) x s ] . (37)</formula>
<text><location><page_11><loc_16><loc_76><loc_56><loc_78></location>This is symmetric in its arguments: σ ( p s , p o ) = σ ( p o , p s ).</text>
<text><location><page_11><loc_16><loc_72><loc_76><loc_76></location>The appearance of B -1 in (37) indicates that σ tends to diverge when its arguments approach conjugate hyperplanes. More specifically, suppose that u s ≈ τ n ( u o ) and I ( τ n ( u o ) , u o ) = 1. Then,</text>
<formula><location><page_11><loc_27><loc_67><loc_76><loc_71></location>σ ( p s , p o ) ≈ -1 2 ( u o -u s u s -τ n ( u o ) ) [ ˆ q n ( u o ) · ( x s -ˆ A n ( u o ) x o )] 2 (38)</formula>
<text><location><page_11><loc_16><loc_60><loc_76><loc_67></location>is an asymptotic approximation for σ if the bracketed term on the right-hand side of this equation is nonzero [23]. It follows from (34) that this expression is valid only when there does not exist any geodesic passing from p o to a point on S τ n ( u o ) with the same transverse coordinates as p s . The equivalent result if p s is near a hyperplane S τ n ( u o ) with multiplicity 2 is</text>
<formula><location><page_11><loc_27><loc_55><loc_76><loc_59></location>σ ( p s , p o ) ≈ -1 2 ( u o -u s u s -τ n ( u o ) ) ∣ ∣ x s -ˆ A n ( u o ) x o ∣ ∣ 2 . (39)</formula>
<text><location><page_11><loc_16><loc_52><loc_76><loc_55></location>Again, this is valid only when there does not exist any geodesic passing from p o to a point on S τ n ( u o ) with the same transverse coordinates as p s .</text>
<section_header_level_1><location><page_11><loc_16><loc_49><loc_31><loc_50></location>3. Image counting</section_header_level_1>
<text><location><page_11><loc_16><loc_40><loc_76><loc_48></location>One of the most basic questions that can be asked regarding a gravitational lens is the number of images that it produces of a particular source. Stated somewhat differently, how many future-directed null geodesics connect a given timelike curve (the source) to a particular spacetime event (the observer at a particular time)? This may be answered using the geodesic structure of plane wave spacetimes summarized above.</text>
<text><location><page_11><loc_16><loc_33><loc_76><loc_40></location>To fix the notation, let p o denote a fixed observation event and Γ the timelike worldline of a point source. Assume that Γ may be parametrized by an everywhereC 1 function γ : R → M . The phase coordinate u serves as a useful 'quasi-time' [21] for plane wave spacetimes ‖ , so let γ satisfy u ( γ ( u s )) = u s for all u s ∈ R . The source's worldline is not required to be geodesic.</text>
<text><location><page_11><loc_16><loc_27><loc_76><loc_33></location>This section establishes that under generic conditions, an ideal observer at p o may see exactly one image of Γ from each of the 'epochs' N n ( u o ) described in Sect. 2.2. Somewhat more precisely, there typically exists exactly one future-directed null geodesic from</text>
<formula><location><page_11><loc_27><loc_24><loc_76><loc_26></location>Γ n := Γ ∩ N n ( u o ) (40)</formula>
<text><location><page_11><loc_16><loc_20><loc_76><loc_24></location>to p o for each n ≤ 0 such that N n ( u o ) exists. See Fig. 2. Recall that the boundaries of N n ( u o ) depend only on the spacetime under consideration, and not at all on the</text>
<figure>
<location><page_12><loc_37><loc_70><loc_54><loc_87></location>
<caption>Figure 2: Schematic illustration of the gravitational lensing problem. Images correspond to future-directed null geodesics from a timelike source Γ to an observation event p o . The arguments of Sect. 3 show that under generic conditions, exactly one image is emitted from each region N n ( u o ) lying before the observation event. The dashed lines correspond to constantu hyperplanes conjugate to S u o .</caption>
</figure>
<text><location><page_12><loc_16><loc_56><loc_76><loc_58></location>behavior of any particular source. This allows generic bounds to be placed on time delays associated with the various images that may be observed.</text>
<text><location><page_12><loc_16><loc_49><loc_76><loc_55></location>Recall from Sect. 2.2 that each point in Γ n is connected to p o via exactly one (not necessarily null) geodesic. Synge's function σ ( γ ( u s ) , p o ) is therefore well-defined and explicitly given by (37) for all u s such that γ ( u s ) ∈ Γ n . Consider instead the rescaled function</text>
<formula><location><page_12><loc_27><loc_45><loc_76><loc_49></location>Σ n ( u s ) := σ ( γ ( u s ) , p o ) u o -u s . (41)</formula>
<text><location><page_12><loc_16><loc_41><loc_76><loc_46></location>If n ≤ 0, the domain of Σ n ( u s ) is equal to all u s such that γ ( u s ) ∈ Γ n . If n = 0, we additionally suppose that u s < u o for reasons of causality. Images of Γ n produced by a plane gravitational wave correspond to the zeros of Σ n .</text>
<text><location><page_12><loc_65><loc_36><loc_65><loc_38></location>glyph[negationslash]</text>
<text><location><page_12><loc_16><loc_35><loc_76><loc_41></location>Recalling the form (37) for σ , it is clear that Σ n depends on Γ as well as the matrices A , B , B -1 , and ∂ (1) B . We have assumed in Sect. 2.1 that A and B are at least C 1 . So is γ . The definition of N n ( u o ) ensures that det B ( u s , u o ) = 0 everywhere Σ n ( u s ) is defined. B -1 is therefore C 1 and Σ n is continuous.</text>
<text><location><page_12><loc_16><loc_32><loc_76><loc_35></location>Σ n is also monotonic. To see this, note that (9), (13), (14), (37), and (41) may be used to show that</text>
<formula><location><page_12><loc_27><loc_29><loc_76><loc_32></location>˙ Σ n = 1 2 [ -˙ γ a ˙ γ a + ∣ ∣ ˙ γ + B -ᵀ ( x o -∂ (1) B ᵀ γ ) ∣ ∣ 2 ] > 0 . (42)</formula>
<text><location><page_12><loc_16><loc_24><loc_76><loc_29></location>Σ n is both continuous and monotonic, so at most one zero can exist for each n . This means that at most one image of a source may reach an observer from each epoch N n ( u o ). Exactly one such image exists if some u ' , u '' are known to satisfy</text>
<formula><location><page_12><loc_27><loc_23><loc_76><loc_24></location>Σ n ( u ' ) < 0 , Σ n ( u '' ) > 0 . (43)</formula>
<text><location><page_12><loc_16><loc_21><loc_41><loc_22></location>Such bounds are easily established.</text>
<section_header_level_1><location><page_12><loc_16><loc_18><loc_47><loc_19></location>3.1. Lensing between conjugate hyperplanes</section_header_level_1>
<text><location><page_12><loc_16><loc_13><loc_76><loc_16></location>The simplest case to consider is one where N n ( u o ) lies 'in between' hyperplanes conjugate to S u o . Suppose that both τ n ( u o ) and τ n -1 ( u o ) exist for some n < 0. This</text>
<text><location><page_13><loc_16><loc_85><loc_76><loc_88></location>is true in Fig. 2 for n = -1. More generally, (38) and (39) imply that if there are no geodesics connecting p o to either γ ( τ n ( u o )) or γ ( τ n -1 ( u o )),</text>
<formula><location><page_13><loc_28><loc_82><loc_76><loc_84></location>lim u s → τ + n -1 ( u o ) Σ n ( u s ) = -∞ , lim u s → τ -n ( u o ) Σ n ( u s ) = ∞ . (44)</formula>
<text><location><page_13><loc_16><loc_76><loc_76><loc_81></location>It follows from these limits together with continuity that Σ n is surjective on R . Since this function is also monotonic, there must exist exactly one emission phase u e ∈ ( τ n -1 ( u o ) , τ n ( u o )) such that p o and γ ( u e ) ∈ Γ n are connected by a null geodesic.</text>
<text><location><page_13><loc_16><loc_68><loc_76><loc_76></location>Under the same assumptions, projectiles moving on timelike geodesics may be thrown from γ ( u s ) ∈ Γ n to p o only if u s < u e . Choosing u s -τ n -1 ( u o ) to be sufficiently small (but positive), these projectiles can require an arbitrarily large amount of proper time to intersect p o . It is somewhat curious that points γ ( u s ) ∈ Γ n satisfying u s > u e cannot be connected to p o by any causal geodesic. Such points may, however, be reached by suitably accelerated curves which are everywhere causal.</text>
<section_header_level_1><location><page_13><loc_16><loc_64><loc_33><loc_66></location>3.2. The youngest image</section_header_level_1>
<text><location><page_13><loc_16><loc_59><loc_76><loc_63></location>Next, consider the case n = 0 when there exists at least one conjugate hyperplane in the observer's past (as occurs in the example illustrated by Fig. 2). The arguments given above imply that if γ ( τ -1 ( u o )) is geodesically disconnected from p o ,</text>
<formula><location><page_13><loc_28><loc_56><loc_76><loc_58></location>lim u s → τ + -1 ( u o ) Σ 0 ( u s ) = -∞ . (45)</formula>
<text><location><page_13><loc_16><loc_54><loc_63><loc_55></location>The other boundary of the domain of Σ 0 occurs at u s = u o . Here,</text>
<formula><location><page_13><loc_28><loc_49><loc_76><loc_54></location>lim u s → u -o Σ 0 ( u s ) = 1 2 lim u s → u -o | x o -γ ( u s ) | 2 u o -u s . (46)</formula>
<text><location><page_13><loc_16><loc_44><loc_76><loc_50></location>This limit clearly tends to + ∞ if γ ( u o ) = x o . Physically, γ ( u o ) = x o implies that p o cannot be connected to γ ( u o ) by any null geodesic. Assuming that this is true, there must exist exactly one u e ∈ ( τ -1 ( u o ) , u o ) such that p o and γ ( u e ) ∈ Γ 0 are connected by a null geodesic.</text>
<text><location><page_13><loc_44><loc_47><loc_44><loc_50></location>glyph[negationslash]</text>
<text><location><page_13><loc_62><loc_47><loc_62><loc_50></location>glyph[negationslash]</text>
<section_header_level_1><location><page_13><loc_16><loc_41><loc_31><loc_42></location>3.3. The oldest image</section_header_level_1>
<text><location><page_13><loc_16><loc_31><loc_76><loc_40></location>The results of Sects. 3.1 and 3.2 typically suffice to describe the images formed in plane wave spacetimes containing an infinite number of conjugate hyperplanes in an observer's past. It is, however, important to consider cases where only a finite number of conjugate points exist. This occurs, e.g., for finite wavepackets where H has compact support. It is also true of linearly polarized vacuum waves that are tame in the sense described in [20].</text>
<text><location><page_13><loc_16><loc_21><loc_76><loc_30></location>If there is at least one conjugate hyperplane in the observer's past, let N denote the smallest negative integer such that τ N ( u o ) exists. If there are no conjugate hyperplanes in the observer's past, set N = 0. The case illustrated in Fig. 2 corresponds to N = -2 if N -2 ( u o ) extends into the infinite past. Regardless, we ask whether there exist any future-directed light rays from Γ N to p o . Unlike in the cases considered above, sources spend an infinite amount of proper time in N N ( u o ).</text>
<text><location><page_13><loc_16><loc_16><loc_76><loc_21></location>Even in flat spacetime, a source that accelerates for an infinitely long time may be causally-disconnected from certain observers via a Rindler horizon. Such phenomena can be ruled out here by supposing that there exists some finite constant κ min > 0 such that</text>
<formula><location><page_13><loc_24><loc_13><loc_76><loc_15></location>-˙ γ a ˙ γ a = 2˙ γ a ( u s ) ∇ a v -[ | ˙ γ ( u s ) | 2 + γ ᵀ ( u s ) H ( u s ) γ ( u s ) ] > κ min (47)</formula>
<text><location><page_14><loc_16><loc_82><loc_76><loc_88></location>for all u s less than some cutoff. As is clear from (25) and (26), the left-hand side of this inequality acts like the square root of a time dilation factor between the coordinate u and the source's proper time. Eq. (47) implies that the source's 4-velocity U a s ∝ ˙ γ a satisfies</text>
<formula><location><page_14><loc_27><loc_78><loc_76><loc_81></location>0 < -glyph[lscript] a U a s < 1 √ κ min (48)</formula>
<text><location><page_14><loc_16><loc_70><loc_76><loc_78></location>sufficiently far in the past. It therefore excludes sources which experience arbitrarily large boosts against the background gravitational wave in the distant past. It is satisfied by, e.g., sources whose motion is geodesic sufficiently far into the past. Assuming that κ min exists, it is clear from (42) that u ' may be chosen sufficiently small that</text>
<formula><location><page_14><loc_27><loc_68><loc_76><loc_70></location>Σ N ( u ' ) < 0 . (49)</formula>
<text><location><page_14><loc_71><loc_65><loc_71><loc_67></location>glyph[negationslash]</text>
<text><location><page_14><loc_16><loc_62><loc_76><loc_67></location>Further assuming that γ ( τ N ( u o )) is geodesically disconnected from p o (if N = 0) or that γ ( u o ) is not null-separated from p o (if N = 0), it follows that there exists exactly one u e smaller than τ N ( u o ) (if N = 0) or u o (if N = 0) such that γ ( u e ) ∈ Γ N is connected to p o via a future-directed null geodesic.</text>
<text><location><page_14><loc_41><loc_62><loc_41><loc_64></location>glyph[negationslash]</text>
<section_header_level_1><location><page_14><loc_16><loc_58><loc_46><loc_60></location>3.4. Total image count for generic sources</section_header_level_1>
<text><location><page_14><loc_45><loc_51><loc_45><loc_53></location>glyph[negationslash]</text>
<text><location><page_14><loc_16><loc_49><loc_76><loc_57></location>The results just described may be summarized as follows: Suppose that there does not exist any past-directed geodesic segment from p o to Γ whose endpoints are conjugate (in the usual sense). Also assume that the source and observer are not instantaneously aligned with the background wave: x o = γ ( u o ). If there exists an 'oldest' phase conjugate to u o , further require that there be some κ min > 0 such that the source's motion is bounded by (48) sufficiently far in the past.</text>
<text><location><page_14><loc_16><loc_36><loc_76><loc_48></location>For each n ≤ 0, these assumptions imply that an observer at p o sees exactly one image of Γ as it appeared in N n ( u o ). This provides a strong bound on the possible emission times of different images. If N N ( u o ) exists for every negative integer N , an infinite number of images are formed. If, however, there is some smallest N ≤ 0 such that N N ( u o ) exists, | N | +1 images appear at p o . Note that these results depend only on the waveform H and the phase coordinate u o associated with the observer. The total number of images is the same for all sources satisfying the hypotheses outlined above.</text>
<text><location><page_14><loc_18><loc_28><loc_18><loc_30></location>glyph[negationslash]</text>
<text><location><page_14><loc_16><loc_21><loc_76><loc_36></location>These hypotheses are generic. Recalling (21), the requirement that Γ exclude any points conjugate to p o along a connecting geodesic is equivalent to demanding that the source's worldline avoid certain well-behaved one- or two-dimensional subsets of the three-dimensional hyperplanes conjugate to S u o . Similarly, the assumption that x o = γ ( u o ) demands only that Γ avoid a certain line on S u o . The bound (48) on a source's asymptotic 4-velocity can fail to hold only for sources which accelerate for an infinitely long time. Violating any of these conditions for a fixed observer would require that a source's worldline be quite exceptional. Moreover, we now show that | N | +1 images appear under even broader (but more difficult to state) conditions than those just discussed.</text>
<section_header_level_1><location><page_14><loc_16><loc_18><loc_34><loc_19></location>3.5. Non-generic imaging</section_header_level_1>
<text><location><page_14><loc_16><loc_14><loc_76><loc_17></location>Despite the comments made above, our assumptions on the behavior of a source's worldline can be violated in certain cases. Suppose, contrary to these assumptions,</text>
<text><location><page_15><loc_16><loc_72><loc_76><loc_88></location>that there exists at least one point γ e ∈ Γ which is conjugate to p o along a geodesic connecting these two points. An infinite number of geodesics then pass between γ e and p o . If the connecting geodesics are null, γ e lies on a caustic of the observer's past light cone. Continuous images of point sources - 'Einstein rings' - are then formed at p o (ignoring the associated breakdown of geometric optics). Such cases are not considered any further here. If conjugate points between the source and observer are associated with non-null geodesics, discrete images of Γ appear at p o . In these cases, the methods used above are easily adapted to find how many images of Γ n arrive at p o . As already mentioned, there can be no more than one root for each Σ n . Depending on the details of the system, the associated image from N n ( u o ) may or may not exist.</text>
<text><location><page_15><loc_16><loc_65><loc_76><loc_72></location>The case considered in Sect. 3.1 where Γ n lies in between successive conjugate hyperplanes is the simplest to analyze. Assuming that Γ does not intersect the caustic of p o , no images are formed of Γ n if either γ ( τ n ( u o )) is connected to p o via a timelike geodesic or γ ( τ n -1 ( u o )) is connected to p o via a spacelike geodesic. Otherwise, exactly one image exists from this region.</text>
<text><location><page_15><loc_16><loc_58><loc_76><loc_65></location>If there exists an oldest conjugate hyperplane S τ N ( u o ) as described in Sect. 3.3, suppose that the source satisfies (47) for some κ min > 0. There are then zero images of Γ N if γ ( τ N ( u o )) is connected to p o via a timelike geodesic. There is exactly one image if these points are either geodesically disconnected or are connected by a spacelike geodesic.</text>
<text><location><page_15><loc_70><loc_55><loc_70><loc_57></location>glyph[negationslash]</text>
<text><location><page_15><loc_16><loc_48><loc_76><loc_57></location>The last cases to consider concern images of Γ 0 . First suppose that x o = γ ( u o ). If there exists at least one conjugate hyperplane in the observer's past, one image is formed of Γ 0 if either γ ( τ -1 ( u o )) is geodesically disconnected from p o or it is connected by a timelike geodesic. No images are formed if γ ( τ -1 ( u o )) and p o are connected by a spacelike geodesic. If x o = γ ( u o ) and there are no conjugate points in the observer's past, condition (47) implies that there exists exactly one image of Γ 0 .</text>
<text><location><page_15><loc_33><loc_49><loc_33><loc_51></location>glyph[negationslash]</text>
<text><location><page_15><loc_16><loc_42><loc_76><loc_48></location>Cases where the source and observer are instantaneously aligned are more interesting. Suppose that x o = γ ( u o ). There then exists one image of Γ 0 with u e = u o . Recalling that u o is not in the domain of Σ 0 , it is possible for a second image to be emitted from Γ 0 if Σ 0 = 0 somewhere. This may be seen by noting that</text>
<formula><location><page_15><loc_28><loc_39><loc_76><loc_42></location>lim u s → u -o Σ 0 = v ( γ ( u o )) -v o . (50)</formula>
<text><location><page_15><loc_16><loc_37><loc_68><loc_39></location>Two images of Γ 0 can therefore exist when v ( γ ( u o )) > v o and x o = γ ( u o ).</text>
<text><location><page_15><loc_16><loc_30><loc_76><loc_37></location>If a source includes points which are conjugate to the observer (in the ordinary sense), there is no simple result for the total number of images formed. Nevertheless, it is always possible to say that the total number of images is less than or equal to | N | +2 if a source does not intersect a caustic of the observer's light cone.</text>
<section_header_level_1><location><page_15><loc_16><loc_28><loc_40><loc_29></location>4. Properties of lensed images</section_header_level_1>
<text><location><page_15><loc_16><loc_21><loc_76><loc_26></location>Plane wave spacetimes typically produce multiple images of each source. Even for sources whose intrinsic properties remain constant, these images can appear with different spectra, brightnesses, etc. We now compute these properties for generic configurations satisfying the hypotheses summarized in Sect. 3.4.</text>
<text><location><page_15><loc_16><loc_16><loc_76><loc_20></location>For each image of a timelike worldline Γ seen at p o , there is an associated null geodesic segment connecting p o to an appropriate emission point γ e = γ ( u e ) ∈ Γ. In terms of Synge's function (36), these points satisfy</text>
<formula><location><page_15><loc_27><loc_14><loc_76><loc_15></location>σ ( γ e , p o ) = 0 . (51)</formula>
<text><location><page_16><loc_16><loc_85><loc_76><loc_88></location>First derivatives of σ are always tangent to the connecting light ray. In particular, the vector</text>
<formula><location><page_16><loc_27><loc_81><loc_76><loc_85></location>r a o := -∇ a σ ( γ e , p o ) u o -u e (52)</formula>
<text><location><page_16><loc_16><loc_75><loc_76><loc_81></location>at p o points along the geodesic which eventually intersects γ e (and is therefore pastdirected). The derivative operator here is understood to act on the second argument of σ . Also note that r a o is normalized such that glyph[lscript] a r a o = 1. Parallel-transporting r a o to the observation point yields</text>
<formula><location><page_16><loc_27><loc_71><loc_76><loc_75></location>r a e = ∇ a σ ( γ e , p o ) u o -u e . (53)</formula>
<text><location><page_16><loc_16><loc_67><loc_76><loc_72></location>The derivative operator in this equation is understood to act on the first argument of σ . Both r a o and r a e may be viewed as (dimensionless) separation vectors between p o and γ e .</text>
<text><location><page_16><loc_16><loc_63><loc_76><loc_67></location>Eq. (37) and the various identities of Sect. 2.1 may be used to compute the explicit coordinate components of r a e and r a o . Components transverse to the direction of wave propagation are</text>
<formula><location><page_16><loc_27><loc_60><loc_76><loc_62></location>r e = B -ᵀ ( u e , u o ) [ x o -∂ (1) B ᵀ ( u e , u o ) γ e ] , (54)</formula>
<text><location><page_16><loc_16><loc_58><loc_76><loc_60></location>where x o = x ( p o ) and γ e = x ( γ ( u e )) denote the transverse coordinates of the observer and source. A similar calculation shows that</text>
<formula><location><page_16><loc_27><loc_55><loc_76><loc_57></location>r o = B -1 ( u e , u o ) [ A ( u e , u o ) x o -γ e ] . (55)</formula>
<text><location><page_16><loc_16><loc_54><loc_33><loc_55></location>r o and r e are related via</text>
<formula><location><page_16><loc_27><loc_51><loc_76><loc_53></location>r e = ∂ (1) B ( u e , u o ) r o -∂ (1) A ( u e , u o ) x o . (56)</formula>
<text><location><page_16><loc_16><loc_48><loc_76><loc_51></location>Much of the discussion below considers sources moving on geodesics. In these cases, use of (23) shows that</text>
<formula><location><page_16><loc_27><loc_46><loc_76><loc_48></location>r o = B -1 ( u e , u o ) A ( u e , u o ) δ x o -˙ γ o . (57)</formula>
<text><location><page_16><loc_16><loc_43><loc_76><loc_46></location>Here, δ x o := x o -γ ( u o ) = x o -γ o . The various identities involving A and B discussed in Sect. 2.1 may also be used to reduce the imaging condition (51) to</text>
<formula><location><page_16><loc_24><loc_40><loc_76><loc_43></location>κ s ( u o -u e ) = 2( ˙ γ o · δ x o -δv o ) -δ x ᵀ o B -1 ( u e , u o ) A ( u e , u o ) δ x o . (58)</formula>
<text><location><page_16><loc_16><loc_35><loc_76><loc_40></location>Here, δv o := v ( p o ) -v ( γ o ). Eq. (58) is a nonlinear relation for the emission 'time' u e in terms of the observer's position p o and the parameters γ o , ˙ γ o , v ( γ o ), κ s describing the source's worldline. As discussed in Sect. 3, there can be many solutions to (58). These correspond to different images.</text>
<text><location><page_16><loc_16><loc_27><loc_76><loc_35></location>Neither u e nor r a o depends on the observer's motion. Nevertheless, redshifts and angles on the observer's sky do depend on that motion (as is true even in flat spacetime). It is often useful to fix this effect by supposing that the observer is instantaneously comoving with the source. Following (32), this is taken to mean that the unit 4-velocities U a s , U a o of the source and observer on S u o satisfy</text>
<formula><location><page_16><loc_27><loc_24><loc_76><loc_27></location>glyph[lscript] a U a s ( u o ) = glyph[lscript] a U a o = -1 √ κ s , (59)</formula>
<formula><location><page_16><loc_27><loc_20><loc_76><loc_23></location>P s ( u o ) = U s ( u o ) = U o = ˙ γ ( u o ) √ κ s = ˙ x o √ κ s . (60)</formula>
<text><location><page_16><loc_16><loc_13><loc_76><loc_20></location>Recall from Sect. 2.3 that the 'transverse momentum' P s ( u o ) is generated by contracting U a s ( u o ) with the two Killing fields equal to X a ( i ) at p o and having vanishing first derivative at that point. Also note that (60) implicitly defines an instantaneous observer velocity ˙ x a o = √ κ s U a o normalized (like ˙ γ a ) such that glyph[lscript] a ˙ x a o = -1.</text>
<section_header_level_1><location><page_17><loc_16><loc_86><loc_30><loc_88></location>4.1. Frequency shifts</section_header_level_1>
<text><location><page_17><loc_16><loc_73><loc_76><loc_85></location>Gravitational lenses typically discussed in astrophysics involve nearly-Newtonian mass distributions which may be regarded as approximately stationary (at least on subcosmological timescales). If both a source and an observer are sufficiently far from such a lens, there can be no significant redshift or blueshift from the gravitational field of that lens. Roughly speaking, a light ray falling into any stationary gravitational potential must climb out of that same potential. This result breaks down if light passes through non-stationary regions of spacetime. Indeed, plane wave spacetimes may produce images with significant frequency shifts [19].</text>
<text><location><page_17><loc_16><loc_69><loc_76><loc_73></location>Consider an approximately monochromatic beam of light emitted from γ e and received at p o . A future-directed tangent vector k a e ∝ -r a e to the emitted light ray may always be chosen such that</text>
<formula><location><page_17><loc_27><loc_66><loc_76><loc_68></location>ω e = -k e · U e (61)</formula>
<text><location><page_17><loc_16><loc_60><loc_76><loc_66></location>is the angular frequency of the light as seen by its source. The frequency ω o of this same light ray as measured by an observer at p o is -k o · U o , where k a o is equal to k a e parallel transported from the source to the observer. The observed and emitted frequencies are therefore related by</text>
<formula><location><page_17><loc_28><loc_55><loc_76><loc_60></location>ω o ω e = U o · k o U e · k e = U o · r o U e · r e = √ κ e κ o ( κ o + | ˙ x o + r o | 2 κ e + | ˙ γ e + r e | 2 ) . (62)</formula>
<text><location><page_17><loc_16><loc_50><loc_76><loc_56></location>Here, κ e := 1 / ( glyph[lscript] · U s ( u e )) 2 and κ o := 1 / ( glyph[lscript] · U o ) 2 . The 2-vectors r e and r o appearing here are determined by the source and observer positions via (37), (51), (54), and (55). The resulting frequency shift is valid for all emission points not contained in a caustic of p o .</text>
<text><location><page_17><loc_16><loc_44><loc_76><loc_49></location>Now suppose that a source moves on a geodesic and that the observer is instantaneously comoving with this geodesic in the sense of (59) and (60). Then κ e = κ s = 1 / ( glyph[lscript] · U s ) 2 doesn't depend on which image is chosen. Eqs. (56), (57), and the symmetry of B -1 A may be used to rewrite (62) as</text>
<formula><location><page_17><loc_28><loc_39><loc_76><loc_43></location>ω o ω e = 1 + ( B -ᵀ δ x o ) ᵀ ( AA ᵀ -δ κ s + | B -ᵀ δ x o | 2 ) ( B -ᵀ δ x o ) . (63)</formula>
<text><location><page_17><loc_16><loc_29><loc_76><loc_39></location>The matrix in parentheses on the right hand side of this equation acts like a metric for the 'separation' 2-vector B -ᵀ ( u e , u o )[ x o -γ ( u o )]. If both eigenvalues of A ( u e , u s ) A ᵀ ( u e , u s ) -δ are negative, the source is necessarily redshifted. Conversely, sources are always blueshifted when this matrix is positive definite. If AA ᵀ -δ has both positive and negative eigenvalues, the sign of the frequency difference depends on the direction of B -ᵀ δ x o . For special configurations, there is no frequency shift at all.</text>
<section_header_level_1><location><page_17><loc_16><loc_26><loc_24><loc_27></location>4.2. Angles</section_header_level_1>
<text><location><page_17><loc_16><loc_19><loc_76><loc_25></location>Various images formed from a single source appear at different points on an observer's sky. Like redshifts, the relative angles between images change depending on an observer's 4-velocity U a o . The angle θ between two images arriving at p o with tangents r a o and r ' a o is</text>
<formula><location><page_17><loc_28><loc_14><loc_76><loc_18></location>cos θ = ( g ab + U o,a U o,b ) r a o r ' b o ( U o · r o )( U o · r ' o ) = 1 + r o · r ' o ( U o · r o )( U o · r ' o ) . (64)</formula>
<text><location><page_18><loc_16><loc_86><loc_24><loc_88></location>Simplifying,</text>
<formula><location><page_18><loc_28><loc_82><loc_76><loc_86></location>cos θ = 1 -2 κ o | r o -r ' o | 2 ( κ o + | ˙ x o + r o | 2 ) ( κ o + | ˙ x o + r ' o | 2 ) . (65)</formula>
<text><location><page_18><loc_16><loc_79><loc_76><loc_82></location>This expression is valid for arbitrary source and observer configurations. Specializing to geodesic sources and comoving observers,</text>
<formula><location><page_18><loc_28><loc_74><loc_76><loc_79></location>cos θ = 1 -2 κ s ∣ ∣ ( B -1 A -B '-1 A ' ) δ x o ∣ ∣ 2 ( κ s + | B -1 A δ x o | 2 )( κ s + | B '-1 A ' δ x o | 2 ) . (66)</formula>
<text><location><page_18><loc_16><loc_71><loc_76><loc_74></location>Here, A = A ( u e , u o ) and A ' = A ( u ' e , u o ). It is evident that angles are largely controlled by the difference between B -1 A at the two emission times.</text>
<text><location><page_18><loc_16><loc_67><loc_76><loc_71></location>Another interesting angle to consider is the observed separation ψ between a single image (emitted at γ e ) and a generator glyph[lscript] a of the background gravitational wave. For arbitrarily moving source and observer configurations,</text>
<formula><location><page_18><loc_28><loc_63><loc_76><loc_66></location>cos ψ = 1 -2 κ o κ o + | ˙ x o + r o | 2 . (67)</formula>
<text><location><page_18><loc_16><loc_61><loc_49><loc_63></location>For observers comoving with geodesic sources,</text>
<formula><location><page_18><loc_28><loc_57><loc_76><loc_61></location>cos ψ = 1 -2 κ s κ s + | B -1 A δ x o | 2 . (68)</formula>
<text><location><page_18><loc_16><loc_55><loc_76><loc_57></location>This may be used to rewrite the angle θ between two different images partially in terms of the angles ψ and ψ ' those images make with glyph[lscript] a . Using (66),</text>
<formula><location><page_18><loc_28><loc_50><loc_76><loc_54></location>cos θ = cos ψ cos ψ ' + ( B -1 A δ x o ) · ( B '-1 A ' δ x o ) | B -1 A δ x o || B '-1 A ' δ x o | sin ψ sin ψ ' . (69)</formula>
<text><location><page_18><loc_16><loc_49><loc_71><loc_50></location>Similarly, the frequency shift (63) of an individual image may be rewritten as</text>
<formula><location><page_18><loc_28><loc_44><loc_76><loc_48></location>ω o ω e = κ s csc 2 ( ψ/ 2) κ s + | B -ᵀ δ x o | 2 . (70)</formula>
<text><location><page_18><loc_16><loc_41><loc_76><loc_44></location>It is evident from this equation that images which appear highly blueshifted to comoving observers must satisfy ψ ≈ 0.</text>
<section_header_level_1><location><page_18><loc_16><loc_38><loc_44><loc_39></location>4.3. Image distortion and magnification</section_header_level_1>
<text><location><page_18><loc_16><loc_22><loc_76><loc_37></location>Thus far, all sources here have been modelled as though they were confined to timelike worldlines. Real objects are not pointlike, however. They form extended worldtubes in spacetime. Images of such worldtubes form null geodesic congruences which converge on p o . These images can be significantly distorted by the curvature of spacetime. It is simplest to quantify such distortions by first fixing a particular null geodesic Z passing between some part of the source and p o . Precisely which geodesic is chosen is not important. Z serves only as an origin from which to discuss nearby light rays connecting p o to other points in the source. Once this origin has been fixed, the image of an extended source may be described entirely using deviation vectors on Z (at least for sufficiently small sources). See Fig. 3.</text>
<text><location><page_18><loc_16><loc_16><loc_76><loc_22></location>Deviation vectors (or Jacobi fields) satisfy the geodesic deviation (or Jacobi) equation along Z . Letting r a denote the past-directed null vector tangent to Z and obtained by parallel-transporting r a o from p o , every deviation vector ξ a is a solution to</text>
<formula><location><page_18><loc_27><loc_13><loc_76><loc_16></location>r b ∇ b ( r c ∇ c ξ a ) = R a bcd r b r c ξ d . (71)</formula>
<figure>
<location><page_19><loc_34><loc_72><loc_58><loc_88></location>
<caption>Figure 3: Imaging for an extended source. The fiducial light ray Z is indicated together with another light ray separated from it by a deviation vector ξ a . The vector r a o is also drawn. This is tangent to Z at the observation point p o .</caption>
</figure>
<text><location><page_19><loc_16><loc_53><loc_76><loc_64></location>This equation is linear, so ξ a must depend linearly on initial data. In particular, all deviation vectors can be written as linear combinations of appropriate bitensors contracted into the initial data ¶ ξ A ( u o ) and ˙ ξ A ( u o ). All light rays observed at p o must necessarily intersect that point, so it suffices to set ξ A ( u o ) = 0. The first derivative of a deviation vector at p o describes an angular deviation between one point of an image and the center associated with Z . We therefore consider deviation vectors ξ a with the form</text>
<formula><location><page_19><loc_27><loc_50><loc_76><loc_53></location>ξ a = B a A ( · , u o ) ˙ ξ A ( u o ) . (72)</formula>
<text><location><page_19><loc_16><loc_49><loc_65><loc_50></location>B a A is known as a Jacobi propagator. It satisfies the Jacobi equation</text>
<formula><location><page_19><loc_27><loc_46><loc_76><loc_48></location>r b ∇ b ( r c ∇ c B a A ) = R a bcd r b r c B d A (73)</formula>
<text><location><page_19><loc_16><loc_44><loc_47><loc_46></location>along Z together with the initial conditions</text>
<formula><location><page_19><loc_28><loc_42><loc_76><loc_44></location>lim u s → u o B a A ( u s , u o ) = 0 , lim u s → u o r b ∇ b B a A ( u s , u o ) = δ a A . (74)</formula>
<text><location><page_19><loc_16><loc_38><loc_76><loc_41></location>Note that B a A is a bitensor. It maps vectors at p o into vectors at others points on Z . The transverse components of B a A are</text>
<formula><location><page_19><loc_27><loc_36><loc_76><loc_38></location>B aA X a ( i ) X A ( j ) = ( B ) ij , (75)</formula>
<text><location><page_19><loc_16><loc_33><loc_76><loc_36></location>where B is the matrix defined in Sect. 2.1. Other components of B a A may be deduced from the eigenvector relations [23]</text>
<formula><location><page_19><loc_21><loc_30><loc_76><loc_32></location>B a A ( u s , u o ) r A o = ( u s -u o ) r a , r a B a A ( u s , u o ) = ( u s -u o ) r A o , (76)</formula>
<formula><location><page_19><loc_21><loc_28><loc_76><loc_30></location>B a A ( u s , u o ) glyph[lscript] A = ( u s -u o ) glyph[lscript] a , glyph[lscript] a B a A ( u s , u o ) = ( u s -u o ) glyph[lscript] A . (77)</formula>
<text><location><page_19><loc_16><loc_23><loc_76><loc_28></location>All parts of an image must arrive at an observer along null geodesics. Additionally, an observer with 4-velocity U a o can only measure angles of vectors orthogonal to U a o . It therefore suffices to restrict attention to deviation vectors satisfying</text>
<formula><location><page_19><loc_27><loc_20><loc_76><loc_23></location>r o · ˙ ξ ( u o ) = U o · ˙ ξ ( u o ) = 0 (78)</formula>
<text><location><page_19><loc_16><loc_17><loc_76><loc_20></location>¶ Capital letters are used in this subsection to denote abstract indices associated with the observation point p o . This is done to avoid confusion when writing down two-point tensors such as B a A [see (72)].</text>
<text><location><page_20><loc_16><loc_85><loc_76><loc_88></location>at p o . These constraints restrict all interesting initial data to a two dimensional space. The orthonormal vectors</text>
<formula><location><page_20><loc_27><loc_80><loc_76><loc_84></location>e A ( i ) = X A ( i ) -r i o glyph[lscript] A -2 ( r i o + ˙ x i o κ o + | ˙ x o + r o | 2 ) r A o (79)</formula>
<text><location><page_20><loc_16><loc_79><loc_48><loc_81></location>form a basis for this space at p o . They satisfy</text>
<formula><location><page_20><loc_27><loc_76><loc_76><loc_78></location>e ( i ) · e ( j ) = δ ij , r o · e ( i ) = U o · e ( i ) = 0 . (80)</formula>
<text><location><page_20><loc_16><loc_74><loc_54><loc_76></location>Parallel-transporting e A ( i ) to another point on Z yields</text>
<formula><location><page_20><loc_27><loc_70><loc_76><loc_74></location>e a ( i ) = X a ( i ) -r i glyph[lscript] a -2 ( r i o + ˙ x i o κ o + | ˙ x o + r o | 2 ) r a , (81)</formula>
<formula><location><page_20><loc_27><loc_66><loc_76><loc_68></location>˙ ξ A ( u o ) = ( U o · r o ) α ( i ) e A ( i ) , (82)</formula>
<text><location><page_20><loc_16><loc_68><loc_76><loc_70></location>which forms a Sachs basis [1] on Z . Initial data appearing in (72) must be of the form</text>
<text><location><page_20><loc_16><loc_59><loc_76><loc_66></location>where α is an unconstrained 2-vector. The factor ( U o · r o ) > 0 is included here so that α is directly related to angles on an observer's sky. A sufficiently small image may be described by a suitable set of 2-vectors α representing the angular locations of each portion of the image with respect to the center defined by Z .</text>
<text><location><page_20><loc_16><loc_53><loc_76><loc_60></location>Each α may be translated into a physical displacement at the source using (72) and (82). First note that for every particular α , (76) implies that r · ξ = 0 throughout Z . Indeed, ξ a is always a linear combination of the e a ( i ) together with r a . Components of ξ a proportional to r a are physically irrelevant, so we consider only the Sachs components</text>
<formula><location><page_20><loc_27><loc_50><loc_76><loc_52></location>ξ ( i ) := e ( i ) · ξ = [( U o · r o ) B aA e a ( i ) e A ( j ) ] α ( j ) . (83)</formula>
<text><location><page_20><loc_16><loc_47><loc_34><loc_50></location>Defining the 2 × 2 matrix</text>
<formula><location><page_20><loc_27><loc_45><loc_76><loc_48></location>D ( i )( j ) ( u s , u o ) := ( U o · r o ) B aA ( u s , u o ) e a ( i ) ( u s ) e A ( j ) ( u o ) , (84)</formula>
<text><location><page_20><loc_16><loc_42><loc_76><loc_45></location>it is then clear that ξ = D α for any α . D is referred to as the Jacobi matrix or Jacobi map [1, 11]. Using (75)-(77), (79), (81), and (84),</text>
<formula><location><page_20><loc_27><loc_39><loc_76><loc_42></location>D ( u s , u o ) = ( U o · r o ) B ( u s , u o ) . (85)</formula>
<text><location><page_20><loc_16><loc_38><loc_76><loc_39></location>If a source moves on a geodesic which is instantaneously comoving with the observer,</text>
<formula><location><page_20><loc_27><loc_34><loc_76><loc_38></location>U o · r o = κ s + | B -1 A δ x o | 2 2 √ κ s . (86)</formula>
<text><location><page_20><loc_16><loc_29><loc_76><loc_34></location>This discussion implies that a portion of an image with angular separation glyph[epsilon1] α from the fiducial direction associated with r a o is spatially separated from the fiducial emission point γ e ∈ Z by</text>
<formula><location><page_20><loc_27><loc_26><loc_76><loc_29></location>glyph[epsilon1] ξ ( u e ) = ( U o · r o ) B ( u e , u o )( glyph[epsilon1] α ) . (87)</formula>
<text><location><page_20><loc_16><loc_15><loc_76><loc_26></location>The factors of glyph[epsilon1] glyph[lessmuch] 1 have been introduced here to emphasize that this description is valid only for infinitesimal deviations. Regardless, (87) shows that up to the time dilation factor ( U o · r o ), the matrix B central to all aspects of plane wave geometry may be physically interpreted as a transformation converting infinitesimal angles on the vertex of a light cone into infinitesimal separations elsewhere on that light cone. B depends only on the u coordinates of the source and emission points, and not on any other aspects of the physical configuration. It may be computed for all possible observer-source pairs directly from the wave profile H .</text>
<text><location><page_21><loc_16><loc_83><loc_76><loc_88></location>Angles of emission (as opposed to observation) of the various light rays travelling from the source to the observer may be found by differentiating (87) and applying the appropriate time dilation factor:</text>
<formula><location><page_21><loc_28><loc_79><loc_76><loc_83></location>˙ ξ ( u e ) U e · r e = ( U o · r o U e · r e ) ∂ (1) B ( u e , u o ) α = ( ω o ω e ) ∂ (1) B ( u e , u o ) α . (88)</formula>
<text><location><page_21><loc_16><loc_77><loc_64><loc_79></location>The last equality here makes use of (62). Applying (87) shows that</text>
<formula><location><page_21><loc_27><loc_75><loc_76><loc_77></location>˙ ξ = ∂ (1) BB -1 ξ . (89)</formula>
<text><location><page_21><loc_16><loc_70><loc_76><loc_75></location>The symmetric matrix ∂ (1) BB -1 / ( U e · r e ) therefore converts spatial locations to emission angles within the source (with the constraint that all light rays intersect p o ). It is referred to as the optical deformation matrix [1, 11].</text>
<text><location><page_21><loc_16><loc_65><loc_76><loc_70></location>Eq. (87) implies that there is a sense in which circles on the observer's sky correspond to ellipses near γ e . This deformation may be parametrized by performing a polar decomposition on D ( u e , u o ):</text>
<formula><location><page_21><loc_27><loc_62><loc_76><loc_65></location>D = R ᵀ β ( D + 0 0 D -) R χ . (90)</formula>
<text><location><page_21><loc_16><loc_55><loc_76><loc_61></location>Here, R β and R χ represent rotation matrices through some angles β and χ . The ratio D + /D -is related to the ellipticity of the aforementioned ellipse. χ represents the angle between the principal axes of that ellipse and the Sachs basis. D ± and χ are referred as shape parameters [1, 43].</text>
<text><location><page_21><loc_16><loc_43><loc_76><loc_55></location>Recalling that polarization vectors are parallel-transported in the geometric optics approximation [3], any polarization vector must have Sachs components which are constant along Z . In principle, the angle χ might therefore be measured by comparing the relative 'rotation' between an object's observed shape and an appropriate polarization angle [1, 44]. For linearly polarized waves where H can be made diagonal by an appropriate coordinate choice, χ = 0 with respect to this coordinate system and the basis (81). It is shown in Sect. 6 that χ also vanishes in a natural way for all sufficiently weak wavepackets which are nonzero only for short times.</text>
<text><location><page_21><loc_16><loc_39><loc_76><loc_43></location>Eq. (87) implies that D converts angles at the observer to separations within the source. The determinant of D must therefore relate solid angles at p o to physical areas near γ e :</text>
<formula><location><page_21><loc_28><loc_35><loc_76><loc_38></location>d A dΩ = | det D | = | D + D -| = ( U o · r o ) 2 | det B | . (91)</formula>
<text><location><page_21><loc_16><loc_33><loc_26><loc_35></location>It follows that</text>
<formula><location><page_21><loc_27><loc_30><loc_76><loc_33></location>d ang := √ | det D | = ( U o · r o ) √ | det B | (92)</formula>
<text><location><page_21><loc_16><loc_22><loc_76><loc_30></location>may be interpreted as an 'angular diameter distance.' Absolute value signs are necessary here because det B changes sign after each pass through a conjugate hyperplane with multiplicity 1. Physically, such sign changes represent parity inversions of the resulting image. Note that d ang does not necessarily increase monotonically with the age of an image (as computed using the source's proper time). Closely related to the angular diameter distance is the luminosity distance</text>
<formula><location><page_21><loc_27><loc_19><loc_76><loc_21></location>d lum := ( ω o /ω e ) -2 d ang . (93)</formula>
<text><location><page_21><loc_16><loc_14><loc_76><loc_19></location>One factor of ω o /ω e arises here from considering light cones emanating from the source instead of the observer. The other factor of ω/ω e is related to the energy change associated with frequency shifts.</text>
<section_header_level_1><location><page_22><loc_16><loc_86><loc_37><loc_88></location>5. Symmetric plane waves</section_header_level_1>
<text><location><page_22><loc_16><loc_75><loc_76><loc_85></location>Now that various optical quantities have been computed for general plane wave spacetimes, we consider their application to various special cases. The simplest nontrivial plane waves are the symmetric waves. These are locally symmetric in the sense that ∇ a R bcd f = 0. It follows from (3) that symmetric plane waves must have constant waveforms. Also note that ( ∂/∂u ) a is Killing in these examples [as well as glyph[lscript] a = ( ∂/∂v ) a , which is Killing in all plane wave spacetimes]. Particular symmetric plane waves may be specified entirely by the (constant) eigenvalues of H .</text>
<text><location><page_22><loc_16><loc_69><loc_76><loc_74></location>Recalling the decomposition (6) of H into h + , h × and h ‖ , a coordinate rotation may always be used to set h × = 0 for symmetric waves. It is then evident that the two eigenvalues of H are given by ± h + -h ‖ . It is always possible to set</text>
<formula><location><page_22><loc_27><loc_65><loc_76><loc_69></location>H = ( -h 1 0 0 -h 2 ) , (94)</formula>
<formula><location><page_22><loc_27><loc_61><loc_76><loc_63></location>h 1 := h ‖ + h + , h 2 := h ‖ -h + . (95)</formula>
<text><location><page_22><loc_16><loc_56><loc_76><loc_61></location>The weak energy condition implies that h ‖ ≥ 0, so at least one eigenvalue of H must be negative (implying that at least one of the h 1 , 2 must be positive). We assume for definiteness that h + ≥ 0. Then,</text>
<formula><location><page_22><loc_27><loc_54><loc_76><loc_56></location>h 1 > 0 , h 1 ≥ | h 2 | . (96)</formula>
<text><location><page_22><loc_16><loc_48><loc_76><loc_54></location>If the vacuum Einstein equation is imposed, h ‖ = 0 and h 2 = -h 1 . For conformallyflat geometries representing spacetimes associated with, e.g., pure electromagnetic plane waves, h + = 0 and h 2 = h 1 . Other cases may be viewed as superpositions of gravitational and ('gravito'-)electromagnetic waves.</text>
<text><location><page_22><loc_16><loc_40><loc_76><loc_48></location>All symmetric waves produce an infinite number of images of almost every source. It is clear from (94) that these waves are also linearly polarized. The angles χ and β appearing in (90) therefore vanish when considering image deformations with respect to the Sachs basis (81). Other lensing properties depend on the sign of h 2 . We call the case h 2 < 0 'gravity-dominated' and the case h 2 > 0 'matter-dominated.'</text>
<section_header_level_1><location><page_22><loc_16><loc_37><loc_45><loc_38></location>5.1. Gravity-dominated symmetric waves</section_header_level_1>
<text><location><page_22><loc_16><loc_30><loc_76><loc_36></location>Consider symmetric plane wave spacetimes where h 2 = h ‖ -h + < 0. Gravitydominated waves such as these generalize the vacuum waves satisfying h 1 = -h 2 . Symmetric vacuum waves arise from, e.g., the Penrose limit of a null geodesic orbiting a Schwarzschild black hole on the light ring.</text>
<text><location><page_22><loc_16><loc_27><loc_76><loc_30></location>For any gravity-dominated symmetric wave, the matrices A and B defined in Sect. 2.1 are</text>
<formula><location><page_22><loc_21><loc_22><loc_76><loc_27></location>A ( u s , u o ) = ( cos h 1 2 1 ( u s -u o ) 0 0 cosh | h 2 | 1 2 ( u s -u o ) ) , (97)</formula>
<formula><location><page_22><loc_21><loc_18><loc_76><loc_23></location>B ( u s , u o ) = ( h -1 2 1 sin h 1 2 1 ( u s -u o ) 0 0 | h 2 | -1 2 sinh | h 2 | 1 2 ( u s -u o ) ) . (98)</formula>
<text><location><page_22><loc_16><loc_14><loc_76><loc_18></location>It is clear that det B ( · , u o ) has an infinite number of zeros for any choice of u o . Each of these zeros represents a phase conjugate to u o . There are an infinite number of such phases in both the past and future of every observer. The discussion in Sect. 3</text>
<text><location><page_22><loc_16><loc_64><loc_20><loc_65></location>where</text>
<text><location><page_23><loc_16><loc_85><loc_76><loc_88></location>therefore implies that under generic conditions, an infinite number of images appear for almost every source. Explicitly, all conjugate phases are given by</text>
<formula><location><page_23><loc_27><loc_82><loc_76><loc_84></location>τ n ( u o ) = u o + nπh -1 2 1 , (99)</formula>
<text><location><page_23><loc_16><loc_71><loc_76><loc_81></location>where n is any nonzero integer. It is evident from (19) that all of these phases have multiplicity 1. For any n < 0 and any observation point p o with u ( p o ) = u o , exactly one image of each source is visible as that source appeared in N n ( u o ). This corresponds to the region between u = u o + nπh -1 2 1 and u = u o +( n -1) πh -1 2 1 . Note that det B ( · , u o ) switches sign on each pass through a conjugate phase. The parity of an image emitted from N n ( u o ) is therefore opposite to the parity of an image emitted from N n -1 ( u o ).</text>
<text><location><page_23><loc_16><loc_68><loc_76><loc_72></location>Specializing to cases where the source is a geodesic and the observer is instantaneously comoving with that source on S u o , some configurations lead to redshifts and others to blueshifts. Using (63),</text>
<formula><location><page_23><loc_28><loc_63><loc_76><loc_67></location>ω o ω e = 1 + δ x ᵀ o H δ x o κ s + | B -ᵀ δ x o | 2 . (100)</formula>
<text><location><page_23><loc_16><loc_62><loc_48><loc_63></location>An image is therefore redshifted if and only if</text>
<formula><location><page_23><loc_27><loc_59><loc_76><loc_61></location>δ x ᵀ o H δ x o = -h 1 ( δx 1 o ) 2 + | h 2 | ( δx 2 o ) 2 < 0 . (101)</formula>
<text><location><page_23><loc_16><loc_58><loc_76><loc_59></location>It is blueshifted when δ x ᵀ o H δ x o > 0. There is no frequency shift at all in cases where</text>
<formula><location><page_23><loc_27><loc_55><loc_76><loc_57></location>( δx 1 o /δx 2 o ) 2 = | h 2 /h 1 | . (102)</formula>
<text><location><page_23><loc_16><loc_49><loc_76><loc_55></location>The direction of the frequency shift clearly depends only on the instantaneous orientation δ x o / | δ x o | of the source and the observer on S u o . In particular, it does not depend on which image is considered. All images of a particular source experience the same type of frequency shift.</text>
<text><location><page_23><loc_16><loc_46><loc_76><loc_49></location>Emission times u e for an observer comoving with a geodesic source may be found by solving (58). For gravity-dominated symmetric waves, this equation reduces to</text>
<formula><location><page_23><loc_27><loc_40><loc_76><loc_45></location>κ s ( u o -u e ) = 2( ˙ γ o · δ x o -δv o ) + h 1 2 1 ( δx 1 o ) 2 cot h 1 2 1 ( u o -u e ) + | h 2 | 1 2 ( δx 2 o ) 2 coth | h 2 | 1 2 ( u o -u e ) . (103)</formula>
<text><location><page_23><loc_16><loc_36><loc_76><loc_41></location>If -n glyph[greatermuch] 1, it is evident that the image from N n ( u o ) must satisfy cot h 1 2 1 ( u o -u e ) glyph[greatermuch] 1. Images from the distant past are therefore emitted at phases u e very nearly conjugate to u o :</text>
<formula><location><page_23><loc_27><loc_31><loc_76><loc_35></location>u e ≈ τ n ( u o ) -h 1 2 1 ( δx 1 o ) 2 | n | πκ s . (104)</formula>
<text><location><page_23><loc_16><loc_28><loc_76><loc_31></location>Substituting this relation into (68) and (100) shows that very old images cluster near glyph[lscript] a on the observer's sky and experience increasingly-negligible frequency shifts:</text>
<formula><location><page_23><loc_27><loc_25><loc_76><loc_27></location>ψ ∝ | n | -1 , | ω o /ω e -1 | ∝ | n | -2 . (105)</formula>
<text><location><page_23><loc_16><loc_22><loc_76><loc_25></location>Old images of slightly extended sources are also highly distorted and demagnified. Their angular diameter and luminosity distances both scale like</text>
<formula><location><page_23><loc_27><loc_19><loc_76><loc_22></location>d ang ∼ d lum ∝ | n | 3 2 exp ( 1 2 √ | h 2 /h 1 || n | π ) . (106)</formula>
<text><location><page_23><loc_16><loc_15><loc_76><loc_18></location>Gravity-dominated symmetric waves therefore produce an infinite number of exponentially dimming images for almost every source.</text>
<section_header_level_1><location><page_24><loc_16><loc_86><loc_44><loc_88></location>5.2. Matter-dominated symmetric waves</section_header_level_1>
<text><location><page_24><loc_16><loc_82><loc_76><loc_85></location>Matter-dominated symmetric waves satisfying h 2 > 0 act somewhat differently than gravity-dominated waves. In these cases,</text>
<formula><location><page_24><loc_27><loc_77><loc_76><loc_82></location>A ( u s , u o ) = ( cos h 1 2 1 ( u s -u o ) 0 0 cos h 1 2 2 ( u s -u o ) ) , (107)</formula>
<formula><location><page_24><loc_27><loc_73><loc_76><loc_77></location>B ( u s , u o ) = ( h -1 2 1 sin h 1 2 1 ( u s -u o ) 0 0 h -1 2 2 sin h 1 2 2 ( u s -u o ) ) . (108)</formula>
<text><location><page_24><loc_16><loc_62><loc_76><loc_73></location>Phases conjugate to u o occur at u o + nπh -1 2 1 and at u o + n ' πh -1 2 2 , where n, n ' are any nonzero integers. If √ h 1 /h 2 is an irrational number, these two families of phases are distinct. Each conjugate pair then has multiplicity 1. If √ h 1 /h 2 is rational, some conjugate pairs have multiplicity 2. In the conformally-flat case where h 1 = h 2 , all conjugate phases have multiplicity 2. In every other case where √ h 1 /h 2 is rational, an infinite number of conjugate phases occur with each multiplicity. Regardless of h 2 , an infinite number of images are formed for almost every source.</text>
<text><location><page_24><loc_16><loc_57><loc_76><loc_61></location>Now consider a luminous source moving on a geodesic. If the source and observer are instantaneously comoving in the sense of (59) and (60), frequency shifts associated with each image are given by (63). Since</text>
<formula><location><page_24><loc_27><loc_54><loc_76><loc_56></location>B -1 ( AA ᵀ -δ ) B -ᵀ = H (109)</formula>
<text><location><page_24><loc_16><loc_53><loc_56><loc_54></location>is negative-definite in this case, all images are redshifted.</text>
<text><location><page_24><loc_16><loc_42><loc_76><loc_53></location>If -n glyph[greatermuch] 1, an image originating from N n ( u o ) must be emitted just before the source intersects S τ n ( u o ) . All such images cluster towards glyph[lscript] a in the observer's sky and have negligible frequency shifts. Images emitted near conjugate hyperplanes with multiplicity 1 are highly distorted. Images emitted near conjugate hyperplanes with multiplicity 2 are not significantly distorted at all. In both cases, however, older images are dimmer (although the rate at which this occurs is much slower than for gravity-dominated symmetric waves).</text>
<text><location><page_24><loc_16><loc_35><loc_76><loc_42></location>Regardless of the sign of h 2 , the oldest images formed by symmetric plane wave spacetimes depend on the spacetime structure at arbitrarily large transverse distances. If the metric is modified so that the wave decays at large distances, only a finite number of images discussed here would be unaffected. The oldest images found in pure symmetric waves likely do not appear at all in perturbed symmetric waves.</text>
<section_header_level_1><location><page_24><loc_16><loc_31><loc_31><loc_33></location>6. Sandwich waves</section_header_level_1>
<text><location><page_24><loc_16><loc_18><loc_76><loc_30></location>Symmetric plane waves are mathematically simple, but are not reasonable models for gravitational radiation emitted from compact sources. More interesting are waves where H is nonzero only for a finite time: Sandwich waves. Suppose, in particular, that there exists some u + > 0 such that H ( u ) = 0 for all u / ∈ [0 , u + ]. It follows from (3) that such spacetimes are locally flat whenever u < 0 or u > u + . The curved region containing the gravitational wave is effectively sandwiched between the two null hyperplanes S 0 and S u + . Every timelike curve eventually passes entirely through such a wave.</text>
<text><location><page_24><loc_16><loc_15><loc_76><loc_18></location>Before an observer interacts with the wave, spacetime is flat and optics is trivial. The case u o > u + where an observer has already passed through the wave is more</text>
<text><location><page_25><loc_16><loc_86><loc_63><loc_88></location>interesting. In this case, A and B reduce to their flat space forms</text>
<formula><location><page_25><loc_27><loc_83><loc_76><loc_86></location>A ( u s , u o ) = δ , B ( u s , u o ) = ( u s -u o ) δ (110)</formula>
<text><location><page_25><loc_16><loc_78><loc_76><loc_84></location>when u s > u + . The forms of A and B inside the wave depend on the details of H , and will not be discussed here. If u s < 0, however, there always exist four constant 2 × 2 matrices α , ˙ α , β and ˙ β such that</text>
<formula><location><page_25><loc_27><loc_77><loc_76><loc_79></location>B ( u s , u o ) = ( α + ˙ α u o ) + ( β + ˙ β u o ) u s . (111)</formula>
<text><location><page_25><loc_16><loc_74><loc_76><loc_77></location>Note that the dots on ˙ α and ˙ β do not refer to derivatives in this case. They are only used as a labelling device. It follows from (17) that A ( u s , u o ) is independent of u o .</text>
<text><location><page_25><loc_16><loc_72><loc_23><loc_73></location>Moreover,</text>
<formula><location><page_25><loc_27><loc_69><loc_76><loc_72></location>A ( u s , u o ) = -˙ α -˙ β u s . (112)</formula>
<text><location><page_25><loc_16><loc_60><loc_76><loc_68></location>In general, ˙ α and ˙ β have a simple physical interpretation. If two geodesics are comoving and have a transverse separation δ x o when u > u + , it follows from (23) that the transverse separation between these geodesics is -˙ α δ x o immediately before they interact with the wave at u = 0. Similarly, the relative transverse velocity of these geodesics is -˙ β δ x o when u < 0.</text>
<text><location><page_25><loc_16><loc_67><loc_57><loc_70></location>If there were no wave at all, α = ˙ β = 0 and -˙ α = β = δ .</text>
<text><location><page_25><loc_16><loc_50><loc_76><loc_60></location>Interpretations for α and β are somewhat less direct. Consider two geodesics which intersect at some time u o > u + after the wave has passed, but which have a relative transverse velocity δ ˙ x o at u = u o . The difference in transverse velocities between these two geodesics is then ( β + u o ˙ β ) δ ˙ x o when u < 0. Similarly, the difference in the transverse positions of these geodesics is ( α + u o ˙ α ) δ ˙ x o at u = 0. It follows that α controls shifts in position that are independent of the time u o at which the two geodesics cross each other.</text>
<text><location><page_25><loc_16><loc_42><loc_76><loc_50></location>In principle, α , ˙ α , β , ˙ β may all be found by solving (9) if H is known. At first glance, this would appear to imply that 4 · 4 = 16 numbers are required to describe observations through a sandwich wave. The actual number of required parameters is somewhat less than this. First note that the Wronskian identity (13) implies that ˙ α ᵀ ˙ β is a symmetric matrix. It also implies that</text>
<formula><location><page_25><loc_27><loc_39><loc_76><loc_42></location>˙ β ᵀ α -˙ α ᵀ β = δ . (113)</formula>
<text><location><page_25><loc_16><loc_37><loc_76><loc_40></location>Further simplifications arise by recalling from (14) that BA ᵀ and B ᵀ ∂ (1) B are symmetric. It follows that</text>
<formula><location><page_25><loc_27><loc_35><loc_76><loc_36></location>α ˙ α ᵀ , β ˙ β ᵀ , α ᵀ β (114)</formula>
<text><location><page_25><loc_16><loc_28><loc_76><loc_34></location>are symmetric as well. These expressions are completely general, and hold for any sandwich wave. They significantly constrain the number of independent parameters needed to specify A and B . Equivalently, they limit the number of parameters that must be extracted from H .</text>
<text><location><page_25><loc_16><loc_19><loc_76><loc_28></location>It follows from the arguments of Sect. 3 that the number of images of a generic source observable in any plane wave spacetime is governed by the number of hypersurfaces conjugate to the u = constant hypersurface S u o containing the observation event p o . Continuing to assume that u o > u + , all phases conjugate to u o must be smaller than u + . It follows from (20) and (111) that conjugate phases occurring before the wave may be found by solving</text>
<formula><location><page_25><loc_28><loc_17><loc_76><loc_19></location>det[( α + ˙ α u o ) + ( β + ˙ β u o ) τ ] = 0 (115)</formula>
<text><location><page_25><loc_16><loc_14><loc_76><loc_16></location>for all τ < 0. This equation is quadratic, so at most two solutions exist. An observer ahead of the wave may therefore see at most three images of a source as it appeared</text>
<text><location><page_26><loc_16><loc_83><loc_76><loc_88></location>behind the wave. There may also be at most one image of a source as it appeared ahead of the wave. In principle, any number of images may arise from inside the wave [where (111) is not valid] if H is sufficiently large.</text>
<text><location><page_26><loc_16><loc_74><loc_76><loc_83></location>Conjugate phases found by solving (115) clearly depend on the observation time u o . Less obviously, the number of conjugate phases can also depend on u o . For an observer moving on a timelike worldline (where u o increases monotonically), new conjugate phases - and therefore new images - sometimes appear at discrete times. These images correspond to observation times where (115) momentarily degenerates to a linear equation. New images can therefore arise when u o = ¯ u o and</text>
<formula><location><page_26><loc_28><loc_72><loc_76><loc_74></location>det( β + ˙ β ¯ u o ) = 0 . (116)</formula>
<text><location><page_26><loc_16><loc_71><loc_52><loc_72></location>If det ˙ β = 0, the two solutions to this equation are</text>
<formula><location><page_26><loc_20><loc_66><loc_76><loc_70></location>¯ u o = -[Tr β Tr ˙ β -Tr( β ˙ β )] ± √ [Tr β Tr ˙ β -Tr( β ˙ β )] 2 -4 det β det ˙ β 2 det ˙ β . (117)</formula>
<text><location><page_26><loc_21><loc_70><loc_21><loc_72></location>glyph[negationslash]</text>
<text><location><page_26><loc_16><loc_63><loc_76><loc_66></location>Only solutions satisfying ¯ u o > u + > 0 are physically relevant. When a conjugate phase of this type first appears, it satisfies</text>
<formula><location><page_26><loc_28><loc_60><loc_76><loc_63></location>lim u o → ¯ u + o τ ( u o ) = -∞ . (118)</formula>
<text><location><page_26><loc_16><loc_56><loc_76><loc_60></location>The associated image therefore provides a picture of the infinitely distant past. Furthermore, an infinite amount of the source's history appears to the observer within a finite amount of proper time. This implies that new images are highly blueshifted.</text>
<text><location><page_26><loc_16><loc_47><loc_76><loc_55></location>No matter how long an observer waits, no conjugate phase can exceed u + . The emission time for an associated image might therefore be expected to tend towards a constant value as u o → ∞ . This means that a very large amount of proper time at the observer corresponds to only a small amount of proper time at a source. Images which appear suddenly and are initially highly blueshifted become highly redshifted at late times.</text>
<text><location><page_26><loc_16><loc_32><loc_76><loc_46></location>It is unclear precisely what these types of images imply. To the extent that geometric optics remains valid, all observers passing through S ¯ u o momentarily see almost the entire universe appear infinitely blueshifted as it was in the infinitely distant past. Furthermore, (70) implies that all of the universe is briefly compressed into a single point on each observer's sky. Of course, such phenomena lie outside the domain of geometric optics. They may even lie outside of the realm of test fields propagating on a fixed background spacetime. Extreme focusing events like these might indicate instabilities inherent in the plane wave geometry itself. It should, however, be noted that all of the infinities just alluded to are likely to have finite cutoffs in 'realistic' plane waves which decay at large transverse distances.</text>
<section_header_level_1><location><page_26><loc_16><loc_28><loc_29><loc_30></location>Weak wavepackets</section_header_level_1>
<text><location><page_26><loc_16><loc_20><loc_76><loc_27></location>One important class of sandwich waves are those that are very weak and last only for a short time. In these cases, α , ˙ α , β , and ˙ β may be expanded as integrals involving successively higher powers of H . To lowest order in such a scheme, A and B are approximately unaffected by the wave while inside of it. Assuming that -u s , u o glyph[greatermuch] u + , the first corrections to this assumption are</text>
<formula><location><page_26><loc_27><loc_17><loc_76><loc_20></location>A ( u s , u o ) ≈ δ -u s ∫ u + 0 d w H ( w ) , (119)</formula>
<formula><location><page_26><loc_27><loc_13><loc_76><loc_17></location>B ( u s , u o ) ≈ ( u s -u o ) δ + u s u o ∫ u + 0 d w H ( w ) . (120)</formula>
<text><location><page_27><loc_16><loc_84><loc_76><loc_88></location>This approximation is consistent with (13) [and therefore (113) as well]. In terms of the matrices appearing in (111), α ≈ 0, -˙ α ≈ β ≈ δ , and</text>
<formula><location><page_27><loc_27><loc_81><loc_76><loc_84></location>˙ β ≈ ∫ u + 0 d w H ( w ) . (121)</formula>
<text><location><page_27><loc_16><loc_76><loc_76><loc_80></location>Note that Eqs. (119) and (120) should be applied with care if H involves many oscillations of an approximately periodic function. In these cases, the integral of H can be very nearly zero. Terms nonlinear in H might then be significant.</text>
<text><location><page_27><loc_16><loc_71><loc_76><loc_76></location>Assuming that (119) and (120) are indeed adequate approximations for A and B , a coordinate rotation may always be used to diagonalize ˙ β . There then exist two constants H 1 and H 2 such that</text>
<text><location><page_27><loc_16><loc_63><loc_76><loc_67></location>In this sense, all sufficiently short gravitational plane waves act as though they are linearly polarized [so β = χ = 0 in (90)]. In terms of the individual wavefunctions appearing in (6),</text>
<formula><location><page_27><loc_27><loc_67><loc_76><loc_71></location>˙ β = ( -H 1 0 0 -H 2 ) . (122)</formula>
<formula><location><page_27><loc_27><loc_59><loc_76><loc_63></location>H 1 = ∫ u + 0 d w [ h ‖ ( w ) + h + ( w )] , (123)</formula>
<formula><location><page_27><loc_27><loc_56><loc_76><loc_59></location>H 2 = ∫ u + 0 d w [ h ‖ ( w ) -h + ( w )] . (124)</formula>
<text><location><page_27><loc_16><loc_54><loc_60><loc_56></location>The transverse coordinates x i have also been chosen such that</text>
<formula><location><page_27><loc_28><loc_51><loc_76><loc_54></location>∫ u + 0 d wh × ( w ) = 0 . (125)</formula>
<text><location><page_27><loc_16><loc_47><loc_76><loc_50></location>If a wave satisfies the vacuum Einstein equation, h ‖ = 0 and H 1 = -H 2 . More generally, it follows from the weak energy condition that</text>
<formula><location><page_27><loc_27><loc_44><loc_76><loc_47></location>H 1 + H 2 ≥ 0 . (126)</formula>
<text><location><page_27><loc_16><loc_42><loc_76><loc_45></location>Now assume that the integral of h + is non-negative, which entails only a minimal loss of generality. Then,</text>
<formula><location><page_27><loc_27><loc_39><loc_76><loc_41></location>H 1 > 0 , H 1 ≥ |H 2 | . (127)</formula>
<text><location><page_27><loc_16><loc_32><loc_76><loc_39></location>We say that a wave is gravity-dominated if H 2 < 0 and matter-dominated if H 2 > 0. These definitions are closely analogous to those used to classify symmetric plane waves in Sect. 5. There, a wave was said to be gravity- or matter-dominated depending on the sign of the constant h 2 = h ‖ -h + appearing in (94).</text>
<text><location><page_27><loc_16><loc_29><loc_76><loc_33></location>For weak gravity-dominated wavepackets, there can be at most one phase conjugate to an observer satisfying u o > u + . If this exists, it evident from (120) that</text>
<formula><location><page_27><loc_27><loc_25><loc_76><loc_29></location>τ -1 ( u o ) = -u o H 1 u o -1 (128)</formula>
<text><location><page_27><loc_16><loc_22><loc_76><loc_25></location>is conjugate to u o with multiplicity 1. This equation is valid only if τ -1 ( u o ) < 0. A conjugate phase therefore exists only for observers satisfying</text>
<formula><location><page_27><loc_27><loc_19><loc_76><loc_22></location>H 1 u o > 1 . (129)</formula>
<text><location><page_27><loc_16><loc_17><loc_76><loc_19></location>Note that H 1 ¯ u o = 1 is the unique physically-relevant solution to (116) in the gravitydominated case.</text>
<text><location><page_27><loc_16><loc_13><loc_76><loc_16></location>Waves that are matter-dominated (so H 2 > 0) also admit the conjugate phase (128) when u o satisfies (129). In the conformally-flat case where H 1 = H 2 , this is the</text>
<figure>
<location><page_28><loc_17><loc_75><loc_75><loc_88></location>
<caption>Figure 4: Example emission versus observation times in the presence of weak wavepackets. The left figure assumes a vacuum wave, while the right figure is a matterdominated wave with H 2 = 4 H 1 / 5. Dotted lines represent τ -1 ( u o ) and τ -2 ( u o ). The wave's location is indicated schematically by a thin grey rectangle. The source and observer are placed on geodesics assumed to be comoving after the wave has passed. In both cases, κ s = 1, H 1 δx 1 o = H 1 δx 2 o = 1 / 2, and H 1 ( ˙ γ o · δ x o -δv o ) = 3.</caption>
</figure>
<text><location><page_28><loc_16><loc_56><loc_76><loc_61></location>only conjugate phase. Unlike in the gravity-dominated case, the multiplicity of τ -1 is equal to 2 for conformally-flat waves. In all other matter-dominated cases, τ -1 has multiplicity 1 and a second conjugate phase is admitted (also with multiplicity 1) for all observers satisfying</text>
<formula><location><page_28><loc_27><loc_53><loc_76><loc_55></location>H 2 u o > 1 . (130)</formula>
<formula><location><page_28><loc_27><loc_47><loc_76><loc_51></location>τ -2 ( u o ) = -u o H 2 u o -1 . (131)</formula>
<text><location><page_28><loc_16><loc_51><loc_26><loc_53></location>This occurs at</text>
<text><location><page_28><loc_16><loc_45><loc_76><loc_48></location>Note that (130) is a more stringent condition than (129). As implied by the notation, τ -2 ( u o ) < τ -1 ( u o ).</text>
<text><location><page_28><loc_16><loc_42><loc_76><loc_45></location>Consider a point source moving on a timelike worldline Γ in a weak sandwich wave spacetime. Such a source appears differently when observed at different times. It is</text>
<text><location><page_28><loc_35><loc_38><loc_36><loc_40></location>ω</text>
<figure>
<location><page_28><loc_32><loc_25><loc_59><loc_39></location>
<caption>Figure 5: Frequency shifts for a vacuum wavepacket. The parameters and dashing used here are the same as those in the left panel of Fig. 4. The younger image (solid) experiences a temporary blueshift and then a redshift before settling down to ω o = ω e . The slight initial blueshift as u o → 0 + is due to the source and observer moving towards each other before interacting with the wave. The older image (dashed) initially appears with infinite blueshift. It then suffers an ever-increasing redshift.</caption>
</figure>
<text><location><page_28><loc_56><loc_24><loc_57><loc_27></location>H</text>
<figure>
<location><page_29><loc_32><loc_75><loc_60><loc_88></location>
<caption>Figure 6: Bending of emission curves for a vacuum wavepacket. All parameters are the same as in the left panel of Fig. 4 except that three different choices are made for the value of δv o . Curves for both images are shifted to the right as δv o is decreased. The younger (solid) curves are almost unaffected by the wave if δv o is sufficiently large. For smaller values of δv o , both curves are strongly bent by the constraint that they can't pass through the dotted curve representing τ -1 .</caption>
</figure>
<text><location><page_29><loc_16><loc_48><loc_76><loc_61></location>clear that when u o < 0, exactly one image of Γ is viewable under generic conditions. As time passes, the wave eventually passes through the observer. A second image then appears when u o = H -1 1 > u + . This image is always emitted before the first. If the wave involves a sufficient amount of Ricci curvature + (from e.g., electromagnetic plane waves) and is not conformally-flat, a third image appears when u o = H -1 2 > H -1 1 . This is emitted before the first two images. All images persist indefinitely once they appear. Sufficiently far in the future, one image is observed of the source as it appeared after interacting with the gravitational wave. All other images predate this interaction. See Fig. 4.</text>
<text><location><page_29><loc_16><loc_34><loc_76><loc_48></location>When the second image first appears at u o = H -1 1 , the new conjugate phase τ -1 ( u o ) is divergent. At all later times, it is finite. The same is also true for the emission times associated with the second image. Almost the entire past history of the source is therefore observable within a finite proper time. This implies an infinite blueshift. At late times, τ -1 → -H -1 1 . The observed evolution of the source via the second image effectively freezes as u e asymptotes to -H -1 1 (which predates the source's interaction with the wave). Images such as these are highly redshifted, as indicated in Fig. 5. Note that a similar transition from infinite blueshift to infinite redshift also applies to the third image if it exists.</text>
<text><location><page_29><loc_16><loc_19><loc_76><loc_34></location>Another qualitative feature of the emission times plotted in Fig. 4 is that there is a sense in which pairs of images can 'switch roles.' Consider, e.g., the left panel of that figure. At late times, the solid curve (corresponding to the younger image) is perfectly linear. Indeed, it remains very nearly linear until H 1 u o ≈ 5. A rapid transition then occurs where the dashed curve effectively takes over this linear behavior while the solid curve strongly deviates from it. In a sense, the two images reverse their roles. This phenomenon occurs one more time (somewhat less sharply) around H 1 u o ≈ 1 when the second image first appears. It arises essentially because u e > τ -1 for the younger image and u e < τ -1 for the older image. These constraints can cause emission curves to bend sharply - with large accompanying frequency shifts - in order to avoid</text>
<figure>
<location><page_30><loc_18><loc_74><loc_74><loc_88></location>
<caption>Figure 8: Angles (in degrees) for images produced by a vacuum wavepacket with the same parameters as those used in the left panel of Fig. 4. θ measures the angle between both images, while ψ measures the angle between each image and glyph[lscript] a on an observer's sky.</caption>
</figure>
<text><location><page_30><loc_41><loc_73><loc_42><loc_75></location>H</text>
<text><location><page_30><loc_70><loc_73><loc_71><loc_75></location>H</text>
<paragraph><location><page_30><loc_16><loc_70><loc_76><loc_73></location>Figure 7: Angular diameter and luminosity distances for a vacuum wavepacket. All parameters are the same as those used in the left panel of Fig. 4.</paragraph>
<text><location><page_30><loc_16><loc_61><loc_76><loc_67></location>intersecting τ -1 . Whether or not this occurs depends on whether the 'average' linear increase of u e ever comes near τ -1 . If it does, this role switching occurs. If not, the younger image is barely affected by the gravitational wave at all. This is illustrated in Fig. 6, where emission curves for several sources are plotted simultaneously.</text>
<text><location><page_30><loc_16><loc_47><loc_76><loc_61></location>To summarize, images which appear at discrete times briefly appear as bright, highly blueshifted 'flashes.' Indeed, Fig. 7 shows that their luminosity distances go to zero. Simultaneously, the angular diameter distance of each new image tends to infinity. It is implied by (70) that all highly blueshifted images make must a very small angle ψ with glyph[lscript] a on the observer's sky. The second (and third) images therefore appear aligned with the direction of propagation of the gravitational wave when they first appear. This direction could be quite different from the location of the other image(s). As time progresses, all images migrate across the observer's sky as illustrated in Fig. 8. Different images may remain separated from each other by large angles at all times.</text>
<section_header_level_1><location><page_30><loc_16><loc_44><loc_26><loc_45></location>7. Discussion</section_header_level_1>
<text><location><page_30><loc_16><loc_38><loc_76><loc_42></location>Despite their simplicity, plane gravitational waves behave in qualitatively different ways from lenses associated with quasi-Newtonian mass distributions. As expected from their dynamic nature, plane waves generically shift the observed frequencies of</text>
<figure>
<location><page_30><loc_17><loc_22><loc_75><loc_36></location>
</figure>
<text><location><page_30><loc_42><loc_21><loc_43><loc_23></location>H</text>
<text><location><page_30><loc_71><loc_21><loc_72><loc_23></location>H</text>
<text><location><page_31><loc_16><loc_85><loc_76><loc_88></location>various images. They may also admit images which appear to move, deform, change brightness, and shift color as time progresses.</text>
<text><location><page_31><loc_16><loc_65><loc_76><loc_85></location>More subtle differences relate to the number of images that are produced of a given source. For example, even numbers of images can appear generically (which has led plane wave spacetimes to be cited [1] as well-behaved examples where the odd number theorem [5, 7] does not apply). Some plane waves can even produce an infinite number of discrete images. Perhaps most striking of all are the bright flashes shown to be produced by generic sandwich waves in Sect. 6. These correspond to individual images which appear at discrete times. More typical gravitational lenses can produce new images if a source crosses an observer's caustic. Individual images then split into two (or vice versa). The flashes produced by sandwich waves are quite different. Their appearance does not require that a source pass through an observer's caustic. Such images appear individually from the infinitely distant past. Initially, they are infinitely bright and infinitely blueshifted points of light appearing in the direction of propagation associated with the gravitational wave.</text>
<text><location><page_31><loc_16><loc_49><loc_76><loc_65></location>Many of these effects depend at least partially on the idealization that a plane wave extends undiminished to infinitely-large transverse distances. Plane wave spacetimes are not asymptotically flat. Despite being topologically trivial and locally well-behaved, they are not even globally hyperbolic: Null geodesics passing between appropriately-chosen pairs of points can extend to arbitrarily large transverse distances in between those points. It is this property which permits the infinite number of images described in Sect. 5 to be produced by symmetric waves. The flashes described in Sect. 6 also depend on the spacetime structure at arbitrarily large distances. This structure likely affects the formation of even numbers of images as well. Indeed, the usual proofs of the odd number theorem require global hyperbolicity, among other assumptions [1, 5] (see, however, [7] for a more general formulation).</text>
<text><location><page_31><loc_16><loc_33><loc_76><loc_48></location>If a spacetime has the geometry of a plane wave only out to some finite transverse distance, all results derived here remain valid if the associated images involve light rays which never extend sufficiently far to interact with any large-distance modifications. The infinite sequence of images formed by a symmetric plane wave would then be expected to become finite for spacetimes which are only approximately plane waves. Calculations involving the oldest images could no longer be trusted in these cases. Similarly, the bright flashes associated with ideal sandwich waves are likely to be somewhat less extreme for waves which decay at infinity. Large brightnesses and large blueshifts can still exist, but these will be cut off at some finite maximum. Such maxima may, however, remain quite large.</text>
<text><location><page_31><loc_16><loc_29><loc_76><loc_33></location>It is reasonably clear that modifications of the geometry at large distances can remove some images. Less obviously, these modification can also introduce new images. Consider, for example, the pp -wave spacetimes obtained by substituting</text>
<formula><location><page_31><loc_27><loc_26><loc_76><loc_28></location>H ij ( u ) x i x j → H ( u, x ) (132)</formula>
<text><location><page_31><loc_16><loc_14><loc_76><loc_26></location>in the metric (1). These generalize the plane wave spacetimes. It has been shown that if H grows subquadratically as | x | → 0, the resulting geometries are globally hyperbolic [45]. Moreover, every pair of points is connected by at least one geodesic in these cases (unlike in pure plane wave spacetimes where the growth of H is precisely quadratic). This implies that modifications of the geometry at large transverse distances can introduce new null geodesics even between points at small transverse distances. It would be interesting to explore these effects in more depth to understand precisely how modifications of this sort (or more general ones) alter the lensing properties</text>
<text><location><page_32><loc_16><loc_82><loc_76><loc_88></location>described here for ideal plane waves. It would also be interesting to better understand what the transient flashes of Sect. 6 imply for waves propagating on plane (or almostplane) wave spacetimes. This can likely be facilitated by the Green functions derived in [23].</text>
<section_header_level_1><location><page_32><loc_16><loc_79><loc_24><loc_80></location>References</section_header_level_1>
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</document>
[
{
"title": "Abraham I. Harte",
"content": "Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut Am M¨uhlenberg 1, D-14476 Golm, Germany E-mail: harte@aei.mpg.de Abstract. Plane-symmetric gravitational waves are considered as gravitational lenses. Numbers of images, frequency shifts, mutual angles, and image distortion parameters are computed exactly in essentially all non-singular plane wave spacetimes. For a fixed observation event in a particular plane wave spacetime, the number of images is found to be the same for almost every source. This number can be any positive integer, including infinity. Wavepackets of finite width are discussed in detail as well as waves which maintain a constant amplitude for all time. Short wavepackets are found to generically produce up to two images of each source which appear (separately) only some time after the wave has passed. They are initially infinitely bright, infinitely blueshifted images of the infinitely distant past. Later, these images become dim and acquire a rapidly-increasing redshift. For sufficiently weak wavepackets, one such 'flash' almost always exists. The appearance of a second flash requires that the Ricci tensor inside the wave exceed a certain threshold. This might occur if a gravitational plane wave is sourced by, e.g., a sufficiently strong electromagnetic plane wave.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The theory of gravitational lensing has by now reached a considerable degree of sophistication [1, 2, 3, 4]. Theorems have been found predicting (or bounding) the number of images in very general systems [5, 6, 7]. Shapes of stable caustics have been exhaustively classified [2, 3, 8, 9], a non-perturbative notion of the lens map has been obtained [10], and various universal behaviors of images have been found for sources lying near caustics [11, 12, 13]. These general results have been complemented by a number of detailed calculations for specific types of lenses. The majority of such calculations have been performed within the quasi-Newtonian viewpoint of gravitational lensing commonly used in astrophysics [3]. While various flavors of this formalism exist, most require that bending angles be small and that all lenses be nearly-Newtonian mass distributions. This is to be contrasted with the more fundamental picture of gravitational lensing where light rays are modelled as null geodesics in a Lorentzian spacetime. Within this context, specific lensing calculations have been performed in Kerr, ReissnerNordstrom, and a handful of other geometries [1]. While curvatures and bending angles may be large in these examples, they all involve (at least conformally) stationary spacetimes. It is of interest to understand if qualitatively new effects appear in dynamical cases. This paper considers the bending of light by (decidedly non-stationary) planesymmetric gravitational waves. Gravitational lensing by gravitational waves has previously been considered by a number of authors, although almost all of this work has been carried out within the weak-field regime [14, 15, 16, 17, 18]. One exception is [19], where redshifts were computed in an exact solution to the vacuum Einstein equation representing a plane gravitational wave. This work considered only very specific waveforms, and was confined to a coordinate patch too small to include caustics and other effects associated with the formation of multiple images. Separately, extensive work has been devoted to non-perturbatively understanding the geodesic structure of generic plane wave spacetimes [20, 21, 22, 23, 24, 25, 26]. This is clearly a subject closely related to gravitational lensing, although few explicit relations between the two subjects appear to have been made (see, however, remarks in [1]). It is the purpose of this work to provide a comprehensive and non-perturbative discussion of lensing in plane wave spacetimes. These geometries are a well-known subclass of pp -waves; plane-fronted waves with parallel rays. Many plane wave spacetimes are exact solutions to the vacuum Einstein equation. Others may be interpreted as, e.g., exact solutions in Einstein-Maxwell theory. While rather idealized from the astrophysical perspective, plane wave spacetimes admit a wide variety of interesting phenomena. Depending on the waveform, any number of astigmatic and anastigmatic caustics may exist. Examples admitting any specified number of images even an infinite number - are easily constructed. In particular, even numbers of images can exist [1]. The image count can also change in time even when a source does not cross an observer's caustic. Despite all of these properties, plane wave spacetimes are geometrically very mild. They are topologically equivalent to R 4 and admit coordinate systems which cover the entire manifold. Aside from their value as models of gravitational radiation, the plane wave spacetimes considered here have also found numerous applications via the Penrose limit. This limit provides a sense in which the metric near any null geodesic in any spacetime is equivalent to the metric of an appropriate plane wave spacetime [28, 29]. It allows problems in relatively complicated spacetimes to be reduced to equivalent problems in plane wave spacetimes (which are often much simpler). This has been particularly valuable within string theory and related fields [30, 31]. Penrose limits have also been applied to ordinary quantum field theory in order to investigate causality and effective indices of refraction for photons and gravitons propagating in curved spacetimes [24, 25, 32]. More recently, Penrose limits were used to deduce the effect of caustics on Green functions associated with the propagation of classical fields in curved spacetimes [23]. Given the content of the Penrose limit, lensing in plane wave spacetimes might be related to lensing in generic spacetimes as seen by ultrarelativistic observers. We make no attempt to justify this conjecture, however. This paper starts by providing a self-contained review of plane wave spacetimes in Sect. 2. Although most of this material is not new [20, 21, 22, 23, 24, 25, 26, 27], it is not widely known. Sect. 3 then derives the number of images of a point source that may be viewed in plane wave spacetimes. Under generic conditions, this is found to depend only on the waveform and a certain time parameter associated with the observation event. The number of images does not depend on any properties of the source. Once this is established, Sect. 4 computes image positions, frequency shifts, angles, and image distortion parameters in general plane wave spacetimes. Sect. 5 applies these results to symmetric plane waves, which have constant waveforms. These geometries produce an infinite number of images of almost every source. Their lensing properties are found to change significantly if the Ricci tensor is increased beyond a certain threshold. Lastly, Sect. 6 discusses 'sandwich waves;' wavepackets with finite width. These spacetimes generically admit images which appear at discrete times and then persist indefinitely. Such images initially provide infinitely blueshifted, infinitely bright pictures into the infinitely distant past. Very quickly, however, such images become highly redshifted and effectively fade away. One of these 'transient flashes' is produced by almost every sufficiently weak vacuum (Ricci-flat) wave. For weak waves, a second flash appears only if the Ricci tensor of the wavepacket exceeds a certain threshold. Throughout this work, the spacetime is assumed to be everywhere transparent. The language used also assumes that the geometric optics approximation [3] holds even in situations where it would be severely strained (such as when light is emitted near an observer's caustic).",
"pages": [
1,
2,
3
]
},
{
"title": "Notation",
"content": "This paper restricts attention only to plane wave spacetimes in four spacetime dimensions. Our sign conventions follow those of Wald [33]. The signature is -+++. Latin letters a, b, . . . (and occasionally A,B,... ) from the beginning of the alphabet are used to denote abstract indices. Greek letters µ, ν, . . . are used to denote fourdimensional coordinate indices. The Latin letters i, j, . . . are instead coordinate indices associated with the two directions transverse to the direction of wave propagation. Objects involving the latter type of index are often written in boldface with all indices suppressed. They are then manipulated using the standard notation of linear algebra [e.g., A ki B kj = ( A ᵀ B ) ij and | x | = √ x i x i ]. Overall, notation related to plane wave spacetimes closely follows the conventions of [23].",
"pages": [
3
]
},
{
"title": "2. Geometry of plane wave spacetimes",
"content": "Plane wave geometries may be interpreted as simple models for gravitational waves emitted from distant sources. Alternatively, they arise as universal limits for the geometries near null geodesics in any spacetime [28, 29]. The typical definition of a plane wave spacetime ( M,g ab ) requires that M = R 4 and that there exist global coordinates ( u, v, x ) = ( u, v, x 1 , x 2 ) : M → R 4 such that the line element takes the form H ij = ( H ) ij is any symmetric 2 × 2 matrix. Its components describe the waveforms associated with a wave's three polarization states ‡ . The u coordinate is interpreted as a phase parameter for the wave, while v affinely parametrizes its rays. The remaining two coordinates x i span spacelike wavefronts transverse to the wave's direction of propagation. Note that if H = 0 in some region, the spacetime is locally flat there. In terms of a Minkowski coordinate system ( t, x 1 , x 2 , x 3 ), u and v satisfy u = ( t + x 3 ) / √ 2 and v = ( t -x 3 ) / √ 2 in such a region. We consider only nontrivial plane waves, so H cannot vanish everywhere. The physical interpretation of the so-called Brinkmann metric (1) as a planesymmetric gravitational wave follows from considering the integral curves of the vector field glyph[lscript] a = ( ∂/∂v ) a . These curves form a null geodesic congruence which may be interpreted as the rays of the gravitational wave. ∇ a glyph[lscript] b = 0, so these rays have vanishing expansion, shear, and twist. There is therefore a sense in which they are everywhere parallel to one another. All rays are also orthogonal to the family of spacelike 2-surfaces generated by the two commuting spacelike vector fields X a ( i ) = ( ∂/∂x i ) a . The induced metric on each such surface is flat: The wavefronts are 2-planes. The curvature is constant on these planes in the sense that the X a ( i ) are curvature collineations: Despite this, the X a ( i ) are not everywhere Killing. There do, however, exist linear combinations of X a ( i ) and glyph[lscript] a which are Killing. glyph[lscript] a itself is also Killing, which may be interpreted as a statement that plane waves do not deform along their characteristics. Plane wave spacetimes admit a minimum of five linearly independent Killing fields. Note that in flat spacetime, five (out of the total of ten) Killing fields are symmetries of all electromagnetic plane waves [27]. Killing fields of plane wave spacetimes are discussed more fully in Sect. 2.3. All non-vanishing coordinate components of the Riemann tensor may be determined from It follows from this that the Ricci tensor is where Tr denotes the ordinary (Euclidean) trace of the 2 × 2 matrix H . The Ricci scalar always vanishes in plane wave spacetimes. More generally, there are no nonzero scalars formed by local contractions of the metric, the curvature, and its derivatives: R ab R ab = R abcd R abcd = R abcd glyph[epsilon1] abfh R fh cd = . . . = 0. This is analogous to the fact that plane electromagnetic waves satisfy, e.g., F ab F ab = glyph[epsilon1] abcd F ab F cd = 0. Note, however, that electromagnetic plane waves are not the only electromagnetic fields with vanishing field scalars. Similarly, plane wave spacetimes are not the only curved geometries with vanishing curvature scalars [35]. It follows from (4) that plane wave spacetimes satisfying the vacuum Einstein equation (and the vacuum equations of many alternative theories of gravity [36]) are characterized by the simple algebraic constraint Tr H = 0. For vacuum waves, there exist two scalar functions h + and h × such that h + and h × describe the waveforms for the two polarization states of a gravitational plane wave propagating in vacuum. A plane wave is said to be linearly polarized if h + and h × are linearly dependent (in which case one of these functions can be eliminated by a suitable rotation of the transverse coordinates x ). If h + and h × have compact support, the geometry is said to be a sandwich wave. This name evinces the image of a curved region of spacetime 'sandwiched' between null hyperplanes in a geometry that is otherwise Minkowski. Physically, it corresponds to a wavepacket of finite length. Note that the planar symmetry considered here is very special in the sense that passing waves do not necessarily leave any 'tail' behind them. After interacting with a sandwich wave, all observers enter a region of spacetime which is perfectly flat. There is a sense in which test fields propagating on plane wave spacetimes also have no tails [23, 37]. A general (not necessarily vacuum) wave profile H may be built by adding to (5) a term proportional to the identity matrix δ . There then exists a third polarization function h ‖ such that If the Ricci tensor of such a wave is associated with a stress-energy tensor via Einstein's equation, that stress-energy tensor obeys the weak energy condition if and only if h ‖ ≥ 0. Assuming this, the stress-energy tensors associated with (6) are very simple. They could be generated by, e.g., electromagnetic plane waves with the form Alternatively, (6) could be associated with the stress-energy tensor of the massless Klein-Gordon plane wave Besides the vacuum case h ‖ = 0, another interesting class of wave profiles are those that are conformally flat. These satisfy h + = h × = 0, so H ∝ δ . As gravitational lenses, all caustics of conformally-flat plane waves are associated with 'perfect' anastigmatic focusing. For more general plane waves, caustics are typically (but not necessarily) associated with astigmatic focusing.",
"pages": [
3,
4,
5
]
},
{
"title": "2.1. The matrices A and B",
"content": "The geometry of plane wave spacetimes has been analyzed in detail by a number of authors [20, 21, 22, 23, 24, 25, 26, 27]. One essential conclusion of this work is that nearly all interesting properties of plane wave spacetimes may be deduced from the properties of 2 × 2 matrices E = E ( u ) satisfying the differential equation This is a 'generalized oscillator equation' with -H acting like a matrix of squared frequencies. Eq. (9) arises when solving for geodesics or Jacobi fields in plane wave spacetimes. Bitensors such as Synge's function and the parallel propagator may be written explicitly in terms of its solutions. The same is also true for a plane wave's Killing vectors. It is convenient to write all possible matrices E in terms of two particular solutions. Fix any § u o ∈ R and define A ( · , u o ) and B ( · , u o ) to be solutions to (9) (where derivatives are applied to the first arguments of A and B ) with the initial conditions Here, the notation ∂ (1) A indicates a partial derivative with respect to the first argument of A . We assume for simplicity that H is a matrix of piecewise-continuous functions and that A and B are at least C 1 (and piecewiseC 2 ) in both of their arguments. Example expressions for A and B are discussed in Sects. 5 and 6. Given any two solutions E 1 and E 2 to (9), it is easily verified that their Wronskian is conserved: Here, ᵀ denotes a matrix transpose. Applying this formula with E 1 → A and E 2 → B shows that Neither A nor B are necessarily symmetric matrices. Nevertheless, (12) and (13) may be used to show that the products are symmetric wherever they exist [23]. Letting E 1 → B ( · , u o ) and E 2 → B ( · , u s ) in (12) shows that for all u s , u o ∈ R . This is essentially Etherington's reciprocity law [1, 3, 38, 39]. A similar calculation may be used to show that as well. It is sometimes useful to consider partial derivatives ∂ (2) with respect to the second arguments of A and B . The resulting matrices remain solutions to (9). Comparing initial conditions shows that A may therefore be derived from B . The opposite is also true wherever det H = 0. Geometrically, B corresponds to the transverse coordinate components of a Jacobi propagator describing the evolution of deviation vectors along geodesics passing between different pairs of points [23]. B is also related to image distortion. Up to an overall time dilation factor, it translates small differences in image position on an observer's sky to spatial deviations from a fiducial source point. In the language of [11], B is proportional to the Jacobi map. The symmetric matrix ∂ (1) BB -1 plays a similar role, but translates source separations to emission (rather than observation) angles. It is proportional to an object typically referred to as the optical deformation matrix. These statements are explained more fully in Sect. 4.3. glyph[negationslash]",
"pages": [
5,
6
]
},
{
"title": "2.2. Conjugate pairs",
"content": "It is often useful when working with plane wave spacetimes to consider hypersurfaces of 'constant phase.' Recalling the interpretation of the u coordinate as a phase, let glyph[negationslash] denote such a hypersurface. Given two points p s ∈ S u s and p o ∈ S u o (with u s = u o ), define the multiplicity or 'index' of these points to be [20] glyph[negationslash] We also set I ( p o , p s ) = 0 whenever u o = u s . It follows from (15) that I ( p s , p o ) = I ( p o , p s ). For most pairs of points, I = 0. Such pairs are said to be 'disconjugate.' Pairs p s , p o satisfying I ( p s , p o ) = 0 are instead said to be conjugate with multiplicity I ( p s , p o ). Similarly, we call the pairs S u s , S u o 'conjugate hyperplanes' and the pairs of real numbers u s , u o 'conjugate phases' when rank B ( u s , u o ) < 2. Despite the appearance of (19), the index map I : M × M →{ 0 , 1 , 2 } describes phenomena which do not depend on any choice of coordinate system. Conjugate pairs as described here are closely related to the conjugate points commonly considered in differential geometry and optics. In general, distinct points p s and p o on a given geodesic are said to be conjugate if and only if there exist nontrivial deviation vectors along that geodesic which vanish at both p o and p s . In plane wave spacetimes, this condition reduces to I ( p s , p o ) > 0. Defining the multiplicity of a pair of conjugate points to be the number of linearly independent deviation vectors which vanish at those points, that multiplicity is equal to I ( p s , p o ). The concept of conjugacy associated with I does not, however, require the specification of any particular geodesic. It is uniquely defined even for pairs of points connected by multiple geodesics or by none. Indeed, these are the only cases where I = 0. All strong lensing effects associated with plane wave spacetimes are related to the existence of conjugate hyperplanes. It follows from (19) that every pair of conjugate phases u s , u o satisfies glyph[negationslash] Finding conjugate pairs and their multiplicities may be viewed as a matter of direct computation once H is specified. Alternatively, various Sturm-type comparison theorems can be used to make general statements regarding the existence and separations of conjugate pairs for various classes of plane wave. See, e.g., Chapt. XI of [40] for results relating to mathematical problems of this type and [20] for an application to 'tame' plane wave spacetimes. More specific examples are discussed in Sects. 5 and 6 below. The qualitative structure of geodesics in plane wave spacetimes is closely related to the index I . First note that every disconjugate pair of points p s , p o is connected by exactly one geodesic. If two points are conjugate, the number of connecting geodesics is either zero or infinity. Sizes of geodesically connected regions may be summarized by A similar relation exists for null cones when u o = u s : glyph[negationslash] dim[(all points connected to p o via null geodesics) ∩ S u s ] + I ( p s , p o ) = 2 . (22) The latter result has been referred to as an 'index theorem' in [20]. Eq. (21) implies that geodesics emanating from p o and intersecting a hyperplane S u s that is disconjugate to S u o form a 3-dimensional region. Indeed, these geodesics fill the entire hyperplane. More interestingly, geodesics intersecting a conjugate hyperplane with multiplicity 1 fill only a 2-dimensional region on that 3-dimensional surface. Geodesics intersecting a hyperplane with multiplicity 2 form a line. Similarly, the null cone of a point reduces to a 1-dimensional curve on every hyperplane with multiplicity 1. A null cone intersecting a hyperplane with multiplicity 2 is focused to a single point on that hyperplane. These two cases correspond to astigmatic and anastigmatic focusing, respectively. Anastigmatic focusing tends to be unstable in the sense that perturbations tend to split a single multiplicity 2 phase into two closely-spaced phases each with multiplicity 1. In many applications, there exists a preferred point p o , or perhaps a preferred hyperplane S u o . p o may, for example, represent the position of an observer at a particular time. Fixing this point, the set of hyperplanes conjugate to S u o divides a plane wave spacetime into a (possibly infinite) number of open regions N n ( u o ). Let N 0 ( u o ) denote the largest connected region containing S u o and excluding any portion of a hyperplane conjugate to S u o . If there exists a smallest τ 1 ( u o ) > u o conjugate to u o , the surface S τ 1 ( u o ) is clearly contained in the boundary of N 0 ( u o ). N 1 ( u o ) may then be defined as the largest connected region which includes S τ 1 ( u o ) as a boundary and contains points p s satisfying u s > τ 1 ( u o ) and I ( p s , p o ) = 0. This continues the spacetime 'above' S τ 1 ( u o ) . Similar constructions may be used to define τ n ( u o ) and N n ( u o ) for values of n other than 1. See Fig. 1. Fixing a particular nonzero integer n and real number u o , it is not necessary that τ n ( u o ) exist at all. In general, the domain of τ n is an open subset of R . This domain can be empty for some n . glyph[negationslash] The geodesic uniqueness result described above can now be reduced to the statement that a point p o is connected to another point p s = p o by exactly one geodesic if and only if there exists some n such that p s ∈ N n ( u o ). Two-point tensors like Synge's world function, the parallel propagator, and the van Vleck determinant may be defined unambiguously throughout ' M × ( ∪ n N n ).' This excludes from M × M only a set of measure zero.",
"pages": [
7,
8
]
},
{
"title": "2.3. Geodesics",
"content": "Beyond the qualitative geodesic structure of plane wave spacetimes discussed above, it is not difficult to obtain explicit coordinate expressions for all geodesics. Let Γ ⊂ M denote some geodesic and γ : R → M an affine parametrization of it. The vector field glyph[lscript] a = ( ∂/∂v ) a generating the characteristics of the gravitational wave is Killing, so ˙ γ a glyph[lscript] a must be constant on Γ. If this constant vanishes, Γ is confined to a hypersurface of constant phase. Such a geodesic has the form of a (Euclidean) straight line in the coordinates ( v, x ). glyph[negationslash] Geodesics satisfying ˙ γ a glyph[lscript] a = 0 are more interesting. In these cases, the affine parameter can always be rescaled such that ˙ γ a glyph[lscript] a = -1. It is then possible to identify that parameter with the phase coordinate u . Doing, so u ( γ ( u s )) = u s for all u s ∈ R . The spatial components γ := x ( γ ) of any geodesic are fixed everywhere once γ o := γ ( u o ) and ˙ γ o := ˙ γ ( u o ) have been specified at some fiducial phase u o . In terms of the matrices A and B defined in Sect. 2.1, The associated v coordinate of Γ may be efficiently derived using that fact that the vector field (2 v∂ v + x i ∂ i ) a is a homothety [23]. This implies that where is a constant. κ s is closely related to the conserved quantity on Γ associated with the Killing field glyph[lscript] a . The (unit) 4-velocity U a s tangent to Γ is related to ˙ γ a via It follows that If the spacetime is nearly flat (so H ≈ 0), κ s reduces to a particle's specific energy minus its specific momentum in the direction of the gravitational wave. The limit -glyph[lscript] a U a → 0 (or κ s → ∞ ) may therefore be interpreted as ultrarelativistic motion in the direction of the gravitational wave. By contrast, the limit -glyph[lscript] a U a → ∞ (or κ s → 0) corresponds to ultrarelativistic motion against the background wave. Other Killing fields present in essentially all plane wave spacetimes may be written as where Ξ i = Ξ i is any 2-vector with the form For each choice of u o , there exists a four-parameter family of such vector fields. Each of these is associated with a conservation law. Two such conserved quantities may be summarized by Fixing any u o , this 2-vector is conserved in the sense that it is independent of u s . Another conserved 2-vector may be defined by In the weak-field limit, P corresponds to the specific momentum transverse to the gravitational wave. In this same context, C may be interpreted as the conserved quantity associated with boosts transverse to the gravitational wave. It constrains transverse displacements. Note that both P and C depend on a choice of u o . This is analogous to the choice of origin necessary to define angular momentum in elementary mechanics. In stationary spacetimes, it is common to discuss various quantities related to gravitational lensing in terms of stationary observers (and often stationary sources). While plane wave spacetimes are not stationary, there does exist sufficient symmetry to define similarly preferred sources and observers. Two geodesics Γ and Γ ' can be said to be 'instantaneously comoving' at u = u o when These conditions imply that the 4-velocities of both geodesics are parallel-transported versions of each other on the constant-phase hyperplane S u o . Note, however, that geodesics which are comoving at one phase are not necessarily comoving at any other phase. As implied by (21), bundles of geodesics are strongly focused on conjugate hyperplanes. Consider such a hyperplane associated with a phase τ n ( u o ) conjugate to u o with multiplicity 1. It is clear from (19) that ˆ B n ( u o ) := B ( τ n ( u o ) , u o ) is a matrix with rank 1. There therefore exists a unit 2-vector ˆ q n ( u o ) such that ˆ q n ( u o ) is unique up to sign. Choosing any ˆ p n ( u o ) orthogonal to ˆ q n ( u o ), the transverse spatial coordinates of all geodesics starting at a given point p o focus to the line as they pass through S τ n ( u 0 ) . Here, w is any real number. All v coordinates may be reached on S τ n ( u 0 ) by appropriate geodesics. Geodesics starting at p o and intersecting a hyperplane S τ n ( u o ) with multiplicity 2 all focus to the single transverse position as they pass through S τ n ( u o ) . As in the multiplicity 1 case, all values of v may be reached by appropriate geodesics. Eqs. (34) and (35) illustrate explicitly how conjugate hyperplanes with multiplicities 1 and 2 are associated with astigmatic and anastigmatic focusing, respectively. The former case involves focusing in only one transverse direction, while the latter case involves simultaneous focusing in both directions transverse to the gravitational wave.",
"pages": [
9,
10
]
},
{
"title": "2.4. Distances",
"content": "As noted above, all pairs of points not lying on conjugate hyperplanes are connected by exactly one geodesic. There is therefore no ambiguity in ascribing geodesic distances to these pairs. In particular, Synge's world function is well-defined whenever its arguments do not lie on conjugate hyperplanes. Plane wave spacetimes constitute one of the few examples where σ is known essentially in closed form: This is symmetric in its arguments: σ ( p s , p o ) = σ ( p o , p s ). The appearance of B -1 in (37) indicates that σ tends to diverge when its arguments approach conjugate hyperplanes. More specifically, suppose that u s ≈ τ n ( u o ) and I ( τ n ( u o ) , u o ) = 1. Then, is an asymptotic approximation for σ if the bracketed term on the right-hand side of this equation is nonzero [23]. It follows from (34) that this expression is valid only when there does not exist any geodesic passing from p o to a point on S τ n ( u o ) with the same transverse coordinates as p s . The equivalent result if p s is near a hyperplane S τ n ( u o ) with multiplicity 2 is Again, this is valid only when there does not exist any geodesic passing from p o to a point on S τ n ( u o ) with the same transverse coordinates as p s .",
"pages": [
10,
11
]
},
{
"title": "3. Image counting",
"content": "One of the most basic questions that can be asked regarding a gravitational lens is the number of images that it produces of a particular source. Stated somewhat differently, how many future-directed null geodesics connect a given timelike curve (the source) to a particular spacetime event (the observer at a particular time)? This may be answered using the geodesic structure of plane wave spacetimes summarized above. To fix the notation, let p o denote a fixed observation event and Γ the timelike worldline of a point source. Assume that Γ may be parametrized by an everywhereC 1 function γ : R → M . The phase coordinate u serves as a useful 'quasi-time' [21] for plane wave spacetimes ‖ , so let γ satisfy u ( γ ( u s )) = u s for all u s ∈ R . The source's worldline is not required to be geodesic. This section establishes that under generic conditions, an ideal observer at p o may see exactly one image of Γ from each of the 'epochs' N n ( u o ) described in Sect. 2.2. Somewhat more precisely, there typically exists exactly one future-directed null geodesic from to p o for each n ≤ 0 such that N n ( u o ) exists. See Fig. 2. Recall that the boundaries of N n ( u o ) depend only on the spacetime under consideration, and not at all on the behavior of any particular source. This allows generic bounds to be placed on time delays associated with the various images that may be observed. Recall from Sect. 2.2 that each point in Γ n is connected to p o via exactly one (not necessarily null) geodesic. Synge's function σ ( γ ( u s ) , p o ) is therefore well-defined and explicitly given by (37) for all u s such that γ ( u s ) ∈ Γ n . Consider instead the rescaled function If n ≤ 0, the domain of Σ n ( u s ) is equal to all u s such that γ ( u s ) ∈ Γ n . If n = 0, we additionally suppose that u s < u o for reasons of causality. Images of Γ n produced by a plane gravitational wave correspond to the zeros of Σ n . glyph[negationslash] Recalling the form (37) for σ , it is clear that Σ n depends on Γ as well as the matrices A , B , B -1 , and ∂ (1) B . We have assumed in Sect. 2.1 that A and B are at least C 1 . So is γ . The definition of N n ( u o ) ensures that det B ( u s , u o ) = 0 everywhere Σ n ( u s ) is defined. B -1 is therefore C 1 and Σ n is continuous. Σ n is also monotonic. To see this, note that (9), (13), (14), (37), and (41) may be used to show that Σ n is both continuous and monotonic, so at most one zero can exist for each n . This means that at most one image of a source may reach an observer from each epoch N n ( u o ). Exactly one such image exists if some u ' , u '' are known to satisfy Such bounds are easily established.",
"pages": [
11,
12
]
},
{
"title": "3.1. Lensing between conjugate hyperplanes",
"content": "The simplest case to consider is one where N n ( u o ) lies 'in between' hyperplanes conjugate to S u o . Suppose that both τ n ( u o ) and τ n -1 ( u o ) exist for some n < 0. This is true in Fig. 2 for n = -1. More generally, (38) and (39) imply that if there are no geodesics connecting p o to either γ ( τ n ( u o )) or γ ( τ n -1 ( u o )), It follows from these limits together with continuity that Σ n is surjective on R . Since this function is also monotonic, there must exist exactly one emission phase u e ∈ ( τ n -1 ( u o ) , τ n ( u o )) such that p o and γ ( u e ) ∈ Γ n are connected by a null geodesic. Under the same assumptions, projectiles moving on timelike geodesics may be thrown from γ ( u s ) ∈ Γ n to p o only if u s < u e . Choosing u s -τ n -1 ( u o ) to be sufficiently small (but positive), these projectiles can require an arbitrarily large amount of proper time to intersect p o . It is somewhat curious that points γ ( u s ) ∈ Γ n satisfying u s > u e cannot be connected to p o by any causal geodesic. Such points may, however, be reached by suitably accelerated curves which are everywhere causal.",
"pages": [
12,
13
]
},
{
"title": "3.2. The youngest image",
"content": "Next, consider the case n = 0 when there exists at least one conjugate hyperplane in the observer's past (as occurs in the example illustrated by Fig. 2). The arguments given above imply that if γ ( τ -1 ( u o )) is geodesically disconnected from p o , The other boundary of the domain of Σ 0 occurs at u s = u o . Here, This limit clearly tends to + ∞ if γ ( u o ) = x o . Physically, γ ( u o ) = x o implies that p o cannot be connected to γ ( u o ) by any null geodesic. Assuming that this is true, there must exist exactly one u e ∈ ( τ -1 ( u o ) , u o ) such that p o and γ ( u e ) ∈ Γ 0 are connected by a null geodesic. glyph[negationslash] glyph[negationslash]",
"pages": [
13
]
},
{
"title": "3.3. The oldest image",
"content": "The results of Sects. 3.1 and 3.2 typically suffice to describe the images formed in plane wave spacetimes containing an infinite number of conjugate hyperplanes in an observer's past. It is, however, important to consider cases where only a finite number of conjugate points exist. This occurs, e.g., for finite wavepackets where H has compact support. It is also true of linearly polarized vacuum waves that are tame in the sense described in [20]. If there is at least one conjugate hyperplane in the observer's past, let N denote the smallest negative integer such that τ N ( u o ) exists. If there are no conjugate hyperplanes in the observer's past, set N = 0. The case illustrated in Fig. 2 corresponds to N = -2 if N -2 ( u o ) extends into the infinite past. Regardless, we ask whether there exist any future-directed light rays from Γ N to p o . Unlike in the cases considered above, sources spend an infinite amount of proper time in N N ( u o ). Even in flat spacetime, a source that accelerates for an infinitely long time may be causally-disconnected from certain observers via a Rindler horizon. Such phenomena can be ruled out here by supposing that there exists some finite constant κ min > 0 such that for all u s less than some cutoff. As is clear from (25) and (26), the left-hand side of this inequality acts like the square root of a time dilation factor between the coordinate u and the source's proper time. Eq. (47) implies that the source's 4-velocity U a s ∝ ˙ γ a satisfies sufficiently far in the past. It therefore excludes sources which experience arbitrarily large boosts against the background gravitational wave in the distant past. It is satisfied by, e.g., sources whose motion is geodesic sufficiently far into the past. Assuming that κ min exists, it is clear from (42) that u ' may be chosen sufficiently small that glyph[negationslash] Further assuming that γ ( τ N ( u o )) is geodesically disconnected from p o (if N = 0) or that γ ( u o ) is not null-separated from p o (if N = 0), it follows that there exists exactly one u e smaller than τ N ( u o ) (if N = 0) or u o (if N = 0) such that γ ( u e ) ∈ Γ N is connected to p o via a future-directed null geodesic. glyph[negationslash]",
"pages": [
13,
14
]
},
{
"title": "3.4. Total image count for generic sources",
"content": "glyph[negationslash] The results just described may be summarized as follows: Suppose that there does not exist any past-directed geodesic segment from p o to Γ whose endpoints are conjugate (in the usual sense). Also assume that the source and observer are not instantaneously aligned with the background wave: x o = γ ( u o ). If there exists an 'oldest' phase conjugate to u o , further require that there be some κ min > 0 such that the source's motion is bounded by (48) sufficiently far in the past. For each n ≤ 0, these assumptions imply that an observer at p o sees exactly one image of Γ as it appeared in N n ( u o ). This provides a strong bound on the possible emission times of different images. If N N ( u o ) exists for every negative integer N , an infinite number of images are formed. If, however, there is some smallest N ≤ 0 such that N N ( u o ) exists, | N | +1 images appear at p o . Note that these results depend only on the waveform H and the phase coordinate u o associated with the observer. The total number of images is the same for all sources satisfying the hypotheses outlined above. glyph[negationslash] These hypotheses are generic. Recalling (21), the requirement that Γ exclude any points conjugate to p o along a connecting geodesic is equivalent to demanding that the source's worldline avoid certain well-behaved one- or two-dimensional subsets of the three-dimensional hyperplanes conjugate to S u o . Similarly, the assumption that x o = γ ( u o ) demands only that Γ avoid a certain line on S u o . The bound (48) on a source's asymptotic 4-velocity can fail to hold only for sources which accelerate for an infinitely long time. Violating any of these conditions for a fixed observer would require that a source's worldline be quite exceptional. Moreover, we now show that | N | +1 images appear under even broader (but more difficult to state) conditions than those just discussed.",
"pages": [
14
]
},
{
"title": "3.5. Non-generic imaging",
"content": "Despite the comments made above, our assumptions on the behavior of a source's worldline can be violated in certain cases. Suppose, contrary to these assumptions, that there exists at least one point γ e ∈ Γ which is conjugate to p o along a geodesic connecting these two points. An infinite number of geodesics then pass between γ e and p o . If the connecting geodesics are null, γ e lies on a caustic of the observer's past light cone. Continuous images of point sources - 'Einstein rings' - are then formed at p o (ignoring the associated breakdown of geometric optics). Such cases are not considered any further here. If conjugate points between the source and observer are associated with non-null geodesics, discrete images of Γ appear at p o . In these cases, the methods used above are easily adapted to find how many images of Γ n arrive at p o . As already mentioned, there can be no more than one root for each Σ n . Depending on the details of the system, the associated image from N n ( u o ) may or may not exist. The case considered in Sect. 3.1 where Γ n lies in between successive conjugate hyperplanes is the simplest to analyze. Assuming that Γ does not intersect the caustic of p o , no images are formed of Γ n if either γ ( τ n ( u o )) is connected to p o via a timelike geodesic or γ ( τ n -1 ( u o )) is connected to p o via a spacelike geodesic. Otherwise, exactly one image exists from this region. If there exists an oldest conjugate hyperplane S τ N ( u o ) as described in Sect. 3.3, suppose that the source satisfies (47) for some κ min > 0. There are then zero images of Γ N if γ ( τ N ( u o )) is connected to p o via a timelike geodesic. There is exactly one image if these points are either geodesically disconnected or are connected by a spacelike geodesic. glyph[negationslash] The last cases to consider concern images of Γ 0 . First suppose that x o = γ ( u o ). If there exists at least one conjugate hyperplane in the observer's past, one image is formed of Γ 0 if either γ ( τ -1 ( u o )) is geodesically disconnected from p o or it is connected by a timelike geodesic. No images are formed if γ ( τ -1 ( u o )) and p o are connected by a spacelike geodesic. If x o = γ ( u o ) and there are no conjugate points in the observer's past, condition (47) implies that there exists exactly one image of Γ 0 . glyph[negationslash] Cases where the source and observer are instantaneously aligned are more interesting. Suppose that x o = γ ( u o ). There then exists one image of Γ 0 with u e = u o . Recalling that u o is not in the domain of Σ 0 , it is possible for a second image to be emitted from Γ 0 if Σ 0 = 0 somewhere. This may be seen by noting that Two images of Γ 0 can therefore exist when v ( γ ( u o )) > v o and x o = γ ( u o ). If a source includes points which are conjugate to the observer (in the ordinary sense), there is no simple result for the total number of images formed. Nevertheless, it is always possible to say that the total number of images is less than or equal to | N | +2 if a source does not intersect a caustic of the observer's light cone.",
"pages": [
14,
15
]
},
{
"title": "4. Properties of lensed images",
"content": "Plane wave spacetimes typically produce multiple images of each source. Even for sources whose intrinsic properties remain constant, these images can appear with different spectra, brightnesses, etc. We now compute these properties for generic configurations satisfying the hypotheses summarized in Sect. 3.4. For each image of a timelike worldline Γ seen at p o , there is an associated null geodesic segment connecting p o to an appropriate emission point γ e = γ ( u e ) ∈ Γ. In terms of Synge's function (36), these points satisfy First derivatives of σ are always tangent to the connecting light ray. In particular, the vector at p o points along the geodesic which eventually intersects γ e (and is therefore pastdirected). The derivative operator here is understood to act on the second argument of σ . Also note that r a o is normalized such that glyph[lscript] a r a o = 1. Parallel-transporting r a o to the observation point yields The derivative operator in this equation is understood to act on the first argument of σ . Both r a o and r a e may be viewed as (dimensionless) separation vectors between p o and γ e . Eq. (37) and the various identities of Sect. 2.1 may be used to compute the explicit coordinate components of r a e and r a o . Components transverse to the direction of wave propagation are where x o = x ( p o ) and γ e = x ( γ ( u e )) denote the transverse coordinates of the observer and source. A similar calculation shows that r o and r e are related via Much of the discussion below considers sources moving on geodesics. In these cases, use of (23) shows that Here, δ x o := x o -γ ( u o ) = x o -γ o . The various identities involving A and B discussed in Sect. 2.1 may also be used to reduce the imaging condition (51) to Here, δv o := v ( p o ) -v ( γ o ). Eq. (58) is a nonlinear relation for the emission 'time' u e in terms of the observer's position p o and the parameters γ o , ˙ γ o , v ( γ o ), κ s describing the source's worldline. As discussed in Sect. 3, there can be many solutions to (58). These correspond to different images. Neither u e nor r a o depends on the observer's motion. Nevertheless, redshifts and angles on the observer's sky do depend on that motion (as is true even in flat spacetime). It is often useful to fix this effect by supposing that the observer is instantaneously comoving with the source. Following (32), this is taken to mean that the unit 4-velocities U a s , U a o of the source and observer on S u o satisfy Recall from Sect. 2.3 that the 'transverse momentum' P s ( u o ) is generated by contracting U a s ( u o ) with the two Killing fields equal to X a ( i ) at p o and having vanishing first derivative at that point. Also note that (60) implicitly defines an instantaneous observer velocity ˙ x a o = √ κ s U a o normalized (like ˙ γ a ) such that glyph[lscript] a ˙ x a o = -1.",
"pages": [
15,
16
]
},
{
"title": "4.1. Frequency shifts",
"content": "Gravitational lenses typically discussed in astrophysics involve nearly-Newtonian mass distributions which may be regarded as approximately stationary (at least on subcosmological timescales). If both a source and an observer are sufficiently far from such a lens, there can be no significant redshift or blueshift from the gravitational field of that lens. Roughly speaking, a light ray falling into any stationary gravitational potential must climb out of that same potential. This result breaks down if light passes through non-stationary regions of spacetime. Indeed, plane wave spacetimes may produce images with significant frequency shifts [19]. Consider an approximately monochromatic beam of light emitted from γ e and received at p o . A future-directed tangent vector k a e ∝ -r a e to the emitted light ray may always be chosen such that is the angular frequency of the light as seen by its source. The frequency ω o of this same light ray as measured by an observer at p o is -k o · U o , where k a o is equal to k a e parallel transported from the source to the observer. The observed and emitted frequencies are therefore related by Here, κ e := 1 / ( glyph[lscript] · U s ( u e )) 2 and κ o := 1 / ( glyph[lscript] · U o ) 2 . The 2-vectors r e and r o appearing here are determined by the source and observer positions via (37), (51), (54), and (55). The resulting frequency shift is valid for all emission points not contained in a caustic of p o . Now suppose that a source moves on a geodesic and that the observer is instantaneously comoving with this geodesic in the sense of (59) and (60). Then κ e = κ s = 1 / ( glyph[lscript] · U s ) 2 doesn't depend on which image is chosen. Eqs. (56), (57), and the symmetry of B -1 A may be used to rewrite (62) as The matrix in parentheses on the right hand side of this equation acts like a metric for the 'separation' 2-vector B -ᵀ ( u e , u o )[ x o -γ ( u o )]. If both eigenvalues of A ( u e , u s ) A ᵀ ( u e , u s ) -δ are negative, the source is necessarily redshifted. Conversely, sources are always blueshifted when this matrix is positive definite. If AA ᵀ -δ has both positive and negative eigenvalues, the sign of the frequency difference depends on the direction of B -ᵀ δ x o . For special configurations, there is no frequency shift at all.",
"pages": [
17
]
},
{
"title": "4.2. Angles",
"content": "Various images formed from a single source appear at different points on an observer's sky. Like redshifts, the relative angles between images change depending on an observer's 4-velocity U a o . The angle θ between two images arriving at p o with tangents r a o and r ' a o is Simplifying, This expression is valid for arbitrary source and observer configurations. Specializing to geodesic sources and comoving observers, Here, A = A ( u e , u o ) and A ' = A ( u ' e , u o ). It is evident that angles are largely controlled by the difference between B -1 A at the two emission times. Another interesting angle to consider is the observed separation ψ between a single image (emitted at γ e ) and a generator glyph[lscript] a of the background gravitational wave. For arbitrarily moving source and observer configurations, For observers comoving with geodesic sources, This may be used to rewrite the angle θ between two different images partially in terms of the angles ψ and ψ ' those images make with glyph[lscript] a . Using (66), Similarly, the frequency shift (63) of an individual image may be rewritten as It is evident from this equation that images which appear highly blueshifted to comoving observers must satisfy ψ ≈ 0.",
"pages": [
17,
18
]
},
{
"title": "4.3. Image distortion and magnification",
"content": "Thus far, all sources here have been modelled as though they were confined to timelike worldlines. Real objects are not pointlike, however. They form extended worldtubes in spacetime. Images of such worldtubes form null geodesic congruences which converge on p o . These images can be significantly distorted by the curvature of spacetime. It is simplest to quantify such distortions by first fixing a particular null geodesic Z passing between some part of the source and p o . Precisely which geodesic is chosen is not important. Z serves only as an origin from which to discuss nearby light rays connecting p o to other points in the source. Once this origin has been fixed, the image of an extended source may be described entirely using deviation vectors on Z (at least for sufficiently small sources). See Fig. 3. Deviation vectors (or Jacobi fields) satisfy the geodesic deviation (or Jacobi) equation along Z . Letting r a denote the past-directed null vector tangent to Z and obtained by parallel-transporting r a o from p o , every deviation vector ξ a is a solution to This equation is linear, so ξ a must depend linearly on initial data. In particular, all deviation vectors can be written as linear combinations of appropriate bitensors contracted into the initial data ¶ ξ A ( u o ) and ˙ ξ A ( u o ). All light rays observed at p o must necessarily intersect that point, so it suffices to set ξ A ( u o ) = 0. The first derivative of a deviation vector at p o describes an angular deviation between one point of an image and the center associated with Z . We therefore consider deviation vectors ξ a with the form B a A is known as a Jacobi propagator. It satisfies the Jacobi equation along Z together with the initial conditions Note that B a A is a bitensor. It maps vectors at p o into vectors at others points on Z . The transverse components of B a A are where B is the matrix defined in Sect. 2.1. Other components of B a A may be deduced from the eigenvector relations [23] All parts of an image must arrive at an observer along null geodesics. Additionally, an observer with 4-velocity U a o can only measure angles of vectors orthogonal to U a o . It therefore suffices to restrict attention to deviation vectors satisfying ¶ Capital letters are used in this subsection to denote abstract indices associated with the observation point p o . This is done to avoid confusion when writing down two-point tensors such as B a A [see (72)]. at p o . These constraints restrict all interesting initial data to a two dimensional space. The orthonormal vectors form a basis for this space at p o . They satisfy Parallel-transporting e A ( i ) to another point on Z yields which forms a Sachs basis [1] on Z . Initial data appearing in (72) must be of the form where α is an unconstrained 2-vector. The factor ( U o · r o ) > 0 is included here so that α is directly related to angles on an observer's sky. A sufficiently small image may be described by a suitable set of 2-vectors α representing the angular locations of each portion of the image with respect to the center defined by Z . Each α may be translated into a physical displacement at the source using (72) and (82). First note that for every particular α , (76) implies that r · ξ = 0 throughout Z . Indeed, ξ a is always a linear combination of the e a ( i ) together with r a . Components of ξ a proportional to r a are physically irrelevant, so we consider only the Sachs components Defining the 2 × 2 matrix it is then clear that ξ = D α for any α . D is referred to as the Jacobi matrix or Jacobi map [1, 11]. Using (75)-(77), (79), (81), and (84), If a source moves on a geodesic which is instantaneously comoving with the observer, This discussion implies that a portion of an image with angular separation glyph[epsilon1] α from the fiducial direction associated with r a o is spatially separated from the fiducial emission point γ e ∈ Z by The factors of glyph[epsilon1] glyph[lessmuch] 1 have been introduced here to emphasize that this description is valid only for infinitesimal deviations. Regardless, (87) shows that up to the time dilation factor ( U o · r o ), the matrix B central to all aspects of plane wave geometry may be physically interpreted as a transformation converting infinitesimal angles on the vertex of a light cone into infinitesimal separations elsewhere on that light cone. B depends only on the u coordinates of the source and emission points, and not on any other aspects of the physical configuration. It may be computed for all possible observer-source pairs directly from the wave profile H . Angles of emission (as opposed to observation) of the various light rays travelling from the source to the observer may be found by differentiating (87) and applying the appropriate time dilation factor: The last equality here makes use of (62). Applying (87) shows that The symmetric matrix ∂ (1) BB -1 / ( U e · r e ) therefore converts spatial locations to emission angles within the source (with the constraint that all light rays intersect p o ). It is referred to as the optical deformation matrix [1, 11]. Eq. (87) implies that there is a sense in which circles on the observer's sky correspond to ellipses near γ e . This deformation may be parametrized by performing a polar decomposition on D ( u e , u o ): Here, R β and R χ represent rotation matrices through some angles β and χ . The ratio D + /D -is related to the ellipticity of the aforementioned ellipse. χ represents the angle between the principal axes of that ellipse and the Sachs basis. D ± and χ are referred as shape parameters [1, 43]. Recalling that polarization vectors are parallel-transported in the geometric optics approximation [3], any polarization vector must have Sachs components which are constant along Z . In principle, the angle χ might therefore be measured by comparing the relative 'rotation' between an object's observed shape and an appropriate polarization angle [1, 44]. For linearly polarized waves where H can be made diagonal by an appropriate coordinate choice, χ = 0 with respect to this coordinate system and the basis (81). It is shown in Sect. 6 that χ also vanishes in a natural way for all sufficiently weak wavepackets which are nonzero only for short times. Eq. (87) implies that D converts angles at the observer to separations within the source. The determinant of D must therefore relate solid angles at p o to physical areas near γ e : It follows that may be interpreted as an 'angular diameter distance.' Absolute value signs are necessary here because det B changes sign after each pass through a conjugate hyperplane with multiplicity 1. Physically, such sign changes represent parity inversions of the resulting image. Note that d ang does not necessarily increase monotonically with the age of an image (as computed using the source's proper time). Closely related to the angular diameter distance is the luminosity distance One factor of ω o /ω e arises here from considering light cones emanating from the source instead of the observer. The other factor of ω/ω e is related to the energy change associated with frequency shifts.",
"pages": [
18,
19,
20,
21
]
},
{
"title": "5. Symmetric plane waves",
"content": "Now that various optical quantities have been computed for general plane wave spacetimes, we consider their application to various special cases. The simplest nontrivial plane waves are the symmetric waves. These are locally symmetric in the sense that ∇ a R bcd f = 0. It follows from (3) that symmetric plane waves must have constant waveforms. Also note that ( ∂/∂u ) a is Killing in these examples [as well as glyph[lscript] a = ( ∂/∂v ) a , which is Killing in all plane wave spacetimes]. Particular symmetric plane waves may be specified entirely by the (constant) eigenvalues of H . Recalling the decomposition (6) of H into h + , h × and h ‖ , a coordinate rotation may always be used to set h × = 0 for symmetric waves. It is then evident that the two eigenvalues of H are given by ± h + -h ‖ . It is always possible to set The weak energy condition implies that h ‖ ≥ 0, so at least one eigenvalue of H must be negative (implying that at least one of the h 1 , 2 must be positive). We assume for definiteness that h + ≥ 0. Then, If the vacuum Einstein equation is imposed, h ‖ = 0 and h 2 = -h 1 . For conformallyflat geometries representing spacetimes associated with, e.g., pure electromagnetic plane waves, h + = 0 and h 2 = h 1 . Other cases may be viewed as superpositions of gravitational and ('gravito'-)electromagnetic waves. All symmetric waves produce an infinite number of images of almost every source. It is clear from (94) that these waves are also linearly polarized. The angles χ and β appearing in (90) therefore vanish when considering image deformations with respect to the Sachs basis (81). Other lensing properties depend on the sign of h 2 . We call the case h 2 < 0 'gravity-dominated' and the case h 2 > 0 'matter-dominated.'",
"pages": [
22
]
},
{
"title": "5.1. Gravity-dominated symmetric waves",
"content": "Consider symmetric plane wave spacetimes where h 2 = h ‖ -h + < 0. Gravitydominated waves such as these generalize the vacuum waves satisfying h 1 = -h 2 . Symmetric vacuum waves arise from, e.g., the Penrose limit of a null geodesic orbiting a Schwarzschild black hole on the light ring. For any gravity-dominated symmetric wave, the matrices A and B defined in Sect. 2.1 are It is clear that det B ( · , u o ) has an infinite number of zeros for any choice of u o . Each of these zeros represents a phase conjugate to u o . There are an infinite number of such phases in both the past and future of every observer. The discussion in Sect. 3 where therefore implies that under generic conditions, an infinite number of images appear for almost every source. Explicitly, all conjugate phases are given by where n is any nonzero integer. It is evident from (19) that all of these phases have multiplicity 1. For any n < 0 and any observation point p o with u ( p o ) = u o , exactly one image of each source is visible as that source appeared in N n ( u o ). This corresponds to the region between u = u o + nπh -1 2 1 and u = u o +( n -1) πh -1 2 1 . Note that det B ( · , u o ) switches sign on each pass through a conjugate phase. The parity of an image emitted from N n ( u o ) is therefore opposite to the parity of an image emitted from N n -1 ( u o ). Specializing to cases where the source is a geodesic and the observer is instantaneously comoving with that source on S u o , some configurations lead to redshifts and others to blueshifts. Using (63), An image is therefore redshifted if and only if It is blueshifted when δ x ᵀ o H δ x o > 0. There is no frequency shift at all in cases where The direction of the frequency shift clearly depends only on the instantaneous orientation δ x o / | δ x o | of the source and the observer on S u o . In particular, it does not depend on which image is considered. All images of a particular source experience the same type of frequency shift. Emission times u e for an observer comoving with a geodesic source may be found by solving (58). For gravity-dominated symmetric waves, this equation reduces to If -n glyph[greatermuch] 1, it is evident that the image from N n ( u o ) must satisfy cot h 1 2 1 ( u o -u e ) glyph[greatermuch] 1. Images from the distant past are therefore emitted at phases u e very nearly conjugate to u o : Substituting this relation into (68) and (100) shows that very old images cluster near glyph[lscript] a on the observer's sky and experience increasingly-negligible frequency shifts: Old images of slightly extended sources are also highly distorted and demagnified. Their angular diameter and luminosity distances both scale like Gravity-dominated symmetric waves therefore produce an infinite number of exponentially dimming images for almost every source.",
"pages": [
22,
23
]
},
{
"title": "5.2. Matter-dominated symmetric waves",
"content": "Matter-dominated symmetric waves satisfying h 2 > 0 act somewhat differently than gravity-dominated waves. In these cases, Phases conjugate to u o occur at u o + nπh -1 2 1 and at u o + n ' πh -1 2 2 , where n, n ' are any nonzero integers. If √ h 1 /h 2 is an irrational number, these two families of phases are distinct. Each conjugate pair then has multiplicity 1. If √ h 1 /h 2 is rational, some conjugate pairs have multiplicity 2. In the conformally-flat case where h 1 = h 2 , all conjugate phases have multiplicity 2. In every other case where √ h 1 /h 2 is rational, an infinite number of conjugate phases occur with each multiplicity. Regardless of h 2 , an infinite number of images are formed for almost every source. Now consider a luminous source moving on a geodesic. If the source and observer are instantaneously comoving in the sense of (59) and (60), frequency shifts associated with each image are given by (63). Since is negative-definite in this case, all images are redshifted. If -n glyph[greatermuch] 1, an image originating from N n ( u o ) must be emitted just before the source intersects S τ n ( u o ) . All such images cluster towards glyph[lscript] a in the observer's sky and have negligible frequency shifts. Images emitted near conjugate hyperplanes with multiplicity 1 are highly distorted. Images emitted near conjugate hyperplanes with multiplicity 2 are not significantly distorted at all. In both cases, however, older images are dimmer (although the rate at which this occurs is much slower than for gravity-dominated symmetric waves). Regardless of the sign of h 2 , the oldest images formed by symmetric plane wave spacetimes depend on the spacetime structure at arbitrarily large transverse distances. If the metric is modified so that the wave decays at large distances, only a finite number of images discussed here would be unaffected. The oldest images found in pure symmetric waves likely do not appear at all in perturbed symmetric waves.",
"pages": [
24
]
},
{
"title": "6. Sandwich waves",
"content": "Symmetric plane waves are mathematically simple, but are not reasonable models for gravitational radiation emitted from compact sources. More interesting are waves where H is nonzero only for a finite time: Sandwich waves. Suppose, in particular, that there exists some u + > 0 such that H ( u ) = 0 for all u / ∈ [0 , u + ]. It follows from (3) that such spacetimes are locally flat whenever u < 0 or u > u + . The curved region containing the gravitational wave is effectively sandwiched between the two null hyperplanes S 0 and S u + . Every timelike curve eventually passes entirely through such a wave. Before an observer interacts with the wave, spacetime is flat and optics is trivial. The case u o > u + where an observer has already passed through the wave is more interesting. In this case, A and B reduce to their flat space forms when u s > u + . The forms of A and B inside the wave depend on the details of H , and will not be discussed here. If u s < 0, however, there always exist four constant 2 × 2 matrices α , ˙ α , β and ˙ β such that Note that the dots on ˙ α and ˙ β do not refer to derivatives in this case. They are only used as a labelling device. It follows from (17) that A ( u s , u o ) is independent of u o . Moreover, In general, ˙ α and ˙ β have a simple physical interpretation. If two geodesics are comoving and have a transverse separation δ x o when u > u + , it follows from (23) that the transverse separation between these geodesics is -˙ α δ x o immediately before they interact with the wave at u = 0. Similarly, the relative transverse velocity of these geodesics is -˙ β δ x o when u < 0. If there were no wave at all, α = ˙ β = 0 and -˙ α = β = δ . Interpretations for α and β are somewhat less direct. Consider two geodesics which intersect at some time u o > u + after the wave has passed, but which have a relative transverse velocity δ ˙ x o at u = u o . The difference in transverse velocities between these two geodesics is then ( β + u o ˙ β ) δ ˙ x o when u < 0. Similarly, the difference in the transverse positions of these geodesics is ( α + u o ˙ α ) δ ˙ x o at u = 0. It follows that α controls shifts in position that are independent of the time u o at which the two geodesics cross each other. In principle, α , ˙ α , β , ˙ β may all be found by solving (9) if H is known. At first glance, this would appear to imply that 4 · 4 = 16 numbers are required to describe observations through a sandwich wave. The actual number of required parameters is somewhat less than this. First note that the Wronskian identity (13) implies that ˙ α ᵀ ˙ β is a symmetric matrix. It also implies that Further simplifications arise by recalling from (14) that BA ᵀ and B ᵀ ∂ (1) B are symmetric. It follows that are symmetric as well. These expressions are completely general, and hold for any sandwich wave. They significantly constrain the number of independent parameters needed to specify A and B . Equivalently, they limit the number of parameters that must be extracted from H . It follows from the arguments of Sect. 3 that the number of images of a generic source observable in any plane wave spacetime is governed by the number of hypersurfaces conjugate to the u = constant hypersurface S u o containing the observation event p o . Continuing to assume that u o > u + , all phases conjugate to u o must be smaller than u + . It follows from (20) and (111) that conjugate phases occurring before the wave may be found by solving for all τ < 0. This equation is quadratic, so at most two solutions exist. An observer ahead of the wave may therefore see at most three images of a source as it appeared behind the wave. There may also be at most one image of a source as it appeared ahead of the wave. In principle, any number of images may arise from inside the wave [where (111) is not valid] if H is sufficiently large. Conjugate phases found by solving (115) clearly depend on the observation time u o . Less obviously, the number of conjugate phases can also depend on u o . For an observer moving on a timelike worldline (where u o increases monotonically), new conjugate phases - and therefore new images - sometimes appear at discrete times. These images correspond to observation times where (115) momentarily degenerates to a linear equation. New images can therefore arise when u o = ¯ u o and If det ˙ β = 0, the two solutions to this equation are glyph[negationslash] Only solutions satisfying ¯ u o > u + > 0 are physically relevant. When a conjugate phase of this type first appears, it satisfies The associated image therefore provides a picture of the infinitely distant past. Furthermore, an infinite amount of the source's history appears to the observer within a finite amount of proper time. This implies that new images are highly blueshifted. No matter how long an observer waits, no conjugate phase can exceed u + . The emission time for an associated image might therefore be expected to tend towards a constant value as u o → ∞ . This means that a very large amount of proper time at the observer corresponds to only a small amount of proper time at a source. Images which appear suddenly and are initially highly blueshifted become highly redshifted at late times. It is unclear precisely what these types of images imply. To the extent that geometric optics remains valid, all observers passing through S ¯ u o momentarily see almost the entire universe appear infinitely blueshifted as it was in the infinitely distant past. Furthermore, (70) implies that all of the universe is briefly compressed into a single point on each observer's sky. Of course, such phenomena lie outside the domain of geometric optics. They may even lie outside of the realm of test fields propagating on a fixed background spacetime. Extreme focusing events like these might indicate instabilities inherent in the plane wave geometry itself. It should, however, be noted that all of the infinities just alluded to are likely to have finite cutoffs in 'realistic' plane waves which decay at large transverse distances.",
"pages": [
24,
25,
26
]
},
{
"title": "Weak wavepackets",
"content": "One important class of sandwich waves are those that are very weak and last only for a short time. In these cases, α , ˙ α , β , and ˙ β may be expanded as integrals involving successively higher powers of H . To lowest order in such a scheme, A and B are approximately unaffected by the wave while inside of it. Assuming that -u s , u o glyph[greatermuch] u + , the first corrections to this assumption are This approximation is consistent with (13) [and therefore (113) as well]. In terms of the matrices appearing in (111), α ≈ 0, -˙ α ≈ β ≈ δ , and Note that Eqs. (119) and (120) should be applied with care if H involves many oscillations of an approximately periodic function. In these cases, the integral of H can be very nearly zero. Terms nonlinear in H might then be significant. Assuming that (119) and (120) are indeed adequate approximations for A and B , a coordinate rotation may always be used to diagonalize ˙ β . There then exist two constants H 1 and H 2 such that In this sense, all sufficiently short gravitational plane waves act as though they are linearly polarized [so β = χ = 0 in (90)]. In terms of the individual wavefunctions appearing in (6), The transverse coordinates x i have also been chosen such that If a wave satisfies the vacuum Einstein equation, h ‖ = 0 and H 1 = -H 2 . More generally, it follows from the weak energy condition that Now assume that the integral of h + is non-negative, which entails only a minimal loss of generality. Then, We say that a wave is gravity-dominated if H 2 < 0 and matter-dominated if H 2 > 0. These definitions are closely analogous to those used to classify symmetric plane waves in Sect. 5. There, a wave was said to be gravity- or matter-dominated depending on the sign of the constant h 2 = h ‖ -h + appearing in (94). For weak gravity-dominated wavepackets, there can be at most one phase conjugate to an observer satisfying u o > u + . If this exists, it evident from (120) that is conjugate to u o with multiplicity 1. This equation is valid only if τ -1 ( u o ) < 0. A conjugate phase therefore exists only for observers satisfying Note that H 1 ¯ u o = 1 is the unique physically-relevant solution to (116) in the gravitydominated case. Waves that are matter-dominated (so H 2 > 0) also admit the conjugate phase (128) when u o satisfies (129). In the conformally-flat case where H 1 = H 2 , this is the only conjugate phase. Unlike in the gravity-dominated case, the multiplicity of τ -1 is equal to 2 for conformally-flat waves. In all other matter-dominated cases, τ -1 has multiplicity 1 and a second conjugate phase is admitted (also with multiplicity 1) for all observers satisfying This occurs at Note that (130) is a more stringent condition than (129). As implied by the notation, τ -2 ( u o ) < τ -1 ( u o ). Consider a point source moving on a timelike worldline Γ in a weak sandwich wave spacetime. Such a source appears differently when observed at different times. It is ω H clear that when u o < 0, exactly one image of Γ is viewable under generic conditions. As time passes, the wave eventually passes through the observer. A second image then appears when u o = H -1 1 > u + . This image is always emitted before the first. If the wave involves a sufficient amount of Ricci curvature + (from e.g., electromagnetic plane waves) and is not conformally-flat, a third image appears when u o = H -1 2 > H -1 1 . This is emitted before the first two images. All images persist indefinitely once they appear. Sufficiently far in the future, one image is observed of the source as it appeared after interacting with the gravitational wave. All other images predate this interaction. See Fig. 4. When the second image first appears at u o = H -1 1 , the new conjugate phase τ -1 ( u o ) is divergent. At all later times, it is finite. The same is also true for the emission times associated with the second image. Almost the entire past history of the source is therefore observable within a finite proper time. This implies an infinite blueshift. At late times, τ -1 → -H -1 1 . The observed evolution of the source via the second image effectively freezes as u e asymptotes to -H -1 1 (which predates the source's interaction with the wave). Images such as these are highly redshifted, as indicated in Fig. 5. Note that a similar transition from infinite blueshift to infinite redshift also applies to the third image if it exists. Another qualitative feature of the emission times plotted in Fig. 4 is that there is a sense in which pairs of images can 'switch roles.' Consider, e.g., the left panel of that figure. At late times, the solid curve (corresponding to the younger image) is perfectly linear. Indeed, it remains very nearly linear until H 1 u o ≈ 5. A rapid transition then occurs where the dashed curve effectively takes over this linear behavior while the solid curve strongly deviates from it. In a sense, the two images reverse their roles. This phenomenon occurs one more time (somewhat less sharply) around H 1 u o ≈ 1 when the second image first appears. It arises essentially because u e > τ -1 for the younger image and u e < τ -1 for the older image. These constraints can cause emission curves to bend sharply - with large accompanying frequency shifts - in order to avoid H H intersecting τ -1 . Whether or not this occurs depends on whether the 'average' linear increase of u e ever comes near τ -1 . If it does, this role switching occurs. If not, the younger image is barely affected by the gravitational wave at all. This is illustrated in Fig. 6, where emission curves for several sources are plotted simultaneously. To summarize, images which appear at discrete times briefly appear as bright, highly blueshifted 'flashes.' Indeed, Fig. 7 shows that their luminosity distances go to zero. Simultaneously, the angular diameter distance of each new image tends to infinity. It is implied by (70) that all highly blueshifted images make must a very small angle ψ with glyph[lscript] a on the observer's sky. The second (and third) images therefore appear aligned with the direction of propagation of the gravitational wave when they first appear. This direction could be quite different from the location of the other image(s). As time progresses, all images migrate across the observer's sky as illustrated in Fig. 8. Different images may remain separated from each other by large angles at all times.",
"pages": [
26,
27,
28,
29,
30
]
},
{
"title": "7. Discussion",
"content": "Despite their simplicity, plane gravitational waves behave in qualitatively different ways from lenses associated with quasi-Newtonian mass distributions. As expected from their dynamic nature, plane waves generically shift the observed frequencies of H H various images. They may also admit images which appear to move, deform, change brightness, and shift color as time progresses. More subtle differences relate to the number of images that are produced of a given source. For example, even numbers of images can appear generically (which has led plane wave spacetimes to be cited [1] as well-behaved examples where the odd number theorem [5, 7] does not apply). Some plane waves can even produce an infinite number of discrete images. Perhaps most striking of all are the bright flashes shown to be produced by generic sandwich waves in Sect. 6. These correspond to individual images which appear at discrete times. More typical gravitational lenses can produce new images if a source crosses an observer's caustic. Individual images then split into two (or vice versa). The flashes produced by sandwich waves are quite different. Their appearance does not require that a source pass through an observer's caustic. Such images appear individually from the infinitely distant past. Initially, they are infinitely bright and infinitely blueshifted points of light appearing in the direction of propagation associated with the gravitational wave. Many of these effects depend at least partially on the idealization that a plane wave extends undiminished to infinitely-large transverse distances. Plane wave spacetimes are not asymptotically flat. Despite being topologically trivial and locally well-behaved, they are not even globally hyperbolic: Null geodesics passing between appropriately-chosen pairs of points can extend to arbitrarily large transverse distances in between those points. It is this property which permits the infinite number of images described in Sect. 5 to be produced by symmetric waves. The flashes described in Sect. 6 also depend on the spacetime structure at arbitrarily large distances. This structure likely affects the formation of even numbers of images as well. Indeed, the usual proofs of the odd number theorem require global hyperbolicity, among other assumptions [1, 5] (see, however, [7] for a more general formulation). If a spacetime has the geometry of a plane wave only out to some finite transverse distance, all results derived here remain valid if the associated images involve light rays which never extend sufficiently far to interact with any large-distance modifications. The infinite sequence of images formed by a symmetric plane wave would then be expected to become finite for spacetimes which are only approximately plane waves. Calculations involving the oldest images could no longer be trusted in these cases. Similarly, the bright flashes associated with ideal sandwich waves are likely to be somewhat less extreme for waves which decay at infinity. Large brightnesses and large blueshifts can still exist, but these will be cut off at some finite maximum. Such maxima may, however, remain quite large. It is reasonably clear that modifications of the geometry at large distances can remove some images. Less obviously, these modification can also introduce new images. Consider, for example, the pp -wave spacetimes obtained by substituting in the metric (1). These generalize the plane wave spacetimes. It has been shown that if H grows subquadratically as | x | → 0, the resulting geometries are globally hyperbolic [45]. Moreover, every pair of points is connected by at least one geodesic in these cases (unlike in pure plane wave spacetimes where the growth of H is precisely quadratic). This implies that modifications of the geometry at large transverse distances can introduce new null geodesics even between points at small transverse distances. It would be interesting to explore these effects in more depth to understand precisely how modifications of this sort (or more general ones) alter the lensing properties described here for ideal plane waves. It would also be interesting to better understand what the transient flashes of Sect. 6 imply for waves propagating on plane (or almostplane) wave spacetimes. This can likely be facilitated by the Green functions derived in [23].",
"pages": [
30,
31,
32
]
}
]
2013CQGra..30g5022B
https://arxiv.org/pdf/1207.2983.pdf
<document>
<section_header_level_1><location><page_1><loc_29><loc_89><loc_71><loc_91></location>Generalized Holographic Cosmology</section_header_level_1>
<text><location><page_1><loc_33><loc_81><loc_68><loc_84></location>Souvik Banerjee † , Samrat Bhowmick † , Anurag Sahay †</text>
<text><location><page_1><loc_39><loc_76><loc_62><loc_79></location>Institute of Physics Bhubaneswar -751 005, India</text>
<text><location><page_1><loc_43><loc_70><loc_57><loc_72></location>George Siopsis /flat</text>
<text><location><page_1><loc_23><loc_64><loc_77><loc_67></location>Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996 - 1200, USA</text>
<section_header_level_1><location><page_1><loc_46><loc_49><loc_54><loc_50></location>Abstract</section_header_level_1>
<text><location><page_1><loc_13><loc_36><loc_88><loc_47></location>We consider general black hole solutions in five-dimensional spacetime in the presence of a negative cosmological constant. We obtain a cosmological evolution via the gravity/gauge theory duality (holography) by defining appropriate boundary conditions on a four-dimensional boundary hypersurface. The standard counterterms are shown to renormalize the bare parameters of the system (the four-dimensional Newton's constant and cosmological constant). We discuss the thermodynamics of cosmological evolution and present various examples. The standard braneworld scenarios are shown to be special cases of our holographic construction.</text>
<unordered_list>
<list_item><location><page_1><loc_13><loc_18><loc_54><loc_20></location>† e-mail address: souvik,samrat,anurag@iopb.res.in</list_item>
<list_item><location><page_1><loc_13><loc_17><loc_44><loc_18></location>/flat e-mail address: siopsis@tennessee.edu</list_item>
</unordered_list>
<section_header_level_1><location><page_2><loc_13><loc_89><loc_31><loc_91></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_13><loc_78><loc_88><loc_86></location>In recent years, understanding cosmology within the framework of string theory, has been an active and interesting field of study. Starting with [1], a substantial amount of research has been based on modeling the Universe by a 3-brane living in a higher-dimensional bulk space (brane world scenario). An incomplete list of references is [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The Hubble equation of cosmological evolution is thus reproduced by the trajectory of the brane.</text>
<text><location><page_2><loc_13><loc_61><loc_88><loc_76></location>A different approach to understanding dynamics starting with a higher-dimensional bulk space is provided by the AdS/CFT correspondence [14]. In this approach, a solution of the Einstein equations in a space with a negative cosmological constant is shown to be dual to a gauge theory of lower dimension living on the boundary (gravity/gauge theory duality). The stress-energy tensor of the gauge theory is constructed using holography [15]. This affords an understanding of the gauge theory at strong coupling, which has been applied to various physical systems, such as the quark-gluon plasma formed in heavy ion collisions and high temperature superconductors. It is possible to understand the motion of a fluid, including time-dependent flows. An application to cosmology, which is our interest here, has also been discussed [16, 17].</text>
<text><location><page_2><loc_13><loc_50><loc_88><loc_59></location>Applying the gravity/gauge theory duality to a cosmological setting is not straightforward due to the fact that the metric on the boundary space in which the gauge theory lives must remain dynamical. This was long thought to be problematic due to the possibility of the fluctuations of the bulk metric corresponding to non-normalizable modes [18, 19, 20, 21, 22, 23, 24]. It was shown in [25], that such problems can be avoided by introducing appropriate counterterms on the boundary needed to cancel the infinities.</text>
<text><location><page_2><loc_13><loc_36><loc_88><loc_48></location>In applications of the gravity/gauge theory duality (holography) to cosmology and other settings, one generally places the boundary at a finite distance r and then takes the limit as the cutoff r → ∞ . The removal of the cutoff introduces infinities, which are canceled by the addition of a local action on the boundary with r -dependent coefficients (counterterms). Unlike in quantum field theory, where counterterms are interpreted as renormalization of the (bare) parameters of the system, it is not clear if counterterms have a similiar physical meaning in a holographic setting.</text>
<text><location><page_2><loc_13><loc_21><loc_88><loc_34></location>Here we generalize the holographic approach to cosmology by placing the boundary hypersurface at a finite distance r and derive expressions for the various physical quantities (e.g., the stress-energy tensor) which are valid for arbitrary r . This leads to a generalized Hubble equation of cosmological evolution. We still need to introduce the standard counterterms to avoid infinities at large r . We show that these counterterms have the usual field theoretic interpretation of renormalizing the (bare) parameters of the system, namely Newton's constant and the cosmological constant. Moreover, we recover the brane-world scenario by fine-tuning Newton's constant. Thus we show that brane-world scenarios are a special case of our generalized holographic approach.</text>
<text><location><page_2><loc_13><loc_13><loc_88><loc_19></location>Our paper is organized as follows. In section 2 we discuss the bulk space concentrating on a time-independent solution (general black hole) of the field equations, and define the boundary hypersurface. In section 3 we introduce the boundary conditions and the counterterms needed to cancel infinities. We calculate the stress-energy tensor and derive the Hubble equation of cosmo-</text>
<text><location><page_3><loc_13><loc_84><loc_88><loc_91></location>logical evolution. In section 4 we discuss the example of a bulk Reissner-Nordstrom black hole including thermodynamics. In section 5 we discuss various examples of cosmological evolution. In particular, we show that the brane-world scenarios are a special case of our holographic approach. Finally in section 6 we conclude.</text>
<section_header_level_1><location><page_3><loc_13><loc_79><loc_27><loc_80></location>2 The Bulk</section_header_level_1>
<text><location><page_3><loc_13><loc_72><loc_88><loc_75></location>We start with a non-extremal black hole in a 4 + 1 dimensional bulk space in the presence of a negative cosmological constant</text>
<formula><location><page_3><loc_46><loc_69><loc_88><loc_72></location>Λ 5 = -6 L 2 . (1)</formula>
<text><location><page_3><loc_13><loc_67><loc_36><loc_69></location>We consider the metric ansatz</text>
<formula><location><page_3><loc_36><loc_64><loc_88><loc_66></location>ds 2 5 = -A ( r ) dt 2 + B ( r ) dr 2 + r 2 d Ω 2 k , (2)</formula>
<text><location><page_3><loc_13><loc_57><loc_88><loc_63></location>r being the radial direction, and k = +1 , 0 , -1 depending on the geometry of the constant ( t, r ) hypersurfaces (spherical, flat, or hyperbolic, respectively). More general metrics are also possible, but will clutter the notation unnecessarily. In section 4, we shall concentrate on the special case of a Reissner-Nordstrom black hole for explicit calculations.</text>
<text><location><page_3><loc_17><loc_53><loc_71><loc_55></location>Asymptotically, we have AdS space of radius L , therefore as r →∞ ,</text>
<formula><location><page_3><loc_42><loc_49><loc_88><loc_53></location>A ( r ) ≈ 1 B ( r ) ≈ r 2 L 2 . (3)</formula>
<text><location><page_3><loc_13><loc_45><loc_88><loc_48></location>We introduce a radial cutoff, r = a and parametrise a and t as a = a ( τ ) and t = t ( τ ) so that a = ˙ adt . Then the metric on the cut-off surface (boundary) takes the form</text>
<formula><location><page_3><loc_32><loc_40><loc_88><loc_44></location>ds 2 4 = [ -A ( a )( dt dτ ) 2 + B ( a )˙ a 2 ] dτ 2 + a 2 ( τ ) d Ω 2 k . (4)</formula>
<text><location><page_3><loc_13><loc_38><loc_61><loc_40></location>In order that the metric on the boundary take the FRW form,</text>
<formula><location><page_3><loc_40><loc_34><loc_88><loc_37></location>ds 2 4 = -dτ 2 + a 2 ( τ ) d Ω 2 k , (5)</formula>
<text><location><page_3><loc_13><loc_33><loc_51><loc_34></location>the metric components should satisfy the relation</text>
<formula><location><page_3><loc_37><loc_27><loc_88><loc_32></location>( dt dτ ) 2 = B A , B = B ( a )˙ a 2 +1 . (6)</formula>
<text><location><page_3><loc_13><loc_24><loc_88><loc_27></location>This in turn fixes our choice of the time parameter τ . Notice also that if T H is the Hawking temperature, then the temperature on the boundary is redshifted,</text>
<formula><location><page_3><loc_46><loc_19><loc_88><loc_23></location>T = T H √ A B . (7)</formula>
<text><location><page_3><loc_13><loc_12><loc_88><loc_19></location>This kind of parametrization has been used before, e.g., in [26, 27, 28]. Note that, while treating τ as a time parameter, we are effectively considering the radial motion of the cut-off surface in the 4+1 dimensional bulk. By adopting appropriate boundary conditions, the cut-off surface can be thought of as the location of a brane, mimicking a moving brane scenario.</text>
<section_header_level_1><location><page_4><loc_13><loc_89><loc_41><loc_91></location>3 Boundary Conditions</section_header_level_1>
<text><location><page_4><loc_13><loc_81><loc_88><loc_86></location>The heart of the construction we are going to elaborate on is based on the observation that the afore-mentioned dynamics of the boundary hypersurface will be captured through the boundary conditions we impose on the system. This approach was first adopted in [17].</text>
<text><location><page_4><loc_17><loc_78><loc_60><loc_80></location>Let us consider a general five dimensional bulk action,</text>
<formula><location><page_4><loc_42><loc_73><loc_88><loc_77></location>S 5 = ∫ M d 5 x √ -g L 5 , (8)</formula>
<text><location><page_4><loc_13><loc_66><loc_88><loc_73></location>where we keep the Lagrangian density L 5 unspecified. In the simplest case, this consists of a fivedimensional Einstein-Hilbert action with a negative cosmological constant (1) plus the requisite Gibbons-Hawking surface term for a well-defined variational principle. If one varies this action with respect to the metric, one obtains a boundary term of the form</text>
<formula><location><page_4><loc_37><loc_61><loc_88><loc_65></location>δS 5 = 1 2 ∫ ∂ M d 4 x √ -γT (CFT) µν δγ µν , (9)</formula>
<text><location><page_4><loc_13><loc_38><loc_88><loc_60></location>where γ µν is the induced boundary metric and γ is its determinant. T (CFT) µν denotes the (bare) stress-energy tensor of the dual conformal field theory that lives on the four-dimensional boundary hypersurface r = a . Generally in the context of the AdS/CFT correspondence, Dirichlet boundary conditions are employed, which fix the boundary metric and consequently eq. (9) vanishes. While this leads to a well-defined variational principle, it does not allow for a dynamical boundary metric. Since we are primarily interested in obtaining a cosmological evolution and hence a dynamical metric on the boundary, we seek different boundary conditions that can be imposed without fixing the metric on the boundary. It was noted in [17] that one could adopt appropriate mixed boundary conditions, which were shown to lead to valid dynamics in [25]. Their definition involves the addition of an appropriate local action, S local , at the boundary. For cosmological evolution, this local action will be chosen as the four-dimensional Einstein-Hilbert action on the boundary with an arbitrary (positive, negative, or vanishing) four-dimensional cosmological constant Λ 4 ,</text>
<formula><location><page_4><loc_33><loc_34><loc_88><loc_38></location>S local = -1 16 πG 4 ∫ ∂ M d 4 x √ -γ ( R [ γ ] -2Λ 4 ) , (10)</formula>
<text><location><page_4><loc_13><loc_29><loc_88><loc_34></location>where R [ γ ] is the Ricci scalar evaluated with the boundary metric which, in our case, is the FRW metric (5). Notice that the cosmological constant may be due, wholly or partly, to a brane of finite tension at the boundary.</text>
<text><location><page_4><loc_13><loc_22><loc_88><loc_27></location>Additionally, to cancel divergences in the limit a → ∞ , it is necessary to introduce counterterms [15]. These are of the same form as the local action and renormalize the four-dimensional physical parameters G 4 and Λ 4 . We have</text>
<formula><location><page_4><loc_34><loc_17><loc_88><loc_21></location>S c . t . = -1 2 ∫ ∂ M d 4 x √ -γ ( κ 1 R [ γ ] + κ 2 ) , (11)</formula>
<text><location><page_4><loc_13><loc_14><loc_88><loc_17></location>which diverges as a →∞ . The parameters κ 1 and κ 2 will be chosen so that physical quantities such as the energy density and pressure remain finite in this limit.</text>
<text><location><page_5><loc_17><loc_89><loc_70><loc_91></location>Putting these pieces together, we define our boundary condition as</text>
<formula><location><page_5><loc_39><loc_86><loc_88><loc_88></location>T (CFT) µν + T (local) µν + T (c . t . ) µν = 0 , (12)</formula>
<text><location><page_5><loc_13><loc_82><loc_88><loc_85></location>where T (CFT) µν is due to the variation δS 5 (eq. (9)), and the other two terms, T (local) µν and T (c . t . ) µν come from the variations</text>
<formula><location><page_5><loc_34><loc_74><loc_88><loc_81></location>δS local = 1 2 ∫ ∂ M d 4 x √ -γT (local) µν δγ µν , δS c . t . = 1 2 ∫ ∂ M d 4 x √ -γT (c . t . ) µν δγ µν , (13)</formula>
<text><location><page_5><loc_13><loc_71><loc_88><loc_74></location>respectively, with respect to the boundary metric, γ µν . Similarly to Dirichlet boundary conditions, the choice (12) leads to a well-defined variational principle with</text>
<formula><location><page_5><loc_40><loc_68><loc_88><loc_69></location>δS 5 + δS local + δS c . t . = 0 . (14)</formula>
<text><location><page_5><loc_13><loc_62><loc_88><loc_67></location>To see the explicit physical content of our mixed boundary conditions (12), we shall derive explicit expressions for each of the three contributing terms. The bare stress-energy tensor on the boundary is given by</text>
<formula><location><page_5><loc_38><loc_59><loc_88><loc_62></location>T (CFT) µν = 1 8 πG 5 ( K µν -K γ µν ) , (15)</formula>
<text><location><page_5><loc_13><loc_53><loc_88><loc_58></location>where K µν is the extrinsic curvature, and K is its trace. The components of this tensor can be evaluated by computing the velocity v µ and unit normal n ν vectors on the boundary hypersurface, r = a ( τ ). For the metric (2), these vectors are given in component form as</text>
<text><location><page_5><loc_13><loc_45><loc_16><loc_47></location>and</text>
<formula><location><page_5><loc_27><loc_46><loc_88><loc_52></location>v µ = √ B A , ˙ a, 0 , 0 , 0 ; v µ = ( -√ A B , B ˙ a, 0 , 0 , 0 ) . (16)</formula>
<formula><location><page_5><loc_22><loc_40><loc_88><loc_46></location>n µ = -√ B A ˙ a, -√ B B , 0 , 0 , 0 ; n µ = ( √ AB ˙ a, -√ B B , 0 , 0 , 0 ) , (17)</formula>
<text><location><page_5><loc_13><loc_37><loc_88><loc_40></location>respectively. The direction of the unit normal vector is taken to be pointing inward, toward the bulk. The extrinsic curvature can be written in terms of the unit normal and velocity vectors as</text>
<formula><location><page_5><loc_35><loc_33><loc_88><loc_36></location>K ij = 1 2 n k ∂ k γ ij K ττ = -∂ τ v t n t . (18)</formula>
<text><location><page_5><loc_13><loc_31><loc_28><loc_32></location>Explicitly, they are</text>
<formula><location><page_5><loc_24><loc_25><loc_88><loc_30></location>K ij = a √ B B γ ij , K ττ = -3 2 aAB ( 2 AB a +( AB ) ' ˙ a + A ' ) , (19)</formula>
<text><location><page_5><loc_13><loc_23><loc_78><loc_25></location>where i, j are indices for the spatial coordinates on the boundary (spanned by Ω k ).</text>
<text><location><page_5><loc_17><loc_20><loc_88><loc_22></location>We deduce the explicit expressions for the components of the bare stress-energy tensor (15),</text>
<formula><location><page_5><loc_28><loc_15><loc_88><loc_19></location>T (CFT) ττ = -3 8 πG 5 a √ B B , (20)</formula>
<formula><location><page_5><loc_27><loc_11><loc_88><loc_15></location>T i (CFT) i = 1 16 πG 5 aA ' B + A [ aB ' ˙ a +2 B ( a a +2˙ a 2 ) +4] aA √ B B , (21)</formula>
<text><location><page_6><loc_13><loc_84><loc_88><loc_91></location>where no summing over the index i is implied. Notice that the energy density T (CFT) ττ obtained above is negative, however we should emphasize that this is only a bare quantity and therefore not physical. It will be corrected by the addition of counter terms resulting into a positive regularized (physical) quantity.</text>
<text><location><page_6><loc_13><loc_79><loc_88><loc_82></location>For the remaining two contributions in (12), we obtain the standard expressions one encounters in Einstein's four-dimensional equations,</text>
<formula><location><page_6><loc_14><loc_75><loc_88><loc_79></location>T (local) µν = -1 8 πG 4 ( R µν -1 2 γ µν RΛ 4 γ µν ) , T (c . t . ) µν = -κ 1 ( R µν -1 2 γ µν R ) -κ 2 γ µν , (22)</formula>
<text><location><page_6><loc_13><loc_64><loc_88><loc_75></location>where R µν ( R ) is the four-dimensional Ricci tensor (scalar) constructed from the four-dimensional boundary metric γ µν . The counter terms diverge in the limit a → ∞ , and the parameters κ 1 and κ 2 will be chosen so that they cancel the divergences in the bare stress-energy tensor T (CFT) µν . Notice that the counter terms are of the same form as the terms coming from the local action. Therefore, they admit the standard interpretation of inducing the renormalization of the physical four-dimensional constants G 4 (Newton's constant) and Λ 4 (cosmological constant).</text>
<text><location><page_6><loc_17><loc_61><loc_55><loc_63></location>The regularized (physical) stress-energy tensor is</text>
<formula><location><page_6><loc_40><loc_58><loc_88><loc_60></location>T (reg) µν = T (CFT) µν + T (c . t . ) µν . (23)</formula>
<text><location><page_6><loc_13><loc_56><loc_57><loc_58></location>We deduce the energy density and pressure, respectively,</text>
<formula><location><page_6><loc_14><loc_50><loc_81><loc_55></location>/epsilon1 = T (reg) ττ = κ 2 + κ 1 ( H 2 + k a 2 ) -3 8 πG 5 a √ B B , 2</formula>
<formula><location><page_6><loc_13><loc_46><loc_88><loc_51></location>p = T i (reg) i = -κ 2 -κ 1 {( H 2 + k a 2 + 2a a )} + 1 16 πG 5 aA ' B + A [ aB ' ˙ a +2 B ( a a +2˙ a ) +4] aA √ B B . (24)</formula>
<text><location><page_6><loc_13><loc_43><loc_54><loc_45></location>where H = ˙ a/a is the Hubble parameter. The choice</text>
<formula><location><page_6><loc_38><loc_39><loc_88><loc_42></location>κ 1 = 3 L 16 πG 5 , κ 2 = 3 8 πG 5 L (25)</formula>
<text><location><page_6><loc_13><loc_35><loc_88><loc_38></location>ensures finiteness in the limit a →∞ . Unlike the bare energy density (20), the regularized energy density /epsilon1 is positive.</text>
<text><location><page_6><loc_17><loc_32><loc_48><loc_34></location>The boundary conditions (12) now read</text>
<formula><location><page_6><loc_35><loc_28><loc_88><loc_31></location>R µν -1 2 γ µν RΛ 4 γ µν = 8 πG 4 T (reg) µν , (26)</formula>
<text><location><page_6><loc_13><loc_26><loc_85><loc_28></location>which are the four-dimensional Einstein equations in the presence of a cosmological constant.</text>
<text><location><page_6><loc_17><loc_23><loc_87><loc_25></location>The cosmological evolution equation is the ττ component of the Einstein equations (26),</text>
<formula><location><page_6><loc_40><loc_19><loc_88><loc_22></location>H 2 + k a 2 -Λ 4 3 = 8 πG 4 3 /epsilon1 , (27)</formula>
<text><location><page_6><loc_13><loc_12><loc_88><loc_19></location>where /epsilon1 is the energy density given in (24) under the condition (25). This is deceptively similar to the standard equation of cosmological evolution. However, it differs in an essential way, because /epsilon1 contains contributions that involve the Hubble parameter H = ˙ a/a , leading to novel cosmological scenarios.</text>
<section_header_level_1><location><page_7><loc_13><loc_89><loc_58><loc_91></location>4 AdS Reissner-Nordstrom black hole</section_header_level_1>
<text><location><page_7><loc_13><loc_83><loc_88><loc_86></location>In this section we take up the example of an asymptotically AdS charged black hole, namely AdS Reissner-Nordstrom black hole for which the functions A and B of (2) are</text>
<formula><location><page_7><loc_36><loc_79><loc_88><loc_82></location>A ( r ) = 1 B ( r ) = r 2 L 2 + k -M r 2 + Q 2 r 4 . (28)</formula>
<text><location><page_7><loc_13><loc_73><loc_88><loc_77></location>The parameters M and Q are related to the mass and charge of the black hole, respectively. k can be +1, 0, or -1 depending on whether the black hole horizon is spherical, flat, or hyperbolic, respectively.</text>
<text><location><page_7><loc_17><loc_69><loc_40><loc_71></location>The Hawking temperature is</text>
<formula><location><page_7><loc_44><loc_65><loc_88><loc_69></location>T H = 2 r 2 + L 2 + k 2 πr + , (29)</formula>
<text><location><page_7><loc_13><loc_63><loc_49><loc_65></location>where r + is the radius of the horizon satisfying</text>
<formula><location><page_7><loc_37><loc_58><loc_88><loc_62></location>A ( r + ) = r 2 + L 2 + k -M r 2 + + Q 2 r 4 + = 0 . (30)</formula>
<text><location><page_7><loc_13><loc_56><loc_24><loc_57></location>The entropy is</text>
<formula><location><page_7><loc_45><loc_53><loc_88><loc_56></location>S = r 3 + 4 G 5 V 3 , (31)</formula>
<text><location><page_7><loc_13><loc_49><loc_88><loc_52></location>where V 3 is the three-dimensional volume spanned by Ω k . Notice that the entropy is independent of a , and therefore constant in time, leading to an adiabatic evolution.</text>
<text><location><page_7><loc_17><loc_46><loc_68><loc_47></location>According to (7), the redshifted temperature on the boundary is</text>
<text><location><page_7><loc_13><loc_39><loc_35><loc_40></location>For large a , it is expanded as</text>
<formula><location><page_7><loc_43><loc_39><loc_88><loc_45></location>T = T H √ ˙ a 2 + A ( a ) . (32)</formula>
<formula><location><page_7><loc_35><loc_34><loc_88><loc_38></location>T = T H L a -T H L 3 2 a ( H 2 + k a 2 ) + . . . . (33)</formula>
<text><location><page_7><loc_13><loc_32><loc_78><loc_33></location>Similarly, we expand the regularized energy density and pressure (24), respectively,</text>
<formula><location><page_7><loc_15><loc_23><loc_68><loc_31></location>/epsilon1 = 3 L 3 64 πG 5 { ( H 2 + k a 2 ) 2 + 4 M L 2 a 4 } -3 L 5 128 πG 5 H 2 + k a 2 3 + 4 kM L 2 a 6 + 8 Q 2 L 4 a 6 + 4 M L 2 H 2 a 4 + . . . ,</formula>
<formula><location><page_7><loc_15><loc_12><loc_88><loc_26></location>{ ( ) } (34) p = L 3 64 πG 5 { ( H 2 + k a 2 ) 2 + 4 M L 2 a 4 -4 ( H 2 + k a 2 ) a a } -3 L 5 128 πG 5 { ( H 2 + k a 2 ) 3 + 4 kM L 2 a 6 + 8 Q 2 L 4 a 6 + 4 M L 2 H 2 a 4 -2 ( H 2 + k a 2 ) 2 a a -8 3 L M a a 5 } + . . . . (35)</formula>
<text><location><page_8><loc_13><loc_89><loc_83><loc_91></location>We deduce the conformal anomaly which is given by the trace of the stress-energy tensor,</text>
<formula><location><page_8><loc_15><loc_76><loc_88><loc_88></location>Tr T = /epsilon1 -3 p = -3 L 3 16 πG 5 ( H 2 + 1 a 2 ) a a -3 L 5 64 πG 5 { ( H 2 + k a 2 ) 3 + 4 kM L 2 a 6 + 8 Q 2 L 4 a 6 + 4 M L 2 H 2 a 4 -3 ( H 2 + k a 2 ) 2 a a -4 L M a a 5 } + . . . . (36)</formula>
<text><location><page_8><loc_13><loc_73><loc_80><loc_75></location>The first term is the standard conformal anomaly one obtains in the large a limit [17].</text>
<text><location><page_8><loc_13><loc_67><loc_88><loc_72></location>As an example, consider the case of a flat static boundary of a Schwarzschild black hole. Then k = 0, Q = 0, and H = 0. The radius of the horizon is r + = ( ML 2 ) 1 / 4 . The expressions for the energy density, pressure and temperature simplify to, respectively,</text>
<formula><location><page_8><loc_13><loc_58><loc_88><loc_65></location>/epsilon1 = T ττ = 3 8 πG 5 L 1 -√ 1 -r 4 + a 4 , p = T i i = 1 8 πG 5 L 3 -r 4 + a 4 √ 1 -r 4 + a 4 -3 , T = r + πLa √ 1 -r 4 + a 4 (37)</formula>
<text><location><page_8><loc_13><loc_56><loc_49><loc_57></location>In the large a limit, we deduce the expansions</text>
<formula><location><page_8><loc_14><loc_51><loc_88><loc_54></location>/epsilon1 = 3 8 πG 5 L ( ( πLT ) 4 2 -7( πLT ) 8 8 + . . . ) , p = 1 8 πG 5 L ( ( πLT ) 4 2 -3( πLT ) 8 8 + . . . ) . (38)</formula>
<text><location><page_8><loc_13><loc_46><loc_88><loc_49></location>Thus, at leading order, we have /epsilon1 = 3 p ∝ T 4 , as expected for a conformal fluid. Including next-order corrections, we no longer have a traceless stress-energy tensor.</text>
<text><location><page_8><loc_17><loc_43><loc_71><loc_44></location>Returning to the general case, we obtain the law of thermodynamics</text>
<formula><location><page_8><loc_40><loc_39><loc_88><loc_41></location>dE = TdS -pdV +Φ dQ , (39)</formula>
<text><location><page_8><loc_13><loc_37><loc_61><loc_38></location>where E = /epsilon1V , V = a 3 V 3 is the volume, and Φ is the potential</text>
<formula><location><page_8><loc_46><loc_32><loc_88><loc_35></location>Φ = Q G 5 a . (40)</formula>
<text><location><page_8><loc_13><loc_27><loc_88><loc_31></location>This is easily verified, e.g., by differentiating with respect to τ , r + , and Q (after using (30) to express M in terms of the other two parameters, r + and Q ).</text>
<section_header_level_1><location><page_8><loc_13><loc_22><loc_44><loc_23></location>5 Cosmological Evolution</section_header_level_1>
<text><location><page_8><loc_13><loc_14><loc_88><loc_19></location>Next, we discuss various explicit examples of cosmological evolution based on an AdS ReissnerNordstrom black hole. For simplicity, in what follows we shall be working with units in which L = 1.</text>
<text><location><page_9><loc_17><loc_89><loc_62><loc_91></location>The Hubble equation (27) can be massaged into the form</text>
<formula><location><page_9><loc_36><loc_83><loc_88><loc_88></location>β ( H 2 + k a 2 ) = 1 L ' -√ H 2 + A ( a ) a 2 , (41)</formula>
<text><location><page_9><loc_13><loc_81><loc_63><loc_83></location>where we introduced the convenient combinations of parameters</text>
<formula><location><page_9><loc_35><loc_76><loc_88><loc_80></location>β = G 5 G 4 -1 2 , 1 L ' = 1 + (1 + 2 β )Λ 4 6 . (42)</formula>
<text><location><page_9><loc_13><loc_74><loc_54><loc_75></location>The Hubble equation can be expanded for large a as</text>
<formula><location><page_9><loc_20><loc_64><loc_88><loc_72></location>H 2 + k a 2 -Λ 4 3 = G 4 L 3 16 G 5 { ( H 2 + 1 a 2 ) 2 + 4 M L 2 a 4 } -G 4 L 4 16 G 5 { ( H 2 + 1 a 2 ) 3 + 4 M L 2 a 6 + 8 Q 2 L 4 a 6 + 4 M L 2 a 4 H 2 } + . . . . (43)</formula>
<text><location><page_9><loc_13><loc_62><loc_61><loc_63></location>At leading order, it coincides with the result obtained in [17].</text>
<text><location><page_9><loc_13><loc_55><loc_88><loc_60></location>After squaring (41), we obtain a quadratic equation for H 2 . However, only one of the two roots is a solution of (41). Let us concentrate on the range of parameters with β > 0, L ' > 0. We obtain</text>
<text><location><page_9><loc_13><loc_44><loc_88><loc_49></location>This can be solved for a = a ( τ ) to obtain the orbit of the boundary hypersurface. Once a solution of (44) is obtained, we still need to verify that it satisfies (41), because the solutions of (41) in general form a subset of the solutions of (44).</text>
<formula><location><page_9><loc_32><loc_48><loc_88><loc_55></location>H 2 = ( 1 L ' -k a 2 β ) 2 -A ( a ) a 2 1 2 + β L ' -kβ 2 a 2 + √ 1 4 + β L ' +( A ( a ) -k ) β 2 a 2 . (44)</formula>
<text><location><page_9><loc_17><loc_41><loc_86><loc_42></location>The fixed points of the orbits are found by setting H = 0 in (41). They are solutions of</text>
<formula><location><page_9><loc_37><loc_36><loc_88><loc_40></location>V ( a ) ≡ 1 L ' -k a 2 β -1 a √ A ( a ) = 0 . (45)</formula>
<text><location><page_9><loc_13><loc_34><loc_78><loc_35></location>These fixed points are also fixed points of (44), but the converse is not always true.</text>
<text><location><page_9><loc_17><loc_31><loc_74><loc_32></location>With the choice of parameters such that β = 0 [29], eq. (44) simplifies to</text>
<formula><location><page_9><loc_40><loc_25><loc_88><loc_29></location>H 2 = ( 1 + Λ 4 6 ) 2 -A ( a ) a 2 , (46)</formula>
<text><location><page_9><loc_13><loc_20><loc_88><loc_25></location>which coincides with the results from a brane world scenario. Thus we recover the evolution of a 3-brane in a five-dimensional bulk space if we fine tune the parameters of our system so that β = 0.</text>
<text><location><page_9><loc_17><loc_17><loc_42><loc_18></location>The fixed points are solutions of</text>
<formula><location><page_9><loc_38><loc_11><loc_88><loc_15></location>V ( a ) ≡ 1 + Λ 4 6 -1 a √ A ( a ) = 0 . (47)</formula>
<text><location><page_10><loc_13><loc_81><loc_88><loc_91></location>Notice that no fixed points exist between the outer and inner horizons (with A ( a ) < 0), because of the square root in the potential V ( a ). Notice also that V ( a ) ≈ Λ 4 6 as a → ∞ , so the sign of the potential is determined by the sign of Λ 4 , and at the horizon, V ( r + ) = 1 L ' > 0. Up to two fixed points can be outside the horizon. However, our classical results likely receive significant quantum corrections as we approach the horizon. Therefore, our results are reliable for orbits away from the horizon, which typically end at infinite distance from the horizon.</text>
<text><location><page_10><loc_49><loc_51><loc_49><loc_53></location>/negationslash</text>
<text><location><page_10><loc_13><loc_49><loc_88><loc_79></location>For Λ 4 = 0, we recover from (46) the brane world scenario of [30]. This scenario is depicted in figure 1a for k = +1 , M = 8 , Q = 1. We notice here that we have only one solution that is bouncing. Of the two turning points, one is inside the inner horizon and the other outside the outer horizon. There is no fixed point between the inner and outer horizons, as noted earlier, because of the presence of the square root in the potential V ( a ) (45). This can be explicitly seen from figure 2a where we see clearly the position of the inner fixed point as the point where the solid line cuts the a -axis. After crossing the turning point outside the outer horizon, the square of the Hubble parameter becomes negative and hence unphysical. The orbit of the bouncing solution is shown in figure 3. Although we reproduce the bouncing cosmology of Mukherji, et al. , through this, as argued in [31] this kind of solution suffers from an instability. Indeed, the inner horizon is the Cauchy horizon for this charged AdS black hole and is unstable under linear fluctuations about the equilibrium black hole space-time. So when the orbit crosses the inner horizon of the black hole, it is not sufficient to consider only the unperturbed background. The backreaction on the background metric due to the fluctuating modes has to be taken into account. This backreaction is significant and may produce a curvature singularity. It should be noted that this pathology occurs only for β = 0. For β = 0, no outward crossing of the horizon occurs. Thus, from our point of view, β acts as a regulator; keeping it small, but finite, is essential for the handling of quantum fluctuations.</text>
<text><location><page_10><loc_13><loc_35><loc_88><loc_47></location>If we now tune Λ 4 to non-zero values, we obtain qualitatively different solutions. In the simplest case, when there is no chemical potential ( Q = 0), for sufficiently small Λ 4 > 0, and k = 1 (spherical geometry) we recover the de Sitter brane scenario of ref. [32]. As an example, set M = 1, Λ 4 = 0 . 5. For β = 0, we obtain two fixed points a = 1 . 13, 2 . 11, outside the outer horizon ( r + = 1 . 03). As we increase β (i.e., G 5 , or equivalently, decrease G 4 ), the larger fixed point increases and the smaller one decreases. After it hits the horizon, the smaller fixed point disappears and we only have one fixed point. No fixed points exist inside the horizon.</text>
<text><location><page_10><loc_13><loc_18><loc_88><loc_33></location>In the same set up and keeping all other parameters fixed to the afore-mentioned values, if we now turn on the chemical potential, we obtain one more fixed point away from the outer horizon. For Q = 1 this is shown in figure 1b. Similarly to the Λ 4 = 0 case, here we also obtain one bouncing solution with two fixed points, one inside the inner horizon (figure 2b) and the other outside the outer horizon. This solution for a ( τ ) is plotted in figure 4a. Additionally, at a = 7 . 09 there is another fixed point. We obtain an accelerating solution from this point (figure 4b). In the region between the first fixed point outside the outer horizon ( a = 3 . 06) and second one at a = 7 . 09, the square of the Hubble parameter is negative, hence there is no physical solution in this region.</text>
<text><location><page_10><loc_68><loc_14><loc_68><loc_17></location>/negationslash</text>
<text><location><page_10><loc_13><loc_14><loc_88><loc_17></location>Comparing the brane world scenario (46) with the general case, β = 0, we observe that there are no qualitative differences in the flat case ( k = 0). In the case of curved horizon (boundary),</text>
<text><location><page_11><loc_13><loc_88><loc_88><loc_91></location>k = ± 1, in general one obtains fixed points other than the ones obtained in the brane world scenario. As an example, consider the choice of parameters</text>
<formula><location><page_11><loc_31><loc_84><loc_88><loc_86></location>k = +1 , M = 8 , Q = 1 , Λ 4 = 0 . 05 , β = 6 . (48)</formula>
<text><location><page_11><loc_13><loc_78><loc_88><loc_83></location>We have only one fixed point in this case, at a = 7 . 705 (figure 1d). The solution is accelerating as shown in figure 5. There is no bouncing solution for any set of parameters once we go away from the special case β = 0.</text>
<text><location><page_11><loc_22><loc_74><loc_22><loc_76></location>/negationslash</text>
<text><location><page_11><loc_13><loc_72><loc_88><loc_76></location>For β = 0, if we set Λ 4 = 0, we do not obtain any physical solution. One such situation is depicted in figure 1c. As we see, the square of the Hubble parameter is imaginary for all values of the cosmic scale a in this case.</text>
<figure>
<location><page_11><loc_17><loc_19><loc_84><loc_69></location>
<caption>Figure 1: Cosmological evolution scenarios for various values of parameters. Solid and dashed lines are plots of ˙ a 2 and a , respectively. Dotted lines denote the black hole potential with its zeros indicating the positions of the inner and outer horizons.</caption>
</figure>
<figure>
<location><page_12><loc_16><loc_67><loc_84><loc_91></location>
<caption>Figure 2: Solid lines are plots of ˙ a 2 whereas dotted lines are plots of the black hole potential for β = 0 and (a) Λ 4 = 0, (b) Λ 4 = 0 . 05. The inner fixed points and the position of the inner horizon are shown.</caption>
</figure>
<figure>
<location><page_12><loc_35><loc_42><loc_66><loc_57></location>
<caption>Figure 3: Plot of a vs τ for β = 0, Λ 4 = 0.</caption>
</figure>
<figure>
<location><page_12><loc_17><loc_20><loc_84><loc_36></location>
<caption>Figure 4: Plots of a vs τ for β = 0, Λ 4 = 0 . 05. In (a) we have a bounce. Initial conditions are chosen as a (0) = 0 . 356. At τ = 3 . 642, a reaches the second fixed point, a = 3 . 059. In (b) we have an accelerating solution, with initial condition chosen as a (0) = 7 . 090.</caption>
</figure>
<figure>
<location><page_13><loc_35><loc_76><loc_65><loc_91></location>
<caption>Figure 5: Plot of a vs τ for β = 6, Λ 4 = 0 . 05. The initial condition is chosen as a (0) = 7 . 705.</caption>
</figure>
<section_header_level_1><location><page_13><loc_13><loc_67><loc_29><loc_69></location>6 Conclusion</section_header_level_1>
<text><location><page_13><loc_13><loc_40><loc_88><loc_64></location>In conclusion, we discussed the cosmological evolution derived from a static bulk solution of the field equations with appropriately defined mixed boundary conditions using the gravity/gauge theory duality (holography). Such an approach was first discussed in [17]. We extended the results of [17] by considering a boundary hypersurface at arbitrary distance. We calculated the general form of the stress-energy tensor and arrived at a generalized form of the Hubble equation of cosmological evolution. We considered various explicit examples in detail based on an AdS Reissner-Nordstrom bulk black hole solution. Interestingly, we obtained the brane-world scenario as a special case, by fine-tuning the parameters of the system, setting β = 0 (eq. (42)). However, keeping β small but finite is important in order to avoid scenarios in which the boundary crosses the event horizon from within [30]. Thus, β acts as a regulator for such problematic solutions for which quantum fluctuations introduce instabilities [31]. Moreover, the counterterms one normally introduces to cancel the infinities were shown to have the usual field theoretic interpretation of renormalizing the bare parameters of the system (Newton's constant and the cosmological constant).</text>
<text><location><page_13><loc_13><loc_27><loc_88><loc_39></location>It would be interesting to explore the parameter space of the cosmological system further to obtain scenarios of cosmological evolution of interest, such as understanding inflation, and phase transitions in general, in a holographic setting. Various extensions are also possible, such as addition of matter fields on the boundary (without gravity duals). Also, anisotropic cosmologies are possible from a static bulk background, if the boundary hypersurface is chosen with a different geometry than the horizon (e.g., flat boundary ( k = 0) in a bulk black hole background of spherical horizon ( k = +1)). Work in this direction is in progress [33].</text>
<section_header_level_1><location><page_13><loc_13><loc_21><loc_33><loc_23></location>Acknowledgments</section_header_level_1>
<text><location><page_13><loc_13><loc_15><loc_88><loc_18></location>We are grateful to Sudipta Mukherji for discussions. G. S. was supported in part by the US Department of Energy under Grant No. DE-FG05-91ER40627.</text>
<section_header_level_1><location><page_14><loc_13><loc_89><loc_25><loc_91></location>References</section_header_level_1>
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</document>
[
{
"title": "Generalized Holographic Cosmology",
"content": "Souvik Banerjee † , Samrat Bhowmick † , Anurag Sahay † Institute of Physics Bhubaneswar -751 005, India George Siopsis /flat Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996 - 1200, USA",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We consider general black hole solutions in five-dimensional spacetime in the presence of a negative cosmological constant. We obtain a cosmological evolution via the gravity/gauge theory duality (holography) by defining appropriate boundary conditions on a four-dimensional boundary hypersurface. The standard counterterms are shown to renormalize the bare parameters of the system (the four-dimensional Newton's constant and cosmological constant). We discuss the thermodynamics of cosmological evolution and present various examples. The standard braneworld scenarios are shown to be special cases of our holographic construction.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "In recent years, understanding cosmology within the framework of string theory, has been an active and interesting field of study. Starting with [1], a substantial amount of research has been based on modeling the Universe by a 3-brane living in a higher-dimensional bulk space (brane world scenario). An incomplete list of references is [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The Hubble equation of cosmological evolution is thus reproduced by the trajectory of the brane. A different approach to understanding dynamics starting with a higher-dimensional bulk space is provided by the AdS/CFT correspondence [14]. In this approach, a solution of the Einstein equations in a space with a negative cosmological constant is shown to be dual to a gauge theory of lower dimension living on the boundary (gravity/gauge theory duality). The stress-energy tensor of the gauge theory is constructed using holography [15]. This affords an understanding of the gauge theory at strong coupling, which has been applied to various physical systems, such as the quark-gluon plasma formed in heavy ion collisions and high temperature superconductors. It is possible to understand the motion of a fluid, including time-dependent flows. An application to cosmology, which is our interest here, has also been discussed [16, 17]. Applying the gravity/gauge theory duality to a cosmological setting is not straightforward due to the fact that the metric on the boundary space in which the gauge theory lives must remain dynamical. This was long thought to be problematic due to the possibility of the fluctuations of the bulk metric corresponding to non-normalizable modes [18, 19, 20, 21, 22, 23, 24]. It was shown in [25], that such problems can be avoided by introducing appropriate counterterms on the boundary needed to cancel the infinities. In applications of the gravity/gauge theory duality (holography) to cosmology and other settings, one generally places the boundary at a finite distance r and then takes the limit as the cutoff r → ∞ . The removal of the cutoff introduces infinities, which are canceled by the addition of a local action on the boundary with r -dependent coefficients (counterterms). Unlike in quantum field theory, where counterterms are interpreted as renormalization of the (bare) parameters of the system, it is not clear if counterterms have a similiar physical meaning in a holographic setting. Here we generalize the holographic approach to cosmology by placing the boundary hypersurface at a finite distance r and derive expressions for the various physical quantities (e.g., the stress-energy tensor) which are valid for arbitrary r . This leads to a generalized Hubble equation of cosmological evolution. We still need to introduce the standard counterterms to avoid infinities at large r . We show that these counterterms have the usual field theoretic interpretation of renormalizing the (bare) parameters of the system, namely Newton's constant and the cosmological constant. Moreover, we recover the brane-world scenario by fine-tuning Newton's constant. Thus we show that brane-world scenarios are a special case of our generalized holographic approach. Our paper is organized as follows. In section 2 we discuss the bulk space concentrating on a time-independent solution (general black hole) of the field equations, and define the boundary hypersurface. In section 3 we introduce the boundary conditions and the counterterms needed to cancel infinities. We calculate the stress-energy tensor and derive the Hubble equation of cosmo- logical evolution. In section 4 we discuss the example of a bulk Reissner-Nordstrom black hole including thermodynamics. In section 5 we discuss various examples of cosmological evolution. In particular, we show that the brane-world scenarios are a special case of our holographic approach. Finally in section 6 we conclude.",
"pages": [
2,
3
]
},
{
"title": "2 The Bulk",
"content": "We start with a non-extremal black hole in a 4 + 1 dimensional bulk space in the presence of a negative cosmological constant We consider the metric ansatz r being the radial direction, and k = +1 , 0 , -1 depending on the geometry of the constant ( t, r ) hypersurfaces (spherical, flat, or hyperbolic, respectively). More general metrics are also possible, but will clutter the notation unnecessarily. In section 4, we shall concentrate on the special case of a Reissner-Nordstrom black hole for explicit calculations. Asymptotically, we have AdS space of radius L , therefore as r →∞ , We introduce a radial cutoff, r = a and parametrise a and t as a = a ( τ ) and t = t ( τ ) so that a = ˙ adt . Then the metric on the cut-off surface (boundary) takes the form In order that the metric on the boundary take the FRW form, the metric components should satisfy the relation This in turn fixes our choice of the time parameter τ . Notice also that if T H is the Hawking temperature, then the temperature on the boundary is redshifted, This kind of parametrization has been used before, e.g., in [26, 27, 28]. Note that, while treating τ as a time parameter, we are effectively considering the radial motion of the cut-off surface in the 4+1 dimensional bulk. By adopting appropriate boundary conditions, the cut-off surface can be thought of as the location of a brane, mimicking a moving brane scenario.",
"pages": [
3
]
},
{
"title": "3 Boundary Conditions",
"content": "The heart of the construction we are going to elaborate on is based on the observation that the afore-mentioned dynamics of the boundary hypersurface will be captured through the boundary conditions we impose on the system. This approach was first adopted in [17]. Let us consider a general five dimensional bulk action, where we keep the Lagrangian density L 5 unspecified. In the simplest case, this consists of a fivedimensional Einstein-Hilbert action with a negative cosmological constant (1) plus the requisite Gibbons-Hawking surface term for a well-defined variational principle. If one varies this action with respect to the metric, one obtains a boundary term of the form where γ µν is the induced boundary metric and γ is its determinant. T (CFT) µν denotes the (bare) stress-energy tensor of the dual conformal field theory that lives on the four-dimensional boundary hypersurface r = a . Generally in the context of the AdS/CFT correspondence, Dirichlet boundary conditions are employed, which fix the boundary metric and consequently eq. (9) vanishes. While this leads to a well-defined variational principle, it does not allow for a dynamical boundary metric. Since we are primarily interested in obtaining a cosmological evolution and hence a dynamical metric on the boundary, we seek different boundary conditions that can be imposed without fixing the metric on the boundary. It was noted in [17] that one could adopt appropriate mixed boundary conditions, which were shown to lead to valid dynamics in [25]. Their definition involves the addition of an appropriate local action, S local , at the boundary. For cosmological evolution, this local action will be chosen as the four-dimensional Einstein-Hilbert action on the boundary with an arbitrary (positive, negative, or vanishing) four-dimensional cosmological constant Λ 4 , where R [ γ ] is the Ricci scalar evaluated with the boundary metric which, in our case, is the FRW metric (5). Notice that the cosmological constant may be due, wholly or partly, to a brane of finite tension at the boundary. Additionally, to cancel divergences in the limit a → ∞ , it is necessary to introduce counterterms [15]. These are of the same form as the local action and renormalize the four-dimensional physical parameters G 4 and Λ 4 . We have which diverges as a →∞ . The parameters κ 1 and κ 2 will be chosen so that physical quantities such as the energy density and pressure remain finite in this limit. Putting these pieces together, we define our boundary condition as where T (CFT) µν is due to the variation δS 5 (eq. (9)), and the other two terms, T (local) µν and T (c . t . ) µν come from the variations respectively, with respect to the boundary metric, γ µν . Similarly to Dirichlet boundary conditions, the choice (12) leads to a well-defined variational principle with To see the explicit physical content of our mixed boundary conditions (12), we shall derive explicit expressions for each of the three contributing terms. The bare stress-energy tensor on the boundary is given by where K µν is the extrinsic curvature, and K is its trace. The components of this tensor can be evaluated by computing the velocity v µ and unit normal n ν vectors on the boundary hypersurface, r = a ( τ ). For the metric (2), these vectors are given in component form as and respectively. The direction of the unit normal vector is taken to be pointing inward, toward the bulk. The extrinsic curvature can be written in terms of the unit normal and velocity vectors as Explicitly, they are where i, j are indices for the spatial coordinates on the boundary (spanned by Ω k ). We deduce the explicit expressions for the components of the bare stress-energy tensor (15), where no summing over the index i is implied. Notice that the energy density T (CFT) ττ obtained above is negative, however we should emphasize that this is only a bare quantity and therefore not physical. It will be corrected by the addition of counter terms resulting into a positive regularized (physical) quantity. For the remaining two contributions in (12), we obtain the standard expressions one encounters in Einstein's four-dimensional equations, where R µν ( R ) is the four-dimensional Ricci tensor (scalar) constructed from the four-dimensional boundary metric γ µν . The counter terms diverge in the limit a → ∞ , and the parameters κ 1 and κ 2 will be chosen so that they cancel the divergences in the bare stress-energy tensor T (CFT) µν . Notice that the counter terms are of the same form as the terms coming from the local action. Therefore, they admit the standard interpretation of inducing the renormalization of the physical four-dimensional constants G 4 (Newton's constant) and Λ 4 (cosmological constant). The regularized (physical) stress-energy tensor is We deduce the energy density and pressure, respectively, where H = ˙ a/a is the Hubble parameter. The choice ensures finiteness in the limit a →∞ . Unlike the bare energy density (20), the regularized energy density /epsilon1 is positive. The boundary conditions (12) now read which are the four-dimensional Einstein equations in the presence of a cosmological constant. The cosmological evolution equation is the ττ component of the Einstein equations (26), where /epsilon1 is the energy density given in (24) under the condition (25). This is deceptively similar to the standard equation of cosmological evolution. However, it differs in an essential way, because /epsilon1 contains contributions that involve the Hubble parameter H = ˙ a/a , leading to novel cosmological scenarios.",
"pages": [
4,
5,
6
]
},
{
"title": "4 AdS Reissner-Nordstrom black hole",
"content": "In this section we take up the example of an asymptotically AdS charged black hole, namely AdS Reissner-Nordstrom black hole for which the functions A and B of (2) are The parameters M and Q are related to the mass and charge of the black hole, respectively. k can be +1, 0, or -1 depending on whether the black hole horizon is spherical, flat, or hyperbolic, respectively. The Hawking temperature is where r + is the radius of the horizon satisfying The entropy is where V 3 is the three-dimensional volume spanned by Ω k . Notice that the entropy is independent of a , and therefore constant in time, leading to an adiabatic evolution. According to (7), the redshifted temperature on the boundary is For large a , it is expanded as Similarly, we expand the regularized energy density and pressure (24), respectively, We deduce the conformal anomaly which is given by the trace of the stress-energy tensor, The first term is the standard conformal anomaly one obtains in the large a limit [17]. As an example, consider the case of a flat static boundary of a Schwarzschild black hole. Then k = 0, Q = 0, and H = 0. The radius of the horizon is r + = ( ML 2 ) 1 / 4 . The expressions for the energy density, pressure and temperature simplify to, respectively, In the large a limit, we deduce the expansions Thus, at leading order, we have /epsilon1 = 3 p ∝ T 4 , as expected for a conformal fluid. Including next-order corrections, we no longer have a traceless stress-energy tensor. Returning to the general case, we obtain the law of thermodynamics where E = /epsilon1V , V = a 3 V 3 is the volume, and Φ is the potential This is easily verified, e.g., by differentiating with respect to τ , r + , and Q (after using (30) to express M in terms of the other two parameters, r + and Q ).",
"pages": [
7,
8
]
},
{
"title": "5 Cosmological Evolution",
"content": "Next, we discuss various explicit examples of cosmological evolution based on an AdS ReissnerNordstrom black hole. For simplicity, in what follows we shall be working with units in which L = 1. The Hubble equation (27) can be massaged into the form where we introduced the convenient combinations of parameters The Hubble equation can be expanded for large a as At leading order, it coincides with the result obtained in [17]. After squaring (41), we obtain a quadratic equation for H 2 . However, only one of the two roots is a solution of (41). Let us concentrate on the range of parameters with β > 0, L ' > 0. We obtain This can be solved for a = a ( τ ) to obtain the orbit of the boundary hypersurface. Once a solution of (44) is obtained, we still need to verify that it satisfies (41), because the solutions of (41) in general form a subset of the solutions of (44). The fixed points of the orbits are found by setting H = 0 in (41). They are solutions of These fixed points are also fixed points of (44), but the converse is not always true. With the choice of parameters such that β = 0 [29], eq. (44) simplifies to which coincides with the results from a brane world scenario. Thus we recover the evolution of a 3-brane in a five-dimensional bulk space if we fine tune the parameters of our system so that β = 0. The fixed points are solutions of Notice that no fixed points exist between the outer and inner horizons (with A ( a ) < 0), because of the square root in the potential V ( a ). Notice also that V ( a ) ≈ Λ 4 6 as a → ∞ , so the sign of the potential is determined by the sign of Λ 4 , and at the horizon, V ( r + ) = 1 L ' > 0. Up to two fixed points can be outside the horizon. However, our classical results likely receive significant quantum corrections as we approach the horizon. Therefore, our results are reliable for orbits away from the horizon, which typically end at infinite distance from the horizon. /negationslash For Λ 4 = 0, we recover from (46) the brane world scenario of [30]. This scenario is depicted in figure 1a for k = +1 , M = 8 , Q = 1. We notice here that we have only one solution that is bouncing. Of the two turning points, one is inside the inner horizon and the other outside the outer horizon. There is no fixed point between the inner and outer horizons, as noted earlier, because of the presence of the square root in the potential V ( a ) (45). This can be explicitly seen from figure 2a where we see clearly the position of the inner fixed point as the point where the solid line cuts the a -axis. After crossing the turning point outside the outer horizon, the square of the Hubble parameter becomes negative and hence unphysical. The orbit of the bouncing solution is shown in figure 3. Although we reproduce the bouncing cosmology of Mukherji, et al. , through this, as argued in [31] this kind of solution suffers from an instability. Indeed, the inner horizon is the Cauchy horizon for this charged AdS black hole and is unstable under linear fluctuations about the equilibrium black hole space-time. So when the orbit crosses the inner horizon of the black hole, it is not sufficient to consider only the unperturbed background. The backreaction on the background metric due to the fluctuating modes has to be taken into account. This backreaction is significant and may produce a curvature singularity. It should be noted that this pathology occurs only for β = 0. For β = 0, no outward crossing of the horizon occurs. Thus, from our point of view, β acts as a regulator; keeping it small, but finite, is essential for the handling of quantum fluctuations. If we now tune Λ 4 to non-zero values, we obtain qualitatively different solutions. In the simplest case, when there is no chemical potential ( Q = 0), for sufficiently small Λ 4 > 0, and k = 1 (spherical geometry) we recover the de Sitter brane scenario of ref. [32]. As an example, set M = 1, Λ 4 = 0 . 5. For β = 0, we obtain two fixed points a = 1 . 13, 2 . 11, outside the outer horizon ( r + = 1 . 03). As we increase β (i.e., G 5 , or equivalently, decrease G 4 ), the larger fixed point increases and the smaller one decreases. After it hits the horizon, the smaller fixed point disappears and we only have one fixed point. No fixed points exist inside the horizon. In the same set up and keeping all other parameters fixed to the afore-mentioned values, if we now turn on the chemical potential, we obtain one more fixed point away from the outer horizon. For Q = 1 this is shown in figure 1b. Similarly to the Λ 4 = 0 case, here we also obtain one bouncing solution with two fixed points, one inside the inner horizon (figure 2b) and the other outside the outer horizon. This solution for a ( τ ) is plotted in figure 4a. Additionally, at a = 7 . 09 there is another fixed point. We obtain an accelerating solution from this point (figure 4b). In the region between the first fixed point outside the outer horizon ( a = 3 . 06) and second one at a = 7 . 09, the square of the Hubble parameter is negative, hence there is no physical solution in this region. /negationslash Comparing the brane world scenario (46) with the general case, β = 0, we observe that there are no qualitative differences in the flat case ( k = 0). In the case of curved horizon (boundary), k = ± 1, in general one obtains fixed points other than the ones obtained in the brane world scenario. As an example, consider the choice of parameters We have only one fixed point in this case, at a = 7 . 705 (figure 1d). The solution is accelerating as shown in figure 5. There is no bouncing solution for any set of parameters once we go away from the special case β = 0. /negationslash For β = 0, if we set Λ 4 = 0, we do not obtain any physical solution. One such situation is depicted in figure 1c. As we see, the square of the Hubble parameter is imaginary for all values of the cosmic scale a in this case.",
"pages": [
8,
9,
10,
11
]
},
{
"title": "6 Conclusion",
"content": "In conclusion, we discussed the cosmological evolution derived from a static bulk solution of the field equations with appropriately defined mixed boundary conditions using the gravity/gauge theory duality (holography). Such an approach was first discussed in [17]. We extended the results of [17] by considering a boundary hypersurface at arbitrary distance. We calculated the general form of the stress-energy tensor and arrived at a generalized form of the Hubble equation of cosmological evolution. We considered various explicit examples in detail based on an AdS Reissner-Nordstrom bulk black hole solution. Interestingly, we obtained the brane-world scenario as a special case, by fine-tuning the parameters of the system, setting β = 0 (eq. (42)). However, keeping β small but finite is important in order to avoid scenarios in which the boundary crosses the event horizon from within [30]. Thus, β acts as a regulator for such problematic solutions for which quantum fluctuations introduce instabilities [31]. Moreover, the counterterms one normally introduces to cancel the infinities were shown to have the usual field theoretic interpretation of renormalizing the bare parameters of the system (Newton's constant and the cosmological constant). It would be interesting to explore the parameter space of the cosmological system further to obtain scenarios of cosmological evolution of interest, such as understanding inflation, and phase transitions in general, in a holographic setting. Various extensions are also possible, such as addition of matter fields on the boundary (without gravity duals). Also, anisotropic cosmologies are possible from a static bulk background, if the boundary hypersurface is chosen with a different geometry than the horizon (e.g., flat boundary ( k = 0) in a bulk black hole background of spherical horizon ( k = +1)). Work in this direction is in progress [33].",
"pages": [
13
]
},
{
"title": "Acknowledgments",
"content": "We are grateful to Sudipta Mukherji for discussions. G. S. was supported in part by the US Department of Energy under Grant No. DE-FG05-91ER40627.",
"pages": [
13
]
}
]
2013CQGra..30h5009R
https://arxiv.org/pdf/1208.3601.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_80><loc_71><loc_82></location>Superradiant scattering of dispersive fields</section_header_level_1>
<text><location><page_1><loc_23><loc_73><loc_82><loc_77></location>Mauricio Richartz 1 , Angus Prain 2 , 3 , Silke Weinfurtner 2 , 3 and Stefano Liberati 2 , 3</text>
<text><location><page_1><loc_23><loc_70><loc_81><loc_73></location>1 Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC (UFABC), 09210-170 Santo Andr´e, SP, Brazil,</text>
<list_item><location><page_1><loc_23><loc_67><loc_84><loc_70></location>2 SISSA - International School for Advanced Studies via Bonomea 265, 34136 Trieste, Italy,</list_item>
<text><location><page_1><loc_23><loc_66><loc_24><loc_66></location>3</text>
<text><location><page_1><loc_25><loc_65><loc_42><loc_66></location>INFN, Sezione di Trieste</text>
<text><location><page_1><loc_23><loc_61><loc_69><loc_64></location>E-mail: mauricio.richartz@ufabc.edu.br , aprain@sissa.it , silkiest@gmail.com , liberati@sissa.it</text>
<section_header_level_1><location><page_1><loc_23><loc_57><loc_31><loc_59></location>Abstract.</section_header_level_1>
<text><location><page_1><loc_23><loc_34><loc_84><loc_57></location>Motivated by analogue models of classical and quantum field theory in curved spacetimes and their recent experimental realizations, we consider wave scattering processes of dispersive fields exhibiting two extra scattering channels. In particular, we investigate how standard superradiant scattering processes are affected by subluminal or superluminal modifications of the dispersion relation. We analyze simple 1+1dimensional toy-models based on fourth-order corrections to the standard second order wave equation and show that low-frequency waves impinging on generic scattering potentials can be amplified during the process. In specific cases, by assuming a simple step potential, we determine quantitatively the deviations in the amplification spectrum that arise due to dispersion, and demonstrate that the amplification can be further enhanced due to the presence of extra scattering channels. We also consider dispersive scattering processes in which the medium where the scattering takes place is moving with respect to the observer and show that superradiance can also be manifest in such situations.</text>
<text><location><page_1><loc_23><loc_29><loc_63><loc_31></location>PACS numbers: 04.62.+v, 04.70.Dy, 47.35.Bb, 03.65.Nk</text>
<section_header_level_1><location><page_2><loc_12><loc_87><loc_42><loc_88></location>1. Introduction and motivation</section_header_level_1>
<text><location><page_2><loc_12><loc_67><loc_84><loc_85></location>Experimental realizations of analogue black holes [1, 2] and their associated effects have drawn a lot of attention in the past few years [3, 4, 5, 6, 7, 8, 9]. Probably the most discussed results have been the first observation of the classical analogue of Hawking radiation in an open channel flow [9] and the still controversial observation of radiation in ultrashort laser pulse filaments [8, 10, 11, 12, 13]. Even though it remains an open question whether or not real black holes emit Hawking radiation, calculations involving analogue black holes suggest that the emission process is most probably unaffected by transplanckian effects that could, in principle, alter or even exclude the radiation process [14, 15, 16].</text>
<text><location><page_2><loc_12><loc_45><loc_84><loc_67></location>Superradiance [17, 18, 19] is another typical phenomenon of black hole physics [20, 21] which is also manifest in analogue models of gravity [22]. In standard scattering processes, the ratio between the reflected and the incident particle number currents [23] (i.e. the reflection coefficient) is smaller than one. This is directly encoded by the fact that the amplitude of the reflected wave is usually smaller than the amplitude of the incident one in non-dispersive normalized scattering processes. However, in some special situations (e.g. wave scattering in a Kerr black hole spacetime), low-frequency incident waves can be amplified in the scattering process. This amplification effect, known as superradiance, was first discovered by Zel'dovich in the context of electromagnetic waves [24], and later shown to be a more general phenomenon in physics [18, 19], in which classical as well as quantum field excitations can be amplified.</text>
<text><location><page_2><loc_12><loc_31><loc_84><loc_45></location>The main goal of our work is to analyze the robustness of the amplification process for dispersive fields. The dispersive fields considered in this paper exhibit two extra scattering channels, such that one can study the overall robustness of the amplification process for multiple superradiant scattering. Additionally, in some simplified cases we also investigate the specific deviations in the amplification spectrum due to dispersion. The examples discussed in this paper are motivated by analogue models of gravity, where fields exhibiting sub or superluminal dispersion relations arise naturally.</text>
<section_header_level_1><location><page_2><loc_12><loc_27><loc_33><loc_29></location>1.1. Superradiant systems</section_header_level_1>
<text><location><page_2><loc_30><loc_17><loc_30><loc_20></location>/negationslash</text>
<text><location><page_2><loc_12><loc_12><loc_84><loc_26></location>Before starting our analysis of superradiance for dispersive fields, it is important to review the basic ingredients that make the phenomenon possible in some simple systems. For example, in a Kerr black hole, incident scalar field modes of frequency ω and azimuthal number m = 0 are known to be superradiant for sufficiently low frequencies. Although the Klein-Gordon equation is very complicated when expressed in standard Boyer-Lindquist coordinates, the radial part of the equation of motion for these modes can be written, after a change of variables, very simply as</text>
<formula><location><page_2><loc_39><loc_8><loc_84><loc_11></location>d 2 u dr ∗ 2 + V ω,m ( r ∗ ) u = 0 , (1)</formula>
<text><location><page_2><loc_12><loc_3><loc_84><loc_8></location>where u is related to the radial part of the separated scalar field and r ∗ is a tortoise-like coordinate (which goes to -∞ at the event horizon and reduces to the standard radial</text>
<text><location><page_3><loc_12><loc_85><loc_84><loc_89></location>coordinate r at spatial infinity). V ω,m ( r ∗ ) is the effective potential and possesses the following asymptotics,</text>
<formula><location><page_3><loc_32><loc_79><loc_84><loc_84></location>V ω,m ( r ∗ ) → { ( ω -m Ω h ) 2 , r ∗ →-∞ , ω 2 , r ∗ → + ∞ , (2)</formula>
<text><location><page_3><loc_12><loc_73><loc_84><loc_78></location>where Ω h is the angular velocity of the black hole at the event horizon. It can be shown that, for 0 < ω < m Ω h , incident modes are superradiantly scattered by the black hole [25, 20]. The corresponding reflection coefficient is given by</text>
<formula><location><page_3><loc_36><loc_69><loc_84><loc_72></location>| R | 2 = 1 -ω -m Ω h ω | T | 2 > 1 , (3)</formula>
<text><location><page_3><loc_12><loc_52><loc_84><loc_68></location>where ω -m Ω h ω | T | 2 is the transmission coefficient. Two factors are responsible for the occurrence of the effect [19]: first, the event horizon behaves as a one-way membrane that allows only ingoing transmitted waves (defined by the group velocity) near the black hole; second, because of the ergoregion of the rotating black hole, low-frequency ingoing waves are associated with an outgoing particle number current at the horizon (the current being proportional to ω -m Ω h ) [23]. This outgoing flux at the horizon is compensated by the superradiantly reflected modes in such a way that the total particle number current is conserved during the process.</text>
<text><location><page_3><loc_12><loc_44><loc_84><loc_52></location>Another simple system exhibiting superradiance is a massless scalar field φ with electric charge e and minimally coupled to an electromagnetic potential A µ = ( V ( x ) , 0) in 1+1-dimensions [17]. The evolution of this system is determined by the Klein-Gordon equation,</text>
<formula><location><page_3><loc_35><loc_41><loc_84><loc_44></location>( ∂ µ -ieA µ ) ( ∂ µ -ieA µ ) φ = 0 , (4)</formula>
<text><location><page_3><loc_12><loc_38><loc_84><loc_41></location>which, after separation of the temporal dependence (i.e. φ = f ( x ) exp( -iωt )), reduces to</text>
<formula><location><page_3><loc_37><loc_34><loc_84><loc_38></location>d 2 f dx 2 +( ω -eV ( x )) 2 f = 0 , (5)</formula>
<text><location><page_3><loc_12><loc_30><loc_84><loc_34></location>where ω is the frequency of the mode in question. If the electromagnetic potential V ( x ) has the following asymptotic behaviour,</text>
<formula><location><page_3><loc_37><loc_24><loc_84><loc_29></location>V ( x ) = { 0 , x →-∞ , e Φ 0 , x → + ∞ , (6)</formula>
<text><location><page_3><loc_12><loc_20><loc_84><loc_24></location>one can calculate the relation between the reflection and transmission coefficients for incident waves [17]:</text>
<formula><location><page_3><loc_38><loc_17><loc_84><loc_20></location>| R | 2 = 1 -ω -e Φ 0 ω | T | 2 . (7)</formula>
<text><location><page_3><loc_12><loc_13><loc_84><loc_17></location>Therefore, low frequency (0 < ω < e Φ 0 ) ‡ right-moving modes originating from -∞ interact with the scattering potential, resulting in transmitted right-moving modes</text>
<text><location><page_4><loc_12><loc_72><loc_84><loc_89></location>and superradiantly reflected ( | R | 2 > 1) left-moving modes. This is basically the Klein paradox [26] - see [17] for a detailed explanation of the relationship between superradiance, the Klein paradox and pair creation. Note that the important ingredients for superradiance are essentially the same as in the rotating black hole case: firstly, there is no left-moving mode at + ∞ (this is similar to the boundary condition at the event horizon of a black hole, where no outgoing solutions are allowed); secondly, the form of the electromagnetic potential allows for low-frequency right-moving modes to be associated with left-moving particle number currents at + ∞ .</text>
<section_header_level_1><location><page_4><loc_12><loc_69><loc_42><loc_70></location>2. Modified dispersion relations</section_header_level_1>
<text><location><page_4><loc_12><loc_57><loc_84><loc_67></location>In order to study how a modified dispersion relation affects superradiance, we shall first generalize the simple, non-dispersive, 1+1-dimensional model above [17] by including fourth-order terms in (4). Inspired by the quartic dispersion relation Ω 2 = k 2 ± k 4 / Λ 2 , where Λ is a dispersive momentum scale and the ± notates super and subluminal dispersion respectively, we propose the following generalization of (4),</text>
<formula><location><page_4><loc_31><loc_52><loc_84><loc_56></location>∓ 1 Λ 2 ∂ 4 x φ +( ∂ µ -ieA µ ) ( ∂ µ -ieA µ ) φ = 0 , (8)</formula>
<text><location><page_4><loc_12><loc_49><loc_78><loc_51></location>which can be obtained from the following action for the complex scalar field φ ,</text>
<formula><location><page_4><loc_20><loc_42><loc_84><loc_48></location>S = ∫ dtdx { [( ∂ µ -ieA µ ) φ ] [( ∂ µ + ieA µ ) φ ∗ ] ± 1 Λ 2 ( ∂ 2 x φ ) ( ∂ 2 x φ ∗ ) } . (9)</formula>
<text><location><page_4><loc_12><loc_41><loc_76><loc_43></location>After separating the temporal dependence in (8), instead of (5), one obtains</text>
<formula><location><page_4><loc_33><loc_36><loc_84><loc_40></location>∓ 1 Λ 2 f '''' + f '' +( ω -eV ( x )) 2 f = 0 , (10)</formula>
<text><location><page_4><loc_12><loc_29><loc_84><loc_35></location>where f represents the field mode with frequency ω > 0 and prime denotes derivative with respect to x . We choose the effective frequency Ω( x ) = ω -eV ( x ) to satisfy asymptotic relations similar to those in (2) and (6),</text>
<formula><location><page_4><loc_30><loc_22><loc_84><loc_27></location>Ω 2 ( x ) = { ω 2 , for x →-∞ , ( ω -e Φ 0 ) 2 , for x → + ∞ , (11)</formula>
<text><location><page_4><loc_12><loc_16><loc_84><loc_21></location>where e Φ 0 is a positive constant. In the asymptotic regions, the solutions of (10) are simple exponentials, exp (i kx ), whose wavenumbers k satisfy the dispersion relations below,</text>
<formula><location><page_4><loc_25><loc_11><loc_84><loc_14></location>∓ k 4 / Λ 2 -k 2 + ω 2 = 0 , x →-∞ , (12)</formula>
<formula><location><page_4><loc_25><loc_9><loc_84><loc_11></location>∓ k 4 / Λ 2 -k 2 +( ω -e Φ 0 ) 2 = 0 , x → + ∞ . (13)</formula>
<text><location><page_4><loc_12><loc_4><loc_84><loc_8></location>In the subluminal case (lower sign), real solutions ( ω, k ) to the dispersion relation correspond to the intersections of a lemniscate (figure-eight) and a straight line, see</text>
<text><location><page_5><loc_12><loc_74><loc_84><loc_89></location>FIG. 1. In the superluminal case (upper sign), real solutions correspond to the intersections of a quartic curve and a straight line, see FIG. 4. In order to relate the asymptotic solutions at + ∞ and -∞ without having to solve the differential equation for all values of x , one needs a conserved quantity analogous to the Wronskian for second order wave equations. Since our model is non-dissipative, we expect such a quantity to exist [27]. In fact, by calculating the x -component of the Noether current associated with the symmetry φ → e iα φ , it is possible to show that the expression</text>
<formula><location><page_5><loc_37><loc_70><loc_84><loc_73></location>Z [ f ] = W 1 + W 2 ∓ Λ 2 W 3 , (14)</formula>
<text><location><page_5><loc_12><loc_68><loc_17><loc_69></location>where</text>
<formula><location><page_5><loc_40><loc_59><loc_84><loc_66></location>W 1 = f '''∗ f -f ∗ f ''' , W 2 = f '∗ f '' -f ''∗ f ' , (15) W 3 = f '∗ f -f ∗ f ' ,</formula>
<text><location><page_5><loc_12><loc_52><loc_84><loc_58></location>generalizes the notion of the Wronskian § to our dispersive model, i.e. dZ/dx = 0 for any solution f of (10). Furthermore, it is convenient to work with the scaled functional X whose action on a function f is defined by</text>
<formula><location><page_5><loc_43><loc_47><loc_84><loc_51></location>X [ f ] = Z [ f ] 2iΛ 2 . (16)</formula>
<text><location><page_5><loc_12><loc_42><loc_84><loc_46></location>In particular, the action of X on a linear combination of 'on shell' plane waves (wavenumbers satisfying the dispersion relation) is simply</text>
<formula><location><page_5><loc_34><loc_35><loc_84><loc_41></location>X [ ∑ n A n e ik n x ] = ∑ n Ω dω dk n | A n | 2 . (17)</formula>
<text><location><page_5><loc_12><loc_25><loc_84><loc_35></location>We would like to emphasize the simplicity of the algebraic expression above, which only depends on the amplitudes, effective frequencies and group velocities of the various scattering channels participating in the scattering process. Throughout the paper, we shall refer to (17) as the particle number current since it generalizes the usual notion of particle number current associated with a complex Klein-Gordon field.</text>
<section_header_level_1><location><page_5><loc_12><loc_21><loc_34><loc_22></location>2.1. Subluminal scattering</section_header_level_1>
<text><location><page_5><loc_12><loc_10><loc_84><loc_20></location>Based on the general notion of superradiance described in the introduction, we will study the scattering process of incident waves originating from x →-∞ in the presence of a subluminal dispersion. In realistic scenarios, we do not expect the dispersion relation Ω 2 = k 2 -k 4 / Λ 2 to have a maximum/minimum value above/below which only imaginary solutions of the dispersion relation are possible. Therefore, in order to guarantee that at</text>
<text><location><page_6><loc_26><loc_84><loc_28><loc_85></location>0.5</text>
<text><location><page_6><loc_25><loc_77><loc_28><loc_78></location>0.25</text>
<text><location><page_6><loc_27><loc_71><loc_28><loc_72></location>0</text>
<text><location><page_6><loc_24><loc_64><loc_27><loc_65></location>-0.25</text>
<text><location><page_6><loc_25><loc_57><loc_27><loc_58></location>-0.5</text>
<figure>
<location><page_6><loc_28><loc_57><loc_68><loc_88></location>
<caption>Figure 1. The dispersion relation for the subluminal case shown in terms of the dimensionless variables k/ Λ and Ω / Λ, where k is the wavenumber and Ω = ω -eV ( x ) is the effective frequency. The green dash-dotted line represents Ω = ω when x →-∞ while the blue dashed line corresponds to Ω = ω -e Φ 0 when x →∞ . Here 0 < ω < e Φ 0 is the fixed lab frequency satisfying the condition for superradiance. We have indicated the incident mode by a black square dot, the transmitted and reflected modes by black circular dots and the remaining solutions of the dispersion relation by white circular dots.</caption>
</figure>
<text><location><page_6><loc_32><loc_55><loc_34><loc_56></location>-1</text>
<text><location><page_6><loc_38><loc_55><loc_41><loc_56></location>-0.5</text>
<text><location><page_6><loc_52><loc_55><loc_55><loc_56></location>0.5</text>
<text><location><page_6><loc_59><loc_55><loc_61><loc_56></location>1</text>
<text><location><page_6><loc_12><loc_13><loc_84><loc_36></location>least one real mode is present in the dispersion relation, we assume that the dispersion parameter Λ is large compared to the electromagnetic interaction, i.e. we assume that e Φ 0 < Λ / 2. In such situations, the behaviour of the solutions to (10) in the asymptotic limits is captured in FIG. 1, where the green dash-dotted (Ω = ω ) and the blue dashed (Ω = ω -e Φ 0 ) lines represent the effective frequency when x → -∞ and x → + ∞ , respectively. Note that there exist four roots, corresponding to four propagating modes as one would expect from a fourth order differential equation. We also note that, when ω < e Φ 0 , the blue dashed line is located below the Ω = 0 axis and the roots in the region x → + ∞ inherit a relative sign between their group and phase velocities with respect to the roots at x → + ∞ (compare the intersections of the green dash-dotted line ( x →-∞ ) and the blue dashed line ( x → + ∞ ) with the red solid curve).</text>
<text><location><page_6><loc_12><loc_6><loc_84><loc_14></location>We label the four roots associated with a frequency 0 < ω < e Φ 0 as k a < k b < k c < k d when x → -∞ and as k A < k B < k C < k D when x → + ∞ (they correspond to the intersections of the straight lines with the figure-eight in FIG. 1). The following table summarizes the character of these modes, where u and v notate right-movers and</text>
<table>
<location><page_7><loc_12><loc_70><loc_37><loc_86></location>
<caption>left-movers respectively,</caption>
</table>
<table>
<location><page_7><loc_39><loc_70><loc_65><loc_86></location>
</table>
<text><location><page_7><loc_12><loc_57><loc_84><loc_68></location>Since, in this setup, there is only one source of waves, located at x →-∞ , and no incoming signal from x → + ∞ , we impose the boundary condition that the modes A and C are unpopulated in the scattering process. Furthermore, we choose the incoming mode at x → -∞ to be entirely composed of low-momentum c modes with no highmomentum a mode component. ‖</text>
<text><location><page_7><loc_16><loc_56><loc_67><loc_58></location>The corresponding solution of (10) in the asymptotic limits is</text>
<formula><location><page_7><loc_29><loc_49><loc_84><loc_55></location>f → { e ik in x + R 1 e ik r 1 x + R 2 e ik r 2 x , x →-∞ , T 1 e ik t 1 x + T 2 e ik t 2 x , x → + ∞ , (18)</formula>
<text><location><page_7><loc_12><loc_41><loc_84><loc_48></location>where the wavenumber of the incident mode is given by k in = k c , the reflected modes are given by k r 1 = k b , k r 2 = k d , and the transmitted modes are k t 1 = k B , k t 2 = k D . In addition, the coefficients R 1 , R 2 , T 1 and T 2 can be related to reflection and transmission coefficients (see (20) below).</text>
<text><location><page_7><loc_12><loc_37><loc_84><loc_40></location>In the asymptotic regions, it is possible to solve exactly the dispersion relation and find the following explicit expressions for the roots,</text>
<formula><location><page_7><loc_33><loc_13><loc_84><loc_34></location>k in = -k r 1 = Λ √ 2 √ 1 -√ 1 -4 ω 2 Λ 2 , k r 2 = Λ √ 2 √ 1 + √ 1 -4 ω 2 Λ 2 , (19) k t 1 = -Λ √ 2 √ 1 -√ 1 -4( ω -e Φ 0 ) 2 Λ 2 , k t 2 = Λ √ 2 √ 1 + √ 1 -4( ω -e Φ 0 ) 2 Λ 2 .</formula>
<text><location><page_8><loc_12><loc_81><loc_84><loc_89></location>In order to compare the particle number current of the reflected waves with the incident and transmitted currents, we substitute (18) into the expression for the conserved generalized Wronskian (16) and find, after straightforward algebraic manipulation, the following relation between the coefficients R 1 , R 2 , T 1 and T 2 ,</text>
<formula><location><page_8><loc_15><loc_73><loc_84><loc_79></location>| R 1 | 2 + ∣ ∣ ∣ k r 2 k in ∣ ∣ ∣ | R 2 | 2 = 1 + √ Λ 2 -4( ω -e Φ 0 ) 2 Λ 2 -4 ω 2 (∣ ∣ ∣ k t 1 k in ∣ ∣ ∣ | T 1 | 2 + ∣ ∣ ∣ k t 2 k in ∣ ∣ ∣ | T 2 | 2 ) > 1 . (20)</formula>
<text><location><page_8><loc_12><loc_43><loc_84><loc_76></location>∣ ∣ ∣ ∣ ∣ ∣ This relation should be compared to the standard result for non-dispersive 1D scattering, | R | 2 = 1 - | T | 2 , and its generalization in the presence of an external potential, | R | 2 = 1 -( ω -e Φ 0 ) | T | 2 /ω , see (7). As discussed above, in the non-dispersive case it is possible to achieve | R | > 1 for sufficiently low frequency scattering with 0 < ω < e Φ 0 . For fields with subluminal dispersion relations, the conclusion is similar. From expression (20) above, which is valid only when 0 < ω < e Φ 0 < Λ / 2, we conclude that the total reflection coefficient (i.e. the ratio between the total reflected particle number current and the incident current) is given by the LHS of (20) and is always greater than one, characterizing a generalization of the usual superradiant scattering which involves extra scattering channels. Note that, in the general case, without an exact solution we have no information about how this total reflection is distributed between the low and high wavenumber channels represented, respectively, by | R 1 | 2 and | k r 2 /k in || R 2 | 2 . However, by looking at the Λ series expansion of the reflected and transmitted wavenumbers, we can draw some interesting conclusions about the regime Λ /greatermuch 1. Using (19), we obtain the following expansions for the relevant wavenumbers,</text>
<formula><location><page_8><loc_32><loc_26><loc_84><loc_43></location>k in = -k r 1 = ω + ω 3 2Λ 2 + O (Λ -3 ) , k r 2 = Λ -ω 2 2Λ + O ( Λ -2 ) , (21) k t 1 = ω -e Φ 0 + ( ω -e Φ 0 ) 3 2Λ 2 + O (Λ -3 ) , k t 2 = Λ -( ω -e Φ 0 ) 2 2Λ + O ( Λ -2 ) .</formula>
<text><location><page_8><loc_12><loc_5><loc_84><loc_10></location>It is also interesting to analyse the case of a non-smooth potential, by solving the idealized problem of a step function, i.e. V ( x ) = e Φ 0 Θ( x ) (see the appendix for a discussion concerning the appropriate boundary conditions used at the discontinuity</text>
<text><location><page_8><loc_12><loc_9><loc_84><loc_26></location>As mentioned before, all possible modes of the system are characterized by the same conserved frequency ω . The momentum of each mode, on the other hand, is determined by the wavenumber k and, therefore, is not the same for every mode. From the expansions above, we note that k t 2 , k r 2 ∼ O (Λ) while k in , k r 1 , k t 1 ∼ O (Λ 0 ). In particular, the difference between the high momentum modes k r 2 and k t 2 is k r 2 -k t 2 ∼ O (Λ -1 ). If the potential is sufficiently smooth, this suggests that the creation of a pair of these modes ( k r 2 , k t 2 ) should be favoured since it requires a negligible momentum change in the system if Λ /greatermuch 1.</text>
<figure>
<location><page_9><loc_28><loc_55><loc_67><loc_88></location>
<caption>Figure 2. The reflection coefficient | R 1 | 2 for the subluminal case as a function of ω/e Φ 0 calculated for the step potential V ( x ) = e Φ 0 Θ( x ) with e Φ 0 = 1. Note the singular behaviour at ω = e Φ 0 / 2, which is present even in the non-dispersive case. This divergence is directly related to the discontinuity in V ( x ) and would be cured by smoothing the step potential at x = 0.</caption>
</figure>
<text><location><page_9><loc_12><loc_36><loc_84><loc_40></location>point x = 0). In such a case, one can show that the reflection and transmission coefficients are given by,</text>
<formula><location><page_9><loc_32><loc_18><loc_84><loc_35></location>R 1 = ( k in -k r 2 )( k in -k t 1 )( k in -k t 2 ) ( k r 2 -k r 1 )( k r 1 -k t 1 )( k r 1 -k t 2 ) , R 2 = ( k in -k r 1 )( k in -k t 1 )( k in -k t 2 ) ( k r 1 -k r 2 )( k r 2 -k t 1 )( k r 2 -k t 2 ) , (22) T 1 = ( k in -k r 1 )( k in -k r 2 )( k in -k t 2 ) ( k r 1 -k t 1 )( k r 2 -k t 1 )( k t 1 -k t 2 ) , T 2 = ( k in -k r 1 )( k in -k r 2 )( k in -k t 1 ) ( k r 1 -k t 2 )( k r 2 -k t 2 )( k t 2 -k t 1 ) ,</formula>
<text><location><page_9><loc_12><loc_4><loc_84><loc_18></location>where the wavenumbers k are explicitly given in (19). Note that the expressions above are valid for any value of Λ, not only in the regime Λ /greatermuch 1. In order to observe the effects of the dispersive parameter, we plot the reflection coefficient | R 1 | 2 as function of ω for different values of Λ, see FIG. 2. Note that as we increase the dispersive effects (i.e. we lower Λ), the reflection coefficient associated with k r 1 decreases. We can also see in FIG. 2 that the R 1 channel even becomes non-superradiant for some frequencies in the range 0 < ω < e Φ 0 . Of course, if we also include the other reflection channel, see FIG. 3,</text>
<figure>
<location><page_10><loc_28><loc_55><loc_67><loc_88></location>
<caption>Figure 3. The total reflection coefficient | R 1 | 2 + | k r 2 /k in || R 2 | 2 as a function of ω/e Φ 0 for a subluminal dispersion relation. Calculations were performed assuming a step potential V ( x ) = e Φ 0 Θ( x ) with e Φ 0 = 1. Note the enhancement effect in the amplification caused by dispersion.</caption>
</figure>
<text><location><page_10><loc_12><loc_34><loc_84><loc_42></location>then the total reflection coefficient is always superradiant, as proven in (20). Note that the presence of extra scattering channels even enhances the amplification process in comparison with the non-dispersive case. This can be understood in the sense that the amplifier is more effective if the number of 'accessible' channels increases.</text>
<text><location><page_10><loc_12><loc_18><loc_84><loc_34></location>Note also that the total reflection coefficient in the dispersive case is not continuously connected with the non-dispersive regime. More precisely, as Λ is increased in FIG. 3, the total reflection coefficient becomes larger and larger and moves away from the non-dispersive coefficient (Λ = ∞ ). A possible explanation for this behaviour resides in the fact that the subluminal dispersion relation itself is not continuously connected to the linear dispersion relation in the limit Λ →∞ . In other words, the modes k r 2 and k t 2 which are always absent in the non-dispersive case, will be present in the dispersive regime no matter how large Λ is.</text>
<section_header_level_1><location><page_10><loc_12><loc_14><loc_35><loc_15></location>2.2. Superluminal scattering</section_header_level_1>
<text><location><page_10><loc_12><loc_5><loc_84><loc_12></location>Let us now turn our attention to the case of a superluminal dispersion. Once again, we can understand much of the scattering process from the dispersion relation, see FIG. 4. The main difference with the subluminal case is that now there are only two real roots in either asymptotic regions, corresponding to one left-moving and one right-moving mode.</text>
<text><location><page_11><loc_27><loc_85><loc_27><loc_86></location>2</text>
<text><location><page_11><loc_27><loc_78><loc_27><loc_79></location>1</text>
<text><location><page_11><loc_26><loc_63><loc_27><loc_64></location>-1</text>
<text><location><page_11><loc_26><loc_56><loc_27><loc_57></location>-2</text>
<figure>
<location><page_11><loc_28><loc_56><loc_69><loc_88></location>
<caption>Figure 4. The dispersion relation for the superluminal case shown in terms of the dimensionless variables k/ Λ and Ω / Λ. We have chosen here a frequency in the superradiant interval 0 < ω < e Φ 0 . The blue dashed line corresponds to the dispersion relation at x → + ∞ while the green dash-dotted line corresponds to the dispersion at x →-∞ . The incident mode is indicated by a black square dot, the transmitted and reflected modes by black circular dots and the remaining solutions of the dispersion relation by white circular dots.</caption>
</figure>
<text><location><page_11><loc_30><loc_54><loc_31><loc_55></location>-1</text>
<text><location><page_11><loc_37><loc_54><loc_40><loc_55></location>-0.5</text>
<text><location><page_11><loc_46><loc_54><loc_47><loc_55></location>0</text>
<text><location><page_11><loc_53><loc_54><loc_55><loc_55></location>0.5</text>
<text><location><page_11><loc_62><loc_54><loc_63><loc_55></location>1</text>
<text><location><page_11><loc_12><loc_21><loc_84><loc_37></location>The other two solutions to the fourth order equation are imaginary roots, corresponding to exponentially decaying and exponentially growing modes. Similarly to the subluminal case, we are interested in the scattering of an incident low-frequency (0 < ω < e Φ 0 ) wave originating from x →-∞ , which is converted into a reflected left-moving mode, a transmitted right-moving mode and exponentially decaying modes (as a boundary condition, we impose the fact that there can be no exponentially growing modes in the asymptotic regions). Therefore, the solution of (10) corresponding to this scattering process is given by</text>
<formula><location><page_11><loc_30><loc_14><loc_84><loc_20></location>f → { e ik in x + R e ik r x + E r e k er x , x →-∞ , T e ik t x + E t e -k et x , x → + ∞ , (23)</formula>
<text><location><page_11><loc_12><loc_8><loc_84><loc_13></location>where the coefficients R and T are related, respectively, to reflection and transmission coefficients (see (25) below), and the coefficients E r and E t are the coefficients of the exponentially decaying modes. The wavenumbers k in (23) are obtained directly from</text>
<text><location><page_11><loc_45><loc_56><loc_46><loc_57></location>-2</text>
<text><location><page_12><loc_12><loc_87><loc_44><loc_89></location>the dispersion relations (12) and (13),</text>
<formula><location><page_12><loc_32><loc_65><loc_84><loc_85></location>k in = -k r = Λ √ 2 √ -1 + √ 1 + 4 ω 2 Λ 2 , k er = Λ √ 2 √ 1 + √ 1 + 4 ω 2 Λ 2 , (24) k t = -Λ √ 2 √ -1 + √ 1 + 4( ω -e Φ 0 ) 2 Λ 2 , k et = Λ √ 2 √ 1 + √ 1 + 4( ω -e Φ 0 ) 2 Λ 2 .</formula>
<text><location><page_12><loc_12><loc_53><loc_84><loc_64></location>The conservation of particle number current can be obtained by inserting (23) into (17) and equating the generalized Wronskian at ±∞ . Compared to the nondispersive result, one might expect extra terms related to the exponentially decaying modes. However, these extra contributions are also exponentially decaying and, therefore, their particle number currents are negligible at ±∞ . As a consequence, one obtains the following reflection coefficient (valid for 0 < ω < e Φ 0 ),</text>
<formula><location><page_12><loc_29><loc_45><loc_84><loc_51></location>| R | 2 = 1 + √ Λ 2 +4( ω -e Φ 0 ) 2 Λ 2 +4 ω 2 ∣ ∣ ∣ k t k in ∣ ∣ ∣ | T | 2 > 1 , (25)</formula>
<text><location><page_12><loc_12><loc_32><loc_84><loc_40></location>Repeating the analysis used in the subluminal case, one can solve exactly the problem for a step potential given by V ( x ) = e Φ 0 Θ( x ). Using appropriate boundary conditions at x = 0 (see the appendix), one can relate the coefficients in (23) to the wavenumbers given in (24),</text>
<text><location><page_12><loc_12><loc_39><loc_84><loc_48></location>∣ ∣ which, similarly to the subluminal case, is always greater than one. Note also that the expression above reduces to the usual non-dispersive reflection coefficient (7) in the limit Λ →∞ .</text>
<formula><location><page_12><loc_32><loc_15><loc_84><loc_31></location>R = ( ik in + k et )( k in + ik er )( k in -k t ) ( k er -ik r )( k r -ik et )( k r -k t ) , E r = ( ik in + k et )( k in -k r )( k in -k t ) ( k er + k et )( k er -ik r )( k er -ik t ) , (26) T = ( ik in -k er )( k et + ik in )( k in -k r ) ( k r -k t )( k t -ik er )( k t -ik et ) , E t = ( ik in -k er )( k in -k r )( k in -k t ) ( k er + k et )( k et + ik r )( k et + k t ) .</formula>
<text><location><page_12><loc_12><loc_10><loc_84><loc_14></location>Once again, it is useful to plot the reflection coefficient as a function of ω for different values of the dispersive parameter Λ, see FIG. 5.</text>
<text><location><page_12><loc_12><loc_4><loc_84><loc_10></location>The results obtained for the scattering of sub and superluminal dispersive fields in our 1+1-dimensional toy-model demonstrate that superradiance is possible and that the amplification is enhanced due to the extra scattering channels in comparison with</text>
<figure>
<location><page_13><loc_28><loc_55><loc_67><loc_88></location>
<caption>Figure 5. The reflection coefficient | R | 2 for the superluminal case as a function of ω/e Φ 0 calculated for the step potential V ( x ) = e Φ 0 Θ( x ) with e Φ 0 = 1. Like in the subluminal case, the divergence at ω = e Φ 0 / 2 is caused by the discontinuity in V ( x ) and would be absent if a smooth potential were used. Note the enhancement in amplification due to dispersion.</caption>
</figure>
<text><location><page_13><loc_12><loc_34><loc_84><loc_40></location>non-dispersive superradiance. These results are interesting for experimental attempts of detecting superradiance, since they indicate that dispersion may increase the amplification rates and, consequently, make the effect easier to observe.</text>
<section_header_level_1><location><page_13><loc_12><loc_28><loc_46><loc_30></location>3. Generalization to non-zero flows</section_header_level_1>
<text><location><page_13><loc_12><loc_17><loc_84><loc_26></location>Having in mind moving media in analogue models of gravity, we extend the ideas of the previous section by including a position-dependent flow velocity W ( x ) in our model. In other words, the medium where the scattering process takes place (fluid frame) is in relative motion with respect to the observer (lab frame). If we require that the dispersion relation be unaltered in the comoving frame of the fluid, the action (9) generalizes to</text>
<formula><location><page_13><loc_23><loc_6><loc_84><loc_15></location>S = ∫ dtdx { [( ∂ t + W∂ x -ieA t ) φ ] [( ∂ t -W∂ x + ieA t ) φ ∗ ] +( ∂ x φ ) ( ∂ x φ ∗ ) ± 1 Λ 2 ( ∂ 2 x φ ) ( ∂ 2 x φ ∗ ) } . (27)</formula>
<text><location><page_14><loc_12><loc_87><loc_70><loc_89></location>Consequently, the modified Klein-Gordon equation (8) generalizes to</text>
<formula><location><page_14><loc_22><loc_80><loc_84><loc_86></location>∓ 1 Λ 2 ∂ 4 x φ + ∂ 2 x φ +( ∂ t -ieA t + ∂ x W ) ( ∂ t -ieA t -W∂ x ) φ = 0 , (28)</formula>
<text><location><page_14><loc_12><loc_69><loc_84><loc_81></location>where the derivative operator ∂ x is understood to act on everything to its right. It is important to remark that the reference velocity c = 1 in such analogue systems corresponds to the velocity of the scalar perturbations (e.g. the sound speed in hydrodynamical systems). Usually, such velocities (and also W ( x )) are much smaller than the speed of light and, therefore, the system is nonrelativistic. For relativistic analogue models of gravity, we refer the reader to [28, 29, 30].</text>
<text><location><page_14><loc_16><loc_67><loc_84><loc_69></location>After separation of variables, φ ( t, x ) = e -iωt f ( x ), the wave equation (28) becomes,</text>
<formula><location><page_14><loc_14><loc_61><loc_84><loc_66></location>∓ 1 Λ 2 d 4 f dx 4 + d 2 f dx 2 + ( ω -eV ( x ) + i d dx W ( x ) )( ω -eV ( x ) + i W ( x ) d dx ) f = 0 , (29)</formula>
<text><location><page_14><loc_12><loc_59><loc_32><loc_60></location>which can be written as</text>
<formula><location><page_14><loc_29><loc_55><loc_84><loc_57></location>f '''' ( x ) + α ( x ) f '' ( x ) + β ( x ) f ' ( x ) + γ ( x ) f = 0 , (30)</formula>
<text><location><page_14><loc_12><loc_52><loc_17><loc_53></location>where</text>
<formula><location><page_14><loc_27><loc_41><loc_84><loc_50></location>α = ± Λ 2 ( 1 -W 2 ( x ) ) β = ∓ Λ 2 [2 W ( x ) ∂ x W ( x ) -2i W ( x ) ( ω -eV ( x ))] (31) γ = ± Λ 2 [ ( ω -eV ( x )) 2 +i ∂ x ( W ( x )( ω -eV ( x ))) ] .</formula>
<text><location><page_14><loc_12><loc_40><loc_84><loc_41></location>Observe that these coefficients are not independent but satisfy the following relations,</text>
<formula><location><page_14><loc_28><loc_34><loc_84><loc_39></location>γ -γ ∗ = 2 i I m ( γ ) = i I m ( ∂ x β ) = ∂ x ( β -β ∗ 2 ) , (32)</formula>
<formula><location><page_14><loc_28><loc_31><loc_84><loc_34></location>R e ( β ) = 1 2 ( β + β ∗ ) = ∂ x α. (33)</formula>
<text><location><page_14><loc_12><loc_28><loc_72><loc_30></location>We choose the functions W ( x ) and V ( x ) to be asymptotically constant,</text>
<formula><location><page_14><loc_37><loc_21><loc_84><loc_26></location>W ( x ) = { 0 , x →-∞ , W 0 , x → + ∞ , (34)</formula>
<formula><location><page_14><loc_37><loc_16><loc_84><loc_21></location>eV ( x ) = { 0 , x →-∞ , e Φ 0 , x → + ∞ , (35)</formula>
<text><location><page_14><loc_12><loc_11><loc_84><loc_15></location>so that, at ±∞ , any solution to (29) can be decomposed into plane waves satisfying the dispersion relation below,</text>
<formula><location><page_14><loc_30><loc_4><loc_84><loc_9></location>k 2 ± k 4 Λ 2 = { ω 2 , x →-∞ , ( ω -e Φ 0 -kW 0 ) 2 , x → + ∞ , (36)</formula>
<text><location><page_15><loc_12><loc_84><loc_84><loc_89></location>where the ± stands for super and subluminal respectively. As previously, we work under the assumption that ω, e Φ 0 /lessmuch Λ.</text>
<text><location><page_15><loc_12><loc_77><loc_84><loc_84></location>Similarly to the zero-flow case, we can obtain a conserved quantity by calculating the x -component of the Noether current associated with the symmetry φ → e iα φ . After separating the temporal dependence in φ , one can show that a modification of (14), namely</text>
<formula><location><page_15><loc_33><loc_74><loc_84><loc_77></location>Z = W 1 + W 2 + αW 3 -i I m ( β ) | f | 2 , (37)</formula>
<text><location><page_15><loc_12><loc_57><loc_84><loc_74></location>and the corresponding scaled quantity X = Z/ (2iΛ 2 ) are independent of x . The action of the functional X on a linear combination of 'on-shell' plane waves takes precisely the same form as in the zero-flow case, see (17) (the only difference is that the effective frequency is now given by Ω = ω -eV -kW instead of Ω = ω -eV ). This result highlights the generality of the algebraic structure of the particle number currents given by (17). Additionally, the existence of superradiance in the zero-flow case was related to the condition that ω -e Φ 0 < 0. We can anticipate that the occurrence of superradiance in non-zero flows will be favoured by modes for which ω -e Φ 0 -kW 0 < 0.</text>
<section_header_level_1><location><page_15><loc_12><loc_54><loc_34><loc_56></location>3.1. Subluminal dispersion</section_header_level_1>
<text><location><page_15><loc_12><loc_27><loc_84><loc_53></location>Let us consider first the case of a subluminal dispersion relation. As shown in FIG. 6 (red solid curve), for fixed e Φ 0 / Λ < W 0 < 1, there are two distinct intervals of frequencies separated by a critical frequency ω crit < e Φ 0 in which we expect superradiance: for 0 < ω < ω crit (region I) two propagating modes are admitted, both right-moving in the lab frame; for ω crit < ω < e Φ 0 (region II) there are four real roots of the dispersion relation corresponding to four propagating modes, three right-moving and one leftmoving (with respect to the lab-frame). Note that the requirement that W 0 is not too small, specifically W 0 > e Φ 0 / Λ, is necessary in order to guarantee that the left-most root in region I (i.e. the circular dot labeled k t 2 in region I of FIG. 6) has positive group velocity and hence defines a true transmitted mode. If 0 < W 0 < e Φ 0 / Λ, the situation is basically the same as the one discussed in section 2.1. On the other hand, if W 0 > 1 (see the black dashed curve in FIG. 6), there is only one possible regime: for all frequencies 0 < ω < e Φ 0 , two right-moving modes are admitted.</text>
<text><location><page_15><loc_12><loc_15><loc_84><loc_27></location>An interesting fact is that, due to the absence of left-moving modes at + ∞ when ω < ω crit , this system is a model for the event horizon of an analogue black hole with modified dispersion relations. However, the precise location of the horizon, besides being ω -dependent, is also rather ill-defined, relying on a global solution to the equation of motion in the vicinity of a classical turning point. This region can be studied by WKB methods and Hamilton-Jacobi theory [31], but it is not of specific interest to us here.</text>
<text><location><page_15><loc_12><loc_9><loc_84><loc_15></location>According to the analysis above, the scattering of an incoming wave from -∞ will result in transmission through two or three channels, depending on whether ω < ω crit or not. An exact solution to the scattering problem can be decomposed as</text>
<formula><location><page_15><loc_29><loc_3><loc_84><loc_8></location>f → { e ik in x + R 1 e ik r 1 x + R 2 e ik r 2 x , x →-∞ , T 1 e ik t 1 x + T 2 e ik t 2 x , x → + ∞ , (38)</formula>
<text><location><page_16><loc_28><loc_72><loc_30><loc_73></location>ω</text>
<figure>
<location><page_16><loc_29><loc_55><loc_67><loc_88></location>
<caption>Figure 6. The subluminal dispersion curve in the lab frame (Λ = 1, e Φ 0 = 0 . 4) at x → + ∞ for two different flow velocities: e Φ 0 / Λ < W 0 < 1 (red solid curve) and W 0 > 1 (black dashed curve). The coloured regions described in the text are associated with the red solid curve: region I (green, 0 < ω < ω crit ), region II (blue, ω crit < ω < e Φ 0 ), and region III (light red, non-superradiantly scattering region). The intersections of the horizontal lines with the red solid curve and the black dashed curve indicate transmitted modes at that frequency. Note that, when x → -∞ , the dispersion is described by the green dash-dotted curve of FIG. 1.</caption>
</figure>
<text><location><page_16><loc_12><loc_34><loc_67><loc_36></location>when 0 < ω < ω crit (if W 0 < 1) or 0 < ω < e Φ 0 (if W 0 > 1) and as</text>
<formula><location><page_16><loc_29><loc_27><loc_84><loc_33></location>f → { e ik in x + R 1 e ik r 1 x + R 2 e ik r 2 x , x →-∞ , T 1 e ik t 1 x + T 2 e ik t 2 x + T 3 e ik t 3 x , x → + ∞ , (39)</formula>
<text><location><page_16><loc_12><loc_15><loc_84><loc_27></location>when ω crit < ω < e Φ 0 (only possible if W 0 < 1). The wavenumbers k in , k r 1 and k r 2 are given by (19) and are labeled in FIG. 1, while the transmitted wavenumbers are not, in general, expressible as simple functions of the parameters. Note that we do not explicitly keep track of possible exponential decaying solutions in (38) (corresponding to complex solutions to the dispersion relation), since they do not contribute directly to (40).</text>
<text><location><page_16><loc_12><loc_9><loc_84><loc_15></location>We insert the solutions (38) and (39) into the functional X of (17) and find, after algebraic manipulations, the following relationship between the transmission and reflection coefficients,</text>
<formula><location><page_16><loc_16><loc_1><loc_84><loc_8></location>| R 1 | 2 + ∣ ∣ ∣ ∣ k r 2 k in ∣ ∣ ∣ ∣ | R 2 | 2 = 1 -Λ √ Λ 2 -4 ω 2 ( ∑ n v g n k in ( ω -e Φ 0 -k t n W 0 ) | T n | 2 ) , (40)</formula>
<text><location><page_17><loc_12><loc_83><loc_84><loc_89></location>where the sum is over all (2 or 3, depending on ω and W 0 ) transmission channels. Here, v g n = v g ( k t n ) are the group velocities of the transmitted modes at + ∞ . The LHS of (40) can be interpreted as the total reflection coefficient associated with the incident modes.</text>
<text><location><page_17><loc_12><loc_61><loc_84><loc_82></location>Since the group velocities v g n of the transmitted modes are, by definition, always positive, the sign of the contribution from each channel k t n to the RHS of (40) is determined by the factor Ω( k ) = ω -e Φ 0 -kW 0 evaluated at k t n , as we anticipated previously. When only two transmission channels are admitted, one of the two factors Ω( k ) is strictly negative while in the case of three transmission channels, two of the three factors Ω( k ) are strictly negative. Therefore, in both situations there is one root which contributes an overall negative amount to the RHS of (40) and thus reduces the magnitude of the total reflection. Because of these troublesome modes, the RHS is not strictly greater than 1 and we cannot straightforwardly conclude superradiance. In general, in order to fully answer the question of superradiance, one would need to specify W ( x ) and V ( x ) in all space and solve for the coefficients R n and T n .</text>
<text><location><page_17><loc_12><loc_46><loc_84><loc_59></location>Large Λ approximation: To better understand the relation between (40) and superradiance, we now focus on small deviations (Λ /greatermuch 1) from the non-dispersive case. For fixed W 0 < 1, the size of region I in FIG. 6 becomes zero when Λ is greater then e Φ 0 [2(1 -W 0 ) / 3] -2 / 3 . We therefore assume that the frequency ω lies in region II, where three transmission and two reflection channels are available, see (39). As explained above, in the most general case one would need to solve the equation of motion for all x in order to conclude superradiance or not from (40).</text>
<text><location><page_17><loc_12><loc_38><loc_84><loc_45></location>Similarly to the zero flow case, by looking at the series expansions of the relevant wavenumbers, we can make useful predictions about the transmission coefficients in this Λ /greatermuch 1 regime. We start the analysis by solving (36) in the asymptotic region x → + ∞ and expressing the obtained transmitted wavenumbers as power series in Λ,</text>
<formula><location><page_17><loc_30><loc_32><loc_84><loc_36></location>k t 1 = ω -e Φ 0 1 + W 0 + 1 2 ( ω -e Φ 0 ) 3 (1 + W 0 ) 4 Λ -2 + O (Λ -4 ) , (41)</formula>
<formula><location><page_17><loc_29><loc_27><loc_84><loc_32></location>k t 2 , 3 = ± Λ √ 1 -W 2 0 + W 0 ω -e Φ 0 1 -W 2 0 + O ( Λ -1 ) . (42)</formula>
<text><location><page_17><loc_12><loc_7><loc_84><loc_27></location>The series expansions of the incident wavenumber k in and of the reflected wavenumbers, k r 1 and k r 2 , are given, as previously, by (21). From these Λ expansions, we note that, unless W 0 ∼ 1, we have k t 2 , 3 , k r 2 ∼ O (Λ) and k in , k r 1 , k t 1 ∼ O (Λ 0 ). However, unlike in the zero-flow case, the difference between the high momentum transmitted modes k t 2 , 3 and the high momentum reflected modes k r 2 is not negligible for Λ /greatermuch 1, being O (Λ). Hence, the appearance of such modes requires a large momentum change in our system. Therefore, if the potential is sufficiently smooth, we expect the conversion of incident modes k in into transmitted modes k t 2 and k t 3 and into reflected modes k r 2 to be disfavoured in comparison with the low momentum transmission/reflection channel involving k t 1 and k r 1 . In other words, we expect that the Wronskian condition (40) will</text>
<text><location><page_18><loc_12><loc_87><loc_49><loc_89></location>include only the low momentum channel, i.e.</text>
<formula><location><page_18><loc_37><loc_82><loc_84><loc_86></location>| R 1 | 2 = 1 -ω -e Φ 0 ω | T 1 | 2 . (43)</formula>
<text><location><page_18><loc_12><loc_65><loc_84><loc_81></location>From this relation for the reflection coefficient, we would conclude that superradiance occurs for all frequencies in region II ( ω crit < ω < e Φ 0 ) when Λ /greatermuch 1. Note that this conclusion certainly does not hold in the case of a general dispersive parameter. If the condition Λ /greatermuch 1 is not satisfied, there can be a mixture between the high and low momentum channels. Consequently, as discussed before, the total reflection coefficient given by (40) is not necessarily larger than one. In such a case, a definite answer about superradiance can only be obtained by solving the differential equations at every spatial point x .</text>
<text><location><page_18><loc_12><loc_55><loc_84><loc_65></location>Having analyzed the case of W 0 < 1, let us now fix W 0 > 1 and assume 0 < ω < e Φ 0 . As discussed previously, there are two transmission and two reflection channels available when W 0 > 1 and the scattering problem is now described by (38). Since we are interested in the regime of large Λ, we calculate the wavenumber of the transmitted modes up to next-to-leading order terms,</text>
<formula><location><page_18><loc_29><loc_45><loc_84><loc_54></location>k t 1 = ω -e Φ 0 1 + W 0 + 1 2 ( ω -e Φ 0 ) 3 (1 + W 0 ) 4 Λ -2 + O (Λ -4 ) , k t 2 = ω -e Φ 0 W 0 -1 -1 2 ( ω -e Φ 0 ) 3 ( W 0 -1) 4 Λ -2 + O (Λ -4 ) . (44)</formula>
<text><location><page_18><loc_12><loc_41><loc_84><loc_45></location>By direct substitution of these expressions into (40), one can straightforwadly determine the reflection coefficient in powers of Λ,</text>
<formula><location><page_18><loc_28><loc_34><loc_84><loc_40></location>| R 1 | 2 + ∣ ∣ ∣ k r 2 k in ∣ ∣ ∣ | R 2 | 2 = 1 -ω -e Φ 0 ω ( | T 1 | 2 -| T 2 | 2 ) , (45)</formula>
<text><location><page_18><loc_12><loc_25><loc_84><loc_37></location>∣ ∣ plus terms of O (Λ -2 ). Note that this second channel T 2 is present even in the absence of dispersion, being an upstream mode which is swept downstream by a superluminal flow. From the equation above, it also becomes evident that the relation between the norms | T 1 | and | T 2 | of the two transmission channels determines the occurrence or not of superradiance.</text>
<text><location><page_18><loc_12><loc_16><loc_84><loc_23></location>Critical case: An interesting situation to be analyzed is the critical case ω = ω crit , which corresponds to the boundary between regions I and II in FIG. 6. In this scenario, the background flow W 0 , when expanded in powers of Λ, relates to the critical frequency according to the following expression,</text>
<formula><location><page_18><loc_30><loc_10><loc_84><loc_15></location>W 0 = 1 -3 2 ( ω crit -e Φ 0 ) 2 3 Λ -2 3 + O ( Λ -4 3 ) . (46)</formula>
<text><location><page_18><loc_12><loc_4><loc_84><loc_10></location>Note that the previous analysis leading to (43) relied on series expansions (see (42)) which are not valid when W 0 -1 ∼ O ( Λ -2 3 ) . Therefore, in order to analyze the possibility of superradiance in the critical case, we cannot use (43); instead, we have to</text>
<text><location><page_19><loc_23><loc_81><loc_29><loc_82></location>k</text>
<text><location><page_19><loc_24><loc_80><loc_27><loc_81></location>0.4</text>
<text><location><page_19><loc_24><loc_75><loc_27><loc_76></location>0.2</text>
<text><location><page_19><loc_24><loc_65><loc_26><loc_66></location>-0.2</text>
<text><location><page_19><loc_24><loc_60><loc_26><loc_62></location>-0.4</text>
<figure>
<location><page_19><loc_27><loc_56><loc_69><loc_88></location>
<caption>Figure 7. The subluminal dispersion relation in the fluid frame at the critical frequency. The blue dashed line represents the effective frequency Ω = ω crit -e Φ 0 -kW 0 at x → + ∞ .</caption>
</figure>
<text><location><page_19><loc_29><loc_54><loc_31><loc_55></location>-1</text>
<text><location><page_19><loc_37><loc_54><loc_39><loc_55></location>-0.5</text>
<text><location><page_19><loc_46><loc_54><loc_47><loc_55></location>0</text>
<text><location><page_19><loc_54><loc_54><loc_56><loc_55></location>0.5</text>
<text><location><page_19><loc_63><loc_54><loc_64><loc_55></location>1</text>
<text><location><page_19><loc_12><loc_30><loc_84><loc_44></location>start from the original Wronskian relation (40). In the critical regime, the dispersion relation (see FIG. 7) has three distinct solutions. Two of these solutions, denoted by k t 1 and k t 2 , have positive group velocities in the lab frame and, therefore, are identified as transmitted modes. The other solution, denoted by k 0 , is a degenerate double root and, consequently, has a vanishing group velocity in the lab frame. In order to obtain the reflection coefficient for the scattering problem, we first expand k t 1 and k t 2 as power series in Λ,</text>
<formula><location><page_19><loc_28><loc_22><loc_84><loc_29></location>k t 1 = ω -e Φ 0 2 + 3 8 ( ω -e Φ 0 ) 5 3 Λ -2 3 + O ( Λ -4 3 ) , (47) k t 2 = -2 ( ω -e Φ 0 ) 1 3 Λ 2 3 + O (1) ,</formula>
<text><location><page_19><loc_12><loc_19><loc_81><loc_21></location>and then substitute the obtained expressions into (40). The final result is given by</text>
<formula><location><page_19><loc_27><loc_11><loc_84><loc_18></location>| R 1 | 2 + ∣ ∣ ∣ k r 2 k in ∣ ∣ ∣ | R 2 | 2 = 1 -ω -e Φ 0 ω ( | T 1 | 2 -3 | T 2 | 2 ) , (48)</formula>
<text><location><page_19><loc_12><loc_3><loc_84><loc_15></location>∣ ∣ plus terms of order O ( Λ -2 3 ) . Observe again the importance of the relative sign of the norm in the two transmission channels. Note that the scattering process converts incident modes with wavenumber k in ≈ ω + O (Λ -1 ) into transmitted modes of large wavenumber k t 2 ≈ O (Λ 2 3 ), which can only be balanced by the high momentum reflected</text>
<text><location><page_20><loc_12><loc_73><loc_84><loc_89></location>modes k r 2 ≈ O (Λ). Another possibility is the conversion of the incident modes k in ≈ ω + O (Λ -1 ) into transmitted modes of wavenumber k t 1 ≈ ( ω -e Φ 0 ) / 2, which is comparable to the low momentum k r 1 channel. Because of the high momentum change required by the second channel (involving k r 2 and k t 2 ), we expect the first channel (involving k r 1 and k t 1 ) to be favoured in our scattering experiment. Since ω crit < e Φ 0 , we therefore deduce that the RHS of (48) is always greater than one in the limit Λ /greatermuch 1. In other words, low-frequency waves in the critical regime are superradiantly scattered in our toy-model if small subluminal corrections are added to the dispersion relation.</text>
<section_header_level_1><location><page_20><loc_12><loc_69><loc_36><loc_70></location>3.2. Superluminal dispersion</section_header_level_1>
<text><location><page_20><loc_12><loc_53><loc_84><loc_67></location>We now turn to superluminal scattering processes, which can be quite different compared to subluminal ones since there does not exist any notion of a horizon or modeindependent blocking region for high momentum incident modes (the group velocity dω/dk is unbounded as a function of k and only the low frequency modes which possess quasilinear dispersion experience a blocking region in such flows). We will follow the standard treatment [31] of analogue black holes with superluminal dispersion and analyze the transmission of an incoming wave from -∞ through to + ∞ .</text>
<text><location><page_20><loc_12><loc_38><loc_84><loc_53></location>The relevant dispersion relation in the superluminal case is depicted in FIG. 8. Given a superluminal flow W 0 > 1 (solid red curve in FIG. 8), there exists an interval of frequencies 0 < ω < ω crit (region I in FIG. 8) for which only two propagating modes are admitted, one right-moving and one left-moving in the lab frame. For ω crit < ω < e Φ 0 (region II in FIG. 8), however, there are four propagating modes, two transmitted rightmovers and two left-movers. The third possibility is a subluminal flow W 0 < 1 with 0 < ω < e Φ 0 , for which there are always two propagating modes (see the black dashed curve in FIG. 8).</text>
<text><location><page_20><loc_12><loc_26><loc_84><loc_37></location>Let us discuss first the cases in which only two propagating modes are available in the asymptotic limit x → + ∞ . Hence, the scattering is produced by an incident wave from -∞ whose frequency ω satisfies 0 < ω < ω crit (if W 0 > 1) or 0 < ω < e Φ 0 (if W 0 < 1). Note that exactly one of these two propagating modes is a left-moving mode. Imposing the boundary condition that no incoming mode is allowed at + ∞ , we obtain the solution of (29) corresponding to the scattering problem,</text>
<formula><location><page_20><loc_34><loc_18><loc_84><loc_24></location>f → { e ik in x + R e ik r x , x →-∞ , T 1 e ik t 1 x , x →∞ , (49)</formula>
<text><location><page_20><loc_12><loc_10><loc_84><loc_18></location>plus exponentially decaying channels which do not contribute directly to the generalized Wronskian current calculated at x →±∞ . Note that the wavenumbers k in and k r are the same ones that appear in (24) for the superluminal W = 0 case and the wavenumber k t represents the only available transmission channel.</text>
<text><location><page_20><loc_12><loc_4><loc_84><loc_10></location>On the other hand, if W 0 > 1 and the frequency ω of the incident wave satisfies ω crit < ω < e Φ 0 (region II in FIG. 8), then there are, in principle, two extra propagating channels available (four in total, as discussed above). However, because of the boundary</text>
<text><location><page_21><loc_28><loc_72><loc_31><loc_73></location>ω</text>
<figure>
<location><page_21><loc_29><loc_55><loc_67><loc_88></location>
<caption>Figure 8. The superluminal dispersion curve in the lab frame (Λ = 1, e Φ 0 = 1 . 5) at x → + ∞ for two different flow velocities: W 0 > 1 (red solid curve) and W 0 < 1 (black dashed curve). The intervals of frequency indicated refer to the red solid curve: region I (green, 0 < ω < ω crit ), region II (blue, ω crit < ω < e Φ 0 ) and region III (light red, non-superradiant region). The intersections of the horizontal lines with the red solid and the black dashed curves indicate transmitted modes at that frequency. Note that, when x →-∞ , the dispersion is described by the green dash-dotted curve of FIG. 4.</caption>
</figure>
<text><location><page_21><loc_12><loc_32><loc_84><loc_37></location>condition imposed at x → + ∞ , only one extra transmission channel has to be considered (the other extra channel is always left-moving at x → + ∞ ). The scattering solution is then given by</text>
<formula><location><page_21><loc_33><loc_26><loc_84><loc_31></location>f → { e ik in x + R e ik r x , x →-∞ , T 1 e ik t 1 x + T 2 e ik t 2 x , x →∞ , (50)</formula>
<text><location><page_21><loc_12><loc_19><loc_84><loc_26></location>where k in and k r are again given by (24) and k t 1 and k t 2 are the wavenumbers of the transmitted modes. Note that we have once again omitted the exponential decaying mode at x →-∞ since it does not affect directly the generalized Wronskian.</text>
<text><location><page_21><loc_12><loc_14><loc_84><loc_20></location>Using (17) to evaluate the functional X in both asymptotic regions, we obtain, similarly to the subluminal case, the following relation between the reflection and transmission coefficients,</text>
<formula><location><page_21><loc_24><loc_8><loc_84><loc_13></location>| R | 2 = 1 -Λ √ Λ 2 +4 ω 2 ( ∑ n v g n k in ( ω -e Φ 0 -k t n W 0 ) | T n | 2 ) , (51)</formula>
<text><location><page_21><loc_12><loc_4><loc_84><loc_8></location>where the sum is over one or two transmission channels, depending on ω and whether W 0 > 1 or W 0 < 1. Here, v g 1 and v g 2 are the group velocities of the transmitted modes</text>
<text><location><page_22><loc_12><loc_77><loc_84><loc_89></location>k t 1 and k t 2 , which are always positive by definition. Furthermore, it is possible to show, for frequencies 0 < ω < e Φ 0 , that the effective frequency Ω = ω -e Φ 0 -k t n W 0 is always negative for k t 1 modes and always positive for k t 2 modes. Therefore, since only the n = 1 transmission channel is available for frequencies lying in region I of FIG. 8, we conclude that the RHS of (51) is greater than 1 and, therefore, the scattering is always superradiant.</text>
<text><location><page_22><loc_12><loc_65><loc_84><loc_76></location>The situation for W 0 < 1 and 0 < ω < e Φ 0 is similar: only the first transmission channel is available and superradiance always occurs. However, for frequencies located in region II, we cannot so easily conclude superradiance since the extra transmission channel k t 2 contributes an overall negative factor in (51). To obtain a conclusive answer, one would need to know the detailed structure of W ( x ) and V ( x ) in the intermediate regime and solve the equations not only in the asymptotic regions but at every point x .</text>
<text><location><page_22><loc_12><loc_56><loc_84><loc_63></location>Large Λ approximation: In order to better understand the scattering of an incident wave whose frequency is located in region II of FIG. 8, we shall consider small deviations from the non-dispersive limit, i.e. Λ /greatermuch 1. In such a case, we can expand the two transmission channels, k t 1 and k t 2 , in powers of Λ,</text>
<formula><location><page_22><loc_29><loc_50><loc_84><loc_54></location>k t 1 = ω -e Φ 0 1 + W 0 -1 2 ( ω -e Φ 0 ) 3 (1 + W 0 ) 4 Λ -2 + O (Λ -4 ) , (52)</formula>
<formula><location><page_22><loc_29><loc_45><loc_84><loc_50></location>k t 2 = ω -e Φ 0 W 0 -1 + 1 2 ( ω -e Φ 0 ) 3 ( W 0 -1) 4 Λ -2 + O (Λ -4 ) , (53)</formula>
<text><location><page_22><loc_12><loc_41><loc_84><loc_45></location>and substitute the obtained wavenumbers into (51) in order to determine the reflection coefficient for the scattering,</text>
<formula><location><page_22><loc_28><loc_34><loc_84><loc_40></location>| R | 2 = 1 -ω -e Φ 0 ω ( | T 1 | 2 -| T 2 | 2 ) + O ( 1 / Λ 2 ) . (54)</formula>
<text><location><page_22><loc_12><loc_19><loc_84><loc_35></location>Since we assume ω < e Φ 0 in region II, this reflection coefficient is larger than 1 whenever | T 2 | < | T 1 | . As explained above, whether this condition is satisfied or not in a general model would depend on the detailed structure of W ( x ) and V ( x ) in the intermediate regime [31]. From a practical point of view, in order to maximize the potential for superradiance in an experiment with a superluminally dispersive medium, one should choose the asymptotic flow W 0 as small as possible while still being superluminal ( W 0 > 1) as this would minimize the size of region II and the extra positive-effectivefrequency transmission channels therein.</text>
<text><location><page_22><loc_12><loc_7><loc_84><loc_19></location>Choosing the flow as such to maximize the T 1 channel is also consistent with our intuition that scattering favors the channel which most closely matches the momentum of the reflected mode; in this case the wavenumber k t 1 is closer to k r than k t 2 is. This prediction is confirmed in the step function model for which V ( x ) = e Φ 0 Θ( x ) and W ( x ) = W 0 Θ( x ). In such a case, one can impose the appropriate boundary conditions at x = 0 discussed in the appendix to obtain the following reflection and transmission</text>
<text><location><page_23><loc_12><loc_87><loc_21><loc_89></location>coefficients,</text>
<formula><location><page_23><loc_34><loc_80><loc_84><loc_86></location>R = e Φ 0 + ω ( W 0 -1) ω ( W 0 +1) -e Φ 0 + O ( Λ -1 ) , (55)</formula>
<formula><location><page_23><loc_34><loc_72><loc_84><loc_77></location>T 2 = ω ( W 0 -1) ω ( W 0 +1) -e Φ 0 + O ( Λ -1 ) . (57)</formula>
<formula><location><page_23><loc_34><loc_76><loc_84><loc_81></location>T 1 = ω ( W 0 +1) ω ( W 0 +1) -e Φ 0 + O ( Λ -1 ) , (56)</formula>
<text><location><page_23><loc_12><loc_68><loc_84><loc_72></location>We can also show that the coefficient correponding to the omitted exponential decaying mode in (50) is of order O (Λ -2 ).</text>
<text><location><page_23><loc_12><loc_59><loc_84><loc_68></location>Comparing T 1 and T 2 above and using the fact that W 0 > 1, one can see that | T 1 | > | T 2 | at zeroth order in Λ, which implies superradiance and confirms our expectations. Alternatively, one can verify the occurence of superradiance by directly analyzing the reflection coefficient R above. It is straightforward to see that | R 2 | > 1 at lowest order in Λ.</text>
<text><location><page_23><loc_12><loc_50><loc_84><loc_57></location>Critical case: Another interesting possibility that we now consider in detail is the critical regime ω = ω crit . This situation corresponds to the boundary between regions I and II in FIG. 8 and is depicted, in the fluid frame, in FIG. 9. The relation between the background flow W 0 and the critical frequency ω crit is given by the following expression,</text>
<formula><location><page_23><loc_30><loc_43><loc_84><loc_48></location>W 0 = 1 + 3 2 ( ω crit -e Φ 0 ) 2 3 Λ -2 3 + O ( Λ -4 3 ) . (58)</formula>
<text><location><page_23><loc_12><loc_35><loc_84><loc_44></location>Note that, since W 0 -1 ∼ O ( Λ -2 3 ) , the Λ expansion (54) of the generalized Wronskian obtained previously is not valid in the present case (check the denominators in (53)). Consequently, we shall need different Λ expansions in order to obtain an appropriate expression for the reflection coefficient.</text>
<text><location><page_23><loc_12><loc_27><loc_84><loc_35></location>Like in the critical subluminal case, the dispersion relation has three distinct roots: the double root k 0 (with vanishing group velocity in the lab frame), a right-moving mode (with negative group velocity in the lab frame) and a transmitted mode (with positive group velocity in the lab frame) whose wavenumber k t is given by</text>
<formula><location><page_23><loc_28><loc_21><loc_84><loc_26></location>k t = ω -e Φ 0 2 -3 8 ( ω -e Φ 0 ) 5 3 Λ -2 3 + O ( Λ -4 3 ) . (59)</formula>
<text><location><page_23><loc_12><loc_16><loc_84><loc_21></location>Applying as a boundary condition the fact that only right moving modes are allowed at + ∞ , we obtain, after substituting the relevant quantities into (51), the relation between the reflection and transmission coefficients for the scattering process,</text>
<formula><location><page_23><loc_33><loc_9><loc_84><loc_14></location>| R | 2 = 1 -ω -e Φ 0 ω | T | 2 + O ( Λ -2 3 ) . (60)</formula>
<text><location><page_23><loc_12><loc_4><loc_84><loc_10></location>Since ω crit < e Φ 0 , we conclude that the RHS of the equation above is always greater than one when Λ /greatermuch 1. In summary, superradiance is expected to occur in the superluminal critical case for small deviations from the non-dispersive regime.</text>
<text><location><page_24><loc_26><loc_83><loc_28><loc_84></location>10</text>
<text><location><page_24><loc_27><loc_76><loc_28><loc_78></location>5</text>
<text><location><page_24><loc_26><loc_64><loc_27><loc_65></location>-5</text>
<text><location><page_24><loc_25><loc_58><loc_27><loc_59></location>-10</text>
<figure>
<location><page_24><loc_28><loc_56><loc_69><loc_88></location>
<caption>Figure 9. The superluminal dispersion relation in the fluid frame at the critical frequency. The blue dashed line represents the effective frequency Ω = ω crit -e Φ 0 -kW 0 at x → + ∞ .</caption>
</figure>
<text><location><page_24><loc_30><loc_54><loc_31><loc_55></location>-3</text>
<text><location><page_24><loc_35><loc_54><loc_37><loc_55></location>-2</text>
<text><location><page_24><loc_41><loc_54><loc_42><loc_55></location>-1</text>
<text><location><page_24><loc_47><loc_54><loc_48><loc_55></location>0</text>
<text><location><page_24><loc_52><loc_54><loc_53><loc_55></location>1</text>
<text><location><page_24><loc_57><loc_54><loc_58><loc_55></location>2</text>
<text><location><page_24><loc_63><loc_54><loc_63><loc_55></location>3</text>
<section_header_level_1><location><page_24><loc_12><loc_42><loc_40><loc_44></location>3.3. Inertial motion superradiance</section_header_level_1>
<text><location><page_24><loc_12><loc_15><loc_84><loc_41></location>Throughout this paper, inspired by the usual condition for rotational superradiance in the black hole case, i.e. ω -m Ω h < 0, we analyzed only scattering problems in which the frequency of the incident mode satisfies ω -e Φ 0 < 0. However, since it is the effective frequency Ω that appears in (17), we conclude that the amplification of an incident mode can also occur for ω -e Φ 0 > 0 given that ω -e Φ 0 -kW 0 < 0. In particular, even when Φ 0 = 0 superradiant scattering will be possible. However, being due exclusively to inertial motion in the system, this kind of superradiance is outside the scope of our work. In fact, inertial motion superradiance has long been known in the literature as the anomalous Doppler effect and the condition for negative effective-frequency modes is referred to as the Ginzburg-Frank condition [32]. Several phenomena in physics, like the Vavilov-Cherenkov effect and the Mach cones (which appear in supersonic airplanes) can be understood in terms of inertial motion superradiance [18]. For a detailed analysis of inertial motion superradiance, we refer the reader to [18].</text>
<section_header_level_1><location><page_24><loc_12><loc_11><loc_49><loc_12></location>4. Applications: axisymmetric systems</section_header_level_1>
<text><location><page_24><loc_12><loc_5><loc_84><loc_9></location>Having analyzed superradiance in simple 1+1-dimensional toy models with modified dispersion relations, we will now discuss how the ideas presented in this paper can</text>
<text><location><page_25><loc_12><loc_83><loc_84><loc_89></location>be generalized to more realistic situations based on analogue models of gravity. Our starting point is a general 2+1-dimensional, axisymmetric and irrotational fluid flow with background velocity v given by</text>
<formula><location><page_25><loc_33><loc_78><loc_84><loc_82></location>v ≡ v r ( r )ˆ r + v φ ( r ) ˆ φ = -A r ˆ r + B r ˆ φ , (61)</formula>
<text><location><page_25><loc_12><loc_67><loc_84><loc_77></location>where A and B are constants and ( r, φ ) are the usual polar coordinates. Velocity perturbations δ v of the background flow can be conveniently described by a scalar field ψ , which relates to δ v through equation δ v = ∇ ψ . We denote the propagation speed of these perturbations by c . The idea of analogue gravity is derived from the observation that the differential equation satisfied by the perturbations ψ , i.e.</text>
<formula><location><page_25><loc_31><loc_63><loc_84><loc_65></location>-( ∂ t + ∇· v ) ( ∂ t + v · ∇ ) ψ + c 2 ∇ 2 ψ = 0 , (62)</formula>
<text><location><page_25><loc_12><loc_56><loc_84><loc_62></location>can be cast into a Klein-Gordon equation in an effectively curved spacetime geometry. This connection between hydrodynamics and gravity is responsible for many important results, see [2] for a detailed review.</text>
<text><location><page_25><loc_12><loc_41><loc_84><loc_56></location>One of the successes of Unruh's [1] original idea of using sound waves to study gravitational phenomena is that it can be extended to many other physical systems, like gravity waves in open channel flows [33] and density perturbations in Bose-Einstein condensates [34]. An important feature of such systems is that (62) is only accurate in certain regimes; at sufficiently small distance scales (e.g. wavelengths comparable to the fluid depth in open channels), the dispersion relation is not linear anymore and (62) has to be replaced by ¶</text>
<formula><location><page_25><loc_27><loc_37><loc_84><loc_41></location>-( ∂ t + ∇· v ) ( ∂ t + v · ∇ ) ψ + c 2 ∇ 2 ψ = ∓ 1 Λ 2 ∇ 4 ψ, (63)</formula>
<text><location><page_25><loc_12><loc_32><loc_84><loc_36></location>where Λ is a dispersive parameter and the upper (lower) sign corresponds to subluminal (superluminal) dispersion.</text>
<text><location><page_25><loc_12><loc_20><loc_84><loc_32></location>Even though superradiance has been subjected to extensive studies in the linear regime of analogue models, it has never been analysed before in the context of modified dispersion relations, as opposed to Hawking radiation (see e.g. [2] and references therein). The remarkable fact about the toy models introduced in this paper is that they can be used to analyze superradiance in realistic dispersive analogue models of gravity satisfying (63).</text>
<text><location><page_25><loc_12><loc_8><loc_84><loc_20></location>Firstly, the electromagnetic interaction term e Φ 0 appearing in our toy models is analogous to the rotational term m Ω in axisymmetric analogue models, where m is the azimuthal number and Ω is the angular velocity. It is interesting to note that this duality is manifest in real black holes: both electromagnetic [35] and rotational [25] superradiance are possible. Another essential ingredient for the occurrence of superradiance in analogue models of gravity is the presence of an event</text>
<text><location><page_26><loc_12><loc_82><loc_84><loc_89></location>horizon, which allows no mode to escape from inside the analogue black hole. In our toy model, such behaviour is mimicked by an appropriate boundary condition imposed in the asymptotic limit x →∞ .</text>
<text><location><page_26><loc_12><loc_71><loc_84><loc_82></location>It is also important to address the usefulness of the generalized Wronskian (37) in the context of axisymmetric systems. More precisely, we are going to show that, if all derivatives with respect to x are replaced by derivatives with respect to r , then the Wronskian (37), when applied to solutions of (63), is independent of r . Indeed, by applying the ansatz ψ = ( H ( r ) / √ r ) e imφ e -iωt , we are able to separate (63) and are left with a radial equation for H ,</text>
<formula><location><page_26><loc_29><loc_67><loc_84><loc_69></location>H '''' ( r ) + α ( r ) H '' ( r ) + β ( r ) H ' ( r ) + γ ( r ) H = 0 , (64)</formula>
<text><location><page_26><loc_12><loc_64><loc_57><loc_66></location>where the coefficients α ( r ), β ( r ) and γ ( r ) are given by</text>
<formula><location><page_26><loc_32><loc_46><loc_84><loc_63></location>α = 1 -4 m 2 2 r 2 ± Λ 2 ( c 2 -v 2 r ) , β = 4 m 2 -1 r 3 ± Λ 2 ( P ( r ) -c 2 -v 2 r r ) , (65) γ = 25 16 r 4 -13 m 2 2 r 4 + m 4 r 4 ± Λ 2 ( 3 4 c 2 -v 2 r r 2 -P ( r ) 2 r + Q ( r ) ) .</formula>
<text><location><page_26><loc_12><loc_40><loc_84><loc_46></location>The functions P ( r ) and Q ( r ) appearing in the coefficients above are, up to a factor ( c 2 -v 2 r ), the same functions P and Q defined in Refs. [36, 37]. They can be expressed as</text>
<formula><location><page_26><loc_30><loc_26><loc_84><loc_39></location>P = c 2 r d dr [ r c 2 ( c 2 -v 2 r ) ] +2 iv r ( ω -mB r 2 ) , Q = ( ω -mB r 2 ) 2 -m 2 c 2 r 2 (66) + i c 2 r d dr [ rv r c 2 ( ω -mB r 2 )] ,</formula>
<text><location><page_26><loc_12><loc_14><loc_84><loc_25></location>where the upper (lower) sign corresponds to subluminal (superluminal) dispersion. Finally, we note that (64) is exactly the same as (30) and, more remarkably, that the coefficients α , β , γ above satisfy conditions (32) and (33). Consequently, the generalized Wronskian defined in (37) can also be used in the context of axisymmetric analogue models of gravity, thus completing the connection between our toy models and realistic physical systems.</text>
<section_header_level_1><location><page_26><loc_12><loc_10><loc_41><loc_11></location>5. Summary and final remarks</section_header_level_1>
<text><location><page_26><loc_12><loc_4><loc_84><loc_8></location>We have proposed idealized systems to investigate multi superradiant scattering processes that are applicable to sub and superluminal dispersive fields. Perhaps the</text>
<text><location><page_27><loc_12><loc_85><loc_84><loc_89></location>most important theoretical result obtained is related to the simplicity of the analytic expression for the particle number currents J n of the scattering channels n (see (17)),</text>
<formula><location><page_27><loc_38><loc_77><loc_84><loc_84></location>J n = | A n | 2 Ω | ±∞ dω dk ∣ ∣ ∣ k n . (67)</formula>
<text><location><page_27><loc_12><loc_65><loc_84><loc_81></location>∣ Notice that the expression above depends only on the amplitude, group velocity and effective frequency of the particular scattering channel. Moreover, this result is universal to all scattering processes discussed in this paper. Note also that, in principle, we could have normalized the modes so that the group velocity in (67) is absorbed into the coefficients A n . Doing that would make the conservation equations (see e.g. (20)) look much simpler. However, their dependence on the dispersive parameter Λ would then also be hidden in these new coefficients.</text>
<text><location><page_27><loc_12><loc_49><loc_84><loc_64></location>Our findings link to standard scattering processes, allowing a deeper insight into superradiance. Let us consider scattering of up to four incident modes by a general scattering potential. There are up to four channels to the left { a, b, c, d } and four to the right { A, B, C, D } of the scattering potential. As a lesson from the analysis carried out before, one needs to be careful when assigning the propagation direction of each mode since its group velocity is dependent on the particular scattering potential and type of dispersion, see FIG. 10. Furthermore, as long as the scattering potential is real, the total current to the left equals the total current to the right of the potential,</text>
<formula><location><page_27><loc_31><loc_45><loc_84><loc_47></location>J a + J b + J c + J d = J A + J B + J C + J D . (68)</formula>
<text><location><page_27><loc_13><loc_42><loc_84><loc_43></location>The scattering potential is amplifying classical and quantum field excitations if, at</text>
<figure>
<location><page_27><loc_34><loc_34><loc_62><loc_40></location>
<caption>Figure 10. The scattering processes for sub and superluminal dispersive fields, with two low (red solid) and two high (blue dashed) momentum degrees on each side of the potential, are represented by a general scattering process whose particle number currents are given by (67).</caption>
</figure>
<text><location><page_27><loc_12><loc_17><loc_84><loc_22></location>the left side of the potential, the total outflux (i.e. the reflected current) is larger then the total in-flux, J ref total > J in total . There are several scattering coefficients that can be considered: the reflection coefficient in each individual scattering channel,</text>
<formula><location><page_27><loc_43><loc_12><loc_84><loc_15></location>P n = J ref n J in total , (69)</formula>
<text><location><page_27><loc_12><loc_9><loc_40><loc_10></location>and the total reflection coefficient,</text>
<formula><location><page_27><loc_38><loc_3><loc_84><loc_7></location>P total = J ref total J in total = ∑ n J ref n J in total . (70)</formula>
<text><location><page_28><loc_12><loc_77><loc_84><loc_89></location>A sufficient, but not necessary, condition for superradiance, is to demand no influx from the right and to strictly require negative effective frequency for the remaining channels to the right -these conditions arise naturally at the event horizon of a rotating black hole. A surprising result of our analysis is that the presence of extra scattering channels can enhance the amplification effect. In FIG. 11, we illustrate the scattering diagrams for some of the examples discussed in the paper.</text>
<figure>
<location><page_28><loc_29><loc_45><loc_67><loc_75></location>
<caption>Figure 11. Schematic representation of some of the scattering processes investigated in the paper. The arrows indicate the group velocity propagation. The particle number current direction is related to the relative sign between this group velocity and the effective frequency Ω. This information, together with (68), can help determine if superradiance occurs or not.</caption>
</figure>
<text><location><page_28><loc_12><loc_8><loc_84><loc_31></location>In addition, the present work can be related to recent experimental realizations of analogue black holes, some of which have even studied the Hawking emission process. In particular, [9] exhibits the first detection of the classical analogue of Hawking radiation using dispersive gravity waves in an open channel flow. One of the most important lessons to be learned from that work is that even though vorticity and viscosity effects cannot be completely removed from the experimental setup, they can be made extremely small. In fact, they can be reduced to the point that the results predicted by the irrotational and inviscid theory match the results obtained experimentally with considerable accuracy. Based on this fact, together with our present results, one might ask whether superradiance occurs (and can be observed) in the laboratory using dispersive gravity waves. The connection between such system and our analysis can be seen directly from the full dispersion relation for gravity waves in a fluid of constant</text>
<text><location><page_29><loc_12><loc_87><loc_22><loc_89></location>depth h [38],</text>
<formula><location><page_29><loc_30><loc_76><loc_84><loc_86></location>ω 2 ( k ) = ( gk + σ ρ k 3 ) tanh ( kh ) = ghk 2 -( gh 3 3 -σh ρ ) k 4 + O ( k 6 ) , (71)</formula>
<text><location><page_29><loc_12><loc_70><loc_84><loc_77></location>where g is the gravitational acceleration and σ and ρ correspond, respectively, to the surface tension and the density of the fluid. If first-order deviations from the shallow water limit ( kh /lessmuch 1) are considered, we recover the quartic dispersion relation analyzed in this paper.</text>
<text><location><page_29><loc_12><loc_55><loc_84><loc_69></location>In other words, non-shallow gravity waves impinging on a rotating analogue black hole (e.g. a draining 'bathtub' vortex) satisfy (63) with c 2 = gh and a dispersive parameter Λ -2 = | gh 3 / 3 -σh/ρ | . Based on the existent analysis of superradiance of linear fields in open channel flows [33, 37] together with our discussion in Section 4, we expect superradiance to also be manifest for non-shallow gravity waves. This relation between dispersion and superradiance in open channels is currently being further investigated by the authors and will be the subject of a future work.</text>
<text><location><page_29><loc_12><loc_41><loc_84><loc_55></location>Another class of analogue black holes, which was recently set up in the laboratory [6] and which might be used in the future to produce superradiant scattering processes, consists of Bose-Einstein condensates [34, 2]. The analogy with gravity arises when one considers the Gross-Pitaevskii equation and uses the Madelung representation of the condensate wave function. If the eikonal approximation is used and axisymmetry is assumed, it is possible to show that perturbations around a background condensate obey a superluminal dispersion relation given by</text>
<formula><location><page_29><loc_35><loc_36><loc_84><loc_40></location>ω 2 = 4 π /planckover2pi1 2 n 0 a m 2 k 2 + ( /planckover2pi1 2 m ) 2 k 4 , (72)</formula>
<text><location><page_29><loc_12><loc_14><loc_84><loc_35></location>where m is the mass of a single boson, a is the scattering length and n 0 ( r ) is the background density. Such perturbations are described by (63) with c 2 = 4 π /planckover2pi1 2 n 0 a/m 2 and Λ = 2 m/ /planckover2pi1 . Based on our work, the most obvious conclusion we can draw is to expect superradiance to be manifest also in low-frequency BEC scattering experiments. However, since the dispersion relation is superluminal, there is no notion of a mode independent blocking region (see section 3.2) and, consequently, it is not clear if the appropriate boundary conditions will be sufficient to guarantee superradiance. Additionally, quantized vortices may be present in such systems, restricting the angular momentum to integer multiples of /planckover2pi1 . The physics of these quantized vortices with respect to superradiance and instabilities is also unclear at this point and more investigation is needed to understand their role in possible BEC scattering processes.</text>
<section_header_level_1><location><page_29><loc_12><loc_10><loc_29><loc_11></location>Acknowledgments</section_header_level_1>
<text><location><page_29><loc_12><loc_4><loc_84><loc_8></location>MR was partially supported by FAPESP. SW was supported by Marie Curie Career Integration Grant (MULTI-QG-2011), the SISSA Young Researchers Grant (Black hole</text>
<text><location><page_30><loc_12><loc_83><loc_84><loc_89></location>horizon effects in fluids and superfluids), and the Fqxi Mini grant (Physics without borders). We wish to thank Carlos Barcel'o for his comments. SW would like to thank Matt Visser for stimulating discussions.</text>
<section_header_level_1><location><page_30><loc_12><loc_79><loc_21><loc_80></location>Appendix</section_header_level_1>
<text><location><page_30><loc_12><loc_59><loc_84><loc_77></location>Throughout this paper we solve simple models based on step functions for the external potential V ( x ) = e Φ 0 Θ( x ) and for the background velocity W ( x ) = W 0 Θ( x ), where Θ( x ) is the Heaviside function. Since this function is characterized by a discontinuity at x = 0, it is important to analyze what happens to the wavefunction, i.e. the solution to (10) or (30), at x = 0. The same problem arises in 1D tunneling problems in quantum mechanics when the potential barrier in the Schrodinger equation is modelled by a step function. In such situations, one has to impose the continuity of the wavefunction and its first derivative at the discontinuity point in order to determine the reflection and the transmission coefficients.</text>
<text><location><page_30><loc_12><loc_47><loc_84><loc_59></location>Let us first consider (10), which is valid only for zero background flows. Following the standard procedure used in quantum mechanics (see e.g. [39]), we integrate the differential equation over a small region ( -/epsilon1, + /epsilon1 ) around the discontinuity x = 0 and then take the limit of the obtained expression as /epsilon1 → 0. Starting from (10) and repeating this procedure three times, one can show that f ( x ) and its derivatives up to third order are all continuous at x = 0.</text>
<text><location><page_30><loc_12><loc_29><loc_84><loc_47></location>We also have to deal with the generalization to non-zero flows, given by (30). The situation now is more complicate since the differential equation involves derivatives of Θ( x ) (i.e. delta functions). Repeating the procedure described above, one can show that the function f and its first order derivative are still continuous at x = 0. The continuity of the second and third order derivatives, because of the delta functions, now depends on Θ(0), i.e. it depends on the choice of the flow velocity and of the external potential exactly at x = 0. Using the half-maximum convention of Θ(0) = 1 / 2, we can further prove that the second order derivative f '' ( x ) is also continuous at x = 0 and that the third order derivative satisfies the following relation,</text>
<formula><location><page_30><loc_33><loc_20><loc_84><loc_25></location>f ''' (0 + ) -f ''' (0 -) = Λ 2 W 2 0 f ' (0) (.1) -i Λ 2 W 0 ( ω -e Φ 0 ) f (0) .</formula>
<section_header_level_1><location><page_30><loc_12><loc_17><loc_22><loc_18></location>References</section_header_level_1>
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</document>
[
{
"title": "Superradiant scattering of dispersive fields",
"content": "Mauricio Richartz 1 , Angus Prain 2 , 3 , Silke Weinfurtner 2 , 3 and Stefano Liberati 2 , 3 1 Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC (UFABC), 09210-170 Santo Andr´e, SP, Brazil, 3 INFN, Sezione di Trieste E-mail: mauricio.richartz@ufabc.edu.br , aprain@sissa.it , silkiest@gmail.com , liberati@sissa.it",
"pages": [
1
]
},
{
"title": "Abstract.",
"content": "Motivated by analogue models of classical and quantum field theory in curved spacetimes and their recent experimental realizations, we consider wave scattering processes of dispersive fields exhibiting two extra scattering channels. In particular, we investigate how standard superradiant scattering processes are affected by subluminal or superluminal modifications of the dispersion relation. We analyze simple 1+1dimensional toy-models based on fourth-order corrections to the standard second order wave equation and show that low-frequency waves impinging on generic scattering potentials can be amplified during the process. In specific cases, by assuming a simple step potential, we determine quantitatively the deviations in the amplification spectrum that arise due to dispersion, and demonstrate that the amplification can be further enhanced due to the presence of extra scattering channels. We also consider dispersive scattering processes in which the medium where the scattering takes place is moving with respect to the observer and show that superradiance can also be manifest in such situations. PACS numbers: 04.62.+v, 04.70.Dy, 47.35.Bb, 03.65.Nk",
"pages": [
1
]
},
{
"title": "1. Introduction and motivation",
"content": "Experimental realizations of analogue black holes [1, 2] and their associated effects have drawn a lot of attention in the past few years [3, 4, 5, 6, 7, 8, 9]. Probably the most discussed results have been the first observation of the classical analogue of Hawking radiation in an open channel flow [9] and the still controversial observation of radiation in ultrashort laser pulse filaments [8, 10, 11, 12, 13]. Even though it remains an open question whether or not real black holes emit Hawking radiation, calculations involving analogue black holes suggest that the emission process is most probably unaffected by transplanckian effects that could, in principle, alter or even exclude the radiation process [14, 15, 16]. Superradiance [17, 18, 19] is another typical phenomenon of black hole physics [20, 21] which is also manifest in analogue models of gravity [22]. In standard scattering processes, the ratio between the reflected and the incident particle number currents [23] (i.e. the reflection coefficient) is smaller than one. This is directly encoded by the fact that the amplitude of the reflected wave is usually smaller than the amplitude of the incident one in non-dispersive normalized scattering processes. However, in some special situations (e.g. wave scattering in a Kerr black hole spacetime), low-frequency incident waves can be amplified in the scattering process. This amplification effect, known as superradiance, was first discovered by Zel'dovich in the context of electromagnetic waves [24], and later shown to be a more general phenomenon in physics [18, 19], in which classical as well as quantum field excitations can be amplified. The main goal of our work is to analyze the robustness of the amplification process for dispersive fields. The dispersive fields considered in this paper exhibit two extra scattering channels, such that one can study the overall robustness of the amplification process for multiple superradiant scattering. Additionally, in some simplified cases we also investigate the specific deviations in the amplification spectrum due to dispersion. The examples discussed in this paper are motivated by analogue models of gravity, where fields exhibiting sub or superluminal dispersion relations arise naturally.",
"pages": [
2
]
},
{
"title": "1.1. Superradiant systems",
"content": "/negationslash Before starting our analysis of superradiance for dispersive fields, it is important to review the basic ingredients that make the phenomenon possible in some simple systems. For example, in a Kerr black hole, incident scalar field modes of frequency ω and azimuthal number m = 0 are known to be superradiant for sufficiently low frequencies. Although the Klein-Gordon equation is very complicated when expressed in standard Boyer-Lindquist coordinates, the radial part of the equation of motion for these modes can be written, after a change of variables, very simply as where u is related to the radial part of the separated scalar field and r ∗ is a tortoise-like coordinate (which goes to -∞ at the event horizon and reduces to the standard radial coordinate r at spatial infinity). V ω,m ( r ∗ ) is the effective potential and possesses the following asymptotics, where Ω h is the angular velocity of the black hole at the event horizon. It can be shown that, for 0 < ω < m Ω h , incident modes are superradiantly scattered by the black hole [25, 20]. The corresponding reflection coefficient is given by where ω -m Ω h ω | T | 2 is the transmission coefficient. Two factors are responsible for the occurrence of the effect [19]: first, the event horizon behaves as a one-way membrane that allows only ingoing transmitted waves (defined by the group velocity) near the black hole; second, because of the ergoregion of the rotating black hole, low-frequency ingoing waves are associated with an outgoing particle number current at the horizon (the current being proportional to ω -m Ω h ) [23]. This outgoing flux at the horizon is compensated by the superradiantly reflected modes in such a way that the total particle number current is conserved during the process. Another simple system exhibiting superradiance is a massless scalar field φ with electric charge e and minimally coupled to an electromagnetic potential A µ = ( V ( x ) , 0) in 1+1-dimensions [17]. The evolution of this system is determined by the Klein-Gordon equation, which, after separation of the temporal dependence (i.e. φ = f ( x ) exp( -iωt )), reduces to where ω is the frequency of the mode in question. If the electromagnetic potential V ( x ) has the following asymptotic behaviour, one can calculate the relation between the reflection and transmission coefficients for incident waves [17]: Therefore, low frequency (0 < ω < e Φ 0 ) ‡ right-moving modes originating from -∞ interact with the scattering potential, resulting in transmitted right-moving modes and superradiantly reflected ( | R | 2 > 1) left-moving modes. This is basically the Klein paradox [26] - see [17] for a detailed explanation of the relationship between superradiance, the Klein paradox and pair creation. Note that the important ingredients for superradiance are essentially the same as in the rotating black hole case: firstly, there is no left-moving mode at + ∞ (this is similar to the boundary condition at the event horizon of a black hole, where no outgoing solutions are allowed); secondly, the form of the electromagnetic potential allows for low-frequency right-moving modes to be associated with left-moving particle number currents at + ∞ .",
"pages": [
2,
3,
4
]
},
{
"title": "2. Modified dispersion relations",
"content": "In order to study how a modified dispersion relation affects superradiance, we shall first generalize the simple, non-dispersive, 1+1-dimensional model above [17] by including fourth-order terms in (4). Inspired by the quartic dispersion relation Ω 2 = k 2 ± k 4 / Λ 2 , where Λ is a dispersive momentum scale and the ± notates super and subluminal dispersion respectively, we propose the following generalization of (4), which can be obtained from the following action for the complex scalar field φ , After separating the temporal dependence in (8), instead of (5), one obtains where f represents the field mode with frequency ω > 0 and prime denotes derivative with respect to x . We choose the effective frequency Ω( x ) = ω -eV ( x ) to satisfy asymptotic relations similar to those in (2) and (6), where e Φ 0 is a positive constant. In the asymptotic regions, the solutions of (10) are simple exponentials, exp (i kx ), whose wavenumbers k satisfy the dispersion relations below, In the subluminal case (lower sign), real solutions ( ω, k ) to the dispersion relation correspond to the intersections of a lemniscate (figure-eight) and a straight line, see FIG. 1. In the superluminal case (upper sign), real solutions correspond to the intersections of a quartic curve and a straight line, see FIG. 4. In order to relate the asymptotic solutions at + ∞ and -∞ without having to solve the differential equation for all values of x , one needs a conserved quantity analogous to the Wronskian for second order wave equations. Since our model is non-dissipative, we expect such a quantity to exist [27]. In fact, by calculating the x -component of the Noether current associated with the symmetry φ → e iα φ , it is possible to show that the expression where generalizes the notion of the Wronskian § to our dispersive model, i.e. dZ/dx = 0 for any solution f of (10). Furthermore, it is convenient to work with the scaled functional X whose action on a function f is defined by In particular, the action of X on a linear combination of 'on shell' plane waves (wavenumbers satisfying the dispersion relation) is simply We would like to emphasize the simplicity of the algebraic expression above, which only depends on the amplitudes, effective frequencies and group velocities of the various scattering channels participating in the scattering process. Throughout the paper, we shall refer to (17) as the particle number current since it generalizes the usual notion of particle number current associated with a complex Klein-Gordon field.",
"pages": [
4,
5
]
},
{
"title": "2.1. Subluminal scattering",
"content": "Based on the general notion of superradiance described in the introduction, we will study the scattering process of incident waves originating from x →-∞ in the presence of a subluminal dispersion. In realistic scenarios, we do not expect the dispersion relation Ω 2 = k 2 -k 4 / Λ 2 to have a maximum/minimum value above/below which only imaginary solutions of the dispersion relation are possible. Therefore, in order to guarantee that at 0.5 0.25 0 -0.25 -0.5 -1 -0.5 0.5 1 least one real mode is present in the dispersion relation, we assume that the dispersion parameter Λ is large compared to the electromagnetic interaction, i.e. we assume that e Φ 0 < Λ / 2. In such situations, the behaviour of the solutions to (10) in the asymptotic limits is captured in FIG. 1, where the green dash-dotted (Ω = ω ) and the blue dashed (Ω = ω -e Φ 0 ) lines represent the effective frequency when x → -∞ and x → + ∞ , respectively. Note that there exist four roots, corresponding to four propagating modes as one would expect from a fourth order differential equation. We also note that, when ω < e Φ 0 , the blue dashed line is located below the Ω = 0 axis and the roots in the region x → + ∞ inherit a relative sign between their group and phase velocities with respect to the roots at x → + ∞ (compare the intersections of the green dash-dotted line ( x →-∞ ) and the blue dashed line ( x → + ∞ ) with the red solid curve). We label the four roots associated with a frequency 0 < ω < e Φ 0 as k a < k b < k c < k d when x → -∞ and as k A < k B < k C < k D when x → + ∞ (they correspond to the intersections of the straight lines with the figure-eight in FIG. 1). The following table summarizes the character of these modes, where u and v notate right-movers and Since, in this setup, there is only one source of waves, located at x →-∞ , and no incoming signal from x → + ∞ , we impose the boundary condition that the modes A and C are unpopulated in the scattering process. Furthermore, we choose the incoming mode at x → -∞ to be entirely composed of low-momentum c modes with no highmomentum a mode component. ‖ The corresponding solution of (10) in the asymptotic limits is where the wavenumber of the incident mode is given by k in = k c , the reflected modes are given by k r 1 = k b , k r 2 = k d , and the transmitted modes are k t 1 = k B , k t 2 = k D . In addition, the coefficients R 1 , R 2 , T 1 and T 2 can be related to reflection and transmission coefficients (see (20) below). In the asymptotic regions, it is possible to solve exactly the dispersion relation and find the following explicit expressions for the roots, In order to compare the particle number current of the reflected waves with the incident and transmitted currents, we substitute (18) into the expression for the conserved generalized Wronskian (16) and find, after straightforward algebraic manipulation, the following relation between the coefficients R 1 , R 2 , T 1 and T 2 , ∣ ∣ ∣ ∣ ∣ ∣ This relation should be compared to the standard result for non-dispersive 1D scattering, | R | 2 = 1 - | T | 2 , and its generalization in the presence of an external potential, | R | 2 = 1 -( ω -e Φ 0 ) | T | 2 /ω , see (7). As discussed above, in the non-dispersive case it is possible to achieve | R | > 1 for sufficiently low frequency scattering with 0 < ω < e Φ 0 . For fields with subluminal dispersion relations, the conclusion is similar. From expression (20) above, which is valid only when 0 < ω < e Φ 0 < Λ / 2, we conclude that the total reflection coefficient (i.e. the ratio between the total reflected particle number current and the incident current) is given by the LHS of (20) and is always greater than one, characterizing a generalization of the usual superradiant scattering which involves extra scattering channels. Note that, in the general case, without an exact solution we have no information about how this total reflection is distributed between the low and high wavenumber channels represented, respectively, by | R 1 | 2 and | k r 2 /k in || R 2 | 2 . However, by looking at the Λ series expansion of the reflected and transmitted wavenumbers, we can draw some interesting conclusions about the regime Λ /greatermuch 1. Using (19), we obtain the following expansions for the relevant wavenumbers, It is also interesting to analyse the case of a non-smooth potential, by solving the idealized problem of a step function, i.e. V ( x ) = e Φ 0 Θ( x ) (see the appendix for a discussion concerning the appropriate boundary conditions used at the discontinuity As mentioned before, all possible modes of the system are characterized by the same conserved frequency ω . The momentum of each mode, on the other hand, is determined by the wavenumber k and, therefore, is not the same for every mode. From the expansions above, we note that k t 2 , k r 2 ∼ O (Λ) while k in , k r 1 , k t 1 ∼ O (Λ 0 ). In particular, the difference between the high momentum modes k r 2 and k t 2 is k r 2 -k t 2 ∼ O (Λ -1 ). If the potential is sufficiently smooth, this suggests that the creation of a pair of these modes ( k r 2 , k t 2 ) should be favoured since it requires a negligible momentum change in the system if Λ /greatermuch 1. point x = 0). In such a case, one can show that the reflection and transmission coefficients are given by, where the wavenumbers k are explicitly given in (19). Note that the expressions above are valid for any value of Λ, not only in the regime Λ /greatermuch 1. In order to observe the effects of the dispersive parameter, we plot the reflection coefficient | R 1 | 2 as function of ω for different values of Λ, see FIG. 2. Note that as we increase the dispersive effects (i.e. we lower Λ), the reflection coefficient associated with k r 1 decreases. We can also see in FIG. 2 that the R 1 channel even becomes non-superradiant for some frequencies in the range 0 < ω < e Φ 0 . Of course, if we also include the other reflection channel, see FIG. 3, then the total reflection coefficient is always superradiant, as proven in (20). Note that the presence of extra scattering channels even enhances the amplification process in comparison with the non-dispersive case. This can be understood in the sense that the amplifier is more effective if the number of 'accessible' channels increases. Note also that the total reflection coefficient in the dispersive case is not continuously connected with the non-dispersive regime. More precisely, as Λ is increased in FIG. 3, the total reflection coefficient becomes larger and larger and moves away from the non-dispersive coefficient (Λ = ∞ ). A possible explanation for this behaviour resides in the fact that the subluminal dispersion relation itself is not continuously connected to the linear dispersion relation in the limit Λ →∞ . In other words, the modes k r 2 and k t 2 which are always absent in the non-dispersive case, will be present in the dispersive regime no matter how large Λ is.",
"pages": [
5,
6,
7,
8,
9,
10
]
},
{
"title": "2.2. Superluminal scattering",
"content": "Let us now turn our attention to the case of a superluminal dispersion. Once again, we can understand much of the scattering process from the dispersion relation, see FIG. 4. The main difference with the subluminal case is that now there are only two real roots in either asymptotic regions, corresponding to one left-moving and one right-moving mode. 2 1 -1 -2 -1 -0.5 0 0.5 1 The other two solutions to the fourth order equation are imaginary roots, corresponding to exponentially decaying and exponentially growing modes. Similarly to the subluminal case, we are interested in the scattering of an incident low-frequency (0 < ω < e Φ 0 ) wave originating from x →-∞ , which is converted into a reflected left-moving mode, a transmitted right-moving mode and exponentially decaying modes (as a boundary condition, we impose the fact that there can be no exponentially growing modes in the asymptotic regions). Therefore, the solution of (10) corresponding to this scattering process is given by where the coefficients R and T are related, respectively, to reflection and transmission coefficients (see (25) below), and the coefficients E r and E t are the coefficients of the exponentially decaying modes. The wavenumbers k in (23) are obtained directly from -2 the dispersion relations (12) and (13), The conservation of particle number current can be obtained by inserting (23) into (17) and equating the generalized Wronskian at ±∞ . Compared to the nondispersive result, one might expect extra terms related to the exponentially decaying modes. However, these extra contributions are also exponentially decaying and, therefore, their particle number currents are negligible at ±∞ . As a consequence, one obtains the following reflection coefficient (valid for 0 < ω < e Φ 0 ), Repeating the analysis used in the subluminal case, one can solve exactly the problem for a step potential given by V ( x ) = e Φ 0 Θ( x ). Using appropriate boundary conditions at x = 0 (see the appendix), one can relate the coefficients in (23) to the wavenumbers given in (24), ∣ ∣ which, similarly to the subluminal case, is always greater than one. Note also that the expression above reduces to the usual non-dispersive reflection coefficient (7) in the limit Λ →∞ . Once again, it is useful to plot the reflection coefficient as a function of ω for different values of the dispersive parameter Λ, see FIG. 5. The results obtained for the scattering of sub and superluminal dispersive fields in our 1+1-dimensional toy-model demonstrate that superradiance is possible and that the amplification is enhanced due to the extra scattering channels in comparison with non-dispersive superradiance. These results are interesting for experimental attempts of detecting superradiance, since they indicate that dispersion may increase the amplification rates and, consequently, make the effect easier to observe.",
"pages": [
10,
11,
12,
13
]
},
{
"title": "3. Generalization to non-zero flows",
"content": "Having in mind moving media in analogue models of gravity, we extend the ideas of the previous section by including a position-dependent flow velocity W ( x ) in our model. In other words, the medium where the scattering process takes place (fluid frame) is in relative motion with respect to the observer (lab frame). If we require that the dispersion relation be unaltered in the comoving frame of the fluid, the action (9) generalizes to Consequently, the modified Klein-Gordon equation (8) generalizes to where the derivative operator ∂ x is understood to act on everything to its right. It is important to remark that the reference velocity c = 1 in such analogue systems corresponds to the velocity of the scalar perturbations (e.g. the sound speed in hydrodynamical systems). Usually, such velocities (and also W ( x )) are much smaller than the speed of light and, therefore, the system is nonrelativistic. For relativistic analogue models of gravity, we refer the reader to [28, 29, 30]. After separation of variables, φ ( t, x ) = e -iωt f ( x ), the wave equation (28) becomes, which can be written as where Observe that these coefficients are not independent but satisfy the following relations, We choose the functions W ( x ) and V ( x ) to be asymptotically constant, so that, at ±∞ , any solution to (29) can be decomposed into plane waves satisfying the dispersion relation below, where the ± stands for super and subluminal respectively. As previously, we work under the assumption that ω, e Φ 0 /lessmuch Λ. Similarly to the zero-flow case, we can obtain a conserved quantity by calculating the x -component of the Noether current associated with the symmetry φ → e iα φ . After separating the temporal dependence in φ , one can show that a modification of (14), namely and the corresponding scaled quantity X = Z/ (2iΛ 2 ) are independent of x . The action of the functional X on a linear combination of 'on-shell' plane waves takes precisely the same form as in the zero-flow case, see (17) (the only difference is that the effective frequency is now given by Ω = ω -eV -kW instead of Ω = ω -eV ). This result highlights the generality of the algebraic structure of the particle number currents given by (17). Additionally, the existence of superradiance in the zero-flow case was related to the condition that ω -e Φ 0 < 0. We can anticipate that the occurrence of superradiance in non-zero flows will be favoured by modes for which ω -e Φ 0 -kW 0 < 0.",
"pages": [
13,
14,
15
]
},
{
"title": "3.1. Subluminal dispersion",
"content": "Let us consider first the case of a subluminal dispersion relation. As shown in FIG. 6 (red solid curve), for fixed e Φ 0 / Λ < W 0 < 1, there are two distinct intervals of frequencies separated by a critical frequency ω crit < e Φ 0 in which we expect superradiance: for 0 < ω < ω crit (region I) two propagating modes are admitted, both right-moving in the lab frame; for ω crit < ω < e Φ 0 (region II) there are four real roots of the dispersion relation corresponding to four propagating modes, three right-moving and one leftmoving (with respect to the lab-frame). Note that the requirement that W 0 is not too small, specifically W 0 > e Φ 0 / Λ, is necessary in order to guarantee that the left-most root in region I (i.e. the circular dot labeled k t 2 in region I of FIG. 6) has positive group velocity and hence defines a true transmitted mode. If 0 < W 0 < e Φ 0 / Λ, the situation is basically the same as the one discussed in section 2.1. On the other hand, if W 0 > 1 (see the black dashed curve in FIG. 6), there is only one possible regime: for all frequencies 0 < ω < e Φ 0 , two right-moving modes are admitted. An interesting fact is that, due to the absence of left-moving modes at + ∞ when ω < ω crit , this system is a model for the event horizon of an analogue black hole with modified dispersion relations. However, the precise location of the horizon, besides being ω -dependent, is also rather ill-defined, relying on a global solution to the equation of motion in the vicinity of a classical turning point. This region can be studied by WKB methods and Hamilton-Jacobi theory [31], but it is not of specific interest to us here. According to the analysis above, the scattering of an incoming wave from -∞ will result in transmission through two or three channels, depending on whether ω < ω crit or not. An exact solution to the scattering problem can be decomposed as ω when 0 < ω < ω crit (if W 0 < 1) or 0 < ω < e Φ 0 (if W 0 > 1) and as when ω crit < ω < e Φ 0 (only possible if W 0 < 1). The wavenumbers k in , k r 1 and k r 2 are given by (19) and are labeled in FIG. 1, while the transmitted wavenumbers are not, in general, expressible as simple functions of the parameters. Note that we do not explicitly keep track of possible exponential decaying solutions in (38) (corresponding to complex solutions to the dispersion relation), since they do not contribute directly to (40). We insert the solutions (38) and (39) into the functional X of (17) and find, after algebraic manipulations, the following relationship between the transmission and reflection coefficients, where the sum is over all (2 or 3, depending on ω and W 0 ) transmission channels. Here, v g n = v g ( k t n ) are the group velocities of the transmitted modes at + ∞ . The LHS of (40) can be interpreted as the total reflection coefficient associated with the incident modes. Since the group velocities v g n of the transmitted modes are, by definition, always positive, the sign of the contribution from each channel k t n to the RHS of (40) is determined by the factor Ω( k ) = ω -e Φ 0 -kW 0 evaluated at k t n , as we anticipated previously. When only two transmission channels are admitted, one of the two factors Ω( k ) is strictly negative while in the case of three transmission channels, two of the three factors Ω( k ) are strictly negative. Therefore, in both situations there is one root which contributes an overall negative amount to the RHS of (40) and thus reduces the magnitude of the total reflection. Because of these troublesome modes, the RHS is not strictly greater than 1 and we cannot straightforwardly conclude superradiance. In general, in order to fully answer the question of superradiance, one would need to specify W ( x ) and V ( x ) in all space and solve for the coefficients R n and T n . Large Λ approximation: To better understand the relation between (40) and superradiance, we now focus on small deviations (Λ /greatermuch 1) from the non-dispersive case. For fixed W 0 < 1, the size of region I in FIG. 6 becomes zero when Λ is greater then e Φ 0 [2(1 -W 0 ) / 3] -2 / 3 . We therefore assume that the frequency ω lies in region II, where three transmission and two reflection channels are available, see (39). As explained above, in the most general case one would need to solve the equation of motion for all x in order to conclude superradiance or not from (40). Similarly to the zero flow case, by looking at the series expansions of the relevant wavenumbers, we can make useful predictions about the transmission coefficients in this Λ /greatermuch 1 regime. We start the analysis by solving (36) in the asymptotic region x → + ∞ and expressing the obtained transmitted wavenumbers as power series in Λ, The series expansions of the incident wavenumber k in and of the reflected wavenumbers, k r 1 and k r 2 , are given, as previously, by (21). From these Λ expansions, we note that, unless W 0 ∼ 1, we have k t 2 , 3 , k r 2 ∼ O (Λ) and k in , k r 1 , k t 1 ∼ O (Λ 0 ). However, unlike in the zero-flow case, the difference between the high momentum transmitted modes k t 2 , 3 and the high momentum reflected modes k r 2 is not negligible for Λ /greatermuch 1, being O (Λ). Hence, the appearance of such modes requires a large momentum change in our system. Therefore, if the potential is sufficiently smooth, we expect the conversion of incident modes k in into transmitted modes k t 2 and k t 3 and into reflected modes k r 2 to be disfavoured in comparison with the low momentum transmission/reflection channel involving k t 1 and k r 1 . In other words, we expect that the Wronskian condition (40) will include only the low momentum channel, i.e. From this relation for the reflection coefficient, we would conclude that superradiance occurs for all frequencies in region II ( ω crit < ω < e Φ 0 ) when Λ /greatermuch 1. Note that this conclusion certainly does not hold in the case of a general dispersive parameter. If the condition Λ /greatermuch 1 is not satisfied, there can be a mixture between the high and low momentum channels. Consequently, as discussed before, the total reflection coefficient given by (40) is not necessarily larger than one. In such a case, a definite answer about superradiance can only be obtained by solving the differential equations at every spatial point x . Having analyzed the case of W 0 < 1, let us now fix W 0 > 1 and assume 0 < ω < e Φ 0 . As discussed previously, there are two transmission and two reflection channels available when W 0 > 1 and the scattering problem is now described by (38). Since we are interested in the regime of large Λ, we calculate the wavenumber of the transmitted modes up to next-to-leading order terms, By direct substitution of these expressions into (40), one can straightforwadly determine the reflection coefficient in powers of Λ, ∣ ∣ plus terms of O (Λ -2 ). Note that this second channel T 2 is present even in the absence of dispersion, being an upstream mode which is swept downstream by a superluminal flow. From the equation above, it also becomes evident that the relation between the norms | T 1 | and | T 2 | of the two transmission channels determines the occurrence or not of superradiance. Critical case: An interesting situation to be analyzed is the critical case ω = ω crit , which corresponds to the boundary between regions I and II in FIG. 6. In this scenario, the background flow W 0 , when expanded in powers of Λ, relates to the critical frequency according to the following expression, Note that the previous analysis leading to (43) relied on series expansions (see (42)) which are not valid when W 0 -1 ∼ O ( Λ -2 3 ) . Therefore, in order to analyze the possibility of superradiance in the critical case, we cannot use (43); instead, we have to k 0.4 0.2 -0.2 -0.4 -1 -0.5 0 0.5 1 start from the original Wronskian relation (40). In the critical regime, the dispersion relation (see FIG. 7) has three distinct solutions. Two of these solutions, denoted by k t 1 and k t 2 , have positive group velocities in the lab frame and, therefore, are identified as transmitted modes. The other solution, denoted by k 0 , is a degenerate double root and, consequently, has a vanishing group velocity in the lab frame. In order to obtain the reflection coefficient for the scattering problem, we first expand k t 1 and k t 2 as power series in Λ, and then substitute the obtained expressions into (40). The final result is given by ∣ ∣ plus terms of order O ( Λ -2 3 ) . Observe again the importance of the relative sign of the norm in the two transmission channels. Note that the scattering process converts incident modes with wavenumber k in ≈ ω + O (Λ -1 ) into transmitted modes of large wavenumber k t 2 ≈ O (Λ 2 3 ), which can only be balanced by the high momentum reflected modes k r 2 ≈ O (Λ). Another possibility is the conversion of the incident modes k in ≈ ω + O (Λ -1 ) into transmitted modes of wavenumber k t 1 ≈ ( ω -e Φ 0 ) / 2, which is comparable to the low momentum k r 1 channel. Because of the high momentum change required by the second channel (involving k r 2 and k t 2 ), we expect the first channel (involving k r 1 and k t 1 ) to be favoured in our scattering experiment. Since ω crit < e Φ 0 , we therefore deduce that the RHS of (48) is always greater than one in the limit Λ /greatermuch 1. In other words, low-frequency waves in the critical regime are superradiantly scattered in our toy-model if small subluminal corrections are added to the dispersion relation.",
"pages": [
15,
16,
17,
18,
19,
20
]
},
{
"title": "3.2. Superluminal dispersion",
"content": "We now turn to superluminal scattering processes, which can be quite different compared to subluminal ones since there does not exist any notion of a horizon or modeindependent blocking region for high momentum incident modes (the group velocity dω/dk is unbounded as a function of k and only the low frequency modes which possess quasilinear dispersion experience a blocking region in such flows). We will follow the standard treatment [31] of analogue black holes with superluminal dispersion and analyze the transmission of an incoming wave from -∞ through to + ∞ . The relevant dispersion relation in the superluminal case is depicted in FIG. 8. Given a superluminal flow W 0 > 1 (solid red curve in FIG. 8), there exists an interval of frequencies 0 < ω < ω crit (region I in FIG. 8) for which only two propagating modes are admitted, one right-moving and one left-moving in the lab frame. For ω crit < ω < e Φ 0 (region II in FIG. 8), however, there are four propagating modes, two transmitted rightmovers and two left-movers. The third possibility is a subluminal flow W 0 < 1 with 0 < ω < e Φ 0 , for which there are always two propagating modes (see the black dashed curve in FIG. 8). Let us discuss first the cases in which only two propagating modes are available in the asymptotic limit x → + ∞ . Hence, the scattering is produced by an incident wave from -∞ whose frequency ω satisfies 0 < ω < ω crit (if W 0 > 1) or 0 < ω < e Φ 0 (if W 0 < 1). Note that exactly one of these two propagating modes is a left-moving mode. Imposing the boundary condition that no incoming mode is allowed at + ∞ , we obtain the solution of (29) corresponding to the scattering problem, plus exponentially decaying channels which do not contribute directly to the generalized Wronskian current calculated at x →±∞ . Note that the wavenumbers k in and k r are the same ones that appear in (24) for the superluminal W = 0 case and the wavenumber k t represents the only available transmission channel. On the other hand, if W 0 > 1 and the frequency ω of the incident wave satisfies ω crit < ω < e Φ 0 (region II in FIG. 8), then there are, in principle, two extra propagating channels available (four in total, as discussed above). However, because of the boundary ω condition imposed at x → + ∞ , only one extra transmission channel has to be considered (the other extra channel is always left-moving at x → + ∞ ). The scattering solution is then given by where k in and k r are again given by (24) and k t 1 and k t 2 are the wavenumbers of the transmitted modes. Note that we have once again omitted the exponential decaying mode at x →-∞ since it does not affect directly the generalized Wronskian. Using (17) to evaluate the functional X in both asymptotic regions, we obtain, similarly to the subluminal case, the following relation between the reflection and transmission coefficients, where the sum is over one or two transmission channels, depending on ω and whether W 0 > 1 or W 0 < 1. Here, v g 1 and v g 2 are the group velocities of the transmitted modes k t 1 and k t 2 , which are always positive by definition. Furthermore, it is possible to show, for frequencies 0 < ω < e Φ 0 , that the effective frequency Ω = ω -e Φ 0 -k t n W 0 is always negative for k t 1 modes and always positive for k t 2 modes. Therefore, since only the n = 1 transmission channel is available for frequencies lying in region I of FIG. 8, we conclude that the RHS of (51) is greater than 1 and, therefore, the scattering is always superradiant. The situation for W 0 < 1 and 0 < ω < e Φ 0 is similar: only the first transmission channel is available and superradiance always occurs. However, for frequencies located in region II, we cannot so easily conclude superradiance since the extra transmission channel k t 2 contributes an overall negative factor in (51). To obtain a conclusive answer, one would need to know the detailed structure of W ( x ) and V ( x ) in the intermediate regime and solve the equations not only in the asymptotic regions but at every point x . Large Λ approximation: In order to better understand the scattering of an incident wave whose frequency is located in region II of FIG. 8, we shall consider small deviations from the non-dispersive limit, i.e. Λ /greatermuch 1. In such a case, we can expand the two transmission channels, k t 1 and k t 2 , in powers of Λ, and substitute the obtained wavenumbers into (51) in order to determine the reflection coefficient for the scattering, Since we assume ω < e Φ 0 in region II, this reflection coefficient is larger than 1 whenever | T 2 | < | T 1 | . As explained above, whether this condition is satisfied or not in a general model would depend on the detailed structure of W ( x ) and V ( x ) in the intermediate regime [31]. From a practical point of view, in order to maximize the potential for superradiance in an experiment with a superluminally dispersive medium, one should choose the asymptotic flow W 0 as small as possible while still being superluminal ( W 0 > 1) as this would minimize the size of region II and the extra positive-effectivefrequency transmission channels therein. Choosing the flow as such to maximize the T 1 channel is also consistent with our intuition that scattering favors the channel which most closely matches the momentum of the reflected mode; in this case the wavenumber k t 1 is closer to k r than k t 2 is. This prediction is confirmed in the step function model for which V ( x ) = e Φ 0 Θ( x ) and W ( x ) = W 0 Θ( x ). In such a case, one can impose the appropriate boundary conditions at x = 0 discussed in the appendix to obtain the following reflection and transmission coefficients, We can also show that the coefficient correponding to the omitted exponential decaying mode in (50) is of order O (Λ -2 ). Comparing T 1 and T 2 above and using the fact that W 0 > 1, one can see that | T 1 | > | T 2 | at zeroth order in Λ, which implies superradiance and confirms our expectations. Alternatively, one can verify the occurence of superradiance by directly analyzing the reflection coefficient R above. It is straightforward to see that | R 2 | > 1 at lowest order in Λ. Critical case: Another interesting possibility that we now consider in detail is the critical regime ω = ω crit . This situation corresponds to the boundary between regions I and II in FIG. 8 and is depicted, in the fluid frame, in FIG. 9. The relation between the background flow W 0 and the critical frequency ω crit is given by the following expression, Note that, since W 0 -1 ∼ O ( Λ -2 3 ) , the Λ expansion (54) of the generalized Wronskian obtained previously is not valid in the present case (check the denominators in (53)). Consequently, we shall need different Λ expansions in order to obtain an appropriate expression for the reflection coefficient. Like in the critical subluminal case, the dispersion relation has three distinct roots: the double root k 0 (with vanishing group velocity in the lab frame), a right-moving mode (with negative group velocity in the lab frame) and a transmitted mode (with positive group velocity in the lab frame) whose wavenumber k t is given by Applying as a boundary condition the fact that only right moving modes are allowed at + ∞ , we obtain, after substituting the relevant quantities into (51), the relation between the reflection and transmission coefficients for the scattering process, Since ω crit < e Φ 0 , we conclude that the RHS of the equation above is always greater than one when Λ /greatermuch 1. In summary, superradiance is expected to occur in the superluminal critical case for small deviations from the non-dispersive regime. 10 5 -5 -10 -3 -2 -1 0 1 2 3",
"pages": [
20,
21,
22,
23,
24
]
},
{
"title": "3.3. Inertial motion superradiance",
"content": "Throughout this paper, inspired by the usual condition for rotational superradiance in the black hole case, i.e. ω -m Ω h < 0, we analyzed only scattering problems in which the frequency of the incident mode satisfies ω -e Φ 0 < 0. However, since it is the effective frequency Ω that appears in (17), we conclude that the amplification of an incident mode can also occur for ω -e Φ 0 > 0 given that ω -e Φ 0 -kW 0 < 0. In particular, even when Φ 0 = 0 superradiant scattering will be possible. However, being due exclusively to inertial motion in the system, this kind of superradiance is outside the scope of our work. In fact, inertial motion superradiance has long been known in the literature as the anomalous Doppler effect and the condition for negative effective-frequency modes is referred to as the Ginzburg-Frank condition [32]. Several phenomena in physics, like the Vavilov-Cherenkov effect and the Mach cones (which appear in supersonic airplanes) can be understood in terms of inertial motion superradiance [18]. For a detailed analysis of inertial motion superradiance, we refer the reader to [18].",
"pages": [
24
]
},
{
"title": "4. Applications: axisymmetric systems",
"content": "Having analyzed superradiance in simple 1+1-dimensional toy models with modified dispersion relations, we will now discuss how the ideas presented in this paper can be generalized to more realistic situations based on analogue models of gravity. Our starting point is a general 2+1-dimensional, axisymmetric and irrotational fluid flow with background velocity v given by where A and B are constants and ( r, φ ) are the usual polar coordinates. Velocity perturbations δ v of the background flow can be conveniently described by a scalar field ψ , which relates to δ v through equation δ v = ∇ ψ . We denote the propagation speed of these perturbations by c . The idea of analogue gravity is derived from the observation that the differential equation satisfied by the perturbations ψ , i.e. can be cast into a Klein-Gordon equation in an effectively curved spacetime geometry. This connection between hydrodynamics and gravity is responsible for many important results, see [2] for a detailed review. One of the successes of Unruh's [1] original idea of using sound waves to study gravitational phenomena is that it can be extended to many other physical systems, like gravity waves in open channel flows [33] and density perturbations in Bose-Einstein condensates [34]. An important feature of such systems is that (62) is only accurate in certain regimes; at sufficiently small distance scales (e.g. wavelengths comparable to the fluid depth in open channels), the dispersion relation is not linear anymore and (62) has to be replaced by ¶ where Λ is a dispersive parameter and the upper (lower) sign corresponds to subluminal (superluminal) dispersion. Even though superradiance has been subjected to extensive studies in the linear regime of analogue models, it has never been analysed before in the context of modified dispersion relations, as opposed to Hawking radiation (see e.g. [2] and references therein). The remarkable fact about the toy models introduced in this paper is that they can be used to analyze superradiance in realistic dispersive analogue models of gravity satisfying (63). Firstly, the electromagnetic interaction term e Φ 0 appearing in our toy models is analogous to the rotational term m Ω in axisymmetric analogue models, where m is the azimuthal number and Ω is the angular velocity. It is interesting to note that this duality is manifest in real black holes: both electromagnetic [35] and rotational [25] superradiance are possible. Another essential ingredient for the occurrence of superradiance in analogue models of gravity is the presence of an event horizon, which allows no mode to escape from inside the analogue black hole. In our toy model, such behaviour is mimicked by an appropriate boundary condition imposed in the asymptotic limit x →∞ . It is also important to address the usefulness of the generalized Wronskian (37) in the context of axisymmetric systems. More precisely, we are going to show that, if all derivatives with respect to x are replaced by derivatives with respect to r , then the Wronskian (37), when applied to solutions of (63), is independent of r . Indeed, by applying the ansatz ψ = ( H ( r ) / √ r ) e imφ e -iωt , we are able to separate (63) and are left with a radial equation for H , where the coefficients α ( r ), β ( r ) and γ ( r ) are given by The functions P ( r ) and Q ( r ) appearing in the coefficients above are, up to a factor ( c 2 -v 2 r ), the same functions P and Q defined in Refs. [36, 37]. They can be expressed as where the upper (lower) sign corresponds to subluminal (superluminal) dispersion. Finally, we note that (64) is exactly the same as (30) and, more remarkably, that the coefficients α , β , γ above satisfy conditions (32) and (33). Consequently, the generalized Wronskian defined in (37) can also be used in the context of axisymmetric analogue models of gravity, thus completing the connection between our toy models and realistic physical systems.",
"pages": [
24,
25,
26
]
},
{
"title": "5. Summary and final remarks",
"content": "We have proposed idealized systems to investigate multi superradiant scattering processes that are applicable to sub and superluminal dispersive fields. Perhaps the most important theoretical result obtained is related to the simplicity of the analytic expression for the particle number currents J n of the scattering channels n (see (17)), ∣ Notice that the expression above depends only on the amplitude, group velocity and effective frequency of the particular scattering channel. Moreover, this result is universal to all scattering processes discussed in this paper. Note also that, in principle, we could have normalized the modes so that the group velocity in (67) is absorbed into the coefficients A n . Doing that would make the conservation equations (see e.g. (20)) look much simpler. However, their dependence on the dispersive parameter Λ would then also be hidden in these new coefficients. Our findings link to standard scattering processes, allowing a deeper insight into superradiance. Let us consider scattering of up to four incident modes by a general scattering potential. There are up to four channels to the left { a, b, c, d } and four to the right { A, B, C, D } of the scattering potential. As a lesson from the analysis carried out before, one needs to be careful when assigning the propagation direction of each mode since its group velocity is dependent on the particular scattering potential and type of dispersion, see FIG. 10. Furthermore, as long as the scattering potential is real, the total current to the left equals the total current to the right of the potential, The scattering potential is amplifying classical and quantum field excitations if, at the left side of the potential, the total outflux (i.e. the reflected current) is larger then the total in-flux, J ref total > J in total . There are several scattering coefficients that can be considered: the reflection coefficient in each individual scattering channel, and the total reflection coefficient, A sufficient, but not necessary, condition for superradiance, is to demand no influx from the right and to strictly require negative effective frequency for the remaining channels to the right -these conditions arise naturally at the event horizon of a rotating black hole. A surprising result of our analysis is that the presence of extra scattering channels can enhance the amplification effect. In FIG. 11, we illustrate the scattering diagrams for some of the examples discussed in the paper. In addition, the present work can be related to recent experimental realizations of analogue black holes, some of which have even studied the Hawking emission process. In particular, [9] exhibits the first detection of the classical analogue of Hawking radiation using dispersive gravity waves in an open channel flow. One of the most important lessons to be learned from that work is that even though vorticity and viscosity effects cannot be completely removed from the experimental setup, they can be made extremely small. In fact, they can be reduced to the point that the results predicted by the irrotational and inviscid theory match the results obtained experimentally with considerable accuracy. Based on this fact, together with our present results, one might ask whether superradiance occurs (and can be observed) in the laboratory using dispersive gravity waves. The connection between such system and our analysis can be seen directly from the full dispersion relation for gravity waves in a fluid of constant depth h [38], where g is the gravitational acceleration and σ and ρ correspond, respectively, to the surface tension and the density of the fluid. If first-order deviations from the shallow water limit ( kh /lessmuch 1) are considered, we recover the quartic dispersion relation analyzed in this paper. In other words, non-shallow gravity waves impinging on a rotating analogue black hole (e.g. a draining 'bathtub' vortex) satisfy (63) with c 2 = gh and a dispersive parameter Λ -2 = | gh 3 / 3 -σh/ρ | . Based on the existent analysis of superradiance of linear fields in open channel flows [33, 37] together with our discussion in Section 4, we expect superradiance to also be manifest for non-shallow gravity waves. This relation between dispersion and superradiance in open channels is currently being further investigated by the authors and will be the subject of a future work. Another class of analogue black holes, which was recently set up in the laboratory [6] and which might be used in the future to produce superradiant scattering processes, consists of Bose-Einstein condensates [34, 2]. The analogy with gravity arises when one considers the Gross-Pitaevskii equation and uses the Madelung representation of the condensate wave function. If the eikonal approximation is used and axisymmetry is assumed, it is possible to show that perturbations around a background condensate obey a superluminal dispersion relation given by where m is the mass of a single boson, a is the scattering length and n 0 ( r ) is the background density. Such perturbations are described by (63) with c 2 = 4 π /planckover2pi1 2 n 0 a/m 2 and Λ = 2 m/ /planckover2pi1 . Based on our work, the most obvious conclusion we can draw is to expect superradiance to be manifest also in low-frequency BEC scattering experiments. However, since the dispersion relation is superluminal, there is no notion of a mode independent blocking region (see section 3.2) and, consequently, it is not clear if the appropriate boundary conditions will be sufficient to guarantee superradiance. Additionally, quantized vortices may be present in such systems, restricting the angular momentum to integer multiples of /planckover2pi1 . The physics of these quantized vortices with respect to superradiance and instabilities is also unclear at this point and more investigation is needed to understand their role in possible BEC scattering processes.",
"pages": [
26,
27,
28,
29
]
},
{
"title": "Acknowledgments",
"content": "MR was partially supported by FAPESP. SW was supported by Marie Curie Career Integration Grant (MULTI-QG-2011), the SISSA Young Researchers Grant (Black hole horizon effects in fluids and superfluids), and the Fqxi Mini grant (Physics without borders). We wish to thank Carlos Barcel'o for his comments. SW would like to thank Matt Visser for stimulating discussions.",
"pages": [
29,
30
]
},
{
"title": "Appendix",
"content": "Throughout this paper we solve simple models based on step functions for the external potential V ( x ) = e Φ 0 Θ( x ) and for the background velocity W ( x ) = W 0 Θ( x ), where Θ( x ) is the Heaviside function. Since this function is characterized by a discontinuity at x = 0, it is important to analyze what happens to the wavefunction, i.e. the solution to (10) or (30), at x = 0. The same problem arises in 1D tunneling problems in quantum mechanics when the potential barrier in the Schrodinger equation is modelled by a step function. In such situations, one has to impose the continuity of the wavefunction and its first derivative at the discontinuity point in order to determine the reflection and the transmission coefficients. Let us first consider (10), which is valid only for zero background flows. Following the standard procedure used in quantum mechanics (see e.g. [39]), we integrate the differential equation over a small region ( -/epsilon1, + /epsilon1 ) around the discontinuity x = 0 and then take the limit of the obtained expression as /epsilon1 → 0. Starting from (10) and repeating this procedure three times, one can show that f ( x ) and its derivatives up to third order are all continuous at x = 0. We also have to deal with the generalization to non-zero flows, given by (30). The situation now is more complicate since the differential equation involves derivatives of Θ( x ) (i.e. delta functions). Repeating the procedure described above, one can show that the function f and its first order derivative are still continuous at x = 0. The continuity of the second and third order derivatives, because of the delta functions, now depends on Θ(0), i.e. it depends on the choice of the flow velocity and of the external potential exactly at x = 0. Using the half-maximum convention of Θ(0) = 1 / 2, we can further prove that the second order derivative f '' ( x ) is also continuous at x = 0 and that the third order derivative satisfies the following relation,",
"pages": [
30
]
}
]
2013CQGra..30j4002N
https://arxiv.org/pdf/1204.1057.pdf
<document>
<section_header_level_1><location><page_1><loc_19><loc_81><loc_81><loc_87></location>State/Operator Correspondence in Higher-Spin DS/CFT</section_header_level_1>
<text><location><page_1><loc_34><loc_76><loc_66><loc_78></location>Gim Seng Ng and Andrew Strominger</text>
<text><location><page_1><loc_23><loc_68><loc_77><loc_72></location>Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA</text>
<section_header_level_1><location><page_1><loc_46><loc_65><loc_54><loc_66></location>Abstract</section_header_level_1>
<text><location><page_1><loc_17><loc_37><loc_83><loc_63></location>A recently conjectured microscopic realization of the dS 4 /CFT 3 correspondence relating Vasiliev's higher-spin gravity on dS 4 to a Euclidean Sp ( N ) CFT 3 is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary. In particular it is argued that states of the boundary CFT 3 on S 2 are holographically dual to bulk states on geodesically complete, spacelike R 3 slices which terminate on an S 2 at future infinity. The dictionary is described in detail for the case of free scalar excitations. The ground states of the free or critical Sp ( N ) model are dual to dS-invariant planewave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT 3 . CFT 3 states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk. A norm is defined on the R 3 bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the Sp ( N ) CFT 3 .</text>
<section_header_level_1><location><page_2><loc_12><loc_91><loc_22><loc_92></location>Contents</section_header_level_1>
<table>
<location><page_2><loc_12><loc_43><loc_88><loc_88></location>
</table>
<section_header_level_1><location><page_2><loc_12><loc_37><loc_30><loc_39></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_12><loc_88><loc_35></location>The conjectured dS/CFT correspondence attempts to adapt the wonderful successes of the AdS/CFT correspondence to universes (possibly like our own) which exponentially expand in the far future. The hope [1, 2, 3, 4, 5] is to define bulk de Sitter (dS) quantum gravity in terms of a holographically dual CFT living at I + of dS, which is the asymptotic conformal boundary at future null infinity. A major obstacle to this program has been the absence of any explicit microscopic realization. This has so far prevented the detailed development of the dS/CFT dictionary. This situation has recently been improved by an explicit proposal [6] relating Vasiliev's higher-spin gravity in dS 4 [7, 8] to the dual Sp ( N ) CFT 3 described in [9]. In this paper we will use this higher-spin context to write some new entries in the dS/CFT dictionary.</text>
<text><location><page_3><loc_12><loc_71><loc_88><loc_92></location>The recent proposal [6] for a microscopic realization of dS/CFT begins with the duality relating the free (critical) O ( N ) CFT 3 to higher-spin gravity on AdS 4 with Neumann (Dirichlet) boundary conditions on the scalar field. Higher-spin gravity - unlike string theory [2] has a simple analytic continuation from negative to positive cosmological constant Λ. Under this continuation, AdS 4 → dS 4 and the (singlet) boundary CFT 3 correlators are simply transformed by the replacement of N →-N . These same transformed correlators arise from the Sp ( N ) models constructed from anticommuting scalars. It follows that the free (critical) Sp ( N ) correlators equal those of higher-spin gravity on dS 4 with future Neumann (Dirichlet) scalar boundary conditions (of the type described in [10]) at I + .</text>
<text><location><page_3><loc_12><loc_60><loc_88><loc_71></location>This mathematical relation between the bulk dS and boundary Sp ( N ) correlators may provide a good starting point for understanding quantum gravity on dS, but so far important physical questions remain unanswered. For example we do not know how to relate these physically un measurable correlators to a set of true physical observables or to the dS horizon entropy. These crucial entries in the dS/CFT dictionary are yet to be written.</text>
<text><location><page_3><loc_12><loc_38><loc_88><loc_59></location>As a step in this direction, in this paper we investigate the relation between quantum states in the bulk higher-spin gravity and those in the boundary CFT 3 . Bulk higher-spin gravity has fields of Φ s with all even spins s = 0 , 2 , .... , which are dual to CFT 3 operators O s with the same spins. In the CFT 3 , we can also associate a state to each operator by the state-operator correspondence. One way to do this is to take the southern hemisphere of S 3 , insert the operator O s at the south pole, and then define a state Ψ s S 2 as a functional of the boundary conditions on the equatorial S 2 . For every object in the CFT 3 , we expect a holographically dual object in the bulk dS 4 theory. This raises the question: what is the bulk representation of the spins state Ψ s S 2 ?</text>
<text><location><page_3><loc_12><loc_29><loc_88><loc_38></location>In Lorentzian AdS 4 holography, the state created by a primary operator O in the CFT 3 on S 2 has, at weak coupling, a bulk representation as the single particle state of the field Φ dual to O with a smooth minimal-energy wavefunction localized near the center of AdS 4 . The form of the wavefunction is dictated by the conformal symmetry.</text>
<text><location><page_3><loc_12><loc_16><loc_88><loc_28></location>In dS 4 holography, the situation is rather different. States in dS 4 quantum gravity are usually thought of as wavefunctions on complete spacelike slices which are topologically S 3 . 1 These do not seem to be good candidates for bulk duals to Ψ s S 2 because, among other reasons, they are not associated to any S 2 in I + . However, dS 4 also has everywhere spacelike and geodesically complete R 3 slices which end at an S 2 in I + . Here we propose a construction of</text>
<text><location><page_4><loc_12><loc_84><loc_88><loc_92></location>the bulk version of Ψ s S 2 on these slices, again as single particle states whose form is dictated by the conformal symmetry. Interestingly, the classical wavefunction for the particle turns out to be the (anti) quasinormal modes for the static patch of de Sitter, as constructed in [11, 12]. 2</text>
<text><location><page_4><loc_12><loc_70><loc_88><loc_83></location>This relation between bulk and boundary states has a potentially profound nonperturbative consequence briefly mentioned in section 4.1 [14]. The operator O 0 dual to a scalar Φ is bilinear in boundary fermions and hence obeys ( O 0 ) N 2 +1 = 0. Under bulk-boundary duality this translates into an N 2 -adicity relation for Φ: one cannot put more than N 2 bulk scalar quanta into the associated quasinormal mode. Further investigation of this dS exclusion principle is deferred to later work.</text>
<text><location><page_4><loc_12><loc_62><loc_88><loc_69></location>We also construct a norm for these bulk states and show that it is the Zamolodchikov norm on S 3 of the CFT 3 operator O s . Explicit formulae are exhibited only for the scalar s = 0 case but we expect the construction to generalize to all s .</text>
<text><location><page_4><loc_12><loc_15><loc_88><loc_62></location>This paper is organized as follows. In section 2 we revisit the issue of the usual global dS-invariant vacua for a free massive scalar field, paying particular attention to the case of m 2 /lscript 2 = 2 (where /lscript is the de Sitter radius) arising in higher spin gravity. The invariant vacua include the familiar Bunch-Davies Euclidean vacuum | 0 E 〉 , as well as a pair of | 0 ± 〉 of in/out vacua with no particle production. As the scalar field acting on | 0 -〉 ( | 0 + 〉 ) obeys Dirichlet (Neumann) boundary conditions on I + , these are related to the critical (free) Sp ( N ) model. Generically all dS-invariant vacua are Bogolyubov transformations of one another, but we find that at m 2 /lscript 2 = 2 the transformation is singular and the in/out vacua are non-normalizable plane-wave type states. In section 3 we use the conformal symmetries to find the classical bulk wavefunctions associated to an operator insertion on I + , and note the relation to (anti) quasinormal modes. The construction uses a rescaled bulk-to-boundary Green function defined with Neumann or Dirichlet I + boundary conditions. We also show that the Klein-Gordon inner product of these wavefunctions agrees with the conformallycovariant CFT 3 operator two-point function. In section 4 we consider the Hilbert space on R 3 slices ending on an S 2 on I + . This Hilbert space was explicitly constructed in [15] for a free scalar on hyperbolic slices ending on I + . There are two such Hilbert spaces, which we denote the northern and southern Hilbert space, which live on spatial R 3 slices extending to the north or south of the S 2 . The northern and southern slices add up to a global S 3 . Hence the tensor product of the northern and southern Hilbert spaces is the global Hilbert space on S 3 , much as the left and right Rindler Hilbert spaces tensor to the global</text>
<text><location><page_5><loc_12><loc_58><loc_88><loc_93></location>Minkowski Hilbert space. We show that the global | 0 ± 〉 vacua are simple tensor products of the northern and southern Dirichlet and Neumann vacua. We then use symmetries to uniquely identify the states of the southern Hilbert space with those of the free and critical Sp ( N ) models on an S 2 . This leads directly to the dS exclusion principle. We further construct an inner product for the southern Hilbert space which agrees, for states dual to I + operator insertions, to the conformal two-point function on I + . In section 5 we discuss the restriction of Euclidean vacuum to a southern state and recall from [15], that this is a mixed state which is thermal with respect to an SO (3 , 1) Casimir. It would be interesting to relate this result to dS entropy in the present context. In section 6 we show that the standard CFT 3 state-operator correspondence maps the known pseudo-unitary C-norm of the Sp ( N ) model to the Zamolodchikov two-point function. This completes the demonstration that the bulk states on R 3 S have the requisite properties to be dual to the boundary Sp ( N ) CFT 3 states on S 2 . Speculations are made on the possible relevance of pseudo-unitarity to the consistency of dS/CFT in general. An appendix gives some explicit formulae for the SO (4 , 1) Killing vectors of dS 4 .</text>
<section_header_level_1><location><page_5><loc_12><loc_52><loc_48><loc_54></location>2 Global dS vacua at m 2 /lscript 2 = 2</section_header_level_1>
<text><location><page_5><loc_12><loc_46><loc_88><loc_50></location>In this section we describe the quantum theory of a free scalar field Φ in dS 4 with wave equation</text>
<formula><location><page_5><loc_43><loc_43><loc_88><loc_46></location>( ∇ 2 -m 2 )Φ = 0 , (2.1)</formula>
<text><location><page_5><loc_12><loc_40><loc_20><loc_42></location>and mass</text>
<formula><location><page_5><loc_46><loc_38><loc_88><loc_40></location>m 2 /lscript 2 = 2 . (2.2)</formula>
<text><location><page_5><loc_12><loc_16><loc_88><loc_36></location>This is the case of interest for Vasiliev's higher-spin gravity. While there have been many general discussions of this problem, peculiar singular behavior as well as simplifications appear at the critical value m 2 /lscript 2 = 2 which are highly relevant to the structure of dS/CFT. A parallel discussion of de Sitter vacua and scalar Green functions in the context of dS/CFT was given in [16]. However that paper in many places specialized to the large mass regime m 2 /lscript 2 > 9 4 , excluding the region of current interest. The behavior in the region m 2 /lscript 2 < 9 4 divides into three cases m 2 /lscript 2 > 2, m 2 /lscript 2 = 2 and m 2 /lscript 2 < 2. Much of the structure we describe below pertains to the entire range m 2 /lscript 2 < 9 4 with an additional branch-cut prescription for the Green functions.</text>
<section_header_level_1><location><page_6><loc_12><loc_91><loc_23><loc_92></location>2.1 Modes</section_header_level_1>
<text><location><page_6><loc_12><loc_87><loc_48><loc_89></location>We will work in the dS 4 global coordinates</text>
<formula><location><page_6><loc_17><loc_81><loc_88><loc_86></location>ds 2 /lscript 2 = -dt 2 +cosh 2 td 2 Ω 3 = -dt 2 +cosh 2 t [ dψ 2 +sin 2 ψ ( dθ 2 +sin 2 θdφ 2 )] , (2.3)</formula>
<text><location><page_6><loc_12><loc_76><loc_88><loc_80></location>where Ω i ∼ ( ψ, θ, φ ) are coordinates on the global S 3 slices. Following the notation of [16] solutions of the wave equation can be expanded in modes</text>
<formula><location><page_6><loc_41><loc_72><loc_88><loc_73></location>φ Lj ( x ) = y L ( t ) Y Lj (Ω) (2.4)</formula>
<text><location><page_6><loc_12><loc_65><loc_88><loc_69></location>of total angular momentum L and spin labeled by the multi-index j . The spherical harmonics Y Lj obey</text>
<formula><location><page_6><loc_27><loc_51><loc_88><loc_63></location>Y ∗ Lj (Ω) = ( -) L Y Lj (Ω) = Y Lj (Ω A ) , D 2 Y Lj (Ω) = -L ( L +2) Y Lj (Ω) , ∫ S 3 √ hd 3 Ω Y ∗ Lj (Ω) Y L ' j ' (Ω) = δ L,L ' δ j,j ' , ∑ Y ∗ Lj (Ω) Y Lj (Ω ' ) = 1 √ h δ 3 (Ω -Ω ' ) , (2.5)</formula>
<text><location><page_6><loc_12><loc_43><loc_88><loc_50></location>where √ h and D 2 are the measure and Laplacian on the unit S 3 , Ω A is the antipodal point of Ω, and here and hereafter ∑ denotes summation over all allowed values of L and j . The time dependence is then governed by the second order ODE</text>
<formula><location><page_6><loc_29><loc_37><loc_88><loc_41></location>∂ 2 t y L +3tanh t∂ t y L + ( m 2 /lscript 2 + L ( L +2) cosh 2 t ) y L = 0 . (2.6)</formula>
<section_header_level_1><location><page_6><loc_12><loc_33><loc_48><loc_35></location>2.1.1 Neumann and Dirichlet modes</section_header_level_1>
<text><location><page_6><loc_12><loc_30><loc_38><loc_32></location>Eq. (2.6) has the real solutions</text>
<formula><location><page_6><loc_20><loc_25><loc_88><loc_28></location>y ± L = 2 L + h ± + 1 2 ( L +1) ± 1 2 cosh L te -( L + h ± ) t F ( L + 3 2 , L + h ± , h ± -1 2 ; -e -2 t ) (2.7)</formula>
<text><location><page_6><loc_12><loc_21><loc_17><loc_23></location>where</text>
<formula><location><page_6><loc_41><loc_18><loc_88><loc_21></location>h ± ≡ 3 2 ± √ 9 4 -m 2 /lscript 2 . (2.8)</formula>
<text><location><page_6><loc_12><loc_15><loc_49><loc_17></location>We are interested in m 2 /lscript 2 = 2, which implies</text>
<formula><location><page_6><loc_42><loc_11><loc_88><loc_12></location>h -= 1 , h + = 2 , (2.9)</formula>
<text><location><page_7><loc_12><loc_91><loc_15><loc_92></location>and</text>
<formula><location><page_7><loc_22><loc_86><loc_88><loc_91></location>y ± L = ( -i ) 1 2 ± 1 2 2 L √ 1 + L cosh L te -( L +1) t [ 1 (1 -ie -t ) 2 L +2 ∓ 1 (1 + ie -t ) 2 L +2 ] . (2.10)</formula>
<text><location><page_7><loc_12><loc_83><loc_42><loc_86></location>The modes behave near I + as e -h ± t</text>
<formula><location><page_7><loc_31><loc_74><loc_88><loc_81></location>t → ∞ , y -L → (2( L +1) -1 2 ) e -t + O ( e -3 t ) Neumann , y + L → (4( L +1) 1 2 ) e -2 t + O ( e -4 t ) Dirichlet . (2.11)</formula>
<text><location><page_7><loc_12><loc_68><loc_88><loc_72></location>Accordingly we refer to the + modes as Dirichlet and the -modes as Neumann. We have normalized so that the Klein-Gordon inner product is</text>
<formula><location><page_7><loc_30><loc_63><loc_88><loc_67></location>〈 φ + Lj | φ -L ' j ' 〉 S 3 ≡ i ∫ S 3 d 3 Σ µ φ + ∗ Lj ←→ ∂ µ φ -L ' j ' = iδ LL ' δ jj ' , (2.12)</formula>
<text><location><page_7><loc_12><loc_59><loc_66><loc_61></location>with d 3 Σ µ the induced measure times the normal to the S 3 slice.</text>
<text><location><page_7><loc_15><loc_57><loc_31><loc_59></location>Under time reversal</text>
<formula><location><page_7><loc_41><loc_54><loc_88><loc_56></location>y ± L ( t ) = ± ( -) L y ± L ( -t ) , (2.13)</formula>
<text><location><page_7><loc_12><loc_51><loc_18><loc_53></location>so that</text>
<formula><location><page_7><loc_36><loc_48><loc_88><loc_51></location>φ ± Lj ( x ) = ± φ ± Lj ( x A ) = ( -) L φ ±∗ Lj ( x ) , (2.14)</formula>
<text><location><page_7><loc_12><loc_38><loc_88><loc_47></location>where the point x A is antipodal to the point x . This implies that an incoming Dirichlet (Neumann) mode propagates to an outgoing Dirichlet (Neumann) mode. This is not the case for generic m 2 and, as will be seen below, allows for Dirichlet and Neumann vacua with no particle production.</text>
<section_header_level_1><location><page_7><loc_12><loc_34><loc_35><loc_35></location>2.1.2 Euclidean modes</section_header_level_1>
<text><location><page_7><loc_12><loc_28><loc_88><loc_32></location>Euclidean modes are defined by the condition that when dS 4 is analytically continued to S 4 they remain nonsingular on the southern hemisphere. In other words</text>
<formula><location><page_7><loc_38><loc_23><loc_88><loc_26></location>y E L ( t = -iπ 2 ) = nonsingular . (2.15)</formula>
<text><location><page_7><loc_12><loc_19><loc_38><loc_21></location>One finds that the combination</text>
<formula><location><page_7><loc_32><loc_13><loc_88><loc_18></location>y E L = y -L + iy + L √ 2 = 2 L +1 √ 2 L +2 cosh L te -( L +1) t (1 -ie -t ) 2 L +2 (2.16)</formula>
<text><location><page_8><loc_12><loc_88><loc_88><loc_93></location>is nonsingular at t = -iπ/ 2. Hence y ∓ L are simply the real and imaginary parts of the y E L . (2.13) and (2.14) imply the relations</text>
<formula><location><page_8><loc_40><loc_83><loc_88><loc_86></location>y E ∗ L ( t ) = ( -) L +1 y E L ( -t ) (2.17)</formula>
<formula><location><page_8><loc_40><loc_79><loc_88><loc_82></location>φ E Lj ( x A ) = ( -) L +1 φ E ∗ Lj ( x ) . (2.18)</formula>
<formula><location><page_8><loc_41><loc_75><loc_88><loc_78></location>〈 φ E Lj | φ E L ' j ' 〉 S 3 = δ LL ' δ jj ' (2.19)</formula>
<section_header_level_1><location><page_8><loc_12><loc_72><loc_23><loc_73></location>2.2 Vacua</section_header_level_1>
<text><location><page_8><loc_12><loc_66><loc_88><loc_70></location>In the quantum theory Φ is promoted to an operator which we denote ˆ Φ obeying the equal time commutation relation</text>
<formula><location><page_8><loc_32><loc_61><loc_88><loc_64></location>[ ˆ Φ(Ω , t ) , ∂ t ˆ Φ(Ω ' , t )] = i √ h cosh 3 t δ 3 (Ω -Ω ' ) . (2.20)</formula>
<text><location><page_8><loc_12><loc_57><loc_49><loc_59></location>Defining annihilation and creation operators</text>
<formula><location><page_8><loc_34><loc_52><loc_88><loc_55></location>a E Lj = 〈 φ E Lj | ˆ Φ 〉 S 3 , a E † Lj = -〈 φ E ∗ Lj | ˆ Φ 〉 S 3 , (2.21)</formula>
<text><location><page_8><loc_12><loc_49><loc_63><loc_50></location>the global Euclidean (or Bunch-Davies) vacuum is defined by</text>
<formula><location><page_8><loc_45><loc_43><loc_88><loc_46></location>a E Lj | 0 E 〉 = 0 . (2.22)</formula>
<text><location><page_8><loc_12><loc_35><loc_88><loc_42></location>We normalize so that 〈 0 E | 0 E 〉 = 1. For any m 2 there is a family of dS-invariant vacua labeled by a complex parameter α . They are annihilated by the normalized Bogolyubov-transformed oscillators</text>
<text><location><page_8><loc_12><loc_22><loc_88><loc_31></location>We are interested in the vacua annihilated by the Dirichlet or Neumann modes for the case of m 2 /lscript 2 = 2, which correspond to e α = ± 1. In that case the Bogolyubov transformation is singular. Nevertheless we can still construct non-normalizable plane-wave type vacua as follows.</text>
<formula><location><page_8><loc_35><loc_31><loc_88><loc_35></location>a α Lj = 1 √ 1 -e α + α ∗ ( a E Lj -e α ∗ a E † Lj ) . (2.23)</formula>
<text><location><page_8><loc_15><loc_20><loc_50><loc_21></location>The field operator may be decomposed as</text>
<formula><location><page_8><loc_44><loc_15><loc_88><loc_17></location>ˆ Φ = ˆ Φ + + ˆ Φ -, (2.24)</formula>
<text><location><page_9><loc_12><loc_90><loc_52><loc_93></location>where ˆ Φ ± ∼ e -h ± t near I + . The squeezed states</text>
<formula><location><page_9><loc_39><loc_85><loc_88><loc_89></location>| 0 ± 〉 = e ± 1 2 ∑ ( -) L ( a E † Lj ) 2 | 0 E 〉 (2.25)</formula>
<text><location><page_9><loc_12><loc_82><loc_20><loc_84></location>then obey</text>
<formula><location><page_9><loc_39><loc_79><loc_88><loc_82></location>ˆ Φ -| 0 -〉 = 0 Dirichlet , (2.26)</formula>
<formula><location><page_9><loc_39><loc_75><loc_88><loc_78></location>ˆ Φ + | 0 + 〉 = 0 Neumann . (2.27)</formula>
<text><location><page_9><loc_12><loc_66><loc_88><loc_75></location>Since only Dirichlet (Neumann) modes act non-trivially on | 0 -〉 ( | 0 + 〉 ) we refer to it as the Dirichlet (Neumann) vacuum. These vacua are dS invariant. With the conventional norm, ˆ Φ ± are hermitian and their eigenstates are non-normalizable. Generalized dS non-invariant plane-wave type Neumann states with nonzero eigenvalues for ˆ Φ +</text>
<formula><location><page_9><loc_42><loc_60><loc_88><loc_63></location>ˆ Φ + | Φ + 〉 = Φ + | Φ + 〉 (2.28)</formula>
<text><location><page_9><loc_12><loc_57><loc_27><loc_59></location>are constructed as</text>
<formula><location><page_9><loc_40><loc_54><loc_88><loc_57></location>| Φ + 〉 = e -〈 Φ + | ˆ Φ -〉 S 3 | 0 + 〉 . (2.29)</formula>
<text><location><page_9><loc_12><loc_48><loc_88><loc_53></location>Φ + here is an arbitrary solution of the classical wave equation, which can be parameterized by an arbitrary function Φ + (Ω) on I +</text>
<formula><location><page_9><loc_36><loc_44><loc_88><loc_46></location>t →∞ , Φ + (Ω , t ) → Φ + (Ω) e -h + t . (2.30)</formula>
<text><location><page_9><loc_12><loc_40><loc_80><loc_42></location>The states are delta-functional normalizable with respect the usual inner product</text>
<formula><location><page_9><loc_39><loc_34><loc_88><loc_38></location>〈 Φ + | Φ + ' 〉 = δ ( Φ + -Φ + ' ) , (2.31)</formula>
<text><location><page_9><loc_12><loc_32><loc_62><loc_33></location>where the delta function integrates to one with the measure</text>
<formula><location><page_9><loc_32><loc_25><loc_88><loc_30></location>D Φ + ≡ ∏ L,j dc + Lj √ π , Φ + ( x ) = ∑ c + Lj φ + Lj ( x ) . (2.32)</formula>
<text><location><page_9><loc_12><loc_19><loc_88><loc_23></location>The c + Lj satisfies the reality condition c + ∗ Lj = c + Lj ( -) L . One may similarly define generalized Dirichlet states obeying</text>
<formula><location><page_9><loc_42><loc_16><loc_88><loc_19></location>ˆ Φ -| Φ -〉 = Φ -| Φ -〉 . (2.33)</formula>
<text><location><page_10><loc_15><loc_90><loc_65><loc_93></location>The Euclidean vacuum can be expressed in terms of | 0 ± 〉 as</text>
<formula><location><page_10><loc_29><loc_85><loc_88><loc_89></location>| 0 E > = ∫ D Φ ± e ∓ 1 16 ∫ d 3 Ω d 3 Ω ' Φ ± (Ω)∆ ∓ (Ω , Ω ' )Φ ± (Ω ' ) | Φ ± 〉 , (2.34)</formula>
<text><location><page_10><loc_12><loc_82><loc_17><loc_83></location>where</text>
<formula><location><page_10><loc_17><loc_72><loc_88><loc_80></location>∆ ± (Ω , Ω ' ) = ∓ ∑ Y ∗ Lj (Ω) Y Lj (Ω ' )(2 L +2) ± 1 = 1 2 2 ∓ 1 π 2 1 (1 -cos Θ 3 ) h ± , cos Θ 3 (Ω , Ω ' ) ≡ cos ψ cos ψ ' +sin ψ sin ψ ' (cos θ cos θ ' +sin θ sin θ ' cos ( φ -φ ' )) . (2.35)</formula>
<text><location><page_10><loc_12><loc_65><loc_88><loc_69></location>∆ ± are the (everywhere positive) two-point functions for a CFT 3 operator with h + = 2 and h -= 1. 3 These satisfy</text>
<formula><location><page_10><loc_28><loc_59><loc_88><loc_63></location>-∫ √ hd 3 Ω '' ∆ + (Ω , Ω '' )∆ -(Ω '' , Ω ' ) = 1 √ h δ 3 (Ω -Ω ' ) . (2.36)</formula>
<text><location><page_10><loc_12><loc_56><loc_34><loc_57></location>We also have the relations</text>
<formula><location><page_10><loc_29><loc_50><loc_88><loc_54></location>| Φ + 〉 = 1 N 0 ∫ D Φ -e 〈 Φ -| Φ + 〉 S 3 | Φ -〉 , N 0 ≡ ∏ L,j √ 2 , (2.37)</formula>
<formula><location><page_10><loc_39><loc_44><loc_88><loc_48></location>〈 Φ -| Φ + 〉 S 3 = 1 N 0 e 〈 Φ -| Φ + 〉 S 3 . (2.38)</formula>
<formula><location><page_10><loc_43><loc_38><loc_88><loc_42></location>〈 0 -| 0 + 〉 S 3 = 1 N 0 . (2.39)</formula>
<text><location><page_10><loc_12><loc_42><loc_22><loc_43></location>In particular</text>
<text><location><page_10><loc_15><loc_36><loc_58><loc_38></location>The Wightman function in the Euclidean vacuum is</text>
<formula><location><page_10><loc_12><loc_24><loc_89><loc_34></location>G E ( x ; x ' ) = ∑ φ E Lj ( x ) φ E ∗ Lj ( x ' ) = ∑ ( -) L +1 φ E Lj ( x ) φ E Lj ( x ' A ) = 1 2 ∑ ( -) L ( φ -Lj ( x ) φ -Lj ( x ' ) + φ + Lj ( x ) φ + Lj ( x ' ) + iφ + Lj ( x ) φ -Lj ( x ' ) -iφ -Lj ( x ) φ + Lj ( x ' ) ) . (2.40)</formula>
<text><location><page_10><loc_12><loc_20><loc_50><loc_21></location>In terms of the dS-invariant distance function</text>
<formula><location><page_10><loc_27><loc_14><loc_88><loc_17></location>P ( t, Ω; t ' , Ω ' ) = cosh t cosh t ' cos Θ 3 (Ω , Ω ' ) -sinh t sinh t ' , (2.41)</formula>
<text><location><page_11><loc_12><loc_91><loc_29><loc_92></location>this becomes simply</text>
<formula><location><page_11><loc_38><loc_86><loc_88><loc_91></location>G E ( x ; x ' ) = 1 8 π 2 1 1 -P ( x ; x ' ) , (2.42)</formula>
<text><location><page_11><loc_12><loc_85><loc_53><loc_86></location>with the usual iε prescription for the singularity.</text>
<section_header_level_1><location><page_11><loc_12><loc_80><loc_67><loc_81></location>3 Boundary operators and quasinormal modes</section_header_level_1>
<text><location><page_11><loc_12><loc_47><loc_88><loc_77></location>According to the dS 4 /CFT 3 dictionary, for every spin zero primary CFT 3 operator O of conformal weight h there is a bulk scalar field Φ with mass m 2 /lscript 2 = h (3 -h ). Boundary correlators of O are then related by a rescaling to bulk Φ correlators whose arguments are pushed to the boundary at I + . As in AdS/CFT, a particular classical bulk wavefunction of Φ can be associated to a boundary insertion of O (at the linearized level) by symmetries: it must scale with weight h under the isometry corresponding to dilations, obey the lowest-weight condition, and be invariant under rotations around the point of the boundary insertion. The resulting wavefunction is a type of bulk-to-boundary Green function. Interestingly[13], the (lowest) highest-weight modes can also be identified as (anti) quasinormal modes for the static patch of de Sitter, as constructed in [11, 12]. In this section we determine this wavefunction explicitly, regulate the singularities, generalize it to multi-particle insertions and define a symplectic product. In the following section we will then use these classical objects to construct the associated dual bulk quantum states and their inner products.</text>
<section_header_level_1><location><page_11><loc_12><loc_43><loc_56><loc_44></location>3.1 Highest and lowest weight wavefunctions</section_header_level_1>
<text><location><page_11><loc_12><loc_30><loc_88><loc_41></location>In this subsection we give expressions for the classical wavefunctions, associated to lowest (highest) weight primary operator insertions at the south (north) pole of I + in terms of rescaled Green functions in the limit that one argument is pushed to I + . These wavefunctions each comes in a Neumann and a Dirichlet flavor, denoted Φ ± lw ( x ) (Φ ± hw ( x )) depending on whether the weight of the dual operator insertion is h + or h -.</text>
<text><location><page_11><loc_15><loc_28><loc_43><loc_29></location>The relevant Green functions are 4</text>
<formula><location><page_11><loc_20><loc_21><loc_88><loc_26></location>G ± ( x ; x ' ) ≡ G E ( x ; x ' ) ± G E ( x ; x ' A ) = 1 8 π 2 ( 1 1 -P ( x ; x ' ) ± 1 1 + P ( x ; x ' ) ) (3.1)</formula>
<text><location><page_11><loc_12><loc_15><loc_88><loc_20></location>with G E the Wightman function for the Euclidean vacuum given in equation (2.42). G -( G + ) obeys Neumann (Dirichlet) boundary conditions at I + away from x = x ' . These are for</text>
<text><location><page_12><loc_12><loc_84><loc_88><loc_93></location>m 2 /lscript 2 = 2 the Green functions with future boundary conditions as discussed in [10]. We have normalized them so that they have the Hadamard form for the short-distance singularity. In the Neumann case we begin with G -, which is (using the mode decomposition (2.40)) given by</text>
<formula><location><page_12><loc_27><loc_79><loc_88><loc_83></location>G -( x ; x ' ) = ∑ ( -) L ( φ -Lj ( x ) φ -Lj ( x ' ) + iφ + Lj ( x ) φ -Lj ( x ' ) ) . (3.2)</formula>
<text><location><page_12><loc_12><loc_78><loc_55><loc_79></location>From this we construct the rescaled Green function</text>
<formula><location><page_12><loc_37><loc_73><loc_88><loc_75></location>Φ -lw ( x ; t ' ) = e h -t ' G -( x ; t ' , Ω SP ) , (3.3)</formula>
<text><location><page_12><loc_12><loc_67><loc_88><loc_71></location>in which the second argument is placed at the south pole Ω SP where ψ ' = 0. One may then check that (ignoring singularity prescriptions)</text>
<formula><location><page_12><loc_28><loc_60><loc_88><loc_65></location>Φ -lw ( x ) ≡ lim t ' →∞ Φ -lw ( x ; t ' ) = 1 2 π 2 (sinh t -cos ψ cosh t ) . (3.4)</formula>
<text><location><page_12><loc_12><loc_57><loc_86><loc_59></location>Using (3.2) and the asymptotics (2.11) one finds that near I + (not ignoring singularities)</text>
<formula><location><page_12><loc_15><loc_48><loc_88><loc_56></location>Φ -lw ( x ) = 8 ∑ ( -) L ( e -t 2 L +2 Y Lj (Ω) Y Lj (Ω SP ) + ie -2 t Y Lj (Ω) Y Lj (Ω SP ) + O ( e -3 t ) ) = 8 e -t ∆ -(Ω , Ω SP ) + 8 i e -2 t √ h δ 3 (Ω -Ω SP ) + O ( e -3 t ) (3.5)</formula>
<text><location><page_12><loc_12><loc_34><loc_88><loc_46></location>Let us now confirm that Φ -lw ( x ) has the same symmetries as an insertion of a primary operator O (Ω SP ) at the south pole of I + . First we note that the choice of a point on I + breaks SO (4 , 1) to SO (3) × SO (1 , 1). Both Φ -lw ( x ) and O (Ω SP ) are manifestly invariant under the SO (3) spatial rotations. The generator of SO (1 , 1) dilations, denoted L 0 , acts on O (Ω SP ) as</text>
<formula><location><page_12><loc_39><loc_32><loc_88><loc_34></location>[ L 0 , O (Ω SP )] = h -O (Ω SP ) . (3.6)</formula>
<text><location><page_12><loc_12><loc_29><loc_56><loc_31></location>In the bulk it is generated by the Killing vector field</text>
<formula><location><page_12><loc_38><loc_24><loc_88><loc_26></location>L 0 = cos ψ∂ t -tanh t sin ψ∂ ψ , (3.7)</formula>
<text><location><page_12><loc_12><loc_21><loc_56><loc_22></location>where the south pole is ψ = 0. dS invariance implies</text>
<formula><location><page_12><loc_40><loc_15><loc_88><loc_18></location>( L 0 -L ' 0 ) G -( x, x ' ) = 0 . (3.8)</formula>
<text><location><page_13><loc_12><loc_91><loc_80><loc_92></location>It follows from this together with the definition (3.4) that the wavefunction obeys</text>
<formula><location><page_13><loc_41><loc_86><loc_88><loc_88></location>L 0 Φ -lw ( x ) = h -Φ -lw ( x ) . (3.9)</formula>
<text><location><page_13><loc_12><loc_82><loc_48><loc_84></location>By construction it obeys the wave equation</text>
<formula><location><page_13><loc_41><loc_77><loc_88><loc_80></location>( ∇ 2 -m 2 )Φ -lw ( x ) = 0 . (3.10)</formula>
<text><location><page_13><loc_12><loc_73><loc_59><loc_75></location>Acting on SO (3) invariant symmetric functions we have</text>
<formula><location><page_13><loc_36><loc_68><loc_88><loc_71></location>/lscript 2 ∇ 2 = -L 0 ( L 0 -3) + M -k M + k , (3.11)</formula>
<text><location><page_13><loc_12><loc_62><loc_88><loc_66></location>where the 6 Killing vector fields M ± k (given in Appendix A) are the raising and lowering operators for L 0 and we sum over k . It then follows that</text>
<formula><location><page_13><loc_28><loc_56><loc_88><loc_60></location>M -k M + k Φ -lw ( x ) = ( m 2 /lscript 2 -h -(3 -h -) ) Φ -lw ( x ) = 0 , (3.12)</formula>
<text><location><page_13><loc_12><loc_54><loc_20><loc_55></location>and hence</text>
<formula><location><page_13><loc_43><loc_51><loc_88><loc_53></location>M + k Φ -lw ( x ) = 0 . (3.13)</formula>
<text><location><page_13><loc_12><loc_47><loc_61><loc_50></location>which corresponds to the lowest-weight condition for the O</text>
<formula><location><page_13><loc_42><loc_43><loc_88><loc_45></location>[ M + k , O (Ω SP )] = 0 . (3.14)</formula>
<text><location><page_13><loc_12><loc_35><loc_88><loc_41></location>It may be shown that these symmetries uniquely determine the solution. Hence Φ -lw is identified as the classical wavefunction associated to the insertion of the primary O at the south pole.</text>
<text><location><page_13><loc_12><loc_30><loc_88><loc_34></location>A parallel argument leads to the dual of a highest weight operator insertion at the north pole . The wavefunction is</text>
<formula><location><page_13><loc_35><loc_25><loc_88><loc_27></location>Φ -hw ( x ; t ' ) = lim t ' →∞ e h -t ' G -( x ; t ' , Ω NP ) . (3.15)</formula>
<text><location><page_13><loc_12><loc_21><loc_32><loc_23></location>This obeys the relations</text>
<formula><location><page_13><loc_31><loc_16><loc_88><loc_19></location>M -k Φ -hw ( x ) = 0 , L 0 Φ -hw ( x ) = -h -Φ -hw ( x ) , (3.16)</formula>
<text><location><page_14><loc_12><loc_91><loc_39><loc_92></location>and has the asymptotic behavior</text>
<formula><location><page_14><loc_25><loc_85><loc_88><loc_89></location>Φ -hw ( x ) = 8 e -t ∆ -(Ω , Ω NP ) + 8 i e -2 t √ h δ 3 (Ω -Ω NP ) + O ( e -3 t ) . (3.17)</formula>
<text><location><page_14><loc_12><loc_78><loc_88><loc_83></location>Similar formulae apply to the Dirichlet case by beginning with G + in the above construction and replacing + ↔-. For example</text>
<formula><location><page_14><loc_24><loc_74><loc_88><loc_77></location>Φ + hw ( x ) = -8 e -2 t ∆ + (Ω , Ω NP ) -8 i e -t √ h δ 3 (Ω -Ω NP ) + O ( e -3 t ) . (3.18)</formula>
<text><location><page_14><loc_12><loc_58><loc_88><loc_72></location>We see from the above that the highest-weight wavefunction is smooth on the future horizon of the southern static patch dS 4 , and hence related to the quasinormal modes found in [11, 12]. The lowest quasinormal mode which is invariant under the SO (3) of the static dS 4 is exactly the Φ -hw with h -= 1 while the second lowest SO (3)-invariant quasinormal mode corresponds to Φ + hw with h + = 2. Lowest weight states are smooth on the past horizon and hence related to anti-quasinormal modes.</text>
<section_header_level_1><location><page_14><loc_12><loc_54><loc_49><loc_55></location>3.2 General multi-operator insertions</section_header_level_1>
<text><location><page_14><loc_12><loc_42><loc_88><loc_52></location>In the preceding subsections we found the bulk duals of primary operators inserted at the north/south pole in the coordinates (2.3). This can be generalized to insertions at an arbitrary point on I + with a general time slicing near I + . Let us introduce coordinates x ∼ ( y i , t ) such that near I +</text>
<formula><location><page_14><loc_32><loc_40><loc_88><loc_43></location>ds 2 4 →-dt 2 + e 2 t h ij ( y ) dy i dy j , i, j = 1 , 2 , 3 . (3.19)</formula>
<text><location><page_14><loc_12><loc_37><loc_88><loc_39></location>The dual wavefunction is then the t ' →∞ limit of the rescaled Green function, denoted by</text>
<formula><location><page_14><loc_37><loc_32><loc_88><loc_35></location>Φ ± y 1 ( x ) = lim t ' →∞ e h ± t ' G ± ( x ; t ' , y 1 ) . (3.20)</formula>
<text><location><page_14><loc_12><loc_23><loc_88><loc_30></location>For the special cases of operator insertions at the north or south pole in global coordinates these reduce to our previous expressions. Note that coordinate transformations of the form t → t + f ( y ) induce a conformal transformation on I +</text>
<formula><location><page_14><loc_32><loc_19><loc_88><loc_22></location>h ij → e 2 f ( y ) h ij , Φ ± y 1 ( x ) → e h ± f ( y ) Φ y 1 ( x ) , (3.21)</formula>
<text><location><page_14><loc_12><loc_12><loc_88><loc_17></location>as appropriate for a conformal field of weight h ± . Hence the relative normalization in (3.4) will depend on the conformal frame at I + .</text>
<text><location><page_14><loc_15><loc_10><loc_88><loc_12></location>One may also consider multi-operator insertions such as O ( y 1 ) O ( y 2 ) in the CFT 3 at I + .</text>
<text><location><page_15><loc_12><loc_88><loc_88><loc_92></location>At the level of free field theory considered here these are associated to a bilocal wavefunction in the product of two bulk scalar fields</text>
<formula><location><page_15><loc_44><loc_84><loc_88><loc_86></location>Φ y 1 ( x 1 )Φ y 2 ( x 2 ) . (3.22)</formula>
<text><location><page_15><loc_12><loc_76><loc_88><loc_81></location>We will use Φ Ω to denote these wavefunctions when working in global coordinates (2.3). We note that in such coordinates near I + for an insertion at a general point</text>
<formula><location><page_15><loc_24><loc_72><loc_88><loc_75></location>Φ ± Ω 1 ( t, Ω) = ∓ 8 e -h ± t ∆ ± (Ω , Ω 1 ) ∓ 8 i e -h ∓ t √ h δ 3 (Ω -Ω 1 ) + O ( e -3 t ) . (3.23)</formula>
<section_header_level_1><location><page_15><loc_12><loc_68><loc_44><loc_70></location>3.3 Klein-Gordon inner product</section_header_level_1>
<text><location><page_15><loc_12><loc_58><loc_88><loc_66></location>We wish to define an inner product between e.g. two Neumann wavefunctions Φ -Ω 1 and Φ -Ω 2 . Later on we will compare this to the inner product on the CFT 3 Hilbert space and the twopoint function of O on S 3 . One choice is to take a global spacelike S 3 slice in the interior and define the Klein-Gordon inner product</text>
<formula><location><page_15><loc_35><loc_51><loc_88><loc_56></location>〈 Φ -Ω 1 | Φ -Ω 2 〉 S 3 ≡ i ∫ S 3 d 3 Σ µ Φ -∗ Ω 1 ←→ ∂ µ Φ -Ω 2 . (3.24)</formula>
<text><location><page_15><loc_12><loc_46><loc_88><loc_50></location>This integral does not depend on the choice of S 3 which can be pushed up to I + . One may then see immediately from (3.23) that there are two nonzero terms proportional to ∆ -giving</text>
<formula><location><page_15><loc_37><loc_40><loc_88><loc_44></location>〈 Φ -Ω 1 | Φ -Ω 2 〉 S 3 = 16∆ -(Ω 1 -Ω 2 ) (3.25)</formula>
<text><location><page_15><loc_12><loc_25><loc_88><loc_39></location>One may also define an inner product not on global spacelike S 3 slices, but on a spacelike R 3 slice which ends on an S 2 on I + . The result is invariant under any deformation of the S 2 which does not cross the insertion point. To be definite, we take the S 2 to be the equator, Ω 1 to be in the northern hemisphere and Ω 2 to be in the southern hemisphere, and the slice to be R 3 S which intersects the south pole. One then finds, pushing R 3 S up to the southern hemisphere of I +</text>
<formula><location><page_15><loc_27><loc_19><loc_88><loc_24></location>〈 Φ -Ω 1 | Φ -Ω 2 〉 R 3 S ≡ i ∫ R 3 S d 3 Σ µ Φ -∗ Ω 1 ←→ ∂ µ Φ -Ω 2 = 8∆ -(Ω 1 -Ω 2 ) . (3.26)</formula>
<text><location><page_15><loc_15><loc_17><loc_79><loc_18></location>Similarly, the inner product between two Dirichlet wavefunctions is given by</text>
<formula><location><page_15><loc_37><loc_10><loc_88><loc_14></location>〈 Φ + Ω 1 | Φ + Ω 2 〉 R 3 S = -8∆ + (Ω 1 -Ω 2 ) . (3.27)</formula>
<section_header_level_1><location><page_16><loc_12><loc_91><loc_47><loc_92></location>4 The southern Hilbert space</section_header_level_1>
<text><location><page_16><loc_12><loc_56><loc_88><loc_88></location>We now turn to the issue of bulk quantum states. Quantum states in dS are often discussed, as in section 2, in terms of a Hilbert space built on the global S 3 slices. The structure of the vacua and Green functions for such states was described in section 2. However dS has the unusual feature that there are geodesically complete topologically R 3 spacelike slices which end on an S 2 in I + , which we will typically take to be the equator. Examples of these are the hyperbolic slices, the quantization on which was studied in detail in [15]. We will see that the quantum states built on these R 3 slices are natural objects in dS/CFT. An S 2 in I + is in general the boundary of a 'northern' slice, denoted R 3 N and a 'southern' slice denoted R 3 S . The topological sum obeys R 3 S ∪ R 3 N = S 3 . Hence the relation of the southern and northern Hilbert spaces on R 3 S and R 3 N to that on S 3 is like that of the left and right Rindler wedges to that of global Minkowski space. It is also like the relation of the Hilbert spaces of the northern and southern causal diamonds to that of global dS. However the diamond Hilbert spaces in dS quantum gravity are problematic in quantum gravity with a fluctuating metric because it is hard to find sensible boundary conditions.</text>
<text><location><page_16><loc_12><loc_35><loc_88><loc_55></location>A strong motivation for considering the R 3 S,N slices comes from the picture of a state in the boundary CFT 3 . The state-operator correspondence in CFT 3 begins with an insertion of a (primary or descendant) operator O at the south pole of S 3 , and then defines a quantum state as a functional of the boundary conditions on an S 2 surrounding the south pole. For every object in the CFT 3 , we expect a holographically dual object in the bulk dS 4 theory. The dual bulk quantum state must somehow depend on the choice of S 2 in I + . Hence it is natural to define the bulk state on the R 3 slice which ends on this S 2 in I + . This is how holography works in AdS/CFT: CFT states live on the boundaries of the spacelike slices used to define the bulk states.</text>
<section_header_level_1><location><page_16><loc_12><loc_30><loc_23><loc_32></location>4.1 States</section_header_level_1>
<text><location><page_16><loc_12><loc_24><loc_88><loc_29></location>In order to define quantum states on R 3 S , we first note that modes of the scalar field operator ˆ Φ(Ω , t ) are labeled by operators ˆ Φ ± (Ω) defined on I + via the relation</text>
<formula><location><page_16><loc_33><loc_19><loc_88><loc_22></location>lim t →∞ ˆ Φ(Ω , t ) = e -h + t ˆ Φ + (Ω) + e -h -t ˆ Φ -(Ω) . (4.1)</formula>
<text><location><page_16><loc_12><loc_16><loc_52><loc_17></location>They satisfy the following commutation relation</text>
<formula><location><page_16><loc_36><loc_10><loc_88><loc_14></location>[ ˆ Φ + (Ω) , ˆ Φ -(Ω ' ) ] = 8 i √ h δ 3 (Ω -Ω ' ) . (4.2)</formula>
<text><location><page_17><loc_12><loc_90><loc_70><loc_93></location>We may then decompose these I + operators as the sum of two terms</text>
<formula><location><page_17><loc_39><loc_86><loc_88><loc_88></location>ˆ Φ ± (Ω) = ˆ Φ ± N (Ω) + ˆ Φ ± S (Ω) (4.3)</formula>
<text><location><page_17><loc_12><loc_80><loc_88><loc_84></location>where the first (second) acts only on R 3 N ( R 3 S ). Defining the northern and southern Dirichlet and Neumann vacua by</text>
<formula><location><page_17><loc_38><loc_76><loc_88><loc_79></location>ˆ Φ ± N | 0 ± N 〉 = 0 , ˆ Φ ± S | 0 ± S > = 0 , (4.4)</formula>
<text><location><page_17><loc_12><loc_72><loc_88><loc_76></location>it follows from the decomposition (4.3) that the global vacua have a simple product decomposition 5</text>
<formula><location><page_17><loc_43><loc_68><loc_88><loc_71></location>| 0 ± 〉 = | 0 ± N 〉| 0 ± S 〉 . (4.5)</formula>
<text><location><page_17><loc_12><loc_63><loc_88><loc_67></location>Excited southern states may then be built by acting on one of these southern vacua with ˆ Φ S . We wish to identify these states with those of the CFT 3 on S 2 .</text>
<text><location><page_17><loc_12><loc_51><loc_88><loc_63></location>In the higher-spin dS/CFT correspondence there are actually two CFT 3 's living on I + : the free Sp ( N ) model, associated to Neumann boundary conditions, and the critical Sp ( N ) model, associated to Dirichlet boundary conditions. Since the field operators ˆ Φ S acting on | 0 + S 〉 ( | 0 -S 〉 ) obeys, according to equation (2.26), Neumann (Dirichlet) boundary conditions near the southern hemisphere of I + , it is natural to identify</text>
<formula><location><page_17><loc_36><loc_44><loc_88><loc_49></location>| 0 + S 〉 ∼ free Sp ( N ) vacuum | 0 -S 〉 ∼ critical Sp ( N ) vacuum . (4.6)</formula>
<text><location><page_17><loc_12><loc_36><loc_88><loc_42></location>Next we want to consider excited states and their duals. To be specific we consider the Neumann theory built on | 0 + S 〉 . Parallel formulae apply to the Dirichlet case. Operator versions of the classical wavefunctions Φ -Ω ( x ) are constructed as</text>
<formula><location><page_17><loc_42><loc_30><loc_88><loc_34></location>ˆ Φ -Ω S ≡ 〈 Φ -Ω S | ˆ Φ 〉 R 3 S , (4.7)</formula>
<text><location><page_17><loc_12><loc_27><loc_86><loc_28></location>where Ω S is presumed to lie on the southern hemisphere. We can make a quantum state</text>
<formula><location><page_17><loc_37><loc_22><loc_88><loc_25></location>| Ω -S 〉 ≡ ˆ Φ -Ω S | 0 + S 〉 = ˆ Φ -(Ω S ) | 0 + S 〉 , (4.8)</formula>
<text><location><page_17><loc_12><loc_18><loc_88><loc_20></location>where in the last line we used (3.23). By construction this will be a lowest weight state, and</text>
<text><location><page_18><loc_12><loc_87><loc_88><loc_92></location>we therefore identify it as the bulk dual to the CFT 3 state created by the primary operator O dual to the field Φ.</text>
<text><location><page_18><loc_12><loc_83><loc_88><loc_88></location>This connection leads to an interesting nonperturbative dS exclusion principle [14]. The operator O has a representation in the Sp ( N ) theory as</text>
<formula><location><page_18><loc_37><loc_78><loc_88><loc_81></location>O = Ω AB η A η B , A, B = 1 , ...N, (4.9)</formula>
<text><location><page_18><loc_12><loc_73><loc_88><loc_77></location>where η A are N anticommuting real scalars and Ω AB is the quadratic form on Sp ( N ). It follows that</text>
<formula><location><page_18><loc_46><loc_69><loc_88><loc_72></location>O N 2 +1 = 0 . (4.10)</formula>
<text><location><page_18><loc_12><loc_65><loc_88><loc_68></location>Bulk-boundary duality and the state-operator relation described above then implies the nonperturbative relation</text>
<formula><location><page_18><loc_43><loc_60><loc_88><loc_64></location>[ ˆ Φ ± (Ω) ] N 2 +1 = 0 . (4.11)</formula>
<text><location><page_18><loc_12><loc_49><loc_88><loc_60></location>Hence the quantum field operators ˆ Φ ± (Ω) are N 2 -adic. One is not allowed to put more than N 2 quanta in any given quasinormal mode. This is similar to the stringy exclusion principle for AdS [20] and may be related to the finiteness of dS entropy. Nonperturbative phenomena due to related finite N effects in the O ( N ) case have been discussed in [21]. We hope to investigate further the consequences of this dS exclusion principle.</text>
<section_header_level_1><location><page_18><loc_12><loc_44><loc_22><loc_46></location>4.2 Norm</section_header_level_1>
<text><location><page_18><loc_12><loc_27><loc_88><loc_42></location>Having identified the bulk duals of the boundary CFT 3 states, we wish to describe the bulk dual of the CFT 3 norm. The standard bulk norm is defined by Φ( x ) = Φ † ( x ). However this norm is not unique. It has been argued for a variety of reasons beginning in [3] that it is appropriate to modify the norm in the context of dS - see also [16, 22]. Here we have the additional problem that this standard norm is divergent for states of the form (4.8). We now construct the modified norm for states on R S 3 by demanding that it is equivalent to the CFT 3 norm. The construction here generalizes to dS 4 the one given in [16] for dS 3 .</text>
<text><location><page_18><loc_12><loc_22><loc_88><loc_26></location>The bulk action of dS Killing vectors K µ A ∂ µ on a scalar field is generated by the integral over any global S 3 slice</text>
<formula><location><page_18><loc_41><loc_18><loc_88><loc_22></location>ˆ L A = ∫ S 3 d 3 Σ µ T µν K ν A , (4.12)</formula>
<text><location><page_18><loc_12><loc_11><loc_88><loc_17></location>where T µν is the bulk stress tensor constructed from the operator ˆ Φ. If we take ˆ Φ † ( x ) = ˆ Φ( x ), then ˆ L A = ˆ L † A which is not what we want. The CFT 3 states are in representations of the SO (3 , 2) conformal group. These arise from analytic continuation of the 10 SO (4 , 1)</text>
<text><location><page_19><loc_12><loc_80><loc_88><loc_93></location>conformal Killing vectors on S 3 which are the boundary restrictions of the bulk dS 4 Killing vectors K µ A ∂ µ . Usually, the standard CFT 3 norm has a self-adjoint dilation operator L 0 generating -i sin ψ∂ ψ as well as 3 self-adjoint SO (3) rotation operators J k . The remaining 6 raising and lowering operators L ± k arising from the Killing vectors iM ± k (described in the appendix) then obey L † ± k = L ∓ k in the conventional CFT 3 norm.</text>
<text><location><page_19><loc_15><loc_79><loc_81><loc_80></location>To obtain an adjoint with the desired properties, we define the modified adjoint</text>
<formula><location><page_19><loc_38><loc_74><loc_88><loc_77></location>ˆ Φ † ( x ) = R ˆ Φ( x ) R = ˆ Φ( Rx ) , (4.13)</formula>
<text><location><page_19><loc_12><loc_61><loc_88><loc_72></location>where here and hereafter † denotes the bulk modified adjoint. The reflection operator R is the discrete isometry of S 3 which reflects through the S 2 equator R ( ψ, θ, φ ) = ( π -ψ, θ, φ ) along with complex conjugation. In particular, it maps the south pole to the north pole while keeping the equator invariant. This implies that L 0 (generating iL 0 ) and J k are self adjoint while</text>
<formula><location><page_19><loc_16><loc_55><loc_88><loc_59></location>L † ± k = -i ∫ S 3 d Σ µ ( x ) T µν ( Rx ) M ν ± k ( x ) = -i ∫ S 3 d Σ µ ( x ) T µν ( x ) M ν ± k ( Rx ) = L ∓ k . (4.14)</formula>
<text><location><page_19><loc_12><loc_50><loc_88><loc_53></location>Hence we have constructed an adjoint admitting the desired SO (3 , 2) action. We do not know whether or not it is unique.</text>
<text><location><page_19><loc_12><loc_44><loc_88><loc_49></location>The action of R maps an operator defined on the southern hemisphere to one defined on the southern hemisphere of I + according to</text>
<formula><location><page_19><loc_42><loc_40><loc_88><loc_42></location>ˆ Φ ±† (Ω) = ˆ Φ ± (Ω R ) , (4.15)</formula>
<text><location><page_19><loc_12><loc_24><loc_88><loc_38></location>Hence the action of R exchanges the northern and southern hemispheres, and maps a southern I + state to a northern one. Therefore it cannot on its own define an adjoint within the southern Hilbert space. For this we must combine (4.13) with a map from the north to the south. Such a map is provided by the Euclidean vacuum. The global Euclidean bra state (constructed with the standard adjoint) can be decomposed in terms of a basis of northern and southern bra states</text>
<formula><location><page_19><loc_40><loc_20><loc_88><loc_24></location>〈 0 E | = ∑ m,n E mn 〈 m S |〈 n N | . (4.16)</formula>
<text><location><page_19><loc_12><loc_16><loc_74><loc_19></location>We then define the modified adjoint of an arbitrary southern state | Ψ S 〉 by</text>
<formula><location><page_19><loc_42><loc_12><loc_88><loc_15></location>| Ψ S 〉 † ≡ 〈 0 E |R| Ψ S 〉 . (4.17)</formula>
<text><location><page_20><loc_12><loc_91><loc_67><loc_92></location>We will denote the corresponding inner product by an S subscript</text>
<formula><location><page_20><loc_40><loc_85><loc_88><loc_88></location>〈 Ψ ' S | Ψ S 〉 S ≡ ( | Ψ ' S 〉 † ) | Ψ S 〉 . (4.18)</formula>
<text><location><page_20><loc_12><loc_82><loc_44><loc_84></location>For example choosing the basis so that</text>
<formula><location><page_20><loc_44><loc_77><loc_88><loc_79></location>R| m S 〉 = | m N 〉 (4.19)</formula>
<text><location><page_20><loc_12><loc_73><loc_18><loc_75></location>we have</text>
<text><location><page_20><loc_12><loc_68><loc_30><loc_69></location>In particular one finds</text>
<formula><location><page_20><loc_36><loc_64><loc_88><loc_67></location>〈 0 + S | 0 + S 〉 S = 〈 0 E | ( | 0 + S 〉R| 0 + S 〉 ) = 1 . (4.21)</formula>
<text><location><page_20><loc_12><loc_58><loc_88><loc_64></location>Let us now compute the norm of the southern state | Ω -S 〉 in (4.8). The action of R gives a northern state which we will denote | R Ω -S 〉 . The norm is then</text>
<formula><location><page_20><loc_25><loc_54><loc_88><loc_57></location>〈 Ω -S | Ω -S 〉 S = 〈 0 E | ( | Ω -S 〉| R Ω -S 〉 ) = 〈 0 E | ˆ Φ -(Ω S ) ˆ Φ -( R Ω S ) | 0 + 〉 . (4.22)</formula>
<text><location><page_20><loc_12><loc_51><loc_27><loc_52></location>Using the relation</text>
<formula><location><page_20><loc_31><loc_47><loc_88><loc_50></location>| 0 E > = N 0 e -1 16 ∫ d 3 Ω d 3 Ω ' ˆ Φ + (Ω)∆ -(Ω , Ω ' ) ˆ Φ + (Ω ' ) | 0 -〉 (4.23)</formula>
<text><location><page_20><loc_12><loc_45><loc_18><loc_47></location>we find</text>
<formula><location><page_20><loc_38><loc_42><loc_88><loc_44></location>〈 Ω -S | Ω -S 〉 S = 8∆ -(Ω S , R Ω S ) . (4.24)</formula>
<text><location><page_20><loc_12><loc_37><loc_88><loc_41></location>This is proportional to the S 3 two-point function of a dimension h -primary at the points Ω S and R Ω S . The analogous computation in the Dirichlet theory gives</text>
<formula><location><page_20><loc_37><loc_31><loc_88><loc_34></location>〈 Ω + S | Ω + S 〉 S = -8∆ + (Ω S , R Ω S ) . (4.25)</formula>
<section_header_level_1><location><page_20><loc_12><loc_27><loc_69><loc_29></location>5 Boundary dual of the bulk Euclidean vacuum</section_header_level_1>
<text><location><page_20><loc_12><loc_12><loc_88><loc_25></location>In the preceding section we have argued that dS/CFT maps CFT 3 states on an S 2 in I + to bulk states on the southern slice ending on the S 2 . A generic state in a global dS slice does not restrict to a pure southern state. However we can always define a density matrix by tracing over the northern Hilbert space. In particular, such a southern density matrix ρ E S can be associated to the global Euclidean vacuum | 0 E 〉 . The choice of an equatorial S 2 in I + breaks the SO (4 , 1) symmetry group down to SO (3 , 1), which also preserves the hyperbolic</text>
<formula><location><page_20><loc_43><loc_70><loc_88><loc_73></location>〈 m S | n S 〉 S = E nm . (4.20)</formula>
<text><location><page_21><loc_12><loc_86><loc_88><loc_93></location>slices ending on the S 2 . ρ S E must be invariant under this SO (3 , 1). In fact ρ S E follows from results in [15]. Writing the quadratic Casimir of SO (3 , 1) as C 2 = -(1 + p 2 ), it was shown, in a basis which diagonalizes p , that</text>
<formula><location><page_21><loc_44><loc_82><loc_88><loc_84></location>ρ S E = N 1 e -2 πp , (5.1)</formula>
<text><location><page_21><loc_12><loc_74><loc_88><loc_79></location>where N 1 is determined by Tr ρ S E = 1. It would be interesting to investigate this further and compute the entropy S = -Tr ρ S E ln ρ S E in the Sp ( N ) model.</text>
<section_header_level_1><location><page_21><loc_12><loc_70><loc_76><loc_72></location>6 Pseudounitarity and the C-norm in the Sp ( N ) CFT 3</section_header_level_1>
<text><location><page_21><loc_12><loc_64><loc_88><loc_67></location>In this section we consider the Sp ( N ) model (where N is even) and compare the norms to those computed above. The action is</text>
<formula><location><page_21><loc_27><loc_57><loc_88><loc_62></location>I Sp ( N ) = 1 8 π ∫ d 3 x [ δ ij δ ab ∂ i ¯ χ a ∂ j χ b + m 2 ¯ χχ + λ (¯ χχ ) 2 ] , (6.1)</formula>
<text><location><page_21><loc_12><loc_45><loc_88><loc_56></location>where χ a ( a = 1 , . . . , N 2 ) is a complex anticommuting scalar and ¯ χχ ≡ δ ab ¯ χ a χ b . This has a global Sp ( N ) symmetry and we restrict to Sp ( N ) singlet operators. 6 For the free theory m = λ = 0 while the critical theory is obtained by flowing to a nontrivial fixed point λ F . The Sp ( N ) theory is not unitary in the sense that in the standard norm following from (6.1) one has that [9]</text>
<formula><location><page_21><loc_47><loc_43><loc_88><loc_44></location>H = H † (6.2)</formula>
<text><location><page_21><loc_49><loc_42><loc_49><loc_44></location>/negationslash</text>
<text><location><page_21><loc_12><loc_37><loc_88><loc_41></location>and 〈 Ψ ' | Ψ 〉 is not preserved. Nevertheless, as detailed in [9], there exists an operator C with the properties</text>
<formula><location><page_21><loc_26><loc_32><loc_88><loc_34></location>C † C = C 2 = 1 , Cχ † C = χ, CH † C = H, C | 0 〉 = | 0 〉 . (6.3)</formula>
<text><location><page_21><loc_12><loc_24><loc_88><loc_30></location>To write it in real fields, for e.g., in the case of Sp (2), writing the real and imaginary part of χ as η 1 and η 2 , the action of C becomes η 2 = Cη † 1 C . One may then define a 'pseudounitary' C -inner product</text>
<formula><location><page_21><loc_42><loc_20><loc_88><loc_23></location>〈 Ψ ' | Ψ 〉 C ≡ 〈 Ψ ' | C | Ψ 〉 (6.4)</formula>
<text><location><page_21><loc_12><loc_15><loc_88><loc_19></location>which is preserved under hamiltonian time evolution. Such hamiltonians are pseudohermitian and are similar to those studied in [23]. We note that the norm is not positive definite.</text>
<text><location><page_22><loc_12><loc_80><loc_88><loc_92></location>Inserting an operator O i constructed from χ a at the south pole gives a functional of the boundary conditions on the equatorial S 2 which we define as the state | O i 〉 . This is the standard state-operator correspondence. An inner product for such states associated to O i and O j can be defined by the two point function with O i at one pole and O j at the other. It follows from (6.3) that this is the C -inner product for the states | O i 〉 and | O j 〉 :</text>
<formula><location><page_22><loc_31><loc_76><loc_88><loc_79></location>〈 O i | O j 〉 C = 〈 O i | C | O j 〉 = 〈 O i † CO j 〉 = 〈 O i O j 〉 . (6.5)</formula>
<text><location><page_22><loc_12><loc_70><loc_88><loc_74></location>In the last line, we used the fact that the (singlet) currents in the Sp ( N ) models satisfy CO i † C = O i since C (¯ χχ ) † C = ¯ χχ . For primary operators of weight h i we then have [6]</text>
<formula><location><page_22><loc_37><loc_65><loc_88><loc_67></location>〈 O i | O i 〉 C = -N ∆ h i (Ω NP , Ω SP ) . (6.6)</formula>
<text><location><page_22><loc_12><loc_50><loc_88><loc_63></location>Hence it is the C-norm which maps under the state-operator correspondence to the Zamolodchikov norm defined as the Euclidean two point function on S 3 . As seen in [6] this C-norm then agrees with the bulk inner product (4.24)-(4.25) of the dual state for the scalar case. 7 Moreover, as the bulk and CFT 3 norms assign the same hermiticity properties to the SO (4 , 1) generators, this result will carry over to descendants of the primaries. A generalization of this construction to all spins seems possible.</text>
<text><location><page_22><loc_12><loc_33><loc_88><loc_49></location>One of the puzzling features of dS/CFT is that the dual CFT cannot be unitary in the ordinary sense. This is not a contradiction of any kind because unitarity of the Euclidean CFT is not directly connected to any spacetime conservation law. At the same time quantum gravity in dS - and its holographic dual - should have some good property replacing unitarity in the AdS case. It is not clear what that good property is. The appearance of a pseudounitary structure in the case of dS/CFT analyzed here is perhaps relevant in this regard.</text>
<section_header_level_1><location><page_22><loc_12><loc_28><loc_34><loc_30></location>Acknowledgements</section_header_level_1>
<text><location><page_22><loc_12><loc_19><loc_88><loc_26></location>It has been a great pleasure discussing this work with Dionysios Anninos, Daniel Harlow, Tom Hartman, Daniel Jafferis, Matt Kleban and Steve Shenker. This work was supported in part by DOE grant DE-FG02-91ER40654 and the Fundamental Laws Initiative at Harvard.</text>
<section_header_level_1><location><page_23><loc_12><loc_91><loc_51><loc_92></location>A Appendix: dS 4 Killing vectors</section_header_level_1>
<text><location><page_23><loc_12><loc_87><loc_47><loc_88></location>The 10 Killing vectors of dS 4 are given by:</text>
<code><location><page_23><loc_16><loc_61><loc_88><loc_84></location>L 0 = cos ψ∂ t -tanh t sin ψ∂ ψ M ∓ 1 = ± sin ψ sin θ sin φ∂ t +(1 ± tanh t cos ψ ) sin θ sin φ∂ ψ +(cot ψ ± tanh t csc ψ ) (cos θ sin φ∂ θ +csc θ cos φ∂ φ ) M ∓ 2 = ± sin ψ sin θ cos φ∂ t +(1 ± tanh t cos ψ ) sin θ cos φ∂ ψ +(cot ψ ± tanh t csc ψ ) (cos θ cos φ∂ θ -csc θ sin φ∂ φ ) M ∓ 3 = ± sin ψ cos θ∂ t +(1 ± tanh t cos ψ ) cos θ∂ ψ -(cot ψ ± tanh t csc ψ ) sin θ∂ θ J 1 = cos φ∂ θ -sin φ cot θ∂ φ J 2 = -sin φ∂ θ -cos φ cot θ∂ φ J 3 = ∂ φ . (A.1)</code>
<text><location><page_23><loc_15><loc_56><loc_70><loc_58></location>For each k , the M ± k and L 0 form a SO (2 , 1) subalgebra satisfying</text>
<formula><location><page_23><loc_25><loc_51><loc_88><loc_53></location>[ M + k , M -k ] = 2 L 0 , [ L 0 , M + k ] = -M + k , [ L 0 , M -k ] = M -k . (A.2)</formula>
<text><location><page_23><loc_12><loc_48><loc_72><loc_49></location>As mentioned in the text, acting on SO (3)-invariant functions, we have</text>
<formula><location><page_23><loc_36><loc_42><loc_88><loc_45></location>/lscript 2 ∇ 2 = -L 0 ( L 0 -3) + M -k M + k , (A.3)</formula>
<text><location><page_23><loc_12><loc_39><loc_41><loc_40></location>where k is summed over k = 1 , 2 , 3.</text>
<text><location><page_23><loc_12><loc_33><loc_88><loc_38></location>The conformal Killing vectors of the S 3 are given by the restriction of dS 4 Killing vectors on I + :</text>
<code><location><page_23><loc_17><loc_13><loc_88><loc_31></location>L 0 = -sin ψ∂ ψ M ∓ 1 = (1 ± cos ψ ) sin θ sin φ∂ ψ +(cot ψ ± csc ψ ) (cos θ sin φ∂ θ +csc θ cos φ∂ φ ) M ∓ 2 = (1 ± cos ψ ) sin θ cos φ∂ ψ +(cot ψ ± csc ψ ) (cos θ cos φ∂ θ -csc θ sin φ∂ φ ) M ∓ 3 = (1 ± cos ψ ) cos θ∂ ψ -(cot ψ ± csc ψ ) sin θ∂ θ . J 1 = cos φ∂ θ -sin φ cot θ∂ φ J 2 = -sin φ∂ θ -cos φ cot θ∂ φ J 3 = ∂ φ . (A.4)</code>
<section_header_level_1><location><page_24><loc_12><loc_91><loc_24><loc_92></location>References</section_header_level_1>
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</document>
[
{
"title": "State/Operator Correspondence in Higher-Spin DS/CFT",
"content": "Gim Seng Ng and Andrew Strominger Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "A recently conjectured microscopic realization of the dS 4 /CFT 3 correspondence relating Vasiliev's higher-spin gravity on dS 4 to a Euclidean Sp ( N ) CFT 3 is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary. In particular it is argued that states of the boundary CFT 3 on S 2 are holographically dual to bulk states on geodesically complete, spacelike R 3 slices which terminate on an S 2 at future infinity. The dictionary is described in detail for the case of free scalar excitations. The ground states of the free or critical Sp ( N ) model are dual to dS-invariant planewave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT 3 . CFT 3 states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk. A norm is defined on the R 3 bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the Sp ( N ) CFT 3 .",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The conjectured dS/CFT correspondence attempts to adapt the wonderful successes of the AdS/CFT correspondence to universes (possibly like our own) which exponentially expand in the far future. The hope [1, 2, 3, 4, 5] is to define bulk de Sitter (dS) quantum gravity in terms of a holographically dual CFT living at I + of dS, which is the asymptotic conformal boundary at future null infinity. A major obstacle to this program has been the absence of any explicit microscopic realization. This has so far prevented the detailed development of the dS/CFT dictionary. This situation has recently been improved by an explicit proposal [6] relating Vasiliev's higher-spin gravity in dS 4 [7, 8] to the dual Sp ( N ) CFT 3 described in [9]. In this paper we will use this higher-spin context to write some new entries in the dS/CFT dictionary. The recent proposal [6] for a microscopic realization of dS/CFT begins with the duality relating the free (critical) O ( N ) CFT 3 to higher-spin gravity on AdS 4 with Neumann (Dirichlet) boundary conditions on the scalar field. Higher-spin gravity - unlike string theory [2] has a simple analytic continuation from negative to positive cosmological constant Λ. Under this continuation, AdS 4 → dS 4 and the (singlet) boundary CFT 3 correlators are simply transformed by the replacement of N →-N . These same transformed correlators arise from the Sp ( N ) models constructed from anticommuting scalars. It follows that the free (critical) Sp ( N ) correlators equal those of higher-spin gravity on dS 4 with future Neumann (Dirichlet) scalar boundary conditions (of the type described in [10]) at I + . This mathematical relation between the bulk dS and boundary Sp ( N ) correlators may provide a good starting point for understanding quantum gravity on dS, but so far important physical questions remain unanswered. For example we do not know how to relate these physically un measurable correlators to a set of true physical observables or to the dS horizon entropy. These crucial entries in the dS/CFT dictionary are yet to be written. As a step in this direction, in this paper we investigate the relation between quantum states in the bulk higher-spin gravity and those in the boundary CFT 3 . Bulk higher-spin gravity has fields of Φ s with all even spins s = 0 , 2 , .... , which are dual to CFT 3 operators O s with the same spins. In the CFT 3 , we can also associate a state to each operator by the state-operator correspondence. One way to do this is to take the southern hemisphere of S 3 , insert the operator O s at the south pole, and then define a state Ψ s S 2 as a functional of the boundary conditions on the equatorial S 2 . For every object in the CFT 3 , we expect a holographically dual object in the bulk dS 4 theory. This raises the question: what is the bulk representation of the spins state Ψ s S 2 ? In Lorentzian AdS 4 holography, the state created by a primary operator O in the CFT 3 on S 2 has, at weak coupling, a bulk representation as the single particle state of the field Φ dual to O with a smooth minimal-energy wavefunction localized near the center of AdS 4 . The form of the wavefunction is dictated by the conformal symmetry. In dS 4 holography, the situation is rather different. States in dS 4 quantum gravity are usually thought of as wavefunctions on complete spacelike slices which are topologically S 3 . 1 These do not seem to be good candidates for bulk duals to Ψ s S 2 because, among other reasons, they are not associated to any S 2 in I + . However, dS 4 also has everywhere spacelike and geodesically complete R 3 slices which end at an S 2 in I + . Here we propose a construction of the bulk version of Ψ s S 2 on these slices, again as single particle states whose form is dictated by the conformal symmetry. Interestingly, the classical wavefunction for the particle turns out to be the (anti) quasinormal modes for the static patch of de Sitter, as constructed in [11, 12]. 2 This relation between bulk and boundary states has a potentially profound nonperturbative consequence briefly mentioned in section 4.1 [14]. The operator O 0 dual to a scalar Φ is bilinear in boundary fermions and hence obeys ( O 0 ) N 2 +1 = 0. Under bulk-boundary duality this translates into an N 2 -adicity relation for Φ: one cannot put more than N 2 bulk scalar quanta into the associated quasinormal mode. Further investigation of this dS exclusion principle is deferred to later work. We also construct a norm for these bulk states and show that it is the Zamolodchikov norm on S 3 of the CFT 3 operator O s . Explicit formulae are exhibited only for the scalar s = 0 case but we expect the construction to generalize to all s . This paper is organized as follows. In section 2 we revisit the issue of the usual global dS-invariant vacua for a free massive scalar field, paying particular attention to the case of m 2 /lscript 2 = 2 (where /lscript is the de Sitter radius) arising in higher spin gravity. The invariant vacua include the familiar Bunch-Davies Euclidean vacuum | 0 E 〉 , as well as a pair of | 0 ± 〉 of in/out vacua with no particle production. As the scalar field acting on | 0 -〉 ( | 0 + 〉 ) obeys Dirichlet (Neumann) boundary conditions on I + , these are related to the critical (free) Sp ( N ) model. Generically all dS-invariant vacua are Bogolyubov transformations of one another, but we find that at m 2 /lscript 2 = 2 the transformation is singular and the in/out vacua are non-normalizable plane-wave type states. In section 3 we use the conformal symmetries to find the classical bulk wavefunctions associated to an operator insertion on I + , and note the relation to (anti) quasinormal modes. The construction uses a rescaled bulk-to-boundary Green function defined with Neumann or Dirichlet I + boundary conditions. We also show that the Klein-Gordon inner product of these wavefunctions agrees with the conformallycovariant CFT 3 operator two-point function. In section 4 we consider the Hilbert space on R 3 slices ending on an S 2 on I + . This Hilbert space was explicitly constructed in [15] for a free scalar on hyperbolic slices ending on I + . There are two such Hilbert spaces, which we denote the northern and southern Hilbert space, which live on spatial R 3 slices extending to the north or south of the S 2 . The northern and southern slices add up to a global S 3 . Hence the tensor product of the northern and southern Hilbert spaces is the global Hilbert space on S 3 , much as the left and right Rindler Hilbert spaces tensor to the global Minkowski Hilbert space. We show that the global | 0 ± 〉 vacua are simple tensor products of the northern and southern Dirichlet and Neumann vacua. We then use symmetries to uniquely identify the states of the southern Hilbert space with those of the free and critical Sp ( N ) models on an S 2 . This leads directly to the dS exclusion principle. We further construct an inner product for the southern Hilbert space which agrees, for states dual to I + operator insertions, to the conformal two-point function on I + . In section 5 we discuss the restriction of Euclidean vacuum to a southern state and recall from [15], that this is a mixed state which is thermal with respect to an SO (3 , 1) Casimir. It would be interesting to relate this result to dS entropy in the present context. In section 6 we show that the standard CFT 3 state-operator correspondence maps the known pseudo-unitary C-norm of the Sp ( N ) model to the Zamolodchikov two-point function. This completes the demonstration that the bulk states on R 3 S have the requisite properties to be dual to the boundary Sp ( N ) CFT 3 states on S 2 . Speculations are made on the possible relevance of pseudo-unitarity to the consistency of dS/CFT in general. An appendix gives some explicit formulae for the SO (4 , 1) Killing vectors of dS 4 .",
"pages": [
2,
3,
4,
5
]
},
{
"title": "2 Global dS vacua at m 2 /lscript 2 = 2",
"content": "In this section we describe the quantum theory of a free scalar field Φ in dS 4 with wave equation and mass This is the case of interest for Vasiliev's higher-spin gravity. While there have been many general discussions of this problem, peculiar singular behavior as well as simplifications appear at the critical value m 2 /lscript 2 = 2 which are highly relevant to the structure of dS/CFT. A parallel discussion of de Sitter vacua and scalar Green functions in the context of dS/CFT was given in [16]. However that paper in many places specialized to the large mass regime m 2 /lscript 2 > 9 4 , excluding the region of current interest. The behavior in the region m 2 /lscript 2 < 9 4 divides into three cases m 2 /lscript 2 > 2, m 2 /lscript 2 = 2 and m 2 /lscript 2 < 2. Much of the structure we describe below pertains to the entire range m 2 /lscript 2 < 9 4 with an additional branch-cut prescription for the Green functions.",
"pages": [
5
]
},
{
"title": "2.1 Modes",
"content": "We will work in the dS 4 global coordinates where Ω i ∼ ( ψ, θ, φ ) are coordinates on the global S 3 slices. Following the notation of [16] solutions of the wave equation can be expanded in modes of total angular momentum L and spin labeled by the multi-index j . The spherical harmonics Y Lj obey where √ h and D 2 are the measure and Laplacian on the unit S 3 , Ω A is the antipodal point of Ω, and here and hereafter ∑ denotes summation over all allowed values of L and j . The time dependence is then governed by the second order ODE",
"pages": [
6
]
},
{
"title": "2.1.1 Neumann and Dirichlet modes",
"content": "Eq. (2.6) has the real solutions where We are interested in m 2 /lscript 2 = 2, which implies and The modes behave near I + as e -h ± t Accordingly we refer to the + modes as Dirichlet and the -modes as Neumann. We have normalized so that the Klein-Gordon inner product is with d 3 Σ µ the induced measure times the normal to the S 3 slice. Under time reversal so that where the point x A is antipodal to the point x . This implies that an incoming Dirichlet (Neumann) mode propagates to an outgoing Dirichlet (Neumann) mode. This is not the case for generic m 2 and, as will be seen below, allows for Dirichlet and Neumann vacua with no particle production.",
"pages": [
6,
7
]
},
{
"title": "2.1.2 Euclidean modes",
"content": "Euclidean modes are defined by the condition that when dS 4 is analytically continued to S 4 they remain nonsingular on the southern hemisphere. In other words One finds that the combination is nonsingular at t = -iπ/ 2. Hence y ∓ L are simply the real and imaginary parts of the y E L . (2.13) and (2.14) imply the relations",
"pages": [
7,
8
]
},
{
"title": "2.2 Vacua",
"content": "In the quantum theory Φ is promoted to an operator which we denote ˆ Φ obeying the equal time commutation relation Defining annihilation and creation operators the global Euclidean (or Bunch-Davies) vacuum is defined by We normalize so that 〈 0 E | 0 E 〉 = 1. For any m 2 there is a family of dS-invariant vacua labeled by a complex parameter α . They are annihilated by the normalized Bogolyubov-transformed oscillators We are interested in the vacua annihilated by the Dirichlet or Neumann modes for the case of m 2 /lscript 2 = 2, which correspond to e α = ± 1. In that case the Bogolyubov transformation is singular. Nevertheless we can still construct non-normalizable plane-wave type vacua as follows. The field operator may be decomposed as where ˆ Φ ± ∼ e -h ± t near I + . The squeezed states then obey Since only Dirichlet (Neumann) modes act non-trivially on | 0 -〉 ( | 0 + 〉 ) we refer to it as the Dirichlet (Neumann) vacuum. These vacua are dS invariant. With the conventional norm, ˆ Φ ± are hermitian and their eigenstates are non-normalizable. Generalized dS non-invariant plane-wave type Neumann states with nonzero eigenvalues for ˆ Φ + are constructed as Φ + here is an arbitrary solution of the classical wave equation, which can be parameterized by an arbitrary function Φ + (Ω) on I + The states are delta-functional normalizable with respect the usual inner product where the delta function integrates to one with the measure The c + Lj satisfies the reality condition c + ∗ Lj = c + Lj ( -) L . One may similarly define generalized Dirichlet states obeying The Euclidean vacuum can be expressed in terms of | 0 ± 〉 as where ∆ ± are the (everywhere positive) two-point functions for a CFT 3 operator with h + = 2 and h -= 1. 3 These satisfy We also have the relations In particular The Wightman function in the Euclidean vacuum is In terms of the dS-invariant distance function this becomes simply with the usual iε prescription for the singularity.",
"pages": [
8,
9,
10,
11
]
},
{
"title": "3 Boundary operators and quasinormal modes",
"content": "According to the dS 4 /CFT 3 dictionary, for every spin zero primary CFT 3 operator O of conformal weight h there is a bulk scalar field Φ with mass m 2 /lscript 2 = h (3 -h ). Boundary correlators of O are then related by a rescaling to bulk Φ correlators whose arguments are pushed to the boundary at I + . As in AdS/CFT, a particular classical bulk wavefunction of Φ can be associated to a boundary insertion of O (at the linearized level) by symmetries: it must scale with weight h under the isometry corresponding to dilations, obey the lowest-weight condition, and be invariant under rotations around the point of the boundary insertion. The resulting wavefunction is a type of bulk-to-boundary Green function. Interestingly[13], the (lowest) highest-weight modes can also be identified as (anti) quasinormal modes for the static patch of de Sitter, as constructed in [11, 12]. In this section we determine this wavefunction explicitly, regulate the singularities, generalize it to multi-particle insertions and define a symplectic product. In the following section we will then use these classical objects to construct the associated dual bulk quantum states and their inner products.",
"pages": [
11
]
},
{
"title": "3.1 Highest and lowest weight wavefunctions",
"content": "In this subsection we give expressions for the classical wavefunctions, associated to lowest (highest) weight primary operator insertions at the south (north) pole of I + in terms of rescaled Green functions in the limit that one argument is pushed to I + . These wavefunctions each comes in a Neumann and a Dirichlet flavor, denoted Φ ± lw ( x ) (Φ ± hw ( x )) depending on whether the weight of the dual operator insertion is h + or h -. The relevant Green functions are 4 with G E the Wightman function for the Euclidean vacuum given in equation (2.42). G -( G + ) obeys Neumann (Dirichlet) boundary conditions at I + away from x = x ' . These are for m 2 /lscript 2 = 2 the Green functions with future boundary conditions as discussed in [10]. We have normalized them so that they have the Hadamard form for the short-distance singularity. In the Neumann case we begin with G -, which is (using the mode decomposition (2.40)) given by From this we construct the rescaled Green function in which the second argument is placed at the south pole Ω SP where ψ ' = 0. One may then check that (ignoring singularity prescriptions) Using (3.2) and the asymptotics (2.11) one finds that near I + (not ignoring singularities) Let us now confirm that Φ -lw ( x ) has the same symmetries as an insertion of a primary operator O (Ω SP ) at the south pole of I + . First we note that the choice of a point on I + breaks SO (4 , 1) to SO (3) × SO (1 , 1). Both Φ -lw ( x ) and O (Ω SP ) are manifestly invariant under the SO (3) spatial rotations. The generator of SO (1 , 1) dilations, denoted L 0 , acts on O (Ω SP ) as In the bulk it is generated by the Killing vector field where the south pole is ψ = 0. dS invariance implies It follows from this together with the definition (3.4) that the wavefunction obeys By construction it obeys the wave equation Acting on SO (3) invariant symmetric functions we have where the 6 Killing vector fields M ± k (given in Appendix A) are the raising and lowering operators for L 0 and we sum over k . It then follows that and hence which corresponds to the lowest-weight condition for the O It may be shown that these symmetries uniquely determine the solution. Hence Φ -lw is identified as the classical wavefunction associated to the insertion of the primary O at the south pole. A parallel argument leads to the dual of a highest weight operator insertion at the north pole . The wavefunction is This obeys the relations and has the asymptotic behavior Similar formulae apply to the Dirichlet case by beginning with G + in the above construction and replacing + ↔-. For example We see from the above that the highest-weight wavefunction is smooth on the future horizon of the southern static patch dS 4 , and hence related to the quasinormal modes found in [11, 12]. The lowest quasinormal mode which is invariant under the SO (3) of the static dS 4 is exactly the Φ -hw with h -= 1 while the second lowest SO (3)-invariant quasinormal mode corresponds to Φ + hw with h + = 2. Lowest weight states are smooth on the past horizon and hence related to anti-quasinormal modes.",
"pages": [
11,
12,
13,
14
]
},
{
"title": "3.2 General multi-operator insertions",
"content": "In the preceding subsections we found the bulk duals of primary operators inserted at the north/south pole in the coordinates (2.3). This can be generalized to insertions at an arbitrary point on I + with a general time slicing near I + . Let us introduce coordinates x ∼ ( y i , t ) such that near I + The dual wavefunction is then the t ' →∞ limit of the rescaled Green function, denoted by For the special cases of operator insertions at the north or south pole in global coordinates these reduce to our previous expressions. Note that coordinate transformations of the form t → t + f ( y ) induce a conformal transformation on I + as appropriate for a conformal field of weight h ± . Hence the relative normalization in (3.4) will depend on the conformal frame at I + . One may also consider multi-operator insertions such as O ( y 1 ) O ( y 2 ) in the CFT 3 at I + . At the level of free field theory considered here these are associated to a bilocal wavefunction in the product of two bulk scalar fields We will use Φ Ω to denote these wavefunctions when working in global coordinates (2.3). We note that in such coordinates near I + for an insertion at a general point",
"pages": [
14,
15
]
},
{
"title": "3.3 Klein-Gordon inner product",
"content": "We wish to define an inner product between e.g. two Neumann wavefunctions Φ -Ω 1 and Φ -Ω 2 . Later on we will compare this to the inner product on the CFT 3 Hilbert space and the twopoint function of O on S 3 . One choice is to take a global spacelike S 3 slice in the interior and define the Klein-Gordon inner product This integral does not depend on the choice of S 3 which can be pushed up to I + . One may then see immediately from (3.23) that there are two nonzero terms proportional to ∆ -giving One may also define an inner product not on global spacelike S 3 slices, but on a spacelike R 3 slice which ends on an S 2 on I + . The result is invariant under any deformation of the S 2 which does not cross the insertion point. To be definite, we take the S 2 to be the equator, Ω 1 to be in the northern hemisphere and Ω 2 to be in the southern hemisphere, and the slice to be R 3 S which intersects the south pole. One then finds, pushing R 3 S up to the southern hemisphere of I + Similarly, the inner product between two Dirichlet wavefunctions is given by",
"pages": [
15
]
},
{
"title": "4 The southern Hilbert space",
"content": "We now turn to the issue of bulk quantum states. Quantum states in dS are often discussed, as in section 2, in terms of a Hilbert space built on the global S 3 slices. The structure of the vacua and Green functions for such states was described in section 2. However dS has the unusual feature that there are geodesically complete topologically R 3 spacelike slices which end on an S 2 in I + , which we will typically take to be the equator. Examples of these are the hyperbolic slices, the quantization on which was studied in detail in [15]. We will see that the quantum states built on these R 3 slices are natural objects in dS/CFT. An S 2 in I + is in general the boundary of a 'northern' slice, denoted R 3 N and a 'southern' slice denoted R 3 S . The topological sum obeys R 3 S ∪ R 3 N = S 3 . Hence the relation of the southern and northern Hilbert spaces on R 3 S and R 3 N to that on S 3 is like that of the left and right Rindler wedges to that of global Minkowski space. It is also like the relation of the Hilbert spaces of the northern and southern causal diamonds to that of global dS. However the diamond Hilbert spaces in dS quantum gravity are problematic in quantum gravity with a fluctuating metric because it is hard to find sensible boundary conditions. A strong motivation for considering the R 3 S,N slices comes from the picture of a state in the boundary CFT 3 . The state-operator correspondence in CFT 3 begins with an insertion of a (primary or descendant) operator O at the south pole of S 3 , and then defines a quantum state as a functional of the boundary conditions on an S 2 surrounding the south pole. For every object in the CFT 3 , we expect a holographically dual object in the bulk dS 4 theory. The dual bulk quantum state must somehow depend on the choice of S 2 in I + . Hence it is natural to define the bulk state on the R 3 slice which ends on this S 2 in I + . This is how holography works in AdS/CFT: CFT states live on the boundaries of the spacelike slices used to define the bulk states.",
"pages": [
16
]
},
{
"title": "4.1 States",
"content": "In order to define quantum states on R 3 S , we first note that modes of the scalar field operator ˆ Φ(Ω , t ) are labeled by operators ˆ Φ ± (Ω) defined on I + via the relation They satisfy the following commutation relation We may then decompose these I + operators as the sum of two terms where the first (second) acts only on R 3 N ( R 3 S ). Defining the northern and southern Dirichlet and Neumann vacua by it follows from the decomposition (4.3) that the global vacua have a simple product decomposition 5 Excited southern states may then be built by acting on one of these southern vacua with ˆ Φ S . We wish to identify these states with those of the CFT 3 on S 2 . In the higher-spin dS/CFT correspondence there are actually two CFT 3 's living on I + : the free Sp ( N ) model, associated to Neumann boundary conditions, and the critical Sp ( N ) model, associated to Dirichlet boundary conditions. Since the field operators ˆ Φ S acting on | 0 + S 〉 ( | 0 -S 〉 ) obeys, according to equation (2.26), Neumann (Dirichlet) boundary conditions near the southern hemisphere of I + , it is natural to identify Next we want to consider excited states and their duals. To be specific we consider the Neumann theory built on | 0 + S 〉 . Parallel formulae apply to the Dirichlet case. Operator versions of the classical wavefunctions Φ -Ω ( x ) are constructed as where Ω S is presumed to lie on the southern hemisphere. We can make a quantum state where in the last line we used (3.23). By construction this will be a lowest weight state, and we therefore identify it as the bulk dual to the CFT 3 state created by the primary operator O dual to the field Φ. This connection leads to an interesting nonperturbative dS exclusion principle [14]. The operator O has a representation in the Sp ( N ) theory as where η A are N anticommuting real scalars and Ω AB is the quadratic form on Sp ( N ). It follows that Bulk-boundary duality and the state-operator relation described above then implies the nonperturbative relation Hence the quantum field operators ˆ Φ ± (Ω) are N 2 -adic. One is not allowed to put more than N 2 quanta in any given quasinormal mode. This is similar to the stringy exclusion principle for AdS [20] and may be related to the finiteness of dS entropy. Nonperturbative phenomena due to related finite N effects in the O ( N ) case have been discussed in [21]. We hope to investigate further the consequences of this dS exclusion principle.",
"pages": [
16,
17,
18
]
},
{
"title": "4.2 Norm",
"content": "Having identified the bulk duals of the boundary CFT 3 states, we wish to describe the bulk dual of the CFT 3 norm. The standard bulk norm is defined by Φ( x ) = Φ † ( x ). However this norm is not unique. It has been argued for a variety of reasons beginning in [3] that it is appropriate to modify the norm in the context of dS - see also [16, 22]. Here we have the additional problem that this standard norm is divergent for states of the form (4.8). We now construct the modified norm for states on R S 3 by demanding that it is equivalent to the CFT 3 norm. The construction here generalizes to dS 4 the one given in [16] for dS 3 . The bulk action of dS Killing vectors K µ A ∂ µ on a scalar field is generated by the integral over any global S 3 slice where T µν is the bulk stress tensor constructed from the operator ˆ Φ. If we take ˆ Φ † ( x ) = ˆ Φ( x ), then ˆ L A = ˆ L † A which is not what we want. The CFT 3 states are in representations of the SO (3 , 2) conformal group. These arise from analytic continuation of the 10 SO (4 , 1) conformal Killing vectors on S 3 which are the boundary restrictions of the bulk dS 4 Killing vectors K µ A ∂ µ . Usually, the standard CFT 3 norm has a self-adjoint dilation operator L 0 generating -i sin ψ∂ ψ as well as 3 self-adjoint SO (3) rotation operators J k . The remaining 6 raising and lowering operators L ± k arising from the Killing vectors iM ± k (described in the appendix) then obey L † ± k = L ∓ k in the conventional CFT 3 norm. To obtain an adjoint with the desired properties, we define the modified adjoint where here and hereafter † denotes the bulk modified adjoint. The reflection operator R is the discrete isometry of S 3 which reflects through the S 2 equator R ( ψ, θ, φ ) = ( π -ψ, θ, φ ) along with complex conjugation. In particular, it maps the south pole to the north pole while keeping the equator invariant. This implies that L 0 (generating iL 0 ) and J k are self adjoint while Hence we have constructed an adjoint admitting the desired SO (3 , 2) action. We do not know whether or not it is unique. The action of R maps an operator defined on the southern hemisphere to one defined on the southern hemisphere of I + according to Hence the action of R exchanges the northern and southern hemispheres, and maps a southern I + state to a northern one. Therefore it cannot on its own define an adjoint within the southern Hilbert space. For this we must combine (4.13) with a map from the north to the south. Such a map is provided by the Euclidean vacuum. The global Euclidean bra state (constructed with the standard adjoint) can be decomposed in terms of a basis of northern and southern bra states We then define the modified adjoint of an arbitrary southern state | Ψ S 〉 by We will denote the corresponding inner product by an S subscript For example choosing the basis so that we have In particular one finds Let us now compute the norm of the southern state | Ω -S 〉 in (4.8). The action of R gives a northern state which we will denote | R Ω -S 〉 . The norm is then Using the relation we find This is proportional to the S 3 two-point function of a dimension h -primary at the points Ω S and R Ω S . The analogous computation in the Dirichlet theory gives",
"pages": [
18,
19,
20
]
},
{
"title": "5 Boundary dual of the bulk Euclidean vacuum",
"content": "In the preceding section we have argued that dS/CFT maps CFT 3 states on an S 2 in I + to bulk states on the southern slice ending on the S 2 . A generic state in a global dS slice does not restrict to a pure southern state. However we can always define a density matrix by tracing over the northern Hilbert space. In particular, such a southern density matrix ρ E S can be associated to the global Euclidean vacuum | 0 E 〉 . The choice of an equatorial S 2 in I + breaks the SO (4 , 1) symmetry group down to SO (3 , 1), which also preserves the hyperbolic slices ending on the S 2 . ρ S E must be invariant under this SO (3 , 1). In fact ρ S E follows from results in [15]. Writing the quadratic Casimir of SO (3 , 1) as C 2 = -(1 + p 2 ), it was shown, in a basis which diagonalizes p , that where N 1 is determined by Tr ρ S E = 1. It would be interesting to investigate this further and compute the entropy S = -Tr ρ S E ln ρ S E in the Sp ( N ) model.",
"pages": [
20,
21
]
},
{
"title": "6 Pseudounitarity and the C-norm in the Sp ( N ) CFT 3",
"content": "In this section we consider the Sp ( N ) model (where N is even) and compare the norms to those computed above. The action is where χ a ( a = 1 , . . . , N 2 ) is a complex anticommuting scalar and ¯ χχ ≡ δ ab ¯ χ a χ b . This has a global Sp ( N ) symmetry and we restrict to Sp ( N ) singlet operators. 6 For the free theory m = λ = 0 while the critical theory is obtained by flowing to a nontrivial fixed point λ F . The Sp ( N ) theory is not unitary in the sense that in the standard norm following from (6.1) one has that [9] /negationslash and 〈 Ψ ' | Ψ 〉 is not preserved. Nevertheless, as detailed in [9], there exists an operator C with the properties To write it in real fields, for e.g., in the case of Sp (2), writing the real and imaginary part of χ as η 1 and η 2 , the action of C becomes η 2 = Cη † 1 C . One may then define a 'pseudounitary' C -inner product which is preserved under hamiltonian time evolution. Such hamiltonians are pseudohermitian and are similar to those studied in [23]. We note that the norm is not positive definite. Inserting an operator O i constructed from χ a at the south pole gives a functional of the boundary conditions on the equatorial S 2 which we define as the state | O i 〉 . This is the standard state-operator correspondence. An inner product for such states associated to O i and O j can be defined by the two point function with O i at one pole and O j at the other. It follows from (6.3) that this is the C -inner product for the states | O i 〉 and | O j 〉 : In the last line, we used the fact that the (singlet) currents in the Sp ( N ) models satisfy CO i † C = O i since C (¯ χχ ) † C = ¯ χχ . For primary operators of weight h i we then have [6] Hence it is the C-norm which maps under the state-operator correspondence to the Zamolodchikov norm defined as the Euclidean two point function on S 3 . As seen in [6] this C-norm then agrees with the bulk inner product (4.24)-(4.25) of the dual state for the scalar case. 7 Moreover, as the bulk and CFT 3 norms assign the same hermiticity properties to the SO (4 , 1) generators, this result will carry over to descendants of the primaries. A generalization of this construction to all spins seems possible. One of the puzzling features of dS/CFT is that the dual CFT cannot be unitary in the ordinary sense. This is not a contradiction of any kind because unitarity of the Euclidean CFT is not directly connected to any spacetime conservation law. At the same time quantum gravity in dS - and its holographic dual - should have some good property replacing unitarity in the AdS case. It is not clear what that good property is. The appearance of a pseudounitary structure in the case of dS/CFT analyzed here is perhaps relevant in this regard.",
"pages": [
21,
22
]
},
{
"title": "Acknowledgements",
"content": "It has been a great pleasure discussing this work with Dionysios Anninos, Daniel Harlow, Tom Hartman, Daniel Jafferis, Matt Kleban and Steve Shenker. This work was supported in part by DOE grant DE-FG02-91ER40654 and the Fundamental Laws Initiative at Harvard.",
"pages": [
22
]
},
{
"title": "A Appendix: dS 4 Killing vectors",
"content": "The 10 Killing vectors of dS 4 are given by: For each k , the M ± k and L 0 form a SO (2 , 1) subalgebra satisfying As mentioned in the text, acting on SO (3)-invariant functions, we have where k is summed over k = 1 , 2 , 3. The conformal Killing vectors of the S 3 are given by the restriction of dS 4 Killing vectors on I + :",
"pages": [
23
]
}
]
2013CQGra..30k5001H
https://arxiv.org/pdf/1211.6079.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_77><loc_67><loc_81></location>Dynamical Excision Boundaries in Spectral Evolutions of Binary Black Hole Spacetimes</section_header_level_1>
<text><location><page_1><loc_45><loc_74><loc_60><loc_75></location>1 2</text>
<text><location><page_1><loc_27><loc_71><loc_74><loc_75></location>Daniel A. Hemberger , Mark A. Scheel , Lawrence E. Kidder 1 , B'ela Szil'agyi 2 , Geoffrey Lovelace 3 , Nicholas W. Taylor 2 , and Saul A. Teukolsky 1</text>
<text><location><page_1><loc_27><loc_68><loc_75><loc_70></location>1 Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853</text>
<text><location><page_1><loc_27><loc_65><loc_75><loc_68></location>2 Theoretical Astrophysics 103-33, California Institute of Technology, Pasadena, CA 91125</text>
<text><location><page_1><loc_27><loc_63><loc_76><loc_65></location>3 Gravitational Wave Physics and Astronomy Center, California State University Fullerton, Fullerton, CA 92831</text>
<text><location><page_1><loc_27><loc_41><loc_76><loc_61></location>Abstract. Simulations of binary black hole systems using the Spectral Einstein Code (SpEC) are done on a computational domain that excises the regions inside the black holes. It is imperative that the excision boundaries are outflow boundaries with respect to the hyperbolic evolution equations used in the simulation. We employ a time-dependent mapping between the fixed computational frame and the inertial frame through which the black holes move. The time-dependent parameters of the mapping are adjusted throughout the simulation by a feedback control system in order to follow the motion of the black holes, to adjust the shape and size of the excision surfaces so that they remain outflow boundaries, and to prevent large distortions of the grid. We describe in detail the mappings and control systems that we use. We show how these techniques have been essential in the evolution of binary black hole systems with extreme configurations, such as large spin magnitudes and high mass ratios, especially during the merger, when apparent horizons are highly distorted and the computational domain becomes compressed. The techniques introduced here may be useful in other applications of partial differential equations that involve time-dependent mappings.</text>
<text><location><page_1><loc_27><loc_36><loc_55><loc_37></location>PACS numbers: 04.25.D-, 04.25.dg, 02.70.Hm</text>
<section_header_level_1><location><page_2><loc_16><loc_86><loc_28><loc_88></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_16><loc_78><loc_76><loc_85></location>Feedback control systems are ubiquitous in technological applications. They are found, for example, in thermostats, autopilots, chemical plants, and cruise control in automobiles. The purpose of a control system is to keep some measured output (such as the temperature in a room) at some desired value by adjusting some input (such as the power to a furnace).</text>
<text><location><page_2><loc_16><loc_73><loc_76><loc_77></location>In the last few years, feedback control systems have also found applications in the field of numerical relativity, particularly in simulations of binary black hole systems that employ spectral methods and excision techniques [1, 2, 3, 4, 5].</text>
<text><location><page_2><loc_16><loc_61><loc_76><loc_73></location>Black hole excision is a means of avoiding the physical singularities that lurk inside black holes. The idea is to solve Einstein's equations only in the region outside apparent horizons, cutting out the region inside the horizons. The boundaries of the excised regions are called excision boundaries. Causality ensures that the excision boundaries and the excised interiors cannot affect the physics of the exterior solution, and an appropriate hyperbolic formulation of Einstein's equations [6, 7] can ensure that gauge and constraint-violating degrees of freedom also do not propagate out of the excised region.</text>
<text><location><page_2><loc_16><loc_41><loc_76><loc_61></location>Excision is straightforward for black holes that remain stationary in the coordinates that are used in the simulation, but excision becomes more complicated when the black holes move or change shape. For numerical methods based on finite differencing, the excision boundaries can be changed at every time step by activating or deactivating appropriate grid points and adjusting differencing stencils [8, 9, 10, 11, 12, 13, 14, 15]. However, for spectral numerical methods, there is no equivalent of deactivating individual grid points; instead, spectral methods are defined in finite extended spatial regions with smooth boundaries. The nearest equivalent to the finite-difference excision approach would be interpolating all variables to a new slightly offset grid at every time step, which would be computationally expensive. Therefore, spectral numerical methods need a different approach to reconcile the need for moving black holes with the need for a fixed excision boundary inside of each black hole.</text>
<text><location><page_2><loc_16><loc_22><loc_76><loc_41></location>The solution [1] to this problem adopted by our group makes use of multiple coordinate systems. We call 'inertial coordinates' those coordinates that asymptotically correspond to an inertial observer; in these coordinates the black holes orbit each other, have distorted shapes, and approach each other as energy is lost to gravitational radiation. Spectral methods are applied in another coordinate system, 'grid coordinates', in which the excision boundaries are spherical and stationary. We connect grid coordinates with inertial coordinates by means of an analytic mapping function M that depends on some set of time-dependent parameters λ ( t ). These parameters must be continually adjusted so that each spherical, stationary grid-frame excision boundary is mapped to a surface in the inertial frame that follows the motion and the shape of the corresponding black hole as the system evolves. It is this adjustment of each parameter λ ( t ) that is accomplished by means of feedback control systems, one control system per parameter.</text>
<text><location><page_2><loc_16><loc_14><loc_76><loc_21></location>In this paper, we describe in detail the mapping functions and the corresponding feedback control systems that we use to handle black hole excision with spectral methods. Some earlier implementations have been described previously [5, 1, 2, 4, 16], but there have been many improvements that now allow spectral excision methods to produce robust simulations of binary black hole systems, including those with unequal</text>
<text><location><page_3><loc_16><loc_77><loc_76><loc_88></location>masses, high spins, and precession. We describe these improvements here. In Sec. 2 we review control theory and present simple examples of control systems. In Sec. 3 we discuss the implementation of control systems in the SpEC [17] code. Sections 4 and 5 detail the coordinate mappings used in SpEC and the feedback control systems used to control them. In Sec. 6 we describe the transition to the post-merger domain with a single excision boundary. In Sec. 7 we describe applications of control systems in SpEC besides the ones used to adjust map parameters. We summarize in Sec. 8.</text>
<section_header_level_1><location><page_3><loc_16><loc_74><loc_31><loc_75></location>2. Control Theory</section_header_level_1>
<text><location><page_3><loc_16><loc_68><loc_76><loc_72></location>To motivate control theory, we begin with a simple example: cruise control in an automobile. Suppose we wish to control the speed v of a car that is driving up an incline of angle θ . The equation of motion for this system is</text>
<formula><location><page_3><loc_37><loc_64><loc_76><loc_67></location>dv dt = -η m v + F m -g sin θ, (1)</formula>
<text><location><page_3><loc_16><loc_57><loc_76><loc_63></location>where m is the mass of the car, g is the gravitational acceleration, η is a drag coefficient, and F is a force supplied by the car's engine. We wish to determine F so as to cause the car to maintain a speed of v = v 0 , even if the angle θ changes as the car climbs the incline. To do this, we choose the force at time t to be:</text>
<formula><location><page_3><loc_34><loc_52><loc_76><loc_56></location>F ( t ) m = K P Q ( t ) + K I ∫ t 0 Q ( τ ) dτ, (2)</formula>
<text><location><page_3><loc_16><loc_46><loc_76><loc_51></location>where the control system is turned on at time t = 0, where K I and K P are constants, and where Q ( t ) = v 0 -v ( t ) is the quantity that we wish to drive to zero. We call Q the control error. Substituting Eq. (2) into Eq. (1) and differentiating with respect to time yields</text>
<formula><location><page_3><loc_31><loc_41><loc_76><loc_46></location>d 2 Q dt 2 + ( K P + η m ) dQ dt + K I Q = g cos θ dθ dt , (3)</formula>
<text><location><page_3><loc_16><loc_38><loc_76><loc_42></location>which is the equation for a damped, forced harmonic oscillator. By choosing K P to produce critical damping, and choosing K I to set a timescale, this choice will drive v toward v 0 as desired.</text>
<text><location><page_3><loc_16><loc_27><loc_76><loc_38></location>The basic structure of a control system is easily understood as a feedback loop (see Fig. 1). The simulation (e.g., a binary black hole simulation, or a car driving up an incline) produces some measure of error, Q ( t ), that defines the deviation from some desired target value. This error acts as the input for the control system, which then computes a control signal U ( t ) (e.g., the derivative of a map parameter, or the force supplied by a car's engine) that will minimize the error Q ( t ) when fed back into the simulation.</text>
<text><location><page_3><loc_16><loc_17><loc_76><loc_27></location>A simple and effective way to compute U ( t ) is to make it a linear combination of the error, Q ( t ), and integrals and/or derivatives of the error. The term proportional to the error acts to reduce the deviation from the desired value, the terms proportional to derivatives of the error act to oppose rapid deviations, and the terms proportional to the integrals act to reduce any persistent deviation or offset that accumulates over time. In the cruise control example, we used a proportional and an integral term only in Eq. (2).</text>
<text><location><page_3><loc_16><loc_14><loc_76><loc_16></location>We now turn to another example of a control system that is more closely related to the way we use control systems in binary black hole simulations. Consider two</text>
<figure>
<location><page_4><loc_35><loc_81><loc_57><loc_88></location>
<caption>Figure 1. A generic control circuit. The simulation outputs a measure of error, Q , which is used by the control system. The control system then outputs a signal, U , which changes the behavior of the simulation.</caption>
</figure>
<text><location><page_4><loc_16><loc_71><loc_62><loc_72></location>coordinate systems, ( x, t ) and (¯ x, ¯ t ), that are related by the map</text>
<formula><location><page_4><loc_41><loc_67><loc_76><loc_69></location>x = ¯ x -V ( t ) ¯ t, (4)</formula>
<formula><location><page_4><loc_41><loc_66><loc_76><loc_68></location>t = ¯ t. (5)</formula>
<text><location><page_4><loc_16><loc_58><loc_76><loc_65></location>We wish to control the parameter V ( t ) so that a wave f (¯ x, ¯ t ) = f (¯ x -c ¯ t ) that propagates at speed ¯ v = c in the (¯ x, ¯ t ) coordinates will propagate at some arbitrary desired speed v d in the ( x, t ) coordinates. According to Eq. (4), the speed of the wave in the ( x, t ) coordinates is v = c -V ( t ). Therefore we define a control error Q to be</text>
<formula><location><page_4><loc_39><loc_55><loc_76><loc_57></location>Q = c -V ( t ) -v d , (6)</formula>
<text><location><page_4><loc_16><loc_49><loc_76><loc_55></location>and we construct a control system that drives this control error to zero. If we choose the control signal U ( t ) to be d 2 V/dt 2 , then the simplest feedback loop that can be constructed uses only a term proportional to the error and amplified by a 'gain' K P . Then the evolution of V ( t ) is given by</text>
<formula><location><page_4><loc_34><loc_44><loc_76><loc_47></location>d 2 V dt 2 = K P Q = K P [ c -V ( t ) -v d ] . (7)</formula>
<text><location><page_4><loc_16><loc_42><loc_47><loc_43></location>The solution to this equation is of the form</text>
<formula><location><page_4><loc_32><loc_38><loc_76><loc_41></location>V ( t ) = c -v d + A 1 sin( αt ) + A 2 cos( αt ) , (8)</formula>
<text><location><page_4><loc_16><loc_33><loc_76><loc_39></location>where α := √ K P , which is oscillatory for K P > 0 and divergent for K P < 0. Thus we see that adding only a proportional term to the control signal U ( t ) is insufficient, since it does not reduce the control error Q .</text>
<text><location><page_4><loc_19><loc_32><loc_64><loc_33></location>However, if we add a derivative term to the feedback equation,</text>
<formula><location><page_4><loc_38><loc_28><loc_76><loc_31></location>d 2 V dt 2 = K P Q + K D dQ dt , (9)</formula>
<text><location><page_4><loc_16><loc_25><loc_38><loc_27></location>then the solution is of the form</text>
<formula><location><page_4><loc_28><loc_22><loc_76><loc_24></location>V ( t ) = c -v d + e -1 2 K D t [ B 1 sin( βt ) + B 2 cos( βt )] , (10)</formula>
<text><location><page_4><loc_16><loc_17><loc_76><loc_21></location>where β := 1 2 √ 4 K P -K 2 D . This solution is stable with an exponentially damped envelope when 4 K P ≥ K 2 D , which will cause v → v d as t →∞ .</text>
<text><location><page_4><loc_16><loc_14><loc_76><loc_18></location>Notice that this control system allows us to choose a V ( t ) such that v is the opposite sign of c and the wave is left-going in the ( x, t ) frame instead of right-going. This behavior can be seen in the example in Fig. 2.</text>
<figure>
<location><page_5><loc_26><loc_70><loc_65><loc_87></location>
<caption>Figure 2. The speed in ( x, t ) coordinates, v , is plotted for a family of gains K P with K D = 1 fixed. The wave velocity is c = -0 . 2 and the desired velocity is v d = 0 . 5 (dashed line). The controller turns on at t = 2. For gains in the stable region, where K P ≥ 0 . 25, v settles down to v d . One overdamped solution with K P = 0 . 01 is plotted for comparison (dotted line).</caption>
</figure>
<text><location><page_5><loc_16><loc_48><loc_76><loc_59></location>The overdamped solution (dotted line in Fig. 2) has a persistent offset that can be ameliorated by adding an integral term to the feedback equation, as was done in the cruise control example. In principle we could continue to add more terms, but in practice, it is usually sufficient to use a PID (proportional-integral-derivative) controller, which has terms proportional to the control error, its integral, and its derivative. When the underlying system is unknown, this is the best controller to use [18].</text>
<section_header_level_1><location><page_5><loc_16><loc_45><loc_39><loc_46></location>3. Control Systems in SpEC</section_header_level_1>
<text><location><page_5><loc_16><loc_24><loc_76><loc_44></location>Control systems are used in SpEC for several purposes. The most important is their role in handling moving, excised black holes in a spectral evolution method. We use a dual-frame method [1] in which the grid is fixed in some coordinates ( t, x i ) but the components of dynamical fields are expressed in a different coordinate system ( ¯ t, ¯ x i ). We call ( t, x i ) the grid coordinates and ( ¯ t, ¯ x i ) the inertial coordinates . Figure 3 shows an example domain decomposition in both coordinate systems. The two coordinate systems are connected by a map M that depends on time-dependent parameters λ ( t ). The excision boundaries are exactly spherical and stationary in grid coordinates. In inertial coordinates, the apparent horizons move and distort as determined by the solution of Einstein's equations supplemented by our gauge conditions. The parameters λ ( t ) need to be controlled so that the excision boundaries in the inertial frame follow the motion and shapes of the apparent horizons. We use control systems to accomplish this.</text>
<text><location><page_5><loc_16><loc_17><loc_76><loc_24></location>The particular maps that we use will be discussed in Sec. 4. In this section we will describe how we construct the control system for a general parameter λ ( t ), including how we define the relationship between the control error Q ( t ) and the control signal U ( t ), how we smooth out noise in the control system, and how we dynamically adjust the feedback parameters and timescales.</text>
<figure>
<location><page_6><loc_30><loc_73><loc_62><loc_88></location>
</figure>
<figure>
<location><page_6><loc_30><loc_58><loc_62><loc_73></location>
<caption>Figure 3. a) Computational domain in grid coordinates; the black hole centers are at rest and the excision boundaries are spherical. b) Same domain in inertial coordinates near merger; the excision boundaries move and distort to track the apparent horizons.</caption>
</figure>
<section_header_level_1><location><page_6><loc_16><loc_47><loc_52><loc_48></location>3.1. Definition of control errors and control signals</section_header_level_1>
<text><location><page_6><loc_16><loc_43><loc_76><loc_46></location>We represent a general time-dependent map parameter λ ( t ) as a polynomial in time with a piecewise constant N th derivative:</text>
<formula><location><page_6><loc_29><loc_37><loc_76><loc_42></location>λ ( t ) = N ∑ n =0 1 n ! ( t -t i ) n λ n i , for t i ≤ t < t i +1 , (11)</formula>
<text><location><page_6><loc_16><loc_35><loc_68><loc_37></location>where for each time interval t i ≤ t < t i +1 the quantities λ n i are constants.</text>
<text><location><page_6><loc_16><loc_23><loc_76><loc_35></location>At the beginning of each new time interval t i , we set the constants λ n i in Eq. (11) as follows. First for n = N we set λ N i = U ( t i ), where U ( t ) is the control signal defined in detail below. For n < N we set λ n i = d n λ/dt n | t = t i , where the derivative is evaluated at the end of the previous time interval. In this way all the derivatives of λ ( t ) except the N th derivative are continuous across intervals. The goal will be to compute the control signal U ( t ) so as to drive the map parameter λ ( t ) to some desired behavior. Before we describe how to compute the control signal U ( t ), we first discuss the control error Q , which will be used in the computation of U ( t ).</text>
<text><location><page_6><loc_16><loc_14><loc_76><loc_23></location>To appropriately define the control error Q , one must answer the question of how a small change in the map parameter corresponds to a change in the observed variables. If the control error is defined to be too large, then the controller will consistently overshoot its target, potentially leading to unstable behavior; conversely, if the control error is defined to be too small, then the controller may never be able to reach its target value.</text>
<text><location><page_7><loc_16><loc_83><loc_76><loc_88></location>If there exists a target value of λ ( t ), call it λ target , that does not depend on the map but may depend on other observable quantities in the system (call them A , B ,. . . ), then we define</text>
<formula><location><page_7><loc_37><loc_81><loc_76><loc_83></location>Q = λ target ( A,B,... ) -λ. (12)</formula>
<text><location><page_7><loc_16><loc_80><loc_59><loc_81></location>The goal is to drive Q to zero and thereby drive λ to λ target .</text>
<text><location><page_7><loc_16><loc_74><loc_76><loc_80></location>If instead, as often happens for nonlinear systems, the target value of λ depends on λ itself, even indirectly, then we define Q differently using a generalization of Eq. (12): we require that λ attains its desired value when Q → 0, and we require that</text>
<formula><location><page_7><loc_40><loc_71><loc_76><loc_74></location>∂Q ∂λ = -1 + O ( Q ) . (13)</formula>
<text><location><page_7><loc_16><loc_64><loc_76><loc_71></location>The primary motivation for this condition is its anticipated use in relating time derivatives of Q to those of λ (see Eq. (18), below). Note that in either Eqs. (12) or (13), Q could in principle be multiplied by an arbitrary factor; however, if this were done, that factor would need to be taken into account in the computation of the control signal U ( t ) below. Without loss of generality we assume no additional scaling.</text>
<text><location><page_7><loc_16><loc_57><loc_76><loc_63></location>In the case of several map parameters λ a ( t ) with corresponding Q a , where a is an index that labels the map parameters, the desired value of some λ a may depend on a different map parameter λ b . In this case, we generalize Eq. (13) and require that each Q a satisfy</text>
<formula><location><page_7><loc_39><loc_54><loc_76><loc_58></location>∂Q a ∂λ b = -δ ab + O ( Q ) , (14)</formula>
<text><location><page_7><loc_16><loc_42><loc_76><loc_54></location>where δ ab is a Kronecker delta. This criterion ensures that we can treat each λ a independently when all control errors are small. A way of understanding Eq. (14) is to consider a set of Q ' a that are obtained via Eq. (13) without regard to coupling between different λ a . Then a set of Q a satisfying Eq. (14) can be obtained by diagonalizing the matrix ∂Q ' a /∂λ b . In the remainder of this section we assume that if there are multiple map parameters λ a , the corresponding control errors Q a satisfy Eq. (14). We therefore drop the a indices and write equations for U ( t ) in terms of a single Q ( t ) satisfying Eq. (13) that represents the control error for a single λ ( t ).</text>
<text><location><page_7><loc_16><loc_28><loc_76><loc_42></location>The control error Q ( t ) is a function of several observables. In the case of the mapping functions that are designed to move and distort the excision boundaries to follow the motion and shapes of the apparent horizons, Q ( t ) is some function of the current position or shape of one or more apparent horizons. The precise definition of Q ( t ) is different for each map parameter, and depends on the details of how each map parameter couples to the observables. We will discuss the control error Q for each of the map parameters in Sec. 4. But it is not necessary to know the exact form of the control error in order to compute the control signal U ( t ); it suffices to know only that U ( t i ) is equal to λ N i in Eq. (11), and that the control error Q obeys Eq. (13).</text>
<text><location><page_7><loc_16><loc_24><loc_76><loc_28></location>We now turn to the computation of the control signal U ( t ). There is some flexibility in the control law determining U ( t ), so long as key feedback mechanisms are in place (as shown in Sec. 2). In SpEC, we use either a standard PID controller,</text>
<formula><location><page_7><loc_32><loc_20><loc_76><loc_23></location>U ( t ) = a 0 ∫ Q ( t ) dt + a 1 Q ( t ) + a 2 dQ dt , (15)</formula>
<text><location><page_7><loc_16><loc_18><loc_53><loc_19></location>or a special PD (proportional-derivative) controller,</text>
<formula><location><page_7><loc_39><loc_12><loc_76><loc_17></location>U ( t ) = K ∑ k =0 a k d k Q dt k , (16)</formula>
<text><location><page_8><loc_16><loc_86><loc_32><loc_88></location>where typically K = 2.</text>
<text><location><page_8><loc_16><loc_80><loc_76><loc_86></location>We set the constants a k so that the system damps Q to zero on some timescale τ d that we choose. We assume that τ d is longer than the interval t i +1 -t i defined in Eq. (11), so that we can approximate Q ( t ) and λ ( t ) as smooth functions rather than as functions with piecewise constant N th derivatives. Under this assumption, we write</text>
<formula><location><page_8><loc_40><loc_78><loc_76><loc_79></location>U ( t ) = d N λ/dt N . (17)</formula>
<text><location><page_8><loc_16><loc_75><loc_25><loc_76></location>We also write</text>
<formula><location><page_8><loc_37><loc_69><loc_76><loc_73></location>dQ/dt = ( ∂Q/∂λ )( dλ/dt ) , = -dλ/dt. (18)</formula>
<text><location><page_8><loc_16><loc_63><loc_76><loc_69></location>In the first line we have neglected the time dependence of other parameters besides λ that enter into Q under the assumption that the control system timescale is shorter than the timescales of the quantities that we want to control. In the second line we have used Eq. (13) and we have assumed that Q is small. Similarly we write</text>
<formula><location><page_8><loc_33><loc_52><loc_76><loc_62></location>d 2 Q/dt 2 = ∂Q ∂λ d 2 λ dt 2 + ∂ 2 Q ∂λ 2 ( dλ dt ) 2 , = ∂Q ∂λ d 2 λ dt 2 , = -d 2 λ dt 2 , (19)</formula>
<text><location><page_8><loc_16><loc_46><loc_76><loc_51></location>where in the second line we have retained only terms of linear order in dQ/dt (and therefore in dλ/dt ). For higher derivatives we continue to retain only terms linear in Q and its derivatives, so from Eq. (17) we obtain</text>
<formula><location><page_8><loc_39><loc_41><loc_76><loc_45></location>U ( t ) = d N λ/dt N , = -d N Q/dt N . (20)</formula>
<text><location><page_8><loc_19><loc_39><loc_70><loc_40></location>For the PID controller and N = 2, combining Eqs. (15) and (20) yields</text>
<formula><location><page_8><loc_32><loc_34><loc_76><loc_38></location>-d 2 Q dt 2 = a 0 ∫ Q ( t ) dt + a 1 Q ( t ) + a 2 dQ dt . (21)</formula>
<text><location><page_8><loc_16><loc_31><loc_76><loc_34></location>If we choose a 0 = 1 /τ 3 d , a 1 = 3 /τ 2 d , and a 2 = 3 /τ d , then the solution to Eq. (21) will be exponentially damped on the timescale τ d ,</text>
<formula><location><page_8><loc_42><loc_27><loc_76><loc_29></location>Q ∝ e -t/τ d . (22)</formula>
<text><location><page_8><loc_16><loc_22><loc_76><loc_26></location>The same exponential damping holds for the PD controller, Eq. (16), for appropriate choices of the parameters a k . For K = 2 and N = 3, the parameters a k are identical to those in the PID case above.</text>
<section_header_level_1><location><page_8><loc_16><loc_19><loc_33><loc_20></location>3.2. Averaging out noise</section_header_level_1>
<text><location><page_8><loc_16><loc_14><loc_76><loc_18></location>The PID controller, Eq. (15), is computed by measuring the control error Q , its time integral, and its time derivative. The PD controller, Eq. (16), may require multiple derivatives of Q ( t ) depending on the order K . Generally only Q , and not its derivatives</text>
<text><location><page_9><loc_16><loc_83><loc_76><loc_88></location>or integrals, is available in the code at any given time step. The simplest way to compute the derivatives of Q is by finite differencing in time, and the simplest way to compute the integral is by a numerical quadrature.</text>
<text><location><page_9><loc_16><loc_79><loc_76><loc_83></location>We measure the control error Q at time intervals of length τ m , where we choose τ m < t i +1 -t i . The measuring time interval τ m is then used as the time step for finite difference stencils and quadratures.</text>
<text><location><page_9><loc_16><loc_64><loc_76><loc_79></location>The measured Q is typically a function of apparent horizon locations or shapes, and this measured Q includes noise caused by the finite resolution of the evolution, and the finite residual and finite number of iterations of the apparent horizon finder. Taking a numerical derivative of Q amplifies the noise, and then this noise is transferred to the control signal via Eqs. (15) or (16), and then to the map. If the noise amplitude is too large, the control system will become unstable. The PID controller generally handles noise better than the PD controller for two reasons. First, each successive numerical derivative amplifies noise even further, so using only one derivative instead of two (or more) results in a more accurate control signal. Second, the inclusion of an integral term acts to further smooth the control signal.</text>
<text><location><page_9><loc_16><loc_53><loc_76><loc_63></location>In some cases, however, the noise in Q can be problematic even for the PID controller. In these cases we implement direct averaging of the control error in one of two ways: 1) we perform a polynomial fit of order N to the previous M measurements of the control error, where M > N , or 2) we perform an exponentially-weighted average, with timescale τ avg , of all previous control error measurements and their derivatives and integrals. The latter is our preferred method, which we describe in detail in Appendix A.</text>
<section_header_level_1><location><page_9><loc_16><loc_50><loc_41><loc_51></location>3.3. Dynamic timescale adjustment</section_header_level_1>
<text><location><page_9><loc_16><loc_32><loc_76><loc_49></location>In the previous section we have identified four timescales relevant for each control system. The first is the damping timescale τ d ; this describes how quickly the control error Q ( t ) falls to zero, and therefore how quickly the map parameter λ ( t ) approaches its desired value. The second timescale is the control interval ∆ t i := t i +1 -t i , which represents how often the N th derivative of the function λ ( t ) in Eq. (11) is updated. The third is the measurement timescale τ m , which indicates how often the control error Q is measured. The fourth is the averaging timescale τ avg , which is used to smooth the control error Q (and its derivatives and integrals) for use in computing the control signal U ( t ). These timescales are not all independent; for example we have assumed ∆ t i < τ d in deriving Eq. (21), and we have assumed τ m < ∆ t i so that we can obtain smooth measurements of the derivatives of Q .</text>
<text><location><page_9><loc_16><loc_26><loc_76><loc_32></location>Because binary black hole evolutions are nonlinear dynamic systems, we adjust the damping timescale, τ d , throughout the simulation. We then set the three other timescales, ∆ t i , τ m , and τ avg , based on the current value of the damping timescale τ d , as we now describe.</text>
<text><location><page_9><loc_16><loc_14><loc_76><loc_26></location>The timescale τ d should be shorter than the timescale on which the physical system changes; otherwise, the control system cannot adjust the map parameters quickly enough to respond to changes in the system. But if the timescale τ d is too small, then the measurement timescale τ m on which we compute the apparent horizon must also be small, meaning that frequent apparent horizon computations are needed; this is undesirable because computing apparent horizons is computationally expensive. We would like to adjust τ d in an automatic way so that it is relatively large during the binary black hole inspiral, decreases during the plunge and merger, and increases</text>
<text><location><page_10><loc_16><loc_83><loc_76><loc_88></location>again as the remnant black hole rings down. In the canonical language of control theory, we would like to implement 'gain scheduling' [19], tuning the behavior of the control system for different operating regimes.</text>
<text><location><page_10><loc_16><loc_76><loc_76><loc_83></location>We do this as follows: For all map parameters (except the /lscript = 0 component of the horizon shape map, which is treated differently; see Sec. 5.3), the damping timescale is a generic function of Q and ˙ Q , i.e., the error in its associated map parameter and its derivative. Whenever we adjust the control signal U ( t ) at interval t i , we also tune the timescale in the following way:</text>
<formula><location><page_10><loc_42><loc_73><loc_76><loc_75></location>τ i +1 d = βτ i d , (23)</formula>
<text><location><page_10><loc_16><loc_71><loc_26><loc_72></location>where typically</text>
<formula><location><page_10><loc_25><loc_65><loc_76><loc_70></location>β = 0 . 99 , if ˙ Q/Q > -1 / 2 τ d and | Q | or | ˙ Qτ d | > Q Max t 1 . 01 , if | Q | < Q Min t and | ˙ Qτ d | < Q Min t 1 , otherwise . (24)</formula>
<text><location><page_10><loc_16><loc_55><loc_76><loc_66></location> Here Q Min t and Q Max t are thresholds for the control error Q . The idea is to keep | Q | < Q Max t so that the map parameters are close to their desired values, but to also keep | Q | > Q Min t because an unnecessarily small | Q | means unnecessarily small timescales and therefore a large computational expense (because the apparent horizon must be found frequently). For binary black hole simulations where the holes have masses M A and M B , we find that the following choices work well:</text>
<formula><location><page_10><loc_36><loc_50><loc_76><loc_54></location>Q Max t = 2 × 10 -3 M A /M B + M B /M A (25)</formula>
<formula><location><page_10><loc_36><loc_47><loc_76><loc_50></location>Q Min t = 1 4 Q Max t . (26)</formula>
<text><location><page_10><loc_16><loc_42><loc_76><loc_46></location>Once we have adjusted the timescale τ d for every control system, we then use these timescales to choose the times t i +1 in Eq. (11) at which we update the polynomial coefficients λ n i in the map parameter expression λ ( t ),</text>
<formula><location><page_10><loc_35><loc_39><loc_76><loc_41></location>∆ t i := t i +1 -t i = α d min( τ d ) , (27)</formula>
<text><location><page_10><loc_16><loc_31><loc_76><loc_38></location>where typically α d = 0 . 3, and the minimum is taken over all map parameters (except for the /lscript = 0 component of the horizon shape map). This ensures that the coefficients λ n i , are updated faster than the physical system is changing, and faster than the control system is damping. For α d too large, we find that the control system becomes unstable.</text>
<text><location><page_10><loc_16><loc_22><loc_76><loc_31></location>We also use the timescale τ d to choose the interval τ m at which we measure the control error. For many map parameters, the associated control error is a function of apparent horizon quantities, which is why we desire τ m to be as large as possible. But a certain number of measurements are needed for each ∆ t i so that the control signals, defined in Eqs. (15) and (16), are sufficiently accurate and the control system is stable. We choose</text>
<formula><location><page_10><loc_41><loc_20><loc_76><loc_22></location>τ m = α m ∆ t i , (28)</formula>
<text><location><page_10><loc_16><loc_13><loc_76><loc_19></location>where α m is typically between 0 . 25 and 0 . 3. In other words, we measure the control error three or four times before we update the control signal. This also ensures that the averaging timescale is greater than the measurement timescale, as we typically choose τ avg ∼ 0 . 25 τ d .</text>
<section_header_level_1><location><page_11><loc_16><loc_86><loc_39><loc_88></location>4. Control Systems for Maps</section_header_level_1>
<text><location><page_11><loc_16><loc_78><loc_76><loc_85></location>In a SpEC evolution, we transform the grid coordinates, x i , into inertial coordinates, ¯ x i , through a series of elementary maps as in [1, 2, 3, 4, 5]. Several maps have been added and many improvements have been made since their original introduction. The full transformation is ¯ x i = M x i , where</text>
<formula><location><page_11><loc_31><loc_74><loc_76><loc_78></location>M = M Translation · M Rotation · M Scaling ·M Skew · M CutX · M Shape . (29)</formula>
<text><location><page_11><loc_16><loc_68><loc_76><loc_73></location>Below we will describe each of these maps and how we measure the error in their parameters. Before we do this, however, we describe our domain decomposition and how we measure apparent horizons, because information from the grid and the horizons is used to determine the maps.</text>
<text><location><page_11><loc_16><loc_59><loc_76><loc_68></location>In the grid coordinates x i , the domain decomposition looks like Fig. 4. There are two excision boundaries, A and B , which are spheres in grid coordinates. The gridcoordinate centers of these excision boundaries we will call C i H , where H is either A or B . The excision boundaries (and therefore C i H ) remain fixed throughout the evolution. The purpose of many of the maps is to move the mapped centers ¯ C i H := M ( C i H ) along with the centers of the apparent horizons.</text>
<text><location><page_11><loc_16><loc_56><loc_76><loc_58></location>We measure the apparent horizons in an intermediate frame ˆ x i , which we call the distorted frame . This frame is connected to the grid frame by the map</text>
<formula><location><page_11><loc_35><loc_52><loc_76><loc_54></location>M Distortion = M CutX · M Shape . (30)</formula>
<text><location><page_11><loc_16><loc_47><loc_76><loc_51></location>We will discuss exactly what M Shape and M CutX are below, but for now we need only demand that M Distortion has two properties: The first is that it leaves the centers of the excision boundaries invariant, i.e.,</text>
<formula><location><page_11><loc_35><loc_43><loc_76><loc_46></location>ˆ C i H := M Distortion ( C i H ) = C i H . (31)</formula>
<text><location><page_11><loc_16><loc_38><loc_76><loc_43></location>The second property is that at each excision boundary H , M Distortion leaves angles invariant. That is, if we define grid-frame polar coordinates ( r H , θ H , φ H ) centered about excised region H in the usual way,</text>
<formula><location><page_11><loc_36><loc_35><loc_76><loc_37></location>x 0 = C 0 H + r H sin θ H cos φ H , (32)</formula>
<formula><location><page_11><loc_36><loc_33><loc_76><loc_35></location>x 1 = C 1 H + r H sin θ H sin φ H , (33)</formula>
<formula><location><page_11><loc_36><loc_31><loc_76><loc_33></location>x 2 = C 2 H + r H cos θ H , (34)</formula>
<text><location><page_11><loc_16><loc_27><loc_76><loc_30></location>and similarly for polar coordinates (ˆ r H , ˆ θ H , ˆ φ H ) centered about ˆ C i H in the ˆ x i frame, then the second property means that</text>
<formula><location><page_11><loc_42><loc_22><loc_76><loc_26></location>ˆ θ H = θ H , ˆ φ H = φ H , (35)</formula>
<text><location><page_11><loc_16><loc_18><loc_76><loc_21></location>on the excision boundary. Note that the index i on spatial coordinates ranges over (0 , 1 , 2) in this paper.</text>
<text><location><page_11><loc_16><loc_15><loc_76><loc_18></location>We find the apparent horizons in the frame ˆ x i by a fast flow method [20]. Each horizon is represented as a smooth surface in this frame, and is parameterized in terms</text>
<text><location><page_12><loc_16><loc_85><loc_76><loc_88></location>of polar coordinates centered around ˆ C i H : the radius of the horizon at each ( ˆ θ H , ˆ φ H ) is given by</text>
<formula><location><page_12><loc_33><loc_81><loc_76><loc_85></location>ˆ r AH H ( ˆ θ H , ˆ φ H ) = ∑ /lscriptm ˆ S H /lscriptm Y /lscriptm ( ˆ θ H , ˆ φ H ) , (36)</formula>
<text><location><page_12><loc_16><loc_77><loc_76><loc_81></location>where again the index H labels which excision boundary is enclosed by the apparent horizon. Here Y /lscriptm ( ˆ θ H , ˆ φ H ) are spherical harmonics and ˆ S H /lscriptm are expansion coefficients that describe the shape of the apparent horizon.</text>
<text><location><page_12><loc_16><loc_69><loc_76><loc_77></location>We search for apparent horizons in the distorted frame ˆ x i rather than in the grid frame x i or the inertial frame ¯ x i because this simplifies the formulae for the control systems. In particular, this choice decouples the control errors of the shape map Q /lscriptm (defined in Sec. 4.5, Eq. (77)) for different values of ( /lscript, m ), and it decouples the errors Q /lscriptm from the control errors of the maps connecting the distorted and inertial frames.</text>
<text><location><page_12><loc_19><loc_68><loc_65><loc_69></location>For each surface H , we define the center of the apparent horizon</text>
<formula><location><page_12><loc_38><loc_61><loc_76><loc_67></location>ˆ ξ i H = ∫ H ˆ x i (ˆ r AH H ) 2 d ˆ Ω ∫ H (ˆ r AH H ) 2 d ˆ Ω , (37)</formula>
<text><location><page_12><loc_16><loc_59><loc_76><loc_62></location>where the integrals are over the surface. In the code, it suffices to use the following approximation of Eq. (37), which becomes exact as the surface becomes spherical:</text>
<formula><location><page_12><loc_36><loc_53><loc_76><loc_58></location>ˆ ξ 0 H = C 0 H -√ 3 / 2 π /Rfractur ( ˆ S H 11 ) , (38) ˆ ξ 1 H = C 1 H + 3 / 2 π /Ifractur ( ˆ S H 11 ) , (39)</formula>
<text><location><page_12><loc_16><loc_40><loc_76><loc_51></location>Here we have used the property ˆ C i H = C i H , Eq. (31). We distinguish the center C i H of the excision boundary from the center ˆ ξ i H of the corresponding apparent horizon. The former is fixed in time in the grid frame, but the latter will change as the metric quantities evolve. The purpose of several of the maps (namely M Scaling , M Rotation , and M Translation described below) is to ensure that ˆ ξ i H -C i H is driven toward zero; i.e., to ensure that the centers of the excision boundaries track the centers of the apparent horizons.</text>
<formula><location><page_12><loc_36><loc_50><loc_76><loc_56></location>√ ˆ ξ 2 H = C 2 H + √ 3 / 4 π ˆ S H 10 . (40)</formula>
<text><location><page_12><loc_16><loc_29><loc_76><loc_40></location>We now describe each of the maps comprising M and how we measure the error in their parameters. The order in which we discuss the maps is not the same as the order in which the maps are composed so that we can discuss the simplest maps first. To produce less cluttered equations in the following descriptions of the maps, we omit accents on variables that represent specific frames (i.e. we just write x instead of ˆ x or ¯ x ) whenever the input or output frame of the map is unambiguous or explicitly stated.</text>
<section_header_level_1><location><page_12><loc_16><loc_26><loc_24><loc_27></location>4.1. Scaling</section_header_level_1>
<text><location><page_12><loc_16><loc_21><loc_76><loc_25></location>The scaling map, M Scaling , causes the grid to shrink or expand (and the excision boundaries to move respectively closer together or farther apart) in the inertial frame, thereby allowing the grid to follow the two black holes as their separation changes.</text>
<text><location><page_12><loc_16><loc_16><loc_76><loc_20></location>This map transforms the radial coordinate with respect to the origin (i.e., the center of the outermost sphere in Fig. 4), such that the region near the black holes is scaled uniformly by a factor a and the outer boundary is scaled by a factor b ,</text>
<formula><location><page_12><loc_36><loc_13><loc_76><loc_15></location>R ↦→ aR +( b -a ) R 3 /R 2 OB . (41)</formula>
<text><location><page_13><loc_16><loc_83><loc_76><loc_88></location>Here R is the radial coordinate, R OB is the radius at the outer boundary, a is a parameter that will be determined by a control system, and b is another parameter that will be determined empirically.</text>
<text><location><page_13><loc_16><loc_73><loc_76><loc_83></location>In [1], the scaling map was simply x i ↦→ a ( t ) x i , which is recovered by Eq. (41) if b = a . In that case, as the black holes inspiral together and a decreases, the outer boundary of the grid decreases as well. For long evolutions, the outer boundary decreases so much that we can no longer extract gravitational radiation far from the hole. The addition of b to the map alleviates this difficulty, allowing the motion of the outer boundary to be decoupled from the motion of the holes. We choose b by an explicit functional form</text>
<formula><location><page_13><loc_35><loc_69><loc_76><loc_72></location>b ( t ) = 1 -10 -6 t 3 / (2500 + t 2 ) , (42)</formula>
<text><location><page_13><loc_16><loc_60><loc_76><loc_69></location>which is designed to keep the outer boundary from shrinking rapidly, but to allow the boundary to move inward at a small speed, so that zero-speed modes are advected off the grid (and thus need no boundary condition imposed on them). We choose a cubic function of time in Eq. (42) because we have found that at least the first two time derivatives of b ( t ) must vanish initially, or else a significant ingoing pulse of constraint violations is produced at the outer boundary.</text>
<text><location><page_13><loc_19><loc_58><loc_65><loc_60></location>We control the scale factor a of this map using the error function</text>
<formula><location><page_13><loc_40><loc_55><loc_76><loc_57></location>Q a = a ( dx 0 -1) , (43)</formula>
<text><location><page_13><loc_16><loc_53><loc_20><loc_54></location>where</text>
<formula><location><page_13><loc_40><loc_49><loc_76><loc_53></location>dx i = ˆ ξ i A -ˆ ξ i B C 0 A -C 0 B . (44)</formula>
<text><location><page_13><loc_16><loc_45><loc_76><loc_49></location>We assume that the separation vector C i A -C i B in the grid frame is parallel to the x -axis. The idea is that the distorted-frame separation of the horizon centers along the x -axis is driven to be the same as the separation of the excision boundary centers.</text>
<text><location><page_13><loc_16><loc_36><loc_76><loc_45></location>To show that the control system for a obeys Eq. (13), we consider the change in Q a under variations of a with all other maps held fixed, and with the inertial-coordinate centers ¯ ξ i H of the horizons held fixed. ‡ This means that the distorted-frame centers of the horizons ˆ ξ i H appearing in Eq. (44) change with variations of a . The second term in Eq. (41) is small when evaluated near the horizon where ( R/R OB ) 2 /lessmuch 1, so we can write the action of the scaling map as</text>
<formula><location><page_13><loc_42><loc_33><loc_76><loc_35></location>¯ ξ i H = a ˆ ξ i H , (45)</formula>
<text><location><page_13><loc_16><loc_30><loc_40><loc_32></location>and therefore under variations δa ,</text>
<formula><location><page_13><loc_37><loc_28><loc_76><loc_29></location>0 = δ ¯ ξ i H = ˆ ξ i H δa + aδ ˆ ξ i H . (46)</formula>
<text><location><page_13><loc_16><loc_23><loc_76><loc_27></location>Taking variations of Eq. (43), using Eq. (46) to substitute for δ ˆ ξ i H , and noting that the excision-boundary centers C i H are constants and do not vary with a , we obtain</text>
<formula><location><page_13><loc_42><loc_20><loc_76><loc_22></location>δQ a = -δa, (47)</formula>
<text><location><page_13><loc_16><loc_18><loc_35><loc_19></location>so that Eq. (13) is satisfied.</text>
<text><location><page_14><loc_16><loc_68><loc_76><loc_88></location>To verify that the more general decoupling equation, Eq. (14), is satisfied for the scaling map, we must show, in addition to Eq. (47), that the quantity dx 0 defined in Eq. (44) is invariant under the action of the other maps, at least in the limit that all the control errors Q are small. We consider each map in turn. The translation map moves both apparent horizons together, so it leaves dx 0 invariant. Changes in the rotation map parameters will change dx 0 only by an amount proportional to the control errors of the rotation map (see Eqs. (53) and (54) below). The skew map (below) leaves the centers of the excision boundaries invariant. Because the intent of the rotation, translation, and scaling maps is to drive the centers of the apparent horizons toward the centers of the excision boundaries, this means that the skew map changes dx 0 by an amount proportional to the control errors of the rotation, translation, and scaling maps. Finally, the shape and CutX maps connect the distorted and grid frames, so they cannot affect dx 0 .</text>
<section_header_level_1><location><page_14><loc_16><loc_65><loc_25><loc_66></location>4.2. Rotation</section_header_level_1>
<text><location><page_14><loc_16><loc_61><loc_76><loc_64></location>The rotation map, M Rotation , is a rigid 3D rotation about the origin that tracks the orbital phase and precession of the system,</text>
<formula><location><page_14><loc_23><loc_54><loc_76><loc_60></location>x i ↦→ R i j x j , where R = cos ϑ cos ϕ -sin ϑ cos ϑ sin ϕ sin ϑ cos ϕ cos ϑ sin ϑ sin ϕ -sin ϕ 0 cos ϕ . (48)</formula>
<text><location><page_14><loc_16><loc_47><loc_76><loc_54></location>The pitch and yaw map parameters ( ϕ, ϑ ) are controlled so as to align the line segment connecting the apparent horizon centers with the distorted-frame x -axis. Note that the map parameters ( ϕ, ϑ ) are functions of time, and are not to be confused with the polar coordinates ( φ H , θ H ) centered about each excision boundary. Then the control error is given by</text>
<formula><location><page_14><loc_38><loc_41><loc_76><loc_45></location>Q ϕ = -ˆ ξ 2 A -ˆ ξ 2 B ˆ ξ 0 A -ˆ ξ 0 B (49)</formula>
<formula><location><page_14><loc_38><loc_37><loc_76><loc_41></location>Q ϑ = ˆ ξ 1 A -ˆ ξ 1 B ( ˆ ξ 0 A -ˆ ξ 0 B ) cos ϕ . (50)</formula>
<text><location><page_14><loc_16><loc_32><loc_76><loc_36></location>In the case of extreme precession where ϕ → π/ 2, these equations are insufficient because Q ϑ diverges. Our solution is to use quaternions, which avoid this singularity (for a complete discussion, see [21]).</text>
<text><location><page_14><loc_16><loc_28><loc_76><loc_32></location>The control errors Q ϑ and Q ϕ obey Eq. (14). To show this, consider variations of ϑ and ϕ with other maps held fixed, and with the inertial-coordinate centers ¯ ξ i H of the horizons held fixed. The rotation map, Eq. (48), implies that under these variations,</text>
<formula><location><page_14><loc_35><loc_24><loc_76><loc_26></location>δ ¯ ξ i H = 0 = R i j δ ˆ ξ j H +( δ R ) i j ˆ ξ j H . (51)</formula>
<text><location><page_14><loc_16><loc_21><loc_47><loc_23></location>Multiplying this equation by R -1 we obtain</text>
<formula><location><page_14><loc_38><loc_18><loc_76><loc_21></location>δ ˆ ξ i H = -( R -1 δ R ) i j ˆ ξ j H , (52)</formula>
<text><location><page_15><loc_16><loc_86><loc_24><loc_88></location>which yields</text>
<formula><location><page_15><loc_19><loc_79><loc_76><loc_85></location>∂ ˆ ξ i H ∂ϕ = A i j ˆ ξ j H , where A := -R -1 ∂ R ∂ϕ = 0 0 -1 0 0 0 1 0 0 (53)</formula>
<text><location><page_15><loc_16><loc_70><loc_76><loc_74></location>We can now verify Eq. (14) for the special case where the indices a and b in Eq. (14) are either ϑ and ϕ . This result is obtained in a straightforward way by differentiating Eqs. (49) or (50) with respect to ϑ or ϕ , and substituting Eqs. (53) or (54).</text>
<formula><location><page_15><loc_19><loc_74><loc_76><loc_80></location>∂ ˆ ξ i H ∂ϑ = B i j ˆ ξ j H , where B := -R -1 ∂ R ∂ϑ = 0 cos ϕ 0 -cos ϕ 0 -sin ϕ 0 sin ϕ 0 . (54)</formula>
<text><location><page_15><loc_16><loc_53><loc_76><loc_70></location>In addition, the control errors Q ϕ and Q ϑ are independent (to leading order in the control errors) of changes in the parameters of the other maps that make up M : Variations of Eqs. (49) and (50) with respect to the scaling map parameter a are zero, because both the numerator and denominator of Q ϕ and Q ϑ scale in the same way with a . Similarly, Q ϕ and Q ϑ are independent of the translation map, since both apparent horizons are translated by the same amount. The skew map can change Q ϕ and Q ϑ , but only by an amount proportional to control errors, because the skew map leaves C i H invariant and other maps ensure that ˆ ξ i H are close (within a control error) to C i H . The shape and CutX maps cannot affect Q ϕ and Q ϑ because those maps connect the grid and distorted frames (and therefore they cannot change the distorted-frame horizon centers ˆ ξ i H ).</text>
<section_header_level_1><location><page_15><loc_16><loc_50><loc_27><loc_51></location>4.3. Translation</section_header_level_1>
<text><location><page_15><loc_16><loc_46><loc_76><loc_49></location>The translation map, M Translation , transforms the Cartesian coordinates, x i , according to</text>
<formula><location><page_15><loc_39><loc_44><loc_76><loc_46></location>x i ↦→ x i + f ( R ) T i , (55)</formula>
<text><location><page_15><loc_16><loc_39><loc_76><loc_44></location>where f ( R ) is a Gaussian centered on the origin with a width set such that f ( R ) falls off to machine precision at the outer boundary radius, and T i ( t ) are translation parameters that are adjusted by a control system.</text>
<text><location><page_15><loc_16><loc_26><loc_76><loc_39></location>The translation map moves the grid to account for any drift of the 'center of mass' of the system (as computed assuming point masses at the apparent horizon centers) in the inertial frame. This drift can be caused by momentum exchange between the black holes and the surrounding gravitational field [22, 23], by anisotropic radiation of linear momentum to infinity, or by linear momentum in the initial data. This is the third map (the other two are rotation and scaling) that drives the centers of the apparent horizons toward the centers of the excision boundaries. The apparent horizon centers ˆ ξ i A and ˆ ξ i B represent six degrees of freedom: one is fixed by the scaling map, two by rotation, and three by translation.</text>
<text><location><page_15><loc_16><loc_21><loc_76><loc_25></location>The control errors will be more complicated than for the other control systems because translation and rotation do not commute. We define the control errors Q i T for each of the translation directions i = 0 , 1 , 2 as</text>
<formula><location><page_15><loc_35><loc_16><loc_76><loc_20></location>Q i T = a R i j [ ˆ ξ j B + P j k ( ˆ ξ k A -ˆ ξ k B ) ] , (56)</formula>
<text><location><page_15><loc_16><loc_13><loc_76><loc_16></location>where R is the rotation matrix in Eq. (48), and P is some matrix yet to be determined, which may depend on the rotation parameters ( ϕ, ϑ ) and on the constants C i H , but</text>
<text><location><page_16><loc_16><loc_83><loc_76><loc_88></location>which may not depend on the translation parameters T i . This control error must have the property that Q i T = 0 when ˆ ξ i H = C i H , so our first restriction on P is that it satisfies</text>
<formula><location><page_16><loc_37><loc_81><loc_76><loc_83></location>0 = C i B + P i j ( C j A -C j B ) . (57)</formula>
<text><location><page_16><loc_16><loc_77><loc_76><loc_81></location>To check Eq. (14), we note that near the black holes we can neglect the last term in Eq. (41) and we can use f ( R ) ∼ 1 in Eq. (55), so that the apparent horizon centers in the inertial and distorted frames are related by</text>
<formula><location><page_16><loc_39><loc_73><loc_76><loc_75></location>a R i j ˆ ξ j H = ¯ ξ i H -T i . (58)</formula>
<text><location><page_16><loc_16><loc_71><loc_46><loc_72></location>Inserting Eq. (58) into Eq. (56), we obtain</text>
<formula><location><page_16><loc_32><loc_67><loc_76><loc_70></location>Q i T = ¯ ξ i B -T i -( RPR -1 ) i j ( ¯ ξ j A -¯ ξ j B ) . (59)</formula>
<text><location><page_16><loc_16><loc_64><loc_76><loc_67></location>The second term in Eq. (59) is the only term that depends on the translation parameter T i , so we have</text>
<formula><location><page_16><loc_42><loc_61><loc_76><loc_64></location>∂Q i T ∂T j = -δ ij (60)</formula>
<text><location><page_16><loc_16><loc_53><loc_76><loc_60></location>When varying map parameters, the inertial-frame horizon centers remain fixed, so the only other term in Eq. (59) that depends on map parameters is the last term, which depends on ( ϑ, ϕ ) because of the rotation matrices and because of the ( ϑ, ϕ ) dependence in P . We can therefore write the variation of Q i T with respect to ( ϑ, ϕ ) as</text>
<formula><location><page_16><loc_32><loc_49><loc_76><loc_52></location>∂Q i T ∂W = -∂ ∂W ( RPR -1 ) i j ( ¯ ξ j A -¯ ξ j B ) , (61)</formula>
<formula><location><page_16><loc_36><loc_46><loc_76><loc_49></location>= -∂ ∂W ( RPR -1 ) i j a R j k ( ˆ ξ k A -ˆ ξ k B ) , (62)</formula>
<text><location><page_16><loc_16><loc_34><loc_76><loc_45></location>where W stands for either ϑ or ϕ , and where in the last line we have used Eq. (58). To obey Eq. (14), ∂Q i T /∂W must either be zero, or on the order of a control error Q . For i = 1 , 2, the quantity ( ˆ ξ i A -ˆ ξ i B ) is proportional to the control error of the rotation map, Eqs. (49) and (50), so these terms can be neglected in Eq. (62). However, for i = 0, ( ˆ ξ i A -ˆ ξ i B ) is not proportional to a control error; instead ( ˆ ξ 0 A -ˆ ξ 0 B ) is driven to a constant finite value of C 0 A -C 0 B by the scaling control system. Therefore, in order to satisfy Eq. (14), we require</text>
<formula><location><page_16><loc_35><loc_27><loc_76><loc_33></location>-∂ ∂W ( RPR -1 ) R 1 0 0 = 0 . (63)</formula>
<text><location><page_16><loc_16><loc_26><loc_62><loc_27></location>We find that we can satisfy both Eqs. (57) and (63) by choosing</text>
<formula><location><page_16><loc_28><loc_18><loc_76><loc_24></location>P = 1 C 0 B -C 0 A C 0 B -C 1 B -C 2 B C 1 B C 0 B + C 2 B tan ϕ 0 C 2 B -C 1 B tan ϕ C 0 B . (64)</formula>
<text><location><page_16><loc_16><loc_15><loc_76><loc_18></location>Here we have again assumed that the separation between the centers of the excision boundaries is parallel to the x -axis, i.e., C 1 A = C 1 B and C 2 A = C 2 B .</text>
<figure>
<location><page_17><loc_26><loc_57><loc_66><loc_88></location>
<caption>Figure 4. Two-dimensional projection of the domain decomposition near the two black holes A and B. Shown are boundaries between subdomains. Each subdomain takes the shape of a spherical shell, a distorted cylindrical shell, or a distorted cylinder. Additional spherical-shell subdomains (not shown) surround the outer boundary of the figure and extend to large radius. This domain decomposition is explained in detail in the Appendix of [5]. The red, magenta, and blue surfaces are those for which the function f A ( r A , θ A , φ A ) from Eq. (72) has a discontinuous gradient, as described in Sec. 4.5.</caption>
</figure>
<section_header_level_1><location><page_17><loc_16><loc_42><loc_23><loc_43></location>4.4. Skew</section_header_level_1>
<text><location><page_17><loc_16><loc_37><loc_76><loc_41></location>In Fig. 4, a prominent feature of the domain decomposition is a plane (a vertical line in the two-dimensional figure) that is perpendicular to the grid-frame x -axis and lies between the two excision boundaries A and B . We call this plane the 'cutting plane'.</text>
<text><location><page_17><loc_16><loc_28><loc_76><loc_37></location>The skew map, M Skew , acts on the distorted-frame coordinates, in which the cutting plane is still perpendicular to the x -axis. The skew map leaves the coordinates y, z unchanged, but changes the x -coordinate in order to give a skewed shape to the cutting plane, as shown in Fig. 5. Let x i C be the intersection point of the line segment connecting the excision boundary centers and the cutting plane. The action of the skew map is defined as</text>
<formula><location><page_17><loc_31><loc_22><loc_76><loc_26></location>x 0 ↦→ x 0 + W ∑ j =1 , 2 tan [ Θ j ( t ) ] · ( x j -x j C ) , (65)</formula>
<formula><location><page_17><loc_31><loc_21><loc_76><loc_23></location>x j ↦→ x j , for j = 1 , 2 (66)</formula>
<text><location><page_17><loc_16><loc_14><loc_76><loc_20></location>where W is a radial Gaussian function centered around x i C , and the angles Θ j ( t ) are the time-dependent map parameters. For j = 1 , 2 the parameter Θ j ( t ) is the angle between the mapped and unmapped x j -axis when projected into the x 3 -j = x 3 -j C plane. Note that the line intersecting x i C and parallel to the x -axis is left invariant by</text>
<figure>
<location><page_18><loc_26><loc_57><loc_65><loc_88></location>
<caption>Figure 5. Snapshot of the grid, viewed in the mapped frame, from a binary black hole evolution with M A = 8 M B shortly before the common horizon forms. Left: without M Skew . Right: with M Skew . The gray areas are excised regions. In the left panel, the grid is compressed as the excision boundaries track increasingly skewed apparent horizons, and the evolution is terminated because of excessively large constraint violation in the compressed region. This problem is resolved by the inclusion of M Skew .</caption>
</figure>
<text><location><page_18><loc_16><loc_40><loc_76><loc_45></location>the skew map § . The width of the Gaussian W is set such that W is below machine precision at the innermost wave extraction sphere. This implies that the sphericalshell subdomains used to evolve the metric in the wave zone will not be affected by the skew map.</text>
<text><location><page_18><loc_16><loc_31><loc_76><loc_39></location>The purpose of the skew map is to (as much as possible) align the cutting plane with the surfaces of the apparent horizons in the region where the surfaces are closest to the cutting plane. We derive the control system in charge of setting the parameters Θ j by the following condition: We demand that the angle between the mapped cutting plane and the x -axis at x i C be driven to the (weighted) average of the angles at which the mapped apparent horizons intersect the same x -axis.</text>
<text><location><page_18><loc_16><loc_18><loc_76><loc_30></location>The input coordinates to the skew map are the coordinates in the distorted frame. For each horizon, we therefore measure an angle in the distorted frame as follows: Let x Int H be the distorted-frame x -coordinate at which apparent horizon H intersects the line segment connecting the centers of the excision boundaries. For j = 1 , 2, we calculate the normal to the surface at this intersection point. This normal is projected into the x 3 -j = x 3 -j C plane. We define Θ j H as the angle between the projected normal and the x -axis. Thus, projecting the normal into the y = const . plane gives Θ z H and vice versa (see Fig. 6). We then compute a weighted average of the Θ j H such that the</text>
<figure>
<location><page_19><loc_33><loc_74><loc_58><loc_88></location>
<caption>Figure 6. A diagram of the skew map in the mapped frame. The horizon surfaces are drawn in black and the excision surfaces are drawn in red. The dotted red line represents the line segment connecting the excision surface centers, which is parallel to the x -axis. The normal to either horizon at the intersection point with this segment is represented by the straight black lines. The cutting plane is represented by the (skewed) blue curve, and the normal to this plane at x i C is represented by the central green line. The green lines near the two excision surfaces are constructed parallel to the central green line, where parallelism is defined assuming a Euclidean background. There are three angles involved in the skew control system - the angle between the normal to the skewed cutting plane and the x -axis, Θ j , and the angle between the normal to either horizon and the normal to the skewed cutting plane, Θ j H . Skew control acts to minimize Θ j H . We measure Θ j H in the unmapped (distorted) frame, but Θ j H is invariant under the skew map for W ≈ 1.</caption>
</figure>
<text><location><page_19><loc_16><loc_50><loc_68><loc_52></location>horizon closer to the cutting plane has a larger effect on the skew angles,</text>
<formula><location><page_19><loc_34><loc_46><loc_76><loc_49></location>Θ j Avg = w A Θ j A + w B Θ j B w A W ( C A ) + w B W ( C B ) , (67)</formula>
<text><location><page_19><loc_16><loc_44><loc_42><loc_45></location>where w A , w B are averaging weights,</text>
<formula><location><page_19><loc_37><loc_38><loc_76><loc_43></location>w H = exp [ -x 0 C -x Int H x 0 C -C 0 H ] . (68)</formula>
<text><location><page_19><loc_16><loc_37><loc_52><loc_38></location>Here C i H is the center of the excision boundary H .</text>
<formula><location><page_19><loc_38><loc_24><loc_76><loc_28></location>∂ Θ j Avg ∂ Θ j = -1 + O ( Q ) , (69)</formula>
<text><location><page_19><loc_16><loc_28><loc_76><loc_37></location>We measure Θ j Avg in the distorted frame, and in this frame the cutting plane is always normal to the x -axis. Therefore, thinking about the desired result in the distorted frame, we see that the control system for the skew map should drive Θ j Avg to zero. This leads us to consider the following control error for the skew angles: Q j Θ = Θ j Avg . Assuming that the function W is unity near the apparent horizons, we find that</text>
<text><location><page_19><loc_16><loc_17><loc_76><loc_24></location>in agreement with Eq. (13). In deriving Eq. (69), it is helpful to observe that the partial derivative in Eq. (69) is taken with the inertial-frame apparent horizon held fixed. For a fixed inertial-frame horizon, the only map that can change the shape (as opposed to merely the center) of the distorted-frame horizon (and thus Θ j Avg ) is the skew map.</text>
<text><location><page_19><loc_16><loc_14><loc_76><loc_17></location>We do not use Q j Θ = Θ j Avg for the entire evolution, however, because at early times, when the coordinate distance between the apparent horizons is larger than</text>
<text><location><page_20><loc_16><loc_79><loc_76><loc_88></location>their combined radii, the skew map is not needed. Furthermore, the skew map can cause difficulties early during the run, especially during the 'junk radiation' phase when the horizons are oscillating in shape. For this reason, we gradually turn on the skew map as the black holes approach each other. This is done by defining a roll-on function g that is zero when the horizons are far apart, and one when they are close together. This roll-on function is defined as</text>
<formula><location><page_20><loc_32><loc_73><loc_76><loc_78></location>g = 1 2 [ 1 -tanh ( 10 x Int A -x Int B C 0 A -C 0 B -5 )] . (70)</formula>
<text><location><page_20><loc_16><loc_69><loc_76><loc_73></location>For values g < 10 -3 the skew map is turned off completely; this is not strictly necessary, but it saves some computation. Given the function g , we define the control error as</text>
<formula><location><page_20><loc_37><loc_66><loc_76><loc_69></location>Q j Θ = g Θ j Avg -(1 -g )Θ j . (71)</formula>
<text><location><page_20><loc_16><loc_63><loc_76><loc_66></location>This control error drives the skew angles to zero when the black holes are far apart and drives Θ j Avg to zero as they approach each other.</text>
<section_header_level_1><location><page_20><loc_16><loc_60><loc_29><loc_61></location>4.5. Shape control</section_header_level_1>
<text><location><page_20><loc_16><loc_57><loc_42><loc_59></location>We define the shape map M Shape as:</text>
<formula><location><page_20><loc_25><loc_52><loc_76><loc_56></location>x i ↦→ x i ( 1 -∑ H f H ( r H , θ H , φ H ) r H ∑ /lscriptm Y /lscriptm ( θ H , φ H ) λ H /lscriptm ( t ) ) . (72)</formula>
<text><location><page_20><loc_16><loc_30><loc_76><loc_51></location>The index H goes over each of the two excised regions A and B , and the map is applied to the grid-frame coordinates. The polar coordinates ( r H , θ H , φ H ) centered about excised region H are defined by Eqs. (32)-(34), the quantities Y /lscriptm ( θ H , φ H ) are spherical harmonics, and λ H /lscriptm ( t ) are expansion coefficients that parameterize the map near excision region H . The function f H ( r H , θ H , φ H ) is chosen to be unity near excision region H and zero near the other excision region, so that the distortion maps for the two black holes are decoupled. Specifically, f H ( r H , θ H , φ H ) is determined as illustrated in Fig. 4. For excision region A in the figure, f A ( r A , θ A , φ A ) = 1 between the excision boundary and the magenta surface, it falls linearly to zero between the magenta and red surfaces, and it is zero everywhere outside the red surface. This means that the gradient of f A ( r A , θ A , φ A ) is discontinuous on the red surface, the magenta surface, and the blue surfaces in the figure. Because we ensure that these discontinuities occur on subdomain boundaries, they cause no difficulty with using spectral methods. Around excision region B , f B ( r B , θ B , φ B ) is chosen similarly.</text>
<text><location><page_20><loc_16><loc_14><loc_76><loc_30></location>In previous implementations of the shape map [3], the functions f H ( r H , θ H , φ H ) were chosen to be smooth Gaussians centered around each excision boundary rather than to be piecewise linear functions. We find smooth Gaussians to be inferior for two reasons. The first is that piecewise linear functions are easier and faster to invert (the inverse map is required for interpolation to trial solutions during apparent horizon finding). The second is that for smooth Gaussians, it is necessary to choose the widths of the Gaussians sufficiently narrow so that the Gaussian for excision region A does not overlap the Gaussian for excision region B and vice versa, so that the maps and control systems for A and B remain decoupled. However, decreasing the width of the Gaussians increases the Jacobians of the map, producing coordinates that are stretched and squeezed nonuniformly. We found that this form of 'grid-stretching'</text>
<text><location><page_21><loc_16><loc_83><loc_76><loc_88></location>significantly increased the computational resources required to resolve the solution to a given level of accuracy. In other words, with smooth Gaussians we were forced to add computational resources just to resolve the large Jacobians.</text>
<text><location><page_21><loc_16><loc_77><loc_76><loc_83></location>A map very similar to Eq. (72) is also described in [4]. The difference compared to this work is the choice of f H ( r H , θ H , φ H ), which corresponds to a different choice of domain decomposition. The control system used to choose the map parameters λ H /lscriptm ( t ) in [4] is also different than what is described here.</text>
<text><location><page_21><loc_16><loc_73><loc_76><loc_77></location>We control the expansion coefficients λ H /lscriptm ( t ) of the shape map, Eq. (72), so that each excision boundary, as measured in the intermediate frame ( t, ˆ x i ), has the same shape as the corresponding apparent horizon. In other words, we desire</text>
<formula><location><page_21><loc_41><loc_68><loc_76><loc_72></location>ˆ r AH 〈 ˆ r AH 〉 = ˆ r EB 〈 ˆ r EB 〉 , (73)</formula>
<text><location><page_21><loc_16><loc_61><loc_76><loc_67></location>where for brevity we have dropped the index H that labels the excision boundary. Here the angle brackets mean averaging over angles, ˆ r AH ( ˆ θ, ˆ φ ) is the radial coordinate of the apparent horizon defined in Eq. (36), and ˆ r EB ( θ, φ ) is the radial coordinate of the excision boundary, which from Eq. (72) can be written</text>
<formula><location><page_21><loc_34><loc_56><loc_76><loc_60></location>ˆ r EB = r EB -∑ /lscriptm Y /lscriptm ( ˆ θ, ˆ φ ) λ /lscriptm ( t ) . (74)</formula>
<text><location><page_21><loc_16><loc_51><loc_76><loc_56></location>Here r EB is the radius of the (spherical) excision boundary in the grid frame. In deriving Eq. (74) we have used the relations M CutX = 1 , f ( r, θ, φ ) = 1, ˆ θ = θ , and ˆ φ = φ , which hold on the excision boundary.</text>
<text><location><page_21><loc_19><loc_50><loc_49><loc_51></location>Combining Eqs. (36), (73), and (74) yields</text>
<formula><location><page_21><loc_28><loc_44><loc_76><loc_48></location>r EB -∑ /lscriptm Y /lscriptm ( ˆ θ, ˆ φ ) λ /lscriptm ( t ) r EB -Y 00 λ 00 ( t ) = ∑ /lscriptm ˆ S /lscriptm Y /lscriptm ( ˆ θ, ˆ φ ) ˆ S 00 Y 00 , (75)</formula>
<text><location><page_21><loc_16><loc_43><loc_44><loc_44></location>which we can satisfy by demanding that</text>
<formula><location><page_21><loc_31><loc_38><loc_76><loc_41></location>λ /lscriptm ( t ) + ˆ S /lscriptm r EB -Y 00 λ 00 ( t ) ˆ S 00 Y 00 = 0 , /lscript > 0 . (76)</formula>
<text><location><page_21><loc_16><loc_34><loc_76><loc_37></location>Therefore, given an apparent horizon and given a value of λ 00 , a control system can be set up for each ( /lscript, m ) pair with /lscript > 0, and the corresponding control errors are</text>
<formula><location><page_21><loc_29><loc_30><loc_76><loc_33></location>Q /lscriptm = -λ /lscriptm ( t ) -ˆ S /lscriptm r EB -Y 00 λ 00 ( t ) ˆ S 00 Y 00 , /lscript > 0 . (77)</formula>
<text><location><page_21><loc_16><loc_26><loc_76><loc_28></location>Driving these Q /lscriptm to zero produces an excision boundary that matches the shape of the corresponding apparent horizon.</text>
<text><location><page_21><loc_16><loc_20><loc_76><loc_25></location>Equation (77) determines λ /lscriptm ( t ) only for /lscript > 0, and Eqs. (73)-(77) can be satisfied for arbitrary values of the remaining undetermined map coefficient λ 00 ( t ). Determination of λ 00 ( t ) is complicated enough that it is described in its own section, Sec. 5.</text>
<text><location><page_21><loc_16><loc_14><loc_76><loc_20></location>Note that Eq. (77) does not satisfy Eq. (14) because ∂Q /lscriptm /∂λ 00 = ˆ S /lscriptm / ˆ S 00 , which does not vanish even if all the control errors are zero. For small distortions, this coupling between λ 00 and Q /lscriptm seems to cause little difficulty. However, at times when the shapes and sizes of the horizons change rapidly (e.g., during the initial</text>
<text><location><page_22><loc_16><loc_80><loc_76><loc_88></location>'junk radiation' phase, after the transition to a single excised region when a common apparent horizon forms, and after rapid gauge changes), simulations using Eq. (77) exhibit relatively large and high-frequency oscillations in Q /lscriptm that usually damp away but occasionally destabilize the evolution. Construction of a control system in which all Q /lscriptm are fully decoupled will be addressed in a future work.</text>
<section_header_level_1><location><page_22><loc_16><loc_77><loc_23><loc_78></location>4.6. CutX</section_header_level_1>
<text><location><page_22><loc_16><loc_72><loc_76><loc_76></location>The map M CutX applies a translation along the grid-frame x -axis in the vicinity of the black holes, but without moving the excision boundaries (or the surrounding spherical shells) themselves. The action of the map is shown in Fig. 7.</text>
<text><location><page_22><loc_16><loc_54><loc_76><loc_71></location>The goal of this map is to allow for a slight motion of the cutting plane toward the smaller excision boundary. This is important for binary black hole systems with mass ratio q /greaterorsimilar 8. As such a binary gets closer to merger, the inertial-frame coordinate distance between each excision boundary and the cutting plane decreases. Eventually, this distance falls to zero for the larger excision boundary, producing a coordinate singularity in which the Jacobian of one of the other maps (often the shape map) becomes infinite. By pushing the cutting plane toward the smaller excision boundary, the map M CutX avoids this singularity. Even for evolutions in which the inertialcoordinate distance between the larger excision boundary and the cutting plane remains finite but becomes small, the map M CutX prevents large Jacobians from developing and thus increases numerical accuracy (because it is no longer necessary to add computational resources to resolve the large Jacobians).</text>
<text><location><page_22><loc_16><loc_42><loc_76><loc_53></location>Figure 8 shows the domain decomposition in the inertial frame for a binary black hole simulation with q = 8. The top panel shows the case without M CutX , and the compressed grid near the larger excision boundary is evident. The lower panel shows the case with M CutX , which removes the extreme grid compression. Looking at the Jacobian of the mapping from the inertial frame to the grid frame shortly before merger, we find that the infinity norm of the determinant of the Jacobian is twice as large in the case without M CutX .</text>
<text><location><page_22><loc_19><loc_41><loc_40><loc_43></location>The map M CutX is written as</text>
<formula><location><page_22><loc_38><loc_38><loc_76><loc_40></location>x 0 ↦→ x 0 + ρ ( x i ) F ( t ) , (78)</formula>
<formula><location><page_22><loc_38><loc_36><loc_76><loc_38></location>x j ↦→ x j , j = 1 , 2 , (79)</formula>
<text><location><page_22><loc_16><loc_34><loc_47><loc_35></location>where F ( t ) is adjusted by a control system.</text>
<text><location><page_22><loc_16><loc_20><loc_76><loc_34></location>We choose ρ ( x i ) to be zero within either of the spherical shell regions, i.e., inside the magenta spheres around A and B in Fig. 7, and it is also zero outside the outer magenta sphere in the same figure. We set ρ ( x i ) = 1 on the solid red boundaries in Fig. 7, and everywhere between the two solid red boundaries that enclose excision boundary B . Every other region on the grid is bounded by a smooth red boundary on one side and a smooth magenta boundary on the other; in these regions ρ varies linearly between zero and one. The solid blue boundaries are locations (in addition to the solid red and solid magenta boundaries) in which the gradient of ρ is discontinuous. Full details for the calculation of ρ can be found in Appendix B.</text>
<text><location><page_22><loc_16><loc_14><loc_76><loc_20></location>As with the skew map, the map M CutX is inactive for most of the inspiral. Let x Exc H be the distorted-frame x -coordinate of the intersection of the excision boundary H with the line segment connecting the centers of the excision boundaries. This is similar to x Int H defined earlier; the difference is that x Int H refers to a point on the</text>
<figure>
<location><page_23><loc_26><loc_57><loc_66><loc_88></location>
<caption>Figure 7. Two-dimensional projection of the domain decomposition near the two black holes A and B. The function ρ in M CutX is zero on the magenta boundaries, one on the red boundaries, and goes linearly between zero and one in the 'radial' direction between these boundaries. The gradient of ρ is discontinuous across the solid magenta, red, and blue boundaries. Dotted lines show the subdomain boundaries under the action of M CutX .</caption>
</figure>
<text><location><page_23><loc_16><loc_43><loc_76><loc_46></location>apparent horizon and x Exc H refers to a point on the excision boundary. The quantity x Exc H is time-dependent because it depends on the shape map.</text>
<text><location><page_23><loc_19><loc_41><loc_56><loc_43></location>The map M CutX is turned off completely as long as</text>
<formula><location><page_23><loc_40><loc_37><loc_76><loc_41></location>x 0 C -x Exc H x Exc H -C 0 H ≥ 1 2 , (80)</formula>
<text><location><page_23><loc_16><loc_33><loc_76><loc_37></location>where C i H are the excision boundary centers, and x i C are the coordinates of the intersection of the cutting plane and the line segment connecting the centers of the excision boundaries, as introduced in Sec. 4.4.</text>
<text><location><page_23><loc_16><loc_29><loc_76><loc_32></location>Let t 0 be the coordinate time at which the map M CutX is activated. ‖ We designate the target position x = x T of the cutting plane as</text>
<text><location><page_23><loc_16><loc_23><loc_20><loc_25></location>where</text>
<formula><location><page_23><loc_33><loc_23><loc_76><loc_29></location>x T = ∆ x A · x Exc A +∆ x B · x Exc B ∆ x A +∆ x B ∣ ∣ ∣ ∣ t = t 0 , (81)</formula>
<text><location><page_23><loc_16><loc_14><loc_76><loc_22></location>∣ ∣ ‖ Both M Skew and M CutX are turned on late in the run, but the condition triggering their activation is different, given the different nature of the problems they address. M Skew is needed for all runs where the horizons eventually intersect the line segment connecting their excision centers at an angle sufficiently different from π/ 2. This will happen essentially for all runs except the simplest head-on collisions. M CutX , on the other hand, is primarily for unequal-mass runs where the larger excision surface encroaches upon the cutting plane near merger.</text>
<formula><location><page_23><loc_34><loc_19><loc_76><loc_23></location>∆ x H = ∣ x Exc H -C 0 H ∣ , H = A,B. (82)</formula>
<figure>
<location><page_24><loc_30><loc_73><loc_62><loc_88></location>
</figure>
<figure>
<location><page_24><loc_30><loc_58><loc_62><loc_73></location>
<caption>Figure 8. Snapshot of the grid, viewed in the inertial frame, from a binary black hole evolution with M A = 8 M B shortly before the common horizon forms. Top: without M CutX . Bottom: with M CutX . The gray areas are excised regions. In the top panel, the grid is compressed as the larger excision boundary approaches the cutting plane; this is especially evident in the long narrow subdomains immediately adjacent to the larger excision boundary. Using M CutX relieves the grid compression.</caption>
</figure>
<text><location><page_24><loc_16><loc_40><loc_76><loc_45></location>Recall that x Exc H is time-dependent (as it is measured in the distorted frame), so we save the value of x T as calculated at the time when the map M CutX is activated, rather than recalculating x T at every measurement time.</text>
<text><location><page_24><loc_16><loc_25><loc_76><loc_40></location>Now that we have a target x T , we could designate x T -x 0 C as the target value to the function F ( t ) by setting Q F = -F + x T -x 0 C and have the control system drive Q F to zero, thus driving the x -coordinate of the cutting plane to x T . However, because we turn on the CutX control system suddenly at time t 0 , we must be more careful. Turning on any control system suddenly will produce some transient oscillations, unless the control error and its relevant derivatives and integrals are all initially zero. In the case of the CutX map, which is turned on during a very dynamic part of the simulation when excision boundaries need to be controlled very tightly, these oscillations can prematurely terminate the run by, for example, pushing an excision boundary outside its accompanying apparent horizon.</text>
<text><location><page_24><loc_16><loc_18><loc_76><loc_25></location>To turn on M CutX gradually, we replace x T by a new time-dependent target function T ( t ) that gradually approaches x T at late times but produces a control error Q F with Q F = ∂Q F /∂t = 0 at the activation time t = t 0 . We start by estimating the time t XC H at which x Exc H will reach the cutting plane, where H is either A or B . This is done by the method described in Appendix C. We then let t XC be the minimum of</text>
<text><location><page_25><loc_16><loc_86><loc_62><loc_88></location>t XC A and t XC B . The smooth target function T ( t ) is then defined by</text>
<formula><location><page_25><loc_29><loc_75><loc_76><loc_85></location>T ( t ; t 0 , x T , t XC ) = x T +exp [ -( t -t 0 0 . 3 · t XC ) 2 ] × [ x 0 C -x T + F ( t 0 ) + ( t -t 0 ) · ∂F ( t ) ∂t ∣ ∣ ∣ ∣ t = t 0 ] . (83)</formula>
<text><location><page_25><loc_16><loc_74><loc_46><loc_76></location>Designating T -x 0 C as the target for F ( t )</text>
<formula><location><page_25><loc_39><loc_71><loc_76><loc_73></location>Q F = T -x 0 C -F (84)</formula>
<formula><location><page_25><loc_35><loc_64><loc_76><loc_69></location>Q F | t = t 0 = 0 , ∂Q F ∂t ∣ ∣ ∣ t = t 0 = 0 , (85)</formula>
<text><location><page_25><loc_16><loc_69><loc_21><loc_70></location>leads to</text>
<text><location><page_25><loc_16><loc_62><loc_76><loc_67></location>∣ so at the activation time the control system does not produce transients. Furthermore, in the limit of small Q F ,</text>
<formula><location><page_25><loc_38><loc_60><loc_76><loc_63></location>x 0 C + F ( t ) | t = t XC ≈ x T , (86)</formula>
<text><location><page_25><loc_16><loc_45><loc_76><loc_60></location>i.e., by the time the excision boundary would have touched the cutting plane (and formed a grid singularity), the smooth target function T ( t ) has approached x T , and the cutting plane will have reached its designated target location, x T . As the run proceeds, the behavior of the cutting plane is determined by the map M CutX while the motion of the excision boundaries is determined by gauge dynamics and the behavior of the other control systems. We continue to monitor the distance between the cutting plane and the excision boundaries, and if it is predicted to touch within a time less than τ d / 0 . 15, the M CutX control system is reset, constructing a new target function T ( t ; t 0 , x T , t XC ), where t 0 is the time of the reset, and x T , t XC are also recalculated based on the state of the grid at this reset time.</text>
<text><location><page_25><loc_16><loc_41><loc_76><loc_45></location>Each time a new target function T is constructed, the damping time τ d of the M CutX control system is set to be t XC / 2.</text>
<text><location><page_25><loc_16><loc_31><loc_76><loc_42></location>The control systems responsible for M CutX and M Shape are decoupled, as M CutX controls the location of the cutting plane, leaving the excision boundary unchanged, while M Shape controls the shape of the excision boundary, leaving the location of the cutting plane unchanged. Recall that M CutX and M Shape define the mapping from the grid frame to the distorted frame. As the apparent horizons are found in the distorted frame, and the other maps only depend upon measurements of the horizons, M CutX and M Shape are decoupled from the other maps.</text>
<section_header_level_1><location><page_25><loc_16><loc_28><loc_28><loc_29></location>5. Size control</section_header_level_1>
<text><location><page_25><loc_16><loc_21><loc_76><loc_27></location>In this section we discuss how we control the spherical part of the map given by Eq. (72), namely, the coefficients λ H 00 for each excision boundary H . We apply the same method to each excision boundary, so for clarity, in this section we again drop the index H from the coefficients λ H 00 and S H /lscriptm , and from the coordinates ( r H , θ H , φ H ).</text>
<section_header_level_1><location><page_25><loc_16><loc_18><loc_39><loc_19></location>5.1. Characteristic speed control</section_header_level_1>
<text><location><page_25><loc_16><loc_14><loc_76><loc_17></location>Controlling the size of the excision boundary is more complicated than simply keeping the excision boundary inside the apparent horizon. This is because black hole excision</text>
<text><location><page_26><loc_16><loc_85><loc_76><loc_88></location>requires conditions on the characteristic speeds of the system, and if these conditions are not enforced they are likely to be violated.</text>
<text><location><page_26><loc_19><loc_83><loc_71><loc_85></location>The minimum characteristic speed at each excision boundary is given by</text>
<formula><location><page_26><loc_37><loc_79><loc_76><loc_82></location>v = -α -¯ β i ¯ n i -¯ n i ∂ ¯ x i ∂t , (87)</formula>
<text><location><page_26><loc_16><loc_69><loc_76><loc_79></location>where α is the lapse, ¯ β i is the shift, and ¯ n i is the normal to the excision boundary pointing out of the computational domain , i.e., toward the center of the hole. Here ( ¯ t, ¯ x i ) are the inertial frame coordinates. The first two terms in Eq. (87) describe the coordinate speed of the ingoing (i.e., directed opposite to ¯ n i ) light cone in the inertial frame, and the last term accounts for the motion of the excision boundary (which is fixed in the grid frame) with respect to the inertial frame.</text>
<text><location><page_26><loc_16><loc_54><loc_76><loc_69></location>In our simulations, we impose no boundary condition whatsoever at each excision boundary. Therefore, well-posedness requires that all of the characteristic speeds, and in particular the minimum speed v , must be non-negative; in other words, characteristics must flow into the hole. In practice, if v becomes negative, the simulation is terminated, because a boundary condition is needed, but we do not have one to impose. This can occur even when the excision boundary is inside the horizon. In this case, one might argue that if the simulation is able to continue without crashing (e.g. by becoming unstable inside the horizon), that the solution outside the horizon would not be contaminated. We have not explored this possibility. Instead, we choose to avoid this situation by terminating the code if negative speeds are detected.</text>
<text><location><page_26><loc_16><loc_50><loc_76><loc_54></location>Therefore, we would like to control λ 00 in such a way that v remains positive. We start by writing v in a way that separates terms that explicitly depend on ˙ λ 00 from terms that do not. To do this we expand the derivative in the last term of Eq. (87) as</text>
<formula><location><page_26><loc_37><loc_39><loc_76><loc_49></location>∂ ¯ x i ∂t = ∂ ¯ x i ∂ ˆ x a ∂ ˆ x a ∂t = ∂ ¯ x i ∂ ˆ t ∂ ˆ t ∂t + ∂ ¯ x i ∂ ˆ x j ∂ ˆ x j ∂t = ∂ ¯ x i ∂ ˆ t + ∂ ¯ x i ∂ ˆ x j ∂ ˆ x j ∂t , (88)</formula>
<text><location><page_26><loc_16><loc_35><loc_76><loc_38></location>where a in the first line of Eq. (88) is a four-dimensional spacetime index, and the last line of Eq. (88) follows from ∂ ˆ t/∂t = 1. Inserting this into Eq. (87) yields</text>
<formula><location><page_26><loc_34><loc_31><loc_76><loc_34></location>v = -α -¯ β i ¯ n i -¯ n i ∂ ¯ x i ∂ ˆ t -ˆ n i ∂ ˆ x i ∂t , (89)</formula>
<text><location><page_26><loc_16><loc_27><loc_76><loc_30></location>where ˆ x i is the frame that is obtained from the grid frame by applying the distortion map; see Eq. (30).</text>
<text><location><page_26><loc_19><loc_26><loc_47><loc_27></location>We then use Eq. (72) to rewrite this as</text>
<formula><location><page_26><loc_36><loc_17><loc_76><loc_25></location>v = -α -¯ β i ¯ n i -¯ n i ∂ ¯ x i ∂ ˆ t + ˆ n i x i r ∑ /lscriptm Y /lscriptm ( θ, φ ) ˙ λ /lscriptm ( t ) , (90)</formula>
<text><location><page_26><loc_16><loc_13><loc_76><loc_16></location>where we have used the relations f ( r, θ, φ ) = 1 and M CutX = 1 when evaluating the distortion map M Distortion on the excision boundary.</text>
<text><location><page_27><loc_16><loc_85><loc_76><loc_88></location>By combining all the terms that do not explicitly depend on ˙ λ 00 into a quantity v 0 , we obtain</text>
<formula><location><page_27><loc_38><loc_82><loc_76><loc_85></location>v = v 0 + ˆ n i x i r Y 00 ˙ λ 00 . (91)</formula>
<text><location><page_27><loc_16><loc_75><loc_76><loc_81></location>Thus, the characteristic speed v can be thought of as consisting of two parts: one part, v 0 , that depends on the position and shape of the excision boundary and the values of the metric quantities there, and another part that depends on the average speed of the excision boundary in the direction of the boundary normal.</text>
<text><location><page_27><loc_16><loc_71><loc_76><loc_75></location>We now construct a control system that drives the characteristic speed v to some target speed v T . This is a control system that controls the derivative quantity ˙ λ 00 , as opposed to directly controlling the map quantity λ 00 . We choose</text>
<formula><location><page_27><loc_37><loc_67><loc_76><loc_69></location>Q = (min( v ) -v T ) / 〈-Ξ 〉 , (92)</formula>
<text><location><page_27><loc_16><loc_65><loc_20><loc_67></location>where</text>
<formula><location><page_27><loc_41><loc_62><loc_76><loc_66></location>Ξ = ˆ n i x i r Y 00 , (93)</formula>
<text><location><page_27><loc_16><loc_56><loc_76><loc_62></location>the minimum is over the excision boundary, and the angle brackets in Eq. (92) refer to an average over the excision boundary. Note that Ξ < 0 because ˆ n i points radially inward and x i /r points radially outward; this means that ˙ Q = -¨ λ 00 , in accordance with our normalization choice for a control system on ˙ λ 00 .</text>
<text><location><page_27><loc_16><loc_52><loc_76><loc_56></location>As in our other controlled map parameters, we demand that ˙ λ 00 is a function with a piecewise-constant second derivative. It is then easy to construct λ 00 as a function with a piecewise-constant third derivative.</text>
<text><location><page_27><loc_16><loc_47><loc_76><loc_51></location>The control system given by Eq. (92) with a hand-chosen value of v T has been used successfully [24, 25] in simulations of high-spin binaries. Figure 9 illustrates why characteristic speed control is crucial for the success of these simulations.</text>
<section_header_level_1><location><page_27><loc_37><loc_43><loc_60><loc_44></location>Minimum characteristic speeds</section_header_level_1>
<figure>
<location><page_27><loc_28><loc_28><loc_64><loc_42></location>
<caption>Figure 9. Minimum characteristic speed on one of the excision surfaces of an equal mass binary with equal aligned spins of χ = 0 . 97, shown just before merger [25]. Two cases are shown: one without characteristic speed control (solid) and one that is restarted from the uncontrolled run at t = 6255 M with characteristic speed control turned on and v T = 5 × 10 -5 (dashed). The uncontrolled run is terminated because of negative characteristic speeds shortly after t = 6290 M , but the controlled run continues through merger and ringdown.</caption>
</figure>
<section_header_level_1><location><page_28><loc_16><loc_86><loc_38><loc_88></location>5.2. Apparent horizon tracking</section_header_level_1>
<text><location><page_28><loc_16><loc_79><loc_76><loc_85></location>The characteristic speed control described above has the disadvantage that it requires a user-specified target value v T . If v T is chosen to be too small, then small fluctuations (due to shape control, the horizon finder, or simply numerical truncation error) can cause the characteristic speed to become negative and spoil the simulation.</text>
<text><location><page_28><loc_16><loc_67><loc_76><loc_79></location>If v T is chosen to be too large, the simulation can also fail. To understand why, recall that characteristic speed control achieves the target characteristic speed by moving the excision boundary and thus changing the velocity term in Eq. (87). So if v > v T the control system moves the excision boundary radially inward, and if v < v T the control system moves the excision boundary radially outward. If v T is too large, the control system can push the excision boundary outward until it crosses the apparent horizon. This halts the evolution because the apparent horizon can no longer be found.</text>
<text><location><page_28><loc_16><loc_63><loc_76><loc_67></location>One way to prevent the excision boundary from crossing the horizon is to drive the excision boundary to some constant fraction of the horizon radius, or in other words, drive the quantity d/dt (∆ r ) to zero, where</text>
<formula><location><page_28><loc_40><loc_58><loc_76><loc_61></location>∆ r = 1 -〈 ˆ r EB 〉 〈 ˆ r AH 〉 (94)</formula>
<text><location><page_28><loc_16><loc_53><loc_76><loc_57></location>is the relative difference between the average radius of the apparent horizon (in the intermediate frame) and the average radius of the excision boundary. Using Eqs. (74) and (36), we can write</text>
<formula><location><page_28><loc_36><loc_49><loc_76><loc_53></location>d dt ∆ r = ˙ λ 00 ˆ S 00 + ˙ ˆ S 00 ˆ S 00 (1 -∆ r ) , (95)</formula>
<text><location><page_28><loc_16><loc_45><loc_76><loc_48></location>and therefore a control system that adjusts ˙ λ 00 to achieve d/dt (∆ r ) = 0 can be obtained by defining</text>
<formula><location><page_28><loc_38><loc_43><loc_76><loc_45></location>Q = ˙ ˆ S 00 (∆ r -1) -˙ λ 00 . (96)</formula>
<text><location><page_28><loc_16><loc_40><loc_76><loc_42></location>A slight generalization of this control system can be obtained by demanding that d/dt (∆ r ) = ˙ r drift , where ˙ r drift is some chosen constant,</text>
<formula><location><page_28><loc_34><loc_36><loc_76><loc_38></location>Q = ˙ ˆ S 00 (∆ r -1) -˙ λ 00 + ˆ S 00 ˙ r drift . (97)</formula>
<text><location><page_28><loc_16><loc_28><loc_76><loc_35></location>We have not found the control systems defined by Eqs. (96) and (97) to be especially useful on their own. One drawback of these systems is that they do not prevent the minimum characteristic speed v from becoming negative. Instead, we use the horizon tracking control systems described here as part of a more sophisticated control system discussed in the next section.</text>
<section_header_level_1><location><page_28><loc_16><loc_24><loc_43><loc_26></location>5.3. Adaptive switching of size control</section_header_level_1>
<text><location><page_28><loc_16><loc_18><loc_76><loc_23></location>Here we introduce a means of controlling λ 00 that combines the best features of characteristic speed control and horizon tracking. The idea is to continuously monitor the state of the system and switch between different control systems as the evolution proceeds.</text>
<text><location><page_28><loc_16><loc_14><loc_76><loc_17></location>At fixed intervals during the simulation that we call 'measurement times,' we monitor the minimum characteristic speed v and the relative distance between the</text>
<text><location><page_29><loc_16><loc_85><loc_76><loc_88></location>horizon and the excision boundary ∆ r . The goal of the control system is to ensure that both of these values remain positive.</text>
<text><location><page_29><loc_16><loc_74><loc_76><loc_85></location>Consider first the characteristic speed v . Using the method described in Appendix C, we determine whether v is in danger of becoming negative in the near future, and if so, we estimate the timescale τ v on which this will occur. Similarly, we also determine whether ∆ r will soon become negative, and if so we estimate a corresponding timescale τ ∆ r . Having estimated both τ v and τ ∆ r , we use these quantities to determine how to control ˙ λ 00 . In particular, we switch between horizon tracking, Eq. (96), and characteristic speed control, Eq. (92), based on τ v and τ ∆ r .</text>
<text><location><page_29><loc_16><loc_61><loc_76><loc_74></location>We do this by the following algorithm, which favors horizon tracking over characteristic speed control unless the latter is essential. Assume that the control system for ˙ λ 00 is currently tracking the horizon, Eq. (96), and that the current damping timescale is some value τ d . If v is in danger of crossing zero according to the above estimate, and if τ v < τ d and τ v < τ ∆ r , we then switch to characteristic speed control, Eq. (92), we set v T = 1 . 01 v where v is the current value of the characteristic speed (the factor 1 . 01 prevents the algorithm from switching back from characteristic speed control to horizon tracking on the very next time step), and we reset the damping time τ d equal to τ v . Otherwise we continue to use Eq. (96), resetting τ d = τ ∆ r if τ ∆ r < τ d .</text>
<text><location><page_29><loc_16><loc_49><loc_76><loc_61></location>Now assume that the control system for ˙ λ 00 is controlling the characteristic speed, Eq. (92). If v is not in danger of crossing zero, so that we no longer need active control of the characteristic speed, then we switch to horizon tracking, Eq. (96), without changing the damping time τ d . If v is in danger of crossing zero, and if ∆ r is either in no danger of crossing zero or if it will cross zero sufficiently far in the future such that τ ∆ r ≥ σ 1 τ d , then we continue to use Eq. (92) with τ d reset to min( τ d , τ v ) so as to maintain control of the characteristic speed. Here σ 1 is a constant that we typically choose to be about 20.</text>
<text><location><page_29><loc_16><loc_41><loc_76><loc_48></location>The more complicated case occurs when both v and ∆ r are in danger of crossing zero and τ ∆ r < σ 1 τ d . In this case, we have two possible methods by which we attempt to prevent ∆ r from becoming negative. We choose between these two methods based on the values of τ ∆ r and τ d , and also by the value of a quantity that we call the comoving characteristic speed :</text>
<formula><location><page_29><loc_28><loc_32><loc_76><loc_40></location>v c = -α -¯ β i ¯ n i -¯ n i ∂ ¯ x i ∂ ˆ t + ˆ n i x i r [ Y 00 ˙ ˆ S 00 (∆ r -1) + ∑ /lscript> 0 Y /lscriptm ( θ, φ ) ˙ λ /lscriptm ( t ) ] . (98)</formula>
<text><location><page_29><loc_16><loc_23><loc_76><loc_31></location>This quantity is the value that the characteristic speed v would have if horizon tracking (Eq. (96)) were in effect and working perfectly. Equation (98) is derived by assuming Q = 0 in Eq. (96), solving for ˙ λ 00 , and substituting this value into Eq. (90). The reason to consider v c is that for min( v c ) < 0 (which can happen temporarily during a simulation), horizon tracking is to be avoided, because horizon tracking will drive min( v ) toward a negative value, namely min( v c ).</text>
<text><location><page_29><loc_16><loc_14><loc_76><loc_22></location>The first method of preventing ∆ r from becoming negative is to switch to horizon tracking, Eq. (96); we do this if min( v c ) > 0 and if τ ∆ r < σ 2 τ d , where σ 2 is a constant that we typically choose to be 5. The second method, which we employ if σ 2 τ d < τ ∆ r < σ 1 τ d or if min( v c ) < 0, is to continue to use characteristic speed control, Eq. (92), but to reduce the target characteristic speed v T by multiplying its current value by some fraction η (typically 0 . 25). By reducing the target speed v T , we reduce</text>
<text><location><page_30><loc_16><loc_71><loc_76><loc_88></location>the outward speed of the excision boundary, and this increases τ ∆ r . The reason we use the latter method for min( v c ) < 0 is that the former method, horizon tracking, will drive v toward v c and we wish to keep v positive. The reason we use the latter method for σ 2 τ d < τ ∆ r < σ 1 τ d is that in this range of timescales the control system is already working hard to keep v positive, therefore a switch to horizon tracking is usually followed by a switch back to characteristic speed control in only a few time steps, and rapid switches between different types of control make it more difficult for the control system to remain locked. The constants η , σ 1 , and σ 2 are adjustable, and although the details of the algorithm (i.e., which particular quantity is controlled at which particular time) are sensitive to the choices of these constants, the overall behavior of the algorithm is not, provided 1 < σ 2 < σ 1 .</text>
<text><location><page_30><loc_16><loc_62><loc_76><loc_71></location>The overall behavior of this algorithm is to cause ˙ λ 00 to settle to a state in which both v > 0 and ∆ r > 0 for a long stretch of the simulation. Occasionally, when either v or ∆ r threatens to cross zero as a result of evolution of the metric or gauge, the algorithm switches between characteristic speed control and horizon tracking several times and then settles into a new near-equilibrium state in which both v > 0 and ∆ r > 0.</text>
<text><location><page_30><loc_16><loc_46><loc_76><loc_62></location>The actual value of ∆ r and of v in the near-equilibrium state is unknown a priori, and in particular this algorithm can in principle settle to values of ∆ r that are either small enough that control system timescales must be very short in order to prevent ∆ r from becoming negative, or large enough that excessive computational resources are needed to resolve the metric quantities deep inside the horizon. To prevent either of these cases from occurring, we introduce a constant correction velocity ˙ r corr , a nominal target value ∆ r target , and a nominal target characteristic speed v target . We currently choose these to be ˙ r corr = 0 . 005, ∆ r target = 0 . 08, and v target = 0 . 08. These quantities are used only to decide whether to replace Eq. (96) by Eq. (97) when the algorithm chooses horizon tracking, as described below; these quantities are not used when the algorithm chooses characteristic speed control.</text>
<text><location><page_30><loc_16><loc_40><loc_76><loc_45></location>First we consider the case where ∆ r is too large during horizon tracking. If ∆ r > 1 . 1∆ r target , then we replace Eq. (96) with Eq. (97), and we use ˙ r drift = -˙ r corr . We continue to use ˙ r drift = -˙ r corr until ∆ r < ∆ r target , at which point we set ˙ r drift = 0.</text>
<text><location><page_30><loc_16><loc_17><loc_76><loc_41></location>The case where ∆ r is too small during horizon tracking is more complicated because we want to be careful so as to not make the characteristic speeds decrease, since it is more difficult to recover from decreasing characteristic speeds when ∆ r is small. In this case, if ∆ r < 0 . 9∆ r target , v < 0 . 9 v target , and 〈 n k ∂ k v c 〉 > 0, then we replace Eq. (96) with Eq. (97), and we use ˙ r drift = v corr . Here v c is the comoving characteristic speed defined in Eq. (98) and the angle brackets refer to an average over the excision boundary. The condition on the gradient of v c ensures that v c will increase if ∆ r is increased; otherwise a positive ˙ r drift will threaten the positivity of the characteristic speeds. We continue to use ˙ r drift = v corr until either v > 0 . 9 v target , ∆ r > 0 . 9∆ r target , 〈 n k ∂ k v c 〉 < 0, or if v will cross v target or ∆ r will cross ∆ r target within a time less than the current damping time τ d (where crossing times are determined by the procedure of Appendix C). When any of these conditions occur, we switch to ˙ r drift = 0, and we prohibit switching back to a positive ˙ r drift until either v > 0 . 99 v target or ∆ r > 0 . 99∆ r target , or until both v and ∆ r stop increasing in time. These last conditions prevent the algorithm from oscillating rapidly between positive and zero values of ˙ r drift .</text>
<section_header_level_1><location><page_31><loc_16><loc_86><loc_36><loc_88></location>6. Merger and ringdown</section_header_level_1>
<text><location><page_31><loc_16><loc_71><loc_76><loc_85></location>The maps and control systems in Sec. 4 were discussed in the context of a domain decomposition with two excision boundaries, one inside each apparent horizon. The skew and CutX maps were introduced specifically to handle difficulties that occur when two distorted excision boundaries approach each other. At some point in the evolution as the two black holes become sufficiently close, a common apparent horizon forms around them. After this occurs, the simulation can be simplified considerably by constructing a new domain decomposition with a single excision boundary placed just inside the common horizon. The evolved variables are interpolated from the old to the new domain decomposition, and the simulation then proceeds on the new one.</text>
<text><location><page_31><loc_16><loc_65><loc_76><loc_71></location>The algorithm for transitioning to a new grid with a single excision boundary is fundamentally the same as that described in [3, 4], but there have been improvements in the maps and the control systems, so for completeness we describe the procedure here.</text>
<text><location><page_31><loc_16><loc_58><loc_76><loc_65></location>At some time t = t m shortly after a common horizon forms, we define a new, post-merger set of grid coordinates ˘ x i and a corresponding new domain decomposition composed only of spherical shells. In ˘ x i coordinates, the new excision boundary is a sphere centered at the origin. We also define a post-merger map ¯ x i = M Ringdown ˘ x i , where</text>
<formula><location><page_31><loc_20><loc_55><loc_76><loc_57></location>M Ringdown = M TranslationRD · M Rotation · M ScalingRD · M ShapeRD . (99)</formula>
<text><location><page_31><loc_16><loc_48><loc_76><loc_54></location>Note that the post-merger grid coordinates differ from the pre-merger grid coordinates, but there is only one set of inertial coordinates ¯ x i . Recall that in the dual-frame picture [1], the inertial-frame components of tensors are stored and evolved, so the evolved variables are continuous at t = t m even though the grid coordinates are not.</text>
<text><location><page_31><loc_16><loc_35><loc_76><loc_48></location>The post-merger translation map M TranslationRD and its control system are the same as discussed in Sec. 4.3, except that M TranslationRD translates with respect to the origin of the ˘ x i coordinate system, which is different from the origin of the x i coordinate system. The post-merger translation control system keeps the common apparent horizon centered at the origin in the post-merger grid frame. Note that the center of the common apparent horizon does not necessarily lie on the same line as the centers of the individual apparent horizons (see [23] for an example). This means that M TranslationRD is not continuous with the pre-merger translation map M Translation , except near the outer boundary where both maps are the identity.</text>
<text><location><page_31><loc_16><loc_21><loc_76><loc_35></location>To evolve a distorted single black hole we do not need a rotation map. However, if rotation is turned off suddenly at t = t m , then the outer boundary condition (which is imposed in inertial coordinates) is not smooth in time, and we see a pulse of gauge and constraint violating modes propagate inward from the outer boundary because of this sharp change in the boundary condition. Therefore, we use the same rotation map M Rotation before and after merger, and we slow down the rotation gradually. To accomplish this, instead of adjusting the map parameters ( ϑ, ϕ ) by a control system, for t ≥ t m we set these parameters equal to prescribed functions that approach a constant at late times. We set</text>
<formula><location><page_31><loc_31><loc_18><loc_76><loc_21></location>ϑ ( t ) = A +[ B + C ( t -t m )] e -( t -t m ) /τ roll , (100)</formula>
<text><location><page_31><loc_16><loc_14><loc_76><loc_18></location>where the constants A , B , and C are determined by demanding that ϑ ( t ) and its first two time derivatives are continuous at t = t m . The parameter ϕ obeys an equation of the same form. We choose τ roll = 20 M .</text>
<text><location><page_32><loc_19><loc_85><loc_56><loc_88></location>The post-merger scaling map M ScalingRD is written</text>
<formula><location><page_32><loc_31><loc_81><loc_76><loc_85></location>˘ R ↦→ [ 1 + b ( t ) -b ( t m ) b ( t m ) sin 2 ( π ˘ R 2 ˘ R OB )] ˘ R, (101)</formula>
<text><location><page_32><loc_16><loc_71><loc_76><loc_80></location>where ˘ R is the post-merger grid-coordinate radius, and ˘ R OB is the outer boundary radius in the post-merger grid coordinates. The function b ( t ) is given by Eq. (42) before and after merger. We set ˘ R OB = b ( t m ) R OB so that at t = t m , M ScalingRD matches M Scaling at the outer boundary. Note that the post-merger scaling map is the identity near the merged black hole; the only purpose of this map is to keep the inertial-coordinate location of the outer boundary smooth in time at t = t m .</text>
<text><location><page_32><loc_16><loc_63><loc_76><loc_70></location>The post-merger shape map M ShapeRD and its control system are the same as discussed in Secs. 4.5 and 5, except the map is centered about the origin of the ˘ x i coordinate system, the sum over excised regions in Eq. (72) runs over only one region (which we call excision region C), and the function f C ( r C , θ C , φ C ) appearing in Eq. (72) is</text>
<formula><location><page_32><loc_20><loc_58><loc_76><loc_62></location>f C ( r C , θ C , φ C ) = { ( r C -r max ) / ( r EB -r max ) , r EB ≤ r C < r max 0 , r C ≥ r max (102)</formula>
<text><location><page_32><loc_16><loc_54><loc_76><loc_57></location>where r max is the radius of a (spherical) subdomain boundary that is well inside the wave zone. We typically choose r max = 32 r EB .</text>
<text><location><page_32><loc_16><loc_43><loc_76><loc_54></location>Once the post-merger map parameters are initialized, as discussed below, we interpolate all variables from the pre-merger grid to the post-merger grid. The inertial coordinates ¯ x i are the same before and after merger, so this interpolation is easily accomplished using both the pre-merger and post-merger maps, e.g. F ( x i ) = F ( M -1 M Ringdown ˘ x i ) for some function F . After interpolating all variables onto the post-merger grid, we continue the evolution. This entire process - from detecting a common apparent horizon to continuing the evolution on the new domain - is done automatically.</text>
<section_header_level_1><location><page_32><loc_16><loc_39><loc_28><loc_40></location>6.1. Initialization</section_header_level_1>
<text><location><page_32><loc_16><loc_32><loc_76><loc_38></location>All that remains for a full specification of M Ringdown is to describe how parameters of the two discontinuous maps M ShapeRD and M TranslationRD are initialized at t = t m . To do this, we first construct a temporary distorted frame with coordinates ' x i , defined by</text>
<formula><location><page_32><loc_30><loc_30><loc_76><loc_32></location>¯ x i = M Translation · M Rotation · M ScalingRD ' x i . (103)</formula>
<text><location><page_32><loc_16><loc_24><loc_76><loc_30></location>The coordinates ' x i are the same as the post-merger distorted coordinates except that the pre-merger translation map M Translation is used in Eq. (103). We then represent the common apparent horizon (which is already known in ¯ x i coordinates) in ' x i coordinates:</text>
<formula><location><page_32><loc_27><loc_19><loc_76><loc_23></location>' x i AH ( ' θ C , ' φ C , t ) = ' x i AH0 ( t ) + ' n i ∑ /lscriptm ' S C /lscriptm ( t ) Y /lscriptm ( ' θ C , ' φ C ) . (104)</formula>
<text><location><page_32><loc_16><loc_14><loc_76><loc_19></location>The expansion coefficients ' S C /lscriptm ( t ) and the horizon expansion center ' x i AH0 ( t ) are computed using the (inertial frame) common horizon surface plus the (time-dependent) map defined in Eq. (103). Here ( ' θ C , ' φ C ) are polar coordinates centered about ' x i AH0 ,</text>
<text><location><page_33><loc_16><loc_79><loc_76><loc_88></location>and ' n i is a unit vector corresponding to the direction ( ' θ C , ' φ C ). Equation (104) is the same expansion as Eq. (36), except here we write the expansion to explicitly include the center ' x i AH0 ( t ). In Eq. (104) we write ' S C /lscriptm ( t ) and ' x i AH0 ( t ) as functions of time because we compute them at several discrete times surrounding the matching time t m . By finite-differencing in time, we then compute first and second time derivatives of ' S C /lscriptm and ' x i AH0 at t = t m .</text>
<text><location><page_33><loc_16><loc_68><loc_76><loc_79></location>Once we have ' S C /lscriptm and its time derivatives, we initialize λ C /lscriptm ( t m ) = -' S C /lscriptm ( t m ), where λ C /lscriptm ( t ) are the map parameters appearing in Eq. (72), and the minus sign accounts for the sign difference between the definitions of Eqs. (72) and (104). We do this for all ( /lscript, m ) except for /lscript = m = 0: we set λ C 00 ( t m ) = 0, thus defining the radius of the common apparent horizon in the post-merger grid frame. For all ( /lscript, m ) including /lscript = m = 0 we set dλ C /lscriptm /dt | t = t m = -d ' S C /lscriptm /dt | t = t m , and similarly for the second derivatives.</text>
<text><location><page_33><loc_16><loc_62><loc_76><loc_68></location>Note that the temporary distorted coordinates ' x i are incompatible with the assumptions of our control system, because the center of the excision boundary ' x i AH0 ( t ) in these coordinates is time-dependent. We therefore define new post-merger distorted coordinates ˆ x i by</text>
<formula><location><page_33><loc_29><loc_58><loc_76><loc_61></location>¯ x i = M TranslationRD · M Rotation · M ScalingRD ˆ x i , (105)</formula>
<text><location><page_33><loc_16><loc_56><loc_20><loc_58></location>where</text>
<formula><location><page_33><loc_39><loc_54><loc_76><loc_56></location>ˆ x i = ' x i -' x i AH0 ( t ) . (106)</formula>
<text><location><page_33><loc_16><loc_50><loc_76><loc_54></location>If we denote the map parameters in M Translation by T i 0 , the map parameters in M TranslationRD by T i , and if we denote M Rotation · M ScalingRD by the matrix M i j , then Eqs. (105) and (106) require</text>
<formula><location><page_33><loc_35><loc_47><loc_76><loc_48></location>T i ( t ) = T i 0 ( t ) + M i j ( t )' x j AH0 ( t ) . (107)</formula>
<text><location><page_33><loc_16><loc_37><loc_76><loc_45></location>Here we have assumed that f ( R ) appearing in Eq. (55) is unity in the vicinity of the horizon, so we can treat the translation map as a rigid translation near the horizon. We initialize T i and its first two time derivatives at t = t m according to Eq. (107). Note that the distorted-frame horizon expansion coefficients ' S C /lscriptm and hence the initialization of λ C /lscriptm ( t ) are unchanged by the change in translation map because ˆ x i and ' x i differ only by an overall translation, Eq. (106), which leaves angles invariant.</text>
<section_header_level_1><location><page_33><loc_16><loc_33><loc_42><loc_34></location>7. Control systems for efficiency</section_header_level_1>
<text><location><page_33><loc_16><loc_26><loc_76><loc_32></location>Although the most important use of control systems in SpEC is to adjust parameters of the mapping between the inertial and grid coordinates, another situation for which control systems are helpful is the approximation of functions that vary slowly in time, are needed frequently during the simulation, but are expensive to compute.</text>
<text><location><page_33><loc_16><loc_17><loc_76><loc_26></location>For example, the average radius of the apparent horizon, r AH , is used in the control system for ˙ λ 00 , which is evaluated at every time step. Evaluation at each time step is necessary to allow the control system for ˙ λ 00 to respond rapidly to sudden changes in the characteristic speed or the size of the excision boundary, as may occur after regridding, after mesh refinement changes, or when other control systems (such as size control) are temporarily out of lock.</text>
<text><location><page_33><loc_16><loc_14><loc_76><loc_17></location>However, computing the apparent horizon (and thus its average radius r AH ) at every time step is prohibitively expensive. To significantly reduce the expense,</text>
<text><location><page_34><loc_16><loc_85><loc_76><loc_88></location>we define a function with piecewise-constant second derivative, r appx AH ( t ), we define a control error</text>
<formula><location><page_34><loc_40><loc_82><loc_76><loc_85></location>Q = r AH -r appx AH , (108)</formula>
<text><location><page_34><loc_16><loc_75><loc_76><loc_83></location>and we define a control system that drives Q to zero. We then pass r appx AH ( t ) instead of r AH to those functions that require the average apparent horizon radius at each time step. The control error Q , and thus the expensive computation of the apparent horizon, needs to be evaluated only infrequently, i.e., on the timescale on which the average horizon radius is changing, which may be tens or hundreds of time steps.</text>
<section_header_level_1><location><page_34><loc_16><loc_72><loc_26><loc_73></location>8. Summary</section_header_level_1>
<text><location><page_34><loc_16><loc_58><loc_76><loc_70></location>In simulations of binary black holes, we use a set of time-dependent coordinate mappings to connect the asymptotically inertial frame (in which the black holes inspiral about one another, merge, and finally ringdown) to the grid frame in which the excision surfaces are stationary and spherical. The maps are described by parameters that are adjusted by a control system to follow the motion of the black holes, to keep the excision surface just inside the apparent horizons of the holes, and also to prevent grid compression. We take care to decouple the control systems and choose stable control timescales.</text>
<text><location><page_34><loc_16><loc_45><loc_76><loc_58></location>The scaling, rotation, and translation maps are used to track the overall motion of the black holes in the inertial frame. These are the most important maps during the inspiral phase of the evolution, as the shapes of the horizons remain fairly constant after the relaxation of the initial data. As the binary approaches merger, the horizon shapes begin to distort, and shape and size control start to become important. Shape and size control are especially crucial for unequal-mass binaries and for black holes with near-extremal spins. In the latter case, the excision surface (which must remain an outflow boundary with respect to the characteristics of the evolution system) needs to be quite close to the horizon.</text>
<text><location><page_34><loc_16><loc_37><loc_76><loc_44></location>In order to decouple the shape maps of the individual black holes, our grid is split by a cutting plane at which the shape maps reduce to the identity map. As the black holes merge, a skew map is introduced in order to align the cutting plane with the excision surfaces (see Fig. 5) and to minimize grid compression between the two black holes.</text>
<text><location><page_34><loc_16><loc_28><loc_76><loc_37></location>Finally, the implementation of CutX control was necessary to complete the merger of high mass-ratio configurations. In these systems, the excision boundaries approach the cutting plane asymmetrically as the black holes merge, thus compressing the grid between the cutting plane and the nearest excision boundary. CutX control translates the cutting plane to keep it centered between the two excision boundaries, alleviating this grid compression.</text>
<text><location><page_34><loc_16><loc_22><loc_76><loc_28></location>We find that we need all of the maps described in this paper with a sufficiently tight control system in order to robustly simulate a wide range of the parameter space of binary black hole systems [4, 24, 5]. Many of these maps and control systems are also used in our simulations of black hole - neutron star binaries [26, 27, 28].</text>
<section_header_level_1><location><page_34><loc_16><loc_19><loc_70><loc_20></location>Appendix A. Implementation of exponentially-weighted averaging</section_header_level_1>
<text><location><page_34><loc_16><loc_13><loc_76><loc_17></location>Our exponentially-weighted averaging scheme uses an averaging timescale τ avg to smooth noisy quantities F that are measured at intervals τ m := t k -t k -1 , where</text>
<text><location><page_35><loc_16><loc_84><loc_76><loc_88></location>F is the measured value of the control error Q , its integral, or its derivatives. Recall from Sec. 3.3 that we typically choose τ avg ∼ 0 . 25 τ d and τ m ∼ 0 . 075 min( τ d ).</text>
<text><location><page_35><loc_16><loc_82><loc_76><loc_85></location>The averaging is implemented by solving the following system of ordinary differential equations (ODEs):</text>
<formula><location><page_35><loc_41><loc_77><loc_76><loc_81></location>d dt W = 1 -W τ avg , (A.1)</formula>
<formula><location><page_35><loc_39><loc_74><loc_76><loc_77></location>d dt ( Wτ ) = t -Wτ τ avg , (A.2)</formula>
<formula><location><page_35><loc_37><loc_71><loc_76><loc_74></location>d dt ( WF avg ) = F -WF avg τ avg , (A.3)</formula>
<text><location><page_35><loc_16><loc_65><loc_76><loc_69></location>where the evolved variables are a weight factor W , the average value F avg , and the effective time τ at which F avg is calculated. The ODEs are solved approximately using backward Euler differencing, which results in the recursive equations</text>
<formula><location><page_35><loc_33><loc_61><loc_76><loc_64></location>W k = 1 D ( τ m + W k -1 ) , (A.4)</formula>
<formula><location><page_35><loc_34><loc_58><loc_76><loc_61></location>τ k = 1 DW k ( τ m t k + W k -1 τ k -1 ) , (A.5)</formula>
<formula><location><page_35><loc_32><loc_53><loc_76><loc_58></location>F k avg = 1 DW k [ τ m F ( t k ) + W k -1 F k -1 avg ] , (A.6)</formula>
<text><location><page_35><loc_16><loc_51><loc_76><loc_54></location>where D = 1 + τ m /τ avg . The recursion is initialized with W 0 = 0, τ 0 = t 0 , and F 0 avg = F ( t 0 ).</text>
<text><location><page_35><loc_16><loc_45><loc_76><loc_51></location>In the control law equations, Eqs. (15) and (16), we want to use the averaged value at t k instead of τ k . Therefore, to adjust for the offset induced by averaging, δt k = t k -τ k , we evaluate at t k an approximate Taylor series for F k avg expanded about τ k</text>
<formula><location><page_35><loc_34><loc_40><loc_76><loc_45></location>F avg ( t k ) = N -n ∑ m =0 δt m m ! ( d m F dt m ) k avg , (A.7)</formula>
<text><location><page_35><loc_16><loc_34><loc_76><loc_40></location>where n is defined by F := d n Q/dt n (for convenience, F represents the integral of Q when n = -1), and N is the same as in Eq. (11), where it is defined as the number of derivatives used to represent the map parameter. Note that this approximation matches the true Taylor series inasmuch as W is constant.</text>
<text><location><page_35><loc_16><loc_31><loc_76><loc_34></location>For the highest order derivative of Q , where n = N , we can no longer directly adjust for the offset because Eq. (A.7) reduces to</text>
<formula><location><page_35><loc_40><loc_28><loc_76><loc_30></location>F avg ( t k ) = F k avg . (A.8)</formula>
<text><location><page_35><loc_16><loc_21><loc_76><loc_27></location>We would need to take additional derivatives of Q to provide a meaningful correction, but this is exactly what we want to avoid because of the noisiness of derivatives. Therefore, to circumvent taking higher order derivatives, we substitute F avg ( t k ) into the control law, e.g. Eq. (21) for PID, and then solve for d N Q/dt N .</text>
<section_header_level_1><location><page_35><loc_16><loc_18><loc_70><loc_19></location>Appendix B. Details on computing the CutX map weight function</section_header_level_1>
<text><location><page_35><loc_16><loc_14><loc_76><loc_16></location>As discussed in Sec. 4.6, we use the map M CutX to control the location of the cutting plane in the last phase of the merger, as the distance between the excision boundaries</text>
<text><location><page_36><loc_16><loc_74><loc_76><loc_88></location>and the cutting plane becomes small. The value of the weight function ρ ( x i ) in Eq. (78) is zero in the spherical shells describing the wave zone and in the shells around the excision boundaries (see Fig. B1). The weight function is unity on the cutting plane, and on the 'cut-sphere' surfaces around either excision boundary. For the smaller excision surface, we have two such cut-sphere surfaces, and the value of the weight function is one within the region enclosed by these two cut-spheres and the cutting plane. The weight function transitions linearly from zero to one (from the magenta curves to the red curves in Fig. B1). In these transitional regions the value of ρ ( x i ) is computed as described below.</text>
<figure>
<location><page_36><loc_34><loc_55><loc_58><loc_73></location>
<caption>Figure B1. Schematic diagram indicating the way the M CutX weight function is defined. The value of the weight function ρ ( x i ) is one on the red curves and in the black region enclosed by red curves, it vanishes in the white regions (inside the two inner magenta curves and outside the outer magenta curve) and it changes linearly from zero to one in the gray regions between the red and magenta curves.</caption>
</figure>
<text><location><page_36><loc_16><loc_36><loc_76><loc_45></location>Consider the region spanning the volume between the spherical shells around the excision boundary H (with H = A,B ) and the cutting plane x = x 0 C , as shown in Fig. B2. (This is referred to in the code as the M region.) In order to calculate the value of ρ ( x i ) in this region, we shoot a ray from C i H in the direction of x i . This ray intersects both the outer spherical boundary of the shells and the cutting plane. The x -coordinate of the intersection with the shells will be</text>
<formula><location><page_36><loc_36><loc_29><loc_76><loc_35></location>x M = C 0 H + ( x 0 -C 0 H ) R H ∑ i ( x i -C i H ) 2 , (B.1)</formula>
<text><location><page_36><loc_16><loc_27><loc_76><loc_30></location>where R H is the outer radius of the spherical region around C i H . Then, for a point x i in the M region ρ is given by</text>
<formula><location><page_36><loc_39><loc_22><loc_76><loc_26></location>ρ ( x i ) = x 0 -x M x 0 C -x M . (B.2)</formula>
<text><location><page_36><loc_16><loc_14><loc_76><loc_21></location>All other transitional regions are delimited on both ends by spheres. Our computational domain has two zones of this type; one is depicted in Fig. B2 (labeled as the E region), while the other is seen in Fig. B3. Let x i P denote the 'center of projection,' i.e., the point toward which the unmapped radial grid lines of the given zone converge, and let x i S denote the center of a sphere of radius R (noting that</text>
<figure>
<location><page_37><loc_35><loc_70><loc_57><loc_88></location>
<caption>Figure B2. Schematic diagram illustrating the algorithm used to compute the M CutX weight function inside the cut-sphere. The point x i represents an arbitrary point in the M region. R H is the radius of the magenta sphere, and x M is the x 0 -component of the point on the magenta sphere that intersects the ray pointing from C i H toward x i . The weight function ρ ( x i ) is zero at the intersection of the ray with the magenta curve, it is unity at the intersection of the ray with the cutting plane, and changes linearly in between. Similarly, for a point in the E region, between the magenta sphere and the spherical part of the red cut-sphere, ρ ( x i ) changes linearly from zero (at the intersection of the ray with the inner, magenta sphere) to unity (at the intersection with the outer, red cut-sphere.)</caption>
</figure>
<text><location><page_37><loc_16><loc_47><loc_76><loc_53></location>the inner and outer spheres may have different centers). In Fig. B2 the center of projection coincides with C i H . In Fig. B3 we choose the center of projection to be on the cutting plane, at its intersection with the line segment connecting the centers of the two excision surfaces, x i C . To compute ρ ( x i ) in this type of region, we shoot a</text>
<figure>
<location><page_37><loc_34><loc_28><loc_58><loc_46></location>
<caption>Figure B3. Schematic diagram illustrating the algorithm used to compute the M CutX weight function in the region between the red cut-sphere and the outer, magenta sphere.</caption>
</figure>
<text><location><page_37><loc_16><loc_14><loc_76><loc_20></location>ray from x i P in the direction of x i . This ray intersects the two spheres that delimit the region. We define r 0 to be the distance between x i P and the point of intersection with the (magenta) sphere, for which ρ = 0; similarly, we define r 1 to be the distance between x i P and the point of intersection with the (red) sphere, for which ρ = 1. Then,</text>
<text><location><page_38><loc_16><loc_86><loc_27><loc_88></location>ρ ( x i ) is given by</text>
<text><location><page_38><loc_16><loc_80><loc_57><loc_83></location>where r = | x i -x i P | . To compute r 0 and r 1 , we must solve</text>
<formula><location><page_38><loc_40><loc_83><loc_76><loc_86></location>ρ ( x i ) = r -r 0 r 1 -r 0 , (B.3)</formula>
<formula><location><page_38><loc_34><loc_76><loc_76><loc_80></location>∑ i [ x i P + η ( X i -x i P ) -x i S ] 2 = R 2 (B.4)</formula>
<text><location><page_38><loc_16><loc_73><loc_76><loc_76></location>for the associated intersection point X i . The parameter η ( x i P , x i S , X i , R ) is defined as the positive solution of the quadratic system,</text>
<text><location><page_38><loc_16><loc_67><loc_20><loc_68></location>where</text>
<formula><location><page_38><loc_36><loc_68><loc_76><loc_72></location>η = B + √ B 2 -C, (B.5)</formula>
<formula><location><page_38><loc_35><loc_61><loc_76><loc_66></location>B = ∑ i ( X i -x i P )( x i S -x i P ) i ( X i -x i P ) 2 , (B.6)</formula>
<text><location><page_38><loc_16><loc_38><loc_76><loc_57></location>The input coordinates to M CutX are the coordinates ˜ x i that are obtained from the grid coordinates x i by the shape map, ˜ x i = M Shape ( x i ). As seen above, computation of the weight function at any point requires knowledge of the region containing that point, which in turn requires knowledge of its grid-frame coordinates x i . Therefore, to compute the weight function one must first compute x i = M -1 Shape (˜ x i ). When computing the forward CutX map, this inverse is done only once. However, the situation is more complicated when computing the inverse CutX map, because the grid-frame coordinates depend upon the inverse CutX map itself, which depends on the weight function. This leads to an iterative algorithm where we must call the inverse map function of M Shape from each iteration of the inverse map function of M CutX . This could easily become an efficiency bottleneck. We mitigate this by keeping M CutX inactive for most of the run, only activating it when it becomes crucial in order to avoid a singular map.</text>
<formula><location><page_38><loc_35><loc_56><loc_76><loc_64></location>∑ C = ∑ i ( x i S -x i P ) 2 -R 2 ∑ i ( X i -x i P ) 2 . (B.7)</formula>
<section_header_level_1><location><page_38><loc_16><loc_35><loc_54><loc_36></location>Appendix C. Estimation of zero-crossing times</section_header_level_1>
<text><location><page_38><loc_16><loc_26><loc_76><loc_33></location>Several of the control systems described here are designed to ensure that some measured quantity remains positive. For example, in Sec. 5.3, we demand that both the characteristic speed v at the excision boundary and the difference ∆ r between the horizon radius and the excision boundary radius remain positive. Similarly, in Sec. 4.6 we demand that each excision boundary does not cross the cutting plane.</text>
<text><location><page_38><loc_16><loc_20><loc_76><loc_25></location>Part of the algorithm for ensuring that some quantity q remains positive is estimating whether q is in danger of becoming negative in the near future, and if so, estimating the timescale τ on which this will occur. In this Appendix we describe a method of obtaining this estimate.</text>
<text><location><page_38><loc_16><loc_13><loc_76><loc_19></location>We assume the quantity q is measured at a set of (not necessarily equally-spaced) measurement times, and we remember the values of q at several (typically 4) previous measurement times. At each measurement time t 0 we fit these remembered values to a line q ( t ) = a + b ( t -t 0 ). The fit gives us not only the slope b and the intercept a , but also</text>
<text><location><page_39><loc_16><loc_79><loc_76><loc_88></location>error bars for these two quantities δa and δb . Assuming that the true q ( t ) lies within the error bars, the earliest time at which q will cross zero is t = t 0 +( -a + δa ) / ( b -δb ), and the latest time is t = t 0 +( -a -δa ) / ( b + δb ). If both of these times are finite and in the future, then we regard q as being in danger of crossing zero, and we estimate the timescale on which this will happen as τ = -a/b .</text>
<section_header_level_1><location><page_39><loc_16><loc_77><loc_30><loc_78></location>Acknowledgments</section_header_level_1>
<text><location><page_39><loc_16><loc_58><loc_76><loc_75></location>We would like to thank Harald Pfeiffer for useful discussions. We would like to thank Abdul Mrou'e for performing simulations that led to the development and improvement of some of the mappings presented in this paper. We gratefully acknowledge support from the Sherman Fairchild Foundation; from NSF grants PHY-0969111 and PHY1005426 at Cornell, and from NSF grants PHY-1068881 and PHY-1005655 at Caltech. Simulations used in this work were computed with the SpEC code [17]. Computations were performed on the Zwicky cluster at Caltech, which is supported by the Sherman Fairchild Foundation and by NSF award PHY-0960291; on the NSF XSEDE network under grant TG-PHY990007N; and on the GPC supercomputer at the SciNet HPC Consortium [29]. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research FundResearch Excellence; and the University of Toronto.</text>
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</document>
[
{
"title": "Dynamical Excision Boundaries in Spectral Evolutions of Binary Black Hole Spacetimes",
"content": "1 2 Daniel A. Hemberger , Mark A. Scheel , Lawrence E. Kidder 1 , B'ela Szil'agyi 2 , Geoffrey Lovelace 3 , Nicholas W. Taylor 2 , and Saul A. Teukolsky 1 1 Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853 2 Theoretical Astrophysics 103-33, California Institute of Technology, Pasadena, CA 91125 3 Gravitational Wave Physics and Astronomy Center, California State University Fullerton, Fullerton, CA 92831 Abstract. Simulations of binary black hole systems using the Spectral Einstein Code (SpEC) are done on a computational domain that excises the regions inside the black holes. It is imperative that the excision boundaries are outflow boundaries with respect to the hyperbolic evolution equations used in the simulation. We employ a time-dependent mapping between the fixed computational frame and the inertial frame through which the black holes move. The time-dependent parameters of the mapping are adjusted throughout the simulation by a feedback control system in order to follow the motion of the black holes, to adjust the shape and size of the excision surfaces so that they remain outflow boundaries, and to prevent large distortions of the grid. We describe in detail the mappings and control systems that we use. We show how these techniques have been essential in the evolution of binary black hole systems with extreme configurations, such as large spin magnitudes and high mass ratios, especially during the merger, when apparent horizons are highly distorted and the computational domain becomes compressed. The techniques introduced here may be useful in other applications of partial differential equations that involve time-dependent mappings. PACS numbers: 04.25.D-, 04.25.dg, 02.70.Hm",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Feedback control systems are ubiquitous in technological applications. They are found, for example, in thermostats, autopilots, chemical plants, and cruise control in automobiles. The purpose of a control system is to keep some measured output (such as the temperature in a room) at some desired value by adjusting some input (such as the power to a furnace). In the last few years, feedback control systems have also found applications in the field of numerical relativity, particularly in simulations of binary black hole systems that employ spectral methods and excision techniques [1, 2, 3, 4, 5]. Black hole excision is a means of avoiding the physical singularities that lurk inside black holes. The idea is to solve Einstein's equations only in the region outside apparent horizons, cutting out the region inside the horizons. The boundaries of the excised regions are called excision boundaries. Causality ensures that the excision boundaries and the excised interiors cannot affect the physics of the exterior solution, and an appropriate hyperbolic formulation of Einstein's equations [6, 7] can ensure that gauge and constraint-violating degrees of freedom also do not propagate out of the excised region. Excision is straightforward for black holes that remain stationary in the coordinates that are used in the simulation, but excision becomes more complicated when the black holes move or change shape. For numerical methods based on finite differencing, the excision boundaries can be changed at every time step by activating or deactivating appropriate grid points and adjusting differencing stencils [8, 9, 10, 11, 12, 13, 14, 15]. However, for spectral numerical methods, there is no equivalent of deactivating individual grid points; instead, spectral methods are defined in finite extended spatial regions with smooth boundaries. The nearest equivalent to the finite-difference excision approach would be interpolating all variables to a new slightly offset grid at every time step, which would be computationally expensive. Therefore, spectral numerical methods need a different approach to reconcile the need for moving black holes with the need for a fixed excision boundary inside of each black hole. The solution [1] to this problem adopted by our group makes use of multiple coordinate systems. We call 'inertial coordinates' those coordinates that asymptotically correspond to an inertial observer; in these coordinates the black holes orbit each other, have distorted shapes, and approach each other as energy is lost to gravitational radiation. Spectral methods are applied in another coordinate system, 'grid coordinates', in which the excision boundaries are spherical and stationary. We connect grid coordinates with inertial coordinates by means of an analytic mapping function M that depends on some set of time-dependent parameters λ ( t ). These parameters must be continually adjusted so that each spherical, stationary grid-frame excision boundary is mapped to a surface in the inertial frame that follows the motion and the shape of the corresponding black hole as the system evolves. It is this adjustment of each parameter λ ( t ) that is accomplished by means of feedback control systems, one control system per parameter. In this paper, we describe in detail the mapping functions and the corresponding feedback control systems that we use to handle black hole excision with spectral methods. Some earlier implementations have been described previously [5, 1, 2, 4, 16], but there have been many improvements that now allow spectral excision methods to produce robust simulations of binary black hole systems, including those with unequal masses, high spins, and precession. We describe these improvements here. In Sec. 2 we review control theory and present simple examples of control systems. In Sec. 3 we discuss the implementation of control systems in the SpEC [17] code. Sections 4 and 5 detail the coordinate mappings used in SpEC and the feedback control systems used to control them. In Sec. 6 we describe the transition to the post-merger domain with a single excision boundary. In Sec. 7 we describe applications of control systems in SpEC besides the ones used to adjust map parameters. We summarize in Sec. 8.",
"pages": [
2,
3
]
},
{
"title": "2. Control Theory",
"content": "To motivate control theory, we begin with a simple example: cruise control in an automobile. Suppose we wish to control the speed v of a car that is driving up an incline of angle θ . The equation of motion for this system is where m is the mass of the car, g is the gravitational acceleration, η is a drag coefficient, and F is a force supplied by the car's engine. We wish to determine F so as to cause the car to maintain a speed of v = v 0 , even if the angle θ changes as the car climbs the incline. To do this, we choose the force at time t to be: where the control system is turned on at time t = 0, where K I and K P are constants, and where Q ( t ) = v 0 -v ( t ) is the quantity that we wish to drive to zero. We call Q the control error. Substituting Eq. (2) into Eq. (1) and differentiating with respect to time yields which is the equation for a damped, forced harmonic oscillator. By choosing K P to produce critical damping, and choosing K I to set a timescale, this choice will drive v toward v 0 as desired. The basic structure of a control system is easily understood as a feedback loop (see Fig. 1). The simulation (e.g., a binary black hole simulation, or a car driving up an incline) produces some measure of error, Q ( t ), that defines the deviation from some desired target value. This error acts as the input for the control system, which then computes a control signal U ( t ) (e.g., the derivative of a map parameter, or the force supplied by a car's engine) that will minimize the error Q ( t ) when fed back into the simulation. A simple and effective way to compute U ( t ) is to make it a linear combination of the error, Q ( t ), and integrals and/or derivatives of the error. The term proportional to the error acts to reduce the deviation from the desired value, the terms proportional to derivatives of the error act to oppose rapid deviations, and the terms proportional to the integrals act to reduce any persistent deviation or offset that accumulates over time. In the cruise control example, we used a proportional and an integral term only in Eq. (2). We now turn to another example of a control system that is more closely related to the way we use control systems in binary black hole simulations. Consider two coordinate systems, ( x, t ) and (¯ x, ¯ t ), that are related by the map We wish to control the parameter V ( t ) so that a wave f (¯ x, ¯ t ) = f (¯ x -c ¯ t ) that propagates at speed ¯ v = c in the (¯ x, ¯ t ) coordinates will propagate at some arbitrary desired speed v d in the ( x, t ) coordinates. According to Eq. (4), the speed of the wave in the ( x, t ) coordinates is v = c -V ( t ). Therefore we define a control error Q to be and we construct a control system that drives this control error to zero. If we choose the control signal U ( t ) to be d 2 V/dt 2 , then the simplest feedback loop that can be constructed uses only a term proportional to the error and amplified by a 'gain' K P . Then the evolution of V ( t ) is given by The solution to this equation is of the form where α := √ K P , which is oscillatory for K P > 0 and divergent for K P < 0. Thus we see that adding only a proportional term to the control signal U ( t ) is insufficient, since it does not reduce the control error Q . However, if we add a derivative term to the feedback equation, then the solution is of the form where β := 1 2 √ 4 K P -K 2 D . This solution is stable with an exponentially damped envelope when 4 K P ≥ K 2 D , which will cause v → v d as t →∞ . Notice that this control system allows us to choose a V ( t ) such that v is the opposite sign of c and the wave is left-going in the ( x, t ) frame instead of right-going. This behavior can be seen in the example in Fig. 2. The overdamped solution (dotted line in Fig. 2) has a persistent offset that can be ameliorated by adding an integral term to the feedback equation, as was done in the cruise control example. In principle we could continue to add more terms, but in practice, it is usually sufficient to use a PID (proportional-integral-derivative) controller, which has terms proportional to the control error, its integral, and its derivative. When the underlying system is unknown, this is the best controller to use [18].",
"pages": [
3,
4,
5
]
},
{
"title": "3. Control Systems in SpEC",
"content": "Control systems are used in SpEC for several purposes. The most important is their role in handling moving, excised black holes in a spectral evolution method. We use a dual-frame method [1] in which the grid is fixed in some coordinates ( t, x i ) but the components of dynamical fields are expressed in a different coordinate system ( ¯ t, ¯ x i ). We call ( t, x i ) the grid coordinates and ( ¯ t, ¯ x i ) the inertial coordinates . Figure 3 shows an example domain decomposition in both coordinate systems. The two coordinate systems are connected by a map M that depends on time-dependent parameters λ ( t ). The excision boundaries are exactly spherical and stationary in grid coordinates. In inertial coordinates, the apparent horizons move and distort as determined by the solution of Einstein's equations supplemented by our gauge conditions. The parameters λ ( t ) need to be controlled so that the excision boundaries in the inertial frame follow the motion and shapes of the apparent horizons. We use control systems to accomplish this. The particular maps that we use will be discussed in Sec. 4. In this section we will describe how we construct the control system for a general parameter λ ( t ), including how we define the relationship between the control error Q ( t ) and the control signal U ( t ), how we smooth out noise in the control system, and how we dynamically adjust the feedback parameters and timescales.",
"pages": [
5
]
},
{
"title": "3.1. Definition of control errors and control signals",
"content": "We represent a general time-dependent map parameter λ ( t ) as a polynomial in time with a piecewise constant N th derivative: where for each time interval t i ≤ t < t i +1 the quantities λ n i are constants. At the beginning of each new time interval t i , we set the constants λ n i in Eq. (11) as follows. First for n = N we set λ N i = U ( t i ), where U ( t ) is the control signal defined in detail below. For n < N we set λ n i = d n λ/dt n | t = t i , where the derivative is evaluated at the end of the previous time interval. In this way all the derivatives of λ ( t ) except the N th derivative are continuous across intervals. The goal will be to compute the control signal U ( t ) so as to drive the map parameter λ ( t ) to some desired behavior. Before we describe how to compute the control signal U ( t ), we first discuss the control error Q , which will be used in the computation of U ( t ). To appropriately define the control error Q , one must answer the question of how a small change in the map parameter corresponds to a change in the observed variables. If the control error is defined to be too large, then the controller will consistently overshoot its target, potentially leading to unstable behavior; conversely, if the control error is defined to be too small, then the controller may never be able to reach its target value. If there exists a target value of λ ( t ), call it λ target , that does not depend on the map but may depend on other observable quantities in the system (call them A , B ,. . . ), then we define The goal is to drive Q to zero and thereby drive λ to λ target . If instead, as often happens for nonlinear systems, the target value of λ depends on λ itself, even indirectly, then we define Q differently using a generalization of Eq. (12): we require that λ attains its desired value when Q → 0, and we require that The primary motivation for this condition is its anticipated use in relating time derivatives of Q to those of λ (see Eq. (18), below). Note that in either Eqs. (12) or (13), Q could in principle be multiplied by an arbitrary factor; however, if this were done, that factor would need to be taken into account in the computation of the control signal U ( t ) below. Without loss of generality we assume no additional scaling. In the case of several map parameters λ a ( t ) with corresponding Q a , where a is an index that labels the map parameters, the desired value of some λ a may depend on a different map parameter λ b . In this case, we generalize Eq. (13) and require that each Q a satisfy where δ ab is a Kronecker delta. This criterion ensures that we can treat each λ a independently when all control errors are small. A way of understanding Eq. (14) is to consider a set of Q ' a that are obtained via Eq. (13) without regard to coupling between different λ a . Then a set of Q a satisfying Eq. (14) can be obtained by diagonalizing the matrix ∂Q ' a /∂λ b . In the remainder of this section we assume that if there are multiple map parameters λ a , the corresponding control errors Q a satisfy Eq. (14). We therefore drop the a indices and write equations for U ( t ) in terms of a single Q ( t ) satisfying Eq. (13) that represents the control error for a single λ ( t ). The control error Q ( t ) is a function of several observables. In the case of the mapping functions that are designed to move and distort the excision boundaries to follow the motion and shapes of the apparent horizons, Q ( t ) is some function of the current position or shape of one or more apparent horizons. The precise definition of Q ( t ) is different for each map parameter, and depends on the details of how each map parameter couples to the observables. We will discuss the control error Q for each of the map parameters in Sec. 4. But it is not necessary to know the exact form of the control error in order to compute the control signal U ( t ); it suffices to know only that U ( t i ) is equal to λ N i in Eq. (11), and that the control error Q obeys Eq. (13). We now turn to the computation of the control signal U ( t ). There is some flexibility in the control law determining U ( t ), so long as key feedback mechanisms are in place (as shown in Sec. 2). In SpEC, we use either a standard PID controller, or a special PD (proportional-derivative) controller, where typically K = 2. We set the constants a k so that the system damps Q to zero on some timescale τ d that we choose. We assume that τ d is longer than the interval t i +1 -t i defined in Eq. (11), so that we can approximate Q ( t ) and λ ( t ) as smooth functions rather than as functions with piecewise constant N th derivatives. Under this assumption, we write We also write In the first line we have neglected the time dependence of other parameters besides λ that enter into Q under the assumption that the control system timescale is shorter than the timescales of the quantities that we want to control. In the second line we have used Eq. (13) and we have assumed that Q is small. Similarly we write where in the second line we have retained only terms of linear order in dQ/dt (and therefore in dλ/dt ). For higher derivatives we continue to retain only terms linear in Q and its derivatives, so from Eq. (17) we obtain For the PID controller and N = 2, combining Eqs. (15) and (20) yields If we choose a 0 = 1 /τ 3 d , a 1 = 3 /τ 2 d , and a 2 = 3 /τ d , then the solution to Eq. (21) will be exponentially damped on the timescale τ d , The same exponential damping holds for the PD controller, Eq. (16), for appropriate choices of the parameters a k . For K = 2 and N = 3, the parameters a k are identical to those in the PID case above.",
"pages": [
6,
7,
8
]
},
{
"title": "3.2. Averaging out noise",
"content": "The PID controller, Eq. (15), is computed by measuring the control error Q , its time integral, and its time derivative. The PD controller, Eq. (16), may require multiple derivatives of Q ( t ) depending on the order K . Generally only Q , and not its derivatives or integrals, is available in the code at any given time step. The simplest way to compute the derivatives of Q is by finite differencing in time, and the simplest way to compute the integral is by a numerical quadrature. We measure the control error Q at time intervals of length τ m , where we choose τ m < t i +1 -t i . The measuring time interval τ m is then used as the time step for finite difference stencils and quadratures. The measured Q is typically a function of apparent horizon locations or shapes, and this measured Q includes noise caused by the finite resolution of the evolution, and the finite residual and finite number of iterations of the apparent horizon finder. Taking a numerical derivative of Q amplifies the noise, and then this noise is transferred to the control signal via Eqs. (15) or (16), and then to the map. If the noise amplitude is too large, the control system will become unstable. The PID controller generally handles noise better than the PD controller for two reasons. First, each successive numerical derivative amplifies noise even further, so using only one derivative instead of two (or more) results in a more accurate control signal. Second, the inclusion of an integral term acts to further smooth the control signal. In some cases, however, the noise in Q can be problematic even for the PID controller. In these cases we implement direct averaging of the control error in one of two ways: 1) we perform a polynomial fit of order N to the previous M measurements of the control error, where M > N , or 2) we perform an exponentially-weighted average, with timescale τ avg , of all previous control error measurements and their derivatives and integrals. The latter is our preferred method, which we describe in detail in Appendix A.",
"pages": [
8,
9
]
},
{
"title": "3.3. Dynamic timescale adjustment",
"content": "In the previous section we have identified four timescales relevant for each control system. The first is the damping timescale τ d ; this describes how quickly the control error Q ( t ) falls to zero, and therefore how quickly the map parameter λ ( t ) approaches its desired value. The second timescale is the control interval ∆ t i := t i +1 -t i , which represents how often the N th derivative of the function λ ( t ) in Eq. (11) is updated. The third is the measurement timescale τ m , which indicates how often the control error Q is measured. The fourth is the averaging timescale τ avg , which is used to smooth the control error Q (and its derivatives and integrals) for use in computing the control signal U ( t ). These timescales are not all independent; for example we have assumed ∆ t i < τ d in deriving Eq. (21), and we have assumed τ m < ∆ t i so that we can obtain smooth measurements of the derivatives of Q . Because binary black hole evolutions are nonlinear dynamic systems, we adjust the damping timescale, τ d , throughout the simulation. We then set the three other timescales, ∆ t i , τ m , and τ avg , based on the current value of the damping timescale τ d , as we now describe. The timescale τ d should be shorter than the timescale on which the physical system changes; otherwise, the control system cannot adjust the map parameters quickly enough to respond to changes in the system. But if the timescale τ d is too small, then the measurement timescale τ m on which we compute the apparent horizon must also be small, meaning that frequent apparent horizon computations are needed; this is undesirable because computing apparent horizons is computationally expensive. We would like to adjust τ d in an automatic way so that it is relatively large during the binary black hole inspiral, decreases during the plunge and merger, and increases again as the remnant black hole rings down. In the canonical language of control theory, we would like to implement 'gain scheduling' [19], tuning the behavior of the control system for different operating regimes. We do this as follows: For all map parameters (except the /lscript = 0 component of the horizon shape map, which is treated differently; see Sec. 5.3), the damping timescale is a generic function of Q and ˙ Q , i.e., the error in its associated map parameter and its derivative. Whenever we adjust the control signal U ( t ) at interval t i , we also tune the timescale in the following way: where typically Here Q Min t and Q Max t are thresholds for the control error Q . The idea is to keep | Q | < Q Max t so that the map parameters are close to their desired values, but to also keep | Q | > Q Min t because an unnecessarily small | Q | means unnecessarily small timescales and therefore a large computational expense (because the apparent horizon must be found frequently). For binary black hole simulations where the holes have masses M A and M B , we find that the following choices work well: Once we have adjusted the timescale τ d for every control system, we then use these timescales to choose the times t i +1 in Eq. (11) at which we update the polynomial coefficients λ n i in the map parameter expression λ ( t ), where typically α d = 0 . 3, and the minimum is taken over all map parameters (except for the /lscript = 0 component of the horizon shape map). This ensures that the coefficients λ n i , are updated faster than the physical system is changing, and faster than the control system is damping. For α d too large, we find that the control system becomes unstable. We also use the timescale τ d to choose the interval τ m at which we measure the control error. For many map parameters, the associated control error is a function of apparent horizon quantities, which is why we desire τ m to be as large as possible. But a certain number of measurements are needed for each ∆ t i so that the control signals, defined in Eqs. (15) and (16), are sufficiently accurate and the control system is stable. We choose where α m is typically between 0 . 25 and 0 . 3. In other words, we measure the control error three or four times before we update the control signal. This also ensures that the averaging timescale is greater than the measurement timescale, as we typically choose τ avg ∼ 0 . 25 τ d .",
"pages": [
9,
10
]
},
{
"title": "4. Control Systems for Maps",
"content": "In a SpEC evolution, we transform the grid coordinates, x i , into inertial coordinates, ¯ x i , through a series of elementary maps as in [1, 2, 3, 4, 5]. Several maps have been added and many improvements have been made since their original introduction. The full transformation is ¯ x i = M x i , where Below we will describe each of these maps and how we measure the error in their parameters. Before we do this, however, we describe our domain decomposition and how we measure apparent horizons, because information from the grid and the horizons is used to determine the maps. In the grid coordinates x i , the domain decomposition looks like Fig. 4. There are two excision boundaries, A and B , which are spheres in grid coordinates. The gridcoordinate centers of these excision boundaries we will call C i H , where H is either A or B . The excision boundaries (and therefore C i H ) remain fixed throughout the evolution. The purpose of many of the maps is to move the mapped centers ¯ C i H := M ( C i H ) along with the centers of the apparent horizons. We measure the apparent horizons in an intermediate frame ˆ x i , which we call the distorted frame . This frame is connected to the grid frame by the map We will discuss exactly what M Shape and M CutX are below, but for now we need only demand that M Distortion has two properties: The first is that it leaves the centers of the excision boundaries invariant, i.e., The second property is that at each excision boundary H , M Distortion leaves angles invariant. That is, if we define grid-frame polar coordinates ( r H , θ H , φ H ) centered about excised region H in the usual way, and similarly for polar coordinates (ˆ r H , ˆ θ H , ˆ φ H ) centered about ˆ C i H in the ˆ x i frame, then the second property means that on the excision boundary. Note that the index i on spatial coordinates ranges over (0 , 1 , 2) in this paper. We find the apparent horizons in the frame ˆ x i by a fast flow method [20]. Each horizon is represented as a smooth surface in this frame, and is parameterized in terms of polar coordinates centered around ˆ C i H : the radius of the horizon at each ( ˆ θ H , ˆ φ H ) is given by where again the index H labels which excision boundary is enclosed by the apparent horizon. Here Y /lscriptm ( ˆ θ H , ˆ φ H ) are spherical harmonics and ˆ S H /lscriptm are expansion coefficients that describe the shape of the apparent horizon. We search for apparent horizons in the distorted frame ˆ x i rather than in the grid frame x i or the inertial frame ¯ x i because this simplifies the formulae for the control systems. In particular, this choice decouples the control errors of the shape map Q /lscriptm (defined in Sec. 4.5, Eq. (77)) for different values of ( /lscript, m ), and it decouples the errors Q /lscriptm from the control errors of the maps connecting the distorted and inertial frames. For each surface H , we define the center of the apparent horizon where the integrals are over the surface. In the code, it suffices to use the following approximation of Eq. (37), which becomes exact as the surface becomes spherical: Here we have used the property ˆ C i H = C i H , Eq. (31). We distinguish the center C i H of the excision boundary from the center ˆ ξ i H of the corresponding apparent horizon. The former is fixed in time in the grid frame, but the latter will change as the metric quantities evolve. The purpose of several of the maps (namely M Scaling , M Rotation , and M Translation described below) is to ensure that ˆ ξ i H -C i H is driven toward zero; i.e., to ensure that the centers of the excision boundaries track the centers of the apparent horizons. We now describe each of the maps comprising M and how we measure the error in their parameters. The order in which we discuss the maps is not the same as the order in which the maps are composed so that we can discuss the simplest maps first. To produce less cluttered equations in the following descriptions of the maps, we omit accents on variables that represent specific frames (i.e. we just write x instead of ˆ x or ¯ x ) whenever the input or output frame of the map is unambiguous or explicitly stated.",
"pages": [
11,
12
]
},
{
"title": "4.1. Scaling",
"content": "The scaling map, M Scaling , causes the grid to shrink or expand (and the excision boundaries to move respectively closer together or farther apart) in the inertial frame, thereby allowing the grid to follow the two black holes as their separation changes. This map transforms the radial coordinate with respect to the origin (i.e., the center of the outermost sphere in Fig. 4), such that the region near the black holes is scaled uniformly by a factor a and the outer boundary is scaled by a factor b , Here R is the radial coordinate, R OB is the radius at the outer boundary, a is a parameter that will be determined by a control system, and b is another parameter that will be determined empirically. In [1], the scaling map was simply x i ↦→ a ( t ) x i , which is recovered by Eq. (41) if b = a . In that case, as the black holes inspiral together and a decreases, the outer boundary of the grid decreases as well. For long evolutions, the outer boundary decreases so much that we can no longer extract gravitational radiation far from the hole. The addition of b to the map alleviates this difficulty, allowing the motion of the outer boundary to be decoupled from the motion of the holes. We choose b by an explicit functional form which is designed to keep the outer boundary from shrinking rapidly, but to allow the boundary to move inward at a small speed, so that zero-speed modes are advected off the grid (and thus need no boundary condition imposed on them). We choose a cubic function of time in Eq. (42) because we have found that at least the first two time derivatives of b ( t ) must vanish initially, or else a significant ingoing pulse of constraint violations is produced at the outer boundary. We control the scale factor a of this map using the error function where We assume that the separation vector C i A -C i B in the grid frame is parallel to the x -axis. The idea is that the distorted-frame separation of the horizon centers along the x -axis is driven to be the same as the separation of the excision boundary centers. To show that the control system for a obeys Eq. (13), we consider the change in Q a under variations of a with all other maps held fixed, and with the inertial-coordinate centers ¯ ξ i H of the horizons held fixed. ‡ This means that the distorted-frame centers of the horizons ˆ ξ i H appearing in Eq. (44) change with variations of a . The second term in Eq. (41) is small when evaluated near the horizon where ( R/R OB ) 2 /lessmuch 1, so we can write the action of the scaling map as and therefore under variations δa , Taking variations of Eq. (43), using Eq. (46) to substitute for δ ˆ ξ i H , and noting that the excision-boundary centers C i H are constants and do not vary with a , we obtain so that Eq. (13) is satisfied. To verify that the more general decoupling equation, Eq. (14), is satisfied for the scaling map, we must show, in addition to Eq. (47), that the quantity dx 0 defined in Eq. (44) is invariant under the action of the other maps, at least in the limit that all the control errors Q are small. We consider each map in turn. The translation map moves both apparent horizons together, so it leaves dx 0 invariant. Changes in the rotation map parameters will change dx 0 only by an amount proportional to the control errors of the rotation map (see Eqs. (53) and (54) below). The skew map (below) leaves the centers of the excision boundaries invariant. Because the intent of the rotation, translation, and scaling maps is to drive the centers of the apparent horizons toward the centers of the excision boundaries, this means that the skew map changes dx 0 by an amount proportional to the control errors of the rotation, translation, and scaling maps. Finally, the shape and CutX maps connect the distorted and grid frames, so they cannot affect dx 0 .",
"pages": [
12,
13,
14
]
},
{
"title": "4.2. Rotation",
"content": "The rotation map, M Rotation , is a rigid 3D rotation about the origin that tracks the orbital phase and precession of the system, The pitch and yaw map parameters ( ϕ, ϑ ) are controlled so as to align the line segment connecting the apparent horizon centers with the distorted-frame x -axis. Note that the map parameters ( ϕ, ϑ ) are functions of time, and are not to be confused with the polar coordinates ( φ H , θ H ) centered about each excision boundary. Then the control error is given by In the case of extreme precession where ϕ → π/ 2, these equations are insufficient because Q ϑ diverges. Our solution is to use quaternions, which avoid this singularity (for a complete discussion, see [21]). The control errors Q ϑ and Q ϕ obey Eq. (14). To show this, consider variations of ϑ and ϕ with other maps held fixed, and with the inertial-coordinate centers ¯ ξ i H of the horizons held fixed. The rotation map, Eq. (48), implies that under these variations, Multiplying this equation by R -1 we obtain which yields We can now verify Eq. (14) for the special case where the indices a and b in Eq. (14) are either ϑ and ϕ . This result is obtained in a straightforward way by differentiating Eqs. (49) or (50) with respect to ϑ or ϕ , and substituting Eqs. (53) or (54). In addition, the control errors Q ϕ and Q ϑ are independent (to leading order in the control errors) of changes in the parameters of the other maps that make up M : Variations of Eqs. (49) and (50) with respect to the scaling map parameter a are zero, because both the numerator and denominator of Q ϕ and Q ϑ scale in the same way with a . Similarly, Q ϕ and Q ϑ are independent of the translation map, since both apparent horizons are translated by the same amount. The skew map can change Q ϕ and Q ϑ , but only by an amount proportional to control errors, because the skew map leaves C i H invariant and other maps ensure that ˆ ξ i H are close (within a control error) to C i H . The shape and CutX maps cannot affect Q ϕ and Q ϑ because those maps connect the grid and distorted frames (and therefore they cannot change the distorted-frame horizon centers ˆ ξ i H ).",
"pages": [
14,
15
]
},
{
"title": "4.3. Translation",
"content": "The translation map, M Translation , transforms the Cartesian coordinates, x i , according to where f ( R ) is a Gaussian centered on the origin with a width set such that f ( R ) falls off to machine precision at the outer boundary radius, and T i ( t ) are translation parameters that are adjusted by a control system. The translation map moves the grid to account for any drift of the 'center of mass' of the system (as computed assuming point masses at the apparent horizon centers) in the inertial frame. This drift can be caused by momentum exchange between the black holes and the surrounding gravitational field [22, 23], by anisotropic radiation of linear momentum to infinity, or by linear momentum in the initial data. This is the third map (the other two are rotation and scaling) that drives the centers of the apparent horizons toward the centers of the excision boundaries. The apparent horizon centers ˆ ξ i A and ˆ ξ i B represent six degrees of freedom: one is fixed by the scaling map, two by rotation, and three by translation. The control errors will be more complicated than for the other control systems because translation and rotation do not commute. We define the control errors Q i T for each of the translation directions i = 0 , 1 , 2 as where R is the rotation matrix in Eq. (48), and P is some matrix yet to be determined, which may depend on the rotation parameters ( ϕ, ϑ ) and on the constants C i H , but which may not depend on the translation parameters T i . This control error must have the property that Q i T = 0 when ˆ ξ i H = C i H , so our first restriction on P is that it satisfies To check Eq. (14), we note that near the black holes we can neglect the last term in Eq. (41) and we can use f ( R ) ∼ 1 in Eq. (55), so that the apparent horizon centers in the inertial and distorted frames are related by Inserting Eq. (58) into Eq. (56), we obtain The second term in Eq. (59) is the only term that depends on the translation parameter T i , so we have When varying map parameters, the inertial-frame horizon centers remain fixed, so the only other term in Eq. (59) that depends on map parameters is the last term, which depends on ( ϑ, ϕ ) because of the rotation matrices and because of the ( ϑ, ϕ ) dependence in P . We can therefore write the variation of Q i T with respect to ( ϑ, ϕ ) as where W stands for either ϑ or ϕ , and where in the last line we have used Eq. (58). To obey Eq. (14), ∂Q i T /∂W must either be zero, or on the order of a control error Q . For i = 1 , 2, the quantity ( ˆ ξ i A -ˆ ξ i B ) is proportional to the control error of the rotation map, Eqs. (49) and (50), so these terms can be neglected in Eq. (62). However, for i = 0, ( ˆ ξ i A -ˆ ξ i B ) is not proportional to a control error; instead ( ˆ ξ 0 A -ˆ ξ 0 B ) is driven to a constant finite value of C 0 A -C 0 B by the scaling control system. Therefore, in order to satisfy Eq. (14), we require We find that we can satisfy both Eqs. (57) and (63) by choosing Here we have again assumed that the separation between the centers of the excision boundaries is parallel to the x -axis, i.e., C 1 A = C 1 B and C 2 A = C 2 B .",
"pages": [
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},
{
"title": "4.4. Skew",
"content": "In Fig. 4, a prominent feature of the domain decomposition is a plane (a vertical line in the two-dimensional figure) that is perpendicular to the grid-frame x -axis and lies between the two excision boundaries A and B . We call this plane the 'cutting plane'. The skew map, M Skew , acts on the distorted-frame coordinates, in which the cutting plane is still perpendicular to the x -axis. The skew map leaves the coordinates y, z unchanged, but changes the x -coordinate in order to give a skewed shape to the cutting plane, as shown in Fig. 5. Let x i C be the intersection point of the line segment connecting the excision boundary centers and the cutting plane. The action of the skew map is defined as where W is a radial Gaussian function centered around x i C , and the angles Θ j ( t ) are the time-dependent map parameters. For j = 1 , 2 the parameter Θ j ( t ) is the angle between the mapped and unmapped x j -axis when projected into the x 3 -j = x 3 -j C plane. Note that the line intersecting x i C and parallel to the x -axis is left invariant by the skew map § . The width of the Gaussian W is set such that W is below machine precision at the innermost wave extraction sphere. This implies that the sphericalshell subdomains used to evolve the metric in the wave zone will not be affected by the skew map. The purpose of the skew map is to (as much as possible) align the cutting plane with the surfaces of the apparent horizons in the region where the surfaces are closest to the cutting plane. We derive the control system in charge of setting the parameters Θ j by the following condition: We demand that the angle between the mapped cutting plane and the x -axis at x i C be driven to the (weighted) average of the angles at which the mapped apparent horizons intersect the same x -axis. The input coordinates to the skew map are the coordinates in the distorted frame. For each horizon, we therefore measure an angle in the distorted frame as follows: Let x Int H be the distorted-frame x -coordinate at which apparent horizon H intersects the line segment connecting the centers of the excision boundaries. For j = 1 , 2, we calculate the normal to the surface at this intersection point. This normal is projected into the x 3 -j = x 3 -j C plane. We define Θ j H as the angle between the projected normal and the x -axis. Thus, projecting the normal into the y = const . plane gives Θ z H and vice versa (see Fig. 6). We then compute a weighted average of the Θ j H such that the horizon closer to the cutting plane has a larger effect on the skew angles, where w A , w B are averaging weights, Here C i H is the center of the excision boundary H . We measure Θ j Avg in the distorted frame, and in this frame the cutting plane is always normal to the x -axis. Therefore, thinking about the desired result in the distorted frame, we see that the control system for the skew map should drive Θ j Avg to zero. This leads us to consider the following control error for the skew angles: Q j Θ = Θ j Avg . Assuming that the function W is unity near the apparent horizons, we find that in agreement with Eq. (13). In deriving Eq. (69), it is helpful to observe that the partial derivative in Eq. (69) is taken with the inertial-frame apparent horizon held fixed. For a fixed inertial-frame horizon, the only map that can change the shape (as opposed to merely the center) of the distorted-frame horizon (and thus Θ j Avg ) is the skew map. We do not use Q j Θ = Θ j Avg for the entire evolution, however, because at early times, when the coordinate distance between the apparent horizons is larger than their combined radii, the skew map is not needed. Furthermore, the skew map can cause difficulties early during the run, especially during the 'junk radiation' phase when the horizons are oscillating in shape. For this reason, we gradually turn on the skew map as the black holes approach each other. This is done by defining a roll-on function g that is zero when the horizons are far apart, and one when they are close together. This roll-on function is defined as For values g < 10 -3 the skew map is turned off completely; this is not strictly necessary, but it saves some computation. Given the function g , we define the control error as This control error drives the skew angles to zero when the black holes are far apart and drives Θ j Avg to zero as they approach each other.",
"pages": [
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},
{
"title": "4.5. Shape control",
"content": "We define the shape map M Shape as: The index H goes over each of the two excised regions A and B , and the map is applied to the grid-frame coordinates. The polar coordinates ( r H , θ H , φ H ) centered about excised region H are defined by Eqs. (32)-(34), the quantities Y /lscriptm ( θ H , φ H ) are spherical harmonics, and λ H /lscriptm ( t ) are expansion coefficients that parameterize the map near excision region H . The function f H ( r H , θ H , φ H ) is chosen to be unity near excision region H and zero near the other excision region, so that the distortion maps for the two black holes are decoupled. Specifically, f H ( r H , θ H , φ H ) is determined as illustrated in Fig. 4. For excision region A in the figure, f A ( r A , θ A , φ A ) = 1 between the excision boundary and the magenta surface, it falls linearly to zero between the magenta and red surfaces, and it is zero everywhere outside the red surface. This means that the gradient of f A ( r A , θ A , φ A ) is discontinuous on the red surface, the magenta surface, and the blue surfaces in the figure. Because we ensure that these discontinuities occur on subdomain boundaries, they cause no difficulty with using spectral methods. Around excision region B , f B ( r B , θ B , φ B ) is chosen similarly. In previous implementations of the shape map [3], the functions f H ( r H , θ H , φ H ) were chosen to be smooth Gaussians centered around each excision boundary rather than to be piecewise linear functions. We find smooth Gaussians to be inferior for two reasons. The first is that piecewise linear functions are easier and faster to invert (the inverse map is required for interpolation to trial solutions during apparent horizon finding). The second is that for smooth Gaussians, it is necessary to choose the widths of the Gaussians sufficiently narrow so that the Gaussian for excision region A does not overlap the Gaussian for excision region B and vice versa, so that the maps and control systems for A and B remain decoupled. However, decreasing the width of the Gaussians increases the Jacobians of the map, producing coordinates that are stretched and squeezed nonuniformly. We found that this form of 'grid-stretching' significantly increased the computational resources required to resolve the solution to a given level of accuracy. In other words, with smooth Gaussians we were forced to add computational resources just to resolve the large Jacobians. A map very similar to Eq. (72) is also described in [4]. The difference compared to this work is the choice of f H ( r H , θ H , φ H ), which corresponds to a different choice of domain decomposition. The control system used to choose the map parameters λ H /lscriptm ( t ) in [4] is also different than what is described here. We control the expansion coefficients λ H /lscriptm ( t ) of the shape map, Eq. (72), so that each excision boundary, as measured in the intermediate frame ( t, ˆ x i ), has the same shape as the corresponding apparent horizon. In other words, we desire where for brevity we have dropped the index H that labels the excision boundary. Here the angle brackets mean averaging over angles, ˆ r AH ( ˆ θ, ˆ φ ) is the radial coordinate of the apparent horizon defined in Eq. (36), and ˆ r EB ( θ, φ ) is the radial coordinate of the excision boundary, which from Eq. (72) can be written Here r EB is the radius of the (spherical) excision boundary in the grid frame. In deriving Eq. (74) we have used the relations M CutX = 1 , f ( r, θ, φ ) = 1, ˆ θ = θ , and ˆ φ = φ , which hold on the excision boundary. Combining Eqs. (36), (73), and (74) yields which we can satisfy by demanding that Therefore, given an apparent horizon and given a value of λ 00 , a control system can be set up for each ( /lscript, m ) pair with /lscript > 0, and the corresponding control errors are Driving these Q /lscriptm to zero produces an excision boundary that matches the shape of the corresponding apparent horizon. Equation (77) determines λ /lscriptm ( t ) only for /lscript > 0, and Eqs. (73)-(77) can be satisfied for arbitrary values of the remaining undetermined map coefficient λ 00 ( t ). Determination of λ 00 ( t ) is complicated enough that it is described in its own section, Sec. 5. Note that Eq. (77) does not satisfy Eq. (14) because ∂Q /lscriptm /∂λ 00 = ˆ S /lscriptm / ˆ S 00 , which does not vanish even if all the control errors are zero. For small distortions, this coupling between λ 00 and Q /lscriptm seems to cause little difficulty. However, at times when the shapes and sizes of the horizons change rapidly (e.g., during the initial 'junk radiation' phase, after the transition to a single excised region when a common apparent horizon forms, and after rapid gauge changes), simulations using Eq. (77) exhibit relatively large and high-frequency oscillations in Q /lscriptm that usually damp away but occasionally destabilize the evolution. Construction of a control system in which all Q /lscriptm are fully decoupled will be addressed in a future work.",
"pages": [
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},
{
"title": "4.6. CutX",
"content": "The map M CutX applies a translation along the grid-frame x -axis in the vicinity of the black holes, but without moving the excision boundaries (or the surrounding spherical shells) themselves. The action of the map is shown in Fig. 7. The goal of this map is to allow for a slight motion of the cutting plane toward the smaller excision boundary. This is important for binary black hole systems with mass ratio q /greaterorsimilar 8. As such a binary gets closer to merger, the inertial-frame coordinate distance between each excision boundary and the cutting plane decreases. Eventually, this distance falls to zero for the larger excision boundary, producing a coordinate singularity in which the Jacobian of one of the other maps (often the shape map) becomes infinite. By pushing the cutting plane toward the smaller excision boundary, the map M CutX avoids this singularity. Even for evolutions in which the inertialcoordinate distance between the larger excision boundary and the cutting plane remains finite but becomes small, the map M CutX prevents large Jacobians from developing and thus increases numerical accuracy (because it is no longer necessary to add computational resources to resolve the large Jacobians). Figure 8 shows the domain decomposition in the inertial frame for a binary black hole simulation with q = 8. The top panel shows the case without M CutX , and the compressed grid near the larger excision boundary is evident. The lower panel shows the case with M CutX , which removes the extreme grid compression. Looking at the Jacobian of the mapping from the inertial frame to the grid frame shortly before merger, we find that the infinity norm of the determinant of the Jacobian is twice as large in the case without M CutX . The map M CutX is written as where F ( t ) is adjusted by a control system. We choose ρ ( x i ) to be zero within either of the spherical shell regions, i.e., inside the magenta spheres around A and B in Fig. 7, and it is also zero outside the outer magenta sphere in the same figure. We set ρ ( x i ) = 1 on the solid red boundaries in Fig. 7, and everywhere between the two solid red boundaries that enclose excision boundary B . Every other region on the grid is bounded by a smooth red boundary on one side and a smooth magenta boundary on the other; in these regions ρ varies linearly between zero and one. The solid blue boundaries are locations (in addition to the solid red and solid magenta boundaries) in which the gradient of ρ is discontinuous. Full details for the calculation of ρ can be found in Appendix B. As with the skew map, the map M CutX is inactive for most of the inspiral. Let x Exc H be the distorted-frame x -coordinate of the intersection of the excision boundary H with the line segment connecting the centers of the excision boundaries. This is similar to x Int H defined earlier; the difference is that x Int H refers to a point on the apparent horizon and x Exc H refers to a point on the excision boundary. The quantity x Exc H is time-dependent because it depends on the shape map. The map M CutX is turned off completely as long as where C i H are the excision boundary centers, and x i C are the coordinates of the intersection of the cutting plane and the line segment connecting the centers of the excision boundaries, as introduced in Sec. 4.4. Let t 0 be the coordinate time at which the map M CutX is activated. ‖ We designate the target position x = x T of the cutting plane as where ∣ ∣ ‖ Both M Skew and M CutX are turned on late in the run, but the condition triggering their activation is different, given the different nature of the problems they address. M Skew is needed for all runs where the horizons eventually intersect the line segment connecting their excision centers at an angle sufficiently different from π/ 2. This will happen essentially for all runs except the simplest head-on collisions. M CutX , on the other hand, is primarily for unequal-mass runs where the larger excision surface encroaches upon the cutting plane near merger. Recall that x Exc H is time-dependent (as it is measured in the distorted frame), so we save the value of x T as calculated at the time when the map M CutX is activated, rather than recalculating x T at every measurement time. Now that we have a target x T , we could designate x T -x 0 C as the target value to the function F ( t ) by setting Q F = -F + x T -x 0 C and have the control system drive Q F to zero, thus driving the x -coordinate of the cutting plane to x T . However, because we turn on the CutX control system suddenly at time t 0 , we must be more careful. Turning on any control system suddenly will produce some transient oscillations, unless the control error and its relevant derivatives and integrals are all initially zero. In the case of the CutX map, which is turned on during a very dynamic part of the simulation when excision boundaries need to be controlled very tightly, these oscillations can prematurely terminate the run by, for example, pushing an excision boundary outside its accompanying apparent horizon. To turn on M CutX gradually, we replace x T by a new time-dependent target function T ( t ) that gradually approaches x T at late times but produces a control error Q F with Q F = ∂Q F /∂t = 0 at the activation time t = t 0 . We start by estimating the time t XC H at which x Exc H will reach the cutting plane, where H is either A or B . This is done by the method described in Appendix C. We then let t XC be the minimum of t XC A and t XC B . The smooth target function T ( t ) is then defined by Designating T -x 0 C as the target for F ( t ) leads to ∣ so at the activation time the control system does not produce transients. Furthermore, in the limit of small Q F , i.e., by the time the excision boundary would have touched the cutting plane (and formed a grid singularity), the smooth target function T ( t ) has approached x T , and the cutting plane will have reached its designated target location, x T . As the run proceeds, the behavior of the cutting plane is determined by the map M CutX while the motion of the excision boundaries is determined by gauge dynamics and the behavior of the other control systems. We continue to monitor the distance between the cutting plane and the excision boundaries, and if it is predicted to touch within a time less than τ d / 0 . 15, the M CutX control system is reset, constructing a new target function T ( t ; t 0 , x T , t XC ), where t 0 is the time of the reset, and x T , t XC are also recalculated based on the state of the grid at this reset time. Each time a new target function T is constructed, the damping time τ d of the M CutX control system is set to be t XC / 2. The control systems responsible for M CutX and M Shape are decoupled, as M CutX controls the location of the cutting plane, leaving the excision boundary unchanged, while M Shape controls the shape of the excision boundary, leaving the location of the cutting plane unchanged. Recall that M CutX and M Shape define the mapping from the grid frame to the distorted frame. As the apparent horizons are found in the distorted frame, and the other maps only depend upon measurements of the horizons, M CutX and M Shape are decoupled from the other maps.",
"pages": [
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},
{
"title": "5. Size control",
"content": "In this section we discuss how we control the spherical part of the map given by Eq. (72), namely, the coefficients λ H 00 for each excision boundary H . We apply the same method to each excision boundary, so for clarity, in this section we again drop the index H from the coefficients λ H 00 and S H /lscriptm , and from the coordinates ( r H , θ H , φ H ).",
"pages": [
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},
{
"title": "5.1. Characteristic speed control",
"content": "Controlling the size of the excision boundary is more complicated than simply keeping the excision boundary inside the apparent horizon. This is because black hole excision requires conditions on the characteristic speeds of the system, and if these conditions are not enforced they are likely to be violated. The minimum characteristic speed at each excision boundary is given by where α is the lapse, ¯ β i is the shift, and ¯ n i is the normal to the excision boundary pointing out of the computational domain , i.e., toward the center of the hole. Here ( ¯ t, ¯ x i ) are the inertial frame coordinates. The first two terms in Eq. (87) describe the coordinate speed of the ingoing (i.e., directed opposite to ¯ n i ) light cone in the inertial frame, and the last term accounts for the motion of the excision boundary (which is fixed in the grid frame) with respect to the inertial frame. In our simulations, we impose no boundary condition whatsoever at each excision boundary. Therefore, well-posedness requires that all of the characteristic speeds, and in particular the minimum speed v , must be non-negative; in other words, characteristics must flow into the hole. In practice, if v becomes negative, the simulation is terminated, because a boundary condition is needed, but we do not have one to impose. This can occur even when the excision boundary is inside the horizon. In this case, one might argue that if the simulation is able to continue without crashing (e.g. by becoming unstable inside the horizon), that the solution outside the horizon would not be contaminated. We have not explored this possibility. Instead, we choose to avoid this situation by terminating the code if negative speeds are detected. Therefore, we would like to control λ 00 in such a way that v remains positive. We start by writing v in a way that separates terms that explicitly depend on ˙ λ 00 from terms that do not. To do this we expand the derivative in the last term of Eq. (87) as where a in the first line of Eq. (88) is a four-dimensional spacetime index, and the last line of Eq. (88) follows from ∂ ˆ t/∂t = 1. Inserting this into Eq. (87) yields where ˆ x i is the frame that is obtained from the grid frame by applying the distortion map; see Eq. (30). We then use Eq. (72) to rewrite this as where we have used the relations f ( r, θ, φ ) = 1 and M CutX = 1 when evaluating the distortion map M Distortion on the excision boundary. By combining all the terms that do not explicitly depend on ˙ λ 00 into a quantity v 0 , we obtain Thus, the characteristic speed v can be thought of as consisting of two parts: one part, v 0 , that depends on the position and shape of the excision boundary and the values of the metric quantities there, and another part that depends on the average speed of the excision boundary in the direction of the boundary normal. We now construct a control system that drives the characteristic speed v to some target speed v T . This is a control system that controls the derivative quantity ˙ λ 00 , as opposed to directly controlling the map quantity λ 00 . We choose where the minimum is over the excision boundary, and the angle brackets in Eq. (92) refer to an average over the excision boundary. Note that Ξ < 0 because ˆ n i points radially inward and x i /r points radially outward; this means that ˙ Q = -¨ λ 00 , in accordance with our normalization choice for a control system on ˙ λ 00 . As in our other controlled map parameters, we demand that ˙ λ 00 is a function with a piecewise-constant second derivative. It is then easy to construct λ 00 as a function with a piecewise-constant third derivative. The control system given by Eq. (92) with a hand-chosen value of v T has been used successfully [24, 25] in simulations of high-spin binaries. Figure 9 illustrates why characteristic speed control is crucial for the success of these simulations.",
"pages": [
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},
{
"title": "5.2. Apparent horizon tracking",
"content": "The characteristic speed control described above has the disadvantage that it requires a user-specified target value v T . If v T is chosen to be too small, then small fluctuations (due to shape control, the horizon finder, or simply numerical truncation error) can cause the characteristic speed to become negative and spoil the simulation. If v T is chosen to be too large, the simulation can also fail. To understand why, recall that characteristic speed control achieves the target characteristic speed by moving the excision boundary and thus changing the velocity term in Eq. (87). So if v > v T the control system moves the excision boundary radially inward, and if v < v T the control system moves the excision boundary radially outward. If v T is too large, the control system can push the excision boundary outward until it crosses the apparent horizon. This halts the evolution because the apparent horizon can no longer be found. One way to prevent the excision boundary from crossing the horizon is to drive the excision boundary to some constant fraction of the horizon radius, or in other words, drive the quantity d/dt (∆ r ) to zero, where is the relative difference between the average radius of the apparent horizon (in the intermediate frame) and the average radius of the excision boundary. Using Eqs. (74) and (36), we can write and therefore a control system that adjusts ˙ λ 00 to achieve d/dt (∆ r ) = 0 can be obtained by defining A slight generalization of this control system can be obtained by demanding that d/dt (∆ r ) = ˙ r drift , where ˙ r drift is some chosen constant, We have not found the control systems defined by Eqs. (96) and (97) to be especially useful on their own. One drawback of these systems is that they do not prevent the minimum characteristic speed v from becoming negative. Instead, we use the horizon tracking control systems described here as part of a more sophisticated control system discussed in the next section.",
"pages": [
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{
"title": "5.3. Adaptive switching of size control",
"content": "Here we introduce a means of controlling λ 00 that combines the best features of characteristic speed control and horizon tracking. The idea is to continuously monitor the state of the system and switch between different control systems as the evolution proceeds. At fixed intervals during the simulation that we call 'measurement times,' we monitor the minimum characteristic speed v and the relative distance between the horizon and the excision boundary ∆ r . The goal of the control system is to ensure that both of these values remain positive. Consider first the characteristic speed v . Using the method described in Appendix C, we determine whether v is in danger of becoming negative in the near future, and if so, we estimate the timescale τ v on which this will occur. Similarly, we also determine whether ∆ r will soon become negative, and if so we estimate a corresponding timescale τ ∆ r . Having estimated both τ v and τ ∆ r , we use these quantities to determine how to control ˙ λ 00 . In particular, we switch between horizon tracking, Eq. (96), and characteristic speed control, Eq. (92), based on τ v and τ ∆ r . We do this by the following algorithm, which favors horizon tracking over characteristic speed control unless the latter is essential. Assume that the control system for ˙ λ 00 is currently tracking the horizon, Eq. (96), and that the current damping timescale is some value τ d . If v is in danger of crossing zero according to the above estimate, and if τ v < τ d and τ v < τ ∆ r , we then switch to characteristic speed control, Eq. (92), we set v T = 1 . 01 v where v is the current value of the characteristic speed (the factor 1 . 01 prevents the algorithm from switching back from characteristic speed control to horizon tracking on the very next time step), and we reset the damping time τ d equal to τ v . Otherwise we continue to use Eq. (96), resetting τ d = τ ∆ r if τ ∆ r < τ d . Now assume that the control system for ˙ λ 00 is controlling the characteristic speed, Eq. (92). If v is not in danger of crossing zero, so that we no longer need active control of the characteristic speed, then we switch to horizon tracking, Eq. (96), without changing the damping time τ d . If v is in danger of crossing zero, and if ∆ r is either in no danger of crossing zero or if it will cross zero sufficiently far in the future such that τ ∆ r ≥ σ 1 τ d , then we continue to use Eq. (92) with τ d reset to min( τ d , τ v ) so as to maintain control of the characteristic speed. Here σ 1 is a constant that we typically choose to be about 20. The more complicated case occurs when both v and ∆ r are in danger of crossing zero and τ ∆ r < σ 1 τ d . In this case, we have two possible methods by which we attempt to prevent ∆ r from becoming negative. We choose between these two methods based on the values of τ ∆ r and τ d , and also by the value of a quantity that we call the comoving characteristic speed : This quantity is the value that the characteristic speed v would have if horizon tracking (Eq. (96)) were in effect and working perfectly. Equation (98) is derived by assuming Q = 0 in Eq. (96), solving for ˙ λ 00 , and substituting this value into Eq. (90). The reason to consider v c is that for min( v c ) < 0 (which can happen temporarily during a simulation), horizon tracking is to be avoided, because horizon tracking will drive min( v ) toward a negative value, namely min( v c ). The first method of preventing ∆ r from becoming negative is to switch to horizon tracking, Eq. (96); we do this if min( v c ) > 0 and if τ ∆ r < σ 2 τ d , where σ 2 is a constant that we typically choose to be 5. The second method, which we employ if σ 2 τ d < τ ∆ r < σ 1 τ d or if min( v c ) < 0, is to continue to use characteristic speed control, Eq. (92), but to reduce the target characteristic speed v T by multiplying its current value by some fraction η (typically 0 . 25). By reducing the target speed v T , we reduce the outward speed of the excision boundary, and this increases τ ∆ r . The reason we use the latter method for min( v c ) < 0 is that the former method, horizon tracking, will drive v toward v c and we wish to keep v positive. The reason we use the latter method for σ 2 τ d < τ ∆ r < σ 1 τ d is that in this range of timescales the control system is already working hard to keep v positive, therefore a switch to horizon tracking is usually followed by a switch back to characteristic speed control in only a few time steps, and rapid switches between different types of control make it more difficult for the control system to remain locked. The constants η , σ 1 , and σ 2 are adjustable, and although the details of the algorithm (i.e., which particular quantity is controlled at which particular time) are sensitive to the choices of these constants, the overall behavior of the algorithm is not, provided 1 < σ 2 < σ 1 . The overall behavior of this algorithm is to cause ˙ λ 00 to settle to a state in which both v > 0 and ∆ r > 0 for a long stretch of the simulation. Occasionally, when either v or ∆ r threatens to cross zero as a result of evolution of the metric or gauge, the algorithm switches between characteristic speed control and horizon tracking several times and then settles into a new near-equilibrium state in which both v > 0 and ∆ r > 0. The actual value of ∆ r and of v in the near-equilibrium state is unknown a priori, and in particular this algorithm can in principle settle to values of ∆ r that are either small enough that control system timescales must be very short in order to prevent ∆ r from becoming negative, or large enough that excessive computational resources are needed to resolve the metric quantities deep inside the horizon. To prevent either of these cases from occurring, we introduce a constant correction velocity ˙ r corr , a nominal target value ∆ r target , and a nominal target characteristic speed v target . We currently choose these to be ˙ r corr = 0 . 005, ∆ r target = 0 . 08, and v target = 0 . 08. These quantities are used only to decide whether to replace Eq. (96) by Eq. (97) when the algorithm chooses horizon tracking, as described below; these quantities are not used when the algorithm chooses characteristic speed control. First we consider the case where ∆ r is too large during horizon tracking. If ∆ r > 1 . 1∆ r target , then we replace Eq. (96) with Eq. (97), and we use ˙ r drift = -˙ r corr . We continue to use ˙ r drift = -˙ r corr until ∆ r < ∆ r target , at which point we set ˙ r drift = 0. The case where ∆ r is too small during horizon tracking is more complicated because we want to be careful so as to not make the characteristic speeds decrease, since it is more difficult to recover from decreasing characteristic speeds when ∆ r is small. In this case, if ∆ r < 0 . 9∆ r target , v < 0 . 9 v target , and 〈 n k ∂ k v c 〉 > 0, then we replace Eq. (96) with Eq. (97), and we use ˙ r drift = v corr . Here v c is the comoving characteristic speed defined in Eq. (98) and the angle brackets refer to an average over the excision boundary. The condition on the gradient of v c ensures that v c will increase if ∆ r is increased; otherwise a positive ˙ r drift will threaten the positivity of the characteristic speeds. We continue to use ˙ r drift = v corr until either v > 0 . 9 v target , ∆ r > 0 . 9∆ r target , 〈 n k ∂ k v c 〉 < 0, or if v will cross v target or ∆ r will cross ∆ r target within a time less than the current damping time τ d (where crossing times are determined by the procedure of Appendix C). When any of these conditions occur, we switch to ˙ r drift = 0, and we prohibit switching back to a positive ˙ r drift until either v > 0 . 99 v target or ∆ r > 0 . 99∆ r target , or until both v and ∆ r stop increasing in time. These last conditions prevent the algorithm from oscillating rapidly between positive and zero values of ˙ r drift .",
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]
},
{
"title": "6. Merger and ringdown",
"content": "The maps and control systems in Sec. 4 were discussed in the context of a domain decomposition with two excision boundaries, one inside each apparent horizon. The skew and CutX maps were introduced specifically to handle difficulties that occur when two distorted excision boundaries approach each other. At some point in the evolution as the two black holes become sufficiently close, a common apparent horizon forms around them. After this occurs, the simulation can be simplified considerably by constructing a new domain decomposition with a single excision boundary placed just inside the common horizon. The evolved variables are interpolated from the old to the new domain decomposition, and the simulation then proceeds on the new one. The algorithm for transitioning to a new grid with a single excision boundary is fundamentally the same as that described in [3, 4], but there have been improvements in the maps and the control systems, so for completeness we describe the procedure here. At some time t = t m shortly after a common horizon forms, we define a new, post-merger set of grid coordinates ˘ x i and a corresponding new domain decomposition composed only of spherical shells. In ˘ x i coordinates, the new excision boundary is a sphere centered at the origin. We also define a post-merger map ¯ x i = M Ringdown ˘ x i , where Note that the post-merger grid coordinates differ from the pre-merger grid coordinates, but there is only one set of inertial coordinates ¯ x i . Recall that in the dual-frame picture [1], the inertial-frame components of tensors are stored and evolved, so the evolved variables are continuous at t = t m even though the grid coordinates are not. The post-merger translation map M TranslationRD and its control system are the same as discussed in Sec. 4.3, except that M TranslationRD translates with respect to the origin of the ˘ x i coordinate system, which is different from the origin of the x i coordinate system. The post-merger translation control system keeps the common apparent horizon centered at the origin in the post-merger grid frame. Note that the center of the common apparent horizon does not necessarily lie on the same line as the centers of the individual apparent horizons (see [23] for an example). This means that M TranslationRD is not continuous with the pre-merger translation map M Translation , except near the outer boundary where both maps are the identity. To evolve a distorted single black hole we do not need a rotation map. However, if rotation is turned off suddenly at t = t m , then the outer boundary condition (which is imposed in inertial coordinates) is not smooth in time, and we see a pulse of gauge and constraint violating modes propagate inward from the outer boundary because of this sharp change in the boundary condition. Therefore, we use the same rotation map M Rotation before and after merger, and we slow down the rotation gradually. To accomplish this, instead of adjusting the map parameters ( ϑ, ϕ ) by a control system, for t ≥ t m we set these parameters equal to prescribed functions that approach a constant at late times. We set where the constants A , B , and C are determined by demanding that ϑ ( t ) and its first two time derivatives are continuous at t = t m . The parameter ϕ obeys an equation of the same form. We choose τ roll = 20 M . The post-merger scaling map M ScalingRD is written where ˘ R is the post-merger grid-coordinate radius, and ˘ R OB is the outer boundary radius in the post-merger grid coordinates. The function b ( t ) is given by Eq. (42) before and after merger. We set ˘ R OB = b ( t m ) R OB so that at t = t m , M ScalingRD matches M Scaling at the outer boundary. Note that the post-merger scaling map is the identity near the merged black hole; the only purpose of this map is to keep the inertial-coordinate location of the outer boundary smooth in time at t = t m . The post-merger shape map M ShapeRD and its control system are the same as discussed in Secs. 4.5 and 5, except the map is centered about the origin of the ˘ x i coordinate system, the sum over excised regions in Eq. (72) runs over only one region (which we call excision region C), and the function f C ( r C , θ C , φ C ) appearing in Eq. (72) is where r max is the radius of a (spherical) subdomain boundary that is well inside the wave zone. We typically choose r max = 32 r EB . Once the post-merger map parameters are initialized, as discussed below, we interpolate all variables from the pre-merger grid to the post-merger grid. The inertial coordinates ¯ x i are the same before and after merger, so this interpolation is easily accomplished using both the pre-merger and post-merger maps, e.g. F ( x i ) = F ( M -1 M Ringdown ˘ x i ) for some function F . After interpolating all variables onto the post-merger grid, we continue the evolution. This entire process - from detecting a common apparent horizon to continuing the evolution on the new domain - is done automatically.",
"pages": [
31,
32
]
},
{
"title": "6.1. Initialization",
"content": "All that remains for a full specification of M Ringdown is to describe how parameters of the two discontinuous maps M ShapeRD and M TranslationRD are initialized at t = t m . To do this, we first construct a temporary distorted frame with coordinates ' x i , defined by The coordinates ' x i are the same as the post-merger distorted coordinates except that the pre-merger translation map M Translation is used in Eq. (103). We then represent the common apparent horizon (which is already known in ¯ x i coordinates) in ' x i coordinates: The expansion coefficients ' S C /lscriptm ( t ) and the horizon expansion center ' x i AH0 ( t ) are computed using the (inertial frame) common horizon surface plus the (time-dependent) map defined in Eq. (103). Here ( ' θ C , ' φ C ) are polar coordinates centered about ' x i AH0 , and ' n i is a unit vector corresponding to the direction ( ' θ C , ' φ C ). Equation (104) is the same expansion as Eq. (36), except here we write the expansion to explicitly include the center ' x i AH0 ( t ). In Eq. (104) we write ' S C /lscriptm ( t ) and ' x i AH0 ( t ) as functions of time because we compute them at several discrete times surrounding the matching time t m . By finite-differencing in time, we then compute first and second time derivatives of ' S C /lscriptm and ' x i AH0 at t = t m . Once we have ' S C /lscriptm and its time derivatives, we initialize λ C /lscriptm ( t m ) = -' S C /lscriptm ( t m ), where λ C /lscriptm ( t ) are the map parameters appearing in Eq. (72), and the minus sign accounts for the sign difference between the definitions of Eqs. (72) and (104). We do this for all ( /lscript, m ) except for /lscript = m = 0: we set λ C 00 ( t m ) = 0, thus defining the radius of the common apparent horizon in the post-merger grid frame. For all ( /lscript, m ) including /lscript = m = 0 we set dλ C /lscriptm /dt | t = t m = -d ' S C /lscriptm /dt | t = t m , and similarly for the second derivatives. Note that the temporary distorted coordinates ' x i are incompatible with the assumptions of our control system, because the center of the excision boundary ' x i AH0 ( t ) in these coordinates is time-dependent. We therefore define new post-merger distorted coordinates ˆ x i by where If we denote the map parameters in M Translation by T i 0 , the map parameters in M TranslationRD by T i , and if we denote M Rotation · M ScalingRD by the matrix M i j , then Eqs. (105) and (106) require Here we have assumed that f ( R ) appearing in Eq. (55) is unity in the vicinity of the horizon, so we can treat the translation map as a rigid translation near the horizon. We initialize T i and its first two time derivatives at t = t m according to Eq. (107). Note that the distorted-frame horizon expansion coefficients ' S C /lscriptm and hence the initialization of λ C /lscriptm ( t ) are unchanged by the change in translation map because ˆ x i and ' x i differ only by an overall translation, Eq. (106), which leaves angles invariant.",
"pages": [
32,
33
]
},
{
"title": "7. Control systems for efficiency",
"content": "Although the most important use of control systems in SpEC is to adjust parameters of the mapping between the inertial and grid coordinates, another situation for which control systems are helpful is the approximation of functions that vary slowly in time, are needed frequently during the simulation, but are expensive to compute. For example, the average radius of the apparent horizon, r AH , is used in the control system for ˙ λ 00 , which is evaluated at every time step. Evaluation at each time step is necessary to allow the control system for ˙ λ 00 to respond rapidly to sudden changes in the characteristic speed or the size of the excision boundary, as may occur after regridding, after mesh refinement changes, or when other control systems (such as size control) are temporarily out of lock. However, computing the apparent horizon (and thus its average radius r AH ) at every time step is prohibitively expensive. To significantly reduce the expense, we define a function with piecewise-constant second derivative, r appx AH ( t ), we define a control error and we define a control system that drives Q to zero. We then pass r appx AH ( t ) instead of r AH to those functions that require the average apparent horizon radius at each time step. The control error Q , and thus the expensive computation of the apparent horizon, needs to be evaluated only infrequently, i.e., on the timescale on which the average horizon radius is changing, which may be tens or hundreds of time steps.",
"pages": [
33,
34
]
},
{
"title": "8. Summary",
"content": "In simulations of binary black holes, we use a set of time-dependent coordinate mappings to connect the asymptotically inertial frame (in which the black holes inspiral about one another, merge, and finally ringdown) to the grid frame in which the excision surfaces are stationary and spherical. The maps are described by parameters that are adjusted by a control system to follow the motion of the black holes, to keep the excision surface just inside the apparent horizons of the holes, and also to prevent grid compression. We take care to decouple the control systems and choose stable control timescales. The scaling, rotation, and translation maps are used to track the overall motion of the black holes in the inertial frame. These are the most important maps during the inspiral phase of the evolution, as the shapes of the horizons remain fairly constant after the relaxation of the initial data. As the binary approaches merger, the horizon shapes begin to distort, and shape and size control start to become important. Shape and size control are especially crucial for unequal-mass binaries and for black holes with near-extremal spins. In the latter case, the excision surface (which must remain an outflow boundary with respect to the characteristics of the evolution system) needs to be quite close to the horizon. In order to decouple the shape maps of the individual black holes, our grid is split by a cutting plane at which the shape maps reduce to the identity map. As the black holes merge, a skew map is introduced in order to align the cutting plane with the excision surfaces (see Fig. 5) and to minimize grid compression between the two black holes. Finally, the implementation of CutX control was necessary to complete the merger of high mass-ratio configurations. In these systems, the excision boundaries approach the cutting plane asymmetrically as the black holes merge, thus compressing the grid between the cutting plane and the nearest excision boundary. CutX control translates the cutting plane to keep it centered between the two excision boundaries, alleviating this grid compression. We find that we need all of the maps described in this paper with a sufficiently tight control system in order to robustly simulate a wide range of the parameter space of binary black hole systems [4, 24, 5]. Many of these maps and control systems are also used in our simulations of black hole - neutron star binaries [26, 27, 28].",
"pages": [
34
]
},
{
"title": "Appendix A. Implementation of exponentially-weighted averaging",
"content": "Our exponentially-weighted averaging scheme uses an averaging timescale τ avg to smooth noisy quantities F that are measured at intervals τ m := t k -t k -1 , where F is the measured value of the control error Q , its integral, or its derivatives. Recall from Sec. 3.3 that we typically choose τ avg ∼ 0 . 25 τ d and τ m ∼ 0 . 075 min( τ d ). The averaging is implemented by solving the following system of ordinary differential equations (ODEs): where the evolved variables are a weight factor W , the average value F avg , and the effective time τ at which F avg is calculated. The ODEs are solved approximately using backward Euler differencing, which results in the recursive equations where D = 1 + τ m /τ avg . The recursion is initialized with W 0 = 0, τ 0 = t 0 , and F 0 avg = F ( t 0 ). In the control law equations, Eqs. (15) and (16), we want to use the averaged value at t k instead of τ k . Therefore, to adjust for the offset induced by averaging, δt k = t k -τ k , we evaluate at t k an approximate Taylor series for F k avg expanded about τ k where n is defined by F := d n Q/dt n (for convenience, F represents the integral of Q when n = -1), and N is the same as in Eq. (11), where it is defined as the number of derivatives used to represent the map parameter. Note that this approximation matches the true Taylor series inasmuch as W is constant. For the highest order derivative of Q , where n = N , we can no longer directly adjust for the offset because Eq. (A.7) reduces to We would need to take additional derivatives of Q to provide a meaningful correction, but this is exactly what we want to avoid because of the noisiness of derivatives. Therefore, to circumvent taking higher order derivatives, we substitute F avg ( t k ) into the control law, e.g. Eq. (21) for PID, and then solve for d N Q/dt N .",
"pages": [
34,
35
]
},
{
"title": "Appendix B. Details on computing the CutX map weight function",
"content": "As discussed in Sec. 4.6, we use the map M CutX to control the location of the cutting plane in the last phase of the merger, as the distance between the excision boundaries and the cutting plane becomes small. The value of the weight function ρ ( x i ) in Eq. (78) is zero in the spherical shells describing the wave zone and in the shells around the excision boundaries (see Fig. B1). The weight function is unity on the cutting plane, and on the 'cut-sphere' surfaces around either excision boundary. For the smaller excision surface, we have two such cut-sphere surfaces, and the value of the weight function is one within the region enclosed by these two cut-spheres and the cutting plane. The weight function transitions linearly from zero to one (from the magenta curves to the red curves in Fig. B1). In these transitional regions the value of ρ ( x i ) is computed as described below. Consider the region spanning the volume between the spherical shells around the excision boundary H (with H = A,B ) and the cutting plane x = x 0 C , as shown in Fig. B2. (This is referred to in the code as the M region.) In order to calculate the value of ρ ( x i ) in this region, we shoot a ray from C i H in the direction of x i . This ray intersects both the outer spherical boundary of the shells and the cutting plane. The x -coordinate of the intersection with the shells will be where R H is the outer radius of the spherical region around C i H . Then, for a point x i in the M region ρ is given by All other transitional regions are delimited on both ends by spheres. Our computational domain has two zones of this type; one is depicted in Fig. B2 (labeled as the E region), while the other is seen in Fig. B3. Let x i P denote the 'center of projection,' i.e., the point toward which the unmapped radial grid lines of the given zone converge, and let x i S denote the center of a sphere of radius R (noting that the inner and outer spheres may have different centers). In Fig. B2 the center of projection coincides with C i H . In Fig. B3 we choose the center of projection to be on the cutting plane, at its intersection with the line segment connecting the centers of the two excision surfaces, x i C . To compute ρ ( x i ) in this type of region, we shoot a ray from x i P in the direction of x i . This ray intersects the two spheres that delimit the region. We define r 0 to be the distance between x i P and the point of intersection with the (magenta) sphere, for which ρ = 0; similarly, we define r 1 to be the distance between x i P and the point of intersection with the (red) sphere, for which ρ = 1. Then, ρ ( x i ) is given by where r = | x i -x i P | . To compute r 0 and r 1 , we must solve for the associated intersection point X i . The parameter η ( x i P , x i S , X i , R ) is defined as the positive solution of the quadratic system, where The input coordinates to M CutX are the coordinates ˜ x i that are obtained from the grid coordinates x i by the shape map, ˜ x i = M Shape ( x i ). As seen above, computation of the weight function at any point requires knowledge of the region containing that point, which in turn requires knowledge of its grid-frame coordinates x i . Therefore, to compute the weight function one must first compute x i = M -1 Shape (˜ x i ). When computing the forward CutX map, this inverse is done only once. However, the situation is more complicated when computing the inverse CutX map, because the grid-frame coordinates depend upon the inverse CutX map itself, which depends on the weight function. This leads to an iterative algorithm where we must call the inverse map function of M Shape from each iteration of the inverse map function of M CutX . This could easily become an efficiency bottleneck. We mitigate this by keeping M CutX inactive for most of the run, only activating it when it becomes crucial in order to avoid a singular map.",
"pages": [
35,
36,
37,
38
]
},
{
"title": "Appendix C. Estimation of zero-crossing times",
"content": "Several of the control systems described here are designed to ensure that some measured quantity remains positive. For example, in Sec. 5.3, we demand that both the characteristic speed v at the excision boundary and the difference ∆ r between the horizon radius and the excision boundary radius remain positive. Similarly, in Sec. 4.6 we demand that each excision boundary does not cross the cutting plane. Part of the algorithm for ensuring that some quantity q remains positive is estimating whether q is in danger of becoming negative in the near future, and if so, estimating the timescale τ on which this will occur. In this Appendix we describe a method of obtaining this estimate. We assume the quantity q is measured at a set of (not necessarily equally-spaced) measurement times, and we remember the values of q at several (typically 4) previous measurement times. At each measurement time t 0 we fit these remembered values to a line q ( t ) = a + b ( t -t 0 ). The fit gives us not only the slope b and the intercept a , but also error bars for these two quantities δa and δb . Assuming that the true q ( t ) lies within the error bars, the earliest time at which q will cross zero is t = t 0 +( -a + δa ) / ( b -δb ), and the latest time is t = t 0 +( -a -δa ) / ( b + δb ). If both of these times are finite and in the future, then we regard q as being in danger of crossing zero, and we estimate the timescale on which this will happen as τ = -a/b .",
"pages": [
38,
39
]
},
{
"title": "Acknowledgments",
"content": "We would like to thank Harald Pfeiffer for useful discussions. We would like to thank Abdul Mrou'e for performing simulations that led to the development and improvement of some of the mappings presented in this paper. We gratefully acknowledge support from the Sherman Fairchild Foundation; from NSF grants PHY-0969111 and PHY1005426 at Cornell, and from NSF grants PHY-1068881 and PHY-1005655 at Caltech. Simulations used in this work were computed with the SpEC code [17]. Computations were performed on the Zwicky cluster at Caltech, which is supported by the Sherman Fairchild Foundation and by NSF award PHY-0960291; on the NSF XSEDE network under grant TG-PHY990007N; and on the GPC supercomputer at the SciNet HPC Consortium [29]. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research FundResearch Excellence; and the University of Toronto.",
"pages": [
39
]
},
{
"title": "References",
"content": "Shoemaker, and Jeffrey Winicour. Grazing collisions of black holes via the excision of singularities. Phys. Rev. Lett. , 85:5496-5499, 2000. Number 6 in Textbooks in Applied Mathematics. Springer, New York, second edition, 1998.",
"pages": [
40
]
}
]
2013CQGra..30p5020S
https://arxiv.org/pdf/1307.2637.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_77><loc_73><loc_82></location>New limits on the violation of local position invariance of gravity</section_header_level_1>
<section_header_level_1><location><page_1><loc_16><loc_73><loc_48><loc_74></location>Lijing Shao 1 , 2 and Norbert Wex 1</section_header_level_1>
<list_item><location><page_1><loc_16><loc_71><loc_80><loc_72></location>1 Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, Germany</list_item>
<list_item><location><page_1><loc_16><loc_69><loc_60><loc_70></location>2 School of Physics, Peking University, Beijing 100871, China</list_item>
<text><location><page_1><loc_16><loc_67><loc_63><loc_68></location>E-mail: lshao@pku.edu.cn (LS) , wex@mpifr-bonn.mpg.de (NW)</text>
<section_header_level_1><location><page_1><loc_16><loc_63><loc_24><loc_64></location>Abstract.</section_header_level_1>
<text><location><page_1><loc_16><loc_47><loc_84><loc_63></location>Within the parameterized post-Newtonian (PPN) formalism, there could be an anisotropy of local gravity induced by an external matter distribution, even for a fully conservative metric theory of gravity. It reflects the breakdown of the local position invariance of gravity and, within the PPN formalism, is characterized by the Whitehead parameter ξ . We present three different kinds of observation, from the Solar system and radio pulsars, to constrain it. The most stringent limit comes from recent results on the extremely stable pulse profiles of solitary millisecond pulsars, that gives | ˆ ξ | < 3 . 9 × 10 -9 (95% CL), where the hat denotes the strong-field generalization of ξ . This limit is six orders of magnitude more constraining than the current best limit from superconducting gravimeter experiments. It can be converted into an upper limit of ∼ 4 × 10 -16 on the spatial anisotropy of the gravitational constant.</text>
<text><location><page_1><loc_16><loc_42><loc_48><loc_43></location>PACS numbers: 04.80.Cc, 96.60.-j, 97.60.Gb</text>
<section_header_level_1><location><page_2><loc_12><loc_87><loc_27><loc_88></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_69><loc_84><loc_85></location>Since the 1960s, advances in technologies are continuously providing a series of formidable tests of gravity theories from on-ground laboratories, the Solar system, various pulsar systems, and also cosmology [42, 43]. Up to now, Einstein's general relativity (GR) passed all experimental tests with flying colors. However, questions related to the nature of dark matter and dark energy, and irreconcilable conflicts between GR and the standard model of particle physics, are strong motivations to study alternative theories of gravity. In addition, gravity as a fundamental interaction of nature deserves most stringent tests from various aspects.</text>
<text><location><page_2><loc_12><loc_55><loc_84><loc_69></location>For tests of gravity theories, one of the most popular frameworks is the parameterized post-Newtonian (PPN) formalism , proposed by Nordtvedt and Will [25, 40, 44, 42]. In the standard PPN gauge, the framework contains ten dimensionless PPN parameters in the metric components as coefficients of various potential forms. These parameters take different values in different gravity theories. Hence, experimental constraints on these parameters can be directly used to test specific gravity theories [30, 42, 43].</text>
<text><location><page_2><loc_12><loc_39><loc_84><loc_55></location>In this paper, we concentrate on one of the ten PPN parameters which characterizes a possible Galaxy-induced anisotropy in the gravitational interaction of localized systems. Such an anisotropy is described by the Whitehead parameter ξ in the weakfield slow-motion limit [41]. We use ˆ ξ to explicitly denote its strong-field generalization. Besides Whitehead's gravity theory [39], ξ is relevant for a class of theories called 'quasilinear' theories of gravity [41]. In GR, the gravitational interaction is local position invariant with ξ = 0, while in Whitehead's gravity, local position invariance (LPI) is violated and ξ = 1 [41, 15].</text>
<text><location><page_2><loc_12><loc_11><loc_84><loc_39></location>An anisotropy of gravitational interaction, induced by the gravitational field of the Galaxy, would lead to anomalous Earth tides at specific frequencies with characteristic phase relations [41, 38]. The ξ -induced Earth tides are caused by a change in the local gravitational attraction on the Earth surface due to the rotation of the Earth with frequencies associated with the sidereal day. By using constraints on ξ from superconducting gravimeter, Will gave the first disproof of Whitehead's parameterfree gravity theory [41] (see [15] for multiple recent disproofs). Later Warburton and Goodkind presented an update on the limit of ξ by using new gravimeter data [38], where they were able to constrain | ξ | to the order of 10 -3 . The uncertainties concerning geophysical perturbations and the imperfect knowledge of the Earth structure limit the precision. Uncertainties include the elastic responses of the Earth, the effects of ocean tides, the effects of atmospheric tides from barometric pressure variation, and the resonances in the liquid core of the Earth [38] (see [16, 36] for recent reviews on superconducting gravimeters).</text>
<text><location><page_2><loc_12><loc_5><loc_84><loc_11></location>Limits from Earth tides are based on periodic terms proportional to ξ , while secular effects in other astrophysical laboratories can be more constraining. Nordtvedt used the close alignment of the Sun's spin with the invariable plane of the Solar system to</text>
<text><location><page_3><loc_12><loc_79><loc_84><loc_89></location>constrain the PPN parameter α 2 , associated with the local Lorentz invariance of gravity down to O (10 -7 ) [28]. In the same publication Nordtvedt pointed out that such a limit is also possible for ξ , as the two terms in the Lagrangian have the same form. However, to our knowledge, no detailed calculations have been published yet. In section 3 we follow Nordtvedt's suggestion and achieve a limit of O (10 -6 ).</text>
<text><location><page_3><loc_12><loc_57><loc_84><loc_79></location>A non-vanishing (strong-field) ˆ ξ would lead to characteristic secular effects in the dynamics of the rotation and orbital motion of radio pulsars. We have presented the methodologies in details to constrain the (strong-field) ˆ α 2 from binary pulsar timing [34] and solitary pulsar profile analysis [33] respectively. By the virtue of the similarity between ˆ α 2 -and ˆ ξ -related effects, in section 4 we extend the analysis in [34, 33] to the case of LPI of gravity. From timing results of PSRs J1012+5307 [19] and J1738+0333 [13], a limit of | ˆ ξ | < 3 . 1 × 10 -4 (95% CL) is achieved for neutron star (NS) white dwarf (WD) systems [35]. As shown in this paper, from the analysis on the pulse profile stability of PSRs B1937+21 and J1744 -1134, a limit of | ˆ ξ | < 3 . 9 × 10 -9 (95% CL) is obtained, utilizing the rotational properties of solitary millisecond pulsars. This limit is six orders of magnitude better than the (weak-field) limit from gravimeter.</text>
<text><location><page_3><loc_12><loc_43><loc_84><loc_56></location>The paper is organized as follows. In the next section, the theoretical framework for tests of LPI of gravity is briefly summarized. In section 3, a limit on ξ from the Solar system is obtained. Then we give limits on ˆ ξ from binary pulsars and solitary pulsars in section 4. In the last section, we discuss issues related to strong-field modifications and conversions from our limits to limits on the anisotropy in the gravitational constant. Comparisons between our tests with other achievable tests from gravimeter and lunar laser ranging (LLR) experiments are also given.</text>
<section_header_level_1><location><page_3><loc_12><loc_39><loc_36><loc_40></location>2. Theoretical framework</section_header_level_1>
<text><location><page_3><loc_12><loc_21><loc_84><loc_36></location>In the PPN formalism, PPN parameters are introduced as dimensionless coefficients in the metric in front of various potential forms [44, 42, 43]. In the standard postNewtonian gauge, ξ appears in the metric components g 00 and g 0 i [44, 42, 43]. However, in most cases, it is relevant only in linear combinations with other PPN parameters like β , γ (see [42, 43] for formalism and details). Due to the limited precision in constraining these PPN parameters (see table 4 in [43] for current constraints on PPN parameters), it is not easy to get an independent stringent limit for ξ . For example, based on the Nordtvedt parameter (see (43) in [15]),</text>
<formula><location><page_3><loc_16><loc_17><loc_84><loc_20></location>η = 4 β -γ -3 -10 3 ξ -α 1 + 2 3 α 2 -2 3 ζ 1 -1 3 ζ 2 , (1)</formula>
<text><location><page_3><loc_12><loc_13><loc_84><loc_16></location>one can only constrain ξ to the order of O (10 -3 ) at most. Nevertheless, in the metric component g 00 , -2 ξ alone appears as the coefficient of the Whitehead potential [41],</text>
<formula><location><page_3><loc_16><loc_7><loc_84><loc_12></location>Φ W ( x ) ≡ G 2 c 2 ∫∫ ρ ( x ' ) ρ ( x '' ) ( x -x ' | x -x ' | 3 ) · ( x ' -x '' | x -x '' | -x -x '' | x ' -x '' | ) d 3 x ' d 3 x '' , (2)</formula>
<text><location><page_3><loc_12><loc_4><loc_84><loc_8></location>where ρ ( x ) is the matter density, G and c are the gravitational constant and the speed of light respectively. This fact provides the possibility to constrain the PPN parameter</text>
<section_header_level_1><location><page_4><loc_12><loc_87><loc_20><loc_89></location>ξ directly.</section_header_level_1>
<text><location><page_4><loc_12><loc_83><loc_84><loc_87></location>Correspondingly, in the PPN n -body Lagrangian, we have a ξ -related term for three-body interactions (see e.g. (6.80) in [42]),</text>
<formula><location><page_4><loc_16><loc_77><loc_84><loc_82></location>L ξ = -ξ 2 G 2 c 2 ∑ i,j m i m j r 3 ij r ij · [ ∑ k m k ( r jk r ik -r ik r jk ) ] , (3)</formula>
<text><location><page_4><loc_12><loc_68><loc_84><loc_77></location>where the summation excludes terms that make any denominators vanish. For our purposes below, we consider the third body being our Galaxy, and only consider a system S (the Solar system or a pulsar binary system or a solitary pulsar) of typical size much less than its distance to the Galactic center R G . Hence the Lagrangian (3) reduces to (dropping a constant factor that rescales G )</text>
<formula><location><page_4><loc_16><loc_62><loc_84><loc_67></location>L ξ = ξ 2 U G c 2 ∑ i,j Gm i m j r 3 ij ( r ij · n G ) 2 , (4)</formula>
<text><location><page_4><loc_12><loc_49><loc_84><loc_62></location>where U G is the Galactic potential at the position of the system S (associated with the mass inside R G ), and n G ≡ R G /R G is a unit vector pointing from S to the Galactic center. In our calculations below we will use U G ∼ v 2 G , where v G is the rotational velocity of the Galaxy at S . Equation (4) is exact, only if the external mass is concentrated at the Galactic center, otherwise a correcting factor has to be applied, which depends on the model for the mass distribution in our Galaxy [21]. At the end of section 5, we show that this factor is close to two, as already estimated in [15].</text>
<text><location><page_4><loc_12><loc_45><loc_84><loc_48></location>From Lagrangian (4), a binary system of mass m 1 and m 2 gets an extra acceleration for the relative movement (see (8.73) in [42] with different sign conventions),</text>
<formula><location><page_4><loc_16><loc_39><loc_84><loc_44></location>a ξ = ξ U G c 2 G ( m 1 + m 2 ) r 2 [ 2( n G · n ) n G -3 n ( n G · n ) 2 ] , (5)</formula>
<text><location><page_4><loc_12><loc_35><loc_84><loc_41></location>where r ≡ r 1 -r 2 and n ≡ r /r . Because of the analogy between the extra acceleration caused by the PPN parameter α 2 (see (8.73) in [42]), the Lagrangian (4) results in similar equations of motion with replacements,</text>
<formula><location><page_4><loc_16><loc_33><loc_84><loc_34></location>w → v G and α 2 →-2 ξ , (6)</formula>
<text><location><page_4><loc_12><loc_24><loc_84><loc_32></location>where v G ≡ v G n G is an effective velocity [35]. With replacements (6), the influence of ξ for an eccentric orbit of a binary system can be read out readily from (17-19) in [34]. As for the α 2 test, in the limit of small eccentricity, ξ induces a precession of the orbital angular momentum around the direction n G with an angular frequency [34],</text>
<formula><location><page_4><loc_16><loc_19><loc_84><loc_24></location>Ω prec = ξ ( 2 π P b ) ( v G c ) 2 cos ψ, (7)</formula>
<text><location><page_4><loc_12><loc_14><loc_84><loc_19></location>where P b is the orbital period, and ψ is the angle between n G and the orbital angular momentum. This precession would introduce observable effects in binary pulsar timing experiments (see section 4.1).</text>
<text><location><page_4><loc_12><loc_8><loc_84><loc_13></location>Similar to the case of a binary system, for an isolated, rotating massive body with internal equilibrium, Nordtvedt showed in [28] that ξ would induce a precession of the spin around n G with an angular frequency (note, in [28] ξ Nordtvedt = -1 2 ξ ),</text>
<formula><location><page_4><loc_16><loc_2><loc_84><loc_7></location>Ω prec = ξ ( 2 π P ) ( v G c ) 2 cos ψ, (8)</formula>
<figure>
<location><page_5><loc_26><loc_70><loc_71><loc_89></location>
<caption>Figure 1. Local position invariance violation causes a precession of the Solar angular momentum S /circledot around the direction of the local Galactic acceleration n G , which causes characteristic changes in the angle θ between S /circledot and the norm of the invariable plane n inv . Due to the movement of the Solar system in the Galaxy, n G is changing periodically with a period of ∼ 250Myr.</caption>
</figure>
<text><location><page_5><loc_12><loc_51><loc_84><loc_57></location>where now ψ stands for the angle between the spin of the body and n G . This precession can be constrained by observables in the Solar system and solitary millisecond pulsars (see section 3 and section 4.2 respectively).</text>
<section_header_level_1><location><page_5><loc_12><loc_47><loc_51><loc_49></location>3. A weak-field limit from the Solar spin</section_header_level_1>
<text><location><page_5><loc_12><loc_28><loc_84><loc_45></location>At the birth of the Solar system ∼ 4 . 6 billion years ago, the angle θ between the Sun's spin S /circledot and the total angular momentum of the Solar system (its direction is represented by the norm of the invariable plane n inv ) were very likely closely aligned, as suggested by our understanding of the formation of planetary systems. After the birth, the Newtonian torque on the Sun produced by the tidal fields of planets is negligibly weak (see (10)). Due to today's observation of θ ∼ 6 · , Nordtvedt suggested to constrain ξ to a high precision through constraining (8) [28]. Based on his α 2 test and an order-of-magnitude estimation, he already concluded ξ /lessorsimilar 10 -7 . Here we slightly improve his method and present detailed calculations to constrain ξ from the Solar spin.</text>
<text><location><page_5><loc_12><loc_13><loc_84><loc_27></location>For directions of S /circledot and n inv , we take the International Celestial Reference Frame (ICRF) equatorial coordinates at epoch J2000.0 from recent reports of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements [32, 1]. The direction of S /circledot is ( α 0 , δ 0 ) /circledot = (286 · . 13 , 63 · . 87) in the Celestial coordinates or ( l, b ) /circledot = (94 · . 45 , 22 · . 77) in the Galactic coordinates. The coordinates of n inv are ( α 0 , δ 0 ) inv = (273 · . 85 , 66 · . 99) or ( l, b ) inv = (96 · . 92 , 28 · . 31). The difference between these two directions is</text>
<formula><location><page_5><loc_16><loc_11><loc_84><loc_12></location>θ | t =0 = 5 · . 97 , (9)</formula>
<text><location><page_5><loc_12><loc_8><loc_44><loc_10></location>where t = 0 denotes the current epoch.</text>
<text><location><page_5><loc_12><loc_4><loc_84><loc_8></location>Assuming that the Sun's spin was closely aligned with n inv right after the formation of the Solar system, 4.6 Gyr in the past, one can convert (9) into a limit for ξ . For this,</text>
<figure>
<location><page_6><loc_22><loc_58><loc_72><loc_87></location>
<caption>Figure 2. Evolutions of the misalignment angle θ ( t ) backward in time with different ξ vaules, which have taken both (8) and (10) into account.</caption>
</figure>
<text><location><page_6><loc_12><loc_39><loc_84><loc_49></location>one has to account for the Solar movement around the Galactic center ( ∼ 20 circles in 4.6 Gyr) when using (8) to properly integrate back in time for a given ξ . We show evolutions of the misalignment angle θ ( t ) in figure 2 for different ξ vaules. In calculations in figure 2, besides the contribution (8), we also include the precession produced by the Newtonian quadrupole coupling with an angular frequency,</text>
<formula><location><page_6><loc_16><loc_33><loc_84><loc_38></location>Ω prec J 2 = 3 2 J 2 GM /circledot R 2 /circledot | S /circledot | ∑ i m i r 3 i , (10)</formula>
<text><location><page_6><loc_12><loc_20><loc_84><loc_33></location>where M /circledot and R /circledot are the Solar mass and the Solar radius, m i and r i are the mass and the orbital size of body i in the Solar system, and J 2 = (2 . 40 ± 0 . 25) × 10 -7 [12]. The main contributions in (10) are coming from Jupiter, Venus and Earth. The coupling is very weak, and (10) has a precession period ∼ 9 × 10 11 yr, hence it precesses ∼ 2 · in 4.6 Gyr (notice a factor of two discrepancy with (15) in [28] mainly due to the use of a modern J 2 value). Such a precession hardly modifies the evolution of θ ( t ); besides, the precession (10) is around n inv which by itself does not change θ .</text>
<text><location><page_6><loc_12><loc_14><loc_84><loc_19></location>In figure 3 we plot the initial misalignment angle at the birth of the Solar system and the angle ∆ χ swept out by S /circledot during the past 4.6 Gyr as functions of ξ . From figure 3 it is obvious that any ξ significantly outside the range</text>
<formula><location><page_6><loc_16><loc_11><loc_84><loc_13></location>| ξ | /lessorsimilar 5 × 10 -6 (11)</formula>
<text><location><page_6><loc_12><loc_4><loc_84><loc_10></location>would contradict the assumption that the Sun was formed spinning in a close alignment with the planetary orbits (say, θ birth /greaterorsimilar 10 · ). Limit (11) is three orders of magnitude better than that from superconducting gravimeter [38].</text>
<figure>
<location><page_7><loc_27><loc_66><loc_68><loc_88></location>
<caption>Figure 3. The initial misalignment angle θ birth and the angle difference ∆ χ between current S /circledot and S /circledot at birth as functions of ξ . They are obtained from evolving S /circledot according to (8) and (10) back in time to the epoch t = -4 . 6Gyr.</caption>
</figure>
<section_header_level_1><location><page_7><loc_12><loc_52><loc_50><loc_54></location>4. Limits from radio millisecond pulsars</section_header_level_1>
<section_header_level_1><location><page_7><loc_12><loc_49><loc_39><loc_50></location>4.1. A limit from binary pulsars</section_header_level_1>
<text><location><page_7><loc_12><loc_36><loc_85><loc_47></location>According to (7), the orbital angular momentum of a binary system with a small eccentricity undergoes a ξ -induced precession around n G (here n G is the direction of the Galactic acceleration at the location of the binary). As mentioned in [35], this precession is analogous to the precession induced by the PPN parameter α 2 [34] with replacements (6). Hence the same analysis done for the ˆ α 2 test in [34] applies to the ˆ ξ test in binary pulsars.</text>
<text><location><page_7><loc_12><loc_23><loc_84><loc_35></location>Using the Galactic potential model in [31] with the distance of the Solar system to the Galactic center ∼ 8 kpc, Shao et al [35] performed 10 7 Monte Carlo simulations to account for measurement uncertainties and the unknown longitude of ascending node (for details, see section 3 of [34]). From a combination of PSRs J1012+5307 and J1738+0333, they got a probabilistic limit (see figure 1 in [35] for probability densities from separated binary pulsars and their combination),</text>
<formula><location><page_7><loc_16><loc_21><loc_84><loc_23></location>| ˆ ξ | < 3 . 1 × 10 -4 , (95% CL) . (12)</formula>
<text><location><page_7><loc_12><loc_14><loc_84><loc_19></location>It is two orders of magnitude weaker than the limit (11) from the Solar spin, but it represents a constraint involving a strongly self-gravitating body, namely, NS-WD binary systems (see section 5).</text>
<section_header_level_1><location><page_7><loc_12><loc_10><loc_40><loc_11></location>4.2. A limit from solitary pulsars</section_header_level_1>
<text><location><page_7><loc_12><loc_5><loc_84><loc_8></location>Similar to the precession of the Solar spin, the spin of a solitary pulsar would undergo a ˆ ξ -induced precession around n G with an angular frequency (8). Such a precession would</text>
<text><location><page_8><loc_12><loc_85><loc_84><loc_89></location>change our line-of-sight cut on the pulsar emission beam, hence change the pulse profile characteristics over time, see figure 1 in [33] for illustrations.</text>
<text><location><page_8><loc_12><loc_73><loc_84><loc_84></location>Recently, to test the local Lorentz invariance of gravity, Shao et al [33] analyzed a large number of pulse profiles from PSRs B1937+21 and J1744 -1134, obtained at the 100-m Effelsberg radio telescope with the same backend, spanning about ∼ 15 years. From various aspects, the pulse profiles are very stable, and no change in the profiles is found (see figures 2-7 in [33] for stabilities of pulse profiles). These results can equally well be used for a test of LPI of gravity.</text>
<text><location><page_8><loc_12><loc_57><loc_84><loc_72></location>By using a simple cone emission model of pulsars [20], one can quantitatively relate a change in the orientation of the pulsar spin with that in the width of the pulse profile (see (10) in [33]). By using the limits on the change of pulse widths in table 1 of [33], we set up 10 7 Monte Carlo simulations to get probability densities of ˆ ξ from PSRs B1937+21 and J1744 -1134. In simulations we use the Galactic potential model in [31] and all other parameters are the same as in [33] with replacements (6). The results are shown in figure 4 for PSRs B1937+21 and J1744 -1134 and their combination. For the individual limits one finds</text>
<formula><location><page_8><loc_16><loc_54><loc_84><loc_56></location>PSR B1937+21: | ˆ ξ | < 2 . 2 × 10 -8 , (95% CL) , (13)</formula>
<formula><location><page_8><loc_16><loc_52><loc_84><loc_54></location>PSR J1744 -1134: | ˆ ξ | < 1 . 2 × 10 -7 , (95% CL) . (14)</formula>
<text><location><page_8><loc_12><loc_41><loc_84><loc_51></location>They are already significantly better than the limit (11) obtained from the Solar spin. Like in [33], the analysis for PSR B1937+21 is based on the main-pulse. Also here, one could use the interpulse to constrain a precession of PSR B1937+21, which again leads to a similar, even slightly more constraining limit. As in [33], we will stay with the more conservative value derived from the main-pulse.</text>
<text><location><page_8><loc_12><loc_33><loc_84><loc_41></location>As explained in details in [34, 33], the combination of two pulsars leads to a significant suppression of the long tails in the probability density function. Assuming that ˆ ξ is only weakly dependent on the pulsar mass, PSRs B1937+21 and J1744 -1134 give a combined limit for strongly self-gravitating bodies of</text>
<formula><location><page_8><loc_16><loc_30><loc_84><loc_32></location>| ˆ ξ | < 3 . 9 × 10 -9 , (95% CL) . (15)</formula>
<text><location><page_8><loc_12><loc_22><loc_84><loc_29></location>The limit (15) is the most constraining one of the three tests presented in this paper. It is more than three orders of magnitude better than the limit (11) from the Solar system and five orders of magnitude better than the limit (12) from binary pulsars. This is in accordance with the α 2 and ˆ α 2 results [28, 34, 33].</text>
<section_header_level_1><location><page_8><loc_12><loc_18><loc_25><loc_19></location>5. Discussions</section_header_level_1>
<text><location><page_8><loc_12><loc_4><loc_84><loc_16></location>Mach's principle states that the inertial mass of a body is determined by the total matter distribution in the Universe, so if the matter distribution is not isotropic, the gravity interaction that a mass feels can depend on its direction of acceleration [7, 8]. The tests presented in this paper are Hughes-Drever-type experiments which originally were conducted to test a possible anisotropy in mass through magnetic resonance measurements in spectroscopy [14, 11]. We note that the constraint on LPI here is</text>
<figure>
<location><page_9><loc_27><loc_65><loc_67><loc_88></location>
<caption>Figure 4. Probability density functions of the strong-field PPN parameter ˆ ξ from PSR B1937+21 (blue dashed histogram), PSR J1744 -1134 (red dotted histogram), and their combination (black solid histogram). All probability density functions are normalized.</caption>
</figure>
<text><location><page_9><loc_12><loc_50><loc_84><loc_54></location>for the gravitational interaction, that is different from the LPI of Einstein's Equivalence Principle related to special relativity, see e.g. [5, 2] and the review article [43].</text>
<text><location><page_9><loc_12><loc_42><loc_84><loc_50></location>Although we express our limits on the anisotropy of gravity in terms of the PPN parameter ξ (or its strong-field generalization ˆ ξ ), it is quite straightforward to convert them into limits on the anisotropy of the gravitational constant. From (6.75) in [42], one has</text>
<formula><location><page_9><loc_16><loc_37><loc_84><loc_42></location>G local = G 0 [ 1 + ξ ( 3 + I MR 2 ) U G + ξ ( e · n G ) 2 ( 1 -3 I MR 2 ) U G ] , (16)</formula>
<text><location><page_9><loc_12><loc_24><loc_84><loc_37></location>where G 0 is the bare gravitational constant; I , M , and R are the moment of inertia, mass and radius of a system S respectively; e is a unit vector pointing from the center of mass of S to the location where G is being measured (see [42]). The first correction only renormalizes the bare gravitational constant and is not relevant here. The second correction contains an anisotropic contribution. For solitary pulsars PSRs B1937+21 and J1744 -1134, they both have v 2 G ∼ 5 × 10 -7 . Hence from (15), by using I/MR 2 /similarequal 0 . 4 for a typical NS [18], one gets</text>
<formula><location><page_9><loc_17><loc_16><loc_84><loc_23></location>∣ ∣ ∆ G G ∣ ∣ ∣ anisotropy < 4 × 10 -16 , (95% CL) (17)</formula>
<text><location><page_9><loc_12><loc_14><loc_84><loc_21></location>∣ ∣ ∣ which is the most constraining limit on the anisotropy of G . It is four orders of magnitude better than that achievable with LLR in the foreseeable future.</text>
<text><location><page_9><loc_12><loc_4><loc_84><loc_14></location>For any 'quasilinear' theory of gravity, the PPN parameters satisfy β = ξ [41]. Hence for such a theory, a limit on β of O (10 -9 ) can be drawn, which is six orders of magnitude more constraining than the limit on β from the anomalous precession of Mercury [43]. Nordtvedt developed an anisotropic PPN framework [27] and suggested to use the binary pulsar PSR B1913+16 [26] and LLR [10, 29] to constrain its</text>
<text><location><page_10><loc_12><loc_77><loc_84><loc_89></location>parameters. Our result shows that careful profile analysis of solitary pulsars can constrain some anisotropic PPN parameters more effectively. The standard model extension of gravity [4, 17] has 20 free parameters in the pure-gravity sector, of which a subset ¯ s jk appears in a Lagrangian term similar to (4) (see (54) in [4]), hence can be constrained tightly through our tests. We expect a combination of ¯ s jk (similar to (97) in [4]) can be constrained to O (10 -15 ) ‡ .</text>
<text><location><page_10><loc_12><loc_61><loc_84><loc_76></location>At this point we would like to elaborate on the distinction between the weakfield PPN parameter ξ and its strong-field generalization ˆ ξ . In GR, ξ = ˆ ξ = 0, but a distinction is necessary for alternative gravity theories. Damour and Esposito-Far'ese explicitly showed that in scalar-tensor theories, the strong gravitational fields of neutron stars can develop nonperturbative effects [9]. Although scalar-tensor theories have no LPI violation, one can imagine that similar nonperturbative strong-field modifications might exist in other theories with LPI violation. If the strong-field modification is perturbative, one may write an expansion like,</text>
<formula><location><page_10><loc_16><loc_58><loc_84><loc_60></location>ˆ ξ = ξ + K 1 C + K 2 C 2 + · · · , (18)</formula>
<text><location><page_10><loc_12><loc_51><loc_84><loc_57></location>where the compactness C (roughly equals the fractional gravitational binding energy) of a NS ( C NS ∼ 0 . 2) is O (10 5 ) times larger than that of the Sun ( C /circledot ∼ 10 -6 ). Hence NSs can probe the coefficients K i 's much more efficiently than the Solar system.</text>
<text><location><page_10><loc_12><loc_9><loc_84><loc_51></location>Let us compare the prospects of different tests of LPI in the future. As mentioned before, the best limit on ξ from superconducting gravimeter [38] is of O (10 -3 ). Modern superconducting gravimeters are more sensitive. They are distributed around the world, where a total of 25 superconducting gravimeters form the Global Geodynamics Project (GGP) network [36]. The sensitivity of a superconducting gravimeter, installed at a quiet site, is better than 1 nGal ≡ 10 -11 ms -2 for a one-year measurement, which is less than the seismic noise level (a few nGal) at the signal frequencies of ξ [36]. However, the test is severely limited by the Earth model and unremovable Earth noises. Even under optimistic estimations for GGP, ξ is expected to be constrained to O (10 -5 ) at best [36], which is four orders of magnitude away from (15). The analysis of LLR data usually does not include the ξ parameter explicitly, but with its analogy with α 2 , one can expect a limit of O (10 -5 ) at best [23]. The Solar limit (11) is based on a long baseline in time (about 4.6 Gyr), hence it is not going to improve anymore. In contrast, the limits (12) and (15) will continuously improve with T -3 / 2 solely based on current pulsars, where T is the observational time span [34, 33]. New telescopes like the Fivehundred-meter Aperture Spherical Telescope (FAST) [24] and the Square Kilometre Array (SKA) [37] will provide better sensitivities in obtaining pulse profiles, that will be very valuable for improving the limit of ˆ ξ (and also ˆ α 2 [33]), especially for the weaker pulsar PSR J1744 -1134. In addition, discoveries of new fast rotating millisecond pulsars through FAST and SKA are expected in the future, which will enrich our set of testing systems and further improve the limits.</text>
<text><location><page_11><loc_12><loc_65><loc_84><loc_89></location>Let us elaborate on a possible correcting factor to our limits on ξ and ˆ ξ , arising from a more rigorous treatment of the Galactic mass distribution. When estimating U G , we have approximated it as U G ∼ v 2 G which, e.g., at the location of the Sun gives U G /c 2 /similarequal 5 . 4 × 10 -7 . Mentock pointed out that the dark matter halo might invalidate such an approximation [21]. However, Gibbons and Will explicitly showed, by using a Galaxy model with spherically symmetric matter distribution, that such a correction is roughly a factor of two [15]. We use the Galaxy potential model in [31] that consists of three components, namely the bulge, the disk and the dark matter halo, and get a factor of 1.86. § The results confirm the correcting factor in [15], and our limits on ξ and ˆ ξ should be weakened by this factor (as well as all previous limits on ξ in literature). Nevertheless, the limit (17) on the anisotropy of G will not change because only the product ξU G enters in (16).</text>
<text><location><page_11><loc_12><loc_61><loc_84><loc_64></location>As a final remark, using the words of [15], also for pulsar astronomers Whitehead's gravity theory [39] ( ξ = 1) is truly dead .</text>
<section_header_level_1><location><page_11><loc_12><loc_57><loc_29><loc_58></location>Acknowledgments</section_header_level_1>
<text><location><page_11><loc_12><loc_47><loc_84><loc_55></location>We thank Nicolas Caballero, David Champion, and Michael Kramer for valuable discussions. We are grateful to Aris Noutsos for reading the manuscript. Lijing Shao is supported by China Scholarship Council (CSC). This research has made use of NASA's Astrophysics Data System.</text>
<section_header_level_1><location><page_11><loc_12><loc_43><loc_22><loc_44></location>References</section_header_level_1>
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</document>
[
{
"title": "Lijing Shao 1 , 2 and Norbert Wex 1",
"content": "E-mail: lshao@pku.edu.cn (LS) , wex@mpifr-bonn.mpg.de (NW)",
"pages": [
1
]
},
{
"title": "Abstract.",
"content": "Within the parameterized post-Newtonian (PPN) formalism, there could be an anisotropy of local gravity induced by an external matter distribution, even for a fully conservative metric theory of gravity. It reflects the breakdown of the local position invariance of gravity and, within the PPN formalism, is characterized by the Whitehead parameter ξ . We present three different kinds of observation, from the Solar system and radio pulsars, to constrain it. The most stringent limit comes from recent results on the extremely stable pulse profiles of solitary millisecond pulsars, that gives | ˆ ξ | < 3 . 9 × 10 -9 (95% CL), where the hat denotes the strong-field generalization of ξ . This limit is six orders of magnitude more constraining than the current best limit from superconducting gravimeter experiments. It can be converted into an upper limit of ∼ 4 × 10 -16 on the spatial anisotropy of the gravitational constant. PACS numbers: 04.80.Cc, 96.60.-j, 97.60.Gb",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Since the 1960s, advances in technologies are continuously providing a series of formidable tests of gravity theories from on-ground laboratories, the Solar system, various pulsar systems, and also cosmology [42, 43]. Up to now, Einstein's general relativity (GR) passed all experimental tests with flying colors. However, questions related to the nature of dark matter and dark energy, and irreconcilable conflicts between GR and the standard model of particle physics, are strong motivations to study alternative theories of gravity. In addition, gravity as a fundamental interaction of nature deserves most stringent tests from various aspects. For tests of gravity theories, one of the most popular frameworks is the parameterized post-Newtonian (PPN) formalism , proposed by Nordtvedt and Will [25, 40, 44, 42]. In the standard PPN gauge, the framework contains ten dimensionless PPN parameters in the metric components as coefficients of various potential forms. These parameters take different values in different gravity theories. Hence, experimental constraints on these parameters can be directly used to test specific gravity theories [30, 42, 43]. In this paper, we concentrate on one of the ten PPN parameters which characterizes a possible Galaxy-induced anisotropy in the gravitational interaction of localized systems. Such an anisotropy is described by the Whitehead parameter ξ in the weakfield slow-motion limit [41]. We use ˆ ξ to explicitly denote its strong-field generalization. Besides Whitehead's gravity theory [39], ξ is relevant for a class of theories called 'quasilinear' theories of gravity [41]. In GR, the gravitational interaction is local position invariant with ξ = 0, while in Whitehead's gravity, local position invariance (LPI) is violated and ξ = 1 [41, 15]. An anisotropy of gravitational interaction, induced by the gravitational field of the Galaxy, would lead to anomalous Earth tides at specific frequencies with characteristic phase relations [41, 38]. The ξ -induced Earth tides are caused by a change in the local gravitational attraction on the Earth surface due to the rotation of the Earth with frequencies associated with the sidereal day. By using constraints on ξ from superconducting gravimeter, Will gave the first disproof of Whitehead's parameterfree gravity theory [41] (see [15] for multiple recent disproofs). Later Warburton and Goodkind presented an update on the limit of ξ by using new gravimeter data [38], where they were able to constrain | ξ | to the order of 10 -3 . The uncertainties concerning geophysical perturbations and the imperfect knowledge of the Earth structure limit the precision. Uncertainties include the elastic responses of the Earth, the effects of ocean tides, the effects of atmospheric tides from barometric pressure variation, and the resonances in the liquid core of the Earth [38] (see [16, 36] for recent reviews on superconducting gravimeters). Limits from Earth tides are based on periodic terms proportional to ξ , while secular effects in other astrophysical laboratories can be more constraining. Nordtvedt used the close alignment of the Sun's spin with the invariable plane of the Solar system to constrain the PPN parameter α 2 , associated with the local Lorentz invariance of gravity down to O (10 -7 ) [28]. In the same publication Nordtvedt pointed out that such a limit is also possible for ξ , as the two terms in the Lagrangian have the same form. However, to our knowledge, no detailed calculations have been published yet. In section 3 we follow Nordtvedt's suggestion and achieve a limit of O (10 -6 ). A non-vanishing (strong-field) ˆ ξ would lead to characteristic secular effects in the dynamics of the rotation and orbital motion of radio pulsars. We have presented the methodologies in details to constrain the (strong-field) ˆ α 2 from binary pulsar timing [34] and solitary pulsar profile analysis [33] respectively. By the virtue of the similarity between ˆ α 2 -and ˆ ξ -related effects, in section 4 we extend the analysis in [34, 33] to the case of LPI of gravity. From timing results of PSRs J1012+5307 [19] and J1738+0333 [13], a limit of | ˆ ξ | < 3 . 1 × 10 -4 (95% CL) is achieved for neutron star (NS) white dwarf (WD) systems [35]. As shown in this paper, from the analysis on the pulse profile stability of PSRs B1937+21 and J1744 -1134, a limit of | ˆ ξ | < 3 . 9 × 10 -9 (95% CL) is obtained, utilizing the rotational properties of solitary millisecond pulsars. This limit is six orders of magnitude better than the (weak-field) limit from gravimeter. The paper is organized as follows. In the next section, the theoretical framework for tests of LPI of gravity is briefly summarized. In section 3, a limit on ξ from the Solar system is obtained. Then we give limits on ˆ ξ from binary pulsars and solitary pulsars in section 4. In the last section, we discuss issues related to strong-field modifications and conversions from our limits to limits on the anisotropy in the gravitational constant. Comparisons between our tests with other achievable tests from gravimeter and lunar laser ranging (LLR) experiments are also given.",
"pages": [
2,
3
]
},
{
"title": "2. Theoretical framework",
"content": "In the PPN formalism, PPN parameters are introduced as dimensionless coefficients in the metric in front of various potential forms [44, 42, 43]. In the standard postNewtonian gauge, ξ appears in the metric components g 00 and g 0 i [44, 42, 43]. However, in most cases, it is relevant only in linear combinations with other PPN parameters like β , γ (see [42, 43] for formalism and details). Due to the limited precision in constraining these PPN parameters (see table 4 in [43] for current constraints on PPN parameters), it is not easy to get an independent stringent limit for ξ . For example, based on the Nordtvedt parameter (see (43) in [15]), one can only constrain ξ to the order of O (10 -3 ) at most. Nevertheless, in the metric component g 00 , -2 ξ alone appears as the coefficient of the Whitehead potential [41], where ρ ( x ) is the matter density, G and c are the gravitational constant and the speed of light respectively. This fact provides the possibility to constrain the PPN parameter",
"pages": [
3
]
},
{
"title": "ξ directly.",
"content": "Correspondingly, in the PPN n -body Lagrangian, we have a ξ -related term for three-body interactions (see e.g. (6.80) in [42]), where the summation excludes terms that make any denominators vanish. For our purposes below, we consider the third body being our Galaxy, and only consider a system S (the Solar system or a pulsar binary system or a solitary pulsar) of typical size much less than its distance to the Galactic center R G . Hence the Lagrangian (3) reduces to (dropping a constant factor that rescales G ) where U G is the Galactic potential at the position of the system S (associated with the mass inside R G ), and n G ≡ R G /R G is a unit vector pointing from S to the Galactic center. In our calculations below we will use U G ∼ v 2 G , where v G is the rotational velocity of the Galaxy at S . Equation (4) is exact, only if the external mass is concentrated at the Galactic center, otherwise a correcting factor has to be applied, which depends on the model for the mass distribution in our Galaxy [21]. At the end of section 5, we show that this factor is close to two, as already estimated in [15]. From Lagrangian (4), a binary system of mass m 1 and m 2 gets an extra acceleration for the relative movement (see (8.73) in [42] with different sign conventions), where r ≡ r 1 -r 2 and n ≡ r /r . Because of the analogy between the extra acceleration caused by the PPN parameter α 2 (see (8.73) in [42]), the Lagrangian (4) results in similar equations of motion with replacements, where v G ≡ v G n G is an effective velocity [35]. With replacements (6), the influence of ξ for an eccentric orbit of a binary system can be read out readily from (17-19) in [34]. As for the α 2 test, in the limit of small eccentricity, ξ induces a precession of the orbital angular momentum around the direction n G with an angular frequency [34], where P b is the orbital period, and ψ is the angle between n G and the orbital angular momentum. This precession would introduce observable effects in binary pulsar timing experiments (see section 4.1). Similar to the case of a binary system, for an isolated, rotating massive body with internal equilibrium, Nordtvedt showed in [28] that ξ would induce a precession of the spin around n G with an angular frequency (note, in [28] ξ Nordtvedt = -1 2 ξ ), where now ψ stands for the angle between the spin of the body and n G . This precession can be constrained by observables in the Solar system and solitary millisecond pulsars (see section 3 and section 4.2 respectively).",
"pages": [
4,
5
]
},
{
"title": "3. A weak-field limit from the Solar spin",
"content": "At the birth of the Solar system ∼ 4 . 6 billion years ago, the angle θ between the Sun's spin S /circledot and the total angular momentum of the Solar system (its direction is represented by the norm of the invariable plane n inv ) were very likely closely aligned, as suggested by our understanding of the formation of planetary systems. After the birth, the Newtonian torque on the Sun produced by the tidal fields of planets is negligibly weak (see (10)). Due to today's observation of θ ∼ 6 · , Nordtvedt suggested to constrain ξ to a high precision through constraining (8) [28]. Based on his α 2 test and an order-of-magnitude estimation, he already concluded ξ /lessorsimilar 10 -7 . Here we slightly improve his method and present detailed calculations to constrain ξ from the Solar spin. For directions of S /circledot and n inv , we take the International Celestial Reference Frame (ICRF) equatorial coordinates at epoch J2000.0 from recent reports of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements [32, 1]. The direction of S /circledot is ( α 0 , δ 0 ) /circledot = (286 · . 13 , 63 · . 87) in the Celestial coordinates or ( l, b ) /circledot = (94 · . 45 , 22 · . 77) in the Galactic coordinates. The coordinates of n inv are ( α 0 , δ 0 ) inv = (273 · . 85 , 66 · . 99) or ( l, b ) inv = (96 · . 92 , 28 · . 31). The difference between these two directions is where t = 0 denotes the current epoch. Assuming that the Sun's spin was closely aligned with n inv right after the formation of the Solar system, 4.6 Gyr in the past, one can convert (9) into a limit for ξ . For this, one has to account for the Solar movement around the Galactic center ( ∼ 20 circles in 4.6 Gyr) when using (8) to properly integrate back in time for a given ξ . We show evolutions of the misalignment angle θ ( t ) in figure 2 for different ξ vaules. In calculations in figure 2, besides the contribution (8), we also include the precession produced by the Newtonian quadrupole coupling with an angular frequency, where M /circledot and R /circledot are the Solar mass and the Solar radius, m i and r i are the mass and the orbital size of body i in the Solar system, and J 2 = (2 . 40 ± 0 . 25) × 10 -7 [12]. The main contributions in (10) are coming from Jupiter, Venus and Earth. The coupling is very weak, and (10) has a precession period ∼ 9 × 10 11 yr, hence it precesses ∼ 2 · in 4.6 Gyr (notice a factor of two discrepancy with (15) in [28] mainly due to the use of a modern J 2 value). Such a precession hardly modifies the evolution of θ ( t ); besides, the precession (10) is around n inv which by itself does not change θ . In figure 3 we plot the initial misalignment angle at the birth of the Solar system and the angle ∆ χ swept out by S /circledot during the past 4.6 Gyr as functions of ξ . From figure 3 it is obvious that any ξ significantly outside the range would contradict the assumption that the Sun was formed spinning in a close alignment with the planetary orbits (say, θ birth /greaterorsimilar 10 · ). Limit (11) is three orders of magnitude better than that from superconducting gravimeter [38].",
"pages": [
5,
6
]
},
{
"title": "4.1. A limit from binary pulsars",
"content": "According to (7), the orbital angular momentum of a binary system with a small eccentricity undergoes a ξ -induced precession around n G (here n G is the direction of the Galactic acceleration at the location of the binary). As mentioned in [35], this precession is analogous to the precession induced by the PPN parameter α 2 [34] with replacements (6). Hence the same analysis done for the ˆ α 2 test in [34] applies to the ˆ ξ test in binary pulsars. Using the Galactic potential model in [31] with the distance of the Solar system to the Galactic center ∼ 8 kpc, Shao et al [35] performed 10 7 Monte Carlo simulations to account for measurement uncertainties and the unknown longitude of ascending node (for details, see section 3 of [34]). From a combination of PSRs J1012+5307 and J1738+0333, they got a probabilistic limit (see figure 1 in [35] for probability densities from separated binary pulsars and their combination), It is two orders of magnitude weaker than the limit (11) from the Solar spin, but it represents a constraint involving a strongly self-gravitating body, namely, NS-WD binary systems (see section 5).",
"pages": [
7
]
},
{
"title": "4.2. A limit from solitary pulsars",
"content": "Similar to the precession of the Solar spin, the spin of a solitary pulsar would undergo a ˆ ξ -induced precession around n G with an angular frequency (8). Such a precession would change our line-of-sight cut on the pulsar emission beam, hence change the pulse profile characteristics over time, see figure 1 in [33] for illustrations. Recently, to test the local Lorentz invariance of gravity, Shao et al [33] analyzed a large number of pulse profiles from PSRs B1937+21 and J1744 -1134, obtained at the 100-m Effelsberg radio telescope with the same backend, spanning about ∼ 15 years. From various aspects, the pulse profiles are very stable, and no change in the profiles is found (see figures 2-7 in [33] for stabilities of pulse profiles). These results can equally well be used for a test of LPI of gravity. By using a simple cone emission model of pulsars [20], one can quantitatively relate a change in the orientation of the pulsar spin with that in the width of the pulse profile (see (10) in [33]). By using the limits on the change of pulse widths in table 1 of [33], we set up 10 7 Monte Carlo simulations to get probability densities of ˆ ξ from PSRs B1937+21 and J1744 -1134. In simulations we use the Galactic potential model in [31] and all other parameters are the same as in [33] with replacements (6). The results are shown in figure 4 for PSRs B1937+21 and J1744 -1134 and their combination. For the individual limits one finds They are already significantly better than the limit (11) obtained from the Solar spin. Like in [33], the analysis for PSR B1937+21 is based on the main-pulse. Also here, one could use the interpulse to constrain a precession of PSR B1937+21, which again leads to a similar, even slightly more constraining limit. As in [33], we will stay with the more conservative value derived from the main-pulse. As explained in details in [34, 33], the combination of two pulsars leads to a significant suppression of the long tails in the probability density function. Assuming that ˆ ξ is only weakly dependent on the pulsar mass, PSRs B1937+21 and J1744 -1134 give a combined limit for strongly self-gravitating bodies of The limit (15) is the most constraining one of the three tests presented in this paper. It is more than three orders of magnitude better than the limit (11) from the Solar system and five orders of magnitude better than the limit (12) from binary pulsars. This is in accordance with the α 2 and ˆ α 2 results [28, 34, 33].",
"pages": [
7,
8
]
},
{
"title": "5. Discussions",
"content": "Mach's principle states that the inertial mass of a body is determined by the total matter distribution in the Universe, so if the matter distribution is not isotropic, the gravity interaction that a mass feels can depend on its direction of acceleration [7, 8]. The tests presented in this paper are Hughes-Drever-type experiments which originally were conducted to test a possible anisotropy in mass through magnetic resonance measurements in spectroscopy [14, 11]. We note that the constraint on LPI here is for the gravitational interaction, that is different from the LPI of Einstein's Equivalence Principle related to special relativity, see e.g. [5, 2] and the review article [43]. Although we express our limits on the anisotropy of gravity in terms of the PPN parameter ξ (or its strong-field generalization ˆ ξ ), it is quite straightforward to convert them into limits on the anisotropy of the gravitational constant. From (6.75) in [42], one has where G 0 is the bare gravitational constant; I , M , and R are the moment of inertia, mass and radius of a system S respectively; e is a unit vector pointing from the center of mass of S to the location where G is being measured (see [42]). The first correction only renormalizes the bare gravitational constant and is not relevant here. The second correction contains an anisotropic contribution. For solitary pulsars PSRs B1937+21 and J1744 -1134, they both have v 2 G ∼ 5 × 10 -7 . Hence from (15), by using I/MR 2 /similarequal 0 . 4 for a typical NS [18], one gets ∣ ∣ ∣ which is the most constraining limit on the anisotropy of G . It is four orders of magnitude better than that achievable with LLR in the foreseeable future. For any 'quasilinear' theory of gravity, the PPN parameters satisfy β = ξ [41]. Hence for such a theory, a limit on β of O (10 -9 ) can be drawn, which is six orders of magnitude more constraining than the limit on β from the anomalous precession of Mercury [43]. Nordtvedt developed an anisotropic PPN framework [27] and suggested to use the binary pulsar PSR B1913+16 [26] and LLR [10, 29] to constrain its parameters. Our result shows that careful profile analysis of solitary pulsars can constrain some anisotropic PPN parameters more effectively. The standard model extension of gravity [4, 17] has 20 free parameters in the pure-gravity sector, of which a subset ¯ s jk appears in a Lagrangian term similar to (4) (see (54) in [4]), hence can be constrained tightly through our tests. We expect a combination of ¯ s jk (similar to (97) in [4]) can be constrained to O (10 -15 ) ‡ . At this point we would like to elaborate on the distinction between the weakfield PPN parameter ξ and its strong-field generalization ˆ ξ . In GR, ξ = ˆ ξ = 0, but a distinction is necessary for alternative gravity theories. Damour and Esposito-Far'ese explicitly showed that in scalar-tensor theories, the strong gravitational fields of neutron stars can develop nonperturbative effects [9]. Although scalar-tensor theories have no LPI violation, one can imagine that similar nonperturbative strong-field modifications might exist in other theories with LPI violation. If the strong-field modification is perturbative, one may write an expansion like, where the compactness C (roughly equals the fractional gravitational binding energy) of a NS ( C NS ∼ 0 . 2) is O (10 5 ) times larger than that of the Sun ( C /circledot ∼ 10 -6 ). Hence NSs can probe the coefficients K i 's much more efficiently than the Solar system. Let us compare the prospects of different tests of LPI in the future. As mentioned before, the best limit on ξ from superconducting gravimeter [38] is of O (10 -3 ). Modern superconducting gravimeters are more sensitive. They are distributed around the world, where a total of 25 superconducting gravimeters form the Global Geodynamics Project (GGP) network [36]. The sensitivity of a superconducting gravimeter, installed at a quiet site, is better than 1 nGal ≡ 10 -11 ms -2 for a one-year measurement, which is less than the seismic noise level (a few nGal) at the signal frequencies of ξ [36]. However, the test is severely limited by the Earth model and unremovable Earth noises. Even under optimistic estimations for GGP, ξ is expected to be constrained to O (10 -5 ) at best [36], which is four orders of magnitude away from (15). The analysis of LLR data usually does not include the ξ parameter explicitly, but with its analogy with α 2 , one can expect a limit of O (10 -5 ) at best [23]. The Solar limit (11) is based on a long baseline in time (about 4.6 Gyr), hence it is not going to improve anymore. In contrast, the limits (12) and (15) will continuously improve with T -3 / 2 solely based on current pulsars, where T is the observational time span [34, 33]. New telescopes like the Fivehundred-meter Aperture Spherical Telescope (FAST) [24] and the Square Kilometre Array (SKA) [37] will provide better sensitivities in obtaining pulse profiles, that will be very valuable for improving the limit of ˆ ξ (and also ˆ α 2 [33]), especially for the weaker pulsar PSR J1744 -1134. In addition, discoveries of new fast rotating millisecond pulsars through FAST and SKA are expected in the future, which will enrich our set of testing systems and further improve the limits. Let us elaborate on a possible correcting factor to our limits on ξ and ˆ ξ , arising from a more rigorous treatment of the Galactic mass distribution. When estimating U G , we have approximated it as U G ∼ v 2 G which, e.g., at the location of the Sun gives U G /c 2 /similarequal 5 . 4 × 10 -7 . Mentock pointed out that the dark matter halo might invalidate such an approximation [21]. However, Gibbons and Will explicitly showed, by using a Galaxy model with spherically symmetric matter distribution, that such a correction is roughly a factor of two [15]. We use the Galaxy potential model in [31] that consists of three components, namely the bulge, the disk and the dark matter halo, and get a factor of 1.86. § The results confirm the correcting factor in [15], and our limits on ξ and ˆ ξ should be weakened by this factor (as well as all previous limits on ξ in literature). Nevertheless, the limit (17) on the anisotropy of G will not change because only the product ξU G enters in (16). As a final remark, using the words of [15], also for pulsar astronomers Whitehead's gravity theory [39] ( ξ = 1) is truly dead .",
"pages": [
8,
9,
10,
11
]
},
{
"title": "Acknowledgments",
"content": "We thank Nicolas Caballero, David Champion, and Michael Kramer for valuable discussions. We are grateful to Aris Noutsos for reading the manuscript. Lijing Shao is supported by China Scholarship Council (CSC). This research has made use of NASA's Astrophysics Data System.",
"pages": [
11
]
},
{
"title": "References",
"content": "pulsars. Classical and Quantum Gravity , 29:215018, 2012.",
"pages": [
13
]
}
]
2013CQGra..30q3001K
https://arxiv.org/pdf/1307.5623.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_80><loc_77><loc_82></location>Quantum cosmology and late-time singularities</section_header_level_1>
<section_header_level_1><location><page_1><loc_23><loc_76><loc_42><loc_77></location>A Yu Kamenshchik</section_header_level_1>
<text><location><page_1><loc_23><loc_72><loc_83><loc_75></location>Dipartimento di Fisica e Astronomia and INFN, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy</text>
<text><location><page_1><loc_23><loc_69><loc_83><loc_71></location>L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences, Kosygin str. 2, 119334 Moscow, Russia</text>
<text><location><page_1><loc_23><loc_66><loc_48><loc_68></location>E-mail: kamenshchik@bo.infn.it</text>
<text><location><page_1><loc_23><loc_38><loc_84><loc_64></location>Abstract. The development of dark energy models has stimulated interest to cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We review a broad class of phenomena connected with soft cosmological singularities in classical and quantum cosmology. We discuss the classification of singularities from the geometrical point of view and from the point of view of the behaviour of finite size objects, crossing such singularities. We discuss in some detail quantum and classical cosmology of models based on perfect fluids (anti-Chaplygin gas and anti-Chaplygin gas plus dust), of models based on the Born-Infeld-type fields and of the model of a scalar field with a potential inversely proportional to the field itself. We dwell also on the phenomenon of the phantom divide line crossing in the scalar field models with cusped potentials. Then we discuss the Friedmann equations modified by quantum corrections to the effective action of the models under considerations and the influence of such modification on the nature and the existence of soft singularities. We review also quantum cosmology of models, where the initial quantum state of the universe is presented by the density matrix (mixed state). Finally, we discuss the exotic singularities arising in the brane-world cosmological models.</text>
<text><location><page_1><loc_12><loc_31><loc_27><loc_32></location>Submitted to: CQG</text>
<section_header_level_1><location><page_2><loc_12><loc_87><loc_27><loc_88></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_71><loc_84><loc_85></location>The problem of cosmological singularities has been attracting the attention of theoreticians working in gravity and cosmology since the early fifties [1, 2, 3]. In the sixties general theorems about the conditions for the appearance of singularities were proven [4, 5] and the oscillatory regime of approaching the singularity [6], called also 'Mixmaster universe' [7] was discovered. Basically, until the end of nineties almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius.</text>
<text><location><page_2><loc_12><loc_41><loc_84><loc_71></location>However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [8], characterized by a diverging a whereas both the scale factor a and ˙ a are finite. Then, the Hubble parameter H = ˙ a/a and the energy density ρ are also finite, while the first derivative of the Hubble parameter and the pressure p diverge. Until recent years, however, the sudden future singularities attracted rather a limited interest of researchers. The situation has changed drastically in the new millennium, when a plenty publications devoted to such singularities have appeared [9]-[25]. The arising interest to their studies is connected basically with two reasons. The recent discovery of the cosmic acceleration [26] has stimulated the elaboration of dark energy models, responsable for such a phenomenon (see e.g. for review [27]). Remarkably in some of these models the sudden singularities arise quite naturally. Another source of the interest to sudden singularities is the development of brane models [10, 11, 18], where also singularity of this kind arise naturally (sometimes the singularities, arising in the brane models are called 'quiescent' [10]). .</text>
<text><location><page_2><loc_12><loc_21><loc_84><loc_41></location>In the investigations devoted to sudden singularities one can distinguish three main topics. First of them deals with the question of the compatibility of the models, possessing soft singularities with observational data [15, 28, 29, 25]. The second direction is connected with the study of quantum effects [11, 17, 30, 31, 32, 33, 34, 35, 36]. Here one can see two subdirections: the study of quantum corrections to effective Friedmann equation, which can eliminate classical singualrities or, at least, change their form [10, 17, 30] and the study of solutions of the Wheeler-DeWitt equation for the quantum state of the universe in the presence of sudden singularities [31, 32, 33, 34, 35]. The third direction is connected with the opportunity of the crossing of sudden singularities in classical cosmology [37, 38, 39, 40, 34].</text>
<text><location><page_2><loc_12><loc_5><loc_84><loc_21></location>A particular feature of the sudden future singularities is their softness [37]. As the Christoffel symbols depend only on the first derivative of the scale factor, they are regular at these singularities. Hence, the geodesics are well behaved and they can cross the singularity [37]. One can argue that the particles crossing the singularity will generate the geometry of the spacetime, providing in such a way a soft rebirth of the universe after the singularity crossing [40]. Note that the opportunity of crossing of some kind of cosmological singularities were noticed already in the early paper by Tipler [41]. Rather a close idea of integrable singularities in black holes, which can give origin to a</text>
<text><location><page_3><loc_12><loc_79><loc_84><loc_89></location>cosmogenesis was recently put forward in [42, 43]. Besides, the results of papers [37, 38] were generalized for the case of general (non-Friedmann) universes in papers [44, 45]. For this purpose was used the formalism of the quasi-isotropic expansion of the solutions of the Einstein equations near cosmological singularities, which has been first proposed in [46]. (For some further developments of this formalism, see [47]).</text>
<text><location><page_3><loc_12><loc_55><loc_84><loc_78></location>The peculiarity of the sudden future singularities makes them to be a good tool for studying some general features of the general relativity, in particular the relations between classical and quantum gravity and cosmology. These relations is the main topic of the present review. We shall also dwell on another aspect of general relativity, which from our point of view is a little bit underestimated. It is the fact the that requirement of the self-consistency of the system of laws of general relativity and particle physics, or, in other words, of the system of Einstein equations and of the equations describing the state or the motion of non-gravitational matter can induce some interesting transformations in the state of matter. Such transformations sometimes occur when the universe passes through the soft singularities, though there are some examples of such transformations which can be observed in the absence of singularities too. We shall consider some of such examples.</text>
<text><location><page_3><loc_12><loc_4><loc_84><loc_54></location>Generally, this review is devoted to three interrelated topics, which are connected in some way with the soft future singularities in cosmology - these are the problem of crossing of such singularities in classical cosmology, the relations between classical and quantum treatments of cosmological singularities and the changes of state of matter, induced by cosmological singularities or other geometrical irregularities in the framework of general relativity. The structure of the paper is the following. In the second section we shall give a brief and convenient classification of the future singularities, following the paper [48]. In the section 3 we present the classification of the types of singularities from the point of view of the finite objects, which approach these singularities. In the section 4 we introduce the toy tachyon model [49] and shall discuss its basic properties. The section 5 is devoted to the cosmological model based on the mixture of the antiChaplygin gas and to the paradox of soft singularity crossing [50]. In sixth section we consider again the paradox of the soft singularity crossing in the presence of dust and shall discuss its possible resolution by introducing some transformation of matter [51]. In seventh section we shall give another example of the transformation of the Lagrangian of a scalar field due to its interaction with geometry, while its potential is not smooth [52]. The eighth section is devoted to the study of classical dynamics of the cosmological model with a scalar field whose potential is inversely proportional to the field, while in the ninth we study its quantum dynamics. The section 10 is devoted to attempts to apply the formalsm of the Wheeler-DeWitt equation to the study of tachyon and pseudotachyon cosmological models. In eleventh section we study Friedmann equations modified by quantum corrections and possible influence of these corrections on soft cosmological singularities [17],[30]. The section 12 is devoted to developing of such notions as density matrix of the universe, quantum consistency and interplay between geometry and matter in quantum cosmology. In the section 13 we</text>
<text><location><page_4><loc_12><loc_85><loc_84><loc_89></location>consider singularities arising in some braneworld models, while the section 14 contains some concluding remarks.</text>
<section_header_level_1><location><page_4><loc_12><loc_81><loc_61><loc_82></location>2. Classification of future cosmological singularities</section_header_level_1>
<text><location><page_4><loc_12><loc_73><loc_84><loc_79></location>In this section we shall present rather a conveninent classification of the future cosmological singularities, following the paper [48]. We shall consider a flat Friedmann universe with the metric</text>
<formula><location><page_4><loc_23><loc_69><loc_84><loc_72></location>ds 2 = dt 2 -a 2 ( t ) dl 2 , (1)</formula>
<text><location><page_4><loc_12><loc_64><loc_84><loc_69></location>where a ( t ) is the cosmological radius (scale factor) and dl 2 is the spatial interval. We shall choose such a normalization of the gravitational constant which provides the follwoing form of the first Friedmann equation is</text>
<formula><location><page_4><loc_23><loc_61><loc_84><loc_63></location>H 2 = ρ, (2)</formula>
<text><location><page_4><loc_12><loc_58><loc_17><loc_60></location>where</text>
<formula><location><page_4><loc_23><loc_54><loc_84><loc_58></location>H ≡ ˙ a a (3)</formula>
<text><location><page_4><loc_12><loc_50><loc_84><loc_54></location>is the Hubble parameter and ρ is the energy density of the universe. The second Friedmann or Raychaudhuri equation is</text>
<formula><location><page_4><loc_24><loc_46><loc_84><loc_50></location>a a = -1 2 ( ρ +3 p ) , (4)</formula>
<text><location><page_4><loc_12><loc_44><loc_68><loc_46></location>where p is the pressure. The energy conservation equation looks as</text>
<formula><location><page_4><loc_23><loc_41><loc_84><loc_43></location>˙ ρ +3 H ( ρ + p ) = 0 . (5)</formula>
<text><location><page_4><loc_12><loc_36><loc_84><loc_40></location>We shall write down also the expressions for the non-vanishing components of the Riemann-Christoffel curvature tensor, defined as [1]</text>
<formula><location><page_4><loc_23><loc_32><loc_84><loc_36></location>R i klm = ∂ Γ i km ∂x i -∂ Γ i kl ∂x m +Γ i nl Γ n km -Γ i nm Γ n kl . (6)</formula>
<text><location><page_4><loc_12><loc_30><loc_42><loc_32></location>These nonvanishing components are</text>
<formula><location><page_4><loc_23><loc_26><loc_84><loc_29></location>R α tβt = -a a δ α β = ( -˙ H + H 2 ) δ α β , (7)</formula>
<text><location><page_4><loc_12><loc_24><loc_37><loc_25></location>where α, β are spatial indices;</text>
<formula><location><page_4><loc_23><loc_21><loc_84><loc_23></location>R 1 212 = R 1 313 = R 2 323 = ˙ a 2 (8)</formula>
<text><location><page_4><loc_12><loc_18><loc_61><loc_20></location>and the corresponding components arising from symmetry.</text>
<text><location><page_4><loc_12><loc_4><loc_84><loc_18></location>The singularities of the type I are the so called Big Rip singularities [54, 55]. A type I singularity arises at some finite moment of the cosmic time t → t BR , when a → ∞ , ˙ a → ∞ , H → ∞ , ρ → ∞ , | p | → ∞ . These singularities are present in the model, where the cosmological evolution is driven by the so called phantom matter [56], when p < 0 , | p | > ρ or, in other words the equation of state parameter w ≡ p ρ < -1. The conditions of arising and avoiding of such singularities were studied in detail in papers [57, 58, 59].</text>
<text><location><page_5><loc_12><loc_71><loc_84><loc_89></location>The singularities of the type II are characterized by the following behaviour of the cosmological parameters: at finite interval of time t = t II a universe arrives with the finite values of the cosmological radius of the time derivative of the cosmological radius, of the Hubble parameter and of the energy density t → t II , a → a II , ˙ a → ˙ a II , H → H II , ρ → ρ II while the acceleration of the universe and the first time derivative of the Hubble parameter tends to minus infinity a → -∞ , ˙ H →-∞ and the pressure tends to plus infinity p → ∞ . A particular case of the type II singularity is the Big Brake singularity, first found in paper [49]. At this singularity, the time derivative of the cosmological radius, the Hubble variable and the energy density are equal exactly to zero.</text>
<text><location><page_5><loc_12><loc_61><loc_84><loc_68></location>The type III singularities are the singularities occuring when the cosmological radius is finite, while its time derivative, the Hubble variable, the energy density and the pressure are divergent. The examples of such singularities were considered, for example in [9, 60].</text>
<text><location><page_5><loc_12><loc_51><loc_84><loc_58></location>The more soft singularities are the singularities of the type IV, at finite value of the cosmological factor both the energy density and the pressure tend to zero and only the higher derivatives of the Hubble parameter H diverge. These singularities sometimes are called Big Separation singularities.</text>
<text><location><page_5><loc_12><loc_41><loc_84><loc_50></location>In paper [61] the type V singularities were added to the scheme proposed in [48]. These are the singularities which like the singularities of the type IV have the pressure and energy density tending to zero, but the higher time derivatives of the Hubble parameter are regular and only the barotropic index (equation of state parameter) w is singular.</text>
<text><location><page_5><loc_12><loc_35><loc_84><loc_38></location>Sometimes the traditional Big Bang and Big Crunch singularities are called type 0 singularities, (see [62]).</text>
<text><location><page_5><loc_12><loc_31><loc_84><loc_34></location>In this review we shall mainly speak about type II singularities and their comparison with type 0 singularities.</text>
<section_header_level_1><location><page_5><loc_12><loc_24><loc_83><loc_28></location>3. The type of the singularity from the point of view of finite size objects, which approach these singularities</section_header_level_1>
<text><location><page_5><loc_12><loc_9><loc_84><loc_22></location>In this section we shall present the classification of singularities, based on the point of view of finite size objects, which approach these singularities. In principle, finite size objects could be destroyed while passing through the singularity due to the occurring infinite tidal forces. A strong curvature singularity is defined by the requirement that an extended finite object is crushed to zero volume by tidal forces. We give below Tipler's [41] and Kr'olak's [63] definitions of strong curvature singularities together with the relative necessary and sufficient conditions.</text>
<text><location><page_5><loc_12><loc_5><loc_84><loc_8></location>First of all we shall write down the geodesics deviation equation. If u i are fourvelocities of test particles and η i is a four-vector separating two spatially close geodesics,</text>
<text><location><page_6><loc_12><loc_87><loc_62><loc_89></location>then the dynamics of this vector is given by the equation [1]</text>
<formula><location><page_6><loc_24><loc_82><loc_84><loc_86></location>D 2 η i ds 2 = R i klm u k u l η m , (9)</formula>
<text><location><page_6><loc_12><loc_77><loc_84><loc_82></location>where D is the covariant derivative along a geodesics. In the case of a flat Friedmann universe (1) for the geodesics of particles, having zero spatial velocities (i.e u α = 0 , u t = 1) Eq. (9) acquires, taking into account Eq. (7), a simple form</text>
<formula><location><page_6><loc_23><loc_72><loc_84><loc_76></location>¨ η α = R α ttβ η β = a a η α . (10)</formula>
<text><location><page_6><loc_12><loc_64><loc_84><loc_72></location>Looking at the above equation one can see that approaching a singularity, chracterized by an infinite value of the deceleration, we experience an infinite force, stopping the farther increase of the separation of geodesics, while geodesics themselves can be quite regular if the the velocity of expansion ˙ a is regular.</text>
<text><location><page_6><loc_12><loc_52><loc_84><loc_64></location>According to Tipler's definition if every volume element, defined by three linearly independent, vorticity-free, geodesic deviation vectors along every causal geodesic through a point P , vanishes, a strong curvature singularity is encountered at the respective point P [41], [37]. The necessary and sufficient condition for a causal geodesic to run into a strong singularity at λ s ( λ is affine parameter of the curve) [64] is that the double integral</text>
<formula><location><page_6><loc_24><loc_45><loc_84><loc_51></location>∫ λ 0 dλ ' ∫ λ ' 0 dλ '' ∣ ∣ R i ajb u a u b ∣ ∣ (11)</formula>
<text><location><page_6><loc_12><loc_43><loc_84><loc_47></location>diverges as λ → λ s . A similar condition is valid for lightlike geodesics, with R i ajb u a u b replacing R ab u a u b in the double integral.</text>
<text><location><page_6><loc_12><loc_31><loc_84><loc_43></location>Kr'olak's definition is less restrictive. A future-endless, future-incomplete null (timelike) geodesic γ is said to terminate in the future at a strong curvature singularity if, for each point P ∈ γ , the expansion of every future-directed congruence of null (timelike) geodesics emanating from P and containing γ becomes negative somewhere on γ [63], [65]. The necessary and sufficient condition for a causal geodesic to run into a strong singularity at λ s [64] is that the integral</text>
<formula><location><page_6><loc_24><loc_25><loc_84><loc_30></location>∫ λ 0 dλ ' ∣ R i ajb u a u b ∣ (12)</formula>
<text><location><page_6><loc_12><loc_22><loc_84><loc_28></location>∣ ∣ diverges as λ → λ s . Again, a similar condition is valid for lightlike geodesics, with R i ajb u a u b replacing R ab u a u b in the integral.</text>
<text><location><page_6><loc_12><loc_14><loc_84><loc_22></location>We conclude this section by mentioning that the singularities of the types 0 and I are strong and the singularities of the types II, IV and V are week according to both the definitions (Tipler's and Kr'olak's ones), while the type III singularities are strong with respect to Kr'olak's definition and week with respect to Tipler's definition [62].</text>
<text><location><page_6><loc_12><loc_6><loc_84><loc_14></location>The weekness of the type II singularities, which we shall study in some details in the next sections of the present review, according to both the definitions, means that although the tidal forces become infinite, the finite objects are not necessarily crushed when reaching the singularity.</text>
<section_header_level_1><location><page_7><loc_12><loc_87><loc_77><loc_88></location>4. The tachyon cosmological model with the trigonometric potential</section_header_level_1>
<text><location><page_7><loc_12><loc_67><loc_84><loc_85></location>The tachyon field, born in the context of the string theory [66], provides an example of matter having a large enough negative pressure to produce an acceleration of the expansion rate of the universe. Such a field is today considered as one of the possible candidates for the role of dark energy and, also for this reason, in the recent years it has been intensively studied. The tachyon models represent a subclass of the models with non-standard kinetic terms [67], which descend from the Born-Infeld model, invented already in thirties [68]. Before considering the model with the trigonometric potential [49], possessing the Big Brake singularity, we write down the general formulae of the tachyon cosmology.</text>
<text><location><page_7><loc_16><loc_65><loc_50><loc_67></location>The Lagrangian of the tachyon field T is</text>
<text><location><page_7><loc_12><loc_60><loc_51><loc_61></location>or, for the spatially homogeneous tachyon field,</text>
<formula><location><page_7><loc_23><loc_60><loc_84><loc_64></location>L = -V ( T ) √ 1 -g µν T ,µ T ,ν (13)</formula>
<formula><location><page_7><loc_23><loc_55><loc_84><loc_59></location>L = -V ( T ) √ 1 -˙ T 2 . (14)</formula>
<text><location><page_7><loc_12><loc_54><loc_66><loc_56></location>The energy density and the pressure of this field are respectively</text>
<text><location><page_7><loc_12><loc_48><loc_15><loc_49></location>and</text>
<formula><location><page_7><loc_23><loc_47><loc_84><loc_54></location>ρ = V ( T ) √ 1 -˙ T 2 (15)</formula>
<formula><location><page_7><loc_23><loc_42><loc_84><loc_47></location>p = -V ( T ) √ 1 -˙ T 2 , (16)</formula>
<text><location><page_7><loc_12><loc_42><loc_33><loc_44></location>while the field equation is</text>
<formula><location><page_7><loc_24><loc_36><loc_84><loc_42></location>T 1 -˙ T 2 +3 H ˙ T + V ,T V ( T ) = 0 . (17)</formula>
<text><location><page_7><loc_16><loc_35><loc_77><loc_37></location>We shall introduce also the pseudo-tachyon field with the Lagrangian [49]</text>
<formula><location><page_7><loc_23><loc_30><loc_84><loc_34></location>L = W ( T ) √ ˙ T 2 -1 (18)</formula>
<text><location><page_7><loc_12><loc_30><loc_35><loc_31></location>and with the energy density</text>
<text><location><page_7><loc_12><loc_23><loc_26><loc_24></location>and the pressure</text>
<formula><location><page_7><loc_23><loc_22><loc_84><loc_29></location>ρ = W ( T ) √ ˙ T 2 -1 (19)</formula>
<formula><location><page_7><loc_23><loc_18><loc_84><loc_22></location>p = W ( T ) √ ˙ T 2 -1 . (20)</formula>
<text><location><page_7><loc_12><loc_17><loc_61><loc_19></location>The Klein-Gordon equation for the pseudo-tachyon field is</text>
<formula><location><page_7><loc_24><loc_11><loc_84><loc_17></location>T 1 -˙ T 2 +3 H ˙ T + W ,T W ( T ) = 0 . (21)</formula>
<text><location><page_7><loc_12><loc_8><loc_84><loc_12></location>We shall also write down the equations for the time derivative of the Hubble parameter in the tachyon and pseudo-tachyons models:</text>
<formula><location><page_7><loc_23><loc_1><loc_84><loc_8></location>˙ H = -3 2 V ( T ) ˙ T 2 √ 1 -˙ T 2 , (22)</formula>
<formula><location><page_8><loc_12><loc_82><loc_84><loc_89></location>˙ H = -3 2 W ( T ) ˙ T 2 √ ˙ T 2 -1 . (23) We see that the Hubble parameter in both these models is decreasing.</formula>
<text><location><page_8><loc_12><loc_74><loc_84><loc_82></location>Note that for the case when the potential of the tachyon field V ( T ) is a constant, the cosmological model with this tachyon coincides with the cosmological model with the Chaplygin gas [69]. The Chaplygin gas is the perfect fluid, satisfying the equation of state</text>
<formula><location><page_8><loc_23><loc_70><loc_84><loc_74></location>p = -A ρ , A > 0 . (24)</formula>
<text><location><page_8><loc_12><loc_62><loc_84><loc_70></location>The cosmological model based on the Chaplygin gas was introduced in [70] and has acquired some popularity as a unified model of dark matter and dark energy [71]. Analogously, the pseudo-tachyon model with the constant potential coincides with the model with a perfect fluid, whose equation of state is .</text>
<formula><location><page_8><loc_23><loc_58><loc_84><loc_61></location>p = + A ρ , A > 0 . (25)</formula>
<text><location><page_8><loc_12><loc_52><loc_84><loc_57></location>This fluid can be called 'anti-Chaplygin gas'. The corresponding model was introduced in [49] and we shall come back to it later. Curiosly, similar equation of motion arises in the theory of wiggly strings [72].</text>
<text><location><page_8><loc_12><loc_48><loc_84><loc_51></location>Now we shall study a very particular tachyon potential depending on the trigonometrical functions which was suggested in the paper [49]. Its form is</text>
<formula><location><page_8><loc_23><loc_37><loc_84><loc_47></location>V ( T ) = Λ sin 2 3 2 √ Λ(1 + k ) T × √ 1 -(1 + k ) cos 2 3 2 √ Λ(1 + k ) T, (26)</formula>
<text><location><page_8><loc_12><loc_20><loc_84><loc_28></location>The origin of the potential (26) is the following one: let us consider a flat Friedmann universe filled with two fluids, one of which is a cosmological constant with the equation of state p = -ρ = -Λ and the second one is a barotropic fluid with the equation of state p = kρ . The Friedmann equation for such a model is exactly solvable and gives</text>
<text><location><page_8><loc_12><loc_27><loc_84><loc_38></location>where Λ is a positive constant and k is a parameter, which is chosen in the interval -1 < k < 1. The case of the positive values of the parameter k is especially interesting. The set of possible cosmological evolutions, is graphically presented in Figure 1, which is the phase portrait of our dynamical system, where the ordinate s is the time derivative of the tachyon field T : s ≡ ˙ T .</text>
<formula><location><page_8><loc_23><loc_16><loc_84><loc_21></location>H ( t ) = √ Λcoth 3 √ Λ( k +1) t 2 . (27)</formula>
<text><location><page_8><loc_12><loc_4><loc_84><loc_16></location>Then using the standard technique of the reconstruction of potentials, which was mainly used for the minimally coupled scalar field [73], but was easily generalized for the cases of non-minimally coupled fields [85, 74] and for tachyons [75, 76, 49, 77] we obtain the expression (26). It is necessary to emphasize that the dynamics of the tachyon model with the potential (26) is much richer than the dynamics of the two fluid model with the unique cosmological evolution given by the expression (27). In paper [49] both the</text>
<figure>
<location><page_9><loc_13><loc_40><loc_93><loc_92></location>
<caption>Figure 1. (Color online) Phase portrait evolution for k > 0 ( k = 0 . 44)</caption>
</figure>
<text><location><page_9><loc_12><loc_27><loc_84><loc_33></location>cases k ≤ 0 and k > 0 were considered. The case k > 0 is of a particular interest, because it reveals two unusual phenomena: a self-transformation of the tachyon into a pseudotachyon field and the appearance of the Big Brake cosmological singularity.</text>
<text><location><page_9><loc_12><loc_20><loc_84><loc_27></location>Let us discuss briefly the classical dynamics of the model with the trigonometric potential for the case k > 0. It is easy to see that the potential (26) is well defined at T 3 ≤ T ≤ T 4 , where</text>
<formula><location><page_9><loc_23><loc_9><loc_84><loc_15></location>T 4 = 2 3 √ (1 + k )Λ ( π -arccos 1 √ 1 + k ) . (29)</formula>
<formula><location><page_9><loc_23><loc_14><loc_84><loc_20></location>T 3 = 2 3 √ (1 + k )Λ arccos 1 √ 1 + k , (28)</formula>
<text><location><page_9><loc_12><loc_4><loc_84><loc_10></location>In turn, the kinetic term √ 1 -˙ T 2 is well defined at -1 ≤ s ≤ 1. In other words, the Lagrangian (14) with the potential (26) is well defined inside the rectangle (see Fig. 1). The analysis of the dynamics of the equation of motion of the tachyon (17) and of the</text>
<text><location><page_10><loc_12><loc_60><loc_84><loc_89></location>Friedmann equations shows that a part of the trajectories end their evolution in the attractive node with the coordinates T 0 = π 3 √ (1+ k )Λ , s 0 = 0, which describes an infinite de Sitter expansion. The upper and lower borders of the rectangle s = 1 , s = -1, excluding the corner points, are the standard Big Bang cosmological singularities, while left and right borders T = T 3 and T = T 4 repel the trajectories. However, another part of the trajectories goes towards the corner points ( T = T 3 , s = -1) and ( T = T 4 , s = 1). These points are regular points from the point of view of the equations of motion of the corresponding dynamical system and besides, the direct calculation shows that there are no cosmological singularities there. Thus, there is no reason which prevents further evolution of the universe through these points. Indeed, one can see also that the equations of motion and their solutions can be continued into the vertical stripes (see Fig. 1). However, to reproduce these equations of motion in the stripes as Euler-Lagrange equations, we should substitute the tachyon Lagrangian (14) by the preudotachyon Lagrangian (19) with the potential</text>
<formula><location><page_10><loc_23><loc_50><loc_84><loc_60></location>W ( T ) = Λ sin 2 3 2 √ Λ(1 + k ) T × √ (1 + k ) cos 2 3 2 √ Λ(1 + k ) T -1 . (30)</formula>
<text><location><page_10><loc_12><loc_40><loc_84><loc_51></location>Thus, we have seen already the first unusual phenomenon - the self-transformation of the tachyon into the pseudotachyon field. Now, the question arises: what happens with the universe after the 'crossing the corner' and the transformation of the tachyon into the pseudotachyon ? The analysis of equations of motion carried out in paper [49] shows that the universe in a finite moment of time t = t BB encounter the singularity, which is characterized by the following values of cosmological parameters:</text>
<formula><location><page_10><loc_23><loc_22><loc_84><loc_38></location>a ( t BB ) = a BB < ∞ , ˙ a ( t BB ) = 0 , a ( t ) →-∞ , at t → t BB , T ( t BB ) = T BB > 0 (in lower left strip) , s ( t ) →-∞ , at t → t BB , (in lower left strip) ρ ( t BB ) = 0 , p ( t ) → + ∞ , at t → t BB . (31)</formula>
<text><location><page_10><loc_12><loc_16><loc_84><loc_22></location>This singularity was called Big Brake singularity [49]. Obviously, it enters into the class II of singularities, according to the classification suggested in paper [48] and recapitulated in Sec. 2 of the present review.</text>
<text><location><page_10><loc_12><loc_4><loc_84><loc_16></location>Now, it is interesting to confront the prediction of this, a little bit artificial, but rather rich model with the observational data coming from the luminosity-redshift relation from Supernovae of type Ia. Such an attempt was undertaken in paper [29], where the set of supernovae studied in [78] was used. The strategy was the following : there were scanned the pairs of present values of the tachyon field and of its time derivative (points in phase space) and then they were propagated backwards in time,</text>
<text><location><page_11><loc_12><loc_69><loc_84><loc_89></location>comparing corresponding luminosity distance - redshift curves with the observational data from SNIa. Then, those pairs of values which appeared to be compatible with the data were chosen as initial conditions for the future cosmological evolution. Though the constraints imposed by the data were rather severe, both evolutions took place: one very similar to ΛCDM and ending in an exponential (de Sitter) expansion; another with the transformation of the tachyon into the pseudotachyon and the successive running towards the Big Brake singularity. It was found that a larger value of the model parameter k enhances the probability to evolve into a Big Brake. The time intervals until the future encounter with the Big Brake were calculated and were found to be compatible with the present age of the universe [29].</text>
<text><location><page_11><loc_12><loc_57><loc_84><loc_68></location>The next question, which arises, is the fate of the universe after the encounter with the Big Brake singularity. As was already told above, this singularity is very soft and the geodesics can be continued across it. Then the matter, passing through the Big Brake singularity reconstructs the spacetime. This process was studied in some detail in paper [40]. The analysis of the equation of motion for the universe approaching the Big Brake singularity gives the following expressions for the basic quantities:</text>
<formula><location><page_11><loc_23><loc_51><loc_84><loc_56></location>T = T BB + ( 4 3 W ( T BB ) ) 1 / 3 ( t BB -t ) 1 / 3 , (32)</formula>
<formula><location><page_11><loc_23><loc_46><loc_84><loc_51></location>s = -( 4 81 W ( T BB ) ) 1 / 3 ( t BB -t ) -2 / 3 , (33)</formula>
<formula><location><page_11><loc_23><loc_41><loc_84><loc_46></location>a = a BB -3 4 a BB ( 9 W 2 ( T BB ) 2 ) 1 / 3 ( t BB -t ) 4 / 3 , (34)</formula>
<formula><location><page_11><loc_23><loc_35><loc_84><loc_40></location>˙ a = a BB ( 9 W 2 ( T BB ) 2 ) 1 / 3 ( t BB -t ) 1 / 3 , (35)</formula>
<formula><location><page_11><loc_23><loc_30><loc_84><loc_35></location>H = ( 9 W 2 ( T BB ) 2 ) 1 / 3 ( t BB -t ) 1 / 3 . (36)</formula>
<text><location><page_11><loc_12><loc_24><loc_84><loc_30></location>The expressions (32)-(36) can be continued into the region where t > t BB , which amounts to crossing the Big Bang singularity. Only the expression for s is singular at t = t BB , but this singularity is integrable and not dangerous.</text>
<text><location><page_11><loc_12><loc_3><loc_84><loc_24></location>Upon reaching the Big Brake, it is impossible for the system to stop there because the infinite deceleration leads to the decrease of the scale factor. This is because after the Big Brake crossing the time derivative of the cosmological radius (35) and of the Hubble variable (36) change their signs. The expansion is then followed by a contraction. Corresponding to given initial conditions, the values of T BB , t BB and a BB were found numerically. Then the numerical integration of the equations of motion describes the contraction of the universe, culminating in the encounter with the Big Crunch singularity. Curiously, the time intervals between the Big Brake and Big Crunch singularities practically do not depend on the initial conditions and are equal approximately to 0 . 3 × 10 9 yrs [40].</text>
<text><location><page_12><loc_12><loc_67><loc_84><loc_89></location>Now, the next question arises: what happens if we consider a little bit more complicated model, adding to the tachyon matter some quantity of dust-like matter ? Obviously, in this case instead of the Big Brake singularity the universe will encounter a soft type II singularity of a more general kind. Namely, due to the presence of dust, the energy density of the expanding universe cannot vanish and, hence, at the moment when the universe experiences an infinite deceleration its expansion should continue. This implies the appearance of some kind of contradictions, which can be resolved by transformation of the pseudotachyon field into another kind of Born-Infeld like field. The corresponding problem was considered in detail in papers [50, 51]. The first of these papers was devoted to a more simple model, based on mixture of the anti-Chaplygin gas with dust. The next section will be devoted to this model.</text>
<section_header_level_1><location><page_12><loc_12><loc_61><loc_84><loc_64></location>5. The cosmological model based on the mixture of the anti-Chaplygin gas and the paradox of soft singularity crossing</section_header_level_1>
<text><location><page_12><loc_12><loc_53><loc_84><loc_59></location>The anti-Chaplygin gas with the equation of state (25) is one of the simplest cosmological models revealing the Big Brake singularity [49]. Indeed, combining the equation of state (25) with the energy conservation equation (5), one obtains immediately</text>
<formula><location><page_12><loc_23><loc_48><loc_84><loc_52></location>ρ = √ B a 6 -A, (37)</formula>
<text><location><page_12><loc_12><loc_42><loc_84><loc_48></location>where B is a positive constant, characterizing the initial condition. Then, when in the process of the cosmological expansion the cosmological radius a arrives to the critical value</text>
<formula><location><page_12><loc_23><loc_37><loc_84><loc_42></location>a S = ( B A ) 1 / 6 (38)</formula>
<text><location><page_12><loc_12><loc_32><loc_84><loc_37></location>the energy density of the universe vanishes while the pressure tends to infinity. Thus, the universe encounters the Big Brake singularity. Then, it begins contraction culminating in the encounter with the Big Crunch singularity.</text>
<text><location><page_12><loc_12><loc_28><loc_84><loc_31></location>Now, let us see what happens if we add some amount of dust with the energy density</text>
<formula><location><page_12><loc_23><loc_24><loc_84><loc_28></location>ρ m = ρ 0 a 3 , (39)</formula>
<text><location><page_12><loc_12><loc_4><loc_84><loc_24></location>where ρ 0 is a positive constant. In this case the traversability of the singularity seems to be obstructed. The main reason for this is that while the energy density of the antiChaplygin gas vanishes at the singularity, the energy density of the matter component does not, leaving the Hubble parameter at the singularity with a finite value. Then some kind of the paradox arises: if the universe continues its expansion, and if the equation of state of the component of matter, responsible for the appearance of the soft singularity (in the simplest case, the anti-Chaplygin gas) is unchanged, then the expression for the energy density of this component becomes imaginary, which is unacceptable. The situation looks rather strange: indeed, the model, including dust should be in some sense more regular, that that, containing only such an exotic fluid as the anti-Chaplygin</text>
<text><location><page_13><loc_12><loc_83><loc_84><loc_89></location>gas. Thus, if the model, based on the pure anti-Chaplygin gas has a traversable Big Brake singularity, than the more general singularity arising in the model, based on the mixture of the anti-Chaplygin gas and dust should also be transversable.</text>
<text><location><page_13><loc_12><loc_65><loc_84><loc_82></location>A possible way of resolution of this paradox, based on use of the distributional cosmological quantities was suggested in paper [50]. Let us suppose that at the moment of the crossing of the soft cosmological singularity the expansion of the universe with the Hubble parameter H is abruptly substituted by the cosmological contraction with the Hubble parameter -H . In this case, the value of the cosmological radius a begins decreasing and the expression (37) for the energy density just like the corresponding expression for the pressure remain well defined. The first Friedmann equation (2) and the energy conservation equation (5) remain also intact. A problem, however, arises with the second Friedmann equation (4). Let us rewrite this equation in the form</text>
<formula><location><page_13><loc_23><loc_61><loc_84><loc_64></location>˙ H = -3 2 ( ρ + p ) . (40)</formula>
<text><location><page_13><loc_12><loc_57><loc_84><loc_60></location>If the Hubble parameter abruptly changes sign at the moment t = t S that means that it contains the term</text>
<formula><location><page_13><loc_23><loc_53><loc_84><loc_55></location>H ( t ) = H S ( θ ( t S -t ) -θ ( t -t S ) , (41)</formula>
<text><location><page_13><loc_12><loc_43><loc_84><loc_53></location>where θ ( x ) is the Heaviside theta function. The derivative of the theta function is equal in the distributional sense to the Dirac delta function (see e.g. [79]). Hence, the lefthand side of Eq. (40) contains the Dirac delta function. Now, let us discuss in more detail the expressions for the Hubble parameter and its time derivative in the vicinity of the singularity. The leading terms of the expression for H ( t ) are</text>
<formula><location><page_13><loc_23><loc_34><loc_84><loc_42></location>H ( t ) = H S sgn ( t S -t ) + √ 3 A 2 H S a 4 S sgn ( t S -t ) √ | t S -t | , (42)</formula>
<formula><location><page_13><loc_23><loc_25><loc_84><loc_32></location>˙ H = -2 H S δ ( t S -t ) -√ 3 A 8 H S a 4 S sgn ( t S -t ) √ | t S -t | . (43)</formula>
<text><location><page_13><loc_12><loc_32><loc_42><loc_34></location>where sgn ( x ) ≡ θ ( x ) -θ ( -x ). Then</text>
<text><location><page_13><loc_12><loc_19><loc_84><loc_27></location>Naturally, the δ -term in ˙ H arises because of the jump in H , as the expansion of the universe is followed by a contraction. To restore the validity of the second Friedmann equation (40) we shall add a singular δ -term to the pressure of the anti-Chaplygin gas, which will acquire the form</text>
<formula><location><page_13><loc_23><loc_13><loc_84><loc_18></location>p ACh = √ A 6 H S | t S -t | + 4 3 H S δ ( t S -t ) . (44)</formula>
<text><location><page_13><loc_12><loc_9><loc_84><loc_13></location>The equation of state of the anti-Chaplygin gas is preserved, if we also modify the expression for its energy density:</text>
<formula><location><page_13><loc_23><loc_2><loc_84><loc_8></location>ρ ACh = A √ A 6 H S | t S -t | + 4 3 H S δ ( t S -t ) . (45)</formula>
<text><location><page_14><loc_12><loc_85><loc_84><loc_89></location>The last expression should be understood in the sense of the composition of distributions (see Appendix A of the paper [50] and references therein).</text>
<text><location><page_14><loc_12><loc_81><loc_84><loc_84></location>In order to prove that p ACh and ρ ACh represent a self-consistent solution of the system of cosmological equations, we shall use the following distributional identities:</text>
<formula><location><page_14><loc_24><loc_77><loc_84><loc_80></location>[ sgn ( τ ) g ( | τ | )] δ ( τ ) = 0 , (46)</formula>
<formula><location><page_14><loc_23><loc_76><loc_84><loc_78></location>[ f ( τ ) + Cδ ( τ )] -1 = f -1 ( τ ) , (47)</formula>
<formula><location><page_14><loc_24><loc_72><loc_84><loc_76></location>d dτ [ f ( τ ) + Cδ ( τ )] -1 = d dτ f -1 ( τ ) . (48)</formula>
<text><location><page_14><loc_12><loc_66><loc_84><loc_72></location>Here g ( | τ | ) is bounded on every finite interval, f ( τ ) > 0 and C > 0 is a constant. These identities were proven in paper [50], where was used the approach to the product and the composition of distributions developed in papers [80].</text>
<text><location><page_14><loc_19><loc_57><loc_19><loc_60></location>/negationslash</text>
<text><location><page_14><loc_28><loc_57><loc_28><loc_60></location>/negationslash</text>
<text><location><page_14><loc_12><loc_56><loc_84><loc_66></location>Due to Eqs. (47)-(48), ρ ACh vanishes at the singularity while still being continuous. The first term in the expression for the pressure (44) diverges at the singularity. Therefore the addition of a Dirac delta term, which is not changing the value of p ACh at any τ = 0 (i.e. t = t S ) does not look too drastic and might be considered as a some kind of renormalization.</text>
<text><location><page_14><loc_12><loc_42><loc_84><loc_56></location>To prove that the first and the second Friedmann equations and the continuity equation are satisfied we must only investigate those terms, appearing in the field equations, which contain Dirac δ -functions. First, we check the continuity equation for the anti-Chaplygin gas. Due to the identities (47)-(48), the δ ( τ )-terms occurring in ρ ACh and ˙ ρ ACh could be dropped. We keep them however in order to have the equation of state explicitly satisfied. Then the δ ( τ )-term appearing in 3 Hp ACh vanishes, because the Hubble parameter changes sign at the singularity (see Eq. (46)).</text>
<text><location><page_14><loc_12><loc_34><loc_84><loc_42></location>The δ ( τ )-term appearing in ρ ACh does not affect the Friedmann equation due to the identity (47). Finally, the δ -term arising in the time derivative of the Hubble parameter in the left-hand side of the Raychaudhuri equation is compensated by the conveniently chosen δ -term in the right-hand side of Eq. (44).</text>
<text><location><page_14><loc_12><loc_20><loc_84><loc_34></location>However, the mathematically self-consistent scenario, based on the use of generalized functions and on the abrupt change of the expansion into a contraction, looks rather counter-intuitive from the physical point of view. Such a behaviour can be compared with the absolutely elastic bounce of a ball from a rigid wall, as studied in classical mechanics. In the latter case the velocity and the momentum of the ball change their direction abruptly. Hence, an infinite force acts from the wall onto the ball during an infinitely small interval of time.</text>
<text><location><page_14><loc_12><loc_4><loc_84><loc_20></location>In reality, the absolutely elastic bounce is an idealization of a process of finite timespan during which inelastic deformations of the ball and of the wall occur. Thus, the continuity of the kinematics of the act of bounce implies a more complex and realistic description of the dynamical process of interaction between the ball and the wall. It is reasonable to think that something similar occurs also in the models, including dust and an anti-Chaplygin gas or a tachyon. The smoothing of the process of a transition from an expanding to a contracting phase should include some (temporary) geometrically induced change of the equation of state of matter or of the form of the Lagrangian. We</text>
<text><location><page_15><loc_12><loc_77><loc_84><loc_89></location>know that such changes do exist in cosmology. In the tachyon model [49], there was the tachyon-pseudotachyon transformation driven by the continuity of the cosmological evolution. In a cosmological model with the phantom filed with a cusped potential [52, 53], the transformations between phantom and standard scalar field were considered. Thus, it is quite natural that the process of crossing of the soft singularity should imply similar transformations.</text>
<text><location><page_15><loc_12><loc_67><loc_84><loc_76></location>However, now the situation is more complicated. It is not enough to require the continuity of evolution of the cosmological radius and of the Hubble parameter. It is necessary also to accept some hypothesis concerning the fate of the change of the equation of state of matter or of the form of the Lagrangian. This problem will be considered in the next section.</text>
<section_header_level_1><location><page_15><loc_12><loc_61><loc_73><loc_64></location>6. Paradox of soft singularity crossing and its resolution due to transformations of matter</section_header_level_1>
<text><location><page_15><loc_12><loc_27><loc_84><loc_59></location>The strategy of the analysis of the the problem of soft singularity crossing in this section is the following [51]. First, we shall consider the model with the anti-Chaplygin gas and dust. We shall require a minimality of the change of the form of the dependence of the energy density and of the pressure, compatible with the continuation of the expansion while crossing the soft singularity. Such a requirement will bring us to the substitution of the anti-Chaplygin gas with the Chaplygin gas with a negative energy density. (Note, that in another context the Chaplygin gas with a negative energy density was considered in paper [81]). Then we shall consider the cosmological model based on the pseudotachyon field with a constant potential and dust. It is known that the energy-momentum tensor for such a pseudotachyon field coincides with that of the anti-Chaplygin gas (this fact relating the Chaplygin gas and the tachyon field with a constant potential was found in paper [69]). Thus, we would like to derive the form of the transformation of the pseudotachyon Lagrangian using its kinship with the antiChaplygin gas. As a result, we shall come to a new type of the Lagrangian, belonging to the 'Born-Infeld family'. Finally, we shall extend the found form of transformation of the pseudotachyon field for the case of the field with the trigonometric potential.</text>
<text><location><page_15><loc_16><loc_25><loc_77><loc_26></location>As follows from Eqs. (25) and (37) the pressure of the anti-Chaplygin gas</text>
<formula><location><page_15><loc_23><loc_18><loc_84><loc_24></location>p = A √ B a 6 -A (49)</formula>
<text><location><page_15><loc_12><loc_13><loc_84><loc_19></location>and it tends to + ∞ when the universe approaches the soft singularity, when the cosmological radius a → a S (see Eq. (38)). If we would like to continue the expansion into the region a > a S , while changing minimally the equation of state we can require</text>
<formula><location><page_15><loc_23><loc_6><loc_84><loc_12></location>p = A √ | B a 6 -A | , (50)</formula>
<text><location><page_16><loc_12><loc_87><loc_27><loc_89></location>or, in other words,</text>
<formula><location><page_16><loc_23><loc_80><loc_84><loc_86></location>p = A √ A -B a 6 , for a > a S . (51)</formula>
<text><location><page_16><loc_12><loc_75><loc_84><loc_81></location>We see that in some 'generalized sense' we conserve the continuity of the pressure crossing the soft singularity. It passes + ∞ conserving its sign. Combining the expression (51) with the energy conservation law (5) we obtain</text>
<formula><location><page_16><loc_23><loc_70><loc_84><loc_74></location>ρ = -√ A -B a 6 for a > a S . (52)</formula>
<text><location><page_16><loc_12><loc_65><loc_84><loc_70></location>Thus, the energy density is also continuous passing through its vanishing value and changing its sign. It is easy to see that the energy density (52) and the pressure (51) satisfy the Chaplygin gas equation of state</text>
<formula><location><page_16><loc_23><loc_60><loc_84><loc_64></location>p = -A ρ . (53)</formula>
<text><location><page_16><loc_12><loc_54><loc_84><loc_60></location>Thus, we have seen the transformation of the anti-Chaplygin gas into the Chaplygin gas with a negative energy density. The Friedmann equation after the crossing the singularity is</text>
<formula><location><page_16><loc_23><loc_48><loc_84><loc_53></location>H 2 = ρ m, 0 a 3 -√ A √ 1 -( a S a ) 6 . (54)</formula>
<text><location><page_16><loc_12><loc_45><loc_84><loc_49></location>It follows immediately from Eq. (54) that after achieving the point of maximal expansion a = a max , where</text>
<formula><location><page_16><loc_23><loc_40><loc_84><loc_45></location>a max = ( ρ 2 m, 0 A + a 6 S ) 1 / 6 , (55)</formula>
<text><location><page_16><loc_12><loc_32><loc_84><loc_40></location>The universe begins contracting. When the contracting universe arrives to a = a S it again stumbles upon a soft singularity and the Chaplygin gas transforms itself into the anti-Chaplygin gas with positive energy density and the contraction continues until hitting the Big Crunch singularity.</text>
<text><location><page_16><loc_12><loc_22><loc_84><loc_32></location>Remember that in the preceding section and in paper [50], the process was described when the universe passed from the expanding to the collapsing phase instantaneously at the singularity causing a jump in the Hubble parameter. Here we showed that the continuos transition to the collapsing phase is possible if the equation of state of antiChaplygin gas has a some kind of a 'phase transition' at the singularity.</text>
<text><location><page_16><loc_12><loc_16><loc_84><loc_22></location>When the potential of the pseudotachyon field is constant, W ( T ) = W 0 , then the energy density (19) and the pressure (20) satisfy the anti-Chaplygin gas equation of state (25) with</text>
<formula><location><page_16><loc_23><loc_13><loc_84><loc_15></location>A = W 2 0 . (56)</formula>
<text><location><page_16><loc_12><loc_9><loc_84><loc_12></location>Solving the equation of motion for the pseudotachyon field (21) with W ( T ) = W 0 one finds</text>
<formula><location><page_16><loc_23><loc_2><loc_84><loc_9></location>˙ T 2 = 1 1 -( a a S ) 6 (57)</formula>
<text><location><page_17><loc_12><loc_86><loc_70><loc_89></location>and we see that the soft singularity arises at a = a S , when ˙ T 2 → + ∞ .</text>
<text><location><page_17><loc_12><loc_81><loc_84><loc_87></location>Now, we would like to change the Lagrangian (18) in such a way that the new Lagrangian gives us the energy density and the pressure satisfying the Chaplygin gas equation with a negative energy density. It is easy to check that the Lagrangian</text>
<text><location><page_17><loc_12><loc_75><loc_17><loc_77></location>giving</text>
<text><location><page_17><loc_12><loc_70><loc_15><loc_71></location>and</text>
<formula><location><page_17><loc_23><loc_76><loc_84><loc_80></location>L = W 0 √ ˙ T 2 +1 (58)</formula>
<formula><location><page_17><loc_23><loc_70><loc_84><loc_74></location>p = W 0 √ ˙ T 2 +1 (59)</formula>
<text><location><page_17><loc_12><loc_63><loc_34><loc_65></location>is what we are looking for.</text>
<formula><location><page_17><loc_23><loc_63><loc_84><loc_69></location>ρ = -W 0 √ ˙ T 2 +1 (60)</formula>
<text><location><page_17><loc_12><loc_55><loc_84><loc_63></location>Note, that the energy density and the pressure, passing through the singularity are continuous in the same sense in which they were continuos in the case of the antiChaplygin gas. Thus, we have introduced a new type of the Born-Infeld field, which can be called 'anti-tachyon'. Generally, its Lagrangian is</text>
<formula><location><page_17><loc_23><loc_50><loc_84><loc_54></location>L = W ( T ) √ ˙ T 2 +1 (61)</formula>
<text><location><page_17><loc_12><loc_49><loc_37><loc_51></location>and the equation of motion is</text>
<formula><location><page_17><loc_24><loc_44><loc_84><loc_49></location>T ˙ T 2 +1 +3 H ˙ T -W ,T W = 0 . (62)</formula>
<text><location><page_17><loc_12><loc_42><loc_60><loc_44></location>For the case W ( T ) = W 0 , the solution of equation (62) is</text>
<formula><location><page_17><loc_23><loc_35><loc_84><loc_42></location>˙ T 2 = 1 ( a a S ) 6 -1 , (63)</formula>
<text><location><page_17><loc_12><loc_34><loc_40><loc_36></location>and the energy density evolves as</text>
<formula><location><page_17><loc_23><loc_28><loc_84><loc_34></location>ρ T = -W 0 √ 1 -( a S a ) 6 (64)</formula>
<text><location><page_17><loc_12><loc_26><loc_84><loc_29></location>and the evolution of the universe repeats that for the model with the anti-Chaplygin gas and dust.</text>
<text><location><page_17><loc_12><loc_18><loc_84><loc_25></location>Let us emphasize once again that to the transformation from the anti-Chaplygin gas to the Chaplygin gas corresponds to the transition from the prseudotachyon field with the Lagrangian (18) to the new type of the Born-Infeld field, which we can call 'quasi-tachyon field' with the Lagrangian (61).</text>
<text><location><page_17><loc_12><loc_7><loc_84><loc_17></location>Now, we shall consider the case of the toy model with the trigonometric potential in the presence of dust. We have seen that the Born-Infeld type pseudotachyon field runs into a soft Big Brake singularity with the expansion of the universe in this model. However, what happens in the presence of dust component? Does the universe still run into soft singularity?</text>
<text><location><page_18><loc_16><loc_87><loc_52><loc_89></location>To answer this question rewrite Eq. (21) as</text>
<formula><location><page_18><loc_23><loc_82><loc_84><loc_86></location>T = ( ˙ T 2 -1) ( 3 H ˙ T + W ,T W ) . (65)</formula>
<text><location><page_18><loc_12><loc_66><loc_84><loc_82></location>It is easy to see that in the left lower and in the right upper stripes (see Fig. 1), where the trajectories describe the expansion of the universe after the transformation of the tachyon into the pseudotachyon field, the signs of T , of ˙ T and of the term W ,T W coincide. The detailed analysis based on this fact was carried out in paper [49] and led to the conclusion that the universe encounters the singularity as T → T S ( T S > 0 or T S > T max ) , | ˙ T | → ∞ . The presence of dust cannot alter this effect because it increases the influence of the term 3 H ˙ T , and hence, accelerates the encounter with the singularity.</text>
<text><location><page_18><loc_12><loc_60><loc_84><loc_66></location>However, the presence of dust changes in an essential way the time dependence of the tachyon field close to the singularity. As it was shown in [40] (see also the Sec. 4 of the present paper)</text>
<formula><location><page_18><loc_23><loc_54><loc_84><loc_59></location>T = T BB + ( 4 3 W ( T BB ) ) 1 / 3 ( t BB -t ) 1 / 3 , (66)</formula>
<text><location><page_18><loc_12><loc_53><loc_43><loc_54></location>while in the presence of dust one has</text>
<formula><location><page_18><loc_23><loc_47><loc_84><loc_52></location>T = T S + √ 2 3 H S √ t S -t, (67)</formula>
<text><location><page_18><loc_12><loc_46><loc_70><loc_48></location>where H S is the nonvanishing value of the Hubble parameter given by</text>
<formula><location><page_18><loc_23><loc_41><loc_84><loc_45></location>H S = √ ρ m, 0 a 3 S . (68)</formula>
<text><location><page_18><loc_12><loc_37><loc_84><loc_41></location>It is easy to see that the smooth continuation of the expression (67) is impossible in contrast to the situation without dust (66) considered in [40].</text>
<text><location><page_18><loc_12><loc_33><loc_84><loc_37></location>Thus, the presence of dust is responsible for the appearance of similar paradoxes in both the anti-Chaplygin gas and tachyon models.</text>
<text><location><page_18><loc_12><loc_21><loc_84><loc_33></location>In the vicinity of the soft singularity, it is the 'friction' term 3 H ˙ T in the equation of motion (21) , which dominates over the potential term W ,T W , hence, the dependence of W ( T ) is not essential and a pseudotachyon field approaching this singularity behaves like one with a constant potential. Thus, it is quite reasonable to suppose that crossing of the soft singularity the pseudotachyon transforms itself into the quasi-tachyon with the Lagrangian (61).</text>
<text><location><page_18><loc_12><loc_15><loc_84><loc_21></location>Now, we can analyze the dynamics of the anti-tachyon field, driven by the equation of motion (62) and by the Friedmann equation, where the right-hand side includes the dust contribution and the anti-tachyon energy density</text>
<formula><location><page_18><loc_23><loc_10><loc_84><loc_14></location>ρ = -W ( T ) ˙ T 2 +1 . (69)</formula>
<text><location><page_18><loc_12><loc_4><loc_84><loc_12></location>√ It is convenient to consider the processes developing in the left lower strip of the phase diagram of the model to facilitate the comparison with the earlier studies of the dynamics of the tachyon model without dust, undertaken in papers [49, 40].</text>
<text><location><page_19><loc_12><loc_55><loc_84><loc_89></location>One can see that the relative sign of the term with the second derivative T with respect to the friction term 3 H ˙ T are oppostite for the pseudotachyons and anti-tachyons. That means that after the crossing of the soft singularity the time derivative ˙ T is growing and its absolute value is diminishing. At the same time the value of the field T is diminishing and the value of the potential W ( T ) is growing. That means that the absolute value of the negative contribution to the energy density of the universe coming from the quasi-tachyon is growing while the energy density of the dust is diminishing due to the expansion of the universe. At some moment this process brings us to the vanishing value of the general energy density and we arrive to the point of maximal expansion of the universe. After that the expansion is replaced by the contraction and the Hubble variable changes sign. The change of sign of the friction term 3 H ˙ T implies the diminishing of the value of ˙ T and at some finite moment of time the universe again encounters the soft singularity when ˙ T → -∞ . Passing this singularity the quasitachyon transforms itself back to the pseudotachyon and the relative sign of the terms with the second and first time derivatives in the equation of motion for this field changes once again. After that the time derivative of the pseudotachyon field begins growing and the universe continues its contraction until it encounters with the Big Crunch singularity.</text>
<text><location><page_19><loc_12><loc_42><loc_84><loc_54></location>It was shown in paper [40] that for the case of the purely tachyon model with the trigonometric potential the encounter of the universe with the Big Crunch singularity occurs at T = 0 and ˙ T = -√ 1+ k k . One can show that the presence of dust does not change these values. Indeed, let us consider the behavior of the pseudotachyon field when T → 0 , ˙ T →-√ 1+ k k . It follows from the expressions (19) and (20) that the ratio between the pressure and the energy density behaves as</text>
<formula><location><page_19><loc_24><loc_38><loc_84><loc_42></location>p ρ = ˙ T 2 -1 → 1 k , (70)</formula>
<text><location><page_19><loc_12><loc_32><loc_84><loc_37></location>i.e. in the vicinity of the Big Crunch singularity the pseudotachyon field behaves as a barotropic fluid with the the equation of state parameter 1 k > 1. That means that the energy density of the pseudotachyon field is growing as</text>
<formula><location><page_19><loc_23><loc_27><loc_84><loc_31></location>ρ ∼ 1 a 3(1+ 1 k ) (71)</formula>
<text><location><page_19><loc_12><loc_19><loc_84><loc_27></location>as a → 0, i.e. much more rapidly than the dust energy density. Thus, one can neglect the contribution of the dust in this regime of approaching the Big Crunch singularity and the description of the evolution of the universe to this point coincides with that of the pure tachyon model [40].</text>
<section_header_level_1><location><page_19><loc_12><loc_13><loc_81><loc_16></location>7. The transformations of the Lagrangian of a scalar field with a cusped potential</section_header_level_1>
<text><location><page_19><loc_12><loc_5><loc_84><loc_11></location>It is well known that the cosmological observations gives as a best fit for the equation of state parameter w = fracpρ a value which is slightly inferior with respect to -1 (see, e.g. [82]). The corresponding type of dark energy was called 'phantom' matter [56].</text>
<text><location><page_20><loc_12><loc_85><loc_84><loc_89></location>Wanting to realize such a dark matter using a minimally coupled scalar field, one has to introduce for the latter a negative kinetic term. Thus, its Lagrangian has the form</text>
<formula><location><page_20><loc_23><loc_81><loc_84><loc_85></location>L = -1 2 g µν φ ,µ φ ,ν -V ( φ ) . (72)</formula>
<text><location><page_20><loc_12><loc_65><loc_84><loc_81></location>Some observations also indicate that the value of the equation of state parameter at some moment in the past has crossed the value w = -1, corresponding to the cosmological constant. Such a phenomenon has received the name of 'phantom divide line crossing' [83]. A minimally coupled scalar field, describing non phantom dark energy has a kinetic term with the positive sign. So, it looks natural, to use two scalar fields, a phantom field with the negative kinetic term and a standard one to describe the phantom divide line crossing [84]. Another posssible way of the phantom divide line crossing, using s scalar field nonminimally coupled to gravity was considered in papers [85].</text>
<text><location><page_20><loc_12><loc_39><loc_84><loc_65></location>However, in papers [52, 53] it was shown that considering potentials with cusps and choosing some particular initial conditions, one can describe the phenomenon of the phantom divide line crossing in the model with one minimally coupled scalar field. Curiously, a passage through the maximum point of the evolution of the Hubble parameter implies the change of sign of the kinetic term. Though a cosmological singularity is absent in these cases, this phenomenon is a close relative of those, considered in the preceding sections, because here also we stumble upon some transformation of matter properties, induced by a change of geometry. One can add that in this aspect the phenomenon of the phantom divide line crossing is the close analog of the transformation between the tachyon and pseudo-tachyon model, in the tachyon model with the trigonometric potential, described in Sec. 4. Here, we shall present a brief sketch of the ideas, described in papers [52, 53], emphasizing the analogy and the differences between different geometrically induced matter transformations.</text>
<text><location><page_20><loc_12><loc_33><loc_84><loc_39></location>We begin with a simple mechanical analog: a particle moving in a potential with a cusp [52]. Let us consider a one-dimensional problem of a classical point particle moving in the potential</text>
<formula><location><page_20><loc_23><loc_29><loc_84><loc_33></location>V ( x ) = V 0 (1 + x 2 / 3 ) 2 , (73)</formula>
<text><location><page_20><loc_12><loc_27><loc_46><loc_29></location>where V 0 > 0. The equation of motion is</text>
<formula><location><page_20><loc_23><loc_23><loc_84><loc_27></location>x -4 V 0 3(1 + x 2 / 3 ) 3 x 1 / 3 = 0 . (74)</formula>
<text><location><page_20><loc_12><loc_12><loc_84><loc_23></location>We consider three classes of possible motions characterized by the value of the energy E . The first class consists of the motions when E < V 0 . Apparently, the particle with x < 0 , ˙ x > 0 or with x > 0 , ˙ x < 0 cannot reach the point x = 0 and stops at the points ∓ (√ V 0 E -1) ) 3 / 2 respectively.</text>
<text><location><page_20><loc_12><loc_10><loc_84><loc_14></location>The second class includes the trajectories when E > V 0 . In this case the particle crosses the point x = 0 with nonvanishing velocity.</text>
<text><location><page_20><loc_12><loc_6><loc_84><loc_10></location>If we have a fine tuning such that E = V 0 , we encounter an exceptional case. Now the trajectory satisfying Eq. (74) in the vicinity of the point x = 0 can behave as</text>
<formula><location><page_20><loc_23><loc_3><loc_84><loc_6></location>x = C ( t 0 -t ) 3 / 2 , (75)</formula>
<text><location><page_21><loc_12><loc_87><loc_17><loc_89></location>where</text>
<formula><location><page_21><loc_23><loc_82><loc_84><loc_87></location>C = ± ( 16 V 0 9 ) 3 / 4 (76)</formula>
<text><location><page_21><loc_12><loc_76><loc_84><loc_82></location>and t ≤ t 0 . It is easy to see that independently of the sign of C in Eq. (76) the signs of the particle coordinate x and of its velocity ˙ x are opposite and hence, the particle can arrive in finite time to the point of the cusp of the potential x = 0.</text>
<text><location><page_21><loc_16><loc_74><loc_37><loc_76></location>Another solution reads as</text>
<formula><location><page_21><loc_23><loc_70><loc_84><loc_73></location>x = C ( t -t 0 ) 3 / 2 , (77)</formula>
<text><location><page_21><loc_12><loc_48><loc_84><loc_70></location>where t ≥ t 0 . This solution describes the particle going away from the point x = 0. Thus, we can combine the branches of the solutions (75) and (77) in four different manners and there is no way to choose if the particle arriving to the point x = 0 should go back or should pass the cusp of the potential (73). It can stop at the top as well. Such a 'degenerate' behaviour of the particle in this third case is connected with the fact that this trajectory is the separatrix between two one-parameter families of trajectories described above. At the moment there is not yet any strict analogy between this separatrix and the cosmological evolution describing the phantom divide line. In order to establish a closer analogy and to understand what is the crucial difference between mechanical consideration and general relativistic one, we can try to introduce a friction term into the Newton equation (74)</text>
<formula><location><page_21><loc_23><loc_44><loc_84><loc_48></location>x + γ ˙ x -4 V 0 3(1 + x 2 / 3 ) 3 x 1 / 3 = 0 . (78)</formula>
<text><location><page_21><loc_12><loc_38><loc_84><loc_43></location>It is easy to check that, if the friction coefficient γ is a constant, one does not have a qualitative change in respect to the discussion above. Let us asuume for γ the dependence</text>
<formula><location><page_21><loc_23><loc_32><loc_84><loc_37></location>γ = 3 √ ˙ x 2 2 + V ( x ) . (79)</formula>
<formula><location><page_21><loc_23><loc_27><loc_84><loc_31></location>˙ γ = -3 2 ˙ x 2 (80)</formula>
<text><location><page_21><loc_12><loc_31><loc_15><loc_32></location>then</text>
<text><location><page_21><loc_12><loc_25><loc_15><loc_27></location>and</text>
<formula><location><page_21><loc_23><loc_21><loc_84><loc_24></location>¨ γ = -3x ˙ x (81)</formula>
<text><location><page_21><loc_12><loc_12><loc_84><loc_21></location>just like in the cosmological case, where the role of the friction coefficient is played by the Hubble parameter. The trajectory arriving to the cusp with vanishing velocity is still described by the solution (75). Consider the particle coming to the cusp from the left ( C < 0. It is easy to see that the value of ˙ γ at the moment t 0 tends to zero, while its second derivative ¨ γ given by Eq. (81) is</text>
<formula><location><page_21><loc_23><loc_7><loc_84><loc_11></location>¨ γ ( t 0 ) = 9 8 C 2 > 0 . (82)</formula>
<text><location><page_22><loc_12><loc_83><loc_84><loc_89></location>Thus, it looks like the friction coefficient γ reaches its minimum value at t = t 0 . Let us suppose now that the particle is coming back to the left from the cusp and its motion is described by Eq. (77) with negative C . A simple check shows that in this case</text>
<formula><location><page_22><loc_23><loc_79><loc_84><loc_82></location>¨ γ ( t 0 ) = -9 8 C 2 < 0 . (83)</formula>
<text><location><page_22><loc_12><loc_63><loc_84><loc_78></location>Thus, from the point of view of the subsequent evolution this point looks as a maximum for the function γ ( t ). In fact, it means simply that the second derivative of the friction coefficient has a jump at the point t = t 0 . It is easy to check that if instead of choosing the motion to the left, we shall move forward our particle to the right from the cusp ( C > 0), the sign of ¨ γ ( t 0 ) remains negative as in Eq. (83) and hence we have the jump of this second derivative again. If one would like to avoid this jump, one should try to change the sign in Eq. (81). To implement it in a self-consistent way one can substitute Eq. (79) by</text>
<formula><location><page_22><loc_23><loc_57><loc_84><loc_62></location>γ = 3 √ -˙ x 2 2 + V ( x ) (84)</formula>
<text><location><page_22><loc_12><loc_56><loc_25><loc_57></location>and Eq. (78) by</text>
<formula><location><page_22><loc_23><loc_51><loc_84><loc_55></location>x + γ ˙ x + 4 V 0 3(1 + x 2 / 3 ) 3 x 1 / 3 = 0 . (85)</formula>
<text><location><page_22><loc_12><loc_39><loc_84><loc_51></location>In fact, it is exactly that what happens automatically in cosmology, when we change the sign of the kinetic energy term for the scalar field, crossing the phantom divide line. Naturally, in cosmology the role of γ is played by the Hubble variable H . The jump of the second derivative of the friction coefficient γ corresponds to the divergence of the third time derivative of the Hubble variable, which represents some kind of very soft cosmological singularity.</text>
<text><location><page_22><loc_12><loc_29><loc_84><loc_39></location>Thus, one seems to confront the problem of choosing between two alternatives: 1) to encounter a weak singularity in the spacetime geometry; 2) to change the sign of the kinetic term for matter field. We have pursued the second alternative insofar as we privilege the smoothness of spacetime geometry and consider equations of motion for matter as less fundamental than the Einstein equations.</text>
<text><location><page_22><loc_12><loc_25><loc_84><loc_28></location>Now, we would like to say that the potential, considered in papers [52, 53] had the general structure</text>
<formula><location><page_22><loc_23><loc_21><loc_84><loc_24></location>V ( φ ) = 1 A + Bφ 2 / 3 . (86)</formula>
<text><location><page_22><loc_12><loc_4><loc_84><loc_20></location>The origin of this structure is the following: one considers the power law expansion of the universe, it is well-known that such an expansion could be provided by an exponential potential [86]. Then one can represent the Friedmann equation for the evolution of the scale factor of the universe as a second-order linear differential equation, where the potential is reperesented as a function of the time parameter [87]. This equation has two independente solutions: one of them is the power-law expansion and other corresponds to an evolution driven by a phantom matter. The linear combination of these two solutions with both nonvanishing coefficients gives an evolution, where a universe crosses</text>
<text><location><page_23><loc_12><loc_81><loc_84><loc_89></location>the phantom divide line. It is impossible to reconstruct the form of the potential as a function of the scalar field, which provides such an evolution explicitly, however, one can study its form around the point where the phantom divide crossing occur and this form is exactly that of Eq. (86) [52].</text>
<text><location><page_23><loc_12><loc_63><loc_84><loc_80></location>At the end of this section, we would like to say that in the Newtonian mechanics there is rather a realistic example of motion when, the dependence of the distance of time is given by some fractional power [88, 89]. Indeed, if one consider the motion of a car with a constant power (which is more realistic than the motion with a constant force, usually presented in textbooks), when the velocity behaves as t 1 / 2 and if the initial value of the coordinate and of the velocity are equal to zero, when the acceleration behaves as t -1 / 2 and at the moment of start is singular. The motion at constant power is an excellent model of drag-car racing [88, 89]. Its analogy with the cosmology at the presence of sudden singularities was noticed in paper [45].</text>
<section_header_level_1><location><page_23><loc_12><loc_57><loc_82><loc_60></location>8. Classical dynamics of the cosmological model with a scalar field whose potential is inversely proportional to the field</section_header_level_1>
<text><location><page_23><loc_12><loc_41><loc_84><loc_55></location>We have considered earlier the simplest model, possessing a soft cosmological singularity (Big Brake) - the model based on the anti-Chaplygin gas. It was noticd that this model is equivalent to the model with the pseudotachyon field with constant potential. Here we would like to study a model, based on a minimally coupled scalar field, which possesses the same evolution as the model based on the anti-Chaplygin gas. Using the standard technique of the reconstruction of potential, the potential of the corresponding scalar field was found in paper [32] and it looks like</text>
<formula><location><page_23><loc_23><loc_35><loc_84><loc_41></location>V ( ϕ ) = ± √ A 2 ( sinh 3 ϕ -1 sinh 3 ϕ ) . (87)</formula>
<text><location><page_23><loc_12><loc_18><loc_84><loc_35></location>As a matter of fact we have two possible potentials, which differs by the general sign. We choose the sign 'plus'. Then, let us remember that the Big Brake occurs when the energy density is equal to zero (the disappearance of the Hubble parameter) and the pressure is positive and infinite (an infinite deceleration). To achieve this condition, in the scalar field model it is necessary to require that the potential is negative and infinite. It is easy to see from Eq. (87) that this occurs when ϕ → 0 being positive. Thus, to have the model with the Big Brake singularity we can consider the scalar field with a potential which is a little bit simpler than that from Eq. (87), but still possesses rather a rich dynamics. Namely we shall study the scalar field with the potential</text>
<formula><location><page_23><loc_23><loc_14><loc_84><loc_17></location>V = -V 0 ϕ , (88)</formula>
<text><location><page_23><loc_12><loc_9><loc_84><loc_13></location>where V 0 is a positive constant. The Klein-Gordon equation for the scalar field with the potential (88) is</text>
<formula><location><page_23><loc_23><loc_5><loc_84><loc_9></location>¨ ϕ +3 H ˙ ϕ + V 0 ϕ 2 = 0 (89)</formula>
<text><location><page_24><loc_12><loc_87><loc_43><loc_89></location>while the first Friedmann equation is</text>
<formula><location><page_24><loc_23><loc_83><loc_84><loc_86></location>H 2 = ˙ ϕ 2 2 -V 0 ϕ . (90)</formula>
<text><location><page_24><loc_12><loc_78><loc_84><loc_82></location>We shall also need the expression for the time derivative of the Hubble parameter, which can be easily obtained from Eqs. (89) and (90):</text>
<formula><location><page_24><loc_23><loc_74><loc_84><loc_78></location>˙ H = -3 2 ˙ ϕ 2 . (91)</formula>
<text><location><page_24><loc_12><loc_70><loc_84><loc_74></location>Now we shall construct the complete classification of the cosmological evolutions (trajectories) of our model, using Eqs. (89)-(91) [34].</text>
<text><location><page_24><loc_16><loc_68><loc_72><loc_70></location>First of all, let us announce briefly the main results of our analysis.</text>
<unordered_list>
<list_item><location><page_24><loc_13><loc_63><loc_84><loc_67></location>(i) The transitions between the positive and negative values of the scalar field are impossible.</list_item>
<list_item><location><page_24><loc_12><loc_57><loc_84><loc_62></location>(ii) All the trajectories (cosmological evolutions) with positive values of the scalar field begin in the Big Bang singularity, then achieve a point of maximal expansion, then contract and end their evolution in the Big Crunch singularity.</list_item>
<list_item><location><page_24><loc_12><loc_48><loc_84><loc_56></location>(iii) All the trajectories with positive values of the scalar field pass through the point where the value of the scalar field is equal to zero. After that the value of the scalar field begin growing. The point ϕ = 0 corresponds to a crossing of the soft singularity.</list_item>
<list_item><location><page_24><loc_12><loc_42><loc_84><loc_48></location>(iv) If the moment when the universe achieves the point of the maximal expansion coincides with the moment of the crossing of the soft singularity then the singularity is the Big Brake.</list_item>
<list_item><location><page_24><loc_12><loc_34><loc_84><loc_41></location>(v) The evolutions with the negative values of the scalar field belong to two classes first, an infinite expansion beginning from the Big Bang and second, the evolutions obtained by the time reversion of those of the first class, which are contracting and end in the Big Crunch singularity.</list_item>
</unordered_list>
<text><location><page_24><loc_12><loc_27><loc_84><loc_32></location>To prove these results, we begin with the consideration of the universe in the vicinity of the point ϕ = 0. We shall look for the leading term of the field ϕ approaching this point in the form</text>
<formula><location><page_24><loc_23><loc_23><loc_84><loc_26></location>ϕ ( t ) = ϕ 1 ( t S -t ) α , (92)</formula>
<text><location><page_24><loc_12><loc_19><loc_84><loc_23></location>where ϕ 1 and α are positive constants and t S is the moment of the soft singularity crossing. The time derivative of the scalar field is now</text>
<formula><location><page_24><loc_23><loc_15><loc_84><loc_18></location>˙ ϕ ( t ) = αϕ 1 ( t S -t ) α -1 . (93)</formula>
<text><location><page_24><loc_12><loc_5><loc_84><loc_15></location>Because of the negativity of the potential (88) at positive values of ϕ , the kinetic term should be stronger than the potential one to satisfy the Friedmann equation (4). That implies that α ≤ 2 3 . However, if α < 2 3 we can neglect the potential term and remain with the massless scalar field. It is easy to show considering the Friedmann (4) and KleinGordon (89) equations that in this case the scalar field behaves like ϕ ∼ ln( t S -t ), which</text>
<text><location><page_25><loc_12><loc_85><loc_84><loc_89></location>is incompatible with the hypothesis of its smallness (92). Thus, one remains with the only choice</text>
<formula><location><page_25><loc_23><loc_81><loc_84><loc_84></location>α = 2 3 . (94)</formula>
<text><location><page_25><loc_12><loc_73><loc_84><loc_80></location>Then, if the coefficient at the leading term in the kinetic energy is greater than that in the potential, it follows from the Friedmann equation (4) that the Hubble parameter behaves as ( t S -t ) -1 3 which is incompatible with Eq. (91). Thus, the leading terms of the potential and kinetic energy should cancel each other:</text>
<formula><location><page_25><loc_24><loc_68><loc_84><loc_72></location>1 2 α 2 ϕ 2 1 ( t S -t ) 2 α -2 = V 0 ϕ 1 ( t S -t ) -α , (95)</formula>
<text><location><page_25><loc_12><loc_66><loc_28><loc_68></location>that for α = 2 3 gives</text>
<formula><location><page_25><loc_23><loc_60><loc_84><loc_65></location>ϕ 1 = ( 9 V 0 2 ) 1 3 . (96)</formula>
<text><location><page_25><loc_12><loc_59><loc_81><loc_60></location>Hence, the leading term for the scalar field in the presence of the soft singularity is</text>
<formula><location><page_25><loc_23><loc_53><loc_84><loc_58></location>ϕ ( t ) = ( 9 V 0 2 ) 1 3 ( t S -t ) 2 3 . (97)</formula>
<text><location><page_25><loc_12><loc_51><loc_42><loc_53></location>Now, integrating Eq. (91) we obtain</text>
<formula><location><page_25><loc_23><loc_45><loc_84><loc_51></location>H ( t ) = 2 ( 9 V 0 2 ) 2 3 ( t S -t ) 1 3 + H S , (98)</formula>
<text><location><page_25><loc_79><loc_37><loc_79><loc_40></location>/negationslash</text>
<text><location><page_25><loc_12><loc_30><loc_84><loc_46></location>where H S is an integration constant giving the value of the Hubble parameter at the moment of the soft singularity crossing. If this constant is equal to zero, H S = 0, the moment of the maximal expansion of the universe coincides with that of the soft singularity crossing and the universe encounters the Big Brake singularity. If H S = 0 we have a more general type of the soft cosmological singularity where the energy density of the matter in the universe is different from zero. The sign of H S can be both, positive or negative, hence, universe can pass through this singularity in the phase of its expansion or of its contraction.</text>
<text><location><page_25><loc_12><loc_22><loc_84><loc_29></location>The form of the leading term for the scalar field in the vicinity of the moment when ϕ = 0 (97) shows that, after passing the zero value, the scalar field begin growing being positive. Thus, it proves the first result from the list presented above about impossibility of the change of the sign of the scalar field in our model.</text>
<text><location><page_25><loc_49><loc_15><loc_49><loc_17></location>/negationslash</text>
<text><location><page_25><loc_12><loc_6><loc_84><loc_21></location>We have already noted that the time derivative of the scalar field had changed the sign crossing the soft singularity. It cannot change the sign in a non-singular way because the conditions ˙ ϕ ( t 0 ) = 0 , ϕ ( t 0 ) = 0 are incompatible with the Friedmann equation (4). It is seen from Eq. (97) that before the crossing of the soft singularity the time derivative of the scalar field is negative and after its crossing it is positive. The impossibility of the changing the sign of the time derivative of the scalar field without the soft singularity crossing implies the inevitability of the approaching of the universe to this soft singularity. Thus, the third result from the list above is proven.</text>
<text><location><page_26><loc_12><loc_73><loc_84><loc_89></location>It is easy to see from Eq. (91) that the value of the Hubble parameter is decreasing during all the evolution. At the same time, the absolute value of its time derivative (proportional to the time derivative squared of the scalar field) is growing after the soft singularity crossing. That means that at some moment the Hubble parameter should change its sign becoming negative. The change of the sign of the Hubble parameter is nothing but the passing through the point of the maximal expansion of the universe, after which it begin contraction culminating in the encounter with the Big Crunch singularity. Thus the second result from the list presented above is proven.</text>
<text><location><page_26><loc_12><loc_31><loc_84><loc_72></location>Summing up, we can say that all the cosmological evolutions where the scalar field has positive values have the following structure: they begin in the Big Bang singularity with an infinite positive value of the scalar field and an infinite negative value of its time derivative, then they pass through the soft singularity where the value of the scalar field is equal to zero and where the derivative of the scalar field changes its sign. All the trajectories also pass through the point of the maximal expansion, and this passage trough the point of the maximal expansion can precede or follow the passage trough the soft singularity: in the case when these two moments coincide ( H S = 0) we have the Big Brake singularity (see the result 4 from the list above). Thus, all the evolutions pass through the soft singularity, but only for one of them this singularity has a character of the Big Brake singularity. The family of the trajectories can be parameterized by the value of the Hubble parameter H S at the moment of the crossing of the soft singularity. There is also another natural parameterization of this family - we can characterize a trajectory by the value of the scalar field ϕ at the moment of the maximal expansion of the universe and by the sign of its time derivative at this moment (if the time derivative of the scalar field is negative that means that the passing through the point of maximal expansion precedes the passing through the soft singularity and if the sign of this time derivative is positive, then passage trough the point of maximal expansion follows the passage through the soft singularity). If at the moment when the universe achieves the point of maximal expansion the value of the scalar field is equal to zero, then it is the exceptional trajectory crossing the Big Brake singularity.</text>
<text><location><page_26><loc_12><loc_4><loc_84><loc_30></location>For completeness, we shall say some words about the result 5, concerning the trajectories with the negative values of the scalar field. Now, both the terms in the right-hand side of the Friedmann equation (4), potential and kinetic, are positive and, hence, the Hubble parameter cannot disappear or change its sign. It can only tends to zero asymptotically while both these terms tend asymptotically to zero. Thus, in this case there are two possible regimes: an infinite expansion which begins with the Big Bang singularity and an infinite contraction which culminates in the encounter with the Big Crunch singularity. The second regime can be obtained by the time reversal of the first one and vice versa. Let us consider the expansion regime. It is easy to check that the scalar field being negative cannot achieve the zero value, because the suggestion ϕ ( t ) = -ϕ 1 ( t 0 -t ) α , where ϕ 1 < 0 , α > 0 is incompatible with the equations (4) and (91). Hence, the potential term is always non-singular and at the birth of the universe from the Big Bang singularity the kinetic term dominates and the dynamics is that of</text>
<text><location><page_27><loc_12><loc_87><loc_53><loc_89></location>the theory with the massless scalar field. Namely</text>
<formula><location><page_27><loc_23><loc_82><loc_84><loc_86></location>ϕ ( t ) = ϕ 0 + √ 2 9 ln t, H ( t ) = 1 3 t , (99)</formula>
<text><location><page_27><loc_12><loc_70><loc_84><loc_82></location>where ϕ 0 is a constant. At the end of the evolution the Hubble parameter tends to zero, while the time grows indefinitely. That means that both the kinetic and potential terms in the right-hand side of Eq. (4) should tend to zero. It is possible if the scalar field tends to infinity while its time derivative tends to zero. The joint analysis of Eqs. (4) and (91) gives the following results for the asymptotic behavior of the scalar field and the Hubble parameter:</text>
<formula><location><page_27><loc_23><loc_65><loc_84><loc_70></location>ϕ ( t ) = ˜ ϕ 0 -( 5 6 ) 2 5 V 1 5 0 t 2 5 , H ( t ) = ( 6 5 ) 1 5 V 2 5 0 t -1 5 , (100)</formula>
<text><location><page_27><loc_12><loc_63><loc_31><loc_65></location>where ˜ ϕ 0 is a constant.</text>
<section_header_level_1><location><page_27><loc_12><loc_57><loc_80><loc_60></location>9. The quantum dynamics of the cosmological model with a scalar field whose potential is inversely proportional to the field</section_header_level_1>
<text><location><page_27><loc_12><loc_43><loc_84><loc_55></location>The introduction of the notion of the quantum state of the universe, satisfying the Wheeler-DeWitt equation [90] has stimulated the diffusion of the hypothesis that in the framework of quantum cosmology the singularities can disappear in some sense. Namely, the probability of finding of the universe with the parameters, which correspond to a classical cosmological singularity can be equal to zero (for a recent treatments see [91, 92, 93]).</text>
<text><location><page_27><loc_12><loc_37><loc_84><loc_43></location>In this section we shall study the quantum dynamics of the model, whose classical dynamics was described in the preceding section. Our presentation follows that of papers [32, 34].</text>
<text><location><page_27><loc_12><loc_31><loc_84><loc_37></location>As usual, we shall use the canonical formalism and the Wheeler-DeWitt equation [90]. For this purpose, instead of the Friedmann metric (1), we shall consider a more general metric,</text>
<formula><location><page_27><loc_23><loc_28><loc_84><loc_30></location>ds 2 = N 2 ( t ) dt 2 -a 2 ( t ) dl 2 , (101)</formula>
<text><location><page_27><loc_12><loc_24><loc_84><loc_27></location>where N is the so-called lapse function. The action of the Friedmann flat model with the minimally coupled scalar field looks now as</text>
<formula><location><page_27><loc_23><loc_19><loc_84><loc_23></location>S = ∫ dt ( a 3 ˙ ϕ 2 2 N -a 3 V ( ϕ ) -a ˙ a 2 N ) . (102)</formula>
<text><location><page_27><loc_12><loc_13><loc_84><loc_19></location>Variating the action (102) with respect to N and putting then N = 1 we come to the standard Friedmann equation. Now, introducing the canonical formalism, we define the canonically conjugated momenta as</text>
<formula><location><page_27><loc_23><loc_9><loc_84><loc_13></location>p ϕ = a 3 ˙ ϕ N (103)</formula>
<formula><location><page_27><loc_23><loc_3><loc_84><loc_7></location>p a = -a ˙ a N . (104)</formula>
<text><location><page_27><loc_12><loc_7><loc_15><loc_9></location>and</text>
<text><location><page_28><loc_12><loc_87><loc_28><loc_89></location>The Hamiltonian is</text>
<formula><location><page_28><loc_23><loc_82><loc_84><loc_87></location>H = N ( -p 2 a 4 a + p 2 ϕ 2 a 3 + V a 3 ) (105)</formula>
<text><location><page_28><loc_12><loc_78><loc_84><loc_82></location>and is proportional to the lapse function. The variation of the action with respect to N gives the constraint</text>
<formula><location><page_28><loc_24><loc_74><loc_84><loc_78></location>-p 2 a 4 a + p 2 ϕ 2 a 3 + V a 3 = 0 , (106)</formula>
<text><location><page_28><loc_12><loc_70><loc_84><loc_73></location>and the implementation of the Dirac quantization procedure, i.e. requirement the that constraint eliminates the quantum state [94], gives the Wheeler-DeWitt equation</text>
<formula><location><page_28><loc_24><loc_64><loc_84><loc_69></location>( -ˆ p 2 a 4 a + ˆ p 2 ϕ 2 a 3 + V a 3 ) ψ ( a, ϕ ) = 0 . (107)</formula>
<text><location><page_28><loc_12><loc_59><loc_84><loc_64></location>Here ψ ( a, φ ) is the wave function of the universe and the hats over the momenta mean that the functions are substituted by the operators. Introducing the differential operators representing the momenta as</text>
<formula><location><page_28><loc_23><loc_54><loc_84><loc_58></location>ˆ p a ≡ ∂ i∂a , ˆ p ϕ ≡ ∂ i∂ϕ (108)</formula>
<text><location><page_28><loc_12><loc_52><loc_83><loc_54></location>and multiplying Eq. (107) by a 3 we obtain the following partial differential equation:</text>
<formula><location><page_28><loc_24><loc_47><loc_84><loc_51></location>( a 2 4 ∂ 2 ∂a 2 -1 2 ∂ 2 ∂ϕ 2 + a 6 V ) ψ ( a, ϕ ) = 0 . (109)</formula>
<text><location><page_28><loc_12><loc_45><loc_74><loc_47></location>Finally, for our potential inversely proportional to the scalar field we have</text>
<formula><location><page_28><loc_24><loc_40><loc_84><loc_45></location>( a 2 4 ∂ 2 ∂a 2 -1 2 ∂ 2 ∂ϕ 2 -a 6 V 0 ϕ ) ψ ( a, ϕ ) = 0 . (110)</formula>
<text><location><page_28><loc_12><loc_18><loc_84><loc_40></location>Note that in the equation (107) and in the subsequent equations we have ignored rather a complicated problem of the choice of the ordering of noncommuting operators, because the specification of such a choice is not essential for our analysis. Moreover, the interpretation of the wave function of the universe is rather an involved question [95, 96, 97]. The point is that to choose the measure in the space of the corresponding Hilbert space we should fix a particular gauge condition, eliminating in such a way the redundant gauge degrees of freedom and introducing a temporal dynamics into the model [96]. We shall not dwell here on this procedure, assuming generally that the cosmological radius a is in some way connected with the chosen time parameter and that the unique physical variable is the scalar field ϕ . Then, it is convenient to represent the solution of Eq. (110) in the form</text>
<formula><location><page_28><loc_23><loc_12><loc_84><loc_18></location>ψ ( a, ϕ ) = ∞ ∑ n =0 C n ( a ) χ n ( a, ϕ ) , (111)</formula>
<text><location><page_28><loc_12><loc_11><loc_48><loc_12></location>where the functions χ n satisfy the equation</text>
<formula><location><page_28><loc_24><loc_5><loc_84><loc_10></location>( -1 2 ∂ 2 ∂ϕ 2 -a 6 V 0 ϕ ) χ ( a, ϕ ) = -E n ( a ) χ n ( a, ϕ ) , (112)</formula>
<text><location><page_29><loc_12><loc_87><loc_50><loc_89></location>while the functions C n ( a ) satisfy the equation</text>
<formula><location><page_29><loc_24><loc_82><loc_84><loc_86></location>a 2 4 ∂ 2 C n ( a ) ∂a 2 = E n ( a ) C n ( a ) , (113)</formula>
<text><location><page_29><loc_12><loc_72><loc_84><loc_82></location>where n = 0 , 1 , . . . . Requiring the normalizability of the functions χ n on the interval 0 ≤ ϕ < ∞ , which, in turn, implies their non-singular behavior at ϕ = 0 and ϕ →∞ , and using the considerations similar to those used in the analysis of the Schrodinger equation for the hydrogen-like atoms, one can show that the acceptable values of the functions E n are</text>
<formula><location><page_29><loc_23><loc_68><loc_84><loc_72></location>E n = V 0 a 12 2( n +1) 2 , (114)</formula>
<text><location><page_29><loc_12><loc_66><loc_47><loc_67></location>while the corresponding eigenfunctions are</text>
<formula><location><page_29><loc_23><loc_61><loc_84><loc_65></location>χ n ( a, ϕ ) = ϕ exp ( -V 0 a 6 ϕ n +1 ) L 1 n ( 2 V 0 a 6 ϕ n +1 ) , (115)</formula>
<text><location><page_29><loc_12><loc_59><loc_54><loc_61></location>where L 1 n are the associated Laguerre polynomials.</text>
<text><location><page_29><loc_12><loc_9><loc_84><loc_58></location>Rather often the fact that the wave function of the universe disappears at the values of the cosmological parameters corresponding to some classical singularity is interpreted as an avoidance of such singularity. However, in the case of the soft singularity considered in the model at hand, such an interpretation does not look too convincing. Indeed, one can have a temptation to think that the probability of finding of the universe in the soft singularity state characterized by the vanishing value of the scalar field is vanishing because the expression for functions (115) entering into the expression for the wave function of the universe (111) is proportional to ϕ . However, the wave function (111) can hardly have a direct probabilistic interpretation. Instead, one should choose some reasonable time-dependent gauge, identifying some combination of variables with an effective time parameter, and interpreting other variables as physical degrees of freedom [96]. The definition of the wave function of the universe in terms of these physical degrees of freedom is rather an involved question; however, we are in a position to make some semi-qualitative considerations. The reduction of the initial set of variables to the smaller set of physical degrees of freedom implies the appearance of the Faddeev-Popov determinant which as usual is equal to the Poisson bracket of the gauge-fixing condition and the constraint [96]. Let us, for example, choose as a gaugefixing condition the identification of the new 'physical' time parameter with the Hubble parameter H taken with the negative sign. Such an identification is reasonable, because as it follows from Eq. (91) the variable H ( t ) is monotonously decreasing. The volume a 3 is the variable canonically conjugated to the Hubble variable. Thus, the Poisson bracket between the gauge-fixing condition χ = H -T phys and the constraint (106) includes the term proportional to the potential of the scalar field, which is inversely proportional to this field itself. Thus, the singularity in ϕ arising in the Faddeev-Popov determinant can cancel zero, arising in (115).</text>
<text><location><page_29><loc_12><loc_5><loc_84><loc_8></location>Let us confront this situation with that of the Big Bang and Big Crunch singularities. As it was seen in Sec. III such singularities classically arise at infinite</text>
<text><location><page_30><loc_12><loc_75><loc_84><loc_89></location>values of the scalar field. To provide the normalizability of the wave function one should have the integral on the values of the scalar field ϕ convergent, when | ϕ | → ∞ . That means that, independently of details connected with the gauge choice, not only the wave function of the universe but also the probability density of scalar field values should decrease rather rapidly when the absolute value of the scalar field is increasing. Thus, in this case, the effect of the quantum avoidance of the classical singularity is present.</text>
<section_header_level_1><location><page_30><loc_12><loc_71><loc_82><loc_72></location>10. The quantum cosmology of the tachyon and the pseudo-tachyon field</section_header_level_1>
<text><location><page_30><loc_12><loc_63><loc_84><loc_69></location>In this section we would like to construct the Hamiltonian formalism for the tachyon and pseudo-tachyon fields. Using the metric (101), one can see that the contribution of the tachyon field into the action is</text>
<formula><location><page_30><loc_23><loc_57><loc_84><loc_62></location>S = -∫ dtNa 3 V ( T ) √ 1 -˙ T 2 N 2 . (116)</formula>
<text><location><page_30><loc_12><loc_55><loc_41><loc_57></location>The conjugate momentum for T is</text>
<formula><location><page_30><loc_23><loc_48><loc_84><loc_55></location>p T = a 3 V ˙ T N √ 1 -˙ T 2 N 2 . (117)</formula>
<text><location><page_30><loc_12><loc_47><loc_45><loc_49></location>and so the velocity can be expressed as</text>
<text><location><page_30><loc_12><loc_40><loc_49><loc_42></location>The Hamiltonian of the tachyon field is now</text>
<formula><location><page_30><loc_23><loc_40><loc_84><loc_47></location>˙ T = Np T √ p 2 T + a 6 V 2 . (118)</formula>
<formula><location><page_30><loc_23><loc_35><loc_84><loc_40></location>H = N √ p 2 T + a 6 V 2 . (119)</formula>
<text><location><page_30><loc_16><loc_35><loc_58><loc_36></location>Analogously, for the pseudo-tachyon field, we have</text>
<formula><location><page_30><loc_23><loc_27><loc_84><loc_34></location>p T = a 3 W ˙ T N √ ˙ T 2 N 2 -1 , (120)</formula>
<formula><location><page_30><loc_23><loc_21><loc_84><loc_28></location>˙ T = Np T √ p 2 T -a 6 W 2 (121)</formula>
<formula><location><page_30><loc_23><loc_16><loc_84><loc_21></location>H = N √ p 2 T -a 6 W 2 . (122)</formula>
<text><location><page_30><loc_12><loc_21><loc_15><loc_23></location>and</text>
<text><location><page_30><loc_12><loc_16><loc_77><loc_17></location>In what follows it will be convenient for us to fix the lapse function as N = 1.</text>
<text><location><page_30><loc_12><loc_9><loc_84><loc_15></location>Now, adding the gravitational part of the Hamiltonians and quantizing the corresponding observables, we obtain the following Wheeler-DeWitt equations for the tachyons</text>
<formula><location><page_30><loc_24><loc_4><loc_84><loc_9></location>( √ ˆ p 2 T + a 6 V 2 -a 2 ˆ p 2 a 4 ) ψ ( a, T ) = 0 (123)</formula>
<text><location><page_31><loc_12><loc_87><loc_35><loc_89></location>and for the pseudo-tachyons</text>
<formula><location><page_31><loc_24><loc_81><loc_84><loc_86></location>( √ ˆ p 2 T -a 6 W 2 -a 2 ˆ p 2 a 4 ) ψ ( a, T ) = 0 . (124)</formula>
<text><location><page_31><loc_12><loc_74><loc_84><loc_82></location>The study of the Wheeler-DeWitt equation for the universe filled with a tachyon or a pseudo-tachyon field is rather a difficult task because the Hamiltonian depends non-polynomially on the conjugate momentum of such fields. However, one can come to interesting conclusions, considering some particular models.</text>
<text><location><page_31><loc_12><loc_64><loc_84><loc_74></location>First of all, let us consider a model with the pseudo-tachyon field having a constant potential. In this case the Hamiltonian in Eq. (124) does not depend on the field T . Thus, it is more convenient to use the representation of the quantum state of the universe where it depends on the coordinate a and the momentum p T . Then the Wheeler-DeWitt equation will have the following form:</text>
<formula><location><page_31><loc_24><loc_58><loc_84><loc_63></location>( √ p 2 T -a 6 W 2 + a 2 4 ∂ 2 ∂a 2 ) ψ ( a, p T ) = 0 . (125)</formula>
<text><location><page_31><loc_12><loc_45><loc_84><loc_59></location>It becomes algebraic in the variable p T . Now, we see that the Hamiltonian is well defined at p 2 T ≥ a 6 W 2 . Looking at the limiting value p 2 T = a 6 W 2 and comparing it with the relation (121) we see that it corresponds to ˙ T 2 →∞ , which, in turn, corresponds to the encounter with the Big Brake singularity as was explained in the section V. The only way to 'neutralize' the values of p T , which imply the negativity of the expression under the square root in the left-hand side of Eq. (125), is to require that the wave function of the universe is such that</text>
<formula><location><page_31><loc_23><loc_41><loc_84><loc_44></location>ψ ( a, p T ) = 0 at p 2 T ≤ a 6 W 2 . (126)</formula>
<text><location><page_31><loc_12><loc_16><loc_84><loc_41></location>The last condition could be considered as a hint on the quantum avoidance of the Big Brake singularity. However, as it was explained in Sec. IV on the example of the scalar field model, to speak about the probabilities in the neighborhood of the point where the wave function of the universe vanishes, it is necessary to realize the procedure of the reduction of the set of variables to a smaller set of physical degrees of freedom. Now, let us suppose that the gauge-fixing condition is chosen in such a way that the role of time is played by a Hubble parameter. In this case the Faddeev-Popov determinant, equal to the Poisson bracket between the gauge-fixing condition and the constraint, will be inversely proportional to the expression √ ˆ p 2 T -a 6 W 2 (see Eq. (122)), which tends to zero at the moment of the encounter with the Big Brake singularity. Thus, in the case of a pseudo-tachyon model, just like in the case of the cosmological model based on the scalar field, the Faddeev-Popov determinant introduces the singular factor, which compensates the vanishing of the wave function of the universe.</text>
<text><location><page_31><loc_12><loc_5><loc_84><loc_15></location>What can we say about the Big Bang and the Big Crunch singularities in this model? It was noticed in the preceding section that at these singularities ˙ T 2 = 1. From the relation (121) it follows that such values of ˙ T correspond to | p T | → ∞ . A general requirement of the normalizability of the wave function of the universe implies the vanishing of ψ ( a, p T ) at p T →±∞ which signifies the quantum avoidance of the Big</text>
<text><location><page_32><loc_12><loc_85><loc_84><loc_89></location>Bang and the Big Crunch singularities. It is quite natural, because these singularity are not traversable in classical cosmology.</text>
<text><location><page_32><loc_12><loc_64><loc_84><loc_84></location>Now we consider the tachyon cosmological model with the trigonometric potential, whose classical dynamics was briefly sketched in the sections 4 and 6. In this case the Hamiltonian depends on both the tachyon field T and its momentum p T . The dependence of the expression under the square root on T is more complicated than that on p T . Hence, it does not make sense to use the representation ψ ( a, p T ) instead of ψ ( a, T ). Now, we have under the square root the second order differential operator -∂ 2 ∂T 2 , which is positively defined, and the function -a 6 W 2 ( T ), which is negatively defined. The complete expression should not be negative, but what does it mean in our case? It means that we should choose such wave functions for which the quantum average of the operator ˆ p 2 T -a 6 W 2 ( T ) is non-negative:</text>
<text><location><page_32><loc_12><loc_40><loc_84><loc_57></location>Here the symbol D T signifies the integration on the tachyon field T with some measure. It is easy to guess that the requirement (127) does not imply the disappearance of the wave function ψ ( a, T ) at some range or at some particular values of the tachyon field, and one can always construct a wave function which is different from zero everywhere and thus does not show the phenomenon of the quantum avoidance of singularity. However, the forms of the potential V ( T ) given by Eq. (26) and of the corresponding potential W ( T ) for the pseudo-tachyon field arising in the same model [49] are too cumbersome to construct such functions explicitly. Thus, to illustrate our statement, we shall consider a more simple toy model.</text>
<formula><location><page_32><loc_23><loc_57><loc_84><loc_64></location>〈 ψ | ˆ p 2 T -a 6 W 2 ( T ) | ψ 〉 = ∫ D Tψ ∗ ( a, T ) ( -∂ 2 ∂T 2 -a 6 W ( T ) 2 ) ψ ( a, T ) ≥ 0 . (127)</formula>
<text><location><page_32><loc_16><loc_38><loc_43><loc_39></location>Let us consider the Hamiltonian</text>
<formula><location><page_32><loc_23><loc_32><loc_84><loc_37></location>H = √ ˆ p 2 -V 0 x 2 , (128)</formula>
<text><location><page_32><loc_12><loc_31><loc_84><loc_34></location>where ˆ p is the conjugate momentum of the coordinate x and V 0 is some positive constant. Let us choose as a wave function a Gaussian function</text>
<formula><location><page_32><loc_23><loc_27><loc_84><loc_30></location>ψ ( x ) = exp( -αx 2 ) , (129)</formula>
<text><location><page_32><loc_12><loc_23><loc_84><loc_27></location>where α is a positive number and we have omitted the normalization factor, which is not essential in the present context. Then the condition (127) will look like</text>
<formula><location><page_32><loc_24><loc_14><loc_84><loc_23></location>∫ dx exp( -αx 2 ) ( -d 2 dx 2 -V 0 x 2 ) exp( -αx 2 ) = √ π 2 ( 3 4 √ α -V 0 2 α 3 2 ) ≥ 0 , (130)</formula>
<text><location><page_32><loc_12><loc_12><loc_37><loc_14></location>which can be easily satisfied if</text>
<formula><location><page_32><loc_23><loc_7><loc_84><loc_12></location>α ≥ √ 2 3 V 0 . (131)</formula>
<text><location><page_32><loc_12><loc_4><loc_84><loc_8></location>Thus, we have seen that for this very simple model one can always choose such a quantum state, which does not disappear at any value of the coordinate x and</text>
<text><location><page_33><loc_12><loc_77><loc_84><loc_89></location>which guarantees the positivity of the quantum average of the operator, which is not generally positively defined. Coming back to our cosmological model we can say that the requirement of the well-definiteness of the pseudo-tachyon part of the Hamiltonian operator in the Wheeler-DeWitt equation does not imply the disappearance of the wave function of the universe at some values of the variables and thus, does not reveal the effect of the quantum avoidance of the cosmological singularity.</text>
<text><location><page_33><loc_37><loc_66><loc_37><loc_68></location>/negationslash</text>
<text><location><page_33><loc_12><loc_53><loc_84><loc_60></location>Finally, summing up the content of the last three sections, devoted to the comparative study of the classical and quantum dynamics in some models with scalar fields and tachyons, revealing soft future singularities, we can make the following remarks.</text>
<text><location><page_33><loc_12><loc_60><loc_84><loc_76></location>At the end of this section we would like also to analyze the Big Bang and Big Crunch singularities in the tachyon model with the trigonometrical potential. As was shown in paper [49] the Big Bang singularity can occur in two occasions (the same is true also for the Big Crunch singularity [40]) - either W ( T ) → ∞ (for example for T → 0) or at ˙ T 2 = 1 , W ( T ) = 0. One can see from Eqs. (15) and (120) that when the universe approaches these singularities the momentum p T tends to infinity. As was explained before, the wave function of the universe in the momentum representation should vanish at | p T | → ∞ and hence, we have the effect of the quantum avoidance.</text>
<text><location><page_33><loc_12><loc_33><loc_84><loc_52></location>It was shown that in the tachyon model with the trigonometrical potential [49] the wave function of the universe is not obliged to vanish in the range of the variables corresponding to the appearance of the classical Big Brake singularity. In a more simple pseudo-tachyon cosmological model the wave function, satisfying the WheelerDeWitt equation and depending on the cosmological radius and the pseudo-tachyon field, disappears at the Big Brake singularity. However, the transition to the wave function depending only on the reduced set of physical degrees of freedom implies the appearance of the Faddeev-Popov factor, which is singular and which singularity compensates the terms, responsible for the vanishing of the wave function of the universe. Thus, in both these cases, the effect of the quantum avoidance of the Big Brake singularity is absent.</text>
<text><location><page_33><loc_12><loc_19><loc_84><loc_32></location>In the case of the scalar field model with the potential inversely proportional to this field, all the classical trajectories pass through a soft singularity (which for one particular trajectory is exactly the Big Brake). The wave function of the universe disappears at the vanishing value of the scalar field which classically corresponds to the soft singularity. However, also in this case the Faddeev-Popov factor arising at the reduction to the physical degrees of freedom provides nonzero value of the probability of finding of the universe at the soft singularity.</text>
<text><location><page_33><loc_12><loc_4><loc_84><loc_18></location>In spite of the fact that we have considered some particular scalar field and tachyonpseudo-tachyon models, our main conclusions were based on rather general properties of these models. Indeed, in the case of the scalar field we have used the fact that its potential at the soft singularity should be negative and divergent, to provide an infinite positive value of the pressure. In the case of the pseudo-tachyon field both the possible vanishing of the wave function of the universe and its 're-emergence' in the process of reduction were connected with the general structure of the contribution of such a field</text>
<text><location><page_34><loc_12><loc_85><loc_84><loc_89></location>into the super-Hamiltonian constraint (122). Note that in the case of the tachyon model with the trigonometric potential, the wave function does not disappear at all.</text>
<text><location><page_34><loc_12><loc_73><loc_84><loc_84></location>On the other hand we have seen that for the Big Bang and Big Crunch singularities not only the wave functions of the universe but also the corresponding probabilities disappear when the universe is approaching to the corresponding values of the fields under consideration, and this fact is also connected with rather general properties of the structure of the Lagrangians of the theories. Thus, in these cases the effect of quantum avoidance of singularities takes place.</text>
<text><location><page_34><loc_12><loc_61><loc_84><loc_72></location>One can say that there is some kind of a classical - quantum correspondence here. The soft singularities are traversable at the classical level (at least for simple homogeneous and isotropic Friedmann models) and the effect of quantum avoidance of singularities is absent. The strong Big Bang and Big Crunch singularities cannot be passed by the universe at the classical level, and the study of the Wheeler-DeWitt equation indicates the presence of the quantum singularity avoidance effect.</text>
<text><location><page_34><loc_12><loc_51><loc_84><loc_60></location>It would be interesting also to find examples of the absence of the effect of the quantum avoidance of singularities, for the singularities of the Big Bang-Big Crunch type. Note that the interest to the study of the possibility of crossing of such singularities is growing and some models treating this phenomenon have been elaborated during last few years [98].</text>
<section_header_level_1><location><page_34><loc_12><loc_45><loc_77><loc_48></location>11. Friedmann equations modified by quantum corrections and soft cosmological singularities</section_header_level_1>
<text><location><page_34><loc_12><loc_27><loc_84><loc_42></location>As we have already mentioned in the Introduction there are two main directions in the study of quantum cosmology of soft future singularities. One is connected with the analysis of the structure of the Wheeler-DeWitt equation and another concentrates of the study of quantum corrections to the Friedmann equations. While in two preceding sections we were studying the Wheeler-DeWitt equation, here we shall dwell on the quantum corrections to the Friedmann equations and on the possible influence of these corrections on the structure of soft singularities. Our presentation will be mainly based on papers [17],[30].</text>
<text><location><page_34><loc_16><loc_25><loc_71><loc_26></location>In paper [17] was considered a cosmological evolution described by</text>
<formula><location><page_34><loc_23><loc_19><loc_84><loc_24></location>a ( t ) = ( t t s ) 1 / 2 ( a s -1) + 1 -( 1 -t t s ) n , (132)</formula>
<text><location><page_34><loc_12><loc_6><loc_84><loc_19></location>where t s is the time, where the sudden singularity occurs, a s is the value of the scale factor in this moment and 1 < n < 2. The matter responsible for this evolution was not specified. It is easy to see that at the beginning of the evolution (132) the universe passes through the radiation-dominated phase of the expansion, while when t → t s it enters into the singular regime. Then it was supposed that a massive scalar field conformally coupled to gravity is present. The general solutions describing behaviour of this scalar field in these two regimes were written down and the requirement of the matching of</text>
<text><location><page_35><loc_12><loc_85><loc_84><loc_89></location>these conditions at the a = a s was imposed. Then, the solution in the first regime is chosen as</text>
<formula><location><page_35><loc_23><loc_81><loc_84><loc_85></location>φ k ( η ) = e ikη √ 2 k , (133)</formula>
<text><location><page_35><loc_12><loc_76><loc_84><loc_80></location>where η is the conformal time parameter. The solution in the regime of approaching the soft singularity will be</text>
<formula><location><page_35><loc_23><loc_74><loc_84><loc_75></location>φ k ( η ) = ξ 01 e i ˜ ωη + ξ 02 e -i ˜ ωη , (134)</formula>
<text><location><page_35><loc_12><loc_71><loc_17><loc_72></location>where</text>
<formula><location><page_35><loc_23><loc_65><loc_84><loc_70></location>˜ ω = √ k 2 + m 2 H 2 s a 4 s (135)</formula>
<formula><location><page_35><loc_23><loc_63><loc_84><loc_66></location>α = √ 2˜ ωξ 01 , β = √ 2˜ ωξ 02 . (136)</formula>
<text><location><page_35><loc_12><loc_65><loc_77><loc_67></location>and the constants ξ 01 and ξ 02 are connected with the Bogoliubov coefficients :</text>
<text><location><page_35><loc_12><loc_58><loc_84><loc_61></location>The matching conditions permit to find the Bogoliubov coefficients and the number of created particles for each mode</text>
<formula><location><page_35><loc_23><loc_52><loc_84><loc_57></location>N k = β k β ∗ k = 1 4 ( 1 -k ˜ ω ) 2 . (137)</formula>
<text><location><page_35><loc_12><loc_51><loc_46><loc_52></location>The total energy of the created particles</text>
<formula><location><page_35><loc_23><loc_45><loc_84><loc_50></location>ρ = ∫ ρ k d 3 k = π ∫ k 2 ˜ ω ( 1 -k ˜ ω ) 2 dk (138)</formula>
<text><location><page_35><loc_12><loc_34><loc_84><loc_45></location>is divergent in the ultraviolet limit. The authors of [17] renormalize the expression (138) using n -wave method [99] and show that the renormalized energy is equal to zero. Thus, they conclude that the quantum phenomena associated with the cosmological dynamics do not change the character of the sudden singularity or prevent its occurrence. Some arguments in favour of the hypothesis that birth of particles of a field which is not conformally invariant cannot change the Friedmann equation are also developed in [17].</text>
<text><location><page_35><loc_12><loc_14><loc_84><loc_33></location>More detailed analysis of the quantum contributions into energy-momentum tensor and, hence, into the Friedmann equations, was undertaken in paper [30]. Here it was noticed that the analysis, presented in paper [17], is applicable only to situations when the frequency of the field under consideration is varying smoothly. Obviously, it is not case here, because two different phases of evolution are considered and a naive matching of the value of the field and of its time derivative at the moment of arrival to the singularity, is required. Moreover, the effect of polarization of the vacuum was not taken into account. Instead, the authors of the paper [30], use the known expressions for the renormalized energy-density and pressure for a massless conformally coupled scalar field [100, 101]:</text>
<formula><location><page_35><loc_23><loc_8><loc_84><loc_13></location>ρ ren = 1 480 π 2 ( 3 H 2 ˙ H + H H -1 2 ˙ H 2 ) + 1 960 π 2 H 4 , (139)</formula>
<formula><location><page_35><loc_23><loc_3><loc_84><loc_8></location>p ren = -1 1440 π 2 ( H +11 H 2 ˙ H +6 H H + 9 2 ˙ H 2 ) -1 960 π 2 H 4 . (140)</formula>
<text><location><page_36><loc_12><loc_85><loc_84><loc_89></location>Then proceeding as in paper [102] the authors of [30] consider the Friedmann semiclassical equation</text>
<formula><location><page_36><loc_23><loc_82><loc_84><loc_84></location>H 2 = ρ + ρ ren , (141)</formula>
<text><location><page_36><loc_12><loc_80><loc_43><loc_81></location>looking for its solution with the form</text>
<formula><location><page_36><loc_23><loc_74><loc_84><loc_79></location>H ( t ) = H s -C ( 1 -t t s ) n ' , (142)</formula>
<text><location><page_36><loc_12><loc_73><loc_75><loc_75></location>where H s , C and n ' are unknown parameters. They find, in particular, that</text>
<formula><location><page_36><loc_23><loc_70><loc_84><loc_72></location>n ' = n +1 . (143)</formula>
<text><location><page_36><loc_12><loc_62><loc_84><loc_69></location>Then, since 3 < n ' < 4, it turns out that ˙ H and H do not diverge at t = t s , which means that, for these kinds of singular solutions, the singularity becomes much milder due to the quantum corrections. In fact, in the absence of the quantum corrections, one can see from Eq. (132) that ˙ H diverges.</text>
<section_header_level_1><location><page_36><loc_12><loc_55><loc_80><loc_59></location>12. Density matrix of the universe, quantum consistency and interplay between geometry and matter in quantum cosmology</section_header_level_1>
<text><location><page_36><loc_12><loc_40><loc_84><loc_53></location>In this section we shall speak about the quantum density matrix of the universe [103, 104, 105, 106, 107] - an approach to quantum cosmology, which permits consideration of mixed quantum states of the universe instead of pure ones. Such and approach is based on rather a delicate interplay between geometry and matter and implies existence of essential restrictions on the basic parameters of the theory. In the framework of this approach as a byproduct arise also some new kinds of soft sudden quantum singularities [106].</text>
<text><location><page_36><loc_12><loc_16><loc_84><loc_39></location>As is well known, quantum cosmology predicts the initial conditions for the cosmological evolution of the universe, defining its quantum state - the wave function of the universe. The connection between the Euclidean quantum theory and the quantum tunneling is used in both the main approaches to the construction of such a function - the no-boundary prescription [108] and the tunneling one[109, 110]. In papers [103, 104] this traditional scheme of quantum cosmology was generalized for the case of fundamental mixed initial quantum states of the universe, in other words instead of wave function of the universe one can consider the density matrix of the universe, possessing some thermodynamical characteristics. Such a mixed state of the universe arises naturally if an instanton with two turning points (surfaces of vanishing external curvature) does exist. (The idea that instead of pure quantum state of the universe one can consider a density matrix of the universe, was suggested already in paper [111]).</text>
<text><location><page_36><loc_12><loc_8><loc_84><loc_15></location>In turn, an instanton with two turning points arises naturally, if we consider a closed Friedmann universe where two essential ingredients are present: an effective cosmological constant and radiation, which corresponds to the presence of the conformally invariant fields. The Euclidean Friedmann equation in this case is written as</text>
<formula><location><page_36><loc_24><loc_3><loc_84><loc_7></location>˙ a 2 a 2 = 1 a 2 -H 2 -C a 4 , (144)</formula>
<figure>
<location><page_37><loc_12><loc_80><loc_33><loc_89></location>
<caption>Figure 2.</caption>
</figure>
<text><location><page_37><loc_12><loc_73><loc_84><loc_77></location>Picture of instanton representing the density matrix. Dashed lines depict the Lorentzian Universe nucleating from the instanton at the minimal surfaces Σ and Σ ' .</text>
<figure>
<location><page_37><loc_12><loc_63><loc_31><loc_72></location>
<caption>Figure 3. Density matrix of the pure Hartle-Hawking state represented by the union of two vacuum instantons.</caption>
</figure>
<text><location><page_37><loc_12><loc_52><loc_84><loc_55></location>where H 2 is an effective cosmological constant and the constant C characterizes the quantity of the radiation in the universe. The turning points are</text>
<formula><location><page_37><loc_23><loc_46><loc_84><loc_51></location>a ± = 1 √ 2 H √ 1 ± (1 -4 CH 2 ) 1 / 2 , 4 CH 2 ≤ 1 . (145)</formula>
<text><location><page_37><loc_12><loc_35><loc_84><loc_46></location>(The same instanton was considered also in paper [112], where the conception of the universe, which gave birth to itself was suggested). Fig. 2 gives the picture of the instanton representing the density matrix of the universe For the pure quantum state [108] the instanton bridge between Σ and Σ ' breaks down (see Fig.3). However, the radiation stress tensor prevents these half instantons from closure. The relevant density matrix is the path integral</text>
<formula><location><page_37><loc_23><loc_29><loc_84><loc_34></location>ρ [ ϕ, ϕ ' ] = e Γ ∫ g, φ | Σ , Σ ' =( ϕ,ϕ ' ) D [ g, φ ] exp ( -S E [ g, φ ]) . (146)</formula>
<text><location><page_37><loc_12><loc_23><loc_84><loc_28></location>with the partition function e -Γ which follows from integrating out the field ϕ in the coincidence ϕ ' = ϕ corresponding to the identification of Σ ' and Σ, the underlying instanton acquiring the toroidal topology.</text>
<text><location><page_37><loc_12><loc_19><loc_84><loc_22></location>The metric of the instanton introduced above is conformally equivalent to the metric of the Einstein static universe:</text>
<formula><location><page_37><loc_23><loc_16><loc_84><loc_18></location>d ¯ s 2 = dη 2 + d 2 Ω (3) , (147)</formula>
<text><location><page_37><loc_12><loc_9><loc_84><loc_15></location>where η is the conformal time parameter. We shall consider conformally invariant fields. As is well known, the quantum effective action for such fields has a conformal anomaly first studied in cosmology in [113, 114]. It has the form</text>
<formula><location><page_37><loc_23><loc_3><loc_84><loc_8></location>g µν δ Γ 1 -loop δg µν = 1 4(4 π ) 2 g 1 / 2 ( α ∆ R + βE + γC 2 µναβ ) , (148)</formula>
<text><location><page_38><loc_12><loc_83><loc_84><loc_89></location>where E = R 2 µναγ -4 R 2 µν + R 2 and ∆ is the four-dimensional Laplacian. This anomaly, when integrated functionally along the orbit of the conformal group, gives the relation between the actions on conformally related backgrounds [115].</text>
<formula><location><page_38><loc_23><loc_80><loc_52><loc_82></location>Γ 1 -loop [ g ] = Γ 1 -loop [ ¯ g ] + δ Γ[ g, ¯ g ] ,</formula>
<formula><location><page_38><loc_23><loc_78><loc_41><loc_80></location>g µν ( x ) = e σ ( x ) ¯ g µν ( x ) ,</formula>
<formula><location><page_38><loc_80><loc_78><loc_84><loc_82></location>(149) (150)</formula>
<text><location><page_38><loc_12><loc_75><loc_17><loc_77></location>where</text>
<formula><location><page_38><loc_23><loc_54><loc_84><loc_75></location>δ Γ[ g, ¯ g ] = 1 2(4 π ) 2 ∫ d 4 x ¯ g 1 / 2 { 1 2 [ γ ¯ C 2 µναβ + β ( ¯ E -2 3 ¯ ∆ ¯ R )] σ + β 2 [ ( ¯ ∆ σ ) 2 + 2 3 ¯ R ( ¯ ∇ µ σ ) 2 ] } -1 2(4 π ) 2 ( α 12 + β 18 ) × ∫ d 4 x ( g 1 / 2 R 2 ( g ) -¯ g 1 / 2 R 2 (¯ g ) ) . (151)</formula>
<text><location><page_38><loc_12><loc_45><loc_84><loc_55></location>One can show that the higher-derivative in σ terms are all proportional to the coefficient α . The α -term can be arbitrarily changed by adding a local counterterm ∼ g 1 / 2 R 2 . We fix this local renormalization ambiguity by an additional criterion of the absence of ghosts. The conformal contribution to the renormalized action on the minisuperspace background equals</text>
<formula><location><page_38><loc_23><loc_35><loc_84><loc_44></location>δ Γ[ g, ¯ g ] ≡ Γ R [ g ] -Γ R [ ¯ g ] = m 2 P B ∫ dτ ( ˙ a 2 a -1 6 ˙ a 4 a ) , (152) 2 P B = 3 β, (153)</formula>
<formula><location><page_38><loc_23><loc_34><loc_32><loc_37></location>m 4</formula>
<text><location><page_38><loc_12><loc_30><loc_84><loc_34></location>with the constant m 2 P B which for scalars, two-component spinors and vectors equals respectively 1 / 240, 11 / 480 and 31 / 120. For a conformal scalar field</text>
<formula><location><page_38><loc_23><loc_24><loc_84><loc_29></location>S [ ¯ g, φ ] = 1 2 ∑ ω ∫ η 0 dη ' ( ( dφ ω dη ' ) 2 + ω 2 φ 2 ω ) , (154)</formula>
<text><location><page_38><loc_12><loc_21><loc_84><loc_24></location>where ω = n , n = 0 , 1 , 2 , ... , labels a set of eigenmodes and eigenvalues of the Laplacian on a unit 3-sphere. Thus</text>
<formula><location><page_38><loc_23><loc_8><loc_84><loc_20></location>e -Γ 1 -loop [ ¯ g ] = ∫ ∏ ω dϕ ω ∫ φ ω ( η )= φ ω (0)= ϕ ω D [ φ ] exp ( -S [ ¯ g, φ ]) = const ∏ ω ( sinh ωη 2 ) -1 , (155)</formula>
<text><location><page_38><loc_12><loc_4><loc_84><loc_8></location>and the effective action equals the sum of contributions of the vacuum energy E 0 and free energy F ( η ) with the inverse temperature played by η - the circumference of the</text>
<text><location><page_39><loc_12><loc_87><loc_51><loc_89></location>toroidal instanton in units of a conformal time,</text>
<formula><location><page_39><loc_23><loc_80><loc_84><loc_87></location>Γ 1 -loop [ ¯ g ] = ∑ ω [ η ω 2 +ln(1 -e -ωη ) ] = m 2 P E 0 η + F ( η ) , (156)</formula>
<formula><location><page_39><loc_23><loc_75><loc_84><loc_80></location>m 2 P E 0 = ∑ ω ω 2 = ∞ ∑ n =1 n 3 2 , (157)</formula>
<formula><location><page_39><loc_24><loc_66><loc_84><loc_71></location>= ∞ ∑ n =1 n 2 ln (1 -e -nη ) . (159)</formula>
<formula><location><page_39><loc_23><loc_71><loc_84><loc_75></location>F ( η ) = ∑ ω ln (1 -e -ωη ) (158)</formula>
<text><location><page_39><loc_12><loc_61><loc_84><loc_66></location>Similar expressions hold for other conformally invariant fields of higher spins. In particular, the vacuum energy (an analog of the Casimir energy) on Einstein static spacetime is</text>
<formula><location><page_39><loc_23><loc_51><loc_84><loc_60></location>m 2 P E 0 = 1 960 × 4 17 88 (160)</formula>
<text><location><page_39><loc_12><loc_46><loc_84><loc_51></location>We should take into account the effect of the finite ghost-avoidance renormalization denoted below by a subscript R , which results in the replacement of E 0 above by a new parameter C 0 :</text>
<text><location><page_39><loc_12><loc_51><loc_51><loc_55></location> respectively for scalar, spinor and vector fields.</text>
<formula><location><page_39><loc_23><loc_43><loc_84><loc_45></location>Γ R [ ¯ g ] = m 2 P C 0 η 0 + F ( η ) , (161)</formula>
<formula><location><page_39><loc_23><loc_39><loc_84><loc_43></location>m 2 P C 0 = m 2 P E 0 + 3 16 α. (162)</formula>
<text><location><page_39><loc_12><loc_35><loc_84><loc_39></location>A direct observation indicates the following universality relation for all conformal fields of low spins</text>
<formula><location><page_39><loc_23><loc_31><loc_84><loc_35></location>m 2 P C 0 = 1 2 m 2 P B. (163)</formula>
<text><location><page_39><loc_12><loc_28><loc_84><loc_31></location>Now we can write down the effective Friedmann equation governing the Euclidean evolution of the universe. First of all, the full conformal time on the instanton is</text>
<formula><location><page_39><loc_23><loc_23><loc_84><loc_27></location>η = 2 ∫ τ + τ -dτ N ( τ ) a ( τ ) , (164)</formula>
<text><location><page_39><loc_12><loc_21><loc_76><loc_23></location>where τ ± label the turning points for a ( τ ) - its minimal and maximal values.</text>
<text><location><page_39><loc_16><loc_18><loc_47><loc_21></location>The effective action is ( m 2 P ≡ 3 / 4 πG )</text>
<formula><location><page_39><loc_23><loc_3><loc_84><loc_18></location>Γ[ a ( τ ) , N ( τ ) ] = 2 m 2 P ∫ τ + τ -dτ ( -a ˙ a 2 N -Na + NH 2 a 3 ) +2 Bm 2 P ∫ τ + τ -dτ ( ˙ a 2 Na -1 6 ˙ a 4 N 3 a ) + F ( 2 ∫ τ + τ -dτ N a ) + Bm 2 P ∫ τ + τ -dτ N a , (165)</formula>
<text><location><page_40><loc_12><loc_87><loc_48><loc_89></location>and the effective Friedmann equation reads</text>
<formula><location><page_40><loc_24><loc_73><loc_84><loc_86></location>δ Γ δN = 2 m 2 P ( a ˙ a 2 N 2 -a + H 2 a 3 ) +2 Bm 2 P ( -˙ a 2 N 2 a + 1 2 ˙ a 4 N 4 a ) + 2 a ( dF ( η ) dη + B 2 m 2 P ) = 0 . (166)</formula>
<text><location><page_40><loc_12><loc_72><loc_49><loc_74></location>In the gauge N = 1 this equation takes form</text>
<formula><location><page_40><loc_24><loc_67><loc_84><loc_71></location>˙ a 2 a 2 + B ( 1 2 ˙ a 4 a 4 -˙ a 2 a 4 ) = 1 a 2 -H 2 -C a 4 , (167)</formula>
<text><location><page_40><loc_12><loc_65><loc_77><loc_67></location>where the amount of radiation constant C is given by the bootstrap equation</text>
<formula><location><page_40><loc_23><loc_59><loc_84><loc_65></location>m 2 P C = m 2 P B 2 + dF ( η ) dη ≡ B 2 m 2 P + ∑ ω ω e ωη -1 . (168)</formula>
<text><location><page_40><loc_16><loc_58><loc_53><loc_60></location>The Friedmann equation can be rewritten as</text>
<formula><location><page_40><loc_23><loc_50><loc_84><loc_57></location>˙ a 2 = √ ( a 2 -B ) 2 B 2 + 2 H 2 B ( a 2 + -a 2 )( a 2 -a 2 -) -( a 2 -B ) B (169)</formula>
<text><location><page_40><loc_12><loc_48><loc_72><loc_49></location>and has the same two turning points a ± as in the classical case provided</text>
<formula><location><page_40><loc_23><loc_44><loc_84><loc_47></location>a 2 -≥ B. (170)</formula>
<text><location><page_40><loc_12><loc_42><loc_39><loc_44></location>This requirement is equivalent to</text>
<formula><location><page_40><loc_23><loc_38><loc_84><loc_42></location>C ≥ B -B 2 H 2 , BH 2 ≤ 1 2 . (171)</formula>
<text><location><page_40><loc_12><loc_36><loc_23><loc_38></location>Together with</text>
<formula><location><page_40><loc_23><loc_32><loc_32><loc_36></location>CH 2 ≤ 1 4 ,</formula>
<text><location><page_40><loc_12><loc_28><loc_84><loc_32></location>the admissible domain for instantons reduces to the curvilinear wedge below the hyperbola and above the straight line to the left of the critical point (see Figure 4)</text>
<formula><location><page_40><loc_23><loc_24><loc_40><loc_28></location>C = B 2 , H 2 = 1 2 B .</formula>
<text><location><page_40><loc_12><loc_10><loc_84><loc_24></location>The suggested approach allows to resolve the problem of the so-called infrared catastrophe for the no-boundary state of the Universe based on the Hartle-Hawking instanton. This problem is related to the fact that the Euclidean action on this instanton is negative and inverse proportional to the value of the effective cosmological constant. This means that the probability of the universe creation with an infinitely big size is infinitely high. We shall show now that the conformal anomaly effect allows one to avoid this counter-intuitive conclusion.</text>
<text><location><page_40><loc_12><loc_4><loc_84><loc_10></location>Indeed, outside of the admissible domain for the instantons with two turning points, obtained above, one can also construct instantons with one turning point which smoothly close at a -= 0 with ˙ a ( τ -) = 1. Such instantons correspond to the Hartle-Hawking pure</text>
<figure>
<location><page_41><loc_12><loc_55><loc_77><loc_89></location>
<caption>Figure 4. The instanton domain in the ( H 2 , C )-plane is located between bold segments of the upper hyperbolic boundary and lower straight line boundary. The first one-parameter family of instantons is labeled by k = 1. Families of garlands are qualitatively shown for k = 2 , 3 , 4. (1 / 2 B,B/ 2) is the critical point of accumulation of the infinite sequence of garland families.</caption>
</figure>
<text><location><page_41><loc_12><loc_39><loc_84><loc_42></location>quantum state. However, in this case the on-shell effective action, which reads for the set of solutions obtained above as</text>
<formula><location><page_41><loc_23><loc_29><loc_84><loc_38></location>Γ 0 = F ( η ) -η dF ( η ) dη +4 m 2 P ∫ a + a -da ˙ a a ( B -a 2 -B ˙ a 2 3 ) , (172)</formula>
<text><location><page_41><loc_12><loc_28><loc_58><loc_30></location>diverges to plus infinity. Indeed, for a -= 0 and ˙ a -= 1</text>
<formula><location><page_41><loc_23><loc_23><loc_84><loc_28></location>η = ∫ a + 0 da ˙ aa = ∞ , F ( ∞ ) = F ' ( ∞ ) = 0 , (173)</formula>
<text><location><page_41><loc_12><loc_20><loc_80><loc_23></location>and hence the effective Euclidean action diverges at the lower limit to + ∞ . Thus,</text>
<formula><location><page_41><loc_23><loc_18><loc_46><loc_20></location>Γ 0 = + ∞ , exp( -Γ 0 ) = 0 ,</formula>
<text><location><page_41><loc_12><loc_12><loc_84><loc_18></location>and this fact completely rules out all pure-state instantons, and only mixed quantum states of the universe, described by the cosmological density matrix appear to be admissible.</text>
<text><location><page_41><loc_12><loc_4><loc_84><loc_12></location>In connection with all said above a natural question arises: where Euclidean quantum gravity comes from? The answer can be formulated briefly as follows: from the Lorentzian quantum gravity (LQG) [105]. Namely, the density matrix of the Universe for the microcanonical ensemble in Lorentzian quantum cosmology of spatially closed</text>
<text><location><page_42><loc_12><loc_75><loc_84><loc_89></location>universes describes an equipartition in the physical phase space of the theory, but in terms of the observable spacetime geometry this ensemble is peaked about a set of cosmological instantons (solutions of the Euclidean quantum cosmology) limited to a bounded range of the cosmological constant. These instantons obtained above as fundamental in Euclidean quantum gravity framework, in fact, turn out to be the saddle points of the LQG path integral, belonging to the imaginary axis in the complex plane of the Lorentzian signature lapse function [105].</text>
<text><location><page_42><loc_12><loc_67><loc_84><loc_74></location>Now let us consider the cosmological evolution of the unverse starting from the initial conditions described above. Making the transition from the Euclidean time to the Lorentzian one, τ = it , we can write the modified Lorentzian Friedmann equation as [106]</text>
<formula><location><page_42><loc_24><loc_61><loc_84><loc_66></location>˙ a 2 a 2 + 1 a 2 = 1 B { 1 -√ 1 -16 πG 3 Bε } , (174)</formula>
<formula><location><page_42><loc_23><loc_56><loc_84><loc_61></location>ε = 3 8 πG ( H 2 + C a 4 ) , (175)</formula>
<formula><location><page_42><loc_23><loc_53><loc_84><loc_57></location>C ≡ C -B 2 , (176)</formula>
<text><location><page_42><loc_12><loc_44><loc_84><loc_53></location>where ε is a total gravitating matter density in the model (including at later stages also the contribution of particles created during inflationary expansion and thermalized at the inflation exit). A remarkable feature of this equation is that the Casimir energy is totally screened here and only the thermal radiation characterized by C weighs.</text>
<text><location><page_42><loc_12><loc_21><loc_84><loc_45></location>If one wants to compare the evolution described by Eq. (176) with the real evoltuion of the universe, first of all it is necessary to have a realistic value for an effective cosmological constant Λ = 3 H 2 . The only way to achieve this goal is to increase the number of conformal fields and the corresponding parameter B , (153), of the conformal anomaly (148). The mechanisms for growing number of the conformal fields exist in some string inspired cosmological models with extra dimensions [105]. If some of these mechanisms work we can encounter an interesting phenomenon: if the B grows with a faster than the rate of decrease of the energy density ε one encounters a new type of the cosmological singularity - Big Boost. This singularity is characterized by finite values of the cosmological radius a BB and of its time derivative ˙ a BB , while the second time variable a has an infinite positive value. The universe reaches this singularity at some finite moment of cosmic time t BB :</text>
<formula><location><page_42><loc_23><loc_17><loc_84><loc_20></location>a ( t BB ) = a BB < ∞ , (177)</formula>
<formula><location><page_42><loc_24><loc_13><loc_84><loc_15></location>lim t → t BB a ( t ) = ∞ . (179)</formula>
<formula><location><page_42><loc_23><loc_15><loc_84><loc_18></location>˙ a ( t BB ) = ˙ a BB < ∞ , (178)</formula>
<text><location><page_42><loc_12><loc_4><loc_84><loc_12></location>In paper [107] it was found that there exist some correspondences between quantum 4-dimensional equations of motion and some classical 5-dimensional equations of motion [106, 107].There were considered two five-dimansional models: the Randall -Sundrum model[116] and the generalized Dvali-Gabadadze-Porrati (DGP) model [117].</text>
<text><location><page_43><loc_12><loc_81><loc_84><loc_89></location>The Randall-Sundrum braneworld model is a 4-dimensional spacetime braneworld embedded into the 5-dimensional anti-de Sitter bulk with the radius L . In the limit of small energy densities the modified quantum Friedmann equations coincide with the modified 4-dimensional Friedmann equations of the Randall-Sundrum model provided</text>
<formula><location><page_43><loc_23><loc_77><loc_84><loc_80></location>βG = πL 2 2 . (180)</formula>
<text><location><page_43><loc_12><loc_71><loc_84><loc_76></location>The 5-dimensional action of the generalized DGP model includes the 5-dimensional curvature term, the 5-dimensional cosmological constant and the 4-dimensinal curvature term on the brane.</text>
<text><location><page_43><loc_12><loc_61><loc_84><loc_70></location>If we require the spherical symmetry, when we have the Schwarzschild-de Sitter solution, which depends also on the Schwarzschil radius R S . The effective 4-dimensional Friedmann equations on the 4-brane coincide with the modified Friedmann equations in quantum model, provided the quantity of the radiation is expressed through the Schwarzschild radius as</text>
<formula><location><page_43><loc_23><loc_58><loc_84><loc_60></location>C = R 2 S (181)</formula>
<text><location><page_43><loc_12><loc_49><loc_84><loc_57></location>If we add the condition of the regularity of the Schwarzschild-de Sitter instanton, (i.e. the condition of the absence of conical singularities), we obtain an additional relation for the parameters of the quantum cosmological model and the set of admissible values for the effective cosmological constant becomes discrete.</text>
<text><location><page_43><loc_12><loc_29><loc_84><loc_48></location>Concluding this section, we would like to say that relaxing the usual tacit requirement of the purity of the quantum state of the universe and imposing the conditions of quantum consistency of the system of equations governing the dynamics of the universe, one comes to non-trivial restrictions on the basic cosmological paprameters. Besides, as a by-product one obtains a particular kind of future soft singularity - Big Boost. Finally, we can note that in the papers, reviewed in this section both the main approaches to the study of quantum effects in cosmology were combined - the study of the modified Friedmann equations and the investigation of the structure of the quantum state of the universe. Usually, these two approaches are separated (see, Sec. 11 and Secs. 9 and 10 of the present review).</text>
<section_header_level_1><location><page_43><loc_12><loc_25><loc_59><loc_26></location>13. Quiescent singularities in braneworld models</section_header_level_1>
<text><location><page_43><loc_12><loc_15><loc_84><loc_23></location>One of the first examples of the soft future singularities in cosmology was presented in paper [10], where some braneworld cosmological models were considered. The higherdimensional models considered there were described by an action, where both the bulk and brane contained the corresponding curvature terms:</text>
<formula><location><page_43><loc_17><loc_9><loc_84><loc_14></location>S = M 3 ∑ i ∫ bulk ( R2Λ i ) -2 ∫ brane K + ∫ brane ( m 2 R -2 σ ) + ∫ brane L ( h αβ , φ ) , (182)</formula>
<text><location><page_43><loc_12><loc_4><loc_84><loc_10></location>where the sum is taken over the bulk components bounded by branes, and Λ i is the cosmological constant on the ith bulk component. The Lagrangian L ( h αβ , φ ) corresponds to the presence of matter fields on the brane interacting with the induced</text>
<text><location><page_44><loc_12><loc_85><loc_84><loc_89></location>metric h αβ , K is the trace of the extrinsic curvature. The Friedmann-type equation has the form</text>
<formula><location><page_44><loc_23><loc_79><loc_84><loc_84></location>H 2 + κ a 2 = ρ + σ 3 m 2 + 2 l 2 [ 1 ± √ 1 + l 2 ( ρ + σ 3 m 2 -Λ 6 -C a 4 ) ] , (183)</formula>
<text><location><page_44><loc_12><loc_71><loc_84><loc_79></location>where ρ is the energy density of the matter on the brane, the integration constant C corresponds to the presence of a black hole in the five-dimensional bulk solution, and the term C/a 4 , sometimes called 'dark radiation', arises due to the projection of the bulk gravitational degrees of freedom onto the brane. The length scale l is defined as</text>
<formula><location><page_44><loc_23><loc_67><loc_84><loc_71></location>l = m 2 M 3 . (184)</formula>
<text><location><page_44><loc_12><loc_61><loc_84><loc_67></location>The appearence of the quiescent singularities is conneced with the fact, that the expression under the square root in (183) turns to zero at some point during the evolution. There are essentially two types of singularities dispaying this behaviour.</text>
<text><location><page_44><loc_12><loc_54><loc_84><loc_61></location>A type 1 singularity (S1) is induced by the presence of the dark radiation term and arises in either of the following two cases: C > 0 and the density of matter increases slower than a -4 as a → 0. An example is provided by dust.</text>
<text><location><page_44><loc_12><loc_53><loc_84><loc_55></location>The energy density of the universe is radiation dominated so that ρ = ρ 0 /a 4 and C > ρ 0 .</text>
<text><location><page_44><loc_12><loc_49><loc_84><loc_52></location>These singularities can take place either in the past of an expanding universe or in the future of a collapsing one.</text>
<text><location><page_44><loc_16><loc_47><loc_44><loc_48></location>A type 2 singularity (S2) arises if</text>
<formula><location><page_44><loc_23><loc_42><loc_84><loc_46></location>l 2 ( σ 3 m 2 -Λ 6 ) < -1 . (185)</formula>
<text><location><page_44><loc_12><loc_36><loc_84><loc_42></location>In this case the combination ρ/ 3 m 2 -C/a 4 decreases monotonically as the universe expands. The expression under the square root of (183) can therefore become zero at suitably late times.</text>
<text><location><page_44><loc_12><loc_22><loc_84><loc_35></location>For both S1 and S2, the scale factor a ( t ) and its first time derivative remain finite, while all the higher time derivatives of a tend to infinity as the singularity is approached. It is important that the energy density and the pressure of the matter in the bulk remain finite. This feature distinguishes these singularities from the singularities considered in the preceding sections, and justifies the special name 'quiescent' [10]. The point is that the existence of these singularities is connected not with special features of the matter on the brane, but with the particularity of the embedding of the brane into the bulk.</text>
<text><location><page_44><loc_12><loc_8><loc_84><loc_21></location>In paper [11] the question of influence of the quantum effects on a braneworld encountering a quiescent singularity during expansion was studied. The matter considered in [11] was constituted from conformally invariant fields. Hence, the particle production was absent and the only quantum effect was connected with the vacuum polarization. It was shown that this effect boils down to the modification of the effective energy density of the matter on the brane. Namely, the quantum correction to this energy density is given by</text>
<formula><location><page_44><loc_23><loc_4><loc_84><loc_7></location>ρ quantum = k 2 H 4 + k 3 (2 HH +6 ˙ HH 2 -˙ H 2 ) . (186)</formula>
<text><location><page_45><loc_12><loc_75><loc_84><loc_89></location>The insertion of this correction to the energy density changes drastically the form of the brane Friedmann-type equation (183) - the original algebraic equation becomes a differential equation. It implies essential changes in the possible behaviour of the universe around singularities. First, the quiscent singularity changes its form and becomes much weaker, in fact, H and ˙ H remain finite and only ··· H → ∞ . Second, vacuum polarization effects can also cause a spatially flat universe to turn around and collapse.</text>
<text><location><page_45><loc_12><loc_45><loc_84><loc_74></location>At the conclusion of this section we would like to mention another type of cosmological singularities, arising in the brane-world context. These are the so called pressure singularities [118, 119]. These singularities arise in the generalized Friedmann branes, which can be asymmetrically embedded into the bulk and can include pullbacks on the brane some non-standard field and geometric configurations, existing in the 5-dimensional bulk [120]. It appears that it is possible to reproduce in this frame work a Swiss cheese Einstein-Strauss model [121]. In this model there pieces of the Schwarzschild regions inserted into a Friedmann universes. At some conditions in the Friedmann regions of such branes the pressure of matter becomes infinite, while the cosmological radius and all its time derivatives remain finite. It was shown also [119] that at some critical value of the assymetry in the embedding of the brane into the bulk, these singularities appear necessarily. It is interesting that these pressure singularities are in a way complementary to the quiescent singularities, discussed above, where the energy density and the pressure are always finite, while the time derivatives of the scale factor become divergent, beginning since the second or some higher-order derivative.</text>
<section_header_level_1><location><page_45><loc_12><loc_40><loc_35><loc_42></location>14. Concluding remarks</section_header_level_1>
<text><location><page_45><loc_12><loc_17><loc_84><loc_38></location>In this review we have considered a broad class of phenomena arising in cosmological models, possessing some exotic cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We have discussed the models, based on standard scalar fields, Born-Infeld-type fields and on perfect fluids, where soft future cosmological singularities exist and are transversable. The crossing of such singularities (or other geometrically peculiar surfaces in the spacetime) can imply such an interesting phenomenon as a transformation of matter properties, which is discussed in some detail here. Another interesting aspect of the study of both soft and 'hard' (Big Bang or Big Crunch) cosmological singularities is the existence of the correspondence between the phenomenon of quantum avoidance (or non-avoidance) of such singularities and the possibility of their crossing (or the absence of such a possibility) in classical cosmology.</text>
<text><location><page_45><loc_12><loc_7><loc_84><loc_16></location>Besides, the quantum cosmological approach, based on the study of the properties of solutions of the Wheeler-DeWitt equation, we have reviewed also some works based on the investigation of the modification of the Friedmann equation due to the quantum corrections and the influence of of these corrections on the structure and the very existence of soft cosmological singularities.</text>
<text><location><page_45><loc_16><loc_5><loc_84><loc_6></location>While the main part of this review deals with the standard Einstein general</text>
<text><location><page_46><loc_12><loc_83><loc_84><loc_89></location>relativity in the presence of non-standard matter, the last section is devoted to the exotic singularities arising in the brane-world cosmological models, which are very close in their nature to the soft sudden singularities arising in the general relativity.</text>
<text><location><page_46><loc_12><loc_63><loc_84><loc_82></location>Generally, we are convinced that the study of exotic singularities in classical and quantum cosmology is a promising branch of the theoretical physics, and nobody can exclude that it can acquire some phenomenological value as well. Here it is necessary to recognize that almost all studies in this field deal only with isotropic and homogeneous Friedmann universes. Thus, the extension of this studies to the anisotropic and inhomogeneous models represents a main challenge for people working in this field. Such an extension can bring some interesting surprises as it was with the study of the Big Bang - Big Crunch singularities, where the consideration of the anisotropic Bianchi models instead of Friedmann models, has given birth to the discovery of the oscillating approach to the singularity (Mixmaster Universe) [6, 7].</text>
<section_header_level_1><location><page_46><loc_12><loc_59><loc_29><loc_60></location>Acknowledgments</section_header_level_1>
<text><location><page_46><loc_12><loc_43><loc_84><loc_57></location>I am grateful to A.M. Akhmeteli, A.A. Andrianov, A.O. Barvinsky, V.A. Belisnky, M. Bouhmadi-Lopez, F. Cannata, S. Cotsakis, M.P. Dabrowski, C. Deffayet, G. Esposito, L.A. Gergely, V. Gorini, D.I. Kazakov, Z. Keresztes, I.M. Khalatnikov, C. Kiefer, V.N. Lukash, S. Manti, P.V. Moniz, U. Moschella, V. Pasquier, D. Polarski, D. Regoli, V.A. Rubakov, B. Sandhofer, D.V. Shirkov, A.A. Starobinsky, O.V. Teryaev, A.V. Toporensky, V. Sahni, G. Venturi, A. Vilenkin and A.V. Yurov for fruitful discussions. This work was partially supported by the RFBR grant 11-02-00643.</text>
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</document>
[
{
"title": "A Yu Kamenshchik",
"content": "Dipartimento di Fisica e Astronomia and INFN, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences, Kosygin str. 2, 119334 Moscow, Russia E-mail: kamenshchik@bo.infn.it Abstract. The development of dark energy models has stimulated interest to cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We review a broad class of phenomena connected with soft cosmological singularities in classical and quantum cosmology. We discuss the classification of singularities from the geometrical point of view and from the point of view of the behaviour of finite size objects, crossing such singularities. We discuss in some detail quantum and classical cosmology of models based on perfect fluids (anti-Chaplygin gas and anti-Chaplygin gas plus dust), of models based on the Born-Infeld-type fields and of the model of a scalar field with a potential inversely proportional to the field itself. We dwell also on the phenomenon of the phantom divide line crossing in the scalar field models with cusped potentials. Then we discuss the Friedmann equations modified by quantum corrections to the effective action of the models under considerations and the influence of such modification on the nature and the existence of soft singularities. We review also quantum cosmology of models, where the initial quantum state of the universe is presented by the density matrix (mixed state). Finally, we discuss the exotic singularities arising in the brane-world cosmological models. Submitted to: CQG",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The problem of cosmological singularities has been attracting the attention of theoreticians working in gravity and cosmology since the early fifties [1, 2, 3]. In the sixties general theorems about the conditions for the appearance of singularities were proven [4, 5] and the oscillatory regime of approaching the singularity [6], called also 'Mixmaster universe' [7] was discovered. Basically, until the end of nineties almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius. However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [8], characterized by a diverging a whereas both the scale factor a and ˙ a are finite. Then, the Hubble parameter H = ˙ a/a and the energy density ρ are also finite, while the first derivative of the Hubble parameter and the pressure p diverge. Until recent years, however, the sudden future singularities attracted rather a limited interest of researchers. The situation has changed drastically in the new millennium, when a plenty publications devoted to such singularities have appeared [9]-[25]. The arising interest to their studies is connected basically with two reasons. The recent discovery of the cosmic acceleration [26] has stimulated the elaboration of dark energy models, responsable for such a phenomenon (see e.g. for review [27]). Remarkably in some of these models the sudden singularities arise quite naturally. Another source of the interest to sudden singularities is the development of brane models [10, 11, 18], where also singularity of this kind arise naturally (sometimes the singularities, arising in the brane models are called 'quiescent' [10]). . In the investigations devoted to sudden singularities one can distinguish three main topics. First of them deals with the question of the compatibility of the models, possessing soft singularities with observational data [15, 28, 29, 25]. The second direction is connected with the study of quantum effects [11, 17, 30, 31, 32, 33, 34, 35, 36]. Here one can see two subdirections: the study of quantum corrections to effective Friedmann equation, which can eliminate classical singualrities or, at least, change their form [10, 17, 30] and the study of solutions of the Wheeler-DeWitt equation for the quantum state of the universe in the presence of sudden singularities [31, 32, 33, 34, 35]. The third direction is connected with the opportunity of the crossing of sudden singularities in classical cosmology [37, 38, 39, 40, 34]. A particular feature of the sudden future singularities is their softness [37]. As the Christoffel symbols depend only on the first derivative of the scale factor, they are regular at these singularities. Hence, the geodesics are well behaved and they can cross the singularity [37]. One can argue that the particles crossing the singularity will generate the geometry of the spacetime, providing in such a way a soft rebirth of the universe after the singularity crossing [40]. Note that the opportunity of crossing of some kind of cosmological singularities were noticed already in the early paper by Tipler [41]. Rather a close idea of integrable singularities in black holes, which can give origin to a cosmogenesis was recently put forward in [42, 43]. Besides, the results of papers [37, 38] were generalized for the case of general (non-Friedmann) universes in papers [44, 45]. For this purpose was used the formalism of the quasi-isotropic expansion of the solutions of the Einstein equations near cosmological singularities, which has been first proposed in [46]. (For some further developments of this formalism, see [47]). The peculiarity of the sudden future singularities makes them to be a good tool for studying some general features of the general relativity, in particular the relations between classical and quantum gravity and cosmology. These relations is the main topic of the present review. We shall also dwell on another aspect of general relativity, which from our point of view is a little bit underestimated. It is the fact the that requirement of the self-consistency of the system of laws of general relativity and particle physics, or, in other words, of the system of Einstein equations and of the equations describing the state or the motion of non-gravitational matter can induce some interesting transformations in the state of matter. Such transformations sometimes occur when the universe passes through the soft singularities, though there are some examples of such transformations which can be observed in the absence of singularities too. We shall consider some of such examples. Generally, this review is devoted to three interrelated topics, which are connected in some way with the soft future singularities in cosmology - these are the problem of crossing of such singularities in classical cosmology, the relations between classical and quantum treatments of cosmological singularities and the changes of state of matter, induced by cosmological singularities or other geometrical irregularities in the framework of general relativity. The structure of the paper is the following. In the second section we shall give a brief and convenient classification of the future singularities, following the paper [48]. In the section 3 we present the classification of the types of singularities from the point of view of the finite objects, which approach these singularities. In the section 4 we introduce the toy tachyon model [49] and shall discuss its basic properties. The section 5 is devoted to the cosmological model based on the mixture of the antiChaplygin gas and to the paradox of soft singularity crossing [50]. In sixth section we consider again the paradox of the soft singularity crossing in the presence of dust and shall discuss its possible resolution by introducing some transformation of matter [51]. In seventh section we shall give another example of the transformation of the Lagrangian of a scalar field due to its interaction with geometry, while its potential is not smooth [52]. The eighth section is devoted to the study of classical dynamics of the cosmological model with a scalar field whose potential is inversely proportional to the field, while in the ninth we study its quantum dynamics. The section 10 is devoted to attempts to apply the formalsm of the Wheeler-DeWitt equation to the study of tachyon and pseudotachyon cosmological models. In eleventh section we study Friedmann equations modified by quantum corrections and possible influence of these corrections on soft cosmological singularities [17],[30]. The section 12 is devoted to developing of such notions as density matrix of the universe, quantum consistency and interplay between geometry and matter in quantum cosmology. In the section 13 we consider singularities arising in some braneworld models, while the section 14 contains some concluding remarks.",
"pages": [
2,
3,
4
]
},
{
"title": "2. Classification of future cosmological singularities",
"content": "In this section we shall present rather a conveninent classification of the future cosmological singularities, following the paper [48]. We shall consider a flat Friedmann universe with the metric where a ( t ) is the cosmological radius (scale factor) and dl 2 is the spatial interval. We shall choose such a normalization of the gravitational constant which provides the follwoing form of the first Friedmann equation is where is the Hubble parameter and ρ is the energy density of the universe. The second Friedmann or Raychaudhuri equation is where p is the pressure. The energy conservation equation looks as We shall write down also the expressions for the non-vanishing components of the Riemann-Christoffel curvature tensor, defined as [1] These nonvanishing components are where α, β are spatial indices; and the corresponding components arising from symmetry. The singularities of the type I are the so called Big Rip singularities [54, 55]. A type I singularity arises at some finite moment of the cosmic time t → t BR , when a → ∞ , ˙ a → ∞ , H → ∞ , ρ → ∞ , | p | → ∞ . These singularities are present in the model, where the cosmological evolution is driven by the so called phantom matter [56], when p < 0 , | p | > ρ or, in other words the equation of state parameter w ≡ p ρ < -1. The conditions of arising and avoiding of such singularities were studied in detail in papers [57, 58, 59]. The singularities of the type II are characterized by the following behaviour of the cosmological parameters: at finite interval of time t = t II a universe arrives with the finite values of the cosmological radius of the time derivative of the cosmological radius, of the Hubble parameter and of the energy density t → t II , a → a II , ˙ a → ˙ a II , H → H II , ρ → ρ II while the acceleration of the universe and the first time derivative of the Hubble parameter tends to minus infinity a → -∞ , ˙ H →-∞ and the pressure tends to plus infinity p → ∞ . A particular case of the type II singularity is the Big Brake singularity, first found in paper [49]. At this singularity, the time derivative of the cosmological radius, the Hubble variable and the energy density are equal exactly to zero. The type III singularities are the singularities occuring when the cosmological radius is finite, while its time derivative, the Hubble variable, the energy density and the pressure are divergent. The examples of such singularities were considered, for example in [9, 60]. The more soft singularities are the singularities of the type IV, at finite value of the cosmological factor both the energy density and the pressure tend to zero and only the higher derivatives of the Hubble parameter H diverge. These singularities sometimes are called Big Separation singularities. In paper [61] the type V singularities were added to the scheme proposed in [48]. These are the singularities which like the singularities of the type IV have the pressure and energy density tending to zero, but the higher time derivatives of the Hubble parameter are regular and only the barotropic index (equation of state parameter) w is singular. Sometimes the traditional Big Bang and Big Crunch singularities are called type 0 singularities, (see [62]). In this review we shall mainly speak about type II singularities and their comparison with type 0 singularities.",
"pages": [
4,
5
]
},
{
"title": "3. The type of the singularity from the point of view of finite size objects, which approach these singularities",
"content": "In this section we shall present the classification of singularities, based on the point of view of finite size objects, which approach these singularities. In principle, finite size objects could be destroyed while passing through the singularity due to the occurring infinite tidal forces. A strong curvature singularity is defined by the requirement that an extended finite object is crushed to zero volume by tidal forces. We give below Tipler's [41] and Kr'olak's [63] definitions of strong curvature singularities together with the relative necessary and sufficient conditions. First of all we shall write down the geodesics deviation equation. If u i are fourvelocities of test particles and η i is a four-vector separating two spatially close geodesics, then the dynamics of this vector is given by the equation [1] where D is the covariant derivative along a geodesics. In the case of a flat Friedmann universe (1) for the geodesics of particles, having zero spatial velocities (i.e u α = 0 , u t = 1) Eq. (9) acquires, taking into account Eq. (7), a simple form Looking at the above equation one can see that approaching a singularity, chracterized by an infinite value of the deceleration, we experience an infinite force, stopping the farther increase of the separation of geodesics, while geodesics themselves can be quite regular if the the velocity of expansion ˙ a is regular. According to Tipler's definition if every volume element, defined by three linearly independent, vorticity-free, geodesic deviation vectors along every causal geodesic through a point P , vanishes, a strong curvature singularity is encountered at the respective point P [41], [37]. The necessary and sufficient condition for a causal geodesic to run into a strong singularity at λ s ( λ is affine parameter of the curve) [64] is that the double integral diverges as λ → λ s . A similar condition is valid for lightlike geodesics, with R i ajb u a u b replacing R ab u a u b in the double integral. Kr'olak's definition is less restrictive. A future-endless, future-incomplete null (timelike) geodesic γ is said to terminate in the future at a strong curvature singularity if, for each point P ∈ γ , the expansion of every future-directed congruence of null (timelike) geodesics emanating from P and containing γ becomes negative somewhere on γ [63], [65]. The necessary and sufficient condition for a causal geodesic to run into a strong singularity at λ s [64] is that the integral ∣ ∣ diverges as λ → λ s . Again, a similar condition is valid for lightlike geodesics, with R i ajb u a u b replacing R ab u a u b in the integral. We conclude this section by mentioning that the singularities of the types 0 and I are strong and the singularities of the types II, IV and V are week according to both the definitions (Tipler's and Kr'olak's ones), while the type III singularities are strong with respect to Kr'olak's definition and week with respect to Tipler's definition [62]. The weekness of the type II singularities, which we shall study in some details in the next sections of the present review, according to both the definitions, means that although the tidal forces become infinite, the finite objects are not necessarily crushed when reaching the singularity.",
"pages": [
5,
6
]
},
{
"title": "4. The tachyon cosmological model with the trigonometric potential",
"content": "The tachyon field, born in the context of the string theory [66], provides an example of matter having a large enough negative pressure to produce an acceleration of the expansion rate of the universe. Such a field is today considered as one of the possible candidates for the role of dark energy and, also for this reason, in the recent years it has been intensively studied. The tachyon models represent a subclass of the models with non-standard kinetic terms [67], which descend from the Born-Infeld model, invented already in thirties [68]. Before considering the model with the trigonometric potential [49], possessing the Big Brake singularity, we write down the general formulae of the tachyon cosmology. The Lagrangian of the tachyon field T is or, for the spatially homogeneous tachyon field, The energy density and the pressure of this field are respectively and while the field equation is We shall introduce also the pseudo-tachyon field with the Lagrangian [49] and with the energy density and the pressure The Klein-Gordon equation for the pseudo-tachyon field is We shall also write down the equations for the time derivative of the Hubble parameter in the tachyon and pseudo-tachyons models: Note that for the case when the potential of the tachyon field V ( T ) is a constant, the cosmological model with this tachyon coincides with the cosmological model with the Chaplygin gas [69]. The Chaplygin gas is the perfect fluid, satisfying the equation of state The cosmological model based on the Chaplygin gas was introduced in [70] and has acquired some popularity as a unified model of dark matter and dark energy [71]. Analogously, the pseudo-tachyon model with the constant potential coincides with the model with a perfect fluid, whose equation of state is . This fluid can be called 'anti-Chaplygin gas'. The corresponding model was introduced in [49] and we shall come back to it later. Curiosly, similar equation of motion arises in the theory of wiggly strings [72]. Now we shall study a very particular tachyon potential depending on the trigonometrical functions which was suggested in the paper [49]. Its form is The origin of the potential (26) is the following one: let us consider a flat Friedmann universe filled with two fluids, one of which is a cosmological constant with the equation of state p = -ρ = -Λ and the second one is a barotropic fluid with the equation of state p = kρ . The Friedmann equation for such a model is exactly solvable and gives where Λ is a positive constant and k is a parameter, which is chosen in the interval -1 < k < 1. The case of the positive values of the parameter k is especially interesting. The set of possible cosmological evolutions, is graphically presented in Figure 1, which is the phase portrait of our dynamical system, where the ordinate s is the time derivative of the tachyon field T : s ≡ ˙ T . Then using the standard technique of the reconstruction of potentials, which was mainly used for the minimally coupled scalar field [73], but was easily generalized for the cases of non-minimally coupled fields [85, 74] and for tachyons [75, 76, 49, 77] we obtain the expression (26). It is necessary to emphasize that the dynamics of the tachyon model with the potential (26) is much richer than the dynamics of the two fluid model with the unique cosmological evolution given by the expression (27). In paper [49] both the cases k ≤ 0 and k > 0 were considered. The case k > 0 is of a particular interest, because it reveals two unusual phenomena: a self-transformation of the tachyon into a pseudotachyon field and the appearance of the Big Brake cosmological singularity. Let us discuss briefly the classical dynamics of the model with the trigonometric potential for the case k > 0. It is easy to see that the potential (26) is well defined at T 3 ≤ T ≤ T 4 , where In turn, the kinetic term √ 1 -˙ T 2 is well defined at -1 ≤ s ≤ 1. In other words, the Lagrangian (14) with the potential (26) is well defined inside the rectangle (see Fig. 1). The analysis of the dynamics of the equation of motion of the tachyon (17) and of the Friedmann equations shows that a part of the trajectories end their evolution in the attractive node with the coordinates T 0 = π 3 √ (1+ k )Λ , s 0 = 0, which describes an infinite de Sitter expansion. The upper and lower borders of the rectangle s = 1 , s = -1, excluding the corner points, are the standard Big Bang cosmological singularities, while left and right borders T = T 3 and T = T 4 repel the trajectories. However, another part of the trajectories goes towards the corner points ( T = T 3 , s = -1) and ( T = T 4 , s = 1). These points are regular points from the point of view of the equations of motion of the corresponding dynamical system and besides, the direct calculation shows that there are no cosmological singularities there. Thus, there is no reason which prevents further evolution of the universe through these points. Indeed, one can see also that the equations of motion and their solutions can be continued into the vertical stripes (see Fig. 1). However, to reproduce these equations of motion in the stripes as Euler-Lagrange equations, we should substitute the tachyon Lagrangian (14) by the preudotachyon Lagrangian (19) with the potential Thus, we have seen already the first unusual phenomenon - the self-transformation of the tachyon into the pseudotachyon field. Now, the question arises: what happens with the universe after the 'crossing the corner' and the transformation of the tachyon into the pseudotachyon ? The analysis of equations of motion carried out in paper [49] shows that the universe in a finite moment of time t = t BB encounter the singularity, which is characterized by the following values of cosmological parameters: This singularity was called Big Brake singularity [49]. Obviously, it enters into the class II of singularities, according to the classification suggested in paper [48] and recapitulated in Sec. 2 of the present review. Now, it is interesting to confront the prediction of this, a little bit artificial, but rather rich model with the observational data coming from the luminosity-redshift relation from Supernovae of type Ia. Such an attempt was undertaken in paper [29], where the set of supernovae studied in [78] was used. The strategy was the following : there were scanned the pairs of present values of the tachyon field and of its time derivative (points in phase space) and then they were propagated backwards in time, comparing corresponding luminosity distance - redshift curves with the observational data from SNIa. Then, those pairs of values which appeared to be compatible with the data were chosen as initial conditions for the future cosmological evolution. Though the constraints imposed by the data were rather severe, both evolutions took place: one very similar to ΛCDM and ending in an exponential (de Sitter) expansion; another with the transformation of the tachyon into the pseudotachyon and the successive running towards the Big Brake singularity. It was found that a larger value of the model parameter k enhances the probability to evolve into a Big Brake. The time intervals until the future encounter with the Big Brake were calculated and were found to be compatible with the present age of the universe [29]. The next question, which arises, is the fate of the universe after the encounter with the Big Brake singularity. As was already told above, this singularity is very soft and the geodesics can be continued across it. Then the matter, passing through the Big Brake singularity reconstructs the spacetime. This process was studied in some detail in paper [40]. The analysis of the equation of motion for the universe approaching the Big Brake singularity gives the following expressions for the basic quantities: The expressions (32)-(36) can be continued into the region where t > t BB , which amounts to crossing the Big Bang singularity. Only the expression for s is singular at t = t BB , but this singularity is integrable and not dangerous. Upon reaching the Big Brake, it is impossible for the system to stop there because the infinite deceleration leads to the decrease of the scale factor. This is because after the Big Brake crossing the time derivative of the cosmological radius (35) and of the Hubble variable (36) change their signs. The expansion is then followed by a contraction. Corresponding to given initial conditions, the values of T BB , t BB and a BB were found numerically. Then the numerical integration of the equations of motion describes the contraction of the universe, culminating in the encounter with the Big Crunch singularity. Curiously, the time intervals between the Big Brake and Big Crunch singularities practically do not depend on the initial conditions and are equal approximately to 0 . 3 × 10 9 yrs [40]. Now, the next question arises: what happens if we consider a little bit more complicated model, adding to the tachyon matter some quantity of dust-like matter ? Obviously, in this case instead of the Big Brake singularity the universe will encounter a soft type II singularity of a more general kind. Namely, due to the presence of dust, the energy density of the expanding universe cannot vanish and, hence, at the moment when the universe experiences an infinite deceleration its expansion should continue. This implies the appearance of some kind of contradictions, which can be resolved by transformation of the pseudotachyon field into another kind of Born-Infeld like field. The corresponding problem was considered in detail in papers [50, 51]. The first of these papers was devoted to a more simple model, based on mixture of the anti-Chaplygin gas with dust. The next section will be devoted to this model.",
"pages": [
7,
8,
9,
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11,
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]
},
{
"title": "5. The cosmological model based on the mixture of the anti-Chaplygin gas and the paradox of soft singularity crossing",
"content": "The anti-Chaplygin gas with the equation of state (25) is one of the simplest cosmological models revealing the Big Brake singularity [49]. Indeed, combining the equation of state (25) with the energy conservation equation (5), one obtains immediately where B is a positive constant, characterizing the initial condition. Then, when in the process of the cosmological expansion the cosmological radius a arrives to the critical value the energy density of the universe vanishes while the pressure tends to infinity. Thus, the universe encounters the Big Brake singularity. Then, it begins contraction culminating in the encounter with the Big Crunch singularity. Now, let us see what happens if we add some amount of dust with the energy density where ρ 0 is a positive constant. In this case the traversability of the singularity seems to be obstructed. The main reason for this is that while the energy density of the antiChaplygin gas vanishes at the singularity, the energy density of the matter component does not, leaving the Hubble parameter at the singularity with a finite value. Then some kind of the paradox arises: if the universe continues its expansion, and if the equation of state of the component of matter, responsible for the appearance of the soft singularity (in the simplest case, the anti-Chaplygin gas) is unchanged, then the expression for the energy density of this component becomes imaginary, which is unacceptable. The situation looks rather strange: indeed, the model, including dust should be in some sense more regular, that that, containing only such an exotic fluid as the anti-Chaplygin gas. Thus, if the model, based on the pure anti-Chaplygin gas has a traversable Big Brake singularity, than the more general singularity arising in the model, based on the mixture of the anti-Chaplygin gas and dust should also be transversable. A possible way of resolution of this paradox, based on use of the distributional cosmological quantities was suggested in paper [50]. Let us suppose that at the moment of the crossing of the soft cosmological singularity the expansion of the universe with the Hubble parameter H is abruptly substituted by the cosmological contraction with the Hubble parameter -H . In this case, the value of the cosmological radius a begins decreasing and the expression (37) for the energy density just like the corresponding expression for the pressure remain well defined. The first Friedmann equation (2) and the energy conservation equation (5) remain also intact. A problem, however, arises with the second Friedmann equation (4). Let us rewrite this equation in the form If the Hubble parameter abruptly changes sign at the moment t = t S that means that it contains the term where θ ( x ) is the Heaviside theta function. The derivative of the theta function is equal in the distributional sense to the Dirac delta function (see e.g. [79]). Hence, the lefthand side of Eq. (40) contains the Dirac delta function. Now, let us discuss in more detail the expressions for the Hubble parameter and its time derivative in the vicinity of the singularity. The leading terms of the expression for H ( t ) are where sgn ( x ) ≡ θ ( x ) -θ ( -x ). Then Naturally, the δ -term in ˙ H arises because of the jump in H , as the expansion of the universe is followed by a contraction. To restore the validity of the second Friedmann equation (40) we shall add a singular δ -term to the pressure of the anti-Chaplygin gas, which will acquire the form The equation of state of the anti-Chaplygin gas is preserved, if we also modify the expression for its energy density: The last expression should be understood in the sense of the composition of distributions (see Appendix A of the paper [50] and references therein). In order to prove that p ACh and ρ ACh represent a self-consistent solution of the system of cosmological equations, we shall use the following distributional identities: Here g ( | τ | ) is bounded on every finite interval, f ( τ ) > 0 and C > 0 is a constant. These identities were proven in paper [50], where was used the approach to the product and the composition of distributions developed in papers [80]. /negationslash /negationslash Due to Eqs. (47)-(48), ρ ACh vanishes at the singularity while still being continuous. The first term in the expression for the pressure (44) diverges at the singularity. Therefore the addition of a Dirac delta term, which is not changing the value of p ACh at any τ = 0 (i.e. t = t S ) does not look too drastic and might be considered as a some kind of renormalization. To prove that the first and the second Friedmann equations and the continuity equation are satisfied we must only investigate those terms, appearing in the field equations, which contain Dirac δ -functions. First, we check the continuity equation for the anti-Chaplygin gas. Due to the identities (47)-(48), the δ ( τ )-terms occurring in ρ ACh and ˙ ρ ACh could be dropped. We keep them however in order to have the equation of state explicitly satisfied. Then the δ ( τ )-term appearing in 3 Hp ACh vanishes, because the Hubble parameter changes sign at the singularity (see Eq. (46)). The δ ( τ )-term appearing in ρ ACh does not affect the Friedmann equation due to the identity (47). Finally, the δ -term arising in the time derivative of the Hubble parameter in the left-hand side of the Raychaudhuri equation is compensated by the conveniently chosen δ -term in the right-hand side of Eq. (44). However, the mathematically self-consistent scenario, based on the use of generalized functions and on the abrupt change of the expansion into a contraction, looks rather counter-intuitive from the physical point of view. Such a behaviour can be compared with the absolutely elastic bounce of a ball from a rigid wall, as studied in classical mechanics. In the latter case the velocity and the momentum of the ball change their direction abruptly. Hence, an infinite force acts from the wall onto the ball during an infinitely small interval of time. In reality, the absolutely elastic bounce is an idealization of a process of finite timespan during which inelastic deformations of the ball and of the wall occur. Thus, the continuity of the kinematics of the act of bounce implies a more complex and realistic description of the dynamical process of interaction between the ball and the wall. It is reasonable to think that something similar occurs also in the models, including dust and an anti-Chaplygin gas or a tachyon. The smoothing of the process of a transition from an expanding to a contracting phase should include some (temporary) geometrically induced change of the equation of state of matter or of the form of the Lagrangian. We know that such changes do exist in cosmology. In the tachyon model [49], there was the tachyon-pseudotachyon transformation driven by the continuity of the cosmological evolution. In a cosmological model with the phantom filed with a cusped potential [52, 53], the transformations between phantom and standard scalar field were considered. Thus, it is quite natural that the process of crossing of the soft singularity should imply similar transformations. However, now the situation is more complicated. It is not enough to require the continuity of evolution of the cosmological radius and of the Hubble parameter. It is necessary also to accept some hypothesis concerning the fate of the change of the equation of state of matter or of the form of the Lagrangian. This problem will be considered in the next section.",
"pages": [
12,
13,
14,
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]
},
{
"title": "6. Paradox of soft singularity crossing and its resolution due to transformations of matter",
"content": "The strategy of the analysis of the the problem of soft singularity crossing in this section is the following [51]. First, we shall consider the model with the anti-Chaplygin gas and dust. We shall require a minimality of the change of the form of the dependence of the energy density and of the pressure, compatible with the continuation of the expansion while crossing the soft singularity. Such a requirement will bring us to the substitution of the anti-Chaplygin gas with the Chaplygin gas with a negative energy density. (Note, that in another context the Chaplygin gas with a negative energy density was considered in paper [81]). Then we shall consider the cosmological model based on the pseudotachyon field with a constant potential and dust. It is known that the energy-momentum tensor for such a pseudotachyon field coincides with that of the anti-Chaplygin gas (this fact relating the Chaplygin gas and the tachyon field with a constant potential was found in paper [69]). Thus, we would like to derive the form of the transformation of the pseudotachyon Lagrangian using its kinship with the antiChaplygin gas. As a result, we shall come to a new type of the Lagrangian, belonging to the 'Born-Infeld family'. Finally, we shall extend the found form of transformation of the pseudotachyon field for the case of the field with the trigonometric potential. As follows from Eqs. (25) and (37) the pressure of the anti-Chaplygin gas and it tends to + ∞ when the universe approaches the soft singularity, when the cosmological radius a → a S (see Eq. (38)). If we would like to continue the expansion into the region a > a S , while changing minimally the equation of state we can require or, in other words, We see that in some 'generalized sense' we conserve the continuity of the pressure crossing the soft singularity. It passes + ∞ conserving its sign. Combining the expression (51) with the energy conservation law (5) we obtain Thus, the energy density is also continuous passing through its vanishing value and changing its sign. It is easy to see that the energy density (52) and the pressure (51) satisfy the Chaplygin gas equation of state Thus, we have seen the transformation of the anti-Chaplygin gas into the Chaplygin gas with a negative energy density. The Friedmann equation after the crossing the singularity is It follows immediately from Eq. (54) that after achieving the point of maximal expansion a = a max , where The universe begins contracting. When the contracting universe arrives to a = a S it again stumbles upon a soft singularity and the Chaplygin gas transforms itself into the anti-Chaplygin gas with positive energy density and the contraction continues until hitting the Big Crunch singularity. Remember that in the preceding section and in paper [50], the process was described when the universe passed from the expanding to the collapsing phase instantaneously at the singularity causing a jump in the Hubble parameter. Here we showed that the continuos transition to the collapsing phase is possible if the equation of state of antiChaplygin gas has a some kind of a 'phase transition' at the singularity. When the potential of the pseudotachyon field is constant, W ( T ) = W 0 , then the energy density (19) and the pressure (20) satisfy the anti-Chaplygin gas equation of state (25) with Solving the equation of motion for the pseudotachyon field (21) with W ( T ) = W 0 one finds and we see that the soft singularity arises at a = a S , when ˙ T 2 → + ∞ . Now, we would like to change the Lagrangian (18) in such a way that the new Lagrangian gives us the energy density and the pressure satisfying the Chaplygin gas equation with a negative energy density. It is easy to check that the Lagrangian giving and is what we are looking for. Note, that the energy density and the pressure, passing through the singularity are continuous in the same sense in which they were continuos in the case of the antiChaplygin gas. Thus, we have introduced a new type of the Born-Infeld field, which can be called 'anti-tachyon'. Generally, its Lagrangian is and the equation of motion is For the case W ( T ) = W 0 , the solution of equation (62) is and the energy density evolves as and the evolution of the universe repeats that for the model with the anti-Chaplygin gas and dust. Let us emphasize once again that to the transformation from the anti-Chaplygin gas to the Chaplygin gas corresponds to the transition from the prseudotachyon field with the Lagrangian (18) to the new type of the Born-Infeld field, which we can call 'quasi-tachyon field' with the Lagrangian (61). Now, we shall consider the case of the toy model with the trigonometric potential in the presence of dust. We have seen that the Born-Infeld type pseudotachyon field runs into a soft Big Brake singularity with the expansion of the universe in this model. However, what happens in the presence of dust component? Does the universe still run into soft singularity? To answer this question rewrite Eq. (21) as It is easy to see that in the left lower and in the right upper stripes (see Fig. 1), where the trajectories describe the expansion of the universe after the transformation of the tachyon into the pseudotachyon field, the signs of T , of ˙ T and of the term W ,T W coincide. The detailed analysis based on this fact was carried out in paper [49] and led to the conclusion that the universe encounters the singularity as T → T S ( T S > 0 or T S > T max ) , | ˙ T | → ∞ . The presence of dust cannot alter this effect because it increases the influence of the term 3 H ˙ T , and hence, accelerates the encounter with the singularity. However, the presence of dust changes in an essential way the time dependence of the tachyon field close to the singularity. As it was shown in [40] (see also the Sec. 4 of the present paper) while in the presence of dust one has where H S is the nonvanishing value of the Hubble parameter given by It is easy to see that the smooth continuation of the expression (67) is impossible in contrast to the situation without dust (66) considered in [40]. Thus, the presence of dust is responsible for the appearance of similar paradoxes in both the anti-Chaplygin gas and tachyon models. In the vicinity of the soft singularity, it is the 'friction' term 3 H ˙ T in the equation of motion (21) , which dominates over the potential term W ,T W , hence, the dependence of W ( T ) is not essential and a pseudotachyon field approaching this singularity behaves like one with a constant potential. Thus, it is quite reasonable to suppose that crossing of the soft singularity the pseudotachyon transforms itself into the quasi-tachyon with the Lagrangian (61). Now, we can analyze the dynamics of the anti-tachyon field, driven by the equation of motion (62) and by the Friedmann equation, where the right-hand side includes the dust contribution and the anti-tachyon energy density √ It is convenient to consider the processes developing in the left lower strip of the phase diagram of the model to facilitate the comparison with the earlier studies of the dynamics of the tachyon model without dust, undertaken in papers [49, 40]. One can see that the relative sign of the term with the second derivative T with respect to the friction term 3 H ˙ T are oppostite for the pseudotachyons and anti-tachyons. That means that after the crossing of the soft singularity the time derivative ˙ T is growing and its absolute value is diminishing. At the same time the value of the field T is diminishing and the value of the potential W ( T ) is growing. That means that the absolute value of the negative contribution to the energy density of the universe coming from the quasi-tachyon is growing while the energy density of the dust is diminishing due to the expansion of the universe. At some moment this process brings us to the vanishing value of the general energy density and we arrive to the point of maximal expansion of the universe. After that the expansion is replaced by the contraction and the Hubble variable changes sign. The change of sign of the friction term 3 H ˙ T implies the diminishing of the value of ˙ T and at some finite moment of time the universe again encounters the soft singularity when ˙ T → -∞ . Passing this singularity the quasitachyon transforms itself back to the pseudotachyon and the relative sign of the terms with the second and first time derivatives in the equation of motion for this field changes once again. After that the time derivative of the pseudotachyon field begins growing and the universe continues its contraction until it encounters with the Big Crunch singularity. It was shown in paper [40] that for the case of the purely tachyon model with the trigonometric potential the encounter of the universe with the Big Crunch singularity occurs at T = 0 and ˙ T = -√ 1+ k k . One can show that the presence of dust does not change these values. Indeed, let us consider the behavior of the pseudotachyon field when T → 0 , ˙ T →-√ 1+ k k . It follows from the expressions (19) and (20) that the ratio between the pressure and the energy density behaves as i.e. in the vicinity of the Big Crunch singularity the pseudotachyon field behaves as a barotropic fluid with the the equation of state parameter 1 k > 1. That means that the energy density of the pseudotachyon field is growing as as a → 0, i.e. much more rapidly than the dust energy density. Thus, one can neglect the contribution of the dust in this regime of approaching the Big Crunch singularity and the description of the evolution of the universe to this point coincides with that of the pure tachyon model [40].",
"pages": [
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},
{
"title": "7. The transformations of the Lagrangian of a scalar field with a cusped potential",
"content": "It is well known that the cosmological observations gives as a best fit for the equation of state parameter w = fracpρ a value which is slightly inferior with respect to -1 (see, e.g. [82]). The corresponding type of dark energy was called 'phantom' matter [56]. Wanting to realize such a dark matter using a minimally coupled scalar field, one has to introduce for the latter a negative kinetic term. Thus, its Lagrangian has the form Some observations also indicate that the value of the equation of state parameter at some moment in the past has crossed the value w = -1, corresponding to the cosmological constant. Such a phenomenon has received the name of 'phantom divide line crossing' [83]. A minimally coupled scalar field, describing non phantom dark energy has a kinetic term with the positive sign. So, it looks natural, to use two scalar fields, a phantom field with the negative kinetic term and a standard one to describe the phantom divide line crossing [84]. Another posssible way of the phantom divide line crossing, using s scalar field nonminimally coupled to gravity was considered in papers [85]. However, in papers [52, 53] it was shown that considering potentials with cusps and choosing some particular initial conditions, one can describe the phenomenon of the phantom divide line crossing in the model with one minimally coupled scalar field. Curiously, a passage through the maximum point of the evolution of the Hubble parameter implies the change of sign of the kinetic term. Though a cosmological singularity is absent in these cases, this phenomenon is a close relative of those, considered in the preceding sections, because here also we stumble upon some transformation of matter properties, induced by a change of geometry. One can add that in this aspect the phenomenon of the phantom divide line crossing is the close analog of the transformation between the tachyon and pseudo-tachyon model, in the tachyon model with the trigonometric potential, described in Sec. 4. Here, we shall present a brief sketch of the ideas, described in papers [52, 53], emphasizing the analogy and the differences between different geometrically induced matter transformations. We begin with a simple mechanical analog: a particle moving in a potential with a cusp [52]. Let us consider a one-dimensional problem of a classical point particle moving in the potential where V 0 > 0. The equation of motion is We consider three classes of possible motions characterized by the value of the energy E . The first class consists of the motions when E < V 0 . Apparently, the particle with x < 0 , ˙ x > 0 or with x > 0 , ˙ x < 0 cannot reach the point x = 0 and stops at the points ∓ (√ V 0 E -1) ) 3 / 2 respectively. The second class includes the trajectories when E > V 0 . In this case the particle crosses the point x = 0 with nonvanishing velocity. If we have a fine tuning such that E = V 0 , we encounter an exceptional case. Now the trajectory satisfying Eq. (74) in the vicinity of the point x = 0 can behave as where and t ≤ t 0 . It is easy to see that independently of the sign of C in Eq. (76) the signs of the particle coordinate x and of its velocity ˙ x are opposite and hence, the particle can arrive in finite time to the point of the cusp of the potential x = 0. Another solution reads as where t ≥ t 0 . This solution describes the particle going away from the point x = 0. Thus, we can combine the branches of the solutions (75) and (77) in four different manners and there is no way to choose if the particle arriving to the point x = 0 should go back or should pass the cusp of the potential (73). It can stop at the top as well. Such a 'degenerate' behaviour of the particle in this third case is connected with the fact that this trajectory is the separatrix between two one-parameter families of trajectories described above. At the moment there is not yet any strict analogy between this separatrix and the cosmological evolution describing the phantom divide line. In order to establish a closer analogy and to understand what is the crucial difference between mechanical consideration and general relativistic one, we can try to introduce a friction term into the Newton equation (74) It is easy to check that, if the friction coefficient γ is a constant, one does not have a qualitative change in respect to the discussion above. Let us asuume for γ the dependence then and just like in the cosmological case, where the role of the friction coefficient is played by the Hubble parameter. The trajectory arriving to the cusp with vanishing velocity is still described by the solution (75). Consider the particle coming to the cusp from the left ( C < 0. It is easy to see that the value of ˙ γ at the moment t 0 tends to zero, while its second derivative ¨ γ given by Eq. (81) is Thus, it looks like the friction coefficient γ reaches its minimum value at t = t 0 . Let us suppose now that the particle is coming back to the left from the cusp and its motion is described by Eq. (77) with negative C . A simple check shows that in this case Thus, from the point of view of the subsequent evolution this point looks as a maximum for the function γ ( t ). In fact, it means simply that the second derivative of the friction coefficient has a jump at the point t = t 0 . It is easy to check that if instead of choosing the motion to the left, we shall move forward our particle to the right from the cusp ( C > 0), the sign of ¨ γ ( t 0 ) remains negative as in Eq. (83) and hence we have the jump of this second derivative again. If one would like to avoid this jump, one should try to change the sign in Eq. (81). To implement it in a self-consistent way one can substitute Eq. (79) by and Eq. (78) by In fact, it is exactly that what happens automatically in cosmology, when we change the sign of the kinetic energy term for the scalar field, crossing the phantom divide line. Naturally, in cosmology the role of γ is played by the Hubble variable H . The jump of the second derivative of the friction coefficient γ corresponds to the divergence of the third time derivative of the Hubble variable, which represents some kind of very soft cosmological singularity. Thus, one seems to confront the problem of choosing between two alternatives: 1) to encounter a weak singularity in the spacetime geometry; 2) to change the sign of the kinetic term for matter field. We have pursued the second alternative insofar as we privilege the smoothness of spacetime geometry and consider equations of motion for matter as less fundamental than the Einstein equations. Now, we would like to say that the potential, considered in papers [52, 53] had the general structure The origin of this structure is the following: one considers the power law expansion of the universe, it is well-known that such an expansion could be provided by an exponential potential [86]. Then one can represent the Friedmann equation for the evolution of the scale factor of the universe as a second-order linear differential equation, where the potential is reperesented as a function of the time parameter [87]. This equation has two independente solutions: one of them is the power-law expansion and other corresponds to an evolution driven by a phantom matter. The linear combination of these two solutions with both nonvanishing coefficients gives an evolution, where a universe crosses the phantom divide line. It is impossible to reconstruct the form of the potential as a function of the scalar field, which provides such an evolution explicitly, however, one can study its form around the point where the phantom divide crossing occur and this form is exactly that of Eq. (86) [52]. At the end of this section, we would like to say that in the Newtonian mechanics there is rather a realistic example of motion when, the dependence of the distance of time is given by some fractional power [88, 89]. Indeed, if one consider the motion of a car with a constant power (which is more realistic than the motion with a constant force, usually presented in textbooks), when the velocity behaves as t 1 / 2 and if the initial value of the coordinate and of the velocity are equal to zero, when the acceleration behaves as t -1 / 2 and at the moment of start is singular. The motion at constant power is an excellent model of drag-car racing [88, 89]. Its analogy with the cosmology at the presence of sudden singularities was noticed in paper [45].",
"pages": [
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},
{
"title": "8. Classical dynamics of the cosmological model with a scalar field whose potential is inversely proportional to the field",
"content": "We have considered earlier the simplest model, possessing a soft cosmological singularity (Big Brake) - the model based on the anti-Chaplygin gas. It was noticd that this model is equivalent to the model with the pseudotachyon field with constant potential. Here we would like to study a model, based on a minimally coupled scalar field, which possesses the same evolution as the model based on the anti-Chaplygin gas. Using the standard technique of the reconstruction of potential, the potential of the corresponding scalar field was found in paper [32] and it looks like As a matter of fact we have two possible potentials, which differs by the general sign. We choose the sign 'plus'. Then, let us remember that the Big Brake occurs when the energy density is equal to zero (the disappearance of the Hubble parameter) and the pressure is positive and infinite (an infinite deceleration). To achieve this condition, in the scalar field model it is necessary to require that the potential is negative and infinite. It is easy to see from Eq. (87) that this occurs when ϕ → 0 being positive. Thus, to have the model with the Big Brake singularity we can consider the scalar field with a potential which is a little bit simpler than that from Eq. (87), but still possesses rather a rich dynamics. Namely we shall study the scalar field with the potential where V 0 is a positive constant. The Klein-Gordon equation for the scalar field with the potential (88) is while the first Friedmann equation is We shall also need the expression for the time derivative of the Hubble parameter, which can be easily obtained from Eqs. (89) and (90): Now we shall construct the complete classification of the cosmological evolutions (trajectories) of our model, using Eqs. (89)-(91) [34]. First of all, let us announce briefly the main results of our analysis. To prove these results, we begin with the consideration of the universe in the vicinity of the point ϕ = 0. We shall look for the leading term of the field ϕ approaching this point in the form where ϕ 1 and α are positive constants and t S is the moment of the soft singularity crossing. The time derivative of the scalar field is now Because of the negativity of the potential (88) at positive values of ϕ , the kinetic term should be stronger than the potential one to satisfy the Friedmann equation (4). That implies that α ≤ 2 3 . However, if α < 2 3 we can neglect the potential term and remain with the massless scalar field. It is easy to show considering the Friedmann (4) and KleinGordon (89) equations that in this case the scalar field behaves like ϕ ∼ ln( t S -t ), which is incompatible with the hypothesis of its smallness (92). Thus, one remains with the only choice Then, if the coefficient at the leading term in the kinetic energy is greater than that in the potential, it follows from the Friedmann equation (4) that the Hubble parameter behaves as ( t S -t ) -1 3 which is incompatible with Eq. (91). Thus, the leading terms of the potential and kinetic energy should cancel each other: that for α = 2 3 gives Hence, the leading term for the scalar field in the presence of the soft singularity is Now, integrating Eq. (91) we obtain /negationslash where H S is an integration constant giving the value of the Hubble parameter at the moment of the soft singularity crossing. If this constant is equal to zero, H S = 0, the moment of the maximal expansion of the universe coincides with that of the soft singularity crossing and the universe encounters the Big Brake singularity. If H S = 0 we have a more general type of the soft cosmological singularity where the energy density of the matter in the universe is different from zero. The sign of H S can be both, positive or negative, hence, universe can pass through this singularity in the phase of its expansion or of its contraction. The form of the leading term for the scalar field in the vicinity of the moment when ϕ = 0 (97) shows that, after passing the zero value, the scalar field begin growing being positive. Thus, it proves the first result from the list presented above about impossibility of the change of the sign of the scalar field in our model. /negationslash We have already noted that the time derivative of the scalar field had changed the sign crossing the soft singularity. It cannot change the sign in a non-singular way because the conditions ˙ ϕ ( t 0 ) = 0 , ϕ ( t 0 ) = 0 are incompatible with the Friedmann equation (4). It is seen from Eq. (97) that before the crossing of the soft singularity the time derivative of the scalar field is negative and after its crossing it is positive. The impossibility of the changing the sign of the time derivative of the scalar field without the soft singularity crossing implies the inevitability of the approaching of the universe to this soft singularity. Thus, the third result from the list above is proven. It is easy to see from Eq. (91) that the value of the Hubble parameter is decreasing during all the evolution. At the same time, the absolute value of its time derivative (proportional to the time derivative squared of the scalar field) is growing after the soft singularity crossing. That means that at some moment the Hubble parameter should change its sign becoming negative. The change of the sign of the Hubble parameter is nothing but the passing through the point of the maximal expansion of the universe, after which it begin contraction culminating in the encounter with the Big Crunch singularity. Thus the second result from the list presented above is proven. Summing up, we can say that all the cosmological evolutions where the scalar field has positive values have the following structure: they begin in the Big Bang singularity with an infinite positive value of the scalar field and an infinite negative value of its time derivative, then they pass through the soft singularity where the value of the scalar field is equal to zero and where the derivative of the scalar field changes its sign. All the trajectories also pass through the point of the maximal expansion, and this passage trough the point of the maximal expansion can precede or follow the passage trough the soft singularity: in the case when these two moments coincide ( H S = 0) we have the Big Brake singularity (see the result 4 from the list above). Thus, all the evolutions pass through the soft singularity, but only for one of them this singularity has a character of the Big Brake singularity. The family of the trajectories can be parameterized by the value of the Hubble parameter H S at the moment of the crossing of the soft singularity. There is also another natural parameterization of this family - we can characterize a trajectory by the value of the scalar field ϕ at the moment of the maximal expansion of the universe and by the sign of its time derivative at this moment (if the time derivative of the scalar field is negative that means that the passing through the point of maximal expansion precedes the passing through the soft singularity and if the sign of this time derivative is positive, then passage trough the point of maximal expansion follows the passage through the soft singularity). If at the moment when the universe achieves the point of maximal expansion the value of the scalar field is equal to zero, then it is the exceptional trajectory crossing the Big Brake singularity. For completeness, we shall say some words about the result 5, concerning the trajectories with the negative values of the scalar field. Now, both the terms in the right-hand side of the Friedmann equation (4), potential and kinetic, are positive and, hence, the Hubble parameter cannot disappear or change its sign. It can only tends to zero asymptotically while both these terms tend asymptotically to zero. Thus, in this case there are two possible regimes: an infinite expansion which begins with the Big Bang singularity and an infinite contraction which culminates in the encounter with the Big Crunch singularity. The second regime can be obtained by the time reversal of the first one and vice versa. Let us consider the expansion regime. It is easy to check that the scalar field being negative cannot achieve the zero value, because the suggestion ϕ ( t ) = -ϕ 1 ( t 0 -t ) α , where ϕ 1 < 0 , α > 0 is incompatible with the equations (4) and (91). Hence, the potential term is always non-singular and at the birth of the universe from the Big Bang singularity the kinetic term dominates and the dynamics is that of the theory with the massless scalar field. Namely where ϕ 0 is a constant. At the end of the evolution the Hubble parameter tends to zero, while the time grows indefinitely. That means that both the kinetic and potential terms in the right-hand side of Eq. (4) should tend to zero. It is possible if the scalar field tends to infinity while its time derivative tends to zero. The joint analysis of Eqs. (4) and (91) gives the following results for the asymptotic behavior of the scalar field and the Hubble parameter: where ˜ ϕ 0 is a constant.",
"pages": [
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},
{
"title": "9. The quantum dynamics of the cosmological model with a scalar field whose potential is inversely proportional to the field",
"content": "The introduction of the notion of the quantum state of the universe, satisfying the Wheeler-DeWitt equation [90] has stimulated the diffusion of the hypothesis that in the framework of quantum cosmology the singularities can disappear in some sense. Namely, the probability of finding of the universe with the parameters, which correspond to a classical cosmological singularity can be equal to zero (for a recent treatments see [91, 92, 93]). In this section we shall study the quantum dynamics of the model, whose classical dynamics was described in the preceding section. Our presentation follows that of papers [32, 34]. As usual, we shall use the canonical formalism and the Wheeler-DeWitt equation [90]. For this purpose, instead of the Friedmann metric (1), we shall consider a more general metric, where N is the so-called lapse function. The action of the Friedmann flat model with the minimally coupled scalar field looks now as Variating the action (102) with respect to N and putting then N = 1 we come to the standard Friedmann equation. Now, introducing the canonical formalism, we define the canonically conjugated momenta as and The Hamiltonian is and is proportional to the lapse function. The variation of the action with respect to N gives the constraint and the implementation of the Dirac quantization procedure, i.e. requirement the that constraint eliminates the quantum state [94], gives the Wheeler-DeWitt equation Here ψ ( a, φ ) is the wave function of the universe and the hats over the momenta mean that the functions are substituted by the operators. Introducing the differential operators representing the momenta as and multiplying Eq. (107) by a 3 we obtain the following partial differential equation: Finally, for our potential inversely proportional to the scalar field we have Note that in the equation (107) and in the subsequent equations we have ignored rather a complicated problem of the choice of the ordering of noncommuting operators, because the specification of such a choice is not essential for our analysis. Moreover, the interpretation of the wave function of the universe is rather an involved question [95, 96, 97]. The point is that to choose the measure in the space of the corresponding Hilbert space we should fix a particular gauge condition, eliminating in such a way the redundant gauge degrees of freedom and introducing a temporal dynamics into the model [96]. We shall not dwell here on this procedure, assuming generally that the cosmological radius a is in some way connected with the chosen time parameter and that the unique physical variable is the scalar field ϕ . Then, it is convenient to represent the solution of Eq. (110) in the form where the functions χ n satisfy the equation while the functions C n ( a ) satisfy the equation where n = 0 , 1 , . . . . Requiring the normalizability of the functions χ n on the interval 0 ≤ ϕ < ∞ , which, in turn, implies their non-singular behavior at ϕ = 0 and ϕ →∞ , and using the considerations similar to those used in the analysis of the Schrodinger equation for the hydrogen-like atoms, one can show that the acceptable values of the functions E n are while the corresponding eigenfunctions are where L 1 n are the associated Laguerre polynomials. Rather often the fact that the wave function of the universe disappears at the values of the cosmological parameters corresponding to some classical singularity is interpreted as an avoidance of such singularity. However, in the case of the soft singularity considered in the model at hand, such an interpretation does not look too convincing. Indeed, one can have a temptation to think that the probability of finding of the universe in the soft singularity state characterized by the vanishing value of the scalar field is vanishing because the expression for functions (115) entering into the expression for the wave function of the universe (111) is proportional to ϕ . However, the wave function (111) can hardly have a direct probabilistic interpretation. Instead, one should choose some reasonable time-dependent gauge, identifying some combination of variables with an effective time parameter, and interpreting other variables as physical degrees of freedom [96]. The definition of the wave function of the universe in terms of these physical degrees of freedom is rather an involved question; however, we are in a position to make some semi-qualitative considerations. The reduction of the initial set of variables to the smaller set of physical degrees of freedom implies the appearance of the Faddeev-Popov determinant which as usual is equal to the Poisson bracket of the gauge-fixing condition and the constraint [96]. Let us, for example, choose as a gaugefixing condition the identification of the new 'physical' time parameter with the Hubble parameter H taken with the negative sign. Such an identification is reasonable, because as it follows from Eq. (91) the variable H ( t ) is monotonously decreasing. The volume a 3 is the variable canonically conjugated to the Hubble variable. Thus, the Poisson bracket between the gauge-fixing condition χ = H -T phys and the constraint (106) includes the term proportional to the potential of the scalar field, which is inversely proportional to this field itself. Thus, the singularity in ϕ arising in the Faddeev-Popov determinant can cancel zero, arising in (115). Let us confront this situation with that of the Big Bang and Big Crunch singularities. As it was seen in Sec. III such singularities classically arise at infinite values of the scalar field. To provide the normalizability of the wave function one should have the integral on the values of the scalar field ϕ convergent, when | ϕ | → ∞ . That means that, independently of details connected with the gauge choice, not only the wave function of the universe but also the probability density of scalar field values should decrease rather rapidly when the absolute value of the scalar field is increasing. Thus, in this case, the effect of the quantum avoidance of the classical singularity is present.",
"pages": [
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},
{
"title": "10. The quantum cosmology of the tachyon and the pseudo-tachyon field",
"content": "In this section we would like to construct the Hamiltonian formalism for the tachyon and pseudo-tachyon fields. Using the metric (101), one can see that the contribution of the tachyon field into the action is The conjugate momentum for T is and so the velocity can be expressed as The Hamiltonian of the tachyon field is now Analogously, for the pseudo-tachyon field, we have and In what follows it will be convenient for us to fix the lapse function as N = 1. Now, adding the gravitational part of the Hamiltonians and quantizing the corresponding observables, we obtain the following Wheeler-DeWitt equations for the tachyons and for the pseudo-tachyons The study of the Wheeler-DeWitt equation for the universe filled with a tachyon or a pseudo-tachyon field is rather a difficult task because the Hamiltonian depends non-polynomially on the conjugate momentum of such fields. However, one can come to interesting conclusions, considering some particular models. First of all, let us consider a model with the pseudo-tachyon field having a constant potential. In this case the Hamiltonian in Eq. (124) does not depend on the field T . Thus, it is more convenient to use the representation of the quantum state of the universe where it depends on the coordinate a and the momentum p T . Then the Wheeler-DeWitt equation will have the following form: It becomes algebraic in the variable p T . Now, we see that the Hamiltonian is well defined at p 2 T ≥ a 6 W 2 . Looking at the limiting value p 2 T = a 6 W 2 and comparing it with the relation (121) we see that it corresponds to ˙ T 2 →∞ , which, in turn, corresponds to the encounter with the Big Brake singularity as was explained in the section V. The only way to 'neutralize' the values of p T , which imply the negativity of the expression under the square root in the left-hand side of Eq. (125), is to require that the wave function of the universe is such that The last condition could be considered as a hint on the quantum avoidance of the Big Brake singularity. However, as it was explained in Sec. IV on the example of the scalar field model, to speak about the probabilities in the neighborhood of the point where the wave function of the universe vanishes, it is necessary to realize the procedure of the reduction of the set of variables to a smaller set of physical degrees of freedom. Now, let us suppose that the gauge-fixing condition is chosen in such a way that the role of time is played by a Hubble parameter. In this case the Faddeev-Popov determinant, equal to the Poisson bracket between the gauge-fixing condition and the constraint, will be inversely proportional to the expression √ ˆ p 2 T -a 6 W 2 (see Eq. (122)), which tends to zero at the moment of the encounter with the Big Brake singularity. Thus, in the case of a pseudo-tachyon model, just like in the case of the cosmological model based on the scalar field, the Faddeev-Popov determinant introduces the singular factor, which compensates the vanishing of the wave function of the universe. What can we say about the Big Bang and the Big Crunch singularities in this model? It was noticed in the preceding section that at these singularities ˙ T 2 = 1. From the relation (121) it follows that such values of ˙ T correspond to | p T | → ∞ . A general requirement of the normalizability of the wave function of the universe implies the vanishing of ψ ( a, p T ) at p T →±∞ which signifies the quantum avoidance of the Big Bang and the Big Crunch singularities. It is quite natural, because these singularity are not traversable in classical cosmology. Now we consider the tachyon cosmological model with the trigonometric potential, whose classical dynamics was briefly sketched in the sections 4 and 6. In this case the Hamiltonian depends on both the tachyon field T and its momentum p T . The dependence of the expression under the square root on T is more complicated than that on p T . Hence, it does not make sense to use the representation ψ ( a, p T ) instead of ψ ( a, T ). Now, we have under the square root the second order differential operator -∂ 2 ∂T 2 , which is positively defined, and the function -a 6 W 2 ( T ), which is negatively defined. The complete expression should not be negative, but what does it mean in our case? It means that we should choose such wave functions for which the quantum average of the operator ˆ p 2 T -a 6 W 2 ( T ) is non-negative: Here the symbol D T signifies the integration on the tachyon field T with some measure. It is easy to guess that the requirement (127) does not imply the disappearance of the wave function ψ ( a, T ) at some range or at some particular values of the tachyon field, and one can always construct a wave function which is different from zero everywhere and thus does not show the phenomenon of the quantum avoidance of singularity. However, the forms of the potential V ( T ) given by Eq. (26) and of the corresponding potential W ( T ) for the pseudo-tachyon field arising in the same model [49] are too cumbersome to construct such functions explicitly. Thus, to illustrate our statement, we shall consider a more simple toy model. Let us consider the Hamiltonian where ˆ p is the conjugate momentum of the coordinate x and V 0 is some positive constant. Let us choose as a wave function a Gaussian function where α is a positive number and we have omitted the normalization factor, which is not essential in the present context. Then the condition (127) will look like which can be easily satisfied if Thus, we have seen that for this very simple model one can always choose such a quantum state, which does not disappear at any value of the coordinate x and which guarantees the positivity of the quantum average of the operator, which is not generally positively defined. Coming back to our cosmological model we can say that the requirement of the well-definiteness of the pseudo-tachyon part of the Hamiltonian operator in the Wheeler-DeWitt equation does not imply the disappearance of the wave function of the universe at some values of the variables and thus, does not reveal the effect of the quantum avoidance of the cosmological singularity. /negationslash Finally, summing up the content of the last three sections, devoted to the comparative study of the classical and quantum dynamics in some models with scalar fields and tachyons, revealing soft future singularities, we can make the following remarks. At the end of this section we would like also to analyze the Big Bang and Big Crunch singularities in the tachyon model with the trigonometrical potential. As was shown in paper [49] the Big Bang singularity can occur in two occasions (the same is true also for the Big Crunch singularity [40]) - either W ( T ) → ∞ (for example for T → 0) or at ˙ T 2 = 1 , W ( T ) = 0. One can see from Eqs. (15) and (120) that when the universe approaches these singularities the momentum p T tends to infinity. As was explained before, the wave function of the universe in the momentum representation should vanish at | p T | → ∞ and hence, we have the effect of the quantum avoidance. It was shown that in the tachyon model with the trigonometrical potential [49] the wave function of the universe is not obliged to vanish in the range of the variables corresponding to the appearance of the classical Big Brake singularity. In a more simple pseudo-tachyon cosmological model the wave function, satisfying the WheelerDeWitt equation and depending on the cosmological radius and the pseudo-tachyon field, disappears at the Big Brake singularity. However, the transition to the wave function depending only on the reduced set of physical degrees of freedom implies the appearance of the Faddeev-Popov factor, which is singular and which singularity compensates the terms, responsible for the vanishing of the wave function of the universe. Thus, in both these cases, the effect of the quantum avoidance of the Big Brake singularity is absent. In the case of the scalar field model with the potential inversely proportional to this field, all the classical trajectories pass through a soft singularity (which for one particular trajectory is exactly the Big Brake). The wave function of the universe disappears at the vanishing value of the scalar field which classically corresponds to the soft singularity. However, also in this case the Faddeev-Popov factor arising at the reduction to the physical degrees of freedom provides nonzero value of the probability of finding of the universe at the soft singularity. In spite of the fact that we have considered some particular scalar field and tachyonpseudo-tachyon models, our main conclusions were based on rather general properties of these models. Indeed, in the case of the scalar field we have used the fact that its potential at the soft singularity should be negative and divergent, to provide an infinite positive value of the pressure. In the case of the pseudo-tachyon field both the possible vanishing of the wave function of the universe and its 're-emergence' in the process of reduction were connected with the general structure of the contribution of such a field into the super-Hamiltonian constraint (122). Note that in the case of the tachyon model with the trigonometric potential, the wave function does not disappear at all. On the other hand we have seen that for the Big Bang and Big Crunch singularities not only the wave functions of the universe but also the corresponding probabilities disappear when the universe is approaching to the corresponding values of the fields under consideration, and this fact is also connected with rather general properties of the structure of the Lagrangians of the theories. Thus, in these cases the effect of quantum avoidance of singularities takes place. One can say that there is some kind of a classical - quantum correspondence here. The soft singularities are traversable at the classical level (at least for simple homogeneous and isotropic Friedmann models) and the effect of quantum avoidance of singularities is absent. The strong Big Bang and Big Crunch singularities cannot be passed by the universe at the classical level, and the study of the Wheeler-DeWitt equation indicates the presence of the quantum singularity avoidance effect. It would be interesting also to find examples of the absence of the effect of the quantum avoidance of singularities, for the singularities of the Big Bang-Big Crunch type. Note that the interest to the study of the possibility of crossing of such singularities is growing and some models treating this phenomenon have been elaborated during last few years [98].",
"pages": [
30,
31,
32,
33,
34
]
},
{
"title": "11. Friedmann equations modified by quantum corrections and soft cosmological singularities",
"content": "As we have already mentioned in the Introduction there are two main directions in the study of quantum cosmology of soft future singularities. One is connected with the analysis of the structure of the Wheeler-DeWitt equation and another concentrates of the study of quantum corrections to the Friedmann equations. While in two preceding sections we were studying the Wheeler-DeWitt equation, here we shall dwell on the quantum corrections to the Friedmann equations and on the possible influence of these corrections on the structure of soft singularities. Our presentation will be mainly based on papers [17],[30]. In paper [17] was considered a cosmological evolution described by where t s is the time, where the sudden singularity occurs, a s is the value of the scale factor in this moment and 1 < n < 2. The matter responsible for this evolution was not specified. It is easy to see that at the beginning of the evolution (132) the universe passes through the radiation-dominated phase of the expansion, while when t → t s it enters into the singular regime. Then it was supposed that a massive scalar field conformally coupled to gravity is present. The general solutions describing behaviour of this scalar field in these two regimes were written down and the requirement of the matching of these conditions at the a = a s was imposed. Then, the solution in the first regime is chosen as where η is the conformal time parameter. The solution in the regime of approaching the soft singularity will be where and the constants ξ 01 and ξ 02 are connected with the Bogoliubov coefficients : The matching conditions permit to find the Bogoliubov coefficients and the number of created particles for each mode The total energy of the created particles is divergent in the ultraviolet limit. The authors of [17] renormalize the expression (138) using n -wave method [99] and show that the renormalized energy is equal to zero. Thus, they conclude that the quantum phenomena associated with the cosmological dynamics do not change the character of the sudden singularity or prevent its occurrence. Some arguments in favour of the hypothesis that birth of particles of a field which is not conformally invariant cannot change the Friedmann equation are also developed in [17]. More detailed analysis of the quantum contributions into energy-momentum tensor and, hence, into the Friedmann equations, was undertaken in paper [30]. Here it was noticed that the analysis, presented in paper [17], is applicable only to situations when the frequency of the field under consideration is varying smoothly. Obviously, it is not case here, because two different phases of evolution are considered and a naive matching of the value of the field and of its time derivative at the moment of arrival to the singularity, is required. Moreover, the effect of polarization of the vacuum was not taken into account. Instead, the authors of the paper [30], use the known expressions for the renormalized energy-density and pressure for a massless conformally coupled scalar field [100, 101]: Then proceeding as in paper [102] the authors of [30] consider the Friedmann semiclassical equation looking for its solution with the form where H s , C and n ' are unknown parameters. They find, in particular, that Then, since 3 < n ' < 4, it turns out that ˙ H and H do not diverge at t = t s , which means that, for these kinds of singular solutions, the singularity becomes much milder due to the quantum corrections. In fact, in the absence of the quantum corrections, one can see from Eq. (132) that ˙ H diverges.",
"pages": [
34,
35,
36
]
},
{
"title": "12. Density matrix of the universe, quantum consistency and interplay between geometry and matter in quantum cosmology",
"content": "In this section we shall speak about the quantum density matrix of the universe [103, 104, 105, 106, 107] - an approach to quantum cosmology, which permits consideration of mixed quantum states of the universe instead of pure ones. Such and approach is based on rather a delicate interplay between geometry and matter and implies existence of essential restrictions on the basic parameters of the theory. In the framework of this approach as a byproduct arise also some new kinds of soft sudden quantum singularities [106]. As is well known, quantum cosmology predicts the initial conditions for the cosmological evolution of the universe, defining its quantum state - the wave function of the universe. The connection between the Euclidean quantum theory and the quantum tunneling is used in both the main approaches to the construction of such a function - the no-boundary prescription [108] and the tunneling one[109, 110]. In papers [103, 104] this traditional scheme of quantum cosmology was generalized for the case of fundamental mixed initial quantum states of the universe, in other words instead of wave function of the universe one can consider the density matrix of the universe, possessing some thermodynamical characteristics. Such a mixed state of the universe arises naturally if an instanton with two turning points (surfaces of vanishing external curvature) does exist. (The idea that instead of pure quantum state of the universe one can consider a density matrix of the universe, was suggested already in paper [111]). In turn, an instanton with two turning points arises naturally, if we consider a closed Friedmann universe where two essential ingredients are present: an effective cosmological constant and radiation, which corresponds to the presence of the conformally invariant fields. The Euclidean Friedmann equation in this case is written as Picture of instanton representing the density matrix. Dashed lines depict the Lorentzian Universe nucleating from the instanton at the minimal surfaces Σ and Σ ' . where H 2 is an effective cosmological constant and the constant C characterizes the quantity of the radiation in the universe. The turning points are (The same instanton was considered also in paper [112], where the conception of the universe, which gave birth to itself was suggested). Fig. 2 gives the picture of the instanton representing the density matrix of the universe For the pure quantum state [108] the instanton bridge between Σ and Σ ' breaks down (see Fig.3). However, the radiation stress tensor prevents these half instantons from closure. The relevant density matrix is the path integral with the partition function e -Γ which follows from integrating out the field ϕ in the coincidence ϕ ' = ϕ corresponding to the identification of Σ ' and Σ, the underlying instanton acquiring the toroidal topology. The metric of the instanton introduced above is conformally equivalent to the metric of the Einstein static universe: where η is the conformal time parameter. We shall consider conformally invariant fields. As is well known, the quantum effective action for such fields has a conformal anomaly first studied in cosmology in [113, 114]. It has the form where E = R 2 µναγ -4 R 2 µν + R 2 and ∆ is the four-dimensional Laplacian. This anomaly, when integrated functionally along the orbit of the conformal group, gives the relation between the actions on conformally related backgrounds [115]. where One can show that the higher-derivative in σ terms are all proportional to the coefficient α . The α -term can be arbitrarily changed by adding a local counterterm ∼ g 1 / 2 R 2 . We fix this local renormalization ambiguity by an additional criterion of the absence of ghosts. The conformal contribution to the renormalized action on the minisuperspace background equals with the constant m 2 P B which for scalars, two-component spinors and vectors equals respectively 1 / 240, 11 / 480 and 31 / 120. For a conformal scalar field where ω = n , n = 0 , 1 , 2 , ... , labels a set of eigenmodes and eigenvalues of the Laplacian on a unit 3-sphere. Thus and the effective action equals the sum of contributions of the vacuum energy E 0 and free energy F ( η ) with the inverse temperature played by η - the circumference of the toroidal instanton in units of a conformal time, Similar expressions hold for other conformally invariant fields of higher spins. In particular, the vacuum energy (an analog of the Casimir energy) on Einstein static spacetime is We should take into account the effect of the finite ghost-avoidance renormalization denoted below by a subscript R , which results in the replacement of E 0 above by a new parameter C 0 : respectively for scalar, spinor and vector fields. A direct observation indicates the following universality relation for all conformal fields of low spins Now we can write down the effective Friedmann equation governing the Euclidean evolution of the universe. First of all, the full conformal time on the instanton is where τ ± label the turning points for a ( τ ) - its minimal and maximal values. The effective action is ( m 2 P ≡ 3 / 4 πG ) and the effective Friedmann equation reads In the gauge N = 1 this equation takes form where the amount of radiation constant C is given by the bootstrap equation The Friedmann equation can be rewritten as and has the same two turning points a ± as in the classical case provided This requirement is equivalent to Together with the admissible domain for instantons reduces to the curvilinear wedge below the hyperbola and above the straight line to the left of the critical point (see Figure 4) The suggested approach allows to resolve the problem of the so-called infrared catastrophe for the no-boundary state of the Universe based on the Hartle-Hawking instanton. This problem is related to the fact that the Euclidean action on this instanton is negative and inverse proportional to the value of the effective cosmological constant. This means that the probability of the universe creation with an infinitely big size is infinitely high. We shall show now that the conformal anomaly effect allows one to avoid this counter-intuitive conclusion. Indeed, outside of the admissible domain for the instantons with two turning points, obtained above, one can also construct instantons with one turning point which smoothly close at a -= 0 with ˙ a ( τ -) = 1. Such instantons correspond to the Hartle-Hawking pure quantum state. However, in this case the on-shell effective action, which reads for the set of solutions obtained above as diverges to plus infinity. Indeed, for a -= 0 and ˙ a -= 1 and hence the effective Euclidean action diverges at the lower limit to + ∞ . Thus, and this fact completely rules out all pure-state instantons, and only mixed quantum states of the universe, described by the cosmological density matrix appear to be admissible. In connection with all said above a natural question arises: where Euclidean quantum gravity comes from? The answer can be formulated briefly as follows: from the Lorentzian quantum gravity (LQG) [105]. Namely, the density matrix of the Universe for the microcanonical ensemble in Lorentzian quantum cosmology of spatially closed universes describes an equipartition in the physical phase space of the theory, but in terms of the observable spacetime geometry this ensemble is peaked about a set of cosmological instantons (solutions of the Euclidean quantum cosmology) limited to a bounded range of the cosmological constant. These instantons obtained above as fundamental in Euclidean quantum gravity framework, in fact, turn out to be the saddle points of the LQG path integral, belonging to the imaginary axis in the complex plane of the Lorentzian signature lapse function [105]. Now let us consider the cosmological evolution of the unverse starting from the initial conditions described above. Making the transition from the Euclidean time to the Lorentzian one, τ = it , we can write the modified Lorentzian Friedmann equation as [106] where ε is a total gravitating matter density in the model (including at later stages also the contribution of particles created during inflationary expansion and thermalized at the inflation exit). A remarkable feature of this equation is that the Casimir energy is totally screened here and only the thermal radiation characterized by C weighs. If one wants to compare the evolution described by Eq. (176) with the real evoltuion of the universe, first of all it is necessary to have a realistic value for an effective cosmological constant Λ = 3 H 2 . The only way to achieve this goal is to increase the number of conformal fields and the corresponding parameter B , (153), of the conformal anomaly (148). The mechanisms for growing number of the conformal fields exist in some string inspired cosmological models with extra dimensions [105]. If some of these mechanisms work we can encounter an interesting phenomenon: if the B grows with a faster than the rate of decrease of the energy density ε one encounters a new type of the cosmological singularity - Big Boost. This singularity is characterized by finite values of the cosmological radius a BB and of its time derivative ˙ a BB , while the second time variable a has an infinite positive value. The universe reaches this singularity at some finite moment of cosmic time t BB : In paper [107] it was found that there exist some correspondences between quantum 4-dimensional equations of motion and some classical 5-dimensional equations of motion [106, 107].There were considered two five-dimansional models: the Randall -Sundrum model[116] and the generalized Dvali-Gabadadze-Porrati (DGP) model [117]. The Randall-Sundrum braneworld model is a 4-dimensional spacetime braneworld embedded into the 5-dimensional anti-de Sitter bulk with the radius L . In the limit of small energy densities the modified quantum Friedmann equations coincide with the modified 4-dimensional Friedmann equations of the Randall-Sundrum model provided The 5-dimensional action of the generalized DGP model includes the 5-dimensional curvature term, the 5-dimensional cosmological constant and the 4-dimensinal curvature term on the brane. If we require the spherical symmetry, when we have the Schwarzschild-de Sitter solution, which depends also on the Schwarzschil radius R S . The effective 4-dimensional Friedmann equations on the 4-brane coincide with the modified Friedmann equations in quantum model, provided the quantity of the radiation is expressed through the Schwarzschild radius as If we add the condition of the regularity of the Schwarzschild-de Sitter instanton, (i.e. the condition of the absence of conical singularities), we obtain an additional relation for the parameters of the quantum cosmological model and the set of admissible values for the effective cosmological constant becomes discrete. Concluding this section, we would like to say that relaxing the usual tacit requirement of the purity of the quantum state of the universe and imposing the conditions of quantum consistency of the system of equations governing the dynamics of the universe, one comes to non-trivial restrictions on the basic cosmological paprameters. Besides, as a by-product one obtains a particular kind of future soft singularity - Big Boost. Finally, we can note that in the papers, reviewed in this section both the main approaches to the study of quantum effects in cosmology were combined - the study of the modified Friedmann equations and the investigation of the structure of the quantum state of the universe. Usually, these two approaches are separated (see, Sec. 11 and Secs. 9 and 10 of the present review).",
"pages": [
36,
37,
38,
39,
40,
41,
42,
43
]
},
{
"title": "13. Quiescent singularities in braneworld models",
"content": "One of the first examples of the soft future singularities in cosmology was presented in paper [10], where some braneworld cosmological models were considered. The higherdimensional models considered there were described by an action, where both the bulk and brane contained the corresponding curvature terms: where the sum is taken over the bulk components bounded by branes, and Λ i is the cosmological constant on the ith bulk component. The Lagrangian L ( h αβ , φ ) corresponds to the presence of matter fields on the brane interacting with the induced metric h αβ , K is the trace of the extrinsic curvature. The Friedmann-type equation has the form where ρ is the energy density of the matter on the brane, the integration constant C corresponds to the presence of a black hole in the five-dimensional bulk solution, and the term C/a 4 , sometimes called 'dark radiation', arises due to the projection of the bulk gravitational degrees of freedom onto the brane. The length scale l is defined as The appearence of the quiescent singularities is conneced with the fact, that the expression under the square root in (183) turns to zero at some point during the evolution. There are essentially two types of singularities dispaying this behaviour. A type 1 singularity (S1) is induced by the presence of the dark radiation term and arises in either of the following two cases: C > 0 and the density of matter increases slower than a -4 as a → 0. An example is provided by dust. The energy density of the universe is radiation dominated so that ρ = ρ 0 /a 4 and C > ρ 0 . These singularities can take place either in the past of an expanding universe or in the future of a collapsing one. A type 2 singularity (S2) arises if In this case the combination ρ/ 3 m 2 -C/a 4 decreases monotonically as the universe expands. The expression under the square root of (183) can therefore become zero at suitably late times. For both S1 and S2, the scale factor a ( t ) and its first time derivative remain finite, while all the higher time derivatives of a tend to infinity as the singularity is approached. It is important that the energy density and the pressure of the matter in the bulk remain finite. This feature distinguishes these singularities from the singularities considered in the preceding sections, and justifies the special name 'quiescent' [10]. The point is that the existence of these singularities is connected not with special features of the matter on the brane, but with the particularity of the embedding of the brane into the bulk. In paper [11] the question of influence of the quantum effects on a braneworld encountering a quiescent singularity during expansion was studied. The matter considered in [11] was constituted from conformally invariant fields. Hence, the particle production was absent and the only quantum effect was connected with the vacuum polarization. It was shown that this effect boils down to the modification of the effective energy density of the matter on the brane. Namely, the quantum correction to this energy density is given by The insertion of this correction to the energy density changes drastically the form of the brane Friedmann-type equation (183) - the original algebraic equation becomes a differential equation. It implies essential changes in the possible behaviour of the universe around singularities. First, the quiscent singularity changes its form and becomes much weaker, in fact, H and ˙ H remain finite and only ··· H → ∞ . Second, vacuum polarization effects can also cause a spatially flat universe to turn around and collapse. At the conclusion of this section we would like to mention another type of cosmological singularities, arising in the brane-world context. These are the so called pressure singularities [118, 119]. These singularities arise in the generalized Friedmann branes, which can be asymmetrically embedded into the bulk and can include pullbacks on the brane some non-standard field and geometric configurations, existing in the 5-dimensional bulk [120]. It appears that it is possible to reproduce in this frame work a Swiss cheese Einstein-Strauss model [121]. In this model there pieces of the Schwarzschild regions inserted into a Friedmann universes. At some conditions in the Friedmann regions of such branes the pressure of matter becomes infinite, while the cosmological radius and all its time derivatives remain finite. It was shown also [119] that at some critical value of the assymetry in the embedding of the brane into the bulk, these singularities appear necessarily. It is interesting that these pressure singularities are in a way complementary to the quiescent singularities, discussed above, where the energy density and the pressure are always finite, while the time derivatives of the scale factor become divergent, beginning since the second or some higher-order derivative.",
"pages": [
43,
44,
45
]
},
{
"title": "14. Concluding remarks",
"content": "In this review we have considered a broad class of phenomena arising in cosmological models, possessing some exotic cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We have discussed the models, based on standard scalar fields, Born-Infeld-type fields and on perfect fluids, where soft future cosmological singularities exist and are transversable. The crossing of such singularities (or other geometrically peculiar surfaces in the spacetime) can imply such an interesting phenomenon as a transformation of matter properties, which is discussed in some detail here. Another interesting aspect of the study of both soft and 'hard' (Big Bang or Big Crunch) cosmological singularities is the existence of the correspondence between the phenomenon of quantum avoidance (or non-avoidance) of such singularities and the possibility of their crossing (or the absence of such a possibility) in classical cosmology. Besides, the quantum cosmological approach, based on the study of the properties of solutions of the Wheeler-DeWitt equation, we have reviewed also some works based on the investigation of the modification of the Friedmann equation due to the quantum corrections and the influence of of these corrections on the structure and the very existence of soft cosmological singularities. While the main part of this review deals with the standard Einstein general relativity in the presence of non-standard matter, the last section is devoted to the exotic singularities arising in the brane-world cosmological models, which are very close in their nature to the soft sudden singularities arising in the general relativity. Generally, we are convinced that the study of exotic singularities in classical and quantum cosmology is a promising branch of the theoretical physics, and nobody can exclude that it can acquire some phenomenological value as well. Here it is necessary to recognize that almost all studies in this field deal only with isotropic and homogeneous Friedmann universes. Thus, the extension of this studies to the anisotropic and inhomogeneous models represents a main challenge for people working in this field. Such an extension can bring some interesting surprises as it was with the study of the Big Bang - Big Crunch singularities, where the consideration of the anisotropic Bianchi models instead of Friedmann models, has given birth to the discovery of the oscillating approach to the singularity (Mixmaster Universe) [6, 7].",
"pages": [
45,
46
]
},
{
"title": "Acknowledgments",
"content": "I am grateful to A.M. Akhmeteli, A.A. Andrianov, A.O. Barvinsky, V.A. Belisnky, M. Bouhmadi-Lopez, F. Cannata, S. Cotsakis, M.P. Dabrowski, C. Deffayet, G. Esposito, L.A. Gergely, V. Gorini, D.I. Kazakov, Z. Keresztes, I.M. Khalatnikov, C. Kiefer, V.N. Lukash, S. Manti, P.V. Moniz, U. Moschella, V. Pasquier, D. Polarski, D. Regoli, V.A. Rubakov, B. Sandhofer, D.V. Shirkov, A.A. Starobinsky, O.V. Teryaev, A.V. Toporensky, V. Sahni, G. Venturi, A. Vilenkin and A.V. Yurov for fruitful discussions. This work was partially supported by the RFBR grant 11-02-00643.",
"pages": [
46
]
},
{
"title": "References",
"content": "[gr-qc]).",
"pages": [
49
]
}
]
2013CQGra..30q5010G
https://arxiv.org/pdf/1206.6216.pdf
<document>
<section_header_level_1><location><page_1><loc_24><loc_79><loc_74><loc_82></location>Rigid spheres in Riemannian spaces</section_header_level_1>
<text><location><page_1><loc_31><loc_73><loc_68><loc_77></location>Hans-Peter Gittel 1 , Jacek Jezierski 2 , Jerzy Kijowski 3 , Szymon Łęski 3 , 4 .</text>
<text><location><page_1><loc_23><loc_67><loc_75><loc_71></location>1 Department of Mathematics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany</text>
<text><location><page_1><loc_24><loc_58><loc_74><loc_64></location>2 Department of Mathematical Methods in Physics Faculty of Physics, University of Warsaw, ul. Hoża 69, Warszawa, Poland</text>
<text><location><page_1><loc_19><loc_52><loc_79><loc_56></location>3 Center for Theoretical Physics, Polish Academy of Sciences al. Lotników 32/46, Warszawa, Poland</text>
<text><location><page_1><loc_28><loc_43><loc_70><loc_50></location>4 Nencki Institute of Experimental Biology, Polish Academy of Sciences ul. Pasteura 3, Warszawa, Poland</text>
<text><location><page_1><loc_42><loc_40><loc_56><loc_42></location>June 13, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_35><loc_53><loc_36></location>Abstract</section_header_level_1>
<text><location><page_1><loc_19><loc_16><loc_79><loc_34></location>We define a special family of topological two-spheres, which we call 'rigid spheres', and prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold whose metric is sufficiently close to the flat metric (e. g. in the external region of an asymptotically flat space). In case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere. The rigid spheres can be used as building blocks for various ('spherical', 'bispherical' etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. [4]) and the foliations used by us (cf. [8]) in the analysis of the two-body problem.</text>
<section_header_level_1><location><page_2><loc_15><loc_85><loc_36><loc_87></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_15><loc_63><loc_84><loc_84></location>In General Relativity Theory, the amount of gravitational energy (mass) contained in a portion V ⊂ Σ of a Cauchy three-surface Σ is assigned to its boundary S = ∂V , rather than to the volume V itself (cf. the notion of a 'quasi-local' mass introduced by Penrose, [10]). The above philosophy was also used in [11], where important quasi-local observables (like, e.g., momentum, angular momentum or center of mass) assigned to a generic 2D surface (whose topology is that of S 2 ) have acquired a hamiltonian interpretation as generators of the corresponding canonical transformations of the (appropriately defined) phase space of gravitational initial data. Recently, we were able to define energy contained in an asymptotically Schwarzschild-de Sitter spacetime (cf. [12]), and again the quasilocal, hamiltonian description of the field dynamics provided an adequate starting point for our analysis.</text>
<text><location><page_2><loc_15><loc_48><loc_84><loc_63></location>Typically, the S 2 -spheres used for the quasi-local purposes come from specific spacetime foliations { t = const.; r = const. } , where a specific choice of coordinates t and r plays the role of a gauge. In literature, gauge conditions based on 3Delliptic problems have been mostly used (see e.g. 'traceless-transversal' condition advocated by J. York (see e.g. [14]) or a 'p-harmonic gauge' analyzed in [2]). Important results have been obtained by Huisken and Ilmanen (cf. [5]) who used a parabolic gauge condition imposed for the radial coordinate r . The same gauge was also used by one of us (J.J., see [6]) to prove stability of the ReissnerNordström solution, together with a version of Penrose's inequality.</text>
<text><location><page_2><loc_15><loc_34><loc_84><loc_48></location>For purposes of the quasi-local analysis, these approaches exhibit an obvious drawback consisting in the fact that we do not control intrinsic properties of the surfaces { r = const. } constructed this way. This feature was partially removed by Huang in [4], where new 3D foliations were thoroughly analyzed. Their fibers { r = const. } are selected by a 2D-elliptic condition: k = const., where k denotes the mean extrinsic curvature. In a generic Riemannian three-manifold Σ , the above equation admits a one-parameter family of 'spheres'. Physically, they are related to the 'center of mass' of the geometry (cf. [4]).</text>
<text><location><page_2><loc_15><loc_22><loc_83><loc_34></location>Unfortunately, the above condition is not stable with respect to small perturbations of the geometry. Indeed, in the (flat) Euclidean space E 3 , this condition admits not 'one-' but a four-parameter family of solutions (parameterized e.g. by the radius R and the three coordinates of a center). Moreover, the exclusive use of the center of mass reference frame is often too restrictive for physical applications. In particular, it does not allow us to describe easily the momentum i.e. the generator of space translations.</text>
<text><location><page_2><loc_15><loc_14><loc_83><loc_22></location>In the present paper we propose a new gauge condition, which is also 2Delliptic but does not exhibit the above drawback. Indeed, in a generic Riemannian three-manifold our condition selects a four-parameter family of solutions, like in the Euclidean space E 3 . Moreover, our condition is weaker than ' k = const.' (equivalent in the non-generic, Euclidean case, only). Topological two-spheres</text>
<text><location><page_3><loc_15><loc_77><loc_84><loc_87></location>satisfying our condition will be called 'rigid spheres'. They can be organized in topologically different ways: not necessarily standard 'nested spheres foliations', but also e.g. 'bispherical foliations' which already proved to be very useful in the analysis of the two body problem 1 (see [8]). We expect that various such arrangements, with rigid spheres used as building blocks, will provide useful gauge conditions in General Relativity Theory.</text>
<text><location><page_3><loc_15><loc_41><loc_84><loc_76></location>The present paper is a part of a bigger project, where we construct 'spheres' which are rigid not only with respect to 3D, but also with respect to 4D deformations. More precisely, an eight-parameter family of similar 'rigid spheres' will be constructed in a generic four-dimensional Lorentzian spacetime. In the present paper we limit ourselves to the 3D Riemannian case. It turns out, however, that our construction can be generalized to the entire pseudo-Riemannian spacetime M , instead of the Riemannian Cauchy three-space Σ ⊂ M . The idea of this extension is to mimic the case of the flat Minkowski space, where all possible round spheres, embedded in all possible flat subspaces Σ of M , form an eightparameter family. All of them can be obtained from a single one by the action of the product of the one-parameter group of dilations (changing the size of S ) and the ten-parameter Poincaré group, quotiented by the three-parameter rotation group. The 4D version of our construction will take into account not only the external curvature of S , but also its torsion (in Section 2.5 we give a short outline of this construction, which will be presented in detail in a subsequent paper). The rigid spheres obtained this way will form an eight-parameter family and will be used to construct useful coordinate systems not only on a given Cauchy surface Σ , but also in the entire spacetime. The main advantage of such a construction consists in its rigidity at infinity. We very much hope to be able to eliminate supertranslations and to reduce the symmetry group of the 'Scri', otherwise infinite dimensional, to the finite dimensional one.</text>
<text><location><page_3><loc_15><loc_22><loc_84><loc_40></location>The construction which we propose in the present paper is based on the following idea. Given a surface S satisfying the rigid sphere condition, consider its infinitesimal deformations. They may be parameterized by sections of the normal bundle T ⊥ S . If we want our condition to admit a four-parameter family of solutions, like in the flat case, its linearization must admit a four-parameter family of deformations. This means that we are not allowed to constrain the complete information about the mean curvature k : four real parameters describing k must be left free. In the flat case these four parameters which have to be left free are: the mean value (or the monopole part ) of k , which is responsible for the size of S , and its dipole part (which vanishes exceptionally in flat case due to GaussCodazzi equations). The dipole part of the deformation is related to the group</text>
<text><location><page_4><loc_15><loc_82><loc_84><loc_87></location>of translations. In fact, possible motions of a metric sphere are described by the group of Euclidean motions, quotiented by the subgroup of rotations which form the group of internal symmetries of every particular sphere S .</text>
<text><location><page_4><loc_15><loc_73><loc_84><loc_82></location>To implement the above idea in a non flat case, an intrinsic, geometric notion of a multipole expansion on an arbitrary Riemannian, topologically S 2 -surface is proposed in Section 2. This construction is our main technical tool and we very much believe in its universal validity, going far beyond the purposes of the present paper. Section 2 is completed with the definition of a rigid sphere.</text>
<text><location><page_4><loc_15><loc_63><loc_84><loc_73></location>Section 3 contains formulation and the proof of theorem 3: a generic Riemannian three-space admits a four-parameter family of rigid spheres. Our proof is relatively simple, but is valid in the 'weak field region' only. This is sufficient for purposes of the quasi-local analysis of gravitational energy (in fact, the idea originates from our analysis of interaction between two black holes, cf. [8]). Further development concerning strong fields will be given elsewhere.</text>
<text><location><page_4><loc_15><loc_58><loc_83><loc_63></location>Finally, discussion concerning less known (but necessary) technical results, like specific spectral properties of the Laplace operator on S 2 or the second variation of area, has been shifted to the Appendix.</text>
<section_header_level_1><location><page_4><loc_15><loc_50><loc_83><loc_55></location>2 Equilibrated spherical coordinates. Multipole calculus on distorted spheres</section_header_level_1>
<section_header_level_1><location><page_4><loc_15><loc_47><loc_60><loc_49></location>2.1 Conformally spherical coordinates</section_header_level_1>
<text><location><page_4><loc_15><loc_34><loc_84><loc_46></location>Let S be a differential two-manifold, diffeomorphic to the two-sphere S 2 ⊂ R 3 and equipped with a (sufficiently smooth) metric g . Coordinates ( ϑ, ϕ ) = ( x A ) , A = 1 , 2 , defined on a dense subset of S \ /lscript , where /lscript is topologically a line interval, will be called conformally spherical coordinates if they have the same range of values as the standard spherical coordinates on S 2 ⊂ R 3 and, moreover, if the corresponding metric tensor g AB is conformally equivalent to the standard round metric on S 2 , i.e. the following formula holds:</text>
<formula><location><page_4><loc_42><loc_30><loc_83><loc_32></location>g AB = ψ · σ AB , (1)</formula>
<text><location><page_4><loc_15><loc_28><loc_60><loc_29></location>where ψ is a (sufficiently smooth) function on S and</text>
<formula><location><page_4><loc_39><loc_23><loc_83><loc_27></location>σ AB = ( 1 0 0 sin 2 ϑ ) . (2)</formula>
<text><location><page_4><loc_15><loc_12><loc_84><loc_22></location>Remark 1. Conformally spherical coordinates always exist (cf. [9]). It is easy to see that there is always a six-parameter freedom in the choice of such coordinates. More precisely, if ( ϑ, ϕ ) are conformally spherical coordinates then ( ˜ ϑ, ˜ ϕ ) are also conformally spherical if and only if they may be obtained from ( ϑ, ϕ ) via a conformal transformation of S 2 ⊂ R 3 .</text>
<text><location><page_5><loc_15><loc_80><loc_84><loc_87></location>Example 1. A 'proper' conformal transformation, i.e. which is not a rotation: Let n ∈ S and τ > 0 be a positive number. Using appropriate rotation, choose conformally spherical coordinates ( ϑ, ϕ ) in such a way that n is a north pole, i.e. the coordinate ϑ vanishes at n . Define</text>
<text><location><page_5><loc_15><loc_74><loc_20><loc_76></location>where</text>
<text><location><page_5><loc_15><loc_69><loc_28><loc_70></location>or, equivalently,</text>
<formula><location><page_5><loc_31><loc_69><loc_83><loc_76></location>˜ ϑ := 2 arctan ( τ · tan ϑ 2 ) , ˜ ϕ := ϕ , (4)</formula>
<formula><location><page_5><loc_40><loc_74><loc_83><loc_79></location>F n ,τ ( ϑ, ϕ ) = ( ˜ ϑ, ˜ ϕ ) , (3)</formula>
<formula><location><page_5><loc_41><loc_65><loc_83><loc_69></location>tan ˜ ϑ 2 = τ · tan ϑ 2 . (5)</formula>
<text><location><page_5><loc_17><loc_63><loc_81><loc_65></location>For the fixed point n these transformations form a one-parameter group 2 :</text>
<formula><location><page_5><loc_40><loc_60><loc_83><loc_62></location>F n ,τ · F n ,σ = F n ,τσ , (6)</formula>
<text><location><page_5><loc_15><loc_58><loc_40><loc_59></location>generated by the vector field:</text>
<formula><location><page_5><loc_22><loc_51><loc_83><loc_57></location>d d t ∣ ∣ ∣ ∣ t =1 F n ,t ( ϑ, ϕ ) = d d t ∣ ∣ ∣ ∣ t =1 [ 2 arctan ( t · tan ϑ 2 )] ∂ ∂ϑ = sin ϑ ∂ ∂ϑ , (7)</formula>
<text><location><page_5><loc_15><loc_50><loc_62><loc_52></location>which is the (minus) gradient of the function z = cos ϑ .</text>
<text><location><page_5><loc_15><loc_47><loc_83><loc_50></location>In particular, F n , 1 = I (the identity map) for every n . Moreover, equation (5) implies the following identity:</text>
<formula><location><page_5><loc_43><loc_43><loc_83><loc_45></location>F -n ,τ = F n , 1 τ . (8)</formula>
<text><location><page_5><loc_17><loc_41><loc_70><loc_42></location>Using (4) and (5) we may easily derive the following formula:</text>
<formula><location><page_5><loc_35><loc_33><loc_83><loc_40></location>d ϑ = d ϑ d ˜ ϑ d ˜ ϑ = τ 1 + tan 2 ˜ ϑ 2 τ 2 +tan 2 ˜ ϑ 2 d ˜ ϑ . (9)</formula>
<text><location><page_5><loc_15><loc_33><loc_36><loc_34></location>Similarly, we may prove:</text>
<formula><location><page_5><loc_21><loc_22><loc_84><loc_31></location>sin ϑ = sin ϑ sin ˜ ϑ sin ˜ ϑ = 2 1 τ tan ˜ ϑ 2 1 + 1 τ 2 tan 2 ˜ ϑ 2 · 1 + tan 2 ˜ ϑ 2 2 tan ˜ ϑ 2 sin ˜ ϑ = τ 1 + tan 2 ˜ ϑ 2 τ 2 +tan 2 ˜ ϑ 2 sin ˜ ϑ . (10)</formula>
<text><location><page_5><loc_15><loc_21><loc_37><loc_22></location>As a conclusion we obtain:</text>
<formula><location><page_5><loc_28><loc_15><loc_84><loc_21></location>( d ϑ ) 2 +sin 2 ϑ ( d ϕ ) 2 = h 2 [ ( d ˜ ϑ ) 2 +sin 2 ˜ ϑ ( d ϕ ) 2 ] , (11)</formula>
<text><location><page_6><loc_15><loc_85><loc_20><loc_87></location>where</text>
<formula><location><page_6><loc_41><loc_81><loc_84><loc_86></location>h = τ 1 + tan 2 ˜ ϑ 2 τ 2 +tan 2 ˜ ϑ 2 , (12)</formula>
<text><location><page_6><loc_15><loc_79><loc_81><loc_80></location>which proves the conformal character of the transformation. Indeed, we have</text>
<formula><location><page_6><loc_18><loc_73><loc_84><loc_78></location>g AB d x A d x B = ψ [ ( d ϑ ) 2 +sin 2 ϑ ( d ϕ ) 2 ] = ψh 2 [ ( d ˜ ϑ ) 2 +sin 2 ˜ ϑ ( d ϕ ) 2 ] . (13)</formula>
<text><location><page_6><loc_15><loc_69><loc_71><loc_73></location>Hence, ( ˜ ϑ, ϕ ) are conformally spherical coordinates if ( ϑ, ϕ ) were.</text>
<section_header_level_1><location><page_6><loc_15><loc_68><loc_73><loc_69></location>2.2 Barycenter of a conformally spherical system</section_header_level_1>
<text><location><page_6><loc_15><loc_63><loc_83><loc_66></location>Given a system of conformally spherical coordinates on S , consider the corresponding three functions:</text>
<formula><location><page_6><loc_41><loc_60><loc_84><loc_61></location>x := sin ϑ cos ϕ , (14)</formula>
<formula><location><page_6><loc_41><loc_58><loc_84><loc_59></location>y := sin ϑ sin ϕ , (15)</formula>
<formula><location><page_6><loc_41><loc_56><loc_84><loc_57></location>z := cos ϑ . (16)</formula>
<text><location><page_6><loc_15><loc_52><loc_69><loc_54></location>We have, therefore, a mapping D : ]0 , π [ × ]0 , 2 π [ ↦→ R 3 , given by:</text>
<text><location><page_6><loc_15><loc_43><loc_32><loc_45></location>The following vector</text>
<formula><location><page_6><loc_33><loc_45><loc_84><loc_52></location>D ( ϑ, ϕ ) = D 1 ( ϑ, ϕ ) D 2 ( ϑ, ϕ ) D 3 ( ϑ, ϕ ) = x y z . (17)</formula>
<formula><location><page_6><loc_39><loc_37><loc_84><loc_44></location>X = < x > < y > < z > ∈ R 3 , (18)</formula>
<text><location><page_6><loc_15><loc_34><loc_83><loc_37></location>where by < f > we denote the average (mean value) of the function f on S , i.e. the number</text>
<text><location><page_6><loc_15><loc_25><loc_83><loc_30></location>will be called a 'barycenter' of the system ( ϑ, ϕ ) on S . Of course, we have ‖ X ‖ ≤ 1 , because of the Hölder inequality:</text>
<formula><location><page_6><loc_38><loc_29><loc_84><loc_35></location>< f > := ∫ S f √ det g d 2 x ∫ S √ det g d 2 x , (19)</formula>
<formula><location><page_6><loc_18><loc_22><loc_80><loc_25></location>‖ X ‖ 2 = < x > 2 + < y > 2 + < z > 2 ≤ < x 2 > + < y 2 > + < z 2 > = 1 .</formula>
<text><location><page_6><loc_15><loc_18><loc_84><loc_22></location>Example 2. Consider the proper conformal transformation (4) and calculate the new barycenter</text>
<formula><location><page_6><loc_39><loc_10><loc_84><loc_19></location>˜ X = < ˜ x > < ˜ y > < ˜ z > ∈ R 3 , (20)</formula>
<text><location><page_7><loc_15><loc_85><loc_20><loc_87></location>where</text>
<text><location><page_7><loc_15><loc_75><loc_38><loc_76></location>The trigonometric identity:</text>
<formula><location><page_7><loc_41><loc_75><loc_57><loc_84></location>˜ x := sin ˜ ϑ cos ϕ , ˜ y := sin ˜ ϑ sin ϕ , ˜ z := cos ˜ ϑ .</formula>
<formula><location><page_7><loc_41><loc_69><loc_84><loc_73></location>cos ϑ = 1 -tan 2 ϑ 2 1 + tan 2 ϑ 2 , (21)</formula>
<formula><location><page_7><loc_37><loc_63><loc_84><loc_67></location>tan 2 ϑ 2 = 1 -cos ϑ 1 + cos ϑ = 1 -z 1 + z . (22)</formula>
<text><location><page_7><loc_15><loc_61><loc_38><loc_63></location>Hence, formula (5) implies:</text>
<text><location><page_7><loc_15><loc_54><loc_28><loc_55></location>or, equivalently,</text>
<formula><location><page_7><loc_42><loc_54><loc_84><loc_60></location>1 -˜ z 1 + ˜ z = τ 2 1 -z 1 + z , (23)</formula>
<text><location><page_7><loc_15><loc_48><loc_48><loc_50></location>Moreover, formula (10) and its inverse:</text>
<formula><location><page_7><loc_39><loc_48><loc_84><loc_54></location>˜ z = 1 + z -τ 2 (1 -z ) 1 + z + τ 2 (1 -z ) . (24)</formula>
<formula><location><page_7><loc_18><loc_41><loc_84><loc_47></location>sin ˜ ϑ = τ 1 + tan 2 ϑ 2 1 + τ 2 tan 2 ϑ 2 sin ϑ = τ 1 + 1 -z 1+ z 1 + τ 2 1 -z 1+ z sin ϑ = 2 τ sin ϑ 1 + z + τ 2 (1 -z ) , (25)</formula>
<text><location><page_7><loc_15><loc_40><loc_18><loc_41></location>give</text>
<formula><location><page_7><loc_37><loc_33><loc_84><loc_39></location>˜ x := 2 τ 1 + z + τ 2 (1 -z ) x , (26)</formula>
<text><location><page_7><loc_15><loc_22><loc_83><loc_30></location>To calculate mean values of the functions (26), (27) and (24) we do not need to pass to new coordinates ( ˜ ϑ, ϕ ) , but we may use, as well, old coordinates ( ϑ, ϕ ) . But we see that for τ → 0 we have ˜ x → 0 , ˜ y → 0 , ˜ z → 1 . The Lebesgue theorem implies, therefore, that for τ → 0 we have</text>
<formula><location><page_7><loc_37><loc_29><loc_84><loc_35></location>˜ y := 2 τ 1 + z + τ 2 (1 -z ) y . (27)</formula>
<formula><location><page_7><loc_32><loc_14><loc_84><loc_23></location>˜ X = < ˜ x > < ˜ y > < ˜ z > -→ < 0 > < 0 > < 1 > = n . (28)</formula>
<text><location><page_7><loc_15><loc_67><loc_21><loc_68></location>implies:</text>
<section_header_level_1><location><page_8><loc_15><loc_85><loc_60><loc_87></location>2.3 Equilibrated spherical coordinates</section_header_level_1>
<text><location><page_8><loc_15><loc_80><loc_84><loc_84></location>Definition 1. Conformally spherical coordinate system ( ϑ, ϕ ) is called equilibrated , if its barycenter vanishes: X = 0 ∈ R 3 .</text>
<text><location><page_8><loc_15><loc_77><loc_84><loc_80></location>Remark 2. If there are two equilibrated spherical systems on S then they are related by a rotation.</text>
<text><location><page_8><loc_15><loc_72><loc_84><loc_75></location>Theorem 1. Each metric tensor on S admits a unique (up to rotations) equilibrated spherical system.</text>
<text><location><page_8><loc_15><loc_65><loc_84><loc_71></location>Proof. Given a metric tensor g on S , choose first any system of conformally spherical coordinates ( ϑ, ϕ ) on S and consider the corresponding identification of its points with the points of S 2 = ∂K (0 , 1) ⊂ R 3 . Consider now the mapping</text>
<formula><location><page_8><loc_30><loc_62><loc_84><loc_65></location>R 3 ⊃ K (0 , 1) /owner N →F ( N ) ∈ K (0 , 1) ⊂ R 3 , (29)</formula>
<text><location><page_8><loc_15><loc_60><loc_50><loc_62></location>given for N = 0 by the following formula</text>
<formula><location><page_8><loc_42><loc_55><loc_84><loc_59></location>F ( N ) := ˜ X n ,τ , (30)</formula>
<text><location><page_8><loc_25><loc_59><loc_25><loc_62></location>/negationslash</text>
<text><location><page_8><loc_15><loc_51><loc_83><loc_56></location>where the latter is the barycenter of the coordinates ( ˜ ϑ, ˜ ϕ ) obtained from ( ϑ, ϕ ) by the proper conformal transformation (3) with</text>
<formula><location><page_8><loc_45><loc_47><loc_84><loc_51></location>n := N ‖ N ‖ (31)</formula>
<formula><location><page_8><loc_43><loc_43><loc_84><loc_45></location>τ := 1 -‖ N ‖ . (32)</formula>
<text><location><page_8><loc_15><loc_19><loc_84><loc_43></location>For N = 0 formula (31) has no sense, but then (32) gives τ = 1 and, whence, equation (5) implies that the corresponding transformation (3) reduces to identity, no matter which vector n do we choose. Consequently, we define F ( 0 ) as the barycenter of the original coordinates ( ϑ, ϕ ) . Obviously, F defined this way is continuous. Moreover, for ‖ N ‖ = 1 we have F ( N ) = N due to (32) and (28). This means that F reduces to the identical mapping when restricted to the boundary S 2 = ∂K (0 , 1) ⊂ K (0 , 1) . Consequently, there must be a point N 0 which solves equation F ( N 0 ) = 0 . This completes the existence proof. To prove the uniqueness, let us suppose that there is another solution: F ( N 1 ) = 0 . Consider now the conformal transformation F n 1 ,τ 1 · F -1 n 0 ,τ 0 . Since the proper conformal transformations do not form any subgroup of the group of all conformal transformations, we cannot assume that it is again a proper transformation. But it may be decomposed into a product of rotations and a proper conformal transformation:</text>
<formula><location><page_8><loc_35><loc_16><loc_84><loc_19></location>F n 1 ,τ 1 · F -1 n 0 ,τ 0 = O 1 · F m ,τ · O 0 , (33)</formula>
<text><location><page_8><loc_15><loc_12><loc_84><loc_16></location>where O 1 and O 0 are rotations. Denote by ( ϑ 0 , ϕ 0 ) the spherical coordinates obtained from ( ϑ, ϕ ) by the transformation F n 0 ,τ 0 and then rotation O -1 0 . Similarly,</text>
<text><location><page_8><loc_15><loc_45><loc_18><loc_47></location>and</text>
<text><location><page_9><loc_15><loc_75><loc_84><loc_87></location>denote by ( ϑ 1 , ϕ 1 ) the ones obtained from ( ϑ, ϕ ) by F n 1 ,τ 1 and then by rotation O -1 1 . Because a rotation does affect equilibration of coordinates, both systems ( ϑ 0 , ϕ 0 ) and ( ϑ 1 , ϕ 1 ) are equilibrated. But the latter may be obtained from the former by a proper conformal transformation F m ,τ . We shall prove that this is impossible unless τ = 1 or, equivalently, transformation F m ,τ is trivial (identical). For this purpose consider, for each value of τ , the linear function z τ . Without any loss of generality we may assume that</text>
<formula><location><page_9><loc_44><loc_68><loc_84><loc_75></location>m = 0 0 1 (34)</formula>
<text><location><page_9><loc_15><loc_65><loc_83><loc_68></location>(if this is not the case, it is sufficient to perform an appropriate rotation of coordinates). Formula (24) implies the following relation:</text>
<formula><location><page_9><loc_38><loc_59><loc_84><loc_64></location>z τ = 1 + z 0 -τ 2 (1 -z 0 ) 1 + z 0 + τ 2 (1 -z 0 ) . (35)</formula>
<formula><location><page_9><loc_34><loc_53><loc_84><loc_58></location>d d τ z τ = -4 τ (1 -z 2 0 ) [1 + z 0 + τ 2 (1 -z 0 )] 2 ≤ 0 , (36)</formula>
<text><location><page_9><loc_15><loc_58><loc_20><loc_59></location>Hence</text>
<text><location><page_9><loc_15><loc_52><loc_79><loc_54></location>and it vanishes only at a single point z 0 = 1 . Consequently, its mean value:</text>
<formula><location><page_9><loc_45><loc_48><loc_84><loc_52></location>< d d τ z τ > (37)</formula>
<text><location><page_9><loc_15><loc_37><loc_84><loc_47></location>is strictly negative. This implies that starting from τ = 1 (which corresponds to the identity mapping F m , 1 ) and moving towards the actual value τ < 1 , the ' z '-component of the vector ˜ X m ,τ is strictly increasing. It vanishes at the beginning because ( ϑ 0 , ϕ 0 ) is equilibrated. Hence, it must be strictly positive at the end. This means that the final system ( ϑ 1 , ϕ 1 ) cannot be equilibrated unless τ = 1 and, therefore, both systems coincide.</text>
<text><location><page_9><loc_63><loc_28><loc_63><loc_31></location>/negationslash</text>
<text><location><page_9><loc_15><loc_26><loc_83><loc_36></location>Different equilibrated spherical systems of coordinates form, therefore, a threedimensional family. They can be parameterized by the position of a fixed point n ∈ S (north pole) and the geographic longitude of a fixed point m ∈ S (Greenwich). More precisely: given two points n , m ∈ S , n = m , there is a unique equilibrated spherical system ( ϑ, ϕ ) of coordinates on S , such that ϑ vanishes at n and ϕ vanishes at m .</text>
<text><location><page_9><loc_15><loc_23><loc_84><loc_26></location>Combining these observations with classical results (cf. [9]), we obtain the following</text>
<text><location><page_9><loc_16><loc_16><loc_16><loc_18></location>/negationslash</text>
<text><location><page_9><loc_15><loc_13><loc_84><loc_22></location>Theorem 2. Let S be a differential two-manifold, diffeomorphic to the two-sphere S 2 ⊂ R 3 and equipped with a metric g of class C ( k,α ) . For every pair n , m ∈ S , n = m , there is a unique equilibrated spherical system ( ϑ, ϕ ) of coordinates on S , such that ϑ vanishes at n and ϕ vanishes at m , and the metric components g AB are of the same class C ( k,α ) .</text>
<text><location><page_10><loc_15><loc_80><loc_84><loc_87></location>Here, C ( k,α ) is a Hölder space C k,α ( S 2 ) , defined for 1 /lessorequalslant k ∈ N and 0 < α < 1 . The space consists of those functions on S 2 which have continuous derivatives up to order k and such that the k -th partial derivatives are Hölder continuous with exponent α . This is a locally convex topological vector space.</text>
<text><location><page_10><loc_17><loc_78><loc_68><loc_80></location>The Hölder coefficient of a function f is defined as follows:</text>
<text><location><page_10><loc_49><loc_73><loc_49><loc_74></location>/negationslash</text>
<formula><location><page_10><loc_35><loc_73><loc_63><loc_77></location>| f | C 0 ,α = sup x,y ∈ S 2 , x = y | f ( x ) -f ( y ) | | x -y | α .</formula>
<text><location><page_10><loc_15><loc_68><loc_83><loc_72></location>The function f is said to be (uniformly) Hölder continuous with exponent α if | f | C 0 ,α is finite. In this case the Hölder coefficient can be used as a seminorm.</text>
<text><location><page_10><loc_15><loc_64><loc_83><loc_69></location>The Hölder space C k,α ( S 2 ) is composed of functions whose derivatives up to order k are bounded and the derivatives of the order k are Hölder continuous. It is a Banach space equipped with the norm</text>
<formula><location><page_10><loc_34><loc_59><loc_64><loc_62></location>‖ f ‖ C k,α = ‖ f ‖ C k +max | β | = k | D β f | C 0 ,α ,</formula>
<text><location><page_10><loc_15><loc_57><loc_47><loc_58></location>where β ranges over multi-indices and</text>
<formula><location><page_10><loc_37><loc_52><loc_61><loc_55></location>‖ f ‖ C k = max | β |≤ k sup x ∈ S 2 | D β f ( x ) | .</formula>
<section_header_level_1><location><page_10><loc_15><loc_49><loc_70><loc_50></location>2.4 Rigid spheres in a Riemannian three-space</section_header_level_1>
<text><location><page_10><loc_15><loc_34><loc_84><loc_47></location>Given a manifold S equipped with a metric tensor g , there is a three-dimensional space of 'linear functions' uniquely defined on S as linear combinations of functions (14-16), calculated in any equilibrated spherical system of coordinates ( ϑ, ϕ ) . We denote this space by M 3 . By M 4 we denote the space spanned by M 3 and the constant functions on S . Linear functions (14-16) on S are eigenfunctions of the Laplace operator 3 ∆ σ , with the eigenvalue equal to -2 , i.e. ∆ σ X i = -2 X i , where we denote x = X 1 , y = X 2 , z = X 3 . Let us denote by d σ := sin ϑ d ϑ d ϕ the measure associated with the metric σ AB .</text>
<text><location><page_10><loc_15><loc_29><loc_83><loc_33></location>Definition 2. Let f ∈ L 2 ( S, d σ ) . The projection of f onto the subspace of constant functions:</text>
<formula><location><page_10><loc_40><loc_26><loc_84><loc_31></location>P m ( f ) := 1 4 π ∫ S f d σ (38)</formula>
<text><location><page_10><loc_15><loc_22><loc_71><loc_26></location>will be called the monopole part of f , whereas the projection onto M 3 = span { X 1 , X 2 , X 3 } :</text>
<formula><location><page_10><loc_36><loc_15><loc_84><loc_22></location>P d ( f ) := 3 ∑ i =1 ( X i ∫ S X i f d σ ∫ S ( X i ) 2 d σ ) (39)</formula>
<text><location><page_11><loc_15><loc_85><loc_60><loc_87></location>will be called the dipole part of f . In addition, we set</text>
<formula><location><page_11><loc_29><loc_81><loc_84><loc_84></location>M 4 := span { 1 } ⊕ M 3 = span { 1 , X 1 , X 2 , X 3 } , (40)</formula>
<text><location><page_11><loc_15><loc_78><loc_76><loc_81></location>and P md ( f ) := P m ( f ) + P d ( f ) ∈ M 4 denotes the mono-dipole part of f .</text>
<text><location><page_11><loc_15><loc_65><loc_84><loc_78></location>The above structure enables us to define the multipole decomposition of the functions defined on a topological sphere S in terms of eigenspaces of the Laplace operator associated with the metric σ AB . If h is a function on S , then by h m := P m ( h ) we denote its monopole (constant) part, by h d := P d ( h ) -the dipole part (projection to the eigenspace of the Laplacian with eigenvalue -2 ). By h w := ( I -P md )( h ) = h -h m -h d we denote the 'wave', or mono-dipole-free, part of h , h dw := ( I -P m )( h ) = h -h m = h d + h w , and finally h md := P md ( h ) = h m + h d .</text>
<text><location><page_11><loc_15><loc_57><loc_84><loc_65></location>Remark 3. Mutually orthogonal projectors P md and P w := ( I -P md ) are, of course, continuous, when considered as operators in the Hilbert space L 2 ( S, d σ ) . For our purposes we have to consider them as operators in the Banach space C ( k,α ) . Here, no 'orthogonality' is defined. Nevertheless, both operators are again continuous projectors. They define an isomorphism:</text>
<formula><location><page_11><loc_39><loc_53><loc_59><loc_56></location>C ( k,α ) ∼ = C ( k,α ) md × C ( k,α ) w ,</formula>
<text><location><page_11><loc_15><loc_47><loc_84><loc_53></location>where C ( k,α ) md = P md ( C ( k,α ) ) ≡ M 4 and C ( k,α ) w = P w ( C ( k,α ) ) . Hence, a function f ∈ C ( k,α ) is uniquely characterized by its mono-dipole part f md and the remaining 'wave' part f w , i.e. we have: f = ( f md , f w ) .</text>
<text><location><page_11><loc_15><loc_40><loc_84><loc_46></location>Definition 3. Let Σ be a Riemannian three-manifold and let S ⊂ Σ be a submanifold homeomorphic with S 2 ⊂ R 3 . We say that S is a rigid sphere if its mean extrinsic curvature k satisfies k ∈ M 4 , i.e. if the following equation holds:</text>
<formula><location><page_11><loc_45><loc_38><loc_84><loc_40></location>k w = 0 . (41)</formula>
<section_header_level_1><location><page_11><loc_15><loc_34><loc_63><loc_36></location>2.5 The 4-D spacetime case - an outline</section_header_level_1>
<text><location><page_11><loc_15><loc_23><loc_83><loc_33></location>Definition of a rigid sphere in a Lorenzian four-manifold is more complicated: to control 'rigidity' of a sphere, we must take into account more geometry. For this purpose we consider the extrinsic curvature vector of S : k a = k a AB g AB , where k a AB denotes the external curvature tensor of S (here, a, b are indices corresponding to the subspace orthogonal to S whereas A, B label coordinates on S ). Moreover, we consider its torsion:</text>
<formula><location><page_11><loc_42><loc_20><loc_84><loc_23></location>/lscript A = ( m |∇ A n ) , (42)</formula>
<formula><location><page_11><loc_44><loc_15><loc_84><loc_19></location>n := k ‖ k ‖ , (43)</formula>
<text><location><page_11><loc_15><loc_19><loc_20><loc_20></location>where</text>
<text><location><page_11><loc_15><loc_11><loc_70><loc_16></location>‖ k ‖ = √ k a g ab k b , and m is a vector orthogonal to both k and S .</text>
<text><location><page_12><loc_15><loc_80><loc_83><loc_87></location>Definition 4. Let M be a Lorenzian four-manifold (a generic curved spacetime) and let S ⊂ M be a spacelike submanifold homeomorphic with S 2 ⊂ R 3 . We say that S is a rigid sphere if k = ( k a ) is spacelike and the following two conditions are satisfied:</text>
<formula><location><page_12><loc_43><loc_76><loc_84><loc_79></location>‖ k ‖ ∈ M 4 , (44)</formula>
<formula><location><page_12><loc_42><loc_74><loc_84><loc_77></location>∇ A /lscript A ∈ M 3 . (45)</formula>
<text><location><page_12><loc_15><loc_70><loc_83><loc_73></location>In this paper we limit ourselves to the purely Riemannian 3D-setting. The general, pseudo-riemannian case will be analyzed in a subsequent paper.</text>
<text><location><page_12><loc_15><loc_64><loc_83><loc_67></location>Example 3. Rigid spheres in a four-dimensional Minkowski spacetime and in Euclidean three-space.</text>
<text><location><page_12><loc_15><loc_58><loc_83><loc_64></location>Let M 0 be the flat Minkowski spacetime, i.e. the space R 4 parameterized by the Lorentzian coordinates ( x α ) = ( x 0 , . . . , x 3 ) and equipped with the metric η = ( η αβ ) = diag( -1 , 1 , 1 , 1) (Greek indices run always from 0 to 3).</text>
<text><location><page_12><loc_15><loc_57><loc_83><loc_58></location>Consider in M 0 a round sphere , i.e. the two-dimensional submanifold defined by</text>
<formula><location><page_12><loc_30><loc_49><loc_68><loc_56></location>S T,R := { x ∈ R 4 ∣ ∣ ∣ x 0 = T , 3 ∑ i =1 ( x i ) 2 = R 2 } ,</formula>
<text><location><page_12><loc_15><loc_47><loc_83><loc_50></location>where the time T ∈ R and the sphere's radius R > 0 are fixed. It may be easily verified that the submanifold fulfills the following conditions:</text>
<formula><location><page_12><loc_38><loc_42><loc_84><loc_46></location>√ k a g ab k b = 2 R ∈ M 4 , (46)</formula>
<formula><location><page_12><loc_42><loc_40><loc_84><loc_43></location>∇ A /lscript A = 0 ∈ M 3 , (47)</formula>
<text><location><page_12><loc_15><loc_25><loc_84><loc_39></location>hence each round sphere S T,R in Minkowski spacetime M 0 is a rigid sphere. Using Poincaré symmetry group of M 0 , it is easy to check that there is an 8-parameter family of such spheres. Indeed, fixing the value of R , a 7-parameter family remains left. All of them may be obtained from a single sphere, say S 0 ,R , by the action of the 10-parameter Poincaré group. Because the three-parameter subgroup of rotations corresponds to internal symmetries of S 0 ,R , we are left with 7 parameters only. The parameter R corresponds to the dilation group. Hence, we have 8 ( = 10 -3 + 1 ) parameters.</text>
<text><location><page_12><loc_15><loc_16><loc_84><loc_26></location>In Euclidean three-space (represented by a slice { x 0 = 0 } in M 0 ) the family of rigid spheres reduces to four-parameter family of such spheres, where 4 = 3+1 - three translations plus dilation (or similarity transformations minus rotations 4 = 7 -3 ). Each round sphere in Euclidean three-space is a rigid sphere because its mean extrinsic curvature k = -2 R ∈ M 4 .</text>
<section_header_level_1><location><page_13><loc_15><loc_85><loc_84><loc_87></location>3 Existence of rigid spheres in a Riemannian space</section_header_level_1>
<text><location><page_13><loc_15><loc_77><loc_83><loc_84></location>Let Σ be a three-dimensional Riemannian manifold. Let S ⊂ Σ be a two-manifold diffeomorphic to the unit sphere S 2 ⊂ R 3 . We consider the following problems: 1) Can we deform S in such a way that the resulting submanifold becomes a rigid sphere? 2) How many of such deformations exist in a vicinity of S ?</text>
<text><location><page_13><loc_15><loc_68><loc_84><loc_77></location>To parameterize these deformations we introduce in a neighbourhood of S a Gaussian system of coordinates ( u, x A ) . Here, by ( x A ) , A = 1 , 2 , we denote any coordinate system on S , whereas u is the arc-length parameter along the ' { x A = const . } ' geodesics starting orthogonally from S . The three-metric takes, therefore, the form</text>
<formula><location><page_13><loc_35><loc_67><loc_84><loc_68></location>g = d u 2 + g AB ( u, x A ) d x A d x B . (48)</formula>
<text><location><page_13><loc_15><loc_63><loc_83><loc_66></location>Suppose, moreover, that coordinates ( x A ) = ( ϑ, ϕ ) are conformal and equilibrated on S . This means that we have</text>
<formula><location><page_13><loc_34><loc_59><loc_84><loc_61></location>˚ g AB d x A d x B = ψ · ( σ AB d x A d x B ) , (49)</formula>
<text><location><page_13><loc_15><loc_57><loc_20><loc_58></location>where</text>
<formula><location><page_13><loc_41><loc_55><loc_84><loc_56></location>˚ g AB := g AB (0 , x A ) (50)</formula>
<text><location><page_13><loc_15><loc_51><loc_83><loc_54></location>is the induced two-metric on S , σ is the 'round' two-metric on the Euclidean unit sphere:</text>
<formula><location><page_13><loc_35><loc_49><loc_84><loc_51></location>σ AB d x A d x B = d ϑ 2 +sin 2 ϑ d ϕ 2 , (51)</formula>
<text><location><page_13><loc_15><loc_45><loc_83><loc_48></location>and the function ψ is dipole-free ( ψ d = 0 ). Second fundamental form of S is given by:</text>
<formula><location><page_13><loc_42><loc_42><loc_84><loc_45></location>˚ k AB = -1 2 g AB,u . (52)</formula>
<text><location><page_13><loc_15><loc_34><loc_83><loc_41></location>Its trace does not need to belong to the space M 4 of mono-dipole-like functions, i.e. the surface S does not need to be a rigid sphere. We are looking for such deformations of S , for which the resulting surface fulfills already the rigidity condition.</text>
<text><location><page_13><loc_15><loc_29><loc_83><loc_34></location>Any deformation of S which is sufficiently small may be uniquely parameterized by a function τ = τ ( x A ) , such that the deformed surface S τ is given by:</text>
<formula><location><page_13><loc_37><loc_27><loc_84><loc_29></location>S τ = { ( u, x A ) | u = τ ( x A ) } . (53)</formula>
<text><location><page_13><loc_15><loc_25><loc_51><loc_27></location>The surface S τ carries the induced metric:</text>
<formula><location><page_13><loc_18><loc_20><loc_84><loc_25></location>g | S τ = [ d τ ( x A ) ] 2 + g AB ( τ ( x C ) , x C ) d x A d x B =: g AB ( x C ) d x A d x B , (54)</formula>
<formula><location><page_13><loc_30><loc_16><loc_84><loc_20></location>g AB ( x C ) = ( ∂ A τ )( ∂ B τ ) + g AB ( τ ( x C ) , x C ) . (55)</formula>
<text><location><page_13><loc_15><loc_19><loc_20><loc_21></location>where</text>
<text><location><page_13><loc_15><loc_13><loc_83><loc_16></location>Here, we use the same coordinate system ( x A ) , which was previously used for S . However, these coordinates do not need to be neither conformally spherical nor</text>
<text><location><page_14><loc_15><loc_75><loc_83><loc_87></location>equilibrated. To verify that the deformation τ was successful, i.e. that S τ is a rigid sphere, we have to pass to an equilibrated system of spherical coordinates, say ̂ x A , on S τ . To make this choice unique, we use the north pole: n := { ϑ = 0 } , and the 'Gulf of Guinea': m := { ϑ = π 2 ; ϕ = 0 } to get rid of the rotation nonuniqueness (cf. Theorem 2). This way we obtain an equilibrated version ̂ g AB of the metric (55). Finally, we calculate the extrinsic curvature k and check whether or not its wave part k w ( S τ ) satisfies condition k w ( S τ ) = 0 .</text>
<text><location><page_14><loc_15><loc_70><loc_83><loc_75></location>The idea of our paper may, therefore, be sketched as follows. We begin with a metric (48) which is of the class C ( k,α ) . The above construction defines a continuous mapping:</text>
<formula><location><page_14><loc_22><loc_66><loc_84><loc_69></location>C ( k +1 ,α ) md × C ( k +1 ,α ) w /owner ( τ md , τ w ) = τ -→ F ( τ ) := k w ∈ C ( k -1 ,α ) w . (56)</formula>
<text><location><page_14><loc_15><loc_55><loc_83><loc_65></location>Indeed, the resulting metric in a neighbourhood of S τ is obtained from g and the first derivatives of τ . The function τ being of the class C ( k +1 ,α ) , the metric obtained this way is again of the class C ( k,α ) . Due to Theorem 2, its equilibrated version ̂ g AB is again of the same class. Finally, the extrinsic curvature k is obtained, using first derivatives of this metric. Hence, the result is of the class C ( k -1 ,α ) and the entire procedure is continuous.</text>
<text><location><page_14><loc_17><loc_53><loc_63><loc_55></location>Now, rigid spheres are those, which satisfy equation:</text>
<formula><location><page_14><loc_45><loc_50><loc_84><loc_52></location>F ( τ ) = 0 . (57)</formula>
<text><location><page_14><loc_15><loc_45><loc_83><loc_48></location>We are going to prove that, for a generic metric g , which is sufficiently close to the flat metric, the above equation defines an implicit function:</text>
<formula><location><page_14><loc_29><loc_41><loc_84><loc_44></location>M 4 ≡ C ( k +1 ,α ) md /owner τ md -→ H ( τ md ) ∈ C ( k +1 ,α ) w , (58)</formula>
<text><location><page_14><loc_15><loc_39><loc_23><loc_40></location>such that</text>
<formula><location><page_14><loc_39><loc_36><loc_84><loc_39></location>F ( τ md , H ( τ md )) ≡ 0 , (59)</formula>
<text><location><page_14><loc_15><loc_33><loc_83><loc_36></location>or, equivalently, that S ( τ md ,H ( τ md )) is a rigid sphere. The main result of our paper follows as:</text>
<text><location><page_14><loc_15><loc_25><loc_83><loc_32></location>Theorem 3. Generically (i. e. if the metric is sufficiently close to the flat metric, e. g. in the external region of an asymptotically flat space) there exists a fourparameter family of rigid spheres in a neighbourhood of a given two-sphere S ⊂ Σ , corresponding to the four-parameter family of mono-dipole functions τ md on S .</text>
<section_header_level_1><location><page_14><loc_15><loc_21><loc_63><loc_23></location>3.1 Infinitesimal deformations of spheres</section_header_level_1>
<text><location><page_14><loc_15><loc_15><loc_84><loc_20></location>To prove existence of the implicit function (59) it is sufficient to show that, given a mono-dipole deformation τ md , the partial derivative of F with respect to the 'wave-like' deformation τ w is an isomorphism of C ( k +1 ,α ) w onto C ( k -1 ,α ) w .</text>
<text><location><page_15><loc_15><loc_81><loc_83><loc_87></location>For this purpose, we analyze the infinitesimal, linear version of the construction discussed above. Consider, therefore, a transversal deformation τ = τ ( x A ) of S ⊂ Σ and a small deformation parameter ε :</text>
<formula><location><page_15><loc_35><loc_78><loc_84><loc_81></location>S /squiggleright S τ = { ( u, x A ) | u = ετ ( x A ) } . (60)</formula>
<text><location><page_15><loc_15><loc_76><loc_80><loc_77></location>Under such transformation the induced metric changes in the following way:</text>
<formula><location><page_15><loc_35><loc_72><loc_84><loc_75></location>g AB -˚ g AB = -2 ετ ˚ k AB + O ( ε 2 ) . (61)</formula>
<text><location><page_15><loc_15><loc_65><loc_83><loc_71></location>Even if the initial system of coordinates was equilibrated, the transformed metric does not need to be conformally spherical. The non-sphericality of the metric must be, therefore, compensated by a change of coordinates. Its infinitesimal version is described by a tangential (with respect to S ) deformation</text>
<formula><location><page_15><loc_42><loc_61><loc_84><loc_64></location>x A = x A -εξ A . (62)</formula>
<text><location><page_15><loc_15><loc_59><loc_74><loc_63></location>̂ Under such coordinate transformation the metric changes as follows:</text>
<formula><location><page_15><loc_39><loc_53><loc_84><loc_57></location>̂ g AB = g AB -£ 2 ε /vector ξ g AB , (63)</formula>
<text><location><page_15><loc_15><loc_46><loc_84><loc_55></location>where the last term represents the Lie derivative of the metric g AB with respect to the vector field ' -εξ A ' on S . But, according to (61), the difference between g AB and ˚ g AB is already of the first order in ε . Hence, if we replace it by the Lie derivative of the metric ˚ g AB , the error will be of the second order in ε . Using the Killing formula for the Lie derivative of the metric, we finally obtain:</text>
<formula><location><page_15><loc_36><loc_40><loc_84><loc_45></location>̂ g AB = g AB +2 εξ ( A || B ) + O ( ε 2 ) , (64)</formula>
<text><location><page_15><loc_15><loc_39><loc_83><loc_42></location>and the covariant derivative || A is taken with respect to the original metric ˚ g AB . Hence, we have:</text>
<text><location><page_15><loc_15><loc_31><loc_84><loc_37></location>̂ g AB -˚ g AB = -2 ετ ˚ k AB +2 εξ ( A || B ) + O ( ε 2 ) . (65) Let us decompose the above equation into the trace and the trace-free parts, calculated with respect to ˚ g AB (we omit the terms of order ε 2 and higher):</text>
<formula><location><page_15><loc_16><loc_24><loc_84><loc_31></location>̂ g AB -˚ g AB = ( εξ C || C -ετ ˚ k ) ˚ g AB -2 ετ ˚ κ AB +2 ε ( ξ ( A || B ) -1 2 ξ C || C ˚ g AB ) , (66)</formula>
<formula><location><page_15><loc_40><loc_21><loc_84><loc_24></location>˚ κ AB := ˚ k AB -1 2 ˚ k ˚ g AB (67)</formula>
<text><location><page_15><loc_15><loc_24><loc_20><loc_25></location>where</text>
<text><location><page_15><loc_15><loc_15><loc_83><loc_20></location>is the traceless part of ˚ k AB . We want ̂ g AB to be conformally spherical, i.e. ̂ g AB = α · ˚ g AB . This implies:</text>
<formula><location><page_15><loc_19><loc_12><loc_84><loc_16></location>( 1 -ετ ˚ k -α + εξ C || C ) ˚ g AB -2 ετ ˚ κ AB +2 εξ ( A || B ) -εξ C || C ˚ g AB = 0 . (68)</formula>
<text><location><page_16><loc_15><loc_85><loc_69><loc_87></location>The trace part of this equation defines uniquely the value of α :</text>
<formula><location><page_16><loc_39><loc_81><loc_84><loc_84></location>α = 1 -ετ ˚ k + εξ C || C , (69)</formula>
<text><location><page_16><loc_15><loc_79><loc_48><loc_80></location>whereas the trace-free part reduces to:</text>
<formula><location><page_16><loc_34><loc_75><loc_84><loc_77></location>ξ A || B + ξ B || A -ξ C || C ˚ g AB = 2 τ ˚ κ AB . (70)</formula>
<text><location><page_16><loc_15><loc_66><loc_83><loc_74></location>It is convenient to rewrite equation (70) in terms of the 'round' unit-sphere geometry σ AB . For this purpose we use the following conventions: components of a vector (i.e. an object having upper indices ) are the same in both geometries σ AB and ˚ g AB = ψσ AB . Components of a co-vector ( lowered indices ) are denoted as follows:</text>
<formula><location><page_16><loc_29><loc_64><loc_84><loc_66></location>ξ σ A = σ AB ξ B , ξ A =˚ g AB ξ B = ψσ AB ξ B = ψξ σ A . (71)</formula>
<text><location><page_16><loc_15><loc_60><loc_83><loc_63></location>The covariant derivative with respect to σ AB will be denoted by /wreathproduct/wreathproduct A , e.g. ξ σ A /wreathproduct/wreathproduct B . Equation (70) can be easily rewritten as:</text>
<formula><location><page_16><loc_34><loc_55><loc_84><loc_59></location>ξ σ A /wreathproduct/wreathproduct B + ξ σ B /wreathproduct/wreathproduct A -ξ C /wreathproduct/wreathproduct C σ AB = 2 τ ψ ˚ κ AB . (72)</formula>
<text><location><page_16><loc_15><loc_41><loc_83><loc_54></location>The left-hand side of this equation defines a mapping from the space of vector fields on the unit sphere to the space of trace-free rank 2 tensor fields. The kernel of this mapping consists of the dipole fields 4 . The 'Fredholm alternative' argument shows that the operator on the left-hand side defines an isomorphism between the space of dipole-free vector fields on the unit sphere and the space of trace-free rank 2 tensor fields (see also [7]). This isomorphism (in metric σ ) will be denoted by i 12 . Hence, the wave part of ξ A is implied uniquely by equation (72) (see Appendix):</text>
<formula><location><page_16><loc_39><loc_37><loc_84><loc_42></location>ξ w A = i -1 12 ( 2 τ ψ ˚ κ AB ) , (73)</formula>
<text><location><page_16><loc_15><loc_35><loc_71><loc_36></location>whereas the dipole part of ξ A , i.e. the field ξ d A , remains arbitrary.</text>
<text><location><page_16><loc_15><loc_21><loc_84><loc_34></location>The above choice of the wave-like component of the tangential deformation ξ w A guarantees that the new coordinate system ̂ x A is conformally spherical. We would like it to be also: 1) equlibrated and 2) satisfying conditions related to the two fixed points n and m . These conditions mean that the field ξ has to vanish at the north pole n and that its ϕ -component vanishes at m . The above 3 + 3 = 6 conditions fix uniquely the total dipole-part of the tangential (to S ) deformation ξ A . This way the continuous mapping which assigns uniquely the tangential deformation ξ A to its transversal component τ has been defined.</text>
<section_header_level_1><location><page_17><loc_15><loc_85><loc_79><loc_87></location>3.2 The infinitesimal change of the extrinsic curvature</section_header_level_1>
<text><location><page_17><loc_15><loc_74><loc_84><loc_84></location>Now, we are going to calculate the infinitesimal change of the wave part k w of the mean curvature 5 , i.e. derivative of the mapping (56) with respect to the 'wavelike' deformation τ w . We have k = ˜ g AB k AB , where ˜ g AB denotes the inverse of the two-metric g AB (whereas g AB denotes the corresponding components of the inverse three-metric.) The simplest way to calculate this change is to use a coordinate system ( ω, x A ) , adapted to the deformed surface:</text>
<formula><location><page_17><loc_31><loc_70><loc_84><loc_73></location>ω = u -ετ ( x A ) , i . e . S τ = { ω = 0 } , (74)</formula>
<text><location><page_17><loc_15><loc_68><loc_29><loc_70></location>and the formula:</text>
<formula><location><page_17><loc_41><loc_65><loc_84><loc_69></location>k AB = 1 √ g ωω Γ ω AB . (75)</formula>
<text><location><page_17><loc_15><loc_63><loc_57><loc_65></location>The three-metric g takes now the following form:</text>
<formula><location><page_17><loc_28><loc_60><loc_84><loc_62></location>g = d ω 2 +2 ετ ,A d ω d x A + g AB d x A d x B + O ( ε 2 ) . (76)</formula>
<text><location><page_17><loc_15><loc_57><loc_56><loc_59></location>This implies g ωω = 1 + O ( ε 2 ) and, consequently,</text>
<formula><location><page_17><loc_16><loc_52><loc_84><loc_56></location>k AB = Γ ωAB + g ωC Γ CAB + O ( ε 2 ) = 1 2 ( g ωA || B + g ωB || A -g AB,ω ) + O ( ε 2 ) , (77)</formula>
<text><location><page_17><loc_15><loc_45><loc_83><loc_52></location>where we treat the 'shift vector' g ωA = ετ ,A as a covector field on S τ . The first two terms combine to ετ || AB , whereas the last one: g AB,ω ( S τ ) can be approximated by the quantity g AB,ω ( S ) = -2 ˚ k AB plus the derivative of this object. Finally, we have</text>
<formula><location><page_17><loc_32><loc_43><loc_84><loc_45></location>k AB = ˚ k AB + ετ ˚ k AB,u + ετ || AB + O ( ε 2 ) . (78)</formula>
<text><location><page_17><loc_15><loc_39><loc_83><loc_43></location>Since the derivative g AB,ω of the metric g AB is described by -2 ˚ k AB , the derivative of its inverse ˜ g AB is described by +2 ˚ k AB . Hence, we have:</text>
<formula><location><page_17><loc_36><loc_35><loc_84><loc_38></location>˜ g AB -˚ g AB = 2 ετ ˚ k AB + O ( ε 2 ) , (79)</formula>
<text><location><page_17><loc_15><loc_33><loc_30><loc_35></location>and, consequently:</text>
<formula><location><page_17><loc_30><loc_30><loc_84><loc_32></location>k = ˜ g AB k AB = ˚ k + ετ∂ u ˚ k + ετ || A A + O ( ε 2 ) . (80)</formula>
<text><location><page_17><loc_15><loc_19><loc_84><loc_29></location>The quantity τ∂ u ˚ k + τ || A A describes already the second variation of area (see Appendix), i.e. the derivative ∇ τ k . However, to calculate the derivative of the mapping (56), we have to select its wave part k w . For this purpose we have to pass to the conformally spherical, equilibrated coordinates ̂ x A , given by formula (62). Infinitesimal change of the scalar function k with respect to this deformation is given by formula:</text>
<formula><location><page_17><loc_39><loc_15><loc_60><loc_19></location>̂ k = k -εξ A k ,A + O ( ε 2 ) .</formula>
<text><location><page_18><loc_15><loc_85><loc_27><loc_87></location>Hence, we get:</text>
<text><location><page_18><loc_15><loc_80><loc_41><loc_81></location>or, equivalently (cf. Appendix),</text>
<formula><location><page_18><loc_31><loc_80><loc_84><loc_85></location>1 ε ( ̂ k -˚ k ) = τ∂ u ˚ k + τ || A A -ξ A ˚ k ,A + O ( ε ) , (81)</formula>
<formula><location><page_18><loc_25><loc_74><loc_84><loc_79></location>1 ε ( ̂ k -˚ k ) = τ ( R u u + ˚ k AB ˚ k AB ) + τ || A A -ξ A ˚ k ,A + O ( ε ) , (82)</formula>
<text><location><page_18><loc_15><loc_73><loc_67><loc_75></location>where R u u = R (d u, ∂ ∂u ) is the component of the Ricci tensor.</text>
<section_header_level_1><location><page_18><loc_15><loc_70><loc_48><loc_71></location>3.3 Proof of the Theorem 3</section_header_level_1>
<text><location><page_18><loc_15><loc_63><loc_83><loc_68></location>The last formula gives, finally, the value of the derivative of the mapping (56). When restricted to the subspace of wave (i.e. mono-dipole-free) deformations, it gives us:</text>
<text><location><page_18><loc_15><loc_55><loc_83><loc_59></location>The above linear operator is, obviously, continuous. In particular, the vector field ξ A is given by formula (73), together with the accompanying vanishing conditions at n and m .</text>
<formula><location><page_18><loc_19><loc_59><loc_84><loc_64></location>C ( k +1 ,α ) w /owner τ ↦→ [ τ ( R u u + ˚ k AB ˚ k AB ) + τ || A A -ξ A ˚ k ,A ] w ∈ C ( k -1 ,α ) w . (83)</formula>
<text><location><page_18><loc_15><loc_51><loc_83><loc_54></location>If the space Σ is flat (Euclidean) and S is a standard (rigid) sphere of radius r , then we have:</text>
<formula><location><page_18><loc_26><loc_48><loc_84><loc_51></location>˚ g AB = r 2 σ AB ; ˚ k AB = -rσ AB ; ˚ k ,A = 0 ; R u u = 0 . (84)</formula>
<text><location><page_18><loc_15><loc_46><loc_47><loc_48></location>Hence, the above operator reduces to:</text>
<formula><location><page_18><loc_20><loc_39><loc_84><loc_46></location>τ w ↦→ [ τ ( R u u + ˚ k AB ˚ k AB ) + τ || A A -ξ A ˚ k ,A ] w = 1 r 2 [(∆ σ +2) τ ] w = 1 r 2 (∆ σ +2)( τ w ) , (85)</formula>
<text><location><page_18><loc_15><loc_30><loc_84><loc_39></location>which is obviously an invertible mapping from C ( k +1 ,α ) w to C ( k -1 ,α ) w . But the mapping (85) depends in a continuous way upon the geometry (metric and curvature) of S . This implies that it remains invertible for sufficiently small deformations of the geometry. This is the case e.g. of a sufficiently big 'coordinate sphere' defined as follows:</text>
<formula><location><page_18><loc_31><loc_24><loc_67><loc_30></location>S /vectorx 0 ,R := { x ∈ R 3 ∣ ∣ ∣ 3 ∑ i =1 ( x i -x i 0 ) 2 = R 2 } ,</formula>
<text><location><page_18><loc_15><loc_23><loc_38><loc_25></location>in an asymptotically flat Σ .</text>
<text><location><page_18><loc_15><loc_19><loc_83><loc_23></location>We say, that Σ is asymptotically flat if there is a coordinate chart ( x k ) covering the exterior of a compact domain D ⊂ Σ and such that</text>
<formula><location><page_18><loc_42><loc_17><loc_56><loc_19></location>g kl = δ kl + h kl ,</formula>
<text><location><page_18><loc_15><loc_13><loc_84><loc_16></location>where h vanishes sufficiently fast at infinity. In that case Σ \ D admits a fourparameter family of rigid spheres, similarly as in the case of the flat metric.</text>
<section_header_level_1><location><page_19><loc_15><loc_85><loc_35><loc_87></location>4 Conclusions</section_header_level_1>
<text><location><page_19><loc_15><loc_62><loc_84><loc_84></location>The main technical ingredient of this paper is the intrinsic, coordinate invariant definition of the 'multipole expansion' of a function defined on a Riemannian twomanifold, diffeomorphic with S 2 . This enables us to select a finite-dimensional family of 'rigid spheres'. The dipole part k d of the curvature parameterizes the position of the center of such a sphere with respect to the center of mass. In particular, k d = 0 corresponds to the spheres, which are centered at the center of mass. Properties of such a foliation have been analyzed in [4]. General topologically spherical coordinates, having property that surfaces { r = const. } are rigid, do not admit supertranslations ambiguity at space infinity. This way symmetries of the 'tangent space at infinity' reduce to a finite-dimensional one. The 4D version of our results, valid for a generic four-dimensional Lorenzian spacetime, which will be presented in the subsequent paper, will do the same job for the symmetry group of the Scri.</text>
<section_header_level_1><location><page_19><loc_15><loc_57><loc_40><loc_59></location>Acknowledgements</section_header_level_1>
<text><location><page_19><loc_15><loc_50><loc_83><loc_55></location>This research was supported by Polish Ministry of Science and Higher Education (grant Nr N N201 372736) and by Narodowe Centrum Nauki (grant DEC2011/03/B/ST1/02625). SŁ was supported by Foundation for Polish Science.</text>
<section_header_level_1><location><page_19><loc_15><loc_45><loc_33><loc_47></location>A Appendix</section_header_level_1>
<section_header_level_1><location><page_19><loc_15><loc_42><loc_72><loc_44></location>A.1 The dipole part of traceless symmetric part</section_header_level_1>
<text><location><page_19><loc_15><loc_39><loc_37><loc_41></location>The kernel of the mapping</text>
<formula><location><page_19><loc_35><loc_35><loc_63><loc_38></location>ξ σ A ↦→ ξ σ A /wreathproduct/wreathproduct B + ξ σ B /wreathproduct/wreathproduct A -σ CD ξ σ C /wreathproduct/wreathproduct D σ AB</formula>
<text><location><page_19><loc_15><loc_31><loc_83><loc_35></location>defined by the left-hand side of the formula (72) consists of the dipole fields. This is a simple consequence of the following observations.</text>
<unordered_list>
<list_item><location><page_19><loc_17><loc_22><loc_84><loc_30></location>· In case of the unit sphere the Hodge decomposition ξ = d α + δβ + h of the covector ξ on a compact manifold does not contain the harmonic part, i.e. harmonic one-form h vanishes ( d h = 0 = δh implies h = 0 ). The topology of the unit sphere (triviality of the corresponding cohomology class) cancels the harmonic part and we can always represent ξ as follows</list_item>
</unordered_list>
<formula><location><page_19><loc_43><loc_18><loc_84><loc_20></location>ξ A = α ,A + ε A B β ,B , (86)</formula>
<text><location><page_19><loc_19><loc_14><loc_83><loc_17></location>where functions α and β are defined up to a constant but their gradients are unique.</text>
<unordered_list>
<list_item><location><page_20><loc_17><loc_84><loc_84><loc_87></location>· The purely dipole covector ξ simply means that the potentials α and β are purely dipole functions: α = a i X i , β = b i X i , where a i , b i are real constants.</list_item>
<list_item><location><page_20><loc_17><loc_78><loc_83><loc_82></location>· Direct computation for dipole functions X i enables one to check the following identity: X i /wreathproduct/wreathproduct AB = -X i σ AB , hence for any dipole function α</list_item>
</unordered_list>
<formula><location><page_20><loc_44><loc_75><loc_84><loc_77></location>α /wreathproduct/wreathproduct AB = -ασ AB . (87)</formula>
<unordered_list>
<list_item><location><page_20><loc_17><loc_71><loc_44><loc_74></location>· Formulae (86) and (87) give</list_item>
</unordered_list>
<formula><location><page_20><loc_41><loc_68><loc_62><loc_71></location>ξ A /wreathproduct/wreathproduct B = -ασ AB -βε AB ,</formula>
<text><location><page_20><loc_19><loc_66><loc_64><loc_68></location>hence the traceless symmetric part of ξ A /wreathproduct/wreathproduct B vanishes.</text>
<section_header_level_1><location><page_20><loc_15><loc_60><loc_84><loc_64></location>A.2 The isomorphism between covector fields and symmetric traceless tensors on ( S 2 , σ AB )</section_header_level_1>
<text><location><page_20><loc_15><loc_57><loc_47><loc_59></location>Let us consider the following diagram:</text>
<formula><location><page_20><loc_21><loc_48><loc_77><loc_56></location>V 0 k +2 ⊕ V 0 k +2 i 01 -→ V 1 k +1 i 12 -→ V 2 k i 21 -→ V 1 k -1 i 10 -→ V 0 k -2 ⊕ V 0 k -2 /arrowbt Fl /arrowbt ˆ /arrowbt ˆ /arrowbt ˆ /arrowbt Fl V 0 k +2 ⊕ V 0 k +2 i 01 -→ V 1 k +1 i 12 -→ V 2 k i 21 -→ V 1 k -1 i 10 -→ V 0 k -2 ⊕ V 0 k -2</formula>
<text><location><page_20><loc_15><loc_46><loc_64><loc_48></location>where the mappings and the spaces are defined as follows:</text>
<formula><location><page_20><loc_38><loc_43><loc_60><loc_45></location>i 01 ( f, g ) = f /wreathproduct/wreathproduct A + ε A B g /wreathproduct/wreathproduct B ,</formula>
<formula><location><page_20><loc_35><loc_39><loc_63><loc_42></location>i 12 ( v ) = v A /wreathproduct/wreathproduct B + v B /wreathproduct/wreathproduct A -σ AB v C /wreathproduct/wreathproduct C ,</formula>
<formula><location><page_20><loc_38><loc_33><loc_60><loc_38></location>i 10 ( v ) = ( v A /wreathproduct/wreathproduct A , ε AB v A /wreathproduct/wreathproduct B ) ,</formula>
<formula><location><page_20><loc_42><loc_38><loc_56><loc_39></location>i 21 ( χ ) = χ A B /wreathproduct/wreathproduct B ,</formula>
<formula><location><page_20><loc_27><loc_33><loc_71><loc_34></location>Fl ( f, g ) = ( g, f ) , ˆ v A = ε A B v B , ˆ χ AB = ε A C χ CB ,</formula>
<text><location><page_20><loc_15><loc_30><loc_60><loc_32></location>V 0 k - scalars on S 2 belonging to Hölder space C k,α ,</text>
<text><location><page_20><loc_15><loc_28><loc_60><loc_30></location>V 1 k - covectors on S 2 belonging to Hölder space C k,α</text>
<text><location><page_20><loc_61><loc_28><loc_61><loc_30></location>,</text>
<text><location><page_20><loc_15><loc_27><loc_76><loc_28></location>V 2 k - symmetric traceless tensors on S 2 belonging to Hölder space C k,α .</text>
<text><location><page_20><loc_15><loc_22><loc_83><loc_27></location>Denote by ∆ σ the Laplace operator on S 2 and by SH l the space of spherical harmonics of degree l , ( f ∈ SH l ⇐⇒ ∆ σ f = -l ( l +1) f ). The following equality</text>
<formula><location><page_20><loc_35><loc_19><loc_63><loc_22></location>i 10 · i 21 · i 12 · i 01 = ∆ σ (∆ σ +2)</formula>
<text><location><page_20><loc_15><loc_13><loc_83><loc_19></location>shows that if we restrict ourselves to the spaces V 0 := V 0 /circleminus [ SH 0 ⊕ SH 1 ] = ( I -P md ) V 0 ( ∆ σ (∆ σ +2) V 0 = V 0 ) and V 1 = V 1 /circleminus [ i 01 ( SH 1 )] ( (∆ σ + I ) V 1 = V 1 ) then all the mappings in the above diagram become isomorphisms.</text>
<section_header_level_1><location><page_21><loc_15><loc_85><loc_70><loc_87></location>A.2.1 Integral operators, generalized Green's functions</section_header_level_1>
<text><location><page_21><loc_15><loc_83><loc_63><loc_84></location>Solution of the Helmholtz equation on a unit sphere S 2 :</text>
<formula><location><page_21><loc_32><loc_79><loc_65><loc_81></location>[∆ σ + l ( l +1)]Ψ l ( n ) = Φ( n ) , n ∈ S 2</formula>
<text><location><page_21><loc_15><loc_77><loc_83><loc_78></location>is given (see e.g. [13]) in terms of the generalized Green's function ¯ G l as follows:</text>
<formula><location><page_21><loc_35><loc_72><loc_84><loc_76></location>Ψ l ( n ) = ∫ S 2 ¯ G l ( n, n ' )Φ( n ' ) d 2 n ' . (88)</formula>
<text><location><page_21><loc_15><loc_61><loc_84><loc_71></location>Here n = D ( ϑ, ϕ ) given by (17) and d 2 n = d σ . The solution Ψ l ( n ) is automatically orthogonal to the space SH l (the kernel of Helmholtz operator ∆ σ + l ( l +1) ) because Green's function is orthogonal to this space. In our case we need to write the inverse of the operator ∆ σ (∆ σ +2) as a double integral with the corresponding kernels ¯ G l for l = 0 and l = 1 . More precisely, the solution g of the equation ∆ σ (∆ σ +2) g = f (with P md f = 0 ) is given in the following form:</text>
<formula><location><page_21><loc_15><loc_48><loc_84><loc_60></location>g ( n ) = P w ∫ S 2 ¯ G 0 ( n, n ') · · [∫ S 2 ¯ G 1 ( n ' , n ' ) f ( n ' ) d 2 n ' -1 4 π ∫ S 2 × S 2 ¯ G 1 ( m,n ' ) f ( n ' ) d 2 n ' d 2 m ] d 2 n ' = ∫ S 2 ¯ G 0 ( n, n ') [∫ S 2 ¯ G 1 ( n ' , n ' ) f ( n ' ) d 2 n ' ] d 2 n ' , (89)</formula>
<text><location><page_21><loc_15><loc_44><loc_84><loc_47></location>where the projection operator P w provides orthogonality 6 of g to the space SH 0 ⊕ SH 1 . The generalized Green's function written in a standard form:</text>
<text><location><page_21><loc_38><loc_38><loc_38><loc_39></location>/negationslash</text>
<formula><location><page_21><loc_25><loc_38><loc_73><loc_42></location>¯ G l ( n, n ' ) = ∞ ∑ i =0 ,i = l i ∑ m = -i Y im ( n ) Y im ( n ' ) l ( l +1) -i ( i +1) , Y im ∈ SH i ,</formula>
<text><location><page_21><loc_15><loc_35><loc_47><loc_36></location>can be simplified as follows (cf. [13]):</text>
<formula><location><page_21><loc_15><loc_29><loc_83><loc_34></location>¯ G l ( n, n ' ) = 1 4 π P l ( n · n ' ) [ ln 1 -n · n ' 2 + c l ] + 1 2 π l -1 ∑ i =0 2 i +1 ( l -i )( l + i +1) P i ( n · n ' ) ,</formula>
<formula><location><page_21><loc_30><loc_23><loc_68><loc_28></location>c l := 1 2 l +1 -2 l -1 ∑ i =0 ( -1) l + i 2 i +1 ( l -i )( l + i +1) ,</formula>
<text><location><page_21><loc_15><loc_17><loc_83><loc_23></location>Y im - spherical harmonics (orthonormal basis in SH i ), n · n ' ∈ [ -1 , 1] is a scalar product of unit vectors in R 3 and P l ( x ) := 1 2 l l ! ( x 2 -1) ( l ) is the Legendre polynomial.</text>
<section_header_level_1><location><page_22><loc_15><loc_85><loc_49><loc_87></location>A.3 Second variation of area</section_header_level_1>
<text><location><page_22><loc_15><loc_83><loc_82><loc_84></location>The Gaussian coordinates (48) and the definition of the Riemann tensor gives</text>
<formula><location><page_22><loc_38><loc_80><loc_84><loc_81></location>R u AuB = ˚ k AB,u + ˚ k A C ˚ k BC . (90)</formula>
<text><location><page_22><loc_15><loc_77><loc_26><loc_78></location>This leads to</text>
<formula><location><page_22><loc_26><loc_73><loc_84><loc_75></location>k AB = ˚ k AB + ετ ( R u AuB -˚ k A C ˚ k BC ) + ετ || AB + O ( ε 2 ) . (91)</formula>
<text><location><page_22><loc_15><loc_71><loc_53><loc_72></location>Taking the trace (and using (79)), we obtain:</text>
<formula><location><page_22><loc_30><loc_68><loc_84><loc_69></location>k = ˚ k + ετ ( R u u + ˚ k AB ˚ k AB ) + ετ || A A + O ( ε 2 ) . (92)</formula>
<text><location><page_22><loc_15><loc_63><loc_83><loc_66></location>The formulae (80) and (92) are equivalent because of the Gauss-Codazzi equations:</text>
<formula><location><page_22><loc_32><loc_60><loc_84><loc_63></location>2 ∂ u ˚ k = R ( g kl ) -R (˚ g AB ) + ˚ k AB ˚ k AB + ˚ k 2 , (93)</formula>
<formula><location><page_22><loc_31><loc_58><loc_84><loc_61></location>2 R u u = R ( g kl ) -R (˚ g AB ) -˚ k AB ˚ k AB + ˚ k 2 , (94)</formula>
<text><location><page_22><loc_15><loc_51><loc_84><loc_58></location>where R ( g kl ) and R (˚ g AB ) are scalar of curvatures of the three-metric g kl and the two-metric ˚ g AB , respectively. Obviously (80), (92) are the first variations of the mean curvature k , which in the literature (see e.g. [3]) are known as the second variations of area. They are usually presented in the following equivalent form:</text>
<formula><location><page_22><loc_18><loc_47><loc_84><loc_51></location>k -˚ k = ε 2 [( R ( g kl ) -R (˚ g AB ) + ˚ k AB ˚ k AB + ˚ k 2 ) τ +2 τ || A A ] + O ( ε 2 ) . (95)</formula>
<section_header_level_1><location><page_22><loc_15><loc_43><loc_29><loc_45></location>References</section_header_level_1>
<text><location><page_22><loc_16><loc_38><loc_84><loc_42></location>[1] H.P. Gittel et al., On the existence of rigid spheres in four-dimensional spacetime manifolds , to be published.</text>
<text><location><page_22><loc_16><loc_13><loc_84><loc_37></location>[2] P. Chruściel, Sur l'existence de solutions singulières d'une équation «condition de coordonnées» utilisé par J. Kijowski dans l'analyse symplectique de la relativité générale , C. R. Acad. Sci. Paris 299I (1984), 891-894; P. Chruściel, Sur les Feuilletages Conformément Minimaux des Variétés Riemaniennes de Dimension Trois , Comptes Rendus de l' Académie des Sciences de Paris (série I) 301, 609-612 (1985); P. Chruściel, Sur les coordonnées p-harmoniques en relativité générale , Comptes Rendus de l' Académie des Sciences de Paris (série I) 305, 797-800 (1987); J. Jezierski, J. Kijowski, Phys. Rev. D 36 (1987), 1041-1044; J. Kijowski, On positivity of gravitational energy , in Proceedings of the fourth Marcel Grossmann meeting on General Relativity, Rome, 1985, ed. R. Ruffini, Elsvier Science Publ. 1986, 1681-1686; J. Jezierski, J. Kijowski, Unconstrained Degrees of Freedom for Gravitational Waves, β -Foliations and Spherically Symmetric Initial Data , http://arxiv.org/abs/gr-qc/0501073v1 , Preprint ESI 1552 (2004).</text>
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<list_item><location><page_23><loc_16><loc_84><loc_83><loc_87></location>[3] I. Chavel, Riemannian geometry, a modern introduction , Cambridge Studies in Advanced Mathematics 98 , Cambridge University Press, 2006</list_item>
<list_item><location><page_23><loc_16><loc_74><loc_84><loc_82></location>[4] Lan-Hsuan Huang, Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics , Commun. Math. Phys. 300 , 331-373 (2010) arXiv:0810.5086v2 [math.DG] 5 May 2010; On the center of mass of isolated systems with general asymptotics , Class. Quantum Grav. 26 (2009), 015012 (25pp).</list_item>
<list_item><location><page_23><loc_16><loc_71><loc_77><loc_73></location>[5] G. Huisken, T. Ilmanen, Journ. Diff. Geometry 59 (2001), 353-437.</list_item>
<list_item><location><page_23><loc_16><loc_67><loc_83><loc_70></location>[6] J. Jezierski, Classical and Quantum Gravity 11 (1994), 1055-1068; Acta Physica Polonica B 25 (1994), 1413-1417.</list_item>
<list_item><location><page_23><loc_16><loc_64><loc_66><loc_66></location>[7] J. Jezierski, Class. Quant. Grav. 19 (2002), 2463-2490.</list_item>
<list_item><location><page_23><loc_16><loc_60><loc_83><loc_63></location>[8] S. Łęski, Phys. Rev. D 71 (2005), 124018; J. Jezierski, J. Kijowski. S. Łęski, Phys. Rev. D 76 (2007), 024014.</list_item>
<list_item><location><page_23><loc_16><loc_47><loc_84><loc_58></location>[9] L. Lichtenstein, Zur Theorie der konformen Abbildung: Konforme Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete , Bull. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Ser. A (1916), 192-217; B. Chow, Journ. Differential Geometry 33 (1991), 325-334; S. Hildebrandt and H. von der Mosel, Calc. Var. 23 (2005), 415-424; X. Chen, P. Lu and G. Tian, Proceedings of the American Mathematical Society, Vol. 134 (2006), 33913393.</list_item>
<list_item><location><page_23><loc_15><loc_39><loc_84><loc_45></location>[10] R. Penrose, Gravitational collapse - the role of general relativity , Riv. del Nuovo Cim. (numero speziale) 1 (1969), 252-276; Quasi-local Mass and Angular Momentum in General Relativity , Proc. Roy. Soc. Lond. A381 (1982), 53-62;</list_item>
<list_item><location><page_23><loc_15><loc_31><loc_84><loc_38></location>[11] J. Kijowski, Gen. Relat. Grav. Journal 29 (1997), 307-343; N. O'Murchadha, L. Szabados and P. Tod, Phys. Rev. Letters 92 (2004), 259001; P. Chruściel, J. Jezierski and J. Kijowski, Hamiltonian field theory in the radiating regime , Springer Lecture Notes in Physics, Monographs, vol. 70 (2002).</list_item>
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<list_item><location><page_23><loc_15><loc_20><loc_83><loc_25></location>[13] R. Szmytkowski, Closed form of the generalized Green's function for the Helmholtz operator on the two-dimensional unit sphere , Journal of Mathematical Physics 47 (2006), 063506.</list_item>
<list_item><location><page_23><loc_15><loc_14><loc_83><loc_19></location>[14] York, James W. Jr. Covariant decompositions of symmetric tensors in the theory of gravitation , Annales de l'institut Henri Poincaré (A) Physique théorique, 21 no. 4 (1974), p. 319-332.</list_item>
</unordered_list>
</document>
[
{
"title": "Rigid spheres in Riemannian spaces",
"content": "Hans-Peter Gittel 1 , Jacek Jezierski 2 , Jerzy Kijowski 3 , Szymon Łęski 3 , 4 . 1 Department of Mathematics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany 2 Department of Mathematical Methods in Physics Faculty of Physics, University of Warsaw, ul. Hoża 69, Warszawa, Poland 3 Center for Theoretical Physics, Polish Academy of Sciences al. Lotników 32/46, Warszawa, Poland 4 Nencki Institute of Experimental Biology, Polish Academy of Sciences ul. Pasteura 3, Warszawa, Poland June 13, 2018",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We define a special family of topological two-spheres, which we call 'rigid spheres', and prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold whose metric is sufficiently close to the flat metric (e. g. in the external region of an asymptotically flat space). In case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere. The rigid spheres can be used as building blocks for various ('spherical', 'bispherical' etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. [4]) and the foliations used by us (cf. [8]) in the analysis of the two-body problem.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "In General Relativity Theory, the amount of gravitational energy (mass) contained in a portion V ⊂ Σ of a Cauchy three-surface Σ is assigned to its boundary S = ∂V , rather than to the volume V itself (cf. the notion of a 'quasi-local' mass introduced by Penrose, [10]). The above philosophy was also used in [11], where important quasi-local observables (like, e.g., momentum, angular momentum or center of mass) assigned to a generic 2D surface (whose topology is that of S 2 ) have acquired a hamiltonian interpretation as generators of the corresponding canonical transformations of the (appropriately defined) phase space of gravitational initial data. Recently, we were able to define energy contained in an asymptotically Schwarzschild-de Sitter spacetime (cf. [12]), and again the quasilocal, hamiltonian description of the field dynamics provided an adequate starting point for our analysis. Typically, the S 2 -spheres used for the quasi-local purposes come from specific spacetime foliations { t = const.; r = const. } , where a specific choice of coordinates t and r plays the role of a gauge. In literature, gauge conditions based on 3Delliptic problems have been mostly used (see e.g. 'traceless-transversal' condition advocated by J. York (see e.g. [14]) or a 'p-harmonic gauge' analyzed in [2]). Important results have been obtained by Huisken and Ilmanen (cf. [5]) who used a parabolic gauge condition imposed for the radial coordinate r . The same gauge was also used by one of us (J.J., see [6]) to prove stability of the ReissnerNordström solution, together with a version of Penrose's inequality. For purposes of the quasi-local analysis, these approaches exhibit an obvious drawback consisting in the fact that we do not control intrinsic properties of the surfaces { r = const. } constructed this way. This feature was partially removed by Huang in [4], where new 3D foliations were thoroughly analyzed. Their fibers { r = const. } are selected by a 2D-elliptic condition: k = const., where k denotes the mean extrinsic curvature. In a generic Riemannian three-manifold Σ , the above equation admits a one-parameter family of 'spheres'. Physically, they are related to the 'center of mass' of the geometry (cf. [4]). Unfortunately, the above condition is not stable with respect to small perturbations of the geometry. Indeed, in the (flat) Euclidean space E 3 , this condition admits not 'one-' but a four-parameter family of solutions (parameterized e.g. by the radius R and the three coordinates of a center). Moreover, the exclusive use of the center of mass reference frame is often too restrictive for physical applications. In particular, it does not allow us to describe easily the momentum i.e. the generator of space translations. In the present paper we propose a new gauge condition, which is also 2Delliptic but does not exhibit the above drawback. Indeed, in a generic Riemannian three-manifold our condition selects a four-parameter family of solutions, like in the Euclidean space E 3 . Moreover, our condition is weaker than ' k = const.' (equivalent in the non-generic, Euclidean case, only). Topological two-spheres satisfying our condition will be called 'rigid spheres'. They can be organized in topologically different ways: not necessarily standard 'nested spheres foliations', but also e.g. 'bispherical foliations' which already proved to be very useful in the analysis of the two body problem 1 (see [8]). We expect that various such arrangements, with rigid spheres used as building blocks, will provide useful gauge conditions in General Relativity Theory. The present paper is a part of a bigger project, where we construct 'spheres' which are rigid not only with respect to 3D, but also with respect to 4D deformations. More precisely, an eight-parameter family of similar 'rigid spheres' will be constructed in a generic four-dimensional Lorentzian spacetime. In the present paper we limit ourselves to the 3D Riemannian case. It turns out, however, that our construction can be generalized to the entire pseudo-Riemannian spacetime M , instead of the Riemannian Cauchy three-space Σ ⊂ M . The idea of this extension is to mimic the case of the flat Minkowski space, where all possible round spheres, embedded in all possible flat subspaces Σ of M , form an eightparameter family. All of them can be obtained from a single one by the action of the product of the one-parameter group of dilations (changing the size of S ) and the ten-parameter Poincaré group, quotiented by the three-parameter rotation group. The 4D version of our construction will take into account not only the external curvature of S , but also its torsion (in Section 2.5 we give a short outline of this construction, which will be presented in detail in a subsequent paper). The rigid spheres obtained this way will form an eight-parameter family and will be used to construct useful coordinate systems not only on a given Cauchy surface Σ , but also in the entire spacetime. The main advantage of such a construction consists in its rigidity at infinity. We very much hope to be able to eliminate supertranslations and to reduce the symmetry group of the 'Scri', otherwise infinite dimensional, to the finite dimensional one. The construction which we propose in the present paper is based on the following idea. Given a surface S satisfying the rigid sphere condition, consider its infinitesimal deformations. They may be parameterized by sections of the normal bundle T ⊥ S . If we want our condition to admit a four-parameter family of solutions, like in the flat case, its linearization must admit a four-parameter family of deformations. This means that we are not allowed to constrain the complete information about the mean curvature k : four real parameters describing k must be left free. In the flat case these four parameters which have to be left free are: the mean value (or the monopole part ) of k , which is responsible for the size of S , and its dipole part (which vanishes exceptionally in flat case due to GaussCodazzi equations). The dipole part of the deformation is related to the group of translations. In fact, possible motions of a metric sphere are described by the group of Euclidean motions, quotiented by the subgroup of rotations which form the group of internal symmetries of every particular sphere S . To implement the above idea in a non flat case, an intrinsic, geometric notion of a multipole expansion on an arbitrary Riemannian, topologically S 2 -surface is proposed in Section 2. This construction is our main technical tool and we very much believe in its universal validity, going far beyond the purposes of the present paper. Section 2 is completed with the definition of a rigid sphere. Section 3 contains formulation and the proof of theorem 3: a generic Riemannian three-space admits a four-parameter family of rigid spheres. Our proof is relatively simple, but is valid in the 'weak field region' only. This is sufficient for purposes of the quasi-local analysis of gravitational energy (in fact, the idea originates from our analysis of interaction between two black holes, cf. [8]). Further development concerning strong fields will be given elsewhere. Finally, discussion concerning less known (but necessary) technical results, like specific spectral properties of the Laplace operator on S 2 or the second variation of area, has been shifted to the Appendix.",
"pages": [
2,
3,
4
]
},
{
"title": "2.1 Conformally spherical coordinates",
"content": "Let S be a differential two-manifold, diffeomorphic to the two-sphere S 2 ⊂ R 3 and equipped with a (sufficiently smooth) metric g . Coordinates ( ϑ, ϕ ) = ( x A ) , A = 1 , 2 , defined on a dense subset of S \\ /lscript , where /lscript is topologically a line interval, will be called conformally spherical coordinates if they have the same range of values as the standard spherical coordinates on S 2 ⊂ R 3 and, moreover, if the corresponding metric tensor g AB is conformally equivalent to the standard round metric on S 2 , i.e. the following formula holds: where ψ is a (sufficiently smooth) function on S and Remark 1. Conformally spherical coordinates always exist (cf. [9]). It is easy to see that there is always a six-parameter freedom in the choice of such coordinates. More precisely, if ( ϑ, ϕ ) are conformally spherical coordinates then ( ˜ ϑ, ˜ ϕ ) are also conformally spherical if and only if they may be obtained from ( ϑ, ϕ ) via a conformal transformation of S 2 ⊂ R 3 . Example 1. A 'proper' conformal transformation, i.e. which is not a rotation: Let n ∈ S and τ > 0 be a positive number. Using appropriate rotation, choose conformally spherical coordinates ( ϑ, ϕ ) in such a way that n is a north pole, i.e. the coordinate ϑ vanishes at n . Define where or, equivalently, For the fixed point n these transformations form a one-parameter group 2 : generated by the vector field: which is the (minus) gradient of the function z = cos ϑ . In particular, F n , 1 = I (the identity map) for every n . Moreover, equation (5) implies the following identity: Using (4) and (5) we may easily derive the following formula: Similarly, we may prove: As a conclusion we obtain: where which proves the conformal character of the transformation. Indeed, we have Hence, ( ˜ ϑ, ϕ ) are conformally spherical coordinates if ( ϑ, ϕ ) were.",
"pages": [
4,
5,
6
]
},
{
"title": "2.2 Barycenter of a conformally spherical system",
"content": "Given a system of conformally spherical coordinates on S , consider the corresponding three functions: We have, therefore, a mapping D : ]0 , π [ × ]0 , 2 π [ ↦→ R 3 , given by: The following vector where by < f > we denote the average (mean value) of the function f on S , i.e. the number will be called a 'barycenter' of the system ( ϑ, ϕ ) on S . Of course, we have ‖ X ‖ ≤ 1 , because of the Hölder inequality: Example 2. Consider the proper conformal transformation (4) and calculate the new barycenter where The trigonometric identity: Hence, formula (5) implies: or, equivalently, Moreover, formula (10) and its inverse: give To calculate mean values of the functions (26), (27) and (24) we do not need to pass to new coordinates ( ˜ ϑ, ϕ ) , but we may use, as well, old coordinates ( ϑ, ϕ ) . But we see that for τ → 0 we have ˜ x → 0 , ˜ y → 0 , ˜ z → 1 . The Lebesgue theorem implies, therefore, that for τ → 0 we have implies:",
"pages": [
6,
7
]
},
{
"title": "2.3 Equilibrated spherical coordinates",
"content": "Definition 1. Conformally spherical coordinate system ( ϑ, ϕ ) is called equilibrated , if its barycenter vanishes: X = 0 ∈ R 3 . Remark 2. If there are two equilibrated spherical systems on S then they are related by a rotation. Theorem 1. Each metric tensor on S admits a unique (up to rotations) equilibrated spherical system. Proof. Given a metric tensor g on S , choose first any system of conformally spherical coordinates ( ϑ, ϕ ) on S and consider the corresponding identification of its points with the points of S 2 = ∂K (0 , 1) ⊂ R 3 . Consider now the mapping given for N = 0 by the following formula /negationslash where the latter is the barycenter of the coordinates ( ˜ ϑ, ˜ ϕ ) obtained from ( ϑ, ϕ ) by the proper conformal transformation (3) with For N = 0 formula (31) has no sense, but then (32) gives τ = 1 and, whence, equation (5) implies that the corresponding transformation (3) reduces to identity, no matter which vector n do we choose. Consequently, we define F ( 0 ) as the barycenter of the original coordinates ( ϑ, ϕ ) . Obviously, F defined this way is continuous. Moreover, for ‖ N ‖ = 1 we have F ( N ) = N due to (32) and (28). This means that F reduces to the identical mapping when restricted to the boundary S 2 = ∂K (0 , 1) ⊂ K (0 , 1) . Consequently, there must be a point N 0 which solves equation F ( N 0 ) = 0 . This completes the existence proof. To prove the uniqueness, let us suppose that there is another solution: F ( N 1 ) = 0 . Consider now the conformal transformation F n 1 ,τ 1 · F -1 n 0 ,τ 0 . Since the proper conformal transformations do not form any subgroup of the group of all conformal transformations, we cannot assume that it is again a proper transformation. But it may be decomposed into a product of rotations and a proper conformal transformation: where O 1 and O 0 are rotations. Denote by ( ϑ 0 , ϕ 0 ) the spherical coordinates obtained from ( ϑ, ϕ ) by the transformation F n 0 ,τ 0 and then rotation O -1 0 . Similarly, and denote by ( ϑ 1 , ϕ 1 ) the ones obtained from ( ϑ, ϕ ) by F n 1 ,τ 1 and then by rotation O -1 1 . Because a rotation does affect equilibration of coordinates, both systems ( ϑ 0 , ϕ 0 ) and ( ϑ 1 , ϕ 1 ) are equilibrated. But the latter may be obtained from the former by a proper conformal transformation F m ,τ . We shall prove that this is impossible unless τ = 1 or, equivalently, transformation F m ,τ is trivial (identical). For this purpose consider, for each value of τ , the linear function z τ . Without any loss of generality we may assume that (if this is not the case, it is sufficient to perform an appropriate rotation of coordinates). Formula (24) implies the following relation: Hence and it vanishes only at a single point z 0 = 1 . Consequently, its mean value: is strictly negative. This implies that starting from τ = 1 (which corresponds to the identity mapping F m , 1 ) and moving towards the actual value τ < 1 , the ' z '-component of the vector ˜ X m ,τ is strictly increasing. It vanishes at the beginning because ( ϑ 0 , ϕ 0 ) is equilibrated. Hence, it must be strictly positive at the end. This means that the final system ( ϑ 1 , ϕ 1 ) cannot be equilibrated unless τ = 1 and, therefore, both systems coincide. /negationslash Different equilibrated spherical systems of coordinates form, therefore, a threedimensional family. They can be parameterized by the position of a fixed point n ∈ S (north pole) and the geographic longitude of a fixed point m ∈ S (Greenwich). More precisely: given two points n , m ∈ S , n = m , there is a unique equilibrated spherical system ( ϑ, ϕ ) of coordinates on S , such that ϑ vanishes at n and ϕ vanishes at m . Combining these observations with classical results (cf. [9]), we obtain the following /negationslash Theorem 2. Let S be a differential two-manifold, diffeomorphic to the two-sphere S 2 ⊂ R 3 and equipped with a metric g of class C ( k,α ) . For every pair n , m ∈ S , n = m , there is a unique equilibrated spherical system ( ϑ, ϕ ) of coordinates on S , such that ϑ vanishes at n and ϕ vanishes at m , and the metric components g AB are of the same class C ( k,α ) . Here, C ( k,α ) is a Hölder space C k,α ( S 2 ) , defined for 1 /lessorequalslant k ∈ N and 0 < α < 1 . The space consists of those functions on S 2 which have continuous derivatives up to order k and such that the k -th partial derivatives are Hölder continuous with exponent α . This is a locally convex topological vector space. The Hölder coefficient of a function f is defined as follows: /negationslash The function f is said to be (uniformly) Hölder continuous with exponent α if | f | C 0 ,α is finite. In this case the Hölder coefficient can be used as a seminorm. The Hölder space C k,α ( S 2 ) is composed of functions whose derivatives up to order k are bounded and the derivatives of the order k are Hölder continuous. It is a Banach space equipped with the norm where β ranges over multi-indices and",
"pages": [
8,
9,
10
]
},
{
"title": "2.4 Rigid spheres in a Riemannian three-space",
"content": "Given a manifold S equipped with a metric tensor g , there is a three-dimensional space of 'linear functions' uniquely defined on S as linear combinations of functions (14-16), calculated in any equilibrated spherical system of coordinates ( ϑ, ϕ ) . We denote this space by M 3 . By M 4 we denote the space spanned by M 3 and the constant functions on S . Linear functions (14-16) on S are eigenfunctions of the Laplace operator 3 ∆ σ , with the eigenvalue equal to -2 , i.e. ∆ σ X i = -2 X i , where we denote x = X 1 , y = X 2 , z = X 3 . Let us denote by d σ := sin ϑ d ϑ d ϕ the measure associated with the metric σ AB . Definition 2. Let f ∈ L 2 ( S, d σ ) . The projection of f onto the subspace of constant functions: will be called the monopole part of f , whereas the projection onto M 3 = span { X 1 , X 2 , X 3 } : will be called the dipole part of f . In addition, we set and P md ( f ) := P m ( f ) + P d ( f ) ∈ M 4 denotes the mono-dipole part of f . The above structure enables us to define the multipole decomposition of the functions defined on a topological sphere S in terms of eigenspaces of the Laplace operator associated with the metric σ AB . If h is a function on S , then by h m := P m ( h ) we denote its monopole (constant) part, by h d := P d ( h ) -the dipole part (projection to the eigenspace of the Laplacian with eigenvalue -2 ). By h w := ( I -P md )( h ) = h -h m -h d we denote the 'wave', or mono-dipole-free, part of h , h dw := ( I -P m )( h ) = h -h m = h d + h w , and finally h md := P md ( h ) = h m + h d . Remark 3. Mutually orthogonal projectors P md and P w := ( I -P md ) are, of course, continuous, when considered as operators in the Hilbert space L 2 ( S, d σ ) . For our purposes we have to consider them as operators in the Banach space C ( k,α ) . Here, no 'orthogonality' is defined. Nevertheless, both operators are again continuous projectors. They define an isomorphism: where C ( k,α ) md = P md ( C ( k,α ) ) ≡ M 4 and C ( k,α ) w = P w ( C ( k,α ) ) . Hence, a function f ∈ C ( k,α ) is uniquely characterized by its mono-dipole part f md and the remaining 'wave' part f w , i.e. we have: f = ( f md , f w ) . Definition 3. Let Σ be a Riemannian three-manifold and let S ⊂ Σ be a submanifold homeomorphic with S 2 ⊂ R 3 . We say that S is a rigid sphere if its mean extrinsic curvature k satisfies k ∈ M 4 , i.e. if the following equation holds:",
"pages": [
10,
11
]
},
{
"title": "2.5 The 4-D spacetime case - an outline",
"content": "Definition of a rigid sphere in a Lorenzian four-manifold is more complicated: to control 'rigidity' of a sphere, we must take into account more geometry. For this purpose we consider the extrinsic curvature vector of S : k a = k a AB g AB , where k a AB denotes the external curvature tensor of S (here, a, b are indices corresponding to the subspace orthogonal to S whereas A, B label coordinates on S ). Moreover, we consider its torsion: where ‖ k ‖ = √ k a g ab k b , and m is a vector orthogonal to both k and S . Definition 4. Let M be a Lorenzian four-manifold (a generic curved spacetime) and let S ⊂ M be a spacelike submanifold homeomorphic with S 2 ⊂ R 3 . We say that S is a rigid sphere if k = ( k a ) is spacelike and the following two conditions are satisfied: In this paper we limit ourselves to the purely Riemannian 3D-setting. The general, pseudo-riemannian case will be analyzed in a subsequent paper. Example 3. Rigid spheres in a four-dimensional Minkowski spacetime and in Euclidean three-space. Let M 0 be the flat Minkowski spacetime, i.e. the space R 4 parameterized by the Lorentzian coordinates ( x α ) = ( x 0 , . . . , x 3 ) and equipped with the metric η = ( η αβ ) = diag( -1 , 1 , 1 , 1) (Greek indices run always from 0 to 3). Consider in M 0 a round sphere , i.e. the two-dimensional submanifold defined by where the time T ∈ R and the sphere's radius R > 0 are fixed. It may be easily verified that the submanifold fulfills the following conditions: hence each round sphere S T,R in Minkowski spacetime M 0 is a rigid sphere. Using Poincaré symmetry group of M 0 , it is easy to check that there is an 8-parameter family of such spheres. Indeed, fixing the value of R , a 7-parameter family remains left. All of them may be obtained from a single sphere, say S 0 ,R , by the action of the 10-parameter Poincaré group. Because the three-parameter subgroup of rotations corresponds to internal symmetries of S 0 ,R , we are left with 7 parameters only. The parameter R corresponds to the dilation group. Hence, we have 8 ( = 10 -3 + 1 ) parameters. In Euclidean three-space (represented by a slice { x 0 = 0 } in M 0 ) the family of rigid spheres reduces to four-parameter family of such spheres, where 4 = 3+1 - three translations plus dilation (or similarity transformations minus rotations 4 = 7 -3 ). Each round sphere in Euclidean three-space is a rigid sphere because its mean extrinsic curvature k = -2 R ∈ M 4 .",
"pages": [
11,
12
]
},
{
"title": "3 Existence of rigid spheres in a Riemannian space",
"content": "Let Σ be a three-dimensional Riemannian manifold. Let S ⊂ Σ be a two-manifold diffeomorphic to the unit sphere S 2 ⊂ R 3 . We consider the following problems: 1) Can we deform S in such a way that the resulting submanifold becomes a rigid sphere? 2) How many of such deformations exist in a vicinity of S ? To parameterize these deformations we introduce in a neighbourhood of S a Gaussian system of coordinates ( u, x A ) . Here, by ( x A ) , A = 1 , 2 , we denote any coordinate system on S , whereas u is the arc-length parameter along the ' { x A = const . } ' geodesics starting orthogonally from S . The three-metric takes, therefore, the form Suppose, moreover, that coordinates ( x A ) = ( ϑ, ϕ ) are conformal and equilibrated on S . This means that we have where is the induced two-metric on S , σ is the 'round' two-metric on the Euclidean unit sphere: and the function ψ is dipole-free ( ψ d = 0 ). Second fundamental form of S is given by: Its trace does not need to belong to the space M 4 of mono-dipole-like functions, i.e. the surface S does not need to be a rigid sphere. We are looking for such deformations of S , for which the resulting surface fulfills already the rigidity condition. Any deformation of S which is sufficiently small may be uniquely parameterized by a function τ = τ ( x A ) , such that the deformed surface S τ is given by: The surface S τ carries the induced metric: where Here, we use the same coordinate system ( x A ) , which was previously used for S . However, these coordinates do not need to be neither conformally spherical nor equilibrated. To verify that the deformation τ was successful, i.e. that S τ is a rigid sphere, we have to pass to an equilibrated system of spherical coordinates, say ̂ x A , on S τ . To make this choice unique, we use the north pole: n := { ϑ = 0 } , and the 'Gulf of Guinea': m := { ϑ = π 2 ; ϕ = 0 } to get rid of the rotation nonuniqueness (cf. Theorem 2). This way we obtain an equilibrated version ̂ g AB of the metric (55). Finally, we calculate the extrinsic curvature k and check whether or not its wave part k w ( S τ ) satisfies condition k w ( S τ ) = 0 . The idea of our paper may, therefore, be sketched as follows. We begin with a metric (48) which is of the class C ( k,α ) . The above construction defines a continuous mapping: Indeed, the resulting metric in a neighbourhood of S τ is obtained from g and the first derivatives of τ . The function τ being of the class C ( k +1 ,α ) , the metric obtained this way is again of the class C ( k,α ) . Due to Theorem 2, its equilibrated version ̂ g AB is again of the same class. Finally, the extrinsic curvature k is obtained, using first derivatives of this metric. Hence, the result is of the class C ( k -1 ,α ) and the entire procedure is continuous. Now, rigid spheres are those, which satisfy equation: We are going to prove that, for a generic metric g , which is sufficiently close to the flat metric, the above equation defines an implicit function: such that or, equivalently, that S ( τ md ,H ( τ md )) is a rigid sphere. The main result of our paper follows as: Theorem 3. Generically (i. e. if the metric is sufficiently close to the flat metric, e. g. in the external region of an asymptotically flat space) there exists a fourparameter family of rigid spheres in a neighbourhood of a given two-sphere S ⊂ Σ , corresponding to the four-parameter family of mono-dipole functions τ md on S .",
"pages": [
13,
14
]
},
{
"title": "3.1 Infinitesimal deformations of spheres",
"content": "To prove existence of the implicit function (59) it is sufficient to show that, given a mono-dipole deformation τ md , the partial derivative of F with respect to the 'wave-like' deformation τ w is an isomorphism of C ( k +1 ,α ) w onto C ( k -1 ,α ) w . For this purpose, we analyze the infinitesimal, linear version of the construction discussed above. Consider, therefore, a transversal deformation τ = τ ( x A ) of S ⊂ Σ and a small deformation parameter ε : Under such transformation the induced metric changes in the following way: Even if the initial system of coordinates was equilibrated, the transformed metric does not need to be conformally spherical. The non-sphericality of the metric must be, therefore, compensated by a change of coordinates. Its infinitesimal version is described by a tangential (with respect to S ) deformation ̂ Under such coordinate transformation the metric changes as follows: where the last term represents the Lie derivative of the metric g AB with respect to the vector field ' -εξ A ' on S . But, according to (61), the difference between g AB and ˚ g AB is already of the first order in ε . Hence, if we replace it by the Lie derivative of the metric ˚ g AB , the error will be of the second order in ε . Using the Killing formula for the Lie derivative of the metric, we finally obtain: and the covariant derivative || A is taken with respect to the original metric ˚ g AB . Hence, we have: ̂ g AB -˚ g AB = -2 ετ ˚ k AB +2 εξ ( A || B ) + O ( ε 2 ) . (65) Let us decompose the above equation into the trace and the trace-free parts, calculated with respect to ˚ g AB (we omit the terms of order ε 2 and higher): where is the traceless part of ˚ k AB . We want ̂ g AB to be conformally spherical, i.e. ̂ g AB = α · ˚ g AB . This implies: The trace part of this equation defines uniquely the value of α : whereas the trace-free part reduces to: It is convenient to rewrite equation (70) in terms of the 'round' unit-sphere geometry σ AB . For this purpose we use the following conventions: components of a vector (i.e. an object having upper indices ) are the same in both geometries σ AB and ˚ g AB = ψσ AB . Components of a co-vector ( lowered indices ) are denoted as follows: The covariant derivative with respect to σ AB will be denoted by /wreathproduct/wreathproduct A , e.g. ξ σ A /wreathproduct/wreathproduct B . Equation (70) can be easily rewritten as: The left-hand side of this equation defines a mapping from the space of vector fields on the unit sphere to the space of trace-free rank 2 tensor fields. The kernel of this mapping consists of the dipole fields 4 . The 'Fredholm alternative' argument shows that the operator on the left-hand side defines an isomorphism between the space of dipole-free vector fields on the unit sphere and the space of trace-free rank 2 tensor fields (see also [7]). This isomorphism (in metric σ ) will be denoted by i 12 . Hence, the wave part of ξ A is implied uniquely by equation (72) (see Appendix): whereas the dipole part of ξ A , i.e. the field ξ d A , remains arbitrary. The above choice of the wave-like component of the tangential deformation ξ w A guarantees that the new coordinate system ̂ x A is conformally spherical. We would like it to be also: 1) equlibrated and 2) satisfying conditions related to the two fixed points n and m . These conditions mean that the field ξ has to vanish at the north pole n and that its ϕ -component vanishes at m . The above 3 + 3 = 6 conditions fix uniquely the total dipole-part of the tangential (to S ) deformation ξ A . This way the continuous mapping which assigns uniquely the tangential deformation ξ A to its transversal component τ has been defined.",
"pages": [
14,
15,
16
]
},
{
"title": "3.2 The infinitesimal change of the extrinsic curvature",
"content": "Now, we are going to calculate the infinitesimal change of the wave part k w of the mean curvature 5 , i.e. derivative of the mapping (56) with respect to the 'wavelike' deformation τ w . We have k = ˜ g AB k AB , where ˜ g AB denotes the inverse of the two-metric g AB (whereas g AB denotes the corresponding components of the inverse three-metric.) The simplest way to calculate this change is to use a coordinate system ( ω, x A ) , adapted to the deformed surface: and the formula: The three-metric g takes now the following form: This implies g ωω = 1 + O ( ε 2 ) and, consequently, where we treat the 'shift vector' g ωA = ετ ,A as a covector field on S τ . The first two terms combine to ετ || AB , whereas the last one: g AB,ω ( S τ ) can be approximated by the quantity g AB,ω ( S ) = -2 ˚ k AB plus the derivative of this object. Finally, we have Since the derivative g AB,ω of the metric g AB is described by -2 ˚ k AB , the derivative of its inverse ˜ g AB is described by +2 ˚ k AB . Hence, we have: and, consequently: The quantity τ∂ u ˚ k + τ || A A describes already the second variation of area (see Appendix), i.e. the derivative ∇ τ k . However, to calculate the derivative of the mapping (56), we have to select its wave part k w . For this purpose we have to pass to the conformally spherical, equilibrated coordinates ̂ x A , given by formula (62). Infinitesimal change of the scalar function k with respect to this deformation is given by formula: Hence, we get: or, equivalently (cf. Appendix), where R u u = R (d u, ∂ ∂u ) is the component of the Ricci tensor.",
"pages": [
17,
18
]
},
{
"title": "3.3 Proof of the Theorem 3",
"content": "The last formula gives, finally, the value of the derivative of the mapping (56). When restricted to the subspace of wave (i.e. mono-dipole-free) deformations, it gives us: The above linear operator is, obviously, continuous. In particular, the vector field ξ A is given by formula (73), together with the accompanying vanishing conditions at n and m . If the space Σ is flat (Euclidean) and S is a standard (rigid) sphere of radius r , then we have: Hence, the above operator reduces to: which is obviously an invertible mapping from C ( k +1 ,α ) w to C ( k -1 ,α ) w . But the mapping (85) depends in a continuous way upon the geometry (metric and curvature) of S . This implies that it remains invertible for sufficiently small deformations of the geometry. This is the case e.g. of a sufficiently big 'coordinate sphere' defined as follows: in an asymptotically flat Σ . We say, that Σ is asymptotically flat if there is a coordinate chart ( x k ) covering the exterior of a compact domain D ⊂ Σ and such that where h vanishes sufficiently fast at infinity. In that case Σ \\ D admits a fourparameter family of rigid spheres, similarly as in the case of the flat metric.",
"pages": [
18
]
},
{
"title": "4 Conclusions",
"content": "The main technical ingredient of this paper is the intrinsic, coordinate invariant definition of the 'multipole expansion' of a function defined on a Riemannian twomanifold, diffeomorphic with S 2 . This enables us to select a finite-dimensional family of 'rigid spheres'. The dipole part k d of the curvature parameterizes the position of the center of such a sphere with respect to the center of mass. In particular, k d = 0 corresponds to the spheres, which are centered at the center of mass. Properties of such a foliation have been analyzed in [4]. General topologically spherical coordinates, having property that surfaces { r = const. } are rigid, do not admit supertranslations ambiguity at space infinity. This way symmetries of the 'tangent space at infinity' reduce to a finite-dimensional one. The 4D version of our results, valid for a generic four-dimensional Lorenzian spacetime, which will be presented in the subsequent paper, will do the same job for the symmetry group of the Scri.",
"pages": [
19
]
},
{
"title": "Acknowledgements",
"content": "This research was supported by Polish Ministry of Science and Higher Education (grant Nr N N201 372736) and by Narodowe Centrum Nauki (grant DEC2011/03/B/ST1/02625). SŁ was supported by Foundation for Polish Science.",
"pages": [
19
]
},
{
"title": "A.1 The dipole part of traceless symmetric part",
"content": "The kernel of the mapping defined by the left-hand side of the formula (72) consists of the dipole fields. This is a simple consequence of the following observations. where functions α and β are defined up to a constant but their gradients are unique. hence the traceless symmetric part of ξ A /wreathproduct/wreathproduct B vanishes.",
"pages": [
19,
20
]
},
{
"title": "A.2 The isomorphism between covector fields and symmetric traceless tensors on ( S 2 , σ AB )",
"content": "Let us consider the following diagram: where the mappings and the spaces are defined as follows: V 0 k - scalars on S 2 belonging to Hölder space C k,α , V 1 k - covectors on S 2 belonging to Hölder space C k,α , V 2 k - symmetric traceless tensors on S 2 belonging to Hölder space C k,α . Denote by ∆ σ the Laplace operator on S 2 and by SH l the space of spherical harmonics of degree l , ( f ∈ SH l ⇐⇒ ∆ σ f = -l ( l +1) f ). The following equality shows that if we restrict ourselves to the spaces V 0 := V 0 /circleminus [ SH 0 ⊕ SH 1 ] = ( I -P md ) V 0 ( ∆ σ (∆ σ +2) V 0 = V 0 ) and V 1 = V 1 /circleminus [ i 01 ( SH 1 )] ( (∆ σ + I ) V 1 = V 1 ) then all the mappings in the above diagram become isomorphisms.",
"pages": [
20
]
},
{
"title": "A.2.1 Integral operators, generalized Green's functions",
"content": "Solution of the Helmholtz equation on a unit sphere S 2 : is given (see e.g. [13]) in terms of the generalized Green's function ¯ G l as follows: Here n = D ( ϑ, ϕ ) given by (17) and d 2 n = d σ . The solution Ψ l ( n ) is automatically orthogonal to the space SH l (the kernel of Helmholtz operator ∆ σ + l ( l +1) ) because Green's function is orthogonal to this space. In our case we need to write the inverse of the operator ∆ σ (∆ σ +2) as a double integral with the corresponding kernels ¯ G l for l = 0 and l = 1 . More precisely, the solution g of the equation ∆ σ (∆ σ +2) g = f (with P md f = 0 ) is given in the following form: where the projection operator P w provides orthogonality 6 of g to the space SH 0 ⊕ SH 1 . The generalized Green's function written in a standard form: /negationslash can be simplified as follows (cf. [13]): Y im - spherical harmonics (orthonormal basis in SH i ), n · n ' ∈ [ -1 , 1] is a scalar product of unit vectors in R 3 and P l ( x ) := 1 2 l l ! ( x 2 -1) ( l ) is the Legendre polynomial.",
"pages": [
21
]
},
{
"title": "A.3 Second variation of area",
"content": "The Gaussian coordinates (48) and the definition of the Riemann tensor gives This leads to Taking the trace (and using (79)), we obtain: The formulae (80) and (92) are equivalent because of the Gauss-Codazzi equations: where R ( g kl ) and R (˚ g AB ) are scalar of curvatures of the three-metric g kl and the two-metric ˚ g AB , respectively. Obviously (80), (92) are the first variations of the mean curvature k , which in the literature (see e.g. [3]) are known as the second variations of area. They are usually presented in the following equivalent form:",
"pages": [
22
]
},
{
"title": "References",
"content": "[1] H.P. Gittel et al., On the existence of rigid spheres in four-dimensional spacetime manifolds , to be published. [2] P. Chruściel, Sur l'existence de solutions singulières d'une équation «condition de coordonnées» utilisé par J. Kijowski dans l'analyse symplectique de la relativité générale , C. R. Acad. Sci. Paris 299I (1984), 891-894; P. Chruściel, Sur les Feuilletages Conformément Minimaux des Variétés Riemaniennes de Dimension Trois , Comptes Rendus de l' Académie des Sciences de Paris (série I) 301, 609-612 (1985); P. Chruściel, Sur les coordonnées p-harmoniques en relativité générale , Comptes Rendus de l' Académie des Sciences de Paris (série I) 305, 797-800 (1987); J. Jezierski, J. Kijowski, Phys. Rev. D 36 (1987), 1041-1044; J. Kijowski, On positivity of gravitational energy , in Proceedings of the fourth Marcel Grossmann meeting on General Relativity, Rome, 1985, ed. R. Ruffini, Elsvier Science Publ. 1986, 1681-1686; J. Jezierski, J. Kijowski, Unconstrained Degrees of Freedom for Gravitational Waves, β -Foliations and Spherically Symmetric Initial Data , http://arxiv.org/abs/gr-qc/0501073v1 , Preprint ESI 1552 (2004).",
"pages": [
22
]
}
]
2013CQGra..30t5002A
https://arxiv.org/pdf/1309.3090.pdf
<document>
<section_header_level_1><location><page_1><loc_24><loc_92><loc_76><loc_93></location>Dynamical Foundations of the Brane Induced Gravity</section_header_level_1>
<section_header_level_1><location><page_1><loc_45><loc_89><loc_56><loc_90></location>Keiichi Akama</section_header_level_1>
<text><location><page_1><loc_24><loc_88><loc_77><loc_89></location>Department of Physics, Saitama Medical University, Saitama, 350-0495, Japan</text>
<section_header_level_1><location><page_1><loc_45><loc_84><loc_56><loc_86></location>Takashi Hattori</section_header_level_1>
<text><location><page_1><loc_24><loc_82><loc_77><loc_84></location>Department of Physics, Kanagawa Dental College, Yokosuka, 238-8580, Japan (Dated: October 8, 2018)</text>
<text><location><page_1><loc_18><loc_71><loc_83><loc_81></location>We present a comprehensive formalism to derive precise expressions for the induced gravity of the braneworld, assuming the dynamics of the Dirac-Nambu-Goto type. The quantum fluctuations of the brane at short distances give rise to divergences, which should be cutoff at the scale of the inverse thickness of the brane. It turns out that the induced-metric formula is converted into an Einstein-like equation via the quantum effects. We determine the coefficients of the induced cosmological and gravity terms, as well as those of the terms including the extrinsic curvature and the normal connection gauge field. The latter is the characteristic of the brane induced gravity theory, distinguished from ordinary none-brane induced gravity.</text>
<text><location><page_1><loc_18><loc_68><loc_51><loc_69></location>PACS numbers: 04.50.-h, 04.62.+v, 11.25.-w, 12.60.Rc</text>
<section_header_level_1><location><page_1><loc_20><loc_64><loc_37><loc_65></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_29><loc_49><loc_62></location>General relativity is based on the premises that the spacetime is curved affected by matter according to the Einstein equation, and that the objects move along the spacetime geodesics. The gravitations are apparent phenomena of the inertial motions in the curved spacetime. This successfully explains why the motion in the 'gravitational' field is universal, i.e. blind to the object properties (mass, charge etc.), and why the 'gravitational' forces are subject to the Newton's law of gravitation (within the ordinary precision). It is supported by further precise observations. It raises, however, another fundamental question why the spacetime is curved so as the Einstein equation indicates. A possible answer was as follows: Suppose our spacetime is a 3+1 dimensional embedded object like domain walls or vortices (braneworld, [1]-[41]) in higher dimensional spacetime, and the gravity is induced through quantum fluctuations (brane induced gravity, [4], [5], [10], [14]-[18], [27], [28], [34], [35]). The cosmological constant should be finely tuned. Then, the brane-gravity field emerges as a composite, which obeys an Einstein-like equation at least at low curvatures, just like in the case of the (non-brane) induced gravity theory [42]-[46].</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_28></location>The ideas of the braneworld and the brane induced gravity have been studied extensively in the last three decades. Physical models were constructed with topological defects in higher dimensional spacetime [5]-[18], and they were realized as 'D-branes' in the superstring theory [19]-[22]. They were applied to the hierarchy problem with large extra dimensions, or with warped extra dimensions [20], [23]-[25]. It was argued that the brane induced gravity would imply infrared modifications of the gravity theory [26]-[28]. The ideas have been studied in wide areas including basic formalism, [29]-[33], brane induced gravity [34]-[35], particle physics phenenomenology [36][38], and cosmology [39]-[41] with many interesting consequences.</text>
<text><location><page_1><loc_52><loc_26><loc_92><loc_65></location>In this paper, we establish a precise formalism to derive the expressions for the quantum induced effects on the brane. For definiteness, we follow the simplest model with the Dirac-Nambu-Goto type dynamics [47]. Such a model was first considered in [4] with scalar fields which we now interpret as the position coordinate of the brane in higher dimensions. Such an interpretation motivated us to consider the model of braneworld as a topological object moving in higher dimensions in [5]. The idea of the brane induced gravity has been considered repeatedly in the literature. They were, however, more or less naive both in model setting and derivation. Here we present a comprehensive formalism from its foundation to the precise outcomes. Among the quantum fluctuations, the only meaningful ones are those transverse to the brane. The quantum loop effects are divergent, which should be cutoff at the scale of the inverse thickness of the brane. We adopt the regularization scheme used in the original work [4], and calculate them at the one-loop level. We determine the coefficients of the induced cosmological and gravity terms, as well as those of the terms including the extrinsic curvature and the normal connection gauge field [48]. The induction of the latter terms is characteristic of the brane induced gravity theory, distinguished from the ordinary (non-brane) induced gravity. It turns out that the induced-metric formula is converted into an Einsteinlike equation via the quantum short-distance effects.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_26></location>The plan of this paper is as follows. First, we define the model (Sec. II), and then we derive the quantum effects (Secs. III-VII). We define the brane fluctuations (Sec. III), formulate the quantum effects (Sec. IV), specify the method to regularize the divergences (Sec. V), classify the possible induced terms according to symmetries (Sec. VI), and calculate them via Feynman diagram method (Sec. VII). Then, we interpret the quantum induced terms, and show why the induced gravity can avoid the problem (Secs. VIII and IX). The cosmological terms are fine-tuned (Sec. VIII), and the Einstein like gravity and other terms are induced (Sec. IX). The final section</text>
<text><location><page_2><loc_9><loc_92><loc_33><loc_93></location>(Sec. X) is devoted to discussions.</text>
<section_header_level_1><location><page_2><loc_21><loc_88><loc_36><loc_89></location>II. THE MODEL</section_header_level_1>
<text><location><page_2><loc_9><loc_68><loc_49><loc_86></location>We consider a quantum theoretical braneworld described by the Dirac-Nambu-Goto type Lagrangian. We will see the quantum effects of the brane fluctuations give rise to effective braneworld gravity. Let X I ( x µ ) ( I = 0 , 1 , · · · , D -1) be the position of our three-brane in the D dimensional spacetime (bulk), parameterized by the brane coordinate x µ ( µ = 0 , 1 , 2 , 3), where I = 0 and µ = 0 indicate the time components. Let G IJ ( X K ) be the bulk metric tensor at the bulk point X K . This is taken to obey some bulk gravity theory. Then we consider a braneworld with dynamics given by the DiracNambu-Goto type Lagrangian (density) [47]:</text>
<formula><location><page_2><loc_13><loc_63><loc_49><loc_67></location>L br = -λ √ -det µν ( ∂X I ∂x µ ∂X J ∂x ν G IJ ( X K ) ) , (1)</formula>
<text><location><page_2><loc_9><loc_61><loc_37><loc_62></location>where λ is a constant. Or we write it as</text>
<formula><location><page_2><loc_22><loc_56><loc_49><loc_60></location>L br = -λ √ -g [ X ] (2)</formula>
<text><location><page_2><loc_9><loc_55><loc_37><loc_57></location>with abbreviations g [ X ] = det g [ X ] µν , and</text>
<formula><location><page_2><loc_19><loc_52><loc_49><loc_54></location>g [ X ] µν = X I ,µ X J ,ν G IJ ( X K ) , (3)</formula>
<text><location><page_2><loc_9><loc_35><loc_49><loc_51></location>where (and hereafter) indices following a comma (,) indicate differentiation with respect to the corresponding coordinate component, and [ X ] is attached to remind that they are abbreviations for expressions written in terms of X I . Note that g [ X ] µν is the induced metric on the brane with (3). We assume that X I appears nowhere other than in L br in the total Lagrangian L tot including the bulk Lagrangian. The system is invariant under the general coordinate transformation of the bulk and the brane separately. The Lagrangian (1) is the simplest among those with this symmetry.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_34></location>Note that we do not a priori have the kinetic term of the metric in the basic Lagrangian (1). It will be induced through quantum effects. The metric emerges as a composite field, and the gravitation is an induced phenomenon but not an elementary one. If this is successful, we achieve a great conceptual advantage, since it means that the familiar and important phenomenon of the gravitation is explained by the more fundamental ingredients. Thus, it is worthwhile and urgent to examine what is true and what is not in this idea. This is the basic spirit of the induced gravity theory [42]-[46]. As for the dynamics of the bulk metric, the situation is more vague and ambiguous. We do not know what is the dynamics, i.e. what kinetic term we have, and it is not clear even whether the kinetic term exists or not, i.e. whether it is dynamical or not. Every case well deserves intensive investigations. It is too naive to take that the bulk Einstein equation provides the brane Einstein equation at the brane, because</text>
<text><location><page_2><loc_52><loc_76><loc_92><loc_93></location>large discrepancies take place due to the extrinsic curvature terms. The brane moves according to its equation of motion and is deformed in manners different from what the naive gravity equation indicates. One may elaborate the elementary brane gravity from the elementary bulk gravity under specific conditions by hand in specific models [25], [39]. On the other hand, many people relied on the mechanism of brane induced gravity to realize gravitation on the brane [5], [10], [14]-[18], [27], [28], [34], [35]. In the spirit of the brane induced gravity, we do not a priori assume elementary gravity, and we take it to be induced from more fundamental ingredients.</text>
<text><location><page_2><loc_53><loc_74><loc_88><loc_76></location>Now, the equation of motion from (1) is given by</text>
<formula><location><page_2><loc_65><loc_72><loc_92><loc_73></location>g [ X ] µν X [ X ] ; µν I = 0 , (4)</formula>
<text><location><page_2><loc_52><loc_66><loc_92><loc_70></location>where X [ X ] ; µν I is the double covariant derivative with respect to both of the general coordinate transformations on the brane and to those in the bulk:</text>
<formula><location><page_2><loc_55><loc_62><loc_92><loc_65></location>X [ X ] ; µν I = X I ,µν -X I ,λ γ [ X ] λ µν + X J ,µ X K ,ν Γ I JK (5)</formula>
<text><location><page_2><loc_52><loc_61><loc_87><loc_62></location>with the affine connections on the brane and bulk</text>
<formula><location><page_2><loc_56><loc_54><loc_92><loc_60></location>γ [ X ] λ µν = 1 2 g [ X ] λρ ( g [ X ] ρµ,ν + g [ X ] ρν,µ -g [ X ] µν,ρ ) , (6) Γ I JK = 1 2 G IL ( G LJ,K + G LK,J -G JK,L ) , (7)</formula>
<text><location><page_2><loc_52><loc_43><loc_92><loc_53></location>respectively. We expect that this gives a good approximation at low curvature limit in many dynamical models of the braneworld (e.g. topological defects, spacetime singularities, D-branes, etc.). It is remarkable that, as we shall see below, this simple model exhibits brane gravity and gauge theory like structure through the quantum effects.</text>
<text><location><page_2><loc_52><loc_40><loc_92><loc_43></location>For convenience of quantum treatments, we consider the following equivalent Lagrangian to (1):</text>
<formula><location><page_2><loc_56><loc_35><loc_92><loc_39></location>L ' br = -λ 2 √ -g [ g µν X I ,µ X J ,ν G IJ ( X K ) -2 ] (8)</formula>
<text><location><page_2><loc_52><loc_28><loc_92><loc_36></location>where g µν is an auxiliary field, g = det g µν , and g µν is the inverse matrix of g µν . Note that g µν , unlike g [ X ] µν above, is treated as a field independent of X I . Then the Euler Lagrange equations with respect to X I and g µν are given by</text>
<formula><location><page_2><loc_62><loc_25><loc_92><loc_27></location>g µν X I ; µν = 0 , (9)</formula>
<formula><location><page_2><loc_62><loc_23><loc_92><loc_25></location>g µν = X I ,µ X J ,ν G IJ ( X K ) , (10)</formula>
<text><location><page_2><loc_52><loc_21><loc_82><loc_22></location>respectively, where the covariant derivative</text>
<formula><location><page_2><loc_56><loc_18><loc_92><loc_20></location>X ; µν I = X I ,µν -X I ,λ γ λ µν + X J ,µ X K ,ν Γ I JK (11)</formula>
<text><location><page_2><loc_52><loc_16><loc_87><loc_17></location>is written in terms of the brane affine connection</text>
<formula><location><page_2><loc_59><loc_12><loc_92><loc_15></location>γ λ µν = 1 2 g λρ ( g ρµ,ν + g ρν,µ -g µν,ρ ) (12)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_11></location>with respect to the auxiliary field g µν . Now g µν in (9) is independent of X I , and, instead, we have an extra</text>
<text><location><page_3><loc_9><loc_82><loc_49><loc_93></location>equation (10), which guarantees that g µν is the induced metric. If we substitute (10) into (9), we obtain the same equation as (4). Thus the systems with the Lagrangians L br and L ' br coincide. Furthermore the argument that their Dirac bracket algebrae coincide [44] indicates their quantum theoretical equivalence. We proceed hereafter based on the Lagrangian L ' br instead of L br .</text>
<section_header_level_1><location><page_3><loc_16><loc_79><loc_41><loc_80></location>III. BRANE FLUCTUATIONS</section_header_level_1>
<text><location><page_3><loc_9><loc_69><loc_49><loc_77></location>In order to extract the quantum effects of L ' br , we deploy a semi-classical method, where we consider those due to small fluctuations of the brane around some classical solution (say Y I ( x µ )) for X I ( x µ ) of the equation of motion (4) [15]. Namely, the solution Y I ( x µ ) obeys the classical equation</text>
<formula><location><page_3><loc_21><loc_66><loc_49><loc_68></location>g µν Y I ; µν = 0 , (13)</formula>
<formula><location><page_3><loc_21><loc_64><loc_49><loc_65></location>g µν = Y I ,µ Y J ,ν G IJ ( Y K ) . (14)</formula>
<text><location><page_3><loc_9><loc_48><loc_49><loc_63></location>In quantum treatment, X I itself in the Lagrangian L ' br does not necessarily obey the equation of motion (4), and may fluctuate from Y I ( x µ ). Among the fluctuations, only those transverse to the brane are physically meaningful , because those along the brane remain within the brane and cause no real fluctuations of the brane. They are absorbed by general coordinate transformations. In order to describe them, we choose D -4 independent normal vectors n m I ( x µ ) ( m = 4 , · · · , D -1) at each point on the brane with the normality condition</text>
<formula><location><page_3><loc_22><loc_46><loc_49><loc_47></location>n m I Y J ,ν G IJ ( Y K ) = 0 . (15)</formula>
<text><location><page_3><loc_9><loc_43><loc_34><loc_45></location>Then we express the fluctuations as</text>
<formula><location><page_3><loc_15><loc_41><loc_49><loc_42></location>X I ( x µ ) = Y I ( x µ ) + φ m ( x µ ) n m I ( x µ ) , (16)</formula>
<text><location><page_3><loc_9><loc_30><loc_49><loc_40></location>where φ m ( x µ ) is the transverse fluctuation along n m I ( x µ ) ( m = 4 , · · · , D -1). The arbitrariness of n m I in choice under the condition (15) gives rise to gauge symmetry under the group GL( D -4) of the general linear transformations of the normal space at each point on the brane. If we define g mn = n I m n J n G IJ ( Y K ) and its inverse g mn , we have the completeness condition,</text>
<formula><location><page_3><loc_16><loc_27><loc_49><loc_29></location>Y I ,µ Y J ,ν g µν + n I m n J n g mn = G IJ ( Y K ) . (17)</formula>
<text><location><page_3><loc_9><loc_13><loc_49><loc_26></location>Bulk coordinate indices I, J, · · · (= 0 , · · · , D -1) are raised and lowered by the metric tensors G IJ and G IJ . We can read off from (8) and (10) that the auxiliary field g µν plays the role the metric tensor on the brane. Hereafter we raise and lower the brane coordinate indices µ, ν, · · · (= 0 , · · · , 3) by g µν and g µν (but not by g [ X ] µν and g [ X ] µν ). We raise and lower the normal space indices m,n, · · · (= 4 , · · · , D -1) by g mn and g mn .</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_14></location>A problem of the definition (16) of φ m is that φ m lacks the bulk general-coordinate invariance. In fact, it is transformed in a complex way under the general coordinate transformations of the bulk. On the contrary,</text>
<text><location><page_3><loc_52><loc_85><loc_92><loc_93></location>the invariant definition of φ m requires a complex formula instead of (16). For quantum treatments, however, it is desirable to have a relation linear in φ m like (16). It requires further careful considerations. Therefore, in this paper, we restrict ourselves to the case where the bulk is flat. Namely, there exists the cartesian frame with</text>
<formula><location><page_3><loc_57><loc_81><loc_92><loc_84></location>G IJ = η IJ ≡ diag { 1 , -1 , -1 , · · · , -1 } , (18)</formula>
<text><location><page_3><loc_52><loc_76><loc_92><loc_81></location>where we have Γ I JK = 0. In this case, we have R I JKL = 0 in any frame, and φ m restores the bulk general-coordinate invariance. The general case of curved bulk will be considered in a separate forthcoming paper.</text>
<text><location><page_3><loc_52><loc_73><loc_92><loc_75></location>Now we substitute (16) into the Lagrangian (8), and obtain</text>
<formula><location><page_3><loc_55><loc_69><loc_92><loc_72></location>L ' br = L ' 0 + L ' φ with (19)</formula>
<formula><location><page_3><loc_55><loc_60><loc_92><loc_67></location>L ' φ = -λ 2 √ -g [ g µν g ρσ φ m φ n B mµρ B nνσ + g µν g mn ( φ m ,µ + g mi φ k A ikµ )( φ n ,ν + g nj φ l A jlν ) ] , (21)</formula>
<formula><location><page_3><loc_55><loc_66><loc_92><loc_70></location>L ' 0 = L ' br | φ =0 = -λ 2 √ -g [ g µν Y I ,µ Y J ,ν G IJ -2 ] , (20)</formula>
<text><location><page_3><loc_52><loc_58><loc_92><loc_61></location>where A mnµ and B mµν are the normal connection and the extrinsic curvature, respectively. They are given by</text>
<formula><location><page_3><loc_64><loc_55><loc_92><loc_57></location>A mnµ = n I m n J n ; µ G IJ , (22)</formula>
<formula><location><page_3><loc_64><loc_53><loc_92><loc_55></location>B mµν = n I m Y J ; µν G IJ , (23)</formula>
<text><location><page_3><loc_52><loc_51><loc_79><loc_52></location>where n J n ; µ is the covariant derivative:</text>
<formula><location><page_3><loc_61><loc_48><loc_92><loc_50></location>n J n ; µ = n J n ,µ + n K n Y M ,µ Γ J KM , (24)</formula>
<text><location><page_3><loc_52><loc_38><loc_92><loc_47></location>which coincides with the ordinary derivative n J n ,µ under the assumption G IJ = η IJ for the present paper. We can see that (21) is the Lagrangian for the quantum scalar fields φ m on the curved brane interacting with the given external fields A mnµ and B mµν . For later use, it is convenient to rewrite (21) into</text>
<formula><location><page_3><loc_57><loc_29><loc_92><loc_37></location>L ' φ = -λ 2 √ -gφ m [ -∂ µ √ -gg µν ∂ ν -∂ µ √ -gA m n µ -√ -gA m n µ ∂ µ -√ -g ( A mkµ A knµ -B mµρ B nµρ ) ] φ n , (25)</formula>
<text><location><page_3><loc_52><loc_29><loc_78><loc_30></location>where total derivatives are neglected.</text>
<section_header_level_1><location><page_3><loc_61><loc_25><loc_83><loc_26></location>IV. QUANTUM EFFECTS</section_header_level_1>
<text><location><page_3><loc_52><loc_19><loc_92><loc_23></location>The quantum effects of the field φ m are described by the effective Lagrangian L eff</text>
<formula><location><page_3><loc_56><loc_15><loc_92><loc_19></location>∫ L eff d 4 x = -i ln ∫ [ dφ m ] exp [ i ∫ L ' φ d 4 x ] , (26)</formula>
<text><location><page_3><loc_52><loc_12><loc_92><loc_15></location>where [ dφ m ] is the path-integration over φ m . To perform it, we rewrite (25) into the form</text>
<formula><location><page_3><loc_60><loc_8><loc_92><loc_11></location>L ' φ = λ 2 φ m ( -δ m n /square + V m n ) φ n , (27)</formula>
<text><location><page_4><loc_9><loc_92><loc_25><loc_93></location>with /square = η µν ∂ µ ∂ ν and</text>
<formula><location><page_4><loc_10><loc_89><loc_49><loc_91></location>V m n = δ m n ∂ µ H µν ∂ ν + ∂ µ A m n µ + A m n µ ∂ µ + Z m n (28)</formula>
<formula><location><page_4><loc_13><loc_85><loc_49><loc_88></location>H µν = η µν -√ -gg µν (29)</formula>
<formula><location><page_4><loc_13><loc_82><loc_49><loc_85></location>Z m n = -√ -g ( A mkµ A knµ -B mµρ B nµρ ) , (31)</formula>
<formula><location><page_4><loc_13><loc_83><loc_49><loc_86></location>A m n µ = -√ -gA m n µ (30)</formula>
<text><location><page_4><loc_9><loc_77><loc_49><loc_81></location>where the differential operator ∂ µ ≡ ∂/∂x µ is taken to operate on the whole expression in its right side in (27). The path-integration in (26) is performed to give</text>
<formula><location><page_4><loc_16><loc_71><loc_49><loc_76></location>∫ L eff d 4 x = ∞ ∑ l =0 1 2 li Tr ( 1 /square V m n ) l , (32)</formula>
<text><location><page_4><loc_9><loc_64><loc_49><loc_71></location>up to additional constants, where Tr indicates the trace over the brane coordinate variable x µ and extra dimension index m . The terms in (32) can be calculated with Feynman-diagram method. In terms of the Fourier transforms</text>
<formula><location><page_4><loc_18><loc_60><loc_49><loc_64></location>˜ H µν ( q i ) = ∫ d 4 x H µν ( x ) e iq i x , (33)</formula>
<formula><location><page_4><loc_17><loc_53><loc_49><loc_57></location>˜ Z m n ( q i ) = ∫ d 4 x Z m n ( x ) e iq i x , (35)</formula>
<formula><location><page_4><loc_16><loc_56><loc_49><loc_61></location>˜ A m n µ ( q i ) = ∫ d 4 x A m n µ ( x ) e iq i x , (34)</formula>
<text><location><page_4><loc_9><loc_51><loc_38><loc_53></location>the effective Lagrangian L eff is written as</text>
<formula><location><page_4><loc_15><loc_46><loc_49><loc_51></location>L eff = ∞ ∑ l =0 1 2 l l ∏ i =1 ∫ d 4 q i (2 π ) 4 e -iq i x G l , (36)</formula>
<formula><location><page_4><loc_15><loc_38><loc_49><loc_42></location>˜ V m n ( p i , q i ) = -δ m n ( p i ) µ ( p i -1 ) ν ˜ H µν ( q i ) -i ( p i + p i -1 ) µ ˜ A m n µ ( q i ) + ˜ Z m n ( q i ) , (38)</formula>
<formula><location><page_4><loc_15><loc_41><loc_49><loc_46></location>G l = ∫ d 4 p i (2 π ) 4 l ∏ i =1 1 -p 2 i ˜ V m i m i -1 ( p i , q i ) , (37)</formula>
<text><location><page_4><loc_9><loc_17><loc_49><loc_37></location>where p i = p + q 1 + · · · + q i and m 0 = m l . The function G l is nothing but the Feynman amplitude for the oneloop diagram with l internal lines of φ m and l vertices of ˜ V m n (FIG. 1). Unfortunately, the p -dependence of the integrand in (36) with (37) indicates that the integration over p diverges at most quartically. The divergences will be regulated in the next section. Then, we can perform the integration over p to obtain the function G l . The q k 's are replaced by differentiation i∂ k of the k -th vertex function according to the inverse Fourier transformation in (36). Collecting all the contributions, which are functions of the fields g µν , A mnµ and B mµν and their derivatives, we can obtain the expression for the effective Lagrangian L eff .</text>
<section_header_level_1><location><page_4><loc_10><loc_13><loc_47><loc_14></location>V. DIVERGENCES AND REGULARIZATION</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_49><loc_11></location>The p -dependence of the integrand in (36) with (37) indicates that the integration over p diverges at most</text>
<figure>
<location><page_4><loc_59><loc_78><loc_84><loc_92></location>
<caption>FIG. 1: The Feynman diagrams. The dashed lines indicate the φ m -propagators, and the wavy lines indicate external fields of A mnµ , B mµν or h µν . The dots indicate an infinite series of diagrams with a dashed-line loop and with more external wavy lines than two. By virtue of the symmetries of the system, we have only to calculate the three diagrams explicitly drawn here.</caption>
</figure>
<text><location><page_4><loc_52><loc_51><loc_92><loc_64></location>quartically. We expect, however, that fluctuations with smaller wave length than the brane thickness are suppressed. Then, the momenta higher than the inverse of the thickness are cut off. In order to model the cutoff without violating full symmetry of L ' br , we introduce three Pauli-Villers regulators Φ m j with very large mass M j ( j =1,2,3), which are taken equal finally, M j → Λ, following the original paper [4]. Precisely, it amounts to consider the regularized effective Lagrangian</text>
<formula><location><page_4><loc_62><loc_45><loc_92><loc_49></location>L reg = L eff + 3 ∑ j =1 C j L eff M j (39)</formula>
<text><location><page_4><loc_52><loc_43><loc_80><loc_44></location>where C j are the coefficients defined by</text>
<formula><location><page_4><loc_59><loc_37><loc_92><loc_41></location>3 ∑ j =1 C j ( M j ) 2 k = -δ 0 k ( k = 0 , 1 , 2) , (40)</formula>
<text><location><page_4><loc_52><loc_31><loc_92><loc_36></location>and L eff M j is the effective Lagrangian for the quantum effects from L ' Φ j which is the same as L ' φ except that φ m is replaced by the regulator field Φ m j with mass M j .</text>
<formula><location><page_4><loc_55><loc_26><loc_92><loc_30></location>∫ L eff M j d 4 x = -i ln ∫ [ d Φ m j ] exp [ i ∫ L ' Φ j d 4 x ] . (41)</formula>
<formula><location><page_4><loc_55><loc_23><loc_92><loc_26></location>L ' Φ j = L ' φ | φ =Φ j + 1 2 λM 2 j √ -g Φ m j Φ n j η mn . (42)</formula>
<text><location><page_4><loc_52><loc_18><loc_92><loc_22></location>Note that the added mass term also preserves the full symmetry of L ' br . Performing the path integration over Φ m j , we have</text>
<formula><location><page_4><loc_57><loc_12><loc_92><loc_17></location>∫ L eff M j d 4 x = ∞ ∑ l =0 1 2 li Tr ( 1 /square + M 2 j V M j m n ) l , (43)</formula>
<formula><location><page_4><loc_57><loc_8><loc_92><loc_11></location>F = 1 -√ -g, (45)</formula>
<formula><location><page_4><loc_57><loc_10><loc_92><loc_12></location>V M j m n = V m n + F M 2 j δ m n , (44)</formula>
<text><location><page_5><loc_9><loc_91><loc_46><loc_93></location>with V m n in (28). In terms of the Fourier transform</text>
<formula><location><page_5><loc_21><loc_87><loc_49><loc_91></location>˜ F ( q i ) = ∫ d 4 x F ( x ) e iq i x . (46)</formula>
<text><location><page_5><loc_9><loc_85><loc_14><loc_86></location>we have</text>
<formula><location><page_5><loc_12><loc_79><loc_49><loc_83></location>L eff M j = ∞ ∑ l =0 1 2 l l ∏ i =1 ∫ d 4 q i (2 π ) 4 e -iq i x G l M j , (47)</formula>
<formula><location><page_5><loc_12><loc_74><loc_49><loc_79></location>G l M j = ∫ d 4 p i (2 π ) 4 l ∏ i =1 1 -p 2 i + M 2 j ˜ V M j m i m i -1 ( p i , q i ) , (48)</formula>
<formula><location><page_5><loc_12><loc_71><loc_49><loc_73></location>˜ V M j m n ( p i , q i ) = ˜ V m n ( p i , q i ) + δ m n M 2 j ˜ F ( q i ) , (49)</formula>
<text><location><page_5><loc_9><loc_65><loc_49><loc_70></location>with ˜ V m n ( p i , q i ) in (38). In dimensional regularization, the divergent parts of the Feynman amplitude G l M j behaves like</text>
<formula><location><page_5><loc_17><loc_62><loc_49><loc_64></location>G l M j ∼ /epsilon1 -1 ( G 4 M 4 j + G 2 M 2 j + G 0 ) , (50)</formula>
<text><location><page_5><loc_9><loc_46><loc_49><loc_61></location>where this is evaluated at the spacetime dimension 4 -2 /epsilon1 , and G 2 k are the appropriate coefficient functions. The singularities at /epsilon1 = 0 reflect the divergences in the p -integration. We can see that, when they are summed with the coefficients C j over j in (39), they cancel out according to (40). Therefore the p -integrations in L reg converge. Any positive power contributions of M j regular at infinity vanish according to (40). The function G l M j involves logarithmic singularities in M j , which tend, in the equal mass limit M j → Λ,</text>
<formula><location><page_5><loc_21><loc_40><loc_49><loc_45></location>3 ∑ j =1 C j M 4 j ln M 2 j →-Λ 4 / 2 (51)</formula>
<formula><location><page_5><loc_21><loc_31><loc_49><loc_36></location>3 ∑ j =1 C j ln M 2 j →-ln Λ 2 (53)</formula>
<formula><location><page_5><loc_21><loc_36><loc_49><loc_40></location>3 ∑ j =1 C j M 2 j ln M 2 j → Λ 2 / 2 (52)</formula>
<section_header_level_1><location><page_5><loc_12><loc_27><loc_45><loc_28></location>VI. CLASSIFICATION OF THE TERMS</section_header_level_1>
<text><location><page_5><loc_9><loc_13><loc_49><loc_25></location>Thus the divergent part L div of the regularized effective Lagrangian L reg consist of the terms which are proportional to Λ 4 , Λ 2 or ln Λ 2 , and are monomials of H µν , F , A m n µ , Z m n , and their derivatives. The expressions H µν , F , A m n µ , and Z m n are written in terms of the fields g µν , A m n µ , and B m µν according to (29), (45), (30), and (31). Introducing the notation h µν ≡ g µν -η µν , we rewrite g µν and √ -g in H µν , F , A m n µ , and Z m n according to</text>
<formula><location><page_5><loc_16><loc_10><loc_49><loc_12></location>g µν = η µν -h µν + h µν (2) + h µν (3) + · · · , (54)</formula>
<formula><location><page_5><loc_15><loc_8><loc_49><loc_11></location>√ -g = 1 + h/ 2 -h (2) / 4 + h 2 / 8 + · · · , (55)</formula>
<table>
<location><page_5><loc_52><loc_65><loc_92><loc_91></location>
<caption>TABLE I: Invariant forms</caption>
</table>
<text><location><page_5><loc_52><loc_61><loc_58><loc_62></location>with [49]</text>
<formula><location><page_5><loc_63><loc_55><loc_92><loc_60></location>h ( n ) µ ν = n ︷ ︸︸ ︷ h µ σ h σ τ · · · h ρ ν , (56) h = h µ µ , h ( n ) = h ( n ) µ µ . (57)</formula>
<text><location><page_5><loc_52><loc_42><loc_92><loc_53></location>Then, L div becomes an infinite sum of monomials of h µν , A m nµ , B m µν , and their derivatives. Let us denote the numbers of h µν , A m nµ , B m µν , and the differential operators in the monomial by N h , N A , N B and N ∂ , respectively. The Lagrangian L reg should have mass dimension 4, while h µν , A m nµ , B m µν , and the differential operator has mass dimension 0, 1, 1, and 1, respectively. Therefore, the numbers N A , N B and N ∂ are restricted by</text>
<formula><location><page_5><loc_61><loc_38><loc_92><loc_40></location>N A + N B + N ∂ ≤ 4 -2 k div , (58)</formula>
<text><location><page_5><loc_52><loc_29><loc_92><loc_38></location>where k div = 2 , 1 , 0 for Λ 4 , Λ 2 , and ln Λ 2 terms, respectively. On the other hand, the number N h of h µν is not restricted. The relation (58) allows only finite numbers of values of N A , N B and N ∂ , according to which we can classify the terms of L div . Each class involves infinitely many terms for arbitrary values of N h .</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_29></location>They are, however, not all independent, because they are related by high symmetry of the system under the general coordinate transformations on the brane and GL(4) gauge transformations of the normal space rotation. Though the original system is invariant under the general coordinate transformations in the bulk also, it is not available here, because we restrict ourselves to the flat bulk in this paper. The general case will be discussed in the forthcoming paper. Owing to the symmetry of the system, only finite number of terms are allowed. The general coordinate transformation symmetry requires that the effective Lagrangian density is proportional to √ -g times a sum of invariant forms. We list the allowed invariant forms in TABLE I. In the table and thereafter,</text>
<text><location><page_6><loc_9><loc_92><loc_34><loc_93></location>we use the following abbreviations.</text>
<formula><location><page_6><loc_17><loc_89><loc_49><loc_91></location>B (2) = B mµν B mµν , B m = B mν ν . (59)</formula>
<formula><location><page_6><loc_17><loc_87><loc_49><loc_89></location>B (4) = B mµν B mρλ B nρλ B nµν . (60)</formula>
<formula><location><page_6><loc_17><loc_85><loc_49><loc_87></location>B m ‖ λ = B m ; λ + A m n λ B n . (61)</formula>
<formula><location><page_6><loc_17><loc_84><loc_49><loc_85></location>B mµν ‖ λ = B mµν ; λ + A m n λ B nµν . (62)</formula>
<formula><location><page_6><loc_17><loc_79><loc_49><loc_83></location>A mnµν = ∂ µ A mnν -∂ ν A mnµ + A mkµ A k nν -A mkν A k nµ , (63)</formula>
<text><location><page_6><loc_9><loc_71><loc_49><loc_79></location>where (61) and (62) are the covariant derivatives of the full symmetry, and (63) is the field strength of the gauge field A mµν . In the table, R 2 , R µν R µν , and R µνρσ R µνρσ are not all independent, but related by Gauss-Bonnet relation, and some other combinations are related due to the Gauss-Codazzi-Ricci formulae.</text>
<section_header_level_1><location><page_6><loc_20><loc_66><loc_37><loc_67></location>VII. CALCULATION</section_header_level_1>
<text><location><page_6><loc_29><loc_56><loc_29><loc_59></location>/negationslash</text>
<text><location><page_6><loc_10><loc_40><loc_41><loc_41></location>From (48), (49) and (38), they are given by</text>
<text><location><page_6><loc_9><loc_40><loc_49><loc_65></location>Thus we can calculate the coefficients of the term √ -g times the invariant forms by calculating the lowest order contributions in h µν . The lowest contributions to the term with N A = N B = N ∂ = 0 are O ( h µν ), while those to N A = N B = 0 and N ∂ = 0 are O (( h µν ) 2 ), because the O ( h µν ) terms are total derivatives. Therefore, their lowest terms are in the one- and two-point functions G 1 and G 2 . We can see from (28) - (31) that B mµν always appears in the combination B mµν B nµν . Therefore, the only possible forms including B mµν are B (2) , RB (2) , ( B (2) ) 2 , and B (4) , among many forms listed in Table I. Their lowest terms are those with O (( h µν ) 0 ) and are also in G 1 and G 2 . The only possible form including A mnµ only is A mnµν A mnµν and its lowest term is of O (( h µν ) 0 ), and it is in G 2 . Thus, it suffices to calculate G 1 and G 2 in order to determine full contributions to L div .</text>
<formula><location><page_6><loc_10><loc_29><loc_49><loc_39></location>G 1 M j = -N ex ˜ H µν I µν + N ex M 2 j ˜ F I + ˜ Z m m I, (64) G 2 M j = N ex ˜ H µν ˜ H λρ J µνλρ +2 i ˜ H µν ˜ A n n ρ J µνρ -( N ex M 2 j ˜ F + ˜ Z m m ) ˜ H µν ( J µν -q µ q ν J ) / 4 -˜ A m n µ ˜ A n m ν J µν -2 i ( M 2 j ˜ F δ m n + Z m n ) ˜ A n m ρ J µ +( N ex M 4 j ˜ F ˜ F + M 2 j ˜ FZ m m + ˜ Z m n ˜ Z n m ) J, (65)</formula>
<text><location><page_6><loc_9><loc_24><loc_49><loc_28></location>where N ex = D -4 is the number of the extra dimensions, q µ is the momentum flowing in and out through the vertices, and</text>
<formula><location><page_6><loc_14><loc_19><loc_49><loc_23></location>I µν = ∫ d 4 p i (2 π ) 4 p µ p ν [ -p 2 + M 2 j ] , (66)</formula>
<formula><location><page_6><loc_11><loc_11><loc_49><loc_16></location>J µνλρ = ∫ d 4 p i (2 π ) 4 ( p + q ) µ p ν p λ ( p + q ) ρ [ -( p + q ) 2 + M 2 j ][ -p 2 + M 2 j ] . (68)</formula>
<formula><location><page_6><loc_14><loc_15><loc_49><loc_19></location>I = ∫ d 4 p i (2 π ) 4 1 [ -p 2 + M 2 j ] , (67)</formula>
<formula><location><page_6><loc_11><loc_8><loc_49><loc_12></location>J µνρ = ∫ d 4 p i (2 π ) 4 ( p + q ) µ p ν (2 p + q ) ρ [ -( p + q ) 2 + M 2 j ][ -p 2 + M 2 j ] , (69)</formula>
<formula><location><page_6><loc_54><loc_89><loc_92><loc_94></location>J µν = ∫ d 4 p i (2 π ) 4 (2 p + q ) µ (2 p + q ) ν [ -( p + q ) 2 + M 2 j ][ -p 2 + M 2 j ] , (70)</formula>
<formula><location><page_6><loc_54><loc_82><loc_92><loc_86></location>J = ∫ d 4 p i (2 π ) 4 1 [ -( p + q ) 2 + M 2 j ][ -p 2 + M 2 j ] , (72)</formula>
<formula><location><page_6><loc_54><loc_86><loc_92><loc_90></location>J µ = ∫ d 4 p i (2 π ) 4 (2 p + q ) µ [ -( p + q ) 2 + M 2 j ][ -p 2 + M 2 j ] , (71)</formula>
<text><location><page_6><loc_52><loc_79><loc_92><loc_82></location>In the dimensional regularization, for large M 2 j , they are calculated to be</text>
<formula><location><page_6><loc_54><loc_76><loc_92><loc_78></location>I µν = -I j M 2 j η µν , I = I j M 2 j , (73)</formula>
<formula><location><page_6><loc_54><loc_64><loc_92><loc_76></location>J µνλρ = I j [( M 4 j 8 -M 2 j q 2 24 + q 4 240 ) S µνλρ -( M 2 j 12 -q 2 60 ) T µνλρ + ( M 2 j 6 -q 2 40 ) T ' µνλρ + 1 30 q µ q ν q λ q ρ ] , (74)</formula>
<formula><location><page_6><loc_54><loc_62><loc_92><loc_64></location>J µν = -I j [2 M 2 j η µν +( q µ q ν -q 2 η µν ) / 3] , (75)</formula>
<formula><location><page_6><loc_54><loc_60><loc_92><loc_62></location>J µνρ = 0 , J µ = 0 , J = I j (76)</formula>
<text><location><page_6><loc_52><loc_57><loc_72><loc_60></location>with I j = M -2 /epsilon1 j / (4 π ) 2 /epsilon1 and</text>
<formula><location><page_6><loc_53><loc_56><loc_92><loc_57></location>S µνλρ = η µν η λρ + η µλ η νρ + η µρ η νλ , (77)</formula>
<formula><location><page_6><loc_53><loc_54><loc_92><loc_55></location>T µνλρ = η µν q λ q ρ + η µρ q λ q ν + η νλ q µ q ρ + η λρ q µ q ν , (78)</formula>
<formula><location><page_6><loc_53><loc_52><loc_92><loc_54></location>T ' µνλρ = η µλ q ν q ρ + η νρ q µ q λ . (79)</formula>
<text><location><page_6><loc_52><loc_44><loc_92><loc_51></location>We substitute (73)-(74) into (64) and (65), and substitute them into (36) to get L eff , and rearrange the terms into a sum of monomials of h µν , A m nµ , B mµν and their derivatives. Each term is proportional to I j M 2 k j ( k = 0 , 1 , 2), which, when regularized via (39), behave as</text>
<formula><location><page_6><loc_63><loc_31><loc_92><loc_43></location>∑ j C j I j → ln Λ 2 (4 π ) 2 , ∑ j C j I j M 2 j →-Λ 2 2(4 π ) 2 , ∑ j C j I j M 4 j → Λ 4 2(4 π ) 2 , (80)</formula>
<text><location><page_6><loc_52><loc_28><loc_92><loc_31></location>for large Λ (=the equal mass limit of M j ). The terms are classified as follows.</text>
<text><location><page_6><loc_52><loc_26><loc_88><loc_28></location>(i) The terms with N A = N B = 0 are given by [49]</text>
<formula><location><page_6><loc_54><loc_12><loc_92><loc_26></location>N ex 32(4 π ) 2 [ Λ 4 2 (4 h -2 h (2) + h 2 ) + Λ 2 3 ( h µν,λ h µν,λ -2 h µν ,ν h µ λ ,λ +2 h µν ,ν h ,µ -h ,µ h ,µ ) + ln Λ 2 15 { h µν,λρ h µν,λρ -2 h µν ,νρ h µ λ ,λρ +4( h µν ,µν ) 2 -6 h µν ,µν h ,λ λ +3( h ,µ µ ) 2 } ] (81)</formula>
<text><location><page_6><loc_52><loc_8><loc_92><loc_13></location>up to total derivatives. Because the full expression should have the symmetry, they should be the lower order expression of √ -g times the invariant forms in table I. The</text>
<text><location><page_7><loc_9><loc_90><loc_49><loc_93></location>terms in (81) are to be compared with the lower contributions for √ -g in (55) and</text>
<formula><location><page_7><loc_12><loc_84><loc_49><loc_89></location>√ -gR = -1 4 ( h µν,λ h µν,λ -2 h µν ,ν h µ λ ,λ +2 h µν ,ν h ,µ -h ,µ h ,µ ) , (82)</formula>
<formula><location><page_7><loc_14><loc_76><loc_47><loc_83></location>-gR µν R µν = -1 4 [ h µν,λρ h µν,λρ -2 h µν ,νρ h µ λ ,λρ +2( h µν ,µν ) 2 -2 h µν ,µν h ,λ λ +( h ,µ µ ) 2 ] ,</formula>
<formula><location><page_7><loc_12><loc_79><loc_49><loc_85></location>√ -gR 2 = ( h µν ,µν ) 2 -2 h µν ,µν h ,λ λ +( h ,µ µ ) 2 , (83) √ (84)</formula>
<text><location><page_7><loc_9><loc_74><loc_49><loc_77></location>where total derivatives are neglected. reg = 0 and</text>
<text><location><page_7><loc_9><loc_73><loc_42><loc_76></location>(ii) The lowest contributions to L with N B N ∂ = 0 are</text>
<text><location><page_7><loc_43><loc_74><loc_43><loc_76></location>/negationslash</text>
<formula><location><page_7><loc_17><loc_68><loc_49><loc_72></location>1 4(4 π ) 2 ( Λ 2 B (2) +lnΛ 2 B (4) ) , (85)</formula>
<text><location><page_7><loc_42><loc_66><loc_49><loc_68></location>√ (2)</text>
<text><location><page_7><loc_45><loc_63><loc_49><loc_65></location>= 2 is</text>
<text><location><page_7><loc_9><loc_63><loc_47><loc_67></location>which are taken as the lowest parts of the forms -gB and √ -gB (4) . (iii) The lowest contribution with N B = 0 and N ∂</text>
<text><location><page_7><loc_36><loc_62><loc_36><loc_65></location>/negationslash</text>
<formula><location><page_7><loc_18><loc_59><loc_49><loc_62></location>ln Λ 2 12(4 π ) 2 ( h µν ,µν + h ,µ µ ) B (2) (86)</formula>
<text><location><page_7><loc_9><loc_55><loc_43><loc_58></location>which is the lowest part of the form √ -gRB (2) . (iv) The lowest contribution with N A = 0 is</text>
<text><location><page_7><loc_36><loc_54><loc_36><loc_56></location>/negationslash</text>
<formula><location><page_7><loc_11><loc_51><loc_49><loc_54></location>1 24(4 π ) 2 ( A mnµ,ν -A mnν,µ )( A mnµ,ν -A mnν,µ ) , (87)</formula>
<text><location><page_7><loc_9><loc_44><loc_49><loc_50></location>which is the lowest part of the form √ -gA mnµν A mnµν with N A = 2. Note that it suffices to determine the coefficient of the form in L reg .</text>
<formula><location><page_7><loc_12><loc_30><loc_49><loc_42></location>L div = √ -g/ (4 π ) 2 × [ N ex { Λ 4 8 -Λ 2 24 R + ln Λ 2 240 ( R 2 +2 R µν R µν ) } + Λ 2 4 B (2) + ln Λ 2 4 B (4) -ln Λ 2 12 RB (2) -ln Λ 2 24 A mnµν A mnµν ] , (88)</formula>
<text><location><page_7><loc_9><loc_41><loc_49><loc_45></location>Collecting the results of (i)-(iv), we finally obtain the expression for the divergent part L div of L reg :</text>
<text><location><page_7><loc_9><loc_22><loc_49><loc_29></location>where B (2) , B (4) and A mnµν are defined in (59), (60) and (63), respectively, and N ex is the number of the extra dimensions. The divergences cannot be renormalized because the original action does not have these terms. They give rise to genuine quantum induced effects.</text>
<section_header_level_1><location><page_7><loc_13><loc_18><loc_44><loc_19></location>VIII. COSMOLOGICAL CONSTANT</section_header_level_1>
<text><location><page_7><loc_9><loc_9><loc_49><loc_16></location>Thus, we have derived the quantum effects of the brane fluctuations. Among them, the Λ 4 term in (88) gives huge a contribution to the cosmological term. To this term, the starting Lagrangian L ' br in (8) by itself also has a contribution. From phenomenological points of view, it</text>
<text><location><page_7><loc_52><loc_89><loc_92><loc_93></location>should be very tiny. Therefore, the large contributions should cancel out each other to give the tiny cosmological term. The condition for the cancellation is</text>
<formula><location><page_7><loc_64><loc_85><loc_92><loc_88></location>λ = -N ex Λ 4 / 128 π. (89)</formula>
<text><location><page_7><loc_52><loc_79><loc_92><loc_85></location>This is, however, an extremely unnatural fine tuning. It is a serious problem common to the quantum theories including gravity in general. The present formulation has no solution to this longstanding problem.</text>
<text><location><page_7><loc_52><loc_73><loc_92><loc_79></location>Furthermore, it may give rise to another contribution which may mimic the cosmological term in the effective equation of motion for g µν . The energy momentum tensor in the equation has the term</text>
<formula><location><page_7><loc_66><loc_70><loc_92><loc_72></location>λY I ,µ Y J ,ν G IJ / 2 , (90)</formula>
<text><location><page_7><loc_52><loc_64><loc_92><loc_69></location>which may look like the cosmological term if the embedding is almost flat. In such cases, we can adjust the cosmological term to the phenomenological tiny value by, for example, adopting the conformally flat embedding</text>
<formula><location><page_7><loc_56><loc_61><loc_92><loc_62></location>Y µ = [1 + N ex Λ 4 / 128 πλ ] 1 / 2 x µ , Y m = 0 (91)</formula>
<text><location><page_7><loc_52><loc_45><loc_92><loc_59></location>instead of the condition (89). This is also an extremely unnatural fine tuning. Thus, the present model is not satisfactory in natural understanding of the cosmological constant. It is, however, not a problem for the model alone, but a serious puzzle for general quantum theoretical models with gravity. It is an open problem, and we wish that it will be solved in the future. The problem will be partly addressed in our forthcoming paper. Here, we phenomenologically adjust the tiny cosmological term via fine tuning of (89) or (91).</text>
<section_header_level_1><location><page_7><loc_61><loc_41><loc_82><loc_42></location>IX. INDUCED GRAVITY</section_header_level_1>
<text><location><page_7><loc_52><loc_11><loc_92><loc_39></location>If the cosmological term is suppressed, the main contribution in the quantum effects (88) comes from the R term. It is nothing but the Einstein-Hilbert action, which supply the kinetic term for the auxiliary field g µν . The sign of the term is right one to give ordinary attractive gravity in accordance with the observation, and its magnitude indicates that the cutoff Λ is order of the Planck scale. The term with ( R 2 + 2 R µν R µν ) gives small corrections of O (log Λ 2 / Λ 2 ) as far as the brane curvature is small. The terms with B (2) , RB (2) and B (4) are the mass and interaction terms of the field B mµν . Note that no kinetic term for B mµν appears. This is because the B mµν interacts with φ m only in the combination B mµν B nµν , but not in single. The term with A mnµν squared gives the kinetic and the interaction terms of the field A mnµ as the gauge field. The fields A mnµ and B mµν appear as fields on the brane. We should, however, be careful because they are not independent and are defined by (22) and (23) in terms of Y I and n m .</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>The quantum induced terms in (88) modify the equations of motion. The equation (9) for Y I is modified</text>
<text><location><page_8><loc_9><loc_77><loc_49><loc_93></location>through the B (2) , RB (2) , B (4) and ( A mnµν ) 2 terms in (88). The classical solution Y I is deformed according to it. The correction terms are suppressed by at least a factor of O (Λ -2 ) for small curvatures. The equation (10) for g µν , the induced-metric formula, is converted into the Einstein equation with the O ( R 2 ) correction terms and the energy momentum tensor for the fields φ m , A mnµ , B mµν , and Y I . The equation (10) holds as operator relation. In classical realizations, however, it suffers from a large quantum corrections. Then, we no longer have the induced-metric formula.</text>
<section_header_level_1><location><page_8><loc_21><loc_73><loc_36><loc_74></location>X. DISCUSSIONS</section_header_level_1>
<text><location><page_8><loc_9><loc_53><loc_49><loc_71></location>The metric g µν emerges in the channel of intermediate quantum states composed of φ m 's, despite its absence in the original setup of the system (1). Hence, it is interpreted as a composite of the brane fluctuation fields φ m . Then the natural question is what is the further quantum effects of the composite metric. Within the semiclassical treatments, it suffices to calculate only the one loop diagrams. The system does not include multi-loop diagrams. This is the virtue of the linear definition (16) of the brane fluctuation. Beyond the semi-classical approximation, however, we should take into account the higher order diagrams with the internal lines of composite metric fields.</text>
<text><location><page_8><loc_9><loc_34><loc_49><loc_52></location>The quantum induction mechanism of composite fields is common phenomena to various composite field theories [50]. A class of non-renormalizable theories with this mechanism becomes equivalent to some remormalizable models (with finite momentum cutoff) under the 'compositeness condition' that the wave-function renormalization constant vanishes [51], [52]. This renders us clues to formulate unambiguously the non-renormalizable theories at higher orders [53]. For example, the NambuJona-Lasinio model is equivalent to the Yukawa model with the vanishing renormalization constants of the scalar and the pseudoscalar fields, and the latter renders a unambiguous higher-order descriptions of the former.</text>
<text><location><page_8><loc_9><loc_15><loc_49><loc_34></location>In the present case, however, the induced composite field theory is the modified Einstein gravity, and is not renormalizable. We have no definite way to calculate the quantum effects due to the metric itself at higher orders. It shares the problems with the general quantum gravity theories. So we cannot apply all the achievements of the composite field theories with the compositeness condition. They are, however, very suggestive in considering properties of the quantum fluctuations. In composite theories, it is plausible that the quantum effects due to the composite would require different treatments. For example, if the cutoff for the composites is much smaller than that for the constituents, the effects can be suppressed</text>
<text><location><page_8><loc_52><loc_88><loc_92><loc_93></location>[54]. Or, the 1/ N expansion would be useful, as is in the various composite field theories [52], [53], [55]. We need further ideas and investigations for the more complete treatments.</text>
<text><location><page_8><loc_52><loc_47><loc_92><loc_87></location>We can see in (88) that the quantum effects give rise to the terms including the extrinsic curvature B mµν and the normal connection A mnµ , in addition to the inducedgravity terms [48]. The induction of these terms is characteristic of the brane induced gravity theory, distinguished from the ordinary (non-brane) induced gravity. The fact was recognized in [14] and [16] in the general braneworld scheme, and they were actually calculated in [18], [34] for the domain-wall type braneworld. The forms of induced terms depend on the brane dynamics. The simplest case of the Nambu-Goto action was considered in [4] within the four dimensional field theory. In the model, however, only the gravity is induced, but no other terms. This is because the spacetime spanned by the scalar fields is not the real one, and hence it assumes no symmetry of the whole spacetime involving the brane. Therefore, we cannot define the normal to the brane. On the contrary, the present model (1) possesses general-coordinate invariance of the bulk, as well as that of the brane. Therefore, the fluctuations along the brane is meaningless, and the only physical ones are those transverse to the brane, as are defined by (15)-(17). This is the origin how it includes the A mnµ and B mµν dependent terms. They should be determined according to the brane dynamics, as is done here. It would be an interesting and urgent subject to derive the induced terms in various brane dynamics, and seek for the models suited for applications.</text>
<section_header_level_1><location><page_8><loc_65><loc_42><loc_79><loc_43></location>Acknowledgments</section_header_level_1>
<text><location><page_8><loc_52><loc_21><loc_92><loc_40></location>We would like to thank Professor G. R. Dvali, Professor G. Gabadadze, Professor M. E. Shaposhnikov, Professor I. Antoniadis, Professor M. Giovannini, Professor S. Randjbar-Daemi, Professor R. Gregory, Professor P. Kanti, Professor G. Gibbons, Professor K. Hashimoto, Professor E. J. Copeland, Professor D. L. Wiltshire, Professor I. P. Neupane, Professor R. R. Volkas, Professor A. Kobakhidze, Professor C. Wetterich, Professor M. Shifman, Professor A. Vainshtein, Professor D. Wands, Professor M. Visser, Professor T. Inami, Professor I. Oda and Professor H. Mukaida for invaluable discussions and their kind hospitality extended to us during our stay in their places.</text>
<text><location><page_8><loc_52><loc_15><loc_92><loc_21></location>This work was supported by Grant-in-Aid for Scientific Research, No. 13640297, 17500601, and 22500819 from Japanese Ministry of Education, Culture, Sports, Science and Technology.</text>
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</document>
[
{
"title": "Keiichi Akama",
"content": "Department of Physics, Saitama Medical University, Saitama, 350-0495, Japan",
"pages": [
1
]
},
{
"title": "Takashi Hattori",
"content": "Department of Physics, Kanagawa Dental College, Yokosuka, 238-8580, Japan (Dated: October 8, 2018) We present a comprehensive formalism to derive precise expressions for the induced gravity of the braneworld, assuming the dynamics of the Dirac-Nambu-Goto type. The quantum fluctuations of the brane at short distances give rise to divergences, which should be cutoff at the scale of the inverse thickness of the brane. It turns out that the induced-metric formula is converted into an Einstein-like equation via the quantum effects. We determine the coefficients of the induced cosmological and gravity terms, as well as those of the terms including the extrinsic curvature and the normal connection gauge field. The latter is the characteristic of the brane induced gravity theory, distinguished from ordinary none-brane induced gravity. PACS numbers: 04.50.-h, 04.62.+v, 11.25.-w, 12.60.Rc",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "General relativity is based on the premises that the spacetime is curved affected by matter according to the Einstein equation, and that the objects move along the spacetime geodesics. The gravitations are apparent phenomena of the inertial motions in the curved spacetime. This successfully explains why the motion in the 'gravitational' field is universal, i.e. blind to the object properties (mass, charge etc.), and why the 'gravitational' forces are subject to the Newton's law of gravitation (within the ordinary precision). It is supported by further precise observations. It raises, however, another fundamental question why the spacetime is curved so as the Einstein equation indicates. A possible answer was as follows: Suppose our spacetime is a 3+1 dimensional embedded object like domain walls or vortices (braneworld, [1]-[41]) in higher dimensional spacetime, and the gravity is induced through quantum fluctuations (brane induced gravity, [4], [5], [10], [14]-[18], [27], [28], [34], [35]). The cosmological constant should be finely tuned. Then, the brane-gravity field emerges as a composite, which obeys an Einstein-like equation at least at low curvatures, just like in the case of the (non-brane) induced gravity theory [42]-[46]. The ideas of the braneworld and the brane induced gravity have been studied extensively in the last three decades. Physical models were constructed with topological defects in higher dimensional spacetime [5]-[18], and they were realized as 'D-branes' in the superstring theory [19]-[22]. They were applied to the hierarchy problem with large extra dimensions, or with warped extra dimensions [20], [23]-[25]. It was argued that the brane induced gravity would imply infrared modifications of the gravity theory [26]-[28]. The ideas have been studied in wide areas including basic formalism, [29]-[33], brane induced gravity [34]-[35], particle physics phenenomenology [36][38], and cosmology [39]-[41] with many interesting consequences. In this paper, we establish a precise formalism to derive the expressions for the quantum induced effects on the brane. For definiteness, we follow the simplest model with the Dirac-Nambu-Goto type dynamics [47]. Such a model was first considered in [4] with scalar fields which we now interpret as the position coordinate of the brane in higher dimensions. Such an interpretation motivated us to consider the model of braneworld as a topological object moving in higher dimensions in [5]. The idea of the brane induced gravity has been considered repeatedly in the literature. They were, however, more or less naive both in model setting and derivation. Here we present a comprehensive formalism from its foundation to the precise outcomes. Among the quantum fluctuations, the only meaningful ones are those transverse to the brane. The quantum loop effects are divergent, which should be cutoff at the scale of the inverse thickness of the brane. We adopt the regularization scheme used in the original work [4], and calculate them at the one-loop level. We determine the coefficients of the induced cosmological and gravity terms, as well as those of the terms including the extrinsic curvature and the normal connection gauge field [48]. The induction of the latter terms is characteristic of the brane induced gravity theory, distinguished from the ordinary (non-brane) induced gravity. It turns out that the induced-metric formula is converted into an Einsteinlike equation via the quantum short-distance effects. The plan of this paper is as follows. First, we define the model (Sec. II), and then we derive the quantum effects (Secs. III-VII). We define the brane fluctuations (Sec. III), formulate the quantum effects (Sec. IV), specify the method to regularize the divergences (Sec. V), classify the possible induced terms according to symmetries (Sec. VI), and calculate them via Feynman diagram method (Sec. VII). Then, we interpret the quantum induced terms, and show why the induced gravity can avoid the problem (Secs. VIII and IX). The cosmological terms are fine-tuned (Sec. VIII), and the Einstein like gravity and other terms are induced (Sec. IX). The final section (Sec. X) is devoted to discussions.",
"pages": [
1,
2
]
},
{
"title": "II. THE MODEL",
"content": "We consider a quantum theoretical braneworld described by the Dirac-Nambu-Goto type Lagrangian. We will see the quantum effects of the brane fluctuations give rise to effective braneworld gravity. Let X I ( x µ ) ( I = 0 , 1 , · · · , D -1) be the position of our three-brane in the D dimensional spacetime (bulk), parameterized by the brane coordinate x µ ( µ = 0 , 1 , 2 , 3), where I = 0 and µ = 0 indicate the time components. Let G IJ ( X K ) be the bulk metric tensor at the bulk point X K . This is taken to obey some bulk gravity theory. Then we consider a braneworld with dynamics given by the DiracNambu-Goto type Lagrangian (density) [47]: where λ is a constant. Or we write it as with abbreviations g [ X ] = det g [ X ] µν , and where (and hereafter) indices following a comma (,) indicate differentiation with respect to the corresponding coordinate component, and [ X ] is attached to remind that they are abbreviations for expressions written in terms of X I . Note that g [ X ] µν is the induced metric on the brane with (3). We assume that X I appears nowhere other than in L br in the total Lagrangian L tot including the bulk Lagrangian. The system is invariant under the general coordinate transformation of the bulk and the brane separately. The Lagrangian (1) is the simplest among those with this symmetry. Note that we do not a priori have the kinetic term of the metric in the basic Lagrangian (1). It will be induced through quantum effects. The metric emerges as a composite field, and the gravitation is an induced phenomenon but not an elementary one. If this is successful, we achieve a great conceptual advantage, since it means that the familiar and important phenomenon of the gravitation is explained by the more fundamental ingredients. Thus, it is worthwhile and urgent to examine what is true and what is not in this idea. This is the basic spirit of the induced gravity theory [42]-[46]. As for the dynamics of the bulk metric, the situation is more vague and ambiguous. We do not know what is the dynamics, i.e. what kinetic term we have, and it is not clear even whether the kinetic term exists or not, i.e. whether it is dynamical or not. Every case well deserves intensive investigations. It is too naive to take that the bulk Einstein equation provides the brane Einstein equation at the brane, because large discrepancies take place due to the extrinsic curvature terms. The brane moves according to its equation of motion and is deformed in manners different from what the naive gravity equation indicates. One may elaborate the elementary brane gravity from the elementary bulk gravity under specific conditions by hand in specific models [25], [39]. On the other hand, many people relied on the mechanism of brane induced gravity to realize gravitation on the brane [5], [10], [14]-[18], [27], [28], [34], [35]. In the spirit of the brane induced gravity, we do not a priori assume elementary gravity, and we take it to be induced from more fundamental ingredients. Now, the equation of motion from (1) is given by where X [ X ] ; µν I is the double covariant derivative with respect to both of the general coordinate transformations on the brane and to those in the bulk: with the affine connections on the brane and bulk respectively. We expect that this gives a good approximation at low curvature limit in many dynamical models of the braneworld (e.g. topological defects, spacetime singularities, D-branes, etc.). It is remarkable that, as we shall see below, this simple model exhibits brane gravity and gauge theory like structure through the quantum effects. For convenience of quantum treatments, we consider the following equivalent Lagrangian to (1): where g µν is an auxiliary field, g = det g µν , and g µν is the inverse matrix of g µν . Note that g µν , unlike g [ X ] µν above, is treated as a field independent of X I . Then the Euler Lagrange equations with respect to X I and g µν are given by respectively, where the covariant derivative is written in terms of the brane affine connection with respect to the auxiliary field g µν . Now g µν in (9) is independent of X I , and, instead, we have an extra equation (10), which guarantees that g µν is the induced metric. If we substitute (10) into (9), we obtain the same equation as (4). Thus the systems with the Lagrangians L br and L ' br coincide. Furthermore the argument that their Dirac bracket algebrae coincide [44] indicates their quantum theoretical equivalence. We proceed hereafter based on the Lagrangian L ' br instead of L br .",
"pages": [
2,
3
]
},
{
"title": "III. BRANE FLUCTUATIONS",
"content": "In order to extract the quantum effects of L ' br , we deploy a semi-classical method, where we consider those due to small fluctuations of the brane around some classical solution (say Y I ( x µ )) for X I ( x µ ) of the equation of motion (4) [15]. Namely, the solution Y I ( x µ ) obeys the classical equation In quantum treatment, X I itself in the Lagrangian L ' br does not necessarily obey the equation of motion (4), and may fluctuate from Y I ( x µ ). Among the fluctuations, only those transverse to the brane are physically meaningful , because those along the brane remain within the brane and cause no real fluctuations of the brane. They are absorbed by general coordinate transformations. In order to describe them, we choose D -4 independent normal vectors n m I ( x µ ) ( m = 4 , · · · , D -1) at each point on the brane with the normality condition Then we express the fluctuations as where φ m ( x µ ) is the transverse fluctuation along n m I ( x µ ) ( m = 4 , · · · , D -1). The arbitrariness of n m I in choice under the condition (15) gives rise to gauge symmetry under the group GL( D -4) of the general linear transformations of the normal space at each point on the brane. If we define g mn = n I m n J n G IJ ( Y K ) and its inverse g mn , we have the completeness condition, Bulk coordinate indices I, J, · · · (= 0 , · · · , D -1) are raised and lowered by the metric tensors G IJ and G IJ . We can read off from (8) and (10) that the auxiliary field g µν plays the role the metric tensor on the brane. Hereafter we raise and lower the brane coordinate indices µ, ν, · · · (= 0 , · · · , 3) by g µν and g µν (but not by g [ X ] µν and g [ X ] µν ). We raise and lower the normal space indices m,n, · · · (= 4 , · · · , D -1) by g mn and g mn . A problem of the definition (16) of φ m is that φ m lacks the bulk general-coordinate invariance. In fact, it is transformed in a complex way under the general coordinate transformations of the bulk. On the contrary, the invariant definition of φ m requires a complex formula instead of (16). For quantum treatments, however, it is desirable to have a relation linear in φ m like (16). It requires further careful considerations. Therefore, in this paper, we restrict ourselves to the case where the bulk is flat. Namely, there exists the cartesian frame with where we have Γ I JK = 0. In this case, we have R I JKL = 0 in any frame, and φ m restores the bulk general-coordinate invariance. The general case of curved bulk will be considered in a separate forthcoming paper. Now we substitute (16) into the Lagrangian (8), and obtain where A mnµ and B mµν are the normal connection and the extrinsic curvature, respectively. They are given by where n J n ; µ is the covariant derivative: which coincides with the ordinary derivative n J n ,µ under the assumption G IJ = η IJ for the present paper. We can see that (21) is the Lagrangian for the quantum scalar fields φ m on the curved brane interacting with the given external fields A mnµ and B mµν . For later use, it is convenient to rewrite (21) into where total derivatives are neglected.",
"pages": [
3
]
},
{
"title": "IV. QUANTUM EFFECTS",
"content": "The quantum effects of the field φ m are described by the effective Lagrangian L eff where [ dφ m ] is the path-integration over φ m . To perform it, we rewrite (25) into the form with /square = η µν ∂ µ ∂ ν and where the differential operator ∂ µ ≡ ∂/∂x µ is taken to operate on the whole expression in its right side in (27). The path-integration in (26) is performed to give up to additional constants, where Tr indicates the trace over the brane coordinate variable x µ and extra dimension index m . The terms in (32) can be calculated with Feynman-diagram method. In terms of the Fourier transforms the effective Lagrangian L eff is written as where p i = p + q 1 + · · · + q i and m 0 = m l . The function G l is nothing but the Feynman amplitude for the oneloop diagram with l internal lines of φ m and l vertices of ˜ V m n (FIG. 1). Unfortunately, the p -dependence of the integrand in (36) with (37) indicates that the integration over p diverges at most quartically. The divergences will be regulated in the next section. Then, we can perform the integration over p to obtain the function G l . The q k 's are replaced by differentiation i∂ k of the k -th vertex function according to the inverse Fourier transformation in (36). Collecting all the contributions, which are functions of the fields g µν , A mnµ and B mµν and their derivatives, we can obtain the expression for the effective Lagrangian L eff .",
"pages": [
3,
4
]
},
{
"title": "V. DIVERGENCES AND REGULARIZATION",
"content": "The p -dependence of the integrand in (36) with (37) indicates that the integration over p diverges at most quartically. We expect, however, that fluctuations with smaller wave length than the brane thickness are suppressed. Then, the momenta higher than the inverse of the thickness are cut off. In order to model the cutoff without violating full symmetry of L ' br , we introduce three Pauli-Villers regulators Φ m j with very large mass M j ( j =1,2,3), which are taken equal finally, M j → Λ, following the original paper [4]. Precisely, it amounts to consider the regularized effective Lagrangian where C j are the coefficients defined by and L eff M j is the effective Lagrangian for the quantum effects from L ' Φ j which is the same as L ' φ except that φ m is replaced by the regulator field Φ m j with mass M j . Note that the added mass term also preserves the full symmetry of L ' br . Performing the path integration over Φ m j , we have with V m n in (28). In terms of the Fourier transform we have with ˜ V m n ( p i , q i ) in (38). In dimensional regularization, the divergent parts of the Feynman amplitude G l M j behaves like where this is evaluated at the spacetime dimension 4 -2 /epsilon1 , and G 2 k are the appropriate coefficient functions. The singularities at /epsilon1 = 0 reflect the divergences in the p -integration. We can see that, when they are summed with the coefficients C j over j in (39), they cancel out according to (40). Therefore the p -integrations in L reg converge. Any positive power contributions of M j regular at infinity vanish according to (40). The function G l M j involves logarithmic singularities in M j , which tend, in the equal mass limit M j → Λ,",
"pages": [
4,
5
]
},
{
"title": "VI. CLASSIFICATION OF THE TERMS",
"content": "Thus the divergent part L div of the regularized effective Lagrangian L reg consist of the terms which are proportional to Λ 4 , Λ 2 or ln Λ 2 , and are monomials of H µν , F , A m n µ , Z m n , and their derivatives. The expressions H µν , F , A m n µ , and Z m n are written in terms of the fields g µν , A m n µ , and B m µν according to (29), (45), (30), and (31). Introducing the notation h µν ≡ g µν -η µν , we rewrite g µν and √ -g in H µν , F , A m n µ , and Z m n according to with [49] Then, L div becomes an infinite sum of monomials of h µν , A m nµ , B m µν , and their derivatives. Let us denote the numbers of h µν , A m nµ , B m µν , and the differential operators in the monomial by N h , N A , N B and N ∂ , respectively. The Lagrangian L reg should have mass dimension 4, while h µν , A m nµ , B m µν , and the differential operator has mass dimension 0, 1, 1, and 1, respectively. Therefore, the numbers N A , N B and N ∂ are restricted by where k div = 2 , 1 , 0 for Λ 4 , Λ 2 , and ln Λ 2 terms, respectively. On the other hand, the number N h of h µν is not restricted. The relation (58) allows only finite numbers of values of N A , N B and N ∂ , according to which we can classify the terms of L div . Each class involves infinitely many terms for arbitrary values of N h . They are, however, not all independent, because they are related by high symmetry of the system under the general coordinate transformations on the brane and GL(4) gauge transformations of the normal space rotation. Though the original system is invariant under the general coordinate transformations in the bulk also, it is not available here, because we restrict ourselves to the flat bulk in this paper. The general case will be discussed in the forthcoming paper. Owing to the symmetry of the system, only finite number of terms are allowed. The general coordinate transformation symmetry requires that the effective Lagrangian density is proportional to √ -g times a sum of invariant forms. We list the allowed invariant forms in TABLE I. In the table and thereafter, we use the following abbreviations. where (61) and (62) are the covariant derivatives of the full symmetry, and (63) is the field strength of the gauge field A mµν . In the table, R 2 , R µν R µν , and R µνρσ R µνρσ are not all independent, but related by Gauss-Bonnet relation, and some other combinations are related due to the Gauss-Codazzi-Ricci formulae.",
"pages": [
5,
6
]
},
{
"title": "VII. CALCULATION",
"content": "/negationslash From (48), (49) and (38), they are given by Thus we can calculate the coefficients of the term √ -g times the invariant forms by calculating the lowest order contributions in h µν . The lowest contributions to the term with N A = N B = N ∂ = 0 are O ( h µν ), while those to N A = N B = 0 and N ∂ = 0 are O (( h µν ) 2 ), because the O ( h µν ) terms are total derivatives. Therefore, their lowest terms are in the one- and two-point functions G 1 and G 2 . We can see from (28) - (31) that B mµν always appears in the combination B mµν B nµν . Therefore, the only possible forms including B mµν are B (2) , RB (2) , ( B (2) ) 2 , and B (4) , among many forms listed in Table I. Their lowest terms are those with O (( h µν ) 0 ) and are also in G 1 and G 2 . The only possible form including A mnµ only is A mnµν A mnµν and its lowest term is of O (( h µν ) 0 ), and it is in G 2 . Thus, it suffices to calculate G 1 and G 2 in order to determine full contributions to L div . where N ex = D -4 is the number of the extra dimensions, q µ is the momentum flowing in and out through the vertices, and In the dimensional regularization, for large M 2 j , they are calculated to be with I j = M -2 /epsilon1 j / (4 π ) 2 /epsilon1 and We substitute (73)-(74) into (64) and (65), and substitute them into (36) to get L eff , and rearrange the terms into a sum of monomials of h µν , A m nµ , B mµν and their derivatives. Each term is proportional to I j M 2 k j ( k = 0 , 1 , 2), which, when regularized via (39), behave as for large Λ (=the equal mass limit of M j ). The terms are classified as follows. (i) The terms with N A = N B = 0 are given by [49] up to total derivatives. Because the full expression should have the symmetry, they should be the lower order expression of √ -g times the invariant forms in table I. The terms in (81) are to be compared with the lower contributions for √ -g in (55) and where total derivatives are neglected. reg = 0 and (ii) The lowest contributions to L with N B N ∂ = 0 are /negationslash √ (2) = 2 is which are taken as the lowest parts of the forms -gB and √ -gB (4) . (iii) The lowest contribution with N B = 0 and N ∂ /negationslash which is the lowest part of the form √ -gRB (2) . (iv) The lowest contribution with N A = 0 is /negationslash which is the lowest part of the form √ -gA mnµν A mnµν with N A = 2. Note that it suffices to determine the coefficient of the form in L reg . Collecting the results of (i)-(iv), we finally obtain the expression for the divergent part L div of L reg : where B (2) , B (4) and A mnµν are defined in (59), (60) and (63), respectively, and N ex is the number of the extra dimensions. The divergences cannot be renormalized because the original action does not have these terms. They give rise to genuine quantum induced effects.",
"pages": [
6,
7
]
},
{
"title": "VIII. COSMOLOGICAL CONSTANT",
"content": "Thus, we have derived the quantum effects of the brane fluctuations. Among them, the Λ 4 term in (88) gives huge a contribution to the cosmological term. To this term, the starting Lagrangian L ' br in (8) by itself also has a contribution. From phenomenological points of view, it should be very tiny. Therefore, the large contributions should cancel out each other to give the tiny cosmological term. The condition for the cancellation is This is, however, an extremely unnatural fine tuning. It is a serious problem common to the quantum theories including gravity in general. The present formulation has no solution to this longstanding problem. Furthermore, it may give rise to another contribution which may mimic the cosmological term in the effective equation of motion for g µν . The energy momentum tensor in the equation has the term which may look like the cosmological term if the embedding is almost flat. In such cases, we can adjust the cosmological term to the phenomenological tiny value by, for example, adopting the conformally flat embedding instead of the condition (89). This is also an extremely unnatural fine tuning. Thus, the present model is not satisfactory in natural understanding of the cosmological constant. It is, however, not a problem for the model alone, but a serious puzzle for general quantum theoretical models with gravity. It is an open problem, and we wish that it will be solved in the future. The problem will be partly addressed in our forthcoming paper. Here, we phenomenologically adjust the tiny cosmological term via fine tuning of (89) or (91).",
"pages": [
7
]
},
{
"title": "IX. INDUCED GRAVITY",
"content": "If the cosmological term is suppressed, the main contribution in the quantum effects (88) comes from the R term. It is nothing but the Einstein-Hilbert action, which supply the kinetic term for the auxiliary field g µν . The sign of the term is right one to give ordinary attractive gravity in accordance with the observation, and its magnitude indicates that the cutoff Λ is order of the Planck scale. The term with ( R 2 + 2 R µν R µν ) gives small corrections of O (log Λ 2 / Λ 2 ) as far as the brane curvature is small. The terms with B (2) , RB (2) and B (4) are the mass and interaction terms of the field B mµν . Note that no kinetic term for B mµν appears. This is because the B mµν interacts with φ m only in the combination B mµν B nµν , but not in single. The term with A mnµν squared gives the kinetic and the interaction terms of the field A mnµ as the gauge field. The fields A mnµ and B mµν appear as fields on the brane. We should, however, be careful because they are not independent and are defined by (22) and (23) in terms of Y I and n m . The quantum induced terms in (88) modify the equations of motion. The equation (9) for Y I is modified through the B (2) , RB (2) , B (4) and ( A mnµν ) 2 terms in (88). The classical solution Y I is deformed according to it. The correction terms are suppressed by at least a factor of O (Λ -2 ) for small curvatures. The equation (10) for g µν , the induced-metric formula, is converted into the Einstein equation with the O ( R 2 ) correction terms and the energy momentum tensor for the fields φ m , A mnµ , B mµν , and Y I . The equation (10) holds as operator relation. In classical realizations, however, it suffers from a large quantum corrections. Then, we no longer have the induced-metric formula.",
"pages": [
7,
8
]
},
{
"title": "X. DISCUSSIONS",
"content": "The metric g µν emerges in the channel of intermediate quantum states composed of φ m 's, despite its absence in the original setup of the system (1). Hence, it is interpreted as a composite of the brane fluctuation fields φ m . Then the natural question is what is the further quantum effects of the composite metric. Within the semiclassical treatments, it suffices to calculate only the one loop diagrams. The system does not include multi-loop diagrams. This is the virtue of the linear definition (16) of the brane fluctuation. Beyond the semi-classical approximation, however, we should take into account the higher order diagrams with the internal lines of composite metric fields. The quantum induction mechanism of composite fields is common phenomena to various composite field theories [50]. A class of non-renormalizable theories with this mechanism becomes equivalent to some remormalizable models (with finite momentum cutoff) under the 'compositeness condition' that the wave-function renormalization constant vanishes [51], [52]. This renders us clues to formulate unambiguously the non-renormalizable theories at higher orders [53]. For example, the NambuJona-Lasinio model is equivalent to the Yukawa model with the vanishing renormalization constants of the scalar and the pseudoscalar fields, and the latter renders a unambiguous higher-order descriptions of the former. In the present case, however, the induced composite field theory is the modified Einstein gravity, and is not renormalizable. We have no definite way to calculate the quantum effects due to the metric itself at higher orders. It shares the problems with the general quantum gravity theories. So we cannot apply all the achievements of the composite field theories with the compositeness condition. They are, however, very suggestive in considering properties of the quantum fluctuations. In composite theories, it is plausible that the quantum effects due to the composite would require different treatments. For example, if the cutoff for the composites is much smaller than that for the constituents, the effects can be suppressed [54]. Or, the 1/ N expansion would be useful, as is in the various composite field theories [52], [53], [55]. We need further ideas and investigations for the more complete treatments. We can see in (88) that the quantum effects give rise to the terms including the extrinsic curvature B mµν and the normal connection A mnµ , in addition to the inducedgravity terms [48]. The induction of these terms is characteristic of the brane induced gravity theory, distinguished from the ordinary (non-brane) induced gravity. The fact was recognized in [14] and [16] in the general braneworld scheme, and they were actually calculated in [18], [34] for the domain-wall type braneworld. The forms of induced terms depend on the brane dynamics. The simplest case of the Nambu-Goto action was considered in [4] within the four dimensional field theory. In the model, however, only the gravity is induced, but no other terms. This is because the spacetime spanned by the scalar fields is not the real one, and hence it assumes no symmetry of the whole spacetime involving the brane. Therefore, we cannot define the normal to the brane. On the contrary, the present model (1) possesses general-coordinate invariance of the bulk, as well as that of the brane. Therefore, the fluctuations along the brane is meaningless, and the only physical ones are those transverse to the brane, as are defined by (15)-(17). This is the origin how it includes the A mnµ and B mµν dependent terms. They should be determined according to the brane dynamics, as is done here. It would be an interesting and urgent subject to derive the induced terms in various brane dynamics, and seek for the models suited for applications.",
"pages": [
8
]
},
{
"title": "Acknowledgments",
"content": "We would like to thank Professor G. R. Dvali, Professor G. Gabadadze, Professor M. E. Shaposhnikov, Professor I. Antoniadis, Professor M. Giovannini, Professor S. Randjbar-Daemi, Professor R. Gregory, Professor P. Kanti, Professor G. Gibbons, Professor K. Hashimoto, Professor E. J. Copeland, Professor D. L. Wiltshire, Professor I. P. Neupane, Professor R. R. Volkas, Professor A. Kobakhidze, Professor C. Wetterich, Professor M. Shifman, Professor A. Vainshtein, Professor D. Wands, Professor M. Visser, Professor T. Inami, Professor I. Oda and Professor H. Mukaida for invaluable discussions and their kind hospitality extended to us during our stay in their places. This work was supported by Grant-in-Aid for Scientific Research, No. 13640297, 17500601, and 22500819 from Japanese Ministry of Education, Culture, Sports, Science and Technology. Filho, Phys. Rev. D 84 , 064019 (2011); D. Maity, arXiv:1209.0862 Phys. Rev. D 86 084056 (2012); L. J. S. Sousa, C. A. S. Silva and C. A. S. Almeida, Phys. Lett. B 718 579-583 (2012).",
"pages": [
8,
10
]
}
]
2013CQGra..30w5031C
https://arxiv.org/pdf/1212.6430.pdf
<document>
<section_header_level_1><location><page_1><loc_30><loc_76><loc_69><loc_78></location>Energy conditions in Jordan frame</section_header_level_1>
<text><location><page_1><loc_23><loc_72><loc_77><loc_74></location>Saugata Chatterjee 1 ∗ , Damien A. Easson 1 † , and Maulik Parikh 1 , 2 ‡</text>
<text><location><page_1><loc_29><loc_67><loc_71><loc_71></location>1 Department of Physics, Arizona State University, Tempe, Arizona 85287, USA</text>
<text><location><page_1><loc_27><loc_63><loc_73><loc_66></location>2 Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe, Arizona 85287, USA</text>
<section_header_level_1><location><page_1><loc_46><loc_56><loc_53><loc_57></location>Abstract</section_header_level_1>
<text><location><page_1><loc_20><loc_46><loc_79><loc_55></location>The null energy condition, in its usual form, can appear to be violated by transformations in the conformal frame of the metric. We propose a generalization of the form of the null energy condition to Jordan frame, in which matter is non-minimally coupled, which reduces to the familiar form in Einstein frame. Using our version of the null energy condition, we provide a direct proof of the second law of black hole thermodynamics in Jordan frame.</text>
<text><location><page_1><loc_19><loc_43><loc_53><loc_45></location>PACS numbers: 98.80.Cq, 04.70.-s, 04.20.-q.</text>
<section_header_level_1><location><page_1><loc_16><loc_39><loc_34><loc_41></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_16><loc_29><loc_84><loc_37></location>By themselves, Einstein's equations impose virtually no restrictions on the kinds of spacetimes that are physically permissible. Any symmetric, suitably differentiable metric that satisfies the boundary conditions is a solution to Einstein's equations since, formally, one can simply equate the energy-momentum tensor to the left-hand side of Einstein's equations.</text>
<text><location><page_1><loc_16><loc_21><loc_84><loc_29></location>To restrict the solution space, one can impose some physical requirements. In particular, energy conditions are imposed on the types of matter considered. The various energy conditions - null (NEC), weak (WEC), dominant (DEC), and strong (SEC) - each express some seemingly reasonable expectation regarding matter, such as that the speed of energy flow be no greater than the speed of light. The energy conditions</text>
<text><location><page_2><loc_16><loc_72><loc_84><loc_83></location>are inequalities that apply locally, and are asserted to hold everywhere in spacetime. They are generalizations of the notion that local energy density be non-negative and are implemented by requiring various linear combinations of components of the matter energy-momentum tensor to be non-negative. The energy conditions are violated by potentially pathological forms of matter, such as certain instances of tachyons (WEC) and ghosts (NEC), and in turn eliminate many of the arbitrary metrics that would otherwise tautologically satisfy Einstein's equations.</text>
<text><location><page_2><loc_16><loc_58><loc_84><loc_71></location>Furthermore, the energy conditions play a critical role in a variety of important theorems in general relativity. They are crucial, for example, to the singularity theorems which indicate that our universe began with a Big Bang singularity; almost all theoretical attempts to evade the Big Bang singularity require the violation of at least some of the energy conditions at some step. Energy conditions are also invoked in the topological censorship theorem, in positive energy theorems, in prohibiting time machines, in the black hole no-hair theorem and, in particular, in the area-law increase for black holes [1, 2, 3].</text>
<text><location><page_2><loc_16><loc_27><loc_84><loc_58></location>Given their importance, it is disturbing that the energy conditions are not derived from any fundamental principles. Indeed, the status of the energy conditions - why or even whether they hold - remains unclear [4]. For example it is unclear what happens to the energy conditions in higher derivative theories of gravity; should they be modified, or do the physical laws that depend on them (viz. the laws of black hole thermodynamics) require modification? In f ( R ) gravity [5] and Brans-Dicke gravity [6] no modification is required. But the question is still open for other higher curvature theories of gravity. The NEC is routinely violated in higher derivative theories [7, 8, 9, 10], in the presence of extra dimensions [11], in ghost condensate models [12], in string cosmology [13, 14] and in other cosmological models which evade the big-bang singularity with a cosmic bounce [15, 16, 17]. The presence of quantum effects and backreaction also tend to introduce ambiguities in the energy conditions [18, 19]. But even if we restrict ourselves to only lowest-curvature tree-level classical actions, ambiguities appear when non-minimal couplings are introduced [20]. Indeed, even simple conformal frame transformations seem to violate the energy conditions [21]. The reason for this is easy to understand: the energy conditions are conditions on the stress tensor of matter but Weyl transformations (local rescalings of the metric) mix matter and gravity, thereby altering the stress tensor nontrivially.</text>
<text><location><page_2><loc_16><loc_15><loc_84><loc_27></location>In this paper, we consider gravity coupled to a scalar field in Einstein frame and show that, after a local rescaling of the metric, the usual form of the null energy condition can appear to be violated. We then propose a new form of the null energy condition, (11), that is valid in all rescaled Weyl frames. Our new null energy condition reduces to the usual form in Einstein frame. We then show that this form is consistent with the invariance of the second law of thermodynamics for black holes under changes in conformal frame. We use our modified NEC to supply a direct proof of the second law</text>
<text><location><page_3><loc_16><loc_82><loc_29><loc_83></location>in Jordan frame.</text>
<text><location><page_3><loc_16><loc_73><loc_84><loc_82></location>The result motivates the following conjecture. The energy conditions, unlike the second law, do not appear to be fundamental. We propose that the second law should be taken as given. Then the correct modification of the NEC in a given theory of gravity is that condition on matter that ensures that a classical black hole solution of the theory has an entropy that grows with time.</text>
<section_header_level_1><location><page_3><loc_16><loc_69><loc_60><loc_71></location>2 Energy conditions in Jordan frame</section_header_level_1>
<text><location><page_3><loc_16><loc_64><loc_84><loc_67></location>Consider a minimally coupled, Einstein-Hilbert scalar field action with canonically normalized scalar field φ :</text>
<formula><location><page_3><loc_30><loc_58><loc_84><loc_63></location>I = ∫ d D x √ -g ( M D -2 P 2 R -1 2 ( ∂φ ) 2 -V ( φ ) ) . (1)</formula>
<text><location><page_3><loc_16><loc_52><loc_84><loc_57></location>Here g ab is the Einstein-frame metric that couples minimally to the metric. The Einsteinframe energy-momentum tensor of the scalar field, T E ab , manifestly satisfies the null energy condition:</text>
<formula><location><page_3><loc_40><loc_50><loc_84><loc_52></location>T E ab k a k b = ( k · ∂φ ) 2 ≥ 0 , (2)</formula>
<text><location><page_3><loc_16><loc_36><loc_84><loc_50></location>for any null vector k a . Let us consider the modification of the Lagrangian due to a Weyl transformation, ˜ g ab = Ω 2 ( x ) g ab . The existence of a well-defined inverse of the metric requires a nowhere-vanishing conformal factor. In addition, we need Ω( x ) to be related to the previously existing fields in Einstein frame; otherwise, we would have introduced a new degree of freedom. We therefore take Ω( x ) = e ζφ ( x ) /µ with µ = M D -2 2 P where ζ is a dimensionless constant, and µ is a dimensionful constant with the same dimensions as φ . We have chosen the conformal factor to be linear in φ for simplicity; our results can trivially be extended to a general conformal factor of the form exp( f ( φ ( x ) /µ )).</text>
<text><location><page_3><loc_16><loc_17><loc_84><loc_35></location>It is important to emphasize here that we have merely redefined field variables from ( g ab ( x ) , φ ( x )) to (˜ g ab ( x ) , φ ( x )). Although the action is typically not invariant under such transformations (unless the metric transformation was induced by a diffeomorphism), the physics is exactly the same [22] provided only that the number of degrees of freedom is unchanged, which is the case here. (Indeed, this is true not only classically, but also quantum-mechanically. In the path integral, the fields are just integration variables. We can redefine these, just as we are free to redefine integration variables in ordinary calculus.) Field redefinitions between the matter and gravitational sectors are used routinely in both string theory and cosmology. In particular, if the energy conditions are to have physical significance, they too must continue to hold after field redefinition. Let us check whether this is the case.</text>
<text><location><page_4><loc_19><loc_82><loc_69><loc_83></location>Using the above choice of conformal factor the action reduces to</text>
<formula><location><page_4><loc_19><loc_75><loc_88><loc_81></location>∫ d D x √ -˜ ge -ζ ( D -2) φ/µ ( M D -2 P 2 ˜ R -1 2 ( 1 -ζ 2 ( D -2)( D -3) ) ( ∂ a φ ) 2 -e -2 ζφ/µ V ( φ ) ) . (3)</formula>
<text><location><page_4><loc_16><loc_67><loc_84><loc_75></location>We will refer to this action as the Jordan-frame action because gravity is non-minimally coupled, via the term in which the Ricci scalar is directly coupled to matter. The various pieces of the Lagrangian no longer split cleanly into a gravity Lagrangian and a matter Lagrangian, and hence it is no longer sensible to define the Jordan-frame stress tensor, T J ab , naively, as the variation of the matter action:</text>
<text><location><page_4><loc_44><loc_62><loc_44><loc_65></location>/negationslash</text>
<formula><location><page_4><loc_41><loc_61><loc_84><loc_66></location>T J ab = -2 √ -˜ g δI matter δ ˜ g ab . (4)</formula>
<text><location><page_4><loc_16><loc_54><loc_84><loc_61></location>However, The equation of motion for ˜ g ab can be rewritten so as to get the Einstein tensor on the left-hand side. Then whatever is on the right-hand side is covariantly conserved as a result of the Bianchi identity. Thus we can simply define the Jordan frame stress tensor as the quantity proportional to the Einstein tensor in the gravitational equations:</text>
<formula><location><page_4><loc_43><loc_50><loc_84><loc_53></location>˜ G ab ≡ 1 M D -2 P T J ab . (5)</formula>
<text><location><page_4><loc_16><loc_47><loc_52><loc_48></location>Explicitly, the stress tensor in Jordan frame is</text>
<formula><location><page_4><loc_19><loc_38><loc_89><loc_46></location>T J ab = (1 + ζ 2 ( D -2))( ∂ a φ )( ∂ b φ ) -˜ g ab ( 1 2 ( 1 + ζ 2 ( D -1)( D -2) ) ( ∂ a φ ) 2 + e -2 ζφ/µ V ( φ ) ) + ζ ( D -2) M D -2 2 P ( ˜ g ab ˜ ∇ 2 -˜ ∇ b ˜ ∇ a ) φ , (6)</formula>
<formula><location><page_4><loc_45><loc_34><loc_84><loc_37></location>˜ ∇ a T J ab = 0 . (7)</formula>
<section_header_level_1><location><page_4><loc_16><loc_31><loc_56><loc_33></location>NEC on the stress tensor in Jordan frame</section_header_level_1>
<text><location><page_4><loc_16><loc_27><loc_84><loc_30></location>The covariantly-conserved stress tensor, (6), can still violate the null energy condition in its Einstein-frame form [21]. To see this, contract T J ab with null vectors:</text>
<formula><location><page_4><loc_24><loc_21><loc_84><loc_25></location>T J ab k a k b = ( 1 + ζ 2 ( D -2) ) ( k a ∂ a φ ) 2 -ζ ( D -2) M D -2 2 P k a k b ˜ ∇ a ˜ ∇ b φ . (8)</formula>
<text><location><page_4><loc_16><loc_14><loc_84><loc_22></location>Notice that the positivity of the last term is ambiguous. If this were the correct form of the NEC, the set of physical solutions ( g ab , φ ) in Einstein frame - namely, those that obey the Einstein frame NEC - would not map to the set of physical solutions (˜ g ab , φ ) in Jordan frame - those that satisfy T J ab k a k b ≥ 0. But the set of physical solutions</text>
<text><location><page_4><loc_16><loc_37><loc_40><loc_38></location>which is covariantly conserved:</text>
<text><location><page_5><loc_16><loc_80><loc_84><loc_83></location>should not change under field redefinitions. It must be, then, that this naive version of the NEC is incorrect in Jordan frame.</text>
<text><location><page_5><loc_19><loc_78><loc_73><loc_80></location>We propose that the correct null energy condition in Jordan frame is</text>
<formula><location><page_5><loc_34><loc_73><loc_84><loc_77></location>[ T J ab + ζ ( D -2) M D -2 2 P ˜ ∇ a ˜ ∇ b φ ] k a k b ≥ 0 . (9)</formula>
<text><location><page_5><loc_16><loc_66><loc_84><loc_72></location>Once this condition is imposed on the matter fields in the Jordan frame the ambiguity is resolved and we have (solution set in Jordan frame) ≡ (solution set in Einstein frame). To see this explicitly, consider our energy-momentum tensor. Using the expression (6) in the relation (9), we find</text>
<formula><location><page_5><loc_25><loc_59><loc_84><loc_64></location>( T J ab + ζ ( D -2) M D -2 2 P ˜ ∇ a ˜ ∇ b φ ) k a k b = ( 1 + ζ 2 ( D -2) ) T E ab k a k b . (10)</formula>
<formula><location><page_5><loc_33><loc_49><loc_84><loc_53></location>[ T J ab +( D -2) M D -2 P ˜ ∇ a ˜ ∇ b ln Ω ] k a k b ≥ 0 . (11)</formula>
<text><location><page_5><loc_16><loc_53><loc_84><loc_59></location>Hence, (modified NEC obeyed in Jordan frame) ⇔ (usual NEC obeyed in Einstein frame). Our expression for the Jordan frame null energy condition can be generalized to arbitrary non-minimal couplings of matter sources to the Ricci scalar. The generalized version is</text>
<text><location><page_5><loc_16><loc_47><loc_84><loc_50></location>This inequality reduces to the usual Einstein frame NEC, (2), in case of minimal coupling (Ω = 1) and to the modified NEC, (9), when Ω = exp( ζφ ( x ) /µ ).</text>
<text><location><page_5><loc_16><loc_40><loc_84><loc_46></location>To summarize, we propose a new form of the null energy condition, (11), that applies to non-minimally coupled scalar fields and whose solution set is the same as that of the usual Einstein frame null energy condition. This is as it should be, since the two frames are related by a field redefinition.</text>
<section_header_level_1><location><page_5><loc_16><loc_35><loc_59><loc_37></location>3 Entropy increase in Jordan frame</section_header_level_1>
<text><location><page_5><loc_16><loc_15><loc_84><loc_34></location>Our proposal for the modified null energy condition in Jordan frame was somewhat ad hoc. It happened to work for non-minimally coupled scalar fields in conformally transformed actions. We will now motivate the prescription by a robust physical principle. If we regard black hole entropy as counting the number of degrees of freedom via the dimensions of its Hilbert space, or alternatively, the number of possible initial configurations from which the black hole could be formed, then the entropy is clearly conformally invariant. If the entropy increases in one frame as a result of imposition of the NEC then it should also increase in the conformal frame without any extra requirement, viz. the modified NEC of the Jordan frame. In this section, we will prove that our modified null energy condition in Jordan frame indeed guarantees that, classically, black hole entropy never decreases.</text>
<text><location><page_6><loc_16><loc_65><loc_84><loc_83></location>Consider a stationary black hole solution to the Einstein equation in D spacetime dimensions. The entropy of black hole solutions of the Einstein-Hilbert action is just proportional to its 'area' [1], by which we mean a (D-2)-dimensional spacelike section of the horizon. But in dealing with spacetimes which are solutions to non-Einstein-Hilbert actions, such as the one in Jordan frame with non-minimal scalar coupling, a notion of entropy is absent. The Wald prescription of entropy [23] identifies a conserved charge with the entropy of the horizon for stationary solutions to these non-Einsteinian theories of gravity. The correct entropy to use for dynamical horizons is Jacobson-Myers entropy [24]. But since the metric has rescaled, we can obtain the entropy just by applying a field redefinition to the Bekentein-Hawking entropy [25]. The black hole entropy in Jordan frame is not simply proportional to the area. Rather it is</text>
<formula><location><page_6><loc_37><loc_59><loc_84><loc_63></location>S = 1 4 G D ∫ d D -2 x √ ˜ γ Ω -( D -2) . (12)</formula>
<text><location><page_6><loc_16><loc_52><loc_84><loc_59></location>The increase of entropy in the Jordan frame has been studied before from the perspective of f ( R ) theories [5] and the second law was proved in Jordan frame in [26] using the Einstein-frame NEC. We provide a proof directly in Jordan frame using our modified Jordan-frame NEC.</text>
<text><location><page_6><loc_16><loc_49><loc_84><loc_52></location>To see how the rate of change of black entropy in the Jordan frame depends on the Jordan frame NEC we first find the expression for the change in the black hole entropy.</text>
<formula><location><page_6><loc_28><loc_39><loc_84><loc_48></location>dS d ˜ λ = 1 4 G D ∫ d D -2 x √ ˜ γ Ω -( D -2) ( ˜ θ -( D -2) d ln Ω d ˜ λ ) ≡ 1 4 G D ∫ d D -2 x √ ˜ γ Ω -( D -2) Θ , (13)</formula>
<text><location><page_6><loc_16><loc_38><loc_33><loc_39></location>where we have defined</text>
<formula><location><page_6><loc_38><loc_33><loc_84><loc_37></location>Θ( ˜ λ ) = ˜ θ -( D -2) d ln Ω d ˜ λ . (14)</formula>
<text><location><page_6><loc_16><loc_29><loc_84><loc_32></location>Here ˜ θ is the expansion scalar for the null generator ˜ k a of the horizon in the Jordan frame:</text>
<formula><location><page_6><loc_41><loc_26><loc_84><loc_30></location>˜ θ = ˜ ∇ a ˜ k a = d (ln √ ˜ γ ) d ˜ λ . (15)</formula>
<text><location><page_6><loc_16><loc_19><loc_84><loc_25></location>Black hole entropy would not decrease if Θ were non-negative, as evident from (13). We will now give a direct proof of the second law of thermodynamics for scalar-tensor theories in Jordan frame. As in the proof that the surface area of black holes in Einstein gravity always increases [2], we will show that, if Θ < 0, then a caustic necessarily forms.</text>
<text><location><page_6><loc_16><loc_14><loc_84><loc_19></location>Notice that the vector ˜ k a = ( d/d ˜ λ ) a is related to k a = ( d/dλ ) a . This is easiest to see for a normalized timelike velocity vector, u a = ( d/dτ ) a . Since dτ 2 = -g ab dx a dx b ,</text>
<text><location><page_7><loc_16><loc_80><loc_84><loc_84></location>rescaling the metric causes τ to scale: ˜ τ = Ω τ . Then d ˜ τ dτ = Ω and hence ˜ u a ≡ ( d/d ˜ τ ) a = dτ d ˜ τ ( d/dτ ) a = (1 / Ω) u a . Similarly, we have</text>
<formula><location><page_7><loc_36><loc_75><loc_84><loc_79></location>˜ k a ≡ ( d d ˜ λ ) a = dλ d ˜ λ ( d dλ ) a = 1 Ω k a . (16)</formula>
<text><location><page_7><loc_16><loc_71><loc_84><loc_74></location>Because Ω( x ) is a function of space and time, ˜ k a is not in general affinely parameterized. Thus</text>
<formula><location><page_7><loc_44><loc_68><loc_84><loc_71></location>˜ k b ∇ b ˜ k a = κ ˜ k a , (17)</formula>
<formula><location><page_7><loc_45><loc_64><loc_84><loc_67></location>κ = d d ˜ λ ln Ω . (18)</formula>
<text><location><page_7><loc_16><loc_67><loc_20><loc_68></location>where</text>
<text><location><page_7><loc_16><loc_62><loc_55><loc_63></location>The corresponding Raychaudhuri equation for ˜ θ is</text>
<formula><location><page_7><loc_30><loc_56><loc_84><loc_60></location>d ˜ θ d ˜ λ = κ ˜ θ -( ˜ θ 2 D -2 + ˜ σ ab ˜ σ ab + ˜ ω ab ˜ ω ab + ˜ R ab ˜ k a ˜ k b ) . (19)</formula>
<text><location><page_7><loc_16><loc_47><loc_84><loc_55></location>The presence of the κ ˜ θ term on the right is a sign that ˜ λ is not an affine parameter. Hypersurface orthogonality of the null generators ˜ k a of the event horizon causes the rotation ( ˜ w = 0) to vanish by the Frobenius theorem. We can now use the equation of motion in the Jordan frame and replace the Ricci tensor with the Jordan frame stress tensor:</text>
<formula><location><page_7><loc_39><loc_43><loc_84><loc_47></location>˜ R ab ˜ k a ˜ k b = 1 M D -2 P T J ab ˜ k a ˜ k b . (20)</formula>
<text><location><page_7><loc_16><loc_41><loc_64><loc_43></location>Taking the derivative of (14) and substituting (19) then gives</text>
<formula><location><page_7><loc_17><loc_26><loc_84><loc_40></location>d Θ d ˜ λ = d ˜ θ d ˜ λ -( D -2) d 2 d ˜ λ 2 ln Ω = κ ˜ θ -( ˜ θ 2 D -2 + ˜ σ 2 ab + ˜ R ab ˜ k a ˜ k b ) -( D -2) d 2 d ˜ λ 2 ln Ω = -Θ 2 D -2 -κ Θ -( D -2) κ 2 -˜ σ 2 ab -˜ k a ˜ k b M D -2 P ( T J ab +( D -2) M D -2 P ˜ ∇ a ˜ ∇ b ln Ω ) . (21)</formula>
<text><location><page_7><loc_16><loc_18><loc_84><loc_25></location>The last line follows from the relation between κ and Ω above (18). Note the presence of the terms in square brackets: this is proportional to precisely the expression that appears in our modified null energy condition in Jordan frame, (11). When that is obeyed we have</text>
<formula><location><page_7><loc_37><loc_14><loc_84><loc_18></location>d Θ d ˜ λ ≤ -Θ 2 D -2 -κ Θ -( D -2) κ 2 . (22)</formula>
<text><location><page_8><loc_16><loc_77><loc_84><loc_83></location>We will now analyze this equation carefully in order to prove that only solutions with Θ > 0 do not have caustics. Cosmic censorship - which in this context means the prohibition of caustics - eliminates all solutions with Θ < 0 and hence, by (13), the second law of black hole thermodynamics holds in Jordan frame.</text>
<text><location><page_8><loc_16><loc_63><loc_84><loc_77></location>Suppose, then, that at some parameter ˜ λ 0 , a pencil of horizon null generators has Θ 0 < 0. For a sufficiently thin pencil, the surface gravity is effectively constant over spacelike sections of the pencil. Therefore, we can regard κ as a function of ˜ λ only. First, consider κ ≤ 0. But then every term on the right-hand side of (22) is nonpositive for Θ < 0. Hence d Θ d ˜ λ ≤ 0. In fact, d Θ d ˜ λ ≤ -Θ 2 D -2 whose solution is Θ( ˜ λ ) ≤ Θ 0 / (1 + ˜ λ -˜ λ 0 D -2 Θ 0 ). For all negative values of Θ 0 , Θ( ˜ λ ) diverges at some finite ˜ λ , resulting in a caustic. Hence, for κ ≤ 0, all solutions with Θ < 0 lead to caustics.</text>
<text><location><page_8><loc_16><loc_39><loc_84><loc_53></location>A monotonically decreasing negative function Θ( ˜ λ ) can have three different asymptotic possibilities. Possibility 1 is that Θ asymptotically and monotonically approaches some finite negative value, Θ min i.e. lim ˜ λ →∞ Θ( ˜ λ ) = Θ min . Possibility 2 is that Θ is unbounded from below but reaches negative infinity only in the infinite future i.e. lim ˜ λ →∞ Θ( ˜ λ ) = -∞ . This is not a caustic because Θ is finite at all finite values of ˜ λ . Possibility 3 is that Θ diverges at some finite ˜ λ c : lim ˜ λ → ˜ λ c Θ( ˜ λ ) = -∞ . This corresponds to a caustic. These three possibilities are illustrated schematically by the curves in Figure 1. We will now show that κ > 0 and Θ < 0 always gives rise to possibility 3.</text>
<text><location><page_8><loc_16><loc_52><loc_84><loc_64></location>Next, consider κ > 0. In this case, the term -κ Θ in (22) is positive for Θ < 0. However, the three terms on the right-hand side together are always negative. To see this, consider the right-hand side of (22) as a quadratic polynomial in Θ; this quadratic has no real roots for κ > 0. Hence again d Θ d ˜ λ ≤ 0; Θ is a monotonically decreasing function of ˜ λ . However, to prove that this inevitably results in a caustic is more subtle because the positivity of the -κ Θ term does not permit us to write d Θ d ˜ λ ≤ -Θ 2 D -2 .</text>
<text><location><page_8><loc_16><loc_30><loc_84><loc_39></location>First we rule out possibility 1; Θ( ˜ λ ) does not asymptotically approach a finite value. For suppose that were true. Then for large values of ˜ λ , we would have Θ ≈ Θ min and d Θ d ˜ λ ≈ 0. But, regarding the right-hand side of (22) as a quadratic in κ , we see that there are no real solutions for κ when Θ = Θ min and d Θ d ˜ λ = 0. Hence, possibility 1 is eliminated and Θ( ˜ λ ) is therefore unbounded from below.</text>
<text><location><page_8><loc_16><loc_25><loc_84><loc_30></location>Since Θ is unbounded from below, consider a very large (negative) value of Θ. Focus on an infinitesimal interval of ˜ λ . In that interval, κ ( ˜ λ ) can be regarded effectively as a constant. Then we can integrate (22) to obtain</text>
<formula><location><page_8><loc_28><loc_16><loc_84><loc_24></location>Θ( ˜ λ ) = √ 3 2 Θ 0 -( Θ 0 + 7 4 ( D -2) κ ) tan ( √ 3 2 κ ( ˜ λ -˜ λ 0 ) ) √ 3 2 + ( Θ 0 κ ( D -2) +1 ) tan ( √ 3 2 κ ( ˜ λ -˜ λ 0 ) ) . (23)</formula>
<figure>
<location><page_9><loc_31><loc_59><loc_68><loc_83></location>
<caption>Figure 1: Possible curves for a negative monotonically decreasing function Θ( ˜ λ ).</caption>
</figure>
<text><location><page_9><loc_16><loc_51><loc_76><loc_53></location>Scrutiny of this reveals that the denominator vanishes for certain values of ˜ λ :</text>
<formula><location><page_9><loc_42><loc_46><loc_84><loc_50></location>˜ λ -˜ λ 0 ≈ -D -2 Θ 0 . (24)</formula>
<text><location><page_9><loc_16><loc_42><loc_84><loc_45></location>Hence if Θ 0 is large and negative, Θ becomes divergently negative in finite time: a caustic.</text>
<text><location><page_9><loc_16><loc_25><loc_84><loc_42></location>We have proven that, whether κ is positive or negative, we always find a caustic in finite parameter ˜ λ whenever Θ 0 < 0. Cosmic censorship bans these solutions leaving only those with Θ ≥ 0. This in turn implies that black hole entropy must be nondecreasing in Jordan frame. Our proof relied crucially on (22), which follows from (21) only when our modification to the Jordan-frame null energy condition is satisfied. A quick check on our result comes from the observation that Θ = θ E / Ω, where θ E is the expansion in Einstein frame. Then, when the usual Einstein-frame null energy condition is satisfied, θ E is positive. This in turn means that Θ must be positive and that the Jordan-frame entropy also increases. Here we have proven that fact directly in Jordan frame without relying on a correspondence with Einstein frame.</text>
<section_header_level_1><location><page_9><loc_16><loc_21><loc_26><loc_23></location>Discussion</section_header_level_1>
<text><location><page_9><loc_16><loc_15><loc_84><loc_20></location>By using a field redefinition, we have seen that the form of the NEC is modified in Jordan frame. Similar rewritings lead to modified versions of the other energy conditions, whose forms are not particularly illuminating. However, field redefinitions from Einstein frame</text>
<text><location><page_10><loc_16><loc_58><loc_84><loc_83></location>do not exist for generic theories, such as most higher-derivative theories. The question naturally arises as to what the appropriate generalization of the NEC is for such theories. A possible clue is to be found in (21). Given only that equation plus the requirement that the second law hold, one could in fact have inferred the modified NEC directly invoking neither a field redefinition nor even the original Einstein-frame NEC. This is because all terms on the right-hand side of (21) need to be negative in order to guarantee the validity of the second law. This is a very interesting observation because, rather than using the modified NEC to prove the second law, we would in this approach take the second law as a given and derive the appropriate condition for matter. This is appealing because the NEC - unlike the second law - does not seem to rest on any fundamental principles of physics. In the same spirit as Jacobson's Einstein equation of state paper [27] (in which thermodynamic laws are taken as axioms rather than as statements to be proved), one should perhaps begin, not end, with the second law. This approach would generalize to other gravitational theories, provided one had the correct formulation of entropy, valid in non-stationary situations.</text>
<section_header_level_1><location><page_10><loc_16><loc_55><loc_32><loc_56></location>Acknowledgments</section_header_level_1>
<text><location><page_10><loc_16><loc_51><loc_84><loc_54></location>It is a pleasure to thank Paul Davies and Jan Pieter van der Schaar for helpful discussions. This work is supported in part by DOE grant de-sc0008016.</text>
<section_header_level_1><location><page_10><loc_16><loc_47><loc_28><loc_49></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_10><loc_17><loc_44><loc_78><loc_45></location>[1] J. D. Bekenstein, 'Black holes and entropy,' Phys. Rev. D 7 , 2333 (1973).</list_item>
<list_item><location><page_10><loc_17><loc_39><loc_84><loc_42></location>[2] J. M. Bardeen, B. Carter and S. W. Hawking, 'The Four laws of black hole mechanics,' Commun. Math. Phys. 31 , 161 (1973).</list_item>
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<list_item><location><page_12><loc_16><loc_71><loc_84><loc_74></location>[23] V. Iyer and R. M. Wald, 'Some properties of Noether charge and a proposal for dynamical black hole entropy,' Phys. Rev. D 50 , 846 (1994) [gr-qc/9403028].</list_item>
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</document>
[
{
"title": "Energy conditions in Jordan frame",
"content": "Saugata Chatterjee 1 ∗ , Damien A. Easson 1 † , and Maulik Parikh 1 , 2 ‡ 1 Department of Physics, Arizona State University, Tempe, Arizona 85287, USA 2 Beyond Center for Fundamental Concepts in Science, Arizona State University, Tempe, Arizona 85287, USA",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "The null energy condition, in its usual form, can appear to be violated by transformations in the conformal frame of the metric. We propose a generalization of the form of the null energy condition to Jordan frame, in which matter is non-minimally coupled, which reduces to the familiar form in Einstein frame. Using our version of the null energy condition, we provide a direct proof of the second law of black hole thermodynamics in Jordan frame. PACS numbers: 98.80.Cq, 04.70.-s, 04.20.-q.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "By themselves, Einstein's equations impose virtually no restrictions on the kinds of spacetimes that are physically permissible. Any symmetric, suitably differentiable metric that satisfies the boundary conditions is a solution to Einstein's equations since, formally, one can simply equate the energy-momentum tensor to the left-hand side of Einstein's equations. To restrict the solution space, one can impose some physical requirements. In particular, energy conditions are imposed on the types of matter considered. The various energy conditions - null (NEC), weak (WEC), dominant (DEC), and strong (SEC) - each express some seemingly reasonable expectation regarding matter, such as that the speed of energy flow be no greater than the speed of light. The energy conditions are inequalities that apply locally, and are asserted to hold everywhere in spacetime. They are generalizations of the notion that local energy density be non-negative and are implemented by requiring various linear combinations of components of the matter energy-momentum tensor to be non-negative. The energy conditions are violated by potentially pathological forms of matter, such as certain instances of tachyons (WEC) and ghosts (NEC), and in turn eliminate many of the arbitrary metrics that would otherwise tautologically satisfy Einstein's equations. Furthermore, the energy conditions play a critical role in a variety of important theorems in general relativity. They are crucial, for example, to the singularity theorems which indicate that our universe began with a Big Bang singularity; almost all theoretical attempts to evade the Big Bang singularity require the violation of at least some of the energy conditions at some step. Energy conditions are also invoked in the topological censorship theorem, in positive energy theorems, in prohibiting time machines, in the black hole no-hair theorem and, in particular, in the area-law increase for black holes [1, 2, 3]. Given their importance, it is disturbing that the energy conditions are not derived from any fundamental principles. Indeed, the status of the energy conditions - why or even whether they hold - remains unclear [4]. For example it is unclear what happens to the energy conditions in higher derivative theories of gravity; should they be modified, or do the physical laws that depend on them (viz. the laws of black hole thermodynamics) require modification? In f ( R ) gravity [5] and Brans-Dicke gravity [6] no modification is required. But the question is still open for other higher curvature theories of gravity. The NEC is routinely violated in higher derivative theories [7, 8, 9, 10], in the presence of extra dimensions [11], in ghost condensate models [12], in string cosmology [13, 14] and in other cosmological models which evade the big-bang singularity with a cosmic bounce [15, 16, 17]. The presence of quantum effects and backreaction also tend to introduce ambiguities in the energy conditions [18, 19]. But even if we restrict ourselves to only lowest-curvature tree-level classical actions, ambiguities appear when non-minimal couplings are introduced [20]. Indeed, even simple conformal frame transformations seem to violate the energy conditions [21]. The reason for this is easy to understand: the energy conditions are conditions on the stress tensor of matter but Weyl transformations (local rescalings of the metric) mix matter and gravity, thereby altering the stress tensor nontrivially. In this paper, we consider gravity coupled to a scalar field in Einstein frame and show that, after a local rescaling of the metric, the usual form of the null energy condition can appear to be violated. We then propose a new form of the null energy condition, (11), that is valid in all rescaled Weyl frames. Our new null energy condition reduces to the usual form in Einstein frame. We then show that this form is consistent with the invariance of the second law of thermodynamics for black holes under changes in conformal frame. We use our modified NEC to supply a direct proof of the second law in Jordan frame. The result motivates the following conjecture. The energy conditions, unlike the second law, do not appear to be fundamental. We propose that the second law should be taken as given. Then the correct modification of the NEC in a given theory of gravity is that condition on matter that ensures that a classical black hole solution of the theory has an entropy that grows with time.",
"pages": [
1,
2,
3
]
},
{
"title": "2 Energy conditions in Jordan frame",
"content": "Consider a minimally coupled, Einstein-Hilbert scalar field action with canonically normalized scalar field φ : Here g ab is the Einstein-frame metric that couples minimally to the metric. The Einsteinframe energy-momentum tensor of the scalar field, T E ab , manifestly satisfies the null energy condition: for any null vector k a . Let us consider the modification of the Lagrangian due to a Weyl transformation, ˜ g ab = Ω 2 ( x ) g ab . The existence of a well-defined inverse of the metric requires a nowhere-vanishing conformal factor. In addition, we need Ω( x ) to be related to the previously existing fields in Einstein frame; otherwise, we would have introduced a new degree of freedom. We therefore take Ω( x ) = e ζφ ( x ) /µ with µ = M D -2 2 P where ζ is a dimensionless constant, and µ is a dimensionful constant with the same dimensions as φ . We have chosen the conformal factor to be linear in φ for simplicity; our results can trivially be extended to a general conformal factor of the form exp( f ( φ ( x ) /µ )). It is important to emphasize here that we have merely redefined field variables from ( g ab ( x ) , φ ( x )) to (˜ g ab ( x ) , φ ( x )). Although the action is typically not invariant under such transformations (unless the metric transformation was induced by a diffeomorphism), the physics is exactly the same [22] provided only that the number of degrees of freedom is unchanged, which is the case here. (Indeed, this is true not only classically, but also quantum-mechanically. In the path integral, the fields are just integration variables. We can redefine these, just as we are free to redefine integration variables in ordinary calculus.) Field redefinitions between the matter and gravitational sectors are used routinely in both string theory and cosmology. In particular, if the energy conditions are to have physical significance, they too must continue to hold after field redefinition. Let us check whether this is the case. Using the above choice of conformal factor the action reduces to We will refer to this action as the Jordan-frame action because gravity is non-minimally coupled, via the term in which the Ricci scalar is directly coupled to matter. The various pieces of the Lagrangian no longer split cleanly into a gravity Lagrangian and a matter Lagrangian, and hence it is no longer sensible to define the Jordan-frame stress tensor, T J ab , naively, as the variation of the matter action: /negationslash However, The equation of motion for ˜ g ab can be rewritten so as to get the Einstein tensor on the left-hand side. Then whatever is on the right-hand side is covariantly conserved as a result of the Bianchi identity. Thus we can simply define the Jordan frame stress tensor as the quantity proportional to the Einstein tensor in the gravitational equations: Explicitly, the stress tensor in Jordan frame is",
"pages": [
3,
4
]
},
{
"title": "NEC on the stress tensor in Jordan frame",
"content": "The covariantly-conserved stress tensor, (6), can still violate the null energy condition in its Einstein-frame form [21]. To see this, contract T J ab with null vectors: Notice that the positivity of the last term is ambiguous. If this were the correct form of the NEC, the set of physical solutions ( g ab , φ ) in Einstein frame - namely, those that obey the Einstein frame NEC - would not map to the set of physical solutions (˜ g ab , φ ) in Jordan frame - those that satisfy T J ab k a k b ≥ 0. But the set of physical solutions which is covariantly conserved: should not change under field redefinitions. It must be, then, that this naive version of the NEC is incorrect in Jordan frame. We propose that the correct null energy condition in Jordan frame is Once this condition is imposed on the matter fields in the Jordan frame the ambiguity is resolved and we have (solution set in Jordan frame) ≡ (solution set in Einstein frame). To see this explicitly, consider our energy-momentum tensor. Using the expression (6) in the relation (9), we find Hence, (modified NEC obeyed in Jordan frame) ⇔ (usual NEC obeyed in Einstein frame). Our expression for the Jordan frame null energy condition can be generalized to arbitrary non-minimal couplings of matter sources to the Ricci scalar. The generalized version is This inequality reduces to the usual Einstein frame NEC, (2), in case of minimal coupling (Ω = 1) and to the modified NEC, (9), when Ω = exp( ζφ ( x ) /µ ). To summarize, we propose a new form of the null energy condition, (11), that applies to non-minimally coupled scalar fields and whose solution set is the same as that of the usual Einstein frame null energy condition. This is as it should be, since the two frames are related by a field redefinition.",
"pages": [
4,
5
]
},
{
"title": "3 Entropy increase in Jordan frame",
"content": "Our proposal for the modified null energy condition in Jordan frame was somewhat ad hoc. It happened to work for non-minimally coupled scalar fields in conformally transformed actions. We will now motivate the prescription by a robust physical principle. If we regard black hole entropy as counting the number of degrees of freedom via the dimensions of its Hilbert space, or alternatively, the number of possible initial configurations from which the black hole could be formed, then the entropy is clearly conformally invariant. If the entropy increases in one frame as a result of imposition of the NEC then it should also increase in the conformal frame without any extra requirement, viz. the modified NEC of the Jordan frame. In this section, we will prove that our modified null energy condition in Jordan frame indeed guarantees that, classically, black hole entropy never decreases. Consider a stationary black hole solution to the Einstein equation in D spacetime dimensions. The entropy of black hole solutions of the Einstein-Hilbert action is just proportional to its 'area' [1], by which we mean a (D-2)-dimensional spacelike section of the horizon. But in dealing with spacetimes which are solutions to non-Einstein-Hilbert actions, such as the one in Jordan frame with non-minimal scalar coupling, a notion of entropy is absent. The Wald prescription of entropy [23] identifies a conserved charge with the entropy of the horizon for stationary solutions to these non-Einsteinian theories of gravity. The correct entropy to use for dynamical horizons is Jacobson-Myers entropy [24]. But since the metric has rescaled, we can obtain the entropy just by applying a field redefinition to the Bekentein-Hawking entropy [25]. The black hole entropy in Jordan frame is not simply proportional to the area. Rather it is The increase of entropy in the Jordan frame has been studied before from the perspective of f ( R ) theories [5] and the second law was proved in Jordan frame in [26] using the Einstein-frame NEC. We provide a proof directly in Jordan frame using our modified Jordan-frame NEC. To see how the rate of change of black entropy in the Jordan frame depends on the Jordan frame NEC we first find the expression for the change in the black hole entropy. where we have defined Here ˜ θ is the expansion scalar for the null generator ˜ k a of the horizon in the Jordan frame: Black hole entropy would not decrease if Θ were non-negative, as evident from (13). We will now give a direct proof of the second law of thermodynamics for scalar-tensor theories in Jordan frame. As in the proof that the surface area of black holes in Einstein gravity always increases [2], we will show that, if Θ < 0, then a caustic necessarily forms. Notice that the vector ˜ k a = ( d/d ˜ λ ) a is related to k a = ( d/dλ ) a . This is easiest to see for a normalized timelike velocity vector, u a = ( d/dτ ) a . Since dτ 2 = -g ab dx a dx b , rescaling the metric causes τ to scale: ˜ τ = Ω τ . Then d ˜ τ dτ = Ω and hence ˜ u a ≡ ( d/d ˜ τ ) a = dτ d ˜ τ ( d/dτ ) a = (1 / Ω) u a . Similarly, we have Because Ω( x ) is a function of space and time, ˜ k a is not in general affinely parameterized. Thus where The corresponding Raychaudhuri equation for ˜ θ is The presence of the κ ˜ θ term on the right is a sign that ˜ λ is not an affine parameter. Hypersurface orthogonality of the null generators ˜ k a of the event horizon causes the rotation ( ˜ w = 0) to vanish by the Frobenius theorem. We can now use the equation of motion in the Jordan frame and replace the Ricci tensor with the Jordan frame stress tensor: Taking the derivative of (14) and substituting (19) then gives The last line follows from the relation between κ and Ω above (18). Note the presence of the terms in square brackets: this is proportional to precisely the expression that appears in our modified null energy condition in Jordan frame, (11). When that is obeyed we have We will now analyze this equation carefully in order to prove that only solutions with Θ > 0 do not have caustics. Cosmic censorship - which in this context means the prohibition of caustics - eliminates all solutions with Θ < 0 and hence, by (13), the second law of black hole thermodynamics holds in Jordan frame. Suppose, then, that at some parameter ˜ λ 0 , a pencil of horizon null generators has Θ 0 < 0. For a sufficiently thin pencil, the surface gravity is effectively constant over spacelike sections of the pencil. Therefore, we can regard κ as a function of ˜ λ only. First, consider κ ≤ 0. But then every term on the right-hand side of (22) is nonpositive for Θ < 0. Hence d Θ d ˜ λ ≤ 0. In fact, d Θ d ˜ λ ≤ -Θ 2 D -2 whose solution is Θ( ˜ λ ) ≤ Θ 0 / (1 + ˜ λ -˜ λ 0 D -2 Θ 0 ). For all negative values of Θ 0 , Θ( ˜ λ ) diverges at some finite ˜ λ , resulting in a caustic. Hence, for κ ≤ 0, all solutions with Θ < 0 lead to caustics. A monotonically decreasing negative function Θ( ˜ λ ) can have three different asymptotic possibilities. Possibility 1 is that Θ asymptotically and monotonically approaches some finite negative value, Θ min i.e. lim ˜ λ →∞ Θ( ˜ λ ) = Θ min . Possibility 2 is that Θ is unbounded from below but reaches negative infinity only in the infinite future i.e. lim ˜ λ →∞ Θ( ˜ λ ) = -∞ . This is not a caustic because Θ is finite at all finite values of ˜ λ . Possibility 3 is that Θ diverges at some finite ˜ λ c : lim ˜ λ → ˜ λ c Θ( ˜ λ ) = -∞ . This corresponds to a caustic. These three possibilities are illustrated schematically by the curves in Figure 1. We will now show that κ > 0 and Θ < 0 always gives rise to possibility 3. Next, consider κ > 0. In this case, the term -κ Θ in (22) is positive for Θ < 0. However, the three terms on the right-hand side together are always negative. To see this, consider the right-hand side of (22) as a quadratic polynomial in Θ; this quadratic has no real roots for κ > 0. Hence again d Θ d ˜ λ ≤ 0; Θ is a monotonically decreasing function of ˜ λ . However, to prove that this inevitably results in a caustic is more subtle because the positivity of the -κ Θ term does not permit us to write d Θ d ˜ λ ≤ -Θ 2 D -2 . First we rule out possibility 1; Θ( ˜ λ ) does not asymptotically approach a finite value. For suppose that were true. Then for large values of ˜ λ , we would have Θ ≈ Θ min and d Θ d ˜ λ ≈ 0. But, regarding the right-hand side of (22) as a quadratic in κ , we see that there are no real solutions for κ when Θ = Θ min and d Θ d ˜ λ = 0. Hence, possibility 1 is eliminated and Θ( ˜ λ ) is therefore unbounded from below. Since Θ is unbounded from below, consider a very large (negative) value of Θ. Focus on an infinitesimal interval of ˜ λ . In that interval, κ ( ˜ λ ) can be regarded effectively as a constant. Then we can integrate (22) to obtain Scrutiny of this reveals that the denominator vanishes for certain values of ˜ λ : Hence if Θ 0 is large and negative, Θ becomes divergently negative in finite time: a caustic. We have proven that, whether κ is positive or negative, we always find a caustic in finite parameter ˜ λ whenever Θ 0 < 0. Cosmic censorship bans these solutions leaving only those with Θ ≥ 0. This in turn implies that black hole entropy must be nondecreasing in Jordan frame. Our proof relied crucially on (22), which follows from (21) only when our modification to the Jordan-frame null energy condition is satisfied. A quick check on our result comes from the observation that Θ = θ E / Ω, where θ E is the expansion in Einstein frame. Then, when the usual Einstein-frame null energy condition is satisfied, θ E is positive. This in turn means that Θ must be positive and that the Jordan-frame entropy also increases. Here we have proven that fact directly in Jordan frame without relying on a correspondence with Einstein frame.",
"pages": [
5,
6,
7,
8,
9
]
},
{
"title": "Discussion",
"content": "By using a field redefinition, we have seen that the form of the NEC is modified in Jordan frame. Similar rewritings lead to modified versions of the other energy conditions, whose forms are not particularly illuminating. However, field redefinitions from Einstein frame do not exist for generic theories, such as most higher-derivative theories. The question naturally arises as to what the appropriate generalization of the NEC is for such theories. A possible clue is to be found in (21). Given only that equation plus the requirement that the second law hold, one could in fact have inferred the modified NEC directly invoking neither a field redefinition nor even the original Einstein-frame NEC. This is because all terms on the right-hand side of (21) need to be negative in order to guarantee the validity of the second law. This is a very interesting observation because, rather than using the modified NEC to prove the second law, we would in this approach take the second law as a given and derive the appropriate condition for matter. This is appealing because the NEC - unlike the second law - does not seem to rest on any fundamental principles of physics. In the same spirit as Jacobson's Einstein equation of state paper [27] (in which thermodynamic laws are taken as axioms rather than as statements to be proved), one should perhaps begin, not end, with the second law. This approach would generalize to other gravitational theories, provided one had the correct formulation of entropy, valid in non-stationary situations.",
"pages": [
9,
10
]
},
{
"title": "Acknowledgments",
"content": "It is a pleasure to thank Paul Davies and Jan Pieter van der Schaar for helpful discussions. This work is supported in part by DOE grant de-sc0008016.",
"pages": [
10
]
}
]
2013CQGra..30x4007S
https://arxiv.org/pdf/1307.3542.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_79><loc_74><loc_81></location>Astrophysics of Super-massive Black Hole Mergers</section_header_level_1>
<section_header_level_1><location><page_1><loc_27><loc_76><loc_47><loc_77></location>Jeremy D. Schnittman 1</section_header_level_1>
<text><location><page_1><loc_27><loc_74><loc_65><loc_75></location>1 NASA Goddard Space Flight Center, Greenbelt, MD 20771</text>
<text><location><page_1><loc_27><loc_58><loc_76><loc_72></location>Abstract. We present here an overview of recent work in the subject of astrophysical manifestations of super-massive black hole (SMBH) mergers. This is a field that has been traditionally driven by theoretical work, but in recent years has also generated a great deal of interest and excitement in the observational astronomy community. In particular, the electromagnetic (EM) counterparts to SMBH mergers provide the means to detect and characterize these highly energetic events at cosmological distances, even in the absence of a spacebased gravitational-wave observatory. In addition to providing a mechanism for observing SMBH mergers, EM counterparts also give important information about the environments in which these remarkable events take place, thus teaching us about the mechanisms through which galaxies form and evolve symbiotically with their central black holes.</text>
<text><location><page_1><loc_27><loc_53><loc_67><loc_54></location>PACS numbers: 95.30.Sf, 98.54.Cm, 98.62.Js, 04.30.Tv, 04.80.Nn</text>
<text><location><page_1><loc_16><loc_48><loc_38><loc_49></location>Submitted to: Clas. Quant. Grav.</text>
<section_header_level_1><location><page_2><loc_16><loc_86><loc_33><loc_88></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_16><loc_67><loc_76><loc_85></location>Following numerical relativity's annus mirabilis of 2006, a deluge of work has explored the astrophysical manifestations of black hole mergers, from both the theoretical and observational perspectives. While the field has traditionally been dominated by applications to the direct detection of gravitational waves (GWs), much of the recent focus of numerical simulations has been on predicting potentially observable electromagnetic (EM) signatures. Of course, the greatest science yield will come from coincident detection of both the GW and EM signature, giving a myriad of observables such as the black hole mass, spins, redshift, and host environment, all with high precision [29]. Yet even in the absence of a direct GW detection (and this indeed is the likely state of affairs for at least the next decade), the EM signal alone may be sufficiently strong to detect with wide-field surveys, and also unique enough to identify unambiguously as a SMBH merger.</text>
<text><location><page_2><loc_16><loc_59><loc_76><loc_67></location>In this article, we review the brief history and astrophysical principles that govern the observable signatures of SMBH mergers. To date, the field has largely been driven by theory, but we also provide a summary of the observational techniques and surveys that have been utilized, including recent claims of potential detections of both SMBH binaries and also post-merger recoiling black holes.</text>
<text><location><page_2><loc_16><loc_44><loc_76><loc_59></location>While the first public use of the term 'black hole' is generally attributed to John Wheeler in 1967, as early as 1964 Edwin Saltpeter proposed that gas accretion onto super-massive black holes provided the tremendous energy source necessary to power the highly luminous quasi-stellar objects (quasars) seen in the centers of some galaxies [216]. Even earlier than that, black holes were understood to be formal mathematical solutions to Einstein's field equations [225], although considered by many to be simply mathematical oddities, as opposed to objects that might actually exist in nature (perhaps most famously, Eddington's stubborn opposition to the possibility of astrophysical black holes probably delayed significant progress in their understanding for decades) [255].</text>
<text><location><page_2><loc_16><loc_31><loc_76><loc_44></location>In 1969, Lynden-Bell outlined the foundations for black hole accretion as the basis for quasar power [160]. The steady-state thin disks of Shakura and Sunyaev [234], along with the relativistic modifications given by Novikov and Thorne [182], are still used as the standard models for accretion disks today. In the following decade, a combination of theoretical work and multi-wavelength observations led to a richer understanding of the wide variety of accretion phenomena in active galactic nuclei (AGN) [204]. In addition to the well-understood thermal disk emission predicted by [234, 182], numerous non-thermal radiative processes such as synchrotron and inverseCompton are also clearly present in a large fraction of AGN [186, 71].</text>
<text><location><page_2><loc_16><loc_16><loc_76><loc_30></location>Peters and Mathews [196] derived the leading-order gravitational wave emission from two point masses more than a decade before Thorne and Braginsky [254] suggested that one of the most promising sources for such a GW signal would be the collapse and formation of a SMBH, or the (near head-on) collision of two such objects in the center of an active galaxy. In that same paper, Thorne and Braginsky build on earlier work by Estabrook and Wahlquist [76] and explore the prospects for a space-based method for direct detection of these GWs via Doppler tracking of inertial spacecraft. They also attempted to estimate event rates for these generic bursts, and arrived at quite a broad range of possibilities, from ∼ < 0 . 01 to ∼ > 50 events per year, numbers that at least bracket our current best-estimates for SMBH mergers [226].</text>
<text><location><page_2><loc_19><loc_14><loc_76><loc_15></location>However it is not apparent that Thorne and Braginsky considered the hierarchical</text>
<text><location><page_3><loc_16><loc_59><loc_76><loc_88></location>merger of galaxies as the driving force behind these SMBH mergers, a concept that was only just emerging at the time [188, 189]. Within the galactic merger context, the seminal paper by Begelman, Blandford, and Rees (BBR) [18] outlines the major stages of the SMBH merger: first the nuclear star clusters merge via dynamical friction on the galactic dynamical time t gal ∼ 10 8 yr; then the SMBHs sink to the center of the new stellar cluster on the stellar dynamical friction time scale t df ∼ 10 6 yr; the two SMBHs form a binary that is initially only loosely bound, and hardens via scattering with the nuclear stars until the loss cone is depleted; further hardening is limited by the diffusive replenishing of the loss cone, until the binary becomes 'hard,' i.e., the binary's orbital velocity is comparable to the local stellar orbital velocity, at which point the evolutionary time scale is t hard ∼ N inf t df , with N inf stars within the influence radius. This is typically much longer than the Hubble time, effectively stalling the binary merger before it can reach the point where gravitational radiation begins to dominate the evolution. Since r hard ∼ 1 pc, and gravitational waves don't take over until r GW ∼ 0 . 01 pc, this loss cone depletion has become known as the 'final parsec problem' [168]. BBR thus propose that there should be a large cosmological population of stalled SMBH binaries with separation of order a parsec, and orbital periods of years to centuries. Yet to date not a single binary system with these subparsec separations has even been unambiguously identified.</text>
<text><location><page_3><loc_16><loc_46><loc_76><loc_59></location>In the decades since BBR, numerous astrophysical mechanisms have been suggested as the solution to the final parsec problem [168]. Yet the very fact that so many different solutions have been proposed and continue to be proposed is indicative of the prevailing opinion that it is still a real impediment to the efficient merger of SMBHs following a galaxy merger. However, the incontrovertible evidence that galaxies regularly undergo minor and major mergers during their lifetimes, coupled with a distinct lack of binary SMBH candidates, strongly suggest that nature has found its own solution to the final parsec problem. Or, as Einstein put it, 'God does not care about mathematical difficulties; He integrates empirically.'</text>
<text><location><page_3><loc_16><loc_29><loc_76><loc_45></location>For incontrovertible evidence of a SMBH binary, nothing can compare with the direct detection of gravitational waves from space. The great irony of gravitationalwave astronomy is that, despite the fact that the peak GW luminosity generated by black hole mergers outshines the entire observable universe , the extremely weak coupling to matter makes both direct and indirect detection exceedingly difficult. For GWs with frequencies less than ∼ 1 Hz, the leading instrumental concept for nearly 25 years now has been a long-baseline laser interferometer with three freefalling test masses housed in drag-free spacecraft [77]. Despite the flurry of recent political and budgetary constraints that have resulted in a number of alternative, less capable designs, we take as our fiducial detector the classic LISA (Laser Interferometer Space Antenna) baseline design [151].</text>
<text><location><page_3><loc_16><loc_15><loc_76><loc_29></location>For SMBHs with masses of 10 6 M /circledot at a redshift of z = 1, LISA should be able to identify the location of the source on the sky within ∼ 10 deg 2 a month before merger, and better than ∼ 0 . 1 deg 2 with the entire waveform, including merger and ringdown [129, 143, 144, 130, 145, 253, 164]. This should almost certainly be sufficient to identify EM counterparts with wide-field surveys such as LSST [1], WFIRST [244], or WFXT [178]. Like the cosmological beacons of gamma-ray bursts and quasars, merging SMBHs can teach us about relativity, high-energy astrophysics, radiation hydrodynamics, dark energy, galaxy formation and evolution, and how they all interact.</text>
<text><location><page_3><loc_19><loc_14><loc_76><loc_15></location>A large variety of potential EM signatures have recently been proposed, almost</text>
<text><location><page_4><loc_16><loc_79><loc_76><loc_88></location>all of which require some significant amount of gas in the near vicinity of the merging black holes [222]. Thus we must begin with the question of whether or not there is any gas present, and if so, what are its properties. Only then can we begin to simulate realistic spectra and light curves, and hope to identify unique observational signatures that will allow us to distinguish these objects from the myriad of other high-energy transients throughout the universe.</text>
<section_header_level_1><location><page_4><loc_16><loc_75><loc_40><loc_77></location>2. CIRCUMBINARY DISKS</section_header_level_1>
<text><location><page_4><loc_16><loc_62><loc_76><loc_74></location>If there is gas present in the vicinity of a SMBH binary, it is likely in the form of an accretion disk, as least at some point in the system's history. Disks are omnipresent in the universe for the simple reason that it is easy to lose energy through dissipative processes, but much more difficult to lose angular momentum. At larger separations, before the SMBHs form a bound binary system, massive gas disks can be quite efficient at bringing the two black holes together [75, 65]. As these massive gas disks are typically self-gravitating, their dynamics can be particularly complicated, and require high-resolution 3D simulations, which will be discussed in more detail in section 3.5.</text>
<text><location><page_4><loc_16><loc_39><loc_76><loc_62></location>Here we focus on the properties of non-self-gravitating circumbinary accretion disks which have traditionally employed the same alpha prescription for pressureviscous stress scaling as in [234]. Much of the early work on this subject was applied to protoplanetary disks around binary stars, or stars with massive planets embedded in their surrounding disks. The classical work on this subject is Pringle (1991) [200], who considered the evolution of a 1D thin disk with an additional torque term added to the inner disk. This source of angular momentum leads to a net outflow of matter, thus giving these systems their common names of 'excretion' or 'decretion' disks. Pringle considered two inner boundary conditions: one for the inflow velocity v r ( R in ) → 0 and one for the surface density Σ( R in ) → 0. For the former case, the torque is applied at a single radius at the inner edge, leading to a surface density profile that increases steadily inwards towards R in . In the latter case, the torque is applied over a finite region in the inner disk, which leads to a relatively large evacuated gap out to ∼ > 6 R in . In both cases, the angular momentum is transferred from the binary outwards through the gas disk, leading to a shrinking of the binary orbit.</text>
<text><location><page_4><loc_16><loc_21><loc_76><loc_39></location>In [8], SPH simulations were utilized to understand in better detail the torquing mechanism between the gas and disk. They find that, in agreement with the linear theory of [89], the vast majority of the binary torque is transmitted to the gas through the ( l, m ) = (1 , 2) outer Lindblad resonance (for more on resonant excitation of spiral density waves, see [247]). The resonant interaction between the gas and eccentric binary ( e = 0 . 1 for the system in [8]) pumps energy and angular momentum into the gas, which gets pulled after the more rapidly rotating interior point mass. This leads to a nearly evacuated disk inside of r ≈ 2 a , where a is the binary's semimajor axis. The interaction with the circumbinary disk not only removes energy and angular momentum from the binary, but it can also increase its eccentricity, and cause the binary pericenter to precess on a similar timescale, all of which could lead to potentially observable effects in GW observations [6, 211, 212].</text>
<text><location><page_4><loc_16><loc_14><loc_76><loc_21></location>In [9, 10], Artymowicz & Lubow expand upon [8] and provide a comprehensive study of the effects of varying the eccentricity, mass ratio, and disk thickness on the behavior of the circumbinary disk and its interaction with the binary. Not surprisingly, they find that the disk truncation radius moves outward with binary eccentricity. Similarly, the mini accretion disks around each of the stars has an outer truncation</text>
<text><location><page_5><loc_16><loc_77><loc_76><loc_88></location>radius that decreases with binary eccentricity. On the other hand, the location of the inner edge of the circumbinary disk appears to be largely insensitive to the binary mass ratio [9]. For relatively thin, cold disks with aspect ratios H/R ≈ 0 . 03, the binary torque is quite effective at preventing accretion, much as in the decretion disks of Pringle [200]. In that case, the gas accretion rate across the inner gap is as much at 10 -100 × smaller than that seen in a single disk, but the authors acknowledge that the low resolution of the SPH simulation makes these estimates inconclusive [9].</text>
<text><location><page_5><loc_16><loc_64><loc_76><loc_77></location>When increasing the disk thickness to H/R ≈ 0 . 1, the gas has a much easier time jumping the gap and streaming onto one of the two stars, typically the smaller one. For H/R ≈ 0 . 1, the gas accretion rate is within a factor of two of the single-disk case [10]. The accretion rate across the gap is strongly modulated at the binary orbital period, although the accretion onto the individual masses can be out of phase with each other. The modulated accretion rate suggests a promising avenue for producing a modulated EM signal in the pre-merger phase, and the very fact that a significant amount of gas can in fact cross the gap is important for setting up a potential prompt signal at the time of merger.</text>
<text><location><page_5><loc_16><loc_40><loc_76><loc_63></location>To adequately resolve the spiral density waves in a thin disk, 2D grid-based calculations are preferable to the inherently noisy and diffusive SPH methods. Armitage and Natarayan [5] take a hybrid approach to the problem, and use a 2D ZEUS [245] hydrodynamics calculation to normalize the torque term in the 1D radial structure equation. Unlike [8], they find almost no leakage across the gap, even for a moderate H/R = 0 . 07. However, they do identify a new effect that is particularly important for binary black holes, as opposed to protoplanetary disks. For a mass ratio of q ≡ m 2 /m 1 = 0 . 01, when a small accretion disk is formed around the primary, the evolution of the secondary due to gravitational radiation can shrink the binary on such short time scales that it plows into the inner accretion disk, building up gas and increasing the mass accretion rate and thus luminosity immediately preceding merger [5]. If robust, this obviously provides a very promising method for generating bright EM counterparts to SMBH mergers. However, recent 2D simulations by [17] suggest that the gas in the inner disk could actually flow across the gap back to the outer disk, like snow flying over the plow. The reverse of this effect, gas piling up in the outer disk before leaking into the inner disk, has recently been explored by [132, 133].</text>
<text><location><page_5><loc_16><loc_27><loc_76><loc_39></location>In the context of T Tau stars, [98, 99] developed a sophisticated simulation tool that combines a polar grid for the outer disk with a Cartesian grid around the binary to best resolve the flow across the gap. They are able to form inner accretion disks around each star, fed by persistent streams from the circumbinary disk. As a test, they compare the inner region to an SPH simulation and find good agreement, but only when the inner disks are artificially fed by some outer source, itself not adequately resolved by the SPH calculation [99]. They also see strong periodic modulation in the accretion rate, due to a relatively large binary eccentricity of e = 0 . 5.</text>
<text><location><page_5><loc_16><loc_14><loc_76><loc_27></location>MacFadyen and Milosavljevic (MM08) [161] also developed a sophisticated gridbased code including adaptive mesh refinement to resolve the flows at the inner edge of the circumbinary disk in the SMBH binary context. However, they excise the inner region entirely to avoid excessive demands on their resolution around each black hole so are unable to study the behavior of mini accretion disks. They also use an alpha prescription for viscosity and find qualitatively similar results to the earlier work described above: a gap with R in ≈ 2 a due to the m = 2 outer Lindblad resonance, spiral density waves in an eccentric disk, highly variable and periodic accretion, and accretion across the gap of ∼ 20% that expected for a single BH accretion disk with</text>
<figure>
<location><page_6><loc_18><loc_69><loc_74><loc_88></location>
<caption>Figure 1. ( left ) Surface density and spiral density wave structure of circumbinary disk with equal-mass BHs on a circular orbit, shown after the disk evolved for 4000 binary periods. The dimensions of the box are x = [ -5 a, 5 a ] and y = [ -5 a, 5 a ]. ( right ) Time-dependent accretion rate across the inner edge of the simulation domain ( r in = a ), normalized by the initial surface density scale Σ 0 . [reproduced from MacFadyen & Milosavljevic (2008), ApJ 672 , 83]</caption>
</figure>
<text><location><page_6><loc_16><loc_51><loc_76><loc_57></location>the same mass [161]. The disk surface density as well as the variable accretion rate are shown in Figure 1. Recent work by the same group carried out a systematic study of the effect of mass ratio and found significant accretion across the gap for all values of q = m 2 /m 1 between 0.01 and 1 [64].</text>
<text><location><page_6><loc_16><loc_36><loc_76><loc_51></location>The net result of these calculations seems to be that circumbinary gas disks are a viable mechanism for driving the SMBH binary through the final parsec to the GWdriven phase, and supplying sufficient accretion power to be observable throughout. Thus it is particularly perplexing that no such systems have been observed with any degree of certainty. According to simple alpha-disk theory, there should also be a point in the GW evolution where the binary separation is shrinking at such a prodigious rate that the circumbinary disk cannot keep up with it, and effectively decouples from the binary. At that point, gas should flow inwards on the relatively slow timescale corresponding to accretion around a single point mass, and a real gap of evacuated space might form around the SMBHs, which then merge in a near vacuum [174].</text>
<section_header_level_1><location><page_6><loc_16><loc_33><loc_44><loc_34></location>3. NUMERICAL SIMULATIONS</section_header_level_1>
<section_header_level_1><location><page_6><loc_16><loc_30><loc_39><loc_31></location>3.1. Vacuum numerical relativity</section_header_level_1>
<text><location><page_6><loc_16><loc_14><loc_76><loc_29></location>In the context of EM counterparts, the numerical simulation of two equal-mass, nonspinning black holes in a vacuum is just about the simplest problem imaginable. Yet the inherent non-linear behavior of Einstein's field equations made this a nearly unsolvable Grand Challenge problem, frustrating generations of relativists from the 3+1 formulation of Arnowitt, Deser, and Misner in 1962 [7], followed shortly by the first attempt at a numerical relativity (NR) simulation on a computer in 1964 [100], decades of uneven progress, slowed in large part by the limited computer power of the day (but also by important fundamental instabilities in the formulation of the field equations), to the ultimate solution by Pretorius in 2005 [198] and subsequent deluge of papers in 2006 from multiple groups around the world (for a much more</text>
<text><location><page_7><loc_16><loc_85><loc_76><loc_88></location>thorough review of this colorful story and the many technical challenges overcome by its participants, see [47]).</text>
<text><location><page_7><loc_16><loc_61><loc_76><loc_85></location>Here we will review just a few highlights from the recent NR results that are most pertinent to our present subject. For the first 50 years since their original conception, black holes (and general relativity as a whole) were largely relegated to mathematicians as a theoretical curiosity with little possibility of application in astronomy. All this changed in the late 1960s and early 70s when both stellar-mass and super-massive black holes were not only observed, but also understood to be critical energy sources and play a major role in the evolution of galaxies and stars [255]. A similar environment was present during the 1990s with regard to binary black holes and gravitational waves. Most believed in their existence, but after decades of false claims and broken promises, the prospect of direct detection of GWs seemed further away than ever. But then in 1999, construction was completed on the two LIGO observatories, and they began taking science data in 2002. At the same time, the space-based LISA concept was formalized with the 'Yellow Book,' a report submitted to ESA in 1996, and together with NASA, an international science team was formed in 2001. Astrophysics theory has long been data-driven, but here was a case where large-scale projects were being proposed and even funded based largely on theoretical predictions.</text>
<text><location><page_7><loc_16><loc_40><loc_76><loc_60></location>The prospect of real observations and data in turn energized the NR community and provided new motivation to finally solve the binary BH merger problem. Longduration, accurate waveforms are necessary for both the detection and characterization of gravitational waves. Generic binary sources are fully described by 17 parameters: the BH masses (2), spin vectors (6), binary orbital elements (6), sky position (2), and distance (1). To adequately cover this huge parameter space requires exceedingly clever algorithms and an efficient method for calculating waveforms. Fortunately, most NR studies to date suggest that even the most non-linear phase of the inspiral and merger process produces a relatively smooth waveform, dominated by the leading quadrupole mode [47]. Additionally, in the early inspiral and late ringdown phases, relatively simple analytic expressions appear to be quite sufficient in matching the waveforms [194]. Even more encouraging is the fact that waveforms from different groups using very different methods agree to a high level of accuracy, thus lending confidence to their value as a description of the real world [13].</text>
<text><location><page_7><loc_16><loc_32><loc_76><loc_39></location>In addition to the waveforms, another valuable result from these first merger simulations was the calculation of the mass and spin of the final black hole, demonstrating that the GWs carried away a full 4% of their initial energy in roughly an orbital time, and leave behind a moderately spinning black hole with a/M = 0 . 7 [11, 44].</text>
<text><location><page_7><loc_16><loc_15><loc_76><loc_32></location>After the initial breakthrough with equal-mass, non-spinning black holes, the remarkably robust 'moving puncture' method was soon applied to a wide variety of systems, including unequal masses [21], eccentric orbits [113], and spinning BHs [45]. As with test particles around Kerr black holes, when the spins are aligned with the orbital angular momentum, the BHs can survive longer before plunging, ultimately producing more GW power and resulting in a larger final spin. This is another critical result for astrophysics, as the spin evolution of SMBHs via mergers and gas accretion episodes is a potentially powerful diagnostic of galaxy evolution [22]. Perhaps the most interesting and unexpected result from the NR bonanza was the first accurate calculation of the gravitational recoil, which will be discussed in more detail in the following section.</text>
<text><location><page_7><loc_19><loc_14><loc_76><loc_15></location>In addition to the widespread moving puncture method, the NR group at</text>
<text><location><page_8><loc_16><loc_68><loc_76><loc_88></location>Cornell/Caltech developed a highly accurate spectral method that is particularly wellsuited for long evolutions [42]. Because it converges exponentially with resolution (as opposed to polynomial convergence for finite-difference methods), the spectral method can generate waveforms with dozens of GW cycles, accurate to a small fraction of phase. These long waveforms are particularly useful for matching the late inspiral to post-Newtonian (PN) equations of motion, the traditional tool of choice for GW data analysis for LIGO and LISA (e.g., [57, 4, 128, 25]). The down side of the spectral method has been its relative lack of flexibility, making it very time consuming to set up simulations of new binary configurations, particularly with arbitrary spins. If this problem can be overcome, spectral waveforms will be especially helpful in guiding the development of more robust semi-analytic tools (e.g., the effective-one-body approach of Buonanno [16]) for calculating the inspiral, merger, and ringdown of binary BHs with arbitrary initial conditions.</text>
<text><location><page_8><loc_16><loc_52><loc_76><loc_68></location>The natural application for long, high-accuracy waveforms is as templates in the matched-filtering approach to GW data analysis. For LIGO, this is critical to detect most BH mergers, where much of the in-band power will come from the final stages of inspiral and merger. The high signal-to-noise expected from SMBHs with LISA means that most events will probably be detected with high significance even when using a primitive template library [82, 58]. However, for parameter estimation , highfidelity waveforms are essential for faithfully reproducing the physical properties of the source. In particular, for spinning BHs, the information contained in the precessing waveform can greatly improve our ability to determine the sky position of the source, and thus improve our prospects for detecting and characterizing any EM counterpart [144, 253, 145].</text>
<section_header_level_1><location><page_8><loc_16><loc_48><loc_33><loc_50></location>3.2. Gravitational recoil</section_header_level_1>
<text><location><page_8><loc_16><loc_31><loc_76><loc_47></location>In the general case where there is some asymmetry between the two black holes (e.g., unequal masses or spins), the GW radiation pattern will have a complicated multipole structure. The beating between these different modes leads to a net asymmetry in the momentum flux from the system, ultimately resulting in a recoil or kick imparted on the final merged black hole [219]. This effect has long been anticipated for any GW source [37, 195, 19], but the specific value of the recoil has been notoriously difficult to calculate using traditional analytic means [266, 81, 24, 59]. Because the vast majority of the recoil is generated during the final merger phase, it is a problem uniquely suited for numerical relativity. Indeed, this was one of the first results published in 2006, for the merger of two non-spinning BHs with mass ratio 3:2, giving a kick of 90 -100 km/s [12].</text>
<text><location><page_8><loc_16><loc_17><loc_76><loc_31></location>Shortly thereafter, a variety of initial configurations were explored, covering a range of mass ratios [112, 90], aligned spins [111, 139], and precessing spins [46, 256]. Arguably the most exciting result came with the discovery of the 'superkick' configuration, where two equal-mass black holes have equal and opposite spins aligned in the orbital plane, leading to kicks of > 3000 km/s [91, 46, 256]. If such a situation were realized in nature, the resulting black hole would certainly be ejected from the host galaxy, leaving behind an empty nuclear host [167]. Some of the many other possible ramifications include offset AGN, displaced star clusters, or unusual accretion modes. These and other signatures are discussed in detail below in section 4.</text>
<text><location><page_8><loc_16><loc_14><loc_76><loc_17></location>Analogous to the PN waveform matching mentioned above, there has been a good deal of analytic modeling of the kicks calculated by the NR simulations</text>
<figure>
<location><page_9><loc_20><loc_74><loc_44><loc_87></location>
</figure>
<figure>
<location><page_9><loc_48><loc_74><loc_73><loc_87></location>
<caption>Figure 2. Magnetic and electric field configurations around binary black hole 40 M ( left ) and 20 M ( right ) before merger. The electric fields get twisted around the black holes, while the magnetic fields remain roughly vertical. [reproduced from Palenzuela et al. 2009, PRL 103 , 081101]</caption>
</figure>
<paragraph><location><page_9><loc_16><loc_54><loc_76><loc_63></location>[217, 219, 41, 202]. Simple empirical fits to the NR data are particularly useful for incorporating the effects of recoil into cosmological N-body simulations that evolve SMBHs along with merging galaxies [14, 46, 158, 259]. While the astrophysical impacts of large kicks are primarily Newtonian in nature (even a kick of v ∼ 3000 km/s is only 1% of the speed of light), the underlying causes, while only imperfectly understood, clearly point to strong non-linear gravitational forces at work [199, 219, 206, 118, 207].</paragraph>
<section_header_level_1><location><page_9><loc_16><loc_51><loc_38><loc_52></location>3.3. Pure electromagnetic fields</section_header_level_1>
<text><location><page_9><loc_16><loc_38><loc_76><loc_50></location>Shortly after the 2006-07 revolution, many groups already began looking for the next big challenge in numerical relativity. One logical direction was the inclusion of electromagnetic fields in the simulations, solving the coupled Einstein-Maxwell equations throughout a black hole merger. The first to do so was Palenzuela et al. [190], who considered an initial condition with zero electric field and a uniform magnetic field surrounding an equal-mass, non-spinning binary a couple orbits before merger. The subsequent evolution generates E-fields twisted around the two BHs, while the B-field remains roughly vertical, although it does experience some amplification (see Fig. 2).</text>
<text><location><page_9><loc_16><loc_26><loc_76><loc_38></location>The EM power from this system was estimated by integrating the radial Poynting flux through a spherical shell at large radius. They found only a modest (30 -40%) increase in EM energy, but there was a clear transient quadrupolar Poynting burst of power coincident with the GW signal, giving one of the first hints of astrophysical EM counterparts from NR simulations. This work was followed up by a more thorough study in [175, 191], which showed that the EM power L EM scaled like the square of the total BH spin and proportional to B 2 , as would be expected for a Poynting flux-powered jet [26].</text>
<section_header_level_1><location><page_9><loc_16><loc_23><loc_35><loc_24></location>3.4. Force-free simulations</section_header_level_1>
<text><location><page_9><loc_16><loc_14><loc_76><loc_22></location>In [192, 193], Palenzuela and collaborators extended their vacuum simulations to include force-free electrodynamics. This is an approximation where a tenuous plasma is present, and can generate currents and magnetic fields, but carries no inertia to push those fields around. They found that any moving, spinning black hole can generate Poynting flux and a Blandford-Znajek-type jet [26]. Compared to the vacuum case,</text>
<text><location><page_10><loc_16><loc_82><loc_76><loc_88></location>force-free simulations of a merging binary predict significant amplification of EM power by a factor of ∼ 10 × , coincident with the peak GW power [193]. For longer simulations run at higher accuracy, [176, 2] found an even greater L EM amplification of ∼ 30 × that of electro-vacuum.</text>
<section_header_level_1><location><page_10><loc_16><loc_79><loc_32><loc_80></location>3.5. M/HD simulations</section_header_level_1>
<text><location><page_10><loc_16><loc_67><loc_76><loc_77></location>As mentioned above in section 2, if there is an appreciable amount of gas around the binary BH, it is likely in the form of a circumbinary disk. This configuration has thus been the subject of most (magneto)hydrodynamical simulations. SPH simulations of disks that are not aligned with the binary orbit show a warped disk that can precess as a rigid body, and generally suffer more gas leakage across the inner gap, modulated at twice the orbital frequency [146, 117, 108]. In many cases, accretion disks can form around the individual BHs [65, 104].</text>
<text><location><page_10><loc_16><loc_55><loc_76><loc_67></location>Massive disks have the ability to drive the binary towards merger on relatively short time scales [75, 65, 56] and also align the BH spins at the same time [32] (although see also [154, 155] for a counter result). Retrograde disks may be even more efficient at shrinking the binary [179] and they may also be quite stable [180]. Recent simulations by [212] show that the binary will evolve due not only to torques from the circumbinary disk, but also from transfer of angular momentum via gas streaming onto the two black holes. They find that the binary does shrink, and eccentricity can still be excited, but not necessarily at the rates predicted by classical theory.</text>
<text><location><page_10><loc_16><loc_45><loc_76><loc_55></location>Following merger, the circumbinary disk can also undergo significant disruption due to the gravitational recoil, as well as the sudden change in potential energy due to the mass loss from gravitational waves. These effects lead to caustics forming in the perturbed disk, in turn leading to shock heating and potentially both prompt and long-lived EM afterglows [184, 165, 214, 55, 268, 197, 213, 269]. Any spin alignment would be critically important for both the character of the prompt EM counterpart, as well as the recoil velocity [159, 23].</text>
<text><location><page_10><loc_16><loc_35><loc_76><loc_44></location>Due to computational limitations, it is generally only possible to include the last few orbits before merger in a full NR simulation. Since there is no time to allow the system to relax into a quasi-steady state, the specific choice of initial conditions is particularly important for these hydrodynamic merger simulations. Some insight can be gained from Newtonian simulations [239] as well as semi-analytic models [153, 203, 236].</text>
<text><location><page_10><loc_16><loc_25><loc_76><loc_35></location>If the disk decouples from the binary well before merger, the gas may be quite hot and diffuse around the black holes [107]. In that case, uniform density diffuse gas may be appropriate. In merger simulations by [78, 30, 35], the diffuse gas experiences Bondi-type accretion onto each of the SMBHs, with a bridge of gas connecting the two before merger. Shock heating of the gas could lead to a strong EM counterpart. As a simple estimate for the EM signal, [35] use bremsstrahlung radiation to predict roughly Eddington luminosity peaking in the hard X-ray band.</text>
<text><location><page_10><loc_16><loc_14><loc_76><loc_25></location>The first hydrodynamic NR simulations with disk-like initial conditions were carried out by [79] by allowing the disk to relax into a quasi-steady state before turning the GR evolution on. They found disk properties qualitatively similar to classical Newtonian results, with a low-density gap threaded by accretion streams at early times, and largely evacuated at late times when the binary decouples from the disk. Due to the low density and high temperatures in the gap, they estimate the EM power will be dominated by synchrotron (peaking in the IR for M = 10 8 M /circledot ), and</text>
<text><location><page_11><loc_16><loc_83><loc_76><loc_88></location>reach Eddington luminosity. An analogous calculation was carried out by [31], yet they find EM luminosity orders of magnitude smaller, perhaps because they do not relax the initial disk for as long.</text>
<text><location><page_11><loc_16><loc_56><loc_76><loc_83></location>Most recently, circumbinary disk simulations have moved from purely hydrodynamic to magneto-hydrodynamic (MHD), which allows them to dispense with alpha prescriptions of viscosity and incorporate the true physical mechanism behind angular momentum in accretion disks: magnetic stresses and the magneto-rotational instability [15]. Newtonian MHD simulations of circumbinary disks find large-scale m = 1 modes growing in the outer disk, modulating the accretion flow across the gap [239]. Similar modes were seen in [181], who used a similar procedure as [79] to construct a quasi-stable state before allowing the binary to merge. They find that the MHD disk is able to follow the inspiraling binary to small separations, showing little evidence for the decoupling predicted by classical disk theory. However, the simulations of [181] use a hybrid space-time based on PN theory [84] that breaks down close to merger. Furthermore, while fully relativistic in its MHD treatment, the individual black holes are excised from the simulation due to computational limitations, making it difficult to estimate EM signatures from the inner flow. Farris et al. [80] have been able to overcome this issue and put the BHs on the grid with the MHD fluid. They find that the disk decouples at a ≈ 10 M , followed by a decrease in luminosity before merger, and then an increase as the gap fills in and resumes normal accretion, as in [174].</text>
<text><location><page_11><loc_16><loc_46><loc_76><loc_56></location>Giacomazzo et al. [88] carried out MHD merger simulations with similar initial conditions to both [191] and [30], with diffuse hot gas threaded by a uniform vertical magnetic field. Unlike in the force-free approximation, the inclusion of significant gas leads to a remarkable amplification of the magnetic field, which is compressed by the accreting fluid. [88] found the B-field increased by of a factor of 100 during merger, corresponding to an increase in synchrotron power by a factor of 10 4 , which could easily lead to super-Eddington luminosities from the IR through hard X-ray bands.</text>
<text><location><page_11><loc_16><loc_38><loc_76><loc_45></location>The near future promises a self-consistent, integrated picture of binary BH-disk evolution. By combining the various methods described above, we can combine multiple MHD simulations at different scales, using the results from one method as initial conditions for another, and evolve a circumbinary disk from the parsec level through merger and beyond.</text>
<section_header_level_1><location><page_11><loc_16><loc_35><loc_33><loc_36></location>3.6. Radiation transport</section_header_level_1>
<text><location><page_11><loc_16><loc_23><loc_76><loc_34></location>Even with high resolution and perfect knowledge of the initial conditions, the value of the GRMHD simulations is limited by the lack of radiation transport and accurate thermodynamics, which have only recently been incorporated into local Newtonian simulations of steady-state accretion disks [114, 115]. Significant future work will be required to incorporate the radiation transport into a fully relativistic global framework, required not just for accurate modeling of the dynamics, but also for the prediction of EM signatures that might be compared directly with observations.</text>
<text><location><page_11><loc_16><loc_14><loc_76><loc_23></location>Some recent progress has been made by using the relativistic Monte Carlo raytracing code Pandurata as a post-processor for MHD simulations of single accretion disks [223, 224], reproducing soft and hard X-ray spectral signatures in agreement with observations of stellar-mass black holes. Applying the same ray-tracing approach to the MHD merger simulations of [88], we can generate light curves and broadband spectra, ranging from synchrotron emission in the IR up through inverse-</text>
<figure>
<location><page_12><loc_26><loc_66><loc_66><loc_88></location>
<caption>Figure 3. A preliminary calculation of the broad-band spectrum produced by the GRMHD merger of [88], sampled near the peak of gravitational wave emission. Synchrotron and bremsstrahlung seeds from the magnetized plasma are ray-traced with Pandurata [224]. Inverse-Compton scattering off hot electrons in a diffuse corona gives a power-law spectrum with cut-off around kT e . The total mass is 10 7 M /circledot and the gas has T e = 100 keV and optical depth of order unity.</caption>
</figure>
<text><location><page_12><loc_16><loc_45><loc_76><loc_54></location>Compton peaking in the X-ray. An example of such a spectrum is shown in Figure 3, corresponding to super-Eddington luminosity at the peak of the EM and GW emission. Since the simulation in [88] does not include a cooling function, we simply estimate the electron temperature as 100 keV, similar to that seen in typical AGN coronas. Future work will explore the effects of radiative cooling within the NR simulations, as well as incorporating the dynamic metric into the ray-tracing analysis.</text>
<text><location><page_12><loc_16><loc_33><loc_76><loc_45></location>Of course, the ultimate goal will be to directly incorporate radiation transport as a dynamical force within the GRMHD simulations. Significant progress has been made recently in developing accurate radiation transport algorithms in a fully covariant framework [187, 121, 215], and we look forward to seeing them mature to the point where they can be integrated into dynamic GRMHD codes. In addition to Pandurata , there are a number of other relativistic ray-tracing codes (e.g., [63, 237]), currently based on the Kerr metric, which may also be adopted to the dynamic space times of merging black holes.</text>
<section_header_level_1><location><page_12><loc_16><loc_30><loc_63><loc_31></location>4. OBSERVATIONS: PAST, PRESENT, AND FUTURE</section_header_level_1>
<text><location><page_12><loc_16><loc_16><loc_76><loc_28></location>One way to categorize EM signatures is by the physical mechanism responsible for the emission: stars, hot diffuse gas, or circumbinary/accretion disks. In Figure 4, we show the diversity of these sources, arranged according the spatial and time scales on which they are likely to occur [222]. Over the course of a typical galaxy merger, we should expect the system to evolve from the upper-left to the lower-center to the upper-right regions of the chart. Sampling over the entire observable universe, the number of objects detected in each source class should be proportional to the product of the lifetime and observable flux of that object.</text>
<text><location><page_12><loc_19><loc_15><loc_76><loc_16></location>Note that most of these effects are fundamentally Newtonian, and many are</text>
<figure>
<location><page_13><loc_21><loc_62><loc_68><loc_88></location>
<caption>Figure 4. Selection of potential EM sources, sorted by timescale, typical size of emission region, and physical mechanism (blue/ italic = stellar; yellow/TimesRoman = accretion disk; green/ bold = diffuse gas/miscellaneous). The evolution of the merger proceeds from the upper-left through the lower-center, to the upperright.</caption>
</figure>
<text><location><page_13><loc_43><loc_62><loc_50><loc_62></location>time since merger</text>
<text><location><page_13><loc_16><loc_33><loc_76><loc_49></location>only indirect evidence of SMBH mergers, as opposed to the prompt EM signatures described above. Yet they are also important in understanding the complete history of binary BHs, as they are crucial for estimating the number of sources one might expect at each stage in a black hole's evolution. If, for example, we predict a large number of bright binary quasars with separations around 0 . 1 pc, and find no evidence for them in any wide-field surveys (as has been the case so far, with limited depth and temporal coverage), we would be forced to revise our theoretical models. But if the same rate calculations accurately predict the number of dual AGN with separations of ∼ 1 -10 kpc, and GW or prompt EM detections are able to confirm the number of actual mergers, then we might infer the lack of binary quasars is due to a lack of observability, as opposed to a lack of existence.</text>
<text><location><page_13><loc_16><loc_25><loc_76><loc_33></location>The long-term goal in observing EM signatures will be to eventually fill out a plot like that of Figure 4, determining event rates for each source class, and checking to make sure we can construct a consistent picture of SMBH-galaxy co-evolution. This is indeed an ambitious goal, but one that has met with reasonable success in other fields, such as stellar evolution or even the fossil record of life on Earth.</text>
<section_header_level_1><location><page_13><loc_16><loc_22><loc_32><loc_23></location>4.1. Stellar Signatures</section_header_level_1>
<text><location><page_13><loc_16><loc_14><loc_76><loc_21></location>On the largest scales, we have strong circumstantial evidence of supermassive BH mergers at the centers of merging galaxies. From large optical surveys of interacting galaxies out to redshifts of z ∼ 1, we can infer that 5 -10% of massive galaxies are merging at any given time, and the majority of galaxies with M gal ∼ > 10 10 M /circledot have experienced a major merger in the past 3 Gyr [20, 162, 61, 43], with even higher</text>
<text><location><page_14><loc_16><loc_70><loc_76><loc_88></location>merger rates at redshifts z ∼ 1 -3 [54]. At the same time, high-resolution observations of nearby galactic nuclei find that every large galaxy hosts a SMBH in its center [140]. Yet we see a remarkably small number of dual AGN [135, 53], and only one known source with an actual binary system where the BHs are gravitationally bound to each other [209, 210]. Taken together, these observations strongly suggest that when galaxies merge, the merger of their central SMBHs inevitably follows, and likely occurs on a relatively short time scale, which would explain the apparent scarcity of binary BHs (although recent estimates by [106] predict as many as 10% of AGNs with M ∼ 10 7 M /circledot might be in close binaries with a ∼ 0 . 01 pc). The famous 'M-sigma' relationship between the SMBH mass and the velocity dispersion of the surrounding bulge also points to a merger-driven history over a wide range of BH masses and galaxy types [96].</text>
<text><location><page_14><loc_16><loc_62><loc_76><loc_69></location>There is additional indirect evidence for SMBH mergers in the stellar distributions of galactic nuclei, with many elliptical galaxies showing light deficits (cores), which correlate strongly with the central BH mass [141]. The cores suggest a history of binary BHs that scour out the nuclear stars via three-body scattering [171, 172, 169], or even post-merger relaxation of recoiling BHs [167, 40, 93, 94].</text>
<text><location><page_14><loc_16><loc_52><loc_76><loc_62></location>While essentially all massive nearby galaxies appear to host central SMBHs, it is quite possible that this is not the case at larger redshifts and smaller masses, where major mergers could lead to the complete ejection of the resulting black hole via large recoils. By measuring the occupation fraction of SMBHs in distant galaxies, one could infer merger rates and the distribution of kick velocities [217, 261, 218, 262, 264]. The occupation fraction will of course also affect the LISA event rates, especially at high redshift [226].</text>
<text><location><page_14><loc_16><loc_41><loc_76><loc_51></location>Another indirect signature of BH mergers comes from the population of stars that remain bound to a recoiling black hole that gets ejected from a galactic nucleus [136, 170, 183]. These stellar systems will appear similar to globular clusters, yet with smaller spatial extent and much larger velocity dispersions, as the potential is completely dominated by the central SMBH. With multi-object spectrometers on large ground-based telescopes, searching for these stellar clusters in the Milky Way halo or nearby galaxy clusters ( d ∼ < 40 Mpc) is technically realistic in the immediate future.</text>
<section_header_level_1><location><page_14><loc_16><loc_38><loc_42><loc_39></location>4.2. Gas Signatures: Accretion Disks</section_header_level_1>
<text><location><page_14><loc_16><loc_23><loc_76><loc_37></location>As discussed above in section 2, circumbinary disks will likely have a low-density gap within r ≈ 2 a , although may still be able to maintain significant gas accretion across this gap, even forming individual accretion disks around each black hole. The most sophisticated GRMHD simulations suggest that this accretion can be maintained even as the binary is rapidly shrinking due to gravitational radiation [181]. If the inner disks can survive long enough, the final inspiral may lead to a rapid enhancement of accretion power as the fossil gas is plowed into the central black hole shortly before merger [5, 48]. For small values of q , a narrow gap could form in the inner disk, changing the AGN spectra in a potentially observable way [97, 163].</text>
<text><location><page_14><loc_16><loc_14><loc_76><loc_23></location>Regardless of how the gas reaches the central BH region, the simulations described above in section 3 all seem to agree that even a modest amount of magnetized gas can lead to a strong EM signature. If the primary energy source for heating the gas is gravitational [258], then typical efficiencies will be on the order of ∼ 1 -10%, comparable to that expected for standard accretion in AGN, although the much shorter timescales could easily lead to super-Eddington transients, depending on the</text>
<text><location><page_15><loc_16><loc_86><loc_54><loc_88></location>optical depth and cooling mechanisms of the gas[142].</text>
<text><location><page_15><loc_16><loc_70><loc_76><loc_86></location>However, if the merging BHs are able to generate strong magnetic fields [190, 175, 192, 88], then hot electrons could easily generate strong synchrotron flux, or highly relativistic jets may be launched along the resulting BH spin axis, converting matter to energy with a Lorentz boost factor of Γ /greatermuch 1. Even with purely hydrodynamic heating, particularly bright and long-lasting afterglows may be produced in the case of large recoil velocities, which effectively can disrupt the entire disk, leading to strong shocks and dissipation [150, 240, 220, 165, 214, 3, 55, 248, 268]. Long-lived afterglows could be discovered in existing multi-wavelength surveys, but successfully identifying them as merger remnants as opposed to obscured AGN or other bright unresolved sources would require improved pipeline analysis of literally millions of point sources, as well as extensive follow-up observations [220].</text>
<text><location><page_15><loc_16><loc_53><loc_76><loc_69></location>For many of these large-kick systems, we may observe quasar activity for millions of years after, with the source displaced from the galactic center, either spatially [125, 156, 263, 52, 70, 122] or spectroscopically [36, 138, 38, 208]. However, large offsets between the redshifts of quasar emission lines and their host galaxies have also been interpreted as evidence of pre-merger binary BHs [34, 66, 252, 69] or due to the large relative velocities in merging galaxies [109, 241, 260, 60], or 'simply' extreme examples of the class of double-peaked emitters, where the line offsets are generally attributed to the disk [85, 72, 242, 51, 86]. An indirect signature for kicked BHs could potentially show up in the statistical properties of active galaxies, in particular in the relative distribution of different classes of AGN in the 'unified model' paradigm [137, 28].</text>
<text><location><page_15><loc_16><loc_38><loc_76><loc_53></location>For systems that open up a gap in the circumbinary disk, another EM signature may take the form of a quasar suddenly turning on as the gas refills the gap, months to years after the BH merger [174, 235, 249]. But again, these sources would be difficult to distinguish from normal AGN variability without known GW counterparts. Some limited searches for this type of variability have recently been carried out in the X-ray band [123], but for large systematic searches, we will need targeted time-domain widefield surveys like PTF, Pan-STARRS, and eventually LSST. One of the most valuable scientific products from these time-domain surveys will be a better understanding of what is the range of variability for normal AGN, which will help us distinguish when an EM signal is most likely due to a binary [251].</text>
<text><location><page_15><loc_16><loc_18><loc_76><loc_38></location>In addition to the many potential prompt and afterglow signals from merging BHs, there has also been a significant amount of theoretical and observational work focusing on the early precursors of mergers. Following the evolutionary trail in Figure 1, we see that shortly after a galaxy merges, dual AGN may form with typical separations of a few kpc [135, 53], sinking to the center of the merged galaxy on a relatively short timescale ( ∼ < 1 Gyr) due to dynamical friction [18]. The galaxy merger process is also expected to funnel a great deal of gas to the galactic center, in turn triggering quasar activity [110, 126, 116, 92]. At separations of ∼ 1 pc, the BH binary (now 'hardened' into a gravitationally bound system) could stall, having depleted its loss cone of stellar scattering and not yet reached the point of gravitational radiation losses [173]. Gas dynamical drag from massive disks ( M disk /greatermuch M BH ) leads to a prompt inspiral ( ∼ 1 -10 Myr), in most cases able to reach sub-parsec separations, depending on the resolution of the simulation [74, 127, 75, 65, 56, 67, 68].</text>
<text><location><page_15><loc_16><loc_14><loc_76><loc_18></location>At this point, a proper binary quasar is formed, with an orbital period of months to decades, which could be identified by periodic accretion [161, 104, 101, 102], density waves in the disk [105], or periodic red-shifted broad emission lines [33, 238, 157, 177].</text>
<text><location><page_16><loc_16><loc_79><loc_76><loc_88></location>If these binary AGN systems do in fact exist, spectroscopic surveys should be able to identify many candidates, which may then be confirmed or ruled out with subsequent observations over relatively short timescales ( ∼ 1 -10 yrs), as the line-of-site velocities to the BHs changes by an observable degree. This approach has been attempted with various initial spectroscopic surveys, but as yet, no objects have been confirmed to be binaries by multi-year spectroscopic monitoring [38, 147, 51, 73].</text>
<section_header_level_1><location><page_16><loc_16><loc_76><loc_46><loc_77></location>4.3. Gas Signatures: Diffuse Gas; 'Other'</section_header_level_1>
<text><location><page_16><loc_16><loc_58><loc_76><loc_74></location>In addition to the many disk-related signatures, there are also a number of potential EM counterparts that are caused by the accretion of diffuse gas in the galaxy. For the Poynting flux generated by the simulations of section 3, transient bursts or modulated jets might be detected in all-sky radio surveys [124, 185]. For BHs that get significant kicks at the time of merger, we expect to see occasional episodes of Bondi accretion as the BH oscillates through the gravitational potential of the galaxy over millions of years, as well as off-center AGN activity [27, 83, 95, 243]. On larger spatial scales, the recoiling BH could also produce trails of over-density in the hot interstellar gas of elliptical galaxies [62]. Also on kpc-Mpc scales, X-shaped radio jets have been seen in a number of galaxies, which could possibly be due to the merger and subsequent spin-flip of the central BHs [166].</text>
<text><location><page_16><loc_16><loc_38><loc_76><loc_58></location>Another potential source of EM counterparts comes not from diffuse gas, or accretion disks, but the occasional capture and tidal disruption of normal stars by the merging BHs. These tidal disruption events (TDEs), which also occurs in 'normal' galaxies [205, 134, 103], may be particularly easy to identify in off-center BHs following a large recoil [136]. TDE rates may be strongly increased prior to the merger [49, 246, 233, 221, 50, 265], but the actual disruption signal may be truncated by the pre-merger binary [152], and post-merger recoil may also reduce the rates [149]. These TDE events are likely to be seen by the dozen in coming years with Pan-STARRS and LSST [87]. In addition to the tidal disruption scenario, in [221] we showed how gas or stars trapped at the stable Lagrange points in a BH binary could evolve during inspiral and eventually lead to enhanced star formation, ejected hyper-velocity stars, highly-shifted narrow emission lines, and short bursts of superEddington accretion coincident with the BH merger.</text>
<text><location><page_16><loc_16><loc_23><loc_76><loc_38></location>A completely different type of EM counterpart can be seen with pulsar timing arrays (PTAs). In this technique, small time delays ( ∼ < 10 ns) in the arrival of pulses from millisecond radio pulsars would be direct evidence of extremely lowfrequency (nano-Hertz) gravitational waves from massive ( ∼ > 10 8 M /circledot ) BH binaries [119, 227, 228, 120, 232, 201, 257, 229]. By cross-correlating the signals from multiple pulsars around the sky, we can effectively make use of a GW detector the size of the entire galaxy. For now, one of the main impediments to GW astronomy with pulsar timing is the relatively small number of known, stable millisecond radio pulsars. Current surveys are working to increase this number and the uniformity of their distribution on the sky [148].</text>
<text><location><page_16><loc_16><loc_14><loc_76><loc_23></location>Even conservative estimates suggest that PTAs are probably only about ten years away from a positive detection of the GW stochastic background signal from the ensemble of SMBH binaries throughout the universe [231]. The probability of resolving an individual source is significantly smaller, but if it were detected, would be close enough ( z ∼ < 1) to allow for extensive EM follow-up, unlike many of the expected LISA sources at z ∼ > 5. Also, unlike LISA sources, PTA sources would be at an earlier stage</text>
<text><location><page_17><loc_16><loc_80><loc_76><loc_88></location>in their inspiral and thus be much longer lived, allowing for even more extensive study. A sufficiently large sample of such sources would even allow us to test whether they are evolving due to GW emission or gas-driven migration [131, 250, 230] (a test that might also be done with LISA with only a single source with sufficient signal-to-noise [267]).</text>
<section_header_level_1><location><page_17><loc_16><loc_77><loc_31><loc_78></location>5. CONCLUSION</section_header_level_1>
<text><location><page_17><loc_16><loc_64><loc_76><loc_75></location>Black holes are fascinating objects. They push our intuition to the limits, and never cease to amaze us with their extreme behavior. For a high-energy theoretical astrophysicist, the only thing more exciting than a real astrophysical black hole is two black holes, destroying everything in their path as they spiral together towards the point of no return. Thus one can easily imagine the frustration that stems from our lack of ability to actually see such an event, despite the fact that it outshines the entire observable universe. And the path forwards does not appear to be a quick one, at least not for gravitational-wave astronomy.</text>
<text><location><page_17><loc_16><loc_51><loc_76><loc_63></location>One important step along this path is the engagement of the broader (EM) astronomy community. Direct detection of gravitational waves will not merely be a confirmation of a century-old theory-one more feather in Einstein's Indian chief head-dress-but the opening of a window through which we can observe the entire universe at once, eagerly listening for the next thing to go bang in the night. And when it does, all our EM eyes can swing over to watch the fireworks go off. With a tool as powerful as coordinated GW/EM observations, we will be able to answer many of the outstanding questions in astrophysics:</text>
<text><location><page_17><loc_16><loc_39><loc_76><loc_51></location>How were the first black holes formed? Where did the first quasars come from? What is the galaxy merger rate as a function of galaxy mass, mass ratio, gas fraction, cluster environment, and redshift? What is the mass function and spin distribution of the central BHs in these merging (and non-merging) galaxies? What is the central environment around the BHs, prior to merger: What is the quantity and quality (temperature, density, composition) of gas? What is the stellar distribution (age, mass function, metallicity)? What are the properties of the circumbinary disk? What is the time delay between galaxy merger and BH merger?</text>
<text><location><page_17><loc_16><loc_36><loc_76><loc_39></location>These are just a few of the mysteries that will be solved with the routine detection and characterization of SMBH mergers, may we witness them speedily in our days!</text>
<text><location><page_17><loc_16><loc_32><loc_76><loc_34></location>We acknowledge helpful conversations with John Baker, Manuela Campanelli, Bruno Giacomazzo, Bernard Kelly, Julian Krolik, Scott Noble, and Cole Miller.</text>
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[
{
"title": "Jeremy D. Schnittman 1",
"content": "1 NASA Goddard Space Flight Center, Greenbelt, MD 20771 Abstract. We present here an overview of recent work in the subject of astrophysical manifestations of super-massive black hole (SMBH) mergers. This is a field that has been traditionally driven by theoretical work, but in recent years has also generated a great deal of interest and excitement in the observational astronomy community. In particular, the electromagnetic (EM) counterparts to SMBH mergers provide the means to detect and characterize these highly energetic events at cosmological distances, even in the absence of a spacebased gravitational-wave observatory. In addition to providing a mechanism for observing SMBH mergers, EM counterparts also give important information about the environments in which these remarkable events take place, thus teaching us about the mechanisms through which galaxies form and evolve symbiotically with their central black holes. PACS numbers: 95.30.Sf, 98.54.Cm, 98.62.Js, 04.30.Tv, 04.80.Nn Submitted to: Clas. Quant. Grav.",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "Following numerical relativity's annus mirabilis of 2006, a deluge of work has explored the astrophysical manifestations of black hole mergers, from both the theoretical and observational perspectives. While the field has traditionally been dominated by applications to the direct detection of gravitational waves (GWs), much of the recent focus of numerical simulations has been on predicting potentially observable electromagnetic (EM) signatures. Of course, the greatest science yield will come from coincident detection of both the GW and EM signature, giving a myriad of observables such as the black hole mass, spins, redshift, and host environment, all with high precision [29]. Yet even in the absence of a direct GW detection (and this indeed is the likely state of affairs for at least the next decade), the EM signal alone may be sufficiently strong to detect with wide-field surveys, and also unique enough to identify unambiguously as a SMBH merger. In this article, we review the brief history and astrophysical principles that govern the observable signatures of SMBH mergers. To date, the field has largely been driven by theory, but we also provide a summary of the observational techniques and surveys that have been utilized, including recent claims of potential detections of both SMBH binaries and also post-merger recoiling black holes. While the first public use of the term 'black hole' is generally attributed to John Wheeler in 1967, as early as 1964 Edwin Saltpeter proposed that gas accretion onto super-massive black holes provided the tremendous energy source necessary to power the highly luminous quasi-stellar objects (quasars) seen in the centers of some galaxies [216]. Even earlier than that, black holes were understood to be formal mathematical solutions to Einstein's field equations [225], although considered by many to be simply mathematical oddities, as opposed to objects that might actually exist in nature (perhaps most famously, Eddington's stubborn opposition to the possibility of astrophysical black holes probably delayed significant progress in their understanding for decades) [255]. In 1969, Lynden-Bell outlined the foundations for black hole accretion as the basis for quasar power [160]. The steady-state thin disks of Shakura and Sunyaev [234], along with the relativistic modifications given by Novikov and Thorne [182], are still used as the standard models for accretion disks today. In the following decade, a combination of theoretical work and multi-wavelength observations led to a richer understanding of the wide variety of accretion phenomena in active galactic nuclei (AGN) [204]. In addition to the well-understood thermal disk emission predicted by [234, 182], numerous non-thermal radiative processes such as synchrotron and inverseCompton are also clearly present in a large fraction of AGN [186, 71]. Peters and Mathews [196] derived the leading-order gravitational wave emission from two point masses more than a decade before Thorne and Braginsky [254] suggested that one of the most promising sources for such a GW signal would be the collapse and formation of a SMBH, or the (near head-on) collision of two such objects in the center of an active galaxy. In that same paper, Thorne and Braginsky build on earlier work by Estabrook and Wahlquist [76] and explore the prospects for a space-based method for direct detection of these GWs via Doppler tracking of inertial spacecraft. They also attempted to estimate event rates for these generic bursts, and arrived at quite a broad range of possibilities, from ∼ < 0 . 01 to ∼ > 50 events per year, numbers that at least bracket our current best-estimates for SMBH mergers [226]. However it is not apparent that Thorne and Braginsky considered the hierarchical merger of galaxies as the driving force behind these SMBH mergers, a concept that was only just emerging at the time [188, 189]. Within the galactic merger context, the seminal paper by Begelman, Blandford, and Rees (BBR) [18] outlines the major stages of the SMBH merger: first the nuclear star clusters merge via dynamical friction on the galactic dynamical time t gal ∼ 10 8 yr; then the SMBHs sink to the center of the new stellar cluster on the stellar dynamical friction time scale t df ∼ 10 6 yr; the two SMBHs form a binary that is initially only loosely bound, and hardens via scattering with the nuclear stars until the loss cone is depleted; further hardening is limited by the diffusive replenishing of the loss cone, until the binary becomes 'hard,' i.e., the binary's orbital velocity is comparable to the local stellar orbital velocity, at which point the evolutionary time scale is t hard ∼ N inf t df , with N inf stars within the influence radius. This is typically much longer than the Hubble time, effectively stalling the binary merger before it can reach the point where gravitational radiation begins to dominate the evolution. Since r hard ∼ 1 pc, and gravitational waves don't take over until r GW ∼ 0 . 01 pc, this loss cone depletion has become known as the 'final parsec problem' [168]. BBR thus propose that there should be a large cosmological population of stalled SMBH binaries with separation of order a parsec, and orbital periods of years to centuries. Yet to date not a single binary system with these subparsec separations has even been unambiguously identified. In the decades since BBR, numerous astrophysical mechanisms have been suggested as the solution to the final parsec problem [168]. Yet the very fact that so many different solutions have been proposed and continue to be proposed is indicative of the prevailing opinion that it is still a real impediment to the efficient merger of SMBHs following a galaxy merger. However, the incontrovertible evidence that galaxies regularly undergo minor and major mergers during their lifetimes, coupled with a distinct lack of binary SMBH candidates, strongly suggest that nature has found its own solution to the final parsec problem. Or, as Einstein put it, 'God does not care about mathematical difficulties; He integrates empirically.' For incontrovertible evidence of a SMBH binary, nothing can compare with the direct detection of gravitational waves from space. The great irony of gravitationalwave astronomy is that, despite the fact that the peak GW luminosity generated by black hole mergers outshines the entire observable universe , the extremely weak coupling to matter makes both direct and indirect detection exceedingly difficult. For GWs with frequencies less than ∼ 1 Hz, the leading instrumental concept for nearly 25 years now has been a long-baseline laser interferometer with three freefalling test masses housed in drag-free spacecraft [77]. Despite the flurry of recent political and budgetary constraints that have resulted in a number of alternative, less capable designs, we take as our fiducial detector the classic LISA (Laser Interferometer Space Antenna) baseline design [151]. For SMBHs with masses of 10 6 M /circledot at a redshift of z = 1, LISA should be able to identify the location of the source on the sky within ∼ 10 deg 2 a month before merger, and better than ∼ 0 . 1 deg 2 with the entire waveform, including merger and ringdown [129, 143, 144, 130, 145, 253, 164]. This should almost certainly be sufficient to identify EM counterparts with wide-field surveys such as LSST [1], WFIRST [244], or WFXT [178]. Like the cosmological beacons of gamma-ray bursts and quasars, merging SMBHs can teach us about relativity, high-energy astrophysics, radiation hydrodynamics, dark energy, galaxy formation and evolution, and how they all interact. A large variety of potential EM signatures have recently been proposed, almost all of which require some significant amount of gas in the near vicinity of the merging black holes [222]. Thus we must begin with the question of whether or not there is any gas present, and if so, what are its properties. Only then can we begin to simulate realistic spectra and light curves, and hope to identify unique observational signatures that will allow us to distinguish these objects from the myriad of other high-energy transients throughout the universe.",
"pages": [
2,
3,
4
]
},
{
"title": "2. CIRCUMBINARY DISKS",
"content": "If there is gas present in the vicinity of a SMBH binary, it is likely in the form of an accretion disk, as least at some point in the system's history. Disks are omnipresent in the universe for the simple reason that it is easy to lose energy through dissipative processes, but much more difficult to lose angular momentum. At larger separations, before the SMBHs form a bound binary system, massive gas disks can be quite efficient at bringing the two black holes together [75, 65]. As these massive gas disks are typically self-gravitating, their dynamics can be particularly complicated, and require high-resolution 3D simulations, which will be discussed in more detail in section 3.5. Here we focus on the properties of non-self-gravitating circumbinary accretion disks which have traditionally employed the same alpha prescription for pressureviscous stress scaling as in [234]. Much of the early work on this subject was applied to protoplanetary disks around binary stars, or stars with massive planets embedded in their surrounding disks. The classical work on this subject is Pringle (1991) [200], who considered the evolution of a 1D thin disk with an additional torque term added to the inner disk. This source of angular momentum leads to a net outflow of matter, thus giving these systems their common names of 'excretion' or 'decretion' disks. Pringle considered two inner boundary conditions: one for the inflow velocity v r ( R in ) → 0 and one for the surface density Σ( R in ) → 0. For the former case, the torque is applied at a single radius at the inner edge, leading to a surface density profile that increases steadily inwards towards R in . In the latter case, the torque is applied over a finite region in the inner disk, which leads to a relatively large evacuated gap out to ∼ > 6 R in . In both cases, the angular momentum is transferred from the binary outwards through the gas disk, leading to a shrinking of the binary orbit. In [8], SPH simulations were utilized to understand in better detail the torquing mechanism between the gas and disk. They find that, in agreement with the linear theory of [89], the vast majority of the binary torque is transmitted to the gas through the ( l, m ) = (1 , 2) outer Lindblad resonance (for more on resonant excitation of spiral density waves, see [247]). The resonant interaction between the gas and eccentric binary ( e = 0 . 1 for the system in [8]) pumps energy and angular momentum into the gas, which gets pulled after the more rapidly rotating interior point mass. This leads to a nearly evacuated disk inside of r ≈ 2 a , where a is the binary's semimajor axis. The interaction with the circumbinary disk not only removes energy and angular momentum from the binary, but it can also increase its eccentricity, and cause the binary pericenter to precess on a similar timescale, all of which could lead to potentially observable effects in GW observations [6, 211, 212]. In [9, 10], Artymowicz & Lubow expand upon [8] and provide a comprehensive study of the effects of varying the eccentricity, mass ratio, and disk thickness on the behavior of the circumbinary disk and its interaction with the binary. Not surprisingly, they find that the disk truncation radius moves outward with binary eccentricity. Similarly, the mini accretion disks around each of the stars has an outer truncation radius that decreases with binary eccentricity. On the other hand, the location of the inner edge of the circumbinary disk appears to be largely insensitive to the binary mass ratio [9]. For relatively thin, cold disks with aspect ratios H/R ≈ 0 . 03, the binary torque is quite effective at preventing accretion, much as in the decretion disks of Pringle [200]. In that case, the gas accretion rate across the inner gap is as much at 10 -100 × smaller than that seen in a single disk, but the authors acknowledge that the low resolution of the SPH simulation makes these estimates inconclusive [9]. When increasing the disk thickness to H/R ≈ 0 . 1, the gas has a much easier time jumping the gap and streaming onto one of the two stars, typically the smaller one. For H/R ≈ 0 . 1, the gas accretion rate is within a factor of two of the single-disk case [10]. The accretion rate across the gap is strongly modulated at the binary orbital period, although the accretion onto the individual masses can be out of phase with each other. The modulated accretion rate suggests a promising avenue for producing a modulated EM signal in the pre-merger phase, and the very fact that a significant amount of gas can in fact cross the gap is important for setting up a potential prompt signal at the time of merger. To adequately resolve the spiral density waves in a thin disk, 2D grid-based calculations are preferable to the inherently noisy and diffusive SPH methods. Armitage and Natarayan [5] take a hybrid approach to the problem, and use a 2D ZEUS [245] hydrodynamics calculation to normalize the torque term in the 1D radial structure equation. Unlike [8], they find almost no leakage across the gap, even for a moderate H/R = 0 . 07. However, they do identify a new effect that is particularly important for binary black holes, as opposed to protoplanetary disks. For a mass ratio of q ≡ m 2 /m 1 = 0 . 01, when a small accretion disk is formed around the primary, the evolution of the secondary due to gravitational radiation can shrink the binary on such short time scales that it plows into the inner accretion disk, building up gas and increasing the mass accretion rate and thus luminosity immediately preceding merger [5]. If robust, this obviously provides a very promising method for generating bright EM counterparts to SMBH mergers. However, recent 2D simulations by [17] suggest that the gas in the inner disk could actually flow across the gap back to the outer disk, like snow flying over the plow. The reverse of this effect, gas piling up in the outer disk before leaking into the inner disk, has recently been explored by [132, 133]. In the context of T Tau stars, [98, 99] developed a sophisticated simulation tool that combines a polar grid for the outer disk with a Cartesian grid around the binary to best resolve the flow across the gap. They are able to form inner accretion disks around each star, fed by persistent streams from the circumbinary disk. As a test, they compare the inner region to an SPH simulation and find good agreement, but only when the inner disks are artificially fed by some outer source, itself not adequately resolved by the SPH calculation [99]. They also see strong periodic modulation in the accretion rate, due to a relatively large binary eccentricity of e = 0 . 5. MacFadyen and Milosavljevic (MM08) [161] also developed a sophisticated gridbased code including adaptive mesh refinement to resolve the flows at the inner edge of the circumbinary disk in the SMBH binary context. However, they excise the inner region entirely to avoid excessive demands on their resolution around each black hole so are unable to study the behavior of mini accretion disks. They also use an alpha prescription for viscosity and find qualitatively similar results to the earlier work described above: a gap with R in ≈ 2 a due to the m = 2 outer Lindblad resonance, spiral density waves in an eccentric disk, highly variable and periodic accretion, and accretion across the gap of ∼ 20% that expected for a single BH accretion disk with the same mass [161]. The disk surface density as well as the variable accretion rate are shown in Figure 1. Recent work by the same group carried out a systematic study of the effect of mass ratio and found significant accretion across the gap for all values of q = m 2 /m 1 between 0.01 and 1 [64]. The net result of these calculations seems to be that circumbinary gas disks are a viable mechanism for driving the SMBH binary through the final parsec to the GWdriven phase, and supplying sufficient accretion power to be observable throughout. Thus it is particularly perplexing that no such systems have been observed with any degree of certainty. According to simple alpha-disk theory, there should also be a point in the GW evolution where the binary separation is shrinking at such a prodigious rate that the circumbinary disk cannot keep up with it, and effectively decouples from the binary. At that point, gas should flow inwards on the relatively slow timescale corresponding to accretion around a single point mass, and a real gap of evacuated space might form around the SMBHs, which then merge in a near vacuum [174].",
"pages": [
4,
5,
6
]
},
{
"title": "3.1. Vacuum numerical relativity",
"content": "In the context of EM counterparts, the numerical simulation of two equal-mass, nonspinning black holes in a vacuum is just about the simplest problem imaginable. Yet the inherent non-linear behavior of Einstein's field equations made this a nearly unsolvable Grand Challenge problem, frustrating generations of relativists from the 3+1 formulation of Arnowitt, Deser, and Misner in 1962 [7], followed shortly by the first attempt at a numerical relativity (NR) simulation on a computer in 1964 [100], decades of uneven progress, slowed in large part by the limited computer power of the day (but also by important fundamental instabilities in the formulation of the field equations), to the ultimate solution by Pretorius in 2005 [198] and subsequent deluge of papers in 2006 from multiple groups around the world (for a much more thorough review of this colorful story and the many technical challenges overcome by its participants, see [47]). Here we will review just a few highlights from the recent NR results that are most pertinent to our present subject. For the first 50 years since their original conception, black holes (and general relativity as a whole) were largely relegated to mathematicians as a theoretical curiosity with little possibility of application in astronomy. All this changed in the late 1960s and early 70s when both stellar-mass and super-massive black holes were not only observed, but also understood to be critical energy sources and play a major role in the evolution of galaxies and stars [255]. A similar environment was present during the 1990s with regard to binary black holes and gravitational waves. Most believed in their existence, but after decades of false claims and broken promises, the prospect of direct detection of GWs seemed further away than ever. But then in 1999, construction was completed on the two LIGO observatories, and they began taking science data in 2002. At the same time, the space-based LISA concept was formalized with the 'Yellow Book,' a report submitted to ESA in 1996, and together with NASA, an international science team was formed in 2001. Astrophysics theory has long been data-driven, but here was a case where large-scale projects were being proposed and even funded based largely on theoretical predictions. The prospect of real observations and data in turn energized the NR community and provided new motivation to finally solve the binary BH merger problem. Longduration, accurate waveforms are necessary for both the detection and characterization of gravitational waves. Generic binary sources are fully described by 17 parameters: the BH masses (2), spin vectors (6), binary orbital elements (6), sky position (2), and distance (1). To adequately cover this huge parameter space requires exceedingly clever algorithms and an efficient method for calculating waveforms. Fortunately, most NR studies to date suggest that even the most non-linear phase of the inspiral and merger process produces a relatively smooth waveform, dominated by the leading quadrupole mode [47]. Additionally, in the early inspiral and late ringdown phases, relatively simple analytic expressions appear to be quite sufficient in matching the waveforms [194]. Even more encouraging is the fact that waveforms from different groups using very different methods agree to a high level of accuracy, thus lending confidence to their value as a description of the real world [13]. In addition to the waveforms, another valuable result from these first merger simulations was the calculation of the mass and spin of the final black hole, demonstrating that the GWs carried away a full 4% of their initial energy in roughly an orbital time, and leave behind a moderately spinning black hole with a/M = 0 . 7 [11, 44]. After the initial breakthrough with equal-mass, non-spinning black holes, the remarkably robust 'moving puncture' method was soon applied to a wide variety of systems, including unequal masses [21], eccentric orbits [113], and spinning BHs [45]. As with test particles around Kerr black holes, when the spins are aligned with the orbital angular momentum, the BHs can survive longer before plunging, ultimately producing more GW power and resulting in a larger final spin. This is another critical result for astrophysics, as the spin evolution of SMBHs via mergers and gas accretion episodes is a potentially powerful diagnostic of galaxy evolution [22]. Perhaps the most interesting and unexpected result from the NR bonanza was the first accurate calculation of the gravitational recoil, which will be discussed in more detail in the following section. In addition to the widespread moving puncture method, the NR group at Cornell/Caltech developed a highly accurate spectral method that is particularly wellsuited for long evolutions [42]. Because it converges exponentially with resolution (as opposed to polynomial convergence for finite-difference methods), the spectral method can generate waveforms with dozens of GW cycles, accurate to a small fraction of phase. These long waveforms are particularly useful for matching the late inspiral to post-Newtonian (PN) equations of motion, the traditional tool of choice for GW data analysis for LIGO and LISA (e.g., [57, 4, 128, 25]). The down side of the spectral method has been its relative lack of flexibility, making it very time consuming to set up simulations of new binary configurations, particularly with arbitrary spins. If this problem can be overcome, spectral waveforms will be especially helpful in guiding the development of more robust semi-analytic tools (e.g., the effective-one-body approach of Buonanno [16]) for calculating the inspiral, merger, and ringdown of binary BHs with arbitrary initial conditions. The natural application for long, high-accuracy waveforms is as templates in the matched-filtering approach to GW data analysis. For LIGO, this is critical to detect most BH mergers, where much of the in-band power will come from the final stages of inspiral and merger. The high signal-to-noise expected from SMBHs with LISA means that most events will probably be detected with high significance even when using a primitive template library [82, 58]. However, for parameter estimation , highfidelity waveforms are essential for faithfully reproducing the physical properties of the source. In particular, for spinning BHs, the information contained in the precessing waveform can greatly improve our ability to determine the sky position of the source, and thus improve our prospects for detecting and characterizing any EM counterpart [144, 253, 145].",
"pages": [
6,
7,
8
]
},
{
"title": "3.2. Gravitational recoil",
"content": "In the general case where there is some asymmetry between the two black holes (e.g., unequal masses or spins), the GW radiation pattern will have a complicated multipole structure. The beating between these different modes leads to a net asymmetry in the momentum flux from the system, ultimately resulting in a recoil or kick imparted on the final merged black hole [219]. This effect has long been anticipated for any GW source [37, 195, 19], but the specific value of the recoil has been notoriously difficult to calculate using traditional analytic means [266, 81, 24, 59]. Because the vast majority of the recoil is generated during the final merger phase, it is a problem uniquely suited for numerical relativity. Indeed, this was one of the first results published in 2006, for the merger of two non-spinning BHs with mass ratio 3:2, giving a kick of 90 -100 km/s [12]. Shortly thereafter, a variety of initial configurations were explored, covering a range of mass ratios [112, 90], aligned spins [111, 139], and precessing spins [46, 256]. Arguably the most exciting result came with the discovery of the 'superkick' configuration, where two equal-mass black holes have equal and opposite spins aligned in the orbital plane, leading to kicks of > 3000 km/s [91, 46, 256]. If such a situation were realized in nature, the resulting black hole would certainly be ejected from the host galaxy, leaving behind an empty nuclear host [167]. Some of the many other possible ramifications include offset AGN, displaced star clusters, or unusual accretion modes. These and other signatures are discussed in detail below in section 4. Analogous to the PN waveform matching mentioned above, there has been a good deal of analytic modeling of the kicks calculated by the NR simulations",
"pages": [
8
]
},
{
"title": "3.3. Pure electromagnetic fields",
"content": "Shortly after the 2006-07 revolution, many groups already began looking for the next big challenge in numerical relativity. One logical direction was the inclusion of electromagnetic fields in the simulations, solving the coupled Einstein-Maxwell equations throughout a black hole merger. The first to do so was Palenzuela et al. [190], who considered an initial condition with zero electric field and a uniform magnetic field surrounding an equal-mass, non-spinning binary a couple orbits before merger. The subsequent evolution generates E-fields twisted around the two BHs, while the B-field remains roughly vertical, although it does experience some amplification (see Fig. 2). The EM power from this system was estimated by integrating the radial Poynting flux through a spherical shell at large radius. They found only a modest (30 -40%) increase in EM energy, but there was a clear transient quadrupolar Poynting burst of power coincident with the GW signal, giving one of the first hints of astrophysical EM counterparts from NR simulations. This work was followed up by a more thorough study in [175, 191], which showed that the EM power L EM scaled like the square of the total BH spin and proportional to B 2 , as would be expected for a Poynting flux-powered jet [26].",
"pages": [
9
]
},
{
"title": "3.4. Force-free simulations",
"content": "In [192, 193], Palenzuela and collaborators extended their vacuum simulations to include force-free electrodynamics. This is an approximation where a tenuous plasma is present, and can generate currents and magnetic fields, but carries no inertia to push those fields around. They found that any moving, spinning black hole can generate Poynting flux and a Blandford-Znajek-type jet [26]. Compared to the vacuum case, force-free simulations of a merging binary predict significant amplification of EM power by a factor of ∼ 10 × , coincident with the peak GW power [193]. For longer simulations run at higher accuracy, [176, 2] found an even greater L EM amplification of ∼ 30 × that of electro-vacuum.",
"pages": [
9,
10
]
},
{
"title": "3.5. M/HD simulations",
"content": "As mentioned above in section 2, if there is an appreciable amount of gas around the binary BH, it is likely in the form of a circumbinary disk. This configuration has thus been the subject of most (magneto)hydrodynamical simulations. SPH simulations of disks that are not aligned with the binary orbit show a warped disk that can precess as a rigid body, and generally suffer more gas leakage across the inner gap, modulated at twice the orbital frequency [146, 117, 108]. In many cases, accretion disks can form around the individual BHs [65, 104]. Massive disks have the ability to drive the binary towards merger on relatively short time scales [75, 65, 56] and also align the BH spins at the same time [32] (although see also [154, 155] for a counter result). Retrograde disks may be even more efficient at shrinking the binary [179] and they may also be quite stable [180]. Recent simulations by [212] show that the binary will evolve due not only to torques from the circumbinary disk, but also from transfer of angular momentum via gas streaming onto the two black holes. They find that the binary does shrink, and eccentricity can still be excited, but not necessarily at the rates predicted by classical theory. Following merger, the circumbinary disk can also undergo significant disruption due to the gravitational recoil, as well as the sudden change in potential energy due to the mass loss from gravitational waves. These effects lead to caustics forming in the perturbed disk, in turn leading to shock heating and potentially both prompt and long-lived EM afterglows [184, 165, 214, 55, 268, 197, 213, 269]. Any spin alignment would be critically important for both the character of the prompt EM counterpart, as well as the recoil velocity [159, 23]. Due to computational limitations, it is generally only possible to include the last few orbits before merger in a full NR simulation. Since there is no time to allow the system to relax into a quasi-steady state, the specific choice of initial conditions is particularly important for these hydrodynamic merger simulations. Some insight can be gained from Newtonian simulations [239] as well as semi-analytic models [153, 203, 236]. If the disk decouples from the binary well before merger, the gas may be quite hot and diffuse around the black holes [107]. In that case, uniform density diffuse gas may be appropriate. In merger simulations by [78, 30, 35], the diffuse gas experiences Bondi-type accretion onto each of the SMBHs, with a bridge of gas connecting the two before merger. Shock heating of the gas could lead to a strong EM counterpart. As a simple estimate for the EM signal, [35] use bremsstrahlung radiation to predict roughly Eddington luminosity peaking in the hard X-ray band. The first hydrodynamic NR simulations with disk-like initial conditions were carried out by [79] by allowing the disk to relax into a quasi-steady state before turning the GR evolution on. They found disk properties qualitatively similar to classical Newtonian results, with a low-density gap threaded by accretion streams at early times, and largely evacuated at late times when the binary decouples from the disk. Due to the low density and high temperatures in the gap, they estimate the EM power will be dominated by synchrotron (peaking in the IR for M = 10 8 M /circledot ), and reach Eddington luminosity. An analogous calculation was carried out by [31], yet they find EM luminosity orders of magnitude smaller, perhaps because they do not relax the initial disk for as long. Most recently, circumbinary disk simulations have moved from purely hydrodynamic to magneto-hydrodynamic (MHD), which allows them to dispense with alpha prescriptions of viscosity and incorporate the true physical mechanism behind angular momentum in accretion disks: magnetic stresses and the magneto-rotational instability [15]. Newtonian MHD simulations of circumbinary disks find large-scale m = 1 modes growing in the outer disk, modulating the accretion flow across the gap [239]. Similar modes were seen in [181], who used a similar procedure as [79] to construct a quasi-stable state before allowing the binary to merge. They find that the MHD disk is able to follow the inspiraling binary to small separations, showing little evidence for the decoupling predicted by classical disk theory. However, the simulations of [181] use a hybrid space-time based on PN theory [84] that breaks down close to merger. Furthermore, while fully relativistic in its MHD treatment, the individual black holes are excised from the simulation due to computational limitations, making it difficult to estimate EM signatures from the inner flow. Farris et al. [80] have been able to overcome this issue and put the BHs on the grid with the MHD fluid. They find that the disk decouples at a ≈ 10 M , followed by a decrease in luminosity before merger, and then an increase as the gap fills in and resumes normal accretion, as in [174]. Giacomazzo et al. [88] carried out MHD merger simulations with similar initial conditions to both [191] and [30], with diffuse hot gas threaded by a uniform vertical magnetic field. Unlike in the force-free approximation, the inclusion of significant gas leads to a remarkable amplification of the magnetic field, which is compressed by the accreting fluid. [88] found the B-field increased by of a factor of 100 during merger, corresponding to an increase in synchrotron power by a factor of 10 4 , which could easily lead to super-Eddington luminosities from the IR through hard X-ray bands. The near future promises a self-consistent, integrated picture of binary BH-disk evolution. By combining the various methods described above, we can combine multiple MHD simulations at different scales, using the results from one method as initial conditions for another, and evolve a circumbinary disk from the parsec level through merger and beyond.",
"pages": [
10,
11
]
},
{
"title": "3.6. Radiation transport",
"content": "Even with high resolution and perfect knowledge of the initial conditions, the value of the GRMHD simulations is limited by the lack of radiation transport and accurate thermodynamics, which have only recently been incorporated into local Newtonian simulations of steady-state accretion disks [114, 115]. Significant future work will be required to incorporate the radiation transport into a fully relativistic global framework, required not just for accurate modeling of the dynamics, but also for the prediction of EM signatures that might be compared directly with observations. Some recent progress has been made by using the relativistic Monte Carlo raytracing code Pandurata as a post-processor for MHD simulations of single accretion disks [223, 224], reproducing soft and hard X-ray spectral signatures in agreement with observations of stellar-mass black holes. Applying the same ray-tracing approach to the MHD merger simulations of [88], we can generate light curves and broadband spectra, ranging from synchrotron emission in the IR up through inverse- Compton peaking in the X-ray. An example of such a spectrum is shown in Figure 3, corresponding to super-Eddington luminosity at the peak of the EM and GW emission. Since the simulation in [88] does not include a cooling function, we simply estimate the electron temperature as 100 keV, similar to that seen in typical AGN coronas. Future work will explore the effects of radiative cooling within the NR simulations, as well as incorporating the dynamic metric into the ray-tracing analysis. Of course, the ultimate goal will be to directly incorporate radiation transport as a dynamical force within the GRMHD simulations. Significant progress has been made recently in developing accurate radiation transport algorithms in a fully covariant framework [187, 121, 215], and we look forward to seeing them mature to the point where they can be integrated into dynamic GRMHD codes. In addition to Pandurata , there are a number of other relativistic ray-tracing codes (e.g., [63, 237]), currently based on the Kerr metric, which may also be adopted to the dynamic space times of merging black holes.",
"pages": [
11,
12
]
},
{
"title": "4. OBSERVATIONS: PAST, PRESENT, AND FUTURE",
"content": "One way to categorize EM signatures is by the physical mechanism responsible for the emission: stars, hot diffuse gas, or circumbinary/accretion disks. In Figure 4, we show the diversity of these sources, arranged according the spatial and time scales on which they are likely to occur [222]. Over the course of a typical galaxy merger, we should expect the system to evolve from the upper-left to the lower-center to the upper-right regions of the chart. Sampling over the entire observable universe, the number of objects detected in each source class should be proportional to the product of the lifetime and observable flux of that object. Note that most of these effects are fundamentally Newtonian, and many are time since merger only indirect evidence of SMBH mergers, as opposed to the prompt EM signatures described above. Yet they are also important in understanding the complete history of binary BHs, as they are crucial for estimating the number of sources one might expect at each stage in a black hole's evolution. If, for example, we predict a large number of bright binary quasars with separations around 0 . 1 pc, and find no evidence for them in any wide-field surveys (as has been the case so far, with limited depth and temporal coverage), we would be forced to revise our theoretical models. But if the same rate calculations accurately predict the number of dual AGN with separations of ∼ 1 -10 kpc, and GW or prompt EM detections are able to confirm the number of actual mergers, then we might infer the lack of binary quasars is due to a lack of observability, as opposed to a lack of existence. The long-term goal in observing EM signatures will be to eventually fill out a plot like that of Figure 4, determining event rates for each source class, and checking to make sure we can construct a consistent picture of SMBH-galaxy co-evolution. This is indeed an ambitious goal, but one that has met with reasonable success in other fields, such as stellar evolution or even the fossil record of life on Earth.",
"pages": [
12,
13
]
},
{
"title": "4.1. Stellar Signatures",
"content": "On the largest scales, we have strong circumstantial evidence of supermassive BH mergers at the centers of merging galaxies. From large optical surveys of interacting galaxies out to redshifts of z ∼ 1, we can infer that 5 -10% of massive galaxies are merging at any given time, and the majority of galaxies with M gal ∼ > 10 10 M /circledot have experienced a major merger in the past 3 Gyr [20, 162, 61, 43], with even higher merger rates at redshifts z ∼ 1 -3 [54]. At the same time, high-resolution observations of nearby galactic nuclei find that every large galaxy hosts a SMBH in its center [140]. Yet we see a remarkably small number of dual AGN [135, 53], and only one known source with an actual binary system where the BHs are gravitationally bound to each other [209, 210]. Taken together, these observations strongly suggest that when galaxies merge, the merger of their central SMBHs inevitably follows, and likely occurs on a relatively short time scale, which would explain the apparent scarcity of binary BHs (although recent estimates by [106] predict as many as 10% of AGNs with M ∼ 10 7 M /circledot might be in close binaries with a ∼ 0 . 01 pc). The famous 'M-sigma' relationship between the SMBH mass and the velocity dispersion of the surrounding bulge also points to a merger-driven history over a wide range of BH masses and galaxy types [96]. There is additional indirect evidence for SMBH mergers in the stellar distributions of galactic nuclei, with many elliptical galaxies showing light deficits (cores), which correlate strongly with the central BH mass [141]. The cores suggest a history of binary BHs that scour out the nuclear stars via three-body scattering [171, 172, 169], or even post-merger relaxation of recoiling BHs [167, 40, 93, 94]. While essentially all massive nearby galaxies appear to host central SMBHs, it is quite possible that this is not the case at larger redshifts and smaller masses, where major mergers could lead to the complete ejection of the resulting black hole via large recoils. By measuring the occupation fraction of SMBHs in distant galaxies, one could infer merger rates and the distribution of kick velocities [217, 261, 218, 262, 264]. The occupation fraction will of course also affect the LISA event rates, especially at high redshift [226]. Another indirect signature of BH mergers comes from the population of stars that remain bound to a recoiling black hole that gets ejected from a galactic nucleus [136, 170, 183]. These stellar systems will appear similar to globular clusters, yet with smaller spatial extent and much larger velocity dispersions, as the potential is completely dominated by the central SMBH. With multi-object spectrometers on large ground-based telescopes, searching for these stellar clusters in the Milky Way halo or nearby galaxy clusters ( d ∼ < 40 Mpc) is technically realistic in the immediate future.",
"pages": [
13,
14
]
},
{
"title": "4.2. Gas Signatures: Accretion Disks",
"content": "As discussed above in section 2, circumbinary disks will likely have a low-density gap within r ≈ 2 a , although may still be able to maintain significant gas accretion across this gap, even forming individual accretion disks around each black hole. The most sophisticated GRMHD simulations suggest that this accretion can be maintained even as the binary is rapidly shrinking due to gravitational radiation [181]. If the inner disks can survive long enough, the final inspiral may lead to a rapid enhancement of accretion power as the fossil gas is plowed into the central black hole shortly before merger [5, 48]. For small values of q , a narrow gap could form in the inner disk, changing the AGN spectra in a potentially observable way [97, 163]. Regardless of how the gas reaches the central BH region, the simulations described above in section 3 all seem to agree that even a modest amount of magnetized gas can lead to a strong EM signature. If the primary energy source for heating the gas is gravitational [258], then typical efficiencies will be on the order of ∼ 1 -10%, comparable to that expected for standard accretion in AGN, although the much shorter timescales could easily lead to super-Eddington transients, depending on the optical depth and cooling mechanisms of the gas[142]. However, if the merging BHs are able to generate strong magnetic fields [190, 175, 192, 88], then hot electrons could easily generate strong synchrotron flux, or highly relativistic jets may be launched along the resulting BH spin axis, converting matter to energy with a Lorentz boost factor of Γ /greatermuch 1. Even with purely hydrodynamic heating, particularly bright and long-lasting afterglows may be produced in the case of large recoil velocities, which effectively can disrupt the entire disk, leading to strong shocks and dissipation [150, 240, 220, 165, 214, 3, 55, 248, 268]. Long-lived afterglows could be discovered in existing multi-wavelength surveys, but successfully identifying them as merger remnants as opposed to obscured AGN or other bright unresolved sources would require improved pipeline analysis of literally millions of point sources, as well as extensive follow-up observations [220]. For many of these large-kick systems, we may observe quasar activity for millions of years after, with the source displaced from the galactic center, either spatially [125, 156, 263, 52, 70, 122] or spectroscopically [36, 138, 38, 208]. However, large offsets between the redshifts of quasar emission lines and their host galaxies have also been interpreted as evidence of pre-merger binary BHs [34, 66, 252, 69] or due to the large relative velocities in merging galaxies [109, 241, 260, 60], or 'simply' extreme examples of the class of double-peaked emitters, where the line offsets are generally attributed to the disk [85, 72, 242, 51, 86]. An indirect signature for kicked BHs could potentially show up in the statistical properties of active galaxies, in particular in the relative distribution of different classes of AGN in the 'unified model' paradigm [137, 28]. For systems that open up a gap in the circumbinary disk, another EM signature may take the form of a quasar suddenly turning on as the gas refills the gap, months to years after the BH merger [174, 235, 249]. But again, these sources would be difficult to distinguish from normal AGN variability without known GW counterparts. Some limited searches for this type of variability have recently been carried out in the X-ray band [123], but for large systematic searches, we will need targeted time-domain widefield surveys like PTF, Pan-STARRS, and eventually LSST. One of the most valuable scientific products from these time-domain surveys will be a better understanding of what is the range of variability for normal AGN, which will help us distinguish when an EM signal is most likely due to a binary [251]. In addition to the many potential prompt and afterglow signals from merging BHs, there has also been a significant amount of theoretical and observational work focusing on the early precursors of mergers. Following the evolutionary trail in Figure 1, we see that shortly after a galaxy merges, dual AGN may form with typical separations of a few kpc [135, 53], sinking to the center of the merged galaxy on a relatively short timescale ( ∼ < 1 Gyr) due to dynamical friction [18]. The galaxy merger process is also expected to funnel a great deal of gas to the galactic center, in turn triggering quasar activity [110, 126, 116, 92]. At separations of ∼ 1 pc, the BH binary (now 'hardened' into a gravitationally bound system) could stall, having depleted its loss cone of stellar scattering and not yet reached the point of gravitational radiation losses [173]. Gas dynamical drag from massive disks ( M disk /greatermuch M BH ) leads to a prompt inspiral ( ∼ 1 -10 Myr), in most cases able to reach sub-parsec separations, depending on the resolution of the simulation [74, 127, 75, 65, 56, 67, 68]. At this point, a proper binary quasar is formed, with an orbital period of months to decades, which could be identified by periodic accretion [161, 104, 101, 102], density waves in the disk [105], or periodic red-shifted broad emission lines [33, 238, 157, 177]. If these binary AGN systems do in fact exist, spectroscopic surveys should be able to identify many candidates, which may then be confirmed or ruled out with subsequent observations over relatively short timescales ( ∼ 1 -10 yrs), as the line-of-site velocities to the BHs changes by an observable degree. This approach has been attempted with various initial spectroscopic surveys, but as yet, no objects have been confirmed to be binaries by multi-year spectroscopic monitoring [38, 147, 51, 73].",
"pages": [
14,
15,
16
]
},
{
"title": "4.3. Gas Signatures: Diffuse Gas; 'Other'",
"content": "In addition to the many disk-related signatures, there are also a number of potential EM counterparts that are caused by the accretion of diffuse gas in the galaxy. For the Poynting flux generated by the simulations of section 3, transient bursts or modulated jets might be detected in all-sky radio surveys [124, 185]. For BHs that get significant kicks at the time of merger, we expect to see occasional episodes of Bondi accretion as the BH oscillates through the gravitational potential of the galaxy over millions of years, as well as off-center AGN activity [27, 83, 95, 243]. On larger spatial scales, the recoiling BH could also produce trails of over-density in the hot interstellar gas of elliptical galaxies [62]. Also on kpc-Mpc scales, X-shaped radio jets have been seen in a number of galaxies, which could possibly be due to the merger and subsequent spin-flip of the central BHs [166]. Another potential source of EM counterparts comes not from diffuse gas, or accretion disks, but the occasional capture and tidal disruption of normal stars by the merging BHs. These tidal disruption events (TDEs), which also occurs in 'normal' galaxies [205, 134, 103], may be particularly easy to identify in off-center BHs following a large recoil [136]. TDE rates may be strongly increased prior to the merger [49, 246, 233, 221, 50, 265], but the actual disruption signal may be truncated by the pre-merger binary [152], and post-merger recoil may also reduce the rates [149]. These TDE events are likely to be seen by the dozen in coming years with Pan-STARRS and LSST [87]. In addition to the tidal disruption scenario, in [221] we showed how gas or stars trapped at the stable Lagrange points in a BH binary could evolve during inspiral and eventually lead to enhanced star formation, ejected hyper-velocity stars, highly-shifted narrow emission lines, and short bursts of superEddington accretion coincident with the BH merger. A completely different type of EM counterpart can be seen with pulsar timing arrays (PTAs). In this technique, small time delays ( ∼ < 10 ns) in the arrival of pulses from millisecond radio pulsars would be direct evidence of extremely lowfrequency (nano-Hertz) gravitational waves from massive ( ∼ > 10 8 M /circledot ) BH binaries [119, 227, 228, 120, 232, 201, 257, 229]. By cross-correlating the signals from multiple pulsars around the sky, we can effectively make use of a GW detector the size of the entire galaxy. For now, one of the main impediments to GW astronomy with pulsar timing is the relatively small number of known, stable millisecond radio pulsars. Current surveys are working to increase this number and the uniformity of their distribution on the sky [148]. Even conservative estimates suggest that PTAs are probably only about ten years away from a positive detection of the GW stochastic background signal from the ensemble of SMBH binaries throughout the universe [231]. The probability of resolving an individual source is significantly smaller, but if it were detected, would be close enough ( z ∼ < 1) to allow for extensive EM follow-up, unlike many of the expected LISA sources at z ∼ > 5. Also, unlike LISA sources, PTA sources would be at an earlier stage in their inspiral and thus be much longer lived, allowing for even more extensive study. A sufficiently large sample of such sources would even allow us to test whether they are evolving due to GW emission or gas-driven migration [131, 250, 230] (a test that might also be done with LISA with only a single source with sufficient signal-to-noise [267]).",
"pages": [
16,
17
]
},
{
"title": "5. CONCLUSION",
"content": "Black holes are fascinating objects. They push our intuition to the limits, and never cease to amaze us with their extreme behavior. For a high-energy theoretical astrophysicist, the only thing more exciting than a real astrophysical black hole is two black holes, destroying everything in their path as they spiral together towards the point of no return. Thus one can easily imagine the frustration that stems from our lack of ability to actually see such an event, despite the fact that it outshines the entire observable universe. And the path forwards does not appear to be a quick one, at least not for gravitational-wave astronomy. One important step along this path is the engagement of the broader (EM) astronomy community. Direct detection of gravitational waves will not merely be a confirmation of a century-old theory-one more feather in Einstein's Indian chief head-dress-but the opening of a window through which we can observe the entire universe at once, eagerly listening for the next thing to go bang in the night. And when it does, all our EM eyes can swing over to watch the fireworks go off. With a tool as powerful as coordinated GW/EM observations, we will be able to answer many of the outstanding questions in astrophysics: How were the first black holes formed? Where did the first quasars come from? What is the galaxy merger rate as a function of galaxy mass, mass ratio, gas fraction, cluster environment, and redshift? What is the mass function and spin distribution of the central BHs in these merging (and non-merging) galaxies? What is the central environment around the BHs, prior to merger: What is the quantity and quality (temperature, density, composition) of gas? What is the stellar distribution (age, mass function, metallicity)? What are the properties of the circumbinary disk? What is the time delay between galaxy merger and BH merger? These are just a few of the mysteries that will be solved with the routine detection and characterization of SMBH mergers, may we witness them speedily in our days! We acknowledge helpful conversations with John Baker, Manuela Campanelli, Bruno Giacomazzo, Bernard Kelly, Julian Krolik, Scott Noble, and Cole Miller.",
"pages": [
17
]
},
{
"title": "References",
"content": "[152] Liu F K, Li S and Chen X 2009 Astroph. J. Lett. 706 133-137 [184] O'Neill S M, Miller M C, Bogdanovic T, Reynolds C S and Schnittman J D 2009 Astroph. J. 700 859-871 [237] Shcherbakov R V and Huang L 2011 Mon. Not. Royal Astron. Soc. 410 1052",
"pages": [
20,
21,
22
]
}
]
2013CeMDA.115..281M
https://arxiv.org/pdf/1205.6719.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_84><loc_83><loc_87></location>Dynamic Limits on Planar Libration-Orbit Coupling Around an Oblate Primary</section_header_level_1>
<text><location><page_1><loc_12><loc_81><loc_26><loc_82></location>Jay W. McMahon</text>
<text><location><page_1><loc_27><loc_81><loc_43><loc_82></location>· Daniel J. Scheeres</text>
<text><location><page_1><loc_12><loc_72><loc_32><loc_73></location>Received: date / Accepted: date</text>
<text><location><page_1><loc_12><loc_55><loc_88><loc_69></location>Abstract This paper explores the dynamic properties of the planar system of an ellipsoidal satellite in an equatorial orbit about an oblate primary. In particular, we investigate the conditions for which the satellite is bound in librational motion or when the satellite will circulate with respect to the primary. We find the existence of stable equilibrium points about which the satellite can librate, and explore both the linearized and non-linear dynamics around these points. Absolute bounds are placed on the phase space of the libration-orbit coupling through the use of zero-velocity curves that exist in the system. These zerovelocity curves are used to derive a sufficient condition for when the satellite's libration is bound to less than 90 · . When this condition is not satisfied so that circulation of the satellite is possible, the initial conditions at zero libration angle are determined which lead to circulation of the satellite. Exact analytical conditions for circulation and the maximum libration angle are derived for the case of a small satellite in orbits of any eccentricity.</text>
<text><location><page_1><loc_12><loc_51><loc_87><loc_53></location>Keywords Libration · Gravity Gradient · Binary Asteroids · Full Two-body Problem · Libration-Orbit Coupling</text>
<section_header_level_1><location><page_1><loc_12><loc_47><loc_23><loc_48></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_34><loc_88><loc_45></location>The investigation of the translational-rotational coupling for a finite orbiting body, referred to in the literature as the Full Two Body Problem, has received renewed attention in recent years [21,11,10,4,16]. Much of this work has been motivated by interest in the dynamic evolution of binary asteroid systems, which comprise 16% of Near-Earth asteroids [12]. Scheeres [18] studied the stability of bodies resting on one-another which can lead to the formation of a binary asteroid system through rotational fission. Bellerose [3] looked at the dynamics and stability in the planar Full Two Body Problem from an energetic standpoint. Fahenstock [6] studied the problem by modeling the gravitational interaction with polyhedral models, which were applied to simulate the dynamics of the near-Earth binary asteroid 1999 KW4 [7].</text>
<text><location><page_1><loc_12><loc_21><loc_88><loc_34></location>Over time, many simplifications have been made to the Full Two Body Problem in order to make analytical progress in the study of specific applications. The most common simplification is to treat the finite bodies to second-order in their mass properties, which allows for the use of inertia matrices for modeling the attitude dynamics [9,2,19,3]. Analytical expressions have been constructed to fourth order [20], however most analytical studies stop at second order. A second simplification comes by reducing the problem to the planar problem. If the second order simplification is made on mass properties, then the in orbit plane libration angle (commonly referred to as the pitch angle) dynamics are decoupled from the out-of-plane (roll/yaw) dynamics if they are initially quiescent [9]. This reduces the dimensionality of the problem from six to three degrees-of-freedom. The third major simplification that is often seen, particularly in the spacecraft community, is decoupling the translational and rotational dynamics [9,2]. In</text>
<text><location><page_1><loc_12><loc_14><loc_23><loc_15></location>Daniel J. Scheeres</text>
<text><location><page_2><loc_12><loc_86><loc_88><loc_91></location>this case, it is assumed that the mass of the orbiting body is insignificant compared to the primary, so that the perturbation to the orbital dynamics is negligible. The orbital dynamics are known from the given Keplerian motion, and these are used as inputs to the attitude dynamics. A notable work which explores the effects of the coupling for the case of spacecraft sized objects was carried out by Mohan [13].</text>
<text><location><page_2><loc_12><loc_76><loc_88><loc_86></location>In this paper, we analyze the problem of a triaxial body in orbit about a spherical primary, or in an equatorial orbit about an oblate primary. These bodies are represented to second order with their moments of inertia, but all coupling between the translation and attitude motions for this case are preserved. This model can be used as a first order representation of common dynamic problems including binary asteroids (secondaries tend to be in near equatorial orbits of oblate primaries [14]), spacecraft orbiters, and planetmoon systems. This paper is closely related to those by Scheeres [19] and Bellerose [3], however we extend the foundation laid by those works.</text>
<text><location><page_2><loc_12><loc_70><loc_88><loc_76></location>The main contribution of this paper is the study of the limits of librational motion for a given system. There is a small amount of literature dedicated to bounding librational motion. Auelmann [1] laid out the bounding conditions for an axially symmetric spacecraft in circular orbit, using the uncoupled attitude dynamics with a zero spin rate about the axis of symmetry. Pringle [15] extended this idea, analyzing the equilibrium cases and librational bounds for all spin states.</text>
<text><location><page_2><loc_12><loc_55><loc_88><loc_69></location>This paper only looks at the planar libration, however we account for coupling, triaxial shapes, and are not limited to circular orbits, or indeed even Keplerian orbits due to the perturbations from the attitude coupling. The methodology used to bound the libration is initially similar to [1,15], in that we use the energy to limit the amount of libration that is dynamically possible by computing zero-velocity curves for the system. We extend this, however, to cases when the energy is high enough so that circulation is possible. In this case, we show that although the energy zero-velocity curve does not bound the libration angle, certain initial conditions can lead to trajectories which have bounded maximum libration angles. In particular, for the case where the ellipsoid is very small compared to the other body (such as a spacecraft in Earth orbit), we can derive analytical expressions which predict the maximum libration amplitude for an orbit of any eccentricity. For cases with significant coupling, we present a method of analysis which determines the structure of bound and unbound trajectories.</text>
<text><location><page_2><loc_12><loc_48><loc_88><loc_54></location>The paper is organized as follows. Section 2 reviews the mathematical description of the system in question as originally derived by Scheeres [19]. Section 3 discusses the location and properties of equilibria in the system. Section 4 presents a sufficient condition for bounded librational motion. The fact that this is only a sufficient condition is shown in Section 5. Finally, Section 6 presents the main results of the paper which describe conditions for unbound libration.</text>
<section_header_level_1><location><page_2><loc_12><loc_43><loc_36><loc_44></location>2 Physical System Description</section_header_level_1>
<text><location><page_2><loc_12><loc_38><loc_88><loc_42></location>In general, we begin with the planar full two-body problem where both bodies are finite, but constrained to have their individual rotation poles perpendicular to the mutual orbit plane. This situation is illustrated in Fig. 1 with the angles of interest and radius defined.</text>
<figure>
<location><page_2><loc_35><loc_15><loc_65><loc_36></location>
<caption>Fig. 1 Definition of the system angles and relative radial vector.</caption>
</figure>
<text><location><page_3><loc_12><loc_87><loc_88><loc_91></location>In this section we present the key mathematical relationships used to describe this system, which are simplifications of the general relationships derived in [19]. Further details of the derivations are available in the Appendix.</text>
<section_header_level_1><location><page_3><loc_12><loc_83><loc_29><loc_84></location>2.1 Equations of Motion</section_header_level_1>
<text><location><page_3><loc_12><loc_79><loc_88><loc_81></location>Scheeres [19] showed that in this case, the potential energy using a second order expansion in the moments of inertia is,</text>
<formula><location><page_3><loc_25><loc_73><loc_88><loc_78></location>V ( r, φ 1 , φ 2 ) = -GM 1 M 2 r { 1 + 1 2 r 2 [ Tr ( I ' 1 ) +Tr ( I ' 2 ) -3 2 ( I ' 1 x + I ' 1 y -( I ' 1 y -I ' 1 x ) cos 2 φ 1 + I ' 2 x + I ' 2 y -( I ' 2 y -I ' 2 x ) cos 2 φ 2 ) ]} (1)</formula>
<text><location><page_3><loc_12><loc_67><loc_88><loc_71></location>where I ' i is the mass normalized (signified by the prime) inertia dyad of body i , and the subscripts on the non-bold versions indicate a principle moment of inertia. G is the gravitational constant, M i the mass of body i , Tr() indicates the trace of the dyad.</text>
<text><location><page_3><loc_12><loc_60><loc_88><loc_67></location>In this paper, we will normalize the lengths with respect to the maximum ellipsoid semi-axis, α ; the time will be normalized by the mean motion of the system at this distance, n = √ G ( M 1 + M 2 ) /α 3 (units of 1/s); and the mass will be normalized by the ellipsoid mass, M 2 . Note that using these normalization factors, the normalized value of µ = G ( M 1 + M 2 ) is 1. All equations from this point on will be written in normalized units.</text>
<text><location><page_3><loc_15><loc_59><loc_43><loc_60></location>First, let us define the reduced mass as,</text>
<formula><location><page_3><loc_45><loc_55><loc_88><loc_58></location>m = M 1 M 2 M 1 + M 2 (2)</formula>
<text><location><page_3><loc_12><loc_53><loc_30><loc_54></location>the mass fraction is then,</text>
<formula><location><page_3><loc_45><loc_50><loc_88><loc_53></location>ν = M 1 M 1 + M 2 (3)</formula>
<text><location><page_3><loc_12><loc_49><loc_85><loc_50></location>which is just the reduced mass in normalized units. Next, we define the general moment of inertia as,</text>
<formula><location><page_3><loc_45><loc_46><loc_88><loc_48></location>I z = I 2 z + νr 2 (4)</formula>
<text><location><page_3><loc_12><loc_42><loc_88><loc_45></location>Note that the general moment of inertia is a function of r . The normalized moment of intertias are computed as</text>
<formula><location><page_3><loc_43><loc_40><loc_88><loc_43></location>I 2 z = I 2 z M 2 α 2 = I ' 2 z α 2 (5)</formula>
<formula><location><page_3><loc_43><loc_36><loc_88><loc_39></location>I 1 z = I 1 z M 1 α 2 = I ' 1 z α 2 (6)</formula>
<text><location><page_3><loc_12><loc_33><loc_88><loc_35></location>For a system with an oblate primary, we have that I 1 x = I 1 y = I s , where I s < I 1 z , so the potential becomes in normalized units,</text>
<formula><location><page_3><loc_21><loc_29><loc_88><loc_32></location>V ( r, φ 2 ) = -ν r { 1 + 1 2 r 2 [ ( I 1 z -I s ) -1 2 I 2 x -1 2 I 2 y + I 2 z + 3 2 ( I 2 y -I 2 x ) cos 2 φ 2 ]} (7)</formula>
<text><location><page_3><loc_12><loc_25><loc_88><loc_28></location>In the case of a spherical primary, which means that I 1 x = I 1 y = I 1 z , and therefore the potential simplifies to,</text>
<formula><location><page_3><loc_24><loc_23><loc_88><loc_25></location>V ( r, φ 2 ) = -ν r { 1 + 1 2 r 2 [ -1 2 I 2 x -1 2 I 2 y + I 2 z + 3 2 ( I 2 y -I 2 x ) cos 2 φ 2 ]} (8)</formula>
<text><location><page_3><loc_12><loc_18><loc_88><loc_22></location>Note that in both the spherical and oblate primary case, the spherical symmetry causes the potential energy to no longer be dependent on the primary's orientation, as represented in the planar problem by φ 1 . Starting from Scheeres' [19] expression, in this case the kinetic energy is,</text>
<formula><location><page_3><loc_28><loc_14><loc_88><loc_17></location>T = 1 2 I 1 z M 1 M 2 ˙ θ 2 1 + 1 2 I 2 z ˙ φ 2 2 + 1 2 ν ˙ r 2 + 1 2 ( I 2 z + νr 2 ) ˙ θ 2 + I 2 z ˙ φ 2 ˙ θ (9)</formula>
<text><location><page_3><loc_12><loc_12><loc_75><loc_13></location>where ˙ θ 1 = ˙ φ 1 + ˙ θ . Note that the dot indicates derivatives with respect to unit-less time.</text>
<section_header_level_1><location><page_4><loc_12><loc_90><loc_39><loc_91></location>2.2 Integrals of Motion for the System</section_header_level_1>
<text><location><page_4><loc_12><loc_85><loc_88><loc_88></location>The system described in Section 2.1 has three integrals of motion that can be used to simplify the problem: the Jacobi constant (or total energy), the total angular momentum ( K tot ), and the inertial angular velocity of the primary ( ˙ θ 1 ). That these quantities are integrals is shown in the Appendix.</text>
<text><location><page_4><loc_15><loc_83><loc_66><loc_84></location>Using the integrals, the kinetic energy from Eq. (9) can be rewritten as</text>
<formula><location><page_4><loc_35><loc_80><loc_88><loc_82></location>T = T 1 + 1 2 I 2 z ˙ φ 2 2 + 1 2 ν ˙ r 2 + 1 2 I z ˙ θ 2 + I 2 z ˙ φ 2 ˙ θ (10)</formula>
<text><location><page_4><loc_12><loc_76><loc_88><loc_79></location>where T 1 = ( M 1 / (2 M 2 )) I 1 z ˙ θ 2 1 is the kinetic energy of the primary, which is constant. Likewise, we can define the free angular momentum from Eq. (11) to be,</text>
<formula><location><page_4><loc_40><loc_74><loc_88><loc_75></location>K = K tot -K 1 = I z ˙ θ + I 2 z ˙ φ 2 (11)</formula>
<text><location><page_4><loc_12><loc_69><loc_88><loc_73></location>where K 1 = ( M 1 /M 2 ) I 1 z ˙ θ 1 is the angular momentum of the primary, which is constant. This relationship between the free angular momentum and the orbit angular velocity will allow us to eliminate ˙ θ from the system. Solving Eq. (11) for the angular velocity gives,</text>
<formula><location><page_4><loc_45><loc_65><loc_88><loc_67></location>˙ θ = K -I 2 z ˙ φ 2 I z (12)</formula>
<text><location><page_4><loc_12><loc_63><loc_66><loc_64></location>Substituting Eq. (12) into Eq. (10) allows us to write the kinetic energy as,</text>
<formula><location><page_4><loc_37><loc_59><loc_88><loc_62></location>T = T 1 + 1 2 K 2 I z + 1 2 ν ˙ r 2 + 1 2 I 2 z νr 2 ˙ φ 2 2 I z (13)</formula>
<text><location><page_4><loc_12><loc_57><loc_53><loc_58></location>so that the total energy of the system can be written as,</text>
<formula><location><page_4><loc_32><loc_53><loc_88><loc_56></location>E tot = T 1 + 1 2 K 2 I z + 1 2 ν ˙ r 2 + 1 2 I 2 z νr 2 ˙ φ 2 2 I z + V ( r, φ 2 ) (14)</formula>
<text><location><page_4><loc_12><loc_51><loc_58><loc_52></location>or alternatively the free energy of the system can be written as,</text>
<formula><location><page_4><loc_31><loc_47><loc_88><loc_50></location>E = E tot -T 1 = 1 2 K 2 I z + 1 2 ν ˙ r 2 + 1 2 I 2 z νr 2 ˙ φ 2 2 I z + V ( r, φ 2 ) (15)</formula>
<text><location><page_4><loc_12><loc_44><loc_88><loc_46></location>It is interesting to note that the description of the free energy of the system does not require any knowledge about the primary spin state as neither φ 1 or ˙ φ 1 appear anywhere in Eq. (15).</text>
<section_header_level_1><location><page_4><loc_12><loc_40><loc_33><loc_41></location>2.3 Dynamic System Analysis</section_header_level_1>
<text><location><page_4><loc_12><loc_36><loc_88><loc_38></location>In order to look at the stability of any relative states for the planar system, we derive the dynamic matrix corresponding to the equations of motion</text>
<formula><location><page_4><loc_45><loc_33><loc_88><loc_35></location>A = ∂ f ( q , ˙ q ) ∂ [ q , ˙ q ] (16)</formula>
<text><location><page_4><loc_12><loc_31><loc_31><loc_32></location>where q = [ r θ φ 1 φ 2 ].</text>
<text><location><page_4><loc_12><loc_27><loc_88><loc_31></location>First, however, we note that we can reduce the order of the system by recalling from Section 2.2 that φ 1 is ignorable, and that through the conservation of angular momentum we can remove θ through the use of Eq. (12). Therefore the equations of motion for the 2 degree-of-freedom system are,</text>
<formula><location><page_4><loc_40><loc_23><loc_88><loc_26></location>r = ( K -I 2 z ˙ φ 2 ) 2 r I 2 z -1 ν ∂V ∂r (17)</formula>
<formula><location><page_4><loc_34><loc_19><loc_88><loc_22></location>¨ φ 2 = -( 1 + νr 2 I 2 z ) 1 νr 2 ∂V ∂φ 2 + 2˙ r ( K -I 2 z ˙ φ 2 ) rI z (18)</formula>
<text><location><page_4><loc_12><loc_17><loc_58><loc_18></location>This system is depicted from an ellipsoid fixed frame in Fig. (2).</text>
<text><location><page_4><loc_12><loc_12><loc_88><loc_17></location>This frame rotates with the orbit, so that if there is no libration of the secondary, the primary location will be fixed. When the secondary is librating, the primary will appear to move relative to the secondary due to the libration, as well as changing the separation distance as energy is traded between the orbit and the secondary libration.</text>
<figure>
<location><page_5><loc_27><loc_74><loc_73><loc_91></location>
<caption>Fig. 2 Definition of the secondary fixed relative frame.</caption>
</figure>
<text><location><page_5><loc_15><loc_68><loc_55><loc_69></location>The dynamic matrix is now a 4x4 matrix with the form,</text>
<formula><location><page_5><loc_41><loc_61><loc_88><loc_67></location>A = 0 0 1 0 0 0 0 1 ∂ r ∂r ∂ r ∂φ 2 0 ∂ r ∂ ˙ φ 2 ∂ ¨ φ 2 ∂r ∂ ¨ φ 2 ∂φ 2 ∂ ¨ φ 2 ∂ ˙ r ∂ ¨ φ 2 ∂ ˙ φ 2 (19)</formula>
<text><location><page_5><loc_12><loc_57><loc_88><loc_60></location>where the state vector has been organized as, x = [ r φ 2 ˙ r ˙ φ 2 ]. This matrix will be used in the following sections to analyze the stability of the system around any relative equilibrium points that may exist.</text>
<section_header_level_1><location><page_5><loc_12><loc_54><loc_32><loc_55></location>3 Equilibrium Conditions</section_header_level_1>
<text><location><page_5><loc_12><loc_45><loc_88><loc_52></location>In this section, we determine the relative equilibria between the two bodies. This means that these equilibria are places where the relative dynamics given by Eqs. (17) and (18) are stationary. However, the primary is free to rotate at a constant rate, and the secondary (ellipsoidal) body will orbit at a constant radius and orbit rate ( ˙ θ ). As in Section 2, many of the relationships given here are simplifications of the general relationships derived in [19]. Further details are available in the Appendix.</text>
<section_header_level_1><location><page_5><loc_12><loc_42><loc_41><loc_43></location>3.1 Relative Equilibrium Point Locations</section_header_level_1>
<text><location><page_5><loc_12><loc_39><loc_74><loc_40></location>The relative equilibrium locations are found by solving the following polynomial for r ,</text>
<formula><location><page_5><loc_26><loc_32><loc_88><loc_38></location>r 6 -K 2 ν 2 r 5 + [ 2 I 2 z ν + 3 2 ( C ± 2 + I 1 z -I s ) ] r 4 + [ I 2 2 z ν 2 + 3 I 2 z ν ( C ± 2 + I 1 z -I s ) ] r 2 + 3 2 I 2 2 z ν 2 ( C ± 2 + I 1 z -I s ) = 0 (20)</formula>
<text><location><page_5><loc_12><loc_30><loc_16><loc_31></location>where</text>
<formula><location><page_5><loc_35><loc_27><loc_88><loc_30></location>C ± 2 = { -2 I 2 x + I 2 y + I 2 z if φ 2 = 0, π I 2 x -2 I 2 y + I 2 z if φ 2 = ± π/ 2 (21)</formula>
<text><location><page_5><loc_12><loc_23><loc_88><loc_27></location>Note that the relative equilibria found here is identical to finding the equilibria of the second-order system given in Eqs. (17) and (18). The stationary conditions for energy also enforce that ˙ r = r = ˙ φ 2 = ¨ φ 2 = 0.</text>
<text><location><page_5><loc_12><loc_19><loc_88><loc_22></location>It is interesting to compare this result to the full case from Scheeres [19] when there is no assumption on the shape of the primary. In that case we have a dependence on φ 1 , and the stationary conditions also tell us that ˙ φ 1 = 0 and φ 1 = 0, ± π/ 2, or π . Recalling the equation from Scheeres [19],</text>
<formula><location><page_5><loc_21><loc_12><loc_88><loc_18></location>r 6 -K 2 tot m 2 µ r 5 + [ 2( I 1 z + I 2 z ) m + 3 2 ( C ± 1 + C ± 2 ) ] r 4 + [ ( I 1 z + I 2 z ) 2 m 2 + 3( I 1 z + I 2 z ) m ( C ± 1 + C ± 2 ) ] r 2 + 3 2 ( I 1 z + I 2 z ) 2 m 2 ( C ± 1 + C ± 2 ) = 0 (22)</formula>
<text><location><page_6><loc_12><loc_82><loc_88><loc_91></location>The differences between Eq. (20) and (22) (aside from the normalization) are that for the case when the primary is not exactly symmetric about its rotation axis, we must consider the entire angular momentum, K tot , and therefore we also see I 1 z appearing in terms with I 2 z . Also, as the primary body becomes spherically symmetric, the dependence on the primary orientation disappears and C ± 1 → I 1 z -I s . This causes the system to decrease the total number of equilibrium configurations from 8 to 4, and since all the dependence on the primary has been removed, the primary can also be spinning at an arbitrary speed, ˙ φ 1 , and the system can still be in a relative equilibrium.</text>
<text><location><page_6><loc_12><loc_79><loc_88><loc_82></location>The location of the equilibrium points depends on three main parameters: the mass ratio, the angular momentum, and the body shapes. The influences of these three parameters are studied in Section 3.5.</text>
<text><location><page_6><loc_12><loc_75><loc_88><loc_79></location>Throughout the remainder of the paper, any reference to a nominal system refers to the system outlined in Table 1, which are based on the binary asteroid system 1999 KW4 [14,7]. The new parameter that appears in Table 1 is</text>
<formula><location><page_6><loc_47><loc_73><loc_88><loc_75></location>χ = R p R e (23)</formula>
<text><location><page_6><loc_12><loc_70><loc_88><loc_72></location>which is the ratio of the polar radius to the equatorial radius of the oblate primary body. This parameter becomes important for scaling the system.</text>
<text><location><page_6><loc_12><loc_66><loc_59><loc_67></location>Table 1 The non-dimensionalized parameters for the nominal test system.</text>
<formula><location><page_6><loc_18><loc_63><loc_82><loc_65></location>ν β γ R p R e χ I 1 z I s I 2 x I 2 y I 2 z 0.9257 0.8109 0.6112 2.3590 2.6182 0.9010 2.4034 2.1175 0.1973 0.2913 0.3434</formula>
<text><location><page_6><loc_12><loc_54><loc_88><loc_60></location>The nominal system has an angular momentum of K = 2 . 8382, which results in equilibria at r + = 9 . 2442 with E + = -0 . 0497, and r -= 9 . 2869 with E -= -0 . 0496 where the superscripts indicate which equilibrium point we are referring to; the + indicates the equilibrium point at φ 2 = 0, and the -refers to the equilibrium at φ 2 = 90 · , as with C ± 2 .</text>
<section_header_level_1><location><page_6><loc_12><loc_50><loc_46><loc_51></location>3.2 Equilibrium Point Linear Stability Analysis</section_header_level_1>
<text><location><page_6><loc_12><loc_46><loc_88><loc_49></location>Given that the linearized system near the equilibrium point is simplified significantly, we can find the eigenvalues of the system analytically. The dynamics matrix around the equilibrium points has the form,</text>
<formula><location><page_6><loc_44><loc_40><loc_88><loc_45></location>A = 0 0 1 0 0 0 0 1 P 0 0 Q 0 R S 0 (24)</formula>
<text><location><page_6><loc_12><loc_36><loc_88><loc_39></location>where the non-zero partials ( ∂ r ∂r , ∂ r ∂ ˙ φ 2 , ∂ ¨ φ 2 ∂φ 2 , and ∂ ¨ φ 2 ∂ ˙ r ) have been represented by simpler variable expressions ( P , Q , R , and S ). The characteristic equation for this matrix is simply,</text>
<formula><location><page_6><loc_39><loc_33><loc_88><loc_35></location>λ 4 -( P + R + QS ) λ 2 + PR = 0 (25)</formula>
<text><location><page_6><loc_12><loc_31><loc_34><loc_32></location>The roots of this equation are,</text>
<formula><location><page_6><loc_34><loc_27><loc_88><loc_30></location>λ 2 = P + R + QS ± √ ( P + R + QS ) 2 -4 PR 2 (26)</formula>
<text><location><page_6><loc_12><loc_24><loc_88><loc_26></location>The determination of the eigenvalues can easily be carried out numerically on a case-by-case basis to determine the dynamic behavior around the equilibrium point.</text>
<text><location><page_6><loc_15><loc_23><loc_79><loc_24></location>The eigenvectors can also be computed analytically by writing the eigenvalue problem as,</text>
<formula><location><page_6><loc_45><loc_20><loc_88><loc_21></location>[ λ I -A ] v = 0 (27)</formula>
<text><location><page_6><loc_12><loc_16><loc_72><loc_18></location>where the eigenvector is v = [ v 1 v 2 v 3 v 4 ] T . The four equations that result are,</text>
<formula><location><page_6><loc_46><loc_14><loc_88><loc_15></location>λv 1 -v 3 = 0 (28)</formula>
<formula><location><page_6><loc_46><loc_12><loc_88><loc_13></location>λv 2 -v 4 = 0 (29)</formula>
<formula><location><page_7><loc_42><loc_90><loc_88><loc_91></location>-Pv 1 + λv 3 -Qv 4 = 0 (30)</formula>
<formula><location><page_7><loc_42><loc_88><loc_88><loc_89></location>-Rv 2 = Sv 3 + λv 4 = 0 (31)</formula>
<text><location><page_7><loc_12><loc_86><loc_58><loc_87></location>The eigenvector can be computed from these relationships to be</text>
<formula><location><page_7><loc_46><loc_80><loc_88><loc_85></location>v = 1 σ λ σλ (32)</formula>
<text><location><page_7><loc_12><loc_77><loc_16><loc_78></location>where</text>
<formula><location><page_7><loc_46><loc_75><loc_88><loc_78></location>σ = λ 2 -P Qλ (33)</formula>
<text><location><page_7><loc_15><loc_73><loc_65><loc_74></location>In the nominal system, the eigenvalues of the equilibrium point r + are</text>
<formula><location><page_7><loc_44><loc_71><loc_88><loc_72></location>λ 1 , 2 = 0 ± 0 . 0302 i (34)</formula>
<formula><location><page_7><loc_44><loc_69><loc_88><loc_70></location>λ 3 , 4 = 0 ± 0 . 0362 i (35)</formula>
<text><location><page_7><loc_12><loc_64><loc_88><loc_68></location>which makes it a center point. However, we will refer to this equilibrium point as stable in the remainder of the study since trajectories can be bound around it, as will be explored. The eigenvalues of the equilibrium point at r -are</text>
<formula><location><page_7><loc_45><loc_62><loc_88><loc_63></location>λ 1 , 2 = ± 0 . 0307 (36)</formula>
<formula><location><page_7><loc_44><loc_60><loc_88><loc_61></location>λ 3 , 4 = 0 ± 0 . 0351 i (37)</formula>
<text><location><page_7><loc_12><loc_55><loc_88><loc_59></location>This equilibrium point has a unstable and stable asymptote associated with the real eigenvalues. It is interesting to note that the eigenstructure of this problem is nearly identical to that of the collinear Lagrange points in the restricted three-body problem studied by Conley [5].</text>
<section_header_level_1><location><page_7><loc_12><loc_51><loc_33><loc_52></location>3.3 Equilibrium Point Energy</section_header_level_1>
<text><location><page_7><loc_12><loc_45><loc_88><loc_49></location>It has been shown in Section 3.1 that the equilibrium points are stationary points for the total energy. In this section, we determine if these stationary points are local maxima, minima, or saddle points for the energy. To find this, we must investigate the second derivatives of the energy with respect to the state,</text>
<formula><location><page_7><loc_38><loc_38><loc_88><loc_44></location>E xx = E rr E rφ 2 E r ˙ r E r ˙ φ 2 E φ 2 r E φ 2 φ 2 E φ 2 ˙ r E φ 2 ˙ φ 2 E ˙ rr E ˙ rφ 2 E ˙ r ˙ r E ˙ r ˙ φ 2 E ˙ φ 2 r E ˙ φ 2 φ 2 E ˙ φ 2 ˙ r E ˙ φ 2 ˙ φ 2 (38)</formula>
<text><location><page_7><loc_12><loc_36><loc_47><loc_37></location>where the subscripts indicate partial derivatives.</text>
<text><location><page_7><loc_15><loc_35><loc_63><loc_36></location>Through investigation of Eqs. (123) - (125), it is quickly clear that,</text>
<formula><location><page_7><loc_36><loc_32><loc_88><loc_34></location>E r ˙ r = E r ˙ φ 2 = E φ 2 ˙ r = E φ 2 ˙ φ 2 = E ˙ r ˙ φ 2 = 0 (39)</formula>
<text><location><page_7><loc_12><loc_28><loc_88><loc_31></location>The only cross second derivative remaining is with respect to r and φ 2 . Looking back at the energy in Eq. (15), we see that the only part of the energy which contains both of these variables is the potential. Therefore,</text>
<formula><location><page_7><loc_43><loc_25><loc_88><loc_27></location>∂ 2 E ∂r∂φ 2 = ∂ 2 V ∂r∂φ 2 = 0 (40)</formula>
<text><location><page_7><loc_12><loc_19><loc_88><loc_24></location>where the equality to zero at the equilibrium point was shown in Eq. (137). We have shown that all of the cross partials are zero, so that E xx is a diagonal matrix. The diagonal entries of the matrix are the eigenvalues, and the definiteness of the matrix is found by looking at the sign of the eigenvalues. The diagonal entries are,</text>
<formula><location><page_7><loc_28><loc_15><loc_88><loc_18></location>E rr = K 2 ν I 3 z ( 2 νr 2 eq -I 2 z ) -2 ν r 3 eq { 1 + 3 r 2 eq [ ( I 1 z -I s ) + C ± 2 ]} (41)</formula>
<formula><location><page_7><loc_38><loc_11><loc_88><loc_14></location>E φ 2 φ 2 = V φ 2 φ 2 = ± 3 ν r 3 eq ( I 2 y -I 2 x ) (42)</formula>
<formula><location><page_8><loc_47><loc_90><loc_88><loc_91></location>E ˙ r ˙ r = ν (43)</formula>
<formula><location><page_8><loc_44><loc_87><loc_88><loc_89></location>E ˙ φ 2 ˙ φ 2 = I 2 z νr 2 eq I z (44)</formula>
<text><location><page_8><loc_12><loc_81><loc_88><loc_86></location>These results are straight forward to interpret. The velocity partials, E ˙ r ˙ r and E ˙ φ 2 ˙ φ 2 are always greater than zero. The angle partial, E φ 2 φ 2 , is positive for the equilibrium points at φ 2 = 0 or π , and is negative for the equilibrium points at φ 2 = ± π/ 2. The position partial, E rr , is much more complicated and can be positive or negative depending on the system parameters and location of the equilibrium point.</text>
<text><location><page_8><loc_12><loc_73><loc_88><loc_81></location>Combined, this tells us that an equilibrium point at φ 2 = 0 or π can be a local minimum or a saddle point depending on the sign of E rr . An equilibrium point at φ 2 = ± π/ 2 is always a saddle point. The fact that the only energetically stable solutions are found at φ 2 = 0 solutions was first shown by [17]. In the nominal case, r + is energetically stable, and is a local minimum in energy. The other equilibrium point, r -, is an energetic saddle, being a local minimum in the radial direction, but a local maximum in the φ 2 direction.</text>
<section_header_level_1><location><page_8><loc_12><loc_69><loc_45><loc_70></location>3.4 Osculating Orbit Elements at Equilibrium</section_header_level_1>
<text><location><page_8><loc_12><loc_62><loc_88><loc_67></location>In considering a planar problem the orbital elements of interest are the semi-major axis, eccentricity, true anomaly, and argument of periapse. The behavior of these elements at equilibrium are discussed in this section. The other orbital elements, namely inclination and argument of the node are effectively meaningless, and won't be discussed here since they are either constant, undefined, and/or always zero.</text>
<text><location><page_8><loc_15><loc_61><loc_60><loc_62></location>We can derive the Keplerian energy and angular momentum as,</text>
<formula><location><page_8><loc_38><loc_58><loc_88><loc_60></location>E K = 1 2 v 2 -1 r = 1 2 r 2 ˙ θ 2 + 1 2 ˙ r 2 -1 r (45)</formula>
<formula><location><page_8><loc_40><loc_56><loc_88><loc_57></location>H K = | H K | = | r × v | = r 2 ˙ θ (46)</formula>
<text><location><page_8><loc_12><loc_54><loc_51><loc_55></location>where v is the inertial velocity vector. At equilibrium,</text>
<formula><location><page_8><loc_36><loc_49><loc_88><loc_53></location>˙ θ = K I z = √ √ √ √ √ 1 r 3 1 + 3 ( I 1 z -I s + C ± 2 ) 2 r 2 (47)</formula>
<text><location><page_8><loc_12><loc_45><loc_88><loc_47></location>which was derived from Eqs. (12) and by setting (126) equal to zero as at equilibrium. The Keplerian energy and angular momentum at equilibrium are then determined as,</text>
<formula><location><page_8><loc_37><loc_41><loc_88><loc_44></location>E K = -1 2 r 1 -3 ( I 1 z -I s + C ± 2 ) 2 r 2 (48)</formula>
<formula><location><page_8><loc_37><loc_35><loc_88><loc_39></location>H K = √ √ √ √ √ r 1 + 3 ( I 1 z -I s + C ± 2 ) 2 r 2 (49)</formula>
<text><location><page_8><loc_12><loc_32><loc_88><loc_34></location>Then the osculating semi-major axis and eccentricity can be computed in terms of the energy and angular momentum as,</text>
<formula><location><page_8><loc_38><loc_24><loc_88><loc_31></location>a = -1 2 E K = r 1 -3 ( I 1 z -I s + C ± 2 ) 2 r 2 -1 (50)</formula>
<formula><location><page_8><loc_29><loc_12><loc_88><loc_23></location>e 2 = 1 + 2 E K ( H K ) 2 = 1 - 1 -3 ( I 1 z -I s + C ± 2 ) 2 r 2 1 + 3 ( I 1 z -I s + C ± 2 ) 2 r 2 = 3 ( I 1 z -I s + C ± 2 ) 2 r 2 2 (51)</formula>
<text><location><page_9><loc_12><loc_90><loc_17><loc_91></location>so that</text>
<formula><location><page_9><loc_42><loc_86><loc_88><loc_90></location>e = 3 ( I 1 z -I s + C ± 2 ) 2 r 2 (52)</formula>
<text><location><page_9><loc_15><loc_84><loc_59><loc_85></location>Using Eqs. (50) and (51), we find that at equilibrium we have</text>
<formula><location><page_9><loc_47><loc_80><loc_88><loc_83></location>a = r 1 -e (53)</formula>
<text><location><page_9><loc_12><loc_74><loc_88><loc_79></location>Due to the relationship in Eq. (51), we can see that e > 0 at the φ = 0 equilibrium points because I 1 z -I s ≥ 0 and C + 2 > 0. At the φ = 90 equilibrium points, e can be positive, negative, or zero because C -2 can be negative. This tells us that at equilbrium the system is always locked at periapse or apoapse, depending on the sign of e .</text>
<text><location><page_9><loc_12><loc_65><loc_88><loc_73></location>However, at a relative equilibria, the orbit rate is given by Eq. (47), and is constant. On a Keplerian orbit, this would imply that the orbit is circular, and this orbit rate is identical to the mean motion. In this case the eccentricity is in general non-zero and constant, and the semi-major axis is constant, along with the radius. Combining these results indicates that in fact the true/mean anomaly are constant (equal to 0 or 180 · as discussed above) and the argument of perigee is precessing at the orbit rate to enforce this condition.</text>
<text><location><page_9><loc_12><loc_63><loc_88><loc_65></location>This can be shown by investigating the evolution of the eccentricity vector, which points to periapse. The eccentricity vector is defined by,</text>
<formula><location><page_9><loc_44><loc_61><loc_88><loc_62></location>e = v × H K -ˆ r (54)</formula>
<text><location><page_9><loc_12><loc_57><loc_88><loc_59></location>and in the secondary fixed frame at equilibrium the eccentricity vector becomes through use of Eqs. (47) and (49)</text>
<formula><location><page_9><loc_43><loc_55><loc_88><loc_56></location>e = ( r 3 ˙ θ 2 -1)ˆ r = e ˆ r (55)</formula>
<text><location><page_9><loc_12><loc_53><loc_35><loc_54></location>where e was defined in Eq. (52).</text>
<text><location><page_9><loc_12><loc_50><loc_88><loc_52></location>The rate of change of the eccentricity vector can be determined by using the transport theorem and resolving in the secondary fixed frame. At the φ 2 = 0 equilibrium point it becomes,</text>
<formula><location><page_9><loc_47><loc_47><loc_88><loc_48></location>˙ e = e ˙ θ ˆ y (56)</formula>
<text><location><page_9><loc_12><loc_44><loc_48><loc_45></location>and at the φ 2 = 90 · equilibrium point it becomes,</text>
<formula><location><page_9><loc_47><loc_41><loc_88><loc_43></location>˙ e = -e ˙ θ ˆ x (57)</formula>
<text><location><page_9><loc_12><loc_38><loc_72><loc_39></location>Given that in this rotating frame, the inertial rate of change of the unit vectors are</text>
<formula><location><page_9><loc_48><loc_36><loc_88><loc_37></location>˙ ˆ x = ˙ θ ˆ y (58)</formula>
<formula><location><page_9><loc_47><loc_32><loc_88><loc_33></location>˙ ˆ y = -˙ θ ˆ x (59)</formula>
<text><location><page_9><loc_12><loc_26><loc_88><loc_31></location>we can see that the angle between an equilibrium point radius vector and the eccentricity vector is constant; therefore the anomalies (true, mean, and eccentric) are fixed at 0 or 180 · . If we assume that the argument of periapse is measured from a fixed direction in the ˆ x -ˆ y plane, then we can see that the rate of change of the argument of periapse is exactly equal to ˙ θ .</text>
<text><location><page_9><loc_12><loc_16><loc_88><loc_25></location>In summary, a system which is at equilibrium will appear to an outside observer to be moving on a circular orbit. Due to the stability of the equilibrium points, this will likely only actually occur at the φ 2 = 0 point, which is commonly referred to as a synchronous orbit. If such a system is observed and fitted to Keplerian dynamics only, modeling each body as a point mass (or sphere), the result would be a circular orbit with an eccentricity of zero. The computed semi-major axis would then imply an incorrect µ , and thus an incorrect mass of the system. Therefore we reiterate that it is crucial to account for the non-spherical shape of both bodies when fitting orbits to celestial objects.</text>
<text><location><page_9><loc_12><loc_12><loc_88><loc_16></location>For reference and later comparison, the nominal system has a semi-major axis of a + = 9 . 3269 and an eccentricity of e + = 8 . 87 × 10 -3 at the φ 2 = 0 equilibrium point. The values at the φ 2 = 90 · equilibrium point are a -= 9 . 3270 and e -= 4 . 31 × 10 -3 .</text>
<text><location><page_10><loc_12><loc_78><loc_88><loc_88></location>The basis for a system to have librational motion is in the properties of the equilibirum points. We have shown in the preceding sections that the equilibrium point at φ 2 = 0 is spectrally stable and an energetic minimum; therefore this is the equilibrium point about with the system will librate. The equilibrium point at φ 2 = 90 · is spectrally unstable and an energetic saddle, which means systems will generally not stay near this equilibrium point. Before moving on to study the actual trajectories of the system, we first study the locations and properties of the equilibrium points for different systems. The parameters that define these systems are the mass ratio, the angular momentum, and the shapes of the bodies. The effect of varying these parameters are discussed in this section.</text>
<text><location><page_10><loc_12><loc_72><loc_88><loc_77></location>First, we investigate how the variation in mass fraction affects the equilibrium points. In order to isolate the effects from varying the mass fraction from the effects of the body shapes, we vary the sizes of the bodies along with the mass fraction to keep the moments of inertia constant as follows. Assuming equal density, the mass fraction is equivalent to the volume fraction,</text>
<formula><location><page_10><loc_41><loc_68><loc_88><loc_71></location>ν = V 1 V 1 + V 2 = R 2 e R p R 2 e R p + βγ (60)</formula>
<text><location><page_10><loc_12><loc_67><loc_61><loc_68></location>This can be solved for the oblate equatorial radius by using χ to get,</text>
<formula><location><page_10><loc_45><loc_63><loc_88><loc_65></location>R 3 e = βγ χ (1 -ν ) (61)</formula>
<text><location><page_10><loc_12><loc_61><loc_69><loc_62></location>Using this equatorial radius, the moments of inertia of the oblate body become,</text>
<formula><location><page_10><loc_46><loc_57><loc_88><loc_60></location>I 1 z = 2 5 R 2 e (62)</formula>
<formula><location><page_10><loc_38><loc_54><loc_88><loc_57></location>I s = 1 5 ( R 2 e + R 2 p ) = 1 5 R 2 e ( 1 + χ 2 ) (63)</formula>
<text><location><page_10><loc_12><loc_47><loc_88><loc_54></location>This paper is mainly concerned with systems where the ellipsoidal body is the smaller body, which would correspond to ν ≥ 0 . 5. However, for the sake of completeness, we show in this section the locations of the equilibrium points for all values of ν . When the ellipsoidal body is larger, our perspective changes and we think of this in terms of studying the location of orbits of an oblate satellite, instead of the libration of an ellipsoidal satellite.</text>
<text><location><page_10><loc_12><loc_39><loc_88><loc_47></location>The locations of the equilibrium points are shown in Fig. 3. The general behavior is that as the mass fraction becomes smaller, the equilibrium points move to larger radii. The second plot shows the difference in radial distance between the two equilibrium points since they are too close to appear as separate lines in the scale of the first figure. The opposite trend is seen here in that the two equilibrium points are further apart radially as ν → 1. At the nominal value of angular momentum, the two equilibrium points are 0.04 apart as was given at the end of Section 3.1.</text>
<text><location><page_10><loc_12><loc_31><loc_88><loc_39></location>The astute reader will notice that this does not seem to match the results from Bellerose [3]; this is because that work included a factor of ν in their computation of angular momentum we do not include (e.g. K Bellerose = νK ). The value of angular momentum used in this paper is absolute for any system. Also note that Bellerose discusses cases for which multiple equilibria appear; we find these cases as well for small values of K . However, since we are interested in studying the outer pair of equilibrium points that have the librational structure, we don't study the inner equilibrium points that may appear.</text>
<text><location><page_10><loc_12><loc_27><loc_88><loc_31></location>The main result of varying the angular momentum as illustrated in Fig. 3 is that for higher values of K , the equilibrium points move to larger radii. This also has the effect of making the absolute difference in energy levels between the equilibrium points smaller, as is seen in Fig. 4.</text>
<text><location><page_10><loc_12><loc_21><loc_88><loc_26></location>The shape of the bodies can have a number of effects on the equilibria. In terms of the oblateness of body 1, the more oblate the body, the larger the difference I 1 z -I s becomes. This generally means that the location of the equilibrium points moves outward with a more oblate body. The effect on the φ 2 = 0 and φ 2 = 90 · equilibria are roughly the same.</text>
<text><location><page_10><loc_12><loc_16><loc_88><loc_21></location>The effects of the secondary shape can be more varied due to the fact that the main asymmetry in this problem is due to the ellipsoid shape/moments of inertia. The effects are encompassed by the differences in C + 2 and C -2 , which can be changed due to variations of any of the three moments of inertia. Recall that changing I 2 z also changes the value of the system moment of inertia, I z .</text>
<text><location><page_10><loc_12><loc_12><loc_88><loc_16></location>It is interesting to recall that the semi-major axis and eccentricity depend largely on the quantity ( I 1 z -I s + C ± 2 ). While the equilibrium point at φ 2 = 0 is always at periapsis, the φ 2 = 90 · equilibrium can be at periapse, apoapse, or on a circular orbit depending completely on the moments of inertia of the</text>
<text><location><page_11><loc_79><loc_76><loc_80><loc_77></location>1</text>
<text><location><page_11><loc_79><loc_59><loc_80><loc_61></location>1</text>
<figure>
<location><page_11><loc_20><loc_58><loc_79><loc_91></location>
<caption>Fig. 3 The equilibrium location for varying values of the mass fraction are shown. Three lines are plotted at different values of angular momentum: K = 2 . 8382, the nominal value in green, K = 3 . 9625 in red, and K = 2 . 1963 in blue. The dotted vertical line indicates the nominal value of ν = 0 . 9257. The lower figure shows the radial difference between the location of the two equilibrium points.</caption>
</figure>
<text><location><page_11><loc_12><loc_46><loc_88><loc_49></location>bodies. Also, due to the same quantity, the φ 2 = 90 · equilibrium point can become a local maximum in the radial direction (see Eq. (41)), although the point will still be an energetic saddle in general.</text>
<text><location><page_11><loc_12><loc_42><loc_88><loc_46></location>In all cases studied here, the results of the stability of the nominal system equilibria holds for the outer set of equilibria points. Namely, the outer equilibrium point at φ 2 = 0 is spectrally stable and an energetic minima, while the outer equilibrium point at φ 2 = 90 · is spectrally unstable and an energetic saddle.</text>
<figure>
<location><page_12><loc_28><loc_67><loc_72><loc_91></location>
<caption>Fig. 4 The difference in energy between the stable and unstable equilibrium points, ∆E = E un -E stab , is plotted in black for the nominal system. The three vertical lines indicate the three different values of angular momentum, classified by the same color scheme as used in Fig. 3.</caption>
</figure>
<section_header_level_1><location><page_13><loc_12><loc_90><loc_46><loc_91></location>4 Sufficient Condition for Bounded Motion</section_header_level_1>
<text><location><page_13><loc_12><loc_81><loc_88><loc_88></location>When the system is not at equilibrium, the trajectory will evolve in r -φ 2 space according to the equations of motion given in Eqns. (17) - (18). As this system is non-integrable, we are particularly interested in finding conditions which classify or restrict the trajectories that will occur. In this section, we present a sufficiency condition for bounded motion, which is taken to mean that the libration angle will be less than 90 · for all time. We then investigate how the sufficiency condition changes for different system parameters. Finally, we look at some properties of the system trajectories that meet the bounded sufficiency conditions.</text>
<section_header_level_1><location><page_13><loc_12><loc_76><loc_27><loc_77></location>4.1 Bounded Motion</section_header_level_1>
<text><location><page_13><loc_12><loc_72><loc_88><loc_75></location>The ZVCs can be used to investigate if the current system can become unbounded by reaching a phi = 90, which leads to the sufficiency condition for bounded motion:</text>
<section_header_level_1><location><page_13><loc_12><loc_70><loc_55><loc_71></location>Theorem 4.1. Sufficient Condition for Bounded Motion</section_header_level_1>
<text><location><page_13><loc_12><loc_68><loc_88><loc_70></location>Given a system with angular momentum K so that a stable equilibrium point exists at φ 2 = 0 , the librational motion is bounded ( φ 2 < 90 · always) if E < E -.</text>
<text><location><page_13><loc_12><loc_66><loc_77><loc_67></location>Proof: The relationship for the free energy of the system, Eq. (15) , can be rearranged to become</text>
<formula><location><page_13><loc_34><loc_62><loc_88><loc_65></location>E -1 2 K 2 I z ( r ) -V ( r, φ 2 ) = 1 2 ν ˙ r 2 + 1 2 I 2 z νr 2 ˙ φ 2 2 I z ( r ) (64)</formula>
<text><location><page_13><loc_12><loc_59><loc_88><loc_61></location>The right hand side of Eq. (64) is always positive, therefore we can state the relationship for a zero-velocity curve (ZVC),</text>
<formula><location><page_13><loc_41><loc_56><loc_88><loc_59></location>E -1 2 K 2 I z ( r ) -V ( r, φ 2 ) ≥ 0 (65)</formula>
<text><location><page_13><loc_12><loc_54><loc_22><loc_55></location>or equivalently,</text>
<formula><location><page_13><loc_42><loc_51><loc_88><loc_54></location>E ≥ 1 2 K 2 I z ( r ) + V ( r, φ 2 ) (66)</formula>
<text><location><page_13><loc_12><loc_45><loc_88><loc_51></location>The stable equilibrium is the minimum energy location, and when E = E + , the ZVC defines only the stable equilibrium. As E is increased, the ZVC will encompass larger areas in r -φ 2 space. If E is increased to E -, the ZVC will touch the unstable equilibria, and because E -is the minimum energy radius with φ 2 = 90 · (see Section 3.3), this is the minimum energy at which the ZVC allows the system to reach φ 2 = 90 · .</text>
<text><location><page_13><loc_15><loc_44><loc_67><loc_45></location>That this is a sufficient, but not necessary, condition is shown in Section 5.1.</text>
<text><location><page_13><loc_12><loc_30><loc_88><loc_44></location>2 A given system will have some area in the r -φ 2 space which it can reside in based upon the free angular momentum and energy. Given a value for the free angular momentum of the system, the entire phase space can be mapped with varying energy levels determined by Eq. (66). The easiest way to visualize this is by looking at the system from a secondary fixed frame, as shown in Fig. 2. For a given value of free energy for the system, there will be some area to which the primary is constrained to reside. Note that due to the fact that this relationship is an inequality, the primary can be anywhere inside the free energy level, not only on the surface. Therefore this relationship clearly doesn't solve the equations of motion to tell us what the state is at any given time, but it does tell us absolutely that the state is always inside the area bounded by that free energy.</text>
<text><location><page_13><loc_12><loc_12><loc_88><loc_29></location>Theorem 4.1 is verified graphically for the nominal system in Figures 5 and 6. The nominal system zero-velocity curves are shown in Fig. 5. This is the typical structure for the zero-velocity curves seen in most situations where the libration between the two outer equilibrium points is being examined. In these types of plots, the stable minimum energy equilibrium point is along the x-axis ( φ 2 = 0), while the unstable equilibrium point is on the y-axis ( φ 2 = 90 · ). As the energy is increased from the minimum at the x-axis equilibrium point, the zero-velocity curves allow for larger libration angles until the unstable equilibrium energy is reached. At energies above the unstable equilibrium point energy the ZVCs open (becoming two separate ZVCs) and the secondary is free to circulate in this area. The black shape around the origin is the projection of the ellipsoid plus the equatorial radius of the oblate body; if the energy is high enough so that this region is inside the ZVC bounds then an impact between the bodies is possible. The color bar lists the values of energy ( E ) corresponding to each ZVC. Recall from Section 3.1 that the stable equilibrium point has energy E + = -0 . 0497 and the unstable equilibrium has an energy of E -= -0 . 496, where the difference between the two is δE = 1 . 500 × 10 -4 .</text>
<figure>
<location><page_14><loc_28><loc_54><loc_72><loc_91></location>
<caption>Fig. 5 Phase space for the nominal test system described in Table 1. The closed ZVCs are at increments of energy of 10% of δE , so that the first close curve has energy E = E + +0 . 1 δE . This first curve limits the libration angle to a maximum of φ 2 = 18 . 4 · .</caption>
</figure>
<text><location><page_14><loc_12><loc_40><loc_88><loc_45></location>In order to make clear the behavior of the energy in the vicinity of the equilibrium points, we plot what is effectively a cross section of Fig. 5 in Fig. 6. This clearly shows the variation of the energy in the radial and circumferential directions. It is clear that both of these equilibrium points are minima in the r direction, however only the x-axis equilibrium point is also a minima in the φ 2 direction.</text>
<text><location><page_14><loc_12><loc_36><loc_88><loc_40></location>It should be noted that the same behavior is seen around the stable equilibrium point at φ 2 = 180 · , and Theorem 4.1 can be applied there as well. In this paper, we generally only look at the φ 2 = 0 equilibrium point for clarity.</text>
<text><location><page_14><loc_12><loc_29><loc_88><loc_35></location>Consider a zero velocity curve with the free energy, E , and the free angular momentum, K . When the inequality in Eq. (65) is precisely equal to zero, we know that ˙ r = 0 and ˙ φ 2 = 0; all of the free kinetic energy in the system has been transferred to the potential energy. At any point inside the zero velocity curve(s) defined by E , the inequality will be greater than zero as there is an excess of energy defined from the ZVC definition,</text>
<formula><location><page_14><loc_30><loc_24><loc_88><loc_27></location>∆E ( r, φ 2 ) = E -1 2 K 2 0 I z -V ( r, φ 2 ) = 1 2 m ˙ r 2 + 1 2 I 2 z mr 2 ˙ φ 2 2 I z (67)</formula>
<text><location><page_14><loc_12><loc_19><loc_88><loc_23></location>This situation is depicted in the cartoon shown in Figure 7. Returning to Eq. (64), it is clear that in this case, the left hand side is greater than zero, and therefore either | ˙ r | > 0, | ˙ φ 2 | > 0, or some combination of the two.</text>
<text><location><page_14><loc_12><loc_12><loc_88><loc_18></location>Note that the excess energy, ∆E is not constant on a given trajectory as it varies with the position variables. The maximum value that ∆E can reach is at the φ 2 = 0 equilibrium point since this is the minimum potential energy location. At any given point on a trajectory, the excess energy can be used to determine the range of possible values of the velocities based on the kinetic energy in the radial and spin components.</text>
<figure>
<location><page_15><loc_12><loc_62><loc_87><loc_91></location>
<caption>Fig. 8 shows the trajectories of three cases for the nominal system with a closed ZVC, each beginning with a different amount of excess energy in the radial and angular forms of the kinetic energy. These results demonstrate that the ZVC is indeed accurate, and that the trajectories stay bound within this area of r -φ 2 space. Furthermore, depending on the initial velocities, the area within the ZVC that is filled by each trajectory is limited further; the trajectory with all the initial velocity in the radial direction (plotted in blue) explores the phase space the furthest in the radial direction, but not as far in the angular direction. The opposite is true for the case with all of the initial velocity in the angular direction, which is plotted in red.</caption>
</figure>
<text><location><page_15><loc_53><loc_61><loc_53><loc_62></location>2</text>
<paragraph><location><page_15><loc_12><loc_54><loc_88><loc_60></location>Fig. 6 Variation of the energy near the equilibrium points for the nominal system. The upper plot shows the variation with respect to the radial distance at φ 2 = 0 and φ 2 = 90 · . The location of the equilibria are marked by the vertical dotted lines. The lower plot shows the variation in the energy at the stable equilibrium point radius as φ 2 is varied. The points at ± π/ 2 on this plot are actually equivalent to where the blue vertical dotted intersects the red line in the upper plot, not precisely at the unstable equilibrium radius. However, the same trend of being a local maximum with respect to variation in φ 2 holds at the unstable equilibrium point radius.</paragraph>
<text><location><page_15><loc_12><loc_48><loc_88><loc_50></location>The split in the kinetic energy will be determined by the factor κ , which dilineates what percentage of the excess energy goes into the radial velocity. The scale factor is therefore bounded such that</text>
<formula><location><page_15><loc_47><loc_45><loc_88><loc_46></location>0 ≤ κ ≤ 1 (68)</formula>
<text><location><page_15><loc_12><loc_43><loc_67><loc_44></location>This means that the radial velocity and ellipsoid rotation rate are defined as,</text>
<formula><location><page_15><loc_46><loc_40><loc_88><loc_42></location>˙ r 2 = 2 κ∆E ν (69)</formula>
<formula><location><page_15><loc_43><loc_36><loc_88><loc_39></location>˙ φ 2 2 = 2(1 -κ ) ∆EI z I 2 z νr 2 (70)</formula>
<text><location><page_15><loc_12><loc_21><loc_88><loc_24></location>The ZVCs are a crucial property of the system which limit the phase space in which the system can reside. Since the conditions in Theorem 4.1 are so important, we investigate how the ZVCs vary when the system parameters are changed.</text>
<figure>
<location><page_16><loc_27><loc_59><loc_73><loc_91></location>
<caption>Fig. 7 The zero-velocity curve for the given energy, E , has zero excess energy, as shown. Moving in toward the equilibrium point increases ∆E until reaching the equilibrium point (marked by the x), where the excess energy is at a maximum. Intervening zero-velocity curves are drawn with dotted lines.</caption>
</figure>
<figure>
<location><page_17><loc_13><loc_53><loc_87><loc_91></location>
<caption>Fig. 8 Three trajectories for the nominal system with an energy of E = E + +0 . 3 δE , which means the zero-velocity curve is closed, as shown in black. All three trajectories are initiated at r + with κ = 0, κ = 0 . 5, and κ = 1.</caption>
</figure>
<section_header_level_1><location><page_18><loc_12><loc_90><loc_47><loc_91></location>4.2 Parameter Influence on Zero-Velocity Curves</section_header_level_1>
<text><location><page_18><loc_12><loc_82><loc_88><loc_88></location>Eq. (65) shows us that the zero-velocity curves are defined first and foremost by the angular momentum and energy in the system. However, the exact form of the zero-velocity curves depends on all the system parameters such as the moments of inertia of the bodies, and the mass fraction of the system. In this section we explore how the nominal zero-velocity curves shown in Fig. 5 change when the system parameters are varied.</text>
<section_header_level_1><location><page_18><loc_12><loc_79><loc_36><loc_80></location>4.2.1 Variation Due to Mass Ratio</section_header_level_1>
<text><location><page_18><loc_12><loc_74><loc_88><loc_77></location>In this section, we look at the ZVCs for three different mass ratio systems pulled from Section 3.5. By comparing these different mass ratio systems to our nominal system in Fig. 5, we get a good idea of the effect of changing the mass ratio on the ZVC structure.</text>
<text><location><page_18><loc_12><loc_65><loc_88><loc_73></location>The first case is the nominal system scaled to a mass ratio of ν = 0 . 5, shown in Fig. 9. Even with the scaling of the bodies to equal masses, the structure of the ZVCs is very similar. The values of the energy and the location of the equilibrium points have changed due to a change in angular momentum, but the interesting point is that the behavior of the system between and around the equilibrium points found in our nominal case of a relatively small ellipsoid orbiting a larger oblate body appears to hold all the way to equal masses.</text>
<figure>
<location><page_18><loc_29><loc_23><loc_72><loc_60></location>
<caption>Fig. 9 Zero-velocity curves for the scaled system with ν = 0 . 5 and K = 1 . 0007. The first 10 zero-velocity curves surrounding the stable equilibrium point are plotted in increments of energy of 0 . 1 δE .</caption>
</figure>
<text><location><page_18><loc_12><loc_12><loc_88><loc_16></location>The second case we look at is with a mass ratio of ν = 0 . 05 shown in Fig. 10. This system was chosen because it illustrates a case discussed by Bellerose [3] where there are two equilibrium points on the x-axis, and the outer point is spectrally stable and an energetic minima, while the inner point is spectrally unstable</text>
<text><location><page_19><loc_12><loc_86><loc_88><loc_91></location>and an energetic saddle. It turns out for these systems that there are also two equilibria on the y-axis. As expected from the systems studied so far, the outer y-axis equilibria is spectrally unstable and an energetic saddle. It is interesting to find that the inner y-axis equilibria is spectrally stable while being an energetic maxima.</text>
<text><location><page_19><loc_12><loc_80><loc_88><loc_85></location>This is interesting, as it implies that as ν → 0, so that the system being studied is that of a small sphere/oblate body orbiting a large ellipsoid, there will be stable orbits along the y-axis around three times the ellipsoid semi-major axis. The orbits that have the small body near the same radius on the x-axis will be unstable.</text>
<figure>
<location><page_19><loc_28><loc_37><loc_72><loc_77></location>
<caption>Fig. 10 The zero-velocity curves for the system with ν = 0 . 05 and K = 0 . 1479. The minimum energy point is the outer equilibrium point on the x-axis, however low energy trajectories can exist inside the inner pair of equilibria, although they would likely quickly impact.</caption>
</figure>
<text><location><page_19><loc_12><loc_20><loc_88><loc_28></location>The final case we show is that with a mass ratio of ν = 0 . 15, shown in Fig. 11. The y-axis equilibrium points keep the same general behavior as those in Fig. 10, however in this case the x-axis equilibria have vanished. This appears to make low energy impacts possible between the two bodies starting from basically anywhere in the system; in other words the energy decreases all the way to the surface along the x-axis, and since a given trajectory can explore the region between ZVCs for its energy, they can reach the surface along the x-axis.</text>
<text><location><page_19><loc_12><loc_12><loc_88><loc_20></location>In the cases in Figs. 10 and 11, it is important to note that there may be some non-trivial errors in the values obtained for the closer equilibria due to the fact that we are using a second order expansion for the potential of the bodies. If accurate results are needed for these cases, we suggest calculating the outcomes with the exact potential expression used by Bellerose [3]. However, the qualitative results match these previous investigations with the exact potential. Furthermore, with the main interest of this paper being the outer set of equilibrium points, these points are far enough from the bodies that the differences</text>
<figure>
<location><page_20><loc_27><loc_51><loc_72><loc_89></location>
<caption>Fig. 11 The zero-velocity curves for the system with ν = 0 . 15 and K = 0 . 3211. At this configuration, the x-axis equilibria have disappeared and the path to impact along the x-axis is open for all energies.</caption>
</figure>
<text><location><page_20><loc_12><loc_41><loc_88><loc_44></location>between our second order approximation and the exact potential should be small enough to make little difference.</text>
<text><location><page_21><loc_12><loc_81><loc_88><loc_88></location>As was alluded to in Section 3.5, the main effect of changing the angular momentum is to move the equilibria in or out. This effect is also seen on the ZVC structure in Fig. 12; the ZVCs have the same structure as the nominal case, they are basically shifted in or out. It was demonstrated in Section 4.2.1 that a combination of low ν and K can cause a second set of equilibrium points to appear, however this situation doesn't happen for larger values of ν due to the fact that these solutions to Eq. (20) are less than the combined radii of the bodies, and usually less than the radius of the secondary.</text>
<figure>
<location><page_21><loc_14><loc_47><loc_86><loc_77></location>
<caption>Fig. 12 The ZVC structure for two cases of the nominal system with different angular momentum values. The case on the left has a lower angular momentum than the nominal at K = 2 . 1963, while the right figure has a higher value of K = 3 . 9625. Recall that the nominal angular momentum is K = 2 . 8382. The main effect of changing the angular momentum is to move the equilibria out for higher values, and in for lower values.</caption>
</figure>
<text><location><page_22><loc_12><loc_78><loc_88><loc_88></location>The degree to which the secondary is triaxial plays a large role in the determination of the zero-velocity curves. This is investigated by tweaking the nominal case with two different values of the intermediate moment of inertia, I 2 y , first so that it is just greater than the I 2 x , and second so that it is just less than I 2 z . The main contribution of modifying the ellipsoid moments of inertia, however, is that the difference ( I 2 y -I 2 x ) controls the dependence of the potential and therefore the ZVC on the libration angle. The ZVCs for these two cases are compared to the nominal case in Fig. 13. The ZVCs of the same color are at the same energy across the three figures, making it easy to see that as the intermediate moment of inertia is made smaller, the body can circulate much easier.</text>
<figure>
<location><page_22><loc_12><loc_44><loc_87><loc_75></location>
<caption>Fig. 13 Zero-velocity curves for the nominal system with three different values of the ellipsoid middle moment of inertia. From left to right, the minimum value is I 2 y =, the nominal value is I 2 y =, and the maximum value is I 2 y =. Each of the three figures has ten ZVCs plotted at the same energy levels, clearly illustrating that it is easier to cause an ellipsoid with a smaller difference ( I 2 y -I 2 x ), the minimum case, to begin circulating.</caption>
</figure>
<text><location><page_22><loc_12><loc_31><loc_88><loc_35></location>Note that, as discussed in Section 3.5, changing the oblateness of the other body will move the location of the equilibrium points in or out. However, there is very little difference beyond this radial shift on the ZVCs, so this effect isn't illustrated.</text>
<section_header_level_1><location><page_23><loc_12><loc_90><loc_48><loc_91></location>4.3 Time-Variation of Osculating Orbital Elements</section_header_level_1>
<text><location><page_23><loc_12><loc_78><loc_88><loc_88></location>The orbital elements of a non-equilibrium trajectory are no longer useful as metrics for the system because they change constantly throughout the course of the trajectory. The groundwork for calculating the orbital elements has already been laid in Sections 3.4 and 4.1. At any given point in r -φ 2 space, given the energy and angular momentum, the possible values of ˙ r and ˙ φ 2 can be calculated using Eqs. (70) and (69). The Keplerian energy and angular momentum can then be calculated from Eqs. (46) and (45), which in turn allows us to calculate the semi-major axis and eccentricity with Eqs. (50) and (51). Note that for these calculations, the sign of ˙ r makes no difference, however the sign of ˙ φ 2 directly effects the value of ˙ θ , which then changes the value of the Keplerian energy and angular momentum.</text>
<text><location><page_23><loc_12><loc_75><loc_88><loc_77></location>The eccentricity vector can again be calculated through Eq. (54). Note that in this case, ˙ r is no longer zero in general, so that</text>
<formula><location><page_23><loc_42><loc_74><loc_88><loc_75></location>e = ( r 3 ˙ θ 2 -1)ˆ r -r 2 ˙ r ˙ θ ˆ θ (71)</formula>
<text><location><page_23><loc_49><loc_72><loc_49><loc_73></location>glyph[negationslash]</text>
<text><location><page_23><loc_12><loc_71><loc_88><loc_73></location>where ˆ θ = ˆ H K × ˆ r . This implies that whenever ˙ r = 0, the eccentricity vector no longer lies along the ˆ r direction, and therefore the system is no longer at an apse.</text>
<text><location><page_23><loc_12><loc_65><loc_88><loc_70></location>To illustrate the behavior the semi-major axis and eccentricity on a non-equilibrium trajectory, the corresponding values of the semi-major axis and eccentricity for each of the trajectories in Fig. 8 are shown in Fig. 14. As with the radial-angular phase space, each trajectory occupies a distinct region of a -e space despite the fact that they each have the same energy.</text>
<figure>
<location><page_23><loc_13><loc_22><loc_87><loc_62></location>
<caption>Fig. 14 Eccentricity versus semi-major axis for the three trajectories shown in Fig. 8.</caption>
</figure>
<text><location><page_23><loc_12><loc_12><loc_88><loc_16></location>To further understand the possible values of semi-major axis and eccentricity that can be reached along the trajectories, the values in the entire region inside a give ZVC can be mapped. Figs. 15 - 16 show these results. Each subplot can be imagined as a cross section of a ZVC plot (e.g. Fig. 7) at a different value</text>
<text><location><page_24><loc_12><loc_89><loc_88><loc_91></location>of φ 2 , where the out of plane axis tells us how the excess energy is apportioned between the radial and angular velocities through the use of the parameter η , which is defined as</text>
<formula><location><page_24><loc_42><loc_84><loc_88><loc_88></location>η = { 1 -κ ˙ φ 2 ≥ 0 κ -1 ˙ φ 2 < 0 (72)</formula>
<text><location><page_24><loc_12><loc_79><loc_88><loc_83></location>so that negative values of η correspond to negative values of ˙ φ 2 . The radial bounds of each subplot correspond to the inner and outer edges of the ZVC for this energy level, which is why at larger values of φ 2 , the radial range is smaller.</text>
<text><location><page_24><loc_12><loc_68><loc_88><loc_79></location>From these figures, we see that the semi-major axis reaches its maximum and minimum values near the stable equilibrium point, with all of the energy in negative or positive angular velocity, respectively. The eccentricity, on the other hand, reaches its maximum value near the inner boundary of the ZVC at φ 2 = 0 with most of the energy in the radial velocity, but some in a negative angular velocity. The minimum eccentricity, which is zero, is seen just outside of the equilibrium point distance with most of the energy in either positive or negative angular velocity. Note that the colorbars near the last subplot are valid for all values of φ 2 . Clearly a wide variety of combinations of semi-major axis and eccentricity can be seen over the course of trajectory.</text>
<figure>
<location><page_24><loc_12><loc_34><loc_88><loc_66></location>
<caption>Fig. 15 Semi-major axis values for various libration angles with the same energy as Fig. 8.</caption>
</figure>
<text><location><page_25><loc_12><loc_86><loc_13><loc_86></location>η</text>
<text><location><page_25><loc_12><loc_75><loc_13><loc_76></location>η</text>
<text><location><page_25><loc_15><loc_89><loc_15><loc_90></location>1</text>
<text><location><page_25><loc_14><loc_88><loc_15><loc_88></location>0.5</text>
<text><location><page_25><loc_15><loc_86><loc_15><loc_86></location>0</text>
<text><location><page_25><loc_14><loc_84><loc_15><loc_84></location>-0.5</text>
<text><location><page_25><loc_14><loc_82><loc_15><loc_83></location>-1</text>
<text><location><page_25><loc_15><loc_79><loc_15><loc_79></location>1</text>
<text><location><page_25><loc_14><loc_77><loc_15><loc_78></location>0.5</text>
<text><location><page_25><loc_15><loc_75><loc_15><loc_76></location>0</text>
<text><location><page_25><loc_14><loc_73><loc_15><loc_74></location>-0.5</text>
<text><location><page_25><loc_14><loc_72><loc_15><loc_72></location>-1</text>
<text><location><page_25><loc_16><loc_82><loc_17><loc_82></location>9</text>
<text><location><page_25><loc_19><loc_82><loc_21><loc_82></location>9.1</text>
<text><location><page_25><loc_23><loc_82><loc_24><loc_82></location>9.2</text>
<text><location><page_25><loc_27><loc_82><loc_28><loc_82></location>9.3</text>
<text><location><page_25><loc_30><loc_82><loc_31><loc_82></location>9.4</text>
<text><location><page_25><loc_34><loc_82><loc_35><loc_82></location>9.5</text>
<text><location><page_25><loc_19><loc_71><loc_20><loc_72></location>9.1</text>
<text><location><page_25><loc_23><loc_71><loc_24><loc_72></location>9.2</text>
<text><location><page_25><loc_27><loc_71><loc_28><loc_72></location>9.3</text>
<text><location><page_25><loc_31><loc_71><loc_32><loc_72></location>9.4</text>
<text><location><page_25><loc_35><loc_71><loc_36><loc_72></location>9.5</text>
<text><location><page_25><loc_23><loc_69><loc_24><loc_70></location>Φ</text>
<text><location><page_25><loc_24><loc_69><loc_28><loc_70></location>= 30 deg.</text>
<figure>
<location><page_25><loc_12><loc_60><loc_40><loc_69></location>
<caption>Fig. 16 Eccentricity values for various libration angles with the same energy as Fig. 8.</caption>
</figure>
<text><location><page_25><loc_25><loc_60><loc_26><loc_60></location>r</text>
<text><location><page_25><loc_25><loc_81><loc_26><loc_81></location>r</text>
<text><location><page_25><loc_24><loc_80><loc_28><loc_80></location>= 15 deg.</text>
<text><location><page_25><loc_24><loc_80><loc_24><loc_80></location>2</text>
<text><location><page_25><loc_23><loc_90><loc_24><loc_91></location>Φ</text>
<text><location><page_25><loc_23><loc_80><loc_24><loc_80></location>Φ</text>
<text><location><page_25><loc_24><loc_90><loc_27><loc_91></location>= 0 deg.</text>
<text><location><page_25><loc_24><loc_90><loc_24><loc_91></location>2</text>
<text><location><page_25><loc_25><loc_70><loc_26><loc_71></location>r</text>
<text><location><page_25><loc_39><loc_86><loc_39><loc_86></location>η</text>
<text><location><page_25><loc_39><loc_75><loc_39><loc_76></location>η</text>
<text><location><page_25><loc_41><loc_89><loc_41><loc_90></location>1</text>
<text><location><page_25><loc_40><loc_88><loc_41><loc_88></location>0.5</text>
<text><location><page_25><loc_41><loc_86><loc_41><loc_86></location>0</text>
<text><location><page_25><loc_40><loc_84><loc_41><loc_84></location>-0.5</text>
<text><location><page_25><loc_40><loc_82><loc_41><loc_83></location>-1</text>
<text><location><page_25><loc_41><loc_79><loc_41><loc_79></location>1</text>
<text><location><page_25><loc_40><loc_77><loc_41><loc_78></location>0.5</text>
<text><location><page_25><loc_41><loc_75><loc_41><loc_76></location>0</text>
<text><location><page_25><loc_40><loc_73><loc_41><loc_74></location>-0.5</text>
<text><location><page_25><loc_40><loc_72><loc_41><loc_72></location>-1</text>
<text><location><page_25><loc_42><loc_82><loc_43><loc_82></location>9</text>
<text><location><page_25><loc_46><loc_82><loc_47><loc_82></location>9.1</text>
<text><location><page_25><loc_49><loc_82><loc_50><loc_82></location>9.2</text>
<text><location><page_25><loc_53><loc_82><loc_54><loc_82></location>9.3</text>
<text><location><page_25><loc_56><loc_82><loc_57><loc_82></location>9.4</text>
<text><location><page_25><loc_60><loc_82><loc_61><loc_82></location>9.5</text>
<text><location><page_25><loc_49><loc_80><loc_50><loc_80></location>Φ</text>
<text><location><page_25><loc_51><loc_81><loc_52><loc_81></location>r</text>
<text><location><page_25><loc_50><loc_80><loc_54><loc_80></location>= 20 deg.</text>
<text><location><page_25><loc_50><loc_80><loc_50><loc_80></location>2</text>
<text><location><page_25><loc_44><loc_71><loc_45><loc_72></location>9.1</text>
<text><location><page_25><loc_49><loc_71><loc_50><loc_72></location>9.2</text>
<text><location><page_25><loc_53><loc_71><loc_54><loc_72></location>9.3</text>
<text><location><page_25><loc_58><loc_71><loc_59><loc_72></location>9.4</text>
<text><location><page_25><loc_49><loc_90><loc_50><loc_91></location>Φ</text>
<text><location><page_25><loc_50><loc_90><loc_54><loc_91></location>= 5 deg.</text>
<text><location><page_25><loc_50><loc_90><loc_50><loc_91></location>2</text>
<text><location><page_25><loc_51><loc_70><loc_52><loc_71></location>r</text>
<text><location><page_25><loc_65><loc_86><loc_66><loc_86></location>η</text>
<text><location><page_25><loc_65><loc_75><loc_66><loc_76></location>η</text>
<text><location><page_25><loc_67><loc_89><loc_68><loc_90></location>1</text>
<text><location><page_25><loc_66><loc_88><loc_68><loc_88></location>0.5</text>
<text><location><page_25><loc_67><loc_86><loc_68><loc_86></location>0</text>
<text><location><page_25><loc_66><loc_84><loc_68><loc_84></location>-0.5</text>
<text><location><page_25><loc_67><loc_82><loc_68><loc_83></location>-1</text>
<text><location><page_25><loc_67><loc_79><loc_68><loc_79></location>1</text>
<text><location><page_25><loc_66><loc_77><loc_68><loc_78></location>0.5</text>
<text><location><page_25><loc_67><loc_75><loc_68><loc_76></location>0</text>
<text><location><page_25><loc_66><loc_73><loc_68><loc_74></location>-0.5</text>
<text><location><page_25><loc_67><loc_72><loc_68><loc_72></location>-1</text>
<text><location><page_25><loc_68><loc_82><loc_68><loc_82></location>9</text>
<text><location><page_25><loc_71><loc_82><loc_73><loc_82></location>9.1</text>
<text><location><page_25><loc_75><loc_82><loc_76><loc_82></location>9.2</text>
<text><location><page_25><loc_79><loc_82><loc_80><loc_82></location>9.3</text>
<text><location><page_25><loc_83><loc_82><loc_84><loc_82></location>9.4</text>
<text><location><page_25><loc_86><loc_82><loc_87><loc_82></location>9.5</text>
<text><location><page_25><loc_68><loc_71><loc_70><loc_72></location>9.1</text>
<text><location><page_25><loc_74><loc_71><loc_75><loc_72></location>9.2</text>
<text><location><page_25><loc_80><loc_71><loc_81><loc_72></location>9.3</text>
<text><location><page_25><loc_85><loc_71><loc_86><loc_72></location>9.4</text>
<text><location><page_25><loc_75><loc_80><loc_76><loc_80></location>Φ</text>
<text><location><page_25><loc_78><loc_81><loc_78><loc_81></location>r</text>
<text><location><page_25><loc_76><loc_80><loc_80><loc_80></location>= 25 deg.</text>
<text><location><page_25><loc_76><loc_80><loc_76><loc_80></location>2</text>
<text><location><page_25><loc_75><loc_90><loc_76><loc_91></location>Φ</text>
<text><location><page_25><loc_76><loc_90><loc_80><loc_91></location>= 10 deg.</text>
<text><location><page_25><loc_76><loc_90><loc_76><loc_91></location>2</text>
<text><location><page_25><loc_78><loc_70><loc_78><loc_71></location>r</text>
<section_header_level_1><location><page_26><loc_12><loc_90><loc_29><loc_91></location>5 Transient Dynamics</section_header_level_1>
<text><location><page_26><loc_12><loc_82><loc_88><loc_89></location>In this section, we explore trajectories for systems with open ZVCs (those with E > E -). Specifically, we explore the existence of trajectories that behave as if they are bounded, but do not meet the sufficiency conditions outlined in Section 4. These results prove that Theorem 4.1 is in fact a sufficient, but not necessary, condition for bounded motion. We also investigate the special case of bounded motion for periodic trajectories, and the families of periodic trajectories that exist as the energy is varied.</text>
<section_header_level_1><location><page_26><loc_12><loc_78><loc_60><loc_79></location>5.1 Existence of Bounded Motion with Open Zero-Velocity Curves</section_header_level_1>
<text><location><page_26><loc_12><loc_66><loc_88><loc_76></location>The existence of bounded trajectories with open ZVCs is shown through numerical examples. Two such trajectories are shown in Fig. 17. Although the ZVC is open, not every trajectory circulates. This is an extension of what was seen above with the closed ZVC trajectories; depending on the initial conditions of the trajectory at φ 2 = 0, the trajectory will only explore a portion of the r -φ 2 phase space inside the ZVC. The existence of this bounded trajectory proves that Theorem 4.1 is a sufficient, but not necessary condition, as this example does not meet the requirement of the theorem. Furthermore, using the ZVC corresponding to the energy of the unstable equilibrium point is a conservative estimate of the energy at which a system will have a circulating, as opposed to librating, secondary body.</text>
<text><location><page_26><loc_52><loc_62><loc_54><loc_63></location>10</text>
<figure>
<location><page_26><loc_16><loc_44><loc_84><loc_63></location>
<caption>Fig. 17 Two trajectories of the nominal system with an energy of E = E -+0 . 1 δE , which is high enough to have an open zero-velocity curve. Although there is no energy barrier to either trajectory circulating, one does and one does not.</caption>
</figure>
<section_header_level_1><location><page_26><loc_12><loc_34><loc_26><loc_35></location>5.2 Periodic Orbits</section_header_level_1>
<text><location><page_26><loc_12><loc_23><loc_88><loc_32></location>Periodic orbits can be found in this system when a given orbit repeats itself exactly after a certain period. These orbits are found when the state transition matrix or monodromy matrix have unity eigenvalues. A detailed methodology for finding periodic orbits in this manner is given in [3]. In this system, due to the symmetry about φ 2 = 0, most of the periodic orbits will have a crossing of this line of symmetry with ˙ r = 0, although this is not strictly required for a periodic orbit. However, it is true that every trajectory that crosses φ 2 = 0 twice with ˙ r = 0 is periodic. The period may be very long, but due to the symmetry of the problem it will eventually repeat.</text>
<text><location><page_26><loc_12><loc_17><loc_88><loc_22></location>Therefore, our search consisted of starting at different radii with ˙ r = 0, and looking at subsequent crossings of φ 2 = 0. The results of this search are combined in a Poincar'e map at of the r -˙ r space at φ 2 = 0. One illustrative case is shown in Fig. 18, where the crossings of several periodic orbits are highlighted.</text>
<text><location><page_26><loc_12><loc_12><loc_88><loc_17></location>The actual trajectories of several periodic orbits in the r -φ 2 space are shown in Fig. 19, corresponding to the highlighted cases from Fig. 18. As alluded to previously, there are actually an infinite number of periodic trajectories in this system, with most of them having very large period ratios such as the T = 49 . 04 and T = 56 . 84 orbits shown. Of particular interest are the two periodic orbits with near T = 1. In the example</text>
<figure>
<location><page_27><loc_12><loc_61><loc_88><loc_92></location>
<caption>Fig. 18 Poincar'e plot of ˙ r and r for the nominal system with E = E + + 0 . 3 δE , the same as in Fig. 8. Five different periodic orbits are highlighted. The small blue dots correspond to many other integrated trajectories that are not periodic, and so they pierce the Poincar'e map at different locations each time.</caption>
</figure>
<text><location><page_27><loc_12><loc_50><loc_88><loc_53></location>shown in Fig. 19, these are the T = 0 . 9896 case, which orbits counter-clockwise, and the T = 1 . 274 case which orbits in a clockwise direction. These two orbits roughly span the space inside the closed ZVC in this case, as they are associated with the eigenmodes of the equilibrium point.</text>
<text><location><page_27><loc_12><loc_36><loc_88><loc_49></location>The evolution of the eigenmode periodic orbits with changing values of energy are explored in Figs. 20 and 21. The first plot shows many different trajectories as the energy is increased. Particularly interesting behavior is seen with the clockwise trajectories. When the energy is approximately 75% of the way to the unstable equilibrium point energy, the shape of these orbits changes from the oval seen in Fig. 19 to a bow shape, and eventually (at the highest energy levels shown) the trajectory crosses itself near φ 2 = ± 50 · . The maximum libration angles reached by these periodic orbits is also very high, approaching 80 · at the energy levels shown. By comparison, the evolution of the counter-clockwise orbits is much more regular. The shape stays basically the same throughout, simply growing in size as the energy grows. Note that the maximum amplitudes reached in these cases is much smaller than the clockwise orbits, getting to a maximum of roughly φ 2 = ± 25 · .</text>
<text><location><page_27><loc_12><loc_29><loc_88><loc_36></location>A different view of the trajectories is shown in Fig. 21. This plot shows the radial locations of the φ 2 = 0 crossings with ˙ r = 0 at each energy level. This clearly shows the more regular evolution of the counter-clockwise orbits. It is especially interesting that these periodic orbits exist well beyond the energy of the unstable equilibrium point, as this gives clear examples of stable, libration bounded non-equilibrium orbits with open ZVCs.</text>
<figure>
<location><page_28><loc_12><loc_61><loc_88><loc_91></location>
</figure>
<text><location><page_28><loc_51><loc_61><loc_52><loc_61></location>r</text>
<figure>
<location><page_28><loc_13><loc_34><loc_50><loc_54></location>
<caption>Fig. 19 The trajectories for five distinct periodic orbits in the r -φ 2 phase plane which were highlighted in Fig. 18. The trajectories are labeled by the ratio of their period with the period of the equilibrium orbit. Note that the T = 56 . 84 trajectory was plotted in cyan in Fig. 18, and the T = 49 . 04 was plotted in magenta.</caption>
</figure>
<text><location><page_28><loc_33><loc_34><loc_33><loc_35></location>r</text>
<figure>
<location><page_28><loc_50><loc_34><loc_86><loc_54></location>
<caption>Fig. 20 The periodic orbit trajectories are shown, with the clockwise orbits ( T = 1 . 274 in Fig. 19) in the left plot and the counter-clockwise ( T = 0 . 9896 in Fig. 19) orbits in the right plot. For this test, the energy was varied from E = E + to E = E -+ δE . Note that the lower energy cases are shown in dark blue, varying through green and yellow, up to the highest energy cases in dark red.</caption>
</figure>
<figure>
<location><page_29><loc_28><loc_67><loc_72><loc_91></location>
<caption>Fig. 21 Radial locations of the φ 2 = 0 crossings with ˙ r = 0 at varying energies. The clockwise orbits are shown in red, and the counter-clockwise are shown in blue. For reference, the stable equilibrium point energy is shown as a vertical dashed line, where the periodic orbit families disappear. The unstable equilibrium point energy is also shown as a dash-dot vertical line.</caption>
</figure>
<section_header_level_1><location><page_30><loc_12><loc_90><loc_48><loc_91></location>6 Sufficient Conditions for Unbounded Motion</section_header_level_1>
<text><location><page_30><loc_12><loc_78><loc_88><loc_88></location>In Section 4 we presented a sufficient condition for bounded librational motion. In Section 5, we proved this condition was only sufficient due to the existence of bounded trajectories when the conditions from Theorem 4.1 are not met. Significantly there are families of periodic orbits that exist even when the ZVCs of the system are open. In this section, we analyze the converse problem of determining when unbounded librational motion occurs. When the system has a mass fraction near unity, conditions for unbounded motion can be derived analytically. For non-unity mass fractions, strict conditions are unavailable. However, we present analysis for these systems that illustrates the structure of the problem and initial conditions that will lead to circulation.</text>
<section_header_level_1><location><page_30><loc_12><loc_74><loc_44><loc_75></location>6.1 Analytical Limits for Systems with ν glyph[similarequal] 1</section_header_level_1>
<text><location><page_30><loc_12><loc_66><loc_88><loc_72></location>In this section, we show sufficient conditions for circulation for systems with mass ratios close to unity. This restriction limits the effect of the coupling on the orbit of the small ellipsoidal body, which allows us to assume that the orbit itself remains unchanged due to the libration of the secondary. However, due to this assumption the results can only be claimed as sufficient conditions for circulation, as shown in the following theorem.</text>
<section_header_level_1><location><page_30><loc_12><loc_64><loc_51><loc_65></location>Theorem 6.1. Sufficient Condition for Circulation</section_header_level_1>
<text><location><page_30><loc_12><loc_60><loc_88><loc_63></location>Given a system with ν glyph[similarequal] 1 , so that the orbit can be considered unperturbed by the ellipsoidal secondary, with some initial secondary spin rate ˙ φ 2 , 0 at a libration angle of φ = 0 · at some location on the orbit ( r 0 ), the secondary body is guaranteed to circulate if the spin rate satisfies the relationship</text>
<formula><location><page_30><loc_29><loc_55><loc_88><loc_58></location>˙ φ 2 2 , 0 ≥ I z, 0 I 2 z r 5 0 [ I 2 y -I 2 x + ( I 2 z -I 2 x + I 1 z -I s ) ( 1 -r 3 0 r 3 a )] (73)</formula>
<text><location><page_30><loc_12><loc_53><loc_41><loc_54></location>where r a is the apoapse radius of the orbit.</text>
<text><location><page_30><loc_12><loc_52><loc_66><loc_53></location>Proof: The free energy of the system is written using Eqs. (7) , (9) and (10) as,</text>
<formula><location><page_30><loc_37><loc_50><loc_88><loc_51></location>E = V + T -T 1 = E orb + E φ + E coup (74)</formula>
<text><location><page_30><loc_12><loc_46><loc_88><loc_48></location>where E orb is the orbit energy that is independent of the librational state, E φ is the librational energy (identical to a pendulum), and E coup is the 'coupling' energy. The first two energy terms are given by relationships</text>
<formula><location><page_30><loc_41><loc_42><loc_88><loc_45></location>E orb = -ν r + ν 2 ˙ r 2 + νK 2 2 I z (75)</formula>
<formula><location><page_30><loc_36><loc_39><loc_88><loc_41></location>E φ = 1 2 I 2 z ˙ φ 2 2 + 3 ν 2 r 3 ( I 2 y -I 2 x ) sin 2 φ 2 (76)</formula>
<text><location><page_30><loc_12><loc_36><loc_88><loc_38></location>In the case of a point mass, the orbit energy collapses to the Keplerian energy of the system. Recalling the trigonometric relationship</text>
<formula><location><page_30><loc_42><loc_33><loc_88><loc_36></location>1 2 cos 2 φ 2 = 1 2 -sin 2 φ 2 (77)</formula>
<text><location><page_30><loc_12><loc_32><loc_75><loc_33></location>allows us to determine the coupling energy from Eq. (74) , using Eqs. (15) , (75) and (76) as</text>
<formula><location><page_30><loc_36><loc_28><loc_88><loc_31></location>E coup = -ν 2 r 3 ( I 1 z -I s + C + 2 ) -I 2 2 z ˙ φ 2 2 2 I z (78)</formula>
<text><location><page_30><loc_12><loc_25><loc_36><loc_27></location>where C + 2 was defined in Eq. (21) .</text>
<text><location><page_30><loc_15><loc_24><loc_64><loc_25></location>If the orbit energy is constant, then the available energy is determined as</text>
<formula><location><page_30><loc_39><loc_22><loc_88><loc_23></location>∆E lib = E -E orb = E φ + E coup (79)</formula>
<text><location><page_30><loc_15><loc_19><loc_74><loc_20></location>Since this value is a constant at any point on the trajectory, we can relate two points by</text>
<formula><location><page_30><loc_33><loc_17><loc_88><loc_18></location>∆E lib ( r 0 , φ 2 = 0 , ˙ φ 2 , 0 ) = ∆E lib ( r m , φ 2 ,m , ˙ φ 2 = 0) (80)</formula>
<text><location><page_30><loc_12><loc_12><loc_88><loc_16></location>This expression relates the available energy for a point on the orbit at radius r 0 with some initial libration rate ˙ φ 2 , 0 and zero libration angle, to another point on the orbit at radius r m with libration angle φ 2 ,max and a libration rate of zero. In other words, this tells us the maximum achievable libration angle at radius r m given a system</text>
<text><location><page_31><loc_12><loc_89><loc_88><loc_91></location>that had an initial libration rate at r 0 . Note that this relationship is independent of ˙ r . Writing out the available energy at these two locations using Eqs. (76) and (78) gives</text>
<formula><location><page_31><loc_19><loc_85><loc_88><loc_88></location>I 2 z νr 2 0 ˙ φ 2 2 , 0 2 I z, 0 -ν 2 r 3 0 ( I 1 z -I s + C + 2 ) = 3 ν 2 r 3 m ( I 2 y -I 2 x ) sin 2 φ 2 ,m -ν 2 r 3 m ( I 1 z -I s + C + 2 ) (81)</formula>
<text><location><page_31><loc_15><loc_83><loc_83><loc_84></location>To determine the bounding condition, we look for the case when φ 2 ,m = 90 · , and solve for ˙ φ 2 , 0 to get</text>
<formula><location><page_31><loc_29><loc_79><loc_88><loc_82></location>˙ φ 2 2 , 0 ≥ I z, 0 I 2 z r 5 0 [ I 2 y -I 2 x + ( I 2 z -I 2 x + I 1 z -I s ) ( 1 -r 3 0 r 3 m )] (82)</formula>
<text><location><page_31><loc_12><loc_77><loc_49><loc_78></location>where the fact that C + 2 = -2 I 2 x + I 2 y + I 2 z was used.</text>
<text><location><page_31><loc_12><loc_72><loc_88><loc_77></location>For any given starting location, the highest ˙ φ 2 , 0 will be required if r m is as large as possible, so r m = r a . Therefore if the the libration rate is high enough to circulate the secondary at apoapse, then the body will circulate no matter what point on the orbit it reaches its maximum libration angle. Thus the bounding condition to ensure circulation occurs when</text>
<formula><location><page_31><loc_29><loc_68><loc_71><loc_71></location>˙ φ 2 2 , 0 ≥ I z, 0 I 2 z r 5 0 [ I 2 y -I 2 x + ( I 2 z -I 2 x + I 1 z -I s ) ( 1 -r 3 0 r 3 a )]</formula>
<text><location><page_31><loc_12><loc_61><loc_88><loc_67></location>2 Theorem 6.1 assumes that the orbit perturbations caused by the libration of the secondary are ignorable. Therefore the orbit can be described using Keplerian orbital elements, which will be constant. We can express the initial orbit radius in terms of the eccentricity, semi-major axis, and true anomaly as</text>
<formula><location><page_31><loc_44><loc_58><loc_88><loc_60></location>r 0 = a (1 -e 2 ) 1 + e cos f (83)</formula>
<text><location><page_31><loc_12><loc_56><loc_34><loc_57></location>and then the apoapsis radius is</text>
<text><location><page_31><loc_12><loc_53><loc_35><loc_54></location>and finally the periapse radius is</text>
<formula><location><page_31><loc_45><loc_51><loc_88><loc_52></location>r p = a (1 -e ) (85)</formula>
<text><location><page_31><loc_12><loc_48><loc_88><loc_51></location>Using these relationships, the condition for circulation can be expressed in terms of the orbital elements as,</text>
<formula><location><page_31><loc_20><loc_45><loc_88><loc_48></location>˙ φ 2 2 , 0 ≥ I z, 0 I 2 z ( 1 + e cos f a (1 -e 2 ) ) 5 [ ( I 2 y -I 2 x ) + ( I 2 z -I 2 x + I 1 z -I s ) [ 1 -(1 -e ) 3 (1 + e cos f ) 3 ]] (86)</formula>
<text><location><page_31><loc_12><loc_41><loc_88><loc_45></location>Theorem 6.1 gives the sufficient condition for circulation for a given starting point, however we can define a single condition for circulation that is sufficient for any location on the orbit by defining the highest value that leads to circulation, which is referred to as the uniform condition.</text>
<section_header_level_1><location><page_31><loc_12><loc_39><loc_58><loc_40></location>Remark 6.1. Uniform Condition for Guaranteed Circulation</section_header_level_1>
<text><location><page_31><loc_12><loc_36><loc_88><loc_39></location>The highest bound for circulation is given by maximizing Eq. (86) in terms of f . The maximum occurs when f = 0 , or r 0 = r p , and the circulation condition becomes</text>
<formula><location><page_31><loc_24><loc_32><loc_88><loc_35></location>˙ φ 2 2 , 0 ≥ I z,p I 2 z a 5 (1 -e ) 5 [ ( I 2 y -I 2 x ) + ( I 2 z -I 2 x + I 1 z -I s ) [ 2 e ( 3 + e 2 ) (1 + e ) 3 ]] (87)</formula>
<text><location><page_31><loc_12><loc_24><loc_88><loc_31></location>2 This condition is considered uniform because it combines the extremes for both r 0 and r m in terms of maximizing the required ˙ φ 2 , 0 . The condition given Eq. (87) is the baseline sufficiency test for circulation. At any point on an orbit, if ˙ φ 2 , 0 meets this condition with φ 2 = 0 · , it is guaranteed to circulate directly, meaning that the current trajectory will pass through ± 90 · before returning to φ 2 = 0 · .</text>
<text><location><page_31><loc_12><loc_12><loc_88><loc_24></location>Depending on the actual location on the orbit, the value of ˙ φ 2 , 0 which is sufficient to cause circulation will generally be lower, and is given by Eq. (73). It is important to note here that Theorem 6.1 is a sufficiency condition only, but it is possible that a lower value of ˙ φ 2 , 0 will lead to circulation. This is because Theorem 6.1 depended on the most extreme value for r m . However, due to the fact that it is easier to circulate at all r m < r a , it is true that lower libration rates will lead to circulation in many cases. This depends on the phasing of the librational motion with the orbit to determine at what radius φ 2 ,max is achieved, and if ˙ φ 2 , 0 was large enough at the φ 2 = 0 crossing preceeding this φ 2 ,max , circulation can occur. Therefore the lowest possible spin rate that can lead to circulation can be determined in the opposite limiting case where ˙ φ 2 , 0 is minimized with respect to r m and r 0 . In fact, this becomes a sufficient condition for bounded libration.</text>
<formula><location><page_31><loc_45><loc_54><loc_88><loc_56></location>r a = a (1 + e ) (84)</formula>
<section_header_level_1><location><page_32><loc_12><loc_90><loc_63><loc_91></location>Corollary 6.1. Sufficient Condition for Bounded Motion with ν glyph[similarequal] 1</section_header_level_1>
<text><location><page_32><loc_12><loc_87><loc_88><loc_90></location>Given the system considered in Theorem 6.1, the lowest value of ˙ φ 2 , 0 which can lead to circulation is computed so that any value below this will not lead to circulation</text>
<formula><location><page_32><loc_29><loc_83><loc_88><loc_86></location>˙ φ 2 2 , 0 < I z, 0 I 2 z r 5 0 [ I 2 y -I 2 x + ( I 2 z -I 2 x + I 1 z -I s ) ( 1 -r 3 0 r 3 p )] (88)</formula>
<text><location><page_32><loc_12><loc_80><loc_42><loc_81></location>where r 0 is found as the real positive root of,</text>
<formula><location><page_32><loc_25><loc_75><loc_88><loc_79></location>r 3 0 - 3 νr 3 p ( I 1 z -I s + C + 2 ) 2 I 2 z ( I 2 z -I 2 x + I 1 z -I s ) r 2 0 - 5 r 3 p ( I 1 z -I s + C + 2 ) 2 ( I 2 z -I 2 x + I 1 z -I s ) = 0 (89)</formula>
<text><location><page_32><loc_12><loc_72><loc_41><loc_73></location>if the resulting r 0 < r a , otherwise r 0 = r a .</text>
<text><location><page_32><loc_12><loc_70><loc_88><loc_72></location>Proof: Eq. 82 is minimized with respect to r 0 and r m . The partial with respect to r m shows that the function monotonically increases with r m , therefore the minima with respect to r m is at the lower boundary, r p .</text>
<text><location><page_32><loc_12><loc_63><loc_88><loc_69></location>There exists an extrema of Eq. 82 with respect to r 0 , which is located at the root of Eq. (89) . Using the Routh criteria it can be proven that there is one and only one real positive root. It can also be shown that this extrema is in fact a minima by investigating the second partial with respect to r 0 . However, the location of this minima depends on the system parameters, and it is not guaranteed to lie within the range of possible radius values given by r p and r a . Therefore, if the minima is found to be greater than r a , the constrained minima is at r 0 = r a .</text>
<text><location><page_32><loc_12><loc_60><loc_13><loc_62></location>2</text>
<text><location><page_32><loc_12><loc_55><loc_88><loc_60></location>These results outline a set of bounds that can be used to determine if a system will circulate or not depending on the libration rate at φ 2 = 0 · . The main boundaries to be tested to give information for an entire orbit are given by Remark 6.1 and Corollary 6.1. If either of these conditions are satisfied, we can immediately state if the body will or will not circulate.</text>
<text><location><page_32><loc_12><loc_47><loc_88><loc_55></location>When the libration rate is between these two boundaries, further inspection must be made. Theorem 6.1 can be used to determine if a given initial condition ( r 0 and ˙ φ 2 , 0 ) will circulate directly. However, this relationship does not tell us if the body will ever circulate; only if it will circulate directly. The difficulty is in the phasing of the libration and the orbit. If the initial libration rate is below the bound given in Theorem 6.1, then the body will not circulate this on this oscillation. However in general the body will pass back through φ 2 = 0 · at a different state, and then the condition must be checked again.</text>
<text><location><page_32><loc_12><loc_43><loc_88><loc_46></location>Corollary 6.1 is an extension of Theorem 4.1 for the specific case when ν glyph[similarequal] 1. If the total energy in the system is low enough so that Theorem 4.1 is applicable, Corollary 6.1 will also be satisfied. However, Corollary 6.1 can be satisfied when Theorem 4.1 is not applicable.</text>
<text><location><page_32><loc_12><loc_40><loc_88><loc_42></location>If circulation does not occur, it is useful to be able to determine the libration amplitudes that can be expected. This is addressed in the following corollary.</text>
<section_header_level_1><location><page_32><loc_12><loc_38><loc_44><loc_38></location>Corollary 6.2. Maximum Libration Angle</section_header_level_1>
<text><location><page_32><loc_12><loc_35><loc_88><loc_37></location>Given the system considered in Theorem 6.1, if ˙ φ 2 , 0 is below the limit for circulation, then the maximum libration angle that can be obtained is given by</text>
<formula><location><page_32><loc_25><loc_30><loc_88><loc_33></location>sin 2 φ 2 ,max = 1 3 ( I 2 y -I 2 x ) [ I 2 z r 5 0 ˙ φ 2 2 , 0 I z, 0 + ( I 1 z -I s + C + 2 ) ( 1 -r 3 p r 3 0 )] (90)</formula>
<text><location><page_32><loc_12><loc_27><loc_58><loc_28></location>Proof: This condition is determined by rearranging Eq. (81) to get</text>
<formula><location><page_32><loc_25><loc_22><loc_88><loc_25></location>sin 2 φ 2 ,max = 1 3 ( I 2 y -I 2 x ) [ I 2 z r 5 0 ˙ φ 2 2 , 0 I z, 0 + ( I 1 z -I s + C + 2 ) ( 1 -r 3 m r 3 0 ) ] (91)</formula>
<text><location><page_32><loc_12><loc_20><loc_72><loc_21></location>This relationship is maximized with a minimum r m , giving r m = r p , the periapse radius.</text>
<text><location><page_32><loc_12><loc_17><loc_13><loc_19></location>2</text>
<text><location><page_32><loc_12><loc_12><loc_88><loc_17></location>If the maximum libration relationship from Corollary 6.2 is tested with ˙ φ 2 , 0 above the circulation limit, the right hand side of Eq. (90) will be greater than one, resulting in an imaginary value for φ 2 ,max . This indicates the body will circulate. The maximum libration angle that would be reached at any point on the orbit can be determined by using a radius other than the periapse radius in Eq. (91) for r m .</text>
<section_header_level_1><location><page_33><loc_12><loc_90><loc_48><loc_91></location>Remark 6.2. Simplification to Classical Results</section_header_level_1>
<text><location><page_33><loc_12><loc_86><loc_88><loc_90></location>Classical gravity gradient results assume that the orbit is circular ( r m = r 0 = r ) and that the system is dominated by the orbit so that νr 2 glyph[greatermuch] I 2 z , and therefore I z → νr 2 , and ν = 1 . Substituting these conditions into Eq. 81 gives the classical result</text>
<formula><location><page_33><loc_38><loc_82><loc_88><loc_86></location>˙ φ 2 2 = 3 r 3 ( I 2 y -I 2 x I 2 z ) sin 2 φ 2 ,max (92)</formula>
<text><location><page_33><loc_12><loc_80><loc_64><loc_81></location>which is recognized as the energy integral of the pendulum equation of motion</text>
<formula><location><page_33><loc_39><loc_76><loc_88><loc_79></location>¨ φ 2 = -3 2 r 3 ( I 2 y -I 2 x I 2 z ) sin 2 φ 2 (93)</formula>
<text><location><page_33><loc_12><loc_71><loc_13><loc_74></location>2</text>
<text><location><page_33><loc_12><loc_68><loc_88><loc_72></location>Derivation of the classical results are given in [2] and [9] among many other sources. Note that Eq. (93) has a slight difference to the referenced results because we have normalized the units so that µ has disappeared.</text>
<text><location><page_33><loc_12><loc_58><loc_88><loc_68></location>These results can be illustrated through several numerical systems. First, a basic example is shown in Fig. 22 with very low eccentricity ( e = 2 . 4 × 10 -6 ). The full equations of motion given in Eqs. (17) - (18) are integrated for two cases, where one trajectory is initialized with 95% of the excess energy is given to ˙ φ 2 , while the other has 70% apportioned to the initial libration rate. Both cases start at r = r + and ˙ r = 0. Using Theorem 6.1, we can see that the former case is above the limit and therefore will circulate, while the latter case is below. The maximum libration angle is calculated for the κ = 0 . 3 case from Eq. (91) with r m = r + , and is found to closely agree with the maximum libration amplitudes seen during the simulation.</text>
<figure>
<location><page_33><loc_27><loc_31><loc_72><loc_55></location>
<caption>Fig. 22 Two different trajectories for the nominal case with energy 10% above the unstable equilibrium energy. The dotted horizontal line indicates the computed φ 2 ,max = 61 . 34 · from Eq. (90) for the κ = 0 . 3 trajectory. Note that on the x-axis, δr = r -r + .</caption>
</figure>
<text><location><page_33><loc_12><loc_12><loc_88><loc_24></location>The results developed in this section can also show when the excitation of the libration amplitude from the eccentricity of the orbit can cause the secondary to begin circulating. Application of Corollary 6.2 for a range of r 0 values is shown in Fig. 23. In this case, every case is tested with φ 2 , 0 = ˙ φ 2 , 0 = 0, so the excitation occurs only from the presence of available energy for this orbit due to variation in the radial position. The results are presented in terms of eccentricity and true anomaly, which determine the extents of r 0 , but are more intuitive for this application. It is shown that for this case, if the eccentricity is less than 1 × 10 -4 the libration is bounded for any initial condition. For larger eccentricities, the initial conditions must be nearer to periapsis in order for the libration to remain bounded. It should be noted that each eccentricity line is at a different energy level, but they all have the same angular momentum value for this study.</text>
<text><location><page_34><loc_12><loc_86><loc_88><loc_91></location>In order to illustrate the validity of these relationships, four different simulations indicated on Fig. 23 with circles were simulated for 500 orbits. These trajectories are plotted in Fig. 24. It is clear that the bounds are accurately predicting the librational motion, and the unbounded case does circulate directly as predicted by Theorem 6.1 for that case.</text>
<figure>
<location><page_34><loc_28><loc_61><loc_73><loc_82></location>
<caption>Fig. 23 Maximum libration amplitudes at different points on a Keplerian orbit with ˙ φ 2 = 0. When e = 0, maximum libration amplitude at apoapsis is only 0 . 34 · and at e = 1 × 10 -5 , the amplitude can reach 1 . 65 · at apoapsis. The unstable eccentricity is between e = 1 × 10 -4 and e = 2 × 10 -4 . For all eccentricities, there are some initial conditions that stay bounded; e.g. for e = 0 . 05, the libration amplitude is bounded for ν < 0 . 0125 · .</caption>
</figure>
<figure>
<location><page_34><loc_27><loc_23><loc_72><loc_50></location>
<caption>Fig. 24 Four trajectories chosen from Fig. 23, integrated for 500 orbits. Integration of the circulating trajectory was stopped upon passing φ 2 = 95 · . These numerical integrations verify the validity of the relationships shown in Fig. 23.</caption>
</figure>
<text><location><page_34><loc_12><loc_12><loc_88><loc_17></location>It is interesting to investigate the case of Saturn's moon Hyperion, which is known to be in chaotic rotation. Using the moments of inertia for Hyperion from [8] (renormalized by α = 180 km for Hyperion), and the known Hyperion orbit and Saturn properties, Eq. (87) is used to find the ˙ φ 2 that will guarantee circulation. We find that if Hyperion were to have | ˙ φ 2 | > 0 . 00524 degrees per second at periapse with zero</text>
<text><location><page_35><loc_12><loc_89><loc_88><loc_91></location>libration angle, it will enter a circulating state. Given the extremely small limit found, it is no surprise that Hyperion is in an unbound chaotic rotation state.</text>
<text><location><page_36><loc_12><loc_82><loc_88><loc_88></location>In the general case with ν < 1, the relationships developed in Section 6.1 can't be used because the orbit energy can not be considered constant due to the coupling from the secondary body's motion. In this section, we investigate which initial conditions at φ 2 = 0 circulate numerically. Our approach is to sample the possible initial conditions at φ 2 = 90 · and propagate them backwards in time to see where they originate at φ 2 = 0.</text>
<text><location><page_36><loc_12><loc_76><loc_88><loc_81></location>By fixing one of the states (we choose to fix φ 2 ), the remaining states are required to lie on the surface of an ellipsoid of constant energy. We use these constant energy surfaces to investigate the states of the simulations at φ 2 = 0 and φ 2 = 90 · . These visualizations are essentially 3-D Poincar'e surfaces. The case investigated here is for the nominal system with E = E -+0 . 1 δE .</text>
<text><location><page_36><loc_12><loc_67><loc_88><loc_76></location>The constant energy surface at φ 2 = 90 · is illustrated in Fig. 25. Initial conditions which sampled the entire surface were tested, with those that propagated back to φ 2 = 0 shown as black dots on the ellipsoid. There were 3362 initial conditions tested, exactly half propagated to φ 2 = 0, the other half propagated toward φ 2 = 180 · and were ignored for this analysis. As seen in Fig. 25, the split is basically associated with the sign of ˙ φ 2 ; those with a positive libration rate when crossing φ 2 = 90 · generally came from φ 2 = 0. However, it is interesting to note that there 22 of the 1681 cases did have negative libration rates, seen as those will small values of r on the ellipsoid.</text>
<figure>
<location><page_36><loc_22><loc_36><loc_78><loc_64></location>
<caption>Fig. 25 Constant energy surface at φ 2 = 90 · for the nominal case with E = E -+0 . 1 δE . The black dots indicate the ICs tested which, when propagated backwards in time, pass through φ 2 = 0.</caption>
</figure>
<text><location><page_36><loc_12><loc_23><loc_88><loc_29></location>The evolution of the backwards propagated trajectories are shown as they cross φ 2 = 0 in Fig. 26. The initial point where the initial conditions reach φ 2 = 0 are labeled as the zeroth crossing, and plotted also as black dots. We then plot each of the next six crossings in different colors, alternating between the north and south poles of the ellipsoid as the trajectories cross in opposite directions. Note there is definite structure to the region of each crossing on the ellipsoid.</text>
<text><location><page_36><loc_12><loc_12><loc_88><loc_22></location>Fig. 27 shows the same thing as in Fig. 26, except that all crossings after the sixth are plotted with red dots. This shows that although the coverage continues to spread and lose some of the distinct structure of the earlier crossings, there are still interesting features. First, there is a large area surrounding the equator where there are no trajectories that are connected to circulation. This tells us that trajectories that reside in this region will stay bounded. We point out that the points around the north pole are biased as a group toward the smaller radius values, while the south pole cases are biased toward the larger radius values. Second, there are some significant areas on the ellipsoids amongst the propagated trajectories that are not filled, such as on the north pole view around ˙ r = 0 and small values of r . These gaps correspond to the</text>
<figure>
<location><page_37><loc_12><loc_72><loc_88><loc_92></location>
<caption>Fig. 26 The north pole (left, positive values of ˙ φ 2 ) and south pole (right) of the constant energy surface at φ 2 = 0 for the nominal case with E = E -+0 . 1 δE . The initial conditions from Fig. 25 are integrated back in time, with each of the first seven crossings of the φ 2 = 0 surface plotted in different colors as indicated.</caption>
</figure>
<text><location><page_37><loc_12><loc_60><loc_88><loc_64></location>periodic trajectories discussed in Section 5.2. The third note about this plot is that currently the region immediately surrounding the south pole is unpopulated. These states would correspond to trajectories that would evolve through φ 2 = -90 · , as opposed to through φ 2 = 90 · as are being analyzed here.</text>
<figure>
<location><page_37><loc_12><loc_39><loc_88><loc_58></location>
<caption>Fig. 27 The north pole (left) and south pole (right) from Fig. 26, with all crossing after the seventh added as red dots.</caption>
</figure>
<text><location><page_37><loc_12><loc_20><loc_88><loc_34></location>Another view of the evolution of the backwards propagated trajectories from Figs. 25 - 27 are shown in Fig. 28. In this figure, we see the absolute value of ˙ φ 2 plotted for every trajectory for each crossing. The lower plot shows how many trajectories still exist. The main takeaway from this plot is that most the trajectories that we propagated backwards don't stay in a librational state for very long; of the 1681 initial trajectories, only about 100 cross φ 2 = 0 more than 100 times. Only 2 trajectories last longer than about 400 crossings. Interestingly, one trajectory lasts for a very long time - approximately 22,000 crossings! This tells us that a large portion of the red points seen in Fig. 27 can be attributed to one very rare case (this trajectory is shown in Fig. 29). The other thing we find from this plot is that, unlike in the ν glyph[similarequal] 1 case, there isn't a clear value of ˙ φ 2 that indicates if circulation will occur or not. This is due to the coupling between the libration and orbital states.</text>
<text><location><page_37><loc_12><loc_12><loc_88><loc_20></location>Finally, it is interesting to see how things change when the energy level is increased. Fig. 30 shows the φ 2 = 0 ellipsoid for three increasingly higher energy levels. These views show that as the energy is increased there are two main impacts. First, the trajectories evolve to much larger areas of ellipsoid, meaning there are smaller sets of initial conditions at φ 2 = 0 that correspond to bounded trajectories. Second, the unbounded trajectories reside for much shorter timespans around φ 2 = 0. In Fig. 31, we show the crossing history, which for that case has all trajectories circulating again after less than 30 crossings. We also show a stable</text>
<figure>
<location><page_38><loc_12><loc_71><loc_88><loc_91></location>
<caption>Fig. 28 Absolute values of ˙ φ 2 for each crossing (those with a negative ˙ φ 2 are plotted in blue), along with the number of trajectories which are bound at that crossing (out of the initial 1681). The left plot is zoomed in on the first 1000 crossings of the right plot; note that after approximately 700 crossings only one sample remains bound. It is clear from the right plot that this one particular long-bound case reaches much lower ˙ φ 2 values during some crossings than any of the other samples.</caption>
</figure>
<figure>
<location><page_38><loc_31><loc_42><loc_69><loc_63></location>
<caption>Fig. 29 The trajectory of the long lived case from Fig. 28 in r -φ 2 space. The initial trajectory from φ 2 = 90 · → 0 is shown in black, and the final portion of the trajectory is shown in red, before it becomes unbound, in this case toward φ 2 = -90 · .</caption>
</figure>
<text><location><page_38><loc_12><loc_32><loc_88><loc_35></location>trajectory, proving that although the ZVC is significantly opened at this energy level, there are still bounded trajectories.</text>
<section_header_level_1><location><page_38><loc_12><loc_28><loc_22><loc_29></location>7 Conclusion</section_header_level_1>
<text><location><page_38><loc_12><loc_12><loc_88><loc_26></location>This paper explored the dynamic system of a triaxial ellipsoid satellite in orbit in the equatorial plane of an oblate body. This system reduces to a 2 degree-of-freedom system in the radial separation and the libration angle of the ellipsoid. The reduced system was then analyzed with the goal of determining limits on the dynamic configurations for which the librational motion is bounded to less than ± 90 · . The relative equilibrium points were found in the ellipsoid-fixed frame, and the stability of these points was discussed. The conservation of energy and angular momentum in the system was exploited to write an expression for zero-velocity curves. These curves are used to determine a sufficiency condition on when the librational motion is bounded. It is shown that bounded trajectories exist beyond these sufficiency conditions, including families of periodic orbits. We addressed the conditions for unbounded motion for all systems. In particular, when the ellipsoid becomes of very small mass compared to the oblate body, analytical relationships were derived that determine the maximum libration angle for any orbit eccentricity.</text>
<figure>
<location><page_39><loc_15><loc_80><loc_85><loc_92></location>
<caption>Fig. 30 Three constant energy ellipsoids at φ 2 = 0 for cases with E = E -+ δE , E = E -+2 δE , and E = E -+2 . 5 δE from left to right.</caption>
</figure>
<figure>
<location><page_39><loc_12><loc_54><loc_88><loc_75></location>
<caption>Fig. 31 An example of a bounded trajectory and the crossing history for the case when E = E -+2 . 5 δE . The trajectory had initial conditions of r glyph[similarequal] 10, φ 2 = 0, ˙ r glyph[similarequal] 0 . 02 and ˙ φ 2 = 0 . 5 deg/s and was integrated for 1000 librational periods.</caption>
</figure>
<text><location><page_39><loc_12><loc_45><loc_88><loc_47></location>Future work includes extending this approach to two ellipsoid systems, non-conservative systems, and coupled out-of-plane motion.</text>
<section_header_level_1><location><page_39><loc_12><loc_41><loc_20><loc_42></location>Appendix</section_header_level_1>
<text><location><page_39><loc_12><loc_35><loc_88><loc_39></location>The mathematical description of the system studied in this paper was determined as a simplified form of the system studied by Scheeres [19]. The simplifications made to obtain the results used in this paper are derived here in order to make the connection to the previous work explicit.</text>
<section_header_level_1><location><page_39><loc_12><loc_31><loc_27><loc_32></location>Equations of Motion</section_header_level_1>
<text><location><page_39><loc_12><loc_28><loc_45><loc_29></location>The Lagrangian, L = T -V for this system is,</text>
<formula><location><page_39><loc_21><loc_25><loc_88><loc_27></location>L = 1 2 I 1 z M 1 M 2 ( ˙ θ + ˙ φ 1 ) 2 + 1 2 I 2 z ˙ φ 2 2 + 1 2 ν ˙ r 2 + 1 2 ( I 2 z + νr 2 ) ˙ θ 2 + I 2 z ˙ φ 2 ˙ θ -V ( r, φ 2 ) (94)</formula>
<text><location><page_39><loc_12><loc_23><loc_53><loc_24></location>Using Lagrange's equations with out any external forces,</text>
<formula><location><page_39><loc_44><loc_19><loc_88><loc_21></location>d dt ( ∂L ∂ ˙ q i ) = ∂L ∂q i (95)</formula>
<text><location><page_39><loc_12><loc_15><loc_88><loc_17></location>the equations of motion for this system with the coordinates r , θ , φ 1 , and φ 2 were found in Scheeres [19] and are rewritten in normalized units to be,</text>
<formula><location><page_39><loc_45><loc_11><loc_88><loc_14></location>r = ˙ θ 2 r -1 ν ∂V ∂r (96)</formula>
<formula><location><page_40><loc_34><loc_89><loc_88><loc_91></location>¨ φ 1 = -( 1 + νr 2 I 1 z ) 1 νr 2 ∂V ∂φ 1 -1 νr 2 ∂V ∂φ 2 +2 ˙ r ˙ θ r (97)</formula>
<formula><location><page_40><loc_34><loc_84><loc_88><loc_87></location>¨ φ 2 = -( 1 + νr 2 I 2 z ) 1 νr 2 ∂V ∂φ 2 -1 νr 2 ∂V ∂φ 1 +2 ˙ r ˙ θ r (98)</formula>
<formula><location><page_40><loc_40><loc_80><loc_88><loc_83></location>¨ θ = 1 νr 2 ∂V ∂φ 1 + 1 νr 2 ∂V ∂φ 2 -2 ˙ r ˙ θ r (99)</formula>
<text><location><page_40><loc_15><loc_78><loc_78><loc_79></location>However, in the case of an oblate body the potential is no longer a function of φ 1 so that</text>
<formula><location><page_40><loc_47><loc_74><loc_88><loc_77></location>∂V ∂φ 1 = 0 (100)</formula>
<text><location><page_40><loc_12><loc_72><loc_49><loc_73></location>and the equations of motion for the angles become,</text>
<formula><location><page_40><loc_42><loc_68><loc_88><loc_70></location>¨ φ 1 = -1 νr 2 ∂V ∂φ 2 +2 ˙ r ˙ θ r (101)</formula>
<formula><location><page_40><loc_38><loc_63><loc_88><loc_66></location>¨ φ 2 = -( 1 + νr 2 I 2 z ) 1 νr 2 ∂V ∂φ 2 +2 ˙ r ˙ θ r (102)</formula>
<formula><location><page_40><loc_43><loc_59><loc_88><loc_62></location>¨ θ = 1 νr 2 ∂V ∂φ 2 -2 ˙ r ˙ θ r (103)</formula>
<text><location><page_40><loc_15><loc_57><loc_54><loc_58></location>The partials of the potential for an oblate primary are,</text>
<formula><location><page_40><loc_23><loc_53><loc_88><loc_55></location>∂V ∂r = ν r 2 { 1 + 3 2 r 2 [ ( I 1 z -I s ) -1 2 I 2 x -1 2 I 2 y + I 2 z + 3 2 ( I 2 y -I 2 x ) cos 2 φ 2 ]} (104)</formula>
<formula><location><page_40><loc_39><loc_48><loc_88><loc_51></location>∂V ∂φ 2 = 3 2 ν r 3 ( I 2 y -I 2 x ) sin(2 φ 2 ) (105)</formula>
<text><location><page_40><loc_12><loc_44><loc_26><loc_45></location>Integrals of Motion</text>
<text><location><page_40><loc_12><loc_40><loc_88><loc_42></location>In the current problem with an oblate primary, we have three integrals of motion. The first is the total energy of the system, which is shown to be the Jacobi integral of this system since it is time invariant,</text>
<formula><location><page_40><loc_42><loc_36><loc_88><loc_38></location>h = ˙ q · ∂L ∂ ˙ q -L = T + V (106)</formula>
<text><location><page_40><loc_12><loc_32><loc_88><loc_34></location>The second integral of motion is the total angular momentum of the system. This is found because the coordinate θ is ignorable, meaning that d / dt( ∂L/∂ ˙ θ ) = 0, so the integral is written as,</text>
<formula><location><page_40><loc_39><loc_25><loc_88><loc_31></location>K tot = ∂L ∂ ˙ θ = I z ˙ θ + I 2 z ˙ φ 2 + M 1 M 2 I 1 z ˙ θ 1 (107)</formula>
<text><location><page_40><loc_15><loc_23><loc_71><loc_24></location>The third integral is found by combining Eqs. (101) and (103), so we find that</text>
<formula><location><page_40><loc_45><loc_20><loc_88><loc_21></location>¨ θ 1 = ¨ φ 1 + ¨ θ = 0 (108)</formula>
<text><location><page_40><loc_12><loc_12><loc_88><loc_19></location>Therefore the inertial angular velocity of the primary, ˙ θ 1 , is an integral of motion. This implies that the terms in the kinetic energy and angular momentum expressions which depend only on ˙ θ 1 are also conserved. This fact makes intuitive sense as a primary that is symmetric about the spin axis can't have any gravitational torques exerted on it from the secondary since the center of mass and the center of gravity (in the secondary's gravity field) are at the same location in the primary.</text>
<text><location><page_41><loc_12><loc_90><loc_31><loc_91></location>Dynamics Matrix Partials</text>
<text><location><page_41><loc_12><loc_87><loc_24><loc_88></location>The partials are,</text>
<formula><location><page_41><loc_14><loc_84><loc_88><loc_87></location>∂ r ∂r = 1 I 2 z [ K 2 -2 KI 2 z ˙ φ 2 + I 2 2 z ˙ φ 2 2 ] -1 I 4 z [( 4 I 2 z νr +4 ν 2 r 3 )( K 2 -2 KI 2 z ˙ φ 2 + I 2 2 z ˙ φ 2 2 ) r ] -1 ν ∂ 2 V ∂r 2 (109)</formula>
<formula><location><page_41><loc_44><loc_80><loc_88><loc_83></location>∂ r ∂φ 2 = -1 ν ∂ 2 V ∂r∂φ 2 (110)</formula>
<formula><location><page_41><loc_40><loc_77><loc_88><loc_79></location>∂ r ∂ ˙ φ 2 = 2 r I 2 z [ -KI 2 z + I 2 2 z ˙ φ 2 ] (111)</formula>
<formula><location><page_41><loc_23><loc_73><loc_88><loc_76></location>∂ ¨ φ 2 ∂r = 2 νr 3 ∂V ∂φ 2 -( 1 + νr 2 I 2 z ) 1 νr 2 ∂ 2 V ∂r∂φ 2 -2˙ r r 2 I 2 z [( K -I 2 z ˙ φ 2 )( I 2 z +3 νr 2 )] (112)</formula>
<formula><location><page_41><loc_40><loc_70><loc_88><loc_73></location>∂ ¨ φ 2 ∂φ 2 = -( 1 νr 2 + 1 I 2 z ) ∂ 2 V ∂φ 2 2 (113)</formula>
<formula><location><page_41><loc_42><loc_66><loc_88><loc_69></location>∂ ¨ φ 2 ∂ ˙ r = 2 rI z ( K -I 2 z ˙ φ 2 ) (114)</formula>
<formula><location><page_41><loc_45><loc_63><loc_88><loc_65></location>∂ ¨ φ 2 ∂ ˙ φ 2 = -2 I 2 z ˙ r rI z (115)</formula>
<text><location><page_41><loc_15><loc_61><loc_53><loc_62></location>And the second partials of the potential are given by,</text>
<formula><location><page_41><loc_22><loc_58><loc_88><loc_60></location>∂ 2 V ∂r 2 = -2 ν r 3 { 1 + 3 r 2 [ ( I 1 z -I s ) -1 2 I 2 x -1 2 I 2 y + I 2 z + 3 2 ( I 2 y -I 2 x ) cos 2 φ 2 ]} (116)</formula>
<formula><location><page_41><loc_40><loc_54><loc_88><loc_57></location>∂ 2 V ∂φ 2 2 = 3 ν r 3 ( I 2 y -I 2 x ) cos 2 φ 2 (117)</formula>
<formula><location><page_41><loc_38><loc_50><loc_88><loc_53></location>∂ 2 V ∂r∂φ 2 = -9 ν 2 r 4 ( I 2 y -I 2 x ) sin 2 φ 2 (118)</formula>
<text><location><page_41><loc_12><loc_47><loc_33><loc_48></location>Relative Equilibria Locations</text>
<text><location><page_41><loc_12><loc_42><loc_88><loc_45></location>Following Scheeres [19], we find the equilibrium points by searching for places where the variations in energy are stationary at a constant value of angular momentum. In other words, we find when the following conditions hold:</text>
<formula><location><page_41><loc_47><loc_39><loc_88><loc_42></location>∂E ∂r = 0 (119)</formula>
<formula><location><page_41><loc_47><loc_37><loc_88><loc_39></location>∂E ∂ ˙ r = 0 (120)</formula>
<formula><location><page_41><loc_47><loc_34><loc_88><loc_37></location>∂E ∂φ 2 = 0 (121)</formula>
<formula><location><page_41><loc_47><loc_31><loc_88><loc_34></location>∂E ∂ ˙ φ 2 = 0 (122)</formula>
<formula><location><page_41><loc_47><loc_26><loc_88><loc_28></location>∂E ∂ ˙ r = ν ˙ r (123)</formula>
<formula><location><page_41><loc_44><loc_23><loc_88><loc_26></location>∂E ∂ ˙ φ 2 = I 2 z νr 2 ˙ φ 2 I z (124)</formula>
<formula><location><page_41><loc_39><loc_20><loc_88><loc_22></location>∂E ∂φ 2 = 3 2 ν r 3 ( I 2 y -I 2 x ) sin(2 φ 2 ) (125)</formula>
<text><location><page_41><loc_12><loc_17><loc_88><loc_20></location>which when combined with the stationarity conditions imply, respectively, that ˙ r = 0, ˙ φ 2 = 0, and φ 2 = 0, ± π/ 2, or π . Using these conditions, we can evaluate the partial with respect to r to be,</text>
<formula><location><page_41><loc_34><loc_13><loc_88><loc_16></location>∂E ∂r = -νr K 2 I 2 z + ν r 2 1 + 3 ( I 1 z -I s + C ± 2 ) 2 r 2 (126)</formula>
<text><location><page_41><loc_12><loc_30><loc_42><loc_31></location>at a given value of K , as seen in Eq. (15).</text>
<text><location><page_41><loc_15><loc_28><loc_42><loc_29></location>The following relationships are found,</text>
<text><location><page_42><loc_12><loc_90><loc_38><loc_91></location>Relative Equilibria Stability Partials</text>
<text><location><page_42><loc_12><loc_86><loc_88><loc_89></location>Recall from Section 3, the equilibrium points must occur at ˙ r = 0, ˙ φ 2 = 0, and φ 2 = 0, ± π/ 2, or π . Therefore the partials for the dynamic matrix are greatly reduced to become,</text>
<formula><location><page_42><loc_29><loc_82><loc_88><loc_85></location>∂ r ∂r = K 2 I 2 z -1 I 4 z [( 4 I 2 z νr eq +4 ν 2 r 3 eq ) K 2 r eq ] -1 ν ( ∂ 2 V ∂r 2 ) eq (127)</formula>
<formula><location><page_42><loc_42><loc_78><loc_88><loc_81></location>∂ r ∂φ 2 = -1 ν ( ∂ 2 V ∂r∂φ 2 ) eq (128)</formula>
<formula><location><page_42><loc_45><loc_74><loc_88><loc_77></location>∂ r ∂ ˙ φ 2 = -2 r eq K I 2 z (129)</formula>
<formula><location><page_42><loc_31><loc_70><loc_88><loc_74></location>∂ ¨ φ 2 ∂r = 2 νr 3 eq ( ∂V ∂φ 2 ) eq -( 1 + νr 2 eq I 2 z ) 1 νr 2 eq ( ∂ 2 V ∂r∂φ 2 ) eq (130)</formula>
<formula><location><page_42><loc_38><loc_67><loc_88><loc_70></location>∂ ¨ φ 2 ∂φ 2 = -( 1 νr 2 eq + 1 I 2 z )( ∂ 2 V ∂φ 2 2 ) eq (131)</formula>
<formula><location><page_42><loc_46><loc_63><loc_88><loc_66></location>∂ ¨ φ 2 ∂ ˙ r = 2 K r eq I z (132)</formula>
<text><location><page_42><loc_47><loc_61><loc_48><loc_62></location>∂</text>
<text><location><page_42><loc_47><loc_59><loc_48><loc_60></location>∂</text>
<text><location><page_42><loc_48><loc_61><loc_49><loc_62></location>¨</text>
<text><location><page_42><loc_48><loc_61><loc_49><loc_62></location>φ</text>
<text><location><page_42><loc_48><loc_59><loc_49><loc_60></location>˙</text>
<text><location><page_42><loc_49><loc_61><loc_50><loc_61></location>2</text>
<text><location><page_42><loc_48><loc_59><loc_49><loc_60></location>φ</text>
<text><location><page_42><loc_49><loc_59><loc_50><loc_60></location>2</text>
<text><location><page_42><loc_12><loc_57><loc_81><loc_58></location>Where the equilibrium point distance is indicated by r eq . Note that from Eq. (105), we see that</text>
<formula><location><page_42><loc_45><loc_53><loc_88><loc_56></location>( ∂V ∂φ 2 ) eq = 0 (134)</formula>
<text><location><page_42><loc_15><loc_51><loc_69><loc_52></location>The second partials of the potential evaluated at the equilibrium points are,</text>
<formula><location><page_42><loc_35><loc_47><loc_88><loc_50></location>∂ 2 V ∂r 2 = -2 ν r 3 eq { 1 + 3 r 2 eq [ ( I 1 z -I s ) + C ± 2 ]} (135)</formula>
<formula><location><page_42><loc_42><loc_43><loc_88><loc_46></location>∂ 2 V ∂φ 2 2 = ± 3 ν r 3 eq ( I 2 y -I 2 x ) (136)</formula>
<formula><location><page_42><loc_46><loc_40><loc_88><loc_43></location>∂ 2 V ∂r∂φ 2 = 0 (137)</formula>
<text><location><page_42><loc_12><loc_37><loc_88><loc_39></location>where C ± 2 was defined in Eq. (21). The plus terms correspond to the case where φ 2 = 0 or π , and the minus terms correspond the the other cases where φ 2 = π/ 2 or 3 π/ 2.</text>
<section_header_level_1><location><page_42><loc_12><loc_33><loc_20><loc_34></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_42><loc_13><loc_30><loc_79><loc_31></location>1. Auelmann, R.R.: Regions of libration for a symmetrical satellite. AIAA Journal 1 (6), 1445-1447 (1963)</list_item>
<list_item><location><page_42><loc_13><loc_28><loc_88><loc_30></location>2. Beletskii, V.V.: Motion of an Artificial Satellite About its Center of Mass. Israel Program for Scientific Translations (1966)</list_item>
<list_item><location><page_42><loc_13><loc_26><loc_88><loc_28></location>3. Bellerose, J.E., Scheeres, D.J.: Energy and stability in the full two body problem. Celestial Mechanics and Dynamical Astronomy 100 (1), 63-91 (2008)</list_item>
<list_item><location><page_42><loc_13><loc_24><loc_88><loc_26></location>4. Cendra, H., Marsden, J.E.: Geometric mechanics and the dynamics of asteroid pairs. Dynamical Systems 20 (1), 3-21 (2005)</list_item>
<list_item><location><page_42><loc_13><loc_22><loc_88><loc_24></location>5. Conley, C.C.: Low energy transit orbits in the restricted three-body problem. SIAM Journal of Applied Mathematics 16 (4), 732-746 (1968)</list_item>
<list_item><location><page_42><loc_13><loc_20><loc_88><loc_21></location>6. Fahnestock, E.G., Scheeres, D.J.: Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach. Celestial Mechanics and Dynamical Astronomy 96 , 317-339 (2006)</list_item>
<list_item><location><page_42><loc_13><loc_17><loc_88><loc_19></location>7. Fahnestock, E.G., Scheeres, D.J.: Simulation and analysis of the dynamics of binary near-earth asteroid (66391) 1999 kw4. Icarus 194 , 410-435 (2008)</list_item>
<list_item><location><page_42><loc_13><loc_15><loc_88><loc_17></location>8. Harbison, R.A., Thomas, P.C., Nicholson, P.C.: Rotational modeling of hyperion. Celestial Mechanics and Dynamical Astronomy 110 , 1-16 (2011)</list_item>
<list_item><location><page_42><loc_13><loc_14><loc_60><loc_15></location>9. Hughes, P.C.: Spacecraft Attitude Dynamics. Dover Publications (2004)</list_item>
<list_item><location><page_42><loc_12><loc_12><loc_88><loc_14></location>10. Koon, W.S., et al.: Geometric mechanics and the dynamics of asteroid pairs. In: E. Belbruno, D. Folta, P. Gurfil (eds.) Astrodynamics, Space Missions, and Chaos, vol. 1017, pp. 11-38. Annals of the New York Academy of Science (2004)</list_item>
</unordered_list>
<text><location><page_42><loc_50><loc_60><loc_53><loc_61></location>= 0</text>
<text><location><page_42><loc_84><loc_60><loc_88><loc_61></location>(133)</text>
<unordered_list>
<list_item><location><page_43><loc_12><loc_89><loc_88><loc_91></location>11. Maciejewski, A.J.: Reduction, relative equilbria and potential in the two rigid bodies problem. Celestial Mechanics and Dynamical Astronomy 63 , 1-28 (1995)</list_item>
<list_item><location><page_43><loc_12><loc_88><loc_79><loc_89></location>12. Margot, J., et al.: Binary asteroids in the near-earth object population. Science 296 , 1445-1448 (2002)</list_item>
<list_item><location><page_43><loc_12><loc_86><loc_88><loc_88></location>13. Mohan, S.N., Breakwell, J.V., Lange, B.O.: Interaction between attitude libration and orbital motion of a rigid body in a near keplerian orbit of low eccentricity. Celestial Mechanics 5 , 157-173 (1972)</list_item>
<list_item><location><page_43><loc_12><loc_85><loc_85><loc_85></location>14. Ostro, S.J., et al.: Radar imaging of binary near-earth asteroid (66391) 1999 kw4. Science 314 , 1276-1280 (2006)</list_item>
<list_item><location><page_43><loc_12><loc_84><loc_77><loc_84></location>15. Pringle, R.: Bounds on the librations of a symmetrical satellite. AIAA Journal 2 (5), 908-912 (1963)</list_item>
<list_item><location><page_43><loc_12><loc_81><loc_88><loc_83></location>16. Scheeres, D.J.: Stability in the full two body problem. Celestial Mechanics and Dynamical Astronomy 83 , 155-169 (2002)</list_item>
<list_item><location><page_43><loc_12><loc_80><loc_87><loc_81></location>17. Scheeres, D.J.: Stability of relative equilibria in the full two-body problem. In: New Trends in Astrodynamics (2003)</list_item>
<list_item><location><page_43><loc_12><loc_79><loc_70><loc_80></location>18. Scheeres, D.J.: Rotational fission of contact binary asteroids. Icarus 189 , 370-385 (2007)</list_item>
<list_item><location><page_43><loc_12><loc_77><loc_88><loc_79></location>19. Scheeres, D.J.: Stability of the planar full 2-body problem. Celestial Mechanics and Dynamical Astronomy 104 , 103-128 (2009)</list_item>
<list_item><location><page_43><loc_12><loc_75><loc_88><loc_77></location>20. Sincarsin, G.B., Hughes, P.C.: Gravitational orbit-attitude coupling for very large spacecraft. Celestial Mechanics 31 , 143-161 (1983)</list_item>
<list_item><location><page_43><loc_12><loc_73><loc_88><loc_75></location>21. Wang, L., Krishnaprasad, P., Maddocks, J.: Hamiltonian dynamics of a rigid body in a central gravitational field. Celestial Mechanics and Dynamical Astronomy 50 , 349-386 (1991)</list_item>
</unordered_list>
</document>
[
{
"title": "Dynamic Limits on Planar Libration-Orbit Coupling Around an Oblate Primary",
"content": "Jay W. McMahon · Daniel J. Scheeres Received: date / Accepted: date Abstract This paper explores the dynamic properties of the planar system of an ellipsoidal satellite in an equatorial orbit about an oblate primary. In particular, we investigate the conditions for which the satellite is bound in librational motion or when the satellite will circulate with respect to the primary. We find the existence of stable equilibrium points about which the satellite can librate, and explore both the linearized and non-linear dynamics around these points. Absolute bounds are placed on the phase space of the libration-orbit coupling through the use of zero-velocity curves that exist in the system. These zerovelocity curves are used to derive a sufficient condition for when the satellite's libration is bound to less than 90 · . When this condition is not satisfied so that circulation of the satellite is possible, the initial conditions at zero libration angle are determined which lead to circulation of the satellite. Exact analytical conditions for circulation and the maximum libration angle are derived for the case of a small satellite in orbits of any eccentricity. Keywords Libration · Gravity Gradient · Binary Asteroids · Full Two-body Problem · Libration-Orbit Coupling",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The investigation of the translational-rotational coupling for a finite orbiting body, referred to in the literature as the Full Two Body Problem, has received renewed attention in recent years [21,11,10,4,16]. Much of this work has been motivated by interest in the dynamic evolution of binary asteroid systems, which comprise 16% of Near-Earth asteroids [12]. Scheeres [18] studied the stability of bodies resting on one-another which can lead to the formation of a binary asteroid system through rotational fission. Bellerose [3] looked at the dynamics and stability in the planar Full Two Body Problem from an energetic standpoint. Fahenstock [6] studied the problem by modeling the gravitational interaction with polyhedral models, which were applied to simulate the dynamics of the near-Earth binary asteroid 1999 KW4 [7]. Over time, many simplifications have been made to the Full Two Body Problem in order to make analytical progress in the study of specific applications. The most common simplification is to treat the finite bodies to second-order in their mass properties, which allows for the use of inertia matrices for modeling the attitude dynamics [9,2,19,3]. Analytical expressions have been constructed to fourth order [20], however most analytical studies stop at second order. A second simplification comes by reducing the problem to the planar problem. If the second order simplification is made on mass properties, then the in orbit plane libration angle (commonly referred to as the pitch angle) dynamics are decoupled from the out-of-plane (roll/yaw) dynamics if they are initially quiescent [9]. This reduces the dimensionality of the problem from six to three degrees-of-freedom. The third major simplification that is often seen, particularly in the spacecraft community, is decoupling the translational and rotational dynamics [9,2]. In Daniel J. Scheeres this case, it is assumed that the mass of the orbiting body is insignificant compared to the primary, so that the perturbation to the orbital dynamics is negligible. The orbital dynamics are known from the given Keplerian motion, and these are used as inputs to the attitude dynamics. A notable work which explores the effects of the coupling for the case of spacecraft sized objects was carried out by Mohan [13]. In this paper, we analyze the problem of a triaxial body in orbit about a spherical primary, or in an equatorial orbit about an oblate primary. These bodies are represented to second order with their moments of inertia, but all coupling between the translation and attitude motions for this case are preserved. This model can be used as a first order representation of common dynamic problems including binary asteroids (secondaries tend to be in near equatorial orbits of oblate primaries [14]), spacecraft orbiters, and planetmoon systems. This paper is closely related to those by Scheeres [19] and Bellerose [3], however we extend the foundation laid by those works. The main contribution of this paper is the study of the limits of librational motion for a given system. There is a small amount of literature dedicated to bounding librational motion. Auelmann [1] laid out the bounding conditions for an axially symmetric spacecraft in circular orbit, using the uncoupled attitude dynamics with a zero spin rate about the axis of symmetry. Pringle [15] extended this idea, analyzing the equilibrium cases and librational bounds for all spin states. This paper only looks at the planar libration, however we account for coupling, triaxial shapes, and are not limited to circular orbits, or indeed even Keplerian orbits due to the perturbations from the attitude coupling. The methodology used to bound the libration is initially similar to [1,15], in that we use the energy to limit the amount of libration that is dynamically possible by computing zero-velocity curves for the system. We extend this, however, to cases when the energy is high enough so that circulation is possible. In this case, we show that although the energy zero-velocity curve does not bound the libration angle, certain initial conditions can lead to trajectories which have bounded maximum libration angles. In particular, for the case where the ellipsoid is very small compared to the other body (such as a spacecraft in Earth orbit), we can derive analytical expressions which predict the maximum libration amplitude for an orbit of any eccentricity. For cases with significant coupling, we present a method of analysis which determines the structure of bound and unbound trajectories. The paper is organized as follows. Section 2 reviews the mathematical description of the system in question as originally derived by Scheeres [19]. Section 3 discusses the location and properties of equilibria in the system. Section 4 presents a sufficient condition for bounded librational motion. The fact that this is only a sufficient condition is shown in Section 5. Finally, Section 6 presents the main results of the paper which describe conditions for unbound libration.",
"pages": [
1,
2
]
},
{
"title": "2 Physical System Description",
"content": "In general, we begin with the planar full two-body problem where both bodies are finite, but constrained to have their individual rotation poles perpendicular to the mutual orbit plane. This situation is illustrated in Fig. 1 with the angles of interest and radius defined. In this section we present the key mathematical relationships used to describe this system, which are simplifications of the general relationships derived in [19]. Further details of the derivations are available in the Appendix.",
"pages": [
2,
3
]
},
{
"title": "2.1 Equations of Motion",
"content": "Scheeres [19] showed that in this case, the potential energy using a second order expansion in the moments of inertia is, where I ' i is the mass normalized (signified by the prime) inertia dyad of body i , and the subscripts on the non-bold versions indicate a principle moment of inertia. G is the gravitational constant, M i the mass of body i , Tr() indicates the trace of the dyad. In this paper, we will normalize the lengths with respect to the maximum ellipsoid semi-axis, α ; the time will be normalized by the mean motion of the system at this distance, n = √ G ( M 1 + M 2 ) /α 3 (units of 1/s); and the mass will be normalized by the ellipsoid mass, M 2 . Note that using these normalization factors, the normalized value of µ = G ( M 1 + M 2 ) is 1. All equations from this point on will be written in normalized units. First, let us define the reduced mass as, the mass fraction is then, which is just the reduced mass in normalized units. Next, we define the general moment of inertia as, Note that the general moment of inertia is a function of r . The normalized moment of intertias are computed as For a system with an oblate primary, we have that I 1 x = I 1 y = I s , where I s < I 1 z , so the potential becomes in normalized units, In the case of a spherical primary, which means that I 1 x = I 1 y = I 1 z , and therefore the potential simplifies to, Note that in both the spherical and oblate primary case, the spherical symmetry causes the potential energy to no longer be dependent on the primary's orientation, as represented in the planar problem by φ 1 . Starting from Scheeres' [19] expression, in this case the kinetic energy is, where ˙ θ 1 = ˙ φ 1 + ˙ θ . Note that the dot indicates derivatives with respect to unit-less time.",
"pages": [
3
]
},
{
"title": "2.2 Integrals of Motion for the System",
"content": "The system described in Section 2.1 has three integrals of motion that can be used to simplify the problem: the Jacobi constant (or total energy), the total angular momentum ( K tot ), and the inertial angular velocity of the primary ( ˙ θ 1 ). That these quantities are integrals is shown in the Appendix. Using the integrals, the kinetic energy from Eq. (9) can be rewritten as where T 1 = ( M 1 / (2 M 2 )) I 1 z ˙ θ 2 1 is the kinetic energy of the primary, which is constant. Likewise, we can define the free angular momentum from Eq. (11) to be, where K 1 = ( M 1 /M 2 ) I 1 z ˙ θ 1 is the angular momentum of the primary, which is constant. This relationship between the free angular momentum and the orbit angular velocity will allow us to eliminate ˙ θ from the system. Solving Eq. (11) for the angular velocity gives, Substituting Eq. (12) into Eq. (10) allows us to write the kinetic energy as, so that the total energy of the system can be written as, or alternatively the free energy of the system can be written as, It is interesting to note that the description of the free energy of the system does not require any knowledge about the primary spin state as neither φ 1 or ˙ φ 1 appear anywhere in Eq. (15).",
"pages": [
4
]
},
{
"title": "2.3 Dynamic System Analysis",
"content": "In order to look at the stability of any relative states for the planar system, we derive the dynamic matrix corresponding to the equations of motion where q = [ r θ φ 1 φ 2 ]. First, however, we note that we can reduce the order of the system by recalling from Section 2.2 that φ 1 is ignorable, and that through the conservation of angular momentum we can remove θ through the use of Eq. (12). Therefore the equations of motion for the 2 degree-of-freedom system are, This system is depicted from an ellipsoid fixed frame in Fig. (2). This frame rotates with the orbit, so that if there is no libration of the secondary, the primary location will be fixed. When the secondary is librating, the primary will appear to move relative to the secondary due to the libration, as well as changing the separation distance as energy is traded between the orbit and the secondary libration. The dynamic matrix is now a 4x4 matrix with the form, where the state vector has been organized as, x = [ r φ 2 ˙ r ˙ φ 2 ]. This matrix will be used in the following sections to analyze the stability of the system around any relative equilibrium points that may exist.",
"pages": [
4,
5
]
},
{
"title": "3 Equilibrium Conditions",
"content": "In this section, we determine the relative equilibria between the two bodies. This means that these equilibria are places where the relative dynamics given by Eqs. (17) and (18) are stationary. However, the primary is free to rotate at a constant rate, and the secondary (ellipsoidal) body will orbit at a constant radius and orbit rate ( ˙ θ ). As in Section 2, many of the relationships given here are simplifications of the general relationships derived in [19]. Further details are available in the Appendix.",
"pages": [
5
]
},
{
"title": "3.1 Relative Equilibrium Point Locations",
"content": "The relative equilibrium locations are found by solving the following polynomial for r , where Note that the relative equilibria found here is identical to finding the equilibria of the second-order system given in Eqs. (17) and (18). The stationary conditions for energy also enforce that ˙ r = r = ˙ φ 2 = ¨ φ 2 = 0. It is interesting to compare this result to the full case from Scheeres [19] when there is no assumption on the shape of the primary. In that case we have a dependence on φ 1 , and the stationary conditions also tell us that ˙ φ 1 = 0 and φ 1 = 0, ± π/ 2, or π . Recalling the equation from Scheeres [19], The differences between Eq. (20) and (22) (aside from the normalization) are that for the case when the primary is not exactly symmetric about its rotation axis, we must consider the entire angular momentum, K tot , and therefore we also see I 1 z appearing in terms with I 2 z . Also, as the primary body becomes spherically symmetric, the dependence on the primary orientation disappears and C ± 1 → I 1 z -I s . This causes the system to decrease the total number of equilibrium configurations from 8 to 4, and since all the dependence on the primary has been removed, the primary can also be spinning at an arbitrary speed, ˙ φ 1 , and the system can still be in a relative equilibrium. The location of the equilibrium points depends on three main parameters: the mass ratio, the angular momentum, and the body shapes. The influences of these three parameters are studied in Section 3.5. Throughout the remainder of the paper, any reference to a nominal system refers to the system outlined in Table 1, which are based on the binary asteroid system 1999 KW4 [14,7]. The new parameter that appears in Table 1 is which is the ratio of the polar radius to the equatorial radius of the oblate primary body. This parameter becomes important for scaling the system. Table 1 The non-dimensionalized parameters for the nominal test system. The nominal system has an angular momentum of K = 2 . 8382, which results in equilibria at r + = 9 . 2442 with E + = -0 . 0497, and r -= 9 . 2869 with E -= -0 . 0496 where the superscripts indicate which equilibrium point we are referring to; the + indicates the equilibrium point at φ 2 = 0, and the -refers to the equilibrium at φ 2 = 90 · , as with C ± 2 .",
"pages": [
5,
6
]
},
{
"title": "3.2 Equilibrium Point Linear Stability Analysis",
"content": "Given that the linearized system near the equilibrium point is simplified significantly, we can find the eigenvalues of the system analytically. The dynamics matrix around the equilibrium points has the form, where the non-zero partials ( ∂ r ∂r , ∂ r ∂ ˙ φ 2 , ∂ ¨ φ 2 ∂φ 2 , and ∂ ¨ φ 2 ∂ ˙ r ) have been represented by simpler variable expressions ( P , Q , R , and S ). The characteristic equation for this matrix is simply, The roots of this equation are, The determination of the eigenvalues can easily be carried out numerically on a case-by-case basis to determine the dynamic behavior around the equilibrium point. The eigenvectors can also be computed analytically by writing the eigenvalue problem as, where the eigenvector is v = [ v 1 v 2 v 3 v 4 ] T . The four equations that result are, The eigenvector can be computed from these relationships to be where In the nominal system, the eigenvalues of the equilibrium point r + are which makes it a center point. However, we will refer to this equilibrium point as stable in the remainder of the study since trajectories can be bound around it, as will be explored. The eigenvalues of the equilibrium point at r -are This equilibrium point has a unstable and stable asymptote associated with the real eigenvalues. It is interesting to note that the eigenstructure of this problem is nearly identical to that of the collinear Lagrange points in the restricted three-body problem studied by Conley [5].",
"pages": [
6,
7
]
},
{
"title": "3.3 Equilibrium Point Energy",
"content": "It has been shown in Section 3.1 that the equilibrium points are stationary points for the total energy. In this section, we determine if these stationary points are local maxima, minima, or saddle points for the energy. To find this, we must investigate the second derivatives of the energy with respect to the state, where the subscripts indicate partial derivatives. Through investigation of Eqs. (123) - (125), it is quickly clear that, The only cross second derivative remaining is with respect to r and φ 2 . Looking back at the energy in Eq. (15), we see that the only part of the energy which contains both of these variables is the potential. Therefore, where the equality to zero at the equilibrium point was shown in Eq. (137). We have shown that all of the cross partials are zero, so that E xx is a diagonal matrix. The diagonal entries of the matrix are the eigenvalues, and the definiteness of the matrix is found by looking at the sign of the eigenvalues. The diagonal entries are, These results are straight forward to interpret. The velocity partials, E ˙ r ˙ r and E ˙ φ 2 ˙ φ 2 are always greater than zero. The angle partial, E φ 2 φ 2 , is positive for the equilibrium points at φ 2 = 0 or π , and is negative for the equilibrium points at φ 2 = ± π/ 2. The position partial, E rr , is much more complicated and can be positive or negative depending on the system parameters and location of the equilibrium point. Combined, this tells us that an equilibrium point at φ 2 = 0 or π can be a local minimum or a saddle point depending on the sign of E rr . An equilibrium point at φ 2 = ± π/ 2 is always a saddle point. The fact that the only energetically stable solutions are found at φ 2 = 0 solutions was first shown by [17]. In the nominal case, r + is energetically stable, and is a local minimum in energy. The other equilibrium point, r -, is an energetic saddle, being a local minimum in the radial direction, but a local maximum in the φ 2 direction.",
"pages": [
7,
8
]
},
{
"title": "3.4 Osculating Orbit Elements at Equilibrium",
"content": "In considering a planar problem the orbital elements of interest are the semi-major axis, eccentricity, true anomaly, and argument of periapse. The behavior of these elements at equilibrium are discussed in this section. The other orbital elements, namely inclination and argument of the node are effectively meaningless, and won't be discussed here since they are either constant, undefined, and/or always zero. We can derive the Keplerian energy and angular momentum as, where v is the inertial velocity vector. At equilibrium, which was derived from Eqs. (12) and by setting (126) equal to zero as at equilibrium. The Keplerian energy and angular momentum at equilibrium are then determined as, Then the osculating semi-major axis and eccentricity can be computed in terms of the energy and angular momentum as, so that Using Eqs. (50) and (51), we find that at equilibrium we have Due to the relationship in Eq. (51), we can see that e > 0 at the φ = 0 equilibrium points because I 1 z -I s ≥ 0 and C + 2 > 0. At the φ = 90 equilibrium points, e can be positive, negative, or zero because C -2 can be negative. This tells us that at equilbrium the system is always locked at periapse or apoapse, depending on the sign of e . However, at a relative equilibria, the orbit rate is given by Eq. (47), and is constant. On a Keplerian orbit, this would imply that the orbit is circular, and this orbit rate is identical to the mean motion. In this case the eccentricity is in general non-zero and constant, and the semi-major axis is constant, along with the radius. Combining these results indicates that in fact the true/mean anomaly are constant (equal to 0 or 180 · as discussed above) and the argument of perigee is precessing at the orbit rate to enforce this condition. This can be shown by investigating the evolution of the eccentricity vector, which points to periapse. The eccentricity vector is defined by, and in the secondary fixed frame at equilibrium the eccentricity vector becomes through use of Eqs. (47) and (49) where e was defined in Eq. (52). The rate of change of the eccentricity vector can be determined by using the transport theorem and resolving in the secondary fixed frame. At the φ 2 = 0 equilibrium point it becomes, and at the φ 2 = 90 · equilibrium point it becomes, Given that in this rotating frame, the inertial rate of change of the unit vectors are we can see that the angle between an equilibrium point radius vector and the eccentricity vector is constant; therefore the anomalies (true, mean, and eccentric) are fixed at 0 or 180 · . If we assume that the argument of periapse is measured from a fixed direction in the ˆ x -ˆ y plane, then we can see that the rate of change of the argument of periapse is exactly equal to ˙ θ . In summary, a system which is at equilibrium will appear to an outside observer to be moving on a circular orbit. Due to the stability of the equilibrium points, this will likely only actually occur at the φ 2 = 0 point, which is commonly referred to as a synchronous orbit. If such a system is observed and fitted to Keplerian dynamics only, modeling each body as a point mass (or sphere), the result would be a circular orbit with an eccentricity of zero. The computed semi-major axis would then imply an incorrect µ , and thus an incorrect mass of the system. Therefore we reiterate that it is crucial to account for the non-spherical shape of both bodies when fitting orbits to celestial objects. For reference and later comparison, the nominal system has a semi-major axis of a + = 9 . 3269 and an eccentricity of e + = 8 . 87 × 10 -3 at the φ 2 = 0 equilibrium point. The values at the φ 2 = 90 · equilibrium point are a -= 9 . 3270 and e -= 4 . 31 × 10 -3 . The basis for a system to have librational motion is in the properties of the equilibirum points. We have shown in the preceding sections that the equilibrium point at φ 2 = 0 is spectrally stable and an energetic minimum; therefore this is the equilibrium point about with the system will librate. The equilibrium point at φ 2 = 90 · is spectrally unstable and an energetic saddle, which means systems will generally not stay near this equilibrium point. Before moving on to study the actual trajectories of the system, we first study the locations and properties of the equilibrium points for different systems. The parameters that define these systems are the mass ratio, the angular momentum, and the shapes of the bodies. The effect of varying these parameters are discussed in this section. First, we investigate how the variation in mass fraction affects the equilibrium points. In order to isolate the effects from varying the mass fraction from the effects of the body shapes, we vary the sizes of the bodies along with the mass fraction to keep the moments of inertia constant as follows. Assuming equal density, the mass fraction is equivalent to the volume fraction, This can be solved for the oblate equatorial radius by using χ to get, Using this equatorial radius, the moments of inertia of the oblate body become, This paper is mainly concerned with systems where the ellipsoidal body is the smaller body, which would correspond to ν ≥ 0 . 5. However, for the sake of completeness, we show in this section the locations of the equilibrium points for all values of ν . When the ellipsoidal body is larger, our perspective changes and we think of this in terms of studying the location of orbits of an oblate satellite, instead of the libration of an ellipsoidal satellite. The locations of the equilibrium points are shown in Fig. 3. The general behavior is that as the mass fraction becomes smaller, the equilibrium points move to larger radii. The second plot shows the difference in radial distance between the two equilibrium points since they are too close to appear as separate lines in the scale of the first figure. The opposite trend is seen here in that the two equilibrium points are further apart radially as ν → 1. At the nominal value of angular momentum, the two equilibrium points are 0.04 apart as was given at the end of Section 3.1. The astute reader will notice that this does not seem to match the results from Bellerose [3]; this is because that work included a factor of ν in their computation of angular momentum we do not include (e.g. K Bellerose = νK ). The value of angular momentum used in this paper is absolute for any system. Also note that Bellerose discusses cases for which multiple equilibria appear; we find these cases as well for small values of K . However, since we are interested in studying the outer pair of equilibrium points that have the librational structure, we don't study the inner equilibrium points that may appear. The main result of varying the angular momentum as illustrated in Fig. 3 is that for higher values of K , the equilibrium points move to larger radii. This also has the effect of making the absolute difference in energy levels between the equilibrium points smaller, as is seen in Fig. 4. The shape of the bodies can have a number of effects on the equilibria. In terms of the oblateness of body 1, the more oblate the body, the larger the difference I 1 z -I s becomes. This generally means that the location of the equilibrium points moves outward with a more oblate body. The effect on the φ 2 = 0 and φ 2 = 90 · equilibria are roughly the same. The effects of the secondary shape can be more varied due to the fact that the main asymmetry in this problem is due to the ellipsoid shape/moments of inertia. The effects are encompassed by the differences in C + 2 and C -2 , which can be changed due to variations of any of the three moments of inertia. Recall that changing I 2 z also changes the value of the system moment of inertia, I z . It is interesting to recall that the semi-major axis and eccentricity depend largely on the quantity ( I 1 z -I s + C ± 2 ). While the equilibrium point at φ 2 = 0 is always at periapsis, the φ 2 = 90 · equilibrium can be at periapse, apoapse, or on a circular orbit depending completely on the moments of inertia of the 1 1 bodies. Also, due to the same quantity, the φ 2 = 90 · equilibrium point can become a local maximum in the radial direction (see Eq. (41)), although the point will still be an energetic saddle in general. In all cases studied here, the results of the stability of the nominal system equilibria holds for the outer set of equilibria points. Namely, the outer equilibrium point at φ 2 = 0 is spectrally stable and an energetic minima, while the outer equilibrium point at φ 2 = 90 · is spectrally unstable and an energetic saddle.",
"pages": [
8,
9,
10,
11
]
},
{
"title": "4 Sufficient Condition for Bounded Motion",
"content": "When the system is not at equilibrium, the trajectory will evolve in r -φ 2 space according to the equations of motion given in Eqns. (17) - (18). As this system is non-integrable, we are particularly interested in finding conditions which classify or restrict the trajectories that will occur. In this section, we present a sufficiency condition for bounded motion, which is taken to mean that the libration angle will be less than 90 · for all time. We then investigate how the sufficiency condition changes for different system parameters. Finally, we look at some properties of the system trajectories that meet the bounded sufficiency conditions.",
"pages": [
13
]
},
{
"title": "4.1 Bounded Motion",
"content": "The ZVCs can be used to investigate if the current system can become unbounded by reaching a phi = 90, which leads to the sufficiency condition for bounded motion:",
"pages": [
13
]
},
{
"title": "Theorem 4.1. Sufficient Condition for Bounded Motion",
"content": "Given a system with angular momentum K so that a stable equilibrium point exists at φ 2 = 0 , the librational motion is bounded ( φ 2 < 90 · always) if E < E -. Proof: The relationship for the free energy of the system, Eq. (15) , can be rearranged to become The right hand side of Eq. (64) is always positive, therefore we can state the relationship for a zero-velocity curve (ZVC), or equivalently, The stable equilibrium is the minimum energy location, and when E = E + , the ZVC defines only the stable equilibrium. As E is increased, the ZVC will encompass larger areas in r -φ 2 space. If E is increased to E -, the ZVC will touch the unstable equilibria, and because E -is the minimum energy radius with φ 2 = 90 · (see Section 3.3), this is the minimum energy at which the ZVC allows the system to reach φ 2 = 90 · . That this is a sufficient, but not necessary, condition is shown in Section 5.1. 2 A given system will have some area in the r -φ 2 space which it can reside in based upon the free angular momentum and energy. Given a value for the free angular momentum of the system, the entire phase space can be mapped with varying energy levels determined by Eq. (66). The easiest way to visualize this is by looking at the system from a secondary fixed frame, as shown in Fig. 2. For a given value of free energy for the system, there will be some area to which the primary is constrained to reside. Note that due to the fact that this relationship is an inequality, the primary can be anywhere inside the free energy level, not only on the surface. Therefore this relationship clearly doesn't solve the equations of motion to tell us what the state is at any given time, but it does tell us absolutely that the state is always inside the area bounded by that free energy. Theorem 4.1 is verified graphically for the nominal system in Figures 5 and 6. The nominal system zero-velocity curves are shown in Fig. 5. This is the typical structure for the zero-velocity curves seen in most situations where the libration between the two outer equilibrium points is being examined. In these types of plots, the stable minimum energy equilibrium point is along the x-axis ( φ 2 = 0), while the unstable equilibrium point is on the y-axis ( φ 2 = 90 · ). As the energy is increased from the minimum at the x-axis equilibrium point, the zero-velocity curves allow for larger libration angles until the unstable equilibrium energy is reached. At energies above the unstable equilibrium point energy the ZVCs open (becoming two separate ZVCs) and the secondary is free to circulate in this area. The black shape around the origin is the projection of the ellipsoid plus the equatorial radius of the oblate body; if the energy is high enough so that this region is inside the ZVC bounds then an impact between the bodies is possible. The color bar lists the values of energy ( E ) corresponding to each ZVC. Recall from Section 3.1 that the stable equilibrium point has energy E + = -0 . 0497 and the unstable equilibrium has an energy of E -= -0 . 496, where the difference between the two is δE = 1 . 500 × 10 -4 . In order to make clear the behavior of the energy in the vicinity of the equilibrium points, we plot what is effectively a cross section of Fig. 5 in Fig. 6. This clearly shows the variation of the energy in the radial and circumferential directions. It is clear that both of these equilibrium points are minima in the r direction, however only the x-axis equilibrium point is also a minima in the φ 2 direction. It should be noted that the same behavior is seen around the stable equilibrium point at φ 2 = 180 · , and Theorem 4.1 can be applied there as well. In this paper, we generally only look at the φ 2 = 0 equilibrium point for clarity. Consider a zero velocity curve with the free energy, E , and the free angular momentum, K . When the inequality in Eq. (65) is precisely equal to zero, we know that ˙ r = 0 and ˙ φ 2 = 0; all of the free kinetic energy in the system has been transferred to the potential energy. At any point inside the zero velocity curve(s) defined by E , the inequality will be greater than zero as there is an excess of energy defined from the ZVC definition, This situation is depicted in the cartoon shown in Figure 7. Returning to Eq. (64), it is clear that in this case, the left hand side is greater than zero, and therefore either | ˙ r | > 0, | ˙ φ 2 | > 0, or some combination of the two. Note that the excess energy, ∆E is not constant on a given trajectory as it varies with the position variables. The maximum value that ∆E can reach is at the φ 2 = 0 equilibrium point since this is the minimum potential energy location. At any given point on a trajectory, the excess energy can be used to determine the range of possible values of the velocities based on the kinetic energy in the radial and spin components. 2 The split in the kinetic energy will be determined by the factor κ , which dilineates what percentage of the excess energy goes into the radial velocity. The scale factor is therefore bounded such that This means that the radial velocity and ellipsoid rotation rate are defined as, The ZVCs are a crucial property of the system which limit the phase space in which the system can reside. Since the conditions in Theorem 4.1 are so important, we investigate how the ZVCs vary when the system parameters are changed.",
"pages": [
13,
14,
15
]
},
{
"title": "4.2 Parameter Influence on Zero-Velocity Curves",
"content": "Eq. (65) shows us that the zero-velocity curves are defined first and foremost by the angular momentum and energy in the system. However, the exact form of the zero-velocity curves depends on all the system parameters such as the moments of inertia of the bodies, and the mass fraction of the system. In this section we explore how the nominal zero-velocity curves shown in Fig. 5 change when the system parameters are varied.",
"pages": [
18
]
},
{
"title": "4.2.1 Variation Due to Mass Ratio",
"content": "In this section, we look at the ZVCs for three different mass ratio systems pulled from Section 3.5. By comparing these different mass ratio systems to our nominal system in Fig. 5, we get a good idea of the effect of changing the mass ratio on the ZVC structure. The first case is the nominal system scaled to a mass ratio of ν = 0 . 5, shown in Fig. 9. Even with the scaling of the bodies to equal masses, the structure of the ZVCs is very similar. The values of the energy and the location of the equilibrium points have changed due to a change in angular momentum, but the interesting point is that the behavior of the system between and around the equilibrium points found in our nominal case of a relatively small ellipsoid orbiting a larger oblate body appears to hold all the way to equal masses. The second case we look at is with a mass ratio of ν = 0 . 05 shown in Fig. 10. This system was chosen because it illustrates a case discussed by Bellerose [3] where there are two equilibrium points on the x-axis, and the outer point is spectrally stable and an energetic minima, while the inner point is spectrally unstable and an energetic saddle. It turns out for these systems that there are also two equilibria on the y-axis. As expected from the systems studied so far, the outer y-axis equilibria is spectrally unstable and an energetic saddle. It is interesting to find that the inner y-axis equilibria is spectrally stable while being an energetic maxima. This is interesting, as it implies that as ν → 0, so that the system being studied is that of a small sphere/oblate body orbiting a large ellipsoid, there will be stable orbits along the y-axis around three times the ellipsoid semi-major axis. The orbits that have the small body near the same radius on the x-axis will be unstable. The final case we show is that with a mass ratio of ν = 0 . 15, shown in Fig. 11. The y-axis equilibrium points keep the same general behavior as those in Fig. 10, however in this case the x-axis equilibria have vanished. This appears to make low energy impacts possible between the two bodies starting from basically anywhere in the system; in other words the energy decreases all the way to the surface along the x-axis, and since a given trajectory can explore the region between ZVCs for its energy, they can reach the surface along the x-axis. In the cases in Figs. 10 and 11, it is important to note that there may be some non-trivial errors in the values obtained for the closer equilibria due to the fact that we are using a second order expansion for the potential of the bodies. If accurate results are needed for these cases, we suggest calculating the outcomes with the exact potential expression used by Bellerose [3]. However, the qualitative results match these previous investigations with the exact potential. Furthermore, with the main interest of this paper being the outer set of equilibrium points, these points are far enough from the bodies that the differences between our second order approximation and the exact potential should be small enough to make little difference. As was alluded to in Section 3.5, the main effect of changing the angular momentum is to move the equilibria in or out. This effect is also seen on the ZVC structure in Fig. 12; the ZVCs have the same structure as the nominal case, they are basically shifted in or out. It was demonstrated in Section 4.2.1 that a combination of low ν and K can cause a second set of equilibrium points to appear, however this situation doesn't happen for larger values of ν due to the fact that these solutions to Eq. (20) are less than the combined radii of the bodies, and usually less than the radius of the secondary. The degree to which the secondary is triaxial plays a large role in the determination of the zero-velocity curves. This is investigated by tweaking the nominal case with two different values of the intermediate moment of inertia, I 2 y , first so that it is just greater than the I 2 x , and second so that it is just less than I 2 z . The main contribution of modifying the ellipsoid moments of inertia, however, is that the difference ( I 2 y -I 2 x ) controls the dependence of the potential and therefore the ZVC on the libration angle. The ZVCs for these two cases are compared to the nominal case in Fig. 13. The ZVCs of the same color are at the same energy across the three figures, making it easy to see that as the intermediate moment of inertia is made smaller, the body can circulate much easier. Note that, as discussed in Section 3.5, changing the oblateness of the other body will move the location of the equilibrium points in or out. However, there is very little difference beyond this radial shift on the ZVCs, so this effect isn't illustrated.",
"pages": [
18,
19,
20,
21,
22
]
},
{
"title": "4.3 Time-Variation of Osculating Orbital Elements",
"content": "The orbital elements of a non-equilibrium trajectory are no longer useful as metrics for the system because they change constantly throughout the course of the trajectory. The groundwork for calculating the orbital elements has already been laid in Sections 3.4 and 4.1. At any given point in r -φ 2 space, given the energy and angular momentum, the possible values of ˙ r and ˙ φ 2 can be calculated using Eqs. (70) and (69). The Keplerian energy and angular momentum can then be calculated from Eqs. (46) and (45), which in turn allows us to calculate the semi-major axis and eccentricity with Eqs. (50) and (51). Note that for these calculations, the sign of ˙ r makes no difference, however the sign of ˙ φ 2 directly effects the value of ˙ θ , which then changes the value of the Keplerian energy and angular momentum. The eccentricity vector can again be calculated through Eq. (54). Note that in this case, ˙ r is no longer zero in general, so that glyph[negationslash] where ˆ θ = ˆ H K × ˆ r . This implies that whenever ˙ r = 0, the eccentricity vector no longer lies along the ˆ r direction, and therefore the system is no longer at an apse. To illustrate the behavior the semi-major axis and eccentricity on a non-equilibrium trajectory, the corresponding values of the semi-major axis and eccentricity for each of the trajectories in Fig. 8 are shown in Fig. 14. As with the radial-angular phase space, each trajectory occupies a distinct region of a -e space despite the fact that they each have the same energy. To further understand the possible values of semi-major axis and eccentricity that can be reached along the trajectories, the values in the entire region inside a give ZVC can be mapped. Figs. 15 - 16 show these results. Each subplot can be imagined as a cross section of a ZVC plot (e.g. Fig. 7) at a different value of φ 2 , where the out of plane axis tells us how the excess energy is apportioned between the radial and angular velocities through the use of the parameter η , which is defined as so that negative values of η correspond to negative values of ˙ φ 2 . The radial bounds of each subplot correspond to the inner and outer edges of the ZVC for this energy level, which is why at larger values of φ 2 , the radial range is smaller. From these figures, we see that the semi-major axis reaches its maximum and minimum values near the stable equilibrium point, with all of the energy in negative or positive angular velocity, respectively. The eccentricity, on the other hand, reaches its maximum value near the inner boundary of the ZVC at φ 2 = 0 with most of the energy in the radial velocity, but some in a negative angular velocity. The minimum eccentricity, which is zero, is seen just outside of the equilibrium point distance with most of the energy in either positive or negative angular velocity. Note that the colorbars near the last subplot are valid for all values of φ 2 . Clearly a wide variety of combinations of semi-major axis and eccentricity can be seen over the course of trajectory. η η 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 9 9.1 9.2 9.3 9.4 9.5 9.1 9.2 9.3 9.4 9.5 Φ = 30 deg. r r = 15 deg. 2 Φ Φ = 0 deg. 2 r η η 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 9 9.1 9.2 9.3 9.4 9.5 Φ r = 20 deg. 2 9.1 9.2 9.3 9.4 Φ = 5 deg. 2 r η η 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 9 9.1 9.2 9.3 9.4 9.5 9.1 9.2 9.3 9.4 Φ r = 25 deg. 2 Φ = 10 deg. 2 r",
"pages": [
23,
24,
25
]
},
{
"title": "5 Transient Dynamics",
"content": "In this section, we explore trajectories for systems with open ZVCs (those with E > E -). Specifically, we explore the existence of trajectories that behave as if they are bounded, but do not meet the sufficiency conditions outlined in Section 4. These results prove that Theorem 4.1 is in fact a sufficient, but not necessary, condition for bounded motion. We also investigate the special case of bounded motion for periodic trajectories, and the families of periodic trajectories that exist as the energy is varied.",
"pages": [
26
]
},
{
"title": "5.1 Existence of Bounded Motion with Open Zero-Velocity Curves",
"content": "The existence of bounded trajectories with open ZVCs is shown through numerical examples. Two such trajectories are shown in Fig. 17. Although the ZVC is open, not every trajectory circulates. This is an extension of what was seen above with the closed ZVC trajectories; depending on the initial conditions of the trajectory at φ 2 = 0, the trajectory will only explore a portion of the r -φ 2 phase space inside the ZVC. The existence of this bounded trajectory proves that Theorem 4.1 is a sufficient, but not necessary condition, as this example does not meet the requirement of the theorem. Furthermore, using the ZVC corresponding to the energy of the unstable equilibrium point is a conservative estimate of the energy at which a system will have a circulating, as opposed to librating, secondary body. 10",
"pages": [
26
]
},
{
"title": "5.2 Periodic Orbits",
"content": "Periodic orbits can be found in this system when a given orbit repeats itself exactly after a certain period. These orbits are found when the state transition matrix or monodromy matrix have unity eigenvalues. A detailed methodology for finding periodic orbits in this manner is given in [3]. In this system, due to the symmetry about φ 2 = 0, most of the periodic orbits will have a crossing of this line of symmetry with ˙ r = 0, although this is not strictly required for a periodic orbit. However, it is true that every trajectory that crosses φ 2 = 0 twice with ˙ r = 0 is periodic. The period may be very long, but due to the symmetry of the problem it will eventually repeat. Therefore, our search consisted of starting at different radii with ˙ r = 0, and looking at subsequent crossings of φ 2 = 0. The results of this search are combined in a Poincar'e map at of the r -˙ r space at φ 2 = 0. One illustrative case is shown in Fig. 18, where the crossings of several periodic orbits are highlighted. The actual trajectories of several periodic orbits in the r -φ 2 space are shown in Fig. 19, corresponding to the highlighted cases from Fig. 18. As alluded to previously, there are actually an infinite number of periodic trajectories in this system, with most of them having very large period ratios such as the T = 49 . 04 and T = 56 . 84 orbits shown. Of particular interest are the two periodic orbits with near T = 1. In the example shown in Fig. 19, these are the T = 0 . 9896 case, which orbits counter-clockwise, and the T = 1 . 274 case which orbits in a clockwise direction. These two orbits roughly span the space inside the closed ZVC in this case, as they are associated with the eigenmodes of the equilibrium point. The evolution of the eigenmode periodic orbits with changing values of energy are explored in Figs. 20 and 21. The first plot shows many different trajectories as the energy is increased. Particularly interesting behavior is seen with the clockwise trajectories. When the energy is approximately 75% of the way to the unstable equilibrium point energy, the shape of these orbits changes from the oval seen in Fig. 19 to a bow shape, and eventually (at the highest energy levels shown) the trajectory crosses itself near φ 2 = ± 50 · . The maximum libration angles reached by these periodic orbits is also very high, approaching 80 · at the energy levels shown. By comparison, the evolution of the counter-clockwise orbits is much more regular. The shape stays basically the same throughout, simply growing in size as the energy grows. Note that the maximum amplitudes reached in these cases is much smaller than the clockwise orbits, getting to a maximum of roughly φ 2 = ± 25 · . A different view of the trajectories is shown in Fig. 21. This plot shows the radial locations of the φ 2 = 0 crossings with ˙ r = 0 at each energy level. This clearly shows the more regular evolution of the counter-clockwise orbits. It is especially interesting that these periodic orbits exist well beyond the energy of the unstable equilibrium point, as this gives clear examples of stable, libration bounded non-equilibrium orbits with open ZVCs. r r",
"pages": [
26,
27,
28
]
},
{
"title": "6 Sufficient Conditions for Unbounded Motion",
"content": "In Section 4 we presented a sufficient condition for bounded librational motion. In Section 5, we proved this condition was only sufficient due to the existence of bounded trajectories when the conditions from Theorem 4.1 are not met. Significantly there are families of periodic orbits that exist even when the ZVCs of the system are open. In this section, we analyze the converse problem of determining when unbounded librational motion occurs. When the system has a mass fraction near unity, conditions for unbounded motion can be derived analytically. For non-unity mass fractions, strict conditions are unavailable. However, we present analysis for these systems that illustrates the structure of the problem and initial conditions that will lead to circulation.",
"pages": [
30
]
},
{
"title": "6.1 Analytical Limits for Systems with ν glyph[similarequal] 1",
"content": "In this section, we show sufficient conditions for circulation for systems with mass ratios close to unity. This restriction limits the effect of the coupling on the orbit of the small ellipsoidal body, which allows us to assume that the orbit itself remains unchanged due to the libration of the secondary. However, due to this assumption the results can only be claimed as sufficient conditions for circulation, as shown in the following theorem.",
"pages": [
30
]
},
{
"title": "Theorem 6.1. Sufficient Condition for Circulation",
"content": "Given a system with ν glyph[similarequal] 1 , so that the orbit can be considered unperturbed by the ellipsoidal secondary, with some initial secondary spin rate ˙ φ 2 , 0 at a libration angle of φ = 0 · at some location on the orbit ( r 0 ), the secondary body is guaranteed to circulate if the spin rate satisfies the relationship where r a is the apoapse radius of the orbit. Proof: The free energy of the system is written using Eqs. (7) , (9) and (10) as, where E orb is the orbit energy that is independent of the librational state, E φ is the librational energy (identical to a pendulum), and E coup is the 'coupling' energy. The first two energy terms are given by relationships In the case of a point mass, the orbit energy collapses to the Keplerian energy of the system. Recalling the trigonometric relationship allows us to determine the coupling energy from Eq. (74) , using Eqs. (15) , (75) and (76) as where C + 2 was defined in Eq. (21) . If the orbit energy is constant, then the available energy is determined as Since this value is a constant at any point on the trajectory, we can relate two points by This expression relates the available energy for a point on the orbit at radius r 0 with some initial libration rate ˙ φ 2 , 0 and zero libration angle, to another point on the orbit at radius r m with libration angle φ 2 ,max and a libration rate of zero. In other words, this tells us the maximum achievable libration angle at radius r m given a system that had an initial libration rate at r 0 . Note that this relationship is independent of ˙ r . Writing out the available energy at these two locations using Eqs. (76) and (78) gives To determine the bounding condition, we look for the case when φ 2 ,m = 90 · , and solve for ˙ φ 2 , 0 to get where the fact that C + 2 = -2 I 2 x + I 2 y + I 2 z was used. For any given starting location, the highest ˙ φ 2 , 0 will be required if r m is as large as possible, so r m = r a . Therefore if the the libration rate is high enough to circulate the secondary at apoapse, then the body will circulate no matter what point on the orbit it reaches its maximum libration angle. Thus the bounding condition to ensure circulation occurs when 2 Theorem 6.1 assumes that the orbit perturbations caused by the libration of the secondary are ignorable. Therefore the orbit can be described using Keplerian orbital elements, which will be constant. We can express the initial orbit radius in terms of the eccentricity, semi-major axis, and true anomaly as and then the apoapsis radius is and finally the periapse radius is Using these relationships, the condition for circulation can be expressed in terms of the orbital elements as, Theorem 6.1 gives the sufficient condition for circulation for a given starting point, however we can define a single condition for circulation that is sufficient for any location on the orbit by defining the highest value that leads to circulation, which is referred to as the uniform condition.",
"pages": [
30,
31
]
},
{
"title": "Remark 6.1. Uniform Condition for Guaranteed Circulation",
"content": "The highest bound for circulation is given by maximizing Eq. (86) in terms of f . The maximum occurs when f = 0 , or r 0 = r p , and the circulation condition becomes 2 This condition is considered uniform because it combines the extremes for both r 0 and r m in terms of maximizing the required ˙ φ 2 , 0 . The condition given Eq. (87) is the baseline sufficiency test for circulation. At any point on an orbit, if ˙ φ 2 , 0 meets this condition with φ 2 = 0 · , it is guaranteed to circulate directly, meaning that the current trajectory will pass through ± 90 · before returning to φ 2 = 0 · . Depending on the actual location on the orbit, the value of ˙ φ 2 , 0 which is sufficient to cause circulation will generally be lower, and is given by Eq. (73). It is important to note here that Theorem 6.1 is a sufficiency condition only, but it is possible that a lower value of ˙ φ 2 , 0 will lead to circulation. This is because Theorem 6.1 depended on the most extreme value for r m . However, due to the fact that it is easier to circulate at all r m < r a , it is true that lower libration rates will lead to circulation in many cases. This depends on the phasing of the librational motion with the orbit to determine at what radius φ 2 ,max is achieved, and if ˙ φ 2 , 0 was large enough at the φ 2 = 0 crossing preceeding this φ 2 ,max , circulation can occur. Therefore the lowest possible spin rate that can lead to circulation can be determined in the opposite limiting case where ˙ φ 2 , 0 is minimized with respect to r m and r 0 . In fact, this becomes a sufficient condition for bounded libration.",
"pages": [
31
]
},
{
"title": "Corollary 6.1. Sufficient Condition for Bounded Motion with ν glyph[similarequal] 1",
"content": "Given the system considered in Theorem 6.1, the lowest value of ˙ φ 2 , 0 which can lead to circulation is computed so that any value below this will not lead to circulation where r 0 is found as the real positive root of, if the resulting r 0 < r a , otherwise r 0 = r a . Proof: Eq. 82 is minimized with respect to r 0 and r m . The partial with respect to r m shows that the function monotonically increases with r m , therefore the minima with respect to r m is at the lower boundary, r p . There exists an extrema of Eq. 82 with respect to r 0 , which is located at the root of Eq. (89) . Using the Routh criteria it can be proven that there is one and only one real positive root. It can also be shown that this extrema is in fact a minima by investigating the second partial with respect to r 0 . However, the location of this minima depends on the system parameters, and it is not guaranteed to lie within the range of possible radius values given by r p and r a . Therefore, if the minima is found to be greater than r a , the constrained minima is at r 0 = r a . 2 These results outline a set of bounds that can be used to determine if a system will circulate or not depending on the libration rate at φ 2 = 0 · . The main boundaries to be tested to give information for an entire orbit are given by Remark 6.1 and Corollary 6.1. If either of these conditions are satisfied, we can immediately state if the body will or will not circulate. When the libration rate is between these two boundaries, further inspection must be made. Theorem 6.1 can be used to determine if a given initial condition ( r 0 and ˙ φ 2 , 0 ) will circulate directly. However, this relationship does not tell us if the body will ever circulate; only if it will circulate directly. The difficulty is in the phasing of the libration and the orbit. If the initial libration rate is below the bound given in Theorem 6.1, then the body will not circulate this on this oscillation. However in general the body will pass back through φ 2 = 0 · at a different state, and then the condition must be checked again. Corollary 6.1 is an extension of Theorem 4.1 for the specific case when ν glyph[similarequal] 1. If the total energy in the system is low enough so that Theorem 4.1 is applicable, Corollary 6.1 will also be satisfied. However, Corollary 6.1 can be satisfied when Theorem 4.1 is not applicable. If circulation does not occur, it is useful to be able to determine the libration amplitudes that can be expected. This is addressed in the following corollary.",
"pages": [
32
]
},
{
"title": "Corollary 6.2. Maximum Libration Angle",
"content": "Given the system considered in Theorem 6.1, if ˙ φ 2 , 0 is below the limit for circulation, then the maximum libration angle that can be obtained is given by Proof: This condition is determined by rearranging Eq. (81) to get This relationship is maximized with a minimum r m , giving r m = r p , the periapse radius. 2 If the maximum libration relationship from Corollary 6.2 is tested with ˙ φ 2 , 0 above the circulation limit, the right hand side of Eq. (90) will be greater than one, resulting in an imaginary value for φ 2 ,max . This indicates the body will circulate. The maximum libration angle that would be reached at any point on the orbit can be determined by using a radius other than the periapse radius in Eq. (91) for r m .",
"pages": [
32
]
},
{
"title": "Remark 6.2. Simplification to Classical Results",
"content": "Classical gravity gradient results assume that the orbit is circular ( r m = r 0 = r ) and that the system is dominated by the orbit so that νr 2 glyph[greatermuch] I 2 z , and therefore I z → νr 2 , and ν = 1 . Substituting these conditions into Eq. 81 gives the classical result which is recognized as the energy integral of the pendulum equation of motion 2 Derivation of the classical results are given in [2] and [9] among many other sources. Note that Eq. (93) has a slight difference to the referenced results because we have normalized the units so that µ has disappeared. These results can be illustrated through several numerical systems. First, a basic example is shown in Fig. 22 with very low eccentricity ( e = 2 . 4 × 10 -6 ). The full equations of motion given in Eqs. (17) - (18) are integrated for two cases, where one trajectory is initialized with 95% of the excess energy is given to ˙ φ 2 , while the other has 70% apportioned to the initial libration rate. Both cases start at r = r + and ˙ r = 0. Using Theorem 6.1, we can see that the former case is above the limit and therefore will circulate, while the latter case is below. The maximum libration angle is calculated for the κ = 0 . 3 case from Eq. (91) with r m = r + , and is found to closely agree with the maximum libration amplitudes seen during the simulation. The results developed in this section can also show when the excitation of the libration amplitude from the eccentricity of the orbit can cause the secondary to begin circulating. Application of Corollary 6.2 for a range of r 0 values is shown in Fig. 23. In this case, every case is tested with φ 2 , 0 = ˙ φ 2 , 0 = 0, so the excitation occurs only from the presence of available energy for this orbit due to variation in the radial position. The results are presented in terms of eccentricity and true anomaly, which determine the extents of r 0 , but are more intuitive for this application. It is shown that for this case, if the eccentricity is less than 1 × 10 -4 the libration is bounded for any initial condition. For larger eccentricities, the initial conditions must be nearer to periapsis in order for the libration to remain bounded. It should be noted that each eccentricity line is at a different energy level, but they all have the same angular momentum value for this study. In order to illustrate the validity of these relationships, four different simulations indicated on Fig. 23 with circles were simulated for 500 orbits. These trajectories are plotted in Fig. 24. It is clear that the bounds are accurately predicting the librational motion, and the unbounded case does circulate directly as predicted by Theorem 6.1 for that case. It is interesting to investigate the case of Saturn's moon Hyperion, which is known to be in chaotic rotation. Using the moments of inertia for Hyperion from [8] (renormalized by α = 180 km for Hyperion), and the known Hyperion orbit and Saturn properties, Eq. (87) is used to find the ˙ φ 2 that will guarantee circulation. We find that if Hyperion were to have | ˙ φ 2 | > 0 . 00524 degrees per second at periapse with zero libration angle, it will enter a circulating state. Given the extremely small limit found, it is no surprise that Hyperion is in an unbound chaotic rotation state. In the general case with ν < 1, the relationships developed in Section 6.1 can't be used because the orbit energy can not be considered constant due to the coupling from the secondary body's motion. In this section, we investigate which initial conditions at φ 2 = 0 circulate numerically. Our approach is to sample the possible initial conditions at φ 2 = 90 · and propagate them backwards in time to see where they originate at φ 2 = 0. By fixing one of the states (we choose to fix φ 2 ), the remaining states are required to lie on the surface of an ellipsoid of constant energy. We use these constant energy surfaces to investigate the states of the simulations at φ 2 = 0 and φ 2 = 90 · . These visualizations are essentially 3-D Poincar'e surfaces. The case investigated here is for the nominal system with E = E -+0 . 1 δE . The constant energy surface at φ 2 = 90 · is illustrated in Fig. 25. Initial conditions which sampled the entire surface were tested, with those that propagated back to φ 2 = 0 shown as black dots on the ellipsoid. There were 3362 initial conditions tested, exactly half propagated to φ 2 = 0, the other half propagated toward φ 2 = 180 · and were ignored for this analysis. As seen in Fig. 25, the split is basically associated with the sign of ˙ φ 2 ; those with a positive libration rate when crossing φ 2 = 90 · generally came from φ 2 = 0. However, it is interesting to note that there 22 of the 1681 cases did have negative libration rates, seen as those will small values of r on the ellipsoid. The evolution of the backwards propagated trajectories are shown as they cross φ 2 = 0 in Fig. 26. The initial point where the initial conditions reach φ 2 = 0 are labeled as the zeroth crossing, and plotted also as black dots. We then plot each of the next six crossings in different colors, alternating between the north and south poles of the ellipsoid as the trajectories cross in opposite directions. Note there is definite structure to the region of each crossing on the ellipsoid. Fig. 27 shows the same thing as in Fig. 26, except that all crossings after the sixth are plotted with red dots. This shows that although the coverage continues to spread and lose some of the distinct structure of the earlier crossings, there are still interesting features. First, there is a large area surrounding the equator where there are no trajectories that are connected to circulation. This tells us that trajectories that reside in this region will stay bounded. We point out that the points around the north pole are biased as a group toward the smaller radius values, while the south pole cases are biased toward the larger radius values. Second, there are some significant areas on the ellipsoids amongst the propagated trajectories that are not filled, such as on the north pole view around ˙ r = 0 and small values of r . These gaps correspond to the periodic trajectories discussed in Section 5.2. The third note about this plot is that currently the region immediately surrounding the south pole is unpopulated. These states would correspond to trajectories that would evolve through φ 2 = -90 · , as opposed to through φ 2 = 90 · as are being analyzed here. Another view of the evolution of the backwards propagated trajectories from Figs. 25 - 27 are shown in Fig. 28. In this figure, we see the absolute value of ˙ φ 2 plotted for every trajectory for each crossing. The lower plot shows how many trajectories still exist. The main takeaway from this plot is that most the trajectories that we propagated backwards don't stay in a librational state for very long; of the 1681 initial trajectories, only about 100 cross φ 2 = 0 more than 100 times. Only 2 trajectories last longer than about 400 crossings. Interestingly, one trajectory lasts for a very long time - approximately 22,000 crossings! This tells us that a large portion of the red points seen in Fig. 27 can be attributed to one very rare case (this trajectory is shown in Fig. 29). The other thing we find from this plot is that, unlike in the ν glyph[similarequal] 1 case, there isn't a clear value of ˙ φ 2 that indicates if circulation will occur or not. This is due to the coupling between the libration and orbital states. Finally, it is interesting to see how things change when the energy level is increased. Fig. 30 shows the φ 2 = 0 ellipsoid for three increasingly higher energy levels. These views show that as the energy is increased there are two main impacts. First, the trajectories evolve to much larger areas of ellipsoid, meaning there are smaller sets of initial conditions at φ 2 = 0 that correspond to bounded trajectories. Second, the unbounded trajectories reside for much shorter timespans around φ 2 = 0. In Fig. 31, we show the crossing history, which for that case has all trajectories circulating again after less than 30 crossings. We also show a stable trajectory, proving that although the ZVC is significantly opened at this energy level, there are still bounded trajectories.",
"pages": [
33,
34,
35,
36,
37,
38
]
},
{
"title": "7 Conclusion",
"content": "This paper explored the dynamic system of a triaxial ellipsoid satellite in orbit in the equatorial plane of an oblate body. This system reduces to a 2 degree-of-freedom system in the radial separation and the libration angle of the ellipsoid. The reduced system was then analyzed with the goal of determining limits on the dynamic configurations for which the librational motion is bounded to less than ± 90 · . The relative equilibrium points were found in the ellipsoid-fixed frame, and the stability of these points was discussed. The conservation of energy and angular momentum in the system was exploited to write an expression for zero-velocity curves. These curves are used to determine a sufficiency condition on when the librational motion is bounded. It is shown that bounded trajectories exist beyond these sufficiency conditions, including families of periodic orbits. We addressed the conditions for unbounded motion for all systems. In particular, when the ellipsoid becomes of very small mass compared to the oblate body, analytical relationships were derived that determine the maximum libration angle for any orbit eccentricity. Future work includes extending this approach to two ellipsoid systems, non-conservative systems, and coupled out-of-plane motion.",
"pages": [
38,
39
]
},
{
"title": "Appendix",
"content": "The mathematical description of the system studied in this paper was determined as a simplified form of the system studied by Scheeres [19]. The simplifications made to obtain the results used in this paper are derived here in order to make the connection to the previous work explicit.",
"pages": [
39
]
},
{
"title": "Equations of Motion",
"content": "The Lagrangian, L = T -V for this system is, Using Lagrange's equations with out any external forces, the equations of motion for this system with the coordinates r , θ , φ 1 , and φ 2 were found in Scheeres [19] and are rewritten in normalized units to be, However, in the case of an oblate body the potential is no longer a function of φ 1 so that and the equations of motion for the angles become, The partials of the potential for an oblate primary are, Integrals of Motion In the current problem with an oblate primary, we have three integrals of motion. The first is the total energy of the system, which is shown to be the Jacobi integral of this system since it is time invariant, The second integral of motion is the total angular momentum of the system. This is found because the coordinate θ is ignorable, meaning that d / dt( ∂L/∂ ˙ θ ) = 0, so the integral is written as, The third integral is found by combining Eqs. (101) and (103), so we find that Therefore the inertial angular velocity of the primary, ˙ θ 1 , is an integral of motion. This implies that the terms in the kinetic energy and angular momentum expressions which depend only on ˙ θ 1 are also conserved. This fact makes intuitive sense as a primary that is symmetric about the spin axis can't have any gravitational torques exerted on it from the secondary since the center of mass and the center of gravity (in the secondary's gravity field) are at the same location in the primary. Dynamics Matrix Partials The partials are, And the second partials of the potential are given by, Relative Equilibria Locations Following Scheeres [19], we find the equilibrium points by searching for places where the variations in energy are stationary at a constant value of angular momentum. In other words, we find when the following conditions hold: which when combined with the stationarity conditions imply, respectively, that ˙ r = 0, ˙ φ 2 = 0, and φ 2 = 0, ± π/ 2, or π . Using these conditions, we can evaluate the partial with respect to r to be, at a given value of K , as seen in Eq. (15). The following relationships are found, Relative Equilibria Stability Partials Recall from Section 3, the equilibrium points must occur at ˙ r = 0, ˙ φ 2 = 0, and φ 2 = 0, ± π/ 2, or π . Therefore the partials for the dynamic matrix are greatly reduced to become, ∂ ∂ ¨ φ ˙ 2 φ 2 Where the equilibrium point distance is indicated by r eq . Note that from Eq. (105), we see that The second partials of the potential evaluated at the equilibrium points are, where C ± 2 was defined in Eq. (21). The plus terms correspond to the case where φ 2 = 0 or π , and the minus terms correspond the the other cases where φ 2 = π/ 2 or 3 π/ 2.",
"pages": [
39,
40,
41,
42
]
},
{
"title": "References",
"content": "= 0 (133)",
"pages": [
42
]
}
]
2013CeMDA.117...59R
https://arxiv.org/pdf/1306.4293.pdf
<document>
<text><location><page_1><loc_13><loc_93><loc_32><loc_94></location>Noname manuscript No.</text>
<text><location><page_1><loc_13><loc_91><loc_35><loc_92></location>(will be inserted by the editor)</text>
<section_header_level_1><location><page_1><loc_12><loc_82><loc_63><loc_85></location>Tidal evolution of close-in exoplanets in co-orbital configurations</section_header_level_1>
<text><location><page_1><loc_12><loc_77><loc_47><loc_80></location>Adri´an Rodr´ıguez · Cristian A. Giuppone · Tatiana A. Michtchenko</text>
<text><location><page_1><loc_12><loc_70><loc_49><loc_71></location>the date of receipt and acceptance should be inserted later</text>
<text><location><page_1><loc_12><loc_51><loc_70><loc_67></location>Abstract In this paper, we study the behavior of a pair of co-orbital planets, both orbiting a central star on the same plane and undergoing tidal interactions. Our goal is to investigate final orbital configurations of the planets, initially involved in the 1/1 mean-motion resonance (MMR), after long-lasting tidal evolution. The study is done in the form of purely numerical simulations of the exact equations of motions accounting for gravitational and tidal forces. The results obtained show that, at least for equal mass planets, the combined effects of the resonant and tidal interactions provoke the orbital instability of the system, often resulting in collision between the planets. We first discuss the case of two hot-super-Earth planets, whose orbital dynamics can be easily understood in the frame of our semianalytical model of the 1/1 MMR. Systems consisting of two hot-Saturn planets are also briefly discussed.</text>
<text><location><page_1><loc_12><loc_49><loc_67><loc_50></location>Keywords Celestial mechanics · Planets and satellites · Tides · Exo-Trojans</text>
<section_header_level_1><location><page_1><loc_12><loc_44><loc_23><loc_45></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_32><loc_70><loc_43></location>In contrast with strong mean-motion resonances (MMR), such as 2/1 and 3/2 MMR (see Beaug'e et al. 2012) there are no known pairs in co-orbital configuration, despite the continuously increasing number of discovered exoplanets. This intriguing fact can be investigated in the context of a past large-scale planetary migration and capture due to, for instance, interactions with the gaseous protoplanetary disk. Several theories about the formation, dissipative evolution and possible detection of exo-Trojans have been developed and will be discussed later in this section.</text>
<text><location><page_1><loc_12><loc_29><loc_56><loc_30></location>Instituto de Astronomia, Geof'ısica e Ciˆencias Atmosf'ericas, IAG - USP,</text>
<text><location><page_1><loc_12><loc_28><loc_43><loc_29></location>Universidade de S˜ao Paulo, S˜ao Paulo, SP, Brazil</text>
<text><location><page_1><loc_12><loc_27><loc_32><loc_28></location>E-mail: adrian@astro.iag.usp.br</text>
<text><location><page_1><loc_12><loc_24><loc_68><loc_25></location>Observatorio Astron'omico, Universidad Nacional de C'ordoba, IATE, C'ordoba, Argentina</text>
<text><location><page_2><loc_12><loc_67><loc_70><loc_89></location>The first detailed analysis of hypothetical co-orbital exoplanets was done in Laughlin & Chambers (2002), focusing the difficulties in fitting of the RV data of these systems. More recently, Hadjidemetriou et al. (2009) studied the topology of the phase space and the long term evolution of two-planet systems in the vicinity of the exact 1/1 MMR in the conservative case and for several planetary mass ratios. Hadjidemetriou & Voyatzis (2011) included a drag force and simulated the dynamical evolution of a two-planet system initially trapped in a stable 1/1 resonant periodic motion. Under this drag force the system migrates along the families of periodic orbits and is finally trapped in a satellite orbit. Giuppone et al. (2010) analyzed the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration using a semi-analytical method. They found two new asymmetric solutions which do not exist in the restricted threebody problem. These solutions were also obtained by Robutel & Pousse (2013), who developed an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance for near coplanar and near circular orbits.</text>
<text><location><page_2><loc_12><loc_42><loc_70><loc_67></location>Another kind of dissipative evolution in planetary systems is due to tidal interactions of short-period planets with the central star, which originate changes in orbital elements and rotational periods of the planets. For a planet orbiting a slow rotating star, the tidal effects lead to orbital decay, orbital circularization and rotation synchronization on timescales which depend on physical parameters and initial orbital configurations of interacting bodies (see Dobbs-Dixon et al. 2004; Ferraz-Mello et al. 2008; Rodr'ıguez et al. 2011a, Michtchenko & Rodr'ıguez 2011 and references therein). Some recent studies address the problem of tidal interaction in two-planet resonant systems (see Papaloizou 2011; Delisle et al. 2012). They show that, as tides tend to circularize the planetary orbits, the planets repel each other, in such a way that the period ratios at the end of the evolution are slightly larger than the corresponding nominal low-order resonant values. This feature seems to be a natural outcome of slow dissipative evolution (Lithwick & Wu 2012; Delisle et al. 2012; Batygin & Morbidelli 2013) and is in agreement with close-in planetary configurations detected by the Kepler mission among KOI's candidates. It should be noted that these works focus the attention on first order MMR(e.g. 2/1, 3/2, 4/3), while the case of co-orbital tidal evolution of exoplanets has not been still explored in the context of the general three-body problem. This is the main task of the present paper.</text>
<section_header_level_1><location><page_2><loc_12><loc_37><loc_44><loc_39></location>1.1 Review of stable co-orbital configurations</section_header_level_1>
<text><location><page_2><loc_12><loc_32><loc_70><loc_36></location>In the following, we describe the most important results concerning the equilibrium points of the co-orbital configurations in the general three-body problem. The reader is referred to Giuppone et al. (2012) for a summary of previous works.</text>
<text><location><page_2><loc_12><loc_25><loc_70><loc_32></location>Particularly, Hadjidemetriou et al. (2009) studied the motion close to a periodic orbit by computing the Poincar'e map on the surfaces of sections. For this task, the symmetric families of stable and unstable motions were constructed and a previously unknown stable configuration was discovered, referred to as the QuasiSatellite ( QS ) solutions.</text>
<text><location><page_2><loc_12><loc_22><loc_70><loc_25></location>Giuppone et al. (2010) constructed the families of periodic orbits in the vicinity of the 1/1 MMR using a semi-analytical method. The authors identified two</text>
<text><location><page_3><loc_12><loc_85><loc_70><loc_89></location>separate regions of stability, symmetric and asymmetric, defined by the behavior of the resonant angles ( σ, ∆/pi1 ) ≡ ( λ 2 -λ 1 , /pi1 2 -/pi1 1 ), where λ i and /pi1 i are mean longitudes and longitudes of pericenter of the planets, respectively. Summarizing:</text>
<unordered_list>
<list_item><location><page_3><loc_13><loc_76><loc_70><loc_84></location>-L 4 and L 5 (asymmetric). Are the classical equilateral Lagrange solution associated to local maxima of the averaged Hamiltonian function. Independently on the mass ratio m 2 /m 1 , where m 1 , m 2 are the masses of the planets, and eccentricities e 1 , e 2 , these solutions are always located at ( σ, ∆/pi1 ) = ( ± 60 · , ± 60 · ). The size of the stable domains around these points decreases rapidly for increasing eccentricities and is practically negligible for e i > 0 . 7.</list_item>
<list_item><location><page_3><loc_13><loc_61><loc_70><loc_75></location>-AL 4 and AL 5 (asymmetric). Anti-Lagrangian solutions which correspond to local minimum of the averaged Hamiltonian function. For low eccentricities, they are located at ( σ, ∆/pi1 ) = ( ± 60 · , ∓ 120 · ). One anti-Lagrangian solution AL i is connected to the corresponding L i solution through the σ -family of periodic orbits in the averaged system (the solutions with zero-amplitudes of the σ -oscillation; for detail, see Giuppone et al. 2010). Contrary to the classical equilateral Lagrange solution, their locations on the plane ( σ, ∆/pi1 ) depend on the planetary mass ratio and eccentricity values. Although their stability domains also shrink with increasing values of e i , these solutions survive at eccentricities as high as ∼ 0 . 7.</list_item>
<list_item><location><page_3><loc_13><loc_52><loc_70><loc_60></location>-Quasi-Satellite (symmetric). Are characterized by oscillations around a fixed point which is always located at ( σ, ∆/pi1 ) = (0 , 180 · ), independently on the planetary mass ratio and eccentricities. In contrast with the L 4 and L 5 configurations, the domain of these orbits increases with increasing eccentricities and it fills a considerable portion of the phase space in the case of moderate to high eccentricities.</list_item>
</unordered_list>
<section_header_level_1><location><page_3><loc_12><loc_47><loc_33><loc_48></location>1.2 Formation of exo-Trojans</section_header_level_1>
<text><location><page_3><loc_12><loc_36><loc_70><loc_45></location>There is a vast literature trying to explain the formation of co-orbital planets and different mechanisms have been proposed. Here we only highlight some of the proposed scenarios: planetesimal and planet-planet scattering (Kortenkamp 2005; Morbidelli et al. 2008), direct collisional emplacement (Beaug'e et al. 2007), in situ accretion (Chiang & Lithwick 2005) and migration in multiple protoplanet systems (Cresswell & Nelson 2006; Hadjidemetriou & Voyatzis 2011; Giuppone et al. 2012).</text>
<text><location><page_3><loc_12><loc_22><loc_70><loc_36></location>Particularly, Giuppone et al. (2012) analyzed whether co-orbital systems may also be formed through the interaction of two planets with a density jump in the protoplanetary disk. The authors considered two planets, initially located farther than the 2/1 MMR (beyond 1 AU, where tidal interaction with the central star is almost negligible) and involved in the inward migration process, which is ended by the capture in the 1/1 MMR. Assuming an isothermal and not self-gravitating disk, the capture of massive planets into the resonance is obtained when the mass ratio is sufficiently close to unity and the surface density of the disk is sufficiently high. Multiple planetary systems with Earth-like planets produced co-orbital configuration after some scattering between them. The final outcome showed co-orbital</text>
<text><location><page_4><loc_12><loc_84><loc_70><loc_89></location>configurations producing more easily solutions around L 4 and L 5 , and less probably the AL 4 and AL 5 configurations. If the planets has large mass ratios, the smaller planet was either pushed inside the cavity or trapped in another meanmotion commensurability outside the density jump.</text>
<text><location><page_4><loc_12><loc_78><loc_70><loc_84></location>In view of the results regarding the formation of co-orbital configurations, we restrict our investigation to the case of equal mass planets, considering super-Earth and Saturn examples. We will investigate which is the subsequent evolution under tidal effects after the 1/1 resonance capture.</text>
<section_header_level_1><location><page_4><loc_12><loc_74><loc_43><loc_75></location>2 Dynamics of the conservative problem</section_header_level_1>
<section_header_level_1><location><page_4><loc_12><loc_71><loc_36><loc_73></location>2.1 Phase space of the 1/1 MMR</section_header_level_1>
<text><location><page_4><loc_12><loc_62><loc_70><loc_70></location>The 1/1 MMR can be analyzed through a Hamiltonian formalism as it has been done for other mean motion resonances (see Michtchenko et al. 2006, 2008 ab). Basically, it requires a transformation to adequate resonant variables and a numerical averaging of the Hamiltonian with respect to short-period terms. The reader is referred to Giuppone et al. (2010) for a detailed description of our semi-analytical approach.</text>
<text><location><page_4><loc_12><loc_40><loc_70><loc_62></location>Using a semi-analytical model we can visualize the phase space of the 1/1 MMR. For this task, we calculate level curves of the Hamiltonian which are plotted on the ( σ, n 1 /n 2 ) representative planes in Fig. 1. Hereafter, we call a i , e i , n i , for i = 1 , 2, the semi-major axes, eccentricities and mean orbital motions of the planets, respectively. In the construction of the top panel we adopted e 1 = e 2 = 0 . 0001 and ∆/pi1 = 60 · , and, for the bottom panel, e 1 = e 2 = 0 . 04 and ∆/pi1 = 180 · ; in each case, the planet masses were m 1 = m 2 = 5 m ⊕ , while the mass of the star was m 0 = 1 m /circledot . The interpretation of the obtained portraits is simple, after some initial considerations. Since the averaged resonant system has two degrees of freedom, the phase space of the problem is four-dimensional and the intersection of one planetary path with the representative plane is generally given by four points belonging to the same energy level. However, for the chosen small values of the planet eccentricities, two degrees of freedom interact weakly each with other; in other words, they are nearly separable. This means that each energy level represents, with a good approximation, one nearly circular planetary path on the ( σ, n 1 /n 2 ) plane.</text>
<text><location><page_4><loc_12><loc_22><loc_70><loc_40></location>The stable stationary orbits of the planets in 1/1 MMR are represented by two fixed points in Fig. 1 (top panel); they correspond to the Lagrangian solutions L 4 and L 5 of the circular problem. Their locations in the phase space are given by n 1 /n 2 = 1 and σ = ± 60 · . The oscillations around L 4 and L 5 points are frequently referred to as tadpole orbits. Their domains are bounded by the separatrix shown by the thick black curves in Fig. 1, which pass through the unstable saddle-like L 3 solution with coordinates n 1 /n 2 = 1 and σ = ± 180 · . Outside the separatrix, there is a zone of horseshoe orbits (large amplitude oscillations of σ around 180 · , encompassing both L 4 and L 5 ), which is extended up to the second separatrix, which contains the Lagrangian saddle-like points L 1 and L 2 (not shown in Fig. 1) located on the vertical axis at σ = 0. Configurations leading to close encounters between the planets are represented with cyan curves. Such configurations are of special interest; indeed, we will show that the width of the regions of close</text>
<figure>
<location><page_5><loc_18><loc_54><loc_64><loc_90></location>
<caption>Fig. 1 Energy levels of the conservative Hamiltonian system composed of the Sun-like star and two co-planar planets of equal masses, m 1 = m 2 = 5 m ⊕ , on the ( σ, n 1 /n 2 ) plane. Top panel: initial values of the planet eccentricities are e 1 = e 2 = 0 . 0001 and ∆/pi1 = 60 · . Bottom panel: same plane (enlarged around the origin), obtained with e 1 = e 2 = 0 . 04 and ∆/pi1 = 180 · . Values of ∆/pi1 and eccentricities were chosen for a better comparison with further results.</caption>
</figure>
<text><location><page_5><loc_12><loc_35><loc_70><loc_44></location>approaches between both planets is correlated with the individual masses and that the short-term mutual interactions inside this region provoke imminent disruptions of the system. However, when the two planets are sufficiently close each to other, they form a specific dipole-like configuration known as quasi-satellite stable orbit. The domain of QS is close to the origin of the plane and can be clearly seen in the amplified frame in Fig. 1 (bottom panel). Finally, the origin is a singular point which corresponds to collision between the planets.</text>
<text><location><page_5><loc_12><loc_22><loc_70><loc_34></location>It is worth noting that the orbits L 4 and L 5 correspond to the maximal values of the energy of the conservative 1/1 resonant system. This fact should be kept in mind when dissipative forces are introduced in the system. In this case, the variation of the energy will dictate the evolution of the resonant system: for slowly increasing energy, the system will converge to one of the L 4 and L 5 solutions, while, for slowly decreasing energy, its trajectory will be a spiral unwinding from L 4 (or L 5 ) solution. In the case of a dissipative evolution, the system starting nearly the L 4 equilibrium point will cross the domains of tadpole and horseshoe orbits, approaching the close encounters region, where it probably will be disrupted.</text>
<figure>
<location><page_6><loc_14><loc_49><loc_66><loc_89></location>
<caption>Fig. 2 Amplitude maps for the resonant angle σ (top panels) and the secular angle ∆/pi1 (bottom panels) on the ( σ, ∆/pi1 )-plane of initial conditions. Results were obtained for a two super-Earth planets orbiting a Sun-like star with a 1 = a 2 = 0 . 04 AU (orbital periods around 2.9 d); the initial eccentricities were chosen as e i =0.1 (left column) and e i =0.0001 (right column). Light-blue domains correspond to nearly zero amplitudes ( σ - and ∆/pi1 -families of periodic orbits), darker regions indicate oscillation amplitudes smaller than ∼ 180 · , while red color indicates collision orbits. For e i =0.1 (left column), the phase space clearly shows σ -families as light-blue vertical strips on the top panel, and ∆/pi1 -families as horizontal strips on the bottom panel. The domains of unstable motion (in red) associated to close encounters between the planets surround the region of QS located at σ = 0 · = 360 · . For nearly circular orbits, with e i =0.0001 (right column), the domains of instability cover the region of QS orbits. The location of the σ -families remains almost unaltered (top panel), in contrast with ∆/pi1 -familie, which almost disappear.</caption>
</figure>
<section_header_level_1><location><page_6><loc_12><loc_29><loc_27><loc_30></location>2.2 Amplitude maps</section_header_level_1>
<text><location><page_6><loc_12><loc_22><loc_70><loc_28></location>In this section, we present dynamical maps of the 1/1 MMR on the ( σ, ∆/pi1 ) representative plane. For this task, we construct grids of initial conditions varying both σ and ∆/pi1 between zero and 360 · . Each point in the grid was then numerically integrated over 5000 years (roughly orbital 450 000 periods) using a Bulirsch-Stoer</text>
<text><location><page_7><loc_12><loc_82><loc_70><loc_89></location>based N-body code. We calculated the amplitudes of oscillation of each angular variable. Initial conditions with zero amplitude in σ correspond to σ -family periodic orbits of the co-orbital system, while solutions with zero amplitude in ∆/pi1 correspond to periodic orbits of the ∆/pi1 -family (see Michtchenko et al. 2008ab). Stationary solutions of the averaged problem are the intersections of both families.</text>
<text><location><page_7><loc_12><loc_63><loc_70><loc_81></location>Fig. 2 shows results obtained for a system composed of a Sun-like star and two hot-super-Earths with masses of 5 m ⊕ and semi-major axes of 0.04 AU. The top and bottom frames show the oscillation amplitude of σ and ∆/pi1 , respectively. Left and right frames were constructed with eccentricities e i = 0 . 1 and e i = 0 . 0001, respectively. The domains in light-blue color correspond to orbits with small amplitudes ( < 5 · ), thus indicating the location of the families of periodic orbits. The σ -families defined by nearly zero amplitudes of oscillation of the resonant angle σ are located on the top frames. The ∆/pi1 -families defined by nearly zero amplitudes of oscillation of the secular angle ∆/pi1 can be observed on the bottom panels. For example, at the initial condition with e i = 0 . 1 located at ( σ, ∆/pi1 )=(120 · ,40 · ), σ oscillates with amplitude 100 · , while ∆/pi1 oscillates with very small amplitude around 40 · . Darker regions correspond to increasing amplitudes and denote initial conditions with quasi-periodic motion. The unstable orbits (collisions) parameterized by amplitudes equal to 180 . 1 · , are shown by red color.</text>
<text><location><page_7><loc_12><loc_58><loc_70><loc_62></location>We observe the domains around the Lagrangian equilateral solutions ( L 4 /L 5 ) at ( σ, ∆/pi1 ) = ( ± 60 · , ± 60 · ), the anti-Lagrangian solutions ( AL 4 /AL 5 ) at ( σ, ∆/pi1 ) /similarequal ( ± 60 · , ∓ 120 · ), and the QS solution ( σ, ∆/pi1 ) = (0 , 180 · ).</text>
<text><location><page_7><loc_12><loc_50><loc_70><loc_57></location>For low eccentricities ( e i = 0 . 1, left frames), we observe the asymmetric solutions connected with the σ -family of periodic orbits (vertical cyan strips) and their intersection with the horizontal ∆/pi1 -family (sinusoidal strip at bottom frame). The unstable orbits are very thin regions (in red) that separate the two regimes of motion (symmetric from asymmetric).</text>
<text><location><page_7><loc_12><loc_41><loc_70><loc_49></location>For initially almost circularized orbits ( e i = 0 . 0001, right frames), the QS region disappears (vertical red strips at σ = 0), meanwhile the low amplitude oscillations for the σ -family in top frame remains almost unaltered as vertical strips. In the bottom right frame, the ∆/pi1 -family shrinks into a concentrated region around the exact location of periodic families. Outside these small domains, ∆/pi1 oscillates with high-amplitude ( > 160 · ).</text>
<text><location><page_7><loc_12><loc_35><loc_70><loc_40></location>The results of the maps show that, during the process of tidal eccentricity damping, we expect an increase in the oscillation amplitude of the angles around the equilibrium points of the 1/1 MMR. This feature will be confirmed through analysis of the numerical simulations of the exact equations of motion.</text>
<text><location><page_7><loc_12><loc_26><loc_70><loc_34></location>Using the semi-analytical Hamiltonian model we constructed the families of periodic orbits following Giuppone et al. (2010). Fig. 3 shows the dependence of the angular coordinates of the equilibrium points on the eccentricities from quasicircular orbits to e i = 0 . 9. This figure will serve us as a guide in the choice of the initial conditions for the numerical simulations of the exact equations of motion (see next section).</text>
<text><location><page_7><loc_12><loc_22><loc_70><loc_25></location>The results shown in this section are qualitatively the same for hot-Saturn planets.</text>
<figure>
<location><page_8><loc_22><loc_69><loc_60><loc_87></location>
<caption>Fig. 3 Equilibrium values of σ and ∆/pi1 for the periodic families as function of the eccentricity for m 2 /m 1 = 1.</caption>
</figure>
<section_header_level_1><location><page_8><loc_12><loc_61><loc_54><loc_62></location>3 The equations of motion of the dissipative problem</section_header_level_1>
<text><location><page_8><loc_12><loc_46><loc_70><loc_60></location>We consider a system composed by a central star and two close-in co-orbital and rotating planets. Due to the short astrocentric distances, we suppose that both planets are deformed by the tides raised by the central star, which is also assumed distorted by the tides raised by the planets. The orbital planes of motion are supposed to be coincident with the reference plane (i.e, zero inclinations). In addition, we assume that the rotation axes are normal to the orbital planes (i.e, zero obliquities). For close-in planets, it is also convenient to consider the forces arising form general relativity, in addition to the mutual gravitational interaction and tidal forces. Remember, we call m 0 , m 1 and m 2 the masses of the star and planets, respectively; whereas the radii are denoted by R 0 , R 1 and R 2 .</text>
<text><location><page_8><loc_12><loc_43><loc_70><loc_46></location>In a reference system centered in the star, where the positions and velocities of the planets are r i and v i , with i = 1 , 2, the equations of motion are given by</text>
<formula><location><page_8><loc_23><loc_27><loc_70><loc_40></location>r 1 = -G ( m 0 + m 1 ) r 3 1 r 1 + Gm 2 ( r 2 -r 1 | r 2 -r 1 | 3 -r 2 r 3 2 ) + ( m 0 + m 1 ) m 0 m 1 ( f 1 -f 01 + g 1 ) + f 2 -f 02 + g 2 m 0 , (1) r 2 = -G ( m 0 + m 2 ) r 3 2 r 2 + Gm 1 ( r 1 -r 2 | r 1 -r 2 | 3 -r 1 r 3 1 ) + ( m 0 + m 2 ) m 0 m 2 ( f 2 -f 02 + g 2 ) + f 1 -f 01 + g 1 m 0 . (2)</formula>
<text><location><page_8><loc_12><loc_22><loc_70><loc_25></location>On one hand, g i are the general relativity contributions acting on the planets, which for i = 1 , 2 are given by</text>
<formula><location><page_9><loc_25><loc_84><loc_70><loc_88></location>g i = Gm i m 0 c 2 r 3 i [( 4 Gm 0 r i -v 2 i ) r i +4( r i · v i ) v i ] (3)</formula>
<text><location><page_9><loc_12><loc_79><loc_70><loc_84></location>where v i = ˙ r i and c is the speed of light (see Beutler 2005). On the other hand, f i are the tidal forces raised by the star acting on the masses m 1 and m 2 due to their deformations, respectively. We use the expression for tidal forces given by Mignard (1979):</text>
<formula><location><page_9><loc_23><loc_75><loc_70><loc_78></location>f i = -3 k i ∆t i Gm 2 0 R 5 i r 10 i [2 r i ( r i · v i ) + r 2 i ( r i × Ω i + v i )] , (4)</formula>
<text><location><page_9><loc_12><loc_65><loc_70><loc_74></location>where Ω i is the rotation angular velocity of the i -planet, for i = 1 , 2. It is worth noting that Mignard's force is given by a closed formula and, therefore, is valid for any value of eccentricity 1 . k 2 i is the second order Love number and ∆t i is the time lag, which can be interpreted as a delay in the deformation of the tidally affected body due to its internal viscosity. The total tidal force on the star is f 0 = f 01 + f 02 , where f 0 i are the individual tidal forces raised by each planet and for i = 1 , 2 are given by</text>
<formula><location><page_9><loc_21><loc_60><loc_70><loc_63></location>-f 0 i = -3 k 0 ∆t 0 Gm 2 i R 5 0 r 10 i [2 r i ( r i · v i ) + r 2 i ( r i × Ω 0 + v i )] , (5)</formula>
<text><location><page_9><loc_12><loc_56><loc_70><loc_60></location>where the subscript '0' stand for star quantities. Note that are the forces -f 0 i which act on the planets and must be considered in the equations of motion of the bodies.</text>
<text><location><page_9><loc_27><loc_55><loc_27><loc_56></location>/negationslash</text>
<text><location><page_9><loc_12><loc_44><loc_70><loc_56></location>The fact that ∆t = 0 introduces energy dissipation in the system, resulting in orbital and rotational evolution due to tidal torques. The tidal model here adopted is a classical linear approach (Darwin, 1880), since it is implicitly assumed that the resulting dissipation is proportional to the tidal frequencies. This tidal model is frequently referred to as a constant time-lag model and, despite that recent works have shown that it could not be appropriate for the study of terrestrial planets, is expected to yield to the approximately correct results. For a review of other tidal models, the reader is referred to Efroimsky & Williams (2009) and Ferraz-Mello (2013).</text>
<section_header_level_1><location><page_9><loc_12><loc_39><loc_31><loc_40></location>4 Numerical simulations</section_header_level_1>
<text><location><page_9><loc_12><loc_27><loc_70><loc_38></location>In this section we show the result of the numerical integration of equations (1)(2) for some particular systems. As shown in Giuppone et al. (2010), large-scale planetary migration should favor the formation of co-orbital configurations more likely for nearly equal mass of the planets (see Sec. 1.2). Hence, we restrict our investigation to the case m 2 /m 1 = 1. We start with the case of two super-Earth planets with m 1 = m 2 = 5 m ⊕ orbiting a Sun-like star with m 0 = m /circledot and R 0 = R /circledot . The radii are such that the mean densities ρ satisfy ρ 1 = ρ 2 = ρ ⊕ , so that R 1 = R 2 = 5 1 / 3 R ⊕ .</text>
<text><location><page_10><loc_12><loc_85><loc_70><loc_89></location>For sake of completeness we also explore the case of individual masses of m Sat , where m Sat is the mass of Saturn, and the radii are computed as in the super-Earth case.</text>
<text><location><page_10><loc_12><loc_66><loc_70><loc_85></location>We are going to investigate the final outcome of a system originally evolving in a co-orbital configuration when the tidal effect acts to change the orbital elements. Thus, we choose the initial values of the angles ( σ, ∆/pi1 ) near to the equilibrium points. The initial semi-major axes and eccentricities are a 1 = a 2 = 0 . 04 AU (orbital periods of /similarequal 2 . 9 d) and e 1 = e 2 = 0 . 1. The initial values of rotation periods are 16 . 7 h for both planets, however, they are not important because the rotation rapidly encounters its stationary value which only depends on the eccentricity within the adopted tidal model (Hut, 1981). We set 19.4 d as the initial value of the stars' rotation period, noting that the stationary value in this case is reached in a much larger timescale than for the planets. In addition, we need to know the value of the moment of inertia around the rotation axis, which is given by ξmR 2 , where ξ is the structure constant. For super-Earths, we adopt Earth values for ξ , namely, ξ = 0 . 33; whereas for hot-Saturn and a Sun-like star we set ξ = 0 . 21 and ξ = 0 . 07, respectively 2 .</text>
<text><location><page_10><loc_12><loc_52><loc_70><loc_66></location>The linear model enable us to relate the time delay ∆t with the quality factor Q through 1 /Q = n∆t , where n is the mean orbital motion (see Correia et al. 2012). However, the quality factor is poorly constrained even for Solar System bodies, although some previous works have brought valuable information. For instance, for the solid Earth we have Q = 370 (Ray et al. 1996), and values of Q/k 2 = 0 . 9 × 10 5 , Q > 1 . 8 × 10 4 and Q/k 2 = 4 . 5 × 10 4 were estimated for Jupiter, Saturn and Neptune, respectively (Lainey et al. 2009; Meyer & Wisdom 2007; Zhang & Hamilton 2008). More recent investigations (Ferraz-Mello 2013) suggest that the hot super-Earth CoRoT-7 b has Q = 49, whereas for a hot Jupiter of 2-3 m Jup in a 5-d orbit, Q ∼ 4 . 2 × 10 5 .</text>
<text><location><page_10><loc_12><loc_45><loc_70><loc_52></location>In this work, we adopt Q = 20, k 2 = 0 . 3 for hot super-Earths and Q = 1 × 10 4 , k 2 = 0 . 34 for hot-Saturn planets, whereas for Sun-like stars we adopt Q = 1 × 10 6 , k 2 = 0 . 34. The corresponding values of k 2 ∆t , where ∆t is obtained through 1 /Q = n∆t , are ( k 2 ∆t ) super -Earth = 600 sec, ( k 2 ∆t ) Saturn = 1 . 37 sec and ( k 2 ∆t ) Sun = 0 . 0137 sec.</text>
<text><location><page_10><loc_12><loc_33><loc_70><loc_45></location>We chose initial values of the angles within 4 degrees around the exact equilibrium solutions (see Fig. 3 for the locations of the exact solutions and Table 1 for initial values chosen). The symmetric and asymmetric periodic solutions for mass ratio close to unity are such that e 1 = e 2 (and also a 1 = a 2 , see Hadjidemetriou et al. 2009; Giuppone et al. 2010). Hence, we start our simulations with equal eccentricities, since we are assuming that the system is evolving under the 1/1 MMR. However, we note that the global results would not be affected by either the initial values of eccentricities or amplitudes of the angles, as long the corresponding configuration is nearby to the exact 1/1 MMR.</text>
<text><location><page_10><loc_12><loc_27><loc_70><loc_33></location>For our stability criterion, we use the critical distance given by d = κ ( R 1 + R 2 ). Hence, we assume that orbital instability would occur whenever the instantaneous mutual distance is equal or smaller than d , that is, for | r 2 -r 1 | ≤ d . We note that κ = 1 implies in the physical collision between the planets 3 . In this work, we use</text>
<table>
<location><page_11><loc_24><loc_83><loc_58><loc_89></location>
<caption>Table 1 Arbitrary initial conditions near the stable periodic solutions in the ( σ, ∆/pi1 ) plane, calculated with the semi-analytical method (see Giuppone et al. 2010). All conditions have a 1 = a 2 = 0 . 04 AU. The same initial co-orbital configurations were chosen for both hot super-Earth and hot-Saturn cases.</caption>
</table>
<text><location><page_11><loc_12><loc_72><loc_70><loc_74></location>κ = 1 and κ = 2, however, we anticipate that the final results are not sensitive (in the sense of collision timescale) to the choice of one of the above values of κ .</text>
<section_header_level_1><location><page_11><loc_12><loc_68><loc_20><loc_69></location>4.1 Results</section_header_level_1>
<section_header_level_1><location><page_11><loc_12><loc_65><loc_27><loc_66></location>4.1.1 Hot super-Earths</section_header_level_1>
<text><location><page_11><loc_12><loc_61><loc_70><loc_63></location>We start with the case in which the individual planets are represented by a 5 m ⊕ super-Earth.</text>
<text><location><page_11><loc_12><loc_56><loc_70><loc_61></location>Figs. 4 shows the time variation of the angles ( σ, ∆/pi1 ) around the symmetric and asymmetric equilibrium points for κ = 1, whereas the evolution of the eccentricities is shown in Fig. 5. A collision between the planets is the final outcome of all simulations .</text>
<text><location><page_11><loc_12><loc_46><loc_70><loc_55></location>For AL 4 /AL 5 configurations, the angle ∆/pi1 oscillates with small amplitudes up to around 35 Myr. However, the amplitudes start to increase and, close to 45 Myr, a new regime of oscillation appears, in which the angles seem to avoid the regions centered near 240 · and 120 · . Note that these positions are the corresponding equilibrium points of ∆/pi1 for AL 4 /AL 5 solutions in the circular case. Moreover, between 60- 65 Myr, a circulation of ∆/pi1 appears and, finally, an oscillation around 180 · with increasing amplitude occurs before the collision between the planets.</text>
<text><location><page_11><loc_12><loc_39><loc_70><loc_46></location>The evolutions of ∆/pi1 along L 4 /L 5 during the first 20 Myr follow close to the stationary values (60 · /300 · ). After that, ∆/pi1 deviates and a time interval of oscillation and circulation occurs, with the angle taking many values in the whole interval between 0 · - 360 · but privileging the oscillation around 0 · . Finally, ∆/pi1 oscillates around 180 · before collision.</text>
<text><location><page_11><loc_12><loc_32><loc_70><loc_39></location>Regarding to the resonant angle σ , we note in Fig. 4 a libration with increasing amplitude around the equilibrium points. When the regime of oscillation/circulation of the angle ∆/pi1 appears, σ librates around 180 · with very high amplitude until the end of the simulation (for all asymmetric configurations). We will return to this discussion later in this section.</text>
<text><location><page_11><loc_12><loc_27><loc_70><loc_32></location>The solution close to QS is the first to become unstable, around 6 Myr of evolution. In this case, the amplitudes of the angles are close to 4 · and e 1 /similarequal e 2 = 0 . 038 before destabilization. Several runs for the QS case were carried out with different values of initial amplitudes (even zero) and the collision was the final</text>
<text><location><page_11><loc_12><loc_22><loc_70><loc_26></location>thus, it is not restricted to impact between the planets. However, close encounters with κ between 1 and 2, may result in tidal disruption of one or both planets rather than ejection into hyperbolic orbit</text>
<figure>
<location><page_12><loc_12><loc_75><loc_41><loc_89></location>
<caption>Fig. 6 shows the time variation of semi-major axes and eccentricities corresponding to the evolution along AL 4 (solid curves), noting that the condition of the equilibrium solution is followed ( a 1 /similarequal a 2 ). In addition, we also plot the same elements when only one planet is present in the system (dashed curves). It is interesting to note that both results are in good agreement. The explanation can be found in the fact that, since the system follows an equilibrium configuration from the beginning provided by the resonant trapping, the tidal evolution acts independently to damp the orbital elements 4 . Moreover, the final value of semi-major axis in the case of a single planet can be found through a fin = a ini exp( e 2 fin -e 2 ini ), where the subscripts ' ini ' and ' fin ' stand for initial and final values, respectively (see Rodr'ıguez et al. 2011b). Replacing numerical values, we obtain a fin = 0 . 0396 AU, in good agreement with the numerical simulations. The above analytical calculation only considers the effect of planetary tides (tides on the planets), indicating</caption>
</figure>
<figure>
<location><page_12><loc_41><loc_75><loc_69><loc_89></location>
</figure>
<text><location><page_12><loc_41><loc_82><loc_42><loc_84></location>[deg]</text>
<text><location><page_12><loc_41><loc_82><loc_42><loc_82></location>σ</text>
<paragraph><location><page_12><loc_12><loc_68><loc_70><loc_74></location>Fig. 4 Time variation of ∆/pi1 and σ (in degrees) for the symmetric ( QS ) and asymmetric ( L 4 /L 5 and AL 4 /AL 5 ) co-orbital configurations, for starting values displayed in Table 1. Several regimes of libration/oscillation/circulation are present along the evolution (see text for detailed discussion). The planets ultimately collide on timescales of tens of millions of years, except for the QS case in which the system destabilizes at approximately 6 Myr.</paragraph>
<text><location><page_12><loc_12><loc_61><loc_70><loc_65></location>outcome in all cases in very similar timescales ( < 10 Myr). Moreover, we tested a run without tides and the system becomes stable at least up to 60 Myr (with constant amplitudes of the angles).</text>
<text><location><page_12><loc_12><loc_54><loc_70><loc_60></location>The evolution of eccentricities follows with e 1 /similarequal e 2 (and thus superimposed in Fig. 5), according to the stationary solutions for equal mass planets. The final result of the tidal evolution is an almost doubly orbital circularization except in the QS case, in which the oscillation amplitude of eccentricities are larger than for other cases.</text>
<text><location><page_12><loc_12><loc_44><loc_70><loc_53></location>If we suppose that the time variation of the eccentricities follows an exponential law (see right panel in Fig. 6), we can estimate the timescale for tidal damping as τ e = ˙ e/e . Using a classical averaged expression for ˙ e obtained from linear tidal models (see Eq. (3.5) in Rodr'ıguez & Ferraz-Mello 2010) we obtain τ e /similarequal 6 Myr. This value gives an estimation of the e-folding of the eccentricity damping. Therefore, the timescales for destabilization shown in Fig. 4 correspond roughly to 10 τ e .</text>
<figure>
<location><page_13><loc_21><loc_69><loc_61><loc_89></location>
<caption>Fig. 5 Time variation of eccentricities for the system with two super-Earths planets. For L 4 and AL 4 co-orbital configurations, the circularization is obtained at the end of the simulations, whereas for the QS case, the final values are close to 0.038. In all cases, the evolution follows close to the stationary solution e 1 = e 2 (the values of the eccentricities are superimposed in the figure).</caption>
</figure>
<text><location><page_13><loc_42><loc_50><loc_43><loc_55></location>ECCENTRICITY</text>
<figure>
<location><page_13><loc_12><loc_45><loc_42><loc_58></location>
<caption>Fig. 7 shows the result of the numerical simulations for L 4 /L 5 in the plane ( σ, n 1 /n 2 ). Here, it is easy to identify the librations regimes of the resonant angle σ within the domain of the 1/1 MMR. The initial small libration amplitude around (60 · , 1) and (300 · , 1) (red and green points) increases as the orbit circularizes due to the tidal evolution. When the amplitude is large enough, the motion occurs</caption>
</figure>
<figure>
<location><page_13><loc_44><loc_45><loc_69><loc_58></location>
<caption>Fig. 6 Time variation of semi-major axes and eccentricities along the AL 4 solution (solid curves) and the comparison with the case of a single-planet (dashed curves), for the two superEarths planet system. Both results are in good agreement, indicating that the evolution around the 1/1 trapping leads the planets to follow independent paths dictated by the tidal effect.</caption>
</figure>
<text><location><page_13><loc_12><loc_33><loc_70><loc_36></location>that, in view of the agreement, the contribution of stellar tides (tides on the star) can be safely neglected 5 .</text>
<figure>
<location><page_14><loc_21><loc_69><loc_61><loc_88></location>
<caption>Fig. 7 The evolution in the plane ( σ, n 1 /n 2 ) for the two super-Earth planets system around the L 4 /L 5 equilibrium points. The structure of the resonant motion is identified through the domains of libration of the angle σ , disconnected by the corresponding separatrix. Similar results can be obtained for the AL 4 /AL 5 equilibrium points.</caption>
</figure>
<text><location><page_14><loc_12><loc_52><loc_70><loc_59></location>around σ = 180 · (blue points) and encompasses both equilateral Lagrangian points (see also Fig. 4, right panel). In analogy with the restricted problem, these types of motion correspond to tadpole and horseshoe co-orbital configurations. In Fig. 7, we clearly see that the tadpole and horseshoe oscillation regimes are disconnected by the separatrix of the 1/1 resonant motion.</text>
<text><location><page_14><loc_12><loc_42><loc_70><loc_51></location>We should note that the condition σ ∼ 0 is necessary for the collision to occur. However, we do not observe in Fig. 7 a separatrix disconnecting the regimes of libration and circulation of σ , and thus the angle seems never takes the zero value. To overcome this situation, we performed a new short numerical simulation starting very close to the configuration in which the collision occurs. We will illustrate these results in the next section in the application for hot-Saturn planets, but the same can be applied for super-Earths.</text>
<text><location><page_14><loc_12><loc_33><loc_70><loc_41></location>It is interesting to compare the numerical results with those predicted by the semi-analytical model (Sec. 2). Indeed, the curve which separates the domains of tadpole and horseshoe motions (black curve on top panel of Fig. 1) agrees with the separatrix appearing in Fig. 7. The oscillation amplitude of n 1 /n 2 in both cases is approximately 0.12, while σ takes values as small as 25 · (for L 4 ), until the horseshoe domain arises.</text>
<text><location><page_14><loc_12><loc_22><loc_70><loc_32></location>The numerical solution for the QS configuration indicates that n 1 /n 2 /similarequal 1 . 02 and σ /similarequal 5 · before destabilization. We thus note an excellent agreement of the above result through an inspection of Fig. 1 (bottom panel), where the point ( σ, n 1 /n 2 ) = (5 · , 1 . 02) belongs to the separatrix of the QS co-orbital configuration (note that Fig. 1 for QS was constructed taking e 1 = e 2 = 0 . 04 for better comparison with the numerical results, in which the eccentricities are close to 0.038 just before collision).</text>
<figure>
<location><page_15><loc_19><loc_67><loc_62><loc_88></location>
<caption>Fig. 8 The analogous to Fig. 7 for the system with two hot-Saturn planets. We can identify the domains of libration around the Lagrangian equilibrium points (tadpole and horseshoe) and also the circulation of σ . The collision takes place when σ becomes very close to zero.</caption>
</figure>
<section_header_level_1><location><page_15><loc_12><loc_58><loc_23><loc_59></location>4.1.2 Hot Saturn</section_header_level_1>
<text><location><page_15><loc_12><loc_50><loc_70><loc_56></location>The numerical exploration of the system with two Saturn planets indicates that, as in the previous system, collision between the planets results in all simulations for κ = 1 and κ = 2. Moreover, all initial co-orbital configurations destabilize in timescales between 55-80 Myr, except the one around QS in which the collision occurs near 430 Kyr.</text>
<text><location><page_15><loc_12><loc_41><loc_70><loc_49></location>In analogy to Fig. 7, Fig. 8 displays the results of the numerical simulations for κ = 1, showing the libration of σ around the equilibrium Lagrangian points (tadpole and horseshoe orbits). In this case, the motion in the horseshoe domain is quite unstable due to the strong mutual interaction between the planets, and the system remains in that libration regime only 166 yr before collision (in the case of the two super-Earth system, the horseshoe regime lasted around 8 Myr).</text>
<text><location><page_15><loc_12><loc_36><loc_70><loc_41></location>In addition to these types of libration, we see the domain in which the angle σ circulates (also in blue points). In the circulation regime, in which the orbits are no longer locked in the 1/1 MMR, σ takes the zero value several times, favoring the orbital conditions for the collision between the planets.</text>
<section_header_level_1><location><page_15><loc_12><loc_31><loc_35><loc_32></location>5 Conclusions and discussion</section_header_level_1>
<text><location><page_15><loc_12><loc_22><loc_70><loc_29></location>We investigated the motion of a two-planet system which evolves under combined effects of mutual and tidal interactions in the vicinity of a 1/1 mean motion resonance (i.e, co-orbital configuration). We considered systems of equal mass planets ( m 2 /m 1 = 1) corresponding to super-Earth (5 m ⊕ ) and Saturn planets ( /similarequal 95 m ⊕ ). Numerical simulations of the exact equations of motion indicate that the colli-</text>
<text><location><page_16><loc_12><loc_87><loc_70><loc_89></location>between the planets is the final outcome of the tidal evolution for all initial co-orbital configurations tested in this work.</text>
<text><location><page_16><loc_12><loc_80><loc_70><loc_86></location>We started our simulations considering initial conditions near the five stationary configurations of the co-orbital motion (namely, L 4 /L 5 , AL 4 /AL 5 and QS ). As tides continually damp the eccentricities, the initially small oscillation amplitudes of the angles ( σ, ∆/pi1 ) around the equilibrium points increases and, ultimately, the instability occurs.</text>
<text><location><page_16><loc_12><loc_67><loc_70><loc_79></location>We identified several libration regimes of the resonant angle σ , including the tadpole and horseshoe co-orbital configurations, in which the motion occurs around σ = 60 · / 300 · and σ = 180 · , respectively. In addition, the motion around the antiLagrangian equilibrium points ( AL 4 /AL 5 ), located close to ( σ, ∆/pi1 ) = (60 · / 300 · , 240 · / 120 · ) for m 2 /m 1 = 1 and small eccentricities, remains stable over a timescale about 1.5 larger than for the Lagrangian points ( L 4 /L 5 ) for both types of systems. Moreover, the stability of the quasi-satellite ( QS ) co-orbital configuration ( σ = 0 · , ∆/pi1 = 180 · ) is restricted to short timescales ( < 10 Myr and < 1 Myr, for super-Earth and Saturn individual masses, respectively).</text>
<text><location><page_16><loc_12><loc_50><loc_70><loc_67></location>The interpretation of the tidal evolution of the 1/1 resonant planet pair and its ultimate disruption is in the following. We know that the L 4 and L 5 stationary solutions correspond to global maxima of the conservative Hamiltonian of the 1/1 MMR, at least for small and moderate eccentricities. This is a consequence of the fact that a mean-motion resonance, acting as a protection mechanism, implies the maximal possible mutual distance between two planets at conjunction, where their closest encounters occur (Michtchenko et al. 2008b). When the tidal interactions are introduced in the system, the energy of the system is dissipated through tidal heating of the planets. In this way, starting near one of the L 4 and L 5 points, the system evolving under dissipation will suffer the increase the oscillation amplitudes of the angles around the exact positions of the equilibrium points. It will cross the domains of the tadpole and horseshoe orbits, as described in Sec. 2. Thus, the system will ultimately tend toward a collisional route.</text>
<text><location><page_16><loc_12><loc_36><loc_70><loc_49></location>Therefore, the results presented in this work suggest that the tidal evolution of close-in planetary systems in the vicinity of the 1/1 MMR is globally unstable, constraining the possible detection of hot exo-Trojans (particularly among KOI's candidates). However, we have to stress that the origin of such close-in systems is unknown. Indeed, as recently shown in Giuppone et al. (2012), the formation of co-orbital configurations at distances of 1 AU is possible, however, the question on how did they reach close-in configurations is still under discussion. Also, it should be kept in mind that, for planets more distant from the central star, we expect a timescale for tidal evolution much longer than the age of the systems, and thus the possibility of radial velocity detections cannot be ruled out.</text>
<text><location><page_16><loc_12><loc_28><loc_70><loc_36></location>As a final considerations, some limitations of our model should be highlighted. First, the applicability for rocky planets of the adopted linear tidal model has been recently questioned in some works (see Efroimsky & Williams 2009; FerrazMello 2013 and references therein). However, we speculate that the adoption of a different tidal model should change the timescales of the tidal evolution but the global result would not be affected.</text>
<text><location><page_16><loc_12><loc_22><loc_70><loc_28></location>Second, we neglected the polar and permanent equatorial deformations of the bodies (i.e, J 2 , C 22 and higher orders terms). Recently, Rodr'ıguez et al. (2012) have shown that the contribution of C 22 can lead to temporary captures in spinorbit resonances of rocky planets (see also Correia & Rodr'ıguez (2013) and Cal-</text>
<text><location><page_17><loc_12><loc_82><loc_70><loc_89></location>ri & Rodr'ıguez (2013) for a calculation of J 2 and C 22 for a specific resonant rotation trapping). Moreover, the orbital decay and the eccentricity damping are larger whenever the planet rotation is trapped in a resonant motion. Hence, the tidal model and the consideration of the equatorial permanent deformation should be taken into account in further investigations.</text>
<text><location><page_17><loc_12><loc_62><loc_70><loc_82></location>Finally, following the suggestion of Dr. M. Efroimsky (private communication), we performed an additional simulation including an indirect tidal term which is described in the following. The tidal effect on the star due to one of the planets creates a (dissipative) torque which affects the orbital and rotational evolution of the tidal raising planet. The other planet will also 'feel' this tidal effect because the star deformation crates a potential in an arbitrary point in the space. The effect of this force may be important for planets trapped in mean-motion resonances. In the co-orbital case, we have a 'frozen' configuration in which the relative positions of the bodies are located in the vertices of an equilateral triangle of variable size. We only consider the static tidal component (i.e., the instantaneous response, independent on ∆t 0 , see Ferraz- Mello et al. 2008), because it is orders of magnitude larger than its dissipative counterpart. The result of the numerical simulation for the two Saturn planets system have shown that the inclusion of these terms only slightly modifies the timescales of the planet evolution, delaying the instability in approximately 10% for a L 4 /L 5 starting configuration.</text>
<text><location><page_17><loc_12><loc_51><loc_70><loc_60></location>Acknowledgements We acknowledge to M. Efroimsky for comments and suggestions. A.R and T.A.M acknowledge the support of this project by FAPESP (2009/16900-5) and CNPq (Brazil). C.A.G acknowledges the support by the Argentinian Research Council, CONICET. This work has made use of the computing facilities of the Laboratory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchase was made possible by the Brazilian agency FAPESP (grant 2009/54006-4) and the INCT-A. Some of the computations were performed on the Blafis cluster at the Aveiro University. We also acknowledge the two anonymous reviewers for their valuable comments and suggestions.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "Noname manuscript No. (will be inserted by the editor)",
"pages": [
1
]
},
{
"title": "Tidal evolution of close-in exoplanets in co-orbital configurations",
"content": "Adri´an Rodr´ıguez · Cristian A. Giuppone · Tatiana A. Michtchenko the date of receipt and acceptance should be inserted later Abstract In this paper, we study the behavior of a pair of co-orbital planets, both orbiting a central star on the same plane and undergoing tidal interactions. Our goal is to investigate final orbital configurations of the planets, initially involved in the 1/1 mean-motion resonance (MMR), after long-lasting tidal evolution. The study is done in the form of purely numerical simulations of the exact equations of motions accounting for gravitational and tidal forces. The results obtained show that, at least for equal mass planets, the combined effects of the resonant and tidal interactions provoke the orbital instability of the system, often resulting in collision between the planets. We first discuss the case of two hot-super-Earth planets, whose orbital dynamics can be easily understood in the frame of our semianalytical model of the 1/1 MMR. Systems consisting of two hot-Saturn planets are also briefly discussed. Keywords Celestial mechanics · Planets and satellites · Tides · Exo-Trojans",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "In contrast with strong mean-motion resonances (MMR), such as 2/1 and 3/2 MMR (see Beaug'e et al. 2012) there are no known pairs in co-orbital configuration, despite the continuously increasing number of discovered exoplanets. This intriguing fact can be investigated in the context of a past large-scale planetary migration and capture due to, for instance, interactions with the gaseous protoplanetary disk. Several theories about the formation, dissipative evolution and possible detection of exo-Trojans have been developed and will be discussed later in this section. Instituto de Astronomia, Geof'ısica e Ciˆencias Atmosf'ericas, IAG - USP, Universidade de S˜ao Paulo, S˜ao Paulo, SP, Brazil E-mail: adrian@astro.iag.usp.br Observatorio Astron'omico, Universidad Nacional de C'ordoba, IATE, C'ordoba, Argentina The first detailed analysis of hypothetical co-orbital exoplanets was done in Laughlin & Chambers (2002), focusing the difficulties in fitting of the RV data of these systems. More recently, Hadjidemetriou et al. (2009) studied the topology of the phase space and the long term evolution of two-planet systems in the vicinity of the exact 1/1 MMR in the conservative case and for several planetary mass ratios. Hadjidemetriou & Voyatzis (2011) included a drag force and simulated the dynamical evolution of a two-planet system initially trapped in a stable 1/1 resonant periodic motion. Under this drag force the system migrates along the families of periodic orbits and is finally trapped in a satellite orbit. Giuppone et al. (2010) analyzed the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration using a semi-analytical method. They found two new asymmetric solutions which do not exist in the restricted threebody problem. These solutions were also obtained by Robutel & Pousse (2013), who developed an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance for near coplanar and near circular orbits. Another kind of dissipative evolution in planetary systems is due to tidal interactions of short-period planets with the central star, which originate changes in orbital elements and rotational periods of the planets. For a planet orbiting a slow rotating star, the tidal effects lead to orbital decay, orbital circularization and rotation synchronization on timescales which depend on physical parameters and initial orbital configurations of interacting bodies (see Dobbs-Dixon et al. 2004; Ferraz-Mello et al. 2008; Rodr'ıguez et al. 2011a, Michtchenko & Rodr'ıguez 2011 and references therein). Some recent studies address the problem of tidal interaction in two-planet resonant systems (see Papaloizou 2011; Delisle et al. 2012). They show that, as tides tend to circularize the planetary orbits, the planets repel each other, in such a way that the period ratios at the end of the evolution are slightly larger than the corresponding nominal low-order resonant values. This feature seems to be a natural outcome of slow dissipative evolution (Lithwick & Wu 2012; Delisle et al. 2012; Batygin & Morbidelli 2013) and is in agreement with close-in planetary configurations detected by the Kepler mission among KOI's candidates. It should be noted that these works focus the attention on first order MMR(e.g. 2/1, 3/2, 4/3), while the case of co-orbital tidal evolution of exoplanets has not been still explored in the context of the general three-body problem. This is the main task of the present paper.",
"pages": [
1,
2
]
},
{
"title": "1.1 Review of stable co-orbital configurations",
"content": "In the following, we describe the most important results concerning the equilibrium points of the co-orbital configurations in the general three-body problem. The reader is referred to Giuppone et al. (2012) for a summary of previous works. Particularly, Hadjidemetriou et al. (2009) studied the motion close to a periodic orbit by computing the Poincar'e map on the surfaces of sections. For this task, the symmetric families of stable and unstable motions were constructed and a previously unknown stable configuration was discovered, referred to as the QuasiSatellite ( QS ) solutions. Giuppone et al. (2010) constructed the families of periodic orbits in the vicinity of the 1/1 MMR using a semi-analytical method. The authors identified two separate regions of stability, symmetric and asymmetric, defined by the behavior of the resonant angles ( σ, ∆/pi1 ) ≡ ( λ 2 -λ 1 , /pi1 2 -/pi1 1 ), where λ i and /pi1 i are mean longitudes and longitudes of pericenter of the planets, respectively. Summarizing:",
"pages": [
2,
3
]
},
{
"title": "1.2 Formation of exo-Trojans",
"content": "There is a vast literature trying to explain the formation of co-orbital planets and different mechanisms have been proposed. Here we only highlight some of the proposed scenarios: planetesimal and planet-planet scattering (Kortenkamp 2005; Morbidelli et al. 2008), direct collisional emplacement (Beaug'e et al. 2007), in situ accretion (Chiang & Lithwick 2005) and migration in multiple protoplanet systems (Cresswell & Nelson 2006; Hadjidemetriou & Voyatzis 2011; Giuppone et al. 2012). Particularly, Giuppone et al. (2012) analyzed whether co-orbital systems may also be formed through the interaction of two planets with a density jump in the protoplanetary disk. The authors considered two planets, initially located farther than the 2/1 MMR (beyond 1 AU, where tidal interaction with the central star is almost negligible) and involved in the inward migration process, which is ended by the capture in the 1/1 MMR. Assuming an isothermal and not self-gravitating disk, the capture of massive planets into the resonance is obtained when the mass ratio is sufficiently close to unity and the surface density of the disk is sufficiently high. Multiple planetary systems with Earth-like planets produced co-orbital configuration after some scattering between them. The final outcome showed co-orbital configurations producing more easily solutions around L 4 and L 5 , and less probably the AL 4 and AL 5 configurations. If the planets has large mass ratios, the smaller planet was either pushed inside the cavity or trapped in another meanmotion commensurability outside the density jump. In view of the results regarding the formation of co-orbital configurations, we restrict our investigation to the case of equal mass planets, considering super-Earth and Saturn examples. We will investigate which is the subsequent evolution under tidal effects after the 1/1 resonance capture.",
"pages": [
3,
4
]
},
{
"title": "2.1 Phase space of the 1/1 MMR",
"content": "The 1/1 MMR can be analyzed through a Hamiltonian formalism as it has been done for other mean motion resonances (see Michtchenko et al. 2006, 2008 ab). Basically, it requires a transformation to adequate resonant variables and a numerical averaging of the Hamiltonian with respect to short-period terms. The reader is referred to Giuppone et al. (2010) for a detailed description of our semi-analytical approach. Using a semi-analytical model we can visualize the phase space of the 1/1 MMR. For this task, we calculate level curves of the Hamiltonian which are plotted on the ( σ, n 1 /n 2 ) representative planes in Fig. 1. Hereafter, we call a i , e i , n i , for i = 1 , 2, the semi-major axes, eccentricities and mean orbital motions of the planets, respectively. In the construction of the top panel we adopted e 1 = e 2 = 0 . 0001 and ∆/pi1 = 60 · , and, for the bottom panel, e 1 = e 2 = 0 . 04 and ∆/pi1 = 180 · ; in each case, the planet masses were m 1 = m 2 = 5 m ⊕ , while the mass of the star was m 0 = 1 m /circledot . The interpretation of the obtained portraits is simple, after some initial considerations. Since the averaged resonant system has two degrees of freedom, the phase space of the problem is four-dimensional and the intersection of one planetary path with the representative plane is generally given by four points belonging to the same energy level. However, for the chosen small values of the planet eccentricities, two degrees of freedom interact weakly each with other; in other words, they are nearly separable. This means that each energy level represents, with a good approximation, one nearly circular planetary path on the ( σ, n 1 /n 2 ) plane. The stable stationary orbits of the planets in 1/1 MMR are represented by two fixed points in Fig. 1 (top panel); they correspond to the Lagrangian solutions L 4 and L 5 of the circular problem. Their locations in the phase space are given by n 1 /n 2 = 1 and σ = ± 60 · . The oscillations around L 4 and L 5 points are frequently referred to as tadpole orbits. Their domains are bounded by the separatrix shown by the thick black curves in Fig. 1, which pass through the unstable saddle-like L 3 solution with coordinates n 1 /n 2 = 1 and σ = ± 180 · . Outside the separatrix, there is a zone of horseshoe orbits (large amplitude oscillations of σ around 180 · , encompassing both L 4 and L 5 ), which is extended up to the second separatrix, which contains the Lagrangian saddle-like points L 1 and L 2 (not shown in Fig. 1) located on the vertical axis at σ = 0. Configurations leading to close encounters between the planets are represented with cyan curves. Such configurations are of special interest; indeed, we will show that the width of the regions of close approaches between both planets is correlated with the individual masses and that the short-term mutual interactions inside this region provoke imminent disruptions of the system. However, when the two planets are sufficiently close each to other, they form a specific dipole-like configuration known as quasi-satellite stable orbit. The domain of QS is close to the origin of the plane and can be clearly seen in the amplified frame in Fig. 1 (bottom panel). Finally, the origin is a singular point which corresponds to collision between the planets. It is worth noting that the orbits L 4 and L 5 correspond to the maximal values of the energy of the conservative 1/1 resonant system. This fact should be kept in mind when dissipative forces are introduced in the system. In this case, the variation of the energy will dictate the evolution of the resonant system: for slowly increasing energy, the system will converge to one of the L 4 and L 5 solutions, while, for slowly decreasing energy, its trajectory will be a spiral unwinding from L 4 (or L 5 ) solution. In the case of a dissipative evolution, the system starting nearly the L 4 equilibrium point will cross the domains of tadpole and horseshoe orbits, approaching the close encounters region, where it probably will be disrupted.",
"pages": [
4,
5
]
},
{
"title": "2.2 Amplitude maps",
"content": "In this section, we present dynamical maps of the 1/1 MMR on the ( σ, ∆/pi1 ) representative plane. For this task, we construct grids of initial conditions varying both σ and ∆/pi1 between zero and 360 · . Each point in the grid was then numerically integrated over 5000 years (roughly orbital 450 000 periods) using a Bulirsch-Stoer based N-body code. We calculated the amplitudes of oscillation of each angular variable. Initial conditions with zero amplitude in σ correspond to σ -family periodic orbits of the co-orbital system, while solutions with zero amplitude in ∆/pi1 correspond to periodic orbits of the ∆/pi1 -family (see Michtchenko et al. 2008ab). Stationary solutions of the averaged problem are the intersections of both families. Fig. 2 shows results obtained for a system composed of a Sun-like star and two hot-super-Earths with masses of 5 m ⊕ and semi-major axes of 0.04 AU. The top and bottom frames show the oscillation amplitude of σ and ∆/pi1 , respectively. Left and right frames were constructed with eccentricities e i = 0 . 1 and e i = 0 . 0001, respectively. The domains in light-blue color correspond to orbits with small amplitudes ( < 5 · ), thus indicating the location of the families of periodic orbits. The σ -families defined by nearly zero amplitudes of oscillation of the resonant angle σ are located on the top frames. The ∆/pi1 -families defined by nearly zero amplitudes of oscillation of the secular angle ∆/pi1 can be observed on the bottom panels. For example, at the initial condition with e i = 0 . 1 located at ( σ, ∆/pi1 )=(120 · ,40 · ), σ oscillates with amplitude 100 · , while ∆/pi1 oscillates with very small amplitude around 40 · . Darker regions correspond to increasing amplitudes and denote initial conditions with quasi-periodic motion. The unstable orbits (collisions) parameterized by amplitudes equal to 180 . 1 · , are shown by red color. We observe the domains around the Lagrangian equilateral solutions ( L 4 /L 5 ) at ( σ, ∆/pi1 ) = ( ± 60 · , ± 60 · ), the anti-Lagrangian solutions ( AL 4 /AL 5 ) at ( σ, ∆/pi1 ) /similarequal ( ± 60 · , ∓ 120 · ), and the QS solution ( σ, ∆/pi1 ) = (0 , 180 · ). For low eccentricities ( e i = 0 . 1, left frames), we observe the asymmetric solutions connected with the σ -family of periodic orbits (vertical cyan strips) and their intersection with the horizontal ∆/pi1 -family (sinusoidal strip at bottom frame). The unstable orbits are very thin regions (in red) that separate the two regimes of motion (symmetric from asymmetric). For initially almost circularized orbits ( e i = 0 . 0001, right frames), the QS region disappears (vertical red strips at σ = 0), meanwhile the low amplitude oscillations for the σ -family in top frame remains almost unaltered as vertical strips. In the bottom right frame, the ∆/pi1 -family shrinks into a concentrated region around the exact location of periodic families. Outside these small domains, ∆/pi1 oscillates with high-amplitude ( > 160 · ). The results of the maps show that, during the process of tidal eccentricity damping, we expect an increase in the oscillation amplitude of the angles around the equilibrium points of the 1/1 MMR. This feature will be confirmed through analysis of the numerical simulations of the exact equations of motion. Using the semi-analytical Hamiltonian model we constructed the families of periodic orbits following Giuppone et al. (2010). Fig. 3 shows the dependence of the angular coordinates of the equilibrium points on the eccentricities from quasicircular orbits to e i = 0 . 9. This figure will serve us as a guide in the choice of the initial conditions for the numerical simulations of the exact equations of motion (see next section). The results shown in this section are qualitatively the same for hot-Saturn planets.",
"pages": [
6,
7
]
},
{
"title": "3 The equations of motion of the dissipative problem",
"content": "We consider a system composed by a central star and two close-in co-orbital and rotating planets. Due to the short astrocentric distances, we suppose that both planets are deformed by the tides raised by the central star, which is also assumed distorted by the tides raised by the planets. The orbital planes of motion are supposed to be coincident with the reference plane (i.e, zero inclinations). In addition, we assume that the rotation axes are normal to the orbital planes (i.e, zero obliquities). For close-in planets, it is also convenient to consider the forces arising form general relativity, in addition to the mutual gravitational interaction and tidal forces. Remember, we call m 0 , m 1 and m 2 the masses of the star and planets, respectively; whereas the radii are denoted by R 0 , R 1 and R 2 . In a reference system centered in the star, where the positions and velocities of the planets are r i and v i , with i = 1 , 2, the equations of motion are given by On one hand, g i are the general relativity contributions acting on the planets, which for i = 1 , 2 are given by where v i = ˙ r i and c is the speed of light (see Beutler 2005). On the other hand, f i are the tidal forces raised by the star acting on the masses m 1 and m 2 due to their deformations, respectively. We use the expression for tidal forces given by Mignard (1979): where Ω i is the rotation angular velocity of the i -planet, for i = 1 , 2. It is worth noting that Mignard's force is given by a closed formula and, therefore, is valid for any value of eccentricity 1 . k 2 i is the second order Love number and ∆t i is the time lag, which can be interpreted as a delay in the deformation of the tidally affected body due to its internal viscosity. The total tidal force on the star is f 0 = f 01 + f 02 , where f 0 i are the individual tidal forces raised by each planet and for i = 1 , 2 are given by where the subscript '0' stand for star quantities. Note that are the forces -f 0 i which act on the planets and must be considered in the equations of motion of the bodies. /negationslash The fact that ∆t = 0 introduces energy dissipation in the system, resulting in orbital and rotational evolution due to tidal torques. The tidal model here adopted is a classical linear approach (Darwin, 1880), since it is implicitly assumed that the resulting dissipation is proportional to the tidal frequencies. This tidal model is frequently referred to as a constant time-lag model and, despite that recent works have shown that it could not be appropriate for the study of terrestrial planets, is expected to yield to the approximately correct results. For a review of other tidal models, the reader is referred to Efroimsky & Williams (2009) and Ferraz-Mello (2013).",
"pages": [
8,
9
]
},
{
"title": "4 Numerical simulations",
"content": "In this section we show the result of the numerical integration of equations (1)(2) for some particular systems. As shown in Giuppone et al. (2010), large-scale planetary migration should favor the formation of co-orbital configurations more likely for nearly equal mass of the planets (see Sec. 1.2). Hence, we restrict our investigation to the case m 2 /m 1 = 1. We start with the case of two super-Earth planets with m 1 = m 2 = 5 m ⊕ orbiting a Sun-like star with m 0 = m /circledot and R 0 = R /circledot . The radii are such that the mean densities ρ satisfy ρ 1 = ρ 2 = ρ ⊕ , so that R 1 = R 2 = 5 1 / 3 R ⊕ . For sake of completeness we also explore the case of individual masses of m Sat , where m Sat is the mass of Saturn, and the radii are computed as in the super-Earth case. We are going to investigate the final outcome of a system originally evolving in a co-orbital configuration when the tidal effect acts to change the orbital elements. Thus, we choose the initial values of the angles ( σ, ∆/pi1 ) near to the equilibrium points. The initial semi-major axes and eccentricities are a 1 = a 2 = 0 . 04 AU (orbital periods of /similarequal 2 . 9 d) and e 1 = e 2 = 0 . 1. The initial values of rotation periods are 16 . 7 h for both planets, however, they are not important because the rotation rapidly encounters its stationary value which only depends on the eccentricity within the adopted tidal model (Hut, 1981). We set 19.4 d as the initial value of the stars' rotation period, noting that the stationary value in this case is reached in a much larger timescale than for the planets. In addition, we need to know the value of the moment of inertia around the rotation axis, which is given by ξmR 2 , where ξ is the structure constant. For super-Earths, we adopt Earth values for ξ , namely, ξ = 0 . 33; whereas for hot-Saturn and a Sun-like star we set ξ = 0 . 21 and ξ = 0 . 07, respectively 2 . The linear model enable us to relate the time delay ∆t with the quality factor Q through 1 /Q = n∆t , where n is the mean orbital motion (see Correia et al. 2012). However, the quality factor is poorly constrained even for Solar System bodies, although some previous works have brought valuable information. For instance, for the solid Earth we have Q = 370 (Ray et al. 1996), and values of Q/k 2 = 0 . 9 × 10 5 , Q > 1 . 8 × 10 4 and Q/k 2 = 4 . 5 × 10 4 were estimated for Jupiter, Saturn and Neptune, respectively (Lainey et al. 2009; Meyer & Wisdom 2007; Zhang & Hamilton 2008). More recent investigations (Ferraz-Mello 2013) suggest that the hot super-Earth CoRoT-7 b has Q = 49, whereas for a hot Jupiter of 2-3 m Jup in a 5-d orbit, Q ∼ 4 . 2 × 10 5 . In this work, we adopt Q = 20, k 2 = 0 . 3 for hot super-Earths and Q = 1 × 10 4 , k 2 = 0 . 34 for hot-Saturn planets, whereas for Sun-like stars we adopt Q = 1 × 10 6 , k 2 = 0 . 34. The corresponding values of k 2 ∆t , where ∆t is obtained through 1 /Q = n∆t , are ( k 2 ∆t ) super -Earth = 600 sec, ( k 2 ∆t ) Saturn = 1 . 37 sec and ( k 2 ∆t ) Sun = 0 . 0137 sec. We chose initial values of the angles within 4 degrees around the exact equilibrium solutions (see Fig. 3 for the locations of the exact solutions and Table 1 for initial values chosen). The symmetric and asymmetric periodic solutions for mass ratio close to unity are such that e 1 = e 2 (and also a 1 = a 2 , see Hadjidemetriou et al. 2009; Giuppone et al. 2010). Hence, we start our simulations with equal eccentricities, since we are assuming that the system is evolving under the 1/1 MMR. However, we note that the global results would not be affected by either the initial values of eccentricities or amplitudes of the angles, as long the corresponding configuration is nearby to the exact 1/1 MMR. For our stability criterion, we use the critical distance given by d = κ ( R 1 + R 2 ). Hence, we assume that orbital instability would occur whenever the instantaneous mutual distance is equal or smaller than d , that is, for | r 2 -r 1 | ≤ d . We note that κ = 1 implies in the physical collision between the planets 3 . In this work, we use κ = 1 and κ = 2, however, we anticipate that the final results are not sensitive (in the sense of collision timescale) to the choice of one of the above values of κ .",
"pages": [
9,
10,
11
]
},
{
"title": "4.1.1 Hot super-Earths",
"content": "We start with the case in which the individual planets are represented by a 5 m ⊕ super-Earth. Figs. 4 shows the time variation of the angles ( σ, ∆/pi1 ) around the symmetric and asymmetric equilibrium points for κ = 1, whereas the evolution of the eccentricities is shown in Fig. 5. A collision between the planets is the final outcome of all simulations . For AL 4 /AL 5 configurations, the angle ∆/pi1 oscillates with small amplitudes up to around 35 Myr. However, the amplitudes start to increase and, close to 45 Myr, a new regime of oscillation appears, in which the angles seem to avoid the regions centered near 240 · and 120 · . Note that these positions are the corresponding equilibrium points of ∆/pi1 for AL 4 /AL 5 solutions in the circular case. Moreover, between 60- 65 Myr, a circulation of ∆/pi1 appears and, finally, an oscillation around 180 · with increasing amplitude occurs before the collision between the planets. The evolutions of ∆/pi1 along L 4 /L 5 during the first 20 Myr follow close to the stationary values (60 · /300 · ). After that, ∆/pi1 deviates and a time interval of oscillation and circulation occurs, with the angle taking many values in the whole interval between 0 · - 360 · but privileging the oscillation around 0 · . Finally, ∆/pi1 oscillates around 180 · before collision. Regarding to the resonant angle σ , we note in Fig. 4 a libration with increasing amplitude around the equilibrium points. When the regime of oscillation/circulation of the angle ∆/pi1 appears, σ librates around 180 · with very high amplitude until the end of the simulation (for all asymmetric configurations). We will return to this discussion later in this section. The solution close to QS is the first to become unstable, around 6 Myr of evolution. In this case, the amplitudes of the angles are close to 4 · and e 1 /similarequal e 2 = 0 . 038 before destabilization. Several runs for the QS case were carried out with different values of initial amplitudes (even zero) and the collision was the final thus, it is not restricted to impact between the planets. However, close encounters with κ between 1 and 2, may result in tidal disruption of one or both planets rather than ejection into hyperbolic orbit [deg] σ outcome in all cases in very similar timescales ( < 10 Myr). Moreover, we tested a run without tides and the system becomes stable at least up to 60 Myr (with constant amplitudes of the angles). The evolution of eccentricities follows with e 1 /similarequal e 2 (and thus superimposed in Fig. 5), according to the stationary solutions for equal mass planets. The final result of the tidal evolution is an almost doubly orbital circularization except in the QS case, in which the oscillation amplitude of eccentricities are larger than for other cases. If we suppose that the time variation of the eccentricities follows an exponential law (see right panel in Fig. 6), we can estimate the timescale for tidal damping as τ e = ˙ e/e . Using a classical averaged expression for ˙ e obtained from linear tidal models (see Eq. (3.5) in Rodr'ıguez & Ferraz-Mello 2010) we obtain τ e /similarequal 6 Myr. This value gives an estimation of the e-folding of the eccentricity damping. Therefore, the timescales for destabilization shown in Fig. 4 correspond roughly to 10 τ e . ECCENTRICITY that, in view of the agreement, the contribution of stellar tides (tides on the star) can be safely neglected 5 . around σ = 180 · (blue points) and encompasses both equilateral Lagrangian points (see also Fig. 4, right panel). In analogy with the restricted problem, these types of motion correspond to tadpole and horseshoe co-orbital configurations. In Fig. 7, we clearly see that the tadpole and horseshoe oscillation regimes are disconnected by the separatrix of the 1/1 resonant motion. We should note that the condition σ ∼ 0 is necessary for the collision to occur. However, we do not observe in Fig. 7 a separatrix disconnecting the regimes of libration and circulation of σ , and thus the angle seems never takes the zero value. To overcome this situation, we performed a new short numerical simulation starting very close to the configuration in which the collision occurs. We will illustrate these results in the next section in the application for hot-Saturn planets, but the same can be applied for super-Earths. It is interesting to compare the numerical results with those predicted by the semi-analytical model (Sec. 2). Indeed, the curve which separates the domains of tadpole and horseshoe motions (black curve on top panel of Fig. 1) agrees with the separatrix appearing in Fig. 7. The oscillation amplitude of n 1 /n 2 in both cases is approximately 0.12, while σ takes values as small as 25 · (for L 4 ), until the horseshoe domain arises. The numerical solution for the QS configuration indicates that n 1 /n 2 /similarequal 1 . 02 and σ /similarequal 5 · before destabilization. We thus note an excellent agreement of the above result through an inspection of Fig. 1 (bottom panel), where the point ( σ, n 1 /n 2 ) = (5 · , 1 . 02) belongs to the separatrix of the QS co-orbital configuration (note that Fig. 1 for QS was constructed taking e 1 = e 2 = 0 . 04 for better comparison with the numerical results, in which the eccentricities are close to 0.038 just before collision).",
"pages": [
11,
12,
13,
14
]
},
{
"title": "4.1.2 Hot Saturn",
"content": "The numerical exploration of the system with two Saturn planets indicates that, as in the previous system, collision between the planets results in all simulations for κ = 1 and κ = 2. Moreover, all initial co-orbital configurations destabilize in timescales between 55-80 Myr, except the one around QS in which the collision occurs near 430 Kyr. In analogy to Fig. 7, Fig. 8 displays the results of the numerical simulations for κ = 1, showing the libration of σ around the equilibrium Lagrangian points (tadpole and horseshoe orbits). In this case, the motion in the horseshoe domain is quite unstable due to the strong mutual interaction between the planets, and the system remains in that libration regime only 166 yr before collision (in the case of the two super-Earth system, the horseshoe regime lasted around 8 Myr). In addition to these types of libration, we see the domain in which the angle σ circulates (also in blue points). In the circulation regime, in which the orbits are no longer locked in the 1/1 MMR, σ takes the zero value several times, favoring the orbital conditions for the collision between the planets.",
"pages": [
15
]
},
{
"title": "5 Conclusions and discussion",
"content": "We investigated the motion of a two-planet system which evolves under combined effects of mutual and tidal interactions in the vicinity of a 1/1 mean motion resonance (i.e, co-orbital configuration). We considered systems of equal mass planets ( m 2 /m 1 = 1) corresponding to super-Earth (5 m ⊕ ) and Saturn planets ( /similarequal 95 m ⊕ ). Numerical simulations of the exact equations of motion indicate that the colli- between the planets is the final outcome of the tidal evolution for all initial co-orbital configurations tested in this work. We started our simulations considering initial conditions near the five stationary configurations of the co-orbital motion (namely, L 4 /L 5 , AL 4 /AL 5 and QS ). As tides continually damp the eccentricities, the initially small oscillation amplitudes of the angles ( σ, ∆/pi1 ) around the equilibrium points increases and, ultimately, the instability occurs. We identified several libration regimes of the resonant angle σ , including the tadpole and horseshoe co-orbital configurations, in which the motion occurs around σ = 60 · / 300 · and σ = 180 · , respectively. In addition, the motion around the antiLagrangian equilibrium points ( AL 4 /AL 5 ), located close to ( σ, ∆/pi1 ) = (60 · / 300 · , 240 · / 120 · ) for m 2 /m 1 = 1 and small eccentricities, remains stable over a timescale about 1.5 larger than for the Lagrangian points ( L 4 /L 5 ) for both types of systems. Moreover, the stability of the quasi-satellite ( QS ) co-orbital configuration ( σ = 0 · , ∆/pi1 = 180 · ) is restricted to short timescales ( < 10 Myr and < 1 Myr, for super-Earth and Saturn individual masses, respectively). The interpretation of the tidal evolution of the 1/1 resonant planet pair and its ultimate disruption is in the following. We know that the L 4 and L 5 stationary solutions correspond to global maxima of the conservative Hamiltonian of the 1/1 MMR, at least for small and moderate eccentricities. This is a consequence of the fact that a mean-motion resonance, acting as a protection mechanism, implies the maximal possible mutual distance between two planets at conjunction, where their closest encounters occur (Michtchenko et al. 2008b). When the tidal interactions are introduced in the system, the energy of the system is dissipated through tidal heating of the planets. In this way, starting near one of the L 4 and L 5 points, the system evolving under dissipation will suffer the increase the oscillation amplitudes of the angles around the exact positions of the equilibrium points. It will cross the domains of the tadpole and horseshoe orbits, as described in Sec. 2. Thus, the system will ultimately tend toward a collisional route. Therefore, the results presented in this work suggest that the tidal evolution of close-in planetary systems in the vicinity of the 1/1 MMR is globally unstable, constraining the possible detection of hot exo-Trojans (particularly among KOI's candidates). However, we have to stress that the origin of such close-in systems is unknown. Indeed, as recently shown in Giuppone et al. (2012), the formation of co-orbital configurations at distances of 1 AU is possible, however, the question on how did they reach close-in configurations is still under discussion. Also, it should be kept in mind that, for planets more distant from the central star, we expect a timescale for tidal evolution much longer than the age of the systems, and thus the possibility of radial velocity detections cannot be ruled out. As a final considerations, some limitations of our model should be highlighted. First, the applicability for rocky planets of the adopted linear tidal model has been recently questioned in some works (see Efroimsky & Williams 2009; FerrazMello 2013 and references therein). However, we speculate that the adoption of a different tidal model should change the timescales of the tidal evolution but the global result would not be affected. Second, we neglected the polar and permanent equatorial deformations of the bodies (i.e, J 2 , C 22 and higher orders terms). Recently, Rodr'ıguez et al. (2012) have shown that the contribution of C 22 can lead to temporary captures in spinorbit resonances of rocky planets (see also Correia & Rodr'ıguez (2013) and Cal- ri & Rodr'ıguez (2013) for a calculation of J 2 and C 22 for a specific resonant rotation trapping). Moreover, the orbital decay and the eccentricity damping are larger whenever the planet rotation is trapped in a resonant motion. Hence, the tidal model and the consideration of the equatorial permanent deformation should be taken into account in further investigations. Finally, following the suggestion of Dr. M. Efroimsky (private communication), we performed an additional simulation including an indirect tidal term which is described in the following. The tidal effect on the star due to one of the planets creates a (dissipative) torque which affects the orbital and rotational evolution of the tidal raising planet. The other planet will also 'feel' this tidal effect because the star deformation crates a potential in an arbitrary point in the space. The effect of this force may be important for planets trapped in mean-motion resonances. In the co-orbital case, we have a 'frozen' configuration in which the relative positions of the bodies are located in the vertices of an equilateral triangle of variable size. We only consider the static tidal component (i.e., the instantaneous response, independent on ∆t 0 , see Ferraz- Mello et al. 2008), because it is orders of magnitude larger than its dissipative counterpart. The result of the numerical simulation for the two Saturn planets system have shown that the inclusion of these terms only slightly modifies the timescales of the planet evolution, delaying the instability in approximately 10% for a L 4 /L 5 starting configuration. Acknowledgements We acknowledge to M. Efroimsky for comments and suggestions. A.R and T.A.M acknowledge the support of this project by FAPESP (2009/16900-5) and CNPq (Brazil). C.A.G acknowledges the support by the Argentinian Research Council, CONICET. This work has made use of the computing facilities of the Laboratory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchase was made possible by the Brazilian agency FAPESP (grant 2009/54006-4) and the INCT-A. Some of the computations were performed on the Blafis cluster at the Aveiro University. We also acknowledge the two anonymous reviewers for their valuable comments and suggestions.",
"pages": [
15,
16,
17
]
}
]
2013CoPhC.184.1333G
https://arxiv.org/pdf/1208.6014.pdf
<document>
<section_header_level_1><location><page_1><loc_19><loc_79><loc_81><loc_84></location>MESAFace, a graphical interface to analyze the MESA output</section_header_level_1>
<section_header_level_1><location><page_1><loc_33><loc_75><loc_67><loc_77></location>M. Giannotti ∗ , M. Wise, A. Mohammed</section_header_level_1>
<text><location><page_1><loc_25><loc_73><loc_74><loc_74></location>Barry University, 11300 NE 2nd Ave., Miami Shores, FL 33161, US</text>
<section_header_level_1><location><page_1><loc_18><loc_65><loc_27><loc_66></location>Abstract</section_header_level_1>
<text><location><page_1><loc_18><loc_57><loc_82><loc_63></location>MESA (Modules for Experiments in Stellar Astrophysics) has become very popular among astrophysicists as a powerful and reliable code to simulate stellar evolution. Analyzing the output data thoroughly may, however, present some challenges and be rather time-consuming.</text>
<text><location><page_1><loc_18><loc_53><loc_82><loc_56></location>Here we describe MESAFace, a graphical and dynamical interface which provides an intuitive, efficient and quick way to analyze the MESA output.</text>
<text><location><page_1><loc_18><loc_50><loc_66><loc_52></location>Keywords: Stellar evolution; MESA; Mathematica; GUI;</text>
<section_header_level_1><location><page_1><loc_18><loc_44><loc_82><loc_47></location>PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY</section_header_level_1>
<text><location><page_1><loc_18><loc_41><loc_81><loc_44></location>Manuscript Title: MESAFace, a graphical interface to analyze the MESA output Authors: M. Giannotti, M. Wise, A. Mohammed</text>
<text><location><page_1><loc_18><loc_39><loc_30><loc_40></location>Program Title:</text>
<text><location><page_1><loc_30><loc_39><loc_39><loc_40></location>MESAFace</text>
<text><location><page_1><loc_18><loc_37><loc_33><loc_38></location>Journal Reference:</text>
<text><location><page_1><loc_18><loc_36><loc_34><loc_37></location>Catalogue identifier:</text>
<text><location><page_1><loc_18><loc_34><loc_34><loc_35></location>Licensing provisions:</text>
<text><location><page_1><loc_35><loc_34><loc_39><loc_35></location>none</text>
<text><location><page_1><loc_18><loc_32><loc_47><loc_33></location>Programming language: Mathematica</text>
<text><location><page_1><loc_18><loc_30><loc_26><loc_32></location>Computer:</text>
<text><location><page_1><loc_27><loc_30><loc_65><loc_32></location>Any computer capable of running Mathematica.</text>
<text><location><page_1><loc_18><loc_27><loc_38><loc_30></location>Operating system: Windows XP, Windows 7.</text>
<text><location><page_1><loc_33><loc_29><loc_82><loc_30></location>Any capable of running Mathematica. Tested on Linux, Mac,</text>
<text><location><page_1><loc_18><loc_22><loc_52><loc_26></location>RAM: Recommended 2 Gigabytes or more. Keywords: Classification:</text>
<text><location><page_1><loc_27><loc_24><loc_62><loc_25></location>Stellar evolution, MESA, Mathematica, GUI.</text>
<text><location><page_1><loc_30><loc_22><loc_63><loc_23></location>1.7 Stars and Stellar Systems, 14 Graphics</text>
<section_header_level_1><location><page_2><loc_18><loc_82><loc_33><loc_84></location>Nature of problem:</section_header_level_1>
<text><location><page_2><loc_18><loc_77><loc_82><loc_82></location>Find a way to quickly and thoroughly analyze the output of a MESA run, including all the profiles, and have an efficient method to produce graphical representations of the data.</text>
<section_header_level_1><location><page_2><loc_18><loc_75><loc_31><loc_77></location>Solution method:</section_header_level_1>
<text><location><page_2><loc_18><loc_62><loc_82><loc_75></location>We created two scripts (to be run consecutively). The first one downloads all the data from a MESA run and organizes the profiles in order of age. All the files are saved as tables or arrays of tables which can then be accessed very quickly by Mathematica. The second script uses the Manipulate function to create a graphical interface which allows the user to choose what to plot from a set of menus and buttons. The information shown is updated in real time. The user can access very quickly all the data from the run under examination and visualize it with plots and tables.</text>
<section_header_level_1><location><page_2><loc_18><loc_60><loc_32><loc_61></location>Unusual features:</section_header_level_1>
<text><location><page_2><loc_18><loc_53><loc_82><loc_60></location>Moving the slides in certain regions may cause an error message. This happens when Mathematica is asked to read nonexistent data. The error message, however, disappears when the slides are moved back. This issue does not preclude the good functioning of the interface.</text>
<section_header_level_1><location><page_2><loc_18><loc_52><loc_35><loc_53></location>Additional comments:</section_header_level_1>
<text><location><page_2><loc_18><loc_46><loc_82><loc_51></location>The program uses the dynamical capabilities of Mathematica. When the program is opened, Mathematica prompts the user to 'Enable Dynamics'. It is necessary to accept before proceeding.</text>
<section_header_level_1><location><page_2><loc_18><loc_45><loc_29><loc_46></location>Running time:</section_header_level_1>
<text><location><page_2><loc_18><loc_33><loc_82><loc_44></location>Depends on the size of the data downloaded, on where the data are stored (harddrive or web), and on the speed of the computer or network connection. In general, downloading the data may take from a minute to several minutes. Loading directly from the web is slower. For example, downloading a 200MB data folder (a total of 102 files) with a dual-core Intel laptop, P8700, 2 GB of RAM, at 2.53 GHz took about a minute from the hard-drive and about 23 minutes from the web (with a basic home wireless connection).</text>
<section_header_level_1><location><page_2><loc_18><loc_27><loc_33><loc_28></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_18><loc_20><loc_82><loc_25></location>The introduction of the new code for stellar evolution, MESA [1] (Modules for Experiments in Stellar Astrophysics) represented somewhat of a revolution in the field of computational stellar evolution.</text>
<text><location><page_2><loc_18><loc_15><loc_82><loc_20></location>MESA is a one-dimensional stellar evolution code, organized in independent modules and continually updated. The code can evolve stars with a very wide range of initial masses and metallicities; it allows the specification</text>
<text><location><page_3><loc_18><loc_79><loc_82><loc_84></location>of many parameters, for example, convection mechanism, mass loss, etc., and it can overcome difficulties typical of the previous stellar evolutionary codes. Notably, it can run a low mass star through the He-flash region.</text>
<text><location><page_3><loc_18><loc_71><loc_82><loc_78></location>All these aspects and the fact that the code is publicly available [2] has made MESA extremely popular among astrophysicists and other physicists interested in stellar evolution and in how stars respond to new models of particle interactions.</text>
<text><location><page_3><loc_18><loc_64><loc_82><loc_71></location>Being publicly available, MESA is widely accessible and is continually tested by hundreds of scientists and students all over the world. In addition, several researchers have created new codes and routines to improve it and to simplify its use [3].</text>
<text><location><page_3><loc_18><loc_53><loc_82><loc_64></location>A MESA run produces a very large amount of data on the life and structure of the star. This information is saved in several files: the file 'star.log' includes global information about the star at different times whereas a large number of profiles (named 'log n .data', where n is an integer), contain detailed information about the profile of the star, each file referring to a specific age.</text>
<text><location><page_3><loc_18><loc_37><loc_82><loc_53></location>Though it is relatively easy, and for some purposes sufficient, to analyze the file 'star.log', a complete analysis of the MESA output, which includes in some cases hundreds of profiles, is much more challenging. In particular, to analyze the detailed structure of a star at a certain age, one should select the corresponding profile by reading the age of each of them (the information is provided in the 3 nd row of 'log n .data'). Alternatively, one should read the model number corresponding to a certain age in the first column of the file 'star.log', and then look for the log number associated with that age in the file 'profile.index'. Both ways are long and, in general, not very practical.</text>
<text><location><page_3><loc_18><loc_28><loc_82><loc_36></location>Another common problem is reading the age of a star from its position in the HR diagram (or any other plot which does not show the age explicitly, e.g., the central temperature-central density diagram). It would be useful to have a quick access to that information and, at the same time, to know which of the many profiles describes the structure of the star at that age.</text>
<text><location><page_3><loc_18><loc_20><loc_82><loc_27></location>Here we describe a graphical interface, MESAFace, which we created to address problems like the ones described above and, in general, to provide fast and simple access to all the information contained in the output of a MESA run, including all the profiles.</text>
<text><location><page_3><loc_18><loc_15><loc_82><loc_20></location>MESAFace is a very intuitive, efficient, and easy-to-use dynamical interface, which allows the selection of the information to be shown through buttons and menus and the age of the star to be selected through a slidebar.</text>
<text><location><page_4><loc_18><loc_79><loc_82><loc_84></location>The code is written in Mathematica and has been tested with Mathematica 7 and 8. However, a knowledge of Mathematica is not strictly necessary to use the interface.</text>
<text><location><page_4><loc_18><loc_71><loc_82><loc_78></location>MESAFace needs to access the MESA output data which, in a standard MESA run, are saved in the folder /work/LOGS. The data can also be downloaded directly from the web. Possibly soon, the results from some standard runs will be available from a dedicated web space.</text>
<text><location><page_4><loc_18><loc_68><loc_82><loc_71></location>After the data has been loaded, MESAFace presents a graphical interface structured in three vertically organized panels:</text>
<unordered_list>
<list_item><location><page_4><loc_21><loc_65><loc_40><loc_66></location>· A history plot panel;</list_item>
<list_item><location><page_4><loc_21><loc_62><loc_40><loc_63></location>· A profile plot panel;</list_item>
<list_item><location><page_4><loc_21><loc_58><loc_41><loc_60></location>· An information panel.</list_item>
</unordered_list>
<text><location><page_4><loc_18><loc_46><loc_82><loc_57></location>All the controls (buttons, menus, check-boxes, etc.) are contained on the left side of the interface. The user can decide what to plot and can change the age of the star by using these controls while the information shown on the right side is dynamically updated. Several aspects of the panels and graphs, for example, the color and style of the lines, can be customized by the user. A separate user manual [4] explains in detail how to do that.</text>
<text><location><page_4><loc_18><loc_43><loc_82><loc_46></location>In this paper, we describe in detail the structure of the MESAFace code and how it accesses, manipulates, and presents the data.</text>
<text><location><page_4><loc_18><loc_32><loc_82><loc_42></location>The structure of the paper is the following: We begin with a brief summary of the MESA output, in section 2. Later, in section 3, we give a basic description of the general structure of the MESAFace code. In section 4, we describe with more details how MESAFace loads and organizes the files. In section 5, we discuss how it creates the dynamical interface and visualizes the data. Finally, in section 6 we conclude with some overall comments.</text>
<text><location><page_4><loc_18><loc_26><loc_82><loc_31></location>A note on the style. We will use the Typewriter style for the content of the Mathematica notebook, 'quotes' for the name of the MESA files, and bold face for the names of the controls in the interface.</text>
<section_header_level_1><location><page_4><loc_18><loc_22><loc_58><loc_23></location>2. Brief description of the MESA output</section_header_level_1>
<text><location><page_4><loc_18><loc_16><loc_82><loc_21></location>MESA is a one-dimensional stellar evolution code. The output data is parameterized by the value of the star age and the radial distance from the star center. Most of the output data is organized in a large number of profiles</text>
<text><location><page_5><loc_18><loc_77><loc_82><loc_84></location>('log n .data' files, with n an integer). The main section of each profile is a table with data referring to a particular age and each row referring to a particular zone in the star. Different zones correspond to different distances from the center.</text>
<text><location><page_5><loc_18><loc_66><loc_82><loc_76></location>Besides the profiles, MESA also produces a 'star.log' file which includes global information about the star at different times, and a 'profiles.index' which provides a useful connection between the profiles and 'star.log'. All these files are created during a MESA run and, by default, are contained in the LOG folder of the star/work directory. Below is a brief description of the MESA output files. For more information, cfr. the MESA literature[1, 2].</text>
<text><location><page_5><loc_18><loc_62><loc_82><loc_65></location>The MESA output folder contains 1 file 'star.log', a large number of profiles , 1 file, 'profiles.index'.</text>
<unordered_list>
<list_item><location><page_5><loc_21><loc_50><loc_82><loc_61></location>· The file 'star.log' includes global information about the star at different times. The second and third row of this file include the names (row 2) and values (row 3) of some physical quantities characterizing the star, for example its initial mass and metallicity. This information is accessible from the interface by selecting the GENERAL radio button in the GENERAL INFO panel of the interface (see section 5.2).</list_item>
<list_item><location><page_5><loc_23><loc_40><loc_82><loc_49></location>The sixth row contains the names of some age-dependent physical quantities, such as effective temperature or central density, and all the rows below that show the values of these quantities at different ages. Each row is indexed by a different model number and represents a different age of the star.</list_item>
<list_item><location><page_5><loc_21><loc_34><loc_82><loc_39></location>· The profiles ('log n .data' files) contain detailed information about the structure of the star, each file referring to a specific age. Here, n is the log file number , also referred to as the log or profile number .</list_item>
<list_item><location><page_5><loc_23><loc_24><loc_82><loc_32></location>The second and third row of these files include the names (row 2) and values (row 3) of some physical quantities characterizing the particular profile, for example the age of the star. This information is accessible from the interface by selecting the DETAILED radio button in the GENERAL INFO panel of the interface (see section 5.2).</list_item>
</unordered_list>
<text><location><page_5><loc_23><loc_18><loc_82><loc_23></location>The sixth row contains the names of some position-dependent physical quantities, such as temperature or density, and all the rows below that show the values of these quantities in the different zones.</text>
<unordered_list>
<list_item><location><page_6><loc_21><loc_73><loc_82><loc_84></location>· The file 'profiles.index' is an index of the 'log n .data' files. It has three columns. The most interesting (for our purpose) are the first and the third. The first contains the model number , which indicates a specific row of the file 'star.log' (each row has a different model number ). The third column has the log file number . In this way 'profiles.index' provides a link among the various 'log n .data' and 'star.log'.</list_item>
</unordered_list>
<text><location><page_6><loc_18><loc_68><loc_82><loc_71></location>The number of different profiles and the amount and kind of information in each file depend on the particular run.</text>
<section_header_level_1><location><page_6><loc_18><loc_64><loc_61><loc_66></location>3. General structure of the MESAFace code</section_header_level_1>
<text><location><page_6><loc_18><loc_59><loc_82><loc_63></location>The MESAFace code contains two scripts, labeled as (* SCRIPT 1 *) and (* SCRIPT 2 *) , which should be run consecutively.</text>
<text><location><page_6><loc_18><loc_50><loc_82><loc_59></location>The first script extracts all the data from a MESA run and organizes them into tables easily accessible by the interface (see section 4). Although extracting all the MESA output at once may be slow, after that, the interface can access all the data very quickly, allowing a dynamical manipulation and visualization of the information.</text>
<text><location><page_6><loc_18><loc_47><loc_82><loc_50></location>The interface is created by the second script, labeled (* SCRIPT 2 *) in the code and described in section 3.2 and 5.</text>
<text><location><page_6><loc_18><loc_36><loc_82><loc_46></location>The choice to have two scripts is a practical one: The second script, which normally runs in no more than a second or so, allows several customizations of the interface, for example it allows the user to set the style of the graphs. Having this script separated from the first one allows the user to change some features in the interface or in the plot style without having to download the data each time.</text>
<section_header_level_1><location><page_6><loc_18><loc_32><loc_45><loc_34></location>3.1. Structure of the first script</section_header_level_1>
<text><location><page_6><loc_21><loc_30><loc_60><loc_32></location>The first script can be divided in two sections:</text>
<unordered_list>
<list_item><location><page_6><loc_20><loc_24><loc_82><loc_29></location>1. The first section, which ends before the definition of the function age , sets the path to the folder with the MESA output data and downloads the data.</list_item>
<list_item><location><page_6><loc_20><loc_22><loc_68><loc_23></location>2. The second section organizes the data (see section 4).</list_item>
</unordered_list>
<text><location><page_6><loc_18><loc_15><loc_82><loc_20></location>The variable slash allows Mathematica to read the location of the output files with different operating systems. If the output data are on the web, the variable slash has to be set equal to "/" .</text>
<section_header_level_1><location><page_7><loc_18><loc_82><loc_47><loc_84></location>3.2. Structure of the second script</section_header_level_1>
<text><location><page_7><loc_21><loc_80><loc_70><loc_81></location>The second script can be divided into three main sections:</text>
<unordered_list>
<list_item><location><page_7><loc_20><loc_70><loc_84><loc_79></location>1. The section contained within the comment lines (* General Settings *) and (* DON'T CHANGE *) at the beginning of the second script allows the user to customize the interface. In this section, it is possible to define the style of the graphs, the labels, and the buttons to show in the interface.</list_item>
<list_item><location><page_7><loc_20><loc_64><loc_82><loc_69></location>2. The section below the comment line (* Plot and Info Section*) produces the information to be presented in the three vertical panels on the right hand side of the interface (see section 5.2).</list_item>
<list_item><location><page_7><loc_20><loc_59><loc_82><loc_64></location>3. The section after the comment line (* Image Size *) creates and manages the controls to be shown on the left hand side of the interface (see section 5.1).</list_item>
</unordered_list>
<section_header_level_1><location><page_7><loc_18><loc_55><loc_43><loc_57></location>3.3. Local and global variables</section_header_level_1>
<text><location><page_7><loc_18><loc_50><loc_82><loc_55></location>Most of the variables used in the scripts are local (they are visible only within the script). A variable is made local by including it in the curly brackets at the beginning of the Module command.</text>
<text><location><page_7><loc_18><loc_44><loc_82><loc_49></location>However, some variables defined in the first script need to be accessed by the second script and, therefore, they cannot be defined locally. These global variables are:</text>
<unordered_list>
<list_item><location><page_7><loc_21><loc_41><loc_80><loc_42></location>· path : it defines the path to the folder with the MESA output data.</list_item>
<list_item><location><page_7><loc_21><loc_38><loc_77><loc_39></location>· star : it is a table and contains the content of the file 'star.log'.</list_item>
<list_item><location><page_7><loc_21><loc_31><loc_82><loc_36></location>· log[logNumber] : it is an array of tables with the content of the profiles. logNumber is an index equal to the log file number (see section 2). The table log[n] corresponds to the file 'log n .data'.</list_item>
<list_item><location><page_7><loc_21><loc_26><loc_82><loc_30></location>· profiles : it is a table and contains the content of the file 'profiles.index'.</list_item>
<list_item><location><page_7><loc_21><loc_21><loc_82><loc_25></location>· logAge : it is a two column table with, respectively, the log file number and the star age of each profile.</list_item>
<list_item><location><page_7><loc_21><loc_17><loc_82><loc_20></location>· maxModelLength : it is an integer equal to the number of lines of the profile which has the largest number of lines.</list_item>
</unordered_list>
<text><location><page_8><loc_18><loc_79><loc_82><loc_84></location>If it is necessary to debug or to use a result outside a MESAFace session, a local variable can be made global by taking it out of the list in the curly brackets at the beginning of the Module command.</text>
<text><location><page_8><loc_18><loc_75><loc_82><loc_78></location>In general, however, it is not recommended to use nonlocal variables outside MESAFace during a session.</text>
<section_header_level_1><location><page_8><loc_18><loc_71><loc_54><loc_72></location>4. Loading and organizing of the files</section_header_level_1>
<text><location><page_8><loc_18><loc_59><loc_82><loc_70></location>The goal of MESAFace is to provide a way to access the data dynamically . In order to do that, MESAFace downloads all the data at once, either from the hard-drive or from the web, and saves it in particular variables. This may be a rather slow process, especially if the data are to be extracted directly from a web location. After the data is downloaded, however, the access to the information is very fast.</text>
<text><location><page_8><loc_18><loc_54><loc_82><loc_59></location>The progress in downloading is indicated: a numerical value represents the actual number of downloaded profiles while a progress-bar shows the relative progress.</text>
<section_header_level_1><location><page_8><loc_18><loc_50><loc_48><loc_52></location>4.1. Extraction of the MESA output</section_header_level_1>
<text><location><page_8><loc_18><loc_39><loc_82><loc_50></location>The job of the first script of MESAFace is just to download and organize the MESA output data. Mathematica looks in the folder defined by the variable path for all the files with names 'star.log', 'profiles.index' and 'log n .data' (it is essential that the output files have those names, as it is for a default run). The variable path can point to a folder on the hard drive or on the web.</text>
<text><location><page_8><loc_18><loc_32><loc_82><loc_39></location>The files 'star.log' and 'profiles.index' are saved (as tables) respectively in the variable star and profiles . Analogously, the 'log n .data' files are saved in the array of tables log[indx] , where indx is the log number n . For example, the file 'log33.data' is saved as log[33] .</text>
<section_header_level_1><location><page_8><loc_18><loc_29><loc_79><loc_30></location>4.2. Extraction of the age of the star and selection of the profile to show</section_header_level_1>
<text><location><page_8><loc_18><loc_21><loc_82><loc_28></location>One of the main features of MESAFace is to allow an easy selection of the profile corresponding to a certain age. To do that, the first script organizes the profiles in order of age in the table logAge , with the log number in column 1 and the star age in column 2.</text>
<text><location><page_8><loc_21><loc_19><loc_36><loc_21></location>The function age :</text>
<text><location><page_8><loc_21><loc_15><loc_80><loc_18></location>age[i ]:= log[i][[1 + Position[log[i], "star age"][[1, 1]], Position[log[i], "star age"][[1, 2]]]];</text>
<text><location><page_9><loc_18><loc_82><loc_59><loc_84></location>has the task of extracting the age of each profile.</text>
<text><location><page_9><loc_21><loc_80><loc_68><loc_82></location>A slide-bar (more precisely, a manipulator ) type control:</text>
<text><location><page_9><loc_21><loc_75><loc_85><loc_79></location>{{ starAge, logAge[[1, 2]], Style["Star Age", 12, { Bold, Blue } ] } , logAge[[All, 2]], ControlType -> Manipulator } ,</text>
<text><location><page_9><loc_18><loc_68><loc_82><loc_73></location>allows the user to set the value of the real variable starAge to any of the lines in the second column of the table logAge (which, as explained, contains the ages of all the profiles).</text>
<text><location><page_9><loc_18><loc_65><loc_82><loc_68></location>Each time starAge is selected, Mathematica uses the table logAge to find the corresponding log file number , and saves it in the variable model :</text>
<text><location><page_9><loc_21><loc_61><loc_86><loc_63></location>model = logAge[[Position[logAge[[All, 2]], starAge][[1, 1]], 1]];</text>
<text><location><page_9><loc_18><loc_56><loc_84><loc_60></location>In addition, if the check-box Show Age in History Plot is checked, MESAFace finds which line of the file 'star.log' corresponds to the age selected:</text>
<text><location><page_9><loc_21><loc_51><loc_82><loc_54></location>modelNumber = profiles[[Position[Drop[profiles, 1][[All, 3]], model][[1, 1]] + 1, 1]];</text>
<text><location><page_9><loc_21><loc_46><loc_78><loc_49></location>indexAge = Position[Drop[star, 6][[All, 1]], modelNumber] [[1, 1]];</text>
<text><location><page_9><loc_18><loc_39><loc_82><loc_44></location>Here, modelNumber is equal to the model number discussed above (see section 2). Its value is reported in the first column of 'profile.index' and in the first column of 'star.log'.</text>
<text><location><page_9><loc_18><loc_30><loc_83><loc_38></location>The variable indexAge , instead, is equal to the the row number of 'star.log' (after its first 6 lines are dropped) corresponding to the model number (and therefore the age of the star). This value is used to identify the position of the age of the current profile in the history plots, providing a link between the History Plots and Profile Plots panels.</text>
<section_header_level_1><location><page_9><loc_18><loc_26><loc_34><loc_27></location>5. The interface</section_header_level_1>
<text><location><page_9><loc_18><loc_19><loc_82><loc_24></location>Running the second script produces a graphical interface in which the controls, located on the right hand side, allow the user to choose what kind of information to show in the visualization panels to the right.</text>
<section_header_level_1><location><page_10><loc_18><loc_82><loc_44><loc_84></location>5.1. Description of the controls</section_header_level_1>
<text><location><page_10><loc_18><loc_75><loc_82><loc_81></location>The controls are coded in the section of the second script which follows the comment line (* Beginning of the proper dynamical part *) . They are described by variables and are normally nested in double curly brackets. For example, the line:</text>
<text><location><page_10><loc_21><loc_70><loc_87><loc_73></location>{{ variable name, Default value, "label" } , list of possible values, ControlType -> PopupMenu } ,</text>
<text><location><page_10><loc_18><loc_66><loc_73><loc_68></location>describes a variable whose value can be set by a popup-menu [5].</text>
<text><location><page_10><loc_18><loc_55><loc_82><loc_66></location>The first controls in each of the three sections of the interface are radio button -type controls which allow the user to choose the kind of plot (top two panels) or information (bottom panel) to show. They are coded in the section of the script below the comment line (* Beginning of the proper dynamical part *) , at the beginning of the (* History Plot Section *) , the (* Profile Plot Section *) , and the (* Info Section *) :</text>
<text><location><page_10><loc_21><loc_48><loc_80><loc_54></location>{{ plotType, "Customized", Style["Type of Plot", 12, Bold] } , { "Customized", "HR", "CentralAbundances", "Burnings" } , ControlType -> RadioButtonBar } ,</text>
<text><location><page_10><loc_21><loc_41><loc_78><loc_46></location>{{ profileType, "Custom", Style["Type of Plot", 12, Bold] } { "Custom", "Abundances", "Reactions" } , ControlType -> RadioButtonBar } ,</text>
<text><location><page_10><loc_21><loc_36><loc_72><loc_39></location>{{ Info, "PATH", "" } , { "PATH", "GENERAL", "DETAILED" } ControlType -> RadioButtonBar } ,</text>
<code><location><page_10><loc_72><loc_38><loc_79><loc_46></location>, ,</code>
<text><location><page_10><loc_18><loc_31><loc_83><loc_34></location>These controls allow the user to set the string value of plotType , profileType , and Info .</text>
<text><location><page_10><loc_18><loc_23><loc_82><loc_30></location>When these variables are set at their default values, the quantities to be plotted are represented by the string-valued variables xH and yH , for the x -and y -coordinates of the history plots and xP and yP , for the x -and y -coordinates of the profile plots.</text>
<text><location><page_10><loc_18><loc_16><loc_82><loc_23></location>The possible values of xH and yH are the elements in the 6 th row of the file 'star.log'. The possible values of xP and yP are the elements in the 6 th row of any of the files 'log n .data'. All these options are accessible to the user through popup menus and buttons. The lists of buttons to show in the</text>
<text><location><page_11><loc_18><loc_82><loc_76><loc_84></location>interface are coded in the (*Buttons to show*) section of the code:</text>
<code><location><page_11><loc_21><loc_57><loc_83><loc_80></location>(*Buttons to show*) HistoryButtonsX = { "star age", "log Teff", "log L", "log R", "log center T", "log center Rho" } ; HistoryButtonsY = { "log Teff", "log L", "log R", "log center T", "log center Rho" } ; HistoryButtonsY2 = { "log center P", "center h1", "center he4", "center c12" } ; ProfileButtonsX = { "mass", "radius", "logR" } ; ProfileButtonsY = { "radius", "mass", "logR", "logT", "logRho", "logP", "h1", "he3", "he4" } ; ProfileButtonsY2 = { "c12", "n14", "o16", "log opacity", "luminosity", "non nuc neu" } ;</code>
<text><location><page_11><loc_18><loc_52><loc_82><loc_55></location>The first element of each list gives the default value of the corresponding variable.</text>
<text><location><page_11><loc_18><loc_45><loc_82><loc_52></location>The plot functions Customized and Custom , plot the values in the two columns defined by xH and yH of the file 'star.log' (after dropping the first 6 rows) and the values in the two columns defined by xP and yP of the current file 'log n .data' (again, after dropping the first 6 rows) respectively.</text>
<text><location><page_11><loc_18><loc_41><loc_82><loc_45></location>The other plot functions show specific columns of the files 'star.log' or 'log n .data' in order to produce some standard plots (see section 5.2).</text>
<text><location><page_11><loc_18><loc_38><loc_82><loc_41></location>Regardless of the value of the plotType and profileType variables, the following parameters define some characteristics of the graphs:</text>
<unordered_list>
<list_item><location><page_11><loc_21><loc_27><loc_82><loc_36></location>· Cx , Cy , Dx , Dy represent the scale factors for, respectively, the x -and y -axis of the history plots and the x -and y -axis of the profile plots. Selecting n in one of the axis scale menu rescales the axis by a factor of 10 -n . For example, if the x -axis represents the age in years, selecting Cx=6 changes the units to Myr.</list_item>
<list_item><location><page_11><loc_21><loc_17><loc_82><loc_26></location>· The string-valued variables funcXH , funcYH , funcXP , and funcYP define the operation to be performed on the data represented by, respectively, xH , yH , xP and yP . Their possible value is selected by popup menu type controls. The expression corresponding to the string is specified in the list DataOperations , in the section (* Operations on the data*) .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_12><loc_21><loc_80><loc_82><loc_84></location>· The real-valued variables offsetXH , offsetYH , offsetXP , and offsetYP define new zeros for the axes.</list_item>
<list_item><location><page_12><loc_21><loc_74><loc_82><loc_79></location>· startH and endH define the plot range for the history plots, that is, the rows of the file 'star.log' to be shown. Changing their value allows to zoom in a particular region of the history plot.</list_item>
<list_item><location><page_12><loc_21><loc_67><loc_82><loc_72></location>· startP and endP define the plot range for the profile plots, that is, the rows of the current profile to be shown. Changing their value allows to zoom in a particular region of the profile plot.</list_item>
</unordered_list>
<text><location><page_12><loc_18><loc_62><loc_82><loc_65></location>All the possible values for these variables can be set from the control section, on the left hand side of the interface.</text>
<text><location><page_12><loc_18><loc_58><loc_82><loc_62></location>The (* Profile Plot Section*) contains two more variables: starAge and goToAge .</text>
<unordered_list>
<list_item><location><page_12><loc_21><loc_44><loc_82><loc_57></location>· starAge can be equal to the age of any of the profiles from the current simulation. Its value can be changed dynamically through a slide-bar and its current value is indicated on top of the profile plot. Changing starAge corresponds to selecting a new 'log n .data' file. The log number corresponding to the current age is also shown on top of the profile plot. When the star age is changed, all the plots and the other information shown are dynamically updated.</list_item>
<list_item><location><page_12><loc_21><loc_30><loc_82><loc_43></location>· goToAge is a logical variable, by default equal to false . If its value is set to true (by checking the box 'Show Age in History Plot' in the profile section of the interface), the age selected will be shown in the history plot through a line or a dot (or both, depending on the kind of plot). The style of the line and of the dot is set by the parameters AgeLineColor , AgeDotSize and AgeDotColor in the (*Graphs colors *) section of the script:</list_item>
</unordered_list>
<code><location><page_12><loc_37><loc_24><loc_65><loc_29></location>AgeLineColor = { Red, Dashed } ; AgeDotSize = 0.01; AgeDotColor = Red;</code>
<section_header_level_1><location><page_12><loc_18><loc_21><loc_43><loc_22></location>5.2. Visualization of the data</section_header_level_1>
<text><location><page_12><loc_18><loc_17><loc_82><loc_20></location>When the second script is executed, Mathematica reads the value of the three string-valued variables plotType , profileType , and Info .</text>
<figure>
<location><page_13><loc_24><loc_57><loc_75><loc_84></location>
<caption>Figure 1: The plots refer to a 1 M glyph[circledot] star, with solar metallicty [6]. The top ones are examples of history plots. The bottom ones of profile plots. The red dot and the dashed line indicate the age corresponding to the profiles shown, in this case about 12.5 Gyr.</caption>
</figure>
<text><location><page_13><loc_21><loc_47><loc_56><loc_48></location>Then, in the following section of the code:</text>
<formula><location><page_13><loc_28><loc_42><loc_69><loc_45></location>Grid[ { (* History Plot *) { Dynamic[ToExpression[plotType]] } ,</formula>
<text><location><page_13><loc_35><loc_38><loc_50><loc_40></location>(* Profile Plot</text>
<text><location><page_13><loc_52><loc_38><loc_54><loc_40></location>*)</text>
<text><location><page_13><loc_35><loc_37><loc_36><loc_38></location>{</text>
<text><location><page_13><loc_36><loc_37><loc_70><loc_38></location>Dynamic[ToExpression[profileType]]</text>
<text><location><page_13><loc_70><loc_37><loc_71><loc_38></location>}</text>
<text><location><page_13><loc_71><loc_37><loc_72><loc_38></location>,</text>
<text><location><page_13><loc_35><loc_33><loc_49><loc_34></location>(* Information</text>
<text><location><page_13><loc_51><loc_33><loc_53><loc_34></location>*)</text>
<text><location><page_13><loc_35><loc_31><loc_66><loc_33></location>{ Dynamic[ToExpression[Info]] }} ,</text>
<text><location><page_13><loc_35><loc_29><loc_67><loc_31></location>Alignment → Left, Frame → All],</text>
<text><location><page_13><loc_18><loc_19><loc_82><loc_27></location>these three strings are transformed into expressions which are evaluated in the section of the code labeled (* Plot and Info Section*) , described below. In addition, the Grid function creates three panels which show the value of these three expressions. Examples of graphs shown in the history and profile panels are given in Fig.1.</text>
<text><location><page_13><loc_18><loc_15><loc_82><loc_18></location>The section (* Plot and Info Section*) , is divided into three subsections, each responsible for one of the three graphical panels:</text>
<unordered_list>
<list_item><location><page_14><loc_20><loc_80><loc_82><loc_84></location>1. (* History Plots Section *) for the top panel. Defines the possible values for the variable plotType ;</list_item>
<list_item><location><page_14><loc_20><loc_77><loc_82><loc_80></location>2. (* Profile Plots Section *) for the middle panel. Defines the possible values for the variable profileType ;</list_item>
<list_item><location><page_14><loc_20><loc_73><loc_82><loc_76></location>3. (* Info Section *) for the bottom panel. Defines the possible values for the variable Info ;</list_item>
</unordered_list>
<text><location><page_14><loc_18><loc_69><loc_82><loc_72></location>Below is the list of the possible values of plotType , with the results that they produce:</text>
<unordered_list>
<list_item><location><page_14><loc_21><loc_58><loc_82><loc_67></location>· Customized is the default choice. In this case, the variables in the x -and y -axes of the graphs (in the top panel) are selected from the buttons or from the popup menus. Each popup menu contains the names of all the variables that are possible to plot (i.e., the names of all the columns in the 'star.log' file).</list_item>
<list_item><location><page_14><loc_21><loc_50><loc_82><loc_57></location>· HR produces the HR diagram, with the logarithm of the effective temperature on the x -axis and the logarithm of the luminosity on the y -axis. The temperature increases toward the left of the x -axis, in line with the standard convention for the HR diagram.</list_item>
<list_item><location><page_14><loc_21><loc_43><loc_83><loc_48></location>· CentralAbundances shows the central abundance of the elements defined in the list CentralAbundancesToShow in the (* General Settings *) section of the code.</list_item>
<list_item><location><page_14><loc_21><loc_36><loc_82><loc_41></location>· Burnings shows the energy released in the nuclear reactions defined in the list CentralAbundancesToShow in the (* General Settings *) section of the code.</list_item>
</unordered_list>
<text><location><page_14><loc_21><loc_33><loc_54><loc_34></location>The possible values of profileType are</text>
<unordered_list>
<list_item><location><page_14><loc_21><loc_28><loc_82><loc_32></location>· Custom is the default choice and works in the same way as for the history plots.</list_item>
<list_item><location><page_14><loc_21><loc_22><loc_83><loc_27></location>· Abundances shows the profile of the abundance of the elements defined in the list ProfileAbundancesToShow in the (* General Settings *) section of the code.</list_item>
<list_item><location><page_14><loc_21><loc_15><loc_82><loc_20></location>· Reactions shows the energy released in the nuclear reactions defined in the list ProfileBurningsToShow in the (* General Settings *) section of the code.</list_item>
</unordered_list>
<text><location><page_15><loc_18><loc_80><loc_82><loc_84></location>All the lists for the abundances and the reactions can be customized according to the user's needs.</text>
<text><location><page_15><loc_21><loc_79><loc_54><loc_80></location>Finally, the possible values of Info are</text>
<unordered_list>
<list_item><location><page_15><loc_21><loc_72><loc_82><loc_77></location>· PATH is the default value. In this case MESAFace shows the value of the variable path and so indicates the working folder or the web location from where Mathematica is extracting the data.</list_item>
<list_item><location><page_15><loc_21><loc_65><loc_82><loc_70></location>· GENERAL shows general information about the star not related to the current profile. The information is taken from the second and third rows of the file 'star.log'.</list_item>
<list_item><location><page_15><loc_21><loc_58><loc_82><loc_64></location>· DETAILED shows detailed information about the current profile. The information is taken from the second and third rows of the current profile.</list_item>
</unordered_list>
<text><location><page_15><loc_18><loc_48><loc_82><loc_57></location>It is possible to add other kinds of plots or information by adding new options for the variables plotType , profileType , and Info in the section below the comment line (* Beginning of the proper dynamical part *) and then define their expression-values in the (* Plot and Info Section*) of the code.</text>
<section_header_level_1><location><page_15><loc_18><loc_44><loc_32><loc_45></location>6. Conclusions</section_header_level_1>
<text><location><page_15><loc_18><loc_37><loc_82><loc_42></location>MESAFace is a user-friendly and efficient interface to analyze the MESA output data. It is written in Mathematica, a very powerful and very welldocumented mathematical software, and it is easily customizable.</text>
<text><location><page_15><loc_18><loc_23><loc_82><loc_37></location>The main purpose of this interface is to provide a tool to quickly analyze all the output data from a MESA run through graphs and tables. Our final objective was to have a very intuitive and easy-to-use instrument, accessible to researchers as well as to students. MESAFace allows the user to choose what to plot through various menus and to scroll the various profiles, organized by age, with a slidebar. The user can also easily zoom in different regions of the plot, change units, set a different zero for each axis, or plot the log (among other functions) of the data.</text>
<text><location><page_15><loc_18><loc_16><loc_82><loc_22></location>The production of high-quality graphs was not our priority, as other tools are available [7]. However, improving the quality and create new options to manipulate the aspect of a graph, is a prominent direction for future improvement of MESAFace.</text>
<text><location><page_16><loc_18><loc_75><loc_82><loc_84></location>This paper is not meant to be a proper user manual. Basic instructions on how to use MESAFace are accessible on-line [4]. Here, instead, we have described in detail the structure of the code and the meaning of the variables and controls. This allows the user not just to use, but if necessary, to modify and possibly improve this code.</text>
<section_header_level_1><location><page_16><loc_18><loc_71><loc_35><loc_72></location>Acknowledgments</section_header_level_1>
<text><location><page_16><loc_18><loc_66><loc_82><loc_69></location>We express our gratitude to Dr. Stan Owocki, who encouraged us to develop this interface.</text>
<section_header_level_1><location><page_16><loc_18><loc_62><loc_28><loc_63></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_16><loc_19><loc_57><loc_82><loc_60></location>[1] B. Paxton, L. Bildsten, A. Dotter, F. Herwig, P. Lesaffre and F. Timmes, Astrophys. J. Suppl. 192 , 3 (2011) [arXiv:1009.1622 [astro-ph.SR]].</list_item>
<list_item><location><page_16><loc_19><loc_52><loc_82><loc_56></location>[2] To download MESA and for instructions on its use, visit http://mesa.sourceforge.net/</list_item>
<list_item><location><page_16><loc_19><loc_47><loc_82><loc_51></location>[3] See http://mesastar.org/. Interesting MESA programming tools can be found following the link Tools & Utilities.</list_item>
<list_item><location><page_16><loc_19><loc_44><loc_61><loc_46></location>[4] See http://www.mgiannotti.com/mesaface.php</list_item>
<list_item><location><page_16><loc_19><loc_39><loc_82><loc_43></location>[5] Many more details on the controls can be found in the Mathematica help for Manipulate .</list_item>
<list_item><location><page_16><loc_19><loc_33><loc_82><loc_38></location>[6] The inlist file to generate the data is available in the MESA test suite folder of MESA. The resulting data folder can also be downloaded directly from [4].</list_item>
<list_item><location><page_16><loc_19><loc_28><loc_66><loc_31></location>[7] An excellent choice is Tioga, available at http://helix.phys.uvic.ca:8080/MESA/tools-utilities.</list_item>
</unordered_list>
</document>
[
{
"title": "M. Giannotti ∗ , M. Wise, A. Mohammed",
"content": "Barry University, 11300 NE 2nd Ave., Miami Shores, FL 33161, US",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "MESA (Modules for Experiments in Stellar Astrophysics) has become very popular among astrophysicists as a powerful and reliable code to simulate stellar evolution. Analyzing the output data thoroughly may, however, present some challenges and be rather time-consuming. Here we describe MESAFace, a graphical and dynamical interface which provides an intuitive, efficient and quick way to analyze the MESA output. Keywords: Stellar evolution; MESA; Mathematica; GUI;",
"pages": [
1
]
},
{
"title": "PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY",
"content": "Manuscript Title: MESAFace, a graphical interface to analyze the MESA output Authors: M. Giannotti, M. Wise, A. Mohammed Program Title: MESAFace Journal Reference: Catalogue identifier: Licensing provisions: none Programming language: Mathematica Computer: Any computer capable of running Mathematica. Operating system: Windows XP, Windows 7. Any capable of running Mathematica. Tested on Linux, Mac, RAM: Recommended 2 Gigabytes or more. Keywords: Classification: Stellar evolution, MESA, Mathematica, GUI. 1.7 Stars and Stellar Systems, 14 Graphics",
"pages": [
1
]
},
{
"title": "Nature of problem:",
"content": "Find a way to quickly and thoroughly analyze the output of a MESA run, including all the profiles, and have an efficient method to produce graphical representations of the data.",
"pages": [
2
]
},
{
"title": "Solution method:",
"content": "We created two scripts (to be run consecutively). The first one downloads all the data from a MESA run and organizes the profiles in order of age. All the files are saved as tables or arrays of tables which can then be accessed very quickly by Mathematica. The second script uses the Manipulate function to create a graphical interface which allows the user to choose what to plot from a set of menus and buttons. The information shown is updated in real time. The user can access very quickly all the data from the run under examination and visualize it with plots and tables.",
"pages": [
2
]
},
{
"title": "Unusual features:",
"content": "Moving the slides in certain regions may cause an error message. This happens when Mathematica is asked to read nonexistent data. The error message, however, disappears when the slides are moved back. This issue does not preclude the good functioning of the interface.",
"pages": [
2
]
},
{
"title": "Additional comments:",
"content": "The program uses the dynamical capabilities of Mathematica. When the program is opened, Mathematica prompts the user to 'Enable Dynamics'. It is necessary to accept before proceeding.",
"pages": [
2
]
},
{
"title": "Running time:",
"content": "Depends on the size of the data downloaded, on where the data are stored (harddrive or web), and on the speed of the computer or network connection. In general, downloading the data may take from a minute to several minutes. Loading directly from the web is slower. For example, downloading a 200MB data folder (a total of 102 files) with a dual-core Intel laptop, P8700, 2 GB of RAM, at 2.53 GHz took about a minute from the hard-drive and about 23 minutes from the web (with a basic home wireless connection).",
"pages": [
2
]
},
{
"title": "1. Introduction",
"content": "The introduction of the new code for stellar evolution, MESA [1] (Modules for Experiments in Stellar Astrophysics) represented somewhat of a revolution in the field of computational stellar evolution. MESA is a one-dimensional stellar evolution code, organized in independent modules and continually updated. The code can evolve stars with a very wide range of initial masses and metallicities; it allows the specification of many parameters, for example, convection mechanism, mass loss, etc., and it can overcome difficulties typical of the previous stellar evolutionary codes. Notably, it can run a low mass star through the He-flash region. All these aspects and the fact that the code is publicly available [2] has made MESA extremely popular among astrophysicists and other physicists interested in stellar evolution and in how stars respond to new models of particle interactions. Being publicly available, MESA is widely accessible and is continually tested by hundreds of scientists and students all over the world. In addition, several researchers have created new codes and routines to improve it and to simplify its use [3]. A MESA run produces a very large amount of data on the life and structure of the star. This information is saved in several files: the file 'star.log' includes global information about the star at different times whereas a large number of profiles (named 'log n .data', where n is an integer), contain detailed information about the profile of the star, each file referring to a specific age. Though it is relatively easy, and for some purposes sufficient, to analyze the file 'star.log', a complete analysis of the MESA output, which includes in some cases hundreds of profiles, is much more challenging. In particular, to analyze the detailed structure of a star at a certain age, one should select the corresponding profile by reading the age of each of them (the information is provided in the 3 nd row of 'log n .data'). Alternatively, one should read the model number corresponding to a certain age in the first column of the file 'star.log', and then look for the log number associated with that age in the file 'profile.index'. Both ways are long and, in general, not very practical. Another common problem is reading the age of a star from its position in the HR diagram (or any other plot which does not show the age explicitly, e.g., the central temperature-central density diagram). It would be useful to have a quick access to that information and, at the same time, to know which of the many profiles describes the structure of the star at that age. Here we describe a graphical interface, MESAFace, which we created to address problems like the ones described above and, in general, to provide fast and simple access to all the information contained in the output of a MESA run, including all the profiles. MESAFace is a very intuitive, efficient, and easy-to-use dynamical interface, which allows the selection of the information to be shown through buttons and menus and the age of the star to be selected through a slidebar. The code is written in Mathematica and has been tested with Mathematica 7 and 8. However, a knowledge of Mathematica is not strictly necessary to use the interface. MESAFace needs to access the MESA output data which, in a standard MESA run, are saved in the folder /work/LOGS. The data can also be downloaded directly from the web. Possibly soon, the results from some standard runs will be available from a dedicated web space. After the data has been loaded, MESAFace presents a graphical interface structured in three vertically organized panels: All the controls (buttons, menus, check-boxes, etc.) are contained on the left side of the interface. The user can decide what to plot and can change the age of the star by using these controls while the information shown on the right side is dynamically updated. Several aspects of the panels and graphs, for example, the color and style of the lines, can be customized by the user. A separate user manual [4] explains in detail how to do that. In this paper, we describe in detail the structure of the MESAFace code and how it accesses, manipulates, and presents the data. The structure of the paper is the following: We begin with a brief summary of the MESA output, in section 2. Later, in section 3, we give a basic description of the general structure of the MESAFace code. In section 4, we describe with more details how MESAFace loads and organizes the files. In section 5, we discuss how it creates the dynamical interface and visualizes the data. Finally, in section 6 we conclude with some overall comments. A note on the style. We will use the Typewriter style for the content of the Mathematica notebook, 'quotes' for the name of the MESA files, and bold face for the names of the controls in the interface.",
"pages": [
2,
3,
4
]
},
{
"title": "2. Brief description of the MESA output",
"content": "MESA is a one-dimensional stellar evolution code. The output data is parameterized by the value of the star age and the radial distance from the star center. Most of the output data is organized in a large number of profiles ('log n .data' files, with n an integer). The main section of each profile is a table with data referring to a particular age and each row referring to a particular zone in the star. Different zones correspond to different distances from the center. Besides the profiles, MESA also produces a 'star.log' file which includes global information about the star at different times, and a 'profiles.index' which provides a useful connection between the profiles and 'star.log'. All these files are created during a MESA run and, by default, are contained in the LOG folder of the star/work directory. Below is a brief description of the MESA output files. For more information, cfr. the MESA literature[1, 2]. The MESA output folder contains 1 file 'star.log', a large number of profiles , 1 file, 'profiles.index'. The sixth row contains the names of some position-dependent physical quantities, such as temperature or density, and all the rows below that show the values of these quantities in the different zones. The number of different profiles and the amount and kind of information in each file depend on the particular run.",
"pages": [
4,
5,
6
]
},
{
"title": "3. General structure of the MESAFace code",
"content": "The MESAFace code contains two scripts, labeled as (* SCRIPT 1 *) and (* SCRIPT 2 *) , which should be run consecutively. The first script extracts all the data from a MESA run and organizes them into tables easily accessible by the interface (see section 4). Although extracting all the MESA output at once may be slow, after that, the interface can access all the data very quickly, allowing a dynamical manipulation and visualization of the information. The interface is created by the second script, labeled (* SCRIPT 2 *) in the code and described in section 3.2 and 5. The choice to have two scripts is a practical one: The second script, which normally runs in no more than a second or so, allows several customizations of the interface, for example it allows the user to set the style of the graphs. Having this script separated from the first one allows the user to change some features in the interface or in the plot style without having to download the data each time.",
"pages": [
6
]
},
{
"title": "3.1. Structure of the first script",
"content": "The first script can be divided in two sections: The variable slash allows Mathematica to read the location of the output files with different operating systems. If the output data are on the web, the variable slash has to be set equal to \"/\" .",
"pages": [
6
]
},
{
"title": "3.2. Structure of the second script",
"content": "The second script can be divided into three main sections:",
"pages": [
7
]
},
{
"title": "3.3. Local and global variables",
"content": "Most of the variables used in the scripts are local (they are visible only within the script). A variable is made local by including it in the curly brackets at the beginning of the Module command. However, some variables defined in the first script need to be accessed by the second script and, therefore, they cannot be defined locally. These global variables are: If it is necessary to debug or to use a result outside a MESAFace session, a local variable can be made global by taking it out of the list in the curly brackets at the beginning of the Module command. In general, however, it is not recommended to use nonlocal variables outside MESAFace during a session.",
"pages": [
7,
8
]
},
{
"title": "4. Loading and organizing of the files",
"content": "The goal of MESAFace is to provide a way to access the data dynamically . In order to do that, MESAFace downloads all the data at once, either from the hard-drive or from the web, and saves it in particular variables. This may be a rather slow process, especially if the data are to be extracted directly from a web location. After the data is downloaded, however, the access to the information is very fast. The progress in downloading is indicated: a numerical value represents the actual number of downloaded profiles while a progress-bar shows the relative progress.",
"pages": [
8
]
},
{
"title": "4.1. Extraction of the MESA output",
"content": "The job of the first script of MESAFace is just to download and organize the MESA output data. Mathematica looks in the folder defined by the variable path for all the files with names 'star.log', 'profiles.index' and 'log n .data' (it is essential that the output files have those names, as it is for a default run). The variable path can point to a folder on the hard drive or on the web. The files 'star.log' and 'profiles.index' are saved (as tables) respectively in the variable star and profiles . Analogously, the 'log n .data' files are saved in the array of tables log[indx] , where indx is the log number n . For example, the file 'log33.data' is saved as log[33] .",
"pages": [
8
]
},
{
"title": "4.2. Extraction of the age of the star and selection of the profile to show",
"content": "One of the main features of MESAFace is to allow an easy selection of the profile corresponding to a certain age. To do that, the first script organizes the profiles in order of age in the table logAge , with the log number in column 1 and the star age in column 2. The function age : age[i ]:= log[i][[1 + Position[log[i], \"star age\"][[1, 1]], Position[log[i], \"star age\"][[1, 2]]]]; has the task of extracting the age of each profile. A slide-bar (more precisely, a manipulator ) type control: {{ starAge, logAge[[1, 2]], Style[\"Star Age\", 12, { Bold, Blue } ] } , logAge[[All, 2]], ControlType -> Manipulator } , allows the user to set the value of the real variable starAge to any of the lines in the second column of the table logAge (which, as explained, contains the ages of all the profiles). Each time starAge is selected, Mathematica uses the table logAge to find the corresponding log file number , and saves it in the variable model : model = logAge[[Position[logAge[[All, 2]], starAge][[1, 1]], 1]]; In addition, if the check-box Show Age in History Plot is checked, MESAFace finds which line of the file 'star.log' corresponds to the age selected: modelNumber = profiles[[Position[Drop[profiles, 1][[All, 3]], model][[1, 1]] + 1, 1]]; indexAge = Position[Drop[star, 6][[All, 1]], modelNumber] [[1, 1]]; Here, modelNumber is equal to the model number discussed above (see section 2). Its value is reported in the first column of 'profile.index' and in the first column of 'star.log'. The variable indexAge , instead, is equal to the the row number of 'star.log' (after its first 6 lines are dropped) corresponding to the model number (and therefore the age of the star). This value is used to identify the position of the age of the current profile in the history plots, providing a link between the History Plots and Profile Plots panels.",
"pages": [
8,
9
]
},
{
"title": "5. The interface",
"content": "Running the second script produces a graphical interface in which the controls, located on the right hand side, allow the user to choose what kind of information to show in the visualization panels to the right.",
"pages": [
9
]
},
{
"title": "5.1. Description of the controls",
"content": "The controls are coded in the section of the second script which follows the comment line (* Beginning of the proper dynamical part *) . They are described by variables and are normally nested in double curly brackets. For example, the line: {{ variable name, Default value, \"label\" } , list of possible values, ControlType -> PopupMenu } , describes a variable whose value can be set by a popup-menu [5]. The first controls in each of the three sections of the interface are radio button -type controls which allow the user to choose the kind of plot (top two panels) or information (bottom panel) to show. They are coded in the section of the script below the comment line (* Beginning of the proper dynamical part *) , at the beginning of the (* History Plot Section *) , the (* Profile Plot Section *) , and the (* Info Section *) : {{ plotType, \"Customized\", Style[\"Type of Plot\", 12, Bold] } , { \"Customized\", \"HR\", \"CentralAbundances\", \"Burnings\" } , ControlType -> RadioButtonBar } , {{ profileType, \"Custom\", Style[\"Type of Plot\", 12, Bold] } { \"Custom\", \"Abundances\", \"Reactions\" } , ControlType -> RadioButtonBar } , {{ Info, \"PATH\", \"\" } , { \"PATH\", \"GENERAL\", \"DETAILED\" } ControlType -> RadioButtonBar } , These controls allow the user to set the string value of plotType , profileType , and Info . When these variables are set at their default values, the quantities to be plotted are represented by the string-valued variables xH and yH , for the x -and y -coordinates of the history plots and xP and yP , for the x -and y -coordinates of the profile plots. The possible values of xH and yH are the elements in the 6 th row of the file 'star.log'. The possible values of xP and yP are the elements in the 6 th row of any of the files 'log n .data'. All these options are accessible to the user through popup menus and buttons. The lists of buttons to show in the interface are coded in the (*Buttons to show*) section of the code: The first element of each list gives the default value of the corresponding variable. The plot functions Customized and Custom , plot the values in the two columns defined by xH and yH of the file 'star.log' (after dropping the first 6 rows) and the values in the two columns defined by xP and yP of the current file 'log n .data' (again, after dropping the first 6 rows) respectively. The other plot functions show specific columns of the files 'star.log' or 'log n .data' in order to produce some standard plots (see section 5.2). Regardless of the value of the plotType and profileType variables, the following parameters define some characteristics of the graphs: All the possible values for these variables can be set from the control section, on the left hand side of the interface. The (* Profile Plot Section*) contains two more variables: starAge and goToAge .",
"pages": [
10,
11,
12
]
},
{
"title": "5.2. Visualization of the data",
"content": "When the second script is executed, Mathematica reads the value of the three string-valued variables plotType , profileType , and Info . Then, in the following section of the code: (* Profile Plot *) { Dynamic[ToExpression[profileType]] } , (* Information *) { Dynamic[ToExpression[Info]] }} , Alignment → Left, Frame → All], these three strings are transformed into expressions which are evaluated in the section of the code labeled (* Plot and Info Section*) , described below. In addition, the Grid function creates three panels which show the value of these three expressions. Examples of graphs shown in the history and profile panels are given in Fig.1. The section (* Plot and Info Section*) , is divided into three subsections, each responsible for one of the three graphical panels: Below is the list of the possible values of plotType , with the results that they produce: The possible values of profileType are All the lists for the abundances and the reactions can be customized according to the user's needs. Finally, the possible values of Info are It is possible to add other kinds of plots or information by adding new options for the variables plotType , profileType , and Info in the section below the comment line (* Beginning of the proper dynamical part *) and then define their expression-values in the (* Plot and Info Section*) of the code.",
"pages": [
12,
13,
14,
15
]
},
{
"title": "6. Conclusions",
"content": "MESAFace is a user-friendly and efficient interface to analyze the MESA output data. It is written in Mathematica, a very powerful and very welldocumented mathematical software, and it is easily customizable. The main purpose of this interface is to provide a tool to quickly analyze all the output data from a MESA run through graphs and tables. Our final objective was to have a very intuitive and easy-to-use instrument, accessible to researchers as well as to students. MESAFace allows the user to choose what to plot through various menus and to scroll the various profiles, organized by age, with a slidebar. The user can also easily zoom in different regions of the plot, change units, set a different zero for each axis, or plot the log (among other functions) of the data. The production of high-quality graphs was not our priority, as other tools are available [7]. However, improving the quality and create new options to manipulate the aspect of a graph, is a prominent direction for future improvement of MESAFace. This paper is not meant to be a proper user manual. Basic instructions on how to use MESAFace are accessible on-line [4]. Here, instead, we have described in detail the structure of the code and the meaning of the variables and controls. This allows the user not just to use, but if necessary, to modify and possibly improve this code.",
"pages": [
15,
16
]
},
{
"title": "Acknowledgments",
"content": "We express our gratitude to Dr. Stan Owocki, who encouraged us to develop this interface.",
"pages": [
16
]
}
]
2013EAS....64...55L
https://arxiv.org/pdf/1310.7503.pdf
<document>
<text><location><page_1><loc_7><loc_84><loc_31><loc_88></location>Title : will be set by the publisher Editors : will be set by the publisher EAS Publications Series, Vol. ?, 2018</text>
<section_header_level_1><location><page_1><loc_12><loc_70><loc_60><loc_73></location>MODELLING BINARY ROTATING STARS BY NEW POPULATION SYNTHESIS CODE BONNFIRES</section_header_level_1>
<text><location><page_1><loc_9><loc_67><loc_62><loc_69></location>Herbert H.B. Lau 1 , Robert G. Izzard 1 and Fabian R.N. Schneider 1</text>
<text><location><page_1><loc_13><loc_50><loc_59><loc_64></location>Abstract. BONNFIRES,a new generation of population synthesis code, can calculate nuclear reaction, various mixing processes and binary interaction in a timely fashion. We use this new population synthesis code to study the interplay between binary mass transfer and rotation. We aim to compare theoretical models with observations, in particular the surface nitrogen abundance and rotational velocity. Preliminary results show binary interactions may explain the formation of nitrogenrich slow rotators and nitrogen-poor fast rotators, but more work needs to be done to estimate whether the observed frequencies of those stars can be matched.</text>
<section_header_level_1><location><page_1><loc_7><loc_46><loc_20><loc_47></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_7><loc_36><loc_65><loc_44></location>Rapid rotation can lead to efficient mixing of the whole star and hence alter the nucleosynthesis process and observed abundances. Hydrogen burning products, such as nitrogen, are a strong indicator of rotational mixing, so nitrogen are expected to be proportional to observed rotational velocities. How the evolution and final fate of the star depends on rotation rate has been recently reviewed by Langer (2012) and Maeder & Meynet (2012).</text>
<text><location><page_1><loc_7><loc_25><loc_65><loc_35></location>However, observations show that the picture of rotational mixing in massive stars are not that simple. Hunter et al. (2008) found a significant fraction of stars that cannot be explained by current single rotating stars models. These are stars with strong nitrogen enhancement without rapid rotations and stars with rapid rotation without nitrogen enhancement. These observations show that other processes must be responsible for either the nitrogen enhancement or the current rotation velocity.</text>
<text><location><page_1><loc_7><loc_21><loc_65><loc_25></location>Sana et al. (2013) shows that almost all massive stars are born in a binary system , so significant fractions of stars observed must have undergone binary interaction to reach their current states. The aim of this work is to investigate</text>
<text><location><page_2><loc_7><loc_81><loc_65><loc_84></location>whether binary interactions can account for the observed anomalies between nitrogen abundances and rotational velocities.</text>
<figure>
<location><page_2><loc_7><loc_48><loc_65><loc_79></location>
<caption>Fig. 1. Evolution of surface nitrogen abundances (by mass fraction) of 20 M /circledot models at solar metallicity (Z=0.02) with different surface rotational velocity from 100 -500kms -1 . Models are produced by the population synthesis code BONNFIRES and stop at the end of main sequence. Fast rotators enhance significantly more nitrogen with longer main sequence lifetimes.</caption>
</figure>
<section_header_level_1><location><page_2><loc_7><loc_36><loc_47><loc_37></location>2 New Population synthesis code BONNFIRES</section_header_level_1>
<text><location><page_2><loc_7><loc_21><loc_65><loc_34></location>BONNFIRES is designed to model the internal mixing and nucleosynthesis of binary stars in a timely fashion. Internal structure variables are interpolated from input models produced by detailed evolutionary code. Binary physics, mixing and nuclear reaction are calculated independently by BONNFIRES to study binary interactions and internal mixing. This will be the first population synthesis code with detailed internal composition profile, mixing processes and nuclear reaction networks. We follow Heger et al. (2000) for treatments of rotational mixing and Brott et al. (2011) for rotational mixing parameters. Evolution of the binary systems are followed till the end of the main-sequence of the primary stars.</text>
<text><location><page_2><loc_7><loc_16><loc_65><loc_20></location>Figure 1 shows 20 M /circledot models with different initial rotational velocities by BONNFIRES. The surface nitrogen enhancement is much stronger for fast rotators. Moreover, the main sequence lifetimes for fast rotators are longer because</text>
<text><location><page_3><loc_7><loc_81><loc_65><loc_84></location>of additional hydrogen that are mixed down to the burning region. This result is consistent with models made by detailed evolutionary codes.</text>
<figure>
<location><page_3><loc_9><loc_47><loc_63><loc_76></location>
<caption>Fig. 2. Evolution of surface nitrogen abundances (by mass fraction) against equatorial rotational velocities for a binary system consisting of a 16 M /circledot primary, a 12 M /circledot secondary with 3 days period. The initial rotational velocity for primary and secondary are 300kms -1 and 200kms -1 respectively. Region between two dotted black lines are what is expected from single star models. After mass transfer, the nitrogen-rich primary (red solid line) spins down while the secondary (blue dashed line) spins up but with significantly less nitrogen enrichment compared to single star models.</caption>
</figure>
<section_header_level_1><location><page_3><loc_7><loc_26><loc_15><loc_28></location>3 Result</section_header_level_1>
<text><location><page_3><loc_7><loc_16><loc_65><loc_25></location>For the short period systems, the primary stars interact with their companions during the main-sequence. Typically, when the primary overflow its Roche Lobe, the star is expanding and hence spin down after mass loss. If the primary is initially rapidly rotating, its surface will be rich in nitrogen but it is now a slow rotating after mass transfer. Alternatively, if the primary suffers heavy mass loss, its inner hydrgeon burning region will be exposed and so the surface will now be</text>
<text><location><page_4><loc_7><loc_81><loc_65><loc_84></location>rich in nitrogen irregardless of any prior rotational mixing. These are possible formation channels of nitrogen-rich slow rotators.</text>
<text><location><page_4><loc_7><loc_66><loc_65><loc_81></location>Materials transferred onto companion stars also carry angular momentum, so the companion stars will be spun up and rotate faster. The surface abundances of the secondary depends on the compositions of the accreted materials as well as any subsequent mixing. Because the secondary is now rapidly rotating it may mix materials from its inner hydrogen-burning region to the surface. However, if the secondary is previously slow rotating, the burning region has a much higher mean molecular weight. This mean molecular weight barrier can suppress rotational mixing, so nitrogen enhancement is much smaller than its single star counterpart that rotates at the same velocity from the beginning. It is therefore possible the secondary is observed as nitrogen-poor fast rotators.</text>
<text><location><page_4><loc_7><loc_52><loc_65><loc_66></location>Figure 2 illustrates a typical short period binary system. This systems consists of a 16 M /circledot primary, a 12 M /circledot secondary with 3 days period. The primary is initially rotating at 300km s -1 and enhanced surface nitrogen to XN = 0 . 029 through rotational mixing, similar to a single star. Then mass transfer from primary to secondary occurs and the primary spins down rapidly to become a nitrogen-rich slow rotators. Surface nitrogen abundance of secondary also increase because the accreted material is enriched in nitrogen, but the secondary is spun up to critical velocity. If we compared to single stars model with the same rotational velocities, the secondary has significantly less nitrogen enhancement on the surface.</text>
<section_header_level_1><location><page_4><loc_7><loc_49><loc_20><loc_50></location>4 Future Work</section_header_level_1>
<text><location><page_4><loc_7><loc_39><loc_65><loc_47></location>In order to statistically compare with observations, we need to take into account the frequency of the above channels and also for how long the stars can be observed nitrogen-rich slow rotators and nitrogen-poor fast rotators. We will produce a fine grid of binary models with various assumptions of tides, efficiency of rotational mixing to determine whether the observed frequency can be reproduced. Results should be expected in the early 2014.</text>
<section_header_level_1><location><page_4><loc_7><loc_35><loc_16><loc_36></location>References</section_header_level_1>
<code><location><page_4><loc_7><loc_24><loc_50><loc_34></location>Brott I., et al. . 2011, A&A, 530, A116 Heger A., Langer N., Woosley S. E., 2000, ApJ, 528, 368 Hunter I., et al. . 2008, ApJL, 676, L29 Langer N., 2012, ARA&A, 50, 107 Maeder A., Meynet G., 2012, Reviews of Modern Physics, 84, 25 Sana H., et al. ., 2013, A&A, 550, A107</code>
</document>
[
{
"title": "ABSTRACT",
"content": "Title : will be set by the publisher Editors : will be set by the publisher EAS Publications Series, Vol. ?, 2018",
"pages": [
1
]
},
{
"title": "MODELLING BINARY ROTATING STARS BY NEW POPULATION SYNTHESIS CODE BONNFIRES",
"content": "Herbert H.B. Lau 1 , Robert G. Izzard 1 and Fabian R.N. Schneider 1 Abstract. BONNFIRES,a new generation of population synthesis code, can calculate nuclear reaction, various mixing processes and binary interaction in a timely fashion. We use this new population synthesis code to study the interplay between binary mass transfer and rotation. We aim to compare theoretical models with observations, in particular the surface nitrogen abundance and rotational velocity. Preliminary results show binary interactions may explain the formation of nitrogenrich slow rotators and nitrogen-poor fast rotators, but more work needs to be done to estimate whether the observed frequencies of those stars can be matched.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Rapid rotation can lead to efficient mixing of the whole star and hence alter the nucleosynthesis process and observed abundances. Hydrogen burning products, such as nitrogen, are a strong indicator of rotational mixing, so nitrogen are expected to be proportional to observed rotational velocities. How the evolution and final fate of the star depends on rotation rate has been recently reviewed by Langer (2012) and Maeder & Meynet (2012). However, observations show that the picture of rotational mixing in massive stars are not that simple. Hunter et al. (2008) found a significant fraction of stars that cannot be explained by current single rotating stars models. These are stars with strong nitrogen enhancement without rapid rotations and stars with rapid rotation without nitrogen enhancement. These observations show that other processes must be responsible for either the nitrogen enhancement or the current rotation velocity. Sana et al. (2013) shows that almost all massive stars are born in a binary system , so significant fractions of stars observed must have undergone binary interaction to reach their current states. The aim of this work is to investigate whether binary interactions can account for the observed anomalies between nitrogen abundances and rotational velocities.",
"pages": [
1,
2
]
},
{
"title": "2 New Population synthesis code BONNFIRES",
"content": "BONNFIRES is designed to model the internal mixing and nucleosynthesis of binary stars in a timely fashion. Internal structure variables are interpolated from input models produced by detailed evolutionary code. Binary physics, mixing and nuclear reaction are calculated independently by BONNFIRES to study binary interactions and internal mixing. This will be the first population synthesis code with detailed internal composition profile, mixing processes and nuclear reaction networks. We follow Heger et al. (2000) for treatments of rotational mixing and Brott et al. (2011) for rotational mixing parameters. Evolution of the binary systems are followed till the end of the main-sequence of the primary stars. Figure 1 shows 20 M /circledot models with different initial rotational velocities by BONNFIRES. The surface nitrogen enhancement is much stronger for fast rotators. Moreover, the main sequence lifetimes for fast rotators are longer because of additional hydrogen that are mixed down to the burning region. This result is consistent with models made by detailed evolutionary codes.",
"pages": [
2,
3
]
},
{
"title": "3 Result",
"content": "For the short period systems, the primary stars interact with their companions during the main-sequence. Typically, when the primary overflow its Roche Lobe, the star is expanding and hence spin down after mass loss. If the primary is initially rapidly rotating, its surface will be rich in nitrogen but it is now a slow rotating after mass transfer. Alternatively, if the primary suffers heavy mass loss, its inner hydrgeon burning region will be exposed and so the surface will now be rich in nitrogen irregardless of any prior rotational mixing. These are possible formation channels of nitrogen-rich slow rotators. Materials transferred onto companion stars also carry angular momentum, so the companion stars will be spun up and rotate faster. The surface abundances of the secondary depends on the compositions of the accreted materials as well as any subsequent mixing. Because the secondary is now rapidly rotating it may mix materials from its inner hydrogen-burning region to the surface. However, if the secondary is previously slow rotating, the burning region has a much higher mean molecular weight. This mean molecular weight barrier can suppress rotational mixing, so nitrogen enhancement is much smaller than its single star counterpart that rotates at the same velocity from the beginning. It is therefore possible the secondary is observed as nitrogen-poor fast rotators. Figure 2 illustrates a typical short period binary system. This systems consists of a 16 M /circledot primary, a 12 M /circledot secondary with 3 days period. The primary is initially rotating at 300km s -1 and enhanced surface nitrogen to XN = 0 . 029 through rotational mixing, similar to a single star. Then mass transfer from primary to secondary occurs and the primary spins down rapidly to become a nitrogen-rich slow rotators. Surface nitrogen abundance of secondary also increase because the accreted material is enriched in nitrogen, but the secondary is spun up to critical velocity. If we compared to single stars model with the same rotational velocities, the secondary has significantly less nitrogen enhancement on the surface.",
"pages": [
3,
4
]
},
{
"title": "4 Future Work",
"content": "In order to statistically compare with observations, we need to take into account the frequency of the above channels and also for how long the stars can be observed nitrogen-rich slow rotators and nitrogen-poor fast rotators. We will produce a fine grid of binary models with various assumptions of tides, efficiency of rotational mixing to determine whether the observed frequency can be reproduced. Results should be expected in the early 2014.",
"pages": [
4
]
}
]
2013EP&S...65.1101M
https://arxiv.org/pdf/1310.8460.pdf
<document>
<text><location><page_1><loc_11><loc_76><loc_11><loc_77></location>4</text>
<section_header_level_1><location><page_1><loc_22><loc_86><loc_78><loc_87></location>Dusty Universe viewed by AKARI far infrared detector</section_header_level_1>
<text><location><page_1><loc_17><loc_82><loc_83><loc_84></location>K. Małek 1 , A. Pollo 2 , 3 , T. T. Takeuchi 1 , E. Giovannoli 4 , 5 , V. Buat 4 , D. Burgarella 4 , M. Malkan 6 , and A. Kurek 3</text>
<text><location><page_1><loc_18><loc_78><loc_82><loc_80></location>1 Department of Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, 464-8602 Nagoya, Japan 2 National Centre for Nuclear Research, ul. Ho˙za 69, 00-681 Warszawa, Poland</text>
<text><location><page_1><loc_24><loc_77><loc_76><loc_78></location>3 The Astronomical Observatory of the Jagiellonian University, ul. Orla 171, 30-244 Krak'ow, Poland</text>
<text><location><page_1><loc_12><loc_76><loc_43><loc_77></location>Laboratoire d'Astrophysique de Marseille, OAMP, Universit</text>
<text><location><page_1><loc_43><loc_76><loc_44><loc_77></location>' e</text>
<text><location><page_1><loc_44><loc_76><loc_61><loc_77></location>Aix-Marseille, CNRS, 38 rue Fr</text>
<text><location><page_1><loc_61><loc_76><loc_61><loc_77></location>'</text>
<text><location><page_1><loc_61><loc_76><loc_61><loc_77></location>e</text>
<text><location><page_1><loc_61><loc_76><loc_62><loc_77></location>d</text>
<text><location><page_1><loc_62><loc_76><loc_63><loc_77></location>' e</text>
<text><location><page_1><loc_63><loc_76><loc_89><loc_77></location>ric Joliot-Curie, 13388 Marseille, cedex 13, France</text>
<text><location><page_1><loc_26><loc_75><loc_74><loc_76></location>5 University of the Western Cape, Private Bag X17, 7535, Bellville, Cape Town, South Africa</text>
<text><location><page_1><loc_26><loc_73><loc_73><loc_75></location>6 Department of Physics and Astronomy, University of California, Los Angeles, CA 90024</text>
<text><location><page_1><loc_21><loc_71><loc_79><loc_72></location>(Received xxxx xx, xxxx; Revised xxxx xx, xxxx; Accepted xxxx xx, xxxx; Online published Xxxxx xx, 2008)</text>
<text><location><page_1><loc_10><loc_54><loc_90><loc_68></location>We present the results of the analysis of multiwavelength Spectral Energy Distributions (SEDs) of far-infrared galaxies detected in the AKARI Deep Field-South (ADF-S) Survey. The analysis uses a carefully selected sample of 186 sources detected at the 90 µ mAKARI band, identified as galaxies with cross-identification in public catalogues. For sources without known spectroscopic redshifts, we estimate photometric redshifts after a test of two independent methods: one based on using mainly the optical - mid infrared range, and one based on the whole range of ultraviolet - far infrared data. We observe a vast improvement in the estimation of photometric redshifts when far infrared data are included, compared with an approach based mainly on the optical - mid infrared range. We discuss the physical properties of our far-infrared-selected sample. We conclude that this sample consists mostly of rich in dust and young stars nearby galaxies, and, furthermore, that almost 25% of these sources are (Ultra)Luminous Infrared Galaxies. Average SEDs normalized at 90 µ m for normal galaxies (138 sources), LIRGs (30 sources), and ULIRGs (18 galaxies) a the significant shift in the peak wavelength of the dust emission, and an increasing ratio between their bolometric and dust luminosities which varies from 0.39 to 0.73.</text>
<text><location><page_1><loc_10><loc_52><loc_70><loc_53></location>Key words: galaxies, starburst galaxies, SED, spectral energy distribution, LIRGs, ULIRGs</text>
<section_header_level_1><location><page_1><loc_7><loc_48><loc_20><loc_49></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_7><loc_13><loc_49><loc_47></location>Star formation (SF) history holds the key to understanding galaxy evolution, and - from a wider perspective - the nature of our Universe. One of the obstacles to observing starburst regions in galaxies using optical telescopes is dust. On the other hand, protostars form from dust clouds and molecular gas. Most of the dust in galaxies is quite cool ( ∼ 10-20 K), and its emission is visible only in the far infrared (Glass, 1999). A warm dust component might be observable during starburst activity (Phillips, 2005). The newly born, blue massive stars are surrounded by gas and dust, which obscure the most interesting regions and, additionally, absorb a part of the ultraviolet (UV) light emitted by stars. Dust heated in this way re-emits the absorbed light in the infrared wavelengths, mostly in the far infrared. Even very careful observations in the UV and optical ranges of wavelengths cannot provide a detailed description of the SF processes in galaxies. The infrared (IR) emission, reflecting the dust-obscured SF activity of galaxies (Genzel & Cesarsky, 2000), combined together with UV and optical data, can give full information about the star formation history and rate. Additionally, the ratio between the UV and far infrared (FIR) emission may serve as an indicator of the dust attenuation in galaxies (e.g., Buat et al., 2005, Takeuchi et al., 2005, Noll et al., 2009). The first all-sky survey at IR was performed by the satellite IRAS ( The Infra-Red Astronom-</text>
<text><location><page_1><loc_51><loc_28><loc_93><loc_49></location>al Satellite , Neugebauer et al., 1984). IRAS mapped the sky at 12 µ m, 15 µ m, 60 µ m, and 100 µ m for 300 days and forever changed our view of the sky (Beichman, 1987). IRAS detected IR emission from about 350 000 astronomical sources, and the number of known astronomical objects went up by 70%. A large fraction of the extragalactic sources detected in the far infrared were spiral galaxies, quasars (QSOs), Seyfert galaxies, and early type galaxies (Genzel & Cesarsky, 2000), but also new classes of galaxies very bright in the IR, such as U LIRGs ( ultraluminous infrared galaxies), were found. IR satellite missions, focusing on selected areas, were followed by MSX ( Midcourse Space Experiment , e.g., Egan et al., 2003), ISO ( Infrared Space Observatory ,e.g., Verma et al., 2005; Genzel & Cesarsky, 2000), and SST ( The Spitzer Space Telescope , e.g., Soifer et al., 2008).</text>
<text><location><page_1><loc_51><loc_7><loc_93><loc_28></location>After more than 20 years, a Japanese spacecraft, AKARI ( akari means warm-light in Japanese), performed a new all-sky survey with a much higher angular resolution than IRAS (Murakami et al., 2007). AKARI has also provided deeper surveys centered on the North and South Ecliptic Poles (e.g., Wada et al., 2008, Takagi et al., 2012, Matsuura et al., 2011). In particular, with the Far-Infrared Surveyor (FIS, Kawada et al., 2007), observations in the four FIR bands were possible (65 µ m, 90 µ m, 140 µ m, and 160 µ m). Among the observed fields, the lowest Galactic cirrus emission density region near the South Ecliptic Pole was selected for deep observation, to provide the best FIR extragalactic image of the Universe. This field is referred to as the AKARI Deep Field South (ADF-S). This survey is unique in having a continuous wavelength coverage with four photometric bands and mapping</text>
<text><location><page_2><loc_7><loc_82><loc_49><loc_91></location>over a wide area (approximately 12 square degrees). In the ADFS, 2 263 infrared sources were detected down to ∼ 20 mJy at the 90 µ m band, and infrared colours for about 400 of these sources were measured. The first analysis of this sample in terms of the nature and properties for 1000 ADF-S objects brighter than 0.301 Jy in the 90 µ m band was presented by Małek et al. (2010).</text>
<text><location><page_2><loc_7><loc_62><loc_49><loc_82></location>In this work, we present a more sophisticated approach to the analysis than the previous analysis (Malek et al., 2010) of the ADF-S sources in multiwavelength studies, based on a sample of 545 identified galaxies. In Section 2, we present our data and sample selection. We discuss the spectroscopic redshifts distribution in Section 3, and a new approach to the estimation of photometric redshifts based on the Le PHARE and CIGALE codes in Section 4. Discussion of physical and statistical properties of the obtained SEDs is presented in Section 5. The basic properties of a sample of galaxies with known spectroscopic redshifts are shown in Section 6. A discussion of (U)LIRGs properties found in our sample is presented in Section 7. Section 8 presents our conclusions.</text>
<text><location><page_2><loc_7><loc_59><loc_49><loc_62></location>In all calculations in this paper we assume the flat model of the Universe, with Ω M = 0.3, Ω Λ = 0.7, and H 0 = 70 km s -1 Mpc -1 .</text>
<section_header_level_1><location><page_2><loc_7><loc_56><loc_14><loc_58></location>2. Data</section_header_level_1>
<text><location><page_2><loc_7><loc_50><loc_49><loc_56></location>The main aim of our work is to build a galaxy sample with high quality fluxes from the UV to the FIR using the ADF-S sample. Redshift information is also needed to obtain physical parameters from the SEDs.</text>
<text><location><page_2><loc_7><loc_28><loc_49><loc_50></location>Our sample is drawn from the AKARI ADF-S catalog presented by Małek et al. (2010), and published at the Centre de Donn'ees astronomiques de Strasbourg, Strasbourg astronomical Data Center (http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/514/A11). The sample consists of the 545 ADF-S sources from the so-called 6 σ catalog (S 90 µ m > 0.0301Jy, which corresponds to the 6 σ detection level) measured by the AKARI FIS detector, for which the optical counterparts were found in public catalogues (Małek et al., 2010). Additional measurements, mostly from WISE (Wright et al., 2010) and GALEX (Dale et al., 2007), and further information from public databases (SIMBAD http://simbad.ustrasbg.fr/simbad/, NED http://ned.ipac.caltech.edu/, and IRSA http://irsa.ipac.caltech.edu/) were used in our present analysis (see also Małek et al., 2013).</text>
<section_header_level_1><location><page_2><loc_7><loc_25><loc_37><loc_26></location>3. Spectroscopic redshifts distribution</section_header_level_1>
<text><location><page_2><loc_7><loc_13><loc_49><loc_24></location>In total (data coming from Małek et al., 2010; the dedicated spectroscopic measurements of selected ADF-S sources performed by Sedgwick et al., 2011; new data from the NED and SIMBAD databases) we have 173 galaxies with known spectroscopic redshifts ( z spectro ). The mean value of the redshift in this sample is equal to 0 . 08 ± 0 . 01 , with a median value of 0.06. This implies, together with AKARI FIR measurements, that our sample mainly consists of nearby FIR-bright galaxies.</text>
<text><location><page_2><loc_7><loc_5><loc_49><loc_12></location>Unfortunately, the majority of our sample with spectroscopic redshift information does not have enough photometric measurements to fit SED models with a high confidence. After a visual inspection of all the spectra, we decided to apply a more stricter selection criterion, and for the next procedure we have chosen</text>
<figure>
<location><page_2><loc_57><loc_77><loc_86><loc_91></location>
<caption>Fig. 1. A typical example of a spectral coverage for a source (here: 2MASX J04421266-5355520) within our sample. Each galaxy from our sample has a measurement in the FIR and optical parts of its spectrum. Nearly 50% of galaxies have additional detections in the UV and MIR.</caption>
</figure>
<figure>
<location><page_2><loc_53><loc_53><loc_89><loc_67></location>
<caption>Fig. 2. The spectroscopic redshift distribution N( z ) in 0.01 z bins. The open histogram corresponds to the distribution of 173 galaxies with known spectroscopic redshifts in our sample. The only one object with a redshift higher than 0.3 is not shown here (quasar HE 0435-5304, located at z=1.232). The filled histogram corresponds to the 95 galaxies used for Spectral Energy Distribution fitting.</caption>
</figure>
<text><location><page_2><loc_51><loc_23><loc_93><loc_39></location>only galaxies with at least six photometric measurements in the whole spectral range. Since our sample selection was based on the WIDE-S 90 µ m AKARI band, each galaxy has at least one measurement in the FIR, at 90 µ m. Additionally, 95% of sources are detected in the MIR, all galaxies have optical information, and half of them were also detected in the UV. The typical spectral coverage of a source from our sample is shown in Fig. 1. After this selection, 95 galaxies from the initial sample of 173 galaxies with a known spectroscopic redshift remained for the subsequent analysis. All the available measurements from the ADF-S database were used for the SED fitting.</text>
<text><location><page_2><loc_51><loc_20><loc_93><loc_23></location>The redshift distribution N ( z ) of the selected sample with spectroscopic redshifts is presented in Fig. 2.</text>
<section_header_level_1><location><page_2><loc_51><loc_17><loc_70><loc_19></location>4. Photometric redshifts</section_header_level_1>
<text><location><page_2><loc_51><loc_5><loc_93><loc_17></location>The information about spectroscopic redshifts is available for 173 galaxies among 545 galaxies identified by Małek et al. (2010). It implies that more than 68% of sources in our sample have an identification in public catalogues as galaxies with photometric data but no redshift. In order to analyse properties of all the identified galaxies, we decided to estimate the photometric redshifts ( z phot ) for all galaxies with at least 6 measurements in the whole spectral range (127 galaxies fulfill this condition).</text>
<text><location><page_3><loc_7><loc_81><loc_49><loc_91></location>We performed a test of the accuracy of the photometric redshift for the sample of 95 galaxies with known z spectro and at least 6 photometric measurements in the whole spectral range (see Fig. 1). As representative parameters describing the accuracy of our method we used the percentage of successfully estimated redshifts, and the percentage of catastrophic errors (hereafter CE), which meet the condition:</text>
<formula><location><page_3><loc_17><loc_77><loc_49><loc_80></location>CE := | z spectro -z photo | (1 + z spectro ) > 0 . 15 , (1)</formula>
<text><location><page_3><loc_7><loc_70><loc_49><loc_76></location>following Ilbert et al. (2006). The fraction of CE (hereafter η ) is defined as the ratio of galaxies for which CE occurred and all the galaxies in the sample. The redshift accuracy σ ∆z / (1+z) was measured using the normalized median absolute deviation:</text>
<formula><location><page_3><loc_15><loc_66><loc_49><loc_69></location>σ ∆z / (1+z) = 1 . 48 · median | ∆z | 1 + z spectro , (2)</formula>
<text><location><page_3><loc_7><loc_62><loc_49><loc_65></location>where ∆ z is the difference between spectroscopic and estimated photometric redshift.</text>
<text><location><page_3><loc_7><loc_59><loc_49><loc_62></location>To estimate the photometric redshifts we used two different codes:</text>
<text><location><page_3><loc_7><loc_38><loc_49><loc_59></location>- Le PHARE: Photometric Analysis for Redshift Estimate version 2.2 (Le PHARE 1 ; Arnouts et al., 1999, Ilbert et al., 2006) code. We tested all the available libraries, one by one, of galaxy SEDs included in the Le PHARE distribution. Using the sources with spectroscopic redshifts available as a test sample, we checked the performance of all the available libraries: the percentage of sources for which the photometric redshift estimation could be performed successfully, and the scatter between estimated photometric and 'real' spectroscopic values of redshifts. The best results were obtained for the Rieke library, which includes eleven SEDs of luminous star forming galaxies constructed by Rieke et al. (2009). This is not very surprising, since it is expected that most of our galaxies belong to this class of galaxies.</text>
<text><location><page_3><loc_7><loc_22><loc_49><loc_38></location>- CIGALE: Considering that a significant part of our data is in the FIR, which is not well covered by the templates included in the Le PHARE distribution, we decided to test a different approach, i.e. to use the Code Investigating GALaxy Emission 2 (CIGALE; Noll et al., 2009) SED fitting program as a tool for the estimation of photometric redshifts. CIGALE was not developed as a tool for the estimation of z phot but since it uses a large number of models covering the wide spectral range, including IR and FIR, it may be expected to provide a better z phot for our FIR-selected sample than the software that mainly uses optical to NIR data.</text>
<text><location><page_3><loc_7><loc_11><loc_49><loc_21></location>In the case of CIGALE, the values of parameters defined above were: η =9.47% and σ ∆z / (1+z) =0.06. We found that less than 10% of z CIGALE suffer from CE. CE occurred in the case of nine galaxies, and the mean value of | z spectro -z CIGALE | / (1 + z spectro ) for the CE subsample was equal to 0.22 ± 0.06, with minimum and maximum values of CE equal to 0.16 and 0.34, respectively.</text>
<text><location><page_3><loc_51><loc_67><loc_93><loc_91></location>Performing the same test with Le PHARE we obtained η =14.73% and σ ∆z / (1+z) =0.05. We were not able to estimate photometric redshifts for 11 galaxies, and, additionally, CE occurred for three galaxies. The mean value of | z spectro -z phot | / (1 + z spectro ) for the CE subsample was equal to 0.27 ± 0.02 (with minimum and maximum values of CE equal to 0.26 and 0.28, respectively). Thus, we conclude that the deviation from the real value for the successful measurements was lower in the case of Le PHARE; however, the percentage of successful measurements was higher when CIGALE was used. Thus, Le PHARE provides higher accuracy while CIGALE assures a higher success rate for our FIR-selected sample. We have also checked that the amplitude of CEs (both mean and median values) was smaller in the case of CIGALE. For example, the minimum value of CE for CIGALE equals 0.16 (two galaxies), and it is very close to the CEs boundary limit.</text>
<text><location><page_3><loc_51><loc_37><loc_93><loc_67></location>This result is most likely related to our sample selection (which consists of galaxies bright at 90 µ m), and its spectral coverage, in particular a small amount of optical and MIR measurements which are needed by Le PHARE to determine the galaxy properties (e.g. Balmer break) properly. The possible explanation of a better performance of CIGALE is the limited number of FIR templates used by Le PHARE. Additionally, according to Ilbert et al. (2006), Le PHARE has the best performance in the redshift range 0.2 ≤ z spectro ≤ 1.5, while our spectroscopic range (0 ≤ z spectro ≤ 0.25) lies outside this redshift range. For galaxies with spectroscopic redshifts lower than 0.2 Ilbert et al. (2006) have found a dramatic increase of the fraction of CE, caused by the mismatch between the Balmer break and the intergalactic Lyman-alpha forest depression at λ < 1216 ˚ A. Yet another reason for the lower percentage of redshifts measured successfully by Le Phare are the spectral types of galaxies which dominate in our sample: photometric redshifts of actively star forming galaxies are less reliable (the fraction of CE increases by a factor of ∼ 5 from the elliptical to the starburst galaxies for a sample used by Ilbert et al., 2006).</text>
<text><location><page_3><loc_51><loc_31><loc_93><loc_36></location>Consequently, for the subsequent analysis we decided to use z CIGALE for sources without a known z spectro . A more detailed comparison of the estimation of photometric redshifts by Le PHARE and CIGALE can be found in Małek et al. (2013).</text>
<section_header_level_1><location><page_3><loc_51><loc_28><loc_62><loc_29></location>5. SED fitting</section_header_level_1>
<text><location><page_3><loc_51><loc_16><loc_93><loc_27></location>To study the physical parameters of the ADF-S sources, we selected galaxies with a known z spectro or z CIGALE , and with the highest quality photometry available. The main selection criteria were to have redshift information, together with at least six measurements in the whole spectral range. As a result, we use 222 galaxies: 95 sources with z spectro and 127 galaxies with z CIGALE . All available photometric measurements for these galaxies were used for the SED fitting with CIGALE.</text>
<text><location><page_3><loc_51><loc_5><loc_93><loc_15></location>CIGALE uses models describing the emission from a galaxy in the wavelength range from the rest-frame far-UV to the restframe far-IR (Noll et al., 2011). The code derives physical parameters of galaxies by fitting their spectral energy distributions (SEDs) to SEDs based on models and templates. CIGALE takes into account both the dust UV attenuation and IR emission. Based on possible values for each physical parameter related to star</text>
<figure>
<location><page_4><loc_53><loc_75><loc_89><loc_90></location>
<caption>Fig. 4. The distribution of redshifts in our final sample (both spectroscopic and estimated by CIGALE).</caption>
</figure>
<text><location><page_4><loc_51><loc_66><loc_86><loc_67></location>population history (SFH 1 ) with exponential decrease:</text>
<formula><location><page_4><loc_64><loc_61><loc_93><loc_64></location>SFR 1 = M gal τ 1 (e 1 /τ 1 -1) (3)</formula>
<text><location><page_4><loc_51><loc_51><loc_93><loc_60></location>In our work we have adopted the box model for the young stellar population history (SFH 2 ), with constant star formation over a limited period of time, starting from 0.0025 to 1 Gyr ago (parameter t 2 in Table 1). In the case of a box model, the SFH 2 is computed as the galaxy mass divided by its age. Thus, CIGALE gives a total value of logSFR, defined as:</text>
<formula><location><page_4><loc_59><loc_48><loc_93><loc_50></location>SFR = (1 -f ySP )SFR 1 +f ySP SFR 2 , (4)</formula>
<text><location><page_4><loc_51><loc_46><loc_88><loc_47></location>where f ySP is the fraction of the young stellar population.</text>
<text><location><page_4><loc_52><loc_44><loc_92><loc_45></location>The list of input parameters of CIGALE is shown in Table 1.</text>
<text><location><page_4><loc_51><loc_28><loc_93><loc_44></location>The reliability of the retrieved parameters for galaxies with known spectroscopic redshifts was checked using the mock catalogue of artificial galaxies (Małek et al., 2013). The comparison between the results from the mock and real catalogues shows that CIGALE gives a very good estimation of stellar masses, star formation rates, ages sensitivity D4000 index, dust attenuations and dust emissions, bolometric and dust luminosities (with values of the linear Pearson moment correlation coefficient, r , higher than 0.8). The accuracy of the relation between the dust mass and the heating intensity, α SED , is estimated with a lower efficiency ( r =0.55).</text>
<section_header_level_1><location><page_4><loc_51><loc_25><loc_82><loc_26></location>6. Physical properties of ADF-S sample</section_header_level_1>
<text><location><page_4><loc_51><loc_22><loc_93><loc_24></location>We restrict the further analysis to the SEDs with a minimum value of χ 2 lower than four.</text>
<text><location><page_4><loc_51><loc_13><loc_93><loc_21></location>This condition was met by 186 galaxies (73 galaxies with z spectro and 113 sources with an estimated z CIGALE ). Consequently, we assume that SEDs were successfully fitted only for this final sample. The redshift distribution of this sample is shown in Fig. 4. Examples of the best fit models obtained from CIGALE are given in Fig. 3.</text>
<text><location><page_4><loc_51><loc_5><loc_93><loc_12></location>The distribution of the main parameters (estimated with the Bayesian analysis for 186 galaxies in our final sample) is plotted in Fig. 5. We found that galaxies in our sample are typically very massive, with a mean value of M star = 10 . 48 ± 0 . 19 · 10 10 [M /circledot ] . Moreover, these galaxies are rather luminous</text>
<figure>
<location><page_4><loc_12><loc_42><loc_43><loc_58></location>
<caption>Fig. 3. Three examples of the best-fit models for the normal galaxy (upper panel), LIRG (middle panel) and ULIRG (bottom panel) sources. Solid lines correspond to the best model obtained from CIGALE code, and the full black circles - observed data used for SED fitting.</caption>
</figure>
<text><location><page_4><loc_7><loc_23><loc_49><loc_30></location>formation history, dust attenuation, and dust emission, CIGALE computes all possible spectra and derives mean fluxes in the observed filters. For each galaxy, the best value for each parameter, as well as the best fitted model, is found using a Bayesian-like statistical analysis (Roehlly et al., 2012).</text>
<text><location><page_4><loc_7><loc_10><loc_49><loc_23></location>Models of stellar emission are given either by Maraston (2005) or Fioc & Rocca-Volmerange (1997). The absorption and scattering of stellar light by dust, the so-called attenuation curves for galaxies, are given by Calzetti et al. (2000). Dust emission is characterized by a power-law model proposed by Dale and Helou (2002), with the slope α SED of the relation between the dust mass and the heating intensity. This is the only dust emission model included in the newest CIGALE distribution (CIGALE version 2013/01/02).</text>
<text><location><page_4><loc_7><loc_5><loc_49><loc_9></location>To reconstruct more accurately Star Formation Rates (SFRs), CIGALE uses the single stellar population of Maraston (2005). For the old stellar population, CIGALE calculates old stellar</text>
<table>
<location><page_5><loc_12><loc_62><loc_88><loc_89></location>
<caption>Table 1. List of the input parameters of CIGALE, based mostly on Buat et al. (2011).</caption>
</table>
<figure>
<location><page_5><loc_11><loc_37><loc_87><loc_59></location>
<caption>Fig. 5. Distribution of the Bayesian estimates of the output parameters: D4000, M star , and logSFR . SSFR values were calculated as the ratio of SFR and M star . L UV , and L TIR were calculated as an integral value of SEDs from 1480 · (1+z) to 1520 · (1+z) ˚ A, and from 8 · (1+z) µ m to 1 · (1+z) mm, respectively.</caption>
</figure>
<text><location><page_5><loc_7><loc_8><loc_49><loc_30></location>L bol = 10 . 81 ± 0 . 93 · 10 10 [L /circledot ] , and also their dust luminosity is high L dust = 10 . 38 ± 1 . 01 · 10 10 [L /circledot ] , but without a precisely defined maximum. A median value of the star formation rate parameter, SFR, is equal to 2.22 [M /circledot yr -1 ] . The estimated value of the heating intensity α SED (Dale & Helou, 2002) implies that the vast majority of analysed galaxies (85.48%) belong to a normal, star-forming galaxy population, with median the value of α SED equal to 2.01. The median value of the A V parameter, describing the effective dust attenuation for the stellar population at a wavelength equal to 5500 ˚ A is 0.47 [mag], and the median value for the attenuation in the FUV (at 1500 ˚ A, A FUV ) is 1.87 [mag]. The parameter A V , ySP , which describes the V-band attenuation for the young stellar population model, spreads almost across the entire range of input parameters from 0.15 to 2.19, with the median value 0.97 [mag].</text>
<section_header_level_1><location><page_5><loc_51><loc_29><loc_83><loc_30></location>7. Average Spectral Energy Distributions</section_header_level_1>
<text><location><page_5><loc_51><loc_21><loc_93><loc_28></location>Using the CIGALE output, we created average SEDs from our 186 galaxies. First, we normalized all SEDs at a rest frame 90 µ m; then we divided them into 3 broad categories: Ultraluminous Infrared Galaxies (ULIRGs), Luminous Infrared Galaxies (LIRGs), and the remaining galaxies.</text>
<text><location><page_5><loc_51><loc_9><loc_93><loc_21></location>Following Sanders & Mirabel (1996), we define ULIRGs as galaxies with a very high IR luminosity, L TIR > 10 12 L /circledot , where L TIR is the total mid- and far-infrared luminosity calculated in the range between 8(1+z) µ mand 1(1+ z ) mm. Sources with less extreme, but still high, IR luminosities 10 11 L /circledot < L TIR < 10 12 L /circledot are classified as LIRGs. In our sample, we found 18 ULIRGs (9.7% of analysed ADF-S sources) and 30 LIRGs (16.1% of the total number of sources).</text>
<text><location><page_5><loc_51><loc_6><loc_93><loc_9></location>Average SEDs for ULIRGs, LIRGs and the remaining galaxies are plotted in Fig. 6.</text>
<figure>
<location><page_6><loc_9><loc_70><loc_45><loc_90></location>
<caption>Fig. 6. The average SEDs, normalized at 90 µ m, of ULIRGs (dashed line), LIRGs (dotted line) and the remaining galaxies (solid line). SEDs were shifted to the rest frame.</caption>
</figure>
<text><location><page_6><loc_7><loc_40><loc_49><loc_61></location>The ratio of bolometric to total mid- and far-infrared luminosity (the integrated luminosities calculated from CIGALE) is higher for the ULIRGs and LIRGs than for the normal galaxies. In the case of the average SEDs, the L bol / L TIR ratio is equal to 0.73 ± 0.16 for the ULIRGs, 0.55 ± 0.16 for the LIRGs, and 0.39 ± 0.22 for the remaining galaxies. Both ULIRGs and LIRGs in our sample contain dust which is cooler than the dust in the remaining galaxies, which can be seen as a shift of the maximal values of the dust components towards longer wavelengths. The brighter the sample is in the IR, the more shifted is the dust peak towards the longer wavelengths. In our sample, the maximum of the dust peak in the spectra normalized to 90 µ m is located at 1.38 · 10 6 ˚ A, 1.21 · 10 6 ˚ A, and 8.65 · 10 5 ˚ A for the ULIRGs, LIRGs and normal galaxies, respectively.</text>
<text><location><page_6><loc_7><loc_25><loc_49><loc_40></location>The median redshift for ULIRGs in our sample is equal to 0.54. The median redshift for LIRGs, and the remaining galaxies was found to be 0.2, and 0.04, respectively. This difference in redshifts is a selection effect, related to the fact that the primary detection limit is in the FIR. The two parameters SFR and L TIR might be correlated because both of them depend on the galaxy mass, but they are estimated independently. For the ULIRGs, the logSFR is very high, with a mean value 2.59 ± 0.32 [M /circledot yr -1 ] . The SFR for the LIRGs is more than ten times lower (the mean logSFR is equal to 1.47 ± 0.28 [M /circledot yr -1 ] ).</text>
<text><location><page_6><loc_7><loc_16><loc_49><loc_24></location>The SFR for the normal galaxies is much lower: we found the logSFR on the level of -0.06 ± 0.62 [M /circledot yr -1 ] . Comparing our results to the sample of normal, nearby galaxies from SINGS (Kennicutt et al., 2003) we found a similar range of SFR: 0-12 [M /circledot yr -1 ] for the ADF-S, and 0-15 [M /circledot yr -1 ] for the SINGS sample.</text>
<text><location><page_6><loc_7><loc_5><loc_49><loc_15></location>Comparing the results obtained by U et al. (2012) for a sample of 53 LIRGs and 11 ULIRGs in the similar redshift range (z between 0.012 and 0.083) from the Great Observatories All-sky LIRG Survey (GOALS), we found a very good agreement of SFR for LIRGs (logSFR LIRGs = 1.57 ± 0.19 [M /circledot yr -1 ] ). The mean value of logSFR ULIRGs in the case of our ADF-S sample is higher than the one found by U et al. (2012) and equals</text>
<text><location><page_6><loc_51><loc_88><loc_93><loc_91></location>logSFR ULIRGs = 2.25 ± 0.16 M /circledot yr -1 , but is still consistent within the error bars.</text>
<text><location><page_6><loc_51><loc_64><loc_93><loc_88></location>The LIRGs in the ADF-S sample have a mean logM star at 11.15 ± 0.49 [M /circledot ] . This value is slightly higher than logM star for LIRGs reported by U et al. 2012 (10.75 ± 0.39 [M /circledot ] ), Giovannoli et al. 2011 (LIRGs sample, with logM star between 10 and 12, and with a peak at 10.8), and Melbourne et al. 2008 (logM star ∼ 10.5, based on a set 15 LIRGs, at redshift ∼ 0.8). Even though our result shows the highest value of logM star , it is still consistent with the values listed above within the error bars. Stellar masses for the ULIRG sample have a mean logM star equal to 11.43 ± 0.31 [M /circledot ] . The mean stellar masses for 11 ULIRGs presented by U et al. (2012) were calculated as logM star =11.00 ± 0.40 [M /circledot ] . The sample of ULIRGs analysed by Howell et al. (2010) is characterized by a mean stellar mass logM star =11.24 ± 0.25 [M /circledot ] . We conclude that the (U)LIRGs in our ADF-S sample are slightly more massive than those in the samples used in previous works, but still all the samples are statistically consistent.</text>
<text><location><page_6><loc_51><loc_41><loc_93><loc_64></location>Figure 7 shows the relation between the SFRs and stellar masses for (U)LIRGs in our ADF-S catalogue. The mean redshift of the (U)LIRG sample is equal to 0.34 (minimum and maximum redshifts are equal to 0.09 and 0.98, respectively - then, the redshift distribution is quite broad). We compared our results with observations at redshifts 0 and 1 (Elbaz et al., 2007), and 2 (Daddi, et al., 2007), and also with other data mentioned above: a sample of nearby (z < 0.032) (U)LIRGs (U et al., 2012), and a sample of LIRGs observed in the GOALS by GALEX and the Spitzer Space Telescope (Howell et al., 2010). We found a rather flat distribution of the SFR parameter in the stellar mass space (similar to Giovannoli et al., 2011, who found for a sample of LIRGs in the Extended Chandra Deep Field South at z = 0.7, selected at 24 µ m by Spitzer). Our results, shown in Fig. 7, confirm a strong correlation between SFR and redshift.</text>
<text><location><page_6><loc_51><loc_17><loc_93><loc_41></location>Based on the physical properties obtained from SED fitting, we computed the Specific Star Formation Rates (SSFR [yr -1 ]; defined as the ratio of SFR and stellar mass) for the ADF-S sample. The SSFR rate is commonly used to analyse star formation history. For ULIRGs, LIRGs, and the rest of our sample, the logarithmic values of the SSFR parameter are equal to -9.00 ± 0.55, -9.68 ± 0.59, and -10.28 ± 0.57, respectively. Howell et al. (2010), and U. et al. (2012) presented the mean and median values of the SSFR for (U)LIRGs sample, without an additional separation. The median value of SSFR for galaxies in the GOALS field (Howell et al., 2010) is equal to -9.41 [yr -1 ], while U et al. (2012) obtained the value of -9.17 [yr -1 ]. The same parameter computed for LIRGs and ULIRGs together in our ADF-S sample is equal to -9.51 [yr -1 ]. We conclude that our results are consistent with the results mentioned in all the other works usedconsidered for comparison.</text>
<text><location><page_6><loc_51><loc_8><loc_93><loc_17></location>However, the dust emission power-law model, given by Dale and Helou (2002) implemented in CIGALE may not be efficient enough to describe the properties of the (U)LIRG sample well. In the future we plan to apply other models (e.g. Siebenmorgen & Krugel, 2007, Chary & Elbaz, 2001, Casey, 2012) to the same (U)LIRGs sample.</text>
<figure>
<location><page_7><loc_10><loc_69><loc_45><loc_90></location>
<caption>Fig. 7. The SFR vs stellar mass relation for ADF-S (U)LIRG sample (black circles). The variable size of points represents the difference in redshift (increasing size from z equal to 0.09 to 0.98). We overplot the Howell et al., (2010) GOALS sample (open blue triangles), and U et al. (2012) (U)LIRG sample (red open squares). The two solid black lines correspond to the SFR-Mstar observation from Elbaz et al. (2007). The violet dashed line indicates the SFR - stellar mass relation for star-forming galaxies at z ∼ 2 defined by Daddi et al. (2007).</caption>
</figure>
<section_header_level_1><location><page_7><loc_7><loc_53><loc_19><loc_54></location>8. Conclusions</section_header_level_1>
<unordered_list>
<list_item><location><page_7><loc_9><loc_48><loc_49><loc_53></location>· For our analysis we used an AKARI ADF-S catalogue (Małek et al., 2010), with additional information about spectroscopic redshifts (Małek et al., 2013)</list_item>
<list_item><location><page_7><loc_9><loc_32><loc_49><loc_47></location>· The CIGALE (Noll et al., 2009), program for fitting SEDs, was used for the first time to estimate photometric redshifts for galaxies without known spectroscopic redshifts (127 galaxies), and then to fit Spectral Energy Distribution models to our whole sample. We conclude that although CIGALE was not designed as a tool for the estimation of z phot , in the case of the ADF-S galaxies it is more efficient than the standard code (Le PHARE), based on the UV and optical data. This observation might be worth taking into account for IR-selected galaxies in the future analysis.</list_item>
<list_item><location><page_7><loc_9><loc_24><loc_49><loc_31></location>· We used CIGALE for the redshift estimation for 127 galaxies with at least 6 photometric measurements in the whole spectral range, from UV to FIR. A satisfactory value of the χ 2 parameter for the fit, lower than 4, was obtained for 186 galaxies.</list_item>
<list_item><location><page_7><loc_9><loc_9><loc_49><loc_23></location>· Based on the physical parameters obtained from SED fitting, we conclude that our FIR selected sample consists mostly of nearby, massive galaxies, bright in the IR, and active in SF (a similar conclusion was found by White et al., 2012). The distribution of the SFR shows that our sample is characterized by a rather high star formation rate, with a the median value equal to 1.96 and 2.56 [ M /circledot yr -1 ], for galaxies with spectroscopic and estimated photometric redshift, respectively,</list_item>
<list_item><location><page_7><loc_9><loc_5><loc_49><loc_8></location>· Average SEDs for ULIRGs, LIRGs and remaining galaxies from our sample, normalized at 90 µ mwere created. Almost</list_item>
</unordered_list>
<text><location><page_7><loc_54><loc_84><loc_93><loc_91></location>25% of our sample are (U)LIRGs, rich in dust and active in star formation processes. For these galaxies we noticed a significant shift in the peak wavelength of the dust emission in the FIR and a different ratio between luminosities in the optical and IR parts of the spectra.</text>
<text><location><page_7><loc_51><loc_76><loc_93><loc_81></location>Acknowledgments. We would like to thank both anonymous Reviewers for their very constructive comments and suggestions which helped to improve the quality of this paper. We thank Olivier Ilbert for useful discussions and kind help in using Le PHARE.</text>
<text><location><page_7><loc_51><loc_51><loc_93><loc_76></location>This work is based on observations with AKARI a JAXA project with the participation of ESA. This research has made use of the SIMBAD and NED databases. KM, AP and AK were financed by the research grant of the Polish Ministry of Science N N203 512938. The collaboration between French and Polish participants was partially supported by the European Associated Laboratory Astrophysics Poland-France HECOLS. This research was partially supported by the project POLISH-SWISS ASTRO PROJECT co-financed by a grant from Switzerland through the Swiss Contribution to the enlarged European Union. KM has been supported from the Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship for Foreign Researchers, P11802. TTT has been supported by the Grant-in-Aid for the Scientific Research Fund (20740105, 23340046, and 24111707) and for the Global COE Program Request for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. VB and DB have been supported by the Centre National des Etudes Spatiales (CNES) and the Programme National Galaxies (PNG). MM acknowledges support from NASA grants NNX08AU59G and NNX09AM45G for analysis of the GALEX data in the Akari Deep Fields.</text>
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</document>
[
{
"title": "ABSTRACT",
"content": "4",
"pages": [
1
]
},
{
"title": "Dusty Universe viewed by AKARI far infrared detector",
"content": "K. Małek 1 , A. Pollo 2 , 3 , T. T. Takeuchi 1 , E. Giovannoli 4 , 5 , V. Buat 4 , D. Burgarella 4 , M. Malkan 6 , and A. Kurek 3 1 Department of Particle and Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, 464-8602 Nagoya, Japan 2 National Centre for Nuclear Research, ul. Ho˙za 69, 00-681 Warszawa, Poland 3 The Astronomical Observatory of the Jagiellonian University, ul. Orla 171, 30-244 Krak'ow, Poland Laboratoire d'Astrophysique de Marseille, OAMP, Universit ' e Aix-Marseille, CNRS, 38 rue Fr ' e d ' e ric Joliot-Curie, 13388 Marseille, cedex 13, France 5 University of the Western Cape, Private Bag X17, 7535, Bellville, Cape Town, South Africa 6 Department of Physics and Astronomy, University of California, Los Angeles, CA 90024 (Received xxxx xx, xxxx; Revised xxxx xx, xxxx; Accepted xxxx xx, xxxx; Online published Xxxxx xx, 2008) We present the results of the analysis of multiwavelength Spectral Energy Distributions (SEDs) of far-infrared galaxies detected in the AKARI Deep Field-South (ADF-S) Survey. The analysis uses a carefully selected sample of 186 sources detected at the 90 µ mAKARI band, identified as galaxies with cross-identification in public catalogues. For sources without known spectroscopic redshifts, we estimate photometric redshifts after a test of two independent methods: one based on using mainly the optical - mid infrared range, and one based on the whole range of ultraviolet - far infrared data. We observe a vast improvement in the estimation of photometric redshifts when far infrared data are included, compared with an approach based mainly on the optical - mid infrared range. We discuss the physical properties of our far-infrared-selected sample. We conclude that this sample consists mostly of rich in dust and young stars nearby galaxies, and, furthermore, that almost 25% of these sources are (Ultra)Luminous Infrared Galaxies. Average SEDs normalized at 90 µ m for normal galaxies (138 sources), LIRGs (30 sources), and ULIRGs (18 galaxies) a the significant shift in the peak wavelength of the dust emission, and an increasing ratio between their bolometric and dust luminosities which varies from 0.39 to 0.73. Key words: galaxies, starburst galaxies, SED, spectral energy distribution, LIRGs, ULIRGs",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Star formation (SF) history holds the key to understanding galaxy evolution, and - from a wider perspective - the nature of our Universe. One of the obstacles to observing starburst regions in galaxies using optical telescopes is dust. On the other hand, protostars form from dust clouds and molecular gas. Most of the dust in galaxies is quite cool ( ∼ 10-20 K), and its emission is visible only in the far infrared (Glass, 1999). A warm dust component might be observable during starburst activity (Phillips, 2005). The newly born, blue massive stars are surrounded by gas and dust, which obscure the most interesting regions and, additionally, absorb a part of the ultraviolet (UV) light emitted by stars. Dust heated in this way re-emits the absorbed light in the infrared wavelengths, mostly in the far infrared. Even very careful observations in the UV and optical ranges of wavelengths cannot provide a detailed description of the SF processes in galaxies. The infrared (IR) emission, reflecting the dust-obscured SF activity of galaxies (Genzel & Cesarsky, 2000), combined together with UV and optical data, can give full information about the star formation history and rate. Additionally, the ratio between the UV and far infrared (FIR) emission may serve as an indicator of the dust attenuation in galaxies (e.g., Buat et al., 2005, Takeuchi et al., 2005, Noll et al., 2009). The first all-sky survey at IR was performed by the satellite IRAS ( The Infra-Red Astronom- al Satellite , Neugebauer et al., 1984). IRAS mapped the sky at 12 µ m, 15 µ m, 60 µ m, and 100 µ m for 300 days and forever changed our view of the sky (Beichman, 1987). IRAS detected IR emission from about 350 000 astronomical sources, and the number of known astronomical objects went up by 70%. A large fraction of the extragalactic sources detected in the far infrared were spiral galaxies, quasars (QSOs), Seyfert galaxies, and early type galaxies (Genzel & Cesarsky, 2000), but also new classes of galaxies very bright in the IR, such as U LIRGs ( ultraluminous infrared galaxies), were found. IR satellite missions, focusing on selected areas, were followed by MSX ( Midcourse Space Experiment , e.g., Egan et al., 2003), ISO ( Infrared Space Observatory ,e.g., Verma et al., 2005; Genzel & Cesarsky, 2000), and SST ( The Spitzer Space Telescope , e.g., Soifer et al., 2008). After more than 20 years, a Japanese spacecraft, AKARI ( akari means warm-light in Japanese), performed a new all-sky survey with a much higher angular resolution than IRAS (Murakami et al., 2007). AKARI has also provided deeper surveys centered on the North and South Ecliptic Poles (e.g., Wada et al., 2008, Takagi et al., 2012, Matsuura et al., 2011). In particular, with the Far-Infrared Surveyor (FIS, Kawada et al., 2007), observations in the four FIR bands were possible (65 µ m, 90 µ m, 140 µ m, and 160 µ m). Among the observed fields, the lowest Galactic cirrus emission density region near the South Ecliptic Pole was selected for deep observation, to provide the best FIR extragalactic image of the Universe. This field is referred to as the AKARI Deep Field South (ADF-S). This survey is unique in having a continuous wavelength coverage with four photometric bands and mapping over a wide area (approximately 12 square degrees). In the ADFS, 2 263 infrared sources were detected down to ∼ 20 mJy at the 90 µ m band, and infrared colours for about 400 of these sources were measured. The first analysis of this sample in terms of the nature and properties for 1000 ADF-S objects brighter than 0.301 Jy in the 90 µ m band was presented by Małek et al. (2010). In this work, we present a more sophisticated approach to the analysis than the previous analysis (Malek et al., 2010) of the ADF-S sources in multiwavelength studies, based on a sample of 545 identified galaxies. In Section 2, we present our data and sample selection. We discuss the spectroscopic redshifts distribution in Section 3, and a new approach to the estimation of photometric redshifts based on the Le PHARE and CIGALE codes in Section 4. Discussion of physical and statistical properties of the obtained SEDs is presented in Section 5. The basic properties of a sample of galaxies with known spectroscopic redshifts are shown in Section 6. A discussion of (U)LIRGs properties found in our sample is presented in Section 7. Section 8 presents our conclusions. In all calculations in this paper we assume the flat model of the Universe, with Ω M = 0.3, Ω Λ = 0.7, and H 0 = 70 km s -1 Mpc -1 .",
"pages": [
1,
2
]
},
{
"title": "2. Data",
"content": "The main aim of our work is to build a galaxy sample with high quality fluxes from the UV to the FIR using the ADF-S sample. Redshift information is also needed to obtain physical parameters from the SEDs. Our sample is drawn from the AKARI ADF-S catalog presented by Małek et al. (2010), and published at the Centre de Donn'ees astronomiques de Strasbourg, Strasbourg astronomical Data Center (http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/514/A11). The sample consists of the 545 ADF-S sources from the so-called 6 σ catalog (S 90 µ m > 0.0301Jy, which corresponds to the 6 σ detection level) measured by the AKARI FIS detector, for which the optical counterparts were found in public catalogues (Małek et al., 2010). Additional measurements, mostly from WISE (Wright et al., 2010) and GALEX (Dale et al., 2007), and further information from public databases (SIMBAD http://simbad.ustrasbg.fr/simbad/, NED http://ned.ipac.caltech.edu/, and IRSA http://irsa.ipac.caltech.edu/) were used in our present analysis (see also Małek et al., 2013).",
"pages": [
2
]
},
{
"title": "3. Spectroscopic redshifts distribution",
"content": "In total (data coming from Małek et al., 2010; the dedicated spectroscopic measurements of selected ADF-S sources performed by Sedgwick et al., 2011; new data from the NED and SIMBAD databases) we have 173 galaxies with known spectroscopic redshifts ( z spectro ). The mean value of the redshift in this sample is equal to 0 . 08 ± 0 . 01 , with a median value of 0.06. This implies, together with AKARI FIR measurements, that our sample mainly consists of nearby FIR-bright galaxies. Unfortunately, the majority of our sample with spectroscopic redshift information does not have enough photometric measurements to fit SED models with a high confidence. After a visual inspection of all the spectra, we decided to apply a more stricter selection criterion, and for the next procedure we have chosen only galaxies with at least six photometric measurements in the whole spectral range. Since our sample selection was based on the WIDE-S 90 µ m AKARI band, each galaxy has at least one measurement in the FIR, at 90 µ m. Additionally, 95% of sources are detected in the MIR, all galaxies have optical information, and half of them were also detected in the UV. The typical spectral coverage of a source from our sample is shown in Fig. 1. After this selection, 95 galaxies from the initial sample of 173 galaxies with a known spectroscopic redshift remained for the subsequent analysis. All the available measurements from the ADF-S database were used for the SED fitting. The redshift distribution N ( z ) of the selected sample with spectroscopic redshifts is presented in Fig. 2.",
"pages": [
2
]
},
{
"title": "4. Photometric redshifts",
"content": "The information about spectroscopic redshifts is available for 173 galaxies among 545 galaxies identified by Małek et al. (2010). It implies that more than 68% of sources in our sample have an identification in public catalogues as galaxies with photometric data but no redshift. In order to analyse properties of all the identified galaxies, we decided to estimate the photometric redshifts ( z phot ) for all galaxies with at least 6 measurements in the whole spectral range (127 galaxies fulfill this condition). We performed a test of the accuracy of the photometric redshift for the sample of 95 galaxies with known z spectro and at least 6 photometric measurements in the whole spectral range (see Fig. 1). As representative parameters describing the accuracy of our method we used the percentage of successfully estimated redshifts, and the percentage of catastrophic errors (hereafter CE), which meet the condition: following Ilbert et al. (2006). The fraction of CE (hereafter η ) is defined as the ratio of galaxies for which CE occurred and all the galaxies in the sample. The redshift accuracy σ ∆z / (1+z) was measured using the normalized median absolute deviation: where ∆ z is the difference between spectroscopic and estimated photometric redshift. To estimate the photometric redshifts we used two different codes: - Le PHARE: Photometric Analysis for Redshift Estimate version 2.2 (Le PHARE 1 ; Arnouts et al., 1999, Ilbert et al., 2006) code. We tested all the available libraries, one by one, of galaxy SEDs included in the Le PHARE distribution. Using the sources with spectroscopic redshifts available as a test sample, we checked the performance of all the available libraries: the percentage of sources for which the photometric redshift estimation could be performed successfully, and the scatter between estimated photometric and 'real' spectroscopic values of redshifts. The best results were obtained for the Rieke library, which includes eleven SEDs of luminous star forming galaxies constructed by Rieke et al. (2009). This is not very surprising, since it is expected that most of our galaxies belong to this class of galaxies. - CIGALE: Considering that a significant part of our data is in the FIR, which is not well covered by the templates included in the Le PHARE distribution, we decided to test a different approach, i.e. to use the Code Investigating GALaxy Emission 2 (CIGALE; Noll et al., 2009) SED fitting program as a tool for the estimation of photometric redshifts. CIGALE was not developed as a tool for the estimation of z phot but since it uses a large number of models covering the wide spectral range, including IR and FIR, it may be expected to provide a better z phot for our FIR-selected sample than the software that mainly uses optical to NIR data. In the case of CIGALE, the values of parameters defined above were: η =9.47% and σ ∆z / (1+z) =0.06. We found that less than 10% of z CIGALE suffer from CE. CE occurred in the case of nine galaxies, and the mean value of | z spectro -z CIGALE | / (1 + z spectro ) for the CE subsample was equal to 0.22 ± 0.06, with minimum and maximum values of CE equal to 0.16 and 0.34, respectively. Performing the same test with Le PHARE we obtained η =14.73% and σ ∆z / (1+z) =0.05. We were not able to estimate photometric redshifts for 11 galaxies, and, additionally, CE occurred for three galaxies. The mean value of | z spectro -z phot | / (1 + z spectro ) for the CE subsample was equal to 0.27 ± 0.02 (with minimum and maximum values of CE equal to 0.26 and 0.28, respectively). Thus, we conclude that the deviation from the real value for the successful measurements was lower in the case of Le PHARE; however, the percentage of successful measurements was higher when CIGALE was used. Thus, Le PHARE provides higher accuracy while CIGALE assures a higher success rate for our FIR-selected sample. We have also checked that the amplitude of CEs (both mean and median values) was smaller in the case of CIGALE. For example, the minimum value of CE for CIGALE equals 0.16 (two galaxies), and it is very close to the CEs boundary limit. This result is most likely related to our sample selection (which consists of galaxies bright at 90 µ m), and its spectral coverage, in particular a small amount of optical and MIR measurements which are needed by Le PHARE to determine the galaxy properties (e.g. Balmer break) properly. The possible explanation of a better performance of CIGALE is the limited number of FIR templates used by Le PHARE. Additionally, according to Ilbert et al. (2006), Le PHARE has the best performance in the redshift range 0.2 ≤ z spectro ≤ 1.5, while our spectroscopic range (0 ≤ z spectro ≤ 0.25) lies outside this redshift range. For galaxies with spectroscopic redshifts lower than 0.2 Ilbert et al. (2006) have found a dramatic increase of the fraction of CE, caused by the mismatch between the Balmer break and the intergalactic Lyman-alpha forest depression at λ < 1216 ˚ A. Yet another reason for the lower percentage of redshifts measured successfully by Le Phare are the spectral types of galaxies which dominate in our sample: photometric redshifts of actively star forming galaxies are less reliable (the fraction of CE increases by a factor of ∼ 5 from the elliptical to the starburst galaxies for a sample used by Ilbert et al., 2006). Consequently, for the subsequent analysis we decided to use z CIGALE for sources without a known z spectro . A more detailed comparison of the estimation of photometric redshifts by Le PHARE and CIGALE can be found in Małek et al. (2013).",
"pages": [
2,
3
]
},
{
"title": "5. SED fitting",
"content": "To study the physical parameters of the ADF-S sources, we selected galaxies with a known z spectro or z CIGALE , and with the highest quality photometry available. The main selection criteria were to have redshift information, together with at least six measurements in the whole spectral range. As a result, we use 222 galaxies: 95 sources with z spectro and 127 galaxies with z CIGALE . All available photometric measurements for these galaxies were used for the SED fitting with CIGALE. CIGALE uses models describing the emission from a galaxy in the wavelength range from the rest-frame far-UV to the restframe far-IR (Noll et al., 2011). The code derives physical parameters of galaxies by fitting their spectral energy distributions (SEDs) to SEDs based on models and templates. CIGALE takes into account both the dust UV attenuation and IR emission. Based on possible values for each physical parameter related to star population history (SFH 1 ) with exponential decrease: In our work we have adopted the box model for the young stellar population history (SFH 2 ), with constant star formation over a limited period of time, starting from 0.0025 to 1 Gyr ago (parameter t 2 in Table 1). In the case of a box model, the SFH 2 is computed as the galaxy mass divided by its age. Thus, CIGALE gives a total value of logSFR, defined as: where f ySP is the fraction of the young stellar population. The list of input parameters of CIGALE is shown in Table 1. The reliability of the retrieved parameters for galaxies with known spectroscopic redshifts was checked using the mock catalogue of artificial galaxies (Małek et al., 2013). The comparison between the results from the mock and real catalogues shows that CIGALE gives a very good estimation of stellar masses, star formation rates, ages sensitivity D4000 index, dust attenuations and dust emissions, bolometric and dust luminosities (with values of the linear Pearson moment correlation coefficient, r , higher than 0.8). The accuracy of the relation between the dust mass and the heating intensity, α SED , is estimated with a lower efficiency ( r =0.55).",
"pages": [
3,
4
]
},
{
"title": "6. Physical properties of ADF-S sample",
"content": "We restrict the further analysis to the SEDs with a minimum value of χ 2 lower than four. This condition was met by 186 galaxies (73 galaxies with z spectro and 113 sources with an estimated z CIGALE ). Consequently, we assume that SEDs were successfully fitted only for this final sample. The redshift distribution of this sample is shown in Fig. 4. Examples of the best fit models obtained from CIGALE are given in Fig. 3. The distribution of the main parameters (estimated with the Bayesian analysis for 186 galaxies in our final sample) is plotted in Fig. 5. We found that galaxies in our sample are typically very massive, with a mean value of M star = 10 . 48 ± 0 . 19 · 10 10 [M /circledot ] . Moreover, these galaxies are rather luminous formation history, dust attenuation, and dust emission, CIGALE computes all possible spectra and derives mean fluxes in the observed filters. For each galaxy, the best value for each parameter, as well as the best fitted model, is found using a Bayesian-like statistical analysis (Roehlly et al., 2012). Models of stellar emission are given either by Maraston (2005) or Fioc & Rocca-Volmerange (1997). The absorption and scattering of stellar light by dust, the so-called attenuation curves for galaxies, are given by Calzetti et al. (2000). Dust emission is characterized by a power-law model proposed by Dale and Helou (2002), with the slope α SED of the relation between the dust mass and the heating intensity. This is the only dust emission model included in the newest CIGALE distribution (CIGALE version 2013/01/02). To reconstruct more accurately Star Formation Rates (SFRs), CIGALE uses the single stellar population of Maraston (2005). For the old stellar population, CIGALE calculates old stellar L bol = 10 . 81 ± 0 . 93 · 10 10 [L /circledot ] , and also their dust luminosity is high L dust = 10 . 38 ± 1 . 01 · 10 10 [L /circledot ] , but without a precisely defined maximum. A median value of the star formation rate parameter, SFR, is equal to 2.22 [M /circledot yr -1 ] . The estimated value of the heating intensity α SED (Dale & Helou, 2002) implies that the vast majority of analysed galaxies (85.48%) belong to a normal, star-forming galaxy population, with median the value of α SED equal to 2.01. The median value of the A V parameter, describing the effective dust attenuation for the stellar population at a wavelength equal to 5500 ˚ A is 0.47 [mag], and the median value for the attenuation in the FUV (at 1500 ˚ A, A FUV ) is 1.87 [mag]. The parameter A V , ySP , which describes the V-band attenuation for the young stellar population model, spreads almost across the entire range of input parameters from 0.15 to 2.19, with the median value 0.97 [mag].",
"pages": [
4,
5
]
},
{
"title": "7. Average Spectral Energy Distributions",
"content": "Using the CIGALE output, we created average SEDs from our 186 galaxies. First, we normalized all SEDs at a rest frame 90 µ m; then we divided them into 3 broad categories: Ultraluminous Infrared Galaxies (ULIRGs), Luminous Infrared Galaxies (LIRGs), and the remaining galaxies. Following Sanders & Mirabel (1996), we define ULIRGs as galaxies with a very high IR luminosity, L TIR > 10 12 L /circledot , where L TIR is the total mid- and far-infrared luminosity calculated in the range between 8(1+z) µ mand 1(1+ z ) mm. Sources with less extreme, but still high, IR luminosities 10 11 L /circledot < L TIR < 10 12 L /circledot are classified as LIRGs. In our sample, we found 18 ULIRGs (9.7% of analysed ADF-S sources) and 30 LIRGs (16.1% of the total number of sources). Average SEDs for ULIRGs, LIRGs and the remaining galaxies are plotted in Fig. 6. The ratio of bolometric to total mid- and far-infrared luminosity (the integrated luminosities calculated from CIGALE) is higher for the ULIRGs and LIRGs than for the normal galaxies. In the case of the average SEDs, the L bol / L TIR ratio is equal to 0.73 ± 0.16 for the ULIRGs, 0.55 ± 0.16 for the LIRGs, and 0.39 ± 0.22 for the remaining galaxies. Both ULIRGs and LIRGs in our sample contain dust which is cooler than the dust in the remaining galaxies, which can be seen as a shift of the maximal values of the dust components towards longer wavelengths. The brighter the sample is in the IR, the more shifted is the dust peak towards the longer wavelengths. In our sample, the maximum of the dust peak in the spectra normalized to 90 µ m is located at 1.38 · 10 6 ˚ A, 1.21 · 10 6 ˚ A, and 8.65 · 10 5 ˚ A for the ULIRGs, LIRGs and normal galaxies, respectively. The median redshift for ULIRGs in our sample is equal to 0.54. The median redshift for LIRGs, and the remaining galaxies was found to be 0.2, and 0.04, respectively. This difference in redshifts is a selection effect, related to the fact that the primary detection limit is in the FIR. The two parameters SFR and L TIR might be correlated because both of them depend on the galaxy mass, but they are estimated independently. For the ULIRGs, the logSFR is very high, with a mean value 2.59 ± 0.32 [M /circledot yr -1 ] . The SFR for the LIRGs is more than ten times lower (the mean logSFR is equal to 1.47 ± 0.28 [M /circledot yr -1 ] ). The SFR for the normal galaxies is much lower: we found the logSFR on the level of -0.06 ± 0.62 [M /circledot yr -1 ] . Comparing our results to the sample of normal, nearby galaxies from SINGS (Kennicutt et al., 2003) we found a similar range of SFR: 0-12 [M /circledot yr -1 ] for the ADF-S, and 0-15 [M /circledot yr -1 ] for the SINGS sample. Comparing the results obtained by U et al. (2012) for a sample of 53 LIRGs and 11 ULIRGs in the similar redshift range (z between 0.012 and 0.083) from the Great Observatories All-sky LIRG Survey (GOALS), we found a very good agreement of SFR for LIRGs (logSFR LIRGs = 1.57 ± 0.19 [M /circledot yr -1 ] ). The mean value of logSFR ULIRGs in the case of our ADF-S sample is higher than the one found by U et al. (2012) and equals logSFR ULIRGs = 2.25 ± 0.16 M /circledot yr -1 , but is still consistent within the error bars. The LIRGs in the ADF-S sample have a mean logM star at 11.15 ± 0.49 [M /circledot ] . This value is slightly higher than logM star for LIRGs reported by U et al. 2012 (10.75 ± 0.39 [M /circledot ] ), Giovannoli et al. 2011 (LIRGs sample, with logM star between 10 and 12, and with a peak at 10.8), and Melbourne et al. 2008 (logM star ∼ 10.5, based on a set 15 LIRGs, at redshift ∼ 0.8). Even though our result shows the highest value of logM star , it is still consistent with the values listed above within the error bars. Stellar masses for the ULIRG sample have a mean logM star equal to 11.43 ± 0.31 [M /circledot ] . The mean stellar masses for 11 ULIRGs presented by U et al. (2012) were calculated as logM star =11.00 ± 0.40 [M /circledot ] . The sample of ULIRGs analysed by Howell et al. (2010) is characterized by a mean stellar mass logM star =11.24 ± 0.25 [M /circledot ] . We conclude that the (U)LIRGs in our ADF-S sample are slightly more massive than those in the samples used in previous works, but still all the samples are statistically consistent. Figure 7 shows the relation between the SFRs and stellar masses for (U)LIRGs in our ADF-S catalogue. The mean redshift of the (U)LIRG sample is equal to 0.34 (minimum and maximum redshifts are equal to 0.09 and 0.98, respectively - then, the redshift distribution is quite broad). We compared our results with observations at redshifts 0 and 1 (Elbaz et al., 2007), and 2 (Daddi, et al., 2007), and also with other data mentioned above: a sample of nearby (z < 0.032) (U)LIRGs (U et al., 2012), and a sample of LIRGs observed in the GOALS by GALEX and the Spitzer Space Telescope (Howell et al., 2010). We found a rather flat distribution of the SFR parameter in the stellar mass space (similar to Giovannoli et al., 2011, who found for a sample of LIRGs in the Extended Chandra Deep Field South at z = 0.7, selected at 24 µ m by Spitzer). Our results, shown in Fig. 7, confirm a strong correlation between SFR and redshift. Based on the physical properties obtained from SED fitting, we computed the Specific Star Formation Rates (SSFR [yr -1 ]; defined as the ratio of SFR and stellar mass) for the ADF-S sample. The SSFR rate is commonly used to analyse star formation history. For ULIRGs, LIRGs, and the rest of our sample, the logarithmic values of the SSFR parameter are equal to -9.00 ± 0.55, -9.68 ± 0.59, and -10.28 ± 0.57, respectively. Howell et al. (2010), and U. et al. (2012) presented the mean and median values of the SSFR for (U)LIRGs sample, without an additional separation. The median value of SSFR for galaxies in the GOALS field (Howell et al., 2010) is equal to -9.41 [yr -1 ], while U et al. (2012) obtained the value of -9.17 [yr -1 ]. The same parameter computed for LIRGs and ULIRGs together in our ADF-S sample is equal to -9.51 [yr -1 ]. We conclude that our results are consistent with the results mentioned in all the other works usedconsidered for comparison. However, the dust emission power-law model, given by Dale and Helou (2002) implemented in CIGALE may not be efficient enough to describe the properties of the (U)LIRG sample well. In the future we plan to apply other models (e.g. Siebenmorgen & Krugel, 2007, Chary & Elbaz, 2001, Casey, 2012) to the same (U)LIRGs sample.",
"pages": [
5,
6
]
},
{
"title": "8. Conclusions",
"content": "25% of our sample are (U)LIRGs, rich in dust and active in star formation processes. For these galaxies we noticed a significant shift in the peak wavelength of the dust emission in the FIR and a different ratio between luminosities in the optical and IR parts of the spectra. Acknowledgments. We would like to thank both anonymous Reviewers for their very constructive comments and suggestions which helped to improve the quality of this paper. We thank Olivier Ilbert for useful discussions and kind help in using Le PHARE. This work is based on observations with AKARI a JAXA project with the participation of ESA. This research has made use of the SIMBAD and NED databases. KM, AP and AK were financed by the research grant of the Polish Ministry of Science N N203 512938. The collaboration between French and Polish participants was partially supported by the European Associated Laboratory Astrophysics Poland-France HECOLS. This research was partially supported by the project POLISH-SWISS ASTRO PROJECT co-financed by a grant from Switzerland through the Swiss Contribution to the enlarged European Union. KM has been supported from the Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship for Foreign Researchers, P11802. TTT has been supported by the Grant-in-Aid for the Scientific Research Fund (20740105, 23340046, and 24111707) and for the Global COE Program Request for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. VB and DB have been supported by the Centre National des Etudes Spatiales (CNES) and the Programme National Galaxies (PNG). MM acknowledges support from NASA grants NNX08AU59G and NNX09AM45G for analysis of the GALEX data in the Akari Deep Fields.",
"pages": [
7
]
},
{
"title": "References",
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"pages": [
7,
8
]
}
]
2013EPJC...73.2274L
https://arxiv.org/pdf/1207.1984.pdf
<document>
<section_header_level_1><location><page_1><loc_21><loc_86><loc_79><loc_91></location>Analytical study of superradiant instability of the five-dimensional Kerr-Godel black hole</section_header_level_1>
<text><location><page_1><loc_46><loc_82><loc_53><loc_83></location>Ran Li ∗</text>
<text><location><page_1><loc_18><loc_79><loc_82><loc_80></location>Department of Physics, Henan Normal University, Xinxiang 453007, China</text>
<section_header_level_1><location><page_1><loc_45><loc_75><loc_54><loc_77></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_56><loc_88><loc_74></location>We present the analytical study of superradiant instability of the rotating asymptotically Godel black hole (Kerr-Godel black hole) in five-dimensional minimal supergravity theory. By employing the matched asymptotic expansion method to solve Klein-Gordon equation of scalar field perturbation, we show that the complex parts of quasinormal frequencies are positive in the regime of superradiance. This implies the growing instability of superradiant modes. The reason for this kind of instability is the Dirichlet boundary condition at asymptotic infinity, which is similar to that of rotating black holes in anti-de Sitter (AdS) spacetime.</text>
<text><location><page_1><loc_12><loc_52><loc_36><loc_54></location>PACS numbers: 04.50.-h, 04.70.-s</text>
<text><location><page_1><loc_12><loc_50><loc_53><loc_51></location>Keywords: Kerr-G¨odel black hole, superradiant instability</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_74><loc_88><loc_86></location>Superradiance is a classical phenomenon associated with ergosphere in a rotating black hole [1-4]. If the modes of impinging bosonic fields Φ ∼ exp {-iωt + imφ } , with frequency ω and angular momentum m , are scattered off by the event horizon of black hole, the requirement of superradiant amplification is 0 < ω < m Ω H , where Ω H is the angular velocity of event horizon.</text>
<text><location><page_2><loc_12><loc_56><loc_88><loc_73></location>The superradiance phenomenon allows extracting the rotational energy efficiently from black hole. It has been proposed by Press and Teukolsky [5] to use superradiance phenomenon to built the black-hole bomb . The essential of black-hole bomb mechanism is to add a reflecting mirror outside the rotating black hole. Then superradiant modes will bounce back and forth between the event horizon and the mirror. Meanwhile, the rotational energy extracted from black hole by means of superradiance process will grow exponentially. This mechanism has been recently restudied by many authors [6-9].</text>
<text><location><page_2><loc_12><loc_35><loc_88><loc_55></location>When the reflecting mirror is not artificial, superradiance amplification of impinging wave can lead to the instability of black hole, which is just called superradiant instability. This kind of instability has been studied extensively in recent years. For example, Kerr and Kerr-Newman black holes [10-13] and Kerr-Newman black hole immersed in magnetic field [14] are all unstable against the massive scalar field perturbations, where the mass terms of perturbations play the role of reflective mirrors. Five-dimensional boosted Kerr black string [15] is also unstable against the massless scalar field where Kaluza-Klein momentum works as reflective mirror.</text>
<text><location><page_2><loc_12><loc_8><loc_88><loc_34></location>For the rotating black holes in AdS space, the boundary at infinity can also work as the reflecting mirror. Small Kerr-AdS black hole in four dimensions is unstable against massless scalar field [16] and gravitational field [17] perturbations. Contrary to four-dimensional case, superradiance instability of five-dimensional rotating charged AdS black hole [18] occurs only when the orbital quantum number is even. More recently, superradiant instability of small Reissner-Nordstrom-anti-de Sitter black hole is investigated analytically and numerically [19]. We also studied the superradiant instability of charged massive scalar field in KerrNewman-anti-de Sitter black hole [20]. In fact, besides AdS space, there are also other cases where the boundary at asymptotic infinity provides the reflecting mirror. For example, the rotating linear dilaton black hole [21] and the charged Myers-Perry black hole in Godel</text>
<text><location><page_3><loc_12><loc_84><loc_88><loc_91></location>universe [22] are unstable due to superradiance. The reason of superradiant instability for these black holes originates from the Dirichlet boundary condition at asymptotic infinity of the perturbation fields.</text>
<text><location><page_3><loc_12><loc_39><loc_88><loc_83></location>As mentioned above, superradiant instability of charged rotating asymptotically Godel black hole has been found by using the numerical methods [22]. In this paper, we will re-investigate the same aspect of this kind of rotating black holes in Godel universe by using the analytical methods. We focus on the rotating asymptotically Godel black hole in five-dimensional minimal supergravity theory. This black hole is also called as KerrGodel black hole in literature. Firstly, by considering the scalar field perturbation in this background, we find that the asymptotically Godel spacetime requires the wave equation to satisfy the Dirichlet boundary condition at asymptotic infinity. Then, we divide the space outside the event horizon of Kerr-Godel black hole into the near-region and the far-region, and employ the matched asymptotic expansion method to solve the wave equation of scalar field perturbation. We only deal with black hole in the limit of small rotating parameter j of Godel universe. The analysis of complex quasinormal modes by imposing the boundary conditions shows that the complex parts are positive in the regime of superradiance, which implies the growing instability of these modes. This is to say that the five-dimensional Kerr-Godel black hole is unstable against the scalar field perturbation. The reason for this instability is just the Dirichlet boundary condition at asymptotic infinity, which is similar to that of the rotating black holes in AdS space.</text>
<text><location><page_3><loc_12><loc_23><loc_88><loc_38></location>The remaining of this paper is arranged as follows. In Section 2, we give a brief review of Kerr-Godel black hole in five-dimensional minimal supergravity theory. In Section 3, we investigate the classical superradiance phenomenon and the boundary condition of scalar field perturbation. In Section 4, the approximated solution of wave equation for scalar field is obtained by using the matched asymptotic expansion method and the superradiant instability is explicitly shown. The last section is devoted to conclusion and discussion.</text>
<section_header_level_1><location><page_3><loc_12><loc_18><loc_66><loc_19></location>II. FIVE-DIMENSIONAL KERR-G ODEL BLACK HOLE</section_header_level_1>
<text><location><page_3><loc_12><loc_11><loc_88><loc_15></location>The bosonic part of five-dimensional minimal supergravity theory consists of the metric and a one-form gauge field, which are governed by Einstein-Maxwell-Chern-Simons (EMCS)</text>
<text><location><page_4><loc_12><loc_89><loc_28><loc_91></location>equations of motion</text>
<formula><location><page_4><loc_32><loc_79><loc_88><loc_88></location>R µν -1 2 Rg µν = 2 ( F µα F α ν -1 4 g µν F αβ F αβ ) , D ν ( F µν + 1 √ 3 √ -g /epsilon1 µναβγ A α F βγ ) = 0 . (1)</formula>
<text><location><page_4><loc_12><loc_74><loc_88><loc_78></location>The five-dimensional Kerr-Godel black hole is a solution to the EMCS equations of motion, the metric of which takes the form [23]</text>
<formula><location><page_4><loc_29><loc_65><loc_88><loc_73></location>ds 2 = -f ( r ) dt 2 -q ( r ) rσ 3 L dt -h ( r ) r 2 ( σ 3 L ) 2 + dr 2 v ( r ) + r 2 4 ( dθ 2 + dψ 2 + dφ 2 +2cos θdψdφ ) , (2)</formula>
<text><location><page_4><loc_12><loc_62><loc_65><loc_64></location>where σ 3 L = dφ +cos θdψ , and the metric functions are given by</text>
<formula><location><page_4><loc_31><loc_46><loc_88><loc_60></location>f ( r ) = 1 -2 M r 2 , q ( r ) = 2 jr + 2 Ma r 3 , h ( r ) = j 2 ( r 2 +2 M ) -Ma 2 2 r 4 , v ( r ) = 1 -2 M r 2 + 8 jM ( a +2 jM ) r 2 + 2 Ma 2 r 4 . (3)</formula>
<text><location><page_4><loc_12><loc_24><loc_88><loc_44></location>The parameters M and a are related to the mass, and the angular momentum of black hole. In this metric, the parameter j defines the scale of the Godel background and is responsible for the rotation of the Godel universe [24]. When a = 0, this solution reduced to the Gimon-Hashimoto solution, i.e. the Schwarzschild black hole in Godel universe [23]. The thermodynamics of this black hole has been studied in [25, 26]. The scalar field perturbation and greybody factor of Hawking radiation of this kind of black holes are also calculated in the limit of small j in [27]. This black hole has also been generalized to being charged [28] and other forms.</text>
<text><location><page_4><loc_12><loc_19><loc_88><loc_23></location>In this paper, we consider the non-extremal black hole case. The metric function v ( r ) has two positive real roots r ± , which are given by</text>
<text><location><page_4><loc_12><loc_11><loc_17><loc_12></location>where</text>
<formula><location><page_4><loc_35><loc_12><loc_88><loc_17></location>r 2 ± = M -4 jMa -8 j 2 M 2 ± √ ξ , (4)</formula>
<formula><location><page_4><loc_33><loc_6><loc_88><loc_9></location>ξ = ( M -4 jMa -8 j 2 M 2 ) 2 -2 Ma 2 . (5)</formula>
<text><location><page_5><loc_12><loc_87><loc_88><loc_91></location>Clearly the non-extremal condition is given by ξ > 0. The event horizon locates at the largest root r + of function v ( r ).</text>
<text><location><page_5><loc_12><loc_81><loc_88><loc_85></location>For latter convenience, we also present the expression of angular momentum at the event horizon</text>
<formula><location><page_5><loc_34><loc_71><loc_88><loc_80></location>Ω H = 2 q ( r + ) r + (1 -4 h ( r + )) = 4( Ma + jr 4 + ) r 4 + -4 j 2 r 4 + ( r 2 + +2 M ) + 2 Ma 2 . (6)</formula>
<text><location><page_5><loc_12><loc_69><loc_50><loc_71></location>By employing the relation of metric functions</text>
<formula><location><page_5><loc_36><loc_64><loc_88><loc_67></location>q 2 ( r ) + f ( r )(1 -4 h ( r )) = v ( r ) , (7)</formula>
<text><location><page_5><loc_12><loc_59><loc_88><loc_63></location>and noting that v ( r ) vanishes at the horizon, another simple expression for the angular momentum can be derived</text>
<formula><location><page_5><loc_41><loc_53><loc_88><loc_57></location>Ω H = 2 M -r 2 + Ma + jr 4 + . (8)</formula>
<text><location><page_5><loc_12><loc_45><loc_88><loc_52></location>In the present paper, we consider the rotating parameters a and j are both positive. From the expression of r 2 + in (4), one can easily find r 2 + < 2 M , which implies that the angular momentum Ω H is always positive when the rotating parameters a and j are positive.</text>
<section_header_level_1><location><page_5><loc_12><loc_40><loc_67><loc_41></location>III. SUPERRADIANCE AND BOUNDARY CONDITION</section_header_level_1>
<text><location><page_5><loc_12><loc_33><loc_88><loc_37></location>Now let us consider the wave equation of massless scalar field perturbation in the background (2), which is given by Klein-Gordon equation</text>
<formula><location><page_5><loc_33><loc_27><loc_88><loc_31></location>∇ µ ∇ µ Φ = 1 √ -g ∂ µ ( g µν √ -g∂ ν Φ) = 0 . (9)</formula>
<text><location><page_5><loc_12><loc_22><loc_88><loc_26></location>Because the metric has the killing vectors ∂ t , ∂ ψ , and ∂ φ , we can take the ansatz of scalar field as</text>
<formula><location><page_5><loc_37><loc_18><loc_88><loc_20></location>Φ = e -iωt + inψ + imφ Θ( θ ) R ( r ) . (10)</formula>
<text><location><page_5><loc_12><loc_11><loc_88><loc_16></location>Substituting this ansatz into the wave equation (9) and separating the variables, we can get the angular equation</text>
<formula><location><page_5><loc_20><loc_6><loc_88><loc_10></location>1 sin θ ∂ θ [sin θ∂ θ Θ( θ )] -( n -m cos θ ) 2 sin 2 θ Θ( θ ) + [ l ( l +1) -m 2 ]Θ( θ ) = 0 , (11)</formula>
<text><location><page_6><loc_12><loc_89><loc_31><loc_91></location>and the radial equation</text>
<formula><location><page_6><loc_23><loc_79><loc_88><loc_88></location>1 4 r ∂ r [ r 3 v ( r ) ∂ r R ( r ) ] + r 2 (1 -4 h ( r )) 4 v ( r ) [ ω -2 mq ( r ) r (1 -4 h ( r )) ] 2 R ( r ) + [ l ( l +1) + 4 m 2 h ( r ) (1 -4 h ( r )) ] R ( r ) = 0 . (12)</formula>
<text><location><page_6><loc_12><loc_68><loc_88><loc_78></location>Obviously, the angular equation (11) is independent of the black hole parameters and is exactly solvable. The solutions for the angular equation are just the spin-weighted spherical harmonics functions, where the integers l = 0 , 1 , 2 , · · · are the separation constants and the modes m = 0 , ± 1 , · · · , ± l .</text>
<text><location><page_6><loc_12><loc_61><loc_88><loc_68></location>In the following, we will specify the appropriate boundary conditions for the instability problem. At the horizon, the third set of terms in radial wave equation (12) can be neglected and this equation can be reduced to the form</text>
<formula><location><page_6><loc_28><loc_56><loc_88><loc_59></location>v∂ r ( v∂ r R ( r )) + (1 -4 h ( r + ))( ω -m Ω H ) 2 R ( r ) = 0 . (13)</formula>
<text><location><page_6><loc_12><loc_50><loc_88><loc_55></location>Near the horizon, we use the approximation v ( r ) ∼ = 2( r 2 + -r 2 -)( r -r + ) /r 3 + . Then the solution of equation (13) satisfying the ingoing boundary condition at the horizon is given by</text>
<formula><location><page_6><loc_34><loc_45><loc_88><loc_48></location>R ( r ) ∼ ( r -r + ) -i/pi1 = e -i/pi1 ln( r -r + ) , (14)</formula>
<text><location><page_6><loc_12><loc_43><loc_30><loc_44></location>where we have defined</text>
<formula><location><page_6><loc_26><loc_36><loc_88><loc_41></location>/pi1 = r + ( r 4 + +2 Ma 2 -4 j 2 r 6 + -8 j 2 Mr 2 + ) 1 / 2 2( r 2 + -r 2 -) ( ω -m Ω H ) . (15)</formula>
<text><location><page_6><loc_12><loc_24><loc_88><loc_36></location>This solution gives us the superradiance condition of five-dimensional Kerr-Godel black hole. When the frequency of the wave is such that /pi1 is negative, i.e. ω < m Ω H , one is in the superradiant regime, and the amplitude of an ingoing bosonic field is amplified after scattering by the event horizon. Meanwhile, for the present purpose, it is enough to consider the frequency ω is positive, which gives the superradiance condition as</text>
<formula><location><page_6><loc_42><loc_20><loc_88><loc_22></location>0 < ω < m Ω H . (16)</formula>
<text><location><page_6><loc_12><loc_13><loc_88><loc_18></location>From this condition, one can see that superradiance will occur only for the positive m . In the following, we will only work with the positive m .</text>
<text><location><page_6><loc_14><loc_11><loc_63><loc_12></location>At infinity, the radial wave equation (12) is dominated by</text>
<formula><location><page_6><loc_32><loc_6><loc_88><loc_9></location>r∂ 2 r R ( r ) + 3 ∂ r R ( r ) -4 j 2 ω 2 r 3 R ( r ) = 0 . (17)</formula>
<text><location><page_7><loc_12><loc_89><loc_32><loc_91></location>The solution is given by</text>
<formula><location><page_7><loc_42><loc_84><loc_88><loc_88></location>R ( r ) ∼ 1 r 2 e -jωr 2 , (18)</formula>
<text><location><page_7><loc_12><loc_79><loc_88><loc_83></location>where we have used the analogy with AdS backgrounds and imposed the Dirichlet boundary conditions at spatial infinity.</text>
<text><location><page_7><loc_12><loc_58><loc_88><loc_78></location>With the boundary conditions that the ingoing wave at the horizon and the Dirichlet boundary condition at the infinity, one can solve the complex quasinormal modes of the massless scalar field in Kerr-Godel background. If the imaginary part of quasinormal mode is negative, it is known that the system is stable against this kind of perturbation. The instability means that the imaginary part is positive. In the next section, we will calculate the quasinormal modes by using the matching technique. It is shown that, in the regime of superradiance, the imaginary part of quasinormal mode is positive. In other words, the superradiant instability of five-dimensional Kerr-Godel black hole can be found analytically.</text>
<section_header_level_1><location><page_7><loc_12><loc_52><loc_82><loc_54></location>IV. ANALYTICAL CALCULATION OF SUPERRADIANT INSTABILITY</section_header_level_1>
<text><location><page_7><loc_12><loc_40><loc_88><loc_49></location>In this section, we will present an analytical calculation of superradiant instability for the massless scalar perturbation. We will adopt the so-called matched asymptotic expansion method to solve the radial wave equation (12). It turns out to be convenient to use the new variable x defined by x = r 2 . Then the radial wave equation can be transformed into</text>
<formula><location><page_7><loc_24><loc_30><loc_88><loc_39></location>∆ ∂ x (∆ ∂ x ) R ( x ) + x 3 4 (1 -4 h ( x )) [ ω -2 mq ( x ) √ x (1 -4 h ( x )) ] 2 R ( x ) +∆ [ l ( l +1) + 4 m 2 h ( x ) 1 -4 h ( x ) ] R ( x ) = 0 , (19)</formula>
<text><location><page_7><loc_12><loc_27><loc_68><loc_30></location>where we have used ∆ = x 2 v ( x ) = ( x -x + )( x -x -) with x ± = r 2 ± .</text>
<text><location><page_7><loc_12><loc_14><loc_88><loc_27></location>In order to employ the matched asymptotic expansion method, we should take the assumption ωM /lessmuch 1, and divide the space outside the event horizon into two regions, namely, a near-region, x -x + /lessmuch 1 /ω , and a far-region, x -x + /greatermuch M . The approximated solution can be obtained by matching the near-region solution and the far-region solution in the overlapping region M /lessmuch x -x + /lessmuch 1 /ω .</text>
<text><location><page_7><loc_12><loc_7><loc_88><loc_14></location>Previous numerical works [22, 29] on the spectrum of asymptotically Godel black holes show a number of common features with the spectrum of AdS spacetime, where the rotational parameter j of Godel universe plays the role of the inverse AdS radius /lscript . Inspired by the</text>
<text><location><page_8><loc_12><loc_80><loc_88><loc_91></location>work of [16], where small AdS black hole are considered, we will deal with the rotating asymptotically Godel black hole in the limit of small rotational parameter j in the following. The small AdS black hole condition implies that r + //lscript /lessmuch 1. For the small Godel black hole, we assume that jr + /lessmuch 1.</text>
<text><location><page_8><loc_12><loc_73><loc_88><loc_80></location>With these assumptions, we can analyse the properties of the solution and study the stability of black hole against the perturbation by imposing the appropriate boundary conditions obtained in the last section.</text>
<section_header_level_1><location><page_8><loc_14><loc_68><loc_37><loc_69></location>A. Near-region solution</section_header_level_1>
<text><location><page_8><loc_12><loc_58><loc_88><loc_65></location>Firstly, Let us focus on the near-region in the vicinity of the event horizon, ω ( x -x + ) /lessmuch 1. For the small j black holes, this means jr + /lessmuch 1. The radial wave function (19) in the nearregion can be reduced to the form</text>
<formula><location><page_8><loc_25><loc_51><loc_88><loc_56></location>∆ ∂ x (∆ ∂ x R ( x )) + [ ( x + -x -) 2 /pi1 2 -l ( l +1)∆ ] R ( r ) = 0 . (20)</formula>
<text><location><page_8><loc_12><loc_49><loc_87><loc_51></location>Noted that the last term in Eq.(19) is neglected because we only consider the case m ∼ ω .</text>
<text><location><page_8><loc_14><loc_47><loc_48><loc_49></location>Introducing the new coordinate variable</text>
<formula><location><page_8><loc_44><loc_41><loc_88><loc_46></location>z = x -x + x -x -, (21)</formula>
<text><location><page_8><loc_12><loc_39><loc_63><loc_40></location>the near-region radial equation can be written in the form of</text>
<formula><location><page_8><loc_28><loc_33><loc_88><loc_37></location>z∂ z ( z∂ z R ( z )) + [ /pi1 2 -l ( l +1) z (1 -z ) 2 ] R ( z ) = 0 , (22)</formula>
<text><location><page_8><loc_12><loc_31><loc_15><loc_32></location>with</text>
<formula><location><page_8><loc_34><loc_24><loc_88><loc_29></location>/pi1 = r + ( r 4 + +2 Ma 2 ) 1 / 2 2( r 2 + -r 2 -) ( ω -m Ω H ) . (23)</formula>
<text><location><page_8><loc_12><loc_22><loc_84><loc_24></location>This expression for /pi1 is coincide with the expression given in (15) in the small j limit.</text>
<text><location><page_8><loc_14><loc_20><loc_29><loc_21></location>Through defining</text>
<formula><location><page_8><loc_39><loc_14><loc_88><loc_17></location>R = z i/pi1 (1 -z ) l +1 F ( z ) , (24)</formula>
<text><location><page_8><loc_12><loc_11><loc_50><loc_13></location>the near-region radial wave equation becomes</text>
<formula><location><page_8><loc_25><loc_6><loc_88><loc_9></location>z (1 -z ) ∂ 2 z F ( z ) + [ c -(1 + a + b )] ∂ z F ( z ) -abF ( z ) = 0 , (25)</formula>
<text><location><page_9><loc_12><loc_89><loc_29><loc_91></location>with the parameters</text>
<formula><location><page_9><loc_42><loc_79><loc_88><loc_87></location>a = l +1+2 i/pi1 , b = l +1 , c = 1 + 2 i/pi1 . (26)</formula>
<text><location><page_9><loc_12><loc_73><loc_88><loc_77></location>In the neighborhood of z = 0, the general solution of the radial wave equation is given in terms of the hypergeometric function</text>
<formula><location><page_9><loc_27><loc_65><loc_88><loc_71></location>R = Az -i/pi1 (1 -z ) l +1 F ( l +1 , l +1 -2 i/pi1, 1 -2 i/pi1, z ) + Bz i/pi1 (1 -z ) l +1 F ( l +1 , l +1+2 i/pi1, 1 + 2 i/pi1, z ) . (27)</formula>
<text><location><page_9><loc_12><loc_52><loc_88><loc_64></location>It is obvious that the first term represents the ingoing wave at the horizon, while the second term represents the outgoing wave at the horizon. Because we are considering the classical superradiance process, the ingoing boundary condition at the horizon should be employed. Then we have to set B = 0. The physical solution of the radial wave equation corresponding to the ingoing wave at the horizon is then given by</text>
<formula><location><page_9><loc_26><loc_47><loc_88><loc_50></location>R = Az -i/pi1 (1 -z ) l +1 F ( l +1 , l +1 -2 i/pi1, 1 -2 i/pi1, z ) . (28)</formula>
<text><location><page_9><loc_12><loc_38><loc_88><loc_46></location>In order to match the far-region solution that will be obtained in the next subsection, we should study the large r , z → 1, behavior of the near-region solution. For the sake of this purpose, we can use the z → 1 -z transformation law for the hypergeometric function</text>
<formula><location><page_9><loc_25><loc_27><loc_88><loc_37></location>F ( a, b, c, z ) = Γ( c )Γ( c -a -b ) Γ( c -a )Γ( c -b ) F ( a, b, a + b -c +1 , 1 -z ) +(1 -z ) c -a -b Γ( c )Γ( a + b -c ) Γ( a )Γ( b ) × F ( c -a, c -b, c -a -b +1 , 1 -z ) . (29)</formula>
<text><location><page_9><loc_12><loc_21><loc_88><loc_26></location>By employing this formula and using the properties of hypergeometric function F ( a, b, c, 0) = 1, we can get the large r behavior of the near-region solution as</text>
<formula><location><page_9><loc_30><loc_11><loc_88><loc_20></location>R ∼ A Γ(1 -2 i/pi1 ) [ ( r 2 + -r 2 -) -l Γ(2 l +1) Γ( l +1)Γ( l +1 -2 i/pi1 ) r 2 l + ( r 2 + -r 2 -) l +1 Γ( -2 l -1) Γ( -l )Γ( -l -2 i/pi1 ) r -2 l -2 ] , (30)</formula>
<text><location><page_9><loc_12><loc_7><loc_88><loc_11></location>where the variable x has been restored to r for later convenience. This solution should be matched with the small r behavior of the far-region solution obtained in the next subsection.</text>
<section_header_level_1><location><page_10><loc_14><loc_89><loc_35><loc_91></location>B. Far-region solution</section_header_level_1>
<text><location><page_10><loc_12><loc_80><loc_88><loc_86></location>In the Far-region, x -x + /greatermuch M , we can neglect the effects induced by the black hole, i.e. we have a ∼ 0 and M ∼ 0. The metric functions can be approximated as v ( x ) = f ( x ) = 1, h ( x ) = j 2 x , and q ( x ) = 2 j √ x . One can deduce the far-region radial wave equation as</text>
<formula><location><page_10><loc_26><loc_73><loc_88><loc_78></location>∂ 2 x ( xR ) + [ -j 2 ω 2 + ω ( ω -8 mj ) 4 x -l ( l +1) x 2 ] ( xR ) = 0 . (31)</formula>
<text><location><page_10><loc_12><loc_69><loc_88><loc_73></location>By defining the new variable ζ = 2 jωx , the far-region radial wave equation can be reduced to</text>
<formula><location><page_10><loc_32><loc_63><loc_88><loc_67></location>∂ 2 ζ ( ζR ) + [ -1 4 + ρ ζ -l ( l +1) ζ 2 ] ( ζR ) = 0 , (32)</formula>
<text><location><page_10><loc_12><loc_59><loc_44><loc_62></location>with the parameter ρ = ( ω -8 mj ) / 8 j .</text>
<text><location><page_10><loc_12><loc_50><loc_88><loc_60></location>This is a standard Whittaker equation ∂ 2 ζ W +[ -1 / 4 + ρ/ζ +(1 / 4 -µ 2 ) /ζ 2 ] W = 0 with W = ζR and µ = l +1 / 2. The general solution is given by W = ζ µ +1 / 2 e -ζ/ 2 [ αM (˜ a, ˜ b, ζ ) + βU (˜ a, ˜ b, ζ )], where M and U are Whittaker's functions with ˜ a = 1 / 2+ µ -ρ and ˜ b = 1+2 µ . So the far-region solution of the radial wave equation is given by</text>
<formula><location><page_10><loc_22><loc_45><loc_88><loc_48></location>R = ζ l e -ζ/ 2 [ αM ( l +1 -ρ, 2 l +2 , ζ ) + βU ( l +1 -ρ, 2 l +2 , ζ )] . (33)</formula>
<text><location><page_10><loc_12><loc_31><loc_88><loc_43></location>Now we want to impose the boundary condition at asymptotic infinity. We are interested in the superradiance region with 0 < ω < m Ω H , so we have ζ = 2 jωr 2 → + ∞ when r → + ∞ . When ζ → + ∞ , by using the properties of the Whittaker's functions M (˜ a, ˜ b, ζ ) ∼ ζ ˜ a -˜ b e ζ Γ( ˜ b ) / Γ(˜ a ) and U (˜ a, ˜ b, ζ ) ∼ ζ -˜ a , one can get the large r behavior of the far-region solution as</text>
<formula><location><page_10><loc_24><loc_25><loc_88><loc_30></location>R ∼ α Γ(2 l +2) Γ( l +1 -ρ ) (2 jωr 2 ) -1 -ρ e jωr 2 + β (2 jωr 2 ) -1 -ρ e -jωr 2 . (34)</formula>
<text><location><page_10><loc_12><loc_18><loc_88><loc_24></location>Obviously the first term is divergent at asymptotic infinity. To match the Dirichlet boundary condition at infinity, we have to set α = 0. Thus the far-region solution with the Dirichlet boundary condition at asymptotic infinity is given by</text>
<formula><location><page_10><loc_29><loc_12><loc_88><loc_15></location>R = β (2 jω ) l r 2 l e -jωr 2 U ( l +1 -ρ, 2 l +2 , 2 jωr 2 ) . (35)</formula>
<text><location><page_10><loc_12><loc_7><loc_88><loc_11></location>This solution is just the solution of scalar field wave equation in the background of the pure five-dimensional Godel spacetime [29-31].</text>
<text><location><page_11><loc_12><loc_76><loc_88><loc_91></location>We assume for a moment that we have no black hole, and calculate the real frequencies that can propagate in the pure five-dimensional Godel spacetime. In this setup, the spacetime geometry is horizon-free, and the solution of the scalar field perturbation in the background of the pure five-dimensional Godel spacetime should be regular at the origin r = 0. When ζ → 0, using the properties of Whittaker's function U (˜ a, ˜ b, ζ ) ∼ ζ 1 -˜ b Γ( ˜ b -1) / Γ(˜ a ), one can get the small r behavior of the far-region solution as</text>
<formula><location><page_11><loc_34><loc_70><loc_88><loc_75></location>R ∼ β (2 jω ) -l -1 Γ(2 l +1) Γ( l +1 -ρ ) r -2 l -2 . (36)</formula>
<text><location><page_11><loc_12><loc_63><loc_88><loc_70></location>So, when r → 0, r -2 l -2 →∞ , and the solution diverges. To have a regular solution at the origin r = 0, we must demand that Γ( l +1 -ρ ) →∞ . This occurs when the argument of the gamma function is a non-positive integer. Therefore, we have the condition</text>
<formula><location><page_11><loc_34><loc_58><loc_88><loc_60></location>l +1 -ρ = -N, N = 0 , 1 , 2 , · · · . (37)</formula>
<text><location><page_11><loc_12><loc_52><loc_88><loc_56></location>So the requirement of the regularity of the wave solution at the origin selects the frequencies of the scalar field that might propagate in the pure five-dimensional Godel spacetime</text>
<formula><location><page_11><loc_38><loc_48><loc_88><loc_49></location>ω N = 8 j ( N + l + m +1) . (38)</formula>
<text><location><page_11><loc_12><loc_39><loc_88><loc_45></location>Now let us come back to the Kerr-Godel black hole case. In the spirit of [16], we expect that there will be a small imaginary part δ in the allowed frequencies induced by the black hole event horizon</text>
<formula><location><page_11><loc_43><loc_35><loc_88><loc_36></location>ω = ω N + iδ . (39)</formula>
<text><location><page_11><loc_12><loc_25><loc_88><loc_32></location>From Ψ ∼ e -iωt , one can see that the small imaginary δ describes the slow growing instability of the modes when δ > 0. Our task is to prove that δ is positive in the regime of superradiance.</text>
<text><location><page_11><loc_12><loc_20><loc_88><loc_24></location>Inserting this expression for the frequency ω , one can get the far-region solution of the radial wave equation as</text>
<formula><location><page_11><loc_27><loc_15><loc_88><loc_18></location>R = β (2 jω ) l r 2 l e -jωr 2 U ( -N -iδ/ 8 j, 2 l +2 , 2 jωr 2 ) . (40)</formula>
<text><location><page_11><loc_12><loc_7><loc_88><loc_14></location>In order to match the far-region solution with the near-region solution, we need to find the small r behavior of the far-region solution. It is known that the Whittaker's function U (˜ a, ˜ b, ζ ) can be expressed in terms of the Whittaker's function M (˜ a, ˜ b, ζ ). By inserting</text>
<text><location><page_12><loc_12><loc_87><loc_88><loc_91></location>this relation on the far-region solution (40), we can show that the far-region solution can be rewritten as</text>
<formula><location><page_12><loc_20><loc_77><loc_88><loc_86></location>R = β (2 jω ) l r 2 l e -jωr 2 π sin π (2 l +2) [ M ( -N -iδ/ 8 j, 2 l +2 , 2 jωr 2 ) Γ( -N -2 l -1 -iδ/ 8 j )Γ(2 l +2) -(2 jω ) -2 l -1 r -4 l -2 M ( -N -2 l -1 -iδ/ 8 j, -2 l, 2 jωr 2 ) Γ( -N -iδ/ 8 j )Γ( -2 l ) ] . (41)</formula>
<text><location><page_12><loc_14><loc_75><loc_81><loc_77></location>Applying to this expression the functional expressions for the gamma functions</text>
<formula><location><page_12><loc_40><loc_67><loc_88><loc_73></location>Γ( n +1) = n ! , Γ( z )Γ(1 -z ) = π sin πz , (42)</formula>
<text><location><page_12><loc_12><loc_65><loc_30><loc_66></location>it is easy to show that</text>
<formula><location><page_12><loc_36><loc_59><loc_88><loc_64></location>1 Γ( -2 l ) = -sin π (2 l +2) π (2 l )! , (43)</formula>
<text><location><page_12><loc_12><loc_58><loc_28><loc_59></location>and for the small δ</text>
<formula><location><page_12><loc_24><loc_48><loc_88><loc_57></location>1 Γ( -N -2 l -1 -iδ/ 8 j ) = ( -1) N sin π (2 l +2) π ( N +2 l +1)! , 1 Γ( -N -iδ/ 8 j ) = ( -1) N +1 N ! iδ/ 8 j . (44)</formula>
<text><location><page_12><loc_12><loc_44><loc_88><loc_48></location>Then by using the property of Whittaker's function M (˜ a, ˜ b, 0) = 1, one can get the small r behavior of the far-region solution as</text>
<formula><location><page_12><loc_21><loc_38><loc_88><loc_42></location>R = β ( -1) N (2 jω N ) l [ ( N +2 l +1)! (2 l +1)! r 2 l -iδ (2 l )! N ! 2 2 l +4 j 2 l +2 ω 2 l +1 N r -2 l -2 ] . (45)</formula>
<section_header_level_1><location><page_12><loc_14><loc_34><loc_55><loc_36></location>C. Matching condition: the unstable modes</section_header_level_1>
<text><location><page_12><loc_12><loc_19><loc_88><loc_31></location>By comparing the large r behavior of the near-region solution with the small r behavior of the far-region solution, one can conclude that there exists the overlapping region M /lessmuch x -x + /lessmuch 1 /ω where the two solutions should match. In this region, the matching of the near-region solution in the large r region (30) and the far-region solution in the small r region (45) yields the allowed values of the small imaginary part δ in the frequency ω</text>
<formula><location><page_12><loc_26><loc_15><loc_88><loc_18></location>δ ∼ = -σ ( ω N -m Ω H ) r + ( r 4 + +2 Ma 2 ) 1 / 2 ( r + -r -) 2 l j 2 l +2 , (46)</formula>
<text><location><page_12><loc_12><loc_12><loc_17><loc_14></location>where</text>
<formula><location><page_12><loc_28><loc_6><loc_88><loc_11></location>σ = 2 2 l +4 ω 2 l +1 N ( l !) 2 (2 l +1+ N )! ((2 l )!(2 l +1)!) 2 N ! [ l ∏ k =1 ( k 2 +4 /pi1 2 ) ] , (47)</formula>
<text><location><page_13><loc_12><loc_88><loc_67><loc_91></location>with /pi1 = ( ω N -m Ω H ) r + ( r 4 + +2 Ma 2 ) 1 / 2 / 2( r 2 + -r 2 -). So, we have</text>
<formula><location><page_13><loc_39><loc_84><loc_88><loc_87></location>δ ∝ -( Re [ ω ] -m Ω H ) . (48)</formula>
<text><location><page_13><loc_12><loc_73><loc_88><loc_82></location>It is easy to see that, in the superradiance regime, Re [ ω ] -m Ω H < 0, the imaginary part of the complex frequency δ > 0. The scalar field has the time dependence e -iωt = e -iω N t e δt , which implies the exponential amplification of superradiance modes. This will lead to the instability of these modes.</text>
<text><location><page_13><loc_12><loc_62><loc_88><loc_72></location>From the normal modes in pure five-dimensional Godel spacetime (38), we can see that Re [ ω ] ∼ j . We have assumed that ωM /lessmuch 1. So we have jM /lessmuch 1, which is consistent with the small Godel black hole assumption jr + /lessmuch 1 because M ∼ r 2 + . This is to say that the two assumptions we have made in this section are consistent with each other.</text>
<text><location><page_13><loc_12><loc_36><loc_88><loc_61></location>Let us make a qualitative comparison of our analytical results with the numerical one in [22]. From Eq.(46), we can see that the growth rate δ of the superradiant modes are propotional to j 2 l +2 . This implies that the larger j corresponds to the higher superradiant instability growth rate, which supports the numerical conclusion in [22]. Next, the superradiant condition that Re [ ω ] -m Ω H < 0 can make a further limit on the parameter space where the superradiance can occur. From Eq.(38), we can see the superradiance condition for the mode of l = m = 1 and N = 0 becomes 24 j < Ω H . Because we are working on the parameter space that jr 2 + /lessmuch 1, we can further take the limit jM /lessmuch 1 in the expression of event horizon Eq.(4) and angular momentum Eq.(8). So the limit on the parameter j can be approximated by the following expression</text>
<formula><location><page_13><loc_36><loc_29><loc_88><loc_35></location>j √ M < 1 -√ 1 -2( a/ √ M ) 2 24( a/ √ M ) , (49)</formula>
<text><location><page_13><loc_12><loc_8><loc_88><loc_29></location>where, the parameter a/ √ M takes the value in the region of (0 , 0 . 71), otherwise there will be no black hole in spacetime. By submitting a/ √ M = 0 . 71 into the inequality, one can see that for j √ M ∼ 0 . 059 there is no superradiance. The numerical one in [22] is j √ M ∼ 0 . 075. The region of the superradiance in the parameter space by using the analytical method roughly coincides with the numerical results in [22]. So, comparing our result with the numerical one in [22], it can be seen that our results are rough and can reproduce the numerical conclusion partly. The analytical method can not solved the superradiance instability as precisely as the numerical method.</text>
<text><location><page_14><loc_12><loc_84><loc_88><loc_91></location>At last, we can conclude that the five-dimensional small Kerr-Godel black hole is unstable against the massless scalar field perturbation. This instability is caused by the superradiance of the scalar field.</text>
<section_header_level_1><location><page_14><loc_12><loc_78><loc_30><loc_80></location>V. CONCLUSION</section_header_level_1>
<text><location><page_14><loc_12><loc_37><loc_88><loc_75></location>This paper is devoted to an analytical study of superradiant instability of five-dimensional small Kerr-Godel black hole. This instability has been found by R. A. Konoplya and A. Zhidenko using the numerical methods previously in [22]. Generally, superradiant instability naturally happens when two conditions are satisfied: (1) Black hole has rotation; (2) There is a natural reflecting mirror outside the black hole. In the present case, Dirichlet boundary condition at infinity for the asymptotically Godel black hole, which is obtained in section 3 by analogy with the AdS background, plays the role of reflecting mirror. We have adopted the analytical methods which is used in [16] to study the superradiant instability of small Kerr-AdS black hole. We assume that the energy of scalar field perturbation is low and the scale of Godel black hole is small. Then we divide the space outside the event horizon into the near-region and the far-region and employ the matched asymptotic expansion method to solve the scalar field wave equation. The analysis of complex quasinormal modes explicitly shows that the complex parts are positive in the regime of superradiance, which implies the growing instability of these modes. This is to say that the five-dimensional small Kerr-Godel black hole is unstable against scalar field perturbation.</text>
<text><location><page_14><loc_12><loc_18><loc_88><loc_36></location>As is well-known, the gravitational perturbation will also undergo an superradiant amplification when scattered by the event horizon. Unlike the simply rotating Kerr-AdS black hole, where the superradiant instability of a massless scalar field implies the gravitational instability [32], we can not obtain the similar conclusion for asymptotically Godel black hole directly. So it will be interesting to study the gravitational (in)stability of five-dimensional Kerr-Godel black hole. Because of the complexity of the metric and the field equation, it will be a challenging project for future work.</text>
<section_header_level_1><location><page_15><loc_14><loc_89><loc_30><loc_91></location>Acknowledgement</section_header_level_1>
<text><location><page_15><loc_12><loc_82><loc_88><loc_86></location>The author would like to thank Ming-Fan Li for reading the manuscript and useful comments. This work was supported by NSFC, China (Grant No. 11147145 and No. 11205048).</text>
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[
{
"title": "Analytical study of superradiant instability of the five-dimensional Kerr-Godel black hole",
"content": "Ran Li ∗ Department of Physics, Henan Normal University, Xinxiang 453007, China",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We present the analytical study of superradiant instability of the rotating asymptotically Godel black hole (Kerr-Godel black hole) in five-dimensional minimal supergravity theory. By employing the matched asymptotic expansion method to solve Klein-Gordon equation of scalar field perturbation, we show that the complex parts of quasinormal frequencies are positive in the regime of superradiance. This implies the growing instability of superradiant modes. The reason for this kind of instability is the Dirichlet boundary condition at asymptotic infinity, which is similar to that of rotating black holes in anti-de Sitter (AdS) spacetime. PACS numbers: 04.50.-h, 04.70.-s Keywords: Kerr-G¨odel black hole, superradiant instability",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Superradiance is a classical phenomenon associated with ergosphere in a rotating black hole [1-4]. If the modes of impinging bosonic fields Φ ∼ exp {-iωt + imφ } , with frequency ω and angular momentum m , are scattered off by the event horizon of black hole, the requirement of superradiant amplification is 0 < ω < m Ω H , where Ω H is the angular velocity of event horizon. The superradiance phenomenon allows extracting the rotational energy efficiently from black hole. It has been proposed by Press and Teukolsky [5] to use superradiance phenomenon to built the black-hole bomb . The essential of black-hole bomb mechanism is to add a reflecting mirror outside the rotating black hole. Then superradiant modes will bounce back and forth between the event horizon and the mirror. Meanwhile, the rotational energy extracted from black hole by means of superradiance process will grow exponentially. This mechanism has been recently restudied by many authors [6-9]. When the reflecting mirror is not artificial, superradiance amplification of impinging wave can lead to the instability of black hole, which is just called superradiant instability. This kind of instability has been studied extensively in recent years. For example, Kerr and Kerr-Newman black holes [10-13] and Kerr-Newman black hole immersed in magnetic field [14] are all unstable against the massive scalar field perturbations, where the mass terms of perturbations play the role of reflective mirrors. Five-dimensional boosted Kerr black string [15] is also unstable against the massless scalar field where Kaluza-Klein momentum works as reflective mirror. For the rotating black holes in AdS space, the boundary at infinity can also work as the reflecting mirror. Small Kerr-AdS black hole in four dimensions is unstable against massless scalar field [16] and gravitational field [17] perturbations. Contrary to four-dimensional case, superradiance instability of five-dimensional rotating charged AdS black hole [18] occurs only when the orbital quantum number is even. More recently, superradiant instability of small Reissner-Nordstrom-anti-de Sitter black hole is investigated analytically and numerically [19]. We also studied the superradiant instability of charged massive scalar field in KerrNewman-anti-de Sitter black hole [20]. In fact, besides AdS space, there are also other cases where the boundary at asymptotic infinity provides the reflecting mirror. For example, the rotating linear dilaton black hole [21] and the charged Myers-Perry black hole in Godel universe [22] are unstable due to superradiance. The reason of superradiant instability for these black holes originates from the Dirichlet boundary condition at asymptotic infinity of the perturbation fields. As mentioned above, superradiant instability of charged rotating asymptotically Godel black hole has been found by using the numerical methods [22]. In this paper, we will re-investigate the same aspect of this kind of rotating black holes in Godel universe by using the analytical methods. We focus on the rotating asymptotically Godel black hole in five-dimensional minimal supergravity theory. This black hole is also called as KerrGodel black hole in literature. Firstly, by considering the scalar field perturbation in this background, we find that the asymptotically Godel spacetime requires the wave equation to satisfy the Dirichlet boundary condition at asymptotic infinity. Then, we divide the space outside the event horizon of Kerr-Godel black hole into the near-region and the far-region, and employ the matched asymptotic expansion method to solve the wave equation of scalar field perturbation. We only deal with black hole in the limit of small rotating parameter j of Godel universe. The analysis of complex quasinormal modes by imposing the boundary conditions shows that the complex parts are positive in the regime of superradiance, which implies the growing instability of these modes. This is to say that the five-dimensional Kerr-Godel black hole is unstable against the scalar field perturbation. The reason for this instability is just the Dirichlet boundary condition at asymptotic infinity, which is similar to that of the rotating black holes in AdS space. The remaining of this paper is arranged as follows. In Section 2, we give a brief review of Kerr-Godel black hole in five-dimensional minimal supergravity theory. In Section 3, we investigate the classical superradiance phenomenon and the boundary condition of scalar field perturbation. In Section 4, the approximated solution of wave equation for scalar field is obtained by using the matched asymptotic expansion method and the superradiant instability is explicitly shown. The last section is devoted to conclusion and discussion.",
"pages": [
2,
3
]
},
{
"title": "II. FIVE-DIMENSIONAL KERR-G ODEL BLACK HOLE",
"content": "The bosonic part of five-dimensional minimal supergravity theory consists of the metric and a one-form gauge field, which are governed by Einstein-Maxwell-Chern-Simons (EMCS) equations of motion The five-dimensional Kerr-Godel black hole is a solution to the EMCS equations of motion, the metric of which takes the form [23] where σ 3 L = dφ +cos θdψ , and the metric functions are given by The parameters M and a are related to the mass, and the angular momentum of black hole. In this metric, the parameter j defines the scale of the Godel background and is responsible for the rotation of the Godel universe [24]. When a = 0, this solution reduced to the Gimon-Hashimoto solution, i.e. the Schwarzschild black hole in Godel universe [23]. The thermodynamics of this black hole has been studied in [25, 26]. The scalar field perturbation and greybody factor of Hawking radiation of this kind of black holes are also calculated in the limit of small j in [27]. This black hole has also been generalized to being charged [28] and other forms. In this paper, we consider the non-extremal black hole case. The metric function v ( r ) has two positive real roots r ± , which are given by where Clearly the non-extremal condition is given by ξ > 0. The event horizon locates at the largest root r + of function v ( r ). For latter convenience, we also present the expression of angular momentum at the event horizon By employing the relation of metric functions and noting that v ( r ) vanishes at the horizon, another simple expression for the angular momentum can be derived In the present paper, we consider the rotating parameters a and j are both positive. From the expression of r 2 + in (4), one can easily find r 2 + < 2 M , which implies that the angular momentum Ω H is always positive when the rotating parameters a and j are positive.",
"pages": [
3,
4,
5
]
},
{
"title": "III. SUPERRADIANCE AND BOUNDARY CONDITION",
"content": "Now let us consider the wave equation of massless scalar field perturbation in the background (2), which is given by Klein-Gordon equation Because the metric has the killing vectors ∂ t , ∂ ψ , and ∂ φ , we can take the ansatz of scalar field as Substituting this ansatz into the wave equation (9) and separating the variables, we can get the angular equation and the radial equation Obviously, the angular equation (11) is independent of the black hole parameters and is exactly solvable. The solutions for the angular equation are just the spin-weighted spherical harmonics functions, where the integers l = 0 , 1 , 2 , · · · are the separation constants and the modes m = 0 , ± 1 , · · · , ± l . In the following, we will specify the appropriate boundary conditions for the instability problem. At the horizon, the third set of terms in radial wave equation (12) can be neglected and this equation can be reduced to the form Near the horizon, we use the approximation v ( r ) ∼ = 2( r 2 + -r 2 -)( r -r + ) /r 3 + . Then the solution of equation (13) satisfying the ingoing boundary condition at the horizon is given by where we have defined This solution gives us the superradiance condition of five-dimensional Kerr-Godel black hole. When the frequency of the wave is such that /pi1 is negative, i.e. ω < m Ω H , one is in the superradiant regime, and the amplitude of an ingoing bosonic field is amplified after scattering by the event horizon. Meanwhile, for the present purpose, it is enough to consider the frequency ω is positive, which gives the superradiance condition as From this condition, one can see that superradiance will occur only for the positive m . In the following, we will only work with the positive m . At infinity, the radial wave equation (12) is dominated by The solution is given by where we have used the analogy with AdS backgrounds and imposed the Dirichlet boundary conditions at spatial infinity. With the boundary conditions that the ingoing wave at the horizon and the Dirichlet boundary condition at the infinity, one can solve the complex quasinormal modes of the massless scalar field in Kerr-Godel background. If the imaginary part of quasinormal mode is negative, it is known that the system is stable against this kind of perturbation. The instability means that the imaginary part is positive. In the next section, we will calculate the quasinormal modes by using the matching technique. It is shown that, in the regime of superradiance, the imaginary part of quasinormal mode is positive. In other words, the superradiant instability of five-dimensional Kerr-Godel black hole can be found analytically.",
"pages": [
5,
6,
7
]
},
{
"title": "IV. ANALYTICAL CALCULATION OF SUPERRADIANT INSTABILITY",
"content": "In this section, we will present an analytical calculation of superradiant instability for the massless scalar perturbation. We will adopt the so-called matched asymptotic expansion method to solve the radial wave equation (12). It turns out to be convenient to use the new variable x defined by x = r 2 . Then the radial wave equation can be transformed into where we have used ∆ = x 2 v ( x ) = ( x -x + )( x -x -) with x ± = r 2 ± . In order to employ the matched asymptotic expansion method, we should take the assumption ωM /lessmuch 1, and divide the space outside the event horizon into two regions, namely, a near-region, x -x + /lessmuch 1 /ω , and a far-region, x -x + /greatermuch M . The approximated solution can be obtained by matching the near-region solution and the far-region solution in the overlapping region M /lessmuch x -x + /lessmuch 1 /ω . Previous numerical works [22, 29] on the spectrum of asymptotically Godel black holes show a number of common features with the spectrum of AdS spacetime, where the rotational parameter j of Godel universe plays the role of the inverse AdS radius /lscript . Inspired by the work of [16], where small AdS black hole are considered, we will deal with the rotating asymptotically Godel black hole in the limit of small rotational parameter j in the following. The small AdS black hole condition implies that r + //lscript /lessmuch 1. For the small Godel black hole, we assume that jr + /lessmuch 1. With these assumptions, we can analyse the properties of the solution and study the stability of black hole against the perturbation by imposing the appropriate boundary conditions obtained in the last section.",
"pages": [
7,
8
]
},
{
"title": "A. Near-region solution",
"content": "Firstly, Let us focus on the near-region in the vicinity of the event horizon, ω ( x -x + ) /lessmuch 1. For the small j black holes, this means jr + /lessmuch 1. The radial wave function (19) in the nearregion can be reduced to the form Noted that the last term in Eq.(19) is neglected because we only consider the case m ∼ ω . Introducing the new coordinate variable the near-region radial equation can be written in the form of with This expression for /pi1 is coincide with the expression given in (15) in the small j limit. Through defining the near-region radial wave equation becomes with the parameters In the neighborhood of z = 0, the general solution of the radial wave equation is given in terms of the hypergeometric function It is obvious that the first term represents the ingoing wave at the horizon, while the second term represents the outgoing wave at the horizon. Because we are considering the classical superradiance process, the ingoing boundary condition at the horizon should be employed. Then we have to set B = 0. The physical solution of the radial wave equation corresponding to the ingoing wave at the horizon is then given by In order to match the far-region solution that will be obtained in the next subsection, we should study the large r , z → 1, behavior of the near-region solution. For the sake of this purpose, we can use the z → 1 -z transformation law for the hypergeometric function By employing this formula and using the properties of hypergeometric function F ( a, b, c, 0) = 1, we can get the large r behavior of the near-region solution as where the variable x has been restored to r for later convenience. This solution should be matched with the small r behavior of the far-region solution obtained in the next subsection.",
"pages": [
8,
9
]
},
{
"title": "B. Far-region solution",
"content": "In the Far-region, x -x + /greatermuch M , we can neglect the effects induced by the black hole, i.e. we have a ∼ 0 and M ∼ 0. The metric functions can be approximated as v ( x ) = f ( x ) = 1, h ( x ) = j 2 x , and q ( x ) = 2 j √ x . One can deduce the far-region radial wave equation as By defining the new variable ζ = 2 jωx , the far-region radial wave equation can be reduced to with the parameter ρ = ( ω -8 mj ) / 8 j . This is a standard Whittaker equation ∂ 2 ζ W +[ -1 / 4 + ρ/ζ +(1 / 4 -µ 2 ) /ζ 2 ] W = 0 with W = ζR and µ = l +1 / 2. The general solution is given by W = ζ µ +1 / 2 e -ζ/ 2 [ αM (˜ a, ˜ b, ζ ) + βU (˜ a, ˜ b, ζ )], where M and U are Whittaker's functions with ˜ a = 1 / 2+ µ -ρ and ˜ b = 1+2 µ . So the far-region solution of the radial wave equation is given by Now we want to impose the boundary condition at asymptotic infinity. We are interested in the superradiance region with 0 < ω < m Ω H , so we have ζ = 2 jωr 2 → + ∞ when r → + ∞ . When ζ → + ∞ , by using the properties of the Whittaker's functions M (˜ a, ˜ b, ζ ) ∼ ζ ˜ a -˜ b e ζ Γ( ˜ b ) / Γ(˜ a ) and U (˜ a, ˜ b, ζ ) ∼ ζ -˜ a , one can get the large r behavior of the far-region solution as Obviously the first term is divergent at asymptotic infinity. To match the Dirichlet boundary condition at infinity, we have to set α = 0. Thus the far-region solution with the Dirichlet boundary condition at asymptotic infinity is given by This solution is just the solution of scalar field wave equation in the background of the pure five-dimensional Godel spacetime [29-31]. We assume for a moment that we have no black hole, and calculate the real frequencies that can propagate in the pure five-dimensional Godel spacetime. In this setup, the spacetime geometry is horizon-free, and the solution of the scalar field perturbation in the background of the pure five-dimensional Godel spacetime should be regular at the origin r = 0. When ζ → 0, using the properties of Whittaker's function U (˜ a, ˜ b, ζ ) ∼ ζ 1 -˜ b Γ( ˜ b -1) / Γ(˜ a ), one can get the small r behavior of the far-region solution as So, when r → 0, r -2 l -2 →∞ , and the solution diverges. To have a regular solution at the origin r = 0, we must demand that Γ( l +1 -ρ ) →∞ . This occurs when the argument of the gamma function is a non-positive integer. Therefore, we have the condition So the requirement of the regularity of the wave solution at the origin selects the frequencies of the scalar field that might propagate in the pure five-dimensional Godel spacetime Now let us come back to the Kerr-Godel black hole case. In the spirit of [16], we expect that there will be a small imaginary part δ in the allowed frequencies induced by the black hole event horizon From Ψ ∼ e -iωt , one can see that the small imaginary δ describes the slow growing instability of the modes when δ > 0. Our task is to prove that δ is positive in the regime of superradiance. Inserting this expression for the frequency ω , one can get the far-region solution of the radial wave equation as In order to match the far-region solution with the near-region solution, we need to find the small r behavior of the far-region solution. It is known that the Whittaker's function U (˜ a, ˜ b, ζ ) can be expressed in terms of the Whittaker's function M (˜ a, ˜ b, ζ ). By inserting this relation on the far-region solution (40), we can show that the far-region solution can be rewritten as Applying to this expression the functional expressions for the gamma functions it is easy to show that and for the small δ Then by using the property of Whittaker's function M (˜ a, ˜ b, 0) = 1, one can get the small r behavior of the far-region solution as",
"pages": [
10,
11,
12
]
},
{
"title": "C. Matching condition: the unstable modes",
"content": "By comparing the large r behavior of the near-region solution with the small r behavior of the far-region solution, one can conclude that there exists the overlapping region M /lessmuch x -x + /lessmuch 1 /ω where the two solutions should match. In this region, the matching of the near-region solution in the large r region (30) and the far-region solution in the small r region (45) yields the allowed values of the small imaginary part δ in the frequency ω where with /pi1 = ( ω N -m Ω H ) r + ( r 4 + +2 Ma 2 ) 1 / 2 / 2( r 2 + -r 2 -). So, we have It is easy to see that, in the superradiance regime, Re [ ω ] -m Ω H < 0, the imaginary part of the complex frequency δ > 0. The scalar field has the time dependence e -iωt = e -iω N t e δt , which implies the exponential amplification of superradiance modes. This will lead to the instability of these modes. From the normal modes in pure five-dimensional Godel spacetime (38), we can see that Re [ ω ] ∼ j . We have assumed that ωM /lessmuch 1. So we have jM /lessmuch 1, which is consistent with the small Godel black hole assumption jr + /lessmuch 1 because M ∼ r 2 + . This is to say that the two assumptions we have made in this section are consistent with each other. Let us make a qualitative comparison of our analytical results with the numerical one in [22]. From Eq.(46), we can see that the growth rate δ of the superradiant modes are propotional to j 2 l +2 . This implies that the larger j corresponds to the higher superradiant instability growth rate, which supports the numerical conclusion in [22]. Next, the superradiant condition that Re [ ω ] -m Ω H < 0 can make a further limit on the parameter space where the superradiance can occur. From Eq.(38), we can see the superradiance condition for the mode of l = m = 1 and N = 0 becomes 24 j < Ω H . Because we are working on the parameter space that jr 2 + /lessmuch 1, we can further take the limit jM /lessmuch 1 in the expression of event horizon Eq.(4) and angular momentum Eq.(8). So the limit on the parameter j can be approximated by the following expression where, the parameter a/ √ M takes the value in the region of (0 , 0 . 71), otherwise there will be no black hole in spacetime. By submitting a/ √ M = 0 . 71 into the inequality, one can see that for j √ M ∼ 0 . 059 there is no superradiance. The numerical one in [22] is j √ M ∼ 0 . 075. The region of the superradiance in the parameter space by using the analytical method roughly coincides with the numerical results in [22]. So, comparing our result with the numerical one in [22], it can be seen that our results are rough and can reproduce the numerical conclusion partly. The analytical method can not solved the superradiance instability as precisely as the numerical method. At last, we can conclude that the five-dimensional small Kerr-Godel black hole is unstable against the massless scalar field perturbation. This instability is caused by the superradiance of the scalar field.",
"pages": [
12,
13,
14
]
},
{
"title": "V. CONCLUSION",
"content": "This paper is devoted to an analytical study of superradiant instability of five-dimensional small Kerr-Godel black hole. This instability has been found by R. A. Konoplya and A. Zhidenko using the numerical methods previously in [22]. Generally, superradiant instability naturally happens when two conditions are satisfied: (1) Black hole has rotation; (2) There is a natural reflecting mirror outside the black hole. In the present case, Dirichlet boundary condition at infinity for the asymptotically Godel black hole, which is obtained in section 3 by analogy with the AdS background, plays the role of reflecting mirror. We have adopted the analytical methods which is used in [16] to study the superradiant instability of small Kerr-AdS black hole. We assume that the energy of scalar field perturbation is low and the scale of Godel black hole is small. Then we divide the space outside the event horizon into the near-region and the far-region and employ the matched asymptotic expansion method to solve the scalar field wave equation. The analysis of complex quasinormal modes explicitly shows that the complex parts are positive in the regime of superradiance, which implies the growing instability of these modes. This is to say that the five-dimensional small Kerr-Godel black hole is unstable against scalar field perturbation. As is well-known, the gravitational perturbation will also undergo an superradiant amplification when scattered by the event horizon. Unlike the simply rotating Kerr-AdS black hole, where the superradiant instability of a massless scalar field implies the gravitational instability [32], we can not obtain the similar conclusion for asymptotically Godel black hole directly. So it will be interesting to study the gravitational (in)stability of five-dimensional Kerr-Godel black hole. Because of the complexity of the metric and the field equation, it will be a challenging project for future work.",
"pages": [
14
]
},
{
"title": "Acknowledgement",
"content": "The author would like to thank Ming-Fan Li for reading the manuscript and useful comments. This work was supported by NSFC, China (Grant No. 11147145 and No. 11205048).",
"pages": [
15
]
}
]
2013EPJC...73.2491J
https://arxiv.org/pdf/1301.4091.pdf
<document>
<section_header_level_1><location><page_1><loc_28><loc_92><loc_72><loc_93></location>Special Relativity induced by Granular Space</section_header_level_1>
<text><location><page_1><loc_36><loc_89><loc_65><loc_90></location>Petr Jizba 1, 2, ∗ and Fabio Scardigli 3, 4, †</text>
<text><location><page_1><loc_19><loc_82><loc_81><loc_88></location>1 FNSPE, Czech Technical University in Prague, Bˇrehov'a 7, 115 19 Praha 1, Czech Republic 2 ITP, Freie Universitat Berlin, Arnimallee 14 D-14195 Berlin, Germany 3 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 4 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_80></location>We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.</text>
<text><location><page_1><loc_18><loc_67><loc_51><loc_68></location>PACS numbers: 03.65.Ca, 03.30.+p, 05.40.-a, 04.60.-m</text>
<text><location><page_1><loc_9><loc_40><loc_49><loc_64></location>Introduction. - The concept of 'emergence' plays an important role in quantum field theory and, in particular, in particle and condensed matter physics, since it embodies the essential feature of systems with several interlocked time scales. In these systems, the observed macroscopic-scale dynamics and related degrees of freedom differ drastically from the actual underlying microscopic-scale physics [1]. Superstatistics provides a specific realization of this paradigm: It predicts that the emergent behavior can be often regarded as a superposition of several statistical systems that operate at different spatio-temporal scales [2, 3]. In particular, many applications have recently been reported, in hydrodynamic turbulence [4], turbulence in quantum liquids [5], pattern forming systems [6] or scattering processes in high-energy physics [7].</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_38></location>The essential assumption of the superstatistics scenario is the existence of sufficient spatio-temporal scale separations between relevant dynamics within the studied system so that the system has enough time to relax to a local equilibrium state and to stay within it for some time. In practical applications one is typically concerned with two scales. Following Ref. [2], we consider an intensive parameter ζ that fluctuates on a much larger time scale than the typical relaxation time of the local dynamics. The random variable ζ can be in practice identified, e.g., with the inverse temperature [2, 3], friction constant [8], volatility [9] or einbein [10]. On intuitive ground, one may understand the superstatistics by using the adiabatic Ansatz. Namely, the system under consideration, during its evolution, travels within its state space X (described by state variable A ∈ X ) which is partitioned into small cells characterized by a sharp value of ζ . Within each cell, the system is described by the conditional distribution p ( A | ζ ). As ζ varies adiabatically from cell to cell, the joint distribution of finding the system with a</text>
<text><location><page_1><loc_52><loc_57><loc_92><loc_64></location>sharp value of ζ in the state A is p ( A,ζ ) = p ( A | ζ ) p ( ζ ) (Bayes theorem). The resulting macro-scale (emergent) statistics p ( A ) for finding system in the state A is obtain by eliminating the nuisance parameter ζ through marginalization, that is</text>
<formula><location><page_1><loc_62><loc_52><loc_92><loc_56></location>p ( A ) = ∫ p ( A | ζ ) p ( ζ ) dζ . (1)</formula>
<text><location><page_1><loc_52><loc_46><loc_92><loc_52></location>Interestingly enough, the sufficient time scale separation between two relevant dynamics in a studied system allows to qualify superstatistics as a form of slow modulation [11].</text>
<text><location><page_1><loc_52><loc_26><loc_92><loc_46></location>In this Letter, we recast the Feynman transition amplitude of a relativistic scalar particle into a form, which (after being analytically continued to imaginary times) coincides with a superstatistics marginal probability (1). The derivation is based on the L'evy-Khinchine theorem for infinitely divisible distributions [12, 13], and for illustration we consider the dynamics and the propagator of a Klein-Gordon (i.e., neutral spin -0) particle. Our reasonings can be also extended to charged spin -0, spin -1 2 , Proca's spin -1 particles and to higherspin particles phrased via the Bargmann-Wigner wave equation [10]. Further generalization to external electromagnetic potential has been reported in Refs. [10, 14].</text>
<text><location><page_1><loc_52><loc_20><loc_92><loc_26></location>We also argue that the above formulation can be looked at as if the particle would randomly propagate (in the sense of Brownian motion) through an inhomogeneous or granular medium ('vacuum') [14].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_20></location>Our argument is based upon a recent observation [9, 10, 14] that the Euclidean path integral (PI) for relativistic particles may be interpreted as describing a doubly-stochastic process that operates at two separate spatio-temporal scales. The short spatial scale, which is much smaller than particle's Compton length λ C = 1 /mc ( /planckover2pi1 = 1), describes a Wiener (i.e., Galilean relativity) process with a sharp (Galilean-invariant) Newtonian mass.</text>
<text><location><page_2><loc_9><loc_65><loc_49><loc_93></location>The large spatial scale, which is of order λ C , corresponds to distances over which the fluctuating Newtonian mass changes appreciably. At scales much larger than λ C the particle evolves according to a genuine relativistic dynamics, with a sharp value of the mass coinciding with the Einstein rest mass. Particularly striking is the fact that when we average the particle's velocity over the structural correlation distance (i.e., over particle's λ C ) we obtain the velocity of light c . So the picture that emerges from this analysis is that the particle (with a non-zero mass!) propagates over the correlation distance λ C with an average velocity c , while at larger distance scales (i.e., when a more coarse grained view is taken) the particle propagates as a relativistic particle with a sharp mass and an average velocity that is subluminal. Quite remarkably, one can observe an identical behavior in the well-known Feynman's checkerboard PI [16, 17] to which the transition amplitude (1) reduces in the case of a relativistic Dirac fermion in 1 + 1 dimensions [10, 14].</text>
<text><location><page_2><loc_9><loc_59><loc_49><loc_64></location>A considerably expanded presentation including the issue of reparametrization invariance, bibliography, and proofs of the main statements and formulas is given in a companion paper [14].</text>
<text><location><page_2><loc_9><loc_53><loc_49><loc_58></location>Superstatistics path integrals. - When a conditional probability density function (PDF) is formulated through a PI, then it satisfies the Einstein-Smoluchowski equation (ESE) for continuous Markovian processes, namely [15]</text>
<formula><location><page_2><loc_12><loc_47><loc_49><loc_51></location>p ( y, t '' | x, t ) = ∫ ∞ -∞ d z p ( y, t '' | z, t ' ) p ( z, t ' | x, t ) , (2)</formula>
<text><location><page_2><loc_9><loc_41><loc_49><loc_47></location>with t ' being any time between t '' and t . Conversely, any transition probability satisfying ESE possesses a PI representation [16]. In physics one often encounters probabilities formulated as a superposition of PI's,</text>
<formula><location><page_2><loc_10><loc_32><loc_49><loc_40></location>℘ ( x ' , t ' | x, t ) = ∫ ∞ 0 d ζ ω ( ζ, T ) ∫ x ( t ' )= x ' x ( t )= x [d x d p ] e ∫ t ' t d τ (i p ˙ x -ζH ( p,x )) . (3)</formula>
<text><location><page_2><loc_9><loc_21><loc_49><loc_31></location>Here ω ( ζ, T ) with T = t ' -t is a normalized PDF defined on R + × R + . The form (3) typically appears in non-perturbative approximations to statistical partition functions, in polymer physics, in financial markets, in systems with reparametrization invariance, etc. The random variable ζ is then related to the inverse temperature, coupling constant, volatility, vielbein, etc.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_20></location>The existence of different time scales and the flow of the information from slow to fast degrees of freedom typically creates the irreversibility on the macroscopical level of the description. The corresponding information thus is not lost, but passes in a form incompatible with the Markovian description. The most general class of distributions ω ( ζ, T ) on R + × R + for which the superposition of Markovian processes remain Markovian, i.e., when also</text>
<text><location><page_2><loc_52><loc_87><loc_92><loc_93></location>℘ ( x ' , t ' | x, t ) satisfies the ESE (2), was found in Ref. [9]. The key is to note that in order to have (2) satisfied by ℘ , the rescaled PDF w ( ζ, T ) ≡ ω ( ζ/T, T ) /T should satisfy the ESE for homogeneous Markovian process</text>
<formula><location><page_2><loc_55><loc_82><loc_92><loc_86></location>w ( ζ, t 1 + t 2 ) = ∫ ζ 0 d ζ ' w ( ζ ' , t 1 ) w ( ζ -ζ ' , t 2 ) . (4)</formula>
<text><location><page_2><loc_52><loc_66><loc_92><loc_81></location>Consequently the Laplace image fulfills the functional equation with t 1 , t 2 ∈ R + . By assuming continuity in T , it follows that the multiplicative semigroup ˜ w ( p ζ , T ) T ≥ 0 satisfies ˜ w ( p ζ , T ) = { ˜ w ( p ζ , 1) } T . From this we see that the distribution of ζ at T is completely determined by the distribution of ζ at T = 1. In addition, because ˜ w ( p ζ , 1) = { ˜ w ( p ζ , 1 /n ) } n for any n ∈ N + , w ( ζ, 1) is infinitely divisible. The L'evy-Khinchine theorem [12, 13] then ensures that log ˜ w ( p ζ , T ) ≡ -TF ( p ζ ) must have the generic form</text>
<formula><location><page_2><loc_53><loc_60><loc_92><loc_64></location>log ˜ w ( p ζ , T ) = -T ( αp ζ + ∫ ∞ 0 (1 -e -p ζ u ) ν (d u ) ) , (5)</formula>
<text><location><page_2><loc_52><loc_49><loc_92><loc_60></location>where α ≥ 0 is a drift constant and ν is some non-negative measure on (0 , ∞ ) satisfying ∫ R + min(1 , u ) ν (d u ) < ∞ . Finally the Laplace inverse of ˜ w ( p ζ , T ) yields ω ( ζ, T ). Once ω ( ζ, T ) is found, then ℘ ( x ' , t ' | x, t ) possesses a PI representation on its own. What is the form of the new Hamiltonian? To this end we rewrite (3) in Dirac operator form as [9]</text>
<formula><location><page_2><loc_57><loc_41><loc_92><loc_48></location>℘ ( x ' , t ' | x, t ) = 〈 x ' | ∫ ∞ 0 d ζ w ( ζ, T ) e -ζ ˆ H | x 〉 = 〈 x ' |{ ˜ w ( ˆ H, 1) } T | x 〉 = 〈 x ' | e -TF ( ˆ H ) | x 〉 . (6)</formula>
<text><location><page_2><loc_52><loc_24><loc_92><loc_40></location>Hence, the identification H ( p , x ) = F ( H ( p , x )) can be made. Here one might worry about the operator-ordering problem. For our purpose it suffices to note that when H is x -independent, the former relation is exact. In more general instances the Weyl ordering is a natural choice because in this case the required mid-point rule follows automatically and one does not need to invoke the gauge invariance [9, 29]. In situations when other non-trivial configuration space symmetries (such as non-holonomic symmetry) are required, other orderings might be more physical [9].</text>
<text><location><page_2><loc_52><loc_18><loc_92><loc_24></location>Emergent Special Relativity. - The Feynman transition amplitudes (or better its Euclidean version - transition probabilities) naturally fits into the structure of superstatistics PI's discussed above.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_17></location>Note first that the choice α = 0 and ν (d u ) = 1 / (2 √ πu 3 / 2 )d u leads to F ( p ζ ) = √ p ζ . This identifies w ( ζ, T ) with the (unshifted) L'evy distribution with the scale parameter T 2 / 2. Moreover, when H ( p , x ) = p 2 c 2 + m 2 c 4 then (3) can be cast into the form (see also Refs. [9, 10, 14])</text>
<formula><location><page_3><loc_10><loc_81><loc_92><loc_91></location>∫ x ( T ) = x ' x (0) = x [d x d p ] exp { ∫ T 0 d τ [ i p · ˙ x -c √ p 2 + m 2 c 2 ] } = ∫ ∞ 0 d m f 1 2 ( m , T c 2 , T c 2 m 2 ) ∫ x ( T ) = x ' x (0) = x [d x d p ] exp { ∫ T 0 d τ [ i p · ˙ x -p 2 2 m -mc 2 ] } , (7)</formula>
<text><location><page_3><loc_9><loc_76><loc_24><loc_78></location>where t ' -t = T , and</text>
<formula><location><page_3><loc_13><loc_72><loc_49><loc_75></location>f p ( z, a, b ) = ( a/b ) p/ 2 2 K p ( √ ab ) z p -1 e -( az + b/z ) / 2 , (8)</formula>
<text><location><page_3><loc_9><loc_58><loc_49><loc_71></location>is the generalized inverse Gaussian distribution [13] ( K p is the modified Bessel function of the second kind with index p ). The LHS of (7) represents the PI for the free spinless relativistic particle in the Newton-Wigner representation [18]. The full Klein-Gordon (KG) kernel which also contains the negative-energy spectrum can be obtained from (7) by considering the Feshbach-Villars representation of the KG equation and making the substitution [10]</text>
<formula><location><page_3><loc_9><loc_52><loc_49><loc_56></location>f 1 2 ( m , tc 2 , tc 2 m 2 ) ↦→ 1+sgn( t ) σ 3 2 f 1 2 ( m , | t | c 2 , | t | c 2 m 2 ) . (9)</formula>
<text><location><page_3><loc_9><loc_42><loc_49><loc_52></location>The matrix nature of the smearing distribution ( σ 3 is the Pauli matrix) naturally includes the FeynmanStuckelberg causal boundary condition and thus treats both particles and antiparticles in a symmetric way [10, 19]. When the partition function is going to be calculated, the trace will get rid of the sgn( t ) term and 1 / 2 is turned to 1.</text>
<text><location><page_3><loc_9><loc_6><loc_49><loc_42></location>The explicit form of the identity (7) indicates that m can be interpreted as a Galilean-invariant Newton-like mass which takes on continuous values distributed according to f 1 2 ( m , T c 2 , T c 2 m 2 ) with 〈 ˜ m 〉 = m +1 /Tc 2 and var( m ) = m/Tc 2 + 2 /T 2 c 4 . Fluctuations of the Newtonian mass can be then depicted as originating from particle's evolution in an inhomogeneous or granular medium. Granularity, as known, for example, from solid-state systems, typically leads to corrections in the local dispersion relation [20] and hence to alterations in the local effective mass . The following picture thus emerges: on the short-distance scale, a non-relativistic particle can be envisaged as propagating via classical Brownian motion through a single grain with a local mass m . This fast-time process has a time scale ∼ 1 / m c 2 . An averaged value of the local time scale represents a transient temporal scale 〈 1 / m c 2 〉 = 1 /mc 2 which coincides with particle's Compton time T C - i.e., the time for light to cross the particle's Compton wavelength. At time scales much longer than T C (large-distance scale), the probability that the particle encounters a grain which endows it with a mass m is f 1 2 ( m , T c 2 , T c 2 m 2 ) . As a result one</text>
<text><location><page_3><loc_52><loc_52><loc_92><loc_78></location>may view a single-particle relativistic theory as a singleparticle non-relativistic theory where the particle's Newtonian mass m represents a fluctuating parameter which approaches on average the Einstein rest mass m in the large t limit. We stress that t should be understood as the observation time , a time after which the observation (position measurement) is made. In particular, during the period t the system remains unperturbed. One can thus justly expect that in the long run all mass fluctuations will be washed out and only a sharp time-independent effective mass will be perceived. The form of 〈 m 〉 identifies the time scale at which this happens with t ∼ 1 /mc 2 , i.e. with the Compton time T C . It should be stressed that above mass fluctuations have nothing to do with the Zitterbewegung which is caused by interference between positive- and negative-energy wave components. In our formulation both regimes are decoupled.</text>
<text><location><page_3><loc_52><loc_49><loc_92><loc_52></location>We may also observe that by coarse-graining the velocity over the correlation time T C we have</text>
<formula><location><page_3><loc_62><loc_43><loc_92><loc_48></location>〈| v |〉 T C = 〈| p |〉 〈 m 〉 ∣ ∣ ∣ T C = c . (10)</formula>
<text><location><page_3><loc_52><loc_24><loc_92><loc_46></location>∣ So on a short-time scale of order λ C the spinless relativistic particle propagates with an averaged velocity which is the speed of light c . But if one checks the particle's position at widely separated intervals (much larger than λ C ), then many directional reversals along a typical PI trajectory will take place, and the particle's net velocity will be then less than c - as it should be for a massive particle. The particle then acquires a sharp mass equal to Einstein's mass, and the process (not being hindered by fluctuating masses) is purely Brownian. This conclusion is in line with the well-known Feynman checkerboard picture [14, 17] to which it reduces in the case of (1+1)D relativistic Dirac particle.</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_23></location>Robustness of emergent special relativity. - Understanding the robustness of the emergent Special Relativity under small variations in the mass-smearing distribution function f 1 2 can guide the study of the relation between Einsteinian SR and other deformed variants of SR, such as Magueijo-Smolin and Amelino-Camelia's doubly special relativity [21, 22], or (quantum) κ -Poincar'e deformation of relativistic kinematics [23]. In DRS models a further invariant scale /lscript is introduced, besides the usual speed of light c , and /lscript is typically considered to be of</text>
<text><location><page_4><loc_9><loc_90><loc_49><loc_93></location>order of the Planck length. A small variation δ f 1 2 of the smearing function originates the new Hamiltonian</text>
<formula><location><page_4><loc_13><loc_85><loc_49><loc_89></location>¯ H = /epsilon1 1 4 + ( 1 + /epsilon1 0 2 ) √ p 2 c 2 + m 2 c 4 + /epsilon1 2 4 , (11)</formula>
<text><location><page_4><loc_9><loc_82><loc_49><loc_85></location>with /epsilon1 1 = -2 (1 + /epsilon1 0 / 2) √ /epsilon1 2 (see Ref. [14] for details). By setting</text>
<formula><location><page_4><loc_10><loc_76><loc_46><loc_80></location>/epsilon1 1 = 2 ( √ 1 1 -c 2 m 2 /lscript 2 -1 ) , /epsilon1 2 = 4 c 6 m 4 /lscript 2 1 -c 2 m 2 /lscript 2 ,</formula>
<text><location><page_4><loc_9><loc_74><loc_46><loc_75></location>the new Hamiltonian ¯ H can be easily identified with</text>
<formula><location><page_4><loc_12><loc_68><loc_49><loc_72></location>¯ H = c -m 2 c 2 /lscript ∓ √ p 2 (1 -m 2 c 2 /lscript 2 ) + m 2 c 2 1 -m 2 c 2 /lscript 2 , (12)</formula>
<text><location><page_4><loc_9><loc_59><loc_49><loc_68></location>which coincides with the Magueijo-Smolin's doubly special relativistic Hamiltonian, in, say, its version [24]. It should be stressed that the Hamiltonian (11) (when also negative energy states are included) violates CPT symmetry. This is a typical byproduct of the Lorentz symmetry violation in many deformed SR systems.</text>
<text><location><page_4><loc_9><loc_48><loc_49><loc_59></location>For the Hamiltonian (12) a relation analog to (7) holds, where now the smearing function has the form f 1 2 ( m , T c 2 λ, T c 2 m 2 λ ) with λ = 1 / (1 -m 2 c 2 /lscript 2 ). The correlation distance 1 /mcλ can be naturally assumed as the minimal size L GRAIN of the 'grain of space' of the polycrystalline medium, which is linked to the new invariant scale /lscript by</text>
<formula><location><page_4><loc_15><loc_44><loc_49><loc_47></location>L GRAIN := 1 mcλ = λ C (1 -m 2 c 2 /lscript 2 ) . (13)</formula>
<text><location><page_4><loc_9><loc_34><loc_49><loc_43></location>By tuning the size L GRAIN of these 'grains of space' it is possible to pass continuously from Lorenz symmetry to other different symmetries, as those enjoyed by DSR models. We can in principle speculate that each large scale symmetry could originate from a specific kind of space(time) granularity.</text>
<text><location><page_4><loc_9><loc_15><loc_49><loc_34></location>Quantum field theory. -The superstatistics transition probability (6) was constructed on the premise that H is associated with a single particle. Of course, a singleparticle relativistic quantum theory is logically untenable, since a multi-particle production is allowed whenever the particle reaches the threshold energy for pair production. In addition, Leutwyler's no-interaction theorem [25] prohibits interaction for any finite number of particles in the context of relativistic mechanics. To evade the no-interaction theorem it is necessary to have an infinite number of degrees of freedom to describe interaction. The latter is typically achieved via local quantum field theories (QFT).</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_14></location>It should be underlined in this context that the PI for a single relativistic particle is still a perfectly legitimate building block even in QFT. Recall that in the standard perturbative treatment of, say, generating functional for</text>
<text><location><page_4><loc_52><loc_89><loc_92><loc_93></location>a scalar field each Feynman diagram is composed of integrals over product of free correlation functions (Feynman's correlators):</text>
<formula><location><page_4><loc_53><loc_84><loc_92><loc_88></location>∆ F ( y , ct y ; z , ct z )= 1 4 ∫ ∞ -∞ d τ sgn( τ -t y ) ℘ ( y , τ | z , t z ) , (14)</formula>
<text><location><page_4><loc_52><loc_65><loc_92><loc_83></location>and may thus be considered as a functional of the PI ℘ ( x ' , t ' | x , t ). In fact, QFT in general, can be viewed as a grand-canonical ensemble of fluctuating particle histories (worldlines) where Feynman diagrammatic representation of quantum fields depicts directly the pictures of the world-lines in a grand-canonical ensemble. This is the so-called 'worldline quantization' of particle physics, and is epitomized, e.g., in Feynman's worldline representation of the one-loop affective action in quantum electrodynamics [26], in Kleinert's disorder field theory [27] or in the Bern-Kosower and Strassler 'string-inspired' approaches to QFT [28].</text>
<text><location><page_4><loc_52><loc_59><loc_92><loc_65></location>Because of (14), the relationship between bosonic Bern-Kosower Green's function G B ( τ 1 , τ 2 ) and the PI ℘ ( x ' , t ' | x , t ) can be found easily through the known functional relation between G B and ∆ F , cf. Refs. [28].</text>
<text><location><page_4><loc_52><loc_50><loc_92><loc_59></location>Gravity and Cosmology. -When spacetime is curved, a metric tensor enters in both PI's in (7) in a different way, yielding different 'counterterms' [15, 29]. For instance, in Bastianelli-van Nieuwenhuizen's time slicing regularization scheme [29] one has (when /planckover2pi1 is reintroduced)</text>
<formula><location><page_4><loc_51><loc_39><loc_92><loc_49></location>p 2 2 m ↦→ g ij p i p j 2 m + /planckover2pi1 2 8 m ( R + g ij Γ m il Γ l jm ) , √ p 2 + m 2 c 2 ↦→ √ g ij p i p j + /planckover2pi1 2 4 ( R + g ij Γ m il Γ l jm ) + m 2 c 2 + /planckover2pi1 4 Φ( R,∂R,∂ 2 R ) + O ( /planckover2pi1 6 ) , (15)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_39></location>where g ij , R , Γ j kl and Φ( . . . ) are the (space-like) pullback metric tensor, the scalar curvature, the Christoffel symbol, and non-vanishing function of R and its first and second derivatives, respectively. This causes the superstatistics identity (7) to break down, as can be explicitly checked to the lowest order in /planckover2pi1 . The respective two cases will thus lead to different physics. Because the Einstein equivalence principle requires that the local spacetime structure can be identified with the Minkowski spacetime possessing Lorentz symmetry, one might assume the validity of (7) at least locally. However, in different space-time points one has, in general, a different typical length scale of the local inertial frames, depending on the gravitational field. The characteristic size of the local inertial (i.e. Minkowski) frame is of order 1 / | K | 1 / 4 where K = R αβγδ R αβγδ is the Kretschmann invariant and R αβγδ is the Riemann curvature. Relation (7) tells us that the special relativistic description breaks down in regions of size smaller than λ C . For curvatures large enough, namely for strong gravitational</text>
<text><location><page_5><loc_9><loc_63><loc_49><loc_93></location>fields, the size of the local inertial frame can become smaller than λ C , that is 1 / | K | 1 / 4 /lessorsimilar λ C . In such regions the special relativistic description is no more valid, and according to (7) must be replaced by a Newtonian description of the events. For instance, in Schwarzschild geometry we have K = 12 r 2 s /r 6 , and the breakdown should be expected at radial distances r /lessorsimilar ( λ 2 C r s ) 1 / 3 ( r s is the Schwarzschild radius) which are - apart from the hypothetical case of micro-black holes (where λ C /similarequal r s ) - always deeply buried below the Schwarzschild event horizon. In the cosmologically relevant FriedmannLemaˆıtre-Robertson-Walker (FLRW) geometry, we have K = 12(˙ a 4 + a 2 a 2 ) / ( ac ) 4 , and the breakdown happens when (˙ a 4 + a 2 a 2 ) /greaterorsimilar ( ac/λ C ) 4 , where a ( t ) is the FLRW scale factor of the Universe and ˙ a = d a/ d t . Applying the well-known Vilenkin-Ford model [30] for inflationary cosmology, where a ( t ) is given by: a ( t ) = A √ sinh( Bt ) with B = 2 c √ Λ / 3 (Λ is the cosmological constant), we obtain a temporal bound on the validity of local Lorentz invariance, which, expressed in FLRW time, is</text>
<formula><location><page_5><loc_12><loc_57><loc_49><loc_62></location>t /lessorsimilar 1 B arcsinh [ Bλ C (8 c 4 -( Bλ C ) 4 ) 1 / 4 ] ≡ ¯ t . (16)</formula>
<text><location><page_5><loc_9><loc_36><loc_49><loc_57></location>By using the presently known [31] value Λ /similarequal 10 -52 m -2 and the τ -lepton Compton's wavelength λ τ C /similarequal 6 . 7 × 10 -16 m (yielding the tightest upper bound on t ), we obtain ¯ t /similarequal 4 × 10 -24 s. Note that, since Bλ C /lessmuch c , then ¯ t /similarequal λ C /c = t C . Such a violation of the local Lorentz invariance naturally breaks the particle-antiparticle symmetry since there is no unified theory of particles and antiparticles in the non-relativistic physics - formally one has two separate theories. If the resulting matterantimatter asymmetry provides a large enough CP asymmetry then this might have essential consequences in the early Universe, e.g., for leptogenesis. In this respect, ¯ t is compatible with the nonthermal leptogenesis period that typically dates between 10 -26 -10 -12 s after the Big Bang.</text>
<text><location><page_5><loc_9><loc_12><loc_49><loc_36></location>Conclusions and perspectives. -The new superstatistics PI representation of a relativistic point particle introduced in this Letter, realizes an explicit quantum mechanical duality between Einsteinian and Galilean relativity. It also makes explicit how the SR invariance is encoded in the grain smearing distribution. Notably, the exact LS of a spacetime has no fundamental significance in our analysis, as it is only an accidental symmetry of the coarse-grained configuration space in which a particle executes a standard Wiener process. In passage from grain to grain particles's Newtonian mass fluctuates according to an inverse Gaussian distribution. The observed inertial mass of the particle is thus not a fundamental constant, but it reflects the particle's interaction with the granular vacuum (cosmic field). This, in a sense, supports Mach's view of the phenomenon of inertia.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_11></location>Interactions can be included in our framework in two different ways. The interaction with a background field</text>
<text><location><page_5><loc_52><loc_69><loc_92><loc_93></location>(such as electromagnetic field) can be directly treated with the superstatistics prescription (7), see [10]. On the other hand, the multi-particle interactions can be consistently formulated by 'embedding' the relativistic PI in QFT via the worldline quantization. Such an embedding may help to study several cosmological implications of systems with granular space. If any of such systems quickly flows to the infrared fixed point, any direct effect due to the space discreteness, and related SR violation, might be insignificant on cosmological scales (where Lorentz and diffeomorphism invariance are restored), while it might be crucial in the early Universe, e.g., for leptogenesis and the ensuing baryogenesis. Consequences on the detailed structure of the Cosmic Microwave Background spectrum will be explored in future work.</text>
<text><location><page_5><loc_52><loc_45><loc_92><loc_69></location>The presented approach implies a preferred frame. In this connection it is worth of noting that, despite the fact that (7) is not manifestly LS invariant, one may use the Stuckelberg trick and introduce a new fictitious variable into the PI (7), in such a way that the new action will have the reparametrization symmetry, but will still be dynamically equivalent to the original action. For relevant details see Ref. [14]. By not knowing the source, one may then view this artificial gauge invariance as being a fundamental or even a defining property of the relativistic theory. One might, however, equally well, proclaim the 'polycrystalline' picture as being a basic (or primitive) edifice of SR and view the reparametrization symmetry as a mere artefact of an artificial redundancy that is allowed in our description. It is this second view that we favored here.</text>
<text><location><page_5><loc_63><loc_40><loc_63><loc_42></location>/negationslash</text>
<text><location><page_5><loc_52><loc_19><loc_92><loc_45></location>The presented scenario cannot directly accommodate the massless particles such as photons (identity (7) holds true only for m = 0). One possibility would be to use the PI representation of Polyakov-Wheeler for massless particles and try to construct a similar superstatistics duality as in the case of massive particles. This procedure is, however, not without technical difficulties and currently is under investigation. Conceptually is far more simpler to assume that the photon has a small mass. At present, there are a number of experimental limits to the mass of the photons [32]. For instance, tests based on Coulomb's law and the galactic vector potential set the upper limit of m γ /lessorsimilar 10 -18 eV/c 2 /similarequal 10 -57 g. This gives the domain correlation distance for the photon /similarequal 1 /m γ c 2 /similarequal 10 43 m which is bigger than the radius of observable Universe ( /similarequal 10 26 m) and so in this picture the photon mass does not fluctuate - it is a quasi-invariant.</text>
<text><location><page_5><loc_52><loc_10><loc_92><loc_19></location>Finally, this approach should reinforce the links between superstatistics paradigm and the approach to quantum gravity based on stochastic quantization [33]. In particular, the outlined granular space could be a natural model for the noise terms in a Parisi-Wu stochasticlike quantization approach to gravity.</text>
<text><location><page_5><loc_53><loc_9><loc_92><loc_10></location>Acknowledgement. - The writers are grateful to</text>
<text><location><page_6><loc_9><loc_89><loc_49><loc_93></location>H. Kleinert, Z. Haba, M. Sakellariadou, and L.S. Schulman for useful feedbacks. This work was supported in part by GA ˇ CR Grant No. P402/12/J077.</text>
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</document>
[
{
"title": "Special Relativity induced by Granular Space",
"content": "Petr Jizba 1, 2, ∗ and Fabio Scardigli 3, 4, † 1 FNSPE, Czech Technical University in Prague, Bˇrehov'a 7, 115 19 Praha 1, Czech Republic 2 ITP, Freie Universitat Berlin, Arnimallee 14 D-14195 Berlin, Germany 3 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy 4 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed. PACS numbers: 03.65.Ca, 03.30.+p, 05.40.-a, 04.60.-m Introduction. - The concept of 'emergence' plays an important role in quantum field theory and, in particular, in particle and condensed matter physics, since it embodies the essential feature of systems with several interlocked time scales. In these systems, the observed macroscopic-scale dynamics and related degrees of freedom differ drastically from the actual underlying microscopic-scale physics [1]. Superstatistics provides a specific realization of this paradigm: It predicts that the emergent behavior can be often regarded as a superposition of several statistical systems that operate at different spatio-temporal scales [2, 3]. In particular, many applications have recently been reported, in hydrodynamic turbulence [4], turbulence in quantum liquids [5], pattern forming systems [6] or scattering processes in high-energy physics [7]. The essential assumption of the superstatistics scenario is the existence of sufficient spatio-temporal scale separations between relevant dynamics within the studied system so that the system has enough time to relax to a local equilibrium state and to stay within it for some time. In practical applications one is typically concerned with two scales. Following Ref. [2], we consider an intensive parameter ζ that fluctuates on a much larger time scale than the typical relaxation time of the local dynamics. The random variable ζ can be in practice identified, e.g., with the inverse temperature [2, 3], friction constant [8], volatility [9] or einbein [10]. On intuitive ground, one may understand the superstatistics by using the adiabatic Ansatz. Namely, the system under consideration, during its evolution, travels within its state space X (described by state variable A ∈ X ) which is partitioned into small cells characterized by a sharp value of ζ . Within each cell, the system is described by the conditional distribution p ( A | ζ ). As ζ varies adiabatically from cell to cell, the joint distribution of finding the system with a sharp value of ζ in the state A is p ( A,ζ ) = p ( A | ζ ) p ( ζ ) (Bayes theorem). The resulting macro-scale (emergent) statistics p ( A ) for finding system in the state A is obtain by eliminating the nuisance parameter ζ through marginalization, that is Interestingly enough, the sufficient time scale separation between two relevant dynamics in a studied system allows to qualify superstatistics as a form of slow modulation [11]. In this Letter, we recast the Feynman transition amplitude of a relativistic scalar particle into a form, which (after being analytically continued to imaginary times) coincides with a superstatistics marginal probability (1). The derivation is based on the L'evy-Khinchine theorem for infinitely divisible distributions [12, 13], and for illustration we consider the dynamics and the propagator of a Klein-Gordon (i.e., neutral spin -0) particle. Our reasonings can be also extended to charged spin -0, spin -1 2 , Proca's spin -1 particles and to higherspin particles phrased via the Bargmann-Wigner wave equation [10]. Further generalization to external electromagnetic potential has been reported in Refs. [10, 14]. We also argue that the above formulation can be looked at as if the particle would randomly propagate (in the sense of Brownian motion) through an inhomogeneous or granular medium ('vacuum') [14]. Our argument is based upon a recent observation [9, 10, 14] that the Euclidean path integral (PI) for relativistic particles may be interpreted as describing a doubly-stochastic process that operates at two separate spatio-temporal scales. The short spatial scale, which is much smaller than particle's Compton length λ C = 1 /mc ( /planckover2pi1 = 1), describes a Wiener (i.e., Galilean relativity) process with a sharp (Galilean-invariant) Newtonian mass. The large spatial scale, which is of order λ C , corresponds to distances over which the fluctuating Newtonian mass changes appreciably. At scales much larger than λ C the particle evolves according to a genuine relativistic dynamics, with a sharp value of the mass coinciding with the Einstein rest mass. Particularly striking is the fact that when we average the particle's velocity over the structural correlation distance (i.e., over particle's λ C ) we obtain the velocity of light c . So the picture that emerges from this analysis is that the particle (with a non-zero mass!) propagates over the correlation distance λ C with an average velocity c , while at larger distance scales (i.e., when a more coarse grained view is taken) the particle propagates as a relativistic particle with a sharp mass and an average velocity that is subluminal. Quite remarkably, one can observe an identical behavior in the well-known Feynman's checkerboard PI [16, 17] to which the transition amplitude (1) reduces in the case of a relativistic Dirac fermion in 1 + 1 dimensions [10, 14]. A considerably expanded presentation including the issue of reparametrization invariance, bibliography, and proofs of the main statements and formulas is given in a companion paper [14]. Superstatistics path integrals. - When a conditional probability density function (PDF) is formulated through a PI, then it satisfies the Einstein-Smoluchowski equation (ESE) for continuous Markovian processes, namely [15] with t ' being any time between t '' and t . Conversely, any transition probability satisfying ESE possesses a PI representation [16]. In physics one often encounters probabilities formulated as a superposition of PI's, Here ω ( ζ, T ) with T = t ' -t is a normalized PDF defined on R + × R + . The form (3) typically appears in non-perturbative approximations to statistical partition functions, in polymer physics, in financial markets, in systems with reparametrization invariance, etc. The random variable ζ is then related to the inverse temperature, coupling constant, volatility, vielbein, etc. The existence of different time scales and the flow of the information from slow to fast degrees of freedom typically creates the irreversibility on the macroscopical level of the description. The corresponding information thus is not lost, but passes in a form incompatible with the Markovian description. The most general class of distributions ω ( ζ, T ) on R + × R + for which the superposition of Markovian processes remain Markovian, i.e., when also ℘ ( x ' , t ' | x, t ) satisfies the ESE (2), was found in Ref. [9]. The key is to note that in order to have (2) satisfied by ℘ , the rescaled PDF w ( ζ, T ) ≡ ω ( ζ/T, T ) /T should satisfy the ESE for homogeneous Markovian process Consequently the Laplace image fulfills the functional equation with t 1 , t 2 ∈ R + . By assuming continuity in T , it follows that the multiplicative semigroup ˜ w ( p ζ , T ) T ≥ 0 satisfies ˜ w ( p ζ , T ) = { ˜ w ( p ζ , 1) } T . From this we see that the distribution of ζ at T is completely determined by the distribution of ζ at T = 1. In addition, because ˜ w ( p ζ , 1) = { ˜ w ( p ζ , 1 /n ) } n for any n ∈ N + , w ( ζ, 1) is infinitely divisible. The L'evy-Khinchine theorem [12, 13] then ensures that log ˜ w ( p ζ , T ) ≡ -TF ( p ζ ) must have the generic form where α ≥ 0 is a drift constant and ν is some non-negative measure on (0 , ∞ ) satisfying ∫ R + min(1 , u ) ν (d u ) < ∞ . Finally the Laplace inverse of ˜ w ( p ζ , T ) yields ω ( ζ, T ). Once ω ( ζ, T ) is found, then ℘ ( x ' , t ' | x, t ) possesses a PI representation on its own. What is the form of the new Hamiltonian? To this end we rewrite (3) in Dirac operator form as [9] Hence, the identification H ( p , x ) = F ( H ( p , x )) can be made. Here one might worry about the operator-ordering problem. For our purpose it suffices to note that when H is x -independent, the former relation is exact. In more general instances the Weyl ordering is a natural choice because in this case the required mid-point rule follows automatically and one does not need to invoke the gauge invariance [9, 29]. In situations when other non-trivial configuration space symmetries (such as non-holonomic symmetry) are required, other orderings might be more physical [9]. Emergent Special Relativity. - The Feynman transition amplitudes (or better its Euclidean version - transition probabilities) naturally fits into the structure of superstatistics PI's discussed above. Note first that the choice α = 0 and ν (d u ) = 1 / (2 √ πu 3 / 2 )d u leads to F ( p ζ ) = √ p ζ . This identifies w ( ζ, T ) with the (unshifted) L'evy distribution with the scale parameter T 2 / 2. Moreover, when H ( p , x ) = p 2 c 2 + m 2 c 4 then (3) can be cast into the form (see also Refs. [9, 10, 14]) where t ' -t = T , and is the generalized inverse Gaussian distribution [13] ( K p is the modified Bessel function of the second kind with index p ). The LHS of (7) represents the PI for the free spinless relativistic particle in the Newton-Wigner representation [18]. The full Klein-Gordon (KG) kernel which also contains the negative-energy spectrum can be obtained from (7) by considering the Feshbach-Villars representation of the KG equation and making the substitution [10] The matrix nature of the smearing distribution ( σ 3 is the Pauli matrix) naturally includes the FeynmanStuckelberg causal boundary condition and thus treats both particles and antiparticles in a symmetric way [10, 19]. When the partition function is going to be calculated, the trace will get rid of the sgn( t ) term and 1 / 2 is turned to 1. The explicit form of the identity (7) indicates that m can be interpreted as a Galilean-invariant Newton-like mass which takes on continuous values distributed according to f 1 2 ( m , T c 2 , T c 2 m 2 ) with 〈 ˜ m 〉 = m +1 /Tc 2 and var( m ) = m/Tc 2 + 2 /T 2 c 4 . Fluctuations of the Newtonian mass can be then depicted as originating from particle's evolution in an inhomogeneous or granular medium. Granularity, as known, for example, from solid-state systems, typically leads to corrections in the local dispersion relation [20] and hence to alterations in the local effective mass . The following picture thus emerges: on the short-distance scale, a non-relativistic particle can be envisaged as propagating via classical Brownian motion through a single grain with a local mass m . This fast-time process has a time scale ∼ 1 / m c 2 . An averaged value of the local time scale represents a transient temporal scale 〈 1 / m c 2 〉 = 1 /mc 2 which coincides with particle's Compton time T C - i.e., the time for light to cross the particle's Compton wavelength. At time scales much longer than T C (large-distance scale), the probability that the particle encounters a grain which endows it with a mass m is f 1 2 ( m , T c 2 , T c 2 m 2 ) . As a result one may view a single-particle relativistic theory as a singleparticle non-relativistic theory where the particle's Newtonian mass m represents a fluctuating parameter which approaches on average the Einstein rest mass m in the large t limit. We stress that t should be understood as the observation time , a time after which the observation (position measurement) is made. In particular, during the period t the system remains unperturbed. One can thus justly expect that in the long run all mass fluctuations will be washed out and only a sharp time-independent effective mass will be perceived. The form of 〈 m 〉 identifies the time scale at which this happens with t ∼ 1 /mc 2 , i.e. with the Compton time T C . It should be stressed that above mass fluctuations have nothing to do with the Zitterbewegung which is caused by interference between positive- and negative-energy wave components. In our formulation both regimes are decoupled. We may also observe that by coarse-graining the velocity over the correlation time T C we have ∣ So on a short-time scale of order λ C the spinless relativistic particle propagates with an averaged velocity which is the speed of light c . But if one checks the particle's position at widely separated intervals (much larger than λ C ), then many directional reversals along a typical PI trajectory will take place, and the particle's net velocity will be then less than c - as it should be for a massive particle. The particle then acquires a sharp mass equal to Einstein's mass, and the process (not being hindered by fluctuating masses) is purely Brownian. This conclusion is in line with the well-known Feynman checkerboard picture [14, 17] to which it reduces in the case of (1+1)D relativistic Dirac particle. Robustness of emergent special relativity. - Understanding the robustness of the emergent Special Relativity under small variations in the mass-smearing distribution function f 1 2 can guide the study of the relation between Einsteinian SR and other deformed variants of SR, such as Magueijo-Smolin and Amelino-Camelia's doubly special relativity [21, 22], or (quantum) κ -Poincar'e deformation of relativistic kinematics [23]. In DRS models a further invariant scale /lscript is introduced, besides the usual speed of light c , and /lscript is typically considered to be of order of the Planck length. A small variation δ f 1 2 of the smearing function originates the new Hamiltonian with /epsilon1 1 = -2 (1 + /epsilon1 0 / 2) √ /epsilon1 2 (see Ref. [14] for details). By setting the new Hamiltonian ¯ H can be easily identified with which coincides with the Magueijo-Smolin's doubly special relativistic Hamiltonian, in, say, its version [24]. It should be stressed that the Hamiltonian (11) (when also negative energy states are included) violates CPT symmetry. This is a typical byproduct of the Lorentz symmetry violation in many deformed SR systems. For the Hamiltonian (12) a relation analog to (7) holds, where now the smearing function has the form f 1 2 ( m , T c 2 λ, T c 2 m 2 λ ) with λ = 1 / (1 -m 2 c 2 /lscript 2 ). The correlation distance 1 /mcλ can be naturally assumed as the minimal size L GRAIN of the 'grain of space' of the polycrystalline medium, which is linked to the new invariant scale /lscript by By tuning the size L GRAIN of these 'grains of space' it is possible to pass continuously from Lorenz symmetry to other different symmetries, as those enjoyed by DSR models. We can in principle speculate that each large scale symmetry could originate from a specific kind of space(time) granularity. Quantum field theory. -The superstatistics transition probability (6) was constructed on the premise that H is associated with a single particle. Of course, a singleparticle relativistic quantum theory is logically untenable, since a multi-particle production is allowed whenever the particle reaches the threshold energy for pair production. In addition, Leutwyler's no-interaction theorem [25] prohibits interaction for any finite number of particles in the context of relativistic mechanics. To evade the no-interaction theorem it is necessary to have an infinite number of degrees of freedom to describe interaction. The latter is typically achieved via local quantum field theories (QFT). It should be underlined in this context that the PI for a single relativistic particle is still a perfectly legitimate building block even in QFT. Recall that in the standard perturbative treatment of, say, generating functional for a scalar field each Feynman diagram is composed of integrals over product of free correlation functions (Feynman's correlators): and may thus be considered as a functional of the PI ℘ ( x ' , t ' | x , t ). In fact, QFT in general, can be viewed as a grand-canonical ensemble of fluctuating particle histories (worldlines) where Feynman diagrammatic representation of quantum fields depicts directly the pictures of the world-lines in a grand-canonical ensemble. This is the so-called 'worldline quantization' of particle physics, and is epitomized, e.g., in Feynman's worldline representation of the one-loop affective action in quantum electrodynamics [26], in Kleinert's disorder field theory [27] or in the Bern-Kosower and Strassler 'string-inspired' approaches to QFT [28]. Because of (14), the relationship between bosonic Bern-Kosower Green's function G B ( τ 1 , τ 2 ) and the PI ℘ ( x ' , t ' | x , t ) can be found easily through the known functional relation between G B and ∆ F , cf. Refs. [28]. Gravity and Cosmology. -When spacetime is curved, a metric tensor enters in both PI's in (7) in a different way, yielding different 'counterterms' [15, 29]. For instance, in Bastianelli-van Nieuwenhuizen's time slicing regularization scheme [29] one has (when /planckover2pi1 is reintroduced) where g ij , R , Γ j kl and Φ( . . . ) are the (space-like) pullback metric tensor, the scalar curvature, the Christoffel symbol, and non-vanishing function of R and its first and second derivatives, respectively. This causes the superstatistics identity (7) to break down, as can be explicitly checked to the lowest order in /planckover2pi1 . The respective two cases will thus lead to different physics. Because the Einstein equivalence principle requires that the local spacetime structure can be identified with the Minkowski spacetime possessing Lorentz symmetry, one might assume the validity of (7) at least locally. However, in different space-time points one has, in general, a different typical length scale of the local inertial frames, depending on the gravitational field. The characteristic size of the local inertial (i.e. Minkowski) frame is of order 1 / | K | 1 / 4 where K = R αβγδ R αβγδ is the Kretschmann invariant and R αβγδ is the Riemann curvature. Relation (7) tells us that the special relativistic description breaks down in regions of size smaller than λ C . For curvatures large enough, namely for strong gravitational fields, the size of the local inertial frame can become smaller than λ C , that is 1 / | K | 1 / 4 /lessorsimilar λ C . In such regions the special relativistic description is no more valid, and according to (7) must be replaced by a Newtonian description of the events. For instance, in Schwarzschild geometry we have K = 12 r 2 s /r 6 , and the breakdown should be expected at radial distances r /lessorsimilar ( λ 2 C r s ) 1 / 3 ( r s is the Schwarzschild radius) which are - apart from the hypothetical case of micro-black holes (where λ C /similarequal r s ) - always deeply buried below the Schwarzschild event horizon. In the cosmologically relevant FriedmannLemaˆıtre-Robertson-Walker (FLRW) geometry, we have K = 12(˙ a 4 + a 2 a 2 ) / ( ac ) 4 , and the breakdown happens when (˙ a 4 + a 2 a 2 ) /greaterorsimilar ( ac/λ C ) 4 , where a ( t ) is the FLRW scale factor of the Universe and ˙ a = d a/ d t . Applying the well-known Vilenkin-Ford model [30] for inflationary cosmology, where a ( t ) is given by: a ( t ) = A √ sinh( Bt ) with B = 2 c √ Λ / 3 (Λ is the cosmological constant), we obtain a temporal bound on the validity of local Lorentz invariance, which, expressed in FLRW time, is By using the presently known [31] value Λ /similarequal 10 -52 m -2 and the τ -lepton Compton's wavelength λ τ C /similarequal 6 . 7 × 10 -16 m (yielding the tightest upper bound on t ), we obtain ¯ t /similarequal 4 × 10 -24 s. Note that, since Bλ C /lessmuch c , then ¯ t /similarequal λ C /c = t C . Such a violation of the local Lorentz invariance naturally breaks the particle-antiparticle symmetry since there is no unified theory of particles and antiparticles in the non-relativistic physics - formally one has two separate theories. If the resulting matterantimatter asymmetry provides a large enough CP asymmetry then this might have essential consequences in the early Universe, e.g., for leptogenesis. In this respect, ¯ t is compatible with the nonthermal leptogenesis period that typically dates between 10 -26 -10 -12 s after the Big Bang. Conclusions and perspectives. -The new superstatistics PI representation of a relativistic point particle introduced in this Letter, realizes an explicit quantum mechanical duality between Einsteinian and Galilean relativity. It also makes explicit how the SR invariance is encoded in the grain smearing distribution. Notably, the exact LS of a spacetime has no fundamental significance in our analysis, as it is only an accidental symmetry of the coarse-grained configuration space in which a particle executes a standard Wiener process. In passage from grain to grain particles's Newtonian mass fluctuates according to an inverse Gaussian distribution. The observed inertial mass of the particle is thus not a fundamental constant, but it reflects the particle's interaction with the granular vacuum (cosmic field). This, in a sense, supports Mach's view of the phenomenon of inertia. Interactions can be included in our framework in two different ways. The interaction with a background field (such as electromagnetic field) can be directly treated with the superstatistics prescription (7), see [10]. On the other hand, the multi-particle interactions can be consistently formulated by 'embedding' the relativistic PI in QFT via the worldline quantization. Such an embedding may help to study several cosmological implications of systems with granular space. If any of such systems quickly flows to the infrared fixed point, any direct effect due to the space discreteness, and related SR violation, might be insignificant on cosmological scales (where Lorentz and diffeomorphism invariance are restored), while it might be crucial in the early Universe, e.g., for leptogenesis and the ensuing baryogenesis. Consequences on the detailed structure of the Cosmic Microwave Background spectrum will be explored in future work. The presented approach implies a preferred frame. In this connection it is worth of noting that, despite the fact that (7) is not manifestly LS invariant, one may use the Stuckelberg trick and introduce a new fictitious variable into the PI (7), in such a way that the new action will have the reparametrization symmetry, but will still be dynamically equivalent to the original action. For relevant details see Ref. [14]. By not knowing the source, one may then view this artificial gauge invariance as being a fundamental or even a defining property of the relativistic theory. One might, however, equally well, proclaim the 'polycrystalline' picture as being a basic (or primitive) edifice of SR and view the reparametrization symmetry as a mere artefact of an artificial redundancy that is allowed in our description. It is this second view that we favored here. /negationslash The presented scenario cannot directly accommodate the massless particles such as photons (identity (7) holds true only for m = 0). One possibility would be to use the PI representation of Polyakov-Wheeler for massless particles and try to construct a similar superstatistics duality as in the case of massive particles. This procedure is, however, not without technical difficulties and currently is under investigation. Conceptually is far more simpler to assume that the photon has a small mass. At present, there are a number of experimental limits to the mass of the photons [32]. For instance, tests based on Coulomb's law and the galactic vector potential set the upper limit of m γ /lessorsimilar 10 -18 eV/c 2 /similarequal 10 -57 g. This gives the domain correlation distance for the photon /similarequal 1 /m γ c 2 /similarequal 10 43 m which is bigger than the radius of observable Universe ( /similarequal 10 26 m) and so in this picture the photon mass does not fluctuate - it is a quasi-invariant. Finally, this approach should reinforce the links between superstatistics paradigm and the approach to quantum gravity based on stochastic quantization [33]. In particular, the outlined granular space could be a natural model for the noise terms in a Parisi-Wu stochasticlike quantization approach to gravity. Acknowledgement. - The writers are grateful to H. Kleinert, Z. Haba, M. Sakellariadou, and L.S. Schulman for useful feedbacks. This work was supported in part by GA ˇ CR Grant No. P402/12/J077.",
"pages": [
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2013EPJC...73.2515M
https://arxiv.org/pdf/1307.6265.pdf
<document>
<section_header_level_1><location><page_1><loc_26><loc_81><loc_74><loc_85></location>Comment on Ricci dark energy in Chern-Simons modified gravity</section_header_level_1>
<text><location><page_1><loc_41><loc_77><loc_59><loc_79></location>Yun Soo Myung a</text>
<text><location><page_1><loc_21><loc_72><loc_79><loc_75></location>Institute of Basic Science and Department of Computer Simulation, Inje University, Gimhae 621-749, Korea</text>
<section_header_level_1><location><page_1><loc_46><loc_69><loc_54><loc_70></location>Abstract</section_header_level_1>
<text><location><page_1><loc_17><loc_55><loc_83><loc_67></location>We revisit Ricci dark energy in Chern-Simons modified gravity. As far as the cosmological evolution, this is nothing but the Ricci dark energy with a minimally coupled scalar without potential which means that the role of Chern-Simons term is suppressed. Using the equation of state parameter, this model is similar to the modified Chaplygin gas model only when two are around the de Sitter universe deriving by the cosmological constant in the future. However, two past evolutions are different.</text>
<text><location><page_1><loc_17><loc_51><loc_70><loc_53></location>Keywords: Ricci dark energy; Chern-Simons modified gravity</text>
<text><location><page_1><loc_17><loc_45><loc_35><loc_46></location>a ysmyung@inje.ac.kr</text>
<text><location><page_2><loc_17><loc_78><loc_83><loc_84></location>Recently, the authors [1] have investigated the Ricci dark energy model in the dynamic Chern-Simons modified gravity which states that its cosmological evolution is similar to that displayed by the modified Chaplygin gas model.</text>
<text><location><page_2><loc_17><loc_64><loc_83><loc_77></location>In this Comment, we wish to draw the reader two important issues: One is that as far as the cosmological evolution, this model is nothing but the Ricci dark energy with a minimally coupled scalar without potential where the role of Chern-Simons term is suppressed totally. The other is that using the equation of state parameter, this model is similar to the modified Chaplygin gas model only when two make turnaround of de Sitter universe deriving by the cosmological constant in the future. In general, two provide different evolutions.</text>
<text><location><page_2><loc_17><loc_61><loc_83><loc_64></location>We start with the dynamic Chern-Simons modified gravity action with Ricci dark energy [1]</text>
<formula><location><page_2><loc_21><loc_54><loc_83><loc_59></location>S = 1 16 πG ∫ d 4 x √ -g [ R -˜ θ 4 ∗ RR -1 2 ∂ µ ˜ θ∂ µ ˜ θ + V ( ˜ θ ) ] + S RDE , (1)</formula>
<text><location><page_2><loc_17><loc_48><loc_83><loc_53></location>where ∗ RR is the Pontryagin term, ˜ θ is a dynamical scalar and S RDE is the action to give the Ricci dark energy. Here, for simplicity, one chooses V ( ˜ θ ) = 0. Their equations are given by</text>
<formula><location><page_2><loc_38><loc_45><loc_83><loc_46></location>G µν + C µν = 8 πGT µν , (2)</formula>
<formula><location><page_2><loc_45><loc_42><loc_83><loc_45></location>2 ˜ θ = 1 ∗ RR, (3)</formula>
<formula><location><page_2><loc_44><loc_41><loc_57><loc_44></location>∇ -64 π</formula>
<text><location><page_2><loc_17><loc_37><loc_83><loc_40></location>where G µν is the Einstein tensor, C µν is the Cotton tensor from the ChernSimons term ' ˜ θ ∗ RR '-term. The energy-momentum tensor is given by</text>
<formula><location><page_2><loc_42><loc_34><loc_83><loc_36></location>T µν = T RDE µν + T ˜ θ µν (4)</formula>
<text><location><page_2><loc_17><loc_31><loc_21><loc_32></location>with</text>
<text><location><page_2><loc_17><loc_26><loc_20><loc_28></location>and</text>
<formula><location><page_2><loc_33><loc_29><loc_83><loc_31></location>T RDE µν = ( ρ RDE + p RDE ) u µ u ν + p RDE g µν (5)</formula>
<formula><location><page_2><loc_38><loc_23><loc_83><loc_27></location>T ˜ θ µν = ∂ µ ˜ θ∂ ν ˜ θ -1 2 g µν ∂ µ ˜ θ∂ µ ˜ θ. (6)</formula>
<text><location><page_2><loc_17><loc_21><loc_69><loc_23></location>Applying ∇ µ to (2) leads to the conservation-law for T µν as</text>
<formula><location><page_2><loc_45><loc_17><loc_83><loc_20></location>∇ µ T µν = 0 , (7)</formula>
<text><location><page_2><loc_17><loc_15><loc_69><loc_17></location>which plays an important role in the cosmological evolution.</text>
<text><location><page_3><loc_17><loc_81><loc_83><loc_84></location>In this work, we consider the flat Friedmann-Robertson-Walker (FRW) spacetimes</text>
<formula><location><page_3><loc_23><loc_75><loc_83><loc_80></location>ds 2 FRW = g µν dx µ dx ν = -dt 2 + a ( t ) 2 ( dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 ) . (8)</formula>
<text><location><page_3><loc_17><loc_74><loc_81><loc_76></location>From (00)-component of (2), we have the Friedmann equation with G = 1</text>
<formula><location><page_3><loc_39><loc_69><loc_83><loc_73></location>H 2 = α (2 H 2 + ˙ H ) + 4 π 3 ˙ ˜ θ. (9)</formula>
<text><location><page_3><loc_17><loc_67><loc_38><loc_68></location>In deriving (9), we used</text>
<formula><location><page_3><loc_19><loc_62><loc_83><loc_65></location>G 00 = 3 H 2 , C 00 = 0 , ρ RDE = -α 16 π R = 6 α 16 π (2 H 2 + ˙ H ) , T θ 00 = 1 2 ˙ ˜ θ. (10)</formula>
<text><location><page_3><loc_17><loc_54><loc_83><loc_61></location>We note here that the Cotton tensor C µν vanishes for the FRW metric (8), implying that ' ˜ θ ∗ RR '-term does not derive any cosmological evolution. Therefore, the whole equations reduce to those of the Ricci dark energy model with a minimally coupled scalar ˜ θ .</text>
<text><location><page_3><loc_17><loc_51><loc_83><loc_54></location>Because of ∗ RR = 0 for the FRW metric (8), equation (3) leads to the conservation-law for ˜ θ</text>
<formula><location><page_3><loc_39><loc_47><loc_83><loc_50></location>∇ 2 ˜ θ = 0 → ¨ ˜ θ +3 H ˙ ˜ θ = 0 , (11)</formula>
<text><location><page_3><loc_17><loc_45><loc_40><loc_46></location>whose solution is given by</text>
<formula><location><page_3><loc_47><loc_41><loc_83><loc_45></location>˙ ˜ θ = C a 3 . (12)</formula>
<text><location><page_3><loc_17><loc_39><loc_53><loc_41></location>Finally, the conservation-law (7) provides</text>
<formula><location><page_3><loc_32><loc_36><loc_83><loc_38></location>˙ ρ RDE +3 H ( ρ RDE + p RDE ) + ¨ ˜ θ +3 H ˙ ˜ θ = 0 , (13)</formula>
<text><location><page_3><loc_17><loc_31><loc_83><loc_35></location>while using the conservation-law for ˜ θ (11), it leads to the conservation-law for the Ricci dark energy</text>
<formula><location><page_3><loc_37><loc_28><loc_83><loc_29></location>˙ ρ RDE +3 H ( ρ RDE + p RDE ) = 0 . (14)</formula>
<text><location><page_3><loc_17><loc_14><loc_83><loc_26></location>Eqs.(9), (11), and (14) state that whole evolution equations amount to the Ricci dark energy with a minimally coupled scalar. This is because the Chern-Simons term of ˜ θ ∗ RR does not contribute to the cosmological evolution. However, the cosmological perturbation will distinguish between Ricci dark energy in Chern-Simons modified gravity and Ricci dark energy with a minimally coupled scalar ˜ θ [3]. At this stage, we wish to mention that the conservation-law (14) might be not useful to see the cosmological evolution</text>
<text><location><page_4><loc_17><loc_81><loc_83><loc_84></location>because the Friedmann equation (9) does not belong to the standard one due to ρ RDE .</text>
<text><location><page_4><loc_17><loc_78><loc_83><loc_81></location>Plugging (12) into (9) and then, expressing it in terms of scale factor a leads to [1]</text>
<formula><location><page_4><loc_38><loc_73><loc_83><loc_78></location>α a a +( α -1) ( ˙ a a ) 2 + β a 6 = 0 (15)</formula>
<text><location><page_4><loc_17><loc_72><loc_70><loc_74></location>with β = 4 πC 2 / 3. For α /similarequal 1 / 2, this was solved for a ( t ) to be</text>
<formula><location><page_4><loc_36><loc_67><loc_83><loc_71></location>a ( t ) = ( 2 β 3 c 1 ) 1 / 6 sinh 1 / 3 [ 3 √ c 1 t ] , (16)</formula>
<text><location><page_4><loc_17><loc_66><loc_60><loc_67></location>where c 1 is an undetermined integration constant.</text>
<text><location><page_4><loc_17><loc_49><loc_83><loc_66></location>The authors [1] insisted that there is a correspondence between the Ricci dark energy in Chern-Simons modified gravity and the modified Chaplygin gas model because the solution (16) was also found in the modified Chaplygin gas model. Aside from the fact that Ricci dark energy in Chern-Simons modified gravity reduces to Ricci dark energy with a minimally coupled scalar, the similarity between two is very restrictive and it is limited to the de Sitter phase derived by the cosmological constant in the future. Therefore, discovering (16) is not sufficient to confirm the correspondence between two models. In order to show this explicitly, we rewrite (9) as the first-order inhomogeneous equation for H 2 with x = ln a instead of scale factor a [2]</text>
<text><location><page_4><loc_17><loc_42><loc_21><loc_43></location>with</text>
<formula><location><page_4><loc_38><loc_43><loc_83><loc_48></location>dH 2 dx + ( 4 -2 α ) H 2 = -2 3 α ρ ˜ θ (17)</formula>
<formula><location><page_4><loc_36><loc_39><loc_83><loc_42></location>ρ ˜ θ = ρ ˜ θ 0 e -6 x , ρ ˜ θ 0 = πC 2 = 3 β 4 . (18)</formula>
<text><location><page_4><loc_17><loc_30><loc_83><loc_39></location>Anew variable x = ln[ a/a 0 ] with a 0 = 1 ranges from -∞ to ∞ which includes the present x = 0 at a = a 0 . It is important to note that ρ ˜ θ plays a role of the stiff matter with its equation of state ω ˜ θ = 1. Eq.(17) could be integrated to give the standard Friedmann equation with a positive integration constant ˜ c 1 as</text>
<text><location><page_4><loc_17><loc_26><loc_21><loc_27></location>with</text>
<formula><location><page_4><loc_46><loc_27><loc_83><loc_31></location>H 2 = ρ t 3 (19)</formula>
<formula><location><page_4><loc_38><loc_23><loc_83><loc_26></location>ρ t = ρ ˜ θ 0 e -6 x α ( α +1) +3˜ c 1 e -(4 -2 α ) . (20)</formula>
<text><location><page_4><loc_17><loc_21><loc_63><loc_22></location>The total energy density is divided into two parts as</text>
<formula><location><page_4><loc_26><loc_15><loc_83><loc_20></location>ρ t = ρ ˜ θ 0 e -6 x + { 1 -α ( α +1) α ( α +1) ρ ˜ θ 0 e -6 x +3˜ c 1 e -(4 -2 α ) x } (21)</formula>
<formula><location><page_4><loc_30><loc_13><loc_83><loc_16></location>≡ ρ ˜ θ + ˜ ρ RDE , (22)</formula>
<text><location><page_5><loc_17><loc_83><loc_66><loc_84></location>where a new scaled Ricci dark energy density is given by</text>
<formula><location><page_5><loc_31><loc_78><loc_83><loc_82></location>˜ ρ RDE = 1 -α ( α +1) α ( α +1) ρ ˜ θ 0 e -6 x +3˜ c 1 e -(4 -2 α ) x . (23)</formula>
<text><location><page_5><loc_17><loc_70><loc_83><loc_77></location>For α ( α +1) < 1, one finds that ˜ ρ RDE > 0. Without the scalar ˜ θ , the pure Ricci dark energy density is given only by the second term in (23) when expressing the standard Friedmann equation like (19) [2]. In this case, its equation of state is given by</text>
<formula><location><page_5><loc_42><loc_64><loc_83><loc_69></location>ω RDE = 1 3 ( 1 -2 α ) (24)</formula>
<formula><location><page_5><loc_43><loc_57><loc_83><loc_60></location>˜ p ˜ θ = -˜ ρ ˜ θ -1 3 d ˜ ρ ˜ θ dx (25)</formula>
<text><location><page_5><loc_17><loc_60><loc_83><loc_65></location>which shows that for α < 1, it describes the dark energy-dominated universe. Also, we note that the energy density ρ ˜ θ in (22) satisfies the conservation-law as</text>
<text><location><page_5><loc_17><loc_53><loc_83><loc_56></location>for ˜ p ˜ θ = ω ˜ θ ˜ ρ ˜ θ with ω ˜ θ = 1, which indicates that the pure kinetic term of ˜ θ plays a role of the stiff matter.</text>
<text><location><page_5><loc_20><loc_51><loc_58><loc_53></location>Substituting ˜ ρ RDE into the conservation-law,</text>
<formula><location><page_5><loc_39><loc_47><loc_83><loc_50></location>˜ p RDE = -˜ ρ RDE -1 3 d ˜ ρ RDE dx , (26)</formula>
<text><location><page_5><loc_17><loc_44><loc_52><loc_46></location>we obtain the Ricci dark energy pressure</text>
<formula><location><page_5><loc_29><loc_40><loc_83><loc_43></location>˜ p RDE = 1 -α ( α +1) α ( α +1) ρ θ 0 e -6 x +(1 -2 α )˜ c 1 e -(4 -2 α ) x . (27)</formula>
<text><location><page_5><loc_18><loc_37><loc_60><loc_38></location>Importantly, its equation of state takes the form</text>
<formula><location><page_5><loc_27><loc_31><loc_83><loc_36></location>˜ ω RDE ≡ ˜ p RDE ˜ ρ RDE = 1 -α ( α +1) α ( α +1) ρ θ 0 e -6 x +(1 -2 α )˜ c 1 e -(4 -2 α ) x 1 -α ( α +1) α ( α +1) ρ θ 0 e -6 x +3˜ c 1 e -(4 -2 α ) x . (28)</formula>
<text><location><page_5><loc_17><loc_27><loc_82><loc_30></location>For α /similarequal 1 / 2 and x > 0, we have an approximate constant equation of state</text>
<formula><location><page_5><loc_39><loc_22><loc_83><loc_27></location>˜ ω RDE /similarequal 1 3 ( 1 -2 α ) →-1 (29)</formula>
<text><location><page_5><loc_17><loc_16><loc_83><loc_23></location>which describes the dark energy-dominated universe deriving by cosmological constant in the future. In order to compare ω RDE with ˜ ω RDE , see Fig. 1. In this case of (4 -2 /α ) x → const, the Friedmann equation (19) takes an approximated from</text>
<formula><location><page_5><loc_46><loc_13><loc_83><loc_16></location>H 2 ≈ ˜ c 1 (30)</formula>
<text><location><page_6><loc_46><loc_84><loc_47><loc_85></location>/OverTilde</text>
<figure>
<location><page_6><loc_28><loc_64><loc_72><loc_85></location>
<caption>Figure 1: Two equation of state parameters as functions of x for ρ ˜ θ 0 = ˜ c 1 = 1 and α = 1 / 2. x = ln[ a/a 0 ] with a 0 = 1 ranges from -∞ to ∞ which includes the present x = 0 at a = a 0 . ω RDE [dotted line] is always -1, whereas ˜ ω RDE [solid curve] changes from 1 to -1 as x increases from the past ( x < 0) to the future ( x > 0).</caption>
</figure>
<text><location><page_6><loc_47><loc_64><loc_48><loc_65></location>/Minus</text>
<text><location><page_6><loc_48><loc_64><loc_49><loc_65></location>3</text>
<text><location><page_6><loc_17><loc_50><loc_53><loc_52></location>which provides the de Sitter-like solution</text>
<formula><location><page_6><loc_44><loc_46><loc_83><loc_50></location>a ( t ) ≈ e √ ˜ c 1 t . (31)</formula>
<text><location><page_6><loc_17><loc_44><loc_62><loc_46></location>Also, this form could be recovered from (16) for t /greatermuch</text>
<text><location><page_6><loc_63><loc_45><loc_66><loc_46></location>1 as</text>
<formula><location><page_6><loc_44><loc_41><loc_83><loc_44></location>a ( t ) ≈ e √ c 1 t . (32)</formula>
<text><location><page_6><loc_17><loc_34><loc_83><loc_41></location>On the other hand, the modified Chaplygin gas model is given by the exotic equation of state of p = -A/ρ α with A > 0 and 0 ≤ α ≤ 1. Here we discuss two saturation bounds only. For α = 1, it provides the Chaplygin gas model whose energy density is given by</text>
<formula><location><page_6><loc_34><loc_29><loc_83><loc_33></location>ρ α =1 = √ A + B a 6 = √ A √ 1 + Be -6 x A , (33)</formula>
<text><location><page_6><loc_17><loc_25><loc_83><loc_28></location>where B is the integration constant [4]. For Be -6 x /A /greatermuch 1, we can approximate ρ α =1 like as</text>
<formula><location><page_6><loc_39><loc_21><loc_83><loc_26></location>ρ α =1 ≈ √ B √ A e -3 x , p α =1 ≈ 0 (34)</formula>
<text><location><page_6><loc_17><loc_16><loc_83><loc_21></location>which describes the dust matter-dominated universe with ω α =1 = 0 in the early stage of the universe. For Be -6 x /A /lessmuch 1, we have the approximated from</text>
<formula><location><page_6><loc_39><loc_13><loc_83><loc_17></location>ρ α =1 ≈ √ A, p α =1 ≈ -√ A, (35)</formula>
<figure>
<location><page_7><loc_21><loc_71><loc_47><loc_84></location>
</figure>
<figure>
<location><page_7><loc_52><loc_71><loc_79><loc_84></location>
<caption>Figure 2: Three energy densities as functions of x for A = B = A 0 = ρ ˜ θ 0 = ˜ c 1 = 1 and α = 1 / 2. x = ln[ a/a 0 ] with a 0 = 1 ranges from -∞ to ∞ which includes the present x = 0 at a = a 0 . On the ρ -axis of left-panel, from top to bottom, the curves represent ρ t , ρ α =0 , and ρ α =1 , respectively. Even though all curves converge on constants for x > 0 [right-panel] which represents de Sitter phase, their past energy densities [left-panel] show different behaviors for x < 0. In this choice of parameters, we note that ρ α =0 /similarequal ρ α =1 .</caption>
</figure>
<text><location><page_7><loc_17><loc_51><loc_83><loc_54></location>which describes the dark energy-dominated universe ω α =1 = -1 in the future. For α = 0 modified Chaplygin gas model [5], its energy density takes</text>
<formula><location><page_7><loc_42><loc_48><loc_83><loc_50></location>ρ α =0 = A + A 0 e -3 x (36)</formula>
<text><location><page_7><loc_17><loc_43><loc_83><loc_47></location>which shows a dust matter-dominated phase for x < 0, while it indicates de Sitter phase deriving by cosmological constant for x > 0.</text>
<text><location><page_7><loc_17><loc_27><loc_83><loc_43></location>Let us compare the total energy density (20) with (33) and (36). From Fig. 2, we observe that their past evolutions appear differently for x < 0, even though they converge on constants for x > 0. The (modified) Chaplygin gas model describes the whole evolution starting from the dust matterdominated universe with ω α =0 , 1 = 0 to the dark energy-dominated universe with ω α =0 , 1 = -1, while the Ricci dark energy in Chern-Simons modified gravity describes the whole evolution starting from the stiff matterdominated universe with ˜ ω RDE = 1 to the dark energy-dominated universe with ˜ ω RDE = -1 as is depicted in Fig. 1.</text>
<text><location><page_7><loc_17><loc_14><loc_83><loc_28></location>Consequently, the claim of Ref. [1] that there is a correspondence between the Ricci dark energy in Chern-Simons modified gravity and the modified Chaplygin gas model might be led to misleading. Aside from the fact that Ricci dark energy in Chern-Simons modified gravity is nothing but Ricci dark energy with a minimally coupled scalar when choosing the FRW metric, the similarity between two is limited to the de Sitter phase derived by the cosmological constant in the future ( x > 0). This similarity can be understood partly by reconstructing the Chaplygin gas model in terms of the scalar [4].</text>
<text><location><page_8><loc_17><loc_81><loc_83><loc_84></location>The Chern-Simons term will participate in the cosmological evolution when choosing the anisotropic metric instead of the isotropic FRW metric (8) [6].</text>
<section_header_level_1><location><page_8><loc_17><loc_76><loc_43><loc_78></location>Acknowledgement</section_header_level_1>
<text><location><page_8><loc_17><loc_72><loc_90><loc_75></location>This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.2012-R1A1A2A10040499).</text>
<section_header_level_1><location><page_8><loc_17><loc_67><loc_33><loc_69></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_18><loc_62><loc_83><loc_65></location>[1] J. G. Silva and A. F. Santos, arXiv:1306.6638 [hep-th], to appear in European Physical Journal C.</list_item>
<list_item><location><page_8><loc_18><loc_57><loc_83><loc_61></location>[2] K. Y. Kim, H. W. Lee and Y. S. Myung, Gen. Rel. Grav. 43 , 1095 (2011) [arXiv:0812.4098 [gr-qc]].</list_item>
<list_item><location><page_8><loc_18><loc_53><loc_83><loc_56></location>[3] S. Alexander and N. Yunes, Phys. Rept. 480 , 1 (2009) [arXiv:0907.2562 [hep-th]].</list_item>
<list_item><location><page_8><loc_18><loc_48><loc_83><loc_51></location>[4] A. Y. .Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511 , 265 (2001) [gr-qc/0103004].</list_item>
<list_item><location><page_8><loc_18><loc_44><loc_83><loc_47></location>[5] J. C. Fabris, S. V. B. Goncalves and R. de Sa Ribeiro, Gen. Rel. Grav. 36 , 211 (2004) [astro-ph/0307028].</list_item>
<list_item><location><page_8><loc_18><loc_39><loc_83><loc_42></location>[6] Y. S. Myung, Y. -W. Kim, W. -S. Son and Y. -J. Park, JHEP 1003 , 085 (2010) [arXiv:1001.3921 [gr-qc]].</list_item>
</unordered_list>
</document>
[
{
"title": "Comment on Ricci dark energy in Chern-Simons modified gravity",
"content": "Yun Soo Myung a Institute of Basic Science and Department of Computer Simulation, Inje University, Gimhae 621-749, Korea",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We revisit Ricci dark energy in Chern-Simons modified gravity. As far as the cosmological evolution, this is nothing but the Ricci dark energy with a minimally coupled scalar without potential which means that the role of Chern-Simons term is suppressed. Using the equation of state parameter, this model is similar to the modified Chaplygin gas model only when two are around the de Sitter universe deriving by the cosmological constant in the future. However, two past evolutions are different. Keywords: Ricci dark energy; Chern-Simons modified gravity a ysmyung@inje.ac.kr Recently, the authors [1] have investigated the Ricci dark energy model in the dynamic Chern-Simons modified gravity which states that its cosmological evolution is similar to that displayed by the modified Chaplygin gas model. In this Comment, we wish to draw the reader two important issues: One is that as far as the cosmological evolution, this model is nothing but the Ricci dark energy with a minimally coupled scalar without potential where the role of Chern-Simons term is suppressed totally. The other is that using the equation of state parameter, this model is similar to the modified Chaplygin gas model only when two make turnaround of de Sitter universe deriving by the cosmological constant in the future. In general, two provide different evolutions. We start with the dynamic Chern-Simons modified gravity action with Ricci dark energy [1] where ∗ RR is the Pontryagin term, ˜ θ is a dynamical scalar and S RDE is the action to give the Ricci dark energy. Here, for simplicity, one chooses V ( ˜ θ ) = 0. Their equations are given by where G µν is the Einstein tensor, C µν is the Cotton tensor from the ChernSimons term ' ˜ θ ∗ RR '-term. The energy-momentum tensor is given by with and Applying ∇ µ to (2) leads to the conservation-law for T µν as which plays an important role in the cosmological evolution. In this work, we consider the flat Friedmann-Robertson-Walker (FRW) spacetimes From (00)-component of (2), we have the Friedmann equation with G = 1 In deriving (9), we used We note here that the Cotton tensor C µν vanishes for the FRW metric (8), implying that ' ˜ θ ∗ RR '-term does not derive any cosmological evolution. Therefore, the whole equations reduce to those of the Ricci dark energy model with a minimally coupled scalar ˜ θ . Because of ∗ RR = 0 for the FRW metric (8), equation (3) leads to the conservation-law for ˜ θ whose solution is given by Finally, the conservation-law (7) provides while using the conservation-law for ˜ θ (11), it leads to the conservation-law for the Ricci dark energy Eqs.(9), (11), and (14) state that whole evolution equations amount to the Ricci dark energy with a minimally coupled scalar. This is because the Chern-Simons term of ˜ θ ∗ RR does not contribute to the cosmological evolution. However, the cosmological perturbation will distinguish between Ricci dark energy in Chern-Simons modified gravity and Ricci dark energy with a minimally coupled scalar ˜ θ [3]. At this stage, we wish to mention that the conservation-law (14) might be not useful to see the cosmological evolution because the Friedmann equation (9) does not belong to the standard one due to ρ RDE . Plugging (12) into (9) and then, expressing it in terms of scale factor a leads to [1] with β = 4 πC 2 / 3. For α /similarequal 1 / 2, this was solved for a ( t ) to be where c 1 is an undetermined integration constant. The authors [1] insisted that there is a correspondence between the Ricci dark energy in Chern-Simons modified gravity and the modified Chaplygin gas model because the solution (16) was also found in the modified Chaplygin gas model. Aside from the fact that Ricci dark energy in Chern-Simons modified gravity reduces to Ricci dark energy with a minimally coupled scalar, the similarity between two is very restrictive and it is limited to the de Sitter phase derived by the cosmological constant in the future. Therefore, discovering (16) is not sufficient to confirm the correspondence between two models. In order to show this explicitly, we rewrite (9) as the first-order inhomogeneous equation for H 2 with x = ln a instead of scale factor a [2] with Anew variable x = ln[ a/a 0 ] with a 0 = 1 ranges from -∞ to ∞ which includes the present x = 0 at a = a 0 . It is important to note that ρ ˜ θ plays a role of the stiff matter with its equation of state ω ˜ θ = 1. Eq.(17) could be integrated to give the standard Friedmann equation with a positive integration constant ˜ c 1 as with The total energy density is divided into two parts as where a new scaled Ricci dark energy density is given by For α ( α +1) < 1, one finds that ˜ ρ RDE > 0. Without the scalar ˜ θ , the pure Ricci dark energy density is given only by the second term in (23) when expressing the standard Friedmann equation like (19) [2]. In this case, its equation of state is given by which shows that for α < 1, it describes the dark energy-dominated universe. Also, we note that the energy density ρ ˜ θ in (22) satisfies the conservation-law as for ˜ p ˜ θ = ω ˜ θ ˜ ρ ˜ θ with ω ˜ θ = 1, which indicates that the pure kinetic term of ˜ θ plays a role of the stiff matter. Substituting ˜ ρ RDE into the conservation-law, we obtain the Ricci dark energy pressure Importantly, its equation of state takes the form For α /similarequal 1 / 2 and x > 0, we have an approximate constant equation of state which describes the dark energy-dominated universe deriving by cosmological constant in the future. In order to compare ω RDE with ˜ ω RDE , see Fig. 1. In this case of (4 -2 /α ) x → const, the Friedmann equation (19) takes an approximated from /OverTilde /Minus 3 which provides the de Sitter-like solution Also, this form could be recovered from (16) for t /greatermuch 1 as On the other hand, the modified Chaplygin gas model is given by the exotic equation of state of p = -A/ρ α with A > 0 and 0 ≤ α ≤ 1. Here we discuss two saturation bounds only. For α = 1, it provides the Chaplygin gas model whose energy density is given by where B is the integration constant [4]. For Be -6 x /A /greatermuch 1, we can approximate ρ α =1 like as which describes the dust matter-dominated universe with ω α =1 = 0 in the early stage of the universe. For Be -6 x /A /lessmuch 1, we have the approximated from which describes the dark energy-dominated universe ω α =1 = -1 in the future. For α = 0 modified Chaplygin gas model [5], its energy density takes which shows a dust matter-dominated phase for x < 0, while it indicates de Sitter phase deriving by cosmological constant for x > 0. Let us compare the total energy density (20) with (33) and (36). From Fig. 2, we observe that their past evolutions appear differently for x < 0, even though they converge on constants for x > 0. The (modified) Chaplygin gas model describes the whole evolution starting from the dust matterdominated universe with ω α =0 , 1 = 0 to the dark energy-dominated universe with ω α =0 , 1 = -1, while the Ricci dark energy in Chern-Simons modified gravity describes the whole evolution starting from the stiff matterdominated universe with ˜ ω RDE = 1 to the dark energy-dominated universe with ˜ ω RDE = -1 as is depicted in Fig. 1. Consequently, the claim of Ref. [1] that there is a correspondence between the Ricci dark energy in Chern-Simons modified gravity and the modified Chaplygin gas model might be led to misleading. Aside from the fact that Ricci dark energy in Chern-Simons modified gravity is nothing but Ricci dark energy with a minimally coupled scalar when choosing the FRW metric, the similarity between two is limited to the de Sitter phase derived by the cosmological constant in the future ( x > 0). This similarity can be understood partly by reconstructing the Chaplygin gas model in terms of the scalar [4]. The Chern-Simons term will participate in the cosmological evolution when choosing the anisotropic metric instead of the isotropic FRW metric (8) [6].",
"pages": [
1,
2,
3,
4,
5,
6,
7,
8
]
},
{
"title": "Acknowledgement",
"content": "This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.2012-R1A1A2A10040499).",
"pages": [
8
]
}
]
2013EPJC...73.2647B
https://arxiv.org/pdf/1310.5333.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_92><loc_93></location>Electromagnetic waves in an axion-active relativistic plasma non-minimally coupled to gravity</section_header_level_1>
<text><location><page_1><loc_22><loc_87><loc_79><loc_89></location>Alexander B. Balakin, 1, ∗ Ruslan K. Muharlyamov, 1, † and Alexei E. Zayats 1, ‡</text>
<text><location><page_1><loc_26><loc_84><loc_75><loc_87></location>1 Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya str. 18, Kazan 420008, Russia</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_83></location>We consider cosmological applications of a new self-consistent system of equations, accounting for a nonminimal coupling of the gravitational, electromagnetic and pseudoscalar (axion) fields in a relativistic plasma. We focus on dispersion relations for electromagnetic perturbations in an initially isotropic ultrarelativistic plasma coupled to the gravitational and axion fields in the framework of isotropic homogeneous cosmological model of the de Sitter type. We classify the longitudinal and transversal electromagnetic modes in an axionically active plasma and distinguish between waves (damping, instable or running), and nonharmonic perturbations (damping or instable). We show that for the special choice of the guiding model parameters the transversal electromagnetic waves in the axionically active plasma, nonminimally coupled to gravity, can propagate with the phase velocity less than speed of light in vacuum, thus displaying a possibility for a new type of resonant particle-wave interactions.</text>
<text><location><page_1><loc_18><loc_66><loc_44><loc_67></location>PACS numbers: 04.40.-b, 52.35.-g, 14.80.Va</text>
<text><location><page_1><loc_18><loc_65><loc_78><loc_66></location>Keywords: Axion field, relativistic plasma, Vlasov model, nonminimal coupling, dispersion relations</text>
<section_header_level_1><location><page_1><loc_20><loc_61><loc_37><loc_62></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_21><loc_49><loc_59></location>Electromagnetic radiation provides the most important channel of information about our Universe. Valuable information about cosmic sources of photons, and about cosmic events accompanying the light propagation, is encoded in the intensity, polarization, phase and spectral characteristics of the electromagnetic radiation. In this sense, reconstruction of the phase and group velocities of observed electromagnetic waves, which travel through plasma and gas in the dark matter environment, gives us the basis for theoretical modeling of the properties of these cosmic substrates. When we deal with plasma, the phase and group velocities are given by V ph = ω k and V gr = ∂ω ∂k , respectively, where ω is the frequency and k is the wave three-vector modulus. Thus, the dependence ω = ω ( k ) obtained from the so-called dispersion relations for the longitudinal and transversal electromagnetic waves in plasma plays an important role in the information decryption. The theory of dispersion relations is well-elaborated for various plasma configurations (see, e.g., [1-4] for details, review and references). To obtain novel results in this scientific sphere we intend to use a new nonminimal Einstein-Maxwell-Vlasov-axion model [5], which deals with the self-consistent theory of tidal-type interactions between gravitational, electromagnetic, pseudoscalar (axion) fields and a relativistic multi-component plasma.</text>
<text><location><page_1><loc_9><loc_17><loc_49><loc_21></location>Cosmological applications of this model seem to be the most interesting, since the study of the interaction of four key players of the nonminimal Einstein-</text>
<text><location><page_1><loc_52><loc_9><loc_92><loc_62></location>Maxwell-Vlasov-axion model (gravitons, photons, electrically charged plasma particles and axions) is important for the description of the history of our Universe. There are at least three motives, which could explain this interest. First, in the early Universe the nonminimal couping of a matter and fields to gravity was important on the stage of inflation, when the space-time curvature was varying catastrophically fast. Late-time accelerated expansion discovered recently [6-8] has revived an interest to inertial, tidal and rip's effects [9-12], for which the curvature coupling could be also important. Second, the cold dark matter, which now is considered to be one of two key elements of the dark fluid, guiding the late-time Universe evolution [13-15], contains (hypothetically) an axion subsystem. In the early Universe the axions (light pseudo-Goldstone bosons) could be created due to the phase transition associated with the Peccei-Quinn symmetry breakdown [16-24]. In the late-time Universe these pseudo-bosons, probably, exist as relic axions, forming various cold dark matter configurations. Third, electrically charged plasma and photons are ubiquitous: one can find plasma configurations and an ocean of photons in every epoch in the cosmic history and in many objects, which form our Universe. Of course, studying the nonminimal Einstein-Maxwell-Vlasov-axion theory, we are restricted to theoretical modeling of the dispersion relations for relativistic homogeneous axionically active plasma. For instance, plasma can be treated as relativistic substratum in the early Universe; however, as was shown in the papers [25-27], the PecceiQuinn phase transition is accompanied by the creation of strongly inhomogeneous axionic primordial configurations, which were indicated (see, e.g., [24, 25]) as 'archioles'. The inhomogeneity of archioles type is frozen at the radiation domination stage and should be inherited in the large scale structure of the modern Universe. Additional problem in the theoretical modeling is con-</text>
<text><location><page_2><loc_9><loc_45><loc_49><loc_93></location>nected with instabilities generated in plasma due to its inhomogeneity (see, e.g., [28]). Self-gravitation of the pseudoscalar (axion) field produces gravitational instability similarly to the instability of scalar fields described in [29], thus making the studying of the electromagnetic waves propagation much more sophisticated. In such situation we need of some toy-model, which provides a balance between complexity of the problem as a whole and mathematical clarity of the simplified model. For the first step we have chosen the approximation, which is standard for homogeneous cosmological model: the spacetime metric and the axionic dark matter distribution are considered to depend on time only. In this case the standard approach to the analysis of waves propagation in relativistic plasma is based on the study of dispersion relations in terms of frequency ω and wave three-vector glyph[vector] k . When we consider homogeneous cosmological models, we can use the standard Fourier transformations with respect to spatial coordinates and can naturally introduce the analog of glyph[vector] k . The Fourier-Laplace transformation with respect to time is, generally, non-effective, since the coefficients in the master equations for the electromagnetic field depend on cosmological time. However, we have found one specific very illustrative model (the plasma is ultrarelativistic, the spacetime is of a constant curvature, the axion field has a constant time derivative), for which the master equations can be effectively transformed into a set of differential equations with constant coefficients. This simplified the analysis of the dispersion relations essentially, and allowed us to interpret the results in the standard terms. As a next step, we plan to consider homogeneous but anisotropic models, and then models with inhomogeneous distribution of the axionic dark matter.</text>
<text><location><page_2><loc_9><loc_21><loc_49><loc_45></location>In this work we obtain and analyze the dispersion relations for perturbations in the axionically active plasma, non-minimally coupled to gravity, classify these perturbations and distinguish the transversal waves, which can propagate with the phase velocity less than the speed of light in vacuum. The work is organized as follows. In Section II we describe the appropriate background state of the system as a whole, and discuss the exact solutions of non-perturbed master equations for the plasma in the gravitational and axionic fields, which were derived in [5]. In Section III we consider the equations for the electromagnetic perturbations in the ( t, glyph[vector]x ) form and then the equations for the corresponding Fourier-Laplace images in the (Ω , glyph[vector] k ) form. We analyze in detail the dispersion relations for longitudinal waves in Section IV, and for transversal waves in Section V. We summarize the results in Section VI (Discussion).</text>
<section_header_level_1><location><page_2><loc_10><loc_15><loc_48><loc_17></location>II. ON THE BACKGROUND SOLUTIONS TO THE NONMINIMAL MASTER EQUATIONS</section_header_level_1>
<text><location><page_2><loc_9><loc_9><loc_49><loc_13></location>Let us consider the background solutions for the nonminimal Einstein-Maxwell-Vlasov-axion model, which satisfy the following four conditions. First , we assume</text>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>that the spacetime is isotropic, spatially homogeneous and is described by the metric</text>
<formula><location><page_2><loc_58><loc_88><loc_92><loc_89></location>ds 2 = a 2 ( τ )[ dτ 2 -( dx 2 + dy 2 + dz 2 )] . (1)</formula>
<text><location><page_2><loc_52><loc_65><loc_92><loc_86></location>Second , we suppose that the pseudoscalar field inherits the spacetime symmetry and depends on time only, φ ( τ ). Third , we assume, that both the background macroscopic (external) and cooperative (internal) electromagnetic fields are absent, i.e., the total Maxwell tensor is equal to zero F ik = 0. Fourth , we consider the plasma to be the test ingredient of the cosmological model; this means that the contribution of the plasma particles into the total stress-energy tensor is negligible in comparison with the dark energy contribution, presented in our model by the Λ term, and with the dark matter one, described by the axion field Ψ 0 φ ( φ is dimensionless pseudoscalar field). These four requirements provide the master equations obtained in [5] to be reduced to the following system.</text>
<section_header_level_1><location><page_2><loc_54><loc_59><loc_90><loc_61></location>A. Background nonminimal equations for the gravity field</section_header_level_1>
<text><location><page_2><loc_52><loc_53><loc_92><loc_57></location>In the absence of the electromagnetic field the nonminimally extended Einstein equations take the form (see Subsection IIIE in [5])</text>
<formula><location><page_2><loc_55><loc_46><loc_92><loc_51></location>R ik -1 2 Rg ik -Λ g ik = = κ Ψ 2 0 [ T (A) ik + η 2 T (5) ik + η 3 T (6) ik + η (A) T (7) ik ] , (2)</formula>
<text><location><page_2><loc_52><loc_43><loc_63><loc_44></location>where the term</text>
<formula><location><page_2><loc_53><loc_39><loc_92><loc_42></location>T (A) ik = { ∇ i φ ∇ k φ -1 2 g ik [ ∇ m φ ∇ m φ -m 2 (A) φ 2 ] } (3)</formula>
<text><location><page_2><loc_52><loc_35><loc_92><loc_37></location>is the stress-energy tensor of the pseudoscalar field. The terms</text>
<formula><location><page_2><loc_53><loc_28><loc_92><loc_34></location>T (5) ik = R ∇ i φ ∇ k φ +( g ik ∇ n ∇ n -∇ i ∇ k ) [ ∇ m φ ∇ m φ ] + + ∇ m φ ∇ m φ ( R ik -1 2 Rg ik ) , (4)</formula>
<formula><location><page_2><loc_59><loc_19><loc_92><loc_25></location>T (6) ik = ∇ m φ [ R m i ∇ k φ + R m k ∇ i φ ] + + 1 2 g ik ( ∇ m ∇ n -R mn ) [ ∇ m φ ∇ n φ ] --∇ m [ ∇ m φ ∇ i ∇ k φ ] . (5)</formula>
<formula><location><page_2><loc_53><loc_12><loc_92><loc_16></location>T (7) ik = ( ∇ i ∇ k -g ik ∇ m ∇ m ) φ 2 -( R ik -1 2 Rg ik ) φ 2 (6)</formula>
<text><location><page_2><loc_52><loc_8><loc_92><loc_11></location>describe the nonminimal contributions associated with coupling constants η 2 , η 3 , and η (A) , respectively.</text>
<text><location><page_3><loc_9><loc_89><loc_49><loc_93></location>We are interested to analyze the specific solution to these equations, which is characterized by the de Sitter metric</text>
<formula><location><page_3><loc_15><loc_85><loc_49><loc_88></location>ds 2 = 1 H 2 τ 2 [ dτ 2 -( dx 2 + dy 2 + dz 2 )] (7)</formula>
<text><location><page_3><loc_9><loc_83><loc_12><loc_84></location>with</text>
<formula><location><page_3><loc_24><loc_79><loc_49><loc_82></location>a ( τ ) = -1 Hτ , (8)</formula>
<text><location><page_3><loc_9><loc_77><loc_49><loc_78></location>where H is a constant. With the transformation of time</text>
<formula><location><page_3><loc_20><loc_73><loc_49><loc_75></location>τ = -1 Ha ( t 0 ) e -H ( t -t 0 ) , (9)</formula>
<text><location><page_3><loc_9><loc_65><loc_49><loc_71></location>which gives the correspondence: τ → 0, when t → ∞ ; τ → -∞ , when t → -∞ ; τ → -1 Ha ( t 0 ) ≡ τ 0 , when t → t 0 , one can obtain the well-known form of the de Sitter metric</text>
<formula><location><page_3><loc_11><loc_63><loc_49><loc_64></location>ds 2 = dt 2 -e 2 H ( t -t 0 ) a 2 ( t 0 )( dx 2 + dy 2 + dz 2 ) . (10)</formula>
<text><location><page_3><loc_9><loc_56><loc_49><loc_61></location>The de Sitter metric (7) describes the spacetime of constant curvature, for which the basic geometric quantities of the Riemann tensor, the Ricci tensor and Ricci scalar take the form</text>
<formula><location><page_3><loc_17><loc_51><loc_49><loc_54></location>R ikmn = -H 2 ( g im g kn -g in g km ) , R im = -3 H 2 g im , R = -12 H 2 , (11)</formula>
<text><location><page_3><loc_9><loc_48><loc_39><loc_49></location>and the nonminimal susceptibility tensors</text>
<formula><location><page_3><loc_10><loc_40><loc_49><loc_47></location>R ikmn ≡ q 1 2 R ( g im g kn -g in g km ) + + q 2 2 ( R im g kn -R in g km + R kn g im -R km g in ) + + q 3 R ikmn , (12)</formula>
<formula><location><page_3><loc_10><loc_29><loc_49><loc_37></location>χ ikmn ( A ) ≡ Q 1 2 R ( g im g kn -g in g km ) + + Q 2 2 ( R im g kn -R in g km + R kn g im -R km g in ) + + Q 3 R ikmn , (13)</formula>
<formula><location><page_3><loc_11><loc_23><loc_49><loc_27></location>glyph[Rfractur] mn (A) ≡ 1 2 η 1 ( F ml R n l + F nl R m l ) + η 2 Rg mn + η 3 R mn (14)</formula>
<text><location><page_3><loc_9><loc_20><loc_31><loc_21></location>introduced in [5] transform into</text>
<formula><location><page_3><loc_10><loc_17><loc_49><loc_18></location>R ikmn = -H 2 (6 q 1 +3 q 2 + q 3 ) ( g im g kn -g in g km ) , (15)</formula>
<formula><location><page_3><loc_10><loc_13><loc_49><loc_14></location>χ ikmn (A) = -H 2 (6 Q 1 +3 Q 2 + Q 3 ) ( g im g kn -g in g km ) , (16)</formula>
<formula><location><page_3><loc_19><loc_8><loc_49><loc_10></location>glyph[Rfractur] mn ( A ) = -3 H 2 (4 η 2 + η 3 ) g mn . (17)</formula>
<text><location><page_3><loc_52><loc_86><loc_92><loc_93></location>One can check directly that the de Sitter-type metric (1) with the scale factor (8) is the exact solution of the equations (2)-(6) with the pseudoscalar field linear in time, i.e., φ ( τ )= ντ , when the following three relationships are satisfied:</text>
<formula><location><page_3><loc_59><loc_82><loc_92><loc_85></location>Λ = 3 H 2 , m 2 (A) = 2 9 Λ 2 (6 η 2 + η 3 ) , (18)</formula>
<formula><location><page_3><loc_62><loc_77><loc_92><loc_80></location>η (A) = -1 6 + 1 9 Λ(9 η 2 +2 η 3 ) . (19)</formula>
<text><location><page_3><loc_52><loc_74><loc_92><loc_76></location>These relations contain neither ν , nor Ψ 0 . In addition, when η 2 = η 3 = 0, we obtain the result</text>
<formula><location><page_3><loc_59><loc_70><loc_92><loc_73></location>Λ = 3 H 2 , m (A) = 0 , η (A) = -1 6 , (20)</formula>
<text><location><page_3><loc_52><loc_58><loc_92><loc_69></location>which is well-known for the massless scalar field conformally coupled to gravity [30]. When the constants of nonminimal coupling η 2 and η 3 are nonvanishing, we can consider the axions to be massive, and the mass m (A) itself is connected not only with these coupling constants, but with the cosmological constant Λ as well. In order to minimize the number of unknown coupling parameters, one can put, for instance,</text>
<formula><location><page_3><loc_56><loc_53><loc_92><loc_56></location>η 3 = -9 2 η 2 , η (A) = -1 6 , η 2 = 3 m 2 (A) Λ 2 , (21)</formula>
<text><location><page_3><loc_52><loc_51><loc_69><loc_52></location>satisfying (18) and (19).</text>
<section_header_level_1><location><page_3><loc_54><loc_45><loc_90><loc_48></location>B. Nonminimal equations for the background pseudoscalar (axion) field</section_header_level_1>
<text><location><page_3><loc_52><loc_39><loc_92><loc_43></location>In the absence of the background electromagnetic field the nonminimally extended master equation (see Eq.(88) in [5]) for the pseudoscalar φ takes the form</text>
<formula><location><page_3><loc_55><loc_32><loc_92><loc_38></location>∇ m [( g mn + glyph[Rfractur] mn (A) ) ∇ n φ ] + [ m 2 (A) + η (A) R ] φ = = -1 Ψ 2 0 ∑ (a) ∫ dPf (a) G (a) . (22)</formula>
<text><location><page_3><loc_52><loc_24><loc_92><loc_30></location>Let us take into account the relations (15)-(17) for glyph[Rfractur] mn (A) and χ ikmn (A) , and suppose that for the background distribution function its zero-order moment in the right-side hand of (22) vanishes, then we obtain the equation</text>
<formula><location><page_3><loc_52><loc_20><loc_92><loc_23></location>[1 -3 H 2 (4 η 2 + η 3 )] ∇ m ∇ m φ +( m 2 A -12 H 2 η (A) ) φ = 0 . (23)</formula>
<text><location><page_3><loc_52><loc_17><loc_92><loc_20></location>The function φ = ντ satisfies (23), if the following relation is valid</text>
<formula><location><page_3><loc_53><loc_14><loc_92><loc_16></location>-2 H 2 [1 -3 H 2 (4 η 2 + η 3 )] + m 2 A -12 H 2 η (A) = 0 . (24)</formula>
<text><location><page_3><loc_52><loc_12><loc_70><loc_13></location>We used here the relation</text>
<formula><location><page_3><loc_59><loc_8><loc_92><loc_11></location>∇ m ∇ m φ = 1 a 4 d dτ ( a 2 ˙ φ ) = -2 H 2 ντ . (25)</formula>
<text><location><page_4><loc_9><loc_85><loc_49><loc_93></location>Clearly, the relation (24) is identically satisfied, when the equations (18), (19) are valid. In other words, we have shown that the master equation for the background pseudoscalar (axion) field admits the existence of the exact solution φ ( τ ) = ντ linear in time, when the spacetime is of the de Sitter type with the metric (7).</text>
<section_header_level_1><location><page_4><loc_9><loc_79><loc_48><loc_82></location>C. Consistency of the background electrodynamic equations</section_header_level_1>
<text><location><page_4><loc_9><loc_67><loc_49><loc_77></location>We assume that the Maxwell tensor describing the background cooperative electromagnetic field in plasma is equal to zero, F ik = 0. It is possible, when the tensor of spontaneous polarization-magnetization H ik vanishes, and when the four-vector of electric current in plasma, I i , is equal to zero. Since φ = φ ( τ ) and R α 0 = 0 ( α = 1 , 2 , 3) the formula (see (87) in [5])</text>
<formula><location><page_4><loc_11><loc_63><loc_49><loc_66></location>H ik ≡ -1 2 η 1 Ψ 2 0 [( R km ∇ i φ -R im ∇ k φ ) ∇ m φ ] , (26)</formula>
<text><location><page_4><loc_9><loc_58><loc_49><loc_62></location>gives H ik = 0. The condition I i = 0 provides the restrictions for the background distribution function f (a) , which we will discuss in the next subsection.</text>
<section_header_level_1><location><page_4><loc_10><loc_54><loc_47><loc_55></location>D. Background solution to the Vlasov equation</section_header_level_1>
<text><location><page_4><loc_9><loc_41><loc_49><loc_51></location>In the context of the Vlasov theory we search for a 8dimensional one-particle distribution function f (a) ( x i , p k ) which describes particles of a sort (a) with the rest mass m (a) , electric charge e (a) . This function of the coordinates x i and of the momentum four-covector p k satisfies the relativistic kinetic equation, which can be presented now in the form</text>
<formula><location><page_4><loc_18><loc_37><loc_49><loc_40></location>p i ( ∂ ∂x i +Γ l ik p l ∂ ∂p k ) f (a) = 0 . (27)</formula>
<text><location><page_4><loc_9><loc_20><loc_49><loc_36></location>We assume here that the background macroscopic cooperative electromagnetic field in plasma is absent, and there are no contact interactions between axions and plasma particles, so that the force F i (a) (see Eq.(29) in [5]) disappears from this equation. In the cosmological context we consider the distribution function depending on the covariant components of the particle fourmomentum, p k in order to simplify the analysis. Here Γ m ik are the Christoffel symbols associated with the spacetime metric g ik . The characteristic equations associated with the kinetic equation (27)</text>
<formula><location><page_4><loc_13><loc_15><loc_49><loc_19></location>dp k ds -Γ l ik p l dx i ds = 0 , dx i ds = 1 m (a) g ij p j , (28)</formula>
<text><location><page_4><loc_9><loc_11><loc_49><loc_14></location>are known to be directly solved for the metric (1). For the cases i = α = 1 , 2 , 3 one obtains immediately that</text>
<formula><location><page_4><loc_26><loc_9><loc_49><loc_10></location>p α = q α , (29)</formula>
<text><location><page_4><loc_52><loc_87><loc_92><loc_93></location>where q α =const are the integrals of motion. The component p 0 of the particle momentum four-vector can be found from the quadratic integral of motion g ij p i p j = m 2 (a) =const:</text>
<formula><location><page_4><loc_53><loc_84><loc_92><loc_86></location>p 0 = √ m 2 (a) a 2 ( τ ) + q 2 , q 2 ≡ q 2 1 + q 2 2 + q 2 3 = const . (30)</formula>
<text><location><page_4><loc_52><loc_81><loc_77><loc_82></location>Other three integrals of motion are</text>
<formula><location><page_4><loc_60><loc_76><loc_92><loc_80></location>X α = x α ( τ ) + q α q 2 √ m 2 (a) H 2 + q 2 τ 2 . (31)</formula>
<text><location><page_4><loc_52><loc_65><loc_92><loc_75></location>Generally, the distribution function, which satisfies kinetic equation (27) can be reconstructed as an arbitrary function of seven integrals of motion q α , X α , √ g ij p i p j , nevertheless, taking into account that the spacetime is isotropic and homogeneous, we require that the distribution function inherits this symmetry and thus it has the form</text>
<formula><location><page_4><loc_55><loc_62><loc_92><loc_65></location>f (a) ( x l , p k ) = f (0) (a) ( q 2 ) δ ( √ g sn p s p n -m (a) ) , (32)</formula>
<text><location><page_4><loc_52><loc_56><loc_92><loc_61></location>where f (0) (a) ( q 2 ) is arbitrary function of one argument, namely, q 2 . We are interested now to calculate the first moment of the distribution function</text>
<formula><location><page_4><loc_58><loc_50><loc_92><loc_55></location>N i (a) ≡ 1 m (a) ∫ d 4 P √ -g g ij p j f (a) ( x l , p k ) × × δ ( √ g sn p s p n -m (a) )Θ( V h p h ) , (33)</formula>
<text><location><page_4><loc_52><loc_38><loc_92><loc_49></location>where d 4 P = dp 0 dp 1 dp 2 dp 3 symbolizes the volume in the four-dimensional momentum space; the delta function guarantees the normalization property of the particle momentum, the Heaviside function Θ( V h p h ) rejects negative values of energy, V h is the velocity four-vector of the system as a whole. In the de Sitter spacetime with V h = δ h 0 1 a ( τ ) the quantities N i (a) can be reduced to</text>
<formula><location><page_4><loc_59><loc_34><loc_92><loc_37></location>N i (a) ( τ ) = ∫ dq 1 dq 2 dq 3 a 2 p 0 g ij p j f (0) (a) ( q 2 ) , (34)</formula>
<text><location><page_4><loc_52><loc_30><loc_92><loc_33></location>therefore, the spatial components N α (a) vanish. The component N 0 (a) reads</text>
<formula><location><page_4><loc_58><loc_24><loc_92><loc_29></location>N 0 (a) ( τ ) = 4 π a 4 ∞ ∫ 0 q 2 dqf (0) (a) ( q 2 ) ≡ N (a) a 4 ( τ ) , (35)</formula>
<text><location><page_4><loc_52><loc_19><loc_92><loc_23></location>where N (a) does not depend on time. This means that the four-vector of the electric current in the electro-neutral plasma</text>
<formula><location><page_4><loc_53><loc_14><loc_92><loc_18></location>I i ( τ ) = ∑ (a) e (a) N i (a) = 1 a 4 ( τ ) δ i 0 ∑ (a) e (a) N (a) = 0 , (36)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_13></location>is equal to zero at arbitrary time moment τ , if it was equal to zero at the initial time moment. This guarantees that the Maxwell equations are self-consistent.</text>
<section_header_level_1><location><page_5><loc_10><loc_88><loc_48><loc_93></location>III. ELECTROMAGNETIC PERTURBATIONS IN AN AXIONICALLY ACTIVE ULTRARELATIVISTIC PLASMA NONMINIMALLY COUPLED TO GRAVITY</section_header_level_1>
<section_header_level_1><location><page_5><loc_18><loc_83><loc_39><loc_84></location>A. Evolutionary equations</section_header_level_1>
<text><location><page_5><loc_9><loc_49><loc_49><loc_78></location>Let us consider now the state of plasma perturbed by a local variation of electric charge. As usual, we assume, first, that the distribution function takes the form f (a) → f (0) (a) ( q 2 ) + δf (a) ( τ, x α , p k ), second, that the tensor F ik describing the variation of the cooperative electromagnetic field in plasma is not equal to zero. Let us stress that the electromagnetic source in the right-hand side of the master equation for the pseudoscalar field (see Eq.(88) in [5]) is quadratic in the Maxwell tensor, and thus, in the linear approximation the background axion field φ ( τ ) = ντ remains unperturbed. Similarly, the nonminimally extended equations for the gravity field (see Eqs.(89)-(100) in [5]) are considered to be unperturbed. This is possible, when the term η 1 T (4) ik , which is in fact the exclusive term linear in the Maxwell tensor, is vanishing. Below we assume, that η 1 = 0, thus guaranteeing that the background de Sitter spacetime is not perturbed in the linear approximation. For the perturbed quantities F ik and δf (a) we obtain the following coupled system of equations:</text>
<formula><location><page_5><loc_17><loc_35><loc_49><loc_43></location>[1 -2 H 2 (6 q 1 +3 q 2 + q 3 )] ∇ k F ik + +[1 -4 H 2 (3 Q 1 -Q 3 )] ∗ F ik ∇ k φ = = -4 π ∑ (a) e (a) ∫ dP δf (a) g ik p k , (37)</formula>
<formula><location><page_5><loc_25><loc_33><loc_49><loc_35></location>∇ k ∗ F ik = 0 , (38)</formula>
<formula><location><page_5><loc_10><loc_28><loc_49><loc_32></location>g ij p j ( ∂ ∂x i +Γ m ik p m ∂ ∂p k ) δf (a) = e (a) p i F i k ∂ ∂p k f (0) (a) . (39)</formula>
<text><location><page_5><loc_9><loc_9><loc_49><loc_22></location>Below we analyze this system for the case, when the plasma particles are ultrarelativistic. The procedure of derivation of the dispersion relations for the case of the ultrarelativistic plasma is very illustrative in our model. Below we assume that in average q 2 glyph[greatermuch] m 2 (a) a 2 ( τ ), so that p 0 can be replaced by p 0 → q in the integrals. Keeping in mind that in the linear approximation φ = ντ = ν Ha ( τ ) , one can check directly that the scale factor a ( τ ) disappears from the reduced electrodynamic equations, which</text>
<text><location><page_5><loc_52><loc_92><loc_69><loc_93></location>take the following form:</text>
<formula><location><page_5><loc_54><loc_87><loc_87><loc_91></location>(1 + 2 K 1 ) η αβ ∂ α F β 0 = 4 π ∑ (a) e (a) ∫ d 3 q δf (a) ,</formula>
<formula><location><page_5><loc_60><loc_74><loc_92><loc_90></location>(40) (1 + 2 K 1 ) [ ∂ 0 F α 0 + η βγ ∂ β F αγ ] --1 2 νglyph[epsilon1] α βγ F βγ (1 + 2 K 2 ) = = -4 π ∑ (a) e (a) ∫ d 3 q q α q δf (a) , (41) ∂ 0 F αβ + ∂ α F β 0 -∂ β F α 0 = 0 , (42) ∂ α F βγ + ∂ β F γα + ∂ γ F αβ = 0 . (43)</formula>
<text><location><page_5><loc_52><loc_72><loc_79><loc_73></location>Here we used the following notations:</text>
<formula><location><page_5><loc_63><loc_67><loc_81><loc_70></location>K 1 ≡ -H 2 (6 q 1 +3 q 2 + q 3 ) , K 2 ≡ -2 H 2 (3 Q 1 -Q 3 ) ,</formula>
<formula><location><page_5><loc_89><loc_67><loc_92><loc_68></location>(44)</formula>
<text><location><page_5><loc_52><loc_64><loc_92><loc_66></location>and introduced the three-dimensional Levi-Civita symbol</text>
<formula><location><page_5><loc_67><loc_62><loc_92><loc_63></location>glyph[epsilon1] αβγ ≡ E 0 αβγ , (45)</formula>
<text><location><page_5><loc_52><loc_56><loc_92><loc_61></location>with glyph[epsilon1] 123 = 1. Here and below for the operation with the indices we use the Minkowski tensor η ik =diag(1 , -1 , -1 , -1).</text>
<text><location><page_5><loc_52><loc_52><loc_92><loc_56></location>The kinetic equation (39) for the ultrarelativistic plasma in the linear approximation can be written in the form</text>
<formula><location><page_5><loc_53><loc_48><loc_92><loc_51></location>q∂ 0 δf (a) + η αβ q α ∂ β δf (a) = e (a) η αβ F β 0 q α df (0) (a) dq . (46)</formula>
<text><location><page_5><loc_52><loc_37><loc_92><loc_46></location>Surprisingly, in the given representation all the electrodynamic equations (40)-(43) and the kinetic equation (46) look like the set of integro-differential equations with the coefficients, which do not depend on time. This fact allows us to apply the method of Fourier transformations, which is widely used in case, when we deal with the Minkowski spacetime [1-4].</text>
<section_header_level_1><location><page_5><loc_56><loc_32><loc_88><loc_33></location>B. Equations for Fourier-Laplace images</section_header_level_1>
<text><location><page_5><loc_52><loc_23><loc_92><loc_30></location>In order to study in detail perturbations arising in plasma we consider the Fourier-Laplace transformations (the Fourier transformation with respect to the spatial coordinates x α and the Laplace transformation for τ ) in the following form:</text>
<formula><location><page_5><loc_54><loc_19><loc_92><loc_22></location>δf (a) = ∫ d Ω d 3 k (2 π ) 4 δϕ (a) (Ω , k γ , q β ) e i ( k α x α -Ω τ ) , (47)</formula>
<formula><location><page_5><loc_58><loc_14><loc_92><loc_17></location>F lk = ∫ d Ω d 3 k (2 π ) 4 F lk (Ω , k γ ) e i ( k α x α -Ω τ ) . (48)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>As usual, we assume that k 1 , k 2 , k 3 are pure real quantities in order to guarantee that both exponential functions, e ik α x α and e -ik α x α , are finite everywhere. The</text>
<text><location><page_6><loc_9><loc_80><loc_49><loc_93></location>quantity Ω is in general case the complex one, Ω= ω + iγ , and, as usual, we assume that perturbations are absent at τ < τ 0 , where τ 0 = -1 Ha ( t 0 ) relates to the moment t = t 0 according to (9). Our choice of the sign minus in the expression for the phase Θ ≡ k α x α -Ω τ in the exponentials in (47), (48) can be motivated as follows. When the value t -t 0 is small, the cosmological time τ (see (9)) can be estimated as τ →-1 Ha ( t 0 ) [1 -H ( t -t 0 )], and thus the phase Θ reads</text>
<formula><location><page_6><loc_10><loc_74><loc_49><loc_77></location>Θ = Θ 0 + k α x α -Ω a ( t 0 ) ( t -t 0 ) , Θ 0 = -Ω Ha ( t 0 ) . (49)</formula>
<text><location><page_6><loc_9><loc_56><loc_49><loc_71></location>Using the notation k 0 = -Ω a ( t 0 ) we obtain from (49) the expression for the phase Θ = Θ 0 + k m x m , which is standard for the case of Minkowski spacetime. Finally, the term e i Θ has the multiplier e -i Ω τ which contains e γτ → e γ a ( t 0 ) ( t -t 0 ) . In other words, both in terms of t and τ the quantity γ has the same sense: when γ < 0 we deal with the plasma-wave damping and | γ | is the decrement of damping; when γ > 0 we deal with increasing of the perturbation in plasma and this positive γ is the increment of instability.</text>
<text><location><page_6><loc_9><loc_45><loc_49><loc_55></location>We should introduce an initial value of the perturbed distribution function at the moment τ = τ 0 , indicated as δf (a) (0 , k α , q γ ); as for initial data for the electromagnetic field, we can put without loss of generality that F α 0 ( τ =0 , k β )=0. Using (40)-(43) and (46) the equations for the Fourier images δϕ (a) (Ω , k γ , q β ) and F lk (Ω , k γ ) can be written as follows:</text>
<formula><location><page_6><loc_13><loc_35><loc_42><loc_43></location>( Ω -k α q α q ) δϕ (a) = = i e (a) F α 0 q α q · df (0) (a) dq + i δf (a) (0 , k α , q γ ) ,</formula>
<formula><location><page_6><loc_46><loc_36><loc_49><loc_37></location>(50)</formula>
<formula><location><page_6><loc_11><loc_31><loc_49><loc_35></location>(1 + 2 K 1 ) k α F α 0 = -4 πi ∑ (a) e (a) ∫ d 3 q δϕ (a) , (51)</formula>
<formula><location><page_6><loc_9><loc_28><loc_48><loc_31></location>(1 + 2 K 1 ) (Ω F α 0 -k γ F αγ ) -i 2 νglyph[epsilon1] α βγ F βγ (1 + 2 K 2 ) =</formula>
<formula><location><page_6><loc_18><loc_24><loc_49><loc_28></location>= -4 πi ∑ (a) e (a) ∫ d 3 q δϕ (a) q α q , (52)</formula>
<formula><location><page_6><loc_18><loc_22><loc_49><loc_24></location>F αβ = Ω -1 ( k α F β 0 -k β F α 0 ) , (53)</formula>
<formula><location><page_6><loc_18><loc_20><loc_49><loc_22></location>k α F βγ + k β F γα + k γ F αβ = 0 . (54)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_17></location>Here and below we use the notation q α ≡ η αβ q β for the sake of simplicity. Clearly, the equation (54) is satisfied identically, if we put F αβ from (53). Then we use the standard method: we take δϕ (a) from (50), put it into (51) and (52), use (53) and obtain, finally, the equations for the Fourier images of the components F β 0 of</text>
<text><location><page_6><loc_52><loc_92><loc_66><loc_93></location>the Maxwell tensor:</text>
<formula><location><page_6><loc_66><loc_88><loc_77><loc_91></location> k α (1 + 2 K 1 ) -</formula>
<formula><location><page_6><loc_54><loc_81><loc_92><loc_86></location>-4 π ∑ (a) e 2 (a) ∫ d 3 q q α ( q Ω -k β q β ) · df (0) (a) dq F α 0 = J 0 , (55)</formula>
<formula><location><page_6><loc_53><loc_74><loc_90><loc_78></location> (1 + 2 K 1 ) [ Ω 2 δ γ α -k 2 Π γ α ] -i νglyph[epsilon1] α βγ k β (1+2 K 2 ) -</formula>
<formula><location><page_6><loc_53><loc_67><loc_92><loc_73></location>-4 π ∑ (a) Ω e 2 (a) ∫ d 3 q q α q γ q ( q Ω -k β q β ) · df (0) (a) dq F γ 0 = Ω J α . (56)</formula>
<text><location><page_6><loc_52><loc_63><loc_92><loc_66></location>The Fourier images of the initial perturbations of the electric current J 0 and J α are defined as follows:</text>
<formula><location><page_6><loc_56><loc_58><loc_92><loc_62></location>J 0 ≡ 4 π ∑ (a) e (a) ∫ d 3 q q δf (a) (0 , k β , q γ ) q Ω -k µ q µ , (57)</formula>
<formula><location><page_6><loc_56><loc_52><loc_92><loc_55></location>J α ≡ 4 π ∑ (a) e (a) ∫ d 3 q q α δf (a) (0 , k β , q γ ) q Ω -k µ q µ . (58)</formula>
<text><location><page_6><loc_52><loc_47><loc_92><loc_50></location>The compatibility condition for the current four-vector ∇ k J k =0, written in terms of Fourier images</text>
<formula><location><page_6><loc_53><loc_41><loc_92><loc_46></location>Ω J 0 -k α J α = 4 π ∑ (a) e (a) ∫ d 3 q δf (a) (0 , k β , q γ ) = 0 , (59)</formula>
<text><location><page_6><loc_52><loc_37><loc_92><loc_41></location>requires in fact that the perturbation in the plasma state does not change the particle number. The term Π γ α is a projector:</text>
<formula><location><page_6><loc_60><loc_33><loc_92><loc_36></location>Π γ α = δ γ α + k α k γ k 2 , Π γ α Π µ γ = Π µ α , (60)</formula>
<text><location><page_6><loc_52><loc_30><loc_70><loc_31></location>it is orthogonal to k α , i.e.,</text>
<formula><location><page_6><loc_65><loc_27><loc_92><loc_29></location>Π γ α k α = 0 = Π γ α k γ . (61)</formula>
<text><location><page_6><loc_52><loc_23><loc_92><loc_26></location>The quantity k 2 is defined as k 2 = -k β k β ; it is real and positive.</text>
<section_header_level_1><location><page_6><loc_62><loc_19><loc_81><loc_20></location>C. Permittivity tensors</section_header_level_1>
<text><location><page_6><loc_52><loc_14><loc_92><loc_17></location>As usual, we introduce the standard permittivity tensor for the spatially isotropic relativistic plasma [1]</text>
<formula><location><page_6><loc_54><loc_9><loc_92><loc_13></location>ε γ α = δ γ α -4 π Ω ∑ (a) e 2 (a) ∫ d 3 q q α q γ q ( q Ω -k β q β ) · df (0) (a) dq , (62)</formula>
<text><location><page_7><loc_9><loc_92><loc_30><loc_93></location>and obtain the decomposition</text>
<formula><location><page_7><loc_16><loc_88><loc_49><loc_91></location>ε γ α = ε ⊥ ( δ γ α + k α k γ k 2 ) -ε || · k α k γ k 2 , (63)</formula>
<text><location><page_7><loc_9><loc_84><loc_49><loc_86></location>where ε ⊥ and ε || are the scalar transversal and longitudinal permittivities, respectively:</text>
<formula><location><page_7><loc_22><loc_79><loc_49><loc_82></location>ε ⊥ ≡ 1 2 ( ε α α -ε || ) , (64)</formula>
<formula><location><page_7><loc_12><loc_73><loc_49><loc_77></location>ε || ≡ 1 + 4 π k 2 ∑ (a) e 2 (a) ∫ d 3 q k α q α ( q Ω -k β q β ) · df (0) (a) dq . (65)</formula>
<text><location><page_7><loc_9><loc_67><loc_49><loc_71></location>Finally, we decompose the Fourier image of the electric field F γ 0 into the longitudinal and transversal components with respect to the wave three-vector</text>
<formula><location><page_7><loc_22><loc_63><loc_49><loc_66></location>F γ 0 = k γ k E || + E ⊥ γ , (66)</formula>
<text><location><page_7><loc_9><loc_61><loc_13><loc_62></location>where</text>
<formula><location><page_7><loc_18><loc_57><loc_49><loc_60></location>E || ≡ -F γ 0 k γ k , E ⊥ γ ≡ F β 0 Π β γ , (67)</formula>
<text><location><page_7><loc_9><loc_51><loc_49><loc_55></location>and obtain the split equations for the Fourier images of the longitudinal and transversal electric field components, respectively:</text>
<formula><location><page_7><loc_21><loc_47><loc_49><loc_50></location>E || = -J 0 k ( ε || +2 K 1 ) , (68)</formula>
<formula><location><page_7><loc_15><loc_38><loc_49><loc_44></location>[( ε ⊥ +2 K 1 -(1+2 K 1 ) k 2 Ω 2 ) δ γ α + + i ν Ω 2 glyph[epsilon1] α γβ k β (1+2 K 2 ) ] E ⊥ γ = 1 Ω Π γ α J γ . (69)</formula>
<text><location><page_7><loc_21><loc_29><loc_21><loc_31></location>glyph[negationslash]</text>
<text><location><page_7><loc_9><loc_20><loc_49><loc_36></location>The second term in the brackets (linear in the wave threevector k β ) describes the well-known effect of optical activity: two transversal components of the field E ⊥ γ are coupled, so that linearly polarized waves do not exist, when ν (1+2 K 2 ) = 0. Since the so-called gyration tensor ν Ω 2 glyph[epsilon1] α γβ (1+2 K 2 ) is proportional to the Levi-Civita symbol glyph[epsilon1] α γβ , we deal with natural optical activity according to the terminology used in [31], which can be described by one (pseudo) scalar quantity. Thus, we can speak about axionically induced optical activity in plasma, and about axionically active plasma itself.</text>
<section_header_level_1><location><page_7><loc_20><loc_16><loc_38><loc_17></location>D. Dispersion relations</section_header_level_1>
<text><location><page_7><loc_9><loc_9><loc_49><loc_14></location>The inverse Fourier-Laplace transformation (48) of the electromagnetic field is associated with the calculation of the residues in the singular points of two principal types. First, one should analyze the poles of the functions J 0</text>
<text><location><page_7><loc_52><loc_87><loc_92><loc_93></location>and J α (see (57) and 58)) describing initial perturbations; the most known among them are the Van Kampen poles Ω= k α q α q . The poles of the second type appear as the roots of the equations</text>
<formula><location><page_7><loc_67><loc_85><loc_92><loc_86></location>ε || +2 K 1 = 0 , (70)</formula>
<text><location><page_7><loc_52><loc_82><loc_54><loc_83></location>and</text>
<formula><location><page_7><loc_58><loc_75><loc_92><loc_81></location>det [( ε ⊥ +2 K 1 -(1+2 K 1 ) k 2 Ω 2 ) δ γ α --i ν Ω 2 glyph[epsilon1] α βγ k β (1+2 K 2 ) ] = 0 . (71)</formula>
<text><location><page_7><loc_52><loc_62><loc_92><loc_73></location>The equation (70) is the nonminimal generalization of the dispersion relations for the longitudinal plasma waves; it includes the spacetime curvature and the constants of nonminimal coupling (via the term K 1 , see (44)), nevertheless, it does not contain any information about the axion field. The equation (71) describes transversal electromagnetic waves in axionically active plasma nonminimally coupled to gravity; it can be transformed into</text>
<formula><location><page_7><loc_52><loc_53><loc_92><loc_61></location>[ ε ⊥ +2 K 1 -(1+2 K 1 ) k 2 Ω 2 ] [ ( ε ⊥ +2 K 1 -(1+2 K 1 ) k 2 Ω 2 ) 2 --ν 2 (1+2 K 2 ) 2 k 2 Ω 4 ] = 0 . (72)</formula>
<text><location><page_7><loc_52><loc_48><loc_92><loc_52></location>This dispersion equation generalizes the one obtained in [32, 33] for a minimal axionic vacuum. Clearly, one should consider two important cases.</text>
<formula><location><page_7><loc_62><loc_44><loc_81><loc_45></location>1. Special case 1 + 2 K 2 = 0</formula>
<text><location><page_7><loc_52><loc_39><loc_92><loc_42></location>The condition 1+2 K 2 = 0 rewritten as 3 Q 1 -Q 3 = 1 4 H 2 , provides the dispersion relations to be of the form</text>
<formula><location><page_7><loc_62><loc_35><loc_92><loc_38></location>[ ε ⊥ +2 K 1 -(1+2 K 1 ) k 2 Ω 2 ] =0 , (73)</formula>
<text><location><page_7><loc_52><loc_31><loc_92><loc_34></location>includes the curvature terms and does not contain the information about the axion field.</text>
<text><location><page_7><loc_79><loc_27><loc_79><loc_28></location>glyph[negationslash]</text>
<formula><location><page_7><loc_62><loc_27><loc_82><loc_28></location>2. General case 1 + 2 K 2 = 0</formula>
<text><location><page_7><loc_52><loc_22><loc_92><loc_25></location>In this case the dispersion relations for the transversal electromagnetic perturbations read</text>
<formula><location><page_7><loc_55><loc_18><loc_92><loc_21></location>ε ⊥ = -2 K 1 + (1 + 2 K 1 ) k 2 ± ν (1 + 2 K 2 ) k Ω 2 , (74)</formula>
<text><location><page_7><loc_62><loc_10><loc_62><loc_11></location>glyph[negationslash]</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_17></location>displaying explicitly the dependence on the axion field strength ν . Two signs, plus and minus, symbolize the difference in the dispersion relations for waves with lefthand and right-hand polarization rotation. In this sense, when 1+2 K 2 = 0, we deal with an axionically active plasma.</text>
<section_header_level_1><location><page_8><loc_10><loc_91><loc_48><loc_93></location>E. Analytical properties of the permittity tensor and the inverse Laplace transformation</section_header_level_1>
<text><location><page_8><loc_9><loc_83><loc_49><loc_88></location>Let us remind three important features concerning the Laplace transformation in the context of relativistic plasma theory. First, as usual, we treat this transformation as a limiting procedure</text>
<formula><location><page_8><loc_15><loc_77><loc_49><loc_81></location>f ( τ ) = 1 2 π lim A → + ∞ + A + iσ ∫ -A + iσ F (Ω) e -i Ω τ d Ω , (75)</formula>
<text><location><page_8><loc_9><loc_55><loc_49><loc_75></location>where a real positive parameter σ exceeds the so-called growth index σ 0 > 0 of the original function f ( τ ). The Laplace image F (Ω), as a function of the complex variable Ω= ω + iγ , is defined and analytic in the domain ImΩ= γ > σ of the plane ω 0 γ . Second, in many interesting cases, two points Ω= ± k happen to be branchpoints of the function F (Ω) (see, e.g., [1, 2] for details); as we will show below, in our case this rule remains valid. Third, in order to use the theorem about residues, we should prolong the integration contour into the domain Im Ω < σ , harboring all the poles of the function F (Ω) and keeping in mind that the branchpoints have to remain the external ones. We use the contour presented on Fig. 1, the radius of the arc being R = √ A 2 + σ 2 .</text>
<figure>
<location><page_8><loc_12><loc_34><loc_45><loc_54></location>
<caption>FIG. 1: Integration contour C R for the inverse Laplace transformation of the electromagnetic field strength. The radius R = √ A 2 + σ 2 tends to infinity, providing that all the poles of the function F lk (Ω , k γ ) are inside. The branchpoints Ω= ± k are excluded by using cuts along the lines ω = ± k , γ = ky , -∞ < y < 0, and infinitely small circles harboring the branchpoints.</caption>
</figure>
<text><location><page_8><loc_9><loc_9><loc_49><loc_21></location>In order to apply the residues theorem to the calculation of the function F ik ( τ, x α ) we should fix one of the analytic continuations of the function F ik (Ω , k α ) into the domain ImΩ < σ . Thus, the function F ik ( τ, x α ) contains contributions of two types: first, the residues in the poles of the function F lk (Ω , k γ ); second, the integrals along the cuts ω = ± k , γ = ky , -∞ < y ≤ 0. The contribution of the second type displays the dependence on time in the form exp ( ikτ ), and looks like the packet</text>
<text><location><page_8><loc_79><loc_65><loc_79><loc_66></location>glyph[negationslash]</text>
<text><location><page_8><loc_52><loc_52><loc_92><loc_93></location>of waves propagating with the phase velocity V ph = c =1. In order to describe the contributions of the first type, below we study in detail the solutions to the dispersion relations for the longitudinal and transversal waves in plasma. Providing the mentioned analytic continuation of the function F lk (Ω , k γ ) into the domain Im Ω < σ , we need to take special attention to the analytic properties of the permittivity scalar ε || (see (65)), which is one of the structural elements of the longitudinal electric field. The discussion of this problem started in [34] and led to the appearance in the scientific lexicon of the term Landau damping [1-4], based on the prediction made in [35]. The results of this discussion briefly can be formulated as follows (we assume here that the growth index vanishes, i.e., σ =0). The most important part of the integral (65) is the integration with respect to the longitudinal velocity v || = k α q α kq , which can be written as ∫ +1 -1 dv || Z ( v || ) ( v || -Ω k ) . When ImΩ= γ =0, this real integral diverges at ω < k , and the function ε || is not defined. When γ = 0, this integral can be rewritten as a contour integral with respect to complex velocity v || = x + iy . Since in the domain γ > 0 the function should be analytic, and the pole v || = Ω k can appear in the lower semi-plane γ < 0 only, we recover (in our terminology) the classical Landau's statement about resonant damping of the longitudinal waves in plasma, which can take place if the plasma particles co-move with the plasma wave and extract the energy from the plasma wave [35].</text>
<section_header_level_1><location><page_8><loc_55><loc_46><loc_89><loc_49></location>IV. ANALYSIS OF THE DISPERSION RELATIONS. I. LONGITUDINAL WAVES</section_header_level_1>
<section_header_level_1><location><page_8><loc_54><loc_42><loc_90><loc_44></location>A. Dispersion equation for the ultrarelativistic plasma</section_header_level_1>
<text><location><page_8><loc_52><loc_36><loc_92><loc_40></location>Below we consider the background state of the ultrarelativistic plasma to be described by the distribution functions</text>
<formula><location><page_8><loc_63><loc_31><loc_92><loc_35></location>f (0) (a) ( q ) = N (a) 8 πT 3 (a) e -q T (a) , (76)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_30></location>where the temperatures for all sorts of particles should coincide if the background state was the equilibrium one. We are interested in the analysis of the solutions Ω( k α , ν )= ω + iγ of the dispersion relations for longitudinal (70) and transversal (74) waves. More precisely, we focus on the classification of the roots of (70) and (74) and search for nonstandard solutions which appear just due to nonminimal interactions and axion-photon couplings. A number of facts, which we discuss below, are well-known in the context of relativistic plasma-wave theory; nevertheless, we prefer to restate them in order to explain properly new results, which appear in the axionically active plasma nonminimally coupled to gravity. We have to stress that in the model under consideration the results of integration are presented in an explicit form,</text>
<text><location><page_9><loc_9><loc_90><loc_49><loc_93></location>and the analytic continuation of all the necessary functions also is made explicitly.</text>
<text><location><page_9><loc_9><loc_86><loc_49><loc_90></location>Since the background state of plasma is spatially isotropic, one can choose the 0 Z axis along the wave vector, i.e., without loss of generality one can put</text>
<formula><location><page_9><loc_10><loc_82><loc_49><loc_85></location>k α = (0 , 0 , -k ) , q α = q (cos ϕ sin θ, sin ϕ sin θ, cos θ ) . (77)</formula>
<text><location><page_9><loc_9><loc_79><loc_49><loc_82></location>Then the longitudinal permittivity scalar (65) can be reduced to the following term:</text>
<formula><location><page_9><loc_21><loc_75><loc_49><loc_78></location>ε || = 1 -3 W 2 k 2 Q 1 ( z ) , (78)</formula>
<text><location><page_9><loc_9><loc_73><loc_34><loc_74></location>where t = cos θ , z ≡ Ω /k , the term</text>
<formula><location><page_9><loc_21><loc_68><loc_49><loc_72></location>W 2 = 4 π 3 ∑ (a) e 2 (a) N (a) T (a) (79)</formula>
<text><location><page_9><loc_9><loc_61><loc_49><loc_67></location>is usually associated with the square of the plasma frequency in the ultrarelativistic approximation [1], and, finally, the Legendre function of the second kind Q 1 ( z ) is given by the integral (see, e.g., [36, 37])</text>
<formula><location><page_9><loc_12><loc_55><loc_49><loc_60></location>Q 1 ( z ) = 1 2 1 ∫ -1 t dt z -t = Re Q 1 ( z ) + i Im Q 1 ( z ) , (80)</formula>
<text><location><page_9><loc_9><loc_53><loc_12><loc_54></location>with</text>
<formula><location><page_9><loc_14><loc_45><loc_49><loc_52></location>Re Q 1 ( z ) = -1 + x 4 log [ ( x +1) 2 + y 2 ( x -1) 2 + y 2 ] --y 2 [ arctan x -1 y -arctan x +1 y ] , (81)</formula>
<formula><location><page_9><loc_16><loc_37><loc_49><loc_43></location>Im Q 1 ( z ) = y 4 log [ ( x +1) 2 + y 2 ( x -1) 2 + y 2 ] + + x 2 [ arctan x -1 y -arctan x +1 y ] . (82)</formula>
<text><location><page_9><loc_9><loc_31><loc_49><loc_35></location>We treat the quantity z = Ω k as a new complex variable z = x + iy , where x = ω k and y = γ k . In terms of the complex variable z this function looks more attractive</text>
<formula><location><page_9><loc_12><loc_27><loc_49><loc_30></location>Q 1 ( z ) = 1 2 z ln ( z +1 z -1 ) -1 , Q 1 (0) = -1 , (83)</formula>
<text><location><page_9><loc_9><loc_16><loc_49><loc_26></location>nevertheless, one should, as usual, clarify analytical properties of this function. Clearly, the points z = ± 1 are the logarithmic branchpoints of the function Q 1 ( z ); in these two points the real part of the Legendre function does not exist. When we cross the line Im z = 0 on the fragment | Re z | < 1 of the real axis, the function Im Q 1 ( z ) experiences the jump, since</text>
<formula><location><page_9><loc_17><loc_9><loc_49><loc_15></location>lim y → 0 +0 { Im Q 1 } = -x π 2 , lim y → 0 -0 { Im Q 1 } = + x π 2 , | x | < 1 . (84)</formula>
<text><location><page_9><loc_82><loc_18><loc_82><loc_20></location>glyph[negationslash]</text>
<text><location><page_9><loc_52><loc_86><loc_92><loc_93></location>When | x | > 1, the function Im Q 1 ( z ) is continuous. In order to obtain analytical function ε || ( z ) we consider the function Q 1 ( z ) to be defined in the domain Im z > 0 and make the analytical extension to the domain Im z < 0 as follows:</text>
<formula><location><page_9><loc_61><loc_78><loc_92><loc_84></location>Q 1 ( z ) → G || ( z ) ≡ 1 2 1 ∫ -1 t dt z -t --iπz Θ( -Im z )Θ(1 -| Re z | ) . (85)</formula>
<text><location><page_9><loc_52><loc_63><loc_92><loc_75></location>Here Θ denotes the Heaviside step function. Let us mention that the function G || ( z ) is analytic everywhere except the lines z = ± 1+ iy with -∞ < y ≤ 0. In fact, to obtain the analytic continuation, we added the residue ( -iπz ) in the singular point t = z into the Legendre function. The same result can be obtained if one deforms the integration contour in (80) so that this contour lies below the singular point t = z and harbors it; in the last case we would repeat the method applied by Landau in [35].</text>
<text><location><page_9><loc_52><loc_59><loc_92><loc_62></location>Let us note that the function G || ( z ) possesses the symmetry</text>
<formula><location><page_9><loc_66><loc_56><loc_92><loc_57></location>G || ( z ) = G || ( -¯ z ) , (86)</formula>
<text><location><page_9><loc_52><loc_48><loc_92><loc_54></location>i.e., it keeps the form with the transformation z → -¯ z , which is equivalent to ω → -ω . Keeping in mind this fact, below we consider ω to be nonnegative without loss of generality.</text>
<text><location><page_9><loc_53><loc_46><loc_66><loc_47></location>Now the function</text>
<formula><location><page_9><loc_65><loc_41><loc_92><loc_44></location>ε || = 1 -3 W 2 k 2 G || ( z ) (87)</formula>
<text><location><page_9><loc_52><loc_34><loc_92><loc_39></location>is defined and is analytical on the complex plane z everywhere except the branchpoints z = ± 1. The corresponding dispersion relation for longitudinal plasma waves can be written as follows:</text>
<formula><location><page_9><loc_64><loc_29><loc_92><loc_32></location>k 2 (1 + 2 K 1 ) 3 W 2 = G || ( z ) . (88)</formula>
<text><location><page_9><loc_70><loc_24><loc_70><loc_25></location>glyph[negationslash]</text>
<text><location><page_9><loc_52><loc_17><loc_92><loc_27></location>When K 1 =0, this equation gives the well-known results (see, e.g., [1]). When K 1 = 0, the results are obtained in [38] for three different cases: 1 + 2 K 1 > 0, 1 + 2 K 1 < 0 and 1+2 K 1 = 0. We recover these results only to demonstrate the method, which we use below for the dispersion equations containing the axionic factor ν = 0. Since the left-hand side of Eq.(88) is real, we require that</text>
<formula><location><page_9><loc_55><loc_12><loc_92><loc_15></location>k 2 (1 + 2 K 1 ) 3 W 2 = Re { G || ( z ) } , Im { G || ( z ) } =0 , (89)</formula>
<text><location><page_9><loc_52><loc_9><loc_64><loc_10></location>or in more detail</text>
<formula><location><page_10><loc_21><loc_88><loc_92><loc_91></location>x 4 log [ ( x +1) 2 + y 2 ( x -1) 2 + y 2 ] -y 2 [ arctan x -1 y -arctan x +1 y ] + πy Θ( -y )Θ(1 -| x | ) = µ, (90)</formula>
<formula><location><page_10><loc_21><loc_82><loc_92><loc_85></location>y 4 log [ ( x +1) 2 + y 2 ( x -1) 2 + y 2 ] + x 2 [ arctan x -1 y -arctan x +1 y ] -πx Θ( -y )Θ(1 -| x | ) = 0 , (91)</formula>
<text><location><page_10><loc_9><loc_77><loc_42><loc_78></location>where we introduced the auxiliary parameter µ</text>
<formula><location><page_10><loc_21><loc_72><loc_49><loc_75></location>µ ≡ 1 + k 2 (1 + 2 K 1 ) 3 W 2 , (92)</formula>
<text><location><page_10><loc_9><loc_68><loc_49><loc_71></location>which is the function of k . There are two explicit solutions to these equations.</text>
<section_header_level_1><location><page_10><loc_20><loc_64><loc_38><loc_65></location>B. Solutions with x = 0</section_header_level_1>
<text><location><page_10><loc_9><loc_56><loc_49><loc_62></location>When x = ω k =0, the equation (91) is satisfied identically, and we can assume that either the frequency vanishes ω =0, or the wavelength becomes infinitely small, k = ∞ . The equation (90) can be written now as</text>
<formula><location><page_10><loc_10><loc_52><loc_49><loc_55></location>µ = y ( arctan 1 y + π Θ( -y ) ) ⇔ arccot y = µ y . (93)</formula>
<text><location><page_10><loc_9><loc_39><loc_49><loc_50></location>Clearly, when µ > 1 (or equivalently, 1+2 K 1 > 0) there are no solutions to this equation. When 1+2 K 1 < 0, i.e., µ < 1, one solution y = y ∗ to the equation (93) appears: it is positive if k 2 < 3 W 2 | 1+2 K 1 | and negative for k 2 > 3 W 2 | 1+2 K 1 | . In the intermediate case µ =1, i.e., when 1+2 K 1 =0, we obtain formally the solution y = ∞ , which we omit as nonphysical.</text>
<section_header_level_1><location><page_10><loc_15><loc_35><loc_42><loc_36></location>C. Solutions with y = 0 and | x | > 1</section_header_level_1>
<text><location><page_10><loc_9><loc_29><loc_49><loc_33></location>When y = 0, we deal with waves propagating without damping/increasing, since γ = 0. The frequency ω can be now found from the equation</text>
<formula><location><page_10><loc_21><loc_24><loc_49><loc_27></location>µ = ω 2 k log ( ω + k ω -k ) . (94)</formula>
<text><location><page_10><loc_9><loc_19><loc_49><loc_23></location>In terms of x and µ introduced in (92) this equation can be rewritten as 1 x =tanh µ x , thus we obtain the following results.</text>
<text><location><page_10><loc_9><loc_13><loc_49><loc_19></location>First, there are no solutions, when µ ≤ 1, i.e., when 1+2 K 1 ≤ 0. Second, there are one positive and one negative roots ( x = ± x ∗ ), when µ > 1, i.e., when 1+2 K 1 is positive.</text>
<text><location><page_10><loc_9><loc_9><loc_49><loc_13></location>To estimate the interval of frequencies, which are allowed by the equation (94), one can use two decompositions for the Legendre function in the case, when y = 0</text>
<text><location><page_10><loc_52><loc_77><loc_59><loc_78></location>and x > 1:</text>
<formula><location><page_10><loc_55><loc_69><loc_92><loc_76></location>Q 1 ( x ) = -1 + x 2 log ( x +1 x -1 ) = ∞ ∑ n =1 x -2 n 2 n +1 = = 1 3 x 2 + 1 5 x 4 + . . . , (95)</formula>
<formula><location><page_10><loc_54><loc_63><loc_92><loc_67></location>( x 2 -1) Q 1 ( x ) = 1 3 -2 ∞ ∑ n =1 x -2 n (2 n +1)(2 n +3) . (96)</formula>
<text><location><page_10><loc_52><loc_56><loc_92><loc_63></location>This means that 1 3 x 2 < Q 1 ( x ) < 1 3( x 2 -1) and thus, taking into account that Q 1 ( x ) = k 2 3 W 2 NM (see (88)), we obtain the inequality W NM < ω < √ W 2 NM + k 2 . In particular, when k glyph[lessmuch] ω , one obtains from (94)</text>
<formula><location><page_10><loc_65><loc_52><loc_92><loc_55></location>ω 2 glyph[similarequal] W 2 NM + 3 5 k 2 , (97)</formula>
<text><location><page_10><loc_52><loc_45><loc_92><loc_51></location>and this result is well-known at K 1 =0 (see, e.g., [1, 3]). Here and below we use the convenient parameter W NM ≡ W √ | 1+2 K 1 | . The plots of the functions ω ( k ), V ph ( k ) and V gr ( k ) are presented on Fig. 2.</text>
<text><location><page_10><loc_52><loc_32><loc_92><loc_44></location>Let us remark, that depending on the value of the wave parameter k two types of longitudinal waves in the relativistic plasma can exist (see, e.g., [1]). The waves of the first type propagate without damping ( γ =0) and with the phase velocity exceeding speed of light in vacuum V ph ( k ) > 1; for instance, the well-known Langmuir waves are described by the dispersion relation ω 2 = ω 2 (p) + k 2 V 2 (s) , thus, V ph ( k ) > 1, when k < ω (p) √ 1 -V 2 (s) (here ω (p) is</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_31></location>the Langmuir plasma frequency, and V (s) is the reduced sound velocity). The longitudinal plasma waves of the second type are characterized by γ < 0 and V ph < 1, thus displaying the well-known Landau damping phenomenon. However, in the ultrarelativistic limit, as it was shown in [1], the plasma waves of the second type are suppressed, and only the waves of the first type exist. In fact, we have shown here that the account for a nonminimal coupling of the electromagnetic field to gravity does not change this result for ultrarelativistic case, if 1+2 K 1 > 0. In the case, when 1+2 K 1 ≤ 0, i.e., when the curvature induced effects are not negligible, the longitudinal waves of the first type in the ultrarelativistic plasma are also suppressed, i.e., no longitudinal waves in the ultrarelativistic plasma can be generated, whatever the value of k is chosen, if 1+2 K 1 ≤ 0.</text>
<figure>
<location><page_11><loc_17><loc_64><loc_41><loc_93></location>
<caption>FIG. 2: The plots of functions ω ( k ), V ph ( k ) and V gr ( k ) for running longitudinal waves in the ultrarelativistic plasma in the case, when 1+2 K 1 > 0. The phase velocity exceeds the speed of light in vacuum, V ph ( k ) > 1, while the group velocity is less than one, V gr ( k ) < 1 for arbitrary k .</caption>
</figure>
<text><location><page_11><loc_23><loc_64><loc_25><loc_65></location>/</text>
<text><location><page_11><loc_29><loc_64><loc_30><loc_65></location></text>
<text><location><page_11><loc_34><loc_64><loc_36><loc_65></location>/</text>
<text><location><page_11><loc_40><loc_64><loc_41><loc_65></location></text>
<section_header_level_1><location><page_11><loc_13><loc_52><loc_44><loc_53></location>D. About solutions with y = 0 and x = 0</section_header_level_1>
<text><location><page_11><loc_35><loc_52><loc_35><loc_53></location>glyph[negationslash]</text>
<text><location><page_11><loc_42><loc_52><loc_42><loc_53></location>glyph[negationslash]</text>
<text><location><page_11><loc_9><loc_45><loc_49><loc_50></location>The question arises, whether the longitudinal wave exists for which both x and y are nonvanishing? In order to answer this question let us transform the pair of equations (90), (91) into</text>
<formula><location><page_11><loc_16><loc_41><loc_49><loc_44></location>µ = ( x 2 + y 2 ) 4 x log [ ( x +1) 2 + y 2 ( x -1) 2 + y 2 ] , (98)</formula>
<formula><location><page_11><loc_13><loc_34><loc_49><loc_39></location>2 µy ( x 2 + y 2 ) + [ arctan x -1 y -arctan x +1 y ] --2 π Θ( -y )Θ(1 -| x | ) = 0 , (99)</formula>
<text><location><page_11><loc_21><loc_32><loc_21><loc_33></location>glyph[negationslash]</text>
<text><location><page_11><loc_29><loc_32><loc_29><loc_33></location>glyph[negationslash]</text>
<text><location><page_11><loc_9><loc_27><loc_49><loc_33></location>assuming that y = 0 and x = 0. Since the expression in the right-hand side of (98) is positive, we should consider the situations with µ > 0 only. We have to distinguish two important cases.</text>
<formula><location><page_11><loc_22><loc_23><loc_36><loc_24></location>1. y < 0 and x 2 < 1</formula>
<text><location><page_11><loc_9><loc_20><loc_41><loc_21></location>In this case the equation (99) takes the form</text>
<formula><location><page_11><loc_11><loc_14><loc_49><loc_19></location>2 µ | y | x 2 + y 2 + [ 2 π +arctan x -1 | y | -arctan x +1 | y | ] = 0 . (100)</formula>
<text><location><page_11><loc_9><loc_9><loc_49><loc_14></location>Since the function arctan U satisfies the condition | arctan U | ≤ π 2 , both the first term and the term in brackets are positive, thus guaranteeing that this equation has no roots in this case.</text>
<formula><location><page_11><loc_64><loc_92><loc_80><loc_93></location>2. y > 0 or/and x 2 > 1</formula>
<text><location><page_11><loc_52><loc_87><loc_92><loc_90></location>In this case we return to the Legendre function and transform its imaginary part as follows</text>
<formula><location><page_11><loc_57><loc_68><loc_92><loc_86></location>Im Q 1 ( z ) = 1 2 i ( Q 1 ( z ) -Q 1 (¯ z )) = = 1 4 i 1 ∫ -1 (¯ z -z ) t dt | z -t | 2 = -1 2 y 1 ∫ -1 t dt | z -t | 2 = = -1 4 y 1 ∫ -1 t dt ( 1 | z -t | 2 -1 | z + t | 2 ) = = -xy 1 ∫ -1 t 2 dt | z 2 -t 2 | 2 . (101)</formula>
<text><location><page_11><loc_52><loc_62><loc_92><loc_67></location>Clearly, the integral in (101) is a positive quantity, thus, Im Q 1 ( z ) = 0 if and only if x · y = 0. In other words, there are no solutions of the dispersion relation under discussion, when both x and y are nonvanishing.</text>
<section_header_level_1><location><page_11><loc_55><loc_56><loc_89><loc_59></location>V. ANALYSIS OF THE DISPERSION RELATIONS. II. TRANSVERSAL WAVES</section_header_level_1>
<section_header_level_1><location><page_11><loc_63><loc_53><loc_81><loc_54></location>A. Dispersion equation</section_header_level_1>
<text><location><page_11><loc_52><loc_46><loc_92><loc_51></location>In analogy with the longitudinal permittivity scalar, the transversal one can be written in terms of reduced frequency z = ( ω + iγ ) k , of the wave three-vector modulus k and the quantity W (see (79)) only:</text>
<formula><location><page_11><loc_64><loc_42><loc_92><loc_45></location>ε ⊥ = -3 W 2 2 z 2 k 2 G ⊥ ( z ) , (102)</formula>
<text><location><page_11><loc_52><loc_40><loc_56><loc_41></location>where</text>
<formula><location><page_11><loc_54><loc_38><loc_92><loc_39></location>G ⊥ ( z ) = [ 1 -( z 2 -1) G || ( z ) ] , G ⊥ (0) = 0 . (103)</formula>
<text><location><page_11><loc_52><loc_33><loc_92><loc_37></location>Here we used the definition (64) of the scalar ε ⊥ and the formulas (62)-(65). The function G ⊥ ( z ) also possesses the property</text>
<formula><location><page_11><loc_65><loc_31><loc_92><loc_32></location>G ⊥ ( z ) = G ⊥ ( -¯ z ) , (104)</formula>
<text><location><page_11><loc_52><loc_24><loc_92><loc_30></location>and is analytic with respect to complex variable z anywhere except the lines z = ± 1+ iy with -∞ < y ≤ 0. In these terms the dispersion relation for transversal waves in ultrarelativistic plasma takes the form</text>
<formula><location><page_11><loc_53><loc_20><loc_92><loc_23></location>k 2 ( z 2 -1) + kp -3 2 W 2 NM G ⊥ ( z ) sgn(1 + 2 K 1 ) = 0 , (105)</formula>
<text><location><page_11><loc_52><loc_19><loc_70><loc_20></location>where the real parameter</text>
<formula><location><page_11><loc_66><loc_15><loc_92><loc_18></location>p = ∓ ν (1 + 2 K 2 ) (1 + 2 K 1 ) (106)</formula>
<text><location><page_11><loc_52><loc_9><loc_92><loc_14></location>appears if and only if the axion field is nonstationary. Since G ⊥ (0)=0, the solution z =0 exists, when k = p ≥ 0. Depending on the value of the parameter (1+2 K 1 ) one obtains three particular cases.</text>
<section_header_level_1><location><page_12><loc_10><loc_91><loc_47><loc_93></location>B. The first case: 1 + 2 K 1 > 0 and x 2 > 1 or/and y > 0</section_header_level_1>
<text><location><page_12><loc_9><loc_83><loc_49><loc_88></location>First, we consider the imaginary part of the equality (105) along the line the analysis for the longitudinal case. When G || ( z )= Q 1 ( z ), i.e. when either Θ( -y )=0 or Θ(1 -| x | )=0, we can show explicitly that</text>
<formula><location><page_12><loc_15><loc_79><loc_43><loc_82></location>Im { k 2 ( z 2 -1) + kp -3 2 W 2 NM G ⊥ ( z ) } =</formula>
<formula><location><page_12><loc_12><loc_73><loc_49><loc_78></location>= x · y 2 k 2 + 3 W 2 NM 2 1 ∫ -1 t 2 ( 1 -t 2 ) dt | z 2 -t 2 | 2 . (107)</formula>
<text><location><page_12><loc_9><loc_70><loc_49><loc_72></location>Clearly, this expression can be equal to zero if and only if x = 0 or y = 0.</text>
<section_header_level_1><location><page_12><loc_15><loc_66><loc_43><loc_67></location>1. On the solutions with y = 0 and x > 1</section_header_level_1>
<text><location><page_12><loc_9><loc_63><loc_38><loc_64></location>When y = 0 the dispersion relation gives</text>
<formula><location><page_12><loc_13><loc_58><loc_49><loc_62></location>k = -p ± √ p 2 +6 W 2 NM ( x 2 -1) G ⊥ ( x ) 2( x 2 -1) , (108)</formula>
<text><location><page_12><loc_9><loc_56><loc_40><loc_58></location>where the function G ⊥ ( x ) can be written as</text>
<formula><location><page_12><loc_12><loc_52><loc_49><loc_56></location>G ⊥ ( x ) = x 2 [ 1 -( x 2 -1) 2 x log ( x +1 x -1 )] . (109)</formula>
<text><location><page_12><loc_9><loc_49><loc_49><loc_52></location>For this function the following decompositions can be found:</text>
<formula><location><page_12><loc_10><loc_43><loc_49><loc_49></location>G ⊥ ( x ) = 1 -( x 2 -1) Q 1 ( x ) = 2 ∞ ∑ n =0 x -2 n (2 n +1)(2 n +3) , (110)</formula>
<formula><location><page_12><loc_9><loc_37><loc_49><loc_43></location>( x 2 -1) G ⊥ ( x ) = 2 x 2 3 -8 ∞ ∑ n =0 x -2 n (2 n +1)(2 n +3)(2 n +5) , (111)</formula>
<text><location><page_12><loc_9><loc_17><loc_49><loc_37></location>from which one can see that G ⊥ (1) = 1 and G ⊥ ( ∞ ) = 2 3 . Being monotonic, the function G ⊥ ( x ) is restricted by the inequalities 2 3 < G ⊥ ( x ) < 1, when x > 1. This means that the square roots in equations (108) is real, the righthand side is positive for arbitrary p , if we use only the root with plus sign in front of the square root. Thus, the equation (108) (with the plus sign in front of the square root) has real solutions Ω = ω ( k ), which describe transversal oscillations without damping/increasing and phase velocity exceeding the speed of light in vacuum. When p is nonpositive, these oscillations have arbitrary wavelength 0 < 1 k < ∞ . When p is positive, k cannot exceed some critical value k crit ≡ 3 W 2 NM 2 p , which can be obtained from the following estimation:</text>
<formula><location><page_12><loc_16><loc_9><loc_49><loc_16></location>k = 3 W 2 NM G ⊥ ( x ) p + √ p 2 +6 W 2 NM ( x 2 -1) G ⊥ ( x ) < 3 W 2 NM G ⊥ ( x ) 2 p < 3 W 2 NM 2 p . (112)</formula>
<text><location><page_12><loc_52><loc_90><loc_92><loc_93></location>In the approximation of long waves, i.e., when k → 0, we obtain</text>
<formula><location><page_12><loc_56><loc_86><loc_92><loc_89></location>ω 2 ( k ) glyph[similarequal] W 2 NM -p k + 6 5 k 2 + p 5 W 2 NM k 3 . (113)</formula>
<text><location><page_12><loc_52><loc_82><loc_92><loc_85></location>For short waves ( k → ∞ ) and p ≤ 0 the square of the frequency reads</text>
<formula><location><page_12><loc_62><loc_78><loc_92><loc_81></location>ω 2 ( k ) glyph[similarequal] k 2 -p k + 3 2 W 2 NM . (114)</formula>
<text><location><page_12><loc_52><loc_70><loc_92><loc_77></location>In Fig. 3 and Fig. 4 we presented the results of numerical calculations: we have chosen four values of the parameter p for illustration of the dependence ω ( k ), and eight values of p for illustration of the behavior of the functions V ph ( k ) and V gr ( k ).</text>
<figure>
<location><page_12><loc_53><loc_48><loc_90><loc_69></location>
<caption>FIG. 3: Plots of the dispersion function ω ( k ) for running transversal waves with γ =0 and ω > k at 1+2 K 1 > 0. When p > 0 the lines ω = ω ( k ) stop on the bisector line ω = k at k = k crit = 3 W 2 NM 2 p , and cannot be prolonged for k > k crit . For negative values of p the curves do not cross and do not touch the bisector ω = k at k → ∞ ; when p =0, the bisector is the asymptote for the corresponding curve.</caption>
</figure>
<text><location><page_12><loc_52><loc_24><loc_92><loc_35></location>Let us remark that, when the parameter p is negative, there exists some critical value of its modulus | p | = p crit , which distinguishes two principally different situations: when | p | < p crit , the group velocity does not exceed the speed of light in vacuum V gr < 1; when | p | > p crit , we obtain V gr > 1 for arbitrary k . This critical behavior of the function V gr ( k ) can be easily illustrated for the case of short waves (see (114)). Indeed, in this case we obtain</text>
<formula><location><page_12><loc_56><loc_19><loc_92><loc_22></location>V gr = dω dk glyph[similarequal] [ 1 + (6 W 2 NM -| p | 2 ) (2 k + | p | ) 2 ] -1 2 , (115)</formula>
<text><location><page_12><loc_52><loc_9><loc_92><loc_18></location>and, clearly, p crit = √ 6 W NM . Another interesting feature is that the function V gr ( k ) can take negative values for some values of the parameter p . When we think about both results: first, that V gr ( k ) is negative for some p , and that V gr ( k ) can exceed the speed of light in vacuum, V gr > 1, we recall the theory of magneto-active plasma, which</text>
<figure>
<location><page_13><loc_13><loc_57><loc_45><loc_94></location>
<caption>FIG. 4: Plots of the phase and group velocities for running transversal waves with γ =0 and ω > k at 1+2 K 1 > 0. When p > 0, both V ph and V gr become equal to the speed of light in vacuum at k = k crit = 3 W 2 NM 2 p ; the solutions of the dispersion equation do not exist for k > k crit . When p =0 (light grey curves), the straight line V =1 is the horizontal asymptote for curves describing both phase and group velocities at k →∞ . When p is negative, and | p | < p crit , one sees that V gr ( k ) < 1 for arbitrary k ; when p < 0 and | p | > p crit , the group velocity exceeds the speed of light in vacuum V gr > 1.</caption>
</figure>
<text><location><page_13><loc_16><loc_57><loc_18><loc_58></location>GLYPH<150></text>
<text><location><page_13><loc_9><loc_34><loc_49><loc_39></location>demonstrates such behavior for the waves of special type (see, e.g., [39]). It can be considered as a supplementary motif to use the term axionically active plasma by the analogy with the term magneto-active plasma.</text>
<section_header_level_1><location><page_13><loc_15><loc_30><loc_43><loc_31></location>2. On the solutions with x = 0 and y > 0</section_header_level_1>
<text><location><page_13><loc_9><loc_25><loc_49><loc_27></location>When x = 0 and y > 0, the solution to the dispersion equation is</text>
<formula><location><page_13><loc_13><loc_20><loc_49><loc_24></location>k = p ± √ p 2 -6 W 2 NM ( y 2 +1) G ⊥ ( iy ) 2( y 2 +1) , (116)</formula>
<text><location><page_13><loc_9><loc_18><loc_13><loc_19></location>where</text>
<formula><location><page_13><loc_14><loc_14><loc_49><loc_17></location>G ⊥ ( iy ) = y 2 [ ( y 2 +1) y arctan 1 y -1 ] . (117)</formula>
<text><location><page_13><loc_9><loc_9><loc_49><loc_13></location>For positive values of the variable y the function G ⊥ ( iy ) is nonnegative ( G ⊥ (0 +0 ) = 0, G ⊥ (+ i ∞ ) = 2 / 3). This means that, first, at p = 0 there are no solutions to (116);</text>
<text><location><page_13><loc_61><loc_92><loc_61><loc_93></location>glyph[negationslash]</text>
<text><location><page_13><loc_52><loc_80><loc_92><loc_93></location>second, at p = 0 the corresponding solutions exist, for which p 2 ≥ 6 W 2 NM ( y 2 +1) G ⊥ ( iy ); third, when p > 0 one can use both signs, plus and minus, in (116); fourth, when p < 0, there are no solutions, since k should be positive. Let us stress the following feature: since G ⊥ ( iy ) > 0 we obtain from (116) the inequality k < p for solutions with x = 0 and y > 0. To conclude, the nonharmonically increasing perturbations exist (Ω = iγ with γ > 0) only in case, when p > k .</text>
<section_header_level_1><location><page_13><loc_53><loc_76><loc_91><loc_77></location>C. The second case: 1 + 2 K 1 > 0 and x 2 < 1 , y < 0</section_header_level_1>
<text><location><page_13><loc_52><loc_70><loc_92><loc_74></location>When y < 0 and | x | < 1, we divide the equation (105) by the quantity ( z 2 -1) and transform the imaginary part of obtained relation into</text>
<formula><location><page_13><loc_61><loc_63><loc_92><loc_70></location>( kp -3 W 2 NM 2 ) 2 x | y | | z 2 -1 | 2 + + 3 W 2 NM 2 [Im Q 1 ( z ) -πx ] = 0 . (118)</formula>
<text><location><page_13><loc_52><loc_52><loc_92><loc_62></location>Keeping in mind the properties of the function Im Q 1 ( z ), which we studied for the case of longitudinal waves, one can state that, when p ≤ 3 W 2 NM 2 k , the left-hand side of the equation (118) is nonpositive, and thus there is only one root of this equation, namely, x = 0. This case can be considered qualitatively. When p > 3 W 2 NM 2 k , we will study solutions numerically.</text>
<section_header_level_1><location><page_13><loc_62><loc_48><loc_82><loc_49></location>1. The case x = 0 and y < 0</section_header_level_1>
<text><location><page_13><loc_53><loc_44><loc_78><loc_45></location>Now the dispersion equation yields</text>
<formula><location><page_13><loc_54><loc_40><loc_92><loc_43></location>k = p ± √ p 2 +6 W 2 NM ( | y | 2 +1)[ -G ⊥ ( -i | y | )] 2( | y | 2 +1) , (119)</formula>
<text><location><page_13><loc_52><loc_38><loc_65><loc_39></location>where the function</text>
<formula><location><page_13><loc_53><loc_33><loc_92><loc_38></location>[ -G ⊥ ( -i | y | )] = | y | 2 [ 1 + ( | y | 2 +1) | y | ( π -arctan 1 | y | )] (120)</formula>
<text><location><page_13><loc_52><loc_26><loc_92><loc_33></location>is positively defined. Again, we should eliminate the sign minus in front of the square root in (119) since k > 0. Thus, for arbitrary p and k > 0 there are solutions Ω = iγ with negative γ , which describe damping transversal nonharmonic perturbations in plasma.</text>
<text><location><page_13><loc_53><loc_24><loc_91><loc_26></location>Two asymptotic decompositions attract an attention</text>
<formula><location><page_13><loc_55><loc_20><loc_92><loc_24></location>γ ( k ) = 4 p 3 πW 2 NM k 2 + . . . , k glyph[lessmuch] W NM , (121)</formula>
<text><location><page_13><loc_52><loc_19><loc_54><loc_20></location>and</text>
<formula><location><page_13><loc_55><loc_15><loc_92><loc_18></location>γ ( k ) = -2 3 πW 2 NM k 3 + . . . , k glyph[greatermuch] W NM . (122)</formula>
<text><location><page_13><loc_52><loc_9><loc_92><loc_14></location>In Fig. 5 we illustrate the dependence γ ( k ) for ten values of the parameter p ; we put together the plots illustrating the solutions discussed in two paragraphs: for x =0, y > 0 and for x =0, y < 0.</text>
<figure>
<location><page_14><loc_10><loc_73><loc_48><loc_94></location>
<caption>FIG. 5: Plots of the function γ ( k ) for various values of the parameter p . Ten curves (indicated from the left to the right) correspond to the values -4 , -3 , . . . , 4 , 5 of the reduced parameter p W NM . Depending on the chosen value of the parameter k = k 0 , one can find one, two or three solutions γ ( k 0 ).</caption>
</figure>
<figure>
<location><page_14><loc_9><loc_41><loc_49><loc_62></location>
<caption>FIG. 6: The curves p = p ( k ) reconstructed using parametric equations (123) and (124). On the left panel we illustrate solutions of the dispersion equation for the transversal waves with Re z > 1 and Im z = 0 (dark grey infinite lines), and the solutions with 0 < Re z < 1 and Im z < 0 (light grey finite lines). On the right panel we display the family of lines related to the solutions with Re z =0 and Im z > 0 (dark grey), and to the solutions with Re z =0 and Im z < 0 (light grey). The curves on the left panel visualize three separatrices; the curves on the right panel add one new separatrix; all these distinguished curves divide the plane p 0 k into seven domains, as it is shown on Fig. 7.</caption>
</figure>
<section_header_level_1><location><page_14><loc_19><loc_17><loc_39><loc_18></location>2. The case x = 0 and y = 0</section_header_level_1>
<text><location><page_14><loc_29><loc_17><loc_29><loc_18></location>glyph[negationslash]</text>
<text><location><page_14><loc_36><loc_17><loc_36><loc_18></location>glyph[negationslash]</text>
<text><location><page_14><loc_9><loc_8><loc_49><loc_14></location>We indicate the solutions of the dispersion equations, for which neither Re z , nor Im z vanishes, as nontrivial solutions, and study them numerically. For this purpose we introduce the dimensionless real quantities ξ = k W NM ,</text>
<text><location><page_14><loc_52><loc_91><loc_90><loc_93></location>η = p W NM and link them by two parametric equations</text>
<formula><location><page_14><loc_66><loc_86><loc_92><loc_89></location>ξ 2 = 3 2 Im G ⊥ ( z ) Im z 2 , (123)</formula>
<formula><location><page_14><loc_61><loc_83><loc_92><loc_86></location>ξη = 3 2 Im [ G ⊥ ( z )( z 2 -1) -1 ] Im[( z 2 -1) -1 ] . (124)</formula>
<text><location><page_14><loc_52><loc_70><loc_92><loc_80></location>We use the following scheme of analysis. We are interested, finally, in the determination of the functions ω = ω ( k, p ) and γ = γ ( k, p ); however, we start with the numerical and qualitative analysis of the inverse functions k = k ( x, y ) and p = p ( x, y ). In other words, first of all, we find the domains on the plane of the parameters ξ and η , in which the corresponding solutions exist.</text>
<text><location><page_14><loc_52><loc_49><loc_92><loc_69></location>In order to illustrate this scheme let us consider auxiliary plots on Fig. 6. On Fig. 6a one can find the lines of two types: first, infinite lines, which relate to solutions with Re z > 1 and Im z = 0; second, finite lines, which correspond to the solutions with 0 < Re z < 1 and Im z < 0. The family of infinite lines visualizes the separatrix in the form of hyperbola. The family of finite lines visualizes two curves on which these lines start or finish. In Fig. 6b one can find infinite lines of two types. First, we see the family of lines, for which Re z = 0 and Im z > 0; they visualize the separatrix in the form of straight line. The second family of lines relates to the solutions with Re z = 0 and Im z < 0; they visualize an envelope line with extreme point.</text>
<text><location><page_14><loc_52><loc_44><loc_92><loc_48></location>We collected the results of numerical analysis on Fig. 7. There are seven domains on the plane η 0 ξ , the corresponding separatrices appeared as follows.</text>
<text><location><page_14><loc_52><loc_29><loc_92><loc_43></location>(c1) The line indicated as (a) is the bisector ξ = η , and the appearance of this formula can be explained as follows. On the one hand, it relates to the straight-line separatrix found by numerical calculations; on the other hand, it corresponds to the special solution z = 0, which appears if k = p > 0. In addition to the static solution ω = γ =0, in the points belonging to the bisector p = k we obtain the running waves with ω > k , γ =0, when the wave number k satisfies the inequality k < √ 3 2 W NM .</text>
<text><location><page_14><loc_52><loc_18><loc_92><loc_29></location>(c2) The line (b) is described by the formula ξη = 3 2 . On the one hand, is relates to the hyperbolic separatrix found numerically; on the other hand it corresponds to the critical case k = k crit = 3 W 2 NM 2 p , found analytically. For the points along this separatrix, the solutions x = 1 exist, or equivalently, ω = k , γ =0; the solutions with x 2 > 1 do not exist.</text>
<text><location><page_14><loc_72><loc_14><loc_72><loc_16></location>glyph[negationslash]</text>
<text><location><page_14><loc_79><loc_14><loc_79><loc_16></location>glyph[negationslash]</text>
<text><location><page_14><loc_52><loc_9><loc_92><loc_17></location>(c3) Keeping in mind that the nontrivial solutions to the dispersion equations with x = 0 and y = 0 exist only for y < 0 and | x | < 1, we associated the numerically found curves, on which the finite lines start or finish, with two limiting lines found analytically (for x ≥ 0). The first curve of this type, the curve (c), corresponds to the limit</text>
<figure>
<location><page_15><loc_13><loc_68><loc_45><loc_94></location>
<caption>FIG. 7: Arrangement of domains on the plane p 0 k with specific types of solutions to the dispersion equation (105) at 1 + 2 K 1 > 0. When the axion field is stationary ( ν =0), the parameter p = ∓ ν (1+2 K 2 ) (1+2 K 1 ) is equal to zero. In the domain I the solutions with γ = 0, ω > k (running transversal waves), and the solutions with ω = 0, γ < 0 (damping nonharmonic perturbations) exist. In the domain II there are running waves with γ = 0, ω > k , and instable perturbations with ω = 0, γ > 0. In the domains III, IV and V there are nonharmonic perturbations only: in III they are damping ( γ < 0), in IV they are instable ( γ > 0), in V both perturbations can be generated. In the domains VI and VII damping waves with ω < k and γ < 0 exist; in addition, there are damping perturbations in VI ( γ < 0) and instable ones in VII ( γ > 0).</caption>
</figure>
<text><location><page_15><loc_9><loc_44><loc_38><loc_45></location>x → 0 and is described parametrically as</text>
<formula><location><page_15><loc_9><loc_32><loc_49><loc_43></location>ξ 2 = 3 2 lim x → 0 Im G ⊥ ( z ) Im z 2 = 9 4 ( 1 + 3 | y | 2 +1 3 | y | arccot( -| y | ) ) , ξη = 3 2 lim x → 0 Im [ G ⊥ ( z )( z 2 -1) -1 ] Im[( z 2 -1) -1 ] = = 3 4 ( | y | 2 +3+ ( | y | 2 +1) 2 | y | arccot( -| y | ) ) . (125)</formula>
<text><location><page_15><loc_9><loc_28><loc_49><loc_31></location>The second curve, the curve (d), relates to the limit x → 1 and has the following parametric representation:</text>
<formula><location><page_15><loc_14><loc_21><loc_49><loc_27></location>ξ 2 = 3 2 lim x → 1 -0 Im G ⊥ ( z ) Im z 2 , ξη = 3 2 lim x → 1 -0 Im [ G ⊥ ( z )( z 2 -1) -1 ] Im[( z 2 -1) -1 ] . (126)</formula>
<text><location><page_15><loc_9><loc_15><loc_49><loc_19></location>Clearly, in the points of the curve (c) the solutions to the dispersion relations have the form ω = 0, γ < 0; as for the curve (d), one obtains that ω = k , γ < 0.</text>
<text><location><page_15><loc_9><loc_12><loc_49><loc_15></location>In the cross-points the solutions inherit the properties, which characterize both separatrices.</text>
<text><location><page_15><loc_9><loc_9><loc_49><loc_12></location>(p1) The lines (a) and (b) cross in the point A with the coordinates η = ξ = √ 3 2 glyph[similarequal] 1 . 225; in this point ω = γ = 0</text>
<formula><location><page_15><loc_52><loc_91><loc_72><loc_93></location>or ω = k = √ 3 2 W NM , γ = 0.</formula>
<unordered_list>
<list_item><location><page_15><loc_52><loc_87><loc_92><loc_91></location>(p2) The lines (a) and (c) cross in the point C with the coordinates η = ξ glyph[similarequal] 2 . 899; in this points there are two solutions: ω = γ = 0 and ω = 0, γ = -2 . 32 W NM .</list_item>
</unordered_list>
<text><location><page_15><loc_52><loc_82><loc_92><loc_87></location>(p3) The extreme point B has the coordinates ξ glyph[similarequal] 2 . 754, η glyph[similarequal] 2 . 613; for this point there exists a solution with ω = 0, γ = -1 . 31 W NM .</text>
<unordered_list>
<list_item><location><page_15><loc_52><loc_77><loc_92><loc_82></location>(p4) Finally, the point D, which marks the minimum of the curve (d) has the coordinates ξ glyph[similarequal] 1 . 342, η glyph[similarequal] 3 . 003; in this point the special solution ω = k , γ = -0 . 303 W NM exists.</list_item>
</unordered_list>
<text><location><page_15><loc_52><loc_72><loc_92><loc_76></location>Let us summarize the properties of solutions, which relate to the domains I, II, . . . , VII in Fig. 7, using the analysis of the formulas presented above.</text>
<text><location><page_15><loc_67><loc_63><loc_67><loc_64></location>glyph[negationslash]</text>
<unordered_list>
<list_item><location><page_15><loc_54><loc_51><loc_92><loc_71></location>· The domain I is characterized by the inequalities η < ξ and η < 3 2 ξ , or in other terms p < k and k < k crit = 3 W 2 NM 2 p ; this domain includes the region p ≤ 0. In this domain, first, the nontrivial solutions with x · y = 0 do not exist; second, there are running waves without damping/increasing ( γ =0), which are characterized by the phase velocity exceeding the speed of light in vacuum ( ω > k ); third, the damping nonharmonic perturbations exist with ω =0 and γ < 0. The line η =0, which belongs this domain, relates to the model without axions ( ν =0 and thus p =0), and in this sense our results recover the well-known ones.</list_item>
<list_item><location><page_15><loc_54><loc_41><loc_92><loc_50></location>· The domain II ( η > ξ and η < 3 2 ξ ) accumulates solutions with p > k and k < k crit ; the corresponding solutions differ from the ones in the domain I by one detail only: the nonharmonic perturbations with ω =0 are instable, since now γ > 0 (see the discussion below the formula (117)).</list_item>
</unordered_list>
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<unordered_list>
<list_item><location><page_15><loc_54><loc_30><loc_92><loc_40></location>· In the domains III, IV and V only nonharmonic perturbations exist: the nontrivial solutions with x = 0, y = 0 are absent, the solutions with x = 0, y =0 are not admissible since k > k crit here. The damping nonharmonic solutions with ω =0 and γ < 0 are admissible in III and V, the instable solutions with γ > 0 exist in IV and V.</list_item>
</unordered_list>
<text><location><page_15><loc_76><loc_27><loc_76><loc_28></location>glyph[negationslash]</text>
<text><location><page_15><loc_81><loc_27><loc_81><loc_28></location>glyph[negationslash]</text>
<unordered_list>
<list_item><location><page_15><loc_54><loc_20><loc_92><loc_28></location>· Non-trivial solutions with x = 0, y = 0 are admissible in the domains VI and VII; here the damping waves exist with ω < k and γ < 0. In addition nonharmonic perturbations are admissible, which are damping ( γ < 0) in VI, and increasing ( γ > 0) in VII.</list_item>
</unordered_list>
<text><location><page_15><loc_52><loc_17><loc_85><loc_19></location>These results are presented shortly in Table I.</text>
<text><location><page_15><loc_52><loc_9><loc_92><loc_17></location>Let us attract the attention to the number of symbols + in the corresponding boxes. For instance, there are three symbols + in the box for the domain III with negative γ , two symbols in the domain V, and one plus in the domains I and VI. This means that for one value of the quantity k we obtain one, two or three values of</text>
<text><location><page_16><loc_9><loc_88><loc_49><loc_93></location>the decrement of damping γ ( k ). This feature can be explained using Fig. 5: depending on the value of the parameter p the vertical straight line k =const can cross the curve γ = γ ( k, p ) one, two or three times, respectively.</text>
<table>
<location><page_16><loc_11><loc_68><loc_46><loc_84></location>
<caption>TABLE I: 1 + 2 K 1 > 0</caption>
</table>
<section_header_level_1><location><page_16><loc_17><loc_61><loc_41><loc_62></location>D. The third case: 1 + 2 K 1 < 0</section_header_level_1>
<text><location><page_16><loc_9><loc_52><loc_49><loc_59></location>This case can be obtained if we make the formal replacement W 2 NM → -W 2 NM in all formulas obtained in the previous subsection. In particular, the dispersion relation for the transversal waves in plasma has now the form</text>
<formula><location><page_16><loc_18><loc_48><loc_49><loc_51></location>ξ 2 ( z 2 -1) + ξη + 3 2 G ⊥ ( z ) = 0 . (127)</formula>
<text><location><page_16><loc_9><loc_39><loc_49><loc_47></location>We do not discuss similar details of the corresponding analysis, and present below the results only. Similarly to the case 1+2 K 1 > 0, numerical modeling of the lines k = k ( p ) displays three separatrices on the plane η 0 ξ ; they are shown in Fig. 8. The line (a), again, is the bisector ξ = η . The parametric representation of the line (b ' ) is</text>
<formula><location><page_16><loc_13><loc_31><loc_49><loc_38></location>ξ 2 = 9 4 [ 3 y 2 +1 3 y arccot( y ) -1 ] , ξη = 3 4 [ ( y 2 +1) 2 y arccot( y ) -y 2 -3 ] . (128)</formula>
<text><location><page_16><loc_9><loc_25><loc_49><loc_29></location>It is obtained using the conditions x =0 and y > 0 (see (125)). The line (c ' ) is described by the parametric equations</text>
<formula><location><page_16><loc_16><loc_21><loc_41><loc_24></location>ξ 2 = 9 4 [ 3 x 2 -1 6 x ln ( x +1 x -1 ) -1 ] ,</formula>
<formula><location><page_16><loc_12><loc_16><loc_49><loc_19></location>ξη = 3 4 [ x 2 -3 -( x 2 -1) 2 2 x ln ( x +1 x -1 )] , (129)</formula>
<text><location><page_16><loc_9><loc_9><loc_49><loc_14></location>and is the envelope of the family of curves p = p ( x, y =0), k = k ( x, y =0), when x > 1. The last line (d ' ) is obtained using the conditions x =1 and y > 0. Contrary to the previous case, the separatrices have neither cross-points,</text>
<table>
<location><page_16><loc_52><loc_79><loc_92><loc_91></location>
<caption>TABLE II: 1 + 2 K 1 < 0</caption>
</table>
<text><location><page_16><loc_52><loc_59><loc_92><loc_75></location>nor extreme points. Thus, in the case of negative constant 1+2 K 1 , the plane ξ 0 η is divided into five domains. In the domain I ' the solutions to the dispersion equation have vanishing real parts ω =0 and negative imaginary parts γ< 0; the domain II ' is characterized by ω =0 and γ> 0. In the domain III ' there exist instable waves with ω < k and γ > 0; the domain IV ' contains waves with ω > k of two types: increasing ( γ > 0) and damping ( γ < 0). The domain V ' describes running waves without damping/increasing, ω > k , γ =0. We summarize these features in Table II.</text>
<figure>
<location><page_16><loc_55><loc_33><loc_88><loc_58></location>
<caption>FIG. 8: Arrangement of domains with specific types of solutions to the dispersion equation (105) at 1 + 2 K 1 < 0. In the domains I ' and II ' all the solutions belong to the class of nonharmonic perturbations with ω = 0; in I ' these perturbations are damping ( γ < 0), while in II ' they are instable ( γ > 0). The domain III ' contains instable waves with ω < k and γ > 0; in the domain IV ' there exist both damping and instable waves with ω > k . The domain V ' contains the solutions for running waves with γ = 0, ω > k .</caption>
</figure>
<text><location><page_16><loc_71><loc_12><loc_71><loc_13></location>glyph[negationslash]</text>
<text><location><page_16><loc_79><loc_12><loc_79><loc_13></location>glyph[negationslash]</text>
<text><location><page_16><loc_52><loc_9><loc_92><loc_17></location>Concerning the principally new result, let us stress, that after the replacement W 2 NM → -W 2 NM in (107) we can see that indeed, in addition to x =0 and y =0, new solutions can exist with x = 0 and y = 0. This means that, when 1+2 K 1 < 0, transversal waves in axionically active plasma can be, first, damping waves with ω > k ,</text>
<table>
<location><page_17><loc_11><loc_79><loc_46><loc_91></location>
<caption>TABLE III: 1 + 2 K 1 = 0</caption>
</table>
<text><location><page_17><loc_9><loc_72><loc_49><loc_75></location>γ < 0, second, can be instable ( γ > 0) both for ω > k and ω < k .</text>
<section_header_level_1><location><page_17><loc_15><loc_68><loc_43><loc_69></location>E. The intermediate case 1+2 K 1 = 0</section_header_level_1>
<text><location><page_17><loc_9><loc_62><loc_49><loc_66></location>To complete our study, let us consider the case K 1 = -1 2 , for which the dispersion relation reduces to the equation</text>
<formula><location><page_17><loc_22><loc_58><loc_49><loc_60></location>kp ∗ = 3 2 W 2 G ⊥ ( z ) , (130)</formula>
<text><location><page_17><loc_9><loc_45><loc_49><loc_56></location>where p ∗ = ∓ ν (1 + 2 K 2 ) and W is given by (79). In analogy with the case 1 + 2 K 1 > 0, four separatrices appear on the plane ξ 0 η (here ξ = k W , η = p ∗ W ). The first separatrix is the hyperbola ξη = 1; it corresponds to G ⊥ ( ∞ ) = 2 3 (minimal value of the function G ⊥ ( x )). The second separatrix ξη = 3 2 corresponds to G ⊥ (1)=1 (maximal value of G ⊥ ( x )). The third separatrix has the following representation:</text>
<formula><location><page_17><loc_24><loc_42><loc_49><loc_43></location>ξη = 2 . 3904 , (131)</formula>
<text><location><page_17><loc_9><loc_38><loc_49><loc_41></location>where the number 2 . 3904 is the maximum of the function 3 2 Re { G ⊥ ( z ) } at the condition that Im { G ⊥ ( z ) } = 0.</text>
<text><location><page_17><loc_9><loc_23><loc_49><loc_38></location>Finally, we visualize the straight line p ∗ =0 as an abscissa on the plane p ∗ 0 k and the last separatrice. Thus, we obtain five domains on the plane ξ 0 η (see Fig. 9). The domain I 0 ( η < 0) is characterized by ω =0 and γ < 0; in the domain II 0 ( ξη < 1, p ∗ > 0) the solutions of the dispersion equations give ω =0 and γ > 0. In the domain III 0 with 1 < ξη < 3 2 we obtain that γ =0 and ω > k . The solutions with ω < k and γ < 0 can appear in the domain IV 0 ; in the domain V 0 there are no solutions to the dispersion equation.</text>
<text><location><page_17><loc_10><loc_22><loc_41><loc_23></location>These results are summarized in Table III.</text>
<section_header_level_1><location><page_17><loc_21><loc_18><loc_36><loc_19></location>VI. DISCUSSION</section_header_level_1>
<text><location><page_17><loc_9><loc_9><loc_49><loc_16></location>In the framework of nonminimal Einstein-MaxwellVlasov-axion model we analyzed the dispersion relations for the perturbations in an initially isotropic and homogeneous axionically active plasma, which expands in the de Sitter-type cosmological background. In this model</text>
<figure>
<location><page_17><loc_56><loc_70><loc_88><loc_94></location>
<caption>FIG. 9: Arrangement of domains with specific types of solutions to the dispersion equation (105) at 1 + 2 K 1 =0. Here p ∗ = ∓ ν (1 + 2 K 2 ) is the modified axionic guiding parameter. In the domain I 0 only solutions with γ < 0 exist; in the domain II 0 the solutions with ω =0, γ > 0 are possible. In the domain III 0 there are running waves with ω > k . The domain IV 0 is characterized by the damping waves with ω < k and γ < 0. In the domain V 0 there are no solutions.</caption>
</figure>
<text><location><page_17><loc_52><loc_34><loc_92><loc_54></location>we take into account, first, the nonminimal interaction of the electromagnetic field with curvature, second, the nonminimal coupling of the axion field to gravity, and axion-photon coupling in the relativistic plasma. The specific choice of the nonstationary background solution to the total system of equations (de Sitter spacetime with constant curvature, axion field linear in time, vanishing initial macroscopic collective electromagnetic field in the electro-neutral ultrarelativistic plasma) allowed us to obtain and study the dispersion relations in the standard (Ω , glyph[vector] k ) form. We consider these dispersion relations as nonminimal axionic extension of the well-known dispersion relations obtained earlier in the framework of the relativistic Maxwell-Vlasov plasma model.</text>
<text><location><page_17><loc_86><loc_9><loc_86><loc_10></location>glyph[negationslash]</text>
<text><location><page_17><loc_52><loc_9><loc_92><loc_34></location>The presence of the pseudoscalar (axion) field provides the plasma to become a gyrotropic medium, which displays the phenomenon of optical activity. When the plasma in the axionic environment is initially spatially isotropic, its three-dimensional gyration tensor, G αβγ is proportional to the three-dimensional Levi-Civita tensor, glyph[epsilon1] αβγ , thus providing the optical activity to be of the natural type [31]. The proportionality coefficient is the pseudoscalar quantity p = ∓ ν (1+2 K 2 ) (1+2 K 1 ) . The upper sign of this coefficient relates to the circularly polarized wave with left-hand rotation; formally speaking, the sign of p depends on the sign of ν = ˙ φ , and on the sign of nonminimal coefficient (1+2 K 2 ) (1+2 K 1 ) . Since ν = ˙ φ , this type of optical activity is induced by the pseudoscalar (axion) field; if the axions form the dark matter, the quantity ν is proportional to the square root of the dark matter energy density (see, e.g., [40, 41] for details). When K 1 = 0 and</text>
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<text><location><page_18><loc_9><loc_66><loc_49><loc_93></location>K 2 = 0 (see their definitions in (44), the gyration coefficient p contains the square of the Hubble function (i.e., the spacetime curvature scalar in the case of the de Sitter model). Thus, we deal with combined axionic-tidal gyration effect. The frequencies of transversal electromagnetic waves are shown to depend not only on the wavelength, but also on the gyration coefficient p (see, e.g., (113)), and this dependence has a critical character. To be more precise, when p =0, the dispersion equations admit some new branches of solutions ω = ω ( k, p ), γ = γ ( k, p ) in addition to the standard ones. If to consider the transversal electromagnetic wave propagation in terms of left- and right-hand rotating components, one can state, that one of the waves (say, with left-hand rotation) can have arbitrary wavelength, while the second wave can possess the wave number less than critical one (see, e.g., (112)); in this sense we deal with some kind of mode suppression caused by the axion-photon interactions.</text>
<text><location><page_18><loc_46><loc_62><loc_46><loc_63></location>glyph[negationslash]</text>
<text><location><page_18><loc_9><loc_46><loc_49><loc_66></location>In order to simplify the classification of the electromagnetic modes in an axionically active plasma, we use the following terminology: damping wave , when ω = 0, γ < 0; instable wave , when ω = 0, γ > 0; running wave , when ω ≥ k , γ = 0; damping nonharmonic perturbation , when ω = 0, γ < 0; instable nonharmonic perturbation , when ω = 0, γ > 0. The results of analytic, qualitative and numerical study are presented on Figs. 7-9 and in Tables I-III. Based on these results, we can illustrate the evolution of types of the transversal electromagnetic perturbations depending on the value of the gyration parameter p for small, medium and large k . For this purpose we draw, first, the horizontal straight line on Fig. 7, and move it from the bottom to upwards.</text>
<text><location><page_18><loc_30><loc_60><loc_30><loc_61></location>glyph[negationslash]</text>
<unordered_list>
<list_item><location><page_18><loc_11><loc_37><loc_49><loc_44></location>· When p ≤ 0, the horizontal straight line does not cross any separatrices; in this case for arbitrary k there exist solutions of two types: first, running waves with ω > k , γ = 0, or in other words V ph > 1; second, damping nonharmonic perturbations.</list_item>
<list_item><location><page_18><loc_11><loc_17><loc_49><loc_36></location>· When 0 < p < √ 3 2 W NM (below the cross-point A), the horizontal straight line crosses the separatrices (a) and (b), thus displaying three zones. In the zone of medium k ( p < k < 3 W 2 NM 2 p ) the running waves with V ph > 1 and damping nonharmonic perturbations happen to be inherited from the zone p ≤ 0; in the zone of small k ( k < p ) the running waves with V ph > 1 are inherited, but damping nonharmonic perturbations convert into the instable ones; in the zone of large k ( k > 3 W 2 NM 2 p ) the running waves convert into damping waves, but damping nonharmonic perturbations are inherited.</list_item>
<list_item><location><page_18><loc_11><loc_8><loc_49><loc_16></location>· When 1 . 225 < p W NM < 2 . 613 (between cross-point A and extreme point B), there are also three zones. When k < 3 W 2 NM 2 p , there exist running waves and instable nonharmonic perturbations; when 3 W 2 NM 2 p <</list_item>
</unordered_list>
<text><location><page_18><loc_56><loc_88><loc_92><loc_93></location>k < p , running waves convert into damping waves and instable nonharmonic perturbations are inherited; when k > p , damping waves and damping perturbations are inherited.</text>
<unordered_list>
<list_item><location><page_18><loc_54><loc_81><loc_92><loc_87></location>· Next interval 2 . 613 < p W NM < 2 . 899 can be describe similarly; new zone indicated as III appears, in which damping waves convert into damping nonharmonic perturbations.</list_item>
<list_item><location><page_18><loc_54><loc_73><loc_92><loc_80></location>· When 2 . 899 < p W NM < 3 . 003, the following new detail appears: the zone of medium k breaks up into three subzones, and in one of them damping waves appear instead of running waves (in the domain VII).</list_item>
<list_item><location><page_18><loc_54><loc_67><loc_92><loc_71></location>· When p > 3 . 003 W NM , the description is similar, but the new subzone appears (see domain IV), in which waves do not exist.</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_9><loc_92><loc_66></location>Similarly, we can illustrate the evolution of the modes for the case 1 + 2 K 1 < 0 and 1 + 2 K 1 =0. Such analysis reveals two interesting features. First, when the gyration parameter p and the nonminimal parameter 1+2 K 1 are fixed, we can find explicitly the range for the wave parameter k , for which the running waves, i.e., non-damping transversal electromagnetic waves, can propagate in the Universe in the axionic dark matter environment and bring us a true information about the Universe structure and history. For instance, when 1+2 K 1 > 0 according to the Table I it is possible in the domains I and II only; thus, if the gyration parameter exceeds, say, the value p = √ 3 2 W NM (see the point A on Fig. 7), the running waves can propagate only in the narrow interval of small k (the interval of long waves). When 1+2 K 1 < 0, according to Table II the running waves can propagate in the domain V ' only; thus, if the parameter p is positive, the running transversal waves are suppressed for any k . Second, we can focus on the problem of existence of transversal electromagnetic waves of a new type, i.e., transversal waves with the phase velocity less that the speed of light in vacuum, ω < k , which can interact in a resonant manner with particles co-moving with them. It is well-known that in the minimal theory ( K 1 = 0) at the absence of axion field ν = 0 the dispersion equations do not admit the transversal waves of this type. Our consideration shows that the damping waves with V ph < 1 exist in the domains VI and VII (see Fig. 7 for the case 1 + 2 K 1 > 0), and the instable waves with V ph < 1 can be generated for the case 1 + 2 K 1 < 0. Since the phase velocity of the transversal electromagnetic wave is less than speed of light in vacuum, we can make the following remark. There are plasma particles, which co-move with this transversal electromagnetic wave, thus providing a resonant interaction; in the case 1 + 2 K 1 > 0 this resonant interaction leads to the Landau-type damping, since the wave transfers the energy to the resonant particles; in the case 1 + 2 K 1 < 0 this resonant interaction provides the Landau-type instability, since now resonant</text>
<text><location><page_19><loc_9><loc_90><loc_49><loc_93></location>particles transfer their energy to the transversal plasma wave.</text>
<text><location><page_19><loc_9><loc_83><loc_49><loc_90></location>In the next paper we plan to study the dispersion relations for the axionically active plasma nonminimally coupled to gravity in the framework of cosmological BianchiI model with initial magnetic field and electric field axionically induced.</text>
<section_header_level_1><location><page_19><loc_22><loc_79><loc_36><loc_80></location>Acknowledgments</section_header_level_1>
<text><location><page_19><loc_9><loc_74><loc_49><loc_77></location>The work was partially supported by the Russian Foundation for Basic Research (Grants Nos. 11-</text>
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</document>
[
{
"title": "Electromagnetic waves in an axion-active relativistic plasma non-minimally coupled to gravity",
"content": "Alexander B. Balakin, 1, ∗ Ruslan K. Muharlyamov, 1, † and Alexei E. Zayats 1, ‡ 1 Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya str. 18, Kazan 420008, Russia We consider cosmological applications of a new self-consistent system of equations, accounting for a nonminimal coupling of the gravitational, electromagnetic and pseudoscalar (axion) fields in a relativistic plasma. We focus on dispersion relations for electromagnetic perturbations in an initially isotropic ultrarelativistic plasma coupled to the gravitational and axion fields in the framework of isotropic homogeneous cosmological model of the de Sitter type. We classify the longitudinal and transversal electromagnetic modes in an axionically active plasma and distinguish between waves (damping, instable or running), and nonharmonic perturbations (damping or instable). We show that for the special choice of the guiding model parameters the transversal electromagnetic waves in the axionically active plasma, nonminimally coupled to gravity, can propagate with the phase velocity less than speed of light in vacuum, thus displaying a possibility for a new type of resonant particle-wave interactions. PACS numbers: 04.40.-b, 52.35.-g, 14.80.Va Keywords: Axion field, relativistic plasma, Vlasov model, nonminimal coupling, dispersion relations",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Electromagnetic radiation provides the most important channel of information about our Universe. Valuable information about cosmic sources of photons, and about cosmic events accompanying the light propagation, is encoded in the intensity, polarization, phase and spectral characteristics of the electromagnetic radiation. In this sense, reconstruction of the phase and group velocities of observed electromagnetic waves, which travel through plasma and gas in the dark matter environment, gives us the basis for theoretical modeling of the properties of these cosmic substrates. When we deal with plasma, the phase and group velocities are given by V ph = ω k and V gr = ∂ω ∂k , respectively, where ω is the frequency and k is the wave three-vector modulus. Thus, the dependence ω = ω ( k ) obtained from the so-called dispersion relations for the longitudinal and transversal electromagnetic waves in plasma plays an important role in the information decryption. The theory of dispersion relations is well-elaborated for various plasma configurations (see, e.g., [1-4] for details, review and references). To obtain novel results in this scientific sphere we intend to use a new nonminimal Einstein-Maxwell-Vlasov-axion model [5], which deals with the self-consistent theory of tidal-type interactions between gravitational, electromagnetic, pseudoscalar (axion) fields and a relativistic multi-component plasma. Cosmological applications of this model seem to be the most interesting, since the study of the interaction of four key players of the nonminimal Einstein- Maxwell-Vlasov-axion model (gravitons, photons, electrically charged plasma particles and axions) is important for the description of the history of our Universe. There are at least three motives, which could explain this interest. First, in the early Universe the nonminimal couping of a matter and fields to gravity was important on the stage of inflation, when the space-time curvature was varying catastrophically fast. Late-time accelerated expansion discovered recently [6-8] has revived an interest to inertial, tidal and rip's effects [9-12], for which the curvature coupling could be also important. Second, the cold dark matter, which now is considered to be one of two key elements of the dark fluid, guiding the late-time Universe evolution [13-15], contains (hypothetically) an axion subsystem. In the early Universe the axions (light pseudo-Goldstone bosons) could be created due to the phase transition associated with the Peccei-Quinn symmetry breakdown [16-24]. In the late-time Universe these pseudo-bosons, probably, exist as relic axions, forming various cold dark matter configurations. Third, electrically charged plasma and photons are ubiquitous: one can find plasma configurations and an ocean of photons in every epoch in the cosmic history and in many objects, which form our Universe. Of course, studying the nonminimal Einstein-Maxwell-Vlasov-axion theory, we are restricted to theoretical modeling of the dispersion relations for relativistic homogeneous axionically active plasma. For instance, plasma can be treated as relativistic substratum in the early Universe; however, as was shown in the papers [25-27], the PecceiQuinn phase transition is accompanied by the creation of strongly inhomogeneous axionic primordial configurations, which were indicated (see, e.g., [24, 25]) as 'archioles'. The inhomogeneity of archioles type is frozen at the radiation domination stage and should be inherited in the large scale structure of the modern Universe. Additional problem in the theoretical modeling is con- nected with instabilities generated in plasma due to its inhomogeneity (see, e.g., [28]). Self-gravitation of the pseudoscalar (axion) field produces gravitational instability similarly to the instability of scalar fields described in [29], thus making the studying of the electromagnetic waves propagation much more sophisticated. In such situation we need of some toy-model, which provides a balance between complexity of the problem as a whole and mathematical clarity of the simplified model. For the first step we have chosen the approximation, which is standard for homogeneous cosmological model: the spacetime metric and the axionic dark matter distribution are considered to depend on time only. In this case the standard approach to the analysis of waves propagation in relativistic plasma is based on the study of dispersion relations in terms of frequency ω and wave three-vector glyph[vector] k . When we consider homogeneous cosmological models, we can use the standard Fourier transformations with respect to spatial coordinates and can naturally introduce the analog of glyph[vector] k . The Fourier-Laplace transformation with respect to time is, generally, non-effective, since the coefficients in the master equations for the electromagnetic field depend on cosmological time. However, we have found one specific very illustrative model (the plasma is ultrarelativistic, the spacetime is of a constant curvature, the axion field has a constant time derivative), for which the master equations can be effectively transformed into a set of differential equations with constant coefficients. This simplified the analysis of the dispersion relations essentially, and allowed us to interpret the results in the standard terms. As a next step, we plan to consider homogeneous but anisotropic models, and then models with inhomogeneous distribution of the axionic dark matter. In this work we obtain and analyze the dispersion relations for perturbations in the axionically active plasma, non-minimally coupled to gravity, classify these perturbations and distinguish the transversal waves, which can propagate with the phase velocity less than the speed of light in vacuum. The work is organized as follows. In Section II we describe the appropriate background state of the system as a whole, and discuss the exact solutions of non-perturbed master equations for the plasma in the gravitational and axionic fields, which were derived in [5]. In Section III we consider the equations for the electromagnetic perturbations in the ( t, glyph[vector]x ) form and then the equations for the corresponding Fourier-Laplace images in the (Ω , glyph[vector] k ) form. We analyze in detail the dispersion relations for longitudinal waves in Section IV, and for transversal waves in Section V. We summarize the results in Section VI (Discussion).",
"pages": [
1,
2
]
},
{
"title": "II. ON THE BACKGROUND SOLUTIONS TO THE NONMINIMAL MASTER EQUATIONS",
"content": "Let us consider the background solutions for the nonminimal Einstein-Maxwell-Vlasov-axion model, which satisfy the following four conditions. First , we assume that the spacetime is isotropic, spatially homogeneous and is described by the metric Second , we suppose that the pseudoscalar field inherits the spacetime symmetry and depends on time only, φ ( τ ). Third , we assume, that both the background macroscopic (external) and cooperative (internal) electromagnetic fields are absent, i.e., the total Maxwell tensor is equal to zero F ik = 0. Fourth , we consider the plasma to be the test ingredient of the cosmological model; this means that the contribution of the plasma particles into the total stress-energy tensor is negligible in comparison with the dark energy contribution, presented in our model by the Λ term, and with the dark matter one, described by the axion field Ψ 0 φ ( φ is dimensionless pseudoscalar field). These four requirements provide the master equations obtained in [5] to be reduced to the following system.",
"pages": [
2
]
},
{
"title": "A. Background nonminimal equations for the gravity field",
"content": "In the absence of the electromagnetic field the nonminimally extended Einstein equations take the form (see Subsection IIIE in [5]) where the term is the stress-energy tensor of the pseudoscalar field. The terms describe the nonminimal contributions associated with coupling constants η 2 , η 3 , and η (A) , respectively. We are interested to analyze the specific solution to these equations, which is characterized by the de Sitter metric with where H is a constant. With the transformation of time which gives the correspondence: τ → 0, when t → ∞ ; τ → -∞ , when t → -∞ ; τ → -1 Ha ( t 0 ) ≡ τ 0 , when t → t 0 , one can obtain the well-known form of the de Sitter metric The de Sitter metric (7) describes the spacetime of constant curvature, for which the basic geometric quantities of the Riemann tensor, the Ricci tensor and Ricci scalar take the form and the nonminimal susceptibility tensors introduced in [5] transform into One can check directly that the de Sitter-type metric (1) with the scale factor (8) is the exact solution of the equations (2)-(6) with the pseudoscalar field linear in time, i.e., φ ( τ )= ντ , when the following three relationships are satisfied: These relations contain neither ν , nor Ψ 0 . In addition, when η 2 = η 3 = 0, we obtain the result which is well-known for the massless scalar field conformally coupled to gravity [30]. When the constants of nonminimal coupling η 2 and η 3 are nonvanishing, we can consider the axions to be massive, and the mass m (A) itself is connected not only with these coupling constants, but with the cosmological constant Λ as well. In order to minimize the number of unknown coupling parameters, one can put, for instance, satisfying (18) and (19).",
"pages": [
2,
3
]
},
{
"title": "B. Nonminimal equations for the background pseudoscalar (axion) field",
"content": "In the absence of the background electromagnetic field the nonminimally extended master equation (see Eq.(88) in [5]) for the pseudoscalar φ takes the form Let us take into account the relations (15)-(17) for glyph[Rfractur] mn (A) and χ ikmn (A) , and suppose that for the background distribution function its zero-order moment in the right-side hand of (22) vanishes, then we obtain the equation The function φ = ντ satisfies (23), if the following relation is valid We used here the relation Clearly, the relation (24) is identically satisfied, when the equations (18), (19) are valid. In other words, we have shown that the master equation for the background pseudoscalar (axion) field admits the existence of the exact solution φ ( τ ) = ντ linear in time, when the spacetime is of the de Sitter type with the metric (7).",
"pages": [
3,
4
]
},
{
"title": "C. Consistency of the background electrodynamic equations",
"content": "We assume that the Maxwell tensor describing the background cooperative electromagnetic field in plasma is equal to zero, F ik = 0. It is possible, when the tensor of spontaneous polarization-magnetization H ik vanishes, and when the four-vector of electric current in plasma, I i , is equal to zero. Since φ = φ ( τ ) and R α 0 = 0 ( α = 1 , 2 , 3) the formula (see (87) in [5]) gives H ik = 0. The condition I i = 0 provides the restrictions for the background distribution function f (a) , which we will discuss in the next subsection.",
"pages": [
4
]
},
{
"title": "D. Background solution to the Vlasov equation",
"content": "In the context of the Vlasov theory we search for a 8dimensional one-particle distribution function f (a) ( x i , p k ) which describes particles of a sort (a) with the rest mass m (a) , electric charge e (a) . This function of the coordinates x i and of the momentum four-covector p k satisfies the relativistic kinetic equation, which can be presented now in the form We assume here that the background macroscopic cooperative electromagnetic field in plasma is absent, and there are no contact interactions between axions and plasma particles, so that the force F i (a) (see Eq.(29) in [5]) disappears from this equation. In the cosmological context we consider the distribution function depending on the covariant components of the particle fourmomentum, p k in order to simplify the analysis. Here Γ m ik are the Christoffel symbols associated with the spacetime metric g ik . The characteristic equations associated with the kinetic equation (27) are known to be directly solved for the metric (1). For the cases i = α = 1 , 2 , 3 one obtains immediately that where q α =const are the integrals of motion. The component p 0 of the particle momentum four-vector can be found from the quadratic integral of motion g ij p i p j = m 2 (a) =const: Other three integrals of motion are Generally, the distribution function, which satisfies kinetic equation (27) can be reconstructed as an arbitrary function of seven integrals of motion q α , X α , √ g ij p i p j , nevertheless, taking into account that the spacetime is isotropic and homogeneous, we require that the distribution function inherits this symmetry and thus it has the form where f (0) (a) ( q 2 ) is arbitrary function of one argument, namely, q 2 . We are interested now to calculate the first moment of the distribution function where d 4 P = dp 0 dp 1 dp 2 dp 3 symbolizes the volume in the four-dimensional momentum space; the delta function guarantees the normalization property of the particle momentum, the Heaviside function Θ( V h p h ) rejects negative values of energy, V h is the velocity four-vector of the system as a whole. In the de Sitter spacetime with V h = δ h 0 1 a ( τ ) the quantities N i (a) can be reduced to therefore, the spatial components N α (a) vanish. The component N 0 (a) reads where N (a) does not depend on time. This means that the four-vector of the electric current in the electro-neutral plasma is equal to zero at arbitrary time moment τ , if it was equal to zero at the initial time moment. This guarantees that the Maxwell equations are self-consistent.",
"pages": [
4
]
},
{
"title": "A. Evolutionary equations",
"content": "Let us consider now the state of plasma perturbed by a local variation of electric charge. As usual, we assume, first, that the distribution function takes the form f (a) → f (0) (a) ( q 2 ) + δf (a) ( τ, x α , p k ), second, that the tensor F ik describing the variation of the cooperative electromagnetic field in plasma is not equal to zero. Let us stress that the electromagnetic source in the right-hand side of the master equation for the pseudoscalar field (see Eq.(88) in [5]) is quadratic in the Maxwell tensor, and thus, in the linear approximation the background axion field φ ( τ ) = ντ remains unperturbed. Similarly, the nonminimally extended equations for the gravity field (see Eqs.(89)-(100) in [5]) are considered to be unperturbed. This is possible, when the term η 1 T (4) ik , which is in fact the exclusive term linear in the Maxwell tensor, is vanishing. Below we assume, that η 1 = 0, thus guaranteeing that the background de Sitter spacetime is not perturbed in the linear approximation. For the perturbed quantities F ik and δf (a) we obtain the following coupled system of equations: Below we analyze this system for the case, when the plasma particles are ultrarelativistic. The procedure of derivation of the dispersion relations for the case of the ultrarelativistic plasma is very illustrative in our model. Below we assume that in average q 2 glyph[greatermuch] m 2 (a) a 2 ( τ ), so that p 0 can be replaced by p 0 → q in the integrals. Keeping in mind that in the linear approximation φ = ντ = ν Ha ( τ ) , one can check directly that the scale factor a ( τ ) disappears from the reduced electrodynamic equations, which take the following form: Here we used the following notations: and introduced the three-dimensional Levi-Civita symbol with glyph[epsilon1] 123 = 1. Here and below for the operation with the indices we use the Minkowski tensor η ik =diag(1 , -1 , -1 , -1). The kinetic equation (39) for the ultrarelativistic plasma in the linear approximation can be written in the form Surprisingly, in the given representation all the electrodynamic equations (40)-(43) and the kinetic equation (46) look like the set of integro-differential equations with the coefficients, which do not depend on time. This fact allows us to apply the method of Fourier transformations, which is widely used in case, when we deal with the Minkowski spacetime [1-4].",
"pages": [
5
]
},
{
"title": "B. Equations for Fourier-Laplace images",
"content": "In order to study in detail perturbations arising in plasma we consider the Fourier-Laplace transformations (the Fourier transformation with respect to the spatial coordinates x α and the Laplace transformation for τ ) in the following form: As usual, we assume that k 1 , k 2 , k 3 are pure real quantities in order to guarantee that both exponential functions, e ik α x α and e -ik α x α , are finite everywhere. The quantity Ω is in general case the complex one, Ω= ω + iγ , and, as usual, we assume that perturbations are absent at τ < τ 0 , where τ 0 = -1 Ha ( t 0 ) relates to the moment t = t 0 according to (9). Our choice of the sign minus in the expression for the phase Θ ≡ k α x α -Ω τ in the exponentials in (47), (48) can be motivated as follows. When the value t -t 0 is small, the cosmological time τ (see (9)) can be estimated as τ →-1 Ha ( t 0 ) [1 -H ( t -t 0 )], and thus the phase Θ reads Using the notation k 0 = -Ω a ( t 0 ) we obtain from (49) the expression for the phase Θ = Θ 0 + k m x m , which is standard for the case of Minkowski spacetime. Finally, the term e i Θ has the multiplier e -i Ω τ which contains e γτ → e γ a ( t 0 ) ( t -t 0 ) . In other words, both in terms of t and τ the quantity γ has the same sense: when γ < 0 we deal with the plasma-wave damping and | γ | is the decrement of damping; when γ > 0 we deal with increasing of the perturbation in plasma and this positive γ is the increment of instability. We should introduce an initial value of the perturbed distribution function at the moment τ = τ 0 , indicated as δf (a) (0 , k α , q γ ); as for initial data for the electromagnetic field, we can put without loss of generality that F α 0 ( τ =0 , k β )=0. Using (40)-(43) and (46) the equations for the Fourier images δϕ (a) (Ω , k γ , q β ) and F lk (Ω , k γ ) can be written as follows: Here and below we use the notation q α ≡ η αβ q β for the sake of simplicity. Clearly, the equation (54) is satisfied identically, if we put F αβ from (53). Then we use the standard method: we take δϕ (a) from (50), put it into (51) and (52), use (53) and obtain, finally, the equations for the Fourier images of the components F β 0 of the Maxwell tensor: The Fourier images of the initial perturbations of the electric current J 0 and J α are defined as follows: The compatibility condition for the current four-vector ∇ k J k =0, written in terms of Fourier images requires in fact that the perturbation in the plasma state does not change the particle number. The term Π γ α is a projector: it is orthogonal to k α , i.e., The quantity k 2 is defined as k 2 = -k β k β ; it is real and positive.",
"pages": [
5,
6
]
},
{
"title": "C. Permittivity tensors",
"content": "As usual, we introduce the standard permittivity tensor for the spatially isotropic relativistic plasma [1] and obtain the decomposition where ε ⊥ and ε || are the scalar transversal and longitudinal permittivities, respectively: Finally, we decompose the Fourier image of the electric field F γ 0 into the longitudinal and transversal components with respect to the wave three-vector where and obtain the split equations for the Fourier images of the longitudinal and transversal electric field components, respectively: glyph[negationslash] The second term in the brackets (linear in the wave threevector k β ) describes the well-known effect of optical activity: two transversal components of the field E ⊥ γ are coupled, so that linearly polarized waves do not exist, when ν (1+2 K 2 ) = 0. Since the so-called gyration tensor ν Ω 2 glyph[epsilon1] α γβ (1+2 K 2 ) is proportional to the Levi-Civita symbol glyph[epsilon1] α γβ , we deal with natural optical activity according to the terminology used in [31], which can be described by one (pseudo) scalar quantity. Thus, we can speak about axionically induced optical activity in plasma, and about axionically active plasma itself.",
"pages": [
6,
7
]
},
{
"title": "D. Dispersion relations",
"content": "The inverse Fourier-Laplace transformation (48) of the electromagnetic field is associated with the calculation of the residues in the singular points of two principal types. First, one should analyze the poles of the functions J 0 and J α (see (57) and 58)) describing initial perturbations; the most known among them are the Van Kampen poles Ω= k α q α q . The poles of the second type appear as the roots of the equations and The equation (70) is the nonminimal generalization of the dispersion relations for the longitudinal plasma waves; it includes the spacetime curvature and the constants of nonminimal coupling (via the term K 1 , see (44)), nevertheless, it does not contain any information about the axion field. The equation (71) describes transversal electromagnetic waves in axionically active plasma nonminimally coupled to gravity; it can be transformed into This dispersion equation generalizes the one obtained in [32, 33] for a minimal axionic vacuum. Clearly, one should consider two important cases. The condition 1+2 K 2 = 0 rewritten as 3 Q 1 -Q 3 = 1 4 H 2 , provides the dispersion relations to be of the form includes the curvature terms and does not contain the information about the axion field. glyph[negationslash] In this case the dispersion relations for the transversal electromagnetic perturbations read glyph[negationslash] displaying explicitly the dependence on the axion field strength ν . Two signs, plus and minus, symbolize the difference in the dispersion relations for waves with lefthand and right-hand polarization rotation. In this sense, when 1+2 K 2 = 0, we deal with an axionically active plasma.",
"pages": [
7
]
},
{
"title": "E. Analytical properties of the permittity tensor and the inverse Laplace transformation",
"content": "Let us remind three important features concerning the Laplace transformation in the context of relativistic plasma theory. First, as usual, we treat this transformation as a limiting procedure where a real positive parameter σ exceeds the so-called growth index σ 0 > 0 of the original function f ( τ ). The Laplace image F (Ω), as a function of the complex variable Ω= ω + iγ , is defined and analytic in the domain ImΩ= γ > σ of the plane ω 0 γ . Second, in many interesting cases, two points Ω= ± k happen to be branchpoints of the function F (Ω) (see, e.g., [1, 2] for details); as we will show below, in our case this rule remains valid. Third, in order to use the theorem about residues, we should prolong the integration contour into the domain Im Ω < σ , harboring all the poles of the function F (Ω) and keeping in mind that the branchpoints have to remain the external ones. We use the contour presented on Fig. 1, the radius of the arc being R = √ A 2 + σ 2 . In order to apply the residues theorem to the calculation of the function F ik ( τ, x α ) we should fix one of the analytic continuations of the function F ik (Ω , k α ) into the domain ImΩ < σ . Thus, the function F ik ( τ, x α ) contains contributions of two types: first, the residues in the poles of the function F lk (Ω , k γ ); second, the integrals along the cuts ω = ± k , γ = ky , -∞ < y ≤ 0. The contribution of the second type displays the dependence on time in the form exp ( ikτ ), and looks like the packet glyph[negationslash] of waves propagating with the phase velocity V ph = c =1. In order to describe the contributions of the first type, below we study in detail the solutions to the dispersion relations for the longitudinal and transversal waves in plasma. Providing the mentioned analytic continuation of the function F lk (Ω , k γ ) into the domain Im Ω < σ , we need to take special attention to the analytic properties of the permittivity scalar ε || (see (65)), which is one of the structural elements of the longitudinal electric field. The discussion of this problem started in [34] and led to the appearance in the scientific lexicon of the term Landau damping [1-4], based on the prediction made in [35]. The results of this discussion briefly can be formulated as follows (we assume here that the growth index vanishes, i.e., σ =0). The most important part of the integral (65) is the integration with respect to the longitudinal velocity v || = k α q α kq , which can be written as ∫ +1 -1 dv || Z ( v || ) ( v || -Ω k ) . When ImΩ= γ =0, this real integral diverges at ω < k , and the function ε || is not defined. When γ = 0, this integral can be rewritten as a contour integral with respect to complex velocity v || = x + iy . Since in the domain γ > 0 the function should be analytic, and the pole v || = Ω k can appear in the lower semi-plane γ < 0 only, we recover (in our terminology) the classical Landau's statement about resonant damping of the longitudinal waves in plasma, which can take place if the plasma particles co-move with the plasma wave and extract the energy from the plasma wave [35].",
"pages": [
8
]
},
{
"title": "A. Dispersion equation for the ultrarelativistic plasma",
"content": "Below we consider the background state of the ultrarelativistic plasma to be described by the distribution functions where the temperatures for all sorts of particles should coincide if the background state was the equilibrium one. We are interested in the analysis of the solutions Ω( k α , ν )= ω + iγ of the dispersion relations for longitudinal (70) and transversal (74) waves. More precisely, we focus on the classification of the roots of (70) and (74) and search for nonstandard solutions which appear just due to nonminimal interactions and axion-photon couplings. A number of facts, which we discuss below, are well-known in the context of relativistic plasma-wave theory; nevertheless, we prefer to restate them in order to explain properly new results, which appear in the axionically active plasma nonminimally coupled to gravity. We have to stress that in the model under consideration the results of integration are presented in an explicit form, and the analytic continuation of all the necessary functions also is made explicitly. Since the background state of plasma is spatially isotropic, one can choose the 0 Z axis along the wave vector, i.e., without loss of generality one can put Then the longitudinal permittivity scalar (65) can be reduced to the following term: where t = cos θ , z ≡ Ω /k , the term is usually associated with the square of the plasma frequency in the ultrarelativistic approximation [1], and, finally, the Legendre function of the second kind Q 1 ( z ) is given by the integral (see, e.g., [36, 37]) with We treat the quantity z = Ω k as a new complex variable z = x + iy , where x = ω k and y = γ k . In terms of the complex variable z this function looks more attractive nevertheless, one should, as usual, clarify analytical properties of this function. Clearly, the points z = ± 1 are the logarithmic branchpoints of the function Q 1 ( z ); in these two points the real part of the Legendre function does not exist. When we cross the line Im z = 0 on the fragment | Re z | < 1 of the real axis, the function Im Q 1 ( z ) experiences the jump, since glyph[negationslash] When | x | > 1, the function Im Q 1 ( z ) is continuous. In order to obtain analytical function ε || ( z ) we consider the function Q 1 ( z ) to be defined in the domain Im z > 0 and make the analytical extension to the domain Im z < 0 as follows: Here Θ denotes the Heaviside step function. Let us mention that the function G || ( z ) is analytic everywhere except the lines z = ± 1+ iy with -∞ < y ≤ 0. In fact, to obtain the analytic continuation, we added the residue ( -iπz ) in the singular point t = z into the Legendre function. The same result can be obtained if one deforms the integration contour in (80) so that this contour lies below the singular point t = z and harbors it; in the last case we would repeat the method applied by Landau in [35]. Let us note that the function G || ( z ) possesses the symmetry i.e., it keeps the form with the transformation z → -¯ z , which is equivalent to ω → -ω . Keeping in mind this fact, below we consider ω to be nonnegative without loss of generality. Now the function is defined and is analytical on the complex plane z everywhere except the branchpoints z = ± 1. The corresponding dispersion relation for longitudinal plasma waves can be written as follows: glyph[negationslash] When K 1 =0, this equation gives the well-known results (see, e.g., [1]). When K 1 = 0, the results are obtained in [38] for three different cases: 1 + 2 K 1 > 0, 1 + 2 K 1 < 0 and 1+2 K 1 = 0. We recover these results only to demonstrate the method, which we use below for the dispersion equations containing the axionic factor ν = 0. Since the left-hand side of Eq.(88) is real, we require that or in more detail where we introduced the auxiliary parameter µ which is the function of k . There are two explicit solutions to these equations.",
"pages": [
8,
9,
10
]
},
{
"title": "B. Solutions with x = 0",
"content": "When x = ω k =0, the equation (91) is satisfied identically, and we can assume that either the frequency vanishes ω =0, or the wavelength becomes infinitely small, k = ∞ . The equation (90) can be written now as Clearly, when µ > 1 (or equivalently, 1+2 K 1 > 0) there are no solutions to this equation. When 1+2 K 1 < 0, i.e., µ < 1, one solution y = y ∗ to the equation (93) appears: it is positive if k 2 < 3 W 2 | 1+2 K 1 | and negative for k 2 > 3 W 2 | 1+2 K 1 | . In the intermediate case µ =1, i.e., when 1+2 K 1 =0, we obtain formally the solution y = ∞ , which we omit as nonphysical.",
"pages": [
10
]
},
{
"title": "C. Solutions with y = 0 and | x | > 1",
"content": "When y = 0, we deal with waves propagating without damping/increasing, since γ = 0. The frequency ω can be now found from the equation In terms of x and µ introduced in (92) this equation can be rewritten as 1 x =tanh µ x , thus we obtain the following results. First, there are no solutions, when µ ≤ 1, i.e., when 1+2 K 1 ≤ 0. Second, there are one positive and one negative roots ( x = ± x ∗ ), when µ > 1, i.e., when 1+2 K 1 is positive. To estimate the interval of frequencies, which are allowed by the equation (94), one can use two decompositions for the Legendre function in the case, when y = 0 and x > 1: This means that 1 3 x 2 < Q 1 ( x ) < 1 3( x 2 -1) and thus, taking into account that Q 1 ( x ) = k 2 3 W 2 NM (see (88)), we obtain the inequality W NM < ω < √ W 2 NM + k 2 . In particular, when k glyph[lessmuch] ω , one obtains from (94) and this result is well-known at K 1 =0 (see, e.g., [1, 3]). Here and below we use the convenient parameter W NM ≡ W √ | 1+2 K 1 | . The plots of the functions ω ( k ), V ph ( k ) and V gr ( k ) are presented on Fig. 2. Let us remark, that depending on the value of the wave parameter k two types of longitudinal waves in the relativistic plasma can exist (see, e.g., [1]). The waves of the first type propagate without damping ( γ =0) and with the phase velocity exceeding speed of light in vacuum V ph ( k ) > 1; for instance, the well-known Langmuir waves are described by the dispersion relation ω 2 = ω 2 (p) + k 2 V 2 (s) , thus, V ph ( k ) > 1, when k < ω (p) √ 1 -V 2 (s) (here ω (p) is the Langmuir plasma frequency, and V (s) is the reduced sound velocity). The longitudinal plasma waves of the second type are characterized by γ < 0 and V ph < 1, thus displaying the well-known Landau damping phenomenon. However, in the ultrarelativistic limit, as it was shown in [1], the plasma waves of the second type are suppressed, and only the waves of the first type exist. In fact, we have shown here that the account for a nonminimal coupling of the electromagnetic field to gravity does not change this result for ultrarelativistic case, if 1+2 K 1 > 0. In the case, when 1+2 K 1 ≤ 0, i.e., when the curvature induced effects are not negligible, the longitudinal waves of the first type in the ultrarelativistic plasma are also suppressed, i.e., no longitudinal waves in the ultrarelativistic plasma can be generated, whatever the value of k is chosen, if 1+2 K 1 ≤ 0. / / ",
"pages": [
10,
11
]
},
{
"title": "D. About solutions with y = 0 and x = 0",
"content": "glyph[negationslash] glyph[negationslash] The question arises, whether the longitudinal wave exists for which both x and y are nonvanishing? In order to answer this question let us transform the pair of equations (90), (91) into glyph[negationslash] glyph[negationslash] assuming that y = 0 and x = 0. Since the expression in the right-hand side of (98) is positive, we should consider the situations with µ > 0 only. We have to distinguish two important cases. In this case the equation (99) takes the form Since the function arctan U satisfies the condition | arctan U | ≤ π 2 , both the first term and the term in brackets are positive, thus guaranteeing that this equation has no roots in this case. In this case we return to the Legendre function and transform its imaginary part as follows Clearly, the integral in (101) is a positive quantity, thus, Im Q 1 ( z ) = 0 if and only if x · y = 0. In other words, there are no solutions of the dispersion relation under discussion, when both x and y are nonvanishing.",
"pages": [
11
]
},
{
"title": "A. Dispersion equation",
"content": "In analogy with the longitudinal permittivity scalar, the transversal one can be written in terms of reduced frequency z = ( ω + iγ ) k , of the wave three-vector modulus k and the quantity W (see (79)) only: where Here we used the definition (64) of the scalar ε ⊥ and the formulas (62)-(65). The function G ⊥ ( z ) also possesses the property and is analytic with respect to complex variable z anywhere except the lines z = ± 1+ iy with -∞ < y ≤ 0. In these terms the dispersion relation for transversal waves in ultrarelativistic plasma takes the form where the real parameter appears if and only if the axion field is nonstationary. Since G ⊥ (0)=0, the solution z =0 exists, when k = p ≥ 0. Depending on the value of the parameter (1+2 K 1 ) one obtains three particular cases.",
"pages": [
11
]
},
{
"title": "B. The first case: 1 + 2 K 1 > 0 and x 2 > 1 or/and y > 0",
"content": "First, we consider the imaginary part of the equality (105) along the line the analysis for the longitudinal case. When G || ( z )= Q 1 ( z ), i.e. when either Θ( -y )=0 or Θ(1 -| x | )=0, we can show explicitly that Clearly, this expression can be equal to zero if and only if x = 0 or y = 0.",
"pages": [
12
]
},
{
"title": "1. On the solutions with y = 0 and x > 1",
"content": "When y = 0 the dispersion relation gives where the function G ⊥ ( x ) can be written as For this function the following decompositions can be found: from which one can see that G ⊥ (1) = 1 and G ⊥ ( ∞ ) = 2 3 . Being monotonic, the function G ⊥ ( x ) is restricted by the inequalities 2 3 < G ⊥ ( x ) < 1, when x > 1. This means that the square roots in equations (108) is real, the righthand side is positive for arbitrary p , if we use only the root with plus sign in front of the square root. Thus, the equation (108) (with the plus sign in front of the square root) has real solutions Ω = ω ( k ), which describe transversal oscillations without damping/increasing and phase velocity exceeding the speed of light in vacuum. When p is nonpositive, these oscillations have arbitrary wavelength 0 < 1 k < ∞ . When p is positive, k cannot exceed some critical value k crit ≡ 3 W 2 NM 2 p , which can be obtained from the following estimation: In the approximation of long waves, i.e., when k → 0, we obtain For short waves ( k → ∞ ) and p ≤ 0 the square of the frequency reads In Fig. 3 and Fig. 4 we presented the results of numerical calculations: we have chosen four values of the parameter p for illustration of the dependence ω ( k ), and eight values of p for illustration of the behavior of the functions V ph ( k ) and V gr ( k ). Let us remark that, when the parameter p is negative, there exists some critical value of its modulus | p | = p crit , which distinguishes two principally different situations: when | p | < p crit , the group velocity does not exceed the speed of light in vacuum V gr < 1; when | p | > p crit , we obtain V gr > 1 for arbitrary k . This critical behavior of the function V gr ( k ) can be easily illustrated for the case of short waves (see (114)). Indeed, in this case we obtain and, clearly, p crit = √ 6 W NM . Another interesting feature is that the function V gr ( k ) can take negative values for some values of the parameter p . When we think about both results: first, that V gr ( k ) is negative for some p , and that V gr ( k ) can exceed the speed of light in vacuum, V gr > 1, we recall the theory of magneto-active plasma, which GLYPH<150> demonstrates such behavior for the waves of special type (see, e.g., [39]). It can be considered as a supplementary motif to use the term axionically active plasma by the analogy with the term magneto-active plasma.",
"pages": [
12,
13
]
},
{
"title": "2. On the solutions with x = 0 and y > 0",
"content": "When x = 0 and y > 0, the solution to the dispersion equation is where For positive values of the variable y the function G ⊥ ( iy ) is nonnegative ( G ⊥ (0 +0 ) = 0, G ⊥ (+ i ∞ ) = 2 / 3). This means that, first, at p = 0 there are no solutions to (116); glyph[negationslash] second, at p = 0 the corresponding solutions exist, for which p 2 ≥ 6 W 2 NM ( y 2 +1) G ⊥ ( iy ); third, when p > 0 one can use both signs, plus and minus, in (116); fourth, when p < 0, there are no solutions, since k should be positive. Let us stress the following feature: since G ⊥ ( iy ) > 0 we obtain from (116) the inequality k < p for solutions with x = 0 and y > 0. To conclude, the nonharmonically increasing perturbations exist (Ω = iγ with γ > 0) only in case, when p > k .",
"pages": [
13
]
},
{
"title": "C. The second case: 1 + 2 K 1 > 0 and x 2 < 1 , y < 0",
"content": "When y < 0 and | x | < 1, we divide the equation (105) by the quantity ( z 2 -1) and transform the imaginary part of obtained relation into Keeping in mind the properties of the function Im Q 1 ( z ), which we studied for the case of longitudinal waves, one can state that, when p ≤ 3 W 2 NM 2 k , the left-hand side of the equation (118) is nonpositive, and thus there is only one root of this equation, namely, x = 0. This case can be considered qualitatively. When p > 3 W 2 NM 2 k , we will study solutions numerically.",
"pages": [
13
]
},
{
"title": "1. The case x = 0 and y < 0",
"content": "Now the dispersion equation yields where the function is positively defined. Again, we should eliminate the sign minus in front of the square root in (119) since k > 0. Thus, for arbitrary p and k > 0 there are solutions Ω = iγ with negative γ , which describe damping transversal nonharmonic perturbations in plasma. Two asymptotic decompositions attract an attention and In Fig. 5 we illustrate the dependence γ ( k ) for ten values of the parameter p ; we put together the plots illustrating the solutions discussed in two paragraphs: for x =0, y > 0 and for x =0, y < 0.",
"pages": [
13
]
},
{
"title": "2. The case x = 0 and y = 0",
"content": "glyph[negationslash] glyph[negationslash] We indicate the solutions of the dispersion equations, for which neither Re z , nor Im z vanishes, as nontrivial solutions, and study them numerically. For this purpose we introduce the dimensionless real quantities ξ = k W NM , η = p W NM and link them by two parametric equations We use the following scheme of analysis. We are interested, finally, in the determination of the functions ω = ω ( k, p ) and γ = γ ( k, p ); however, we start with the numerical and qualitative analysis of the inverse functions k = k ( x, y ) and p = p ( x, y ). In other words, first of all, we find the domains on the plane of the parameters ξ and η , in which the corresponding solutions exist. In order to illustrate this scheme let us consider auxiliary plots on Fig. 6. On Fig. 6a one can find the lines of two types: first, infinite lines, which relate to solutions with Re z > 1 and Im z = 0; second, finite lines, which correspond to the solutions with 0 < Re z < 1 and Im z < 0. The family of infinite lines visualizes the separatrix in the form of hyperbola. The family of finite lines visualizes two curves on which these lines start or finish. In Fig. 6b one can find infinite lines of two types. First, we see the family of lines, for which Re z = 0 and Im z > 0; they visualize the separatrix in the form of straight line. The second family of lines relates to the solutions with Re z = 0 and Im z < 0; they visualize an envelope line with extreme point. We collected the results of numerical analysis on Fig. 7. There are seven domains on the plane η 0 ξ , the corresponding separatrices appeared as follows. (c1) The line indicated as (a) is the bisector ξ = η , and the appearance of this formula can be explained as follows. On the one hand, it relates to the straight-line separatrix found by numerical calculations; on the other hand, it corresponds to the special solution z = 0, which appears if k = p > 0. In addition to the static solution ω = γ =0, in the points belonging to the bisector p = k we obtain the running waves with ω > k , γ =0, when the wave number k satisfies the inequality k < √ 3 2 W NM . (c2) The line (b) is described by the formula ξη = 3 2 . On the one hand, is relates to the hyperbolic separatrix found numerically; on the other hand it corresponds to the critical case k = k crit = 3 W 2 NM 2 p , found analytically. For the points along this separatrix, the solutions x = 1 exist, or equivalently, ω = k , γ =0; the solutions with x 2 > 1 do not exist. glyph[negationslash] glyph[negationslash] (c3) Keeping in mind that the nontrivial solutions to the dispersion equations with x = 0 and y = 0 exist only for y < 0 and | x | < 1, we associated the numerically found curves, on which the finite lines start or finish, with two limiting lines found analytically (for x ≥ 0). The first curve of this type, the curve (c), corresponds to the limit x → 0 and is described parametrically as The second curve, the curve (d), relates to the limit x → 1 and has the following parametric representation: Clearly, in the points of the curve (c) the solutions to the dispersion relations have the form ω = 0, γ < 0; as for the curve (d), one obtains that ω = k , γ < 0. In the cross-points the solutions inherit the properties, which characterize both separatrices. (p1) The lines (a) and (b) cross in the point A with the coordinates η = ξ = √ 3 2 glyph[similarequal] 1 . 225; in this point ω = γ = 0 (p3) The extreme point B has the coordinates ξ glyph[similarequal] 2 . 754, η glyph[similarequal] 2 . 613; for this point there exists a solution with ω = 0, γ = -1 . 31 W NM . Let us summarize the properties of solutions, which relate to the domains I, II, . . . , VII in Fig. 7, using the analysis of the formulas presented above. glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] These results are presented shortly in Table I. Let us attract the attention to the number of symbols + in the corresponding boxes. For instance, there are three symbols + in the box for the domain III with negative γ , two symbols in the domain V, and one plus in the domains I and VI. This means that for one value of the quantity k we obtain one, two or three values of the decrement of damping γ ( k ). This feature can be explained using Fig. 5: depending on the value of the parameter p the vertical straight line k =const can cross the curve γ = γ ( k, p ) one, two or three times, respectively.",
"pages": [
14,
15,
16
]
},
{
"title": "D. The third case: 1 + 2 K 1 < 0",
"content": "This case can be obtained if we make the formal replacement W 2 NM → -W 2 NM in all formulas obtained in the previous subsection. In particular, the dispersion relation for the transversal waves in plasma has now the form We do not discuss similar details of the corresponding analysis, and present below the results only. Similarly to the case 1+2 K 1 > 0, numerical modeling of the lines k = k ( p ) displays three separatrices on the plane η 0 ξ ; they are shown in Fig. 8. The line (a), again, is the bisector ξ = η . The parametric representation of the line (b ' ) is It is obtained using the conditions x =0 and y > 0 (see (125)). The line (c ' ) is described by the parametric equations and is the envelope of the family of curves p = p ( x, y =0), k = k ( x, y =0), when x > 1. The last line (d ' ) is obtained using the conditions x =1 and y > 0. Contrary to the previous case, the separatrices have neither cross-points, nor extreme points. Thus, in the case of negative constant 1+2 K 1 , the plane ξ 0 η is divided into five domains. In the domain I ' the solutions to the dispersion equation have vanishing real parts ω =0 and negative imaginary parts γ< 0; the domain II ' is characterized by ω =0 and γ> 0. In the domain III ' there exist instable waves with ω < k and γ > 0; the domain IV ' contains waves with ω > k of two types: increasing ( γ > 0) and damping ( γ < 0). The domain V ' describes running waves without damping/increasing, ω > k , γ =0. We summarize these features in Table II. glyph[negationslash] glyph[negationslash] Concerning the principally new result, let us stress, that after the replacement W 2 NM → -W 2 NM in (107) we can see that indeed, in addition to x =0 and y =0, new solutions can exist with x = 0 and y = 0. This means that, when 1+2 K 1 < 0, transversal waves in axionically active plasma can be, first, damping waves with ω > k , γ < 0, second, can be instable ( γ > 0) both for ω > k and ω < k .",
"pages": [
16,
17
]
},
{
"title": "E. The intermediate case 1+2 K 1 = 0",
"content": "To complete our study, let us consider the case K 1 = -1 2 , for which the dispersion relation reduces to the equation where p ∗ = ∓ ν (1 + 2 K 2 ) and W is given by (79). In analogy with the case 1 + 2 K 1 > 0, four separatrices appear on the plane ξ 0 η (here ξ = k W , η = p ∗ W ). The first separatrix is the hyperbola ξη = 1; it corresponds to G ⊥ ( ∞ ) = 2 3 (minimal value of the function G ⊥ ( x )). The second separatrix ξη = 3 2 corresponds to G ⊥ (1)=1 (maximal value of G ⊥ ( x )). The third separatrix has the following representation: where the number 2 . 3904 is the maximum of the function 3 2 Re { G ⊥ ( z ) } at the condition that Im { G ⊥ ( z ) } = 0. Finally, we visualize the straight line p ∗ =0 as an abscissa on the plane p ∗ 0 k and the last separatrice. Thus, we obtain five domains on the plane ξ 0 η (see Fig. 9). The domain I 0 ( η < 0) is characterized by ω =0 and γ < 0; in the domain II 0 ( ξη < 1, p ∗ > 0) the solutions of the dispersion equations give ω =0 and γ > 0. In the domain III 0 with 1 < ξη < 3 2 we obtain that γ =0 and ω > k . The solutions with ω < k and γ < 0 can appear in the domain IV 0 ; in the domain V 0 there are no solutions to the dispersion equation. These results are summarized in Table III.",
"pages": [
17
]
},
{
"title": "VI. DISCUSSION",
"content": "In the framework of nonminimal Einstein-MaxwellVlasov-axion model we analyzed the dispersion relations for the perturbations in an initially isotropic and homogeneous axionically active plasma, which expands in the de Sitter-type cosmological background. In this model we take into account, first, the nonminimal interaction of the electromagnetic field with curvature, second, the nonminimal coupling of the axion field to gravity, and axion-photon coupling in the relativistic plasma. The specific choice of the nonstationary background solution to the total system of equations (de Sitter spacetime with constant curvature, axion field linear in time, vanishing initial macroscopic collective electromagnetic field in the electro-neutral ultrarelativistic plasma) allowed us to obtain and study the dispersion relations in the standard (Ω , glyph[vector] k ) form. We consider these dispersion relations as nonminimal axionic extension of the well-known dispersion relations obtained earlier in the framework of the relativistic Maxwell-Vlasov plasma model. glyph[negationslash] The presence of the pseudoscalar (axion) field provides the plasma to become a gyrotropic medium, which displays the phenomenon of optical activity. When the plasma in the axionic environment is initially spatially isotropic, its three-dimensional gyration tensor, G αβγ is proportional to the three-dimensional Levi-Civita tensor, glyph[epsilon1] αβγ , thus providing the optical activity to be of the natural type [31]. The proportionality coefficient is the pseudoscalar quantity p = ∓ ν (1+2 K 2 ) (1+2 K 1 ) . The upper sign of this coefficient relates to the circularly polarized wave with left-hand rotation; formally speaking, the sign of p depends on the sign of ν = ˙ φ , and on the sign of nonminimal coefficient (1+2 K 2 ) (1+2 K 1 ) . Since ν = ˙ φ , this type of optical activity is induced by the pseudoscalar (axion) field; if the axions form the dark matter, the quantity ν is proportional to the square root of the dark matter energy density (see, e.g., [40, 41] for details). When K 1 = 0 and glyph[negationslash] glyph[negationslash] K 2 = 0 (see their definitions in (44), the gyration coefficient p contains the square of the Hubble function (i.e., the spacetime curvature scalar in the case of the de Sitter model). Thus, we deal with combined axionic-tidal gyration effect. The frequencies of transversal electromagnetic waves are shown to depend not only on the wavelength, but also on the gyration coefficient p (see, e.g., (113)), and this dependence has a critical character. To be more precise, when p =0, the dispersion equations admit some new branches of solutions ω = ω ( k, p ), γ = γ ( k, p ) in addition to the standard ones. If to consider the transversal electromagnetic wave propagation in terms of left- and right-hand rotating components, one can state, that one of the waves (say, with left-hand rotation) can have arbitrary wavelength, while the second wave can possess the wave number less than critical one (see, e.g., (112)); in this sense we deal with some kind of mode suppression caused by the axion-photon interactions. glyph[negationslash] In order to simplify the classification of the electromagnetic modes in an axionically active plasma, we use the following terminology: damping wave , when ω = 0, γ < 0; instable wave , when ω = 0, γ > 0; running wave , when ω ≥ k , γ = 0; damping nonharmonic perturbation , when ω = 0, γ < 0; instable nonharmonic perturbation , when ω = 0, γ > 0. The results of analytic, qualitative and numerical study are presented on Figs. 7-9 and in Tables I-III. Based on these results, we can illustrate the evolution of types of the transversal electromagnetic perturbations depending on the value of the gyration parameter p for small, medium and large k . For this purpose we draw, first, the horizontal straight line on Fig. 7, and move it from the bottom to upwards. glyph[negationslash] k < p , running waves convert into damping waves and instable nonharmonic perturbations are inherited; when k > p , damping waves and damping perturbations are inherited. Similarly, we can illustrate the evolution of the modes for the case 1 + 2 K 1 < 0 and 1 + 2 K 1 =0. Such analysis reveals two interesting features. First, when the gyration parameter p and the nonminimal parameter 1+2 K 1 are fixed, we can find explicitly the range for the wave parameter k , for which the running waves, i.e., non-damping transversal electromagnetic waves, can propagate in the Universe in the axionic dark matter environment and bring us a true information about the Universe structure and history. For instance, when 1+2 K 1 > 0 according to the Table I it is possible in the domains I and II only; thus, if the gyration parameter exceeds, say, the value p = √ 3 2 W NM (see the point A on Fig. 7), the running waves can propagate only in the narrow interval of small k (the interval of long waves). When 1+2 K 1 < 0, according to Table II the running waves can propagate in the domain V ' only; thus, if the parameter p is positive, the running transversal waves are suppressed for any k . Second, we can focus on the problem of existence of transversal electromagnetic waves of a new type, i.e., transversal waves with the phase velocity less that the speed of light in vacuum, ω < k , which can interact in a resonant manner with particles co-moving with them. It is well-known that in the minimal theory ( K 1 = 0) at the absence of axion field ν = 0 the dispersion equations do not admit the transversal waves of this type. Our consideration shows that the damping waves with V ph < 1 exist in the domains VI and VII (see Fig. 7 for the case 1 + 2 K 1 > 0), and the instable waves with V ph < 1 can be generated for the case 1 + 2 K 1 < 0. Since the phase velocity of the transversal electromagnetic wave is less than speed of light in vacuum, we can make the following remark. There are plasma particles, which co-move with this transversal electromagnetic wave, thus providing a resonant interaction; in the case 1 + 2 K 1 > 0 this resonant interaction leads to the Landau-type damping, since the wave transfers the energy to the resonant particles; in the case 1 + 2 K 1 < 0 this resonant interaction provides the Landau-type instability, since now resonant particles transfer their energy to the transversal plasma wave. In the next paper we plan to study the dispersion relations for the axionically active plasma nonminimally coupled to gravity in the framework of cosmological BianchiI model with initial magnetic field and electric field axionically induced.",
"pages": [
17,
18,
19
]
},
{
"title": "Acknowledgments",
"content": "The work was partially supported by the Russian Foundation for Basic Research (Grants Nos. 11- 02-01162 and 11-05-97518), by the Federal Targeted Program N14.T37.21.0668 and the State Assignment N5.2971.2011.",
"pages": [
19
]
}
]
2013EPJWC..4604003L
https://arxiv.org/pdf/1305.0850.pdf
<document>
<text><location><page_1><loc_8><loc_88><loc_53><loc_92></location>EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c GLYPH<13> Owned by the authors, published by EDP Sciences, 2021</text>
<section_header_level_1><location><page_1><loc_8><loc_76><loc_82><loc_78></location>Elliptic and magneto-elliptic instabilities of disk vortices</section_header_level_1>
<text><location><page_1><loc_8><loc_72><loc_25><loc_73></location>Wladimir Lyra 1 ; 2 ; 3 ; a</text>
<text><location><page_1><loc_8><loc_67><loc_92><loc_71></location>1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA, 91109 2 Division of Geological & Planetary Sciences, California Institute of Technology, 1200 E California Blvd MC 150-21, Pasadena, CA 91125</text>
<text><location><page_1><loc_8><loc_65><loc_19><loc_66></location>3 Sagan fellow</text>
<text><location><page_1><loc_17><loc_52><loc_83><loc_62></location>Abstract. Vortices are the fundamental units of turbulent flow. Understanding their stability properties therefore provides fundamental insights on the nature of turbulence itself. In this contribution I briefly review the phenomenological aspects of the instability of elliptic streamlines, in the hydro (elliptic instability) and hydromagnetic (magneto-elliptic instability) regimes. Vortex survival in disks is a balance between vortex destruction by these mechanisms, and vortex production by others, namely, the Rossby wave instability and the baroclinic instability.</text>
<section_header_level_1><location><page_1><loc_8><loc_45><loc_24><loc_47></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_8><loc_39><loc_92><loc_43></location>The first documented observation of non-laminar motion in fluids is present in the work of Leonardo da Vinci [1]. He called the phenomenon turbolenza , after the Latin word for swirl (turbo). Sketching the flow, he wrote</text>
<text><location><page_1><loc_13><loc_34><loc_87><loc_37></location>[...] the smallest eddies are almost numberless, and large things are rotated only by large eddies and not by small ones, and small things are turned by small eddies and large.</text>
<text><location><page_1><loc_8><loc_20><loc_92><loc_30></location>Though written around 1500, this passage reads surprisingly modern, containing the seeds of concepts such as power spectrum and locality of the cascade. It contains also the insight that vortices are the fundamental unit of turbulent flow. The stability of vortices is thus a problem of paramount importance in fluid mechanics. Unveiling the mechanism that renders them unstable should provide vital insights into the nature of turbulence itself. The instabilities of magnetized vortices should likewise provide a similar framework when it comes to MHD turbulence.</text>
<text><location><page_1><loc_8><loc_10><loc_92><loc_20></location>Phenomenologically, turbulence can be described as a series of bifurcations, starting with a primary instability that converts shear into vorticity, creating vortices. This is followed by another bifurcation, a secondary instability, to break these vortices into lesser vortical structures. These in turn shall experience a sequence of 'inertial instabilities', leading to a cascade. Though the Kelvin-Helmholtz instability and the Rayleigh-Taylor instability are well established as examples of primary instabilities, the highly successful theory of the turbulent cascade put forth by Kolmogorov [2] rested on a</text>
<section_header_level_1><location><page_2><loc_41><loc_91><loc_59><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<figure>
<location><page_2><loc_10><loc_68><loc_49><loc_87></location>
<caption>Figure 1. Illustration of the circularly polarized tilted Kelvin wave, a mode supported by fluids in uniform rotation. The contour in red traces the wave. As the wave is transverse, fluid parcels execute motion along the blue 'arms' of the figure, oscillating both in-plane and vertically. Scan the QR code for an animation of the wave.</caption>
</figure>
<text><location><page_2><loc_8><loc_55><loc_92><loc_64></location>heuristic picture of secondary instability, established by early experiments [3]. It was not until the 80's [4, 5] that the elliptic instability was introduced as a mechanism for the secondary instability. A fluid in rigid rotation supports a spectrum of stable inertial waves, the simplest case being circularly polarized transverse plane waves oscillating at twice the frequency of the base flow [6]. Strain is introduced when the streamlines pass from circular to elliptical, and some modes find resonance with the strain field, leading to de-stabilization.</text>
<text><location><page_2><loc_8><loc_46><loc_92><loc_54></location>The cascade reversal from inverse to direct when passing from 2D to 3D is a result of the di GLYPH<11> erent properties of 2D and 3D vortices. Two-dimensional vortices do not decay, merging viscously and growing to the integral scale; three-dimensional vortices also merge viscously, but before that they generally fall prey to the elliptic instability, which does not exist in two dimensions. Vortex survival depends on a balance between production and destruction.</text>
<section_header_level_1><location><page_2><loc_8><loc_41><loc_30><loc_43></location>2 Elliptic instability</section_header_level_1>
<text><location><page_2><loc_8><loc_23><loc_92><loc_39></location>A column of fluid under rigid rotation supports stable oscillations in the form of circularly polarized transverse plane waves (Fig. 1). Restored by the Coriolis force and propagating along a waveguide, these waves have also been called Kelvin waves . As the waves are transverse, if the direction of propagation coincides with the rotation axis, the action of the wave is that fluid parcels will execute in-plane epicyclic oscillations. The propagation vector may also have an angle GLYPH<18> with the rotation axis, in which case the fluid motion is no longer in-plane, executing both epicyclic and vertical motions, well known in galactic dynamics [7]. Destabilization occurs when strain in introduced; steepening gradients and providing a source of free energy. Instability occurs when a mode or pair of modes find resonance with the rotating strain field, which is to say when a multiple of the rotation frequency matches the frequency of the inertial waves.</text>
<text><location><page_2><loc_8><loc_15><loc_92><loc_23></location>The elliptic motion U = GLYPH<10> [ GLYPH<0> (1 GLYPH<0> " ) y; (1 GLYPH<0> " ) x ], where 0 GLYPH<20> " GLYPH<20> 1 is the ellipticity, is readily decomposed U = GLYPH<10> ( R + S ) into rigid rotation R = [ GLYPH<0> y; x ] and the strain field S = GLYPH<0> " [ y; x ]. The growth rates, reproduced from [8], are shown in Fig. 2 in the GLYPH<31> -GLYPH<18> plane. The aspect ratio GLYPH<31> of the streamlines is a measure of the strain. As the instability grows the vortex coherence is destroyed, energy cascades forward and dissipates; the flow relaminarizes.</text>
<text><location><page_2><loc_8><loc_8><loc_92><loc_14></location>As seen in Fig. 2, the instability is elliptical : there is no growth for GLYPH<31> = 1, i.e., circular streamlines in rigid rotation. The instability is also inherently three-dimensional : there is no growth for GLYPH<18> = 0, i.e., in-plane fluid oscillations. This later behavior is at the heart of the di GLYPH<11> erence between the cascade in 2D and 3D. A stirring or primary instability, such as Rayleigh-Taylor or Kelvin-Helmholtz, generates</text>
<figure>
<location><page_3><loc_9><loc_65><loc_48><loc_88></location>
<caption>Figure 2. Growth rates for perturbations in closed elliptic streamlines, in the simplest case of absent background rotation or magnetic fields. In the x -axis GLYPH<31> is the aspect ratio of the base vortex, and GLYPH<18> the angle between the propagation vector of the inertial wave and the rotation axis of the vortex. There is no instability for circular streamlines ( GLYPH<31> = 1): the instability is elliptic . No instability exists either for GLYPH<18> = 0, i.e., in-plane motion. The elliptic instability is inherently three-dimensional. Color-coded is the logarithm of the growth rate. Reproduced from [8].</caption>
</figure>
<text><location><page_3><loc_8><loc_55><loc_92><loc_61></location>the first eddies. The elliptic instability generates 3D turbulence out of this 2D motion, breaking the eddies. The growth rates are of the order of the turnover frequency, which explains why vortices have lifetimes of this order. As elliptic destruction occurs faster than viscous merging, the 3D cascade is direct. In two dimensions, there is no elliptic instability, and the eddies simply merge viscously.</text>
<text><location><page_3><loc_8><loc_39><loc_92><loc_54></location>Adding a background rotation has significant e GLYPH<11> ects for the stability. The system now has two timescales: the turnover vortex time and that of the background flow. Not only that, the motions can be either aligned (cyclonic vortex) or anti-aligned (anti-cyclonic). Not surprisingly, the strongest instability occurs for anticyclonic motion, as the two flows rotating in opposite directions greatly enhances the e GLYPH<11> ective shear on a fluid parcel. In this case, the in-plane, horizontal, motion gets de-stabilized. This instability is not of resonant nature, but centrifugal, appearing as exponential growth of epicyclic disturbances. This behaviour invites a connection with the Rayleigh (centrifugal) instability, and indeed the mechanism is similar [8], suggesting that the Rayleigh instability is a limit of the elliptic instability in the presence of rotation.</text>
<section_header_level_1><location><page_3><loc_8><loc_34><loc_41><loc_35></location>3 Magneto-Elliptic Instability</section_header_level_1>
<text><location><page_3><loc_8><loc_21><loc_92><loc_31></location>When magnetic fields are introduced in the problem, the addition of Alfvén waves enriches the families of unstable modes. In the absence of rotation, the magneto-elliptic instability is a parametric instability as well [9], have three unstable branches in the GLYPH<31> -GLYPH<18> plane. The first two are 'hydrodynamic' and 'magnetic', corresponding to resonant destabilization of Kelvin and Alfvén waves, respectively. A third, 'mixed' mode occurs through instability of a pair of modes when one is hydrodynamic and the other is magnetic, or a Kelvin-Alfvén instability.</text>
<text><location><page_3><loc_8><loc_8><loc_92><loc_21></location>When background rotation is introduced, the same as had occurred to the elliptic instability ensues. When this background rotation runs opposite to the rotation of the vortex (anti-cyclonic motion), a fluid parcel becomes subject to an intense e GLYPH<11> ective shear. Since the magnetic tension resists shear, leading to instability [10] a powerful unstable in-plane mode appears [11]. This instability is of course the magneto-rotational instability, in generalized form. Indeed, the dispersion relation of this horizontal mode reduces to that of the MRI in the pure shear limit of " = 1 (Fig. 4, [12, 13]). This provides an interesting unification, explaining the magneto-elliptic and magneto-rotational instabilities as di GLYPH<11> erent manifestations of the same magneto-elliptic-rotational instability .</text>
<section_header_level_1><location><page_4><loc_41><loc_91><loc_59><loc_92></location>EPJ Web of Conferences</section_header_level_1>
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<location><page_4><loc_9><loc_68><loc_49><loc_88></location>
</figure>
<section_header_level_1><location><page_4><loc_19><loc_63><loc_37><loc_65></location>Elliptic-Rotational</section_header_level_1>
<figure>
<location><page_4><loc_10><loc_42><loc_48><loc_62></location>
<caption>Figure 3. Growth rates of the horizontal ( GLYPH<18> = 0) mode of the magneto-elliptic instability in the presence of rotation, for di GLYPH<11> erent aspect ratios of the base vortex. The quantity in the x -axis is q = k = k BH, where k BH is the Balbus-Hawley wavenumber. As GLYPH<31> GLYPH<29> 1 we approach the pure shear limit ( " = 1) and the MRI growth rate is recovered: the MRI is a limiting case of the more general magneto-elliptic-rotational instability. Notice that for q = 0 (no magnetic field), there is growth for GLYPH<31> = 2 and GLYPH<31> = 3, recovering the result of Rayleigh instability of horizontal modes of the hydro elliptic instability in the presence of rotation. Reproduced from [12].</caption>
</figure>
<section_header_level_1><location><page_4><loc_57><loc_64><loc_83><loc_66></location>Magneto-Elliptic-Rotational</section_header_level_1>
<figure>
<location><page_4><loc_50><loc_43><loc_90><loc_63></location>
<caption>Figure 4. The four identified modes of vortex destruction. In the hydro case (left panel, reproduced from [8]), the Rayleigh instability of horizontal modes, and the parametric resonant tuning of 3D inertial waves with the underlying strain field. In the magnetic case (right panel, reproduced from [12]), the magneto-rotational instability of horizontal modes, and resonant tuning of Alfvén waves. These processes tend to oppose vortex coherence, as growing perturbations extract kinetic energy from the vortical motion. Vorticity is scattered, cascading forward, and eventually being removed by viscosity. They thus behave as vorticity sinks. A vortex survives if there is a mechanism to counteract these losses, by injecting vorticity.</caption>
</figure>
<section_header_level_1><location><page_4><loc_8><loc_21><loc_63><loc_22></location>4 Formation and destruction of vortices in disks</section_header_level_1>
<text><location><page_4><loc_8><loc_13><loc_92><loc_19></location>Four ways of destroying vortices in disks are identified in Fig. 4. In the non-magnetic case (EI), these are the Rayleigh instability for the horizontal mode, and the resonant destabilization of Kelvin waves. In the magnetic case (MEI), these are the MRI for the horizontal modes, and resonant destabilization of Alfvén waves.</text>
<text><location><page_4><loc_8><loc_8><loc_92><loc_12></location>Counterbalancing these there are two mechanics to inject vorticity in the flow. These are the Rossby wave instability (RWI, [14-19]), and the Baroclinic Instability (BI, [12, 20-22]). The former is a linear instability powered by a modification of the shear profile, that behaves as an external</text>
<figure>
<location><page_5><loc_9><loc_74><loc_92><loc_88></location>
<caption>Figure 5. In non-magnetized flows, the vorticity injected by the baroclinic instability is able to counteract the vorticity lost the elliptic instability, at least the parametric, non-horizontal, version. In the magnetized case, however, the MEI growing inside the vortex core is more than the vortex can withstand. The conclusion is that baroclinic vortices are restricted to the dead zones of accretion disks.</caption>
</figure>
<text><location><page_5><loc_8><loc_57><loc_92><loc_60></location>reservoir of vorticity. The latter is a nonlinear instability powered by buoyancy and thermal di GLYPH<11> usion, that establish a nonzero baroclinic source term.</text>
<text><location><page_5><loc_8><loc_50><loc_92><loc_56></location>As long as these vorticity sources (RWI and BI) are more powerful than the vorticity sinks (EI and MEI), a vortex can survive. It has been shown in the literature [12, 22] that despite the elliptic instability, the baroclinic instability keeps 3D vortices coherent. The result is core turbulence only, as the vorticity lost by the EI is replenished by that injected by the BI.</text>
<text><location><page_5><loc_8><loc_33><loc_92><loc_50></location>It has also been shown [12] that the same is not true for the MEI, with a magnetized baroclinic vortex getting quickly destroyed by the strong excitation of magneto-elliptic modes Fig. 5. Yet, magnetized vortices are seen in global, high-resolution simulations [19]. These were triggered by the RWI, albeit artificially, due to an intense peak in magnetic pressure in the initial condition, leading to a non-Keplerian shear profile in the active zone. The cylindrical simulation was performed at very high resolution, Nr ; N GLYPH<30> ; Nz = (768, 1536, 128) 1 ranging GLYPH<25> in azimuth and 0.4 to 2.5 in radius. This translates into 40 grid points per scale height, more than enough to resolve the unstable magnetoelliptic modes. So, either the said phenomenon is not a vortex but a zonal flow, or the RWI provides vorticity faster than the MEI can destroy it, as is the case with the BI and EI. If that is the case, we can group the processes in order of strength as EI < BI < MEI < RWI.</text>
<section_header_level_1><location><page_5><loc_8><loc_29><loc_21><loc_30></location>References</section_header_level_1>
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<text><location><page_5><loc_8><loc_8><loc_92><loc_10></location>1 It also made use of high-end computing, being calculated in over 18 000 processors at the NICS-Kraken cluster. We used the P encil C ode , with which we achieve nearly linear scalability in NICS-Kraken, up to 70 000 processors.</text>
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<list_item><location><page_6><loc_8><loc_81><loc_63><loc_82></location>[12] Lyra, W. & Klahr H., Astron. Astrophys. 527 138-157 (2011)</list_item>
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</document>
[
{
"title": "ABSTRACT",
"content": "EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c GLYPH<13> Owned by the authors, published by EDP Sciences, 2021",
"pages": [
1
]
},
{
"title": "Elliptic and magneto-elliptic instabilities of disk vortices",
"content": "Wladimir Lyra 1 ; 2 ; 3 ; a 1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA, 91109 2 Division of Geological & Planetary Sciences, California Institute of Technology, 1200 E California Blvd MC 150-21, Pasadena, CA 91125 3 Sagan fellow Abstract. Vortices are the fundamental units of turbulent flow. Understanding their stability properties therefore provides fundamental insights on the nature of turbulence itself. In this contribution I briefly review the phenomenological aspects of the instability of elliptic streamlines, in the hydro (elliptic instability) and hydromagnetic (magneto-elliptic instability) regimes. Vortex survival in disks is a balance between vortex destruction by these mechanisms, and vortex production by others, namely, the Rossby wave instability and the baroclinic instability.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The first documented observation of non-laminar motion in fluids is present in the work of Leonardo da Vinci [1]. He called the phenomenon turbolenza , after the Latin word for swirl (turbo). Sketching the flow, he wrote [...] the smallest eddies are almost numberless, and large things are rotated only by large eddies and not by small ones, and small things are turned by small eddies and large. Though written around 1500, this passage reads surprisingly modern, containing the seeds of concepts such as power spectrum and locality of the cascade. It contains also the insight that vortices are the fundamental unit of turbulent flow. The stability of vortices is thus a problem of paramount importance in fluid mechanics. Unveiling the mechanism that renders them unstable should provide vital insights into the nature of turbulence itself. The instabilities of magnetized vortices should likewise provide a similar framework when it comes to MHD turbulence. Phenomenologically, turbulence can be described as a series of bifurcations, starting with a primary instability that converts shear into vorticity, creating vortices. This is followed by another bifurcation, a secondary instability, to break these vortices into lesser vortical structures. These in turn shall experience a sequence of 'inertial instabilities', leading to a cascade. Though the Kelvin-Helmholtz instability and the Rayleigh-Taylor instability are well established as examples of primary instabilities, the highly successful theory of the turbulent cascade put forth by Kolmogorov [2] rested on a",
"pages": [
1
]
},
{
"title": "EPJ Web of Conferences",
"content": "heuristic picture of secondary instability, established by early experiments [3]. It was not until the 80's [4, 5] that the elliptic instability was introduced as a mechanism for the secondary instability. A fluid in rigid rotation supports a spectrum of stable inertial waves, the simplest case being circularly polarized transverse plane waves oscillating at twice the frequency of the base flow [6]. Strain is introduced when the streamlines pass from circular to elliptical, and some modes find resonance with the strain field, leading to de-stabilization. The cascade reversal from inverse to direct when passing from 2D to 3D is a result of the di GLYPH<11> erent properties of 2D and 3D vortices. Two-dimensional vortices do not decay, merging viscously and growing to the integral scale; three-dimensional vortices also merge viscously, but before that they generally fall prey to the elliptic instability, which does not exist in two dimensions. Vortex survival depends on a balance between production and destruction.",
"pages": [
2
]
},
{
"title": "2 Elliptic instability",
"content": "A column of fluid under rigid rotation supports stable oscillations in the form of circularly polarized transverse plane waves (Fig. 1). Restored by the Coriolis force and propagating along a waveguide, these waves have also been called Kelvin waves . As the waves are transverse, if the direction of propagation coincides with the rotation axis, the action of the wave is that fluid parcels will execute in-plane epicyclic oscillations. The propagation vector may also have an angle GLYPH<18> with the rotation axis, in which case the fluid motion is no longer in-plane, executing both epicyclic and vertical motions, well known in galactic dynamics [7]. Destabilization occurs when strain in introduced; steepening gradients and providing a source of free energy. Instability occurs when a mode or pair of modes find resonance with the rotating strain field, which is to say when a multiple of the rotation frequency matches the frequency of the inertial waves. The elliptic motion U = GLYPH<10> [ GLYPH<0> (1 GLYPH<0> \" ) y; (1 GLYPH<0> \" ) x ], where 0 GLYPH<20> \" GLYPH<20> 1 is the ellipticity, is readily decomposed U = GLYPH<10> ( R + S ) into rigid rotation R = [ GLYPH<0> y; x ] and the strain field S = GLYPH<0> \" [ y; x ]. The growth rates, reproduced from [8], are shown in Fig. 2 in the GLYPH<31> -GLYPH<18> plane. The aspect ratio GLYPH<31> of the streamlines is a measure of the strain. As the instability grows the vortex coherence is destroyed, energy cascades forward and dissipates; the flow relaminarizes. As seen in Fig. 2, the instability is elliptical : there is no growth for GLYPH<31> = 1, i.e., circular streamlines in rigid rotation. The instability is also inherently three-dimensional : there is no growth for GLYPH<18> = 0, i.e., in-plane fluid oscillations. This later behavior is at the heart of the di GLYPH<11> erence between the cascade in 2D and 3D. A stirring or primary instability, such as Rayleigh-Taylor or Kelvin-Helmholtz, generates the first eddies. The elliptic instability generates 3D turbulence out of this 2D motion, breaking the eddies. The growth rates are of the order of the turnover frequency, which explains why vortices have lifetimes of this order. As elliptic destruction occurs faster than viscous merging, the 3D cascade is direct. In two dimensions, there is no elliptic instability, and the eddies simply merge viscously. Adding a background rotation has significant e GLYPH<11> ects for the stability. The system now has two timescales: the turnover vortex time and that of the background flow. Not only that, the motions can be either aligned (cyclonic vortex) or anti-aligned (anti-cyclonic). Not surprisingly, the strongest instability occurs for anticyclonic motion, as the two flows rotating in opposite directions greatly enhances the e GLYPH<11> ective shear on a fluid parcel. In this case, the in-plane, horizontal, motion gets de-stabilized. This instability is not of resonant nature, but centrifugal, appearing as exponential growth of epicyclic disturbances. This behaviour invites a connection with the Rayleigh (centrifugal) instability, and indeed the mechanism is similar [8], suggesting that the Rayleigh instability is a limit of the elliptic instability in the presence of rotation.",
"pages": [
2,
3
]
},
{
"title": "3 Magneto-Elliptic Instability",
"content": "When magnetic fields are introduced in the problem, the addition of Alfvén waves enriches the families of unstable modes. In the absence of rotation, the magneto-elliptic instability is a parametric instability as well [9], have three unstable branches in the GLYPH<31> -GLYPH<18> plane. The first two are 'hydrodynamic' and 'magnetic', corresponding to resonant destabilization of Kelvin and Alfvén waves, respectively. A third, 'mixed' mode occurs through instability of a pair of modes when one is hydrodynamic and the other is magnetic, or a Kelvin-Alfvén instability. When background rotation is introduced, the same as had occurred to the elliptic instability ensues. When this background rotation runs opposite to the rotation of the vortex (anti-cyclonic motion), a fluid parcel becomes subject to an intense e GLYPH<11> ective shear. Since the magnetic tension resists shear, leading to instability [10] a powerful unstable in-plane mode appears [11]. This instability is of course the magneto-rotational instability, in generalized form. Indeed, the dispersion relation of this horizontal mode reduces to that of the MRI in the pure shear limit of \" = 1 (Fig. 4, [12, 13]). This provides an interesting unification, explaining the magneto-elliptic and magneto-rotational instabilities as di GLYPH<11> erent manifestations of the same magneto-elliptic-rotational instability .",
"pages": [
3
]
},
{
"title": "4 Formation and destruction of vortices in disks",
"content": "Four ways of destroying vortices in disks are identified in Fig. 4. In the non-magnetic case (EI), these are the Rayleigh instability for the horizontal mode, and the resonant destabilization of Kelvin waves. In the magnetic case (MEI), these are the MRI for the horizontal modes, and resonant destabilization of Alfvén waves. Counterbalancing these there are two mechanics to inject vorticity in the flow. These are the Rossby wave instability (RWI, [14-19]), and the Baroclinic Instability (BI, [12, 20-22]). The former is a linear instability powered by a modification of the shear profile, that behaves as an external reservoir of vorticity. The latter is a nonlinear instability powered by buoyancy and thermal di GLYPH<11> usion, that establish a nonzero baroclinic source term. As long as these vorticity sources (RWI and BI) are more powerful than the vorticity sinks (EI and MEI), a vortex can survive. It has been shown in the literature [12, 22] that despite the elliptic instability, the baroclinic instability keeps 3D vortices coherent. The result is core turbulence only, as the vorticity lost by the EI is replenished by that injected by the BI. It has also been shown [12] that the same is not true for the MEI, with a magnetized baroclinic vortex getting quickly destroyed by the strong excitation of magneto-elliptic modes Fig. 5. Yet, magnetized vortices are seen in global, high-resolution simulations [19]. These were triggered by the RWI, albeit artificially, due to an intense peak in magnetic pressure in the initial condition, leading to a non-Keplerian shear profile in the active zone. The cylindrical simulation was performed at very high resolution, Nr ; N GLYPH<30> ; Nz = (768, 1536, 128) 1 ranging GLYPH<25> in azimuth and 0.4 to 2.5 in radius. This translates into 40 grid points per scale height, more than enough to resolve the unstable magnetoelliptic modes. So, either the said phenomenon is not a vortex but a zonal flow, or the RWI provides vorticity faster than the MEI can destroy it, as is the case with the BI and EI. If that is the case, we can group the processes in order of strength as EI < BI < MEI < RWI.",
"pages": [
4,
5
]
},
{
"title": "References",
"content": "1 It also made use of high-end computing, being calculated in over 18 000 processors at the NICS-Kraken cluster. We used the P encil C ode , with which we achieve nearly linear scalability in NICS-Kraken, up to 70 000 processors.",
"pages": [
5
]
}
]
<document>
<text><location><page_1><loc_8><loc_88><loc_53><loc_92></location>EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c GLYPH<13> Owned by the authors, published by EDP Sciences, 2018</text>
<section_header_level_1><location><page_1><loc_8><loc_76><loc_65><loc_78></location>The Next Generation Transit Survey (NGTS)</section_header_level_1>
<text><location><page_1><loc_8><loc_65><loc_92><loc_73></location>Peter J. Wheatley 1 ; a , Don L. Pollacco 1 , Didier Queloz 2 , Heike Rauer 3 ; 4 , Christopher A. Watson 5 , Richard G. West 6 , Bruno Chazelas 2 , Tom M. Louden 1 , Simon Walker 1 , Nigel Bannister 6 , Joao Bento 1 , Matthew Burleigh 6 , Juan Cabrera 3 , Philipp Eigmüller 3 , Anders Erikson 3 , Ludovic Genolet 2 , Michael Goad 6 , Andrew Grange 6 , Andrés Jordán 7 , Katherine Lawrie 6 , James McCormac 8 , and Marion Neveu 2</text>
<unordered_list>
<list_item><location><page_1><loc_8><loc_63><loc_64><loc_64></location>1 Department of Physics, University of Warwick, Coventry CV4 7AL, UK</list_item>
<list_item><location><page_1><loc_8><loc_61><loc_74><loc_63></location>2 Observatoire Astronomique de l'Université de Genève, 1290 Sauverny, Switzerland</list_item>
<list_item><location><page_1><loc_8><loc_60><loc_86><loc_61></location>3 Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, 12489 Berlin, Germany</list_item>
<list_item><location><page_1><loc_8><loc_58><loc_83><loc_60></location>4 Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, 10623 Berlin, Germany</list_item>
<list_item><location><page_1><loc_8><loc_57><loc_72><loc_58></location>5 Astrophysics Research Centre, Queen's University Belfast, Belfast BT7 1NN, UK</list_item>
<list_item><location><page_1><loc_8><loc_55><loc_76><loc_57></location>6 Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK</list_item>
<list_item><location><page_1><loc_8><loc_54><loc_87><loc_55></location>7 Departamento de Astronomía y Astrofísica, Pontificia Universidad Católica de Chile, Santiago, Chile</list_item>
<list_item><location><page_1><loc_8><loc_52><loc_80><loc_53></location>8 Isaac Newton Group of Telescopes, 38700 Santa Cruz de la Palma, Canary Islands, Spain</list_item>
</unordered_list>
<text><location><page_1><loc_17><loc_31><loc_83><loc_49></location>Abstract. The Next Generation Transit Survey (NGTS) is a new ground-based sky survey designed to find transiting Neptunes and super-Earths. By covering at least sixteen times the sky area of Kepler , we will find small planets around stars that are su GLYPH<14> ciently bright for radial velocity confirmation, mass determination and atmospheric characterisation. The NGTS instrument will consist of an array of twelve independently pointed 20 cm telescopes fitted with red-sensitive CCD cameras. It will be constructed at the ESO Paranal Observatory, thereby benefiting from the very best photometric conditions as well as follow up synergy with the VLT and E-ELT. Our design has been verified through the operation of two prototype instruments, demonstrating white noise characteristics to sub-mmag photometric precision. Detailed simulations show that about thirty bright super-Earths and up to two hundred Neptunes could be discovered. Our science operations are due to begin in 2014.</text>
<section_header_level_1><location><page_1><loc_8><loc_25><loc_24><loc_26></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_8><loc_10><loc_92><loc_23></location>Ground-based transit surveys such as WASP [1] and HAT [2] have discovered an extraordinary variety of mainly Jupiter and Saturn-sized exoplanets that are challenging our understanding of giant planet formation and migration. Discoveries have included planets that are variously highly-inflated, extremely close-in, or in retrograde orbits. Meanwhile, space-based transit surveys such as CoRoT [3] and Kepler [4] have extended our sensitivity to smaller planets, finding the first rocky exoplanets as well as a startling variety of multiple systems. However, due to the relatively narrow fields of view of space-based surveys, most small planet candidates orbit stars that are too faint for radial-velocity confirmation. As a consequence, their masses are often unknown or poorly known, placing only weak</text>
<section_header_level_1><location><page_2><loc_41><loc_91><loc_59><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<text><location><page_2><loc_8><loc_83><loc_92><loc_88></location>constraints on their bulk composition. Most of these planets are also too faint for atmospheric characterisation, even with future instrumentation such as the E-ELT, JWST, EChO or FINESSE. There is thus a strong scientific imperative to find small exoplanets around brighter stars.</text>
<text><location><page_2><loc_8><loc_72><loc_92><loc_83></location>Building on experience gained in the WASP project, we have designed the Next Generation Transit Survey (NGTS) with the primary aim of discovering transiting Neptunes and Super-Earths around bright stars from the ground. Our objective is to find a statistically significant sample of such systems that are bright enough for radial-velocity confirmation and hence mass and density determination. By constraining the bulk compositions of our sample we will test population synthesis models of planet formation and evolution [e.g. 5]. Our brightest Neptunes and Super-Earths will also be suitable for atmospheric characterisation by transmission spectroscopy and secondary eclipse observations.</text>
<section_header_level_1><location><page_2><loc_8><loc_67><loc_35><loc_69></location>2 The NGTS instrument</section_header_level_1>
<text><location><page_2><loc_8><loc_58><loc_92><loc_66></location>Our scientific goals require the detection of transits with depths at the 1 mmag level. While this level of accuracy is routinely achieved from the ground in narrow-field observations of individual objects, it is unprecedented for a ground-based wide-field survey. To reach this level of accuracy we have drawn on experience from the WASP project in order to minimise known sources of red noise related to imprecise pointing, focus and flat fielding.</text>
<text><location><page_2><loc_8><loc_48><loc_92><loc_58></location>The NGTS facility will be an array of twelve 20cm f / 2.8 telescopes on independent equatorial mounts, each fitted with a red-optimised large-format CCD camera. The facility will be sited at the ESO Paranal Observatory (Chile) in order to benefit from the world's best photometric conditions. There is also excellent synergy with ESO facilities, including HARPS and ESPRESSO for radialvelocity confirmation, SPHERE for imaging wider-separation companions, and a variety of VLT and planned E-ELT instruments for atmospheric characterisation.</text>
<text><location><page_2><loc_8><loc_36><loc_92><loc_48></location>The telescopes are astrographs of a custom design by ASA in Austria. They have a carbon fibre tube, hyperbolic primary mirror and custom corrector optics that provide a uniform point spread function across the entire 3 deg field of view (FWHM of 1.1 pixels). The telescopes are fitted with a 400 mm ba GLYPH<15> e in order to minimise sensitivity to scattered moonlight. The CCD cameras are also of custom design, by Andor Technology in the UK, employing e2v CCDs with back-illuminated deepdepletion silicon and anti-fringing technology that is optimised for the far red of the optical spectrum. This improves our sensitivity to smaller stars (K and M types) and hence smaller planets.</text>
<text><location><page_2><loc_8><loc_26><loc_92><loc_36></location>The telescopes and cameras are mounted on independent equatorial fork mounts by OMI in the USA, and will be located in a single enclosure with a slide o GLYPH<11> -roof by GR PRO in the UK. The enclosure is designed to shelter the telescope units from the prevailing Northerly wind, while allowing each unit to point independently without obscuring the view of any other. The full telescope array has an instantaneous field of view of 96 square degrees, and we intend to observe around four fields each year. The instrument design is described in more detail in [6].</text>
<text><location><page_2><loc_8><loc_18><loc_92><loc_26></location>The deployment phase of NGTS is fully funded by our consortium institutes (DLR, Geneva, Leicester, QUB, Warwick) and most of the equipment has been procured. A complete telescope unit has been assembled and tested in Geneva, shown in the left hand panel of Fig. 1. On-site construction is due to begin in summer 2013, with science operations from 2014. Our data will be made publicly available after a proprietary period through the normal ESO archive.</text>
<section_header_level_1><location><page_2><loc_8><loc_14><loc_35><loc_15></location>3 The NGTS prototypes</section_header_level_1>
<text><location><page_2><loc_8><loc_8><loc_92><loc_12></location>In order to develop and verify our instrument design we have deployed two prototype instruments. The first was operated on La Palma in 2010, and it verified the photometric performance of the backilluminated deep-depletion e2v CCDs, and demonstrated the value of precise autoguiding. The second</text>
<figure>
<location><page_3><loc_16><loc_64><loc_36><loc_85></location>
</figure>
<section_header_level_1><location><page_3><loc_40><loc_91><loc_60><loc_92></location>Hot Planets and Cool Stars</section_header_level_1>
<figure>
<location><page_3><loc_38><loc_65><loc_84><loc_88></location>
<caption>Figure 1. Left: the first of twelve NGTS telescope units. This unit was operated at Geneva during summer 2012. Right: the noise characteristics of the NGTS telescope unit measured in Geneva. These noise measurements have been made using an ensemble of bright stars from across the whole field of view. The results show white noise behaviour to well below 1 mmag precision.</caption>
</figure>
<text><location><page_3><loc_8><loc_37><loc_92><loc_52></location>prototype was the complete NGTS telescope unit shown in Fig. 1. Tests with this unit have verified the optical performance of the telescope, as well as the pointing precision of the mount. The right hand panel of Fig. 1 shows the results of end-to-end tests of our photometry from Geneva on the Kepler field. The median fractional RMS variability of an ensemble of bright stars from across the field of view is plotted as a function of binned exposure time. The measured noise is consistent with the expected scintillation level for the Geneva site [7]. We initially found a noise floor of 3-mmag that we ascribe to variable water vapour extinction at the Geneva site, but since this was strongly correlated between all stars it was readily detrended using the standard SysRem algorithm [8]. Fig. 1 then shows purely white-noise down to sub-mmag photometry.</text>
<section_header_level_1><location><page_3><loc_8><loc_32><loc_36><loc_33></location>4 Simulated planet catch</section_header_level_1>
<text><location><page_3><loc_8><loc_8><loc_92><loc_29></location>We have performed detailed simulations of the NGTS planet catch for a four year baseline survey that covers sixteen times the area of the Kepler field. A population of host stars was drawn from the Besancon model of the Galaxy [9], and assigned a population of planets based on the planet size distributions and occurrence rates from Kepler [10]. This population was then sampled with the known characteristics of the NGTS prototype instruments, accounting for realistic source and background spectra, weather statistics for Paranal and scintillation. The resulting planet candidate population was then filtered with the sensitivity limits of the HARPS and ESPRESSO radial-velocity spectrographs, and the final predicted confirmable population is shown in Fig. 2 assuming a total of 10 h exposure time per candidate. We find that HARPS / HARPS-N observations are capable of confirming 37 Neptunes from NGTS compared with only 7 from Kepler . Allowing the HARPS-N exposure times to increase to 20 h per candidate retrieves 21 Neptunes from Kepler , but still only 1 super-Earth. In contrast, follow up of NGTS candidates with ESPRESSO on the VLT is sensitive to 231 Neptunes and 39 super-Earths (with only 10 h total exposure time per candidate).</text>
<figure>
<location><page_4><loc_9><loc_62><loc_60><loc_88></location>
<caption>Figure 2. Our simulated population of NGTS planets that can be confirmed in 10 h with HARPS or ESPRESSO (blue). These are compared with the known transiting planets with radial-velocity confirmation (green) and the Kepler candidates that are confirmable with HARPS-N (red). This simulation shows a total of 39 confirmable super-Earths from NGTS and 231 Neptunes.</caption>
</figure>
<text><location><page_4><loc_8><loc_51><loc_92><loc_59></location>The smallest NGTS confirmed planets will be prime targets for the ESA S Mission CHEOPS, which will provide precise radii and hence densities of our super-Earths. As well as testing models of bulk composition of super-Earths, this will allow us to prioritise objects by scale height for atmospheric follow up with VLT and HST , and eventually E-ELT and JWST (as well as dedicated missions such as EChO or FINESSE).</text>
<section_header_level_1><location><page_4><loc_8><loc_47><loc_21><loc_48></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_4><loc_9><loc_41><loc_92><loc_46></location>[1] D.L. Pollacco, I. Skillen, A. Collier Cameron, D.J. Christian, C. Hellier, J. Irwin, T.A. Lister, R.A. Street, R.G. West, D. Anderson et al., PASP 118 , 1407 (2006), arXiv:astro-ph/0608454</list_item>
<list_item><location><page_4><loc_9><loc_37><loc_92><loc_40></location>[2] G. Bakos, R.W. Noyes, G. Kovács, K.Z. Stanek, D.D. Sasselov, I. Domsa, PASP 116 , 266 (2004), arXiv:astro-ph/0401219</list_item>
<list_item><location><page_4><loc_9><loc_34><loc_92><loc_37></location>[3] M. Auvergne, P. Bodin, L. Boisnard, J.T. Buey, S. Chaintreuil, G. Epstein, M. Jouret, T. LamTrong, P. Levacher, A. Magnan et al., A&A 506 , 411 (2009), 0901.2206</list_item>
<list_item><location><page_4><loc_9><loc_30><loc_92><loc_33></location>[4] W.J. Borucki, D. Koch, G. Basri, N. Batalha, T. Brown, D. Caldwell, J. Caldwell, J. ChristensenDalsgaard, W.D. Cochran, E. DeVore et al., Science 327 , 977 (2010)</list_item>
<list_item><location><page_4><loc_9><loc_27><loc_92><loc_30></location>[5] C. Mordasini, Y. Alibert, C. Georgy, K.M. Dittkrist, H. Klahr, T. Henning, A&A 547 , A112 (2012), 1206.3303</list_item>
<list_item><location><page_4><loc_9><loc_18><loc_92><loc_26></location>[6] B. Chazelas, D. Pollacco, D. Queloz, H. Rauer, P.J. Wheatley, R. West, J. Da Silva Bento, M. Burleigh, J. McCormac, P. Eigmüller et al., NGTS: a robotic transit survey to detect Neptune and super-Earth mass planets , in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2012), Vol. 8444 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series</list_item>
<list_item><location><page_4><loc_9><loc_17><loc_73><loc_18></location>[7] D. Dravins, L. Lindegren, E. Mezey, A.T. Young, PASP 110 , 610 (1998)</list_item>
<list_item><location><page_4><loc_9><loc_15><loc_87><loc_16></location>[8] O. Tamuz, T. Mazeh, S. Zucker, MNRAS 356 , 1466 (2005), arXiv:astro-ph/0502056</list_item>
<list_item><location><page_4><loc_9><loc_11><loc_92><loc_14></location>[9] A.C. Robin, C. Reylé, S. Derrière, S. Picaud, A&A 416 , 157 (2004), arXiv:astro-ph/0401052</list_item>
<list_item><location><page_4><loc_8><loc_8><loc_92><loc_11></location>[10] A.W. Howard, G.W. Marcy, S.T. Bryson, J.M. Jenkins, J.F. Rowe, N.M. Batalha, W.J. Borucki, D.G. Koch, E.W. Dunham, T.N. Gautier, III et al., ApJS 201 , 15 (2012), 1103.2541</list_item>
</document>
[
{
"title": "ABSTRACT",
"content": "EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c GLYPH<13> Owned by the authors, published by EDP Sciences, 2018",
"pages": [
1
]
},
{
"title": "The Next Generation Transit Survey (NGTS)",
"content": "Peter J. Wheatley 1 ; a , Don L. Pollacco 1 , Didier Queloz 2 , Heike Rauer 3 ; 4 , Christopher A. Watson 5 , Richard G. West 6 , Bruno Chazelas 2 , Tom M. Louden 1 , Simon Walker 1 , Nigel Bannister 6 , Joao Bento 1 , Matthew Burleigh 6 , Juan Cabrera 3 , Philipp Eigmüller 3 , Anders Erikson 3 , Ludovic Genolet 2 , Michael Goad 6 , Andrew Grange 6 , Andrés Jordán 7 , Katherine Lawrie 6 , James McCormac 8 , and Marion Neveu 2 Abstract. The Next Generation Transit Survey (NGTS) is a new ground-based sky survey designed to find transiting Neptunes and super-Earths. By covering at least sixteen times the sky area of Kepler , we will find small planets around stars that are su GLYPH<14> ciently bright for radial velocity confirmation, mass determination and atmospheric characterisation. The NGTS instrument will consist of an array of twelve independently pointed 20 cm telescopes fitted with red-sensitive CCD cameras. It will be constructed at the ESO Paranal Observatory, thereby benefiting from the very best photometric conditions as well as follow up synergy with the VLT and E-ELT. Our design has been verified through the operation of two prototype instruments, demonstrating white noise characteristics to sub-mmag photometric precision. Detailed simulations show that about thirty bright super-Earths and up to two hundred Neptunes could be discovered. Our science operations are due to begin in 2014.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Ground-based transit surveys such as WASP [1] and HAT [2] have discovered an extraordinary variety of mainly Jupiter and Saturn-sized exoplanets that are challenging our understanding of giant planet formation and migration. Discoveries have included planets that are variously highly-inflated, extremely close-in, or in retrograde orbits. Meanwhile, space-based transit surveys such as CoRoT [3] and Kepler [4] have extended our sensitivity to smaller planets, finding the first rocky exoplanets as well as a startling variety of multiple systems. However, due to the relatively narrow fields of view of space-based surveys, most small planet candidates orbit stars that are too faint for radial-velocity confirmation. As a consequence, their masses are often unknown or poorly known, placing only weak",
"pages": [
1
]
},
{
"title": "EPJ Web of Conferences",
"content": "constraints on their bulk composition. Most of these planets are also too faint for atmospheric characterisation, even with future instrumentation such as the E-ELT, JWST, EChO or FINESSE. There is thus a strong scientific imperative to find small exoplanets around brighter stars. Building on experience gained in the WASP project, we have designed the Next Generation Transit Survey (NGTS) with the primary aim of discovering transiting Neptunes and Super-Earths around bright stars from the ground. Our objective is to find a statistically significant sample of such systems that are bright enough for radial-velocity confirmation and hence mass and density determination. By constraining the bulk compositions of our sample we will test population synthesis models of planet formation and evolution [e.g. 5]. Our brightest Neptunes and Super-Earths will also be suitable for atmospheric characterisation by transmission spectroscopy and secondary eclipse observations.",
"pages": [
2
]
},
{
"title": "2 The NGTS instrument",
"content": "Our scientific goals require the detection of transits with depths at the 1 mmag level. While this level of accuracy is routinely achieved from the ground in narrow-field observations of individual objects, it is unprecedented for a ground-based wide-field survey. To reach this level of accuracy we have drawn on experience from the WASP project in order to minimise known sources of red noise related to imprecise pointing, focus and flat fielding. The NGTS facility will be an array of twelve 20cm f / 2.8 telescopes on independent equatorial mounts, each fitted with a red-optimised large-format CCD camera. The facility will be sited at the ESO Paranal Observatory (Chile) in order to benefit from the world's best photometric conditions. There is also excellent synergy with ESO facilities, including HARPS and ESPRESSO for radialvelocity confirmation, SPHERE for imaging wider-separation companions, and a variety of VLT and planned E-ELT instruments for atmospheric characterisation. The telescopes are astrographs of a custom design by ASA in Austria. They have a carbon fibre tube, hyperbolic primary mirror and custom corrector optics that provide a uniform point spread function across the entire 3 deg field of view (FWHM of 1.1 pixels). The telescopes are fitted with a 400 mm ba GLYPH<15> e in order to minimise sensitivity to scattered moonlight. The CCD cameras are also of custom design, by Andor Technology in the UK, employing e2v CCDs with back-illuminated deepdepletion silicon and anti-fringing technology that is optimised for the far red of the optical spectrum. This improves our sensitivity to smaller stars (K and M types) and hence smaller planets. The telescopes and cameras are mounted on independent equatorial fork mounts by OMI in the USA, and will be located in a single enclosure with a slide o GLYPH<11> -roof by GR PRO in the UK. The enclosure is designed to shelter the telescope units from the prevailing Northerly wind, while allowing each unit to point independently without obscuring the view of any other. The full telescope array has an instantaneous field of view of 96 square degrees, and we intend to observe around four fields each year. The instrument design is described in more detail in [6]. The deployment phase of NGTS is fully funded by our consortium institutes (DLR, Geneva, Leicester, QUB, Warwick) and most of the equipment has been procured. A complete telescope unit has been assembled and tested in Geneva, shown in the left hand panel of Fig. 1. On-site construction is due to begin in summer 2013, with science operations from 2014. Our data will be made publicly available after a proprietary period through the normal ESO archive.",
"pages": [
2
]
},
{
"title": "3 The NGTS prototypes",
"content": "In order to develop and verify our instrument design we have deployed two prototype instruments. The first was operated on La Palma in 2010, and it verified the photometric performance of the backilluminated deep-depletion e2v CCDs, and demonstrated the value of precise autoguiding. The second",
"pages": [
2
]
},
{
"title": "Hot Planets and Cool Stars",
"content": "prototype was the complete NGTS telescope unit shown in Fig. 1. Tests with this unit have verified the optical performance of the telescope, as well as the pointing precision of the mount. The right hand panel of Fig. 1 shows the results of end-to-end tests of our photometry from Geneva on the Kepler field. The median fractional RMS variability of an ensemble of bright stars from across the field of view is plotted as a function of binned exposure time. The measured noise is consistent with the expected scintillation level for the Geneva site [7]. We initially found a noise floor of 3-mmag that we ascribe to variable water vapour extinction at the Geneva site, but since this was strongly correlated between all stars it was readily detrended using the standard SysRem algorithm [8]. Fig. 1 then shows purely white-noise down to sub-mmag photometry.",
"pages": [
3
]
},
{
"title": "4 Simulated planet catch",
"content": "We have performed detailed simulations of the NGTS planet catch for a four year baseline survey that covers sixteen times the area of the Kepler field. A population of host stars was drawn from the Besancon model of the Galaxy [9], and assigned a population of planets based on the planet size distributions and occurrence rates from Kepler [10]. This population was then sampled with the known characteristics of the NGTS prototype instruments, accounting for realistic source and background spectra, weather statistics for Paranal and scintillation. The resulting planet candidate population was then filtered with the sensitivity limits of the HARPS and ESPRESSO radial-velocity spectrographs, and the final predicted confirmable population is shown in Fig. 2 assuming a total of 10 h exposure time per candidate. We find that HARPS / HARPS-N observations are capable of confirming 37 Neptunes from NGTS compared with only 7 from Kepler . Allowing the HARPS-N exposure times to increase to 20 h per candidate retrieves 21 Neptunes from Kepler , but still only 1 super-Earth. In contrast, follow up of NGTS candidates with ESPRESSO on the VLT is sensitive to 231 Neptunes and 39 super-Earths (with only 10 h total exposure time per candidate). The smallest NGTS confirmed planets will be prime targets for the ESA S Mission CHEOPS, which will provide precise radii and hence densities of our super-Earths. As well as testing models of bulk composition of super-Earths, this will allow us to prioritise objects by scale height for atmospheric follow up with VLT and HST , and eventually E-ELT and JWST (as well as dedicated missions such as EChO or FINESSE).",
"pages": [
3,
4
]
}
]
2013EPJWC..5301007A
https://arxiv.org/pdf/1306.6090.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_79><loc_67><loc_80></location>Air Shower Simulation and Hadronic Interactions</section_header_level_1>
<text><location><page_1><loc_15><loc_75><loc_81><loc_77></location>Je ff Allen 1 , Antonella Castellina 2 , Ralph Engel 3 , Katsuaki Kasahara 4 , Stanislav Knurenko 5 , Tanguy Pierog 3 , Artem Sabourov 5 , Benjamin T. Stokes 6 , Ralf Ulrich 3 ,</text>
<text><location><page_1><loc_15><loc_72><loc_62><loc_74></location>for the Pierre Auger, Telescope Array, and Yakutsk Collaborations, and</text>
<text><location><page_1><loc_15><loc_70><loc_42><loc_72></location>Sergey Ostapchenko 7 and Takashi Sako 8</text>
<unordered_list>
<list_item><location><page_1><loc_15><loc_68><loc_59><loc_69></location>1 New York University, 4 Washington Place, New York, NY, USA</list_item>
<list_item><location><page_1><loc_15><loc_67><loc_80><loc_68></location>2 Osservatorio Astrofisico di Torino (INAF), Universit'a di Torino and Sezione INFN, Torino, Italy</list_item>
<list_item><location><page_1><loc_15><loc_65><loc_69><loc_67></location>3 Karlsruhe Institute of Technology, Institut fur Kernphysik, Karlsruhe, Germany</list_item>
<list_item><location><page_1><loc_15><loc_64><loc_43><loc_65></location>4 RISE, Waseda University, Tokyo, Japan</list_item>
<list_item><location><page_1><loc_15><loc_62><loc_71><loc_64></location>5 Yu.G. Shafer Institute of Cosmophysical Research and Aeronomy, Yakutsk, Russia</list_item>
<list_item><location><page_1><loc_15><loc_60><loc_82><loc_62></location>6 High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, UT, USA</list_item>
<list_item><location><page_1><loc_15><loc_58><loc_52><loc_59></location>7 NTNU, Institutt for fysikk, 7491 Trondheim, Norway</list_item>
<list_item><location><page_1><loc_15><loc_57><loc_61><loc_58></location>8 Solar-Terrestrial Environment Laboratory, Nagoya University, Japan</list_item>
</unordered_list>
<text><location><page_1><loc_22><loc_40><loc_75><loc_53></location>Abstract. The aim of this report of the Working Group on Hadronic Interactions and Air Shower Simulation is to give an overview of the status of the field, emphasizing open questions and a comparison of relevant results of the di ff erent experiments. It is shown that an approximate overall understanding of extensive air showers and the corresponding hadronic interactions has been reached. The simulations provide a qualitative description of the bulk of the air shower observables. Discrepancies are however found when the correlation between measurements of the longitudinal shower profile are compared to that of the lateral particle distributions at ground. The report concludes with a list of important problems that should be addressed to make progress in understanding hadronic interactions and, hence, improve the reliability of air shower simulations.</text>
<section_header_level_1><location><page_1><loc_15><loc_36><loc_28><loc_37></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_16><loc_82><loc_34></location>After the first interaction of a cosmic ray particle of very high energy in the atmosphere a multitude of subsequent interactions, leading to particle multiplication, and decay processes give rise to a cascade of secondary particles called extensive air shower (EAS) [1]. With the electromagnetic and weak interactions being well described by perturbative calculations within the Standard Model of Particle Physics, the limited understanding of strong interactions becomes the dominant source of uncertainties of shower predictions. Even though Quantum Chromodynamics (QCD) is the well-established and experimentally confirmed theory of strong interactions, only processes with very large momentum transfer can be predicted from first principles until now. It is still not possible to calculate the bulk properties of multiparticle production as needed for air shower simulation. Additional, simplifying assumptions as well as phenomenological and empirical parametrizations are needed to develop models for hadronic interactions describing various particle production processes. These additional assumptions need to be verified, parametrizations constrained, and parameters tuned by comparisons to accelerator data.</text>
<text><location><page_1><loc_15><loc_11><loc_82><loc_15></location>The simulation of extensive air showers forms one of the pillars on which the data analysis of modern experiments for ultra-high energy cosmic rays (UHECRs) rests. Improving the understanding and modeling of hadronic particle production is one of the important prerequisites for a reliable interpreta-</text>
<section_header_level_1><location><page_2><loc_40><loc_90><loc_57><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<text><location><page_2><loc_15><loc_83><loc_82><loc_88></location>tion of UHECR data. Even though calorimetric techniques based on the measurement of fluorescence and Cherenkov light have been developed for a nearly model-independent energy determination of extensive air showers, determining the type of the primary particles and, in particular, estimating the mass of the primary particle can only be done with the help of sets of simulated reference showers. 1</text>
<text><location><page_2><loc_15><loc_71><loc_82><loc_82></location>Much e ff ort has been devoted to improving both the simulation of extensive air showers in the atmosphere and the understanding of the corresponding, and typically very complex detector response function. In the following we will concentrate on discussing the simulation of extensive air showers as this aspect of the overall data simulation chain is the same in all UHECR experiments. We will assume that uncertainties arising from simulating the detector response due to the shower particles are much smaller and can be disregarded here. This is certainly not guaranteed per se and it is the task of each collaboration to verify the quality of the detector simulation before attributing possible discrepancies to, for example, shortcomings in the understanding of air showers or hadronic multiparticle production.</text>
<text><location><page_2><loc_15><loc_60><loc_82><loc_71></location>After recalling some basic features of air showers (Sec. 2) we will give an overview of the most frequently used code packages for air shower simulation in Sec. 3. The LHC data allow us to test the hadronic interaction models employed in these code packages, for the first time, at equivalent energies beyond that of the knee in the cosmic ray spectrum. In Sec. 4 some representative model predictions are compared to LHC data and implications are discussed. A good overall bracketing of the LHC data by model predictions is found, even though each of the models will have to be improved to obtain a satisfactory description of the LHC data. Therefore it is not surprising that the distributions of most of the shower observables are well reproduced by shower simulations (Sec. 5).</text>
<text><location><page_2><loc_15><loc_49><loc_82><loc_60></location>Taking advantage of the hybrid detection setups of the latest generation of UHECR detectors, longitudinal and lateral particle distributions can be measured shower-by-shower. A comparison of the particle densities at ground with the ones expected according to the shower longitudinal profile reveals possible shortcomings of the shower predictions that are not yet understood. The overall data set of such comparisons is discussed in Sec. 6 and the results from the Auger, TA, and Yakutsk Collaborations are compared. There are strong indications that a significant part of the shortcomings of the shower predictions are related to the muonic shower component. Therefore we summarize recent developments aiming at better understanding muon production in air showers in Sec. 7.</text>
<text><location><page_2><loc_15><loc_40><loc_82><loc_48></location>Given the increased awareness of apparent deficits of current air shower predictions and related theory developments, the wealth of new data from the LHC, and the high-quality measurements of the current generation of air shower observatories, significant progress can be expected in the reliability of air shower predictions in the next few years. Open questions that need to be addressed to optimally benefit from these developments to improve the interpretation of EAS data and to solve some outstanding problems will be listed in the concluding section of this report.</text>
<section_header_level_1><location><page_2><loc_15><loc_35><loc_38><loc_37></location>2 Physics of Air Showers</section_header_level_1>
<text><location><page_2><loc_15><loc_28><loc_82><loc_33></location>Reviews of the physics of extensive air showers and hadronic interactions can be found in, for example, [3-6] and a very instructive extension of Heitler's cascade model to hadronic showers is given in [7]. Here we give only a qualitative introduction to some aspects of shower physics that will be needed in the discussions later on.</text>
<text><location><page_2><loc_15><loc_15><loc_82><loc_28></location>With pions being the most abundant secondary particles of hadronic interactions at high energy, the di ff erence in lifetime and decay products of neutral and charged pions lead to fundamentally di ff erent ranges of interaction energies that give rise to the electromagnetic and muonic shower components. In air showers, neutral pions decay almost always before interacting again (except E π 0 > 10 19 eV) and provide high-energy photons that feed the electromagnetic shower component. In contrast, the energy of charged pions has to be degraded down to some 30 -100GeV before the probability to decay will exceed that of another interaction. In first approximation, the depth profile of charged shower particles, being dominated by e + e -of the electromagnetic component, is linked to the secondary particles of the first few interactions in the hadronic core of the shower. In contrast, the muons at ground stem from</text>
<section_header_level_1><location><page_3><loc_39><loc_90><loc_58><loc_92></location>Conference Title, to be filled</section_header_level_1>
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<location><page_3><loc_29><loc_47><loc_65><loc_66></location>
<caption>Fig. 1. Longitudinal shower profiles of electrons and muons. Shown are also the sub-showers produced by the secondary particles of the first 100 highest-energy interactions and the cumulative profile of these sub-showers [8]. The simulation was done with a modified version of CONEX [9] for a proton shower of 10 19 eV.</caption>
</figure>
<text><location><page_3><loc_15><loc_29><loc_82><loc_38></location>a chain of 8 -10 successive hadronic interactions with the energy of the last interaction producing pions or kaons that lead to observed muons being in the range of 20 -200GeV [10,11]. This is illustrated in Fig. 1 showing the sub-showers produced by the secondary particles of the first 100 highest-energy interactions in an air shower [8]. While more than 50% of the electromagnetic shower particles are produced by decaying neutral pions of the first 100 interactions in an air shower, only a negligible fraction of the muonic component stems from decaying charged pions and kaons of the same interactions.</text>
<text><location><page_3><loc_15><loc_24><loc_82><loc_28></location>The longitudinal profile of showers as well as the energy and angular distributions of charged particles in a shower exhibit universality features [12-17] if considered as a function of shower age, here approximated by</text>
<formula><location><page_3><loc_43><loc_21><loc_82><loc_24></location>s = 3 X X + 2 X max , (1)</formula>
<text><location><page_3><loc_15><loc_11><loc_82><loc_21></location>with X being the slant depth along the shower axis and X max the depth of shower maximum. Universality features are routinely used to estimate the Cherenkov light signal of showers, see, for example, [15, 18]. Not only the electromagnetic particles but also the production depth as well as the energy and transverse momentum distributions of muons can be described by functions [19,20] that depend only weakly on the assumed hadronic interaction model. Breaking down the overall shower signal at ground into di ff erent muonic and electromagnetic components, the description of showers can be improved considerably [21-23].</text>
<section_header_level_1><location><page_4><loc_40><loc_90><loc_57><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<text><location><page_4><loc_15><loc_71><loc_82><loc_88></location>While the bulk of charged particles is considered for longitudinal profiles of the electromagnetic and muonic shower components, lateral particle distributions at ground are only measured at large distances from the shower axis. Particle densities at, for example, 600 or 1000 m from the shower core are only indirectly related to the overall particle multiplicity of a shower at ground. This is reflected in the fact that, for example, the last interaction for producing a muon observable at ground has a median energy of ∼ 100GeV for the KASCADE array (typical primary energy 3 × 10 15 eV; lateral distance 40 -200m, see [11]) and only 20 -30GeV for the Auger array (typical primary energy 10 19 eV; lateral distance 1000 m, see [24]). Still universality arguments can also be applied to particle densities far away from the core [21] but the model-related di ff erences between the predictions are larger. In particular, low-energy interactions can lead to a model-dependent violation of universality profiles that is reflected in both the muonic and electromagnetic shower content at large lateral distance [16, 23].</text>
<section_header_level_1><location><page_4><loc_15><loc_67><loc_72><loc_68></location>3 Shower Simulation Packages and Hadronic Interaction Models</section_header_level_1>
<text><location><page_4><loc_15><loc_45><loc_82><loc_65></location>The most frequently used code packages for the simulation of air showers for the Auger, TA and Yakutsk experiments are AIRES [25], CORSIKA [26], and COSMOS [27]. In these packages, the Monte Carlo method is applied to simulate the evolution of air showers with all secondary particles above a user-specified energy threshold. The large number of secondary particles that would have to be followed in the simulation of ultra-high energy showers makes the simulations so time consuming that thinning techniques are applied. Only a representative set of particles is included in the simulation below a pre-defined energy threshold. The discarded particles are accounted for by increasing the statistical weight of the particles remaining in the simulation [28,29]. The main drawbacks of this method are artificial fluctuations related to particles with large statistical weight [30,31] and the need for treating weighted particles in the detector simulations. The Auger and TA Collaborations have developed di ff erent methods for de-thinning simulated showers, see [32] and [33], respectively. Also a limited number of fully simulated showers of the highest energies are available in each collaboration (see, for example, [34]) but the statistics is very limited and there is a lack of methods to work with such large amounts of data in an e ffi cient way.</text>
<text><location><page_4><loc_15><loc_35><loc_82><loc_45></location>The classic Monte Carlo codes are complemented by hybrid simulation programs that combine the Monte Carlo technique with either numerical solutions of cascade equations (CONEX [9] and SENECA [35]) or libraries of pre-simulated air showers (COSMOS [27]). The advantage of the hybrid codes is the much reduced CPU time needed for simulating showers at very high energy and the possibility to simulate showers either without thinning or with very low statistical weights. But there is always a remaining risk that rare shower fluctuations or local correlations might not be correctly described.</text>
<text><location><page_4><loc_15><loc_27><loc_82><loc_35></location>Di ff erent codes are applied for the simulation of the electromagnetic shower component. The code used in AIRES is based on the one developed by Hillas for MOCCA [28,36]. CONEX, CORSIKA, and SENECA are interfaced to modified versions of EGS4 [37]. Similar to AIRES, COSMOS comes with a custom-developed code for electromagnetic interactions called EPICS [27]. The suppression of high-energy interactions of photons and electrons due to the Landau-Pomeranchuk-Migdal (LPM) e ff ect (for example, see [38]) is taken into account in all code packages.</text>
<text><location><page_4><loc_15><loc_17><loc_82><loc_26></location>Due to the di ff erent underlying approaches for modeling hadronic interactions at low ( E < 80 -200GeV) and high energies typically always two hadronic interaction models are employed in air shower simulations. The low-energy models GHEISHA [39], FLUKA [40], and UrQMD [41] are available in CORSIKA while AIRES and COSMOS employ custom-made codes. Typical high-energy interaction models are DPMJET II [42] and III [43,44], EPOS [45-47], QGSJET 01 [48,49] and II [50-52], and SIBYLL [53-55]. A compilation of simulation results obtained with many of these low- and high-energy models can be found in [56-58].</text>
<text><location><page_4><loc_15><loc_11><loc_82><loc_17></location>Several comparisons of the predictions of shower simulation packages for the same interaction models are available in the literature, see [4] and [61,62,59,60,63]. In the ideal case, the predictions for air showers of a given energy and primary particle depend only on the user-selected energy thresholds and on the choice of hadronic interaction models selected for the simulation. While it is possible to</text>
<section_header_level_1><location><page_5><loc_39><loc_90><loc_58><loc_92></location>Conference Title, to be filled</section_header_level_1>
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<location><page_5><loc_30><loc_67><loc_66><loc_87></location>
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<caption>Fig. 2. Longitudinal shower profiles of electrons and muons. Predictions calculated with CORSIKA and COSMOS are compared for vertical proton showers of 10 19 eV [59,60]. High-energy interactions were simulated with QGSJET II.03 for energies above 80 GeV.</caption>
</figure>
<text><location><page_5><loc_15><loc_24><loc_82><loc_36></location>chose a high-energy interaction model that is supported in all shower simulation codes, less flexibility is available for the low-energy interaction models, which are often di ff erent in each of the simulation packages. Therefore the results of these comparisons typically show a good agreement of the longitudinal shower profiles, which are mainly related to hadronic interactions at high energy. Di ff erences are found for observables that are sensitive to low-energy interactions, which are, for example, the total number of muons and particle densities at large lateral distances. The di ff erences between the predictions of di ff erent simulation packages are typically of the order of 5%, increasing in some phase space regions up to ∼ 10%. For example, longitudinal shower profiles from a recent comparison between CORSIKA and COSMOS [59,60] are shown in Fig.2.</text>
<section_header_level_1><location><page_5><loc_15><loc_20><loc_56><loc_21></location>4 Performance of Hadronic Interaction Models</section_header_level_1>
<text><location><page_5><loc_15><loc_11><loc_82><loc_18></location>Most of the interaction models used in air shower simulations are not commonly applied in high energy physics (HEP) simulations and, conversely, HEP models are not used for air shower simulations. This is related to the fact that HEP models are typically limited to a set of primary particles that are available in accelerator experiments and optimized only for collider energies. Similarly, cosmic ray interaction models are not routinely used by HEP collaborations for acceptance correction calculations including</text>
<figure>
<location><page_6><loc_27><loc_63><loc_69><loc_86></location>
<caption>Fig. 3. Pseudorapidity distribution of charged particles measured at LHC. CMS data [64] are shown together with model predictions. Similar data sets are available from the ATLAS and ALICE Collaborations [65-67] (not shown). Below 2 TeV center-of-mass energy the models were tuned to previously available collider data from Tevatron and SPS.</caption>
</figure>
<text><location><page_6><loc_15><loc_42><loc_82><loc_52></location>the comparison of raw, uncorrected data as this can be done with the HEP event generators as well. Furthermore dedicated HEP event generators describe highp ⊥ and electroweak physics processes in much more detail and contain many more parameters for improving the description of particular distributions than any of the cosmic ray or general purpose models. In contrast, interaction models for cosmic ray physics are tuned to describe data of accelerator measurements over a wide range of energies, often sacrificing a perfect description of some distributions in favour of a good overall reproduction of the whole data set.</text>
<text><location><page_6><loc_15><loc_33><loc_82><loc_42></location>Hence comparisons of model predictions to data from fixed-target and collider experiments are typically made by the authors of the models and using only fully acceptance corrected data. However, over the last years much progress has been made regarding the interaction between the cosmic ray and high energy physics communities. A number of workshops and meetings (including this one) and the direct involvement of cosmic ray physicists in HEP collaborations have lead to a much higher level of awareness and intensified communication of both communities.</text>
<text><location><page_6><loc_15><loc_19><loc_82><loc_33></location>Comparisons of high-energy interaction models used for air shower simulations with LHC data can be found in, for example, [69-71]. Here only some representative examples can be included for illustration. The pseudorapidity distribution of charged particles is shown together with model predictions in Fig. 3. Not only most central particle distributions are reasonably well bracketed by the model predictions, also the predicted energy flow at larger pseudorapidity is in good agreement with the LHC data. This can be seen in Fig. 4 where the energy flow measured by CMS [68] is compared to model predictions. Note that all these model results are true predictions as the models were tuned years before LHC data became available. Nevertheless, in many cases, the minimum bias data are better described by interaction models developed primarily for cosmic ray physics than the various tunes of standard HEP models [68,69].</text>
<text><location><page_6><loc_15><loc_11><loc_82><loc_18></location>On the other hand, there are also some important distributions measured at LHC that are not well described or bracketed by these pre-LHC interaction models. Examples of large deviations from measurements are multiplicity distributions of charged particles and particle production spectra at large pseudorapidity and Feynmanx . The possible impact of these deviations on air shower predictions is subject to ongoing research and not yet fully understood. In Fig. 5 the Feynmanx distribution of</text>
<figure>
<location><page_7><loc_22><loc_63><loc_75><loc_88></location>
<caption>Fig. 4. Energy flow as function of pseudorapidity in forward direction. CMS data [68] are shown together with model predictions calculated by the CMS Collaboration. The left (right) panel shows the measurements for 900 GeV (7 TeV) c.m.s. energy.</caption>
</figure>
<figure>
<location><page_7><loc_16><loc_23><loc_78><loc_53></location>
<caption>Fig. 5. Photon spectra in very forward direction as measured by the LHCf Collaboration [72]. Superimposed are the predictions of hadronic interaction models used for air shower simulations. The lower panels show the relative di ff erence between data and model predictions.</caption>
</figure>
<text><location><page_7><loc_15><loc_11><loc_82><loc_14></location>photons produced in forward direction is shown [72]. The data have been obtained within the LHCf experiment that is specifically built for measuring particles in the very forward direction.</text>
<section_header_level_1><location><page_8><loc_40><loc_90><loc_57><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<text><location><page_8><loc_15><loc_78><loc_82><loc_88></location>In general, the predictions of the di ff erent cosmic ray interaction models bracket the LHC data reasonably well [69]. This observation supports the hope that simulating air showers with di ff erent interaction models should also bracket the correct, but unknown predictions for air showers. On the other hand, this success does not mean that there is no need to improve the interaction models. There is not a single interaction model that reproduces most of new LHC data well and, hence, would be much better than the others. Each of the models needs to be re-tuned or underlying ideas be extended to obtain a better description of the data.</text>
<text><location><page_8><loc_15><loc_70><loc_82><loc_78></location>Manynewdata sets measured in LHC experiments and also the fixed-target experiments NA49 [73] and NA61 [74,75] became available within the last 2 -3 years and the process of tuning and modifying the interaction models to obtain an improved description is ongoing. At the time of writing these proceedings the new model versions QGSJET II.04 [70] and EPOS LHC [71] have been released as the first versions of interaction models tuned to LHC data. In addition, work is in progress to develop improved versions of SIBYLL and DPMJET that are also tuned to the new LHC data.</text>
<text><location><page_8><loc_15><loc_60><loc_82><loc_70></location>Finally it should be mentioned that, in addition to accelerator data, also air shower measurements provide important information on hadronic interactions. For example, cross section measurements made with air shower observables [76] can be used to estimate not only the rise of the cross section but also to verify the model approximations used to calculate the proton-air cross section from the data on proton-proton cross sections. Even though the systematic uncertainties of air shower measurements of this kind are much higher than that of accelerator data, such measurements are the only way to study particle production at interaction energies well beyond the range of colliders [77-80].</text>
<section_header_level_1><location><page_8><loc_15><loc_56><loc_58><loc_57></location>5 Overall Description of Shower Characteristics</section_header_level_1>
<text><location><page_8><loc_15><loc_45><loc_82><loc_54></location>Simulating air showers with a realistic energy spectrum, primary mass composition, and arrival direction distribution, the predicted and observed distributions of the various observables of reconstructed showers can be compared. Such comparisons are very important as they are end-to-end tests of the simulation chain for the experiments. Only if good agreement is found one can cross-check the impact of quality cuts applied in the process of data analysis and avoid unexpected biases. The results of such end-to-end simulations depend, of course, on the assumed mass composition of the primary particles and the employed hadronic interaction models.</text>
<text><location><page_8><loc_15><loc_31><loc_82><loc_45></location>Examples of such comparisons for fluorescence detectors can be found in [81-84]. A number of unpublished comparisons of typical observables measured with the Auger and TA surface detector arrays (zenith and azimuth angles, number of stations per event, signal size distribution) have been presented at this meeting. They all show very good agreement between the measured distributions and the corresponding Monte Carlo predictions. While the TA simulations were done for proton primaries, the Auger simulations were based on a 50 / 50 proton-iron mixture. Both collaborations used QGSJET II.03 as high-energy interaction model. The TA data are better described by a primary composition of only protons than using 100% iron. In case of the Auger simulations, which were also made assuming only proton or iron primaries, it was found that most of the surface detector observables exhibit only a very limited sensitivity to the primary mass composition.</text>
<text><location><page_8><loc_15><loc_19><loc_82><loc_30></location>It should be considered as an important success of hadronic interaction models and modern air shower simulation packages that such a good overall description of the general features of the observed showers is reached. However, di ff erent combinations of primary mass composition and shower energy can lead to very similar surface detector signals (for example, the signal at 600 or 1000 m in units of that of vertical muons). Therefore it is very important to use additional information to determine, for example, the primary energy in a composition-independent way. This is done with fluorescence telescopes in the case of the Auger Observatory [85] and Telescope Array [86], and with non-imaging Cherenkov light measurements in the Yakutsk setup [87].</text>
<section_header_level_1><location><page_8><loc_15><loc_15><loc_52><loc_16></location>6 Detailed Comparison with Shower Data</section_header_level_1>
<text><location><page_8><loc_15><loc_11><loc_82><loc_14></location>Detecting air showers simultaneously with fluorescence telescopes and an array of surface detectors at ground is often referred to as hybrid measurements. One of the first setups of this type was the</text>
<section_header_level_1><location><page_9><loc_39><loc_90><loc_58><loc_92></location>Conference Title, to be filled</section_header_level_1>
<text><location><page_9><loc_15><loc_78><loc_82><loc_88></location>prototype instrument of the High Resolution Fly's Eye (HiRes) [88] that was operated in coincidence with the CASA-MIA array [89] in Utah. Already in this prototype experiment strong indications for a discrepancy between the measured density of muons at 600 m from the shower core and the one expected from simulations were found, if the composition inferred from the measurements of the longitudinal shower profiles was used [90]. While the data on the depth of shower maximum suggested the conclusion that a transition from a heavy to a light, almost proton dominated composition is seen, the density of muons stayed above or at the level of the predictions for iron primaries.</text>
<figure>
<location><page_9><loc_16><loc_57><loc_81><loc_75></location>
<caption>Fig. 6. Longitudinal and lateral profiles of one high-energy shower observed simultaneously with the fluorescence and the surface detectors of the Pierre Auger Observatory [91]. The data are compared with simulated showers for proton and iron primaries that predict the same longitudinal profiles. The simulations were made with SENECA and QGSJET II.03 / FLUKA as high-/ low-energy interaction models.</caption>
</figure>
<text><location><page_9><loc_15><loc_33><loc_82><loc_47></location>Covering energies higher than those of the early HiRes-MIA measurements, the data of the Auger Collaboration also indicate a discrepancy between the observed and predicted number of muons in air showers [91]. Di ff erent methods have been applied to obtain either the expected surface detector signal S (1000) or the muon density at 1000m from the shower core for a given primary energy or even an observed longitudinal shower profile (see [92] for a review given at this meeting). This discrepancy is most directly displayed in a shower-by-shower comparison of simulated profiles with those obtained from hybrid measurements, see Fig. 6. Compared to proton showers as reference, the measured values of S (1000) are about 1.5 to 2 times higher that the simulated ones. Studying inclined showers of about 10 19 eV with zenith angles larger than 60 · , whose particle content at ground is completely dominated by muons, gives a ratio of [91]</text>
<formula><location><page_9><loc_33><loc_28><loc_82><loc_32></location>N µ, data N µ, MC ∣ ∣ ∣ ∣ ∣ ∣ QGSJET , p = 2 . 13 ± 0 . 04(stat) ± 0 . 11(sys) , (2)</formula>
<text><location><page_9><loc_15><loc_21><loc_82><loc_27></location>where the systematic uncertainty due to the 22% energy scale uncertainty is not included. This ratio is about 1 . 2 for iron showers simulated with EPOS 1.99. The data on inclined showers imply that the observed discrepancy is closely related to, if not dominated by a deficiency in simulating the muon component of air showers.</text>
<text><location><page_9><loc_15><loc_11><loc_82><loc_21></location>The detectors of the Auger, TA and Yakutsk air shower arrays are di ff ering in their sensitivities to muons and electromagnetic particles, and the Yakutsk array even contains a number of shielded muon detectors. For example, the contribution of muons to the scintillator signal of the TA surface detectors will never exceed 30% for showers with zenith angles less than 45 · . In comparison, a signal fraction of 30 -80%is expected due to muons for the Auger water Cherenkov tanks in the zenith angle range from 0 -60 · . Therefore similar studies of the TA and Yakutsk Collaborations can provide a very important, independent information on a possible muon deficit of air shower simulations.</text>
<section_header_level_1><location><page_10><loc_40><loc_90><loc_57><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<text><location><page_10><loc_15><loc_83><loc_82><loc_88></location>The Yakutsk Collaboration compared the predicted and measured charged particle and muon lateral distributions for a number of high energy showers. Within the uncertainties implied by the unknown primary composition, no significant discrepancies between the predictions obtained with both QGSJET II / FLUKA and EPOS / UrQMD and the data are found [93].</text>
<text><location><page_10><loc_15><loc_74><loc_82><loc_82></location>The TA Collaboration studied the di ff erence between the energy one would assign to a shower based on the calorimetric fluorescence light measurement and that derived from the comparison with simulations of the surface detector signal. It was found that the fluorescence-assigned energy is about 27% lower than the energy estimated from the comparison with simulated showers [94]. Given the limited sensitivity of the TA surface detector stations, no direct study of the muon contribution to the overall signal has been done so far.</text>
<text><location><page_10><loc_15><loc_59><loc_82><loc_74></location>The observations of all three collaborations can be brought into qualitative agreement if the di ff erent energy scales of the experiments are taken into account. The factor with which the energy scales need to be re-scaled are listed in the report of the Spectrum Working Group [95]. For example, for the same shower, the Yakutsk simulations are done at an energy about two times higher than that of the Auger Observatory. Hence, also about a factor two more muons are predicted in the Yakutsk simulations in comparison to Auger simulations. Given that the Yakutsk simulations agree with their measurements, this implies two times more muons in the Yakutsk data than predicted by simulations if the Auger energy scale is used. A similar estimate can be done for the TA-Auger comparison, where one has to keep in mind that only ∼ 20%of the scintillator signal stems from muons. If the discrepancy observed by the Auger Collaboration would be attributed to the muon component only, the expected energy rescaling would be somewhat smaller than the ∼ 27% reported by the TA Collaboration [94].</text>
<text><location><page_10><loc_15><loc_51><loc_82><loc_58></location>It remains to be shown how much of the apparent discrepancy between the surface detector signals and the predictions based on calorimetric shower energy determination can be cured by adopting di ff erent energy scales in the experiments. A reduction of the systematic uncertainties of the energy assignment to air showers will be needed to quantify more reliably possible discrepancies between shower data and predictions.</text>
<section_header_level_1><location><page_10><loc_15><loc_47><loc_69><loc_48></location>7 Hadronic Interactions and Muon Production in Air Showers</section_header_level_1>
<text><location><page_10><loc_15><loc_34><loc_82><loc_45></location>The observation of a possible muon deficit in simulated showers in comparison to Auger measurements, first reported already in 2007 [96], has triggered a number of theoretical studies searching for modifications of hadronic interactions that could result in an enhancement of the muonic shower component. Changes of the inelastic cross section, inelasticity of interactions, secondary particle multiplicity at high energy, and many other parameters lead only to moderate changes of the predicted number of muons (see, for example, [97,98]). Also scenarios with new physics processes such as string percolation [99] or the drastic change of interaction properties due to, for example, chiral symmetry restoration [100] have been discussed.</text>
<text><location><page_10><loc_15><loc_31><loc_82><loc_34></location>So far all proposed changes that lead to a significant increase of the number of muons in air showers are directly or indirectly based on either one or both of the following e ff ects:</text>
<text><location><page_10><loc_15><loc_24><loc_82><loc_31></location>(i) An increase of the production rate of particles that do not decay, for example baryon-antibaryon pairs, leads to higher muon multiplicities of showers since these particles will stay being part of the hadronic shower component and loose their energy only by producing further hadronic particles. This e ff ect has been discussed first in [101] and is one of the di ff erences of the EPOS model with respect to the other models [102].</text>
<text><location><page_10><loc_15><loc_11><loc_82><loc_24></location>(ii) A change of the type of the leading particle produced in inelastic interactions can also be very e ffi cient in reducing the energy that is transferred to the electromagnetic shower component [103]. The majority of sub-showers in an extensive air shower are initiated by charged pions. The chance probability of producing a charged pion or a neutral pion as leading particle in charged pion interactions is about 2 : 1. Replacing all leading π 0 by ρ 0 mesons, which have the same valence quarks but are spin 1 particles, leads to a drastic enhancement of muon production since neutral ρ mesons decay immediately into two charged pions [104]. Indeed, fixed-target data [105-107] indicate that, in contrast to conventional model predictions, the production of ρ 0 dominates that of π 0 for Feynmanx larger than 0 . 5.</text>
<section_header_level_1><location><page_11><loc_39><loc_90><loc_58><loc_92></location>Conference Title, to be filled</section_header_level_1>
<text><location><page_11><loc_15><loc_81><loc_82><loc_88></location>Both e ff ects are the reason for the higher muon multiplicities predicted with EPOS. Leading rho mesons are explicitly generated for pion interactions in the new version II.04 of QGSJET, boosting the muon multiplicity in air showers significantly. Depending on the relative importance of e ff ect (i) and energy distribution of the baryon pairs, the energy spectrum of the additionally produced muons is very soft and might be in contradiction to shower attenuation data.</text>
<text><location><page_11><loc_15><loc_78><loc_82><loc_81></location>Finally it should be mentioned that also an increase of the production of kaons could lead to a higher muon multiplicity and also a harder muon energy spectrum [103,99].</text>
<section_header_level_1><location><page_11><loc_15><loc_74><loc_40><loc_75></location>8 Conclusions and Outlook</section_header_level_1>
<text><location><page_11><loc_15><loc_68><loc_82><loc_72></location>Over the last two decades the quality and predictive power of air shower simulations has improved significantly and a very good overall description of most of the shower features observed in experiments has been reached.</text>
<text><location><page_11><loc_15><loc_57><loc_82><loc_68></location>The predictions of independently developed shower simulation packages agree reasonably well with each other if the same hadronic interaction models are used for the simulations. While there is very good agreement for the electromagnetic shower component, the di ff erences between the predictions for muon multiplicities and lateral distributions can be as large as ∼ 5 -10%. These di ff erences are most likely related to the use of di ff erent low-energy interaction models. A comprehensive comparison of the di ff erent shower simulation packages, similar to the recent study of CORSIKA and COSMOS predictions [60], should be made to quantify the systematic uncertainties of the predictions and possibly also to identify and address shortcomings of the simulation packages.</text>
<text><location><page_11><loc_15><loc_48><loc_82><loc_56></location>The limited theoretical understanding of modeling hadronic multiparticle production together with the limitations of accelerator measurements in energy, covered phase space, and projectile-target combinations are the dominating source of systematic uncertainties of air shower predictions. Because of this, the systematic uncertainties of the model predictions cannot be estimated reliably. More work is needed to improve QCD calculations in the lowp ⊥ domain, in particular to understand screening and saturation e ff ects, and to develop alternative models to describe particle production.</text>
<text><location><page_11><loc_15><loc_41><loc_82><loc_48></location>The new LHC data provide extremely useful input for tuning hadronic interaction models. Even though the first LHC data on multiplicities and cross sections were well bracketed by the predictions of interaction models used for air shower simulations, the comparison to data revealed the need for further model developments and tuning. Improved and re-tuned versions of EPOS and QGSJET are already available and similar versions of DPMJET and SIBYLL are in preparation.</text>
<text><location><page_11><loc_15><loc_32><loc_82><loc_41></location>The interaction between the cosmic ray and high energy physics communities has intensified and the direct engagement of cosmic ray physicists in LHC and fixed-target experiments has lead to a much better understanding of the needs of the cosmic ray community. There is also very large interest from the side of LHC communities to have cosmic ray physicists being involved in the analysis of accelerator data. One example of the achieved progress is the use of cosmic ray interaction models by LHC collaborations to compare data with predictions in publications.</text>
<text><location><page_11><loc_15><loc_23><loc_82><loc_32></location>Both the Auger and TA Collaborations have found indications for a discrepancy between the expected and observed surface detector signals for showers with fluorescence energy measurement. Accounting for the di ff erent energy scales of the Auger, TA, and Yakutsk experiments the observed discrepancies are consistent with each other. Most likely, the dominating sources of the discrepancies are shortcomings of the simulation of muon production in air showers. The systematic uncertainties of the overall energy scales of the experiments of 20 -30% will have to be reduced significantly to be able to combine data from di ff erent experiments for a stronger test of the shower predictions.</text>
<text><location><page_11><loc_15><loc_11><loc_82><loc_22></location>While the production of electromagnetic particles is clearly dominated by the electromagnetic cascade induced by photons of very high energy due to neutral pion decay early on in the shower evolution, the muonic component receives contributions from all high-energy interactions above ∼ 20GeV lab. energy. Di ff erent modifications of the simulation of particle production in low- and high-energy interactions have been considered to increase the number of muons in air showers. Progress has been made by understanding that both production of stable or long-lived hadrons such as baryon pairs and modifications to the particle types generated as leading particles are e ffi cient mechanisms to increase the muon multiplicity without changing significantly the longitudinal shower profile. It remains to</text>
<section_header_level_1><location><page_12><loc_40><loc_90><loc_57><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<text><location><page_12><loc_15><loc_83><loc_82><loc_88></location>be shown whether the discrepancies between the observed and predicted surface detector signals for a given shower energy can be resolved by improving low- and high-energy interaction models using conventional physics assumptions or whether these observations indeed indicate a change of the characteristics of hadronic interactions at energies beyond that of LHC.</text>
<section_header_level_1><location><page_12><loc_15><loc_79><loc_26><loc_80></location>References</section_header_level_1>
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</document>
[
{
"title": "Air Shower Simulation and Hadronic Interactions",
"content": "Je ff Allen 1 , Antonella Castellina 2 , Ralph Engel 3 , Katsuaki Kasahara 4 , Stanislav Knurenko 5 , Tanguy Pierog 3 , Artem Sabourov 5 , Benjamin T. Stokes 6 , Ralf Ulrich 3 , for the Pierre Auger, Telescope Array, and Yakutsk Collaborations, and Sergey Ostapchenko 7 and Takashi Sako 8 Abstract. The aim of this report of the Working Group on Hadronic Interactions and Air Shower Simulation is to give an overview of the status of the field, emphasizing open questions and a comparison of relevant results of the di ff erent experiments. It is shown that an approximate overall understanding of extensive air showers and the corresponding hadronic interactions has been reached. The simulations provide a qualitative description of the bulk of the air shower observables. Discrepancies are however found when the correlation between measurements of the longitudinal shower profile are compared to that of the lateral particle distributions at ground. The report concludes with a list of important problems that should be addressed to make progress in understanding hadronic interactions and, hence, improve the reliability of air shower simulations.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "After the first interaction of a cosmic ray particle of very high energy in the atmosphere a multitude of subsequent interactions, leading to particle multiplication, and decay processes give rise to a cascade of secondary particles called extensive air shower (EAS) [1]. With the electromagnetic and weak interactions being well described by perturbative calculations within the Standard Model of Particle Physics, the limited understanding of strong interactions becomes the dominant source of uncertainties of shower predictions. Even though Quantum Chromodynamics (QCD) is the well-established and experimentally confirmed theory of strong interactions, only processes with very large momentum transfer can be predicted from first principles until now. It is still not possible to calculate the bulk properties of multiparticle production as needed for air shower simulation. Additional, simplifying assumptions as well as phenomenological and empirical parametrizations are needed to develop models for hadronic interactions describing various particle production processes. These additional assumptions need to be verified, parametrizations constrained, and parameters tuned by comparisons to accelerator data. The simulation of extensive air showers forms one of the pillars on which the data analysis of modern experiments for ultra-high energy cosmic rays (UHECRs) rests. Improving the understanding and modeling of hadronic particle production is one of the important prerequisites for a reliable interpreta-",
"pages": [
1
]
},
{
"title": "EPJ Web of Conferences",
"content": "be shown whether the discrepancies between the observed and predicted surface detector signals for a given shower energy can be resolved by improving low- and high-energy interaction models using conventional physics assumptions or whether these observations indeed indicate a change of the characteristics of hadronic interactions at energies beyond that of LHC.",
"pages": [
12
]
},
{
"title": "2 Physics of Air Showers",
"content": "Reviews of the physics of extensive air showers and hadronic interactions can be found in, for example, [3-6] and a very instructive extension of Heitler's cascade model to hadronic showers is given in [7]. Here we give only a qualitative introduction to some aspects of shower physics that will be needed in the discussions later on. With pions being the most abundant secondary particles of hadronic interactions at high energy, the di ff erence in lifetime and decay products of neutral and charged pions lead to fundamentally di ff erent ranges of interaction energies that give rise to the electromagnetic and muonic shower components. In air showers, neutral pions decay almost always before interacting again (except E π 0 > 10 19 eV) and provide high-energy photons that feed the electromagnetic shower component. In contrast, the energy of charged pions has to be degraded down to some 30 -100GeV before the probability to decay will exceed that of another interaction. In first approximation, the depth profile of charged shower particles, being dominated by e + e -of the electromagnetic component, is linked to the secondary particles of the first few interactions in the hadronic core of the shower. In contrast, the muons at ground stem from",
"pages": [
2
]
},
{
"title": "Conference Title, to be filled",
"content": "Both e ff ects are the reason for the higher muon multiplicities predicted with EPOS. Leading rho mesons are explicitly generated for pion interactions in the new version II.04 of QGSJET, boosting the muon multiplicity in air showers significantly. Depending on the relative importance of e ff ect (i) and energy distribution of the baryon pairs, the energy spectrum of the additionally produced muons is very soft and might be in contradiction to shower attenuation data. Finally it should be mentioned that also an increase of the production of kaons could lead to a higher muon multiplicity and also a harder muon energy spectrum [103,99].",
"pages": [
11
]
},
{
"title": "3 Shower Simulation Packages and Hadronic Interaction Models",
"content": "The most frequently used code packages for the simulation of air showers for the Auger, TA and Yakutsk experiments are AIRES [25], CORSIKA [26], and COSMOS [27]. In these packages, the Monte Carlo method is applied to simulate the evolution of air showers with all secondary particles above a user-specified energy threshold. The large number of secondary particles that would have to be followed in the simulation of ultra-high energy showers makes the simulations so time consuming that thinning techniques are applied. Only a representative set of particles is included in the simulation below a pre-defined energy threshold. The discarded particles are accounted for by increasing the statistical weight of the particles remaining in the simulation [28,29]. The main drawbacks of this method are artificial fluctuations related to particles with large statistical weight [30,31] and the need for treating weighted particles in the detector simulations. The Auger and TA Collaborations have developed di ff erent methods for de-thinning simulated showers, see [32] and [33], respectively. Also a limited number of fully simulated showers of the highest energies are available in each collaboration (see, for example, [34]) but the statistics is very limited and there is a lack of methods to work with such large amounts of data in an e ffi cient way. The classic Monte Carlo codes are complemented by hybrid simulation programs that combine the Monte Carlo technique with either numerical solutions of cascade equations (CONEX [9] and SENECA [35]) or libraries of pre-simulated air showers (COSMOS [27]). The advantage of the hybrid codes is the much reduced CPU time needed for simulating showers at very high energy and the possibility to simulate showers either without thinning or with very low statistical weights. But there is always a remaining risk that rare shower fluctuations or local correlations might not be correctly described. Di ff erent codes are applied for the simulation of the electromagnetic shower component. The code used in AIRES is based on the one developed by Hillas for MOCCA [28,36]. CONEX, CORSIKA, and SENECA are interfaced to modified versions of EGS4 [37]. Similar to AIRES, COSMOS comes with a custom-developed code for electromagnetic interactions called EPICS [27]. The suppression of high-energy interactions of photons and electrons due to the Landau-Pomeranchuk-Migdal (LPM) e ff ect (for example, see [38]) is taken into account in all code packages. Due to the di ff erent underlying approaches for modeling hadronic interactions at low ( E < 80 -200GeV) and high energies typically always two hadronic interaction models are employed in air shower simulations. The low-energy models GHEISHA [39], FLUKA [40], and UrQMD [41] are available in CORSIKA while AIRES and COSMOS employ custom-made codes. Typical high-energy interaction models are DPMJET II [42] and III [43,44], EPOS [45-47], QGSJET 01 [48,49] and II [50-52], and SIBYLL [53-55]. A compilation of simulation results obtained with many of these low- and high-energy models can be found in [56-58]. Several comparisons of the predictions of shower simulation packages for the same interaction models are available in the literature, see [4] and [61,62,59,60,63]. In the ideal case, the predictions for air showers of a given energy and primary particle depend only on the user-selected energy thresholds and on the choice of hadronic interaction models selected for the simulation. While it is possible to",
"pages": [
4
]
},
{
"title": "4 Performance of Hadronic Interaction Models",
"content": "Most of the interaction models used in air shower simulations are not commonly applied in high energy physics (HEP) simulations and, conversely, HEP models are not used for air shower simulations. This is related to the fact that HEP models are typically limited to a set of primary particles that are available in accelerator experiments and optimized only for collider energies. Similarly, cosmic ray interaction models are not routinely used by HEP collaborations for acceptance correction calculations including the comparison of raw, uncorrected data as this can be done with the HEP event generators as well. Furthermore dedicated HEP event generators describe highp ⊥ and electroweak physics processes in much more detail and contain many more parameters for improving the description of particular distributions than any of the cosmic ray or general purpose models. In contrast, interaction models for cosmic ray physics are tuned to describe data of accelerator measurements over a wide range of energies, often sacrificing a perfect description of some distributions in favour of a good overall reproduction of the whole data set. Hence comparisons of model predictions to data from fixed-target and collider experiments are typically made by the authors of the models and using only fully acceptance corrected data. However, over the last years much progress has been made regarding the interaction between the cosmic ray and high energy physics communities. A number of workshops and meetings (including this one) and the direct involvement of cosmic ray physicists in HEP collaborations have lead to a much higher level of awareness and intensified communication of both communities. Comparisons of high-energy interaction models used for air shower simulations with LHC data can be found in, for example, [69-71]. Here only some representative examples can be included for illustration. The pseudorapidity distribution of charged particles is shown together with model predictions in Fig. 3. Not only most central particle distributions are reasonably well bracketed by the model predictions, also the predicted energy flow at larger pseudorapidity is in good agreement with the LHC data. This can be seen in Fig. 4 where the energy flow measured by CMS [68] is compared to model predictions. Note that all these model results are true predictions as the models were tuned years before LHC data became available. Nevertheless, in many cases, the minimum bias data are better described by interaction models developed primarily for cosmic ray physics than the various tunes of standard HEP models [68,69]. On the other hand, there are also some important distributions measured at LHC that are not well described or bracketed by these pre-LHC interaction models. Examples of large deviations from measurements are multiplicity distributions of charged particles and particle production spectra at large pseudorapidity and Feynmanx . The possible impact of these deviations on air shower predictions is subject to ongoing research and not yet fully understood. In Fig. 5 the Feynmanx distribution of photons produced in forward direction is shown [72]. The data have been obtained within the LHCf experiment that is specifically built for measuring particles in the very forward direction.",
"pages": [
5,
6,
7
]
},
{
"title": "5 Overall Description of Shower Characteristics",
"content": "Simulating air showers with a realistic energy spectrum, primary mass composition, and arrival direction distribution, the predicted and observed distributions of the various observables of reconstructed showers can be compared. Such comparisons are very important as they are end-to-end tests of the simulation chain for the experiments. Only if good agreement is found one can cross-check the impact of quality cuts applied in the process of data analysis and avoid unexpected biases. The results of such end-to-end simulations depend, of course, on the assumed mass composition of the primary particles and the employed hadronic interaction models. Examples of such comparisons for fluorescence detectors can be found in [81-84]. A number of unpublished comparisons of typical observables measured with the Auger and TA surface detector arrays (zenith and azimuth angles, number of stations per event, signal size distribution) have been presented at this meeting. They all show very good agreement between the measured distributions and the corresponding Monte Carlo predictions. While the TA simulations were done for proton primaries, the Auger simulations were based on a 50 / 50 proton-iron mixture. Both collaborations used QGSJET II.03 as high-energy interaction model. The TA data are better described by a primary composition of only protons than using 100% iron. In case of the Auger simulations, which were also made assuming only proton or iron primaries, it was found that most of the surface detector observables exhibit only a very limited sensitivity to the primary mass composition. It should be considered as an important success of hadronic interaction models and modern air shower simulation packages that such a good overall description of the general features of the observed showers is reached. However, di ff erent combinations of primary mass composition and shower energy can lead to very similar surface detector signals (for example, the signal at 600 or 1000 m in units of that of vertical muons). Therefore it is very important to use additional information to determine, for example, the primary energy in a composition-independent way. This is done with fluorescence telescopes in the case of the Auger Observatory [85] and Telescope Array [86], and with non-imaging Cherenkov light measurements in the Yakutsk setup [87].",
"pages": [
8
]
},
{
"title": "6 Detailed Comparison with Shower Data",
"content": "Detecting air showers simultaneously with fluorescence telescopes and an array of surface detectors at ground is often referred to as hybrid measurements. One of the first setups of this type was the",
"pages": [
8
]
},
{
"title": "7 Hadronic Interactions and Muon Production in Air Showers",
"content": "The observation of a possible muon deficit in simulated showers in comparison to Auger measurements, first reported already in 2007 [96], has triggered a number of theoretical studies searching for modifications of hadronic interactions that could result in an enhancement of the muonic shower component. Changes of the inelastic cross section, inelasticity of interactions, secondary particle multiplicity at high energy, and many other parameters lead only to moderate changes of the predicted number of muons (see, for example, [97,98]). Also scenarios with new physics processes such as string percolation [99] or the drastic change of interaction properties due to, for example, chiral symmetry restoration [100] have been discussed. So far all proposed changes that lead to a significant increase of the number of muons in air showers are directly or indirectly based on either one or both of the following e ff ects: (i) An increase of the production rate of particles that do not decay, for example baryon-antibaryon pairs, leads to higher muon multiplicities of showers since these particles will stay being part of the hadronic shower component and loose their energy only by producing further hadronic particles. This e ff ect has been discussed first in [101] and is one of the di ff erences of the EPOS model with respect to the other models [102]. (ii) A change of the type of the leading particle produced in inelastic interactions can also be very e ffi cient in reducing the energy that is transferred to the electromagnetic shower component [103]. The majority of sub-showers in an extensive air shower are initiated by charged pions. The chance probability of producing a charged pion or a neutral pion as leading particle in charged pion interactions is about 2 : 1. Replacing all leading π 0 by ρ 0 mesons, which have the same valence quarks but are spin 1 particles, leads to a drastic enhancement of muon production since neutral ρ mesons decay immediately into two charged pions [104]. Indeed, fixed-target data [105-107] indicate that, in contrast to conventional model predictions, the production of ρ 0 dominates that of π 0 for Feynmanx larger than 0 . 5.",
"pages": [
10
]
},
{
"title": "8 Conclusions and Outlook",
"content": "Over the last two decades the quality and predictive power of air shower simulations has improved significantly and a very good overall description of most of the shower features observed in experiments has been reached. The predictions of independently developed shower simulation packages agree reasonably well with each other if the same hadronic interaction models are used for the simulations. While there is very good agreement for the electromagnetic shower component, the di ff erences between the predictions for muon multiplicities and lateral distributions can be as large as ∼ 5 -10%. These di ff erences are most likely related to the use of di ff erent low-energy interaction models. A comprehensive comparison of the di ff erent shower simulation packages, similar to the recent study of CORSIKA and COSMOS predictions [60], should be made to quantify the systematic uncertainties of the predictions and possibly also to identify and address shortcomings of the simulation packages. The limited theoretical understanding of modeling hadronic multiparticle production together with the limitations of accelerator measurements in energy, covered phase space, and projectile-target combinations are the dominating source of systematic uncertainties of air shower predictions. Because of this, the systematic uncertainties of the model predictions cannot be estimated reliably. More work is needed to improve QCD calculations in the lowp ⊥ domain, in particular to understand screening and saturation e ff ects, and to develop alternative models to describe particle production. The new LHC data provide extremely useful input for tuning hadronic interaction models. Even though the first LHC data on multiplicities and cross sections were well bracketed by the predictions of interaction models used for air shower simulations, the comparison to data revealed the need for further model developments and tuning. Improved and re-tuned versions of EPOS and QGSJET are already available and similar versions of DPMJET and SIBYLL are in preparation. The interaction between the cosmic ray and high energy physics communities has intensified and the direct engagement of cosmic ray physicists in LHC and fixed-target experiments has lead to a much better understanding of the needs of the cosmic ray community. There is also very large interest from the side of LHC communities to have cosmic ray physicists being involved in the analysis of accelerator data. One example of the achieved progress is the use of cosmic ray interaction models by LHC collaborations to compare data with predictions in publications. Both the Auger and TA Collaborations have found indications for a discrepancy between the expected and observed surface detector signals for showers with fluorescence energy measurement. Accounting for the di ff erent energy scales of the Auger, TA, and Yakutsk experiments the observed discrepancies are consistent with each other. Most likely, the dominating sources of the discrepancies are shortcomings of the simulation of muon production in air showers. The systematic uncertainties of the overall energy scales of the experiments of 20 -30% will have to be reduced significantly to be able to combine data from di ff erent experiments for a stronger test of the shower predictions. While the production of electromagnetic particles is clearly dominated by the electromagnetic cascade induced by photons of very high energy due to neutral pion decay early on in the shower evolution, the muonic component receives contributions from all high-energy interactions above ∼ 20GeV lab. energy. Di ff erent modifications of the simulation of particle production in low- and high-energy interactions have been considered to increase the number of muons in air showers. Progress has been made by understanding that both production of stable or long-lived hadrons such as baryon pairs and modifications to the particle types generated as leading particles are e ffi cient mechanisms to increase the muon multiplicity without changing significantly the longitudinal shower profile. It remains to",
"pages": [
11
]
}
]
2013EPJWC..5308013S
https://arxiv.org/pdf/1210.1427.pdf
<document>
<section_header_level_1><location><page_1><loc_16><loc_80><loc_61><loc_82></location>Radar reflection off extensive air showers</section_header_level_1>
<text><location><page_1><loc_16><loc_73><loc_84><loc_79></location>J. Stasielak 1 , a , S. Baur 2 , M. Bertaina 3 , J. Blumer 2 , A. Chiavassa 3 , R. Engel 2 , A. Haungs 2 , T. Huege 2 , K.-H. Kampert 4 , H. Klages 2 , M. Kleifges 2 , O. Kromer 2 , M. Ludwig 2 , S. Mathys 4 , P. Neunteufel 2 , J. Pekala 1 , J. Rautenberg 4 , M. Riegel 2 , M. Roth 2 , F. Salamida 5 , H. Schieler 2 , R. ˇ Sm'ıda 2 , M. Unger 2 , M. Weber 2 , F. Werner 2 , H. Wilczy'nski 1 , and J. Wochele 2</text>
<unordered_list>
<list_item><location><page_1><loc_16><loc_71><loc_51><loc_73></location>1 Institute of Nuclear Physics PAN, Krakow, Poland</list_item>
<list_item><location><page_1><loc_16><loc_70><loc_59><loc_71></location>2 Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany</list_item>
<list_item><location><page_1><loc_16><loc_69><loc_53><loc_70></location>3 Universit'a di Torino and Sezione INFN, Torino, Italy</list_item>
<list_item><location><page_1><loc_16><loc_67><loc_54><loc_69></location>4 Bergische Universitat Wuppertal, Wuppertal, Germany</list_item>
<list_item><location><page_1><loc_16><loc_66><loc_50><loc_67></location>5 Universit'a dell'Aquila and INFN, L'Aquila, Italy</list_item>
</unordered_list>
<text><location><page_1><loc_23><loc_55><loc_77><loc_63></location>Abstract. We investigate the possibility of detecting extensive air showers by the radar technique. Considering a bistatic radar system and di ff erent shower geometries, we simulate reflection of radio waves o ff the static plasma produced by the shower in the air. Using the Thomson cross-section for radio wave reflection, we obtain the time evolution of the signal received by the antennas. The frequency upshift of the radar echo and the power received are studied to verify the feasibility of the radar detection technique.</text>
<section_header_level_1><location><page_1><loc_16><loc_51><loc_29><loc_53></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_16><loc_42><loc_84><loc_49></location>Traditional techniques of extensive air shower (EAS) detection include counting the shower particles at the ground level and measuring fluorescence light from the excited nitrogen in the atmosphere. Detecting radio emission from the shower particles is a promising new technique that is currently in development. An alternative method is the radar technique, where a ground-based radio transmitter illuminates the ionization trail left behind the shower front and another ground-based antenna receives the scattered radio signal with its upshifted frequency.</text>
<text><location><page_1><loc_16><loc_34><loc_84><loc_41></location>The concept of implementing a radar for cosmic ray detection was first introduced about 70 years ago by Blackett and Lovell [1]. However due to the negative results of dedicated experiments, this concept was forgotten for several decades. In recent years renewed attention has been attracted to this topic, see e.g. [2,3,4,5]. Experimental e ff orts to detect EAS using the radar technique were made by several groups, e.g. [6], the LAAS group [7], [8], the MARIACHI experiment [9], or the TARA project [10], however no detection was confirmed so far.</text>
<text><location><page_1><loc_16><loc_22><loc_84><loc_33></location>In this paper we investigate the possibility of detecting EAS by the bistatic radar system. Especially, we are interested in verifying the feasibility of supplementing the CROME (Cosmic-Ray Observation via Microwave Emission) detector [11,12] with a radio transmitter and using its existing antennas as receivers for the radar echo. The CROME antenna setup consists of several microwave receivers for the frequency ranges 1 . 2 -1 . 7 (L band), 3 . 4 -4 . 2 (extended C band), and 10 . 7 -11 . 7 GHz (low Ku band). Since we expect large frequency upshifts of the radar echo and the CROME antennas are fine tuned to the GHz frequency band, we can narrow down our interest to the frequency range of 1 -100 MHz of the emitted radio waves.</text>
<text><location><page_1><loc_16><loc_19><loc_84><loc_23></location>The concept of detecting EAS using radar technique is based on the principle of scattering radio waves o ff the static plasma produced in the atmosphere by the energetic particles of the shower. The locally produced plasma decays exponentially with the lifetime τ . According to Vidmar [13], for the</text>
<section_header_level_1><location><page_2><loc_42><loc_91><loc_58><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<figure>
<location><page_2><loc_31><loc_66><loc_71><loc_89></location>
<caption>Fig. 1. A schematic diagram representing a bistatic radar system and reflection from the static plasma disk produced by the EAS in the atmosphere. See the text for a detailed explanation.</caption>
</figure>
<text><location><page_2><loc_71><loc_66><loc_72><loc_66></location>R</text>
<text><location><page_2><loc_16><loc_52><loc_84><loc_59></location>plasma densities relevant for EAS and at low altitudes, the three-body attachment to oxygen dominates the deionization process as it depends quadratically on the oxygen density. This leads to the plasma lifetime of 10 ns at sea level and about 100 ns at altitude of 10 km. Since the received power of the radar echo is strongly diminished by the geometrical factor, the strongest signal will be obtained from the altitudes close to the ground level. Thus we can safely assume τ = 10 ns.</text>
<text><location><page_2><loc_16><loc_43><loc_84><loc_52></location>The ionization trail that results from meteor or lightning is traditionally divided into the underdense and overdense regions, depending on the local plasma frequency ν p . If the electron density is high enough that the plasma frequency exceeds the radar frequency, then the region is overdense and the radio wave is reflected from its interface. In contrast, if the local plasma frequency is lower than the frequency of the incoming radio wave, then the region is underdense and the radio wave can penetrate the ionized region. In such a case the reflections are caused by the Thomson scattering of the radio wave on individual free electrons.</text>
<text><location><page_2><loc_16><loc_35><loc_84><loc_43></location>If we consider the radar frequency in the range 1 -100 MHz and shower energies up to 10 20 eV, then the maximum size of the overdense plasma region will be several meters. Now, if we calculate the attenuation length of the radio wave in the plasma with frequency ν p = 100 MHz and ν p = 1 MHz one obtains ∼ 300 m and ∼ 3000 km, respectively. Those lengths are much larger than the size of the overdense plasma, thus the radio wave can easily penetrate the whole volume of the plasma disk produced by the EAS.</text>
<text><location><page_2><loc_16><loc_30><loc_84><loc_35></location>Using the Thomson cross-section for radio wave reflection seems to be justified from the above arguments. An exact calculation of the total power received requires the integration of the contributions from each individual electron in the shower. The final result will then depend on the individual phase factors of the scattering electrons.</text>
<section_header_level_1><location><page_2><loc_16><loc_25><loc_32><loc_26></location>2 Radar reflection</section_header_level_1>
<text><location><page_2><loc_16><loc_16><loc_84><loc_23></location>A schematic diagram representing a bistatic radar system is shown in Figure 1. A ground-based radio transmitter (T) irradiates a disk-like shower. The front edge of the disk is an ionization front moving towards the ground with the speed of light and leaving created static plasma behind it. The radio signal is scattered by the free electrons in the ionization trail and subsequently received by the groundbased antenna (R). The geometry of the bistatic radar system is set up by the polar coordinates of the</text>
<figure>
<location><page_3><loc_26><loc_56><loc_74><loc_89></location>
<caption>Fig. 2. The time evolution of the region from which the scattered radio waves arrive simultaneously to the detector. The arrival time t = 0 corresponds to the moment at which the shower hits the ground. The considered shower has energy of 10 18 eV and is heading towards the transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude.</caption>
</figure>
<text><location><page_3><loc_16><loc_44><loc_84><loc_47></location>transmitter and the receiver, i.e. by the distances from the shower core to the transmitter ( dT ) and to the receiver ( dR ) together with the angles ϕ t and ϕ r .</text>
<text><location><page_3><loc_16><loc_41><loc_84><loc_44></location>Let us consider the radar reflection from the disk element with coordinates ( rL , ϕ ), altitude h, and electron density ne . Its contribution to the radar echo at the receiver at time t can be expressed by</text>
<formula><location><page_3><loc_27><loc_37><loc_84><loc_41></location>Urc v ( t , s , rL , ϕ ) = UT √ GTe i ω t + ϕ 0 e -i ∫ r n k · d r e -i ∫ rsc n ksc · d rsc | r || rsc | sin α √ d σ T d Ω ne , (1)</formula>
<text><location><page_3><loc_16><loc_29><loc_84><loc_37></location>where GT is the transmitter antenna gain, UT [V] is the magnitude of the transmitted field, k and ksc are the wave vectors of incoming and scattered radio wave, ϕ 0 is the initial phase of the emitted signal, d σ T / d Ω is the Thomson cross-section, s is the projection of the distance between the shower core and the considered disk element on the shower axis, α is the inclination angle of the reflected radio wave, and n is the refractive index of the air. By analogy to the setup of the CROME detector, we assume that the receiver is oriented vertically upwards.</text>
<text><location><page_3><loc_16><loc_23><loc_84><loc_29></location>The signal received by the antenna at a given time t is sum of the signals scattered at di ff erent times and from di ff erent parts of the plasma disk. These individual contributions interfere with each other and only an integral over the whole volume v ol ( t ) from which they arrive simultaneously gives us the correct value. The total electric field strength of the radio wave at the receiver is given by</text>
<formula><location><page_3><loc_36><loc_20><loc_84><loc_23></location>Urc v ( t ) = ∫ v ol ( t ) Urc v ( t , s , rL , ϕ ) rLdrLd ϕ ds . (2)</formula>
<text><location><page_3><loc_16><loc_16><loc_84><loc_19></location>Note that time t is defined in such a way that t = 0 coincides with the moment at which the shower hits the ground.</text>
<section_header_level_1><location><page_4><loc_42><loc_91><loc_58><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<figure>
<location><page_4><loc_19><loc_72><loc_78><loc_89></location>
<caption>Fig. 3. Upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d .</caption>
</figure>
<text><location><page_4><loc_16><loc_60><loc_84><loc_64></location>Figure 2 shows the time evolution of the region of integration v ol ( t ) (see equation (2)) for a shower with an energy of 10 18 eV heading towards transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude.</text>
<text><location><page_4><loc_16><loc_55><loc_84><loc_60></location>As we can see, the volume over which we integrate the signal spans usually over the wide range of altitudes. This implies a high variation of the air density. Therefore, the refractive index and the Cherenkov angle can vary significantly over the considered region. This makes the whole analysis very complex.</text>
<section_header_level_1><location><page_4><loc_16><loc_50><loc_34><loc_52></location>3 Frequency upshift</section_header_level_1>
<text><location><page_4><loc_16><loc_43><loc_84><loc_49></location>Despite the fact that the radio wave is scattered on the static plasma, the region in which the plasma is created moves with relativistic velocity. Therefore, we can observe the Doppler shift of the received radar echo. Figure 3 shows the upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d .</text>
<text><location><page_4><loc_16><loc_37><loc_84><loc_42></location>The frequency upshift depends on the wave direction and the refractive index of air. It has the highest value for the case in which the viewing angle coincides with the Cherenkov cone. The typical fr is high enough to upshift the MHz signal into the GHz range. Note that fr only weakly depends on the distance from the shower core to the transmitter dT .</text>
<section_header_level_1><location><page_4><loc_16><loc_33><loc_40><loc_34></location>4 Details of the simulation</section_header_level_1>
<text><location><page_4><loc_16><loc_30><loc_57><loc_31></location>The main assumptions and approximations are the following:</text>
<unordered_list>
<list_item><location><page_4><loc_17><loc_28><loc_84><loc_29></location>-The shower longitudinal profile is taken to be a fit to the Gaisser-Hillas function for proton showers.</list_item>
<list_item><location><page_4><loc_17><loc_27><loc_67><loc_28></location>-The lateral distribution of the shower is given by the Gora function [14].</list_item>
<list_item><location><page_4><loc_17><loc_26><loc_84><loc_27></location>-Static plasma produced by the shower decays exponentially with the characteristic time τ = 10 ns.</list_item>
<list_item><location><page_4><loc_17><loc_20><loc_84><loc_25></location>-We consider radar reflections only from the part of the ionization trail which is up to 40 ns behind the shower front. In other words, the length of the plasma disk produced by the extensive air shower is constrained to 4 τ . This leads to less than 2% of the total plasma electrons not being accounted for.</list_item>
<list_item><location><page_4><loc_17><loc_19><loc_48><loc_20></location>-We use the US Standard Atmosphere model.</list_item>
<list_item><location><page_4><loc_17><loc_16><loc_84><loc_19></location>-We assume that the initial phase of the emitted signal is ϕ 0 = 0. Choosing di ff erent values of ϕ 0 can lead to slightly di ff erent values of the received power.</list_item>
</unordered_list>
<figure>
<location><page_5><loc_19><loc_71><loc_80><loc_89></location>
<caption>Fig. 4. The waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases a vertical shower with energy 10 18 eV heading towards the transmitter is considered.</caption>
</figure>
<unordered_list>
<list_item><location><page_5><loc_17><loc_61><loc_84><loc_64></location>-The e ff ective area of the receiver antenna is AR = 1 m 2 and the transmitter emits signal over the whole upper hemisphere ( GT = 2).</list_item>
<list_item><location><page_5><loc_17><loc_59><loc_84><loc_61></location>-The receiver points vertically and its sensitivity is independent of the direction and frequency of the radar echo. The detector e ffi ciency is 100%.</list_item>
</unordered_list>
<section_header_level_1><location><page_5><loc_16><loc_55><loc_53><loc_56></location>5 Power received by the detector antenna</section_header_level_1>
<text><location><page_5><loc_16><loc_51><loc_84><loc_54></location>The ratio of the 'instantaneous' power PR ( t ) received by the detector antenna to the power emitted by the transmitter PT is given by</text>
<formula><location><page_5><loc_44><loc_50><loc_84><loc_51></location>PR ( t ) / PT = R 2 ( t ) (3)</formula>
<text><location><page_5><loc_16><loc_48><loc_20><loc_49></location>where</text>
<formula><location><page_5><loc_41><loc_45><loc_84><loc_48></location>R ( t ) = 1 √ 4 π ReUrcv(t) UT √ AR (4)</formula>
<text><location><page_5><loc_16><loc_39><loc_84><loc_45></location>The term R ( t ) is proportional to the electric field strength detected by the receiver. It can be used in the Fourier analysis to obtain the power spectrum of the recorded signal. The 'real' power received by the detector PR can be obtained by averaging R 2 ( t ) (according to the time resolution of the detector). The ratio PR / PT ∼ GTAR .</text>
<section_header_level_1><location><page_5><loc_16><loc_36><loc_24><loc_37></location>6 Results</section_header_level_1>
<text><location><page_5><loc_16><loc_30><loc_84><loc_35></location>Figure 4 shows the waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases the receiver is outside the Cherenkov cone (which is the most common case) and vertical showers heading towards the transmitter are considered. The time t = 0 coincides with the moment at which the shower hits the ground.</text>
<text><location><page_5><loc_16><loc_26><loc_84><loc_29></location>As we can see, the frequency of the received signal is higher than that of the emitted one. Moreover, the amplitude grows with time, which is the geometrical e ff ect of the reflections from the lower parts of the atmosphere.</text>
<text><location><page_5><loc_16><loc_22><loc_84><loc_25></location>For showers, where the receiver is inside the Cherenkov cone, the time sequence is reversed and the lower part of the shower is seen first. Therefore, the amplitude decreases with time. However, this is the case only for very inclined showers with cores up to several meters away from the receiver.</text>
<text><location><page_5><loc_16><loc_16><loc_84><loc_21></location>The ratios of the power received by the detector to the emitted one ( PR / PT ) for vertical showers are given in table 1. It is evident that the strength of the signal decreases with the frequency ν . It is simply because the size of the region from which we get the coherent signal decreases with the wavelength and the negative interference cancels out the signal.</text>
<section_header_level_1><location><page_6><loc_42><loc_91><loc_58><loc_92></location>EPJ Web of Conferences</section_header_level_1>
<table>
<location><page_6><loc_16><loc_74><loc_84><loc_83></location>
<caption>Table 1. The ratios of the power received by the detector to the emitted one ( PR / PT ) in dB for di ff erent frequencies ν of the emitted radio wave, di ff erent transmitter-receiver distances dR and di ff erent shower energy E . In all cases a vertical shower heading towards the transmitter is considered.</caption>
</table>
<section_header_level_1><location><page_6><loc_16><loc_70><loc_29><loc_71></location>7 Conclusions</section_header_level_1>
<text><location><page_6><loc_16><loc_59><loc_84><loc_68></location>The typical ratio of the received to the emitted power is in the range of 10 -10 - 10 -13 , which makes the detection of the radar reflection plausible. The C band ( ∼ 3 GHz) is particularly well-suited for this purpose because of very low noise ( < 10 K) in this frequency range. The CROME antennas supplemented with commercial high power MHz transmitter would create a radar system dedicated to EAS detection. A note should be added, that the shown time traces are for an infinite bandwidth detector -a realistic detector would only be able to detect the signal in narrow frequency range. Moreover, it will only see the shower for a certain fraction of its development.</text>
<text><location><page_6><loc_16><loc_56><loc_84><loc_59></location>The performed analysis will help to choose the frequency of the emitted wave to optimize the detection of the radar echo. The work is in progress.</text>
<section_header_level_1><location><page_6><loc_16><loc_52><loc_34><loc_53></location>8 Acknowledgments</section_header_level_1>
<text><location><page_6><loc_16><loc_46><loc_84><loc_51></location>It is our pleasure to acknowledge the interaction and collaboration with many colleagues from the KASCADE-Grande and Pierre Auger Collaboration. This work has been supported in part by the KIT start-up grant 2066995641,the ASPERA project BMBF 05A11VKA, the Helmholtz-University Young Investigators Group VH-NG-413 and the National Centre for Research and Development (NCBiR).</text>
<section_header_level_1><location><page_6><loc_16><loc_41><loc_26><loc_43></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_6><loc_16><loc_39><loc_68><loc_40></location>1. P. M. S. Blackett and A. C. B. Lovell, Proc. Royal. Soc. A177 , 183 (1940)</list_item>
<list_item><location><page_6><loc_16><loc_37><loc_53><loc_39></location>2. P. W. Gorham, Astroparticle Physics 15 , 177 (2001)</list_item>
<list_item><location><page_6><loc_16><loc_36><loc_59><loc_37></location>3. P. W. Gorham, AIP Conference proceedings 579 , 253 (2001)</list_item>
<list_item><location><page_6><loc_16><loc_35><loc_57><loc_36></location>4. M. I. Bakunov et al., Astroparticle Physics 33 , 335 (2010)</list_item>
<list_item><location><page_6><loc_16><loc_33><loc_59><loc_35></location>5. H. Takai et al., Proceedings of the 32th ICRC, Beijing (2011)</list_item>
<list_item><location><page_6><loc_16><loc_32><loc_60><loc_33></location>6. T. Matano et al., Canadian Journal of Physics 46 , S255 (1968)</list_item>
<list_item><location><page_6><loc_16><loc_31><loc_61><loc_32></location>7. A. Lyono et al., Proceedings of the 28th ICRC, Tsukuba (2003)</list_item>
<list_item><location><page_6><loc_16><loc_29><loc_60><loc_31></location>8. T. Terasawa et al., Proceedings of the 31st ICRC, Lodz (2009)</list_item>
<list_item><location><page_6><loc_16><loc_27><loc_84><loc_29></location>9. M. F. Bugallo et al., Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 2661 (2008)</list_item>
<list_item><location><page_6><loc_16><loc_24><loc_84><loc_27></location>10. J. Belz et al., Proceedings of the International Symposium on Future Directions in UHECR Physics, CERN (2012) (to be published)</list_item>
<list_item><location><page_6><loc_16><loc_23><loc_73><loc_24></location>11. R. ˇ Sm'ıda et al., Proceedings of the 32th ICRC, Beijing (2011), arXiv:1108.0588</list_item>
<list_item><location><page_6><loc_16><loc_20><loc_84><loc_23></location>12. R. ˇ Sm'ıda et al., Proceedings of the International Symposium on Future Directions in UHECR Physics, CERN (2012) (to be published)</list_item>
<list_item><location><page_6><loc_16><loc_19><loc_64><loc_20></location>13. R. J. Vidmar, IEEE Transactions On Plasma Science 18 , 733 (1990)</list_item>
<list_item><location><page_6><loc_16><loc_18><loc_54><loc_19></location>14. D. Gora, et al., Astroparticle Physics 24 , 484 (2006)</list_item>
</document>
[
{
"title": "Radar reflection off extensive air showers",
"content": "J. Stasielak 1 , a , S. Baur 2 , M. Bertaina 3 , J. Blumer 2 , A. Chiavassa 3 , R. Engel 2 , A. Haungs 2 , T. Huege 2 , K.-H. Kampert 4 , H. Klages 2 , M. Kleifges 2 , O. Kromer 2 , M. Ludwig 2 , S. Mathys 4 , P. Neunteufel 2 , J. Pekala 1 , J. Rautenberg 4 , M. Riegel 2 , M. Roth 2 , F. Salamida 5 , H. Schieler 2 , R. ˇ Sm'ıda 2 , M. Unger 2 , M. Weber 2 , F. Werner 2 , H. Wilczy'nski 1 , and J. Wochele 2 Abstract. We investigate the possibility of detecting extensive air showers by the radar technique. Considering a bistatic radar system and di ff erent shower geometries, we simulate reflection of radio waves o ff the static plasma produced by the shower in the air. Using the Thomson cross-section for radio wave reflection, we obtain the time evolution of the signal received by the antennas. The frequency upshift of the radar echo and the power received are studied to verify the feasibility of the radar detection technique.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Traditional techniques of extensive air shower (EAS) detection include counting the shower particles at the ground level and measuring fluorescence light from the excited nitrogen in the atmosphere. Detecting radio emission from the shower particles is a promising new technique that is currently in development. An alternative method is the radar technique, where a ground-based radio transmitter illuminates the ionization trail left behind the shower front and another ground-based antenna receives the scattered radio signal with its upshifted frequency. The concept of implementing a radar for cosmic ray detection was first introduced about 70 years ago by Blackett and Lovell [1]. However due to the negative results of dedicated experiments, this concept was forgotten for several decades. In recent years renewed attention has been attracted to this topic, see e.g. [2,3,4,5]. Experimental e ff orts to detect EAS using the radar technique were made by several groups, e.g. [6], the LAAS group [7], [8], the MARIACHI experiment [9], or the TARA project [10], however no detection was confirmed so far. In this paper we investigate the possibility of detecting EAS by the bistatic radar system. Especially, we are interested in verifying the feasibility of supplementing the CROME (Cosmic-Ray Observation via Microwave Emission) detector [11,12] with a radio transmitter and using its existing antennas as receivers for the radar echo. The CROME antenna setup consists of several microwave receivers for the frequency ranges 1 . 2 -1 . 7 (L band), 3 . 4 -4 . 2 (extended C band), and 10 . 7 -11 . 7 GHz (low Ku band). Since we expect large frequency upshifts of the radar echo and the CROME antennas are fine tuned to the GHz frequency band, we can narrow down our interest to the frequency range of 1 -100 MHz of the emitted radio waves. The concept of detecting EAS using radar technique is based on the principle of scattering radio waves o ff the static plasma produced in the atmosphere by the energetic particles of the shower. The locally produced plasma decays exponentially with the lifetime τ . According to Vidmar [13], for the",
"pages": [
1
]
},
{
"title": "EPJ Web of Conferences",
"content": "Figure 2 shows the time evolution of the region of integration v ol ( t ) (see equation (2)) for a shower with an energy of 10 18 eV heading towards transmitter (x = 0 m, y = 0 m). The receiver (x = 200 m, y = 0 m) is placed 200 m away from the transmitter. The colors scale with the altitude. As we can see, the volume over which we integrate the signal spans usually over the wide range of altitudes. This implies a high variation of the air density. Therefore, the refractive index and the Cherenkov angle can vary significantly over the considered region. This makes the whole analysis very complex.",
"pages": [
4
]
},
{
"title": "2 Radar reflection",
"content": "A schematic diagram representing a bistatic radar system is shown in Figure 1. A ground-based radio transmitter (T) irradiates a disk-like shower. The front edge of the disk is an ionization front moving towards the ground with the speed of light and leaving created static plasma behind it. The radio signal is scattered by the free electrons in the ionization trail and subsequently received by the groundbased antenna (R). The geometry of the bistatic radar system is set up by the polar coordinates of the transmitter and the receiver, i.e. by the distances from the shower core to the transmitter ( dT ) and to the receiver ( dR ) together with the angles ϕ t and ϕ r . Let us consider the radar reflection from the disk element with coordinates ( rL , ϕ ), altitude h, and electron density ne . Its contribution to the radar echo at the receiver at time t can be expressed by where GT is the transmitter antenna gain, UT [V] is the magnitude of the transmitted field, k and ksc are the wave vectors of incoming and scattered radio wave, ϕ 0 is the initial phase of the emitted signal, d σ T / d Ω is the Thomson cross-section, s is the projection of the distance between the shower core and the considered disk element on the shower axis, α is the inclination angle of the reflected radio wave, and n is the refractive index of the air. By analogy to the setup of the CROME detector, we assume that the receiver is oriented vertically upwards. The signal received by the antenna at a given time t is sum of the signals scattered at di ff erent times and from di ff erent parts of the plasma disk. These individual contributions interfere with each other and only an integral over the whole volume v ol ( t ) from which they arrive simultaneously gives us the correct value. The total electric field strength of the radio wave at the receiver is given by Note that time t is defined in such a way that t = 0 coincides with the moment at which the shower hits the ground.",
"pages": [
2,
3
]
},
{
"title": "3 Frequency upshift",
"content": "Despite the fact that the radio wave is scattered on the static plasma, the region in which the plasma is created moves with relativistic velocity. Therefore, we can observe the Doppler shift of the received radar echo. Figure 3 shows the upshifts fr of the radio wave scattered on di ff erent parts of the shower disk, which is heading vertically towards the transmitter. The altitude of the disk element is given by h , whereas its distance to the receiver in the horizontal plane is d . The frequency upshift depends on the wave direction and the refractive index of air. It has the highest value for the case in which the viewing angle coincides with the Cherenkov cone. The typical fr is high enough to upshift the MHz signal into the GHz range. Note that fr only weakly depends on the distance from the shower core to the transmitter dT .",
"pages": [
4
]
},
{
"title": "4 Details of the simulation",
"content": "The main assumptions and approximations are the following:",
"pages": [
4
]
},
{
"title": "5 Power received by the detector antenna",
"content": "The ratio of the 'instantaneous' power PR ( t ) received by the detector antenna to the power emitted by the transmitter PT is given by where The term R ( t ) is proportional to the electric field strength detected by the receiver. It can be used in the Fourier analysis to obtain the power spectrum of the recorded signal. The 'real' power received by the detector PR can be obtained by averaging R 2 ( t ) (according to the time resolution of the detector). The ratio PR / PT ∼ GTAR .",
"pages": [
5
]
},
{
"title": "6 Results",
"content": "Figure 4 shows the waveforms R(t) of the radar echoes for di ff erent frequencies ν of the emitted radio wave and di ff erent shower core-receiver distances dR . In all cases the receiver is outside the Cherenkov cone (which is the most common case) and vertical showers heading towards the transmitter are considered. The time t = 0 coincides with the moment at which the shower hits the ground. As we can see, the frequency of the received signal is higher than that of the emitted one. Moreover, the amplitude grows with time, which is the geometrical e ff ect of the reflections from the lower parts of the atmosphere. For showers, where the receiver is inside the Cherenkov cone, the time sequence is reversed and the lower part of the shower is seen first. Therefore, the amplitude decreases with time. However, this is the case only for very inclined showers with cores up to several meters away from the receiver. The ratios of the power received by the detector to the emitted one ( PR / PT ) for vertical showers are given in table 1. It is evident that the strength of the signal decreases with the frequency ν . It is simply because the size of the region from which we get the coherent signal decreases with the wavelength and the negative interference cancels out the signal.",
"pages": [
5
]
},
{
"title": "7 Conclusions",
"content": "The typical ratio of the received to the emitted power is in the range of 10 -10 - 10 -13 , which makes the detection of the radar reflection plausible. The C band ( ∼ 3 GHz) is particularly well-suited for this purpose because of very low noise ( < 10 K) in this frequency range. The CROME antennas supplemented with commercial high power MHz transmitter would create a radar system dedicated to EAS detection. A note should be added, that the shown time traces are for an infinite bandwidth detector -a realistic detector would only be able to detect the signal in narrow frequency range. Moreover, it will only see the shower for a certain fraction of its development. The performed analysis will help to choose the frequency of the emitted wave to optimize the detection of the radar echo. The work is in progress.",
"pages": [
6
]
},
{
"title": "8 Acknowledgments",
"content": "It is our pleasure to acknowledge the interaction and collaboration with many colleagues from the KASCADE-Grande and Pierre Auger Collaboration. This work has been supported in part by the KIT start-up grant 2066995641,the ASPERA project BMBF 05A11VKA, the Helmholtz-University Young Investigators Group VH-NG-413 and the National Centre for Research and Development (NCBiR).",
"pages": [
6
]
}
]
2013EPSC....8..507W
https://arxiv.org/pdf/1302.6425.pdf
<document>
<section_header_level_1><location><page_1><loc_7><loc_80><loc_63><loc_84></location>Photometric stability analysis of the Exoplanet Characterisation Observatory</section_header_level_1>
<text><location><page_1><loc_7><loc_75><loc_86><loc_77></location>I. P. Waldmann, 1 glyph[star] E. Pascale, 2 B. Swinyard, 1 , 3 G. Tinetti, 1 A. Amaral-Rogers 2</text>
<text><location><page_1><loc_73><loc_73><loc_74><loc_75></location>7</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_73><loc_73><loc_75></location>L. Spencer, 2 M. Tessenyi, 1 M. Ollivier 5 and V. Coud'e du Foresto 6 ,</list_item>
<list_item><location><page_1><loc_7><loc_72><loc_68><loc_73></location>1 Dept. Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK</list_item>
<list_item><location><page_1><loc_7><loc_70><loc_56><loc_71></location>2 School of Physics & Astronomy, Cardiff University, Cardiff, CF24 3AA, UK</list_item>
<list_item><location><page_1><loc_7><loc_69><loc_58><loc_70></location>3 STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, OX11 0QX, UK</list_item>
<list_item><location><page_1><loc_7><loc_68><loc_56><loc_69></location>4 Dept. Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK</list_item>
<list_item><location><page_1><loc_7><loc_67><loc_71><loc_68></location>5 Institut d'Astrophysique Spatiale, Btiment 121, Universit'e de Paris-Sud, 91405 ORSAY Cedex, France</list_item>
<list_item><location><page_1><loc_7><loc_65><loc_58><loc_66></location>6 Observatoire de Paris (LESIA), 5 place Jules Janssen, F-92190 Meudon, France</list_item>
<list_item><location><page_1><loc_7><loc_64><loc_69><loc_65></location>7 Center for Space and Habitability, University of Bern, Sidlerstrasse 5, CH-3012, Bern, Switzerland</list_item>
</unordered_list>
<section_header_level_1><location><page_1><loc_28><loc_57><loc_38><loc_58></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_28><loc_46><loc_89><loc_56></location>Photometric stability is a key requirement for time-resolved spectroscopic observations of transiting extrasolar planets. In the context of the Exoplanet Characterisation Observatory ( EChO ) mission design, we here present and investigate means of translating space-craft pointing instabilities as well as temperature fluctuation of its optical chain into an overall error budget of the exoplanetary spectrum to be retrieved. Given the instrument specifications as of date, we investigate the magnitudes of these photometric instabilities in the context of simulated observations of the exoplanet HD189733b secondary eclipse.</text>
<text><location><page_1><loc_28><loc_42><loc_89><loc_44></location>Key words: space vehicles: instruments - instrumentation: spectrographs - techniques: spectroscopic - stars:planetary systems</text>
<section_header_level_1><location><page_1><loc_7><loc_36><loc_24><loc_37></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_16><loc_46><loc_35></location>The last decade has seen a surge in exoplanetary discoveries, with ∼ 850 planets confirmed (Schneider et al. 2011) and over two thousand Kepler candidates (Borucki et al. 2011; Batalha et al. 2013) waiting for confirmation. With the precise measurements of their masses and radii we have gained a staggering wealth of information for a plethora of targets and planetary types. In this large and growing consensus of foreign worlds, some afford us the opportunity of further characterisation. By studying the exoplanets' atmospheres, we can not only infer their chemical make-up but also constrain their climates, thermodynamical processes and formation histories. The sum total of this knowledge will allow us to understand planetary science and our own solar system in the context of a much larger picture.</text>
<text><location><page_1><loc_7><loc_5><loc_46><loc_15></location>Using Hubble , Spitzer as well as ground based facilities, the success of exoplanetary spectroscopy of transiting (e.g. Beaulieu et al. 2010, 2011; Berta et al. 2012; Charbonneau et al. 2002, 2008; Deming et al. 2005, 2007; Grillmair et al. 2007, 2008; Richardson et al. 2007; Snellen et al. 2008, 2010a,b; Bean et al. 2011; Stevenson et al. 2010; Swain et al. 2008, 2009a,b; Tinetti et al. 2007a,b, 2010a; Thatte et al. 2010; Mandell et al. 2011; Sing et al. 2009, 2011; Pont</text>
<text><location><page_1><loc_50><loc_30><loc_89><loc_37></location>et al. 2008; Burke et al. 2007, 2010; D'esert et al. 2011; Redfield et al. 2008; Waldmann et al. 2012, 2013; Crouzet et al. 2012; Brogi et al. 2012) as well as non-transiting planets (e.g. Janson et al. 2010; Currie et al. 2011) has been remarkable in recent years.</text>
<text><location><page_1><loc_50><loc_1><loc_89><loc_30></location>Following these recent successes and in the frame of ESA's Cosmic Vision programme, the Exoplanet Characterisation Observatory ( EChO ) has been considered as mediumsized M3 mission candidate for launch in the 2022 - 2024 timeframe (Tinetti et al. 2012b). The current 'Phase-A study' space-mission concept is a 1.2 metre class telescope, passively cooled to ∼ 50 K and orbiting around the second Lagrangian Point (L2). The current baseline for the payload consists of four integrated spectrographs providing continuous spectral coverage from 0.5 - 16 µ m at resolution ranging from R ∼ 300 to 30. For a detailed description of the telescope and current payload design studied by our instrument consortium, we refer the reader to the literature (Puig et al. 2012; Tinetti et al. 2012a,b; Swinyard et al. 2012; Eccleston et al. 2012; Reess et al. 2012; Adriani et al. 2012; Zapata et al. 2012; Pascale et al. 2012; Focardi et al. 2012; Tessenyi et al. 2012). In order to observe the spectrum of the extrasolar planet, the EChO mission uses the time-resolved spectrophotometry method to observe transiting exoplanets. The spectrum of the planet is either seen in transmission when the planet transits in front of the star</text>
<text><location><page_2><loc_7><loc_83><loc_46><loc_87></location>along our line of sight or in emission when the planet's thermal contribution is lost during secondary eclipse when the planet disappears behind its host.</text>
<text><location><page_2><loc_7><loc_53><loc_46><loc_83></location>By taking consecutive short observations we follow the transit or secondary eclipse event of an extrasolar planet. This approach results in an exoplanetary lightcurve for each individual wavelength channel. The spectrum of the planet is then derived by model fitting said lightcurves and recording the eclipse depths at each wavelength bin. The amplitudes of these modulations on the mean of the exoplanetary lightcurve depths are unsurprisingly small. Taking the case of the secondary eclipse emission spectrum we typically find the planetary contrast to be of the order of 10 -3 for hotJuptiers in an orbit around a typical K0 star, and a much less favourable 10 -5 contrast for so called 'temperate SuperEarths' orbiting M-dwarf stars (Tessenyi et al. 2012). A mission such as EChO hence needs to provide a sensitivity that either matches or exceeds these contrasts over the duration of a transit/eclipse observation lasting one (e.g. GJ1214b, Charbonneau et al. 2009) to tens of hours (e.g. HD80606b, Fossey et al. 2009). The required instrument stability over said time spans can be achieved by providing a high photometric stability. There are three main factors that may introduce photometric variability over time and may limit the photometric stability. These are:</text>
<section_header_level_1><location><page_2><loc_7><loc_50><loc_31><loc_51></location>(i) Pointing stability of the telescope:</section_header_level_1>
<text><location><page_2><loc_7><loc_25><loc_46><loc_50></location>The 1 σ pointing jitter of the satellite is currently baselined to be of the order of 10 milli-arcsec from 90s to 10h of continuous observation 1 . Effectively this defines the maximum Performance Reproducibility Error (PRE) over ten hours, specifying the reproducibility of the experiment. These pointing drifts manifest themselves in the observed data product via two mechanisms: 1) the drifting of the spectrum along the spectral axis of the detector, from here on referred to as 'spectral jitter'; 2) the drift of the spectrum along the spatial direction (or 'spatial jitter'). The effect of pointing jitter on the observed time series manifests as nonGaussian noise correlated among all detectors in all focal planes of the payload and is characterised by the powerspectrum of the telescope pointing. The amplitude of the resultant photometric scatter depends on the pointing jitter power spectrum, the PSF of the instruments, the detector intra-pixel response and the amplitude of the inter- pixel variations.</text>
<text><location><page_2><loc_7><loc_24><loc_43><loc_25></location>(ii) Thermal stability of the optical-bench and mirrors:</text>
<text><location><page_2><loc_7><loc_6><loc_46><loc_24></location>Thermal emission of the instrument, baselined at 45K, is a source of photon noise and does hence not directly contribute to the photometric stability budge beyond an achievable signal to noise ratio (SNR) of the observation. However fluctuations in thermal emissions constitute a source of correlated noise in the observations and need to be maintained at amplitudes small compared to the science signal observed. Given a ∼ 45K black-body peaks in the far-IR, we are dominated by the Wien tail of the black body distribution, resulting in steep temperature gradients and stringent requirements on thermal stability in the long wavelength instruments of the EChO payload. Additionally to variable photon noise contribution, thermal variations may impact</text>
<text><location><page_2><loc_50><loc_85><loc_89><loc_87></location>dark currents and responsivities of the detector which must be taken into account.</text>
<text><location><page_2><loc_50><loc_83><loc_84><loc_84></location>(iii) Stellar noise and other temporal noise sources:</text>
<text><location><page_2><loc_50><loc_72><loc_89><loc_83></location>Whilst beyond the control of the instrument design, stellar noise is an important source of temporal instability in exoplanetary time series measurements (Ballerini et al. 2011). This is particularly true for M dwarf host stars as well as many non-main sequence stars. Correction mechanisms of said fluctuations must and will be an integral part of the EChO science study but goes beyond the instrument photometric stability discussion presented here.</text>
<text><location><page_2><loc_50><loc_67><loc_89><loc_71></location>Along with the achievable SNR, the photometric stability of the instrument is the deciding factor for the success of missions or facilities aiming at transit spectroscopy.</text>
<text><location><page_2><loc_50><loc_63><loc_89><loc_67></location>In this paper we study the effects of the spatial/spectral jitter and thermal variability on the photometric stability budget of the instrument and telescope.</text>
<section_header_level_1><location><page_2><loc_50><loc_59><loc_71><loc_60></location>2 ECHO AND ECHOSIM</section_header_level_1>
<text><location><page_2><loc_50><loc_47><loc_89><loc_58></location>Given EChO is currently in its Phase-A study phase, we concede that a noise budget and stability analysis, such as this one, can only be preliminary. Here and in Pascale et al. (in prep.), we present methodology used for the testing and optimisation of the current instrument design. The study presented in this paper draws on the end-to-end mission simulator EChOSim , which is discussed in detail in Pascale et al. (in prep.).</text>
<text><location><page_2><loc_50><loc_37><loc_89><loc_47></location>In the simulations of this paper we assume EChO to have a 1.2 metre diameter primary mirror off-axis telescope. The light beam is simultaneously fed into five spectrographs via dichroic beam splitters: Visible (Vis), short wavelength IR (SWIR), mid-wavelength IR (MWIR1 and MWIR2) and long-wavelength IR (LWIR) instruments covering the spectral range from 0.5 - 16 µ m.</text>
<text><location><page_2><loc_50><loc_12><loc_89><loc_37></location>In order to provide an end-to-end simulation of the observation, EChOSim incorporates a full simulation of the science payload including the telescope, as well as the astrophysical scene including Zodi emissions. EChOSim uses realistic mirror reflectivities and estimates the instrument transmission function for each channel as a function of wavelength. This includes transmission, optical throughput and spatial modulation transfer function. Using tabulated emissivity values as a function of wavelength, EChOSim also estimates the thermal emission spectrum of the several optical elements of the telescope. The transmission through the dichroic chain is simulated, and the incoming radiation is split among the 5 instruments assuming realistic transmittance and reflectivity data (Pascale et al., in prep.). Dispersion and detection by the focal plane array are simulated. The dispersion of the light over the focal plane is modelled by a linear dispersion law, which is related to the sampled spectral resolving power R .</text>
<text><location><page_2><loc_50><loc_1><loc_89><loc_12></location>The full focal plane illumination due to the backgrounds and the science signal's point spread function (PSF) is calculated and convolved with realistic intra-pixel variations (see section 5). Photon noise, read noise and pointing jitter noise are calculated on a pixel by pixel basis and time series of the observation are generated. These time series are then analysed and model fitted using Mandel & Agol (2002) analytical solutions and an adaptive Metropolis-Hastings Markov</text>
<figure>
<location><page_3><loc_8><loc_70><loc_43><loc_86></location>
<caption>Figure 1. Secondary eclipse lightcurve of a hot-Jupiter type exoplanet with eclipse duration of 720min (Mandel & Agol 2002). Noise at 10 -4 level was added.</caption>
</figure>
<figure>
<location><page_3><loc_8><loc_48><loc_42><loc_62></location>
<caption>Figure 2. Time series of 6 orbits of a hot-Jupiter (akin to HD189733b, Torres et al. 2008). The deep troughs are limbdarkened transits (Mandel & Agol 2002; Claret 2000), smaller troughs are secondary eclipses and sinusoidal variations are due to the planetary phase curve as the planetary day-side rotates in and out of view. White noise of the level of 10 -4 was added.</caption>
</figure>
<text><location><page_3><loc_7><loc_32><loc_46><loc_37></location>Chain Monte Carlo (MCMC) algorithm (Haario et al. 2001, 2006; Hastings 1970). For a more detailed list of the instrument parameters assumed in these simulations see table 1 and Pascale et al. (in prep.).</text>
<section_header_level_1><location><page_3><loc_7><loc_27><loc_40><loc_28></location>3 FREQUENCY BANDS OF INTEREST</section_header_level_1>
<text><location><page_3><loc_7><loc_24><loc_46><loc_26></location>Figure 1 shows an example of a secondary eclipse of a typical hot-Jupiter planet. Figure 2 shows the signal observed</text>
<figure>
<location><page_3><loc_10><loc_8><loc_42><loc_21></location>
<caption>Figure 3. Power spectra of time series shown in figure 2 for different orbital periods. Blue: Period = 120 days, Green = 2.21 days (akin to HD189733b), Red = 0.4 days. The sensitive frequency range extends from 1.9x10 -4 - 1.7x10 -3 Hz.</caption>
</figure>
<text><location><page_3><loc_50><loc_55><loc_89><loc_87></location>by EChO over the duration of 6 planetary orbits of a hotJupiter. From these figures it can easily be seen that timecorrelated noise has the greatest impact on the retrieved science at temporal variation frequencies comparable to those of the transit/eclipse event, or multiples thereof. Figure 3 shows the frequency domain representation of Figure 2 given a variety of orbital periods. Here the desired signal is contained in discrete frequencies and their respective overtones. Frequency ranges beyond these shown in Figure 3 can safely be filtered out using pass-band filters, or normalisation using low order polynomials in the time domain, without impairing the transit morphology. Given the range of transit periods observed and the goal of accurate ingress and egress mapping, we find the 'crucial frequency band' to be from 1.9x10 -4 to 1.7x10 -3 Hz, outside of which slow moving trends and high-frequency noise can effectively be filtered. This approach of slow moving trend removal is well tested for Kepler data (e.g. Gilliland et al. 2010). The overall critical frequency band for EChO is determined by the longest observation expected and the need to Nyquist sample the highest expected frequencies. We hence limit our simulations of pointing jitter or temperature fluctuations to this frequency range of interest.</text>
<section_header_level_1><location><page_3><loc_50><loc_51><loc_69><loc_52></location>4 SPECTRAL JITTER</section_header_level_1>
<text><location><page_3><loc_50><loc_27><loc_89><loc_50></location>As described in the introduction, the telescope pointing jitter has the effect of translating the spectrum on the detector along both, the spatial and spectral directions. Whilst the real translation is a combination of these orthogonal components, we will consider the effect of both these translations independently of each other. With the observing approach of taking consecutive spectra over a given length of time, we obtain a flux time series for each pixel or resolution element ∆ λ . Should, during the course of the observation, the stellar spectra be shifted along the detector the same pixel will not observe the same stellar flux but that corresponding to the shifted stellar spectrum. If we are to construct a time series for each resolution element of the detector, we consequently need to re-sample the stellar spectra to a common grid. The ability to re-sample the individual stellar spectra to a uniform wavelength grid depends on how well we can determine these relative shifts.</text>
<text><location><page_3><loc_50><loc_11><loc_89><loc_26></location>In this section we will simulate a series of consecutively observed stellar spectra and shift each one of them according to a pointing jitter distribution derived from the Herschel Space Telescope. We will fit stellar absorption lines of these spectra and use the derived centroids to determine the accuracy with which we can determine the spectral drift. This is translated into a final post-correction flux error. It is worth noting that inter and intra pixel variations are less critical in this case as the the Nyquist sampling adopted at instrument level spreads each resolution element over at least two adjacent pixels.</text>
<section_header_level_1><location><page_3><loc_50><loc_6><loc_86><loc_9></location>4.1 Synthesising pointing jitter from Herschel observations</section_header_level_1>
<text><location><page_3><loc_50><loc_1><loc_89><loc_5></location>In order to simulate the pointing jitter of the EChO mission as realistically as possible, we synthesised the pointing patterns of the Herschel Space Telescope for which real pointing</text>
<table>
<location><page_4><loc_12><loc_69><loc_84><loc_84></location>
<caption>Table 1. Simulation parameters for EChOSim</caption>
</table>
<figure>
<location><page_4><loc_7><loc_40><loc_46><loc_66></location>
<caption>Figure 4. Herschel pointing over a three hour continuous observation.</caption>
</figure>
<text><location><page_4><loc_7><loc_25><loc_46><loc_32></location>information exists (Swinyard private communication). Figure 4 shows Herschel pointings recorded for a three hour time period with a sampling frequency of ∼ 1 second. The pointing jitter distance for Herschel, ∆ Her , is given by the Pythagorean argument relative to their means:</text>
<formula><location><page_4><loc_19><loc_22><loc_46><loc_23></location>∆ Her = √ Ra 2 + Dec 2 . (1)</formula>
<text><location><page_4><loc_7><loc_1><loc_46><loc_20></location>Assuming the pointing jitter in Ra and Dec are normally (Gaussian) distributed, we can state that the probability distribution function (pdf) of ∆ Her , P (∆ Her ), is given by a Rayleigh distribution. The Rayleigh distribution is the pythagorean argument of two orthogonal Gaussian distributions. However, for a real system such as Herschel, the pointing jitter is described by a Gaussian and a non-Gaussian component. These non-Gaussian components propagate to P (∆ Her ) in the form of an increased skew and broader wings than those of a pure Rayleigh distribution. We can approximate P (∆ Her ) using the more general Weibull distribution of which the Rayleigh distribution is a special case. The Weibull distribution interpolates between a Rayleigh and an Exponential distribution and is hence ideally suited to de-</text>
<figure>
<location><page_4><loc_51><loc_53><loc_86><loc_67></location>
<caption>Figure 5. Probability distribution function of the pointing jitter length of Herschel</caption>
</figure>
<text><location><page_4><loc_50><loc_44><loc_89><loc_47></location>he broader wings and skew of the Herschel pointing jitter pdf. The Weibull distribution is given by</text>
<formula><location><page_4><loc_52><loc_39><loc_89><loc_42></location>P wbl ( x ; τ, κ ) = { τ κ ( x κ ) κ -1 e -( x/τ ) κ x glyph[greaterorequalslant] 0 , 0 x < 0 , (2)</formula>
<text><location><page_4><loc_50><loc_32><loc_89><loc_38></location>where τ is known as the amplitude coefficient and κ as the distribution shape coefficient. Equation 2 reduces to an Exponential and a Rayleigh distribution for κ = 1 and κ = 2 respectively. The mean and the variance of the Weibull distribution are given by:</text>
<formula><location><page_4><loc_63><loc_28><loc_89><loc_29></location>µ wbl = τ Γ(1 + 1 /τ ) (3)</formula>
<formula><location><page_4><loc_61><loc_23><loc_89><loc_25></location>σ 2 wbl = τ 2 Γ(1 + 2 /τ ) -µ 2 (4)</formula>
<text><location><page_4><loc_50><loc_21><loc_84><loc_22></location>where Γ is the gamma function (Riley et al. 2002).</text>
<section_header_level_1><location><page_4><loc_50><loc_17><loc_82><loc_18></location>4.1.1 Scaling the Herschel pointing distribution</section_header_level_1>
<text><location><page_4><loc_50><loc_4><loc_89><loc_16></location>Figure 5 shows P (∆ Her ) (blue) obtained from the pointing information in figure 4 and the best fitting Weibull distribution, P wbl ( · ), with τ = 247 . 71 and κ = 1 . 6547. The mean of the distribution is µ = 221 . 45 mas. In order to obtain the scaled distribution to a jitter amplitude of 10 mas predicted for EChO, we maintain the shape parameter at κ = 1 . 6547 and re-derive the scaling parameter, τ , for µ = 10 mas. This yields the pointing jitter distribution P (∆ Echo ) shown in figure 6.</text>
<text><location><page_4><loc_53><loc_1><loc_89><loc_2></location>We now randomly sample from P (∆ Echo ) to obtain</text>
<figure>
<location><page_5><loc_9><loc_73><loc_42><loc_86></location>
<caption>Figure 6. Synthesised probability distribution for EChO pointing jitter.</caption>
</figure>
<figure>
<location><page_5><loc_9><loc_48><loc_45><loc_66></location>
<caption>Figure 7. Stellar spectrum between 1.0 and 2.5 µ m. Red: down sampled spectrum at R = 300; Blue: interpolated spectrum (see step 4); Inset: spectral line used to fit for the wavelength jitter; G1: flux gradient of stellar 'black-body' at 1.29 - 1.40 µ m; G2: flux gradient at 1.46 - 1.64 µ m.</caption>
</figure>
<text><location><page_5><loc_7><loc_28><loc_46><loc_37></location>∆ Echo for M number of observations in our simulated observing run. Assuming a resolving power of R = 300 for the NIR channel (1.0-2.5 µ m), we can calculate the spectral resolving power to be 0.0159nm/mas. Hence, we can express ∆ Echo as function of spectral wavelength drift and from hence forth denote the EChO jitter as ∆ m ( λ ), where m is the m'th spectrum observed.</text>
<section_header_level_1><location><page_5><loc_7><loc_24><loc_43><loc_25></location>4.2 Applying pointing jitter to stellar spectra</section_header_level_1>
<text><location><page_5><loc_7><loc_21><loc_46><loc_23></location>Using the Phoenix 2 code we generated a stellar spectrum of a sun analogue and proceeded with the following steps:</text>
<unordered_list>
<list_item><location><page_5><loc_7><loc_13><loc_46><loc_20></location>(i) The input spectrum was trimmed to a wavelength range of 1.0 - 2.5 µ m and re-sampled to a resolution of R = 300 at a central wavelength λ c = 1 . 75 µ m. This yields a wavelength coverage of δλ = 5 . 8 nm per pixel. We denote the spectrum by F star ( λ ), where λ is the wavelength.</list_item>
</unordered_list>
<text><location><page_5><loc_7><loc_8><loc_46><loc_13></location>(ii) A single stellar absorption line was now selected (wavelength range: 1.19 - 1.21 µ m), and the spectrum consequently trimmed. We denote the trimmed spectrum as F line ( λ ) encompassing N data points in λ .</text>
<unordered_list>
<list_item><location><page_5><loc_7><loc_5><loc_46><loc_7></location>(iii) Normally distributed random noise was added to the spectrum with the signal-to-noise (SNR) chosen to be the</list_item>
</unordered_list>
<text><location><page_5><loc_50><loc_85><loc_89><loc_87></location>fraction of the maximal line amplitude over the standard deviation of the Gaussian noise component</text>
<formula><location><page_5><loc_61><loc_80><loc_89><loc_82></location>SNR = max | ( F line ( λ )) | σ gauss (5)</formula>
<text><location><page_5><loc_50><loc_73><loc_89><loc_78></location>The resulting spectrum, F final ( λ ), is hence F final ( λ ) = F line ( λ )+ N ( σ, µ ), where N ( σ, µ ) is the Gaussian noise with σ variance and µ = 0 mean. We chose a SNR of 500 for this study.</text>
<unordered_list>
<list_item><location><page_5><loc_50><loc_65><loc_89><loc_73></location>(iv) The spectrum F final ( λ ) was now interpolated by a factor S using a 'piecewise cubic Hermite interpolation' (Press et al. 2007) which assures the best fit to the original time series. Here we chose S = 100. This step is required to numerically implement sub-pixel drifts and does not impair or bias the results.</list_item>
<list_item><location><page_5><loc_50><loc_57><loc_89><loc_65></location>(v) Steps iii & iv were repeated M times to create the M × N dimensional matrix X , where N is the number of points in λ and M was taken to be 100. One can think of M being the time axis containing spectra F final (1 , λ ) , F final (2 , λ ) . . . F final ( M,λ ), and N being the spectral axis of the matrix X .</list_item>
<list_item><location><page_5><loc_50><loc_53><loc_89><loc_56></location>(vi) Each spectrum in X was now shifted by ∆ m ( λ ) along the wavelength axis randomly towards either the blue or red part of the spectrum.</list_item>
</unordered_list>
<section_header_level_1><location><page_5><loc_50><loc_48><loc_72><loc_49></location>4.3 Fitting 'jittered' spectra</section_header_level_1>
<unordered_list>
<list_item><location><page_5><loc_50><loc_42><loc_89><loc_47></location>(i) Now, each spectrum F final ( m,λ ) was fitted with a Voigt profile using a Levenberg-Marquardt minimisation algorithm (Press et al. 2007) and the resulting centroids, C m ( λ ), were recorded.</list_item>
<list_item><location><page_5><loc_50><loc_39><loc_89><loc_41></location>(ii) The recorded centroids were subtracted from the motion jitter to give the fitting residual</list_item>
<list_item><location><page_5><loc_50><loc_36><loc_89><loc_39></location>R m ( λ ) = √ (∆ m ( λ ) -C m ( λ )) 2 and the standard deviation of the residual σ R , which was recorded.</list_item>
<list_item><location><page_5><loc_50><loc_33><loc_89><loc_36></location>(iii) Steps i to ii were repeated 100 times to obtain the sampled probability distribution of P( σ R ).</list_item>
</unordered_list>
<section_header_level_1><location><page_5><loc_50><loc_28><loc_89><loc_29></location>4.4 Translating fitting residuals to total flux error</section_header_level_1>
<text><location><page_5><loc_50><loc_24><loc_89><loc_27></location>We can now use the fitting residual R m ( λ ) to calculate the observed flux, F obs ( m,λ ), per wavelength range δλ and time step m :</text>
<formula><location><page_5><loc_56><loc_17><loc_89><loc_21></location>F obs ( m,λ ) = λ + δλ + R m ( λ ) ∫ λ + R m ( λ ) F final ( m,λ )d λ (6)</formula>
<text><location><page_5><loc_50><loc_10><loc_89><loc_16></location>above we can see that R m ( λ ) constitutes a change in the integration interval of each pixel with respect to the stellar spectrum. For the case where δλ → 0, we can calculate F obs ( m,λ ) using the linear approximation</text>
<formula><location><page_5><loc_53><loc_7><loc_89><loc_8></location>F obs ( m,λ ) = G ( R m ( λ ) -¯ R m ( λ )) + F final ( m,λ ) (7)</formula>
<formula><location><page_5><loc_64><loc_1><loc_89><loc_3></location>G = d F star ( λ ) d λ (8)</formula>
<figure>
<location><page_6><loc_7><loc_63><loc_45><loc_86></location>
<caption>Figure 8. Snapshot of EChOSim output for the MWIR detector. Top: Focal plane illumination including the science spectrum, instrumental and astrophysical backgrounds. Bottom: Plot of the total and individually contributing fluxes.</caption>
</figure>
<text><location><page_6><loc_7><loc_47><loc_46><loc_54></location>where ¯ R m ( λ ) is the mean of the fitting residual and was subtracted to account for equal amounts of positive and negative drifts, and G is the local gradient of the stellar spectrum. We can now express the resulting flux error due to residual drifts along the spectral direction as</text>
<formula><location><page_6><loc_16><loc_43><loc_46><loc_45></location>F err ( m,λ ) = G ( R m ( λ ) -¯ R m ( λ )) (9)</formula>
<text><location><page_6><loc_7><loc_40><loc_46><loc_42></location>In section 8 we show the residual flux error F err for the gradients G1 and G2 shown in figure 7.</text>
<section_header_level_1><location><page_6><loc_7><loc_34><loc_24><loc_35></location>5 SPATIAL JITTER</section_header_level_1>
<text><location><page_6><loc_7><loc_5><loc_46><loc_33></location>We have investigated the effect of spatial jitter as well as thermal variability. Whereas inter/intra pixel variations can largely be ignored in the case of spectral jitter, the detector pixel responses and variations can play a significant role in the computation of the spatial jitter which is caused by the movement of the spectral PSF along the spatial direction of the detector. EChOSim fully simulates the effect of inter and intra pixel variations on a spatially resolved detector grid. We here outline the spatial jitter calculations but refer the reader to Pascale et al. (in prep.) for a more detailed discussion. The dispersed signals are sampled by each detector assuming a wavelength-depended and instrument specific PSF which is convolved with the intra-pixel dependent response of the detector. Figure 8 is a snapshot of a standard diagnostic diagram returned by EChOSim . The top panel shows the focal plane illumination of the detector array. The bottom panel shows the individual flux contributions: astrophysical ( Q point ), zodiacal light ( Q zodi ), instrument thermal emissions ( Q optics ), and their combined total ( Q tot ).</text>
<text><location><page_6><loc_7><loc_1><loc_46><loc_5></location>We can describe the focal plane illumination of a detector with a monochromatic point source using the diffraction PSF pattern (Marc Ferlet priv. com.; Pascale et al. in prep.)</text>
<formula><location><page_6><loc_57><loc_82><loc_89><loc_85></location>p ( x, y, λ ) = 1 2 πσ x σ y e -y 2 2 σ 2 y e -[ x -x 0 ( λ )] 2 2 σ 2 x (10)</formula>
<formula><location><page_6><loc_56><loc_78><loc_89><loc_80></location>σ x = F # λ π √ 2 /K x σ y = F # λ π √ 2 /K y (11)</formula>
<text><location><page_6><loc_50><loc_71><loc_89><loc_77></location>where x and y are detector coordinates along the spectral and spatial axes respectively, K x and K y are parameters accounting for spatial aberrations (we assume no aberrations K x = K y ) and F # is the ratio between the effective focal length and the effective telescope diameter (see table 1).</text>
<text><location><page_6><loc_50><loc_68><loc_89><loc_70></location>The one dimensional PSF for the spatial direction can now be written as</text>
<formula><location><page_6><loc_61><loc_63><loc_89><loc_66></location>p y ( y, λ ) = 1 σ y √ 2 π e -y 2 2 σy . (12)</formula>
<text><location><page_6><loc_50><loc_57><loc_89><loc_62></location>Barron et al. (2007) studied the intra-pixel response of IR detectors, and their best-fit model is analytically implemented in EChOSim . The cross section of the response along the spatial axis of the detector array is hence given by:</text>
<formula><location><page_6><loc_55><loc_49><loc_89><loc_55></location>F ( y ) = arctan { tanh [ 1 2 l d ( y + ∆ pix 2 )]} (13) -arctan { tanh [ 1 2 l d ( y -∆ pix 2 )]}</formula>
<text><location><page_6><loc_50><loc_45><loc_89><loc_48></location>where l d is the diffusion length. The sampled PSF along the spatial axis is then given by the convolution</text>
<formula><location><page_6><loc_61><loc_42><loc_89><loc_43></location>p s ( y, λ ) = p y ( y, λ ) × F ( y ) (14)</formula>
<text><location><page_6><loc_50><loc_36><loc_89><loc_41></location>Finally the signal is sampled by the detector. The detector response is given by the convolution of the point source flux of the star-planet system, Q point ( λ, t ), with the spectrally dispersed PSFs</text>
<formula><location><page_6><loc_54><loc_32><loc_89><loc_33></location>Q point ( i, j, t ) = QE ( λ ) Q point ( λ, t ) × p s ( y, λ ) (15)</formula>
<text><location><page_6><loc_50><loc_24><loc_89><loc_31></location>where i and j are the detector pixel indices. Note that optical efficiencies don't explicitly appear in this equation as Q point ( λ, t ) already accounts the throughput budget, with the exception of the quantum efficiency, QE ( λ ) (Pascale et al., in prep.).</text>
<formula><location><page_6><loc_60><loc_20><loc_89><loc_22></location>σ p ( i, j, t ) glyph[similarequal] σ p f eff p s ∂p s ∂y Q point (16)</formula>
<text><location><page_6><loc_50><loc_7><loc_89><loc_19></location>where f eff is the plate scale (in µ m rad -1 ) and σ p is the pointing jitter (in radians per second) randomly sampled from the pointing distribution P (∆ Her ). Variations in photometric errors are estimated by consecutive runs for a range of pointing jitter amplitudes from zero to 200 milli-arcsec over a ten hour observing window. The observed spectrum is reconstructed using the EChOSim pipeline and the uncertainties on the reconstructed spectra are shown in figures 12 & 13 for two cases where PSFs of different sizes are used.</text>
<text><location><page_6><loc_50><loc_1><loc_89><loc_6></location>Given the current uncertainty over the exact nature of inter and intra-pixel variations we have not attempted to decorrelate the pointing jitter in post processing (e.g. Swain et al. 2008; Burke et al. 2010; Crouzet et al. 2012) which</text>
<text><location><page_7><loc_7><loc_83><loc_46><loc_87></location>would reduce the uncertainties reported in figures 12 & 13. We must hence consider this analysis as a conservative worst case scenario.</text>
<section_header_level_1><location><page_7><loc_7><loc_78><loc_28><loc_79></location>6 THERMAL STABILITY</section_header_level_1>
<text><location><page_7><loc_7><loc_64><loc_46><loc_77></location>We have also investigated the impact of thermal emissions and fluctuations of the optics on the photometric stability of the reconstructed spectrum. Whilst thermal emissions are not directly a problem to photometric stability (source of white noise), they pose constraints on the temperature fluctuations allowed over the time span of an exoplanetary transit/eclipse in the reddest wavelengths. Here varying thermal emissions may produce detector counts that are equivalent or exceed the signal amplitude expected from a planetary eclipse. We calculate the thermal contribution using</text>
<formula><location><page_7><loc_10><loc_56><loc_46><loc_61></location>Q thermal ( i, j, t ) = π 4 ∆ 2 p f 2 # λ i + 1 2 L ∆ p /LD ∫ λ i + 1 2 L ∆ p /LD BB λ,T d λ (17)</formula>
<text><location><page_7><loc_7><loc_32><loc_46><loc_55></location>where L is the image size of the spectrometer slit in number of pixels, LD the lateral dispersion in mm, ∆ p is the pixel size, f # is the f-number and BB is the Planck function for a given temperature T and wavelength λ . EChOSim calculates the thermal emission of the instrument optics as well as the primary, secondary and tertiary mirrors separately and coherently propagates the resulting emission to detector counts. We have explored a temperature regime ranging from 40 - 60K with a grid size of 0.2K for both, the optical bench and the mirror temperatures. This provides us with an absolute scale of the thermal contributions, see results in section 8. We have now calculated the expected signal strength of a secondary eclipse of a HD189733b like hot-Jupiter, F planet , and for a given base temperature, T 0 , calculated the temperature variation, ∆ T , required to produce a detector signal of the same strength as F planet for a given wavelength λ :</text>
<formula><location><page_7><loc_14><loc_29><loc_46><loc_30></location>∆ F planet ( λ ) = f λ ( T 0 +∆ T ) -f λ ( T 0 ) (18)</formula>
<text><location><page_7><loc_7><loc_21><loc_46><loc_27></location>where f λ ( T ) is the functional form of the thermal emission/temperature relation for a given wavelength seen in figures 14 & 15. This can either be numerically approximated by interpolation to a fine enough grid or by fitting a high-order polynomial function to the data.</text>
<section_header_level_1><location><page_7><loc_7><loc_16><loc_19><loc_17></location>7 TESTCASE</section_header_level_1>
<text><location><page_7><loc_7><loc_5><loc_46><loc_15></location>We have now calculated the spectral and spatial jitter contributions for the hot-Jupiter HD189733b (Torres et al. 2008). We currently have insufficient information on the expected thermal stability in the longest wavelength range so we do not include thermal stability in our test-case calculations. This said, as seen in section 8, the overall thermal stability is not a decisive factor for wavelengths shorter than ∼ 14 µ m.</text>
<text><location><page_7><loc_7><loc_1><loc_46><loc_5></location>For our simulations we use realistic Phoenix stellar models for the host stars. The planetary emission model is taken from Tessenyi et al. (2012). EChOSim fully solves</text>
<figure>
<location><page_7><loc_50><loc_73><loc_84><loc_87></location>
<caption>Figure 9. Residual centroid fitting error</caption>
</figure>
<figure>
<location><page_7><loc_50><loc_48><loc_84><loc_67></location>
<caption>Figure 10. Probability distribution of fitting residual standard deviation σ R</caption>
</figure>
<text><location><page_7><loc_50><loc_39><loc_89><loc_41></location>the dynamical star-planet system and computes the time resolved spectra by model fitting the generated time series.</text>
<text><location><page_7><loc_50><loc_32><loc_89><loc_38></location>We have now computed the emission spectra of HD189733b with the full noise contribution (shot noise, read noise, spatial/spectral jitter) as well as for the spatial jitter and spectral jitter only. This allows for the direct comparability of the noise contributions.</text>
<section_header_level_1><location><page_7><loc_50><loc_27><loc_60><loc_28></location>8 RESULTS</section_header_level_1>
<text><location><page_7><loc_50><loc_22><loc_89><loc_26></location>As described in section 4 the 'jittered' stellar absorption line was fitted using a Voigt profile and the centroids recorded. The residual between the real and fitted spectral shifts are</text>
<figure>
<location><page_7><loc_51><loc_7><loc_84><loc_21></location>
<caption>Figure 11. Fraction of residual fitting flux F err over the total flux of the star F star for both spectral flux gradients shown in figure 7. G1 = -760 and G2 = 24 Wm -2 nm -1 .</caption>
</figure>
<figure>
<location><page_8><loc_9><loc_74><loc_43><loc_86></location>
<caption>Figure 12. Simulation using EChOSim showing spatial jitter noise as fraction of total stellar flux, studied as a function of spectral wavelength and pointing jitter amplitude ranging from 0 200 milli-arcseconds. A PSF FWHM is assumed of 0.7 and 0.5 of the detector-pixel size. Figure 13 shows the same simulation with a PSF twice this size.</caption>
</figure>
<figure>
<location><page_8><loc_8><loc_50><loc_43><loc_62></location>
<caption>Figure 13. Simulation using EChOSim showing spatial jitter noise as fraction of total stellar flux, studied as a function of spectral wavelength and pointing jitter amplitude ranging from 0 200 milli-arcseconds. A PSF FWHM is assumed of 1.4 and 1.0 of the detector-pixel size and twice the detector pixel size as stated in table 1.</caption>
</figure>
<text><location><page_8><loc_7><loc_24><loc_46><loc_39></location>shown in figure 9. The Monte-Carlo analysis of the standard deviation of the fitting residual is shown in figure 10. As expected, the probability distribution on the parameter σ R is largely Gaussian as we randomly sample from the pointing jitter distribution P (∆ Echo ). This also shows that the centroid fitting does not introduce biases in the pointing jitter correction. The fitting residual can now be translated to a total flux error using equation 9. Taking the ratio F err /F star we can derive the relative error due to residual jitter, figure 11. The flux error is dependent on the local stellar flux gradient. From figure 7, we derived two gradients: G1 = -</text>
<figure>
<location><page_8><loc_8><loc_6><loc_41><loc_21></location>
<caption>Figure 14. Detector counts (in e-/s) due to the telescope emission, as a function of temperature and wavelength.</caption>
</figure>
<figure>
<location><page_8><loc_50><loc_71><loc_86><loc_87></location>
<caption>Figure 15. Detector counts (in e-/s) due to the optical bench emission, as a function of temperature and wavelength.</caption>
</figure>
<figure>
<location><page_8><loc_50><loc_52><loc_86><loc_64></location>
<caption>Figure 16. The temperature fluctuation required (∆ T mirror ) for a given mirror temperature to mimic the variation of counts observed by a secondary eclipse of a HD189733b like hot-Jupiter</caption>
</figure>
<text><location><page_8><loc_50><loc_38><loc_89><loc_44></location>760 Wm -2 nm -1 and G2 = +24 Wm -2 nm -1 . For these two gradients, figure 11 shows that the relative flux error lies between 10 -6 ∼ 2 × 10 -5 . These values are of course larger for the Wien tail of the stellar black body with spectral jitter errors of ∼ 10 -4 at places (see figures 18 & 19).</text>
<text><location><page_8><loc_50><loc_22><loc_89><loc_37></location>Figures 12 & 13 show the simulated spatial jitter contributions for a pointing jitter amplitude range or 0 - 200 milliarcseconds. As the photometric error resulting from spatial jitter is largely dependent on the PSF of the instrument and the pixel dimensions, we have calculated the photometric error for a PSF FWHM of 0.7 and 0.5 of the detector-pixel size (figure 12) and that for double these values (figure 13). As seen in the figures the spatial jitter is of the order of ∼ 10 -5 -10 -4 but significantly higher for the spectral ranges of the NIR instrument (2 - 5 µ m). This is to be expected as the SWIR instrument features a smaller pixel size.</text>
<text><location><page_8><loc_53><loc_21><loc_89><loc_22></location>Figures 14 & 15 show the thermal contribution of the</text>
<figure>
<location><page_8><loc_50><loc_7><loc_87><loc_19></location>
<caption>Figure 17. The temperature fluctuation required (∆ T optics ) for a given optical bench temperature to mimic the variation of counts observed by a secondary eclipse of a HD189733b like hot-Jupiter</caption>
</figure>
<figure>
<location><page_9><loc_7><loc_67><loc_48><loc_86></location>
<caption>Figure 18. Observations of a single secondary eclipse observation of HD189733b binned to 0.1 µ m bins. Inset: native resolution of the instrument.</caption>
</figure>
<figure>
<location><page_9><loc_10><loc_42><loc_45><loc_59></location>
<caption>Figure 19. Error budget per wavelength in terms of planetary contrast ratio of HD189733b for spectrum in figure 18. Red: total flux error; blue: spectral jitter, green: spatial jitter.</caption>
</figure>
<text><location><page_9><loc_7><loc_14><loc_46><loc_34></location>telescope (mirrors and optics) and the optical bench temperatures respectively. This can largely be regarded as negligible for wavelengths shortward of ∼ 14 µ m but significant at longer wavelengths and temperatures above 50K. As described in section 6, this thermal emission was translated into a temperature tolerance, ∆ T optics , showing the temperature change necessary to mimic the transit depth of a secondary eclipse feature of hot-Jupiter HD189733b at a given wavelength. These tolerances are shown in figures 16 & 17 for the telescope and optical bench temperatures respectively. We find that when the telescope and optical bench are at 45K, temperature fluctuations need to be below 6K for the telescope and 500mK for the bench in order to observe a hot Jupiter. More restrictive constraints would be required when fainter signals are involved.</text>
<text><location><page_9><loc_7><loc_1><loc_46><loc_13></location>Having investigated the pointing jitter contributions for given wavelengths and pointing jitter amplitudes, we show the simulated 'observations' of a single eclipse event of HD189733b in figure 18. The blue spectra's error-bars contain the full noise contribution. Figure 19 shows a breakdown of the individual error contributions. Here red stands for the total noise contribution (including read and shot noise), blue for the spectral jitter as calculated in section 4 and green for the spatial jitter contribution as calculated in section 5.</text>
<text><location><page_9><loc_50><loc_80><loc_89><loc_87></location>It can easily be seen that spectral jitter is important at the Wien tail of the stellar black body distribution and less so at the Rayleigh-Jeans tail. The spatial jitter noise depends on the individual detectors with the effect being strongest for the SWIR detector.</text>
<section_header_level_1><location><page_9><loc_50><loc_75><loc_63><loc_76></location>9 DISCUSSION</section_header_level_1>
<text><location><page_9><loc_50><loc_53><loc_89><loc_74></location>One way to reduce spatial jitter noise is to use pixels that are small compared to the PSF. In this case the effect of the spatial jitter is only governed by the intra-pixel response, and using many pixels to sample the PSF will washout both the inter and intra response variations. A higher sampling of the PSF using more pixels will also positively impact the spectral jitter as tracking of the stellar lines becomes easier. However, this will only be realistic if the noise from the detectors is sufficiently low but will allow a significant amount of de-correlation as the PSF centroid can be tracked across the spatial dimension of the array. The simulations take into account realistic intra-pixel responses in the spectral band from MWIR and LWIR. At shorter wavelength, the simulations currently have to be considered worst case scenarios because the system is not diffraction limited at these wavelength.</text>
<text><location><page_9><loc_50><loc_26><loc_89><loc_52></location>Figures 14 & 15 show the detected radiation in electrons/second from the instrument and telescope (separately) for a temperature range of 40 - 60K. Further studies will include additional instrumental effects such as mechanical vibrations, thermo-mechanical distortions, variable detector dark currents (assumed fixed with temperature in this study), detector responsitivity drifts and the effect of cosmic ray impacts. These studies will look at the extent the effect can be de-correlated from the timelines when additional information is available for data processing, such as temperature sensors monitoring the optics and telescope temperatures. We also plan to provide off-axis detectors to monitor the non stellar backgrounds and therefore provide a means of directly removing the background signals. Such background removal is particularly important for fainter sources. Whereas HD189733b is photon limited, observations of faint super-earths will likely be background limited and thermal and zodiacal light emissions need to be carefully accounted for.</text>
<section_header_level_1><location><page_9><loc_50><loc_21><loc_65><loc_22></location>10 CONCLUSION</section_header_level_1>
<text><location><page_9><loc_50><loc_1><loc_89><loc_20></location>In this paper we present the methodology used for a photometric stability analysis of the EChO mission and asses the photometric stability given its current 'Phase-A' design specifications. We describe how spectral and spatial jitter due to space-craft pointing uncertainties are propagated to an uncertainty on the exoplanetary spectrum measured by EChO . We furthermore investigate tolerances on the thermal stability of the space-craft's optical path. The photometric stability error budget was estimated for a simulated secondary eclipse observations of the hot-Jupiter HD189733b. As the instrument parameters are not set in stone as of date, we have throughout considered the 'worst-case' assumptions only and photometric stability errors may significantly decrease as the instrument definition phase proceeds.</text>
<section_header_level_1><location><page_10><loc_7><loc_86><loc_26><loc_87></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_10><loc_7><loc_82><loc_46><loc_85></location>This work is supported by STFC, NERC, UKSA, UCL and the Royal Society.</text>
<section_header_level_1><location><page_10><loc_7><loc_78><loc_19><loc_79></location>REFERENCES</section_header_level_1>
<table>
<location><page_10><loc_7><loc_1><loc_46><loc_78></location>
</table>
<table>
<location><page_10><loc_50><loc_1><loc_89><loc_87></location>
</table>
<table>
<location><page_11><loc_7><loc_47><loc_46><loc_87></location>
</table>
</document>
[
{
"title": "ABSTRACT",
"content": "Photometric stability is a key requirement for time-resolved spectroscopic observations of transiting extrasolar planets. In the context of the Exoplanet Characterisation Observatory ( EChO ) mission design, we here present and investigate means of translating space-craft pointing instabilities as well as temperature fluctuation of its optical chain into an overall error budget of the exoplanetary spectrum to be retrieved. Given the instrument specifications as of date, we investigate the magnitudes of these photometric instabilities in the context of simulated observations of the exoplanet HD189733b secondary eclipse. Key words: space vehicles: instruments - instrumentation: spectrographs - techniques: spectroscopic - stars:planetary systems",
"pages": [
1
]
},
{
"title": "Photometric stability analysis of the Exoplanet Characterisation Observatory",
"content": "I. P. Waldmann, 1 glyph[star] E. Pascale, 2 B. Swinyard, 1 , 3 G. Tinetti, 1 A. Amaral-Rogers 2 7",
"pages": [
1
]
},
{
"title": "1 INTRODUCTION",
"content": "The last decade has seen a surge in exoplanetary discoveries, with ∼ 850 planets confirmed (Schneider et al. 2011) and over two thousand Kepler candidates (Borucki et al. 2011; Batalha et al. 2013) waiting for confirmation. With the precise measurements of their masses and radii we have gained a staggering wealth of information for a plethora of targets and planetary types. In this large and growing consensus of foreign worlds, some afford us the opportunity of further characterisation. By studying the exoplanets' atmospheres, we can not only infer their chemical make-up but also constrain their climates, thermodynamical processes and formation histories. The sum total of this knowledge will allow us to understand planetary science and our own solar system in the context of a much larger picture. Using Hubble , Spitzer as well as ground based facilities, the success of exoplanetary spectroscopy of transiting (e.g. Beaulieu et al. 2010, 2011; Berta et al. 2012; Charbonneau et al. 2002, 2008; Deming et al. 2005, 2007; Grillmair et al. 2007, 2008; Richardson et al. 2007; Snellen et al. 2008, 2010a,b; Bean et al. 2011; Stevenson et al. 2010; Swain et al. 2008, 2009a,b; Tinetti et al. 2007a,b, 2010a; Thatte et al. 2010; Mandell et al. 2011; Sing et al. 2009, 2011; Pont et al. 2008; Burke et al. 2007, 2010; D'esert et al. 2011; Redfield et al. 2008; Waldmann et al. 2012, 2013; Crouzet et al. 2012; Brogi et al. 2012) as well as non-transiting planets (e.g. Janson et al. 2010; Currie et al. 2011) has been remarkable in recent years. Following these recent successes and in the frame of ESA's Cosmic Vision programme, the Exoplanet Characterisation Observatory ( EChO ) has been considered as mediumsized M3 mission candidate for launch in the 2022 - 2024 timeframe (Tinetti et al. 2012b). The current 'Phase-A study' space-mission concept is a 1.2 metre class telescope, passively cooled to ∼ 50 K and orbiting around the second Lagrangian Point (L2). The current baseline for the payload consists of four integrated spectrographs providing continuous spectral coverage from 0.5 - 16 µ m at resolution ranging from R ∼ 300 to 30. For a detailed description of the telescope and current payload design studied by our instrument consortium, we refer the reader to the literature (Puig et al. 2012; Tinetti et al. 2012a,b; Swinyard et al. 2012; Eccleston et al. 2012; Reess et al. 2012; Adriani et al. 2012; Zapata et al. 2012; Pascale et al. 2012; Focardi et al. 2012; Tessenyi et al. 2012). In order to observe the spectrum of the extrasolar planet, the EChO mission uses the time-resolved spectrophotometry method to observe transiting exoplanets. The spectrum of the planet is either seen in transmission when the planet transits in front of the star along our line of sight or in emission when the planet's thermal contribution is lost during secondary eclipse when the planet disappears behind its host. By taking consecutive short observations we follow the transit or secondary eclipse event of an extrasolar planet. This approach results in an exoplanetary lightcurve for each individual wavelength channel. The spectrum of the planet is then derived by model fitting said lightcurves and recording the eclipse depths at each wavelength bin. The amplitudes of these modulations on the mean of the exoplanetary lightcurve depths are unsurprisingly small. Taking the case of the secondary eclipse emission spectrum we typically find the planetary contrast to be of the order of 10 -3 for hotJuptiers in an orbit around a typical K0 star, and a much less favourable 10 -5 contrast for so called 'temperate SuperEarths' orbiting M-dwarf stars (Tessenyi et al. 2012). A mission such as EChO hence needs to provide a sensitivity that either matches or exceeds these contrasts over the duration of a transit/eclipse observation lasting one (e.g. GJ1214b, Charbonneau et al. 2009) to tens of hours (e.g. HD80606b, Fossey et al. 2009). The required instrument stability over said time spans can be achieved by providing a high photometric stability. There are three main factors that may introduce photometric variability over time and may limit the photometric stability. These are:",
"pages": [
1,
2
]
},
{
"title": "(i) Pointing stability of the telescope:",
"content": "The 1 σ pointing jitter of the satellite is currently baselined to be of the order of 10 milli-arcsec from 90s to 10h of continuous observation 1 . Effectively this defines the maximum Performance Reproducibility Error (PRE) over ten hours, specifying the reproducibility of the experiment. These pointing drifts manifest themselves in the observed data product via two mechanisms: 1) the drifting of the spectrum along the spectral axis of the detector, from here on referred to as 'spectral jitter'; 2) the drift of the spectrum along the spatial direction (or 'spatial jitter'). The effect of pointing jitter on the observed time series manifests as nonGaussian noise correlated among all detectors in all focal planes of the payload and is characterised by the powerspectrum of the telescope pointing. The amplitude of the resultant photometric scatter depends on the pointing jitter power spectrum, the PSF of the instruments, the detector intra-pixel response and the amplitude of the inter- pixel variations. (ii) Thermal stability of the optical-bench and mirrors: Thermal emission of the instrument, baselined at 45K, is a source of photon noise and does hence not directly contribute to the photometric stability budge beyond an achievable signal to noise ratio (SNR) of the observation. However fluctuations in thermal emissions constitute a source of correlated noise in the observations and need to be maintained at amplitudes small compared to the science signal observed. Given a ∼ 45K black-body peaks in the far-IR, we are dominated by the Wien tail of the black body distribution, resulting in steep temperature gradients and stringent requirements on thermal stability in the long wavelength instruments of the EChO payload. Additionally to variable photon noise contribution, thermal variations may impact dark currents and responsivities of the detector which must be taken into account. (iii) Stellar noise and other temporal noise sources: Whilst beyond the control of the instrument design, stellar noise is an important source of temporal instability in exoplanetary time series measurements (Ballerini et al. 2011). This is particularly true for M dwarf host stars as well as many non-main sequence stars. Correction mechanisms of said fluctuations must and will be an integral part of the EChO science study but goes beyond the instrument photometric stability discussion presented here. Along with the achievable SNR, the photometric stability of the instrument is the deciding factor for the success of missions or facilities aiming at transit spectroscopy. In this paper we study the effects of the spatial/spectral jitter and thermal variability on the photometric stability budget of the instrument and telescope.",
"pages": [
2
]
},
{
"title": "2 ECHO AND ECHOSIM",
"content": "Given EChO is currently in its Phase-A study phase, we concede that a noise budget and stability analysis, such as this one, can only be preliminary. Here and in Pascale et al. (in prep.), we present methodology used for the testing and optimisation of the current instrument design. The study presented in this paper draws on the end-to-end mission simulator EChOSim , which is discussed in detail in Pascale et al. (in prep.). In the simulations of this paper we assume EChO to have a 1.2 metre diameter primary mirror off-axis telescope. The light beam is simultaneously fed into five spectrographs via dichroic beam splitters: Visible (Vis), short wavelength IR (SWIR), mid-wavelength IR (MWIR1 and MWIR2) and long-wavelength IR (LWIR) instruments covering the spectral range from 0.5 - 16 µ m. In order to provide an end-to-end simulation of the observation, EChOSim incorporates a full simulation of the science payload including the telescope, as well as the astrophysical scene including Zodi emissions. EChOSim uses realistic mirror reflectivities and estimates the instrument transmission function for each channel as a function of wavelength. This includes transmission, optical throughput and spatial modulation transfer function. Using tabulated emissivity values as a function of wavelength, EChOSim also estimates the thermal emission spectrum of the several optical elements of the telescope. The transmission through the dichroic chain is simulated, and the incoming radiation is split among the 5 instruments assuming realistic transmittance and reflectivity data (Pascale et al., in prep.). Dispersion and detection by the focal plane array are simulated. The dispersion of the light over the focal plane is modelled by a linear dispersion law, which is related to the sampled spectral resolving power R . The full focal plane illumination due to the backgrounds and the science signal's point spread function (PSF) is calculated and convolved with realistic intra-pixel variations (see section 5). Photon noise, read noise and pointing jitter noise are calculated on a pixel by pixel basis and time series of the observation are generated. These time series are then analysed and model fitted using Mandel & Agol (2002) analytical solutions and an adaptive Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm (Haario et al. 2001, 2006; Hastings 1970). For a more detailed list of the instrument parameters assumed in these simulations see table 1 and Pascale et al. (in prep.).",
"pages": [
2,
3
]
},
{
"title": "3 FREQUENCY BANDS OF INTEREST",
"content": "Figure 1 shows an example of a secondary eclipse of a typical hot-Jupiter planet. Figure 2 shows the signal observed by EChO over the duration of 6 planetary orbits of a hotJupiter. From these figures it can easily be seen that timecorrelated noise has the greatest impact on the retrieved science at temporal variation frequencies comparable to those of the transit/eclipse event, or multiples thereof. Figure 3 shows the frequency domain representation of Figure 2 given a variety of orbital periods. Here the desired signal is contained in discrete frequencies and their respective overtones. Frequency ranges beyond these shown in Figure 3 can safely be filtered out using pass-band filters, or normalisation using low order polynomials in the time domain, without impairing the transit morphology. Given the range of transit periods observed and the goal of accurate ingress and egress mapping, we find the 'crucial frequency band' to be from 1.9x10 -4 to 1.7x10 -3 Hz, outside of which slow moving trends and high-frequency noise can effectively be filtered. This approach of slow moving trend removal is well tested for Kepler data (e.g. Gilliland et al. 2010). The overall critical frequency band for EChO is determined by the longest observation expected and the need to Nyquist sample the highest expected frequencies. We hence limit our simulations of pointing jitter or temperature fluctuations to this frequency range of interest.",
"pages": [
3
]
},
{
"title": "4 SPECTRAL JITTER",
"content": "As described in the introduction, the telescope pointing jitter has the effect of translating the spectrum on the detector along both, the spatial and spectral directions. Whilst the real translation is a combination of these orthogonal components, we will consider the effect of both these translations independently of each other. With the observing approach of taking consecutive spectra over a given length of time, we obtain a flux time series for each pixel or resolution element ∆ λ . Should, during the course of the observation, the stellar spectra be shifted along the detector the same pixel will not observe the same stellar flux but that corresponding to the shifted stellar spectrum. If we are to construct a time series for each resolution element of the detector, we consequently need to re-sample the stellar spectra to a common grid. The ability to re-sample the individual stellar spectra to a uniform wavelength grid depends on how well we can determine these relative shifts. In this section we will simulate a series of consecutively observed stellar spectra and shift each one of them according to a pointing jitter distribution derived from the Herschel Space Telescope. We will fit stellar absorption lines of these spectra and use the derived centroids to determine the accuracy with which we can determine the spectral drift. This is translated into a final post-correction flux error. It is worth noting that inter and intra pixel variations are less critical in this case as the the Nyquist sampling adopted at instrument level spreads each resolution element over at least two adjacent pixels.",
"pages": [
3
]
},
{
"title": "4.1 Synthesising pointing jitter from Herschel observations",
"content": "In order to simulate the pointing jitter of the EChO mission as realistically as possible, we synthesised the pointing patterns of the Herschel Space Telescope for which real pointing information exists (Swinyard private communication). Figure 4 shows Herschel pointings recorded for a three hour time period with a sampling frequency of ∼ 1 second. The pointing jitter distance for Herschel, ∆ Her , is given by the Pythagorean argument relative to their means: Assuming the pointing jitter in Ra and Dec are normally (Gaussian) distributed, we can state that the probability distribution function (pdf) of ∆ Her , P (∆ Her ), is given by a Rayleigh distribution. The Rayleigh distribution is the pythagorean argument of two orthogonal Gaussian distributions. However, for a real system such as Herschel, the pointing jitter is described by a Gaussian and a non-Gaussian component. These non-Gaussian components propagate to P (∆ Her ) in the form of an increased skew and broader wings than those of a pure Rayleigh distribution. We can approximate P (∆ Her ) using the more general Weibull distribution of which the Rayleigh distribution is a special case. The Weibull distribution interpolates between a Rayleigh and an Exponential distribution and is hence ideally suited to de- he broader wings and skew of the Herschel pointing jitter pdf. The Weibull distribution is given by where τ is known as the amplitude coefficient and κ as the distribution shape coefficient. Equation 2 reduces to an Exponential and a Rayleigh distribution for κ = 1 and κ = 2 respectively. The mean and the variance of the Weibull distribution are given by: where Γ is the gamma function (Riley et al. 2002).",
"pages": [
3,
4
]
},
{
"title": "4.1.1 Scaling the Herschel pointing distribution",
"content": "Figure 5 shows P (∆ Her ) (blue) obtained from the pointing information in figure 4 and the best fitting Weibull distribution, P wbl ( · ), with τ = 247 . 71 and κ = 1 . 6547. The mean of the distribution is µ = 221 . 45 mas. In order to obtain the scaled distribution to a jitter amplitude of 10 mas predicted for EChO, we maintain the shape parameter at κ = 1 . 6547 and re-derive the scaling parameter, τ , for µ = 10 mas. This yields the pointing jitter distribution P (∆ Echo ) shown in figure 6. We now randomly sample from P (∆ Echo ) to obtain ∆ Echo for M number of observations in our simulated observing run. Assuming a resolving power of R = 300 for the NIR channel (1.0-2.5 µ m), we can calculate the spectral resolving power to be 0.0159nm/mas. Hence, we can express ∆ Echo as function of spectral wavelength drift and from hence forth denote the EChO jitter as ∆ m ( λ ), where m is the m'th spectrum observed.",
"pages": [
4,
5
]
},
{
"title": "4.2 Applying pointing jitter to stellar spectra",
"content": "Using the Phoenix 2 code we generated a stellar spectrum of a sun analogue and proceeded with the following steps: (ii) A single stellar absorption line was now selected (wavelength range: 1.19 - 1.21 µ m), and the spectrum consequently trimmed. We denote the trimmed spectrum as F line ( λ ) encompassing N data points in λ . fraction of the maximal line amplitude over the standard deviation of the Gaussian noise component The resulting spectrum, F final ( λ ), is hence F final ( λ ) = F line ( λ )+ N ( σ, µ ), where N ( σ, µ ) is the Gaussian noise with σ variance and µ = 0 mean. We chose a SNR of 500 for this study.",
"pages": [
5
]
},
{
"title": "4.4 Translating fitting residuals to total flux error",
"content": "We can now use the fitting residual R m ( λ ) to calculate the observed flux, F obs ( m,λ ), per wavelength range δλ and time step m : above we can see that R m ( λ ) constitutes a change in the integration interval of each pixel with respect to the stellar spectrum. For the case where δλ → 0, we can calculate F obs ( m,λ ) using the linear approximation where ¯ R m ( λ ) is the mean of the fitting residual and was subtracted to account for equal amounts of positive and negative drifts, and G is the local gradient of the stellar spectrum. We can now express the resulting flux error due to residual drifts along the spectral direction as In section 8 we show the residual flux error F err for the gradients G1 and G2 shown in figure 7.",
"pages": [
5,
6
]
},
{
"title": "5 SPATIAL JITTER",
"content": "We have investigated the effect of spatial jitter as well as thermal variability. Whereas inter/intra pixel variations can largely be ignored in the case of spectral jitter, the detector pixel responses and variations can play a significant role in the computation of the spatial jitter which is caused by the movement of the spectral PSF along the spatial direction of the detector. EChOSim fully simulates the effect of inter and intra pixel variations on a spatially resolved detector grid. We here outline the spatial jitter calculations but refer the reader to Pascale et al. (in prep.) for a more detailed discussion. The dispersed signals are sampled by each detector assuming a wavelength-depended and instrument specific PSF which is convolved with the intra-pixel dependent response of the detector. Figure 8 is a snapshot of a standard diagnostic diagram returned by EChOSim . The top panel shows the focal plane illumination of the detector array. The bottom panel shows the individual flux contributions: astrophysical ( Q point ), zodiacal light ( Q zodi ), instrument thermal emissions ( Q optics ), and their combined total ( Q tot ). We can describe the focal plane illumination of a detector with a monochromatic point source using the diffraction PSF pattern (Marc Ferlet priv. com.; Pascale et al. in prep.) where x and y are detector coordinates along the spectral and spatial axes respectively, K x and K y are parameters accounting for spatial aberrations (we assume no aberrations K x = K y ) and F # is the ratio between the effective focal length and the effective telescope diameter (see table 1). The one dimensional PSF for the spatial direction can now be written as Barron et al. (2007) studied the intra-pixel response of IR detectors, and their best-fit model is analytically implemented in EChOSim . The cross section of the response along the spatial axis of the detector array is hence given by: where l d is the diffusion length. The sampled PSF along the spatial axis is then given by the convolution Finally the signal is sampled by the detector. The detector response is given by the convolution of the point source flux of the star-planet system, Q point ( λ, t ), with the spectrally dispersed PSFs where i and j are the detector pixel indices. Note that optical efficiencies don't explicitly appear in this equation as Q point ( λ, t ) already accounts the throughput budget, with the exception of the quantum efficiency, QE ( λ ) (Pascale et al., in prep.). where f eff is the plate scale (in µ m rad -1 ) and σ p is the pointing jitter (in radians per second) randomly sampled from the pointing distribution P (∆ Her ). Variations in photometric errors are estimated by consecutive runs for a range of pointing jitter amplitudes from zero to 200 milli-arcsec over a ten hour observing window. The observed spectrum is reconstructed using the EChOSim pipeline and the uncertainties on the reconstructed spectra are shown in figures 12 & 13 for two cases where PSFs of different sizes are used. Given the current uncertainty over the exact nature of inter and intra-pixel variations we have not attempted to decorrelate the pointing jitter in post processing (e.g. Swain et al. 2008; Burke et al. 2010; Crouzet et al. 2012) which would reduce the uncertainties reported in figures 12 & 13. We must hence consider this analysis as a conservative worst case scenario.",
"pages": [
6,
7
]
},
{
"title": "6 THERMAL STABILITY",
"content": "We have also investigated the impact of thermal emissions and fluctuations of the optics on the photometric stability of the reconstructed spectrum. Whilst thermal emissions are not directly a problem to photometric stability (source of white noise), they pose constraints on the temperature fluctuations allowed over the time span of an exoplanetary transit/eclipse in the reddest wavelengths. Here varying thermal emissions may produce detector counts that are equivalent or exceed the signal amplitude expected from a planetary eclipse. We calculate the thermal contribution using where L is the image size of the spectrometer slit in number of pixels, LD the lateral dispersion in mm, ∆ p is the pixel size, f # is the f-number and BB is the Planck function for a given temperature T and wavelength λ . EChOSim calculates the thermal emission of the instrument optics as well as the primary, secondary and tertiary mirrors separately and coherently propagates the resulting emission to detector counts. We have explored a temperature regime ranging from 40 - 60K with a grid size of 0.2K for both, the optical bench and the mirror temperatures. This provides us with an absolute scale of the thermal contributions, see results in section 8. We have now calculated the expected signal strength of a secondary eclipse of a HD189733b like hot-Jupiter, F planet , and for a given base temperature, T 0 , calculated the temperature variation, ∆ T , required to produce a detector signal of the same strength as F planet for a given wavelength λ : where f λ ( T ) is the functional form of the thermal emission/temperature relation for a given wavelength seen in figures 14 & 15. This can either be numerically approximated by interpolation to a fine enough grid or by fitting a high-order polynomial function to the data.",
"pages": [
7
]
},
{
"title": "7 TESTCASE",
"content": "We have now calculated the spectral and spatial jitter contributions for the hot-Jupiter HD189733b (Torres et al. 2008). We currently have insufficient information on the expected thermal stability in the longest wavelength range so we do not include thermal stability in our test-case calculations. This said, as seen in section 8, the overall thermal stability is not a decisive factor for wavelengths shorter than ∼ 14 µ m. For our simulations we use realistic Phoenix stellar models for the host stars. The planetary emission model is taken from Tessenyi et al. (2012). EChOSim fully solves the dynamical star-planet system and computes the time resolved spectra by model fitting the generated time series. We have now computed the emission spectra of HD189733b with the full noise contribution (shot noise, read noise, spatial/spectral jitter) as well as for the spatial jitter and spectral jitter only. This allows for the direct comparability of the noise contributions.",
"pages": [
7
]
},
{
"title": "8 RESULTS",
"content": "As described in section 4 the 'jittered' stellar absorption line was fitted using a Voigt profile and the centroids recorded. The residual between the real and fitted spectral shifts are shown in figure 9. The Monte-Carlo analysis of the standard deviation of the fitting residual is shown in figure 10. As expected, the probability distribution on the parameter σ R is largely Gaussian as we randomly sample from the pointing jitter distribution P (∆ Echo ). This also shows that the centroid fitting does not introduce biases in the pointing jitter correction. The fitting residual can now be translated to a total flux error using equation 9. Taking the ratio F err /F star we can derive the relative error due to residual jitter, figure 11. The flux error is dependent on the local stellar flux gradient. From figure 7, we derived two gradients: G1 = - 760 Wm -2 nm -1 and G2 = +24 Wm -2 nm -1 . For these two gradients, figure 11 shows that the relative flux error lies between 10 -6 ∼ 2 × 10 -5 . These values are of course larger for the Wien tail of the stellar black body with spectral jitter errors of ∼ 10 -4 at places (see figures 18 & 19). Figures 12 & 13 show the simulated spatial jitter contributions for a pointing jitter amplitude range or 0 - 200 milliarcseconds. As the photometric error resulting from spatial jitter is largely dependent on the PSF of the instrument and the pixel dimensions, we have calculated the photometric error for a PSF FWHM of 0.7 and 0.5 of the detector-pixel size (figure 12) and that for double these values (figure 13). As seen in the figures the spatial jitter is of the order of ∼ 10 -5 -10 -4 but significantly higher for the spectral ranges of the NIR instrument (2 - 5 µ m). This is to be expected as the SWIR instrument features a smaller pixel size. Figures 14 & 15 show the thermal contribution of the telescope (mirrors and optics) and the optical bench temperatures respectively. This can largely be regarded as negligible for wavelengths shortward of ∼ 14 µ m but significant at longer wavelengths and temperatures above 50K. As described in section 6, this thermal emission was translated into a temperature tolerance, ∆ T optics , showing the temperature change necessary to mimic the transit depth of a secondary eclipse feature of hot-Jupiter HD189733b at a given wavelength. These tolerances are shown in figures 16 & 17 for the telescope and optical bench temperatures respectively. We find that when the telescope and optical bench are at 45K, temperature fluctuations need to be below 6K for the telescope and 500mK for the bench in order to observe a hot Jupiter. More restrictive constraints would be required when fainter signals are involved. Having investigated the pointing jitter contributions for given wavelengths and pointing jitter amplitudes, we show the simulated 'observations' of a single eclipse event of HD189733b in figure 18. The blue spectra's error-bars contain the full noise contribution. Figure 19 shows a breakdown of the individual error contributions. Here red stands for the total noise contribution (including read and shot noise), blue for the spectral jitter as calculated in section 4 and green for the spatial jitter contribution as calculated in section 5. It can easily be seen that spectral jitter is important at the Wien tail of the stellar black body distribution and less so at the Rayleigh-Jeans tail. The spatial jitter noise depends on the individual detectors with the effect being strongest for the SWIR detector.",
"pages": [
7,
8,
9
]
},
{
"title": "9 DISCUSSION",
"content": "One way to reduce spatial jitter noise is to use pixels that are small compared to the PSF. In this case the effect of the spatial jitter is only governed by the intra-pixel response, and using many pixels to sample the PSF will washout both the inter and intra response variations. A higher sampling of the PSF using more pixels will also positively impact the spectral jitter as tracking of the stellar lines becomes easier. However, this will only be realistic if the noise from the detectors is sufficiently low but will allow a significant amount of de-correlation as the PSF centroid can be tracked across the spatial dimension of the array. The simulations take into account realistic intra-pixel responses in the spectral band from MWIR and LWIR. At shorter wavelength, the simulations currently have to be considered worst case scenarios because the system is not diffraction limited at these wavelength. Figures 14 & 15 show the detected radiation in electrons/second from the instrument and telescope (separately) for a temperature range of 40 - 60K. Further studies will include additional instrumental effects such as mechanical vibrations, thermo-mechanical distortions, variable detector dark currents (assumed fixed with temperature in this study), detector responsitivity drifts and the effect of cosmic ray impacts. These studies will look at the extent the effect can be de-correlated from the timelines when additional information is available for data processing, such as temperature sensors monitoring the optics and telescope temperatures. We also plan to provide off-axis detectors to monitor the non stellar backgrounds and therefore provide a means of directly removing the background signals. Such background removal is particularly important for fainter sources. Whereas HD189733b is photon limited, observations of faint super-earths will likely be background limited and thermal and zodiacal light emissions need to be carefully accounted for.",
"pages": [
9
]
},
{
"title": "10 CONCLUSION",
"content": "In this paper we present the methodology used for a photometric stability analysis of the EChO mission and asses the photometric stability given its current 'Phase-A' design specifications. We describe how spectral and spatial jitter due to space-craft pointing uncertainties are propagated to an uncertainty on the exoplanetary spectrum measured by EChO . We furthermore investigate tolerances on the thermal stability of the space-craft's optical path. The photometric stability error budget was estimated for a simulated secondary eclipse observations of the hot-Jupiter HD189733b. As the instrument parameters are not set in stone as of date, we have throughout considered the 'worst-case' assumptions only and photometric stability errors may significantly decrease as the instrument definition phase proceeds.",
"pages": [
9
]
},
{
"title": "ACKNOWLEDGMENTS",
"content": "This work is supported by STFC, NERC, UKSA, UCL and the Royal Society.",
"pages": [
10
]
}
]
2013ExA....35....1B
https://arxiv.org/pdf/1208.0447.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_84><loc_60><loc_85></location>The Astro-WISE datacentric information system</section_header_level_1>
<text><location><page_1><loc_12><loc_79><loc_46><loc_82></location>K. Begeman, A.N. Belikov, D.R. Boxhoorn, E.A. Valentijn</text>
<text><location><page_1><loc_12><loc_70><loc_32><loc_71></location>Received: date / Accepted: date</text>
<text><location><page_1><loc_12><loc_59><loc_70><loc_67></location>Abstract In this paper we present the various concepts behind the Astro-WISE Information System. The concepts form a blueprint for general scientific information systems (WISE) which can satisfy a wide and challenging range of requirements for the data dissemination, storage and processing for various fields in science. We review the main features of the information system and its practical implementation.</text>
<text><location><page_1><loc_12><loc_57><loc_67><loc_58></location>Keywords Data Grid · Grid Computing · Information System · Middleware</text>
<section_header_level_1><location><page_1><loc_12><loc_52><loc_23><loc_53></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_40><loc_70><loc_51></location>Digital astronomical catalogues have been built from the very first moment information technology enabled this, e.g. the first Abell catalogue of clusters of galaxies was digitally prepared and printed in the days minus signs were not available in print [1]. As soon as digital scanning devices became available photographic material was scanned and first large image surveys were digitally published, such as ESO-LV [2] and the digitized sky surveys like DPOSS [3], succeeded by the Palomar-Quest Survey [4], in the 90's followed by CCD-based surveys such as the 2MASS survey [5], the Sloan Digital Sky Survey SDSS [6].</text>
<text><location><page_1><loc_12><loc_34><loc_70><loc_40></location>The rapid accumulation of astronomical digital data and its public dissemination was, compared to other disciplines, achieved at an early stage, thanks to an open and collaborative astronomical world community who adopted the FITS image data format as early as 1979 [7].</text>
<text><location><page_1><loc_12><loc_27><loc_70><loc_34></location>The CDS, initially Centre de Donn'ees Stellaires and later renamed into Centre de Donn'ees astronomiques de Strasbourg took the lead in Europe to collect and disseminate the ever growing data sets of a zoo of astronomical observatories and projects. VizieR webservices nowadays provide access to over 9000 catalogues. Numerous astronomical data centers followed, developing specialized services, for</text>
<text><location><page_2><loc_12><loc_87><loc_70><loc_89></location>example, the Infrared Processing and Analysis Center and the bibliographical SAO/NASA Astrophysics Data System.</text>
<text><location><page_2><loc_12><loc_74><loc_70><loc_86></location>In the early 2000's it was realized that the ever growing data volumes require new approaches: the community becomes the data provider and the International Virtual Observatory and its European branch, the European Virtual Observatory, developed standards and interfaces to allow individual data centers to publish their catalogs and images in a common framework providing worldwide access to users who can query, cross-match and visualize multiple databases via the internet. A highlight formed the overplotting of data from different experiments, like Xray satellites and optical ground-based observatories with some keystrokes on the Aladin interactive software sky atlas.</text>
<text><location><page_2><loc_12><loc_57><loc_70><loc_74></location>While the Virtual Observatory focused on the dissemination of published data it was also realized in the early 2000's that the upcoming data deluge required a new approach to the handling of the data stream from the telescope to the science-ready result. Modern experiments not only involve more data, but also require increasing precision on the various calibrations. The dependency of the final result (sometimes only a few numbers, like cosmological parameter values) on time-variable calibrations of very large data sets, involving the evaluation by large research teams distributed in smaller groups over various sites sets the basic requirements to the system handling the data. While in classical systems data is delivered in various releases to the public, often re-processing the whole set with higher versions of the code, the high data rates of modern experiments require an alternative approach, where the up-to-date result is derived on demand by the user.</text>
<text><location><page_2><loc_12><loc_46><loc_70><loc_56></location>Thus, we set out to design and implement an integrated datacentric system Astro-WISE in which the processing, storage and administration is integrated in a single environment, providing a living system to both the data producers and the customers. Early reports of this development and implementation have been given in [8] and [9]. To reach this goal, traceability of each individual data item handled by pipelines or any piece of code is carefully maintained, every data item beyond pixel value is kept as Metadata and made persistent and distributed in a relational database with an object-oriented view, mapping all the dependencies.</text>
<section_header_level_1><location><page_2><loc_12><loc_41><loc_46><loc_43></location>2 Datacentric Approach to Data Processing</section_header_level_1>
<text><location><page_2><loc_12><loc_29><loc_70><loc_40></location>The datacentric approach facilitates scientists to cooperate in data analysis and mining by means of the internet, nowadays referred to as e-science. The AstroWISE information system connects to an own developed distributed grid processing system (based on Distributed Processing Unit, DPU), but has been also connected to the EGEE/EGI/BiGGrid-Grid [10], particularly for the operations of the Lofar radio telescope. Nowadays, this aspect of the system is referred to as cloud computing, but the Astro-WISE datacentric approach at the same time facilitates the sharing and web hosting of all the data handled by the system.</text>
<text><location><page_2><loc_12><loc_22><loc_70><loc_29></location>Today, we are just taking the first small steps towards the development of approaches and systems who could handle the upcoming data deluge. The way of managing and administration the data will be a key, and will require ever improving, refining and self-organizing approaches. The key notion is that a fruitful approach towards handling the data deluge focuses on the administration, modeling,</text>
<text><location><page_3><loc_12><loc_70><loc_70><loc_89></location>standardizing and tracking of the data rather than on the processing of the data by CPUs. This datacentric approach, in which the design of operational systems is driven by data management and data standard issues rather than data processing/cpu/pipelining issues. Processing systems turn into information systems. First attempts to combine archiving and processing in the 1980s include ESO's MIDAS table file system [11], integrating common tables both accessible in source code and by user interfaces like the MIDAS monitor/prompt. The MIDAS table file system allows both users and programmers to change its content and also registering the modifications in table keywords leading to archives ready for further review and (trend) analysis. Its development was essential for the production and photometric calibration of the images of the ESO-LV galaxies. However, the 600,000 fore- and background objects visible on the images, stars and galaxies, could only be handled with great difficulties inside the system, response times running into hours per operations, demonstrating the importance of built-in scalability.</text>
<text><location><page_3><loc_12><loc_65><loc_70><loc_70></location>This MIDAS Table file system marked a stepping stone to current datacentric approaches which are enabled by the introduction of modern object-oriented programming languages, data modeling tools like UML and XML and relational databases.</text>
<text><location><page_3><loc_12><loc_58><loc_70><loc_65></location>Wehave combined these views, systems and requirements in order to design and build integrated information systems which have the potential for self-organizing and enrichment with new data. Both self-organizing and enrichment are processes in time. In fact, the datacentric approach implies the detailed modeling and awareness of how things change in time. Important changes range from</text>
<unordered_list>
<list_item><location><page_3><loc_13><loc_56><loc_44><loc_57></location>I. new data entering the system (ingest) to</list_item>
<list_item><location><page_3><loc_12><loc_53><loc_70><loc_56></location>II. new or modified source code handling these data. In turn, the new data entering the system might be</list_item>
<list_item><location><page_3><loc_11><loc_50><loc_70><loc_53></location>III. subject to physical changes (e.g. the gain of a sensor), while source code is often modified on the basis of</list_item>
<list_item><location><page_3><loc_12><loc_46><loc_70><loc_50></location>IV. advancements of human understanding of the physical changes - our model of the world also changes (e.g. the cause and modeling of the gain variations of the sensor).</list_item>
</unordered_list>
<text><location><page_3><loc_12><loc_42><loc_70><loc_45></location>The ideal information system would seamlessly cope with all these changes in time, thus creating a living long term digital preservation environment. Astro-WISE is one of the first systems attempting to reach this goal.</text>
<text><location><page_3><loc_12><loc_32><loc_70><loc_41></location>For several years many authors predicted the upcoming data avalanche due to the advancements in digital sensor technology. Obviously, this has now started, both scientific experiments such as LHC and Lofar are running into the tens of Petabytes data acquisition regimes, while text, imaging, genome and internet gathered data collections like e-Bay are exceeding similar volumes. This triggered the development of Grid infrastructure that is able to handle Petabytes of data ([12] and [13]).</text>
<text><location><page_3><loc_12><loc_27><loc_70><loc_32></location>All this has happened in an approximate 20 year time span when the first large digital imaging collection was published, ESO-LV [2], containing scans of 32,000 galaxy images, 4 Gbyte of data filling a room full with tapes and populating the very first version of optical media.</text>
<text><location><page_3><loc_12><loc_22><loc_70><loc_26></location>By now, the data avalanche is multidisciplinary and worldwide and involves all aspects of our society, from science to commerce, public services and health care. Digital data is gathered, processed, distributed and accessed at a continuously</text>
<text><location><page_4><loc_12><loc_80><loc_70><loc_89></location>growing rate. And this is only the beginning: in, say, two decades, the same room full with 4 Gbyte of tapes 20 years ago, and 40 Petabyte of tapes today could possibly contain 40 Zetabytes (40 million Petabytes) of data. The growth of the data volume produced by various types of sensor networks met with the growth of the disk capacity and search for new methods in the increasing of the memory storage density (see [14]). In order to be useful all this data has to be organized, administrated, managed and distributed.</text>
<text><location><page_4><loc_12><loc_70><loc_70><loc_79></location>It took a billion years for cells to develop into extremely complex information systems around the DNA; it has resulted into systems and approaches with an incredible complexity with memory mapping, copying, filtering and distribution mechanisms of data including feedback. Compared to this, our present data organizing mechanisms in IT are still very simple and rudimentary and we are just making the first steps to living self-organizing systems and archives required to deal with and optimally use the ocean of digital data to come.</text>
<text><location><page_4><loc_12><loc_62><loc_70><loc_70></location>However, also in the cell stochastic processes are important, and absence of organization 'from above' are countered by survival of the fittest type of mechanisms, which appear as self-organizing. Also in human endeavored IT, standards seldom come from above, an exception being the ASCII character standards, mandated in 1968 by U.S. President Lyndon B. Johnson to be used in all computers purchased by the United States federal government.</text>
<text><location><page_4><loc_12><loc_55><loc_70><loc_62></location>I have also approved recommendations of the Secretary of Commerce regarding standards for recording the Standard Code for Information Interchange on magnetic tapes and paper tapes when they are used in computer operations. All computers and related equipment configurations brought into the Federal Government inventory on and after July 1, 1969, must have the capability to use the Standard Code for Information.</text>
<text><location><page_4><loc_12><loc_50><loc_70><loc_55></location>Interchange and the formats prescribed by the magnetic tape and paper tape standards when these media are used.In practice, standards are often invented by companies or scientific communities which are discriminated in a complex survival of the fittest battle, which involves that many factors that it appears stochastic.</text>
<text><location><page_4><loc_12><loc_44><loc_70><loc_49></location>The view, or better requirement, for future self-organizing information systems and its archives is that it organizes itself for all these changes in time. It could be seen as a step towards the extremely complex and self-organizing information systems in a cell.</text>
<text><location><page_4><loc_12><loc_33><loc_70><loc_44></location>The core problem for the definition of an information system is the definition of information itself. The information can be an input data flow from sensor detectors, an archive of transactions, trends deduced from the analysis of some data and even results of the data modeling or simulations. As result, information systems will host a variety of subsystems defined by its purpose. Examples are a Decision Support System gathering and analyzing trends for a specific business use case (see [15]) or Geographical Information Systems with focus on geospatial data analysis and a visualization of the data.</text>
<text><location><page_4><loc_12><loc_25><loc_70><loc_33></location>Nevertheless, it is possible to define primary components of any Information System following simple requirements on storage, processing and sharing data. Such a system should include a data model, storage and data processing facilities together with user interfaces. In Astro-WISE we have decided to separate and store all data beyond pixel/measurement data in a database and all pixel/measurement data in files on independent storage media.</text>
<text><location><page_4><loc_12><loc_22><loc_70><loc_25></location>This paper does not touch details of the implementation of Astro-WISE information system which are described in other papers but describes the general</text>
<text><location><page_5><loc_12><loc_81><loc_70><loc_89></location>outline of the Astro-WISE information system and specify features which make Astro-WISE unique. For the more in-depth review of particular aspects of AstroWISE we recommend to read corresponding papers on the optical pipeline implemented in Astro-WISE [16], quality control realised in Astro-WISE [17], the system of user interfaces and services [18], and the hardware solution for Astro-WISE and information systems created on the basis of Astro-WISE [19].</text>
<section_header_level_1><location><page_5><loc_12><loc_76><loc_45><loc_77></location>3 Building a Scientific Information System</section_header_level_1>
<text><location><page_5><loc_12><loc_64><loc_70><loc_75></location>The development of the Astro-WISE information system, the first one which implemented WISE technology, started from the very practical challenge: enable a community of researchers distributed over the world to process the data of astronomical imaging survey of the OmegaCAM 256 Megapixel camera on ESO's VST telescope at Cerro Paranal (Chile). These scientists should be able to evaluate the quality of the data, apply a number of calibrations, share the data within the team and employ distributed resources of a Petabytes scale of data storage and Teraflops capacity in data processing.</text>
<text><location><page_5><loc_15><loc_62><loc_58><loc_64></location>From this our basic requirements on the system are derived:</text>
<unordered_list>
<list_item><location><page_5><loc_13><loc_49><loc_70><loc_61></location>* Scalability of the system: any part of the system, i.e., data storage, data processing, metadata management, should be scalable with the increase of incoming data and a number of users involved in the data processing. The system should be scalable with respect to the data processing algorithms and pipelines, allowing the implementation of new pipelines and derive improved results from the same raw or intermediate data with new algorithms. The scalability of data mining should be possible, i.e. the system should satisfy all possible kinds of requests; from the retrieval of a single data item by identifier to a complicated archive study involving multiple complex queries.</list_item>
<list_item><location><page_5><loc_13><loc_45><loc_70><loc_49></location>* Distributed system: any derivation of a result or search for a result should be possible by different users at different sites where the system is implemented. This makes it possible to optimally use shared resources.</list_item>
<list_item><location><page_5><loc_13><loc_40><loc_70><loc_45></location>* Traceability: all activity in the system should leave a clear footprint so that it will be possible to trace the origin of any changes in the data and find an algorithm, program and user who created a data item. This allows expert knowledge to be shared amongst all users of the system.</list_item>
<list_item><location><page_5><loc_13><loc_36><loc_70><loc_40></location>* Adaptability. The system should be possible to adopt for a number of different scientific use-cases, providing resources, pipelines and expertise to perform a data processing according to user's interests on the same data set.</list_item>
</unordered_list>
<text><location><page_5><loc_12><loc_22><loc_70><loc_34></location>The most general requirements listed above results in a set of more detailed and specific requirements to different components of the information system. First, the ability to share data among a number of users becomes more valuable when all these users are working with the same set of standards for the data. Second, the requirement to trace any changes in the data processed by different users with different programs and pipelines implies that the system has a common approach to treat the data items on different stages of the data processing and keeps the record of changes in the data. As result the basic requirements to the system dictate the use of a common data model and the common standards for the data</text>
<text><location><page_6><loc_12><loc_87><loc_70><loc_89></location>storage. Below we will describe the common data model features defined from the requirements to the system.</text>
<section_header_level_1><location><page_6><loc_15><loc_85><loc_41><loc_86></location>Requirements to the Data Model:</section_header_level_1>
<unordered_list>
<list_item><location><page_6><loc_13><loc_68><loc_70><loc_84></location>-Petabyte systems can not work when anarchy is allowed at the data acquisition. Users should operate with the same standards for the raw measurements data acquired at the experiment. Moreover, this implies a careful definition of the data taking scenarios, observing templates in astronomy, scan scenarios on sensor systems both for the scientific measurements data and for the calibrations. The raw data ingested in the system should be fully and completely described in the system so that users do not need to involve detailed knowledge about the acquired data and can share it with other users of the system. The data model must provide complete data provenance , i.e., both external data ingested in the system and the data derived inside the system should be described with sufficient level of detail to be able to rederive them in the environment external to their original one.</list_item>
<list_item><location><page_6><loc_13><loc_60><loc_70><loc_68></location>-The data model should describe all the data products, on all stages of the data processing For the user it should be possible to understand the origin of each data item and to trace the data item back to the raw data. The data model must provide full data lineage , i.e., each data product and literally each bit of information must be traced back to the origin and creator of the information.</list_item>
<list_item><location><page_6><loc_13><loc_50><loc_70><loc_60></location>-The model should match all scientific use-cases and should allow the flexibility to modify source code without changing the model as long as this makes sense. Obviously, here boundaries will have to be determined by means of inside, thus human, knowledge. This is the most tedious part in the design of the information system, but in practice it turns out that the various processes are well defined and dictated by physics. The data model must be flexible to the changes in the data processing .</list_item>
<list_item><location><page_6><loc_13><loc_37><loc_70><loc_50></location>-Requirements to trace changes in the data triggered by different versions of the processing software and the necessity to repeat the data processing due to the changes in the pipeline implies that all parameters that affect the derived result must be preserved in the system. The user should be able to reproduce the data item and to find a trend in the data due to the changes in the processing parameters or algorithm. At the same time the user should be able to reproduce already existing item to ensure the preservation of the data and software used previously. The data model must allow both data reprocessing and data reproduction , and the reprocessing should be done with the preservation of the previously produced versions of the data.</list_item>
</unordered_list>
<text><location><page_6><loc_12><loc_33><loc_70><loc_36></location>When properly defined the data model allows to code, instantiate and trace the data processing for all scientific use-cases.</text>
<text><location><page_6><loc_12><loc_31><loc_70><loc_33></location>Also, the data model should allow to accommodate external data, as well as provide necessary comparison and checking with these external data sets.</text>
<text><location><page_6><loc_12><loc_22><loc_70><loc_30></location>When the data model is carefully defined our approach is to convert this into a pipeline model, in turn translated into a hierarchy of Classes. These Classes are then mapped into a database. We originally did this with object-oriented databases like Objectivity, but currently we do this in a relational database and build an object oriented view in order to maintain transparency in the class model, and its dependencies.</text>
<figure>
<location><page_7><loc_15><loc_69><loc_67><loc_90></location>
<caption>Fig. 1 Top-level definition of Astro-WISE information system with the specification for each layer.</caption>
</figure>
<text><location><page_7><loc_12><loc_57><loc_70><loc_64></location>Requirements to Infrastructure. The data model should be shared among geographically remote sites of the system, as research groups at various locations are collaboratively adding and extracting data from the distributed system. The basic requirement is that in fact any component (database, file system, code base, CPUs) should possibly be present at different sites, for a number of reasons:</text>
<unordered_list>
<list_item><location><page_7><loc_13><loc_55><loc_52><loc_57></location>-hardware resources between the sites can be shared,</list_item>
<list_item><location><page_7><loc_13><loc_53><loc_70><loc_55></location>-avoiding single centralized components prevent an overload of resources and duties at that node,</list_item>
<list_item><location><page_7><loc_13><loc_51><loc_53><loc_52></location>-redundancy of the system to single-site malfunctions,</list_item>
<list_item><location><page_7><loc_13><loc_49><loc_70><loc_51></location>-sharing of resources between partners allows to reduce the overall cost of the system.</list_item>
</unordered_list>
<text><location><page_7><loc_12><loc_44><loc_70><loc_48></location>For example, the data ingestion to the system can be done in Garching, Germany, the data processing in Groningen, The Netherlands and the image analysis in Napoli, Italy.</text>
<text><location><page_7><loc_12><loc_38><loc_70><loc_44></location>Requirements to the User Interfaces. The user should be able to trace the data processing, to retrieve data items and identify all its dependencies on other items (mostly calibrations) in the system, to initiate data processing of particular data items and to find a quality estimation for the data.</text>
<text><location><page_7><loc_12><loc_34><loc_70><loc_38></location>As we can see, all these requirements put together are more challenging than requirements on common business information systems. The key difference is that we allow everything, including the code base to change in time.</text>
<text><location><page_7><loc_12><loc_30><loc_70><loc_34></location>In the following chapter we outline how our requirements formed a layout for a scientific information system capable to share a huge data volume, to process these data and fulfill scientific use-cases for a number of users.</text>
<section_header_level_1><location><page_7><loc_12><loc_26><loc_25><loc_28></location>4 WISE Concept</section_header_level_1>
<text><location><page_7><loc_12><loc_22><loc_70><loc_25></location>First of all we have to clearly distinguish between two classes of data storage systems in Astronomy: proper archives containing fixed published data and infor-</text>
<text><location><page_8><loc_12><loc_82><loc_70><loc_89></location>mation systems. The main difference can be found in an ability to work with the data and to change it during the storage. For an archive stored data is a stable entity which is provided to the user as it is, meanwhile the information system allows to the user to modify their own data creating user-specific version of an archive and share this version with other users.</text>
<text><location><page_8><loc_12><loc_79><loc_70><loc_82></location>The Astro-WISE approach, and its general multi-discipline WISE approach implements a scientific information system.</text>
<text><location><page_8><loc_12><loc_60><loc_70><loc_78></location>The usual approach to building an information system is to perceive three components: the data layer, business rules and the interfaces (see Fig. 1). In our case, we separate the data layer in two parts: the pure measurement data layer, hereafter data layer, and everything beyond that, hereafter Metadata layer, ranging from file sizes, to statistics of pixel values and detected events in the measurements (galaxy, particle, word). The Metadata layer allows to implement a data model, the data layer allows to store the data as files in a standard format, while business rules implemented in Python classes bind the metadata and the data layers. Interfaces provide user access to both the business rules and the data. In the case of AstroWISE business rules present in the Metadata layer (Data Definition Language used to create a data model), data layer (the on-the-fly compression of the data on data storage grid) and-the most apparent to the user-in a number of pipelines and programs which the user defines to process the data. This last part of business rules components we will call data processing layer.</text>
<text><location><page_8><loc_12><loc_48><loc_70><loc_59></location>To implement Astro-WISE we use abstractions of storage, processing and database capabilities as a basis for the infrastructure for each of the layers of the system. The metadata layer is realized in a relational DBMS through an abstraction of the required database functionality, the data layer is put on the Astro-WISE data storage grid through an abstraction of storage and the processing grid is used to connect the user with the data and metadata layers by a number of interfaces for the data processing layer. Separation of these three infrastructures plays a key role in the flexibility of the system, as we will see below.</text>
<text><location><page_8><loc_15><loc_46><loc_62><loc_48></location>The metadata layer implements a list of necessary functionalities:</text>
<unordered_list>
<list_item><location><page_8><loc_13><loc_40><loc_70><loc_44></location>1. Inheritance of data objects . Using object-oriented programming, all objects within the system can inherit key properties of the parent object. All these properties are made persistent.</list_item>
<list_item><location><page_8><loc_13><loc_31><loc_70><loc_40></location>2. Full lineage . The linking (associations or references, or joins) between object instances in the database is maintained completely. Each data item in the system can be traced back to its origin. The tracing of the data object can be both forward and backward. For example, it is possible to find which raw frames were used to find magnitudes, shapes and position for this particular source and, at the same time, which sources were extracted from that particular raw frame.</list_item>
<list_item><location><page_8><loc_13><loc_25><loc_70><loc_30></location>3. Consistency . At each processing step, all processing parameters and the inputs which are used are kept within the system. Astro-WISE keeps the old versions of all data items along with all parameters used to produce them and all dependencies between objects.</list_item>
<list_item><location><page_8><loc_13><loc_22><loc_70><loc_25></location>4. Embarrassingly parallel and distributed processing, the administration of asynchronous processing is recorded in the metadata layer in a natural way.</list_item>
</unordered_list>
<text><location><page_9><loc_12><loc_85><loc_70><loc_89></location>Our requirements on distribution and multiple users propagate as key principles of the realization of metadata and data layers and business rules which form the core of the WISE approach:</text>
<unordered_list>
<list_item><location><page_9><loc_13><loc_79><loc_70><loc_84></location>1. Component based software engineering (CBSE) . This is a modular approach to software development, each module can be developed independently and wrapped in the base language of the system (Python) to form a pipeline or workflow.</list_item>
<list_item><location><page_9><loc_13><loc_74><loc_70><loc_79></location>2. An object-oriented common data model used throughout the system . This means that each module, application and pipeline will deal with the unified data model for the whole cycle of data processing from the raw data to the final data product.</list_item>
<list_item><location><page_9><loc_13><loc_68><loc_70><loc_73></location>3. Persistence of all the data model objects . Each data product in the data processing chain is described as an object of a certain class and saved in the archive of the specific project along with the parameters used for data processing.</list_item>
</unordered_list>
<text><location><page_9><loc_12><loc_55><loc_70><loc_67></location>The Astro-WISE system is realized in the Python programming language. It allows to wrap any program into a Python module, library or class. The use of Python also allows to combine the principles of modular programming with objectoriented programming, so that each package in the system can be built and run independently with an object-oriented data model serving as glue between modules. At the same time, the logic behind pipelines and workflows in Astro-WISE allows the execution of part of the processing chain independently from the other parts. We will describe this approach in more detail in the example of optical image processing in Section 4.1.</text>
<text><location><page_9><loc_12><loc_50><loc_70><loc_55></location>The conceptual difference between Astro-WISE and other existing systems is that Astro-WISE moves from the usual for astronomy processing-centric approach to a data-centric approach. The data processing itself becomes an integral part of the archive.</text>
<text><location><page_9><loc_12><loc_40><loc_70><loc_49></location>The typical solution for the data processing and storage for Astronomy handles the data processing and the final data product delivered to the user as two completely separated entities. The data product usually is a result of the processing for the whole survey with fixed processing parameters used for the whole set of images. The user has access to reduced images and the catalog, both data products are stable within a 'release', which usually refers to the sky coverage performed by the survey.</text>
<text><location><page_9><loc_12><loc_36><loc_70><loc_40></location>In this way the survey data center provides the user with a specific version of the data product that will not change over time, which covers most of the science use-cases.</text>
<text><location><page_9><loc_12><loc_25><loc_70><loc_36></location>Nevertheless there are a number of use-cases which can not be satisfied by the 'standard' version. For example, in the search for objects like brown dwarfs or quasars it is important to lower the detection threshold which implies the reprocessing of the data. The user himself has to care about such reprocessing thereby reinventing the whole data processing system for the survey and involving his own resources. The Astro-WISE system allows to work on any use-case using the standard pipelines and performing programming on a minimal level - if this is necessary at all.</text>
<text><location><page_9><loc_12><loc_22><loc_70><loc_25></location>Figure 1 shows principles of WISE concept in the binding together all layers of the information system. Data processing pipelines define use-cases and data model</text>
<figure>
<location><page_10><loc_18><loc_64><loc_65><loc_90></location>
<caption>Fig. 2 Implementation of WISE concept in the data storage. Each persistent Python class inherits interfaces to the metadata and data layers.</caption>
</figure>
<text><location><page_10><loc_12><loc_54><loc_70><loc_59></location>for the system, which is implemented in the Python classes which wrap pipeline and define the common persistent data model objects. The data layer uses this data model, and interfaces are providing an access to the system for users. The common data model effectively brings together all layers of the information system.</text>
<text><location><page_10><loc_12><loc_48><loc_70><loc_53></location>The common data model can be modified by a mutial agreement of all users of the system. Any user can propose changes in the data model, and, if changes were accepted, the system administrator implements these changes in corresponding Python classes and in the database scheme.</text>
<text><location><page_10><loc_12><loc_45><loc_70><loc_48></location>In the next sections we will review in detail the infrastructural layers of AstroWISE and their implementation.</text>
<section_header_level_1><location><page_10><loc_12><loc_41><loc_26><loc_43></location>4.1 Metadata Layer</section_header_level_1>
<text><location><page_10><loc_12><loc_35><loc_70><loc_40></location>The bulk of the data stored in Astro-WISE is stored in files of some format (FITS). Each file is registered in the Astro-WISE metadata database with the unique filename. Apart from just registering each file in the system the metadata database implements the important part of WISE approach - the common data model.</text>
<text><location><page_10><loc_12><loc_27><loc_70><loc_34></location>Figure 2 shows the general outline for all data models implemented in informations systems based on WISE concept. two core classes DBObject and DataObject are parent classes for the class which is storing metadata persistently and the class which is storing the data in the file. These classes include interfaces to the metadata database and the data storage, the physical implementation of the metadata database and the data (files) storage can be different for different systems.</text>
<text><location><page_10><loc_12><loc_22><loc_70><loc_26></location>Let us suppose that we wish to use Astro-WISE for the data processing of optical images. Figure 3 shows typical classes of data items used by the optical data processing pipeline to reduce the data from raw images to the final science ready</text>
<figure>
<location><page_11><loc_15><loc_57><loc_67><loc_90></location>
<caption>Fig. 3 A set of Astro-WISE classes for an optical data processing pipeline. Each class is a persistent one stored in the metadata database and linked to the file stored in Astro-WISE system.</caption>
</figure>
<figure>
<location><page_11><loc_15><loc_40><loc_67><loc_50></location>
<caption>Fig. 4 Input, intermediate and final data products of the typical processing pipeline.</caption>
</figure>
<text><location><page_11><loc_12><loc_32><loc_70><loc_35></location>catalogues. This data model, deduced from the pipeline and enhanced keeping in mind possible scientific use-cases, is a central part of the metadata layer.</text>
<text><location><page_11><loc_12><loc_22><loc_70><loc_32></location>The data model is implemented both in the relational database (currently Oracle 11g RAC is used) and in the hierarchy of Python classes. All core classes are made persistent, i.e., any change in the object is mirrored in the corresponding tables of the database. The method used for the implementing of the data lineage is the Persistent Object Hierarchy (Fig. 2). According to this method objects of Astro-WISE are made persistent recursively, all operations and attributes of the object are saved in the metadata database.</text>
<text><location><page_12><loc_12><loc_85><loc_70><loc_89></location>As an example let us explore the data reduction task - we will create a reduced image from the raw image on one CCD chip. First of all, we will select an image we wish to process:</text>
<text><location><page_12><loc_12><loc_83><loc_76><loc_84></location>awe> raw = (RawScienceFrame.filename == 'WFI.2000-01-01T08:57:15.410_3.fits')[0]</text>
<text><location><page_12><loc_12><loc_74><loc_71><loc_82></location>The image is selected by browsing the metadata database for an object of class RawScienceFrame with filename WFI.2000-01-01T08:57:15.410_3.fits . The returned object raw was retrieved from the metadata database with all attributes. In the next step we will retrieve from the database all necessary calibration objects. The logic in the function select_for_raw allows to take the calibration files for the same night of observation ( 2000-01-01T08:57:15.410 ) or closest to this night.</text>
<text><location><page_12><loc_12><loc_72><loc_46><loc_73></location>awe> hot = HotPixelMap.select_for_raw(raw)</text>
<text><location><page_12><loc_12><loc_71><loc_47><loc_72></location>awe> cold = ColdPixelMap.select_for_raw(raw)</text>
<text><location><page_12><loc_12><loc_70><loc_49><loc_71></location>awe> flat = MasterFlatFrame.select_for_raw(raw)</text>
<text><location><page_12><loc_12><loc_68><loc_44><loc_69></location>awe> bias = BiasFrame.select_for_raw(raw)</text>
<text><location><page_12><loc_12><loc_65><loc_70><loc_67></location>Now we can instantiate a new object - our target - a reduced image which will be based on all the images above:</text>
<code><location><page_12><loc_12><loc_56><loc_40><loc_64></location>awe> reduced = ReducedScienceFrame() awe> reduced.raw = raw awe> reduced.hot = hot awe> reduced.cold = cold awe> reduced.bias = bias awe> reduced.flat = flat</code>
<text><location><page_12><loc_12><loc_51><loc_70><loc_55></location>Please, note that all images which will be used to create a new reduced image reduced are referenced through attributes of the new image. This new image should get a unique ID, which is a name of the file where the image will be stored.</text>
<text><location><page_12><loc_12><loc_49><loc_33><loc_50></location>awe> reduced.set_filename()</text>
<text><location><page_12><loc_12><loc_46><loc_72><loc_48></location>Now the image can be created invoking the make() method of the ReducedScienceFrame class.</text>
<text><location><page_12><loc_12><loc_44><loc_27><loc_45></location>awe> reduced.make()</text>
<text><location><page_12><loc_12><loc_40><loc_70><loc_43></location>The make() produced a file with a new image which will be stored on one of the dataservers of Astro-WISE (see Section 4.2):</text>
<text><location><page_12><loc_12><loc_38><loc_28><loc_39></location>awe> reduced.store()</text>
<text><location><page_12><loc_12><loc_35><loc_70><loc_37></location>Finally, the metadata database will be updated and the new image will become a part of the metadata layer.</text>
<text><location><page_12><loc_12><loc_33><loc_29><loc_34></location>awe> reduced.commit()</text>
<text><location><page_12><loc_12><loc_22><loc_70><loc_32></location>The sequence described above is a general way to create a new data entity in Astro-WISE: to combine a set of references to entities which will be used to create a new one, to invoke a pipeline or a part of pipeline which will generate a new entity, save the created data in the data layer, commit a new data entity to the metadata layer. The processing parameters, for example, the method used for the overscan of the image, will be saved as a persistent attribute of the new data entity as well.</text>
<figure>
<location><page_13><loc_18><loc_67><loc_64><loc_90></location>
<caption>Fig. 5 Data storage network of Astro-WISE. Dataservers are grouped geographically with one dataserver dedicated to the external data exchange. The user requests a file from a local dataserver and, if the file is not found in the local group, the local dataserver will request the file from the other dataserver groups. As soon as file is found it is copied to the dataserver that the user contacted and provided to the user.</caption>
</figure>
<text><location><page_13><loc_12><loc_47><loc_70><loc_58></location>The important feature of keeping the full data lineage in the system is an ability to avoid unnecessary reprocessing. The execution of the make() method of reduced object actually will start with the search in the metadata database for an object with the same attributes, and if such an object exists (and the user is allowed to retrieve it according to the user's permissions) the user will be redirected to an already existing object and the processing part of the method will be skipped. The data lineage allows to avoid unnecessary reprocessing as well as to use forward and backward chaining in the dependencies of the data items in the system.</text>
<text><location><page_13><loc_12><loc_41><loc_70><loc_47></location>Objects can be deleted from the database under the following restrictions: every user can delete the data he created at the privileges level 1 (see 4.5). For higher privileges levels only the project manager is allowed to delete. To delete a data object myobject from the database Context class is used (see [16] for the full description):</text>
<section_header_level_1><location><page_13><loc_12><loc_38><loc_37><loc_39></location>awe> Context().delete(myobject)</section_header_level_1>
<text><location><page_13><loc_12><loc_31><loc_70><loc_37></location>Objects can be deleted if they are not referenced by other data objects. If they are, it might be desirable to invalidate the object, so that the object will stay in the system but will not be used for the data processing (unless the user specifically ask for invalidated objects). The user-friendly service for the validation of data objects is described in [17].</text>
<section_header_level_1><location><page_13><loc_12><loc_27><loc_23><loc_28></location>4.2 Data Layer</section_header_level_1>
<text><location><page_13><loc_12><loc_22><loc_70><loc_25></location>Each data entity in Astro-WISE has two parts: a data part which is stored in a file and a metadata which is a part of the metadata database and which keeps all</text>
<text><location><page_14><loc_12><loc_82><loc_70><loc_89></location>dependencies of the data entity. The data part of the entity is stored as a file on one of the Astro-WISE dataservers. The predefined file format can be changed, but for most files the format is FITS. Nevertheless the dataserver is not limited to a single file format and can store files of any type. The dataserver is a specific solution which in its functionality is closests to the storage element of the Grid.</text>
<text><location><page_14><loc_12><loc_74><loc_70><loc_82></location>The main requirement to the dataservers as storage space is, apart from the safety, scalable size. The size should be scalable at the Terabyte level allowing to increase the storage volume from a few Terabytes to almost a Petabyte (this is the typical scale for the KIDS data archive from a few raw images at the beginning of the survey to hundreds of Terabytes with all intermediate data products at the end of observations).</text>
<text><location><page_14><loc_12><loc_66><loc_70><loc_74></location>Dataservers are organized in geographically close clusters, within each cluster all dataservers know about the existence of all other dataservers in the cluster. A dataserver in one cluster can contact other clusters via a dedicated dataserver in another cluster (see Fig. 5). All communications are done using a standard HTTP protocol. Two types of requests are available: to retrieve a file and to store a new file.</text>
<text><location><page_14><loc_12><loc_63><loc_70><loc_66></location>The unique name of the file is used as a unique identifier and this unique name is stored in the database as part of the metadata.</text>
<text><location><page_14><loc_12><loc_54><loc_70><loc_63></location>The user of the system does not know the actual location of the file on the underlying file system and operates with the URL of the file only. The URL has the form http://<data server address>/<file name> . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers.</text>
<text><location><page_14><loc_12><loc_35><loc_76><loc_54></location>On the request from the user to retrieve the file the dataserver can then either return the requested file or redirect the client to the dataserver that has the file. Each dataserver has a permanent data storage and a cache, the last one is used to temporarily store a file retrieved from the other dataserver. By user request, for example, http://ds.astro.rug.astro-wise.org:8000/WFI.2000-09-28T02:22:37.466_8.fits the dataserver ds.astro.rug.astro-wise.org will check cache space for the file WFI.2000-09-28T02:22:37.466_8.fits . If the file is not found, the dataserver will check it's own permanent storage space, and if there is no such file, all dataservers in a 'local' cluster ( astro.rug.astro-wise.org ) will be requested. If there is still no file found, all other clusters will be requested, and as soon as the file is located it will be copied to the cache of the ds.astro.rug.astro-wise.org and returned to the user. In addition, a file can be compressed or decompressed on-the-fly during retrieval, and a slice of the data can be retrieved by specifying the slice as part of the URL.</text>
<text><location><page_14><loc_12><loc_31><loc_70><loc_35></location>The dataserver is written in Python and can be installed on any operating system and underlying filesystem that can support long case-sensitive filenames. For the Astro-WISE Linux filesystems, XFS and GPFS are currently in use.</text>
<section_header_level_1><location><page_14><loc_12><loc_26><loc_27><loc_28></location>4.3 Processing Layer</section_header_level_1>
<text><location><page_14><loc_12><loc_22><loc_70><loc_25></location>The example in the Section 4.1 involves the processing facilities of Astro-WISE, which are distributed and combined from processing facilities of all Astro-WISE</text>
<text><location><page_15><loc_12><loc_85><loc_70><loc_89></location>partners. The user has the choice to send the job to one of the processing elements of Astro-WISE (including Grid computing elements) or to use the processing power at the user's disposal, for example, PC or even notebook.</text>
<text><location><page_15><loc_12><loc_69><loc_70><loc_85></location>In the core of the processing layer of Astro-WISE is a distributed processing unit (DPU). This is a three-component middleware which consists of the DPU server, DPU client and the so called DPU runner. The DPU server is a front-end to any processing system and interacts with the processing system's native queuing system, allowing the user to submit jobs to this particular queuing system, inspect or cancel them. The DPU client includes all functions and methods the user can call to interact with DPU server. The DPU runner is a program which is run on the remote processing facility and checks availability of all necessary software to run an Astro-WISE job, installing required packages if necessary. The system runs on openpbs or under its own queue management software. The DPU itself allows synchronizations of jobs as well and can also transfer parts of a sequence of jobs to other DPU's.</text>
<text><location><page_15><loc_12><loc_62><loc_70><loc_68></location>In the case of Grid computing element the DPU server will check a user's identity and will use the user's credentials (Grid certificate) to mediate with the Virtual Organization Management System. Currently OmegaCEN is using omegac Virtual Organization with an ability to submit jobs to Grid computing elements in Amsterdam and Groningen.</text>
<section_header_level_1><location><page_15><loc_12><loc_57><loc_22><loc_58></location>4.4 Interfaces</section_header_level_1>
<text><location><page_15><loc_12><loc_48><loc_70><loc_56></location>The description of Astro-WISE system would be incomplete without the description of a number of interfaces provided for the user. The user can write his own applications in the Python language calling Astro-WISE libraries or can involve Astro-WISE services. The first case requires the use of the Astro-WISE Command Line Interface called AWE (Astro-WISE Environment), which can be installed on any site, PC or notebook</text>
<text><location><page_15><loc_12><loc_37><loc_70><loc_47></location>The Command Line Interface - CLI - supposes that the user writes his own programs using Python, but to browse the data or even to process observations there is no need to use the CLI. All operations required to perform this activity are possible with a set of standard web services of Astro-WISE. Of course, the use of the CLI gives to the user much more freedom in data processing. The CLI is more useful for experienced users working on a particular use-case, meanwhile web services are developed for routine operations during the data processing of the surveys.</text>
<text><location><page_15><loc_12><loc_34><loc_70><loc_36></location>The web interfaces are divided into two types: data browsing/exploration and data processing/qualification. The first group includes:</text>
<unordered_list>
<list_item><location><page_15><loc_13><loc_30><loc_70><loc_33></location>* dbviewer 1 - the metadata database interface which allows browsing and querying of all attributes of all persistent classes stored in the system,</list_item>
<list_item><location><page_15><loc_13><loc_26><loc_70><loc_30></location>* quick data search 2 - allows querying on a limited subset of attributes of the data model (coordinate range and object name), and provides results of all projects in the database,</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_16><loc_13><loc_84><loc_70><loc_89></location>* image cut out service 3 and color image maker 4 - these two services are for the astronomical image data type and allow to create a cut out of the image or to create a pseudo-color RGB image from three different images of the same part of the sky,</list_item>
<list_item><location><page_16><loc_13><loc_81><loc_70><loc_84></location>* GMap 5 - exploration tool of the Astro-WISE system using the GoogleSky interface.</list_item>
</unordered_list>
<section_header_level_1><location><page_16><loc_12><loc_79><loc_45><loc_80></location>Data processing / qualification interfaces are:</section_header_level_1>
<unordered_list>
<list_item><location><page_16><loc_13><loc_73><loc_70><loc_78></location>* target processing 6 - the main web tool to process the data in Astro-WISE. This web interface allows users to go through pre-defined processing chains, submitting jobs on the Astro-WISE computing resources with the ability to select the computing node of Astro-WISE,</list_item>
<list_item><location><page_16><loc_13><loc_70><loc_70><loc_73></location>* quality service 7 - allows to estimate the quality of the data processing and set a flag highlighting the quality of the data,</list_item>
<list_item><location><page_16><loc_13><loc_69><loc_65><loc_70></location>* CalTS 8 - web interface for identifying and qualifying calibration data.</list_item>
</unordered_list>
<text><location><page_16><loc_12><loc_64><loc_70><loc_68></location>All web services are built using a set of Python classes developed for AstroWISE as a basis and uses a modular principle, which allows to create a new web service using components of older ones.</text>
<section_header_level_1><location><page_16><loc_12><loc_60><loc_45><loc_61></location>4.5 Authorization and Authentication System</section_header_level_1>
<text><location><page_16><loc_12><loc_53><loc_70><loc_58></location>Astro-WISE is a multi-user system which must accommodate sharing data between scientists and at the same time protect the private data of each user. Each user in the Astro-WISE system has an identity protected by a password, and, optionally, if the user wants to submit job to Grid resources he has to get a Grid certificate.</text>
<text><location><page_16><loc_12><loc_42><loc_70><loc_52></location>The authorization and authentication system is implemented on the level of the metadata database. As soon as a user logged in with his username/password, the user's privileges are checked in the database, allowing the user to browse the data according to the user's privileges while obeying the privileges of other users. The data in the metadata database are grouped by projects . A Project is a collection of resources which is associated with a group of users who can access these resources, usually one of these users has the role of Project Manager who has privileges to include new users, remove users and publish the data to the wider community.</text>
<text><location><page_16><loc_12><loc_31><loc_70><loc_42></location>The system of access to the data is based on three attributes which any data entity in Astro-WISE has: user , project and privileges . The first one identifies the user that created the data entity. The second one defines to which project the data entity belongs, the third one defines who is able to use this data entity. All these attributes are initialized the first time the data entity is made persistent in the Astro-WISE system (including the case that the entity is created by one of the Astro-WISE pipelines) and they are both persistent attributes, i.e., stored in the metadata database.</text>
<table>
<location><page_17><loc_12><loc_79><loc_52><loc_87></location>
<caption>Table 1 Privileges system of Astro-WISE</caption>
</table>
<text><location><page_17><loc_12><loc_66><loc_70><loc_77></location>Table 1 shows the range of values for the privileges attribute. In the case of privileges=1 only the creator of the data item can see it, the creator can raise privileges to privileges=2 , in this case all the users of the project will be able to browse this data entity. Raising privileges to 3 the user makes the data item accessible for all Astro-WISE users in all projects, and with priveleges=4 the anonymous user (a user with minimal read-only privileges in the system) can see it as well. Finally, with priveleges=5 the data item is accessible via Astro-WISE Virtual Observatory interfaces.</text>
<text><location><page_17><loc_12><loc_57><loc_70><loc_66></location>The special Python class Context was created to handle authorization and authentication. The Context allows to change privileges of all user's data items, during the change Context is checking for dependencies to prevent inconsistency in the dependencies due to the different privileges of data items. For example, if the raw image was published by the user to the project scope and some other user has created a reduced image from this raw image, the original user will not be able to downgrade privileges on the raw image to the private data scope.</text>
<section_header_level_1><location><page_17><loc_12><loc_53><loc_28><loc_54></location>4.6 WISE architecture</section_header_level_1>
<text><location><page_17><loc_12><loc_41><loc_70><loc_52></location>We described above three infrastructural components of Astro-WISE: the relational DBMS (the metadata layer), dataserver (the data layer) and DPU (the processing layer). Combined with user interfaces these components build an AstroWISE node - a detached Astro-WISE site which can operate independently from other sites. Multiple nodes can be combined to form a distributed system. Each node is independent from others in the sense that it is administrated independently and can handle both the data distributed over other nodes and the data restricted to this node only and shielded from other nodes.</text>
<text><location><page_17><loc_15><loc_40><loc_44><loc_41></location>Figure 6 shows the elements of the node:</text>
<unordered_list>
<list_item><location><page_17><loc_13><loc_38><loc_40><loc_39></location>* Data storage servers (dataservers).</list_item>
<list_item><location><page_17><loc_13><loc_35><loc_70><loc_38></location>* A metadata database to store a full description of each data file with links and references to other objects.</list_item>
<list_item><location><page_17><loc_13><loc_32><loc_70><loc_35></location>* Astro-WISE programming environment (CLI) along to web services which give the user an ability to access to the stored data and launch a data processing.</list_item>
<list_item><location><page_17><loc_13><loc_30><loc_70><loc_32></location>* Computing nodes for the data processing with DPU servers on top of their quing system.</list_item>
<list_item><location><page_17><loc_13><loc_27><loc_70><loc_29></location>* A version control system for developers to include new modules, classes and libraries into the system.</list_item>
</unordered_list>
<text><location><page_17><loc_12><loc_22><loc_70><loc_26></location>All these elements are optional, an Astro-WISE node can be installed without dataservers (dataservers from other nodes are used), metadata database or processing facilities. In fact an Astro-WISE node can be installed on a notebook as</text>
<figure>
<location><page_18><loc_16><loc_59><loc_68><loc_90></location>
<caption>Fig. 6 Present-day Astro-WISE nodes with a typical composition of a node in a full deployment node.</caption>
</figure>
<text><location><page_18><loc_12><loc_51><loc_70><loc_54></location>Astro-WISE environment only giving the user access to the system - if the user is not satisfied with web services on the remote nodes.</text>
<text><location><page_18><loc_12><loc_46><loc_70><loc_51></location>Presently Astro-WISE includes sites at Groningen University, Leiden University and Nijmegen University (The Netherlands), Argelander-Institut fur Astronomie, Bonn and Universitats-Sternwarte Munchen (Germany) and Osservatorio Astronomico di Capodimonte, Napoli (Italy).</text>
<section_header_level_1><location><page_18><loc_12><loc_41><loc_45><loc_42></location>5 Astro-WISE: migration to other systems</section_header_level_1>
<text><location><page_18><loc_12><loc_33><loc_70><loc_40></location>The WISE concept for an information system is a set of principles described in Section 4, Astro-WISE is the first system which realizes this concept and serves as a basis for the further development of information systems. In this section we describe the adaptivity of the parent system to new tasks and challenges on the example of LOFAR Long Term Archive and Molgenis system.</text>
<text><location><page_18><loc_12><loc_25><loc_70><loc_33></location>The importance of the LOFAR Long Term Archive (LTA) for the development of WISE concept is a necessity to use an external infrastructure which should be included in the architecture to form a complete information system. We preserved all principles of Astro-WISE adding storage and processing which is not controlled by the system itself (BiGGrid 9 ). Additionally we integrated three different systems of Authorization and Authentication to make it possible for the LOFAR observa-</text>
<text><location><page_19><loc_12><loc_87><loc_70><loc_89></location>tory to create new users and control resources in the system. The LOFAR LTA design is described in [20] and the architecture and infrastructure in [21].</text>
<text><location><page_19><loc_12><loc_72><loc_70><loc_86></location>Another significant development was achieved with the adaptation of Molgenis 10 . Molgenis is a framework written in Java to build user interfaces and databases from definitions that are written in XML. From these definitions the database tables and webserver are generated. Its origins lie in biomedical applications. Historically Molgenis focused more on interaction with the user to streamline the 'protocols' (recipes for analysis) of researchers to keep track of analyses of results. On the other hand, the WISE technology focused more on results from a processing perspective. This lead to the idea to combine the strengths of both frameworks and use some models that already existed in XML for Molgenis to generate a datamodel in Python for Astro-Wise. Then Molgenis could be extended to use the WISE infrastructure for distributed storage and processing.</text>
<text><location><page_19><loc_12><loc_66><loc_70><loc_71></location>The existing Molgenis XML model is organized in a common fashion through 'module', 'entity' and 'field' where each field has a type of int , float , str or can be a reference type such as mref or xref . Most of these have a counterparts in the WISE framework which made it possible to write a conversion tool.</text>
<text><location><page_19><loc_12><loc_54><loc_70><loc_66></location>The webserver that is generated by Molgenis has to communicate to a database that it usually creates itself. However, in this case the database had to be generated by Astro-Wise because the frameworks did not have a backend for a common database (Oracle vs. Non-Oracle). Instead of writing a new database backend for either framework it was decided to write an xmlrpc interface to encapsulate database queries and return their results. Since client software for all database flavours does not have to be present this functionality can be extended, e.g., since this is xmlrpc, a programming language independent implementation could also use the database in a way that the WISE framework dictates.</text>
<text><location><page_19><loc_12><loc_47><loc_70><loc_54></location>The approach developed and tested in the case of Molgenis allows to create a new information system based on Astro-WISE with all services starting from the data model coded in XML and to do this automatically. This approach allowed to decrease the time and resources spent for the developing and implementation of a new system significantly.</text>
<text><location><page_19><loc_12><loc_44><loc_70><loc_47></location>The next system which will be created with the approach tested on Molgenis is a data processing system for Multi Unit Spectroscopic Explorer 11 (MUSE).</text>
<section_header_level_1><location><page_19><loc_12><loc_41><loc_36><loc_42></location>6 Conclusion and Future Work</section_header_level_1>
<text><location><page_19><loc_12><loc_32><loc_70><loc_39></location>The Astro-WISE information system, the first information system in Astronomy, proved to be a reliable and flexible tool for the data processing. Originally developed to process the data of KIlo Degree Survey (KIDS 12 ) it triggered development of the unique approach to the architecture of scientific information systems (WISE approach).</text>
<text><location><page_19><loc_12><loc_28><loc_70><loc_32></location>Both Astro-WISE and the WISE approach are living systems which are open to improvements. For the last 2 years further development of the WISE approach is hosted by Target Holding 13 . Target Holding is an expertise center in the Northern</text>
<text><location><page_20><loc_12><loc_82><loc_70><loc_89></location>Netherlands which is building a cluster of sensor network information systems and provides cooperation between a number of scientific projects and business partners like IBM and Oracle. Target creates and supports a hardware infrastructure for hosting tens of Petabytes of data for projects in astronomy, medicine, artificial intellegence and biology.</text>
<text><location><page_20><loc_12><loc_78><loc_70><loc_82></location>In this development the WISE approach was used to create new information systems extending the original Astro-WISE on new data models, new data storage and processing capacities and new fields.</text>
<text><location><page_20><loc_12><loc_69><loc_70><loc_76></location>Acknowledgements Astro-WISE is an on-going project which started from a FP5 RTD programme funded by the EC Action 'Enhancing Access to Research Infrastructures'. This work was performed as part of the Target project. Target project is supported by Samenwerkingsverband Noord Nederland. It operates under the auspices of Sensor Universe. It is also financially supported by the European fund for Regional Development and the Dutch Ministry of Economic Affairs, Pieken in de Delta, the Province of Groningen and the Province of Drenthe.</text>
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</document>
[
{
"title": "The Astro-WISE datacentric information system",
"content": "K. Begeman, A.N. Belikov, D.R. Boxhoorn, E.A. Valentijn Received: date / Accepted: date Abstract In this paper we present the various concepts behind the Astro-WISE Information System. The concepts form a blueprint for general scientific information systems (WISE) which can satisfy a wide and challenging range of requirements for the data dissemination, storage and processing for various fields in science. We review the main features of the information system and its practical implementation. Keywords Data Grid · Grid Computing · Information System · Middleware",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Digital astronomical catalogues have been built from the very first moment information technology enabled this, e.g. the first Abell catalogue of clusters of galaxies was digitally prepared and printed in the days minus signs were not available in print [1]. As soon as digital scanning devices became available photographic material was scanned and first large image surveys were digitally published, such as ESO-LV [2] and the digitized sky surveys like DPOSS [3], succeeded by the Palomar-Quest Survey [4], in the 90's followed by CCD-based surveys such as the 2MASS survey [5], the Sloan Digital Sky Survey SDSS [6]. The rapid accumulation of astronomical digital data and its public dissemination was, compared to other disciplines, achieved at an early stage, thanks to an open and collaborative astronomical world community who adopted the FITS image data format as early as 1979 [7]. The CDS, initially Centre de Donn'ees Stellaires and later renamed into Centre de Donn'ees astronomiques de Strasbourg took the lead in Europe to collect and disseminate the ever growing data sets of a zoo of astronomical observatories and projects. VizieR webservices nowadays provide access to over 9000 catalogues. Numerous astronomical data centers followed, developing specialized services, for example, the Infrared Processing and Analysis Center and the bibliographical SAO/NASA Astrophysics Data System. In the early 2000's it was realized that the ever growing data volumes require new approaches: the community becomes the data provider and the International Virtual Observatory and its European branch, the European Virtual Observatory, developed standards and interfaces to allow individual data centers to publish their catalogs and images in a common framework providing worldwide access to users who can query, cross-match and visualize multiple databases via the internet. A highlight formed the overplotting of data from different experiments, like Xray satellites and optical ground-based observatories with some keystrokes on the Aladin interactive software sky atlas. While the Virtual Observatory focused on the dissemination of published data it was also realized in the early 2000's that the upcoming data deluge required a new approach to the handling of the data stream from the telescope to the science-ready result. Modern experiments not only involve more data, but also require increasing precision on the various calibrations. The dependency of the final result (sometimes only a few numbers, like cosmological parameter values) on time-variable calibrations of very large data sets, involving the evaluation by large research teams distributed in smaller groups over various sites sets the basic requirements to the system handling the data. While in classical systems data is delivered in various releases to the public, often re-processing the whole set with higher versions of the code, the high data rates of modern experiments require an alternative approach, where the up-to-date result is derived on demand by the user. Thus, we set out to design and implement an integrated datacentric system Astro-WISE in which the processing, storage and administration is integrated in a single environment, providing a living system to both the data producers and the customers. Early reports of this development and implementation have been given in [8] and [9]. To reach this goal, traceability of each individual data item handled by pipelines or any piece of code is carefully maintained, every data item beyond pixel value is kept as Metadata and made persistent and distributed in a relational database with an object-oriented view, mapping all the dependencies.",
"pages": [
1,
2
]
},
{
"title": "2 Datacentric Approach to Data Processing",
"content": "The datacentric approach facilitates scientists to cooperate in data analysis and mining by means of the internet, nowadays referred to as e-science. The AstroWISE information system connects to an own developed distributed grid processing system (based on Distributed Processing Unit, DPU), but has been also connected to the EGEE/EGI/BiGGrid-Grid [10], particularly for the operations of the Lofar radio telescope. Nowadays, this aspect of the system is referred to as cloud computing, but the Astro-WISE datacentric approach at the same time facilitates the sharing and web hosting of all the data handled by the system. Today, we are just taking the first small steps towards the development of approaches and systems who could handle the upcoming data deluge. The way of managing and administration the data will be a key, and will require ever improving, refining and self-organizing approaches. The key notion is that a fruitful approach towards handling the data deluge focuses on the administration, modeling, standardizing and tracking of the data rather than on the processing of the data by CPUs. This datacentric approach, in which the design of operational systems is driven by data management and data standard issues rather than data processing/cpu/pipelining issues. Processing systems turn into information systems. First attempts to combine archiving and processing in the 1980s include ESO's MIDAS table file system [11], integrating common tables both accessible in source code and by user interfaces like the MIDAS monitor/prompt. The MIDAS table file system allows both users and programmers to change its content and also registering the modifications in table keywords leading to archives ready for further review and (trend) analysis. Its development was essential for the production and photometric calibration of the images of the ESO-LV galaxies. However, the 600,000 fore- and background objects visible on the images, stars and galaxies, could only be handled with great difficulties inside the system, response times running into hours per operations, demonstrating the importance of built-in scalability. This MIDAS Table file system marked a stepping stone to current datacentric approaches which are enabled by the introduction of modern object-oriented programming languages, data modeling tools like UML and XML and relational databases. Wehave combined these views, systems and requirements in order to design and build integrated information systems which have the potential for self-organizing and enrichment with new data. Both self-organizing and enrichment are processes in time. In fact, the datacentric approach implies the detailed modeling and awareness of how things change in time. Important changes range from The ideal information system would seamlessly cope with all these changes in time, thus creating a living long term digital preservation environment. Astro-WISE is one of the first systems attempting to reach this goal. For several years many authors predicted the upcoming data avalanche due to the advancements in digital sensor technology. Obviously, this has now started, both scientific experiments such as LHC and Lofar are running into the tens of Petabytes data acquisition regimes, while text, imaging, genome and internet gathered data collections like e-Bay are exceeding similar volumes. This triggered the development of Grid infrastructure that is able to handle Petabytes of data ([12] and [13]). All this has happened in an approximate 20 year time span when the first large digital imaging collection was published, ESO-LV [2], containing scans of 32,000 galaxy images, 4 Gbyte of data filling a room full with tapes and populating the very first version of optical media. By now, the data avalanche is multidisciplinary and worldwide and involves all aspects of our society, from science to commerce, public services and health care. Digital data is gathered, processed, distributed and accessed at a continuously growing rate. And this is only the beginning: in, say, two decades, the same room full with 4 Gbyte of tapes 20 years ago, and 40 Petabyte of tapes today could possibly contain 40 Zetabytes (40 million Petabytes) of data. The growth of the data volume produced by various types of sensor networks met with the growth of the disk capacity and search for new methods in the increasing of the memory storage density (see [14]). In order to be useful all this data has to be organized, administrated, managed and distributed. It took a billion years for cells to develop into extremely complex information systems around the DNA; it has resulted into systems and approaches with an incredible complexity with memory mapping, copying, filtering and distribution mechanisms of data including feedback. Compared to this, our present data organizing mechanisms in IT are still very simple and rudimentary and we are just making the first steps to living self-organizing systems and archives required to deal with and optimally use the ocean of digital data to come. However, also in the cell stochastic processes are important, and absence of organization 'from above' are countered by survival of the fittest type of mechanisms, which appear as self-organizing. Also in human endeavored IT, standards seldom come from above, an exception being the ASCII character standards, mandated in 1968 by U.S. President Lyndon B. Johnson to be used in all computers purchased by the United States federal government. I have also approved recommendations of the Secretary of Commerce regarding standards for recording the Standard Code for Information Interchange on magnetic tapes and paper tapes when they are used in computer operations. All computers and related equipment configurations brought into the Federal Government inventory on and after July 1, 1969, must have the capability to use the Standard Code for Information. Interchange and the formats prescribed by the magnetic tape and paper tape standards when these media are used.In practice, standards are often invented by companies or scientific communities which are discriminated in a complex survival of the fittest battle, which involves that many factors that it appears stochastic. The view, or better requirement, for future self-organizing information systems and its archives is that it organizes itself for all these changes in time. It could be seen as a step towards the extremely complex and self-organizing information systems in a cell. The core problem for the definition of an information system is the definition of information itself. The information can be an input data flow from sensor detectors, an archive of transactions, trends deduced from the analysis of some data and even results of the data modeling or simulations. As result, information systems will host a variety of subsystems defined by its purpose. Examples are a Decision Support System gathering and analyzing trends for a specific business use case (see [15]) or Geographical Information Systems with focus on geospatial data analysis and a visualization of the data. Nevertheless, it is possible to define primary components of any Information System following simple requirements on storage, processing and sharing data. Such a system should include a data model, storage and data processing facilities together with user interfaces. In Astro-WISE we have decided to separate and store all data beyond pixel/measurement data in a database and all pixel/measurement data in files on independent storage media. This paper does not touch details of the implementation of Astro-WISE information system which are described in other papers but describes the general outline of the Astro-WISE information system and specify features which make Astro-WISE unique. For the more in-depth review of particular aspects of AstroWISE we recommend to read corresponding papers on the optical pipeline implemented in Astro-WISE [16], quality control realised in Astro-WISE [17], the system of user interfaces and services [18], and the hardware solution for Astro-WISE and information systems created on the basis of Astro-WISE [19].",
"pages": [
2,
3,
4,
5
]
},
{
"title": "3 Building a Scientific Information System",
"content": "The development of the Astro-WISE information system, the first one which implemented WISE technology, started from the very practical challenge: enable a community of researchers distributed over the world to process the data of astronomical imaging survey of the OmegaCAM 256 Megapixel camera on ESO's VST telescope at Cerro Paranal (Chile). These scientists should be able to evaluate the quality of the data, apply a number of calibrations, share the data within the team and employ distributed resources of a Petabytes scale of data storage and Teraflops capacity in data processing. From this our basic requirements on the system are derived: The most general requirements listed above results in a set of more detailed and specific requirements to different components of the information system. First, the ability to share data among a number of users becomes more valuable when all these users are working with the same set of standards for the data. Second, the requirement to trace any changes in the data processed by different users with different programs and pipelines implies that the system has a common approach to treat the data items on different stages of the data processing and keeps the record of changes in the data. As result the basic requirements to the system dictate the use of a common data model and the common standards for the data storage. Below we will describe the common data model features defined from the requirements to the system.",
"pages": [
5,
6
]
},
{
"title": "Requirements to the Data Model:",
"content": "When properly defined the data model allows to code, instantiate and trace the data processing for all scientific use-cases. Also, the data model should allow to accommodate external data, as well as provide necessary comparison and checking with these external data sets. When the data model is carefully defined our approach is to convert this into a pipeline model, in turn translated into a hierarchy of Classes. These Classes are then mapped into a database. We originally did this with object-oriented databases like Objectivity, but currently we do this in a relational database and build an object oriented view in order to maintain transparency in the class model, and its dependencies. Requirements to Infrastructure. The data model should be shared among geographically remote sites of the system, as research groups at various locations are collaboratively adding and extracting data from the distributed system. The basic requirement is that in fact any component (database, file system, code base, CPUs) should possibly be present at different sites, for a number of reasons: For example, the data ingestion to the system can be done in Garching, Germany, the data processing in Groningen, The Netherlands and the image analysis in Napoli, Italy. Requirements to the User Interfaces. The user should be able to trace the data processing, to retrieve data items and identify all its dependencies on other items (mostly calibrations) in the system, to initiate data processing of particular data items and to find a quality estimation for the data. As we can see, all these requirements put together are more challenging than requirements on common business information systems. The key difference is that we allow everything, including the code base to change in time. In the following chapter we outline how our requirements formed a layout for a scientific information system capable to share a huge data volume, to process these data and fulfill scientific use-cases for a number of users.",
"pages": [
6,
7
]
},
{
"title": "4 WISE Concept",
"content": "First of all we have to clearly distinguish between two classes of data storage systems in Astronomy: proper archives containing fixed published data and infor- mation systems. The main difference can be found in an ability to work with the data and to change it during the storage. For an archive stored data is a stable entity which is provided to the user as it is, meanwhile the information system allows to the user to modify their own data creating user-specific version of an archive and share this version with other users. The Astro-WISE approach, and its general multi-discipline WISE approach implements a scientific information system. The usual approach to building an information system is to perceive three components: the data layer, business rules and the interfaces (see Fig. 1). In our case, we separate the data layer in two parts: the pure measurement data layer, hereafter data layer, and everything beyond that, hereafter Metadata layer, ranging from file sizes, to statistics of pixel values and detected events in the measurements (galaxy, particle, word). The Metadata layer allows to implement a data model, the data layer allows to store the data as files in a standard format, while business rules implemented in Python classes bind the metadata and the data layers. Interfaces provide user access to both the business rules and the data. In the case of AstroWISE business rules present in the Metadata layer (Data Definition Language used to create a data model), data layer (the on-the-fly compression of the data on data storage grid) and-the most apparent to the user-in a number of pipelines and programs which the user defines to process the data. This last part of business rules components we will call data processing layer. To implement Astro-WISE we use abstractions of storage, processing and database capabilities as a basis for the infrastructure for each of the layers of the system. The metadata layer is realized in a relational DBMS through an abstraction of the required database functionality, the data layer is put on the Astro-WISE data storage grid through an abstraction of storage and the processing grid is used to connect the user with the data and metadata layers by a number of interfaces for the data processing layer. Separation of these three infrastructures plays a key role in the flexibility of the system, as we will see below. The metadata layer implements a list of necessary functionalities: Our requirements on distribution and multiple users propagate as key principles of the realization of metadata and data layers and business rules which form the core of the WISE approach: The Astro-WISE system is realized in the Python programming language. It allows to wrap any program into a Python module, library or class. The use of Python also allows to combine the principles of modular programming with objectoriented programming, so that each package in the system can be built and run independently with an object-oriented data model serving as glue between modules. At the same time, the logic behind pipelines and workflows in Astro-WISE allows the execution of part of the processing chain independently from the other parts. We will describe this approach in more detail in the example of optical image processing in Section 4.1. The conceptual difference between Astro-WISE and other existing systems is that Astro-WISE moves from the usual for astronomy processing-centric approach to a data-centric approach. The data processing itself becomes an integral part of the archive. The typical solution for the data processing and storage for Astronomy handles the data processing and the final data product delivered to the user as two completely separated entities. The data product usually is a result of the processing for the whole survey with fixed processing parameters used for the whole set of images. The user has access to reduced images and the catalog, both data products are stable within a 'release', which usually refers to the sky coverage performed by the survey. In this way the survey data center provides the user with a specific version of the data product that will not change over time, which covers most of the science use-cases. Nevertheless there are a number of use-cases which can not be satisfied by the 'standard' version. For example, in the search for objects like brown dwarfs or quasars it is important to lower the detection threshold which implies the reprocessing of the data. The user himself has to care about such reprocessing thereby reinventing the whole data processing system for the survey and involving his own resources. The Astro-WISE system allows to work on any use-case using the standard pipelines and performing programming on a minimal level - if this is necessary at all. Figure 1 shows principles of WISE concept in the binding together all layers of the information system. Data processing pipelines define use-cases and data model for the system, which is implemented in the Python classes which wrap pipeline and define the common persistent data model objects. The data layer uses this data model, and interfaces are providing an access to the system for users. The common data model effectively brings together all layers of the information system. The common data model can be modified by a mutial agreement of all users of the system. Any user can propose changes in the data model, and, if changes were accepted, the system administrator implements these changes in corresponding Python classes and in the database scheme. In the next sections we will review in detail the infrastructural layers of AstroWISE and their implementation.",
"pages": [
7,
8,
9,
10
]
},
{
"title": "4.1 Metadata Layer",
"content": "The bulk of the data stored in Astro-WISE is stored in files of some format (FITS). Each file is registered in the Astro-WISE metadata database with the unique filename. Apart from just registering each file in the system the metadata database implements the important part of WISE approach - the common data model. Figure 2 shows the general outline for all data models implemented in informations systems based on WISE concept. two core classes DBObject and DataObject are parent classes for the class which is storing metadata persistently and the class which is storing the data in the file. These classes include interfaces to the metadata database and the data storage, the physical implementation of the metadata database and the data (files) storage can be different for different systems. Let us suppose that we wish to use Astro-WISE for the data processing of optical images. Figure 3 shows typical classes of data items used by the optical data processing pipeline to reduce the data from raw images to the final science ready catalogues. This data model, deduced from the pipeline and enhanced keeping in mind possible scientific use-cases, is a central part of the metadata layer. The data model is implemented both in the relational database (currently Oracle 11g RAC is used) and in the hierarchy of Python classes. All core classes are made persistent, i.e., any change in the object is mirrored in the corresponding tables of the database. The method used for the implementing of the data lineage is the Persistent Object Hierarchy (Fig. 2). According to this method objects of Astro-WISE are made persistent recursively, all operations and attributes of the object are saved in the metadata database. As an example let us explore the data reduction task - we will create a reduced image from the raw image on one CCD chip. First of all, we will select an image we wish to process: awe> raw = (RawScienceFrame.filename == 'WFI.2000-01-01T08:57:15.410_3.fits')[0] The image is selected by browsing the metadata database for an object of class RawScienceFrame with filename WFI.2000-01-01T08:57:15.410_3.fits . The returned object raw was retrieved from the metadata database with all attributes. In the next step we will retrieve from the database all necessary calibration objects. The logic in the function select_for_raw allows to take the calibration files for the same night of observation ( 2000-01-01T08:57:15.410 ) or closest to this night. awe> hot = HotPixelMap.select_for_raw(raw) awe> cold = ColdPixelMap.select_for_raw(raw) awe> flat = MasterFlatFrame.select_for_raw(raw) awe> bias = BiasFrame.select_for_raw(raw) Now we can instantiate a new object - our target - a reduced image which will be based on all the images above: Please, note that all images which will be used to create a new reduced image reduced are referenced through attributes of the new image. This new image should get a unique ID, which is a name of the file where the image will be stored. awe> reduced.set_filename() Now the image can be created invoking the make() method of the ReducedScienceFrame class. awe> reduced.make() The make() produced a file with a new image which will be stored on one of the dataservers of Astro-WISE (see Section 4.2): awe> reduced.store() Finally, the metadata database will be updated and the new image will become a part of the metadata layer. awe> reduced.commit() The sequence described above is a general way to create a new data entity in Astro-WISE: to combine a set of references to entities which will be used to create a new one, to invoke a pipeline or a part of pipeline which will generate a new entity, save the created data in the data layer, commit a new data entity to the metadata layer. The processing parameters, for example, the method used for the overscan of the image, will be saved as a persistent attribute of the new data entity as well. The important feature of keeping the full data lineage in the system is an ability to avoid unnecessary reprocessing. The execution of the make() method of reduced object actually will start with the search in the metadata database for an object with the same attributes, and if such an object exists (and the user is allowed to retrieve it according to the user's permissions) the user will be redirected to an already existing object and the processing part of the method will be skipped. The data lineage allows to avoid unnecessary reprocessing as well as to use forward and backward chaining in the dependencies of the data items in the system. Objects can be deleted from the database under the following restrictions: every user can delete the data he created at the privileges level 1 (see 4.5). For higher privileges levels only the project manager is allowed to delete. To delete a data object myobject from the database Context class is used (see [16] for the full description):",
"pages": [
10,
11,
12,
13
]
},
{
"title": "awe> Context().delete(myobject)",
"content": "Objects can be deleted if they are not referenced by other data objects. If they are, it might be desirable to invalidate the object, so that the object will stay in the system but will not be used for the data processing (unless the user specifically ask for invalidated objects). The user-friendly service for the validation of data objects is described in [17].",
"pages": [
13
]
},
{
"title": "4.2 Data Layer",
"content": "Each data entity in Astro-WISE has two parts: a data part which is stored in a file and a metadata which is a part of the metadata database and which keeps all dependencies of the data entity. The data part of the entity is stored as a file on one of the Astro-WISE dataservers. The predefined file format can be changed, but for most files the format is FITS. Nevertheless the dataserver is not limited to a single file format and can store files of any type. The dataserver is a specific solution which in its functionality is closests to the storage element of the Grid. The main requirement to the dataservers as storage space is, apart from the safety, scalable size. The size should be scalable at the Terabyte level allowing to increase the storage volume from a few Terabytes to almost a Petabyte (this is the typical scale for the KIDS data archive from a few raw images at the beginning of the survey to hundreds of Terabytes with all intermediate data products at the end of observations). Dataservers are organized in geographically close clusters, within each cluster all dataservers know about the existence of all other dataservers in the cluster. A dataserver in one cluster can contact other clusters via a dedicated dataserver in another cluster (see Fig. 5). All communications are done using a standard HTTP protocol. Two types of requests are available: to retrieve a file and to store a new file. The unique name of the file is used as a unique identifier and this unique name is stored in the database as part of the metadata. The user of the system does not know the actual location of the file on the underlying file system and operates with the URL of the file only. The URL has the form http:// / . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers. . Each Astro-WISE service has pre-defined dataservers which are used retrieve data. Usually the administrator of the local Astro-WISE node assigns this dataserver selecting 'geographycally closest' dataserver. The user can change this assignment and use the dataserver the user prefers. On the request from the user to retrieve the file the dataserver can then either return the requested file or redirect the client to the dataserver that has the file. Each dataserver has a permanent data storage and a cache, the last one is used to temporarily store a file retrieved from the other dataserver. By user request, for example, http://ds.astro.rug.astro-wise.org:8000/WFI.2000-09-28T02:22:37.466_8.fits the dataserver ds.astro.rug.astro-wise.org will check cache space for the file WFI.2000-09-28T02:22:37.466_8.fits . If the file is not found, the dataserver will check it's own permanent storage space, and if there is no such file, all dataservers in a 'local' cluster ( astro.rug.astro-wise.org ) will be requested. If there is still no file found, all other clusters will be requested, and as soon as the file is located it will be copied to the cache of the ds.astro.rug.astro-wise.org and returned to the user. In addition, a file can be compressed or decompressed on-the-fly during retrieval, and a slice of the data can be retrieved by specifying the slice as part of the URL. The dataserver is written in Python and can be installed on any operating system and underlying filesystem that can support long case-sensitive filenames. For the Astro-WISE Linux filesystems, XFS and GPFS are currently in use.",
"pages": [
13,
14
]
},
{
"title": "4.3 Processing Layer",
"content": "The example in the Section 4.1 involves the processing facilities of Astro-WISE, which are distributed and combined from processing facilities of all Astro-WISE partners. The user has the choice to send the job to one of the processing elements of Astro-WISE (including Grid computing elements) or to use the processing power at the user's disposal, for example, PC or even notebook. In the core of the processing layer of Astro-WISE is a distributed processing unit (DPU). This is a three-component middleware which consists of the DPU server, DPU client and the so called DPU runner. The DPU server is a front-end to any processing system and interacts with the processing system's native queuing system, allowing the user to submit jobs to this particular queuing system, inspect or cancel them. The DPU client includes all functions and methods the user can call to interact with DPU server. The DPU runner is a program which is run on the remote processing facility and checks availability of all necessary software to run an Astro-WISE job, installing required packages if necessary. The system runs on openpbs or under its own queue management software. The DPU itself allows synchronizations of jobs as well and can also transfer parts of a sequence of jobs to other DPU's. In the case of Grid computing element the DPU server will check a user's identity and will use the user's credentials (Grid certificate) to mediate with the Virtual Organization Management System. Currently OmegaCEN is using omegac Virtual Organization with an ability to submit jobs to Grid computing elements in Amsterdam and Groningen.",
"pages": [
14,
15
]
},
{
"title": "4.4 Interfaces",
"content": "The description of Astro-WISE system would be incomplete without the description of a number of interfaces provided for the user. The user can write his own applications in the Python language calling Astro-WISE libraries or can involve Astro-WISE services. The first case requires the use of the Astro-WISE Command Line Interface called AWE (Astro-WISE Environment), which can be installed on any site, PC or notebook The Command Line Interface - CLI - supposes that the user writes his own programs using Python, but to browse the data or even to process observations there is no need to use the CLI. All operations required to perform this activity are possible with a set of standard web services of Astro-WISE. Of course, the use of the CLI gives to the user much more freedom in data processing. The CLI is more useful for experienced users working on a particular use-case, meanwhile web services are developed for routine operations during the data processing of the surveys. The web interfaces are divided into two types: data browsing/exploration and data processing/qualification. The first group includes:",
"pages": [
15
]
},
{
"title": "Data processing / qualification interfaces are:",
"content": "All web services are built using a set of Python classes developed for AstroWISE as a basis and uses a modular principle, which allows to create a new web service using components of older ones.",
"pages": [
16
]
},
{
"title": "4.5 Authorization and Authentication System",
"content": "Astro-WISE is a multi-user system which must accommodate sharing data between scientists and at the same time protect the private data of each user. Each user in the Astro-WISE system has an identity protected by a password, and, optionally, if the user wants to submit job to Grid resources he has to get a Grid certificate. The authorization and authentication system is implemented on the level of the metadata database. As soon as a user logged in with his username/password, the user's privileges are checked in the database, allowing the user to browse the data according to the user's privileges while obeying the privileges of other users. The data in the metadata database are grouped by projects . A Project is a collection of resources which is associated with a group of users who can access these resources, usually one of these users has the role of Project Manager who has privileges to include new users, remove users and publish the data to the wider community. The system of access to the data is based on three attributes which any data entity in Astro-WISE has: user , project and privileges . The first one identifies the user that created the data entity. The second one defines to which project the data entity belongs, the third one defines who is able to use this data entity. All these attributes are initialized the first time the data entity is made persistent in the Astro-WISE system (including the case that the entity is created by one of the Astro-WISE pipelines) and they are both persistent attributes, i.e., stored in the metadata database. Table 1 shows the range of values for the privileges attribute. In the case of privileges=1 only the creator of the data item can see it, the creator can raise privileges to privileges=2 , in this case all the users of the project will be able to browse this data entity. Raising privileges to 3 the user makes the data item accessible for all Astro-WISE users in all projects, and with priveleges=4 the anonymous user (a user with minimal read-only privileges in the system) can see it as well. Finally, with priveleges=5 the data item is accessible via Astro-WISE Virtual Observatory interfaces. The special Python class Context was created to handle authorization and authentication. The Context allows to change privileges of all user's data items, during the change Context is checking for dependencies to prevent inconsistency in the dependencies due to the different privileges of data items. For example, if the raw image was published by the user to the project scope and some other user has created a reduced image from this raw image, the original user will not be able to downgrade privileges on the raw image to the private data scope.",
"pages": [
16,
17
]
},
{
"title": "4.6 WISE architecture",
"content": "We described above three infrastructural components of Astro-WISE: the relational DBMS (the metadata layer), dataserver (the data layer) and DPU (the processing layer). Combined with user interfaces these components build an AstroWISE node - a detached Astro-WISE site which can operate independently from other sites. Multiple nodes can be combined to form a distributed system. Each node is independent from others in the sense that it is administrated independently and can handle both the data distributed over other nodes and the data restricted to this node only and shielded from other nodes. Figure 6 shows the elements of the node: All these elements are optional, an Astro-WISE node can be installed without dataservers (dataservers from other nodes are used), metadata database or processing facilities. In fact an Astro-WISE node can be installed on a notebook as Astro-WISE environment only giving the user access to the system - if the user is not satisfied with web services on the remote nodes. Presently Astro-WISE includes sites at Groningen University, Leiden University and Nijmegen University (The Netherlands), Argelander-Institut fur Astronomie, Bonn and Universitats-Sternwarte Munchen (Germany) and Osservatorio Astronomico di Capodimonte, Napoli (Italy).",
"pages": [
17,
18
]
},
{
"title": "5 Astro-WISE: migration to other systems",
"content": "The WISE concept for an information system is a set of principles described in Section 4, Astro-WISE is the first system which realizes this concept and serves as a basis for the further development of information systems. In this section we describe the adaptivity of the parent system to new tasks and challenges on the example of LOFAR Long Term Archive and Molgenis system. The importance of the LOFAR Long Term Archive (LTA) for the development of WISE concept is a necessity to use an external infrastructure which should be included in the architecture to form a complete information system. We preserved all principles of Astro-WISE adding storage and processing which is not controlled by the system itself (BiGGrid 9 ). Additionally we integrated three different systems of Authorization and Authentication to make it possible for the LOFAR observa- tory to create new users and control resources in the system. The LOFAR LTA design is described in [20] and the architecture and infrastructure in [21]. Another significant development was achieved with the adaptation of Molgenis 10 . Molgenis is a framework written in Java to build user interfaces and databases from definitions that are written in XML. From these definitions the database tables and webserver are generated. Its origins lie in biomedical applications. Historically Molgenis focused more on interaction with the user to streamline the 'protocols' (recipes for analysis) of researchers to keep track of analyses of results. On the other hand, the WISE technology focused more on results from a processing perspective. This lead to the idea to combine the strengths of both frameworks and use some models that already existed in XML for Molgenis to generate a datamodel in Python for Astro-Wise. Then Molgenis could be extended to use the WISE infrastructure for distributed storage and processing. The existing Molgenis XML model is organized in a common fashion through 'module', 'entity' and 'field' where each field has a type of int , float , str or can be a reference type such as mref or xref . Most of these have a counterparts in the WISE framework which made it possible to write a conversion tool. The webserver that is generated by Molgenis has to communicate to a database that it usually creates itself. However, in this case the database had to be generated by Astro-Wise because the frameworks did not have a backend for a common database (Oracle vs. Non-Oracle). Instead of writing a new database backend for either framework it was decided to write an xmlrpc interface to encapsulate database queries and return their results. Since client software for all database flavours does not have to be present this functionality can be extended, e.g., since this is xmlrpc, a programming language independent implementation could also use the database in a way that the WISE framework dictates. The approach developed and tested in the case of Molgenis allows to create a new information system based on Astro-WISE with all services starting from the data model coded in XML and to do this automatically. This approach allowed to decrease the time and resources spent for the developing and implementation of a new system significantly. The next system which will be created with the approach tested on Molgenis is a data processing system for Multi Unit Spectroscopic Explorer 11 (MUSE).",
"pages": [
18,
19
]
},
{
"title": "6 Conclusion and Future Work",
"content": "The Astro-WISE information system, the first information system in Astronomy, proved to be a reliable and flexible tool for the data processing. Originally developed to process the data of KIlo Degree Survey (KIDS 12 ) it triggered development of the unique approach to the architecture of scientific information systems (WISE approach). Both Astro-WISE and the WISE approach are living systems which are open to improvements. For the last 2 years further development of the WISE approach is hosted by Target Holding 13 . Target Holding is an expertise center in the Northern Netherlands which is building a cluster of sensor network information systems and provides cooperation between a number of scientific projects and business partners like IBM and Oracle. Target creates and supports a hardware infrastructure for hosting tens of Petabytes of data for projects in astronomy, medicine, artificial intellegence and biology. In this development the WISE approach was used to create new information systems extending the original Astro-WISE on new data models, new data storage and processing capacities and new fields. Acknowledgements Astro-WISE is an on-going project which started from a FP5 RTD programme funded by the EC Action 'Enhancing Access to Research Infrastructures'. This work was performed as part of the Target project. Target project is supported by Samenwerkingsverband Noord Nederland. It operates under the auspices of Sensor Universe. It is also financially supported by the European fund for Regional Development and the Dutch Ministry of Economic Affairs, Pieken in de Delta, the Province of Groningen and the Province of Drenthe.",
"pages": [
19,
20
]
}
]
2013ExA....36..451C
https://arxiv.org/pdf/1305.3789.pdf
<document>
<text><location><page_1><loc_13><loc_92><loc_48><loc_94></location>Experimental Astronomy manuscript No.</text>
<text><location><page_1><loc_13><loc_91><loc_35><loc_92></location>(will be inserted by the editor)</text>
<section_header_level_1><location><page_1><loc_12><loc_82><loc_69><loc_85></location>Background simulations for the Large Area Detector onboard LOFT</section_header_level_1>
<text><location><page_1><loc_12><loc_79><loc_43><loc_80></location>Riccardo Campana · Marco Feroci ·</text>
<text><location><page_1><loc_12><loc_77><loc_27><loc_78></location>Ettore Del Monte</text>
<text><location><page_1><loc_28><loc_77><loc_29><loc_78></location>·</text>
<text><location><page_1><loc_30><loc_77><loc_43><loc_78></location>Teresa Mineo ·</text>
<text><location><page_1><loc_12><loc_76><loc_39><loc_77></location>Niels Lund · George W. Fraser</text>
<text><location><page_1><loc_12><loc_70><loc_32><loc_71></location>Received: date / Accepted: date</text>
<text><location><page_1><loc_12><loc_42><loc_69><loc_67></location>Abstract The Large Observatory For X-ray Timing (LOFT), currently in an assessment phase in the framework the ESA M3 Cosmic Vision programme, is an innovative medium-class mission specifically designed to answer fundamental questions about the behaviour of matter, in the very strong gravitational and magnetic fields around compact objects and in supranuclear density conditions. Having an effective area of ∼ 10 m 2 at 8 keV, LOFT will be able to measure with high sensitivity very fast variability in the X-ray fluxes and spectra. A good knowledge of the in-orbit background environment is essential to assess the scientific performance of the mission and optimize the design of its main instrument, the Large Area Detector (LAD). In this paper the results of an extensive Geant-4 simulation of the instrument will be discussed, showing the main contributions to the background and the design solutions for its reduction and control. Our results show that the current LOFT/LAD design is expected to meet its scientific requirement of a background rate equivalent to 10 mCrab in 2-30 keV, achieving about 5 mCrab in the most important 2-10 keV energy band. Moreover, simulations show an anticipated modulation of the background rate as small as 10% over the orbital timescale. The intrinsic photonic origin of the largest background component also allows for</text>
<text><location><page_1><loc_12><loc_40><loc_36><loc_41></location>On behalf of the LOFT collaboration.</text>
<text><location><page_1><loc_12><loc_38><loc_36><loc_39></location>R. Campana, M. Feroci, E. Del Monte</text>
<text><location><page_1><loc_12><loc_37><loc_51><loc_37></location>INAF/IAPS, Via Fosso del Cavaliere 100, I-00133, Roma, Italy</text>
<text><location><page_1><loc_12><loc_35><loc_65><loc_36></location>and INFN/Sezione di Roma 2, viale della Ricerca Scientifica 1, I-00133, Roma, Italy.</text>
<text><location><page_1><loc_12><loc_34><loc_33><loc_35></location>E-mail: riccardo.campana@inaf.it</text>
<text><location><page_1><loc_12><loc_33><loc_32><loc_34></location>Present address of R. Campana:</text>
<text><location><page_1><loc_12><loc_32><loc_51><loc_33></location>INAF/IASF-Bologna, via Gobetti 101, I-40129, Bologna, Italy.</text>
<text><location><page_1><loc_12><loc_31><loc_18><loc_32></location>T. Mineo</text>
<text><location><page_1><loc_12><loc_30><loc_55><loc_31></location>INAF/IASF-Palermo, Via Ugo La Malfa 153, I-90146, Palermo, Italy.</text>
<text><location><page_1><loc_12><loc_29><loc_17><loc_30></location>N. Lund</text>
<text><location><page_1><loc_12><loc_28><loc_54><loc_28></location>DTU-Space, Elektrovej Bld. 327, DK-2800, Kgs. Lyngby, Denmark.</text>
<text><location><page_1><loc_12><loc_26><loc_18><loc_27></location>G. Fraser</text>
<text><location><page_1><loc_12><loc_24><loc_69><loc_26></location>Space Research Centre, Dept. of Physics and Astronomy, University of Leicester, LE17RH, Leicester, UK.</text>
<text><location><page_2><loc_12><loc_85><loc_69><loc_89></location>an efficient modelling, supported by an in-flight active monitoring, allowing to predict systematic residuals significantly better than the requirement of 1%, and actually meeting the 0.25% science goal.</text>
<text><location><page_2><loc_12><loc_81><loc_65><loc_84></location>Keywords X-ray astronomy · Instrumental background · Montecarlo simulations</text>
<text><location><page_2><loc_12><loc_79><loc_40><loc_80></location>PACS 07.85.Fv · 07.87.+v · 95.40.+s</text>
<section_header_level_1><location><page_2><loc_12><loc_74><loc_24><loc_75></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_54><loc_69><loc_72></location>Astrophysical observations in the X-ray domain are often dominated by the background. In order to assess and maximize the scientific performance of a satellite-borne instrument during its design phase, is essential to have an early estimation of the background level and to identify its origin. This will allow to drive the optimization of the experiment design itself. Moreover, a reliable estimation of the background level is important to assess the scientific objectives and performance of the mission. The most convenient and consolidated way to compute such a background is by means of a Monte Carlo simulation, using e.g. the Geant-4 toolkit [1], in which the interactions of the space environment with a somewhat simplified mass model of the spacecraft and of the instruments are sampled and studied. A correct evaluation of the radiation environment surrounding the experiment is of course the most important ingredient in the background simulation.</text>
<text><location><page_2><loc_12><loc_45><loc_69><loc_54></location>In this paper we will show Monte Carlo simulations of the instrumental background for the Large Area Detector (LAD) instrument onboard the proposed LOFT (Large Observatory for X-ray Timing) satellite [2,3], that is a candidate for the third slot of medium-class missions (M3) of the Cosmic Vision 2015-2025 programme of the European Space Agency, with a possible launch in the 2022-24 timeframe.</text>
<text><location><page_2><loc_12><loc_32><loc_69><loc_45></location>LOFT (Figure 1) has been specially designed to answer the 'Fundamental Question' #3.3 of the Cosmic Vision programme, Matter under extreme conditions . In the aim of investigating the behaviour of the matter in the most extreme physical conditions that can be found in the gravitational and magnetic fields around neutron stars and black holes, LOFT will study the rapid variability of the X-ray emission from these objects, both in the spectral and temporal domains. Moreover, LOFT will be an observatory for virtually all classes of bright X-ray sources: among them, X-ray bursters, high mass X-ray binaries, cataclysmic variables, magnetars and active galactic nuclei.</text>
<text><location><page_2><loc_12><loc_24><loc_69><loc_32></location>The main experiment onboard LOFT is the LAD [3] (Sect. 2), a collimated instrument with a collecting area 20 times bigger than its immediate predecessor (PCA, the Proportional Counter Array onboard the Rossi X-ray Timing Explorer, RXTE), and based on large-area Silicon Drift Detectors (SDDs) coupled to lead-glass Micro Channel Plate (MCP) collimators. In order to survey a large fraction of the sky simultaneously, and to trigger follow-up observations</text>
<text><location><page_3><loc_12><loc_84><loc_69><loc_89></location>with the main instrument, LOFT will host also a coded-mask Wide Field Monitor (WFM, [4]). This latter instrument, sensitive in the same energy range of LAD, will use basically the same SDD detectors, with some design differences to optimize the detector response for imaging [5,6,7].</text>
<text><location><page_3><loc_12><loc_55><loc_69><loc_83></location>The background level required to satisfy the LOFT scientific objectives, in particular regarding the relatively faint (1-10 mCrab) sources like most Active Galactic Nuclei (AGNs) and some accretion-powered X-ray pulsars, is ≤ 10 mCrab for the 2-30 keV LAD standard band. Being a collimated instrument, the LAD will not measure simultaneously and independently the source flux and the background. While some collimated instruments (e.g. BeppoSAX/PDS and RXTE/HEXTE, both at hard X-rays) employ a 'rocking' strategy, where the same instrument unit alternates observations of the source and of the background, others (e.g. RXTE/PCA, at soft X-rays, similar to LAD) rely on a background model obtained by means of dedicated blank-field pointings and by using onboard ratemeters [8]. As we will discuss in the following, LAD will adopt an approach similar to PCA. To ensure the fulfillment of the scientific objectives, in particular for low flux sources, an accurate estimation and modelling of the background level for this instrument is of particular importance. The background variations that have a significant impact on the scientific performance should be minimized, and controlled to a high degree of accuracy. The LOFT science case resulted in a requirement on background knowledge at 1%, with a goal at 0.25%. As a comparison, RXTE/PCA reached a background residual systematic level of ∼ 0.5%-1% [8,9], while BeppoSAX/PDS of ∼ 1% [10].</text>
<text><location><page_3><loc_12><loc_47><loc_69><loc_55></location>This paper is structured as follows. In Section 2 we describe the LAD instrument. In Section 3 we introduce the space environment for LOFT and discuss the main contributions to the instrumental background, while in Section 4 the main features of the LAD Monte Carlo mass model and simulator are described. The simulation results are shown in Section 5, and in Section 6 we draw our conclusions.</text>
<section_header_level_1><location><page_3><loc_12><loc_42><loc_35><loc_43></location>2 The Large Area Detector</section_header_level_1>
<text><location><page_3><loc_12><loc_29><loc_69><loc_41></location>The Large Area Detector is an array of 2016 individual large-area Silicon Drift Detectors, sensitive to the 2-30 keV emission (with a extended 30-80 keV band) collimated in a ∼ 1 · field of view by means of a lead-glass microcapillary collimator plate. The instrument design is modular: an array of 4 × 4 SDDs and their front-end electronics (FEE) are placed in a module . Each module has its independent back-end electronics (MBEE) that powers and configure the FEE, manages the event readout and monitors the detector health. A panel is composed of an array of 21 modules, placed on a support grid (Figure 2).</text>
<text><location><page_3><loc_12><loc_24><loc_69><loc_29></location>The current LAD design by the LOFT consortium envisages 6 panels (each about 350 cm × 90 cm), that stems from the satellite bus and are deployed once the spacecraft reaches its orbit (Figure 1). The collimators, made of lead glass, have the same footprint as the SDDs (about 11 cm × 7 cm), a</text>
<figure>
<location><page_4><loc_25><loc_67><loc_60><loc_89></location>
<caption>Fig. 1 A sketch of the LOFT satellite and its instruments.</caption>
</figure>
<figure>
<location><page_4><loc_12><loc_53><loc_69><loc_61></location>
<caption>Fig. 2 Left : a LAD module, showing the mounting of the collimator, SDD and the FEE. Center : Back-side view of the module, showing the radiative surface and the back-end electronics. Right : a LAD Detector Panel.</caption>
</figure>
<text><location><page_4><loc_12><loc_40><loc_69><loc_45></location>thickness of 6 mm, and 100 µ m wide square holes separated with 20 µ m thick walls (and thus having an open area ratio , f OAR , of ∼ 70%). An aluminium frame couples the collimators with the detectors frame and ensures the tight alignment constraints.</text>
<text><location><page_4><loc_12><loc_24><loc_69><loc_39></location>The working principle of the SDDs is the following: a photon is absorbed in the depleted silicon bulk, and produces a cloud of electrons whose number is proportional to the incident energy. The cloud drifts towards an array of read-out anodes, by means of a high-voltage field applied between the middle section and the two detector edges, hosting two independent rows of anodes. During the drift, the cloud size increases due to diffusion. The number of anodes that will read-out the charge cloud thus depends on the absorption point distance and anode pitch [5]. For the LAD detectors, that have a pitch of 970 µ m, the charge cloud originating from the absorption of a X-ray photon in the 2-80 keV band is at most 0.5 mm (FWHM) thus read-out by 1 or 2 anodes, depending on the impact point and the energy. At 6 keV, for example,</text>
<text><location><page_5><loc_12><loc_85><loc_69><loc_89></location>∼ 40% of the events are collected by one anode and ∼ 60% by two anodes. The event multiplicity is read-out and transmitted to the ground, enabling a selection of 'single' and 'double' events, with different energy resolutions.</text>
<section_header_level_1><location><page_5><loc_12><loc_80><loc_33><loc_82></location>3 Sources of background</section_header_level_1>
<text><location><page_5><loc_12><loc_73><loc_69><loc_79></location>In this section we discuss the main sources of the LOFT background. We describe the environment encountered in the baseline low-Earth orbit, with its hadronic, leptonic and photon components. Moreover, we describe the internal source of background, i.e. the natural radioactivity of the collimator material.</text>
<section_header_level_1><location><page_5><loc_12><loc_69><loc_27><loc_70></location>3.1 The LOFT orbit</section_header_level_1>
<text><location><page_5><loc_12><loc_57><loc_69><loc_67></location>The LOFT orbit requirement is at an altitude of h = 600 km with an inclination of 5 · . Likewise, lower-altitude orbits (down to 550 km) and/or inclinations (down to 0 · ) are considered, with small improvements for what regards the conclusions about the background rate and properties. The lower altitude and inclinations, however, impacts on the long-term radiation damage on the detectors. For these orbits, the geomagnetic latitude range is usually θ M < 0 . 3 radians.</text>
<figure>
<location><page_5><loc_17><loc_29><loc_66><loc_54></location>
<caption>Fig. 3 Values of the trapped proton intensity (AP8MIN model) for the LOFT baseline orbit. Data obtained from the SPENVIS software ( http://www.spenvis.oma.be ). The South Atlantic Anomaly is well apparent above South America.</caption>
</figure>
<text><location><page_6><loc_12><loc_64><loc_69><loc_89></location>In the current work we neglected the possible background component due to the activation of the instrument and satellite materials. Based on the arguments below, we estimated it to be a second-order effect. The South Atlantic Anomaly, a region of trapped high-energy protons and electrons, is grazed by the LOFT baseline orbit only in its outermost regions (Figure 3), with a shallower passage for lower altitudes and inclinations. Therefore, the effect on background due to the activation of materials by this intense radiation is expected to be negligible with respect to other sources, as confirmed by preliminary evaluations and heritage from past missions in similar orbits (e.g., BeppoSAX and AGILE). It should also be noted that in contrast to previous experiments, the LAD instrument has a very light structure per unit volume and the relatively small spacecraft assembly is seen at a small viewing angle, due to the tower supporting the LAD panels (see Figure 1). In addition to that, as for example observed by Swift/XRT [11], Silicon detectors show marginal activation effects in LEO, even when operated as focal plane instruments (more local mass) and on more inclined orbits (up to ∼ 20 · ). However, given our goal of controlling the LAD background to the highest possible accuracy, we plan to study this aspect in great detail in a future work.</text>
<section_header_level_1><location><page_6><loc_12><loc_60><loc_30><loc_61></location>3.2 Primary cosmic rays</section_header_level_1>
<text><location><page_6><loc_12><loc_53><loc_69><loc_58></location>Following the approach of Mizuno et al. [12], we assume that the primary cosmic ray spectrum is expressed as F U ( R ), a function of the particle magnetic rigidity, R = pc/Ze , where Z is the atomic number and p the particle momentum.</text>
<text><location><page_6><loc_12><loc_50><loc_69><loc_53></location>The full spectrum at a given location in the magnetosphere and a given phase of the solar cycle will be given by:</text>
<formula><location><page_6><loc_20><loc_48><loc_69><loc_49></location>F ( E ) = F U ( E + Zeφ ) × F M ( E,M,Z,φ ) × C ( R,h,θ M ) (1)</formula>
<text><location><page_6><loc_12><loc_43><loc_69><loc_47></location>where M and Ze are the mass and charge of the particle, E its kinetic energy, φ is a solar modulation factor, h is the orbit height and θ M is the geomagnetic latitude.</text>
<text><location><page_6><loc_12><loc_40><loc_69><loc_42></location>The second term, that includes the effects of the solar modulation on the cosmic ray particles, is given by [13]:</text>
<formula><location><page_6><loc_23><loc_36><loc_69><loc_39></location>F M ( E,M,Z,φ ) = ( E + Mc 2 ) 2 -( Mc 2 ) 2 ( E + Zeφ + Mc 2 ) 2 -( Mc 2 ) 2 (2)</formula>
<text><location><page_6><loc_12><loc_34><loc_49><loc_35></location>where the solar modulation potential φ is given by:</text>
<formula><location><page_6><loc_28><loc_32><loc_69><loc_33></location>φ = 0 . 55 GV at solar minimum (3)</formula>
<formula><location><page_6><loc_28><loc_30><loc_69><loc_31></location>φ = 1 . 10 GV at solar maximum (4)</formula>
<text><location><page_6><loc_12><loc_26><loc_69><loc_29></location>The geomagnetic cutoff function is given, for vertically incident particles, by [12]:</text>
<formula><location><page_6><loc_29><loc_23><loc_69><loc_26></location>C ( R,h,θ M ) = 1 1 + ( R/R cut ) -r (5)</formula>
<text><location><page_7><loc_12><loc_88><loc_64><loc_89></location>in which the cutoff rigidity is obtained from the Stormer equation [14]:</text>
<formula><location><page_7><loc_26><loc_83><loc_69><loc_87></location>R cut = 14 . 5 × ( 1 + h R E ) -2 cos 4 θ M GV (6)</formula>
<text><location><page_7><loc_12><loc_80><loc_69><loc_82></location>where R E = 6371 km is the Earth radius. Moreover, the r exponent in Eq. 5 is:</text>
<formula><location><page_7><loc_34><loc_78><loc_69><loc_79></location>r = 12 for protons (7)</formula>
<formula><location><page_7><loc_28><loc_76><loc_69><loc_77></location>r = 6 for electrons and positrons (8)</formula>
<text><location><page_7><loc_12><loc_72><loc_69><loc_75></location>For the 600 km, 5 · inclination LOFT orbit, the cutoff rigidity therefore is found to be in the range R cut ∼ 10.1-12.1 GV.</text>
<text><location><page_7><loc_12><loc_67><loc_69><loc_72></location>Since we are interested mostly in the average background fluxes, we discard at the present stage the east-west effect for which particles coming from different directions have different cutoff rigidities. We therefore assume that the general primary spectrum, now expressed as:</text>
<formula><location><page_7><loc_23><loc_62><loc_69><loc_66></location>F ( R ) = F U ( E + Zeφ ) × ( E + Mc 2 ) 2 -( Mc 2 ) 2 ( E + Zeφ + Mc 2 ) 2 -( Mc 2 ) 2 × 1 1+( R/R cut ) -r (9)</formula>
<text><location><page_7><loc_12><loc_56><loc_69><loc_60></location>has an uniform angular distribution with respect to the zenith angle, up to the Earth horizon, i.e. for zenith angles θ from 0 · to θ cut , where the latter is given by:</text>
<formula><location><page_7><loc_31><loc_53><loc_69><loc_56></location>θ cut = arcsin ( 1 h R E +1 ) (10)</formula>
<text><location><page_7><loc_12><loc_49><loc_69><loc_52></location>For h = 600 km, θ cut ∼ 114 · . The solid angle subtended by the Earth at an altitude h is given by:</text>
<formula><location><page_7><loc_26><loc_44><loc_69><loc_48></location>Ω E = 2 π ( 1 -1 R E /h +1 √ 1 + 2 R E h ) (11)</formula>
<text><location><page_7><loc_12><loc_41><loc_69><loc_43></location>and therefore the accessibile sky subtends a solid angle Ω = 4 π -Ω E (Figure 4). For h = 600 km, the Earth blocks about 30% of the sky.</text>
<section_header_level_1><location><page_7><loc_12><loc_38><loc_28><loc_39></location>3.2.1 Primary protons</section_header_level_1>
<text><location><page_7><loc_12><loc_33><loc_69><loc_36></location>The unmodulated value of the cosmic ray electron spectrum is given by the BESS [15] and AMS measurements [16]:</text>
<formula><location><page_7><loc_16><loc_28><loc_69><loc_32></location>F U ( E ) = 23 . 9 × [ R ( E ) GV ] -2 . 83 particles m -2 s -1 sr -1 MeV -1 (12)</formula>
<text><location><page_7><loc_12><loc_24><loc_69><loc_28></location>In Figure 5 the primary proton spectra for different values of the solar modulation and for the LOFT orbit is reported. The variation in flux between solar minimum and maximum, at the peak flux, is ∼ 15%.</text>
<figure>
<location><page_8><loc_17><loc_63><loc_62><loc_87></location>
<caption>Fig. 4 The fraction of the sky blocked by the Earth, for a given orbit altitude h .</caption>
</figure>
<figure>
<location><page_8><loc_17><loc_32><loc_61><loc_57></location>
<caption>Fig. 5 Model spectrum of the primary protons for the LOFT orbit. The red data points are the AMS-01 measurements of the proton spectrum, primary and secondary, for the Shuttle orbit (380 km) and low magnetic latitude ( θ M < 0 . 2), from [16].</caption>
</figure>
<section_header_level_1><location><page_9><loc_12><loc_88><loc_40><loc_89></location>3.2.2 Primary electrons and positrons</section_header_level_1>
<text><location><page_9><loc_12><loc_84><loc_69><loc_86></location>The unmodulated value of the cosmic ray electron spectrum is given by [17, 18,12]:</text>
<formula><location><page_9><loc_18><loc_79><loc_69><loc_83></location>F U ( E ) = 0 . 65 × [ R ( E ) GV ] -3 . 3 particles m -2 s -1 sr -1 MeV -1 (13)</formula>
<text><location><page_9><loc_12><loc_73><loc_69><loc_79></location>The fraction of positrons to electrons, usually given by the ratio e + / ( e + + e -), is found to be rather independent of the energy [19], i.e. the spectrum of primary positrons has the same slope of the electron one, but a different normalization:</text>
<formula><location><page_9><loc_16><loc_69><loc_69><loc_73></location>F U ( E ) = 0 . 051 × [ R ( E ) GV ] -3 . 3 particles m -2 s -1 sr -1 MeV -1 (14)</formula>
<text><location><page_9><loc_12><loc_64><loc_69><loc_68></location>In Figure 6 the primary e -and e + spectra for different values of the solar modulation and for the LOFT orbit are reported. The difference in flux between solar minumum and maximum is about 20% at the peak.</text>
<figure>
<location><page_9><loc_16><loc_36><loc_62><loc_60></location>
<caption>Fig. 6 Model spectrum of the primary electrons and positrons for the LOFT orbit.</caption>
</figure>
<section_header_level_1><location><page_9><loc_12><loc_28><loc_33><loc_29></location>3.2.3 Primary alpha particles</section_header_level_1>
<text><location><page_9><loc_12><loc_24><loc_69><loc_26></location>We consider only the primary helium nuclei, because the contribution of heavier nuclei is negligible with respect to the other primary particles flux [20].</text>
<text><location><page_10><loc_12><loc_86><loc_69><loc_89></location>The unmodulated flux, as given by Mizuno et al. [12] on the basis of AMS and BESS data is:</text>
<formula><location><page_10><loc_18><loc_81><loc_69><loc_85></location>F U ( E ) = 1 . 5 × [ R ( E ) GV ] -2 . 77 particles m -2 s -1 sr -1 MeV -1 (15)</formula>
<text><location><page_10><loc_12><loc_79><loc_31><loc_81></location>and is shown in Figure 7.</text>
<text><location><page_10><loc_61><loc_52><loc_61><loc_52></location>5</text>
<figure>
<location><page_10><loc_16><loc_50><loc_61><loc_75></location>
<caption>Fig. 7 Model spectrum of the primary helium nuclei for the LOFT orbit.</caption>
</figure>
<section_header_level_1><location><page_10><loc_12><loc_41><loc_29><loc_42></location>3.3 Secondary particles</section_header_level_1>
<section_header_level_1><location><page_10><loc_12><loc_38><loc_22><loc_39></location>3.3.1 Protons</section_header_level_1>
<text><location><page_10><loc_12><loc_24><loc_69><loc_36></location>For the low altitude equatorial Earth orbits considered, the impinging proton spectrum outside the trapped particle belt, i.e. the South Atlantic Anomaly, consists of the primary component discussed in Sect. 3.2 and of a secondary, quasi-trapped component, originating from and impacting to the Earth atmosphere (sometimes in the literature they are referred to as the 'splash' and 'reentrant' albedo components). AMS measurements [16] showed that this secondary component is composed of a short-lived and a long-lived particle population, both originating from the regions near the geomagnetic equator. Mizuno et al. [12] model the secondary equatorial proton spectrum as a cutoff</text>
<text><location><page_11><loc_12><loc_88><loc_20><loc_89></location>power-law:</text>
<formula><location><page_11><loc_12><loc_82><loc_69><loc_86></location>F ( E ) = 0 . 123 × ( E 1 GeV ) -0 . 155 e -( E/ 0 . 51) 0 . 845 particles m -2 s -1 sr -1 MeV -1</formula>
<text><location><page_11><loc_66><loc_82><loc_69><loc_83></location>(16)</text>
<text><location><page_11><loc_12><loc_80><loc_45><loc_82></location>and a power law for energies below 100 MeV:</text>
<formula><location><page_11><loc_16><loc_75><loc_69><loc_79></location>F ( E ) = 0 . 136 × ( E 100 MeV ) -1 particles m -2 s -1 sr -1 MeV -1 (17)</formula>
<text><location><page_11><loc_12><loc_70><loc_69><loc_74></location>This spectrum is shown in Figure 8 together with the AMS measurements. According to the measurements performed with the NINA and NINA-II instruments [21], the extrapolation below 100 MeV is likely an overestimate.</text>
<text><location><page_11><loc_12><loc_61><loc_69><loc_70></location>We do not consider here the soft (10 keV-1 MeV), highly directional equatorial proton population [22,23], since these particles are most efficiently stopped by all the spacecraft structures surrounding the detectors. Even the small fraction that impinges on the detector through the capillary holes, albeit significant when considering the long-term radiation damage on the SDDs, leaves a negligible background signal.</text>
<figure>
<location><page_11><loc_16><loc_32><loc_61><loc_56></location>
<caption>Fig. 8 Secondary proton spectrum. The red data points are the AMS-01 measurements of the proton spectrum, both primary and secondary, for the Shuttle orbit (380 km) and low magnetic latitude ( θ M < 0 . 2, from [16]), while the black line is the Mizuno et al. [12] analytic form.</caption>
</figure>
<section_header_level_1><location><page_12><loc_12><loc_88><loc_33><loc_89></location>3.3.2 Electrons and positrons</section_header_level_1>
<text><location><page_12><loc_12><loc_84><loc_69><loc_86></location>The equatorial secondary electron spectrum can be approximated with a broken power law [12]:</text>
<formula><location><page_12><loc_19><loc_78><loc_69><loc_82></location>F ( E ) = 0 . 3 × ( E 100 MeV ) -2 . 2 for 100 MeV < E < 3 GeV (18)</formula>
<formula><location><page_12><loc_18><loc_71><loc_69><loc_75></location>F ( E ) = 0 . 3 × ( 3 GeV 100 MeV ) -2 . 2 ( E 3 GeV ) -4 for E ≥ 3 GeV (19)</formula>
<text><location><page_12><loc_12><loc_70><loc_45><loc_71></location>and a power law for energies below 100 MeV:</text>
<formula><location><page_12><loc_16><loc_64><loc_69><loc_68></location>F ( E ) = 0 . 3 × ( E 100 MeV ) -1 particles m -2 s -1 sr -1 MeV -1 (20)</formula>
<text><location><page_12><loc_12><loc_58><loc_69><loc_62></location>At variance with respect to the primary particles, in the geomagnetic equatorial region the positrons are predominant with respect to the electrons. The spectrum is the same, but the ratio e + /e -is about 3.3 (Figure 9).</text>
<figure>
<location><page_12><loc_16><loc_29><loc_61><loc_53></location>
<caption>Fig. 9 Secondary electron and positron spectrum. Analytic form of Mizuno et al. [12].</caption>
</figure>
<section_header_level_1><location><page_13><loc_12><loc_88><loc_29><loc_89></location>3.4 Photon background</section_header_level_1>
<text><location><page_13><loc_12><loc_81><loc_69><loc_86></location>As we will see in the following Sections, the main contribution to the overall background for the LAD instrument is due to the high energy X-ray emission transmitted and scattered in the lead-glass collimators. The main contribution comes from the CXB and the Earth albedo emission.</text>
<section_header_level_1><location><page_13><loc_12><loc_78><loc_36><loc_79></location>3.4.1 Cosmic photon background</section_header_level_1>
<text><location><page_13><loc_12><loc_70><loc_69><loc_76></location>For the cosmic X-ray and γ -ray diffuse background we assume the Gruber et al. [24] analytic form, derived from HEAO-1 A4 and COMPTEL measurements, and fitted in the energy range between 3 keV and 100 GeV. This spectrum is usually used as a standard reference [25,26], and is plotted in Figure 10.</text>
<text><location><page_13><loc_15><loc_69><loc_43><loc_70></location>The first branch is valid below 60 keV:</text>
<formula><location><page_13><loc_13><loc_64><loc_69><loc_68></location>F ( E ) = 7 . 877 × ( E 1 keV ) -1 . 29 e -E/ 41 . 13 photons cm -2 s -1 sr -1 keV -1 (21)</formula>
<text><location><page_13><loc_15><loc_62><loc_38><loc_64></location>while for energies above 60 keV:</text>
<formula><location><page_13><loc_26><loc_53><loc_69><loc_61></location>F ( E ) = 0 . 0004317 × ( E 60 keV ) -6 . 5 +0 . 0084 × ( E 60 keV ) -2 . 58 (22) +0 . 00048 × ( E 60 keV ) -2 . 05 photons cm -2 s -1 sr -1 keV -1</formula>
<text><location><page_13><loc_12><loc_47><loc_69><loc_52></location>Other measurements of the CXB have been performed by, e.g., BeppoSAX [25], INTEGRAL [27,28] and BAT [26]. These measurements are consistent with the HEAO-1 results, albeit some of these seems to indicate a slightly higher normalization ( ∼ 8%) for the > 10 keV spectrum.</text>
<section_header_level_1><location><page_13><loc_12><loc_43><loc_34><loc_45></location>3.4.2 Albedo γ -ray background</section_header_level_1>
<text><location><page_13><loc_12><loc_31><loc_69><loc_42></location>The secondary photon background is due to the interactions of cosmic rays (proton and leptonic components) with the atmosphere, and to the reflection of CXB emission. As such, it has a strong zenith dependence [29]. This albedo component has a higher flux, for unit of solid angle, than the CXB for energies above ∼ 70 keV. We assume the albedo spectrum as measured by BAT [26], that agrees in the range above 50 keV, after some corrections, with previous measurements [30,31]. This spectrum can be parameterized by the function [32,26]:</text>
<formula><location><page_13><loc_14><loc_24><loc_69><loc_29></location>F ( E ) = 0 . 0148 ( E 33 . 7 keV ) -5 + ( E 33 . 7 keV ) 1 . 72 photons cm -2 s -1 sr -1 keV -1 (23)</formula>
<text><location><page_13><loc_12><loc_24><loc_32><loc_25></location>and is plotted in Figure 11.</text>
<figure>
<location><page_14><loc_17><loc_63><loc_61><loc_87></location>
<caption>Fig. 10 The cosmic X-ray diffuse background as derived from HEAO-1 A4 measurements [24] (solid line). CXB observations performed by PDS [25] and BAT [26] are also shown.</caption>
</figure>
<figure>
<location><page_14><loc_17><loc_31><loc_61><loc_56></location>
<caption>Fig. 11 Albedo γ -ray observations performed by BAT [26] and its parameterisation (Eq. 23).</caption>
</figure>
<section_header_level_1><location><page_15><loc_12><loc_88><loc_31><loc_89></location>3.5 Earth neutron albedo</section_header_level_1>
<text><location><page_15><loc_12><loc_74><loc_69><loc_86></location>As reported by the official ESA ECSS documents 1 , there is presently no model for atmospheric albedo neutron fluxes considered mature enough to be used as a standard. To account for the flux of neutrons produced by cosmic-ray interactions in the Earth atmosphere, we however used the results of the QinetiQ Atmospheric Radiation Model (QARM, [33,34]), that uses a response function approach based on Monte Carlo radiation transport codes to generate directional fluxes of atmospheric secondary radiation. The resulting spectrum, shown in Figure 12, is also consistent with the Monte Carlo simulations of Armstrong & Colborn [35,36].</text>
<text><location><page_15><loc_61><loc_46><loc_61><loc_47></location>3</text>
<figure>
<location><page_15><loc_16><loc_45><loc_61><loc_69></location>
<caption>Fig. 12 Model spectrum of the Earth albedo neutrons for the LOFT orbit.</caption>
</figure>
<section_header_level_1><location><page_15><loc_12><loc_36><loc_30><loc_37></location>3.6 Natural radioactivity</section_header_level_1>
<text><location><page_15><loc_12><loc_26><loc_69><loc_34></location>The lead-glass used in the microcapillary plate collimators contains potassium, in a fraction of approximately 7.2% by weight. The activity due to the naturally occurring long-lived radioactive isotope 40 K (with an abundance of 0.0117% with respect to natural potassium) is to be taken into account. This isotope decays with the emission of β rays having a continuum of energies up to 1.31 MeV ( ∼ 89% branching ratio) and of 1.46 MeV photons ( ∼ 11% b.r.) due to</text>
<text><location><page_16><loc_12><loc_86><loc_69><loc_89></location>electronic capture. Let L , ρ , f OAR be the thickness, density and open fraction of the collimator. Therefore, the collimator activity for unit area is:</text>
<formula><location><page_16><loc_34><loc_84><loc_69><loc_85></location>A coll = A K · M K (24)</formula>
<text><location><page_16><loc_12><loc_80><loc_69><loc_83></location>where A K ∼ 30 Bq/g is the specific activity of natural potassium due to 40 K, and M K is the total potassium mass per unit area in the collimator, i.e.</text>
<formula><location><page_16><loc_32><loc_78><loc_69><loc_79></location>M K = L (1 -f OAR ) f K (25)</formula>
<text><location><page_16><loc_12><loc_65><loc_69><loc_76></location>where f K is the fraction by weight of potassium in the collimator glass. For the MCP lead glass, we have f K ∼ 7 . 2%. Assuming for the MCP collimator L = 6 mm, ρ = 3 . 3 g/cm 3 , f OAR = 69 . 44% we obtain M K = 436 g/m 2 . Therefore, we have A coll = 13100 Bq/m 2 . For a total geometrical LAD collimator area of ∼ 15 m 2 we have thus ∼ 196500 decays/s. Considering the branching ratio, we have therefore that every second in the collimator bulk are uniformly and isotropically generated ∼ 175000 electrons with a continuum of energies and ∼ 21000 monochromatic photons with a 1.46 MeV energy.</text>
<section_header_level_1><location><page_16><loc_12><loc_61><loc_46><loc_62></location>4 The GEANT-4 LOFT/LAD simulator</section_header_level_1>
<section_header_level_1><location><page_16><loc_12><loc_59><loc_27><loc_60></location>4.1 The mass model</section_header_level_1>
<text><location><page_16><loc_12><loc_49><loc_69><loc_57></location>Simulations were performed using the Geant-4 Monte Carlo toolkit [1] (version 4.9.4). Geant-4 allows to describe the geometry and the materials of the instruments and of the satellite bus. Moreover, the code enables to follow the various physical interactions along the path of a primary event through the various components of the geometry, evaluating the secondary particles and energy deposits generated.</text>
<text><location><page_16><loc_12><loc_32><loc_69><loc_48></location>For the description of the electromagnetic interactions, the Low Energy Livermore library 2 is used, that trace the interactions of electrons and photons with matter down to about 250 eV, using interpolated data tables. Physical processes like photoelectric effect, Compton and Rayleigh scattering, pair production, bremsstrahlung, multiple scattering and annihilation are simulated, optionally taking into account also the effect of photon polarization. Fluorescence X-rays and Auger electrons from the various chemical elements are included. The hadronic physics, instead, is described using different models in different energy ranges (e.g. Bertini Cascade model, Quark-Gluon String model, etc. 3 ). Both elastic and inelastic processes are treated, the latter describing e.g. nuclear capture, fragmentation, de-excitation and scattering of the relevant stable and long-lived nucleons and mesons.</text>
<text><location><page_16><loc_12><loc_27><loc_69><loc_31></location>In Figure 13 is shown the mass model geometry used in the simulations for the LAD instrument. A single LAD panel is simulated, with the actual dimensions ( ∼ 350 cm × 90 cm) but with a simplified design that consists of a</text>
<text><location><page_17><loc_12><loc_84><loc_69><loc_89></location>stacked-layers geometry (sketched in Figure 14). A simplified geometry for the satellite bus, the structural tower and the other five panels is assumed, using aluminium as the material with an 'effective' density that takes into account the total mass and volume.</text>
<figure>
<location><page_17><loc_23><loc_52><loc_58><loc_78></location>
<caption>Fig. 13 The Geant-4 LOFT/LAD mass model.</caption>
</figure>
<text><location><page_17><loc_12><loc_24><loc_69><loc_43></location>The SDD array is represented as a 450 µ m thick slab of silicon, subdivided in 0.97 mm × 35 mm pixels. On its surface are placed various passive layers (Al cathode implants, SiO 2 passivation layer, undepleted Silicon bulk), the MCP collimator and the optical/thermal filter (1 µ m kapton coupled with a 80 nm Al layer). The MCP collimator has an 'effective', reduced density, to take into account the holes without resorting to a more accurate but much more timeconsuming geometrical description. Simulations on a smaller geometrical size have been performed to compare the 'effective' collimator with a complete geometrical capillary plate description (see Sect. 5.2), giving consistent results. Underneath the detectors, the FEE board is simulated as a 2 mm thick FR4 slab with an interposed 100 µ mCu conductive layer and a 100 µ mAl shielding layer, a 1 mm thick Al radiator and a 500 µ m Pb backshield. An aluminium frame encloses the module sides, while the panel structure consists of a carbonfiber reinforced plastic grid frame.</text>
<figure>
<location><page_18><loc_13><loc_61><loc_66><loc_87></location>
<caption>Fig. 14 Sketch of the simulated Geant-4 LOFT/LAD module geometry.</caption>
</figure>
<section_header_level_1><location><page_18><loc_12><loc_54><loc_33><loc_55></location>4.2 Primary event generation</section_header_level_1>
<text><location><page_18><loc_12><loc_43><loc_69><loc_52></location>For the generation of primary events, the General Particle Source 4 (GPS) approach is followed, that allows to produce arbitrary spectra in rather complex geometries. The initial events (particles or photons) are generated on the surface of a sphere of radius R , with a cosine-biased emission angle to ensure an isotropic flux. The emission angle is further restricted between 0 (normal to the spherical surface) to θ max : the emission cone then subtends a smaller sphere of radius r that surrounds the experiment. We have:</text>
<formula><location><page_18><loc_33><loc_38><loc_69><loc_42></location>θ max = arctan ( r R ) . (26)</formula>
<text><location><page_18><loc_12><loc_36><loc_69><loc_39></location>Let Φ be the energy-integrated flux, between E min and E max , expressed in particles cm -2 s -1 sr -1 . The total rate is therefore:</text>
<formula><location><page_18><loc_32><loc_34><loc_69><loc_35></location>N r = Φ 4 π 2 R 2 sin 2 θ max (27)</formula>
<text><location><page_18><loc_12><loc_30><loc_69><loc_33></location>This derives from the integration over the 2 π emission angle for a point on the spherical surface, biased with the cosine law:</text>
<formula><location><page_18><loc_31><loc_26><loc_69><loc_29></location>2 π 0 dφ π/ 2 0 dθ cos θ sin θ = π (28)</formula>
<formula><location><page_18><loc_30><loc_25><loc_36><loc_29></location>∫ ∫</formula>
<text><location><page_19><loc_12><loc_86><loc_69><loc_89></location>and integrated over the source sphere surface S = 4 πR 2 . The restriction on the emission angle introduces the further factor sin 2 θ max .</text>
<text><location><page_19><loc_12><loc_84><loc_54><loc_86></location>Therefore the simulation time that corresponds to N events is:</text>
<text><location><page_19><loc_55><loc_85><loc_69><loc_86></location>generated primary</text>
<formula><location><page_19><loc_32><loc_81><loc_69><loc_84></location>τ = N Φ 4 π 2 R 2 sin 2 θ max (29)</formula>
<text><location><page_19><loc_12><loc_78><loc_69><loc_80></location>If we detect C i counts in the energy bin i , the corresponding measured flux (in counts cm -2 s -1 ), i.e. convolved with the detector response, would be:</text>
<formula><location><page_19><loc_36><loc_74><loc_69><loc_77></location>F i = C i τA det (30)</formula>
<text><location><page_19><loc_12><loc_72><loc_42><loc_73></location>where A det is the detector sensitive area.</text>
<section_header_level_1><location><page_19><loc_12><loc_67><loc_39><loc_69></location>4.3 Anode multiplicity rejection filter</section_header_level_1>
<text><location><page_19><loc_12><loc_56><loc_69><loc_66></location>For the LAD, an anode multiplicity rejection algorithm is implemented to filter out ionization streaks from charged particles, that leave an energy deposit on more than 2 adjacent anodes. More in detail, for each of the two independent halves of a SDD tile, events are rejected if they trigger more than two adjacent anodes. Likewise, events are rejected if there is a group of non-adjacent triggering anodes within a 'distance' of 32 channels in the same half-SDD (Figure 15). Energy deposits at a further distance are treated as independent.</text>
<text><location><page_19><loc_12><loc_39><loc_69><loc_56></location>Since the average energy loss of minimum ionizing particles (MIPs) in silicon is about 3.7 MeV/cm, their total energy deposit is usually above ∼ 150 keV. The anode multiplicity rejection filter, combined together with an upper threshold on the reconstructed event signal, is very effective in the suppression of the particle-induced background: a threshold of 80 keV allow to filter out 94%96% of the particle background, depending on the incoming particle type and spectrum. Of course, since this event elaboration is performed in the frontend electronics, dead-time calculations should be performed using the total incoming background rate. The acceptance efficiency of this filter, defined as the fraction of photons from a 'true', on-axis X-ray source that survive the multiplicity rejection, is very high. For a Crab-like spectrum, the efficiency is 99.98%.</text>
<section_header_level_1><location><page_19><loc_12><loc_35><loc_29><loc_36></location>5 Simulation results</section_header_level_1>
<text><location><page_19><loc_12><loc_24><loc_69><loc_33></location>The simulations have been performed by generating an isotropic flux of primary events (photons or particles, the latter evaluated at the Solar minimum, the worst case condition) on a sphere surrounding the experiment, recording the energy deposits in the detector pixels and then applying the proper filter for the event multiplicity (Sect. 4.3). The resulting counts have been then properly normalized, taking into account the energy spectra and solid angles of the simulated background contributions.</text>
<figure>
<location><page_20><loc_18><loc_62><loc_63><loc_89></location>
<caption>Fig. 15 The anode multiplicity rejection filter. Four SDDs are conceptually shown, each subdivided into two independent halves. Darker rectangles represent triggering anodes. Events a and b are single-anode energy deposits, while c and d are double-anode events: all of these are accepted by the filter. On the contrary, triple-anode events ( e ) or disconnected multiple-anode events ( f, g, h ) are rejected.</caption>
</figure>
<text><location><page_20><loc_12><loc_44><loc_69><loc_50></location>Before discussing the general results for the LAD background and a preliminary assessment of its stability, in the next two subsections we deal with some subleties that required a more detailed geometrical description and event processing.</text>
<section_header_level_1><location><page_20><loc_12><loc_38><loc_29><loc_39></location>5.1 Capillary reflections</section_header_level_1>
<text><location><page_20><loc_12><loc_24><loc_69><loc_36></location>A standalone, more detailed raytracing simulator has been built for the MCP to evaluate the effects of the grazing incidence angle reflectivity from the pore inner walls for the CXB photons collected in the ∼ 1 · × 1 · field of view. The code tracks photons from their incoming direction up to the detector plane modeling their interaction with the optical/thermal filter and the collimator structure. The simulated collimator has a full geometrical description, 6 mm thick filled with square pores 100 µ m wide and 20 µ m thick walls. The absorption and transmission probability of the optical/thermal filter elements are computed from tables derived from the database of the National Insti-</text>
<text><location><page_21><loc_12><loc_75><loc_69><loc_89></location>tandards and Technology 5 ; the lead glass attenuation coefficient is derived from tabulated values (G. Fraser, priv. comm.). Reflectivity from the pore walls, as derived from a set of laboratory measurements (G. Fraser, priv. comm.) is well modeled by data derived from CXRO database 6 corresponding to lead glass with a 11.8 nm surface micro-roughness value. Results from this code, when reflection is not taken into account, are well in agreement with those obtained by the Geant-4 simulator (Figure 16). The effect of the wall reflectivity produces an increase of the aperture CXB (Mineo et al., in preparation), going from 40% at energies lower than 3 keV, down to ∼ 10% between 5 and 10 keV and below 5% above 10 keV.</text>
<figure>
<location><page_21><loc_14><loc_43><loc_64><loc_71></location>
<caption>Fig. 16 Comparison of results between the Geant-4 code and a stand-alone raytracing simulator developed to evaluate the effects of the reflection of grazing-incident photons on the inner walls of the collimator.</caption>
</figure>
<section_header_level_1><location><page_21><loc_12><loc_32><loc_30><loc_33></location>5.2 Internal 40 K activity</section_header_level_1>
<text><location><page_21><loc_12><loc_28><loc_69><loc_30></location>Not all the potassium decay products (see Sect. 3.6) will leave a signal in the detector: the emission is isotropic, and the photon (or the electron) has to</text>
<text><location><page_22><loc_12><loc_84><loc_69><loc_89></location>cross different thicknesses of lead glass, at various pitch angles, before reaching the SDD. Moreover, the efficiency of the silicon detector should be taken into account; eventually, only a fraction of the detected events will have a pulse height amplitude corresponding to the standard LAD energy band (2-30 keV).</text>
<text><location><page_22><loc_12><loc_67><loc_69><loc_83></location>Primary particles (1.46 MeV monochromatic photons and electrons having energies between 1 keV and 1311 keV) have been generated randomly in the lead-glass walls of a ∼ 1 . 2 × 1 . 2 cm complete model of a MCP geometry and mass (6 mm thickness, 100 µ m square holes, 20 µ m wall thickness, ∼ 70% OAR) coupled to a SDD detector. It is found that the decay of one 40 K atom generating a 1.46 MeV photon has a very small probability to generate a signal with equivalent amplitude between 2 and 30 keV, obtaining we obtain a background rate of only ∼ 5 × 10 -5 counts cm -2 s -1 . Therefore, for all practical purposes, we can discard the effect of the 40 K electronic capture decay channel. Electrons produced by the β decay, on the contrary, have a higher chance to cause a signal, producing a background rate of ∼ 2 × 10 -3 counts cm -2 s -1 .</text>
<section_header_level_1><location><page_22><loc_12><loc_62><loc_31><loc_63></location>5.3 The LAD background</section_header_level_1>
<text><location><page_22><loc_12><loc_57><loc_69><loc_61></location>The total resulting LAD background is shown in Figure 17, while the breakdown of the count rate in its various components is shown in Table 1. No detector energy resolution smoothing of the spectra was included here.</text>
<text><location><page_22><loc_12><loc_51><loc_69><loc_56></location>The main background contribution is due to the high-energy photons from the diffuse cosmic X-ray background and the Earth albedo that leaks from and are scattered in the collimators. These two components alone accounts for about 70% of the background count rate in the 2-30 keV band.</text>
<text><location><page_22><loc_12><loc_39><loc_69><loc_51></location>The diffuse emission collected in the field of view (even including the effect of capillary reflectivity), the particle-induced and internal backgrounds are a smaller contribution to the total count rate. While the CXB emission collected in the aperture field of view has a significant contribution only below ∼ 10 keV, cosmic-ray particles and neutrons produce almost flat spectra, similarly to the internal activity background, and these components become dominant only above ∼ 20 keV. The dips in the neutron-induced spectrum are due to inelastic scattering resonances.</text>
<text><location><page_22><loc_12><loc_27><loc_69><loc_39></location>Fluorescence emission from the Pb contained in the collimator glass ( L -shell lines at 10.55 and 12.61 keV) is well apparent, artificially emphasized by the non-inclusion of the energy resolution in the plot. Analysis are ongoing to evaluate whether these lines are to be shielded or used as in-flight calibration lines. In the plot is also shown, as a reference to the LAD science requirement, the spectrum of a 10 mCrab point-like source (dashed line). The total background count rate in the 2-30 keV band corresponds to a flux of ∼ 9 mCrab, thus ensuring the fulfillment of the scientific requirements, and it is below 5 mCrab in the most important 2-10 keV band.</text>
<text><location><page_22><loc_12><loc_24><loc_69><loc_26></location>We can estimate a margin of error on the background rate, taking into account that the CXB emission and the Earth albedo fluxes are affected by</text>
<text><location><page_23><loc_12><loc_74><loc_69><loc_89></location>a maximum error of ∼ 10%-20% on their normalization [26,28]. The particle components are more uncertain [12,16] (see also Sect. 3.2). However, these have been conservatively assumed at the Solar minimum, while LOFT, if selected, will fly around the next Solar maximum ( ∼ 2024), when the particle flux is expected to be approximately 20% less intense. We consider a ∼ 50% error on these fluxes. Weighting the above uncertainties on the LAD background components, we thus estimate an overall conservative maximum margin of error on the total background rate of ∼ 20%, given the present geometrical model. Future developments in the mission design will allow for a more accurate mass model and consequently to refine these results. Therefore, we can conclude that the LOFT scientific requirements are expected to be met.</text>
<figure>
<location><page_23><loc_14><loc_42><loc_64><loc_70></location>
<caption>Fig. 17 The LAD total background and its various components discussed in the text. The spectrum corresponding to a 10 mCrab source is shown as a dashed line. Note that the latter spectrum and the aperture CXB one have been normalized relatively to the other curves according to the reduced effective area that includes the collimator OAR (i.e. ∼ 10 m 2 at 8 keV).</caption>
</figure>
<section_header_level_1><location><page_23><loc_12><loc_29><loc_31><loc_31></location>5.4 Background variability</section_header_level_1>
<text><location><page_23><loc_12><loc_24><loc_69><loc_28></location>The main components of the LAD background are shown in Figure 17. The high energy photons from CXB emission and the Earth γ -ray albedo leaking through the collimator represent the dominant background component.</text>
<table>
<location><page_24><loc_14><loc_74><loc_67><loc_86></location>
<caption>Table 1 The LAD background contributions. The last line shows the LAD requirement for the total background level in the 2-30 keV band.</caption>
</table>
<text><location><page_24><loc_12><loc_52><loc_69><loc_70></location>Although they are intrinsically steady and predictable, due to their different intensity and spectra a varying relative orientation of the LAD in their radiation environment will cause a small and smooth modulation of the detected background, on the orbital timescales ( ∼ 90 minutes). This is due to the varying viewing geometry along the orbit and for different attitudes. This effect has been studied through simulations, finding that the maximum expected modulation of the background is less than 10%. This value has to be compared to a factor of a few for instruments dominated by particle-induced background. For example, RXTE/PCA had up to a ∼ 250% variation on orbital timescales [8]. In the LOFT case, the effect of the other potentially varying sources, i.e. particle induced background, is greatly reduced by the very stable environment offered by the low Earth equatorial orbit and by the fact that this component accounts for less than 6% of the overall background.</text>
<text><location><page_24><loc_12><loc_24><loc_69><loc_52></location>In Figure 18 the background rate is shown as a function of the angle between the LAD pointing direction and the center of the Earth. θ E = 0 · represents the Earth center aligned with the field of view, while θ E = 180 · stands for the Earth at the instrument nadir. In practice, this corresponds to the orbital modulation for a low-declination source. The curve representing the total background (black symbols) shows a maximum modulation of ∼ 8%. The modulation of the background rate is entirely due to geometrical effect and it can be predicted and modeled, thanks to the intrinsic steadiness of the relevant sources. Each background component is well modeled as a function of the Earth location with respect to the pointing direction by a sum of two Gaussian distributions, centered at about θ E = 0 · and 180 · for the Earthoriginated components and at about θ E = 120 · and 240 · for the diffuse sky components. These distributions, also reported in Figure 18, arise from the convolution of the directional 'transparency' of the LAD instrument with the Earth-occulted field of view. Of course, when the detector points towards the zenith ( θ E = 180 · ) the contribution from the leaking CXB emission is maximum. The overall convolution of these out-of-phase components is to give a very small fluctuation of the total background. Moreover, for an actual pointing towards an astrophysical source, the range of possible Earth angles θ E is restricted, from a θ E = 90 · for a source at the orbital pole (that is nearly coin-</text>
<text><location><page_25><loc_12><loc_75><loc_69><loc_89></location>pole) to the full 0 · -180 · range for equatorial sources. This lowers the background modulation further below the maximum during the observation of a realistic scientific target. The geometrical model, properly calibrated using in-orbit flat fields, is anticipated to allow for a background prediction at a level significantly better than 1% in the 2-10 keV band, which is the LAD science requirement. Such a level of systematic uncertainty was indeed already reached by the past experiment RXTE/PCA, which, in the presence of a much more variable background level (250% vs 8%) and less predictable background sources, with an appropriate modeling reached the level of ∼ 1% [8,9].</text>
<text><location><page_25><loc_12><loc_52><loc_69><loc_75></location>However, as some of the LOFT science cases (in particular the extragalactic science) will benefit from reaching a background knowledge significantly better than the requirement, an active background monitoring was designed for the LAD, to further improve the modeling. Due to the slow and smooth background variation, there is no need for a high-statistics, instantaneous monitoring of the rate. Rather, a continuous benchmark of the slow modulation will allow for a real-time verification of the background model. This active background monitoring is achieved by the introduction of a 'blocked collimator' (a collimator with the same stopping power but no holes) for an area corresponding to one Module of the LAD. This will enable the continuous monitor of all components of the LAD background, with the exception of the aperture background, accounting for ∼ 90% of the total background, and also evaluating the long-term variations (e.g. linked to the Solar cycle). Preliminary simulations for different targets (i.e., attitude configurations) show that the accuracy in the background modeling during a typical observation can be improved down to ∼ 0.1%-0.3% by using these additional data.</text>
<text><location><page_25><loc_12><loc_45><loc_69><loc_52></location>The subject of time-dependent modeling of the background is being further studied in depth to definitely assess the ultimate instrument capability for weak sources. Here we just reported a few preliminary results allowing to identify the range of interest. An exhaustive report and discussion is deferred to a dedicated paper, currently in preparation.</text>
<section_header_level_1><location><page_25><loc_12><loc_41><loc_66><loc_42></location>5.5 Absolute background level: cosmic variance and source contamination</section_header_level_1>
<text><location><page_25><loc_12><loc_24><loc_69><loc_39></location>In the previous section we preliminarily addressed the issue of background stability and modeling during a single observation. This is the key parameter for most of the science driver of observations of weak sources, where relative variations of source features will be studied (e.g., the Fe line in AGNs). Anyway, a few science cases may require an estimate of the absolute value of the background. This faces an intrinsic physical property of the CXB, known as cosmic variance [37]: the flux of the CXB is not perfectly isotropic but varies on different angular scales. The variation was measured [37,38,39] as ∼ 7% (rms) on the angular scale of 1 · and ∼ 2% on 20 · -40 · . In the case of the LAD, this affects only the aperture background, as the CXB/albedo contamination is seen from all directions. The expected variation in the LAD background</text>
<figure>
<location><page_26><loc_18><loc_63><loc_61><loc_87></location>
<caption>Fig. 18 Non-aperture background modulation due to the varying position of the Earth with respect to the pointing direction. θ E is the polar angle between the LAD pointing axis and the center of the Earth. Blue points are the CXB-induced contribution, orange points the Earth γ -ray albedo, cyan points the albedo neutrons and red points the particles. The black triangles represent the total background, while the black continuous curve is the background model. All quantities have been scaled to the average isotropic background count rate in the 2-30 keV band.</caption>
</figure>
<text><location><page_26><loc_12><loc_39><loc_69><loc_50></location>due to the cosmic variance is given by the fraction of the total rate due to the aperture CXB (13%) weighted by the variance over the field of view ( ∼ 7%), which is about 0.9%. Given its intrinsic astrophysical nature, the only way of measuring this component is through a local blank field, just as for any other imaging or non-imaging experiment. This is indeed the plan for the few science cases requiring the absolute knowledge of the background rate rather than its stability. It is worth stressing that the cosmic variance does not affect at all the background stability in time for a given target observation.</text>
<text><location><page_26><loc_12><loc_24><loc_69><loc_38></location>A further small contribution to the 'local background' (i.e., for a given target/attitude) is given by the possible contamination due to bright and hard sources outside the field of view. Similar to the leaking of diffuse CXB/albedo photons through the collimator, the same effect can occur to photons from point-like sources. We investigated this effect by simulations, and the maximum contribution to the overall background is of the order of 1-3% (Crab-like sources, in 2-30 keV) down to 0.1% (softer sources, e.g. Sco X-1). This additional minor contribution will be monitored by the active background monitoring system, as well as by the Wide Field Monitor. The quantitative assessment of the small effect on the background stability when the contaminating sources</text>
<text><location><page_27><loc_12><loc_86><loc_69><loc_89></location>are variable on the time scale of interest will be extensively addressed in the paper in preparation mentioned earlier.</text>
<section_header_level_1><location><page_27><loc_12><loc_82><loc_24><loc_83></location>6 Conclusions</section_header_level_1>
<text><location><page_27><loc_12><loc_66><loc_69><loc_80></location>LOFT will be an innovative mission that will observe compact Galactic and extragalactic objects in both the spectral and the temporal domains. The unprecedented sensitive area of the LAD instrument will open new windows in the study of the fundamental physics allowed by these natural laboratories. The scientific objectives of the mission require an accurate knowledge and minimisation of the detector background. To this end, an extensive massmodel for the LOFT/LAD experiment has been developed, using the standard Geant-4 toolkit (Sect. 4), and all the main components (photonic, leptonic, hadronic and internal, Sect. 3) of the orbital background environment have been simulated.</text>
<text><location><page_27><loc_12><loc_51><loc_69><loc_66></location>The main contribution to the overall background is found to be due to the diffuse X-rays that 'leaks' from the lead-glass collimators of the instrument. This emission originates from the diffuse cosmic X-ray background and from the Earth albedo. The particle-induced background is minimised mainly thanks to the LOFT low-inclination, low-altitude orbit and small mass for unit area of the LAD experiment, becoming dominant only at high energies, above 30 keV. A further suppression of the particle background is enabled by the particular signature that these events leave on the Silicon Drift Detectors (Sect. 4.3). The simulations show the feasibility for the current LAD instrument design to fulfill the required background level of 10 mCrab in the 2-30 keV band.</text>
<text><location><page_27><loc_12><loc_38><loc_69><loc_51></location>Background variations on orbital timescales are mostly induced by the varying geometry between the position of the Earth and the pointing direction, As such, they can be modelled and accounted for. The use of special detector units ('blocked' module) and a carefully planned blank-field pointing strategy, are foreseen to monitor these background variations enabling to reach a residual systematics level better than 0.3%. The residual contribution of strong (and variable) off-axis sources, above 30-50 keV, if needed, can be further modelled using observations performed with the other instrument onboard LOFT, the WFM.</text>
<text><location><page_27><loc_12><loc_28><loc_69><loc_38></location>An increase in the accuracy of the background determination for the LAD will be provided by further improvements in the geometrical description of the instrument itself and of the spacecraft bus structures. Furthermore, results will be refined by taking into account a complete determination of the residual activation of the detector materials following the grazing passages in the highbackground environment of the South Atlantic Anomaly, where however no scientific observations are conducted.</text>
<text><location><page_27><loc_12><loc_24><loc_69><loc_26></location>Acknowledgements We are grateful to the anonymous referee whose helpful comments greatly improved the text, and to Alessandra De Rosa for useful discussions. We acknowl-</text>
<text><location><page_28><loc_12><loc_87><loc_69><loc_89></location>edge financial support from ASI/INAF contract I/021/12/0. RC furthermore acknowledge support from INAF Tecno-PRIN 2009 grant.</text>
<section_header_level_1><location><page_28><loc_12><loc_82><loc_21><loc_84></location>References</section_header_level_1>
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</document>
[
{
"title": "ABSTRACT",
"content": "Experimental Astronomy manuscript No. (will be inserted by the editor)",
"pages": [
1
]
},
{
"title": "Background simulations for the Large Area Detector onboard LOFT",
"content": "Riccardo Campana · Marco Feroci · Ettore Del Monte · Teresa Mineo · Niels Lund · George W. Fraser Received: date / Accepted: date Abstract The Large Observatory For X-ray Timing (LOFT), currently in an assessment phase in the framework the ESA M3 Cosmic Vision programme, is an innovative medium-class mission specifically designed to answer fundamental questions about the behaviour of matter, in the very strong gravitational and magnetic fields around compact objects and in supranuclear density conditions. Having an effective area of ∼ 10 m 2 at 8 keV, LOFT will be able to measure with high sensitivity very fast variability in the X-ray fluxes and spectra. A good knowledge of the in-orbit background environment is essential to assess the scientific performance of the mission and optimize the design of its main instrument, the Large Area Detector (LAD). In this paper the results of an extensive Geant-4 simulation of the instrument will be discussed, showing the main contributions to the background and the design solutions for its reduction and control. Our results show that the current LOFT/LAD design is expected to meet its scientific requirement of a background rate equivalent to 10 mCrab in 2-30 keV, achieving about 5 mCrab in the most important 2-10 keV energy band. Moreover, simulations show an anticipated modulation of the background rate as small as 10% over the orbital timescale. The intrinsic photonic origin of the largest background component also allows for On behalf of the LOFT collaboration. R. Campana, M. Feroci, E. Del Monte INAF/IAPS, Via Fosso del Cavaliere 100, I-00133, Roma, Italy and INFN/Sezione di Roma 2, viale della Ricerca Scientifica 1, I-00133, Roma, Italy. E-mail: riccardo.campana@inaf.it Present address of R. Campana: INAF/IASF-Bologna, via Gobetti 101, I-40129, Bologna, Italy. T. Mineo INAF/IASF-Palermo, Via Ugo La Malfa 153, I-90146, Palermo, Italy. N. Lund DTU-Space, Elektrovej Bld. 327, DK-2800, Kgs. Lyngby, Denmark. G. Fraser Space Research Centre, Dept. of Physics and Astronomy, University of Leicester, LE17RH, Leicester, UK. an efficient modelling, supported by an in-flight active monitoring, allowing to predict systematic residuals significantly better than the requirement of 1%, and actually meeting the 0.25% science goal. Keywords X-ray astronomy · Instrumental background · Montecarlo simulations PACS 07.85.Fv · 07.87.+v · 95.40.+s",
"pages": [
1,
2
]
},
{
"title": "1 Introduction",
"content": "Astrophysical observations in the X-ray domain are often dominated by the background. In order to assess and maximize the scientific performance of a satellite-borne instrument during its design phase, is essential to have an early estimation of the background level and to identify its origin. This will allow to drive the optimization of the experiment design itself. Moreover, a reliable estimation of the background level is important to assess the scientific objectives and performance of the mission. The most convenient and consolidated way to compute such a background is by means of a Monte Carlo simulation, using e.g. the Geant-4 toolkit [1], in which the interactions of the space environment with a somewhat simplified mass model of the spacecraft and of the instruments are sampled and studied. A correct evaluation of the radiation environment surrounding the experiment is of course the most important ingredient in the background simulation. In this paper we will show Monte Carlo simulations of the instrumental background for the Large Area Detector (LAD) instrument onboard the proposed LOFT (Large Observatory for X-ray Timing) satellite [2,3], that is a candidate for the third slot of medium-class missions (M3) of the Cosmic Vision 2015-2025 programme of the European Space Agency, with a possible launch in the 2022-24 timeframe. LOFT (Figure 1) has been specially designed to answer the 'Fundamental Question' #3.3 of the Cosmic Vision programme, Matter under extreme conditions . In the aim of investigating the behaviour of the matter in the most extreme physical conditions that can be found in the gravitational and magnetic fields around neutron stars and black holes, LOFT will study the rapid variability of the X-ray emission from these objects, both in the spectral and temporal domains. Moreover, LOFT will be an observatory for virtually all classes of bright X-ray sources: among them, X-ray bursters, high mass X-ray binaries, cataclysmic variables, magnetars and active galactic nuclei. The main experiment onboard LOFT is the LAD [3] (Sect. 2), a collimated instrument with a collecting area 20 times bigger than its immediate predecessor (PCA, the Proportional Counter Array onboard the Rossi X-ray Timing Explorer, RXTE), and based on large-area Silicon Drift Detectors (SDDs) coupled to lead-glass Micro Channel Plate (MCP) collimators. In order to survey a large fraction of the sky simultaneously, and to trigger follow-up observations with the main instrument, LOFT will host also a coded-mask Wide Field Monitor (WFM, [4]). This latter instrument, sensitive in the same energy range of LAD, will use basically the same SDD detectors, with some design differences to optimize the detector response for imaging [5,6,7]. The background level required to satisfy the LOFT scientific objectives, in particular regarding the relatively faint (1-10 mCrab) sources like most Active Galactic Nuclei (AGNs) and some accretion-powered X-ray pulsars, is ≤ 10 mCrab for the 2-30 keV LAD standard band. Being a collimated instrument, the LAD will not measure simultaneously and independently the source flux and the background. While some collimated instruments (e.g. BeppoSAX/PDS and RXTE/HEXTE, both at hard X-rays) employ a 'rocking' strategy, where the same instrument unit alternates observations of the source and of the background, others (e.g. RXTE/PCA, at soft X-rays, similar to LAD) rely on a background model obtained by means of dedicated blank-field pointings and by using onboard ratemeters [8]. As we will discuss in the following, LAD will adopt an approach similar to PCA. To ensure the fulfillment of the scientific objectives, in particular for low flux sources, an accurate estimation and modelling of the background level for this instrument is of particular importance. The background variations that have a significant impact on the scientific performance should be minimized, and controlled to a high degree of accuracy. The LOFT science case resulted in a requirement on background knowledge at 1%, with a goal at 0.25%. As a comparison, RXTE/PCA reached a background residual systematic level of ∼ 0.5%-1% [8,9], while BeppoSAX/PDS of ∼ 1% [10]. This paper is structured as follows. In Section 2 we describe the LAD instrument. In Section 3 we introduce the space environment for LOFT and discuss the main contributions to the instrumental background, while in Section 4 the main features of the LAD Monte Carlo mass model and simulator are described. The simulation results are shown in Section 5, and in Section 6 we draw our conclusions.",
"pages": [
2,
3
]
},
{
"title": "2 The Large Area Detector",
"content": "The Large Area Detector is an array of 2016 individual large-area Silicon Drift Detectors, sensitive to the 2-30 keV emission (with a extended 30-80 keV band) collimated in a ∼ 1 · field of view by means of a lead-glass microcapillary collimator plate. The instrument design is modular: an array of 4 × 4 SDDs and their front-end electronics (FEE) are placed in a module . Each module has its independent back-end electronics (MBEE) that powers and configure the FEE, manages the event readout and monitors the detector health. A panel is composed of an array of 21 modules, placed on a support grid (Figure 2). The current LAD design by the LOFT consortium envisages 6 panels (each about 350 cm × 90 cm), that stems from the satellite bus and are deployed once the spacecraft reaches its orbit (Figure 1). The collimators, made of lead glass, have the same footprint as the SDDs (about 11 cm × 7 cm), a thickness of 6 mm, and 100 µ m wide square holes separated with 20 µ m thick walls (and thus having an open area ratio , f OAR , of ∼ 70%). An aluminium frame couples the collimators with the detectors frame and ensures the tight alignment constraints. The working principle of the SDDs is the following: a photon is absorbed in the depleted silicon bulk, and produces a cloud of electrons whose number is proportional to the incident energy. The cloud drifts towards an array of read-out anodes, by means of a high-voltage field applied between the middle section and the two detector edges, hosting two independent rows of anodes. During the drift, the cloud size increases due to diffusion. The number of anodes that will read-out the charge cloud thus depends on the absorption point distance and anode pitch [5]. For the LAD detectors, that have a pitch of 970 µ m, the charge cloud originating from the absorption of a X-ray photon in the 2-80 keV band is at most 0.5 mm (FWHM) thus read-out by 1 or 2 anodes, depending on the impact point and the energy. At 6 keV, for example, ∼ 40% of the events are collected by one anode and ∼ 60% by two anodes. The event multiplicity is read-out and transmitted to the ground, enabling a selection of 'single' and 'double' events, with different energy resolutions.",
"pages": [
3,
4,
5
]
},
{
"title": "3 Sources of background",
"content": "In this section we discuss the main sources of the LOFT background. We describe the environment encountered in the baseline low-Earth orbit, with its hadronic, leptonic and photon components. Moreover, we describe the internal source of background, i.e. the natural radioactivity of the collimator material.",
"pages": [
5
]
},
{
"title": "3.1 The LOFT orbit",
"content": "The LOFT orbit requirement is at an altitude of h = 600 km with an inclination of 5 · . Likewise, lower-altitude orbits (down to 550 km) and/or inclinations (down to 0 · ) are considered, with small improvements for what regards the conclusions about the background rate and properties. The lower altitude and inclinations, however, impacts on the long-term radiation damage on the detectors. For these orbits, the geomagnetic latitude range is usually θ M < 0 . 3 radians. In the current work we neglected the possible background component due to the activation of the instrument and satellite materials. Based on the arguments below, we estimated it to be a second-order effect. The South Atlantic Anomaly, a region of trapped high-energy protons and electrons, is grazed by the LOFT baseline orbit only in its outermost regions (Figure 3), with a shallower passage for lower altitudes and inclinations. Therefore, the effect on background due to the activation of materials by this intense radiation is expected to be negligible with respect to other sources, as confirmed by preliminary evaluations and heritage from past missions in similar orbits (e.g., BeppoSAX and AGILE). It should also be noted that in contrast to previous experiments, the LAD instrument has a very light structure per unit volume and the relatively small spacecraft assembly is seen at a small viewing angle, due to the tower supporting the LAD panels (see Figure 1). In addition to that, as for example observed by Swift/XRT [11], Silicon detectors show marginal activation effects in LEO, even when operated as focal plane instruments (more local mass) and on more inclined orbits (up to ∼ 20 · ). However, given our goal of controlling the LAD background to the highest possible accuracy, we plan to study this aspect in great detail in a future work.",
"pages": [
5,
6
]
},
{
"title": "3.2 Primary cosmic rays",
"content": "Following the approach of Mizuno et al. [12], we assume that the primary cosmic ray spectrum is expressed as F U ( R ), a function of the particle magnetic rigidity, R = pc/Ze , where Z is the atomic number and p the particle momentum. The full spectrum at a given location in the magnetosphere and a given phase of the solar cycle will be given by: where M and Ze are the mass and charge of the particle, E its kinetic energy, φ is a solar modulation factor, h is the orbit height and θ M is the geomagnetic latitude. The second term, that includes the effects of the solar modulation on the cosmic ray particles, is given by [13]: where the solar modulation potential φ is given by: The geomagnetic cutoff function is given, for vertically incident particles, by [12]: in which the cutoff rigidity is obtained from the Stormer equation [14]: where R E = 6371 km is the Earth radius. Moreover, the r exponent in Eq. 5 is: For the 600 km, 5 · inclination LOFT orbit, the cutoff rigidity therefore is found to be in the range R cut ∼ 10.1-12.1 GV. Since we are interested mostly in the average background fluxes, we discard at the present stage the east-west effect for which particles coming from different directions have different cutoff rigidities. We therefore assume that the general primary spectrum, now expressed as: has an uniform angular distribution with respect to the zenith angle, up to the Earth horizon, i.e. for zenith angles θ from 0 · to θ cut , where the latter is given by: For h = 600 km, θ cut ∼ 114 · . The solid angle subtended by the Earth at an altitude h is given by: and therefore the accessibile sky subtends a solid angle Ω = 4 π -Ω E (Figure 4). For h = 600 km, the Earth blocks about 30% of the sky.",
"pages": [
6,
7
]
},
{
"title": "3.2.1 Primary protons",
"content": "The unmodulated value of the cosmic ray electron spectrum is given by the BESS [15] and AMS measurements [16]: In Figure 5 the primary proton spectra for different values of the solar modulation and for the LOFT orbit is reported. The variation in flux between solar minimum and maximum, at the peak flux, is ∼ 15%.",
"pages": [
7
]
},
{
"title": "3.2.2 Primary electrons and positrons",
"content": "The unmodulated value of the cosmic ray electron spectrum is given by [17, 18,12]: The fraction of positrons to electrons, usually given by the ratio e + / ( e + + e -), is found to be rather independent of the energy [19], i.e. the spectrum of primary positrons has the same slope of the electron one, but a different normalization: In Figure 6 the primary e -and e + spectra for different values of the solar modulation and for the LOFT orbit are reported. The difference in flux between solar minumum and maximum is about 20% at the peak.",
"pages": [
9
]
},
{
"title": "3.2.3 Primary alpha particles",
"content": "We consider only the primary helium nuclei, because the contribution of heavier nuclei is negligible with respect to the other primary particles flux [20]. The unmodulated flux, as given by Mizuno et al. [12] on the basis of AMS and BESS data is: and is shown in Figure 7. 5",
"pages": [
9,
10
]
},
{
"title": "3.3.1 Protons",
"content": "For the low altitude equatorial Earth orbits considered, the impinging proton spectrum outside the trapped particle belt, i.e. the South Atlantic Anomaly, consists of the primary component discussed in Sect. 3.2 and of a secondary, quasi-trapped component, originating from and impacting to the Earth atmosphere (sometimes in the literature they are referred to as the 'splash' and 'reentrant' albedo components). AMS measurements [16] showed that this secondary component is composed of a short-lived and a long-lived particle population, both originating from the regions near the geomagnetic equator. Mizuno et al. [12] model the secondary equatorial proton spectrum as a cutoff power-law: (16) and a power law for energies below 100 MeV: This spectrum is shown in Figure 8 together with the AMS measurements. According to the measurements performed with the NINA and NINA-II instruments [21], the extrapolation below 100 MeV is likely an overestimate. We do not consider here the soft (10 keV-1 MeV), highly directional equatorial proton population [22,23], since these particles are most efficiently stopped by all the spacecraft structures surrounding the detectors. Even the small fraction that impinges on the detector through the capillary holes, albeit significant when considering the long-term radiation damage on the SDDs, leaves a negligible background signal.",
"pages": [
10,
11
]
},
{
"title": "3.3.2 Electrons and positrons",
"content": "The equatorial secondary electron spectrum can be approximated with a broken power law [12]: and a power law for energies below 100 MeV: At variance with respect to the primary particles, in the geomagnetic equatorial region the positrons are predominant with respect to the electrons. The spectrum is the same, but the ratio e + /e -is about 3.3 (Figure 9).",
"pages": [
12
]
},
{
"title": "3.4 Photon background",
"content": "As we will see in the following Sections, the main contribution to the overall background for the LAD instrument is due to the high energy X-ray emission transmitted and scattered in the lead-glass collimators. The main contribution comes from the CXB and the Earth albedo emission.",
"pages": [
13
]
},
{
"title": "3.4.1 Cosmic photon background",
"content": "For the cosmic X-ray and γ -ray diffuse background we assume the Gruber et al. [24] analytic form, derived from HEAO-1 A4 and COMPTEL measurements, and fitted in the energy range between 3 keV and 100 GeV. This spectrum is usually used as a standard reference [25,26], and is plotted in Figure 10. The first branch is valid below 60 keV: while for energies above 60 keV: Other measurements of the CXB have been performed by, e.g., BeppoSAX [25], INTEGRAL [27,28] and BAT [26]. These measurements are consistent with the HEAO-1 results, albeit some of these seems to indicate a slightly higher normalization ( ∼ 8%) for the > 10 keV spectrum.",
"pages": [
13
]
},
{
"title": "3.4.2 Albedo γ -ray background",
"content": "The secondary photon background is due to the interactions of cosmic rays (proton and leptonic components) with the atmosphere, and to the reflection of CXB emission. As such, it has a strong zenith dependence [29]. This albedo component has a higher flux, for unit of solid angle, than the CXB for energies above ∼ 70 keV. We assume the albedo spectrum as measured by BAT [26], that agrees in the range above 50 keV, after some corrections, with previous measurements [30,31]. This spectrum can be parameterized by the function [32,26]: and is plotted in Figure 11.",
"pages": [
13
]
},
{
"title": "3.5 Earth neutron albedo",
"content": "As reported by the official ESA ECSS documents 1 , there is presently no model for atmospheric albedo neutron fluxes considered mature enough to be used as a standard. To account for the flux of neutrons produced by cosmic-ray interactions in the Earth atmosphere, we however used the results of the QinetiQ Atmospheric Radiation Model (QARM, [33,34]), that uses a response function approach based on Monte Carlo radiation transport codes to generate directional fluxes of atmospheric secondary radiation. The resulting spectrum, shown in Figure 12, is also consistent with the Monte Carlo simulations of Armstrong & Colborn [35,36]. 3",
"pages": [
15
]
},
{
"title": "3.6 Natural radioactivity",
"content": "The lead-glass used in the microcapillary plate collimators contains potassium, in a fraction of approximately 7.2% by weight. The activity due to the naturally occurring long-lived radioactive isotope 40 K (with an abundance of 0.0117% with respect to natural potassium) is to be taken into account. This isotope decays with the emission of β rays having a continuum of energies up to 1.31 MeV ( ∼ 89% branching ratio) and of 1.46 MeV photons ( ∼ 11% b.r.) due to electronic capture. Let L , ρ , f OAR be the thickness, density and open fraction of the collimator. Therefore, the collimator activity for unit area is: where A K ∼ 30 Bq/g is the specific activity of natural potassium due to 40 K, and M K is the total potassium mass per unit area in the collimator, i.e. where f K is the fraction by weight of potassium in the collimator glass. For the MCP lead glass, we have f K ∼ 7 . 2%. Assuming for the MCP collimator L = 6 mm, ρ = 3 . 3 g/cm 3 , f OAR = 69 . 44% we obtain M K = 436 g/m 2 . Therefore, we have A coll = 13100 Bq/m 2 . For a total geometrical LAD collimator area of ∼ 15 m 2 we have thus ∼ 196500 decays/s. Considering the branching ratio, we have therefore that every second in the collimator bulk are uniformly and isotropically generated ∼ 175000 electrons with a continuum of energies and ∼ 21000 monochromatic photons with a 1.46 MeV energy.",
"pages": [
15,
16
]
},
{
"title": "4.1 The mass model",
"content": "Simulations were performed using the Geant-4 Monte Carlo toolkit [1] (version 4.9.4). Geant-4 allows to describe the geometry and the materials of the instruments and of the satellite bus. Moreover, the code enables to follow the various physical interactions along the path of a primary event through the various components of the geometry, evaluating the secondary particles and energy deposits generated. For the description of the electromagnetic interactions, the Low Energy Livermore library 2 is used, that trace the interactions of electrons and photons with matter down to about 250 eV, using interpolated data tables. Physical processes like photoelectric effect, Compton and Rayleigh scattering, pair production, bremsstrahlung, multiple scattering and annihilation are simulated, optionally taking into account also the effect of photon polarization. Fluorescence X-rays and Auger electrons from the various chemical elements are included. The hadronic physics, instead, is described using different models in different energy ranges (e.g. Bertini Cascade model, Quark-Gluon String model, etc. 3 ). Both elastic and inelastic processes are treated, the latter describing e.g. nuclear capture, fragmentation, de-excitation and scattering of the relevant stable and long-lived nucleons and mesons. In Figure 13 is shown the mass model geometry used in the simulations for the LAD instrument. A single LAD panel is simulated, with the actual dimensions ( ∼ 350 cm × 90 cm) but with a simplified design that consists of a stacked-layers geometry (sketched in Figure 14). A simplified geometry for the satellite bus, the structural tower and the other five panels is assumed, using aluminium as the material with an 'effective' density that takes into account the total mass and volume. The SDD array is represented as a 450 µ m thick slab of silicon, subdivided in 0.97 mm × 35 mm pixels. On its surface are placed various passive layers (Al cathode implants, SiO 2 passivation layer, undepleted Silicon bulk), the MCP collimator and the optical/thermal filter (1 µ m kapton coupled with a 80 nm Al layer). The MCP collimator has an 'effective', reduced density, to take into account the holes without resorting to a more accurate but much more timeconsuming geometrical description. Simulations on a smaller geometrical size have been performed to compare the 'effective' collimator with a complete geometrical capillary plate description (see Sect. 5.2), giving consistent results. Underneath the detectors, the FEE board is simulated as a 2 mm thick FR4 slab with an interposed 100 µ mCu conductive layer and a 100 µ mAl shielding layer, a 1 mm thick Al radiator and a 500 µ m Pb backshield. An aluminium frame encloses the module sides, while the panel structure consists of a carbonfiber reinforced plastic grid frame.",
"pages": [
16,
17
]
},
{
"title": "4.2 Primary event generation",
"content": "For the generation of primary events, the General Particle Source 4 (GPS) approach is followed, that allows to produce arbitrary spectra in rather complex geometries. The initial events (particles or photons) are generated on the surface of a sphere of radius R , with a cosine-biased emission angle to ensure an isotropic flux. The emission angle is further restricted between 0 (normal to the spherical surface) to θ max : the emission cone then subtends a smaller sphere of radius r that surrounds the experiment. We have: Let Φ be the energy-integrated flux, between E min and E max , expressed in particles cm -2 s -1 sr -1 . The total rate is therefore: This derives from the integration over the 2 π emission angle for a point on the spherical surface, biased with the cosine law: and integrated over the source sphere surface S = 4 πR 2 . The restriction on the emission angle introduces the further factor sin 2 θ max . Therefore the simulation time that corresponds to N events is: generated primary If we detect C i counts in the energy bin i , the corresponding measured flux (in counts cm -2 s -1 ), i.e. convolved with the detector response, would be: where A det is the detector sensitive area.",
"pages": [
18,
19
]
},
{
"title": "4.3 Anode multiplicity rejection filter",
"content": "For the LAD, an anode multiplicity rejection algorithm is implemented to filter out ionization streaks from charged particles, that leave an energy deposit on more than 2 adjacent anodes. More in detail, for each of the two independent halves of a SDD tile, events are rejected if they trigger more than two adjacent anodes. Likewise, events are rejected if there is a group of non-adjacent triggering anodes within a 'distance' of 32 channels in the same half-SDD (Figure 15). Energy deposits at a further distance are treated as independent. Since the average energy loss of minimum ionizing particles (MIPs) in silicon is about 3.7 MeV/cm, their total energy deposit is usually above ∼ 150 keV. The anode multiplicity rejection filter, combined together with an upper threshold on the reconstructed event signal, is very effective in the suppression of the particle-induced background: a threshold of 80 keV allow to filter out 94%96% of the particle background, depending on the incoming particle type and spectrum. Of course, since this event elaboration is performed in the frontend electronics, dead-time calculations should be performed using the total incoming background rate. The acceptance efficiency of this filter, defined as the fraction of photons from a 'true', on-axis X-ray source that survive the multiplicity rejection, is very high. For a Crab-like spectrum, the efficiency is 99.98%.",
"pages": [
19
]
},
{
"title": "5 Simulation results",
"content": "The simulations have been performed by generating an isotropic flux of primary events (photons or particles, the latter evaluated at the Solar minimum, the worst case condition) on a sphere surrounding the experiment, recording the energy deposits in the detector pixels and then applying the proper filter for the event multiplicity (Sect. 4.3). The resulting counts have been then properly normalized, taking into account the energy spectra and solid angles of the simulated background contributions. Before discussing the general results for the LAD background and a preliminary assessment of its stability, in the next two subsections we deal with some subleties that required a more detailed geometrical description and event processing.",
"pages": [
19,
20
]
},
{
"title": "5.1 Capillary reflections",
"content": "A standalone, more detailed raytracing simulator has been built for the MCP to evaluate the effects of the grazing incidence angle reflectivity from the pore inner walls for the CXB photons collected in the ∼ 1 · × 1 · field of view. The code tracks photons from their incoming direction up to the detector plane modeling their interaction with the optical/thermal filter and the collimator structure. The simulated collimator has a full geometrical description, 6 mm thick filled with square pores 100 µ m wide and 20 µ m thick walls. The absorption and transmission probability of the optical/thermal filter elements are computed from tables derived from the database of the National Insti- tandards and Technology 5 ; the lead glass attenuation coefficient is derived from tabulated values (G. Fraser, priv. comm.). Reflectivity from the pore walls, as derived from a set of laboratory measurements (G. Fraser, priv. comm.) is well modeled by data derived from CXRO database 6 corresponding to lead glass with a 11.8 nm surface micro-roughness value. Results from this code, when reflection is not taken into account, are well in agreement with those obtained by the Geant-4 simulator (Figure 16). The effect of the wall reflectivity produces an increase of the aperture CXB (Mineo et al., in preparation), going from 40% at energies lower than 3 keV, down to ∼ 10% between 5 and 10 keV and below 5% above 10 keV.",
"pages": [
20,
21
]
},
{
"title": "5.2 Internal 40 K activity",
"content": "Not all the potassium decay products (see Sect. 3.6) will leave a signal in the detector: the emission is isotropic, and the photon (or the electron) has to cross different thicknesses of lead glass, at various pitch angles, before reaching the SDD. Moreover, the efficiency of the silicon detector should be taken into account; eventually, only a fraction of the detected events will have a pulse height amplitude corresponding to the standard LAD energy band (2-30 keV). Primary particles (1.46 MeV monochromatic photons and electrons having energies between 1 keV and 1311 keV) have been generated randomly in the lead-glass walls of a ∼ 1 . 2 × 1 . 2 cm complete model of a MCP geometry and mass (6 mm thickness, 100 µ m square holes, 20 µ m wall thickness, ∼ 70% OAR) coupled to a SDD detector. It is found that the decay of one 40 K atom generating a 1.46 MeV photon has a very small probability to generate a signal with equivalent amplitude between 2 and 30 keV, obtaining we obtain a background rate of only ∼ 5 × 10 -5 counts cm -2 s -1 . Therefore, for all practical purposes, we can discard the effect of the 40 K electronic capture decay channel. Electrons produced by the β decay, on the contrary, have a higher chance to cause a signal, producing a background rate of ∼ 2 × 10 -3 counts cm -2 s -1 .",
"pages": [
21,
22
]
},
{
"title": "5.3 The LAD background",
"content": "The total resulting LAD background is shown in Figure 17, while the breakdown of the count rate in its various components is shown in Table 1. No detector energy resolution smoothing of the spectra was included here. The main background contribution is due to the high-energy photons from the diffuse cosmic X-ray background and the Earth albedo that leaks from and are scattered in the collimators. These two components alone accounts for about 70% of the background count rate in the 2-30 keV band. The diffuse emission collected in the field of view (even including the effect of capillary reflectivity), the particle-induced and internal backgrounds are a smaller contribution to the total count rate. While the CXB emission collected in the aperture field of view has a significant contribution only below ∼ 10 keV, cosmic-ray particles and neutrons produce almost flat spectra, similarly to the internal activity background, and these components become dominant only above ∼ 20 keV. The dips in the neutron-induced spectrum are due to inelastic scattering resonances. Fluorescence emission from the Pb contained in the collimator glass ( L -shell lines at 10.55 and 12.61 keV) is well apparent, artificially emphasized by the non-inclusion of the energy resolution in the plot. Analysis are ongoing to evaluate whether these lines are to be shielded or used as in-flight calibration lines. In the plot is also shown, as a reference to the LAD science requirement, the spectrum of a 10 mCrab point-like source (dashed line). The total background count rate in the 2-30 keV band corresponds to a flux of ∼ 9 mCrab, thus ensuring the fulfillment of the scientific requirements, and it is below 5 mCrab in the most important 2-10 keV band. We can estimate a margin of error on the background rate, taking into account that the CXB emission and the Earth albedo fluxes are affected by a maximum error of ∼ 10%-20% on their normalization [26,28]. The particle components are more uncertain [12,16] (see also Sect. 3.2). However, these have been conservatively assumed at the Solar minimum, while LOFT, if selected, will fly around the next Solar maximum ( ∼ 2024), when the particle flux is expected to be approximately 20% less intense. We consider a ∼ 50% error on these fluxes. Weighting the above uncertainties on the LAD background components, we thus estimate an overall conservative maximum margin of error on the total background rate of ∼ 20%, given the present geometrical model. Future developments in the mission design will allow for a more accurate mass model and consequently to refine these results. Therefore, we can conclude that the LOFT scientific requirements are expected to be met.",
"pages": [
22,
23
]
},
{
"title": "5.4 Background variability",
"content": "The main components of the LAD background are shown in Figure 17. The high energy photons from CXB emission and the Earth γ -ray albedo leaking through the collimator represent the dominant background component. Although they are intrinsically steady and predictable, due to their different intensity and spectra a varying relative orientation of the LAD in their radiation environment will cause a small and smooth modulation of the detected background, on the orbital timescales ( ∼ 90 minutes). This is due to the varying viewing geometry along the orbit and for different attitudes. This effect has been studied through simulations, finding that the maximum expected modulation of the background is less than 10%. This value has to be compared to a factor of a few for instruments dominated by particle-induced background. For example, RXTE/PCA had up to a ∼ 250% variation on orbital timescales [8]. In the LOFT case, the effect of the other potentially varying sources, i.e. particle induced background, is greatly reduced by the very stable environment offered by the low Earth equatorial orbit and by the fact that this component accounts for less than 6% of the overall background. In Figure 18 the background rate is shown as a function of the angle between the LAD pointing direction and the center of the Earth. θ E = 0 · represents the Earth center aligned with the field of view, while θ E = 180 · stands for the Earth at the instrument nadir. In practice, this corresponds to the orbital modulation for a low-declination source. The curve representing the total background (black symbols) shows a maximum modulation of ∼ 8%. The modulation of the background rate is entirely due to geometrical effect and it can be predicted and modeled, thanks to the intrinsic steadiness of the relevant sources. Each background component is well modeled as a function of the Earth location with respect to the pointing direction by a sum of two Gaussian distributions, centered at about θ E = 0 · and 180 · for the Earthoriginated components and at about θ E = 120 · and 240 · for the diffuse sky components. These distributions, also reported in Figure 18, arise from the convolution of the directional 'transparency' of the LAD instrument with the Earth-occulted field of view. Of course, when the detector points towards the zenith ( θ E = 180 · ) the contribution from the leaking CXB emission is maximum. The overall convolution of these out-of-phase components is to give a very small fluctuation of the total background. Moreover, for an actual pointing towards an astrophysical source, the range of possible Earth angles θ E is restricted, from a θ E = 90 · for a source at the orbital pole (that is nearly coin- pole) to the full 0 · -180 · range for equatorial sources. This lowers the background modulation further below the maximum during the observation of a realistic scientific target. The geometrical model, properly calibrated using in-orbit flat fields, is anticipated to allow for a background prediction at a level significantly better than 1% in the 2-10 keV band, which is the LAD science requirement. Such a level of systematic uncertainty was indeed already reached by the past experiment RXTE/PCA, which, in the presence of a much more variable background level (250% vs 8%) and less predictable background sources, with an appropriate modeling reached the level of ∼ 1% [8,9]. However, as some of the LOFT science cases (in particular the extragalactic science) will benefit from reaching a background knowledge significantly better than the requirement, an active background monitoring was designed for the LAD, to further improve the modeling. Due to the slow and smooth background variation, there is no need for a high-statistics, instantaneous monitoring of the rate. Rather, a continuous benchmark of the slow modulation will allow for a real-time verification of the background model. This active background monitoring is achieved by the introduction of a 'blocked collimator' (a collimator with the same stopping power but no holes) for an area corresponding to one Module of the LAD. This will enable the continuous monitor of all components of the LAD background, with the exception of the aperture background, accounting for ∼ 90% of the total background, and also evaluating the long-term variations (e.g. linked to the Solar cycle). Preliminary simulations for different targets (i.e., attitude configurations) show that the accuracy in the background modeling during a typical observation can be improved down to ∼ 0.1%-0.3% by using these additional data. The subject of time-dependent modeling of the background is being further studied in depth to definitely assess the ultimate instrument capability for weak sources. Here we just reported a few preliminary results allowing to identify the range of interest. An exhaustive report and discussion is deferred to a dedicated paper, currently in preparation.",
"pages": [
23,
24,
25
]
},
{
"title": "5.5 Absolute background level: cosmic variance and source contamination",
"content": "In the previous section we preliminarily addressed the issue of background stability and modeling during a single observation. This is the key parameter for most of the science driver of observations of weak sources, where relative variations of source features will be studied (e.g., the Fe line in AGNs). Anyway, a few science cases may require an estimate of the absolute value of the background. This faces an intrinsic physical property of the CXB, known as cosmic variance [37]: the flux of the CXB is not perfectly isotropic but varies on different angular scales. The variation was measured [37,38,39] as ∼ 7% (rms) on the angular scale of 1 · and ∼ 2% on 20 · -40 · . In the case of the LAD, this affects only the aperture background, as the CXB/albedo contamination is seen from all directions. The expected variation in the LAD background due to the cosmic variance is given by the fraction of the total rate due to the aperture CXB (13%) weighted by the variance over the field of view ( ∼ 7%), which is about 0.9%. Given its intrinsic astrophysical nature, the only way of measuring this component is through a local blank field, just as for any other imaging or non-imaging experiment. This is indeed the plan for the few science cases requiring the absolute knowledge of the background rate rather than its stability. It is worth stressing that the cosmic variance does not affect at all the background stability in time for a given target observation. A further small contribution to the 'local background' (i.e., for a given target/attitude) is given by the possible contamination due to bright and hard sources outside the field of view. Similar to the leaking of diffuse CXB/albedo photons through the collimator, the same effect can occur to photons from point-like sources. We investigated this effect by simulations, and the maximum contribution to the overall background is of the order of 1-3% (Crab-like sources, in 2-30 keV) down to 0.1% (softer sources, e.g. Sco X-1). This additional minor contribution will be monitored by the active background monitoring system, as well as by the Wide Field Monitor. The quantitative assessment of the small effect on the background stability when the contaminating sources are variable on the time scale of interest will be extensively addressed in the paper in preparation mentioned earlier.",
"pages": [
25,
26,
27
]
},
{
"title": "6 Conclusions",
"content": "LOFT will be an innovative mission that will observe compact Galactic and extragalactic objects in both the spectral and the temporal domains. The unprecedented sensitive area of the LAD instrument will open new windows in the study of the fundamental physics allowed by these natural laboratories. The scientific objectives of the mission require an accurate knowledge and minimisation of the detector background. To this end, an extensive massmodel for the LOFT/LAD experiment has been developed, using the standard Geant-4 toolkit (Sect. 4), and all the main components (photonic, leptonic, hadronic and internal, Sect. 3) of the orbital background environment have been simulated. The main contribution to the overall background is found to be due to the diffuse X-rays that 'leaks' from the lead-glass collimators of the instrument. This emission originates from the diffuse cosmic X-ray background and from the Earth albedo. The particle-induced background is minimised mainly thanks to the LOFT low-inclination, low-altitude orbit and small mass for unit area of the LAD experiment, becoming dominant only at high energies, above 30 keV. A further suppression of the particle background is enabled by the particular signature that these events leave on the Silicon Drift Detectors (Sect. 4.3). The simulations show the feasibility for the current LAD instrument design to fulfill the required background level of 10 mCrab in the 2-30 keV band. Background variations on orbital timescales are mostly induced by the varying geometry between the position of the Earth and the pointing direction, As such, they can be modelled and accounted for. The use of special detector units ('blocked' module) and a carefully planned blank-field pointing strategy, are foreseen to monitor these background variations enabling to reach a residual systematics level better than 0.3%. The residual contribution of strong (and variable) off-axis sources, above 30-50 keV, if needed, can be further modelled using observations performed with the other instrument onboard LOFT, the WFM. An increase in the accuracy of the background determination for the LAD will be provided by further improvements in the geometrical description of the instrument itself and of the spacecraft bus structures. Furthermore, results will be refined by taking into account a complete determination of the residual activation of the detector materials following the grazing passages in the highbackground environment of the South Atlantic Anomaly, where however no scientific observations are conducted. Acknowledgements We are grateful to the anonymous referee whose helpful comments greatly improved the text, and to Alessandra De Rosa for useful discussions. We acknowl- edge financial support from ASI/INAF contract I/021/12/0. RC furthermore acknowledge support from INAF Tecno-PRIN 2009 grant.",
"pages": [
27,
28
]
},
{
"title": "References",
"content": "2. B. Bertucci, A. Biland, S. Bizzaglia, S. Blasko, G. Boella, M. Boschini, M. Bourquin, L. Brocco, G. Bruni, M. Buenerd, J.D. Burger, W.J. Burger, X.D. Cai, C. Camps, P. Cannarsa, M. Capell, D. Casadei, J. Casaus, G. Castellini, C. Cecchi, Y.H. Chang, H.F. Chen, H.S. Chen, Z.G. Chen, N.A. Chernoplekov, T.H. Chiueh, Y.L. Chuang, F. Cindolo, V. Commichau, A. Contin, P. Crespo, M. Cristinziani, J.P. da Cunha, T.S. Dai, J.D. Deus, N. Dinu, L. Djambazov, I. D'Antone, Z.R. Dong, P. Emonet, J. Engelberg, F.J. Eppling, T. Eronen, G. Esposito, P. Extermann, J. Favier, E. Fiandrini, P.H. Fisher, G. Fluegge, N. Fouque, Y. Galaktionov, M. Gervasi, P. Giusti, D. Grandi, O. Grimm, W.Q. Gu, K. Hangarter, A. Hasan, V. Hermel, H. Hofer, M.A. Huang, W. Hungerford, M. Ionica, R. Ionica, M. Jongmanns, K. Karlamaa, W. Karpinski, G. Kenney, J. Kenny, W. Kim, A. Klimentov, R. Kossakowski, V. Koutsenko, M. Kraeber, G. Laborie, T. Laitinen, G. Lamanna, G. Laurenti, A. Lebedev, S.C. Lee, G. Levi, P. Levtchenko, C.L. Liu, H.T. Liu, I. Lopes, G. Lu, Y.S. Lu, K. Lubelsmeyer, D. Luckey, W. Lustermann, C. Ma˜na, A. Margotti, F. Mayet, R.R. McNeil, B. Meillon, M. Menichelli, A. Mihul, A. Mourao, A. Mujunen, F. Palmonari, A. Papi, I.H. Park, M. Pauluzzi, F. Pauss, E. Perrin, A. Pesci, A. Pevsner, M. Pimenta, V. Plyaskin, V. Pojidaev, M. Pohl, V. Postolache, N. Produit, P.G. Rancoita, D. Rapin, F. Raupach, D. Ren, Z. Ren, M. Ribordy, J.P. Richeux, E. Riihonen, J. Ritakari, U. Roeser, C. Roissin, R. Sagdeev, G. Sartorelli, A. Schultz von Dratzig, G. Schwering, G. Scolieri, E.S. Seo, V. Shoutko, E. Shoumilov, R. Siedling, D. Son, T. Song, M. Steuer, G.S. Sun, H. Suter, X.W. Tang, S.C.C. Ting, S.M. Ting, M. Tornikoski, J. Torsti, J. Trumper, J. Ulbricht, S. Urpo, I. Usoskin, E. Valtonen, J. Vandenhirtz, F. Velcea, E. Velikhov, B. Verlaat, I. Vetlitsky, F. Vezzu, J.P. Vialle, G. Viertel, D. Vit'e, H. Von Gunten, S. Waldmeier Wicki, W. Wallraff, B.C. Wang, J.Z. Wang, Y.H. Wang, K. Wiik, C. Williams, S.X. Wu, P.C. Xia, J.L. Yan, L.G. Yan, C.G. Yang, M. Yang, S.W. Ye, P. Yeh, Z.Z. Xu, H.Y. Zhang, Z.P. Zhang, D.X. Zhao, G.Y. Zhu, W.Z. Zhu, H.L. Zhuang, A. Zichichi, B. Zimmermann, Physics Letters B 490 , 27 (2000)",
"pages": [
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2013FoPh...43.1478B
https://arxiv.org/pdf/1301.5471.pdf
<document>
<section_header_level_1><location><page_1><loc_22><loc_78><loc_78><loc_80></location>A new approach to modifying theories of gravity</section_header_level_1>
<text><location><page_1><loc_26><loc_72><loc_74><loc_77></location>Christian G. Bohmer ∗ and Nicola Tamanini † Department of Mathematics, University College London Gower Street, London, WC1E 6BT, UK</text>
<text><location><page_1><loc_43><loc_69><loc_57><loc_70></location>October 3, 2018</text>
<section_header_level_1><location><page_1><loc_46><loc_64><loc_53><loc_65></location>Abstract</section_header_level_1>
<text><location><page_1><loc_25><loc_55><loc_75><loc_63></location>We propose a new point of view for interpreting Newton's and Einstein's theories of gravity. By taking inspiration from Continuum Mechanics and its treatment of anisotropies, we formulate new gravitational actions for modified theories of gravity. These models are simple and natural generalisations with many interesting properties. Above all, their precise form can, in principle, be determined experimentally.</text>
<section_header_level_1><location><page_1><loc_21><loc_51><loc_40><loc_53></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_21><loc_32><loc_79><loc_50></location>Since Newton's formulation of the law of universal gravitation in the Principia 1687, the theory has been unchanged. Very few attempts have been made to modify Newtonian gravity, modified Newtonian dynamics being probably the one exception [1]. It took almost 330 years until a generalisation of Newton's theory was successfully constructed. Einstein's theory of gravity was radically different to any previous physical theory. It abandoned the absolute notion of time and replaced the concept of force by the curvature of a four dimensional space. However, it took only a few years when the first modifications and extensions of Einstein's theory appeared in the literature, see [2] for an excellent historical review. Ever since, modified theories of gravity have enjoyed a prominent role in theoretical physics. It is the aim of the current work to propose modifications of Newton's and Einstein's theories of gravity by applying the same idea to both of them.</text>
<text><location><page_1><loc_21><loc_29><loc_79><loc_31></location>Let us start with the actions of Newtonian gravity and its relativistic analogue, the Einstein-Hilbert action</text>
<formula><location><page_1><loc_32><loc_25><loc_79><loc_28></location>S Newton = ∫ [ -ρϕ + 1 8 πG δ ij ∂ i ϕ∂ j ϕ ] d 3 x, (1)</formula>
<formula><location><page_1><loc_35><loc_21><loc_79><loc_25></location>S EH = ∫ [ L matter + c 4 16 πG g µν R µν ] √ -g d 4 x. (2)</formula>
<text><location><page_1><loc_21><loc_18><loc_79><loc_21></location>Variation of the Newtonian action with respect to the gravitation potential ϕ yields the well-known Poisson equation ∆ ϕ = 4 πGρ where ρ is a given matter</text>
<text><location><page_2><loc_21><loc_81><loc_79><loc_85></location>distribution. Likewise, variations of the Einstein-Hilbert action with respect to the metric (which contains the gravitational potentials) gives the famous Einstein field equations G αβ = 8 πG/c 4 T αβ .</text>
<text><location><page_2><loc_21><loc_65><loc_79><loc_80></location>When comparing the Einstein-Hilbert action (2) with other models in physics, it appears to be somewhat unnatural as one generally considers potential energies quadratic in the field strength, Hooke's law probably being the best known. However, when looking at the Einstein-Hilbert action, there are various very good reason for its form. Some of these can be motivated mathematically while others are simply observational. The field equations derived from it are close to Newton's theory of gravity and where the solutions deviate they are doing so in precisely the way to be in agreement with observations. However, there are observational facts which strongly indicate that our understanding of the gravitational force is far from complete. Dark matter and dark energy are two unknown forms of matter which are required to make the Universe work.</text>
<text><location><page_2><loc_21><loc_32><loc_79><loc_65></location>Looking at (1) and (2) we note that both actions contain a contraction with respect to the metric. In the Newtonian model we simply have the flat metric for Cartesian 3-space. We are now taking some inspiration from Continuum Mechanics and in particular constitutive equations which specify the various different models studied in this field. In Maxwell's electromagnetism the constitutive equations for instance define the form of the dielectric tensor D = ε E where ε is a rank 2 tensor in general and we should really write D i = ε i j E j . When working with the Faraday tensor and its corresponding excitation, one writes H ij = 1 / 4 /epsilon1 ijmn χ mnkl F kl where for Maxwell's theory in vacuum χ mnkl = √ -g ( g mk g nl -g nk g ml ), see [3]. In many simple applications ε i j is taken to be proportional to the Kronecker delta which corresponds to a simple isotropic medium with dielectric constant /epsilon1 , namely ε i j = /epsilon1δ j i . Sometimes /epsilon1 is allowed to vary throughout the medium. In other words, a rank 2 isotropic tensor is proportional to the metric tensor. We are using this observation to argue that the Newton and Einstein-Hilbert actions are based on the assumption of an isotropic medium. Since Nature has a strong tendency to be anisotropic, this appears to be rather unnatural. By doing so, we are changing the interpretation of the terms δ ij and g µν in (1) and (2), respectively. We now view them much like the material tensors specified by constitutive equations which define the properties of the material. It is an interesting historical fact that F. Klein noted in a letter to Hilbert, as early as 1917, that the Ricci scalar in the Einstein-Hilbert action can be written as χ mnkl R mnkl , see [4]. For a recent paper inspired by similar thoughts, see [5].</text>
<text><location><page_2><loc_21><loc_24><loc_79><loc_32></location>Our approach follows the principal ideas of Brans and Dicke [6]. They suggested a model where the gravitational coupling was allowed to vary in space and time and it was viewed as an additional dynamical degree of freedom in the theory. The original Brans-Dicke model can be viewed as the isotropic limit of out model, we simply choose C µν = φ ( x α ) g µν with the main difference that we do not treat the new degrees of freedom as dynamical.</text>
<text><location><page_2><loc_21><loc_14><loc_79><loc_24></location>We might want to, for sake of concreteness, speak of the properties of the vacuum here. However, we would like to be very careful and point out that this change of viewpoint has a variety of philosophical implications when interpreting the modified theories. We would like to keep these issues aside for now and proceed with the formulation of the theories. The assumption of an isotropic vacuum is certainly well supported by a host of experimental observations. However, there is clearly room for some improvement, in particular</text>
<text><location><page_3><loc_21><loc_79><loc_79><loc_85></location>in general relativity. Many of the well-known modifications are is some sense rather severe. Field equations become higher than second order, new fields are introduced, locality is broken, local Lorentz invariance is broken etc. We will show that our theories are completely harmless and retain all desired properties.</text>
<section_header_level_1><location><page_3><loc_21><loc_75><loc_78><loc_77></location>2 Modified gravity using continuum mechanics</section_header_level_1>
<text><location><page_3><loc_21><loc_71><loc_79><loc_74></location>One of the most conservative modifications of Newton's and Einstein's theories we can think of is therefore the following</text>
<formula><location><page_3><loc_32><loc_68><loc_79><loc_71></location>S Newton = ∫ [ -ρϕ + 1 8 πG C ij ∂ i ϕ∂ j ϕ ] d 3 x, (3)</formula>
<formula><location><page_3><loc_35><loc_65><loc_79><loc_68></location>S EH = ∫ [ L matter + c 4 16 πG C µν R µν ] √ -g d 4 x, (4)</formula>
<text><location><page_3><loc_21><loc_53><loc_79><loc_64></location>where C ij and C µν are symmetric rank 2 tensors which contain information about the underlying structure of the theory. In Continuum Mechanics such objects are often referred to as material tensors or elastic coefficients [7]. We will stick with this well established notation and should clearly distinguish it from the energy-momentum tensor T αβ . We should emphasise that in this approach to gravity we are able to assume any symmetry for the metric, for instance spherical symmetry or homogeneity and isotropy. However, the symmetry of the metric is independent of the symmetry of the elastic coefficients.</text>
<text><location><page_3><loc_21><loc_50><loc_79><loc_53></location>Let us firstly consider variations of the modified Newtonian theory (3) with respect to the potential φ . We find</text>
<formula><location><page_3><loc_42><loc_47><loc_79><loc_49></location>∂ i ( C ij ∂ j ϕ ) = 4 πGρ, (5)</formula>
<text><location><page_3><loc_21><loc_36><loc_79><loc_47></location>which reduces to the Poisson equation if we choose C ij = δ ij . As C ij can in principle be an arbitrary tensor, solutions to this equation may be quite different. As a simple example, let us consider the case when C ij = diag( c 1 , c 2 , c 3 ) with the c i being constants. In continuum mechanics or solid state physics one would speak of a crystal with three principal propagation directions. In this case, the field equation (5) can be reduced to the Poisson equation by rescaling the coordinates. Looking for a radially symmetric solution gives the interesting result</text>
<formula><location><page_3><loc_40><loc_31><loc_79><loc_36></location>ϕ ∝ 1 √ c 1 x 2 + c 2 y 2 + c 3 z 2 . (6)</formula>
<text><location><page_3><loc_21><loc_25><loc_79><loc_32></location>In this case the gravitational field of a massive body is ellipsoidal instead of being spherical. However, the strength of this effect depends on the values of the constants. If for instance, the numerical values of the three constants c 1 , c 2 , c 3 are very close to each other, then the gravitational field will look spherical unless very large distances are taking into account.</text>
<text><location><page_3><loc_21><loc_18><loc_79><loc_25></location>It should also be noted that the components of C ij do not have to be constants. They can be functions of the coordinates. Thus, the form of the gravitational field may be different in different regions of space, however, the gravitational law (5) would still be universal. If we consider C ij = χ ( t, x, y, z ) δ ij , the field equations are</text>
<formula><location><page_3><loc_41><loc_14><loc_79><loc_17></location>∆ ϕ = 4 πG χ ρ -∂ i ϕ∂ i χ χ . (7)</formula>
<text><location><page_4><loc_21><loc_78><loc_79><loc_85></location>If the function χ is slowly varying in space | ∂ i χ | /lessmuch 1, we would have a theory which would be in very good agreement with Newton's theory on smaller scales, like the solar system. Note also that assuming a time varying χ would correspond to a time dependent gravitational constant, compare with Dirac's large number hypothesis [8].</text>
<text><location><page_4><loc_21><loc_68><loc_79><loc_78></location>Let us now consider the action (4). We assume that the tensor C µν may depend on the metric, this is important so that the limit of general relativity can be recovered. Note that the variation of the Ricci tensor does not give a surface term anymore because C µν is not the metric. One has to integrate by parts twice (the Ricci tensor contains second derivatives of the metric) and move the derivatives action on the metric variations to act on the matter tensor. Denoting</text>
<formula><location><page_4><loc_43><loc_64><loc_79><loc_67></location>Σ µναβ = -δC µν δg αβ , (8)</formula>
<text><location><page_4><loc_21><loc_62><loc_55><loc_63></location>the whole calculation yields the field equation</text>
<formula><location><page_4><loc_23><loc_55><loc_79><loc_61></location>Σ µναβ R µν -1 2 g αβ C µν R µν + 1 2 /square C αβ + 1 2 g αβ ∇ µ ∇ ν C µν -∇ µ ∇ ( α C β ) µ = 8 πG c 4 T αβ . (9)</formula>
<text><location><page_4><loc_21><loc_54><loc_72><loc_55></location>where we have also included the energy-momentum tensor of matter.</text>
<text><location><page_4><loc_21><loc_46><loc_79><loc_53></location>Recall that in General Relativity the field equations imply the conservation equations by virtue of the twice contracted Bianchi identities. This does no longer hold due to the presence of the material tensor C µν in the field equations. Nonetheless, we can take the covariant derivative of the field equation (9) with respect to ∇ α and find the following conservation equation</text>
<formula><location><page_4><loc_23><loc_40><loc_79><loc_45></location>∇ α Σ µναβ R µν +Σ µναβ ∇ α R µν -1 2 g αβ R µν ∇ α C µν -C ασ ∇ α R β σ -R β σ ∇ α C ασ = 8 πG c 4 ∇ α T αβ . (10)</formula>
<text><location><page_4><loc_21><loc_32><loc_79><loc_39></location>This implies that the conservation equations for T αβ is no longer a direct consequence of the gravitational field equations alone. However, if we assume that the energy-momentum tensor is derived from a diffeomorphism invariant matter action, then the energy-momentum tensor is covariantly conserved in view of Noether's theorem</text>
<formula><location><page_4><loc_45><loc_29><loc_79><loc_31></location>∇ α T αβ = 0 . (11)</formula>
<text><location><page_4><loc_21><loc_24><loc_79><loc_29></location>We will assume this henceforth. Therefore, the conservation equation (11) imposes constraints on the components of C µν via (10) and thus its components cannot be chosen completely arbitrarily. For concreteness, we define</text>
<formula><location><page_4><loc_23><loc_18><loc_79><loc_23></location>J β = ∇ α Σ µναβ R µν +Σ µναβ ∇ α R µν -1 2 g αβ R µν ∇ α C µν -C ασ ∇ α R β σ -R β σ ∇ α C ασ , (12)</formula>
<text><location><page_4><loc_21><loc_17><loc_56><loc_18></location>which gives the additional consistency equation</text>
<formula><location><page_4><loc_47><loc_14><loc_79><loc_16></location>J β = 0 . (13)</formula>
<text><location><page_5><loc_21><loc_74><loc_79><loc_85></location>It should also be emphasised that one has to be careful when choosing the tensor C µν and the derived quantity Σ µναβ . Namely, there is a conceptual difference between prescribing the tensor C µ ν from prescribing C µν . In the former case, we find that C µν = g µσ C ν σ and thus C µν has an explicit dependence on the metric which in turn will yield a non-trivial form for Σ µναβ . On the other, if we specify C µν a priori , then this tensor does not depend on the metric and hence the resulting variational derivative would be zero. One has to carefully distinguish both cases when analysing the field equations.</text>
<text><location><page_5><loc_21><loc_68><loc_79><loc_74></location>It is very difficult to analyse the field equations (9) given general Σ µναβ and C µν . It is also not clear what would constitute a good choice for those quantities. As a first attempt to understand the theory, we assume C µν to be conformally related to the inverse metric</text>
<formula><location><page_5><loc_43><loc_65><loc_79><loc_67></location>C µν = φ ( x α ) g µν , (14)</formula>
<text><location><page_5><loc_21><loc_61><loc_79><loc_64></location>where φ ( x α ) is a scalar function depending on the coordinates. This choice simplifies the field equation considerably and they are now given by</text>
<formula><location><page_5><loc_36><loc_57><loc_79><loc_60></location>φG αβ + g αβ /square φ -∇ α ∇ β φ = 8 πG c 4 T αβ . (15)</formula>
<text><location><page_5><loc_21><loc_54><loc_79><loc_56></location>Upon division by the field φ and solving for the Einstein tensor, the resulting equation</text>
<formula><location><page_5><loc_32><loc_49><loc_79><loc_53></location>G αβ = φ -1 8 πG c 4 T αβ + φ -1 ( ∇ α ∇ β φ -g αβ /square φ ) . (16)</formula>
<text><location><page_5><loc_21><loc_33><loc_80><loc_49></location>shows similarities with non-minimally coupled scalar field theories [6, see Eq. (11)]. The main difference between the current theory and most other approaches is that φ is not a dynamical degree of freedom because C µν is a prescribed tensor and has not corresponding equations of motion. The conformal model appears to agree with [6] in the limit when the Brans-Dicke parameter ω → 0. However, as variations with respect to φ are not considered, there is no propagation equation for the scalar field. By substituting (14) back into the action (4), we note that this simply corresponds to assuming the gravitational constant to be varying in time and space, see for instance [8]. Varying constants models are generally based on non-minimally coupled scalar field with kinetic term similar to Brans-Dicke theory, see [9].</text>
<section_header_level_1><location><page_5><loc_21><loc_30><loc_60><loc_31></location>3 A Schwarzschild like solution</section_header_level_1>
<text><location><page_5><loc_21><loc_26><loc_79><loc_28></location>Let us start by considering a static and spherically symmetric vacuum spacetime described by the metric</text>
<formula><location><page_5><loc_37><loc_22><loc_79><loc_24></location>ds 2 = -e ν ( r ) dt 2 + e µ ( r ) dr 2 + r 2 d Ω 2 , (17)</formula>
<text><location><page_5><loc_21><loc_16><loc_79><loc_22></location>where d Ω 2 is the usual line element of the two-sphere. We also assume φ = e ξ ( r ) . When the analogue situation is analysed in General Relativity where ξ ≡ 0, one finds two independent equations which determine the two unknown functions ν ( r ) and µ ( r ). In this model, there is the additional degree of freedom ξ and</text>
<text><location><page_6><loc_21><loc_84><loc_76><loc_85></location>fortunately, there are now three independent equations. These are given by</text>
<formula><location><page_6><loc_33><loc_80><loc_79><loc_82></location>-e µ r 2 + 1 r 2 -1 2 µ ' ξ ' -µ ' r + ξ '' +( ξ ' ) 2 +2 ξ ' r = 0 , (18)</formula>
<formula><location><page_6><loc_42><loc_77><loc_79><loc_80></location>-e µ r 2 + 1 r 2 + 1 2 ν ' ξ ' + ν ' r +2 ξ ' r = 0 , (19)</formula>
<formula><location><page_6><loc_32><loc_71><loc_79><loc_76></location>-1 4 µ ' ν ' -1 2 µ ' ξ ' -µ ' 2 r + ν '' 2 + 1 2 ν ' ξ ' + 1 4 ( ν ' ) 2 + ν ' 2 r + ξ '' +( ξ ' ) 2 + ξ ' r = 0 . (20)</formula>
<text><location><page_6><loc_21><loc_60><loc_79><loc_70></location>Eqn. (19) can be solved for the function µ which can then be substituted into the other two equations. Combining those linearly gives the condition ν ' ∝ ξ ' which then allows us to reduce this problem to a single differential equation. While separation of variable and subsequent integration is possible, the resulting equation cannot be solved analytically for the unknown functions due to its high nonlinearity. However, by assuming ξ ' = -C 2 ν ' with C /lessmuch 1 we can find an approximate solution to the field equations which is given by</text>
<formula><location><page_6><loc_40><loc_56><loc_79><loc_59></location>e ν = 1 -C r + O ( C 4 ) , (21)</formula>
<formula><location><page_6><loc_39><loc_53><loc_79><loc_56></location>e -µ = 1 -C r + 2 C 3 r + O ( C 4 ) , (22)</formula>
<formula><location><page_6><loc_40><loc_50><loc_79><loc_53></location>e ξ = 1 + C 3 r + O ( C 4 ) . (23)</formula>
<text><location><page_6><loc_21><loc_41><loc_79><loc_49></location>One easily verifies that this solution satisfies the field equations (18)-(20) up to O ( C 4 ). By choosing C = 2 GM we arrive at a Schwarzschild like solution with only a small difference. Clearly, this difference of the order of C 3 and therefore it would be very difficult to distinguish between this metric and the Schwarzschild metric using solar system tests. This is a promising result which indicates that this theory can pass solar system tests without great difficulty.</text>
<section_header_level_1><location><page_6><loc_21><loc_37><loc_38><loc_38></location>4 Cosmology</section_header_level_1>
<section_header_level_1><location><page_6><loc_21><loc_34><loc_43><loc_35></location>4.1 Conformal model</section_header_level_1>
<text><location><page_6><loc_21><loc_30><loc_79><loc_33></location>Next, we want to study the cosmological implications of field equations (9). Similar to the above we assume C µν = φ ( t ) g µν and consider a FLRW universe</text>
<formula><location><page_6><loc_36><loc_24><loc_79><loc_29></location>ds 2 = -dt 2 + a 2 ( t ) dx 2 + dy 2 + dz 2 ( 1 + k 4 r 2 ) 2 , (24)</formula>
<text><location><page_6><loc_21><loc_22><loc_79><loc_24></location>where a ( t ) is the scale factor and r 2 = x 2 + y 2 + z 2 . This yields the following cosmological field equations</text>
<formula><location><page_6><loc_43><loc_18><loc_79><loc_20></location>3 H 2 φ +3 H ˙ φ +3 kφ a 2 = 8 πG c 4 ρ, (25)</formula>
<formula><location><page_6><loc_35><loc_15><loc_79><loc_18></location>-¨ φ -2 φ a a -2 H ˙ φ -H 2 φ -kφ a 2 = 8 πG c 4 p , (26)</formula>
<text><location><page_7><loc_21><loc_81><loc_79><loc_85></location>where an overdot denotes differentiation with respect to time derivative and H is the Hubble parameter H = ˙ a/a . Moreover, the conservation equation (11) gives</text>
<formula><location><page_7><loc_43><loc_78><loc_79><loc_79></location>˙ ρ +3 H ( ρ + p ) = 0 , (27)</formula>
<text><location><page_7><loc_21><loc_76><loc_58><loc_77></location>while the consistency equation (13) for J 0 becomes</text>
<formula><location><page_7><loc_41><loc_71><loc_79><loc_74></location>3 ˙ φ ( ˙ H +2 H 2 + k a 2 ) = 0 . (28)</formula>
<text><location><page_7><loc_21><loc_69><loc_69><loc_71></location>The other three components vanish identically J 1 = J 2 = J 3 = 0.</text>
<text><location><page_7><loc_21><loc_65><loc_79><loc_69></location>Irrespective of the choice of matter in this model, the consistency equation (13) either implies that ˙ φ = 0 which is equivalent to General Relativity, or</text>
<formula><location><page_7><loc_43><loc_61><loc_79><loc_64></location>˙ H +2 H 2 + k a 2 = 0 . (29)</formula>
<text><location><page_7><loc_21><loc_59><loc_61><loc_60></location>The solutions to this differential equation are given by</text>
<formula><location><page_7><loc_36><loc_56><loc_79><loc_59></location>a ( t ) = a 0 √ t -t 0 k = 0 , (30)</formula>
<formula><location><page_7><loc_36><loc_53><loc_79><loc_56></location>a ( t ) = √ a 2 0 -k ( t -t 0 ) 2 k = ± 1 , (31)</formula>
<text><location><page_7><loc_21><loc_47><loc_79><loc_53></location>and correspond to a radiation dominated universe. Next, we have to check whether these solutions are consistent with the remaining equations (25), (26) and (27). Firstly, we start looking for vacuum solutions, ρ = p = 0. The conservation equation (27) is trivially satisfied and one verifies that</text>
<formula><location><page_7><loc_36><loc_42><loc_79><loc_46></location>φ ( t ) = φ 0 √ ( t -t 0 ) k = 0 , (32)</formula>
<formula><location><page_7><loc_36><loc_38><loc_79><loc_42></location>φ ( t ) = φ 0 ( t -t 0 ) √ a 2 0 -k ( t -t 0 ) 2 k = ± 1 , (33)</formula>
<text><location><page_7><loc_21><loc_37><loc_54><loc_38></location>is a solution to the remaining field equations.</text>
<text><location><page_7><loc_21><loc_27><loc_79><loc_37></location>Under the assumption that C µν = φ ( t ) g µν and ρ = p = 0 we cannot find an accelerating solution to the field equations. We could, of course, add regular matter to the field equations and seek other solutions. Let us briefly consider the situation where we include an incompressible perfect fluid or dust, for simplicity we consider k = 0 only. The scale factor is unchanged and given by (30). The conservation equation (27) implies the standard relation ρ ∝ 1 /a 3 and it turns out that</text>
<formula><location><page_7><loc_33><loc_22><loc_79><loc_26></location>ρ ( t ) = ρ 0 ( t -t 0 ) 3 / 2 , φ ( t ) = 16 πρ 0 t +3 φ 0 3 √ t -t 0 , (34)</formula>
<text><location><page_7><loc_21><loc_21><loc_46><loc_22></location>is a solution to the field equations.</text>
<section_header_level_1><location><page_7><loc_21><loc_18><loc_42><loc_19></location>4.2 Fluid like model</section_header_level_1>
<text><location><page_7><loc_21><loc_14><loc_79><loc_17></location>However, we are particularly interested in the vacuum equations without ordinary matter. The reason for this is simply that we want to show that an</text>
<text><location><page_8><loc_21><loc_81><loc_79><loc_85></location>additional non-dynamical structure in the theory suffices to get a dynamical universe. One can easily find such solutions by introducing an extra degree of freedom in the material tensor. Let us choose the elastic coefficients to be</text>
<formula><location><page_8><loc_41><loc_78><loc_79><loc_80></location>C µ ν = diag( -/rho1, σ, σ, σ ) , (35)</formula>
<text><location><page_8><loc_21><loc_75><loc_79><loc_78></location>which means that C µν = 1 2 ( g µσ C ν σ + g νσ C µ σ ) and thus we have an explicit dependence on the metric, implying a non-zero Σ µναβ given by</text>
<formula><location><page_8><loc_30><loc_71><loc_79><loc_74></location>Σ µναβ = 1 4 ( g µα C νβ + g να C µβ + g µβ C να + g νβ C µα ) . (36)</formula>
<text><location><page_8><loc_21><loc_69><loc_79><loc_71></location>One can interpret the quantities /rho1 and σ as the energy density and pressure of the vacuum, thereby specifying its internal structure.</text>
<text><location><page_8><loc_21><loc_66><loc_79><loc_68></location>The cosmological field equations, the conservation equation and the consistency equation of this model are given by</text>
<formula><location><page_8><loc_39><loc_62><loc_79><loc_65></location>-H 2 /rho1 -1 2 H ( ˙ /rho1 -˙ σ ) + kσ a 2 = 8 πG 3 c 4 ρ, (37)</formula>
<formula><location><page_8><loc_34><loc_59><loc_79><loc_62></location>3 H 2 /rho1 +2( /rho1H )˙ + 1 2 (¨ /rho1 -¨ σ ) -kσ a 2 = 8 πG c 4 p , (38)</formula>
<formula><location><page_8><loc_36><loc_57><loc_79><loc_59></location>H 2 ( ˙ /rho1 -3 ˙ σ ) + ˙ H ( ˙ /rho1 -˙ σ ) -2 k ˙ σ a 2 = 0 , (39)</formula>
<formula><location><page_8><loc_48><loc_55><loc_79><loc_56></location>˙ ρ +3 H ( ρ + p ) = 0 , (40)</formula>
<text><location><page_8><loc_21><loc_49><loc_79><loc_54></location>where the conservation equation can be derived from the first three equations. In the analogue situation in General Relativity we would have two independent equations for three unknown functions while for this model we have one additional equation and two additional degrees of freedom.</text>
<text><location><page_8><loc_21><loc_43><loc_79><loc_49></location>To begin with, we consider the vacuum case ρ = p = 0 which eliminates one equation and two degrees of freedom. Thus, we are left with two independent equations and three unknown functions. In order to close this system, we choose a linear equation of state σ = w/rho1 for the spacetime structure.</text>
<text><location><page_8><loc_21><loc_36><loc_79><loc_43></location>Now, we can solve Eq. (37) for ˙ /rho1 and substitute this result into Eq. (38) to arrive at a single differential equation in a ( t ). One can also check that substitution of ˙ /rho1 into Eq. (39) leads to the same differential equation, confirming that these equations are indeed not independent. This differential equation is given by</text>
<formula><location><page_8><loc_35><loc_33><loc_79><loc_35></location>(˙ a 2 -kw ) ( ( w -1) a a +2 w ( k + ˙ a 2 ) ) = 0 , (41)</formula>
<text><location><page_8><loc_49><loc_30><loc_49><loc_32></location>/negationslash</text>
<text><location><page_8><loc_55><loc_30><loc_55><loc_32></location>/negationslash</text>
<text><location><page_8><loc_63><loc_30><loc_63><loc_32></location>/negationslash</text>
<formula><location><page_8><loc_43><loc_25><loc_79><loc_29></location>a ( t ) = √ kw ( t -t 0 ) , (42)</formula>
<text><location><page_8><loc_21><loc_28><loc_79><loc_32></location>where in the derivation we assumed /rho1 = 0, w = 1 and ˙ a = 0. This differential equation is the product of two equations and thus can be solved by finding a solution to either of the two equations. The first one is easily solved by</text>
<text><location><page_8><loc_21><loc_24><loc_41><loc_25></location>and is valid only if kw > 0.</text>
<text><location><page_8><loc_21><loc_20><loc_79><loc_24></location>The second differential equation in (41) cannot be solved analytically for arbitrary k and w due to the non-linear nature of the equation. For w = 1 / 3 for instance we can find the two solutions</text>
<formula><location><page_8><loc_37><loc_17><loc_79><loc_20></location>a ( t ) = 1 2 ( 1 α 2 e α ( t -t 0 ) + ke -α ( t -t 0 ) ) , (43)</formula>
<formula><location><page_8><loc_37><loc_14><loc_79><loc_17></location>a ( t ) = 1 2 ( ke α ( t -t 0 ) + 1 α 2 e -α ( t -t 0 ) ) , (44)</formula>
<text><location><page_9><loc_21><loc_81><loc_79><loc_85></location>which are valid for all k . For w = 0 one only finds one solution a ( t ) = a 0 t . For k = 0 one can solve the differential equation for all w and its solution is given by</text>
<formula><location><page_9><loc_42><loc_77><loc_79><loc_79></location>a ( t ) = a 0 ( t -t 0 ) w -1 3 w -1 , (45)</formula>
<text><location><page_9><loc_50><loc_74><loc_50><loc_77></location>/negationslash</text>
<text><location><page_9><loc_21><loc_70><loc_79><loc_77></location>which is well defined provided that w = 1 / 3 and corresponds to a power-law solution. For 0 < w < 1 / 3 this would correspond to an accelerated solution. The case w = 1 / 3 needs to be treated separately. As the exponent becomes very large, one would expect this to correspond to exponential functions, and indeed in this case</text>
<formula><location><page_9><loc_46><loc_67><loc_79><loc_68></location>a = a 0 e λt . (46)</formula>
<text><location><page_9><loc_21><loc_60><loc_79><loc_66></location>Thus, we were able to find solutions of the field equations modelling a universe which can undergo periods of accelerated expansion, without the need to introduce any forms of matter. All we have done is to add an additional non-dynamical structure to the theory on a very fundamental level.</text>
<section_header_level_1><location><page_9><loc_21><loc_57><loc_45><loc_58></location>4.3 Kasner type model</section_header_level_1>
<text><location><page_9><loc_21><loc_55><loc_61><loc_56></location>We are now considering a Kasner type metric given by</text>
<formula><location><page_9><loc_35><loc_51><loc_79><loc_53></location>ds 2 = -dt 2 + t 2 p 1 dx 2 + t 2 p 2 dy 2 + t 2 p 3 dz 2 , (47)</formula>
<text><location><page_9><loc_21><loc_49><loc_60><loc_51></location>and assume the material tensor C µ ν to be of the form</text>
<formula><location><page_9><loc_38><loc_46><loc_79><loc_48></location>C µ ν = diag( -1 , c 1 ( t ) , c 2 ( t ) , c 3 ( t )) . (48)</formula>
<text><location><page_9><loc_21><loc_41><loc_79><loc_45></location>The field equations of this system are quite complicated. However, one notes that all field equations contain terms of the form tc ' i ( t ) and t 2 c '' i ( t ) which indicate that one can arrive at algebraic equations by choosing</text>
<formula><location><page_9><loc_43><loc_39><loc_79><loc_40></location>c i ( t ) = 2 γ i log( t ) , (49)</formula>
<text><location><page_9><loc_21><loc_35><loc_79><loc_37></location>where the γ i are constants. With this additional assumption (49), and considering a vacuum ρ = p = 0, the field equations are given by</text>
<formula><location><page_9><loc_33><loc_31><loc_79><loc_33></location>p 1 γ 1 + p 2 γ 2 + p 3 γ 3 -p 1 p 2 -p 1 p 3 -p 2 p 3 = 0 , (50)</formula>
<formula><location><page_9><loc_36><loc_28><loc_79><loc_30></location>( p 1 + p 2 + p 3 -1)( γ 1 -γ 3 + p 1 -p 3 ) = 0 , (52)</formula>
<formula><location><page_9><loc_36><loc_30><loc_79><loc_32></location>( p 1 + p 2 + p 3 -1)( γ 1 -γ 2 + p 1 -p 2 ) = 0 , (51)</formula>
<formula><location><page_9><loc_36><loc_26><loc_79><loc_28></location>( p 1 + p 2 + p 3 -1)( γ 2 -γ 3 + p 2 -p 3 ) = 0 , (53)</formula>
<formula><location><page_9><loc_35><loc_24><loc_79><loc_26></location>( p 1 + p 2 + p 3 -1)( p 1 γ 1 + p 2 γ 2 + p 3 γ 3 ) = 0 . (54)</formula>
<text><location><page_9><loc_21><loc_17><loc_79><loc_24></location>The structure of these equations is quite interesting as 4 of the 5 equations can be solved immediately by assuming the Kasner condition p 1 + p 2 + p 3 = 1. Note that in General Relativity this condition is necessary to solve the field equations. The condition p 1 + p 2 + p 3 = 1 and the remaining equation (50) give two algebraic relation for the 6 free parameters.</text>
<text><location><page_9><loc_21><loc_14><loc_79><loc_17></location>However, we can also find solutions to Eqs. (50)-(54) without the Kasner condition. We start by assuming that p 1 + p 2 + p 3 = 1 which allows us to</text>
<text><location><page_9><loc_61><loc_13><loc_61><loc_15></location>/negationslash</text>
<text><location><page_10><loc_21><loc_81><loc_79><loc_85></location>divide Eqs. (51)-(54) by the factor ( p 1 + p 2 + p 3 -1). We note that the three Eqs. (51)-(53) are not independent as (51) -(52) + (53) = 0 . Hence, we are left with four independent equations.</text>
<text><location><page_10><loc_23><loc_79><loc_68><loc_80></location>One easily verifies that a solution can be written in the form</text>
<formula><location><page_10><loc_31><loc_75><loc_79><loc_78></location>γ 1 = p 2 2 p 1 + p 2 , γ 2 = p 2 1 p 1 + p 2 , γ 3 = p 1 + p 2 , (55)</formula>
<formula><location><page_10><loc_45><loc_72><loc_79><loc_75></location>p 3 = -p 1 p 2 p 1 + p 2 . (56)</formula>
<text><location><page_10><loc_21><loc_66><loc_79><loc_71></location>We should remark that Eq. (56) does imply that one of the three p i has to be negative. While the additional structure due to the material tensor C µ ν changes the underlying conceptual physics substantially, the solution shows many similarities with General Relativity.</text>
<section_header_level_1><location><page_10><loc_21><loc_62><loc_39><loc_63></location>5 Conclusions</section_header_level_1>
<text><location><page_10><loc_21><loc_48><loc_79><loc_60></location>It is tempting to argue that we introduced a form of matter through the back door by choosing our material tensor, the elastic coefficients. However, it is far from clear whether this is indeed the case. Note that C µν is not a dynamical variable and thus it cannot be interpreted simply as matter. In Continuum Mechanics and when working with crystal symmetries, the tensor C µν is said to encode the symmetry properties of the material, in our case the vacuum. We simply say that the vacuum as we know it may have an internal structure which is specified by C µν . We are breaking away from the assumption that the vacuum is isotropic and structureless.</text>
<text><location><page_10><loc_21><loc_35><loc_79><loc_48></location>The form of the elastic coefficients can in principle be determined observationally. In the context of cosmology, one could start with the specific C µν given by (35) and assume it to be close to the metric g µν . When one considers models where deviations from General Relativity will vary with cosmological time, it would be most interesting to see how observational data would determine the form of C µν which provides the best fit to the data. Using our approach to gravity, we will be able to use observations directly to specify the model instead of guessing new theories and deriving their implications on a case by case basis. Ultimately, experiments and observations will determine the correct theory.</text>
<section_header_level_1><location><page_10><loc_21><loc_32><loc_40><loc_33></location>Acknowledgements</section_header_level_1>
<text><location><page_10><loc_21><loc_27><loc_79><loc_31></location>We thank Friedrich Hehl and Dmitri Vassiliev for useful discussions. We would also like to thank the anonymous referees who have given us very valuable feedback.</text>
<section_header_level_1><location><page_10><loc_21><loc_23><loc_34><loc_24></location>References</section_header_level_1>
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<list_item><location><page_10><loc_22><loc_20><loc_79><loc_21></location>[1] M. Milgrom, Astrophys. J. 270 , 365 (1983); Astrophys. J. 270 , 371 (1983).</list_item>
<list_item><location><page_10><loc_22><loc_18><loc_60><loc_19></location>[2] H. F. M. Goenner, Living Rev. Rel. 7 , 2 (2004).</list_item>
<list_item><location><page_10><loc_22><loc_14><loc_79><loc_17></location>[3] F. W. Hehl and Y. N. Obukhov, Foundations of Classical Electrodynamics , Birkhauser, Boston (2003).</list_item>
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<list_item><location><page_11><loc_22><loc_78><loc_79><loc_85></location>[4] F. Klein, Nachrichten der Kgl. Gesellschaft der Wissenschaften zu Gottingen. Mathematisch-physikalische Klasse. (1917.) Vorgelegt in der Sitzung vom 25. Januar 1918. In Felix Klein Gesammelte Mathematische Abhandlungen, Erster Band , Section XXXI, R. Fricke and A. Ostrowski (Eds.), Springer, Berlin (1921).</list_item>
<list_item><location><page_11><loc_22><loc_74><loc_79><loc_77></location>[5] R. A. Battye and J. A. Pearson, 'Massive gravity, the elasticity of spacetime and perturbations in the dark sector,' arXiv:1301.5042 [astro-ph.CO].</list_item>
<list_item><location><page_11><loc_22><loc_72><loc_65><loc_73></location>[6] C. Brans and R. H. Dicke, Phys. Rev. 124 , 925 (1961).</list_item>
<list_item><location><page_11><loc_22><loc_68><loc_79><loc_71></location>[7] A. E. Green and W. Zerna, Theoretical Elasticity , Oxford University Press, London (1963).</list_item>
<list_item><location><page_11><loc_22><loc_66><loc_67><loc_67></location>[8] P. A. M. Dirac, Proc. Roy. Soc. Lond. A 165 , 199 (1938).</list_item>
<list_item><location><page_11><loc_22><loc_62><loc_79><loc_64></location>[9] J. D. Barrow and P. Parsons, Phys. Rev. D 55 , 1906 (1997) [gr-qc/9607072]; J. -P. Uzan, Living Rev. Rel. 14 , 2 (2011) [arXiv:1009.5514 [astro-ph.CO]].</list_item>
</unordered_list>
</document>
[
{
"title": "A new approach to modifying theories of gravity",
"content": "Christian G. Bohmer ∗ and Nicola Tamanini † Department of Mathematics, University College London Gower Street, London, WC1E 6BT, UK October 3, 2018",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We propose a new point of view for interpreting Newton's and Einstein's theories of gravity. By taking inspiration from Continuum Mechanics and its treatment of anisotropies, we formulate new gravitational actions for modified theories of gravity. These models are simple and natural generalisations with many interesting properties. Above all, their precise form can, in principle, be determined experimentally.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Since Newton's formulation of the law of universal gravitation in the Principia 1687, the theory has been unchanged. Very few attempts have been made to modify Newtonian gravity, modified Newtonian dynamics being probably the one exception [1]. It took almost 330 years until a generalisation of Newton's theory was successfully constructed. Einstein's theory of gravity was radically different to any previous physical theory. It abandoned the absolute notion of time and replaced the concept of force by the curvature of a four dimensional space. However, it took only a few years when the first modifications and extensions of Einstein's theory appeared in the literature, see [2] for an excellent historical review. Ever since, modified theories of gravity have enjoyed a prominent role in theoretical physics. It is the aim of the current work to propose modifications of Newton's and Einstein's theories of gravity by applying the same idea to both of them. Let us start with the actions of Newtonian gravity and its relativistic analogue, the Einstein-Hilbert action Variation of the Newtonian action with respect to the gravitation potential ϕ yields the well-known Poisson equation ∆ ϕ = 4 πGρ where ρ is a given matter distribution. Likewise, variations of the Einstein-Hilbert action with respect to the metric (which contains the gravitational potentials) gives the famous Einstein field equations G αβ = 8 πG/c 4 T αβ . When comparing the Einstein-Hilbert action (2) with other models in physics, it appears to be somewhat unnatural as one generally considers potential energies quadratic in the field strength, Hooke's law probably being the best known. However, when looking at the Einstein-Hilbert action, there are various very good reason for its form. Some of these can be motivated mathematically while others are simply observational. The field equations derived from it are close to Newton's theory of gravity and where the solutions deviate they are doing so in precisely the way to be in agreement with observations. However, there are observational facts which strongly indicate that our understanding of the gravitational force is far from complete. Dark matter and dark energy are two unknown forms of matter which are required to make the Universe work. Looking at (1) and (2) we note that both actions contain a contraction with respect to the metric. In the Newtonian model we simply have the flat metric for Cartesian 3-space. We are now taking some inspiration from Continuum Mechanics and in particular constitutive equations which specify the various different models studied in this field. In Maxwell's electromagnetism the constitutive equations for instance define the form of the dielectric tensor D = ε E where ε is a rank 2 tensor in general and we should really write D i = ε i j E j . When working with the Faraday tensor and its corresponding excitation, one writes H ij = 1 / 4 /epsilon1 ijmn χ mnkl F kl where for Maxwell's theory in vacuum χ mnkl = √ -g ( g mk g nl -g nk g ml ), see [3]. In many simple applications ε i j is taken to be proportional to the Kronecker delta which corresponds to a simple isotropic medium with dielectric constant /epsilon1 , namely ε i j = /epsilon1δ j i . Sometimes /epsilon1 is allowed to vary throughout the medium. In other words, a rank 2 isotropic tensor is proportional to the metric tensor. We are using this observation to argue that the Newton and Einstein-Hilbert actions are based on the assumption of an isotropic medium. Since Nature has a strong tendency to be anisotropic, this appears to be rather unnatural. By doing so, we are changing the interpretation of the terms δ ij and g µν in (1) and (2), respectively. We now view them much like the material tensors specified by constitutive equations which define the properties of the material. It is an interesting historical fact that F. Klein noted in a letter to Hilbert, as early as 1917, that the Ricci scalar in the Einstein-Hilbert action can be written as χ mnkl R mnkl , see [4]. For a recent paper inspired by similar thoughts, see [5]. Our approach follows the principal ideas of Brans and Dicke [6]. They suggested a model where the gravitational coupling was allowed to vary in space and time and it was viewed as an additional dynamical degree of freedom in the theory. The original Brans-Dicke model can be viewed as the isotropic limit of out model, we simply choose C µν = φ ( x α ) g µν with the main difference that we do not treat the new degrees of freedom as dynamical. We might want to, for sake of concreteness, speak of the properties of the vacuum here. However, we would like to be very careful and point out that this change of viewpoint has a variety of philosophical implications when interpreting the modified theories. We would like to keep these issues aside for now and proceed with the formulation of the theories. The assumption of an isotropic vacuum is certainly well supported by a host of experimental observations. However, there is clearly room for some improvement, in particular in general relativity. Many of the well-known modifications are is some sense rather severe. Field equations become higher than second order, new fields are introduced, locality is broken, local Lorentz invariance is broken etc. We will show that our theories are completely harmless and retain all desired properties.",
"pages": [
1,
2,
3
]
},
{
"title": "2 Modified gravity using continuum mechanics",
"content": "One of the most conservative modifications of Newton's and Einstein's theories we can think of is therefore the following where C ij and C µν are symmetric rank 2 tensors which contain information about the underlying structure of the theory. In Continuum Mechanics such objects are often referred to as material tensors or elastic coefficients [7]. We will stick with this well established notation and should clearly distinguish it from the energy-momentum tensor T αβ . We should emphasise that in this approach to gravity we are able to assume any symmetry for the metric, for instance spherical symmetry or homogeneity and isotropy. However, the symmetry of the metric is independent of the symmetry of the elastic coefficients. Let us firstly consider variations of the modified Newtonian theory (3) with respect to the potential φ . We find which reduces to the Poisson equation if we choose C ij = δ ij . As C ij can in principle be an arbitrary tensor, solutions to this equation may be quite different. As a simple example, let us consider the case when C ij = diag( c 1 , c 2 , c 3 ) with the c i being constants. In continuum mechanics or solid state physics one would speak of a crystal with three principal propagation directions. In this case, the field equation (5) can be reduced to the Poisson equation by rescaling the coordinates. Looking for a radially symmetric solution gives the interesting result In this case the gravitational field of a massive body is ellipsoidal instead of being spherical. However, the strength of this effect depends on the values of the constants. If for instance, the numerical values of the three constants c 1 , c 2 , c 3 are very close to each other, then the gravitational field will look spherical unless very large distances are taking into account. It should also be noted that the components of C ij do not have to be constants. They can be functions of the coordinates. Thus, the form of the gravitational field may be different in different regions of space, however, the gravitational law (5) would still be universal. If we consider C ij = χ ( t, x, y, z ) δ ij , the field equations are If the function χ is slowly varying in space | ∂ i χ | /lessmuch 1, we would have a theory which would be in very good agreement with Newton's theory on smaller scales, like the solar system. Note also that assuming a time varying χ would correspond to a time dependent gravitational constant, compare with Dirac's large number hypothesis [8]. Let us now consider the action (4). We assume that the tensor C µν may depend on the metric, this is important so that the limit of general relativity can be recovered. Note that the variation of the Ricci tensor does not give a surface term anymore because C µν is not the metric. One has to integrate by parts twice (the Ricci tensor contains second derivatives of the metric) and move the derivatives action on the metric variations to act on the matter tensor. Denoting the whole calculation yields the field equation where we have also included the energy-momentum tensor of matter. Recall that in General Relativity the field equations imply the conservation equations by virtue of the twice contracted Bianchi identities. This does no longer hold due to the presence of the material tensor C µν in the field equations. Nonetheless, we can take the covariant derivative of the field equation (9) with respect to ∇ α and find the following conservation equation This implies that the conservation equations for T αβ is no longer a direct consequence of the gravitational field equations alone. However, if we assume that the energy-momentum tensor is derived from a diffeomorphism invariant matter action, then the energy-momentum tensor is covariantly conserved in view of Noether's theorem We will assume this henceforth. Therefore, the conservation equation (11) imposes constraints on the components of C µν via (10) and thus its components cannot be chosen completely arbitrarily. For concreteness, we define which gives the additional consistency equation It should also be emphasised that one has to be careful when choosing the tensor C µν and the derived quantity Σ µναβ . Namely, there is a conceptual difference between prescribing the tensor C µ ν from prescribing C µν . In the former case, we find that C µν = g µσ C ν σ and thus C µν has an explicit dependence on the metric which in turn will yield a non-trivial form for Σ µναβ . On the other, if we specify C µν a priori , then this tensor does not depend on the metric and hence the resulting variational derivative would be zero. One has to carefully distinguish both cases when analysing the field equations. It is very difficult to analyse the field equations (9) given general Σ µναβ and C µν . It is also not clear what would constitute a good choice for those quantities. As a first attempt to understand the theory, we assume C µν to be conformally related to the inverse metric where φ ( x α ) is a scalar function depending on the coordinates. This choice simplifies the field equation considerably and they are now given by Upon division by the field φ and solving for the Einstein tensor, the resulting equation shows similarities with non-minimally coupled scalar field theories [6, see Eq. (11)]. The main difference between the current theory and most other approaches is that φ is not a dynamical degree of freedom because C µν is a prescribed tensor and has not corresponding equations of motion. The conformal model appears to agree with [6] in the limit when the Brans-Dicke parameter ω → 0. However, as variations with respect to φ are not considered, there is no propagation equation for the scalar field. By substituting (14) back into the action (4), we note that this simply corresponds to assuming the gravitational constant to be varying in time and space, see for instance [8]. Varying constants models are generally based on non-minimally coupled scalar field with kinetic term similar to Brans-Dicke theory, see [9].",
"pages": [
3,
4,
5
]
},
{
"title": "3 A Schwarzschild like solution",
"content": "Let us start by considering a static and spherically symmetric vacuum spacetime described by the metric where d Ω 2 is the usual line element of the two-sphere. We also assume φ = e ξ ( r ) . When the analogue situation is analysed in General Relativity where ξ ≡ 0, one finds two independent equations which determine the two unknown functions ν ( r ) and µ ( r ). In this model, there is the additional degree of freedom ξ and fortunately, there are now three independent equations. These are given by Eqn. (19) can be solved for the function µ which can then be substituted into the other two equations. Combining those linearly gives the condition ν ' ∝ ξ ' which then allows us to reduce this problem to a single differential equation. While separation of variable and subsequent integration is possible, the resulting equation cannot be solved analytically for the unknown functions due to its high nonlinearity. However, by assuming ξ ' = -C 2 ν ' with C /lessmuch 1 we can find an approximate solution to the field equations which is given by One easily verifies that this solution satisfies the field equations (18)-(20) up to O ( C 4 ). By choosing C = 2 GM we arrive at a Schwarzschild like solution with only a small difference. Clearly, this difference of the order of C 3 and therefore it would be very difficult to distinguish between this metric and the Schwarzschild metric using solar system tests. This is a promising result which indicates that this theory can pass solar system tests without great difficulty.",
"pages": [
5,
6
]
},
{
"title": "4.1 Conformal model",
"content": "Next, we want to study the cosmological implications of field equations (9). Similar to the above we assume C µν = φ ( t ) g µν and consider a FLRW universe where a ( t ) is the scale factor and r 2 = x 2 + y 2 + z 2 . This yields the following cosmological field equations where an overdot denotes differentiation with respect to time derivative and H is the Hubble parameter H = ˙ a/a . Moreover, the conservation equation (11) gives while the consistency equation (13) for J 0 becomes The other three components vanish identically J 1 = J 2 = J 3 = 0. Irrespective of the choice of matter in this model, the consistency equation (13) either implies that ˙ φ = 0 which is equivalent to General Relativity, or The solutions to this differential equation are given by and correspond to a radiation dominated universe. Next, we have to check whether these solutions are consistent with the remaining equations (25), (26) and (27). Firstly, we start looking for vacuum solutions, ρ = p = 0. The conservation equation (27) is trivially satisfied and one verifies that is a solution to the remaining field equations. Under the assumption that C µν = φ ( t ) g µν and ρ = p = 0 we cannot find an accelerating solution to the field equations. We could, of course, add regular matter to the field equations and seek other solutions. Let us briefly consider the situation where we include an incompressible perfect fluid or dust, for simplicity we consider k = 0 only. The scale factor is unchanged and given by (30). The conservation equation (27) implies the standard relation ρ ∝ 1 /a 3 and it turns out that is a solution to the field equations.",
"pages": [
6,
7
]
},
{
"title": "4.2 Fluid like model",
"content": "However, we are particularly interested in the vacuum equations without ordinary matter. The reason for this is simply that we want to show that an additional non-dynamical structure in the theory suffices to get a dynamical universe. One can easily find such solutions by introducing an extra degree of freedom in the material tensor. Let us choose the elastic coefficients to be which means that C µν = 1 2 ( g µσ C ν σ + g νσ C µ σ ) and thus we have an explicit dependence on the metric, implying a non-zero Σ µναβ given by One can interpret the quantities /rho1 and σ as the energy density and pressure of the vacuum, thereby specifying its internal structure. The cosmological field equations, the conservation equation and the consistency equation of this model are given by where the conservation equation can be derived from the first three equations. In the analogue situation in General Relativity we would have two independent equations for three unknown functions while for this model we have one additional equation and two additional degrees of freedom. To begin with, we consider the vacuum case ρ = p = 0 which eliminates one equation and two degrees of freedom. Thus, we are left with two independent equations and three unknown functions. In order to close this system, we choose a linear equation of state σ = w/rho1 for the spacetime structure. Now, we can solve Eq. (37) for ˙ /rho1 and substitute this result into Eq. (38) to arrive at a single differential equation in a ( t ). One can also check that substitution of ˙ /rho1 into Eq. (39) leads to the same differential equation, confirming that these equations are indeed not independent. This differential equation is given by /negationslash /negationslash /negationslash where in the derivation we assumed /rho1 = 0, w = 1 and ˙ a = 0. This differential equation is the product of two equations and thus can be solved by finding a solution to either of the two equations. The first one is easily solved by and is valid only if kw > 0. The second differential equation in (41) cannot be solved analytically for arbitrary k and w due to the non-linear nature of the equation. For w = 1 / 3 for instance we can find the two solutions which are valid for all k . For w = 0 one only finds one solution a ( t ) = a 0 t . For k = 0 one can solve the differential equation for all w and its solution is given by /negationslash which is well defined provided that w = 1 / 3 and corresponds to a power-law solution. For 0 < w < 1 / 3 this would correspond to an accelerated solution. The case w = 1 / 3 needs to be treated separately. As the exponent becomes very large, one would expect this to correspond to exponential functions, and indeed in this case Thus, we were able to find solutions of the field equations modelling a universe which can undergo periods of accelerated expansion, without the need to introduce any forms of matter. All we have done is to add an additional non-dynamical structure to the theory on a very fundamental level.",
"pages": [
7,
8,
9
]
},
{
"title": "4.3 Kasner type model",
"content": "We are now considering a Kasner type metric given by and assume the material tensor C µ ν to be of the form The field equations of this system are quite complicated. However, one notes that all field equations contain terms of the form tc ' i ( t ) and t 2 c '' i ( t ) which indicate that one can arrive at algebraic equations by choosing where the γ i are constants. With this additional assumption (49), and considering a vacuum ρ = p = 0, the field equations are given by The structure of these equations is quite interesting as 4 of the 5 equations can be solved immediately by assuming the Kasner condition p 1 + p 2 + p 3 = 1. Note that in General Relativity this condition is necessary to solve the field equations. The condition p 1 + p 2 + p 3 = 1 and the remaining equation (50) give two algebraic relation for the 6 free parameters. However, we can also find solutions to Eqs. (50)-(54) without the Kasner condition. We start by assuming that p 1 + p 2 + p 3 = 1 which allows us to /negationslash divide Eqs. (51)-(54) by the factor ( p 1 + p 2 + p 3 -1). We note that the three Eqs. (51)-(53) are not independent as (51) -(52) + (53) = 0 . Hence, we are left with four independent equations. One easily verifies that a solution can be written in the form We should remark that Eq. (56) does imply that one of the three p i has to be negative. While the additional structure due to the material tensor C µ ν changes the underlying conceptual physics substantially, the solution shows many similarities with General Relativity.",
"pages": [
9,
10
]
},
{
"title": "5 Conclusions",
"content": "It is tempting to argue that we introduced a form of matter through the back door by choosing our material tensor, the elastic coefficients. However, it is far from clear whether this is indeed the case. Note that C µν is not a dynamical variable and thus it cannot be interpreted simply as matter. In Continuum Mechanics and when working with crystal symmetries, the tensor C µν is said to encode the symmetry properties of the material, in our case the vacuum. We simply say that the vacuum as we know it may have an internal structure which is specified by C µν . We are breaking away from the assumption that the vacuum is isotropic and structureless. The form of the elastic coefficients can in principle be determined observationally. In the context of cosmology, one could start with the specific C µν given by (35) and assume it to be close to the metric g µν . When one considers models where deviations from General Relativity will vary with cosmological time, it would be most interesting to see how observational data would determine the form of C µν which provides the best fit to the data. Using our approach to gravity, we will be able to use observations directly to specify the model instead of guessing new theories and deriving their implications on a case by case basis. Ultimately, experiments and observations will determine the correct theory.",
"pages": [
10
]
},
{
"title": "Acknowledgements",
"content": "We thank Friedrich Hehl and Dmitri Vassiliev for useful discussions. We would also like to thank the anonymous referees who have given us very valuable feedback.",
"pages": [
10
]
}
]
2013FrPhy...8..630Z
https://arxiv.org/pdf/1302.5485.pdf
<document>
<section_header_level_1><location><page_1><loc_32><loc_92><loc_69><loc_93></location>Black Hole Binaries and Microquasars</section_header_level_1>
<text><location><page_1><loc_42><loc_89><loc_59><loc_90></location>Shuang-Nan Zhang 1 , 2 ∗</text>
<text><location><page_1><loc_31><loc_84><loc_69><loc_89></location>1 Laboratory for Particle Astrophysics, Institute of High Energy Physics,Beijing 100049, China. 2 National Astronomical Observatories, Chinese Academy Of Sciences, Beijing 100012, China.</text>
<text><location><page_1><loc_42><loc_81><loc_59><loc_82></location>(Dated: October 31, 2018)</text>
<text><location><page_1><loc_18><loc_66><loc_83><loc_80></location>This is a general review on the observations and physics of black hole X-ray binaries and microquasars, with the emphasize on recent developments in the high energy regime. The focus is put on understanding the accretion flows and measuring the parameters of black holes in them. It includes mainly two parts: (1) Brief review of several recent review article on this subject; (2) Further development on several topics, including black hole spin measurements, hot accretion flows, corona formation, state transitions and thermal stability of standard think disk. This is thus not a regular bottom-up approach, which I feel not necessary at this stage. Major effort is made in making and incorporating from many sources useful plots and illustrations, in order to make this article more comprehensible to non-expert readers. In the end I attempt to make a unification scheme on the accretion-outflow (wind/jet) connections of all types of accreting BHs of all accretion rates and all BH mass scales, and finally provide a brief outlook.</text>
<section_header_level_1><location><page_1><loc_18><loc_64><loc_30><loc_65></location>PACS numbers:</section_header_level_1>
<section_header_level_1><location><page_1><loc_24><loc_60><loc_34><loc_61></location>CONTENTS</section_header_level_1>
<table>
<location><page_1><loc_9><loc_13><loc_49><loc_59></location>
</table>
<text><location><page_1><loc_52><loc_59><loc_88><loc_61></location>VIII. Further developments on thermal stability of SSD</text>
<text><location><page_1><loc_90><loc_59><loc_92><loc_60></location>23</text>
<unordered_list>
<list_item><location><page_1><loc_53><loc_56><loc_92><loc_57></location>IX. Unification and Outlook 25</list_item>
<list_item><location><page_1><loc_56><loc_53><loc_92><loc_54></location>References 27</list_item>
</unordered_list>
<section_header_level_1><location><page_1><loc_66><loc_48><loc_78><loc_49></location>I. SYNOPSIS</section_header_level_1>
<text><location><page_1><loc_52><loc_38><loc_92><loc_46></location>I will start by defining what I mean by black hole binaries (BHBs) and microquasars in this article. I decide to restrict myself to only a subclass of BHBs, namely, BH X-ray binaries (BHXBs), since these are the only class of BHBs known observationally. I will then simply refer microquasars as BHXBs for reasons discussed in Section II.</text>
<text><location><page_1><loc_52><loc_10><loc_92><loc_37></location>Since many excellent, comprehensive and quite up-todate review articles on BHXBs are readily available in literature, I feel it is not necessary to write another bottomup and comprehensive review article on the same subjects at this stage. I will thus take quite an unusual approach in this article. I will first give some concise guides on several representative review articles [1-4], with some necessary updates. I will then focus on the further developments on several topics I feel deserve more discussions, i.e., BH spin measurements (Section IV), hot accretion flows (Section V), corona formation (Section VI), state transitions (Section VII) and thermal stability of SSD (Section VIII). The emphasis is thus put on understanding the accretion flows and measuring the parameters of black holes in them. Some rather general issues on BH astrophysics, such as what astrophysical BHs are and how to identify them observationally, are not discussed here but can be found from my recent book chapter entitled 'Astrophysical Black Holes in the Physical Universe' [5].</text>
<text><location><page_1><loc_53><loc_9><loc_92><loc_10></location>The usual practice of writing a review article is to end</text>
<figure>
<location><page_2><loc_15><loc_80><loc_43><loc_94></location>
<caption>FIG. 1. Illustration of a BHXB and microquasar. X-ray emission is produced from the central hot accretion disk. A jet is normally observed in the radio band. The companion star may produce optical emission lines. The disk wind may be observed with UV/X-ray absorption lines.</caption>
</figure>
<text><location><page_2><loc_9><loc_49><loc_49><loc_69></location>by listing some outstanding issues and major unsolved problems, and then to propose some possible approaches to them. I initially did not do this in the first draft. The history of astronomy tells us that major progress is almost always made by unexpected discoveries and research results; unpredictability is an essential nature of astronomy. This article is not intended to be read by funding agencies or proposal reviewers, so I thought I did not have to do it. In astronomy, knowing what has happened, but looking and doing it differently are far more important and effective than following other people's advises. However, the editors of this book suggested me to write a brief 'outlook' in the end. I thus did it nevertheless.</text>
<section_header_level_1><location><page_2><loc_13><loc_44><loc_45><loc_45></location>II. ACRONYMS AND TERMINOLOGY</section_header_level_1>
<text><location><page_2><loc_9><loc_39><loc_49><loc_42></location>In Table I, I list all acronyms used in this article; most of these are quite commonly used in this community.</text>
<text><location><page_2><loc_9><loc_15><loc_49><loc_39></location>A BH binary (BHB) is a gravitationally bound binary system in which one of the objects is a stellar mass BH with mass from several to tens of solar masses ( M glyph[circledot] ); the other object, i.e. its companion, can be either a normal star, a white dwarf, or a neutron star (NS). In case a binary system consists of two BHs, it is is referred to as a binary BH system, which is not covered in this article. When the companion in a BHB is a normal star, the gas from the star may be accreted to the BH and X-rays are produced, as a consequence of the heating by converting the gravitational potential energy into the kinetic energy of the gas, and a BH X-ray binary (BHXB) is referred to as such a binary system, as shown in Figure 1. The possible existence of BHXBs was first suggested by Zel'dovich & Novikov [6, 7]. The first BHXB found is Cygnus X-1 [8], now a well-studied system among many others found subsequently in the Milky Way and nearby galaxies.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_14></location>The terminology of microquasar has some twisting in it. Historically it was first referred to the BHXB 1E1740.7 -2942 in the Galactic center region, because a double-sided jet was detected from it, mimicking some</text>
<table>
<location><page_2><loc_55><loc_27><loc_89><loc_91></location>
<caption>TABLE I. List of Acronyms.</caption>
</table>
<text><location><page_2><loc_52><loc_9><loc_92><loc_23></location>quasars with similar radio lobes, which have much larger scales [9]. Soon after, superluminal jets are observed from a BHXB GRS 1915+105 [10], which is also referred to as a microquasar. Nevertheless both BHXBs are quite unusual compared to many others, thus microquasars were considered quite unusual. However it became clear that microquasars may be quite common among BHXBs, since the discovery of a normal BHXB GRO J1655 -40 [11], whose superluminal jets were observed [12] and showed some correlations with its X-ray</text>
<text><location><page_3><loc_9><loc_77><loc_49><loc_93></location>emission [13]. Subsequently several more BHXBs have been observed with superluminal jets. At this stage microquasars began to be referred to as BHXBs with relativistic jets (with bulk motion of about or larger than 90% of the speed of light), to distinguish them from NS X-ray binaries (NSXBs) that only have mildly relativistic jets (with bulk motion of about or smaller than 50% of the speed of light) [14]. However the discovery of relativistic jets from a NSXB Circinus X-1 made the situation complicated: relativistic jets are no longer uniquely linked to BHs [15].</text>
<text><location><page_3><loc_9><loc_56><loc_49><loc_77></location>Now in retrospect, a microquasar can be literally and easily understood as the micro version of a quasar; however a quasar may or may not be observed with collimated jets. A quasar has been already understood as a special galaxy centered by an actively accreting supermassive BH with a mass from millions to billions of M glyph[circledot] , and thus its total light output is dominated by the BH's accretion process, in a similar way as in BHXBs. The production or lack of relativistic jets may have similar or even the same underlying physical mechanisms in BHXBs and AGNs, though their surrounding environments may modify their observed morphologies [16]. It is therefore more natural to simply refer microquasars as BHXBs; in the rest of this article, microquasar and BHXB are used interchangeably.</text>
<text><location><page_3><loc_9><loc_41><loc_49><loc_55></location>We therefore will focus on BHXBs and thus will not discuss binary systems producing gamma-rays and sometimes radio jets, which are most likely high-mass NSXBs and in which jets or pulsars' winds interact with the wind of its high-mass companion to produce the observed gamma-rays [17-23]. Such systems show very different observational characteristics, e.g. the long (1667 days) super-orbital modulation with phase offset (about 280 days) between its X-ray and radio light curves found in LS I+61 · 303 [24].</text>
<section_header_level_1><location><page_3><loc_17><loc_37><loc_41><loc_38></location>III. REVIEW OF REVIEWS</section_header_level_1>
<text><location><page_3><loc_9><loc_23><loc_49><loc_35></location>Here I attempt to review the four recent review articles [1-4] I consider most useful to readers. Additional information and updates are provided when necessary. Some overlaps exists between these review articles, as expected and inevitable in bottom-up review articles. To avoid repetitions as much as possible in this article, I thus put different emphasizes on different articles, with of course my personal tastes and perceptions.</text>
<section_header_level_1><location><page_3><loc_14><loc_19><loc_44><loc_20></location>A. The most recent Science Collection</section_header_level_1>
<text><location><page_3><loc_9><loc_9><loc_49><loc_17></location>The recent collection of perspectives and reviews in the Science magazine provides excellent introductions to and concise summaries of the current state of our understanding of BH physics and astrophysics [1, 25-27]. To the subjects of this article, the most relevant article in this collection is the one by Fender and Belloni enti-</text>
<figure>
<location><page_3><loc_58><loc_67><loc_85><loc_93></location>
<caption>FIG. 2. A typical Hardness-Intensity-Diagram (HID) of spectral evolution of a BHXB, following the A → B → C → D → E → F cycle (top). Steady or transient jets are present during the A → B or B → C → D stage. No jets are observed, but hot disk winds are ubiquitous during the D → E stage. (Figure 2 in Ref.[1].)</caption>
</figure>
<text><location><page_3><loc_52><loc_32><loc_92><loc_53></location>tled 'Stellar-Mass Black Holes and Ultraluminous X-ray Sources', which is focused on the observational characteristics of BHXBs [1]. In particular, the article provides an excellent description of the general picture of spectral evolutions of BHXBs with the Hardness-IntensityDiagram (HID), which is found to be well correlated with the observed jets from BHXBs, as shown in Figure 2. Actually this cycle is also well tracked by their flux variability, represented by the measured root-mean-squares (rms) above its average flux, as shown by the RMSIntensity-Diagram (RID) [28] of the BHXB GX 339-4 in Figure 3, which also shows additional horizontal tracks at intermediate intensities. Sometimes, a full HID cycle does not go into the soft state at all (Figure 4), perhaps due to a failed outburst [29].</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_31></location>The basic scenario is as follows. During the initial stage of an X-ray outburst of a BHXB (A → B), which is triggered by a sudden increase of accretion rate onto the BH, its spectrum is normally hard and steady jets are always observed. After reaching its peak luminosity, its spectrum begins to soften in a chaotic way and transient jets are normally observed (B → C → D). After this transition, the system calms down with a soft spectrum and no jets are present (D → E). Finally the system returns to its quiescent state with a hard spectrum accompanied with the reappearance of jets (E → F). Throughout this cycle, the presence of hot accretion disk winds appears to be anti-correlated with its spectral hardness and jet ejection. This empirical pattern appears to be quite universal for all BHXBs with very few exceptions, though the underlying physics is still not well understood yet.</text>
<figure>
<location><page_4><loc_9><loc_76><loc_49><loc_94></location>
<caption>FIG. 3. The Hardness-Intensity-Diagram (HID) and RMSIntensity-Diagram (RID) of the BHXB GX 339-4. The main difference from Figure 2 is that there are additional horizontal tracks at intermediate intensities, a phenomenon known as hysteresis of state transitions, which will be discussed in Section VII. (Adapted from Figure 1 in Ref.[28].)</caption>
</figure>
<figure>
<location><page_4><loc_12><loc_45><loc_46><loc_64></location>
<caption>FIG. 4. The Hardness-Intensity-Diagram (HID) of H1743322. The main difference from Figure 3 is that there is a complete track at low intensity from the 2008 outburst, which did not go into the soft state at all. (Adapted from Figure 7 in Ref.[29].)</caption>
</figure>
<text><location><page_4><loc_9><loc_26><loc_49><loc_33></location>Nevertheless putting together the above scenario is a very significant progress in this field over the last ten years. Due to the conciseness of this article, some other important subjects on BHXBs are not fully discussed and many original references are also missing.</text>
<text><location><page_4><loc_9><loc_9><loc_49><loc_26></location>Other articles in this collection [25-27] are less relevant to the subjects of this article, but are still quite interesting to read, within the context of BH astrophysics and physics. The only other type of astrophysical BHs known to exist in the physical universe are supermassive BHs in the center of almost each galaxy. Volonteri [27] concisely summarized our current understanding on how they are formed and grow over the cosmic time; merging of two BHs is a key process here. Thorne [25] focused on what happens when BHs merge together to produce gravitational waves, which might be used as a new laboratory for studying gravitational physics and a new window for</text>
<text><location><page_4><loc_52><loc_80><loc_92><loc_93></location>exploring the universe. Witten [26] then explained the quantum properties of BHs, in particular the basic ideas behind Hawking radiation, which might not be important at all for astrophysical BHs. Nevertheless the understanding gained through studying the quantum mechanics of BHs actually plays very important roles in developing other theories of physics, such as that in heavy ion collisions and high-temperature superconductors, as vividly described by Witten [26].</text>
<section_header_level_1><location><page_4><loc_53><loc_75><loc_91><loc_77></location>B. The most recent ARA&A article on BHXBs, with some updates</section_header_level_1>
<text><location><page_4><loc_52><loc_47><loc_92><loc_73></location>The most recent ARA&A article on BHXBs by Remillard and McClintock entitled 'X-Ray Properties of Black-Hole Binaries' [2] (referred to as RM06 hereafter) provides the most complete, comprehensive and accurate review of BHXBs, which actually covers subjects much beyond just the X-ray properties of BHXBs. The Introduction of RM06 highlights the initial theoretical and observational studies of BHXBs, followed by brief comments on several main review articles on BHXBs preceding this one and the basic properties of BHs within the context of general relativity (GR). All main properties of BHXBs known at the time are summarized in the first table and figure there. In Table II, I compile the most updated data on all BHXBs currently known, including three BHXBs outside the Galaxy and not in the Large Magellanic Cloud (LMC); the currently available spin measurements for these BHs are also included for completeness. In Table II,</text>
<formula><location><page_4><loc_54><loc_44><loc_92><loc_46></location>f ( M ) ≡ P orb K 3 C / 2 πG = M BH sin 3 i/ (1 + q ) 2 , (1)</formula>
<text><location><page_4><loc_52><loc_35><loc_92><loc_43></location>where P orb is the orbital period, K C is the semiamplitude of the velocity curve of the companion star, M BH is BH mass, i is the the orbital inclination angle, and q ≡ M C /M BH is the mass ratio. In Figure 5, I show the updated graphical representation of most of the BHXBs listed in Table II.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_34></location>The X-ray properties of BHXBs are characterized by their X-ray light curves, timing and spectra. Essentially all known BHXBs were discovered initially as bright Xray sources, and the majority of them were detected as transient X-ray sources with X-ray all-sky monitors. The transient properties of some of BHXBs can be interpreted by the disk instability model (DIM) (see Ref.[49] and references therein), which assumes a constant mass transfer rate from the mass donor to the accretion disk. However the accretion rate from the disk to the compact object, i.e. a white dwarf (WD), a neutron star (NS), or a BH, is normally lower than the mass supply rate in the disk, so mass is accumulated in the disk. When the accumulated mass exceeds a certain critical value, a sudden increase of accretion rate results in a nova-like outburst. DIM is most successful in explaining an outburst with fast rise and exponential decay (FRED). However as shown in RM06, many observed X-ray light curves of BHXBs</text>
<table>
<location><page_5><loc_20><loc_54><loc_81><loc_89></location>
<caption>TABLE II. Twenty four confirmed BHXBs and their BH masses and spins. Except those marked with references, all other data are taken from Remillard & McClintock (2006) [2] (references therein).</caption>
</table>
<text><location><page_5><loc_9><loc_47><loc_91><loc_54></location>a : Reported in the first paper on systematic BH spin measurements [47]; b : Reported in the most recent literature; c: Postulated to be extreme retrograde Kerr BH, due to the lack of the thermal disk component above 2 keV (GRO J1719-24 also belongs to this class) [47]; d : Postulated to be non-spinning BH, due to the observed low disk temperature; e : BH mass of 7 M glyph[circledot] assumed [47]; f : Based on the inner disk radius decrease by a factor of two from the hard state to the soft state transition [47].</text>
<text><location><page_5><loc_9><loc_38><loc_49><loc_44></location>are far more complicated than just FRED and their recurrent time scales are also not compatible with DIM. Perhaps disk truncation (a subject to be discussed extensively more later) and mass transfer instability are additional ingredients [50].</text>
<text><location><page_5><loc_9><loc_14><loc_49><loc_37></location>X-ray emission from a BHXB is variable at all time scales, from the free-fall or Keplerian orbital time scale of milliseconds near the BH, to the various oscillations (some are even related to GR) in the disk with time scales from milliseconds to minutes, to the viscous time scales of minutes, and to the various instabilities of different time scales. Therefore timing studies of BHXBs can probe the geometry and dynamics in BHXBs. However, lack of coherent signals, such as that observed from pulsars, makes it difficult to unambiguously identify the underlying mechanisms from the detected X-ray variabilities. Nevertheless power density spectra (PDS) and rms still allow us to make progress in understanding the general characteristics of a BHXB, in particular when combined with its spectral behaviors, as already discussed briefly above.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_13></location>The most successful understanding of BHXBs so far is the description of their thermal X-ray spectral component by the classical Shakura-Sunyaev Disk (SSD) model</text>
<text><location><page_5><loc_52><loc_29><loc_92><loc_44></location>[51]. After applying GR to SSD, the temperature distribution in the disk can be obtained [52]. Applying this model to BHXBs, one can even measure the spin of their BHs [47], by assuming that the inner accretion disk boundary is the inner-most stable circular orbit (ISCO) of the BH (a subject to be discussed more in Section IV). However a power-law (PL) component is almost ubiquitous in the spectra of BHXBs. The interplay between these two components results in various spectral states, which are also found to be well correlated with their timing properties.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_27></location>Historically these spectral states have been named in many different ways, reflecting mostly how they were identified with the observations available at the times. In RM06, three states are defined, which show distinctively different spectral and timing behaviors, as shown in Figure 6 based on the RXTE data on the BHXB GRO J1655-40. The thermal state has its X-ray spectrum dominated by the thermal disk component and very little variability. The hard state has its X-ray spectrum dominated by the a PL component and strong variability. The steep power-law (SPL) state is almost a combination of the above two states, but the PL is steeper. I will keep using these definitions throughout this article for consis-</text>
<figure>
<location><page_6><loc_10><loc_59><loc_48><loc_94></location>
<caption>FIG. 5. Schematic diagram of the dynamically confirmed BHXBs, maintained by Dr. Orosz (http://mintaka.sdsu.edu/faculty/orosz/web/). The color scale for the 17 objects with low mass companions (i.e. stars with masses less than about 3 M glyph[circledot] ) represents the temperature of the star. However, the high mass companions in Cyg X-1, LMC X-1, LMC X-3, and M33 X-7 are considerably hotter, and are thus not well represented in color scale. (Adapted from the plot maintained by Dr. Orosz (http://mintaka.sdsu.edu/faculty/orosz/web/ and Figure 1 in Ref.[48])</caption>
</figure>
<text><location><page_6><loc_9><loc_34><loc_49><loc_39></location>tency. Please refer to Table 2 in RM06 for quantitative descriptions of these three states. When a BHXB is in quiescence, i.e. not in an outburst, its spectral shape is similar to the hard state spectrum.</text>
<text><location><page_6><loc_9><loc_15><loc_49><loc_33></location>Referring back to Figure 2, RM06 found that the HID can be reasonably well understood in terms of transitions between the above three states; I summarize this in Table III. After examining the extensive RXTE data on spectral evolutions of six BHXBs, the above conclusion seems to be valid generally. Thus a coherent picture seems to emerge: jets are produced only when the PL component in the spectrum is strong and winds are present only when the PL component disappears; jets and winds appear to be mutually exclusive. This suggests that transient winds should appear in the SPL state, anti-phased with transient jets, if winds quench jets as suggested recently [53, 54].</text>
<text><location><page_6><loc_9><loc_9><loc_49><loc_14></location>The robust link between the PL component and jets provides an important clue to the jet production mechanism. The PL component is believed to be produced in an optically thin and geometrically thick corona, in</text>
<figure>
<location><page_6><loc_53><loc_63><loc_91><loc_94></location>
<caption>FIG. 6. Characteristic spectra and PDS of the BHXB GRO J1655-40 in three different states, namely, steep powerlaw (SPL), thermal and hard states. Please refer to Figure 2 in RM06 for details. (Figure 2 in Ref.[2].)</caption>
</figure>
<table>
<location><page_6><loc_55><loc_43><loc_88><loc_51></location>
<caption>TABLE III. Relations between HID, spectral states, jets and wind.</caption>
</table>
<text><location><page_6><loc_52><loc_29><loc_92><loc_40></location>which hot electrons up-scatter the thermal photons from the disk. As discussed in RM06, the inner region of the disk appears to be truncated quite far away from the BH in the hard state; this should be the primary reason for the low luminosity of this state, as evidence for the existence of BHs in BHXBs. However it is still not understood how the corona is formed; I will discuss this later in Section VI.</text>
<text><location><page_6><loc_52><loc_16><loc_92><loc_29></location>In the final part, RM06 discussed the exciting possibilities of using BHXBs as probes of strong gravity, including confirming these BHXBs contain true BHs, measuring the spins of BHs, relating BH spin to the Penrose process and other phenomena, and finally carrying out tests of the Kerr Metric. They also discussed in length how to measure the spins of BHs and commented the various methods of doing so. In Section IV I will re-visit the BH spin measurements in details.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_16></location>Lacking of a solid surface, a BHXB is not expected to produce and actually never observed to have coherent pulses. However quasi-periodic oscillations (QPOs) are frequently detected in their X-ray light curves; a QPO is defined as a 'bump' feature in the PDS, if Q = ν/ ∆ ν > 2,</text>
<text><location><page_7><loc_9><loc_75><loc_49><loc_93></location>where ν and ∆ ν are the peak frequency and FWHM of the bump, respectively. The observed QPOs are further divided into low frequency (LF) (0.1-30 Hz) and high frequency (HF) ( > 30 Hz) QPOs. Typical LFQPOs can be found as the peak around several Hz in the top and bottom PDS of Figure 6. In literature, LFQPOs are further divided into several subclasses and their underlying mechanisms are far from clear at this stage, though many models have been proposed and briefly discussed in RM06. HFQPOs have been detected from several sources at 40-450 Hz. Of particular interests are their stable nature when detected, which may be linked to either the mass or spin or both of the BH in a BHXB.</text>
<text><location><page_7><loc_9><loc_54><loc_49><loc_74></location>In some cases a 3:2 frequency ratio is found. Although the 3:2 HFQPO pairs could be interpreted in some epicyclic resonance models [55, 56], there remain serious uncertainties as to whether epicyclic resonance could overcome the severe damping forces and emit Xrays with sufficient amplitude and coherence to produce the HFQPOs. A revised model applies epicyclic resonances to the magnetic coupling (MC) of a BH's accretion disk to interpret the HFQPOs [57]. This model naturally explains the association of the 3:2 HFQPO pairs with the steep power-law states and finds that the severe damping can be overcome by transferring energy and angular momentum from a spinning BH to the inner disk in the MC process.</text>
<section_header_level_1><location><page_7><loc_12><loc_49><loc_46><loc_51></location>C. Two comprehensive and long articles on modeling accretion flows in BHXBs</section_header_level_1>
<text><location><page_7><loc_9><loc_37><loc_49><loc_46></location>Page limitations to the above review articles did not allow in depth discussions on the detailed processes and models of accretion flows in BHXBs, which are responsible for the above described energy spectra, PDS, state transitions, jets, and wind. Here I introduce two comprehensive and long articles just on this, by Done [3] and Done, Gierli'nski & Kubota [4].</text>
<section_header_level_1><location><page_7><loc_21><loc_32><loc_36><loc_33></location>1. A beginner's guide</section_header_level_1>
<text><location><page_7><loc_9><loc_9><loc_49><loc_30></location>The first is intended to readers who just start research in this field [3]. It started with the basic tools in plotting spectra and variability, and then described the basic ideas and methods used to infer the inner accretion disk radius when a BHXB is in a thermal state, in order to measure the BH spin [47]. A useful introduction is made on how to make various corrections to account for various effects, including color correction, special and general relativistic effects, starting from the original work [47]. Several commonly use fitting models, i.e. DISKBB , BHSPEC and KERRBB , in the XSPEC package are also briefly introduced. An example is given to demonstrate that the expected relation L disk ∝ T 4 in [47] agrees with the data from the BHXB GX 339-4 [58], where L disk and T in are the disk's total luminosity and temperature at the inner</text>
<text><location><page_7><loc_52><loc_92><loc_71><loc_93></location>disk boundary, respectively.</text>
<text><location><page_7><loc_52><loc_70><loc_92><loc_91></location>The hard state is then briefly touched upon, using the Advection Dominated Accretion Flow (ADAF) model [59]. ADAF naturally explains the hot corona required by the observed PL component, especially at very low accretion rate. Evaporation at low accretion rate in the inner disk region has been proposed as the mechanism producing a geometry with a radially truncated disk and a hot inner flow; the latter might be the ADAF [60]. This also means that when a significant PL component is present in the spectrum, BH spin measurement cannot be done with the inferred inner disk boundary (see, however, counter evidence discussed in Sections IV and VI). This geometry is considered a paradigm that can account for many of the observed diverse phenomena from spectral evolution to timing properties.</text>
<text><location><page_7><loc_52><loc_55><loc_92><loc_69></location>A brief, yet interesting discussion is given on scaling up the above models to Active Galactic Nuclei (AGNs), which host actively accreting supermassive BHs in the centers of galaxies. Applying the insights learnt from BHXBs on their spectral evolution, changing disktemperature with accretion rate and BH mass, and diskjet connections, one might be able to understand many phenomena beyond the simple AGN unification scheme, in which the different observational appearance is all attributed to an viewing angle difference.</text>
<text><location><page_7><loc_52><loc_29><loc_92><loc_55></location>The continuum emissions of both the disk thermal and PL components are modified by absorptions and added by additional spectral features along the line of sight (LOS). Absorptions in neutral media produce various photoelectric absorption edges, but in ionized media result in both absorption edges and lines. In addition to recovering the original X-ray emission of a BHXB, modeling the absorption features is important in learning the physical properties of the absorption media, such as column density, ionization and velocity along LOS. Winds from BHXBs discussed above are always detected this way. Several XSPEC fitting models for various kinds of absorption edges and lines are also introduced here, i.e. TBABS , ZTHABS , TBVARABS , ZVPHABS , TBNEW , ABSORI and WARMABS . Figure 7 shows the calculated absorption structures with ABSORI and WARMABS ; note that ABSORI does not include absorption lines, but WARMABS does.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_29></location>X-rays interacting with the surrounding medium can ionize it and heat the free electrons up by Compton scattering; hot electrons can also loose energy by interacting with lower energy photons. A Compton temperature of the plasma is reached when the above two processes reaches an equilibrium. The temperature is determined by only the spectral shape of the continuum, so the heated plasma can escape as winds from the disk if the velocity of the ions exceed that of the escape velocity at that radius, which is usually very far away from the inner disk boundary. As the continuum luminosity approaches the Eddington limit, the radiation pressure reduces the escape velocity substantially in the inner disk region such that continuum driven winds can be launched</text>
<figure>
<location><page_8><loc_12><loc_52><loc_45><loc_94></location>
<caption>FIG. 7. Calculated absorption structures of a column density N H = 10 23 cm -2 for different ionization parameter ξ = L/nr 2 , where L is the illuminating luminosity, n is the density of the medium, and r is the distance from the radiation source. Top panel : with the ABSORI model that does not include absorption lines. Bottom panel : for ξ = 100 with both ABSORI and WARMABS, the latter is based on the XSTAR photo-ionization package thus absorption lines are taken into account properly.(Adapted from Figure 17 in Ref.[3].)</caption>
</figure>
<text><location><page_8><loc_9><loc_15><loc_49><loc_35></location>almost everywhere in the disk of a BHXB, forming a radiation driven wind, which is also called thermal wind; alternatively winds may also be driven magnetically in the inner disk region, but this is much less understood yet. In contrast, in an AGN the peak continuum emission is in the UV band, which has a much larger opacity than Compton scattering in neutral or weakly ionized medium during both photoelectric and line absorptions. This means the effective Eddington luminosity is reduced by large factors and UV line driven wind is easily produced at high velocity. This explains the relatively lower velocity (hundreds km/s) and highly ionized winds from BHXBs, but much higher velocity (thousands km/s to 0.2 c ) and weakly ionized winds from AGNs.</text>
<text><location><page_8><loc_9><loc_9><loc_49><loc_14></location>Material illuminated by X-rays produce both fluorescence lines and reflection features, which depends on both the continuum spectral shape and ionization state of the material, as shown in Figure 8 calculated with</text>
<figure>
<location><page_8><loc_53><loc_68><loc_91><loc_94></location>
<caption>FIG. 8. Ionized reflections from a constant density slab, calculated with the XSPEC ATABLE model, REFLIONX.MOD , which includes the self-consistent line and recombination continuum emission, for different values of ξ . The inset shows a detailed view of the iron line region. For neutral material, around 1/3 of the line photons are scattered in the cool upper layers of the disk before escaping, forming a Compton down-scattered shoulder to the line. For highly ionized reflection, the upper layers of the disk are heated to the Compton temperature so that the spectral features are broadened. (Adapted from Figure 22 in Ref.[3].)</caption>
</figure>
<text><location><page_8><loc_52><loc_19><loc_92><loc_48></location>the XSPEC ATABLE model REFLIONX.MOD, which includes the self-consistent line and recombination continuum emission. Note that for highly ionized reflection, the Compton heated upper layers of the disk broadens the spectral features; these effects are not included in the simpler XSPEC model PEXRIV . Replacing the stationary slab by a disk around a BH, both the special and general relativistic effects will further smear (broaden) the spectral features, as shown in Figure 9 for the iron line region with different values of ξ , inner disk radius r in , and viewing angle i ; these are calculated with the REFLIONX.MOD and then convolved with KDBLUR in the XSPEC package. Other similar XSPEC fitting models are DISKLINE , LAOR , and KY . The inner disk radius can in principle be determined by modeling the observed broad iron line features, that in turn can be used to measure BH spins, as will be discussed briefly in Section IV. Figure 10 shows all components of a broad band spectrum of a BHXBs, including interstellar absorption.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_18></location>We thus have two ways discussed so far to determine r in of BHXBs by either fitting its thermal continuum or the fluorescent iron line feature. However results are not always consistent, obtained with different methods, even using the same data by different authors. The author believes that r in increases at low accrete rate, to make room for the corona occupying this space. Nevertheless</text>
<figure>
<location><page_9><loc_10><loc_48><loc_47><loc_94></location>
<caption>FIG. 9. Relativistic smearing of the iron line region. The sharp peak shown in solid grey lines are the original features. Other colored solid lines have different r in marked in units of r g = GM/c 2 for i = 30 · (solid lines); similarly those for i = 60 · are shown in dotted lines. (Adapted from Figure 25 in Ref.[3].)</caption>
</figure>
<figure>
<location><page_9><loc_10><loc_16><loc_47><loc_37></location>
<caption>FIG. 10. Illustration of all components of the broad band energy spectrum of a BHXBs; interstellar absorption is normally prominent at low energies. (Adapted from Figure 3b in Ref.[61].)</caption>
</figure>
<text><location><page_9><loc_52><loc_89><loc_92><loc_93></location>a fairly good summary on the current conflicts and confrontations is made on this issue, which I will discuss further in Section IV.</text>
<section_header_level_1><location><page_9><loc_63><loc_84><loc_80><loc_85></location>2. An expert's handbook</section_header_level_1>
<text><location><page_9><loc_52><loc_65><loc_92><loc_82></location>The second one can be considered as a handbook on accretion flows on BHXBs, and really lives up to its subtitle 'Everything you always wanted to know about accretion but were afraid to ask' [4] (DGK07 hereafter). Besides its much longer length of 66 pages, the main difference from the above review articles is that it is focused on confrontations between theories and observations and intends to depict a coherence picture of the accretion physics in BHXBs. In the following I will summarize briefly the main points and conclusions reached in DGK07; those I have reviewed above and will discuss more later will be skipped for brevity.</text>
<text><location><page_9><loc_52><loc_38><loc_92><loc_65></location>The underlying physics of DIM for triggering the outbursts in BHXBs discussed above is the hydrogenionization instability, which produces the so-called 'S'curve, as shown in Figure 11; irradiation by the inner hot disk to keep the outer disk hot is required to produce the slow flux decays, e.g. the exponential decays, frequently observed in them. The outer disk radius is obviously another key parameter, which is determined by the tidal instability in a binary system. This mechanism can explain why BHXBs with high mass companions are all persistently bright, since their outer disks are always in the upper branch due to the combination of their higher average mass transfer rate and inner disk irradiation. Similarly it also explains the differences and similarities between the light curve properties of neutron star X-ray binaries (NSXBs) and BHXBs, since a NS has a lower mass. It should be noted that the additional surface emission from the NS may also help to maintain the outer disk hot and stay in the upper stable branch [62].</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_37></location>The SSD prescription assumes that the stress is proportional to pressure. Because the gas pressure is P gas ∝ T but radiation pressure is P rad ∝ T 4 , so a small temperature increase causes a large pressure increase when P rad ≥ P gas , and thus large stress increase, which in turn heats the disk even more. The opacity cannot decrease effectively to cool it down, so the disk becomes unstable when L ≥ 0 . 06 L Edd , where L Edd is the Eddington luminosity of a BHXB. However in most BHXBs their disk emissions appear to be stable up to around 0 . 5 L Edd . One way out is to assume that the stress is proportional to √ P rad P gas ∝ T 5 / 2 , so the stress increases slower in the radiation pressure dominated regime. Beyond this the disk should become unstable, as evidenced by the sometimes 'heart-beat' bursts of the super-Eddington BHXB GRS 1915+105; other BHXBs with L max > L Edd (e.g. V404 Cyg and V4641) were not observed with such instabilities, perhaps due to the combination of insufficient observational coverage and sensitivity. However similar 'heart-beat' bursts were also observed re-</text>
<figure>
<location><page_10><loc_9><loc_72><loc_45><loc_93></location>
<caption>FIG. 11. 'S'-curve due to the Hydrogen-ionization instability. At a radius R , the SSD solution gives kT ∝ ˙ m 1 / 4 , where T is the local disk temperature and ˙ m is the local mass accretion rate. A positive slope means cooling can balance heating, so T changes slowly and the disk is stable. In the unstable middle branch, thermally emitted photons are absorbed to ionize the hydrogen atoms and are thus trapped in the disk, causing a large opacity ( σ ). If the mass supply rate ˙ m in happens in the middle range, then the disk experiences a limit-cycle instability from (1) to (4): (1) T < 10 4 K, neutral hydrogen has low σ , ˙ m in > ˙ m , so the disk is built up and T increases slowly; (2) T ∼ 10 4 K, hydrogen atoms begin to be ionized and so has high σ , T increases rapidly until hydrogen is fully ionized; (3) T > 10 5 K, ionized hydrogen has modest σ , ˙ m in < ˙ m , so the disk is eaten out and T decreases slowly; (4) T ∼ 10 5 K, protons and electrons begin to recombine, T decreases rapidly until reaching the lower stable branch again. (Adapted from Figure 1 in Ref.[4].)</caption>
</figure>
<text><location><page_10><loc_9><loc_38><loc_49><loc_42></location>IGR J17091-3624, which is likely substantially sub-Eddington unless it is located much beyond 20 kpc and/or its BH mass is quite small [63].</text>
<text><location><page_10><loc_9><loc_23><loc_49><loc_37></location>Super-Eddington accretion flow (SEAF) can become stable again, if the radiation instability is overcome by an optically thick ADAF, i.e. the slim disk model, in which the trapped photons in the flow is advected inwards, thus balancing the heating generated by viscosity. Strong radiation driven winds can be easily produced; this can happen in BHXBs and NSXBs (e.g. Z-sources). Evidence exists that truncated inner disk (TID) is quite common in SEAF and outflow even dominates over inflow in SEAF [64].</text>
<text><location><page_10><loc_9><loc_9><loc_49><loc_23></location>The observed PL component in BHXBs cannot be explained by the SSD-like models and thus requires a hot accretion flow (HAF). At low accretion luminosity (e.g. the quiescent state), the HAF may be the advection dominated accretion flow (ADAF), the convection dominated accretion flow (CDAF), or the advection dominated accretion inflow/outflow solution (ADIOS). At higher luminosity (e.g. the hard state), the original hot and optically thin disk solution (i.e. the SLE solution) is unstable, because the electron heating efficiency by the Coulomb</text>
<text><location><page_10><loc_52><loc_82><loc_92><loc_93></location>coupling between protons and electrons is too low. The luminous HAF (LHAF), however, has the advection as a heating source to electrons, so the heating efficiency increases and thus electrons can cool the flow more effectively. Outflows can be produced in ADAF/ADIOS; collimated jets can also be produced if magnetic fields are involved, so an accretion flow may even be jet dominated (i.e. JDAF).</text>
<text><location><page_10><loc_52><loc_60><loc_92><loc_81></location>The interplays between the SSD and HAF may be responsible for the observed different states in BHXBs discussed above. If the PL component is produced by Compton up-scattering, then the combination of the optical depth τ and L h / L s , where L h is the heating power in electrons and L s is the cooling power in seed photons, can describe the observed variety of spectra. For example, the hard state has L h / L s glyph[greatermuch] 1, but the thermal and SPL state have L h / L s ≤ 1. The location of r in is proposed to be closely related to L h / L s , as illustrated in Figure 12. As discussed above, the steeper PL in the thermal and SPL states is mostly non-thermal in nature, so non-thermal Comptonization is required. Detailed comparisons with data suggest thermal Comptonization also cannot be ignored even in the thermal and SPL states.</text>
<text><location><page_10><loc_52><loc_22><loc_92><loc_59></location>In Figure 12, the hot inner flow (HIF) and patchy corona are responsible for thermal and non-thermal Comptonization, respectively. When a source transits from hard to SPL state, r in decreases, HIF is reduced but the patchy corona becomes dominant. The transition from SPL to soft state is then marked by the disappearance of HIF and significant reduction of the patchy corona. The above scenario obviously depends on two fundamental assumptions: (1) slab-HIF produces the non-thermal PL component; (2) slab-HIF is mostly located between the TID and the BH. Both assumptions are examined exhaustively based on the existing observations and their spectral modeling. Slab-HIF is found to be consistent with essentially all data. An alternative to slab-HIF is that the PL is produced from the jet base and beamed away from the disk; however observations suggest the PL emission is quite isotropic, thus in conflict with this. TID is found also consistent with data when the PL component becomes important, since the observed L disk ∼ T 4 in deviates significantly from a linear relation, suggesting r in is larger when the PL component becomes important. However some studies showed that r in is unchanged in the initial hard state, if the Compton up-scattering process is treated properly in Monte-Carlo simulations to recover the lost disk photons [65]; this issue will be further discussed in Section IV.</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_21></location>The above developed HIF/TID model based on the spectral evolutions of BHXBs can also be applied to explain the majority of time variability PDS of BHXBs consistently. The TID acts as a low pass filter as it cannot response effectively to variations in the HIF, so r in controls the low frequency break of the PDS: ν LFB ∼ 0 . 2( r/ 6) -3 / 2 ( m/ 10) -1 Hz, where r = r in /r g and m = M BH /M glyph[circledot] ; this predicts that ν LFB changes from 0.03 to 0.2 Hz as observed during transitions from the</text>
<figure>
<location><page_11><loc_23><loc_71><loc_77><loc_93></location>
<caption>FIG. 12. Left panel: a selection of spectra in different states taken from the 2005 outburst of GRO J1655-40; state names in parentheses are defined in RM06. Right panel: proposed accretion flow structures in these different spectra; note the SSD is truncated away from ISCO in the bottom two states. (Adapted from Figure 9 in Ref.[4].)</caption>
</figure>
<text><location><page_11><loc_9><loc_39><loc_49><loc_62></location>hard to SPL and soft states if r decreases from 20 to 6. The HIF/TID model can also explain the observed LFQPO variations with luminosity, if the LFQPOs are some kinds of characteristic frequencies related to r in . To some extend, the shape of the PDS can also be explained by this model. However the observed variability rms ∼ flux linear correlation and log-normal distribution of fluctuations may need additional ingredient, such as the proposed propagation fluctuation model. However I point out that such correlations and distributions are also observed in the light curves of gamma-ray bursts and solar flares [66, 67], which can be produced by the generic self-organized criticality mechanism [66]. Similar rms ∼ flux correlation has also been found for blazars, in agreement with the minijets-in-a-jet statistical model [68].</text>
<text><location><page_11><loc_9><loc_9><loc_49><loc_39></location>The spectral and timing properties of weakly magnetized NSXBs are known to have many similarities and differences from BHXBs; this is particularly true for the atoll sources that have similar ranges of L/L Edd to BHXBs. The essential distinction is that a NS has a solid surface, but a BH does not. Observationally the continuum spectra of atoll sources can be modeled as composed of SSD, PL emission, and blackbody emission from the NS surface (or the boundary layer between the TID and NS surface); the former two components are quite similar to BHXBs. The spectral evolutions of these NSXBs are thus driven similarly by the combinations of τ and L h / L s . However the blackbody emission is an additional source of L s , so the PL is not as hard as that in BHXBs and r in variations are less effective in changing L h / L s ; the latter means it is more difficult to find evidence of TID from modeling only the spectral evolutions in NSXBs. On the other hand, the TID/HIF model in NSXBs can produce timing behaviors in the same way as in BHXBs discussed above, consistent with observations; additional timing behaviors, such as coherent X-ray pulsations and</text>
<text><location><page_11><loc_52><loc_60><loc_92><loc_62></location>kilo-Hz QPOs observed in these NSXBs, are caused by the rapid spins of the hard surfaces of the NSs.</text>
<section_header_level_1><location><page_11><loc_52><loc_54><loc_91><loc_56></location>IV. FURTHER DEVELOPMENTS ON BH SPIN MEASUREMENTS</section_header_level_1>
<text><location><page_11><loc_52><loc_32><loc_92><loc_52></location>A BH predicted in GR can only possess three parameters, namely, mass, spin and electric charge, known as the so-called BH no hair theorem. Even if a BH was born with net electric charge, its electric charge can be rapidly neutralized by attracting the opposite charge around it in any astrophysical setting, because the strength of electromagnetic interaction is many orders of magnitude stronger than that of gravitational interaction. Therefore an astrophysical BH may only have two measurable properties, namely, mass and spin, making BHs the simplest macroscopic objects in the universe. Practically, only Newtonian gravity is needed in measuring the BH mass in a binary system. However, GR is needed in measuring the BH spin.</text>
<text><location><page_11><loc_52><loc_17><loc_92><loc_32></location>The mass and spin of a BH has different astrophysical meanings. Its mass can be used to address the question of ' How much matter (and energy) has plunged into the BH?'. However its spin can be used to address the question of ' How did the matter (and energy) plunge into the BH?'. This is because matter and energy plunged into a BH can carry angular momentum, which is a vector with respect to the spin axis of the BH. In order to increase the total gravitating mass-energy from M i with zero spin to M f , the added rest-mass must be [69, 70]</text>
<formula><location><page_11><loc_57><loc_15><loc_92><loc_17></location>∆ M = 3 M i [sin -1 ( M f / 3 M i ) -sin -1 (1 / 3)] , (2)</formula>
<text><location><page_11><loc_52><loc_13><loc_70><loc_14></location>and its final spin becomes</text>
<formula><location><page_11><loc_53><loc_8><loc_92><loc_12></location>a ∗ ≡ cJ/GM 2 f = ( 2 3 ) 1 / 2 M i M f [ 4 -( 18 M 2 i M 2 f -2 ) 1 / 2 ] , (3)</formula>
<text><location><page_12><loc_9><loc_59><loc_49><loc_93></location>where J is the BH's angular momentum. Clearly we have a ∗ = 1 when M f /M i = 6 1 / 2 ; further accretion simply maintains this state [70]. Therefore the required additional rest-mass to spin a BH from zero to maximum spin is ∆ M glyph[similarequal] 1 . 85 M i = 0 . 75 M f ; this is a lower limit to the accreted mass [71]. Figure 13 shows a ∗ as a function of ∆ M . Ignoring Hawking radiation of a macroscopic BH, the only way to extract the energy of a BH and thus reducing its gravitating mass is by extracting its spin energy. Recently, evidence of BH spin energy extraction to power relativistic jets has been found, from the observed correlation between the maximum radio luminosity and its BH spin of a microquasar [72, 73]; however the average jet power is not correlated with BH spin [74]. This indicates that jets may be produced by both BlandfordPayne (BP) [75] and Blandford-Znajek (BZ) [76] mechanisms; but the BZ mechanism is more powerful and responsible for producing the peak radio luminosity. This provides another possible way to estimate a BH's spin [73], similar to a recent proposal of using the peak luminosity of the disk emission to estimate a BH's spin [77]. However there is so far no independent demonstration of validness of either of the above two new methods, which are thus not discussed further in this section.</text>
<text><location><page_12><loc_9><loc_35><loc_49><loc_54></location>Figure 14 shows the mass reduction, ∆ m = ( M max -M f ) /M max ( M max and M f are the BH's maximum mass and final mass, respectively), as a function of extracting efficiency glyph[epsilon1] of an extreme Kerr BH. For a 10 M glyph[circledot] BH with glyph[epsilon1] = 1 (∆ m ∼ 0 . 3), the total extracted energy is ∆ mM f c 2 glyph[similarequal] 10 54 erg and its total gravitating mass is reduced to M max / √ 2 [71]; this energy could be sufficient to power gamma-ray bursts (GRBs). It is thus plausible that supercritical accretion onto a newly born BH may spin it up and extract its spin energy to power ultrarelativistic jets; multiple spin-up and spin-down cycles may also happen during one GRB, if the collapsing material is clumpy.</text>
<text><location><page_12><loc_9><loc_9><loc_49><loc_30></location>As shown in Figure 15, the radius of the ISCO of the BH, R ISCO , is a monotonic function of the BH spin [78], beyond which radius a test particle will plunge into the BH under any perturbation; however, in Newtonian gravity a stable circular orbit can be found at any radius. It is thus reasonable to assume that the accretion disk around a BH terminates at this radius, i.e., r in = R ISCO . Therefore, a ∗ can be inferred if one can measure R ISCO in units of its gravitational radius r g = GM/c 2 . Currently three methods have been proposed to measure the BH spin in BHXBs, and all these methods rely essentially on measuring R ISCO . In case the radiative efficiency ( η ≡ L/ ˙ Mc 2 ) can be measured, a ∗ can also be determined this way, as shown in Figure 16. Actually η is a very simple function of R ISCO , i.e, η ∼ 1 /R ISCO , as shown in Figure 17.</text>
<figure>
<location><page_12><loc_56><loc_67><loc_88><loc_94></location>
<caption>FIG. 13. BH spin a ∗ vs. accreted rest-mass ∆ M , in units the final mass M f ( a ∗ = 1) (solid line, bottom axis), and in units of the initial mass M i ( a ∗ = 0) (dashed, top axis). (Adapted slightly from Figure 3 Ref.[71].</caption>
</figure>
<figure>
<location><page_12><loc_57><loc_36><loc_87><loc_59></location>
<caption>FIG. 14. Extracted fractional energy ∆ m = ( M max -M f ) /M max as a function of efficiency glyph[epsilon1] of extraction of a BH's rotational energy. (Figure 4 in Ref.[71].</caption>
</figure>
<section_header_level_1><location><page_12><loc_54><loc_25><loc_90><loc_27></location>A. The first black hole spin measurement with X-ray spectral continuum fitting</section_header_level_1>
<text><location><page_12><loc_52><loc_9><loc_92><loc_23></location>In 1997, I and my colleagues proposed a method of measuring the BH spin in BHXBs. It started when we tried to measure the mass of the BH in GRO J1655-40 [11] by measuring r in from its X-ray continuum spectral fitting and assuming the BH is not spinning; we found the mass of the BH is around 4 M glyph[circledot] [79]. Coincidentally just when this paper was going to press, Orosz and Bailyn [80] announced their accurate measurement of the BH mass of GRO J1655-40, which is significantly larger than what we found. A co-author of our paper, Rashid</text>
<figure>
<location><page_13><loc_13><loc_72><loc_45><loc_94></location>
<caption>FIG. 17. The solid line shows numerical curve of the radiative efficiency η as a function of R ISCO . The dashed line shows η ∼ 1 /R ISCO , which approximates the numerical curve.</caption>
</figure>
<figure>
<location><page_13><loc_56><loc_72><loc_88><loc_94></location>
<caption>FIG. 15. The radius of the innermost stable circular orbit ( R ISCO ) of a BH as a function of the spin parameter ( a ∗ ) of the BH, i.e., the dimensionless angular momentum; a negative value of a ∗ represents the case that the angular momentum of the disk is opposite to that of the BH, i.e., the disk is in a retrograde mode. Therefore a ∗ can be measured by determining the inner accretion disk radius, if the inner boundary of the disk is the ISCO of the BH.</caption>
</figure>
<figure>
<location><page_13><loc_13><loc_36><loc_45><loc_58></location>
<caption>FIG. 16. Radiative efficiency η as a function of BH spin parameter a ∗ .</caption>
</figure>
<text><location><page_13><loc_9><loc_13><loc_49><loc_29></location>Sunyaev urged me to resolve this apparent discrepancy. I immediately realized that a spinning BH of 7 M glyph[circledot] in GRO J1655-40 would be consistent with the inferred with Xray data, and added a note in proof in the paper suggesting this possibility [79]. This turns out to be the first BH spin measurement on record. I then invited two of my close friends and collaborators, Wei Cui and Wan Chen, to join me to apply this method systematically to other BHXBs; a new main conclusion in this work was that the first microquasar in the Milky Way, GRS 1915+105, also contains a spinning BH [47].</text>
<text><location><page_13><loc_9><loc_9><loc_49><loc_13></location>At the time DISKBB was the only available fitting model in the XSPEC package for determining r in from an observed X-ray continuum spectra, if the disk contin-</text>
<text><location><page_13><loc_52><loc_32><loc_92><loc_64></location>uum is described by the SSD model: L disk = 4 πσr 2 in T 2 in , where T in in DISKBB is the disk temperature at r in (i.e. the peak disk temperature in SSD) and L disk can be calculated from the disk flux (after the correction to absorption), the distance to the source and disk inclination (an issue to be discussed later). To determine the physical inner disk radius from the DISKBB parameter r in , several effects must be considered: (1) electron scattering in the disk modifies the observed X-ray spectrum; (2) the temperature distribution in the disk is not accurately described by the Newtonian gravity as assumed in SSD; (3) the observed temperature distribution is different from the locally emitted one; (4) the observed flux is different from the locally emitted one. The latter three are all due to GR effects [52]. For each of the above effects, we introduced a correction factor, using the best available knowledge at the time. Since then, several improvements have been made to correct for these effects and this continuum fitting (CF) method is now quite mature in making accurate BH spin measurements, given sufficiently high quality X-ray continuum spectral measurements and accurate system parameters of the observed BHXB.</text>
<section_header_level_1><location><page_13><loc_53><loc_26><loc_91><loc_29></location>B. Further developments and applications of the continuum fitting method</section_header_level_1>
<text><location><page_13><loc_52><loc_9><loc_92><loc_24></location>This CF method of measuring BH spin has since been applied widely to essentially every BHXB with a well measured X-ray continuum spectrum showing a prominent thermal accretion disk component. In particular, this method has been improved and incorporated into the widely used X-ray spectral fitting package XSPEC, e.g., KERRBB [81], BHSPEC [82], and KERRBB2 [35, 37]. Both the KERRBB and BHSPEC are relativistic models, but they have their own drawbacks and advantages (See Ref.[35] for a detailed comparison). KERRBB includes all the relativistic effects, but it requires to fix the</text>
<text><location><page_14><loc_9><loc_73><loc_49><loc_93></location>spectral hardening factor. In contrast, BHSPEC could calculate the spectral hardening factor on its own; however, it does not include the returning radiation effect, which turns out to be an important factor in BH spin determination in BHXBs. KERRBB2 combines both models by generating the spectral hardening factor table from BHSPEC and using the table as the input for KERRBB . The research group led by Ramesh Narayan of Harvard University, Jeffrey McClintock at Smithsonian Astronomical Observatory (SAO), and Ronald Remillard of Massachusetts Institute of Technology (MIT) [83] has since applied this method and contributed to most of the BH spin measurements available in the community, as shown in Table II.</text>
<text><location><page_14><loc_9><loc_55><loc_49><loc_73></location>The CF method relies on two fundamental assumptions: (1) The measured r in is uniquely related to R ISCO of the BH. (2) There is no or negligible X-ray radiation from the plunging matter onto the BH beyond R ISCO . The latter has been studied with numerical simulations that include the full physics of the magnetized flow, which predict that a small fraction of the disks total luminosity emanates from the plunging region [84]. However, in the context of BH spin estimation, it has been found that the neglected inner light in the CF method only has a modest effect, i.e., this bias is less than typical observational systematic errors [85, 86].</text>
<text><location><page_14><loc_9><loc_19><loc_49><loc_55></location>The first assumption above requires that the measured r in remains stable as a BHXB changes its spectral state and luminosity. However it was noticed that r in measured is usually much smaller, sometimes even smaller than R ISCO of a extreme Kerr BH in a prograde orbit, when the X-ray spectrum contains a significant hard PL component, which is believed to be produced by inverse Compton scattering of the thermal disk photons in a hot corona. We realized that the inferred smaller r in could be due to the lost thermal disk photons in the scattering process. We then investigated this problem and confirmed that the inferred r in can be made consistent with r in inferred from the thermal disk component dominated spectrum, if the scattered photons are recovered properly by doing detailed radiative transfer in the corona [65]; the same conclusion was also reached by the Harvard/SAO/MIT group independently without knowing our much earlier results [87]. Therefore the method of BH spin measurement by X-ray continuum fitting can also be applied to some SPL state with strong PL component. The stable nature of the measured r in is proven with the textbook case of LMC X-3, when its X-ray luminosity varied over more than one order of magnitude observed in nearly two decades with many different X-ray instruments, as shown in Figure 18 [88].</text>
<text><location><page_14><loc_9><loc_9><loc_49><loc_18></location>However, when ordered by the observed disk luminosity l D = L disk /L Edd , the measured r in shows a clear increasing trend when l D > 0 . 3, as shown in Figure 19 [88]. The similar trend has also been found in another BHXB GRS 1915+105 [35] and two NSXBs [64, 89]. It was found that r = r in /r g indeed increases physically when l D > 0 . 3, by comparing the evolution of r as a function</text>
<figure>
<location><page_14><loc_54><loc_73><loc_90><loc_94></location>
<caption>FIG. 18. The measured r = r in /r g and disk luminosity l D = L disk /L Edd (assuming M BH = 10 M glyph[circledot] ) as functions of observation time, for the BHXB LMC X-3. (Slightly adapted from Figure 1 in Ref.[88])</caption>
</figure>
<text><location><page_14><loc_52><loc_37><loc_92><loc_62></location>of l D over a large range for several BHXBs and NSXBs. Using the blackbody surface emissions of the NSs in these NSXBs is critical in evaluating any possible disk thickening due to high luminosity that would block at least part of the NS surface emission, as well as determining the actual NS mass accretion rate, which turns out to be much less than the disk mass accretion rate; this suggests that the increased radiation pressure is responsible for the increase of r and significant outflow when l D > 0 . 3 [64]. The same trend is much more pronounced in the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2, as shown in Figure 20 together with the data from other BHXBs and NSXBs [90]. However the exact value of r obtained this way should be taken with caution, since the non-negligible energy advection at high accretion rate can modify the disk structure in non-trivial ways, thus making the SSD prescription inaccurate in this case [91].</text>
<text><location><page_14><loc_52><loc_22><loc_92><loc_36></location>Figure 20 also shows that as l D decreases, r again starts to increases. However r increases at higher l D and with a different slope for a NSXB than for a BHXB, which can be naturally explained as due to the 'propeller' effect of the interaction between the NS's magnetosphere and its accretion disk [92] and the 'no-hair' of the BH [90]. Figure 21 shows the radiative efficiencies of various systems; BHXBs may have either higher or lower efficiencies than NSXBs, because a BH has neither solid surface nor magnetic field [5].</text>
<text><location><page_14><loc_52><loc_9><loc_92><loc_21></location>It is interesting to compare the BH spin results in our first paper [47] and the most recent literature for the same BHXBs as listed in Table II: (1) For GRS 1915+105, A0620-003 and LMC X-3, both results are fully consistent; (2) For GRO J1655-40, the original result points to an extreme Kerr BH ( a ∗ ∼ 0 . 93) [47], somewhat different from the most recent result of mildly spinning BH ( a ∗ = 0 . 65 -0 . 75) [32]. However, the original result was obtained using the BH mass of 7 M glyph[circledot] , about 10% larger</text>
<figure>
<location><page_15><loc_11><loc_71><loc_47><loc_93></location>
<caption>FIG. 19. The measured r = r in /r g as a function of disk luminosity l D = L disk /L Edd (assuming M BH = 10 M glyph[circledot] ), for the BHXB LMC X-3. (Slightly adapted from Figure 2 in Ref.[88])</caption>
</figure>
<figure>
<location><page_15><loc_10><loc_40><loc_48><loc_62></location>
<caption>FIG. 20. The measured r = r in /r g as a function of disk luminosity l D = L disk /L Edd , for several NSXBs and BHXBs, and the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2. All data points are taken from Ref.[64], except for that on XTE J1118+480 [93] and NGC1313 X-2 [90].</caption>
</figure>
<text><location><page_15><loc_9><loc_9><loc_49><loc_29></location>than the currently best estimate that was used to obtain an updated BH spin in the most recent literature [32]. From Figure 15, it can be seen that a ∗ would be decreased from 0.93 to 0.87, if the BH mass is decreased by about 10%; actually a ∗ = 0 . 85 is also allowed in the new estimate [32]; (3) For Cygnus X-1, our original conclusion that a ∗ = 0 . 75 in the high/soft state and a ∗ = -0 . 75 in the low/hard state was based on the assumption that r in decreased by a factor of two when the source made a transition from its normal low/hard state to the unusual high/soft state [94]. However the more realistic constraint is that r in changed by more than a factor of 1.8-3.2 during the state transition [94], which implies that a ∗ switched from | a ∗ | > 0 . 85, fully compatible with the</text>
<figure>
<location><page_15><loc_53><loc_69><loc_91><loc_93></location>
<caption>FIG. 21. The diagonal line shows a 1 /r scaling, calibrated to take the value of 0.057 when r = 6. The thick black line is for a BH accreting systems. The range of r = 1 -9 corresponds to the ISCO of a BH with different a ∗ ; the radiative efficiency ranges between a few to several tens of percents, far exceeding the p-p fusion radiative efficiency taking place in the Sun. The case for r > 9 corresponds to TID, whose radiative efficiency can be extremely low, because energy is lost into the event horizon of the BH. The thin solid black horizontal line is for the 10% efficiency when matter hits the surface of a neutron star where all gravitational energy is released as radiation. The thin solid black diagonal line above the point marked for 'Kerr BH' is for a speculated 'naked' compact object [5], whose surface radius is extremely small and thus the radiative efficiency can be extremely high. (Figure 10.7 in Ref.[5]</caption>
</figure>
<text><location><page_15><loc_52><loc_42><loc_89><loc_43></location>latest result of a ∗ > 0 . 95 in the high/soft state [37].</text>
<text><location><page_15><loc_52><loc_9><loc_92><loc_42></location>The above would suggest that a very low temperature disk component exists in its hard state spectrum, if Cygnus X-1 indeed harbors a Kerr BH and the accretion disk switches from a retrograde mode to a prograde mode when it makes the hard-to-soft state transition. For a supermassive BH in the center of a galaxy, it is well understood that its mass is mostly gained through accretion in its AGN phase [95]. Therefore random accretion (between prograde and retrograde modes) tends to make its final BH spin close to zero, regardless its initial BH spin. For Cygnus X-1, its current BH mass of around 10-20 M glyph[circledot] cannot be gained through post-formation accretion, since its observed average mass accretion rate ˙ M ∼ 2 × 10 -9 M glyph[circledot] /yr and the age of its companion is much less that 10 8 yr; actually the age of the companion is estimated to range from 4.6 and 7.8 million years [96]. This means that its post-formation BH mass growth is much less than a fraction of its current mass and thus its current BH spin must be quite close to that at birth. Even if accreting at Eddington rate, to grow its BH spin from 0 and the final mass to be the current observed value, the timescale is around 3 . 1 × 10 7 yr and the accreted mass is roughly 7.3 M glyph[circledot] [37]. If its accretions al-</text>
<text><location><page_16><loc_9><loc_83><loc_49><loc_93></location>tes between prograde and retrograde modes, then its current BH spin should be even closer to its initial spin than that in low mass BHXBs that only stay in one accretion mode due to roche-lobe overflows. Therefore the current high BH spin in Cyg X-1 must be natal; this conclusion is also true for the other highly spinning BHs in other BHXBs [48].</text>
<text><location><page_16><loc_9><loc_59><loc_49><loc_83></location>In summary, BH spin in BHXBs can now be measured reliably with the CF method, when the luminosity of a BHXB is between ∼ 0 . 02 and ∼ 0 . 3 in Eddington unit and their system parameters are well-known a prior . The observed thermal disk spectrum can be modeled directly to obtain the BH spin with the available KERRBB2 model in XSPEC when the X-ray spectrum in dominated by this component, i.e., the source is in the thermal state. When a significant power-law component is present, the inverse Compton scattering process must be taken into account to recover the disk photons scattered into this PL component, with for example the SIMPL/SIMPLR model [37, 97] in XSPEC. Recently another way to measure BH spin using the outburst properties of BHXBs has been proposed [77], which is mentioned briefly in Section VIII; however the effectiveness of this method needs to be tested.</text>
<section_header_level_1><location><page_16><loc_10><loc_54><loc_48><loc_55></location>C. Uncertainties of the continuum fitting method</section_header_level_1>
<text><location><page_16><loc_9><loc_36><loc_49><loc_52></location>In spite of the tremendous progress made so far on BH spin measurements in BHXBs, most of these BH spin measurements suffer from considerable uncertainties. Actually the major source of these uncertainties comes primarily from the uncertainties in their BH masses, accretion disk inclination angles, and distances. The accurate BH mass measurement is required because R ISCO must be in units of r g = GM/c 2 in Figure 15. The disk inclination angle and distance are also required because the total luminosity of the disk emission L disk is needed, in order to estimate the absolute value of a ∗ .</text>
<text><location><page_16><loc_9><loc_12><loc_49><loc_36></location>So far, all BH masses in X-ray binaries (XRBs) have been estimated using the Kepler's 3rd law of stellar motion, expressed in the so-called the mass function given in Equation (1). Since the only direct observables are P orb and K C , both M C and i have to be determined indirectly in order to obtain the BH mass estimate reliably. The companion's mass M C can be determined relatively reliably by the observed spectral type of the companion star and i can be estimated by modeling the observed ellipsoidal modulation of the companion's optical or infrared light curve. The observed ellipsoidal modulation is a consequence of exposing different parts of the pearshaped companion star to the observer at different orbital phases (see Figs. 1 and 5); the pear-shape is caused by the tidal force of the compact star, which also heats the side of the companion star facing it. For details of BH mass estimates using this method, please refer to [2].</text>
<text><location><page_16><loc_9><loc_9><loc_49><loc_11></location>However model dependence and other uncertainties (such as accretion disk contamination) cannot be circum-</text>
<figure>
<location><page_16><loc_54><loc_69><loc_89><loc_94></location>
<caption>FIG. 22. Phased VIH ellipsoidal light curves corresponding to the three states, namely 'passive', 'loop' and 'active' states; the data in each state and each wave band are plotted twice for clarity. Typical differential photometric error bars are shown in the upper right corner of the passive-state light curve for each band. To produce purer ellipsoidal light curves, variability on timescales greater than 10 days has been removed from loop- and active-state data. (Figure 2 in Ref.[98].)</caption>
</figure>
<text><location><page_16><loc_52><loc_22><loc_92><loc_53></location>vented completely and thus systematic error may exist in determining their system parameters. For example, three optical states, namely 'passive', 'loop' and 'active' states, have been identified in the normally called 'quiescent' state of A0620-003 when its X-ray luminosity is very low; only during the passive state its optical light curve modulation is purely ellipsoidal, i.e., accretion disk contamination is completely negligible, as shown in Figure 22 [98]. This means that considerable systematic errors in determining its inclination angle may occur unless only the 'passive' state data are used. Unfortunately previous observations of BHXBs used to determine their inclinations did not always occur during the passive state, thus systematic errors may be common in previous results [99]. Even for GRO J1655-40, of which all previous observations were made during its passive state [99], its BH mass measured with different observations, or the same data analyzed by different groups are not exactly the same, and even not completely consistent between them, as shown in Figure 23, which show a scatter of about 20-30% to the estimated BH mass, much larger than its statistical error of a few percents.</text>
<text><location><page_16><loc_52><loc_9><loc_92><loc_21></location>It should be pointed out that the inclination i in Equation (1) refers to that of the orbital plane of the binary system. Because the accreting material initially carries the angular momentum inherited from the companion star, the formed disk should be co-planar with the orbital plane of the binary system. However, the BH spin in a BHXB cannot be changed significantly by accretion [71] (and see discussion above on Cygnus X-1). Therefore its spin axis may not coincide with the normal direc-</text>
<figure>
<location><page_17><loc_13><loc_60><loc_45><loc_94></location>
<caption>FIG. 23. The BH mass and system inclination of GRO J165540 reported at different times, ordered by their publication dates [80, 100-105]. Note that the third and fourth reports [101, 102] adopted the inclination in the second report [100], and the last (seventh) report [105] adopted the inclination in the fifth report [103].</caption>
</figure>
<text><location><page_17><loc_9><loc_14><loc_49><loc_47></location>tion of the orbital plane of the binary system. In this case the Bardeen-Petterson effect [106], due to frame-dragging by the spinning BH, can rapidly align the normal axis of the inner disk region with the spin axis of the BH, making the inner disk and binary system mis-aligned. Circumstantial evidence already exists for mis-alignment between the two axes, because the orbital inclination of GRO J1655-40 shown in the upper panel of Figure 23 is significantly different from ∼ 80 · inferred from its relativistic jets [107], if the jets are powered by extracting the spin energy of the BH via the Blandford-Znajek mechanism [76]. Nevertheless, the orbital inclination is normally used in place of the disk inclination, which is needed in BH spin measurement using the CF method, in order to infer the total disk luminosity and calculate GR correction factors. However, the inner disk inclination is currently not available essentially for any of the known BHXBs. For example, a mis-alignment of 10 · from i = 70 · can cause nearly 50% error to the total disk luminosity, which will translate into nearly 30% error in r in . A Schwarzschild BH may be estimated to have a ∗ falling any where between [ -0 . 5 , 0 . 5], if r in is uncertain within about 30%, according to Fig.(15).</text>
<text><location><page_17><loc_9><loc_9><loc_49><loc_14></location>Accurate determinations of distances of astrophysical objects in the Milky Way are difficult, e.g., for BHXBs that are not standard candles. Normally some absorption features in their spectra, in conjunction with their posi-</text>
<text><location><page_17><loc_52><loc_70><loc_92><loc_93></location>tions in the galactic coordinates, are used to infer their distances. For example, the distance of GRO J1655-40 is commonly taken as 3 . 2 ± 0 . 2 kpc, based primarily on observed absorption lines and somewhat on the dynamics of the observed jets. Critical examinations of all available data related to its distance, however, are in favor of a much closer distance of less than 2 kpc and more likely just 1 kpc [108, 109]. Similar conclusion is also reached to the distance of A0620-003, revising its distance from the commonly accepted 1 . 2 ± 0 . 4 kpc to ∼ 0 . 4 kpc, making its possibly the closest BHXB known so far [109]; however, a distance of 1 . 06 ± 0 . 12 pc was preferred in a more recent study [30]. Similarly the currently adopted distances of many other BHXBs may also have considerable systematic errors. If true, this would change significantly the current estimates on their masses and spins.</text>
<text><location><page_17><loc_52><loc_64><loc_92><loc_70></location>Therefore future improvements of the continuum fitting method depend upon the improved measurements on their BH masses, accretion disk inclination, and distances.</text>
<section_header_level_1><location><page_17><loc_53><loc_58><loc_91><loc_61></location>D. Future improvements of the continuum fitting method</section_header_level_1>
<text><location><page_17><loc_52><loc_54><loc_92><loc_56></location>The mass ratio q can be determined directly according to the law of momentum conservation, i.e.,</text>
<formula><location><page_17><loc_64><loc_51><loc_92><loc_52></location>M C /M BH = K BH /K C , (4)</formula>
<text><location><page_17><loc_52><loc_42><loc_92><loc_49></location>if the semi-amplitude of the velocity curve of the BH K BH can be observed directly, as illustrated in Figure 1. The orbital inclination i can be then calculated using Equation (1), avoiding any systematics related to the ellipsoidal light curve modeling.</text>
<text><location><page_17><loc_52><loc_9><loc_92><loc_42></location>Since a BH is not directly observable, we can only hope to observe any emission or absorption line feature comoving with it. The accretion disk certainly moves with the accreting BH. However any line feature of the inner accretion disk suffers from the broadening of disk's orbital motion and distortions by relativistic effects around the BH, thus making it practically impossible, or difficult at least, for detecting the binary orbital motion of the BH. Orbital motion of double-peaked disk emission lines were observed for NSXB Sco X-1 [110], the BHXBs A0620-003 [111, 112]) and GRS 1124-68 [112]. Unfortunately a significant phase offset of velocity modulation was found from that expected based on the observed orbital motion of the companion, though the velocity semi-amplitude is consistent with the expected mass ratio [112]. Soria et al. (1998) observed the orbital motion of the double-peaked disk emission line He II λ 4686 from GRO J1655-40, and found its velocity modulation phase and semi-amplitude in agreement with the kinematic and dynamical parameters of the system [113]. However one major problem in accurately measuring the orbital motion of the primary from the observed double-peaked emission lines is how to determine reliably the line cen-</text>
<figure>
<location><page_18><loc_13><loc_76><loc_45><loc_94></location>
<caption>FIG. 24. Velocity curve of the 39 observed X-ray absorption lines with Chandra from GRO J1655-40, after subtracting the line of sight intrinsic velocity of each line at each orbital phase. The upper panel marks the velocity of each line with its 1σ error bar, slightly shifted horizontally for visual clarity. The bottom panel shows the weighted average velocity of all lines in the upper panel at each phase; the solid curve is the fitted velocity curve with its orbital period and phase fixed at the values observed previously. (Slightly adapted from Figures 2 in Ref.[114])</caption>
</figure>
<text><location><page_18><loc_9><loc_56><loc_49><loc_58></location>er, because the lines are typically asymmetric and also variable.</text>
<text><location><page_18><loc_9><loc_33><loc_49><loc_56></location>We have recently proposed to observe the Doppler shifts of the absorption lines of the accretion disk winds co-rotating with the BH around its companion star [114], since in many XRBs accretion disk winds are ubiquitous and appear to be rather stable when observed (e.g., in Ref.[115]). We verified this method using Chandra and HST high resolution spectroscopic observations of GRO J1655-40 (shown in Figure 24) and LMC X-3. Unfortunately the currently available data only covered small portions of their orbital phases and thus do not allow better constraints to their system parameters. Future more observations of these two sources and other sources with detectable absorption lines from their accretion disk winds will allow reliable and precise measurements of the BH masses and orbital inclination angles in accreting BHXBs.</text>
<text><location><page_18><loc_9><loc_15><loc_49><loc_32></location>Since the accretion disk has very high density and is ionized near the BH, where the majority of the observed disk emission is produced due to the very deep gravitational potential near the BH, scattering of the primary disk emission is inevitable. The scattered light is polarized and its polarization fraction and position angle depend on the viewing direction (inclination ), scattering optical depth and the radius where the scattering occurs [116-119]. Ignoring many details, it can be shown that the polarization fraction, P ( i ), of the observed disk photons (initial disk emission plus the scattered emission) is given by</text>
<formula><location><page_18><loc_21><loc_11><loc_49><loc_14></location>1 P ( i ) = 1 + A cos i 1 -cos 2 i , (5)</formula>
<text><location><page_18><loc_9><loc_9><loc_49><loc_10></location>where A is a constant depending upon the scattering op-</text>
<figure>
<location><page_18><loc_55><loc_72><loc_88><loc_94></location>
<caption>FIG. 25. Polarization fraction of observed accretion disk emission as a function of its inclination.</caption>
</figure>
<text><location><page_18><loc_52><loc_41><loc_92><loc_65></location>tical depth; detailed calculations made by Chandrasekhar [120] gives P (75 · ) = 0 . 04. Note that here the disk photons are from the Raleigh-Jeans part of the multicolor blackbody spectrum with a characteristic shape of f ( ν ) ∝ ν 1 / 3 , i.e., no GR effect is included. We can therefore find the disk inclination angle by measuring the polarization fraction of this part of the disk emission, as shown in Figure 25. At energies above the Raleigh-Jeans part of the multi-color blackbody spectrum, the polarization is strongly effected by both the inclination and BH spin, as shown in Figure 26. The continuum spectra are clearly degenerated for the different combinations of inclination and BH spin, but the polarization fraction and angle as functions of energy can clearly distinguish between them [118]. Therefore X-ray spectra-polarimetry observations of BHXBs will certainly make important progresses in measuring BH spin.</text>
<text><location><page_18><loc_52><loc_10><loc_92><loc_40></location>Besides using polarization measurements to obtain inner disk inclination (and BH spin), the broad iron Kalpha measurements can also be used to do so [121-123], because iron K-alpha emission is believed to come from the fluorescent emission of the disk as a result of illumination by a hard X-ray source above the disk. Naturally this emission is sensitive to the disk inclination (and BH spin). However, compared to the polarization measurement method, this method is less straight forward and may suffer from systematic uncertainties in deriving the disk inclination (and BH spin), because complicated modeling of the hard X-ray component and line emissivity from the disk is required. Therefore it is essential to use all methods discussed above to measure both inclination angles (orbital plane and inner disk) and BH spin. Studying the relationship between the results obtained with different methods is also important in its own rights, in order to understand accretion disk physics, its interaction with the BH in its center, production of relativistic jets, the origin of the BH spin, and ultimately the formation mechanisms of BHs and BHXB systems.</text>
<text><location><page_18><loc_53><loc_9><loc_92><loc_10></location>The recent dispute on the distances of GRO J1655-40</text>
<figure>
<location><page_19><loc_11><loc_16><loc_47><loc_94></location>
<caption>FIG. 26. Disk continuum flux (a), polarization fraction (b) and polarization angle (c) as functions of photon energy for different model parameters. i , inclination angle of the inner disk in degrees; ˙ m , mass accretion rate in units of Eddington rate by assuming a 10% conversion efficiency from rest mass to radiation in all cases. Other parameters: mass of the BH is 10 M glyph[circledot] ; distance to the BH is 10 kpc; spectral hardening factor is 1 . 6. (Adapted from Figures 4-6 and Table 1 in Ref.[118])</caption>
</figure>
<text><location><page_19><loc_52><loc_67><loc_92><loc_93></location>and A0620-003 exemplifies the difficulty of determining the distances of BHXBs using mostly absorption lines [108, 109]. We have recently suggested a method of using the delay time between the X-ray fluxes of an XRB and its X-ray scattering halo by interstellar dust to infer its distance [124, 125]. However this method may suffer from our incomplete knowledge of the distribution of interstellar medium. Ideally precise astrometry can determine their distances model-independently, by measuring their trigonometric parallaxes. Recently the distance to Cygnus X-1 was determined reliably and accurately this way (1 . 86 +0 . 12 -0 . 11 kpc; [126]); which is key to the consequent measurement of its BH mass and spin [36, 37]. Currently it remains challenging to measure the trigonometric parallaxes of objects at distances beyond several kpc where most BHXBs are located. Future high precision astrometry missions are expected to improve the distance estimates to these BHXBs significantly.</text>
<text><location><page_19><loc_52><loc_61><loc_92><loc_67></location>Therefore we expect that the improved measurements discussed above on their BH masses, inner disk inclination, and distances will allow future improvements of BH spin measurement with the CF method.</text>
<section_header_level_1><location><page_19><loc_53><loc_55><loc_90><loc_58></location>E. Possible application of the continuum fitting method to AGNs</section_header_level_1>
<text><location><page_19><loc_52><loc_26><loc_92><loc_53></location>More recently, the CF method is also applied to constrain the BH spin in an active galactic nucleus (AGN), which is powered by matter accretion onto the central supermassive BH in its center [127]. However the BH spin inferred this way for an supermassive BH is quite uncertain, because: (1) The peak temperature of the accretion disk inversely increases with the mass of a BH. Therefore, for a supermassive, its temperature would be in the ultraviolet energy range which will be strongly absorbed and hard to observe; (2) The system parameters of a supermassive BH (e.g., the mass of the BH and the inclination angle of the accretion disk) have larger uncertainties; (3) The uncertain mechanism for some components (e.g., the soft X-ray excess) in AGN spectra also increases the difficulty; (4) Some emission and absorption lines may distort the continuum spectrum substantially; and (5) In some cases the contribution of its host galaxy to the observed total continuum spectrum cannot be removed satisfactorily.</text>
<text><location><page_19><loc_52><loc_9><loc_92><loc_26></location>The polarized continuum of an AGN should be a pure accretion disk continuum, at least in the optical to near infrared band [128]. One possible way to measure the BH spin in an AGN with the CF method is to combine the observed polarized optical to near infrared continuum spectrum with the observed total UV continuum spectrum to get a broad band continuum spectrum of an AGN [129]. In principle the broadened Balmer edge features and the total UV spectrum can be used to constrain the disk inclination angle and fraction of host galaxy contamination, respectively [129]. However the quality of the currently available data is still insufficient to allow</text>
<text><location><page_20><loc_9><loc_90><loc_49><loc_93></location>accurate determination of BH spin in AGNs with this method.</text>
<text><location><page_20><loc_9><loc_76><loc_49><loc_90></location>Nevertheless the principal method of measuring BH spin in AGNs should be using the reflected broad iron line and continuum components [121-123]. The main reason is that i can also be determined simultaneously and r in obtained this way is already in units of r g , thus avoiding naturally the uncertainties caused by the BH mass and inclination in the CF method. However cross calibration can be done between the CF and reflection fitting methods if both can be applied to the same supermassive BH.</text>
<section_header_level_1><location><page_20><loc_11><loc_70><loc_46><loc_72></location>V. FURTHER DEVELOPMENTS ON HOT ACCRETION FLOWS</section_header_level_1>
<text><location><page_20><loc_9><loc_31><loc_49><loc_68></location>In this section I briefly summarize some further developments on hot accretion flows, which are believed to be responsible for the PL component of the spectra in BHXBs. The multi-waveband spectra of the hard state of the BHXB XTE J1118+480 was modeled with the TID with hot accretion flow (ADAF) geometry (panel (a) in Figure 27) and TID (with ADAF) plus jet model (panel (b) in Figure 27), with ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D [93]. The steep UV spectrum provides clear evidence for a large truncation radius for the SSD ( r = 600), and the radio to infrared spectrum dominates the jet emission, which also contribute to the hard PL component [93]. Such low accretion rate gives very low SSD luminosity l D ∼ 5 × 10 -4 (since the disk radiation efficiency η ∼ 1 /r ; see Figure 17), which is far below the turn-over luminosity of l D ∼ 10 -2 shown in Figure 20. Therefore the inferred truncation radius of XTE J1118+480 agrees with the extrapolation of data points of XTE J1817-330 down to very low disk luminosity. Large truncation radii are also reported from several other sources in the hard state (e.g., in Ref.[130]). However, the exact values of these truncation radii may have large uncertainties, since no direct detection of the inner disk peak emission was available, unlike the strong case of XTE J1118+480 [93]. For this reason I did not include these reported values in Figure 20.</text>
<text><location><page_20><loc_9><loc_13><loc_49><loc_30></location>Panel (b) in Figure 27 also shows that the radiation from both the hot accretion flow and the jet contribute to the X-ray emission. However the former is roughly proportional to ˙ m 2 , whereas the latter to ˙ m . Thus with the decrease of ˙ m , the contribution from the jet becomes more and more important, thus the X-ray radiation will be dominated by the jet (when l glyph[lessorsimilar] 10 -5 -10 -6 ), as shown in Figure 28 [93]. The observational data of very low-luminosity AGNs clearly show a correlation between radio and X-ray with a correlation index of ∼ 1 . 2 [131], in excellent agreement with the prediction shown in Figure 28 [93].</text>
<text><location><page_20><loc_9><loc_9><loc_49><loc_13></location>It is well known that the highest luminosity an ADAF can produce is only about 3% L Edd . However the observed highest luminosity a hard state can reach can</text>
<figure>
<location><page_20><loc_53><loc_53><loc_91><loc_94></location>
<caption>FIG. 27. Spectral modeling results for XTE J1118+480. (a) Outer cool disk plus inner hot accretion flow model. (b) An additional jet component is included. ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D . (Adapted from Figures 1-2 in Ref.[93])</caption>
</figure>
<text><location><page_20><loc_52><loc_33><loc_92><loc_43></location>be 10% L Edd or even higher, which can be described by LHAF shown in Figure 29, where several previously known solutions of accretion flows are all unified in a single scheme [133]. The hard state spectrum of XTE J1550-564 with L ∼ 6% L Edd is explained by LHAF very well, including the X-ray spectral slope and the value of the cutoff energy [134].</text>
<section_header_level_1><location><page_20><loc_52><loc_28><loc_92><loc_30></location>VI. FURTHER DEVELOPMENTS ON CORONA FORMATION</section_header_level_1>
<text><location><page_20><loc_52><loc_9><loc_92><loc_26></location>While the formation of the SSD in a BHXB is reasonably well understood, the formation of the corona, which is the hot accretion flow discussed in Section V, remains less understood. Over the last more than 10 years, Liu and her collaborators have developed a model to explain the formation and evolution of the corona in a BHXB or AGN, in which the mass accretion rate ˙ m in units of the Eddington ratio drives the variations of the complex accretion flows by the interaction between the cold SSD and the hot corona [60, 135-139]. Specifically, the coupling between the hot corona and the cold disk leads to mass exchange between them. The gas in the thin</text>
<figure>
<location><page_21><loc_14><loc_77><loc_44><loc_93></location>
<caption>FIG. 28. Radio (8.6 GHz)CX-ray (2C11 keV) correlation for BHXBs. The observed correlation is shown by the segment AB. Segments BCD show the predicted correlation at lower luminosities, which approaches that of a pure-jet model, as shown by the segment DE. Note that below the point C ( ∼ 10 -6 l D ), the X-ray emission is dominated by the jet and the correlation steepens. (Figures 1 in Ref.[132])</caption>
</figure>
<figure>
<location><page_21><loc_14><loc_44><loc_44><loc_64></location>
<caption>FIG. 29. The thermal equilibrium curve of various accretion solutions. The accretion rate is in units of ˙ M Edd ≡ 10 L Edd /c 2 and the units of Σ is g cm -2 . The parameters are M BH = 10 M glyph[circledot] , α = 0 . 1, and r = 10. (Figures 1 in Ref.[133])</caption>
</figure>
<text><location><page_21><loc_9><loc_10><loc_49><loc_33></location>disk is heated up and evaporates into the corona as a consequence of thermal conduction from the hot corona, or the corona gas condenses into the disk as a result of overcooling by, for example, external inverse Compton scattering. If ˙ m is low, evaporation occurs and can completely remove the thin disk, leaving only the hot corona in the inner region and a truncated thin disk in outer region; this provides a mechanism for ADAF at low ˙ m . If ˙ m is high, the gas in the corona partially condenses to the disk due to strong Compton cooling, resulting in disk dominant accretion. The model naturally explains the different structures of accretion flow in different spectral states as shown in Figure 30 [60, 135, 137-144]. The hysteresis observed in spectral state transitions can also be explained by different irradiations from different evolution history under the same scenario [145-147].</text>
<text><location><page_21><loc_10><loc_9><loc_49><loc_10></location>Figure 30 is significantly different from the illustration</text>
<figure>
<location><page_21><loc_55><loc_62><loc_89><loc_93></location>
<caption>FIG. 30. A schematic description of the accretion flow structures in different spectral states as consequences of diskcorona interaction, which is primarily driven by mass accretion rate dotm . SPL: steep power-law state; VHS: very high state; soft: soft state; H/S: high/soft state; hard: hard state; L/H: low/hard state; Q: quiescent state. (Adapted from a similar figure provided by Prof. Bifang Liu.)</caption>
</figure>
<text><location><page_21><loc_52><loc_10><loc_92><loc_48></location>of accretion flow structures in different spectral state in Figure 12 on three aspects: (1) At intermediate ˙ m , the SSD is broken by ADAF into two parts, an outer disk and an inner disk; (2) The inner disk boundary here is always located very close to the BH, except in the very low ˙ m hard or quiescent state, which is very different from the TID scenario depicted in Figure 12; (3) The corona here covers essentially the whole accretion disk, especially the inner disk region, whereas in Figure 12 the corona is mostly located inward from the inner disk boundary. The observed soft X-ray component in the low/hard state can be explained by the existence of the inner disk [139, 140, 142, 144]. As I have discussed above, the inner disk boundary radius inferred with the CF method in the presence of a strong PL component is actually consistent with that in the soft state when the PL component is weak, after taking into account the Compton scattering in the corona [65, 87]. Actually the essential assumption behind the broad iron line/reflection fitting method of determining BH spin is that the inner disk boundary is at the ISCO when the PL component is strong. All these tend to support the existence of the inner disk at intermediate ˙ m . However it remains to be demonstrated that if the whole SSD is indeed broken into the two parts at intermediate ˙ m , as illustrated in Figure 30.</text>
<text><location><page_21><loc_53><loc_9><loc_92><loc_10></location>Recently it has been suggested that the ADAF and</text>
<figure>
<location><page_22><loc_9><loc_75><loc_49><loc_93></location>
<caption>FIG. 31. Solid lines: accretion disk models at different accretion rates. Clumpy structures in either ADAF or corona (shown as insets) may be developed at intermediate accretion rates, corresponding to different spectral states. (Adapted from Figures 1 and 2 in Ref.[148])</caption>
</figure>
<text><location><page_22><loc_9><loc_53><loc_49><loc_64></location>corona shown in Figure 30 may have clumpy structures, as shown in Figure 31 [148]. The 'clumpy' model (Figure 31) has mainly two different consequences from the 'uniform' model (Figure 30): (1) The inner disk is transient in the 'clumpy' model; (2) The 'clumpy' model can explain the variabilities observed in X-ray binaries (such as the state transitions discussed in Section VII) and radio-loud AGNs (such as BL Lac objects).</text>
<text><location><page_22><loc_9><loc_13><loc_49><loc_52></location>Determining the structure of the corona in an BHXB observationally remains difficult; the fundamental issue on whether the corona covers mostly the accretion disk or the central compact object still remains unclear so far. Since the observed similar spectral and state transitions between some NS and BH XRBs are quite similar [149], it is reasonable to assume that they have similar coronae. Recently we have used type I X-ray bursts from low-mass NSXBs to show that X-ray bursts experience negligible Comptonization and that the corona cools rapidly during the rising phase of X-ray bursts and is then heated up rapidly during the rising phase of X-ray bursts, as shown in the upper panel of Figure 32 for IGR J1747721 [150]. These results suggest that the corona cannot cover the central compact object completely (lower panel of Figure 32) and that the destruction and formation time scales of the corona are as short as seconds; such short time scales are quite difficult to understand in the above discussed evaporation model, in which the time scales are related to the viscous time scales of the accretion disk. However, this short time scale is consistent with a corona produced by magnetic reconnections in the accretion disk, in a similar way to the solar corona heating [151]; this conclusion was based on the inferred accretion flow structure of a BHXB shown in panel (A) of Figure 33, in comparison with the atmospheric structure of the Sun shown in panel (B) of Figure 33.</text>
<text><location><page_22><loc_9><loc_9><loc_49><loc_13></location>It is worth noticing that the purported coronae in both XRBs and AGNs have strikingly similar properties, e.g., they all have electron temperatures of the order of hun-</text>
<figure>
<location><page_22><loc_57><loc_65><loc_87><loc_93></location>
<caption>FIG. 32. Upper panel: Anti-correlation between the observed type I X-ray bursts from surface of the NS in IGR J1747-721 (Adapted from Figures 3 and 2 in Ref.[150]).</caption>
</figure>
<figure>
<location><page_22><loc_60><loc_32><loc_84><loc_58></location>
<caption>FIG. 33. Schematic diagrams of the solar atmosphere (A) and accretion disk structure (B). (Figure 2 in Ref.[151]).</caption>
</figure>
<text><location><page_22><loc_52><loc_9><loc_92><loc_24></location>dreds keV, in spite that the temperatures, inner disk radii and variability time scales of their cold accretion disks all scale with their BH masses (and accretion rates) as predicted in the SSD model. This means that their coronae are scale independent. It is perhaps not coincidental that the electrons' velocities in a corona are approximately the same as the Keplerian orbital velocities of the inner disk, which are also roughly the same as the launching velocities of jets. Of course these velocities are also the varialized velocities of the central BHs. It is plausible that turbulent small scale magnetic fields lifts</text>
<figure>
<location><page_23><loc_10><loc_71><loc_48><loc_94></location>
<caption>FIG. 34. The clumps in the clumpy corona/ADAF of a BHXB are channeled into the rotating and wound-up magnetic field lines and then ejected from the system. The launched jets should be episodic and have knotted structures.</caption>
</figure>
<text><location><page_23><loc_9><loc_43><loc_49><loc_61></location>the plasma in the 'warm layer' shown in Figure 33 to form the corona, which is then launched into the jets by the rotating large scale magnetic fields through either the BP or BZ mechanism; during these processes the magnetic fields are mostly responsible for changing the directions of motion of plasmas by the Lorentz force. If the corona is clumped as discussed in Section VI [148], then the plasma channeled into the jets should be clumpy and thus the ejected jets should be episodic and have knotted structures, in agreement with observations; this scenario is illustrated in Figure 34. Alternatively episodic ejections may also occur in the disk, similar to the coronal mass ejection on the Sun [152].</text>
<text><location><page_23><loc_9><loc_27><loc_49><loc_42></location>Interestingly, the soft X-ray excess (SXE) frequently observed in AGNs can be interpreted as the warm layer found in XRBs, since the inferred plasmas parameters, with electron temperature of 0.1 to 0.3 keV and Compton scattering optical depth of around 10, are similar to that of XRBs and universal among AGNs with different BHmasses, but seem to be related only with ˙ m [153-155]. Therefore their warm layers are also scale independent. Indeed, magnetic flares, perhaps caused by magnetic reconnections in the cold disk, can produce the warm layers in both XRBs and AGNs [156, 157].</text>
<section_header_level_1><location><page_23><loc_10><loc_21><loc_48><loc_23></location>VII. FURTHER DEVELOPMENTS ON STATE TRANSITIONS</section_header_level_1>
<text><location><page_23><loc_9><loc_9><loc_49><loc_18></location>To further understand the mechanism of the spectral state transitions, it is important to study those spectral state transitions in individual BHXBs during different outbursts. The advantages are obvious - uncertainties in our estimates of black hole mass, the binary properties such as the orbital period, or the source distance will not play any role in producing the observed diverse transi-</text>
<text><location><page_23><loc_52><loc_63><loc_92><loc_93></location>tion properties in individual BHXBs, which can only be driven by the accretion process under different initial conditions. The sources which firstly allowed such a study were the BH transients GX 339-4 and XTE J1550-564, the NS transient Aquila X-1, and the flaring NS low-mass XRB (LMXB) 4U 1705-44, in which a remarkable correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the following soft state was found [158-160]. More recently, a comprehensive study of the hard-to-soft spectral state transitions detected in all the bright XRBs in a period of about five years with simultaneous X-ray monitoring observations with the RXTE/ASM and the Swift/BAT confirmed the correlation between the hard-to-soft transition and the peak luminosity of the following soft state [161, 162], as shown in the upper panel of Figure 35 [161]. More important was the discovery of the correlation between the transition luminosity and the rate-of-change of the luminosity during the rising phase of an outburst of transients or a flare of persistent sources, as shown in the lower panel of Figure 35 [161].</text>
<text><location><page_23><loc_52><loc_44><loc_92><loc_63></location>The above correlation implies that in most cases it is the rate-of-increase of the mass accretion rate, rather than the mass accretion rate itself, determines the hardto-soft spectral transition; this is depicted in Figure 36 [161]. In addition, the discovery of the relation between the hard X-ray peak flux and the waiting time of transient outburst in the BH transient GX 339-4 (as shown in Figure 37) supports that the total mass in the disk determines the peak soft state luminosity of the following outburst [159]. Therefore spectral states should be understood in the non-stationary accretion regime, which would be described by both ˙ m and m , as well as the total mass in the disk before an outburst.</text>
<section_header_level_1><location><page_23><loc_55><loc_38><loc_88><loc_40></location>VIII. FURTHER DEVELOPMENTS ON THERMAL STABILITY OF SSD</section_header_level_1>
<text><location><page_23><loc_52><loc_9><loc_92><loc_36></location>The SSD model predicts that when the accretion rate is over a small fraction of the Eddington rate, which corresponds to L glyph[greaterorsimilar] 0 . 06 L Edd , the inner region of the disk is radiation-pressure-dominated and then both secularly [163] thermally unstable [164, 165]. However, observations of the high/soft state of black hole X-ray binaries with luminosity well within this regime (0 . 01 L Edd glyph[lessorsimilar] L glyph[lessorsimilar] 0 . 5 L Edd ) indicate that the disk has very little variability, i.e., quite stable [166]. It has been well established that the accretion flow in this state is described by the SSD model [4, 167]. Radiation magnetohydrodynamic simulations of a vertically stratified shearing box have confirmed the absence of the thermal instability [168]. Recently, the thermal stability is revisited by linear analysis, by taking into account the role of magnetic field in the accretion flow [169]. By assuming that the field responds negatively to a positive temperature perturbation, it was found that the threshold of accretion rate above which the disk becomes thermally unstable increases signifi-</text>
<figure>
<location><page_24><loc_9><loc_42><loc_50><loc_69></location>
<caption>FIG. 35. Upper panel: correlation between the transition luminosity (15-50 keV) and the peak luminosity of the following soft state (2-12 keV) in Eddington units. Lower panel: correlation between the peak luminosity of the soft state and the maximum rate-of-increase of the X-ray luminosity around the hard-to-soft transition. (Adapted from Figures 24 and 27 in Ref.[161])</caption>
</figure>
<text><location><page_24><loc_9><loc_15><loc_49><loc_28></location>cantly, compared with the case of not considering the role of magnetic field. This accounts for the stability of the observed sources with high luminosities, as shown in Figure 38. If the magnetic pressure is less than about 24% of the total pressure, then this model can explain the 'heart-beat' limit-cycle instability observed in GRS 1915+105 at its highest luminosity; this peculiar source holds the highest accretion rate (or luminosity) among BHXBs.</text>
<text><location><page_24><loc_9><loc_9><loc_49><loc_14></location>Observations of GRS 1915+105 showed that t high (the duration of the outburst phase) is comparable to t low (duration of the quiescent phase) and L high is 3 to 20 times larger than L low [170]. However, numerical calculations</text>
<figure>
<location><page_24><loc_59><loc_70><loc_85><loc_94></location>
<caption>FIG. 36. Correlation between the peak fluxes of the initial low/hard states in the outbursts of GX 339-4 and the time since the latest low/hard state peak in the previous outburst. The dashed line passes the origin and the data point of maximal peak flux, showing an example of a linear relation. (Adapted from Figure 3 in Ref.[159])</caption>
</figure>
<figure>
<location><page_24><loc_60><loc_43><loc_84><loc_58></location>
<caption>FIG. 37. Aschematic picture of the regimes of the hard state. Two assumed transient outbursts of different peak luminosities are shown. When a source is under stationary accretion, spectral transitions between the hard state and the soft state occurs at a nearly a constant luminosity L 0 . When a source is undergoing an outburst or flare, the hard-to-soft transition occurs at a luminosity above L 0 . The additional luminosity roughly proportional to ∆ L ∆ T . The soft-to-hard transitions are expected to occur around L 0 . (Figures 28 in Ref.[161])</caption>
</figure>
<text><location><page_24><loc_52><loc_9><loc_92><loc_26></location>showed that t high is less than 5 percent of t low , and L high is around two orders of magnitude larger than L low . Some efforts have been made to improve the theory in order to explain observations either by some artificial viscosity prescription [171] or by additional assumption of the energy exchange between the disk and corona [172]. Taking into account the stress evolution process, it was found that the growth rate of thermally unstable modes can decrease significantly owing to the stress delay, which may help to understand the 'heart-beat' limit-cycle variability of GRS 1915+105 [173]. The limit-cycle properties are found to be dominated by the mass-supply rate (accre-</text>
<text><location><page_25><loc_14><loc_87><loc_15><loc_87></location>·</text>
<text><location><page_25><loc_14><loc_85><loc_15><loc_85></location>·</text>
<figure>
<location><page_25><loc_14><loc_77><loc_42><loc_93></location>
<caption>FIG. 38. The thermal equilibrium curves of a thin disk at 10 r g for different Φ 1 , which is defined as Φ 1 ≡ B ϕ H and is in unit of Gs · cm. Other model parameters are BH mass M BH = 10 M glyph[circledot] and viscosity parameter α = 0 . 1 . (Figure 2 in Ref.[169])</caption>
</figure>
<text><location><page_25><loc_9><loc_53><loc_49><loc_66></location>tion rate at the outer boundary) and the value of the α -viscosity parameter in the SSD model that assumes that the viscous torque is proportional to the total pressure [77]. It was also found that only the maximal outburst luminosity (in Eddington units) is positively correlated with the spin of a BH, providing another way to probe BH spin [77]; this is mainly due to the smaller inner disk radius and thus higher radiative efficiency for larger a ∗ as shown in Figures 15, 16 and 17.</text>
<section_header_level_1><location><page_25><loc_14><loc_48><loc_44><loc_49></location>IX. UNIFICATION AND OUTLOOK</section_header_level_1>
<text><location><page_25><loc_9><loc_22><loc_49><loc_46></location>Despite of the many progresses made over the last decades on the study of BHXBs and microquasars, there are still many outstanding and unresolved issues, which can be pierced by two big pictures. One big picture is related to the astrophysics of BHXBs and microquasars, which is centered on understanding the accretion-outflow (wind or jet) connections at different accretion rates for different types of astrophysical systems involving BHs. The other big picture is related to the fundamental physics of BHXBs and microquasars, which is centered on identifying astrophysical BHs and testing theories of strong gravity. These two pictures are actually also entangled together, because a strong gravity theory, such as GR, is needed to describe the astrophysical aspects of these systems, and we need to understand the astrophysics of BHXBs and microquasars before we can start to test any strong gravity theory[174].</text>
<text><location><page_25><loc_9><loc_9><loc_49><loc_21></location>Figure 39 shows the relation between the collimationcorrected gamma-ray luminosity and the kinetic power for AGNs and gamma-ray bursts (GRBs)[175]. For at least some of GRBs, a super-Eddington accreting compact stellar object, probably a BH with around 10 M glyph[circledot] , is believed to be responsible for powering the highly relativistic jets, which produces intense gamma-rays through violent collisions of blobs in the jets[176]. The two types of AGNs shown here, flat-spectrum radio quasars (FS-</text>
<figure>
<location><page_25><loc_58><loc_68><loc_86><loc_93></location>
<caption>FIG. 39. The relation between the collimation-corrected gamma-ray luminosity and the kinetic power for AGNs and GRBs. The shaded regions display the 2 σ confidence band of the fits. The blazar and GRB best-fit models (dashed and dotted lines, respectively) follow correlations which are consistent, within the uncertainties, with the best-fit model obtained from the joint data set (solid line).(Figure 3 in Ref.[175])</caption>
</figure>
<text><location><page_25><loc_52><loc_41><loc_92><loc_53></location>RQs) and BL Lacs, have all been observed to have mildly relativistic jets. Their central engines are supermassive BHs with around 10 7 -9 M glyph[circledot] with accretion rates just below (for FSRQs) or far below (for BL Lacs) Eddington rate[177]. The good correlation over about 10 orders of magnitudes between these very different systems with very different accretion rates suggests that there must be some common mechanisms responsible for the accretionoutflow (wind/jet) connections for all accreting BHs[178].</text>
<text><location><page_25><loc_52><loc_17><loc_92><loc_40></location>Putting together all related phenomenologies and some theoretical modeling of BHXBs and microquasars discussed in this article, an unification scheme is illustrated in Figure 40 with the major elements in the accretionoutflow connections in different types of astrophysical systems harboring both stellar mass BHs and supermassive BHs with accretion rates over several orders of magnitudes. The types of accreting stellar mass BHs include BHXBs and microquasars in different spectral states, as well as ultra-luminous X-ray sources (ULXs) (some of which are most likely ultra-Eddington accreting BHs[180]), and GRBs. The types of accreting supermassive mass BHs include low-luminosity AGNs (LLAGNs), BL Lac objects, normal radio quiet quasars (RQQs), FSRQs and broad-line-less luminous quasars (BL3Qs) [179] (I create the acronym 'BL3Q' here just for fun).</text>
<text><location><page_25><loc_52><loc_9><loc_92><loc_17></location>This scheme is not a theoretical model, or even toy model yet. It, however, can be used as one possible chain to pierce many observed phenomenologies together, providing a possible frame work for further theoretical developments on accretion-outflow connections. Future observations will scrutinize this unification scheme and</text>
<figure>
<location><page_26><loc_9><loc_36><loc_47><loc_93></location>
<caption>FIG. 40. Unification scheme of accretion-outflow connections of accreting BHs for different ranges of accretion rates, corresponding to different spectral states or different types of astrophysical systems. From very low to very high accretion rates in units of the Eddington rate ˙ m , these states are: hard/quiescent state, hard/low state, thermal-dominated soft state, steep power-law state, slim disk, and NDAF disk; different types of AGNs with similar accretion-outflow structures are also labeled for comparison. The ubiquitous dusty tori and BLRs are absent in all BHXBs and microquasars. The lack of dusty tori can be easily understood since there is no dust supply in them. It is then plausible that the lack of BLRs is the consequence of no dusty tori, suggesting that a BLR may be formed out of the evaporated inner dusty torus by the anisotropic radiation of the accretion disk in an AGN[179].</caption>
</figure>
<section_header_level_1><location><page_26><loc_49><loc_92><loc_88><loc_93></location>Some necessary elaborations and points:</section_header_level_1>
<unordered_list>
<list_item><location><page_26><loc_50><loc_80><loc_89><loc_89></location>· ˙ m glyph[greatermuch] 10: the two-phase clumpy thick disk is in the form of neutrino dominated accretion flow (NDAF), which can extend to ISCO due to the extremely low opacity of neutrinos. The large scale magnetic field lines are wound-up by the spinning BH and rotating disk. The discontinuous jet is produced by the BZ mechanism. Most likely no such ultra-Eddington AGN exists.</list_item>
<list_item><location><page_26><loc_50><loc_65><loc_89><loc_79></location>· ˙ m ∼ 1: the thin disk is truncated by radiation pressure, which drives near-spherical winds out. BL3Q is predicted as the early phase of a quasar, when the radiation pressure drives all gas in the radiation cone out so that no broadline region (BLR) can be formed[179]. Actually all gas mixed with dust in the radiation cone can be blown out at substantially below Eddington rate, since the dust has a very high opacity to emissions from visible to UV. There BL3Q should be generic in the early phase of a quasar's active cycle, if the quasar activity is triggered by enhanced mass inflow consisting of a mixture of gas and dust.</list_item>
<list_item><location><page_26><loc_50><loc_59><loc_89><loc_64></location>· ˙ m ∼ 0 . 3: the thin disk extends to around ISCO and the two-phase corona is clumpy, so the jet is produced mostly via the BZ mechanism and is discontinuous. The dusty torus and BLR for in a FSRQ are not shown.</list_item>
<list_item><location><page_26><loc_50><loc_52><loc_89><loc_58></location>· ˙ m ∼ 0 . 1: the thin disk extends to around ISCO and nearequatorial wind is thermally driven out; small scale and turbulent magnetic fields may be responsible for launching the plasma out of the disk via magnetic reconnections. The dusty torus and BLR in a RQQ are not shown.</list_item>
<list_item><location><page_26><loc_50><loc_45><loc_89><loc_51></location>· ˙ m ∼ 0 . 03: the thin disk is made of an inner and outer part; the inner disk extends to around ISCO. The twophase ADAF is clumpy, so the jet produced via a mixture of the BP and BZ mechanisms is discontinuous. The dusty torus and BLR in a BL Lac are not shown.</list_item>
<list_item><location><page_26><loc_50><loc_35><loc_89><loc_44></location>· ˙ m< 0 . 01: 'Q' refers to the very low luminosity quiescent state that normally displays a hard power-law spectrum. The large scale magnetic fields rotating with the truncated thin disk and thick ADAF channel the continuous plasma to form collimated and continuous jets via the BP mechanism. There is neither dusty torus nor BLR in LLAGNs [179].</list_item>
</unordered_list>
<text><location><page_26><loc_9><loc_9><loc_49><loc_20></location>revise it inevitably. Of particular importance is the reliable determinations of the mass and spin of accreting BHs. The BH mass allows us to determine the accretion rate in units of Eddington rate, i.e., ˙ m , which is a key parameter to the unification scheme of accretion-outflow connections. The BH spin is of course another key parameter, because it can determine the radiative efficiency of the disk and jet power in the BZ mechanism.</text>
<text><location><page_26><loc_52><loc_9><loc_92><loc_20></location>Once we have well determined BH parameters and a consistent and predictable theory describing the observed accretion-outflow connections, we are then ready to study the fundamental physics of BHs with BHXBs and microquasars, such as the properties of event horizon, spacetime around Schwarzschild and Kerr BHs, BH spin energy (and mass) extraction. For example, both the broad iron line and CF fitting methods can be used to measure</text>
<text><location><page_27><loc_9><loc_79><loc_49><loc_93></location>a BH's spin. However the two methods are equivalent only if the metric is accurately described with Kerr metric. Different spin measurements for a BH with the two different methods would invalidate Kerr metric, provided that we understand the accretion disk physics thoroughly. On the other hand, a correct metric is of course required to describe accurately the accretion flow around BHs. This is one example of the entanglement or interplay between astrophysics and fundamental physics of accreting BHs. The stake is high, but the job is difficult.</text>
<text><location><page_27><loc_52><loc_77><loc_92><loc_93></location>Acknowledgment. I appreciate inputs from Profs. Lijun Gou, Weimin Gu, Lixin Li, Bifang Liu, Dingxiong Wang, Wenfei Yu, Feng Yuan, and Shu Zhang. The editors of this book are thanked for inviting me to write this article, as well as their patience, persistency, proof reading it, and offering comments and suggestions to improve it in the end. This work is partially supported with funding by the 973 Program of China under grant 2009CB824800, by the National Natural Science Foundation of China under grant Nos. 11133002 and 10725313, and by the Qianren start-up grant 292012312D1117210.</text>
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</document>
[
{
"title": "Black Hole Binaries and Microquasars",
"content": "Shuang-Nan Zhang 1 , 2 ∗ 1 Laboratory for Particle Astrophysics, Institute of High Energy Physics,Beijing 100049, China. 2 National Astronomical Observatories, Chinese Academy Of Sciences, Beijing 100012, China. (Dated: October 31, 2018) This is a general review on the observations and physics of black hole X-ray binaries and microquasars, with the emphasize on recent developments in the high energy regime. The focus is put on understanding the accretion flows and measuring the parameters of black holes in them. It includes mainly two parts: (1) Brief review of several recent review article on this subject; (2) Further development on several topics, including black hole spin measurements, hot accretion flows, corona formation, state transitions and thermal stability of standard think disk. This is thus not a regular bottom-up approach, which I feel not necessary at this stage. Major effort is made in making and incorporating from many sources useful plots and illustrations, in order to make this article more comprehensible to non-expert readers. In the end I attempt to make a unification scheme on the accretion-outflow (wind/jet) connections of all types of accreting BHs of all accretion rates and all BH mass scales, and finally provide a brief outlook.",
"pages": [
1
]
},
{
"title": "CONTENTS",
"content": "VIII. Further developments on thermal stability of SSD 23",
"pages": [
1
]
},
{
"title": "I. SYNOPSIS",
"content": "I will start by defining what I mean by black hole binaries (BHBs) and microquasars in this article. I decide to restrict myself to only a subclass of BHBs, namely, BH X-ray binaries (BHXBs), since these are the only class of BHBs known observationally. I will then simply refer microquasars as BHXBs for reasons discussed in Section II. Since many excellent, comprehensive and quite up-todate review articles on BHXBs are readily available in literature, I feel it is not necessary to write another bottomup and comprehensive review article on the same subjects at this stage. I will thus take quite an unusual approach in this article. I will first give some concise guides on several representative review articles [1-4], with some necessary updates. I will then focus on the further developments on several topics I feel deserve more discussions, i.e., BH spin measurements (Section IV), hot accretion flows (Section V), corona formation (Section VI), state transitions (Section VII) and thermal stability of SSD (Section VIII). The emphasis is thus put on understanding the accretion flows and measuring the parameters of black holes in them. Some rather general issues on BH astrophysics, such as what astrophysical BHs are and how to identify them observationally, are not discussed here but can be found from my recent book chapter entitled 'Astrophysical Black Holes in the Physical Universe' [5]. The usual practice of writing a review article is to end by listing some outstanding issues and major unsolved problems, and then to propose some possible approaches to them. I initially did not do this in the first draft. The history of astronomy tells us that major progress is almost always made by unexpected discoveries and research results; unpredictability is an essential nature of astronomy. This article is not intended to be read by funding agencies or proposal reviewers, so I thought I did not have to do it. In astronomy, knowing what has happened, but looking and doing it differently are far more important and effective than following other people's advises. However, the editors of this book suggested me to write a brief 'outlook' in the end. I thus did it nevertheless.",
"pages": [
1,
2
]
},
{
"title": "II. ACRONYMS AND TERMINOLOGY",
"content": "In Table I, I list all acronyms used in this article; most of these are quite commonly used in this community. A BH binary (BHB) is a gravitationally bound binary system in which one of the objects is a stellar mass BH with mass from several to tens of solar masses ( M glyph[circledot] ); the other object, i.e. its companion, can be either a normal star, a white dwarf, or a neutron star (NS). In case a binary system consists of two BHs, it is is referred to as a binary BH system, which is not covered in this article. When the companion in a BHB is a normal star, the gas from the star may be accreted to the BH and X-rays are produced, as a consequence of the heating by converting the gravitational potential energy into the kinetic energy of the gas, and a BH X-ray binary (BHXB) is referred to as such a binary system, as shown in Figure 1. The possible existence of BHXBs was first suggested by Zel'dovich & Novikov [6, 7]. The first BHXB found is Cygnus X-1 [8], now a well-studied system among many others found subsequently in the Milky Way and nearby galaxies. The terminology of microquasar has some twisting in it. Historically it was first referred to the BHXB 1E1740.7 -2942 in the Galactic center region, because a double-sided jet was detected from it, mimicking some quasars with similar radio lobes, which have much larger scales [9]. Soon after, superluminal jets are observed from a BHXB GRS 1915+105 [10], which is also referred to as a microquasar. Nevertheless both BHXBs are quite unusual compared to many others, thus microquasars were considered quite unusual. However it became clear that microquasars may be quite common among BHXBs, since the discovery of a normal BHXB GRO J1655 -40 [11], whose superluminal jets were observed [12] and showed some correlations with its X-ray emission [13]. Subsequently several more BHXBs have been observed with superluminal jets. At this stage microquasars began to be referred to as BHXBs with relativistic jets (with bulk motion of about or larger than 90% of the speed of light), to distinguish them from NS X-ray binaries (NSXBs) that only have mildly relativistic jets (with bulk motion of about or smaller than 50% of the speed of light) [14]. However the discovery of relativistic jets from a NSXB Circinus X-1 made the situation complicated: relativistic jets are no longer uniquely linked to BHs [15]. Now in retrospect, a microquasar can be literally and easily understood as the micro version of a quasar; however a quasar may or may not be observed with collimated jets. A quasar has been already understood as a special galaxy centered by an actively accreting supermassive BH with a mass from millions to billions of M glyph[circledot] , and thus its total light output is dominated by the BH's accretion process, in a similar way as in BHXBs. The production or lack of relativistic jets may have similar or even the same underlying physical mechanisms in BHXBs and AGNs, though their surrounding environments may modify their observed morphologies [16]. It is therefore more natural to simply refer microquasars as BHXBs; in the rest of this article, microquasar and BHXB are used interchangeably. We therefore will focus on BHXBs and thus will not discuss binary systems producing gamma-rays and sometimes radio jets, which are most likely high-mass NSXBs and in which jets or pulsars' winds interact with the wind of its high-mass companion to produce the observed gamma-rays [17-23]. Such systems show very different observational characteristics, e.g. the long (1667 days) super-orbital modulation with phase offset (about 280 days) between its X-ray and radio light curves found in LS I+61 · 303 [24].",
"pages": [
2,
3
]
},
{
"title": "III. REVIEW OF REVIEWS",
"content": "Here I attempt to review the four recent review articles [1-4] I consider most useful to readers. Additional information and updates are provided when necessary. Some overlaps exists between these review articles, as expected and inevitable in bottom-up review articles. To avoid repetitions as much as possible in this article, I thus put different emphasizes on different articles, with of course my personal tastes and perceptions.",
"pages": [
3
]
},
{
"title": "A. The most recent Science Collection",
"content": "The recent collection of perspectives and reviews in the Science magazine provides excellent introductions to and concise summaries of the current state of our understanding of BH physics and astrophysics [1, 25-27]. To the subjects of this article, the most relevant article in this collection is the one by Fender and Belloni enti- tled 'Stellar-Mass Black Holes and Ultraluminous X-ray Sources', which is focused on the observational characteristics of BHXBs [1]. In particular, the article provides an excellent description of the general picture of spectral evolutions of BHXBs with the Hardness-IntensityDiagram (HID), which is found to be well correlated with the observed jets from BHXBs, as shown in Figure 2. Actually this cycle is also well tracked by their flux variability, represented by the measured root-mean-squares (rms) above its average flux, as shown by the RMSIntensity-Diagram (RID) [28] of the BHXB GX 339-4 in Figure 3, which also shows additional horizontal tracks at intermediate intensities. Sometimes, a full HID cycle does not go into the soft state at all (Figure 4), perhaps due to a failed outburst [29]. The basic scenario is as follows. During the initial stage of an X-ray outburst of a BHXB (A → B), which is triggered by a sudden increase of accretion rate onto the BH, its spectrum is normally hard and steady jets are always observed. After reaching its peak luminosity, its spectrum begins to soften in a chaotic way and transient jets are normally observed (B → C → D). After this transition, the system calms down with a soft spectrum and no jets are present (D → E). Finally the system returns to its quiescent state with a hard spectrum accompanied with the reappearance of jets (E → F). Throughout this cycle, the presence of hot accretion disk winds appears to be anti-correlated with its spectral hardness and jet ejection. This empirical pattern appears to be quite universal for all BHXBs with very few exceptions, though the underlying physics is still not well understood yet. Nevertheless putting together the above scenario is a very significant progress in this field over the last ten years. Due to the conciseness of this article, some other important subjects on BHXBs are not fully discussed and many original references are also missing. Other articles in this collection [25-27] are less relevant to the subjects of this article, but are still quite interesting to read, within the context of BH astrophysics and physics. The only other type of astrophysical BHs known to exist in the physical universe are supermassive BHs in the center of almost each galaxy. Volonteri [27] concisely summarized our current understanding on how they are formed and grow over the cosmic time; merging of two BHs is a key process here. Thorne [25] focused on what happens when BHs merge together to produce gravitational waves, which might be used as a new laboratory for studying gravitational physics and a new window for exploring the universe. Witten [26] then explained the quantum properties of BHs, in particular the basic ideas behind Hawking radiation, which might not be important at all for astrophysical BHs. Nevertheless the understanding gained through studying the quantum mechanics of BHs actually plays very important roles in developing other theories of physics, such as that in heavy ion collisions and high-temperature superconductors, as vividly described by Witten [26].",
"pages": [
3,
4
]
},
{
"title": "B. The most recent ARA&A article on BHXBs, with some updates",
"content": "The most recent ARA&A article on BHXBs by Remillard and McClintock entitled 'X-Ray Properties of Black-Hole Binaries' [2] (referred to as RM06 hereafter) provides the most complete, comprehensive and accurate review of BHXBs, which actually covers subjects much beyond just the X-ray properties of BHXBs. The Introduction of RM06 highlights the initial theoretical and observational studies of BHXBs, followed by brief comments on several main review articles on BHXBs preceding this one and the basic properties of BHs within the context of general relativity (GR). All main properties of BHXBs known at the time are summarized in the first table and figure there. In Table II, I compile the most updated data on all BHXBs currently known, including three BHXBs outside the Galaxy and not in the Large Magellanic Cloud (LMC); the currently available spin measurements for these BHs are also included for completeness. In Table II, where P orb is the orbital period, K C is the semiamplitude of the velocity curve of the companion star, M BH is BH mass, i is the the orbital inclination angle, and q ≡ M C /M BH is the mass ratio. In Figure 5, I show the updated graphical representation of most of the BHXBs listed in Table II. The X-ray properties of BHXBs are characterized by their X-ray light curves, timing and spectra. Essentially all known BHXBs were discovered initially as bright Xray sources, and the majority of them were detected as transient X-ray sources with X-ray all-sky monitors. The transient properties of some of BHXBs can be interpreted by the disk instability model (DIM) (see Ref.[49] and references therein), which assumes a constant mass transfer rate from the mass donor to the accretion disk. However the accretion rate from the disk to the compact object, i.e. a white dwarf (WD), a neutron star (NS), or a BH, is normally lower than the mass supply rate in the disk, so mass is accumulated in the disk. When the accumulated mass exceeds a certain critical value, a sudden increase of accretion rate results in a nova-like outburst. DIM is most successful in explaining an outburst with fast rise and exponential decay (FRED). However as shown in RM06, many observed X-ray light curves of BHXBs a : Reported in the first paper on systematic BH spin measurements [47]; b : Reported in the most recent literature; c: Postulated to be extreme retrograde Kerr BH, due to the lack of the thermal disk component above 2 keV (GRO J1719-24 also belongs to this class) [47]; d : Postulated to be non-spinning BH, due to the observed low disk temperature; e : BH mass of 7 M glyph[circledot] assumed [47]; f : Based on the inner disk radius decrease by a factor of two from the hard state to the soft state transition [47]. are far more complicated than just FRED and their recurrent time scales are also not compatible with DIM. Perhaps disk truncation (a subject to be discussed extensively more later) and mass transfer instability are additional ingredients [50]. X-ray emission from a BHXB is variable at all time scales, from the free-fall or Keplerian orbital time scale of milliseconds near the BH, to the various oscillations (some are even related to GR) in the disk with time scales from milliseconds to minutes, to the viscous time scales of minutes, and to the various instabilities of different time scales. Therefore timing studies of BHXBs can probe the geometry and dynamics in BHXBs. However, lack of coherent signals, such as that observed from pulsars, makes it difficult to unambiguously identify the underlying mechanisms from the detected X-ray variabilities. Nevertheless power density spectra (PDS) and rms still allow us to make progress in understanding the general characteristics of a BHXB, in particular when combined with its spectral behaviors, as already discussed briefly above. The most successful understanding of BHXBs so far is the description of their thermal X-ray spectral component by the classical Shakura-Sunyaev Disk (SSD) model [51]. After applying GR to SSD, the temperature distribution in the disk can be obtained [52]. Applying this model to BHXBs, one can even measure the spin of their BHs [47], by assuming that the inner accretion disk boundary is the inner-most stable circular orbit (ISCO) of the BH (a subject to be discussed more in Section IV). However a power-law (PL) component is almost ubiquitous in the spectra of BHXBs. The interplay between these two components results in various spectral states, which are also found to be well correlated with their timing properties. Historically these spectral states have been named in many different ways, reflecting mostly how they were identified with the observations available at the times. In RM06, three states are defined, which show distinctively different spectral and timing behaviors, as shown in Figure 6 based on the RXTE data on the BHXB GRO J1655-40. The thermal state has its X-ray spectrum dominated by the thermal disk component and very little variability. The hard state has its X-ray spectrum dominated by the a PL component and strong variability. The steep power-law (SPL) state is almost a combination of the above two states, but the PL is steeper. I will keep using these definitions throughout this article for consis- tency. Please refer to Table 2 in RM06 for quantitative descriptions of these three states. When a BHXB is in quiescence, i.e. not in an outburst, its spectral shape is similar to the hard state spectrum. Referring back to Figure 2, RM06 found that the HID can be reasonably well understood in terms of transitions between the above three states; I summarize this in Table III. After examining the extensive RXTE data on spectral evolutions of six BHXBs, the above conclusion seems to be valid generally. Thus a coherent picture seems to emerge: jets are produced only when the PL component in the spectrum is strong and winds are present only when the PL component disappears; jets and winds appear to be mutually exclusive. This suggests that transient winds should appear in the SPL state, anti-phased with transient jets, if winds quench jets as suggested recently [53, 54]. The robust link between the PL component and jets provides an important clue to the jet production mechanism. The PL component is believed to be produced in an optically thin and geometrically thick corona, in which hot electrons up-scatter the thermal photons from the disk. As discussed in RM06, the inner region of the disk appears to be truncated quite far away from the BH in the hard state; this should be the primary reason for the low luminosity of this state, as evidence for the existence of BHs in BHXBs. However it is still not understood how the corona is formed; I will discuss this later in Section VI. In the final part, RM06 discussed the exciting possibilities of using BHXBs as probes of strong gravity, including confirming these BHXBs contain true BHs, measuring the spins of BHs, relating BH spin to the Penrose process and other phenomena, and finally carrying out tests of the Kerr Metric. They also discussed in length how to measure the spins of BHs and commented the various methods of doing so. In Section IV I will re-visit the BH spin measurements in details. Lacking of a solid surface, a BHXB is not expected to produce and actually never observed to have coherent pulses. However quasi-periodic oscillations (QPOs) are frequently detected in their X-ray light curves; a QPO is defined as a 'bump' feature in the PDS, if Q = ν/ ∆ ν > 2, where ν and ∆ ν are the peak frequency and FWHM of the bump, respectively. The observed QPOs are further divided into low frequency (LF) (0.1-30 Hz) and high frequency (HF) ( > 30 Hz) QPOs. Typical LFQPOs can be found as the peak around several Hz in the top and bottom PDS of Figure 6. In literature, LFQPOs are further divided into several subclasses and their underlying mechanisms are far from clear at this stage, though many models have been proposed and briefly discussed in RM06. HFQPOs have been detected from several sources at 40-450 Hz. Of particular interests are their stable nature when detected, which may be linked to either the mass or spin or both of the BH in a BHXB. In some cases a 3:2 frequency ratio is found. Although the 3:2 HFQPO pairs could be interpreted in some epicyclic resonance models [55, 56], there remain serious uncertainties as to whether epicyclic resonance could overcome the severe damping forces and emit Xrays with sufficient amplitude and coherence to produce the HFQPOs. A revised model applies epicyclic resonances to the magnetic coupling (MC) of a BH's accretion disk to interpret the HFQPOs [57]. This model naturally explains the association of the 3:2 HFQPO pairs with the steep power-law states and finds that the severe damping can be overcome by transferring energy and angular momentum from a spinning BH to the inner disk in the MC process.",
"pages": [
4,
5,
6,
7
]
},
{
"title": "C. Two comprehensive and long articles on modeling accretion flows in BHXBs",
"content": "Page limitations to the above review articles did not allow in depth discussions on the detailed processes and models of accretion flows in BHXBs, which are responsible for the above described energy spectra, PDS, state transitions, jets, and wind. Here I introduce two comprehensive and long articles just on this, by Done [3] and Done, Gierli'nski & Kubota [4].",
"pages": [
7
]
},
{
"title": "1. A beginner's guide",
"content": "The first is intended to readers who just start research in this field [3]. It started with the basic tools in plotting spectra and variability, and then described the basic ideas and methods used to infer the inner accretion disk radius when a BHXB is in a thermal state, in order to measure the BH spin [47]. A useful introduction is made on how to make various corrections to account for various effects, including color correction, special and general relativistic effects, starting from the original work [47]. Several commonly use fitting models, i.e. DISKBB , BHSPEC and KERRBB , in the XSPEC package are also briefly introduced. An example is given to demonstrate that the expected relation L disk ∝ T 4 in [47] agrees with the data from the BHXB GX 339-4 [58], where L disk and T in are the disk's total luminosity and temperature at the inner disk boundary, respectively. The hard state is then briefly touched upon, using the Advection Dominated Accretion Flow (ADAF) model [59]. ADAF naturally explains the hot corona required by the observed PL component, especially at very low accretion rate. Evaporation at low accretion rate in the inner disk region has been proposed as the mechanism producing a geometry with a radially truncated disk and a hot inner flow; the latter might be the ADAF [60]. This also means that when a significant PL component is present in the spectrum, BH spin measurement cannot be done with the inferred inner disk boundary (see, however, counter evidence discussed in Sections IV and VI). This geometry is considered a paradigm that can account for many of the observed diverse phenomena from spectral evolution to timing properties. A brief, yet interesting discussion is given on scaling up the above models to Active Galactic Nuclei (AGNs), which host actively accreting supermassive BHs in the centers of galaxies. Applying the insights learnt from BHXBs on their spectral evolution, changing disktemperature with accretion rate and BH mass, and diskjet connections, one might be able to understand many phenomena beyond the simple AGN unification scheme, in which the different observational appearance is all attributed to an viewing angle difference. The continuum emissions of both the disk thermal and PL components are modified by absorptions and added by additional spectral features along the line of sight (LOS). Absorptions in neutral media produce various photoelectric absorption edges, but in ionized media result in both absorption edges and lines. In addition to recovering the original X-ray emission of a BHXB, modeling the absorption features is important in learning the physical properties of the absorption media, such as column density, ionization and velocity along LOS. Winds from BHXBs discussed above are always detected this way. Several XSPEC fitting models for various kinds of absorption edges and lines are also introduced here, i.e. TBABS , ZTHABS , TBVARABS , ZVPHABS , TBNEW , ABSORI and WARMABS . Figure 7 shows the calculated absorption structures with ABSORI and WARMABS ; note that ABSORI does not include absorption lines, but WARMABS does. X-rays interacting with the surrounding medium can ionize it and heat the free electrons up by Compton scattering; hot electrons can also loose energy by interacting with lower energy photons. A Compton temperature of the plasma is reached when the above two processes reaches an equilibrium. The temperature is determined by only the spectral shape of the continuum, so the heated plasma can escape as winds from the disk if the velocity of the ions exceed that of the escape velocity at that radius, which is usually very far away from the inner disk boundary. As the continuum luminosity approaches the Eddington limit, the radiation pressure reduces the escape velocity substantially in the inner disk region such that continuum driven winds can be launched almost everywhere in the disk of a BHXB, forming a radiation driven wind, which is also called thermal wind; alternatively winds may also be driven magnetically in the inner disk region, but this is much less understood yet. In contrast, in an AGN the peak continuum emission is in the UV band, which has a much larger opacity than Compton scattering in neutral or weakly ionized medium during both photoelectric and line absorptions. This means the effective Eddington luminosity is reduced by large factors and UV line driven wind is easily produced at high velocity. This explains the relatively lower velocity (hundreds km/s) and highly ionized winds from BHXBs, but much higher velocity (thousands km/s to 0.2 c ) and weakly ionized winds from AGNs. Material illuminated by X-rays produce both fluorescence lines and reflection features, which depends on both the continuum spectral shape and ionization state of the material, as shown in Figure 8 calculated with the XSPEC ATABLE model REFLIONX.MOD, which includes the self-consistent line and recombination continuum emission. Note that for highly ionized reflection, the Compton heated upper layers of the disk broadens the spectral features; these effects are not included in the simpler XSPEC model PEXRIV . Replacing the stationary slab by a disk around a BH, both the special and general relativistic effects will further smear (broaden) the spectral features, as shown in Figure 9 for the iron line region with different values of ξ , inner disk radius r in , and viewing angle i ; these are calculated with the REFLIONX.MOD and then convolved with KDBLUR in the XSPEC package. Other similar XSPEC fitting models are DISKLINE , LAOR , and KY . The inner disk radius can in principle be determined by modeling the observed broad iron line features, that in turn can be used to measure BH spins, as will be discussed briefly in Section IV. Figure 10 shows all components of a broad band spectrum of a BHXBs, including interstellar absorption. We thus have two ways discussed so far to determine r in of BHXBs by either fitting its thermal continuum or the fluorescent iron line feature. However results are not always consistent, obtained with different methods, even using the same data by different authors. The author believes that r in increases at low accrete rate, to make room for the corona occupying this space. Nevertheless a fairly good summary on the current conflicts and confrontations is made on this issue, which I will discuss further in Section IV.",
"pages": [
7,
8,
9
]
},
{
"title": "2. An expert's handbook",
"content": "The second one can be considered as a handbook on accretion flows on BHXBs, and really lives up to its subtitle 'Everything you always wanted to know about accretion but were afraid to ask' [4] (DGK07 hereafter). Besides its much longer length of 66 pages, the main difference from the above review articles is that it is focused on confrontations between theories and observations and intends to depict a coherence picture of the accretion physics in BHXBs. In the following I will summarize briefly the main points and conclusions reached in DGK07; those I have reviewed above and will discuss more later will be skipped for brevity. The underlying physics of DIM for triggering the outbursts in BHXBs discussed above is the hydrogenionization instability, which produces the so-called 'S'curve, as shown in Figure 11; irradiation by the inner hot disk to keep the outer disk hot is required to produce the slow flux decays, e.g. the exponential decays, frequently observed in them. The outer disk radius is obviously another key parameter, which is determined by the tidal instability in a binary system. This mechanism can explain why BHXBs with high mass companions are all persistently bright, since their outer disks are always in the upper branch due to the combination of their higher average mass transfer rate and inner disk irradiation. Similarly it also explains the differences and similarities between the light curve properties of neutron star X-ray binaries (NSXBs) and BHXBs, since a NS has a lower mass. It should be noted that the additional surface emission from the NS may also help to maintain the outer disk hot and stay in the upper stable branch [62]. The SSD prescription assumes that the stress is proportional to pressure. Because the gas pressure is P gas ∝ T but radiation pressure is P rad ∝ T 4 , so a small temperature increase causes a large pressure increase when P rad ≥ P gas , and thus large stress increase, which in turn heats the disk even more. The opacity cannot decrease effectively to cool it down, so the disk becomes unstable when L ≥ 0 . 06 L Edd , where L Edd is the Eddington luminosity of a BHXB. However in most BHXBs their disk emissions appear to be stable up to around 0 . 5 L Edd . One way out is to assume that the stress is proportional to √ P rad P gas ∝ T 5 / 2 , so the stress increases slower in the radiation pressure dominated regime. Beyond this the disk should become unstable, as evidenced by the sometimes 'heart-beat' bursts of the super-Eddington BHXB GRS 1915+105; other BHXBs with L max > L Edd (e.g. V404 Cyg and V4641) were not observed with such instabilities, perhaps due to the combination of insufficient observational coverage and sensitivity. However similar 'heart-beat' bursts were also observed re- IGR J17091-3624, which is likely substantially sub-Eddington unless it is located much beyond 20 kpc and/or its BH mass is quite small [63]. Super-Eddington accretion flow (SEAF) can become stable again, if the radiation instability is overcome by an optically thick ADAF, i.e. the slim disk model, in which the trapped photons in the flow is advected inwards, thus balancing the heating generated by viscosity. Strong radiation driven winds can be easily produced; this can happen in BHXBs and NSXBs (e.g. Z-sources). Evidence exists that truncated inner disk (TID) is quite common in SEAF and outflow even dominates over inflow in SEAF [64]. The observed PL component in BHXBs cannot be explained by the SSD-like models and thus requires a hot accretion flow (HAF). At low accretion luminosity (e.g. the quiescent state), the HAF may be the advection dominated accretion flow (ADAF), the convection dominated accretion flow (CDAF), or the advection dominated accretion inflow/outflow solution (ADIOS). At higher luminosity (e.g. the hard state), the original hot and optically thin disk solution (i.e. the SLE solution) is unstable, because the electron heating efficiency by the Coulomb coupling between protons and electrons is too low. The luminous HAF (LHAF), however, has the advection as a heating source to electrons, so the heating efficiency increases and thus electrons can cool the flow more effectively. Outflows can be produced in ADAF/ADIOS; collimated jets can also be produced if magnetic fields are involved, so an accretion flow may even be jet dominated (i.e. JDAF). The interplays between the SSD and HAF may be responsible for the observed different states in BHXBs discussed above. If the PL component is produced by Compton up-scattering, then the combination of the optical depth τ and L h / L s , where L h is the heating power in electrons and L s is the cooling power in seed photons, can describe the observed variety of spectra. For example, the hard state has L h / L s glyph[greatermuch] 1, but the thermal and SPL state have L h / L s ≤ 1. The location of r in is proposed to be closely related to L h / L s , as illustrated in Figure 12. As discussed above, the steeper PL in the thermal and SPL states is mostly non-thermal in nature, so non-thermal Comptonization is required. Detailed comparisons with data suggest thermal Comptonization also cannot be ignored even in the thermal and SPL states. In Figure 12, the hot inner flow (HIF) and patchy corona are responsible for thermal and non-thermal Comptonization, respectively. When a source transits from hard to SPL state, r in decreases, HIF is reduced but the patchy corona becomes dominant. The transition from SPL to soft state is then marked by the disappearance of HIF and significant reduction of the patchy corona. The above scenario obviously depends on two fundamental assumptions: (1) slab-HIF produces the non-thermal PL component; (2) slab-HIF is mostly located between the TID and the BH. Both assumptions are examined exhaustively based on the existing observations and their spectral modeling. Slab-HIF is found to be consistent with essentially all data. An alternative to slab-HIF is that the PL is produced from the jet base and beamed away from the disk; however observations suggest the PL emission is quite isotropic, thus in conflict with this. TID is found also consistent with data when the PL component becomes important, since the observed L disk ∼ T 4 in deviates significantly from a linear relation, suggesting r in is larger when the PL component becomes important. However some studies showed that r in is unchanged in the initial hard state, if the Compton up-scattering process is treated properly in Monte-Carlo simulations to recover the lost disk photons [65]; this issue will be further discussed in Section IV. The above developed HIF/TID model based on the spectral evolutions of BHXBs can also be applied to explain the majority of time variability PDS of BHXBs consistently. The TID acts as a low pass filter as it cannot response effectively to variations in the HIF, so r in controls the low frequency break of the PDS: ν LFB ∼ 0 . 2( r/ 6) -3 / 2 ( m/ 10) -1 Hz, where r = r in /r g and m = M BH /M glyph[circledot] ; this predicts that ν LFB changes from 0.03 to 0.2 Hz as observed during transitions from the hard to SPL and soft states if r decreases from 20 to 6. The HIF/TID model can also explain the observed LFQPO variations with luminosity, if the LFQPOs are some kinds of characteristic frequencies related to r in . To some extend, the shape of the PDS can also be explained by this model. However the observed variability rms ∼ flux linear correlation and log-normal distribution of fluctuations may need additional ingredient, such as the proposed propagation fluctuation model. However I point out that such correlations and distributions are also observed in the light curves of gamma-ray bursts and solar flares [66, 67], which can be produced by the generic self-organized criticality mechanism [66]. Similar rms ∼ flux correlation has also been found for blazars, in agreement with the minijets-in-a-jet statistical model [68]. The spectral and timing properties of weakly magnetized NSXBs are known to have many similarities and differences from BHXBs; this is particularly true for the atoll sources that have similar ranges of L/L Edd to BHXBs. The essential distinction is that a NS has a solid surface, but a BH does not. Observationally the continuum spectra of atoll sources can be modeled as composed of SSD, PL emission, and blackbody emission from the NS surface (or the boundary layer between the TID and NS surface); the former two components are quite similar to BHXBs. The spectral evolutions of these NSXBs are thus driven similarly by the combinations of τ and L h / L s . However the blackbody emission is an additional source of L s , so the PL is not as hard as that in BHXBs and r in variations are less effective in changing L h / L s ; the latter means it is more difficult to find evidence of TID from modeling only the spectral evolutions in NSXBs. On the other hand, the TID/HIF model in NSXBs can produce timing behaviors in the same way as in BHXBs discussed above, consistent with observations; additional timing behaviors, such as coherent X-ray pulsations and kilo-Hz QPOs observed in these NSXBs, are caused by the rapid spins of the hard surfaces of the NSs.",
"pages": [
9,
10,
11
]
},
{
"title": "IV. FURTHER DEVELOPMENTS ON BH SPIN MEASUREMENTS",
"content": "A BH predicted in GR can only possess three parameters, namely, mass, spin and electric charge, known as the so-called BH no hair theorem. Even if a BH was born with net electric charge, its electric charge can be rapidly neutralized by attracting the opposite charge around it in any astrophysical setting, because the strength of electromagnetic interaction is many orders of magnitude stronger than that of gravitational interaction. Therefore an astrophysical BH may only have two measurable properties, namely, mass and spin, making BHs the simplest macroscopic objects in the universe. Practically, only Newtonian gravity is needed in measuring the BH mass in a binary system. However, GR is needed in measuring the BH spin. The mass and spin of a BH has different astrophysical meanings. Its mass can be used to address the question of ' How much matter (and energy) has plunged into the BH?'. However its spin can be used to address the question of ' How did the matter (and energy) plunge into the BH?'. This is because matter and energy plunged into a BH can carry angular momentum, which is a vector with respect to the spin axis of the BH. In order to increase the total gravitating mass-energy from M i with zero spin to M f , the added rest-mass must be [69, 70] and its final spin becomes where J is the BH's angular momentum. Clearly we have a ∗ = 1 when M f /M i = 6 1 / 2 ; further accretion simply maintains this state [70]. Therefore the required additional rest-mass to spin a BH from zero to maximum spin is ∆ M glyph[similarequal] 1 . 85 M i = 0 . 75 M f ; this is a lower limit to the accreted mass [71]. Figure 13 shows a ∗ as a function of ∆ M . Ignoring Hawking radiation of a macroscopic BH, the only way to extract the energy of a BH and thus reducing its gravitating mass is by extracting its spin energy. Recently, evidence of BH spin energy extraction to power relativistic jets has been found, from the observed correlation between the maximum radio luminosity and its BH spin of a microquasar [72, 73]; however the average jet power is not correlated with BH spin [74]. This indicates that jets may be produced by both BlandfordPayne (BP) [75] and Blandford-Znajek (BZ) [76] mechanisms; but the BZ mechanism is more powerful and responsible for producing the peak radio luminosity. This provides another possible way to estimate a BH's spin [73], similar to a recent proposal of using the peak luminosity of the disk emission to estimate a BH's spin [77]. However there is so far no independent demonstration of validness of either of the above two new methods, which are thus not discussed further in this section. Figure 14 shows the mass reduction, ∆ m = ( M max -M f ) /M max ( M max and M f are the BH's maximum mass and final mass, respectively), as a function of extracting efficiency glyph[epsilon1] of an extreme Kerr BH. For a 10 M glyph[circledot] BH with glyph[epsilon1] = 1 (∆ m ∼ 0 . 3), the total extracted energy is ∆ mM f c 2 glyph[similarequal] 10 54 erg and its total gravitating mass is reduced to M max / √ 2 [71]; this energy could be sufficient to power gamma-ray bursts (GRBs). It is thus plausible that supercritical accretion onto a newly born BH may spin it up and extract its spin energy to power ultrarelativistic jets; multiple spin-up and spin-down cycles may also happen during one GRB, if the collapsing material is clumpy. As shown in Figure 15, the radius of the ISCO of the BH, R ISCO , is a monotonic function of the BH spin [78], beyond which radius a test particle will plunge into the BH under any perturbation; however, in Newtonian gravity a stable circular orbit can be found at any radius. It is thus reasonable to assume that the accretion disk around a BH terminates at this radius, i.e., r in = R ISCO . Therefore, a ∗ can be inferred if one can measure R ISCO in units of its gravitational radius r g = GM/c 2 . Currently three methods have been proposed to measure the BH spin in BHXBs, and all these methods rely essentially on measuring R ISCO . In case the radiative efficiency ( η ≡ L/ ˙ Mc 2 ) can be measured, a ∗ can also be determined this way, as shown in Figure 16. Actually η is a very simple function of R ISCO , i.e, η ∼ 1 /R ISCO , as shown in Figure 17.",
"pages": [
11,
12
]
},
{
"title": "A. The first black hole spin measurement with X-ray spectral continuum fitting",
"content": "In 1997, I and my colleagues proposed a method of measuring the BH spin in BHXBs. It started when we tried to measure the mass of the BH in GRO J1655-40 [11] by measuring r in from its X-ray continuum spectral fitting and assuming the BH is not spinning; we found the mass of the BH is around 4 M glyph[circledot] [79]. Coincidentally just when this paper was going to press, Orosz and Bailyn [80] announced their accurate measurement of the BH mass of GRO J1655-40, which is significantly larger than what we found. A co-author of our paper, Rashid Sunyaev urged me to resolve this apparent discrepancy. I immediately realized that a spinning BH of 7 M glyph[circledot] in GRO J1655-40 would be consistent with the inferred with Xray data, and added a note in proof in the paper suggesting this possibility [79]. This turns out to be the first BH spin measurement on record. I then invited two of my close friends and collaborators, Wei Cui and Wan Chen, to join me to apply this method systematically to other BHXBs; a new main conclusion in this work was that the first microquasar in the Milky Way, GRS 1915+105, also contains a spinning BH [47]. At the time DISKBB was the only available fitting model in the XSPEC package for determining r in from an observed X-ray continuum spectra, if the disk contin- uum is described by the SSD model: L disk = 4 πσr 2 in T 2 in , where T in in DISKBB is the disk temperature at r in (i.e. the peak disk temperature in SSD) and L disk can be calculated from the disk flux (after the correction to absorption), the distance to the source and disk inclination (an issue to be discussed later). To determine the physical inner disk radius from the DISKBB parameter r in , several effects must be considered: (1) electron scattering in the disk modifies the observed X-ray spectrum; (2) the temperature distribution in the disk is not accurately described by the Newtonian gravity as assumed in SSD; (3) the observed temperature distribution is different from the locally emitted one; (4) the observed flux is different from the locally emitted one. The latter three are all due to GR effects [52]. For each of the above effects, we introduced a correction factor, using the best available knowledge at the time. Since then, several improvements have been made to correct for these effects and this continuum fitting (CF) method is now quite mature in making accurate BH spin measurements, given sufficiently high quality X-ray continuum spectral measurements and accurate system parameters of the observed BHXB.",
"pages": [
12,
13
]
},
{
"title": "B. Further developments and applications of the continuum fitting method",
"content": "This CF method of measuring BH spin has since been applied widely to essentially every BHXB with a well measured X-ray continuum spectrum showing a prominent thermal accretion disk component. In particular, this method has been improved and incorporated into the widely used X-ray spectral fitting package XSPEC, e.g., KERRBB [81], BHSPEC [82], and KERRBB2 [35, 37]. Both the KERRBB and BHSPEC are relativistic models, but they have their own drawbacks and advantages (See Ref.[35] for a detailed comparison). KERRBB includes all the relativistic effects, but it requires to fix the spectral hardening factor. In contrast, BHSPEC could calculate the spectral hardening factor on its own; however, it does not include the returning radiation effect, which turns out to be an important factor in BH spin determination in BHXBs. KERRBB2 combines both models by generating the spectral hardening factor table from BHSPEC and using the table as the input for KERRBB . The research group led by Ramesh Narayan of Harvard University, Jeffrey McClintock at Smithsonian Astronomical Observatory (SAO), and Ronald Remillard of Massachusetts Institute of Technology (MIT) [83] has since applied this method and contributed to most of the BH spin measurements available in the community, as shown in Table II. The CF method relies on two fundamental assumptions: (1) The measured r in is uniquely related to R ISCO of the BH. (2) There is no or negligible X-ray radiation from the plunging matter onto the BH beyond R ISCO . The latter has been studied with numerical simulations that include the full physics of the magnetized flow, which predict that a small fraction of the disks total luminosity emanates from the plunging region [84]. However, in the context of BH spin estimation, it has been found that the neglected inner light in the CF method only has a modest effect, i.e., this bias is less than typical observational systematic errors [85, 86]. The first assumption above requires that the measured r in remains stable as a BHXB changes its spectral state and luminosity. However it was noticed that r in measured is usually much smaller, sometimes even smaller than R ISCO of a extreme Kerr BH in a prograde orbit, when the X-ray spectrum contains a significant hard PL component, which is believed to be produced by inverse Compton scattering of the thermal disk photons in a hot corona. We realized that the inferred smaller r in could be due to the lost thermal disk photons in the scattering process. We then investigated this problem and confirmed that the inferred r in can be made consistent with r in inferred from the thermal disk component dominated spectrum, if the scattered photons are recovered properly by doing detailed radiative transfer in the corona [65]; the same conclusion was also reached by the Harvard/SAO/MIT group independently without knowing our much earlier results [87]. Therefore the method of BH spin measurement by X-ray continuum fitting can also be applied to some SPL state with strong PL component. The stable nature of the measured r in is proven with the textbook case of LMC X-3, when its X-ray luminosity varied over more than one order of magnitude observed in nearly two decades with many different X-ray instruments, as shown in Figure 18 [88]. However, when ordered by the observed disk luminosity l D = L disk /L Edd , the measured r in shows a clear increasing trend when l D > 0 . 3, as shown in Figure 19 [88]. The similar trend has also been found in another BHXB GRS 1915+105 [35] and two NSXBs [64, 89]. It was found that r = r in /r g indeed increases physically when l D > 0 . 3, by comparing the evolution of r as a function of l D over a large range for several BHXBs and NSXBs. Using the blackbody surface emissions of the NSs in these NSXBs is critical in evaluating any possible disk thickening due to high luminosity that would block at least part of the NS surface emission, as well as determining the actual NS mass accretion rate, which turns out to be much less than the disk mass accretion rate; this suggests that the increased radiation pressure is responsible for the increase of r and significant outflow when l D > 0 . 3 [64]. The same trend is much more pronounced in the super-Eddington accreting ultra-luminous X-ray source NGC1313 X-2, as shown in Figure 20 together with the data from other BHXBs and NSXBs [90]. However the exact value of r obtained this way should be taken with caution, since the non-negligible energy advection at high accretion rate can modify the disk structure in non-trivial ways, thus making the SSD prescription inaccurate in this case [91]. Figure 20 also shows that as l D decreases, r again starts to increases. However r increases at higher l D and with a different slope for a NSXB than for a BHXB, which can be naturally explained as due to the 'propeller' effect of the interaction between the NS's magnetosphere and its accretion disk [92] and the 'no-hair' of the BH [90]. Figure 21 shows the radiative efficiencies of various systems; BHXBs may have either higher or lower efficiencies than NSXBs, because a BH has neither solid surface nor magnetic field [5]. It is interesting to compare the BH spin results in our first paper [47] and the most recent literature for the same BHXBs as listed in Table II: (1) For GRS 1915+105, A0620-003 and LMC X-3, both results are fully consistent; (2) For GRO J1655-40, the original result points to an extreme Kerr BH ( a ∗ ∼ 0 . 93) [47], somewhat different from the most recent result of mildly spinning BH ( a ∗ = 0 . 65 -0 . 75) [32]. However, the original result was obtained using the BH mass of 7 M glyph[circledot] , about 10% larger than the currently best estimate that was used to obtain an updated BH spin in the most recent literature [32]. From Figure 15, it can be seen that a ∗ would be decreased from 0.93 to 0.87, if the BH mass is decreased by about 10%; actually a ∗ = 0 . 85 is also allowed in the new estimate [32]; (3) For Cygnus X-1, our original conclusion that a ∗ = 0 . 75 in the high/soft state and a ∗ = -0 . 75 in the low/hard state was based on the assumption that r in decreased by a factor of two when the source made a transition from its normal low/hard state to the unusual high/soft state [94]. However the more realistic constraint is that r in changed by more than a factor of 1.8-3.2 during the state transition [94], which implies that a ∗ switched from | a ∗ | > 0 . 85, fully compatible with the latest result of a ∗ > 0 . 95 in the high/soft state [37]. The above would suggest that a very low temperature disk component exists in its hard state spectrum, if Cygnus X-1 indeed harbors a Kerr BH and the accretion disk switches from a retrograde mode to a prograde mode when it makes the hard-to-soft state transition. For a supermassive BH in the center of a galaxy, it is well understood that its mass is mostly gained through accretion in its AGN phase [95]. Therefore random accretion (between prograde and retrograde modes) tends to make its final BH spin close to zero, regardless its initial BH spin. For Cygnus X-1, its current BH mass of around 10-20 M glyph[circledot] cannot be gained through post-formation accretion, since its observed average mass accretion rate ˙ M ∼ 2 × 10 -9 M glyph[circledot] /yr and the age of its companion is much less that 10 8 yr; actually the age of the companion is estimated to range from 4.6 and 7.8 million years [96]. This means that its post-formation BH mass growth is much less than a fraction of its current mass and thus its current BH spin must be quite close to that at birth. Even if accreting at Eddington rate, to grow its BH spin from 0 and the final mass to be the current observed value, the timescale is around 3 . 1 × 10 7 yr and the accreted mass is roughly 7.3 M glyph[circledot] [37]. If its accretions al- tes between prograde and retrograde modes, then its current BH spin should be even closer to its initial spin than that in low mass BHXBs that only stay in one accretion mode due to roche-lobe overflows. Therefore the current high BH spin in Cyg X-1 must be natal; this conclusion is also true for the other highly spinning BHs in other BHXBs [48]. In summary, BH spin in BHXBs can now be measured reliably with the CF method, when the luminosity of a BHXB is between ∼ 0 . 02 and ∼ 0 . 3 in Eddington unit and their system parameters are well-known a prior . The observed thermal disk spectrum can be modeled directly to obtain the BH spin with the available KERRBB2 model in XSPEC when the X-ray spectrum in dominated by this component, i.e., the source is in the thermal state. When a significant power-law component is present, the inverse Compton scattering process must be taken into account to recover the disk photons scattered into this PL component, with for example the SIMPL/SIMPLR model [37, 97] in XSPEC. Recently another way to measure BH spin using the outburst properties of BHXBs has been proposed [77], which is mentioned briefly in Section VIII; however the effectiveness of this method needs to be tested.",
"pages": [
13,
14,
15,
16
]
},
{
"title": "C. Uncertainties of the continuum fitting method",
"content": "In spite of the tremendous progress made so far on BH spin measurements in BHXBs, most of these BH spin measurements suffer from considerable uncertainties. Actually the major source of these uncertainties comes primarily from the uncertainties in their BH masses, accretion disk inclination angles, and distances. The accurate BH mass measurement is required because R ISCO must be in units of r g = GM/c 2 in Figure 15. The disk inclination angle and distance are also required because the total luminosity of the disk emission L disk is needed, in order to estimate the absolute value of a ∗ . So far, all BH masses in X-ray binaries (XRBs) have been estimated using the Kepler's 3rd law of stellar motion, expressed in the so-called the mass function given in Equation (1). Since the only direct observables are P orb and K C , both M C and i have to be determined indirectly in order to obtain the BH mass estimate reliably. The companion's mass M C can be determined relatively reliably by the observed spectral type of the companion star and i can be estimated by modeling the observed ellipsoidal modulation of the companion's optical or infrared light curve. The observed ellipsoidal modulation is a consequence of exposing different parts of the pearshaped companion star to the observer at different orbital phases (see Figs. 1 and 5); the pear-shape is caused by the tidal force of the compact star, which also heats the side of the companion star facing it. For details of BH mass estimates using this method, please refer to [2]. However model dependence and other uncertainties (such as accretion disk contamination) cannot be circum- vented completely and thus systematic error may exist in determining their system parameters. For example, three optical states, namely 'passive', 'loop' and 'active' states, have been identified in the normally called 'quiescent' state of A0620-003 when its X-ray luminosity is very low; only during the passive state its optical light curve modulation is purely ellipsoidal, i.e., accretion disk contamination is completely negligible, as shown in Figure 22 [98]. This means that considerable systematic errors in determining its inclination angle may occur unless only the 'passive' state data are used. Unfortunately previous observations of BHXBs used to determine their inclinations did not always occur during the passive state, thus systematic errors may be common in previous results [99]. Even for GRO J1655-40, of which all previous observations were made during its passive state [99], its BH mass measured with different observations, or the same data analyzed by different groups are not exactly the same, and even not completely consistent between them, as shown in Figure 23, which show a scatter of about 20-30% to the estimated BH mass, much larger than its statistical error of a few percents. It should be pointed out that the inclination i in Equation (1) refers to that of the orbital plane of the binary system. Because the accreting material initially carries the angular momentum inherited from the companion star, the formed disk should be co-planar with the orbital plane of the binary system. However, the BH spin in a BHXB cannot be changed significantly by accretion [71] (and see discussion above on Cygnus X-1). Therefore its spin axis may not coincide with the normal direc- tion of the orbital plane of the binary system. In this case the Bardeen-Petterson effect [106], due to frame-dragging by the spinning BH, can rapidly align the normal axis of the inner disk region with the spin axis of the BH, making the inner disk and binary system mis-aligned. Circumstantial evidence already exists for mis-alignment between the two axes, because the orbital inclination of GRO J1655-40 shown in the upper panel of Figure 23 is significantly different from ∼ 80 · inferred from its relativistic jets [107], if the jets are powered by extracting the spin energy of the BH via the Blandford-Znajek mechanism [76]. Nevertheless, the orbital inclination is normally used in place of the disk inclination, which is needed in BH spin measurement using the CF method, in order to infer the total disk luminosity and calculate GR correction factors. However, the inner disk inclination is currently not available essentially for any of the known BHXBs. For example, a mis-alignment of 10 · from i = 70 · can cause nearly 50% error to the total disk luminosity, which will translate into nearly 30% error in r in . A Schwarzschild BH may be estimated to have a ∗ falling any where between [ -0 . 5 , 0 . 5], if r in is uncertain within about 30%, according to Fig.(15). Accurate determinations of distances of astrophysical objects in the Milky Way are difficult, e.g., for BHXBs that are not standard candles. Normally some absorption features in their spectra, in conjunction with their posi- tions in the galactic coordinates, are used to infer their distances. For example, the distance of GRO J1655-40 is commonly taken as 3 . 2 ± 0 . 2 kpc, based primarily on observed absorption lines and somewhat on the dynamics of the observed jets. Critical examinations of all available data related to its distance, however, are in favor of a much closer distance of less than 2 kpc and more likely just 1 kpc [108, 109]. Similar conclusion is also reached to the distance of A0620-003, revising its distance from the commonly accepted 1 . 2 ± 0 . 4 kpc to ∼ 0 . 4 kpc, making its possibly the closest BHXB known so far [109]; however, a distance of 1 . 06 ± 0 . 12 pc was preferred in a more recent study [30]. Similarly the currently adopted distances of many other BHXBs may also have considerable systematic errors. If true, this would change significantly the current estimates on their masses and spins. Therefore future improvements of the continuum fitting method depend upon the improved measurements on their BH masses, accretion disk inclination, and distances.",
"pages": [
16,
17
]
},
{
"title": "D. Future improvements of the continuum fitting method",
"content": "The mass ratio q can be determined directly according to the law of momentum conservation, i.e., if the semi-amplitude of the velocity curve of the BH K BH can be observed directly, as illustrated in Figure 1. The orbital inclination i can be then calculated using Equation (1), avoiding any systematics related to the ellipsoidal light curve modeling. Since a BH is not directly observable, we can only hope to observe any emission or absorption line feature comoving with it. The accretion disk certainly moves with the accreting BH. However any line feature of the inner accretion disk suffers from the broadening of disk's orbital motion and distortions by relativistic effects around the BH, thus making it practically impossible, or difficult at least, for detecting the binary orbital motion of the BH. Orbital motion of double-peaked disk emission lines were observed for NSXB Sco X-1 [110], the BHXBs A0620-003 [111, 112]) and GRS 1124-68 [112]. Unfortunately a significant phase offset of velocity modulation was found from that expected based on the observed orbital motion of the companion, though the velocity semi-amplitude is consistent with the expected mass ratio [112]. Soria et al. (1998) observed the orbital motion of the double-peaked disk emission line He II λ 4686 from GRO J1655-40, and found its velocity modulation phase and semi-amplitude in agreement with the kinematic and dynamical parameters of the system [113]. However one major problem in accurately measuring the orbital motion of the primary from the observed double-peaked emission lines is how to determine reliably the line cen- er, because the lines are typically asymmetric and also variable. We have recently proposed to observe the Doppler shifts of the absorption lines of the accretion disk winds co-rotating with the BH around its companion star [114], since in many XRBs accretion disk winds are ubiquitous and appear to be rather stable when observed (e.g., in Ref.[115]). We verified this method using Chandra and HST high resolution spectroscopic observations of GRO J1655-40 (shown in Figure 24) and LMC X-3. Unfortunately the currently available data only covered small portions of their orbital phases and thus do not allow better constraints to their system parameters. Future more observations of these two sources and other sources with detectable absorption lines from their accretion disk winds will allow reliable and precise measurements of the BH masses and orbital inclination angles in accreting BHXBs. Since the accretion disk has very high density and is ionized near the BH, where the majority of the observed disk emission is produced due to the very deep gravitational potential near the BH, scattering of the primary disk emission is inevitable. The scattered light is polarized and its polarization fraction and position angle depend on the viewing direction (inclination ), scattering optical depth and the radius where the scattering occurs [116-119]. Ignoring many details, it can be shown that the polarization fraction, P ( i ), of the observed disk photons (initial disk emission plus the scattered emission) is given by where A is a constant depending upon the scattering op- tical depth; detailed calculations made by Chandrasekhar [120] gives P (75 · ) = 0 . 04. Note that here the disk photons are from the Raleigh-Jeans part of the multicolor blackbody spectrum with a characteristic shape of f ( ν ) ∝ ν 1 / 3 , i.e., no GR effect is included. We can therefore find the disk inclination angle by measuring the polarization fraction of this part of the disk emission, as shown in Figure 25. At energies above the Raleigh-Jeans part of the multi-color blackbody spectrum, the polarization is strongly effected by both the inclination and BH spin, as shown in Figure 26. The continuum spectra are clearly degenerated for the different combinations of inclination and BH spin, but the polarization fraction and angle as functions of energy can clearly distinguish between them [118]. Therefore X-ray spectra-polarimetry observations of BHXBs will certainly make important progresses in measuring BH spin. Besides using polarization measurements to obtain inner disk inclination (and BH spin), the broad iron Kalpha measurements can also be used to do so [121-123], because iron K-alpha emission is believed to come from the fluorescent emission of the disk as a result of illumination by a hard X-ray source above the disk. Naturally this emission is sensitive to the disk inclination (and BH spin). However, compared to the polarization measurement method, this method is less straight forward and may suffer from systematic uncertainties in deriving the disk inclination (and BH spin), because complicated modeling of the hard X-ray component and line emissivity from the disk is required. Therefore it is essential to use all methods discussed above to measure both inclination angles (orbital plane and inner disk) and BH spin. Studying the relationship between the results obtained with different methods is also important in its own rights, in order to understand accretion disk physics, its interaction with the BH in its center, production of relativistic jets, the origin of the BH spin, and ultimately the formation mechanisms of BHs and BHXB systems. The recent dispute on the distances of GRO J1655-40 and A0620-003 exemplifies the difficulty of determining the distances of BHXBs using mostly absorption lines [108, 109]. We have recently suggested a method of using the delay time between the X-ray fluxes of an XRB and its X-ray scattering halo by interstellar dust to infer its distance [124, 125]. However this method may suffer from our incomplete knowledge of the distribution of interstellar medium. Ideally precise astrometry can determine their distances model-independently, by measuring their trigonometric parallaxes. Recently the distance to Cygnus X-1 was determined reliably and accurately this way (1 . 86 +0 . 12 -0 . 11 kpc; [126]); which is key to the consequent measurement of its BH mass and spin [36, 37]. Currently it remains challenging to measure the trigonometric parallaxes of objects at distances beyond several kpc where most BHXBs are located. Future high precision astrometry missions are expected to improve the distance estimates to these BHXBs significantly. Therefore we expect that the improved measurements discussed above on their BH masses, inner disk inclination, and distances will allow future improvements of BH spin measurement with the CF method.",
"pages": [
17,
18,
19
]
},
{
"title": "E. Possible application of the continuum fitting method to AGNs",
"content": "More recently, the CF method is also applied to constrain the BH spin in an active galactic nucleus (AGN), which is powered by matter accretion onto the central supermassive BH in its center [127]. However the BH spin inferred this way for an supermassive BH is quite uncertain, because: (1) The peak temperature of the accretion disk inversely increases with the mass of a BH. Therefore, for a supermassive, its temperature would be in the ultraviolet energy range which will be strongly absorbed and hard to observe; (2) The system parameters of a supermassive BH (e.g., the mass of the BH and the inclination angle of the accretion disk) have larger uncertainties; (3) The uncertain mechanism for some components (e.g., the soft X-ray excess) in AGN spectra also increases the difficulty; (4) Some emission and absorption lines may distort the continuum spectrum substantially; and (5) In some cases the contribution of its host galaxy to the observed total continuum spectrum cannot be removed satisfactorily. The polarized continuum of an AGN should be a pure accretion disk continuum, at least in the optical to near infrared band [128]. One possible way to measure the BH spin in an AGN with the CF method is to combine the observed polarized optical to near infrared continuum spectrum with the observed total UV continuum spectrum to get a broad band continuum spectrum of an AGN [129]. In principle the broadened Balmer edge features and the total UV spectrum can be used to constrain the disk inclination angle and fraction of host galaxy contamination, respectively [129]. However the quality of the currently available data is still insufficient to allow accurate determination of BH spin in AGNs with this method. Nevertheless the principal method of measuring BH spin in AGNs should be using the reflected broad iron line and continuum components [121-123]. The main reason is that i can also be determined simultaneously and r in obtained this way is already in units of r g , thus avoiding naturally the uncertainties caused by the BH mass and inclination in the CF method. However cross calibration can be done between the CF and reflection fitting methods if both can be applied to the same supermassive BH.",
"pages": [
19,
20
]
},
{
"title": "V. FURTHER DEVELOPMENTS ON HOT ACCRETION FLOWS",
"content": "In this section I briefly summarize some further developments on hot accretion flows, which are believed to be responsible for the PL component of the spectra in BHXBs. The multi-waveband spectra of the hard state of the BHXB XTE J1118+480 was modeled with the TID with hot accretion flow (ADAF) geometry (panel (a) in Figure 27) and TID (with ADAF) plus jet model (panel (b) in Figure 27), with ˙ m D = 0 . 05 and ˙ m jet = 5 × 10 -3 ˙ m D [93]. The steep UV spectrum provides clear evidence for a large truncation radius for the SSD ( r = 600), and the radio to infrared spectrum dominates the jet emission, which also contribute to the hard PL component [93]. Such low accretion rate gives very low SSD luminosity l D ∼ 5 × 10 -4 (since the disk radiation efficiency η ∼ 1 /r ; see Figure 17), which is far below the turn-over luminosity of l D ∼ 10 -2 shown in Figure 20. Therefore the inferred truncation radius of XTE J1118+480 agrees with the extrapolation of data points of XTE J1817-330 down to very low disk luminosity. Large truncation radii are also reported from several other sources in the hard state (e.g., in Ref.[130]). However, the exact values of these truncation radii may have large uncertainties, since no direct detection of the inner disk peak emission was available, unlike the strong case of XTE J1118+480 [93]. For this reason I did not include these reported values in Figure 20. Panel (b) in Figure 27 also shows that the radiation from both the hot accretion flow and the jet contribute to the X-ray emission. However the former is roughly proportional to ˙ m 2 , whereas the latter to ˙ m . Thus with the decrease of ˙ m , the contribution from the jet becomes more and more important, thus the X-ray radiation will be dominated by the jet (when l glyph[lessorsimilar] 10 -5 -10 -6 ), as shown in Figure 28 [93]. The observational data of very low-luminosity AGNs clearly show a correlation between radio and X-ray with a correlation index of ∼ 1 . 2 [131], in excellent agreement with the prediction shown in Figure 28 [93]. It is well known that the highest luminosity an ADAF can produce is only about 3% L Edd . However the observed highest luminosity a hard state can reach can be 10% L Edd or even higher, which can be described by LHAF shown in Figure 29, where several previously known solutions of accretion flows are all unified in a single scheme [133]. The hard state spectrum of XTE J1550-564 with L ∼ 6% L Edd is explained by LHAF very well, including the X-ray spectral slope and the value of the cutoff energy [134].",
"pages": [
20
]
},
{
"title": "VI. FURTHER DEVELOPMENTS ON CORONA FORMATION",
"content": "While the formation of the SSD in a BHXB is reasonably well understood, the formation of the corona, which is the hot accretion flow discussed in Section V, remains less understood. Over the last more than 10 years, Liu and her collaborators have developed a model to explain the formation and evolution of the corona in a BHXB or AGN, in which the mass accretion rate ˙ m in units of the Eddington ratio drives the variations of the complex accretion flows by the interaction between the cold SSD and the hot corona [60, 135-139]. Specifically, the coupling between the hot corona and the cold disk leads to mass exchange between them. The gas in the thin disk is heated up and evaporates into the corona as a consequence of thermal conduction from the hot corona, or the corona gas condenses into the disk as a result of overcooling by, for example, external inverse Compton scattering. If ˙ m is low, evaporation occurs and can completely remove the thin disk, leaving only the hot corona in the inner region and a truncated thin disk in outer region; this provides a mechanism for ADAF at low ˙ m . If ˙ m is high, the gas in the corona partially condenses to the disk due to strong Compton cooling, resulting in disk dominant accretion. The model naturally explains the different structures of accretion flow in different spectral states as shown in Figure 30 [60, 135, 137-144]. The hysteresis observed in spectral state transitions can also be explained by different irradiations from different evolution history under the same scenario [145-147]. Figure 30 is significantly different from the illustration of accretion flow structures in different spectral state in Figure 12 on three aspects: (1) At intermediate ˙ m , the SSD is broken by ADAF into two parts, an outer disk and an inner disk; (2) The inner disk boundary here is always located very close to the BH, except in the very low ˙ m hard or quiescent state, which is very different from the TID scenario depicted in Figure 12; (3) The corona here covers essentially the whole accretion disk, especially the inner disk region, whereas in Figure 12 the corona is mostly located inward from the inner disk boundary. The observed soft X-ray component in the low/hard state can be explained by the existence of the inner disk [139, 140, 142, 144]. As I have discussed above, the inner disk boundary radius inferred with the CF method in the presence of a strong PL component is actually consistent with that in the soft state when the PL component is weak, after taking into account the Compton scattering in the corona [65, 87]. Actually the essential assumption behind the broad iron line/reflection fitting method of determining BH spin is that the inner disk boundary is at the ISCO when the PL component is strong. All these tend to support the existence of the inner disk at intermediate ˙ m . However it remains to be demonstrated that if the whole SSD is indeed broken into the two parts at intermediate ˙ m , as illustrated in Figure 30. Recently it has been suggested that the ADAF and corona shown in Figure 30 may have clumpy structures, as shown in Figure 31 [148]. The 'clumpy' model (Figure 31) has mainly two different consequences from the 'uniform' model (Figure 30): (1) The inner disk is transient in the 'clumpy' model; (2) The 'clumpy' model can explain the variabilities observed in X-ray binaries (such as the state transitions discussed in Section VII) and radio-loud AGNs (such as BL Lac objects). Determining the structure of the corona in an BHXB observationally remains difficult; the fundamental issue on whether the corona covers mostly the accretion disk or the central compact object still remains unclear so far. Since the observed similar spectral and state transitions between some NS and BH XRBs are quite similar [149], it is reasonable to assume that they have similar coronae. Recently we have used type I X-ray bursts from low-mass NSXBs to show that X-ray bursts experience negligible Comptonization and that the corona cools rapidly during the rising phase of X-ray bursts and is then heated up rapidly during the rising phase of X-ray bursts, as shown in the upper panel of Figure 32 for IGR J1747721 [150]. These results suggest that the corona cannot cover the central compact object completely (lower panel of Figure 32) and that the destruction and formation time scales of the corona are as short as seconds; such short time scales are quite difficult to understand in the above discussed evaporation model, in which the time scales are related to the viscous time scales of the accretion disk. However, this short time scale is consistent with a corona produced by magnetic reconnections in the accretion disk, in a similar way to the solar corona heating [151]; this conclusion was based on the inferred accretion flow structure of a BHXB shown in panel (A) of Figure 33, in comparison with the atmospheric structure of the Sun shown in panel (B) of Figure 33. It is worth noticing that the purported coronae in both XRBs and AGNs have strikingly similar properties, e.g., they all have electron temperatures of the order of hun- dreds keV, in spite that the temperatures, inner disk radii and variability time scales of their cold accretion disks all scale with their BH masses (and accretion rates) as predicted in the SSD model. This means that their coronae are scale independent. It is perhaps not coincidental that the electrons' velocities in a corona are approximately the same as the Keplerian orbital velocities of the inner disk, which are also roughly the same as the launching velocities of jets. Of course these velocities are also the varialized velocities of the central BHs. It is plausible that turbulent small scale magnetic fields lifts the plasma in the 'warm layer' shown in Figure 33 to form the corona, which is then launched into the jets by the rotating large scale magnetic fields through either the BP or BZ mechanism; during these processes the magnetic fields are mostly responsible for changing the directions of motion of plasmas by the Lorentz force. If the corona is clumped as discussed in Section VI [148], then the plasma channeled into the jets should be clumpy and thus the ejected jets should be episodic and have knotted structures, in agreement with observations; this scenario is illustrated in Figure 34. Alternatively episodic ejections may also occur in the disk, similar to the coronal mass ejection on the Sun [152]. Interestingly, the soft X-ray excess (SXE) frequently observed in AGNs can be interpreted as the warm layer found in XRBs, since the inferred plasmas parameters, with electron temperature of 0.1 to 0.3 keV and Compton scattering optical depth of around 10, are similar to that of XRBs and universal among AGNs with different BHmasses, but seem to be related only with ˙ m [153-155]. Therefore their warm layers are also scale independent. Indeed, magnetic flares, perhaps caused by magnetic reconnections in the cold disk, can produce the warm layers in both XRBs and AGNs [156, 157].",
"pages": [
20,
21,
22,
23
]
},
{
"title": "VII. FURTHER DEVELOPMENTS ON STATE TRANSITIONS",
"content": "To further understand the mechanism of the spectral state transitions, it is important to study those spectral state transitions in individual BHXBs during different outbursts. The advantages are obvious - uncertainties in our estimates of black hole mass, the binary properties such as the orbital period, or the source distance will not play any role in producing the observed diverse transi- tion properties in individual BHXBs, which can only be driven by the accretion process under different initial conditions. The sources which firstly allowed such a study were the BH transients GX 339-4 and XTE J1550-564, the NS transient Aquila X-1, and the flaring NS low-mass XRB (LMXB) 4U 1705-44, in which a remarkable correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the following soft state was found [158-160]. More recently, a comprehensive study of the hard-to-soft spectral state transitions detected in all the bright XRBs in a period of about five years with simultaneous X-ray monitoring observations with the RXTE/ASM and the Swift/BAT confirmed the correlation between the hard-to-soft transition and the peak luminosity of the following soft state [161, 162], as shown in the upper panel of Figure 35 [161]. More important was the discovery of the correlation between the transition luminosity and the rate-of-change of the luminosity during the rising phase of an outburst of transients or a flare of persistent sources, as shown in the lower panel of Figure 35 [161]. The above correlation implies that in most cases it is the rate-of-increase of the mass accretion rate, rather than the mass accretion rate itself, determines the hardto-soft spectral transition; this is depicted in Figure 36 [161]. In addition, the discovery of the relation between the hard X-ray peak flux and the waiting time of transient outburst in the BH transient GX 339-4 (as shown in Figure 37) supports that the total mass in the disk determines the peak soft state luminosity of the following outburst [159]. Therefore spectral states should be understood in the non-stationary accretion regime, which would be described by both ˙ m and m , as well as the total mass in the disk before an outburst.",
"pages": [
23
]
},
{
"title": "VIII. FURTHER DEVELOPMENTS ON THERMAL STABILITY OF SSD",
"content": "The SSD model predicts that when the accretion rate is over a small fraction of the Eddington rate, which corresponds to L glyph[greaterorsimilar] 0 . 06 L Edd , the inner region of the disk is radiation-pressure-dominated and then both secularly [163] thermally unstable [164, 165]. However, observations of the high/soft state of black hole X-ray binaries with luminosity well within this regime (0 . 01 L Edd glyph[lessorsimilar] L glyph[lessorsimilar] 0 . 5 L Edd ) indicate that the disk has very little variability, i.e., quite stable [166]. It has been well established that the accretion flow in this state is described by the SSD model [4, 167]. Radiation magnetohydrodynamic simulations of a vertically stratified shearing box have confirmed the absence of the thermal instability [168]. Recently, the thermal stability is revisited by linear analysis, by taking into account the role of magnetic field in the accretion flow [169]. By assuming that the field responds negatively to a positive temperature perturbation, it was found that the threshold of accretion rate above which the disk becomes thermally unstable increases signifi- cantly, compared with the case of not considering the role of magnetic field. This accounts for the stability of the observed sources with high luminosities, as shown in Figure 38. If the magnetic pressure is less than about 24% of the total pressure, then this model can explain the 'heart-beat' limit-cycle instability observed in GRS 1915+105 at its highest luminosity; this peculiar source holds the highest accretion rate (or luminosity) among BHXBs. Observations of GRS 1915+105 showed that t high (the duration of the outburst phase) is comparable to t low (duration of the quiescent phase) and L high is 3 to 20 times larger than L low [170]. However, numerical calculations showed that t high is less than 5 percent of t low , and L high is around two orders of magnitude larger than L low . Some efforts have been made to improve the theory in order to explain observations either by some artificial viscosity prescription [171] or by additional assumption of the energy exchange between the disk and corona [172]. Taking into account the stress evolution process, it was found that the growth rate of thermally unstable modes can decrease significantly owing to the stress delay, which may help to understand the 'heart-beat' limit-cycle variability of GRS 1915+105 [173]. The limit-cycle properties are found to be dominated by the mass-supply rate (accre- · · tion rate at the outer boundary) and the value of the α -viscosity parameter in the SSD model that assumes that the viscous torque is proportional to the total pressure [77]. It was also found that only the maximal outburst luminosity (in Eddington units) is positively correlated with the spin of a BH, providing another way to probe BH spin [77]; this is mainly due to the smaller inner disk radius and thus higher radiative efficiency for larger a ∗ as shown in Figures 15, 16 and 17.",
"pages": [
23,
24,
25
]
},
{
"title": "IX. UNIFICATION AND OUTLOOK",
"content": "Despite of the many progresses made over the last decades on the study of BHXBs and microquasars, there are still many outstanding and unresolved issues, which can be pierced by two big pictures. One big picture is related to the astrophysics of BHXBs and microquasars, which is centered on understanding the accretion-outflow (wind or jet) connections at different accretion rates for different types of astrophysical systems involving BHs. The other big picture is related to the fundamental physics of BHXBs and microquasars, which is centered on identifying astrophysical BHs and testing theories of strong gravity. These two pictures are actually also entangled together, because a strong gravity theory, such as GR, is needed to describe the astrophysical aspects of these systems, and we need to understand the astrophysics of BHXBs and microquasars before we can start to test any strong gravity theory[174]. Figure 39 shows the relation between the collimationcorrected gamma-ray luminosity and the kinetic power for AGNs and gamma-ray bursts (GRBs)[175]. For at least some of GRBs, a super-Eddington accreting compact stellar object, probably a BH with around 10 M glyph[circledot] , is believed to be responsible for powering the highly relativistic jets, which produces intense gamma-rays through violent collisions of blobs in the jets[176]. The two types of AGNs shown here, flat-spectrum radio quasars (FS- RQs) and BL Lacs, have all been observed to have mildly relativistic jets. Their central engines are supermassive BHs with around 10 7 -9 M glyph[circledot] with accretion rates just below (for FSRQs) or far below (for BL Lacs) Eddington rate[177]. The good correlation over about 10 orders of magnitudes between these very different systems with very different accretion rates suggests that there must be some common mechanisms responsible for the accretionoutflow (wind/jet) connections for all accreting BHs[178]. Putting together all related phenomenologies and some theoretical modeling of BHXBs and microquasars discussed in this article, an unification scheme is illustrated in Figure 40 with the major elements in the accretionoutflow connections in different types of astrophysical systems harboring both stellar mass BHs and supermassive BHs with accretion rates over several orders of magnitudes. The types of accreting stellar mass BHs include BHXBs and microquasars in different spectral states, as well as ultra-luminous X-ray sources (ULXs) (some of which are most likely ultra-Eddington accreting BHs[180]), and GRBs. The types of accreting supermassive mass BHs include low-luminosity AGNs (LLAGNs), BL Lac objects, normal radio quiet quasars (RQQs), FSRQs and broad-line-less luminous quasars (BL3Qs) [179] (I create the acronym 'BL3Q' here just for fun). This scheme is not a theoretical model, or even toy model yet. It, however, can be used as one possible chain to pierce many observed phenomenologies together, providing a possible frame work for further theoretical developments on accretion-outflow connections. Future observations will scrutinize this unification scheme and",
"pages": [
25
]
},
{
"title": "Some necessary elaborations and points:",
"content": "revise it inevitably. Of particular importance is the reliable determinations of the mass and spin of accreting BHs. The BH mass allows us to determine the accretion rate in units of Eddington rate, i.e., ˙ m , which is a key parameter to the unification scheme of accretion-outflow connections. The BH spin is of course another key parameter, because it can determine the radiative efficiency of the disk and jet power in the BZ mechanism. Once we have well determined BH parameters and a consistent and predictable theory describing the observed accretion-outflow connections, we are then ready to study the fundamental physics of BHs with BHXBs and microquasars, such as the properties of event horizon, spacetime around Schwarzschild and Kerr BHs, BH spin energy (and mass) extraction. For example, both the broad iron line and CF fitting methods can be used to measure a BH's spin. However the two methods are equivalent only if the metric is accurately described with Kerr metric. Different spin measurements for a BH with the two different methods would invalidate Kerr metric, provided that we understand the accretion disk physics thoroughly. On the other hand, a correct metric is of course required to describe accurately the accretion flow around BHs. This is one example of the entanglement or interplay between astrophysics and fundamental physics of accreting BHs. The stake is high, but the job is difficult. Acknowledgment. I appreciate inputs from Profs. Lijun Gou, Weimin Gu, Lixin Li, Bifang Liu, Dingxiong Wang, Wenfei Yu, Feng Yuan, and Shu Zhang. The editors of this book are thanked for inviting me to write this article, as well as their patience, persistency, proof reading it, and offering comments and suggestions to improve it in the end. This work is partially supported with funding by the 973 Program of China under grant 2009CB824800, by the National Natural Science Foundation of China under grant Nos. 11133002 and 10725313, and by the Qianren start-up grant 292012312D1117210. Orford, J L Osborne, M Ouchrif, M Panter, G Pelletier, S Pita, G Puhlhofer, M Punch, B C Raubenheimer, M Raue, J Raux, S M Rayner, A Reimer, O Reimer, J Ripken, L Rob, L Rolland, G Rowell, V Sahakian, L Saug'e, S Schlenker, R Schlickeiser, C Schuster, U Schwanke, M Siewert, H Sol, D Spangler, R Steenkamp, C Stegmann, J-P Tavernet, R Terrier, C G Th'eoret, M Tluczykont, G Vasileiadis, C Venter, P Vincent, H J Volk, and S J Wagner. Discovery of very high energy gamma rays associated with an x-ray binary. Science (New York, N.Y.) , 309(5735):746-9, July 2005. Flow above a Thin Disk. The Astrophysical Journal , 575(1):117-126, August 2002.",
"pages": [
26,
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32
]
}
]
2013GApFD.107..185P
https://arxiv.org/pdf/1205.4835.pdf
<document>
<text><location><page_1><loc_8><loc_94><loc_31><loc_95></location>Geophysical and Astrophysical Fluid Dynamics Vol. 00, No. 00, Month 2013, 1-19</text>
<section_header_level_1><location><page_1><loc_8><loc_82><loc_89><loc_84></location>Helicity-vorticity turbulent pumping of magnetic fields in the solar convection zone</section_header_level_1>
<section_header_level_1><location><page_1><loc_44><loc_79><loc_53><loc_80></location>V. V. PIPIN</section_header_level_1>
<section_header_level_1><location><page_1><loc_31><loc_77><loc_67><loc_78></location>Institute Solar-Terrestrial Physics, Irkutsk, Russia</section_header_level_1>
<text><location><page_1><loc_39><loc_74><loc_59><loc_75></location>( July 31, 2021, Revision: 1.20 )</text>
<text><location><page_1><loc_8><loc_62><loc_90><loc_72></location>We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker-Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. We estimate the magnitude of the drift velocity and find that it can be several m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. We find that the helicity-vorticity pumping effect can have an influence on the features of the sunspot time-latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.</text>
<section_header_level_1><location><page_1><loc_8><loc_57><loc_21><loc_58></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_8><loc_34><loc_90><loc_55></location>It is believed that the evolution of the large-scale magnetic field of the Sun is governed by the interplay between large-scale motions, like differential rotation and meridional circulation, turbulent convection flows and magnetic fields. One of the most important issues in solar dynamo theory is related to the origin of the equatorial drift of sunspot activity in the equatorial regions and, simultaneously at high latitudes, the poleward drift of the location of large-scale unipolar regions and quiet prominences. Parker (1955) and Yoshimura (1975) suggested that the evolution of large-scale magnetic activity of the Sun can be interpreted as dynamo waves propagating perpendicular to the direction of shear from the differential rotation. They found that the propagation can be considered as a diffusion process, which follows the iso-rotation surfaces of angular velocity in the solar convection zone. The direction of propagation can be modified by meridional circulation, anisotropic diffusion and the effects of turbulent pumping (see, e.g., Choudhuri et al. 1995, Kitchatinov 2002, Guerrero and de Gouveia Dal Pino 2008). The latter induces an effective drift of the large-scale magnetic field even though the mean flow of the turbulent medium may be zero.</text>
<text><location><page_1><loc_8><loc_24><loc_90><loc_34></location>The turbulent pumping effects can be equally important both for dynamos without meridional circulation and for the meridional circulation-dominated dynamo regimes. For the latter case the velocity of turbulent pumping has to be comparable to the meridional circulation speed. It is known that an effect of this magnitude can be produced by diamagnetic pumping and perhaps by so-called topological pumping. Both effects produce pumping in the radial direction and have not a direct impact on the latitudinal drift of the large-scale magnetic field.</text>
<text><location><page_1><loc_8><loc_8><loc_90><loc_24></location>Recently (Pipin 2008, Mitra et al. 2009, Leprovost and Kim 2010), it has been found that the helical convective motions and the helical turbulent magnetic fields interacting with large-scale magnetic fields and differential rotation can produce effective pumping in the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. It is believed that these regions, namely the tachocline beneath the solar convection zone and the subsurface shear layer, are important for the solar dynamo. Figure 1 illustrates the principal processes that induce the helicity-vorticity pumping effect. It is suggested that this effect produces an anisotropic drift of the large-scale magnetic field, which means that the different components of the large-scale magnetic field drift in different directions. Earlier work, e.g. by Kichatinov (1991) and Kleeorin and Rogachevskii (2003), suggests that the effect of anisotropy in the</text>
<text><location><page_2><loc_8><loc_96><loc_9><loc_96></location>2</text>
<figure>
<location><page_2><loc_24><loc_74><loc_73><loc_93></location>
<caption>Figure 1. The field lines of the large-scale magnetic field, B ( T ) , are transformed by the helical motions to a twisted Ω -like shape. This loop is folded by the large-scale shear, V ( T ) , into the direction of the background large-scale magnetic field, B ( T ) . The induced electromotive force has a component, E ( P ) , which is perpendicular to the field B ( T ) . The resulting effect is identical to the effective drift of the large-scale magnetic field along the x -axis, in the direction opposite to the large-scale vorticity vector W = ∇× V ( T ) , i.e., E ( P ) ∼ -W × B ( T ) .</caption>
</figure>
<text><location><page_2><loc_8><loc_45><loc_90><loc_56></location>transport of mean-field is related to nonlinear effects of the global Coriolis force on the convection. Also, nonlinear effects of the large-scale magnetic field result in an anisotropy of turbulent pumping (Kleeorin et al. 1996). It is noteworthy, that the helicity-vorticity effect produces an anisotropy of the large-scale magnetic field drift already in the case of slow rotation and a weak magnetic field. A comprehensive study of the linear helicity-vorticity pumping effect for the case of weak shear and slow rotation was given by Rogachevskii et al. (2011) and their results were extended by DNS with a more general test-field method Brandenburg et al. (2012).</text>
<text><location><page_2><loc_8><loc_34><loc_90><loc_45></location>In this paper we analytically estimate the helicity-vorticity pumping effect taking into account the Coriolis force due to global rotation. The calculations were done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. The results are applied to mean field dynamo models, which are used to examine this effect on the dynamo. The paper is structured as follows. In the next section we briefly outline the basic equations and assumptions, and consider the results of calculations. Next, we apply the results to the solar dynamo. In Section 3 we summarize the main results of the paper. The details of analytical calculations are given in the Appendices A and B.</text>
<section_header_level_1><location><page_2><loc_8><loc_24><loc_23><loc_25></location>2 Basic equations</section_header_level_1>
<text><location><page_2><loc_8><loc_7><loc_90><loc_22></location>In the spirit of mean-field magnetohydrodynamics, we split the physical quantities of the turbulent conducting fluid into mean and fluctuating parts where the mean part is defined as an ensemble average. One assumes the validity of the Reynolds rules. The magnetic field B and the velocity V are decomposed as B = B + b and V = V + u , respectively. Hereafter, we use small letters for the fluctuating parts and capital letters with an overbar for mean fields. Angle brackets are used for ensemble averages of products. We use the two-scale approximation (Roberts and Soward 1975, Krause and Radler 1980) and assume that mean fields vary over much larger scales (both in time and in space) than fluctuating fields. The average effect of MHD-turbulence on the large-scale magnetic field (LSMF) evolution is described by the mean-electromotive force (MEMF), E = 〈 u × b 〉 . The governing equations for fluctuating magnetic field</text>
<text><location><page_3><loc_8><loc_92><loc_60><loc_94></location>and velocity are written in a rotating coordinate system as follows:</text>
<formula><location><page_3><loc_33><loc_87><loc_90><loc_91></location>∂ b ∂t = ∇× ( u × ¯ B + ¯ V × b ) + η ∇ 2 b + G , (1)</formula>
<formula><location><page_3><loc_22><loc_83><loc_90><loc_87></location>∂u i ∂t +2( Ω × u ) i = -∇ i ( p + ( b · ¯ B ) µ ) + ν ∆ u i (2)</formula>
<formula><location><page_3><loc_36><loc_79><loc_76><loc_82></location>+ 1 µ ∇ j ( ¯ B j b i + ¯ B i b j ) -∇ j ( ¯ V j u i + ¯ V i u j ) + f i + F i ,</formula>
<text><location><page_3><loc_8><loc_63><loc_90><loc_77></location>where G , F stand for nonlinear contributions to the fluctuating fields, p is the fluctuating pressure, Ω is the angular velocity responsible for the Coriolis force, ¯ V is mean flow which is a weakly variable in space, and f is the random force driving the turbulence. Equations (1) and (2) are used to compute the mean-electromotive force E = 〈 u × b 〉 . It was computed with the help of the equations for the second moments of fluctuating velocity and magnetic fields using the double-scale Fourier transformation and the minimal τ -approximations and for a given model of background turbulence. To simplify the estimation of nonlinear effects due to global rotation, we use scale-independent background turbulence spectra and correlation time. Details of the calculations are given in Appendix A. In what follows we discuss only those parts of the mean-electromotive force which are related to shear and the pumping effect.</text>
<section_header_level_1><location><page_3><loc_8><loc_58><loc_18><loc_59></location>2.1 Results</section_header_level_1>
<text><location><page_3><loc_8><loc_49><loc_90><loc_57></location>The large-scale shear flow is described by the tensor V i,j = ∇ V i . It can be decomposed into a sum of strain and vorticity tensors, ∇ j V i = 1 2 ( V i,j + V j,i ) -1 2 ε ijp W p , where W = ∇ × V is the large-scale vorticity vector. The joint effect of large-scale shear, helical turbulent flows and magnetic fields can be expressed by the following contributions to the mean-electromotive force (omitting the α -effect):</text>
<formula><location><page_3><loc_16><loc_41><loc_90><loc_47></location>E ( H ) = ( W × B ) ( f ( γ ) 2 h C + f ( γ ) 1 h K ) τ 2 c + ˜ V ( B ) ( f ( γ ) 4 h C + f ( γ ) 3 h K ) τ 2 c (3) + e [( e × W ) · B ] ( f ( γ ) 6 h C + f ( γ ) 5 h K ) τ 2 c + ( e × W ) ( e · B ) ( f ( γ ) 8 h C + f ( γ ) 7 h K ) τ 2 c ,</formula>
<text><location><page_3><loc_8><loc_35><loc_90><loc_39></location>where ˜ V ( B ) = B j 2 ( V i,j + V j,i ) , e = Ω | Ω | is the unit vector along the rotation axis, τ c is the typical</text>
<text><location><page_3><loc_8><loc_28><loc_90><loc_36></location>relaxation time of turbulent flows and magnetic fields, h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇ × b (0) 〉 µρ are kinetic and current helicity of the background turbulence. These parameters are assumed to be known in advance. Functions f ( γ ) n ( Ω ∗ ) are given in Appendix B, they depend on the Coriolis number Ω ∗ = 2 Ω 0 τ c and describe the nonlinear effect due the Coriolis force, and Ω 0 is the global rotation rate.</text>
<text><location><page_3><loc_9><loc_25><loc_73><loc_28></location>For slow rotation, Ω ∗ glyph[lessmuch] 1, we perform a Taylor expansion of f ( γ ) n ( Ω ∗ ) and obtain</text>
<formula><location><page_3><loc_26><loc_21><loc_90><loc_24></location>E ( H ) = τ 2 c 2 ( W × B ) ( h C -h K ) + τ 2 c 5 ˜ V ( B ) ( 3 h K -13 3 h C ) . (4)</formula>
<text><location><page_3><loc_8><loc_7><loc_90><loc_19></location>The coefficients in the kinetic part of Eq. (4) are two times larger than those found by Rogachevskii et al. (2011). This difference results from our assumption that the background turbulence spectra and the correlation time are scale-independent. The results for the magnetic part are in agreement with our earlier findings (see Pipin 2008). The first term in Eq. (4) describes turbulent pumping with an effective velocity τ 2 c W 2 ( h C -h K ) and the second term describes anisotropic turbulent pumping. Its structure depends on the geometry of the shear flow. For large Coriolis numbers, Ω ∗ glyph[greatermuch] 1, only the kinetic helicity contributions</text>
<text><location><page_4><loc_8><loc_96><loc_9><loc_96></location>4</text>
<text><location><page_4><loc_45><loc_96><loc_52><loc_96></location>V. V. Pipin</text>
<figure>
<location><page_4><loc_12><loc_72><loc_85><loc_94></location>
<caption>Figure 2. The dependence of the pumping effects on the Coriolis number. Solid lines show contributions from kinetic helicity and dashed lines the same for current helicity.</caption>
</figure>
<text><location><page_4><loc_8><loc_65><loc_14><loc_66></location>survive:</text>
<formula><location><page_4><loc_33><loc_59><loc_90><loc_63></location>E ( H ) = -τ 2 c 6 ( W × B ) h K + τ 2 c 5 ˜ V ( B ) h K . (5)</formula>
<text><location><page_4><loc_8><loc_47><loc_90><loc_58></location>Figure 2 show the dependence of the pumping effects on the Coriolis number. We observe that for the terms ( W × B ) and ˜ V ( B ) the effects of kinetic helicity are non-monotonic and have a maximum at Ω ∗ ≈ 1. The effects of current helicity for these terms are monotonically quenched with increasing values of Ω ∗ . The additional contributions in Eq. (3) are rather small in comparison with the main terms. Thus, we can conclude that the first line in Eq. (3) describes the leading effect of pumping due to the helicity of turbulent flows and magnetic field. Below, we drop the contributions from the second line in Eq. (3) from our analysis.</text>
<section_header_level_1><location><page_4><loc_8><loc_41><loc_58><loc_43></location>2.2 Helicity-vorticity pumping in the solar convection zone</section_header_level_1>
<text><location><page_4><loc_8><loc_30><loc_90><loc_40></location>2.2.1 The dynamo model. To estimate the impact of this pumping effect on the dynamo we consider the example of a dynamo model which takes into account contributions of the mean electromotive force given by Eq. (3). The dynamo model employed in this paper has been described in detail by Pipin and Kosovichev (2011a,c). This type of dynamo was proposed originally by Brandenburg (2005). The reader may find the discussion for different types of mean-field dynamos in Brandenburg and Subramanian (2005) and Tobias and Weiss (2007).</text>
<text><location><page_4><loc_9><loc_29><loc_79><loc_30></location>We study the standard mean-field induction equation in a perfectly conducting medium:</text>
<formula><location><page_4><loc_39><loc_23><loc_90><loc_26></location>∂ B ∂t = ∇ × ( E + U × B ) , (6)</formula>
<text><location><page_4><loc_8><loc_16><loc_90><loc_21></location>where E = u × b is the mean electromotive force, with u, b being fluctuating velocity and magnetic field, respectively, U is the mean velocity (differential rotation and meridional circulation), and the axisymmetric magnetic field is:</text>
<formula><location><page_4><loc_40><loc_11><loc_58><loc_14></location>B = e φ B + ∇× A e φ r sin θ ,</formula>
<text><location><page_4><loc_8><loc_8><loc_90><loc_9></location>where θ is the polar angle. The expression for the mean electromotive force E is given by Pipin (2008). It</text>
<text><location><page_5><loc_8><loc_92><loc_26><loc_94></location>is expressed as follows:</text>
<formula><location><page_5><loc_33><loc_88><loc_90><loc_91></location>E i = ( α ij + γ ( Λ ) ij ) B -η ijk ∇ j B k + E ( H ) i . (7)</formula>
<text><location><page_5><loc_8><loc_83><loc_90><loc_86></location>The new addition due to helicity and mean vorticity effects is marked by E H . The tensor α ij represents the α -effect. It includes hydrodynamic and magnetic helicity contributions,</text>
<formula><location><page_5><loc_27><loc_79><loc_90><loc_81></location>α ij = C α sin 2 θα ( H ) ij + α ( M ) ij , (8)</formula>
<formula><location><page_5><loc_26><loc_76><loc_90><loc_78></location>α ( H ) ij = δ ij { 3 η T ( f ( a ) 10 ( e · Λ ( ρ ) ) + f ( a ) 11 ( e · Λ ( u ) ))} + (9)</formula>
<formula><location><page_5><loc_30><loc_73><loc_90><loc_75></location>+ e i e j { 3 η T ( f ( a ) 5 ( e · Λ ( ρ ) ) + f ( a ) 4 ( e · Λ ( u ) ))} + (10)</formula>
<formula><location><page_5><loc_30><loc_69><loc_90><loc_71></location>3 η T {( e i Λ ( ρ ) j + e j Λ ( ρ ) i ) f ( a ) 6 + ( e i Λ ( u ) j + e j Λ ( u ) i ) f ( a ) 8 } , (11)</formula>
<text><location><page_5><loc_8><loc_57><loc_90><loc_67></location>where the hydrodynamic part of the α -effect is defined by α ( H ) ij , Λ ( ρ ) = ∇ log ρ quantifies the density stratification, Λ ( u ) = ∇ log ( η (0) T ) quantifies the turbulent diffusivity variation, and e = Ω / | Ω | is a unit vector along the axis of rotation. The turbulent pumping, γ ( Λ ) ij , depends on mean density and turbulent diffusivity stratification, and on the Coriolis number Ω ∗ = 2 τ c Ω 0 where τ c is the typical convective turnover time and Ω 0 is the global angular velocity. Following the results of Pipin (2008), γ ( Λ ) ij is expressed as follows:</text>
<formula><location><page_5><loc_22><loc_52><loc_90><loc_55></location>γ ( Λ ) ij = 3 η T { f ( a ) 3 Λ ( ρ ) n + f ( a ) 1 ( e · Λ ( ρ ) ) e n } ε inj -3 η T f ( a ) 1 e j ε inm e n Λ ( ρ ) m , (12)</formula>
<formula><location><page_5><loc_26><loc_49><loc_90><loc_51></location>-3 η T ( ε -1) { f ( a ) 2 Λ ( u ) n + f ( a ) 1 ( e · Λ ( u ) ) e n } ε inj . (13)</formula>
<text><location><page_5><loc_8><loc_46><loc_80><loc_47></location>The effect of turbulent diffusivity, which is anisotropic due to the Coriolis force, is given by:</text>
<formula><location><page_5><loc_24><loc_41><loc_90><loc_44></location>η ijk = 3 η T {( 2 f ( a ) 1 -f ( d ) 2 ) ε ijk -2 f ( a ) 1 e i e n ε njk + εC ω f ( d ) 4 e j δ ik } . (14)</formula>
<text><location><page_5><loc_8><loc_36><loc_90><loc_39></location>The last term in Eq. (14) describes Radler's Ω × J effect. The functions f ( a,d ) { 1 -11 } depend on the Coriolis number. They can be found in Pipin (2008); see also Pipin and Kosovichev (2011a) or Pipin and Sokoloff</text>
<text><location><page_5><loc_42><loc_33><loc_90><loc_35></location>b 2 , which measures the ratio between magnetic and kinetic</text>
<text><location><page_5><loc_8><loc_32><loc_45><loc_34></location>(2011)). In the model, the parameter ε = µ 0 ρ u 2</text>
<text><location><page_5><loc_8><loc_22><loc_90><loc_32></location>energies of the fluctuations in the background turbulence, is assumed to be equal to 1. This corresponds to perfect energy equipartition. The ε contribution in the second line of Eq. (12) describes the paramagnetic effect (Kleeorin and Rogachevskii 2003). In the state of perfect energy equipartition the effect of diamagnetic pumping is compensated by the paramagnetic effect. We can, formally, skip the second line in Eq. (12) from our consideration if ε = 1. To compare the magnitude of the helicity-vorticity pumping effect with the diamagnetic effect we will show results for the pumping velocity distribution with ε = 0.</text>
<text><location><page_5><loc_8><loc_19><loc_90><loc_22></location>The contribution of small-scale magnetic helicity χ = a · b ( a is the fluctuating vector-potential of the magnetic field) to the α -effect is defined as</text>
<formula><location><page_5><loc_33><loc_14><loc_90><loc_17></location>α ( M ) ij = 2 f ( a ) 2 δ ij χτ c µ 0 ρglyph[lscript] 2 -2 f ( a ) 1 e i e j χτ c µ 0 ρglyph[lscript] 2 . (15)</formula>
<text><location><page_5><loc_8><loc_8><loc_90><loc_12></location>The nonlinear feedback of the large-scale magnetic field to the α -effect is described by a dynamical quenching due to the constraint of magnetic helicity conservation. The magnetic helicity, χ , subject to a conservation law, is described by the following equation (Kleeorin and Rogachevskii 1999, Subramanian and</text>
<text><location><page_6><loc_8><loc_96><loc_9><loc_96></location>6</text>
<text><location><page_6><loc_45><loc_96><loc_52><loc_96></location>V. V. Pipin</text>
<text><location><page_6><loc_8><loc_92><loc_23><loc_94></location>Brandenburg 2004):</text>
<formula><location><page_6><loc_34><loc_85><loc_90><loc_88></location>∂χ ∂t = -2 ( E· B ) -χ R χ τ c + ∇· ( η χ ∇ ¯ χ ) , (16)</formula>
<text><location><page_6><loc_8><loc_73><loc_90><loc_81></location>where τ c is a typical convective turnover time. The parameter R χ controls the helicity dissipation rate without specifying the nature of the loss. The turnover time τ c decreases from about 2 months at the bottom of the integration domain, which is located at 0 . 71 R glyph[circledot] , to several hours at the top boundary located at 0 . 99 R glyph[circledot] . It seems reasonable that the helicity dissipation is most efficient near the surface. The last term in Eq. (16) describes a turbulent diffusive flux of magnetic helicity (Mitra et al. 2010).</text>
<text><location><page_6><loc_8><loc_59><loc_90><loc_73></location>We use the solar convection zone model computed by Stix (2002), in which the mixing-length is defined as glyph[lscript] = α MLT ∣ ∣ Λ ( p ) ∣ ∣ -1 , where Λ ( p ) = ∇ log p quantifies the pressure variation, and α MLT = 2. The turbulent diffusivity is parameterized in the form, η T = C η η (0) T , where η (0) T = u ' glyph[lscript] 3 is the characteristic mixinglength turbulent diffusivity, glyph[lscript] is the typical correlation length of the turbulence, and C η is a constant to control the efficiency of large-scale magnetic field dragging by the turbulent flow. Currently, this parameter cannot be introduced in the mean-field theory in a consistent way. In this paper we use C η = 0 . 05. The differential rotation profile, Ω = Ω 0 f Ω ( x, µ ) (shown in Fig.3a) is a slightly modified version of the analytic approximation proposed by Antia et al. (1998):</text>
<formula><location><page_6><loc_27><loc_41><loc_90><loc_55></location>f Ω ( x, µ ) = 1 Ω 0 [ Ω 0 +55( x -0 . 7) φ ( x, x 0 ) φ ( -x, -0 . 96) (17) -200 ( x -0 . 95) φ ( x, 0 . 96)) + (21 P 3 ( µ ) + 3 P 5 ( µ )] ( µ 2 j p ( x ) + 1 -µ 2 j e ( x ) ) /Ω 0 , j p = 1 1 + exp ( 0 . 709 -x 0 . 02 ) , j e = 1 1 + exp ( 0 . 692 -x 0 . 01 ) ,</formula>
<text><location><page_6><loc_8><loc_33><loc_90><loc_37></location>where Ω 0 = 2 . 87 · 10 -6 s -1 is the equatorial angular velocity of the Sun at the surface, x = r/R glyph[circledot] , φ ( x, x 0 ) = 0 . 5 [1 + tanh [100( x -x 0 )]], and x 0 = 0 . 71.</text>
<text><location><page_6><loc_8><loc_25><loc_90><loc_29></location>2.2.2 Pumping effects in the solar convection zone. The components of the strain tensor ˜ V in a spherical coordinate system are given by the matrix:</text>
<formula><location><page_6><loc_38><loc_16><loc_60><loc_21></location>˜ V = 0 0 ˜ V ( r,ϕ ) 0 0 ˜ V ( θ,ϕ ) ˜ V ( r,ϕ ) ˜ V ( θ,ϕ ) 0 ,</formula>
<text><location><page_6><loc_8><loc_8><loc_90><loc_12></location>where we take into account only the azimuthal component of the large-scale flow, ˜ V ( r,ϕ ) = r sin θ∂ r Ω ( r, θ ), ˜ V ( θ,ϕ ) = sin θ∂ θ Ω ( r, θ ), so ˆ V ( B ) = ( B ˜ V ( r,ϕ ) , B ˜ V ( θ,ϕ ) , B p i ˜ V ( i,ϕ ) ) . Substituting this into Eq. (3) we find the</text>
<figure>
<location><page_7><loc_8><loc_55><loc_90><loc_94></location>
<caption>Figure 3. Distributions of the angular velocity and the turbulent parameters, and the kinetic helicity inside the solar convection zone. The bottom panel shows the patterns of the pumping velocity fields for the toroidal magnetic field(left) and for the poloidal field(right). They were computed on the basis of Eqs. (18,19,20).</caption>
</figure>
<text><location><page_7><loc_8><loc_48><loc_74><loc_49></location>components of the mean-electromotive force for the helicity-vorticity pumping effect,</text>
<formula><location><page_7><loc_12><loc_43><loc_86><loc_46></location>E ( H ) r = Ω ∗ τ c 2 sin θ { [ h K ( f ( γ ) 3 -f ( γ ) 1 ) + h C ( f ( γ ) 4 -f ( γ ) 2 )] x ∂ ˜ Ω ∂x -2 ( ˜ Ω -1 )[ h K f ( γ ) 1 + h C f ( γ ) 2 ] } B,</formula>
<formula><location><page_7><loc_17><loc_27><loc_61><loc_34></location>--( ˜ Ω -1 ) Ω ∗ τ c x sin θ [ h K f ( γ ) 1 + h C f ( γ ) 2 ] ( µ ∂A ∂x + sin 2 θ x ∂A ∂µ ) ,</formula>
<formula><location><page_7><loc_12><loc_32><loc_90><loc_45></location>(18) E ( H ) θ = Ω ∗ τ c 2 { sin 2 θ [ h K ( f ( γ ) 3 -f ( γ ) 1 ) + h C ( f ( γ ) 4 -f ( γ ) 2 )] ∂ ˜ Ω ∂µ -2 µ ( ˜ Ω -1 )[ h K f ( γ ) 1 + h C f ( γ ) 2 ] } B, (19) E ( H ) φ = Ω ∗ τ c 2 sin θ x [ h K ( f ( γ ) 3 + f ( γ ) 1 ) + h C ( f ( γ ) 4 + f ( γ ) 2 )] ∂ ( ˜ Ω,A ) ∂ ( x, µ ) (20)</formula>
<text><location><page_7><loc_8><loc_21><loc_90><loc_25></location>where h C = C C χ µ 0 ρglyph[lscript] 2 . It remains to define the kinetic helicity distribution. We use a formula proposed in our earlier study (see Kuzanyan et al. 2006),</text>
<formula><location><page_7><loc_32><loc_15><loc_66><loc_19></location>h K = C η C K u (0)2 2 ∂ ∂r log ( ρ √ u (0)2 ) F 1 cos θ,</formula>
<text><location><page_7><loc_8><loc_8><loc_90><loc_14></location>where F 1 ( Ω ∗ ) was defined in the above cited paper. The radial profile of h K cos θ is shown in Figure 3. The radial profile of kinetic helicity is shown in Figure 3a of the above cited paper. The parameters C K , C are introduced to switch on/off the pumping effects in the model.</text>
<figure>
<location><page_8><loc_12><loc_72><loc_85><loc_93></location>
<caption>Figure 4. The patterns of the total (including the diamagnetic and the density gradient effects) pumping velocity fields for the toroidal magnetic field(left) and for the poloidal field(right).</caption>
</figure>
<text><location><page_8><loc_8><loc_56><loc_90><loc_67></location>The expressions given by Eq. (3) are valid for the case of weak shear, when τ c max (∣ ∣ ∇ i V j ∣ ∣ ) glyph[lessmuch] 1. In terms of the strain tensor ˜ V this condition of weak shear implies Ω glyph[star] max (∣ ∣ ∣ r∂ r ˜ Ω ∣ ∣ ∣ , ∣ ∣ ∣ ∂ θ ˜ Ω ∣ ∣ ∣ ) glyph[lessmuch] 1. This is not valid at the bottom of the solar convection zone where the radial gradient of the angular velocity is strong and Ω glyph[star] glyph[greatermuch] 1 and τ c max (∣ ∣ ∇ i V j ∣ ∣ ) ≈ 2. Leprovost and Kim (2010) suggested that this pumping effect is quenched with increasing shear inversely proportional to ( τ c max (∣ ∣ ∇ i V j ∣ ∣ )) 1 ... 2 . Therefore, we introduce an ad-hoc quenching function for the pumping effect:</text>
<formula><location><page_8><loc_34><loc_49><loc_90><loc_54></location>f ( S ) = 1 1 + C S Ω ∗ s (∣ ∣ ∣ ∣ ∣ r ∂ ˜ Ω ∂r ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∂ ˜ Ω ∂θ ∣ ∣ ∣ ∣ ∣ ) s , (21)</formula>
<text><location><page_8><loc_8><loc_41><loc_90><loc_47></location>where C S is a constant to control the magnitude of the quenching, and s = 1. Results by Leprovost and Kim (2010) suggest 1 < s < 2 in relation to geometry of the large-scale shear. We find that for the solar convection zone the amplitude of the pumping effect does not change very much ( ∼ 1 m / s) with s varying in the range 1 . . . 2.</text>
<text><location><page_8><loc_8><loc_27><loc_90><loc_40></location>From the given relations, using E ( H ) = U (eff) × B , we find the effective drift velocity, U (eff) , due to the helicity-vorticity pumping effect. Taking into account the variation of turbulence parameters in the solar convection zone we compute U (eff) . The bottom panel of Figure 3 shows the distribution of the velocity field U ( eff ) for the helicity-vorticity pumping effect for the toroidal and poloidal components of the large-scale magnetic field. The maximum velocity drift occurs in the middle and at the bottom of the convection zone. The direction of drift has equatorial and polar cells corresponding to two regions in the solar convection zone with different signs of the radial gradient of the angular velocity. The anisotropy in transport of the toroidal and poloidal components of the large-scale magnetic field is clearly seen.</text>
<text><location><page_8><loc_8><loc_8><loc_90><loc_27></location>The other important pumping effects are due to mean density and turbulence intensity gradients (Zeldovich 1957, Kichatinov 1991, Kichatinov and Rudiger 1992, Tobias et al. 2001). These effects were estimated using Eq. (12). For these calculations we put C η = 1, ε = 0, C K = 1 and χ = 0. Figure 4 shows the sum of the pumping effects for the toroidal and poloidal components of mean magnetic fields including the helicity-vorticity pumping effect. In agreement with previous studies, it is found that the radial direction is the principal direction of mean-field transport in the solar convection zone. In its upper part the transport is downward because of pumping due to the density gradient (Kichatinov 1991). At the bottom of the convection zone the diamagnetic pumping effect produces downward transport as well (Kichatinov 1991, Rudiger and Brandenburg 1995). The diamagnetic pumping is quenched inversely proportional to the Coriolis number (e.g., Kichatinov 1991, Pipin 2008) and it has the same order of magnitude as the helicity-vorticity pumping effect. The latter effect modifies the direction of effective drift of the toroidal magnetic field near the bottom of the convection zone. There is also upward drift of the toroidal field at</text>
<text><location><page_9><loc_23><loc_82><loc_23><loc_85></location>·</text>
<table>
<location><page_9><loc_8><loc_83><loc_51><loc_92></location>
<caption>Table 1. The parameters of the models. Here, B max is the maximum of the toroidal magnetic field strength inside the convection zone, P is the dynamo period of the model.</caption>
</table>
<text><location><page_9><loc_8><loc_73><loc_90><loc_81></location>low latitudes in the middle of the convection zone. It results from the combined effects of density gradient and global rotation (Kichatinov 1991, Krivodubskij 2004). For the poloidal magnetic field the transport is downward everywhere in the convection zone. At the bottom of the convection zone the action of the diamagnetic pumping on the meridional component of the large-scale magnetic field is amplified due to the helicity-vorticity pumping effect.</text>
<text><location><page_9><loc_8><loc_68><loc_90><loc_72></location>The obtained pattern of large-scale magnetic field drift in the solar convection zone does not take into account nonlinear effects, e.g., because of magnetic buoyancy. The effect of mean-field buoyancy is rather small compared with flux-tube buoyancy (Kichatinov and Pipin 1993, cf. Guerrero and Kapyla 2011).</text>
<text><location><page_9><loc_8><loc_59><loc_90><loc_68></location>To find out the current helicity counterpart of the pumping effect we analyze dynamo models by solving Eqs. (6, 16). The governing parameters of the model are C η = 0 . 05, C ω = 1 3 C α . We discuss the choice of the governing parameters later. The other parameters of the model are given in the Table 1. Because of the weakening factor C η the magnitude of the pumping velocity is about one order of magnitude smaller than what is shown in Figure 4.</text>
<text><location><page_9><loc_8><loc_56><loc_90><loc_58></location>Following Pipin and Kosovichev (2011b), we use a combination of 'open' and 'closed' boundary conditions at the top, controlled by a parameter δ = 0 . 95, with</text>
<formula><location><page_9><loc_40><loc_51><loc_90><loc_54></location>δ η T r e B +(1 -δ ) E θ = 0 . (22)</formula>
<text><location><page_9><loc_8><loc_45><loc_90><loc_49></location>This is similar to the boundary condition discussed by Kitchatinov et al. (2000). For the poloidal field we apply a combination of the local condition A = 0 and the requirement of a smooth transition from the internal poloidal field to the external potential (vacuum) field:</text>
<formula><location><page_9><loc_31><loc_38><loc_90><loc_43></location>δ ( ∂A ∂r ∣ ∣ ∣ ∣ r = r e -∂A ( vac ) ∂r ∣ ∣ ∣ ∣ ∣ r = r e ) +(1 -δ ) A = 0 . (23)</formula>
<text><location><page_9><loc_8><loc_32><loc_90><loc_37></location>We assume perfect conductivity at the bottom boundary with standard boundary conditions. For the magnetic helicity, similar to Guerrero et al. (2010), we put ∇ r ¯ χ = 0 at the bottom of the domain and ¯ χ = 0 at the top of the convection zone.</text>
<text><location><page_9><loc_8><loc_13><loc_90><loc_32></location>In this paper we study dynamo models which include Radler's Ω × J dynamo effect due to a large-scale current and global rotation (Radler 1969). There is also a dynamo effect due to large-scale shear and current (Rogachevskii and Kleeorin 2003). The motivation to consider these addional turbulent sources in the mean-field dynamo comes from DNS dynamo experiments (Brandenburg and Kapyla 2007, Kapyla et al. 2008, Hughes and Proctor 2009, Kapyla et al. 2009) and from our earlier studies (Pipin and Seehafer 2009, Seehafer and Pipin 2009). The dynamo effect due to large-scale current gives an additional source of large-scale poloidal magnetic field. This can help to solve the issue with the dynamo period being otherwise too short. Also, in the models the large-scale current dynamo effect produces less overlapping cycles than dynamo models with α -effect alone. The choice of parameters in the dynamo is justified by our previous studies (Pipin and Seehafer 2009, Pipin and Kosovichev 2011c), where we showed that solar-types dynamos can be obtained for C α /C ω > 2. In those papers we find the approximate threshold to be C α ≈ 0 . 02 for a given diffusivity dilution factor of C η = 0 . 05.</text>
<text><location><page_9><loc_8><loc_8><loc_90><loc_12></location>As follows from the results given in Fig.4, the kinetic helicity-vorticity pumping effect has a negligible contribution in the near-surface layers, where downward pumping due to density stratification dominates. Therefore, it is expected that the surface dynamo waves are not affected if we discard magnetic helicity</text>
<figure>
<location><page_10><loc_7><loc_42><loc_79><loc_94></location>
<caption>Figure 5. The time-latitude diagrams for the toroidal and radial magnetic fields for the models D1 and D2: a) the model D1, the toroidal field (iso-contours, ± . 25 KG ) near the surface and the radial field (gray-scale density plot); b) the model D1, the toroidal field at the bottom of the solar convection zone, the contours drawn in the range ± . 5 KG ; c) the same as for item a) for the model D2; d) the same as for item b) for the model D2.</caption>
</figure>
<text><location><page_10><loc_8><loc_24><loc_90><loc_35></location>from the dynamo equations. Figure 5 shows time-latitude diagrams for toroidal and radial magnetic fields at the surface and for toroidal magnetic field at the bottom of the convection zone for two dynamo models D1 and D2 with and without the helicity-vorticity pumping effect, but magnetic helicity is taken into account as the main dynamo quenching effect. To compare with observational data from a time-latitude diagram of sunspot area (e.g., Hathaway 2011), we multiply the toroidal field component B by factor sin θ . This gives a quantity, which is proportional to the flux of large-scale toroidal field at colatitude θ . We further assume that the sunspot area is related to this flux.</text>
<text><location><page_10><loc_8><loc_13><loc_90><loc_24></location>Near the surface, models D1 and D2 give similar patterns of magnetic field evolution. At the bottom of the convection zone model D1 shows both poleward and equatorward branches of the dynamo wave propagation that is in agreement with the Parker-Yoshimura rule. Both branches have nearly the same time scale that equals glyph[similarequal] 16 years. The results from model D2 show that at the bottom of the convection zone the poleward branch of the dynamo wave dominates. Thus we conclude that the helicity-vorticity pumping effect alters the propagation of the dynamo wave near the bottom of the solar convection zone. We find that models with magnetic helicity contributions to the pumping effect do not change this conclusion.</text>
<text><location><page_10><loc_8><loc_8><loc_90><loc_12></location>Figure 6 shows a typical snapshot of the magnetic helicity distribution in the northern hemisphere for all our models. The helicity has a negative sign in the bulk of the solar convection zone. Regions with positive current helicity roughly correspond to domains of the negative large-scale current helicity concentration.</text>
<figure>
<location><page_11><loc_8><loc_81><loc_27><loc_94></location>
<caption>Figure 8 shows in more detail the latitudinal drift of the maximum of the toroidal magnetic field evolution during the cycle (left panel in the Figure 8),</caption>
</figure>
<figure>
<location><page_11><loc_30><loc_81><loc_49><loc_94></location>
</figure>
<text><location><page_11><loc_33><loc_81><loc_34><loc_82></location>×</text>
<paragraph><location><page_11><loc_8><loc_77><loc_90><loc_80></location>Figure 6. Snapshots for the mean magnetic field and the current helicity distributions at the north hemisphere in the model D4. Left panel shows the field lines of the poloidal component of the mean magnetic field. The right panel shows the toroidal magnetic field (iso-contours ± 500G) and the current helicity (gray scale density plot).</paragraph>
<text><location><page_11><loc_8><loc_64><loc_90><loc_73></location>They are located in the middle of the solar convection zone and at the high and low latitudes near the top of the solar convection zone. As follows from Fig. 6, the pumping effect due to current helicity may be efficient in the upper part of the solar convection zone where it might intensify the equatorial drift of the dynamo wave along iso-surfaces of the angular velocity. We find that the pumping effect that results from magnetic helicity is rather small in our models. This may be due to the weakness of the magnetic field. Observations (Zhang et al. 2010) give about one order magnitude larger current helicity than what</text>
<text><location><page_11><loc_8><loc_60><loc_90><loc_64></location>is shown in Fig. 6. In the model we estimate the current helicity as H C = χ µ 0 glyph[lscript] 2 . This result depends</text>
<text><location><page_11><loc_8><loc_51><loc_90><loc_60></location>essentially on the mixing length parameter glyph[lscript] . The stronger helicity is concentrated to the surface, the larger H C . In observations, we do not know from were the helical magnetic structures come from. In view of the given uncertainties we estimate the probable effect of a larger magnitude of magnetic helicity in the model by increasing the parameter C C to 10 (model D3). In addition, we consider the results for the nonlinear model D4. It has a higher C α and a lower R χ to increase the nonlinear impact of the magnetic helicity on the large-scale magnetic field evolution.</text>
<text><location><page_11><loc_8><loc_40><loc_90><loc_51></location>The top panel of Figure 7 shows a time-latitude diagram of toroidal magnetic field and current helicity evolution near the surface for model D4. We find a positive sign of current helicity at the decay edges of the toroidal magnetic field butterfly diagram. There are also areas with positive magnetic helicity at high latitudes at the growing edges of the toroidal magnetic field butterfly diagram. The induced pumping velocity is about 1 cm s -1 . The increase of the magnetic helicity pumping effect by a factor of 10 (model D3) shifts the latitude of the maximum of the toroidal magnetic field by about 5 · toward the equator. The induced pumping velocity is about 5 cm s -1 .</text>
<text><location><page_11><loc_8><loc_33><loc_90><loc_39></location>Stronger nonlinearity (model D4) and a stronger magnetic helicity pumping effect (model D3) modify the butterfly diagram in different ways. Model D3 shows a simple shift of the maximum of toroidal magnetic field toward the equator. Model D4 shows a fast drift of large-scale toroidal field at the beginning of a cycle and a slow-down of the drift velocity as the cycle progresses.</text>
<formula><location><page_11><loc_35><loc_25><loc_90><loc_27></location>λ max ( t ) = 90 · -max θ> 45 · ( | B S ( θ ) | sin θ ) , (24)</formula>
<text><location><page_11><loc_8><loc_21><loc_88><loc_23></location>and the latitudinal drift of the centroid position of the toroidal magnetic field flux (cf. Hathaway 2011)</text>
<formula><location><page_11><loc_35><loc_15><loc_90><loc_19></location>λ C ( t ) = 90 · -∫ π/ 2 0 θB S ( θ ) sin θdθ ∫ π/ 2 0 B S ( θ ) sin θdθ , (25)</formula>
<text><location><page_11><loc_8><loc_8><loc_90><loc_12></location>where B S ( θ ) = 〈 B ( r, θ ) 〉 (0 . 9 , 0 . 99) R is the toroidal magnetic field, which is averaged over the surface layers. Note that the overlap between subsequent cycles influences the value of λ C more than the value of λ max . The behaviour of λ max in models D1,D2 and D3 reproduces qualitatively the exponential drift of maximum</text>
<figure>
<location><page_12><loc_8><loc_69><loc_78><loc_94></location>
</figure>
<text><location><page_12><loc_76><loc_85><loc_79><loc_86></location>×</text>
<text><location><page_12><loc_77><loc_84><loc_78><loc_85></location>C</text>
<figure>
<location><page_12><loc_8><loc_37><loc_89><loc_61></location>
<caption>Figure 7. Top, the near-surface time-latitude diagrams for the toroidal magnetic field and the current helicity for the models D4. Bottom, the near-surface time-latitude diagrams for the toroidal magnetic field and the latitudinal component of the drift velocity induced by the magnetic helicity for the model D3.Figure 8. The drift of the latitude of maximum (left) and the centroid position of the magnetic flux at the near-surface layer in the models D(1-4). The dash-dotted line shows results for the model D1, the red dashed line - for the model D2, the solid black line - for the model D3, the black dashed line - for the model D4 and the solid green line shows the exponential law of the sunspot area centroid drift, as suggested by Hathaway (2011).</caption>
</figure>
<text><location><page_12><loc_8><loc_27><loc_41><loc_28></location>latitude as suggested by Hathaway (2011):</text>
<formula><location><page_12><loc_39><loc_21><loc_59><loc_24></location>λ C ( t ) = 28 · exp ( -12 t 90 ) ,</formula>
<text><location><page_12><loc_8><loc_8><loc_90><loc_19></location>where t is time measured in years. Model D4 shows a change between fast (nearly steady dynamo wave) drift at the beginning of the cycle to slow drift at the decaying phase of the cycle. The overlap between subsequent cycles is growing from model D1 to model D4. In all the models the highest latitude of the centroid position of the toroidal magnetic flux is below 30 · . Models D3 and D4 have nearly equal starting latitude of the centroid position. It is about 24 · . This means that a model with increased magnetic helicity pumping produces nearly the same effect for the shift of the centroid position as a model with a strong nonlinear effect of magnetic helicity.</text>
<section_header_level_1><location><page_13><loc_8><loc_92><loc_33><loc_94></location>3 Discussion and conclusions</section_header_level_1>
<text><location><page_13><loc_8><loc_83><loc_90><loc_90></location>We have shown that the interaction of helical convective motions and differential rotation in the solar convection zone produces a turbulent drift of large-scale magnetic field. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. The large-scale vorticity vector roughly follows to iso-surfaces of angular velocity. Since the direction of the drift depends on the sign of helicity, the pumping effect is governed by the Parker-Yoshimura rule (Parker 1955, Yoshimura 1975).</text>
<text><location><page_13><loc_8><loc_66><loc_90><loc_82></location>The effect is computed within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. In the calculations, we have assumed that the turbulent kinetic and current helicities are given. The calculations were done for arbitrary Coriolis number. In agreement with Mitra et al. (2009) and Rogachevskii et al. (2011), the analytical calculations show that the leading effect of pumping is described by a large-scale magnetic drift in the direction of the large-scale vorticity vector and by anisotropic pumping which produces a drift of toroidal and poloidal components of the field in opposite directions. The component of the drift that is induced by global rotation and helicity (second line in Eq. (3)) is rather small compared to the main effect. The latter conclusion should be checked separately for a different model of background turbulence, taking into account the generation of kinetic helicity due to global rotation and stratification in a turbulent medium.</text>
<text><location><page_13><loc_8><loc_48><loc_90><loc_66></location>We have estimated the pumping effect for the solar convection zone and compared it with other turbulent pumping effects including diamagnetic pumping and turbulent pumping that results from magnetic fluctuations in stratified turbulence (Kichatinov 1991, Pipin 2008). The latter is sometimes referred to as 'density-gradient pumping effect' (Krivodubskij 2004). The diamagnetic pumping is upward in the upper part of the convection zone and downward near the bottom. The velocity field of density-gradient pumping is more complicated (see Figure 4). However, its major effect is concentrated near the surface. Both diamagnetic pumping and density-gradient pumping effects are quenched inversely proportional to the Coriolis number (Kichatinov 1991, Pipin 2008). The helicity-vorticity pumping effect modifies the direction of large-scale magnetic drift at the bottom of the convection zone. This effect was illustrated by a dynamo model that shows a dominant poleward branch of the dynamo wave at the bottom of the convection zone.</text>
<text><location><page_13><loc_8><loc_37><loc_90><loc_48></location>It is found that the magnetic helicity contribution of the pumping effect can be important for explaining the fine structure of the sunspot butterfly diagram. In particular, the magnetic helicity contribution results in a slow-down of equatorial propagation of the dynamo wave. The slow-down starts just before the maximum of the cycle. Observations indicate a similar behavior in sunspot activity (Ternullo 2007, Hathaway 2011). A behavior like this can be seen in flux-transport models as well (Rempel 2006). For the time being it is unclear what are the differences between different dynamo models and how well do they reproduce the observations. A more detailed analysis is needed.</text>
<section_header_level_1><location><page_13><loc_8><loc_32><loc_21><loc_33></location>REFERENCES</section_header_level_1>
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<list_item><location><page_14><loc_8><loc_17><loc_90><loc_20></location>V. V. Pipin and A. G. Kosovichev. The Subsurface-shear-shaped Solar αΩ Dynamo. ApJL , 727:L45, 2011b.</list_item>
<list_item><location><page_14><loc_8><loc_14><loc_90><loc_17></location>V. V. Pipin and A. G. Kosovichev. Mean-field Solar Dynamo Models with a Strong Meridional Flow at the Bottom of the Convection Zone. Astrophys. J. , 738:104, 2011c.</list_item>
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<text><location><page_15><loc_10><loc_92><loc_16><loc_94></location>065903 .</text>
<text><location><page_15><loc_8><loc_89><loc_90><loc_92></location>Radler, K.-H.. On the electrodynamics of turbulent fields under the influence of corilois forces. Monats. Dt. Akad. Wiss., 11:, 194-201, 1969.</text>
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<section_header_level_1><location><page_15><loc_8><loc_33><loc_18><loc_34></location>Appendix A</section_header_level_1>
<text><location><page_15><loc_8><loc_29><loc_69><loc_31></location>To compute E it is convenient to write equations (1) and (2) in Fourier space:</text>
<formula><location><page_15><loc_21><loc_8><loc_90><loc_27></location>( ∂ ∂t + ηz 2 ) ˆ b j = i µ z l ˆ u j ( z ) ¯ B l + (26) + iz l ∫ [ ˆ b l ( z -q ) ̂ ¯ V j ( q ) -ˆ b j ( z -q ) ̂ ¯ V l ( q ) ] d q + ̂ G j . ( ∂ ∂t + νz 2 ) ˆ u i = ˆ f i + ̂ F i -2 ( Ω ̂ z ) ( ̂ z × ̂ u ) i (27) -iπ if ( z ) z l ∫ [ ˆ u l ( z -q ) ̂ ¯ V f ( q ) + ˆ u f ( z -q ) ̂ ¯ V l ( q ) ] d q + i µ ˆ b i ( z ) ( z · ¯ B ) ,</formula>
<text><location><page_16><loc_8><loc_86><loc_90><loc_94></location>where the turbulent pressure was excluded from (2) by convolution with tensor π ij ( z ) = δ ij -ˆ z i ˆ z j , δ ij is the Kronecker symbol and ̂ z is a unit wave vector. The equations for the second-order moments that make contributions to the mean-electro-motive force(MEMF) can be found directly from (26, 27). As the preliminary step we write the equations for the second-order products of the fluctuating fields, and make the ensemble averaging of them,</text>
<formula><location><page_16><loc_12><loc_47><loc_90><loc_84></location>∂ ∂t 〈 ˆ u i ( z ) ˆ b j ( z ' ) 〉 = i z ' l ¯ B l 〈 ˆ u i ( z ) ˆ u j ( z ' )〉 -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z ) ˆ b j ( z ' ) 〉 (28) + i z ' l ∫ [〈 ˆ u i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ u i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V l ( q ) ] d q + i µ ¯ B l z l 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 + Th κ ij ( z , z ' ) -( ηz ' 2 + νz 2 ) 〈 ˆ u i ( z ) ˆ b j ( z ' ) 〉 , ∂ ∂t 〈 ˆ u i ( z ) ˆ u j ( z ' )〉 = -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z )ˆ u j ( z ' ) 〉 -2 ( Ω ̂ z ' ) ε jln ̂ z ' l 〈 ˆ u i ( z )ˆ u n ( z ' ) 〉 (29) -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V l ( q ) ] d q -i π jf ( z ' ) z ' l ∫ [〈 ˆ u i ( z ) ˆ u l ( z -q ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u i ( z ) ˆ u f ( z -q ) 〉 ̂ ¯ V l ( q ) ] d q + Th v ij ( z , z ' ) -ν ( z ' 2 + z 2 ) 〈 ˆ u i ( z ) ˆ u j ( z ' )〉 , ∂ ∂t 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 = Th h ij ( z , z ' ) -( ηz ' 2 + ηz 2 ) 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 (30) + i z ' l ∫ [〈 ˆ b i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ b i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q ,</formula>
<text><location><page_16><loc_8><loc_40><loc_90><loc_45></location>where, the terms Th ( κ ,v,h ) ij involve the third-order moments of fluctuating fields and second-order moments of them with the forcing term. Next, we apply the τ -approximation, substituting the Th ( κ ,v,h ) ij -terms by the corresponding τ relaxation terms of the second-order contributions,</text>
<formula><location><page_16><loc_30><loc_34><loc_90><loc_37></location>Th ( κ ) ij →-〈 ˆ m i ( z ) ˆ b j ( z ' ) 〉 /τ c , (31)</formula>
<formula><location><page_16><loc_30><loc_30><loc_90><loc_33></location>Th ( v ) ij →-〈 ˆ m i ( z ) ˆ m j ( z ' ) 〉 - 〈 ˆ m i ( z ) ˆ m j ( z ' ) 〉 (0) τ c , (32)</formula>
<formula><location><page_16><loc_30><loc_24><loc_90><loc_29></location>Th ( h ) ij →-〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 -〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 (0) τ c , (33)</formula>
<text><location><page_16><loc_8><loc_8><loc_90><loc_22></location>where the superscript . . . (0) denotes the moments of the background turbulence. Approximating these complicated contributions by the simple relaxation terms has to be considered as a questionable assumption. It involves additional assumptions (see Radler and Rheinhardt 2007), e.g., it is assumed that the second-order correlations in Eq. (8) do not vary significantly on the time scale of τ c . This assumption is consistent with scale separation between the mean and fluctuating quantities in the mean-field magneto hydrodynamics. The reader can find a comprehensive discussion of the τ -approximation in the above cited papers. Furthermore, we restrict ourselves to the high Reynolds numbers limit and discard the microscopic diffusion terms. The contributions of the mean magnetic field in the turbulent stresses will be neglected because they give the nonlinear terms in the cross helicity tensor. Also, τ c is independent on k (cf, Radler</text>
<text><location><page_17><loc_8><loc_88><loc_90><loc_94></location>et al. 2003, Rogachevskii and Kleeorin 2003, Brandenburg and Subramanian 2005) and it is independent on the mean fields as well. This should be taken into account in considering the nonlinear effects due to rotation. Taking all the above assumptions into account, we get the system of equations for the moments for the stationary case:</text>
<formula><location><page_17><loc_13><loc_65><loc_14><loc_67></location>〈</formula>
<formula><location><page_17><loc_14><loc_43><loc_90><loc_84></location>〈 ˆ u i ( z ) ˆ b j ( z ' ) 〉 τ c = -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z ) ˆ b j ( z ' ) 〉 + i z ' l ¯ B l ˆ u i ( z ) ˆ u j ( z ' ) 〉 (34) + i z ' l ∫ [〈 ˆ u i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ u i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ b j ( z ' ) 〉 ̂ ¯ V l ( q ) ] d q + i µ ¯ B l z l 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 , ˆ m i ( z ) ˆ m j ( z ' ) 〉 τ c = -2 ( Ω ̂ z ) ε iln ̂ z l 〈 ˆ u n ( z )ˆ u j ( z ' ) 〉 -2 ( Ω ̂ z ' ) ε jln ̂ z ' l 〈 ˆ u i ( z )ˆ u n ( z ' ) 〉 (35) -i π if ( z ) z l ∫ [〈 ˆ u l ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V f ( q ) + 〈 ˆ u f ( z -q ) ˆ u j ( z ' )〉 ̂ ¯ V l ( q ) ] d q -i π jf ( z ' ) z ' l ∫ [〈 ˆ u i ( z ) ˆ u l ( z -q ) 〉 ̂ ¯ V f ( q ) + 〈 ˆ u i ( z ) ˆ u f ( z -q ) 〉 ̂ ¯ V l ( q ) ] d q + 〈 ˆ m i ( z ) ˆ m j ( z ' ) 〉 (0) τ c 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 τ c = 〈 ˆ b i ( z ) ˆ b j ( z ' ) 〉 (0) τ c (36) + i z ' l ∫ [〈 ˆ b i ( z ) ˆ b l ( z ' -q ) 〉 ̂ ¯ V j ( q ) -〈 ˆ b i ( z ) ˆ b j ( z ' -q ) 〉 ̂ ¯ V l ( q ) ] d q ,</formula>
<text><location><page_17><loc_8><loc_36><loc_90><loc_41></location>To proceed further, we have to introduce some conventions and notations that are widely used in the literature. The double Fourier transformation of an ensemble average of two fluctuating quantities, say f and g , taken at equal times and at the different positions x , x ' , is given by</text>
<formula><location><page_17><loc_27><loc_32><loc_90><loc_35></location>〈 f ( x ) g ( x ' )〉 = ∫ ∫ 〈 ˆ f ( z ) ˆ g ( z ' ) 〉 e i ( z · x + z ' · x ' ) d 3 z d 3 z ' . (37)</formula>
<text><location><page_17><loc_8><loc_25><loc_90><loc_29></location>In the spirit of the general formalism of the two-scale approximation (Roberts and Soward 1975) we introduce 'fast' and 'slow' variables. They are defined by the relative r = x -x ' and the mean R = 1 2 ( x + x ' ) coordinates, respectively. Then, Eq. (37) can be written in the form</text>
<formula><location><page_17><loc_20><loc_20><loc_90><loc_23></location>〈 f ( x ) g ( x ' )〉 = ∫ ∫ 〈 ˆ f ( k + 1 2 K ) ˆ g ( -k + 1 2 K )〉 e i ( K · R + k · r ) d 3 K d 3 k , (38)</formula>
<text><location><page_17><loc_8><loc_13><loc_90><loc_18></location>where we have introduced the wave vectors k = 1 2 ( z -z ' ) and K = z + z ' . Then, following to Subramanian and Brandenburg (2004), we define the correlation function of ̂ f and ̂ g obtained from (38) by integration with respect to K ,</text>
<formula><location><page_17><loc_24><loc_8><loc_90><loc_11></location>Φ ( ˆ f, ˆ g, k , R ) = ∫ 〈 ˆ f ( k + 1 2 K ) ˆ g ( -k + 1 2 K )〉 e i ( K · R ) d 3 K . (39)</formula>
<text><location><page_18><loc_8><loc_91><loc_90><loc_94></location>For further convenience we define the second order correlations of momentum density, magnetic fluctuations and the cross-correlations of momentum and magnetic fluctuations via</text>
<formula><location><page_18><loc_27><loc_87><loc_90><loc_89></location>ˆ v ij ( k , R ) = Φ(ˆ u i , ˆ u j , k , R ) , 〈 u 2 〉 ( R ) = ∫ ˆ v ii ( k , R ) d 3 k , (40)</formula>
<formula><location><page_18><loc_26><loc_83><loc_90><loc_85></location>ˆ h ij ( k , R ) = Φ( ˆ b i , ˆ b j , k , R ) , 〈 b 2 〉 ( R ) = ∫ ˆ h ii ( k , R ) d 3 k , (41)</formula>
<formula><location><page_18><loc_26><loc_78><loc_90><loc_81></location>ˆ κ ij ( k , R ) = Φ(ˆ u i , ˆ b j , k , R ) , E i ( R ) = ε ijk ∫ ˆ κ jk ( k , R ) d 3 k . (42)</formula>
<text><location><page_18><loc_8><loc_70><loc_90><loc_76></location>We now return to equations (34), (35) and (36). As the first step, we solve these equations about Ω (nonlinear effects of the Coriolis force) and make the Taylor expansion with respect to the 'slow' variables and take the Fourier transformation, (39), about them. The details of this procedure can be found in (Subramanian and Brandenburg 2004). In result we get the following equations for the second moments</text>
<formula><location><page_18><loc_25><loc_60><loc_73><loc_67></location>ˆ κ ij τ c = -ι D (0) if ( B · k ) ( v fj -m fj µ ) + D (0) if ¯ V j,l ˆ κ fl -D (0) if ¯ V f,l ˆ κ lj + + 2 D (0) ip ˆ k p ˆ k f ˆ κ lj ¯ V f,l + D (0) if k l ¯ V f,l ∂ ˆ κ fj ∂k f ,</formula>
<formula><location><page_18><loc_24><loc_55><loc_63><loc_59></location>D (0) if = δ if + ψ Ω ˆ k p ε ifp + ψ 2 Ω ˆ k i ˆ k f 1 + ψ 2 Ω , ψ Ω = 2 ( Ω · ˆ k ) τ c</formula>
<formula><location><page_18><loc_18><loc_48><loc_81><loc_51></location>ˆ v ij τ c = T (0) ijnm ( ˆ v (0) nm τ c +2 ˆ k f ¯ V f,l ( ˆ k n ˆ v lm + ˆ k m ˆ v nl ) -¯ V n,l ˆ v lm -¯ V m,l ˆ v nl + k l ¯ V f,l ∂ ˆ v nm ∂k f ) ,</formula>
<formula><location><page_18><loc_28><loc_37><loc_29><loc_38></location>Ω</formula>
<formula><location><page_18><loc_16><loc_37><loc_90><loc_50></location>(44) T (0) ijnm = δ in δ jm + ψ Ω ˆ k p M ( ε inp δ jm + ε jmp δ in ) -ψ 2 Ω M ( δ ij π nm -δ nm ˆ k i ˆ k j + δ im ˆ k n ˆ k j + δ nj ˆ k i ˆ k m -2 δ n [ i δ j ] m ) , M = 1 + 4 ψ 2</formula>
<formula><location><page_18><loc_31><loc_30><loc_90><loc_33></location>ˆ h ij = ˆ h (0) ij + τ c ˆ h il ¯ V j,l + τ c ˆ h lj ¯ V i,l + τ c k l ¯ V f,l ∂ ˆ h ij ∂k f (45)</formula>
<text><location><page_18><loc_8><loc_22><loc_90><loc_28></location>These equations were solved with respect to the shear tensor, V i,j = ∇ j V i , by means of perturbation procedure. One remains to define the spectra of the background turbulence. We will adopt the isotropic form of the spectra (Roberts and Soward 1975). Additionally, the background magnetic fluctuations are helical while there is no prescribed kinetic helicity in the background turbulence:</text>
<formula><location><page_18><loc_31><loc_17><loc_90><loc_20></location>ˆ v (0) ij = { π ij ( k ) E ( k, R ) 8 πk 2 -i ε ijp k p H ( k, R ) 8 πk 4 } , (46)</formula>
<formula><location><page_18><loc_31><loc_12><loc_90><loc_16></location>ˆ h (0) ij = { π ij ( k ) B ( k, R ) 8 πk 2 -i ε ijp k p N ( k, R ) 8 πk 4 } , (47)</formula>
<text><location><page_18><loc_8><loc_8><loc_90><loc_11></location>where, the spectral functions E ( k, R ) , B ( k, R ) , N ( k, R ) define, respectively, the intensity of the velocity fluctuations, the intensity of the magnetic fluctuations and amount of current helicity in the background</text>
<formula><location><page_18><loc_87><loc_65><loc_90><loc_66></location>(43)</formula>
<text><location><page_19><loc_8><loc_92><loc_33><loc_94></location>turbulence. They are defined via</text>
<formula><location><page_19><loc_27><loc_83><loc_90><loc_91></location>〈 u (0)2 〉 = ∫ E ( k, R ) 4 πk 2 d 3 k , 〈 b (0)2 〉 = ∫ B ( k, R ) 4 πk 2 d 3 k , (48) h (0) K = ∫ H ( k, R ) 4 πk 2 d 3 k , h (0) C = 1 µρ ∫ N ( k, R ) 4 πk 2 d 3 k ,</formula>
<text><location><page_19><loc_8><loc_75><loc_90><loc_82></location>where h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇× b (0) 〉 µρ . In final results we use the relation between intensities of magnetic and kinetic fluctuations which is defined via B ( k, R ) = εµ ¯ ρE ( k, R ). The state with ε = 1 means equipartition between energies of magnetic an kinetic fluctuations in the background turbulence.</text>
<section_header_level_1><location><page_19><loc_8><loc_70><loc_18><loc_71></location>Appendix B</section_header_level_1>
<formula><location><page_19><loc_11><loc_14><loc_87><loc_67></location>f ( γ ) 1 = 1 (24 Ω ∗ ) 2 ( ( 1300 Ω ∗ 2 +391 ) arctan (2 Ω ∗ ) 2 Ω ∗ -1456 ( Ω ∗ 2 +1 ) arctan ( Ω ∗ ) Ω ∗ -3(32 Ω ∗ 2 -355) ) f ( γ ) 2 = 3 4 Ω ∗ 2 ( ( Ω ∗ 2 +1) arctan ( Ω ∗ ) Ω ∗ -1 ) f ( γ ) 3 = -1 36540 Ω ∗ 4 ( 5 (( 10672 Ω ∗ 2 +3872 ) Ω ∗ 2 +337 ) arctan (2 Ω ∗ ) 2 Ω ∗ -320 (( 515 Ω ∗ 2 -128 ) Ω ∗ 2 -895 ) arctan ( Ω ∗ ) Ω ∗ +3 (( 2304 Ω ∗ 2 +3380 ) Ω ∗ 2 -48295 ) ) f ( γ ) 4 = -1 24 Ω ∗ 4 ( 3 (( 11 Ω ∗ 2 +8 ) Ω ∗ 2 -7 ) ) arctan ( Ω ∗ ) Ω ∗ -( 31 Ω ∗ 2 -21 ) ) f ( γ ) 5 = -1 55296 Ω ∗ 4 ( ( 7472 Ω ∗ 4 +5016 Ω ∗ 2 -1685 ) arctan (2 Ω ∗ ) 2 Ω ∗ -32(79 Ω ∗ 4 +1410 Ω ∗ 2 +4475) arctan ( Ω ∗ ) Ω ∗ + 3 ( 9872 Ω ∗ 6 +183632 Ω ∗ 4 +238183 Ω ∗ 2 +48295 ) ( Ω ∗ 2 +1)(4 Ω ∗ 2 +1) ) f ( γ ) 6 = 1 32 Ω ∗ 4 ( (( 9 Ω ∗ 2 +30 ) Ω ∗ 2 -35 ) arctan ( Ω ∗ ) Ω ∗ -(( 101 Ω ∗ 2 +20 ) Ω ∗ 2 -105 ) 3 ( Ω ∗ 2 +1) ) f ( γ ) 7 = -1 55296 Ω ∗ 4 ( ( 20528 Ω ∗ 4 +16536 Ω ∗ 2 -1685 ) arctan (2 Ω ∗ ) 2 Ω ∗ -32 ( 577 Ω ∗ 4 +3660 Ω ∗ 2 +4475 ) arctan ( Ω ∗ ) Ω ∗ + 3 ( 80528 Ω ∗ 6 +282512 Ω ∗ 4 +258343 Ω ∗ 2 +48295 ) ( Ω ∗ 2 +1)(4 Ω ∗ 2 +1) ) )</formula>
<formula><location><page_19><loc_11><loc_11><loc_57><loc_14></location>f ( γ ) 8 = -( Ω ∗ 4 +7) 32 Ω ∗ 4 ( ( Ω ∗ 2 +5) arctan ( Ω ∗ ) Ω ∗ -( 13 Ω ∗ 2 +15 ) 3( Ω ∗ 2 +1)</formula>
</document>
[
{
"title": "ABSTRACT",
"content": "Geophysical and Astrophysical Fluid Dynamics Vol. 00, No. 00, Month 2013, 1-19",
"pages": [
1
]
},
{
"title": "Institute Solar-Terrestrial Physics, Irkutsk, Russia",
"content": "( July 31, 2021, Revision: 1.20 ) We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker-Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. We estimate the magnitude of the drift velocity and find that it can be several m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. We find that the helicity-vorticity pumping effect can have an influence on the features of the sunspot time-latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.",
"pages": [
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{
"title": "1 Introduction",
"content": "It is believed that the evolution of the large-scale magnetic field of the Sun is governed by the interplay between large-scale motions, like differential rotation and meridional circulation, turbulent convection flows and magnetic fields. One of the most important issues in solar dynamo theory is related to the origin of the equatorial drift of sunspot activity in the equatorial regions and, simultaneously at high latitudes, the poleward drift of the location of large-scale unipolar regions and quiet prominences. Parker (1955) and Yoshimura (1975) suggested that the evolution of large-scale magnetic activity of the Sun can be interpreted as dynamo waves propagating perpendicular to the direction of shear from the differential rotation. They found that the propagation can be considered as a diffusion process, which follows the iso-rotation surfaces of angular velocity in the solar convection zone. The direction of propagation can be modified by meridional circulation, anisotropic diffusion and the effects of turbulent pumping (see, e.g., Choudhuri et al. 1995, Kitchatinov 2002, Guerrero and de Gouveia Dal Pino 2008). The latter induces an effective drift of the large-scale magnetic field even though the mean flow of the turbulent medium may be zero. The turbulent pumping effects can be equally important both for dynamos without meridional circulation and for the meridional circulation-dominated dynamo regimes. For the latter case the velocity of turbulent pumping has to be comparable to the meridional circulation speed. It is known that an effect of this magnitude can be produced by diamagnetic pumping and perhaps by so-called topological pumping. Both effects produce pumping in the radial direction and have not a direct impact on the latitudinal drift of the large-scale magnetic field. Recently (Pipin 2008, Mitra et al. 2009, Leprovost and Kim 2010), it has been found that the helical convective motions and the helical turbulent magnetic fields interacting with large-scale magnetic fields and differential rotation can produce effective pumping in the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. It is believed that these regions, namely the tachocline beneath the solar convection zone and the subsurface shear layer, are important for the solar dynamo. Figure 1 illustrates the principal processes that induce the helicity-vorticity pumping effect. It is suggested that this effect produces an anisotropic drift of the large-scale magnetic field, which means that the different components of the large-scale magnetic field drift in different directions. Earlier work, e.g. by Kichatinov (1991) and Kleeorin and Rogachevskii (2003), suggests that the effect of anisotropy in the 2 transport of mean-field is related to nonlinear effects of the global Coriolis force on the convection. Also, nonlinear effects of the large-scale magnetic field result in an anisotropy of turbulent pumping (Kleeorin et al. 1996). It is noteworthy, that the helicity-vorticity effect produces an anisotropy of the large-scale magnetic field drift already in the case of slow rotation and a weak magnetic field. A comprehensive study of the linear helicity-vorticity pumping effect for the case of weak shear and slow rotation was given by Rogachevskii et al. (2011) and their results were extended by DNS with a more general test-field method Brandenburg et al. (2012). In this paper we analytically estimate the helicity-vorticity pumping effect taking into account the Coriolis force due to global rotation. The calculations were done within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. The results are applied to mean field dynamo models, which are used to examine this effect on the dynamo. The paper is structured as follows. In the next section we briefly outline the basic equations and assumptions, and consider the results of calculations. Next, we apply the results to the solar dynamo. In Section 3 we summarize the main results of the paper. The details of analytical calculations are given in the Appendices A and B.",
"pages": [
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{
"title": "2 Basic equations",
"content": "In the spirit of mean-field magnetohydrodynamics, we split the physical quantities of the turbulent conducting fluid into mean and fluctuating parts where the mean part is defined as an ensemble average. One assumes the validity of the Reynolds rules. The magnetic field B and the velocity V are decomposed as B = B + b and V = V + u , respectively. Hereafter, we use small letters for the fluctuating parts and capital letters with an overbar for mean fields. Angle brackets are used for ensemble averages of products. We use the two-scale approximation (Roberts and Soward 1975, Krause and Radler 1980) and assume that mean fields vary over much larger scales (both in time and in space) than fluctuating fields. The average effect of MHD-turbulence on the large-scale magnetic field (LSMF) evolution is described by the mean-electromotive force (MEMF), E = 〈 u × b 〉 . The governing equations for fluctuating magnetic field and velocity are written in a rotating coordinate system as follows: where G , F stand for nonlinear contributions to the fluctuating fields, p is the fluctuating pressure, Ω is the angular velocity responsible for the Coriolis force, ¯ V is mean flow which is a weakly variable in space, and f is the random force driving the turbulence. Equations (1) and (2) are used to compute the mean-electromotive force E = 〈 u × b 〉 . It was computed with the help of the equations for the second moments of fluctuating velocity and magnetic fields using the double-scale Fourier transformation and the minimal τ -approximations and for a given model of background turbulence. To simplify the estimation of nonlinear effects due to global rotation, we use scale-independent background turbulence spectra and correlation time. Details of the calculations are given in Appendix A. In what follows we discuss only those parts of the mean-electromotive force which are related to shear and the pumping effect.",
"pages": [
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{
"title": "2.1 Results",
"content": "The large-scale shear flow is described by the tensor V i,j = ∇ V i . It can be decomposed into a sum of strain and vorticity tensors, ∇ j V i = 1 2 ( V i,j + V j,i ) -1 2 ε ijp W p , where W = ∇ × V is the large-scale vorticity vector. The joint effect of large-scale shear, helical turbulent flows and magnetic fields can be expressed by the following contributions to the mean-electromotive force (omitting the α -effect): where ˜ V ( B ) = B j 2 ( V i,j + V j,i ) , e = Ω | Ω | is the unit vector along the rotation axis, τ c is the typical relaxation time of turbulent flows and magnetic fields, h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇ × b (0) 〉 µρ are kinetic and current helicity of the background turbulence. These parameters are assumed to be known in advance. Functions f ( γ ) n ( Ω ∗ ) are given in Appendix B, they depend on the Coriolis number Ω ∗ = 2 Ω 0 τ c and describe the nonlinear effect due the Coriolis force, and Ω 0 is the global rotation rate. For slow rotation, Ω ∗ glyph[lessmuch] 1, we perform a Taylor expansion of f ( γ ) n ( Ω ∗ ) and obtain The coefficients in the kinetic part of Eq. (4) are two times larger than those found by Rogachevskii et al. (2011). This difference results from our assumption that the background turbulence spectra and the correlation time are scale-independent. The results for the magnetic part are in agreement with our earlier findings (see Pipin 2008). The first term in Eq. (4) describes turbulent pumping with an effective velocity τ 2 c W 2 ( h C -h K ) and the second term describes anisotropic turbulent pumping. Its structure depends on the geometry of the shear flow. For large Coriolis numbers, Ω ∗ glyph[greatermuch] 1, only the kinetic helicity contributions 4 V. V. Pipin survive: Figure 2 show the dependence of the pumping effects on the Coriolis number. We observe that for the terms ( W × B ) and ˜ V ( B ) the effects of kinetic helicity are non-monotonic and have a maximum at Ω ∗ ≈ 1. The effects of current helicity for these terms are monotonically quenched with increasing values of Ω ∗ . The additional contributions in Eq. (3) are rather small in comparison with the main terms. Thus, we can conclude that the first line in Eq. (3) describes the leading effect of pumping due to the helicity of turbulent flows and magnetic field. Below, we drop the contributions from the second line in Eq. (3) from our analysis.",
"pages": [
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{
"title": "2.2 Helicity-vorticity pumping in the solar convection zone",
"content": "2.2.1 The dynamo model. To estimate the impact of this pumping effect on the dynamo we consider the example of a dynamo model which takes into account contributions of the mean electromotive force given by Eq. (3). The dynamo model employed in this paper has been described in detail by Pipin and Kosovichev (2011a,c). This type of dynamo was proposed originally by Brandenburg (2005). The reader may find the discussion for different types of mean-field dynamos in Brandenburg and Subramanian (2005) and Tobias and Weiss (2007). We study the standard mean-field induction equation in a perfectly conducting medium: where E = u × b is the mean electromotive force, with u, b being fluctuating velocity and magnetic field, respectively, U is the mean velocity (differential rotation and meridional circulation), and the axisymmetric magnetic field is: where θ is the polar angle. The expression for the mean electromotive force E is given by Pipin (2008). It is expressed as follows: The new addition due to helicity and mean vorticity effects is marked by E H . The tensor α ij represents the α -effect. It includes hydrodynamic and magnetic helicity contributions, where the hydrodynamic part of the α -effect is defined by α ( H ) ij , Λ ( ρ ) = ∇ log ρ quantifies the density stratification, Λ ( u ) = ∇ log ( η (0) T ) quantifies the turbulent diffusivity variation, and e = Ω / | Ω | is a unit vector along the axis of rotation. The turbulent pumping, γ ( Λ ) ij , depends on mean density and turbulent diffusivity stratification, and on the Coriolis number Ω ∗ = 2 τ c Ω 0 where τ c is the typical convective turnover time and Ω 0 is the global angular velocity. Following the results of Pipin (2008), γ ( Λ ) ij is expressed as follows: The effect of turbulent diffusivity, which is anisotropic due to the Coriolis force, is given by: The last term in Eq. (14) describes Radler's Ω × J effect. The functions f ( a,d ) { 1 -11 } depend on the Coriolis number. They can be found in Pipin (2008); see also Pipin and Kosovichev (2011a) or Pipin and Sokoloff b 2 , which measures the ratio between magnetic and kinetic (2011)). In the model, the parameter ε = µ 0 ρ u 2 energies of the fluctuations in the background turbulence, is assumed to be equal to 1. This corresponds to perfect energy equipartition. The ε contribution in the second line of Eq. (12) describes the paramagnetic effect (Kleeorin and Rogachevskii 2003). In the state of perfect energy equipartition the effect of diamagnetic pumping is compensated by the paramagnetic effect. We can, formally, skip the second line in Eq. (12) from our consideration if ε = 1. To compare the magnitude of the helicity-vorticity pumping effect with the diamagnetic effect we will show results for the pumping velocity distribution with ε = 0. The contribution of small-scale magnetic helicity χ = a · b ( a is the fluctuating vector-potential of the magnetic field) to the α -effect is defined as The nonlinear feedback of the large-scale magnetic field to the α -effect is described by a dynamical quenching due to the constraint of magnetic helicity conservation. The magnetic helicity, χ , subject to a conservation law, is described by the following equation (Kleeorin and Rogachevskii 1999, Subramanian and 6 V. V. Pipin Brandenburg 2004): where τ c is a typical convective turnover time. The parameter R χ controls the helicity dissipation rate without specifying the nature of the loss. The turnover time τ c decreases from about 2 months at the bottom of the integration domain, which is located at 0 . 71 R glyph[circledot] , to several hours at the top boundary located at 0 . 99 R glyph[circledot] . It seems reasonable that the helicity dissipation is most efficient near the surface. The last term in Eq. (16) describes a turbulent diffusive flux of magnetic helicity (Mitra et al. 2010). We use the solar convection zone model computed by Stix (2002), in which the mixing-length is defined as glyph[lscript] = α MLT ∣ ∣ Λ ( p ) ∣ ∣ -1 , where Λ ( p ) = ∇ log p quantifies the pressure variation, and α MLT = 2. The turbulent diffusivity is parameterized in the form, η T = C η η (0) T , where η (0) T = u ' glyph[lscript] 3 is the characteristic mixinglength turbulent diffusivity, glyph[lscript] is the typical correlation length of the turbulence, and C η is a constant to control the efficiency of large-scale magnetic field dragging by the turbulent flow. Currently, this parameter cannot be introduced in the mean-field theory in a consistent way. In this paper we use C η = 0 . 05. The differential rotation profile, Ω = Ω 0 f Ω ( x, µ ) (shown in Fig.3a) is a slightly modified version of the analytic approximation proposed by Antia et al. (1998): where Ω 0 = 2 . 87 · 10 -6 s -1 is the equatorial angular velocity of the Sun at the surface, x = r/R glyph[circledot] , φ ( x, x 0 ) = 0 . 5 [1 + tanh [100( x -x 0 )]], and x 0 = 0 . 71. 2.2.2 Pumping effects in the solar convection zone. The components of the strain tensor ˜ V in a spherical coordinate system are given by the matrix: where we take into account only the azimuthal component of the large-scale flow, ˜ V ( r,ϕ ) = r sin θ∂ r Ω ( r, θ ), ˜ V ( θ,ϕ ) = sin θ∂ θ Ω ( r, θ ), so ˆ V ( B ) = ( B ˜ V ( r,ϕ ) , B ˜ V ( θ,ϕ ) , B p i ˜ V ( i,ϕ ) ) . Substituting this into Eq. (3) we find the components of the mean-electromotive force for the helicity-vorticity pumping effect, where h C = C C χ µ 0 ρglyph[lscript] 2 . It remains to define the kinetic helicity distribution. We use a formula proposed in our earlier study (see Kuzanyan et al. 2006), where F 1 ( Ω ∗ ) was defined in the above cited paper. The radial profile of h K cos θ is shown in Figure 3. The radial profile of kinetic helicity is shown in Figure 3a of the above cited paper. The parameters C K , C are introduced to switch on/off the pumping effects in the model. The expressions given by Eq. (3) are valid for the case of weak shear, when τ c max (∣ ∣ ∇ i V j ∣ ∣ ) glyph[lessmuch] 1. In terms of the strain tensor ˜ V this condition of weak shear implies Ω glyph[star] max (∣ ∣ ∣ r∂ r ˜ Ω ∣ ∣ ∣ , ∣ ∣ ∣ ∂ θ ˜ Ω ∣ ∣ ∣ ) glyph[lessmuch] 1. This is not valid at the bottom of the solar convection zone where the radial gradient of the angular velocity is strong and Ω glyph[star] glyph[greatermuch] 1 and τ c max (∣ ∣ ∇ i V j ∣ ∣ ) ≈ 2. Leprovost and Kim (2010) suggested that this pumping effect is quenched with increasing shear inversely proportional to ( τ c max (∣ ∣ ∇ i V j ∣ ∣ )) 1 ... 2 . Therefore, we introduce an ad-hoc quenching function for the pumping effect: where C S is a constant to control the magnitude of the quenching, and s = 1. Results by Leprovost and Kim (2010) suggest 1 < s < 2 in relation to geometry of the large-scale shear. We find that for the solar convection zone the amplitude of the pumping effect does not change very much ( ∼ 1 m / s) with s varying in the range 1 . . . 2. From the given relations, using E ( H ) = U (eff) × B , we find the effective drift velocity, U (eff) , due to the helicity-vorticity pumping effect. Taking into account the variation of turbulence parameters in the solar convection zone we compute U (eff) . The bottom panel of Figure 3 shows the distribution of the velocity field U ( eff ) for the helicity-vorticity pumping effect for the toroidal and poloidal components of the large-scale magnetic field. The maximum velocity drift occurs in the middle and at the bottom of the convection zone. The direction of drift has equatorial and polar cells corresponding to two regions in the solar convection zone with different signs of the radial gradient of the angular velocity. The anisotropy in transport of the toroidal and poloidal components of the large-scale magnetic field is clearly seen. The other important pumping effects are due to mean density and turbulence intensity gradients (Zeldovich 1957, Kichatinov 1991, Kichatinov and Rudiger 1992, Tobias et al. 2001). These effects were estimated using Eq. (12). For these calculations we put C η = 1, ε = 0, C K = 1 and χ = 0. Figure 4 shows the sum of the pumping effects for the toroidal and poloidal components of mean magnetic fields including the helicity-vorticity pumping effect. In agreement with previous studies, it is found that the radial direction is the principal direction of mean-field transport in the solar convection zone. In its upper part the transport is downward because of pumping due to the density gradient (Kichatinov 1991). At the bottom of the convection zone the diamagnetic pumping effect produces downward transport as well (Kichatinov 1991, Rudiger and Brandenburg 1995). The diamagnetic pumping is quenched inversely proportional to the Coriolis number (e.g., Kichatinov 1991, Pipin 2008) and it has the same order of magnitude as the helicity-vorticity pumping effect. The latter effect modifies the direction of effective drift of the toroidal magnetic field near the bottom of the convection zone. There is also upward drift of the toroidal field at · low latitudes in the middle of the convection zone. It results from the combined effects of density gradient and global rotation (Kichatinov 1991, Krivodubskij 2004). For the poloidal magnetic field the transport is downward everywhere in the convection zone. At the bottom of the convection zone the action of the diamagnetic pumping on the meridional component of the large-scale magnetic field is amplified due to the helicity-vorticity pumping effect. The obtained pattern of large-scale magnetic field drift in the solar convection zone does not take into account nonlinear effects, e.g., because of magnetic buoyancy. The effect of mean-field buoyancy is rather small compared with flux-tube buoyancy (Kichatinov and Pipin 1993, cf. Guerrero and Kapyla 2011). To find out the current helicity counterpart of the pumping effect we analyze dynamo models by solving Eqs. (6, 16). The governing parameters of the model are C η = 0 . 05, C ω = 1 3 C α . We discuss the choice of the governing parameters later. The other parameters of the model are given in the Table 1. Because of the weakening factor C η the magnitude of the pumping velocity is about one order of magnitude smaller than what is shown in Figure 4. Following Pipin and Kosovichev (2011b), we use a combination of 'open' and 'closed' boundary conditions at the top, controlled by a parameter δ = 0 . 95, with This is similar to the boundary condition discussed by Kitchatinov et al. (2000). For the poloidal field we apply a combination of the local condition A = 0 and the requirement of a smooth transition from the internal poloidal field to the external potential (vacuum) field: We assume perfect conductivity at the bottom boundary with standard boundary conditions. For the magnetic helicity, similar to Guerrero et al. (2010), we put ∇ r ¯ χ = 0 at the bottom of the domain and ¯ χ = 0 at the top of the convection zone. In this paper we study dynamo models which include Radler's Ω × J dynamo effect due to a large-scale current and global rotation (Radler 1969). There is also a dynamo effect due to large-scale shear and current (Rogachevskii and Kleeorin 2003). The motivation to consider these addional turbulent sources in the mean-field dynamo comes from DNS dynamo experiments (Brandenburg and Kapyla 2007, Kapyla et al. 2008, Hughes and Proctor 2009, Kapyla et al. 2009) and from our earlier studies (Pipin and Seehafer 2009, Seehafer and Pipin 2009). The dynamo effect due to large-scale current gives an additional source of large-scale poloidal magnetic field. This can help to solve the issue with the dynamo period being otherwise too short. Also, in the models the large-scale current dynamo effect produces less overlapping cycles than dynamo models with α -effect alone. The choice of parameters in the dynamo is justified by our previous studies (Pipin and Seehafer 2009, Pipin and Kosovichev 2011c), where we showed that solar-types dynamos can be obtained for C α /C ω > 2. In those papers we find the approximate threshold to be C α ≈ 0 . 02 for a given diffusivity dilution factor of C η = 0 . 05. As follows from the results given in Fig.4, the kinetic helicity-vorticity pumping effect has a negligible contribution in the near-surface layers, where downward pumping due to density stratification dominates. Therefore, it is expected that the surface dynamo waves are not affected if we discard magnetic helicity from the dynamo equations. Figure 5 shows time-latitude diagrams for toroidal and radial magnetic fields at the surface and for toroidal magnetic field at the bottom of the convection zone for two dynamo models D1 and D2 with and without the helicity-vorticity pumping effect, but magnetic helicity is taken into account as the main dynamo quenching effect. To compare with observational data from a time-latitude diagram of sunspot area (e.g., Hathaway 2011), we multiply the toroidal field component B by factor sin θ . This gives a quantity, which is proportional to the flux of large-scale toroidal field at colatitude θ . We further assume that the sunspot area is related to this flux. Near the surface, models D1 and D2 give similar patterns of magnetic field evolution. At the bottom of the convection zone model D1 shows both poleward and equatorward branches of the dynamo wave propagation that is in agreement with the Parker-Yoshimura rule. Both branches have nearly the same time scale that equals glyph[similarequal] 16 years. The results from model D2 show that at the bottom of the convection zone the poleward branch of the dynamo wave dominates. Thus we conclude that the helicity-vorticity pumping effect alters the propagation of the dynamo wave near the bottom of the solar convection zone. We find that models with magnetic helicity contributions to the pumping effect do not change this conclusion. Figure 6 shows a typical snapshot of the magnetic helicity distribution in the northern hemisphere for all our models. The helicity has a negative sign in the bulk of the solar convection zone. Regions with positive current helicity roughly correspond to domains of the negative large-scale current helicity concentration. × They are located in the middle of the solar convection zone and at the high and low latitudes near the top of the solar convection zone. As follows from Fig. 6, the pumping effect due to current helicity may be efficient in the upper part of the solar convection zone where it might intensify the equatorial drift of the dynamo wave along iso-surfaces of the angular velocity. We find that the pumping effect that results from magnetic helicity is rather small in our models. This may be due to the weakness of the magnetic field. Observations (Zhang et al. 2010) give about one order magnitude larger current helicity than what is shown in Fig. 6. In the model we estimate the current helicity as H C = χ µ 0 glyph[lscript] 2 . This result depends essentially on the mixing length parameter glyph[lscript] . The stronger helicity is concentrated to the surface, the larger H C . In observations, we do not know from were the helical magnetic structures come from. In view of the given uncertainties we estimate the probable effect of a larger magnitude of magnetic helicity in the model by increasing the parameter C C to 10 (model D3). In addition, we consider the results for the nonlinear model D4. It has a higher C α and a lower R χ to increase the nonlinear impact of the magnetic helicity on the large-scale magnetic field evolution. The top panel of Figure 7 shows a time-latitude diagram of toroidal magnetic field and current helicity evolution near the surface for model D4. We find a positive sign of current helicity at the decay edges of the toroidal magnetic field butterfly diagram. There are also areas with positive magnetic helicity at high latitudes at the growing edges of the toroidal magnetic field butterfly diagram. The induced pumping velocity is about 1 cm s -1 . The increase of the magnetic helicity pumping effect by a factor of 10 (model D3) shifts the latitude of the maximum of the toroidal magnetic field by about 5 · toward the equator. The induced pumping velocity is about 5 cm s -1 . Stronger nonlinearity (model D4) and a stronger magnetic helicity pumping effect (model D3) modify the butterfly diagram in different ways. Model D3 shows a simple shift of the maximum of toroidal magnetic field toward the equator. Model D4 shows a fast drift of large-scale toroidal field at the beginning of a cycle and a slow-down of the drift velocity as the cycle progresses. and the latitudinal drift of the centroid position of the toroidal magnetic field flux (cf. Hathaway 2011) where B S ( θ ) = 〈 B ( r, θ ) 〉 (0 . 9 , 0 . 99) R is the toroidal magnetic field, which is averaged over the surface layers. Note that the overlap between subsequent cycles influences the value of λ C more than the value of λ max . The behaviour of λ max in models D1,D2 and D3 reproduces qualitatively the exponential drift of maximum × C latitude as suggested by Hathaway (2011): where t is time measured in years. Model D4 shows a change between fast (nearly steady dynamo wave) drift at the beginning of the cycle to slow drift at the decaying phase of the cycle. The overlap between subsequent cycles is growing from model D1 to model D4. In all the models the highest latitude of the centroid position of the toroidal magnetic flux is below 30 · . Models D3 and D4 have nearly equal starting latitude of the centroid position. It is about 24 · . This means that a model with increased magnetic helicity pumping produces nearly the same effect for the shift of the centroid position as a model with a strong nonlinear effect of magnetic helicity.",
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{
"title": "3 Discussion and conclusions",
"content": "We have shown that the interaction of helical convective motions and differential rotation in the solar convection zone produces a turbulent drift of large-scale magnetic field. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. The large-scale vorticity vector roughly follows to iso-surfaces of angular velocity. Since the direction of the drift depends on the sign of helicity, the pumping effect is governed by the Parker-Yoshimura rule (Parker 1955, Yoshimura 1975). The effect is computed within the framework of mean-field magnetohydrodynamics using the minimal τ -approximation. In the calculations, we have assumed that the turbulent kinetic and current helicities are given. The calculations were done for arbitrary Coriolis number. In agreement with Mitra et al. (2009) and Rogachevskii et al. (2011), the analytical calculations show that the leading effect of pumping is described by a large-scale magnetic drift in the direction of the large-scale vorticity vector and by anisotropic pumping which produces a drift of toroidal and poloidal components of the field in opposite directions. The component of the drift that is induced by global rotation and helicity (second line in Eq. (3)) is rather small compared to the main effect. The latter conclusion should be checked separately for a different model of background turbulence, taking into account the generation of kinetic helicity due to global rotation and stratification in a turbulent medium. We have estimated the pumping effect for the solar convection zone and compared it with other turbulent pumping effects including diamagnetic pumping and turbulent pumping that results from magnetic fluctuations in stratified turbulence (Kichatinov 1991, Pipin 2008). The latter is sometimes referred to as 'density-gradient pumping effect' (Krivodubskij 2004). The diamagnetic pumping is upward in the upper part of the convection zone and downward near the bottom. The velocity field of density-gradient pumping is more complicated (see Figure 4). However, its major effect is concentrated near the surface. Both diamagnetic pumping and density-gradient pumping effects are quenched inversely proportional to the Coriolis number (Kichatinov 1991, Pipin 2008). The helicity-vorticity pumping effect modifies the direction of large-scale magnetic drift at the bottom of the convection zone. This effect was illustrated by a dynamo model that shows a dominant poleward branch of the dynamo wave at the bottom of the convection zone. It is found that the magnetic helicity contribution of the pumping effect can be important for explaining the fine structure of the sunspot butterfly diagram. In particular, the magnetic helicity contribution results in a slow-down of equatorial propagation of the dynamo wave. The slow-down starts just before the maximum of the cycle. Observations indicate a similar behavior in sunspot activity (Ternullo 2007, Hathaway 2011). A behavior like this can be seen in flux-transport models as well (Rempel 2006). For the time being it is unclear what are the differences between different dynamo models and how well do they reproduce the observations. A more detailed analysis is needed.",
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},
{
"title": "REFERENCES",
"content": "065903 . Radler, K.-H.. On the electrodynamics of turbulent fields under the influence of corilois forces. Monats. Dt. Akad. Wiss., 11:, 194-201, 1969.",
"pages": [
15
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},
{
"title": "Appendix A",
"content": "To compute E it is convenient to write equations (1) and (2) in Fourier space: where the turbulent pressure was excluded from (2) by convolution with tensor π ij ( z ) = δ ij -ˆ z i ˆ z j , δ ij is the Kronecker symbol and ̂ z is a unit wave vector. The equations for the second-order moments that make contributions to the mean-electro-motive force(MEMF) can be found directly from (26, 27). As the preliminary step we write the equations for the second-order products of the fluctuating fields, and make the ensemble averaging of them, where, the terms Th ( κ ,v,h ) ij involve the third-order moments of fluctuating fields and second-order moments of them with the forcing term. Next, we apply the τ -approximation, substituting the Th ( κ ,v,h ) ij -terms by the corresponding τ relaxation terms of the second-order contributions, where the superscript . . . (0) denotes the moments of the background turbulence. Approximating these complicated contributions by the simple relaxation terms has to be considered as a questionable assumption. It involves additional assumptions (see Radler and Rheinhardt 2007), e.g., it is assumed that the second-order correlations in Eq. (8) do not vary significantly on the time scale of τ c . This assumption is consistent with scale separation between the mean and fluctuating quantities in the mean-field magneto hydrodynamics. The reader can find a comprehensive discussion of the τ -approximation in the above cited papers. Furthermore, we restrict ourselves to the high Reynolds numbers limit and discard the microscopic diffusion terms. The contributions of the mean magnetic field in the turbulent stresses will be neglected because they give the nonlinear terms in the cross helicity tensor. Also, τ c is independent on k (cf, Radler et al. 2003, Rogachevskii and Kleeorin 2003, Brandenburg and Subramanian 2005) and it is independent on the mean fields as well. This should be taken into account in considering the nonlinear effects due to rotation. Taking all the above assumptions into account, we get the system of equations for the moments for the stationary case: To proceed further, we have to introduce some conventions and notations that are widely used in the literature. The double Fourier transformation of an ensemble average of two fluctuating quantities, say f and g , taken at equal times and at the different positions x , x ' , is given by In the spirit of the general formalism of the two-scale approximation (Roberts and Soward 1975) we introduce 'fast' and 'slow' variables. They are defined by the relative r = x -x ' and the mean R = 1 2 ( x + x ' ) coordinates, respectively. Then, Eq. (37) can be written in the form where we have introduced the wave vectors k = 1 2 ( z -z ' ) and K = z + z ' . Then, following to Subramanian and Brandenburg (2004), we define the correlation function of ̂ f and ̂ g obtained from (38) by integration with respect to K , For further convenience we define the second order correlations of momentum density, magnetic fluctuations and the cross-correlations of momentum and magnetic fluctuations via We now return to equations (34), (35) and (36). As the first step, we solve these equations about Ω (nonlinear effects of the Coriolis force) and make the Taylor expansion with respect to the 'slow' variables and take the Fourier transformation, (39), about them. The details of this procedure can be found in (Subramanian and Brandenburg 2004). In result we get the following equations for the second moments These equations were solved with respect to the shear tensor, V i,j = ∇ j V i , by means of perturbation procedure. One remains to define the spectra of the background turbulence. We will adopt the isotropic form of the spectra (Roberts and Soward 1975). Additionally, the background magnetic fluctuations are helical while there is no prescribed kinetic helicity in the background turbulence: where, the spectral functions E ( k, R ) , B ( k, R ) , N ( k, R ) define, respectively, the intensity of the velocity fluctuations, the intensity of the magnetic fluctuations and amount of current helicity in the background turbulence. They are defined via where h (0) K = 〈 u (0) ·∇ × u (0) 〉 and h (0) C = 〈 b (0) ·∇× b (0) 〉 µρ . In final results we use the relation between intensities of magnetic and kinetic fluctuations which is defined via B ( k, R ) = εµ ¯ ρE ( k, R ). The state with ε = 1 means equipartition between energies of magnetic an kinetic fluctuations in the background turbulence.",
"pages": [
15,
16,
17,
18,
19
]
}
]
2013GReGr..45.1005N
https://arxiv.org/pdf/1201.2806.pdf
<document>
<section_header_level_1><location><page_1><loc_42><loc_92><loc_58><loc_93></location>Dragged metrics</section_header_level_1>
<text><location><page_1><loc_38><loc_89><loc_63><loc_90></location>M. Novello ∗† and E. Bittencourt ‡</text>
<text><location><page_1><loc_26><loc_85><loc_74><loc_89></location>Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: November 15, 2018)</text>
<text><location><page_1><loc_18><loc_76><loc_83><loc_84></location>We show that the path of any accelerated body in an arbitrary space-time geometry g µν can be described as geodesics in a dragged metric ˆ q µν that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kind of non-gravitational forces as modifications of the metric of space-time. This method of effective elimination of the forces by a change of the metric of the substratum can be understood as a generalization of the d'Alembert principle applied to all relativistic processes.</text>
<text><location><page_1><loc_18><loc_74><loc_33><loc_75></location>PACS numbers: 04.20.-q</text>
<section_header_level_1><location><page_1><loc_20><loc_70><loc_37><loc_71></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_59><loc_49><loc_68></location>In 1923 Gordon [1] made a seminal suggestion to treat the propagation of electromagnetic waves in a moving dielectric as a modification of the metric structure of the background. He showed that the waves propagate as geodesics not in the geometry η µν but instead in the dragged metric</text>
<text><location><page_1><loc_9><loc_50><loc_49><loc_58></location>̂ q µν = η µν +( /epsilon1µ -1) v µ v ν , (1) where /epsilon1 and µ are constant parameters that characterize the dielectric. Latter it was recognized that this interpretation can be used to describe non-linear structures when /epsilon1 and µ depends on the intensity of the field [2].</text>
<text><location><page_1><loc_9><loc_41><loc_49><loc_50></location>In recent years an intense activity concerning properties of Riemannian geometries similar to the one described by Gordon has been done [3]. In particular those that allows a binomial form for both the metric and its inverse, that is its covariant and the corresponding contravariant expressions</text>
<text><location><page_1><loc_9><loc_36><loc_11><loc_37></location>and</text>
<formula><location><page_1><loc_22><loc_35><loc_49><loc_40></location>̂ q µν = η µν + b Φ µν , (2)</formula>
<text><location><page_1><loc_9><loc_31><loc_43><loc_32></location>Thus, the tensor Φ µν must satisfy the condition</text>
<formula><location><page_1><loc_21><loc_30><loc_49><loc_35></location>̂ q µν = Aη µν + B Φ µν . (3)</formula>
<formula><location><page_1><loc_20><loc_28><loc_49><loc_30></location>Φ µν Φ νλ = mδ λ µ + n Φ λ µ . (4)</formula>
<text><location><page_1><loc_9><loc_21><loc_49><loc_27></location>In the present paper we limit our analysis only to the simplest dragged form by setting Φ µν = v µ v ν . In this case the coefficients of the covariant form of the metric are given by</text>
<formula><location><page_1><loc_20><loc_16><loc_38><loc_19></location>A = 1; B = -b 1 + b .</formula>
<text><location><page_1><loc_52><loc_52><loc_92><loc_71></location>The origin of the dragging effect in the case of Gordon's metric is due to the modifications of the path of the electromagnetic waves inside the moving dielectric. Then we face the question: could such particular description of the electromagnetic waves in moving bodies be generalized for other cases, independently of the electromagnetic forces? In other words, could such geometrized paths be used to describe other kinds of forces? We shall see that the answer is yes. Indeed, we will show that it is possible to geometrize different kinds of forces by the introduction of a dragged metric ̂ q µν such that in this geometry the accelerated body follows the free path of geodesics.</text>
<text><location><page_1><loc_52><loc_26><loc_92><loc_53></location>Let us emphasize that we deal here with any kind of force that has a non-gravitational character. It is precisely the consequences of such non-gravitational force that we describe in terms of a modified dragged metric. This means that the observable effects of any force can be interpreted as nothing but a modification of the geometry of space-time. In other words the motion of any accelerated body can be described as a free body following geodesics in a modified metric. This procedures generalizes d'Alembert principle of classical mechanics [4, 5] which states that it is possible to transform a dynamical problem into a static one, where the body is free of any force. Going from the background metric - where an accelerated body experiences a non-gravitational force - to a dragged metric where the body follows a geodesics and become free of non-gravitational forces is the relativistic expression of this principle. In this way we produce a geometric description of all kind of motion whatever the force that originates it.</text>
<section_header_level_1><location><page_1><loc_63><loc_21><loc_81><loc_22></location>II. A SPECIAL CASE</section_header_level_1>
<text><location><page_1><loc_52><loc_17><loc_92><loc_19></location>We claim that accelerated bodies in a flat Minkowski space-time 1 can be equivalently described as free bodies</text>
<text><location><page_2><loc_9><loc_88><loc_49><loc_93></location>following geodesics in an associated dragged geometry. In order to simplify our calculation we restrict this section to the case in which the acceleration vector a µ is the gradient of a function, that is 2</text>
<formula><location><page_2><loc_25><loc_84><loc_49><loc_85></location>a µ = ∂ µ Ψ . (5)</formula>
<text><location><page_2><loc_9><loc_78><loc_49><loc_83></location>Thus the force acting on the body under observation comes from a potential V in the Lagrangian formalism, i.e., F µ = ∂ µ V.</text>
<text><location><page_2><loc_10><loc_77><loc_39><loc_78></location>We write the dragged metric in the form</text>
<text><location><page_2><loc_9><loc_71><loc_43><loc_76></location>̂ q µν = η µν + b v µ v ν . The associated covariant derivative is defined by</text>
<text><location><page_2><loc_9><loc_67><loc_48><loc_69></location>where the corresponding Christoffel symbol is given by</text>
<formula><location><page_2><loc_21><loc_67><loc_37><loc_71></location>v α ; µ = v α , µ + ̂ Γ α µν v ν ,</formula>
<text><location><page_2><loc_9><loc_57><loc_49><loc_67></location>̂ Γ /epsilon1 µν = 1 2 ( η /epsilon1α + b v /epsilon1 v α ) ( ̂ q αµ,ν + ̂ q αν,µ -̂ q µν,α ) , (6) where we are using a comma to denote simple derivative, that is A ,µ ≡ ∂ µ A. The description of an accelerated curve in a flat space-time as a geodesics in a dragged metric is possible if the following condition is satisfied</text>
<formula><location><page_2><loc_21><loc_51><loc_49><loc_57></location>( v µ,ν -̂ Γ /epsilon1 µν v /epsilon1 ) v ν = 0 , (7)</formula>
<formula><location><page_2><loc_21><loc_44><loc_49><loc_50></location>( v µ,ν -̂ Γ /epsilon1 µν v /epsilon1 ) v ν = 0 . (8)</formula>
<text><location><page_2><loc_9><loc_47><loc_49><loc_56></location>̂ where we have used the dragged metric to write ̂ v µ ≡ ̂ q µν v ν . Or, equivalently,</text>
<text><location><page_2><loc_9><loc_41><loc_49><loc_46></location>Then noting that the acceleration in the background is defined by a µ = v µ,ν v ν and using equation (5) the condition of geodesics in the dragged geometry takes the form</text>
<text><location><page_2><loc_9><loc_34><loc_15><loc_36></location>We have</text>
<formula><location><page_2><loc_23><loc_34><loc_49><loc_38></location>∂ µ Ψ = ̂ Γ /epsilon1 µν v /epsilon1 v ν . (9)</formula>
<formula><location><page_2><loc_19><loc_28><loc_49><loc_34></location>̂ Γ /epsilon1 µν v /epsilon1 v ν = 1 + b 2 v α v ν ̂ q αν,µ . (10)</formula>
<formula><location><page_2><loc_22><loc_23><loc_35><loc_26></location>a µ + ∂ µ b 2(1 + b ) = 0 ,</formula>
<text><location><page_2><loc_9><loc_25><loc_49><loc_30></location>Using the expression of ̂ q αν and combining with condition (8) it follows</text>
<text><location><page_2><loc_9><loc_18><loc_49><loc_22></location>that is, the expression of the coefficient b of the dragged metric is given in terms of the potential of the acceleration</text>
<formula><location><page_2><loc_24><loc_16><loc_49><loc_17></location>1 + b = e -2Ψ . (11)</formula>
<section_header_level_1><location><page_2><loc_56><loc_92><loc_88><loc_93></location>A. The curvature of the dragged metric</section_header_level_1>
<text><location><page_2><loc_52><loc_84><loc_92><loc_90></location>In the case the background metric is not flat or if we use an arbitrary coordinate system the connection is given by the sum of the corresponding background one and a tensor, that is</text>
<text><location><page_2><loc_52><loc_76><loc_92><loc_82></location>̂ Γ /epsilon1 µν = Γ /epsilon1 µν + K /epsilon1 µν . (12) In the case of the Minkowski background a direct calculation gives for the connection Γ /epsilon1 µν the form</text>
<formula><location><page_2><loc_59><loc_73><loc_92><loc_78></location>̂ K /epsilon1 µν = v /epsilon1 ( a µ v ν + a ν v µ ) -a /epsilon1 v µ v ν . (13)</formula>
<text><location><page_2><loc_52><loc_71><loc_56><loc_72></location>Then,</text>
<formula><location><page_2><loc_68><loc_68><loc_75><loc_70></location>K /epsilon1 µ/epsilon1 = a µ .</formula>
<text><location><page_2><loc_52><loc_66><loc_88><loc_67></location>The contracted Ricci curvature has the expression</text>
<formula><location><page_2><loc_53><loc_58><loc_92><loc_63></location>̂ R µν = a µ,ν -a µ a ν +( ω + a α ,α ) v µ v ν , (14) Noting that a µ = a ν q µν = a µ it follows that</formula>
<text><location><page_2><loc_52><loc_50><loc_79><loc_55></location>The scalar of curvature ̂ R = ̂ R µν ̂ q µν is</text>
<formula><location><page_2><loc_58><loc_53><loc_86><loc_60></location>̂ ̂ ω ≡ a µ a µ = a µ a ν η µν = a µ a ν ̂ q µν = ̂ ω.</formula>
<text><location><page_2><loc_52><loc_45><loc_92><loc_51></location>̂ R = (2 + b ) a α ,α . (15) These expressions can be re-written in a covariant way by noting that</text>
<text><location><page_2><loc_52><loc_40><loc_60><loc_41></location>which yields</text>
<formula><location><page_2><loc_64><loc_40><loc_79><loc_44></location>a µ ; ν ≡ a µ,ν -̂ Γ /epsilon1 µν a /epsilon1 ,</formula>
<formula><location><page_2><loc_58><loc_33><loc_92><loc_37></location>̂ R µν = a µ ; ν -a µ a ν -( ω -a α ; α ) v µ v ν , (16)</formula>
<formula><location><page_2><loc_64><loc_26><loc_92><loc_31></location>̂ R = (2 + b ) ( a α ; α -ω ) . (17)</formula>
<text><location><page_2><loc_52><loc_30><loc_73><loc_35></location>and for the scalar ̂ R the form</text>
<section_header_level_1><location><page_2><loc_64><loc_25><loc_79><loc_26></location>B. Analog gravity</section_header_level_1>
<text><location><page_2><loc_52><loc_9><loc_92><loc_23></location>Suppose that an observer following a path with fourvelocity v µ and acceleration a µ in the flat Minkowski space-time background is not able to identify the origin of the force that is acting on him. In other words he is going to believe that only long-range gravitational forces are constraining his motion. Let us assume that he knows that gravity does not accelerate any curve but instead change the metric of the background according to the principles of general relativity. This means that if he is able to represent his motion as a geodesics in a dragged</text>
<text><location><page_3><loc_9><loc_88><loc_49><loc_93></location>metric ̂ q µν he will consider that the origin of such curved metric is nothing but a consequence of a distribution of energy which he will describe by using the equation</text>
<formula><location><page_3><loc_21><loc_81><loc_49><loc_87></location>̂ R µν -1 2 ̂ R ̂ q µν = -̂ T µν . (18)</formula>
<formula><location><page_3><loc_23><loc_75><loc_35><loc_77></location>ˆ u α = √ 1 + b v α .</formula>
<text><location><page_3><loc_9><loc_75><loc_49><loc_83></location>He will identify the different terms of the source through his own motion. From his velocity v µ he defines the normalized four-velocity ˆ u α in the ̂ Q metric by setting</text>
<text><location><page_3><loc_9><loc_71><loc_49><loc_73></location>He then proceed to characterize the origin of the curved metric using the standard decomposition</text>
<unordered_list>
<list_item><location><page_3><loc_9><loc_68><loc_24><loc_69></location>(a) density of energy</list_item>
<list_item><location><page_3><loc_9><loc_62><loc_24><loc_64></location>(b) isotropic pressure</list_item>
<list_item><location><page_3><loc_9><loc_56><loc_18><loc_57></location>(c) heat flux</list_item>
</unordered_list>
<formula><location><page_3><loc_24><loc_62><loc_49><loc_67></location>̂ ρ = ̂ T µν ˆ u µ ˆ u ν ; (19)</formula>
<formula><location><page_3><loc_23><loc_56><loc_49><loc_61></location>̂ p = -1 3 ̂ T µν h µν ; (20)</formula>
<unordered_list>
<list_item><location><page_3><loc_9><loc_50><loc_26><loc_51></location>(d) anisotropic pressure</list_item>
</unordered_list>
<formula><location><page_3><loc_23><loc_49><loc_49><loc_54></location>̂ q λ = ̂ T αβ ˆ u β h α λ ; (21)</formula>
<text><location><page_3><loc_10><loc_44><loc_31><loc_45></location>In these expressions we used</text>
<formula><location><page_3><loc_20><loc_44><loc_49><loc_49></location>̂ π µν = ̂ T αβ h α µ h β ν + ̂ p h µν . (22)</formula>
<text><location><page_3><loc_9><loc_38><loc_38><loc_43></location>̂ h µν = ̂ q µν -̂ u µ ̂ u ν . Note that h µν = h µν . Thus, he will write</text>
<text><location><page_3><loc_9><loc_30><loc_49><loc_37></location>̂ ̂ ̂ ̂ ̂ ̂ From this decomposition, using equation (18) and the curvature (16) he will identify the energy-momentum distribution as</text>
<formula><location><page_3><loc_11><loc_35><loc_49><loc_40></location>̂ T µν = ρ ˆ u µ ˆ u ν -p h µν + q µ ˆ u ν + q ν ˆ u µ + π µν . (23)</formula>
<text><location><page_3><loc_9><loc_17><loc_13><loc_18></location>where</text>
<formula><location><page_3><loc_15><loc_16><loc_49><loc_30></location>̂ ρ = b 2 Q, ̂ p = ( 2 3 + b 2 ) Q, (24) ̂ q µ = 0 , ̂ π µν = -a µ ; ν + a µ a ν -Q 3 ̂ q µν + Q 3 ̂ u µ ˆ u ν ,</formula>
<formula><location><page_3><loc_24><loc_13><loc_34><loc_15></location>Q = ω -a α ; α ,</formula>
<text><location><page_3><loc_9><loc_5><loc_49><loc_13></location>Summarizing, we can say that this observer will state that there is a gravitational field represented by the metric ̂ q µν produced by the distribution of energy given</text>
<text><location><page_3><loc_52><loc_82><loc_92><loc_93></location>by equation (24). We note that this reduction of the dragged metric to the framework of general relativity is not mandatory. Indeed, we deal here precisely with some accelerated paths that are not reduced to the gravitational force in the standard theory. This will become more clear when we present examples of accelerated curves in specific solutions of general relativity in the next sections.</text>
<text><location><page_3><loc_52><loc_67><loc_92><loc_82></location>Indeed, let us present some clarifying examples. The first one considers the motion of rotating bodies in Minkowski space-time. The other ones take into account the general relativity effects. We shall see that it is possible to produce what could be called double gravity , if the origin of the curvature of the dragged metric is identified to an effective energy-momentum tensor satisfying the equations of general relativity. However there is no reason for this restriction. We will come back to this question elsewhere.</text>
<section_header_level_1><location><page_3><loc_56><loc_62><loc_88><loc_64></location>III. ACCELERATION IN MINKOWSKI SPACE-TIME</section_header_level_1>
<text><location><page_3><loc_52><loc_54><loc_92><loc_60></location>Let us consider a simple example concerning the acceleration of a body in flat Minkowski space-time written in non-stationary cylindrical coordinate system ( t, r, φ, z ) to express the following line element</text>
<formula><location><page_3><loc_53><loc_49><loc_92><loc_52></location>ds 2 = a 2 [ dt 2 -dr 2 -dz 2 + g ( r ) dφ 2 +2 h ( r ) dφdt ] , (25)</formula>
<text><location><page_3><loc_52><loc_47><loc_92><loc_49></location>where a is a constant. We choose the following local tetrad frame given implicitly by the 1-forms</text>
<formula><location><page_3><loc_65><loc_35><loc_92><loc_44></location>θ 0 = a ( dt + hdφ ) , θ 1 = adr, θ 2 = a ∆ dφ, θ 3 = adz, (26)</formula>
<text><location><page_3><loc_52><loc_30><loc_92><loc_35></location>where we define ∆ = √ h 2 -g . The unique nonidentically null components of the Ricci tensor R AB in the tetrad frame are</text>
<formula><location><page_3><loc_57><loc_14><loc_92><loc_28></location>R 00 = 1 2 a 2 ( h ' ∆ ) , R 11 = 1 a 2 ( ∆ '' ∆ -1 2 h ' 2 ∆ 2 ) = R 22 = R 33 , R 02 = 1 2 a 2 ( -h '' ∆ + h ' ∆ ∆ 2 ) . (27)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_14></location>where a prime means derivative with respect to coordinate r. The equations of general relativity for this geometry have two simple solutions that we shall analyze below.</text>
<text><location><page_4><loc_10><loc_92><loc_32><loc_93></location>In the case of R AB =0, we get</text>
<formula><location><page_4><loc_21><loc_88><loc_37><loc_90></location>h ' = 0; ∆ '' = 0 .</formula>
<text><location><page_4><loc_9><loc_86><loc_32><loc_87></location>Solving these equations, we find</text>
<formula><location><page_4><loc_19><loc_82><loc_39><loc_84></location>h ≡ const ; ∆ ≡ ω 2 r 2 ,</formula>
<text><location><page_4><loc_9><loc_79><loc_49><loc_82></location>where ω is a constant. Therefore, Eq. (25) takes the form</text>
<formula><location><page_4><loc_9><loc_74><loc_49><loc_77></location>ds 2 = a 2 [ dt 2 -dr 2 -dz 2 +( h 2 -ω 2 r 2 ) dφ 2 +2 h ( r ) dφdt ] . (28)</formula>
<text><location><page_4><loc_10><loc_73><loc_33><loc_74></location>If we consider the observer field</text>
<formula><location><page_4><loc_21><loc_67><loc_37><loc_71></location>v µ = 1 a √ h 2 -ω 2 r 2 δ µ 2 .</formula>
<text><location><page_4><loc_9><loc_66><loc_44><loc_67></location>This path corresponds to an acceleration given by</text>
<formula><location><page_4><loc_19><loc_61><loc_39><loc_66></location>a µ = ( 0 , ω 2 r ( h 2 -ω 2 r 2 ) , 0 , 0 ) .</formula>
<text><location><page_4><loc_9><loc_60><loc_33><loc_61></location>This means that a µ = ∂ µ Ψ , where</text>
<formula><location><page_4><loc_21><loc_57><loc_37><loc_59></location>2Ψ = -ln( h 2 -ω 2 r 2 ) .</formula>
<text><location><page_4><loc_9><loc_53><loc_49><loc_57></location>We are in a situation similar to the previous section since the acceleration is a gradient. The parameter b of the dragged metric is given by the expression (11)</text>
<formula><location><page_4><loc_22><loc_48><loc_35><loc_51></location>1 + b = h 2 -ω 2 r 2 ,</formula>
<text><location><page_4><loc_9><loc_47><loc_35><loc_48></location>and for the dragged metric the form</text>
<formula><location><page_4><loc_16><loc_38><loc_49><loc_45></location>ds 2 a 2 = ω 4 r 4 -ω 2 r 2 h 2 +1 ( h 2 -ω 2 r 2 ) 2 dt 2 + dφ 2 + 2 h h 2 -ω 2 r 2 dφ dt -dr 2 -dz 2 (29)</formula>
<formula><location><page_4><loc_20><loc_25><loc_37><loc_31></location>̂ ω 2 r 2 ± = h 2 ± √ h 4 -4 2 .</formula>
<text><location><page_4><loc_9><loc_29><loc_49><loc_38></location>Note that we are dealing with the case in which h 2 -ω 2 r 2 > 0 . This allows the presence of accelerated closed time-like curves (CTC) in the original background that will be mapped into closed time-like geodesics (CTG) in the dragged geometry. Note that there exists a real singularity in r = h/ω and that q 00 change sign where</text>
<section_header_level_1><location><page_4><loc_14><loc_21><loc_43><loc_23></location>IV. ACCELERATION IN CURVED SPACE-TIMES</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_49><loc_19></location>Let us present now how our dragged metric approach works in some solutions of the equations of general relativity. We choose three well-known geometries: Schwarzschild, Godel universe and the Kerr solution. In these Riemannian manifolds we analyze some examples of accelerated paths that are interpreted as geodesics in the associated dragged metrics.</text>
<section_header_level_1><location><page_4><loc_61><loc_92><loc_83><loc_93></location>A. Schwarzschild geometry</section_header_level_1>
<text><location><page_4><loc_52><loc_87><loc_92><loc_90></location>We set the Schwarzschild metric in the ( t, r, θ, ϕ ) coordinate system</text>
<formula><location><page_4><loc_52><loc_81><loc_92><loc_86></location>ds 2 = (1 -r H r ) dt 2 -1 (1 -r H r ) dr 2 -r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (30)</formula>
<text><location><page_4><loc_52><loc_80><loc_85><loc_81></location>Choose the path described by the four-velocity</text>
<formula><location><page_4><loc_65><loc_75><loc_78><loc_80></location>v µ = √ 1 -r H r δ µ 0 .</formula>
<text><location><page_4><loc_52><loc_73><loc_76><loc_74></location>The corresponding acceleration is</text>
<formula><location><page_4><loc_61><loc_68><loc_83><loc_73></location>a µ = ( 0 , -r H 2( r 2 -r r H ) , 0 , 0 ) .</formula>
<text><location><page_4><loc_52><loc_65><loc_92><loc_67></location>In this case the acceleration is the gradient of function Ψ given by</text>
<formula><location><page_4><loc_64><loc_59><loc_79><loc_62></location>Ψ = -1 2 ln(1 -r H r ) .</formula>
<text><location><page_4><loc_53><loc_57><loc_86><loc_58></location>The factor b of the dragged metric is given by</text>
<formula><location><page_4><loc_68><loc_53><loc_75><loc_56></location>b = -r H r ,</formula>
<text><location><page_4><loc_52><loc_51><loc_79><loc_52></location>and the dragged metric takes the form</text>
<formula><location><page_4><loc_53><loc_45><loc_92><loc_48></location>ds 2 = dt 2 -1 (1 -r H r ) dr 2 -r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (31)</formula>
<text><location><page_4><loc_52><loc_39><loc_92><loc_44></location>The only non-null Ricci curvature of this ̂ q µν metric are</text>
<formula><location><page_4><loc_68><loc_36><loc_75><loc_39></location>R 1 1 = r H r 3 ;</formula>
<formula><location><page_4><loc_65><loc_31><loc_79><loc_34></location>R 2 2 = R 3 3 = -1 2 R 1 1 .</formula>
<text><location><page_4><loc_52><loc_27><loc_92><loc_30></location>All 14 Debever invariants are finite in all points except at the origin r = 0 .</text>
<section_header_level_1><location><page_4><loc_63><loc_23><loc_80><loc_24></location>B. Godel's geometry</section_header_level_1>
<text><location><page_4><loc_52><loc_15><loc_92><loc_21></location>Let us now turn our analysis to the Godel geometry. In the cylindrical coordinate system this metric is given by Eq. (25), where a is a constant related to the vorticity a = 2 /ω 2 and</text>
<formula><location><page_4><loc_65><loc_13><loc_78><loc_15></location>h ( r ) = √ 2 sinh 2 r ;</formula>
<formula><location><page_4><loc_62><loc_8><loc_82><loc_10></location>g ( r ) = sinh 2 r (sinh 2 r -1) .</formula>
<text><location><page_5><loc_9><loc_90><loc_49><loc_93></location>For completeness we note the non-trivial contravariant terms of the metric:</text>
<formula><location><page_5><loc_20><loc_80><loc_49><loc_90></location>g 00 = 1 -sinh 2 r a 2 cosh 2 r , g 02 = √ 2 a 2 cosh 2 r , g 22 = -1 a 2 sinh 2 r cosh 2 r . (32)</formula>
<text><location><page_5><loc_9><loc_75><loc_49><loc_79></location>In [6] it was pointed out the acausal properties of a particle moving into a circular orbit around the z -axis with four-velocity</text>
<formula><location><page_5><loc_15><loc_68><loc_42><loc_75></location>v µ = ( 0 , 0 , 1 a sinh r √ sinh 2 r -1 , 0 ) .</formula>
<text><location><page_5><loc_9><loc_68><loc_44><loc_69></location>This path corresponds to an acceleration given by</text>
<formula><location><page_5><loc_15><loc_63><loc_42><loc_68></location>a µ = ( 0 , cosh r [2 sinh 2 r -1] a 2 sinh r [sinh 2 r -1] , 0 , 0 ) .</formula>
<text><location><page_5><loc_9><loc_61><loc_33><loc_63></location>This means that a µ = ∂ µ Ψ , where</text>
<formula><location><page_5><loc_18><loc_57><loc_40><loc_61></location>Ψ = -ln(sinh r √ sinh 2 r -1) .</formula>
<text><location><page_5><loc_9><loc_52><loc_49><loc_58></location>Again, we are in a situation where the acceleration is a gradient. Therefore, the parameter b of the dragged metric is given by the expression (11) which in the Godel's background takes the form</text>
<formula><location><page_5><loc_19><loc_47><loc_39><loc_50></location>1 + b = sinh 2 r (sinh 2 r -1) ,</formula>
<text><location><page_5><loc_9><loc_46><loc_42><loc_47></location>and the dragged metric has the following form</text>
<formula><location><page_5><loc_11><loc_38><loc_49><loc_44></location>d ̂ s 2 a 2 = 3 -sinh 4 r (sinh 2 r -1) 2 dt 2 + dφ 2 +2 √ 2 sinh 2 r -1 dφ dt , -dr 2 -dz 2 (33)</formula>
<text><location><page_5><loc_9><loc_19><loc_49><loc_37></location>From the analysis of geodesics in Godel geometry the domain r < r c where sinh 2 r c = 1 separates causal from non-causal regions of the space-time. This is related to the fact that a geodesic that pass the value r = 0 will be confined within the domain Ω i defined by the values of coordinate r in the region 0 < r < r c . See [7] for details. However, the gravitational field is finite in the region r = r c . Nothing similar in the dragged metric, once at sinh 2 r = 1 there exists a real singularity in the dragged metric. Only the exterior domain is allowed. This means that for this kind of accelerated path in Godel geometry the allowed domain for the dragged metric is precisely the whole acausal region.</text>
<section_header_level_1><location><page_5><loc_23><loc_15><loc_35><loc_16></location>C. Kerr metric</section_header_level_1>
<text><location><page_5><loc_9><loc_9><loc_49><loc_13></location>Let us turn now to the dragged metric approach in the case the background is the Kerr metric. In the BoyerLindquist coordinate system this metric is given by</text>
<formula><location><page_5><loc_56><loc_81><loc_92><loc_92></location>ds 2 = ( 1 -2 Mr ρ 2 ) dt 2 -ρ 2 Σ dr 2 -ρ 2 dθ 2 + + 4 Mra sin 2 θ ρ 2 dtdφ + -[ ( r 2 + a 2 ) sin 2 θ + 2 Mra 2 sin 4 θ ρ 2 ] dφ 2 , (34)</formula>
<text><location><page_5><loc_52><loc_76><loc_92><loc_80></location>where Σ = r 2 + a 2 -2 Mr and ρ 2 = r 2 + a 2 cos 2 θ . On equatorial plane ( θ = π/ 2) consider the following vector field</text>
<formula><location><page_5><loc_58><loc_67><loc_85><loc_74></location>v µ = ( 0 , 0 , 0 , r √ -( r 2 + a 2 ) 2 + a 2 Σ ) .</formula>
<text><location><page_5><loc_52><loc_67><loc_87><loc_68></location>This path corresponds to an acceleration given by</text>
<formula><location><page_5><loc_58><loc_61><loc_85><loc_65></location>a µ = ( 0 , -r 3 -Ma 2 r 4 + r 2 a 2 +2 Mra 2 , 0 , 0 ) .</formula>
<text><location><page_5><loc_52><loc_59><loc_76><loc_60></location>This means that a µ = ∂ µ Ψ , where</text>
<formula><location><page_5><loc_59><loc_53><loc_85><loc_57></location>2Ψ = -ln [ -( r 2 + a 2 + 2 Ma 2 r )] .</formula>
<text><location><page_5><loc_52><loc_48><loc_92><loc_52></location>Once more we choose an accelerated path that can be represented by a gradient. The parameter b of the dragged metric is given by the expression (11)</text>
<formula><location><page_5><loc_61><loc_41><loc_83><loc_46></location>1 + b = -( r 2 + a 2 + 2 Ma 2 r ) ,</formula>
<text><location><page_5><loc_52><loc_38><loc_92><loc_41></location>and for the dragged metric, on the equatorial plane, the form</text>
<formula><location><page_5><loc_54><loc_30><loc_92><loc_37></location>ds 2 a 2 = 1 (1 + b ) 2 ( 1 -2 M r -b 4 M 2 a 2 r 2 ) dt 2 + dφ 2 + 4 Ma (1 + b ) r dtdφ -r 2 ∆ dr 2 , (35)</formula>
<text><location><page_5><loc_52><loc_19><loc_92><loc_29></location>These two last cases (Godel and Kerr) show a very curious and intriguing property: the accelerated CTC's at their respective metrics) are transformed in curves that are geodesics, that is CTG's (at their corresponding dragged metric). Besides, in both cases, the dragged metrics display a real singularity excluding the causal domain.</text>
<section_header_level_1><location><page_5><loc_63><loc_15><loc_81><loc_16></location>V. GENERAL CASE</section_header_level_1>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>In the precedent sections we limited our analysis to the case in which the acceleration is given by a unique function. Let us now pass to more general situation. In</text>
<text><location><page_6><loc_9><loc_86><loc_49><loc_93></location>order to geometrize any kind of force we must deal with a larger class of geometries. The most general form of dragged metric that allows the description of accelerated bodies as true geodesics in a modified geometry has the form</text>
<text><location><page_6><loc_9><loc_78><loc_49><loc_85></location>̂ q µν = g µν + b v µ v ν + ma µ a ν + n ( v µ a ν + a µ v ν ) . (36) The three parameters b, m, n are related to the three degrees of freedom of the acceleration vector. The corresponding covariant form of the metric is given by</text>
<text><location><page_6><loc_9><loc_69><loc_49><loc_75></location>̂ q µν = g µν + Bv µ v ν + Ma µ a ν + N ( v µ a ν + a µ v ν ) . (37) in which B,M,N are given in terms of the parameters b, m, n by the relations</text>
<formula><location><page_6><loc_17><loc_64><loc_40><loc_68></location>B = -b (1 + mω ) -n 2 ω (1 + b ) (1 + mω ) -n 2 ω ,</formula>
<formula><location><page_6><loc_11><loc_58><loc_47><loc_63></location>M = 1 1 + mω ( -m + n 2 (1 + b ) (1 + mω ) -n 2 ω ) ,</formula>
<formula><location><page_6><loc_17><loc_53><loc_40><loc_57></location>N = -n (1 + b ) (1 + mω ) -n 2 ω .</formula>
<text><location><page_6><loc_9><loc_50><loc_49><loc_52></location>In this case the equation that generalizes the condition of geodesics (8) has the form</text>
<formula><location><page_6><loc_15><loc_43><loc_49><loc_49></location>a µ = 1 2 ( (1 + b ) v λ v ν + na λ a ν ) [ ̂ q λµ,ν + q λν , µ -q µν , λ ] . (38)</formula>
<text><location><page_6><loc_9><loc_40><loc_49><loc_45></location>̂ ̂ This equation can be cast in the following formal expression</text>
<formula><location><page_6><loc_17><loc_37><loc_49><loc_38></location>a µ = Ψ 1 ∂ µ b +Ψ 2 ∂ µ m +Ψ 3 ∂ µ n, (39)</formula>
<text><location><page_6><loc_9><loc_33><loc_49><loc_36></location>where each term Ψ 1 , Ψ 2 and Ψ 3 depends on all three functions m,b and n.</text>
<text><location><page_6><loc_9><loc_30><loc_49><loc_33></location>Solving this equation for these three functions provide the most general expression for any acceleration.</text>
<unordered_list>
<list_item><location><page_6><loc_9><loc_23><loc_43><loc_24></location>[1] W. Gordon Ann. Phys. (Leipzig) 72 421 (1923);</list_item>
<list_item><location><page_6><loc_9><loc_21><loc_49><loc_23></location>[2] M. Novello, V.A. De Lorenci, J. M. Salim and R. Klippert Phys. Rev. D 61 045001 (2000);</list_item>
<list_item><location><page_6><loc_9><loc_18><loc_49><loc_21></location>[3] M. Novello and E. Goulart Class. Quantum Grav. 28 145022 (2011) and references therein;</list_item>
<list_item><location><page_6><loc_9><loc_15><loc_49><loc_18></location>[4] C. Lanczos, The Variational Principles of Mechanics , Ed. 4, Dover, New York (1970);</list_item>
</unordered_list>
<text><location><page_6><loc_52><loc_85><loc_92><loc_93></location>With these results, we have transformed the path of any particle submitted to any kind of force as a geodetic motion in the dragged metric. This result is the extension of the d'Alembert principle, corresponding to all types of motion i.e. the acceleration is geometrized through the dragged metric.</text>
<section_header_level_1><location><page_6><loc_64><loc_80><loc_80><loc_81></location>VI. CONCLUSION</section_header_level_1>
<text><location><page_6><loc_52><loc_75><loc_92><loc_78></location>We summarize the novelty of our analysis in the following steps:</text>
<unordered_list>
<list_item><location><page_6><loc_54><loc_69><loc_92><loc_73></location>· Let v µ represent the four-vector that describes the kinematics of a body in an arbitrary space-time endowed with a geometry g µν ;</list_item>
<list_item><location><page_6><loc_54><loc_66><loc_92><loc_69></location>· If the body experiences a non-gravitational force it acquires an acceleration a µ ;</list_item>
<list_item><location><page_6><loc_54><loc_58><loc_92><loc_65></location>· It is always possible to define an associated dragged metric ̂ q µν given by (37) such that in this metric the acceleration is removed. That is, the path is represented as a free particle that follows geodesics in the dragged geometry.</list_item>
</unordered_list>
<text><location><page_6><loc_52><loc_53><loc_92><loc_56></location>We have shown by a constructive operation that the following conjecture is true:</text>
<text><location><page_6><loc_52><loc_39><loc_92><loc_52></location>Conjecture. For any accelerated path Γ described by four-vector velocity v µ and acceleration a µ in a given Riemannian geometry g µν we can always construct another geometry ̂ Q endowed with a dragged metric ̂ q µν which depends only on g µν , v µ and a µ such that the path Γ is a geodesic in ̂ Q.</text>
<section_header_level_1><location><page_6><loc_59><loc_38><loc_85><loc_39></location>VII. ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_6><loc_52><loc_30><loc_92><loc_36></location>We would like to thank Dr Ivano Dami˜ao Soares for his comments in a previous version of this paper. We would like to thank FINEP, CNPq and FAPERJ and EB CNPq for their financial support.</text>
<unordered_list>
<list_item><location><page_6><loc_52><loc_22><loc_92><loc_24></location>[5] E. Mach, The Science of Mechanics , Ed. 6, The Open Court Publishing CO., Illinois (1960);</list_item>
<list_item><location><page_6><loc_52><loc_19><loc_92><loc_22></location>[6] M. Novello, N.F. Svaiter and M. E. X. Guimares Mod. Phys. Lett. A 7 5 381 (1992);</list_item>
<list_item><location><page_6><loc_52><loc_17><loc_92><loc_19></location>[7] M. Novello, I. Dami˜ao Soares and J. Tiomno, Phys. Rev. D 27 4 779 (1983).</list_item>
</document>
[
{
"title": "Dragged metrics",
"content": "M. Novello ∗† and E. Bittencourt ‡ Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: November 15, 2018) We show that the path of any accelerated body in an arbitrary space-time geometry g µν can be described as geodesics in a dragged metric ˆ q µν that depends only on the background metric and on the motion of the body. Such procedure allows the interpretation of all kind of non-gravitational forces as modifications of the metric of space-time. This method of effective elimination of the forces by a change of the metric of the substratum can be understood as a generalization of the d'Alembert principle applied to all relativistic processes. PACS numbers: 04.20.-q",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "In 1923 Gordon [1] made a seminal suggestion to treat the propagation of electromagnetic waves in a moving dielectric as a modification of the metric structure of the background. He showed that the waves propagate as geodesics not in the geometry η µν but instead in the dragged metric ̂ q µν = η µν +( /epsilon1µ -1) v µ v ν , (1) where /epsilon1 and µ are constant parameters that characterize the dielectric. Latter it was recognized that this interpretation can be used to describe non-linear structures when /epsilon1 and µ depends on the intensity of the field [2]. In recent years an intense activity concerning properties of Riemannian geometries similar to the one described by Gordon has been done [3]. In particular those that allows a binomial form for both the metric and its inverse, that is its covariant and the corresponding contravariant expressions and Thus, the tensor Φ µν must satisfy the condition In the present paper we limit our analysis only to the simplest dragged form by setting Φ µν = v µ v ν . In this case the coefficients of the covariant form of the metric are given by The origin of the dragging effect in the case of Gordon's metric is due to the modifications of the path of the electromagnetic waves inside the moving dielectric. Then we face the question: could such particular description of the electromagnetic waves in moving bodies be generalized for other cases, independently of the electromagnetic forces? In other words, could such geometrized paths be used to describe other kinds of forces? We shall see that the answer is yes. Indeed, we will show that it is possible to geometrize different kinds of forces by the introduction of a dragged metric ̂ q µν such that in this geometry the accelerated body follows the free path of geodesics. Let us emphasize that we deal here with any kind of force that has a non-gravitational character. It is precisely the consequences of such non-gravitational force that we describe in terms of a modified dragged metric. This means that the observable effects of any force can be interpreted as nothing but a modification of the geometry of space-time. In other words the motion of any accelerated body can be described as a free body following geodesics in a modified metric. This procedures generalizes d'Alembert principle of classical mechanics [4, 5] which states that it is possible to transform a dynamical problem into a static one, where the body is free of any force. Going from the background metric - where an accelerated body experiences a non-gravitational force - to a dragged metric where the body follows a geodesics and become free of non-gravitational forces is the relativistic expression of this principle. In this way we produce a geometric description of all kind of motion whatever the force that originates it.",
"pages": [
1
]
},
{
"title": "II. A SPECIAL CASE",
"content": "We claim that accelerated bodies in a flat Minkowski space-time 1 can be equivalently described as free bodies following geodesics in an associated dragged geometry. In order to simplify our calculation we restrict this section to the case in which the acceleration vector a µ is the gradient of a function, that is 2 Thus the force acting on the body under observation comes from a potential V in the Lagrangian formalism, i.e., F µ = ∂ µ V. We write the dragged metric in the form ̂ q µν = η µν + b v µ v ν . The associated covariant derivative is defined by where the corresponding Christoffel symbol is given by ̂ Γ /epsilon1 µν = 1 2 ( η /epsilon1α + b v /epsilon1 v α ) ( ̂ q αµ,ν + ̂ q αν,µ -̂ q µν,α ) , (6) where we are using a comma to denote simple derivative, that is A ,µ ≡ ∂ µ A. The description of an accelerated curve in a flat space-time as a geodesics in a dragged metric is possible if the following condition is satisfied ̂ where we have used the dragged metric to write ̂ v µ ≡ ̂ q µν v ν . Or, equivalently, Then noting that the acceleration in the background is defined by a µ = v µ,ν v ν and using equation (5) the condition of geodesics in the dragged geometry takes the form We have Using the expression of ̂ q αν and combining with condition (8) it follows that is, the expression of the coefficient b of the dragged metric is given in terms of the potential of the acceleration",
"pages": [
1,
2
]
},
{
"title": "A. The curvature of the dragged metric",
"content": "In the case the background metric is not flat or if we use an arbitrary coordinate system the connection is given by the sum of the corresponding background one and a tensor, that is ̂ Γ /epsilon1 µν = Γ /epsilon1 µν + K /epsilon1 µν . (12) In the case of the Minkowski background a direct calculation gives for the connection Γ /epsilon1 µν the form Then, The contracted Ricci curvature has the expression The scalar of curvature ̂ R = ̂ R µν ̂ q µν is ̂ R = (2 + b ) a α ,α . (15) These expressions can be re-written in a covariant way by noting that which yields and for the scalar ̂ R the form",
"pages": [
2
]
},
{
"title": "B. Analog gravity",
"content": "Suppose that an observer following a path with fourvelocity v µ and acceleration a µ in the flat Minkowski space-time background is not able to identify the origin of the force that is acting on him. In other words he is going to believe that only long-range gravitational forces are constraining his motion. Let us assume that he knows that gravity does not accelerate any curve but instead change the metric of the background according to the principles of general relativity. This means that if he is able to represent his motion as a geodesics in a dragged metric ̂ q µν he will consider that the origin of such curved metric is nothing but a consequence of a distribution of energy which he will describe by using the equation He will identify the different terms of the source through his own motion. From his velocity v µ he defines the normalized four-velocity ˆ u α in the ̂ Q metric by setting He then proceed to characterize the origin of the curved metric using the standard decomposition In these expressions we used ̂ h µν = ̂ q µν -̂ u µ ̂ u ν . Note that h µν = h µν . Thus, he will write ̂ ̂ ̂ ̂ ̂ ̂ From this decomposition, using equation (18) and the curvature (16) he will identify the energy-momentum distribution as where Summarizing, we can say that this observer will state that there is a gravitational field represented by the metric ̂ q µν produced by the distribution of energy given by equation (24). We note that this reduction of the dragged metric to the framework of general relativity is not mandatory. Indeed, we deal here precisely with some accelerated paths that are not reduced to the gravitational force in the standard theory. This will become more clear when we present examples of accelerated curves in specific solutions of general relativity in the next sections. Indeed, let us present some clarifying examples. The first one considers the motion of rotating bodies in Minkowski space-time. The other ones take into account the general relativity effects. We shall see that it is possible to produce what could be called double gravity , if the origin of the curvature of the dragged metric is identified to an effective energy-momentum tensor satisfying the equations of general relativity. However there is no reason for this restriction. We will come back to this question elsewhere.",
"pages": [
2,
3
]
},
{
"title": "III. ACCELERATION IN MINKOWSKI SPACE-TIME",
"content": "Let us consider a simple example concerning the acceleration of a body in flat Minkowski space-time written in non-stationary cylindrical coordinate system ( t, r, φ, z ) to express the following line element where a is a constant. We choose the following local tetrad frame given implicitly by the 1-forms where we define ∆ = √ h 2 -g . The unique nonidentically null components of the Ricci tensor R AB in the tetrad frame are where a prime means derivative with respect to coordinate r. The equations of general relativity for this geometry have two simple solutions that we shall analyze below. In the case of R AB =0, we get Solving these equations, we find where ω is a constant. Therefore, Eq. (25) takes the form If we consider the observer field This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where We are in a situation similar to the previous section since the acceleration is a gradient. The parameter b of the dragged metric is given by the expression (11) and for the dragged metric the form Note that we are dealing with the case in which h 2 -ω 2 r 2 > 0 . This allows the presence of accelerated closed time-like curves (CTC) in the original background that will be mapped into closed time-like geodesics (CTG) in the dragged geometry. Note that there exists a real singularity in r = h/ω and that q 00 change sign where",
"pages": [
3,
4
]
},
{
"title": "IV. ACCELERATION IN CURVED SPACE-TIMES",
"content": "Let us present now how our dragged metric approach works in some solutions of the equations of general relativity. We choose three well-known geometries: Schwarzschild, Godel universe and the Kerr solution. In these Riemannian manifolds we analyze some examples of accelerated paths that are interpreted as geodesics in the associated dragged metrics.",
"pages": [
4
]
},
{
"title": "A. Schwarzschild geometry",
"content": "We set the Schwarzschild metric in the ( t, r, θ, ϕ ) coordinate system Choose the path described by the four-velocity The corresponding acceleration is In this case the acceleration is the gradient of function Ψ given by The factor b of the dragged metric is given by and the dragged metric takes the form The only non-null Ricci curvature of this ̂ q µν metric are All 14 Debever invariants are finite in all points except at the origin r = 0 .",
"pages": [
4
]
},
{
"title": "B. Godel's geometry",
"content": "Let us now turn our analysis to the Godel geometry. In the cylindrical coordinate system this metric is given by Eq. (25), where a is a constant related to the vorticity a = 2 /ω 2 and For completeness we note the non-trivial contravariant terms of the metric: In [6] it was pointed out the acausal properties of a particle moving into a circular orbit around the z -axis with four-velocity This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where Again, we are in a situation where the acceleration is a gradient. Therefore, the parameter b of the dragged metric is given by the expression (11) which in the Godel's background takes the form and the dragged metric has the following form From the analysis of geodesics in Godel geometry the domain r < r c where sinh 2 r c = 1 separates causal from non-causal regions of the space-time. This is related to the fact that a geodesic that pass the value r = 0 will be confined within the domain Ω i defined by the values of coordinate r in the region 0 < r < r c . See [7] for details. However, the gravitational field is finite in the region r = r c . Nothing similar in the dragged metric, once at sinh 2 r = 1 there exists a real singularity in the dragged metric. Only the exterior domain is allowed. This means that for this kind of accelerated path in Godel geometry the allowed domain for the dragged metric is precisely the whole acausal region.",
"pages": [
4,
5
]
},
{
"title": "C. Kerr metric",
"content": "Let us turn now to the dragged metric approach in the case the background is the Kerr metric. In the BoyerLindquist coordinate system this metric is given by where Σ = r 2 + a 2 -2 Mr and ρ 2 = r 2 + a 2 cos 2 θ . On equatorial plane ( θ = π/ 2) consider the following vector field This path corresponds to an acceleration given by This means that a µ = ∂ µ Ψ , where Once more we choose an accelerated path that can be represented by a gradient. The parameter b of the dragged metric is given by the expression (11) and for the dragged metric, on the equatorial plane, the form These two last cases (Godel and Kerr) show a very curious and intriguing property: the accelerated CTC's at their respective metrics) are transformed in curves that are geodesics, that is CTG's (at their corresponding dragged metric). Besides, in both cases, the dragged metrics display a real singularity excluding the causal domain.",
"pages": [
5
]
},
{
"title": "V. GENERAL CASE",
"content": "In the precedent sections we limited our analysis to the case in which the acceleration is given by a unique function. Let us now pass to more general situation. In order to geometrize any kind of force we must deal with a larger class of geometries. The most general form of dragged metric that allows the description of accelerated bodies as true geodesics in a modified geometry has the form ̂ q µν = g µν + b v µ v ν + ma µ a ν + n ( v µ a ν + a µ v ν ) . (36) The three parameters b, m, n are related to the three degrees of freedom of the acceleration vector. The corresponding covariant form of the metric is given by ̂ q µν = g µν + Bv µ v ν + Ma µ a ν + N ( v µ a ν + a µ v ν ) . (37) in which B,M,N are given in terms of the parameters b, m, n by the relations In this case the equation that generalizes the condition of geodesics (8) has the form ̂ ̂ This equation can be cast in the following formal expression where each term Ψ 1 , Ψ 2 and Ψ 3 depends on all three functions m,b and n. Solving this equation for these three functions provide the most general expression for any acceleration. With these results, we have transformed the path of any particle submitted to any kind of force as a geodetic motion in the dragged metric. This result is the extension of the d'Alembert principle, corresponding to all types of motion i.e. the acceleration is geometrized through the dragged metric.",
"pages": [
5,
6
]
},
{
"title": "VI. CONCLUSION",
"content": "We summarize the novelty of our analysis in the following steps: We have shown by a constructive operation that the following conjecture is true: Conjecture. For any accelerated path Γ described by four-vector velocity v µ and acceleration a µ in a given Riemannian geometry g µν we can always construct another geometry ̂ Q endowed with a dragged metric ̂ q µν which depends only on g µν , v µ and a µ such that the path Γ is a geodesic in ̂ Q.",
"pages": [
6
]
},
{
"title": "VII. ACKNOWLEDGEMENTS",
"content": "We would like to thank Dr Ivano Dami˜ao Soares for his comments in a previous version of this paper. We would like to thank FINEP, CNPq and FAPERJ and EB CNPq for their financial support.",
"pages": [
6
]
}
]
2013GReGr..45.2441P
https://arxiv.org/pdf/1310.0159.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_82><loc_70><loc_85></location>Scalar filed evolution and Area Spectrum for Lovelock-AdS Black Holes</section_header_level_1>
<text><location><page_1><loc_12><loc_79><loc_22><loc_80></location>C.B Prasobh</text>
<text><location><page_1><loc_23><loc_79><loc_36><loc_80></location>· V.C Kuriakose</text>
<text><location><page_1><loc_12><loc_70><loc_45><loc_71></location>Received: 25 June 2013 / Accepted: 30 August 2013</text>
<text><location><page_1><loc_12><loc_58><loc_70><loc_67></location>Abstract We study the modes of evolution of massless scalar fields in the asymptotically AdS spacetime surrounding maximally symmetric black holes of large and intermediate size in the Lovelock model. It is observed that all modes are purely damped at higher orders. Also, the rate of damping is seen to be independent of order at higher dimensions. The asymptotic form of these frequencies for the case of large black holes is found analytically. Finally, the area spectrum for such black holes is found from these asymptotic modes.</text>
<text><location><page_1><loc_12><loc_56><loc_70><loc_57></location>Keywords Quasinormal Modes Lovelock AdS Area Spectrum Scalar Field</text>
<text><location><page_1><loc_35><loc_55><loc_61><loc_57></location>· · · ·</text>
<text><location><page_1><loc_12><loc_53><loc_42><loc_55></location>PACS PACS code1 · PACS code2 · more</text>
<section_header_level_1><location><page_1><loc_12><loc_50><loc_23><loc_51></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_31><loc_70><loc_49></location>Gauge-gravity dualities like the AdS/CFT correspondence [1] make it possible to study the properties of conformal fields in a particular dimension d by studying the evolution of fields in a black hole spacetime that is asymptotically AdS in ( d +1) dimensions and this has led to considerable interest in the study of asymptotically AdS black hole spacetimes. The main difficulty in studying field evolution in such spacetimes is that the stability of the spacetime against perturbations is not always guaranteed, unlike in the case of asymptotically flat spacetimes in first order theories in four dimensions. The instability of linear perturbations in higher-order and higher dimensional theories has already been investigated [2,3, 4]. The instability also extends to the thermodynamics of the black hole when we consider AdS spacetimes, with the well-known Hawking-Page phase transitions [5] signaling a transition between the black hole spacetime and thermal AdS spacetime. Recent studies [6,7,8] on the Lovelock model have confirmed the existence of</text>
<section_header_level_1><location><page_1><loc_12><loc_29><loc_20><loc_30></location>C.B Prasobh</section_header_level_1>
<text><location><page_1><loc_12><loc_27><loc_70><loc_29></location>Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: prasobhcb@outlook.com</text>
<text><location><page_1><loc_12><loc_25><loc_21><loc_26></location>V.C Kuriakose</text>
<text><location><page_1><loc_12><loc_23><loc_70><loc_25></location>Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: vck@cusat.ac.in</text>
<text><location><page_2><loc_12><loc_80><loc_70><loc_89></location>dynamic instability against metric perturbations in asymptotically flat black hole spacetimes. Dynamic instability means that the solutions to the equation for the metric perturbation become unstable outside the event horizon for large values of their eigenvalue. This instability exists for all types of metric perturbations-tensor, vector and scalar. These instabilities occur when the mass of the black hole falls below a lower critical bound that depends on the coupling constants, the dimension of the spacetime and the order of the theory.</text>
<text><location><page_2><loc_12><loc_35><loc_70><loc_78></location>Quasi normal modes are damped oscillations (having complex frequencies), known to dominate the intermediate stage of the evolution of small perturbations of a black hole spacetime and have been studied extensively. Detailed reviews and methods of calculation of quasi normal modes are found in numerous papers that include [9,10,11,12,13,14]. Quasi normal modes are known to depend only on the parameters of the black hole, such as mass, charge and angular momentum, and be completely independent of the type of the agent that caused it. These modes are obtained as the solution of the respective field equation, when solved with respect to the metric that describes the particular black hole spacetime of interest. In the case of asymptotically flat spacetimes, the effective potential that is perceived by the field in the spacetime often resembles a finite potential barrier such that the potential vanishes at infinity. This leads to the possibility of obtaining solutions that are purely ingoing at the event horizon of the black hole and purely outgoing at infinity, both resembling plane waves when expressed in terms of a scaled co-ordinate called the tortoise co-ordinate. These modes may be observed in the future with the aid of gravitational detectors. In the case of asymptotically AdS spacetimes, however, the potential grows indefinitely at infinity. Therefore, one usually looks for solutions that vanish at infinity, while the boundary condition at the event horizon unchanged [15]. This is motivated by the case of pure AdS spacetime, where the potential effectively confines the field as if 'in a box' and solutions exist only with a discrete spectrum of real frequencies. Even though black holes in asymptotically AdS spacetimes are not believed to exist in nature, interest in studying their quasi normal modes stems from the above-mentioned AdS/CFT correspondence [1]. According to the AdS/CFT correspondence, these perturbations correspond to perturbations of the thermal state of the strongly coupled conformal field at the boundary of the spacetime and the quasi normal modes correspond to the return to thermal equilibrium, so that the quasi normal frequencies give a measure of the time scale for the relaxation, which is difficult to compute directly. This provides the motivation to study the quasi normal modes of various fields in asymptotically AdS spacetimes. Earlier work the quasi normal modes of Schwarzschild-AdS black holes [15] have proved that the modes scale with the temperature of the event horizon.</text>
<text><location><page_2><loc_12><loc_22><loc_70><loc_33></location>Complete quantization of gravity is one of the major goals of modern theoretical physics. Despite decades of research by physicists all over the world, it is yet to be achieved with complete success. One often takes clues from the classical theory of a system when attempting its quantization and gravity needs to be no different. Quantization of the black hole horizon area is expected to be a major feature of any successful quantum theory of gravity. The original 'classical' theory of gravity, namely the General Theory of Relativity (GTR) does provide us with the tools necessary to estimate the value of the area quantum. It is known</text>
<text><location><page_3><loc_12><loc_61><loc_70><loc_89></location>from field theory that the presence of a periodicity in the classical theory of a system points to the existence of an adiabatic invariant with a discrete spectrum in the corresponding quantum theory. Interestingly, it has been observed in GTR that the numerical value of these frequencies, in the limit of 'large' frequencies, follow a distinct pattern, with the real part approaching a fixed value. These are termed asymptotic frequencies. Suggestions have been made that the fixed value of the real part can be viewed as a physically relevant periodicity in the (classical) black hole system which would then lead to the existence of certain adiabatic invariant quantity, which in turn would possess equally spaced spectrum according to Bohr-Sommerfeld quantization. Once we read it together with Bekenstein's original proposal [16] that the black hole entropy is an adiabatic invariant with a discrete, equally spaced spectrum, we come to the conclusion that the entropy spectrum (and, by extension, the area spectrum) can be deduced from the asymptotic value of the quasi normal frequencies. The connection between the fixed asymptotic frequencies and the quantized area spectrum was made by Hod [17]. Dreyer [18] recovered Hod's result in the Loop Quantum Gravity. A new interpretation for the quasi normal modes ω = ω R + iω I of perturbed black holes as equivalent to that of a collection of damped harmonic oscillators with real frequency ω 0 = √ ω 2 R + ω 2 I was introduced by Maggiore [19] and used in conjunction with Hod's method in order to compute the area spectrum of Schwarzschild black holes.</text>
<text><location><page_3><loc_12><loc_39><loc_70><loc_60></location>As with the case of quasi normal modes, it is the AdS/CFT correspondence that provides the motivation for studying the area spectrum of black holes in asymptotically AdS spacetimes. Recent studies ([20,21,22,23,24] and references therein) suggest that the gravitational dual of the holographic entanglement entropy in quantum field theories is the area of minimal-area surfaces in AdS spaces. The entropy can be used to study phase transitions between various states of the field. An area-entanglement entropy relation of the form S A = Area ( γ A ) 4 G ( d -2) N has been proposed [20,25] which is very similar to the familiar area-entropy relation in the General Theory of Relativity. Here, γ A is the d -dimensional minimal area surface in AdS d +2 and G ( d -2) N is the ( d +2)-dimensional gravitational constant of the AdS gravity. Although the above relation was originally proposed for AdS spaces, it is equally applicable to any asymptotically AdS space, including one containing a black hole. In that case, the minimal surface tends to wrap the horizon and thus we can use the area of the event horizon in order to compute the entanglement entropy in the conformal field theory.</text>
<text><location><page_3><loc_12><loc_22><loc_70><loc_37></location>In this work, we numerically compute the quasi normal frequencies for massless scalar field perturbations in asymptotically AdS, spherically symmetric spacetimes in Lovelock model using the metric derived in [26]. We analytically find out the asymptotic form of the frequencies following [27] and use it to deduce the area spectrum of large black holes in the model. A brief outline of the paper is as follows: in Sect. 2, we explain the maximally symmetric Lovelock model and the resulting metric for the spacetime as given in [26]. Details of the Horowitz-Hubeny numerical method of computing the quasi normal frequencies in AdS black hole spacetimes to the case of massless scalar fields in the vicinity of such a black hole in the Lovelock-AdS model are also given in the same section. The results of the numerical procedure are presented and analyzed in Sect. 3. In Sect. 4, we</text>
<text><location><page_4><loc_12><loc_84><loc_70><loc_89></location>analytically determine the asymptotic quasi normal frequencies of the field for the case of large back holes following [27]. The results of that analysis are used in order to find the area spectrum of the black hole using the Kunstatter's method [28] in Sect. 4.1. The results are summarized in Sect. 5.</text>
<section_header_level_1><location><page_4><loc_12><loc_80><loc_70><loc_81></location>2 The metric and the numerical computation of Quasi normal Frequencies</section_header_level_1>
<text><location><page_4><loc_12><loc_65><loc_70><loc_78></location>The Lovelock model [29] is considered to be the most natural generalization of GTR. The Lovelock Lagrangian is a polynomial which consists of dimensionally continued higher order curvature terms. The most striking property of this Lagrangian is that it yields field equations that are in second order in the metric although the Lagrangian itself may contain higher order terms. Also, the theory is known to give solutions that are free of ghosts. The maximum order of terms in the action, k , is fixed by the number of dimensions of the spacetime d in Lovelock model according to the relation k = [ d -1 2 ] where [ x ] denotes the integer part of x . The action is written in terms of the Riemann curvature R ab = dω ab + ω a c ω cb and the vielbein e a as</text>
<formula><location><page_4><loc_34><loc_59><loc_70><loc_64></location>I G = κ ∫ k ∑ p =0 α p L ( p ) , (1)</formula>
<text><location><page_4><loc_12><loc_58><loc_63><loc_59></location>where α p are arbitrary (positive) coupling constants, and L ( p ) , given by</text>
<formula><location><page_4><loc_26><loc_54><loc_70><loc_57></location>L ( p ) = /epsilon1 a 1 ··· a d R a 1 a 2 ··· R a 2 p -1 a 2 p e a 2 p +1 ··· e a d , (2)</formula>
<text><location><page_4><loc_12><loc_27><loc_70><loc_55></location>is the p th order dimensionally continued term in the Lagrangian. The difficulty with the Lagrangian given above, with arbitrary values for α p , is that it becomes very difficult (if not impossible) to study the evolution of fields, since it is not at all clear whether the operator representing the evolution is Hermitian or not. As mentioned in the previous section, this problem, for the case of metric perturbations, has been analyzed in [6]. Although a general instability depending on the black hole mass has been established in that work, the numerical value for the critical mass has not been calculated. Also, it is very difficult to predict whether the presence of a cosmological constant raises or lowers the critical mass, as long as we consider a model with arbitrary α p . Moreover, for the same case, the the existence of negative energy solutions with horizons and positive energy solutions with naked singularities for (1) has been pointed out earlier [30,31]. These difficulties bring out the necessity of selecting suitable values for the coupling coefficients in order to have models that support maximally symmetric solutions and external perturbations. Maximally symmetric solutions to Lovelock model have long been known [26], which are derived by requiring that the theories must possess a unique cosmological constant (and consequently a unique AdS radius R ) for all orders. The resulting set of coupling constants are seen to be labeled by the order k and the gravitational constant G k . The metric describing the spherically symmetric black hole spacetime is derived from the action given in (1) with the choice</text>
<formula><location><page_4><loc_31><loc_20><loc_70><loc_26></location>α p = R 2( p -k ) ( d -2 p ) ( k p ) , p ≤ k 0 , p > k (3)</formula>
<text><location><page_5><loc_15><loc_87><loc_62><loc_89></location>where 1 ≤ k ≤ [ d -1 2 ]. The resulting field equations are of the form</text>
<formula><location><page_5><loc_27><loc_83><loc_70><loc_85></location>/epsilon1 ba 1 ··· a d -1 ¯ R a 1 a 2 ··· ¯ R a 2 k -1 a 2 k e a 2 k +1 ··· e a d -1 = 0 (4)</formula>
<formula><location><page_5><loc_23><loc_82><loc_70><loc_84></location>/epsilon1 aba 3 ··· a d ¯ R a 3 a 4 ··· ¯ R a 2 k -1 a 2 k T a 2 k +1 e a 2 k +2 ··· e a d -1 = 0 (5)</formula>
<text><location><page_5><loc_12><loc_77><loc_70><loc_81></location>Here, ¯ R ab := R ab + 1 R 2 e a e b . Such theories are labeled by k and have two fundamental constants, κ and R , related to the gravitational constant G k and the cosmological constant Λ respectively through the relations</text>
<formula><location><page_5><loc_33><loc_71><loc_70><loc_75></location>κ = 1 2( d -2)! Ω d -2 G k , (6)</formula>
<formula><location><page_5><loc_33><loc_69><loc_70><loc_72></location>Λ = -( d -1)( d -2) 2 R 2 , (7)</formula>
<text><location><page_5><loc_12><loc_64><loc_70><loc_68></location>Ω d -2 being the volume of the ( d -2) dimensional spherically symmetric tangent space. The static and spherically symmetric solutions to (4), written in Schwarzschild-like coordinates, take the form</text>
<formula><location><page_5><loc_30><loc_60><loc_70><loc_63></location>ds 2 = f ( r ) dt 2 + dr 2 f ( r ) + r 2 dΩ 2 d -2 , (8)</formula>
<text><location><page_5><loc_15><loc_58><loc_30><loc_59></location>where f ( r ) is given by</text>
<formula><location><page_5><loc_29><loc_53><loc_70><loc_57></location>f ( r ) = 1 + r 2 R 2 -σ ( C 1 r d -2 k -1 ) 1 /k . (9)</formula>
<text><location><page_5><loc_15><loc_52><loc_55><loc_53></location>We take σ = 1. The integration constant C 1 is written as</text>
<formula><location><page_5><loc_34><loc_49><loc_70><loc_50></location>C 1 = 2 G k ( M + C 0 ) , (10)</formula>
<text><location><page_5><loc_12><loc_45><loc_70><loc_48></location>where M stands for the mass of the black hole. The constant C 0 is chosen so that the horizon shrinks to a point for M → 0, as</text>
<formula><location><page_5><loc_35><loc_42><loc_70><loc_44></location>C 0 = 1 2 G k δ d -2 k, 1 . (11)</formula>
<text><location><page_5><loc_69><loc_28><loc_69><loc_30></location>/negationslash</text>
<text><location><page_5><loc_12><loc_26><loc_70><loc_41></location>It is interesting that the exponent of ( 1 r ) in (9) is proportional to ( d -2 k -1). Since k = [ d -1 2 ], ( d -2 k -1) = 0 in odd dimensions and ( d -2 k -1) = 1 in even dimensions. Thus the solution in even dimensions resemble the Schwarzschild-AdS solution. The ( d -2 k -1) = 0 cases correspond to Chern-Simmons theories which have a vacuum that is different from AdS [26]. Their quasi normal modes, mass and area spectra have already been computed [32]. What makes it interesting is the fact that recent studies [6] on the stability of metric perturbations in Lovelock model also point out that it is possible to predict the (in)stability of the perturbations only in even dimensions. The present work is limited to the cases where d -2 k -1 = 0. Then we have C 0 = 0 and C 1 = 2 G k M . Consider the scalar field Φ ( r, t, x i ) that obeys the Klein-Gordon equation given by</text>
<formula><location><page_5><loc_33><loc_22><loc_70><loc_25></location>1 √ g ∂ A √ gg AB ∂ B Φ = 0 , (12)</formula>
<text><location><page_6><loc_12><loc_82><loc_70><loc_89></location>x i being the co-ordinates in the spherically symmetric tangent space and g AB being components of the metric tensor. We impose the boundary conditions of ingoing plane wave solution at the event horizon and vanishing field at the boundary. The boundary condition at the horizon suggests the ansatz Φ = e -iω ( t + r ∗ ) where r ∗ is the tortoise coordinate defined by</text>
<formula><location><page_6><loc_37><loc_79><loc_70><loc_81></location>dr ∗ = dr f ( r ) . (13)</formula>
<text><location><page_6><loc_15><loc_77><loc_47><loc_78></location>In the ( v = t + r ∗ , r ) system, the metric reads</text>
<formula><location><page_6><loc_29><loc_73><loc_70><loc_75></location>ds 2 = -f ( r ) dv 2 +2 dvdr + r 2 dΩ 2 d -2 , (14)</formula>
<text><location><page_6><loc_15><loc_72><loc_31><loc_73></location>and we take the ansatz</text>
<formula><location><page_6><loc_29><loc_69><loc_70><loc_71></location>Φ ( v, r, x i ) = r 2 -d 2 ψ ( r ) Y ( x i ) e -iωv , (15)</formula>
<text><location><page_6><loc_15><loc_67><loc_30><loc_68></location>so that (12) becomes</text>
<formula><location><page_6><loc_23><loc_63><loc_70><loc_66></location>f ( r ) d 2 dr 2 ψ ( r ) + [ f ' ( r ) -2 iω ] d dr ψ ( r ) -V ( r ) ψ ( r ) = 0 , (16)</formula>
<text><location><page_6><loc_15><loc_61><loc_34><loc_63></location>with the effective potential</text>
<formula><location><page_6><loc_22><loc_58><loc_70><loc_60></location>V ( r ) = ( d -2)( d -4) 4 r 2 f ( r ) + d -2 2 r f ' ( r ) + l ( l + d -3) r 2 . (17)</formula>
<text><location><page_6><loc_12><loc_48><loc_70><loc_57></location>Here, l represents the eigenvalue of the operator on the LHS of (12) acting on the functions Y ( x i ) in the spherically symmetric tangent space. In order to numerically calculate the quasi normal frequencies for (16), we expand the field Φ as a power series about the horizon and impose the vanishing boundary condition at infinity. We change the variable from r to x = 1 r in order to map the range r + < r < ∞ to a finite range. In terms of x , (16) becomes</text>
<formula><location><page_6><loc_22><loc_45><loc_70><loc_48></location>s ( x ) d 2 dx 2 ψ ( x ) + t ( x ) x -x + d dx ψ ( x ) + u ( x ) ( x -x + ) 2 ψ ( x ) = 0 , (18)</formula>
<text><location><page_6><loc_15><loc_43><loc_19><loc_44></location>where</text>
<text><location><page_6><loc_15><loc_35><loc_17><loc_37></location>and</text>
<formula><location><page_6><loc_24><loc_28><loc_70><loc_33></location>V ( x ) = ( d -2)( d -4) x 2 f ( x ) 4 -( d -2) x 3 f ' ( x ) 2 + l ( l + d -3) x 2 . (20)</formula>
<text><location><page_6><loc_12><loc_22><loc_70><loc_28></location>In the numerical procedure, we find out the coefficients of the expansion of the functions s ( x ) , t ( x ) and u ( x ) as power series in ( x -x + ) using a computer with s i , t i and u i denoting the coefficient for the i th term. Then an expansion for ψ ( x ) of the form</text>
<formula><location><page_6><loc_22><loc_37><loc_70><loc_41></location>s ( x ) = -x 4 f ( x ) , t ( x ) = -x 4 f ' ( x ) -2 x 3 f ( x ) -2 iωx 2 , u ( x ) = ( x -x + ) V ( x ) , (19)</formula>
<formula><location><page_7><loc_33><loc_84><loc_70><loc_88></location>ψ ( x ) = ∞ ∑ n =0 a n ( x -x + ) n , (21)</formula>
<text><location><page_7><loc_15><loc_83><loc_40><loc_84></location>is substituted into (18) which yields</text>
<formula><location><page_7><loc_24><loc_77><loc_70><loc_81></location>a n = -1 P n n -1 ∑ k =0 [ k ( k -1) s n -k + kt n -k + u n -k ] a k , (22)</formula>
<text><location><page_7><loc_15><loc_76><loc_19><loc_77></location>where</text>
<formula><location><page_7><loc_33><loc_72><loc_70><loc_74></location>P n = n ( n -1) s 0 + nt 0 . (23)</formula>
<text><location><page_7><loc_12><loc_51><loc_70><loc_72></location>We fix a 0 and numerically calculate the coefficients in (19) and (22) to different orders and compute the value of the filed ψ ( x ) as given in (21). Since we wish to impose the boundary condition of vanishing field at infinity, we solve the equation ψ (0) = 0 for ω . It is observed, by comparison with the values in [15], that the quasi normal frequencies are one of the solutions of the equation ψ (0) = 0, solved for ω , after ψ has been computed using (21) for some reasonably high value of n , which should be fixed by trial and error. We assume the same to be true in higher orders as well and look for the value of the quasi normal modes among the set of discrete values of ω that are obtained. For each value of ω obtained, we evaluate the absolute value of the LHS of (18) at a point close to the event horizon, since we have assumed plane wave solutions there. We choose that value of ω as the quasi normal frequency for which the absolute value comes closest to zero (the assumption here is that (21) is satisfied exactly only for the quasi normal frequency, which we seek, and not by other roots of ψ = 0.). We increase the number of terms to which ψ ( x ) and the coefficients are calculated until the required precision is attained.</text>
<section_header_level_1><location><page_7><loc_12><loc_47><loc_54><loc_48></location>3 Discussion on Results of the Numerical Calculation</section_header_level_1>
<text><location><page_7><loc_32><loc_38><loc_32><loc_39></location>/negationslash</text>
<text><location><page_7><loc_12><loc_26><loc_70><loc_45></location>We implement the procedure outlined above after fixing the value of the constants a 0 and R to 1. We investigate the quasi normal frequencies of large ( r h /greatermuch R ) and intermediate ( r + ∼ R ) black holes and set ω = ω R -iω I as done in [15] since we are interested in damped modes. As mentioned before, the analysis is limited to the cases where d -2 k -1 = 0. The results for the lowest ( l = 0) modes of the massless scalar field for first order theories have been summarized in TABLE 1. TABLE 2 contains the same for higher orders. Figures 1 to 6 show the results in detail. As evident from Figure 1 and Figure 2, both ω R and ω I show linear dependence on r + for the case of large black holes in first order theories. The linearity seems to be broken when we move to the intermediate-sized black holes, the results for which have been plotted in Figure 3 and Figure 4. Although the plots for intermediate-sized black holes look linear, the ω R -r + dependence for their case rather resembles an ( x, y = x + 1 x ) relation. The temperature of the event horizon for the metric (8), given by</text>
<formula><location><page_7><loc_27><loc_21><loc_70><loc_25></location>T = 1 4 πκ B k ( ( d -1) r + R 2 + d -2 k -1 r + ) , (24)</formula>
<text><location><page_8><loc_19><loc_84><loc_20><loc_85></location>d</text>
<text><location><page_8><loc_21><loc_84><loc_23><loc_85></location>= 4</text>
<text><location><page_8><loc_23><loc_84><loc_24><loc_85></location>, k</text>
<text><location><page_8><loc_25><loc_84><loc_27><loc_85></location>= 1</text>
<text><location><page_8><loc_33><loc_84><loc_34><loc_85></location>d</text>
<text><location><page_8><loc_34><loc_84><loc_37><loc_85></location>= 5</text>
<text><location><page_8><loc_37><loc_84><loc_38><loc_85></location>, k</text>
<text><location><page_8><loc_38><loc_84><loc_41><loc_85></location>= 1</text>
<text><location><page_8><loc_47><loc_84><loc_48><loc_85></location>d</text>
<text><location><page_8><loc_48><loc_84><loc_50><loc_85></location>= 6</text>
<text><location><page_8><loc_50><loc_84><loc_52><loc_85></location>, k</text>
<text><location><page_8><loc_52><loc_84><loc_54><loc_85></location>= 1</text>
<table>
<location><page_8><loc_12><loc_72><loc_58><loc_84></location>
<caption>Table 2 Variation of the frequency of the lowest ( l = 0) massless ( m = 0) mode in higher orders.</caption>
</table>
<table>
<location><page_8><loc_12><loc_53><loc_63><loc_66></location>
<caption>Table 1 Variation of the frequency of the lowest ( l = 0) massless ( m = 0) mode for first order theories.</caption>
</table>
<text><location><page_8><loc_12><loc_37><loc_70><loc_50></location>where κ B denotes the Boltzmann's constant, also depends on r + in the same manner. Since ( x, y = x + 1 x ) ∼ x for large x , we conclude that both ω R and ω I scale with the temperature for large as well as intermediate-sized black holes for first order theories, in agreement with earlier works [15]. Another observation is that ω I seems to be independent of dimension d in first order theories. When we consider higher order theories, the numerical results for which have been plotted in Figure 5 and Figure 6, we observe that all modes are purely damped ones. We have only plotted ω I vs r + in the case of higher order theories for this reason. There, we observe that, both for large and intermediate-sized black holes, ω I is independent of the order of the theory when the dimension d stays the same.</text>
<section_header_level_1><location><page_8><loc_12><loc_32><loc_70><loc_33></location>4 Asymptotic quasi normal modes and area spectrum of large black holes</section_header_level_1>
<text><location><page_8><loc_12><loc_22><loc_70><loc_30></location>We analytically find the asymptotic form of the quasi normal frequencies in the large black hole limit following the method of perturbative expansion of the wave equation in the dimensionless parameter ω/T H that has earlier been employed in the case of d -dimensional SAdS black holes [27]. Here, T H is the Hawking temperature of the horizon and ω is the frequency of the mode. We take the metric to be of the form given in (8). For large black holes, the metric gets approximated as,</text>
<text><location><page_9><loc_13><loc_80><loc_14><loc_80></location>ω</text>
<figure>
<location><page_9><loc_14><loc_71><loc_45><loc_88></location>
<caption>Fig. 1 ω R vs r + plot for large black holes in first order theories</caption>
</figure>
<figure>
<location><page_9><loc_13><loc_49><loc_45><loc_66></location>
<caption>Fig. 2 ω I vs r + plot for large black holes in first order theories</caption>
</figure>
<formula><location><page_9><loc_28><loc_40><loc_70><loc_43></location>ds 2 = ˆ f ( r ) dt 2 + dr 2 ˆ f ( r ) + r 2 ds 2 ( E d -2 ) , (25)</formula>
<text><location><page_9><loc_15><loc_37><loc_18><loc_39></location>with</text>
<formula><location><page_9><loc_32><loc_32><loc_70><loc_36></location>ˆ f ( r ) = r 2 R 2 -( 2 G k M r d -2 k -1 ) . (26)</formula>
<text><location><page_9><loc_15><loc_31><loc_50><loc_32></location>It is easily seen that the event horizon is given by</text>
<formula><location><page_9><loc_33><loc_25><loc_70><loc_29></location>r h = R [ 2 G k M R d -2 k -1 ] 1 d -1 . (27)</formula>
<text><location><page_9><loc_12><loc_22><loc_70><loc_25></location>In terms of the new metric (25), the Klein-Gordon field equation (12) for m = 0 becomes</text>
<text><location><page_10><loc_14><loc_80><loc_15><loc_80></location>ω</text>
<figure>
<location><page_10><loc_14><loc_71><loc_44><loc_88></location>
<caption>Fig. 3 ω R vs r + plot for intermediate black holes in first order theories</caption>
</figure>
<text><location><page_10><loc_13><loc_58><loc_14><loc_59></location>ω</text>
<figure>
<location><page_10><loc_14><loc_49><loc_44><loc_66></location>
<caption>Fig. 4 ω I vs r + plot for intermediate black holes in first order theories</caption>
</figure>
<text><location><page_10><loc_30><loc_49><loc_31><loc_49></location>+</text>
<formula><location><page_10><loc_24><loc_41><loc_70><loc_43></location>1 r d -2 ∂ r ( r d A ( r ) ∂ r Φ ) -R 4 r 2 A ( r ) ∂ 2 t Φ -R 2 r 2 ∇ 2 Φ = 0 , (28)</formula>
<text><location><page_10><loc_15><loc_39><loc_19><loc_40></location>where</text>
<text><location><page_10><loc_15><loc_33><loc_31><loc_34></location>We write the field Φ as</text>
<formula><location><page_10><loc_34><loc_34><loc_70><loc_38></location>A ( r ) = 1 -( r h r ) d -1 k . (29)</formula>
<formula><location><page_10><loc_32><loc_30><loc_70><loc_32></location>Φ ( t, r, x i ) = e i ( ωt -p · x ) Ψ ( r ) , (30)</formula>
<text><location><page_10><loc_15><loc_29><loc_39><loc_30></location>and change the variable from r to</text>
<formula><location><page_10><loc_36><loc_24><loc_70><loc_27></location>y = ( r r h ) d -1 2 k , (31)</formula>
<text><location><page_10><loc_15><loc_22><loc_30><loc_24></location>so that (28) becomes</text>
<text><location><page_11><loc_12><loc_80><loc_14><loc_80></location>ω</text>
<figure>
<location><page_11><loc_13><loc_71><loc_45><loc_88></location>
<caption>Fig. 5 ω I vs r + plot for large black holes in higher order theories</caption>
</figure>
<text><location><page_11><loc_13><loc_58><loc_14><loc_58></location>ω</text>
<figure>
<location><page_11><loc_13><loc_49><loc_44><loc_66></location>
<caption>Fig. 6 ω I vs r + plot for intermediate black holes in first order theories</caption>
</figure>
<formula><location><page_11><loc_30><loc_35><loc_70><loc_41></location>y Q ( y 2 -1) ( y 2 k -1 ( y 2 -1) Ψ ' ) ' + [ ˆ ω 2 A 2 y 2 -ˆ p 2 A 2 ( y 2 -1) ] Ψ = 0 , (32)</formula>
<text><location><page_11><loc_15><loc_33><loc_47><loc_34></location>where the parameters ˆ ω and ˆ p are defined as</text>
<formula><location><page_11><loc_33><loc_28><loc_70><loc_31></location>ˆ ω = ωR 2 r h , ˆ p = | p | R r h , (33)</formula>
<text><location><page_11><loc_15><loc_26><loc_17><loc_27></location>and</text>
<formula><location><page_11><loc_28><loc_21><loc_70><loc_24></location>Q = 6 k -(2 k -1) d -1 d -1 , A = d -1 2 k . (34)</formula>
<text><location><page_12><loc_12><loc_85><loc_70><loc_89></location>We investigate the behavior of (32) near the boundaries y → 1 and y →∞ and the point y →-1 in order to develop an ansatz for Ψ ( y ). The following solutions are obtained:</text>
<formula><location><page_12><loc_30><loc_79><loc_70><loc_84></location>Ψ ∼ y -2 k , y →∞ ( y -1) ± i ˆ ω/ 2 A , y → 1 ( y +1) ± ˆ ω/ 2 A , y →-1 (35)</formula>
<text><location><page_12><loc_12><loc_74><loc_70><loc_81></location> Since we demand ingoing plane wave like solutions at the horizon, we take the form Ψ ∼ ( y -1) -i ˆ ω/ 2 A near the horizon ( y = 1). We isolate the solutions near y = ± 1 and write</text>
<formula><location><page_12><loc_27><loc_72><loc_70><loc_74></location>Ψ ( y ) = ( y -1) -i ˆ ω/ 2 A ( y +1) ± ˆ ω/ 2 A N ( y ) . (36)</formula>
<text><location><page_12><loc_15><loc_70><loc_65><loc_72></location>Substituting (36) in (32), we deduce the equation satisfied by N ( y ) as</text>
<formula><location><page_12><loc_23><loc_55><loc_70><loc_68></location>y ( y 2 -1) N '' -ˆ ω 2 y 2 A 2 ( y 2 -1) N + + { ˆ ω A ( ∓ i ˆ ω 2 A ± k -ik ) y -( i ± 1)(2 k -1) ˆ ω 2 A } N {( 2 k +1 -i ∓ 1 A ˆ ω ) y 2 -i ± 1 A ˆ ωy -(2 k -1) } N ' + 1 y Q +2 k -2 ( ˆ ω 2 y 2 A 2 ( y 2 -1) -ˆ p 2 A 2 ) N = 0 . (37)</formula>
<text><location><page_12><loc_12><loc_51><loc_70><loc_55></location>We consider (37) in the range of large ˆ ω and large y , so that y 2 ≈ y 2 -1 and the terms proportional to 1 / ( y Q +2 k -2 ) may be dropped along with the constant terms. Then (37) reduces to</text>
<formula><location><page_12><loc_25><loc_42><loc_70><loc_49></location>( y 2 -1) N '' + ˆ ω A ( ∓ i ˆ ω 2 A ± k -ik ) N + {( (2 k +1) -i ∓ 1 A ˆ ω ) y -i ± 1 A ˆ ω } N ' = 0 , (38)</formula>
<text><location><page_12><loc_15><loc_41><loc_55><loc_42></location>which is the Hypergeometric equation with the solution</text>
<formula><location><page_12><loc_31><loc_38><loc_70><loc_39></location>N ( y ) = 2 F 1 ( a, b ; c ; ( y +1) / 2) , (39)</formula>
<text><location><page_12><loc_15><loc_36><loc_19><loc_38></location>where</text>
<formula><location><page_12><loc_23><loc_33><loc_70><loc_35></location>a = k -i ∓ 1 2 A ˆ ω + k, b = -i ∓ 1 2 A ˆ ω, c = 2 k +1 2 ± ˆ ω A . (40)</formula>
<text><location><page_12><loc_12><loc_28><loc_70><loc_32></location>In order to match the behavior of the solution (36) at infinity with that demanded by (35), we demand that N ( y ) be a polynomial as y →∞ . That condition is satisfied when</text>
<formula><location><page_12><loc_34><loc_25><loc_70><loc_27></location>a = -n, n = 1 , 2 , ... (41)</formula>
<text><location><page_12><loc_12><loc_22><loc_70><loc_25></location>If a = -n , then, according to the property of the hypergeometric equation, N ( y ) ∼ y n = y -a , so that, according to (36),</text>
<formula><location><page_13><loc_28><loc_83><loc_70><loc_87></location>Ψ ∼ ( y -1) -i ˆ ω/ 2 A ( y +1) ± ˆ ω/ 2 A y -a ≈ y -i ˆ ω/ 2 A y ± ˆ ω/ 2 A y -a = y -2 k , (42)</formula>
<text><location><page_13><loc_12><loc_80><loc_70><loc_82></location>as required. We deduce the expression for the asymptotic form of quasi normal frequencies from (41) as follows:</text>
<formula><location><page_13><loc_30><loc_73><loc_70><loc_78></location>a = -n ⇒ 2 k -i ∓ 1 2 A ˆ ω = -n ⇒ ˆ ω asy = A ( n +2 k )( ± 1 -i ) , (43)</formula>
<text><location><page_13><loc_15><loc_71><loc_29><loc_72></location>so that (33) implies</text>
<formula><location><page_13><loc_30><loc_67><loc_70><loc_70></location>ω asy = A ( r h R 2 ) ( n +2 k )( ± 1 -i ) , (44)</formula>
<text><location><page_13><loc_12><loc_62><loc_70><loc_67></location>which gives the asymptotic form of the quasi normal frequencies for large maximally symmetric AdS black holes in the Lovelock model. We observe that the high-overtone quasi normal frequencies are equispaced, which is in agreement with earlier observations [15,9,12].</text>
<section_header_level_1><location><page_13><loc_12><loc_58><loc_26><loc_59></location>4.1 Area Spectrum</section_header_level_1>
<text><location><page_13><loc_12><loc_50><loc_70><loc_56></location>We calculate the area spectrum of the large AdS black holes in Lovelock model using the new physical interpretation of the quasi normal modes proposed by Maggiore [19] and following Kunstatter's method [28]. According to the first law of black hole thermodynamics, for a black hole system with energy E (or M ) with Hawking temperature T H and horizon area A , the following relation holds:</text>
<formula><location><page_13><loc_36><loc_46><loc_70><loc_49></location>dM = 1 4 T H dA. (45)</formula>
<text><location><page_13><loc_12><loc_43><loc_70><loc_45></location>According to Kunstatter, if the frequency of oscillation of the system is ω ( E ), then the quantity</text>
<formula><location><page_13><loc_37><loc_38><loc_70><loc_42></location>I = ∫ dE ω ( E ) , (46)</formula>
<text><location><page_13><loc_12><loc_33><loc_70><loc_38></location>is to be taken as the corresponding adiabatic invariant. According to Maggiore [19], the black hole system has to be modeled by a collection of damped harmonic oscillators. If the system has a quasi normal frequency ω = ω R + iω I , then the corresponding vibrational frequency according to the model is to be taken as</text>
<formula><location><page_13><loc_35><loc_28><loc_70><loc_32></location>ω 0 = √ ω 2 R + ω 2 I . (47)</formula>
<text><location><page_13><loc_12><loc_22><loc_70><loc_29></location>In the highly damped and highly excited cases, ω 0 can be approximated by ω I and ω R respectively. Here, we see that for higher values of the number n , the both ω R and ω I increase. Therefore we consider transitions between two adjacent energy levels of the system and take the physical frequency as equal to the difference in ω 0 for the two systems; that is, we take</text>
<formula><location><page_14><loc_30><loc_86><loc_70><loc_88></location>ω ( E ) = ∆ω = ( ω 0 ) n -( ω 0 ) n -1 . (48)</formula>
<text><location><page_14><loc_12><loc_83><loc_70><loc_85></location>We deduce the area spectrum from (46) using the relations (27) and (44). The expression for the adiabatic invariant now reads</text>
<formula><location><page_14><loc_25><loc_78><loc_70><loc_82></location>I = ∫ dE ω ( E ) = ∫ dM ∆ω = ∫ ( dM dr h )( 1 ∆ω ) dr h . (49)</formula>
<text><location><page_14><loc_12><loc_76><loc_71><loc_78></location>Using (27), (44), (47) and (48) in (49), it is easy to see that the Bohr-Sommerfeld quantization condition, namely I = n /planckover2pi1 now reads</text>
<formula><location><page_14><loc_27><loc_71><loc_70><loc_74></location>( d -1 d -2 )( 1 2 √ 2 AG k R 2 k -2 ) r d -2 h = n /planckover2pi1 , (50)</formula>
<text><location><page_14><loc_15><loc_69><loc_66><loc_71></location>which, in terms of the area A of the horizon, can be written in the form</text>
<formula><location><page_14><loc_38><loc_66><loc_70><loc_68></location>A = γn /planckover2pi1 , (51)</formula>
<text><location><page_14><loc_12><loc_65><loc_16><loc_66></location>where</text>
<formula><location><page_14><loc_21><loc_59><loc_70><loc_63></location>γ = [ Γ ( d -1 2 )( d -1 d -2 )( 1 4 √ 2 π ( d -1) / 2 AG k R 2 k -2 ) ] -1 . (52)</formula>
<text><location><page_14><loc_12><loc_52><loc_70><loc_59></location>Thus the area spectrum, and consequently the entropy spectrum, of large black holes in asymptotically AdS Lovelock spacetimes is seen to depend on the parameters G k and the AdS radius R of the theory. The dependence of A on R is observed only when higher orders are considered. A is also dependent on the dimension d of the spacetime, and consequently on the order of the theory.</text>
<section_header_level_1><location><page_14><loc_12><loc_48><loc_22><loc_49></location>5 Conclusion</section_header_level_1>
<text><location><page_14><loc_12><loc_33><loc_70><loc_47></location>We have analyzed the evolution of massless Klein-Gordon field in the maximally symmetric asymptotically AdS spacetime surrounding a black hole in the Lovelock model. We have used the form of the metric that has been derived in [26] in order to compute the quasi normal frequencies using the Horowitz-Hubeny method. The results of the numerical computation show that the modes in the case of higher order theories are purely damped. For the case of large as well as intermediate black holes, the frequencies are observed to scale linearly with the temperature of the event horizon. When we consider higher order theories, the imaginary part of the quasi normal frequencies is observed to be independent of the order of the theory in higher dimensions. They appear to be dependent on the dimension only.</text>
<text><location><page_14><loc_12><loc_22><loc_70><loc_33></location>The asymptotic form of the quasi normal frequencies for the case of very large black holes has been analytically determined using the method of perturbative expansion of the wave equation in terms of ω/T H , as developed in [27]. We find that the asymptotic modes are equispaced, in agreement with previous results. We have also calculated the area spectrum spacing and found it to be dependent on the value of R , d and k . This is also in contrast to the case of first order theories where we always obtain area spectra that are equidistant even when the parameters of the black hole spacetime change.</text>
<section_header_level_1><location><page_15><loc_12><loc_88><loc_28><loc_89></location>6 Acknowledgement</section_header_level_1>
<text><location><page_15><loc_12><loc_81><loc_70><loc_86></location>The authors would like to acknowledge financial assistance from the Council of Scientific and Industrial Research (CSIR), India during the work under the CSIR Emeritus Scientistship Scheme sanctioned to one of the authors (VCK). VCK would also like to acknowledge Associateship of IUCAA, Pune, India.</text>
<section_header_level_1><location><page_15><loc_12><loc_77><loc_20><loc_78></location>References</section_header_level_1>
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</document>
[
{
"title": "Scalar filed evolution and Area Spectrum for Lovelock-AdS Black Holes",
"content": "C.B Prasobh · V.C Kuriakose Received: 25 June 2013 / Accepted: 30 August 2013 Abstract We study the modes of evolution of massless scalar fields in the asymptotically AdS spacetime surrounding maximally symmetric black holes of large and intermediate size in the Lovelock model. It is observed that all modes are purely damped at higher orders. Also, the rate of damping is seen to be independent of order at higher dimensions. The asymptotic form of these frequencies for the case of large black holes is found analytically. Finally, the area spectrum for such black holes is found from these asymptotic modes. Keywords Quasinormal Modes Lovelock AdS Area Spectrum Scalar Field · · · · PACS PACS code1 · PACS code2 · more",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Gauge-gravity dualities like the AdS/CFT correspondence [1] make it possible to study the properties of conformal fields in a particular dimension d by studying the evolution of fields in a black hole spacetime that is asymptotically AdS in ( d +1) dimensions and this has led to considerable interest in the study of asymptotically AdS black hole spacetimes. The main difficulty in studying field evolution in such spacetimes is that the stability of the spacetime against perturbations is not always guaranteed, unlike in the case of asymptotically flat spacetimes in first order theories in four dimensions. The instability of linear perturbations in higher-order and higher dimensional theories has already been investigated [2,3, 4]. The instability also extends to the thermodynamics of the black hole when we consider AdS spacetimes, with the well-known Hawking-Page phase transitions [5] signaling a transition between the black hole spacetime and thermal AdS spacetime. Recent studies [6,7,8] on the Lovelock model have confirmed the existence of",
"pages": [
1
]
},
{
"title": "C.B Prasobh",
"content": "Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: prasobhcb@outlook.com V.C Kuriakose Department of Physics, Cochin University of Science and Technology, Cochin 682202, India E-mail: vck@cusat.ac.in dynamic instability against metric perturbations in asymptotically flat black hole spacetimes. Dynamic instability means that the solutions to the equation for the metric perturbation become unstable outside the event horizon for large values of their eigenvalue. This instability exists for all types of metric perturbations-tensor, vector and scalar. These instabilities occur when the mass of the black hole falls below a lower critical bound that depends on the coupling constants, the dimension of the spacetime and the order of the theory. Quasi normal modes are damped oscillations (having complex frequencies), known to dominate the intermediate stage of the evolution of small perturbations of a black hole spacetime and have been studied extensively. Detailed reviews and methods of calculation of quasi normal modes are found in numerous papers that include [9,10,11,12,13,14]. Quasi normal modes are known to depend only on the parameters of the black hole, such as mass, charge and angular momentum, and be completely independent of the type of the agent that caused it. These modes are obtained as the solution of the respective field equation, when solved with respect to the metric that describes the particular black hole spacetime of interest. In the case of asymptotically flat spacetimes, the effective potential that is perceived by the field in the spacetime often resembles a finite potential barrier such that the potential vanishes at infinity. This leads to the possibility of obtaining solutions that are purely ingoing at the event horizon of the black hole and purely outgoing at infinity, both resembling plane waves when expressed in terms of a scaled co-ordinate called the tortoise co-ordinate. These modes may be observed in the future with the aid of gravitational detectors. In the case of asymptotically AdS spacetimes, however, the potential grows indefinitely at infinity. Therefore, one usually looks for solutions that vanish at infinity, while the boundary condition at the event horizon unchanged [15]. This is motivated by the case of pure AdS spacetime, where the potential effectively confines the field as if 'in a box' and solutions exist only with a discrete spectrum of real frequencies. Even though black holes in asymptotically AdS spacetimes are not believed to exist in nature, interest in studying their quasi normal modes stems from the above-mentioned AdS/CFT correspondence [1]. According to the AdS/CFT correspondence, these perturbations correspond to perturbations of the thermal state of the strongly coupled conformal field at the boundary of the spacetime and the quasi normal modes correspond to the return to thermal equilibrium, so that the quasi normal frequencies give a measure of the time scale for the relaxation, which is difficult to compute directly. This provides the motivation to study the quasi normal modes of various fields in asymptotically AdS spacetimes. Earlier work the quasi normal modes of Schwarzschild-AdS black holes [15] have proved that the modes scale with the temperature of the event horizon. Complete quantization of gravity is one of the major goals of modern theoretical physics. Despite decades of research by physicists all over the world, it is yet to be achieved with complete success. One often takes clues from the classical theory of a system when attempting its quantization and gravity needs to be no different. Quantization of the black hole horizon area is expected to be a major feature of any successful quantum theory of gravity. The original 'classical' theory of gravity, namely the General Theory of Relativity (GTR) does provide us with the tools necessary to estimate the value of the area quantum. It is known from field theory that the presence of a periodicity in the classical theory of a system points to the existence of an adiabatic invariant with a discrete spectrum in the corresponding quantum theory. Interestingly, it has been observed in GTR that the numerical value of these frequencies, in the limit of 'large' frequencies, follow a distinct pattern, with the real part approaching a fixed value. These are termed asymptotic frequencies. Suggestions have been made that the fixed value of the real part can be viewed as a physically relevant periodicity in the (classical) black hole system which would then lead to the existence of certain adiabatic invariant quantity, which in turn would possess equally spaced spectrum according to Bohr-Sommerfeld quantization. Once we read it together with Bekenstein's original proposal [16] that the black hole entropy is an adiabatic invariant with a discrete, equally spaced spectrum, we come to the conclusion that the entropy spectrum (and, by extension, the area spectrum) can be deduced from the asymptotic value of the quasi normal frequencies. The connection between the fixed asymptotic frequencies and the quantized area spectrum was made by Hod [17]. Dreyer [18] recovered Hod's result in the Loop Quantum Gravity. A new interpretation for the quasi normal modes ω = ω R + iω I of perturbed black holes as equivalent to that of a collection of damped harmonic oscillators with real frequency ω 0 = √ ω 2 R + ω 2 I was introduced by Maggiore [19] and used in conjunction with Hod's method in order to compute the area spectrum of Schwarzschild black holes. As with the case of quasi normal modes, it is the AdS/CFT correspondence that provides the motivation for studying the area spectrum of black holes in asymptotically AdS spacetimes. Recent studies ([20,21,22,23,24] and references therein) suggest that the gravitational dual of the holographic entanglement entropy in quantum field theories is the area of minimal-area surfaces in AdS spaces. The entropy can be used to study phase transitions between various states of the field. An area-entanglement entropy relation of the form S A = Area ( γ A ) 4 G ( d -2) N has been proposed [20,25] which is very similar to the familiar area-entropy relation in the General Theory of Relativity. Here, γ A is the d -dimensional minimal area surface in AdS d +2 and G ( d -2) N is the ( d +2)-dimensional gravitational constant of the AdS gravity. Although the above relation was originally proposed for AdS spaces, it is equally applicable to any asymptotically AdS space, including one containing a black hole. In that case, the minimal surface tends to wrap the horizon and thus we can use the area of the event horizon in order to compute the entanglement entropy in the conformal field theory. In this work, we numerically compute the quasi normal frequencies for massless scalar field perturbations in asymptotically AdS, spherically symmetric spacetimes in Lovelock model using the metric derived in [26]. We analytically find out the asymptotic form of the frequencies following [27] and use it to deduce the area spectrum of large black holes in the model. A brief outline of the paper is as follows: in Sect. 2, we explain the maximally symmetric Lovelock model and the resulting metric for the spacetime as given in [26]. Details of the Horowitz-Hubeny numerical method of computing the quasi normal frequencies in AdS black hole spacetimes to the case of massless scalar fields in the vicinity of such a black hole in the Lovelock-AdS model are also given in the same section. The results of the numerical procedure are presented and analyzed in Sect. 3. In Sect. 4, we analytically determine the asymptotic quasi normal frequencies of the field for the case of large back holes following [27]. The results of that analysis are used in order to find the area spectrum of the black hole using the Kunstatter's method [28] in Sect. 4.1. The results are summarized in Sect. 5.",
"pages": [
1,
2,
3,
4
]
},
{
"title": "2 The metric and the numerical computation of Quasi normal Frequencies",
"content": "The Lovelock model [29] is considered to be the most natural generalization of GTR. The Lovelock Lagrangian is a polynomial which consists of dimensionally continued higher order curvature terms. The most striking property of this Lagrangian is that it yields field equations that are in second order in the metric although the Lagrangian itself may contain higher order terms. Also, the theory is known to give solutions that are free of ghosts. The maximum order of terms in the action, k , is fixed by the number of dimensions of the spacetime d in Lovelock model according to the relation k = [ d -1 2 ] where [ x ] denotes the integer part of x . The action is written in terms of the Riemann curvature R ab = dω ab + ω a c ω cb and the vielbein e a as where α p are arbitrary (positive) coupling constants, and L ( p ) , given by is the p th order dimensionally continued term in the Lagrangian. The difficulty with the Lagrangian given above, with arbitrary values for α p , is that it becomes very difficult (if not impossible) to study the evolution of fields, since it is not at all clear whether the operator representing the evolution is Hermitian or not. As mentioned in the previous section, this problem, for the case of metric perturbations, has been analyzed in [6]. Although a general instability depending on the black hole mass has been established in that work, the numerical value for the critical mass has not been calculated. Also, it is very difficult to predict whether the presence of a cosmological constant raises or lowers the critical mass, as long as we consider a model with arbitrary α p . Moreover, for the same case, the the existence of negative energy solutions with horizons and positive energy solutions with naked singularities for (1) has been pointed out earlier [30,31]. These difficulties bring out the necessity of selecting suitable values for the coupling coefficients in order to have models that support maximally symmetric solutions and external perturbations. Maximally symmetric solutions to Lovelock model have long been known [26], which are derived by requiring that the theories must possess a unique cosmological constant (and consequently a unique AdS radius R ) for all orders. The resulting set of coupling constants are seen to be labeled by the order k and the gravitational constant G k . The metric describing the spherically symmetric black hole spacetime is derived from the action given in (1) with the choice where 1 ≤ k ≤ [ d -1 2 ]. The resulting field equations are of the form Here, ¯ R ab := R ab + 1 R 2 e a e b . Such theories are labeled by k and have two fundamental constants, κ and R , related to the gravitational constant G k and the cosmological constant Λ respectively through the relations Ω d -2 being the volume of the ( d -2) dimensional spherically symmetric tangent space. The static and spherically symmetric solutions to (4), written in Schwarzschild-like coordinates, take the form where f ( r ) is given by We take σ = 1. The integration constant C 1 is written as where M stands for the mass of the black hole. The constant C 0 is chosen so that the horizon shrinks to a point for M → 0, as /negationslash It is interesting that the exponent of ( 1 r ) in (9) is proportional to ( d -2 k -1). Since k = [ d -1 2 ], ( d -2 k -1) = 0 in odd dimensions and ( d -2 k -1) = 1 in even dimensions. Thus the solution in even dimensions resemble the Schwarzschild-AdS solution. The ( d -2 k -1) = 0 cases correspond to Chern-Simmons theories which have a vacuum that is different from AdS [26]. Their quasi normal modes, mass and area spectra have already been computed [32]. What makes it interesting is the fact that recent studies [6] on the stability of metric perturbations in Lovelock model also point out that it is possible to predict the (in)stability of the perturbations only in even dimensions. The present work is limited to the cases where d -2 k -1 = 0. Then we have C 0 = 0 and C 1 = 2 G k M . Consider the scalar field Φ ( r, t, x i ) that obeys the Klein-Gordon equation given by x i being the co-ordinates in the spherically symmetric tangent space and g AB being components of the metric tensor. We impose the boundary conditions of ingoing plane wave solution at the event horizon and vanishing field at the boundary. The boundary condition at the horizon suggests the ansatz Φ = e -iω ( t + r ∗ ) where r ∗ is the tortoise coordinate defined by In the ( v = t + r ∗ , r ) system, the metric reads and we take the ansatz so that (12) becomes with the effective potential Here, l represents the eigenvalue of the operator on the LHS of (12) acting on the functions Y ( x i ) in the spherically symmetric tangent space. In order to numerically calculate the quasi normal frequencies for (16), we expand the field Φ as a power series about the horizon and impose the vanishing boundary condition at infinity. We change the variable from r to x = 1 r in order to map the range r + < r < ∞ to a finite range. In terms of x , (16) becomes where and In the numerical procedure, we find out the coefficients of the expansion of the functions s ( x ) , t ( x ) and u ( x ) as power series in ( x -x + ) using a computer with s i , t i and u i denoting the coefficient for the i th term. Then an expansion for ψ ( x ) of the form is substituted into (18) which yields where We fix a 0 and numerically calculate the coefficients in (19) and (22) to different orders and compute the value of the filed ψ ( x ) as given in (21). Since we wish to impose the boundary condition of vanishing field at infinity, we solve the equation ψ (0) = 0 for ω . It is observed, by comparison with the values in [15], that the quasi normal frequencies are one of the solutions of the equation ψ (0) = 0, solved for ω , after ψ has been computed using (21) for some reasonably high value of n , which should be fixed by trial and error. We assume the same to be true in higher orders as well and look for the value of the quasi normal modes among the set of discrete values of ω that are obtained. For each value of ω obtained, we evaluate the absolute value of the LHS of (18) at a point close to the event horizon, since we have assumed plane wave solutions there. We choose that value of ω as the quasi normal frequency for which the absolute value comes closest to zero (the assumption here is that (21) is satisfied exactly only for the quasi normal frequency, which we seek, and not by other roots of ψ = 0.). We increase the number of terms to which ψ ( x ) and the coefficients are calculated until the required precision is attained.",
"pages": [
4,
5,
6,
7
]
},
{
"title": "3 Discussion on Results of the Numerical Calculation",
"content": "/negationslash We implement the procedure outlined above after fixing the value of the constants a 0 and R to 1. We investigate the quasi normal frequencies of large ( r h /greatermuch R ) and intermediate ( r + ∼ R ) black holes and set ω = ω R -iω I as done in [15] since we are interested in damped modes. As mentioned before, the analysis is limited to the cases where d -2 k -1 = 0. The results for the lowest ( l = 0) modes of the massless scalar field for first order theories have been summarized in TABLE 1. TABLE 2 contains the same for higher orders. Figures 1 to 6 show the results in detail. As evident from Figure 1 and Figure 2, both ω R and ω I show linear dependence on r + for the case of large black holes in first order theories. The linearity seems to be broken when we move to the intermediate-sized black holes, the results for which have been plotted in Figure 3 and Figure 4. Although the plots for intermediate-sized black holes look linear, the ω R -r + dependence for their case rather resembles an ( x, y = x + 1 x ) relation. The temperature of the event horizon for the metric (8), given by d = 4 , k = 1 d = 5 , k = 1 d = 6 , k = 1 where κ B denotes the Boltzmann's constant, also depends on r + in the same manner. Since ( x, y = x + 1 x ) ∼ x for large x , we conclude that both ω R and ω I scale with the temperature for large as well as intermediate-sized black holes for first order theories, in agreement with earlier works [15]. Another observation is that ω I seems to be independent of dimension d in first order theories. When we consider higher order theories, the numerical results for which have been plotted in Figure 5 and Figure 6, we observe that all modes are purely damped ones. We have only plotted ω I vs r + in the case of higher order theories for this reason. There, we observe that, both for large and intermediate-sized black holes, ω I is independent of the order of the theory when the dimension d stays the same.",
"pages": [
7,
8
]
},
{
"title": "4 Asymptotic quasi normal modes and area spectrum of large black holes",
"content": "We analytically find the asymptotic form of the quasi normal frequencies in the large black hole limit following the method of perturbative expansion of the wave equation in the dimensionless parameter ω/T H that has earlier been employed in the case of d -dimensional SAdS black holes [27]. Here, T H is the Hawking temperature of the horizon and ω is the frequency of the mode. We take the metric to be of the form given in (8). For large black holes, the metric gets approximated as, ω with It is easily seen that the event horizon is given by In terms of the new metric (25), the Klein-Gordon field equation (12) for m = 0 becomes ω ω + where We write the field Φ as and change the variable from r to so that (28) becomes ω ω where the parameters ˆ ω and ˆ p are defined as and We investigate the behavior of (32) near the boundaries y → 1 and y →∞ and the point y →-1 in order to develop an ansatz for Ψ ( y ). The following solutions are obtained: Since we demand ingoing plane wave like solutions at the horizon, we take the form Ψ ∼ ( y -1) -i ˆ ω/ 2 A near the horizon ( y = 1). We isolate the solutions near y = ± 1 and write Substituting (36) in (32), we deduce the equation satisfied by N ( y ) as We consider (37) in the range of large ˆ ω and large y , so that y 2 ≈ y 2 -1 and the terms proportional to 1 / ( y Q +2 k -2 ) may be dropped along with the constant terms. Then (37) reduces to which is the Hypergeometric equation with the solution where In order to match the behavior of the solution (36) at infinity with that demanded by (35), we demand that N ( y ) be a polynomial as y →∞ . That condition is satisfied when If a = -n , then, according to the property of the hypergeometric equation, N ( y ) ∼ y n = y -a , so that, according to (36), as required. We deduce the expression for the asymptotic form of quasi normal frequencies from (41) as follows: so that (33) implies which gives the asymptotic form of the quasi normal frequencies for large maximally symmetric AdS black holes in the Lovelock model. We observe that the high-overtone quasi normal frequencies are equispaced, which is in agreement with earlier observations [15,9,12].",
"pages": [
8,
9,
10,
11,
12,
13
]
},
{
"title": "4.1 Area Spectrum",
"content": "We calculate the area spectrum of the large AdS black holes in Lovelock model using the new physical interpretation of the quasi normal modes proposed by Maggiore [19] and following Kunstatter's method [28]. According to the first law of black hole thermodynamics, for a black hole system with energy E (or M ) with Hawking temperature T H and horizon area A , the following relation holds: According to Kunstatter, if the frequency of oscillation of the system is ω ( E ), then the quantity is to be taken as the corresponding adiabatic invariant. According to Maggiore [19], the black hole system has to be modeled by a collection of damped harmonic oscillators. If the system has a quasi normal frequency ω = ω R + iω I , then the corresponding vibrational frequency according to the model is to be taken as In the highly damped and highly excited cases, ω 0 can be approximated by ω I and ω R respectively. Here, we see that for higher values of the number n , the both ω R and ω I increase. Therefore we consider transitions between two adjacent energy levels of the system and take the physical frequency as equal to the difference in ω 0 for the two systems; that is, we take We deduce the area spectrum from (46) using the relations (27) and (44). The expression for the adiabatic invariant now reads Using (27), (44), (47) and (48) in (49), it is easy to see that the Bohr-Sommerfeld quantization condition, namely I = n /planckover2pi1 now reads which, in terms of the area A of the horizon, can be written in the form where Thus the area spectrum, and consequently the entropy spectrum, of large black holes in asymptotically AdS Lovelock spacetimes is seen to depend on the parameters G k and the AdS radius R of the theory. The dependence of A on R is observed only when higher orders are considered. A is also dependent on the dimension d of the spacetime, and consequently on the order of the theory.",
"pages": [
13,
14
]
},
{
"title": "5 Conclusion",
"content": "We have analyzed the evolution of massless Klein-Gordon field in the maximally symmetric asymptotically AdS spacetime surrounding a black hole in the Lovelock model. We have used the form of the metric that has been derived in [26] in order to compute the quasi normal frequencies using the Horowitz-Hubeny method. The results of the numerical computation show that the modes in the case of higher order theories are purely damped. For the case of large as well as intermediate black holes, the frequencies are observed to scale linearly with the temperature of the event horizon. When we consider higher order theories, the imaginary part of the quasi normal frequencies is observed to be independent of the order of the theory in higher dimensions. They appear to be dependent on the dimension only. The asymptotic form of the quasi normal frequencies for the case of very large black holes has been analytically determined using the method of perturbative expansion of the wave equation in terms of ω/T H , as developed in [27]. We find that the asymptotic modes are equispaced, in agreement with previous results. We have also calculated the area spectrum spacing and found it to be dependent on the value of R , d and k . This is also in contrast to the case of first order theories where we always obtain area spectra that are equidistant even when the parameters of the black hole spacetime change.",
"pages": [
14
]
},
{
"title": "6 Acknowledgement",
"content": "The authors would like to acknowledge financial assistance from the Council of Scientific and Industrial Research (CSIR), India during the work under the CSIR Emeritus Scientistship Scheme sanctioned to one of the authors (VCK). VCK would also like to acknowledge Associateship of IUCAA, Pune, India.",
"pages": [
15
]
}
]
2013Galax...2....1A
https://arxiv.org/pdf/1303.1394.pdf
<document>
<section_header_level_1><location><page_1><loc_18><loc_84><loc_80><loc_86></location>Galaxy Rotation Curves in Covariant Hoˇrava-Lifshitz Gravity</section_header_level_1>
<section_header_level_1><location><page_1><loc_34><loc_81><loc_64><loc_82></location>J. Alexandre 1 , M. Kostacinska 2</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_24><loc_79><loc_75><loc_81></location>1 King's College London, Department of Physics, WC2R 2LS</list_item>
<list_item><location><page_1><loc_25><loc_77><loc_74><loc_79></location>2 Imperial College London, Theoretical Physics, SW7 2AZ</list_item>
</unordered_list>
<section_header_level_1><location><page_1><loc_45><loc_68><loc_53><loc_70></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_56><loc_87><loc_64></location>Using the multiplicity of solutions for the projectable case of the covariant extension of Hoˇrava-Lifshitz Gravity, we show that an appropriate choice for the auxiliary field allows for an effective description of galaxy rotation curves. This description is based on static and spherically symmetric solutions of covariant Hoˇrava-Lifshitz Gravity and does not require Dark Matter.</text>
<section_header_level_1><location><page_1><loc_12><loc_50><loc_34><loc_52></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_21><loc_35><loc_21><loc_38></location>/negationslash</text>
<text><location><page_1><loc_12><loc_29><loc_87><loc_49></location>Although General Relativity (GR) has been accurately checked in the solar system, the small distance behaviour of gravity is still not understood, and several alternatives to GR have been proposed. Amongst these modified theories is Hoˇrava-Lifshitz (HL) gravity [1], which is based on the Lifshitz approach and consists of introducing an anisotropy between space and time: when space is rescaled as x → bx , time is rescaled as t → b z t , where z is an integer. In Lifshitz models, Lorentz invariance must be recovered for z = 1, and the situation z = 1 leads to new renormalizable interactions: the convergence of loop integrals is improved by inclusion of higher orders in space derivatives, without introducing ghosts, since the number of time derivatives remains the minimum. A review of Lifshitz-type quantum field theories in flat space time can be found in [2], and the HL power-counting renormalizable theory of gravity is reviewed in [3], [4] and [5].</text>
<text><location><page_1><loc_12><loc_13><loc_87><loc_29></location>A fundamental problem of the original model of HL gravity is the existence of an additional scalar degree of freedom for the metric, which can be understood as a Goldstone mode arising from breaking of 4-dimensional diffeomorphism [6], [7]. A solution to this problem has been proposed in [8], where an auxiliary field A is introduced, such that its 'equation of motion' leads to an additional constraint, eliminating the unwanted scalar degree of freedom of the metric. The resulting theory is invariant under a new Abelian gauge symmetry U Σ (1) which involves the metric components, the auxiliary field A and an additional auxiliary field ν . This gauge symmetry, together with the 3-dimensional diffeomorphism of HL gravity, can be shown to be equivalent to a 4-dimensional diffeomorphism</text>
<text><location><page_2><loc_12><loc_75><loc_87><loc_86></location>at the lowest order in a post-Newtonian expansion, showing the equivalence with GR at long distances. For this reason, this modified version of HL gravity is called covariant HL gravity. We note however, that the long-distance limit is not obviously recovered: it has been shown in [9] that the equivalence principle is not automatically retrieved in the infrared, and that the meaning of the auxiliary field A and its coupling to matter are still open questions.</text>
<text><location><page_2><loc_12><loc_50><loc_87><loc_75></location>Because of the anisotropy between space and time, HL gravity is naturally described in terms of the Arnowitt-Deser-Misner (ADM) decomposition of the metric, which expresses a space-time foliation. An important consequences of space-time anisotropy is the possibility of imposing the lapse function N to depend on time only. This situation is called the projectable case and leads to an interesting feature: the solutions of the equations of motion are not unique. Indeed, the Hamiltonian constraint, obtained by variation of the action with respect to N , leads to an integral equation, which does not have a unique solution, as will be seen in the present article. This multiplicity of solutions is independent of the above mentioned new gauge symmetry U Σ (1), and the different solutions for A in the projectable case belong to different gauge orbits. On the other hand, in the nonprojectable case where N depends on both space and time, the Hamiltonian constraint leads to a differential equation, which has a unique solution, after fixing the constants of integration. The non-projectable case has been studied in [10] for static spherically symmetric solutions of covariant HL gravity.</text>
<text><location><page_2><loc_12><loc_26><loc_87><loc_48></location>In this article we use the freedom of choice of the auxiliary field A in the projectable case to study the possibility of fitting galaxy rotations curves without the need for Dark Matter. One can argue that the ambiguity in the choice of A leaves its physical interpretation unclear. This may be understood as one of the incomplete aspects of covariant HL gravity, and one could conjecture the existence of an additional symmetry which would restrict the set of solutions for the auxiliary field A , but this still remains to be studied. We note that the multiplicity of solutions, arising from a Hamiltonian constraint expressed in terms of an integral over space, has been discussed in [11] as an alternative to Dark Matter, in the context of the original HL gravity. Fits to galaxy rotation curves have been studied in the framework of the original HL gravity in [12] and [13], where a detailed comparison with experimental data is presented. Also, an interesting analogy between HL gravity for z = 0 and Modified Newtonian Dynamics theories has been described in [14].</text>
<text><location><page_2><loc_12><loc_8><loc_87><loc_24></location>In section 2 we review the static and spherically symmetric solutions of covariant HL gravity, and in section 3 we show how galaxy rotation curves can be described by the covariant extension of HL gravity. Although the curves we use were derived from Dark Matter models, we use them as experimental data fits. It is interesting to note that we obtain exact solutions for the auxiliary field A . Another exact solution in the Lifshitz context has been derived in [15], where the exact effective potential for a Liouville scalar theory, renormalizable in 3+1 dimensions with anisotropic scaling z = 3, proves to be an exponential, as in the usual 1+1 dimensional relativistic Liouville theory. Finally, we conclude by discussing the possiblity to describe the solar system with our solution.</text>
<section_header_level_1><location><page_3><loc_12><loc_82><loc_87><loc_86></location>2 Static and spherically symmetric solutions of covariant HL gravity</section_header_level_1>
<text><location><page_3><loc_12><loc_76><loc_87><loc_80></location>The static and spherically symmetric solutions of covariant HL gravity were derived in [25] and [26], and we review here the aspects relevant to our present study.</text>
<section_header_level_1><location><page_3><loc_12><loc_72><loc_25><loc_74></location>2.1 Action</section_header_level_1>
<text><location><page_3><loc_12><loc_69><loc_38><loc_71></location>The ADM metric we consider is</text>
<formula><location><page_3><loc_28><loc_64><loc_87><loc_68></location>ds 2 = -c 2 N 2 dt 2 + g ij ( dx i + N i dt ) ( dx j + N j dt ) , (1)</formula>
<text><location><page_3><loc_12><loc_53><loc_87><loc_64></location>where N and N i are the lapse and shift functions respectively, and g ij is the threedimensional space metric. We consider the anisotropic scaling with z = 3, for which the different operators, allowed for the model to be power-counting renormalizable, have the mass dimension of 6, at most. We note that in this case the mass dimension of the speed of light is [ c ] = z -1 = 2, and we will keep c in the different expression for the sake of clarity.</text>
<text><location><page_3><loc_12><loc_52><loc_79><loc_53></location>The covariant Hoˇrava-Lifshitz action, in the absence of cosmological constant, is</text>
<formula><location><page_3><loc_17><loc_46><loc_87><loc_51></location>S = ∫ dtd 3 x √ g ( N [ K ij K ij -K 2 -V + ν Θ ij (2 K ij + ∇ i ∇ j ν ) ] -AR (3) ) , (2)</formula>
<text><location><page_3><loc_12><loc_39><loc_87><loc_46></location>where g is the determinant of the three-dimensional metric g ij , with curvature scalar R (3) . A is an auxiliary field (without kinetic term) of mass dimension z + 1 = 4 and ν is an auxiliary field of mass dimension z -2 = 1. In the above expression, the extrinsic curvature is</text>
<formula><location><page_3><loc_29><loc_36><loc_87><loc_39></location>K ij = 1 2 N { ˙ g ij -∇ i N j -∇ j N i } , i, j = 1 , 2 , 3 , (3)</formula>
<text><location><page_3><loc_12><loc_34><loc_48><loc_35></location>where a dot denotes a time derivative, and</text>
<formula><location><page_3><loc_39><loc_29><loc_87><loc_32></location>Θ ij = R (3) ij -1 2 R (3) g ij , (4)</formula>
<text><location><page_3><loc_36><loc_12><loc_36><loc_15></location>/negationslash</text>
<text><location><page_3><loc_12><loc_8><loc_87><loc_28></location>where where R (3) ij is the Ricci tensor corresponding to the three-dimensional space metric g ij . Note that a more general kinetic term K ij K ij -λK 2 can be considered, with λ a generic parameter. The latter should be equal to 1 if one wishes to recover GR in the infrared, but in the framework of HL gravity, no symmetry imposes this parameter to be equal to 1. Constraints on λ resulting from observations of Type Ia Supernovae, Baryon Acoustic Oscillations, CMB and the requirement of Big Bang Nucleosynthesis all point towards a value near the GR parameter λ = 1 [16], while it has also been shown that covariant HL gravity with λ = 1 leads to inconsistencies, if one compares the predictions of the model with solar system tests [17]. We therefore focus here on the situation where λ = 1 only. Independently of these tests, the mathematical consistency of covariant HL gravity for λ = 1 has been studied in [18], and the corresponding Hamiltonian structure in</text>
<text><location><page_3><loc_23><loc_7><loc_23><loc_9></location>/negationslash</text>
<text><location><page_4><loc_12><loc_75><loc_87><loc_86></location>[19]. We stress again here that the purpose of this covariant extension of HL gravity is to eliminate the unwanted additional scalar graviton from the spectrum. As shown in [8], no singularity arises in the limit λ → 1, and the strong coupling of the scalar graviton does not exist anymore, as it is the case for the original HL gravity. Besides [6] and [7], one can find detailed analysis of the strong coupling problem in [20], [21], [22], and also related discussions in [23] on a extension of the original HL gravity [24].</text>
<text><location><page_4><loc_15><loc_73><loc_79><loc_75></location>The potential term V contains up to six spatial derivatives of the metric g ij :</text>
<formula><location><page_4><loc_19><loc_69><loc_87><loc_72></location>V = -c 2 R (3) -α 1 ( R (3) ) 2 -α 2 R (3) ij R (3) ij -β 1 ( R (3) ) 3 -β 2 R (3) R (3) ij R (3) ij (5)</formula>
<formula><location><page_4><loc_26><loc_66><loc_87><loc_69></location>-β 3 R (3) i j R (3) j k R (3) k i -β 4 R (3) ∇ 2 R (3) -β 5 ∇ i R (3) jk ∇ i R (3) jk , (6)</formula>
<text><location><page_4><loc_12><loc_64><loc_59><loc_66></location>where the mass dimensions of the different couplings are</text>
<formula><location><page_4><loc_19><loc_60><loc_87><loc_62></location>[ α 1 ] = [ α 2 ] = z -1 = 2 , [ β 1 ] = [ β 2 ] = [ β 3 ] = [ β 4 ] = [ β 5 ] = z -3 = 0 . (7)</formula>
<text><location><page_4><loc_12><loc_57><loc_79><loc_59></location>We consider the most general, static, spherically symmetric metric, of the form:</text>
<formula><location><page_4><loc_24><loc_53><loc_87><loc_56></location>ds 2 = -c 2 N 2 dt 2 + 1 f ( r ) ( dr + n ( r ) dt ) 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (8)</formula>
<text><location><page_4><loc_12><loc_47><loc_87><loc_51></location>where n ( r ) = N r ( r ) is the radial component of shift function, and N r = n ( r ) /f ( r ) since g rr = 1 /f ( r ). Note that n has mass dimension z -1 = 2.</text>
<section_header_level_1><location><page_4><loc_12><loc_43><loc_60><loc_45></location>2.2 Constraints and equations of motion</section_header_level_1>
<text><location><page_4><loc_12><loc_37><loc_87><loc_42></location>The variation of the action (2) with respect to the different degrees of freedom leads to the following constraints, or equations of motion, where a prime denotes a derivative with respect to the radial coordinate r .</text>
<unordered_list>
<list_item><location><page_4><loc_15><loc_32><loc_69><loc_35></location>· The variation with respect to A gives R (3) = 0, or equivalently</list_item>
</unordered_list>
<formula><location><page_4><loc_45><loc_29><loc_87><loc_32></location>rf ' + f -1 = 0 , (9)</formula>
<text><location><page_4><loc_17><loc_27><loc_63><loc_28></location>which imposes the function f ( r ) in the metric (8) to be</text>
<formula><location><page_4><loc_45><loc_22><loc_87><loc_25></location>f ( r ) = 1 -2 B r , (10)</formula>
<text><location><page_4><loc_17><loc_19><loc_47><loc_21></location>where B is a constant of integration.</text>
<unordered_list>
<list_item><location><page_4><loc_15><loc_15><loc_47><loc_18></location>· The variation with respect to ν gives</list_item>
</unordered_list>
<formula><location><page_4><loc_42><loc_12><loc_87><loc_15></location>Θ ij ∇ i ∇ j ν +Θ ij K ij = 0 , (11)</formula>
<text><location><page_4><loc_17><loc_8><loc_87><loc_11></location>In what follows we will assume the Gauge fixing of ν = 0, then the above constraint gives Θ ij K ij = 0, which is satisfied for spherically symmetric solutions.</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_83><loc_71><loc_86></location>· The variation with respect to N gives the Hamiltonian constraint</list_item>
</unordered_list>
<formula><location><page_5><loc_36><loc_77><loc_87><loc_83></location>∫ ∞ 0 dr r 2 √ f ( r ) ( K ij K ij -K 2 + V ) = 0 , (12)</formula>
<text><location><page_5><loc_17><loc_76><loc_72><loc_78></location>which is an space-integral equation since N depends on time only.</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_72><loc_48><loc_75></location>· The variation with respect to n gives</list_item>
</unordered_list>
<formula><location><page_5><loc_46><loc_70><loc_87><loc_72></location>f ' ( r ) n ( r ) = 0 , (13)</formula>
<text><location><page_5><loc_17><loc_67><loc_58><loc_69></location>such that either f is a constant ( B = 0) or n = 0.</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_63><loc_48><loc_66></location>· The variation with respect to f gives</list_item>
</unordered_list>
<formula><location><page_5><loc_18><loc_58><loc_87><loc_64></location>A ' + A 2 r ( 1 -1 f ) +4 fn ( √ rn ) ' √ r = { r 4 f -√ f 2 r 3 ∑ n =0 ( -1) n d n dr n ( r 2 √ f ∂ ∂f ( n ) )} V, (14)</formula>
<text><location><page_5><loc_17><loc_56><loc_59><loc_58></location>where the time has been rescaled such that N = 1.</text>
<section_header_level_1><location><page_5><loc_12><loc_52><loc_28><loc_54></location>2.3 Solutions</section_header_level_1>
<text><location><page_5><loc_12><loc_47><loc_87><loc_51></location>The different solutions of the previous set of equations were derived in [25], [26], and here we shortly review the different cases.</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_43><loc_87><loc_46></location>· n = 0: the solution of the differential equation (14) gives short-distance corrections to the Schwarzschild solution, details of which can be found in [25], [26];</list_item>
</unordered_list>
<text><location><page_5><loc_18><loc_39><loc_18><loc_41></location>/negationslash</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_37><loc_87><loc_41></location>· n = 0 and A = 0: leads to the Schwarzschild solution in the Painleve-Gullstrand coordinates ( n ∝ r -1 / 2 ), and the Hamiltonian constraint is automatically satisfied;</list_item>
</unordered_list>
<text><location><page_5><loc_18><loc_34><loc_18><loc_36></location>/negationslash</text>
<text><location><page_5><loc_28><loc_34><loc_28><loc_36></location>/negationslash</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_31><loc_87><loc_36></location>· n = 0 and A = 0: corresponds to the situation with multiple solutions for A and n , on which this article focuses. Since B = 0, we have f = 1 and V = 0, such that the equations (12) and (14) give respectively</list_item>
</unordered_list>
<formula><location><page_5><loc_43><loc_25><loc_87><loc_30></location>∫ ∞ 0 dr ( rn 2 ) ' = 0 rA ' +2( rn 2 ) ' = 0 , (15)</formula>
<text><location><page_5><loc_17><loc_22><loc_28><loc_23></location>with solutions</text>
<text><location><page_5><loc_17><loc_17><loc_75><loc_18></location>where C is a constant of integration and the auxiliary field A satisfies</text>
<formula><location><page_5><loc_36><loc_18><loc_87><loc_22></location>n 2 ( r ) = C r -1 2 A ( r ) + 1 2 r ∫ r 0 dρ A ( ρ ) , (16)</formula>
<formula><location><page_5><loc_44><loc_12><loc_87><loc_16></location>∫ ∞ 0 drrA ' ( r ) = 0 . (17)</formula>
<text><location><page_5><loc_17><loc_8><loc_87><loc_11></location>The flexibility in the choice of solution for A will help describe galaxy rotation curves, as explained in the next section.</text>
<section_header_level_1><location><page_6><loc_12><loc_84><loc_48><loc_86></location>3 Galaxy rotation curves</section_header_level_1>
<text><location><page_6><loc_53><loc_73><loc_53><loc_75></location>/negationslash</text>
<text><location><page_6><loc_62><loc_73><loc_62><loc_75></location>/negationslash</text>
<text><location><page_6><loc_12><loc_72><loc_87><loc_83></location>In what follows, we consider each star in the spiral arms of the galaxy as a test particle, moving on a circular trajectory under the influence of the potential φ ( r ), generated by the centre of the galaxy, which is assumed to be static and spherically symmetric. Our approach is to start from a stellar velocity distribution and derive the corresponding expression for the auxiliary field A , in the above situation with n = 0 and A = 0 , where multiple solutions for A are allowed. We note the following few points:</text>
<text><location><page_6><loc_12><loc_59><loc_87><loc_72></location>(i) We consider vacuum solutions of covariant HL gravity, for which one cannot in principle describe the region in the centre of the galaxy. Nevertheless, as we will see in case of the following velocity profiles, a consistent solution for A can be found for 0 ≤ r ≤ R , where R is a typical radial length describing the galaxy. To be consistent with the usual studies of galaxy rotations, we will choose here R = r 200 , corresponding to the virial radius of the galaxy. The latter is defined as the distance from the centre of the galaxy, where the density ρ 200 is 200 times the critical density ρ c of the Universe</text>
<formula><location><page_6><loc_39><loc_54><loc_87><loc_58></location>ρ 200 = 200 ρ c = 200 3 H 2 8 πG , (18)</formula>
<text><location><page_6><loc_12><loc_52><loc_39><loc_53></location>where H is the Hubble constant;</text>
<text><location><page_6><loc_12><loc_43><loc_87><loc_51></location>(ii) For r > r 200 , we assume the solution A = 0, which leads to the usual Schwarzschild solution outside the galaxy. Indeed, far from the galaxy, one expects to see the Newtonian potential. This solution implies that the Hamiltonian constraint is expressed in terms of an integral over the finite range [0 , r 200 ] of radial coordinate r , which we can impose to vanish by fixing the constants of integration;</text>
<text><location><page_6><loc_12><loc_32><loc_87><loc_42></location>(iii) For r ≤ r 200 , since the shift A → A +constant does not have a physical implication, one can expect to find a consistent solution for A which is continuous at r = r 200 , i.e. A ( r 200 ) = 0. On the other hand, one cannot expect the auxiliary field to be differentiable at r = r 200 , but this is not necessary for the consistency of covariant HL gravity: the second derivative of A does not appear in any equation of motion or constraint, so the first derivative can be discontinuous.</text>
<section_header_level_1><location><page_6><loc_12><loc_27><loc_44><loc_29></location>3.1 Gravitational potential</section_header_level_1>
<text><location><page_6><loc_12><loc_23><loc_87><loc_26></location>On a circular trajectory, the relation between the speed v of a star and the Newtonian potential φ is</text>
<formula><location><page_6><loc_45><loc_19><loc_87><loc_23></location>v 2 r = dφ dr , (19)</formula>
<text><location><page_6><loc_12><loc_17><loc_77><loc_19></location>where φ is obtained from the g 00 component of the low energy effective theory</text>
<formula><location><page_6><loc_39><loc_13><loc_87><loc_16></location>-c 2 + n 2 = -c 2 (1 + 2 φ ) , (20)</formula>
<text><location><page_6><loc_12><loc_11><loc_29><loc_12></location>and is thus given by</text>
<formula><location><page_6><loc_43><loc_7><loc_87><loc_11></location>φ ( r ) = -n 2 ( r ) 2 c 2 . (21)</formula>
<text><location><page_7><loc_12><loc_83><loc_87><loc_86></location>We note here that, in the situation where the second auxiliary field ν does not vanish, the Newtonian potential is instead given by</text>
<formula><location><page_7><loc_38><loc_78><loc_87><loc_82></location>φ ( r ) = -( n ( r ) -∇ r ν ( r )) 2 2 c 2 , (22)</formula>
<text><location><page_7><loc_12><loc_69><loc_87><loc_77></location>and φ is independent of the U Σ (1) gauge choice for ν and A . Also, φ is dimensionless, so that the speed v we use in this article is also dimensionless, which corresponds to a usual definition of speed, for isotropic space-time. The corresponding 'Lifshitz velocity' is cv , with mass dimension z -1 = 2.</text>
<text><location><page_7><loc_12><loc_68><loc_87><loc_70></location>From the relations (16) and (19), the speed v and the auxiliary field A are then related by</text>
<formula><location><page_7><loc_28><loc_63><loc_87><loc_68></location>rA ' ( r ) 4 c 2 = d dr ( r ∫ r dρ v 2 ( ρ ) ρ ) = v 2 ( r ) + ∫ r dρ v 2 ( ρ ) ρ , (23)</formula>
<text><location><page_7><loc_12><loc_60><loc_87><loc_63></location>and we have to check that the Hamiltonian constraint (17) is satisfied, in order to show the consistency of the approach.</text>
<text><location><page_7><loc_12><loc_58><loc_62><loc_59></location>We also note from eqs.(16,19) that the solution A = 0 gives,</text>
<formula><location><page_7><loc_43><loc_53><loc_87><loc_57></location>v ( r ) = √ C 2 c 2 r , (24)</formula>
<text><location><page_7><loc_12><loc_49><loc_87><loc_52></location>which is expected from Newtonian mechanics outside the galaxy, for r > r 200 . If one fixes the constant of integration C to the value</text>
<formula><location><page_7><loc_42><loc_46><loc_87><loc_48></location>C = 2 c 2 r 200 v 2 200 , (25)</formula>
<text><location><page_7><loc_12><loc_40><loc_87><loc_45></location>where v 200 = v ( r 200 ), one obtains a continuous speed at r = r 200 . Finally a different choice of the auxiliary field ν would modify the field A in such a way that physical results would not be changed.</text>
<section_header_level_1><location><page_7><loc_12><loc_36><loc_56><loc_37></location>3.2 Navarro, Frenk and White profile</section_header_level_1>
<text><location><page_7><loc_12><loc_24><loc_87><loc_34></location>This profile originates from Cold Dark Matter halo models and is characterised by the 'cusp' shape of the density distribution. Its validity has been tested on variety of observational results, including Low Surface Brightness galaxies in [27] and spherical galaxies and clusters in [28]. For our purposes, we use the circular velocity profile derived from the mass density profile in [29], based on the assumption of a Dark Matter halo and ignoring the contribution of baryons,</text>
<formula><location><page_7><loc_31><loc_19><loc_87><loc_23></location>v ( r ) = v 0 √ r 200 r ln ( 1 + ar r 200 ) -a 1 + ar/r 200 , (26)</formula>
<text><location><page_7><loc_12><loc_17><loc_48><loc_18></location>where a is the concentration parameter and</text>
<formula><location><page_7><loc_37><loc_10><loc_87><loc_16></location>v 0 = v 200 √ ln (1 + a ) -a/ (1 + a ) (27)</formula>
<text><location><page_7><loc_12><loc_8><loc_87><loc_11></location>where v 200 , the virial velocity, is the circular velocity at the virial radius. Some typical values are [30]</text>
<table>
<location><page_8><loc_26><loc_79><loc_72><loc_86></location>
</table>
<text><location><page_8><loc_42><loc_78><loc_43><loc_81></location>±</text>
<text><location><page_8><loc_54><loc_78><loc_55><loc_81></location>±</text>
<text><location><page_8><loc_67><loc_78><loc_69><loc_81></location>±</text>
<text><location><page_8><loc_12><loc_76><loc_43><loc_77></location>With the profile (26), eq.(23) leads to</text>
<formula><location><page_8><loc_38><loc_71><loc_87><loc_75></location>rA ' ( r ) 4 c 2 v 2 0 = φ 1 -a 1 + ar/r 200 , (28)</formula>
<text><location><page_8><loc_12><loc_68><loc_55><loc_70></location>where φ 1 is a constant of integration, and therefore</text>
<formula><location><page_8><loc_28><loc_63><loc_87><loc_67></location>A ( r ) 4 c 2 v 2 0 = ( φ 1 -a ) ln ( r r 200 ) + a ln ( 1 + a r r 200 ) + A 1 , (29)</formula>
<text><location><page_8><loc_12><loc_58><loc_87><loc_62></location>where A 1 is a constant of integration. Since we consider the solution A ( r ) = 0 for r > r 200 and impose A ( r ) to be continuous at r = r 200 , then A 1 = -a ln(1 + a ) and</text>
<formula><location><page_8><loc_21><loc_52><loc_87><loc_58></location>A ( r ) 4 c 2 v 2 0 = ( φ 1 -a ) ln ( r r 200 ) + a ln ( 1 + ar/r 200 1 + a ) if 0 ≤ r ≤ r 200 (30) A ( r ) = 0 if r > r 200 .</formula>
<text><location><page_8><loc_12><loc_49><loc_41><loc_50></location>Finally, the Hamiltonian constraint</text>
<formula><location><page_8><loc_26><loc_43><loc_87><loc_47></location>0 = ∫ ∞ 0 dr rA ' ( r ) = 4 c 2 v 2 0 ∫ r 200 0 dr ( φ 1 -a 1 + ar/r 200 ) , (31)</formula>
<text><location><page_8><loc_12><loc_41><loc_22><loc_42></location>is satisfied if</text>
<formula><location><page_8><loc_43><loc_39><loc_87><loc_41></location>φ 1 = ln(1 + a ) . (32)</formula>
<text><location><page_8><loc_12><loc_35><loc_87><loc_38></location>We note that the solution (30) is consistent for any relevant values of the parameters a, v 0 , r 200 and therefore allows the description of a whole range of galaxies.</text>
<section_header_level_1><location><page_8><loc_12><loc_30><loc_46><loc_32></location>3.3 Pseudoisothermal profile</section_header_level_1>
<text><location><page_8><loc_12><loc_20><loc_87><loc_29></location>The pseudoisothermal mass density profile assumes an existence of a 'cored' dark matter halo component in the galaxy [31] , with mass density of an approximately constant value in the central region of the galaxy, for r ≤ R c . Empirically motivated, the model is often contrasted with the aforementioned NFW 'cuspy' profile, while evidence shows that it provides better fit of the galaxy rotation velocities [27]. The velocity profile is</text>
<formula><location><page_8><loc_35><loc_15><loc_87><loc_19></location>v ( r ) = v 0 √ 1 -R C r arctan ( r R C ) , (33)</formula>
<text><location><page_8><loc_12><loc_12><loc_29><loc_14></location>where v 0 is given by</text>
<text><location><page_8><loc_12><loc_8><loc_76><loc_9></location>and ρ 0 , the central density, is the density within R C . Typical values are [30]</text>
<formula><location><page_8><loc_42><loc_9><loc_87><loc_12></location>v 0 = √ 4 πGρ 0 R 2 C , (34)</formula>
<table>
<location><page_9><loc_23><loc_79><loc_75><loc_86></location>
</table>
<text><location><page_9><loc_39><loc_78><loc_41><loc_81></location>±</text>
<text><location><page_9><loc_51><loc_78><loc_52><loc_81></location>±</text>
<text><location><page_9><loc_67><loc_78><loc_69><loc_81></location>±</text>
<text><location><page_9><loc_12><loc_74><loc_87><loc_77></location>We note that in the case of NCG 2403, both the NFW and the present profile can be shown to produce fits of comparably good quality.</text>
<text><location><page_9><loc_12><loc_72><loc_43><loc_74></location>With the profile (33), eq.(23) leads to</text>
<formula><location><page_9><loc_36><loc_67><loc_87><loc_71></location>rA ' ( r ) 4 c 2 v 2 0 = φ 2 + 1 2 ln ( 1 + r 2 R 2 c ) , (35)</formula>
<text><location><page_9><loc_12><loc_63><loc_87><loc_66></location>where φ 2 is a constant of integration. The same steps as those describe for the previous profile lead to the solution</text>
<formula><location><page_9><loc_20><loc_56><loc_87><loc_61></location>A ( r ) 4 c 2 v 2 0 = φ 2 ln ( r r 200 ) + 1 2 ∫ r/R c 0 dt t ln(1 + t 2 ) + A 2 if 0 ≤ r ≤ r 200 (36) A ( r ) = 0 if r > r 200 ,</formula>
<text><location><page_9><loc_12><loc_53><loc_44><loc_54></location>where the constants of integration are</text>
<formula><location><page_9><loc_36><loc_47><loc_87><loc_51></location>A 2 = -1 2 ∫ r 200 /R c 0 dt t ln(1 + t 2 ) , (37)</formula>
<text><location><page_9><loc_12><loc_45><loc_47><loc_47></location>for the continuity of A ( r ) at r = r 200 , and</text>
<formula><location><page_9><loc_29><loc_40><loc_87><loc_44></location>φ 2 = -1 2 ln ( 1 + r 2 200 R 2 c ) +1 -R c r 200 arctan ( r 200 R c ) , (38)</formula>
<text><location><page_9><loc_12><loc_37><loc_50><loc_39></location>for the Hamiltonian constraint to be satisfied.</text>
<section_header_level_1><location><page_9><loc_12><loc_33><loc_75><loc_34></location>3.4 Vacuum energy contribution of the auxiliary field</section_header_level_1>
<text><location><page_9><loc_12><loc_23><loc_87><loc_32></location>Since the present model does not require Dark Matter to explain galaxy rotation curves, the resulting mass content of the Universe might therefore be reduced. But the static configuration of A actually results in an effective vacuum energy. This can be seen from the original action (2), after integrating by parts the term -√ gAR (3) , which involve secondorder space derivatives of the metric. This leads to the following contribution to the</text>
<text><location><page_9><loc_12><loc_21><loc_22><loc_22></location>Hamiltonian</text>
<formula><location><page_9><loc_29><loc_17><loc_87><loc_21></location>M ∝ 1 c ∫ d 3 x √ gg ij ∇ i ∇ j A = 4 π c ∫ r 200 0 r 2 dr ∂ 2 r A , (39)</formula>
<text><location><page_9><loc_12><loc_14><loc_87><loc_17></location>where the factor 1 /c arises from the absence of integration over time, and restores the correct dimension.</text>
<text><location><page_9><loc_12><loc_12><loc_74><loc_14></location>In the case of the Navarro, Frenk and White profile, the energy (39) reads</text>
<formula><location><page_9><loc_33><loc_7><loc_87><loc_11></location>M 1 = 16 πcv 2 0 r 200 [ ln(1 + a ) -a 1 + a ] , (40)</formula>
<text><location><page_10><loc_12><loc_83><loc_64><loc_86></location>which is always positive, for any value of the parameter a ≥ 0.</text>
<text><location><page_10><loc_12><loc_83><loc_57><loc_84></location>For the Pseudoisothermal case, we obtain from eq.(39)</text>
<formula><location><page_10><loc_31><loc_77><loc_87><loc_81></location>M 2 = 16 πcv 2 0 r 200 [ 1 -R c r 200 arctan ( r 200 R c )] , (41)</formula>
<text><location><page_10><loc_12><loc_73><loc_65><loc_76></location>which is also always positive, for any value of the ratio r 200 /R c . The auxilliary field A</text>
<text><location><page_10><loc_12><loc_71><loc_87><loc_74></location>has therefore a cosmological role, by contributing to the vacuum energy of the Universe, consistently with its observed acceleration.</text>
<section_header_level_1><location><page_10><loc_12><loc_66><loc_32><loc_68></location>4 Conclusion</section_header_level_1>
<text><location><page_10><loc_12><loc_50><loc_87><loc_64></location>The multiplicity of solutions in covariant HL gravity has been used to describe galaxy rotation curves, without the need for Dark Matter. This multiplicity is a consequence of the breaking of 4-dimensional diffeomorphism, in the projectable case, where the lapse function is imposed to depend on time only. We considered two specific velocity profiles and showed in each case that a corresponding consistent solution can be derived for the auxiliary field A . Clearly, similar results can be expected with other profiles. A more detailed analysis would consist in taking into account the finite star density at the centre of the galaxy, as well as its rotation.</text>
<text><location><page_10><loc_12><loc_39><loc_87><loc_49></location>One should note that the solutions found, for the auxilliary field A , depend on parameters wich cannot appear in the Lagrangian in a universal way. These parameters arise as constant of integrations, as a result of the integral form of the Hamiltonian constraint, and they have to be fitted to each galaxy which is studied. This is a bit similar to the Schwazschild metric in General Relativity, involving a mass which is a constant of integration and depends on the star/black hole under consideration.</text>
<text><location><page_10><loc_12><loc_28><loc_87><loc_38></location>It is interesting to note that the solutions (30) and (36) actually do not only describe galaxy rotations curves, but also the usual Schwarzschild solutions for the Solar System, by taking the limit r 200 → 0 such that A = 0 for all r . In this sense these solutions provide an effective unified description of gravitational effects, from the solar system scale to the galaxy scale, where local Lorentz violation appears gradually, as the observational length increases from the the size of the solar system to the size of the galaxy.</text>
<text><location><page_10><loc_12><loc_13><loc_87><loc_27></location>We finally note that, although HL gravity was initially developed to modify ultraviolet behaviour of gravity, our present work uses another consequence of local Lorentz symmetry violation, which is the existence of a non-unique solution to the Hamiltonian constraint. Since it has been shown that canonical Hoˇrava - Lifshitz gravity provides a description of Cosmology which is consistent with constraints from observational data [32], it would be interesting to see if the non-unique time-dependent solutions of the covariant version of the theory could also allow for an effective description of Cosmology, including an alternative to Dark Energy.</text>
<section_header_level_1><location><page_11><loc_12><loc_84><loc_27><loc_86></location>References</section_header_level_1>
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<list_item><location><page_11><loc_12><loc_34><loc_87><loc_37></location>[10] K. Lin and A. Wang, 'Static post-Newtonian limits in non-projectable Hoˇrava-Lifshitz gravity with an extra U(1) symmetry,' arXiv:1212.6794 [hep-th].</list_item>
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</document>
[
{
"title": "Abstract",
"content": "Using the multiplicity of solutions for the projectable case of the covariant extension of Hoˇrava-Lifshitz Gravity, we show that an appropriate choice for the auxiliary field allows for an effective description of galaxy rotation curves. This description is based on static and spherically symmetric solutions of covariant Hoˇrava-Lifshitz Gravity and does not require Dark Matter.",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "/negationslash Although General Relativity (GR) has been accurately checked in the solar system, the small distance behaviour of gravity is still not understood, and several alternatives to GR have been proposed. Amongst these modified theories is Hoˇrava-Lifshitz (HL) gravity [1], which is based on the Lifshitz approach and consists of introducing an anisotropy between space and time: when space is rescaled as x → bx , time is rescaled as t → b z t , where z is an integer. In Lifshitz models, Lorentz invariance must be recovered for z = 1, and the situation z = 1 leads to new renormalizable interactions: the convergence of loop integrals is improved by inclusion of higher orders in space derivatives, without introducing ghosts, since the number of time derivatives remains the minimum. A review of Lifshitz-type quantum field theories in flat space time can be found in [2], and the HL power-counting renormalizable theory of gravity is reviewed in [3], [4] and [5]. A fundamental problem of the original model of HL gravity is the existence of an additional scalar degree of freedom for the metric, which can be understood as a Goldstone mode arising from breaking of 4-dimensional diffeomorphism [6], [7]. A solution to this problem has been proposed in [8], where an auxiliary field A is introduced, such that its 'equation of motion' leads to an additional constraint, eliminating the unwanted scalar degree of freedom of the metric. The resulting theory is invariant under a new Abelian gauge symmetry U Σ (1) which involves the metric components, the auxiliary field A and an additional auxiliary field ν . This gauge symmetry, together with the 3-dimensional diffeomorphism of HL gravity, can be shown to be equivalent to a 4-dimensional diffeomorphism at the lowest order in a post-Newtonian expansion, showing the equivalence with GR at long distances. For this reason, this modified version of HL gravity is called covariant HL gravity. We note however, that the long-distance limit is not obviously recovered: it has been shown in [9] that the equivalence principle is not automatically retrieved in the infrared, and that the meaning of the auxiliary field A and its coupling to matter are still open questions. Because of the anisotropy between space and time, HL gravity is naturally described in terms of the Arnowitt-Deser-Misner (ADM) decomposition of the metric, which expresses a space-time foliation. An important consequences of space-time anisotropy is the possibility of imposing the lapse function N to depend on time only. This situation is called the projectable case and leads to an interesting feature: the solutions of the equations of motion are not unique. Indeed, the Hamiltonian constraint, obtained by variation of the action with respect to N , leads to an integral equation, which does not have a unique solution, as will be seen in the present article. This multiplicity of solutions is independent of the above mentioned new gauge symmetry U Σ (1), and the different solutions for A in the projectable case belong to different gauge orbits. On the other hand, in the nonprojectable case where N depends on both space and time, the Hamiltonian constraint leads to a differential equation, which has a unique solution, after fixing the constants of integration. The non-projectable case has been studied in [10] for static spherically symmetric solutions of covariant HL gravity. In this article we use the freedom of choice of the auxiliary field A in the projectable case to study the possibility of fitting galaxy rotations curves without the need for Dark Matter. One can argue that the ambiguity in the choice of A leaves its physical interpretation unclear. This may be understood as one of the incomplete aspects of covariant HL gravity, and one could conjecture the existence of an additional symmetry which would restrict the set of solutions for the auxiliary field A , but this still remains to be studied. We note that the multiplicity of solutions, arising from a Hamiltonian constraint expressed in terms of an integral over space, has been discussed in [11] as an alternative to Dark Matter, in the context of the original HL gravity. Fits to galaxy rotation curves have been studied in the framework of the original HL gravity in [12] and [13], where a detailed comparison with experimental data is presented. Also, an interesting analogy between HL gravity for z = 0 and Modified Newtonian Dynamics theories has been described in [14]. In section 2 we review the static and spherically symmetric solutions of covariant HL gravity, and in section 3 we show how galaxy rotation curves can be described by the covariant extension of HL gravity. Although the curves we use were derived from Dark Matter models, we use them as experimental data fits. It is interesting to note that we obtain exact solutions for the auxiliary field A . Another exact solution in the Lifshitz context has been derived in [15], where the exact effective potential for a Liouville scalar theory, renormalizable in 3+1 dimensions with anisotropic scaling z = 3, proves to be an exponential, as in the usual 1+1 dimensional relativistic Liouville theory. Finally, we conclude by discussing the possiblity to describe the solar system with our solution.",
"pages": [
1,
2
]
},
{
"title": "2 Static and spherically symmetric solutions of covariant HL gravity",
"content": "The static and spherically symmetric solutions of covariant HL gravity were derived in [25] and [26], and we review here the aspects relevant to our present study.",
"pages": [
3
]
},
{
"title": "2.1 Action",
"content": "The ADM metric we consider is where N and N i are the lapse and shift functions respectively, and g ij is the threedimensional space metric. We consider the anisotropic scaling with z = 3, for which the different operators, allowed for the model to be power-counting renormalizable, have the mass dimension of 6, at most. We note that in this case the mass dimension of the speed of light is [ c ] = z -1 = 2, and we will keep c in the different expression for the sake of clarity. The covariant Hoˇrava-Lifshitz action, in the absence of cosmological constant, is where g is the determinant of the three-dimensional metric g ij , with curvature scalar R (3) . A is an auxiliary field (without kinetic term) of mass dimension z + 1 = 4 and ν is an auxiliary field of mass dimension z -2 = 1. In the above expression, the extrinsic curvature is where a dot denotes a time derivative, and /negationslash where where R (3) ij is the Ricci tensor corresponding to the three-dimensional space metric g ij . Note that a more general kinetic term K ij K ij -λK 2 can be considered, with λ a generic parameter. The latter should be equal to 1 if one wishes to recover GR in the infrared, but in the framework of HL gravity, no symmetry imposes this parameter to be equal to 1. Constraints on λ resulting from observations of Type Ia Supernovae, Baryon Acoustic Oscillations, CMB and the requirement of Big Bang Nucleosynthesis all point towards a value near the GR parameter λ = 1 [16], while it has also been shown that covariant HL gravity with λ = 1 leads to inconsistencies, if one compares the predictions of the model with solar system tests [17]. We therefore focus here on the situation where λ = 1 only. Independently of these tests, the mathematical consistency of covariant HL gravity for λ = 1 has been studied in [18], and the corresponding Hamiltonian structure in /negationslash [19]. We stress again here that the purpose of this covariant extension of HL gravity is to eliminate the unwanted additional scalar graviton from the spectrum. As shown in [8], no singularity arises in the limit λ → 1, and the strong coupling of the scalar graviton does not exist anymore, as it is the case for the original HL gravity. Besides [6] and [7], one can find detailed analysis of the strong coupling problem in [20], [21], [22], and also related discussions in [23] on a extension of the original HL gravity [24]. The potential term V contains up to six spatial derivatives of the metric g ij : where the mass dimensions of the different couplings are We consider the most general, static, spherically symmetric metric, of the form: where n ( r ) = N r ( r ) is the radial component of shift function, and N r = n ( r ) /f ( r ) since g rr = 1 /f ( r ). Note that n has mass dimension z -1 = 2.",
"pages": [
3,
4
]
},
{
"title": "2.2 Constraints and equations of motion",
"content": "The variation of the action (2) with respect to the different degrees of freedom leads to the following constraints, or equations of motion, where a prime denotes a derivative with respect to the radial coordinate r . which imposes the function f ( r ) in the metric (8) to be where B is a constant of integration. In what follows we will assume the Gauge fixing of ν = 0, then the above constraint gives Θ ij K ij = 0, which is satisfied for spherically symmetric solutions. which is an space-integral equation since N depends on time only. such that either f is a constant ( B = 0) or n = 0. where the time has been rescaled such that N = 1.",
"pages": [
4,
5
]
},
{
"title": "2.3 Solutions",
"content": "The different solutions of the previous set of equations were derived in [25], [26], and here we shortly review the different cases. /negationslash /negationslash /negationslash with solutions where C is a constant of integration and the auxiliary field A satisfies The flexibility in the choice of solution for A will help describe galaxy rotation curves, as explained in the next section.",
"pages": [
5
]
},
{
"title": "3 Galaxy rotation curves",
"content": "/negationslash /negationslash In what follows, we consider each star in the spiral arms of the galaxy as a test particle, moving on a circular trajectory under the influence of the potential φ ( r ), generated by the centre of the galaxy, which is assumed to be static and spherically symmetric. Our approach is to start from a stellar velocity distribution and derive the corresponding expression for the auxiliary field A , in the above situation with n = 0 and A = 0 , where multiple solutions for A are allowed. We note the following few points: (i) We consider vacuum solutions of covariant HL gravity, for which one cannot in principle describe the region in the centre of the galaxy. Nevertheless, as we will see in case of the following velocity profiles, a consistent solution for A can be found for 0 ≤ r ≤ R , where R is a typical radial length describing the galaxy. To be consistent with the usual studies of galaxy rotations, we will choose here R = r 200 , corresponding to the virial radius of the galaxy. The latter is defined as the distance from the centre of the galaxy, where the density ρ 200 is 200 times the critical density ρ c of the Universe where H is the Hubble constant; (ii) For r > r 200 , we assume the solution A = 0, which leads to the usual Schwarzschild solution outside the galaxy. Indeed, far from the galaxy, one expects to see the Newtonian potential. This solution implies that the Hamiltonian constraint is expressed in terms of an integral over the finite range [0 , r 200 ] of radial coordinate r , which we can impose to vanish by fixing the constants of integration; (iii) For r ≤ r 200 , since the shift A → A +constant does not have a physical implication, one can expect to find a consistent solution for A which is continuous at r = r 200 , i.e. A ( r 200 ) = 0. On the other hand, one cannot expect the auxiliary field to be differentiable at r = r 200 , but this is not necessary for the consistency of covariant HL gravity: the second derivative of A does not appear in any equation of motion or constraint, so the first derivative can be discontinuous.",
"pages": [
6
]
},
{
"title": "3.1 Gravitational potential",
"content": "On a circular trajectory, the relation between the speed v of a star and the Newtonian potential φ is where φ is obtained from the g 00 component of the low energy effective theory and is thus given by We note here that, in the situation where the second auxiliary field ν does not vanish, the Newtonian potential is instead given by and φ is independent of the U Σ (1) gauge choice for ν and A . Also, φ is dimensionless, so that the speed v we use in this article is also dimensionless, which corresponds to a usual definition of speed, for isotropic space-time. The corresponding 'Lifshitz velocity' is cv , with mass dimension z -1 = 2. From the relations (16) and (19), the speed v and the auxiliary field A are then related by and we have to check that the Hamiltonian constraint (17) is satisfied, in order to show the consistency of the approach. We also note from eqs.(16,19) that the solution A = 0 gives, which is expected from Newtonian mechanics outside the galaxy, for r > r 200 . If one fixes the constant of integration C to the value where v 200 = v ( r 200 ), one obtains a continuous speed at r = r 200 . Finally a different choice of the auxiliary field ν would modify the field A in such a way that physical results would not be changed.",
"pages": [
6,
7
]
},
{
"title": "3.2 Navarro, Frenk and White profile",
"content": "This profile originates from Cold Dark Matter halo models and is characterised by the 'cusp' shape of the density distribution. Its validity has been tested on variety of observational results, including Low Surface Brightness galaxies in [27] and spherical galaxies and clusters in [28]. For our purposes, we use the circular velocity profile derived from the mass density profile in [29], based on the assumption of a Dark Matter halo and ignoring the contribution of baryons, where a is the concentration parameter and where v 200 , the virial velocity, is the circular velocity at the virial radius. Some typical values are [30] ± ± ± With the profile (26), eq.(23) leads to where φ 1 is a constant of integration, and therefore where A 1 is a constant of integration. Since we consider the solution A ( r ) = 0 for r > r 200 and impose A ( r ) to be continuous at r = r 200 , then A 1 = -a ln(1 + a ) and Finally, the Hamiltonian constraint is satisfied if We note that the solution (30) is consistent for any relevant values of the parameters a, v 0 , r 200 and therefore allows the description of a whole range of galaxies.",
"pages": [
7,
8
]
},
{
"title": "3.3 Pseudoisothermal profile",
"content": "The pseudoisothermal mass density profile assumes an existence of a 'cored' dark matter halo component in the galaxy [31] , with mass density of an approximately constant value in the central region of the galaxy, for r ≤ R c . Empirically motivated, the model is often contrasted with the aforementioned NFW 'cuspy' profile, while evidence shows that it provides better fit of the galaxy rotation velocities [27]. The velocity profile is where v 0 is given by and ρ 0 , the central density, is the density within R C . Typical values are [30] ± ± ± We note that in the case of NCG 2403, both the NFW and the present profile can be shown to produce fits of comparably good quality. With the profile (33), eq.(23) leads to where φ 2 is a constant of integration. The same steps as those describe for the previous profile lead to the solution where the constants of integration are for the continuity of A ( r ) at r = r 200 , and for the Hamiltonian constraint to be satisfied.",
"pages": [
8,
9
]
},
{
"title": "3.4 Vacuum energy contribution of the auxiliary field",
"content": "Since the present model does not require Dark Matter to explain galaxy rotation curves, the resulting mass content of the Universe might therefore be reduced. But the static configuration of A actually results in an effective vacuum energy. This can be seen from the original action (2), after integrating by parts the term -√ gAR (3) , which involve secondorder space derivatives of the metric. This leads to the following contribution to the Hamiltonian where the factor 1 /c arises from the absence of integration over time, and restores the correct dimension. In the case of the Navarro, Frenk and White profile, the energy (39) reads which is always positive, for any value of the parameter a ≥ 0. For the Pseudoisothermal case, we obtain from eq.(39) which is also always positive, for any value of the ratio r 200 /R c . The auxilliary field A has therefore a cosmological role, by contributing to the vacuum energy of the Universe, consistently with its observed acceleration.",
"pages": [
9,
10
]
},
{
"title": "4 Conclusion",
"content": "The multiplicity of solutions in covariant HL gravity has been used to describe galaxy rotation curves, without the need for Dark Matter. This multiplicity is a consequence of the breaking of 4-dimensional diffeomorphism, in the projectable case, where the lapse function is imposed to depend on time only. We considered two specific velocity profiles and showed in each case that a corresponding consistent solution can be derived for the auxiliary field A . Clearly, similar results can be expected with other profiles. A more detailed analysis would consist in taking into account the finite star density at the centre of the galaxy, as well as its rotation. One should note that the solutions found, for the auxilliary field A , depend on parameters wich cannot appear in the Lagrangian in a universal way. These parameters arise as constant of integrations, as a result of the integral form of the Hamiltonian constraint, and they have to be fitted to each galaxy which is studied. This is a bit similar to the Schwazschild metric in General Relativity, involving a mass which is a constant of integration and depends on the star/black hole under consideration. It is interesting to note that the solutions (30) and (36) actually do not only describe galaxy rotations curves, but also the usual Schwarzschild solutions for the Solar System, by taking the limit r 200 → 0 such that A = 0 for all r . In this sense these solutions provide an effective unified description of gravitational effects, from the solar system scale to the galaxy scale, where local Lorentz violation appears gradually, as the observational length increases from the the size of the solar system to the size of the galaxy. We finally note that, although HL gravity was initially developed to modify ultraviolet behaviour of gravity, our present work uses another consequence of local Lorentz symmetry violation, which is the existence of a non-unique solution to the Hamiltonian constraint. Since it has been shown that canonical Hoˇrava - Lifshitz gravity provides a description of Cosmology which is consistent with constraints from observational data [32], it would be interesting to see if the non-unique time-dependent solutions of the covariant version of the theory could also allow for an effective description of Cosmology, including an alternative to Dark Energy.",
"pages": [
10
]
}
]
2013GeCoA.101..285K
https://arxiv.org/pdf/1210.1674.pdf
<document>
<section_header_level_1><location><page_1><loc_11><loc_82><loc_89><loc_86></location>Ab initio prediction of equilibrium boron isotope fractionation between minerals and aqueous fluids at high P and T</section_header_level_1>
<text><location><page_1><loc_31><loc_79><loc_69><loc_80></location>Piotr M. Kowalski 1 , 2 ∗ , Bernd Wunder 1 and Sandro Jahn 1</text>
<text><location><page_1><loc_25><loc_77><loc_75><loc_78></location>1 GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany</text>
<text><location><page_1><loc_16><loc_75><loc_84><loc_76></location>2 Forschungszentrum Julich, Institute of Energy and Climate Research (IEK-6), Wilhelm-Johnen-Strasse, 52425 Julich, Germany</text>
<section_header_level_1><location><page_1><loc_11><loc_67><loc_17><loc_68></location>Abstract</section_header_level_1>
<text><location><page_1><loc_11><loc_42><loc_89><loc_66></location>Over the last decade experimental studies have shown a large B isotope fractionation between materials carrying boron incorporated in trigonally and tetrahedrally coordinated sites, but the mechanisms responsible for producing the observed isotopic signatures are poorly known. In order to understand the boron isotope fractionation processes and to obtain a better interpretation of the experimental data and isotopic signatures observed in natural samples, we use first principles calculations based on density functional theory in conjunction with ab initio molecular dynamics and a new pseudofrequency analysis method to investigate the B isotope fractionation between B-bearing minerals (such as tourmaline and micas) and aqueous fluids containing H3BO3 and H4BO -4 species. We confirm the experimental finding that the isotope fractionation is mainly driven by the coordination of the fractionating boron atoms and have found in addition that the strength of the produced isotopic signature is strongly correlated with the B-O bond length. We also demonstrate the ability of our computational scheme to predict the isotopic signatures of fluids at extreme pressures by showing the consistency of computed pressure-dependent β factors with the measured pressure shifts of the B-O vibrational frequencies of H3BO3 and H4BO -4 in aqueous fluid. The comparison of the predicted with measured fractionation factors between boromuscovite and neutral fluid confirms the existence of the admixture of tetrahedral boron species in neutral fluid at high P and T found experimentally, which also explains the inconsistency between the various measurements on the tourmaline-mica system reported in the literature. Our investigation shows that the calculated equilibrium isotope fractionation factors have an accuracy comparable to the experiments and give unique and valuable insight into the processes governing the isotope fractionation mechanisms on the atomic scale.</text>
<text><location><page_1><loc_11><loc_40><loc_71><loc_41></location>Keywords: stable isotope fractionation, boron, fluid speciation, DFT, molecular dynamics</text>
<section_header_level_1><location><page_1><loc_11><loc_36><loc_22><loc_37></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_11><loc_18><loc_48><loc_33></location>The use of isotopes as geochemical tracers depends upon the existence of reliable P -T -dependent equilibrium isotope fractionation data between solids, fluids and melts. The common method used for determination of such data is to conduct experiments, where the phases of interest are equilibrated at a range of di ff erent conditions and individually measured for their equilibrium isotopic compositions. Recently, with the development of computational methods, software and increase in hardware performance it is also possible to simulate and compute the isotope fractionation with ab initio</text>
<text><location><page_1><loc_52><loc_13><loc_89><loc_37></location>methods. These computational considerations complement the experimental e ff ort and provide information on the mechanisms governing the equilibrium isotope fractionation processes on the atomic scale. By establishing an e ffi cient computational approach for materials at high P and T and testing its reliability by computing Li isotope fractionation between minerals and aqueous fluids, Kowalski & Jahn (2011) have shown that ab initio methods can provide a reliable estimation of equilibrium isotope fractionation factors at an accuracy level comparable to experiments. Motivated by the encouraging results for the fractionation of lithium isotopes we applied our new method to study boron isotope fractionation. The main goals of the present study are to make theoretical predictions and obtain a better understanding of the B isotope fractionation process between tourmaline, B-bearing mica and flu-</text>
<text><location><page_2><loc_11><loc_75><loc_48><loc_86></location>ids at various pressures and temperatures, to compare these data to results from recent in situ experimental studies (Wunder et al., 2005; Meyer et al., 2008) and measurements of isotopic signatures in natural samples (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002) and to investigate the underlying mechanisms driving the boron isotope fractionation processes between the considered materials.</text>
<text><location><page_2><loc_11><loc_54><loc_48><loc_75></location>With two stable isotopes 10 B and 11 B, of relatively large mass di ff erence of about 10 %, boron isotopes strongly fractionate during geological processes, thus leading to natural δ 11 B-variations ranging from -30 to + 60 /permil (Barth, 1993). Therefore, B isotopes are ideal for the distinction of di ff erent geological environments and for quantifying mass transfer processes, e.g. in the range of subduction zones. First-order criteria driving isotope fractionation in Earth materials are di ff erences in coordination and in the bonding environments of coexisting phases. The lighter isotope usually preferentially occupies the higher coordinated site, which is generally accompanied with a longer cation-anion bond length and weaker bond strength (Schauble et al., 2009; Wunder et al., 2011).</text>
<text><location><page_2><loc_11><loc_27><loc_48><loc_54></location>Tourmaline has an extensive chemical variability and is the most widespread borosilicate in various rocks over a large range of bulk compositions. It has a large P -T -stability which ranges from surface conditions to high pressures and temperatures of at least 7.0 GPa and about 1000 o C as determined experimentally for dravitic tourmaline (Krosse, 1995). Most tourmaline minerals contain 3 boron atoms per formula unit (pfu) with B in trigonal planar coordination (B [3] ). High pressure combined with an Al-rich environment can lead to the formation of olenitic tourmaline with significant amounts of excess B substituting for Si at the tetrahedral site (B [4] ). The highest amounts of up to 1.2 B [4] pfu have been found in olenitic tourmaline from Koralpe, Austria (Ertl et al., 1997). Al-rich tourmaline with up to 2.2 B [4] pfu was synthesized experimentally at 2 . 5 GPa, 600 o C (Schreyer et al., 2000). The maximum possible amount of B [4] in olenitic tourmaline is limited to three B [4] pfu, due to structural and crystal chemical reasons.</text>
<text><location><page_2><loc_11><loc_13><loc_48><loc_27></location>As tourmaline is not stable at basic pH (Morgan & London, 1989), the dominant B-species of fluids coexisting with tourmaline is B(OH)3. Therefore, due to the absence of change in boron coordination, B-isotopic fractionation between B [4] -free tourmaline and fluid ( ∆ 11 B(tour-fl)) should be small. However, at 0 . 2 GPa, the experimentally determined ∆ 11 B(tour-fl)values of -2 . 5 ± 0 . 4 /permil at 400 o C and -0 . 4 ± 0 . 4 /permil at 700 o C (Meyer et al., 2008) suggest small but significant di ff erences in the B-O(H) bond strength between</text>
<text><location><page_2><loc_52><loc_75><loc_89><loc_87></location>tourmaline and the neutral fluids. Incorporation of B [4] into tourmaline is expected to significantly increase the fractionation of boron isotopes between the mineral and aqueous fluid (i.e. increase of | ∆ 11 B | (tour-fl)). In this contribution we present calculated P -T -dependent data on B-isotope fractionation between B [4] -bearing tourmaline and fluid, which so far are not available from experiments.</text>
<text><location><page_2><loc_52><loc_36><loc_89><loc_75></location>Despite of its low boron content of up to maximum values of 270 ppm (Domanik et al., 1993), potassic white mica is probably the main host of boron in metasedimentary and metabasaltic blueschists and eclogites, because of its high modal abundance and B-incompatibility in all other stable minerals of these rocks (Brenan et al., 1998). During subduction the modal amount of mica continuously decreases by dehydration reactions and the chemistry of residual micas is shifted towards phengitic compositions. Phengitic mica has an extended stability, extending as deep as 300 km within cold subduction zones (Schmidt, 1996). In contrast to most tourmalines, boron is tetrahedrally coordinated in mica, where it substitutes for aluminum. Due to the coordination change from mostly three-fold coordinated in near-neutral fluids (Schmidt et al., 2005) to B [4] in mica, B-isotopic fractionation between Bbearing mica and such fluids is much larger than for tourmaline - fluid. ∆ 11 B(mica-fl)-values determined experimentally at P = 3 . 0 GPa, are -10 . 9 ± 1 . 3 /permil at 500 o C and -7 . 1 ± 0 . 5 /permil at 700 o C (Wunder et al., 2005). Such a strong B-isotope fractionation and its pronounced T -dependence, in combination with the continuous dehydration of micas during ongoing subduction and boron transport via fluids into mantle wedge regions of arc magma-formation, probably determines the boron variations and B-isotopic signatures in volcanic arcs (Wunder et al., 2005).</text>
<text><location><page_2><loc_52><loc_13><loc_89><loc_35></location>Using in situ Raman spectroscopic measurements of near-neutral B-bearing fluids, Schmidt et al. (2005) observed a significant amount of 4-fold B-species at high P and T . The abundance of these species increases with temperature and pressure and at T = 800 K and P = 1 . 9 GPa there should be a considerable amount (15 -30%) of B [4] species in the fluid. Such a significant amount of tetrahedrally coordinated B-species in high temperature and pressure fluids should a ff ect solid-fluid B-isotopic fractionation, which we investigate in our calculations. In the light of this, we also discuss recently determined B-isotope data from coexisting natural tourmaline and B-bearing mica (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002), which show slight inconsistency with in situ measurements of Wunder et al. (2005) and Meyer et al. (2008).</text>
<text><location><page_3><loc_11><loc_21><loc_48><loc_86></location>Reliable ab initio computational methods to predict isotope fractionation factors have been established recently. Several groups have proved that such calculations can contribute towards understanding geochemical mechanisms responsible for production of isotope signatures (Driesner, 1997; Yamaji et al., 2001; Schauble, 2004; Domagal-Goldman et al., 2008; Hill & Schauble, 2008; Meheut et al., 2009, 2007; Schauble et al., 2009; Zeebe, 2009; Hill et al., 2010; Rustad et al., 2010a; Rustad et al, 2010b; Zeebe, 2009, 2010; Kowalski & Jahn, 2011). The majority of these works, however, concentrate on the computation of the stable isotope fractionation between various, mostly simple crystalline minerals, and the aqueous solutions are usually computed using an isolated cluster containing fractionating species and a hydration shell (e.g. Zeebe (2010, 2009); Hill et al. (2010); Rustad et al. (2010a); Domagal-Goldman et al. (2008); Hill & Schauble (2008); Rustad et al (2010b); Zeebe (2005)). However, aqueous solutions at high pressure and temperature must be computed with caution as discussed in Kowalski & Jahn (2011). This is because the distribution of cation coordination and cation-oxygen bond lengths that a ff ect the isotope fractionation (Bigeleisen & Mayer, 1947) is driven by the dynamics of the system and change under compression (Jahn & Wunder, 2009; Wunder et al., 2011; Kowalski & Jahn, 2011). The only recent ab initio work, besides Kowalski & Jahn (2011), that accounts for the dynamical e ff ects on the isotope fractionation in fluid is by Rustad & Bylaska (2007) who considered boron equilibrium isotope fractionation between B(OH)3 and B(OH) -4 species in aqueous solution. They performed ab initio molecular dynamics simulations of these fluids and attempted to use the vibrational density of states, derived through the Fourier transform of the velocity auto-correlation function, as an input for the calculation of the 11 B / 10 B isotope fractionation coe ffi -cient. However, the resulting fractionation factor α = 0 . 86 happened to be much lower than the experimental value of α = 1 . 028. Interestingly, the discrepancy between experiment and theory is cured by quenching the selected configurations along the molecular dynamics trajectory and computing the harmonic frequencies. The fractionation factor derived using these frequencies exactly reproduces the experimental value.</text>
<text><location><page_3><loc_11><loc_13><loc_48><loc_21></location>In our approach both solids and fluids are treated as extended systems by application of periodic boundary conditions in all three spatial directions, which is crucial to model high pressure materials. Large enough supercells are chosen to avoid significant interaction between atoms and their periodic images, as well as to reduce the</text>
<text><location><page_3><loc_52><loc_50><loc_89><loc_86></location>number of k-points and q-vectors for sampling the Brillouin zones and the phonon spectra of the crystals, respectively (for a liquid or fluid, both the Brillouin zone and phonons are not defined). In our investigation we use cells of at least 7 Å width in each spatial dimension. A representative statistical sampling of the fluid structure is obtained by performing Car-Parrinello molecular dynamics simulations (Car & Parrinello, 1985). For the calculation of the isotope fractionation factors in fluids, several random snapshots from the simulation runs are chosen. The force constants acting on the fractionating element and the resulting fractionation factors are then obtained for each ionic configuration, and the relevant fractionation factor for the boron species in the fluid is computed as an average over the whole set of considered geometries. In Kowalski & Jahn (2011) it was shown that in line with the Bigeleisen & Mayer (1947) approximation, considering the force constants acting on the fractionating atom only leads to a satisfactory estimation of the Li isotope fractionation factors for high temperature fluids and minerals. Here, we will show that approximating the vibrational spectrum by the three pseudofrequencies derived from the force constants allows for further improvement of the accuracy of the predicted isotope fractionation factors, especially at lower temperatures.</text>
<text><location><page_3><loc_52><loc_33><loc_89><loc_49></location>In this contribution we present the theoretical prediction of B isotope fractionation factors between B bearing aqueous fluids and solids, specifically tourmaline and B-muscovite, for which experimental data and measurements on natural samples are available for comparison (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011). We will show that the application of ab initio methods to B-bearing crystalline solids and fluids not only provides unique insight into the mechanisms driving equilibrium B-isotope fractionation on the atomic scale, but helps in proper interpretation of the data.</text>
<section_header_level_1><location><page_3><loc_52><loc_29><loc_72><loc_30></location>2. Computational approach</section_header_level_1>
<section_header_level_1><location><page_3><loc_52><loc_25><loc_68><loc_27></location>2.1. Theoretical model</section_header_level_1>
<text><location><page_3><loc_52><loc_22><loc_89><loc_24></location>2.1.1. The single atom approximation: &</text>
<text><location><page_3><loc_57><loc_22><loc_82><loc_23></location>Bigeleisen Mayer (1947) approach</text>
<text><location><page_3><loc_52><loc_13><loc_89><loc_21></location>Mass-dependent equilibrium isotope fractionation is driven by the change in molecular and crystalline vibration frequencies resulting from the di ff erent masses of the isotopes. The fractionation between species and an ideal monoatomic gas is called the β factor. In the harmonic approximation it is given by the formula</text>
<table>
<location><page_4><loc_13><loc_59><loc_87><loc_83></location>
<caption>Table 1: The computed and measured vibrational frequencies for H3BO3 and H4BO -4 molecules reported for two di ff erent boron isotopes B 11 / B 10 . Only the frequencies a ff ected by the isotopic substitution are reported. The units are cm -1 .</caption>
</table>
<text><location><page_4><loc_11><loc_54><loc_89><loc_59></location>References: 1 Zeebe (2005), 2 Liu & Tossell (2005), 3 Rustad et al (2010b), 4 Sanchez-Valle et al. (2005), 5 Gilson (1991), 6 Oi (2000); Liu & Tossell (2005), 7 Andrews & Burkholder (1992), 8 Ogden & Young (1988), 9 Zeebe (2005) and references hereafter.</text>
<text><location><page_4><loc_11><loc_49><loc_48><loc_52></location>(Bigeleisen & Mayer, 1947; Urey, 1947; Chacko et al., 2001):</text>
<formula><location><page_4><loc_16><loc_45><loc_48><loc_48></location>β = Ndof ∏ i = 1 u ∗ i ui exp [ ( ui -u ∗ i ) 2 ] 1 -exp( -ui ) 1 -exp( -u ∗ i ) , (1)</formula>
<text><location><page_4><loc_11><loc_28><loc_48><loc_44></location>where u = h ν i / kBT , h is the Planck constant, ν i is the vibrational frequency of the i -th degree of freedom, kB is the Boltzmann constant, Ndof is the number of degrees of freedom, which for N being the number of atoms in the considered system (molecule, mineral or fluid) is equal to 3 N -5 for a diatomic molecule, 3 N -6 for multiatomic molecules and 3 N for crystals, and a star symbol marks the heavier isotope. The fractionation factor between two substances A and B, α A -B is computed as the ratio of the relevant β factors, which is well approximated by the di ff erences in the β factors:</text>
<formula><location><page_4><loc_12><loc_24><loc_48><loc_27></location>α A -B = β A /β B , ∆ A -B /simequal 1000 ln β A -1000 ln β B [ /permil ] . (2)</formula>
<text><location><page_4><loc_11><loc_13><loc_48><loc_24></location>The calculation of the β factor requires only knowledge of the vibrational properties of the considered system computed for the two di ff erent isotopes. However, computation of the whole vibrational spectra of complex, multiparticle minerals or fluids requires substantial computational resources and is currently limited to systems containing a few dozens of atoms or less. In our recent work (Kowalski & Jahn, 2011) we proposed</text>
<text><location><page_4><loc_52><loc_42><loc_89><loc_52></location>to use an e ffi cient method for computing the high temperature isotope fractionation factors between complex materials such as fluids and crystalline solids, which requires the knowledge of the force constants acting upon the fractionating element only. The β factor (Eq. 1) can be then approximated by (Bigeleisen & Mayer, 1947; Kowalski & Jahn, 2011):</text>
<formula><location><page_4><loc_54><loc_38><loc_89><loc_41></location>β /similarequal 1 + Ndof ∑ i = 1 u 2 i -u ∗ 2 i 24 = 1 + ∆ m mm ∗ /planckover2pi1 2 24 k 2 B T 2 3 ∑ i = 1 Ai , (3)</formula>
<text><location><page_4><loc_52><loc_13><loc_89><loc_37></location>where Ai are the force constants acting on the isotopic atom in the three perpendicular spatial directions (x, y and z), ∆ m = m ∗ -m , where m and m ∗ are the masses of the lighter and heavier isotopes of the fractionating element. As the computation of the β factors from formula 3 requires the knowledge of properties of the fractionating element only we will call such an approach the single atom approximation throughout the paper. The validity criteria restricts the usage of the formula to frequencies ν [cm -1 ] /lessorsimilar 1 . 39 T [K] (assuming u < 2, see Fig. 1 of Bigeleisen & Mayer (1947)). We are interested in temperature range 800-1000 K. The highest vibrational frequency of the modes involving movement of B atoms for H3BO3 is ∼ 1400 cm -1 and of H4BO -1 4 is < 1200 cm -1 (Table 1). In case of H3BO3, the single atom approximation may produce an error of 2 . 2 /permil in the β factor at T = 800 K. The relevant error for</text>
<table>
<location><page_5><loc_11><loc_65><loc_93><loc_81></location>
<caption>Table 2: 1000( β -1) factors for isolated H3BO3 and H4BO -4 molecules at T = 300K obtained using three methods: (1) the full frequency spectrum and Equation 1, (2) the single atom approximation of Kowalski & Jahn (2011), (3) the single atom approximation with the pseudofrequencies and equation 1. The units are /permil . The ∆ β/β BM is the check of condition given by Eq. 15 and indicates the improvement of the method (3) over method (2) expressed in %.</caption>
</table>
<text><location><page_5><loc_11><loc_60><loc_48><loc_63></location>H4BO -1 4 is 0 . 8 /permil (Table 2). A further improvement to the method is therefore desired.</text>
<section_header_level_1><location><page_5><loc_11><loc_56><loc_48><loc_58></location>2.1.2. The single atom approximation with pseudofrequencies: our improvement</section_header_level_1>
<text><location><page_5><loc_11><loc_42><loc_48><loc_55></location>We will show that the error of the single atom approximation can be substantially reduced if one uses the three frequencies ¯ ν i derived from the force constants acting on the fractionating element (¯ ν 2 i = Ai / 4 π 2 m ). We call them 'pseudofrequencies' , and compute the β factors using formula 1. In the following we present the formal justification of such an approach. According to Bigeleisen & Mayer (1947), equation 1 for small ∆ ui = ui -u ∗ i reduces to (Bigeleisen & Mayer (1947), Eq. 11a):</text>
<formula><location><page_5><loc_16><loc_37><loc_48><loc_40></location>β = 1 + Ndof ∑ i = 1 ( 1 2 -1 ui + 1 exp( ui ) -1 ) ∆ ui . (4)</formula>
<text><location><page_5><loc_11><loc_33><loc_48><loc_36></location>The Taylor expansion of the function appearing under the summation sign is:</text>
<formula><location><page_5><loc_17><loc_25><loc_48><loc_32></location>G ( u ) = 1 2 -1 u + 1 exp( ui ) -1 = u 12 -u 3 720 + u 5 30240 -u 7 1209600 + .... (5)</formula>
<text><location><page_5><loc_11><loc_22><loc_48><loc_25></location>When we consider just the first term of the expansion the β factor is:</text>
<formula><location><page_5><loc_18><loc_12><loc_48><loc_21></location>β = 1 + Ndof ∑ i ui 12 ∆ ui = 1 + Ndof ∑ i = 1 ∆ u 2 i 24 = 1 + Ndof ∑ i = 1 u 2 i -u ∗ 2 i 24 , (6)</formula>
<table>
<location><page_5><loc_52><loc_52><loc_93><loc_59></location>
<caption>Table 3: The three pseudofrequencies of the B atom derived for two selected molecules and two crystalline solids. The units are cm -1 .</caption>
</table>
<text><location><page_5><loc_52><loc_48><loc_71><loc_49></location>which is exactly equation 3.</text>
<text><location><page_5><loc_52><loc_45><loc_89><loc_48></location>Let us consider the Taylor expansion of the di ff erent estimations of β factors. Equation 4 then reads:</text>
<formula><location><page_5><loc_57><loc_39><loc_89><loc_44></location>β = β exact = 1 + Ndof ∑ i = 1 ui 12 -u 3 i 720 + ... ∆ ui . (7)</formula>
<text><location><page_5><loc_52><loc_36><loc_89><loc_39></location>The Bigeleisen & Mayer (1947) approximation given by equation 3 reads:</text>
<formula><location><page_5><loc_62><loc_31><loc_89><loc_35></location>β ∼ β BM = 1 + Ndof ∑ i = 1 ui 12 ∆ ui , (8)</formula>
<text><location><page_5><loc_52><loc_27><loc_89><loc_30></location>and the proposed approximation based on pseudofrequencies is:</text>
<formula><location><page_5><loc_56><loc_21><loc_89><loc_26></location>β ∼ β pseudo = 1 + 3 ∑ i = 1 ¯ ui 12 -¯ u 3 i 720 + ... ∆ ¯ ui . (9)</formula>
<text><location><page_5><loc_52><loc_13><loc_89><loc_21></location>In the above equation ¯ ui = h ¯ ν i / kBT . Next we check how this approximation compares to the Bigeleisen & Mayer (1947) approximation given by equations 3 and 8. In order to make a comparison we derive the di ff erences between the two approximate expressions β BM , β pseudo and the exact one β exact (Eq. 7). In the case of the</text>
<figure>
<location><page_6><loc_14><loc_53><loc_79><loc_84></location>
<caption>Figure 1: β factors for isolated H3BO3 and H4BO -4 molecules as well as dravite and boromuscovite 1M crystalline solids. The lines represent the results obtained using: (1) the full frequency spectrum and Eq. 1 (solid), (2) the single atom approximation of Bigeleisen & Mayer (1947); Kowalski & Jahn (2011), Eq. 3 (dotted) and (3) the method described in the text with the pseudofrequencies and Eq. 1 (dashed lines).</caption>
</figure>
<text><location><page_6><loc_11><loc_45><loc_46><loc_46></location>Bigeleisen & Mayer (1947) approximation we have:</text>
<formula><location><page_6><loc_11><loc_39><loc_50><loc_44></location>∆ β BM = β BM -β exact = Ndof ∑ i ui 12 -Ndof ∑ i ui 12 -u 3 i 720 + ... ∆ ui</formula>
<text><location><page_6><loc_11><loc_34><loc_37><loc_35></location>and having from equations 3 and 8 that</text>
<formula><location><page_6><loc_23><loc_34><loc_48><loc_39></location>= Ndof ∑ i u 3 i 720 -... ∆ ui (10)</formula>
<formula><location><page_6><loc_22><loc_29><loc_48><loc_32></location>Ndof ∑ i ui 12 ∆ ui = 3 ∑ i ¯ ui 12 ∆ ¯ ui (11)</formula>
<text><location><page_6><loc_11><loc_27><loc_44><loc_28></location>in the case of the proposed approximation we get:</text>
<formula><location><page_6><loc_13><loc_11><loc_48><loc_25></location>∆ β pseudo = β pseudo -β exact = 3 ∑ i = 1 -¯ u 3 i 720 + ... ∆ ¯ ui + Ndof ∑ i u 3 i 720 -... ∆ ui = = 3 ∑ i = 1 -¯ u 3 i 720 + ... ∆ ¯ ui + ∆ β BM (12)</formula>
<text><location><page_6><loc_52><loc_44><loc_89><loc_47></location>Because relation G ( u ) < u 12 holds for any u (see Bigeleisen & Mayer (1947), Fig. 1), the function</text>
<formula><location><page_6><loc_53><loc_38><loc_89><loc_43></location>3 ∑ i = 1 -¯ u 3 i 720 + ... ∆ ui = 3 ∑ i = 1 ( G ( ¯ ui ) -¯ ui 12 ) ∆ ¯ ui < 0 (13)</formula>
<text><location><page_6><loc_52><loc_13><loc_89><loc_38></location>and ∆ β pseudo < ∆ β BM . On the other hand the expression for ∆ β pseudo is given by a di ff erence of the higher order terms of the Taylor expansions of the two expressions for the β factor. In the considered cases the values of pseudofrequencies are similar to the real frequencies that are a ff ected upon B isotope substitution in a given B-bearing system. This can be seen by comparing the pseudofrequencies computed for the selected cases of B-bearing molecules and crystalline solids considered here and reported in Table 3 with the real frequencies given in Table 1. This indicates that the two terms of opposite signs in Eq. 12 should be similar in value and cancel out to a great extent, so | ∆ β pseudo | << ∆ β BM . Therefore the approach proposed here to compute the β factor based on pseudofrequencies and Eq. 1 should give a better approximation to the exact β factors than equation 3, which we will show in section 3.1. The difference to the Bigeleisen & Mayer (1947) approxima-</text>
<table>
<location><page_7><loc_24><loc_73><loc_76><loc_85></location>
<caption>Table 4: The lattice parameters of the investigated B-bearing crystalline solids. Natoms is the number of atoms in the modeled supercell.</caption>
</table>
<text><location><page_7><loc_26><loc_71><loc_74><loc_73></location>References: 1 Liang et al. (1995), 2 Ertl et al. (2010), 3 Marler et al. (2002)</text>
<text><location><page_7><loc_11><loc_68><loc_22><loc_69></location>tion is given by:</text>
<formula><location><page_7><loc_20><loc_63><loc_48><loc_66></location>∆ β = 1 + 3 ∑ i = 1 ¯ ui 12 ∆ ¯ ui -β pseudo (14)</formula>
<text><location><page_7><loc_11><loc_56><loc_48><loc_62></location>and can be easily computed for any considered system. We assume that the pseudofrequency-based approach to the computation of β factors is applicable if it is just a correction to equation 3, i.e. when:</text>
<formula><location><page_7><loc_23><loc_52><loc_48><loc_55></location>∆ β << 3 ∑ i = 1 ¯ ui 12 ∆ ¯ ui . (15)</formula>
<text><location><page_7><loc_11><loc_45><loc_48><loc_51></location>We also note that the proposed approach satisfies the Redlich-Teller product rule (Redlich, 1935) when the mass of considered isotope m is much smaller than the mass of the whole considered system M , namely,</text>
<formula><location><page_7><loc_14><loc_41><loc_48><loc_44></location>Ndof ∏ i = 1 u ∗ i ui = ( M ∗ M m m ∗ ) 3 / 2 /approxorequal ( m m ∗ ) 3 / 2 = 3 ∏ i = 1 ¯ u ∗ i ¯ ui . (16)</formula>
<text><location><page_7><loc_11><loc_29><loc_48><loc_40></location>We notice that we could force the strict conservation of the Redlich-Teller product by just adjusting the ratios of ¯ ui / ¯ u ∗ i . However, such a modification would not preserve relation 11 and the pseudofrequency approach would not recover exactly the high temperature limit (Eq. 3) of the exact solution (Eq. 1), which is a more important constraint to fulfill strictly by the proposed approximation.</text>
<section_header_level_1><location><page_7><loc_11><loc_26><loc_30><loc_27></location>2.2. Representation of solids</section_header_level_1>
<text><location><page_7><loc_11><loc_13><loc_48><loc_25></location>In this paper we investigate the boron isotope fractionation between dravite, olenite and boromuscovite minerals and aqueous fluids. The solids were represented by large cells containing at least 84 atoms. The number of atoms used in the crystal calculations together with the lattice parameters of modeled crystals are summarized in Table 4. The lattice parameters and chemical compositions of the modeled crystalline solids are the experimental values measured at</text>
<text><location><page_7><loc_52><loc_42><loc_89><loc_69></location>ambient conditions found in the literature. Dravite is the crystalline solid which was used in the experiments on tourmaline by Meyer et al. (2008). The chemical composition of the supercell used in the investigation is Na3Mg9Al18(Si18O54)(B [3] O3)9(OH)12 with structural data of Marler et al. (2002). Olenite can contain B in both trigonal and tetragonal sites. The modeled structure is that of Ertl et al. (2010). The chemical composition of the unit cell used in the investigation is NaAl3Al6(Si4B [4] 2 O18)(B [3] O3)3(OH)3O. For boromuscovite, the 1M and 2M1 crystal structures of Liang et al. (1995) were used. In the isotope fractionation experiments of Wunder et al. (2005) boromuscovite forms two polytypes, 1M and 2M1, with relative abundances of 10% and 90% respectively. In boromuscovite B occupies the 4-fold coordinated site occupied mainly by Si atoms. The constructed model constitutes a 2x1x1 supercell of elementary chemical composition KAl2(B [4] Si3O10)(OH)2.</text>
<section_header_level_1><location><page_7><loc_52><loc_38><loc_79><loc_39></location>2.3. Representation of aqueous solution</section_header_level_1>
<text><location><page_7><loc_52><loc_13><loc_89><loc_35></location>The aqueous solution was represented by a periodically repeated box containing up to 64 water molecules and one H3BO3 or H4BO -4 molecule. The pressure and temperature conditions were chosen to be close to the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The pressure of the aqueous solution for a given temperature and volume was calculated according to the equation of state of Wagner & Pruss (2002). The ab initio molecular dynamics simulations (AIMD) of aqueous fluids were performed for fixed temperature and volume using the Car-Parrinello scheme (Car & Parrinello, 1985). The temperature during each run was controlled by a Nos'e-Hoover chain thermostat (Nos'e & Klein, 1983; Hoover, 1985). For each T -V conditions at least 10 ps long trajectories were generated with an integration step of 0 . 12 fs.</text>
<figure>
<location><page_8><loc_15><loc_61><loc_44><loc_85></location>
</figure>
<figure>
<location><page_8><loc_46><loc_61><loc_75><loc_85></location>
<caption>Figure 2: Left panel: β factors for H3BO3 (solid line) and H4BO -4 (dashed lines) in the gas phase. Right panel: fractionation factors between H3BO3 and H4BO -4 in gas phase. The thick lines represent our results, while the thin lines represent the results obtained using frequencies of Zeebe (2005).</caption>
</figure>
<section_header_level_1><location><page_8><loc_11><loc_52><loc_31><loc_53></location>2.4. Computational technique</section_header_level_1>
<text><location><page_8><loc_11><loc_21><loc_48><loc_51></location>The calculations of pseudofrequencies and β factors for solids and aqueous solutions were performed by applying density functional theory (DFT) methods, which are currently the most e ffi cient methods allowing for treating extended many particle systems quantummechanically. We used the planewave DFT code CPMD (Marx & Hutter, 2000), which is especially suited for ab initio simulations of fluids, the BLYP exchangecorrelation functional (Becke, 1988; Lee et al., 1988) and norm-conserving Goedecker pseudopotentials for the description of the core electrons (Goedecker et al., 1996). One advantage of using the BLYP functional is that it usually gives harmonic frequencies that most closely resemble the observed frequencies of benchmark chemical systems (Finley & Stephens, 1995; Alecu et al., 2010) 1 . The energy cut-o ff for the plane wave basis set was 70 Ryd for geometry relaxations and molecular dynamics simulations and 140 Ryd for computation of vibrational frequencies. Periodic boundary conditions were applied for both crystalline solids and aqueous solutions to preserve the continuity of the media.</text>
<text><location><page_8><loc_52><loc_22><loc_89><loc_53></location>The force constants and frequencies needed for the computation of the β factors were computed using the finite displacement scheme. Before performing the calculations of the crystal structures all atomic positions were relaxed to the equilibrium positions to minimize the forces acting on the atoms. We note that to compute the β factors for crystals one formally should account for phonon dispersion. Here we use large supercells and restrict our calculations to a single phonon wave-vector ( Γ ). Schauble (2011) has shown recently for 26 Mg / 24 Mg fractionation in Mg-bearing minerals that supercells containing more than 20 atoms are su ffi -cient to get very accurate β factors even at T = 300 K (error of 0.1 /permil ). At T = 1000 K the error is in the order of 0.01 /permil . The accuracy of the high temperature isotope fractionation factors computed on a single phonon wave-vector is also demonstrated for ironbearing minerals by Blanchard et al. (2009) and confirmed with good agreement of the predicted with the measured Li isotope fractionation factors between staurolite, spodumene, micas and aqueous fluid presented in our previous work (Kowalski & Jahn, 2011).</text>
<text><location><page_8><loc_52><loc_13><loc_89><loc_21></location>Prior to the computation of the force constants and frequencies of boron atoms in the fluids the positions of all the atoms constituting the boron-carrying molecule (H3BO3 or H4BO -4 ) were relaxed to the equilibrium positions, while all other atomic positions remained unchanged. The full normal mode analyzes were per-</text>
<figure>
<location><page_9><loc_15><loc_61><loc_44><loc_85></location>
</figure>
<figure>
<location><page_9><loc_46><loc_61><loc_75><loc_85></location>
<caption>Figure 3: Left panel: β factors for H3BO3 (solid line) and H4BO -4 (dashed lines) in aqueous solution. Right panel: fractionation factors between H3BO3 and H4BO -4 in aqueous solution. The thick lines represent our results, while the thin lines represent the results of Sanchez-Valle et al. (2005) obtained using harmonic frequencies (their Table 2). The dotted line represents the corrected Sanchez-Valle et al. (2005) results. The correction is made by comparing the work of Rustad et al (2010b) and the correction derivation procedure is discussed in the text.</caption>
</figure>
<table>
<location><page_9><loc_18><loc_43><loc_82><loc_50></location>
<caption>Table 5: The 1000( β -1), α = β H3BO3 /β H4BO -4 and ∆ = 1000(ln β H3BO3 -ln β H4BO -4 ) factors computed for isolated H3BO3 and H4BO -4 at T = 300 K by Zeebe (2005) using di ff erent basis sets and our result obtained using the full normal mode spectrum and equation 1. The units are /permil .</caption>
</table>
<text><location><page_9><loc_11><loc_14><loc_48><loc_41></location>formed using the same method, but displacing all the atoms constituting the considered system. In the latter case the frequencies were obtained through the diagonalization of the full dynamical matrix (Schauble, 2004) as implemented in CPMD code. The e ff ect of the various approximations on the derived fractionation factors was studied by additional computations of H3BO3 and H4BO -4 isolated clusters. For that purpose we used a large, isolated simulation box with a cell length of 16 Å, forcing the charge density to be zero at the boundary, as implemented in CPMD code. In order to compute the β factors of boron species in the aqueous fluid we apply the same method as in our recent work on Li isotopes (Kowalski & Jahn, 2011), with the exception that we use the pseudofrequencies, i.e. the frequencies obtained from the three force constants acting on the fractionating element, and formula 1 for calculation of β factors, as discussed in section 2.1.2. In order to fully account for the spatial continuity of the fluid and</text>
<text><location><page_9><loc_52><loc_27><loc_89><loc_41></location>its dynamical motion we produced 10 ps long molecular dynamics trajectories of systems consisting of 64 H2O molecules and one H3BO3 or H4BO -4 molecule for different T = 1000 K, 800 K and 600 K and pressure of 0 . 5 GPa, which closely resembles the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The corresponding simulation box length is 13 . 75 Å at T = 1000 K. The β factors were computed on the ionic configuration snapshots extracted uniformly in 0 . 1 ps intervals along the molecular dynamics trajectories.</text>
<section_header_level_1><location><page_9><loc_52><loc_24><loc_73><loc_25></location>2.5. Error estimation technique</section_header_level_1>
<text><location><page_9><loc_52><loc_13><loc_89><loc_24></location>The errors in the computed value of the ( β -1) and ∆ fractionation factors were estimated from an average error of vibrational frequencies computed using the chosen DFT method. Finley & Stephens (1995), Menconi & Tozer (2002) and Alecu et al. (2010) estimated the errors made in calculations of vibrational frequencies of small molecules using di ff erent DFT functionals. According to these works the BLYP functional</text>
<text><location><page_10><loc_11><loc_74><loc_48><loc_86></location>systematically overestimates the harmonic frequencies by ∼ 3 . 5 %, with a deviation from the mean o ff set of ∼ 1 %. Therefore, we expect that using BLYP functional the ( β -1) and ∆ values are systematically overestimated by 7 % and that in addition there is a 2 % error in derived ( β -1) factors. Similar errors result from using other functionals or even more sophisticated and time consuming post-Hartree-Fock methods such as MP2 (Finley & Stephens, 1995; Alecu et al., 2010).</text>
<section_header_level_1><location><page_10><loc_11><loc_71><loc_29><loc_72></location>3. Results and discussion</section_header_level_1>
<section_header_level_1><location><page_10><loc_11><loc_68><loc_36><loc_69></location>3.1. Test of the computational method</section_header_level_1>
<text><location><page_10><loc_11><loc_21><loc_48><loc_68></location>First, we illustrate the performance of the approximation proposed in section 2.1.2 by computing the β factors for the isolated H3BO3 and H4BO -4 molecules and selected crystalline solids. In Figure 1 we present three sets of calculations of β factors: (1) the 'exact' result obtained from a full normal mode analysis and formula 1, (2) the results obtained applying Kowalski & Jahn (2011) method based on Eq. 3, (3) the results obtained using pseudofrequencies computed for the fractionating element and Eq. 1 for the estimation of the β factor. The numerical values for selected temperatures are reported in Table 2. Approach (3) results in much better agreement with the 'exact' result. For H3BO3, the β factor is overestimated by only 0 . 5 /permil and 1 . 5 /permil for temperatures of 800 K and 600 K respectively. Applying method (2), the error is more pronounced, 2 . 2 /permil and 6 . 7 /permil respectively. In the case of molecular H4BO -4 , the errors using method (3) for the same temperatures are only 0 . 1 /permil and 0 . 5 /permil respectively. The same behavior is shown for dravite and boromuscovite crystalline solids that contain boron in the coordinative arrangement that resemble the configurations of aforementioned B-bearing molecules. For T > 600 K the proposed method represents only a few percent correction to the approximation given by equation (3), so the relation (15) is satisfied. It is evident that for B-bearing materials considered here the improvement made by using the pseudofrequencies based approach is substantial. It corrects for about 75% of error of the Bigeleisen & Mayer (1947) approximation (Eq. 3). However, the question of general applicability of the proposed method to other isotopic systems would require careful testing on a large set of materials, which is well beyond the scope of the current paper.</text>
<section_header_level_1><location><page_10><loc_11><loc_16><loc_48><loc_20></location>3.2. B isotope fractionation in gas and fluid phases 3.2.1. B isotope fractionation between H3BO3 and H4BO -4 in the gas phase</section_header_level_1>
<text><location><page_10><loc_11><loc_13><loc_48><loc_15></location>In a first step of our investigation of boron-rich aqueous fluids we derived the full frequency spectra of</text>
<text><location><page_10><loc_52><loc_34><loc_89><loc_86></location>molecules in the gas phase (isolated molecules). The relevant β factors were computed using Equation 1. These studies were performed in order to compare our results with the published values of Zeebe (2005), both computed using the same DFT BLYP functional. In Table 1 we report the computed frequencies that are affected by the di ff erent B isotope substitutions along with other theoretical estimations and experimental measurements. The computed frequencies are in good agreement with earlier theoretical predictions and show similar agreement with the experimental measurements. The results in terms of computed β factors for the two considered species are reported in Figure 2, where we compare our results with the values computed using frequencies of Zeebe (2005). The comparison of the two sets of calculations reveals that our β factors for both species are smaller by ∼ 1 /permil at 600 K -1000 K than the values of Zeebe (2005). However, the di ff erence between β factors of H3BO3 and H4BO -4 remains nearly identical in both sets of calculations and the agreement is nearly perfect for higher temperatures. We note, that for the comparison we used the frequencies of Zeebe (2005) computed using 6-31 + G(d) basis set, as only these are provided by the authors. β and α factors obtained at T = 300 K using a more extended 6311 + G(d,p) basis set indicate that the β factors using 6-31 + G(d) basis set are not fully converged. In particular, the ( β -1) factor of H4BO -4 computed with 6311 + G(d,p) basis set is 3 . 9 /permil smaller than the one derived using 6-31 + G(d). In Table 5, we compare these results with the results of our calculation. It is clearly seen that for lower temperatures such as T = 300 K the values computed with 6-311 + G(d,p) basis set are in better agreement with our results indicating that planewave based DFT approach we use provides adequate vibrational frequencies and resulting isotope fractionation factors.</text>
<section_header_level_1><location><page_10><loc_52><loc_30><loc_89><loc_33></location>3.2.2. B isotope fractionation between H3BO3 and H4BO -4 in aqueous fluid</section_header_level_1>
<text><location><page_10><loc_52><loc_14><loc_89><loc_30></location>In order to obtain the temperature dependent β factor for aqueous fluids we fitted the function 1 + A / T 2 + B / T 4 to the computed values using the least squares minimization procedure. The computed β values for H3BO3 in fluid are: 1 . 02366 ± 0 . 00012, 1 . 03624 ± 0 . 00018 and 1 . 06262 ± 0 . 00010and for H4BO -4 in fluid are: 1 . 01745 ± 0 . 00005, 1 . 02650 ± 0 . 00010and 1 . 04597 ± 0 . 00015, for the temperatures of 1000 K, 800 K and 600 K respectively. The resulting temperature dependent β factor for H3BO3 is β = 1 + 2 . 416 · 10 4 / T 2 -5 . 823 · 10 8 / T 4 and for H4BO -4 is β = 1 + 1 . 772 · 10 4 / T 2 -4 . 234 · 10 8 / T 4 .</text>
<text><location><page_10><loc_54><loc_13><loc_89><loc_14></location>The results for H3BO3 and H4BO -4 in aqueous so-</text>
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</figure>
<figure>
<location><page_11><loc_52><loc_62><loc_83><loc_86></location>
<caption>Figure 4: Left panel: pressure dependence of the β factor of neutral fluid (H3BO3) at T = 1000K. Dashed line is the linear regression fit to the calculated values (points): 1000( β -1) = 23 . 6 + 0 . 28 P (GPa); Right panel: The computed change of the β factor with pressure (symbols) in comparison to the increase in the β factor derived from the frequency shifts of the 666 cm -1 and 1454 cm -1 lines measured by Sanchez-Valle et al. (2005) (dashed line). The comparison is made assuming that ( β -1) ∼ ν 2 .</caption>
</figure>
<text><location><page_11><loc_11><loc_14><loc_48><loc_52></location>lutions are shown in Figure 3. As was observed for the isolated molecules, the β factor of H3BO3-bearing fluid is substantially larger than the one for the H4BO -4 . This can be understood in terms of the substantial difference in the B-O bond lengths exhibited by the two considered species. In case of isolated molecules our calculations indicate a B-O bond length of 1 . 40 Å for H3BO3 and 1 . 51 Å for H 4BO -4 . We compared our β factors with the values computed by Sanchez-Valle et al. (2005), which were derived by the combination of force field methods and experimental data to derive accurate vibrational frequencies. For H3BO3 we got a nearly identical result. In case of H4BO -4 our calculation predicts a value which is lower by 2 -4 /permil . However, Rustad et al (2010b) and Rustad & Bylaska (2007) revealed the improper assignment of a major fractionating vibrational mode of H4BO -4 in the force field by Sanchez-Valle et al. (2005). This leads to the underestimation of the fractionation factor between aqueous H3BO3 and H4BO -4 by Sanchez-Valle et al. (2005). Assuming that α ∝ T -2 and having the di ff erence between BLYP calculations of Rustad et al (2010b) and Sanchez-Valle et al. (2005) of ∆ α = 16 . 4 /permil at T = 300 K, the value reported by Sanchez-Valle et al. (2005) should be underestimated by ∆ α = 16 . 4 · 2(300 / T ) 2 /permil , which results in ∆ α ∼ 1 . 5 /permil at T = 1000 K. Corrected in such as way result of Sanchez-Valle et al. (2005) is</text>
<text><location><page_11><loc_52><loc_50><loc_89><loc_52></location>also plotted in Figure 3. It is now very consistent with our prediction.</text>
<section_header_level_1><location><page_11><loc_52><loc_47><loc_80><loc_48></location>3.2.3. Discussion of computational errors</section_header_level_1>
<text><location><page_11><loc_52><loc_13><loc_89><loc_47></location>Most previous computational studies of boron isotope fractionation in aqueous solutions concentrate on the computation of the isotope fractionation at ambient conditions (Rustad et al, 2010b; Rustad & Bylaska, 2007; Liu & Tossell, 2005; Zeebe, 2005). Rustad et al (2010b) performed detailed analysis of impact of the chosen computational method (HF, MP2, di ff erent DFT functionals) and size of the basis set on the calculated fractionation factors between H3BO3 and H4BO -4 . They found that DFT methods are not performing well for the borate system and concluded that DFT 'is of limited usefulness in chemically accurate predictions of isotope fractionation in aqueous systems' (Rustad et al, 2010b). The empirically derived error of the derived fractionation factor is of the order of 5 -10 /permil for a total fractionation of ∼ 30 /permil . We note that this is expected and clearly visible if we apply the error estimation procedure outlined in section 2.5. For instance, at room temperature the derived beta factors using the BLYP functional are 213 . 6 /permil and 173 . 3 /permil respectively (Rustad et al (2010b), Table 2). This gives a fractionation factor of 1.0343. Following our error estimation scheme, the absolute error of the fractionation factor is 10 . 5 /permil , and the properly reported computed value</text>
<figure>
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</figure>
<figure>
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<caption>Figure 5: Left panel: pressure dependence of the β factor of strongly basic fluid (H4BO -4 ) at T = 1000 K. Dashed line is the regression fit to the calculated values (points): 1000( β -1) = 17 . 15 + 0 . 754 P -0 . 027 P 2 , where pressure is given in GPa; Right panel: The computed change of the β factor with pressure in comparison to the increase in the β factor derived from the frequency shift of the 975 cm -1 line measured by Sanchez-Valle et al. (2005) (dashed line). The comparison is made assuming that ( β -1) ∼ ν 2 .</caption>
</figure>
<text><location><page_12><loc_11><loc_14><loc_48><loc_52></location>is α = 1 . 034 ± 0 . 011. When one corrects for the systematic error of 7% and assumes 2% of statistical error on β factors, then the value of α decreases and the error is slightly smaller, i.e. α = 1 . 032 ± 0 . 008. This is in good agreement with the experimental data reported in Rustad et al (2010b) and explains the spread of the values computed using di ff erent methods and reported in that paper. It is very di ffi cult to get the fractionation factors for ambient conditions, as the fractionation factor is often just a small fraction of the relevant ( β -1) factors, ( α -1) ∼ 0 . 15( β -1) in the considered case. Assuming that ( α -1) = 0 . 15( β -1), a 2% error in the ( β -1) factors leads to an absolute error in ( α -1) of 0 . 04( β -1) = 0 . 04( α -1) / 0 . 15 ∼ 0 . 27( α -1), i.e. ∼ 27% of relative error in the derived fractionation factor ( α -1). On the other hand, we note that such a big error is not substantially larger than the uncertainties in the experimental data reported by Rustad et al (2010b) in their Figure 2. Thus, the case of boron fractionation in aqueous fluid at ambient conditions does not necessarily show the limited usefulness of DFT in the prediction of isotope fractionation factors, but only reflects the fact that precise estimation or measurement of the B isotopes fractionation factors at ambient condition requires unprecedented accuracy of both experimental or computational techniques. For instance, in order to get the value of ( α -1) with a relative error of 5% (at ambient</text>
<text><location><page_12><loc_52><loc_30><loc_89><loc_52></location>conditions) one needs to estimate the ( β -1) factors or measure relevant quantities with precision of less than 1%. At higher temperatures the situation is di ff erent. Looking just at the fractionation between H3BO3 and H4BO -4 in the gas phase or the aqueous solution one can see that for T > 600 K the fractionation factor between the two substances, ( α -1), is at least 25% of the ( β -1) factor. This results in smaller 0 . 04 / 0 . 25 ∼ 16% for T = 600 K and 0 . 04 / 0 . 36 ∼ 11% for T = 1000 K relative error, which is acceptable in our calculations. Nevertheless, this case shows the importance of proper error estimation on the computed fractionation factors. Such an estimation is usually omitted or not provided explicitly, which can lead to wrong conclusions when the theoretical prediction is confronted with the measured data.</text>
<section_header_level_1><location><page_12><loc_52><loc_26><loc_89><loc_28></location>3.2.4. Pressure dependence of the fluid fractionation factor</section_header_level_1>
<text><location><page_12><loc_52><loc_13><loc_89><loc_25></location>In our recent paper (Kowalski & Jahn, 2011) we have shown that due to compression the β factor of Li in aqueous fluid increases with increase in pressure (for P > 2 GPa). The same should happen for H3BO3 and H4BO -4 aqueous fluid as the vibrational frequencies of boron species in aqueous fluid increase with increase in pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). Having the experimental data we checked whether the derived pressure-dependent β</text>
<text><location><page_13><loc_11><loc_30><loc_48><loc_86></location>factors are consistent with the pressure shifts of vibrational frequencies of considered boron species measured by Sanchez-Valle et al. (2005). For that purpose we performed a set of calculations using supercells containing 8 water molecules and the relevant boron species. We note that in line with our previous results for Li (Kowalski & Jahn, 2011), the obtained values of ( β -1) at P = 0 . 5 GPa are within 0 . 1 /permil in agreement with the values obtained for supercells containing 64 water molecules. The results are given in Figures 4 and 5. The computed ( β -1) values for H3BO3 fluid show a linear dependence in pressure, ( β -1) = 23 . 60 + 0 . 28 P (GPa) /permil . This is expected as ( β -1) ∝ ν 2 ∼ ν 2 0 + 2 ν 0 ∆ ν (Schauble, 2004) and ∆ ν is a linear function of pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). In case of H4BO -4 the pressure-dependence is linear up to P ∼ 2 -3 GPa and it becomes less steep at higher pressures. In order to quantitatively check the consistency of our prediction with the measured vibrational frequency shifts of Sanchez-Valle et al. (2005) we derived the relative shifts in the ( β -1) factor assuming that ( β -1) ∝ ν 2 and the measured pressure dependence of the frequency shifts: ∆ ν = 2 . 15 cm -1 · P (GPa) and ∆ ν = 3 . 50 cm -1 · P (GPa) for 1454 cm -1 and 666 cm -1 vibrational frequencies of H3BO3 and ∆ ν = 6 . 47 cm -1 · P (GPa) for the 975 cm -1 vibrational frequency of H4BO -4 . The chosen vibrational frequencies are the ones a ff ected by the di ff erent B isotope substitution. Our predicted shift of ( β -1) matches well the shifts derived from the measured frequency shifts. Such a good agreement with the experimental data validates further our computational approach and shows that ab initio calculations can be successfully used in derivation of the pressure dependence of the fractionation factors and pressure-induced vibrational frequencies shifts. Moreover, first principles calculations can be useful in extrapolation of the experimental values for β and ∆ ν to more extreme conditions, which otherwise are extremely di ffi cult to reach by experimental techniques.</text>
<section_header_level_1><location><page_13><loc_11><loc_27><loc_33><loc_28></location>3.3. Fluid-mineral fractionation</section_header_level_1>
<text><location><page_13><loc_11><loc_19><loc_48><loc_27></location>Next we present the results of the fractionation between boron bearing fluids and minerals such as dravite, olenite and boromuscovite. The aim of these studies is to investigate the mechanisms driving the fractionation process, the role of coordination and the B-O bond length. Below we discuss each case separately.</text>
<section_header_level_1><location><page_13><loc_11><loc_16><loc_32><loc_17></location>3.3.1. Tourmaline-neutral fluid</section_header_level_1>
<text><location><page_13><loc_11><loc_13><loc_48><loc_15></location>Meyer et al. (2008) measured the boron isotope fractionation between tourmaline and neutral fluid at T =</text>
<figure>
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<caption>Figure 6: The fractionation factors between tourmaline and aqueous fluid. The solid line represents our prediction for the fractionation between dravite and H3BO3 neutral fluid and the shadowed region represent the computational uncertainty. The dotted line represents our prediction for the fractionation between olenite containing B [3] species only and neutral fluid. The computational error is similar in both cases. The data points are the values measured for dravite-fluid system by Meyer et al. (2008).</caption>
</figure>
<text><location><page_13><loc_52><loc_34><loc_89><loc_56></location>400 -700 o C and P = 0 . 2 GPa. In the experiment the tourmaline was represented by dravite. In contrast to the former measurements of Palmer et al. (1992) the measured fractionation is very small and does not exceed 2 . 5 /permil at 400 o C. Our calculated fractionation curve together with the experimental data are given in Figure 6. Our result correctly reproduces the experimental measurements within the computational accuracy. The dravite-fluid fractionation is small as the two materials contain boron in BO3 units. We also predict a small fractionation between olenite carrying 3-fold coordinated boron only and aqueous fluid, although the olenite-fluid fractionation is positive because of the shorter B-O bond lengths for olenite (1 . 378 Å vs. 1 . 397 Å).</text>
<section_header_level_1><location><page_13><loc_52><loc_32><loc_80><loc_33></location>3.3.2. Boromuscovite-strongly basic fluid</section_header_level_1>
<text><location><page_13><loc_52><loc_13><loc_89><loc_31></location>Boromuscovite synthesized in the experiments of Wunder et al. (2005) consisted of two type of polytypes, 1M ( ∼ 10 %) and 2M1 ( ∼ 90 %). In order to be consistent with the experimental conditions, we derived the β factors for both polytypes and computed their weighted average. We note that the β factors for both polytypes of mica are similar with the di ff erence in ( β -1) not larger than 3 %. This is consistent with similar B-O bond lengths (1 . 532 Å) found in both polytypes. Boromuscovite contains boron in tetrahedral sites. Therefore, in order to investigate the impact of the B-O bond length on the fractionation we first compare the fractionation between the mineral and a strongly basic fluid contain-</text>
<figure>
<location><page_14><loc_10><loc_68><loc_47><loc_85></location>
<caption>Figure 7: The fractionation factors between boromuscovite and basic aqueous fluid (fluid containing H4BO -4 ). The lines represent our results assuming the presence of boron species in form of H4BO -4 only (solid line) and mixture of 90% of H4BO -4 and 10% of H3BO3 (dotted line) in the fluid. The data points are the values measured by Wunder et al. (2005). The uncertainty of calculated values is indicated by shadowed area and is similar for both curves.</caption>
</figure>
<text><location><page_14><loc_11><loc_30><loc_48><loc_56></location>n in H4BO -4 . The result, together with the measurements of Wunder et al. (2005) of the fractionation between boromuscovite and strongly basic fluid, is summarized in Figure 7. Our calculations predict a negative fractionation between mica and the H4BO -4 fluid. The agreement of our prediction with the experimental measurements is relatively good; however, the experimental data indicate slightly stronger fractionation. We note that the experimental conditions of Wunder et al. (2005) do not assure that the measured basic fluid contained four-fold coordinated boron species, i.e. H4BO -4 , only. As we indicate in the Figure 7, the presence of as little as 10% of H3BO3 in the measured basic fluid brings the prediction and measurements into much better agreement. It makes sense that the β factor of boromuscovite is smaller than for aqueous H4BO -4 as the average B-O bond length in mica is 1 . 532 Å, while it is 1 . 513 Å and therefore shorter in case of aqueous H4BO -4 .</text>
<text><location><page_14><loc_11><loc_13><loc_48><loc_30></location>We note that in our derivation we assumed that the fluid consists mostly of [4] B species. As we already mentioned, although Wunder et al. (2005) call the fluid 'strongly basic' its exact composition, especially the amount of [3] B species is unknown. However, if for instance the [4] B to [3] B ratio was 1, then the predicted mica-basic fluid fractionation at 800 K would be 5 /permil larger than measured. This would result in a large inconsistency between the computed values and the experimental data. On the other hand, good agreement between the prediction and the measurements indicates that the strongly basic fluid was dominated by H4BO -4</text>
<figure>
<location><page_14><loc_51><loc_68><loc_89><loc_85></location>
<caption>Figure 8: The fractionation factors between boromuscovite and aqueous fluid. The data points are the values measured by Wunder et al. (2005). The solid line represents the result obtained for ambient pressure. The dashed line represents the result for fluid containing H3BO3 only obtained for P = 3GPa accounting for compression and thermal expansion, with the uncertainties in calculated values indicated by shadowed area. The dotted lines represent the results assuming different admixture of four-fold coordinated boron species (represented by H4BO -4 with abundance indicated in the figure) to the fluid. The computational error is comparable for all the results.</caption>
</figure>
<text><location><page_14><loc_52><loc_50><loc_89><loc_53></location>species, which is in line with previous studies (Zeebe, 2005; Sanchez-Valle et al., 2005).</text>
<section_header_level_1><location><page_14><loc_52><loc_47><loc_75><loc_48></location>3.3.3. Boromuscovite-neutral fluid</section_header_level_1>
<text><location><page_14><loc_52><loc_13><loc_89><loc_47></location>The fractionation between boromuscovite and neutral fluid involves a change in coordination from B [4] in boromuscovite to B [3] in neutral fluid. Wunder et al. (2005) measured the fractionation between the two materials at 3 GPa, shown in Figure 8. The predicted fractionation is about 3 ± 2 . 5 /permil larger than the measured value. Looking for potential sources of this discrepancy, we have checked for the e ff ect of the change in lattice parameters due to combined thermal expansion and compression. For that purpose we applied the EOS of Holland and Powell (2011) for muscovite, which gives 4 . 4%, 3 . 8% and 3 . 1% decrease in volume for T = 600 K, 800 K and 1000 K respectively and P = 3 GPa. As muscovites show highly anisotropic compressibility patterns, in line with Comodi & Zanazzi (1995) we applied the T and P driven change in volume assuming the 16%, 19% and 65% contribution to compression along the /vector a , /vector b , /vector c lattice vectors. On the other hand ( β -1) factors of H3BO3 and H4BO -4 in aqueous solutions at P = 3 GPa increase by 3 . 5% and 11 . 8% respectively (Figures 4 and 5), leading to pressure-induced increase in the boromuscovite-aqueous fluid fractionation at the experimental pressure. The ∆ factor, corrected for e ff ects of thermal expansion and compression of</text>
<text><location><page_15><loc_11><loc_74><loc_48><loc_86></location>boromuscovite and compression of fluid, is also given in Figure 8. Because the high T and P e ff ects result in similar increases in the β factors for both solid and aqueous fluid, the resulting fractionation factor between these two phases is close to the one derived at ambient conditions. Therefore, thermal expansion and compression e ff ects cannot explain the observed discrepancy between prediction and the measurements of Wunder et al. (2005).</text>
<text><location><page_15><loc_11><loc_56><loc_48><loc_74></location>On the other hand, the comparison of our results with the experimental data suggests that the fractionation between boromuscovite and fluid is the same as between H3BO3 and H4BO -4 fluids (see Figure 3), which is at odds with the non-negligible and negative fractionation between boromuscovite and a strongly basic fluid. In section 3.3.2 we have shown that we are able to correctly reproduce the fractionation between boromuscovite and strongly basic fluid, which indicates that our result for boromuscovite is reliable. This suggests that another, unaccounted e ff ect leads to the decrease of the boron isotope fractionation between mica and neutral fluid in the experiments of Wunder et al. (2005).</text>
<text><location><page_15><loc_11><loc_29><loc_48><loc_55></location>One possible solution for the discrepancy is a nonnegligible amount of boron residing in four-fold coordinated configurations in neutral solution. This is in line with the Raman spectroscopy measurements of Schmidt et al. (2005), who detected a broad peak in the Raman spectra of neutral H3BO3-dominated fluid and attributed it to B [4] species. The integrated area of this peak, compared to the peak of the Raman 877 cm -1 line of B [3] species, indicates the presence of at least 15 -30 % of B [4] species by mole fraction. Assuming that there is 15 -30 % of B [4] species present in the fluid and that the β factor of these species is similar to that of H4BO -4 , the fractionation factor between boromuscovite and H3BO3 aqueous fluid decreases bringing the theory and the experiment to better agreement, which is illustrated in Figure 8. If this interpretation is true, it suggests that boron isotope fractionation could be used to gather information on the speciation of B in aqueous fluids.</text>
<section_header_level_1><location><page_15><loc_11><loc_26><loc_42><loc_27></location>3.4. B isotope fractionation between minerals</section_header_level_1>
<text><location><page_15><loc_11><loc_13><loc_48><loc_25></location>The boron isotope fractionation between B-bearing crystalline solids has received considerable attention recently (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002). We focus here on the investigation of boron isotope fractionation between mica and tourmaline as boron atoms in these minerals occupy sites of di ff erent coordination, which should result in a large B isotope fractionation between these two minerals. In mi-</text>
<figure>
<location><page_15><loc_51><loc_60><loc_88><loc_85></location>
<caption>Figure 9: The fractionation factors between mica and tourmaline. The solid line represents our value for fractionation between Bmuscovite and dravite. The dashed line is the experimental fractionation factor between tourmaline and mica determined by Wunder et al. (2005) and Meyer et al. (2008). The experimental error is 2 /permil . The dotted line is the experimental fractionation factor of Wunder et al. (2005) and Meyer et al. (2008) but corrected for the presence of B [4] species in the neutral fluid in the high P experiments of Wunder et al. (2005), as is discussed in the text. The diamonds are the data from natural samples taken from Klemme et al. (2011) and references herein. The uncertainties in calculated values are indicated by shadowed area.</caption>
</figure>
<text><location><page_15><loc_52><loc_13><loc_89><loc_42></location>cas boron substitutes for silicon in the four-fold coordinated site (Wunder et al., 2005), while in tourmaline (dravite) it is incorporated in the three fold coordinated site (Meyer et al., 2008). Comparing the B isotope fractionation between di ff erent minerals, melts and fluids Wunder et al. (2005) have shown that the fractionation between two materials of di ff erent B coordination is large, reaching 5 /permil at 1000 K and much higher values at lower temperatures. Klemme et al. (2011),Marschall (2005) and Hervig et al. (2002) measured the fractionation between coexisting phases of the two minerals in natural samples. The fractionation between these two minerals is also derived from experimental isotopic fractionation data of B-muscovite-fluid (Wunder et al., 2005) and tourmaline-fluid (Meyer et al., 2008) systems. The results of these measurements and our computed T -dependent fractionation curve are given in Figure 9. The first striking observation is that our predicted fractionation factors are much larger (taking the absolute value) than the experimental values (Wunder et al., 2005; Meyer et al., 2008). The latter are also incon-</text>
<text><location><page_16><loc_11><loc_46><loc_48><loc_86></location>sistent with the natural samples data of Klemme et al. (2011) and previous studies discussed in that paper (Marschall, 2005; Hervig et al., 2002). On the other hand, the measurements on natural samples are consistent with our calculated values, which tends to validate our predictions. We notice that the most recent measurements of boron isotope signatures of tourmaline and white mica from the Broken Hill area in Australia by Klemme et al. (2011) indicate for the assumed temperature of 600 o C that the fractionation factor between the two phases is 10.4 ± 2 . 7 /permil , which is in good agreement with our computed value of 10 . 7 ± 1 . 8 /permil . The experimental mica-tourmaline B isotope fractionation factors of Wunder et al. (2005) and Meyer et al. (2008) are 2 /permil and 6 /permil smaller at temperatures of 1000 K and 800 K respectively, with an experimental uncertainty of 2 /permil . However, this discrepancy can be resolved by assuming that in the experiments of Wunder et al. (2005) the fluid contained a significant admixture of B [4] species, which leads to the underestimation of the experimental boromuscovite-fluid fractionation factor by ∼ 2 /permil at 1000 K and ∼ 3 . 5 /permil at 1000 K, as is seen in Figure 8. The experimental mica-tourmaline fractionation factor corrected for the presence of B [4] species is also plotted in Figure 9. It is now more consistent with the natural data. We note that this result independently supports the conclusion underlined in section 3.3.3 and result of Schmidt et al. (2005) that highP , B-bearing neutral fluids contain significant admixtures of B [4] species.</text>
<text><location><page_16><loc_11><loc_13><loc_48><loc_42></location>Olenite is a mineral which can incorporate boron in both trigonal and tetrahedral sites as it substitutes for both Al and Si atoms. It is therefore interesting to check the fractionation of boron isotopes between the two differently coordinated sites in one mineral and compare it with the above result for mica and tourmaline. The computed fractionation between the trigonal and tetrahedral sites at 600 o C is 10 . 6 ± 1 . 9 /permil , which is consistent with the fractionation between tourmaline and mica, indicating that the coordination of the B atom is the driving factor for the fractionation of the B isotopes. Similarly, we computed the boron isotopes fractionation between trigonal and tetragonal boron sites in dravite. In order to create the tetragonal B site we replaced one Si atom with B and we added one H atom forming an additional OH group to compensate the charge. The computed fractionation between the sites at 600 o C is 8 . 9 ± 1 . 7 /permil , which is also in agreement with the aforementioned results. Next, we will show that the value of the β factor depends not only on coordination but is also strongly correlated with B-O bond length.</text>
<figure>
<location><page_16><loc_52><loc_65><loc_89><loc_86></location>
<caption>Figure 10: β factors for various considered materials as a function of temperature. The materials are indicated on the right side. Their order reflects the value of the β factor at 1000 K from the largest (top) to the smallest (bottom).</caption>
</figure>
<section_header_level_1><location><page_16><loc_52><loc_56><loc_87><loc_57></location>3.5. Fractionation between B [3] and B [4] materials</section_header_level_1>
<text><location><page_16><loc_52><loc_29><loc_89><loc_55></location>The β factors computed for all the considered materials are grouped together in Figure 10. β factors can be grouped into two sets, one that includes materials with boron in three-fold coordination and another one that includes materials having boron in four-fold coordination. For olenite and dravite we also computed the β factors with boron sitting on four-fold coordinated site. The β factor for these crystalline solids with given B [3] / B [4] ratio can be derived as a weighted average of the β factors obtained for boron sitting on the two di ff erently coordinated sites. The fractionation factor between materials of di ff erent boron coordination is ∼ 8 /permil on average at T = 1000 K. We note that it is ∼ 3 /permil larger than the one deduced by Wunder et al. (2005) from measurements performed on solids, silicate melts and fluids, but this can be attributed to the underestimation of the fractionation factors for boromuscovite-fluid system by Wunder et al. (2005) due to potential admixture of B [4] species in the investigated fluid.</text>
<text><location><page_16><loc_52><loc_13><loc_89><loc_28></location>Our results show also a substantial spread of β factors of substances containing boron of a given coordination. The spread is at least 4 /permil and results from di ff erent B-O bond lengths. We illustrate this in Figure 11 by plotting together the β factors derived for all considered materials at T = 1000 K as a function of B-O bond length. It is clearly seen that there is a roughly linear correlation between the β factor and BO bond length, which is especially evident comparing the results for crystalline solids. For instance, out of the considered B [4] -bearing minerals boromuscovite has the</text>
<figure>
<location><page_17><loc_10><loc_60><loc_46><loc_86></location>
<caption>Figure 11: The β factor at T = 1000K for various considered materials as a function of B-O bond length. Filled circles represent the values obtained for boron in trigonal sites and open circles represent the values obtained for tetragonal sites.</caption>
</figure>
<text><location><page_17><loc_11><loc_31><loc_48><loc_51></location>longest B-O bond length of 1 . 525 Å (1M) and 1 . 516 Å (2M1), followed by dravite 1 . 514 Å and olenite with BObond length of 1 . 502 Å. This tracks the di ff erences in the β factors derived for these materials. In addition the materials having B [3] species only exhibit shorter bond lengths of ∼ 1 . 37 Å and higher β factors, while the materials containing B [4] species having bond lengths of about ∼ 1 . 52 Å show much smaller β factors. Therefore, the tighter bonding of B [3] species likely explains why the heavy B isotope prefers the less coordinated phases. This clearly shows that the change in the B-O bond length during an isotope exchange is the leading factor driving the production of the boron equilibrium isotope signatures at high T .</text>
<section_header_level_1><location><page_17><loc_11><loc_27><loc_22><loc_28></location>4. Conclusions</section_header_level_1>
<text><location><page_17><loc_11><loc_13><loc_48><loc_25></location>In this work we have presented a detailed analysis of boron isotope fractionation between boronbearing crystalline solids and aqueous fluids at high T and P conditions. In order to perform our investigation we have applied and extended a computationally e ffi cient approach for the computation of isotope fractionation factors for complex minerals and fluids at high temperatures and pressures presented by Kowalski & Jahn (2011). As an extension to the</text>
<text><location><page_17><loc_52><loc_74><loc_89><loc_86></location>Bigeleisen & Mayer (1947) 'single atom approximation' method we demonstrated that using the pseudofrequencies derived from the force constants acting on the fractionating element together with the full formula for computation of the reduced partition function ratios results in significant improvement in the accuracy of the computed fractionation factors, which is essential when lower temperature materials and high vibrational frequency complexes are considered.</text>
<text><location><page_17><loc_52><loc_27><loc_89><loc_72></location>In order to understand the fractionation between Bbearing crystalline solids and aqueous fluids we performed a set of calculations of β factors for dravite, olenite, boromuscovite and aqueous solutions of H3BO3 and H4BO -4 . In agreement with the experimental findings we show that the fractionation strongly correlates with coordination through the change in the B-O bond length. The lower trigonal coordination BO3 arrangement results in higher 11 B / 10 B (by ∼ 8 /permil at T = 1000 K) than the tetrahedrally coordinated boron complexes, which exhibit ∼ 0 . 15 Å longer B-O bonds. The computed fractionation between minerals and fluids of the same coordination are in good agreement with experiments. However, we predict larger isotope fractionation between boromuscovite and H3BO3 fluid (by at least a few /permil ) than was measured in situ at high P by Wunder et al. (2005) and Meyer et al. (2008), but that is consistent with measurements on natural samples. We note that the presence of B [4] in highP fluid could reconcile the in situ experimental results with our prediction and other measurements. This is expected from the experiments of Schmidt et al. (2005), but requires further experimental confirmation. If true, this would open the possibility for using the isotope fractionation techniques as a tool to measure the speciation of boron in fluids and crystalline solids. We have also demonstrated that with our computational approach we are able to correctly predict the pressure-induced isotope fractionation for compressed aqueous fluids, which indicates the ability of ab initio methods to predict the isotopic signatures of highly compressed materials, even those that are difficult to investigate experimentally.</text>
<text><location><page_17><loc_52><loc_13><loc_89><loc_25></location>Our study confirms that ab initio computer simulations are a useful tool not only for prediction but also understanding the equilibrium stable isotope fractionation processes between various phases, including aqueous solutions, at high pressures and temperatures. They can nicely complement experimental e ff orts, provide unique insight into the isotope fractionation process on the atomic scale and deliver data for conditions that are inaccessible by the current experimental techniques.</text>
<section_header_level_1><location><page_18><loc_11><loc_85><loc_25><loc_86></location>Acknowledgements</section_header_level_1>
<text><location><page_18><loc_11><loc_73><loc_48><loc_84></location>The authors wish to acknowledge financial support in the framework of DFG project no. JA 1469 / 4-1. Part of the calculations were performed on the IBM BlueGene / P JUGENE of the John von Neumann Institute for Computing (NIC). We are also grateful the associate editor Edwin A. Schauble and anonymous referees for constructive comments that helped improving the manuscript.</text>
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<text><location><page_19><loc_11><loc_39><loc_48><loc_45></location>Sanchez-Valle C., Reynard B., Daniel I., Lecuyer C., Martinez I., and Chervin J. C. (2005) Boron isotopic fractionation between minerals and fluids; new insights from in situ high pressure-high temperature vibrational spectroscopic data Geochim. Cosmochim. Acta 69 , 4301-4313</text>
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<text><location><page_19><loc_52><loc_73><loc_89><loc_77></location>Yamaji K., Makita Y., Watanabe H., Sonoda A., Kanoh H., Hirotsu T. and Ooi K. (2001) Theoretical Estimation of Lithium Isotopic Reduced Partition Function Ratio for Lithium Ions in Aqueous Solution. J. Phys. Chem. A 105 , 602613.</text>
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</document>
[
{
"title": "Ab initio prediction of equilibrium boron isotope fractionation between minerals and aqueous fluids at high P and T",
"content": "Piotr M. Kowalski 1 , 2 ∗ , Bernd Wunder 1 and Sandro Jahn 1 1 GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany 2 Forschungszentrum Julich, Institute of Energy and Climate Research (IEK-6), Wilhelm-Johnen-Strasse, 52425 Julich, Germany",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "Over the last decade experimental studies have shown a large B isotope fractionation between materials carrying boron incorporated in trigonally and tetrahedrally coordinated sites, but the mechanisms responsible for producing the observed isotopic signatures are poorly known. In order to understand the boron isotope fractionation processes and to obtain a better interpretation of the experimental data and isotopic signatures observed in natural samples, we use first principles calculations based on density functional theory in conjunction with ab initio molecular dynamics and a new pseudofrequency analysis method to investigate the B isotope fractionation between B-bearing minerals (such as tourmaline and micas) and aqueous fluids containing H3BO3 and H4BO -4 species. We confirm the experimental finding that the isotope fractionation is mainly driven by the coordination of the fractionating boron atoms and have found in addition that the strength of the produced isotopic signature is strongly correlated with the B-O bond length. We also demonstrate the ability of our computational scheme to predict the isotopic signatures of fluids at extreme pressures by showing the consistency of computed pressure-dependent β factors with the measured pressure shifts of the B-O vibrational frequencies of H3BO3 and H4BO -4 in aqueous fluid. The comparison of the predicted with measured fractionation factors between boromuscovite and neutral fluid confirms the existence of the admixture of tetrahedral boron species in neutral fluid at high P and T found experimentally, which also explains the inconsistency between the various measurements on the tourmaline-mica system reported in the literature. Our investigation shows that the calculated equilibrium isotope fractionation factors have an accuracy comparable to the experiments and give unique and valuable insight into the processes governing the isotope fractionation mechanisms on the atomic scale. Keywords: stable isotope fractionation, boron, fluid speciation, DFT, molecular dynamics",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The use of isotopes as geochemical tracers depends upon the existence of reliable P -T -dependent equilibrium isotope fractionation data between solids, fluids and melts. The common method used for determination of such data is to conduct experiments, where the phases of interest are equilibrated at a range of di ff erent conditions and individually measured for their equilibrium isotopic compositions. Recently, with the development of computational methods, software and increase in hardware performance it is also possible to simulate and compute the isotope fractionation with ab initio methods. These computational considerations complement the experimental e ff ort and provide information on the mechanisms governing the equilibrium isotope fractionation processes on the atomic scale. By establishing an e ffi cient computational approach for materials at high P and T and testing its reliability by computing Li isotope fractionation between minerals and aqueous fluids, Kowalski & Jahn (2011) have shown that ab initio methods can provide a reliable estimation of equilibrium isotope fractionation factors at an accuracy level comparable to experiments. Motivated by the encouraging results for the fractionation of lithium isotopes we applied our new method to study boron isotope fractionation. The main goals of the present study are to make theoretical predictions and obtain a better understanding of the B isotope fractionation process between tourmaline, B-bearing mica and flu- ids at various pressures and temperatures, to compare these data to results from recent in situ experimental studies (Wunder et al., 2005; Meyer et al., 2008) and measurements of isotopic signatures in natural samples (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002) and to investigate the underlying mechanisms driving the boron isotope fractionation processes between the considered materials. With two stable isotopes 10 B and 11 B, of relatively large mass di ff erence of about 10 %, boron isotopes strongly fractionate during geological processes, thus leading to natural δ 11 B-variations ranging from -30 to + 60 /permil (Barth, 1993). Therefore, B isotopes are ideal for the distinction of di ff erent geological environments and for quantifying mass transfer processes, e.g. in the range of subduction zones. First-order criteria driving isotope fractionation in Earth materials are di ff erences in coordination and in the bonding environments of coexisting phases. The lighter isotope usually preferentially occupies the higher coordinated site, which is generally accompanied with a longer cation-anion bond length and weaker bond strength (Schauble et al., 2009; Wunder et al., 2011). Tourmaline has an extensive chemical variability and is the most widespread borosilicate in various rocks over a large range of bulk compositions. It has a large P -T -stability which ranges from surface conditions to high pressures and temperatures of at least 7.0 GPa and about 1000 o C as determined experimentally for dravitic tourmaline (Krosse, 1995). Most tourmaline minerals contain 3 boron atoms per formula unit (pfu) with B in trigonal planar coordination (B [3] ). High pressure combined with an Al-rich environment can lead to the formation of olenitic tourmaline with significant amounts of excess B substituting for Si at the tetrahedral site (B [4] ). The highest amounts of up to 1.2 B [4] pfu have been found in olenitic tourmaline from Koralpe, Austria (Ertl et al., 1997). Al-rich tourmaline with up to 2.2 B [4] pfu was synthesized experimentally at 2 . 5 GPa, 600 o C (Schreyer et al., 2000). The maximum possible amount of B [4] in olenitic tourmaline is limited to three B [4] pfu, due to structural and crystal chemical reasons. As tourmaline is not stable at basic pH (Morgan & London, 1989), the dominant B-species of fluids coexisting with tourmaline is B(OH)3. Therefore, due to the absence of change in boron coordination, B-isotopic fractionation between B [4] -free tourmaline and fluid ( ∆ 11 B(tour-fl)) should be small. However, at 0 . 2 GPa, the experimentally determined ∆ 11 B(tour-fl)values of -2 . 5 ± 0 . 4 /permil at 400 o C and -0 . 4 ± 0 . 4 /permil at 700 o C (Meyer et al., 2008) suggest small but significant di ff erences in the B-O(H) bond strength between tourmaline and the neutral fluids. Incorporation of B [4] into tourmaline is expected to significantly increase the fractionation of boron isotopes between the mineral and aqueous fluid (i.e. increase of | ∆ 11 B | (tour-fl)). In this contribution we present calculated P -T -dependent data on B-isotope fractionation between B [4] -bearing tourmaline and fluid, which so far are not available from experiments. Despite of its low boron content of up to maximum values of 270 ppm (Domanik et al., 1993), potassic white mica is probably the main host of boron in metasedimentary and metabasaltic blueschists and eclogites, because of its high modal abundance and B-incompatibility in all other stable minerals of these rocks (Brenan et al., 1998). During subduction the modal amount of mica continuously decreases by dehydration reactions and the chemistry of residual micas is shifted towards phengitic compositions. Phengitic mica has an extended stability, extending as deep as 300 km within cold subduction zones (Schmidt, 1996). In contrast to most tourmalines, boron is tetrahedrally coordinated in mica, where it substitutes for aluminum. Due to the coordination change from mostly three-fold coordinated in near-neutral fluids (Schmidt et al., 2005) to B [4] in mica, B-isotopic fractionation between Bbearing mica and such fluids is much larger than for tourmaline - fluid. ∆ 11 B(mica-fl)-values determined experimentally at P = 3 . 0 GPa, are -10 . 9 ± 1 . 3 /permil at 500 o C and -7 . 1 ± 0 . 5 /permil at 700 o C (Wunder et al., 2005). Such a strong B-isotope fractionation and its pronounced T -dependence, in combination with the continuous dehydration of micas during ongoing subduction and boron transport via fluids into mantle wedge regions of arc magma-formation, probably determines the boron variations and B-isotopic signatures in volcanic arcs (Wunder et al., 2005). Using in situ Raman spectroscopic measurements of near-neutral B-bearing fluids, Schmidt et al. (2005) observed a significant amount of 4-fold B-species at high P and T . The abundance of these species increases with temperature and pressure and at T = 800 K and P = 1 . 9 GPa there should be a considerable amount (15 -30%) of B [4] species in the fluid. Such a significant amount of tetrahedrally coordinated B-species in high temperature and pressure fluids should a ff ect solid-fluid B-isotopic fractionation, which we investigate in our calculations. In the light of this, we also discuss recently determined B-isotope data from coexisting natural tourmaline and B-bearing mica (Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002), which show slight inconsistency with in situ measurements of Wunder et al. (2005) and Meyer et al. (2008). Reliable ab initio computational methods to predict isotope fractionation factors have been established recently. Several groups have proved that such calculations can contribute towards understanding geochemical mechanisms responsible for production of isotope signatures (Driesner, 1997; Yamaji et al., 2001; Schauble, 2004; Domagal-Goldman et al., 2008; Hill & Schauble, 2008; Meheut et al., 2009, 2007; Schauble et al., 2009; Zeebe, 2009; Hill et al., 2010; Rustad et al., 2010a; Rustad et al, 2010b; Zeebe, 2009, 2010; Kowalski & Jahn, 2011). The majority of these works, however, concentrate on the computation of the stable isotope fractionation between various, mostly simple crystalline minerals, and the aqueous solutions are usually computed using an isolated cluster containing fractionating species and a hydration shell (e.g. Zeebe (2010, 2009); Hill et al. (2010); Rustad et al. (2010a); Domagal-Goldman et al. (2008); Hill & Schauble (2008); Rustad et al (2010b); Zeebe (2005)). However, aqueous solutions at high pressure and temperature must be computed with caution as discussed in Kowalski & Jahn (2011). This is because the distribution of cation coordination and cation-oxygen bond lengths that a ff ect the isotope fractionation (Bigeleisen & Mayer, 1947) is driven by the dynamics of the system and change under compression (Jahn & Wunder, 2009; Wunder et al., 2011; Kowalski & Jahn, 2011). The only recent ab initio work, besides Kowalski & Jahn (2011), that accounts for the dynamical e ff ects on the isotope fractionation in fluid is by Rustad & Bylaska (2007) who considered boron equilibrium isotope fractionation between B(OH)3 and B(OH) -4 species in aqueous solution. They performed ab initio molecular dynamics simulations of these fluids and attempted to use the vibrational density of states, derived through the Fourier transform of the velocity auto-correlation function, as an input for the calculation of the 11 B / 10 B isotope fractionation coe ffi -cient. However, the resulting fractionation factor α = 0 . 86 happened to be much lower than the experimental value of α = 1 . 028. Interestingly, the discrepancy between experiment and theory is cured by quenching the selected configurations along the molecular dynamics trajectory and computing the harmonic frequencies. The fractionation factor derived using these frequencies exactly reproduces the experimental value. In our approach both solids and fluids are treated as extended systems by application of periodic boundary conditions in all three spatial directions, which is crucial to model high pressure materials. Large enough supercells are chosen to avoid significant interaction between atoms and their periodic images, as well as to reduce the number of k-points and q-vectors for sampling the Brillouin zones and the phonon spectra of the crystals, respectively (for a liquid or fluid, both the Brillouin zone and phonons are not defined). In our investigation we use cells of at least 7 Å width in each spatial dimension. A representative statistical sampling of the fluid structure is obtained by performing Car-Parrinello molecular dynamics simulations (Car & Parrinello, 1985). For the calculation of the isotope fractionation factors in fluids, several random snapshots from the simulation runs are chosen. The force constants acting on the fractionating element and the resulting fractionation factors are then obtained for each ionic configuration, and the relevant fractionation factor for the boron species in the fluid is computed as an average over the whole set of considered geometries. In Kowalski & Jahn (2011) it was shown that in line with the Bigeleisen & Mayer (1947) approximation, considering the force constants acting on the fractionating atom only leads to a satisfactory estimation of the Li isotope fractionation factors for high temperature fluids and minerals. Here, we will show that approximating the vibrational spectrum by the three pseudofrequencies derived from the force constants allows for further improvement of the accuracy of the predicted isotope fractionation factors, especially at lower temperatures. In this contribution we present the theoretical prediction of B isotope fractionation factors between B bearing aqueous fluids and solids, specifically tourmaline and B-muscovite, for which experimental data and measurements on natural samples are available for comparison (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011). We will show that the application of ab initio methods to B-bearing crystalline solids and fluids not only provides unique insight into the mechanisms driving equilibrium B-isotope fractionation on the atomic scale, but helps in proper interpretation of the data.",
"pages": [
1,
2,
3
]
},
{
"title": "2.1. Theoretical model",
"content": "2.1.1. The single atom approximation: & Bigeleisen Mayer (1947) approach Mass-dependent equilibrium isotope fractionation is driven by the change in molecular and crystalline vibration frequencies resulting from the di ff erent masses of the isotopes. The fractionation between species and an ideal monoatomic gas is called the β factor. In the harmonic approximation it is given by the formula References: 1 Zeebe (2005), 2 Liu & Tossell (2005), 3 Rustad et al (2010b), 4 Sanchez-Valle et al. (2005), 5 Gilson (1991), 6 Oi (2000); Liu & Tossell (2005), 7 Andrews & Burkholder (1992), 8 Ogden & Young (1988), 9 Zeebe (2005) and references hereafter. (Bigeleisen & Mayer, 1947; Urey, 1947; Chacko et al., 2001): where u = h ν i / kBT , h is the Planck constant, ν i is the vibrational frequency of the i -th degree of freedom, kB is the Boltzmann constant, Ndof is the number of degrees of freedom, which for N being the number of atoms in the considered system (molecule, mineral or fluid) is equal to 3 N -5 for a diatomic molecule, 3 N -6 for multiatomic molecules and 3 N for crystals, and a star symbol marks the heavier isotope. The fractionation factor between two substances A and B, α A -B is computed as the ratio of the relevant β factors, which is well approximated by the di ff erences in the β factors: The calculation of the β factor requires only knowledge of the vibrational properties of the considered system computed for the two di ff erent isotopes. However, computation of the whole vibrational spectra of complex, multiparticle minerals or fluids requires substantial computational resources and is currently limited to systems containing a few dozens of atoms or less. In our recent work (Kowalski & Jahn, 2011) we proposed to use an e ffi cient method for computing the high temperature isotope fractionation factors between complex materials such as fluids and crystalline solids, which requires the knowledge of the force constants acting upon the fractionating element only. The β factor (Eq. 1) can be then approximated by (Bigeleisen & Mayer, 1947; Kowalski & Jahn, 2011): where Ai are the force constants acting on the isotopic atom in the three perpendicular spatial directions (x, y and z), ∆ m = m ∗ -m , where m and m ∗ are the masses of the lighter and heavier isotopes of the fractionating element. As the computation of the β factors from formula 3 requires the knowledge of properties of the fractionating element only we will call such an approach the single atom approximation throughout the paper. The validity criteria restricts the usage of the formula to frequencies ν [cm -1 ] /lessorsimilar 1 . 39 T [K] (assuming u < 2, see Fig. 1 of Bigeleisen & Mayer (1947)). We are interested in temperature range 800-1000 K. The highest vibrational frequency of the modes involving movement of B atoms for H3BO3 is ∼ 1400 cm -1 and of H4BO -1 4 is < 1200 cm -1 (Table 1). In case of H3BO3, the single atom approximation may produce an error of 2 . 2 /permil in the β factor at T = 800 K. The relevant error for H4BO -1 4 is 0 . 8 /permil (Table 2). A further improvement to the method is therefore desired.",
"pages": [
3,
4,
5
]
},
{
"title": "2.1.2. The single atom approximation with pseudofrequencies: our improvement",
"content": "We will show that the error of the single atom approximation can be substantially reduced if one uses the three frequencies ¯ ν i derived from the force constants acting on the fractionating element (¯ ν 2 i = Ai / 4 π 2 m ). We call them 'pseudofrequencies' , and compute the β factors using formula 1. In the following we present the formal justification of such an approach. According to Bigeleisen & Mayer (1947), equation 1 for small ∆ ui = ui -u ∗ i reduces to (Bigeleisen & Mayer (1947), Eq. 11a): The Taylor expansion of the function appearing under the summation sign is: When we consider just the first term of the expansion the β factor is: which is exactly equation 3. Let us consider the Taylor expansion of the di ff erent estimations of β factors. Equation 4 then reads: The Bigeleisen & Mayer (1947) approximation given by equation 3 reads: and the proposed approximation based on pseudofrequencies is: In the above equation ¯ ui = h ¯ ν i / kBT . Next we check how this approximation compares to the Bigeleisen & Mayer (1947) approximation given by equations 3 and 8. In order to make a comparison we derive the di ff erences between the two approximate expressions β BM , β pseudo and the exact one β exact (Eq. 7). In the case of the Bigeleisen & Mayer (1947) approximation we have: and having from equations 3 and 8 that in the case of the proposed approximation we get: Because relation G ( u ) < u 12 holds for any u (see Bigeleisen & Mayer (1947), Fig. 1), the function and ∆ β pseudo < ∆ β BM . On the other hand the expression for ∆ β pseudo is given by a di ff erence of the higher order terms of the Taylor expansions of the two expressions for the β factor. In the considered cases the values of pseudofrequencies are similar to the real frequencies that are a ff ected upon B isotope substitution in a given B-bearing system. This can be seen by comparing the pseudofrequencies computed for the selected cases of B-bearing molecules and crystalline solids considered here and reported in Table 3 with the real frequencies given in Table 1. This indicates that the two terms of opposite signs in Eq. 12 should be similar in value and cancel out to a great extent, so | ∆ β pseudo | << ∆ β BM . Therefore the approach proposed here to compute the β factor based on pseudofrequencies and Eq. 1 should give a better approximation to the exact β factors than equation 3, which we will show in section 3.1. The difference to the Bigeleisen & Mayer (1947) approxima- References: 1 Liang et al. (1995), 2 Ertl et al. (2010), 3 Marler et al. (2002) tion is given by: and can be easily computed for any considered system. We assume that the pseudofrequency-based approach to the computation of β factors is applicable if it is just a correction to equation 3, i.e. when: We also note that the proposed approach satisfies the Redlich-Teller product rule (Redlich, 1935) when the mass of considered isotope m is much smaller than the mass of the whole considered system M , namely, We notice that we could force the strict conservation of the Redlich-Teller product by just adjusting the ratios of ¯ ui / ¯ u ∗ i . However, such a modification would not preserve relation 11 and the pseudofrequency approach would not recover exactly the high temperature limit (Eq. 3) of the exact solution (Eq. 1), which is a more important constraint to fulfill strictly by the proposed approximation.",
"pages": [
5,
6,
7
]
},
{
"title": "2.2. Representation of solids",
"content": "In this paper we investigate the boron isotope fractionation between dravite, olenite and boromuscovite minerals and aqueous fluids. The solids were represented by large cells containing at least 84 atoms. The number of atoms used in the crystal calculations together with the lattice parameters of modeled crystals are summarized in Table 4. The lattice parameters and chemical compositions of the modeled crystalline solids are the experimental values measured at ambient conditions found in the literature. Dravite is the crystalline solid which was used in the experiments on tourmaline by Meyer et al. (2008). The chemical composition of the supercell used in the investigation is Na3Mg9Al18(Si18O54)(B [3] O3)9(OH)12 with structural data of Marler et al. (2002). Olenite can contain B in both trigonal and tetragonal sites. The modeled structure is that of Ertl et al. (2010). The chemical composition of the unit cell used in the investigation is NaAl3Al6(Si4B [4] 2 O18)(B [3] O3)3(OH)3O. For boromuscovite, the 1M and 2M1 crystal structures of Liang et al. (1995) were used. In the isotope fractionation experiments of Wunder et al. (2005) boromuscovite forms two polytypes, 1M and 2M1, with relative abundances of 10% and 90% respectively. In boromuscovite B occupies the 4-fold coordinated site occupied mainly by Si atoms. The constructed model constitutes a 2x1x1 supercell of elementary chemical composition KAl2(B [4] Si3O10)(OH)2.",
"pages": [
7
]
},
{
"title": "2.3. Representation of aqueous solution",
"content": "The aqueous solution was represented by a periodically repeated box containing up to 64 water molecules and one H3BO3 or H4BO -4 molecule. The pressure and temperature conditions were chosen to be close to the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The pressure of the aqueous solution for a given temperature and volume was calculated according to the equation of state of Wagner & Pruss (2002). The ab initio molecular dynamics simulations (AIMD) of aqueous fluids were performed for fixed temperature and volume using the Car-Parrinello scheme (Car & Parrinello, 1985). The temperature during each run was controlled by a Nos'e-Hoover chain thermostat (Nos'e & Klein, 1983; Hoover, 1985). For each T -V conditions at least 10 ps long trajectories were generated with an integration step of 0 . 12 fs.",
"pages": [
7
]
},
{
"title": "2.4. Computational technique",
"content": "The calculations of pseudofrequencies and β factors for solids and aqueous solutions were performed by applying density functional theory (DFT) methods, which are currently the most e ffi cient methods allowing for treating extended many particle systems quantummechanically. We used the planewave DFT code CPMD (Marx & Hutter, 2000), which is especially suited for ab initio simulations of fluids, the BLYP exchangecorrelation functional (Becke, 1988; Lee et al., 1988) and norm-conserving Goedecker pseudopotentials for the description of the core electrons (Goedecker et al., 1996). One advantage of using the BLYP functional is that it usually gives harmonic frequencies that most closely resemble the observed frequencies of benchmark chemical systems (Finley & Stephens, 1995; Alecu et al., 2010) 1 . The energy cut-o ff for the plane wave basis set was 70 Ryd for geometry relaxations and molecular dynamics simulations and 140 Ryd for computation of vibrational frequencies. Periodic boundary conditions were applied for both crystalline solids and aqueous solutions to preserve the continuity of the media. The force constants and frequencies needed for the computation of the β factors were computed using the finite displacement scheme. Before performing the calculations of the crystal structures all atomic positions were relaxed to the equilibrium positions to minimize the forces acting on the atoms. We note that to compute the β factors for crystals one formally should account for phonon dispersion. Here we use large supercells and restrict our calculations to a single phonon wave-vector ( Γ ). Schauble (2011) has shown recently for 26 Mg / 24 Mg fractionation in Mg-bearing minerals that supercells containing more than 20 atoms are su ffi -cient to get very accurate β factors even at T = 300 K (error of 0.1 /permil ). At T = 1000 K the error is in the order of 0.01 /permil . The accuracy of the high temperature isotope fractionation factors computed on a single phonon wave-vector is also demonstrated for ironbearing minerals by Blanchard et al. (2009) and confirmed with good agreement of the predicted with the measured Li isotope fractionation factors between staurolite, spodumene, micas and aqueous fluid presented in our previous work (Kowalski & Jahn, 2011). Prior to the computation of the force constants and frequencies of boron atoms in the fluids the positions of all the atoms constituting the boron-carrying molecule (H3BO3 or H4BO -4 ) were relaxed to the equilibrium positions, while all other atomic positions remained unchanged. The full normal mode analyzes were per- formed using the same method, but displacing all the atoms constituting the considered system. In the latter case the frequencies were obtained through the diagonalization of the full dynamical matrix (Schauble, 2004) as implemented in CPMD code. The e ff ect of the various approximations on the derived fractionation factors was studied by additional computations of H3BO3 and H4BO -4 isolated clusters. For that purpose we used a large, isolated simulation box with a cell length of 16 Å, forcing the charge density to be zero at the boundary, as implemented in CPMD code. In order to compute the β factors of boron species in the aqueous fluid we apply the same method as in our recent work on Li isotopes (Kowalski & Jahn, 2011), with the exception that we use the pseudofrequencies, i.e. the frequencies obtained from the three force constants acting on the fractionating element, and formula 1 for calculation of β factors, as discussed in section 2.1.2. In order to fully account for the spatial continuity of the fluid and its dynamical motion we produced 10 ps long molecular dynamics trajectories of systems consisting of 64 H2O molecules and one H3BO3 or H4BO -4 molecule for different T = 1000 K, 800 K and 600 K and pressure of 0 . 5 GPa, which closely resembles the experimental conditions of Wunder et al. (2005) and Meyer et al. (2008). The corresponding simulation box length is 13 . 75 Å at T = 1000 K. The β factors were computed on the ionic configuration snapshots extracted uniformly in 0 . 1 ps intervals along the molecular dynamics trajectories.",
"pages": [
8,
9
]
},
{
"title": "2.5. Error estimation technique",
"content": "The errors in the computed value of the ( β -1) and ∆ fractionation factors were estimated from an average error of vibrational frequencies computed using the chosen DFT method. Finley & Stephens (1995), Menconi & Tozer (2002) and Alecu et al. (2010) estimated the errors made in calculations of vibrational frequencies of small molecules using di ff erent DFT functionals. According to these works the BLYP functional systematically overestimates the harmonic frequencies by ∼ 3 . 5 %, with a deviation from the mean o ff set of ∼ 1 %. Therefore, we expect that using BLYP functional the ( β -1) and ∆ values are systematically overestimated by 7 % and that in addition there is a 2 % error in derived ( β -1) factors. Similar errors result from using other functionals or even more sophisticated and time consuming post-Hartree-Fock methods such as MP2 (Finley & Stephens, 1995; Alecu et al., 2010).",
"pages": [
9,
10
]
},
{
"title": "3.1. Test of the computational method",
"content": "First, we illustrate the performance of the approximation proposed in section 2.1.2 by computing the β factors for the isolated H3BO3 and H4BO -4 molecules and selected crystalline solids. In Figure 1 we present three sets of calculations of β factors: (1) the 'exact' result obtained from a full normal mode analysis and formula 1, (2) the results obtained applying Kowalski & Jahn (2011) method based on Eq. 3, (3) the results obtained using pseudofrequencies computed for the fractionating element and Eq. 1 for the estimation of the β factor. The numerical values for selected temperatures are reported in Table 2. Approach (3) results in much better agreement with the 'exact' result. For H3BO3, the β factor is overestimated by only 0 . 5 /permil and 1 . 5 /permil for temperatures of 800 K and 600 K respectively. Applying method (2), the error is more pronounced, 2 . 2 /permil and 6 . 7 /permil respectively. In the case of molecular H4BO -4 , the errors using method (3) for the same temperatures are only 0 . 1 /permil and 0 . 5 /permil respectively. The same behavior is shown for dravite and boromuscovite crystalline solids that contain boron in the coordinative arrangement that resemble the configurations of aforementioned B-bearing molecules. For T > 600 K the proposed method represents only a few percent correction to the approximation given by equation (3), so the relation (15) is satisfied. It is evident that for B-bearing materials considered here the improvement made by using the pseudofrequencies based approach is substantial. It corrects for about 75% of error of the Bigeleisen & Mayer (1947) approximation (Eq. 3). However, the question of general applicability of the proposed method to other isotopic systems would require careful testing on a large set of materials, which is well beyond the scope of the current paper.",
"pages": [
10
]
},
{
"title": "3.2. B isotope fractionation in gas and fluid phases 3.2.1. B isotope fractionation between H3BO3 and H4BO -4 in the gas phase",
"content": "In a first step of our investigation of boron-rich aqueous fluids we derived the full frequency spectra of molecules in the gas phase (isolated molecules). The relevant β factors were computed using Equation 1. These studies were performed in order to compare our results with the published values of Zeebe (2005), both computed using the same DFT BLYP functional. In Table 1 we report the computed frequencies that are affected by the di ff erent B isotope substitutions along with other theoretical estimations and experimental measurements. The computed frequencies are in good agreement with earlier theoretical predictions and show similar agreement with the experimental measurements. The results in terms of computed β factors for the two considered species are reported in Figure 2, where we compare our results with the values computed using frequencies of Zeebe (2005). The comparison of the two sets of calculations reveals that our β factors for both species are smaller by ∼ 1 /permil at 600 K -1000 K than the values of Zeebe (2005). However, the di ff erence between β factors of H3BO3 and H4BO -4 remains nearly identical in both sets of calculations and the agreement is nearly perfect for higher temperatures. We note, that for the comparison we used the frequencies of Zeebe (2005) computed using 6-31 + G(d) basis set, as only these are provided by the authors. β and α factors obtained at T = 300 K using a more extended 6311 + G(d,p) basis set indicate that the β factors using 6-31 + G(d) basis set are not fully converged. In particular, the ( β -1) factor of H4BO -4 computed with 6311 + G(d,p) basis set is 3 . 9 /permil smaller than the one derived using 6-31 + G(d). In Table 5, we compare these results with the results of our calculation. It is clearly seen that for lower temperatures such as T = 300 K the values computed with 6-311 + G(d,p) basis set are in better agreement with our results indicating that planewave based DFT approach we use provides adequate vibrational frequencies and resulting isotope fractionation factors.",
"pages": [
10
]
},
{
"title": "3.2.2. B isotope fractionation between H3BO3 and H4BO -4 in aqueous fluid",
"content": "In order to obtain the temperature dependent β factor for aqueous fluids we fitted the function 1 + A / T 2 + B / T 4 to the computed values using the least squares minimization procedure. The computed β values for H3BO3 in fluid are: 1 . 02366 ± 0 . 00012, 1 . 03624 ± 0 . 00018 and 1 . 06262 ± 0 . 00010and for H4BO -4 in fluid are: 1 . 01745 ± 0 . 00005, 1 . 02650 ± 0 . 00010and 1 . 04597 ± 0 . 00015, for the temperatures of 1000 K, 800 K and 600 K respectively. The resulting temperature dependent β factor for H3BO3 is β = 1 + 2 . 416 · 10 4 / T 2 -5 . 823 · 10 8 / T 4 and for H4BO -4 is β = 1 + 1 . 772 · 10 4 / T 2 -4 . 234 · 10 8 / T 4 . The results for H3BO3 and H4BO -4 in aqueous so- lutions are shown in Figure 3. As was observed for the isolated molecules, the β factor of H3BO3-bearing fluid is substantially larger than the one for the H4BO -4 . This can be understood in terms of the substantial difference in the B-O bond lengths exhibited by the two considered species. In case of isolated molecules our calculations indicate a B-O bond length of 1 . 40 Å for H3BO3 and 1 . 51 Å for H 4BO -4 . We compared our β factors with the values computed by Sanchez-Valle et al. (2005), which were derived by the combination of force field methods and experimental data to derive accurate vibrational frequencies. For H3BO3 we got a nearly identical result. In case of H4BO -4 our calculation predicts a value which is lower by 2 -4 /permil . However, Rustad et al (2010b) and Rustad & Bylaska (2007) revealed the improper assignment of a major fractionating vibrational mode of H4BO -4 in the force field by Sanchez-Valle et al. (2005). This leads to the underestimation of the fractionation factor between aqueous H3BO3 and H4BO -4 by Sanchez-Valle et al. (2005). Assuming that α ∝ T -2 and having the di ff erence between BLYP calculations of Rustad et al (2010b) and Sanchez-Valle et al. (2005) of ∆ α = 16 . 4 /permil at T = 300 K, the value reported by Sanchez-Valle et al. (2005) should be underestimated by ∆ α = 16 . 4 · 2(300 / T ) 2 /permil , which results in ∆ α ∼ 1 . 5 /permil at T = 1000 K. Corrected in such as way result of Sanchez-Valle et al. (2005) is also plotted in Figure 3. It is now very consistent with our prediction.",
"pages": [
10,
11
]
},
{
"title": "3.2.3. Discussion of computational errors",
"content": "Most previous computational studies of boron isotope fractionation in aqueous solutions concentrate on the computation of the isotope fractionation at ambient conditions (Rustad et al, 2010b; Rustad & Bylaska, 2007; Liu & Tossell, 2005; Zeebe, 2005). Rustad et al (2010b) performed detailed analysis of impact of the chosen computational method (HF, MP2, di ff erent DFT functionals) and size of the basis set on the calculated fractionation factors between H3BO3 and H4BO -4 . They found that DFT methods are not performing well for the borate system and concluded that DFT 'is of limited usefulness in chemically accurate predictions of isotope fractionation in aqueous systems' (Rustad et al, 2010b). The empirically derived error of the derived fractionation factor is of the order of 5 -10 /permil for a total fractionation of ∼ 30 /permil . We note that this is expected and clearly visible if we apply the error estimation procedure outlined in section 2.5. For instance, at room temperature the derived beta factors using the BLYP functional are 213 . 6 /permil and 173 . 3 /permil respectively (Rustad et al (2010b), Table 2). This gives a fractionation factor of 1.0343. Following our error estimation scheme, the absolute error of the fractionation factor is 10 . 5 /permil , and the properly reported computed value is α = 1 . 034 ± 0 . 011. When one corrects for the systematic error of 7% and assumes 2% of statistical error on β factors, then the value of α decreases and the error is slightly smaller, i.e. α = 1 . 032 ± 0 . 008. This is in good agreement with the experimental data reported in Rustad et al (2010b) and explains the spread of the values computed using di ff erent methods and reported in that paper. It is very di ffi cult to get the fractionation factors for ambient conditions, as the fractionation factor is often just a small fraction of the relevant ( β -1) factors, ( α -1) ∼ 0 . 15( β -1) in the considered case. Assuming that ( α -1) = 0 . 15( β -1), a 2% error in the ( β -1) factors leads to an absolute error in ( α -1) of 0 . 04( β -1) = 0 . 04( α -1) / 0 . 15 ∼ 0 . 27( α -1), i.e. ∼ 27% of relative error in the derived fractionation factor ( α -1). On the other hand, we note that such a big error is not substantially larger than the uncertainties in the experimental data reported by Rustad et al (2010b) in their Figure 2. Thus, the case of boron fractionation in aqueous fluid at ambient conditions does not necessarily show the limited usefulness of DFT in the prediction of isotope fractionation factors, but only reflects the fact that precise estimation or measurement of the B isotopes fractionation factors at ambient condition requires unprecedented accuracy of both experimental or computational techniques. For instance, in order to get the value of ( α -1) with a relative error of 5% (at ambient conditions) one needs to estimate the ( β -1) factors or measure relevant quantities with precision of less than 1%. At higher temperatures the situation is di ff erent. Looking just at the fractionation between H3BO3 and H4BO -4 in the gas phase or the aqueous solution one can see that for T > 600 K the fractionation factor between the two substances, ( α -1), is at least 25% of the ( β -1) factor. This results in smaller 0 . 04 / 0 . 25 ∼ 16% for T = 600 K and 0 . 04 / 0 . 36 ∼ 11% for T = 1000 K relative error, which is acceptable in our calculations. Nevertheless, this case shows the importance of proper error estimation on the computed fractionation factors. Such an estimation is usually omitted or not provided explicitly, which can lead to wrong conclusions when the theoretical prediction is confronted with the measured data.",
"pages": [
11,
12
]
},
{
"title": "3.2.4. Pressure dependence of the fluid fractionation factor",
"content": "In our recent paper (Kowalski & Jahn, 2011) we have shown that due to compression the β factor of Li in aqueous fluid increases with increase in pressure (for P > 2 GPa). The same should happen for H3BO3 and H4BO -4 aqueous fluid as the vibrational frequencies of boron species in aqueous fluid increase with increase in pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). Having the experimental data we checked whether the derived pressure-dependent β factors are consistent with the pressure shifts of vibrational frequencies of considered boron species measured by Sanchez-Valle et al. (2005). For that purpose we performed a set of calculations using supercells containing 8 water molecules and the relevant boron species. We note that in line with our previous results for Li (Kowalski & Jahn, 2011), the obtained values of ( β -1) at P = 0 . 5 GPa are within 0 . 1 /permil in agreement with the values obtained for supercells containing 64 water molecules. The results are given in Figures 4 and 5. The computed ( β -1) values for H3BO3 fluid show a linear dependence in pressure, ( β -1) = 23 . 60 + 0 . 28 P (GPa) /permil . This is expected as ( β -1) ∝ ν 2 ∼ ν 2 0 + 2 ν 0 ∆ ν (Schauble, 2004) and ∆ ν is a linear function of pressure (Sanchez-Valle et al., 2005; Schmidt et al., 2005). In case of H4BO -4 the pressure-dependence is linear up to P ∼ 2 -3 GPa and it becomes less steep at higher pressures. In order to quantitatively check the consistency of our prediction with the measured vibrational frequency shifts of Sanchez-Valle et al. (2005) we derived the relative shifts in the ( β -1) factor assuming that ( β -1) ∝ ν 2 and the measured pressure dependence of the frequency shifts: ∆ ν = 2 . 15 cm -1 · P (GPa) and ∆ ν = 3 . 50 cm -1 · P (GPa) for 1454 cm -1 and 666 cm -1 vibrational frequencies of H3BO3 and ∆ ν = 6 . 47 cm -1 · P (GPa) for the 975 cm -1 vibrational frequency of H4BO -4 . The chosen vibrational frequencies are the ones a ff ected by the di ff erent B isotope substitution. Our predicted shift of ( β -1) matches well the shifts derived from the measured frequency shifts. Such a good agreement with the experimental data validates further our computational approach and shows that ab initio calculations can be successfully used in derivation of the pressure dependence of the fractionation factors and pressure-induced vibrational frequencies shifts. Moreover, first principles calculations can be useful in extrapolation of the experimental values for β and ∆ ν to more extreme conditions, which otherwise are extremely di ffi cult to reach by experimental techniques.",
"pages": [
12,
13
]
},
{
"title": "3.3. Fluid-mineral fractionation",
"content": "Next we present the results of the fractionation between boron bearing fluids and minerals such as dravite, olenite and boromuscovite. The aim of these studies is to investigate the mechanisms driving the fractionation process, the role of coordination and the B-O bond length. Below we discuss each case separately.",
"pages": [
13
]
},
{
"title": "3.3.1. Tourmaline-neutral fluid",
"content": "Meyer et al. (2008) measured the boron isotope fractionation between tourmaline and neutral fluid at T = 400 -700 o C and P = 0 . 2 GPa. In the experiment the tourmaline was represented by dravite. In contrast to the former measurements of Palmer et al. (1992) the measured fractionation is very small and does not exceed 2 . 5 /permil at 400 o C. Our calculated fractionation curve together with the experimental data are given in Figure 6. Our result correctly reproduces the experimental measurements within the computational accuracy. The dravite-fluid fractionation is small as the two materials contain boron in BO3 units. We also predict a small fractionation between olenite carrying 3-fold coordinated boron only and aqueous fluid, although the olenite-fluid fractionation is positive because of the shorter B-O bond lengths for olenite (1 . 378 Å vs. 1 . 397 Å).",
"pages": [
13
]
},
{
"title": "3.3.2. Boromuscovite-strongly basic fluid",
"content": "Boromuscovite synthesized in the experiments of Wunder et al. (2005) consisted of two type of polytypes, 1M ( ∼ 10 %) and 2M1 ( ∼ 90 %). In order to be consistent with the experimental conditions, we derived the β factors for both polytypes and computed their weighted average. We note that the β factors for both polytypes of mica are similar with the di ff erence in ( β -1) not larger than 3 %. This is consistent with similar B-O bond lengths (1 . 532 Å) found in both polytypes. Boromuscovite contains boron in tetrahedral sites. Therefore, in order to investigate the impact of the B-O bond length on the fractionation we first compare the fractionation between the mineral and a strongly basic fluid contain- n in H4BO -4 . The result, together with the measurements of Wunder et al. (2005) of the fractionation between boromuscovite and strongly basic fluid, is summarized in Figure 7. Our calculations predict a negative fractionation between mica and the H4BO -4 fluid. The agreement of our prediction with the experimental measurements is relatively good; however, the experimental data indicate slightly stronger fractionation. We note that the experimental conditions of Wunder et al. (2005) do not assure that the measured basic fluid contained four-fold coordinated boron species, i.e. H4BO -4 , only. As we indicate in the Figure 7, the presence of as little as 10% of H3BO3 in the measured basic fluid brings the prediction and measurements into much better agreement. It makes sense that the β factor of boromuscovite is smaller than for aqueous H4BO -4 as the average B-O bond length in mica is 1 . 532 Å, while it is 1 . 513 Å and therefore shorter in case of aqueous H4BO -4 . We note that in our derivation we assumed that the fluid consists mostly of [4] B species. As we already mentioned, although Wunder et al. (2005) call the fluid 'strongly basic' its exact composition, especially the amount of [3] B species is unknown. However, if for instance the [4] B to [3] B ratio was 1, then the predicted mica-basic fluid fractionation at 800 K would be 5 /permil larger than measured. This would result in a large inconsistency between the computed values and the experimental data. On the other hand, good agreement between the prediction and the measurements indicates that the strongly basic fluid was dominated by H4BO -4 species, which is in line with previous studies (Zeebe, 2005; Sanchez-Valle et al., 2005).",
"pages": [
13,
14
]
},
{
"title": "3.3.3. Boromuscovite-neutral fluid",
"content": "The fractionation between boromuscovite and neutral fluid involves a change in coordination from B [4] in boromuscovite to B [3] in neutral fluid. Wunder et al. (2005) measured the fractionation between the two materials at 3 GPa, shown in Figure 8. The predicted fractionation is about 3 ± 2 . 5 /permil larger than the measured value. Looking for potential sources of this discrepancy, we have checked for the e ff ect of the change in lattice parameters due to combined thermal expansion and compression. For that purpose we applied the EOS of Holland and Powell (2011) for muscovite, which gives 4 . 4%, 3 . 8% and 3 . 1% decrease in volume for T = 600 K, 800 K and 1000 K respectively and P = 3 GPa. As muscovites show highly anisotropic compressibility patterns, in line with Comodi & Zanazzi (1995) we applied the T and P driven change in volume assuming the 16%, 19% and 65% contribution to compression along the /vector a , /vector b , /vector c lattice vectors. On the other hand ( β -1) factors of H3BO3 and H4BO -4 in aqueous solutions at P = 3 GPa increase by 3 . 5% and 11 . 8% respectively (Figures 4 and 5), leading to pressure-induced increase in the boromuscovite-aqueous fluid fractionation at the experimental pressure. The ∆ factor, corrected for e ff ects of thermal expansion and compression of boromuscovite and compression of fluid, is also given in Figure 8. Because the high T and P e ff ects result in similar increases in the β factors for both solid and aqueous fluid, the resulting fractionation factor between these two phases is close to the one derived at ambient conditions. Therefore, thermal expansion and compression e ff ects cannot explain the observed discrepancy between prediction and the measurements of Wunder et al. (2005). On the other hand, the comparison of our results with the experimental data suggests that the fractionation between boromuscovite and fluid is the same as between H3BO3 and H4BO -4 fluids (see Figure 3), which is at odds with the non-negligible and negative fractionation between boromuscovite and a strongly basic fluid. In section 3.3.2 we have shown that we are able to correctly reproduce the fractionation between boromuscovite and strongly basic fluid, which indicates that our result for boromuscovite is reliable. This suggests that another, unaccounted e ff ect leads to the decrease of the boron isotope fractionation between mica and neutral fluid in the experiments of Wunder et al. (2005). One possible solution for the discrepancy is a nonnegligible amount of boron residing in four-fold coordinated configurations in neutral solution. This is in line with the Raman spectroscopy measurements of Schmidt et al. (2005), who detected a broad peak in the Raman spectra of neutral H3BO3-dominated fluid and attributed it to B [4] species. The integrated area of this peak, compared to the peak of the Raman 877 cm -1 line of B [3] species, indicates the presence of at least 15 -30 % of B [4] species by mole fraction. Assuming that there is 15 -30 % of B [4] species present in the fluid and that the β factor of these species is similar to that of H4BO -4 , the fractionation factor between boromuscovite and H3BO3 aqueous fluid decreases bringing the theory and the experiment to better agreement, which is illustrated in Figure 8. If this interpretation is true, it suggests that boron isotope fractionation could be used to gather information on the speciation of B in aqueous fluids.",
"pages": [
14,
15
]
},
{
"title": "3.4. B isotope fractionation between minerals",
"content": "The boron isotope fractionation between B-bearing crystalline solids has received considerable attention recently (Wunder et al., 2005; Meyer et al., 2008; Klemme et al., 2011; Marschall, 2005; Hervig et al., 2002). We focus here on the investigation of boron isotope fractionation between mica and tourmaline as boron atoms in these minerals occupy sites of di ff erent coordination, which should result in a large B isotope fractionation between these two minerals. In mi- cas boron substitutes for silicon in the four-fold coordinated site (Wunder et al., 2005), while in tourmaline (dravite) it is incorporated in the three fold coordinated site (Meyer et al., 2008). Comparing the B isotope fractionation between di ff erent minerals, melts and fluids Wunder et al. (2005) have shown that the fractionation between two materials of di ff erent B coordination is large, reaching 5 /permil at 1000 K and much higher values at lower temperatures. Klemme et al. (2011),Marschall (2005) and Hervig et al. (2002) measured the fractionation between coexisting phases of the two minerals in natural samples. The fractionation between these two minerals is also derived from experimental isotopic fractionation data of B-muscovite-fluid (Wunder et al., 2005) and tourmaline-fluid (Meyer et al., 2008) systems. The results of these measurements and our computed T -dependent fractionation curve are given in Figure 9. The first striking observation is that our predicted fractionation factors are much larger (taking the absolute value) than the experimental values (Wunder et al., 2005; Meyer et al., 2008). The latter are also incon- sistent with the natural samples data of Klemme et al. (2011) and previous studies discussed in that paper (Marschall, 2005; Hervig et al., 2002). On the other hand, the measurements on natural samples are consistent with our calculated values, which tends to validate our predictions. We notice that the most recent measurements of boron isotope signatures of tourmaline and white mica from the Broken Hill area in Australia by Klemme et al. (2011) indicate for the assumed temperature of 600 o C that the fractionation factor between the two phases is 10.4 ± 2 . 7 /permil , which is in good agreement with our computed value of 10 . 7 ± 1 . 8 /permil . The experimental mica-tourmaline B isotope fractionation factors of Wunder et al. (2005) and Meyer et al. (2008) are 2 /permil and 6 /permil smaller at temperatures of 1000 K and 800 K respectively, with an experimental uncertainty of 2 /permil . However, this discrepancy can be resolved by assuming that in the experiments of Wunder et al. (2005) the fluid contained a significant admixture of B [4] species, which leads to the underestimation of the experimental boromuscovite-fluid fractionation factor by ∼ 2 /permil at 1000 K and ∼ 3 . 5 /permil at 1000 K, as is seen in Figure 8. The experimental mica-tourmaline fractionation factor corrected for the presence of B [4] species is also plotted in Figure 9. It is now more consistent with the natural data. We note that this result independently supports the conclusion underlined in section 3.3.3 and result of Schmidt et al. (2005) that highP , B-bearing neutral fluids contain significant admixtures of B [4] species. Olenite is a mineral which can incorporate boron in both trigonal and tetrahedral sites as it substitutes for both Al and Si atoms. It is therefore interesting to check the fractionation of boron isotopes between the two differently coordinated sites in one mineral and compare it with the above result for mica and tourmaline. The computed fractionation between the trigonal and tetrahedral sites at 600 o C is 10 . 6 ± 1 . 9 /permil , which is consistent with the fractionation between tourmaline and mica, indicating that the coordination of the B atom is the driving factor for the fractionation of the B isotopes. Similarly, we computed the boron isotopes fractionation between trigonal and tetragonal boron sites in dravite. In order to create the tetragonal B site we replaced one Si atom with B and we added one H atom forming an additional OH group to compensate the charge. The computed fractionation between the sites at 600 o C is 8 . 9 ± 1 . 7 /permil , which is also in agreement with the aforementioned results. Next, we will show that the value of the β factor depends not only on coordination but is also strongly correlated with B-O bond length.",
"pages": [
15,
16
]
},
{
"title": "3.5. Fractionation between B [3] and B [4] materials",
"content": "The β factors computed for all the considered materials are grouped together in Figure 10. β factors can be grouped into two sets, one that includes materials with boron in three-fold coordination and another one that includes materials having boron in four-fold coordination. For olenite and dravite we also computed the β factors with boron sitting on four-fold coordinated site. The β factor for these crystalline solids with given B [3] / B [4] ratio can be derived as a weighted average of the β factors obtained for boron sitting on the two di ff erently coordinated sites. The fractionation factor between materials of di ff erent boron coordination is ∼ 8 /permil on average at T = 1000 K. We note that it is ∼ 3 /permil larger than the one deduced by Wunder et al. (2005) from measurements performed on solids, silicate melts and fluids, but this can be attributed to the underestimation of the fractionation factors for boromuscovite-fluid system by Wunder et al. (2005) due to potential admixture of B [4] species in the investigated fluid. Our results show also a substantial spread of β factors of substances containing boron of a given coordination. The spread is at least 4 /permil and results from di ff erent B-O bond lengths. We illustrate this in Figure 11 by plotting together the β factors derived for all considered materials at T = 1000 K as a function of B-O bond length. It is clearly seen that there is a roughly linear correlation between the β factor and BO bond length, which is especially evident comparing the results for crystalline solids. For instance, out of the considered B [4] -bearing minerals boromuscovite has the longest B-O bond length of 1 . 525 Å (1M) and 1 . 516 Å (2M1), followed by dravite 1 . 514 Å and olenite with BObond length of 1 . 502 Å. This tracks the di ff erences in the β factors derived for these materials. In addition the materials having B [3] species only exhibit shorter bond lengths of ∼ 1 . 37 Å and higher β factors, while the materials containing B [4] species having bond lengths of about ∼ 1 . 52 Å show much smaller β factors. Therefore, the tighter bonding of B [3] species likely explains why the heavy B isotope prefers the less coordinated phases. This clearly shows that the change in the B-O bond length during an isotope exchange is the leading factor driving the production of the boron equilibrium isotope signatures at high T .",
"pages": [
16,
17
]
},
{
"title": "4. Conclusions",
"content": "In this work we have presented a detailed analysis of boron isotope fractionation between boronbearing crystalline solids and aqueous fluids at high T and P conditions. In order to perform our investigation we have applied and extended a computationally e ffi cient approach for the computation of isotope fractionation factors for complex minerals and fluids at high temperatures and pressures presented by Kowalski & Jahn (2011). As an extension to the Bigeleisen & Mayer (1947) 'single atom approximation' method we demonstrated that using the pseudofrequencies derived from the force constants acting on the fractionating element together with the full formula for computation of the reduced partition function ratios results in significant improvement in the accuracy of the computed fractionation factors, which is essential when lower temperature materials and high vibrational frequency complexes are considered. In order to understand the fractionation between Bbearing crystalline solids and aqueous fluids we performed a set of calculations of β factors for dravite, olenite, boromuscovite and aqueous solutions of H3BO3 and H4BO -4 . In agreement with the experimental findings we show that the fractionation strongly correlates with coordination through the change in the B-O bond length. The lower trigonal coordination BO3 arrangement results in higher 11 B / 10 B (by ∼ 8 /permil at T = 1000 K) than the tetrahedrally coordinated boron complexes, which exhibit ∼ 0 . 15 Å longer B-O bonds. The computed fractionation between minerals and fluids of the same coordination are in good agreement with experiments. However, we predict larger isotope fractionation between boromuscovite and H3BO3 fluid (by at least a few /permil ) than was measured in situ at high P by Wunder et al. (2005) and Meyer et al. (2008), but that is consistent with measurements on natural samples. We note that the presence of B [4] in highP fluid could reconcile the in situ experimental results with our prediction and other measurements. This is expected from the experiments of Schmidt et al. (2005), but requires further experimental confirmation. If true, this would open the possibility for using the isotope fractionation techniques as a tool to measure the speciation of boron in fluids and crystalline solids. We have also demonstrated that with our computational approach we are able to correctly predict the pressure-induced isotope fractionation for compressed aqueous fluids, which indicates the ability of ab initio methods to predict the isotopic signatures of highly compressed materials, even those that are difficult to investigate experimentally. Our study confirms that ab initio computer simulations are a useful tool not only for prediction but also understanding the equilibrium stable isotope fractionation processes between various phases, including aqueous solutions, at high pressures and temperatures. They can nicely complement experimental e ff orts, provide unique insight into the isotope fractionation process on the atomic scale and deliver data for conditions that are inaccessible by the current experimental techniques.",
"pages": [
17
]
},
{
"title": "Acknowledgements",
"content": "The authors wish to acknowledge financial support in the framework of DFG project no. JA 1469 / 4-1. Part of the calculations were performed on the IBM BlueGene / P JUGENE of the John von Neumann Institute for Computing (NIC). We are also grateful the associate editor Edwin A. Schauble and anonymous referees for constructive comments that helped improving the manuscript.",
"pages": [
18
]
},
{
"title": "References",
"content": "Alecu I. M., Zheng J. , Zhao Y. and Truhlar D. G. (2010) Computational Thermochemistry: Scale Factor Databases and Scale Factors for Vibrational Frequencies Obtained from Electronic Model Chemistries. J. Chem. Theory Comput. 6 , 2872-2887. Andrews, L., Burkholder, T.R., (1992) Infrared spectra of molecular B(OH)3 and HOBO in solid argon, J. Chem. Phys. , 97 , 10, 72037210. Barth S. (1993) Boron isotope variations in nature: a synthesis. Geol. Rundsch. 82 , 640-651. Becke A. D. (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38 , 3098-3100. Bigeleisen J. and Mayer M. G. (1947) Calculation of Equilibrium Constants for Isotopic Exchange Reactions. J. Chem. Phys. 15 , 262-267. Blanchard M., Poitrasson F., Meheut M., Lazzeri M., Mauri F., and Balan E. (2009) Iron isotope fractionation between pyrite (FeS2), hematite (Fe2O3) and siderite (FeCO3): A first-principles density functional theory study. Geochim. Cosmochim. Acta 73 , 65656578. Brenan J.M., Ryerson F.J. and Shaw H.F. (1998) The role of aqueous fluids in the slab-to-mantle transfer of boron, beryllium and lithium during subduction: experiments and models. Geochim. Cosmochim. Acta 62 , 3337-3347. Domagal-Goldman S. D. and Kubicki J. D. (2008) Density functional theory predictions of equilibrium isotope fractionation of iron due to redox changes and organic complexation. Geochim. Cosmochim. Acta 72 , 5201-5216. Domanik K.J., Hervig R.L. and Peacock S.M. (1993) Beryllium and boron in subduction zone minerals: An ion microprobe study. Geochim. Cosmochim. Acta 57 , 4997-5010. Marler B., Borowski M., Wodara U., Schreyer W. (2002) Synthetic tourmaline (olenite) with excess boron replacing silicon in the tetrahedral site: II. Structure analysis Eur. J. Mineral. 14 , 763-771. Marschall H. R. (2005): Lithium, Beryllium, and Boron in High-Pressure Metamorphic Rocks from Syros (Greece). PhD thesis, Universitt Heidelberg, Heidelberg, Germany [http: // www.ub.uni-heidelberg.de / archiv / 5634]. Marx D. and Hutter J. (2000) Ab initio molecular dynamics: Theory and Implementation. Modern Methods and Algorithms of Quantum Chemistry 1 edited by J. Grotendorst, NIC, FZ Julich, 301-449 ; CPMD code: J. Hutter et al. , see: www.cpmd.org . Meheut M., Lazzeri M., Balan E. and Mauri F. (2007) Equilibrium isotopic fractionation in the kaolinite, quartz, water system: Prediction from first-principles density-functional theory. Geochim. Cosmochim. Acta 71 , 3170-3181. Chem. Geol. 258 , 28-37. Menconi G. and Tozer D. J. (2002) Diatomic bond lengths and vibrational frequencies: assesment of recently developed exchangecorrelation functionals. Chem. Phys. Lett. 360 , 38-46. Meyer Ch., Wunder B., Meixner A., Romer R. L., and Heinrich, W. (2008) Boron isotope fractionation between tourmaline and fluid; an experimental re-investigation. Contrib. Mineral. Petrol. 156 , 259-267. Morgan VI, G.B. and London, D. (1989) Experimental reactions of amphibolite with boron-bearing aqueous fluids at 200 MPa: implications for tourmaline stability and partial melting in mafic rocks. Contrib. Mineral. Petrol. 102 , 281-298. Redlich, O. (1935). Eine allgemeine Beziehung zwichen den Schwingungsfrequenzen isotoper Molekeln. Z. Physikal. Chem. B 28 , 371-382. Rustad J. R, Casey W. H., Yin Q.-Z., Bylaska E. J., Felmy, A. R., Bogatko S. A., Jackson V. E., Dixon D. A. (2010) Isotopic fractionation of Mg2 + (aq), Ca2 + (aq), and Fe2 + (aq) with carbonate minerals, Geochim. Cosmochim. Acta 74 , 6301-6323. Rustad J. R. and Bylaska E. J. (2007) Ab Initio Calculation of Isotopic Fractionation in B(OH)3(aq) and BOH -4 (aq). J. Am. Chem. Soc. (Communication) 129 , 2222-2223. Sanchez-Valle C., Reynard B., Daniel I., Lecuyer C., Martinez I., and Chervin J. C. (2005) Boron isotopic fractionation between minerals and fluids; new insights from in situ high pressure-high temperature vibrational spectroscopic data Geochim. Cosmochim. Acta 69 , 4301-4313 Schauble E. A. (2004) Applying Stable Isotope Fractionation Theory to New Systems. Rev. Mineral. Geochem. 55 , 65-111. Schreyer W., Wodara U., Marler B., van Aken P.A., Seifert F. and Robert J.-L. (2000) Synthetic tourmaline (olenite) with excess boron replacing silicon in the tetrahedral site: I. Synthesis conditions, chemical and spectroscopic evidence. Eur. J. Mineral. 12 , 529-541. Urey, H. C. (1947). The Thermodynamic Properties of Isotopic Substances. J. Chem. Soc. , 562581. and scientific use. J. Phys. Chem. Ref. Data 31 , 387-535. Wunder B., Meixner A., Romer R. L., Wirth R., and Heinrich W. (2005) The geochemical cycle of boron; constraints from boron isotope partitioning experiments between mica and fluid Lithos 84 , 206-216 Yamaji K., Makita Y., Watanabe H., Sonoda A., Kanoh H., Hirotsu T. and Ooi K. (2001) Theoretical Estimation of Lithium Isotopic Reduced Partition Function Ratio for Lithium Ions in Aqueous Solution. J. Phys. Chem. A 105 , 602613.",
"pages": [
18,
19
]
}
]
2013GrCo...19..275C
https://arxiv.org/pdf/1212.0484.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_82><loc_87><loc_87></location>On stability of power-law solution in multidimensional Gauss-Bonnet cosmology</section_header_level_1>
<text><location><page_1><loc_37><loc_78><loc_65><loc_79></location>D.M.Chirkov 1,2 ∗ , A.V.Toporensky 1 †</text>
<text><location><page_1><loc_45><loc_74><loc_56><loc_76></location>July 16, 2018</text>
<text><location><page_1><loc_27><loc_67><loc_74><loc_71></location>1 Sternberg Astronomical Institute, Moscow State University, Universitetsky pr., 13, Moscow, 119991, Russia</text>
<text><location><page_1><loc_33><loc_65><loc_34><loc_65></location>2</text>
<text><location><page_1><loc_34><loc_62><loc_68><loc_65></location>Faculty of Physics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia</text>
<section_header_level_1><location><page_1><loc_47><loc_59><loc_54><loc_60></location>Abstract</section_header_level_1>
<text><location><page_1><loc_13><loc_48><loc_88><loc_57></location>We consider dynamics of a flat anisotropic multidimensional cosmological model in Gauss-Bonnet gravity in the presence of a homogeneous magnetic field. In particular, we find conditions under which the known power-law vacuum solution can be an attractor for the case with non-zero magnetic field. We also describe a particular class of numerical solution in (5 + 1) -dimensional case which does not approach the power-law regime.</text>
<section_header_level_1><location><page_1><loc_9><loc_43><loc_28><loc_45></location>I. Introduction.</section_header_level_1>
<text><location><page_1><loc_9><loc_23><loc_92><loc_41></location>The fact that structure of initial cosmological singularity can be rather complicated have been recognized since 60-th of the last century when the conception of BKL chaos have been presented [1]. It appears that Kasner solution being a general solution for a vacuum Bianchi I Universe becomes unstable in the case of metric with positive spatial curvature (belonging to Bianchi IX class) and is replaced by a complicated sequence of transient "Kasner epochs". Later it was found that some classes of an anisotropic matter can induce similar type of cosmological behavior even in flat Bianchi I case. This can be shown for a magnetic field by LeBlanc [2] and for a general vector field by Kirillov [3]. The BKL analogs for the flat magnetic Universe is a typical behavior when off-diagonal terms in the metric in the frame determined by magnetic field are present. For a diagonal case see for example [4].</text>
<text><location><page_1><loc_9><loc_14><loc_92><loc_22></location>Possible generalisation of these results may involve studies in the framework of multidimensional cosmology as well as alternative gravity theories. Analogs of magnetic field for multidimensional space-times have been studied in [13, 14, 15]. If we want to introduce some corrections to Einstein gravity, the first goal is to find a solution which replaces Kasner solution for a flat anisotropic Universe. In the present paper</text>
<text><location><page_2><loc_9><loc_68><loc_92><loc_93></location>we consider Lovelock gravity as a generalization of General Relativity. The principal feature of Lovelock gravity is that this theory keeps the order of equations of motion the same as in GR, while other theories (like popular now f ( R ) -theory) results in increasing of the number of derivatives in equations of motion. Another important property of Lovelock gravity is that it gives corrections to GR only in higher-dimensional space-time, so it is natural to consider it in the framework of multidimensional cosmology. As the number of non-Einstein terms in Lovelock gravity is finite for any given dimensionality of space-time, it is possible to consider regimes when these terms are not only small corrections to Einstein gravity. In particular, it is reasonable to expect that the highest order Lovelock term, consisting of highest power of curvature invariants, dominates near a cosmological singularity. In the present paper we consider the second Lovelock term, which is the famous Gauss-Bonnet combination. It is the highest possible Lovelock term for (4 + 1) and (5 + 1) dimensional spacetimes. As cosmology in the (4 + 1) dimensional case has some pathological features, we restrict ourselves by the ( N +1) dimensional case with N /greaterorequalslant 5 in the present paper.</text>
<text><location><page_2><loc_9><loc_47><loc_92><loc_67></location>In late 80-th some vacuum solutions for a flat anisotropic Universe in Gauss-Bonnet gravity for N = 4 , 5 have been found [5, 6]. They replace Kasner solution of GR. Later these solutions have been rediscovered and verified for N = 6 , 7 in [7]; after that this solution have been generalized for all N and also to the general Lovelock gravity [8]. The main goal of this paper is to address question of its stability near the initial singularity. It is known that in the presence of an ordinary isotropic matter these solutions are stable to the past if the matter has the equation of state with w < 1 / 3 , otherwise the solution tends to isotropic one [9, 10]. For more interesting regimes near a singularity it is necessary to consider either anisotropic matter or curved geometries. As introduction of spatial curvature usually leads to very cumbersome equations of motion, we have chosen a magnetic fields as a possible source of instability. In this paper we restrict ourselves by the diagonal case.</text>
<text><location><page_2><loc_9><loc_43><loc_92><loc_46></location>It is also known that in contrast to Einstein gravity certain initial conditions lead to exponential solutions instead to those of power-law behavior [9, 11, 12], we do not consider them in the present paper.</text>
<text><location><page_2><loc_9><loc_28><loc_92><loc_42></location>The structure of the paper is as follows: In Sec. II. we describe the metric and matter content of the model studied, in Sec. III. the model is investigated in the framework of Einstein relativity. We present results on stability of Kasner solution in the presence of magnetic field. The same model in Gauss-Bonnet gravity is studied in Sec. IV. with presentation made as parallel as possible to presentation in the preceding section in order to compare Einstein and Gauss-Bonnet cases. In Sec. V. we describe a particular regime existing in the zone of instability of power-law regimes in (5+1) -dimensional Gauss-Bonnet gravity. Sec. VI. contains a brief summary of results obtained.</text>
<section_header_level_1><location><page_2><loc_9><loc_23><loc_30><loc_24></location>II. Preliminaries.</section_header_level_1>
<text><location><page_2><loc_9><loc_19><loc_54><loc_21></location>In what follows we use a reference system chosen so that 1 :</text>
<formula><location><page_2><loc_34><loc_15><loc_92><loc_17></location>g 00 = 1 , g kk = -e 2 a k ( t ) , g ij = 0 , i = j (1)</formula>
<text><location><page_2><loc_65><loc_15><loc_65><loc_17></location>/negationslash</text>
<text><location><page_3><loc_9><loc_90><loc_92><loc_93></location>In this section we looking for the energy-momentum tensor of a pure magnetic field in ( N +1) -dimensional space-time. In the general case components of the energy-momentum tensor of the electromagnetic field are</text>
<formula><location><page_3><loc_37><loc_84><loc_92><loc_89></location>T µ ν = 1 4 π ( F νγ F γµ + 1 4 δ µ ν F αβ F αβ ) (2)</formula>
<text><location><page_3><loc_9><loc_82><loc_89><loc_84></location>where F αβ is the Faraday tensor. The components of the Faraday tensor obey the following equations:</text>
<formula><location><page_3><loc_43><loc_77><loc_92><loc_82></location>{ ∇ µ F µν = 0 , dF = 0 , (3)</formula>
<text><location><page_3><loc_9><loc_74><loc_23><loc_75></location>or, in more detail:</text>
<formula><location><page_3><loc_36><loc_70><loc_92><loc_75></location>{ D µ F µν +Γ µ µλ F λν +Γ ν µσ F µσ = 0 , ( dF ) αβγ = 0 , α < β < γ , (4)</formula>
<text><location><page_3><loc_9><loc_67><loc_34><loc_69></location>where D µ = ∂ ∂x µ . As it is known</text>
<formula><location><page_3><loc_31><loc_63><loc_92><loc_65></location>( dF ) α 1 α 2 α 3 = D α 1 F α 2 α 3 -D α 2 F α 1 α 3 + D α 3 F α 1 α 2 , (5)</formula>
<text><location><page_3><loc_9><loc_60><loc_46><loc_62></location>therefore we may rewrite system (4) in the form</text>
<formula><location><page_3><loc_32><loc_54><loc_92><loc_60></location>{ D µ F µν +Γ µ µλ F λν +Γ ν µσ F µσ = 0 , D α F βγ -D β F αγ + D γ F αβ = 0 , α < β < γ (6)</formula>
<text><location><page_3><loc_9><loc_52><loc_55><loc_53></location>The components of the Riemann connection have the form:</text>
<formula><location><page_3><loc_36><loc_48><loc_92><loc_50></location>Γ i i 0 = Γ i 0 i = ˙ a i ( t ) , Γ 0 jj = ˙ a j ( t ) e 2 a j ( t ) (7)</formula>
<text><location><page_3><loc_9><loc_43><loc_92><loc_47></location>Its other components are equal to zero. Since we consider the case of a homogenous space, Faraday tensor depends on time t only; then, in view of (7) the system (6) takes the form:</text>
<text><location><page_3><loc_9><loc_34><loc_21><loc_36></location>Its solutions are</text>
<formula><location><page_3><loc_41><loc_34><loc_92><loc_43></location> ˙ F 0 i + F 0 i ∑ k ˙ a k = 0 , ˙ F ij = 0 (8)</formula>
<text><location><page_3><loc_9><loc_27><loc_92><loc_29></location>Hereinafter we will be interested in the case of a pure magnetic field; so that φ m = 0 and, as a consequence,</text>
<formula><location><page_3><loc_37><loc_27><loc_92><loc_35></location> F 0 m = φ m e -∑ k a k , φ m = const , F ij = ψ ij , ψ ij = const (9)</formula>
<formula><location><page_3><loc_45><loc_24><loc_92><loc_25></location>F 0 m = F 0 m = 0 (10)</formula>
<text><location><page_3><loc_9><loc_18><loc_92><loc_22></location>In what follows we will deal with diagonal energy-momentum tensor; in view of (1),(2),(9) and (10) it implies that</text>
<text><location><page_3><loc_31><loc_15><loc_31><loc_18></location>/negationslash</text>
<text><location><page_3><loc_54><loc_15><loc_54><loc_18></location>/negationslash</text>
<text><location><page_3><loc_9><loc_12><loc_68><loc_14></location>Functions e -2 a l are linearly independent, therefore from (11) it follows that</text>
<formula><location><page_3><loc_23><loc_14><loc_92><loc_19></location>T µ ν = 0 , µ = ν ⇐⇒ F νγ F γµ = 0 , µ = ν ⇐⇒ ∑ l e -2 a l ψ il ψ lk = 0 (11)</formula>
<formula><location><page_3><loc_38><loc_9><loc_92><loc_10></location>ψ il ψ lk = 0 , i, k, l = 1 , N, i = k (12)</formula>
<text><location><page_3><loc_60><loc_8><loc_60><loc_10></location>/negationslash</text>
<text><location><page_4><loc_9><loc_92><loc_92><loc_93></location>Each ψ ij is multiplied by all that ψ kl , which has one of the indices k, l coincident with one of the indices i, j .</text>
<text><location><page_4><loc_89><loc_89><loc_89><loc_91></location>/negationslash</text>
<text><location><page_4><loc_9><loc_85><loc_92><loc_91></location>Let us fix pair ( i, j ) ; the number of combinations, in which ψ ij is found, equals to 2( N -2) . Let ψ ij = 0 ; then the other 2( N -2) quantities ψ ik ( ψ jk ), which are multiplied by ψ ij , must be equals to zero; indeed, if ψ ik = 0 ( k = i, j ) , then ψ ij ψ ik = 0 , but that contradict (12).</text>
<text><location><page_4><loc_12><loc_83><loc_75><loc_85></location>Continuing the argument, we find the number of zero magnetic field components:</text>
<text><location><page_4><loc_12><loc_84><loc_12><loc_87></location>/negationslash</text>
<text><location><page_4><loc_17><loc_84><loc_17><loc_87></location>/negationslash</text>
<text><location><page_4><loc_33><loc_84><loc_33><loc_87></location>/negationslash</text>
<formula><location><page_4><loc_11><loc_78><loc_92><loc_82></location>2( N -2) + 2( N -4) + . . . = 2 ( N -2) + ( N -2 N -1 2 ) 2 N -1 2 = ( N -1) 2 2 for an odd dimensions (13)</formula>
<formula><location><page_4><loc_13><loc_73><loc_92><loc_77></location>2( N -2) + 2( N -4) + . . . = 2 ( N -2) + ( N -2 N 2 ) 2 N 2 = N ( N -2) 2 for an even dimensions (14)</formula>
<text><location><page_4><loc_9><loc_69><loc_92><loc_73></location>We took into account that the number of pairs of indices without the same elements equals to N -1 2 for an odd dimensions and to N 2 for an even dimensions.</text>
<text><location><page_4><loc_9><loc_65><loc_92><loc_69></location>The total number of components of the magnetic field is N ( N -1) 2 ; then the number χ of non-zero magnetic field components is</text>
<formula><location><page_4><loc_27><loc_60><loc_92><loc_64></location>χ = N ( N -1) 2 -( N -1) 2 2 = N -1 2 for an odd dimensions , (15)</formula>
<formula><location><page_4><loc_28><loc_56><loc_92><loc_59></location>χ = N ( N -1) 2 -N ( N -2) 2 = N 2 for an even dimensions (16)</formula>
<text><location><page_4><loc_9><loc_49><loc_92><loc_55></location>In what follows they are exactly ψ 12 , ψ 34 , . . . , ψ 2 n -1 , 2 n , n = 1 , χ that we set to be non-zero; other components of the Faraday tensor are assumed to be zero. Thus, components of the energy-momentum tensor of the electromagnetic field take the form:</text>
<formula><location><page_4><loc_45><loc_47><loc_92><loc_49></location>T µ ν = 0 , µ = ν, (17)</formula>
<text><location><page_4><loc_62><loc_42><loc_62><loc_43></location>/negationslash</text>
<formula><location><page_4><loc_10><loc_42><loc_92><loc_47></location>T 0 0 = ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( a 2 i -1 + a 2 i ) , T n n = -ψ 2 2 n -1 , 2 n e -2( a 2 n -1 + a 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (18)</formula>
<text><location><page_4><loc_53><loc_46><loc_53><loc_49></location>/negationslash</text>
<text><location><page_4><loc_9><loc_40><loc_54><loc_41></location>Hereinafter it will be convenient to use notations like this:</text>
<text><location><page_4><loc_9><loc_33><loc_21><loc_34></location>We assume that</text>
<formula><location><page_4><loc_38><loc_33><loc_92><loc_39></location>< n > = { 2 n -1 , 2 n } , n = 1 , χ (19)</formula>
<formula><location><page_4><loc_17><loc_28><loc_92><loc_30></location>i = < n > ⇐⇒ i = 2 n -1 ∧ i = 2 n, k = < n > ⇐⇒ k = 2 n -1 ∨ k = 2 n (20)</formula>
<text><location><page_4><loc_18><loc_28><loc_18><loc_30></location>/negationslash</text>
<text><location><page_4><loc_33><loc_28><loc_33><loc_30></location>/negationslash</text>
<text><location><page_4><loc_44><loc_28><loc_44><loc_30></location>/negationslash</text>
<section_header_level_1><location><page_4><loc_9><loc_24><loc_53><loc_26></location>III. Stability of the Kasner solutions.</section_header_level_1>
<section_header_level_1><location><page_4><loc_9><loc_21><loc_31><loc_22></location>III.1. Field equations.</section_header_level_1>
<text><location><page_4><loc_9><loc_18><loc_22><loc_19></location>The action reads:</text>
<formula><location><page_4><loc_35><loc_13><loc_92><loc_19></location>S = 1 16 π ∫ d N +1 x √ | det( g ) | ( L E + L m ) (21)</formula>
<formula><location><page_4><loc_41><loc_11><loc_92><loc_14></location>L E = R, L m = F αβ F αβ (22)</formula>
<text><location><page_5><loc_9><loc_92><loc_92><loc_93></location>where R and F αβ are scalar curvature and Faraday tensor respectively. Here and after we use Planck units:</text>
<formula><location><page_5><loc_42><loc_87><loc_92><loc_91></location>c = 1 , G = 1 m N -2 Pl = 1 , (23)</formula>
<text><location><page_5><loc_9><loc_85><loc_59><loc_87></location>m Pl is the Planck mass. The gravitational equations is given by</text>
<formula><location><page_5><loc_46><loc_82><loc_92><loc_84></location>G µ ν = 8 πT µ ν (24)</formula>
<text><location><page_5><loc_9><loc_80><loc_13><loc_81></location>where</text>
<formula><location><page_5><loc_44><loc_76><loc_92><loc_80></location>G µ ν = R µ ν -1 2 δ µ ν R (25)</formula>
<text><location><page_5><loc_9><loc_75><loc_67><loc_76></location>are the components of the Einstein tensor. Taking into account (1) we get:</text>
<formula><location><page_5><loc_45><loc_72><loc_92><loc_74></location>G µ ν = 0 , µ = ν (26)</formula>
<text><location><page_5><loc_47><loc_68><loc_47><loc_69></location>/negationslash</text>
<formula><location><page_5><loc_27><loc_67><loc_92><loc_72></location>G 0 0 = ∑ i<j ˙ a i ˙ a j , G n n = ∑ i = n ( a i + ˙ a 2 i ) + ∑ i<j i,j = n ˙ a i ˙ a j , n = 1 , N (27)</formula>
<text><location><page_5><loc_60><loc_67><loc_60><loc_68></location>/negationslash</text>
<text><location><page_5><loc_9><loc_63><loc_92><loc_66></location>As we consider the diagonal case here, then in view of (17),(18),(24),(27) for an even number of dimensions the gravitational equations can be written as</text>
<text><location><page_5><loc_13><loc_58><loc_13><loc_59></location>/negationslash</text>
<formula><location><page_5><loc_13><loc_57><loc_92><loc_63></location>∑ i = <n> ( a i + ˙ a 2 i ) + ∑ i<j i,j = <n> ˙ a i ˙ a j = -ψ 2 2 n -1 , 2 n e -2( a 2 n -1 + a 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (28)</formula>
<text><location><page_5><loc_59><loc_58><loc_59><loc_58></location>/negationslash</text>
<formula><location><page_5><loc_35><loc_52><loc_92><loc_57></location>∑ i<j ˙ a i ˙ a j -∑ j =1 ,χ ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) = 0 (29)</formula>
<text><location><page_5><loc_9><loc_47><loc_92><loc_52></location>Note that each expression like (28) describes two equations with numbers 2 n -1 and 2 n simultaneously (we use the notation (19) here). In the case of an odd number of dimensions there is one more equation in addition to these:</text>
<text><location><page_5><loc_27><loc_42><loc_27><loc_43></location>/negationslash</text>
<formula><location><page_5><loc_26><loc_41><loc_92><loc_47></location>∑ i = N ( a i + ˙ a 2 i ) + ∑ i<j i,j = N ˙ a i ˙ a j = ∑ j =1 ,χ ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (30)</formula>
<text><location><page_5><loc_40><loc_42><loc_40><loc_42></location>/negationslash</text>
<text><location><page_5><loc_9><loc_39><loc_56><loc_41></location>The left side of the expression (29) is a first integral of (28).</text>
<section_header_level_1><location><page_5><loc_9><loc_35><loc_33><loc_37></location>III.2. Vacuum solution.</section_header_level_1>
<text><location><page_5><loc_9><loc_32><loc_56><loc_34></location>When there is no any matter the equations (28)-(30) lead to:</text>
<text><location><page_5><loc_36><loc_28><loc_36><loc_29></location>/negationslash</text>
<formula><location><page_5><loc_35><loc_27><loc_92><loc_32></location>∑ i = n ( a i + ˙ a 2 i ) + ∑ i<j i,j = n ˙ a i ˙ a j = 0 , n = 1 , N (31)</formula>
<text><location><page_5><loc_9><loc_21><loc_46><loc_22></location>These equations has the following solutions [16]:</text>
<text><location><page_5><loc_49><loc_27><loc_49><loc_28></location>/negationslash</text>
<formula><location><page_5><loc_46><loc_22><loc_92><loc_27></location>∑ i<j ˙ a i ˙ a j = 0 (32)</formula>
<formula><location><page_5><loc_23><loc_16><loc_92><loc_21></location>a k = p k ln( t ) + C k , C k = const , k = 1 , N, ∑ n p 2 n = 1 , ∑ n p n = 1 (33)</formula>
<text><location><page_5><loc_9><loc_15><loc_65><loc_16></location>As a consequence, components of the metric tensor follows a power law:</text>
<text><location><page_5><loc_9><loc_9><loc_50><loc_10></location>This is well-known multidimensional Kasner solution.</text>
<formula><location><page_5><loc_31><loc_8><loc_92><loc_13></location>g kk ( t ) = e 2 a k ( t ) = ˜ C k t 2 p k , ˜ C k = const , k = 1 , N (34)</formula>
<text><location><page_5><loc_54><loc_71><loc_54><loc_73></location>/negationslash</text>
<text><location><page_5><loc_28><loc_58><loc_28><loc_58></location>/negationslash</text>
<section_header_level_1><location><page_6><loc_9><loc_92><loc_35><loc_93></location>III.3. Stability conditions.</section_header_level_1>
<text><location><page_6><loc_9><loc_78><loc_92><loc_90></location>Metric ceases to obey a power law when there is a matter; but it turns out that the metric of the space filled with the magnetic field may get close to the Kasner metric and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (28) that get close to the Kasner solutions (33) and converge to them as time tends to the point t = 0 . Considering an influence of the magnetic field as a perturbation we will search for solutions of the equations (28) in the form:</text>
<formula><location><page_6><loc_38><loc_76><loc_92><loc_77></location>a k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (35)</formula>
<text><location><page_6><loc_9><loc_71><loc_92><loc_75></location>where a 0 k is the Kasner solution (33), ϕ k ∈ C 2 ( R ) is a perturbation. We will say that solutions a 0 k are asymptotically stable if</text>
<formula><location><page_6><loc_40><loc_68><loc_92><loc_70></location>lim t → 0 a k ( t ) = a 0 k ( t ) , k = 1 , N (36)</formula>
<text><location><page_6><loc_9><loc_65><loc_61><loc_67></location>In other words, solutions a 0 k are asymptotically stable if and only if</text>
<formula><location><page_6><loc_41><loc_61><loc_92><loc_64></location>lim t → 0 ϕ k ( t ) = 0 , k = 1 , N (37)</formula>
<text><location><page_6><loc_9><loc_59><loc_75><loc_60></location>Now we will find out conditions which specify asymptotically stable Kasner solutions.</text>
<text><location><page_6><loc_9><loc_56><loc_92><loc_58></location>Proposition 1. Kasner solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 1 , n = 1 , χ .</text>
<unordered_list>
<list_item><location><page_6><loc_9><loc_54><loc_84><loc_55></location>/square We consider the space of an even dimension; that of an odd dimension is treated similar way.</list_item>
</unordered_list>
<text><location><page_6><loc_9><loc_48><loc_92><loc_53></location>1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (31)-(32) given by (33); then there exists solution { a 1 , . . . , a N } of the equations (28),(29) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that</text>
<formula><location><page_6><loc_38><loc_45><loc_92><loc_47></location>a k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (38)</formula>
<formula><location><page_6><loc_41><loc_42><loc_92><loc_44></location>lim t → 0 ϕ k ( t ) = 0 , k = 1 , N (39)</formula>
<text><location><page_6><loc_9><loc_39><loc_57><loc_41></location>Substitution (33),(38) to the equations (28) and (29) leads to:</text>
<text><location><page_6><loc_9><loc_33><loc_9><loc_34></location>/negationslash</text>
<text><location><page_6><loc_34><loc_33><loc_34><loc_34></location>/negationslash</text>
<text><location><page_6><loc_66><loc_33><loc_66><loc_34></location>/negationslash</text>
<formula><location><page_6><loc_9><loc_21><loc_93><loc_39></location>∑ i = <n> p i ( p i -1)+ ∑ j<k j,k = <n> p j p k + t ∑ i = <n> p i ˙ ϕ i + ∑ j<k j,k = <n> ( p j ˙ ϕ k + p k ˙ ϕ j ) + t 2 ∑ i = <n> ( ¨ ϕ i + ˙ ϕ 2 i ) + ∑ j<k j,k = <n> ˙ ϕ j ˙ ϕ k = = -ψ 2 2 n -1 , 2 n e -2( ϕ 2 n -1 + ϕ 2 n ) t 2 -2( p 2 n -1 + p 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( ϕ 2 j -1 + ϕ 2 j ) t 2 -2( p 2 j -1 + p 2 j ) (40) ∑ i<j p i p j + ∑ i<j ( p i ˙ ϕ j + p j ˙ ϕ i ) + ∑ i<j ˙ ϕ i ˙ ϕ j = ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( ϕ 2 j -1 + ϕ 2 j ) t 2 -2( p 2 i -1 + p 2 i ) (41)</formula>
<text><location><page_6><loc_23><loc_32><loc_23><loc_33></location>/negationslash</text>
<text><location><page_6><loc_9><loc_20><loc_28><loc_21></location>It follows from (39) that</text>
<text><location><page_6><loc_9><loc_13><loc_63><loc_16></location>Let t → 0 ; in view of (33),(39),(42) equations (40)-(41) take the form:</text>
<text><location><page_6><loc_54><loc_27><loc_54><loc_27></location>/negationslash</text>
<formula><location><page_6><loc_32><loc_15><loc_92><loc_20></location>lim t → 0 ( t ˙ ϕ k ( t ) ) = 0 , lim t → 0 ( t 2 ¨ ϕ k ( t ) ) = 0 , k = 1 , N (42)</formula>
<text><location><page_6><loc_54><loc_9><loc_54><loc_9></location>/negationslash</text>
<formula><location><page_6><loc_29><loc_9><loc_92><loc_14></location>0 = -ψ 2 2 n -1 , 2 n t 2 -2( p 2 n -1 + p 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 2 -2( p 2 j -1 + p 2 j ) (43)</formula>
<text><location><page_6><loc_45><loc_32><loc_45><loc_33></location>/negationslash</text>
<text><location><page_6><loc_82><loc_32><loc_82><loc_33></location>/negationslash</text>
<formula><location><page_7><loc_39><loc_89><loc_92><loc_94></location>0 = ∑ i =1 ,χ ψ 2 2 i -1 , 2 i t 2 -2( p 2 i -1 + p 2 i ) (44)</formula>
<text><location><page_7><loc_9><loc_88><loc_48><loc_89></location>Equalities (43) and (44) are satisfied if and only if</text>
<formula><location><page_7><loc_40><loc_84><loc_92><loc_86></location>p 2 n -1 + p 2 n < 1 , n = 1 , χ (45)</formula>
<text><location><page_7><loc_9><loc_81><loc_13><loc_83></location>q.e.d.</text>
<text><location><page_7><loc_9><loc_76><loc_92><loc_81></location>2. Let { a 0 1 , . . . , a 0 N } be the Kasner solution given by (33) such that p 2 n -1 + p 2 n < 1 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have:</text>
<formula><location><page_7><loc_38><loc_74><loc_92><loc_76></location>a k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (46)</formula>
<text><location><page_7><loc_9><loc_71><loc_50><loc_72></location>Let t = t 0 > 0 be the initial moment; we assume that</text>
<formula><location><page_7><loc_31><loc_64><loc_92><loc_70></location>| ϕ i ( t 0 ) | /lessmuch | a 0 k ( t 0 ) | , | ˙ ϕ i ( t 0 ) | /lessmuch ∣ ∣ ˙ a 0 k ( t 0 ) ∣ ∣ , i = 1 , N (47)</formula>
<text><location><page_7><loc_53><loc_53><loc_53><loc_53></location>/negationslash</text>
<formula><location><page_7><loc_37><loc_47><loc_92><loc_49></location>A Φ( t ) = R ( t ) , t ∈ ( t 0 -ε, t 0 + ε ) (49)</formula>
<formula><location><page_7><loc_29><loc_43><loc_92><loc_45></location>A k k = 0 , A k j = 1 , k = j ; Φ k ( t ) = ¨ ϕ k ( t ) , j, k = 1 , N (50)</formula>
<text><location><page_7><loc_55><loc_38><loc_55><loc_39></location>/negationslash</text>
<formula><location><page_7><loc_19><loc_38><loc_92><loc_43></location>R 2 n -1 ( t ) = R 2 n ( t ) = ψ 2 2 n -1 , 2 n t -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t -2( p 2 j -1 + p 2 j ) , n = 1 , χ (51)</formula>
<text><location><page_7><loc_9><loc_36><loc_57><loc_37></location>It is easy to check that matrix A is nonsingular, so we obtain:</text>
<formula><location><page_7><loc_44><loc_33><loc_92><loc_34></location>Φ( t ) = A -1 R ( t ) , (52)</formula>
<text><location><page_7><loc_9><loc_29><loc_11><loc_31></location>or,</text>
<formula><location><page_7><loc_21><loc_25><loc_92><loc_30></location>¨ ϕ k ( t ) = ∑ m =1 ,χ η km t λ m , η km = const , λ m = -2( p 2 m -1 + p 2 m ) , k = 1 , N (53)</formula>
<text><location><page_7><loc_9><loc_23><loc_26><loc_25></location>Hence we deduce that</text>
<formula><location><page_7><loc_24><loc_18><loc_92><loc_23></location>˙ ϕ k ( t ) = ∑ m =1 ,χ η km λ m +1 t λ m +1 , ϕ k ( t ) = ∑ m =1 ,χ η km λ m +2 t λ m +2 , k = 1 , N (54)</formula>
<text><location><page_7><loc_9><loc_16><loc_63><loc_17></location>It follows from p 2 m -1 + p 2 m < 1 , m = 1 , χ that λ m +2 > 0 ; therefore</text>
<formula><location><page_7><loc_41><loc_12><loc_92><loc_14></location>lim t → 0 ϕ k ( t ) = 0 , k = 1 , N (55)</formula>
<text><location><page_7><loc_9><loc_9><loc_69><loc_11></location>and solutions (33) with p 2 m -1 + p 2 m < 1 , m = 1 , χ are asymptotically stable.</text>
<text><location><page_7><loc_71><loc_9><loc_72><loc_11></location>/squaresolid</text>
<text><location><page_7><loc_45><loc_42><loc_45><loc_44></location>/negationslash</text>
<text><location><page_7><loc_9><loc_58><loc_92><loc_66></location>Substitution (33),(46) to the equations (28) and (29) give us equations (40)-(41). In view of (47) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j in the lhs of the equations (40)-(41) and terms contained ϕ i in the rhs of that equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (40) can be written in the form:</text>
<text><location><page_7><loc_23><loc_54><loc_23><loc_54></location>/negationslash</text>
<formula><location><page_7><loc_21><loc_52><loc_92><loc_58></location>t 2 ∑ i = <n> ¨ ϕ i = -ψ 2 2 n -1 , 2 n t 2 -2( p 2 n -1 + p 2 n ) + ∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 2 -2( p 2 j -1 + p 2 j ) , n = 1 , χ (48)</formula>
<text><location><page_7><loc_9><loc_50><loc_27><loc_51></location>or, in matrix notations,</text>
<text><location><page_7><loc_9><loc_45><loc_13><loc_47></location>where</text>
<section_header_level_1><location><page_8><loc_9><loc_92><loc_18><loc_93></location>Examples.</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_12><loc_82><loc_92><loc_90></location>· (3+1)-dimensional space-time; in view of (15) we have a single non-zero component of the Faraday tensor; according to our convention (see Sec.II.) we denote this component by ψ 12 ; under the Proposition 1 it follows that stable Kasner solutions are characterized by the condition p 1 + p 2 < 1 ; it implies that p 3 > 0 -the space expands along the magnetic field.</list_item>
<list_item><location><page_8><loc_12><loc_74><loc_92><loc_80></location>· (4+1)-dimensional space-time; in that case we have a couple non-zero components of the Faraday tensor: ψ 12 and ψ 34 ; in accordance with the Proposition 1 it follows that stable Kasner solutions are specified by the criteria p 1 + p 2 < 1 and p 3 + p 4 < 1 .</list_item>
</unordered_list>
<section_header_level_1><location><page_8><loc_9><loc_70><loc_39><loc_72></location>III.4. Numerical calculations.</section_header_level_1>
<text><location><page_8><loc_9><loc_63><loc_92><loc_69></location>The result presented above have been verified numerically. We have got a few hundreds numerical solutions with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found a lot of solutions that converge to the Kasner ones. Example of such solution is presented on the Fig. 1 (a).</text>
<text><location><page_8><loc_9><loc_54><loc_92><loc_62></location>It should be note that looking for asymptotically stable solutions numerically using the synchronous time t has a number of drawbacks; particularly, numerical solutions of the equations (68)-(69) increase too rapidly impeding computations as t goes to zero. To avoid these problem we have introduced new time coordinate τ by means of the relation</text>
<formula><location><page_8><loc_44><loc_52><loc_92><loc_54></location>dτ = e -∑ j a j ( t ) dt (56)</formula>
<text><location><page_8><loc_9><loc_47><loc_36><loc_50></location>The change of variables from t to τ Indeed, (33) and (56) leads to</text>
<text><location><page_8><loc_37><loc_49><loc_92><loc_50></location>results in transformation of logarithmic functions (33) into linear ones.</text>
<formula><location><page_8><loc_38><loc_43><loc_92><loc_47></location>dτ = C t dt, C = e -∑ j C j = const (57)</formula>
<text><location><page_8><loc_9><loc_41><loc_20><loc_43></location>Consequently,</text>
<formula><location><page_8><loc_36><loc_33><loc_92><loc_41></location>τ = C ln( t ) + ˜ C, ˜ C = const (58) a k ( τ ) = p k C τ + C k -p k ˜ C C , k = 1 , N (59)</formula>
<text><location><page_8><loc_9><loc_33><loc_38><loc_34></location>Without lack of generality we can set</text>
<text><location><page_8><loc_9><loc_26><loc_28><loc_27></location>Then, as was announced,</text>
<formula><location><page_8><loc_41><loc_26><loc_92><loc_31></location>C 1 = . . . = C N = ˜ C = 0 (60)</formula>
<formula><location><page_8><loc_41><loc_23><loc_92><loc_25></location>a k ( τ ) = p k τ, k = 1 , N (61)</formula>
<text><location><page_8><loc_9><loc_18><loc_92><loc_22></location>Such a way, looking for asymptotically stable solutions numerically, we should expect to get quasi-linear solutions like ones that shown on the Fig. 1 (a).</text>
<figure>
<location><page_9><loc_15><loc_72><loc_44><loc_93></location>
</figure>
<figure>
<location><page_9><loc_58><loc_73><loc_86><loc_93></location>
<caption>Figure 1: Numerical solutions for (5+1)-dimension space-time (Einstein gravity). The figure a) illustrates stability of the Kasner solution near the initial singularity. The figure b) shows decreasing of the determinant of the metric; it implies that any given element of volume dV = √ | det( g ) | d N x tend to zero and therefore we go to the initial singularity.</caption>
</figure>
<section_header_level_1><location><page_9><loc_9><loc_62><loc_80><loc_64></location>IV. Stability of power-law solutions in Gauss-Bonnet models.</section_header_level_1>
<section_header_level_1><location><page_9><loc_9><loc_59><loc_31><loc_60></location>IV.1. Field equations.</section_header_level_1>
<text><location><page_9><loc_9><loc_55><loc_50><loc_57></location>Action of the theory under consideration is given by:</text>
<formula><location><page_9><loc_35><loc_49><loc_92><loc_55></location>S = 1 16 π ∫ d N +1 x √ | det( g ) | ( L GB + L m ) (62)</formula>
<formula><location><page_9><loc_29><loc_47><loc_92><loc_49></location>L GB = R αβγδ R αβγδ -4 R αβ R αβ + R 2 , L m = F αβ F αβ (63)</formula>
<text><location><page_9><loc_9><loc_43><loc_92><loc_46></location>where R,R αβ , R αβγδ , F αβ are the ( N + 1) -dimensional scalar curvature, Ricci tensor, Riemann tensor and Faraday tensor respectively. The gravitational equations has the form:</text>
<formula><location><page_9><loc_46><loc_39><loc_92><loc_41></location>H µ ν = 8 πT µ ν , (64)</formula>
<text><location><page_9><loc_9><loc_36><loc_13><loc_37></location>where</text>
<formula><location><page_9><loc_14><loc_30><loc_92><loc_35></location>H µ ν = 2 RR µ ν -4 R µ γ R γ ν -4 R αβ R µ . ανβ +2 R µαβγ R ναβγ -1 2 δ µ ν ( R αβγδ R αβγδ -4 R αβ R αβ + R 2 ) (65)</formula>
<text><location><page_9><loc_9><loc_29><loc_72><loc_30></location>are the components of the Gauss-Bonnet tensor. Taking into account (1) we get:</text>
<formula><location><page_9><loc_45><loc_25><loc_92><loc_27></location>H µ ν = 0 , µ = ν (66)</formula>
<text><location><page_9><loc_55><loc_19><loc_55><loc_20></location>/negationslash</text>
<text><location><page_9><loc_69><loc_19><loc_69><loc_20></location>/negationslash</text>
<text><location><page_9><loc_54><loc_24><loc_54><loc_27></location>/negationslash</text>
<formula><location><page_9><loc_12><loc_19><loc_92><loc_24></location>H 0 0 = -12 ∑ i<j<k<l ˙ a i ˙ a j ˙ a k ˙ a l , H n n = -4 ∑ i = n ( a i + ˙ a 2 i ) ∑ j<k j,k = i,n ˙ a j ˙ a k -12 ∑ i<j<k<l i,j,k,l = n ˙ a i ˙ a j ˙ a k ˙ a l , n = 1 , N (67)</formula>
<text><location><page_9><loc_9><loc_15><loc_92><loc_18></location>For the diagonal case in view of (17),(18),(64),(67) for an even number of dimensions the gravitational equations can be written as:</text>
<text><location><page_9><loc_31><loc_10><loc_31><loc_11></location>/negationslash</text>
<formula><location><page_9><loc_31><loc_9><loc_71><loc_14></location>∑ i = <n> ( a i + ˙ a 2 i ) ∑ j<k j,k = i,<n> ˙ a j ˙ a k +3 ∑ i<j<k<l i,j,k,l = <n> ˙ a i ˙ a j ˙ a k ˙ a l =</formula>
<text><location><page_9><loc_44><loc_9><loc_44><loc_10></location>/negationslash</text>
<text><location><page_9><loc_59><loc_9><loc_59><loc_10></location>/negationslash</text>
<text><location><page_9><loc_44><loc_20><loc_44><loc_21></location>/negationslash</text>
<text><location><page_10><loc_33><loc_75><loc_33><loc_75></location>/negationslash</text>
<text><location><page_10><loc_47><loc_75><loc_47><loc_75></location>/negationslash</text>
<formula><location><page_10><loc_25><loc_82><loc_92><loc_94></location>= 1 4 ψ 2 2 n -1 , 2 n e -2( a 2 n -1 + a 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( a 2 j -1 + a 2 j ) , n = 1 , χ (68) ∑ i<j<k<l ˙ a i ˙ a j ˙ a k ˙ a l + 1 12 ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( a 2 i -1 + a 2 i ) = 0 , (69)</formula>
<text><location><page_10><loc_50><loc_88><loc_50><loc_88></location>/negationslash</text>
<text><location><page_10><loc_9><loc_80><loc_81><loc_82></location>There is one more equation in addition to these in the case of an odd number of dimensions:</text>
<text><location><page_10><loc_22><loc_75><loc_22><loc_76></location>/negationslash</text>
<formula><location><page_10><loc_22><loc_74><loc_92><loc_79></location>∑ i = N ( a i + ˙ a 2 i ) ∑ j<k j,k = i,N ˙ a j ˙ a k +3 ∑ i<j<k<l i,j,k,l = N ˙ a i ˙ a j ˙ a k ˙ a l = -1 4 ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( a 2 i -1 + a 2 i ) (70)</formula>
<text><location><page_10><loc_9><loc_72><loc_56><loc_73></location>The left side of the expression (69) is a first integral of (68).</text>
<section_header_level_1><location><page_10><loc_9><loc_67><loc_33><loc_69></location>IV.2. Vacuum solution.</section_header_level_1>
<text><location><page_10><loc_9><loc_64><loc_56><loc_66></location>When there is no any matter the equations (68)-(70) lead to:</text>
<text><location><page_10><loc_29><loc_60><loc_29><loc_60></location>/negationslash</text>
<formula><location><page_10><loc_28><loc_58><loc_92><loc_64></location>∑ i = n ( a i + ˙ a 2 i ) ∑ j<k j,k = i,n ˙ a j ˙ a k +3 ∑ i<j<k<l i,j,k,l = n ˙ a i ˙ a j ˙ a k ˙ a l = 0 , n = 1 , N (71)</formula>
<text><location><page_10><loc_39><loc_59><loc_39><loc_59></location>/negationslash</text>
<text><location><page_10><loc_53><loc_59><loc_53><loc_59></location>/negationslash</text>
<formula><location><page_10><loc_43><loc_53><loc_92><loc_58></location>∑ i<j<k<l ˙ a i ˙ a j ˙ a k ˙ a l = 0 (72)</formula>
<text><location><page_10><loc_9><loc_51><loc_47><loc_53></location>These equations has the following solutions [5, 6]:</text>
<formula><location><page_10><loc_19><loc_45><loc_92><loc_51></location>a k = p k ln( t ) + C k , C k = const , k = 1 , N, ∑ i<j<l<m p i p j p l p m = 0 , ∑ n p n = 3 (73)</formula>
<text><location><page_10><loc_9><loc_43><loc_65><loc_45></location>As a consequence, components of the metric tensor follows a power law:</text>
<formula><location><page_10><loc_31><loc_40><loc_92><loc_42></location>g kk ( t ) = e 2 a k ( t ) = C k t 2 p k , C k = const , k = 1 , N (74)</formula>
<text><location><page_10><loc_9><loc_36><loc_85><loc_38></location>Thus by analogy with (33)-(34) we will call solutions (73) and metric (74) Kasner-like for brevity.</text>
<section_header_level_1><location><page_10><loc_9><loc_32><loc_35><loc_34></location>IV.3. Stability conditions.</section_header_level_1>
<text><location><page_10><loc_9><loc_19><loc_92><loc_31></location>As well as in the case of the Einstein gravity (see III.3.), metric ceases to obey a power law when there is a matter; it appears however that the metric of the space filled with the magnetic field may get close to the Kasner-like metric (74) and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (68) that get close to the Kasnerlike solutions (73) and converge to them as time goes to the point t = 0 . We will looking for solutions of the equations (68) in the form:</text>
<formula><location><page_10><loc_38><loc_16><loc_92><loc_18></location>a k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (75)</formula>
<text><location><page_10><loc_9><loc_11><loc_92><loc_15></location>where a 0 k is the Kasner-like solution (73), ϕ k ∈ C 2 ( R ) . We will say that solutions a 0 k are asymptotically stable if</text>
<formula><location><page_10><loc_40><loc_9><loc_92><loc_11></location>lim t → 0 a k ( t ) = a 0 k ( t ) , k = 1 , N (76)</formula>
<text><location><page_11><loc_9><loc_91><loc_61><loc_93></location>In other words, solutions a 0 k are asymptotically stable if and only if</text>
<formula><location><page_11><loc_41><loc_87><loc_92><loc_90></location>lim t → 0 ϕ k ( t ) = 0 , k = 1 , N (77)</formula>
<text><location><page_11><loc_9><loc_85><loc_79><loc_86></location>Now we will find out conditions which specify asymptotically stable Kasner-like solutions.</text>
<text><location><page_11><loc_9><loc_82><loc_94><loc_84></location>Proposition 2 . Kasner-like solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 2 , n = 1 , χ .</text>
<unordered_list>
<list_item><location><page_11><loc_9><loc_74><loc_92><loc_81></location>/square We consider the space of an even dimension; that of an odd dimension is treated similar way. 1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (71)-(72) given by (73); then there exists solution { a 1 , . . . , a N } of the equations (68),(69) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that</list_item>
</unordered_list>
<formula><location><page_11><loc_38><loc_71><loc_92><loc_73></location>a k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (78)</formula>
<formula><location><page_11><loc_41><loc_67><loc_92><loc_69></location>lim t → 0 ϕ k ( t ) = 0 , k = 1 , N (79)</formula>
<text><location><page_11><loc_9><loc_65><loc_40><loc_67></location>Let us introduce the following notation:</text>
<formula><location><page_11><loc_19><loc_61><loc_92><loc_63></location>A ( i 1 B i 2 · . . . · B i N ) = A i 1 B i 2 · . . . · B i N + A i 2 B i 1 · . . . · B i N + . . . + A i N B i 2 · . . . · B i 1 (80)</formula>
<text><location><page_11><loc_9><loc_56><loc_92><loc_60></location>where A i 1 , . . . , A i N , B i 1 , . . . , B i N are any indexed mathematical objects. Substitution (73),(78) to the equations (68) and (69) leads to:</text>
<text><location><page_11><loc_31><loc_51><loc_31><loc_52></location>/negationslash</text>
<formula><location><page_11><loc_31><loc_50><loc_61><loc_55></location>∑ ∑ ∑</formula>
<text><location><page_11><loc_47><loc_44><loc_47><loc_44></location>/negationslash</text>
<formula><location><page_11><loc_16><loc_7><loc_92><loc_54></location>i = <n> p i ( p i -1) j<k j,k = i,<n> p j p k +3 i<j<k<l i,j,k,l = <n> p i p j p k p l + + t ∑ i = <n> p i ( p i -1) ∑ j<k j,k = i,<n> p ( j ˙ ϕ k ) +2 ∑ i = <n> p i ˙ ϕ i ∑ j<k j,k = i,<n> p j p k +3 ∑ i<j<k<l i,j,k,l = <n> ˙ ϕ ( i p j p k p l ) + + t 2 ∑ i = <n> p i ( p i -1) ∑ j<k j,k = i,<n> ˙ ϕ j ˙ ϕ k +2 ∑ i = <n> p i ˙ ϕ i ∑ j<k j,k = i,<n> p ( j ˙ ϕ k ) + t 2 ∑ i = <n> ( ¨ ϕ i + ˙ ϕ 2 i ) ∑ j<k j,k = i,<n> p j p k +3 ∑ i<j<k<l i,j,k,l = <n> ( ˙ ϕ ( i p k p l ) ˙ ϕ j + ˙ ϕ i ˙ ϕ ( j p k p l ) ) + t 3 2 ∑ i = <n> p i ˙ ϕ i ∑ j<k j,k = i,<n> ˙ ϕ j ˙ ϕ k + ∑ i = <n> ( ¨ ϕ i + ˙ ϕ 2 i ) ∑ j<k j,k = i,<n> p ( j ˙ ϕ k ) +3 ∑ i<j<k<l i,j,k,l = <n> p ( i ˙ ϕ j ˙ ϕ k ˙ ϕ l ) + + t 4 ∑ i = <n> ( ¨ ϕ i + ˙ ϕ 2 i ) ∑ j<k j,k = i,<n> ˙ ϕ j ˙ ϕ k +3 ∑ i<j<k<l i,j,k,l = <n> ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l = = 1 4 ψ 2 2 n -1 , 2 n e -2( ϕ 2 n -1 + ϕ 2 n ) t 4 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j e -2( ϕ 2 j -1 + ϕ 2 j ) t 4 -2( p 2 j -1 + p 2 j ) (81)</formula>
<text><location><page_11><loc_54><loc_9><loc_54><loc_9></location>/negationslash</text>
<formula><location><page_11><loc_15><loc_25><loc_16><loc_26></location>+</formula>
<text><location><page_11><loc_20><loc_44><loc_20><loc_44></location>/negationslash</text>
<text><location><page_11><loc_21><loc_23><loc_21><loc_24></location>/negationslash</text>
<text><location><page_11><loc_25><loc_30><loc_25><loc_31></location>/negationslash</text>
<text><location><page_11><loc_29><loc_37><loc_29><loc_38></location>/negationslash</text>
<text><location><page_11><loc_30><loc_23><loc_30><loc_23></location>/negationslash</text>
<text><location><page_11><loc_31><loc_16><loc_31><loc_17></location>/negationslash</text>
<text><location><page_11><loc_33><loc_43><loc_33><loc_44></location>/negationslash</text>
<text><location><page_11><loc_38><loc_29><loc_38><loc_30></location>/negationslash</text>
<text><location><page_11><loc_53><loc_29><loc_53><loc_30></location>/negationslash</text>
<text><location><page_11><loc_41><loc_23><loc_41><loc_24></location>/negationslash</text>
<text><location><page_11><loc_43><loc_36><loc_43><loc_37></location>/negationslash</text>
<text><location><page_11><loc_44><loc_50><loc_44><loc_51></location>/negationslash</text>
<text><location><page_11><loc_59><loc_50><loc_59><loc_51></location>/negationslash</text>
<text><location><page_11><loc_45><loc_16><loc_45><loc_16></location>/negationslash</text>
<text><location><page_11><loc_60><loc_16><loc_60><loc_16></location>/negationslash</text>
<text><location><page_11><loc_55><loc_23><loc_55><loc_23></location>/negationslash</text>
<text><location><page_11><loc_71><loc_23><loc_71><loc_23></location>/negationslash</text>
<text><location><page_11><loc_55><loc_37><loc_55><loc_38></location>/negationslash</text>
<text><location><page_11><loc_56><loc_43><loc_56><loc_44></location>/negationslash</text>
<text><location><page_11><loc_71><loc_43><loc_71><loc_44></location>/negationslash</text>
<text><location><page_11><loc_64><loc_36><loc_64><loc_37></location>/negationslash</text>
<text><location><page_12><loc_9><loc_14><loc_13><loc_15></location>where</text>
<text><location><page_12><loc_19><loc_84><loc_19><loc_85></location>/negationslash</text>
<text><location><page_12><loc_38><loc_84><loc_38><loc_85></location>/negationslash</text>
<formula><location><page_12><loc_12><loc_84><loc_92><loc_94></location>∑ i<j<k<l i,j,k,l = <n> p i p j p k p l + t ∑ i<j<k<l i,j,k,l = <n> ˙ ϕ ( i p j p k p l ) + t 2 ∑ i<j<k<l i,j,k,l = <n> ( ˙ ϕ ( i p k p l ) ˙ ϕ j + ˙ ϕ i ˙ ϕ ( j p k p l ) ) + + t 3 ∑ i<j<k<l i,j,k,l = <n> p ( i ˙ ϕ j ˙ ϕ k ˙ ϕ l ) + t 4 ∑ i<j<k<l i,j,k,l = <n> ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l = -1 12 ∑ i =1 ,χ ψ 2 2 i -1 , 2 i e -2( ϕ 2 j -1 + ϕ 2 j ) t 4 -2( p 2 i -1 + p 2 i ) (82)</formula>
<text><location><page_12><loc_9><loc_81><loc_28><loc_83></location>It follows from (79) that</text>
<text><location><page_12><loc_9><loc_75><loc_69><loc_78></location>In the limit t → 0 in view of (73),(79),(83) equations (81)-(82) take the form:</text>
<formula><location><page_12><loc_32><loc_77><loc_92><loc_82></location>lim t → 0 ( t ˙ ϕ k ( t ) ) = 0 , lim t → 0 ( t 2 ¨ ϕ k ( t ) ) = 0 , k = 1 , N (83)</formula>
<text><location><page_12><loc_55><loc_69><loc_55><loc_70></location>/negationslash</text>
<formula><location><page_12><loc_36><loc_63><loc_92><loc_68></location>0 = -1 12 lim t → 0 ∑ i =1 ,χ ψ 2 2 i -1 , 2 i t 4 -2( p 2 i -1 + p 2 i ) (85)</formula>
<formula><location><page_12><loc_26><loc_67><loc_92><loc_76></location>0 = 1 4 lim t → 0 ψ 2 2 n -1 , 2 n t 4 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 4 -2( p 2 j -1 + p 2 j ) (84)</formula>
<text><location><page_12><loc_9><loc_61><loc_48><loc_62></location>Equalities (84) and (85) are satisfied if and only if</text>
<formula><location><page_12><loc_40><loc_57><loc_92><loc_59></location>p 2 n -1 + p 2 n < 2 , n = 1 , χ (86)</formula>
<text><location><page_12><loc_9><loc_54><loc_13><loc_55></location>q.e.d.</text>
<text><location><page_12><loc_9><loc_49><loc_92><loc_53></location>2. Let { a 0 1 , . . . , a 0 N } be the Kasner-like solution given by (73) such that p 2 n -1 + p 2 n < 2 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have:</text>
<formula><location><page_12><loc_38><loc_46><loc_92><loc_48></location>a k ( t ) = a 0 k ( t ) + ϕ k ( t ) , k = 1 , N (87)</formula>
<text><location><page_12><loc_9><loc_42><loc_50><loc_44></location>Let t = t 0 > 0 be the initial moment; we assume that</text>
<formula><location><page_12><loc_31><loc_36><loc_92><loc_42></location>| ϕ i ( t 0 ) | /lessmuch | a 0 k ( t 0 ) | , | ˙ ϕ i ( t 0 ) | /lessmuch ∣ ∣ ˙ a 0 k ( t 0 ) ∣ ∣ , i = 1 , N (88)</formula>
<text><location><page_12><loc_57><loc_22><loc_57><loc_22></location>/negationslash</text>
<text><location><page_12><loc_9><loc_29><loc_92><loc_37></location>Substitution (73),(87) to the equations (68) and (69) give us equations (81)-(82). In view of (88) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j , t 3 ˙ ϕ i ˙ ϕ j ˙ ϕ k , t 4 ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l in the lhs of the equations (81)-(82) and terms contained ϕ i in the rhs of these equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (81) can be written in the form:</text>
<text><location><page_12><loc_15><loc_23><loc_15><loc_23></location>/negationslash</text>
<formula><location><page_12><loc_13><loc_20><loc_92><loc_29></location>t 2 ∑ i = <n> ¨ ϕ i ∑ j<k j,k = i,<n> p j p k = 1 4 ψ 2 2 n -1 , 2 n t 4 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 4 -2( p 2 j -1 + p 2 j ) , n = 1 , χ (89)</formula>
<text><location><page_12><loc_23><loc_22><loc_23><loc_23></location>/negationslash</text>
<text><location><page_12><loc_9><loc_19><loc_27><loc_20></location>or, in matrix notations,</text>
<formula><location><page_12><loc_37><loc_16><loc_92><loc_18></location>A Φ( t ) = R ( t ) , t ∈ ( t 0 -ε, t 0 + ε ) (90)</formula>
<text><location><page_12><loc_27><loc_9><loc_27><loc_10></location>/negationslash</text>
<text><location><page_12><loc_29><loc_9><loc_29><loc_10></location>/negationslash</text>
<text><location><page_12><loc_46><loc_9><loc_46><loc_10></location>/negationslash</text>
<text><location><page_12><loc_47><loc_9><loc_47><loc_10></location>/negationslash</text>
<formula><location><page_12><loc_19><loc_9><loc_92><loc_14></location>A k 2 n -1 = ∑ i<j i,j = k =2 n -1 p i p j , A k 2 n = ∑ i<j i,j = k =2 n p i p j , Φ k ( t ) = ¨ ϕ k ( t ) , n = 1 , χ, k = 1 , N (91)</formula>
<text><location><page_12><loc_22><loc_89><loc_22><loc_90></location>/negationslash</text>
<text><location><page_12><loc_38><loc_89><loc_38><loc_90></location>/negationslash</text>
<text><location><page_12><loc_57><loc_89><loc_57><loc_90></location>/negationslash</text>
<text><location><page_13><loc_9><loc_85><loc_49><loc_86></location>Assuming that matrix A is nonsingular 2 , we obtain:</text>
<formula><location><page_13><loc_16><loc_86><loc_92><loc_94></location>R 2 n -1 ( t ) = R 2 n ( t ) = 1 4 ψ 2 2 n -1 , 2 n t 2 -2( p 2 n -1 + p 2 n ) -∑ j =1 ,χ j = n ψ 2 2 j -1 , 2 j t 2 -2( p 2 j -1 + p 2 j ) , n = 1 , χ (92)</formula>
<formula><location><page_13><loc_44><loc_81><loc_92><loc_83></location>Φ( t ) = A -1 R ( t ) , (93)</formula>
<formula><location><page_13><loc_21><loc_73><loc_92><loc_78></location>¨ ϕ k ( t ) = ∑ m =1 ,χ δ km t γ m , δ km = const , γ m = 2 -2( p 2 m -1 + p 2 m ) , k = 1 , N (94)</formula>
<text><location><page_13><loc_9><loc_78><loc_11><loc_79></location>or,</text>
<text><location><page_13><loc_9><loc_72><loc_26><loc_73></location>Hence we deduce that</text>
<formula><location><page_13><loc_24><loc_66><loc_92><loc_71></location>˙ ϕ k ( t ) = ∑ m =1 ,χ δ km γ m +1 t γ m +1 , ϕ k ( t ) = ∑ m =1 ,χ δ km γ m +2 t γ m +2 , k = 1 , N (95)</formula>
<text><location><page_13><loc_9><loc_63><loc_63><loc_65></location>It follows from p 2 m -1 + p 2 m < 2 , m = 1 , χ that γ m +2 > 0 ; therefore</text>
<formula><location><page_13><loc_41><loc_59><loc_92><loc_61></location>lim t → 0 ϕ k ( t ) = 0 , k = 1 , N (96)</formula>
<text><location><page_13><loc_9><loc_56><loc_72><loc_57></location>and solutions (73) with p 2 m -1 + p 2 m < 2 , m = 1 , χ are asymptotically stable. /squaresolid</text>
<text><location><page_13><loc_9><loc_47><loc_92><loc_55></location>Example. Let us consider (5+1)-dimensional space-time. According to (15) there are pair of non-zero components of the Faraday tensor; taking into account our convention (see II. for details) we denote them ψ 12 and ψ 34 . Then under the Proposition 2 it follows that stable Kasner-like solutions are described by the conditions p 1 + p 2 < 2 and p 3 + p 4 < 2 .</text>
<section_header_level_1><location><page_13><loc_9><loc_43><loc_38><loc_45></location>IV.4. Numerical calculations.</section_header_level_1>
<text><location><page_13><loc_9><loc_34><loc_92><loc_42></location>The result obtained have been confirmed by numerical calculations. As well as in the study of the stability of the Kasner solutions (see III.3.) we have received a few hundreds solutions numerically with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found numerous solutions that converge to the Kasner-like ones. Example of such solution is presented on the Fig. 2 (a).</text>
<text><location><page_13><loc_9><loc_30><loc_92><loc_33></location>For the reasons mentioned above (see III.4.) one should use special time coordinate for numerical calculations. Let us introduce new time coordinate τ by the following way:</text>
<formula><location><page_13><loc_44><loc_26><loc_92><loc_28></location>dτ = e -1 3 ∑ j a j ( t ) dt (97)</formula>
<text><location><page_13><loc_9><loc_20><loc_92><loc_24></location>The change of variables from t to τ results in transformation of logarithmic functions (73) into linear ones. Indeed, (73) and (97) give us:</text>
<formula><location><page_13><loc_37><loc_16><loc_92><loc_20></location>dτ = C t dt, C = e -1 3 ∑ j C j = const (98)</formula>
<text><location><page_13><loc_56><loc_88><loc_56><loc_88></location>/negationslash</text>
<figure>
<location><page_14><loc_15><loc_72><loc_44><loc_93></location>
</figure>
<figure>
<location><page_14><loc_58><loc_73><loc_86><loc_93></location>
<caption>Figure 2: Numerical solutions for (5+1)-dimension space-time (Gauss-Bonnet gravity). The figure a) illustrates stability of the Kasner-like regime near the initial singularity. The figure b) shows decreasing of the determinant of the metric; it implies that any given element of volume dV = √ | det( g ) | d N x tend to zero and, consequently, we go to the initial singularity.</caption>
</figure>
<text><location><page_14><loc_9><loc_62><loc_20><loc_63></location>Consequently,</text>
<formula><location><page_14><loc_36><loc_54><loc_92><loc_62></location>τ = C ln( t ) + ˜ C, ˜ C = const (99) a k ( τ ) = p k C τ + C k -p k ˜ C C , k = 1 , N (100)</formula>
<text><location><page_14><loc_9><loc_53><loc_38><loc_55></location>Without loss of generality we can set</text>
<text><location><page_14><loc_9><loc_48><loc_28><loc_50></location>Then, as was announced,</text>
<formula><location><page_14><loc_41><loc_48><loc_92><loc_53></location>C 1 = . . . = C N = ˜ C = 0 (101)</formula>
<formula><location><page_14><loc_41><loc_46><loc_92><loc_48></location>a k ( τ ) = p k τ, k = 1 , N (102)</formula>
<text><location><page_14><loc_9><loc_41><loc_92><loc_45></location>Thus, searching for asymptotically stable solutions numerically we should expect to get quasi-linear solutions like ones that shown on the Fig. 2 (a).</text>
<section_header_level_1><location><page_14><loc_9><loc_36><loc_36><loc_38></location>V. Oscillatory regime.</section_header_level_1>
<text><location><page_14><loc_9><loc_13><loc_92><loc_34></location>Results of the preceding section pose an interesting question about the fate of trajectories in the instability zone. They can either reach stable Kasner-like attractor or represent some other type of dynamics different from Kasner-like behavior. In this section we show that the latter alternative can be realized. In our numerical calculations we have revealed a specific class of solutions of the equations (68). The discovered solutions possess the following properties: two of N functions a 1 , . . . , a N oscillate and has opposite signs, the other ones vary slowly; owing to phase difference between the oscillating functions volume element √ | det( g ) | d N x oscillates too. In view of limited computational capabilities we have obtained numerical solutions of the equations (68) only for the cases N = 5 , 6 , 7 , 8 ; for each of that dimensions we have found out sufficiently large number of oscillatory solutions. Example of such solution is presented on the Fig. 3 (a); the corresponding oscillations of the volume element is shown on the Fig. 3 (b).</text>
<text><location><page_14><loc_9><loc_9><loc_92><loc_13></location>For N = 5 we have been able to derive an analytical approximation to such solutions on a certain interval of time. Below we describe the procedure of derivation of this approximation. Henceforth we use</text>
<figure>
<location><page_15><loc_15><loc_40><loc_86><loc_93></location>
<caption>Figure 3: Numerical solutions for (5+1)-dimension space-time (Gauss-Bonnet gravity). The figure a) depicts oscillatory solution which does not tend to Kasner-like solutions both to the past and to the future. The figure b) shows the oscillatory behavior of the determinant of the metric; it means that any given volume element dV = √ | det( g ) | d N x oscillates in this regime.</caption>
</figure>
<text><location><page_15><loc_9><loc_30><loc_81><loc_31></location>time coordinate τ introduced by (97). In the (5+1)-dimensional case equations (68) become:</text>
<formula><location><page_15><loc_20><loc_23><loc_92><loc_28></location>( a ' 2 [ a ' 3 a ' 4 + a ' 3 a ' 5 + a ' 4 a ' 5 ] + a ' 3 a ' 4 a ' 5 ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 12 e -2 a 1 -2 a 2 -2 ψ 2 34 e -2 a 3 -2 a 4 ) (103)</formula>
<formula><location><page_15><loc_21><loc_12><loc_92><loc_18></location>( a ' 1 a ' 2 [ a ' 4 + a ' 5 ] + a ' 4 a ' 5 [ a ' 1 + a ' 2 ] ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 34 e -2 a 3 -2 a 4 -2 ψ 2 12 e -2 a 1 -2 a 2 ) (105)</formula>
<formula><location><page_15><loc_20><loc_17><loc_92><loc_23></location>( a ' 1 [ a ' 3 a ' 4 + a ' 3 a ' 5 + a ' 4 a ' 5 ] + a ' 3 a ' 4 a ' 5 ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 12 e -2 a 1 -2 a 2 -2 ψ 2 34 e -2 a 3 -2 a 4 ) (104)</formula>
<formula><location><page_15><loc_21><loc_8><loc_92><loc_13></location>( a ' 1 a ' 2 [ a ' 3 + a ' 5 ] + a ' 3 a ' 5 [ a ' 1 + a ' 2 ] ) ' = 1 6 e 4 3 5 ∑ j =1 a j ( ψ 2 34 e -2 a 3 -2 a 4 -2 ψ 2 12 e -2 a 1 -2 a 2 ) (106)</formula>
<text><location><page_16><loc_9><loc_24><loc_14><loc_25></location>so that</text>
<text><location><page_16><loc_9><loc_16><loc_85><loc_21></location>Let us decompose functions f, g, h with respect to ( 4 3 v -2 3 u ) and ( 4 3 u -2 3 v ) up to the first order:</text>
<formula><location><page_16><loc_21><loc_88><loc_92><loc_94></location>( a ' 1 a ' 2 [ a ' 3 + a ' 4 ] + a ' 3 a ' 4 [ a ' 1 + a ' 2 ] ) ' = -1 3 e 4 3 ∑ j a j ( ψ 2 12 e -2 a 1 -2 a 2 + ψ 2 34 e -2 a 3 -2 a 4 ) (107)</formula>
<text><location><page_16><loc_9><loc_88><loc_26><loc_89></location>with the first integral</text>
<formula><location><page_16><loc_10><loc_81><loc_92><loc_86></location>a ' 1 a ' 2 a ' 3 a ' 4 + a ' 1 a ' 2 a ' 3 a ' 5 + a ' 1 a ' 2 a ' 4 a ' 5 + a ' 1 a ' 3 a ' 4 a ' 5 + a ' 2 a ' 3 a ' 4 a ' 5 = -1 12 e 4 3 ∑ j a j ( ψ 2 12 e -2 a 1 -2 a 2 + ψ 2 34 e -2 a 3 -2 a 4 ) (108)</formula>
<text><location><page_16><loc_9><loc_78><loc_92><loc_82></location>The dash denotes derivative with respect to τ . Analysis of numerical solutions shows that there exists τ 0 > 0 ( τ = 0 is accepted as the initial moment) such that</text>
<formula><location><page_16><loc_37><loc_74><loc_92><loc_76></location>a ' 5 ( τ ) ≈ 0 , a ' 3 ( τ ) ≈ a ' 4 ( τ ) , τ > τ 0 (109)</formula>
<text><location><page_16><loc_9><loc_71><loc_60><loc_73></location>(see Fig. 6). In view of (109) equations (103)-(108) take the form:</text>
<formula><location><page_16><loc_24><loc_65><loc_92><loc_70></location>( a ' 1 a ' 3 a ' 4 ) ' = f 2 , ( a ' 2 a ' 3 a ' 4 ) ' = f 2 , ( a ' 1 a ' 2 a ' 3 ) ' = g 2 , ( a ' 1 a ' 2 a ' 4 ) ' = g 2 (110)</formula>
<formula><location><page_16><loc_45><loc_59><loc_92><loc_62></location>a ' 1 a ' 2 a ' 3 a ' 4 = h 4 (112)</formula>
<formula><location><page_16><loc_36><loc_61><loc_92><loc_66></location>( a ' 1 a ' 2 [ a ' 3 + a ' 4 ] + a ' 3 a ' 4 [ a ' 1 + a ' 2 ] ) ' = h (111)</formula>
<text><location><page_16><loc_9><loc_57><loc_13><loc_58></location>where</text>
<formula><location><page_16><loc_10><loc_51><loc_92><loc_56></location>f = ( κ 1 e 4 3 v -2 3 u -2 κ 2 e 4 3 u -2 3 v ) , g = ( -2 κ 1 e 4 3 v -2 3 u + κ 2 e 4 3 u -2 3 v ) , h = -( κ 1 e 4 3 v -2 3 u + κ 2 e 4 3 u -2 3 v ) (113)</formula>
<formula><location><page_16><loc_40><loc_49><loc_92><loc_51></location>v = a 3 + a 4 , u = a 1 + a 2 (114)</formula>
<formula><location><page_16><loc_42><loc_45><loc_92><loc_49></location>κ 1 = ψ 2 12 3 , κ 2 = ψ 2 34 3 (115)</formula>
<text><location><page_16><loc_9><loc_43><loc_88><loc_45></location>Equation (111) is the sum of the equations (110); equations (110) can be replaced by their equivalent:</text>
<formula><location><page_16><loc_12><loc_37><loc_92><loc_43></location>( [ a ' 1 + a ' 2 ] a ' 3 a ' 4 ) ' = f, ( [ a ' 1 -a ' 2 ] a ' 3 a ' 4 ) ' = 0 , ( a ' 1 a ' 2 [ a ' 3 + a ' 4 ] ) ' = g, ( a ' 1 a ' 2 [ a ' 3 -a ' 4 ] ) ' = 0 (116)</formula>
<text><location><page_16><loc_9><loc_36><loc_89><loc_37></location>In view of (109) the last equation from (116) is satisfied automatically, and the other equations become</text>
<formula><location><page_16><loc_30><loc_30><loc_92><loc_35></location>( u ' v ' 2 ) ' = 4 f, ( a ' 1 a ' 2 v ' ) ' = g, ( [ a ' 1 -a ' 2 ] v ' 2 ) ' = 0 (117)</formula>
<text><location><page_16><loc_9><loc_29><loc_35><loc_30></location>The last of equations (117) gives:</text>
<formula><location><page_16><loc_40><loc_26><loc_92><loc_29></location>a ' 1 -a ' 2 = C v ' 2 , C = const (118)</formula>
<formula><location><page_16><loc_35><loc_20><loc_92><loc_25></location>a ' 1 = 1 2 ( u ' + C v ' 2 ) , a ' 2 = 1 2 ( u ' -C v ' 2 ) (119)</formula>
<formula><location><page_16><loc_33><loc_12><loc_92><loc_17></location>f ≈ [ κ 1 -2 κ 2 ] -2 3 [ κ 1 +4 κ 2 ] u + 4 3 [ κ 1 + κ 2 ] v (120)</formula>
<formula><location><page_16><loc_33><loc_7><loc_92><loc_13></location>g ≈ [ κ 2 -2 κ 1 ] + 4 3 [ κ 1 + κ 2 ] u -2 3 [ κ 2 +4 κ 1 ] v (121)</formula>
<text><location><page_17><loc_9><loc_15><loc_20><loc_16></location>Consequently,</text>
<formula><location><page_17><loc_32><loc_89><loc_92><loc_94></location>h ≈ -[ κ 1 + κ 2 ] + 2 3 [ κ 1 -2 κ 2 ] u + 2 3 [ κ 2 -2 κ 1 ] u (122)</formula>
<text><location><page_17><loc_9><loc_88><loc_92><loc_90></location>Numerical results show that such assumption is valid in a certain interval of time. Taking into account (119)-</text>
<text><location><page_17><loc_9><loc_86><loc_19><loc_88></location>(122) we get:</text>
<formula><location><page_17><loc_29><loc_81><loc_92><loc_87></location>( u ' v ' 2 ) ' = 4 [ κ 1 -2 κ 2 ] -8 3 [ κ 1 +4 κ 2 ] u + 16 3 [ κ 1 + κ 2 ] v (123)</formula>
<text><location><page_17><loc_9><loc_74><loc_92><loc_78></location>Let us consider the equation (124). Numerical calculations show that in the regime under consideration v '' / ( v ' ) 4 is the leading term in the lhs of the equation (124); then:</text>
<formula><location><page_17><loc_25><loc_77><loc_92><loc_83></location>([ ( u ' ) 2 -C 2 v ' 4 ] v ' ) ' = 4 [ κ 2 -2 κ 1 ] + 16 3 [ κ 1 + κ 2 ] u -8 3 [ κ 2 +4 κ 1 ] v (124)</formula>
<formula><location><page_17><loc_30><loc_68><loc_92><loc_73></location>C 2 v '' v ' 4 = 4 [ κ 2 -2 κ 1 ] + 16 3 [ κ 1 + κ 2 ] u -8 3 [ κ 2 +4 κ 1 ] v (125)</formula>
<text><location><page_17><loc_9><loc_67><loc_92><loc_68></location>We can obtain analytical solution of the equation (125) by omitting the term containing u in the rhs of (125):</text>
<formula><location><page_17><loc_38><loc_60><loc_92><loc_66></location>v '' v ' 4 = 4 [ κ 2 -2 κ 1 ] -8 3 [ κ 2 +4 κ 1 ] v (126)</formula>
<text><location><page_17><loc_9><loc_59><loc_43><loc_61></location>Solution of the equation (126) has the form</text>
<formula><location><page_17><loc_39><loc_56><loc_92><loc_58></location>v ( τ ) = η √ τ + λ, η, λ = const (127)</formula>
<text><location><page_17><loc_9><loc_52><loc_77><loc_54></location>Let us consider the equation (123). We omit the term, containing v in the rhs of (123):</text>
<formula><location><page_17><loc_36><loc_46><loc_92><loc_52></location>( u ' v ' 2 ) ' = 4 [ κ 1 -2 κ 2 ] -8 3 [ κ 1 +4 κ 2 ] u (128)</formula>
<text><location><page_17><loc_9><loc_45><loc_47><loc_47></location>Under substitution of (127) into (128) we obtain:</text>
<formula><location><page_17><loc_38><loc_41><loc_92><loc_43></location>τu '' ( τ ) -u ' ( τ ) + ξ 2 τ 2 u ( τ ) = ζτ 2 (129)</formula>
<text><location><page_17><loc_9><loc_38><loc_38><loc_40></location>We have used the following notations:</text>
<formula><location><page_17><loc_35><loc_33><loc_92><loc_38></location>ξ 2 = 32 [ κ 1 +4 κ 2 ] 3 η 2 , ζ = 16 [ κ 1 -2 κ 2 ] η 2 (130)</formula>
<text><location><page_17><loc_9><loc_30><loc_42><loc_32></location>Equation (129) has the following solution:</text>
<formula><location><page_17><loc_24><loc_25><loc_92><loc_30></location>u ( τ ) = τC 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + τC 2 Y 2 3 ( 2 ξτ 3 / 2 3 ) + ζ ξ 2 , C 1 , C 2 = const (131)</formula>
<text><location><page_17><loc_9><loc_20><loc_92><loc_23></location>Here J 2 3 , Y 2 3 are Bessel functions of the first and the second kind respectively. Going back to the initial variables we get:</text>
<formula><location><page_17><loc_32><loc_17><loc_92><loc_20></location>a ' 1 -a ' 2 = C v ' 2 , a 3 + a 4 = v, a ' 3 = a ' 4 , a ' 5 = 0 (132)</formula>
<formula><location><page_17><loc_21><loc_8><loc_92><loc_13></location>a 1 ( τ ) -a 2 ( τ ) = 2 C η 2 τ 2 + C, a 3 ( τ ) -˜ C = a 4 ( τ ) = η √ τ + λ, C, ˜ C = const (133)</formula>
<text><location><page_18><loc_9><loc_92><loc_18><loc_93></location>and, finally:</text>
<text><location><page_18><loc_9><loc_77><loc_13><loc_78></location>where</text>
<formula><location><page_18><loc_26><loc_88><loc_92><loc_93></location>a 1 ( τ ) ≈ -C 0 τ 2 + τ [ C 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + C 2 Y 2 3 ( 2 ξτ 3 / 2 3 )] + D 1 (134)</formula>
<formula><location><page_18><loc_27><loc_82><loc_92><loc_87></location>a 2 ( τ ) ≈ C 0 τ 2 + τ [ C 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + C 2 Y 2 3 ( 2 ξτ 3 / 2 3 )] + D 2 (135)</formula>
<formula><location><page_18><loc_30><loc_79><loc_92><loc_82></location>a 3 ( τ ) ≈ η √ τ + D 3 , a 4 ( τ ) ≈ η √ τ + λ, a 5 ( τ ) ≈ D, (136)</formula>
<formula><location><page_18><loc_41><loc_75><loc_92><loc_76></location>C 0 , D 1 , D 2 , D 3 , D = const (137)</formula>
<text><location><page_18><loc_9><loc_69><loc_92><loc_74></location>To demonstrate oscillations of volume element √ | det( g ) | d N x one should write down expression for the determinant of the metric:</text>
<formula><location><page_18><loc_14><loc_64><loc_92><loc_70></location>| det( g ) | = e 2( a 1 + ... + a 5 ) = exp { 4 ( η √ τ + τ [ C 1 J 2 3 ( 2 ξτ 3 / 2 3 ) + C 2 Y 2 3 ( 2 ξτ 3 / 2 3 )] + D )} (138)</formula>
<formula><location><page_18><loc_36><loc_61><loc_92><loc_63></location>D = D + D 1 + D 2 + D 3 + λ = const (139)</formula>
<text><location><page_18><loc_9><loc_52><loc_92><loc_60></location>Despite the fact that functions (134)-(136) are the solution of the simplified system (126),(128) they catch correctly qualitative features of solution of the full system (103)-(107) which can be shown by comparing functions (134)-(136) with numerical solution of the system (103)-(107). Figures 4,5 below depicts pairs ( a 2 , A 2 ); ( a 4 , A 4 ) of numerical and analytical solutions; pairs ( a 1 , A 1 ); ( a 3 , A 3 ) behave themselves analogously.</text>
<text><location><page_18><loc_9><loc_46><loc_92><loc_51></location>This regime is not similar to Kasner-like one (Kasner-like solution has linear behavior on chosen time coordinate τ (see IV.4. for details), the leading asymptotic of the oscillatory solution has quadratic behaviour w.r.t. τ ).</text>
<figure>
<location><page_19><loc_9><loc_20><loc_92><loc_87></location>
<caption>Figure 4: a) Comparison of numerical ( a 2 ) and analytical ( A 2 ) solutions. b) Relative difference of numerical ( a 2 ) and analytical ( A 2 ) solutions. A 1 ( τ ) = -τ 2 +0 . 7 τJ 2 3 ( 10 τ 3 2 ) +0 . 7 τY 2 3 ( 10 τ 3 2 ) +15 . 2 .</caption>
</figure>
<figure>
<location><page_20><loc_9><loc_18><loc_92><loc_88></location>
<caption>Figure 5: a) Comparison of numerical ( a 4 ) and analytical ( A 4 ) solutions. b) Relative difference of numerical ( a 4 ) and analytical ( A 4 ) solutions. A 4 ( τ ) = 0 . 741 √ τ -0 . 749 .</caption>
</figure>
<figure>
<location><page_21><loc_9><loc_19><loc_92><loc_88></location>
<caption>Figure 6: a) The figure shows that the derivative a ' 5 ( τ ) asymptotically tends to zero. b) The figure indicates that the difference a ' 3 ( τ ) -a ' 4 ( τ ) asymptotically tends to zero.</caption>
</figure>
<section_header_level_1><location><page_22><loc_9><loc_92><loc_29><loc_93></location>VI. Conclusions</section_header_level_1>
<text><location><page_22><loc_9><loc_73><loc_92><loc_90></location>In this paper we provided an analysis of stability of known power-law solutions in Gauss-Bonnet anisotropic cosmology in the presence of a homogeneous magnetic field in the case when metric is diagonal in the frame determined by magnetic field. The relative simplicity of equations of motion in this case allows us to find analytically a simple condition for the particular power-law solution to be stable. It appears that a set of initial conditions for which the power-law solution can not be an attractor near a cosmological singularity has non-zero measure (though does not coincide with the whole initial conditions space). The result has the same structure as in the Einstein gravity (particular sums of Kasner indices should be restricted by some number), the only difference is the exact value of this number.</text>
<text><location><page_22><loc_9><loc_67><loc_92><loc_72></location>The fate of trajectories near singularity from this unstable domain requires future investigations. In the present paper we describe one particular class of trajectories found by numerical studies. The volume element experiences oscillations in the presented regime, making it qualitatively different from BKL oscillations.</text>
<text><location><page_22><loc_9><loc_52><loc_92><loc_66></location>We can expect that some other regimes which do not tend to Kasner-like power-law regime exist in the system under investigation. Full description of them (the task which we leave for a future work) in this particular rather simple model can be useful in studies of cosmological singularity in Lovelock gravity in the presence of spatial curvature. The latter model usually lead to very cumbersome equations of motion, so understanding the nature of possible regimes in the instability zone for the Kasner-like solution in the present model would help in finding similar solution in more general and more technically complex situations of curved multidimensional geometries.</text>
<section_header_level_1><location><page_22><loc_9><loc_47><loc_29><loc_49></location>Acknowledgments</section_header_level_1>
<text><location><page_22><loc_9><loc_41><loc_92><loc_45></location>The authors are grateful to Vladimir Ivashchuk for helpful discussion. The work is partially supported by the RFBR grant 11-02-00643.</text>
<section_header_level_1><location><page_22><loc_9><loc_36><loc_21><loc_38></location>References</section_header_level_1>
<code><location><page_22><loc_10><loc_10><loc_78><loc_34></location>[1] V.A.Belinskii, I.M.Khalatnikov and E.M.Lifshitz, Adv.Phys. 19, 525 (1970). 1 [2] Victor G. LeBlanc, Class.Quantum.Grav. 14, 2281 (1997). 1 [3] R. Benini, A.A. Kirillov, G. Montani, Class.Quant.Grav. 22, 1483-1491 (2005). 1 [4] G. Rosen, J.Math.Phys. 3, 313 (1962). 1 [5] N. Deruelle, Nucl.Phys. B 327, 253-266 (1989). 2, 10 [6] N. Deruelle, L. Farina Busto, Phys.Rev. D 41, 3696 (1990). 2, 10 [7] A. Toporensky, P.Tretyakov, Grav.Cosmol. 13, 207-210 (2007); arXive: 0705.1346. 2 [8] S.A. Pavluchenko, Phys.Rev. D 80, 107501 (2009); arXiv: 0906.0141. 2</code>
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<list_item><location><page_23><loc_10><loc_90><loc_92><loc_93></location>[9] I. Kirnos, A. Makarenko, S. Pavluchenko and A. Toporensky, Gen.Rel.Grav. 42, 2633 (2010) arXive: 0906.0140. 2</list_item>
<list_item><location><page_23><loc_9><loc_86><loc_91><loc_88></location>[10] I.Kirnos, S. Pavluchenko and A. Toporensky, Grav. Cosmol., 16, 274-282 (2010) arXive: 1002.4488. 2</list_item>
<list_item><location><page_23><loc_9><loc_83><loc_66><loc_84></location>[11] V. Ivashchuk, Grav.Cosmol., 16, 118-125 (2010) arXive: 0910.3426. 2</list_item>
<list_item><location><page_23><loc_9><loc_80><loc_77><loc_81></location>[12] V. Ivashchuk, Int.J.Geom.Meth.Mod.Phys., 7, 797-819 (2010) arXive: 0910.3426. 2</list_item>
<list_item><location><page_23><loc_9><loc_76><loc_68><loc_78></location>[13] N.V. Mitskievich, Rev.Mex.Fis., 49S2, 39-51 (2003) arXive: 0202032. 1</list_item>
<list_item><location><page_23><loc_9><loc_71><loc_92><loc_75></location>[14] N.V. Mitskievich, Relativistic Physics in Arbitrary Reference Frames , Nova Science Publishers, 2006 arXive: 9606051 1</list_item>
<list_item><location><page_23><loc_9><loc_66><loc_92><loc_69></location>[15] N.V. Mitskievich, Electromagnetism and perfect fluids interplay in multidimensional spacetimes , in Proc. MG11 (2006) arXive: 0707.3190 1</list_item>
<list_item><location><page_23><loc_9><loc_62><loc_60><loc_64></location>[16] Kasner E., American Journal of Math., 43, 217-221 (1921). 5</list_item>
</unordered_list>
</document>
[
{
"title": "On stability of power-law solution in multidimensional Gauss-Bonnet cosmology",
"content": "D.M.Chirkov 1,2 ∗ , A.V.Toporensky 1 † July 16, 2018 1 Sternberg Astronomical Institute, Moscow State University, Universitetsky pr., 13, Moscow, 119991, Russia 2 Faculty of Physics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "We consider dynamics of a flat anisotropic multidimensional cosmological model in Gauss-Bonnet gravity in the presence of a homogeneous magnetic field. In particular, we find conditions under which the known power-law vacuum solution can be an attractor for the case with non-zero magnetic field. We also describe a particular class of numerical solution in (5 + 1) -dimensional case which does not approach the power-law regime.",
"pages": [
1
]
},
{
"title": "I. Introduction.",
"content": "The fact that structure of initial cosmological singularity can be rather complicated have been recognized since 60-th of the last century when the conception of BKL chaos have been presented [1]. It appears that Kasner solution being a general solution for a vacuum Bianchi I Universe becomes unstable in the case of metric with positive spatial curvature (belonging to Bianchi IX class) and is replaced by a complicated sequence of transient \"Kasner epochs\". Later it was found that some classes of an anisotropic matter can induce similar type of cosmological behavior even in flat Bianchi I case. This can be shown for a magnetic field by LeBlanc [2] and for a general vector field by Kirillov [3]. The BKL analogs for the flat magnetic Universe is a typical behavior when off-diagonal terms in the metric in the frame determined by magnetic field are present. For a diagonal case see for example [4]. Possible generalisation of these results may involve studies in the framework of multidimensional cosmology as well as alternative gravity theories. Analogs of magnetic field for multidimensional space-times have been studied in [13, 14, 15]. If we want to introduce some corrections to Einstein gravity, the first goal is to find a solution which replaces Kasner solution for a flat anisotropic Universe. In the present paper we consider Lovelock gravity as a generalization of General Relativity. The principal feature of Lovelock gravity is that this theory keeps the order of equations of motion the same as in GR, while other theories (like popular now f ( R ) -theory) results in increasing of the number of derivatives in equations of motion. Another important property of Lovelock gravity is that it gives corrections to GR only in higher-dimensional space-time, so it is natural to consider it in the framework of multidimensional cosmology. As the number of non-Einstein terms in Lovelock gravity is finite for any given dimensionality of space-time, it is possible to consider regimes when these terms are not only small corrections to Einstein gravity. In particular, it is reasonable to expect that the highest order Lovelock term, consisting of highest power of curvature invariants, dominates near a cosmological singularity. In the present paper we consider the second Lovelock term, which is the famous Gauss-Bonnet combination. It is the highest possible Lovelock term for (4 + 1) and (5 + 1) dimensional spacetimes. As cosmology in the (4 + 1) dimensional case has some pathological features, we restrict ourselves by the ( N +1) dimensional case with N /greaterorequalslant 5 in the present paper. In late 80-th some vacuum solutions for a flat anisotropic Universe in Gauss-Bonnet gravity for N = 4 , 5 have been found [5, 6]. They replace Kasner solution of GR. Later these solutions have been rediscovered and verified for N = 6 , 7 in [7]; after that this solution have been generalized for all N and also to the general Lovelock gravity [8]. The main goal of this paper is to address question of its stability near the initial singularity. It is known that in the presence of an ordinary isotropic matter these solutions are stable to the past if the matter has the equation of state with w < 1 / 3 , otherwise the solution tends to isotropic one [9, 10]. For more interesting regimes near a singularity it is necessary to consider either anisotropic matter or curved geometries. As introduction of spatial curvature usually leads to very cumbersome equations of motion, we have chosen a magnetic fields as a possible source of instability. In this paper we restrict ourselves by the diagonal case. It is also known that in contrast to Einstein gravity certain initial conditions lead to exponential solutions instead to those of power-law behavior [9, 11, 12], we do not consider them in the present paper. The structure of the paper is as follows: In Sec. II. we describe the metric and matter content of the model studied, in Sec. III. the model is investigated in the framework of Einstein relativity. We present results on stability of Kasner solution in the presence of magnetic field. The same model in Gauss-Bonnet gravity is studied in Sec. IV. with presentation made as parallel as possible to presentation in the preceding section in order to compare Einstein and Gauss-Bonnet cases. In Sec. V. we describe a particular regime existing in the zone of instability of power-law regimes in (5+1) -dimensional Gauss-Bonnet gravity. Sec. VI. contains a brief summary of results obtained.",
"pages": [
1,
2
]
},
{
"title": "II. Preliminaries.",
"content": "In what follows we use a reference system chosen so that 1 : /negationslash In this section we looking for the energy-momentum tensor of a pure magnetic field in ( N +1) -dimensional space-time. In the general case components of the energy-momentum tensor of the electromagnetic field are where F αβ is the Faraday tensor. The components of the Faraday tensor obey the following equations: or, in more detail: where D µ = ∂ ∂x µ . As it is known therefore we may rewrite system (4) in the form The components of the Riemann connection have the form: Its other components are equal to zero. Since we consider the case of a homogenous space, Faraday tensor depends on time t only; then, in view of (7) the system (6) takes the form: Its solutions are Hereinafter we will be interested in the case of a pure magnetic field; so that φ m = 0 and, as a consequence, In what follows we will deal with diagonal energy-momentum tensor; in view of (1),(2),(9) and (10) it implies that /negationslash /negationslash Functions e -2 a l are linearly independent, therefore from (11) it follows that /negationslash Each ψ ij is multiplied by all that ψ kl , which has one of the indices k, l coincident with one of the indices i, j . /negationslash Let us fix pair ( i, j ) ; the number of combinations, in which ψ ij is found, equals to 2( N -2) . Let ψ ij = 0 ; then the other 2( N -2) quantities ψ ik ( ψ jk ), which are multiplied by ψ ij , must be equals to zero; indeed, if ψ ik = 0 ( k = i, j ) , then ψ ij ψ ik = 0 , but that contradict (12). Continuing the argument, we find the number of zero magnetic field components: /negationslash /negationslash /negationslash We took into account that the number of pairs of indices without the same elements equals to N -1 2 for an odd dimensions and to N 2 for an even dimensions. The total number of components of the magnetic field is N ( N -1) 2 ; then the number χ of non-zero magnetic field components is In what follows they are exactly ψ 12 , ψ 34 , . . . , ψ 2 n -1 , 2 n , n = 1 , χ that we set to be non-zero; other components of the Faraday tensor are assumed to be zero. Thus, components of the energy-momentum tensor of the electromagnetic field take the form: /negationslash /negationslash Hereinafter it will be convenient to use notations like this: We assume that /negationslash /negationslash /negationslash",
"pages": [
2,
3,
4
]
},
{
"title": "III.1. Field equations.",
"content": "The action reads: where R and F αβ are scalar curvature and Faraday tensor respectively. Here and after we use Planck units: m Pl is the Planck mass. The gravitational equations is given by where are the components of the Einstein tensor. Taking into account (1) we get: /negationslash /negationslash As we consider the diagonal case here, then in view of (17),(18),(24),(27) for an even number of dimensions the gravitational equations can be written as /negationslash /negationslash Note that each expression like (28) describes two equations with numbers 2 n -1 and 2 n simultaneously (we use the notation (19) here). In the case of an odd number of dimensions there is one more equation in addition to these: /negationslash /negationslash The left side of the expression (29) is a first integral of (28).",
"pages": [
4,
5
]
},
{
"title": "III.2. Vacuum solution.",
"content": "When there is no any matter the equations (28)-(30) lead to: /negationslash These equations has the following solutions [16]: /negationslash As a consequence, components of the metric tensor follows a power law: This is well-known multidimensional Kasner solution. /negationslash /negationslash",
"pages": [
5
]
},
{
"title": "III.3. Stability conditions.",
"content": "Metric ceases to obey a power law when there is a matter; but it turns out that the metric of the space filled with the magnetic field may get close to the Kasner metric and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (28) that get close to the Kasner solutions (33) and converge to them as time tends to the point t = 0 . Considering an influence of the magnetic field as a perturbation we will search for solutions of the equations (28) in the form: where a 0 k is the Kasner solution (33), ϕ k ∈ C 2 ( R ) is a perturbation. We will say that solutions a 0 k are asymptotically stable if In other words, solutions a 0 k are asymptotically stable if and only if Now we will find out conditions which specify asymptotically stable Kasner solutions. Proposition 1. Kasner solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 1 , n = 1 , χ . 1. Let { a 0 1 , . . . , a 0 N } be an asymptotically stable solution of the equations (31)-(32) given by (33); then there exists solution { a 1 , . . . , a N } of the equations (28),(29) and functions ϕ 1 , . . . , ϕ N ∈ C 2 ( R ) such that Substitution (33),(38) to the equations (28) and (29) leads to: /negationslash /negationslash /negationslash /negationslash It follows from (39) that Let t → 0 ; in view of (33),(39),(42) equations (40)-(41) take the form: /negationslash /negationslash /negationslash /negationslash Equalities (43) and (44) are satisfied if and only if q.e.d. 2. Let { a 0 1 , . . . , a 0 N } be the Kasner solution given by (33) such that p 2 n -1 + p 2 n < 1 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have: Let t = t 0 > 0 be the initial moment; we assume that /negationslash /negationslash It is easy to check that matrix A is nonsingular, so we obtain: or, Hence we deduce that It follows from p 2 m -1 + p 2 m < 1 , m = 1 , χ that λ m +2 > 0 ; therefore and solutions (33) with p 2 m -1 + p 2 m < 1 , m = 1 , χ are asymptotically stable. /squaresolid /negationslash Substitution (33),(46) to the equations (28) and (29) give us equations (40)-(41). In view of (47) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j in the lhs of the equations (40)-(41) and terms contained ϕ i in the rhs of that equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (40) can be written in the form: /negationslash or, in matrix notations, where",
"pages": [
6,
7
]
},
{
"title": "III.4. Numerical calculations.",
"content": "The result presented above have been verified numerically. We have got a few hundreds numerical solutions with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found a lot of solutions that converge to the Kasner ones. Example of such solution is presented on the Fig. 1 (a). It should be note that looking for asymptotically stable solutions numerically using the synchronous time t has a number of drawbacks; particularly, numerical solutions of the equations (68)-(69) increase too rapidly impeding computations as t goes to zero. To avoid these problem we have introduced new time coordinate τ by means of the relation The change of variables from t to τ Indeed, (33) and (56) leads to results in transformation of logarithmic functions (33) into linear ones. Consequently, Without lack of generality we can set Then, as was announced, Such a way, looking for asymptotically stable solutions numerically, we should expect to get quasi-linear solutions like ones that shown on the Fig. 1 (a).",
"pages": [
8
]
},
{
"title": "IV.1. Field equations.",
"content": "Action of the theory under consideration is given by: where R,R αβ , R αβγδ , F αβ are the ( N + 1) -dimensional scalar curvature, Ricci tensor, Riemann tensor and Faraday tensor respectively. The gravitational equations has the form: where are the components of the Gauss-Bonnet tensor. Taking into account (1) we get: /negationslash /negationslash /negationslash For the diagonal case in view of (17),(18),(64),(67) for an even number of dimensions the gravitational equations can be written as: /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash There is one more equation in addition to these in the case of an odd number of dimensions: /negationslash The left side of the expression (69) is a first integral of (68).",
"pages": [
9,
10
]
},
{
"title": "IV.2. Vacuum solution.",
"content": "When there is no any matter the equations (68)-(70) lead to: /negationslash /negationslash /negationslash These equations has the following solutions [5, 6]: As a consequence, components of the metric tensor follows a power law: Thus by analogy with (33)-(34) we will call solutions (73) and metric (74) Kasner-like for brevity.",
"pages": [
10
]
},
{
"title": "IV.3. Stability conditions.",
"content": "As well as in the case of the Einstein gravity (see III.3.), metric ceases to obey a power law when there is a matter; it appears however that the metric of the space filled with the magnetic field may get close to the Kasner-like metric (74) and converge to it when moving towards the initial singularity. Namely, numerical calculations shows that there exists solutions of the equations (68) that get close to the Kasnerlike solutions (73) and converge to them as time goes to the point t = 0 . We will looking for solutions of the equations (68) in the form: where a 0 k is the Kasner-like solution (73), ϕ k ∈ C 2 ( R ) . We will say that solutions a 0 k are asymptotically stable if In other words, solutions a 0 k are asymptotically stable if and only if Now we will find out conditions which specify asymptotically stable Kasner-like solutions. Proposition 2 . Kasner-like solutions are asymptotically stable for t → 0 if and only if p 2 n -1 + p 2 n < 2 , n = 1 , χ . Let us introduce the following notation: where A i 1 , . . . , A i N , B i 1 , . . . , B i N are any indexed mathematical objects. Substitution (73),(78) to the equations (68) and (69) leads to: /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash where /negationslash /negationslash It follows from (79) that In the limit t → 0 in view of (73),(79),(83) equations (81)-(82) take the form: /negationslash Equalities (84) and (85) are satisfied if and only if q.e.d. 2. Let { a 0 1 , . . . , a 0 N } be the Kasner-like solution given by (73) such that p 2 n -1 + p 2 n < 2 , n = 1 , χ , and ϕ 1 ( t ) , . . . , ϕ N ( t ) ∈ C 2 ( R ) be a small deviations from that solution. We have: Let t = t 0 > 0 be the initial moment; we assume that /negationslash Substitution (73),(87) to the equations (68) and (69) give us equations (81)-(82). In view of (88) we can neglect terms contained factors like t ˙ ϕ i , t 2 ˙ ϕ i ˙ ϕ j , t 3 ˙ ϕ i ˙ ϕ j ˙ ϕ k , t 4 ˙ ϕ i ˙ ϕ j ˙ ϕ k ˙ ϕ l in the lhs of the equations (81)-(82) and terms contained ϕ i in the rhs of these equations. Namely, let ε be a small positive real number; then in the ε -vicinity of the point t 0 equations (81) can be written in the form: /negationslash /negationslash or, in matrix notations, /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash /negationslash Assuming that matrix A is nonsingular 2 , we obtain: or, Hence we deduce that It follows from p 2 m -1 + p 2 m < 2 , m = 1 , χ that γ m +2 > 0 ; therefore and solutions (73) with p 2 m -1 + p 2 m < 2 , m = 1 , χ are asymptotically stable. /squaresolid Example. Let us consider (5+1)-dimensional space-time. According to (15) there are pair of non-zero components of the Faraday tensor; taking into account our convention (see II. for details) we denote them ψ 12 and ψ 34 . Then under the Proposition 2 it follows that stable Kasner-like solutions are described by the conditions p 1 + p 2 < 2 and p 3 + p 4 < 2 .",
"pages": [
10,
11,
12,
13
]
},
{
"title": "IV.4. Numerical calculations.",
"content": "The result obtained have been confirmed by numerical calculations. As well as in the study of the stability of the Kasner solutions (see III.3.) we have received a few hundreds solutions numerically with a random sets of initial conditions for each of the dimensions N = 5 , 6 , 7 , 8 and found numerous solutions that converge to the Kasner-like ones. Example of such solution is presented on the Fig. 2 (a). For the reasons mentioned above (see III.4.) one should use special time coordinate for numerical calculations. Let us introduce new time coordinate τ by the following way: The change of variables from t to τ results in transformation of logarithmic functions (73) into linear ones. Indeed, (73) and (97) give us: /negationslash Consequently, Without loss of generality we can set Then, as was announced, Thus, searching for asymptotically stable solutions numerically we should expect to get quasi-linear solutions like ones that shown on the Fig. 2 (a).",
"pages": [
13,
14
]
},
{
"title": "V. Oscillatory regime.",
"content": "Results of the preceding section pose an interesting question about the fate of trajectories in the instability zone. They can either reach stable Kasner-like attractor or represent some other type of dynamics different from Kasner-like behavior. In this section we show that the latter alternative can be realized. In our numerical calculations we have revealed a specific class of solutions of the equations (68). The discovered solutions possess the following properties: two of N functions a 1 , . . . , a N oscillate and has opposite signs, the other ones vary slowly; owing to phase difference between the oscillating functions volume element √ | det( g ) | d N x oscillates too. In view of limited computational capabilities we have obtained numerical solutions of the equations (68) only for the cases N = 5 , 6 , 7 , 8 ; for each of that dimensions we have found out sufficiently large number of oscillatory solutions. Example of such solution is presented on the Fig. 3 (a); the corresponding oscillations of the volume element is shown on the Fig. 3 (b). For N = 5 we have been able to derive an analytical approximation to such solutions on a certain interval of time. Below we describe the procedure of derivation of this approximation. Henceforth we use time coordinate τ introduced by (97). In the (5+1)-dimensional case equations (68) become: so that Let us decompose functions f, g, h with respect to ( 4 3 v -2 3 u ) and ( 4 3 u -2 3 v ) up to the first order: with the first integral The dash denotes derivative with respect to τ . Analysis of numerical solutions shows that there exists τ 0 > 0 ( τ = 0 is accepted as the initial moment) such that (see Fig. 6). In view of (109) equations (103)-(108) take the form: where Equation (111) is the sum of the equations (110); equations (110) can be replaced by their equivalent: In view of (109) the last equation from (116) is satisfied automatically, and the other equations become The last of equations (117) gives: Consequently, Numerical results show that such assumption is valid in a certain interval of time. Taking into account (119)- (122) we get: Let us consider the equation (124). Numerical calculations show that in the regime under consideration v '' / ( v ' ) 4 is the leading term in the lhs of the equation (124); then: We can obtain analytical solution of the equation (125) by omitting the term containing u in the rhs of (125): Solution of the equation (126) has the form Let us consider the equation (123). We omit the term, containing v in the rhs of (123): Under substitution of (127) into (128) we obtain: We have used the following notations: Equation (129) has the following solution: Here J 2 3 , Y 2 3 are Bessel functions of the first and the second kind respectively. Going back to the initial variables we get: and, finally: where To demonstrate oscillations of volume element √ | det( g ) | d N x one should write down expression for the determinant of the metric: Despite the fact that functions (134)-(136) are the solution of the simplified system (126),(128) they catch correctly qualitative features of solution of the full system (103)-(107) which can be shown by comparing functions (134)-(136) with numerical solution of the system (103)-(107). Figures 4,5 below depicts pairs ( a 2 , A 2 ); ( a 4 , A 4 ) of numerical and analytical solutions; pairs ( a 1 , A 1 ); ( a 3 , A 3 ) behave themselves analogously. This regime is not similar to Kasner-like one (Kasner-like solution has linear behavior on chosen time coordinate τ (see IV.4. for details), the leading asymptotic of the oscillatory solution has quadratic behaviour w.r.t. τ ).",
"pages": [
14,
15,
16,
17,
18
]
},
{
"title": "VI. Conclusions",
"content": "In this paper we provided an analysis of stability of known power-law solutions in Gauss-Bonnet anisotropic cosmology in the presence of a homogeneous magnetic field in the case when metric is diagonal in the frame determined by magnetic field. The relative simplicity of equations of motion in this case allows us to find analytically a simple condition for the particular power-law solution to be stable. It appears that a set of initial conditions for which the power-law solution can not be an attractor near a cosmological singularity has non-zero measure (though does not coincide with the whole initial conditions space). The result has the same structure as in the Einstein gravity (particular sums of Kasner indices should be restricted by some number), the only difference is the exact value of this number. The fate of trajectories near singularity from this unstable domain requires future investigations. In the present paper we describe one particular class of trajectories found by numerical studies. The volume element experiences oscillations in the presented regime, making it qualitatively different from BKL oscillations. We can expect that some other regimes which do not tend to Kasner-like power-law regime exist in the system under investigation. Full description of them (the task which we leave for a future work) in this particular rather simple model can be useful in studies of cosmological singularity in Lovelock gravity in the presence of spatial curvature. The latter model usually lead to very cumbersome equations of motion, so understanding the nature of possible regimes in the instability zone for the Kasner-like solution in the present model would help in finding similar solution in more general and more technically complex situations of curved multidimensional geometries.",
"pages": [
22
]
},
{
"title": "Acknowledgments",
"content": "The authors are grateful to Vladimir Ivashchuk for helpful discussion. The work is partially supported by the RFBR grant 11-02-00643.",
"pages": [
22
]
}
]
2013IAUS..289...74N
https://arxiv.org/pdf/1411.5999.pdf
<document>
<section_header_level_1><location><page_1><loc_11><loc_86><loc_70><loc_90></location>Determining distances to stars statistically from photometry</section_header_level_1>
<section_header_level_1><location><page_1><loc_32><loc_83><loc_49><loc_85></location>Heidi Jo Newberg 1</section_header_level_1>
<text><location><page_1><loc_9><loc_80><loc_72><loc_83></location>1 Rensselaer Polytechnic Institute, Dept. of Physics, Applied Physics & Astronomy, 110 8 th St., Troy, NY 12180, USA</text>
<text><location><page_1><loc_33><loc_79><loc_48><loc_80></location>email: newbeh@rpi.edu</text>
<text><location><page_1><loc_9><loc_63><loc_72><loc_76></location>Abstract. In determining the distances to stars within the Milky Way galaxy, one often uses photometric or spectroscopic parallax. In these methods, the type of each individual star is determined, and the absolute magnitude of that star type is compared with the measured apparent magnitude to determine individual distances. In this article, we define the term statistical photometric parallax , in which statistical knowledge of the absolute magnitudes of stellar populations is used to determine the underlying density distributions of those stars. This technique has been used to determine the density distribution of the Milky Way stellar halo and its component tidal streams, using very large samples of stars from the Sloan Digital Sky Survey. Most recently, the volunteer computing platform MilkyWay@home has been used to find the best fit model parameters for the density of these halo stars.</text>
<text><location><page_1><loc_9><loc_59><loc_72><loc_62></location>Keywords. methods: data analysis, methods: statistical, stars: distances, stars: statistics, Galaxy: globular clusters: general, Galaxy: stellar content, Galaxy: structure</text>
<section_header_level_1><location><page_1><loc_9><loc_54><loc_23><loc_55></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_34><loc_72><loc_53></location>We often use the stars in the Milky Way to trace its structure. The brightest stars, which can be used to trace Galactic structure to the largest distances, include blue horizontal branch stars; O, B and A main sequence stars; RR Lyraes; Cepheid variables; red clump stars; K giants; and M giants (see Figure 1). All of these stellar types are reasonably well calibrated as distance indicators. K giant and red clump stars almost always require spectroscopy to accurately distinguish them from the more numerous main sequence stars of the same color, and RR Lyrae stars are usually identified from multiple epochs of photometry. All of these bright stars are relatively rare in any stellar population, and many of them are only observed in certain populations. For example, blue horizontal branch stars and RR Lyrae stars are only found in old populations, M giants are only found in relatively metal-rich populations, and O, B, and A main sequence stars are only found in very young populations (though A stars with main sequence gravities are found in some old populations as blue stragglers).</text>
<text><location><page_1><loc_9><loc_19><loc_72><loc_34></location>Deep, large sky area, multicolor surveys with high accuracy calibrations, like the Sloan Digital Sky Survey (SDSS; York et al. 2000) enable new methods for studying the structure of the Milky Way. For example, Ivezi'c et al. (2008) derived a formula that allows us to estimate the temperature and metallicity of main sequence stars of type G and later from SDSS photometry. Since these lower mass main sequence stars do not evolve in the age of the Universe, their absolute magnitudes are independent of age. Photometry cannot tell us the surface gravities of these types of stars (Lenz et al. 1998), but since the vast majority of the red stars in the photometric survey are main sequence stars, this is not a major obstacle. Juri'c et al. (2009) used this technique to measure the distances to 48 million stars, and then used these distances to determine the density distributions</text>
<text><location><page_2><loc_9><loc_88><loc_72><loc_94></location>of the disks and halo at 100 pc to 20 kpc from the Sun, over 6500 square degrees of sky. Though this analysis used a large number of stars, and resulted in the measurement of stellar density, the Juri'c analysis was using photometric parallax; they were determining the distance to each star individually using photometry.</text>
<text><location><page_2><loc_9><loc_69><loc_72><loc_88></location>In this article, we introduce the concept of statistical photometric parallax . We have used this technique most successfully to study the structure of the Galaxy using turnoff stars (Newberg et al. 2002). These stars are by definition brighter than main sequence stars such as those used by Juri'c et al., so they can trace the structure of the Milky Way 30 kpc or more from the Sun. However, the turnoff stars in a single stellar population can differ in absolute magnitude by two magnitudes (producing a distance error of a factor of 2.5). We do not have a way to determine the distance to a single turnoff star with reasonable accuracy. However, it has been shown that the absolute magnitude distribution of turnoff stars in halo globular clusters are surprisingly similar to each other (Newby et al. 2011), over a metallicity range -2.3 dex < [Fe/H] < -1.2 and ages ranging from 9 to 13.5 Gyr. Recently, Grabowski, Newby, & Newberg (2012) showed that this similarity even holds for the globular cluster Whiting 1, which is thought to be only 6.5 Gyrs old (Carraro, Zinn & Moni Bidin 2007).</text>
<text><location><page_2><loc_9><loc_55><loc_72><loc_68></location>This striking similarity between the absolute magnitude distributions of turnoff stars was not expected. One expects that younger globular clusters would have brighter, bluer turnoff stars. Also, one expects that more metal-rich clusters will have dimmer, redder turnoff stars. As it turns out, older stars in the Milky Way generally have lower metallicity, and the two effects cancel each other. This appears to be an unanticipated consequence of the Milky Way's Age-Metallicity Relationship (AMR - Muratov & Gnedin 2010, Dotter; Sarajedini & Anderson 2011). Apparently, the absolute magnitude distribution of turnoff stars is similar over the full age and metallicity range of typical stellar populations in the Milky Way halo.</text>
<text><location><page_2><loc_9><loc_43><loc_72><loc_55></location>We will describe here the general techique of statistical photometric parallax, which can be used to statistically account for the effects of a range of intrinsic brightnesses of the stellar population which is being used to trace Milky Way density structure, and can also statistically account for the observational biases in a survey such as the Sloan Digital Sky Survey. We have implemented this technique as a search for the density parameters with the highest likelihood of matching the observed data. Because this parameter search can be computationally expensive, we have employed supercomputers and a large volunteer computing platform, MilkyWay@home, that was built to solve this problem.</text>
<section_header_level_1><location><page_2><loc_9><loc_39><loc_46><loc_40></location>2. Using Statistical Photometric Parallax</section_header_level_1>
<text><location><page_2><loc_9><loc_29><loc_72><loc_38></location>We are defining the term statistical photometric parallax here for the first time, and it is intended to apply in general for any case where the statistical distribution of absolute magnitudes is used to find the underlying density distribution of stars. However, we will describe here as an example the application of this technique to determine halo substructure using color-selected F turnoff stars, as used by Newberg et al. (2002), Cole et al. (2008), and Newby et al. (2012).</text>
<text><location><page_2><loc_9><loc_19><loc_72><loc_29></location>In Newberg et al. (2002), stars with colors 0 . 1 < ( g -r ) 0 < 0 . 3 and ( u -g ) 0 > 0 . 4 were selected as turnoff stars. The color range was chosen to be bluer than the turnoff of the thick disk, so that halo stars would preferentially be selected. In this paper, only the simplest form of statistical photometric parallax was employed. The distance to the Sagittarius dwarf tidal stream was determined by assuming the center of the absolute magnitude distribution of turnoff stars in the g filter was M g = 4 . 2. This number was calculated by comparing the apparent magnitude of the turnoff to the apparent</text>
<text><location><page_3><loc_9><loc_90><loc_72><loc_94></location>magnitude of RR Lyrae stars in the same stellar population. In this example, distances to single stars were not calculated; instead we made a more accurate determination of distance by looking at the distribution of apparent magnitudes of a particular set of stars.</text>
<text><location><page_3><loc_9><loc_82><loc_72><loc_89></location>In Cole et al. (2008), an algorithm was presented that allowed us to determine not just the distance to the center of a stream, but the three dimensional density of turnoff stars. The technique used maximum likelihood to find the model parameters /vector Q that make the observed star positions ( l i , b i , g i ) the most likely. The Likelihood L is given by the product of the probability density functions (PDFs) evaluated at all of the star positions:</text>
<formula><location><page_3><loc_32><loc_79><loc_49><loc_81></location>L = ∏ PDF( l i , b i , g i | /vector Q ) .</formula>
<text><location><page_3><loc_9><loc_76><loc_43><loc_78></location>The PDF is constructed by the following steps:</text>
<unordered_list>
<list_item><location><page_3><loc_9><loc_73><loc_72><loc_76></location>( a ) For each stellar component, one assumes a parameterized model (for example a double exponential, NFW, Hernquist, etc.) for the spatial density.</list_item>
<list_item><location><page_3><loc_9><loc_70><loc_72><loc_73></location>( b ) This spatial density is transformed to ( l, b, g ) coordinates, assuming that the absolute magnitude of each of the stars is the average for the population.</list_item>
<list_item><location><page_3><loc_9><loc_67><loc_72><loc_70></location>( c ) This density is convolved with the absolute magnitude distribution of the tracers, so that we produce the distribution that we expect to observe.</list_item>
<list_item><location><page_3><loc_9><loc_64><loc_72><loc_67></location>( d ) This expected distribution is multiplied by the completeness for observing stars of a given apparent magnitude in a given survey, as a function of apparent magnitude.</list_item>
<list_item><location><page_3><loc_9><loc_61><loc_72><loc_64></location>( e ) The resulting distribution is normalized so that the integrated probability of finding a star in the entire volume observed is one.</list_item>
<list_item><location><page_3><loc_9><loc_55><loc_72><loc_61></location>( f ) The final PDF is the sum of the fraction of stars in each component times the normalized distribution, summed over the number of components in the model. The fraction of stars in each component are also parameters that are fit in the maximum likelihood optimization.</list_item>
<list_item><location><page_3><loc_9><loc_49><loc_72><loc_55></location>Of these steps, the most time-consuming is the calculation of the integral over the volume. Contrary to first impressions, the time to calculate the likelihood depends more heavily on the number of sub-volumes into which the survey space needs to be divided to achieve an accurate result, than on the number of stars in the dataset.</list_item>
</unordered_list>
<text><location><page_3><loc_9><loc_41><loc_72><loc_49></location>One then uses an optimization technique to find the model parameters that produce the highest likelihood. When using a supercomputer, we usually use conjugate gradient descent. This algorithm is sequential; one evaluates the likelihood and the derivatives with respect to each parameter, chooses a direction, then decides how far in that direction to go before repeating that process.</text>
<text><location><page_3><loc_9><loc_19><loc_72><loc_41></location>Newby et al. (2012) applied this technique to all of the data available in SDSS DR7 to find the density of the Sagittarius dwarf tidal stream in the North Galactic Cap, and in three SDSS stripes in the South Galactic Cap. One of the advantages of this probablistic technique is that we are able to extract from the sample of 1.7 million turnoff stars a set of 200 , 000 stars that have the spatial characteristics of the Sagittarius dwarf tidal stream. This is accomplished by generating a random mumber for each star, and using that random number to place it either in the Sagittarius dwarf tidal stream catalog, with the probability that a star at that position in the Galaxy is in the Sagittarius dwarf tidal stream; or in the catalog of non-Sagittarius halo stars, with the probability that a star at that position in the Galaxy is not in the Sagittarius dwaf tidal stream. Note that if you wanted to find actual stars in the Sagittarius dwarf tidal stream, say for spectroscopic follow-up, you should use the original catalog of stars and select those with the highest probability of being in the Sagittarius dwarf tidal stream. However, the statistically separated catalog we generated facilitates the study of density substructures in the halo. In particular, we can remove the Sagittarius stream from the original stellar</text>
<text><location><page_4><loc_9><loc_90><loc_72><loc_94></location>sample so we can study the smaller tidal streams, and the density structure of the smooth component of the halo. An example stripe analyzed in Newby et al. (2012) is shown in Figure 2.</text>
<text><location><page_4><loc_9><loc_72><loc_72><loc_89></location>In Cole et al. (2008) and Newby et al. (2012) we modeled the distribution of turnoff star absolute magnitudes as a Gaussian centered at M g = 4 . 2 with a width of σ = 0 . 6 magnitudes. Since then, we have recognized that the distribution is asymmetric; there are more stars fainter than the maximum than there are brighter than the maximum. More importantly we have learned that, due to larger color errors at fainter magnitudes, the absolute magnitude distribution of color-selected stars is different near the survey limit than it is for the brighter stars. This effect is much larger than we expected it to be. Near the survey limit, the majority of the stars are not turnoff stars, but are fainter main sequence stars that have scattered into our color selection limits due to large measurement errors. Because we are using a statistical approach, this effect can be included in the analysis by varying the absolute magnitude distribution as a function of apparent magnitude. We plan to include this in future analyses.</text>
<section_header_level_1><location><page_4><loc_9><loc_66><loc_70><loc_69></location>3. Processing time and the MilkyWay@home Volunteer Computing Platform</section_header_level_1>
<text><location><page_4><loc_9><loc_52><loc_72><loc_65></location>We originally tried to implement the search for maximumum likelihood on a single CPU. In a single 2 . 5 · -wide SDSS data stripe, we fit 2 parameters to a smooth halo with a Hernquist profile, and 6 parameters per tidal debris stream. If there were three tidal streams in a single stripe, there would be 20 parameters. Evaluating the likelihood for one guess for the model parameters currently takes about 4 hours, with most of the time spent integrating the PDF over the survey volume. In order to optimize 20 parameters requires about 50 likelihood evaluations per conjugate gradient descent step and 50 steps per maximum likelihood evaluation. This totals ten thousand hours per optimization (over a year).</text>
<text><location><page_4><loc_9><loc_35><loc_72><loc_52></location>Luckily the optimization is embarrassingly parallel. It is possible to parallelize the integral, since each integral volume calculation is completely independent of the others. We are able to run this algorithm on a 256 node rack of a Blue Gene/L supercomputer. Parallelizing the integral over 256 nodes cuts the time per likelihood calculation down under a minute. A conjugate gradient step can then be accomplished in 47 minutes, and ten iterations can be accomplished in under eight hours (which is comfortably less the the maximum job size allowed in our queue). In practice, we need to try several conjugate gradient descents to approximate the best parameters. Once they are known approximately, we run of order ten conjugate gradient descents starting near the best values. Including submitting jobs a few at a time and waiting for queue time, this process can take a couple of weeks to validate the results for one SDSS stripe.</text>
<text><location><page_4><loc_9><loc_19><loc_72><loc_35></location>Currently our best method for computing the parameters is the volunteer computing platform MilkyWay@home. This project is part of the Berkeley Open Infrastructure for Network Computing (BOINC; Anderson, Korpela & Walton 2005) group of volunteer computing platforms. The first and most famous of these is SETI@home. BOINC offers us a template server and database application, and an infrastructure for volunteers to donate their time to our server. We implemented our own server, including the maximum likelihood algorithm, a set of optimization routines that will run in a heterogeneous, asynchronous parallel computing environment (Desell et al. 2010b), and a modified server application that sends out 'work units' to the volunteers, collects the results, and validates that the results are not in error (Desell et al. 2010a). One of the surprises in operating a BOINC server is that some of the results sent back from the</text>
<text><location><page_5><loc_9><loc_87><loc_72><loc_94></location>volunteers are not correct, either because their hardware is malfunctioning, they did not update the software correctly, or they purposely sent back a wrong answer quickly so that they can accumulate BOINC 'credit' more quickly. It is impossible to overestimate how important it is to our volunteers that they get credit for the work units their computers crunch, and that credit is apportioned fairly between the volunteers.</text>
<text><location><page_5><loc_9><loc_70><loc_72><loc_86></location>MilkyWay@home is currently delivering 0.5 PetaFLOPS of computing power from 25,000 active volunteers giving us access to over 35,000 CPUs or GPUs. The majority of the computing power comes from the GPUs, the best of which can process our likelihood calculations about 100 times faster than the CPUs (Desell et al. 2009). It is not easy to parallelize the computation of the integral on BOINC, because that would require communication between the processors, which is not possible at this time. Instead, we parallelize the calculations by sending a single likelihood calculation (including the whole integral for a given /vector Q ) to each volunteer. These work units can take a couple minutes (if the work unit is sent to a GPU) or four or more hours (if it goes to a CPU), but it might take minutes, days, or weeks for the likelihood to be returned depending on how much the volunteer is using the computer for her own purposes, and whether she turns it off.</text>
<text><location><page_5><loc_9><loc_37><loc_72><loc_70></location>It takes far more computing power to calculate best fit parameters on the MilkyWay@home computing system than on a supercomputer. There are four factors responsible for this: (1) We cannot use sequential searches like conjugate gradient descent. Instead, we use 'particle swarm' or a genetic search algorithm. In the particle swarm technique, we send out a random set of guesses that span the parameter space. As the likelihood results from the volunteers come back, we send out more work units with guesses that are closer to the higher likelihoods. This search method requires many more steps, but produces more accurate results. (2) A fraction (about 10%) of the work units, selected at random, are sent out five times so that we can validate that they are correct. If three or more are returned with the same answer, then the result is validated. We also choose to validate the best likelihoods (since those influence our future guesses), and the likelihoods of users that have submitted previous results that did not validate correctly. (3) Some of the work units that are sent out are never returned. (4) Because we can, we run the searches for a longer period of time over a wider range of of parameter space and we get better global values for the parameters (Newby et al. 2012). To optimize one stripe takes 1-2 weeks, and hundreds of thousands of likelihood calculations. There are enough volunteers that we can optimize 4-5 stripes at the same time and still get the results within the same time scale. By putting more jobs on MilkyWay@home at the same time, we decrease the number of work units that are sent out at the same time. By waiting a little longer to send out more work units, the results of previous searches can be used to make better guesses of the parameters, so adding more jobs increases the computing time at a rate that is less than linear in the number of jobs.</text>
<section_header_level_1><location><page_5><loc_9><loc_33><loc_35><loc_34></location>4. Discussion and Conclusion</section_header_level_1>
<text><location><page_5><loc_9><loc_19><loc_72><loc_32></location>The purpose of this conference contribution is to define the term statistical photometric parallax , which allows us to determine the density of a population of stars, even if we cannot determine the distance to each individual star in the population. We find the most likely parameters for the density distribution, given that we know the absolute magnitude distribution and the observational constraints of the observed sample of stars. If the population of stars is all at the same distance (for example in a globular cluster or tidal stream), then statistical photometric parallax can be used to determine the distance to the stellar population. Because the SDSS made available a large, well calibrated sample of stars with multi-color photometry, this technique has recently become feasible.</text>
<text><location><page_6><loc_9><loc_85><loc_72><loc_94></location>The example I present is the use of turnoff stars to determine the density distribution of stars in the stellar halo of the Milky Way. By apparent coincidence, the distribution of absolute magnitudes of turnoff stars is very similar for all stellar populations in the age and metallicity range of halo stars. This appears to be a result of the Milky Way age-metallicity relationship. We are in the process of using this technique to accurately map the density distribution of the entire Milky Way stellar halo.</text>
<text><location><page_6><loc_9><loc_78><loc_72><loc_85></location>The drawback to this techique is that to use it one must use a maximum likelihood algorithm that can in some cases require high performance parallel computing to get an accurate measurement of the parameters in the density function. In the process of learning to use this method, we created a large volunteer computing platform called MilkyWay@home.</text>
<section_header_level_1><location><page_6><loc_9><loc_75><loc_24><loc_76></location>Acknowledgement</section_header_level_1>
<text><location><page_6><loc_9><loc_62><loc_72><loc_74></location>I would like to thank Matthew Newby and Brian Yanny for helping me obtain the data for the figures in this publication. I also thank Matthew Newby and Jeff Carlin for their help in proofreading. This paper is based upon work supported by the National Science Foundation under Grant No. AST 10-09670. I also would like to thank the MilkyWay@home volunteers for providing us with computing power at no cost, and the Marvin Clan for their support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.</text>
<section_header_level_1><location><page_6><loc_9><loc_58><loc_18><loc_60></location>References</section_header_level_1>
<text><location><page_6><loc_9><loc_57><loc_72><loc_58></location>Anderson, D. P., Korpela, E., and Walton, R. 2005, The First International Conf. on e-Science</text>
<code><location><page_6><loc_9><loc_26><loc_72><loc_56></location>and Grid Tech. (e-Science 2005), 196 Carraro, G., Zinn, R., & Moni Bidin, C. 2007, A&A , 466, 181 Cole, N., Newberg, H. J., Magdon-Ismail, M., et al. 2008, ApJ , 683, 750 Desell, T., Waters, A., Magdon-Ismail, M., et al. 2009, PPAM, Pt I, Lecture Notes in Computer Science, 6067, 276 Desell, T., Magdon-Ismail, M., Szymanski, B., Varela, C., Newberg, H., & Anderson, D. 2010a, DAIS, Lecture Notes in Computer Science, 6115, 29 Desell, T., Anderson, D., Magdon-Ismail, M., Szymanski, B., Newberg, H. J., Varela, C. 2010b, The 2010 IEEE congress on evolutionary computation (IEEE CEC) Grabowski, K., Newby, M., and Newberg, H. J. 2012, J. of Undergrad. Res. in Phys. , submittted Ivezi'c, ˇ Z., Sesar, B., Juri'c, M., et al. 2008, ApJ , 684, 287 Juri'c, M., Ivezi'c, ˇ Z., Brooks, A., et al. 2008, ApJ , 673, 864 Lenz, D. D., Newberg, J., Rosner, R., Richards, G. T., & Stoughton, C. 1998, ApJ (Supplement), 119, 121 Muratov, A. L., & Gnedin, O. Y. 2010, ApJ , 718, 1266 Newberg, H. J., Yanny, B., Rockosi, C., et al. 2002, ApJ , 569, 245 Newby, M., Newberg, H. J., Simones, J., Cole, N., & Monaco, M. 2011, ApJ , 743, 187 Newby, M., Cole, N., Newberg, H. J., Desell, T., Magdon-Ismail, M., Szymanski, B., Varela, C., Willett, B., & Yanny, B. 2012, ApJ , submitted Rave, H. A., Zhao, C., Newberg, H. J., et al. 2003, ApJ (Supplement), 145, 245 SDSS-III Collaboration, Ahn, C. P., et al. 2012, arXiv:1207.7137 Yanny, B., Newberg, H. J., Johnson, J. A., et al. 2009, ApJ , 700, 1282 York, D. G., Adelman, J., Anderson, J. E., Jr., et al. 2000, AJ , 120, 1579</code>
<figure>
<location><page_7><loc_17><loc_43><loc_64><loc_94></location>
<caption>Figure 1. Color-magnitude diagram of stars withing 4 arcminutes of the globular cluster Palomar 5. All stars within four arcminutes of Pal5 were selected from SDSS DR9 (SDSS III Collaboration et al. 2012). We illustrate how sparse the brighter stars are in a stellar population, compared to the main sequence and turnoff. Most of the selection boxes for each type of star were drawn to include the representative stars in Palomar 5. The RR Lyrae selection box was taken from Rave et al. (2003). The M giant selection box was estimated from Yanny et al. (2009) measurements of M giants in the Sagittarius dwarf tidal stream. There are no representatives of these types of stars in the Palomar 5 globular cluster. Although giant stars of all types can be observed at larger distances, many types of giant stars are specific to particular populations. K giants are seen in all but the youngest stellar populations, but these stars are more difficult to separate from dwarf K stars that have the same colors. Also, K giant stars are more difficult to use as distance indicators because their absolute magnitude is a strong function of color (temperature), and is also sensitive to metallicity and age. The drawback to using turnoff stars as distance indicators is that they vary in absolute magnitude by two magnitudes. Statistical photometric parallax allows us to used them effectively to determine the underlying density of stars even though we cannot accurately measure the distance to each individual star.</caption>
</figure>
<figure>
<location><page_8><loc_9><loc_58><loc_72><loc_94></location>
<caption>Figure 2. The A and F star distribution on the Celestial Equator. All of the stars in this figure were selected within 0.1 degrees of the Celestial Equator, and 145 · < RA < 265 · . The top left panel shows photometrically selected A stars from SDSS DR9 (SDSS-III Collaboration et al. 2012). The stars were selected with -0 . 3 < ( g -r ) 0 < 0 . 0, 0 . 8 < ( u -g ) 0 < 1 . 5, and 14 < g 0 < 22 . 5, where the subscript indicates that the values have been corrected for reddening using the extinction calculated in the database. The top right panel shows photometrically selected turnoff stars, selected from SDSS DR9. The stars were selected with 0 . 2 < ( g -r ) 0 < 0 . 4 and 14 < g 0 < 23. Notice that even though we selected only the bluest turnoff stars, there are very many more blue turnoff stars than stars with the colors of A stars. However, the wide range of g 0 magnitudes observed in the globular cluster Palomar 5 at RA = 229 · indicates that the range of absolute magnitudes of turnoff stars is at least two magnitudes. Even though the range of absolute magnitudes is large, we can see the Palomar 5 globular cluster, the Sagittarius dwarf galaxy tidal stream at ( RA,g 0 ) = (210 · , 22 . 5), the Virgo Overdensity at ( RA,g 0 ) = (180 · , 21), and stars that are presumably part of the smooth spheroid near the Galactic center at ( RA,g 0 ) = (245 · , 19). The substructure in A-type stars is not as evident partially because there are many fewer A-type stars, partially because we have included both blue stragglers and blue horizontal branch stars, and partially because not all stellar populations have A-type stars. Blue horizontal branch stars are about 1.5 magnitudes brighter than turnoff stars, and blue stragglers are about 3.5 magnitudes brigher than turnoff stars. A small number of blue stragglers from the Sagittarius dwarf tidal stream are evident at g 0 = 21 in the A star panel, but Palomar 5 blue stragglers are not evident. We do not see blue horizontal branch stars from either Palomar 5 or the Sagittarius dwarf tidal stream in this plot. One can see the advantage of using turnoff stars in identifying the Sagittarius dwarf tidal stream. The lower two panels show the stars selected by Newby et al. (2012), from this small region of the sky, that have the density distribution of the Sagittarius dwarf tidal stream (left) and the density distribution over everything else (right). The region around the globular cluster Palomar 5 was removed from the data before fitting, so that this globular cluster would not affect the results.</caption>
</figure>
</document>
[
{
"title": "Heidi Jo Newberg 1",
"content": "1 Rensselaer Polytechnic Institute, Dept. of Physics, Applied Physics & Astronomy, 110 8 th St., Troy, NY 12180, USA email: newbeh@rpi.edu Abstract. In determining the distances to stars within the Milky Way galaxy, one often uses photometric or spectroscopic parallax. In these methods, the type of each individual star is determined, and the absolute magnitude of that star type is compared with the measured apparent magnitude to determine individual distances. In this article, we define the term statistical photometric parallax , in which statistical knowledge of the absolute magnitudes of stellar populations is used to determine the underlying density distributions of those stars. This technique has been used to determine the density distribution of the Milky Way stellar halo and its component tidal streams, using very large samples of stars from the Sloan Digital Sky Survey. Most recently, the volunteer computing platform MilkyWay@home has been used to find the best fit model parameters for the density of these halo stars. Keywords. methods: data analysis, methods: statistical, stars: distances, stars: statistics, Galaxy: globular clusters: general, Galaxy: stellar content, Galaxy: structure",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "We often use the stars in the Milky Way to trace its structure. The brightest stars, which can be used to trace Galactic structure to the largest distances, include blue horizontal branch stars; O, B and A main sequence stars; RR Lyraes; Cepheid variables; red clump stars; K giants; and M giants (see Figure 1). All of these stellar types are reasonably well calibrated as distance indicators. K giant and red clump stars almost always require spectroscopy to accurately distinguish them from the more numerous main sequence stars of the same color, and RR Lyrae stars are usually identified from multiple epochs of photometry. All of these bright stars are relatively rare in any stellar population, and many of them are only observed in certain populations. For example, blue horizontal branch stars and RR Lyrae stars are only found in old populations, M giants are only found in relatively metal-rich populations, and O, B, and A main sequence stars are only found in very young populations (though A stars with main sequence gravities are found in some old populations as blue stragglers). Deep, large sky area, multicolor surveys with high accuracy calibrations, like the Sloan Digital Sky Survey (SDSS; York et al. 2000) enable new methods for studying the structure of the Milky Way. For example, Ivezi'c et al. (2008) derived a formula that allows us to estimate the temperature and metallicity of main sequence stars of type G and later from SDSS photometry. Since these lower mass main sequence stars do not evolve in the age of the Universe, their absolute magnitudes are independent of age. Photometry cannot tell us the surface gravities of these types of stars (Lenz et al. 1998), but since the vast majority of the red stars in the photometric survey are main sequence stars, this is not a major obstacle. Juri'c et al. (2009) used this technique to measure the distances to 48 million stars, and then used these distances to determine the density distributions of the disks and halo at 100 pc to 20 kpc from the Sun, over 6500 square degrees of sky. Though this analysis used a large number of stars, and resulted in the measurement of stellar density, the Juri'c analysis was using photometric parallax; they were determining the distance to each star individually using photometry. In this article, we introduce the concept of statistical photometric parallax . We have used this technique most successfully to study the structure of the Galaxy using turnoff stars (Newberg et al. 2002). These stars are by definition brighter than main sequence stars such as those used by Juri'c et al., so they can trace the structure of the Milky Way 30 kpc or more from the Sun. However, the turnoff stars in a single stellar population can differ in absolute magnitude by two magnitudes (producing a distance error of a factor of 2.5). We do not have a way to determine the distance to a single turnoff star with reasonable accuracy. However, it has been shown that the absolute magnitude distribution of turnoff stars in halo globular clusters are surprisingly similar to each other (Newby et al. 2011), over a metallicity range -2.3 dex < [Fe/H] < -1.2 and ages ranging from 9 to 13.5 Gyr. Recently, Grabowski, Newby, & Newberg (2012) showed that this similarity even holds for the globular cluster Whiting 1, which is thought to be only 6.5 Gyrs old (Carraro, Zinn & Moni Bidin 2007). This striking similarity between the absolute magnitude distributions of turnoff stars was not expected. One expects that younger globular clusters would have brighter, bluer turnoff stars. Also, one expects that more metal-rich clusters will have dimmer, redder turnoff stars. As it turns out, older stars in the Milky Way generally have lower metallicity, and the two effects cancel each other. This appears to be an unanticipated consequence of the Milky Way's Age-Metallicity Relationship (AMR - Muratov & Gnedin 2010, Dotter; Sarajedini & Anderson 2011). Apparently, the absolute magnitude distribution of turnoff stars is similar over the full age and metallicity range of typical stellar populations in the Milky Way halo. We will describe here the general techique of statistical photometric parallax, which can be used to statistically account for the effects of a range of intrinsic brightnesses of the stellar population which is being used to trace Milky Way density structure, and can also statistically account for the observational biases in a survey such as the Sloan Digital Sky Survey. We have implemented this technique as a search for the density parameters with the highest likelihood of matching the observed data. Because this parameter search can be computationally expensive, we have employed supercomputers and a large volunteer computing platform, MilkyWay@home, that was built to solve this problem.",
"pages": [
1,
2
]
},
{
"title": "2. Using Statistical Photometric Parallax",
"content": "We are defining the term statistical photometric parallax here for the first time, and it is intended to apply in general for any case where the statistical distribution of absolute magnitudes is used to find the underlying density distribution of stars. However, we will describe here as an example the application of this technique to determine halo substructure using color-selected F turnoff stars, as used by Newberg et al. (2002), Cole et al. (2008), and Newby et al. (2012). In Newberg et al. (2002), stars with colors 0 . 1 < ( g -r ) 0 < 0 . 3 and ( u -g ) 0 > 0 . 4 were selected as turnoff stars. The color range was chosen to be bluer than the turnoff of the thick disk, so that halo stars would preferentially be selected. In this paper, only the simplest form of statistical photometric parallax was employed. The distance to the Sagittarius dwarf tidal stream was determined by assuming the center of the absolute magnitude distribution of turnoff stars in the g filter was M g = 4 . 2. This number was calculated by comparing the apparent magnitude of the turnoff to the apparent magnitude of RR Lyrae stars in the same stellar population. In this example, distances to single stars were not calculated; instead we made a more accurate determination of distance by looking at the distribution of apparent magnitudes of a particular set of stars. In Cole et al. (2008), an algorithm was presented that allowed us to determine not just the distance to the center of a stream, but the three dimensional density of turnoff stars. The technique used maximum likelihood to find the model parameters /vector Q that make the observed star positions ( l i , b i , g i ) the most likely. The Likelihood L is given by the product of the probability density functions (PDFs) evaluated at all of the star positions: The PDF is constructed by the following steps: One then uses an optimization technique to find the model parameters that produce the highest likelihood. When using a supercomputer, we usually use conjugate gradient descent. This algorithm is sequential; one evaluates the likelihood and the derivatives with respect to each parameter, chooses a direction, then decides how far in that direction to go before repeating that process. Newby et al. (2012) applied this technique to all of the data available in SDSS DR7 to find the density of the Sagittarius dwarf tidal stream in the North Galactic Cap, and in three SDSS stripes in the South Galactic Cap. One of the advantages of this probablistic technique is that we are able to extract from the sample of 1.7 million turnoff stars a set of 200 , 000 stars that have the spatial characteristics of the Sagittarius dwarf tidal stream. This is accomplished by generating a random mumber for each star, and using that random number to place it either in the Sagittarius dwarf tidal stream catalog, with the probability that a star at that position in the Galaxy is in the Sagittarius dwarf tidal stream; or in the catalog of non-Sagittarius halo stars, with the probability that a star at that position in the Galaxy is not in the Sagittarius dwaf tidal stream. Note that if you wanted to find actual stars in the Sagittarius dwarf tidal stream, say for spectroscopic follow-up, you should use the original catalog of stars and select those with the highest probability of being in the Sagittarius dwarf tidal stream. However, the statistically separated catalog we generated facilitates the study of density substructures in the halo. In particular, we can remove the Sagittarius stream from the original stellar sample so we can study the smaller tidal streams, and the density structure of the smooth component of the halo. An example stripe analyzed in Newby et al. (2012) is shown in Figure 2. In Cole et al. (2008) and Newby et al. (2012) we modeled the distribution of turnoff star absolute magnitudes as a Gaussian centered at M g = 4 . 2 with a width of σ = 0 . 6 magnitudes. Since then, we have recognized that the distribution is asymmetric; there are more stars fainter than the maximum than there are brighter than the maximum. More importantly we have learned that, due to larger color errors at fainter magnitudes, the absolute magnitude distribution of color-selected stars is different near the survey limit than it is for the brighter stars. This effect is much larger than we expected it to be. Near the survey limit, the majority of the stars are not turnoff stars, but are fainter main sequence stars that have scattered into our color selection limits due to large measurement errors. Because we are using a statistical approach, this effect can be included in the analysis by varying the absolute magnitude distribution as a function of apparent magnitude. We plan to include this in future analyses.",
"pages": [
2,
3,
4
]
},
{
"title": "3. Processing time and the MilkyWay@home Volunteer Computing Platform",
"content": "We originally tried to implement the search for maximumum likelihood on a single CPU. In a single 2 . 5 · -wide SDSS data stripe, we fit 2 parameters to a smooth halo with a Hernquist profile, and 6 parameters per tidal debris stream. If there were three tidal streams in a single stripe, there would be 20 parameters. Evaluating the likelihood for one guess for the model parameters currently takes about 4 hours, with most of the time spent integrating the PDF over the survey volume. In order to optimize 20 parameters requires about 50 likelihood evaluations per conjugate gradient descent step and 50 steps per maximum likelihood evaluation. This totals ten thousand hours per optimization (over a year). Luckily the optimization is embarrassingly parallel. It is possible to parallelize the integral, since each integral volume calculation is completely independent of the others. We are able to run this algorithm on a 256 node rack of a Blue Gene/L supercomputer. Parallelizing the integral over 256 nodes cuts the time per likelihood calculation down under a minute. A conjugate gradient step can then be accomplished in 47 minutes, and ten iterations can be accomplished in under eight hours (which is comfortably less the the maximum job size allowed in our queue). In practice, we need to try several conjugate gradient descents to approximate the best parameters. Once they are known approximately, we run of order ten conjugate gradient descents starting near the best values. Including submitting jobs a few at a time and waiting for queue time, this process can take a couple of weeks to validate the results for one SDSS stripe. Currently our best method for computing the parameters is the volunteer computing platform MilkyWay@home. This project is part of the Berkeley Open Infrastructure for Network Computing (BOINC; Anderson, Korpela & Walton 2005) group of volunteer computing platforms. The first and most famous of these is SETI@home. BOINC offers us a template server and database application, and an infrastructure for volunteers to donate their time to our server. We implemented our own server, including the maximum likelihood algorithm, a set of optimization routines that will run in a heterogeneous, asynchronous parallel computing environment (Desell et al. 2010b), and a modified server application that sends out 'work units' to the volunteers, collects the results, and validates that the results are not in error (Desell et al. 2010a). One of the surprises in operating a BOINC server is that some of the results sent back from the volunteers are not correct, either because their hardware is malfunctioning, they did not update the software correctly, or they purposely sent back a wrong answer quickly so that they can accumulate BOINC 'credit' more quickly. It is impossible to overestimate how important it is to our volunteers that they get credit for the work units their computers crunch, and that credit is apportioned fairly between the volunteers. MilkyWay@home is currently delivering 0.5 PetaFLOPS of computing power from 25,000 active volunteers giving us access to over 35,000 CPUs or GPUs. The majority of the computing power comes from the GPUs, the best of which can process our likelihood calculations about 100 times faster than the CPUs (Desell et al. 2009). It is not easy to parallelize the computation of the integral on BOINC, because that would require communication between the processors, which is not possible at this time. Instead, we parallelize the calculations by sending a single likelihood calculation (including the whole integral for a given /vector Q ) to each volunteer. These work units can take a couple minutes (if the work unit is sent to a GPU) or four or more hours (if it goes to a CPU), but it might take minutes, days, or weeks for the likelihood to be returned depending on how much the volunteer is using the computer for her own purposes, and whether she turns it off. It takes far more computing power to calculate best fit parameters on the MilkyWay@home computing system than on a supercomputer. There are four factors responsible for this: (1) We cannot use sequential searches like conjugate gradient descent. Instead, we use 'particle swarm' or a genetic search algorithm. In the particle swarm technique, we send out a random set of guesses that span the parameter space. As the likelihood results from the volunteers come back, we send out more work units with guesses that are closer to the higher likelihoods. This search method requires many more steps, but produces more accurate results. (2) A fraction (about 10%) of the work units, selected at random, are sent out five times so that we can validate that they are correct. If three or more are returned with the same answer, then the result is validated. We also choose to validate the best likelihoods (since those influence our future guesses), and the likelihoods of users that have submitted previous results that did not validate correctly. (3) Some of the work units that are sent out are never returned. (4) Because we can, we run the searches for a longer period of time over a wider range of of parameter space and we get better global values for the parameters (Newby et al. 2012). To optimize one stripe takes 1-2 weeks, and hundreds of thousands of likelihood calculations. There are enough volunteers that we can optimize 4-5 stripes at the same time and still get the results within the same time scale. By putting more jobs on MilkyWay@home at the same time, we decrease the number of work units that are sent out at the same time. By waiting a little longer to send out more work units, the results of previous searches can be used to make better guesses of the parameters, so adding more jobs increases the computing time at a rate that is less than linear in the number of jobs.",
"pages": [
4,
5
]
},
{
"title": "4. Discussion and Conclusion",
"content": "The purpose of this conference contribution is to define the term statistical photometric parallax , which allows us to determine the density of a population of stars, even if we cannot determine the distance to each individual star in the population. We find the most likely parameters for the density distribution, given that we know the absolute magnitude distribution and the observational constraints of the observed sample of stars. If the population of stars is all at the same distance (for example in a globular cluster or tidal stream), then statistical photometric parallax can be used to determine the distance to the stellar population. Because the SDSS made available a large, well calibrated sample of stars with multi-color photometry, this technique has recently become feasible. The example I present is the use of turnoff stars to determine the density distribution of stars in the stellar halo of the Milky Way. By apparent coincidence, the distribution of absolute magnitudes of turnoff stars is very similar for all stellar populations in the age and metallicity range of halo stars. This appears to be a result of the Milky Way age-metallicity relationship. We are in the process of using this technique to accurately map the density distribution of the entire Milky Way stellar halo. The drawback to this techique is that to use it one must use a maximum likelihood algorithm that can in some cases require high performance parallel computing to get an accurate measurement of the parameters in the density function. In the process of learning to use this method, we created a large volunteer computing platform called MilkyWay@home.",
"pages": [
5,
6
]
},
{
"title": "Acknowledgement",
"content": "I would like to thank Matthew Newby and Brian Yanny for helping me obtain the data for the figures in this publication. I also thank Matthew Newby and Jeff Carlin for their help in proofreading. This paper is based upon work supported by the National Science Foundation under Grant No. AST 10-09670. I also would like to thank the MilkyWay@home volunteers for providing us with computing power at no cost, and the Marvin Clan for their support. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.",
"pages": [
6
]
},
{
"title": "References",
"content": "Anderson, D. P., Korpela, E., and Walton, R. 2005, The First International Conf. on e-Science",
"pages": [
6
]
}
]
2013IAUS..289..209W
https://arxiv.org/pdf/1210.7307.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_86><loc_67><loc_90></location>Asymptotic Giant Branch Variables as Extragalactic Distance Indicators</section_header_level_1>
<section_header_level_1><location><page_1><loc_30><loc_83><loc_51><loc_85></location>Patricia A. Whitelock 1 , 2</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_11><loc_82><loc_12><loc_83></location>1</list_item>
<list_item><location><page_1><loc_11><loc_79><loc_70><loc_83></location>South African Astronomical Observatory, P. O. Box 9, 7935 Observatory, South Africa 2 Astronomy, Cosmology and Gravity Centre, Astronomy Department, University of Cape Town, 7701 Rondebosch, South Africa</list_item>
</unordered_list>
<text><location><page_1><loc_9><loc_68><loc_72><loc_77></location>Abstract. Large-amplitude asymptotic giant branch variables potentially rival Cepheid variables as fundamental calibrators of the distance scale, particularly if observations are made in the infrared, or where there is substantial interstellar obscuration. They are particularly useful for probing somewhat older populations, such as those found in dwarf spheroidal galaxies, elliptical galaxies or in the halos of spirals. Calibration data from the Galaxy and new observations of various Local Group galaxies are described and the outlook for the future, with a calibration from Gaia and observations from the next generation of infrared telescopes, is discussed.</text>
<text><location><page_1><loc_9><loc_63><loc_72><loc_67></location>Keywords. stars: late-type, stars: distances, stars: carbon, stars: AGB and post-AGB, galaxies: distances and redshifts, galaxies: dwarf, galaxies: individual (NGC 6822, Fornax dSph, Leo i , Sculptor, Phoenix), galaxies: stellar content</text>
<section_header_level_1><location><page_1><loc_9><loc_57><loc_23><loc_58></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_48><loc_72><loc_56></location>The focus of this paper is on the large-amplitude, or Mira, asymptotic giant branch (AGB) variables. Because of their high luminosity, and particularly their high infrared luminosity, stars in this group have huge potential as extragalactic distance indicators. I briefly describe what we know about the Mira period-luminosity (PL) relationship, before going on to review recent work on Miras in Local Group galaxies, contrasting those found in dwarf spheroidals with those in the dwarf irregular galaxy NGC 6822.</text>
<text><location><page_1><loc_9><loc_36><loc_72><loc_47></location>Miras are very large-amplitude (∆ V > 2 . 5, ∆ K > 0 . 4 mag), long-period ( P > 100 days) variables. They are close to the maximum bolometric luminosity that they will ever achieve and they are cool, with spectra dominated by molecular absorption. Stars with initial masses in the range from 0.8 to 8 M /circledot , possibly higher, are thought to go through the Mira evolutionary phase. For most of their AGB lifetime, their nuclear energy comes from burning of the hydrogen shell. However, from time to time stars on the upper AGB experience helium-shell flashes with far-reaching consequences for the stellar atmospheres.</text>
<text><location><page_1><loc_9><loc_27><loc_72><loc_35></location>It is following these flashes that dredge-up can occur, bringing material from the core into the convective regime, and the atmospheric abundance can change from oxygen (O)to carbon (C)-rich. In relatively high-mass AGB stars, hot-bottom burning can also affect abundances, turning carbon to nitrogen and producing lithium, among other things. The relative fraction of O- and C-rich Miras in any population is a function of both age and metallicity.</text>
<section_header_level_1><location><page_1><loc_9><loc_22><loc_38><loc_23></location>2. Period-Luminosity Relations</section_header_level_1>
<text><location><page_1><loc_9><loc_19><loc_72><loc_21></location>Early work on the Mira PL relation (e.g., at K or M bol ) in the Large Magellanic Cloud (LMC) was published by Feast et al. (1989) and Hughes et al. (1990). However,</text>
<text><location><page_2><loc_9><loc_84><loc_72><loc_94></location>our understanding of this was greatly advanced when Wood (2000) demonstrated that most AGB and upper-giant-branch variables followed various parallel PL sequences. The Mira PL relation, which had been known for some while, was just one of them. Wood showed that the Mira sequence corresponds to fundamental pulsation and includes most of the large-amplitude variables as well as a few of the low-amplitude, or semi-regular, variables. Some of the O-rich large-amplitude, long-period ( P > 400 days) Miras lay above the fundamental sequence, as had also been known for some while.</text>
<text><location><page_2><loc_9><loc_72><loc_72><loc_83></location>The early studies of the multiple sequences used periods determined from microlensing studies, e.g., from macho and ogle at V and I , and these results are now being refined as more systematic observations are made at infrared wavelengths. Among other things, the situation is obviously confused by the fact that many of the Miras have circumstellar shells. In some cases these are so thick that they affect the position of the star on the PL( K ) relation (e.g., Ita & Matsunaga 2011). In those cases, it is preferable to work with bolometric magnitudes, if they can be reliably estimated, although it is sometimes possible to correct for circumstellar extinction.</text>
<text><location><page_2><loc_9><loc_66><loc_72><loc_71></location>The Miras which lie above the PL relation are an interesting subgroup that may be experiencing hot-bottom burning; there is certainly evidence for lithium and other enrichment (Whitelock et al. 2003; Whitelock 2012; and references therein). It also seems possible that they are pulsating in the first overtone (Feast 2009).</text>
<text><location><page_2><loc_9><loc_58><loc_72><loc_65></location>From the distance-scale perspective, the multiple PL sequences of the small-amplitude variables are not very useful, since it is unclear which sequence any particular star will be on. It is therefore only the Miras, i.e., the large-amplitude variables, that we consider useful for distance-scale studies and, where possible, Miras with short periods ( P < 400 days) and thin dust shells are easier to work with.</text>
<text><location><page_2><loc_9><loc_47><loc_72><loc_58></location>Whitelock et al. (2008) discuss the Galactic calibration of the PL( K ) relation and show that it is consistent with the LMC relation. Rejkuba (2004) demonstrated that the PL( K ) relation for O-rich Miras in the LMC also fitted similar stars in NGC 5128. This, together with recent work on Miras in dwarf spheroidals and NGC 6822 (see below) is consistent with the same PL relation applying everywhere. It remains possible that there are metallicity effects, but these are unlikely to be significantly greater than 0.1 mag for stars from populations considered so far (see Matsunaga 2012).</text>
<section_header_level_1><location><page_2><loc_9><loc_43><loc_57><loc_45></location>3. Which Mira PL relation for distance-scale studies?</section_header_level_1>
<text><location><page_2><loc_9><loc_32><loc_72><loc_43></location>For most purposes at the present time it is preferable to work with the PL( K ) relation, primarily because the K magnitude is easy to measure, the pulsation amplitude at K is lower than at shorter wavelengths and K is not often severely affected by circumstellar emission or absorption. However, pulsation drives mass loss, and there are Miras with thick shells and severe circumstellar reddening. This is particularly true of the very longperiod O-rich Miras, known as OH/IR stars, but it is also the case for a variety of C-rich stars.</text>
<text><location><page_2><loc_9><loc_28><loc_72><loc_32></location>In principle using bolometric magnitudes avoids the problem with circumstellar extinction and there are three related ways of deriving such magnitudes, all of which have drawbacks:</text>
<unordered_list>
<list_item><location><page_2><loc_9><loc_23><loc_72><loc_28></location>( a ) Ideally, one obtains bolometric magnitudes derived from multi-epoch measurements across a wide wavelength range extending at least into the mid-infrared regime. Such data are rarely available for many stars.</list_item>
<list_item><location><page_2><loc_9><loc_19><loc_72><loc_23></location>( b ) An alternative approach uses well-calibrated colour-dependent (e.g., J -K ) bolometric corrections, derived for stars that are similar to those of interest, and for which multi-wavelength observations are available. A potential problem is that the observed</list_item>
</unordered_list>
<figure>
<location><page_3><loc_20><loc_63><loc_58><loc_92></location>
<caption>Figure 1. PL( K ) relation for the Miras in the dwarf spheroidals, coded as follows: Fornax (cyan), Leo i (magenta), Sculptor (green), Phoenix (black). The line is at the slope of the LMC PL relation and assumes a distance modulus to the LMC of 18.39 mag (changing this distance modulus will not alter the loci of the points and the line, only their place relative to the magnitude scale).</caption>
</figure>
<text><location><page_3><loc_9><loc_46><loc_72><loc_51></location>colour is a combination of the intrinsic colour of the star and the reddening. The same ( J -K ) colour can be due to different combinations of these two effects. Thus, ( J -K ) may not necessarily be uniquely related to the true bolometric correction.</text>
<text><location><page_3><loc_9><loc_39><loc_72><loc_46></location>( c ) It is sometimes possible to make a correction for the circumstellar reddening (e.g., Matsunaga et al. 2009), which is similar to applying a colour-dependent bolometric correction. The weakness of this is that it requires adoption of an intrinsic colour for the star and we have little evidence of the way this colour correlates with period, particularly for long-period stars, almost all of which have shells.</text>
<text><location><page_3><loc_9><loc_26><loc_72><loc_38></location>Of course, non-uniform shells, which are the obvious consequence of non-uniform mass loss, present obvious problems to any method for estimating the bolometric magnitude. Different methods of calculating the bolometric magnitude give very significantly different results. For example, Groenewegen et al. (2007) and Kamath et al. (2010) derive bolometric magnitudes for several pulsating stars in the Small Magellanic Cloud (SMC) cluster NGC 419 using slightly different JHKL values but the same Spitzer data. Their bolometric magnitudes differ by amounts that range from -0 . 1 to 0.4 mag for the same star (see also Kerschbaum et al. 2010).</text>
<text><location><page_3><loc_9><loc_19><loc_72><loc_26></location>Noting the challenge of determining accurate bolometric magnitudes, estimated values can still provide useful distances, provided a very systematic approach is followed. In practice, this means determining the bolometric magnitude of the star for which the distance is required in exactly the same way as for the calibrators used to define the PL relation.</text>
<figure>
<location><page_4><loc_20><loc_63><loc_58><loc_92></location>
<caption>Figure 2. Bolometric PL relation for the Miras in the dwarf spheroidal galaxies, colours as in Fig. 1.</caption>
</figure>
<section_header_level_1><location><page_4><loc_9><loc_55><loc_35><loc_56></location>4. Globular Cluster: Lyng˚a 7</section_header_level_1>
<text><location><page_4><loc_9><loc_45><loc_72><loc_54></location>Matsunaga (2006) discovered a Mira variable, V1, in Lyng˚a 7 (an old, metal-rich Galactic bulge cluster) with a period of 551 days, a large amplitude, ∆ K = 1 . 22 mag, and red colour, ( J -K ) = 4 . 1 mag. Sloan et al. (2010) showed, based on a Spitzer spectrum, that it is carbon-rich and Feast et al. (2012a) used a spectrum from the Southern African Large Telescope ( SALT ) to demonstrate that V1 is a radial-velocity member of the cluster.</text>
<text><location><page_4><loc_9><loc_39><loc_72><loc_45></location>Assuming a distance modulus of 14.55 mag (Sarajedini et al. 2007), this Mira has M bol = -5 . 0 mag, in agreement with the PL-relation value of M bol = -5 . 2 mag. Such a luminous star must have had an initial mass M i ∼ 1 . 5M /circledot and cannot be a normal member of the cluster. It therefore must have formed from a stellar merger.</text>
<text><location><page_4><loc_9><loc_33><loc_72><loc_39></location>To the best of our knowledge, this is the first ever demonstration of a star that was produced by the merger of two others, but that nevertheless obeys a PL relation. It is an interesting result and we might well expect there to be remnants of other mergers in the dense environment of the Galactic bulge.</text>
<section_header_level_1><location><page_4><loc_9><loc_27><loc_31><loc_28></location>5. Local Group Galaxies</section_header_level_1>
<text><location><page_4><loc_9><loc_19><loc_72><loc_26></location>A group of us from South Africa and Japan have used the Infrared Survey Facility ( IRSF ) at the South African Astronomical Observatory to survey a variety of Local Group galaxies for AGB variables. The several-year survey uses observations made with the SIRIUS camera, which simultaneously gives J , H and K s photometry over a 7 × 7 arcmin 2 field.</text>
<section_header_level_1><location><page_5><loc_29><loc_93><loc_51><loc_94></location>5.1. Dwarf Spheroidal Galaxies</section_header_level_1>
<text><location><page_5><loc_9><loc_86><loc_72><loc_92></location>Results so far have been published for a total of 17 Miras from Fornax, Leo i , Sculptor and Phoenix (Menzies et al. 2008, 2010, 2011; Whitelock et al. 2009). Where spectral types are available for these stars, they show them to be C-rich and we assume that they are all C-type stars.</text>
<text><location><page_5><loc_9><loc_80><loc_72><loc_86></location>Fig. 1 shows the absolute K magnitudes on a PL( K ) relation for all the Miras in dwarf spheroidals. The large scatter is very striking and the distance below the LMC's PL( K ) relation is a function of the ( J -K ) colour, indicating that the stars below the line are there because of their thick circumstellar shells.</text>
<text><location><page_5><loc_9><loc_67><loc_72><loc_80></location>Bolometric magnitudes can be estimated using a ( J -K )-dependent bolometric correction (Whitelock et al. 2009) and the results are shown in a PL relation in Fig. 2. The scatter is vastly reduced, although there are still two stars which lie well below the mean relation. Two possible explanations have been offered for these faint points. It may be that the bolometric correction does not apply to these intrinsically relatively blue, shortperiod, stars (see point [ b ] in Sect. 3), since it was derived for significantly longer-period stars. Alternatively, they are undergoing obscuration events of the type that are common among C-rich Miras in the Galaxy and the LMC (e.g., Whitelock et al. 2006) and have non-uniform shells.</text>
<section_header_level_1><location><page_5><loc_35><loc_64><loc_46><loc_65></location>5.2. NGC6822</section_header_level_1>
<text><location><page_5><loc_9><loc_53><loc_72><loc_64></location>NGC6822 is an isolated barred dwarf galaxy, similar to the SMC, but with slightly higher metallicity. It has been examined for AGB variables in the same way as the dwarf spheroidals over 3.5 years and has had numerous Miras catalogued (Whitelock et al. 2012; see also Battinelli & Demers 2011). Fig. 3 shows the variables in a colour-magnitude diagram. Spectral types are available for only very few stars and the split into O- and C-rich assumes that all very red stars are C-rich. Several of the large-amplitude stars without measured periods are probably also Miras and all very red stars are variable.</text>
<text><location><page_5><loc_9><loc_41><loc_72><loc_53></location>Fig. 4 shows the NGC 6822 Miras on a PL( K ) relation. Most of the longer-period Orich stars fall above the PL relation and are probably similar to the stars in the LMC (mentioned above) that may be hot-bottom burning (Whitelock et al. 2003). Feast (2009) suggested that these stars may be pulsating in the first overtone. Many of the C-rich Miras fall well below the line as the result of thick circumstellar shells. Fig. 5 shows the same stars on a bolometric PL relation, and we see that the C stars scatter around a relation that is very similar to the one obeyed by LMC Miras. The slope is almost identical to the slope of the LMC line, within the uncertainties.</text>
<text><location><page_5><loc_9><loc_31><loc_72><loc_41></location>Using an LMC distance modulus of 18.5 mag, we determine from the C-rich Miras that ( m -M ) 0 = 23 . 56 ± 0 . 03 mag for NGC 6822. This may be compared to 23 . 40 ± 0 . 05 mag derived from Cepheid variables (Feast et al. 2012b) and 23 . 49 ± 0 . 03 mag from RR Lyrae variables (Clementini et al. 2003). Note that all errors quoted here are internal, but there are systematic uncertainties in all of the measurements. The agreement is reasonable and certainly shows that Miras offer a viable alternative to the more conventional distance indicators.</text>
<section_header_level_1><location><page_5><loc_35><loc_28><loc_46><loc_29></location>5.3. Challenges</section_header_level_1>
<text><location><page_5><loc_9><loc_19><loc_72><loc_28></location>However, I should note that there remain challenges in using Miras as distance indicators. One of the most serious of these is ensuring that measurements made with different photometric systems give the same result. Battinelli & Demers (2011) have 16 largeamplitude variables in common with Whitelock et al. (2012) in NGC 6822. The periods determined for these agree well, but the mean magnitudes differ by ∆ K = 0 . 25 mag. Both groups know that their photometry of normal stars, i.e., those with ( J -K ) < 1 . 0 mag,</text>
<figure>
<location><page_6><loc_12><loc_50><loc_67><loc_92></location>
<caption>Figure 3. Colour-magnitude diagram for the variables in NGC 6822, showing Cepheids (magenta), large-amplitude variables with measured periods, which are presumed O-rich (blue) or C-rich (red), and variables without measured periods that have large (yellow) or small (green) amplitudes.</caption>
</figure>
<text><location><page_6><loc_9><loc_38><loc_72><loc_40></location>is on the 2mass system. Dealing with very red stars, of which there are no non-variable examples, will require considerably more effort.</text>
<text><location><page_6><loc_9><loc_35><loc_72><loc_37></location>It also remains possible that we will find metallicity effects as the PL relationships become better defined.</text>
<section_header_level_1><location><page_6><loc_9><loc_30><loc_34><loc_31></location>6. Cepheids and/or Miras?</section_header_level_1>
<text><location><page_6><loc_9><loc_25><loc_72><loc_29></location>Cepheids have long provided a vital step on the distance-scale ladder linking the Galaxy to distant supernovae. However, Mira variables offer a viable alternative which may well be preferable for the following reasons:</text>
<unordered_list>
<list_item><location><page_6><loc_9><loc_22><loc_72><loc_24></location>· Miras are of comparable brightness to Cepheids at K and brighter at longer wavelengths (see Table 1 and Whitelock 2012).</list_item>
<list_item><location><page_6><loc_9><loc_19><loc_72><loc_21></location>· Miras are found in galaxies which do not host Cepheids, such as dwarf spheroidals, and will be found in ellipticals.</list_item>
</unordered_list>
<figure>
<location><page_7><loc_20><loc_74><loc_61><loc_94></location>
<caption>Figure 4. PL( K ) relation for NGC 6822 showing O-rich (open circles) and C-rich Miras. The line has the slope of the LMC PL( K ) relation and has been fit to the four short-period O-rich Miras.</caption>
</figure>
<figure>
<location><page_7><loc_20><loc_48><loc_61><loc_68></location>
<caption>Figure 5. Bolometric PL relation for NGC 6822, showing the same stars as in Fig. 5. The dashed line is the LMC PL relation, while the solid line is the relation fitted to these data.</caption>
</figure>
<unordered_list>
<list_item><location><page_7><loc_9><loc_40><loc_72><loc_43></location>· Miras are found in the haloes of spiral galaxies, where they may be less confused, and therefore more easily observable at large distances, than stars in the spiral arms.</list_item>
<list_item><location><page_7><loc_9><loc_37><loc_72><loc_40></location>· Miras are best observed in the infrared and are therefore not severely affected by interstellar extinction.</list_item>
<list_item><location><page_7><loc_9><loc_33><loc_72><loc_37></location>· The next generation of space telescopes, as well as large ground-based telescopes equipped with adaptive optics, will primarily work in the infrared. They will be ideally suited to observing distant Mira variables.</list_item>
</unordered_list>
<section_header_level_1><location><page_7><loc_9><loc_28><loc_21><loc_30></location>7. Conclusion</section_header_level_1>
<text><location><page_7><loc_9><loc_19><loc_72><loc_28></location>Large-amplitude AGB variables offer a viable alternative to Cepheids for distance-scale studies, which will be particularly valuable when infrared observations are available. There remain, however, calibration issues that must be resolved if observations from different instruments are to be combined reliably. The Gaia satellite will provide a vital Galactic calibration that will put the Mira absolute-magnitude scale on a new footing (see Whitelock 2012).</text>
<table>
<location><page_8><loc_27><loc_79><loc_54><loc_90></location>
<caption>Table 1. Comparison of the absolute magnitudes of Cepheid and Mira variables at 2.2 and 8 µ m (Whitelock 2012 and Feast 2010, unpublished).</caption>
</table>
<section_header_level_1><location><page_8><loc_9><loc_76><loc_25><loc_77></location>Acknowledgements</section_header_level_1>
<text><location><page_8><loc_9><loc_71><loc_72><loc_75></location>I am grateful to my colleagues for allowing me to discuss our results and particularly to Michael Feast and John Menzies for their comments on this manuscript. I acknowledge a grant from the South African National Research Foundation.</text>
<section_header_level_1><location><page_8><loc_9><loc_67><loc_18><loc_68></location>References</section_header_level_1>
<text><location><page_8><loc_9><loc_27><loc_72><loc_66></location>Battinelli, P., & Demers, S. 2011, A&A , 525, 69 Clementini, G., Held, E.V., Baldacci, L., & Rizzi, L. 2003, ApJ , 588, L85 Feast, M.W., Glass, I.S., Whitelock, P.A., & Catchpole, R.M. 1989, MNRAS , 241, 375 Feast, M.W. 2009, in: AGB stars and related phenomena (Ueta, T., Matsunaga, N., & Ita, Y., eds.), AGB stars and related phenomena , p. 48 Feast, M.W., Menzies, J.W., & Whitelock, P.A. 2012a, MNRAS , in press (arXiv:1210.0415) Feast, M.W., Whitelock, P.A., Menzies, J.W., & Matsunaga, N. 2012b, MNRAS , 421, 2998 Groenewegen M.A.T., et al. 2007, MNRAS , 376, 313 Hughes, S.M.G., & Wood, P.R 1990, AJ , 99, 784 Ita, Y., & Matsunaga, N. 2011, MNRAS , 412, 2345 Kamath, D., Wood, P.R., Soszy´nski, I., & Lebzelter, T. 2010, MNRAS , 408, 522 Kerschbaum, F., Lebzelter, T., & Makul, L. 2010, A&A , 524, A87 Matsunaga, N. 2012, Ap&SS , 341, 93 Matsunaga, N., 2006, Ph.D. Thesis , University of Tokyo (Japan) Matsunaga, N., Kawadu, T., Nishiyama, S., Nagayama, T., Hatano, H., Tamura, M., Glass, I.S., & Nagata, T. 2009, MNRAS , 399, 1709 Menzies, J.W., Feast, M.W., Whitelock, P.A., Olivier, E., Matsunaga, N., & da Costa, G. 2008, MNRAS , 385, 1045 Menzies, J.W., Whitelock, P.A., Feast, M.W., & Matsunaga, N. 2010, MNRAS , 406, 86 Menzies, J.W., Feast, M.W., Whitelock, P.A., & Matsunaga, N. 2011, MNRAS , 414, 3492 Rejkuba, M. 2004, A&A , 413, 903 Sarajedini, A., et al. 2007, AJ , 133, 1658 Sloan, G.C., Matsunaga, N., Matsuura, M., et al. 2010, ApJ , 719, 1274 Whitelock, P.A. 2012, Ap&SS , 341, 123 Whitelock, P.A., Feast, M.W., van Loon, J.Th., & Zijlstra, A.A. 2003, MNRAS , 342, 86 Whitelock, P.A., Feast, M.W., Marang, F., & Groenewegen, M.A.T. 2006, MNRAS , 369, 751 Whitelock, P.A., Feast, M.W., & van Leeuwen, F. 2008, MNRAS , 386, 313 Whitelock, P.A., Menzies, J.W., Feast, M.W., Matsunaga, N., Tanab´e, T., & Ita, Y. 2009, MNRAS , 394, 795</text>
<text><location><page_8><loc_9><loc_24><loc_72><loc_27></location>Whitelock, P.A., Menzies, J.W., Feast, M.W., Nsengiyumva, F., & Matsunaga, N. 2012, MNRAS , in press (arXiv:1210.3695)</text>
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</document>
[
{
"title": "Patricia A. Whitelock 1 , 2",
"content": "Abstract. Large-amplitude asymptotic giant branch variables potentially rival Cepheid variables as fundamental calibrators of the distance scale, particularly if observations are made in the infrared, or where there is substantial interstellar obscuration. They are particularly useful for probing somewhat older populations, such as those found in dwarf spheroidal galaxies, elliptical galaxies or in the halos of spirals. Calibration data from the Galaxy and new observations of various Local Group galaxies are described and the outlook for the future, with a calibration from Gaia and observations from the next generation of infrared telescopes, is discussed. Keywords. stars: late-type, stars: distances, stars: carbon, stars: AGB and post-AGB, galaxies: distances and redshifts, galaxies: dwarf, galaxies: individual (NGC 6822, Fornax dSph, Leo i , Sculptor, Phoenix), galaxies: stellar content",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The focus of this paper is on the large-amplitude, or Mira, asymptotic giant branch (AGB) variables. Because of their high luminosity, and particularly their high infrared luminosity, stars in this group have huge potential as extragalactic distance indicators. I briefly describe what we know about the Mira period-luminosity (PL) relationship, before going on to review recent work on Miras in Local Group galaxies, contrasting those found in dwarf spheroidals with those in the dwarf irregular galaxy NGC 6822. Miras are very large-amplitude (∆ V > 2 . 5, ∆ K > 0 . 4 mag), long-period ( P > 100 days) variables. They are close to the maximum bolometric luminosity that they will ever achieve and they are cool, with spectra dominated by molecular absorption. Stars with initial masses in the range from 0.8 to 8 M /circledot , possibly higher, are thought to go through the Mira evolutionary phase. For most of their AGB lifetime, their nuclear energy comes from burning of the hydrogen shell. However, from time to time stars on the upper AGB experience helium-shell flashes with far-reaching consequences for the stellar atmospheres. It is following these flashes that dredge-up can occur, bringing material from the core into the convective regime, and the atmospheric abundance can change from oxygen (O)to carbon (C)-rich. In relatively high-mass AGB stars, hot-bottom burning can also affect abundances, turning carbon to nitrogen and producing lithium, among other things. The relative fraction of O- and C-rich Miras in any population is a function of both age and metallicity.",
"pages": [
1
]
},
{
"title": "2. Period-Luminosity Relations",
"content": "Early work on the Mira PL relation (e.g., at K or M bol ) in the Large Magellanic Cloud (LMC) was published by Feast et al. (1989) and Hughes et al. (1990). However, our understanding of this was greatly advanced when Wood (2000) demonstrated that most AGB and upper-giant-branch variables followed various parallel PL sequences. The Mira PL relation, which had been known for some while, was just one of them. Wood showed that the Mira sequence corresponds to fundamental pulsation and includes most of the large-amplitude variables as well as a few of the low-amplitude, or semi-regular, variables. Some of the O-rich large-amplitude, long-period ( P > 400 days) Miras lay above the fundamental sequence, as had also been known for some while. The early studies of the multiple sequences used periods determined from microlensing studies, e.g., from macho and ogle at V and I , and these results are now being refined as more systematic observations are made at infrared wavelengths. Among other things, the situation is obviously confused by the fact that many of the Miras have circumstellar shells. In some cases these are so thick that they affect the position of the star on the PL( K ) relation (e.g., Ita & Matsunaga 2011). In those cases, it is preferable to work with bolometric magnitudes, if they can be reliably estimated, although it is sometimes possible to correct for circumstellar extinction. The Miras which lie above the PL relation are an interesting subgroup that may be experiencing hot-bottom burning; there is certainly evidence for lithium and other enrichment (Whitelock et al. 2003; Whitelock 2012; and references therein). It also seems possible that they are pulsating in the first overtone (Feast 2009). From the distance-scale perspective, the multiple PL sequences of the small-amplitude variables are not very useful, since it is unclear which sequence any particular star will be on. It is therefore only the Miras, i.e., the large-amplitude variables, that we consider useful for distance-scale studies and, where possible, Miras with short periods ( P < 400 days) and thin dust shells are easier to work with. Whitelock et al. (2008) discuss the Galactic calibration of the PL( K ) relation and show that it is consistent with the LMC relation. Rejkuba (2004) demonstrated that the PL( K ) relation for O-rich Miras in the LMC also fitted similar stars in NGC 5128. This, together with recent work on Miras in dwarf spheroidals and NGC 6822 (see below) is consistent with the same PL relation applying everywhere. It remains possible that there are metallicity effects, but these are unlikely to be significantly greater than 0.1 mag for stars from populations considered so far (see Matsunaga 2012).",
"pages": [
1,
2
]
},
{
"title": "3. Which Mira PL relation for distance-scale studies?",
"content": "For most purposes at the present time it is preferable to work with the PL( K ) relation, primarily because the K magnitude is easy to measure, the pulsation amplitude at K is lower than at shorter wavelengths and K is not often severely affected by circumstellar emission or absorption. However, pulsation drives mass loss, and there are Miras with thick shells and severe circumstellar reddening. This is particularly true of the very longperiod O-rich Miras, known as OH/IR stars, but it is also the case for a variety of C-rich stars. In principle using bolometric magnitudes avoids the problem with circumstellar extinction and there are three related ways of deriving such magnitudes, all of which have drawbacks: colour is a combination of the intrinsic colour of the star and the reddening. The same ( J -K ) colour can be due to different combinations of these two effects. Thus, ( J -K ) may not necessarily be uniquely related to the true bolometric correction. ( c ) It is sometimes possible to make a correction for the circumstellar reddening (e.g., Matsunaga et al. 2009), which is similar to applying a colour-dependent bolometric correction. The weakness of this is that it requires adoption of an intrinsic colour for the star and we have little evidence of the way this colour correlates with period, particularly for long-period stars, almost all of which have shells. Of course, non-uniform shells, which are the obvious consequence of non-uniform mass loss, present obvious problems to any method for estimating the bolometric magnitude. Different methods of calculating the bolometric magnitude give very significantly different results. For example, Groenewegen et al. (2007) and Kamath et al. (2010) derive bolometric magnitudes for several pulsating stars in the Small Magellanic Cloud (SMC) cluster NGC 419 using slightly different JHKL values but the same Spitzer data. Their bolometric magnitudes differ by amounts that range from -0 . 1 to 0.4 mag for the same star (see also Kerschbaum et al. 2010). Noting the challenge of determining accurate bolometric magnitudes, estimated values can still provide useful distances, provided a very systematic approach is followed. In practice, this means determining the bolometric magnitude of the star for which the distance is required in exactly the same way as for the calibrators used to define the PL relation.",
"pages": [
2,
3
]
},
{
"title": "4. Globular Cluster: Lyng˚a 7",
"content": "Matsunaga (2006) discovered a Mira variable, V1, in Lyng˚a 7 (an old, metal-rich Galactic bulge cluster) with a period of 551 days, a large amplitude, ∆ K = 1 . 22 mag, and red colour, ( J -K ) = 4 . 1 mag. Sloan et al. (2010) showed, based on a Spitzer spectrum, that it is carbon-rich and Feast et al. (2012a) used a spectrum from the Southern African Large Telescope ( SALT ) to demonstrate that V1 is a radial-velocity member of the cluster. Assuming a distance modulus of 14.55 mag (Sarajedini et al. 2007), this Mira has M bol = -5 . 0 mag, in agreement with the PL-relation value of M bol = -5 . 2 mag. Such a luminous star must have had an initial mass M i ∼ 1 . 5M /circledot and cannot be a normal member of the cluster. It therefore must have formed from a stellar merger. To the best of our knowledge, this is the first ever demonstration of a star that was produced by the merger of two others, but that nevertheless obeys a PL relation. It is an interesting result and we might well expect there to be remnants of other mergers in the dense environment of the Galactic bulge.",
"pages": [
4
]
},
{
"title": "5. Local Group Galaxies",
"content": "A group of us from South Africa and Japan have used the Infrared Survey Facility ( IRSF ) at the South African Astronomical Observatory to survey a variety of Local Group galaxies for AGB variables. The several-year survey uses observations made with the SIRIUS camera, which simultaneously gives J , H and K s photometry over a 7 × 7 arcmin 2 field.",
"pages": [
4
]
},
{
"title": "5.1. Dwarf Spheroidal Galaxies",
"content": "Results so far have been published for a total of 17 Miras from Fornax, Leo i , Sculptor and Phoenix (Menzies et al. 2008, 2010, 2011; Whitelock et al. 2009). Where spectral types are available for these stars, they show them to be C-rich and we assume that they are all C-type stars. Fig. 1 shows the absolute K magnitudes on a PL( K ) relation for all the Miras in dwarf spheroidals. The large scatter is very striking and the distance below the LMC's PL( K ) relation is a function of the ( J -K ) colour, indicating that the stars below the line are there because of their thick circumstellar shells. Bolometric magnitudes can be estimated using a ( J -K )-dependent bolometric correction (Whitelock et al. 2009) and the results are shown in a PL relation in Fig. 2. The scatter is vastly reduced, although there are still two stars which lie well below the mean relation. Two possible explanations have been offered for these faint points. It may be that the bolometric correction does not apply to these intrinsically relatively blue, shortperiod, stars (see point [ b ] in Sect. 3), since it was derived for significantly longer-period stars. Alternatively, they are undergoing obscuration events of the type that are common among C-rich Miras in the Galaxy and the LMC (e.g., Whitelock et al. 2006) and have non-uniform shells.",
"pages": [
5
]
},
{
"title": "5.2. NGC6822",
"content": "NGC6822 is an isolated barred dwarf galaxy, similar to the SMC, but with slightly higher metallicity. It has been examined for AGB variables in the same way as the dwarf spheroidals over 3.5 years and has had numerous Miras catalogued (Whitelock et al. 2012; see also Battinelli & Demers 2011). Fig. 3 shows the variables in a colour-magnitude diagram. Spectral types are available for only very few stars and the split into O- and C-rich assumes that all very red stars are C-rich. Several of the large-amplitude stars without measured periods are probably also Miras and all very red stars are variable. Fig. 4 shows the NGC 6822 Miras on a PL( K ) relation. Most of the longer-period Orich stars fall above the PL relation and are probably similar to the stars in the LMC (mentioned above) that may be hot-bottom burning (Whitelock et al. 2003). Feast (2009) suggested that these stars may be pulsating in the first overtone. Many of the C-rich Miras fall well below the line as the result of thick circumstellar shells. Fig. 5 shows the same stars on a bolometric PL relation, and we see that the C stars scatter around a relation that is very similar to the one obeyed by LMC Miras. The slope is almost identical to the slope of the LMC line, within the uncertainties. Using an LMC distance modulus of 18.5 mag, we determine from the C-rich Miras that ( m -M ) 0 = 23 . 56 ± 0 . 03 mag for NGC 6822. This may be compared to 23 . 40 ± 0 . 05 mag derived from Cepheid variables (Feast et al. 2012b) and 23 . 49 ± 0 . 03 mag from RR Lyrae variables (Clementini et al. 2003). Note that all errors quoted here are internal, but there are systematic uncertainties in all of the measurements. The agreement is reasonable and certainly shows that Miras offer a viable alternative to the more conventional distance indicators.",
"pages": [
5
]
},
{
"title": "5.3. Challenges",
"content": "However, I should note that there remain challenges in using Miras as distance indicators. One of the most serious of these is ensuring that measurements made with different photometric systems give the same result. Battinelli & Demers (2011) have 16 largeamplitude variables in common with Whitelock et al. (2012) in NGC 6822. The periods determined for these agree well, but the mean magnitudes differ by ∆ K = 0 . 25 mag. Both groups know that their photometry of normal stars, i.e., those with ( J -K ) < 1 . 0 mag, is on the 2mass system. Dealing with very red stars, of which there are no non-variable examples, will require considerably more effort. It also remains possible that we will find metallicity effects as the PL relationships become better defined.",
"pages": [
5,
6
]
},
{
"title": "6. Cepheids and/or Miras?",
"content": "Cepheids have long provided a vital step on the distance-scale ladder linking the Galaxy to distant supernovae. However, Mira variables offer a viable alternative which may well be preferable for the following reasons:",
"pages": [
6
]
},
{
"title": "7. Conclusion",
"content": "Large-amplitude AGB variables offer a viable alternative to Cepheids for distance-scale studies, which will be particularly valuable when infrared observations are available. There remain, however, calibration issues that must be resolved if observations from different instruments are to be combined reliably. The Gaia satellite will provide a vital Galactic calibration that will put the Mira absolute-magnitude scale on a new footing (see Whitelock 2012).",
"pages": [
7
]
},
{
"title": "Acknowledgements",
"content": "I am grateful to my colleagues for allowing me to discuss our results and particularly to Michael Feast and John Menzies for their comments on this manuscript. I acknowledge a grant from the South African National Research Foundation.",
"pages": [
8
]
},
{
"title": "References",
"content": "Battinelli, P., & Demers, S. 2011, A&A , 525, 69 Clementini, G., Held, E.V., Baldacci, L., & Rizzi, L. 2003, ApJ , 588, L85 Feast, M.W., Glass, I.S., Whitelock, P.A., & Catchpole, R.M. 1989, MNRAS , 241, 375 Feast, M.W. 2009, in: AGB stars and related phenomena (Ueta, T., Matsunaga, N., & Ita, Y., eds.), AGB stars and related phenomena , p. 48 Feast, M.W., Menzies, J.W., & Whitelock, P.A. 2012a, MNRAS , in press (arXiv:1210.0415) Feast, M.W., Whitelock, P.A., Menzies, J.W., & Matsunaga, N. 2012b, MNRAS , 421, 2998 Groenewegen M.A.T., et al. 2007, MNRAS , 376, 313 Hughes, S.M.G., & Wood, P.R 1990, AJ , 99, 784 Ita, Y., & Matsunaga, N. 2011, MNRAS , 412, 2345 Kamath, D., Wood, P.R., Soszy´nski, I., & Lebzelter, T. 2010, MNRAS , 408, 522 Kerschbaum, F., Lebzelter, T., & Makul, L. 2010, A&A , 524, A87 Matsunaga, N. 2012, Ap&SS , 341, 93 Matsunaga, N., 2006, Ph.D. Thesis , University of Tokyo (Japan) Matsunaga, N., Kawadu, T., Nishiyama, S., Nagayama, T., Hatano, H., Tamura, M., Glass, I.S., & Nagata, T. 2009, MNRAS , 399, 1709 Menzies, J.W., Feast, M.W., Whitelock, P.A., Olivier, E., Matsunaga, N., & da Costa, G. 2008, MNRAS , 385, 1045 Menzies, J.W., Whitelock, P.A., Feast, M.W., & Matsunaga, N. 2010, MNRAS , 406, 86 Menzies, J.W., Feast, M.W., Whitelock, P.A., & Matsunaga, N. 2011, MNRAS , 414, 3492 Rejkuba, M. 2004, A&A , 413, 903 Sarajedini, A., et al. 2007, AJ , 133, 1658 Sloan, G.C., Matsunaga, N., Matsuura, M., et al. 2010, ApJ , 719, 1274 Whitelock, P.A. 2012, Ap&SS , 341, 123 Whitelock, P.A., Feast, M.W., van Loon, J.Th., & Zijlstra, A.A. 2003, MNRAS , 342, 86 Whitelock, P.A., Feast, M.W., Marang, F., & Groenewegen, M.A.T. 2006, MNRAS , 369, 751 Whitelock, P.A., Feast, M.W., & van Leeuwen, F. 2008, MNRAS , 386, 313 Whitelock, P.A., Menzies, J.W., Feast, M.W., Matsunaga, N., Tanab´e, T., & Ita, Y. 2009, MNRAS , 394, 795 Whitelock, P.A., Menzies, J.W., Feast, M.W., Nsengiyumva, F., & Matsunaga, N. 2012, MNRAS , in press (arXiv:1210.3695) Wood, P.R. 2000, Publ. Astron. Soc. Aus. , 17, 18",
"pages": [
8
]
}
]
2013IAUS..290..163B
https://arxiv.org/pdf/1211.1905.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_86><loc_68><loc_90></location>LOFT: the Large Observatory For X-ray Timing</section_header_level_1>
<section_header_level_1><location><page_1><loc_11><loc_82><loc_69><loc_85></location>Tomaso M. Belloni 1 and Enrico Bozzo 2 (on behalf of the LOFT Consortium)</section_header_level_1>
<text><location><page_1><loc_25><loc_79><loc_55><loc_81></location>1 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy</text>
<text><location><page_1><loc_27><loc_73><loc_53><loc_79></location>email: tomaso.belloni@brera.inaf.it email:</text>
<text><location><page_1><loc_19><loc_73><loc_62><loc_77></location>2 ISDC, Data Center for Astrophysics of the University of Geneva chemin d' ´ Ecogia, 16 1290 Versoix, Switzerland enrico.bozzo@unige.ch</text>
<text><location><page_1><loc_9><loc_59><loc_72><loc_71></location>Abstract. LOFT, the large observatory for X-ray timing, is a new mission concept competing with other four candidates for a launch opportunity in 2022-2024. LOFT will be performing high-time resolution X-ray observations of compact objects, combining for the first time an unprecedented large collecting area for X-ray photons and a spectral resolution approaching that of CCD-based X-ray instruments (down to 200 eV FWHM at 6 keV). The operating energy range is 2-80 keV. The main science goals of LOFT are the measurement of the neutron stars equation of states and the test of General Relativity in the strong field regime. The breakthrough capabilities of the instruments on-board LOFT will permit to open also new discovery windows for a wide range of Galactic and extragalactic X-ray sources.</text>
<text><location><page_1><loc_9><loc_56><loc_72><loc_59></location>In this contribution, we provide a general description of the mission concept and summarize its main scientific capabilities.</text>
<text><location><page_1><loc_9><loc_53><loc_72><loc_55></location>Keywords. instrumentation: detectors, X-rays: binaries, X-rays: galaxies, relativity, equation of state</text>
<section_header_level_1><location><page_1><loc_9><loc_48><loc_23><loc_49></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_32><loc_72><loc_47></location>LOFT, the Large Observatory for X-ray Timing, is one of the four mission candidates currently competing for a launch opportunity in 2022-2024. The four missions were selected by the European Space Agency (ESA) in February 2011 within the framework of the Cosmic Vision program. LOFT was specifically conceived to investigate the behavior of matter in presence of strong gravitational fields and at supranuclear densities by performing high-time-resolution X-ray observations of compact objects. Measurements of the rapid X-ray flux and spectral variability in these sources can indeed provide constraints on the equation of states of neutron stars (NS) and measurements of black holes masses and spins (see Sect. 3). A representation of the current view of the LOFT satellite is provided in Fig. 1.</text>
<text><location><page_1><loc_9><loc_20><loc_72><loc_32></location>The Large Area Detector (LAD) is the main instrument on board LOFT. It is a collimated experiment with a field-of-view (FOV) of about 1 degree, reaching a peak X-ray sensitive effective area of ∼ 10 m 2 at 8 keV (see Fig. 3). The expected scientific throughput is of ∼ 240,000 cts/s for a source with a flux of 1 Crab in the 2-80 keV energy range. The main operating energy band of the LAD is 2-30 keV, where the expected spectral resolution is < 260 eV (200 eV for single anode events, i.e. about 40% of the total counts). In the range 30-80 keV, only a coarse energy resolution of ∼ 2 keV will be available. The time resolution of the LAD is 10 µ s.</text>
<text><location><page_1><loc_11><loc_19><loc_72><loc_20></location>The second instrument completing the LOFT payload is the Wide Field Monitor</text>
<figure>
<location><page_2><loc_10><loc_72><loc_71><loc_93></location>
<caption>Figure 1. Left : an artist representation of the LOFT satellite. The LAD instrument comprises 6 panels connected to the optical bench of the spacecraft; the WFM is located on the top of the optical bench. Right : Breakdown of the LAD instrument. Each panel of the LAD (6 in total) comprises 21 modules and each module is equipped with 16 SDDs.</caption>
</figure>
<text><location><page_2><loc_9><loc_57><loc_72><loc_65></location>(WFM). This is a coded mask imager whose main goal is to detect new transient sources and state changes of any X-ray emitter that can be suitable for observations with the LAD. The WFM will be observing about 1/3 of the sky at once and will be capable to provide data for each source in this FOV with a spectral and timing resolution similar to that of the LAD.</text>
<text><location><page_2><loc_9><loc_53><loc_72><loc_57></location>We provide in Sect. 2 a detailed description of the LOFT payload, and describe in Sect. 3 some of the most recent improvements in the instruments design. All images and simulations used in the following sections are provided by the LOFT teams † .</text>
<section_header_level_1><location><page_2><loc_9><loc_48><loc_46><loc_50></location>2. LOFT Spacecraft and Instrumentation</section_header_level_1>
<text><location><page_2><loc_9><loc_42><loc_72><loc_48></location>In the current configuration of the LOFT mission (see Fig. 1), the LAD instrument comprises six panels attached to the spacecraft optical bench. On the top of it, the five units (10 cameras) of the WFM are placed in a way to maximize the sky coverage (Brandt et al. 2012).</text>
<text><location><page_2><loc_9><loc_22><loc_72><loc_42></location>The LAD is the prime instrument onboard LOFT. Its geometric area is ∼ 15 m 2 and provides a peak effective area for X-ray observations that is a factor of 20 larger than that of any previously flown X-ray experiment (see left panel of Fig. 3). The LAD achieves an energy resolution (FWHM) better than 260 eV at 6 keV (end-of-life, EoL). The two enabling technologies of the LAD are the large area Silicon drift detectors developed originally for the ALICE experiment at CERN (Vacchi et al. 1991; Rashevski et al. 2002) and the capillary plates collimators designed for the ESA BepiColombo mission (Fraser et al. 2010). The SDDs have been optimized for LOFT to reach a configuration in which each detector is now 450 µ m thick and provides a total active area of 76 cm 2 , readout by two rows of anodes with a pitch of 970 µ m and a drift channel of 35 mm. The capillary-plate collimator was chosen to make advantage of the lead glass with millions of micro-pores, already proven to be efficient in collimating X-ray photons within a field of view (FOV) of ∼ 1 degree and energies up to 30-40 keV. In the current LOFT design,</text>
<text><location><page_2><loc_9><loc_19><loc_72><loc_21></location>† See http://www.isdc.unige.ch/loft/index.php/preliminar-response-files-and-simulatedbackground and http://www.isdc.unige.ch/ loft/index.php/instruments-on-board-loft.</text>
<figure>
<location><page_3><loc_9><loc_75><loc_40><loc_93></location>
</figure>
<figure>
<location><page_3><loc_40><loc_75><loc_71><loc_94></location>
<caption>Figure 2. Left : detailed view of the WFM located at the top of the optical bench. The insert shows the concept of a WFM unit comprising two orthogonal cameras to achieve a full 2D imaging capable system. Right: The WFM FOV expected when the instrument points toward the Galactic center. The color bar at the bottom gives the effective area in units of cm 2 .</caption>
</figure>
<text><location><page_3><loc_9><loc_58><loc_72><loc_67></location>each of the SDD is equipped with a single collimator tile ( ∼ 80 cm 2 in area). There are in total 6 Detector Panels, each hosting 21 Detector Modules. 16 SDDs are allocated within each module (see also Zane et al. 2012). A total of 2016 SDDs are needed for the complete LAD instrument. The 6 LAD panels will be initially stowed in the launcher, and then deployed once in space (Feroci et al. 2011). A summary of the LAD capabilities is provided in Table 2.</text>
<text><location><page_3><loc_9><loc_28><loc_72><loc_58></location>The WFM is a wide field coded mask imager working mainly in the 2-50 keV energy range (see Fig. 2). The instrument is designed from the heritage of the SuperAGILE experiment (Feroci et al. 2007), but presents very notable improvements in the low energy threshold, energy resolution and imaging capabilities. These are provided by the usage of similar SDDs to that employed for the LAD but optimized for imaging purposes (Campana et al. 2011). Due to this optimization, the time resolution is kept of the order of ∼ 10 µ s, whereas the spectral resolution is somewhat lower than that achieved by the LAD (300 eV FWHM at 6 keV). The WFM SDDs are able to measure the impact point of a photon with an accuracy that corresponds to few arcmin along the anode direction and few degrees along the drift direction (Evangelista et al. (2012)). For this reason, each of the WFM unit (5 in total, see Fig. 2) comprises two orthogonal cameras to achieve a full 2D imaging capability (Brandt et al. (2012)). This design has also the advantage of providing a strong redundancy in case of failure of one camera. In that case, indeed, a unit will be left with a fine resolution in one direction and a coarse resolution in the other, still ensuring some coverage of the sky in the corresponding region). The imaging properties of the WFM are extensively discussed in Donnarumma et al. 2012. In Fig. 2 we show as an example the WFM FOV during a pointing toward the Galactic Center. The WFM will cover more than 1/3 of the sky at once, corresponding to 50% of the sky accessible to the LAD at any time. A summary of the WFM capabilities is provided in Table 2.</text>
<section_header_level_1><location><page_3><loc_28><loc_25><loc_52><loc_26></location>2.1. Latest mission improvements</section_header_level_1>
<text><location><page_3><loc_9><loc_19><loc_72><loc_24></location>Due to its large field of view, wide energy coverage and imaging capabilities, the WFM is expected to detect and localize a large number of gamma-ray bursts and other fast transients every year (Brandt et al. 2012). The orientation of one of the WFM unit in the anti-Sun direction makes also particularly favorable the follow-up of any interesting event</text>
<table>
<location><page_4><loc_42><loc_80><loc_71><loc_93></location>
<caption>WFM</caption>
</table>
<text><location><page_4><loc_42><loc_79><loc_66><loc_80></location>a : Events in the 30-80 keV energy range are used</text>
<unordered_list>
<list_item><location><page_4><loc_42><loc_78><loc_65><loc_79></location>to monitor contamination from bright sources.</list_item>
</unordered_list>
<paragraph><location><page_4><loc_21><loc_77><loc_60><loc_78></location>Table 1. Scientific requirements and goals of the LAD and WFM.</paragraph>
<text><location><page_4><loc_9><loc_65><loc_72><loc_75></location>detected by this instrument with ground-based facility that will be operating at the time of LOFT. In order to optimize the synergy between LOFT and these facilities, the WFM onboard data processing is endowed with a triggering and imaging system to calculate the coordinates of the transient event in real-time on-board (LBAS, i.e. LOFT Burst Alert System). This information is broadcast to the ground through a VHF transmission system to within /lessorsimilar 30 s from the detection. Due to possible telemetry limitations, the maximum sustainable rate of triggers might be of one per orbit (i.e. one every 90 min.).</text>
<text><location><page_4><loc_9><loc_51><loc_72><loc_65></location>Some effort has been recently devoted to increase the fraction of the sky accessible to the LAD at any time with respect to that defined in the original LOFT configuration † . This will improve the flexibility of the mission in response to requests for target of opportunity observations (ToOs). However, pointings performed outside the nominal sky region might be affected by a somewhat degraded spectral resolution (a factor 1.5 worse than the nominal value) due to the more challenging thermal environment at which the SDDs have to be operated. This is, however, not expected to be an issue for all those observations that do not deal with narrow spectral features and do not require the maximum achievable spectral resolution.</text>
<section_header_level_1><location><page_4><loc_9><loc_47><loc_24><loc_48></location>3. LOFT Science</section_header_level_1>
<text><location><page_4><loc_9><loc_27><loc_72><loc_46></location>The main science goals of LOFT are the study of matter in presence of extreme densities and strong gravitational fields. Through its unique timing capabilities and fine spectral resolution, LOFT will be able to provide unprecedented constraints to the neutron star equation of state (EoS) and test general relativistic effects that are expected to affect X-rays emitted to within a few gravitational radii ( r g ) from a central black hole (so far General Relativity has been tested only in the weak-field regime, i.e. for r g ∼ 10 5 -10 6 ). Beside this, the unique capabilities of the LAD and WFM will also contribute to dramatically deepen our understanding of the physics of a wide range of Galactic and extra-galactic sources. The two instruments will permit spectroscopic and variability studies down to previously unexplored small timescales (few µ s), and provide prompt fine spectral and timing resolution data for any impulsive bright events originating from Galactic sources as well as cosmic gamma-ray bursts. About 50% of the total LOFT observational time will be devoted to this 'observatory science'.</text>
<section_header_level_1><location><page_4><loc_32><loc_24><loc_49><loc_26></location>3.1. Main Science Goals</section_header_level_1>
<text><location><page_4><loc_9><loc_21><loc_72><loc_24></location>Obtaining sufficiently accurate (within a few %) measurements of neutron stars masses and radii is one of the most direct probes we have to test our currently understanding of</text>
<text><location><page_4><loc_17><loc_19><loc_65><loc_20></location>† See http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=49448</text>
<table>
<location><page_4><loc_10><loc_80><loc_38><loc_93></location>
<caption>LAD</caption>
</table>
<unordered_list>
<list_item><location><page_4><loc_10><loc_79><loc_26><loc_80></location>a : Refers to single-anode events.</list_item>
</unordered_list>
<figure>
<location><page_5><loc_17><loc_76><loc_62><loc_93></location>
<caption>Figure 3. Left : Plot of the LAD effective area as a function of the energy according to the instrument requirements (see Table 2). The effective area of some other instruments is reported for comparison. Right : A simulated power-spectrum of the black hole candidate XTE J1550-564 as observed by RXTE/PCA (Miller et al. 2001). For the simulation it is assumed that the two QPOs have frequencies of ν 1 =188 Hz, ν 2 =268 Hz and the fractional rms are 2.8% and 6.2%, respectively. The source flux is 1 Crab (3-20 keV). The exposure time used for the LAD simulation is 1 ks. The improvement in the S/N of this detection should be compared by looking at Fig. 4 in Miller et al. (2001), where the original RXTE/PCA exposure time of 54 ks permitted only a detection at 3-4 σ confidence level.</caption>
</figure>
<text><location><page_5><loc_9><loc_50><loc_72><loc_61></location>matter at supranuclear densities, as this can constraint the different EoS models proposed for these objects. LOFT will be able to measure the neutron star mass and radius down to an accuracy of a few percent independently by using different techniques (e.g., pulse profile fitting in millisecond X-ray pulsars and spectral fitting of the type-I X-ray bursts observed from neutron star low mass X-ray binaries; Feroci et al. 2011). The top-level science goals of LOFT related to the determination of dense matter EOS are summarized in Feroci et al. (2012).</text>
<text><location><page_5><loc_9><loc_19><loc_72><loc_50></location>By making advantage of its unique throughput, LOFT will also be able to test General Relativistic effects produced by the strong gravitational field of a black hole onto accreting matter orbiting close-by (a few r g ). One of the most promising way to carry out such investigations is to observe quasi-periodic oscillations (QPOs) arising from the X-ray emission of the accretion flow. The dynamical time-scales of the inner accretion flow are typically of few milliseconds, and thus the interpretation of the highest frequency QPOs involves fundamental frequencies of the motion of matter orbiting in disk regions dominated by the strong black hole gravitational field (e.g., the relativistic radial and vertical epicyclic frequencies or the relativistic nodal and periastron precession). Distinguishing among these possibilities have been proved impossible so far, due to the limited observation capabilities of the previous generation of X-ray timing instruments. The enhanced capabilities of the LAD will permit to dramatically increase the S/N of any of the QPO feature (see right panel of Fig. 3), allowing to measure QPOs accurately within their coherence time and follow their evolution in time with the source X-ray flux (see Fig. 4). This will secure access to still untested General relativistic effects, such as frame-dragging, strong-field periastron precession, and the existence of an innermost stable orbit around black holes. An independent investigation of the motion of the accretion flow orbiting close to the central black hole can be carried out through the observation of the fluorescence Fe K line profile emitted by this material at different orbital phases, by making advantages of the fine resolution of the LAD. Variation in the line profile are expected as a consequence of the Lense-Thirring precession of the inner disk (at ∼ r g ),</text>
<figure>
<location><page_6><loc_14><loc_64><loc_66><loc_94></location>
<caption>Figure 4. The goal of the LAD is not only to detect QPOs with higher S/N, as shown in Fig. 3 (right panel). In this figure we show that the LAD will also be able to follow the evolution in time of the QPOs, removing the observational constraints of its predecessor the RXTE/PCA (represented with a blue obscuring rectangular in the upper panel). In the bottom two panels we show a simulation in which the same QPO in Fig. 3 is evolved following the prediction of the epicyclic resonant model (Abramowicz et al. 2001) and the relativistic precession model (Stella et al. 1999). In the simulations we assumed: flux of 1 Crab and fractional rms 2.8% and 6.2% for the blue QPOs, flux of 400 mCrab and fractional rms 1.4% and 3.1% for the green QPOs, flux of 300 mCrab and fractional rms 0.7% and 1.5% for the magenta QPOs. The exposure time is 16 ks in all cases.</caption>
</figure>
<text><location><page_6><loc_9><loc_43><loc_72><loc_48></location>and provide a tool to measure mass and spin of Galactic and extra-galactic black holes. In Fig. 5, we show a simulation of such a study in the Active Galactic Nuclei (AGN) MGC6-30. A summary of the top-level science goals of LOFT with respect to the measure of General Relativistic effects can be found in Feroci et al. (2012).</text>
<section_header_level_1><location><page_6><loc_31><loc_40><loc_49><loc_41></location>3.2. Observatory Science</section_header_level_1>
<text><location><page_6><loc_9><loc_19><loc_72><loc_40></location>About 50% of the total LOFT observational time will be made available for observatory science. The breakthrough capabilities of the LAD will revolutionize X-ray observations of any relatively bright source down to a limiting flux of few mCrab. Providing an overview of the entire potential discovery space of the LAD is not possible in this relatively short contribution. Instead we provide here some more examples of the capabilities of the WFM for the observatory science. This instrument combines a very wide FOV, covering more than 1/3 of the sky at once, together with a fine timing (few µ s) and spectral resolution ( < 300 eV). Through the LBAS system, the position and trigger time of bright events can be broadcast to the ground within a delay of /lessorsimilar 30 s, allowing for quick follow-up with ground-based facilities. Due to the large FOV, about 150 GRB and thousands typeI X-ray bursts are expected to be detected with the WFM per year. The low energy threshold and fine spectral resolution of the instrument will allow to investigate in detail any spectral feature expected during these events, including absorption edges. We show two specific simulations in Fig. 6. As for the LAD, the range of scientific investigations</text>
<figure>
<location><page_7><loc_19><loc_68><loc_58><loc_91></location>
<caption>Figure 5. Simulation of the detection with the LAD of the iron line in four different orbital phases of the accretion flow in the AGN MGC 6-30. The simulation assumes an average flux of 3 mCrab and an exposure time of 16 ks for each of the orbital phases (each phase is observed twice; courtesy of A. de Rosa).</caption>
</figure>
<figure>
<location><page_7><loc_11><loc_43><loc_70><loc_60></location>
<caption>Figure 6. Left : Absorption edges in energetic type-I X-ray bursts were predicted theoretically by Weinberg et al. (2006) and the only observational evidence of them was reported so far by in't Zand et al. (2010). Following their results we simulated here an X-ray burst with a flux of 5 Crab with the WFM including a τ =1 absorption edge at 6 keV. The edge is clearly detected already with 1 s exposure (courtesy of WFM-SIM group). Right : The only example in the literature of narrow spectral features in the prompt emission of a GRB is reported by Amati et al. (2000). Here we simulated an observation of that GRB with the WFM using an exposure time of 13 s. The edge at 3.8 keV is clearly detected (courtesy of WFM-SIM group).</caption>
</figure>
<text><location><page_7><loc_9><loc_25><loc_72><loc_30></location>available to the WFM cannot be sufficiently summarized here. As an example, we refer the reader to Romano et al. (2012) and Ferrigno et al. (2012) for a description of the WFM contribution to the improvement in our knowledge of the Be X-ray binaries and Supergiant Fast X-ray Transients.</text>
<section_header_level_1><location><page_7><loc_9><loc_21><loc_18><loc_22></location>References</section_header_level_1>
<text><location><page_7><loc_9><loc_19><loc_35><loc_20></location>Amati, L. et al. 2000, Science, 290, 953</text>
<text><location><page_8><loc_9><loc_75><loc_63><loc_94></location>Abramowicz, M.A. & Kluzniak, W. 2001, A&A, 374, 19 Brandt, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-88, 2012 Campana, R. et al. 2011, NIMPA, 633, 22 Campana, R. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-209, 2012 Donnarumma, I. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-211, 2012 Evangelista, Y. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-85, 2012 Feroci, M. et al. 2007, NIMPA, 581, 728 Feroci, M. et al. 2011, ExA, in press (arXiv:1107.0436) Feroci, M. et al. 2012, Proceedings of SPIE, Vol. 8443, Paper No. 8443-210, 2012 Ferrigno, C. et al. 2012, Proceedings of COSPAR, in preparation in't Zand, J. et al. 2010, A&A, 520, 81 Miller, J.M. et al. 2001, ApJ, 563, 928 Romano, P. et al. 2012, AIP, in preparation Stella, L. & Vietri 1999, ApJ, 524, 63</text>
<text><location><page_8><loc_9><loc_73><loc_57><loc_74></location>Zane, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-87, 2012</text>
<text><location><page_8><loc_9><loc_72><loc_35><loc_73></location>Weimberg, . et al., 2012 ApJ, 639, 1018</text>
</document>
[
{
"title": "Tomaso M. Belloni 1 and Enrico Bozzo 2 (on behalf of the LOFT Consortium)",
"content": "1 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy email: tomaso.belloni@brera.inaf.it email: 2 ISDC, Data Center for Astrophysics of the University of Geneva chemin d' ´ Ecogia, 16 1290 Versoix, Switzerland enrico.bozzo@unige.ch Abstract. LOFT, the large observatory for X-ray timing, is a new mission concept competing with other four candidates for a launch opportunity in 2022-2024. LOFT will be performing high-time resolution X-ray observations of compact objects, combining for the first time an unprecedented large collecting area for X-ray photons and a spectral resolution approaching that of CCD-based X-ray instruments (down to 200 eV FWHM at 6 keV). The operating energy range is 2-80 keV. The main science goals of LOFT are the measurement of the neutron stars equation of states and the test of General Relativity in the strong field regime. The breakthrough capabilities of the instruments on-board LOFT will permit to open also new discovery windows for a wide range of Galactic and extragalactic X-ray sources. In this contribution, we provide a general description of the mission concept and summarize its main scientific capabilities. Keywords. instrumentation: detectors, X-rays: binaries, X-rays: galaxies, relativity, equation of state",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "LOFT, the Large Observatory for X-ray Timing, is one of the four mission candidates currently competing for a launch opportunity in 2022-2024. The four missions were selected by the European Space Agency (ESA) in February 2011 within the framework of the Cosmic Vision program. LOFT was specifically conceived to investigate the behavior of matter in presence of strong gravitational fields and at supranuclear densities by performing high-time-resolution X-ray observations of compact objects. Measurements of the rapid X-ray flux and spectral variability in these sources can indeed provide constraints on the equation of states of neutron stars (NS) and measurements of black holes masses and spins (see Sect. 3). A representation of the current view of the LOFT satellite is provided in Fig. 1. The Large Area Detector (LAD) is the main instrument on board LOFT. It is a collimated experiment with a field-of-view (FOV) of about 1 degree, reaching a peak X-ray sensitive effective area of ∼ 10 m 2 at 8 keV (see Fig. 3). The expected scientific throughput is of ∼ 240,000 cts/s for a source with a flux of 1 Crab in the 2-80 keV energy range. The main operating energy band of the LAD is 2-30 keV, where the expected spectral resolution is < 260 eV (200 eV for single anode events, i.e. about 40% of the total counts). In the range 30-80 keV, only a coarse energy resolution of ∼ 2 keV will be available. The time resolution of the LAD is 10 µ s. The second instrument completing the LOFT payload is the Wide Field Monitor (WFM). This is a coded mask imager whose main goal is to detect new transient sources and state changes of any X-ray emitter that can be suitable for observations with the LAD. The WFM will be observing about 1/3 of the sky at once and will be capable to provide data for each source in this FOV with a spectral and timing resolution similar to that of the LAD. We provide in Sect. 2 a detailed description of the LOFT payload, and describe in Sect. 3 some of the most recent improvements in the instruments design. All images and simulations used in the following sections are provided by the LOFT teams † .",
"pages": [
1,
2
]
},
{
"title": "2. LOFT Spacecraft and Instrumentation",
"content": "In the current configuration of the LOFT mission (see Fig. 1), the LAD instrument comprises six panels attached to the spacecraft optical bench. On the top of it, the five units (10 cameras) of the WFM are placed in a way to maximize the sky coverage (Brandt et al. 2012). The LAD is the prime instrument onboard LOFT. Its geometric area is ∼ 15 m 2 and provides a peak effective area for X-ray observations that is a factor of 20 larger than that of any previously flown X-ray experiment (see left panel of Fig. 3). The LAD achieves an energy resolution (FWHM) better than 260 eV at 6 keV (end-of-life, EoL). The two enabling technologies of the LAD are the large area Silicon drift detectors developed originally for the ALICE experiment at CERN (Vacchi et al. 1991; Rashevski et al. 2002) and the capillary plates collimators designed for the ESA BepiColombo mission (Fraser et al. 2010). The SDDs have been optimized for LOFT to reach a configuration in which each detector is now 450 µ m thick and provides a total active area of 76 cm 2 , readout by two rows of anodes with a pitch of 970 µ m and a drift channel of 35 mm. The capillary-plate collimator was chosen to make advantage of the lead glass with millions of micro-pores, already proven to be efficient in collimating X-ray photons within a field of view (FOV) of ∼ 1 degree and energies up to 30-40 keV. In the current LOFT design, † See http://www.isdc.unige.ch/loft/index.php/preliminar-response-files-and-simulatedbackground and http://www.isdc.unige.ch/ loft/index.php/instruments-on-board-loft. each of the SDD is equipped with a single collimator tile ( ∼ 80 cm 2 in area). There are in total 6 Detector Panels, each hosting 21 Detector Modules. 16 SDDs are allocated within each module (see also Zane et al. 2012). A total of 2016 SDDs are needed for the complete LAD instrument. The 6 LAD panels will be initially stowed in the launcher, and then deployed once in space (Feroci et al. 2011). A summary of the LAD capabilities is provided in Table 2. The WFM is a wide field coded mask imager working mainly in the 2-50 keV energy range (see Fig. 2). The instrument is designed from the heritage of the SuperAGILE experiment (Feroci et al. 2007), but presents very notable improvements in the low energy threshold, energy resolution and imaging capabilities. These are provided by the usage of similar SDDs to that employed for the LAD but optimized for imaging purposes (Campana et al. 2011). Due to this optimization, the time resolution is kept of the order of ∼ 10 µ s, whereas the spectral resolution is somewhat lower than that achieved by the LAD (300 eV FWHM at 6 keV). The WFM SDDs are able to measure the impact point of a photon with an accuracy that corresponds to few arcmin along the anode direction and few degrees along the drift direction (Evangelista et al. (2012)). For this reason, each of the WFM unit (5 in total, see Fig. 2) comprises two orthogonal cameras to achieve a full 2D imaging capability (Brandt et al. (2012)). This design has also the advantage of providing a strong redundancy in case of failure of one camera. In that case, indeed, a unit will be left with a fine resolution in one direction and a coarse resolution in the other, still ensuring some coverage of the sky in the corresponding region). The imaging properties of the WFM are extensively discussed in Donnarumma et al. 2012. In Fig. 2 we show as an example the WFM FOV during a pointing toward the Galactic Center. The WFM will cover more than 1/3 of the sky at once, corresponding to 50% of the sky accessible to the LAD at any time. A summary of the WFM capabilities is provided in Table 2.",
"pages": [
2,
3
]
},
{
"title": "2.1. Latest mission improvements",
"content": "Due to its large field of view, wide energy coverage and imaging capabilities, the WFM is expected to detect and localize a large number of gamma-ray bursts and other fast transients every year (Brandt et al. 2012). The orientation of one of the WFM unit in the anti-Sun direction makes also particularly favorable the follow-up of any interesting event a : Events in the 30-80 keV energy range are used detected by this instrument with ground-based facility that will be operating at the time of LOFT. In order to optimize the synergy between LOFT and these facilities, the WFM onboard data processing is endowed with a triggering and imaging system to calculate the coordinates of the transient event in real-time on-board (LBAS, i.e. LOFT Burst Alert System). This information is broadcast to the ground through a VHF transmission system to within /lessorsimilar 30 s from the detection. Due to possible telemetry limitations, the maximum sustainable rate of triggers might be of one per orbit (i.e. one every 90 min.). Some effort has been recently devoted to increase the fraction of the sky accessible to the LAD at any time with respect to that defined in the original LOFT configuration † . This will improve the flexibility of the mission in response to requests for target of opportunity observations (ToOs). However, pointings performed outside the nominal sky region might be affected by a somewhat degraded spectral resolution (a factor 1.5 worse than the nominal value) due to the more challenging thermal environment at which the SDDs have to be operated. This is, however, not expected to be an issue for all those observations that do not deal with narrow spectral features and do not require the maximum achievable spectral resolution.",
"pages": [
3,
4
]
},
{
"title": "3. LOFT Science",
"content": "The main science goals of LOFT are the study of matter in presence of extreme densities and strong gravitational fields. Through its unique timing capabilities and fine spectral resolution, LOFT will be able to provide unprecedented constraints to the neutron star equation of state (EoS) and test general relativistic effects that are expected to affect X-rays emitted to within a few gravitational radii ( r g ) from a central black hole (so far General Relativity has been tested only in the weak-field regime, i.e. for r g ∼ 10 5 -10 6 ). Beside this, the unique capabilities of the LAD and WFM will also contribute to dramatically deepen our understanding of the physics of a wide range of Galactic and extra-galactic sources. The two instruments will permit spectroscopic and variability studies down to previously unexplored small timescales (few µ s), and provide prompt fine spectral and timing resolution data for any impulsive bright events originating from Galactic sources as well as cosmic gamma-ray bursts. About 50% of the total LOFT observational time will be devoted to this 'observatory science'.",
"pages": [
4
]
},
{
"title": "3.1. Main Science Goals",
"content": "Obtaining sufficiently accurate (within a few %) measurements of neutron stars masses and radii is one of the most direct probes we have to test our currently understanding of † See http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=49448 matter at supranuclear densities, as this can constraint the different EoS models proposed for these objects. LOFT will be able to measure the neutron star mass and radius down to an accuracy of a few percent independently by using different techniques (e.g., pulse profile fitting in millisecond X-ray pulsars and spectral fitting of the type-I X-ray bursts observed from neutron star low mass X-ray binaries; Feroci et al. 2011). The top-level science goals of LOFT related to the determination of dense matter EOS are summarized in Feroci et al. (2012). By making advantage of its unique throughput, LOFT will also be able to test General Relativistic effects produced by the strong gravitational field of a black hole onto accreting matter orbiting close-by (a few r g ). One of the most promising way to carry out such investigations is to observe quasi-periodic oscillations (QPOs) arising from the X-ray emission of the accretion flow. The dynamical time-scales of the inner accretion flow are typically of few milliseconds, and thus the interpretation of the highest frequency QPOs involves fundamental frequencies of the motion of matter orbiting in disk regions dominated by the strong black hole gravitational field (e.g., the relativistic radial and vertical epicyclic frequencies or the relativistic nodal and periastron precession). Distinguishing among these possibilities have been proved impossible so far, due to the limited observation capabilities of the previous generation of X-ray timing instruments. The enhanced capabilities of the LAD will permit to dramatically increase the S/N of any of the QPO feature (see right panel of Fig. 3), allowing to measure QPOs accurately within their coherence time and follow their evolution in time with the source X-ray flux (see Fig. 4). This will secure access to still untested General relativistic effects, such as frame-dragging, strong-field periastron precession, and the existence of an innermost stable orbit around black holes. An independent investigation of the motion of the accretion flow orbiting close to the central black hole can be carried out through the observation of the fluorescence Fe K line profile emitted by this material at different orbital phases, by making advantages of the fine resolution of the LAD. Variation in the line profile are expected as a consequence of the Lense-Thirring precession of the inner disk (at ∼ r g ), and provide a tool to measure mass and spin of Galactic and extra-galactic black holes. In Fig. 5, we show a simulation of such a study in the Active Galactic Nuclei (AGN) MGC6-30. A summary of the top-level science goals of LOFT with respect to the measure of General Relativistic effects can be found in Feroci et al. (2012).",
"pages": [
4,
5,
6
]
},
{
"title": "3.2. Observatory Science",
"content": "About 50% of the total LOFT observational time will be made available for observatory science. The breakthrough capabilities of the LAD will revolutionize X-ray observations of any relatively bright source down to a limiting flux of few mCrab. Providing an overview of the entire potential discovery space of the LAD is not possible in this relatively short contribution. Instead we provide here some more examples of the capabilities of the WFM for the observatory science. This instrument combines a very wide FOV, covering more than 1/3 of the sky at once, together with a fine timing (few µ s) and spectral resolution ( < 300 eV). Through the LBAS system, the position and trigger time of bright events can be broadcast to the ground within a delay of /lessorsimilar 30 s, allowing for quick follow-up with ground-based facilities. Due to the large FOV, about 150 GRB and thousands typeI X-ray bursts are expected to be detected with the WFM per year. The low energy threshold and fine spectral resolution of the instrument will allow to investigate in detail any spectral feature expected during these events, including absorption edges. We show two specific simulations in Fig. 6. As for the LAD, the range of scientific investigations available to the WFM cannot be sufficiently summarized here. As an example, we refer the reader to Romano et al. (2012) and Ferrigno et al. (2012) for a description of the WFM contribution to the improvement in our knowledge of the Be X-ray binaries and Supergiant Fast X-ray Transients.",
"pages": [
6,
7
]
},
{
"title": "References",
"content": "Amati, L. et al. 2000, Science, 290, 953 Abramowicz, M.A. & Kluzniak, W. 2001, A&A, 374, 19 Brandt, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-88, 2012 Campana, R. et al. 2011, NIMPA, 633, 22 Campana, R. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-209, 2012 Donnarumma, I. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-211, 2012 Evangelista, Y. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-85, 2012 Feroci, M. et al. 2007, NIMPA, 581, 728 Feroci, M. et al. 2011, ExA, in press (arXiv:1107.0436) Feroci, M. et al. 2012, Proceedings of SPIE, Vol. 8443, Paper No. 8443-210, 2012 Ferrigno, C. et al. 2012, Proceedings of COSPAR, in preparation in't Zand, J. et al. 2010, A&A, 520, 81 Miller, J.M. et al. 2001, ApJ, 563, 928 Romano, P. et al. 2012, AIP, in preparation Stella, L. & Vietri 1999, ApJ, 524, 63 Zane, S. et al., Proceedings of SPIE, Vol. 8443, Paper No. 8443-87, 2012 Weimberg, . et al., 2012 ApJ, 639, 1018",
"pages": [
7,
8
]
}
]
2013IAUS..290..237K
https://arxiv.org/pdf/1304.7581.pdf
<document>
<figure>
<location><page_1><loc_14><loc_90><loc_24><loc_95></location>
</figure>
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<location><page_1><loc_64><loc_91><loc_85><loc_95></location>
</figure>
<section_header_level_1><location><page_1><loc_14><loc_77><loc_85><loc_82></location>The X-ray spectro-timing properties of a major radio flare episode in Cygnus X-3</section_header_level_1>
<section_header_level_1><location><page_1><loc_14><loc_69><loc_28><loc_71></location>Karri Koljonen ∗ †</section_header_level_1>
<text><location><page_1><loc_14><loc_68><loc_47><loc_69></location>Aalto University Metsähovi Observatory, Finland</text>
<text><location><page_1><loc_14><loc_66><loc_44><loc_67></location>E-mail: karri.koljonen@gmail.com</text>
<section_header_level_1><location><page_1><loc_14><loc_63><loc_32><loc_65></location>Michael McCollough</section_header_level_1>
<text><location><page_1><loc_14><loc_62><loc_44><loc_63></location>Smithsonian Astrophysical Observatory, USA</text>
<text><location><page_1><loc_14><loc_60><loc_52><loc_61></location>E-mail: mmccollough@head.cfa.harvard.edu</text>
<section_header_level_1><location><page_1><loc_14><loc_58><loc_30><loc_59></location>Diana Hannikainen</section_header_level_1>
<text><location><page_1><loc_14><loc_56><loc_39><loc_57></location>Florida Institute of Technology, USA</text>
<text><location><page_1><loc_14><loc_55><loc_38><loc_56></location>E-mail: ddcarina@gmail.com</text>
<section_header_level_1><location><page_1><loc_14><loc_52><loc_28><loc_53></location>Robert Droulans</section_header_level_1>
<text><location><page_1><loc_14><loc_51><loc_42><loc_52></location>Lycée classique d'Echternach, Luxemburg</text>
<text><location><page_1><loc_14><loc_49><loc_48><loc_50></location>E-mail: robert.droulans@education.lu</text>
<text><location><page_1><loc_18><loc_32><loc_82><loc_47></location>We have performed a principal component analysis on the X-ray spectra of the microquasar Cygnus X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event in 2006 MayJuly. The analysis showed that there are two main variability components in play, i.e. two principal components explained almost all the variability in the X-ray lightcurves. According to the spectral shape of these components and spectral fits to the original data, the most probable emission components corresponding to the principal components are inverse-Compton scattering and bremsstrahlung. We find that these components form a double-peaked profile when phase-folded with the peaks occurring in opposite phases. This could be due to an asymmetrical wind around the companion star with which the compact object is interacting.</text>
<text><location><page_1><loc_14><loc_22><loc_85><loc_24></location>An INTEGRAL view of the high-energy sky (the first 10 years) - 9th INTEGRAL Workshop and celebration of the 10th anniversary of the launch</text>
<text><location><page_1><loc_14><loc_20><loc_28><loc_21></location>15-19 October 2012</text>
<text><location><page_1><loc_14><loc_19><loc_47><loc_20></location>Bibliotheque Nationale de France, Paris, France</text>
<section_header_level_1><location><page_2><loc_14><loc_87><loc_27><loc_89></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_14><loc_74><loc_86><loc_85></location>Modeling the X-ray spectra of X-ray binaries (XRBs) often leads to a problem of degeneracy, i.e. multiple distinct models fit the observed data equally well. A striking example can be seen e.g. in [1], where three very different models are fitted equally well to the same data set of Cygnus X-1, despite the excellent quality of the data that was obtained by all the X-ray satellites in orbit at the time. Similar spectral degeneracy was observed for Cygnus X-3 in [2]. Therefore, even if an apparently good fit is obtained between the data and the model, it does not necessarily imply a match between theory and physical reality.</text>
<text><location><page_2><loc_14><loc_63><loc_86><loc_73></location>In order to make sense of this degeneracy we need to take other data dimensions into account, namely timing and/or polarization. While we are just reaching the point where the polarization dimension can be explored with very long exposure observations [3], X-ray timing data is readily available, and several methods have been developed to combine spectral and timing analyses (e.g. [4]). These comparisons, however, do not reveal what the actual spectral components causing the variability are .</text>
<text><location><page_2><loc_14><loc_54><loc_86><loc_62></location>One way of revealing the variability components of X-ray spectra is to employ principal component analysis (PCA), one of the standard tools of time series analysis that has been introduced in the analysis of the X-ray data of XRBs by [5] and references therein, and further refined in [6] for Cyg X-3 to single out individual emission components causing the variability in the X-ray lightcurves.</text>
<section_header_level_1><location><page_2><loc_14><loc_50><loc_42><loc_51></location>2. Principal component analysis</section_header_level_1>
<text><location><page_2><loc_14><loc_26><loc_86><loc_48></location>Below, we summarize the main points of the PCA. More detail can be found from [5] and [6]. The first step of the PCA is to arrange a number of spectra measured at different times, i.e. a time series of X-ray spectra, to a data matrix. Then the eigenvectors of the covariance matrix calculated from the original data matrix form the principal components that are responsible for the spectral variation. The accompanying eigenvalue states the proportion of variance of a particular eigenvector, i.e. the highest eigenvalue and accompanying eigenvector is the first principal component of the data set etc. One can then form a linear decomposition of the data set using these eigenvectors, ordered by the proportion of variance. The components producing only a small fraction of the overall variance can then be dropped, reducing the dimensionality, i.e. choosing a small number of eigenvectors for the linear composition. The PCA can be exploited in different ways using both spectral and timing information. In the spectral domain this includes producing the variability spectra and log-eigenvalue diagrams. In the timing domain we can follow the evolution of individual principal components by tracking the value of its normalization.</text>
<section_header_level_1><location><page_2><loc_14><loc_22><loc_46><loc_23></location>3. Applying PCA to Cygnus X-3 data</section_header_level_1>
<text><location><page_2><loc_14><loc_13><loc_86><loc_20></location>Cyg X-3 is one of the most peculiar sources amongst microquasars. It is known for massive outbursts that emit radiation from radio to g -rays and produce major radio flaring episodes usually with multiple flares that peak up to 20 Jy [7], making it the most radio luminous single object in our Galaxy. The binary components of Cyg X-3 orbit each other in a tight 4.8-hour period [8], typical</text>
<figure>
<location><page_3><loc_36><loc_69><loc_64><loc_89></location>
<caption>Figure 1: The figure show the one-day integrated Ryle/AMI-LA 15 GHz lightcurves from the May 2006 major radio flare. The top part of the panel shows the epochs of the RXTE (black), Swift (orange) and INTEGRAL (magenta) data that are examined here.</caption>
</figure>
<text><location><page_3><loc_14><loc_47><loc_86><loc_59></location>for XRBs with a low-mass companion. However, infrared spectral observations suggest that the mass-donating companion in the binary is a massive Wolf-Rayet star [9]. Due to this discrepancy Cyg X-3 is by definition a unique source and similar sources have only been found in other nearby galaxies [10]. The X-ray spectral and timing properties of Cyg X-3 show a disparity with other XRBs/microquasars increasing the difficulty of interpreting the nature of the system. This disparity could be due to the interaction of the strong stellar wind of the Wolf-Rayet companion with the compact object.</text>
<text><location><page_3><loc_14><loc_35><loc_86><loc_46></location>We have performed a PCA to the X-ray spectra of Cyg X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event. The X-ray observations were obtained during the 2006 May major radio flaring episode, that consisted of two major radio flares with peak flux densities of 13.8 Jy (15 GHz) and 11.2 Jy (11.2 GHz) at MJD 53865 and MJD 53942, respectively (see Fig. 1 for the May 2006 flare). For RXTE and Swift data we used the consecutive observations taken right after a flare and for INTEGRAL we used revolutions 437, 438 and 462. The radio data are from [11].</text>
<section_header_level_1><location><page_3><loc_14><loc_31><loc_35><loc_33></location>3.1 PCA tools of the trade</section_header_level_1>
<text><location><page_3><loc_14><loc_13><loc_86><loc_30></location>The standard tool for deciding how many principal components one should retain is called the log-eigenvalue (LEV) diagram (upper panels in Fig. 2). If the principal components decay in geometric progression in the data, the corresponding eigenvalues will appear as a straight line in the LEV diagram and thus signal the start of the 'noise" components. The variance spectrum (lower panels in Fig. 2) is a graph that shows the measured variance as a function of energy. It can be plotted for all the principal components, i.e. showing the overall variance across the energy range of the data, or for each principal component independently thus showing the spectral shape of the varying principal components. From Fig. 2 we see that there are two main variability components in play in the Cyg X-3 data, i.e. two principal components explain almost all the variability in the X-ray lightcurves.</text>
<figure>
<location><page_4><loc_14><loc_63><loc_85><loc_89></location>
<caption>Figure 2: Upper panels: The LEV diagrams of the principal components from all X-ray observatories (left: Swift , center: RXTE , right: INTEGRAL ). The panels show the proportion of variance attributed to each principal component. Lower panels: The variance spectra of all observations (black solid line with data points) from all observatories. The colored curves and areas in the figure show the contributions of the principal components a 1 (red), a 2 (blue), a 3 (green) and the remainder of the components totaling to noise and systematic errors (orange).</caption>
</figure>
<section_header_level_1><location><page_4><loc_14><loc_46><loc_41><loc_47></location>3.2 The nature of the components</section_header_level_1>
<text><location><page_4><loc_14><loc_19><loc_86><loc_45></location>According to the variance spectral shape of the principal components, a number of models can be 'guessed' and then fitted to the time-averaged data. The first principal component resembles a Comptonized component and the second a thermal component 1 . Since most of the Cyg X-3 spectra are well fitted by hybrid Comptonization (e.g. [11, 2]) we assume that the first principal component is also coupled to this model. However, we need a second spectral component to fit all the spectra successfully and to satisfy the PCA. As the overall effect of the second component in the PCA and X-ray spectra is smaller, multiple components will fit the spectra. We found good fits when using reflection, multicolor disc blackbody, thermal bremsstrahlung, or other thermal Comptonization models in addition to hybrid Comptonization. These best-fitting models and their spectral component normalizations are then compared to the time-averaged evolution of the principal component normalizations. This imposes a second requirement for the X-ray spectral fits, so that in addition to fitting the spectra acceptably, the resulting fits also have to satisfy the spectral evolution inferred from the PCA. This extra requirement reduces greatly, if not completely, the degeneracy of simply using the results from the spectral fits in determining the emission components of the system. Thus, the most probable emission components are those producing the best correlations to the principal</text>
<text><location><page_5><loc_14><loc_87><loc_52><loc_89></location>component evolution and fits well the X-ray spectra.</text>
<text><location><page_5><loc_14><loc_77><loc_86><loc_87></location>To summarize, the best-fitting model in spectral and variability terms has optically thin, rather thermal, Comptonization dominating the variability throughout the X-ray regime and a thermal, rather hot ( ∼ 5 keV) plasma component producing variability in the ∼ 10-20 keV regime. We model the hybrid Comptonization component with BELM [12] and the thermal component as thermal bremsstrahlung (Fig. 3). We add to the final model a photoionized emission line model (Savolainen et al., in prep.) and multiply all with a simple absorption model from [11].</text>
<figure>
<location><page_5><loc_19><loc_58><loc_55><loc_75></location>
</figure>
<figure>
<location><page_5><loc_57><loc_59><loc_80><loc_75></location>
<caption>Figure 3: Left: An example of two energy spectra with the best-fitting model (black, solid line) and the individual components (Comptonization: blue; bremsstrahlung: green; iron line: orange) overlaid and labelled. Right: The scatter plot matrix of the best-fitting model satisfying the principal component evolution. The grid shows the spectral model normalizations NBELM and NBremss and the first two principal components ( a 1, a 2) with robust correlations drawn and written in the appropriate grid cells.</caption>
</figure>
<section_header_level_1><location><page_5><loc_14><loc_43><loc_25><loc_44></location>4. Discussion</section_header_level_1>
<text><location><page_5><loc_14><loc_31><loc_86><loc_41></location>The most plausible origin of the thermal component is a plasma cloud that forms as a result of the compact object colliding with the WR stellar wind [13]. This model was evoked to explain Cyg X-3's lack of high frequencies in the power spectra and the peculiar hard state X-ray spectra with ∼ 30 keV cut-off by Compton downscattering. For the plasma parameters found in [13] the thermal bremsstrahlung emission becomes a substantial source for photons which get upscattered by Comptonization in the plasma cloud.</text>
<text><location><page_5><loc_14><loc_20><loc_86><loc_30></location>Due to the short period of the system we can track the principal components through phase. Both components show a double-peaked profile (Fig. 4). When relating the second principal component to the bremsstrahlung normalization (see Fig. 3) which is proportional to n e n i V , we see that a change in the density and/or the volume of the bremsstrahlung-emitting plasma is observed along the orbit. The peaks are formed opposite each other, which can be explained by a disk-like shape of the plasma and could arise if the stellar wind is asymmetric.</text>
<text><location><page_5><loc_14><loc_13><loc_86><loc_20></location>Similar thermal and hot components have also been found in other microquasars such as GRS 1915 + 105 [14, 15], SS 433 [16], and in several XRBs [17], thus raising questions such as could the thermal, hot component be something intrinsic to microquasars/XRBs. Furthermore, could the emission mechanism be the same and could this scenario be extended to disk winds.</text>
<figure>
<location><page_6><loc_21><loc_71><loc_77><loc_89></location>
<caption>Figure 4: Left: The first and second principal component phase-folded through all the RXTE data. Right: A sketch depicting the geometry of the system, with the Wolf-Rayet companion surrounded by a disc-like stellar wind and the companion object orbiting it (shown in four different orbital phases). Depending on the orbital phase the compact object is either inside (enhanced bremsstrahlung emission) or outside the stellar wind.</caption>
</figure>
<section_header_level_1><location><page_6><loc_14><loc_57><loc_23><loc_58></location>References</section_header_level_1>
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</document>
[
{
"title": "Karri Koljonen ∗ †",
"content": "Aalto University Metsähovi Observatory, Finland E-mail: karri.koljonen@gmail.com",
"pages": [
1
]
},
{
"title": "Michael McCollough",
"content": "Smithsonian Astrophysical Observatory, USA E-mail: mmccollough@head.cfa.harvard.edu",
"pages": [
1
]
},
{
"title": "Diana Hannikainen",
"content": "Florida Institute of Technology, USA E-mail: ddcarina@gmail.com",
"pages": [
1
]
},
{
"title": "Robert Droulans",
"content": "Lycée classique d'Echternach, Luxemburg E-mail: robert.droulans@education.lu We have performed a principal component analysis on the X-ray spectra of the microquasar Cygnus X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event in 2006 MayJuly. The analysis showed that there are two main variability components in play, i.e. two principal components explained almost all the variability in the X-ray lightcurves. According to the spectral shape of these components and spectral fits to the original data, the most probable emission components corresponding to the principal components are inverse-Compton scattering and bremsstrahlung. We find that these components form a double-peaked profile when phase-folded with the peaks occurring in opposite phases. This could be due to an asymmetrical wind around the companion star with which the compact object is interacting. An INTEGRAL view of the high-energy sky (the first 10 years) - 9th INTEGRAL Workshop and celebration of the 10th anniversary of the launch 15-19 October 2012 Bibliotheque Nationale de France, Paris, France",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Modeling the X-ray spectra of X-ray binaries (XRBs) often leads to a problem of degeneracy, i.e. multiple distinct models fit the observed data equally well. A striking example can be seen e.g. in [1], where three very different models are fitted equally well to the same data set of Cygnus X-1, despite the excellent quality of the data that was obtained by all the X-ray satellites in orbit at the time. Similar spectral degeneracy was observed for Cygnus X-3 in [2]. Therefore, even if an apparently good fit is obtained between the data and the model, it does not necessarily imply a match between theory and physical reality. In order to make sense of this degeneracy we need to take other data dimensions into account, namely timing and/or polarization. While we are just reaching the point where the polarization dimension can be explored with very long exposure observations [3], X-ray timing data is readily available, and several methods have been developed to combine spectral and timing analyses (e.g. [4]). These comparisons, however, do not reveal what the actual spectral components causing the variability are . One way of revealing the variability components of X-ray spectra is to employ principal component analysis (PCA), one of the standard tools of time series analysis that has been introduced in the analysis of the X-ray data of XRBs by [5] and references therein, and further refined in [6] for Cyg X-3 to single out individual emission components causing the variability in the X-ray lightcurves.",
"pages": [
2
]
},
{
"title": "2. Principal component analysis",
"content": "Below, we summarize the main points of the PCA. More detail can be found from [5] and [6]. The first step of the PCA is to arrange a number of spectra measured at different times, i.e. a time series of X-ray spectra, to a data matrix. Then the eigenvectors of the covariance matrix calculated from the original data matrix form the principal components that are responsible for the spectral variation. The accompanying eigenvalue states the proportion of variance of a particular eigenvector, i.e. the highest eigenvalue and accompanying eigenvector is the first principal component of the data set etc. One can then form a linear decomposition of the data set using these eigenvectors, ordered by the proportion of variance. The components producing only a small fraction of the overall variance can then be dropped, reducing the dimensionality, i.e. choosing a small number of eigenvectors for the linear composition. The PCA can be exploited in different ways using both spectral and timing information. In the spectral domain this includes producing the variability spectra and log-eigenvalue diagrams. In the timing domain we can follow the evolution of individual principal components by tracking the value of its normalization.",
"pages": [
2
]
},
{
"title": "3. Applying PCA to Cygnus X-3 data",
"content": "Cyg X-3 is one of the most peculiar sources amongst microquasars. It is known for massive outbursts that emit radiation from radio to g -rays and produce major radio flaring episodes usually with multiple flares that peak up to 20 Jy [7], making it the most radio luminous single object in our Galaxy. The binary components of Cyg X-3 orbit each other in a tight 4.8-hour period [8], typical for XRBs with a low-mass companion. However, infrared spectral observations suggest that the mass-donating companion in the binary is a massive Wolf-Rayet star [9]. Due to this discrepancy Cyg X-3 is by definition a unique source and similar sources have only been found in other nearby galaxies [10]. The X-ray spectral and timing properties of Cyg X-3 show a disparity with other XRBs/microquasars increasing the difficulty of interpreting the nature of the system. This disparity could be due to the interaction of the strong stellar wind of the Wolf-Rayet companion with the compact object. We have performed a PCA to the X-ray spectra of Cyg X-3 from RXTE , INTEGRAL and Swift during a major flare ejection event. The X-ray observations were obtained during the 2006 May major radio flaring episode, that consisted of two major radio flares with peak flux densities of 13.8 Jy (15 GHz) and 11.2 Jy (11.2 GHz) at MJD 53865 and MJD 53942, respectively (see Fig. 1 for the May 2006 flare). For RXTE and Swift data we used the consecutive observations taken right after a flare and for INTEGRAL we used revolutions 437, 438 and 462. The radio data are from [11].",
"pages": [
2,
3
]
},
{
"title": "3.1 PCA tools of the trade",
"content": "The standard tool for deciding how many principal components one should retain is called the log-eigenvalue (LEV) diagram (upper panels in Fig. 2). If the principal components decay in geometric progression in the data, the corresponding eigenvalues will appear as a straight line in the LEV diagram and thus signal the start of the 'noise\" components. The variance spectrum (lower panels in Fig. 2) is a graph that shows the measured variance as a function of energy. It can be plotted for all the principal components, i.e. showing the overall variance across the energy range of the data, or for each principal component independently thus showing the spectral shape of the varying principal components. From Fig. 2 we see that there are two main variability components in play in the Cyg X-3 data, i.e. two principal components explain almost all the variability in the X-ray lightcurves.",
"pages": [
3
]
},
{
"title": "3.2 The nature of the components",
"content": "According to the variance spectral shape of the principal components, a number of models can be 'guessed' and then fitted to the time-averaged data. The first principal component resembles a Comptonized component and the second a thermal component 1 . Since most of the Cyg X-3 spectra are well fitted by hybrid Comptonization (e.g. [11, 2]) we assume that the first principal component is also coupled to this model. However, we need a second spectral component to fit all the spectra successfully and to satisfy the PCA. As the overall effect of the second component in the PCA and X-ray spectra is smaller, multiple components will fit the spectra. We found good fits when using reflection, multicolor disc blackbody, thermal bremsstrahlung, or other thermal Comptonization models in addition to hybrid Comptonization. These best-fitting models and their spectral component normalizations are then compared to the time-averaged evolution of the principal component normalizations. This imposes a second requirement for the X-ray spectral fits, so that in addition to fitting the spectra acceptably, the resulting fits also have to satisfy the spectral evolution inferred from the PCA. This extra requirement reduces greatly, if not completely, the degeneracy of simply using the results from the spectral fits in determining the emission components of the system. Thus, the most probable emission components are those producing the best correlations to the principal component evolution and fits well the X-ray spectra. To summarize, the best-fitting model in spectral and variability terms has optically thin, rather thermal, Comptonization dominating the variability throughout the X-ray regime and a thermal, rather hot ( ∼ 5 keV) plasma component producing variability in the ∼ 10-20 keV regime. We model the hybrid Comptonization component with BELM [12] and the thermal component as thermal bremsstrahlung (Fig. 3). We add to the final model a photoionized emission line model (Savolainen et al., in prep.) and multiply all with a simple absorption model from [11].",
"pages": [
4,
5
]
},
{
"title": "4. Discussion",
"content": "The most plausible origin of the thermal component is a plasma cloud that forms as a result of the compact object colliding with the WR stellar wind [13]. This model was evoked to explain Cyg X-3's lack of high frequencies in the power spectra and the peculiar hard state X-ray spectra with ∼ 30 keV cut-off by Compton downscattering. For the plasma parameters found in [13] the thermal bremsstrahlung emission becomes a substantial source for photons which get upscattered by Comptonization in the plasma cloud. Due to the short period of the system we can track the principal components through phase. Both components show a double-peaked profile (Fig. 4). When relating the second principal component to the bremsstrahlung normalization (see Fig. 3) which is proportional to n e n i V , we see that a change in the density and/or the volume of the bremsstrahlung-emitting plasma is observed along the orbit. The peaks are formed opposite each other, which can be explained by a disk-like shape of the plasma and could arise if the stellar wind is asymmetric. Similar thermal and hot components have also been found in other microquasars such as GRS 1915 + 105 [14, 15], SS 433 [16], and in several XRBs [17], thus raising questions such as could the thermal, hot component be something intrinsic to microquasars/XRBs. Furthermore, could the emission mechanism be the same and could this scenario be extended to disk winds.",
"pages": [
5
]
}
]
2013IAUS..291...61W
https://arxiv.org/pdf/1210.1910.pdf
<document>
<section_header_level_1><location><page_1><loc_23><loc_89><loc_58><loc_90></location>Structure of Quark Stars</section_header_level_1>
<section_header_level_1><location><page_1><loc_9><loc_84><loc_71><loc_87></location>Fridolin Weber 1 , Milva Orsaria 2 , † Hilario Rodrigues 3 ‡ and Shu-Hua Yang 4 ¶</section_header_level_1>
<text><location><page_1><loc_12><loc_81><loc_69><loc_83></location>Department of Physics, San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA 1</text>
<text><location><page_1><loc_31><loc_80><loc_51><loc_81></location>email: fweber@mail.sdsu.edu</text>
<text><location><page_1><loc_24><loc_75><loc_56><loc_79></location>2 email: morsaria@fcaglp.fcaglp.unlp.edu.ar 3 email: harg@cefet-rj.br 4 email: ysh@phy.ccnu.edu.cn</text>
<text><location><page_1><loc_9><loc_62><loc_72><loc_73></location>Abstract. This paper gives an brief overview of the structure of hypothetical strange quarks stars (quark stars, for short), which are made of absolutely stable 3-flavor strange quark matter. Such objects can be either bare or enveloped in thin nuclear crusts, which consist of heavy ions immersed in an electron gas. In contrast to neutron stars, the structure of quark stars is determined by two (rather than one) parameters, the central star density and the density at the base of the crust. If bare, quark stars possess ultra-high electric fields on the order of 10 18 to 10 19 V/cm. These features render the properties of quark stars more multifaceted than those of neutron stars and may allow one to observationally distinguish quark stars from neutron stars.</text>
<text><location><page_1><loc_9><loc_60><loc_67><loc_61></location>Keywords. neutron stars, quark stars, pulsars, strange quark matter, equation of state</text>
<section_header_level_1><location><page_1><loc_9><loc_54><loc_23><loc_56></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_34><loc_72><loc_54></location>The theoretical possibility that quark matter made of up, down and strange quarks (so-called strange quark matter (Farhi & Jaffe 1984)) may be more stable than ordinary nuclear matter has been pointed out by Bodmer (1971), Terazawa (1979), and Witten (1984). This so-called strange matter hypothesis constitutes one of the most startling possibilities regarding the behavior of superdense matter, which, if true, would have implications of fundamental importance for cosmology, the early universe, its evolution to the present day, and astrophysical compact objects such as neutron stars and white dwarfs (see Alcock & Farhi 1986, Alcock & Olinto 1988, Aarhus 1991, Weber 1999, Madsen 1999, Glendenning 2000, Weber 2005, Page & Reddy 2006, Sagert et al. 2006, and references therein). The properties of quark stars are compared with those of neutron stars in Table 1 and Fig. 1. Even to the present day there is no sound scientific basis on which one can either confirm or reject the hypothesis so that it is a serious possibility of fundamental significance for various (astro) physical phenomena.</text>
<text><location><page_1><loc_9><loc_28><loc_72><loc_34></location>The multifaceted properties of these objects are reviewed in this paper. Particular emphasis is is put on stellar properties such as rapid rotation, ultra-high electric surface fields, and rotational vortex expulsion, which may allow one to observationally discriminate between quark stars and neutron stars and-ultimately-prove or disprove the strange</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_24><loc_72><loc_27></location>† Home address: CONICET, Rivadavia 1917, 1033 Buenos Aires; Gravitation, Astrophysics and Cosmology Group, Facultad de Ciencias Astron'omicas y Geofisicas, Paseo del Bosque S/N (1900), Universidad Nacional de La Plata UNLP, La Plata, Argentina.</list_item>
<list_item><location><page_1><loc_9><loc_21><loc_72><loc_24></location>‡ Home address: Centro Federal de Educa¸c˜ao Tecnol'ogica do Rio de Janeiro, Av Maracan˜ a 249, 20271-110, Rio de Janeiro, RJ, Brazil.</list_item>
<list_item><location><page_1><loc_9><loc_19><loc_72><loc_21></location>¶ Home address: Institute of Astrophysics, Huazhong Normal University, Wuhan, 430079, P. R. China.</list_item>
</unordered_list>
<table>
<location><page_2><loc_10><loc_59><loc_70><loc_92></location>
<caption>Table 1. Theoretical properties of quark stars and neutron stars compared.</caption>
</table>
<text><location><page_2><loc_9><loc_52><loc_72><loc_56></location>quark matter hypothesis. Futher information on the existence of qark stars may come from quark novae, hypothetical types of supernovae which could occur if neutron stars spontaneously collapse to quark stars (Ouyed et al. 2002).</text>
<section_header_level_1><location><page_2><loc_9><loc_48><loc_50><loc_49></location>2. Quark-Lepton Composition of Quark Stars</section_header_level_1>
<text><location><page_2><loc_9><loc_19><loc_72><loc_47></location>Quark star matter is composed of the three lightest quark flavor states (up, down, and strange quarks). Per hypothesis, the energy per baryon of such matter is lower than the energy per baryon of the most stable atomic nucleus, 56 Fe. Since stars in their lowest energy state are electrically charge neutral to very high precision, any net positive quark charge must be balanced by electrons. The concentration of electrons is largest at low densities due to the finite strange-quark mass, which leads to a deficit of net negative quark charge. If quark star matter forms a color superconductor (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008, and references therein) in the Color-Flavor-Locked (CFL) phase the interiors of quarks stars will be rigorously electrically neutral with no need for electrons, as shown by Rajagopal & Wilczek (2001). For sufficiently large strange quark masses, however, the low density regime of quark star matter is rather expected to form other condensation patterns (e.g. 2SC, CFLK 0 , CFLK + , CFLπ 0 , -) in which electrons will be present (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008). The presence of electrons in quark star matter is crucial for the possible existence of a nuclear crust on quark stars. As shown by Alcock et al. (1986), Kettner et al. (1995), and Alcock & Olinto (1988), the electrons, because they are bound to strange matter by the Coulomb force rather than the strong force, extend several hundred fermi beyond the surface of the strange star. Associated with this electron displacement is a electric dipole layer which can support, out of contact with the surface of the strange star, a crust</text>
<figure>
<location><page_3><loc_24><loc_76><loc_54><loc_93></location>
<caption>Figure 1. Schematic structures of quark stars and neutron stars.</caption>
</figure>
<text><location><page_3><loc_9><loc_67><loc_72><loc_72></location>of nuclear material, which it polarizes (Alcock et al. 1986, Alcock & Olinto 1988). The maximum possible density at the base of the crust (inner crust density) is determined by neutron drip, which occurs at about 4 . 3 × 10 11 g / cm 3 .</text>
<section_header_level_1><location><page_3><loc_9><loc_63><loc_61><loc_64></location>3. Bare versus Dressed Quark Stars and Eddington Limit</section_header_level_1>
<text><location><page_3><loc_9><loc_34><loc_72><loc_62></location>A bare quark star differs qualitatively from a neutron star which has a density at the surface of about 0.1 to 1 g / cm 3 . The thickness of the quark surface is just ∼ 1 fm, the length scale of the strong interaction. The electrons at the surface of a quark star are held to quark matter electrostatically, and the thickness of the electron surface is several hundred fermis. Since neither component, electrons and quark matter, is held in place gravitationally, the Eddington limit to the luminosity that a static surface may emit does not apply, so that bare quark stars may have photon luminosities much greater than 10 38 erg / s. It was shown by Usov (1998) that this value may be exceeded by many orders of magnitude by the luminosity of e + e -pairs produced by the Coulomb barrier at the surface of a hot strange star. For a surface temperature of ∼ 10 11 K, the luminosity in the outflowing pair plasma was calculated to be as high as ∼ 3 × 10 51 erg / s. Such an effect may be a good observational signature of bare strange stars (Usov 2001a, Usov 2001b, Usov 1998, and Cheng & Harko 2003). If the strange star is enveloped in a nuclear crust, however, which is gravitationally bound to the strange star, the surface, made of ordinary atomic matter, would be subject to the Eddington limit. Hence the photon emissivity of such a 'dressed' quark star would be the same as for an ordinary neutron star. If quark matter at the stellar surface is in the CFL phase the process of e + e -pair creation at the stellar quark matter surface may be turned off. This may be different for the early stages of a very hot CFL quark star (Vogt et al. 2004).</text>
<section_header_level_1><location><page_3><loc_9><loc_30><loc_49><loc_31></location>4. Mass-Radius Relationship of Quark Stars</section_header_level_1>
<text><location><page_3><loc_9><loc_20><loc_72><loc_29></location>The mass-radius relationship of bare quark stars is shown in Fig. 2. In contrast to neutron stars, the radii of self-bound quark stars decrease the lighter the stars, according to M ∝ R 3 . The existence of nuclear crusts on quark stars changes the situation drastically (Glendenning et al. 1995, Weber 1999, and Weber 2005). Since the crust is bound gravitationally, the mass-radius relationship of quark stars with crusts is then qualitatively similar to neutron stars.</text>
<text><location><page_3><loc_11><loc_19><loc_72><loc_20></location>In general, quark stars with or without nuclear crusts possess smaller radii than neutron</text>
<text><location><page_4><loc_42><loc_93><loc_42><loc_94></location>/s32</text>
<figure>
<location><page_4><loc_24><loc_75><loc_57><loc_93></location>
<caption>Figure 2. Mass-radius relationship of bare quark stars (from Orsaria et al. 2011).</caption>
</figure>
<text><location><page_4><loc_9><loc_58><loc_72><loc_70></location>stars. This feature implies that quark stars posses smaller mass shedding periods than neutron stars. Due to the smaller radii of quarks stars, the complete sequence of such objects-and not just those close to the mass peak, as it is the case for neutron stars-can sustain extremely rapid rotation (Glendenning et al. 1995, Weber 1999, and Weber 2005). In particular, a strange star with a typical pulsar mass of around 1 . 45 M /circledot can rotate at Kepler (mass shedding) periods as small as 0 . 55 < ∼ P K / msec < ∼ 0 . 8 (Glendenning & Weber 1992, and Glendenning et al. 1995). This range is to be compared with P K ∼ 1 msec obtained for neutron stars of the same mass (Weber 1999).</text>
<text><location><page_4><loc_9><loc_45><loc_72><loc_58></location>Another novelty of the strange quark matter hypothesis concerns the existence of a new class of white-dwarf-like objects, referred to as strange (quark matter) dwarfs (Glendenning et al. 1995). The mass-radius relationship of the latter may differs somewhat from the mass-radius relationship of ordinary white-dwarf, which may be testable in the future. Until recently, only rather vague tests of the theoretical mass-radius relation of white dwarfs were possible. This has changed dramatically because of the availability of new data emerging from the Hipparcos project (Provencal 1998). These data allow the first accurate measurements of white dwarf distances and, as a result, establishing the mass-radius relation of such objects empirically.</text>
<section_header_level_1><location><page_4><loc_9><loc_40><loc_25><loc_42></location>5. Pulsar Glitches</section_header_level_1>
<text><location><page_4><loc_9><loc_19><loc_72><loc_40></location>Of considerable relevance for the viability of the strange matter hypothesis is the question of whether strange stars can exhibit glitches in rotation frequency. From the study performed by Glendenning & Weber (1992) and Zdunik et al. (2001) it is known that the ratio of the crustal moment of inertia to the total moment of inertia varies between 10 -3 and ∼ 10 -5 . If the angular momentum of the pulsar is conserved in a stellar quake one obtains for the change of the star's frequency ∆Ω / Ω ∼ (10 -5 -10 -3 ) f , where 0 < f < 1 (Glendenning & Weber 1992). The factor f represents the fraction of the crustal moment of inertia that is altered in the quake. Since the observed glitches have relative frequency changes ∆Ω / Ω = (10 -9 -10 -6 ), a change in the crustal moment of inertia of f < ∼ 0 . 1 would cause a giant glitch (Glendenning & Weber 1992). Moreover it turns out that the observed range of the fractional change in the spin-down rate, ˙ Ω, is consistent with the crust having a small moment of inertia and the quake involving only a small fraction f of that. For this purpose we write ∆ ˙ Ω / ˙ Ω > (10 -1 to 10) f (Glendenning & Weber 1992). This relation yields a small f value, i.e., f < (10 -4 to 10 -1 ), in agree-</text>
<text><location><page_4><loc_57><loc_84><loc_57><loc_84></location>/s32</text>
<text><location><page_5><loc_9><loc_91><loc_72><loc_94></location>ment with f < ∼ 0 . 1 established just above. For these estimates, the measured values of (∆Ω / Ω) / (∆ ˙ Ω / ˙ Ω) ∼ 10 -6 to 10 -4 for Crab and Vela, respectively, have been used.</text>
<section_header_level_1><location><page_5><loc_9><loc_86><loc_38><loc_87></location>6. Possible Connection to CCOs</section_header_level_1>
<text><location><page_5><loc_9><loc_72><loc_72><loc_86></location>One of the most amazing features of quark stars concerns the possible existence of ultra-high electric fields on their surfaces, which, for ordinary quark matter, is around 10 18 V/cm. If strange matter forms a color superconductor, as expected for such matter, the strength of the electric field may increase to values that exceed 10 19 V/cm. The energy density associated with such huge electric fields is on the same order of magnitude as the energy density of strange matter itself, which alters the masses and radii of strange quark stars at the 15% and 5% level, respectively (Negreiros et al. 2009). Such mass increases facilitate the interpretation of massive compact stars, with masses of around 2 M /circledot , as strange quark stars (see also Rodrigues et al. 2011).</text>
<text><location><page_5><loc_9><loc_62><loc_72><loc_72></location>The electrons at the surface of a quark star are not necessarily in a fixed position but may rotate with respect to the quark matter star (Negreiros et al. 2010). In this event magnetic fields can be generated which, for moderate effective rotational frequencies between the electron layer and the stellar body, agree with the magnetic fields inferred for several Compact Central Objects (CCOs). These objects could thus be interpreted as quark stars whose electron atmospheres rotate at frequencies that are moderately different ( ∼ 10 Hz) from the rotational frequency of the quark star itself.</text>
<text><location><page_5><loc_9><loc_45><loc_72><loc_61></location>Last but not least, we mention that the electron surface layer may be strongly affected by the magnetic field of a quark star in such a way that the electron layer performs vortex hydrodynamical oscillations (Xu et al. 2012). The frequency spectrum of these oscillations has been derived in analytic form by Xu et al. (2012). If the thermal Xray spectra of quark stars are modulated by vortex hydrodynamical oscillations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model (Xu et al. 2012).</text>
<text><location><page_5><loc_9><loc_39><loc_72><loc_45></location>A study which looks at the thermal evolution of CCOs is presently being carried out by Yang et al. (2012). Preliminary results indicate that the observed temperatures of CCOs can be well reproduced if one assumes that these objects are small quark matter objects with radii less than around 3 km.</text>
<section_header_level_1><location><page_5><loc_9><loc_34><loc_56><loc_35></location>7. Possible Connection to SGRs, AXPs, and XDINs</section_header_level_1>
<text><location><page_5><loc_9><loc_19><loc_72><loc_34></location>If quarks stars are made of color superconducting quark matter rather than normal non-superconducting quark matter. If rotating, superconducting quark stars ought to be threaded with rotational vortex lines, within which the star's interior magnetic field is at least partially confined. The vortices (and thus magnetic flux) would be expelled from the star during stellar spin-down, leading to magnetic reconnection at the surface of the star and the prolific production of thermal energy. Niebergal et al. (2010) have shown that this energy release can re-heat quark stars to exceptionally high temperatures, such as observed for Soft Gamma Repeaters (SGRs), Anomalous X-Ray pulsars (AXPs), and X-ray dim isolated neutron stars (XDINs). Moreover, numerical investigations of the temperature evolution, spin-down rate, and magnetic field behavior of such superconducting</text>
<text><location><page_6><loc_9><loc_91><loc_72><loc_94></location>quark stars suggest that SGRs, AXPs, and XDINs may be linked ancestrally (Niebergal et al 2010).</text>
<section_header_level_1><location><page_6><loc_9><loc_88><loc_25><loc_90></location>Acknowledgements</section_header_level_1>
<text><location><page_6><loc_9><loc_82><loc_72><loc_88></location>M. Orsaria thanks CONICET for financial support. H. Rodrigues thanks CAPES for financial support under contract number BEX 6379/10-9. F. Weber is supported by the National Science Foundation (USA) under Grant PHY-0854699. S.-H. Yang is supported by the China Scholarship Council (CSC) and by NFSC under Grant No. 11147170.</text>
<section_header_level_1><location><page_6><loc_9><loc_78><loc_18><loc_79></location>References</section_header_level_1>
<text><location><page_6><loc_9><loc_75><loc_72><loc_77></location>Proc. Int. Workshop on Strange Quark Matter in Physics and Astrophysics 1991, J. Madsen and P. Haensel (eds.), Nucl. Phys. B (Proc. Suppl.) , 24B</text>
<text><location><page_6><loc_9><loc_24><loc_72><loc_75></location>Alcock, C., Farhi, E., & Olinto, A. V. 1986, ApJ , 310, 261 Alcock, C., & Olinto, A.V. 1988, Ann. Rev. Nucl. Part. Sci. , 38, 161 Alford, M. 2001, Ann. Rev. Nucl. Part. Sci. , 51, 131 Alford, M.G., Schmitt, A., Rajagopal, K., & Schafer, T. 2008, Rev. Mod. Phys. , 80, 1455 Bodmer, A.R. 1971, Phys. Rev. D , 4, 1601 Cheng, K.S. & Harko, T. 2003, ApJ , 596, 451 Farhi, E., & Jaffe, R. L. 1984, Phys. Rev. D , 30, 2379 Glendenning, N.K., & Weber, F. 1992, ApJ , 400, 647 Glendenning, N.K., Kettner, Ch., & Weber, F. 1995, ApJ , 450, 253 Glendenning, N.K. 2000, Compact Stars, Nuclear Physics, Particle Physics, and General Relativity , 2nd ed. (Springer-Verlag, New York) Kettner, Ch., Weber, F., Weigel, M.K., & Glendenning, N.K. 1995, Phys. Rev. D , 51, 1440 Madsen, J. 1999, Lecture Notes in Physics , 516, 162 Negreiros, R., Weber, F., Malheiro, M., & Usov, V. 2009, Phys. Rev. D , 80, 083006 Negreiros, R.P. , Mishustin, I.N., Schramm, S., & Weber, F. 2010, Phys. Rev. D , 82, 103010 Niebergal, B., Ouyed, R., Negreiros, R., & Weber, F. 2010, Phys. Rev. D , 81, 043005 Orsaria, M, Ranea-Sandoval, I.F., & Vucetich, H. 2011, ApJ , 734, 41 Ouyed, R., Dey, J., & Dey, M. 2002, A&A , 390, L39 Page D., & Reddy, S. 2006, Ann. Rev. Nucl. Part. Sci. , 56, 327 Provencal, J.L., Shipman, H.L., Hog, E., and Thejll, P. 1998, ApJ , 494, 759 Rajagopal K. & Wilczek, F. 2001, The Condensed Matter Physics of QCD , At the Frontier of Particle Physics/Handbook of QCD, ed. M. Shifman (World Scientific) Rajagopal, K., & Wilczek, F. 2001, Phys. Rev. Lett. , 86, 3492 Rodrigues, H., Duarte, S.B., & de Oliveira, J.C.T. 2011, ApJ , 730, 31 Sagert, I., Wietoska, M., & Schaffner-Bielich, J. 2006 J. Phys. G. , 32, S241 Terazawa, H. 1979, INS-Report-338 (INS, Univ. of Tokyo ; 1989 J. Phys. Soc. Japan , 58, 3555; 1989 ibid. , 58, 4388; 1990 ibid. , 59, 1199 Usov, V.V. 1998, Phys. Rev. Lett. , 80, 230 Usov, V.V. 2001, ApJ , 550, L179 Usov, V.V. 2001, ApJ , 559, L137 Vogt, C., Rapp, R., & Ouyed, R. 2004, Nucl. Phys. , A735, 543 Weber, F. 1999, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics , (IOP Publishing, Bristol, Great Britain). Weber, F. 2005, Prog. Part. Nucl. Phys. , 54, 193 Witten, E. 1984, Phys. Rev. D , 30, 272 Xu, R.X., Bastrukov, S.I., Weber, F., Yu, J.W, & Molodtsova, I.V. 2012, Phys. Rev. D , 85, 023008</text>
<text><location><page_6><loc_9><loc_22><loc_72><loc_24></location>Yang, S.-H., Weber, F., Negreiros, R., & Becker, W. 2012, Cooling Simulations of CCOs (in preparation)</text>
<text><location><page_6><loc_9><loc_20><loc_54><loc_22></location>Zdunik, J. L., Haensel, E., & Gourgoulhon, E. 2001, A&A , 372, 535</text>
</document>
[
{
"title": "Fridolin Weber 1 , Milva Orsaria 2 , † Hilario Rodrigues 3 ‡ and Shu-Hua Yang 4 ¶",
"content": "Department of Physics, San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA 1 email: fweber@mail.sdsu.edu 2 email: morsaria@fcaglp.fcaglp.unlp.edu.ar 3 email: harg@cefet-rj.br 4 email: ysh@phy.ccnu.edu.cn Abstract. This paper gives an brief overview of the structure of hypothetical strange quarks stars (quark stars, for short), which are made of absolutely stable 3-flavor strange quark matter. Such objects can be either bare or enveloped in thin nuclear crusts, which consist of heavy ions immersed in an electron gas. In contrast to neutron stars, the structure of quark stars is determined by two (rather than one) parameters, the central star density and the density at the base of the crust. If bare, quark stars possess ultra-high electric fields on the order of 10 18 to 10 19 V/cm. These features render the properties of quark stars more multifaceted than those of neutron stars and may allow one to observationally distinguish quark stars from neutron stars. Keywords. neutron stars, quark stars, pulsars, strange quark matter, equation of state",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The theoretical possibility that quark matter made of up, down and strange quarks (so-called strange quark matter (Farhi & Jaffe 1984)) may be more stable than ordinary nuclear matter has been pointed out by Bodmer (1971), Terazawa (1979), and Witten (1984). This so-called strange matter hypothesis constitutes one of the most startling possibilities regarding the behavior of superdense matter, which, if true, would have implications of fundamental importance for cosmology, the early universe, its evolution to the present day, and astrophysical compact objects such as neutron stars and white dwarfs (see Alcock & Farhi 1986, Alcock & Olinto 1988, Aarhus 1991, Weber 1999, Madsen 1999, Glendenning 2000, Weber 2005, Page & Reddy 2006, Sagert et al. 2006, and references therein). The properties of quark stars are compared with those of neutron stars in Table 1 and Fig. 1. Even to the present day there is no sound scientific basis on which one can either confirm or reject the hypothesis so that it is a serious possibility of fundamental significance for various (astro) physical phenomena. The multifaceted properties of these objects are reviewed in this paper. Particular emphasis is is put on stellar properties such as rapid rotation, ultra-high electric surface fields, and rotational vortex expulsion, which may allow one to observationally discriminate between quark stars and neutron stars and-ultimately-prove or disprove the strange quark matter hypothesis. Futher information on the existence of qark stars may come from quark novae, hypothetical types of supernovae which could occur if neutron stars spontaneously collapse to quark stars (Ouyed et al. 2002).",
"pages": [
1,
2
]
},
{
"title": "2. Quark-Lepton Composition of Quark Stars",
"content": "Quark star matter is composed of the three lightest quark flavor states (up, down, and strange quarks). Per hypothesis, the energy per baryon of such matter is lower than the energy per baryon of the most stable atomic nucleus, 56 Fe. Since stars in their lowest energy state are electrically charge neutral to very high precision, any net positive quark charge must be balanced by electrons. The concentration of electrons is largest at low densities due to the finite strange-quark mass, which leads to a deficit of net negative quark charge. If quark star matter forms a color superconductor (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008, and references therein) in the Color-Flavor-Locked (CFL) phase the interiors of quarks stars will be rigorously electrically neutral with no need for electrons, as shown by Rajagopal & Wilczek (2001). For sufficiently large strange quark masses, however, the low density regime of quark star matter is rather expected to form other condensation patterns (e.g. 2SC, CFLK 0 , CFLK + , CFLπ 0 , -) in which electrons will be present (Rajagopal & Wilczek 2001, Alford 2001, Alford et al. 2008). The presence of electrons in quark star matter is crucial for the possible existence of a nuclear crust on quark stars. As shown by Alcock et al. (1986), Kettner et al. (1995), and Alcock & Olinto (1988), the electrons, because they are bound to strange matter by the Coulomb force rather than the strong force, extend several hundred fermi beyond the surface of the strange star. Associated with this electron displacement is a electric dipole layer which can support, out of contact with the surface of the strange star, a crust of nuclear material, which it polarizes (Alcock et al. 1986, Alcock & Olinto 1988). The maximum possible density at the base of the crust (inner crust density) is determined by neutron drip, which occurs at about 4 . 3 × 10 11 g / cm 3 .",
"pages": [
2,
3
]
},
{
"title": "3. Bare versus Dressed Quark Stars and Eddington Limit",
"content": "A bare quark star differs qualitatively from a neutron star which has a density at the surface of about 0.1 to 1 g / cm 3 . The thickness of the quark surface is just ∼ 1 fm, the length scale of the strong interaction. The electrons at the surface of a quark star are held to quark matter electrostatically, and the thickness of the electron surface is several hundred fermis. Since neither component, electrons and quark matter, is held in place gravitationally, the Eddington limit to the luminosity that a static surface may emit does not apply, so that bare quark stars may have photon luminosities much greater than 10 38 erg / s. It was shown by Usov (1998) that this value may be exceeded by many orders of magnitude by the luminosity of e + e -pairs produced by the Coulomb barrier at the surface of a hot strange star. For a surface temperature of ∼ 10 11 K, the luminosity in the outflowing pair plasma was calculated to be as high as ∼ 3 × 10 51 erg / s. Such an effect may be a good observational signature of bare strange stars (Usov 2001a, Usov 2001b, Usov 1998, and Cheng & Harko 2003). If the strange star is enveloped in a nuclear crust, however, which is gravitationally bound to the strange star, the surface, made of ordinary atomic matter, would be subject to the Eddington limit. Hence the photon emissivity of such a 'dressed' quark star would be the same as for an ordinary neutron star. If quark matter at the stellar surface is in the CFL phase the process of e + e -pair creation at the stellar quark matter surface may be turned off. This may be different for the early stages of a very hot CFL quark star (Vogt et al. 2004).",
"pages": [
3
]
},
{
"title": "4. Mass-Radius Relationship of Quark Stars",
"content": "The mass-radius relationship of bare quark stars is shown in Fig. 2. In contrast to neutron stars, the radii of self-bound quark stars decrease the lighter the stars, according to M ∝ R 3 . The existence of nuclear crusts on quark stars changes the situation drastically (Glendenning et al. 1995, Weber 1999, and Weber 2005). Since the crust is bound gravitationally, the mass-radius relationship of quark stars with crusts is then qualitatively similar to neutron stars. In general, quark stars with or without nuclear crusts possess smaller radii than neutron /s32 stars. This feature implies that quark stars posses smaller mass shedding periods than neutron stars. Due to the smaller radii of quarks stars, the complete sequence of such objects-and not just those close to the mass peak, as it is the case for neutron stars-can sustain extremely rapid rotation (Glendenning et al. 1995, Weber 1999, and Weber 2005). In particular, a strange star with a typical pulsar mass of around 1 . 45 M /circledot can rotate at Kepler (mass shedding) periods as small as 0 . 55 < ∼ P K / msec < ∼ 0 . 8 (Glendenning & Weber 1992, and Glendenning et al. 1995). This range is to be compared with P K ∼ 1 msec obtained for neutron stars of the same mass (Weber 1999). Another novelty of the strange quark matter hypothesis concerns the existence of a new class of white-dwarf-like objects, referred to as strange (quark matter) dwarfs (Glendenning et al. 1995). The mass-radius relationship of the latter may differs somewhat from the mass-radius relationship of ordinary white-dwarf, which may be testable in the future. Until recently, only rather vague tests of the theoretical mass-radius relation of white dwarfs were possible. This has changed dramatically because of the availability of new data emerging from the Hipparcos project (Provencal 1998). These data allow the first accurate measurements of white dwarf distances and, as a result, establishing the mass-radius relation of such objects empirically.",
"pages": [
3,
4
]
},
{
"title": "5. Pulsar Glitches",
"content": "Of considerable relevance for the viability of the strange matter hypothesis is the question of whether strange stars can exhibit glitches in rotation frequency. From the study performed by Glendenning & Weber (1992) and Zdunik et al. (2001) it is known that the ratio of the crustal moment of inertia to the total moment of inertia varies between 10 -3 and ∼ 10 -5 . If the angular momentum of the pulsar is conserved in a stellar quake one obtains for the change of the star's frequency ∆Ω / Ω ∼ (10 -5 -10 -3 ) f , where 0 < f < 1 (Glendenning & Weber 1992). The factor f represents the fraction of the crustal moment of inertia that is altered in the quake. Since the observed glitches have relative frequency changes ∆Ω / Ω = (10 -9 -10 -6 ), a change in the crustal moment of inertia of f < ∼ 0 . 1 would cause a giant glitch (Glendenning & Weber 1992). Moreover it turns out that the observed range of the fractional change in the spin-down rate, ˙ Ω, is consistent with the crust having a small moment of inertia and the quake involving only a small fraction f of that. For this purpose we write ∆ ˙ Ω / ˙ Ω > (10 -1 to 10) f (Glendenning & Weber 1992). This relation yields a small f value, i.e., f < (10 -4 to 10 -1 ), in agree- /s32 ment with f < ∼ 0 . 1 established just above. For these estimates, the measured values of (∆Ω / Ω) / (∆ ˙ Ω / ˙ Ω) ∼ 10 -6 to 10 -4 for Crab and Vela, respectively, have been used.",
"pages": [
4,
5
]
},
{
"title": "6. Possible Connection to CCOs",
"content": "One of the most amazing features of quark stars concerns the possible existence of ultra-high electric fields on their surfaces, which, for ordinary quark matter, is around 10 18 V/cm. If strange matter forms a color superconductor, as expected for such matter, the strength of the electric field may increase to values that exceed 10 19 V/cm. The energy density associated with such huge electric fields is on the same order of magnitude as the energy density of strange matter itself, which alters the masses and radii of strange quark stars at the 15% and 5% level, respectively (Negreiros et al. 2009). Such mass increases facilitate the interpretation of massive compact stars, with masses of around 2 M /circledot , as strange quark stars (see also Rodrigues et al. 2011). The electrons at the surface of a quark star are not necessarily in a fixed position but may rotate with respect to the quark matter star (Negreiros et al. 2010). In this event magnetic fields can be generated which, for moderate effective rotational frequencies between the electron layer and the stellar body, agree with the magnetic fields inferred for several Compact Central Objects (CCOs). These objects could thus be interpreted as quark stars whose electron atmospheres rotate at frequencies that are moderately different ( ∼ 10 Hz) from the rotational frequency of the quark star itself. Last but not least, we mention that the electron surface layer may be strongly affected by the magnetic field of a quark star in such a way that the electron layer performs vortex hydrodynamical oscillations (Xu et al. 2012). The frequency spectrum of these oscillations has been derived in analytic form by Xu et al. (2012). If the thermal Xray spectra of quark stars are modulated by vortex hydrodynamical oscillations, the thermal spectra of compact stars, foremost central compact objects (CCOs) and X-ray dim isolated neutron stars (XDINSs), could be used to verify the existence of these vibrational modes observationally. The central compact object 1E 1207.4-5209 appears particularly interesting in this context, since its absorption features at 0.7 keV and 1.4 keV can be comfortably explained in the framework of the hydro-cyclotron oscillation model (Xu et al. 2012). A study which looks at the thermal evolution of CCOs is presently being carried out by Yang et al. (2012). Preliminary results indicate that the observed temperatures of CCOs can be well reproduced if one assumes that these objects are small quark matter objects with radii less than around 3 km.",
"pages": [
5
]
},
{
"title": "7. Possible Connection to SGRs, AXPs, and XDINs",
"content": "If quarks stars are made of color superconducting quark matter rather than normal non-superconducting quark matter. If rotating, superconducting quark stars ought to be threaded with rotational vortex lines, within which the star's interior magnetic field is at least partially confined. The vortices (and thus magnetic flux) would be expelled from the star during stellar spin-down, leading to magnetic reconnection at the surface of the star and the prolific production of thermal energy. Niebergal et al. (2010) have shown that this energy release can re-heat quark stars to exceptionally high temperatures, such as observed for Soft Gamma Repeaters (SGRs), Anomalous X-Ray pulsars (AXPs), and X-ray dim isolated neutron stars (XDINs). Moreover, numerical investigations of the temperature evolution, spin-down rate, and magnetic field behavior of such superconducting quark stars suggest that SGRs, AXPs, and XDINs may be linked ancestrally (Niebergal et al 2010).",
"pages": [
5,
6
]
},
{
"title": "Acknowledgements",
"content": "M. Orsaria thanks CONICET for financial support. H. Rodrigues thanks CAPES for financial support under contract number BEX 6379/10-9. F. Weber is supported by the National Science Foundation (USA) under Grant PHY-0854699. S.-H. Yang is supported by the China Scholarship Council (CSC) and by NFSC under Grant No. 11147170.",
"pages": [
6
]
},
{
"title": "References",
"content": "Proc. Int. Workshop on Strange Quark Matter in Physics and Astrophysics 1991, J. Madsen and P. Haensel (eds.), Nucl. Phys. B (Proc. Suppl.) , 24B Alcock, C., Farhi, E., & Olinto, A. V. 1986, ApJ , 310, 261 Alcock, C., & Olinto, A.V. 1988, Ann. Rev. Nucl. Part. Sci. , 38, 161 Alford, M. 2001, Ann. Rev. Nucl. Part. Sci. , 51, 131 Alford, M.G., Schmitt, A., Rajagopal, K., & Schafer, T. 2008, Rev. Mod. Phys. , 80, 1455 Bodmer, A.R. 1971, Phys. Rev. D , 4, 1601 Cheng, K.S. & Harko, T. 2003, ApJ , 596, 451 Farhi, E., & Jaffe, R. L. 1984, Phys. Rev. D , 30, 2379 Glendenning, N.K., & Weber, F. 1992, ApJ , 400, 647 Glendenning, N.K., Kettner, Ch., & Weber, F. 1995, ApJ , 450, 253 Glendenning, N.K. 2000, Compact Stars, Nuclear Physics, Particle Physics, and General Relativity , 2nd ed. (Springer-Verlag, New York) Kettner, Ch., Weber, F., Weigel, M.K., & Glendenning, N.K. 1995, Phys. Rev. D , 51, 1440 Madsen, J. 1999, Lecture Notes in Physics , 516, 162 Negreiros, R., Weber, F., Malheiro, M., & Usov, V. 2009, Phys. Rev. D , 80, 083006 Negreiros, R.P. , Mishustin, I.N., Schramm, S., & Weber, F. 2010, Phys. Rev. D , 82, 103010 Niebergal, B., Ouyed, R., Negreiros, R., & Weber, F. 2010, Phys. Rev. D , 81, 043005 Orsaria, M, Ranea-Sandoval, I.F., & Vucetich, H. 2011, ApJ , 734, 41 Ouyed, R., Dey, J., & Dey, M. 2002, A&A , 390, L39 Page D., & Reddy, S. 2006, Ann. Rev. Nucl. Part. Sci. , 56, 327 Provencal, J.L., Shipman, H.L., Hog, E., and Thejll, P. 1998, ApJ , 494, 759 Rajagopal K. & Wilczek, F. 2001, The Condensed Matter Physics of QCD , At the Frontier of Particle Physics/Handbook of QCD, ed. M. Shifman (World Scientific) Rajagopal, K., & Wilczek, F. 2001, Phys. Rev. Lett. , 86, 3492 Rodrigues, H., Duarte, S.B., & de Oliveira, J.C.T. 2011, ApJ , 730, 31 Sagert, I., Wietoska, M., & Schaffner-Bielich, J. 2006 J. Phys. G. , 32, S241 Terazawa, H. 1979, INS-Report-338 (INS, Univ. of Tokyo ; 1989 J. Phys. Soc. Japan , 58, 3555; 1989 ibid. , 58, 4388; 1990 ibid. , 59, 1199 Usov, V.V. 1998, Phys. Rev. Lett. , 80, 230 Usov, V.V. 2001, ApJ , 550, L179 Usov, V.V. 2001, ApJ , 559, L137 Vogt, C., Rapp, R., & Ouyed, R. 2004, Nucl. Phys. , A735, 543 Weber, F. 1999, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics , (IOP Publishing, Bristol, Great Britain). Weber, F. 2005, Prog. Part. Nucl. Phys. , 54, 193 Witten, E. 1984, Phys. Rev. D , 30, 272 Xu, R.X., Bastrukov, S.I., Weber, F., Yu, J.W, & Molodtsova, I.V. 2012, Phys. Rev. D , 85, 023008 Yang, S.-H., Weber, F., Negreiros, R., & Becker, W. 2012, Cooling Simulations of CCOs (in preparation) Zdunik, J. L., Haensel, E., & Gourgoulhon, E. 2001, A&A , 372, 535",
"pages": [
6
]
}
]
2013IAUS..291..356C
https://arxiv.org/pdf/1304.6907.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_84><loc_68><loc_90></location>Does a hadron-quark phase transition in dense matter preclude the existence of massive neutron stars ?</section_header_level_1>
<section_header_level_1><location><page_1><loc_14><loc_81><loc_66><loc_83></location>N. Chamel 1 , A.F. Fantina 1 , J.M. Pearson 2 and S. Goriely 1</section_header_level_1>
<text><location><page_1><loc_11><loc_78><loc_69><loc_80></location>1 Institut d'Astronomie et d'Astrophysique, CP-226, Universit'e Libre de Bruxelles, 1050 Brussels, Belgium</text>
<text><location><page_1><loc_13><loc_76><loc_68><loc_78></location>2 D'ept. de Physique, Universit'e de Montr'eal, Montr'eal (Qu'ebec), H3C 3J7 Canada</text>
<text><location><page_1><loc_9><loc_65><loc_72><loc_74></location>Abstract. We study the impact of a hadron-quark phase transition on the maximum neutronstar mass. The hadronic part of the equation of state relies on the most up-to-date Skyrme nuclear energy density functionals, fitted to essentially all experimental nuclear mass data and constrained to reproduce the properties of infinite nuclear matter as obtained from microscopic calculations using realistic forces. We show that the softening of the dense matter equation of state due to the phase transition is not necessarily incompatible with the existence of massive neutron stars like PSR J1614 -2230.</text>
<text><location><page_1><loc_9><loc_63><loc_67><loc_64></location>Keywords. stars: neutron, dense matter, equation of state, stars: interiors, gravitation</text>
<section_header_level_1><location><page_1><loc_9><loc_57><loc_23><loc_58></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_37><loc_72><loc_57></location>Neutron stars (NSs) result from the gravitational collapse of massive stars with M /greaterorsimilar 8 M /circledot at the end point of their evolution. They are among the most compact objects in the universe, with a central density which can reach several times the nuclear saturation density. At least, three different regions can be identified in the interior of a NS: (i) the 'outer crust', at densities above ∼ 10 4 g cm -3 , composed of fully ionized atoms, arranged in a Coulomb lattice of nuclei, neutralized by a degenerate electron gas, (ii) the 'inner crust', at densities above ∼ 4 × 10 11 g cm -3 , composed of neutron-proton clusters and unbound neutrons, neutralized by a degenerate electron gas, and (iii) the core, at densities above ∼ 10 14 g cm -3 . The precise measurement of the mass of pulsar PSR J1614 -2230 by Demorest et al. (2010) has revived the question of the composition of the core. Just below the crust, the matter consists of a mixture of neutrons, protons, electrons and possibly muons. The composition of the central region of a NS is still a matter of debate (see e.g. Haensel et al. 2007).</text>
<text><location><page_1><loc_9><loc_33><loc_72><loc_37></location>In the present work, we study the impact of a hadron-quark phase transition in dense matter on the maximum mass of cold isolated NSs (see Chamel et al. 2013 for a general discussion of the maximum mass of hybrid stars).</text>
<section_header_level_1><location><page_1><loc_9><loc_28><loc_36><loc_29></location>2. Hadronic equation of state</section_header_level_1>
<text><location><page_1><loc_9><loc_19><loc_72><loc_28></location>The global structure of a NS is determined by the equation of state (EoS), i.e. the relation between the matter pressure P and the mass-energy density ρ . Before considering the possibility of a phase transition from hadronic to quark matter in the core of NSs, we will begin with the hadronic EoSs. A good starting point is the family of three EoSs, BSk19, BSk20 and BSk21, which have been developed to provide a unified treatment of all regions of a NS (see Pearson et al. 2011, Pearson et al. 2012). These EoSs are based on</text>
<text><location><page_2><loc_9><loc_82><loc_72><loc_94></location>nuclear energy-density functionals derived from generalized Skyrme forces (in that they contain additional momentum- and density-dependent terms), which fit essentially all measured masses of atomic nuclei with an rms deviation of 0.58 MeV for all three models. Moreover, these functionals were constrained to reproduce three different neutron matter EoSs, as obtained from microscopic calculations (see Goriely et al. 2010). All three EoSs assume that the core of a NS is made of nucleons and leptons. The BSk19 EoS was found to be too soft to support NSs as massive as PSR J1614 -2230 (Chamel et al. 2011) and therefore, it will not be considered here.</text>
<section_header_level_1><location><page_2><loc_9><loc_78><loc_39><loc_79></location>3. Hadron-quark phase transition</section_header_level_1>
<text><location><page_2><loc_9><loc_61><loc_72><loc_77></location>Given the uncertainties in the composition of dense matter in NSs, we will simply suppose that above the average baryon density n N , matter undergoes a first-order phase transition to deconfined quark matter subject to the following restrictions: (i) for the transition to occur the energy density of the quark phase must be lower than that of the hadronic phase, (ii) according to perturbative quantum chromodynamics (QCD) calculations (e.g. Kurkela et al. 2010), the speed of sound in quark matter cannot exceed c/ √ 3 where c is the speed of light. At densities below n N , matter is purely hadronic while a pure quark phase is found at densities above some density n X . In the intermediate region ( n N < n < n X ) where the two phases can coexist, the pressure and the chemical potential of the two phases are equal: P quark ( n ) = P hadron ( n N ) and µ quark ( n ) = µ hadron ( n N ). The EoS of the quark phase at n > n X is given by:</text>
<formula><location><page_2><loc_21><loc_57><loc_72><loc_60></location>P quark ( n ) = 1 3 ( E quark ( n ) -E quark ( n X )) + P hadron ( n N ) . (3.1)</formula>
<text><location><page_2><loc_9><loc_46><loc_72><loc_57></location>We set the density n N to lie above the highest density found in nuclei as predicted by Hartree-Fock-Bogoliubov calculations, namely n N = 0 . 2 fm -3 (BRUSLIB). The density n X is adjusted to optimize the maximum mass under the conditions mentioned above. Eq. (3.1) turns out to be very similar to that obtained within the simple MIT bag model, which has been widely applied to describe quark matter in compact stars (see e.g. Haensel et al. 2007). The effective bag constant B associated with the BSk21 hadronic EoS is 56 . 7 MeV fm -3 .</text>
<section_header_level_1><location><page_2><loc_9><loc_42><loc_26><loc_43></location>4. Maximum mass</section_header_level_1>
<text><location><page_2><loc_9><loc_28><loc_73><loc_41></location>Considering the stiffest hadronic EoS (BSk21), we have solved the Tolman-OppenheimerVolkoff equations (Tolman 1939, Oppenheimer & Volkoff 1939) in order to determine the global structure of a nonrotating NS. The effect of rotation on the maximum mass was found to be very small for stars with spin-periods comparable to that of PSR J1614 -2230 (Chamel et al. 2011); we therefore neglect it. The gravitational mass versus circumferential radius relation is shown in Fig. 1. We have considered two cases: a purely hadronic NS described by our BSk21 EoS (dashed line) and a hybrid star with a quark core (solid line). The corresponding maximum masses are 2.28 M /circledot and 2.02 M /circledot respectively. In both cases, the existence of two-solar mass NSs is therefore allowed.</text>
<section_header_level_1><location><page_2><loc_9><loc_24><loc_22><loc_25></location>5. Conclusions</section_header_level_1>
<text><location><page_2><loc_9><loc_19><loc_72><loc_23></location>The presence of a deconfined quark-matter phase in NS cores leads to a maximum mass of about 2 M /circledot , which is still compatible with the mass measurement of PSR J1614 -2230 by Demorest et al. (2010), but which could be challenged by observations of significantly</text>
<figure>
<location><page_3><loc_22><loc_73><loc_58><loc_94></location>
<caption>Figure 1. Gravitational mass versus circumferential radius with (solid line) and without (dashed line) a quark-matter core. See the text for detail.</caption>
</figure>
<text><location><page_3><loc_9><loc_64><loc_72><loc_68></location>more massive NSs (see Clark et al. 2002, Freire et al. 2008, van Kerkwijk et al. 2011) unless the sound speed in quark matter is significantly larger than that predicted by perturbative QCD calculations (Kurkela et al. 2010).</text>
<section_header_level_1><location><page_3><loc_9><loc_61><loc_25><loc_62></location>Acknowledgements</section_header_level_1>
<text><location><page_3><loc_9><loc_57><loc_72><loc_60></location>FNRS (Belgium), NSERC (Canada) and CompStar, a Research Networking Programme of the European Science Foundation are gratefully acknowledged.</text>
<section_header_level_1><location><page_3><loc_9><loc_53><loc_18><loc_55></location>References</section_header_level_1>
<text><location><page_3><loc_9><loc_31><loc_72><loc_53></location>BRUSLIB http://www.astro.ulb.ac.be/bruslib Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2011, Phys. Rev. C , 84, 062802 Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2013, Astron. Astrophys. , 553, A22 Clark, J. S., Goodwin, S. P., Crowther, P. A., Kaper, L, Fairbairn, M., Langer, N., Brocksopp, C. 2002, Astron. Astrophys. , 392, 909 Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Nature , 467, 1081 Freire, P. C. C., Ransom, S. M., B'egin, S., Stairs, I. H., Hessels, J. W. T., Frey, L. H., Camilo, F. 2008, Astrophys. J. , 675, 670 Goriely, S., Chamel, N., & Pearson, J. M. 2010, Phys. Rev. C , 82, 035804 Haensel, P., Potekhin, A. Y., & Yakovlev, D. G. 2007, Astrophys. Space Sc. L. , 326 Kurkela, A., Romatschke, P., & Vuorinen, A. 2010, Phys. Rev. D , 81, 105021 Oppenheimer, J. R., & Volkoff, G. M. 1939, Phys. Rev. , 55, 374 Pearson, J. M., Goriely, S., & Chamel, N. 2011, Phys. Rev. C , 83, 065810 Pearson, J. M., Chamel, N., Goriely, S., & Ducoin, C. 2012, Phys. Rev. C , 85, 065803 Tolman, R. C. 1939, Phys. Rev. , 55, 364</text>
<text><location><page_3><loc_9><loc_30><loc_61><loc_31></location>van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, Astrophys. J. 728, 95</text>
</document>
[
{
"title": "N. Chamel 1 , A.F. Fantina 1 , J.M. Pearson 2 and S. Goriely 1",
"content": "1 Institut d'Astronomie et d'Astrophysique, CP-226, Universit'e Libre de Bruxelles, 1050 Brussels, Belgium 2 D'ept. de Physique, Universit'e de Montr'eal, Montr'eal (Qu'ebec), H3C 3J7 Canada Abstract. We study the impact of a hadron-quark phase transition on the maximum neutronstar mass. The hadronic part of the equation of state relies on the most up-to-date Skyrme nuclear energy density functionals, fitted to essentially all experimental nuclear mass data and constrained to reproduce the properties of infinite nuclear matter as obtained from microscopic calculations using realistic forces. We show that the softening of the dense matter equation of state due to the phase transition is not necessarily incompatible with the existence of massive neutron stars like PSR J1614 -2230. Keywords. stars: neutron, dense matter, equation of state, stars: interiors, gravitation",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Neutron stars (NSs) result from the gravitational collapse of massive stars with M /greaterorsimilar 8 M /circledot at the end point of their evolution. They are among the most compact objects in the universe, with a central density which can reach several times the nuclear saturation density. At least, three different regions can be identified in the interior of a NS: (i) the 'outer crust', at densities above ∼ 10 4 g cm -3 , composed of fully ionized atoms, arranged in a Coulomb lattice of nuclei, neutralized by a degenerate electron gas, (ii) the 'inner crust', at densities above ∼ 4 × 10 11 g cm -3 , composed of neutron-proton clusters and unbound neutrons, neutralized by a degenerate electron gas, and (iii) the core, at densities above ∼ 10 14 g cm -3 . The precise measurement of the mass of pulsar PSR J1614 -2230 by Demorest et al. (2010) has revived the question of the composition of the core. Just below the crust, the matter consists of a mixture of neutrons, protons, electrons and possibly muons. The composition of the central region of a NS is still a matter of debate (see e.g. Haensel et al. 2007). In the present work, we study the impact of a hadron-quark phase transition in dense matter on the maximum mass of cold isolated NSs (see Chamel et al. 2013 for a general discussion of the maximum mass of hybrid stars).",
"pages": [
1
]
},
{
"title": "2. Hadronic equation of state",
"content": "The global structure of a NS is determined by the equation of state (EoS), i.e. the relation between the matter pressure P and the mass-energy density ρ . Before considering the possibility of a phase transition from hadronic to quark matter in the core of NSs, we will begin with the hadronic EoSs. A good starting point is the family of three EoSs, BSk19, BSk20 and BSk21, which have been developed to provide a unified treatment of all regions of a NS (see Pearson et al. 2011, Pearson et al. 2012). These EoSs are based on nuclear energy-density functionals derived from generalized Skyrme forces (in that they contain additional momentum- and density-dependent terms), which fit essentially all measured masses of atomic nuclei with an rms deviation of 0.58 MeV for all three models. Moreover, these functionals were constrained to reproduce three different neutron matter EoSs, as obtained from microscopic calculations (see Goriely et al. 2010). All three EoSs assume that the core of a NS is made of nucleons and leptons. The BSk19 EoS was found to be too soft to support NSs as massive as PSR J1614 -2230 (Chamel et al. 2011) and therefore, it will not be considered here.",
"pages": [
1,
2
]
},
{
"title": "3. Hadron-quark phase transition",
"content": "Given the uncertainties in the composition of dense matter in NSs, we will simply suppose that above the average baryon density n N , matter undergoes a first-order phase transition to deconfined quark matter subject to the following restrictions: (i) for the transition to occur the energy density of the quark phase must be lower than that of the hadronic phase, (ii) according to perturbative quantum chromodynamics (QCD) calculations (e.g. Kurkela et al. 2010), the speed of sound in quark matter cannot exceed c/ √ 3 where c is the speed of light. At densities below n N , matter is purely hadronic while a pure quark phase is found at densities above some density n X . In the intermediate region ( n N < n < n X ) where the two phases can coexist, the pressure and the chemical potential of the two phases are equal: P quark ( n ) = P hadron ( n N ) and µ quark ( n ) = µ hadron ( n N ). The EoS of the quark phase at n > n X is given by: We set the density n N to lie above the highest density found in nuclei as predicted by Hartree-Fock-Bogoliubov calculations, namely n N = 0 . 2 fm -3 (BRUSLIB). The density n X is adjusted to optimize the maximum mass under the conditions mentioned above. Eq. (3.1) turns out to be very similar to that obtained within the simple MIT bag model, which has been widely applied to describe quark matter in compact stars (see e.g. Haensel et al. 2007). The effective bag constant B associated with the BSk21 hadronic EoS is 56 . 7 MeV fm -3 .",
"pages": [
2
]
},
{
"title": "4. Maximum mass",
"content": "Considering the stiffest hadronic EoS (BSk21), we have solved the Tolman-OppenheimerVolkoff equations (Tolman 1939, Oppenheimer & Volkoff 1939) in order to determine the global structure of a nonrotating NS. The effect of rotation on the maximum mass was found to be very small for stars with spin-periods comparable to that of PSR J1614 -2230 (Chamel et al. 2011); we therefore neglect it. The gravitational mass versus circumferential radius relation is shown in Fig. 1. We have considered two cases: a purely hadronic NS described by our BSk21 EoS (dashed line) and a hybrid star with a quark core (solid line). The corresponding maximum masses are 2.28 M /circledot and 2.02 M /circledot respectively. In both cases, the existence of two-solar mass NSs is therefore allowed.",
"pages": [
2
]
},
{
"title": "5. Conclusions",
"content": "The presence of a deconfined quark-matter phase in NS cores leads to a maximum mass of about 2 M /circledot , which is still compatible with the mass measurement of PSR J1614 -2230 by Demorest et al. (2010), but which could be challenged by observations of significantly more massive NSs (see Clark et al. 2002, Freire et al. 2008, van Kerkwijk et al. 2011) unless the sound speed in quark matter is significantly larger than that predicted by perturbative QCD calculations (Kurkela et al. 2010).",
"pages": [
2,
3
]
},
{
"title": "Acknowledgements",
"content": "FNRS (Belgium), NSERC (Canada) and CompStar, a Research Networking Programme of the European Science Foundation are gratefully acknowledged.",
"pages": [
3
]
},
{
"title": "References",
"content": "BRUSLIB http://www.astro.ulb.ac.be/bruslib Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2011, Phys. Rev. C , 84, 062802 Chamel, N., Fantina, A. F., Pearson, J. M., & Goriely, S. 2013, Astron. Astrophys. , 553, A22 Clark, J. S., Goodwin, S. P., Crowther, P. A., Kaper, L, Fairbairn, M., Langer, N., Brocksopp, C. 2002, Astron. Astrophys. , 392, 909 Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Nature , 467, 1081 Freire, P. C. C., Ransom, S. M., B'egin, S., Stairs, I. H., Hessels, J. W. T., Frey, L. H., Camilo, F. 2008, Astrophys. J. , 675, 670 Goriely, S., Chamel, N., & Pearson, J. M. 2010, Phys. Rev. C , 82, 035804 Haensel, P., Potekhin, A. Y., & Yakovlev, D. G. 2007, Astrophys. Space Sc. L. , 326 Kurkela, A., Romatschke, P., & Vuorinen, A. 2010, Phys. Rev. D , 81, 105021 Oppenheimer, J. R., & Volkoff, G. M. 1939, Phys. Rev. , 55, 374 Pearson, J. M., Goriely, S., & Chamel, N. 2011, Phys. Rev. C , 83, 065810 Pearson, J. M., Chamel, N., Goriely, S., & Ducoin, C. 2012, Phys. Rev. C , 85, 065803 Tolman, R. C. 1939, Phys. Rev. , 55, 364 van Kerkwijk, M. H., Breton, R. P., Kulkarni, S. R. 2011, Astrophys. J. 728, 95",
"pages": [
3
]
}
]
2013IAUS..291..496S
https://arxiv.org/pdf/1209.5171.pdf
<document>
<section_header_level_1><location><page_1><loc_9><loc_86><loc_71><loc_90></location>New Constraints on Preferred Frame Effects from Binary Pulsars</section_header_level_1>
<section_header_level_1><location><page_1><loc_20><loc_83><loc_61><loc_85></location>Lijing Shao 1 , Norbert Wex 2 , Michael Kramer 3</section_header_level_1>
<text><location><page_1><loc_12><loc_79><loc_69><loc_83></location>1 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany School of Physics, Peking University, Beijing 100871, China email: lshao@pku.edu.cn</text>
<text><location><page_1><loc_12><loc_76><loc_69><loc_79></location>2 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany email: wex@mpifr-bonn.mpg.de</text>
<text><location><page_1><loc_12><loc_71><loc_69><loc_76></location>3 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, M13 9PL, UK email: mkramer@mpifr-bonn.mpg.de</text>
<text><location><page_1><loc_9><loc_58><loc_72><loc_69></location>Abstract. Preferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, ˆ α 1 and ˆ α 2 . In the limit of a small orbital eccentricity, ˆ α 1 and ˆ α 2 contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of ˆ α 1 and ˆ α 2 in the strong-field regime. The constraint on ˆ α 1 also surpasses its counterpart in the weak-field regime.</text>
<text><location><page_1><loc_9><loc_56><loc_68><loc_57></location>Keywords. Gravitation, pulsars: general, pulsars: individual (J1012+5307, J1738+0333)</text>
<section_header_level_1><location><page_1><loc_9><loc_51><loc_23><loc_52></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_39><loc_72><loc_50></location>Pulsars are extremely stable electromagnetic emitters, and along with their extreme physical properties and surrounding environments, they provide useful astrophysical laboratories to study fundamental physics (Lorimer & Kramer 2005). Radio timing of binary pulsars maps out the binary orbital dynamics through recording the time-of-arrivals of the pulsar signals at the telescope. For millisecond pulsars this can be done with high precision, providing a powerful tool to probe gravity (see e.g., Stairs (2003) and Kramer et al. (2006)). In this work, we summarize new results on testing the local Lorentz invariance (LLI) of gravity from binary pulsars obtained by Shao & Wex (2012).</text>
<section_header_level_1><location><page_1><loc_9><loc_35><loc_47><loc_36></location>2. New Limits on Preferred Frame Effects</section_header_level_1>
<text><location><page_1><loc_9><loc_22><loc_72><loc_34></location>Non-gravitational LLI is an important ingredient of the Einstein equivalence principle (EEP) (Will 1993, 2006). But even metric gravity, which fulfills the EEP exactly, could still exhibit a violation of LLI in the gravitational sector (Will & Nordtvedt 1972, Nordtvedt & Will 1972, Damour & Esposito-Far'ese 1992, Will 1993). Such a violation of LLI induces preferred frame effects (PFEs) in the orbital dynamics of a binary system that moves with respect to the preferred frame. In the parametrized post-Newtonian (PPN) formalism, PFEs of a semi-conservative gravity theory are described by two parameters, ˆ α 1 and ˆ α 2 . †</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_19><loc_72><loc_21></location>† To distinguish from their weak-field counterparts ( α 1 and α 2 ), here 'hat' indicates possible modifications by strong-field effects.</list_item>
</unordered_list>
<text><location><page_2><loc_9><loc_87><loc_72><loc_94></location>The orbital dynamics of binary pulsars with non-vanishing ˆ α 1 and ˆ α 2 are obtained from a generic semi-conservative Lagrangian (Damour & Esposito-Far'ese 1992). It is found that in the limit of small orbital eccentricity, PFEs induced by ˆ α 1 and ˆ α 2 decouple, and lead to separable effects in the timing observations. Hence they can be tested independently using observations of only one binary pulsar (Shao & Wex 2012).</text>
<unordered_list>
<list_item><location><page_2><loc_9><loc_62><loc_72><loc_86></location>· A non-zero ˆ α 1 induces a polarization of the eccentricity vector towards a direction in the orbital plane perpendicular to the velocity of the binary system with respect to the preferred frame, w . † The observed eccentricity vector e ( t ) is a vectorial superposition of a 'relativistically rotating' eccentricity e R ( t ) (of constant length) and a 'fixed eccentricity' e F ∝ ˆ α 1 : e ( t ) = e R ( t ) + e F (Damour & Esposito-Far'ese 1992). The effect is graphically illustrated in the left panel of Fig. 1. Previous methods use the smallness of the observed eccentricity, combined with probabilistic considerations concerning the unknown angle θ (the angle between e R and e F ) to constrain ˆ α 1 (Damour & Esposito-Far'ese 1992, Wex 2000). The method developed in Shao & Wex (2012) is an extension of the method by Damour & Esposito-Far'ese (1992) that does not require any probabilistic considerations concerning θ . It is applicable to binary pulsars of short orbital period that have been observed for a long enough time, during which the periastron has advanced significantly. Even if the advance of periastron is not resolved in the timing observation, the constraints on the (observed) eccentricity vector can be converted into a limit on ˆ α 1 . From the 10 years of timing and the optical observations of PSR J1738+0333 (Antoniadis et al. 2012, Freire et al. 2012) one obtains the most constraining limit,</list_item>
</unordered_list>
<formula><location><page_2><loc_27><loc_60><loc_72><loc_62></location>ˆ α 1 = -0 . 4 +3 . 7 -3 . 1 × 10 -5 (95% C.L.) , (2.1)</formula>
<text><location><page_2><loc_9><loc_56><loc_72><loc_59></location>which is significantly better than the current best limits from both weak field (Muller et al. 2008) and strong field (Wex 2000).</text>
<unordered_list>
<list_item><location><page_2><loc_9><loc_49><loc_72><loc_56></location>· A non-vanishing ˆ α 2 induces a precession of the orbital angular momentum around the direction of w (see the right panel of Fig. 1), which changes the observed orbital inclination. Using the long-term timing results on PSRs J1012+5307 (Lazaridis et al. 2009) and J1738+0333 (Antoniadis et al. 2012, Freire et al. 2012), Shao & Wex (2012) find an upper limit</list_item>
</unordered_list>
<formula><location><page_2><loc_29><loc_47><loc_72><loc_48></location>| ˆ α 2 | < 1 . 8 × 10 -4 (95% C.L.) , (2.2)</formula>
<text><location><page_2><loc_9><loc_45><loc_72><loc_46></location>which is better than the current best limit for strongly self-gravitating bodies (Wex &</text>
<unordered_list>
<list_item><location><page_2><loc_9><loc_42><loc_72><loc_44></location>† Here we choose the isotropic cosmic microwave background as the preferred frame; nevertheless, see Shao & Wex (2012) for constraints on other preferred frames.</list_item>
</unordered_list>
<figure>
<location><page_2><loc_20><loc_27><loc_59><loc_39></location>
<caption>Figure 1. Illustration of preferred frame effects in the orbital dynamics of small-eccentricity binary pulsars; see Shao & Wex (2012) for details. Left : ˆ α 1 tends to polarize the orbital eccentricity vector e ( t ) towards the direction perpendicular to w ⊥ (Damour & Esposito-Far'ese 1992); right : ˆ α 2 induces a precession of the orbital angular momentum around the direction of w .</caption>
</figure>
<text><location><page_3><loc_9><loc_91><loc_72><loc_94></location>Kramer 2007) by more than three orders of magnitude, but still considerably weaker than the weak-field limit of α 2 by Nordtvedt (1987).</text>
<section_header_level_1><location><page_3><loc_9><loc_87><loc_20><loc_88></location>3. Summary</section_header_level_1>
<text><location><page_3><loc_9><loc_75><loc_72><loc_86></location>We summarize results presented in Shao & Wex (2012) that proposed new tests of LLI. These yield improved constraints on PFEs from binary pulsar experiments. Specifically, limits on parameters ˆ α 1 and ˆ α 2 are obtained from long-term timing of two binary pulsars with short orbital period (see Eqs. (2.1) and (2.2)). Our extended ˆ α 1 test no longer requires probabilistic considerations related to unknown angles. The proposed tests have the advantage that they continuously improve with time, and will benefit greatly from the next generation of radio telescopes, like FAST (Nan et al. 2011) and SKA (Smits et al. 2009).</text>
<section_header_level_1><location><page_3><loc_9><loc_70><loc_24><loc_72></location>Acknowledgment</section_header_level_1>
<text><location><page_3><loc_11><loc_69><loc_55><loc_70></location>Lijing Shao is supported by China Scholarship Council (CSC).</text>
<section_header_level_1><location><page_3><loc_9><loc_65><loc_18><loc_66></location>References</section_header_level_1>
<text><location><page_3><loc_9><loc_62><loc_72><loc_64></location>Antoniadis, J., van Kerkwijk, M. H., Koester, D., Freire, P. C. C., Wex, N., Tauris, T. M., Kramer, M., & Bassa, C. G. 2012, MNRAS , 423, 3316</text>
<text><location><page_3><loc_9><loc_60><loc_51><loc_61></location>Damour, T. & Esposito-Far'ese, G. 1992, Phys. Rev. D , 46, 4128</text>
<text><location><page_3><loc_9><loc_57><loc_72><loc_60></location>Freire, P. C. C., Wex, N., Esposito-Far'ese, G., Verbiest, J. P. W., Bailes, M., Jacoby, B. A., Kramer, M., Stairs, I. H., Antoniadis, J., & Janssen., G. H. 2012, MNRAS , 423, 3328</text>
<text><location><page_3><loc_9><loc_54><loc_72><loc_57></location>Kramer, M., Stairs, I. H., Manchester, R. N., McLaughlin, M. A., Lyne, A. G., Ferdman, R. D., Burgay, M., Lorimer, D. R., Possenti, A., D'Amico, N., Sarkissian, J. M., Hobbs, G. B., Reynolds, J. E., Freire, P. C. C., & Camilo, F. 2006, Science , 314, 97</text>
<text><location><page_3><loc_9><loc_50><loc_72><loc_53></location>Lazaridis, K., Wex, N., Jessner, A., Kramer, M., Stappers, B. W., Janssen, G. H., Desvignes, G., Purver, M. B., Cognard, I., Theureau, G., Lyne, A. G., Jordan, C. A., & Zensus, J. A. 2009, MNRAS , 400, 805</text>
<text><location><page_3><loc_9><loc_47><loc_72><loc_49></location>Lorimer, D. R. & Kramer, M. 2005, Handbook of Pulsar Astronomy (Cambridge University Press)</text>
<text><location><page_3><loc_9><loc_43><loc_72><loc_47></location>Muller, J., Williams, J. G., & Turyshev, S. G. 2008, in: H. Dittus, C. Lammerzahl, & S. G. Turyshev (eds.), Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space 349 (Astrophysics and Space Science Library: Springer), p. 457</text>
<text><location><page_3><loc_9><loc_41><loc_72><loc_43></location>Nan, R., Li, D., Jin, C., Wang, Q., Zhu, L., Zhu, W., Zhang, H., Yue, Y., & Qian, L. 2011, International Journal of Modern Physics D , 20, 989</text>
<text><location><page_3><loc_9><loc_39><loc_32><loc_40></location>Nordtvedt, K. 1987, ApJ , 320, 871</text>
<text><location><page_3><loc_9><loc_38><loc_41><loc_39></location>Nordtvedt, K. & Will, C. M. 1972, ApJ , 177, 775</text>
<text><location><page_3><loc_9><loc_36><loc_60><loc_37></location>Shao, L., & Wex, N. 2012, Class. Quantum Grav. , accepted [arXiv:1209.4503]</text>
<text><location><page_3><loc_9><loc_34><loc_72><loc_36></location>Smits, R., Kramer, M., Stappers, B., Lorimer, D. R., Cordes, J., & Faulkner, A. 2009, A&A , 493, 1161</text>
<text><location><page_3><loc_9><loc_31><loc_72><loc_33></location>Stairs, I. H. 2003, 'Testing General Relativity with Pulsar Timing' , Living Rev. Relativity 6, 5. URL (cited on 2012-09-11): http://www.livingreviews.org/lrr-2003-5</text>
<text><location><page_3><loc_9><loc_27><loc_72><loc_31></location>Wex, N. 2000, in: M. Kramer, N. Wex, & R. Wielebinski (eds.), IAU Colloq. 177: Pulsar Astronomy - 2000 and Beyond 202 (Astronomical Society of the Pacific Conference Series: San Francisco), p. 113</text>
<text><location><page_3><loc_9><loc_26><loc_41><loc_27></location>Wex, N., & Kramer, M. 2007, MNRAS , 380, 455</text>
<text><location><page_3><loc_9><loc_24><loc_72><loc_26></location>Will, C. M. 1993, Theory and Experiment in Gravitational Physics (Cambridge University Press)</text>
<text><location><page_3><loc_9><loc_23><loc_72><loc_24></location>Will, C. M. 2006, 'The Confrontation between General Relativity and Experiment' , Living Rev.</text>
<text><location><page_3><loc_12><loc_22><loc_68><loc_23></location>Relativity 9, 3. URL (cited on 2012-09-06): http://www.livingreviews.org/lrr-2006-3</text>
<text><location><page_3><loc_9><loc_20><loc_42><loc_22></location>Will, C. M., & Nordtvedt, K. 1972, ApJ , 177, 757</text>
</document>
[
{
"title": "Lijing Shao 1 , Norbert Wex 2 , Michael Kramer 3",
"content": "1 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany School of Physics, Peking University, Beijing 100871, China email: lshao@pku.edu.cn 2 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany email: wex@mpifr-bonn.mpg.de 3 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, M13 9PL, UK email: mkramer@mpifr-bonn.mpg.de Abstract. Preferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, ˆ α 1 and ˆ α 2 . In the limit of a small orbital eccentricity, ˆ α 1 and ˆ α 2 contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of ˆ α 1 and ˆ α 2 in the strong-field regime. The constraint on ˆ α 1 also surpasses its counterpart in the weak-field regime. Keywords. Gravitation, pulsars: general, pulsars: individual (J1012+5307, J1738+0333)",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Pulsars are extremely stable electromagnetic emitters, and along with their extreme physical properties and surrounding environments, they provide useful astrophysical laboratories to study fundamental physics (Lorimer & Kramer 2005). Radio timing of binary pulsars maps out the binary orbital dynamics through recording the time-of-arrivals of the pulsar signals at the telescope. For millisecond pulsars this can be done with high precision, providing a powerful tool to probe gravity (see e.g., Stairs (2003) and Kramer et al. (2006)). In this work, we summarize new results on testing the local Lorentz invariance (LLI) of gravity from binary pulsars obtained by Shao & Wex (2012).",
"pages": [
1
]
},
{
"title": "2. New Limits on Preferred Frame Effects",
"content": "Non-gravitational LLI is an important ingredient of the Einstein equivalence principle (EEP) (Will 1993, 2006). But even metric gravity, which fulfills the EEP exactly, could still exhibit a violation of LLI in the gravitational sector (Will & Nordtvedt 1972, Nordtvedt & Will 1972, Damour & Esposito-Far'ese 1992, Will 1993). Such a violation of LLI induces preferred frame effects (PFEs) in the orbital dynamics of a binary system that moves with respect to the preferred frame. In the parametrized post-Newtonian (PPN) formalism, PFEs of a semi-conservative gravity theory are described by two parameters, ˆ α 1 and ˆ α 2 . † The orbital dynamics of binary pulsars with non-vanishing ˆ α 1 and ˆ α 2 are obtained from a generic semi-conservative Lagrangian (Damour & Esposito-Far'ese 1992). It is found that in the limit of small orbital eccentricity, PFEs induced by ˆ α 1 and ˆ α 2 decouple, and lead to separable effects in the timing observations. Hence they can be tested independently using observations of only one binary pulsar (Shao & Wex 2012). which is significantly better than the current best limits from both weak field (Muller et al. 2008) and strong field (Wex 2000). which is better than the current best limit for strongly self-gravitating bodies (Wex & Kramer 2007) by more than three orders of magnitude, but still considerably weaker than the weak-field limit of α 2 by Nordtvedt (1987).",
"pages": [
1,
2,
3
]
},
{
"title": "3. Summary",
"content": "We summarize results presented in Shao & Wex (2012) that proposed new tests of LLI. These yield improved constraints on PFEs from binary pulsar experiments. Specifically, limits on parameters ˆ α 1 and ˆ α 2 are obtained from long-term timing of two binary pulsars with short orbital period (see Eqs. (2.1) and (2.2)). Our extended ˆ α 1 test no longer requires probabilistic considerations related to unknown angles. The proposed tests have the advantage that they continuously improve with time, and will benefit greatly from the next generation of radio telescopes, like FAST (Nan et al. 2011) and SKA (Smits et al. 2009).",
"pages": [
3
]
},
{
"title": "Acknowledgment",
"content": "Lijing Shao is supported by China Scholarship Council (CSC).",
"pages": [
3
]
},
{
"title": "References",
"content": "Antoniadis, J., van Kerkwijk, M. H., Koester, D., Freire, P. C. C., Wex, N., Tauris, T. M., Kramer, M., & Bassa, C. G. 2012, MNRAS , 423, 3316 Damour, T. & Esposito-Far'ese, G. 1992, Phys. Rev. D , 46, 4128 Freire, P. C. C., Wex, N., Esposito-Far'ese, G., Verbiest, J. P. W., Bailes, M., Jacoby, B. A., Kramer, M., Stairs, I. H., Antoniadis, J., & Janssen., G. H. 2012, MNRAS , 423, 3328 Kramer, M., Stairs, I. H., Manchester, R. N., McLaughlin, M. A., Lyne, A. G., Ferdman, R. D., Burgay, M., Lorimer, D. R., Possenti, A., D'Amico, N., Sarkissian, J. M., Hobbs, G. B., Reynolds, J. E., Freire, P. C. C., & Camilo, F. 2006, Science , 314, 97 Lazaridis, K., Wex, N., Jessner, A., Kramer, M., Stappers, B. W., Janssen, G. H., Desvignes, G., Purver, M. B., Cognard, I., Theureau, G., Lyne, A. G., Jordan, C. A., & Zensus, J. A. 2009, MNRAS , 400, 805 Lorimer, D. R. & Kramer, M. 2005, Handbook of Pulsar Astronomy (Cambridge University Press) Muller, J., Williams, J. G., & Turyshev, S. G. 2008, in: H. Dittus, C. Lammerzahl, & S. G. Turyshev (eds.), Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space 349 (Astrophysics and Space Science Library: Springer), p. 457 Nan, R., Li, D., Jin, C., Wang, Q., Zhu, L., Zhu, W., Zhang, H., Yue, Y., & Qian, L. 2011, International Journal of Modern Physics D , 20, 989 Nordtvedt, K. 1987, ApJ , 320, 871 Nordtvedt, K. & Will, C. M. 1972, ApJ , 177, 775 Shao, L., & Wex, N. 2012, Class. Quantum Grav. , accepted [arXiv:1209.4503] Smits, R., Kramer, M., Stappers, B., Lorimer, D. R., Cordes, J., & Faulkner, A. 2009, A&A , 493, 1161 Stairs, I. H. 2003, 'Testing General Relativity with Pulsar Timing' , Living Rev. Relativity 6, 5. URL (cited on 2012-09-11): http://www.livingreviews.org/lrr-2003-5 Wex, N. 2000, in: M. Kramer, N. Wex, & R. Wielebinski (eds.), IAU Colloq. 177: Pulsar Astronomy - 2000 and Beyond 202 (Astronomical Society of the Pacific Conference Series: San Francisco), p. 113 Wex, N., & Kramer, M. 2007, MNRAS , 380, 455 Will, C. M. 1993, Theory and Experiment in Gravitational Physics (Cambridge University Press) Will, C. M. 2006, 'The Confrontation between General Relativity and Experiment' , Living Rev. Relativity 9, 3. URL (cited on 2012-09-06): http://www.livingreviews.org/lrr-2006-3 Will, C. M., & Nordtvedt, K. 1972, ApJ , 177, 757",
"pages": [
3
]
}
]
2013IAUS..291..521T
https://arxiv.org/pdf/1210.4310.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_89><loc_69><loc_90></location>What can Fermi tell us about magnetars?</section_header_level_1>
<section_header_level_1><location><page_1><loc_29><loc_86><loc_51><loc_87></location>H. Tong 1 and R. X. Xu 2</section_header_level_1>
<text><location><page_1><loc_9><loc_83><loc_71><loc_85></location>1 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China Email: tonghao@xao.ac.cn</text>
<text><location><page_1><loc_11><loc_80><loc_70><loc_82></location>2 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Email: r.x.xu@pku.edu.cn</text>
<section_header_level_1><location><page_1><loc_9><loc_77><loc_16><loc_78></location>Abstract.</section_header_level_1>
<text><location><page_1><loc_9><loc_63><loc_72><loc_76></location>We have analyzed the physical implications of Fermi observations of magnetars. Observationally, no significant detection is reported in Fermi observations of all magnetars. Then there are conflicts between outer gap model in the case of magnetars and Fermi observations. One possible explanation is that magnetars are wind braking instead of magnetic dipole braking. In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. A magnetism-powered pulsar wind nebula and a braking index smaller than three are the two predictions of wind braking of magnetars. Future deeper Fermi observations will help us make clear whether they are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.</text>
<text><location><page_1><loc_9><loc_61><loc_50><loc_62></location>Keywords. pulsars: general, stars: magnetars, stars: neutron</text>
<section_header_level_1><location><page_1><loc_9><loc_55><loc_23><loc_57></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_40><loc_72><loc_55></location>Anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are peculiar pulsar-like objects. They are commonly assumed to be magnetars, magnetism-powered neutron stars. The traditional model of magnetars is that they are young neutron stars with both strong dipole field and strong multipole field (Duncan & Thompson 1992; Thompson et al. 2002). The presence of strong dipole field ensures that they can also accelerate particles to very high energy via the outer gap mechanism. Therefore, magnetar may emit high-energy gamma-rays detectable by Fermi -LAT (Cheng & Zhang 2001). These high-energy gamma-rays are rotation-powered in nature (Zhang 2003). Detection of rotation-powered activities in mangetars will help bridge the gap between magnetars and normal pulsars.</text>
<text><location><page_1><loc_9><loc_34><loc_72><loc_40></location>Fermi -LAT has observed the whole magnetar class. No significant detection is reported (Sasmaz Mus & Gogus 2010; Abdo et al. 2010). Then they are conflicts between the outer gap model in the case of magnetars and Fermi observations (Tong, Song, & Xu 2010, 2011). Below we will show what can Fermi tell us about magnetars.</text>
<section_header_level_1><location><page_1><loc_9><loc_30><loc_55><loc_31></location>2. Implications of Fermi observations of magentars</section_header_level_1>
<text><location><page_1><loc_9><loc_19><loc_72><loc_29></location>We have analyzed the implications of Fermi observation of AXP 4U 0142+61 (Sasmaz Mus, & Gogus 2010; Tong, Song, & Xu 2010). It is shown that there are conflicts between outer gap model in the case of AXP 4U 0142+61 and Fermi observations. The fallback disk model for AXPs and SGRs can still not be ruled out. In Fermi observations of the whole magnetar class, still no significant detection is reported. This is consistent with out previous analysis (Abdo et al. 2010; Tong, Song, & Xu 2011). The upper limit of AXP 4U 0142+62 lies already below the theoretical calculations for some parameter space (Tong,</text>
<text><location><page_2><loc_9><loc_91><loc_72><loc_94></location>Song, & Xu 2011). Future deeper Fermi observations will help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.</text>
<section_header_level_1><location><page_2><loc_9><loc_87><loc_34><loc_88></location>3. Solutions and predictions</section_header_level_1>
<text><location><page_2><loc_9><loc_81><loc_72><loc_86></location>There are two possible explanations to the non-detection in Fermi observations of magnetars. One possibility is that AXPs and SGRs are accretion-powered systems. Then it is natural that they are not high-energy gamam-ray emitters. Various observations of AXPs and SGRs can be explained naturally in the accretion scenario (Tong & Xu 2011).</text>
<text><location><page_2><loc_9><loc_73><loc_72><loc_80></location>The other possibility is: magnetars are wind braking. If magnetars are wind braking instead of mangetic dipole braking, then their magnetosphere structure is different from that of normal pulsars. Vacuum gaps (including outer gap) may not exist in magnetars. This may explain the non-detection in Fermi observations of magnetars (section 4.2 in Tong, Xu, Song, & Qiao 2012).</text>
<text><location><page_2><loc_9><loc_68><loc_72><loc_73></location>In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. Figure 1 shows the correspond dipole field in the case of wind braking and that of magnetic dipole braking</text>
<figure>
<location><page_2><loc_16><loc_30><loc_65><loc_66></location>
<caption>Figure 1. Dipole magnetic field in the case of wind braking versus that in the case of magnetic dipole braking. A wind luminosity L p = 10 35 erg s -1 is assumed. The solid, dashed, and dotted lines are for B dip , w = B dip , d , 0 . 1 B dip , d , 0 . 01 B dip , d , respectively. The dot-dashed line marks the position of quantum critical magnetic field B QED = 4 . 4 × 10 13 G. See figure 2 and corresponding text in Tong, Xu, Song, & Qiao (2012) for details.</caption>
</figure>
<text><location><page_2><loc_9><loc_19><loc_72><loc_21></location>Recent challenging observations of magnetars may be explained naturally in the wind braking scenario: (1) The supernova energies of magnetars are of normal value; (2) The</text>
<text><location><page_3><loc_9><loc_81><loc_72><loc_94></location>problem posed by the low-magnetic field soft gamma-ray repeater; (3) The relation between magnetars and high magnetic field pulsars ; (4) A decreasing period derivative during magnetar outbursts etc. A magnetism-powered (instead of rotation-powered) pulsar wind nebula will be one of the consequences of wind braking. For a magnetismpowered pulsar wind nebula, we should see a correlation between the nebula luminosity and the magnetar luminosity. The extended emission around AXP 1E 1547.0-5408 may be a magnetism-powered pulsar wind nebula. Under the wind braking scenario, a braking index smaller than three is expected. More details are presented in Tong, Xu, Song, & Qiao (2012).</text>
<text><location><page_3><loc_9><loc_69><loc_73><loc_80></location>Considering that magnetars are wind braking (both a rotation-powered and a magnetismpowered particle wind), many aspects of magnetars can be reinterpreted. For example, the 'low magnetic field' magnetar SGR 0418+5729 may actually be a normal magnetar (Rea et al. 2010; Tong & Xu 2012). It is a little special since it has a special geometry, e.g. a small magnetic inclination angle. Another example is: low luminosity magnetars are more likely to have radio emissions. The reason is that low luminosity magnears may have similar magnetospheric structure to that of normal radio pulsars (Rea et al. 2012; Liu, Tong, & Yuan 2012).</text>
<section_header_level_1><location><page_3><loc_9><loc_64><loc_22><loc_66></location>4. Conclusions</section_header_level_1>
<text><location><page_3><loc_9><loc_55><loc_72><loc_64></location>Fermi observations of magnetars tell us that magnetars may be wind braking instead of magnetic dipole braking. Future deeper Fermi -LAT observations will help us make clear whether magnetars are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Therefore, deeper Fermi -LAT observations of the magnetar class in the future are highly recommended.</text>
<section_header_level_1><location><page_3><loc_9><loc_51><loc_25><loc_52></location>Acknowledgments</section_header_level_1>
<text><location><page_3><loc_9><loc_46><loc_74><loc_50></location>This work is supported by the National Basic Research Program of China (2009CB824800), the National Natural Science Foundation of China (11103021, 10935001, 10973002), and the John Templeton Foundation.</text>
<section_header_level_1><location><page_3><loc_9><loc_42><loc_18><loc_43></location>References</section_header_level_1>
<text><location><page_3><loc_9><loc_40><loc_54><loc_41></location>Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ , 725, L73</text>
<text><location><page_3><loc_9><loc_39><loc_40><loc_40></location>Cheng, K. S., & Zhang, L. 2001, ApJ , 562, 918</text>
<text><location><page_3><loc_9><loc_37><loc_43><loc_38></location>Duncan, R. C., & Thompson, C. 1992, ApJ , 392, L9</text>
<text><location><page_3><loc_9><loc_36><loc_47><loc_37></location>Liu, Z. Y., Tong, H., & Yuan, J. P. 2012, arXiv:1210.2799</text>
<text><location><page_3><loc_9><loc_35><loc_51><loc_36></location>Rea, N., Esposito, P., Turolla, R., et al. 2010, Science , 330, 944</text>
<text><location><page_3><loc_9><loc_33><loc_50><loc_34></location>Rea, N., Pons, J. A., Torres, D. F., et al. 2012, ApJ , 748, L12</text>
<text><location><page_3><loc_9><loc_32><loc_42><loc_33></location>Sasmaz Mus, S., & Gogus, E. 2010, ApJ , 723, 100</text>
<text><location><page_3><loc_9><loc_30><loc_54><loc_32></location>Thompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, ApJ , 574, 332</text>
<text><location><page_3><loc_9><loc_29><loc_47><loc_30></location>Tong, H., Song, L. M., & Xu, R. X. 2010, ApJ , 725, L196</text>
<text><location><page_3><loc_9><loc_28><loc_45><loc_29></location>Tong, H., Song L. M., & Xu, R. X. 2011, ApJ , 738, 31</text>
<text><location><page_3><loc_9><loc_26><loc_49><loc_27></location>Tong, H., & Xu, R. X. 2011, Int. Jour. Mod. Phys. E , 20, 15</text>
<text><location><page_3><loc_9><loc_25><loc_38><loc_26></location>Tong, H., & Xu, R. X. 2012, ApJ , 757, L10</text>
<text><location><page_3><loc_9><loc_24><loc_56><loc_25></location>Tong, H., Xu, R. X., Song, L. M., & Qiao, G. J. 2012, arXiv:1205.1626</text>
<text><location><page_3><loc_9><loc_22><loc_65><loc_23></location>Zhang, B. 2003, Astrophysics and Space Science Library , 298, 27 (astro-ph/0212016)</text>
</document>
[
{
"title": "H. Tong 1 and R. X. Xu 2",
"content": "1 Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, China Email: tonghao@xao.ac.cn 2 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Email: r.x.xu@pku.edu.cn",
"pages": [
1
]
},
{
"title": "Abstract.",
"content": "We have analyzed the physical implications of Fermi observations of magnetars. Observationally, no significant detection is reported in Fermi observations of all magnetars. Then there are conflicts between outer gap model in the case of magnetars and Fermi observations. One possible explanation is that magnetars are wind braking instead of magnetic dipole braking. In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. A magnetism-powered pulsar wind nebula and a braking index smaller than three are the two predictions of wind braking of magnetars. Future deeper Fermi observations will help us make clear whether they are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Keywords. pulsars: general, stars: magnetars, stars: neutron",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are peculiar pulsar-like objects. They are commonly assumed to be magnetars, magnetism-powered neutron stars. The traditional model of magnetars is that they are young neutron stars with both strong dipole field and strong multipole field (Duncan & Thompson 1992; Thompson et al. 2002). The presence of strong dipole field ensures that they can also accelerate particles to very high energy via the outer gap mechanism. Therefore, magnetar may emit high-energy gamma-rays detectable by Fermi -LAT (Cheng & Zhang 2001). These high-energy gamma-rays are rotation-powered in nature (Zhang 2003). Detection of rotation-powered activities in mangetars will help bridge the gap between magnetars and normal pulsars. Fermi -LAT has observed the whole magnetar class. No significant detection is reported (Sasmaz Mus & Gogus 2010; Abdo et al. 2010). Then they are conflicts between the outer gap model in the case of magnetars and Fermi observations (Tong, Song, & Xu 2010, 2011). Below we will show what can Fermi tell us about magnetars.",
"pages": [
1
]
},
{
"title": "2. Implications of Fermi observations of magentars",
"content": "We have analyzed the implications of Fermi observation of AXP 4U 0142+61 (Sasmaz Mus, & Gogus 2010; Tong, Song, & Xu 2010). It is shown that there are conflicts between outer gap model in the case of AXP 4U 0142+61 and Fermi observations. The fallback disk model for AXPs and SGRs can still not be ruled out. In Fermi observations of the whole magnetar class, still no significant detection is reported. This is consistent with out previous analysis (Abdo et al. 2010; Tong, Song, & Xu 2011). The upper limit of AXP 4U 0142+62 lies already below the theoretical calculations for some parameter space (Tong, Song, & Xu 2011). Future deeper Fermi observations will help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs.",
"pages": [
1,
2
]
},
{
"title": "3. Solutions and predictions",
"content": "There are two possible explanations to the non-detection in Fermi observations of magnetars. One possibility is that AXPs and SGRs are accretion-powered systems. Then it is natural that they are not high-energy gamam-ray emitters. Various observations of AXPs and SGRs can be explained naturally in the accretion scenario (Tong & Xu 2011). The other possibility is: magnetars are wind braking. If magnetars are wind braking instead of mangetic dipole braking, then their magnetosphere structure is different from that of normal pulsars. Vacuum gaps (including outer gap) may not exist in magnetars. This may explain the non-detection in Fermi observations of magnetars (section 4.2 in Tong, Xu, Song, & Qiao 2012). In the wind braking scenario, magnetars are neutron stars with strong multipole field. A strong dipole field is no longer required. Figure 1 shows the correspond dipole field in the case of wind braking and that of magnetic dipole braking Recent challenging observations of magnetars may be explained naturally in the wind braking scenario: (1) The supernova energies of magnetars are of normal value; (2) The problem posed by the low-magnetic field soft gamma-ray repeater; (3) The relation between magnetars and high magnetic field pulsars ; (4) A decreasing period derivative during magnetar outbursts etc. A magnetism-powered (instead of rotation-powered) pulsar wind nebula will be one of the consequences of wind braking. For a magnetismpowered pulsar wind nebula, we should see a correlation between the nebula luminosity and the magnetar luminosity. The extended emission around AXP 1E 1547.0-5408 may be a magnetism-powered pulsar wind nebula. Under the wind braking scenario, a braking index smaller than three is expected. More details are presented in Tong, Xu, Song, & Qiao (2012). Considering that magnetars are wind braking (both a rotation-powered and a magnetismpowered particle wind), many aspects of magnetars can be reinterpreted. For example, the 'low magnetic field' magnetar SGR 0418+5729 may actually be a normal magnetar (Rea et al. 2010; Tong & Xu 2012). It is a little special since it has a special geometry, e.g. a small magnetic inclination angle. Another example is: low luminosity magnetars are more likely to have radio emissions. The reason is that low luminosity magnears may have similar magnetospheric structure to that of normal radio pulsars (Rea et al. 2012; Liu, Tong, & Yuan 2012).",
"pages": [
2,
3
]
},
{
"title": "4. Conclusions",
"content": "Fermi observations of magnetars tell us that magnetars may be wind braking instead of magnetic dipole braking. Future deeper Fermi -LAT observations will help us make clear whether magnetars are wind braking or magnetic dipole braking. It will also help us to distinguish between the magnetar model and the accretion model for AXPs and SGRs. Therefore, deeper Fermi -LAT observations of the magnetar class in the future are highly recommended.",
"pages": [
3
]
},
{
"title": "Acknowledgments",
"content": "This work is supported by the National Basic Research Program of China (2009CB824800), the National Natural Science Foundation of China (11103021, 10935001, 10973002), and the John Templeton Foundation.",
"pages": [
3
]
},
{
"title": "References",
"content": "Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ , 725, L73 Cheng, K. S., & Zhang, L. 2001, ApJ , 562, 918 Duncan, R. C., & Thompson, C. 1992, ApJ , 392, L9 Liu, Z. Y., Tong, H., & Yuan, J. P. 2012, arXiv:1210.2799 Rea, N., Esposito, P., Turolla, R., et al. 2010, Science , 330, 944 Rea, N., Pons, J. A., Torres, D. F., et al. 2012, ApJ , 748, L12 Sasmaz Mus, S., & Gogus, E. 2010, ApJ , 723, 100 Thompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, ApJ , 574, 332 Tong, H., Song, L. M., & Xu, R. X. 2010, ApJ , 725, L196 Tong, H., Song L. M., & Xu, R. X. 2011, ApJ , 738, 31 Tong, H., & Xu, R. X. 2011, Int. Jour. Mod. Phys. E , 20, 15 Tong, H., & Xu, R. X. 2012, ApJ , 757, L10 Tong, H., Xu, R. X., Song, L. M., & Qiao, G. J. 2012, arXiv:1205.1626 Zhang, B. 2003, Astrophysics and Space Science Library , 298, 27 (astro-ph/0212016)",
"pages": [
3
]
}
]
2013IAUS..292...87T
https://arxiv.org/pdf/1211.2170.pdf
<document>
<section_header_level_1><location><page_1><loc_13><loc_89><loc_67><loc_90></location>Modes of star formation from Herschel</section_header_level_1>
<text><location><page_1><loc_21><loc_86><loc_60><loc_87></location>L. Testi 1 , 2 , E. Bressert 1 , 3 and S. Longmore 1</text>
<text><location><page_1><loc_12><loc_81><loc_69><loc_85></location>1 ESO, Karl Schwarzschild srt. 2, D-85748 Garching, Germany, email: ltesti@eso.org 2 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 3 School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK</text>
<text><location><page_1><loc_9><loc_59><loc_72><loc_78></location>Abstract. We summarize some of the results obtained from Herschel surveys of the nearby star forming regions and the Galactic plane. We show that in the nearby star forming regions the starless core spatial surface density distribution is very similar to that of the young stellar objects. This, taken together with the similarity between the core mass function and the initial mass function for stars and the relationship between the amount of dense gas and star formation rate, suggest that the cloud fragmentation process defines the global outcome of star formation. This 'simple' view of star formation may not hold on all scales. In particular dynamical interactions are expected to become important at the conditions required to form young massive clusters. We describe the successes of a simple criterion to identify young massive cluster precursors in our Galaxy based on (sub-)millimetre wide area surveys. We further show that in the location of our Galaxy where the best candidate for a precursor of a young massive cluster is found, the 'simple' scaling relationship between dense gas and star formation rate appear to break down. We suggest that in regions where the conditions approach those of the central molecular zone of our Galaxy it may be necessary to revise the scaling laws for star formation.</text>
<text><location><page_1><loc_9><loc_57><loc_47><loc_58></location>Keywords. stars: formation, ISM: clouds, Galaxy: center</text>
<section_header_level_1><location><page_1><loc_9><loc_51><loc_23><loc_53></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_39><loc_72><loc_51></location>The Herschel Space Observatory multiband wide area surveys of our own Galaxy in the far infrared and submillimeter, when combined with surveys at other wavelengths allow us, for the first time, to obtain a detailed view of the relationship between the physical structure of gas and dust in molecular clouds and cores and the young stellar populations that are produced in these clouds. In this contribution we focus on three results using data from the Herschel Gould Belt Survey and the HIGAL survey. Detailed descriptions of these surveys are given elsewhere in this volume by Di Francesco and Molinari see also Molinari et al. (2010) and Andr'e et al. (2010).</text>
<text><location><page_1><loc_9><loc_31><loc_72><loc_39></location>We will focus on the origin of the observed ranges of young stellar objects (YSO) surface densities in the nearby star forming regions, on the possible identification of precursors of the most massive clusters in our own Galaxy and the special conditions of star formation in the Central Molecular Zone (CMZ) of our Galaxy as compared to the Galactic disk, and in particular the star forming regions in the solar neighborhood.</text>
<section_header_level_1><location><page_1><loc_9><loc_27><loc_53><loc_28></location>2. The starless cores surface density distribution</section_header_level_1>
<text><location><page_1><loc_9><loc_19><loc_72><loc_26></location>Following the extensive Spitzer surveys in nearby star forming regions, it has become clear that young stellar objects are found in a continuous range of stellar densities rather than showing a clear bimodal distribution as postulated by the paradigm of clustered vs. distributed star formation. In particular Bressert et al. (2010) showed that the distribution of surface YSO densities in the regions surveyed by Spitzer is scale-free and the</text>
<text><location><page_2><loc_9><loc_91><loc_72><loc_94></location>definition of the percentage of stars forming in clusters is arbitrary if based on specific thresholds on the local surface density.</text>
<text><location><page_2><loc_9><loc_76><loc_72><loc_91></location>Thanks to the wide area surveyed and the large samples of starless cores detected, the Herschel surveys of nearby star forming regions allow us to investigate whether this continuous distribution of surface densities are an imprint of the molecular cloud fragmentation. In Bressert et al. (2012b) we combined the Herschel data in the Perseus West and Serpens clouds with infrared data from the Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010) to select samples of starless cores and compare their spatial distribution with that of the YSOs. We employed WISE color-color diagrams to identify YSOs and remove extragalactic contamination (following the method of Koenig et al. 2012), this method was cross checked, in the overlap region, with the YSOs classification in the Spitzer c2d survey (Evans et al. 2009).</text>
<text><location><page_2><loc_9><loc_60><loc_72><loc_76></location>The spatial distribution of cores and young stellar object are being compared using two separate methodologies. We first compare the cumulative distributions of surface densities of the neighbours of each YSO or core and we show that, as found by Bressert et al. (2010) for YSOs, the core distribution is scale free and matches closely that of the formed stars (see Fig. 1). In addition, we employed a minimum spanning tree method to identify groups of YSOs or cores, following the method of Gutermuth et al. (2009). As discussed above, given the lack of preferential clustering scales, the identification of groups is based on an arbitrary criterion, but if we use the same criterion for YSOs and cores, we find that the groups of the two populations tend to overlap spatially and the average densities that we derive are consistent within the uncertainties (albeit affected by very large errors).</text>
<text><location><page_2><loc_9><loc_49><loc_72><loc_59></location>These results strongly favors the view in which the density distribution of YSOs in these regions are the result of the imprint of the cloud fragmentation process, rather than dynamical evolution. Taken together with the fact that the core mass function in nearby star forming regions closely resembles the initial mass function for stars (Testi & Sargent 1998; Motte, Andr'e & Neri 1998; Konyves et al. 2010), these results strongly support the view that in these regions the global outcome of the star formation process is mainly determined by cloud fragmentation process rather than by dynamical interactions.</text>
<section_header_level_1><location><page_2><loc_9><loc_45><loc_54><loc_46></location>3. Looking for Young Massive Clusters precursors</section_header_level_1>
<text><location><page_2><loc_9><loc_23><loc_72><loc_44></location>Nearby star forming regions do not have the conditions to form very massive and dense stellar clusters, and have very modest capabilities of forming high mass stars (if at all). Nevertheless, the Galaxy is known to host a number of very massive and dense clusters, which are also prominently seen in extragalactic starbursts. These clusters are normally called Young Massive Clusters (YMC) or, in the most extreme cases, Super Star Clusters and have total stellar masses and stellar densities that in some cases approaching those of globular clusters (see Portegies Zwart et al. 2010, for a recent review). The conditions to form these systems are expected to be very different than those found in nearby star forming regions and dynamical interactions during formation may be the dominant mechanism (Smith et al. 2009). Given the importance of these systems for extragalactic studies, constraining the physical conditions for their formation is especially important. The precursors of YMCs are expected to be very rare in our galaxy, but wide area surveys of our Galaxy at far infrared and millimetre wavelengths offer a unique possibility to search for these rare objects.</text>
<text><location><page_2><loc_9><loc_19><loc_72><loc_23></location>In Bressert et al. (2012a) we developed and tested a set of guidelines to identify good candidates of YMC precursors. The criteria that we developed are based on the fact that these precursors not only need to contain enough mass to form a YMC, but also need to</text>
<figure>
<location><page_3><loc_24><loc_79><loc_57><loc_94></location>
<caption>Fig. 3. The surface densities ( Σ ) of the cores and YSOs in western Perseus and Serpens combined. The red dotted line and the blue dashed line represents the cores and the WISE YSOs, respectively. The solid black line is the all the cores and WISE YSOs combined. The light grey solid line is the profile shown in follows: 1. The ratio of YSOs to cores in Perseus-W and Serpens is 1 . 4 and 4 . 6, respectively. While the ratio of Class II to Class I objects is 1.2 and 3.2. This implies that the Perseus-W region is younger than Serpens. Figure 1. Cumulative distribution of surface densities of cores and YSOs in Perseus and Serpens combined. Red, dotted line is for Herschel starless cores, blue, dashed line for the WISE YSOs. Solid black line is for the two samples combined. The thin grey line is the distribution for YSOs in nearby star forming regions (Bressert et al. 2010), this reaches higher surface densities but has a shape consistent with the other distributions. Figure adapted from Bressert et al. (2012b). - 8 -</caption>
</figure>
<text><location><page_3><loc_23><loc_70><loc_58><loc_71></location>Bressert et al. (2010) for nearly all YSOs in the solar neighbour-</text>
<text><location><page_3><loc_62><loc_93><loc_95><loc_94></location>The similarity between the YSOs and cores spatial distri-</text>
<text><location><page_3><loc_60><loc_91><loc_95><loc_93></location>butions suggest that these are set by the cloud fragmentation</text>
<text><location><page_3><loc_60><loc_90><loc_95><loc_91></location>process. This result, together with the similarity between the</text>
<text><location><page_3><loc_60><loc_89><loc_95><loc_90></location>cores mass function and the stellar IMF in these regions (Testi &</text>
<text><location><page_3><loc_60><loc_88><loc_95><loc_89></location>Sargent 1998; Sandell & Knee 2001), suggest that the dynamical</text>
<text><location><page_3><loc_60><loc_87><loc_95><loc_88></location>processes in early stellar evolution is globally less important than</text>
<text><location><page_3><loc_60><loc_86><loc_93><loc_87></location>cloud fragmentation for low-mass star forming environments.</text>
<text><location><page_3><loc_60><loc_82><loc_68><loc_83></location>4. Summary</text>
<text><location><page_3><loc_60><loc_80><loc_95><loc_81></location>Combining Herschel, WISE, and Spitzer data we presented a</text>
<text><location><page_3><loc_60><loc_79><loc_95><loc_80></location>comparison of the spatial distribution of cores and YSOs in</text>
<text><location><page_3><loc_60><loc_78><loc_95><loc_79></location>Perseus-W and Serpens star-forming regions. The results are as</text>
<text><location><page_3><loc_23><loc_69><loc_58><loc_70></location>hood using several Spitzer Legacy surveys. The data is hinting</text>
<text><location><page_3><loc_56><loc_56><loc_56><loc_57></location>/</text>
<text><location><page_3><loc_57><loc_56><loc_58><loc_57></location>2</text>
<text><location><page_3><loc_56><loc_56><loc_57><loc_57></location>pc</text>
<text><location><page_3><loc_60><loc_70><loc_95><loc_71></location>2. Using the minimum spanning tree method to identify groups</text>
<text><location><page_3><loc_62><loc_68><loc_95><loc_70></location>of cores and YSOs, we found that they tend to be associated</text>
<text><location><page_3><loc_62><loc_67><loc_95><loc_69></location>rather than not. In Perseus-W we find a few core groups that</text>
<text><location><page_3><loc_62><loc_66><loc_95><loc_67></location>are not directly associated with YSO groups. This finding is</text>
<text><location><page_3><loc_62><loc_65><loc_92><loc_66></location>consistent with Perseus-W being younger than Serpens.</text>
<text><location><page_3><loc_60><loc_64><loc_95><loc_65></location>3. Using the nearest neighbour method we found that the cu-</text>
<text><location><page_3><loc_62><loc_63><loc_95><loc_64></location>mulative surface density distribution of the Herschel cores</text>
<text><location><page_3><loc_62><loc_61><loc_95><loc_63></location>and WISE YSOs in the solar neighbourhood have similar</text>
<text><location><page_3><loc_62><loc_60><loc_95><loc_62></location>profiles. In agreement with Bressert et al. (2010), the cumu-</text>
<text><location><page_3><loc_62><loc_59><loc_95><loc_60></location>lative density profiles are smooth and imply that there are no</text>
<text><location><page_3><loc_62><loc_58><loc_83><loc_59></location>fixed scales for clustering (scale-free).</text>
<text><location><page_3><loc_60><loc_57><loc_95><loc_58></location>4. The similarity with the cores spatial distribution suggests</text>
<text><location><page_3><loc_62><loc_56><loc_95><loc_57></location>that the spatial distribution of YSOs is already set during the</text>
<text><location><page_3><loc_62><loc_54><loc_95><loc_56></location>cloud fragmentation process, and supports the idea that dy-</text>
<text><location><page_3><loc_60><loc_45><loc_95><loc_46></location>other star forming regions are required to put our findings on</text>
<text><location><page_3><loc_60><loc_44><loc_95><loc_45></location>solid statistical grounds and to verify if the global densities of</text>
<text><location><page_3><loc_9><loc_19><loc_95><loc_34></location>This is a result of Spitzer having higher angular resolution ( ∼ 5) than WISE and Herschel and for the YSOs a higher sensitivity limit than WISE. The similarity between the WISE YSOs and the cores implies that the YSOs are likely tracing the substructured and filamentary natal gas that they formed from, like their earlier predecessors, and that, even at the earliest stages, star formation is scale-free. The velocity dispersion of cores and YSOs is typically low (e.g.. v core /lessorsimilar 0 . 4 km s -1 in Ophiuchus and v YSO ∼ 1 . 4 kms -1 , Andr'e et al. 2007; Covey et al. 2006), which is consistent with our results where the Σ YSO and Σ core distributions are similar. If there was significant dynamical evolution in the YSO population, the Σ distribution between the YSOs and cores should di ff er. Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); IFSI, OAP / AOT, OAA / CAISMI, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESAPRODEX (Belgium), CEA / CNES (France), DLR (Germany), ASI (Italy), and CICT / MCT (Spain). These criteria successfully confirm the earlier identifications of candidate YMC precursor identified combining ALMA and VLT data in the Antennae Galaxies (Herrera et al. 2012) as well as the candidate precursor G 0 . 25+0 . 02 identified in the HOPS and HIGAL surveys by Longmore et al. (2012a). A few more candidates, identified by Ginsburg et al. (2012), need further study. G 0 . 25 + 0 . 02 in our Galaxy offer a unique opportunity to study in detail the initial conditions for YMC formation. The initial results seem to suggest that this YMC precursor is a highly turbulent cloud (Rathborne et al. 2012). While the properties and internal structure of the cloud are still being investigated, the origin of the cloud is unclear. The cloud is located in the CMZ of our own Galaxy and the special conditions near the Galactic Centre may be responsible for its formation. How-</text>
<text><location><page_3><loc_29><loc_54><loc_30><loc_56></location>±</text>
<figure>
<location><page_3><loc_25><loc_55><loc_55><loc_70></location>
</figure>
<text><location><page_3><loc_23><loc_54><loc_27><loc_56></location>and 17</text>
<text><location><page_3><loc_27><loc_55><loc_27><loc_56></location>.</text>
<text><location><page_3><loc_27><loc_54><loc_28><loc_56></location>28</text>
<text><location><page_3><loc_30><loc_54><loc_32><loc_56></location>13</text>
<text><location><page_3><loc_32><loc_55><loc_32><loc_56></location>.</text>
<text><location><page_3><loc_32><loc_54><loc_58><loc_56></location>85, respectively. Within the uncertainties the</text>
<paragraph><location><page_3><loc_9><loc_46><loc_95><loc_55></location>groups of di ff erent populations have similar densities. The closer connection between cores groups and YSO groups observed in Serpens as compared to Perseus-W is an additional indication that star formation is more advanced in the Serpens cloud. This is in agreement and reinforces the conclusions based on the statistics of cores and YSO classes presented in Sect. 2.2. namical processes in early stellar evolution is globally less important than cloud fragmentation for low-mass star forming environments. The cores and YSO groups in Perseus-W and Serpens have similar mean densities within the error (see § 3.1). Due to the large spread in the values of the mean densities, observations of Fig. 1.- The mass-radius parameter-space for clumps partitioned by radii for r Ω (solid blue) and rvir (solid black). MPC candidates are defined with the following properties (green shaded region): a minimum mass of 3 × 10 4 M /circledot , r > r Ω for Mclump < 8 . 4 × 10 4 M /circledot , and r > rvir for Mclump > 8 . 4 × 10 4 M /circledot . Clump masses and sizes are plotted on top from three different data catalogs: IRDCs (Rathborne et al. 2006), HOPS clumps (Walsh et al. 2011), and YMCs (Portegies Zwart et al. 2010). The YMCs are converted to their possible clump progenitors by assuming that SFE is ∼ 30%, which boosts the mass of the systems by a factor of 10 / 3. The scaled YMC progenitors happen to lie near the critical r Ω line without any tweaking of parameters. Two published sources that have radii less than both their respective r Ω and rvir are G0.253+0.016 (L12; Longmore et al. 2012) and an extragalactic massive proto-cluster candidate reported in Herrera et al. (H12; 2012). Figure 2. Criteria to identify precursors of YMCs on the radius-mass plane. The values for r Ω and r vir are shown as a thin blue line and a thick black line respectively (see text for definitions). The open symbols represent candidates YMC progenitors: triangle is a YMC candidate progenitor in the Antennae Galaxies (Herrera et al. 2012); the star is G 0 . 25 + 0 . 02 near the Galactic Centre (Longmore et al. 2012a); the squares are additional candidates in our Galaxy from Ginsburg et al. (2012). Figure adapted from Bressert et al. (2012a)</paragraph>
<text><location><page_3><loc_25><loc_46><loc_52><loc_46></location>The MPC candidates reported in Ginsburg et al. (GS; submitted) are shown as squares.</text>
<text><location><page_3><loc_9><loc_34><loc_95><loc_44></location>3.2. Surface densities Bressert et al. (2010) showed that the cumulative surface density of nearly all the YSOs in the solar neighbourhood exhibits a smooth and scale-free profile. In Fig. 3 we similarly investigate how the core population in Perseus-W and Serpens compare to the WISE YSO populations. The cumulative surface density profiles of the YSOs and cores are similar, but reach lower surface densities as compared to the Bressert et al. (2010) profile. the groups are distinct or not. Acknowledgments SPIRE has been developed by a consortium of institutes led by Cardi ff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); be very compact to allow for rapid and continuous star formation even under the effect of strong feedback from forming very massive stars. The final criteria to form a bound YMC with M tot glyph[greaterorequalslant] 10 4 M glyph[circledot] , requires objects to have more than 3 × 10 4 M glyph[circledot] (assuming 30% star formation efficiency), and a radius smaller than the minimum between r Ω , the maximum radius needed to gravitationally bind a fully ionised core, and r vir , the maximum radius that would allow a crossing time of ∼ 1 Myr (the typical observed value for YMCs, Portegies Zwart et al. 2010).</text>
<text><location><page_3><loc_23><loc_16><loc_23><loc_17></location>4</text>
<text><location><page_4><loc_9><loc_91><loc_72><loc_94></location>ever, it should be noted that YMCs are not only found in the CMZ in our own Galaxy. Further studies of this and other candidates will help clarify their nature.</text>
<section_header_level_1><location><page_4><loc_9><loc_87><loc_49><loc_88></location>4. Star formation and dense gas in the CMZ</section_header_level_1>
<text><location><page_4><loc_9><loc_73><loc_72><loc_86></location>In the previous section we have discussed a unique massive core in the CMZ of our Galaxy, and we speculated that in the CMZ the conditions may be favorable for the assembly of such extreme objects. In the solar neighborhood, several different arguments suggest that the process of star formation is controlled by the amount of dense molecular gas. Lada et al. (2012) and Krumholz et al. (2012) presented the latest incarnation of the views that the star formation rates observed in star forming regions and entire galaxies could be understood in terms of simple scaling relations with the amount of dense gas above a certain column density threshold (Lada's formulation) or the total local volumetric gas density (Krumholz's formulation).</text>
<text><location><page_4><loc_9><loc_52><loc_72><loc_73></location>We used data from HOPS and HIGAL Galactic plane surveys as well as some other datasets to show that the relationships between star formation rate and dense gas that appear to hold throughout the Galactic disk do appear to break down in the CMZ (Longmore et al. 2012b). More details on the methodology and results are given by Longmore in this volume, but the basic result is that while ∼ 80% of the dense gas in our own Galaxy is concentrated in the CMZ, there is no similar enhancement in the star formation rate. Similarly, the predictions of the volumetric gas density relations exceed by at least a factor of ten the observed star formation rate. These findings suggest that the prescriptions based on a single parameter (the total gas density or the dense gas column density) break down in the CMZ and a more complicated formulation is needed to reconcile this region with the rest of the Galaxy. The large shear and increased turbulence in the CMZ need to be looked at as the possible causes for the increased stability of cloud cores in that region. These processes may also favor the formation of more massive stable clumps that once they became unstable and collapse may form YMCs.</text>
<section_header_level_1><location><page_4><loc_9><loc_48><loc_18><loc_49></location>References</section_header_level_1>
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<text><location><page_4><loc_9><loc_37><loc_58><loc_38></location>Koenig, X. P., Leisawitz, D. T., Benford, D. J., et al. 2012, ApJ , 744, 130</text>
<text><location><page_4><loc_9><loc_35><loc_57><loc_36></location>K¨onyves, V., Andr´e, Ph., Men'shchikov, A., et al. 2010, A&A , 518, L106</text>
<text><location><page_4><loc_9><loc_34><loc_49><loc_35></location>Krumholz, M.R., Dekel, A., McKee, C.F. 2012, ApJ , 745, 69</text>
<text><location><page_4><loc_9><loc_32><loc_56><loc_33></location>Lada, C.J., Forbrich, J., Lombardi, M., Alves, J.F. 2012, ApJ , 745, 190</text>
<text><location><page_4><loc_9><loc_31><loc_57><loc_32></location>Longmore,S.N., Bally, J., Testi, L., et al. 2012, MNRAS , arXiv:1208.4256</text>
<text><location><page_4><loc_9><loc_30><loc_54><loc_31></location>Longmore,S.N., Rathborne,J., Bastian,N., et al. 2012, ApJ , 746, 117</text>
<text><location><page_4><loc_9><loc_28><loc_52><loc_29></location>Molinari, S., Swinyard, B., Bally, J., et al. 2012, A&A , 518, L100</text>
<text><location><page_4><loc_9><loc_27><loc_46><loc_28></location>Motte, F., Andr´e, Ph., & Neri, R. 1998, A&A , 336, 150</text>
<text><location><page_4><loc_9><loc_25><loc_60><loc_26></location>Portegies Zwart, S.F., McMillan, S.L.W., & Gieles, M. 2010, ARA&A 48, 431</text>
<text><location><page_4><loc_9><loc_24><loc_60><loc_25></location>Rathborne, J.M., Longmore, S.N., Jackson, J.M., et al. 2012, ApJ , submitted</text>
<text><location><page_4><loc_9><loc_23><loc_53><loc_24></location>Smith, R.J., Longmore, S.N., Bonnell, I. 2009, MNRAS , 400, 1775</text>
<text><location><page_4><loc_9><loc_21><loc_39><loc_22></location>Testi, L. & Sargent, A.I. 1998, ApJ , 508, L91</text>
<text><location><page_4><loc_9><loc_20><loc_61><loc_21></location>Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ , 140, 1868</text>
</document>
[
{
"title": "Modes of star formation from Herschel",
"content": "L. Testi 1 , 2 , E. Bressert 1 , 3 and S. Longmore 1 1 ESO, Karl Schwarzschild srt. 2, D-85748 Garching, Germany, email: ltesti@eso.org 2 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 3 School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Abstract. We summarize some of the results obtained from Herschel surveys of the nearby star forming regions and the Galactic plane. We show that in the nearby star forming regions the starless core spatial surface density distribution is very similar to that of the young stellar objects. This, taken together with the similarity between the core mass function and the initial mass function for stars and the relationship between the amount of dense gas and star formation rate, suggest that the cloud fragmentation process defines the global outcome of star formation. This 'simple' view of star formation may not hold on all scales. In particular dynamical interactions are expected to become important at the conditions required to form young massive clusters. We describe the successes of a simple criterion to identify young massive cluster precursors in our Galaxy based on (sub-)millimetre wide area surveys. We further show that in the location of our Galaxy where the best candidate for a precursor of a young massive cluster is found, the 'simple' scaling relationship between dense gas and star formation rate appear to break down. We suggest that in regions where the conditions approach those of the central molecular zone of our Galaxy it may be necessary to revise the scaling laws for star formation. Keywords. stars: formation, ISM: clouds, Galaxy: center",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The Herschel Space Observatory multiband wide area surveys of our own Galaxy in the far infrared and submillimeter, when combined with surveys at other wavelengths allow us, for the first time, to obtain a detailed view of the relationship between the physical structure of gas and dust in molecular clouds and cores and the young stellar populations that are produced in these clouds. In this contribution we focus on three results using data from the Herschel Gould Belt Survey and the HIGAL survey. Detailed descriptions of these surveys are given elsewhere in this volume by Di Francesco and Molinari see also Molinari et al. (2010) and Andr'e et al. (2010). We will focus on the origin of the observed ranges of young stellar objects (YSO) surface densities in the nearby star forming regions, on the possible identification of precursors of the most massive clusters in our own Galaxy and the special conditions of star formation in the Central Molecular Zone (CMZ) of our Galaxy as compared to the Galactic disk, and in particular the star forming regions in the solar neighborhood.",
"pages": [
1
]
},
{
"title": "2. The starless cores surface density distribution",
"content": "Following the extensive Spitzer surveys in nearby star forming regions, it has become clear that young stellar objects are found in a continuous range of stellar densities rather than showing a clear bimodal distribution as postulated by the paradigm of clustered vs. distributed star formation. In particular Bressert et al. (2010) showed that the distribution of surface YSO densities in the regions surveyed by Spitzer is scale-free and the definition of the percentage of stars forming in clusters is arbitrary if based on specific thresholds on the local surface density. Thanks to the wide area surveyed and the large samples of starless cores detected, the Herschel surveys of nearby star forming regions allow us to investigate whether this continuous distribution of surface densities are an imprint of the molecular cloud fragmentation. In Bressert et al. (2012b) we combined the Herschel data in the Perseus West and Serpens clouds with infrared data from the Wide-Field Infrared Survey Explorer (WISE, Wright et al. 2010) to select samples of starless cores and compare their spatial distribution with that of the YSOs. We employed WISE color-color diagrams to identify YSOs and remove extragalactic contamination (following the method of Koenig et al. 2012), this method was cross checked, in the overlap region, with the YSOs classification in the Spitzer c2d survey (Evans et al. 2009). The spatial distribution of cores and young stellar object are being compared using two separate methodologies. We first compare the cumulative distributions of surface densities of the neighbours of each YSO or core and we show that, as found by Bressert et al. (2010) for YSOs, the core distribution is scale free and matches closely that of the formed stars (see Fig. 1). In addition, we employed a minimum spanning tree method to identify groups of YSOs or cores, following the method of Gutermuth et al. (2009). As discussed above, given the lack of preferential clustering scales, the identification of groups is based on an arbitrary criterion, but if we use the same criterion for YSOs and cores, we find that the groups of the two populations tend to overlap spatially and the average densities that we derive are consistent within the uncertainties (albeit affected by very large errors). These results strongly favors the view in which the density distribution of YSOs in these regions are the result of the imprint of the cloud fragmentation process, rather than dynamical evolution. Taken together with the fact that the core mass function in nearby star forming regions closely resembles the initial mass function for stars (Testi & Sargent 1998; Motte, Andr'e & Neri 1998; Konyves et al. 2010), these results strongly support the view that in these regions the global outcome of the star formation process is mainly determined by cloud fragmentation process rather than by dynamical interactions.",
"pages": [
1,
2
]
},
{
"title": "3. Looking for Young Massive Clusters precursors",
"content": "Nearby star forming regions do not have the conditions to form very massive and dense stellar clusters, and have very modest capabilities of forming high mass stars (if at all). Nevertheless, the Galaxy is known to host a number of very massive and dense clusters, which are also prominently seen in extragalactic starbursts. These clusters are normally called Young Massive Clusters (YMC) or, in the most extreme cases, Super Star Clusters and have total stellar masses and stellar densities that in some cases approaching those of globular clusters (see Portegies Zwart et al. 2010, for a recent review). The conditions to form these systems are expected to be very different than those found in nearby star forming regions and dynamical interactions during formation may be the dominant mechanism (Smith et al. 2009). Given the importance of these systems for extragalactic studies, constraining the physical conditions for their formation is especially important. The precursors of YMCs are expected to be very rare in our galaxy, but wide area surveys of our Galaxy at far infrared and millimetre wavelengths offer a unique possibility to search for these rare objects. In Bressert et al. (2012a) we developed and tested a set of guidelines to identify good candidates of YMC precursors. The criteria that we developed are based on the fact that these precursors not only need to contain enough mass to form a YMC, but also need to Bressert et al. (2010) for nearly all YSOs in the solar neighbour- The similarity between the YSOs and cores spatial distri- butions suggest that these are set by the cloud fragmentation process. This result, together with the similarity between the cores mass function and the stellar IMF in these regions (Testi & Sargent 1998; Sandell & Knee 2001), suggest that the dynamical processes in early stellar evolution is globally less important than cloud fragmentation for low-mass star forming environments. 4. Summary Combining Herschel, WISE, and Spitzer data we presented a comparison of the spatial distribution of cores and YSOs in Perseus-W and Serpens star-forming regions. The results are as hood using several Spitzer Legacy surveys. The data is hinting / 2 pc 2. Using the minimum spanning tree method to identify groups of cores and YSOs, we found that they tend to be associated rather than not. In Perseus-W we find a few core groups that are not directly associated with YSO groups. This finding is consistent with Perseus-W being younger than Serpens. 3. Using the nearest neighbour method we found that the cu- mulative surface density distribution of the Herschel cores and WISE YSOs in the solar neighbourhood have similar profiles. In agreement with Bressert et al. (2010), the cumu- lative density profiles are smooth and imply that there are no fixed scales for clustering (scale-free). 4. The similarity with the cores spatial distribution suggests that the spatial distribution of YSOs is already set during the cloud fragmentation process, and supports the idea that dy- other star forming regions are required to put our findings on solid statistical grounds and to verify if the global densities of This is a result of Spitzer having higher angular resolution ( ∼ 5) than WISE and Herschel and for the YSOs a higher sensitivity limit than WISE. The similarity between the WISE YSOs and the cores implies that the YSOs are likely tracing the substructured and filamentary natal gas that they formed from, like their earlier predecessors, and that, even at the earliest stages, star formation is scale-free. The velocity dispersion of cores and YSOs is typically low (e.g.. v core /lessorsimilar 0 . 4 km s -1 in Ophiuchus and v YSO ∼ 1 . 4 kms -1 , Andr'e et al. 2007; Covey et al. 2006), which is consistent with our results where the Σ YSO and Σ core distributions are similar. If there was significant dynamical evolution in the YSO population, the Σ distribution between the YSOs and cores should di ff er. Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KUL, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); IFSI, OAP / AOT, OAA / CAISMI, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESAPRODEX (Belgium), CEA / CNES (France), DLR (Germany), ASI (Italy), and CICT / MCT (Spain). These criteria successfully confirm the earlier identifications of candidate YMC precursor identified combining ALMA and VLT data in the Antennae Galaxies (Herrera et al. 2012) as well as the candidate precursor G 0 . 25+0 . 02 identified in the HOPS and HIGAL surveys by Longmore et al. (2012a). A few more candidates, identified by Ginsburg et al. (2012), need further study. G 0 . 25 + 0 . 02 in our Galaxy offer a unique opportunity to study in detail the initial conditions for YMC formation. The initial results seem to suggest that this YMC precursor is a highly turbulent cloud (Rathborne et al. 2012). While the properties and internal structure of the cloud are still being investigated, the origin of the cloud is unclear. The cloud is located in the CMZ of our own Galaxy and the special conditions near the Galactic Centre may be responsible for its formation. How- ± and 17 . 28 13 . 85, respectively. Within the uncertainties the The MPC candidates reported in Ginsburg et al. (GS; submitted) are shown as squares. 3.2. Surface densities Bressert et al. (2010) showed that the cumulative surface density of nearly all the YSOs in the solar neighbourhood exhibits a smooth and scale-free profile. In Fig. 3 we similarly investigate how the core population in Perseus-W and Serpens compare to the WISE YSO populations. The cumulative surface density profiles of the YSOs and cores are similar, but reach lower surface densities as compared to the Bressert et al. (2010) profile. the groups are distinct or not. Acknowledgments SPIRE has been developed by a consortium of institutes led by Cardi ff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); be very compact to allow for rapid and continuous star formation even under the effect of strong feedback from forming very massive stars. The final criteria to form a bound YMC with M tot glyph[greaterorequalslant] 10 4 M glyph[circledot] , requires objects to have more than 3 × 10 4 M glyph[circledot] (assuming 30% star formation efficiency), and a radius smaller than the minimum between r Ω , the maximum radius needed to gravitationally bind a fully ionised core, and r vir , the maximum radius that would allow a crossing time of ∼ 1 Myr (the typical observed value for YMCs, Portegies Zwart et al. 2010). 4 ever, it should be noted that YMCs are not only found in the CMZ in our own Galaxy. Further studies of this and other candidates will help clarify their nature.",
"pages": [
2,
3,
4
]
},
{
"title": "4. Star formation and dense gas in the CMZ",
"content": "In the previous section we have discussed a unique massive core in the CMZ of our Galaxy, and we speculated that in the CMZ the conditions may be favorable for the assembly of such extreme objects. In the solar neighborhood, several different arguments suggest that the process of star formation is controlled by the amount of dense molecular gas. Lada et al. (2012) and Krumholz et al. (2012) presented the latest incarnation of the views that the star formation rates observed in star forming regions and entire galaxies could be understood in terms of simple scaling relations with the amount of dense gas above a certain column density threshold (Lada's formulation) or the total local volumetric gas density (Krumholz's formulation). We used data from HOPS and HIGAL Galactic plane surveys as well as some other datasets to show that the relationships between star formation rate and dense gas that appear to hold throughout the Galactic disk do appear to break down in the CMZ (Longmore et al. 2012b). More details on the methodology and results are given by Longmore in this volume, but the basic result is that while ∼ 80% of the dense gas in our own Galaxy is concentrated in the CMZ, there is no similar enhancement in the star formation rate. Similarly, the predictions of the volumetric gas density relations exceed by at least a factor of ten the observed star formation rate. These findings suggest that the prescriptions based on a single parameter (the total gas density or the dense gas column density) break down in the CMZ and a more complicated formulation is needed to reconcile this region with the rest of the Galaxy. The large shear and increased turbulence in the CMZ need to be looked at as the possible causes for the increased stability of cloud cores in that region. These processes may also favor the formation of more massive stable clumps that once they became unstable and collapse may form YMCs.",
"pages": [
4
]
},
{
"title": "References",
"content": "Andr´e, Ph., Menshchikov, A., Bontemps, S., et al. 2010, A&A , 518, L102 Bressert, E., Bastian, N., Gutermuth, R., et al. 2010, MNRAS , 409, L54 Bressert, E., Ginsburg, A., Bally, J., et al. 2012, ApJL , in press Bressert, E., Testi, L., Facchini, A., et al. 2012, A&A , submitted Evans, N. J., Dunham, M. M., Jrgensen, J. K., et al. 2009, ApJS , 181, 321 Ginsburg, A., Bressert, E., Bally, J., Battersby, C., et al. 2012, ApJ , in press Herrera,C.N., Boulanger,F., Nesvadba,N.P.H., Falgarone,E. 2012, A&A , 538, L9 Koenig, X. P., Leisawitz, D. T., Benford, D. J., et al. 2012, ApJ , 744, 130 K¨onyves, V., Andr´e, Ph., Men'shchikov, A., et al. 2010, A&A , 518, L106 Krumholz, M.R., Dekel, A., McKee, C.F. 2012, ApJ , 745, 69 Lada, C.J., Forbrich, J., Lombardi, M., Alves, J.F. 2012, ApJ , 745, 190 Longmore,S.N., Bally, J., Testi, L., et al. 2012, MNRAS , arXiv:1208.4256 Longmore,S.N., Rathborne,J., Bastian,N., et al. 2012, ApJ , 746, 117 Molinari, S., Swinyard, B., Bally, J., et al. 2012, A&A , 518, L100 Motte, F., Andr´e, Ph., & Neri, R. 1998, A&A , 336, 150 Portegies Zwart, S.F., McMillan, S.L.W., & Gieles, M. 2010, ARA&A 48, 431 Rathborne, J.M., Longmore, S.N., Jackson, J.M., et al. 2012, ApJ , submitted Smith, R.J., Longmore, S.N., Bonnell, I. 2009, MNRAS , 400, 1775 Testi, L. & Sargent, A.I. 1998, ApJ , 508, L91 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ , 140, 1868",
"pages": [
4
]
}
]
2013IAUS..294..577T
https://arxiv.org/pdf/1306.3116.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_84><loc_67><loc_90></location>An Peculiar Microwave Quasi-periodic Pulsations with Zigzag Pattern in a CME-related Flare on 2005-01-15</section_header_level_1>
<section_header_level_1><location><page_1><loc_35><loc_81><loc_46><loc_83></location>Baolin Tan 1</section_header_level_1>
<text><location><page_1><loc_11><loc_77><loc_69><loc_80></location>1 Key Laboratory of Solar Activity, National Astronomical Observatories of the Chinese Academy of Sciences, Beijing 100012, China. email: bltan@nao.cas.cn</text>
<section_header_level_1><location><page_1><loc_9><loc_73><loc_16><loc_74></location>Abstract.</section_header_level_1>
<text><location><page_1><loc_9><loc_61><loc_72><loc_73></location>A microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in a solar flare on 200501-15 is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at 1.10-1.34 GHz. The zigzag pulsation occurred just in the early rising phase of the flare with weakly right-handed circular polarization. Its period is only several decades millisecond. Particularly, before and after the pulsation, there are many spectral fine structures, such as zebra patterns, fibers, and millisecond spikes. The microwave Z-QPP can provide some kinematic information of the source region in the early rising phase of the flare, and the source width changes from ∼ 1000 km to 3300 km, even if we have no imaging observations. The abundant spectral fine structures possibly reflect the dynamic features of non-thermal particles.</text>
<text><location><page_1><loc_9><loc_59><loc_47><loc_60></location>Keywords. solar flare, microwave bursts, fine structures</text>
<text><location><page_1><loc_9><loc_50><loc_72><loc_56></location>Solar microwave spectral fine structures are the most important and interesting phenomena, which can provide many intrinsic features of solar eruptions. This work reports first time a peculiar fine structure: a microwave quasi-periodic pulsation (QPP) with zigzag pattern (abbreviated as Z-QPP, hereafter) in a flare on 2005-01-15.</text>
<figure>
<location><page_1><loc_10><loc_26><loc_69><loc_48></location>
<caption>Figure 1. Left panels are the profiles of the M8.6 long-duration flare on 2005-1-15. Soft X-ray intensity observed by GOES (upper), microwave emission of left-handed circular polarization at 1.20 GHz (middle), and the microwave emission of right-handed circular polarization at 1.20 GHz (bottom). Right panels are the spectrogram of microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in the early rising phase of the flare.</caption>
</figure>
<text><location><page_2><loc_9><loc_66><loc_72><loc_94></location>The flare is an M8.6 class long-duration event accompanying with a powerful CME in active region AR10720 with location of N16E04, very close to the center of solar disk. It is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at frequency of 1.10-1.34 GHz (4.0 MHz resolution, 1.25 ms cadence). The left panels of Figure 1 show that the flare lasts from 05:54 UT to 07:17 UT, for about 83 minutes (Cheng et al, 2010). The Z-QPP occurred just in the early rising phase of the flare. The right panels of present the Z-QPP with left- and right-handed circular polarization spectrogram. The Z-QPP is weakly right-handed circular polarization, its period is only several decades of milliseconds, which belongs to very short period pulsation. We may partition the Z-QPP into three paragraphs. The first paragraph starts at 06:04:32 UT, ends at 06:04:43 UT, the global frequency drifting rate is about -16.5 MHz/s, the pulse frequency drifting rate is 2.30-9.60 GHz/s, the period is 40-85 ms with averaged value of 60 ms. The bandwidth increases slowly from 60 MHz to 160 MHz. The second paragraph starts at 06:04:43 UT and ends at 06:04:47 UT, its global frequency drifting rate is about 0 and the pulse frequency drifting rate is 9.60-14.40 GHz/s, the period is 48-106 ms, and the bandwidth 140-200 MHz, few variations. The third paragraph starts at 06:04:47 UT and ends at 06:04:47 UT, its global frequency drifting rate is about -5.6 MHz/s and the pulse frequency drifting rate is 6.7-20 GHz/s, the period is 38-87 ms, and the bandwidth increases slowly from 60 to 140 MHz, and possibly beyond frequency domain (1.34 GHz).</text>
<text><location><page_2><loc_9><loc_55><loc_72><loc_65></location>The frequency distance between the two adjacent strips of the concomitant zebra pattern is about 80 MHz, which implies that the magnetic field in the source region is about 71.5 Gs from DPR model. The weakly polarization and strongly emission intensity imply that plasma emission is most possibly the emission mechanism. The Z-QPP may reflect the variations of the physical conditions and the dynamic processes in source region (Aschwanden & Benz, 1986; Tan, 2008). The velocities of source motion or energetic particles ( v ), and the width of source region (L) can be estimated:</text>
<formula><location><page_2><loc_29><loc_50><loc_72><loc_53></location>v = 2 df fdt H n , L = 2 /triangle f f H n . (0.1)</formula>
<text><location><page_2><loc_9><loc_47><loc_72><loc_50></location>H n is the scale length of plasma density in the background for the source motion and in the source region for the energetic particles.</text>
<text><location><page_2><loc_9><loc_37><loc_72><loc_47></location>The above estimation of the Z-QPP indicates the source region moves in a speed from 275-93 km/s upwards, the source width expands from 1000 km to 3300 km, and the associated speeds of energetic particles is about 0.13 0.53 c , 0.53-0.8 c , and 0.36-0.9 c in the three paragraphs, respectively. Here, c is the light speed. Before and after the Z-QPP, there are many spectral fine structures, such as zebra patterns, fibers and millisecond spikes, etc. Some of them are marked in Figure 1. The abundant spectral fine structures reflect the dynamic features of the non-thermal particles (Huang & Tan, 2012).</text>
<section_header_level_1><location><page_2><loc_9><loc_34><loc_25><loc_35></location>Acknowledgements</section_header_level_1>
<text><location><page_2><loc_9><loc_30><loc_72><loc_33></location>This work is supported by NSFC Grant No. 11273030, 10921303, MOST Grant No. 2011CB811401, and the National Major Scientific Equipment R&D Project ZDYZ2009-3.</text>
<section_header_level_1><location><page_2><loc_9><loc_27><loc_18><loc_28></location>References</section_header_level_1>
<text><location><page_2><loc_9><loc_21><loc_53><loc_26></location>Aschwanden, M.J., & Benz A.O. 1986, Astron Astrophys , 158, 102 Cheng, X., Ding, M.D., & Guo, Y., et al. 2010, ApJ , 716, L68 Huang, J., & Tan, B.L. 2012, ApJ , 745, 186 Qin, Z.H., Li, C.S., & Fu, Q.J., et al 1996, Sol. Phys. , 163, 383</text>
<text><location><page_2><loc_9><loc_19><loc_33><loc_20></location>Tan, B.L. 2008, Sol. Phys. , 253, 117</text>
</document>
[
{
"title": "Baolin Tan 1",
"content": "1 Key Laboratory of Solar Activity, National Astronomical Observatories of the Chinese Academy of Sciences, Beijing 100012, China. email: bltan@nao.cas.cn",
"pages": [
1
]
},
{
"title": "Abstract.",
"content": "A microwave quasi-periodic pulsation with zigzag pattern (Z-QPP) in a solar flare on 200501-15 is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at 1.10-1.34 GHz. The zigzag pulsation occurred just in the early rising phase of the flare with weakly right-handed circular polarization. Its period is only several decades millisecond. Particularly, before and after the pulsation, there are many spectral fine structures, such as zebra patterns, fibers, and millisecond spikes. The microwave Z-QPP can provide some kinematic information of the source region in the early rising phase of the flare, and the source width changes from ∼ 1000 km to 3300 km, even if we have no imaging observations. The abundant spectral fine structures possibly reflect the dynamic features of non-thermal particles. Keywords. solar flare, microwave bursts, fine structures Solar microwave spectral fine structures are the most important and interesting phenomena, which can provide many intrinsic features of solar eruptions. This work reports first time a peculiar fine structure: a microwave quasi-periodic pulsation (QPP) with zigzag pattern (abbreviated as Z-QPP, hereafter) in a flare on 2005-01-15. The flare is an M8.6 class long-duration event accompanying with a powerful CME in active region AR10720 with location of N16E04, very close to the center of solar disk. It is observed by the Chinese Solar Broadband Spectrometer in Huairou (SBRS/Huairou) at frequency of 1.10-1.34 GHz (4.0 MHz resolution, 1.25 ms cadence). The left panels of Figure 1 show that the flare lasts from 05:54 UT to 07:17 UT, for about 83 minutes (Cheng et al, 2010). The Z-QPP occurred just in the early rising phase of the flare. The right panels of present the Z-QPP with left- and right-handed circular polarization spectrogram. The Z-QPP is weakly right-handed circular polarization, its period is only several decades of milliseconds, which belongs to very short period pulsation. We may partition the Z-QPP into three paragraphs. The first paragraph starts at 06:04:32 UT, ends at 06:04:43 UT, the global frequency drifting rate is about -16.5 MHz/s, the pulse frequency drifting rate is 2.30-9.60 GHz/s, the period is 40-85 ms with averaged value of 60 ms. The bandwidth increases slowly from 60 MHz to 160 MHz. The second paragraph starts at 06:04:43 UT and ends at 06:04:47 UT, its global frequency drifting rate is about 0 and the pulse frequency drifting rate is 9.60-14.40 GHz/s, the period is 48-106 ms, and the bandwidth 140-200 MHz, few variations. The third paragraph starts at 06:04:47 UT and ends at 06:04:47 UT, its global frequency drifting rate is about -5.6 MHz/s and the pulse frequency drifting rate is 6.7-20 GHz/s, the period is 38-87 ms, and the bandwidth increases slowly from 60 to 140 MHz, and possibly beyond frequency domain (1.34 GHz). The frequency distance between the two adjacent strips of the concomitant zebra pattern is about 80 MHz, which implies that the magnetic field in the source region is about 71.5 Gs from DPR model. The weakly polarization and strongly emission intensity imply that plasma emission is most possibly the emission mechanism. The Z-QPP may reflect the variations of the physical conditions and the dynamic processes in source region (Aschwanden & Benz, 1986; Tan, 2008). The velocities of source motion or energetic particles ( v ), and the width of source region (L) can be estimated: H n is the scale length of plasma density in the background for the source motion and in the source region for the energetic particles. The above estimation of the Z-QPP indicates the source region moves in a speed from 275-93 km/s upwards, the source width expands from 1000 km to 3300 km, and the associated speeds of energetic particles is about 0.13 0.53 c , 0.53-0.8 c , and 0.36-0.9 c in the three paragraphs, respectively. Here, c is the light speed. Before and after the Z-QPP, there are many spectral fine structures, such as zebra patterns, fibers and millisecond spikes, etc. Some of them are marked in Figure 1. The abundant spectral fine structures reflect the dynamic features of the non-thermal particles (Huang & Tan, 2012).",
"pages": [
1,
2
]
},
{
"title": "Acknowledgements",
"content": "This work is supported by NSFC Grant No. 11273030, 10921303, MOST Grant No. 2011CB811401, and the National Major Scientific Equipment R&D Project ZDYZ2009-3.",
"pages": [
2
]
},
{
"title": "References",
"content": "Aschwanden, M.J., & Benz A.O. 1986, Astron Astrophys , 158, 102 Cheng, X., Ding, M.D., & Guo, Y., et al. 2010, ApJ , 716, L68 Huang, J., & Tan, B.L. 2012, ApJ , 745, 186 Qin, Z.H., Li, C.S., & Fu, Q.J., et al 1996, Sol. Phys. , 163, 383 Tan, B.L. 2008, Sol. Phys. , 253, 117",
"pages": [
2
]
}
]
2013ICRC...33..980E
https://arxiv.org/pdf/1308.0161.pdf
<document>
<figure>
<location><page_1><loc_69><loc_88><loc_89><loc_94></location>
</figure>
<section_header_level_1><location><page_1><loc_10><loc_85><loc_84><loc_86></location>Diffuse TeV Gamma-Ray Emission in the H.E.S.S. Galactic Plane Survey</section_header_level_1>
<text><location><page_1><loc_10><loc_81><loc_90><loc_84></location>K. EGBERTS 1 , F. BRUN 2 , S. CASANOVA 3 , 2 , W. HOFMANN 2 , M. DE NAUROIS 4 , O. REIMER 1 , Q. WEITZEL 2 FOR THE H.E.S.S. COLLABORATION, AND Y. FUKUI 5</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_79><loc_75><loc_81></location>1 Institut fur Astro- und Teilchenphysik, Leopold-Franzens-Universitat Innsbruck, A-6020 Innsbruck, Austria</list_item>
<list_item><location><page_1><loc_10><loc_78><loc_62><loc_79></location>2 Max-Planck-Institut fur Kernphysik, P.O. Box 103980, D 69029 Heidelberg, Germany</list_item>
<list_item><location><page_1><loc_10><loc_77><loc_60><loc_78></location>3 Unit for Space Physics, North-West University, Potchefstroom 2520, South Africa</list_item>
<list_item><location><page_1><loc_9><loc_76><loc_68><loc_77></location>4 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France</list_item>
<list_item><location><page_1><loc_10><loc_74><loc_63><loc_76></location>5 Department of Astrophysics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_72><loc_28><loc_73></location>kathrin.egberts@uibk.ac.at</text>
<text><location><page_1><loc_15><loc_52><loc_91><loc_71></location>Abstract: Diffuse γ -ray emission has long been established as the most prominent feature in the GeV sky. Although the imaging atmospheric Cherenkov technique has been successful in revealing a large population of discrete TeV γ -ray sources, a thorough investigation of diffuse emission at TeV energies is still pending. Data from the Galactic Plane Survey (GPS) obtained by the High Energy Stereoscopic System (H.E.S.S.) have now achieved a sensitivity and coverage adequate for probing signatures of diffuse emission in the energy range of ∼ 100 GeV to a few TeV. γ -rays are produced in cosmic-ray interactions with the interstellar medium (aka 'sea of cosmic rays') and in inverse Compton scattering on cosmic photon fields. This inevitably leads to guaranteed γ -ray emission related to the gas content along the line-of-sight. Further contributions relate to those γ -ray sources that fall below the current detection threshold and the aforementioned inverse Compton emission. Based on the H.E.S.S. GPS, we present the first observational assessment of diffuse TeV γ -ray emission. The observation is compared with corresponding flux predictions based on the HI (LAB data) and CO (as a tracer of H2, NANTEN data) gas distributions. Consequences for unresolved source contributions and the anticipated level of inverse Compton emission are discussed.</text>
<text><location><page_1><loc_16><loc_49><loc_69><loc_51></location>Keywords: diffuse γ -ray emission, imaging atmospheric Cherenkov telescopes</text>
<section_header_level_1><location><page_1><loc_10><loc_45><loc_23><loc_46></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_5><loc_49><loc_45></location>The Galactic diffuse γ -ray emission, observed by several space-born γ -ray experiments like SAS-2 [1], COS-B [2], EGRET[3], and Fermi-LAT [4], is the most prominent feature of the sky at GeV energies. The processes leading to this emission are cosmic-ray interactions: bremsstrahlung and π 0 production in the interstellar medium and inverse Compton scattering on radiation fields. The contributions differ in their relative amplitude on different energy scales and vary in their spatial extension depending on the interaction targets as the matter density and the respective radiation fields. When moving to higher energies, discrete γ -ray sources become more prominent compared to cosmicray induced processes (at TeV energies, inverse Compton and π 0 decay) and dominate the celestial γ -ray emission. The TeV energy regime is the realm of ground-based observations. Diffuse γ -ray emission has been reported by the Milagro experiment [5] at a median energy of 15 TeV, and by ARGO-YBJ [6]. Common to both experiments is that while providing a very good duty cycle and a large field of view, they suffer from a rather poor angular resolution, which challenges the identification of discrete γ -ray sources. With a lower energy threshold and arc-minute angular resolution, imaging atmospheric Cherenkov telescopes have the potential to improve on these measurements at energies starting at ∼ 100 GeV and thereby improve our understanding of the Galactic diffuse γ -ray emission and its underlying production mechanisms. The most suited experiment for a measurement of diffuse emission in the Galactic Plane is the High Energy Stereoscopic System (H.E.S.S.): with its location in Namibia it provides</text>
<text><location><page_1><loc_51><loc_31><loc_91><loc_46></location>ideal viewing conditions on the central part of the Galactic Plane and features a comparatively large field of view for imaging atmospheric Cherenkov telescopes of 5 · in diameter. The H.E.S.S. experiment has performed a survey of the Galactic Plane and has from 2004 to 2013 accumulated about 2800 observation hours of good-quality data. Although the H.E.S.S. Galactic Plane Survey (GPS) has revealed a wealth of new sources [7], and already resolved diffuse emission in the Galactic Center region on sub-degree scales [8], a measurement of the large-scale diffuse γ -ray emission is at the edge of current instrumental sensitivity and a challenge for analysis methodology.</text>
<section_header_level_1><location><page_1><loc_52><loc_27><loc_66><loc_28></location>2 Methodology</section_header_level_1>
<section_header_level_1><location><page_1><loc_52><loc_25><loc_75><loc_26></location>2.1 Background subtraction</section_header_level_1>
<text><location><page_1><loc_51><loc_5><loc_91><loc_25></location>A measurement of diffuse γ -ray emission is challenging for imaging atmospheric Cherenkov telescopes because of their restricted capabilities of γ -hadron separation and the limited field of view of a few degrees. The limitations in the γ -hadron separation require a subtraction of the background. In order to minimize systematic effects due to changing atmospheric or instrument conditions, the background level is determined from measurement, usually within the field of view, in regions of no known γ -ray sources. This has proven to be the most reliable and failsafe background-subtraction technique [9] and is therefore used throughout the community. However, this method places severe constraints on the size of emission that can be probed. Furthermore, for emission larger than the field of view, the procedure results necessarily in a subtraction</text>
<figure>
<location><page_2><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<figure>
<location><page_2><loc_10><loc_79><loc_90><loc_90></location>
<caption>Fig. 1 : The region in Galactic longitude and latitude used for the measurement of diffuse γ -ray emission. Red denotes exclusion from the analysis because of the presence of significant γ -ray sources, white is in the following denoted 'diffuse analysis region'.</caption>
</figure>
<text><location><page_2><loc_10><loc_66><loc_49><loc_71></location>of part of the signal together with the background. Therefore, special care needs to be taken in the treatment of the background and the interpretation of signals obtained after background subtraction.</text>
<text><location><page_2><loc_10><loc_55><loc_49><loc_65></location>For the current analysis a dataset of 1926 hours of deadtime corrected observations are used, covering a region of -75 · < l < 60 · in longitude and -2 · < b < 2 · in latitude. The data were analysed using the Model analysis technique [10] with standard cuts for event reconstruction and background reduction. The remaining background was measured in regions that do not meet any of the following exclusion criteria:</text>
<unordered_list>
<list_item><location><page_2><loc_12><loc_42><loc_49><loc_54></location>1. Any region is excluded that contains a γ -ray signal with significance of > 4 σ in the chosen analysis bin and a significance in one neighbouring bin of > 4 . 5 σ or lies within 0 . 2 · of such a signal (in order to include also tails in the point spread function used to describe γ -ray sources). Significances are considered for standard and hard cuts [10] and regions are determined iteratively. The resulting regions are visualized in Fig. 1.</list_item>
<list_item><location><page_2><loc_12><loc_36><loc_49><loc_40></location>2. Galactic longitudes within a latitude range of | b | < 1 . 2 · are excluded in order to study a diffuse signal close to the Galactic Plane.</list_item>
</unordered_list>
<text><location><page_2><loc_9><loc_17><loc_49><loc_34></location>The first criterion is a very conservative approach assuring a minimum of γ -ray-source contaminations. The choice of the latitude range in the second criterion is a compromise between a desired large excluded region in order to avoid a contamination of the background estimate on the one hand and the need for statistics and reduction of systematics in the background measurement on the other hand. It is further motivated by the scale height in the distribution of interstellar gas, which is expected to correlate to a certain level with p-p interaction induced γ -ray emission. An adaptive ring background subtraction method has been chosen [7] for an optimal treatment of these constraints in the choice of background regions.</text>
<text><location><page_2><loc_10><loc_9><loc_49><loc_17></location>It is worth noting here that a subtraction of the background measured at regions with | b | > 1 . 2 · also means a subtraction of any large-scale diffuse signal that extends signifcantly beyond | b | > 1 . 2 · . Therefore, any signal measured at | b | < 1 . 2 · is a signal with respect to the baseline of a potential flux outside the region.</text>
<text><location><page_2><loc_9><loc_5><loc_49><loc_9></location>After background subtraction the γ -ray excess events are folded with the exposure to obtain a 2-dimensional representation of flux in the Galactic Plane [7].</text>
<section_header_level_1><location><page_2><loc_52><loc_69><loc_73><loc_71></location>2.2 Discrete γ -ray sources</section_header_level_1>
<text><location><page_2><loc_51><loc_52><loc_91><loc_69></location>As the flux in the Galactic Plane is completely dominated by many, mostly extended γ -ray sources, a measurement of diffuse γ -ray emission needs to exclude these sources from the analysis. Due to their spatial extension and the limitations in properly characterizing their flux distributions, a modelling of the sources turns out to be presently not feasible. Therefore, source locations are identified and excluded from the analysis by applying a cut in the observed detection significance. The same criterion (1) as for the choice of the background regions is applied. The remaining regions in the sky are denoted as 'diffuse analysis region' in the following. Results are stable with respect to the details in the choice of the analysis region.</text>
<text><location><page_2><loc_51><loc_46><loc_91><loc_52></location>The procedure of spatially excluding any significant signal results by definition in fluxes (as function of position) that are individually not significant. However, by investigating profiles of the flux distribution, the cummulative projected signal results in a notable flux excess.</text>
<section_header_level_1><location><page_2><loc_52><loc_42><loc_75><loc_44></location>3 The diffuse γ -ray signal</section_header_level_1>
<text><location><page_2><loc_51><loc_35><loc_91><loc_41></location>The latitude flux profile of the Galactic Plane for a longitude range of -75 · < l < 60 · is shown in Fig. 2 for both, fluxes including γ -ray sources and fluxes of the diffuse analysis region only. Errors are 1 σ statistical errors and do not account for systematics.</text>
<text><location><page_2><loc_51><loc_23><loc_91><loc_35></location>Both distributions are characterized by a clear excess in the proximity of the Galactic Plane. Notebly, also the excess of the diffuse analysis region is significant. It peaks not exactly at b = 0 · but slightly offset at around b = -0 . 25 · with a peak value of around 2 . 5 × 10 -9 cm -2 s -1 TeV -1 sr -1 . The signal accumulates over the considered longitude range and consists of larger contributions from longitude values of the Galactic centre region and smaller ones from the outskirts of the observed region.</text>
<text><location><page_2><loc_51><loc_19><loc_91><loc_22></location>The observed diffuse emission can be interpreted as a combination of contributions from</text>
<unordered_list>
<list_item><location><page_2><loc_54><loc_17><loc_72><loc_19></location>· unresolved γ -ray sources</list_item>
<list_item><location><page_2><loc_54><loc_13><loc_91><loc_16></location>· γ -rays resulting from cosmic-ray interactions with the interstellar medium ( π 0 decay)</list_item>
<list_item><location><page_2><loc_54><loc_10><loc_91><loc_13></location>· γ -rays resulting from cosmic-ray interactions with radiation fields (inverse Compton scattering).</list_item>
</unordered_list>
<text><location><page_2><loc_51><loc_5><loc_91><loc_9></location>The estimation of the individual contributions to the observed signal is non-trivial and further complicated by the relatively small signal. A large part of the signal can be ex-</text>
<text><location><page_3><loc_9><loc_85><loc_49><loc_90></location>pected to stem from faint sources that are unresolved because of their low fluxes (avoiding significant detection) or very large extension not properly handled in the standard analysis.</text>
<text><location><page_3><loc_9><loc_81><loc_49><loc_85></location>A minimum level of cosmic-ray induced contribution can be estimated from p-p interactions with interstellar matter. The γ -ray flux can be calculated following [11]</text>
<formula><location><page_3><loc_11><loc_77><loc_49><loc_80></location>dN γ dAdE γ dtd Ω = ∫ dl d ∫ d σ p -→ γ dEp n ( l , b , l d ) J ( Ep ) dEp (1)</formula>
<text><location><page_3><loc_9><loc_43><loc_49><loc_76></location>with l d being the line-of-sight, Ep the proton energy, J ( Ep ) the cosmic-ray spectrum, d σ p -→ γ dEp the interaction cross section for the γ -ray producing interaction, and n ( l , b , l d ) being the column density of the gas of the interstellar medium. The cosmic-ray spectrum used for the calculation is the one measured locally at Earth, taken from [12]. The interaction cross section is a parametrization of the SIBYLL interaction code following [13]. The interstellar matter that constitutes the target material are HI and H2. HI data are measurements from the Leiden/Argentine/Bonn survey (LAB [14]) assuming a spin temperature of T = 125 K, the H2 column density is obtained using NANTEN CO data [15] as tracer of H2. A conversion factor of XCO = 2 × 10 20 cm -2 K -1 km -1 s [16] has been used to convert the velocity-integrated NANTEN data to H2 column density. The results of these calculations can be seen in the model curves of Fig. 2. In order to assure comparability, the same regions in the sky have been used for the calculation of the expected γ -ray signal and the analysis, i.e. positions of γ -ray sources have been excluded from the model calculation for the diffuse analysis region as well. Note that due to the poor angular resolution of the HI data sources are excluded on scales smaller than the HI bin size, which is justified only by the apparent lack of correlation between γ -ray sources and HI densities.</text>
<text><location><page_3><loc_10><loc_31><loc_49><loc_43></location>The calculated γ -ray emission from p-p interactions has to be treated as a minimal or guaranteed level of the anticipated diffuse emission signal. The calculation uses the locally measured cosmic-ray spectrum, while the flux is expected to vary throughout the Galaxy. The flux observed at Earth, in no immediate proximity of any cosmic-ray accelerator is assumed to be the minimum level of cosmic rays in the Galaxy ('sea of cosmic rays'), while close to accelerators the level can be significantly enhanced.</text>
<text><location><page_3><loc_10><loc_23><loc_49><loc_31></location>Further simplifying assumptions that are invoked (most of them reducing the estimated contribution) include a constant XCO (as opposed to some functional dependence on Galactocentric distance), the limitation to hydrogen contributions and usage of p-p cross section also for heavier cosmic rays.</text>
<text><location><page_3><loc_9><loc_5><loc_49><loc_23></location>As the contribution of neutral-pion-decay γ -rays is neatly localized along the Galactic Plane, a comparison of the contribution can be made with the H.E.S.S. data in a first approximation without considering the issue of background subtraction and the corresponding reduction of the signal. In the region of | b | < 1 · in the diffuse analysis region the integrated contribution of the calculated γ -ray signal originating from π 0 decay is ∼ 25% of the H.E.S.S. measurement (as seen in Fig. 2 bottom panel), thereby limiting the contribution of unresolved sources (also expected to concentrate close to the Plane) to less than 75%. In comparison, the calculated contribution from neutral-piondecay γ -rays to the total flux including γ -ray sources is for the region | b | < 1 · less than 10% (as in Fig. 2 top panel).</text>
<figure>
<location><page_3><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<figure>
<location><page_3><loc_54><loc_78><loc_85><loc_90></location>
</figure>
<figure>
<location><page_3><loc_54><loc_66><loc_85><loc_77></location>
<caption>Fig. 2 : The latitude profile of γ -ray flux (shown is the differential flux at an energy of 1 TeV), covering a longitude range of -75 · < l < 60 · , for the total flux including γ -ray sources (top panel) and for the diffuse analysis region only as defined in Fig. 1 (bottom panel). H.E.S.S. measurements (black data points) are compared with the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 (solid red line, the individual components are blue dotted for H2 and green dot-dashed for HI interactions).</caption>
</figure>
<text><location><page_3><loc_51><loc_25><loc_91><loc_47></location>The second cosmic-ray interaction component is inverse Compton scattering. Cosmic-ray electrons upscatter photons of optical starlight, infrared dust emission and the cosmic microwave background. The cosmic-ray electrons that are responsible for the inverse Compton emission at TeV energies have, depending on the radiation field they are interacting with, energies between a few and a few hundred TeV. At these energies, energy losses are severe and their lifetimes and propagation distances are limited. Therefore, the electrons are found close to their production sites, which are either homogeneously distributed in case of secondary production of electrons via hadronic cosmicray interactions, or highly inhomogeneously localized as for primary cosmic-ray electrons [17]. This makes an estimation of a contribution to the diffuse γ -ray emission at TeV energies challenging and dependent on source model assumptions.</text>
<text><location><page_3><loc_52><loc_17><loc_91><loc_25></location>Since the radiation fields as target for inverse Compton scattering extend to higher latitudes than the bulk of the gas distribution, the inverse Compton component to be contained in the diffuse γ -ray signal will be subject to background subtraction and only gradients will be measurable in the observed signal.</text>
<section_header_level_1><location><page_3><loc_52><loc_13><loc_64><loc_15></location>4 Conclusion</section_header_level_1>
<text><location><page_3><loc_51><loc_5><loc_91><loc_13></location>We present the first ever investigation of large-scale diffuse emission with imaging atmospheric Cherenkov telescopes. Imaging atmospheric Cherenkov telescopes have the advantage of a rather precise direction reconstruction, which puts them into advantage compared to other instruments that measure at TeV energies in the identification</text>
<text><location><page_4><loc_9><loc_76><loc_49><loc_90></location>and exclusion of γ -ray sources as dominant 'background' of a diffuse γ -ray analysis. The H.E.S.S. data reveal an excess around a latitude of about b = -0 . 25 · over the baseline at latitude extensions larger 1 . 2 · in the Galactic Plane. This signal exceeds the guaranteed contribution stemming from the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 via π 0 decay. Additional contributions to the observed signal relate to unresolved γ -ray sources and inverse Compton scattering of cosmic-ray electrons on interstellar radiation fields.</text>
<text><location><page_4><loc_10><loc_54><loc_49><loc_73></location>The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.</text>
<section_header_level_1><location><page_4><loc_10><loc_50><loc_19><loc_51></location>References</section_header_level_1>
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</figure>
</document>
[
{
"title": "Diffuse TeV Gamma-Ray Emission in the H.E.S.S. Galactic Plane Survey",
"content": "K. EGBERTS 1 , F. BRUN 2 , S. CASANOVA 3 , 2 , W. HOFMANN 2 , M. DE NAUROIS 4 , O. REIMER 1 , Q. WEITZEL 2 FOR THE H.E.S.S. COLLABORATION, AND Y. FUKUI 5 kathrin.egberts@uibk.ac.at Abstract: Diffuse γ -ray emission has long been established as the most prominent feature in the GeV sky. Although the imaging atmospheric Cherenkov technique has been successful in revealing a large population of discrete TeV γ -ray sources, a thorough investigation of diffuse emission at TeV energies is still pending. Data from the Galactic Plane Survey (GPS) obtained by the High Energy Stereoscopic System (H.E.S.S.) have now achieved a sensitivity and coverage adequate for probing signatures of diffuse emission in the energy range of ∼ 100 GeV to a few TeV. γ -rays are produced in cosmic-ray interactions with the interstellar medium (aka 'sea of cosmic rays') and in inverse Compton scattering on cosmic photon fields. This inevitably leads to guaranteed γ -ray emission related to the gas content along the line-of-sight. Further contributions relate to those γ -ray sources that fall below the current detection threshold and the aforementioned inverse Compton emission. Based on the H.E.S.S. GPS, we present the first observational assessment of diffuse TeV γ -ray emission. The observation is compared with corresponding flux predictions based on the HI (LAB data) and CO (as a tracer of H2, NANTEN data) gas distributions. Consequences for unresolved source contributions and the anticipated level of inverse Compton emission are discussed. Keywords: diffuse γ -ray emission, imaging atmospheric Cherenkov telescopes",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "The Galactic diffuse γ -ray emission, observed by several space-born γ -ray experiments like SAS-2 [1], COS-B [2], EGRET[3], and Fermi-LAT [4], is the most prominent feature of the sky at GeV energies. The processes leading to this emission are cosmic-ray interactions: bremsstrahlung and π 0 production in the interstellar medium and inverse Compton scattering on radiation fields. The contributions differ in their relative amplitude on different energy scales and vary in their spatial extension depending on the interaction targets as the matter density and the respective radiation fields. When moving to higher energies, discrete γ -ray sources become more prominent compared to cosmicray induced processes (at TeV energies, inverse Compton and π 0 decay) and dominate the celestial γ -ray emission. The TeV energy regime is the realm of ground-based observations. Diffuse γ -ray emission has been reported by the Milagro experiment [5] at a median energy of 15 TeV, and by ARGO-YBJ [6]. Common to both experiments is that while providing a very good duty cycle and a large field of view, they suffer from a rather poor angular resolution, which challenges the identification of discrete γ -ray sources. With a lower energy threshold and arc-minute angular resolution, imaging atmospheric Cherenkov telescopes have the potential to improve on these measurements at energies starting at ∼ 100 GeV and thereby improve our understanding of the Galactic diffuse γ -ray emission and its underlying production mechanisms. The most suited experiment for a measurement of diffuse emission in the Galactic Plane is the High Energy Stereoscopic System (H.E.S.S.): with its location in Namibia it provides ideal viewing conditions on the central part of the Galactic Plane and features a comparatively large field of view for imaging atmospheric Cherenkov telescopes of 5 · in diameter. The H.E.S.S. experiment has performed a survey of the Galactic Plane and has from 2004 to 2013 accumulated about 2800 observation hours of good-quality data. Although the H.E.S.S. Galactic Plane Survey (GPS) has revealed a wealth of new sources [7], and already resolved diffuse emission in the Galactic Center region on sub-degree scales [8], a measurement of the large-scale diffuse γ -ray emission is at the edge of current instrumental sensitivity and a challenge for analysis methodology.",
"pages": [
1
]
},
{
"title": "2.1 Background subtraction",
"content": "A measurement of diffuse γ -ray emission is challenging for imaging atmospheric Cherenkov telescopes because of their restricted capabilities of γ -hadron separation and the limited field of view of a few degrees. The limitations in the γ -hadron separation require a subtraction of the background. In order to minimize systematic effects due to changing atmospheric or instrument conditions, the background level is determined from measurement, usually within the field of view, in regions of no known γ -ray sources. This has proven to be the most reliable and failsafe background-subtraction technique [9] and is therefore used throughout the community. However, this method places severe constraints on the size of emission that can be probed. Furthermore, for emission larger than the field of view, the procedure results necessarily in a subtraction of part of the signal together with the background. Therefore, special care needs to be taken in the treatment of the background and the interpretation of signals obtained after background subtraction. For the current analysis a dataset of 1926 hours of deadtime corrected observations are used, covering a region of -75 · < l < 60 · in longitude and -2 · < b < 2 · in latitude. The data were analysed using the Model analysis technique [10] with standard cuts for event reconstruction and background reduction. The remaining background was measured in regions that do not meet any of the following exclusion criteria: The first criterion is a very conservative approach assuring a minimum of γ -ray-source contaminations. The choice of the latitude range in the second criterion is a compromise between a desired large excluded region in order to avoid a contamination of the background estimate on the one hand and the need for statistics and reduction of systematics in the background measurement on the other hand. It is further motivated by the scale height in the distribution of interstellar gas, which is expected to correlate to a certain level with p-p interaction induced γ -ray emission. An adaptive ring background subtraction method has been chosen [7] for an optimal treatment of these constraints in the choice of background regions. It is worth noting here that a subtraction of the background measured at regions with | b | > 1 . 2 · also means a subtraction of any large-scale diffuse signal that extends signifcantly beyond | b | > 1 . 2 · . Therefore, any signal measured at | b | < 1 . 2 · is a signal with respect to the baseline of a potential flux outside the region. After background subtraction the γ -ray excess events are folded with the exposure to obtain a 2-dimensional representation of flux in the Galactic Plane [7].",
"pages": [
1,
2
]
},
{
"title": "2.2 Discrete γ -ray sources",
"content": "As the flux in the Galactic Plane is completely dominated by many, mostly extended γ -ray sources, a measurement of diffuse γ -ray emission needs to exclude these sources from the analysis. Due to their spatial extension and the limitations in properly characterizing their flux distributions, a modelling of the sources turns out to be presently not feasible. Therefore, source locations are identified and excluded from the analysis by applying a cut in the observed detection significance. The same criterion (1) as for the choice of the background regions is applied. The remaining regions in the sky are denoted as 'diffuse analysis region' in the following. Results are stable with respect to the details in the choice of the analysis region. The procedure of spatially excluding any significant signal results by definition in fluxes (as function of position) that are individually not significant. However, by investigating profiles of the flux distribution, the cummulative projected signal results in a notable flux excess.",
"pages": [
2
]
},
{
"title": "3 The diffuse γ -ray signal",
"content": "The latitude flux profile of the Galactic Plane for a longitude range of -75 · < l < 60 · is shown in Fig. 2 for both, fluxes including γ -ray sources and fluxes of the diffuse analysis region only. Errors are 1 σ statistical errors and do not account for systematics. Both distributions are characterized by a clear excess in the proximity of the Galactic Plane. Notebly, also the excess of the diffuse analysis region is significant. It peaks not exactly at b = 0 · but slightly offset at around b = -0 . 25 · with a peak value of around 2 . 5 × 10 -9 cm -2 s -1 TeV -1 sr -1 . The signal accumulates over the considered longitude range and consists of larger contributions from longitude values of the Galactic centre region and smaller ones from the outskirts of the observed region. The observed diffuse emission can be interpreted as a combination of contributions from The estimation of the individual contributions to the observed signal is non-trivial and further complicated by the relatively small signal. A large part of the signal can be ex- pected to stem from faint sources that are unresolved because of their low fluxes (avoiding significant detection) or very large extension not properly handled in the standard analysis. A minimum level of cosmic-ray induced contribution can be estimated from p-p interactions with interstellar matter. The γ -ray flux can be calculated following [11] with l d being the line-of-sight, Ep the proton energy, J ( Ep ) the cosmic-ray spectrum, d σ p -→ γ dEp the interaction cross section for the γ -ray producing interaction, and n ( l , b , l d ) being the column density of the gas of the interstellar medium. The cosmic-ray spectrum used for the calculation is the one measured locally at Earth, taken from [12]. The interaction cross section is a parametrization of the SIBYLL interaction code following [13]. The interstellar matter that constitutes the target material are HI and H2. HI data are measurements from the Leiden/Argentine/Bonn survey (LAB [14]) assuming a spin temperature of T = 125 K, the H2 column density is obtained using NANTEN CO data [15] as tracer of H2. A conversion factor of XCO = 2 × 10 20 cm -2 K -1 km -1 s [16] has been used to convert the velocity-integrated NANTEN data to H2 column density. The results of these calculations can be seen in the model curves of Fig. 2. In order to assure comparability, the same regions in the sky have been used for the calculation of the expected γ -ray signal and the analysis, i.e. positions of γ -ray sources have been excluded from the model calculation for the diffuse analysis region as well. Note that due to the poor angular resolution of the HI data sources are excluded on scales smaller than the HI bin size, which is justified only by the apparent lack of correlation between γ -ray sources and HI densities. The calculated γ -ray emission from p-p interactions has to be treated as a minimal or guaranteed level of the anticipated diffuse emission signal. The calculation uses the locally measured cosmic-ray spectrum, while the flux is expected to vary throughout the Galaxy. The flux observed at Earth, in no immediate proximity of any cosmic-ray accelerator is assumed to be the minimum level of cosmic rays in the Galaxy ('sea of cosmic rays'), while close to accelerators the level can be significantly enhanced. Further simplifying assumptions that are invoked (most of them reducing the estimated contribution) include a constant XCO (as opposed to some functional dependence on Galactocentric distance), the limitation to hydrogen contributions and usage of p-p cross section also for heavier cosmic rays. As the contribution of neutral-pion-decay γ -rays is neatly localized along the Galactic Plane, a comparison of the contribution can be made with the H.E.S.S. data in a first approximation without considering the issue of background subtraction and the corresponding reduction of the signal. In the region of | b | < 1 · in the diffuse analysis region the integrated contribution of the calculated γ -ray signal originating from π 0 decay is ∼ 25% of the H.E.S.S. measurement (as seen in Fig. 2 bottom panel), thereby limiting the contribution of unresolved sources (also expected to concentrate close to the Plane) to less than 75%. In comparison, the calculated contribution from neutral-piondecay γ -rays to the total flux including γ -ray sources is for the region | b | < 1 · less than 10% (as in Fig. 2 top panel). The second cosmic-ray interaction component is inverse Compton scattering. Cosmic-ray electrons upscatter photons of optical starlight, infrared dust emission and the cosmic microwave background. The cosmic-ray electrons that are responsible for the inverse Compton emission at TeV energies have, depending on the radiation field they are interacting with, energies between a few and a few hundred TeV. At these energies, energy losses are severe and their lifetimes and propagation distances are limited. Therefore, the electrons are found close to their production sites, which are either homogeneously distributed in case of secondary production of electrons via hadronic cosmicray interactions, or highly inhomogeneously localized as for primary cosmic-ray electrons [17]. This makes an estimation of a contribution to the diffuse γ -ray emission at TeV energies challenging and dependent on source model assumptions. Since the radiation fields as target for inverse Compton scattering extend to higher latitudes than the bulk of the gas distribution, the inverse Compton component to be contained in the diffuse γ -ray signal will be subject to background subtraction and only gradients will be measurable in the observed signal.",
"pages": [
2,
3
]
},
{
"title": "4 Conclusion",
"content": "We present the first ever investigation of large-scale diffuse emission with imaging atmospheric Cherenkov telescopes. Imaging atmospheric Cherenkov telescopes have the advantage of a rather precise direction reconstruction, which puts them into advantage compared to other instruments that measure at TeV energies in the identification and exclusion of γ -ray sources as dominant 'background' of a diffuse γ -ray analysis. The H.E.S.S. data reveal an excess around a latitude of about b = -0 . 25 · over the baseline at latitude extensions larger 1 . 2 · in the Galactic Plane. This signal exceeds the guaranteed contribution stemming from the calculated level of γ -ray emission due to p-p interaction of the locally measured cosmic-ray spectrum with HI and H2 via π 0 decay. Additional contributions to the observed signal relate to unresolved γ -ray sources and inverse Compton scattering of cosmic-ray electrons on interstellar radiation fields. The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.",
"pages": [
3,
4
]
}
]
2013ICRC...33.1032L
https://arxiv.org/pdf/1308.0475.pdf
<document>
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<section_header_level_1><location><page_1><loc_10><loc_83><loc_91><loc_86></location>Discovery of the mysterious gamma-ray source HESSJ1832-093 in the vicinity of SNR G22.7-0.2</section_header_level_1>
<text><location><page_1><loc_10><loc_80><loc_82><loc_82></location>H. LAFFON 1 , 2 , F. ACERO 3 , F. BRUN 4 , B. KH'ELIFI 2 , G. P UHLHOFER 5 , R. TERRIER 6 , FOR THE H.E.S.S. COLLABORATION.</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_78><loc_57><loc_79></location>1 Centre d'Etudes Nucl'eaires de Bordeaux-Gradignan, CNRS/IN2P3, Universit'e Bordeaux 1, France</list_item>
<list_item><location><page_1><loc_10><loc_77><loc_50><loc_78></location>2 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France</list_item>
<list_item><location><page_1><loc_10><loc_76><loc_35><loc_77></location>3 NASA Goddard Space Flight Center, Greenbelt, USA</list_item>
<list_item><location><page_1><loc_9><loc_75><loc_38><loc_76></location>4 Max-Planck-Institut fur Kernphysik, Heidelberg, Germany</list_item>
<list_item><location><page_1><loc_10><loc_74><loc_44><loc_75></location>5 Institut fur Astronomie und Astrophysik, Universitat Tubingen, Germany</list_item>
<list_item><location><page_1><loc_10><loc_73><loc_47><loc_74></location>6 Astroparticules et Cosmologie, CNRS/IN2P3, Universit'e Paris 7, Paris, France</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_71><loc_25><loc_72></location>laffon@cenbg.in2p3.fr</text>
<text><location><page_1><loc_15><loc_56><loc_91><loc_69></location>Abstract: Thanks to the use of advanced analysis techniques, the H.E.S.S. imaging Cherenkov telescope array has now reached a sensitivity level allowing the detection of sources with fluxes around 1% of the Crab with a limited observation time (less than ∼ 100 hours). 67 hours of observations in the region of the supernova remnant G22.7-0.2 thus yielded the detection of the faint point-like source HESS J1832-093 in spatial coincidence with a part of the radio shell of the supernova remnant. A multi-wavelength search for counterparts was then performed and led to the elaboration of various scenarii in order to explain the origin of the very high energy excess. The presence of molecular clouds in the line of sight could indicate a hadronic origin through π 0 production and decay. However, the discovery of an X-ray point source and its potential infrared counterpart give rise to other possibilities such as a pulsar wind nebula nature or a binary system. The latest results on this source are presented as well as the different scenarii brought by the multi-wavelength observations.</text>
<text><location><page_1><loc_16><loc_53><loc_23><loc_54></location>Keywords:</text>
<text><location><page_1><loc_24><loc_53><loc_83><loc_54></location>HESS J1832-093, SNR G22.7-0.2, molecular clouds, pulsar wind nebula, binary system</text>
<section_header_level_1><location><page_1><loc_10><loc_48><loc_23><loc_50></location>1 Introduction</section_header_level_1>
<section_header_level_1><location><page_1><loc_52><loc_48><loc_73><loc_50></location>2 H.E.S.S. observations</section_header_level_1>
<text><location><page_1><loc_10><loc_36><loc_49><loc_48></location>H.E.S.S. (High Energy Stereoscopic System) is an array of four imaging atmospheric Cherenkov telescopes located 1800 m above sea level in the Khomas Highland of Namibia and has been fully operational since 2004 [1]. The H.E.S.S. collaboration has been conducting a systematic scan of the Galactic Plane, which led to the discovery of a rich population of very high energy (VHE) gamma-ray sources such as in the region around the supernova remnant (SNR) W41.</text>
<text><location><page_1><loc_9><loc_5><loc_49><loc_36></location>As a consequence, a dedicated observation campaign was launched by H.E.S.S. to study those TeV sources in detail. These observations led to the discovery of HESS J1832 -093 in spatial coincidence with the edge of SNR G22.7 -0.2. The latter shows a non-thermal shell of 26 ' diameter in radio [2] and partially overlaps the neighbouring remnant W41. A search for multiwavelength counterparts was performed in order to identify the nature of HESS J1832 -093. A proposal for observations in X-rays at the position of HESS J1832 -093 was submitted to the XMMNewton satellite and allowed the discovery of a pointlike source in X-rays near the center of the TeV source. Apart from the radio emission of the SNR, no radio pointlike counterpart was found but measurements of the 13 CO (J=1 → 0) transition line show the presence of molecular structures on the line of sight. However, an infrared pointlike source lying less than 2 '' away from the center of the Xray source was found in the 2MASS catalog. The features of the TeV emission discovered by H.E.S.S. are described in section 2. Details on the X-ray, IR and radio counterparts are given in section 3 and different considered scenarios to explain the VHE gamma-ray emission are presented in section 4.</text>
<section_header_level_1><location><page_1><loc_52><loc_47><loc_76><loc_48></location>2.1 Detection and morphology</section_header_level_1>
<text><location><page_1><loc_51><loc_40><loc_91><loc_46></location>A standard run selection procedure is used to remove bad quality observations in order to study the field of view. This results in a data set comprising 67 hours live time of observations taken from 2004 to 2011.</text>
<text><location><page_1><loc_52><loc_26><loc_91><loc_40></location>A standard H.E.S.S. analysis frame is adopted with the Hillas reconstruction method using the weighted intersections of the main axis of the images to reconstruct the direction of the events [1]. A recently developped multi-variate analysis is used to provide a better discrimination between hadrons and gamma-rays [3]. A minimum charge of 110 photo-electrons in the shower images is applied to the data, resulting in an energy threshold of about 450 GeV. The background is estimated with the ring-background model, as described in [1].</text>
<text><location><page_1><loc_51><loc_5><loc_91><loc_25></location>Using these techniques, HESS J1832 -093 is detected with a peak significance of 7 . 9 σ pre-trials, resulting in a post-trial detection significance of 5 . 6 σ , following the approach described in [4]. The corresponding excess map of the field of view showing the new detected source is presented in Fig. 1. The average angular resolution (r68) for this selected data set and assuming a power-law spectral index of 2.3 is 0 . 081 · at the source position. A twodimensional symmetrical Gaussian function is used to determine the position and size of the TeV emission with a χ 2 minimization. The best-fit position obtained is RA = 18 h 32 m 50 s ± 32 s stat ± 36 s syst , Dec = -9 · 22 ' 36 '' ± 32 '' stat ± 36 '' syst ( J2000 ) ( χ 2 /ndf=0.89). No significant extension was found for the source.</text>
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<caption>Figure 1 : H.E.S.S. excess map oversampled with the r68 value of 0.081 · (dashed circle) and smoothed with a 2Dgaussian with the same width in units of counts per integration area. The newly discovered source HESS J1832 -093 is shown with its best-fit position and corresponding statistical errors (black cross). The SNR G22.7 -0.2 observed in radio [5] is represented by the white contours. The emission seen on the upper left is a small part of HESS J1834 -087 [4], the TeV source in spatial coincidence with SNR W41.</caption>
</figure>
<section_header_level_1><location><page_2><loc_10><loc_45><loc_27><loc_46></location>2.2 Spectral analysis</section_header_level_1>
<text><location><page_2><loc_9><loc_27><loc_49><loc_45></location>To produce the energy spectrum, only the highest quality data are used, corresponding to a data set of 59 hours live time. In order to broaden the accessible energy range, the charge cut is lowered to a minimum of 80 photo-electrons, resulting in an energy threshold of ∼ 400 GeV. The background is estimated with the reflected background model, which is better suited for spectral studies [1]. The forwardfolding method described in [1] is applied to the data to derive the spectrum. Source counts are extracted from a circular region of 0.1 · radius around the best fit position of HESS J1832 -093, a size optimized for point source studies with the applied cuts. The corresponding excess is 152 ± 24 gamma-like events.</text>
<text><location><page_2><loc_9><loc_16><loc_49><loc_27></location>The spectrum obtained between 400 GeV and 5 TeV (displayed on Fig. 2) can be described by a power-law d Φ dE = Φ 0 ( E 1TeV ) -Γ , with an index Γ = 2 . 6 ± 0 . 3stat ± 0 . 3syst and a differential flux normalisation at 1 TeV of Φ 0 = ( 4 . 8 ± 0 . 8stat ± 0 . 9syst ) × 10 -13 cm -2 s -1 TeV -1 . A light curve was produced with the available observations and no significant temporal variability was detected in the H.E.S.S. data set.</text>
<section_header_level_1><location><page_2><loc_10><loc_13><loc_39><loc_14></location>3 Multi-wavelength observations</section_header_level_1>
<section_header_level_1><location><page_2><loc_10><loc_11><loc_45><loc_12></location>3.1 X-Ray observations with XMMNewton</section_header_level_1>
<text><location><page_2><loc_9><loc_5><loc_49><loc_10></location>In order to constrain the nature of the source HESS J1832 -093, a dedicated XMMNewton observation (ID: 0654480101) was performed in March 2011 for 17 ks, targeted at the position of the gamma-ray emission. After fil-</text>
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<caption>Figure 2 : Top panel: VHE gamma-ray spectrum observed from HESS J1832 -093 obtained with the forward-folding method. Bottom panel: corresponding residuals. The green contour represents the 1 σ confidence level of the fitted spectrum using a power-law hypothesis.</caption>
</figure>
<text><location><page_2><loc_52><loc_54><loc_91><loc_58></location>tering out proton flare contamination, 13 ks and 7 ks of exposure time remained for the two EPIC-MOS cameras and for the EPIC-pn camera respectively.</text>
<text><location><page_2><loc_51><loc_48><loc_91><loc_54></location>The data were processed using the XMMNewton Science Analysis System (v10.0). The instrumental background was derived from a compilation of blank sky observations [6], renormalized to the actual exposure using the count rate in the 10-12 keV energy band.</text>
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<caption>Figure 3 : X-ray count map in the 2-10 keV band smoothed with a Gaussian of 15 '' width (HPD of the EPIC cameras PSF) of the field of view around HESS J1832 -093 as seen by XMMNewton . The white cross symbolizes the best-fit position of HESS J1832 -093 with corresponding statistical errors and the white contours represent the H.E.S.S. excess.</caption>
</figure>
<text><location><page_2><loc_52><loc_5><loc_91><loc_10></location>The brightest object in the XMMNewton field of view is a point-like source located at RA = 18 h 32 m 45 s . 04 , Dec = -09 · 21 ' 53 '' . 9, 1.4 ' away from the best-fit position of the H.E.S.S. excess, as shown in Fig. 3. This new source,</text>
<text><location><page_3><loc_10><loc_87><loc_49><loc_90></location>dubbed XMMU J183245 -0921539, lies within the 2 σ uncertainty of the VHE gamma-ray centroid.</text>
<text><location><page_3><loc_9><loc_65><loc_49><loc_87></location>Spectra from the three instruments were extracted from a 15 '' radius circular region centered on XMMU J183245 -0921539. As the statistics are low ( ∼ 500 counts when summing the three EPIC instruments), the data were fitted in Xspec (v12.5) using Cash statistics and the spectra were not rebinned for the fitting process. The pvalue corresponding to the null-hypothesis probability is derived using the goodness command in XSPEC which performs Monte Carlo simulations of the goodness-of-fit. The best fit parameters for a power-law assumption are a column density NH = 10 . 5 + 3 . 1 -2 . 7 × 10 22 cm -2 , a photon index Γ = 1 . 3 + 0 . 5 -0 . 4 and an unabsorbed energy flux Φ (2-10 keV) = 6 . 9 + 1 . 7 -2 . 8 × 10 -13 ergcm -2 s -1 with a p-value of 0.75. A search for pulsations was performed at the position of XMMUJ183245 -0921539, but no pulsations were found in the data.</text>
<text><location><page_3><loc_9><loc_51><loc_49><loc_64></location>A comparison of the absorption along the line of sight obtained from the X-ray spectral model with the column depth derived from the atomic (HI) and molecular ( 12 CO, J=1 → 0 transition line) gas can be used to provide a lower limit on the distance to XMMU J183245 -0921539, assuming that all absorbing material is at the near distance allowed by the Galactic rotation curve. The Galactic rotation curve model of [7] is used to translate the measured velocities into distances. A lower limit of about 5 kpc on the distance to XMMU J183245 -0921539 is thus derived.</text>
<section_header_level_1><location><page_3><loc_10><loc_47><loc_35><loc_49></location>3.2 IR counterparts to XMMU J183245 -0921539</section_header_level_1>
<text><location><page_3><loc_9><loc_30><loc_49><loc_47></location>The 2MASScatalog 1 shows one infrared source around the position of XMMU J183245 -0921539 within the systematic pointing error of XMMNewton ( ∼ 2 '' ). This source, 2MASS J18324516 -0921545, lies 1.9 '' away from the center of gravity of the X-ray source. No optical counterpart is found, likely due to strong extinction in the galactic plane. The apparent magnitudes observed in the J, H, K bands are mJ = 15 . 52 ± 0 . 06, mH = 13 . 26 ± 0 . 04 and mK = 12 . 17 ± 0 . 02, respectively. The IR emission could originate from a massive companion to the X-ray compact source, thus forming a binary system, as discussed in section 4.2.</text>
<section_header_level_1><location><page_3><loc_10><loc_27><loc_28><loc_29></location>3.3 13 COobservations</section_header_level_1>
<text><location><page_3><loc_9><loc_5><loc_49><loc_27></location>The Galactic Ring Survey (GRS) performed with the Boston University FCRAO telescopes [8] provides measurements of the 13 CO (J=1 → 0) transition line covering the velocity range from -5 to 135 km s -1 in this region. The detection of this line is evidence for the presence of dense molecular clouds that are known to be targets for cosmic-rays and hence gamma-ray emitters via neutral pion production and decay or bremsstrahlung emission. A 0.2 · -side square region is defined around the source HESS J1832 -093 to look for molecular clouds traced by the 13 CO transition line. Several molecular clouds measured at different radial velocities are found in this region. The two structures that show the best spatial coincidence with the TeV excess are selected. Their velocity ranges are 26 to 31 kms -1 and 73 to 81 kms -1 respectively. The antenna temperature of each molecular cloud is integrated on</text>
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<text><location><page_3><loc_51><loc_76><loc_91><loc_90></location>the corresponding range. The same Galactic rotation curve model as used in section 3.1 [7] is assumed to translate the measured velocities into distances, each velocity corresponding to two possible distances given our position in the Milky Way. However, in case of an association with the SNR, only the near distance (2.3 kpc and 4.5 kpc respectively) would be compatible with the distance estimates to the remnant. Following the approach described in [9], the integrated antenna temperatures are used to derive the gas mass of each structure and the corresponding gas densities (20 cm -3 and 62 cm -3 respectively).</text>
<section_header_level_1><location><page_3><loc_52><loc_72><loc_64><loc_73></location>4 Discussion</section_header_level_1>
<section_header_level_1><location><page_3><loc_52><loc_70><loc_67><loc_71></location>4.1 PWNscenario</section_header_level_1>
<text><location><page_3><loc_52><loc_41><loc_91><loc_69></location>Dedicated XMMNewton observations at the position of HESS J1832 -093 have revealed the presence of the pointlike source XMMU J183245 -0921539 showing a hard spectrum ( Γ = 1 . 3 + 0 . 5 -0 . 4 ). The source position in the galactic plane points towards a location inside our galaxy, at a distance ≥ 5 kpc as deduced from the X-ray power-law model. XMMU J183245 -0921539 is a serious counterpart candidate for HESS J1832 -093 because of its proximity with the best-fit position of the TeV emission and its hard spectrum. A likely scenario would be that both the X-ray and TeV sources stem from a pulsar wind nebula (PWN) powered by a yet unknown pulsar. Even if the non-thermal aspect of the X-ray emission is not well determined, its hard spectral index is indicative of an emission from the vicinity of a pulsar (magnetospheric, striped wind [10],...). Therefore, despite the lack of observed pulsations in the object, we will consider a pulsar origin for XMMU J183245 -0921539 in the following. It can be tested whether energetically a PWN scenario plausibly matches with the population of known TeVemitting PWNe, under the hypothesis that the X-ray emission comes from the pulsar's magnetosphere.</text>
<text><location><page_3><loc_52><loc_35><loc_91><loc_41></location>The unabsorbed flux Φ X (2-10 keV) of the point source for the power-law model is used to compute the corresponding luminosity in the same energy band for a distance of 5 kpc: LX ( 2 -10keV ) /similarequal 2 × 10 33 ergs -1 .</text>
<text><location><page_3><loc_52><loc_30><loc_91><loc_35></location>It can then be translated to an estimate of the ˙ E of the hypothetical pulsar using the relation provided by [11]. The estimated spin-down luminosity is about 1 . 5 × 10 37 ergs -1 for the same distance.</text>
<text><location><page_3><loc_51><loc_21><loc_91><loc_30></location>If we now compute the ˙ E/d 2 we obtain a value of 6 × 10 35 ergs -1 kpc -2 , corresponding to the band for which 70% of the PWNe are detected by H.E.S.S. [12]. Therefore, if the putative pulsar powers a TeV PWN, it should be detectable by the H.E.S.S. array. Together with the absence of detected X-ray pulsations, the PWN scenario remains unconfirmed but is energetically possible.</text>
<section_header_level_1><location><page_3><loc_52><loc_18><loc_68><loc_19></location>4.2 Binary scenario</section_header_level_1>
<text><location><page_3><loc_51><loc_8><loc_91><loc_17></location>The infrared source 2MASS J18324516 -0921545 discovered in spatial coindidence with XMMU J183245 -0921539 could suggest that the X-ray source resides in a binary system around a massive star. The magnitude extinction number in the optical band expected from the X-ray column density is AV=59 + 17 -15 , using the NH -A V relation given by [13]. This would result in an absolute magnitude MJ ≤ -</text>
<text><location><page_4><loc_9><loc_79><loc_49><loc_90></location>14.6 for a distance ≥ 5 kpc [14]. This value is excluded but the NH -A V relation cannot apply in case of strong local absorption. Therefore, a binary scenario with 2MASS J18324516 -0921545 as optical companion could only work if the X-ray absorption arises mainly locally around XMMU J183245 -0921539. In the absence of orbitally modulated X-ray or TeV emission, the binary possibility remains unconfirmed for the moment.</text>
<section_header_level_1><location><page_4><loc_10><loc_77><loc_46><loc_78></location>4.3 SNR-molecular cloud interaction scenario</section_header_level_1>
<text><location><page_4><loc_9><loc_52><loc_49><loc_76></location>Despite the lack of X-ray emission from the SNR shell that could be due to synchrotron radiation of high energy electrons, or thermal X-ray emission which could stem from shock-gas interactions seen frequently in middleaged SNRs, the observed VHE emission might still come from particles accelerated in the remnant. Those particles would then interact with localized target material via neutral pion production and decay or bremsstrahlung emission. Indeed, 13 CO measurements show the presence of structures around HESS J1832 -093, while at corresponding distance ranges lower gas densities are seen in other portions of the SNR shell. There is, however, no support of the association of these gas structures with G22.7 -0.2, e.g. through maser emission from shock-cloud interaction. An identification with the TeV source might therefore be due to chance coincidence. Further investigation on a possible CRpropagationto a nearby MC at the origin of the gammaray emission thus needs to be performed.</text>
<section_header_level_1><location><page_4><loc_10><loc_49><loc_22><loc_50></location>5 Conclusion</section_header_level_1>
<text><location><page_4><loc_9><loc_24><loc_49><loc_48></location>Observations in the field of view of SNR G22.7 -0.2 have led to the discovery of the VHE source HESS J1832 -093 lying on the edge of the SNR radio rim. The available multi-wavelength data, using archival radio and infrared data, as well as a dedicated XMMNewton X-ray pointing towards the source, do not permit to unambiguously determine the nature of the object that gives rise to the VHE emission. A compelling X-ray counterpart, XMMU J183245 -0921539, has been discovered. The nature of this X-ray source could, however, not be established from the X-ray data alone. Together with the TeV emission and the infrared point source 2MASS J18324516 -0921545, plausible object classifications are a pulsar wind nebula or a binary system. Cosmic-rays accelerated in the SNR G22.7 -0.2 interacting with dense gas material could also result in TeV emission but the interaction between the SNR and a molecular cloud is not supported by observational evidence such as maser emission.</text>
<text><location><page_4><loc_10><loc_17><loc_49><loc_24></location>The nature of the gamma-ray source HESS J1832 -093 presented here remains, therefore, undetermined. More data in X-rays and radio to look for variability and faint diffuse emission could help to constrain the nature of the source.</text>
<section_header_level_1><location><page_4><loc_10><loc_14><loc_25><loc_15></location>Acknowledgments</section_header_level_1>
<text><location><page_4><loc_9><loc_5><loc_49><loc_13></location>The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Tech-</text>
<figure>
<location><page_4><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<text><location><page_4><loc_51><loc_80><loc_91><loc_90></location>nology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.</text>
<section_header_level_1><location><page_4><loc_52><loc_77><loc_61><loc_78></location>References</section_header_level_1>
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<list_item><location><page_4><loc_52><loc_71><loc_90><loc_74></location>[2] Shaver, P. A. and Goss, W. M., AJPAS 14 (1970) p.133 Bib. code:1970AuJPA..14..133S</list_item>
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<list_item><location><page_4><loc_52><loc_67><loc_83><loc_69></location>[4] H.E.S.S. collaboration, ApJ 636 (2006) 777-797 doi:10.1086/498013</list_item>
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<text><location><page_4><loc_53><loc_50><loc_72><loc_51></location>1166-1176 doi:10.1086/589495</text>
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<list_item><location><page_4><loc_52><loc_45><loc_88><loc_48></location>[13] Predehl, P. and Schmitt, J. H. M. M., A&A 293 (1995) 889-905 Bib. code:1995A&A...293..889P</list_item>
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<text><location><page_4><loc_52><loc_44><loc_89><loc_45></location>[14] Cardelli, J. A. and Clayton, G. C. and Mathis, J. S., proc.</text>
<text><location><page_4><loc_53><loc_43><loc_80><loc_44></location>IAU (1989) Bib. code:1989IAUS..135P...5C</text>
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</document>
[
{
"title": "Discovery of the mysterious gamma-ray source HESSJ1832-093 in the vicinity of SNR G22.7-0.2",
"content": "H. LAFFON 1 , 2 , F. ACERO 3 , F. BRUN 4 , B. KH'ELIFI 2 , G. P UHLHOFER 5 , R. TERRIER 6 , FOR THE H.E.S.S. COLLABORATION. laffon@cenbg.in2p3.fr Abstract: Thanks to the use of advanced analysis techniques, the H.E.S.S. imaging Cherenkov telescope array has now reached a sensitivity level allowing the detection of sources with fluxes around 1% of the Crab with a limited observation time (less than ∼ 100 hours). 67 hours of observations in the region of the supernova remnant G22.7-0.2 thus yielded the detection of the faint point-like source HESS J1832-093 in spatial coincidence with a part of the radio shell of the supernova remnant. A multi-wavelength search for counterparts was then performed and led to the elaboration of various scenarii in order to explain the origin of the very high energy excess. The presence of molecular clouds in the line of sight could indicate a hadronic origin through π 0 production and decay. However, the discovery of an X-ray point source and its potential infrared counterpart give rise to other possibilities such as a pulsar wind nebula nature or a binary system. The latest results on this source are presented as well as the different scenarii brought by the multi-wavelength observations. Keywords: HESS J1832-093, SNR G22.7-0.2, molecular clouds, pulsar wind nebula, binary system",
"pages": [
1
]
},
{
"title": "2 H.E.S.S. observations",
"content": "H.E.S.S. (High Energy Stereoscopic System) is an array of four imaging atmospheric Cherenkov telescopes located 1800 m above sea level in the Khomas Highland of Namibia and has been fully operational since 2004 [1]. The H.E.S.S. collaboration has been conducting a systematic scan of the Galactic Plane, which led to the discovery of a rich population of very high energy (VHE) gamma-ray sources such as in the region around the supernova remnant (SNR) W41. As a consequence, a dedicated observation campaign was launched by H.E.S.S. to study those TeV sources in detail. These observations led to the discovery of HESS J1832 -093 in spatial coincidence with the edge of SNR G22.7 -0.2. The latter shows a non-thermal shell of 26 ' diameter in radio [2] and partially overlaps the neighbouring remnant W41. A search for multiwavelength counterparts was performed in order to identify the nature of HESS J1832 -093. A proposal for observations in X-rays at the position of HESS J1832 -093 was submitted to the XMMNewton satellite and allowed the discovery of a pointlike source in X-rays near the center of the TeV source. Apart from the radio emission of the SNR, no radio pointlike counterpart was found but measurements of the 13 CO (J=1 → 0) transition line show the presence of molecular structures on the line of sight. However, an infrared pointlike source lying less than 2 '' away from the center of the Xray source was found in the 2MASS catalog. The features of the TeV emission discovered by H.E.S.S. are described in section 2. Details on the X-ray, IR and radio counterparts are given in section 3 and different considered scenarios to explain the VHE gamma-ray emission are presented in section 4.",
"pages": [
1
]
},
{
"title": "2.1 Detection and morphology",
"content": "A standard run selection procedure is used to remove bad quality observations in order to study the field of view. This results in a data set comprising 67 hours live time of observations taken from 2004 to 2011. A standard H.E.S.S. analysis frame is adopted with the Hillas reconstruction method using the weighted intersections of the main axis of the images to reconstruct the direction of the events [1]. A recently developped multi-variate analysis is used to provide a better discrimination between hadrons and gamma-rays [3]. A minimum charge of 110 photo-electrons in the shower images is applied to the data, resulting in an energy threshold of about 450 GeV. The background is estimated with the ring-background model, as described in [1]. Using these techniques, HESS J1832 -093 is detected with a peak significance of 7 . 9 σ pre-trials, resulting in a post-trial detection significance of 5 . 6 σ , following the approach described in [4]. The corresponding excess map of the field of view showing the new detected source is presented in Fig. 1. The average angular resolution (r68) for this selected data set and assuming a power-law spectral index of 2.3 is 0 . 081 · at the source position. A twodimensional symmetrical Gaussian function is used to determine the position and size of the TeV emission with a χ 2 minimization. The best-fit position obtained is RA = 18 h 32 m 50 s ± 32 s stat ± 36 s syst , Dec = -9 · 22 ' 36 '' ± 32 '' stat ± 36 '' syst ( J2000 ) ( χ 2 /ndf=0.89). No significant extension was found for the source.",
"pages": [
1
]
},
{
"title": "2.2 Spectral analysis",
"content": "To produce the energy spectrum, only the highest quality data are used, corresponding to a data set of 59 hours live time. In order to broaden the accessible energy range, the charge cut is lowered to a minimum of 80 photo-electrons, resulting in an energy threshold of ∼ 400 GeV. The background is estimated with the reflected background model, which is better suited for spectral studies [1]. The forwardfolding method described in [1] is applied to the data to derive the spectrum. Source counts are extracted from a circular region of 0.1 · radius around the best fit position of HESS J1832 -093, a size optimized for point source studies with the applied cuts. The corresponding excess is 152 ± 24 gamma-like events. The spectrum obtained between 400 GeV and 5 TeV (displayed on Fig. 2) can be described by a power-law d Φ dE = Φ 0 ( E 1TeV ) -Γ , with an index Γ = 2 . 6 ± 0 . 3stat ± 0 . 3syst and a differential flux normalisation at 1 TeV of Φ 0 = ( 4 . 8 ± 0 . 8stat ± 0 . 9syst ) × 10 -13 cm -2 s -1 TeV -1 . A light curve was produced with the available observations and no significant temporal variability was detected in the H.E.S.S. data set.",
"pages": [
2
]
},
{
"title": "3.1 X-Ray observations with XMMNewton",
"content": "In order to constrain the nature of the source HESS J1832 -093, a dedicated XMMNewton observation (ID: 0654480101) was performed in March 2011 for 17 ks, targeted at the position of the gamma-ray emission. After fil- tering out proton flare contamination, 13 ks and 7 ks of exposure time remained for the two EPIC-MOS cameras and for the EPIC-pn camera respectively. The data were processed using the XMMNewton Science Analysis System (v10.0). The instrumental background was derived from a compilation of blank sky observations [6], renormalized to the actual exposure using the count rate in the 10-12 keV energy band. The brightest object in the XMMNewton field of view is a point-like source located at RA = 18 h 32 m 45 s . 04 , Dec = -09 · 21 ' 53 '' . 9, 1.4 ' away from the best-fit position of the H.E.S.S. excess, as shown in Fig. 3. This new source, dubbed XMMU J183245 -0921539, lies within the 2 σ uncertainty of the VHE gamma-ray centroid. Spectra from the three instruments were extracted from a 15 '' radius circular region centered on XMMU J183245 -0921539. As the statistics are low ( ∼ 500 counts when summing the three EPIC instruments), the data were fitted in Xspec (v12.5) using Cash statistics and the spectra were not rebinned for the fitting process. The pvalue corresponding to the null-hypothesis probability is derived using the goodness command in XSPEC which performs Monte Carlo simulations of the goodness-of-fit. The best fit parameters for a power-law assumption are a column density NH = 10 . 5 + 3 . 1 -2 . 7 × 10 22 cm -2 , a photon index Γ = 1 . 3 + 0 . 5 -0 . 4 and an unabsorbed energy flux Φ (2-10 keV) = 6 . 9 + 1 . 7 -2 . 8 × 10 -13 ergcm -2 s -1 with a p-value of 0.75. A search for pulsations was performed at the position of XMMUJ183245 -0921539, but no pulsations were found in the data. A comparison of the absorption along the line of sight obtained from the X-ray spectral model with the column depth derived from the atomic (HI) and molecular ( 12 CO, J=1 → 0 transition line) gas can be used to provide a lower limit on the distance to XMMU J183245 -0921539, assuming that all absorbing material is at the near distance allowed by the Galactic rotation curve. The Galactic rotation curve model of [7] is used to translate the measured velocities into distances. A lower limit of about 5 kpc on the distance to XMMU J183245 -0921539 is thus derived.",
"pages": [
2,
3
]
},
{
"title": "3.2 IR counterparts to XMMU J183245 -0921539",
"content": "The 2MASScatalog 1 shows one infrared source around the position of XMMU J183245 -0921539 within the systematic pointing error of XMMNewton ( ∼ 2 '' ). This source, 2MASS J18324516 -0921545, lies 1.9 '' away from the center of gravity of the X-ray source. No optical counterpart is found, likely due to strong extinction in the galactic plane. The apparent magnitudes observed in the J, H, K bands are mJ = 15 . 52 ± 0 . 06, mH = 13 . 26 ± 0 . 04 and mK = 12 . 17 ± 0 . 02, respectively. The IR emission could originate from a massive companion to the X-ray compact source, thus forming a binary system, as discussed in section 4.2.",
"pages": [
3
]
},
{
"title": "3.3 13 COobservations",
"content": "The Galactic Ring Survey (GRS) performed with the Boston University FCRAO telescopes [8] provides measurements of the 13 CO (J=1 → 0) transition line covering the velocity range from -5 to 135 km s -1 in this region. The detection of this line is evidence for the presence of dense molecular clouds that are known to be targets for cosmic-rays and hence gamma-ray emitters via neutral pion production and decay or bremsstrahlung emission. A 0.2 · -side square region is defined around the source HESS J1832 -093 to look for molecular clouds traced by the 13 CO transition line. Several molecular clouds measured at different radial velocities are found in this region. The two structures that show the best spatial coincidence with the TeV excess are selected. Their velocity ranges are 26 to 31 kms -1 and 73 to 81 kms -1 respectively. The antenna temperature of each molecular cloud is integrated on the corresponding range. The same Galactic rotation curve model as used in section 3.1 [7] is assumed to translate the measured velocities into distances, each velocity corresponding to two possible distances given our position in the Milky Way. However, in case of an association with the SNR, only the near distance (2.3 kpc and 4.5 kpc respectively) would be compatible with the distance estimates to the remnant. Following the approach described in [9], the integrated antenna temperatures are used to derive the gas mass of each structure and the corresponding gas densities (20 cm -3 and 62 cm -3 respectively).",
"pages": [
3
]
},
{
"title": "4.1 PWNscenario",
"content": "Dedicated XMMNewton observations at the position of HESS J1832 -093 have revealed the presence of the pointlike source XMMU J183245 -0921539 showing a hard spectrum ( Γ = 1 . 3 + 0 . 5 -0 . 4 ). The source position in the galactic plane points towards a location inside our galaxy, at a distance ≥ 5 kpc as deduced from the X-ray power-law model. XMMU J183245 -0921539 is a serious counterpart candidate for HESS J1832 -093 because of its proximity with the best-fit position of the TeV emission and its hard spectrum. A likely scenario would be that both the X-ray and TeV sources stem from a pulsar wind nebula (PWN) powered by a yet unknown pulsar. Even if the non-thermal aspect of the X-ray emission is not well determined, its hard spectral index is indicative of an emission from the vicinity of a pulsar (magnetospheric, striped wind [10],...). Therefore, despite the lack of observed pulsations in the object, we will consider a pulsar origin for XMMU J183245 -0921539 in the following. It can be tested whether energetically a PWN scenario plausibly matches with the population of known TeVemitting PWNe, under the hypothesis that the X-ray emission comes from the pulsar's magnetosphere. The unabsorbed flux Φ X (2-10 keV) of the point source for the power-law model is used to compute the corresponding luminosity in the same energy band for a distance of 5 kpc: LX ( 2 -10keV ) /similarequal 2 × 10 33 ergs -1 . It can then be translated to an estimate of the ˙ E of the hypothetical pulsar using the relation provided by [11]. The estimated spin-down luminosity is about 1 . 5 × 10 37 ergs -1 for the same distance. If we now compute the ˙ E/d 2 we obtain a value of 6 × 10 35 ergs -1 kpc -2 , corresponding to the band for which 70% of the PWNe are detected by H.E.S.S. [12]. Therefore, if the putative pulsar powers a TeV PWN, it should be detectable by the H.E.S.S. array. Together with the absence of detected X-ray pulsations, the PWN scenario remains unconfirmed but is energetically possible.",
"pages": [
3
]
},
{
"title": "4.2 Binary scenario",
"content": "The infrared source 2MASS J18324516 -0921545 discovered in spatial coindidence with XMMU J183245 -0921539 could suggest that the X-ray source resides in a binary system around a massive star. The magnitude extinction number in the optical band expected from the X-ray column density is AV=59 + 17 -15 , using the NH -A V relation given by [13]. This would result in an absolute magnitude MJ ≤ - 14.6 for a distance ≥ 5 kpc [14]. This value is excluded but the NH -A V relation cannot apply in case of strong local absorption. Therefore, a binary scenario with 2MASS J18324516 -0921545 as optical companion could only work if the X-ray absorption arises mainly locally around XMMU J183245 -0921539. In the absence of orbitally modulated X-ray or TeV emission, the binary possibility remains unconfirmed for the moment.",
"pages": [
3,
4
]
},
{
"title": "4.3 SNR-molecular cloud interaction scenario",
"content": "Despite the lack of X-ray emission from the SNR shell that could be due to synchrotron radiation of high energy electrons, or thermal X-ray emission which could stem from shock-gas interactions seen frequently in middleaged SNRs, the observed VHE emission might still come from particles accelerated in the remnant. Those particles would then interact with localized target material via neutral pion production and decay or bremsstrahlung emission. Indeed, 13 CO measurements show the presence of structures around HESS J1832 -093, while at corresponding distance ranges lower gas densities are seen in other portions of the SNR shell. There is, however, no support of the association of these gas structures with G22.7 -0.2, e.g. through maser emission from shock-cloud interaction. An identification with the TeV source might therefore be due to chance coincidence. Further investigation on a possible CRpropagationto a nearby MC at the origin of the gammaray emission thus needs to be performed.",
"pages": [
4
]
},
{
"title": "5 Conclusion",
"content": "Observations in the field of view of SNR G22.7 -0.2 have led to the discovery of the VHE source HESS J1832 -093 lying on the edge of the SNR radio rim. The available multi-wavelength data, using archival radio and infrared data, as well as a dedicated XMMNewton X-ray pointing towards the source, do not permit to unambiguously determine the nature of the object that gives rise to the VHE emission. A compelling X-ray counterpart, XMMU J183245 -0921539, has been discovered. The nature of this X-ray source could, however, not be established from the X-ray data alone. Together with the TeV emission and the infrared point source 2MASS J18324516 -0921545, plausible object classifications are a pulsar wind nebula or a binary system. Cosmic-rays accelerated in the SNR G22.7 -0.2 interacting with dense gas material could also result in TeV emission but the interaction between the SNR and a molecular cloud is not supported by observational evidence such as maser emission. The nature of the gamma-ray source HESS J1832 -093 presented here remains, therefore, undetermined. More data in X-rays and radio to look for variability and faint diffuse emission could help to constrain the nature of the source.",
"pages": [
4
]
},
{
"title": "Acknowledgments",
"content": "The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the German Research Foundation (DFG), the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Tech- nology Facilities Council (STFC), the IPNP of the Charles University, the Czech Science Foundation, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.",
"pages": [
4
]
},
{
"title": "References",
"content": "1166-1176 doi:10.1086/589495 [14] Cardelli, J. A. and Clayton, G. C. and Mathis, J. S., proc. IAU (1989) Bib. code:1989IAUS..135P...5C",
"pages": [
4
]
}
]
2013ICRC...33.1128B
https://arxiv.org/pdf/1308.1512.pdf
<document>
<figure>
<location><page_1><loc_69><loc_88><loc_89><loc_94></location>
</figure>
<section_header_level_1><location><page_1><loc_10><loc_85><loc_78><loc_86></location>FACT - The First G-APD Cherenkov Telescope: Status and Results</section_header_level_1>
<text><location><page_1><loc_10><loc_73><loc_91><loc_84></location>T. BRETZ 1 , H. ANDERHUB 1 , M. BACKES 2 , A. BILAND 1 ,A V. BOCCONE 3 , I. BRAUN 1 , J. BUSS 2 , F. CADOUX 3 , V. COMMICHAU 1 , L. DJAMBAZOV 1 , D. DORNER 4 , S. EINECKE 2 , D. EISENACHER 4 , A. GENDOTTI 1 , O. GRIMM 1 , H. VON GUNTEN 1 , C. HALLER 1 , D. HILDEBRAND 1 , U. HORISBERGER 1 , B. HUBER 1 a , K.-S. KIM 1 b , M. L. KNOETIG 1 , J.-H. K OHNE 2 , T. KR AHENB UHL 1 , B. KRUMM 2 , M. LEE 1 b , E. LORENZ 1 c , W. LUSTERMANN 1 , E. LYARD 3 , K. MANNHEIM 4 , M. MEHARGA 3 , K. MEIER 4 , T. MONTARULI 3 , D. NEISE 2 , F. NESSI-TEDALDI 1 , A.-K. OVERKEMPING 2 , A. PARAVAC 4 , F. PAUSS 1 , D. RENKER 1 d , W. RHODE 2 , M. RIBORDY 5 , U. R OSER 1 , J.-P. STUCKI 1 , J. SCHNEIDER 1 , T. STEINBRING 4 , F. TEMME 2 , J. THAELE 2 , S. TOBLER 1 , G. VIERTEL 1 , P. VOGLER 1 , R. WALTER 3 , K. WARDA 2 , Q. WEITZEL 1 , M. Z ANGLEIN 4 (FACT COLLABORATION)</text>
<text><location><page_1><loc_9><loc_71><loc_64><loc_72></location>1 ETH Zurich, Switzerland -Institute for Particle Physics, Schafmattstr. 20, 8093 Zurich</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_70><loc_76><loc_71></location>2 Technische Universitat Dortmund, Germany -Experimental Physics 5, Otto-Hahn-Str. 4, 44221 Dortmund</list_item>
<list_item><location><page_1><loc_10><loc_68><loc_90><loc_70></location>3 University of Geneva, Switzerland -ISDC, Chemin d'Ecogia 16, 1290 Versoix -DPNC, Quai Ernest-Ansermet 24, 1211 Geneva</list_item>
<list_item><location><page_1><loc_9><loc_67><loc_87><loc_68></location>4 Universitat Wurzburg, Germany -Institute for Theoretical Physics and Astrophysics, Emil-Fischer-Str. 31, 97074 Wurzburg</list_item>
<list_item><location><page_1><loc_10><loc_66><loc_62><loc_67></location>5 EPF Lausanne, Switzerland -Laboratory for High Energy Physics, 1015 Lausanne</list_item>
<list_item><location><page_1><loc_10><loc_65><loc_43><loc_66></location>a Also at: University of Zurich, Physik-Institut, 8057 Zurich, Switzerland</list_item>
<list_item><location><page_1><loc_10><loc_64><loc_56><loc_65></location>b Also at: Kyungpook National University, Center for High Energy Physics, 702-701 Daegu, Korea</list_item>
<list_item><location><page_1><loc_10><loc_63><loc_42><loc_64></location>c Also at: Max-Planck-Institut fur Physik, D-80805 Munich, Germany</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_62><loc_44><loc_63></location>d Also at: Technische Universitat Munchen, D-85748 Garching, Germany</text>
<text><location><page_1><loc_10><loc_60><loc_28><loc_61></location>thomas.bretz@phys.ethz.ch</text>
<text><location><page_1><loc_15><loc_44><loc_91><loc_58></location>Abstract: The First G-APD Cherenkov telescope (FACT) is the first telescope using silicon photon detectors (G-APD aka. SiPM). It is built on the mount of the HEGRA CT3 telescope, still located at the Observatorio del Roque de los Muchachos, and it is successfully in operation since Oct. 2011. The use of Silicon devices promises a higher photon detection efficiency, more robustness and higher precision than photo-multiplier tubes. The FACT collaboration is investigating with which precision these devices can be operated on the long-term. Currently, the telescope is successfully operated from remote and robotic operation is under development. During the past months of operation, the foreseen monitoring program of the brightest known TeV blazars has been carried out, and first physics results have been obtained including a strong flare of Mrk501. An instantaneous flare alert system is already in a testing phase. This presentation will give an overview of the project and summarize its goals, status and first results.</text>
<text><location><page_1><loc_16><loc_41><loc_56><loc_42></location>Keywords: FACT, G-APD, silicon photo sensor, focal plane</text>
<section_header_level_1><location><page_1><loc_10><loc_37><loc_23><loc_38></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_21><loc_49><loc_36></location>Since Oct. 2011, the FACT Collaboration is operating the First G-APD Cherenkov telescope [1] at the Observatorio del Roque de los Muchachos on the Canary Island of La Palma. The telescope's camera is the first focal plane installation using Geiger-mode avalanche photo-diodes (G-APD) for photo detection. Comprising 1440 channels individually read out, each pixel has a field-of-view of 0.11° yielding a total field-of-view of 4.5°. The camera is installed on the mount of the former HEGRA CT 3 telescope. After replacement of the old disc shaped mirrors with refurbished hexagonal mirrors, it has now a total reflective surface of 9.5 m 2 . A picture of the telescope is shown in Fig. 1.</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_21></location>With this novel camera, silicon photo sensors have started to replace photo multiplier tubes (PMT) still widely used for photo detection. They offer a high gain (10 5 to 10 6 ) and are robust enough to be operated under moon light conditions. Their single photon counting capability, their compactness and their low operational voltage ( < 100 V) simplifies the camera design. The photon detection efficiency (PDE) of commercially available G-APDs is already at the level of the best PMTs, and will still significantly improve in the future.</text>
<text><location><page_1><loc_10><loc_5><loc_49><loc_7></location>The challenge of the project is to understand and operate the silicon based photo sensors under changing environ-</text>
<text><location><page_1><loc_52><loc_33><loc_91><loc_38></location>mental conditions, such as changing auxiliary temperature or changing photon flux from the diffuse night-sky background. A system to keep the G-APD response stable under these conditions has been developed.</text>
<text><location><page_1><loc_52><loc_25><loc_91><loc_33></location>The telescope is dedicated to long-term monitoring of the brightest known TeV blazars. These highly violent objects show variability on time scales of seconds to years. To study their behavior on the longer time scales, a complete data sample obtained from continuous measurements on long time scales is necessary.</text>
<section_header_level_1><location><page_1><loc_52><loc_21><loc_69><loc_23></location>2 System overview</section_header_level_1>
<text><location><page_1><loc_51><loc_5><loc_91><loc_21></location>The camera of the telescope is compiled from 1440 channels. Each channel is equipped with a G-APD (Hamamatsu MPPC S10362-33-50C) and a solid light concentrator. The solid light concentrators have the advantage, as compared to hollow ones, to feature a better compression ratio between entrance window and exit and thus balance the relatively small size of the sensors (9 mm 2 ). Sensors, cones and a protective window were glued together using optical glue. All channels are individually read out by the Domino Ring Sampling chip (DRS 4). The data is transferred by a TCP/IP connection via Ethernet to a data acquisition PC. The summed signal of nine channels form a trigger signal which</text>
<figure>
<location><page_2><loc_10><loc_69><loc_48><loc_90></location>
<caption>Figure 1 : Picture of the telescope</caption>
</figure>
<text><location><page_2><loc_9><loc_54><loc_49><loc_63></location>is discriminated by a comparator. Each trigger patch is divided in two bias voltage channels with four and five GAPDs, resp. Although all read out electronics was integrated into the camera, the bias voltage supply is located in the counting hut. Each bias voltage channel has its own current readout. The precision of the voltage application is 12 bit up to 90 V and 12 bit up to 5 mA for the current readout.</text>
<text><location><page_2><loc_10><loc_48><loc_49><loc_54></location>The total power consumption of the camera electronics during operation is about 570 W. About 100 W are dissipated in the supply lines and another 100 W due to the limited efficiency of the DC-DC converters, yielding a total of 370 W consumed by the electronics, ≈ 260 mW per channel.</text>
<text><location><page_2><loc_10><loc_41><loc_48><loc_47></location>To get rid of the waste heat produced by the camera electronics, a passive water cooling is applied. To avoid the sensors being heat up by the electronics, a thermal insulation is installed in between. In total, 31 temperature sensors measure the temperature of the sensor plane.</text>
<text><location><page_2><loc_10><loc_39><loc_48><loc_41></location>All details about the camera construction, the electronics and control software can be found in [1].</text>
<section_header_level_1><location><page_2><loc_10><loc_35><loc_34><loc_36></location>3 Status and achievements</section_header_level_1>
<section_header_level_1><location><page_2><loc_10><loc_33><loc_22><loc_34></location>3.1 Operation</section_header_level_1>
<text><location><page_2><loc_10><loc_10><loc_49><loc_32></location>For a high duty cycle and consistent and stable data taking, a robust slow control and data acquisition software is needed. To avoid the need of shift crew on site, full remote operation must be possible. The camera is now operated since 20 months and its auxiliary hardware has been upgraded gradually, to allow for full remote control. This includes an interlock system, which purpose is to protect the camera from a cooling failure, Ethernet switchable 220 V plugs and the Ethernet access to all power supplies and a remote controllable lids. More details on that can be found in [2]. In parallel, a fully automatic control system has been developed. While all important parameters of the system are available via Web interface (see Fig. 2), the control of the system is achieved from a JavaScript engine reading the schedule from a database. Since a couple of weeks, this system is operating, and user interaction is usually only required in case of unexpected weather conditions.</text>
<text><location><page_2><loc_10><loc_5><loc_49><loc_10></location>Currently, a fully automatic analysis is also in the testing phase. It is applied on the data immediately after the data has been recorded and the file has been closed which typically happens every 5 min for physics data. The analysis takes</text>
<figure>
<location><page_2><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<text><location><page_2><loc_52><loc_87><loc_90><loc_90></location>about the same time than data taking and will allow prompt flare alerts for other telescopes in the near future.</text>
<section_header_level_1><location><page_2><loc_52><loc_85><loc_62><loc_86></location>3.2 Stability</section_header_level_1>
<text><location><page_2><loc_51><loc_63><loc_91><loc_84></location>One of the main goals of the project was the prove of the applicability of G-APDs during standard operation conditions. Since the gain and all other properties of GAPDs depend on their temperature and the voltage applied, a correcting element is necessary. By design, the temperature gradient in the focal plane is small enough allowing several G-APDs to be connected to a single bias voltage channel. In total, 31 temperature sensors installed in the focal plane allow to counteract temperature changes by adapting the applied bias voltage accordingly. The power supply of each sensor is equipped with a filter network of resistors. Due to the changing light conditions induced by changes of the atmospheric conditions or moon light, the current drawn by the G-APDs is varying inducing a voltage drop in the resistances. To correct for this voltage drop, the current of each channel is measured and the voltage corrected accordingly.</text>
<text><location><page_2><loc_52><loc_60><loc_91><loc_62></location>To measure the stability of the gain with these two feedback loops applied, three different methods are available:</text>
<text><location><page_2><loc_52><loc_49><loc_90><loc_58></location>External light pulser An external temperature stabilized light pulser triggers the camera. Since its light yield is stable on average, the gain can directly be deduced from the average pulse height. However, the precision of this system measuring the gain of the G-APD is limited by the fact that also the transmission of the entrance window and the cones could change with time.</text>
<text><location><page_2><loc_51><loc_38><loc_91><loc_47></location>Dark count spectrum A direct and mostly unbiased method to extract the gain, is the extraction of the dark count spectrum. It only depends on the performance of the G-APDs and the readout chain. Its drawback is that it can only be determined under light conditions, i.e. count rates, which still allow for extraction of single pulses. This is the case with closed lids or at dark time conditions.</text>
<text><location><page_2><loc_51><loc_18><loc_91><loc_36></location>Ratescans To measure the stability of the gain in moon lit nights, so called ratescans are used. Ratescans measure the trigger response as a function of the applied trigger threshold. While at low thresholds, the very high rates are dominated by random coincidences of background photons, at high thresholds, the rate represents the number of triggers from coincident light flashed, induced mainly by hadronic showers. Although this method depends on the performance of the whole system, it allows to measure the response of the system even at the brightest light conditions. To ensure an unbiased measurement, only ratescans taken during good atmospheric conditions and weather conditions must be taken into account. A more detailed discussion of this can be found in [9].</text>
<text><location><page_2><loc_52><loc_5><loc_91><loc_17></location>All three methods have been applied almost daily since the beginning of operation. This allows to compare the response at any possible light condition and all temperatures occurring over the seasons. All methods have shown consistently that with the applied feedback system, a stable gain up to full moon conditions can be achieved. From the dark count spectrum, a stability of better than ± 3% over time and 4% pixel-to-pixel variations was deduced. Currently, the limiting factor of the system is the calibration of the</text>
<text><location><page_3><loc_10><loc_87><loc_48><loc_90></location>applied bias voltage which is under investigation and will be improved soon.</text>
<text><location><page_3><loc_9><loc_85><loc_48><loc_87></location>More details about the stability can be found in [3] and will be available soon in [4].</text>
<figure>
<location><page_3><loc_11><loc_38><loc_47><loc_83></location>
</figure>
<figure>
<location><page_3><loc_11><loc_31><loc_47><loc_38></location>
<caption>Figure 2 : Web interface. To achieve low network traffic, only the average signal is displayed for four and five pixels. http://www.fact-project.org/smartfact/</caption>
</figure>
<section_header_level_1><location><page_3><loc_10><loc_20><loc_26><loc_21></location>3.3 Data acquisition</section_header_level_1>
<text><location><page_3><loc_9><loc_5><loc_49><loc_20></location>The Ethernet connection between the forty readout boards and the data acquisition PC is routed over four Ethernet switches, two located in the camera and two located in the control room. In total, four Ethernet lines are available, one Ethernet line per ten boards. A total maximum transfer rate of about 1.9 Gbps is achieved. This is limited by the maximum throughput of 50 Mbps of each of the Ethernetchips (Wiznet 5300) on-board of the forty data acquisition boards. The transfer rate corresponds to a trigger rate of ∼ 250 Hz for a readout window of 300 samples and ∼ 80 Hz for the readout of the full DRS pipeline (1024 samples).</text>
<section_header_level_1><location><page_3><loc_52><loc_89><loc_75><loc_90></location>3.4 Trigger threshold setting</section_header_level_1>
<text><location><page_3><loc_51><loc_59><loc_91><loc_88></location>One of the most important aspects during data taking is the correct setting of the trigger thresholds. To avoid high rates from random coincidences of background photons, the need to be set above the noise level. Since the noise level, mainly defined by the diffuse light from the night sky background, is not necessarily constant, some contingency is added to avoid strong rate fluctuations during data taking. At the same time, they must be set to as low as possible to achieve the lowest energy threshold and thus the highest integral sensitivity. Since the number of breakdowns in the G-APD, i.e. the number of detected photons, is proportional to the measured current, the rate induced from random coincidences as a function of the trigger threshold can directly be determined from the current. From ratescans taken at different light conditions over more than a year, a parametrization for the rate induced from background light and for the rate induced by hadronic showers under best conditions, was derived. Since several weeks, the trigger threshold of the system is now directly deduced from the measured current resulting in very stable data taking conditions. To allow for easy analysis, it is kept constant during every 5 min run.</text>
<section_header_level_1><location><page_3><loc_52><loc_57><loc_87><loc_58></location>3.5 Current and energy threshold prediction</section_header_level_1>
<text><location><page_3><loc_51><loc_41><loc_91><loc_56></location>To allow for an efficient schedule, the prediction of the system response under changing light conditions is mandatory. Therefore, the measured current has been correlated with moon conditions, i.e. moon brightness and zenith angle, which leads to a very precise prediction of the measured current. Since the trigger threshold is in the first order proportional to the light yield of the triggered shower, the change in energy threshold can directly be deduced. Including the change of the light yield with zenith angle, gives a very good estimate of the raise in energy threshold, i.e. loss in sensitivity, for each observation.</text>
<text><location><page_3><loc_51><loc_39><loc_90><loc_41></location>These topics are discussed in more details in [5, 6] and will be available soon in [4].</text>
<section_header_level_1><location><page_3><loc_52><loc_35><loc_67><loc_36></location>4 Physics results</section_header_level_1>
<text><location><page_3><loc_51><loc_23><loc_91><loc_34></location>Very soon after the first observations, first results could be presented [7]. From the measurements of the Crab Nebula, a sensitivity of 8% Crab in 50 h could be derived. This is based on the analysis of data taking during the first months of operation and based on an analysis primarily developed for MAGIC not yet necessarily optimized for the current system. From the rate of excess events an energy threshold between 400 GeV and 700 GeV can be derived taking the known spectrum of the source into account</text>
<text><location><page_3><loc_52><loc_20><loc_90><loc_22></location>The analysis, especially the automatic analysis, and first results are discussed in [8].</text>
<section_header_level_1><location><page_3><loc_52><loc_16><loc_65><loc_18></location>5 Conclusions</section_header_level_1>
<text><location><page_3><loc_51><loc_5><loc_91><loc_16></location>The camera of the First G-APD Cherenkov Telescope (FACT) has proven the applicability of silicon photo sensors (G-APD) in focal plane installations, in particular as detectors in Cherenkov telescopes. The camera is now in operation since more than one and a half years, and although there were hardware problems during that time, none of the was related to the G-APDs at all. So far, no decrease in performance could be detected. Neither any hint for aging of</text>
<figure>
<location><page_3><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<figure>
<location><page_4><loc_10><loc_64><loc_48><loc_90></location>
<caption>Figure 3 : An exemplary event. The color scale is proportional to the signal amplitude detected in each pixel.</caption>
</figure>
<text><location><page_4><loc_10><loc_55><loc_49><loc_57></location>the G-APDs nor of the transmission of cones and window has been found.</text>
<text><location><page_4><loc_9><loc_48><loc_49><loc_54></location>With different types of measurements, it was shown that a stability to on the few percent level, independent of temperature and light conditions can be achieved. Further improvements are expected by an improvement of the bias voltage calibration.</text>
<text><location><page_4><loc_10><loc_40><loc_49><loc_48></location>An important result derived from the current study is that the application of a feedback system which keeps the G-APD's bias voltage constant, renders the need for an external calibration device obsolete. The combination of the extraction of the dark count spectrum and the current measurement, is enough for the operation of such a device.</text>
<text><location><page_4><loc_10><loc_34><loc_49><loc_40></location>Recently, Hamamatsu presented a new generation of G-APD sensors with significantly lower dark count rates, significantly reduced optical crosstalk and significantly reduced afterpulse probability, expected to be available on the market within the next weeks [10].</text>
<text><location><page_4><loc_10><loc_28><loc_49><loc_33></location>Although, it was shown that these improvements are not obligatory for the operation in a Cherenkov telescope, especially the reduction in optical crosstalk guarantees that the influence on image reconstruction is negligible.</text>
<text><location><page_4><loc_9><loc_18><loc_49><loc_28></location>Not only, can the FACT telescope serve as an ideal instrument for long-term monitoring of bright blazars, the application of G-APDs in focal plane installations will significantly reduce construction and installation costs and improve stability of operation. This will yield a significantly impact on future projects like The Cherenkov Telescope Array (CTA) and makes small telescopes for monitoring purposes affordable.</text>
<text><location><page_4><loc_9><loc_5><loc_49><loc_16></location>Acknowledgment The important contributions from ETH Zurich grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) are gratefully acknowledged. We thank the Instituto de Astrofisica de Canarias allowing us to operate the telescope at the Observatorio Roque de los Muchachos in La Palma, the Max-Planck-Institut fur Physik for providing us with the mount of the former HEGRA CT 3 telescope, and the MAGIC collaboration for their support. We also thank the group of Marinella Tose from the College of Engineering and Technology at Western</text>
<figure>
<location><page_4><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<text><location><page_4><loc_52><loc_88><loc_90><loc_90></location>Mindanao State University, Philippines, for providing us with the scheduling web-interface.</text>
<section_header_level_1><location><page_4><loc_52><loc_84><loc_61><loc_85></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_4><loc_52><loc_81><loc_89><loc_83></location>[1] H. Anderhub et al. (FACT Collaboration), 2013, JINST 8 P06008 [arXiv:1304.1710].</list_item>
<list_item><location><page_4><loc_52><loc_80><loc_89><loc_81></location>[2] A. Biland et al. (FACT Collaboration), these proc., ID 708.</list_item>
<list_item><location><page_4><loc_52><loc_79><loc_88><loc_80></location>[3] T. Bretz et al. (FACT Collaboration), these proc., ID 683.</list_item>
<list_item><location><page_4><loc_52><loc_78><loc_72><loc_79></location>[4] FACT Collaboration, in prep.</list_item>
<list_item><location><page_4><loc_52><loc_77><loc_88><loc_78></location>[5] T. Bretz et al. (FACT Collaboration), these proc., ID 720.</list_item>
<list_item><location><page_4><loc_52><loc_76><loc_90><loc_77></location>[6] M. Koetig et al. (FACT Collaboration), these proc., ID 695.</list_item>
<list_item><location><page_4><loc_52><loc_75><loc_91><loc_76></location>[7] Bretz, T. et al. (FACT Collaboration) 2012, AIPC, 1505, 773.</list_item>
<list_item><location><page_4><loc_52><loc_74><loc_90><loc_75></location>[8] D. Dorner et al. (FACT Collaboration), these proc., ID 686.</list_item>
<list_item><location><page_4><loc_52><loc_72><loc_87><loc_74></location>[9] D. Hildebrand et al. (FACT Collaboration), these proc., ID709.</list_item>
</unordered_list>
<text><location><page_4><loc_52><loc_71><loc_89><loc_72></location>[10] http://kicp-workshops.uchicago.edu/ieu2013 .</text>
</document>
[
{
"title": "FACT - The First G-APD Cherenkov Telescope: Status and Results",
"content": "T. BRETZ 1 , H. ANDERHUB 1 , M. BACKES 2 , A. BILAND 1 ,A V. BOCCONE 3 , I. BRAUN 1 , J. BUSS 2 , F. CADOUX 3 , V. COMMICHAU 1 , L. DJAMBAZOV 1 , D. DORNER 4 , S. EINECKE 2 , D. EISENACHER 4 , A. GENDOTTI 1 , O. GRIMM 1 , H. VON GUNTEN 1 , C. HALLER 1 , D. HILDEBRAND 1 , U. HORISBERGER 1 , B. HUBER 1 a , K.-S. KIM 1 b , M. L. KNOETIG 1 , J.-H. K OHNE 2 , T. KR AHENB UHL 1 , B. KRUMM 2 , M. LEE 1 b , E. LORENZ 1 c , W. LUSTERMANN 1 , E. LYARD 3 , K. MANNHEIM 4 , M. MEHARGA 3 , K. MEIER 4 , T. MONTARULI 3 , D. NEISE 2 , F. NESSI-TEDALDI 1 , A.-K. OVERKEMPING 2 , A. PARAVAC 4 , F. PAUSS 1 , D. RENKER 1 d , W. RHODE 2 , M. RIBORDY 5 , U. R OSER 1 , J.-P. STUCKI 1 , J. SCHNEIDER 1 , T. STEINBRING 4 , F. TEMME 2 , J. THAELE 2 , S. TOBLER 1 , G. VIERTEL 1 , P. VOGLER 1 , R. WALTER 3 , K. WARDA 2 , Q. WEITZEL 1 , M. Z ANGLEIN 4 (FACT COLLABORATION) 1 ETH Zurich, Switzerland -Institute for Particle Physics, Schafmattstr. 20, 8093 Zurich d Also at: Technische Universitat Munchen, D-85748 Garching, Germany thomas.bretz@phys.ethz.ch Abstract: The First G-APD Cherenkov telescope (FACT) is the first telescope using silicon photon detectors (G-APD aka. SiPM). It is built on the mount of the HEGRA CT3 telescope, still located at the Observatorio del Roque de los Muchachos, and it is successfully in operation since Oct. 2011. The use of Silicon devices promises a higher photon detection efficiency, more robustness and higher precision than photo-multiplier tubes. The FACT collaboration is investigating with which precision these devices can be operated on the long-term. Currently, the telescope is successfully operated from remote and robotic operation is under development. During the past months of operation, the foreseen monitoring program of the brightest known TeV blazars has been carried out, and first physics results have been obtained including a strong flare of Mrk501. An instantaneous flare alert system is already in a testing phase. This presentation will give an overview of the project and summarize its goals, status and first results. Keywords: FACT, G-APD, silicon photo sensor, focal plane",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "Since Oct. 2011, the FACT Collaboration is operating the First G-APD Cherenkov telescope [1] at the Observatorio del Roque de los Muchachos on the Canary Island of La Palma. The telescope's camera is the first focal plane installation using Geiger-mode avalanche photo-diodes (G-APD) for photo detection. Comprising 1440 channels individually read out, each pixel has a field-of-view of 0.11° yielding a total field-of-view of 4.5°. The camera is installed on the mount of the former HEGRA CT 3 telescope. After replacement of the old disc shaped mirrors with refurbished hexagonal mirrors, it has now a total reflective surface of 9.5 m 2 . A picture of the telescope is shown in Fig. 1. With this novel camera, silicon photo sensors have started to replace photo multiplier tubes (PMT) still widely used for photo detection. They offer a high gain (10 5 to 10 6 ) and are robust enough to be operated under moon light conditions. Their single photon counting capability, their compactness and their low operational voltage ( < 100 V) simplifies the camera design. The photon detection efficiency (PDE) of commercially available G-APDs is already at the level of the best PMTs, and will still significantly improve in the future. The challenge of the project is to understand and operate the silicon based photo sensors under changing environ- mental conditions, such as changing auxiliary temperature or changing photon flux from the diffuse night-sky background. A system to keep the G-APD response stable under these conditions has been developed. The telescope is dedicated to long-term monitoring of the brightest known TeV blazars. These highly violent objects show variability on time scales of seconds to years. To study their behavior on the longer time scales, a complete data sample obtained from continuous measurements on long time scales is necessary.",
"pages": [
1
]
},
{
"title": "2 System overview",
"content": "The camera of the telescope is compiled from 1440 channels. Each channel is equipped with a G-APD (Hamamatsu MPPC S10362-33-50C) and a solid light concentrator. The solid light concentrators have the advantage, as compared to hollow ones, to feature a better compression ratio between entrance window and exit and thus balance the relatively small size of the sensors (9 mm 2 ). Sensors, cones and a protective window were glued together using optical glue. All channels are individually read out by the Domino Ring Sampling chip (DRS 4). The data is transferred by a TCP/IP connection via Ethernet to a data acquisition PC. The summed signal of nine channels form a trigger signal which is discriminated by a comparator. Each trigger patch is divided in two bias voltage channels with four and five GAPDs, resp. Although all read out electronics was integrated into the camera, the bias voltage supply is located in the counting hut. Each bias voltage channel has its own current readout. The precision of the voltage application is 12 bit up to 90 V and 12 bit up to 5 mA for the current readout. The total power consumption of the camera electronics during operation is about 570 W. About 100 W are dissipated in the supply lines and another 100 W due to the limited efficiency of the DC-DC converters, yielding a total of 370 W consumed by the electronics, ≈ 260 mW per channel. To get rid of the waste heat produced by the camera electronics, a passive water cooling is applied. To avoid the sensors being heat up by the electronics, a thermal insulation is installed in between. In total, 31 temperature sensors measure the temperature of the sensor plane. All details about the camera construction, the electronics and control software can be found in [1].",
"pages": [
1,
2
]
},
{
"title": "3.1 Operation",
"content": "For a high duty cycle and consistent and stable data taking, a robust slow control and data acquisition software is needed. To avoid the need of shift crew on site, full remote operation must be possible. The camera is now operated since 20 months and its auxiliary hardware has been upgraded gradually, to allow for full remote control. This includes an interlock system, which purpose is to protect the camera from a cooling failure, Ethernet switchable 220 V plugs and the Ethernet access to all power supplies and a remote controllable lids. More details on that can be found in [2]. In parallel, a fully automatic control system has been developed. While all important parameters of the system are available via Web interface (see Fig. 2), the control of the system is achieved from a JavaScript engine reading the schedule from a database. Since a couple of weeks, this system is operating, and user interaction is usually only required in case of unexpected weather conditions. Currently, a fully automatic analysis is also in the testing phase. It is applied on the data immediately after the data has been recorded and the file has been closed which typically happens every 5 min for physics data. The analysis takes about the same time than data taking and will allow prompt flare alerts for other telescopes in the near future.",
"pages": [
2
]
},
{
"title": "3.2 Stability",
"content": "One of the main goals of the project was the prove of the applicability of G-APDs during standard operation conditions. Since the gain and all other properties of GAPDs depend on their temperature and the voltage applied, a correcting element is necessary. By design, the temperature gradient in the focal plane is small enough allowing several G-APDs to be connected to a single bias voltage channel. In total, 31 temperature sensors installed in the focal plane allow to counteract temperature changes by adapting the applied bias voltage accordingly. The power supply of each sensor is equipped with a filter network of resistors. Due to the changing light conditions induced by changes of the atmospheric conditions or moon light, the current drawn by the G-APDs is varying inducing a voltage drop in the resistances. To correct for this voltage drop, the current of each channel is measured and the voltage corrected accordingly. To measure the stability of the gain with these two feedback loops applied, three different methods are available: External light pulser An external temperature stabilized light pulser triggers the camera. Since its light yield is stable on average, the gain can directly be deduced from the average pulse height. However, the precision of this system measuring the gain of the G-APD is limited by the fact that also the transmission of the entrance window and the cones could change with time. Dark count spectrum A direct and mostly unbiased method to extract the gain, is the extraction of the dark count spectrum. It only depends on the performance of the G-APDs and the readout chain. Its drawback is that it can only be determined under light conditions, i.e. count rates, which still allow for extraction of single pulses. This is the case with closed lids or at dark time conditions. Ratescans To measure the stability of the gain in moon lit nights, so called ratescans are used. Ratescans measure the trigger response as a function of the applied trigger threshold. While at low thresholds, the very high rates are dominated by random coincidences of background photons, at high thresholds, the rate represents the number of triggers from coincident light flashed, induced mainly by hadronic showers. Although this method depends on the performance of the whole system, it allows to measure the response of the system even at the brightest light conditions. To ensure an unbiased measurement, only ratescans taken during good atmospheric conditions and weather conditions must be taken into account. A more detailed discussion of this can be found in [9]. All three methods have been applied almost daily since the beginning of operation. This allows to compare the response at any possible light condition and all temperatures occurring over the seasons. All methods have shown consistently that with the applied feedback system, a stable gain up to full moon conditions can be achieved. From the dark count spectrum, a stability of better than ± 3% over time and 4% pixel-to-pixel variations was deduced. Currently, the limiting factor of the system is the calibration of the applied bias voltage which is under investigation and will be improved soon. More details about the stability can be found in [3] and will be available soon in [4].",
"pages": [
2,
3
]
},
{
"title": "3.3 Data acquisition",
"content": "The Ethernet connection between the forty readout boards and the data acquisition PC is routed over four Ethernet switches, two located in the camera and two located in the control room. In total, four Ethernet lines are available, one Ethernet line per ten boards. A total maximum transfer rate of about 1.9 Gbps is achieved. This is limited by the maximum throughput of 50 Mbps of each of the Ethernetchips (Wiznet 5300) on-board of the forty data acquisition boards. The transfer rate corresponds to a trigger rate of ∼ 250 Hz for a readout window of 300 samples and ∼ 80 Hz for the readout of the full DRS pipeline (1024 samples).",
"pages": [
3
]
},
{
"title": "3.4 Trigger threshold setting",
"content": "One of the most important aspects during data taking is the correct setting of the trigger thresholds. To avoid high rates from random coincidences of background photons, the need to be set above the noise level. Since the noise level, mainly defined by the diffuse light from the night sky background, is not necessarily constant, some contingency is added to avoid strong rate fluctuations during data taking. At the same time, they must be set to as low as possible to achieve the lowest energy threshold and thus the highest integral sensitivity. Since the number of breakdowns in the G-APD, i.e. the number of detected photons, is proportional to the measured current, the rate induced from random coincidences as a function of the trigger threshold can directly be determined from the current. From ratescans taken at different light conditions over more than a year, a parametrization for the rate induced from background light and for the rate induced by hadronic showers under best conditions, was derived. Since several weeks, the trigger threshold of the system is now directly deduced from the measured current resulting in very stable data taking conditions. To allow for easy analysis, it is kept constant during every 5 min run.",
"pages": [
3
]
},
{
"title": "3.5 Current and energy threshold prediction",
"content": "To allow for an efficient schedule, the prediction of the system response under changing light conditions is mandatory. Therefore, the measured current has been correlated with moon conditions, i.e. moon brightness and zenith angle, which leads to a very precise prediction of the measured current. Since the trigger threshold is in the first order proportional to the light yield of the triggered shower, the change in energy threshold can directly be deduced. Including the change of the light yield with zenith angle, gives a very good estimate of the raise in energy threshold, i.e. loss in sensitivity, for each observation. These topics are discussed in more details in [5, 6] and will be available soon in [4].",
"pages": [
3
]
},
{
"title": "4 Physics results",
"content": "Very soon after the first observations, first results could be presented [7]. From the measurements of the Crab Nebula, a sensitivity of 8% Crab in 50 h could be derived. This is based on the analysis of data taking during the first months of operation and based on an analysis primarily developed for MAGIC not yet necessarily optimized for the current system. From the rate of excess events an energy threshold between 400 GeV and 700 GeV can be derived taking the known spectrum of the source into account The analysis, especially the automatic analysis, and first results are discussed in [8].",
"pages": [
3
]
},
{
"title": "5 Conclusions",
"content": "The camera of the First G-APD Cherenkov Telescope (FACT) has proven the applicability of silicon photo sensors (G-APD) in focal plane installations, in particular as detectors in Cherenkov telescopes. The camera is now in operation since more than one and a half years, and although there were hardware problems during that time, none of the was related to the G-APDs at all. So far, no decrease in performance could be detected. Neither any hint for aging of the G-APDs nor of the transmission of cones and window has been found. With different types of measurements, it was shown that a stability to on the few percent level, independent of temperature and light conditions can be achieved. Further improvements are expected by an improvement of the bias voltage calibration. An important result derived from the current study is that the application of a feedback system which keeps the G-APD's bias voltage constant, renders the need for an external calibration device obsolete. The combination of the extraction of the dark count spectrum and the current measurement, is enough for the operation of such a device. Recently, Hamamatsu presented a new generation of G-APD sensors with significantly lower dark count rates, significantly reduced optical crosstalk and significantly reduced afterpulse probability, expected to be available on the market within the next weeks [10]. Although, it was shown that these improvements are not obligatory for the operation in a Cherenkov telescope, especially the reduction in optical crosstalk guarantees that the influence on image reconstruction is negligible. Not only, can the FACT telescope serve as an ideal instrument for long-term monitoring of bright blazars, the application of G-APDs in focal plane installations will significantly reduce construction and installation costs and improve stability of operation. This will yield a significantly impact on future projects like The Cherenkov Telescope Array (CTA) and makes small telescopes for monitoring purposes affordable. Acknowledgment The important contributions from ETH Zurich grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) are gratefully acknowledged. We thank the Instituto de Astrofisica de Canarias allowing us to operate the telescope at the Observatorio Roque de los Muchachos in La Palma, the Max-Planck-Institut fur Physik for providing us with the mount of the former HEGRA CT 3 telescope, and the MAGIC collaboration for their support. We also thank the group of Marinella Tose from the College of Engineering and Technology at Western Mindanao State University, Philippines, for providing us with the scheduling web-interface.",
"pages": [
3,
4
]
},
{
"title": "References",
"content": "[10] http://kicp-workshops.uchicago.edu/ieu2013 .",
"pages": [
4
]
}
]
2013ICRC...33.1856M
https://arxiv.org/pdf/1307.3880.pdf
<document>
<figure>
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</figure>
<section_header_level_1><location><page_1><loc_9><loc_83><loc_91><loc_86></location>All Sky Cameras for the characterization of the Cherenkov Telescope Array candidate sites</section_header_level_1>
<text><location><page_1><loc_10><loc_80><loc_91><loc_82></location>DUSAN MANDAT 1 , MIROSLAV PECH 1 , JAN EBR 1 , MIROSLAV HRABOVSKY 1 , MICHAEL PROUZA 1 , TOMASZ BULIK 2 , INGOMAR ALLEKOTTE 3 , FOR THE CTA CONSORTIUM.</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_78><loc_60><loc_79></location>1 Institute of Physics of Academy of Science of The Czech Republic, Czech Republic.</list_item>
<list_item><location><page_1><loc_10><loc_77><loc_45><loc_78></location>2 Astronomical Observatory University of Warsaw, Poland.</list_item>
<list_item><location><page_1><loc_10><loc_75><loc_74><loc_76></location>3 Centro At'omico Bariloche, Comisi'on Nacional de Energ'ıa At'omica, San Carlos de Bariloche, Argentina.</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_73><loc_20><loc_74></location>mandat@fzu.cz</text>
<text><location><page_1><loc_15><loc_62><loc_91><loc_71></location>Abstract: The All Sky Camera (ASC) was developed as a universal device for the monitoring of the night sky quality. Eight ASCs are already installed and measure night sky parameters at eight of the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The ACS system consists of an astronomical CCD camera, a fish eye lens, a control computer and associated electronics. The measurement is carried out during astronomical night. The images are automatically taken every 5 minutes and automatically processed using the control computer of the device. The analysis results are the cloud fraction (the percentage of the sky covered by clouds) and night sky brightness (in mag/arcsec 2 ).</text>
<text><location><page_1><loc_16><loc_59><loc_60><loc_60></location>Keywords: All Sky Camera, CTA, candidate sites, cloud fraction.</text>
<section_header_level_1><location><page_1><loc_10><loc_53><loc_43><loc_56></location>1 Cloudiness characterization of CTA candidate sites</section_header_level_1>
<text><location><page_1><loc_9><loc_34><loc_49><loc_52></location>The future CTA observatory [1] will include two observatories. One on the Southern and one on the Northern Hemisphere. There are nine candidate sites for the future CTA observatories - five at the Southern and four at the Northern Hemisphere. The goal of Site Selection Work Package [2] (SITE WP) is to characterize the candidate sites and produce relevant data for the Site Selection Committee. One of the important characteristic of the candidate site is the cloud fraction of the night sky and the amount of cloudless nights or more specifically cloudless time in a year. The ASC is an automated instrument for measurement of this parameter. Starting in November 2011 the following sites are being characterized using ASCs :</text>
<table>
<location><page_1><loc_10><loc_18><loc_48><loc_32></location>
<caption>Table 1 : Candidate sites for CTA. Aar - Aar farm, Namibia, SAC - San Antonio de Los Cobres, Argentina, SPM SAN Pedro Martir, Mexico, AZE - Meteor Crater, USA, AZW- Yavapai ranch, USA, TEN - Tenerife Spain, CAS CASLEO observatory, Argentina, ARM - Cerro Armazones, Chile .</caption>
</table>
<figure>
<location><page_1><loc_55><loc_46><loc_70><loc_56></location>
</figure>
<figure>
<location><page_1><loc_70><loc_46><loc_87><loc_56></location>
<caption>Fig. 1 : Electronics and the inside of the camera box.</caption>
</figure>
<section_header_level_1><location><page_1><loc_52><loc_38><loc_71><loc_39></location>1.1 System Description</section_header_level_1>
<text><location><page_1><loc_51><loc_23><loc_91><loc_38></location>The basis of the system is an astronomical camera G1 -2000 (Moravian Instruments a.s., Czech Republic, www.mii.cz), which is equipped with a CCD chip ICX274AL by SONY company. This chip has quantum efficiency higher than 50% at 450-550 nm and its spectral sensitivy covers the range from approximately 400 to 900 nm. It has low read-out noise, a 16-bit ADC and a resolution of 1600 × 1200 pixels. The camera is equipped with a fish-eye varifocal lens (the field of view is 185 degrees) and an electronic iris control. This setup is capable of detecting a star with visual magnitude 6.3 in zenith.</text>
<text><location><page_1><loc_52><loc_5><loc_91><loc_23></location>A miniPC computer with USB I / O controls the system and processes the data. The I / O card controls the iris and the power switch of the camera. All electronics and a camera body (see Figure 1) are weatherproof. The system power is supplied by a solar power system. The switching electronics turns the system ON after sunset and keeps it going during astronomical night. The system is OFF during the day (to save energy). Temperature of the electronics is controlled and stabilized using internal heating system during winter time to protect the ASC system. The camera is a part of an Atmoscope instrument. The Atmoscope measures local weather conditions and night sky brightness parameters. The Atmoscope is connected to central data storage using GPRS or WIFI connection depending on</text>
<figure>
<location><page_2><loc_11><loc_64><loc_47><loc_90></location>
<caption>Fig. 2 : Image of clear night sky from San Pedro Martir.</caption>
</figure>
<text><location><page_2><loc_9><loc_54><loc_49><loc_57></location>Atmoscope local position and nearby facilities. The quality of the link limits the amount and speed of the data transfer.</text>
<section_header_level_1><location><page_2><loc_10><loc_52><loc_34><loc_53></location>1.2 Calibration of the camera</section_header_level_1>
<text><location><page_2><loc_9><loc_30><loc_49><loc_52></location>The fish-eye varifocal lens has 185 degrees field of view (FOV) and its aberration distorts the image. The correct transformation of the incident angle to the imaged pixel position is very important. The mechanism of the calibration is as follows: the light source (a spot as small as possible) is placed at a defined distance (at least 10 m), then the body of camera is rotated around the axis such that the incident light angle (corresponding to the FOV angle) varies between 0 and 90 degrees. The position of the light spot is analyzed for each anglular step. This procedure is repeated for different plane cuts of the lens FOV. The final data are analyzed and fitted with a polynomial function. This calibration is regularly checked on-site by comparing the position of detected stars with the stars in a catalogue. Onsite measurements are also used to determine the variation of the sensitivty of the system with the zenith angle.</text>
<section_header_level_1><location><page_2><loc_10><loc_28><loc_47><loc_29></location>1.3 Data acquisition and principles of analysis</section_header_level_1>
<text><location><page_2><loc_9><loc_16><loc_49><loc_27></location>The ASC system takes full sky images each 5 minutes (resp. 10 minutes - Aar and SAC). Examples of clear and partly cloudy night sky are shown in Figure 2 and 6. The onsite analysis (raw cloudiness analysis) follows immediately, the result is saved to two files. The first file consists of a compressed image and online analysis results, the second one consists of a RAW image and relevant data (UTC time, temperature of CCD, Moon and Sun position).</text>
<text><location><page_2><loc_9><loc_5><loc_49><loc_16></location>The measurement process consists of the following steps. Initially, every exposure of a night sky is followed by a dark frame image so that subtraction of one from the other enables significant reduction of noise in the image. The horizontal coordinates of each image pixel are then calculated with the calibration data and calibration constants used to determine the horizontal coordinates, azimuth and elevation of each image pixel. The total charge of each</text>
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</figure>
<figure>
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<caption>Fig. 3 : Fog or haze affects the visibility of night sky, the nearby tower is not visible (the visibility fades with the fog/haze).</caption>
</figure>
<text><location><page_2><loc_52><loc_44><loc_91><loc_55></location>pixel is proportional to the light intensity, but the intensity of the final image varies with respect to zenith angle due to vignetting and distortion of the lens. The calibration of this zenith angle intensity dependence is obtained by considering extinction in air masses, optical distortions and vignetting variation as a function of zenith angle. The resulting attenuation at, for example, 60 degrees zenith angle corresponds to 2 mag.</text>
<text><location><page_2><loc_51><loc_38><loc_91><loc_44></location>To estimate the cloud fraction, we investigate the presence of stars in the field of view of the camera. We separate the full sky to approximately 70 segments and analyze each segment of the sky individually to obtain the cloudiness. As a star catalogue we use Yale Bright Star Catalog BSC5</text>
<figure>
<location><page_2><loc_53><loc_9><loc_89><loc_35></location>
<caption>Fig. 4 : Lens of the ASC covered with water drops.</caption>
</figure>
<figure>
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</figure>
<figure>
<location><page_3><loc_10><loc_60><loc_90><loc_90></location>
<caption>Fig. 5 : An example of analysis of partly cloudy night sky. The image on the left shows RAW image of the sky. The image on the right shows the analysis results, the yellow crosses indicate the detected stars, green circles catalogue stars and red circles represent catalogue stars without detected pairs - the region covered by cloudiness. The RED circle indicates the limiting zenith angle (60 · ).</caption>
</figure>
<text><location><page_3><loc_10><loc_40><loc_49><loc_51></location>[3], particularly the visual magnitues of the stars. First we estimate the position of each segment on the sky using the data from the calibration. This information is then further improved using the brightest stars from the catalogue in each segement. Once the position of a segment on the sky is fixed, the stars for the segment are read from the catalogue and selected according to the expected sensitivty at the given zenith angle. For every star in the catalogue, we</text>
<figure>
<location><page_3><loc_10><loc_10><loc_48><loc_37></location>
<caption>Fig. 6 : Image of partly cloudy night sky from San Pedro Martir. Moon and clouds are visible close to horizon.</caption>
</figure>
<text><location><page_3><loc_51><loc_35><loc_91><loc_51></location>look for a detected star within the angular limit of 1 deg. If a star that fits this criteria is found, then the catalogue and detected stars are flagged as paired. Figure 5 shows a result from cloudiness analysis of partly cloudy night sky. The ratio of paired / unpaired stars (for all segments) gives final cloudiness of the night sky. Typically, we check about 900 catalogue stars for the whole observed sky (up to 60 · which is the maximal operating zenith angle of Cherenkov telescopes). The error of the algorithm was calculated using artificial cloud simulations and the total error of the cloudiness calculation is ± 5%. The cloudiness of the night sky is computed up to the limiting angle with 10 · steps.</text>
<text><location><page_3><loc_52><loc_25><loc_91><loc_35></location>Brightness of the night sky is evaluated directly from a light flux. The system has been calibrated using a darkness sky measurement tool Unihedron SQM-LE [4]. The night sky was scanned overnight and recorded data both from the camera and SQM instrument were compared to each other. Subsequently the conversion formula is obtained. The data represent the brightness of the night sky in mag / arcsec 2 and in visible light spectrum.</text>
<section_header_level_1><location><page_3><loc_52><loc_22><loc_81><loc_23></location>1.4 Limits of the cloudiness analysis</section_header_level_1>
<text><location><page_3><loc_51><loc_5><loc_91><loc_22></location>The measurement using the ASC is limited by local weather conditions, camera misting and Moon position. The local weather limitations mean for example rain or snow, fog, haze and water condensation on the camera. Examples of rain, fog and snow are shown in Figures 4, 3 and 7 respectively. The algorithm usually fails if the lens of the ASC is obscured in some way, for example by snow, water resulting from rain fall, haze or fog surrounding the Atmoscope, or even bird droppings from birds found locally at the Aar and Tenerife sites. The presence of the Moon saturates the image and increases the background in the image, and consequently the algorithm for cloud detection could not be used at such times.</text>
<figure>
<location><page_4><loc_11><loc_64><loc_47><loc_90></location>
<caption>Fig. 7 : Snow covers the whole FOV of the ASC and no star is visible in this case.</caption>
</figure>
<section_header_level_1><location><page_4><loc_10><loc_55><loc_46><loc_57></location>2 Applications, operation and advantages</section_header_level_1>
<text><location><page_4><loc_9><loc_25><loc_49><loc_55></location>The main advantage of the all-sky camera is that it is a fully autonomous and self-reliant tool which can be used to monitor the night sky in even the world's most remote places, which happen to be just the ones most promising for astronomically related applications. When using solar power, the only requirements for the deployment of the system are the ability to transport the device and the solar power system to the location. The availability of internet connection is useful in order to ensure fast response to possible issues, but is by no means a requirement. The autonomy of the camera significantly contributes to the extremely low cost for its operation. The costs involved are concentrated mainly in the deployment phase, namely the costs of the parts and assembly of the camera, shipping (varies depending on destination, the shippable weight being 10 kg), the solar power system, if there is no pre-existing power on-site (the 180 W required by the system) and installation (two people working for two or three days). After installation, the camera can operate without maintenance for years. However, it is advantageous to have local support for the case of unexpected external hindrances, but in most cases, a nonspecialist local personnel is sufficient to fix most of the possible issues.</text>
<section_header_level_1><location><page_4><loc_10><loc_21><loc_23><loc_22></location>3 Conclusions</section_header_level_1>
<text><location><page_4><loc_9><loc_10><loc_49><loc_20></location>ASC camera for night sky measurement was presented as a versatile remote system for cloudiness and night sky brightness characterization. Eight ASC's are being used to measure night sky parameters at the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The data will be used to characterize the candidate site and to use the data in the selection process for Southern and Northern CTA site.</text>
<text><location><page_4><loc_10><loc_5><loc_49><loc_8></location>Acknowledgment: This work is supported by Ministry of Education of the Czech Republic, under the projects EUPROII LE13012 and Mobility 7AMB12AR013. We gratefully acknowl-</text>
<figure>
<location><page_4><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<text><location><page_4><loc_52><loc_88><loc_90><loc_90></location>edge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22.</text>
<section_header_level_1><location><page_4><loc_52><loc_84><loc_61><loc_85></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_4><loc_52><loc_81><loc_83><loc_83></location>[1] CTA consortium, Astropart. Phys. 43 , (2013), 3, doi:10.1016/j.astropartphys.2013.01.007.</list_item>
<list_item><location><page_4><loc_52><loc_78><loc_87><loc_81></location>[2] T. Bulik, I. Puerto-Gimenez, 32nd INTERNATIONAL COSMIC RAY CONFERENCE,BEIJING 2011, doi:10.7529/ICRC2011/V09/1257.</list_item>
<list_item><location><page_4><loc_52><loc_77><loc_83><loc_78></location>[3] http://tdc-www.harvard.edu/catalogs/bsc5.html.</list_item>
<list_item><location><page_4><loc_52><loc_76><loc_81><loc_77></location>[4] http://www.unihedron.com/projects/darksky/.</list_item>
</document>
[
{
"title": "All Sky Cameras for the characterization of the Cherenkov Telescope Array candidate sites",
"content": "DUSAN MANDAT 1 , MIROSLAV PECH 1 , JAN EBR 1 , MIROSLAV HRABOVSKY 1 , MICHAEL PROUZA 1 , TOMASZ BULIK 2 , INGOMAR ALLEKOTTE 3 , FOR THE CTA CONSORTIUM. mandat@fzu.cz Abstract: The All Sky Camera (ASC) was developed as a universal device for the monitoring of the night sky quality. Eight ASCs are already installed and measure night sky parameters at eight of the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The ACS system consists of an astronomical CCD camera, a fish eye lens, a control computer and associated electronics. The measurement is carried out during astronomical night. The images are automatically taken every 5 minutes and automatically processed using the control computer of the device. The analysis results are the cloud fraction (the percentage of the sky covered by clouds) and night sky brightness (in mag/arcsec 2 ). Keywords: All Sky Camera, CTA, candidate sites, cloud fraction.",
"pages": [
1
]
},
{
"title": "1 Cloudiness characterization of CTA candidate sites",
"content": "The future CTA observatory [1] will include two observatories. One on the Southern and one on the Northern Hemisphere. There are nine candidate sites for the future CTA observatories - five at the Southern and four at the Northern Hemisphere. The goal of Site Selection Work Package [2] (SITE WP) is to characterize the candidate sites and produce relevant data for the Site Selection Committee. One of the important characteristic of the candidate site is the cloud fraction of the night sky and the amount of cloudless nights or more specifically cloudless time in a year. The ASC is an automated instrument for measurement of this parameter. Starting in November 2011 the following sites are being characterized using ASCs :",
"pages": [
1
]
},
{
"title": "1.1 System Description",
"content": "The basis of the system is an astronomical camera G1 -2000 (Moravian Instruments a.s., Czech Republic, www.mii.cz), which is equipped with a CCD chip ICX274AL by SONY company. This chip has quantum efficiency higher than 50% at 450-550 nm and its spectral sensitivy covers the range from approximately 400 to 900 nm. It has low read-out noise, a 16-bit ADC and a resolution of 1600 × 1200 pixels. The camera is equipped with a fish-eye varifocal lens (the field of view is 185 degrees) and an electronic iris control. This setup is capable of detecting a star with visual magnitude 6.3 in zenith. A miniPC computer with USB I / O controls the system and processes the data. The I / O card controls the iris and the power switch of the camera. All electronics and a camera body (see Figure 1) are weatherproof. The system power is supplied by a solar power system. The switching electronics turns the system ON after sunset and keeps it going during astronomical night. The system is OFF during the day (to save energy). Temperature of the electronics is controlled and stabilized using internal heating system during winter time to protect the ASC system. The camera is a part of an Atmoscope instrument. The Atmoscope measures local weather conditions and night sky brightness parameters. The Atmoscope is connected to central data storage using GPRS or WIFI connection depending on Atmoscope local position and nearby facilities. The quality of the link limits the amount and speed of the data transfer.",
"pages": [
1,
2
]
},
{
"title": "1.2 Calibration of the camera",
"content": "The fish-eye varifocal lens has 185 degrees field of view (FOV) and its aberration distorts the image. The correct transformation of the incident angle to the imaged pixel position is very important. The mechanism of the calibration is as follows: the light source (a spot as small as possible) is placed at a defined distance (at least 10 m), then the body of camera is rotated around the axis such that the incident light angle (corresponding to the FOV angle) varies between 0 and 90 degrees. The position of the light spot is analyzed for each anglular step. This procedure is repeated for different plane cuts of the lens FOV. The final data are analyzed and fitted with a polynomial function. This calibration is regularly checked on-site by comparing the position of detected stars with the stars in a catalogue. Onsite measurements are also used to determine the variation of the sensitivty of the system with the zenith angle.",
"pages": [
2
]
},
{
"title": "1.3 Data acquisition and principles of analysis",
"content": "The ASC system takes full sky images each 5 minutes (resp. 10 minutes - Aar and SAC). Examples of clear and partly cloudy night sky are shown in Figure 2 and 6. The onsite analysis (raw cloudiness analysis) follows immediately, the result is saved to two files. The first file consists of a compressed image and online analysis results, the second one consists of a RAW image and relevant data (UTC time, temperature of CCD, Moon and Sun position). The measurement process consists of the following steps. Initially, every exposure of a night sky is followed by a dark frame image so that subtraction of one from the other enables significant reduction of noise in the image. The horizontal coordinates of each image pixel are then calculated with the calibration data and calibration constants used to determine the horizontal coordinates, azimuth and elevation of each image pixel. The total charge of each pixel is proportional to the light intensity, but the intensity of the final image varies with respect to zenith angle due to vignetting and distortion of the lens. The calibration of this zenith angle intensity dependence is obtained by considering extinction in air masses, optical distortions and vignetting variation as a function of zenith angle. The resulting attenuation at, for example, 60 degrees zenith angle corresponds to 2 mag. To estimate the cloud fraction, we investigate the presence of stars in the field of view of the camera. We separate the full sky to approximately 70 segments and analyze each segment of the sky individually to obtain the cloudiness. As a star catalogue we use Yale Bright Star Catalog BSC5 [3], particularly the visual magnitues of the stars. First we estimate the position of each segment on the sky using the data from the calibration. This information is then further improved using the brightest stars from the catalogue in each segement. Once the position of a segment on the sky is fixed, the stars for the segment are read from the catalogue and selected according to the expected sensitivty at the given zenith angle. For every star in the catalogue, we look for a detected star within the angular limit of 1 deg. If a star that fits this criteria is found, then the catalogue and detected stars are flagged as paired. Figure 5 shows a result from cloudiness analysis of partly cloudy night sky. The ratio of paired / unpaired stars (for all segments) gives final cloudiness of the night sky. Typically, we check about 900 catalogue stars for the whole observed sky (up to 60 · which is the maximal operating zenith angle of Cherenkov telescopes). The error of the algorithm was calculated using artificial cloud simulations and the total error of the cloudiness calculation is ± 5%. The cloudiness of the night sky is computed up to the limiting angle with 10 · steps. Brightness of the night sky is evaluated directly from a light flux. The system has been calibrated using a darkness sky measurement tool Unihedron SQM-LE [4]. The night sky was scanned overnight and recorded data both from the camera and SQM instrument were compared to each other. Subsequently the conversion formula is obtained. The data represent the brightness of the night sky in mag / arcsec 2 and in visible light spectrum.",
"pages": [
2,
3
]
},
{
"title": "1.4 Limits of the cloudiness analysis",
"content": "The measurement using the ASC is limited by local weather conditions, camera misting and Moon position. The local weather limitations mean for example rain or snow, fog, haze and water condensation on the camera. Examples of rain, fog and snow are shown in Figures 4, 3 and 7 respectively. The algorithm usually fails if the lens of the ASC is obscured in some way, for example by snow, water resulting from rain fall, haze or fog surrounding the Atmoscope, or even bird droppings from birds found locally at the Aar and Tenerife sites. The presence of the Moon saturates the image and increases the background in the image, and consequently the algorithm for cloud detection could not be used at such times.",
"pages": [
3
]
},
{
"title": "2 Applications, operation and advantages",
"content": "The main advantage of the all-sky camera is that it is a fully autonomous and self-reliant tool which can be used to monitor the night sky in even the world's most remote places, which happen to be just the ones most promising for astronomically related applications. When using solar power, the only requirements for the deployment of the system are the ability to transport the device and the solar power system to the location. The availability of internet connection is useful in order to ensure fast response to possible issues, but is by no means a requirement. The autonomy of the camera significantly contributes to the extremely low cost for its operation. The costs involved are concentrated mainly in the deployment phase, namely the costs of the parts and assembly of the camera, shipping (varies depending on destination, the shippable weight being 10 kg), the solar power system, if there is no pre-existing power on-site (the 180 W required by the system) and installation (two people working for two or three days). After installation, the camera can operate without maintenance for years. However, it is advantageous to have local support for the case of unexpected external hindrances, but in most cases, a nonspecialist local personnel is sufficient to fix most of the possible issues.",
"pages": [
4
]
},
{
"title": "3 Conclusions",
"content": "ASC camera for night sky measurement was presented as a versatile remote system for cloudiness and night sky brightness characterization. Eight ASC's are being used to measure night sky parameters at the candidate sites of the Cherenkov Telescope Array (CTA) gamma-ray observatory. The data will be used to characterize the candidate site and to use the data in the selection process for Southern and Northern CTA site. Acknowledgment: This work is supported by Ministry of Education of the Czech Republic, under the projects EUPROII LE13012 and Mobility 7AMB12AR013. We gratefully acknowl- edge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22.",
"pages": [
4
]
}
]
2013ICRC...33.2823K
https://arxiv.org/pdf/1307.3386.pdf
<document>
<figure>
<location><page_1><loc_69><loc_88><loc_89><loc_94></location>
</figure>
<section_header_level_1><location><page_1><loc_10><loc_83><loc_91><loc_86></location>Development of the Photomultiplier-Tube Readout System for the CTA Large Size Telescope</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_10><loc_81><loc_84><loc_82></location>H. KUBO 1 , R. PAOLETTI 2 , Y. AWANE 1 , A. BAMBA 3 , M. BARCELO 4 , J.A. BARRIO 5 , O. BLANCH 4 , J. BOIX 4 ,</list_item>
</unordered_list>
<text><location><page_1><loc_9><loc_72><loc_88><loc_81></location>C. DELGADO 6 , D. FINK 7 , D. GASCON 8 , S. GUNJI 9 , R. HAGIWARA 9 , Y. HANABATA 10 , K. HATANAKA 1 , M. HAYASHIDA 10 , M. IKENO 11 , S. KABUKI 12 , H. KATAGIRI 13 , J. KATAOKA 14 , Y. KONNO 1 , S. KOYAMA 15 , T. KISHIMOTO 1 , J. KUSHIDA 16 , G. MART'INEZ 6 , S. MASUDA 1 , J.M. MIRANDA 17 , R. MIRZOYAN 7 , T. MIZUNO 18 , T. NAGAYOSHI 15 , D. NAKAJIMA 7 , T. NAKAMORI 9 , H. OHOKA 10 , A. OKUMURA 19 , R. ORITO 20 , T. SAITO 1 , A. SANUY 8 , H. SASAKI 21 , M. SAWADA 3 , T. SCHWEIZER 7 , R. SUGAWARA 20 , K.-H. SULANKE 22 , H. TAJIMA 19 , M. TANAKA 11 , S. TANAKA 13 , L.A. TEJEDOR 5 , Y. TERADA 15 , M. TESHIMA 7 , 10 , F. TOKANAI 9 , Y. TSUCHIYA 1 , 11 15 13 21</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_71><loc_73><loc_72></location>T. UCHIDA , H. UENO , K. UMEHARA , T. YAMAMOTO FOR THE CTA CONSORTIUM.</list_item>
</unordered_list>
<text><location><page_1><loc_9><loc_70><loc_10><loc_70></location>1</text>
<text><location><page_1><loc_10><loc_69><loc_71><loc_70></location>Department of Physics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_68><loc_44><loc_69></location>2 Universit'a di Siena, and INFN Pisa, I-53100 Siena, Italy</list_item>
<list_item><location><page_1><loc_10><loc_67><loc_78><loc_68></location>3 Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa, 252-5258, Japan</list_item>
<list_item><location><page_1><loc_9><loc_65><loc_61><loc_67></location>4 Institut de Fsica d'Altes Energies, Edifici Cn, Campus UAB, 08193 Bellaterra, Spain</list_item>
<list_item><location><page_1><loc_10><loc_64><loc_74><loc_65></location>5 Grupo de Altas Energas, Universidad Complutense de Madrid, Av Complutense s/n, 28040 Madrid, Spain</list_item>
<list_item><location><page_1><loc_10><loc_63><loc_43><loc_64></location>6 CIEMAT, Avda. Complutense 22, 28040 Madrid, Spain</list_item>
<list_item><location><page_1><loc_9><loc_62><loc_59><loc_63></location>7 Max-Planck-Institut fur Physik, Fohringer Ring 6, D 80805 Munchen, Germany</list_item>
<list_item><location><page_1><loc_10><loc_60><loc_59><loc_62></location>8 Universitat de Barcelona (ICC-UB), Mart i Franqus, 1, 08028, Barcelona, Spain</list_item>
<list_item><location><page_1><loc_10><loc_59><loc_73><loc_60></location>9 Department of Physics, Faculty of Science, Yamagata University, Yamagata, Yamagata 990-8560, Japan</list_item>
<list_item><location><page_1><loc_9><loc_58><loc_67><loc_59></location>10 Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan</list_item>
<list_item><location><page_1><loc_9><loc_56><loc_60><loc_58></location>11 Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki 305-0801, Japan</list_item>
<list_item><location><page_1><loc_9><loc_55><loc_79><loc_56></location>12 Department of Radiation Oncology, Tokai University, 143 Shimokasuya, Isehara-shi, Kanagawa 259-1193, Japan</list_item>
<list_item><location><page_1><loc_9><loc_54><loc_53><loc_55></location>13 College of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan</list_item>
<list_item><location><page_1><loc_9><loc_53><loc_83><loc_54></location>14 Research Institute for Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan</list_item>
<list_item><location><page_1><loc_9><loc_51><loc_85><loc_53></location>15 Graduate School of Science and Engineering, Saitama University, 255 Simo-Ohkubo, Sakura-ku, Saitama 338-8570, Japan</list_item>
<list_item><location><page_1><loc_9><loc_50><loc_72><loc_51></location>16 Department of Physics, Tokai University, 4-1-1 Kita-Kaname, Hiratsuka, Kanagawa 259-1292, Japan</list_item>
<list_item><location><page_1><loc_9><loc_49><loc_73><loc_50></location>17 Grupo de Electronica, Universidad Complutense de Madrid, Av. Complutense s/n, 28040 Madrid, Spain</list_item>
<list_item><location><page_1><loc_9><loc_48><loc_84><loc_49></location>18 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-hiroshima, 739-8526 Japan</list_item>
<list_item><location><page_1><loc_9><loc_46><loc_70><loc_47></location>19 Solar-Terrestrial Environment Laboratory, Nagoya University, Chikusa, Nagoya 464-8601, Japan</list_item>
<list_item><location><page_1><loc_10><loc_45><loc_70><loc_46></location>20 Faculty of Integrated Arts and Sciences, The University of Tokushima, Tokushima 770-8502, Japan</list_item>
<list_item><location><page_1><loc_10><loc_44><loc_75><loc_45></location>21 Department of Physics, Konan University, 8-9-1 Okamoto Higashinada-ku Kobe, Hyogo, 658-8501 Japan</list_item>
<list_item><location><page_1><loc_10><loc_42><loc_58><loc_44></location>22 Deutsches Elektronen-Synchrotron, Platanenallee 6, 15738 Zeuthen, Germany</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_40><loc_30><loc_41></location>kubo@cr.scphys.kyoto-u.ac.jp</text>
<text><location><page_1><loc_15><loc_22><loc_91><loc_38></location>Abstract: We have developed a prototype of the photomultiplier tube (PMT) readout system for the Cherenkov Telescope Array (CTA) Large Size Telescope (LST). Two thousand PMTs along with their readout systems are arranged on the focal plane of each telescope, with one readout system per 7-PMT cluster. The Cherenkov light pulses generated by the air showers are detected by the PMTs and amplified in a compact, low noise and wide dynamic range gain block. The output of this block is then digitized at a sampling rate of the order of GHz using the Domino Ring Sampler DRS4, an analog memory ASIC developed at Paul Scherrer Institute. The sampler has 1,024 capacitors per channel and four channels are cascaded for increased depth. After a trigger is generated in the system, the charges stored in the capacitors are digitized by an external slow sampling ADC and then transmitted via Gigabit Ethernet. An onboard FPGA controls the DRS4, trigger threshold, and Ethernet transfer. In addition, the control and monitoring of the Cockcroft-Walton circuit that provides high voltage for the 7-PMT cluster are performed by the same FPGA. A prototype named Dragon has been developed that has successfully sampled PMT signals at a rate of 2 GHz, and generated single photoelectron spectra.</text>
<text><location><page_1><loc_16><loc_19><loc_72><loc_20></location>Keywords: Imaging Atmospheric Cherenkov Telescope, Gamma-rays, Electronics</text>
<section_header_level_1><location><page_1><loc_10><loc_15><loc_23><loc_16></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_5><loc_49><loc_14></location>A ground-based imaging atmospheric Cherenkov telescope (IACT) measures Cherenkov light from an extended air shower (EAS) generated by the interaction between very high energy (VHE) gamma rays and the upper atmosphere. Night sky background (NSB) also enters a pixel photon sensor of the focal plane of the IACT with a rate of the order of 10-100 MHz, depending on the mirror size</text>
<text><location><page_1><loc_51><loc_5><loc_91><loc_16></location>and the pixel size. In some region of the Galactic plane, the NSB can reach on average up to several hundred MHz. The NSB therefore becomes noise that affects the sensitivity and the energy threshold. Given this NSB pollution, and the fact that the duration of Cherenkov light from EAS is a few nanoseconds, a fast digitization speed of the readout system coupled to a fast photosensor like a photomultiplier tube (PMT) is beneficial to increase the pixels' signal-</text>
<text><location><page_2><loc_10><loc_79><loc_49><loc_90></location>to-noise ratio. In addition, this system should be compact and have low cost and low power consumption because each IACT possesses several thousand photon sensor pixels and the readout system attached to the sensors is in a camera container located at the focal position. Furthermore, a wide dynamic range of more than 8-10 bits is required to resolve a single photoelectron and have a wider energy range.</text>
<text><location><page_2><loc_9><loc_60><loc_49><loc_79></location>A commercial flash analog-to-digital converter (FADC) satisfies the requirement of a wide dynamic range. However, it is costly and consumes relatively high power of a few watts per channel. On the other hand, an analog memory application specific integrated circuit (ASIC), that consists of several hundred to several thousand switched capacitor arrays (SCA) per channel, can sample a signal at the order of GHz, with a wide dynamic range and lower power consumption. Several types of analog memory ASICs have been developed for applications in particle physics and cosmic ray physics. With respect to IACTs, a modified version of ARS0 [1], Swift Analog Memory (SAM) [2], and Domino Ring Sampler (DRS) [3] chips are used in the H.E.S.S.-I, H.E.S.S.-II, and MAGIC experiments, respectively.</text>
<text><location><page_2><loc_9><loc_39><loc_49><loc_59></location>Cherenkov Telescope Array (CTA) [4] is the next generation VHE gamma-ray observatory, which improves the sensitivity by a factor of 10 in the range 100 GeV-10 TeV and an extension to energies well below 100 GeV and above 100 TeV. CTA consists of telescopes having mirrors with size 23 m, 12 m, 3.5-7 m, and ∼ 50 m 2 , which are called large size telescope (LST), medium size telescope (MST), small size telescope (SST), and SchwarzschildCouder telescope (SCT), respectively. Several types of readout systems are developed for CTA with different analog memories for the requirements of LST, MST, SST, and SCT. At the same time, there has been progress in the development of photon sensors such as PMT and a Geigermode avalanche photodiode (or silicon photomultiplier) for CTA. At this time, the primary candidate for LST is a PMT.</text>
<text><location><page_2><loc_9><loc_24><loc_49><loc_38></location>Using the analog memory DRS version 4 (hereafter DRS4), we have so far developed three versions of prototypes for the PMT readout system for CTA, named Dragon . Using the first version of the prototype, we demonstrated that the waveform of a PMT signal can be well digitized with the DRS4 chip. The second version was developed with improvements made to the sampling depth of DRS4 and a trigger [5]. The third version was developed to be downsized and reduce the cost for mass production. In this paper, we report the design and performance of the third version of the prototype.</text>
<section_header_level_1><location><page_2><loc_10><loc_20><loc_35><loc_22></location>2 Design of Readout System</section_header_level_1>
<section_header_level_1><location><page_2><loc_10><loc_18><loc_21><loc_20></location>2.1 Overview</section_header_level_1>
<text><location><page_2><loc_9><loc_5><loc_49><loc_18></location>Two thousand PMTs and their readout systems are arranged on the focal plane of each LST telescope [6], with one readout system per 7-PMT cluster. We have developed the third prototype of the PMT readout system. Figures 1 and 2 show a photograph and block diagram of the prototype, respectively. The prototype consists of a 7-PMT cluster, a slow control board and a DRS4 readout board. The total size is 14 cm × 48 cm. The 7-PMT cluster and the slow control board are described in detail in reference [7]. In this paper, a brief description is provided.</text>
<figure>
<location><page_2><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<text><location><page_2><loc_51><loc_76><loc_91><loc_90></location>Our 7-PMT cluster consists of seven head-on type PMTs with a super-bialkali photocathode and 8-stage dynodes (Hamamatsu Photonics, R11920-100 with a diameter of 38 mm) [8], a Cockcroft-Walton (CW) circuit for high voltage supply to the PMTs (designed by Hamamatsu Photonics), and a preamplifier board. A signal from the PMT is amplified by a preamplifier, and fed to the DRS4 readout board. A commercial preamplifier, Mini-circuits LEE-39+, is used in the current Dragon (Fig. 3). A preamplifier board with an ASIC designed for CTA, PACTA [9], has been recently developed and will be used in the next Dragon .</text>
<figure>
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<caption>Figure 1 : Photograph of the 7-PMT cluster and readout system (ver. 3).</caption>
</figure>
<figure>
<location><page_2><loc_52><loc_39><loc_89><loc_54></location>
<caption>Figure 2 : Block diagram of the 7-PMT cluster and readout system (ver. 3).</caption>
</figure>
<text><location><page_2><loc_51><loc_14><loc_91><loc_35></location>Onthe DRS4 readout board the preamplified signal is divided into three lines: a high gain channel, a low gain channel, and a trigger channel. The high and low gain channels are connected to DRS4 chips. The signal is sampled at a rate of the order of GHz and the waveform is stored in a SCA in DRS4. When a trigger is generated in the trigger circuit, the voltages stored in the capacitor array are sequentially output and then digitized by an external slow sampling ( ∼ 30 MHz) ADC. The digitized data is sent to a field programmable gate array (FPGA) and then transmitted to a Gigabit Ethernet transceiver and a backplane via a data input/output (I/O) connector. The FPGA controls a static random access memory (SRAM) that stores large amounts of data before transmission and a digital-toanalog converter (DAC) used for thresholding in the trigger circuit.</text>
<text><location><page_2><loc_52><loc_5><loc_91><loc_14></location>The slow control board (Fig.1) is equipped with a generator for test pulses that are fed to the preamplifier, a temperature and humidity sensor with I2C interface, a DAC for setting the voltage of the CW high voltage circuit, and an ADC for monitoring both the CW circuit and the DC anode current. These devices on the slow control board are controlled by a complex programmable logic device</text>
<figure>
<location><page_3><loc_12><loc_83><loc_46><loc_91></location>
<caption>Figure 5 : Block diagram of the main amplifiers.</caption>
</figure>
<figure>
<location><page_3><loc_59><loc_83><loc_82><loc_91></location>
<caption>Figure 3 : Preamplifier board with a commercial amplifier (left) and the PACTA (right).</caption>
</figure>
<text><location><page_3><loc_9><loc_70><loc_49><loc_78></location>(CPLD). Since the CPLD communicates with the FPGA on the DRS4 readout board, the data to and from the CPLD is sent via the Ethernet. The power supply to the DRS4 readout board and the slow control board is ± 3.3V and +5V. The total power consumption is ∼ 2W per readout channel.</text>
<section_header_level_1><location><page_3><loc_10><loc_67><loc_31><loc_69></location>2.2 DRS4 Readout Board</section_header_level_1>
<text><location><page_3><loc_9><loc_46><loc_49><loc_67></location>Figure 4 shows a photograph of the DRS4 readout board with a size of 14 cm × 30 cm, 10 cm shorter than the version 2. The slow control board attached to the 7-PMT cluster is connected to the DRS4 readout board from the right side via two card-edge connectors. The DRS4 board has main amplifiers, eight DRS4 chips, ADCs for digitizing a signal stored in the capacitor array in DRS4 at a sampling frequency of 33 MHz, a DAC to control the DRS4, an FPGA, a 18Mbit SRAM, a Gigabit Ethernet transceiver, and a data I/O connector to the backplane. A Xilinx Vertex4 FPGA was adopted in the version 2. In the version 3, a Xilinx Spartan-6 FPGA is used to reduce the cost for mass production. In addition, analog level 0 (L0) and 1 (L1) trigger mezzanines or a digital level 0 (L0) trigger mezzanine are mounted on the DRS4 board. Details of these parts are described in the following subsections.</text>
<figure>
<location><page_3><loc_10><loc_29><loc_49><loc_46></location>
<caption>Figure 4 : Photograph of the DRS4 readout board (ver. 3).</caption>
</figure>
<section_header_level_1><location><page_3><loc_10><loc_24><loc_26><loc_25></location>2.3 Main Amplifier</section_header_level_1>
<text><location><page_3><loc_9><loc_5><loc_49><loc_23></location>Figure 5 shows a block diagram of the main amplifiers. It is designed to have a bandwidth greater than 250 MHz for the high gain channel, lower power consumption, and a dynamic range of 0.2-1000 photoelectrons for LST. A preamplified signal from a PMT with a typical gain of 4 × 10 4 is fed to the main amplifier. The signal is amplified using two differential amplifiers to meet the requirement for bandwidth and power consumption. For the high gain channel, it is amplified by a gain of 9 using Analog Devices, ADA4927. For the trigger, it is amplified by a gain of 4 using Analog Devices, ADA4927 and ADA4950. For the low gain channel, the preamplified signal is attenuated by 1/4 and then buffered with a differential amplifier using Analog Devices, ADA4950.</text>
<section_header_level_1><location><page_3><loc_52><loc_77><loc_61><loc_78></location>2.4 Trigger</section_header_level_1>
<text><location><page_3><loc_51><loc_71><loc_91><loc_76></location>The DRS4 readout board has been designed to host an analog trigger [10] or digital trigger [5]. The different elements of the trigger system will be placed at different locations on the readout electronics, as shown in Fig. 4.</text>
<text><location><page_3><loc_52><loc_57><loc_91><loc_71></location>For the analog trigger, the decision boards (L0/L1) are placed on the readout board itself, while for the digital trigger the L0 board is placed in the frontend and the L1 in the backplane, keeping both analog and digital L0 as close as possible after the signals from PMTs. The distribution boards are placed at the backplane of the readout board, in order to send and receive trigger signals efficiently among neighbouring clusters. The choice of using trigger mezzanines and sharing of connectors has allowed the independent study of both analog and digital solution in the development phase.</text>
<section_header_level_1><location><page_3><loc_52><loc_54><loc_65><loc_55></location>2.5 DRS4 Chip</section_header_level_1>
<text><location><page_3><loc_51><loc_37><loc_91><loc_54></location>The DRS chip is being developed at the Paul Scherrer Institute (PSI), Switzerland, for the MEG experiment [3]. The latest version, DRS4, is also used in the MAGIC experiment. The DRS4 chip includes nine differential input channels at a sampling speed of 700 MS/s-5 GS/s, with a bandwidth of 950 MHz, and a low noise of 0.35 mV after offset correction. The analog waveform is stored in 1,024 sampling capacitors per channel and the waveform can be read out after sampling via a shift register that is clocked at a maximum of 33 MHz for digitization using an external ADC. The power consumption of the DRS4 chip is 17.5 mWper channel at 2 GS/s sampling rate. The chip is fabricated using the 0.25 µ mCMOStechnology.</text>
<text><location><page_3><loc_52><loc_28><loc_91><loc_37></location>It is possible to cascade two or more channels to obtain deeper sampling depth. In the Dragon , four channels were cascaded, leading to a sampling depth of 4,096 for each PMT signal, which corresponds to a depth of 4 µ s at 1 GS/s sampling rate. It enables continuous sampling until the stereo coincidence trigger confirms the local telescope trigger, and initializes the camera readout.</text>
<section_header_level_1><location><page_3><loc_52><loc_25><loc_86><loc_26></location>2.6 FPGA-based Gigabit Ethernet (SiTCP)</section_header_level_1>
<text><location><page_3><loc_51><loc_12><loc_91><loc_25></location>The digitized waveform data and the monitor/control data are transmitted via Ethernet with only two devices: FPGA and Gigabit Ethernet transceiver (PHY). This simple composition is available on a hardware-based TCP processor, SiTCP [11]. The circuit size of SiTCP in the FPGA is ∼ 3000 slices, which is small enough to allow implementation with user circuits on a single FPGA. In addition, the SiTCP has an advantage in that the throughputs in both directions can simultaneously reach the theoretical upper limits of Gigabit Ethernet.</text>
<section_header_level_1><location><page_3><loc_52><loc_9><loc_82><loc_10></location>2.7 Mini-Camera with Three Clusters</section_header_level_1>
<text><location><page_3><loc_51><loc_5><loc_91><loc_9></location>A prototype with three clusters of PMTs and their readout boards was constructed to develop the trigger system between multiple clusters (Fig. 6) [10].</text>
<figure>
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</figure>
<figure>
<location><page_4><loc_9><loc_80><loc_29><loc_90></location>
<caption>Figure 8 shows normalized gains for high and low gain channels as a function of frequency. The bandwidths for high and low gain channels are 260 MHz and 190 MHz at -3dB, respectively while those of the DRS4 readout board combined with a preamplifier (LEE-39+) were 250 MHz and 180 MHz at -3dB for high and low gain channels, respectively.</caption>
</figure>
<figure>
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</figure>
<figure>
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<caption>Figure 6 : Prototype with three clusters of PMTs and their readout boards.</caption>
</figure>
<section_header_level_1><location><page_4><loc_10><loc_72><loc_40><loc_74></location>3 Performance of Readout System</section_header_level_1>
<text><location><page_4><loc_9><loc_65><loc_49><loc_72></location>The performance of the DRS4 readout system Dragon was measured. Figure 7 shows that dynamic ranges of high and low gain channels are 60 p.e. and 2500 p.e., which satisfied the requirement of > 1000 photoelectrons. Measured noise level for the high gain channel is 0.1 p.e. (rms).</text>
<text><location><page_4><loc_9><loc_36><loc_49><loc_56></location>After the above measurements, the performance of the DRS4 readout board attached to a PMT was investigated. Figure 9 shows a pulse shape of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with a UV laser and the DRS4 readout system at a sampling rate of 2 GS/s. The PMT signal having a width of ∼ 3 ns (FWHM) and a height corresponding to ∼ 3 photoelectrons was successfully digitized. Measured dead time for readout of 60 cells in the DRS4 chip is 0.9% and 5.4% at 1 kHz and 7 kHz, respectively. Figure 10 shows a single photoelectron spectrum of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with Dragon and an LED at a sampling rate of 2 GS/s. In the figure, a single photoelectron peak is clearly seen. The signal to noise ratio defined as (single-photoelectron mean - pedestal mean) / pedestal r.m.s. is 3.6.</text>
<figure>
<location><page_4><loc_16><loc_22><loc_41><loc_34></location>
<caption>Figure7 : Dynamic ranges of the DRS4 readout board.</caption>
</figure>
<section_header_level_1><location><page_4><loc_10><loc_15><loc_22><loc_16></location>4 Conclusion</section_header_level_1>
<text><location><page_4><loc_9><loc_5><loc_49><loc_14></location>We have developed a prototype 7-PMT cluster readout system for the large-size telescope of the next generation VHE gammaray observatory,CTA. In the readout system named Dragon , a PMT signal is amplified, and its waveform is then digitized at a sampling rate of the order of GHz using an analog memory ASIC DRS4 that has 4,096 capacitors per readout channel. Using the prototype system at-</text>
<figure>
<location><page_4><loc_58><loc_62><loc_84><loc_75></location>
<caption>Figure 8 : Bandwidths of high-(square) and low-(triangle) gain channels.Figure 9 : Pulse shape of the PMT signal measured with the DRS4 readout system at a sampling rate of 2 GS/s.</caption>
</figure>
<figure>
<location><page_4><loc_59><loc_46><loc_82><loc_58></location>
<caption>Figure 10 : Single photoelectron spectrum of the PMT signal measured with the DRS4 readout system. The horizontal axis is in units of mV × ns.</caption>
</figure>
<text><location><page_4><loc_52><loc_28><loc_91><loc_39></location>tached to a PMT with a Cockcroft-Walton circuit, we successfully obtained a pulse shape of the signal of the PMT detecting a UV light at a sampling rate of 2 GS/s, and also a single photoelectron spectrum. Evaluation of the readout system's performance is ongoing, and a prototype with several dozen clusters and their readout boards will be constructed as the next step for the development of a full telescope camera.</text>
<text><location><page_4><loc_52><loc_23><loc_91><loc_26></location>Acknowledgment: We gratefully acknowledge support from the agencies and organisations listed in this page: http://www.ctaobservatory.org/?q=node/22</text>
<section_header_level_1><location><page_4><loc_52><loc_20><loc_61><loc_21></location>References</section_header_level_1>
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<list_item><location><page_4><loc_52><loc_18><loc_86><loc_19></location>[1] D. Lachartre, F. Feinstein, NIMA 442 (2000) 99-104</list_item>
<list_item><location><page_4><loc_52><loc_17><loc_81><loc_18></location>[2] E. Delagnes et. al., NIMA 567 (2006) 21-26</list_item>
<list_item><location><page_4><loc_52><loc_15><loc_86><loc_17></location>[3] S. Ritt, R. Dinapoli, U. Hartmann, NIMA 623 (2010) 486-488</list_item>
<list_item><location><page_4><loc_52><loc_13><loc_90><loc_15></location>[4] The CTA Consortium, Experimental Astronomy 32 (2011) 193-316</list_item>
<list_item><location><page_4><loc_52><loc_12><loc_89><loc_13></location>[5] H. Kubo et al., proceeding of ICRC2011 9 (2011) 179-182</list_item>
<list_item><location><page_4><loc_52><loc_11><loc_80><loc_12></location>[6] O. Blanch et al., these proceedings, ID-776</list_item>
<list_item><location><page_4><loc_52><loc_9><loc_90><loc_10></location>[7] R. Orito et. al., proceeding of ICRC2011 9 (2011) 171-174</list_item>
<list_item><location><page_4><loc_52><loc_8><loc_80><loc_9></location>[8] T. Toyama et al., these proceedings, ID-684</list_item>
<list_item><location><page_4><loc_52><loc_7><loc_78><loc_8></location>[9] A. Sanuy et al., JINST 7 (2012) C01100</list_item>
<list_item><location><page_4><loc_52><loc_6><loc_81><loc_7></location>[10] J.A. Barrio et al., these proceedings, ID-396</list_item>
<list_item><location><page_4><loc_52><loc_5><loc_81><loc_6></location>[11] T. Uchida, IEEE TNS 55 (2008) 1631-1637</list_item>
</unordered_list>
<figure>
<location><page_4><loc_79><loc_92><loc_91><loc_96></location>
</figure>
</document>
[
{
"title": "Development of the Photomultiplier-Tube Readout System for the CTA Large Size Telescope",
"content": "C. DELGADO 6 , D. FINK 7 , D. GASCON 8 , S. GUNJI 9 , R. HAGIWARA 9 , Y. HANABATA 10 , K. HATANAKA 1 , M. HAYASHIDA 10 , M. IKENO 11 , S. KABUKI 12 , H. KATAGIRI 13 , J. KATAOKA 14 , Y. KONNO 1 , S. KOYAMA 15 , T. KISHIMOTO 1 , J. KUSHIDA 16 , G. MART'INEZ 6 , S. MASUDA 1 , J.M. MIRANDA 17 , R. MIRZOYAN 7 , T. MIZUNO 18 , T. NAGAYOSHI 15 , D. NAKAJIMA 7 , T. NAKAMORI 9 , H. OHOKA 10 , A. OKUMURA 19 , R. ORITO 20 , T. SAITO 1 , A. SANUY 8 , H. SASAKI 21 , M. SAWADA 3 , T. SCHWEIZER 7 , R. SUGAWARA 20 , K.-H. SULANKE 22 , H. TAJIMA 19 , M. TANAKA 11 , S. TANAKA 13 , L.A. TEJEDOR 5 , Y. TERADA 15 , M. TESHIMA 7 , 10 , F. TOKANAI 9 , Y. TSUCHIYA 1 , 11 15 13 21 1 Department of Physics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan kubo@cr.scphys.kyoto-u.ac.jp Abstract: We have developed a prototype of the photomultiplier tube (PMT) readout system for the Cherenkov Telescope Array (CTA) Large Size Telescope (LST). Two thousand PMTs along with their readout systems are arranged on the focal plane of each telescope, with one readout system per 7-PMT cluster. The Cherenkov light pulses generated by the air showers are detected by the PMTs and amplified in a compact, low noise and wide dynamic range gain block. The output of this block is then digitized at a sampling rate of the order of GHz using the Domino Ring Sampler DRS4, an analog memory ASIC developed at Paul Scherrer Institute. The sampler has 1,024 capacitors per channel and four channels are cascaded for increased depth. After a trigger is generated in the system, the charges stored in the capacitors are digitized by an external slow sampling ADC and then transmitted via Gigabit Ethernet. An onboard FPGA controls the DRS4, trigger threshold, and Ethernet transfer. In addition, the control and monitoring of the Cockcroft-Walton circuit that provides high voltage for the 7-PMT cluster are performed by the same FPGA. A prototype named Dragon has been developed that has successfully sampled PMT signals at a rate of 2 GHz, and generated single photoelectron spectra. Keywords: Imaging Atmospheric Cherenkov Telescope, Gamma-rays, Electronics",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "A ground-based imaging atmospheric Cherenkov telescope (IACT) measures Cherenkov light from an extended air shower (EAS) generated by the interaction between very high energy (VHE) gamma rays and the upper atmosphere. Night sky background (NSB) also enters a pixel photon sensor of the focal plane of the IACT with a rate of the order of 10-100 MHz, depending on the mirror size and the pixel size. In some region of the Galactic plane, the NSB can reach on average up to several hundred MHz. The NSB therefore becomes noise that affects the sensitivity and the energy threshold. Given this NSB pollution, and the fact that the duration of Cherenkov light from EAS is a few nanoseconds, a fast digitization speed of the readout system coupled to a fast photosensor like a photomultiplier tube (PMT) is beneficial to increase the pixels' signal- to-noise ratio. In addition, this system should be compact and have low cost and low power consumption because each IACT possesses several thousand photon sensor pixels and the readout system attached to the sensors is in a camera container located at the focal position. Furthermore, a wide dynamic range of more than 8-10 bits is required to resolve a single photoelectron and have a wider energy range. A commercial flash analog-to-digital converter (FADC) satisfies the requirement of a wide dynamic range. However, it is costly and consumes relatively high power of a few watts per channel. On the other hand, an analog memory application specific integrated circuit (ASIC), that consists of several hundred to several thousand switched capacitor arrays (SCA) per channel, can sample a signal at the order of GHz, with a wide dynamic range and lower power consumption. Several types of analog memory ASICs have been developed for applications in particle physics and cosmic ray physics. With respect to IACTs, a modified version of ARS0 [1], Swift Analog Memory (SAM) [2], and Domino Ring Sampler (DRS) [3] chips are used in the H.E.S.S.-I, H.E.S.S.-II, and MAGIC experiments, respectively. Cherenkov Telescope Array (CTA) [4] is the next generation VHE gamma-ray observatory, which improves the sensitivity by a factor of 10 in the range 100 GeV-10 TeV and an extension to energies well below 100 GeV and above 100 TeV. CTA consists of telescopes having mirrors with size 23 m, 12 m, 3.5-7 m, and ∼ 50 m 2 , which are called large size telescope (LST), medium size telescope (MST), small size telescope (SST), and SchwarzschildCouder telescope (SCT), respectively. Several types of readout systems are developed for CTA with different analog memories for the requirements of LST, MST, SST, and SCT. At the same time, there has been progress in the development of photon sensors such as PMT and a Geigermode avalanche photodiode (or silicon photomultiplier) for CTA. At this time, the primary candidate for LST is a PMT. Using the analog memory DRS version 4 (hereafter DRS4), we have so far developed three versions of prototypes for the PMT readout system for CTA, named Dragon . Using the first version of the prototype, we demonstrated that the waveform of a PMT signal can be well digitized with the DRS4 chip. The second version was developed with improvements made to the sampling depth of DRS4 and a trigger [5]. The third version was developed to be downsized and reduce the cost for mass production. In this paper, we report the design and performance of the third version of the prototype.",
"pages": [
1,
2
]
},
{
"title": "2.1 Overview",
"content": "Two thousand PMTs and their readout systems are arranged on the focal plane of each LST telescope [6], with one readout system per 7-PMT cluster. We have developed the third prototype of the PMT readout system. Figures 1 and 2 show a photograph and block diagram of the prototype, respectively. The prototype consists of a 7-PMT cluster, a slow control board and a DRS4 readout board. The total size is 14 cm × 48 cm. The 7-PMT cluster and the slow control board are described in detail in reference [7]. In this paper, a brief description is provided. Our 7-PMT cluster consists of seven head-on type PMTs with a super-bialkali photocathode and 8-stage dynodes (Hamamatsu Photonics, R11920-100 with a diameter of 38 mm) [8], a Cockcroft-Walton (CW) circuit for high voltage supply to the PMTs (designed by Hamamatsu Photonics), and a preamplifier board. A signal from the PMT is amplified by a preamplifier, and fed to the DRS4 readout board. A commercial preamplifier, Mini-circuits LEE-39+, is used in the current Dragon (Fig. 3). A preamplifier board with an ASIC designed for CTA, PACTA [9], has been recently developed and will be used in the next Dragon . Onthe DRS4 readout board the preamplified signal is divided into three lines: a high gain channel, a low gain channel, and a trigger channel. The high and low gain channels are connected to DRS4 chips. The signal is sampled at a rate of the order of GHz and the waveform is stored in a SCA in DRS4. When a trigger is generated in the trigger circuit, the voltages stored in the capacitor array are sequentially output and then digitized by an external slow sampling ( ∼ 30 MHz) ADC. The digitized data is sent to a field programmable gate array (FPGA) and then transmitted to a Gigabit Ethernet transceiver and a backplane via a data input/output (I/O) connector. The FPGA controls a static random access memory (SRAM) that stores large amounts of data before transmission and a digital-toanalog converter (DAC) used for thresholding in the trigger circuit. The slow control board (Fig.1) is equipped with a generator for test pulses that are fed to the preamplifier, a temperature and humidity sensor with I2C interface, a DAC for setting the voltage of the CW high voltage circuit, and an ADC for monitoring both the CW circuit and the DC anode current. These devices on the slow control board are controlled by a complex programmable logic device (CPLD). Since the CPLD communicates with the FPGA on the DRS4 readout board, the data to and from the CPLD is sent via the Ethernet. The power supply to the DRS4 readout board and the slow control board is ± 3.3V and +5V. The total power consumption is ∼ 2W per readout channel.",
"pages": [
2,
3
]
},
{
"title": "2.2 DRS4 Readout Board",
"content": "Figure 4 shows a photograph of the DRS4 readout board with a size of 14 cm × 30 cm, 10 cm shorter than the version 2. The slow control board attached to the 7-PMT cluster is connected to the DRS4 readout board from the right side via two card-edge connectors. The DRS4 board has main amplifiers, eight DRS4 chips, ADCs for digitizing a signal stored in the capacitor array in DRS4 at a sampling frequency of 33 MHz, a DAC to control the DRS4, an FPGA, a 18Mbit SRAM, a Gigabit Ethernet transceiver, and a data I/O connector to the backplane. A Xilinx Vertex4 FPGA was adopted in the version 2. In the version 3, a Xilinx Spartan-6 FPGA is used to reduce the cost for mass production. In addition, analog level 0 (L0) and 1 (L1) trigger mezzanines or a digital level 0 (L0) trigger mezzanine are mounted on the DRS4 board. Details of these parts are described in the following subsections.",
"pages": [
3
]
},
{
"title": "2.3 Main Amplifier",
"content": "Figure 5 shows a block diagram of the main amplifiers. It is designed to have a bandwidth greater than 250 MHz for the high gain channel, lower power consumption, and a dynamic range of 0.2-1000 photoelectrons for LST. A preamplified signal from a PMT with a typical gain of 4 × 10 4 is fed to the main amplifier. The signal is amplified using two differential amplifiers to meet the requirement for bandwidth and power consumption. For the high gain channel, it is amplified by a gain of 9 using Analog Devices, ADA4927. For the trigger, it is amplified by a gain of 4 using Analog Devices, ADA4927 and ADA4950. For the low gain channel, the preamplified signal is attenuated by 1/4 and then buffered with a differential amplifier using Analog Devices, ADA4950.",
"pages": [
3
]
},
{
"title": "2.4 Trigger",
"content": "The DRS4 readout board has been designed to host an analog trigger [10] or digital trigger [5]. The different elements of the trigger system will be placed at different locations on the readout electronics, as shown in Fig. 4. For the analog trigger, the decision boards (L0/L1) are placed on the readout board itself, while for the digital trigger the L0 board is placed in the frontend and the L1 in the backplane, keeping both analog and digital L0 as close as possible after the signals from PMTs. The distribution boards are placed at the backplane of the readout board, in order to send and receive trigger signals efficiently among neighbouring clusters. The choice of using trigger mezzanines and sharing of connectors has allowed the independent study of both analog and digital solution in the development phase.",
"pages": [
3
]
},
{
"title": "2.5 DRS4 Chip",
"content": "The DRS chip is being developed at the Paul Scherrer Institute (PSI), Switzerland, for the MEG experiment [3]. The latest version, DRS4, is also used in the MAGIC experiment. The DRS4 chip includes nine differential input channels at a sampling speed of 700 MS/s-5 GS/s, with a bandwidth of 950 MHz, and a low noise of 0.35 mV after offset correction. The analog waveform is stored in 1,024 sampling capacitors per channel and the waveform can be read out after sampling via a shift register that is clocked at a maximum of 33 MHz for digitization using an external ADC. The power consumption of the DRS4 chip is 17.5 mWper channel at 2 GS/s sampling rate. The chip is fabricated using the 0.25 µ mCMOStechnology. It is possible to cascade two or more channels to obtain deeper sampling depth. In the Dragon , four channels were cascaded, leading to a sampling depth of 4,096 for each PMT signal, which corresponds to a depth of 4 µ s at 1 GS/s sampling rate. It enables continuous sampling until the stereo coincidence trigger confirms the local telescope trigger, and initializes the camera readout.",
"pages": [
3
]
},
{
"title": "2.6 FPGA-based Gigabit Ethernet (SiTCP)",
"content": "The digitized waveform data and the monitor/control data are transmitted via Ethernet with only two devices: FPGA and Gigabit Ethernet transceiver (PHY). This simple composition is available on a hardware-based TCP processor, SiTCP [11]. The circuit size of SiTCP in the FPGA is ∼ 3000 slices, which is small enough to allow implementation with user circuits on a single FPGA. In addition, the SiTCP has an advantage in that the throughputs in both directions can simultaneously reach the theoretical upper limits of Gigabit Ethernet.",
"pages": [
3
]
},
{
"title": "2.7 Mini-Camera with Three Clusters",
"content": "A prototype with three clusters of PMTs and their readout boards was constructed to develop the trigger system between multiple clusters (Fig. 6) [10].",
"pages": [
3
]
},
{
"title": "3 Performance of Readout System",
"content": "The performance of the DRS4 readout system Dragon was measured. Figure 7 shows that dynamic ranges of high and low gain channels are 60 p.e. and 2500 p.e., which satisfied the requirement of > 1000 photoelectrons. Measured noise level for the high gain channel is 0.1 p.e. (rms). After the above measurements, the performance of the DRS4 readout board attached to a PMT was investigated. Figure 9 shows a pulse shape of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with a UV laser and the DRS4 readout system at a sampling rate of 2 GS/s. The PMT signal having a width of ∼ 3 ns (FWHM) and a height corresponding to ∼ 3 photoelectrons was successfully digitized. Measured dead time for readout of 60 cells in the DRS4 chip is 0.9% and 5.4% at 1 kHz and 7 kHz, respectively. Figure 10 shows a single photoelectron spectrum of the high gain channel of the PMT signal with a gain of 5 × 10 4 , which was measured with Dragon and an LED at a sampling rate of 2 GS/s. In the figure, a single photoelectron peak is clearly seen. The signal to noise ratio defined as (single-photoelectron mean - pedestal mean) / pedestal r.m.s. is 3.6.",
"pages": [
4
]
},
{
"title": "4 Conclusion",
"content": "We have developed a prototype 7-PMT cluster readout system for the large-size telescope of the next generation VHE gammaray observatory,CTA. In the readout system named Dragon , a PMT signal is amplified, and its waveform is then digitized at a sampling rate of the order of GHz using an analog memory ASIC DRS4 that has 4,096 capacitors per readout channel. Using the prototype system at- tached to a PMT with a Cockcroft-Walton circuit, we successfully obtained a pulse shape of the signal of the PMT detecting a UV light at a sampling rate of 2 GS/s, and also a single photoelectron spectrum. Evaluation of the readout system's performance is ongoing, and a prototype with several dozen clusters and their readout boards will be constructed as the next step for the development of a full telescope camera. Acknowledgment: We gratefully acknowledge support from the agencies and organisations listed in this page: http://www.ctaobservatory.org/?q=node/22",
"pages": [
4
]
}
]
2013ICRC...33.2840N
https://arxiv.org/pdf/1307.4189.pdf
<document>
<figure>
<location><page_1><loc_69><loc_88><loc_89><loc_94></location>
</figure>
<section_header_level_1><location><page_1><loc_10><loc_83><loc_91><loc_86></location>Single-Mirror Small-Size Telescope Structure for the Cherenkov Telescope Array</section_header_level_1>
<text><location><page_1><loc_10><loc_80><loc_89><loc_83></location>JACEK NIEMIEC, JERZY MICHAŁOWSKI, MICHAŁ DYRDA, WOJCIECH KOCHA'NSKI, JAROMIR LUDWIN, MAREK STODULSKI, PAWEŁ ZI'OŁKOWSKI, PAWEŁ ˙ ZYCHOWSKI, FOR THE CTA CONSORTIUM.</text>
<text><location><page_1><loc_9><loc_78><loc_72><loc_79></location>Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krak'ow, Poland</text>
<text><location><page_1><loc_10><loc_76><loc_26><loc_77></location>jacek.niemiec@ifj.edu.pl</text>
<text><location><page_1><loc_15><loc_62><loc_91><loc_74></location>Abstract: A single-mirror small-size (1M-SST) Davies-Cotton telescope has been proposed for the southern observatory of the Cherenkov Telescope Array (CTA) by a consortium of scientific institutions from Poland, Switzerland, and Germany. The telescope has a 4 m diameter reflector and will be equipped with a fully digital camera based on Geiger avalanche photodiodes (APDs). Such a design is particularly interesting for CTA because it represents a very simple, reliable, and cheap solution for a SST. Here we present the design and the characteristics of the mechanical structure of the 1M-SST telescope and its drive system. We also discuss the results of a finite element method analysis in order to demonstrate the conformance of the design with the CTA specifications and scientific objectives. In addition, we report on the current status of the construction of a aprototype telescope structure at the Institute of Nuclear Physics PAS in Krakow.</text>
<text><location><page_1><loc_16><loc_59><loc_67><loc_60></location>Keywords: CTA, imaging atmospheric Cherenkov telescopes, FEM analysis</text>
<section_header_level_1><location><page_1><loc_10><loc_55><loc_23><loc_56></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_30><loc_49><loc_54></location>A consortium of scientific institutions from Poland, Switzerland, and Germany has recently proposed a singlemirror small-size (1M-SST) telescope for the southern observatory of the Cherenkov Telescope Array (CTA) [1]. The telescope utilizes a Davies-Cotton design - a classic telescope type used in very high-energy ground-based astronomy. 1M-SST will be equipped with a fully digital and lightweight camera based on Geiger avalanche photodiodes (APDs) with the physical plate scale size of 24 mm (see [2]). With the assumed angular pixel pitch p = 0 . 25 deg, the telescope focal length, i.e., the distance from the center of the dish to the camera pixel plane, is thus f = 5 . 6 m. To assure that the optical point-spread function of the telescope is smaller than 0.25 deg at 4 deg off-axis, a reflective dish with diameter of 3.98 m is needed. This results in the camera field-of-view of 9 deg. The telescope's effective mirror area, corrected for the effect of shadowing by the camera and the mast, is 7.6 m 2 .</text>
<text><location><page_1><loc_9><loc_14><loc_49><loc_30></location>The 1M-SST telescope thus represents a compact structure. The use of silicon photomultipliers (PMTs) results in a low weight of the photodetector plane of 30 kg and the full camera weight of less than 300 kg. Hence, a number of engineering problems, challenged in earlier designs (e.g., H.E.S.S. or VERITAS telescopes [3, 4]) are largely relaxed. The latter were caused by a necessity of supporting a heavy and large camera based on analog PMTs on the mast of a telescope with a large focal length ( f > 10 m). A design of a low-cost solution for a 1M-SST that can be easily replicated to form a high-energy sub-array of the CTA is therefore feasible.</text>
<text><location><page_1><loc_9><loc_5><loc_49><loc_14></location>The design of the 1M-SST structure is described in this work. In sections 2.1 and 2.2 the telescope's mechanical structure and drive system are presented, respectively. The results of the finite element method (FEM) analysis are shown in section 2.3. Finally, in section 3 a current status of the construction of a prototype telescope structure at the Institute of Nuclear Physics PAS in Krakow is reported.</text>
<section_header_level_1><location><page_1><loc_52><loc_55><loc_71><loc_56></location>2 Telescope Structure</section_header_level_1>
<section_header_level_1><location><page_1><loc_52><loc_53><loc_61><loc_54></location>2.1 Frame</section_header_level_1>
<text><location><page_1><loc_51><loc_41><loc_91><loc_52></location>The telescope frame consists of several sub-systems as shown in figure 1. The mast (1) is directly connected to the dish support structure (2), to which the counterweight (3) is also attached. The rigidity of the mast is increased by the use of thin pre-tensioned steel rods. The mast positions the camera (4) with respect to the mirrors (5) mounted on the dish (6). The latter is fixed to the dish support structure mounted on the telescope support (tower, 7). Finally, the dock station (8) locks the telescope in a parking position.</text>
<text><location><page_1><loc_51><loc_24><loc_91><loc_40></location>The telescope is a compact and lightweight structure. Its hight is about 5 m, measured from the base of the tower, its width is 3.3 m, and the total length is 9.3 m. To cut the transportation costs a division of the telescope structure into smaller units that fit the standard size of a shipping container is applied. With steel used as a basic construction material, the telescope weighs less than 9 tons, including counterweights and the components of the drive system. The low weight, the fact that all steel profiles and tubes can be obtained as off-the-shelf products from the industry and that the design uses as many as possible identical components lowers down the cost of manufacturing and guarantees a low price of the telescope frame.</text>
<text><location><page_1><loc_52><loc_8><loc_91><loc_23></location>The layout of 18 mirror tiles that make up the reflecting dish of 4 m diameter is shown in figure 2 (left). Hexagonal mirrors of size 0.78 m flat-to-flat are used, with space between the tiles of 2 cm. The radius of curvature of a single mirror is 11.2 m. The dish structure is built of square steel profiles. It is to be connected to the dish support structure at four points, visible in figure 2 as pads. The required spherical shape of the dish is obtained with two hexagonal sections welded to eight straight star-forming profiles. A precise positioning of the mirrors with respect to the camera is accomplished through a use of dedicated mirror fixtures (see figures 2 and 3).</text>
<text><location><page_1><loc_52><loc_5><loc_91><loc_7></location>The mirror fixture design is presented in figure 3. The fixture consists of an aluminium triangular frame with</text>
<figure>
<location><page_2><loc_79><loc_92><loc_91><loc_96></location>
<caption>Figure 2 : Layout of 18 mirror tiles on the dish in the rear/back view (left). The dish connected to the dish support structure (right). /g4/g19/g2 /g6/g5/g11/g16/g1/g29/g24/g1/g47 /g6/g24/g29/g39/g29/g34/g33/g1/g47/g1/g64/g48/g65 /g5/g21/g39/g25 /g14/g37/g25/g31/g29/g32/g29/g33/g21/g37/g44/g1/g5/g21/g41/g29/g25/g38 /g4/g34/g39/g39/g34/g33 /g14/g25/g37/g26/g34/g37/g32/g21/g33/g23/g25/g1/g15/g25/g36/g40/g29/g37/g25/g32/g25/g33/g39/g38/g1/g37/g25/g41/g29/g25/g42</caption>
</figure>
<figure>
<location><page_2><loc_10><loc_65><loc_90><loc_90></location>
<caption>Figure 1 : 1M-SST sub-systems. Drawings in figures 2, 4, and 5 are based on a technical documentation of the telescope structure. See text for the explanation of figure markings.</caption>
</figure>
<text><location><page_2><loc_10><loc_55><loc_49><loc_58></location>three fixing points and a steel threaded rod with a sphere at one end and a threaded flange at the other end. The flanges are bolted to the dish. The use of the threaded rod enables</text>
<figure>
<location><page_2><loc_10><loc_32><loc_24><loc_52></location>
</figure>
<figure>
<location><page_2><loc_25><loc_32><loc_48><loc_52></location>
</figure>
<text><location><page_2><loc_36><loc_31><loc_40><loc_31></location>/g4/g17/g2/g1/g15/g25/g26/g1/g47</text>
<text><location><page_2><loc_36><loc_26><loc_40><loc_26></location>/g14/g21/g27/g25/g1/g47/g1/g65/g72</text>
<text><location><page_2><loc_41><loc_26><loc_42><loc_26></location>/g68/g64</text>
<figure>
<location><page_2><loc_11><loc_9><loc_47><loc_24></location>
<caption>Figure 7.1.9: Mirror fixture with interfaces (left) and mirror attached (right) Mast and dish support structure Figure 3 : Mirror fixture with interfaces (left) and the mirror attached (right).</caption>
</figure>
<text><location><page_2><loc_52><loc_51><loc_91><loc_58></location>a precise positioning of a mirror at the focal length distance from the telescope's optical center. The sphere of the threaded rod is tightened in the central fixing point of the triangular frame to create a hinge that allows for additional degrees of freedom during a preadjustment of a mirror orientation.</text>
<text><location><page_2><loc_51><loc_34><loc_91><loc_51></location>The final precise alignment of the mirrors is to be performed by means of the automatic motor controllers (AMCs), called actuators. Each mirror fixture houses three interfaces that provide 3-point isostatic support to a mirror. One interface is a so-called fixed point support (1, figure 3). Two other interfaces are actuators (2), which give a mirror facet two rotational degrees of freedom. The mirrors are fixed to these interfaces via steel pads (3) glued on the mirrors' rear surfaces. Note, that the design of the mirror fixtures applied here does not only enable an arcminutelevel stable alignment of the mirror facets relative to the camera, but it also allows for a simple and easy mounting/dismounting of the mirrors on/from the dish.</text>
<text><location><page_2><loc_51><loc_6><loc_91><loc_34></location>The design of the mast (see (1) in figure 1) guarantees a proper setup of the camera with respect to the reflecting mirror surface at the telescope's focal distance of 5.6 m. The mast is built of eight circular tubes that are bolted together into four longer beams. The beams are connected with the camera housing at one end, and with the dish support structure at the other end. In this way the mast does not incur any mechanical stresses to the dish so that all mast deformations are practically disconnected from the deformations of the dish. A rectangular frame roughly in the middle of the mast increases the stiffness of the mast structure and also facilitates its transport. The rigidity of the mast is additionally increased by the use of sixteen steel rods. The rods are pre-stressed with a force of 2000 N and fixed to the long tubes by means of turnbuckles. To assure a proper geometry of the mast, eight special-design interfaces between the rectangular frame and the beams must be used. Their shape is identical but requires computer numerical control (CNC) machining. Note, that such interfaces are not needed at points in which the mast is connected to the dish support structure.</text>
<text><location><page_2><loc_53><loc_5><loc_91><loc_6></location>The dish support structure is shown in figure 4. It is com-</text>
<text><location><page_2><loc_1><loc_4><loc_55><loc_5></location>The design of the mast guarantees a proper setup of the camera with respect to reflecting mirror surface at</text>
<text><location><page_2><loc_1><loc_3><loc_55><loc_4></location>the focal distance of 5.6 m. The main components of the mast, Fig. 7.1.10, are eight circular steel tubes (1)</text>
<text><location><page_2><loc_1><loc_2><loc_10><loc_3></location>OD= 0.06 m and</text>
<text><location><page_2><loc_11><loc_2><loc_55><loc_3></location>m thick bolted together into four longer beams (2) to provide the required focal length.</text>
<text><location><page_2><loc_1><loc_1><loc_55><loc_2></location>At one end the four tubes are connected with the camera interface (3) that is made of aluminium sheet. At</text>
<text><location><page_2><loc_1><loc_0><loc_55><loc_1></location>the second end the tubes are bolted to the dish support structure (4). The applied solution practically</text>
<text><location><page_3><loc_67><loc_95><loc_69><loc_96></location>/g4/g19/g2</text>
<figure>
<location><page_3><loc_10><loc_66><loc_48><loc_90></location>
<caption>Figure 4 : Design of the dish support structure.</caption>
</figure>
<text><location><page_3><loc_9><loc_48><loc_49><loc_60></location>posed of various components. Two forks/Y-shaped frames (1) are box structures welded out of steel sheets and tubes. The forks are spaced with four tubes (2). As mentioned above, the beams of the mast are bolted to the four ends of the forks. Counterweights are fixed at their two remaining ends. Four struts (3) and steel sheets (wings) (4) provide the necessary stiffness of the structure. The latter also improve the resistance to the wind blows. There are also two circular tubes with pads (5) incorporated in the forks.</text>
<figure>
<location><page_3><loc_10><loc_8><loc_48><loc_46></location>
<caption>Figure5 : Design of the telescope support.</caption>
</figure>
<text><location><page_3><loc_77><loc_96><loc_81><loc_96></location>/g4/g17/g2/g1/g15/g25/g26/g1/g47</text>
<section_header_level_1><location><page_3><loc_77><loc_92><loc_82><loc_95></location>/g6/g5/g11/g16/g1/g29/g24/g1/g47 /g6/g24/g29/g39/g29/g34/g33/g1/g47/g1/g64/g48/g65 /g5/g21/g39/g25 /g14/g21/g27/g25/g1/g47/g1/g67/g64 /g68/g64</section_header_level_1>
<figure>
<location><page_3><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<figure>
<location><page_3><loc_52><loc_80><loc_91><loc_90></location>
<caption>Figure 8.2.1: The drive system movement range in elevation (left) and azimuth (right) 8.2.1 Drive system requirements Figure 6 : 1M-SST drive system movement range in elevation (left) and azimuth (right).</caption>
</figure>
<text><location><page_3><loc_50><loc_74><loc_92><loc_74></location>The drive system parameters have been determined according to the CTA requirements for 1M SST</text>
<text><location><page_3><loc_50><loc_73><loc_92><loc_74></location>available at the time the calculations were made. The assumed environmental conditions specify the</text>
<text><location><page_3><loc_50><loc_72><loc_92><loc_73></location>telescope site altitude at 2500 m above sea level and the operation temperature range between 20 C and</text>
<text><location><page_3><loc_50><loc_67><loc_92><loc_72></location>+30 C. The observation mode assumes the average maximum wind velocity of 50 km/h and the emergency mode allows for a safe parking of the telescope in conditions with the maximum wind velocity reaching 100 km/h, including gusts. The emergency mode also includes the possibility of reaching the parking position at the decreased angular acceleration in the case of a failure of one of the drives. It is assumed that in the operation mode the telescope can fast slew to any defined point in the sky within 1 minute, including 6 seconds for both the acceleration at a constant rate from the start out position and the deceleration. Resulting maximum angular velocities are max,a=6 /sec and max,e=2 /sec with respect The pads represent the interfaces to which the elevation drive system components (slew drive and the ball-bearing mounted on the sides of the head of the telescope support) are bolted.</text>
<text><location><page_3><loc_50><loc_47><loc_92><loc_67></location>to the azimuth and the elevation axis, respectively. The positioning accuracy is taken within 7 arcsec and the tracking accuracy is assumed not to exceed 5 arcmin 8.2.2. Drive mechanical torque calculations For the conditions thus specified mechanical torques and power requirements have been calculated that allowed for the selection of the drive system components. The calculations follow the guidelines of the 'Drive Power Estimates for CTA Telescopes' document, provided by the Project Office. The method uses a simplified approach that assumes the mechanical shaft power (the power required to move the telescope at the elevation and azimuth axes) to comprise of three components: the friction power, the wind power and the power required to accelerate the telescope. The total mechanical power is calculated as the product of the required total mechanical torque and the telescope angular velocity. The maximum values of the latter are given above. Tables 8.2.1 and 8.2.2 below present the maximum total torques about each axis and their components. Details of the calculations are provided in a separate document. The 1M-SST telescope support is built of two main components: the tower (1) and the head (2), figure 5. The elements of the azimuth drive system (3) are incorporated into the tower, while these of the elevation drive system (4) into the head. The tower must be bolted to a foundation in which a steel anchor grid structure with reinforcements is embedded. The slew drive (drawn in grey in figure 5) realizing the azimuth rotation of the telescope is fixed to the conical part of the tower (the tower cap). Inside the tower, there is a radial ball-bearing connected to the azimuth slew drive with the tube, that stabilizes the whole structure. The housing of the azimuth ball bearing inside the tower can be accessed for installation and maintenance purposes via two openings.</text>
<text><location><page_3><loc_51><loc_39><loc_91><loc_47></location>The head is rotated by the azimuth slew drive. The elevation drive system components, including the slew drive and the ball-bearing, figure 5, are bolted to the pads of the dish support structure (see figure 4). The slew drives and the bearings of both the azimuth and elevation drive systems are identical.</text>
<section_header_level_1><location><page_3><loc_52><loc_37><loc_66><loc_38></location>2.2 Drive System</section_header_level_1>
<text><location><page_3><loc_51><loc_5><loc_91><loc_36></location>As introduced in section 2.1, a positioning and tracking system of the telescope is realized with two independent drive axes: the azimuth axis and the elevation axis. The range for the telescope movement in elevation and azimuth is shown in figure 6. The drive system of each axis is based on an IMO [5] slew drive and a radial ball-bearing. An IMO slew drive is a compact system that combines a warm gear with a motor and also a roller bearing, thus enabling transmission of both radial and axial forces. It has a fully enclosed and self-supporting housing. Rotating components are fixed to the housing with bolts. Each IMO slew drive selected for the 1M-SST contains a twin warm gear with two servo-motors. Such a solution helps to increase the torque capacity and to eliminate backlash that is needed to achieve a required pointing accuracy of 7 arcsec and the tracking accuracy less than 5 arcmin. The drive system of the medium-size telescope (MST) for CTA is also based on the IMO slew drives and the choice of this type of drives for the 1M-SST has been partially done for the sake of the uniformity among the various CTA telescope array subcomponents. For this reason the concepts for control software and structure and safety will also be elaborated based on the MST solutions.</text>
<text><location><page_4><loc_49><loc_24><loc_50><loc_25></location>/g88/g85/g1</text>
<section_header_level_1><location><page_4><loc_10><loc_89><loc_25><loc_90></location>2.3 FEMAnalysis</section_header_level_1>
<text><location><page_4><loc_9><loc_50><loc_49><loc_88></location>The 1M-SST telescope structure design has been optimized based on the static FEM analysis. In order to perform such analysis a simplified computer-aided design (CAD) model of the telescope structure was built with ANSYS [6]. The model allows for an investigation of the structure deformations and mechanical stresses under gravity, wind, snow, and ice loads, and also for the earthquake conditions. The results show that the design of 1M-SST meets CTA optical requirement that the maximum displacement of the camera with respect to the dish is smaller than 1/3 of the pixel size for the extreme observing mode conditions. The latter assume the maximum wind velocity with gusts of 50 km/h (the average velocity of 36 km/h) during which observation runs can still be taken. The resulting maximum displacement is 8.2 mm for the back-wind conditions and the telescope at 60 deg in the elevation angle. The maximum equivalent stresses in the structure are also well below the elasticity limit for the steel. With Re = 240 MPa the safety factor is about 3. Similar FEM analysis was conducted for the survival mode conditions. They allow for the wind velocity to reach 200 km/h while the telescope is locked in the parking position. Also in this case all stresses are well below the plasticity of the materials used for the mechanical construction. Preliminary FEM calculations show that the same is true if the earthquake loads are considered. Finally, the modal analysis, figure 7, demonstrates that the lowest eigenfrequency of the structure is 3.8 Hz. Thus it is well above the minimum value of 2.5 Hz specified for CTA telescopes.</text>
<text><location><page_4><loc_10><loc_46><loc_49><loc_49></location>Note, that the FEM calculations performed demonstrate a full conformance of the design with the CTA specications and scientic objectives. /g7/g6/g11/g1/g21/g19/g31/g21/g42/g31/g19/g40/g34/g33/g38/g1</text>
<figure>
<location><page_4><loc_10><loc_29><loc_23><loc_44></location>
<caption>Figure 7 : Results of the modal analysis showing the three lowest eigenfrequencies and their corresponding modes. The elevation angle is given in parantheses.</caption>
</figure>
<figure>
<location><page_4><loc_38><loc_29><loc_49><loc_44></location>
</figure>
<text><location><page_4><loc_24><loc_45><loc_33><loc_46></location>/g20/g23/g21/g20/g29/g40/g16/g26/g39/g20/g36/g1</text>
<figure>
<location><page_4><loc_25><loc_29><loc_36><loc_44></location>
<caption>/g1/g1/g1/g1/g1/g1/g1/g10/g1/g76/g1/g88/g70/g93/g1/g9/g63/g1/g1/g77/g88/g85/g82/g78/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g10/g10/g1/g76/g1/g89/g70/g90/g1/g9/g63/g1/g77/g94/g85/g82/g78</caption>
</figure>
<paragraph><location><page_4><loc_35><loc_27><loc_47><loc_28></location>/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g1/g10/g10/g10/g1/g76/g1/g86/g86/g70/g89/g1/g9/g63/g1/g77/g88/g85/g82/g78/g1</paragraph>
<section_header_level_1><location><page_4><loc_10><loc_16><loc_37><loc_17></location>3 Towards 1M-SST Prototype</section_header_level_1>
<text><location><page_4><loc_9><loc_5><loc_49><loc_15></location>Based on the design presented in section 2 a prototype telescope structure with drive system will be constructed at the Institute of Nuclear Physics PAS in Krak'ow. The elements of the drive system have already been ordered. The technical design of the telescope frame has recently been prepared by an industrial partner. A call for tender for manufacturing of the telescope components and preinstallation at the INP PAS site will soon be placed. It is</text>
<figure>
<location><page_4><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<text><location><page_4><loc_52><loc_87><loc_91><loc_90></location>foreseen that the 1M-SST prototype will be ready for tests by the end of 2013.</text>
<text><location><page_4><loc_52><loc_79><loc_91><loc_86></location>Acknowledgment: This work was supported by The National Centre for Research and Development through project ERANET-ASPERA/01/10 and The National Science Centre through project DEC-2011/01/M/ST9/01891. We also gratefully acknowledge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22 .</text>
<section_header_level_1><location><page_4><loc_52><loc_75><loc_61><loc_77></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_4><loc_52><loc_73><loc_83><loc_75></location>[1] R. Moderski, et al., contribution no. 0840, these proceedings.</list_item>
<list_item><location><page_4><loc_52><loc_71><loc_90><loc_73></location>[2] V. Boccone, et al., contribution no. 0234, these proceedings. [3] K. Bernloehr, et al., Astroparticle Physics 20 (2003)</list_item>
<list_item><location><page_4><loc_53><loc_70><loc_59><loc_71></location>111-128.</list_item>
<list_item><location><page_4><loc_52><loc_69><loc_89><loc_70></location>[4] J. Holder, et al., Astroparticle Physics 25 (2006) 391-401.</list_item>
<list_item><location><page_4><loc_52><loc_68><loc_82><loc_69></location>[5] http://www.goimo.eu/Home.217.0.html</list_item>
<list_item><location><page_4><loc_52><loc_67><loc_71><loc_68></location>[6] http://www.ansys.com/</list_item>
</document>
[
{
"title": "Single-Mirror Small-Size Telescope Structure for the Cherenkov Telescope Array",
"content": "JACEK NIEMIEC, JERZY MICHAŁOWSKI, MICHAŁ DYRDA, WOJCIECH KOCHA'NSKI, JAROMIR LUDWIN, MAREK STODULSKI, PAWEŁ ZI'OŁKOWSKI, PAWEŁ ˙ ZYCHOWSKI, FOR THE CTA CONSORTIUM. Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krak'ow, Poland jacek.niemiec@ifj.edu.pl Abstract: A single-mirror small-size (1M-SST) Davies-Cotton telescope has been proposed for the southern observatory of the Cherenkov Telescope Array (CTA) by a consortium of scientific institutions from Poland, Switzerland, and Germany. The telescope has a 4 m diameter reflector and will be equipped with a fully digital camera based on Geiger avalanche photodiodes (APDs). Such a design is particularly interesting for CTA because it represents a very simple, reliable, and cheap solution for a SST. Here we present the design and the characteristics of the mechanical structure of the 1M-SST telescope and its drive system. We also discuss the results of a finite element method analysis in order to demonstrate the conformance of the design with the CTA specifications and scientific objectives. In addition, we report on the current status of the construction of a aprototype telescope structure at the Institute of Nuclear Physics PAS in Krakow. Keywords: CTA, imaging atmospheric Cherenkov telescopes, FEM analysis",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "A consortium of scientific institutions from Poland, Switzerland, and Germany has recently proposed a singlemirror small-size (1M-SST) telescope for the southern observatory of the Cherenkov Telescope Array (CTA) [1]. The telescope utilizes a Davies-Cotton design - a classic telescope type used in very high-energy ground-based astronomy. 1M-SST will be equipped with a fully digital and lightweight camera based on Geiger avalanche photodiodes (APDs) with the physical plate scale size of 24 mm (see [2]). With the assumed angular pixel pitch p = 0 . 25 deg, the telescope focal length, i.e., the distance from the center of the dish to the camera pixel plane, is thus f = 5 . 6 m. To assure that the optical point-spread function of the telescope is smaller than 0.25 deg at 4 deg off-axis, a reflective dish with diameter of 3.98 m is needed. This results in the camera field-of-view of 9 deg. The telescope's effective mirror area, corrected for the effect of shadowing by the camera and the mast, is 7.6 m 2 . The 1M-SST telescope thus represents a compact structure. The use of silicon photomultipliers (PMTs) results in a low weight of the photodetector plane of 30 kg and the full camera weight of less than 300 kg. Hence, a number of engineering problems, challenged in earlier designs (e.g., H.E.S.S. or VERITAS telescopes [3, 4]) are largely relaxed. The latter were caused by a necessity of supporting a heavy and large camera based on analog PMTs on the mast of a telescope with a large focal length ( f > 10 m). A design of a low-cost solution for a 1M-SST that can be easily replicated to form a high-energy sub-array of the CTA is therefore feasible. The design of the 1M-SST structure is described in this work. In sections 2.1 and 2.2 the telescope's mechanical structure and drive system are presented, respectively. The results of the finite element method (FEM) analysis are shown in section 2.3. Finally, in section 3 a current status of the construction of a prototype telescope structure at the Institute of Nuclear Physics PAS in Krakow is reported.",
"pages": [
1
]
},
{
"title": "2.1 Frame",
"content": "The telescope frame consists of several sub-systems as shown in figure 1. The mast (1) is directly connected to the dish support structure (2), to which the counterweight (3) is also attached. The rigidity of the mast is increased by the use of thin pre-tensioned steel rods. The mast positions the camera (4) with respect to the mirrors (5) mounted on the dish (6). The latter is fixed to the dish support structure mounted on the telescope support (tower, 7). Finally, the dock station (8) locks the telescope in a parking position. The telescope is a compact and lightweight structure. Its hight is about 5 m, measured from the base of the tower, its width is 3.3 m, and the total length is 9.3 m. To cut the transportation costs a division of the telescope structure into smaller units that fit the standard size of a shipping container is applied. With steel used as a basic construction material, the telescope weighs less than 9 tons, including counterweights and the components of the drive system. The low weight, the fact that all steel profiles and tubes can be obtained as off-the-shelf products from the industry and that the design uses as many as possible identical components lowers down the cost of manufacturing and guarantees a low price of the telescope frame. The layout of 18 mirror tiles that make up the reflecting dish of 4 m diameter is shown in figure 2 (left). Hexagonal mirrors of size 0.78 m flat-to-flat are used, with space between the tiles of 2 cm. The radius of curvature of a single mirror is 11.2 m. The dish structure is built of square steel profiles. It is to be connected to the dish support structure at four points, visible in figure 2 as pads. The required spherical shape of the dish is obtained with two hexagonal sections welded to eight straight star-forming profiles. A precise positioning of the mirrors with respect to the camera is accomplished through a use of dedicated mirror fixtures (see figures 2 and 3). The mirror fixture design is presented in figure 3. The fixture consists of an aluminium triangular frame with three fixing points and a steel threaded rod with a sphere at one end and a threaded flange at the other end. The flanges are bolted to the dish. The use of the threaded rod enables /g4/g17/g2/g1/g15/g25/g26/g1/g47 /g14/g21/g27/g25/g1/g47/g1/g65/g72 /g68/g64 a precise positioning of a mirror at the focal length distance from the telescope's optical center. The sphere of the threaded rod is tightened in the central fixing point of the triangular frame to create a hinge that allows for additional degrees of freedom during a preadjustment of a mirror orientation. The final precise alignment of the mirrors is to be performed by means of the automatic motor controllers (AMCs), called actuators. Each mirror fixture houses three interfaces that provide 3-point isostatic support to a mirror. One interface is a so-called fixed point support (1, figure 3). Two other interfaces are actuators (2), which give a mirror facet two rotational degrees of freedom. The mirrors are fixed to these interfaces via steel pads (3) glued on the mirrors' rear surfaces. Note, that the design of the mirror fixtures applied here does not only enable an arcminutelevel stable alignment of the mirror facets relative to the camera, but it also allows for a simple and easy mounting/dismounting of the mirrors on/from the dish. The design of the mast (see (1) in figure 1) guarantees a proper setup of the camera with respect to the reflecting mirror surface at the telescope's focal distance of 5.6 m. The mast is built of eight circular tubes that are bolted together into four longer beams. The beams are connected with the camera housing at one end, and with the dish support structure at the other end. In this way the mast does not incur any mechanical stresses to the dish so that all mast deformations are practically disconnected from the deformations of the dish. A rectangular frame roughly in the middle of the mast increases the stiffness of the mast structure and also facilitates its transport. The rigidity of the mast is additionally increased by the use of sixteen steel rods. The rods are pre-stressed with a force of 2000 N and fixed to the long tubes by means of turnbuckles. To assure a proper geometry of the mast, eight special-design interfaces between the rectangular frame and the beams must be used. Their shape is identical but requires computer numerical control (CNC) machining. Note, that such interfaces are not needed at points in which the mast is connected to the dish support structure. The dish support structure is shown in figure 4. It is com- The design of the mast guarantees a proper setup of the camera with respect to reflecting mirror surface at the focal distance of 5.6 m. The main components of the mast, Fig. 7.1.10, are eight circular steel tubes (1) OD= 0.06 m and m thick bolted together into four longer beams (2) to provide the required focal length. At one end the four tubes are connected with the camera interface (3) that is made of aluminium sheet. At the second end the tubes are bolted to the dish support structure (4). The applied solution practically /g4/g19/g2 posed of various components. Two forks/Y-shaped frames (1) are box structures welded out of steel sheets and tubes. The forks are spaced with four tubes (2). As mentioned above, the beams of the mast are bolted to the four ends of the forks. Counterweights are fixed at their two remaining ends. Four struts (3) and steel sheets (wings) (4) provide the necessary stiffness of the structure. The latter also improve the resistance to the wind blows. There are also two circular tubes with pads (5) incorporated in the forks. /g4/g17/g2/g1/g15/g25/g26/g1/g47",
"pages": [
1,
2,
3
]
},
{
"title": "/g6/g5/g11/g16/g1/g29/g24/g1/g47 /g6/g24/g29/g39/g29/g34/g33/g1/g47/g1/g64/g48/g65 /g5/g21/g39/g25 /g14/g21/g27/g25/g1/g47/g1/g67/g64 /g68/g64",
"content": "The drive system parameters have been determined according to the CTA requirements for 1M SST available at the time the calculations were made. The assumed environmental conditions specify the telescope site altitude at 2500 m above sea level and the operation temperature range between 20 C and +30 C. The observation mode assumes the average maximum wind velocity of 50 km/h and the emergency mode allows for a safe parking of the telescope in conditions with the maximum wind velocity reaching 100 km/h, including gusts. The emergency mode also includes the possibility of reaching the parking position at the decreased angular acceleration in the case of a failure of one of the drives. It is assumed that in the operation mode the telescope can fast slew to any defined point in the sky within 1 minute, including 6 seconds for both the acceleration at a constant rate from the start out position and the deceleration. Resulting maximum angular velocities are max,a=6 /sec and max,e=2 /sec with respect The pads represent the interfaces to which the elevation drive system components (slew drive and the ball-bearing mounted on the sides of the head of the telescope support) are bolted. to the azimuth and the elevation axis, respectively. The positioning accuracy is taken within 7 arcsec and the tracking accuracy is assumed not to exceed 5 arcmin 8.2.2. Drive mechanical torque calculations For the conditions thus specified mechanical torques and power requirements have been calculated that allowed for the selection of the drive system components. The calculations follow the guidelines of the 'Drive Power Estimates for CTA Telescopes' document, provided by the Project Office. The method uses a simplified approach that assumes the mechanical shaft power (the power required to move the telescope at the elevation and azimuth axes) to comprise of three components: the friction power, the wind power and the power required to accelerate the telescope. The total mechanical power is calculated as the product of the required total mechanical torque and the telescope angular velocity. The maximum values of the latter are given above. Tables 8.2.1 and 8.2.2 below present the maximum total torques about each axis and their components. Details of the calculations are provided in a separate document. The 1M-SST telescope support is built of two main components: the tower (1) and the head (2), figure 5. The elements of the azimuth drive system (3) are incorporated into the tower, while these of the elevation drive system (4) into the head. The tower must be bolted to a foundation in which a steel anchor grid structure with reinforcements is embedded. The slew drive (drawn in grey in figure 5) realizing the azimuth rotation of the telescope is fixed to the conical part of the tower (the tower cap). Inside the tower, there is a radial ball-bearing connected to the azimuth slew drive with the tube, that stabilizes the whole structure. The housing of the azimuth ball bearing inside the tower can be accessed for installation and maintenance purposes via two openings. The head is rotated by the azimuth slew drive. The elevation drive system components, including the slew drive and the ball-bearing, figure 5, are bolted to the pads of the dish support structure (see figure 4). The slew drives and the bearings of both the azimuth and elevation drive systems are identical.",
"pages": [
3
]
},
{
"title": "2.2 Drive System",
"content": "As introduced in section 2.1, a positioning and tracking system of the telescope is realized with two independent drive axes: the azimuth axis and the elevation axis. The range for the telescope movement in elevation and azimuth is shown in figure 6. The drive system of each axis is based on an IMO [5] slew drive and a radial ball-bearing. An IMO slew drive is a compact system that combines a warm gear with a motor and also a roller bearing, thus enabling transmission of both radial and axial forces. It has a fully enclosed and self-supporting housing. Rotating components are fixed to the housing with bolts. Each IMO slew drive selected for the 1M-SST contains a twin warm gear with two servo-motors. Such a solution helps to increase the torque capacity and to eliminate backlash that is needed to achieve a required pointing accuracy of 7 arcsec and the tracking accuracy less than 5 arcmin. The drive system of the medium-size telescope (MST) for CTA is also based on the IMO slew drives and the choice of this type of drives for the 1M-SST has been partially done for the sake of the uniformity among the various CTA telescope array subcomponents. For this reason the concepts for control software and structure and safety will also be elaborated based on the MST solutions. /g88/g85/g1",
"pages": [
3,
4
]
},
{
"title": "2.3 FEMAnalysis",
"content": "The 1M-SST telescope structure design has been optimized based on the static FEM analysis. In order to perform such analysis a simplified computer-aided design (CAD) model of the telescope structure was built with ANSYS [6]. The model allows for an investigation of the structure deformations and mechanical stresses under gravity, wind, snow, and ice loads, and also for the earthquake conditions. The results show that the design of 1M-SST meets CTA optical requirement that the maximum displacement of the camera with respect to the dish is smaller than 1/3 of the pixel size for the extreme observing mode conditions. The latter assume the maximum wind velocity with gusts of 50 km/h (the average velocity of 36 km/h) during which observation runs can still be taken. The resulting maximum displacement is 8.2 mm for the back-wind conditions and the telescope at 60 deg in the elevation angle. The maximum equivalent stresses in the structure are also well below the elasticity limit for the steel. With Re = 240 MPa the safety factor is about 3. Similar FEM analysis was conducted for the survival mode conditions. They allow for the wind velocity to reach 200 km/h while the telescope is locked in the parking position. Also in this case all stresses are well below the plasticity of the materials used for the mechanical construction. Preliminary FEM calculations show that the same is true if the earthquake loads are considered. Finally, the modal analysis, figure 7, demonstrates that the lowest eigenfrequency of the structure is 3.8 Hz. Thus it is well above the minimum value of 2.5 Hz specified for CTA telescopes. Note, that the FEM calculations performed demonstrate a full conformance of the design with the CTA specications and scientic objectives. /g7/g6/g11/g1/g21/g19/g31/g21/g42/g31/g19/g40/g34/g33/g38/g1 /g20/g23/g21/g20/g29/g40/g16/g26/g39/g20/g36/g1",
"pages": [
4
]
},
{
"title": "3 Towards 1M-SST Prototype",
"content": "Based on the design presented in section 2 a prototype telescope structure with drive system will be constructed at the Institute of Nuclear Physics PAS in Krak'ow. The elements of the drive system have already been ordered. The technical design of the telescope frame has recently been prepared by an industrial partner. A call for tender for manufacturing of the telescope components and preinstallation at the INP PAS site will soon be placed. It is foreseen that the 1M-SST prototype will be ready for tests by the end of 2013. Acknowledgment: This work was supported by The National Centre for Research and Development through project ERANET-ASPERA/01/10 and The National Science Centre through project DEC-2011/01/M/ST9/01891. We also gratefully acknowledge support from the agencies and organizations listed in this page: http://www.cta-observatory.org/?q=node/22 .",
"pages": [
4
]
}
]
2013ICRC...33.3036G
https://arxiv.org/pdf/1307.4545.pdf
<document>
<figure>
<location><page_1><loc_69><loc_88><loc_89><loc_94></location>
</figure>
<section_header_level_1><location><page_1><loc_9><loc_85><loc_43><loc_86></location>The NectarCAM camera project</section_header_level_1>
<text><location><page_1><loc_10><loc_71><loc_91><loc_84></location>J-F.GLICENSTEIN 1 , M.BARCELO 11 , J-A. BARRIO 12 , O.BLANCH 11 , J.BOIX 11 , J.BOLMONT 4 , C.BOUTONNET 2 S.CAZAUX 1 , E.CHABANNE 7 , C.CHAMPION 2 , F.CHATEAU 1 , S.COLONGES 2 , P.CORONA 4 , S.COUTURIER 5 , B.COURTY 2 , E.DELAGNES 1 , C.DELGADO 10 , J-P.ERNENWEIN 6 , S.FEGAN 5 , O.FERREIRA 5 , M.FESQUET 1 , G.FONTAINE 5 , N.FOUQUE 7 , F.HENAULT 8 , D.GASC 'ON 13 , D.HERRANZ 12 , R.HERMEL 7 , D.HOFFMANN 6 , J.HOULES 6 , S.KARKAR 4 , B.KHELIFI 5 , J.KN ODLSEDER 3 , G.MARTINEZ 10 , K.LACOMBE 3 , G.LAMANNA 7 , T.LEFLOUR 7 , R.LOPEZCOTO 11 , F.LOUIS 1 , A.MATHIEU 5 , E.MOULIN 1 , P.NAYMAN 4 , F.NUNIO 1 , J-F. OLIVE 3 , J-L. PANAZOL 7 , P-O. PETRUCCI 8 , M.PUNCH 2 , J.PRAST 7 , P.RAMON 3 , M.RIALLOT 1 , M.RIB 'O 13 , S.ROSIER-LEES 7 , A.SANUY 13 , J.SIERO 13 , J-P.TAVERNET 4 , L.A.TEJEDOR 12 , F.TOUSSENEL 455 , G.VASILEIADIS 9 , V.VOISIN 4 , V.WAEGEBERT 3 , C.ZURBACH 9 , FOR THE CTA CONSORTIUM.</text>
<text><location><page_1><loc_9><loc_70><loc_45><loc_71></location>1 DSM/IRFU, CEA-Saclay, F91191 Gif-sur-Yvette, France</text>
<text><location><page_1><loc_10><loc_67><loc_91><loc_70></location>2 APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_66><loc_84><loc_67></location>3 Institut de Recherche en Astrophysique et Planetologie, 9 av. colonel Roche, BP 44 346, 31028 Toulouse Cedex 4, France</list_item>
<list_item><location><page_1><loc_9><loc_64><loc_90><loc_66></location>4 LPNHE, Universite Pierre et Marie Curie Paris 6, Universite Denis Diderot Paris 7, CNRS/IN2P3, 4 Place Jussieu, F-75252, Paris Cedex 5, France</list_item>
<list_item><location><page_1><loc_10><loc_63><loc_68><loc_64></location>5 Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France</list_item>
<list_item><location><page_1><loc_10><loc_61><loc_78><loc_62></location>6 Centre de Physique des Particules de Marseille, CNRS/IN2P3, 168 Avenue de Luminy, 13009 Marseille, France</list_item>
<list_item><location><page_1><loc_9><loc_60><loc_79><loc_61></location>7 LAPP, Universite de Savoie, CNRS/IN2P3, 9 Chemin de Bellevue - BP 110, 74941 Annecy-le-Vieux Cedex, France</list_item>
<list_item><location><page_1><loc_10><loc_59><loc_90><loc_60></location>8 UJF-Grenoble 1 CNRS/INSU, Institut de Planetologie et d'Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble, F-38041, France</list_item>
<list_item><location><page_1><loc_9><loc_56><loc_90><loc_59></location>9 Laboratoire Univers et Particules de Montpellier, Universite Montpellier 2, CNRS/IN2P3, CC 72, Place Eugene Bataillon, F-34095 Montpellier Cedex 5, France</list_item>
<list_item><location><page_1><loc_9><loc_55><loc_44><loc_56></location>10 CIEMAT, Avda. Complutense 22, 28040 Madrid, Spain</list_item>
<list_item><location><page_1><loc_9><loc_54><loc_67><loc_55></location>11 Institut de Fisica d'Altes Energies (IFAE) Edifici Cn, Campus UAB, 08193 Bellaterra, Spain</list_item>
<list_item><location><page_1><loc_9><loc_53><loc_74><loc_54></location>12 Grupo de Altas Energias, Universidad Complutense de Madrid, Av Complutenses, 28040 Madrid, Spain</list_item>
<list_item><location><page_1><loc_9><loc_51><loc_87><loc_52></location>13 Departament d'Astronomia i Meteorologia (ICC-UB), Universitat de Barcelona, Marti i Franques, 1, 08028, Barcelona, Spain</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_49><loc_20><loc_50></location>glicens@cea.fr</text>
<text><location><page_1><loc_15><loc_26><loc_91><loc_47></location>Abstract: In the framework of the next generation of Cherenkov telescopes, the Cherenkov Telescope Array (CTA), NectarCAM is a camera designed for the medium size telescopes covering the central energy range of 100 GeV to 30 TeV. NectarCAM will be finely pixelated ( ∼ 1800 pixels for a 8 o field of view, FoV) in order to image atmospheric Cherenkov showers by measuring the charge deposited within a few nanoseconds time-window. It will have additional features like the capacity to record the full waveform with GHz sampling for every pixel and to measure event times with nanosecond accuracy. An array of a few tens of medium size telescopes, equipped with NectarCAMs, will achieve up to a factor of ten improvement in sensitivity over existing instruments in the energy range of 100 GeV to 10 TeV. The camera is made of roughly 250 independent read-out modules, each composed of seven photo-multipliers, with their associated high voltage base and control, a read-out board and a multi-service backplane board. The read-out boards use NECTAr (New Electronics for the Cherenkov Telescope Array) ASICs which have the dual functionality of analogue memories and Analogue to Digital Converter (ADC). The camera trigger to be used will be flexible so as to minimize the read-out dead-time of the NECTAr chips. We present the camera concept and the design and tests of the various subcomponents. The design includes the mechanical parts, the cooling of the electronics, the readout, the data acquisition, the trigger, the monitoring and services.</text>
<text><location><page_1><loc_16><loc_23><loc_52><loc_25></location>Keywords: Methods, techniques and instrumentation</text>
<section_header_level_1><location><page_1><loc_10><loc_19><loc_23><loc_20></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_11><loc_48><loc_18></location>NectarCAM is a new camera design for the medium size telescopes (MST) of the planned CTA 1 array of Imaging Atmospheric Cerenkov telescopes (IACTs). The CTA will have several sizes of single or dual-mirror telescopes. The single-mirror MST has a 12-meter diameter dish. Its camera will face new challenges:</text>
<unordered_list>
<list_item><location><page_1><loc_12><loc_5><loc_49><loc_10></location>1. The fields of view of camera are expected to be larger (e.g. at least 7 o compared to 5 o for the cameras of H.E.S.S.-1, an previous generation array of IACTs). As a consequence more distant showers will be seen</list_item>
</unordered_list>
<text><location><page_1><loc_56><loc_8><loc_91><loc_20></location>in the cameras. The arrival of the photons from a distant shower can last more than 100 nanoseconds. However, the signal to noise ratio in a pixel is optimized by recording and later integrating the signal in a much smaller window ( ∼ 10 ns). If the camera trigger is not flexible enough (e.g is similar to the H.E.S.S. camera trigger), the event energies would be systematically underestimated. Special trigger strategies have been designed to trigger different pixels at different moments (see Figure 16 of [1]). These</text>
<figure>
<location><page_2><loc_10><loc_79><loc_48><loc_90></location>
<caption>Figure 1 : Early version (V0) of the NECTAr readout module used in the NectarCAM camera.</caption>
</figure>
<text><location><page_2><loc_14><loc_66><loc_49><loc_72></location>strategies still need to be tested. The trigger rate is also expected to increase by a factor of two compared to the previous generation of Cherenkov telescopes, as result of the larger field of view, potetially leading to a larger dead time of the instrument.</text>
<unordered_list>
<list_item><location><page_2><loc_12><loc_43><loc_49><loc_65></location>2. Large timing gradients are expected in high-energy gamma-ray events. Measuring these gradients improves the reconstruction of the events. Analyzing the whole waveform also helps improve the hadron rejection (see e.g., Aliu et al. [3]). It has been estimated that a convenient reconstruction of the time gradient could be achieved with a time resolution of 2 ns per pixel. To achieve a proper time resolution, it may be necessary to analyze the whole waveform sampled at 1 GHz. In the H.E.S.S. camera, an average charge, time and time spread per pixel was calculated on-board for every triggered event. Sending the whole waveform of the event to the ground implies a data rate per event roughly 5 times larger than that of H.E.S.S. Taking into account the increase in trigger rate, the data rate will increase by an order of magnitude compared to H.E.S.S.</list_item>
<list_item><location><page_2><loc_12><loc_34><loc_49><loc_42></location>3. To achieve the aforementionned 2 ns time resolution per pixel, the systematics on the time measurement such as the analogue bandwidth of the read-out system, or the delay between pixels due to the transit time spread of phototubes have to be carefully evaluated and corrected whenever possible.</list_item>
</unordered_list>
<text><location><page_2><loc_9><loc_26><loc_49><loc_33></location>As a result of the larger field of view and better timing capabilities of the future Cherenkov cameras, it will be possible to improve the reconstruction of the events and to have a better hadron background rejection. However, these new capabilities present new challenges for Cherenkov telescopes.</text>
<text><location><page_2><loc_9><loc_5><loc_49><loc_26></location>Two starting points for the NectarCAM are the architecture of the H.E.S.S.-2 camera and a read-out module: the NECTAr module [1]. Figure 1 shows an early version of the NECTAr module. This module converts the light of a set of photomultiplier tubes (PMTs) into a digital signal when some triggering conditions are fulfilled. The modular architecture of the H.E.S.S.-2 camera has the advantage of avoiding costly cables and allowing for easy maintenance. The modular mechanical structure is described in Section 2. For the NectarCAM, modules are groups of seven pixels, with the associated front-end board, which are completely autonomous in what concerns the power and control electronics. The only cables that go to the ground are the input power voltage and optical fibers for network communications and array trigger connections. Preliminary versions of the readout module exist. The photo-sensors, the readout</text>
<figure>
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<figure>
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<caption>Figure 2 : Exploded view of the NectarCAM camera mechanics</caption>
</figure>
<text><location><page_2><loc_52><loc_51><loc_91><loc_58></location>and trigger system, the acquisition system, the monitoring and the power supplies are described in Sections 3, 4, 5 and 6 respectively. The tests of a camera demonstrator, composed of 7 modules, and the future plans are presented in Section 7.</text>
<section_header_level_1><location><page_2><loc_52><loc_48><loc_82><loc_49></location>2 Global structure and mechanics</section_header_level_1>
<text><location><page_2><loc_51><loc_20><loc_91><loc_47></location>All the scientific equipment and internal mechanical pieces are contained within a global structure, called the skeleton, that also provides the rigidity for the structure (Figure 2). The camera sealing is realized by aluminium honeycomb plates, creating the so-called skin, enclosing the camera equipment. The front part of the camera is sealed against rain by motorized lids controlled either remotely or locally. These lids have a secondary function to hold equipment for the electronics calibration (e.g. a Mylar plate for the single photo-electron calibration) and for the pointing calibration (positioning LEDs, reflecting screen). An additional plexiglass plate is placed between the lid and the entrance of the focal plane to avoid dust contamination. The camera will have ∼ 1800 PMTs grouped in ∼ 250 modules. These modules are inserted into a global structure called the sandwich. The rest of the internal equipment (services, communication and electrical interface with the exterior, cables) are held by mechanical fixtures such as small racks, wiring ducts and electrical cabinets. The weight of the camera is less than 2 tons. The total power consumption is of the order of 7.4 kW.</text>
<text><location><page_2><loc_51><loc_15><loc_90><loc_20></location>Since the camera is sealed, proper care has to be taken to avoid overheating of the camera. Air-based cooling systems have been studied (Figure 3), and merging these with a water-cooling system is under study.</text>
<section_header_level_1><location><page_2><loc_52><loc_11><loc_82><loc_12></location>3 Photodetector and detector unit</section_header_level_1>
<text><location><page_2><loc_51><loc_5><loc_91><loc_10></location>The focal plane of the NectarCAM camera is equipped with detector units . These detector units are composed of a photo-detector, the associated high voltage power supply, and an amplifier. CTA has decided to use PMTs for the</text>
<figure>
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<caption>Figure 3 : Simulation of a cooling airflow in the NectarCAM sandwich. The sandwich hosts ∼ 250 NECTAr modules.</caption>
</figure>
<figure>
<location><page_3><loc_10><loc_42><loc_48><loc_60></location>
<caption>Figure 4 : Principle of NECTAr readout modules.</caption>
</figure>
<text><location><page_3><loc_9><loc_20><loc_49><loc_36></location>photodetectors of their single mirror telescopes. A candidate PMT is the R11920-100 developed by Hamamatsu 2 . The R11920-100 PMT has a single photon electron signal FWHMof 2.5 to 3 ns. The light from the dead spaces between PMTs will be collected by custom designed Winston cones or lenses. The high voltage of the PMTs will be obtained from an external 24-V supply with an ASIC, similar to a chip designed for the KM3Net experiment [2]. The output from the PMTs is amplified by a wideband (450 MHz) 16-bit amplifier called PACTA [4]. The large bandwidth of the PACTA allows a minimal distortion of the PMT signal, and thus a better measurement of the arrival time of detected photons.</text>
<section_header_level_1><location><page_3><loc_10><loc_16><loc_20><loc_17></location>4 Readout</section_header_level_1>
<text><location><page_3><loc_9><loc_5><loc_49><loc_15></location>As illustrated in Figure 4, NECTAr modules are composed of 3 blocks. The block on the left hand side is composed of 7 PMTs, each associated with its detector unit, as described in Section 3. The central block is the front-end board. Signals from the detector units are amplified again in an ACTA[5] ASIC. At this level, the signal is divided into a low-gain, high-gain (relative gain 16) and a trigger channel. The trigger strategy is described in Section 5. The two first levels</text>
<figure>
<location><page_3><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<figure>
<location><page_3><loc_52><loc_71><loc_90><loc_90></location>
<caption>Figure 5 : Single photoelectron spectrum obtained with the NectarCAM V1 module. The PMT signal is amplified by a PACTA and an ACTA amplifier.</caption>
</figure>
<text><location><page_3><loc_51><loc_18><loc_91><loc_63></location>of the trigger (L0 and L1) are temporarily implemented as mezzanines on the read-out board. The outputs of the low-gain and high-gain channels of the ACTA are sent to a NECTAr chip[6]. The NECTAr chip has the dual functionality of switched capacitor array and analog-todigital convertor. The switched capacitor array, which acts as a circular buffer, has a depth of 1024 samples and can be operated between 500 MHz and 3.2 GHz. The NECTAr chip has a bandwidth of more than 400 MHz and a dynamic range of 11.3 bits. The power consumption is 210 mW. The dead time is 2 m s for the readout of 16 cells. The NECTAr chip is thus a low power, cheap alternative to the use of a Flash Analogue to Digital Convert (FADC). Once a L1 trigger signal is emitted, the data from a 1620 ns time interval around the date of trigger are read out. They are retrieved from the NECTAr chip and sent to a FPGA located on the front end board. This FPGA has several functions: interface to the ethernet, control of the NECTAr chip, of the L0, L1 trigger configurations, and the HV power supply. High level quantites such as the integrated charge over the readout window and the arrival time of the signal can be calculated inside the FPGA. It is then possible to send these quantities over the ethernet connection instead of the full event sample. The last block, on the right hand side of Figure 4 is a backplane board. It is used for the clock and synchronisation signal distribution, the low voltage (24 V) power distribution, and the L0/L1 trigger distribution. In some trigger schemes, one of the backplane boards could be used to make the L1 (camera) trigger decision. The readout board has been tested with photon and electrical signals. Its charge response is linear. It allows the measurement of signals between 1 and 3000 photo-electrons. The single photoelectron peak can be measured for calibration purposes (see Figure 5).</text>
<section_header_level_1><location><page_3><loc_52><loc_15><loc_79><loc_16></location>5 Data acquisition and trigger</section_header_level_1>
<text><location><page_3><loc_51><loc_8><loc_91><loc_14></location>The NectarCAM camera is triggered in a multilevel scheme, shown in Figure 6. The first level trigger (L0) is a modulelevel trigger. The information from several modules is combined to realize a camera-level trigger. The L0 and L1 trigger can be implemented with an 'analogue'[7] or a</text>
<figure>
<location><page_4><loc_10><loc_71><loc_48><loc_90></location>
<caption>Figure 6 : Triggering scheme of the NectarCAM camera.</caption>
</figure>
<text><location><page_4><loc_9><loc_40><loc_49><loc_65></location>'digital' solution. The latency of the camera trigger is less than 400 nanoseconds. This is much less than the depth of the switched capacitor array in the NECTAr chip, so that trigged events can be read back from the past. The trigged events are time-stamped and sent to a camera server by ethernet. Events on the camera server are accepted only if they are coincident with triggers from one or several other telescopes. The typical trigger rate of a single telescope is 5 kHz. The latency of the array trigger is a few m s. Single telescope events can be time stamped with an accuracy of a few ( ∼ 2) nanoseconds. The data acquisition follows the model described in [9]. Before being sent to the camera server, data from NECTAr modules are concentrated in ∼ 7 switches. The switches are linked to the camera server through three 10 Gbit connections. The data rate between a NECTAr module and the switches is 2 Mbit/s, if only the total charge and arrival time of pixels are transferred. It is ∼ 40 Mbit/s if all the samples in the region of interest are transferred.</text>
<section_header_level_1><location><page_4><loc_10><loc_36><loc_34><loc_37></location>6 Slow control and services</section_header_level_1>
<text><location><page_4><loc_9><loc_19><loc_49><loc_35></location>NectarCAM will have sensors to monitor the temperature, pressure and humidity inside the camera. Its safety will be ensured with ambient light sensors, smoke detectors and by tracking the position of the lids and back doors. Most subsystems of the camera are controlled remotely. This is the case for the calibration-related hardware (positionning leds), for the cooling system, for the DAQ crate and the clock distribution board. The monitoring/slow control of the NectarCAM will use either industrial solutions such as Programmable Logic Controllers, or custom made boards with FPGAs. The latter solution, where the sensors are accessed by industrial buses such as I2C, has been used for the H.E.S.S-2 telescope.</text>
<text><location><page_4><loc_10><loc_15><loc_49><loc_18></location>NectarCAM is powered by industrial low voltage supplies, since the High Voltage for operating the PMTs is created in the detector units (Section 3).</text>
<section_header_level_1><location><page_4><loc_10><loc_11><loc_32><loc_12></location>7 Camera demonstrators</section_header_level_1>
<text><location><page_4><loc_9><loc_5><loc_49><loc_10></location>The performance of the NectarCAM, especially the timing and triggering performances, the cooling and data acquisition will be tested with demonstrators. A first demonstrator with 7 NECTAr modules (Figure 7) is readily available.</text>
<figure>
<location><page_4><loc_79><loc_92><loc_91><loc_96></location>
</figure>
<figure>
<location><page_4><loc_52><loc_76><loc_90><loc_90></location>
<caption>Figure 7 : A 7-module demonstrator of the NectarCAM camera.</caption>
</figure>
<text><location><page_4><loc_51><loc_63><loc_91><loc_69></location>It will enable to test the performance of the analogue and digital triggers. The construction of a second demonstrator with 19 modules should start by the end of the year, with the aim of testing the integration of the components and the industrial processes.</text>
<section_header_level_1><location><page_4><loc_52><loc_59><loc_71><loc_60></location>8 Acknowledgements</section_header_level_1>
<text><location><page_4><loc_51><loc_55><loc_91><loc_59></location>We gratefully acknowledge support from the agencies and organizations listed in this page http: http://www.ctaobservatory.org/?q=node/22.</text>
<section_header_level_1><location><page_4><loc_52><loc_52><loc_61><loc_53></location>References</section_header_level_1>
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<list_item><location><page_4><loc_52><loc_48><loc_90><loc_51></location>[1] C. Naumann et al (the NECTAr consortium), Nuclear Instruments and Methods in Physics Research A 695 (2012), 44-51 doi:10.1016/j.nima.2011.11.008</list_item>
<list_item><location><page_4><loc_52><loc_46><loc_85><loc_48></location>[2] D. Gajanana, V. Gromov and P. Timmer, Journal of Instrumentation 8 (2013), C02030</list_item>
<list_item><location><page_4><loc_53><loc_45><loc_76><loc_46></location>doi:10.1088/1748-0221/8/02/C02030</list_item>
<list_item><location><page_4><loc_52><loc_43><loc_90><loc_45></location>[3] E.Aliu et al (the MAGIC collaboration), Astroparticle Physics 30 (2009), 293-305 doi:10.1016/j.astropartphys.2008.10.003</list_item>
<list_item><location><page_4><loc_52><loc_40><loc_91><loc_43></location>[4] A. Sanuy, D. Gascon, J-M. Paredes, L. Garrido, M. Rib and J. Sieiro, Journal of Instrumentation 7 (2012), C01100 doi:10.1088/1748-0221/7/01/C01100</list_item>
<list_item><location><page_4><loc_52><loc_35><loc_89><loc_39></location>[5] A. Sanuy, E.Delagnes, D.Gascon, X. Sieiro et al (the NECTAr consortium), Nuclear Instruments and Methods in Physics Research A 695 (2012), 385-389 doi:10.1016/j.nima.2011.12.025</list_item>
<list_item><location><page_4><loc_52><loc_31><loc_90><loc_35></location>[6] E.Delagnes et al. (the NECTAr consortium), Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (2011), 1457-1462 doi:10.1109/NSSMIC.2011.6154348</list_item>
<list_item><location><page_4><loc_52><loc_30><loc_81><loc_31></location>[7] M.Barcelo et al, these proceedings, ID-0396</list_item>
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</document>
[
{
"title": "The NectarCAM camera project",
"content": "J-F.GLICENSTEIN 1 , M.BARCELO 11 , J-A. BARRIO 12 , O.BLANCH 11 , J.BOIX 11 , J.BOLMONT 4 , C.BOUTONNET 2 S.CAZAUX 1 , E.CHABANNE 7 , C.CHAMPION 2 , F.CHATEAU 1 , S.COLONGES 2 , P.CORONA 4 , S.COUTURIER 5 , B.COURTY 2 , E.DELAGNES 1 , C.DELGADO 10 , J-P.ERNENWEIN 6 , S.FEGAN 5 , O.FERREIRA 5 , M.FESQUET 1 , G.FONTAINE 5 , N.FOUQUE 7 , F.HENAULT 8 , D.GASC 'ON 13 , D.HERRANZ 12 , R.HERMEL 7 , D.HOFFMANN 6 , J.HOULES 6 , S.KARKAR 4 , B.KHELIFI 5 , J.KN ODLSEDER 3 , G.MARTINEZ 10 , K.LACOMBE 3 , G.LAMANNA 7 , T.LEFLOUR 7 , R.LOPEZCOTO 11 , F.LOUIS 1 , A.MATHIEU 5 , E.MOULIN 1 , P.NAYMAN 4 , F.NUNIO 1 , J-F. OLIVE 3 , J-L. PANAZOL 7 , P-O. PETRUCCI 8 , M.PUNCH 2 , J.PRAST 7 , P.RAMON 3 , M.RIALLOT 1 , M.RIB 'O 13 , S.ROSIER-LEES 7 , A.SANUY 13 , J.SIERO 13 , J-P.TAVERNET 4 , L.A.TEJEDOR 12 , F.TOUSSENEL 455 , G.VASILEIADIS 9 , V.VOISIN 4 , V.WAEGEBERT 3 , C.ZURBACH 9 , FOR THE CTA CONSORTIUM. 1 DSM/IRFU, CEA-Saclay, F91191 Gif-sur-Yvette, France 2 APC, AstroParticule et Cosmologie, Universite Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cite, 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France glicens@cea.fr Abstract: In the framework of the next generation of Cherenkov telescopes, the Cherenkov Telescope Array (CTA), NectarCAM is a camera designed for the medium size telescopes covering the central energy range of 100 GeV to 30 TeV. NectarCAM will be finely pixelated ( ∼ 1800 pixels for a 8 o field of view, FoV) in order to image atmospheric Cherenkov showers by measuring the charge deposited within a few nanoseconds time-window. It will have additional features like the capacity to record the full waveform with GHz sampling for every pixel and to measure event times with nanosecond accuracy. An array of a few tens of medium size telescopes, equipped with NectarCAMs, will achieve up to a factor of ten improvement in sensitivity over existing instruments in the energy range of 100 GeV to 10 TeV. The camera is made of roughly 250 independent read-out modules, each composed of seven photo-multipliers, with their associated high voltage base and control, a read-out board and a multi-service backplane board. The read-out boards use NECTAr (New Electronics for the Cherenkov Telescope Array) ASICs which have the dual functionality of analogue memories and Analogue to Digital Converter (ADC). The camera trigger to be used will be flexible so as to minimize the read-out dead-time of the NECTAr chips. We present the camera concept and the design and tests of the various subcomponents. The design includes the mechanical parts, the cooling of the electronics, the readout, the data acquisition, the trigger, the monitoring and services. Keywords: Methods, techniques and instrumentation",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "NectarCAM is a new camera design for the medium size telescopes (MST) of the planned CTA 1 array of Imaging Atmospheric Cerenkov telescopes (IACTs). The CTA will have several sizes of single or dual-mirror telescopes. The single-mirror MST has a 12-meter diameter dish. Its camera will face new challenges: in the cameras. The arrival of the photons from a distant shower can last more than 100 nanoseconds. However, the signal to noise ratio in a pixel is optimized by recording and later integrating the signal in a much smaller window ( ∼ 10 ns). If the camera trigger is not flexible enough (e.g is similar to the H.E.S.S. camera trigger), the event energies would be systematically underestimated. Special trigger strategies have been designed to trigger different pixels at different moments (see Figure 16 of [1]). These strategies still need to be tested. The trigger rate is also expected to increase by a factor of two compared to the previous generation of Cherenkov telescopes, as result of the larger field of view, potetially leading to a larger dead time of the instrument. As a result of the larger field of view and better timing capabilities of the future Cherenkov cameras, it will be possible to improve the reconstruction of the events and to have a better hadron background rejection. However, these new capabilities present new challenges for Cherenkov telescopes. Two starting points for the NectarCAM are the architecture of the H.E.S.S.-2 camera and a read-out module: the NECTAr module [1]. Figure 1 shows an early version of the NECTAr module. This module converts the light of a set of photomultiplier tubes (PMTs) into a digital signal when some triggering conditions are fulfilled. The modular architecture of the H.E.S.S.-2 camera has the advantage of avoiding costly cables and allowing for easy maintenance. The modular mechanical structure is described in Section 2. For the NectarCAM, modules are groups of seven pixels, with the associated front-end board, which are completely autonomous in what concerns the power and control electronics. The only cables that go to the ground are the input power voltage and optical fibers for network communications and array trigger connections. Preliminary versions of the readout module exist. The photo-sensors, the readout and trigger system, the acquisition system, the monitoring and the power supplies are described in Sections 3, 4, 5 and 6 respectively. The tests of a camera demonstrator, composed of 7 modules, and the future plans are presented in Section 7.",
"pages": [
1,
2
]
},
{
"title": "2 Global structure and mechanics",
"content": "All the scientific equipment and internal mechanical pieces are contained within a global structure, called the skeleton, that also provides the rigidity for the structure (Figure 2). The camera sealing is realized by aluminium honeycomb plates, creating the so-called skin, enclosing the camera equipment. The front part of the camera is sealed against rain by motorized lids controlled either remotely or locally. These lids have a secondary function to hold equipment for the electronics calibration (e.g. a Mylar plate for the single photo-electron calibration) and for the pointing calibration (positioning LEDs, reflecting screen). An additional plexiglass plate is placed between the lid and the entrance of the focal plane to avoid dust contamination. The camera will have ∼ 1800 PMTs grouped in ∼ 250 modules. These modules are inserted into a global structure called the sandwich. The rest of the internal equipment (services, communication and electrical interface with the exterior, cables) are held by mechanical fixtures such as small racks, wiring ducts and electrical cabinets. The weight of the camera is less than 2 tons. The total power consumption is of the order of 7.4 kW. Since the camera is sealed, proper care has to be taken to avoid overheating of the camera. Air-based cooling systems have been studied (Figure 3), and merging these with a water-cooling system is under study.",
"pages": [
2
]
},
{
"title": "3 Photodetector and detector unit",
"content": "The focal plane of the NectarCAM camera is equipped with detector units . These detector units are composed of a photo-detector, the associated high voltage power supply, and an amplifier. CTA has decided to use PMTs for the photodetectors of their single mirror telescopes. A candidate PMT is the R11920-100 developed by Hamamatsu 2 . The R11920-100 PMT has a single photon electron signal FWHMof 2.5 to 3 ns. The light from the dead spaces between PMTs will be collected by custom designed Winston cones or lenses. The high voltage of the PMTs will be obtained from an external 24-V supply with an ASIC, similar to a chip designed for the KM3Net experiment [2]. The output from the PMTs is amplified by a wideband (450 MHz) 16-bit amplifier called PACTA [4]. The large bandwidth of the PACTA allows a minimal distortion of the PMT signal, and thus a better measurement of the arrival time of detected photons.",
"pages": [
2,
3
]
},
{
"title": "4 Readout",
"content": "As illustrated in Figure 4, NECTAr modules are composed of 3 blocks. The block on the left hand side is composed of 7 PMTs, each associated with its detector unit, as described in Section 3. The central block is the front-end board. Signals from the detector units are amplified again in an ACTA[5] ASIC. At this level, the signal is divided into a low-gain, high-gain (relative gain 16) and a trigger channel. The trigger strategy is described in Section 5. The two first levels of the trigger (L0 and L1) are temporarily implemented as mezzanines on the read-out board. The outputs of the low-gain and high-gain channels of the ACTA are sent to a NECTAr chip[6]. The NECTAr chip has the dual functionality of switched capacitor array and analog-todigital convertor. The switched capacitor array, which acts as a circular buffer, has a depth of 1024 samples and can be operated between 500 MHz and 3.2 GHz. The NECTAr chip has a bandwidth of more than 400 MHz and a dynamic range of 11.3 bits. The power consumption is 210 mW. The dead time is 2 m s for the readout of 16 cells. The NECTAr chip is thus a low power, cheap alternative to the use of a Flash Analogue to Digital Convert (FADC). Once a L1 trigger signal is emitted, the data from a 1620 ns time interval around the date of trigger are read out. They are retrieved from the NECTAr chip and sent to a FPGA located on the front end board. This FPGA has several functions: interface to the ethernet, control of the NECTAr chip, of the L0, L1 trigger configurations, and the HV power supply. High level quantites such as the integrated charge over the readout window and the arrival time of the signal can be calculated inside the FPGA. It is then possible to send these quantities over the ethernet connection instead of the full event sample. The last block, on the right hand side of Figure 4 is a backplane board. It is used for the clock and synchronisation signal distribution, the low voltage (24 V) power distribution, and the L0/L1 trigger distribution. In some trigger schemes, one of the backplane boards could be used to make the L1 (camera) trigger decision. The readout board has been tested with photon and electrical signals. Its charge response is linear. It allows the measurement of signals between 1 and 3000 photo-electrons. The single photoelectron peak can be measured for calibration purposes (see Figure 5).",
"pages": [
3
]
},
{
"title": "5 Data acquisition and trigger",
"content": "The NectarCAM camera is triggered in a multilevel scheme, shown in Figure 6. The first level trigger (L0) is a modulelevel trigger. The information from several modules is combined to realize a camera-level trigger. The L0 and L1 trigger can be implemented with an 'analogue'[7] or a 'digital' solution. The latency of the camera trigger is less than 400 nanoseconds. This is much less than the depth of the switched capacitor array in the NECTAr chip, so that trigged events can be read back from the past. The trigged events are time-stamped and sent to a camera server by ethernet. Events on the camera server are accepted only if they are coincident with triggers from one or several other telescopes. The typical trigger rate of a single telescope is 5 kHz. The latency of the array trigger is a few m s. Single telescope events can be time stamped with an accuracy of a few ( ∼ 2) nanoseconds. The data acquisition follows the model described in [9]. Before being sent to the camera server, data from NECTAr modules are concentrated in ∼ 7 switches. The switches are linked to the camera server through three 10 Gbit connections. The data rate between a NECTAr module and the switches is 2 Mbit/s, if only the total charge and arrival time of pixels are transferred. It is ∼ 40 Mbit/s if all the samples in the region of interest are transferred.",
"pages": [
3,
4
]
},
{
"title": "6 Slow control and services",
"content": "NectarCAM will have sensors to monitor the temperature, pressure and humidity inside the camera. Its safety will be ensured with ambient light sensors, smoke detectors and by tracking the position of the lids and back doors. Most subsystems of the camera are controlled remotely. This is the case for the calibration-related hardware (positionning leds), for the cooling system, for the DAQ crate and the clock distribution board. The monitoring/slow control of the NectarCAM will use either industrial solutions such as Programmable Logic Controllers, or custom made boards with FPGAs. The latter solution, where the sensors are accessed by industrial buses such as I2C, has been used for the H.E.S.S-2 telescope. NectarCAM is powered by industrial low voltage supplies, since the High Voltage for operating the PMTs is created in the detector units (Section 3).",
"pages": [
4
]
},
{
"title": "7 Camera demonstrators",
"content": "The performance of the NectarCAM, especially the timing and triggering performances, the cooling and data acquisition will be tested with demonstrators. A first demonstrator with 7 NECTAr modules (Figure 7) is readily available. It will enable to test the performance of the analogue and digital triggers. The construction of a second demonstrator with 19 modules should start by the end of the year, with the aim of testing the integration of the components and the industrial processes.",
"pages": [
4
]
},
{
"title": "8 Acknowledgements",
"content": "We gratefully acknowledge support from the agencies and organizations listed in this page http: http://www.ctaobservatory.org/?q=node/22.",
"pages": [
4
]
}
]
2013IJAA....3....1N
https://arxiv.org/pdf/1207.6846.pdf
<document>
<section_header_level_1><location><page_1><loc_12><loc_77><loc_80><loc_82></location>Wave optics and image formation in gravitational lensing</section_header_level_1>
<section_header_level_1><location><page_1><loc_23><loc_73><loc_40><loc_74></location>Yasusada Nambu</section_header_level_1>
<text><location><page_1><loc_23><loc_69><loc_81><loc_72></location>Department of Physics, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan</text>
<text><location><page_1><loc_23><loc_67><loc_29><loc_68></location>E-mail:</text>
<text><location><page_1><loc_29><loc_67><loc_58><loc_68></location>nambu@gravity.phys.nagoya-u.ac.jp</text>
<text><location><page_1><loc_23><loc_57><loc_84><loc_64></location>Abstract. We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing.</text>
<text><location><page_1><loc_23><loc_51><loc_48><loc_53></location>PACS numbers: 04.20.-q, 42.25.Fx</text>
<text><location><page_1><loc_12><loc_46><loc_52><loc_48></location>wave optics; image formation; gravitational lens</text>
<section_header_level_1><location><page_2><loc_12><loc_87><loc_27><loc_88></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_49><loc_84><loc_85></location>Gravitational lensing is one of the prediction of Einstein's general theory of relativity and many samples of images caused by gravitational lensing have been obtained observationally [1]. Light rays obey null geodesics in curved spacetime and they are deflected by gravitational potential of lens objects. In weak gravitational field with thin lens approximation, a path of a light ray obeys so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing effect are carried out based on this equation. Especially, we can obtain images of source objects by solving the lens equation using a ray tracing method. As a path of light ray is derived as the high frequency limit of electromagnetic wave, wave effects of gravitational lensing become important when the wavelength is not so much smaller than the size of lens objects and in such a situation, we must take into account of wave effects. For example, when we consider gravitational wave is scattered by gravitational lens objects, the wave effect gives significant impact on the amplification factor of intensity for waves [2, 3, 4]. Another example is direct detection of black holes via imaging their shadows [5, 6]. The apparent angular size of black hole shadows are so small that their detectability depends on angular resolution of telescopes which is determined by diffraction limit of image formation system. Thus it is important to investigate wave effects on images for successful detection of black hole shadows.</text>
<text><location><page_2><loc_12><loc_25><loc_84><loc_49></location>Although interference and diffraction of waves by gravitational lensing has been discussed in connection with amplification of waves, a little was discussed about how images by gravitational lensing are obtained based on wave optics. For electromagnetic wave, E. Herlt and H. Stephani [7] discussed the position of images by a spherical gravitational lens evaluating the Poynting flux of scattered wave at an observer. They claimed that there is a disagreement between wave optics and geometrical optics concerning the position of double images of a point source. But they have not presented complete understanding of image formation. In this paper, we consider image formation in gravitational lensing using wave optics and aim to understand how images by gravitational lensing are obtained in terms of waves. For this purpose, we adopt the diffraction theory of image formation in wave optics [8], which explains image formation in optical systems in terms of diffraction of waves.</text>
<section_header_level_1><location><page_2><loc_12><loc_21><loc_49><loc_22></location>2. Wave optics in gravitational lensing</section_header_level_1>
<text><location><page_2><loc_12><loc_14><loc_84><loc_19></location>We review the basic formalism of gravitational lensing based on wave optics [1]. In this paper, we does not consider polarization of waves and treat scalar waves as a model for electromagnetic waves.</text>
<figure>
<location><page_3><loc_20><loc_68><loc_75><loc_87></location>
<caption>Figure 1. The gravitational lens geometry of the source, the lens and the observer. α /lessmuch 1 is the deflection angle. r L is the distance from the observer to the lens object and r S is the distance from the observer to the source. r LS = r S -r L .</caption>
</figure>
<text><location><page_3><loc_12><loc_54><loc_84><loc_57></location>Let us consider waves propagating under the influence of the gravitational potential of a lens object. The background metric is assumed to be</text>
<formula><location><page_3><loc_23><loc_50><loc_84><loc_53></location>ds 2 = g µν dx µ dx ν = -(1 + 2 U ( r )) dt 2 +(1 -2 U ( r )) d r 2 , (1)</formula>
<text><location><page_3><loc_12><loc_44><loc_84><loc_50></location>where U ( r ) is the gravitational potential of the lens object with the condition | U | /lessmuch 1. The scalar wave propagation in this curved spacetime is described by the following wave equation:</text>
<text><location><page_3><loc_12><loc_39><loc_63><loc_40></location>and for a monochromatic wave with the angular frequency ω ,</text>
<formula><location><page_3><loc_23><loc_39><loc_84><loc_44></location>∂ µ ( √ -gg µν ∂ ν Φ ) = 0 , (2)</formula>
<formula><location><page_3><loc_23><loc_35><loc_84><loc_38></location>( ∇ 2 + ω 2 )Φ = 4 ω 2 U ( r )Φ , (3)</formula>
<text><location><page_3><loc_12><loc_32><loc_43><loc_35></location>where ∇ 2 is the flat space Laplacian.</text>
<text><location><page_3><loc_12><loc_19><loc_84><loc_33></location>We show the configuration of the gravitational lensing system considering here (Figure 1). The wave is emitted by a point source, scattered by the gravitational potential of the lens object and reaches the observer. We assume the wave scattering occurs in a small spatial region around the lens object and outside of this region, the wave propagates in a flat space. With the assumptions of the eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula provides the following amplitude of the wave at the observer [1, 3]</text>
<formula><location><page_3><loc_23><loc_14><loc_84><loc_19></location>Φ( η , ∆ ) = ωa 0 2 πir LS r L ∫ d 2 ξ exp [ iωS ( η , ξ , ∆ )] (4)</formula>
<text><location><page_3><loc_12><loc_11><loc_84><loc_14></location>where S ( η , ξ , ∆ ) is the effective path length (eikonal) along a path from the source position η to the observer position ∆ via a point ξ on the lens plane</text>
<formula><location><page_3><loc_23><loc_3><loc_84><loc_10></location>S ( η , ξ , ∆ ) = [ ( ξ -η ) 2 + r 2 LS ] 1 / 2 + [ ( ξ -∆ ) 2 + r 2 L ] 1 / 2 -ˆ ψ ( ξ ) ≈ ( η -∆ ) 2 2 r S + r S + r L r S 2 r LS ( ξ -∆ r L -η -∆ r S ) 2 -ˆ ψ ( ξ ) (5)</formula>
<text><location><page_4><loc_12><loc_83><loc_84><loc_89></location>and we assume that | η -∆ | /lessmuch r S and | ξ -∆ | /lessmuch r L . A constant a 0 represents the intensity of a point source. The two dimensional gravitational potential is introduced by</text>
<formula><location><page_4><loc_23><loc_78><loc_84><loc_83></location>ˆ ψ ( ξ ) = 2 ∫ ∞ -∞ dzU ( ξ , z ) . (6)</formula>
<text><location><page_4><loc_12><loc_76><loc_66><loc_78></location>Then the wave amplitude at the observer can be written as [1, 2]</text>
<formula><location><page_4><loc_23><loc_74><loc_84><loc_75></location>Φ( η , ∆ ) = Φ 0 ( η , ∆ ) F ( η , ∆ ) (7)</formula>
<text><location><page_4><loc_12><loc_69><loc_84><loc_73></location>where Φ 0 is the wave amplitude at the observer in the absence of the gravitational potential U :</text>
<formula><location><page_4><loc_23><loc_65><loc_84><loc_69></location>Φ 0 ( η , ∆ ) = a 0 r S exp [ iωS 0 ( η , ∆ )] , S 0 ( η , ∆ ) = ( η -∆ ) 2 2 r S + r S . (8)</formula>
<text><location><page_4><loc_12><loc_61><loc_84><loc_64></location>S 0 ( η , ∆ ) is the path length along a straight path from η to ∆ . The amplification factor F is given by the following form of a diffraction integral</text>
<formula><location><page_4><loc_23><loc_56><loc_84><loc_60></location>F ( η , ∆ ) = r S r L r LS ω 2 πi ∫ d 2 ξ exp [ iωS 1 ( η , ξ , ∆ )] , (9)</formula>
<text><location><page_4><loc_12><loc_42><loc_84><loc_52></location>where S 1 ( η , ξ , ∆ ) is the Fermat potential along a path from the source position η to the observer position ∆ via a point ξ on the lens plane. The first term in S 1 is the difference of the geometric time delay between a straight path from the source to the observer and a deflected path. The second term is the time delay due to the gravitational potential of the lens object. Now we introduce the following dimensionless variables:</text>
<formula><location><page_4><loc_28><loc_51><loc_84><loc_56></location>S 1 ( η , ξ , ∆ ) = r L r S 2 r LS ( ξ -∆ r L -η -∆ r S ) 2 -ˆ ψ ( ξ ) (10)</formula>
<formula><location><page_4><loc_14><loc_37><loc_84><loc_41></location>x = ξ ξ 0 , y = r L r S η ξ 0 , d = ( 1 -r L r S ) ∆ ξ 0 , w = r S ξ 2 0 r LS r L ω, ψ = r L r LS r S ξ 2 0 ˆ ψ (11)</formula>
<text><location><page_4><loc_12><loc_35><loc_30><loc_37></location>where we choose ξ 0 as</text>
<formula><location><page_4><loc_23><loc_30><loc_84><loc_35></location>ξ 0 = r L θ E , θ E = √ 4 Mr LS r L r S . (12)</formula>
<text><location><page_4><loc_12><loc_27><loc_84><loc_30></location>M is the mass of the gravitational source, ξ 0 and θ E represent the Einstein radius and the Einstein angle, respectively. Using these dimensionless variables,</text>
<formula><location><page_4><loc_23><loc_22><loc_84><loc_26></location>F ( y , d ) = w 2 πi ∫ d 2 x exp [ iw ( 1 2 ( x -y -d ) 2 -ψ ( x ) )] , (13)</formula>
<formula><location><page_4><loc_23><loc_17><loc_84><loc_22></location>Φ 0 ( y , d ) = a 0 r S exp [ iw ( r LS 2 r L ( y -r L r LS d ) 2 + r LS r L ξ 2 0 )] . (14)</formula>
<text><location><page_4><loc_12><loc_11><loc_84><loc_17></location>In the geometrical optics limit w /greatermuch 1, the diffraction integral (13) can be evaluated around the stationary points of the phase function in the integrand. The stationary points are determined by the solution of the following equation:</text>
<formula><location><page_4><loc_23><loc_8><loc_84><loc_10></location>x -y -d -∇ x ψ ( x ) = 0 . (15)</formula>
<text><location><page_4><loc_12><loc_4><loc_84><loc_8></location>This is the lens equation for gravitational lensing and determines the location of the image x for given source position y . As the specific model of gravitational lensing,</text>
<text><location><page_5><loc_12><loc_85><loc_84><loc_89></location>we consider a point mass as a gravitational source. In this case, the two dimensional gravitational potential is</text>
<formula><location><page_5><loc_23><loc_81><loc_84><loc_84></location>ψ ( x ) = ln | x | (16)</formula>
<text><location><page_5><loc_12><loc_79><loc_43><loc_81></location>and the deflection angle is given by ‡</text>
<formula><location><page_5><loc_23><loc_76><loc_84><loc_79></location>α = |∇ ξ ˆ ψ | = 4 M ξ . (17)</formula>
<text><location><page_5><loc_12><loc_74><loc_58><loc_75></location>For y = d = 0, the solution of the lens equation (15) is</text>
<formula><location><page_5><loc_23><loc_70><loc_84><loc_73></location>| x | = 1 (18)</formula>
<text><location><page_5><loc_12><loc_64><loc_84><loc_71></location>and represents the Einstein ring with the apparent angular radius θ E defined by (12). We show an example of images obtained as solutions of the lens equation (15) (Figure 2). To produce these images, we have assumed an extended source with Gaussian distribution of intensity.</text>
<figure>
<location><page_5><loc_12><loc_49><loc_30><loc_63></location>
</figure>
<figure>
<location><page_5><loc_66><loc_49><loc_84><loc_63></location>
</figure>
<figure>
<location><page_5><loc_30><loc_49><loc_48><loc_63></location>
</figure>
<figure>
<location><page_5><loc_48><loc_49><loc_66><loc_63></location>
<caption>Figure 2. Images of gravitational lensing by a point mass. The source is assumed to have the intensity with Gaussian distribution. From the left to the right panels, the source positions are y = 0 . 0 , 0 . 5 , 1 . 0 , 1 . 5.</caption>
</figure>
<text><location><page_5><loc_12><loc_37><loc_84><loc_40></location>The wave property is obtained by evaluating the diffraction integral (13). For a point mass lens potential (16), the integral can be obtained exactly</text>
<formula><location><page_5><loc_23><loc_31><loc_85><loc_36></location>F ( y ) = e ( i/ 2) w ( | y | 2 +ln( w/ 2)) e π/ 4 Γ ( 1 -i 2 w ) 1 F 1 ( 1 -i 2 w, 1 , -i 2 w | y | 2 ) . (19)</formula>
<text><location><page_5><loc_12><loc_30><loc_67><loc_32></location>On the observer plane, an interference pattern appears (Figure 3).</text>
<figure>
<location><page_5><loc_30><loc_10><loc_67><loc_29></location>
<caption>Figure 3. Amplification factor for w = 10.</caption>
</figure>
<text><location><page_5><loc_12><loc_3><loc_68><loc_6></location>‡ Using the original variables, the lens equation is ξ -∆ r L -η -∆ r S -r LS r S ∇ ξ ˆ ψ = 0 .</text>
<text><location><page_6><loc_12><loc_87><loc_78><loc_89></location>Near y = 0, the distance between adjacent fringes of the interference pattern is</text>
<formula><location><page_6><loc_23><loc_82><loc_84><loc_86></location>∆ y ∼ √ 2 π 4 Mω . (20)</formula>
<text><location><page_6><loc_12><loc_64><loc_84><loc_82></location>This fringe pattern is interpreted as interference between double images of a point source by the gravitational lensing. The question we aim to raise in this paper is how the interference pattern on the observer plane is related to the images of gravitational lensing in the geometrical optics limit. The wave amplitude on the observer plane does not make the image of the source and we have to transform the wave function to extract images. To answer this question, we introduce a 'telescope' in the gravitational lensing system and simulate observation of a star (a point source) using the telescope. With this setup, it is possible to understand how images of a source are formed in the framework of wave optics.</text>
<section_header_level_1><location><page_6><loc_12><loc_60><loc_45><loc_61></location>3. Image formation in wave optics</section_header_level_1>
<text><location><page_6><loc_12><loc_50><loc_84><loc_58></location>To establish relation between the interference pattern of the wave and the images of the source in the gravitational lensing system, we first consider an image formation system composed of a single convex lens and review how images of source objects appear in the framework of wave optics [8].</text>
<section_header_level_1><location><page_6><loc_12><loc_46><loc_44><loc_48></location>3.1. Image formation by a convex lens</section_header_level_1>
<text><location><page_6><loc_12><loc_39><loc_84><loc_45></location>Let us Φ in ( x ) is the incident wave from a point source in front of a thin convex lens and Φ t ( x ) is the transmitted wave by the lens (Figure 4). They are connected by the following relation</text>
<formula><location><page_6><loc_23><loc_36><loc_84><loc_38></location>Φ t ( x ) = T ( x )Φ in ( x ) , T ( x ) = e -iω | x | 2 2 f (21)</formula>
<text><location><page_6><loc_12><loc_29><loc_84><loc_35></location>where T ( x ) is called a lens transformation function. The action of a convex lens is to modify the phase of the incident wave. For a point source placed at z = -f (front focal point), the incident wave and the transmitted wave are</text>
<formula><location><page_6><loc_27><loc_23><loc_70><loc_28></location>Φ in ( x ) = e iωr r ≈ e iω ( f + | x | 2 2 f ) f , Φ t ( x ) = T Φ in = e iωf ,</formula>
<text><location><page_6><loc_12><loc_17><loc_84><loc_21></location>where we have used r = √ f 2 + | x | 2 ≈ f + | x | 2 / (2 f ) assuming | x | /lessmuch f . Thus, a convex lens converts a spherical wave front to a plane wave front.</text>
<figure>
<location><page_7><loc_34><loc_74><loc_62><loc_88></location>
<caption>Figure 4. Wave front modification by a convex lens.</caption>
</figure>
<text><location><page_7><loc_12><loc_63><loc_84><loc_68></location>Using this action of a convex lens for the incident wave and the transmitted wave, we can demonstrate the image formation by a convex lens in the framework of wave optics. Let us consider the configuration of the lens system shown in Figure 5.</text>
<figure>
<location><page_7><loc_31><loc_45><loc_65><loc_60></location>
<caption>Figure 5. One lens image formation system.</caption>
</figure>
<text><location><page_7><loc_12><loc_33><loc_84><loc_38></location>We assume the distribution of the source field on the object plane z = -a as Φ 0 ( x 0 ). Using the Fresnel-Kirchhoff diffraction formula, the amplitude of the wave in front of the lens is given by</text>
<formula><location><page_7><loc_23><loc_26><loc_73><loc_31></location>Φ 1 ( x ) ∝ ∫ d 2 x 0 Φ 0 ( x 0 ) e iωr 1 ≈ ∫ d 2 x 0 Φ 0 ( x 0 ) e iω ( a + | x 1 -x 0 | 2 2 a )</formula>
<text><location><page_7><loc_12><loc_20><loc_84><loc_26></location>where r 1 is the path length from a point on the object plane to a point on the lens plane and we have assumed | x 1 -x 0 | /lessmuch a . The amplitude of the wave just behind the lens is given by the relation (21)</text>
<formula><location><page_7><loc_35><loc_16><loc_61><loc_19></location>Φ 1 ' ( x 1 ) = t L ( x 1 ) e -iω | x 1 | 2 2 f Φ 1 ( x 1 )</formula>
<text><location><page_7><loc_12><loc_8><loc_84><loc_14></location>where t L is the aperture function of the lens defined by t L ( x ) = 1 for 0 ≤ | x | ≤ D and t L ( x ) = 0 for D < | x | . D represents a radius of the lens. With the assumption | x 1 -x | /lessmuch b , the amplitude of the wave on the z = b plane behind the lens is</text>
<formula><location><page_7><loc_12><loc_3><loc_35><loc_8></location>Φ 2 ( x ) ∝ ∫ d 2 x 1 Φ 1 ' ( x 1 ) e iωr 2</formula>
<formula><location><page_8><loc_17><loc_76><loc_84><loc_89></location>∝ ∫ d 2 x 0 d 2 x 1 Φ 0 ( x 0 ) t L ( x 1 ) e i ω 2 a | x 1 -x 0 | 2 e -i ω 2 f | x 1 | 2 e i ω 2 b | x 1 -x | 2 = ∫ d 2 x 0 d 2 x 1 Φ 0 ( x 0 ) t L ( x 1 ) exp [ iω { 1 2 ( 1 a + 1 b -1 f ) | x 1 | 2 -( x 0 a + x b ) · x 1 }] × exp [ i ω 2 ( | x 0 | 2 a + | x | 2 b )] . (22)</formula>
<text><location><page_8><loc_12><loc_73><loc_84><loc_76></location>For a value of b satisfying the following relation (the lens equation for a convex thin lens),</text>
<formula><location><page_8><loc_24><loc_69><loc_84><loc_72></location>1 a + 1 b = 1 f , (23)</formula>
<text><location><page_8><loc_12><loc_66><loc_36><loc_68></location>the wave amplitude becomes</text>
<formula><location><page_8><loc_12><loc_57><loc_84><loc_66></location>Φ 2 ( x ) ∝ ∫ d 2 x 0 d 2 x 1 Φ 0 ( x 0 ) t L ( x 1 ) exp [ -iω ( x 0 a + x b ) · x 1 ] exp [ iω | x 0 | 2 2 a ] ∝ ∫ d 2 x 0 Φ 0 ( x 0 ) ( 2 J 1 ( ωD | x /b + x 0 /a | ) ωD | x /b + x 0 /a | ) exp [ iω | x 0 | 2 2 a ] . (24)</formula>
<text><location><page_8><loc_12><loc_53><loc_84><loc_57></location>For ωD → ∞ limit, the Bessel function in (24) becomes the delta function and we obtains the following wave amplitude on z = b :</text>
<formula><location><page_8><loc_23><loc_48><loc_84><loc_53></location>Φ 2 ( x ) ∝ ∫ d 2 x 0 Φ 0 ( x 0 ) × δ 2 [ x 0 a + x b ] = Φ 0 ( -a b x ) . (25)</formula>
<text><location><page_8><loc_12><loc_41><loc_84><loc_48></location>Thus, a magnified image of the source field appears on the z = b plane. This reproduces the result of image formation in geometric optics; we have shown that an inverted images with magnification b/a of a source object appears on z = b satisfying the lens equation (23).</text>
<text><location><page_8><loc_12><loc_37><loc_84><loc_40></location>If we do not take ωD →∞ limit, due to the diffraction effect, an image of a point source has finite size on the image plane called the Airy disk [8]. Its size is given by</text>
<formula><location><page_8><loc_23><loc_32><loc_84><loc_36></location>∆ x Airy ∼ bλ D , λ = 2 π ω . (26)</formula>
<text><location><page_8><loc_12><loc_26><loc_84><loc_32></location>This value determines the resolving power of image formation system. For two point sources at x 0 = -d / 2 , d / 2, their separation on the image plane is bd/a . To resolve them, their separation must be larger than the size of the Airy disk:</text>
<formula><location><page_8><loc_24><loc_22><loc_84><loc_26></location>d a > λ D ≡ θ 0 . (27)</formula>
<text><location><page_8><loc_12><loc_18><loc_84><loc_22></location>The lefthand side of this inequality is the angular separation of the sources and θ 0 determines the resolving power of the image formation system.</text>
<section_header_level_1><location><page_8><loc_12><loc_14><loc_53><loc_16></location>3.2. Image formation in gravitational lens system</section_header_level_1>
<text><location><page_8><loc_12><loc_7><loc_84><loc_13></location>As we have observed that a convex lens can be a device for image formation in wave optics, we combine it with a gravitational lensing system and obtain images by gravitational lensing. We consider a configuration of the gravitational lens system shown in Figure 6 and examine how the images of the source object appear using wave optics.</text>
<figure>
<location><page_9><loc_21><loc_71><loc_74><loc_89></location>
<caption>Figure 6. Configuration of a gravitational lens with a convex lens system. Thin orange lines represent paths that contribute to diffraction integrals.</caption>
</figure>
<text><location><page_9><loc_12><loc_59><loc_84><loc_62></location>As the source object, we assume a point source of wave. The amplitude of the wave just in front of a convex lens is</text>
<formula><location><page_9><loc_23><loc_56><loc_84><loc_58></location>Φ L ( y , d ' ) = Φ 0 ( y , d ' ) F ( y , d ' ) . (28)</formula>
<text><location><page_9><loc_12><loc_51><loc_84><loc_55></location>This equation is the same as (7). After passing through the convex lens, the wave amplitude on the image plane z 2 is given by</text>
<formula><location><page_9><loc_12><loc_46><loc_84><loc_50></location>Φ I ( η , ∆ ) = ∫ | ∆ ' |≤ ∆ 0 d 2 ∆ ' Φ L ( η , ∆ ' ) exp [ -iω 2 f ∆ ' 2 ] exp [ iω 2 z 2 ( ∆ -∆ ' ) 2 ] (29)</formula>
<text><location><page_9><loc_12><loc_42><loc_84><loc_46></location>where ∆ 0 denotes the aperture of the convex lens. Using dimensionless variables, the wave amplitude on the image plane is</text>
<formula><location><page_9><loc_12><loc_33><loc_87><loc_42></location>Φ I ( y , d ) = a 0 r S ∫ | d ' |≤ d 0 d 2 d ' F ( y + d ' ) (30) × exp [ iw { r L r S 2 r LS ( 1 r S + 1 z 2 -1 f ) d ' 2 -( y + r L r S r LS z 2 d ) · d ' + 1 2 ( r LS r L y 2 + r L r S r LS z 2 d 2 )}] .</formula>
<text><location><page_9><loc_12><loc_29><loc_84><loc_33></location>If we choose the location of the image plane z 2 to satisfy the following 'lens equation' for a convex lens,</text>
<formula><location><page_9><loc_43><loc_26><loc_54><loc_29></location>1 r S + 1 z 2 = 1 f ,</formula>
<text><location><page_9><loc_12><loc_24><loc_57><loc_25></location>then the wave amplitude on the image plane becomes</text>
<formula><location><page_9><loc_12><loc_18><loc_84><loc_23></location>Φ I ( y , d ) = a 0 r S ∫ | d ' |≤ d 0 d 2 d ' F ( y + d ' ) exp [ -iw ( y + r L r S r LS f d ) · d ' ] × e i ( w/ 2) g ( y , d ) (31)</formula>
<text><location><page_9><loc_12><loc_10><loc_84><loc_18></location>where g = ( r LS /r L ) y 2 +( r L r S /r LS z 2 ) d 2 . Thus the wave amplitude on the image plane is the Fourier transform of the amplification factor F which gives the interference fringe pattern. Under the geometrical optics limit w /greatermuch 1, x integral in the amplification factor (13) can be approximated by the WKB form</text>
<formula><location><page_9><loc_33><loc_6><loc_63><loc_9></location>F ( y + d ' ) ≈ Ae iw [ 1 2 ( x ∗ -y -d ' ) 2 -ψ ( x ∗ ) ] ,</formula>
<text><location><page_10><loc_12><loc_87><loc_54><loc_89></location>where x ∗ ( y , d ' ) is the solution of the lens equation</text>
<formula><location><page_10><loc_23><loc_83><loc_84><loc_86></location>0 = x -y -d ' -∇ x ψ ( x ) ≈ x -y -∇ x ψ ( x ) . (32)</formula>
<text><location><page_10><loc_12><loc_77><loc_84><loc_83></location>We have assumed that the aperture of the convex lens is sufficiently smaller than the size of the gravitational lensing system and | d ' | ≤ d 0 /lessmuch 1 holds. Then, the wave amplitude on the image plane is</text>
<formula><location><page_10><loc_12><loc_68><loc_84><loc_77></location>Φ I ( y , d ) ∝ Ae iw [ 1 2 ( x ∗ -y ) 2 -ψ ( x ∗ ) ] e iwg ( y , d ) / 2 ∫ | d ' |≤ d 0 d 2 d ' exp [ -iw ( x ∗ + r L r S r LS f d ) · d ' ] = Ae iw [ 1 2 ( x ∗ -y ) 2 -ψ ( x ∗ ) ] e iwg ( y , d ) / 2 × 2 πd 2 0 J 1 ( w | x ∗ + β d | d 0 ) w | x ∗ + β d | d 0 , β ≡ r L r S r LS f . (33)</formula>
<formula><location><page_10><loc_23><loc_61><loc_84><loc_66></location>Φ I ( y , d ) ∝ δ 2 [ x ∗ ( y ) + r L r S r LS f d ] (34)</formula>
<text><location><page_10><loc_12><loc_65><loc_80><loc_68></location>For wd 0 →∞ limit (large lens aperture limit or high frequency limit), we obtains</text>
<text><location><page_10><loc_12><loc_58><loc_84><loc_61></location>and the image of the point source appears at the following location on the image plane determined by the lens equation (32):</text>
<formula><location><page_10><loc_23><loc_53><loc_84><loc_57></location>d = -r LS f r L r S × x ∗ ( y ) . (35)</formula>
<text><location><page_10><loc_12><loc_45><loc_84><loc_52></location>(34) and (35) reproduce the same result of image formation in the geometrical optics (ray tracing) in terms of the wave optics. This is what we aim to clarify in this paper. If the lens equation (32) has multiple solutions x ( j ) ∗ , j = 1 , 2 , · · · , the wave amplitude on the image plane becomes</text>
<formula><location><page_10><loc_23><loc_39><loc_84><loc_44></location>Φ I ( y , d ) ∝ ∑ j A j 2 J 1 ( w | x ( j ) ∗ + β d | d 0 ) w | x ( j ) ∗ + β d | d 0 (36)</formula>
<text><location><page_10><loc_12><loc_37><loc_32><loc_39></location>where A j are constants.</text>
<text><location><page_10><loc_12><loc_26><loc_84><loc_37></location>As an example of image formation in a gravitational lensing system using wave optics, we present the wave optical images of a point source by the gravitational lensing of a point mass (Figure 7). They are obtained by Fourier transformation of the amplification factor F (equation (31)) and teh lens equation for gravitational lensing (15) has not been used. This procedure corresponds to image formation by a convex lens. These images correspond to images obtained by geometric optics (Figure 2). We can observe wave effect in these images.</text>
<figure>
<location><page_10><loc_12><loc_10><loc_30><loc_24></location>
</figure>
<figure>
<location><page_10><loc_66><loc_10><loc_84><loc_24></location>
</figure>
<figure>
<location><page_10><loc_48><loc_10><loc_66><loc_24></location>
</figure>
<figure>
<location><page_10><loc_30><loc_10><loc_48><loc_24></location>
<caption>Figure 7. Wave optical images of a point source by the gravitational lensing of a point mass. Parameters are w = 40 , d 0 = 0 . 5(aperture of a convex lens), y = 0 , 0 . 5 , 1 , 1 . 5.</caption>
</figure>
<text><location><page_11><loc_12><loc_71><loc_84><loc_89></location>In each images, we can observe concentric interference pattern which is caused by finite size of the lens aperture and this is not intrinsic feature of the gravitational lensing system. We can also observe radial non-concentric patterns. They are caused by interference between double images and represent the intrinsic feature of the gravitational lensing system. For y = 0 case which corresponds to the Einstein ring in the geometrical optics limit, we can observe a bright spot at the center of the ring, which is the result of constructive interference and does not appear in geometric optics. For sufficiently large values of wd 0 , the wave amplitude at the observer coincides with the result obtained by geometric optics.</text>
<section_header_level_1><location><page_11><loc_12><loc_67><loc_24><loc_68></location>4. Summary</section_header_level_1>
<text><location><page_11><loc_12><loc_45><loc_84><loc_65></location>We investigated image formation in gravitational lensing system based on wave optics. Instead of using a ray tracing method, we obtained images directly from wave functions at the observer without using a lens equation of gravitational lensing. For this purpose, we introduced a 'telescope' with a single convex thin lens, which acts as a Fourier transformer for the interference pattern formed at an observer. The analysis in this paper relates the wave amplitude and images of the gravitational lensing directly. In the geometric optics limit of waves, images by lens systems are obtained by a lens equation which determines paths of each light rays. As light rays are trajectories of massless test particles (photon), expressing image in terms of wave is to express particle motion in terms of waves.</text>
<text><location><page_11><loc_12><loc_27><loc_84><loc_45></location>As an application and extension of analysis presented in this paper, we plan to investigate gravitational lensing by a black hole and obtain wave optical images of black holes. This subject is related to observation of black hole shadows [5, 6]. As the apparent angular sizes of black hole shadows are so small, the diffraction effect on images are crucial to resolve black hole shadows in observation using radio interferometer. For SgrA ∗ , which is the black hole candidate at Galactic center, the apparent angular size of its shadow is estimated to be ∼ 30 µ arc seconds and this value is the largest among black hole candidates. For a sub-mm VLBI with a baseline length D , using (27), the condition to resolve the shadow becomes</text>
<section_header_level_1><location><page_11><loc_42><loc_23><loc_54><loc_25></location>D > 1000 km ,</section_header_level_1>
<text><location><page_11><loc_12><loc_14><loc_84><loc_22></location>and this requirement shows the possibility to detect the black hole shadow of SgrA ∗ using the present day technology of VLBI telescope. Thus, analysis of black hole shadows based on wave optics is an important task to evaluate detectability of shadows and determination of black hole parameters via imaging of black holes.</text>
<text><location><page_11><loc_12><loc_4><loc_84><loc_14></location>The topic of wave optical image formation in black hole spacetimes belongs to a classical problem of wave scattering in black hole spacetimes [9]. As is well known, waves incident to a rotating black hole are amplified by the superradiance [10] due to dragging of spacetimes. This effect enables waves to extract the rotation energy of black holes. On the other hand, it is known that particles can also extract the rotation energy of</text>
<text><location><page_12><loc_12><loc_83><loc_84><loc_89></location>black holes via so called Penrose process. By investigating images of scattered waves by a rotating black hole, we expect to find out new aspect or interpretation of phenomena associated with superradiance in connection with the Penrose process.</text>
<section_header_level_1><location><page_12><loc_12><loc_79><loc_29><loc_80></location>Acknowledgments</section_header_level_1>
<text><location><page_12><loc_12><loc_71><loc_84><loc_77></location>This work was supported in part by the JSPS Grant-In-Aid for Scientific Research (C) (23540297). The author thanks all member of 'black hole horizon project meeting' in which the preliminary version of this paper was presented.</text>
<section_header_level_1><location><page_12><loc_12><loc_67><loc_22><loc_68></location>References</section_header_level_1>
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<list_item><location><page_12><loc_13><loc_61><loc_84><loc_63></location>[2] Nakamura T T and Deguchi S 1999 Wave Optics in Gravitational Lensing Prog. Theor. Phys. Suppl. 133 137-153</list_item>
<list_item><location><page_12><loc_13><loc_57><loc_84><loc_60></location>[3] Baraldo C and Hosoya A 1999 Gravitationally induced interference of gravitational waves by a rotating massive object Phys. Rev. D 59 083001</list_item>
<list_item><location><page_12><loc_13><loc_54><loc_84><loc_57></location>[4] Matsunaga N and Yamamoto K 2006 The finite source size effect and wave optics in gravitational lensing JCAP 01 023</list_item>
<list_item><location><page_12><loc_13><loc_51><loc_84><loc_54></location>[5] Falcke H, Melia F and Agol E 2000 Viewing the shadow of the black hole at the galactic center Astrophys. J. 528 L13-L16</list_item>
<list_item><location><page_12><loc_13><loc_47><loc_84><loc_50></location>[6] Miyoshi M, Ishituska K, Kameno S and Shen Z Q 2004 Direct imaging of the black hole, SgrA ∗ Prog. Theor. Phys. Suppl. 155 186-189</list_item>
<list_item><location><page_12><loc_13><loc_44><loc_84><loc_47></location>[7] Herlt E and Stephani H 1976 Wave Optics of the Spherical Gravitational Lens Part I: Diffraction of a Plane Electromagnetic Wave by a Large Star Int. J. of Theor. Phys. 15 45-65</list_item>
<list_item><location><page_12><loc_13><loc_43><loc_67><loc_44></location>[8] Sharma K K Optics: principles and applications (Academic Press, 2006)</list_item>
<list_item><location><page_12><loc_13><loc_39><loc_84><loc_42></location>[9] Futterman J A H, Handler F A and Matzner R A Scattering from black holes (Cambridge Univ. Press, 1988)</list_item>
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</document>
[
{
"title": "Yasusada Nambu",
"content": "Department of Physics, Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan E-mail: nambu@gravity.phys.nagoya-u.ac.jp Abstract. We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing. PACS numbers: 04.20.-q, 42.25.Fx wave optics; image formation; gravitational lens",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Gravitational lensing is one of the prediction of Einstein's general theory of relativity and many samples of images caused by gravitational lensing have been obtained observationally [1]. Light rays obey null geodesics in curved spacetime and they are deflected by gravitational potential of lens objects. In weak gravitational field with thin lens approximation, a path of a light ray obeys so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing effect are carried out based on this equation. Especially, we can obtain images of source objects by solving the lens equation using a ray tracing method. As a path of light ray is derived as the high frequency limit of electromagnetic wave, wave effects of gravitational lensing become important when the wavelength is not so much smaller than the size of lens objects and in such a situation, we must take into account of wave effects. For example, when we consider gravitational wave is scattered by gravitational lens objects, the wave effect gives significant impact on the amplification factor of intensity for waves [2, 3, 4]. Another example is direct detection of black holes via imaging their shadows [5, 6]. The apparent angular size of black hole shadows are so small that their detectability depends on angular resolution of telescopes which is determined by diffraction limit of image formation system. Thus it is important to investigate wave effects on images for successful detection of black hole shadows. Although interference and diffraction of waves by gravitational lensing has been discussed in connection with amplification of waves, a little was discussed about how images by gravitational lensing are obtained based on wave optics. For electromagnetic wave, E. Herlt and H. Stephani [7] discussed the position of images by a spherical gravitational lens evaluating the Poynting flux of scattered wave at an observer. They claimed that there is a disagreement between wave optics and geometrical optics concerning the position of double images of a point source. But they have not presented complete understanding of image formation. In this paper, we consider image formation in gravitational lensing using wave optics and aim to understand how images by gravitational lensing are obtained in terms of waves. For this purpose, we adopt the diffraction theory of image formation in wave optics [8], which explains image formation in optical systems in terms of diffraction of waves.",
"pages": [
2
]
},
{
"title": "2. Wave optics in gravitational lensing",
"content": "We review the basic formalism of gravitational lensing based on wave optics [1]. In this paper, we does not consider polarization of waves and treat scalar waves as a model for electromagnetic waves. Let us consider waves propagating under the influence of the gravitational potential of a lens object. The background metric is assumed to be where U ( r ) is the gravitational potential of the lens object with the condition | U | /lessmuch 1. The scalar wave propagation in this curved spacetime is described by the following wave equation: and for a monochromatic wave with the angular frequency ω , where ∇ 2 is the flat space Laplacian. We show the configuration of the gravitational lensing system considering here (Figure 1). The wave is emitted by a point source, scattered by the gravitational potential of the lens object and reaches the observer. We assume the wave scattering occurs in a small spatial region around the lens object and outside of this region, the wave propagates in a flat space. With the assumptions of the eikonal and the thin lens approximation, the Fresnel-Kirchhoff diffraction formula provides the following amplitude of the wave at the observer [1, 3] where S ( η , ξ , ∆ ) is the effective path length (eikonal) along a path from the source position η to the observer position ∆ via a point ξ on the lens plane and we assume that | η -∆ | /lessmuch r S and | ξ -∆ | /lessmuch r L . A constant a 0 represents the intensity of a point source. The two dimensional gravitational potential is introduced by Then the wave amplitude at the observer can be written as [1, 2] where Φ 0 is the wave amplitude at the observer in the absence of the gravitational potential U : S 0 ( η , ∆ ) is the path length along a straight path from η to ∆ . The amplification factor F is given by the following form of a diffraction integral where S 1 ( η , ξ , ∆ ) is the Fermat potential along a path from the source position η to the observer position ∆ via a point ξ on the lens plane. The first term in S 1 is the difference of the geometric time delay between a straight path from the source to the observer and a deflected path. The second term is the time delay due to the gravitational potential of the lens object. Now we introduce the following dimensionless variables: where we choose ξ 0 as M is the mass of the gravitational source, ξ 0 and θ E represent the Einstein radius and the Einstein angle, respectively. Using these dimensionless variables, In the geometrical optics limit w /greatermuch 1, the diffraction integral (13) can be evaluated around the stationary points of the phase function in the integrand. The stationary points are determined by the solution of the following equation: This is the lens equation for gravitational lensing and determines the location of the image x for given source position y . As the specific model of gravitational lensing, we consider a point mass as a gravitational source. In this case, the two dimensional gravitational potential is and the deflection angle is given by ‡ For y = d = 0, the solution of the lens equation (15) is and represents the Einstein ring with the apparent angular radius θ E defined by (12). We show an example of images obtained as solutions of the lens equation (15) (Figure 2). To produce these images, we have assumed an extended source with Gaussian distribution of intensity. The wave property is obtained by evaluating the diffraction integral (13). For a point mass lens potential (16), the integral can be obtained exactly On the observer plane, an interference pattern appears (Figure 3). ‡ Using the original variables, the lens equation is ξ -∆ r L -η -∆ r S -r LS r S ∇ ξ ˆ ψ = 0 . Near y = 0, the distance between adjacent fringes of the interference pattern is This fringe pattern is interpreted as interference between double images of a point source by the gravitational lensing. The question we aim to raise in this paper is how the interference pattern on the observer plane is related to the images of gravitational lensing in the geometrical optics limit. The wave amplitude on the observer plane does not make the image of the source and we have to transform the wave function to extract images. To answer this question, we introduce a 'telescope' in the gravitational lensing system and simulate observation of a star (a point source) using the telescope. With this setup, it is possible to understand how images of a source are formed in the framework of wave optics.",
"pages": [
2,
3,
4,
5,
6
]
},
{
"title": "3. Image formation in wave optics",
"content": "To establish relation between the interference pattern of the wave and the images of the source in the gravitational lensing system, we first consider an image formation system composed of a single convex lens and review how images of source objects appear in the framework of wave optics [8].",
"pages": [
6
]
},
{
"title": "3.1. Image formation by a convex lens",
"content": "Let us Φ in ( x ) is the incident wave from a point source in front of a thin convex lens and Φ t ( x ) is the transmitted wave by the lens (Figure 4). They are connected by the following relation where T ( x ) is called a lens transformation function. The action of a convex lens is to modify the phase of the incident wave. For a point source placed at z = -f (front focal point), the incident wave and the transmitted wave are where we have used r = √ f 2 + | x | 2 ≈ f + | x | 2 / (2 f ) assuming | x | /lessmuch f . Thus, a convex lens converts a spherical wave front to a plane wave front. Using this action of a convex lens for the incident wave and the transmitted wave, we can demonstrate the image formation by a convex lens in the framework of wave optics. Let us consider the configuration of the lens system shown in Figure 5. We assume the distribution of the source field on the object plane z = -a as Φ 0 ( x 0 ). Using the Fresnel-Kirchhoff diffraction formula, the amplitude of the wave in front of the lens is given by where r 1 is the path length from a point on the object plane to a point on the lens plane and we have assumed | x 1 -x 0 | /lessmuch a . The amplitude of the wave just behind the lens is given by the relation (21) where t L is the aperture function of the lens defined by t L ( x ) = 1 for 0 ≤ | x | ≤ D and t L ( x ) = 0 for D < | x | . D represents a radius of the lens. With the assumption | x 1 -x | /lessmuch b , the amplitude of the wave on the z = b plane behind the lens is For a value of b satisfying the following relation (the lens equation for a convex thin lens), the wave amplitude becomes For ωD → ∞ limit, the Bessel function in (24) becomes the delta function and we obtains the following wave amplitude on z = b : Thus, a magnified image of the source field appears on the z = b plane. This reproduces the result of image formation in geometric optics; we have shown that an inverted images with magnification b/a of a source object appears on z = b satisfying the lens equation (23). If we do not take ωD →∞ limit, due to the diffraction effect, an image of a point source has finite size on the image plane called the Airy disk [8]. Its size is given by This value determines the resolving power of image formation system. For two point sources at x 0 = -d / 2 , d / 2, their separation on the image plane is bd/a . To resolve them, their separation must be larger than the size of the Airy disk: The lefthand side of this inequality is the angular separation of the sources and θ 0 determines the resolving power of the image formation system.",
"pages": [
6,
7,
8
]
},
{
"title": "3.2. Image formation in gravitational lens system",
"content": "As we have observed that a convex lens can be a device for image formation in wave optics, we combine it with a gravitational lensing system and obtain images by gravitational lensing. We consider a configuration of the gravitational lens system shown in Figure 6 and examine how the images of the source object appear using wave optics. As the source object, we assume a point source of wave. The amplitude of the wave just in front of a convex lens is This equation is the same as (7). After passing through the convex lens, the wave amplitude on the image plane z 2 is given by where ∆ 0 denotes the aperture of the convex lens. Using dimensionless variables, the wave amplitude on the image plane is If we choose the location of the image plane z 2 to satisfy the following 'lens equation' for a convex lens, then the wave amplitude on the image plane becomes where g = ( r LS /r L ) y 2 +( r L r S /r LS z 2 ) d 2 . Thus the wave amplitude on the image plane is the Fourier transform of the amplification factor F which gives the interference fringe pattern. Under the geometrical optics limit w /greatermuch 1, x integral in the amplification factor (13) can be approximated by the WKB form where x ∗ ( y , d ' ) is the solution of the lens equation We have assumed that the aperture of the convex lens is sufficiently smaller than the size of the gravitational lensing system and | d ' | ≤ d 0 /lessmuch 1 holds. Then, the wave amplitude on the image plane is For wd 0 →∞ limit (large lens aperture limit or high frequency limit), we obtains and the image of the point source appears at the following location on the image plane determined by the lens equation (32): (34) and (35) reproduce the same result of image formation in the geometrical optics (ray tracing) in terms of the wave optics. This is what we aim to clarify in this paper. If the lens equation (32) has multiple solutions x ( j ) ∗ , j = 1 , 2 , · · · , the wave amplitude on the image plane becomes where A j are constants. As an example of image formation in a gravitational lensing system using wave optics, we present the wave optical images of a point source by the gravitational lensing of a point mass (Figure 7). They are obtained by Fourier transformation of the amplification factor F (equation (31)) and teh lens equation for gravitational lensing (15) has not been used. This procedure corresponds to image formation by a convex lens. These images correspond to images obtained by geometric optics (Figure 2). We can observe wave effect in these images. In each images, we can observe concentric interference pattern which is caused by finite size of the lens aperture and this is not intrinsic feature of the gravitational lensing system. We can also observe radial non-concentric patterns. They are caused by interference between double images and represent the intrinsic feature of the gravitational lensing system. For y = 0 case which corresponds to the Einstein ring in the geometrical optics limit, we can observe a bright spot at the center of the ring, which is the result of constructive interference and does not appear in geometric optics. For sufficiently large values of wd 0 , the wave amplitude at the observer coincides with the result obtained by geometric optics.",
"pages": [
8,
9,
10,
11
]
},
{
"title": "4. Summary",
"content": "We investigated image formation in gravitational lensing system based on wave optics. Instead of using a ray tracing method, we obtained images directly from wave functions at the observer without using a lens equation of gravitational lensing. For this purpose, we introduced a 'telescope' with a single convex thin lens, which acts as a Fourier transformer for the interference pattern formed at an observer. The analysis in this paper relates the wave amplitude and images of the gravitational lensing directly. In the geometric optics limit of waves, images by lens systems are obtained by a lens equation which determines paths of each light rays. As light rays are trajectories of massless test particles (photon), expressing image in terms of wave is to express particle motion in terms of waves. As an application and extension of analysis presented in this paper, we plan to investigate gravitational lensing by a black hole and obtain wave optical images of black holes. This subject is related to observation of black hole shadows [5, 6]. As the apparent angular sizes of black hole shadows are so small, the diffraction effect on images are crucial to resolve black hole shadows in observation using radio interferometer. For SgrA ∗ , which is the black hole candidate at Galactic center, the apparent angular size of its shadow is estimated to be ∼ 30 µ arc seconds and this value is the largest among black hole candidates. For a sub-mm VLBI with a baseline length D , using (27), the condition to resolve the shadow becomes",
"pages": [
11
]
},
{
"title": "D > 1000 km ,",
"content": "and this requirement shows the possibility to detect the black hole shadow of SgrA ∗ using the present day technology of VLBI telescope. Thus, analysis of black hole shadows based on wave optics is an important task to evaluate detectability of shadows and determination of black hole parameters via imaging of black holes. The topic of wave optical image formation in black hole spacetimes belongs to a classical problem of wave scattering in black hole spacetimes [9]. As is well known, waves incident to a rotating black hole are amplified by the superradiance [10] due to dragging of spacetimes. This effect enables waves to extract the rotation energy of black holes. On the other hand, it is known that particles can also extract the rotation energy of black holes via so called Penrose process. By investigating images of scattered waves by a rotating black hole, we expect to find out new aspect or interpretation of phenomena associated with superradiance in connection with the Penrose process.",
"pages": [
11,
12
]
},
{
"title": "Acknowledgments",
"content": "This work was supported in part by the JSPS Grant-In-Aid for Scientific Research (C) (23540297). The author thanks all member of 'black hole horizon project meeting' in which the preliminary version of this paper was presented.",
"pages": [
12
]
}
]
2013IJAsB..12...63B
https://arxiv.org/pdf/1206.0953.pdf
<document>
<section_header_level_1><location><page_1><loc_26><loc_86><loc_80><loc_89></location>Galactic exploration by directed self-replicating probes, and its implications for the Fermi paradox</section_header_level_1>
<text><location><page_1><loc_46><loc_82><loc_59><loc_83></location>Martin T. Barlow 1</text>
<text><location><page_1><loc_27><loc_79><loc_78><loc_81></location>Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. Email: barlow@math.ubc.ca</text>
<section_header_level_1><location><page_1><loc_14><loc_71><loc_21><loc_73></location>Abstract</section_header_level_1>
<text><location><page_1><loc_14><loc_65><loc_91><loc_70></location>This paper proposes a long term scheme for robotic exploration of the galaxy, and then considers the implications in terms of the 'Fermi paradox' and our search for ETI. We discuss the 'galactic ecology' of civilizations in terms of the parameters T (time between ET civilizations arising) and L , the lifetime of these civilizations. Six di ff erent regions are described.</text>
<text><location><page_1><loc_14><loc_63><loc_85><loc_64></location>Keywords: Self-Replicating Probes, Galactic Exploration, Search for Extraterrestrial Life, Fermi Paradox</text>
<section_header_level_1><location><page_1><loc_14><loc_59><loc_26><loc_60></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_14><loc_45><loc_51><loc_57></location>The proposal to explore the galaxy by selfreplicating probes goes back at least as far as Freitas 1980a and Tipler 1980. Numerous criticisms have been made, notably by Sagan and Newman 1983. A recent paper (Wiley 2012) reconsiders the topic, and finds some of these criticisms without much merit. This paper outlines in more detail one exploration scheme and its benefits, and then considers the implication for the galactic 'ecology' of intelligent species.</text>
<section_header_level_1><location><page_1><loc_14><loc_40><loc_51><loc_42></location>2. Exploration of the galaxy by self replicating probes</section_header_level_1>
<text><location><page_1><loc_14><loc_33><loc_51><loc_39></location>Our starting point is as in Freitas 1980b and Tipler 1980. This relies on two technologies which we do not have at present, but which it is reasonable to suppose we will attain within the next few hundred years:</text>
<unordered_list>
<list_item><location><page_1><loc_14><loc_30><loc_51><loc_32></location>(T1) A propulsion system capable of sending probes to nearby stars, at say 0.01 c.</list_item>
<list_item><location><page_1><loc_14><loc_27><loc_51><loc_30></location>(T2) An AI system which, in total, is able, with the resources found in most star systems, of replicating itself.</list_item>
</unordered_list>
<text><location><page_1><loc_14><loc_20><loc_51><loc_26></location>Propulsion systems. While no existing system can reach these speeds, various proposals for 100 year probes to Alpha Centauri have been made, such as Project Daedalus. A speed of 0.01c does not seem unduly optimistic, and in fact many proposals for galactic</text>
<text><location><page_1><loc_54><loc_57><loc_90><loc_60></location>exploration, such Bjørk (2007), consider probes which travel at 0.1c.</text>
<text><location><page_1><loc_54><loc_51><loc_91><loc_57></location>AIsystemsAgain, we do not have self-replicating systems at present, and what such a system would be like must rely largely on conjecture. In this context note the following remark in Wiley (2012):</text>
<text><location><page_1><loc_58><loc_45><loc_87><loc_50></location>One point we take issue with is an inherent and frequently unconscious biological bias that pervades consideration of computerized intelligence, including SRPs.</text>
<text><location><page_1><loc_54><loc_34><loc_91><loc_44></location>A common tendency has been to imagine a selfreplicating machine as being rather like a bacterium: that is a single machine which (somehow, almost magically) is able to move around in its environment and replicate itself. If we start with present technology, we are forced to imagine something rather di ff erent. The system as a whole might consist of 3 parts:</text>
<unordered_list>
<list_item><location><page_1><loc_54><loc_26><loc_91><loc_34></location>(A) A number of robots and probes, of several di ff erent types, which are together capable of exploring a solar system, and gathering resources (metals, volatiles etc). (B) A 'slow assembler' which would be able refine these materials into components which would make the final factory (C).</list_item>
<list_item><location><page_1><loc_54><loc_20><loc_91><loc_25></location>(C) A large scale factory, or collection of factories, which would be able to manufacture copies of (A) and (B), as well as additional surveying and communication devices.</list_item>
</unordered_list>
<text><location><page_1><loc_54><loc_14><loc_91><loc_20></location>The payload of the probe would consist of (A + B), together with enough raw materials (fuel etc.) to get started in the new system. Once (C) was made, resources would be gathered for as long as was necessary,</text>
<text><location><page_2><loc_14><loc_83><loc_51><loc_90></location>and a number of probes would then be sent to nearby stars. If we take this view, then a 'self-replicating space probe' (SRP) would not be a single machine, but rather a collection of di ff erent machines with an overall capability of replication.</text>
<text><location><page_2><loc_14><loc_68><loc_51><loc_83></location>See Freitas 1980b for a much more detailed description of such a probe, with the probe (A + B) plus the fuel for the voyage having a mass of of around 10 10 kg. The factory described there only makes 1 new probe every 500 years, but (see Section 7.1) using a longer period for the initial construction gives a larger factory which can create 1000 new probes in 1500-200 years. For simplicity I will take the reproduction time, between the arrival of the initial probe in a star system, and the completed factory (C) sending out new probes, to be TR = 1000.</text>
<text><location><page_2><loc_14><loc_53><loc_51><loc_67></location>The AI needed for such a system far exceeds what is possible at present. But while the kinds of decisions necessary for the AI (e.g. 'what kind of material is present in this asteroid?', 'can it be transported to the factory?') would require a very high level of skill, this would be within fairly narrow parameters, and a human level of overall initiative and judgement would not be required. Even if machines with a human intelligence can be constructed, it might be desirable to limit the intelligence of the SRPs.</text>
<text><location><page_2><loc_14><loc_35><loc_51><loc_53></location>As Tipler 1980 notes, there are reasons other than stellar exploration to develop these technologies. Progress in AI has been far slower than supposed by early optimists, but it still seems reasonable to suppose that, within a few hundred years, we will be able to build such SRPs. The development of such machines would, at least for a while, introduce an age of plenty, since it would open up the resources of the solar system for our exploitation. Some idealistic individuals or groups might then be willing to invest the resources in making a number of the probes (A + B), and send them to nearby stars. (See Mathews 2011 for another proposal to explore the galaxy by SRPs.)</text>
<section_header_level_1><location><page_2><loc_14><loc_32><loc_32><loc_33></location>2.1. Exploration strategy</section_header_level_1>
<text><location><page_2><loc_14><loc_15><loc_51><loc_31></location>I now propose a scheme for galactic exploration, assuming that we do develop the technologies (T1) and (T2) described above. The first step would be to send probes to the 10-100 nearest stars, which are all within about 20 ly of Earth. (Long before the start of such a mission we will have very good data on the planetary systems of these stars.) The probes would arrive at their destinations 400-2000 years after the mission start. They would remain in radio contact with Earth, (with a time lag of 40 years or less), would report on their discoveries, and would be able to receive updates on strategy. (Among the exploration devices (A) would</text>
<text><location><page_2><loc_54><loc_86><loc_91><loc_90></location>be systems able to transmit and receive narrow band radio or laser communication over a distance of say 100 ly.)</text>
<text><location><page_2><loc_54><loc_62><loc_91><loc_86></location>I will call the initial star systems Level 1 'colonies', though there is no suggestion that they would have a human population. After the construction of the factory (C) on a Level 1 colony, the colony would send out SRPs (let us say about 1,000 - 10,000) to create colonies at 'Level 2'. I have suggested an initial 'hop size' of 10-20 ly, since the number of probes that could be sent out from our Solar System might be limited by resource constraints. However once a colony at Level 1 or higher had a working factory, there would be no such limit on the number of probes that could be sent out, and it would be sensible to send as many as was necessary to explore every star system within the second 'maximum hop size'. There are about 15,000 stars within 100 ly of Earth, so with some useful duplication the Level 1 systems would together be able to send probes to every star in this region.</text>
<text><location><page_2><loc_54><loc_53><loc_91><loc_62></location>The maximum hop size hm would be the greatest distance such a probe could be sent with a probability greater than 90% of arriving. I will take hm = 100; this also needs to be less than the maximum distance for radio or laser communication, but this is much greater than 100 ly.</text>
<text><location><page_2><loc_54><loc_42><loc_91><loc_53></location>The Probes from the Level 2 colonies would then establish Level 3 colonies, and so on. Each colony at Level n would report back to its Level n -1 ancestor, and receive updating instructions from it. While it would be desirable for the Level 1 colonies to produce manyprobes, as the radius of the exploration sphere became larger, and so the curvature of its surface became less, fewer new probes per colony would be needed.</text>
<text><location><page_2><loc_54><loc_31><loc_91><loc_42></location>Within a few thousand years of the mission start our descendants on earth (if they still existed) would be receiving a flood of information from the exploration of hundreds of star systems. The Great Pyramid was built around 4500 years ago; 4500 years after its start the mission would be well under way, and have given us detailed data on every star system within about 30 ly of Earth.</text>
<text><location><page_2><loc_54><loc_24><loc_91><loc_30></location>The overall mission would continue until the planetary system of every star in the galaxy had been explored. Let vp = 0 . 01 c be the speed of the probes, and ve be the propagation speed of the exploration front. Then</text>
<formula><location><page_2><loc_60><loc_20><loc_84><loc_23></location>ve = hm TR + hm / vp = vp 1 1 + vpTR / hm .</formula>
<text><location><page_2><loc_54><loc_15><loc_91><loc_19></location>So TR = 1000 and hm = 100 give ve / vp = 10 / 11, the exploration front travels at nearly the same speed as the probes, and the total time to explore the galaxy</text>
<text><location><page_3><loc_14><loc_86><loc_51><loc_90></location>is around 10 7 years. This compares with exploration times of the order of 10 8 years given by Bjørk 2007, using probes which travel at 0.1c, but do not replicate.</text>
<section_header_level_1><location><page_3><loc_14><loc_83><loc_26><loc_84></location>2.2. Refinements</section_header_level_1>
<text><location><page_3><loc_14><loc_80><loc_51><loc_83></location>While the basic strategy is as above, it is necessary to consider a number of refinements.</text>
<text><location><page_3><loc_14><loc_64><loc_51><loc_79></location>(a)Resourceusewithinsystem. The best place for the construction of the factory (C) might be the moons of a planet in the outer reaches of the star system. Assuming a mass of 10 10 kg for the probes (A + B) and fuel, the construction of (C) plus say 10,000 probes would use at most a handful of minor planets and comets. This would leave plenty of material behind, even on the Level 1 colonies, and it would not be necessary to to 'strip mine' the galaxy in order to complete this exploration. One would only need a few probes per star one plus a margin for accidents.</text>
<unordered_list>
<list_item><location><page_3><loc_14><loc_60><loc_51><loc_63></location>(b) Systems with planets with life. In systems with planets with complex life, a di ff erent procedure should be followed. Two possibilities would be:</list_item>
<list_item><location><page_3><loc_14><loc_58><loc_51><loc_59></location>(i) Report, build a factory (C), explore the system thor-</list_item>
<list_item><location><page_3><loc_14><loc_54><loc_51><loc_58></location>oughly, and then await instructions from Earth, (ii) Report, do nothing, and await instructions from Earth.</list_item>
</unordered_list>
<text><location><page_3><loc_14><loc_45><loc_51><loc_53></location>The first and second level probes would provide enough data to refine this strategy, at an early stage of the overall mission. In the very unlikely event that more than 90% of systems have planets with complex life, a modification of (ii) would be needed so that a reasonable proportion of colonies did send out probes.</text>
<text><location><page_3><loc_14><loc_21><loc_51><loc_45></location>(c) Extinctionof the humanrace. Once set in motion, the exploration could continue without any further human intervention. However, this proposal envisions continued interaction and direction: Earth would receive data from the probes, and based on this revised instructions on exploration (as well as possible system upgrades) would be sent out. What however if humanity becomes extinct, or just loses interest in the mission? There are many possible procedures which could be followed, of which the simplest are: (i) Continue anyway, (ii) Abandon further exploration. For simplicity I propose (ii), and suggest that every 100 years the Level 1 colonies would ask Earth 'Shall we continue?' If 1000 years went by with no positive response, the project would be mothballed, and instructions would be sent through the communication tree that no further SRPs were to be built.</text>
<text><location><page_3><loc_14><loc_15><loc_51><loc_20></location>(d) Communication and direction. A key part of this exploration scheme is that the SRPs are not autonomous, but that the whole exploration process is directed, ultimately from Earth. The first requirement</text>
<text><location><page_3><loc_54><loc_83><loc_91><loc_90></location>here is that Level n colonies be able to communicate (over a distance of say 100 ly) with their Level n -1 ancestors. Even with present technology, we could build transmitters and receivers capable of working over these distances.</text>
<text><location><page_3><loc_54><loc_59><loc_91><loc_83></location>While all the Level 1 colonies would send out probes, this would not be necessary for higher level colonies, and stars could be divided into two groups. For the first, 'end nodes', a factory C would be built, the star system explored, but no probes would be sent out. The second, 'branch nodes' would send out probes. Among the pieces of infrastructure built in each colony would be telescopes to survey the stellar neighbourhood, and using this data, nearby Level n colonies would coordinate the exploration of their neighbourhoods. (Nearby colonies would be 10-100 ly apart, so communication time would be small compared with the time to build probes, or for the journeys.) While the algorithm to coordinate this process may appear complicated, it is well within our current capabilities - unlike the AI needed for robotic exploration of a planetary system</text>
<text><location><page_3><loc_54><loc_38><loc_91><loc_58></location>(e) Mission creep and machine mutation. A widely voiced concern with SRPs has been that they might mutate, run amok, and eat up the galaxy - see Sagan and Newman 1983. However Wiley 2012 argues that it should be possible to build in su ffi cient reliability to avoid this outcome - note also his comment above on inappropriate biological analogies. In the exploration scheme proposed here about 1000 generations would be needed to explore the galaxy - fewer if hops longer than 100 ly are feasible. The total number of replicators (C) built would be of the same order as the number of stars in the galaxy, that is about 4 × 10 11 . Wiley 2012 points out that this is much less than the number of cell divisions within a human lifetime, which is of the order of 10 16 .</text>
<text><location><page_3><loc_54><loc_15><loc_91><loc_37></location>If necessary, further steps could be taken to reduce the overall risk. In the initial stages of the exploration we would want every planetary system to be explored carefully. However, it seems likely that after the first million or so systems had been explored, we would have a good understanding of the processes underlying the formation and development of planetary systems, and might only be interested in those systems which had life, complex life, or other exceptional features. The later phases of the exploration could therefore proceed as follows. Each 'branch node' would first send out about 10 new full probes (A + B) to establish the next generation of colonies. Next, it would send out reduced probes, just consisting of (A), to all the stars in its exploration patch. These would explore the target system, and report back to the send-</text>
<text><location><page_4><loc_14><loc_75><loc_51><loc_90></location>ing branch node. Without the reproductive capacity (B) these probes would ultimately run out of fuel and become inactive. A full probe (A + B) would then be sent to any system that merited further attention. Assuming that 'interesting' systems are rare, this modification would reduce the number of full replications by a factor of 1000 or so. Further safety mechanisms could also be built in, such as deeply embedded software constraints on the total number of probes that the factories (C) could make, or on the total number of permitted generations.</text>
<text><location><page_4><loc_14><loc_47><loc_51><loc_74></location>(f)Crossinglargespacesandpercolation. Landis 1998 has suggested a percolation model for the spread of a species through the galaxy, and shown that in some cases this leads to large vacant (unexplored) regions. However bond percolation on the lattice is a poor model for the type of exploration proposed above, since each 'branch node' would send out rather more than 5 probes. Further, the communication envisaged between colonies would mean that colonies would become aware of interstellar voids (with no or few stars), and regions that, perhaps because of the failure of a number of probes, were remaining unexplored. They could then send additional probes to explore these regions. If we consider the mathematical graph whose vertices are the stars, and join by edges all pairs of stars within 100 ly., then the exploration scheme proposed here will explore all stars in the connected component containing our sun, and it seems overwhelmingly likely that this spans most of the galaxy.</text>
<text><location><page_4><loc_14><loc_33><loc_51><loc_47></location>(g) System updates. We would want to be able to incorporate updates into the systems (A,B,C). It is possible that this could be done by radio, but the available bandwidth might be too small for the necessary amount of data. One can imagine a system of 'fast packets' - small probes carrying data, which travel at say 0.1c between colonies with the infrastructure to send and receive them. However one disadvantage of allowing such updates is that it would make the colonies more vulnerable to mutations or computer viruses.</text>
<text><location><page_4><loc_14><loc_17><loc_51><loc_32></location>(h)Contactwithextraterrestrialintelligence(ETI).Detailed thought would need to be given on what course of action should be taken if either ETI were found, or traces of them. There would be time to refine strategies in the first few millenia of the mission, as data on frequency and type of life in other star systems accumulated. The number of possible actions the probes could take is large, and a full discussion of this is beyond the scope of this article. The simplest (but not the quickest) option would be for the the probe to report back, and take no further action until instructed.</text>
<section_header_level_1><location><page_4><loc_54><loc_89><loc_88><loc_90></location>3. Implications for SETI, and Fermi's question</section_header_level_1>
<text><location><page_4><loc_54><loc_72><loc_91><loc_88></location>Let us now make the hypothesis (H) that the technolgies (T1) and (T2) can be attained, and explore the consequences. The exploration scheme outlined above, using these technologies is one which, if it survives long enough, the human race might adopt - no doubt with a number of improvements. The payo ff is that with a relatively low initial cost our descendants would obtain detailed data about every star system in the galaxy. In particular they would learn how many planets support life, what kind of life it is, and just how rare complex or intelligent life is.</text>
<text><location><page_4><loc_54><loc_64><loc_91><loc_72></location>If there have been technological ETI in the galaxy, then they would also have had this option. So - this is Fermi's question 'Where are they?' (This is often called the 'Fermi paradox', but it is only a paradox if one begins with the assumption that intelligent life is common. In fact we have no information on this.)</text>
<text><location><page_4><loc_54><loc_61><loc_91><loc_63></location>Let us recall the Drake equation, slightly modified for our purposes:</text>
<formula><location><page_4><loc_63><loc_58><loc_81><loc_60></location>N = R ∗ · fp · ne · f /lscript · fci · L ;</formula>
<text><location><page_4><loc_54><loc_42><loc_91><loc_57></location>here N is the number of existing civilizations sending out SRPs, R ∗ is the rate of star formation per year in the galaxy, fp is the fraction of those stars that have planets, ne is the average number of planets that can potentially support life per star that has planets, fl is the fraction of these that develop life, fci is the fraction of these that develop civilizations that send out SRPs, and L is the average lifetime of such civilizations. This lifetime is the time that either the civilization itself, or its SRPs, remain active. (From now on, I will use the term 'civilization' for 'civilizations that send out SRPs').</text>
<text><location><page_4><loc_54><loc_29><loc_91><loc_41></location>We do have estimates of at least the order of magnitude of some of the early terms in this expression: for example R ∗ /similarequal 7, and data from the Kepler satellite suggests that fp /similarequal 0 . 5, while ne is quite small. (Out of about 10,000 systems surveyed, only a handful have planets which look really promising from the point of finding earthlike life.) At present fl and fci are utterly unknown, though estimates of fl may at some point become available via spectroscopic search for oxygen.</text>
<text><location><page_4><loc_54><loc_16><loc_91><loc_28></location>I have given the Drake equation in a simple form. A more realistic equation would take account of randomness, and the fact that these factors are not constant in time - see for example Glade et. al. 2012. However, the uncertainty in our knowledge of the parameters in the equation is so great that these refinements seem to the author of this article to add little to what can be achieved with a simple 'back of an envelope' calculation.</text>
<text><location><page_5><loc_1><loc_87><loc_15><loc_88></location>PSfrag replacements</text>
<figure>
<location><page_5><loc_15><loc_72><loc_49><loc_90></location>
<caption>Figure 1: Galactic ecology parameter space</caption>
</figure>
<text><location><page_5><loc_17><loc_65><loc_27><loc_66></location>Let us now set</text>
<formula><location><page_5><loc_23><loc_62><loc_43><loc_64></location>λ = R ∗ · fp · ne · f /lscript · fci = N / L ;</formula>
<text><location><page_5><loc_14><loc_53><loc_51><loc_61></location>so that λ is the number of civilizations arising per year in the galaxy. As an upper bound, if fl = fci = 1 (surely very unlikely) and ne = 0 . 03, we obtain λ ≤ 0 . 1. Let us set T = 1 /λ to be the average length of time in years between successive civilizations arising in the galaxy; the estimates above suggest it is unlikely that T < 10.</text>
<text><location><page_5><loc_14><loc_33><loc_51><loc_52></location>Let us now consider the 'galactic ecology' in terms of the two parameters T = λ -1 and L . While a better model would allow for randomness of L , a simple mean model already yields useful insights. Figure 1 shows a plot of log L against log T . Since the galaxy is about 10 10 years old, we have log L ≤ 10, and it seems reasonable to take also log L ≥ 2. The estimates above give log T ≥ 1. We have no upper bound on T : it is not legitimate to use the Copernican principle to assert that because there is at least one potential civilization in the galaxy (us) then T ≤ L . Civilizations might only arise in one galaxy in a billion, and those that arose would still observe themselves to be in a galaxy. In the diagram I take 1 ≤ log T ≤ 14.</text>
<text><location><page_5><loc_14><loc_26><loc_51><loc_32></location>Let us now consider the various regions of the diagram. The descriptive statements for the regions apply to typical points in the region - naturally these will become weaker if the point (log T , log L ) is close to the boundary between regions.</text>
<text><location><page_5><loc_14><loc_20><loc_51><loc_25></location>( R 1) ('Alone') If log T > 10 then probably no other civilization has arisen in the galaxy. (A more accurate statement would be that the mean number of such civilizations is less than 1.)</text>
<text><location><page_5><loc_14><loc_15><loc_51><loc_19></location>( R 2) ('Pompeii') If log T ≤ 10 and log L < log T then N < 1 and there is no other civilization existing now. However, 10 10 / T ≥ 1 civilizations have existed, and</text>
<text><location><page_5><loc_54><loc_88><loc_91><loc_90></location>their ruins await discovery - except that we may not last long enough to find them.</text>
<text><location><page_5><loc_54><loc_74><loc_91><loc_87></location>( R 3) ('Galactic hegemony') log L ≥ log T ≥ 7. We have seen above that in a time of about te = 10 7 years a civilization can explore the galaxy via SRPs. If this civilization lasts longer than that, and no other civilization arises during the exploration period, then the exploring civilization would attain 'galactic hegemony'. It would know of the existence of any other civilization that might arise, and would be able to control their growth and activities.</text>
<text><location><page_5><loc_54><loc_59><loc_91><loc_74></location>In the remaining parts of the diagram there are many civilizations in the galaxy. Assume for simplicity that the galaxy is a uniform disk of thickness hG = 1000 ly and radius RG = 50 , 000 ly, that civilizations arise uniformly in the galaxy at rate λ , start exploring the galaxy by SRPs with an exploration speed of ve = 0 . 01 c , and continue to do so until the civilization (and the SRPs) end L years after the start of the exploration. (A more detailed analysis would take account of the likely existence of a galactic habitable zone described by Lineweaver et. al. 2004.)</text>
<text><location><page_5><loc_54><loc_49><loc_91><loc_58></location>If a civilization starts at position x 0 and time t 0, then the space-time region explored will be the cone consisting of the points ( x , t ) such that t 0 ≤ t ≤ t 0 + L , and | x -x 0 | ≤ vet . (This neglects for the moment the hard question of interaction between civilizations.) A point ( x , t ) will be explored by some civilization if any civilization starts in the space-time region</text>
<formula><location><page_5><loc_57><loc_46><loc_88><loc_47></location>CP ( x , t ) = { ( y , t -s ) : 0 ≤ s ≤ L , | x -y | ≤ ves } .</formula>
<text><location><page_5><loc_54><loc_37><loc_91><loc_45></location>The space volume explored will initially grow cubically with L , but with a transition to quadratic growth at the time tw = hG / ve taken to cross the thickness of the galactic disc. We have tw = 10 5 , and it turns out that it is the case L ≥ tw which is of interest. The (space-time) volume of CP ( x , t ) is of the order of</text>
<formula><location><page_5><loc_67><loc_33><loc_78><loc_36></location>WC = π 3 hGv 2 e L 3 ;</formula>
<text><location><page_5><loc_54><loc_27><loc_91><loc_33></location>the exact value will depend on its location within the galaxy. The volume of the galaxy is VG = π R 2 G hG , and so the mean number of civilizations arising in the region CP ( x , t ) is</text>
<formula><location><page_5><loc_61><loc_23><loc_84><loc_26></location>M = λ WC VG = λ hGv 2 e L 3 3 hGR 2 G = L 3 T v 2 e 3 R 2 G .</formula>
<text><location><page_5><loc_54><loc_19><loc_91><loc_22></location>Taking 3 R 2 G = 7 . 5 ∗ 10 9 /similarequal 10 10 ly 3 , we have M ≥ 1 when</text>
<formula><location><page_5><loc_65><loc_18><loc_79><loc_19></location>3 log L ≥ 14 + log T .</formula>
<text><location><page_5><loc_54><loc_14><loc_91><loc_17></location>(Note that log T ≥ 1 then gives L ≥ tw .) If M /greatermuch 1 then a typical space-time point in the galaxy will lie</text>
<text><location><page_6><loc_14><loc_85><loc_51><loc_90></location>in the exploration cone of many civilizations, and so these cones will cover most of the galaxy, while if M /lessmuch 1 then there will be substantial vacant unexplored regions.</text>
<text><location><page_6><loc_14><loc_77><loc_51><loc_84></location>( R 4) ('Multiple zones') In the region 3 log L ≥ 14 + log T , log T ≤ 7 we therefore expect that the galaxy will covered by the zones of control of more than civilization. How these civilizations might interact is considered briefly below.</text>
<text><location><page_6><loc_14><loc_59><loc_51><loc_77></location>If 3 log L ≤ 14 + log T then civilizations are too rare and short-lived for their SRPs to cover the galaxy, but we can still ask about their radio signals. Let us begin by considering the conditions for 2-way communication by radio with an ETI. The same analysis as with the SRPs applies in this case, but with ve replaced by the speed of light vc = 1. Assume for simplicity that the time between a civilization starting to send out radio transmissions and sending out SRPs is small, and that radio transmissions continue for the lifetime of a civilization. Then the mean number of civilization still extant whose broadcasts can be accessed at a point ( t , x ) will be</text>
<formula><location><page_6><loc_26><loc_56><loc_40><loc_59></location>M ' = L 3 T v 2 c 3 R 2 G = M v 2 e .</formula>
<text><location><page_6><loc_14><loc_50><loc_51><loc_55></location>Thus M ' ≥ 1 if 3 log L ≥ 10 + log T . (If log T ≥ 1 then this condition gives L ≥ 10 11 / 3 > 1000, so the case when we need to consider zones with radius less than hG does not arise.)</text>
<text><location><page_6><loc_14><loc_38><loc_51><loc_49></location>( R 5) ('2 way SETI') If 10 + log T ≤ 3 log L ≤ 14 + log T and log T ≥ 1 then a typical point will be able to receive radio signals from a civilization which is still extant, but will not be visited by SRPs. There is therefore the possibility of 2-way communication by radio between two civilizations, possibly continuing until one becomes extinct. This is the situation envisaged in much of the early SETI literature.</text>
<text><location><page_6><loc_14><loc_31><loc_51><loc_37></location>( R 6) ('1-way SETI'). If 3 log L ≤ 10 + log T and log T ≥ 1 then a typical point can only receive signals from extinct civilizations. A point ( t , x ) will be able to receive signals from a civilization if that civilization arose in the region</text>
<formula><location><page_6><loc_17><loc_28><loc_49><loc_29></location>CS ( t , x ) = { ( y , s ) : t - | x -y | -L ≤ st - | x -y |} .</formula>
<text><location><page_6><loc_14><loc_20><loc_51><loc_27></location>This has space time volume LVG , and so the mean number of such civilizations is λ LVG / VG = L / T . Thus L ≥ T is (not surprisingly) also the condition for there to be some civilization in the galaxy within our light cone.</text>
<text><location><page_6><loc_14><loc_14><loc_51><loc_20></location>The space of galactic ecologies is therefore divided into six regions. For regions R 1 and R 2 there is little more to be said, but some other cases deserve further attention.</text>
<text><location><page_6><loc_54><loc_85><loc_91><loc_90></location>In region R 4 a typical point in the galaxy could be explored by SRPs from many civilizations, and it is necessary to consider how such civilizations might interact. One can identify three broad possibilities:</text>
<text><location><page_6><loc_54><loc_82><loc_91><loc_84></location>(i) No interaction, and mutual interpenetration between explored regions of di ff erent civilizations,</text>
<text><location><page_6><loc_54><loc_79><loc_91><loc_82></location>(ii) Civilizations establish boundaries between their di ff erent 'zones of control',</text>
<text><location><page_6><loc_54><loc_78><loc_89><loc_79></location>(iii) Civilizations (or their SRPs) engage in warfare.</text>
<text><location><page_6><loc_54><loc_73><loc_91><loc_77></location>In case (i) we would expect to see many probes within our solar system, and our failure to do so tends towards excluding this possibility.</text>
<text><location><page_6><loc_54><loc_59><loc_91><loc_73></location>For case (ii), consider the arrival of a SRP from Civilization X in a star system already containing infrastructure built by Civilization Y. The probe would need to decelerate from 0.01c, and this would require the expenditure of large amounts of energy over a significant period, making the arrival detectable by Y. On arrival the SRP would have limited fuel and resources, and could be quarantined or neutralised by Y. A (lengthy) period of negotiation might then lead to agreed boundaries between X and Y.</text>
<text><location><page_6><loc_54><loc_31><loc_91><loc_59></location>If negotiation failed then war might ensue, which is case (iii). In the case of all out war, constraints on the number of SRPs built would be dropped, and all available material would be used. If it is the case that the material in stars and gas giants is too tightly bound gravitationally to be used to make SRPs, then the effects of such a war on other star systems might not be detectable to us at present. However, two pieces of evidence support the conclusion that such a war has never occurred in our galaxy. The first is that the solar system has not been mined in this way. Second, if SRPs can only utilize smaller planets then the total mass usable for SRPs in a typical stellar system would be around 10 22 -10 23 kg. However, a protostellar nebula contains a mass of around 10 30 kg, which is not be tightly bound gravitationally. Such nebulae would be major military prizes, and their continued existence in our galaxy, as well as that of recently formed stars, suggests that our galaxy has seen neither an all out war, nor an arms race. (This applies also to other galaxies.)</text>
<section_header_level_1><location><page_6><loc_54><loc_27><loc_64><loc_28></location>4. Conclusion</section_header_level_1>
<text><location><page_6><loc_54><loc_20><loc_91><loc_26></location>Under our hypothesis that the technologies (T1) and (T2) can be attained, consideration of the points above, and Figure 1, leads to three broad categories of answer to Fermi's question:</text>
<text><location><page_6><loc_54><loc_17><loc_91><loc_20></location>(F1) They have not visited us because they do not exist. (Regions R 1 and R 2.)</text>
<text><location><page_6><loc_54><loc_14><loc_91><loc_17></location>(F2) The 'zoo hypothesis': their probes are watching us now (Regions R 3 and R 4.)</text>
<text><location><page_7><loc_14><loc_87><loc_51><loc_90></location>(F3) They have not visited us because civilizations are all too short lived (Regions R 5 and R 6).</text>
<text><location><page_7><loc_14><loc_78><loc_51><loc_87></location>Of these, possibility (F3) relies all all civilizations being short lived, while the zoo hypothesis appears to be deeply unpopular (partly I suspect because it compromises human dignity.) The analysis above reduces the force of some of the objections that have been made to the zoo hypothesis, since in both cases R 3 and R 4(ii) we would lie in the zone of control of just one ETI.</text>
<text><location><page_7><loc_14><loc_63><loc_51><loc_77></location>If we exclude (F2) and (F3), then we are left with (F1), to which there are no objections except that it is uninteresting. It is worth noting that while astronomers have frequently given rather large values to fci - typically in the range 0 . 01-0 . 1, many evolutionary biologists have been much more pessimistic. Even if one is not convinced by all the arguments in Ward and Brownlee 2000, it seems very possible that the development of intelligent life requires evolution to pass through several gateways, and hence that fci is very small.</text>
<section_header_level_1><location><page_7><loc_14><loc_60><loc_30><loc_61></location>5. Acknowledgements</section_header_level_1>
<text><location><page_7><loc_14><loc_56><loc_51><loc_58></location>This research was partially supported by NSERC (Canada) and Trinity College Cambridge (UK).</text>
<section_header_level_1><location><page_7><loc_14><loc_52><loc_22><loc_53></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_7><loc_14><loc_49><loc_51><loc_51></location>[1] Bjørk, R., 2007. Exploring the galaxy using space probes . International J. Astrobiology. 6 , 89-93.</list_item>
<list_item><location><page_7><loc_14><loc_47><loc_51><loc_49></location>[2] Freitas, R. A. Jr., 1980a. Interstellar probes: A new approach to SETI, J. Brit. Interplanet. Soc. 33 , 95-100.</list_item>
<list_item><location><page_7><loc_14><loc_45><loc_51><loc_47></location>[3] Freitas, R. A. Jr., 1980b. A self-reproducing interstellar probe, J. Brit. Interplanet. Soc. , 33 , 251-264.</list_item>
<list_item><location><page_7><loc_14><loc_41><loc_51><loc_44></location>[4] Glade, N., Ballet, P., Bastien, O., 2012. A stochastic process approach of the drake equation parameters. International J. Astrobiology. 11 , 103-108.</list_item>
<list_item><location><page_7><loc_14><loc_39><loc_51><loc_41></location>[5] Landis, G.A., 1998. The fermi paradox: An approach based on percolation theory. J. Brit. Interplanet. Soc. 51 , 163-166.</list_item>
<list_item><location><page_7><loc_14><loc_36><loc_51><loc_39></location>[6] Lineweaver, C.H., Fenner, Y and Gibson, B.K., 2004. The Galactical Habitable Zone and the age distribution of complex life in the Milky Way. Science 303 , 59-62.</list_item>
<list_item><location><page_7><loc_14><loc_33><loc_51><loc_35></location>[7] Mathews, J.D., 2011. From here to ET. J. Brit. Interplanet. Soc. 64 , 234-241.</list_item>
<list_item><location><page_7><loc_14><loc_31><loc_51><loc_33></location>[8] Sagan, C., Newman, W. 1983., The solipsist approach to extraterrestial intelligence. Quart. J. Royal Ast. Soc. 24 , 113-121.</list_item>
<list_item><location><page_7><loc_14><loc_29><loc_51><loc_31></location>[9] Tipler, F.J. 1980., Extraterrestial intelligent beings do not exist. Quart. J. Royal Ast. Soc. 21 , 267-281.</list_item>
<list_item><location><page_7><loc_14><loc_27><loc_51><loc_29></location>[10] Ward, R., Brownlee D., 2000. Rare Earth: Why Complex Life is Uncommon in the Universe. Springer, Berlin.</list_item>
<list_item><location><page_7><loc_14><loc_24><loc_51><loc_26></location>[11] Wiley, K.B. 2012., The Fermi Paradox, self-replicating probes, and the interstellar transportation bandwidth. arXiv:1111.6131v1.</list_item>
</unordered_list>
</document>
[
{
"title": "Galactic exploration by directed self-replicating probes, and its implications for the Fermi paradox",
"content": "Martin T. Barlow 1 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. Email: barlow@math.ubc.ca",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "This paper proposes a long term scheme for robotic exploration of the galaxy, and then considers the implications in terms of the 'Fermi paradox' and our search for ETI. We discuss the 'galactic ecology' of civilizations in terms of the parameters T (time between ET civilizations arising) and L , the lifetime of these civilizations. Six di ff erent regions are described. Keywords: Self-Replicating Probes, Galactic Exploration, Search for Extraterrestrial Life, Fermi Paradox",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "The proposal to explore the galaxy by selfreplicating probes goes back at least as far as Freitas 1980a and Tipler 1980. Numerous criticisms have been made, notably by Sagan and Newman 1983. A recent paper (Wiley 2012) reconsiders the topic, and finds some of these criticisms without much merit. This paper outlines in more detail one exploration scheme and its benefits, and then considers the implication for the galactic 'ecology' of intelligent species.",
"pages": [
1
]
},
{
"title": "2. Exploration of the galaxy by self replicating probes",
"content": "Our starting point is as in Freitas 1980b and Tipler 1980. This relies on two technologies which we do not have at present, but which it is reasonable to suppose we will attain within the next few hundred years: Propulsion systems. While no existing system can reach these speeds, various proposals for 100 year probes to Alpha Centauri have been made, such as Project Daedalus. A speed of 0.01c does not seem unduly optimistic, and in fact many proposals for galactic exploration, such Bjørk (2007), consider probes which travel at 0.1c. AIsystemsAgain, we do not have self-replicating systems at present, and what such a system would be like must rely largely on conjecture. In this context note the following remark in Wiley (2012): One point we take issue with is an inherent and frequently unconscious biological bias that pervades consideration of computerized intelligence, including SRPs. A common tendency has been to imagine a selfreplicating machine as being rather like a bacterium: that is a single machine which (somehow, almost magically) is able to move around in its environment and replicate itself. If we start with present technology, we are forced to imagine something rather di ff erent. The system as a whole might consist of 3 parts: The payload of the probe would consist of (A + B), together with enough raw materials (fuel etc.) to get started in the new system. Once (C) was made, resources would be gathered for as long as was necessary, and a number of probes would then be sent to nearby stars. If we take this view, then a 'self-replicating space probe' (SRP) would not be a single machine, but rather a collection of di ff erent machines with an overall capability of replication. See Freitas 1980b for a much more detailed description of such a probe, with the probe (A + B) plus the fuel for the voyage having a mass of of around 10 10 kg. The factory described there only makes 1 new probe every 500 years, but (see Section 7.1) using a longer period for the initial construction gives a larger factory which can create 1000 new probes in 1500-200 years. For simplicity I will take the reproduction time, between the arrival of the initial probe in a star system, and the completed factory (C) sending out new probes, to be TR = 1000. The AI needed for such a system far exceeds what is possible at present. But while the kinds of decisions necessary for the AI (e.g. 'what kind of material is present in this asteroid?', 'can it be transported to the factory?') would require a very high level of skill, this would be within fairly narrow parameters, and a human level of overall initiative and judgement would not be required. Even if machines with a human intelligence can be constructed, it might be desirable to limit the intelligence of the SRPs. As Tipler 1980 notes, there are reasons other than stellar exploration to develop these technologies. Progress in AI has been far slower than supposed by early optimists, but it still seems reasonable to suppose that, within a few hundred years, we will be able to build such SRPs. The development of such machines would, at least for a while, introduce an age of plenty, since it would open up the resources of the solar system for our exploitation. Some idealistic individuals or groups might then be willing to invest the resources in making a number of the probes (A + B), and send them to nearby stars. (See Mathews 2011 for another proposal to explore the galaxy by SRPs.)",
"pages": [
1,
2
]
},
{
"title": "2.1. Exploration strategy",
"content": "I now propose a scheme for galactic exploration, assuming that we do develop the technologies (T1) and (T2) described above. The first step would be to send probes to the 10-100 nearest stars, which are all within about 20 ly of Earth. (Long before the start of such a mission we will have very good data on the planetary systems of these stars.) The probes would arrive at their destinations 400-2000 years after the mission start. They would remain in radio contact with Earth, (with a time lag of 40 years or less), would report on their discoveries, and would be able to receive updates on strategy. (Among the exploration devices (A) would be systems able to transmit and receive narrow band radio or laser communication over a distance of say 100 ly.) I will call the initial star systems Level 1 'colonies', though there is no suggestion that they would have a human population. After the construction of the factory (C) on a Level 1 colony, the colony would send out SRPs (let us say about 1,000 - 10,000) to create colonies at 'Level 2'. I have suggested an initial 'hop size' of 10-20 ly, since the number of probes that could be sent out from our Solar System might be limited by resource constraints. However once a colony at Level 1 or higher had a working factory, there would be no such limit on the number of probes that could be sent out, and it would be sensible to send as many as was necessary to explore every star system within the second 'maximum hop size'. There are about 15,000 stars within 100 ly of Earth, so with some useful duplication the Level 1 systems would together be able to send probes to every star in this region. The maximum hop size hm would be the greatest distance such a probe could be sent with a probability greater than 90% of arriving. I will take hm = 100; this also needs to be less than the maximum distance for radio or laser communication, but this is much greater than 100 ly. The Probes from the Level 2 colonies would then establish Level 3 colonies, and so on. Each colony at Level n would report back to its Level n -1 ancestor, and receive updating instructions from it. While it would be desirable for the Level 1 colonies to produce manyprobes, as the radius of the exploration sphere became larger, and so the curvature of its surface became less, fewer new probes per colony would be needed. Within a few thousand years of the mission start our descendants on earth (if they still existed) would be receiving a flood of information from the exploration of hundreds of star systems. The Great Pyramid was built around 4500 years ago; 4500 years after its start the mission would be well under way, and have given us detailed data on every star system within about 30 ly of Earth. The overall mission would continue until the planetary system of every star in the galaxy had been explored. Let vp = 0 . 01 c be the speed of the probes, and ve be the propagation speed of the exploration front. Then So TR = 1000 and hm = 100 give ve / vp = 10 / 11, the exploration front travels at nearly the same speed as the probes, and the total time to explore the galaxy is around 10 7 years. This compares with exploration times of the order of 10 8 years given by Bjørk 2007, using probes which travel at 0.1c, but do not replicate.",
"pages": [
2,
3
]
},
{
"title": "2.2. Refinements",
"content": "While the basic strategy is as above, it is necessary to consider a number of refinements. (a)Resourceusewithinsystem. The best place for the construction of the factory (C) might be the moons of a planet in the outer reaches of the star system. Assuming a mass of 10 10 kg for the probes (A + B) and fuel, the construction of (C) plus say 10,000 probes would use at most a handful of minor planets and comets. This would leave plenty of material behind, even on the Level 1 colonies, and it would not be necessary to to 'strip mine' the galaxy in order to complete this exploration. One would only need a few probes per star one plus a margin for accidents. The first and second level probes would provide enough data to refine this strategy, at an early stage of the overall mission. In the very unlikely event that more than 90% of systems have planets with complex life, a modification of (ii) would be needed so that a reasonable proportion of colonies did send out probes. (c) Extinctionof the humanrace. Once set in motion, the exploration could continue without any further human intervention. However, this proposal envisions continued interaction and direction: Earth would receive data from the probes, and based on this revised instructions on exploration (as well as possible system upgrades) would be sent out. What however if humanity becomes extinct, or just loses interest in the mission? There are many possible procedures which could be followed, of which the simplest are: (i) Continue anyway, (ii) Abandon further exploration. For simplicity I propose (ii), and suggest that every 100 years the Level 1 colonies would ask Earth 'Shall we continue?' If 1000 years went by with no positive response, the project would be mothballed, and instructions would be sent through the communication tree that no further SRPs were to be built. (d) Communication and direction. A key part of this exploration scheme is that the SRPs are not autonomous, but that the whole exploration process is directed, ultimately from Earth. The first requirement here is that Level n colonies be able to communicate (over a distance of say 100 ly) with their Level n -1 ancestors. Even with present technology, we could build transmitters and receivers capable of working over these distances. While all the Level 1 colonies would send out probes, this would not be necessary for higher level colonies, and stars could be divided into two groups. For the first, 'end nodes', a factory C would be built, the star system explored, but no probes would be sent out. The second, 'branch nodes' would send out probes. Among the pieces of infrastructure built in each colony would be telescopes to survey the stellar neighbourhood, and using this data, nearby Level n colonies would coordinate the exploration of their neighbourhoods. (Nearby colonies would be 10-100 ly apart, so communication time would be small compared with the time to build probes, or for the journeys.) While the algorithm to coordinate this process may appear complicated, it is well within our current capabilities - unlike the AI needed for robotic exploration of a planetary system (e) Mission creep and machine mutation. A widely voiced concern with SRPs has been that they might mutate, run amok, and eat up the galaxy - see Sagan and Newman 1983. However Wiley 2012 argues that it should be possible to build in su ffi cient reliability to avoid this outcome - note also his comment above on inappropriate biological analogies. In the exploration scheme proposed here about 1000 generations would be needed to explore the galaxy - fewer if hops longer than 100 ly are feasible. The total number of replicators (C) built would be of the same order as the number of stars in the galaxy, that is about 4 × 10 11 . Wiley 2012 points out that this is much less than the number of cell divisions within a human lifetime, which is of the order of 10 16 . If necessary, further steps could be taken to reduce the overall risk. In the initial stages of the exploration we would want every planetary system to be explored carefully. However, it seems likely that after the first million or so systems had been explored, we would have a good understanding of the processes underlying the formation and development of planetary systems, and might only be interested in those systems which had life, complex life, or other exceptional features. The later phases of the exploration could therefore proceed as follows. Each 'branch node' would first send out about 10 new full probes (A + B) to establish the next generation of colonies. Next, it would send out reduced probes, just consisting of (A), to all the stars in its exploration patch. These would explore the target system, and report back to the send- ing branch node. Without the reproductive capacity (B) these probes would ultimately run out of fuel and become inactive. A full probe (A + B) would then be sent to any system that merited further attention. Assuming that 'interesting' systems are rare, this modification would reduce the number of full replications by a factor of 1000 or so. Further safety mechanisms could also be built in, such as deeply embedded software constraints on the total number of probes that the factories (C) could make, or on the total number of permitted generations. (f)Crossinglargespacesandpercolation. Landis 1998 has suggested a percolation model for the spread of a species through the galaxy, and shown that in some cases this leads to large vacant (unexplored) regions. However bond percolation on the lattice is a poor model for the type of exploration proposed above, since each 'branch node' would send out rather more than 5 probes. Further, the communication envisaged between colonies would mean that colonies would become aware of interstellar voids (with no or few stars), and regions that, perhaps because of the failure of a number of probes, were remaining unexplored. They could then send additional probes to explore these regions. If we consider the mathematical graph whose vertices are the stars, and join by edges all pairs of stars within 100 ly., then the exploration scheme proposed here will explore all stars in the connected component containing our sun, and it seems overwhelmingly likely that this spans most of the galaxy. (g) System updates. We would want to be able to incorporate updates into the systems (A,B,C). It is possible that this could be done by radio, but the available bandwidth might be too small for the necessary amount of data. One can imagine a system of 'fast packets' - small probes carrying data, which travel at say 0.1c between colonies with the infrastructure to send and receive them. However one disadvantage of allowing such updates is that it would make the colonies more vulnerable to mutations or computer viruses. (h)Contactwithextraterrestrialintelligence(ETI).Detailed thought would need to be given on what course of action should be taken if either ETI were found, or traces of them. There would be time to refine strategies in the first few millenia of the mission, as data on frequency and type of life in other star systems accumulated. The number of possible actions the probes could take is large, and a full discussion of this is beyond the scope of this article. The simplest (but not the quickest) option would be for the the probe to report back, and take no further action until instructed.",
"pages": [
3,
4
]
},
{
"title": "3. Implications for SETI, and Fermi's question",
"content": "Let us now make the hypothesis (H) that the technolgies (T1) and (T2) can be attained, and explore the consequences. The exploration scheme outlined above, using these technologies is one which, if it survives long enough, the human race might adopt - no doubt with a number of improvements. The payo ff is that with a relatively low initial cost our descendants would obtain detailed data about every star system in the galaxy. In particular they would learn how many planets support life, what kind of life it is, and just how rare complex or intelligent life is. If there have been technological ETI in the galaxy, then they would also have had this option. So - this is Fermi's question 'Where are they?' (This is often called the 'Fermi paradox', but it is only a paradox if one begins with the assumption that intelligent life is common. In fact we have no information on this.) Let us recall the Drake equation, slightly modified for our purposes: here N is the number of existing civilizations sending out SRPs, R ∗ is the rate of star formation per year in the galaxy, fp is the fraction of those stars that have planets, ne is the average number of planets that can potentially support life per star that has planets, fl is the fraction of these that develop life, fci is the fraction of these that develop civilizations that send out SRPs, and L is the average lifetime of such civilizations. This lifetime is the time that either the civilization itself, or its SRPs, remain active. (From now on, I will use the term 'civilization' for 'civilizations that send out SRPs'). We do have estimates of at least the order of magnitude of some of the early terms in this expression: for example R ∗ /similarequal 7, and data from the Kepler satellite suggests that fp /similarequal 0 . 5, while ne is quite small. (Out of about 10,000 systems surveyed, only a handful have planets which look really promising from the point of finding earthlike life.) At present fl and fci are utterly unknown, though estimates of fl may at some point become available via spectroscopic search for oxygen. I have given the Drake equation in a simple form. A more realistic equation would take account of randomness, and the fact that these factors are not constant in time - see for example Glade et. al. 2012. However, the uncertainty in our knowledge of the parameters in the equation is so great that these refinements seem to the author of this article to add little to what can be achieved with a simple 'back of an envelope' calculation. PSfrag replacements Let us now set so that λ is the number of civilizations arising per year in the galaxy. As an upper bound, if fl = fci = 1 (surely very unlikely) and ne = 0 . 03, we obtain λ ≤ 0 . 1. Let us set T = 1 /λ to be the average length of time in years between successive civilizations arising in the galaxy; the estimates above suggest it is unlikely that T < 10. Let us now consider the 'galactic ecology' in terms of the two parameters T = λ -1 and L . While a better model would allow for randomness of L , a simple mean model already yields useful insights. Figure 1 shows a plot of log L against log T . Since the galaxy is about 10 10 years old, we have log L ≤ 10, and it seems reasonable to take also log L ≥ 2. The estimates above give log T ≥ 1. We have no upper bound on T : it is not legitimate to use the Copernican principle to assert that because there is at least one potential civilization in the galaxy (us) then T ≤ L . Civilizations might only arise in one galaxy in a billion, and those that arose would still observe themselves to be in a galaxy. In the diagram I take 1 ≤ log T ≤ 14. Let us now consider the various regions of the diagram. The descriptive statements for the regions apply to typical points in the region - naturally these will become weaker if the point (log T , log L ) is close to the boundary between regions. ( R 1) ('Alone') If log T > 10 then probably no other civilization has arisen in the galaxy. (A more accurate statement would be that the mean number of such civilizations is less than 1.) ( R 2) ('Pompeii') If log T ≤ 10 and log L < log T then N < 1 and there is no other civilization existing now. However, 10 10 / T ≥ 1 civilizations have existed, and their ruins await discovery - except that we may not last long enough to find them. ( R 3) ('Galactic hegemony') log L ≥ log T ≥ 7. We have seen above that in a time of about te = 10 7 years a civilization can explore the galaxy via SRPs. If this civilization lasts longer than that, and no other civilization arises during the exploration period, then the exploring civilization would attain 'galactic hegemony'. It would know of the existence of any other civilization that might arise, and would be able to control their growth and activities. In the remaining parts of the diagram there are many civilizations in the galaxy. Assume for simplicity that the galaxy is a uniform disk of thickness hG = 1000 ly and radius RG = 50 , 000 ly, that civilizations arise uniformly in the galaxy at rate λ , start exploring the galaxy by SRPs with an exploration speed of ve = 0 . 01 c , and continue to do so until the civilization (and the SRPs) end L years after the start of the exploration. (A more detailed analysis would take account of the likely existence of a galactic habitable zone described by Lineweaver et. al. 2004.) If a civilization starts at position x 0 and time t 0, then the space-time region explored will be the cone consisting of the points ( x , t ) such that t 0 ≤ t ≤ t 0 + L , and | x -x 0 | ≤ vet . (This neglects for the moment the hard question of interaction between civilizations.) A point ( x , t ) will be explored by some civilization if any civilization starts in the space-time region The space volume explored will initially grow cubically with L , but with a transition to quadratic growth at the time tw = hG / ve taken to cross the thickness of the galactic disc. We have tw = 10 5 , and it turns out that it is the case L ≥ tw which is of interest. The (space-time) volume of CP ( x , t ) is of the order of the exact value will depend on its location within the galaxy. The volume of the galaxy is VG = π R 2 G hG , and so the mean number of civilizations arising in the region CP ( x , t ) is Taking 3 R 2 G = 7 . 5 ∗ 10 9 /similarequal 10 10 ly 3 , we have M ≥ 1 when (Note that log T ≥ 1 then gives L ≥ tw .) If M /greatermuch 1 then a typical space-time point in the galaxy will lie in the exploration cone of many civilizations, and so these cones will cover most of the galaxy, while if M /lessmuch 1 then there will be substantial vacant unexplored regions. ( R 4) ('Multiple zones') In the region 3 log L ≥ 14 + log T , log T ≤ 7 we therefore expect that the galaxy will covered by the zones of control of more than civilization. How these civilizations might interact is considered briefly below. If 3 log L ≤ 14 + log T then civilizations are too rare and short-lived for their SRPs to cover the galaxy, but we can still ask about their radio signals. Let us begin by considering the conditions for 2-way communication by radio with an ETI. The same analysis as with the SRPs applies in this case, but with ve replaced by the speed of light vc = 1. Assume for simplicity that the time between a civilization starting to send out radio transmissions and sending out SRPs is small, and that radio transmissions continue for the lifetime of a civilization. Then the mean number of civilization still extant whose broadcasts can be accessed at a point ( t , x ) will be Thus M ' ≥ 1 if 3 log L ≥ 10 + log T . (If log T ≥ 1 then this condition gives L ≥ 10 11 / 3 > 1000, so the case when we need to consider zones with radius less than hG does not arise.) ( R 5) ('2 way SETI') If 10 + log T ≤ 3 log L ≤ 14 + log T and log T ≥ 1 then a typical point will be able to receive radio signals from a civilization which is still extant, but will not be visited by SRPs. There is therefore the possibility of 2-way communication by radio between two civilizations, possibly continuing until one becomes extinct. This is the situation envisaged in much of the early SETI literature. ( R 6) ('1-way SETI'). If 3 log L ≤ 10 + log T and log T ≥ 1 then a typical point can only receive signals from extinct civilizations. A point ( t , x ) will be able to receive signals from a civilization if that civilization arose in the region This has space time volume LVG , and so the mean number of such civilizations is λ LVG / VG = L / T . Thus L ≥ T is (not surprisingly) also the condition for there to be some civilization in the galaxy within our light cone. The space of galactic ecologies is therefore divided into six regions. For regions R 1 and R 2 there is little more to be said, but some other cases deserve further attention. In region R 4 a typical point in the galaxy could be explored by SRPs from many civilizations, and it is necessary to consider how such civilizations might interact. One can identify three broad possibilities: (i) No interaction, and mutual interpenetration between explored regions of di ff erent civilizations, (ii) Civilizations establish boundaries between their di ff erent 'zones of control', (iii) Civilizations (or their SRPs) engage in warfare. In case (i) we would expect to see many probes within our solar system, and our failure to do so tends towards excluding this possibility. For case (ii), consider the arrival of a SRP from Civilization X in a star system already containing infrastructure built by Civilization Y. The probe would need to decelerate from 0.01c, and this would require the expenditure of large amounts of energy over a significant period, making the arrival detectable by Y. On arrival the SRP would have limited fuel and resources, and could be quarantined or neutralised by Y. A (lengthy) period of negotiation might then lead to agreed boundaries between X and Y. If negotiation failed then war might ensue, which is case (iii). In the case of all out war, constraints on the number of SRPs built would be dropped, and all available material would be used. If it is the case that the material in stars and gas giants is too tightly bound gravitationally to be used to make SRPs, then the effects of such a war on other star systems might not be detectable to us at present. However, two pieces of evidence support the conclusion that such a war has never occurred in our galaxy. The first is that the solar system has not been mined in this way. Second, if SRPs can only utilize smaller planets then the total mass usable for SRPs in a typical stellar system would be around 10 22 -10 23 kg. However, a protostellar nebula contains a mass of around 10 30 kg, which is not be tightly bound gravitationally. Such nebulae would be major military prizes, and their continued existence in our galaxy, as well as that of recently formed stars, suggests that our galaxy has seen neither an all out war, nor an arms race. (This applies also to other galaxies.)",
"pages": [
4,
5,
6
]
},
{
"title": "4. Conclusion",
"content": "Under our hypothesis that the technologies (T1) and (T2) can be attained, consideration of the points above, and Figure 1, leads to three broad categories of answer to Fermi's question: (F1) They have not visited us because they do not exist. (Regions R 1 and R 2.) (F2) The 'zoo hypothesis': their probes are watching us now (Regions R 3 and R 4.) (F3) They have not visited us because civilizations are all too short lived (Regions R 5 and R 6). Of these, possibility (F3) relies all all civilizations being short lived, while the zoo hypothesis appears to be deeply unpopular (partly I suspect because it compromises human dignity.) The analysis above reduces the force of some of the objections that have been made to the zoo hypothesis, since in both cases R 3 and R 4(ii) we would lie in the zone of control of just one ETI. If we exclude (F2) and (F3), then we are left with (F1), to which there are no objections except that it is uninteresting. It is worth noting that while astronomers have frequently given rather large values to fci - typically in the range 0 . 01-0 . 1, many evolutionary biologists have been much more pessimistic. Even if one is not convinced by all the arguments in Ward and Brownlee 2000, it seems very possible that the development of intelligent life requires evolution to pass through several gateways, and hence that fci is very small.",
"pages": [
6,
7
]
},
{
"title": "5. Acknowledgements",
"content": "This research was partially supported by NSERC (Canada) and Trinity College Cambridge (UK).",
"pages": [
7
]
}
]
<document>
<section_header_level_1><location><page_1><loc_14><loc_90><loc_87><loc_93></location>On Spin-Statistics and Bogoliubov Transformations in Flat Spacetime With Acceleration Conditions</section_header_level_1>
<text><location><page_1><loc_43><loc_87><loc_58><loc_88></location>Michael R.R. Good ∗</text>
<text><location><page_1><loc_37><loc_86><loc_64><loc_87></location>Institute of Advanced Studies, Singapore</text>
<text><location><page_1><loc_41><loc_84><loc_60><loc_86></location>(Dated: September 26, 2018)</text>
<text><location><page_1><loc_18><loc_72><loc_83><loc_83></location>A single real scalar field of spin zero obeying the Klein-Gordon equation in flat spacetime under external conditions is considered in the context of the spin-statistics connection. An imposed accelerated boundary on the field is made to become, in the far future, (1) asymptotically inertial and (2) asymptotically non-inertial (with an infinite acceleration). The constant acceleration Unruh effect is also considered. The systems involving non-trivial Bogoliubov transformations contain dynamics which point to commutation relations. Particles described by in-modes obey the same statistics as particles described by out-modes. It is found in the non-trivial systems that the spin-statistics connection can be manifest from the acceleration. The equation of motion for the boundary which forever emits thermal radiation is revealed.</text>
<text><location><page_1><loc_18><loc_70><loc_45><loc_70></location>PACS numbers: 03.70.+k, 04.62.+v, 04.60.-m</text>
<text><location><page_1><loc_18><loc_68><loc_71><loc_69></location>Keywords: dynamical Casimir effect, Unruh effect, external conditions, moving mirrors</text>
<section_header_level_1><location><page_1><loc_20><loc_64><loc_37><loc_65></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_48><loc_49><loc_62></location>Parker demonstrated that the spin-statistics connection is evident from the dynamics of curved spacetime.[1] [2] [3] It was first shown using a quantized real scalar field of spin zero which obeys the curved spacetime KleinGordon equation[1]. It was then found that consistency resulted when commutation relations imposed on the creation and annihilation operators for the in-region, were also imposed for the creation and annihilation operators for the out-region. It is not consistent with the dynamics to impose anti-commutation relations for the out-region.</text>
<text><location><page_1><loc_9><loc_35><loc_49><loc_47></location>Parker's derivation was extended to fermions of spin1/2 which obey the Dirac equation and satisfy anticommutation relations [2]. Higher spin and parastatistics was treated [3], the statistics were generalized [4], [5], [6], and ghost fields [7] were examined. It is assumed [8] that a connection between statistics and dynamics is not present in Minkowski spacetime. It is also generally accepted that particles at late times should obey the same statistics as at early times.</text>
<text><location><page_1><loc_9><loc_24><loc_49><loc_34></location>Recently, researchers may have presented the first experimental evidence for observation of the dynamical Casimir effect [9]. Within the context of an imposed boundary in flat spacetime, the question arises whether the connection between spin and statistics is revealed from the dynamics of a field-boundary system in the same way as from the dynamics of curved spacetime.</text>
<text><location><page_1><loc_9><loc_14><loc_49><loc_24></location>In this note, it is shown that if a quantized real scalar field of spin-0 obeys the flat spacetime Klein-Gordon equation with an imposed accelerated boundary condition, then commutation relations are the admissible algebra that may be imposed upon the creation and annihilation operators on the in-region and the out-region. It will not be consistent with the dynamics to establish</text>
<text><location><page_1><loc_52><loc_63><loc_92><loc_66></location>anti-commutation relations upon this field-boundary system.</text>
<text><location><page_1><loc_52><loc_34><loc_92><loc_63></location>As is pointed out by Wald [10], the spin-statistics results from curved spacetime depend on the conserved inner product. Wald emphasized that no reasonable quantum field theory in curved spacetime (or in flat spacetime with an external potential), can have Bose-Einstein statistics if the inner product is positive definite for all positive and negative frequency solutions. Likewise, it's unreasonable to have Fermi-Dirac statistics if the inner product is positive definite only on positive frequency solutions. In this note, it is emphasized that acceleration induces the appearance of distinct mode solutions with their respective positive and negative frequency pieces. These distinct modes give rise to a non-trivial Bogoliubov transformation which dictates the commutation relations. In the following three sections, it is shown that an acceleration ultimately gives rise to the spin-statistics connection in three salient examples: an asymptotically inertial mirror in Section II A (the most simple case), an infinitely accelerated mirror in Section II B and the Unruh effect in Section II C.</text>
<section_header_level_1><location><page_1><loc_58><loc_29><loc_85><loc_30></location>II. DYNAMICS TO STATISTICS</section_header_level_1>
<section_header_level_1><location><page_1><loc_58><loc_26><loc_86><loc_27></location>A. Asymptotically Inertial Mirror</section_header_level_1>
<text><location><page_1><loc_52><loc_8><loc_92><loc_24></location>The moving mirror model is the most simple example of the dynamical Casimir effect [11] [12]. Consider a moving mirror (an external boundary condition in 1+1 dimensions) which does not accelerate forever. This mirror starts (ends) at time-like past (future) infinity, asymptotically having zero acceleration in the far past (future). As such, an asymptotically inertial mirror will contain no horizons or pathological acceleration singularities, as seen in Figure 1 and Figure 2. The use of an asymptotically inertial mirror is chosen for clarity of derivation. Relaxing the condition of asymptotically</text>
<text><location><page_2><loc_9><loc_82><loc_49><loc_93></location>inertial motion will alter the form of the derivation and the complexity of the relevant identities. The connection between acceleration and the spin-statistics connection remains. The use of the forever accelerating observer in the Unruh effect, or a forever accelerating mirror that results in a null horizon, requires the appropriate left-right construction (for example, [13]) to incorporate complete Cauchy surface information.</text>
<figure>
<location><page_2><loc_12><loc_53><loc_46><loc_79></location>
<caption>FIG. 1: An asymptotically inertial trajectory in a Penrose diagram. In this example, z ( t ) = -1 2 sinh -1 ( e t ).</caption>
</figure>
<text><location><page_2><loc_9><loc_26><loc_49><loc_45></location>A moving mirror in flat spacetime reveals a connection between statistics and dynamics that is not present in Minkowski spacetime without acceleration. Statistics can be derived from the dynamics of the moving mirror as an accelerated mirror gives a beta Bogoliubov transformation coefficient which is non-zero. The statistics are determined by the algebra of the creation operators. The associated relations of the creation operators are commuting for Bose-Einstein statistics and anticommuting for Fermi-Dirac statistics. The following derivation shows that for the spin-zero field, only Bose-Einstein statistics are invariable with the flat spacetime dynamics that include an accelerated external boundary condition.</text>
<text><location><page_2><loc_9><loc_19><loc_49><loc_24></location>The field equation of motion is the Klein-Gordon equation /square ψ = 0. The moving mirror is imposed such that at the position of the mirror the field is zero, ψ | z = 0. This system has modes that can be used to expand the field,</text>
<formula><location><page_2><loc_17><loc_14><loc_49><loc_17></location>ψ = ∫ ∞ 0 dω ' [ a ω ' φ ω ' + a † ω ' φ ∗ ω ' ] , (1)</formula>
<formula><location><page_2><loc_19><loc_8><loc_49><loc_12></location>ψ = ∫ ∞ 0 dω [ b ω χ ω + b † ω χ ∗ ω ] . (2)</formula>
<figure>
<location><page_2><loc_55><loc_67><loc_89><loc_94></location>
<caption>FIG. 2: An asymptotically inertial trajectory with a final coasting speed of half the speed of light displayed in the usual spacetime diagram. The gray dashed lines represent the light cone, and the black dotted-dashed line shows the asymptote of the mirror trajectory. The trajectory example here is the same as Figure 1.</caption>
</figure>
<text><location><page_2><loc_52><loc_51><loc_92><loc_54></location>The modes are orthonormal and complete. They take the null coordinate form,</text>
<formula><location><page_2><loc_59><loc_47><loc_92><loc_50></location>φ ω ' = (4 πω ' ) -1 / 2 [ e -iω ' v -e -iω ' p ( u ) ] , (3)</formula>
<formula><location><page_2><loc_60><loc_43><loc_92><loc_45></location>χ ω = (4 πω ) -1 / 2 [ e -iωf ( v ) -e -iωu ] . (4)</formula>
<text><location><page_2><loc_52><loc_41><loc_89><loc_42></location>The modes can be expanded in terms of each other,</text>
<formula><location><page_2><loc_60><loc_36><loc_92><loc_40></location>φ ω ' = ∫ ∞ 0 dω [ α ω ' ω χ ω + β ω ' ω χ ∗ ω ] , (5)</formula>
<formula><location><page_2><loc_59><loc_31><loc_92><loc_34></location>χ ω = ∫ ∞ 0 dω ' [ α ∗ ω ' ω φ ω ' -β ω ' ω φ ∗ ω ' ] . (6)</formula>
<text><location><page_2><loc_52><loc_28><loc_82><loc_30></location>by introduction of Bogoliubov coefficients,</text>
<formula><location><page_2><loc_57><loc_25><loc_92><loc_27></location>α ω ' ω = ( φ ω ' , χ ω ) , β ω ' ω = -( φ ω ' , χ ∗ ω ) , (7)</formula>
<text><location><page_2><loc_52><loc_21><loc_92><loc_24></location>defined by the flat-space scalar product, in spacetime coordinates,</text>
<formula><location><page_2><loc_61><loc_17><loc_92><loc_20></location>( φ ω ' , χ ω ) = i ∫ ∞ -∞ dx φ ∗ ω ' ↔ ∂ t χ ω , (8)</formula>
<text><location><page_2><loc_52><loc_14><loc_67><loc_16></location>or in null coordinates,</text>
<formula><location><page_2><loc_53><loc_9><loc_92><loc_13></location>( φ ω ' , χ ω ) = i ∫ ∞ -∞ du φ ∗ ω ' ↔ ∂ u χ ω + i ∫ ∞ -∞ dv φ ∗ ω ' ↔ ∂ v χ ω . (9)</formula>
<text><location><page_3><loc_9><loc_87><loc_49><loc_93></location>The field equation combined with the moving mirror boundary condition imply that the Bogoliubov coefficients α ω ' ω and β ω ' ω relate the operators a ω ' and a † ω ' to the operators b ω and b † ω :</text>
<formula><location><page_3><loc_18><loc_83><loc_49><loc_86></location>a ω ' = ∫ dω [ α ∗ ω ' ω b ω -β ∗ ω ' ω b † ω ] , (10)</formula>
<formula><location><page_3><loc_17><loc_78><loc_49><loc_81></location>b ω = ∫ dω ' [ α ω ' ω a ω ' + β ∗ ω ' ω a † ω ' ] . (11)</formula>
<text><location><page_3><loc_9><loc_76><loc_36><loc_77></location>The modes are orthonormal such that,</text>
<formula><location><page_3><loc_10><loc_72><loc_49><loc_75></location>( φ ω , φ ω ' ) = -( φ ∗ ω , φ ∗ ω ' ) = δ ( ω -ω ' ) , ( φ ω , φ ∗ ω ' ) = 0 , (12)</formula>
<formula><location><page_3><loc_10><loc_69><loc_49><loc_71></location>( χ ω , χ ω ' ) = -( χ ∗ ω , χ ∗ ω ' ) = δ ( ω -ω ' ) , ( χ ω , χ ∗ ω ' ) = 0 . (13)</formula>
<text><location><page_3><loc_9><loc_63><loc_49><loc_68></location>Since the modes are related by a Bogoliubov transformation, the above scalar products imply several identities. Using (12), (5), and (13) the following Wronskian identity holds,</text>
<formula><location><page_3><loc_11><loc_59><loc_49><loc_62></location>∫ dω '' [ α ωω '' α ∗ ω ' ω '' -β ωω '' β ∗ ω ' ω '' ] = δ ( ω -ω ' ) . (14)</formula>
<text><location><page_3><loc_9><loc_54><loc_49><loc_58></location>Note that this relation is a derived consequence only of the field equation and the imposed asymptotically inertial mirror. This relation is a result of the dynamics.</text>
<text><location><page_3><loc_9><loc_48><loc_49><loc_52></location>Consider now, the possible commutation relations of the creation operators, a ω ' , a † ω ' and b ω , b † ω (which are associated with the φ ω ' and χ ω modes, respectively),</text>
<formula><location><page_3><loc_10><loc_44><loc_49><loc_47></location>[ a ω , a † ω ' ] ± = δ ( ω ' -ω ) , [ a ω , a ω ' ] ± = [ a † ω , a † ω ' ] ± = 0 , (15)</formula>
<formula><location><page_3><loc_10><loc_40><loc_49><loc_43></location>[ b ω , b † ω ' ] ± = δ ( ω ' -ω ) , [ b ω , b ω ' ] ± = [ b † ω , b † ω ' ] ± = 0 , (16)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_40></location>Here the + sign corresponds to Fermi-Dirac statistics from the anticommutator, while the -sign corresponds to Bose-Einstein statistics from the commutator. The relations associated with the φ ω ' modes may be expressed in terms of Bogoliubov coefficients using (10),</text>
<formula><location><page_3><loc_11><loc_29><loc_49><loc_32></location>[ a ω , a † ω ' ] ± = ∫ dω '' [ α ω '' ω α ∗ ω '' ω ' ± β ω '' ω ' β ∗ ω '' ω ] . (17)</formula>
<text><location><page_3><loc_9><loc_25><loc_49><loc_28></location>The relations associated with the χ ω modes may be expressed in terms of Bogoliubov coefficients using (11),</text>
<formula><location><page_3><loc_11><loc_21><loc_49><loc_24></location>[ b ω , b † ω ' ] ± = ∫ dω '' [ α ω '' ω α ∗ ω '' ω ' ± β ω '' ω ' β ∗ ω '' ω ] . (18)</formula>
<text><location><page_3><loc_9><loc_11><loc_49><loc_20></location>The dynamics-only result (14) dictates which sign to use in (17) and (18). Since the dynamics indicates the choice of the -sign, the election of Bose-Einstein statistics is straightforward. Similar identities to (14), using (12) and (13), dictate the correct sign to be used in (15) and (16) which give, in total,</text>
<formula><location><page_3><loc_10><loc_8><loc_49><loc_10></location>[ a ω , a † ω ' ] -= δ ( ω ' -ω ) , [ a ω , a ω ' ] -= [ a † ω , a † ω ' ] -= 0 , (19)</formula>
<formula><location><page_3><loc_53><loc_91><loc_92><loc_93></location>[ b ω , b † ω ' ] -= δ ( ω ' -ω ) , [ b ω , b ω ' ] -= [ b † ω , b † ω ' ] -= 0 . (20)</formula>
<text><location><page_3><loc_52><loc_70><loc_92><loc_91></location>Particles at different times, (those associated with b † ω b ω and a † ω ' a ω ' ) obey the same statistics only in the BoseEinstein case. The Bose-Einstein case is then fixed with the dynamics of the field and mirror. When the mirror has acceleration, the β Bogoliubov transformation coefficient is not zero. If the mirror is not accelerating, then β = 0, and this connection between dynamics and statistics is absent, despite the presence of a boundary and the abandonment of Minkowski spacetime. The spinstatistics-dynamics connection in flat spacetime relies on a non-zero acceleration. The derivation presented above is an 'accelerated boundary condition in flat spacetime' -moving mirror- derivation of the relationship between spin and statistics.</text>
<section_header_level_1><location><page_3><loc_59><loc_66><loc_85><loc_67></location>B. Infinitely Accelerated Mirror</section_header_level_1>
<text><location><page_3><loc_52><loc_49><loc_92><loc_64></location>Consider a moving mirror which accelerates for all time. When a mirror in flat spacetime accelerates forever, a horizon singularity is formed. The horizon can be seen most easily in a Penrose diagram, (see Figure 3 or in a spacetime diagram in Figure 4). It is reasonable to ask if the physically pathological singularity breaks the dynamics-statistics connection. As it turns out, the acceleration singularity complicates the mathematics but does not ruin the dynamics-statistics link. The spin-statistics connection is apparent in the case of an infinitely accelerated boundary.</text>
<figure>
<location><page_3><loc_55><loc_21><loc_89><loc_47></location>
<caption>FIG. 3: Asymptotically infinite accelerating trajectories in a Penrose diagram. In this figure, all mirrors emit thermal radiation at all times with T = κ/ 2 π along the trajectory z ( t ) = -t -1 κ W ( e -2 κt ), where κ = π/ 8 , π/ 4 , π/ 2 , π, 2 π .</caption>
</figure>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>In this section only, we adopt Carlitz-Willey's normalization, transformation conventions and left-right con-</text>
<figure>
<location><page_4><loc_12><loc_67><loc_46><loc_93></location>
<caption>FIG. 4: The asymptotically infinite accelerating trajectory of the Carlitz-Willey mirror in a spacetime diagram. The mirror emits thermal radiation at all times with T = κ/ 2 π along the trajectory z ( t ) = -t -1 κ W ( e -2 κt ). Notice the Carlitz-Willey mirror does not start statically, and also approaches the speed of light in the future with an acceleration singularity, resulting in a horizon.</caption>
</figure>
<text><location><page_4><loc_9><loc_47><loc_49><loc_53></location>struction [13]. The advantage here is the simplicity of the explicit mode forms. The field motion is the KleinGordon equation /square ψ ( u, v ) = 0, and the field is written as</text>
<formula><location><page_4><loc_16><loc_41><loc_49><loc_45></location>ψ = 1 4 π ∫ ∞ 0 dω ' ω ' [ a ω ' φ ω ' + a † ω ' φ ∗ ω ' ] (21)</formula>
<formula><location><page_4><loc_16><loc_35><loc_49><loc_39></location>ψ = 1 4 π ∑ I ∫ ∞ 0 dω ω [ b I ω χ I ω + b I † ω χ I ∗ ω ] (22)</formula>
<text><location><page_4><loc_9><loc_32><loc_47><loc_34></location>where I = R,L . The modes take the simplified form,</text>
<formula><location><page_4><loc_20><loc_27><loc_49><loc_30></location>φ ω ' = e -iω ' v -e -iω ' p ( u ) (23)</formula>
<formula><location><page_4><loc_17><loc_22><loc_49><loc_25></location>{ χ R ω = e -iωf R ( v ) Θ( -v ) -e -iωu χ L ω = e -iωf L ( v ) Θ( v ) (24)</formula>
<text><location><page_4><loc_9><loc_18><loc_49><loc_21></location>while the ray-tracing functions for the specified CarlitzWilley trajectory assume</text>
<formula><location><page_4><loc_24><loc_14><loc_49><loc_17></location>p ( u ) = -e -κu κ (25)</formula>
<formula><location><page_4><loc_17><loc_8><loc_49><loc_11></location>{ f R ( v ) = -κ -1 ln( -κv ) v < 0 f L ( v ) = + κ -1 ln(+ κv ) v > 0 . (26)</formula>
<text><location><page_4><loc_52><loc_87><loc_92><loc_93></location>Carlitz and Willey point out that a constant energy flux emitted by the mirror, κ 2 48 π , is obtained by the ray tracing function of (25). Substituting p ( u ) = t + z ( t ) into (25) gives the resulting condition on the trajectory,</text>
<formula><location><page_4><loc_63><loc_83><loc_92><loc_86></location>t + z ( t ) = -1 κ e -κt + κz ( t ) . (27)</formula>
<text><location><page_4><loc_52><loc_77><loc_92><loc_82></location>Carlitz and Willey did not provide an analytic solution to this equation. However one can be obtained using the Lambert W function (the product logarithm) with the result that</text>
<formula><location><page_4><loc_63><loc_73><loc_92><loc_76></location>z ( t ) = -t -1 κ W ( e -2 κt ) . (28)</formula>
<text><location><page_4><loc_52><loc_66><loc_92><loc_72></location>Figure 3 and Figure 4 are plots of this trajectory. It is straightforward to show that ˙ z → ∓ 1 in the limits t → ±∞ and that z < 0 for all time. The proper acceleration is</text>
<formula><location><page_4><loc_63><loc_61><loc_92><loc_66></location>α ( t ) = -κ 2 √ W ( e -2 κt ) . (29)</formula>
<text><location><page_4><loc_52><loc_53><loc_92><loc_61></location>Note that the proper acceleration is not constant even though the energy flux is. The acceleration 29 is required to have thermal emission, for all times, in the moving mirror model. Contrast this to the Unruh effect, where time-independent acceleration, κ , is responsible for thermal emission.</text>
<text><location><page_4><loc_52><loc_48><loc_92><loc_53></location>The dynamics of the field-mirror system give rise to the Bogoliubov coefficients α I ω ' ω and β I ω ' ω which relate the operators a ω ' and a † ω ' to the operators b I ω and b I † ω :</text>
<formula><location><page_4><loc_58><loc_42><loc_92><loc_46></location>a ω ' = 1 4 π ∑ I ∫ dω ω [ α I ω ' ω b I ω + β I ω ' ω b I † ω ] (30)</formula>
<text><location><page_4><loc_52><loc_36><loc_92><loc_41></location>Since the operators are related by a Bogoliubov transformation, it now pays to examine the the conserved scalar products for the two distinct modes φ and χ , (here J = L, R ). Specifically,</text>
<formula><location><page_4><loc_60><loc_31><loc_92><loc_35></location>i ∫ ∞ -∞ dvφ ∗ ω ' ↔ ∂ v φ ω = 4 πωδ ( ω -ω ' ) (31)</formula>
<formula><location><page_4><loc_52><loc_24><loc_92><loc_29></location>i ∫ ∞ 0 dvχ I ∗ ω ' ↔ ∂ v χ J ω + i ∫ ∞ -∞ duχ I ∗ ω ' ↔ ∂ u χ J ω = 4 πωδ ( ω -ω ' ) δ IJ (32)</formula>
<text><location><page_4><loc_52><loc_21><loc_92><loc_24></location>The conserved scalar products imply several identities, as first shown by [13], in particular:</text>
<formula><location><page_4><loc_52><loc_14><loc_92><loc_19></location>1 4 π ∑ I ∫ dω ω [ α I ω ' ω α I ∗ ω '' ω -β I ω ' ω β I ∗ ω '' ω ] = 4 πω ' δ ( ω '' -ω ' ) (33)</formula>
<formula><location><page_4><loc_56><loc_8><loc_92><loc_12></location>1 4 π ∑ I ∫ dω ω [ α I ω ' ω β I ω '' ω -β I ω ' ω α I ω '' ω ] = 0 (34)</formula>
<formula><location><page_5><loc_9><loc_87><loc_49><loc_91></location>1 4 π ∫ dω ' ω ' [ α I ∗ ω ' ω α J ω ' ω '' -β I ω ' ω β J ∗ ω ' ω '' ] = 4 πωδ ( ω -ω '' ) δ IJ (35)</formula>
<formula><location><page_5><loc_15><loc_81><loc_49><loc_84></location>1 4 π ∫ dω ' ω ' [ α I ∗ ω ' ω β J ω ' ω '' -β I ω ' ω α J ∗ ωω '' ] = 0 (36)</formula>
<text><location><page_5><loc_9><loc_76><loc_49><loc_80></location>These relations are derived consequences of the field equation and the imposed Carlitz-Willey mirror. Now consider the possible commutations relations</text>
<formula><location><page_5><loc_20><loc_73><loc_49><loc_75></location>[ a ω , a † ω ' ] ± = 4 πωδ ( ω -ω ' ) (37)</formula>
<formula><location><page_5><loc_19><loc_69><loc_49><loc_71></location>[ a ω , a ω ' ] ± = [ a † ω , a † ω ' ] ± = 0 (38)</formula>
<text><location><page_5><loc_9><loc_63><loc_49><loc_68></location>As before, the + sign corresponds to FD statistics, while the -sign corresponds to BE statistics. The possible commutation relations in terms of the transformation coefficients are</text>
<formula><location><page_5><loc_10><loc_57><loc_49><loc_60></location>[ a ω , a † ω ' ] ± = 1 4 π ∫ dω '' ω '' [ α I ∗ ω '' ω α J ω '' ω ' -β I ω '' ω β J ∗ ω '' ω ' ] (39)</formula>
<text><location><page_5><loc_9><loc_50><loc_49><loc_56></location>The dynamics result (35) can be used to determine which of the possible commutation relations are to be used in (39). The dynamics dictates the commutation relations use the -sign,</text>
<formula><location><page_5><loc_20><loc_47><loc_49><loc_49></location>[ a ω , a † ω ' ] -= 4 πωδ ( ω -ω ' ) (40)</formula>
<text><location><page_5><loc_9><loc_44><loc_49><loc_47></location>The acceleration singularity does not corrupt the spinstatistics-dynamics connection.</text>
<section_header_level_1><location><page_5><loc_22><loc_40><loc_36><loc_41></location>C. Unruh Effect</section_header_level_1>
<text><location><page_5><loc_9><loc_19><loc_49><loc_38></location>Consider the spin-statistics connection in the setting of the Unruh effect. We specify to the right movers in the right quadrant of the Rindler wedge, x > 0 and x > | t | . The left movers follow suit. Further consideration to the left Rindler wedge is similar to the right Rindler wedge [14]. To satisfy completeness on the full Cauchy surface, both the left and right movers, and both the left and right Rindler wedges must be treated. It is found that correct commutations relations are given due to the presence of acceleration for the Rindler observer. We start with the Minkowski metric, ds 2 = -dt 2 + dx 2 , and KleinGordon field equation ( -∂ 2 t + ∂ 2 x ) ψ = 0. Transformation to Rindler coordinates,</text>
<formula><location><page_5><loc_19><loc_17><loc_49><loc_18></location>t = ρ sinh κτ, x = ρ cosh κτ (41)</formula>
<text><location><page_5><loc_9><loc_13><loc_49><loc_15></location>where ρ > 0 and -∞ < τ < ∞ , gives the metric and field equation,</text>
<formula><location><page_5><loc_21><loc_8><loc_49><loc_10></location>ds 2 = -ρ 2 κ 2 dτ 2 + dρ 2 , (42)</formula>
<formula><location><page_5><loc_60><loc_88><loc_92><loc_91></location>[ -1 κ 2 ∂ 2 ( ∂τ ) 2 + ∂ 2 [ ∂ (ln κρ )] 2 ] ψ = 0 . (43)</formula>
<text><location><page_5><loc_52><loc_79><loc_92><loc_86></location>The mode functions that are right-movers, (dependent only on U ) and the right quadrant, R, are shown graphically in Figure 5. The left-movers are dependent on V and follow the analogous forthcoming procedure. Specifying to these right-moving eigenmodes and the R</text>
<figure>
<location><page_5><loc_55><loc_51><loc_89><loc_77></location>
<caption>FIG. 5: Right movers in the right-quadrant Rindler wedge. In this shaded spacetime region, U < 0, and V > 0. The hyperbolas are constant τ values, while the radial lines are constant ρ values.</caption>
</figure>
<text><location><page_5><loc_52><loc_38><loc_92><loc_41></location>spacetime region is clarifying. This is done by the use of null coordinates,</text>
<formula><location><page_5><loc_66><loc_34><loc_92><loc_37></location>u v } = τ ∓ ln κρ κ , (44)</formula>
<text><location><page_5><loc_52><loc_31><loc_54><loc_33></location>and</text>
<formula><location><page_5><loc_62><loc_27><loc_92><loc_30></location>U V } = t ∓ x = { -κ -1 e -κu κ -1 e κv . (45)</formula>
<text><location><page_5><loc_52><loc_25><loc_88><loc_26></location>Consider now, the two representations of the field:</text>
<formula><location><page_5><loc_55><loc_20><loc_92><loc_24></location>ψ ( U ) = ψ R ( U ) = ∫ ∞ 0 dω ' [ a ω ' φ ω ' + a † ω ' φ ∗ ω ' ] (46)</formula>
<text><location><page_5><loc_52><loc_18><loc_54><loc_19></location>and</text>
<formula><location><page_5><loc_56><loc_14><loc_92><loc_17></location>ψ R ( U ) = θ ( -U ) ∫ ∞ 0 dω [ b R ω χ R ω + b R † ω χ R ∗ ω ] (47)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>Notice that we have specified to the right quadrant by the use of θ ( -U ) and restricted our concern to right movers by utilizing the single U coordinate dependence. The</text>
<text><location><page_6><loc_9><loc_89><loc_49><loc_93></location>advantage here is that left and right movers in the null coordinates U and V do not mix under the Bogoliubov transformation.</text>
<text><location><page_6><loc_10><loc_88><loc_29><loc_89></location>The modes take the form,</text>
<formula><location><page_6><loc_22><loc_83><loc_49><loc_86></location>φ ω ' = 1 √ 4 πω ' e -iω ' U , (48)</formula>
<formula><location><page_6><loc_19><loc_78><loc_49><loc_81></location>χ R ω = θ ( -U ) 1 √ 4 πω ( -κU ) iω κ . (49)</formula>
<text><location><page_6><loc_9><loc_74><loc_49><loc_77></location>The Bogoliubov transformation for the operator associated with the φ ω ' takes the R-form,</text>
<formula><location><page_6><loc_17><loc_70><loc_49><loc_73></location>a ω ' = ∫ ∞ 0 dw [ α R ∗ ω ' ω b R ω -β R ∗ ω ' ω b R † ω ] (50)</formula>
<text><location><page_6><loc_9><loc_67><loc_39><loc_69></location>and the operator associated with the χ R ω is</text>
<formula><location><page_6><loc_17><loc_63><loc_49><loc_66></location>b R ω = ∫ ∞ 0 dw ' [ α R ω ' ω a ω ' + β R ∗ ω ' ω a † ω ' ] (51)</formula>
<text><location><page_6><loc_9><loc_59><loc_49><loc_62></location>The modes are non-trivially expanded in terms of each other,</text>
<formula><location><page_6><loc_16><loc_54><loc_49><loc_58></location>φ ω ' = ∫ ∞ 0 dω [ α R ω ' ω χ R ω + β R ω ' ω χ R ∗ ω ] , (52)</formula>
<formula><location><page_6><loc_16><loc_49><loc_49><loc_52></location>χ R ω = ∫ ∞ 0 dω ' [ α R ∗ ω ' ω φ ω ' -β R ω ' ω φ ∗ ω ' ] . (53)</formula>
<text><location><page_6><loc_9><loc_47><loc_39><loc_48></location>by introduction of Bogoliubov coefficients,</text>
<formula><location><page_6><loc_12><loc_43><loc_49><loc_46></location>α R ω ' ω = ( φ ω ' , χ R ω ) , β R ω ' ω = -( φ ω ' , χ R ∗ ω ) , (54)</formula>
<text><location><page_6><loc_9><loc_40><loc_49><loc_43></location>defined by the flat-space scalar product, in spacetime coordinates,</text>
<formula><location><page_6><loc_18><loc_36><loc_49><loc_39></location>( φ ω ' , χ R ω ) = i ∫ ∞ 0 dx φ ∗ ω ' ↔ ∂ t χ R ω , (55)</formula>
<text><location><page_6><loc_9><loc_33><loc_25><loc_35></location>or in null coordinates,</text>
<formula><location><page_6><loc_17><loc_29><loc_49><loc_32></location>( φ ω ' , χ R ω ) = i ∫ 0 -∞ dU φ ∗ ω ' ↔ ∂ U χ R ω (56)</formula>
<text><location><page_6><loc_9><loc_23><loc_49><loc_28></location>Expanding ( χ R ω , χ R ω ' ) in terms of φ ω ' using the transformation (53) and utilizing the orthonormality of the φ ω ' modes, the R-completeness relation holds</text>
<formula><location><page_6><loc_10><loc_18><loc_49><loc_22></location>∫ ∞ 0 dω '' [ α R ωω '' α R ∗ ω ' ω '' -β R ωω '' β R ∗ ω ' ω '' ] = δ ( ω -ω ' ) . (57)</formula>
<text><location><page_6><loc_9><loc_15><loc_49><loc_18></location>This is contingent on the orthonormality of both sets of distinct modes by the conserved inner product,</text>
<formula><location><page_6><loc_16><loc_12><loc_49><loc_14></location>( φ ω , φ ω ' ) = -( φ ∗ ω , φ ∗ ω ' ) = δ ( ω -ω ' ) , (58)</formula>
<formula><location><page_6><loc_16><loc_8><loc_49><loc_10></location>( χ R ω , χ R ω ' ) = -( χ R ∗ ω , χ R ∗ ω ' ) = δ ( ω -ω ' ) . (59)</formula>
<text><location><page_6><loc_52><loc_89><loc_92><loc_93></location>and where ( φ ω , φ ∗ ω ' ) = ( χ R ω , χ R ∗ ω ' ) = 0. As we have seen in a complementary fashion in Section II A, the possible commutation relations are</text>
<formula><location><page_6><loc_53><loc_84><loc_92><loc_88></location>[ a ω , a † ω ' ] ± = ∫ ∞ 0 dω '' [ α R ωω '' α R ∗ ω ' ω '' ± β R ωω '' β R ∗ ω ' ω '' ] , (60)</formula>
<formula><location><page_6><loc_53><loc_77><loc_92><loc_81></location>[ b R ω , b R † ω ' ] ± = ∫ ∞ 0 dω '' [ α R ωω '' α R ∗ ω ' ω '' ± β R ωω '' β R ∗ ω ' ω '' ] . (61)</formula>
<text><location><page_6><loc_52><loc_71><loc_92><loc_76></location>The Wronskian R-completeness relation (57) from the acceleration dynamics connects to the spin-statistics by necessitating the use of the negative sign in the commutation relations (60) and (61). It is dictated that</text>
<formula><location><page_6><loc_60><loc_66><loc_92><loc_69></location>[ a ω , a † ω ' ] -= [ b R ω , b R † ω ' ] -= δ ( ω -ω ' ) (62)</formula>
<text><location><page_6><loc_52><loc_60><loc_92><loc_65></location>and similarly, the zero commuting relations stem from the analogous identities to (57). Therefore, in totality, the accelerated Rindler observer entails the spin-statistics connection.</text>
<section_header_level_1><location><page_6><loc_63><loc_54><loc_81><loc_55></location>III. CONCLUSIONS</section_header_level_1>
<text><location><page_6><loc_52><loc_23><loc_92><loc_52></location>The crux of these results are dependent on the presence of acceleration. In the cases presented, acceleration arises as an asymptotically inertial mirror, an infinitely accelerated mirror and the constant acceleration Unruh effect. That is, in the far future these examples correspond to a proper acceleration α → 0 , ∞ , κ , respectively. These new results where found: 1) The assumption of particles at early times obeying late time statistics is un-necessary, as only commutation relations are acceptable in either region, as shown in the asymptotically inertial mirror case. 2) In a non-trivial system, the presence of external acceleration conditions is enough to provide the link to commutation relations. This is in contrast to the specification to dynamic curved spacetime. Flat-spacetime acceleration conditions allowing non-trivial Bogoliubov transformations establishes the spin-statistics connection. 3) With regards to thermal emission, the time-dependent expression for acceleration in the moving mirror case was explicitly revealed. This clarifies a salient difference between the Unruh effect and the dynamical Casimir effect.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_23></location>An acceleration in these above situations gives rise to distinct eigenmode solutions that can be used to represent the field. One set of eigenmodes can be expressed in mixed positive and negative frequencies pieces of the other set of eigenmodes. The spin-statistics connection is possible to obtain because of the properties of the conserved inner product and of the existence of these sets of eigenmodes in the first place. If there was zero acceleration, the Bogoliubov transformation would be trivial and the distinct sets of eigenmodes would not exist.</text>
<section_header_level_1><location><page_7><loc_22><loc_92><loc_36><loc_93></location>Acknowledgments</section_header_level_1>
<text><location><page_7><loc_9><loc_87><loc_49><loc_90></location>MRRG is grateful for discussions with Paul R. Anderson, Charles R. Evans and Xiong Chi. MRRG appre-</text>
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</document>
[
{
"title": "On Spin-Statistics and Bogoliubov Transformations in Flat Spacetime With Acceleration Conditions",
"content": "Michael R.R. Good ∗ Institute of Advanced Studies, Singapore (Dated: September 26, 2018) A single real scalar field of spin zero obeying the Klein-Gordon equation in flat spacetime under external conditions is considered in the context of the spin-statistics connection. An imposed accelerated boundary on the field is made to become, in the far future, (1) asymptotically inertial and (2) asymptotically non-inertial (with an infinite acceleration). The constant acceleration Unruh effect is also considered. The systems involving non-trivial Bogoliubov transformations contain dynamics which point to commutation relations. Particles described by in-modes obey the same statistics as particles described by out-modes. It is found in the non-trivial systems that the spin-statistics connection can be manifest from the acceleration. The equation of motion for the boundary which forever emits thermal radiation is revealed. PACS numbers: 03.70.+k, 04.62.+v, 04.60.-m Keywords: dynamical Casimir effect, Unruh effect, external conditions, moving mirrors",
"pages": [
1
]
},
{
"title": "I. INTRODUCTION",
"content": "Parker demonstrated that the spin-statistics connection is evident from the dynamics of curved spacetime.[1] [2] [3] It was first shown using a quantized real scalar field of spin zero which obeys the curved spacetime KleinGordon equation[1]. It was then found that consistency resulted when commutation relations imposed on the creation and annihilation operators for the in-region, were also imposed for the creation and annihilation operators for the out-region. It is not consistent with the dynamics to impose anti-commutation relations for the out-region. Parker's derivation was extended to fermions of spin1/2 which obey the Dirac equation and satisfy anticommutation relations [2]. Higher spin and parastatistics was treated [3], the statistics were generalized [4], [5], [6], and ghost fields [7] were examined. It is assumed [8] that a connection between statistics and dynamics is not present in Minkowski spacetime. It is also generally accepted that particles at late times should obey the same statistics as at early times. Recently, researchers may have presented the first experimental evidence for observation of the dynamical Casimir effect [9]. Within the context of an imposed boundary in flat spacetime, the question arises whether the connection between spin and statistics is revealed from the dynamics of a field-boundary system in the same way as from the dynamics of curved spacetime. In this note, it is shown that if a quantized real scalar field of spin-0 obeys the flat spacetime Klein-Gordon equation with an imposed accelerated boundary condition, then commutation relations are the admissible algebra that may be imposed upon the creation and annihilation operators on the in-region and the out-region. It will not be consistent with the dynamics to establish anti-commutation relations upon this field-boundary system. As is pointed out by Wald [10], the spin-statistics results from curved spacetime depend on the conserved inner product. Wald emphasized that no reasonable quantum field theory in curved spacetime (or in flat spacetime with an external potential), can have Bose-Einstein statistics if the inner product is positive definite for all positive and negative frequency solutions. Likewise, it's unreasonable to have Fermi-Dirac statistics if the inner product is positive definite only on positive frequency solutions. In this note, it is emphasized that acceleration induces the appearance of distinct mode solutions with their respective positive and negative frequency pieces. These distinct modes give rise to a non-trivial Bogoliubov transformation which dictates the commutation relations. In the following three sections, it is shown that an acceleration ultimately gives rise to the spin-statistics connection in three salient examples: an asymptotically inertial mirror in Section II A (the most simple case), an infinitely accelerated mirror in Section II B and the Unruh effect in Section II C.",
"pages": [
1
]
},
{
"title": "A. Asymptotically Inertial Mirror",
"content": "The moving mirror model is the most simple example of the dynamical Casimir effect [11] [12]. Consider a moving mirror (an external boundary condition in 1+1 dimensions) which does not accelerate forever. This mirror starts (ends) at time-like past (future) infinity, asymptotically having zero acceleration in the far past (future). As such, an asymptotically inertial mirror will contain no horizons or pathological acceleration singularities, as seen in Figure 1 and Figure 2. The use of an asymptotically inertial mirror is chosen for clarity of derivation. Relaxing the condition of asymptotically inertial motion will alter the form of the derivation and the complexity of the relevant identities. The connection between acceleration and the spin-statistics connection remains. The use of the forever accelerating observer in the Unruh effect, or a forever accelerating mirror that results in a null horizon, requires the appropriate left-right construction (for example, [13]) to incorporate complete Cauchy surface information. A moving mirror in flat spacetime reveals a connection between statistics and dynamics that is not present in Minkowski spacetime without acceleration. Statistics can be derived from the dynamics of the moving mirror as an accelerated mirror gives a beta Bogoliubov transformation coefficient which is non-zero. The statistics are determined by the algebra of the creation operators. The associated relations of the creation operators are commuting for Bose-Einstein statistics and anticommuting for Fermi-Dirac statistics. The following derivation shows that for the spin-zero field, only Bose-Einstein statistics are invariable with the flat spacetime dynamics that include an accelerated external boundary condition. The field equation of motion is the Klein-Gordon equation /square ψ = 0. The moving mirror is imposed such that at the position of the mirror the field is zero, ψ | z = 0. This system has modes that can be used to expand the field, The modes are orthonormal and complete. They take the null coordinate form, The modes can be expanded in terms of each other, by introduction of Bogoliubov coefficients, defined by the flat-space scalar product, in spacetime coordinates, or in null coordinates, The field equation combined with the moving mirror boundary condition imply that the Bogoliubov coefficients α ω ' ω and β ω ' ω relate the operators a ω ' and a † ω ' to the operators b ω and b † ω : The modes are orthonormal such that, Since the modes are related by a Bogoliubov transformation, the above scalar products imply several identities. Using (12), (5), and (13) the following Wronskian identity holds, Note that this relation is a derived consequence only of the field equation and the imposed asymptotically inertial mirror. This relation is a result of the dynamics. Consider now, the possible commutation relations of the creation operators, a ω ' , a † ω ' and b ω , b † ω (which are associated with the φ ω ' and χ ω modes, respectively), Here the + sign corresponds to Fermi-Dirac statistics from the anticommutator, while the -sign corresponds to Bose-Einstein statistics from the commutator. The relations associated with the φ ω ' modes may be expressed in terms of Bogoliubov coefficients using (10), The relations associated with the χ ω modes may be expressed in terms of Bogoliubov coefficients using (11), The dynamics-only result (14) dictates which sign to use in (17) and (18). Since the dynamics indicates the choice of the -sign, the election of Bose-Einstein statistics is straightforward. Similar identities to (14), using (12) and (13), dictate the correct sign to be used in (15) and (16) which give, in total, Particles at different times, (those associated with b † ω b ω and a † ω ' a ω ' ) obey the same statistics only in the BoseEinstein case. The Bose-Einstein case is then fixed with the dynamics of the field and mirror. When the mirror has acceleration, the β Bogoliubov transformation coefficient is not zero. If the mirror is not accelerating, then β = 0, and this connection between dynamics and statistics is absent, despite the presence of a boundary and the abandonment of Minkowski spacetime. The spinstatistics-dynamics connection in flat spacetime relies on a non-zero acceleration. The derivation presented above is an 'accelerated boundary condition in flat spacetime' -moving mirror- derivation of the relationship between spin and statistics.",
"pages": [
1,
2,
3
]
},
{
"title": "B. Infinitely Accelerated Mirror",
"content": "Consider a moving mirror which accelerates for all time. When a mirror in flat spacetime accelerates forever, a horizon singularity is formed. The horizon can be seen most easily in a Penrose diagram, (see Figure 3 or in a spacetime diagram in Figure 4). It is reasonable to ask if the physically pathological singularity breaks the dynamics-statistics connection. As it turns out, the acceleration singularity complicates the mathematics but does not ruin the dynamics-statistics link. The spin-statistics connection is apparent in the case of an infinitely accelerated boundary. In this section only, we adopt Carlitz-Willey's normalization, transformation conventions and left-right con- struction [13]. The advantage here is the simplicity of the explicit mode forms. The field motion is the KleinGordon equation /square ψ ( u, v ) = 0, and the field is written as where I = R,L . The modes take the simplified form, while the ray-tracing functions for the specified CarlitzWilley trajectory assume Carlitz and Willey point out that a constant energy flux emitted by the mirror, κ 2 48 π , is obtained by the ray tracing function of (25). Substituting p ( u ) = t + z ( t ) into (25) gives the resulting condition on the trajectory, Carlitz and Willey did not provide an analytic solution to this equation. However one can be obtained using the Lambert W function (the product logarithm) with the result that Figure 3 and Figure 4 are plots of this trajectory. It is straightforward to show that ˙ z → ∓ 1 in the limits t → ±∞ and that z < 0 for all time. The proper acceleration is Note that the proper acceleration is not constant even though the energy flux is. The acceleration 29 is required to have thermal emission, for all times, in the moving mirror model. Contrast this to the Unruh effect, where time-independent acceleration, κ , is responsible for thermal emission. The dynamics of the field-mirror system give rise to the Bogoliubov coefficients α I ω ' ω and β I ω ' ω which relate the operators a ω ' and a † ω ' to the operators b I ω and b I † ω : Since the operators are related by a Bogoliubov transformation, it now pays to examine the the conserved scalar products for the two distinct modes φ and χ , (here J = L, R ). Specifically, The conserved scalar products imply several identities, as first shown by [13], in particular: These relations are derived consequences of the field equation and the imposed Carlitz-Willey mirror. Now consider the possible commutations relations As before, the + sign corresponds to FD statistics, while the -sign corresponds to BE statistics. The possible commutation relations in terms of the transformation coefficients are The dynamics result (35) can be used to determine which of the possible commutation relations are to be used in (39). The dynamics dictates the commutation relations use the -sign, The acceleration singularity does not corrupt the spinstatistics-dynamics connection.",
"pages": [
3,
4,
5
]
},
{
"title": "C. Unruh Effect",
"content": "Consider the spin-statistics connection in the setting of the Unruh effect. We specify to the right movers in the right quadrant of the Rindler wedge, x > 0 and x > | t | . The left movers follow suit. Further consideration to the left Rindler wedge is similar to the right Rindler wedge [14]. To satisfy completeness on the full Cauchy surface, both the left and right movers, and both the left and right Rindler wedges must be treated. It is found that correct commutations relations are given due to the presence of acceleration for the Rindler observer. We start with the Minkowski metric, ds 2 = -dt 2 + dx 2 , and KleinGordon field equation ( -∂ 2 t + ∂ 2 x ) ψ = 0. Transformation to Rindler coordinates, where ρ > 0 and -∞ < τ < ∞ , gives the metric and field equation, The mode functions that are right-movers, (dependent only on U ) and the right quadrant, R, are shown graphically in Figure 5. The left-movers are dependent on V and follow the analogous forthcoming procedure. Specifying to these right-moving eigenmodes and the R spacetime region is clarifying. This is done by the use of null coordinates, and Consider now, the two representations of the field: and Notice that we have specified to the right quadrant by the use of θ ( -U ) and restricted our concern to right movers by utilizing the single U coordinate dependence. The advantage here is that left and right movers in the null coordinates U and V do not mix under the Bogoliubov transformation. The modes take the form, The Bogoliubov transformation for the operator associated with the φ ω ' takes the R-form, and the operator associated with the χ R ω is The modes are non-trivially expanded in terms of each other, by introduction of Bogoliubov coefficients, defined by the flat-space scalar product, in spacetime coordinates, or in null coordinates, Expanding ( χ R ω , χ R ω ' ) in terms of φ ω ' using the transformation (53) and utilizing the orthonormality of the φ ω ' modes, the R-completeness relation holds This is contingent on the orthonormality of both sets of distinct modes by the conserved inner product, and where ( φ ω , φ ∗ ω ' ) = ( χ R ω , χ R ∗ ω ' ) = 0. As we have seen in a complementary fashion in Section II A, the possible commutation relations are The Wronskian R-completeness relation (57) from the acceleration dynamics connects to the spin-statistics by necessitating the use of the negative sign in the commutation relations (60) and (61). It is dictated that and similarly, the zero commuting relations stem from the analogous identities to (57). Therefore, in totality, the accelerated Rindler observer entails the spin-statistics connection.",
"pages": [
5,
6
]
},
{
"title": "III. CONCLUSIONS",
"content": "The crux of these results are dependent on the presence of acceleration. In the cases presented, acceleration arises as an asymptotically inertial mirror, an infinitely accelerated mirror and the constant acceleration Unruh effect. That is, in the far future these examples correspond to a proper acceleration α → 0 , ∞ , κ , respectively. These new results where found: 1) The assumption of particles at early times obeying late time statistics is un-necessary, as only commutation relations are acceptable in either region, as shown in the asymptotically inertial mirror case. 2) In a non-trivial system, the presence of external acceleration conditions is enough to provide the link to commutation relations. This is in contrast to the specification to dynamic curved spacetime. Flat-spacetime acceleration conditions allowing non-trivial Bogoliubov transformations establishes the spin-statistics connection. 3) With regards to thermal emission, the time-dependent expression for acceleration in the moving mirror case was explicitly revealed. This clarifies a salient difference between the Unruh effect and the dynamical Casimir effect. An acceleration in these above situations gives rise to distinct eigenmode solutions that can be used to represent the field. One set of eigenmodes can be expressed in mixed positive and negative frequencies pieces of the other set of eigenmodes. The spin-statistics connection is possible to obtain because of the properties of the conserved inner product and of the existence of these sets of eigenmodes in the first place. If there was zero acceleration, the Bogoliubov transformation would be trivial and the distinct sets of eigenmodes would not exist.",
"pages": [
6
]
},
{
"title": "Acknowledgments",
"content": "MRRG is grateful for discussions with Paul R. Anderson, Charles R. Evans and Xiong Chi. MRRG appre- ciates the financial support and hospitality of the IASSingapore.",
"pages": [
7
]
}
]
2013IJMPA..2850148C
https://arxiv.org/pdf/1205.3138.pdf
<document>
<section_header_level_1><location><page_1><loc_21><loc_77><loc_82><loc_82></location>An Alternative String Landscape Cosmology: Eliminating Bizarreness</section_header_level_1>
<text><location><page_1><loc_34><loc_71><loc_70><loc_73></location>L. Clavelli ∗† and Gary R. Goldstein ‡</text>
<text><location><page_1><loc_16><loc_69><loc_87><loc_71></location>Dept. of Physics and Astronomy, Tufts University, Medford MA 02155</text>
<section_header_level_1><location><page_1><loc_48><loc_61><loc_56><loc_62></location>Abstract</section_header_level_1>
<text><location><page_1><loc_18><loc_37><loc_85><loc_59></location>In what has become a standard eternal inflation picture of the string landscape there are many problematic consequences and a difficulty defining probabilities for the occurrence of each type of universe. One feature in particular that might be philosophically disconcerting is the infinite cloning of each individual and each civilization in infinite numbers of separated regions of the multiverse. Even if this is not ruled out due to causal separation one should ask whether the infinite cloning is a universal prediction of string landscape models or whether there are scenarios in which it is avoided. If a viable alternative cosmology can be constructed one might search for predictions that might allow one to discriminate experimentally between the models. We present one such scenario although, in doing so, we are forced to give up several popular presuppositions including the absence of a preferred frame and the homogeneity of matter in the universe. The model also has several ancillary advantages. We also consider the future lifetime of the current universe before becoming a light trapping region.</text>
<text><location><page_1><loc_13><loc_33><loc_90><loc_34></location>keywords: String Landscape, String cosmology, Multiverse, Supersymmetry, preferred frame</text>
<section_header_level_1><location><page_1><loc_13><loc_27><loc_36><loc_29></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_13><loc_14><loc_90><loc_24></location>By now it is widely known that Einstein's greatest blunder was not, as he thought, the introduction of the cosmological constant but rather his uncritical presupposition that the universe was static and eternal into the past and into the future. It might be that the eternal inflation (or chaotic inflation) model [1] rests on a similar untested presupposition that the universe has neither beginning nor end and is an isotropic, chaotically bubbling but otherwise homogeneous vat of varying vacuum energies and matter content.</text>
<text><location><page_2><loc_13><loc_71><loc_90><loc_88></location>In the standard picture it is difficult to define probabilities [2] and difficult to prevent the occurrence of infinite cloning of each individual as well as the production of disembodied brains (Boltzmann brains [3]) and other bizarre entities. In a homogeneous and isotropic multiverse, everything that can exist does exist in infinite numbers. There is also an entropy problem in which, if the universe had an infinite past, one might wonder why our universe is not totally featureless. We ask whether one can construct a viable landscape scenario that avoids the paradoxes and bizarre features of the eternal inflation scenario. For example, can one avoid an infinite number of human, quasi-human, and monster populations in the string landscape and can one avoid the other problems mentioned above?</text>
<text><location><page_2><loc_13><loc_63><loc_90><loc_70></location>If, as is commonly assumed, the discovery of dark energy is due to our existence in a De Sitter space-time, the universe is homogeneous and isotropic with a scale factor growing at large times as an exponential of t. In this world the matter density is homogeneous and varies as the inverse cube of the scale factor a ( t ).</text>
<formula><location><page_2><loc_46><loc_57><loc_90><loc_59></location>lim t →∞ a ( t ) = e Ht (1.1)</formula>
<formula><location><page_2><loc_44><loc_52><loc_90><loc_55></location>ρ m ( t ) ≈ a ( t ) -3 . (1.2)</formula>
<text><location><page_2><loc_13><loc_42><loc_90><loc_51></location>A homogeneous De Sitter space, of course, is not compatible with the notion that each universe within the multiverse is a spherical bubble with a center. We seek to construct a bubble universe that retains, in an adequate approximation, the results of eq. 1.2. This necessarily involves a preferred frame, the bubble center, and implies some amount of matter inhomogeneity.</text>
<text><location><page_2><loc_13><loc_30><loc_90><loc_41></location>If our model, like the standard cosmology, is not in conflict with observations, each physicist is entitled to assume as a presupposition either model based on perceived aesthetic differences keeping in mind that the alternative is not ruled out. In addition to being consistent with current observations we point out several phenomenological advantages of our picture including the banishing of infinite numbers of monsters from the multiverse. In this we may be pursuing a time-honored role of science.</text>
<text><location><page_2><loc_13><loc_20><loc_90><loc_29></location>Next we briefly enumerate nine assumptions. The number of assumptions may seem too great to be justified by the mere avoidance of infinite cloning and other landscape paradoxes. However, if these assumptions are not ruled out and can be distinguished in their consequences from those of the standard inflation picture, we feel that they should be critically considered.</text>
<text><location><page_2><loc_13><loc_14><loc_90><loc_19></location>In a subsequent section we elaborate on the assumptions and examine their consequences relative to the more standard string landscape picture. Finally, in a concluding section, we summarize the advantages of our proposed model.</text>
<section_header_level_1><location><page_2><loc_16><loc_11><loc_28><loc_13></location>We assume:</section_header_level_1>
<unordered_list>
<list_item><location><page_2><loc_16><loc_5><loc_90><loc_9></location>1. The multiverse is not infinite into the past but originated at some time t = -t 0 .</list_item>
<list_item><location><page_3><loc_16><loc_84><loc_90><loc_88></location>2. There is a finite rest energy of matter Mc 2 in the multiverse centered on an origin.</list_item>
<list_item><location><page_3><loc_16><loc_79><loc_90><loc_83></location>3. At time t = -t 0 the matter density is proportional to a delta function at the spatial origin.</list_item>
<list_item><location><page_3><loc_16><loc_74><loc_90><loc_78></location>4. The multiverse begins at t = -t 0 in a state of high vacuum energy density. It then rapidly cascades down to the present state of vacuum energy</list_item>
</unordered_list>
<formula><location><page_3><loc_42><loc_70><loc_90><loc_73></location>ρ v = 4 . 0 ± 0 . 2 GeV/c 2 /m 3 . (1.3)</formula>
<unordered_list>
<list_item><location><page_3><loc_16><loc_65><loc_90><loc_69></location>5. These transitions are governed by the Coleman-De Luccia thin wall formulae and transitions up to higher vacuum energy are of negligible probability.</list_item>
<list_item><location><page_3><loc_16><loc_59><loc_90><loc_64></location>6. Since such a phase transition does not create particles only a transition to our ρ v near the peak of the matter distribution ( r = 0 ) will lead to galaxies, planets, and people.</list_item>
<list_item><location><page_3><loc_16><loc_54><loc_90><loc_57></location>7. We are protected from being enveloped in a black hole like region by a future phase transition to zero vacuum energy.</list_item>
<list_item><location><page_3><loc_16><loc_47><loc_90><loc_52></location>8. Eventually, a bubble of zero vacuum energy should form within our universe and expand to convert the entire universe to this presumably supersymmetric state.</list_item>
<list_item><location><page_3><loc_16><loc_44><loc_84><loc_45></location>9. This zero vacuum energy state is the ground state of the multiverse.</list_item>
</unordered_list>
<section_header_level_1><location><page_3><loc_13><loc_38><loc_33><loc_40></location>2 Discussion</section_header_level_1>
<text><location><page_3><loc_13><loc_32><loc_90><loc_35></location>We now reiterate the assumptions and explain their relevance and effect to compare with the standard eternal inflation picture.</text>
<unordered_list>
<list_item><location><page_3><loc_16><loc_15><loc_90><loc_29></location>1. The multiverse is not infinite into the past but originated at some time t = -t 0 . There is a growing consensus that the alternative leads to numerous thorny paradoxes [4]. The maintainance of a universe from the infinite past requires that there is an equilibrium between transitions to lower vacuum energy and transitions to higher vacuum energy. However, in a thermodynamic system, transitions to lower energy must be enormously favored over the entropy decreasing transitions to higher states and asymptotically in time the system should preferentially settle into the lowest available state.</list_item>
<list_item><location><page_3><loc_16><loc_6><loc_90><loc_13></location>2. There is a finite rest energy of matter Mc 2 in the multiverse centered on an origin. This assumption may be initially repugnant to most cosmologists but is not ruled out. It suggests, contrary to what physics has held for a century, that there is a preferred frame in the multiverse. The assumption suggests that inertial frames are</list_item>
</unordered_list>
<text><location><page_4><loc_18><loc_48><loc_90><loc_88></location>those that are unaccelerated relative to this origin. This allows rotating coordinating systems to be inequivalent. In classical physics and in general relativity it is taught that the laws of physics hold in inertial frames but no prescription is given a priori for defining an inertial frame. The best that can be done is to state that, if a frame is found in which Newton's laws hold, they will also hold in any frame traveling relative to that frame with constant velocity. Newton, himself, was aware of the puzzle and posed the famous 'bucket problem'. A rotating bucket of water has a concave surface even when viewed from a co-rotating frame. The current assumption is analogous to Mach's Principle which states that the matter distribution of the universe defines a preferred frame, in his case the rest frame of the distant stars. Given a finite total mass, there can be no infinite cloning of individuals. Similarly, the Boltzmann Brain problem of standard string cosmology is largely due to the infinite number of universes like ours. Thermal fluctuations are a property of material substances. Given that only a vanishing fraction of the multiverse contains appreciable matter in our model, thermal fluctuations are unlikely to create Boltzmann Brains and quantum fluctuations are unlikely to produce such macroscopic complex objects in the short time available. Given the normal laws of evolution, it is easier to create a complete human being than a disconnected brain and this preference may become absolute if there is only a limited amount of matter available. The number of Boltzmann Brains with macroscopic baryon number produced in our model in matter-containing regions due to quantum fluctuations in the short time since the big bang should not be expected to be large compared to the vanishing number we have already observed.</text>
<unordered_list>
<list_item><location><page_4><loc_16><loc_41><loc_90><loc_46></location>3. In the limit t →-t 0 the matter density is proportional to a delta function at the spatial origin. With two free parameters, M and R , we may write as a non-unique example</list_item>
</unordered_list>
<formula><location><page_4><loc_39><loc_36><loc_90><loc_40></location>ρ m ( r, t ) = M ( R √ π a ( t )) 3 e -r 2 / ( Ra ( t )) 2 . (2.4)</formula>
<text><location><page_4><loc_18><loc_31><loc_90><loc_35></location>The scale factor a ( t ) is an increasing function of time that vanishes at t = -t 0 and that can be taken equal to unity at the current time, t = 0. The equation of continuity</text>
<formula><location><page_4><loc_44><loc_27><loc_90><loc_30></location>/vector ∇· ( ρ m /vectorv ( r )) + ∂ρ m ∂t = 0 (2.5)</formula>
<text><location><page_4><loc_18><loc_24><loc_48><loc_26></location>has, independent of R , the solution</text>
<formula><location><page_4><loc_50><loc_19><loc_90><loc_23></location>/vectorv ( r ) = /vectorr ˙ a a (2.6)</formula>
<text><location><page_4><loc_18><loc_6><loc_90><loc_18></location>in agreement with Hubble's law. This justifies the identification of a ( t ) with the scale factor. An important open question left to future study is the effect of curvature corrections to these equations and other results of this work. In classical and quantum physics the initial conditions do not follow from physical law although they must be among the possible states of the system. Thus one cannot require a justification for initial conditions such as we propose in the preceeding assumptions. In standard cosmology there is no comparably clear statement of initial conditions.</text>
<text><location><page_5><loc_18><loc_84><loc_90><loc_88></location>As t approaches -t 0 the matter density of 2.4 approaches a spatial delta function at the origin</text>
<formula><location><page_5><loc_46><loc_80><loc_90><loc_83></location>ρ m ( /vectorr , -t 0 )= Mδ 3 ( /vectorr ) (2.7)</formula>
<text><location><page_5><loc_18><loc_78><loc_56><loc_80></location>and it integrates to a time independent mass</text>
<formula><location><page_5><loc_45><loc_73><loc_90><loc_77></location>∫ d 3 r ρ m ( r, t ) = M . (2.8)</formula>
<text><location><page_5><loc_18><loc_55><loc_90><loc_72></location>ρ m can be taken to be a neutral, flavor and color singlet density of quarks, leptons, and gauge bosons together with their broken susy partners. Apart from pair production at positive times, there is no anti-matter in the universe. This possible solution to the baryon asymmetry problem is not available in the standard picture where matter is pair produced at the end of the inflationary era. Of course, one is free to consider, in our model also, the possibility that the matter density at the origin of time is a CP symmetric state, thus preserving a possible role for CP violation at later times and the interesting problem of insuring an asymmetric survival of a sufficient number of baryons.</text>
<text><location><page_5><loc_18><loc_31><loc_90><loc_55></location>The matter distribution is isotropic relative to its origin. At r/R << a ( t ) and at r/R >> a ( t ) the matter density is also homogeneous so the metric approaches a Friedmann, Robertson, Walker (FRW) metric and the matter density satisfies eq. 1.2. At sufficiently large t , the density is homogeneous (and for large r negligible) around all fixed positions. A prediction of this picture is that significant deviations from homogeneity should exist near r/R ≈ a ( t ). Observations from Earth cannot probe inhomogeneities at scales larger than the Hubble length, the distance light has travelled since the big bang. Thus if R , is much greater than the Hubble length, the proposed matter distribution is indistinguishable from the homogeneous distribution of the standard cosmological model. This will be our default assumption although, in future work, one might ask whether smaller values of R might also be observationally viable. Of course if we choose R to be beyond current limits of observability, the attractiveness of the model, though not its viability, must depend on other conceptual advantages.</text>
<text><location><page_5><loc_18><loc_18><loc_90><loc_30></location>The cosmic background radiation is isotropic in its rest frame. Relative to this frame the earth frame is presently moving with a speed of 369 ± 1 km/s. At present the origins of these two frames are 5 . 1 Mpc apart. One way to subtract the dipole term from the CBR is to cite all measurements as they would appear in the CBR rest frame. If the CBR rest frame is the r = 0 position of our model the universe is isotropic relative to this frame but not homogeneous except on scales small compared to our R parameter.</text>
<text><location><page_5><loc_18><loc_10><loc_90><loc_17></location>On the other hand if the r = 0 position is far from the center of our sphere of last scattering, the universe would still appear isotropic and homogeneous if R is much greater than the Hubble length, 4 . 2 Gpc. A direct measure of how homogeneous the universe is on large scales is a complicated issue [5].</text>
<text><location><page_5><loc_18><loc_6><loc_90><loc_9></location>If the sun is close to the r = 0 position, it could provide a basis for understanding the remarkable large angle correlations [6] between the ecliptic plane and the CBR.</text>
<text><location><page_6><loc_18><loc_73><loc_90><loc_88></location>(The great statistical significance of these correlations, however, could be somewhat weakened by systematic effects [7]). Then, if R is in the Gpc range one would have to wonder whether some selection effect led to the closeness of the solar system to the origin or, equivalently, to the smallness of the solar velocity relative to the CBR. E.g. perhaps the larger collision rate of fast moving inhomogeneities with the matter background inhibits the rise of life. Alternatively it could be related to galactic merger rates as a function of velocity. These could be residual effects related to the anthropic discrimination against high inflation rates [8].</text>
<text><location><page_6><loc_18><loc_62><loc_90><loc_72></location>Given a small velocity, the parameters, M and R , in eq. 2.4 are constrained by the observed current matter density and degree of matter homogeneity. If at the current time the averaged matter distribution is homogeneous out to a Hubble length and falls off rapidly thereafter, the current value of the matter density, ρ m (0) = Ω m (0) ρ c = 1 . 4 ± 0 . 2 GeV/c 2 /m 3 then implies that the rest energy of the universe is</text>
<formula><location><page_6><loc_36><loc_59><loc_90><loc_62></location>Mc 2 ≈ ρ m (0) c 2 π 3 / 2 R 3 > 2 . 6 · 10 76 GeV . (2.9)</formula>
<text><location><page_6><loc_18><loc_49><loc_90><loc_58></location>Newton's laws hold to an excellent approximation implying, in the current context, that the earth and the milky way are presently moving with negligible acceleration and with some velocity, /vectorv relative to the origin. The magnitude of /vectorv could be related to the dipole term that is evident in the cosmic background radiation (CBR) or could be some other speed. The matter distribution relative to earth is then</text>
<formula><location><page_6><loc_35><loc_44><loc_90><loc_47></location>ρ m ( /vectorr , t ) = M ( R √ π a ( t )) 3 e -( /vectorr + /vectorv ( t + t 0 )) 2 / ( Ra ( t )) 2 . (2.10)</formula>
<text><location><page_6><loc_18><loc_37><loc_90><loc_42></location>Here /vectorv is not to be confused with the Hubble flow of eq. 2.6. Small deviations from isotropy beyond the dipole term can be obtained by multiplying this equation or that of eq. 2.4 by the spherical harmonic expansion</text>
<formula><location><page_6><loc_38><loc_32><loc_90><loc_35></location>F ( r, t ) = 1 + ∑ l,m f l,m ( r, t ) Y l,m ( θ, φ ) . (2.11)</formula>
<text><location><page_6><loc_18><loc_22><loc_90><loc_30></location>The acoustic peak analysis suggests that the lowest values of l for which f l,m are appreciable are near l ≈ 100. Expanding eq. 2.10 in the velocity /vectorv produces correlated low multipole moments suppressed by inverse powers of R . The quadrupole moment, for instance, is suppressed in agreement with observation whereas this suppression is problematic in the standard cosmology.</text>
<text><location><page_6><loc_18><loc_17><loc_90><loc_21></location>If the f l,m vanish at r = 0 and are sufficiently well behaved in t and at large r , the properties of eqs. 2.7 and 2.8 are preserved.</text>
<unordered_list>
<list_item><location><page_6><loc_16><loc_12><loc_90><loc_16></location>4. The multiverse begins at t = -t 0 in a state of high vacuum energy density, ( M I c 2 ) 4 / (¯ hc ) 3 . It then rapidly cascades down to the present state of vacuum energy</list_item>
</unordered_list>
<formula><location><page_6><loc_42><loc_8><loc_90><loc_11></location>ρ v = 4 . 0 ± 0 . 2 GeV/c 2 /m 3 . (2.12)</formula>
<unordered_list>
<list_item><location><page_7><loc_16><loc_73><loc_90><loc_88></location>5. These transitions are governed by the Coleman-De Luccia [9] thin wall formulae and transitions up to higher vacuum energy are of negligible probability. In the thin wall approximation, the energy released in the transition to a lower vacuum energy does not create particles but goes into the surface energy. In the words of Coleman [10] 'This refutes the naive expectation that the decay of the false vacuum would leave behind it a roiling sea of mesons'. The thin wall assumption thus implies that galaxies, planets, and people come not from vacuum decay but from a primordial matter distribution such as that of eq. 2.4.</list_item>
</unordered_list>
<text><location><page_7><loc_18><loc_69><loc_90><loc_72></location>The probability per unit time per unit volume to nucleate a bubble of critical size destined to take over the universe is</text>
<formula><location><page_7><loc_48><loc_64><loc_90><loc_68></location>dP dt d 3 r = Ae -B (2.13)</formula>
<text><location><page_7><loc_18><loc_62><loc_22><loc_63></location>with</text>
<formula><location><page_7><loc_47><loc_57><loc_90><loc_61></location>B = 27 π 2 S 4 2¯ hc ( c 2 ∆ ρ ) 3 (2.14)</formula>
<text><location><page_7><loc_18><loc_54><loc_61><loc_56></location>where S is the energy per unit area on the surface.</text>
<text><location><page_7><loc_18><loc_45><loc_90><loc_53></location>The two parameters A and S must at this point be considered free since they are determined by the exact shape of the unknown effective potential. One can speculate about various values for these parameters. For example, if supernovae Ia are triggered by a transition [11] to exact supersymmetry (zero vacuum energy) within a white dwarf one arrives at the estimates</text>
<formula><location><page_7><loc_44><loc_38><loc_90><loc_43></location>S ≈ 4 . 76 · 10 34 GeV/m 2 A -1 ≈ 7 . 89 · 10 36 m 3 s . (2.15)</formula>
<text><location><page_7><loc_18><loc_32><loc_90><loc_37></location>However, it has been suggested [12] that A -1 depends on ∆ ρ . The wall thickness in the transition out of inflation can be estimated by equating the energy released in a shell of thickness δr to the increase of energy stored in the wall.</text>
<formula><location><page_7><loc_44><loc_29><loc_90><loc_30></location>4 πr 2 ∆ ρ c 2 δr = 4 πr 2 S . (2.16)</formula>
<text><location><page_7><loc_18><loc_24><loc_90><loc_27></location>If ∆ ρ is anywhere near the GUT density M G 4 c 3 / ¯ h 3 and S is near the estimate of eq. 2.15, the wall thickness is extremely small.</text>
<formula><location><page_7><loc_45><loc_19><loc_90><loc_22></location>δr ≈ 3 . 64 · 10 -77 m . (2.17)</formula>
<text><location><page_7><loc_18><loc_15><loc_90><loc_19></location>At the time of bubble nucleation this is 1 / 3 of the bubble radius and becomes rapidly negligible in comparison as the bubble grows.</text>
<text><location><page_7><loc_18><loc_6><loc_90><loc_15></location>Also, in the case of the estimates of [11], B is small for the initial transition. If B remains small until the vacuum approaches our vacuum, the situation is one of rapid cascading with each intermediate bubble carried away by the higher vacuum energy background. Thus the inflating multiverse may decay explosively everywhere to our current low-lying metastable positive vacuum energy universe.</text>
<unordered_list>
<list_item><location><page_8><loc_16><loc_79><loc_90><loc_88></location>6. Since such a phase transition does not create particles only a transition to our ρ v near the peak of the pre-existing matter distribution ( r = 0 ) will lead to galaxies, planets, and people. Bubbles nucleated at r significantly different from zero will have insufficient matter to generate life. It is possible that only one or only very few bubbles contain intelligent life.</list_item>
</unordered_list>
<text><location><page_8><loc_18><loc_71><loc_90><loc_78></location>Even at r = 0 the matter distribution falls off as a ( t ) -3 so there is, perhaps, an anthropic understanding of an early end to inflation. Neighboring bubbles have very close to the same vacuum energy so there is little effect from bubble collisions contrary to the prediction of the chaotic inflation picture [13].</text>
<text><location><page_8><loc_18><loc_65><loc_90><loc_70></location>Our assumption is contrary to a common untested presupposition (often referred to as the cosmological principle) that our universe does not occupy a privileged position in the multiverse.</text>
<text><location><page_8><loc_18><loc_63><loc_77><loc_64></location>The root mean square radius of the matter distribution from eq. 2.4 is</text>
<formula><location><page_8><loc_44><loc_58><loc_90><loc_62></location>√ < r 2 > = 3 2 a ( t ) R . (2.18)</formula>
<unordered_list>
<list_item><location><page_8><loc_16><loc_36><loc_90><loc_57></location>7. We are protected from being enveloped in a black hole like region by a future phase transition to zero vacuum energy As each bubble grows the energy enclosed approaches the value at which light is trapped. If there were no energy outside, this critical mass would be the mass of a black hole of that radius and the metric outside would be that of Schwarzschild. In the standard cosmology every trajectory through the multiverse, except for a set of measure zero, ends on a black hole. This and other problematic features discussed in [14] derive primarily from the asymptotic features of eternal inflation. Banks has shown that transitions between symmetric (constant curvature) spaces of differing vacuum energy do not occur contrary to the usual multiverse picture. This leaves open the possibility that inhomogeneous models such as ours could support vacuum transitions although bubble nucleation in regions of negligible matter density could be suppressed.</list_item>
</unordered_list>
<text><location><page_8><loc_18><loc_33><loc_90><loc_35></location>The finite age of the multiverse allows that the limiting radius has not yet been reached.</text>
<formula><location><page_8><loc_34><loc_28><loc_90><loc_32></location>∫ r 0 4 πr ' 2 dr ' ( ρ m ( r ' , t ) + ρ v ) + 4 πr 2 S/c 2 < rc 2 2 G N . (2.19)</formula>
<text><location><page_8><loc_18><loc_15><loc_90><loc_28></location>Here S is the energy per unit area on the bubble surface. If it is as small as the estimate of eq. 2.15 the surface term can be neglected. The time at which the visible universe saturates this inequality is given by putting r = ct max . The effect of pressure may be important but has not yet been analyzed. The condition that this limit has not yet been reached puts a tight limit on the current matter density of the universe. Whether reaching the limit is incompatible with the continuation of life in the universe is not clear.</text>
<formula><location><page_8><loc_44><loc_11><loc_90><loc_15></location>4 π 3 ρ c t 3 max = t max / (2 G N ) (2.20)</formula>
<text><location><page_8><loc_18><loc_10><loc_20><loc_11></location>or</text>
<formula><location><page_8><loc_34><loc_5><loc_90><loc_9></location>t max = √ 3 8 πρ c G N = 1 H 0 = (13 . 58 ± 0 . 27 ) Gyr . (2.21)</formula>
<text><location><page_9><loc_18><loc_79><loc_90><loc_88></location>Here H 0 is the current value of Hubble's constant which is determined observationally by the ratio of a galaxy's recession speed to its distance from us. The critical density, ρ c , is determined by the central equation of eq. 2.21. The numerical value for H 0 given in eq. 2.21 is from the latest WMAP compilation including other relevant data. t max should be compared to the current age of the universe, t 0 .</text>
<formula><location><page_9><loc_39><loc_74><loc_90><loc_78></location>t 0 = 1 H 0 F (Ω m (0) , Ω Λ (0) , Ω γ (0) , .. ) . (2.22)</formula>
<text><location><page_9><loc_18><loc_63><loc_90><loc_73></location>The correction factor, F , which depends on the current values of various densities is obtained [15] by integrating back from the current time t = 0 to the point at which the scale factor goes to zero. Since the scale factor is only determined up to a constant multiplicative factor we may take a (0) = 1. The current values of the Hubble constant and the matter density of the universe taken from the 2011 Particle Data Group compilation [16] are</text>
<formula><location><page_9><loc_41><loc_56><loc_90><loc_59></location>h =0 . 702 ± 0 . 014 (2.23)</formula>
<formula><location><page_9><loc_38><loc_53><loc_90><loc_57></location>H 0 -1 = 9 . 777752 Gyr h = 13 . 58 ± 0 . 27 Gyr (2.24)</formula>
<formula><location><page_9><loc_37><loc_51><loc_90><loc_53></location>Ω m (0)=0 . 26 ± 0 . 02 . (2.25)</formula>
<text><location><page_9><loc_18><loc_47><loc_90><loc_50></location>For r very small and very large we have approximately a De Sitter space so we can use, as an approximation, the usual equations for the scale factor. Since</text>
<formula><location><page_9><loc_50><loc_43><loc_90><loc_46></location>˙ a a = H ( t ) (2.26)</formula>
<text><location><page_9><loc_18><loc_40><loc_43><loc_42></location>and, with a sum over species,</text>
<formula><location><page_9><loc_35><loc_35><loc_90><loc_39></location>a a = -4 πG N 3 ∑ ( ρ i +3 p i ) = H 0 2 ( -ρ m 2 ρ c + ρ v ρ c ) (2.27)</formula>
<text><location><page_9><loc_18><loc_33><loc_25><loc_35></location>we have</text>
<text><location><page_9><loc_18><loc_27><loc_30><loc_28></location>We may write</text>
<formula><location><page_9><loc_48><loc_22><loc_90><loc_26></location>da aH ( t ) = dt . (2.29)</formula>
<text><location><page_9><loc_18><loc_19><loc_51><loc_21></location>The age of the universe, t 0 , is therefore</text>
<formula><location><page_9><loc_49><loc_14><loc_90><loc_18></location>t 0 = ∫ 1 0 da aH (2.30)</formula>
<text><location><page_9><loc_18><loc_10><loc_90><loc_14></location>where we have written H parametrically as a function of a . Equivalently, t 0 can also be defined by</text>
<formula><location><page_9><loc_45><loc_5><loc_90><loc_9></location>∫ 0 -t 0 dtH ( t ) = ∫ 1 0 da/a (2.31)</formula>
<formula><location><page_9><loc_45><loc_29><loc_90><loc_32></location>dH dt = -3 2 H 0 2 Ω m ( t ) . (2.28)</formula>
<text><location><page_10><loc_18><loc_82><loc_90><loc_88></location>where H ( -t 0 ) = ∞ . If we are not already living in a light trapping region t 0 must be less than the t max of eq. 2.21, i.e. the correction factor F in eq. 2.22 must be less than unity. The time from now at which our universe becomes light trapping is</text>
<formula><location><page_10><loc_36><loc_77><loc_90><loc_81></location>∆ t = 1 H 0 (1 -F (Ω m (0) , Ω Λ (0) , Ω γ (0) , .. )) . (2.32)</formula>
<text><location><page_10><loc_18><loc_65><loc_90><loc_76></location>Assuming that Ω Λ (0) = 1 -Ω m (0) and that the photon and neutrino contributions to the energy content of the universe can be neglected, ∆ t is plotted against Ω m (0) in fig. 1. To be consistent with the experimental value, Ω m (0) must also be less than 0 . 28 and the future lifetime of the universe in the present phase must be less than 0 . 23 Gyr. This estimate is two orders of magnitude lower than another estimate [17] which suggests that the future lifetime against vacuum decay could be of order 20 Gyr.</text>
<unordered_list>
<list_item><location><page_10><loc_16><loc_46><loc_90><loc_64></location>8. Eventually, a bubble of zero vacuum energy should form within our universe and expand to convert the entire universe to this presumably supersymmetric state. At this point the interior of the bubble may never reach the light trapping limit since the first term in eq. 2.19 is limited by M and the vacuum energy term drops to zero. The surface term can never contribute to light trapping since it is itself moving out at the speed of light. Other regions devoid of significant matter densities will undergo the transition to zero vacuum energy without producing matter. If we are not destined to be enveloped in a light trapping region, the current model predicts a cosmologically proximate end to the current era in a transition to a universe of zero vacuum energy.</list_item>
<list_item><location><page_10><loc_16><loc_30><loc_90><loc_44></location>9. The zero vacuum energy state is the ground state of the universe. If there are local minima of negative vacuum energy with the same number of degrees of freedom as our universe, there might eventually be a transition to one or more of these leading to a big crunch. However, Banks [14] has given strong arguments that this will not happen. In the standard picture where the mass is not finite, the eventual formation of a light trapping region is then unavoidable. Assumption 9 and the previous one are motivated by avoiding the light trapping limit although it is not clear whether such a limit would be inimical to life or pose other problems to the theory.</list_item>
</unordered_list>
<section_header_level_1><location><page_10><loc_13><loc_23><loc_33><loc_25></location>3 Conclusion</section_header_level_1>
<text><location><page_10><loc_13><loc_6><loc_90><loc_20></location>In the current model we have introduced several free parameters for which we have only given preliminary estimates. The model can be falsified if inconsistencies develop and could be confirmed if systematic matter inhomogeneities are found within the Hubble length. The thesis of this paper is that what is called the cosmological principle may be an overgeneralization of observations at lower distance scales. Our proposal is anti-Copernican in the sense that our inhabited bubble universe is at the center of the multiverse. As R → ∞ , the model approaches the standard landscape cosmology. However, for any finite R , the model has the following advantages over the standard picture.</text>
<figure>
<location><page_11><loc_38><loc_67><loc_62><loc_85></location>
<caption>Figure 1: The time from now before our universe becomes light trapping is plotted against the current matter density, Ω m (0). The experimental range of this quantity is shown by the dashed line. As indicated, if the universe is not already light trapping, (∆ t > 0), Ω m (0) must be greater than 0 . 2645.</caption>
</figure>
<text><location><page_11><loc_51><loc_66><loc_52><loc_67></location>m</text>
<unordered_list>
<list_item><location><page_11><loc_16><loc_49><loc_90><loc_54></location>· we can avoid an infinite number of human, quasi-human, and monster populations in the string landscape and we have, at least, a reduced production of Boltzmann Brains and other bizarres entities,</list_item>
<list_item><location><page_11><loc_16><loc_45><loc_84><loc_47></location>· with a finite amount of matter in the multiverse we have no measure problem,</list_item>
<list_item><location><page_11><loc_16><loc_42><loc_88><loc_44></location>· we can unify an inhomogeneous matter distribution with an apparent FRW metric,</list_item>
<list_item><location><page_11><loc_16><loc_39><loc_71><loc_41></location>· a simple solution to the baryon asymmetry problem is possible,</list_item>
<list_item><location><page_11><loc_16><loc_35><loc_57><loc_38></location>· a clear definition of inertial frames is obtained,</list_item>
<list_item><location><page_11><loc_16><loc_32><loc_57><loc_35></location>· the earth rest frame is approximately inertial,</list_item>
<list_item><location><page_11><loc_16><loc_29><loc_90><loc_32></location>· we can understand the current order of magnitude equality between matter and dark energy densities, and</list_item>
<list_item><location><page_11><loc_16><loc_24><loc_85><loc_27></location>· we can at least potentially avoid becoming enveloped by a black hole like state.</list_item>
</unordered_list>
<text><location><page_11><loc_13><loc_8><loc_90><loc_23></location>At present it may be considered a matter of individual taste as to whether these advantages outweigh the required paradigm shift. In particular, many cosmologists seem at peace with the prospect that we and the entire human history of life on earth may be playing out an infinite number of times in other parts of the multiverse. At a minimum the present paper could be considered a challenge to cosmologists to prove the uniqueness of the infinite cloning prediction or to find other alternative scenarios which avoid the infinite cloning and preserve some or all of the conceptional advantages of our model while also preserving the string landscape explanation for the smallness of the vacuum energy.</text>
<text><location><page_12><loc_13><loc_81><loc_90><loc_88></location>It is left to future work to consider the general relativistic corrections to the picture presented here and to incorporate the effects of non-zero pressure. In the process, modifications to the matter density of eq. 2.4 that preserve the properties of eqs. 2.7 and 2.8 may become evident.</text>
<text><location><page_12><loc_13><loc_74><loc_90><loc_79></location>Acknowledgements We acknowledge useful discussions of the matter presented here with Larry Ford at Tufts University. The research of LC was supported in part by the DOE under grant DE-FG02-10ER41714 and that of GG under DOE grant DE-FG02-92ER40702.</text>
<section_header_level_1><location><page_12><loc_13><loc_68><loc_28><loc_70></location>References</section_header_level_1>
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</document>
[
{
"title": "An Alternative String Landscape Cosmology: Eliminating Bizarreness",
"content": "L. Clavelli ∗† and Gary R. Goldstein ‡ Dept. of Physics and Astronomy, Tufts University, Medford MA 02155",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "In what has become a standard eternal inflation picture of the string landscape there are many problematic consequences and a difficulty defining probabilities for the occurrence of each type of universe. One feature in particular that might be philosophically disconcerting is the infinite cloning of each individual and each civilization in infinite numbers of separated regions of the multiverse. Even if this is not ruled out due to causal separation one should ask whether the infinite cloning is a universal prediction of string landscape models or whether there are scenarios in which it is avoided. If a viable alternative cosmology can be constructed one might search for predictions that might allow one to discriminate experimentally between the models. We present one such scenario although, in doing so, we are forced to give up several popular presuppositions including the absence of a preferred frame and the homogeneity of matter in the universe. The model also has several ancillary advantages. We also consider the future lifetime of the current universe before becoming a light trapping region. keywords: String Landscape, String cosmology, Multiverse, Supersymmetry, preferred frame",
"pages": [
1
]
},
{
"title": "1 Introduction",
"content": "By now it is widely known that Einstein's greatest blunder was not, as he thought, the introduction of the cosmological constant but rather his uncritical presupposition that the universe was static and eternal into the past and into the future. It might be that the eternal inflation (or chaotic inflation) model [1] rests on a similar untested presupposition that the universe has neither beginning nor end and is an isotropic, chaotically bubbling but otherwise homogeneous vat of varying vacuum energies and matter content. In the standard picture it is difficult to define probabilities [2] and difficult to prevent the occurrence of infinite cloning of each individual as well as the production of disembodied brains (Boltzmann brains [3]) and other bizarre entities. In a homogeneous and isotropic multiverse, everything that can exist does exist in infinite numbers. There is also an entropy problem in which, if the universe had an infinite past, one might wonder why our universe is not totally featureless. We ask whether one can construct a viable landscape scenario that avoids the paradoxes and bizarre features of the eternal inflation scenario. For example, can one avoid an infinite number of human, quasi-human, and monster populations in the string landscape and can one avoid the other problems mentioned above? If, as is commonly assumed, the discovery of dark energy is due to our existence in a De Sitter space-time, the universe is homogeneous and isotropic with a scale factor growing at large times as an exponential of t. In this world the matter density is homogeneous and varies as the inverse cube of the scale factor a ( t ). A homogeneous De Sitter space, of course, is not compatible with the notion that each universe within the multiverse is a spherical bubble with a center. We seek to construct a bubble universe that retains, in an adequate approximation, the results of eq. 1.2. This necessarily involves a preferred frame, the bubble center, and implies some amount of matter inhomogeneity. If our model, like the standard cosmology, is not in conflict with observations, each physicist is entitled to assume as a presupposition either model based on perceived aesthetic differences keeping in mind that the alternative is not ruled out. In addition to being consistent with current observations we point out several phenomenological advantages of our picture including the banishing of infinite numbers of monsters from the multiverse. In this we may be pursuing a time-honored role of science. Next we briefly enumerate nine assumptions. The number of assumptions may seem too great to be justified by the mere avoidance of infinite cloning and other landscape paradoxes. However, if these assumptions are not ruled out and can be distinguished in their consequences from those of the standard inflation picture, we feel that they should be critically considered. In a subsequent section we elaborate on the assumptions and examine their consequences relative to the more standard string landscape picture. Finally, in a concluding section, we summarize the advantages of our proposed model.",
"pages": [
1,
2
]
},
{
"title": "2 Discussion",
"content": "We now reiterate the assumptions and explain their relevance and effect to compare with the standard eternal inflation picture. those that are unaccelerated relative to this origin. This allows rotating coordinating systems to be inequivalent. In classical physics and in general relativity it is taught that the laws of physics hold in inertial frames but no prescription is given a priori for defining an inertial frame. The best that can be done is to state that, if a frame is found in which Newton's laws hold, they will also hold in any frame traveling relative to that frame with constant velocity. Newton, himself, was aware of the puzzle and posed the famous 'bucket problem'. A rotating bucket of water has a concave surface even when viewed from a co-rotating frame. The current assumption is analogous to Mach's Principle which states that the matter distribution of the universe defines a preferred frame, in his case the rest frame of the distant stars. Given a finite total mass, there can be no infinite cloning of individuals. Similarly, the Boltzmann Brain problem of standard string cosmology is largely due to the infinite number of universes like ours. Thermal fluctuations are a property of material substances. Given that only a vanishing fraction of the multiverse contains appreciable matter in our model, thermal fluctuations are unlikely to create Boltzmann Brains and quantum fluctuations are unlikely to produce such macroscopic complex objects in the short time available. Given the normal laws of evolution, it is easier to create a complete human being than a disconnected brain and this preference may become absolute if there is only a limited amount of matter available. The number of Boltzmann Brains with macroscopic baryon number produced in our model in matter-containing regions due to quantum fluctuations in the short time since the big bang should not be expected to be large compared to the vanishing number we have already observed. The scale factor a ( t ) is an increasing function of time that vanishes at t = -t 0 and that can be taken equal to unity at the current time, t = 0. The equation of continuity has, independent of R , the solution in agreement with Hubble's law. This justifies the identification of a ( t ) with the scale factor. An important open question left to future study is the effect of curvature corrections to these equations and other results of this work. In classical and quantum physics the initial conditions do not follow from physical law although they must be among the possible states of the system. Thus one cannot require a justification for initial conditions such as we propose in the preceeding assumptions. In standard cosmology there is no comparably clear statement of initial conditions. As t approaches -t 0 the matter density of 2.4 approaches a spatial delta function at the origin and it integrates to a time independent mass ρ m can be taken to be a neutral, flavor and color singlet density of quarks, leptons, and gauge bosons together with their broken susy partners. Apart from pair production at positive times, there is no anti-matter in the universe. This possible solution to the baryon asymmetry problem is not available in the standard picture where matter is pair produced at the end of the inflationary era. Of course, one is free to consider, in our model also, the possibility that the matter density at the origin of time is a CP symmetric state, thus preserving a possible role for CP violation at later times and the interesting problem of insuring an asymmetric survival of a sufficient number of baryons. The matter distribution is isotropic relative to its origin. At r/R << a ( t ) and at r/R >> a ( t ) the matter density is also homogeneous so the metric approaches a Friedmann, Robertson, Walker (FRW) metric and the matter density satisfies eq. 1.2. At sufficiently large t , the density is homogeneous (and for large r negligible) around all fixed positions. A prediction of this picture is that significant deviations from homogeneity should exist near r/R ≈ a ( t ). Observations from Earth cannot probe inhomogeneities at scales larger than the Hubble length, the distance light has travelled since the big bang. Thus if R , is much greater than the Hubble length, the proposed matter distribution is indistinguishable from the homogeneous distribution of the standard cosmological model. This will be our default assumption although, in future work, one might ask whether smaller values of R might also be observationally viable. Of course if we choose R to be beyond current limits of observability, the attractiveness of the model, though not its viability, must depend on other conceptual advantages. The cosmic background radiation is isotropic in its rest frame. Relative to this frame the earth frame is presently moving with a speed of 369 ± 1 km/s. At present the origins of these two frames are 5 . 1 Mpc apart. One way to subtract the dipole term from the CBR is to cite all measurements as they would appear in the CBR rest frame. If the CBR rest frame is the r = 0 position of our model the universe is isotropic relative to this frame but not homogeneous except on scales small compared to our R parameter. On the other hand if the r = 0 position is far from the center of our sphere of last scattering, the universe would still appear isotropic and homogeneous if R is much greater than the Hubble length, 4 . 2 Gpc. A direct measure of how homogeneous the universe is on large scales is a complicated issue [5]. If the sun is close to the r = 0 position, it could provide a basis for understanding the remarkable large angle correlations [6] between the ecliptic plane and the CBR. (The great statistical significance of these correlations, however, could be somewhat weakened by systematic effects [7]). Then, if R is in the Gpc range one would have to wonder whether some selection effect led to the closeness of the solar system to the origin or, equivalently, to the smallness of the solar velocity relative to the CBR. E.g. perhaps the larger collision rate of fast moving inhomogeneities with the matter background inhibits the rise of life. Alternatively it could be related to galactic merger rates as a function of velocity. These could be residual effects related to the anthropic discrimination against high inflation rates [8]. Given a small velocity, the parameters, M and R , in eq. 2.4 are constrained by the observed current matter density and degree of matter homogeneity. If at the current time the averaged matter distribution is homogeneous out to a Hubble length and falls off rapidly thereafter, the current value of the matter density, ρ m (0) = Ω m (0) ρ c = 1 . 4 ± 0 . 2 GeV/c 2 /m 3 then implies that the rest energy of the universe is Newton's laws hold to an excellent approximation implying, in the current context, that the earth and the milky way are presently moving with negligible acceleration and with some velocity, /vectorv relative to the origin. The magnitude of /vectorv could be related to the dipole term that is evident in the cosmic background radiation (CBR) or could be some other speed. The matter distribution relative to earth is then Here /vectorv is not to be confused with the Hubble flow of eq. 2.6. Small deviations from isotropy beyond the dipole term can be obtained by multiplying this equation or that of eq. 2.4 by the spherical harmonic expansion The acoustic peak analysis suggests that the lowest values of l for which f l,m are appreciable are near l ≈ 100. Expanding eq. 2.10 in the velocity /vectorv produces correlated low multipole moments suppressed by inverse powers of R . The quadrupole moment, for instance, is suppressed in agreement with observation whereas this suppression is problematic in the standard cosmology. If the f l,m vanish at r = 0 and are sufficiently well behaved in t and at large r , the properties of eqs. 2.7 and 2.8 are preserved. The probability per unit time per unit volume to nucleate a bubble of critical size destined to take over the universe is with where S is the energy per unit area on the surface. The two parameters A and S must at this point be considered free since they are determined by the exact shape of the unknown effective potential. One can speculate about various values for these parameters. For example, if supernovae Ia are triggered by a transition [11] to exact supersymmetry (zero vacuum energy) within a white dwarf one arrives at the estimates However, it has been suggested [12] that A -1 depends on ∆ ρ . The wall thickness in the transition out of inflation can be estimated by equating the energy released in a shell of thickness δr to the increase of energy stored in the wall. If ∆ ρ is anywhere near the GUT density M G 4 c 3 / ¯ h 3 and S is near the estimate of eq. 2.15, the wall thickness is extremely small. At the time of bubble nucleation this is 1 / 3 of the bubble radius and becomes rapidly negligible in comparison as the bubble grows. Also, in the case of the estimates of [11], B is small for the initial transition. If B remains small until the vacuum approaches our vacuum, the situation is one of rapid cascading with each intermediate bubble carried away by the higher vacuum energy background. Thus the inflating multiverse may decay explosively everywhere to our current low-lying metastable positive vacuum energy universe. Even at r = 0 the matter distribution falls off as a ( t ) -3 so there is, perhaps, an anthropic understanding of an early end to inflation. Neighboring bubbles have very close to the same vacuum energy so there is little effect from bubble collisions contrary to the prediction of the chaotic inflation picture [13]. Our assumption is contrary to a common untested presupposition (often referred to as the cosmological principle) that our universe does not occupy a privileged position in the multiverse. The root mean square radius of the matter distribution from eq. 2.4 is The finite age of the multiverse allows that the limiting radius has not yet been reached. Here S is the energy per unit area on the bubble surface. If it is as small as the estimate of eq. 2.15 the surface term can be neglected. The time at which the visible universe saturates this inequality is given by putting r = ct max . The effect of pressure may be important but has not yet been analyzed. The condition that this limit has not yet been reached puts a tight limit on the current matter density of the universe. Whether reaching the limit is incompatible with the continuation of life in the universe is not clear. or Here H 0 is the current value of Hubble's constant which is determined observationally by the ratio of a galaxy's recession speed to its distance from us. The critical density, ρ c , is determined by the central equation of eq. 2.21. The numerical value for H 0 given in eq. 2.21 is from the latest WMAP compilation including other relevant data. t max should be compared to the current age of the universe, t 0 . The correction factor, F , which depends on the current values of various densities is obtained [15] by integrating back from the current time t = 0 to the point at which the scale factor goes to zero. Since the scale factor is only determined up to a constant multiplicative factor we may take a (0) = 1. The current values of the Hubble constant and the matter density of the universe taken from the 2011 Particle Data Group compilation [16] are For r very small and very large we have approximately a De Sitter space so we can use, as an approximation, the usual equations for the scale factor. Since and, with a sum over species, we have We may write The age of the universe, t 0 , is therefore where we have written H parametrically as a function of a . Equivalently, t 0 can also be defined by where H ( -t 0 ) = ∞ . If we are not already living in a light trapping region t 0 must be less than the t max of eq. 2.21, i.e. the correction factor F in eq. 2.22 must be less than unity. The time from now at which our universe becomes light trapping is Assuming that Ω Λ (0) = 1 -Ω m (0) and that the photon and neutrino contributions to the energy content of the universe can be neglected, ∆ t is plotted against Ω m (0) in fig. 1. To be consistent with the experimental value, Ω m (0) must also be less than 0 . 28 and the future lifetime of the universe in the present phase must be less than 0 . 23 Gyr. This estimate is two orders of magnitude lower than another estimate [17] which suggests that the future lifetime against vacuum decay could be of order 20 Gyr.",
"pages": [
3,
4,
5,
6,
7,
8,
9,
10
]
},
{
"title": "3 Conclusion",
"content": "In the current model we have introduced several free parameters for which we have only given preliminary estimates. The model can be falsified if inconsistencies develop and could be confirmed if systematic matter inhomogeneities are found within the Hubble length. The thesis of this paper is that what is called the cosmological principle may be an overgeneralization of observations at lower distance scales. Our proposal is anti-Copernican in the sense that our inhabited bubble universe is at the center of the multiverse. As R → ∞ , the model approaches the standard landscape cosmology. However, for any finite R , the model has the following advantages over the standard picture. m At present it may be considered a matter of individual taste as to whether these advantages outweigh the required paradigm shift. In particular, many cosmologists seem at peace with the prospect that we and the entire human history of life on earth may be playing out an infinite number of times in other parts of the multiverse. At a minimum the present paper could be considered a challenge to cosmologists to prove the uniqueness of the infinite cloning prediction or to find other alternative scenarios which avoid the infinite cloning and preserve some or all of the conceptional advantages of our model while also preserving the string landscape explanation for the smallness of the vacuum energy. It is left to future work to consider the general relativistic corrections to the picture presented here and to incorporate the effects of non-zero pressure. In the process, modifications to the matter density of eq. 2.4 that preserve the properties of eqs. 2.7 and 2.8 may become evident. Acknowledgements We acknowledge useful discussions of the matter presented here with Larry Ford at Tufts University. The research of LC was supported in part by the DOE under grant DE-FG02-10ER41714 and that of GG under DOE grant DE-FG02-92ER40702.",
"pages": [
10,
11,
12
]
}
]
2013IJMPA..2850160K
https://arxiv.org/pdf/1209.3612.pdf
<document>
<section_header_level_1><location><page_1><loc_14><loc_71><loc_78><loc_79></location>Note About Hamiltonian Formalism for General Non-Linear Massive Gravity Action in Stuckelberg Formalism</section_header_level_1>
<section_header_level_1><location><page_1><loc_14><loc_62><loc_21><loc_64></location>J. Klusoˇn</section_header_level_1>
<text><location><page_1><loc_16><loc_54><loc_53><loc_62></location>Department of Theoretical Physics and Astrophysics Faculty of Science, Masaryk University Kotl´aˇrsk´a 2, 611 37, Brno Czech Republic E-mail: klu@physics.muni.cz</text>
<text><location><page_1><loc_14><loc_28><loc_84><loc_50></location>Abstract: In this note we try to prove the absence of the ghosts in case of the general non-linear massive gravity action in Stuckelberg formalism. We argue that in order to find the explicit form of the Hamiltonian it is natural to start with the general non-linear massive gravity action found in arXiv:1106.3344 [hep-th]. We perform the complete Hamiltonian analysis of the Stuckelberg form of the minimal the non-linear gravity action in this formulation and show that the constraint structure is so rich that it is possible to eliminate non-physical modes. Then we extend this analysis to the case of the general non-linear massive gravity action. We find the corresponding Hamiltonian and collection of the primary constraints. Unfortunately we are not able to finish the complete analysis of the stability of all constraints due to the complex form of one primary constraint so that we are not able to determine the conditions under which given constraint is preserved during the time evolution of the system.</text>
<text><location><page_1><loc_14><loc_24><loc_37><loc_26></location>Keywords: Massive Gravity.</text>
<section_header_level_1><location><page_2><loc_14><loc_83><loc_22><loc_84></location>Contents</section_header_level_1>
<table>
<location><page_2><loc_13><loc_68><loc_84><loc_81></location>
</table>
<section_header_level_1><location><page_2><loc_14><loc_63><loc_42><loc_64></location>1. Introduction and Summary</section_header_level_1>
<text><location><page_2><loc_14><loc_50><loc_84><loc_61></location>The first formulation of the massive gravity was performed by [1] at least in its linearized level as the propagation of the massive graviton above the flat background 1 . Even if the theory seems to be well defined at the linearized level there is a Boulware-Deser ghost [3] in the naive non-linear extension of the Fierz-Pauli formulation. On the other hand recently there has been much progress in the non-linear formulation of the massive gravity without the Boulware-Deser ghost [4, 5] and also [37, 38] 2 .</text>
<text><location><page_2><loc_14><loc_32><loc_84><loc_50></location>The Hamiltonian treatment of non-linear massive gravity theory was performed in many papers with emphasis on the general proof of the absence of the ghosts in given theory. The first attempt for the analysis of the constraint structure of the non-linear massive gravity was performed in [47]. However it turned out that this analysis was not complete and the wrong conclusion was reached as was then shown in the fundamental paper [40] where the complete Hamiltonian analysis of the gauge fixed form of the general non-linear massive gravity was performed. The fundamental result of given paper is the proof of the existence of two additional constraints in the theory which are crucial for the elimination of non physical modes and hence for the consistency of the non-linear massive gravity at least at the classical level.</text>
<text><location><page_2><loc_14><loc_20><loc_84><loc_31></location>Then the Hamiltonian analysis of the non-linear massive gravity in the Stuckelberg formulation was performed in [48, 49]. Unfortunately the wrong conclusion was again reached in the first versions of given papers as was then shown in [50] where the absence of the ghosts in the minimal version of non-linear massive gravity was proven for the first time. Then an independent proof of the absence of the ghosts in the minimal version of non-linear massive gravity in the Stuckelberg formulation was presented in [51].</text>
<text><location><page_2><loc_14><loc_15><loc_84><loc_20></location>However the proof of the absence of the ghosts in the general form of the non-linear massive gravity in the Stuckelberg formulation is still lacking. The difficulty with the possible Hamiltonian formulation of given theory is that the action depends on the kinetic</text>
<text><location><page_3><loc_14><loc_74><loc_84><loc_89></location>terms of the Stuckelberg fields in a highly non-linear way so that it seems to be impossible to find an explicit relation between canonical conjugate momenta and the time derivatives of the Stuckelberg fields. On the other hand there exists the formulation of the non-linear massive gravity action with the linear dependence on the kinetic term for the Stuckelberg fields. This is the form of the non-linear massive gravity action that arises from the original one when the redefinition of the shift functions is performed [38, 37, 41, 42]. The goal of this paper is to perform the Hamiltonian analysis of given action and try to identify all constraints.</text>
<text><location><page_3><loc_14><loc_35><loc_84><loc_74></location>As we argued above the main advantage of the formulation of the non-linear massive gravity with redefined shift function is that the kinetic term for the Stuckelberg fields appears linearly and hence it is easy to find the corresponding Hamiltonian even for the general form of the non-linear massive gravity action. Further it is possible to identify four primary constraints of the theory where the three ones are parts of the generators of the spatial diffeomorphism. Note that the presence of the diffeomorphism constraints is the reflection of the fact that we have manifestly diffeomorphism invariant theory. On the other hand the fourth primary constraint could be responsible for the elimination of the additional non physical mode. However this claim is only true when the requirement of the preservation of given constraint during the time development of the system generates another additional constraint. Unfortunately we find that the original primary constraint cannot provide such additional constraint due to the fact that the Poisson bracket between the primary constraints defined at different space points is non zero. For that reason we should find another constraint that obeys the property that Poisson bracket between these constraints defined at different space points is zero. We find such a constraint in the case of the minimal non-linear massive gravity action and we show that this constraint has the same form as the primary constraint found in [51]. Then we will be able to show that the requirement of the preservation of given constraint during the time evolution of the system implies the additional constraint and these two constraints together allow to eliminate two non-physical modes. This result agrees with previous two independent analysis performed in [50] and in [51].</text>
<text><location><page_3><loc_14><loc_10><loc_84><loc_34></location>Unfortunately we are not able to reach the main goal of this paper which is the proof of the absence of the ghosts for the general non-linear massive gravity theory in Stuckelberg formalism. The reason is that we are not able to find the primary constraint that has vanishing Poisson bracket between these constraints defined at different space points and that has the Poisson brackets with another constraints that vanish on the constraint surface. This is crucial condition for the existence of the additional constraint. It is rather worrying that we are not able to finish the Hamiltonian analysis for the general non-linear massive gravity action especially in the light of the very nice proof of the absence of the ghosts in case of the gauge fixed non-linear massive gravity action [40]. However there is a possibility that the proof of the absence of the ghosts for general non-linear massive gravity action in Stuckelberg formalism could be found in the very elegant formulation of the massive and multi metric theories of gravity presented in [23]. We hope to return to this problem in future.</text>
<text><location><page_3><loc_17><loc_9><loc_84><loc_10></location>The structure of this paper is as follows. In the next section (2) we introduce the</text>
<text><location><page_4><loc_14><loc_88><loc_84><loc_89></location>non-linear massive gravity action and perform the field redefinition of the shift function.</text>
<text><location><page_4><loc_14><loc_78><loc_84><loc_87></location>We also we find the Hamiltonian formulation of the minimal form of the non-lineal massive gravity in Stuckelberg formulation with redefined shift functions and we find that given theory is free from the ghosts. Then we extend this approach to the case of the general non-linear massive gravity theory in the section ( ?? ). We identify all primary constraints and discuss the difficulties that prevent us to finish the complete Hamiltonian analysis.</text>
<section_header_level_1><location><page_4><loc_14><loc_74><loc_73><loc_76></location>2. Non-linear Massive Gravity with Redefined Shift Functions</section_header_level_1>
<text><location><page_4><loc_14><loc_63><loc_84><loc_72></location>As we stressed in the introduction section the goal of this paper is to perform the Hamiltonian analysis of the general non-linear massive gravity with presence of the Stuckelberg fields. It turns out that it is useful to consider this action with redefined shift functions [38, 37, 41, 42]. More explicitly, let us begin with following general form of the non-linear massive gravity action</text>
<formula><location><page_4><loc_29><loc_57><loc_84><loc_62></location>S = M 2 p ∫ d 4 x √ -ˆ g [ (4) R +2 m 2 3 ∑ n =0 β n e n ( √ ˆ g -1 f )] , (2.1)</formula>
<text><location><page_4><loc_14><loc_52><loc_84><loc_56></location>where e k ( A ) are elementary symmetric polynomials of the eigenvalues of A . For generic 4 × 4 matrix they are given by</text>
<formula><location><page_4><loc_24><loc_34><loc_84><loc_51></location>e 0 ( A ) = 1 , e 1 ( A ) = [ A ] , e 2 ( A ) = 1 2 ([ A ] 2 -[ A 2 ]) , e 3 ( A ) = 1 6 ( [ A ] 3 -3[ A ][ A 2 ] + 2[ A 3 ] ) , e 4 ( A ) = 1 24 ( [ A ] 4 -6[ A ] 2 [ A 2 ] + 3[ A 2 ] 2 +8[ A ][ A 3 ] -6[ A 4 ] ) , e k ( A ) = 0 , for k > 4 , (2.2)</formula>
<text><location><page_4><loc_14><loc_29><loc_43><loc_32></location>where A µ ν is 4 × 4 matrix and where</text>
<formula><location><page_4><loc_43><loc_27><loc_84><loc_29></location>[ A ] = Tr A µ µ . (2.3)</formula>
<text><location><page_4><loc_14><loc_18><loc_84><loc_25></location>Of the four β n two combinations are related to the mass and the cosmological constant while the remaining two combinations are free parameters. If we consider the case when the cosmological constant is zero and the parameter m is mass, the four β n are parameterized in terms of the α 3 and α 4 as [5]</text>
<formula><location><page_4><loc_29><loc_12><loc_84><loc_17></location>β n = ( -1) n ( 1 2 (4 -n )(3 -n ) -(4 -n ) α 3 + α 4 ) . (2.4)</formula>
<text><location><page_4><loc_14><loc_8><loc_84><loc_12></location>The minimal action corresponds to β 2 = β 3 = 0 that implies α 3 = α 4 = 1 and consequently β 0 = 3 , β 1 = -1.</text>
<text><location><page_5><loc_14><loc_86><loc_84><loc_89></location>We consider the massive gravity with that is manifestly diffeomorphism invariant. This can be ensured with the help of 4 scalar fields φ A , A = 0 , 1 , 2 , 3 so that</text>
<formula><location><page_5><loc_39><loc_83><loc_84><loc_84></location>ˆ g µν f νρ = ˆ g µν ∂ ν φ A ∂ ρ φ A . (2.5)</formula>
<text><location><page_5><loc_14><loc_79><loc_24><loc_81></location>Then we have</text>
<formula><location><page_5><loc_18><loc_74><loc_84><loc_78></location>N 2 ˆ g -1 f = ( -f 00 + N l f l 0 -f 0 j + N l f lj N 2 g il f l 0 -N i ( -f 00 + N l f l 0 ) N 2 g il f lj -N i ( -f 0 j + N l f lj ) ) , (2.6)</formula>
<text><location><page_5><loc_14><loc_71><loc_78><loc_73></location>where we also used 3 + 1 decomposition of the four dimensional metric ˆ g µν [45, 46]</text>
<formula><location><page_5><loc_29><loc_63><loc_84><loc_70></location>ˆ g 00 = -N 2 + N i g ij N j , ˆ g 0 i = N i , ˆ g ij = g ij , ˆ g 00 = -1 N 2 , ˆ g 0 i = N i N 2 , ˆ g ij = g ij -N i N j N 2 . (2.7)</formula>
<text><location><page_5><loc_14><loc_58><loc_84><loc_61></location>Let us now perform the redefinition of the shift function N i that was introduced in [37, 38, 41, 42]</text>
<formula><location><page_5><loc_36><loc_56><loc_84><loc_58></location>N i = M ˜ n i + f ik f 0 k + N ˜ D i j ˜ n j , (2.8)</formula>
<text><location><page_5><loc_14><loc_53><loc_18><loc_55></location>where</text>
<formula><location><page_5><loc_32><loc_50><loc_84><loc_53></location>˜ x = 1 -˜ n i f ij ˜ n j , M 2 = -f 00 + f 0 k f kl f l 0 (2.9)</formula>
<text><location><page_5><loc_14><loc_49><loc_60><loc_50></location>and where we defined f ij as the inverse to f ij in the sense 3</text>
<formula><location><page_5><loc_43><loc_45><loc_84><loc_47></location>f ik f kj = δ j i . (2.10)</formula>
<text><location><page_5><loc_14><loc_42><loc_54><loc_44></location>Finally note that the matrix ˜ D i j obeys the equation</text>
<formula><location><page_5><loc_34><loc_36><loc_84><loc_41></location>√ ˜ x ˜ D i j = √ ( g ik -˜ D i m ˜ n m ˜ D k n ˜ n n ) f kj (2.11)</formula>
<text><location><page_5><loc_14><loc_33><loc_43><loc_35></location>and also following important identity</text>
<formula><location><page_5><loc_41><loc_28><loc_84><loc_32></location>f ik ˜ D k j = f jk ˜ D k i . (2.12)</formula>
<text><location><page_5><loc_14><loc_23><loc_84><loc_26></location>Let us now concentrate on the minimal form of the non-linear massive gravity action. Using the redefinition (2.8) we find that it takes the form</text>
<formula><location><page_5><loc_15><loc_16><loc_84><loc_22></location>S = M 2 p ∫ d 3 x dt [ N √ g ˜ K ij G ijkl ˜ K kl + N √ gR -√ gMU -2 m 2 ( N √ g √ ˜ xD i i -3 N √ g )] , (2.13)</formula>
<text><location><page_5><loc_14><loc_13><loc_18><loc_15></location>where</text>
<formula><location><page_5><loc_43><loc_11><loc_84><loc_14></location>U = 2 m 2 √ ˜ x , (2.14)</formula>
<text><location><page_6><loc_14><loc_88><loc_79><loc_89></location>and where we used the 3 + 1 decomposition of the four dimensional scalar curvature</text>
<formula><location><page_6><loc_39><loc_83><loc_84><loc_86></location>(4) R = ˜ K ij G ijkl ˜ K kl + R , (2.15)</formula>
<text><location><page_6><loc_14><loc_81><loc_58><loc_82></location>where R is three dimensional scalar curvature and where</text>
<formula><location><page_6><loc_36><loc_76><loc_84><loc_80></location>G ijkl = 1 2 ( g ik g jl + g il g jk ) -g ij g kl (2.16)</formula>
<text><location><page_6><loc_14><loc_74><loc_23><loc_75></location>with inverse</text>
<formula><location><page_6><loc_19><loc_69><loc_84><loc_73></location>G ijkl = 1 2 ( g ik g jl + g il g jk ) -1 2 g ij g kl , G ijkl G klmn = 1 2 ( δ m i δ n j + δ n i δ m j ) . (2.17)</formula>
<text><location><page_6><loc_14><loc_65><loc_84><loc_68></location>Note that in (2.15) we ignored the terms containing total derivatives. Finally note that ˜ K ij is defined as</text>
<formula><location><page_6><loc_31><loc_61><loc_84><loc_65></location>˜ K ij = 1 2 N ( ∂ t g ij -∇ i N j (˜ n, g ) -∇ j N i (˜ n, g )) , (2.18)</formula>
<text><location><page_6><loc_14><loc_59><loc_58><loc_61></location>where N i depends on ˜ n i and g through the relation (2.8).</text>
<text><location><page_6><loc_14><loc_38><loc_84><loc_58></location>At this point we should stress the reason why we consider the non-linear massive gravity action in the form (2.13). The reason is that we want to perform the Hamiltonian analysis for the general non-linear massive gravity action written in the Stuckelberg formalism. It turns out that the action (2.13) has formally the same form as in case of the general nonlinear massive gravity action when we replace U with more general form whose explicit form was determined in [37, 38, 41, 42]. On the other hand the main advantage of the action (2.13) is that it depends on the time derivatives of φ A through the term M and that this term appears linearly in the action (2.13). We should compare this fact with the original form of the non-linear massive gravity action where the dependence on the time derivatives of φ A is highly non-linear and hence it is very difficult to find corresponding Hamiltonian.</text>
<text><location><page_6><loc_17><loc_36><loc_72><loc_38></location>Explicitly, from (2.13) we find the momenta conjugate to N, ˜ n i and g ij</text>
<formula><location><page_6><loc_34><loc_32><loc_84><loc_35></location>π N ≈ 0 , π i ≈ 0 , π ij = M 2 p √ g G ijkl ˜ K kl (2.19)</formula>
<text><location><page_6><loc_14><loc_29><loc_42><loc_31></location>and the momentum conjugate to φ A</text>
<formula><location><page_6><loc_28><loc_24><loc_68><loc_28></location>p A = -( δM δ∂ t φ A ˜ n i + f ij ∂ j φ A ) R i -M 2 p √ g δM ∂ t φ A U ,</formula>
<text><location><page_6><loc_79><loc_23><loc_84><loc_24></location>(2.20)</text>
<text><location><page_6><loc_14><loc_19><loc_18><loc_21></location>where</text>
<formula><location><page_6><loc_41><loc_16><loc_84><loc_19></location>R i = -2 g ik ∇ j π kj . (2.21)</formula>
<text><location><page_6><loc_14><loc_14><loc_55><loc_16></location>It turns out that it is useful to write M 2 in the form</text>
<formula><location><page_6><loc_26><loc_10><loc_70><loc_13></location>M 2 = -∂ t φ A M AB ∂ t φ B , M AB = η AB -∂ i φ A f ij ∂ j φ B ,</formula>
<text><location><page_6><loc_79><loc_9><loc_84><loc_10></location>(2.22)</text>
<text><location><page_7><loc_14><loc_87><loc_62><loc_89></location>where by definition the matrix M AB obeys following relations</text>
<formula><location><page_7><loc_37><loc_85><loc_60><loc_86></location>BC A</formula>
<formula><location><page_7><loc_32><loc_82><loc_84><loc_86></location>M AB η M CD = M AD , det M B = 1 (2.23)</formula>
<text><location><page_7><loc_14><loc_79><loc_24><loc_81></location>together with</text>
<formula><location><page_7><loc_31><loc_75><loc_84><loc_78></location>∂ i φ A M AB = ∂ i φ B -∂ i φ A ∂ k φ A f kl ∂ l φ B = 0 . (2.24)</formula>
<text><location><page_7><loc_14><loc_73><loc_43><loc_75></location>With the help of these results we find</text>
<formula><location><page_7><loc_28><loc_69><loc_84><loc_72></location>p A + R i f ij ∂ j φ A = (˜ n i R i + M 2 p √ gU ) 1 M M AB ∂ t φ B (2.25)</formula>
<text><location><page_7><loc_14><loc_67><loc_27><loc_68></location>and consequently</text>
<formula><location><page_7><loc_14><loc_60><loc_84><loc_66></location>M 2 = -∂ t φ A M AB ∂ t φ B = -M 2 (˜ n i R i + M 2 p √ gU ) 2 ( p A + R i f ij ∂ j φ A ) η AB ( p B + R i f ij ∂ j φ B ) (2.26)</formula>
<text><location><page_7><loc_14><loc_57><loc_54><loc_59></location>which however implies following primary constraint</text>
<formula><location><page_7><loc_20><loc_53><loc_84><loc_57></location>Σ p = (˜ n i R i + M 2 p √ gU ) 2 +( p A + R i f ij ∂ j φ A )( p A + R i f ij ∂ j φ A ) ≈ 0 . (2.27)</formula>
<text><location><page_7><loc_14><loc_51><loc_69><loc_53></location>Note that using (2.24) we obtain another set of the primary constraints</text>
<formula><location><page_7><loc_34><loc_47><loc_61><loc_50></location>∂ i φ A Π A = ∂ i φ A p A + R i = Σ i ≈ 0 .</formula>
<text><location><page_7><loc_79><loc_46><loc_84><loc_47></location>(2.28)</text>
<text><location><page_7><loc_14><loc_43><loc_44><loc_44></location>Observe that using (2.28) we can write</text>
<formula><location><page_7><loc_31><loc_39><loc_84><loc_42></location>p A + R i f ij ∂ j φ A = M AC η CB p B +Σ i f ij ∂ j φ A (2.29)</formula>
<text><location><page_7><loc_14><loc_37><loc_45><loc_38></location>so that we can rewrite Σ p into the form</text>
<formula><location><page_7><loc_30><loc_33><loc_84><loc_36></location>Σ p = (˜ n i R i + M 2 p √ gU ) 2 + p A M AB p B + H i Σ i , (2.30)</formula>
<text><location><page_7><loc_14><loc_29><loc_84><loc_33></location>where H i are functions of the phase space variables. As a result we see that it is natural to consider following independent constraint Σ p</text>
<formula><location><page_7><loc_32><loc_25><loc_84><loc_29></location>Σ p = (˜ n i R i + M 2 p √ gU ) 2 + p A M AB p B ≈ 0 . (2.31)</formula>
<text><location><page_7><loc_14><loc_23><loc_55><loc_25></location>We return to the analysis of the constraint Σ p below.</text>
<text><location><page_7><loc_14><loc_19><loc_84><loc_23></location>Now we are ready to write the extended Hamiltonian which includes all the primary constraints</text>
<text><location><page_7><loc_14><loc_15><loc_18><loc_16></location>where</text>
<formula><location><page_7><loc_29><loc_16><loc_84><loc_20></location>H E = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p Σ p +Ω i ˜ Σ i ) , (2.32)</formula>
<formula><location><page_7><loc_17><loc_9><loc_84><loc_14></location>C 0 = 1 √ gM 2 p π ij G ijkl π kl -M 2 p √ gR +2 m 2 M 2 p √ g √ ˜ x ˜ D i i -6 m 2 M 2 p √ g + ˜ D i j ˜ n j R i (2.33)</formula>
<text><location><page_8><loc_14><loc_88><loc_56><loc_89></location>and where we introduced the constraints ˜ Σ i defined as</text>
<formula><location><page_8><loc_37><loc_84><loc_84><loc_86></location>˜ Σ i = Σ i + ∂ i ˜ n i π i + ∂ j (˜ n j π i ) . (2.34)</formula>
<text><location><page_8><loc_14><loc_78><loc_84><loc_83></location>Note that ˜ Σ i is defined as linear combination of the constraints Σ i ≈ 0 together with the constraints π i ≈ 0.</text>
<text><location><page_8><loc_14><loc_72><loc_84><loc_79></location>To proceed further we have to check the stability of all constraints. To do this we have to calculate the Poisson brackets between all constraints and the Hamiltonian H E . Note that we have following set of the canonical variables g ij , π ij , φ A , p A , ˜ n i , π i and N,π N with non-zero Poisson brackets</text>
<formula><location><page_8><loc_18><loc_63><loc_84><loc_71></location>{ g ij ( x ) , π kl ( y ) } = 1 2 ( δ k i δ l j + δ l i δ k j ) δ ( x -y ) , { φ A ( x ) , p B ( y ) } = δ A B δ ( x -y ) , { N ( x ) , π N ( y ) } = δ ( x -y ) , { ˜ n i ( x ) , π j ( y ) } = δ i j δ ( x -y ) . (2.35)</formula>
<text><location><page_8><loc_14><loc_59><loc_61><loc_62></location>Now we show that the smeared form of the constraint ˜ Σ i ≈ 0</text>
<formula><location><page_8><loc_40><loc_55><loc_84><loc_59></location>T S ( ζ i ) = ∫ d 3 x ζ i ˜ Σ i (2.36)</formula>
<text><location><page_8><loc_14><loc_53><loc_75><loc_54></location>is the generator of the spatial diffeomorphism. First of all using (2.35) we find</text>
<formula><location><page_8><loc_35><loc_47><loc_84><loc_52></location>{ T S ( ζ i ) , ˜ n k } = -ζ i ∂ i ˜ n k + ˜ n j ∂ j ζ k (2.37)</formula>
<text><location><page_8><loc_14><loc_44><loc_71><loc_46></location>which is the correct transformation rule for ˜ n i . Then using (2.35) we find</text>
<formula><location><page_8><loc_28><loc_25><loc_84><loc_42></location>{ T S ( N i ) , R j } = -∂ i N i R j -N i ∂ i R j -R i ∂ j N i , { T S ( N i ) , p A } = -N i ∂ i p A -∂ i N i p A , { T S ( N i ) , φ A } = -N i ∂ i φ A , { T S ( N i ) , g ij } = -N k ∂ k g ij -∂ i N k g kj -g ik ∂ j N k , { T S ( N i ) , π ij } = -∂ k ( N k π ij ) + ∂ k N i π kj + π ik ∂ k N j , { T S ( N i ) , f ij } = -N k ∂ k f ij -∂ i N k f kj -f ik ∂ j N k , { T S ( N i ) , π i } = -∂ i N i π j -N i ∂ i π j + ∂ j N i π j , (2.38)</formula>
<text><location><page_8><loc_14><loc_18><loc_84><loc_23></location>that are the correct transformation rules of the canonical variables under spatial diffeomorphism. To proceed further we need the Poisson bracket between T S ( N i ) and ˜ D i j . It turns out that it is convenient to know the explicit form of the matrix ˜ D i j [37, 38, 41, 42]</text>
<text><location><page_8><loc_14><loc_11><loc_18><loc_12></location>where</text>
<formula><location><page_8><loc_37><loc_12><loc_84><loc_16></location>˜ D i j = √ g im f mn Q n p ( Q -1 ) p j , (2.39)</formula>
<formula><location><page_8><loc_27><loc_8><loc_84><loc_11></location>Q i j = ˜ xδ i j + ˜ n i ˜ n k f kj , ( Q -1 ) p q = 1 ˜ x ( δ p q -˜ n p ˜ n m f mq ) . (2.40)</formula>
<text><location><page_9><loc_14><loc_85><loc_84><loc_89></location>Using this expression we can easily determine the Poisson brackets between T S ( N i ) and ˜ D k j . In fact, by definition we have</text>
<formula><location><page_9><loc_20><loc_77><loc_84><loc_84></location>{ T S ( N i ) , Q k l } = -N m ∂ m Q k l + ∂ m N k Q m l -Q k n ∂ l N n , { T S ( N i ) , Q i j Q j k } = -N m ∂ m ( Q i j Q j k ) + ∂ m N i Q i j Q j k -Q i j Q j m ∂ k N m . (2.41)</formula>
<text><location><page_9><loc_14><loc_73><loc_53><loc_75></location>Using (2.39) and the results derived above we find</text>
<formula><location><page_9><loc_26><loc_67><loc_84><loc_72></location>{ T S ( N i ) , ˜ D i j } = -N m ∂ m ˜ D i j + ∂ m N i ˜ D m j -˜ D i m ∂ j N m . (2.42)</formula>
<text><location><page_9><loc_14><loc_66><loc_62><loc_68></location>Collecting all these results and after some calculations we find</text>
<formula><location><page_9><loc_32><loc_58><loc_84><loc_64></location>{ T S ( N i ) , C 0 } = -N m ∂ m C 0 -∂ m N m C 0 , { T S ( N i ) , Σ p } = -N m ∂ m Σ p -∂ m N m Σ p . (2.43)</formula>
<text><location><page_9><loc_14><loc_55><loc_68><loc_56></location>Note also that it is easy to show that following Poisson bracket holds</text>
<formula><location><page_9><loc_30><loc_49><loc_84><loc_53></location>{ T S ( N i ) , T S ( M j ) } = T S ( N j ∂ j M i -M j ∂ j N i ) . (2.44)</formula>
<text><location><page_9><loc_14><loc_39><loc_84><loc_50></location>Now we are ready to analyze the stability of all primary constraints. As usual the requirement of the preservation of the constraint π N ≈ 0 implies an existence of the secondary constraint C 0 ≈ 0. However the fact that C 0 is the constraint immediately implies that the constraint ˜ Σ i ≈ 0 is preserved during the time evolution of the system, using (2.43) and (2.44). Now we analyze the requirement of the preservation of the constraints π i ≈ 0 during the time evolution of the system</text>
<formula><location><page_9><loc_19><loc_33><loc_84><loc_38></location>∂ t π i = { π i , H E } = -( Ω p δ k i + ∂ ( ˜ D k j ˜ n j ) ∂ ˜ n i ) ( R k -2 m 2 M 2 p √ g √ ˜ x f km ˜ n m ) = 0 . (2.45)</formula>
<text><location><page_9><loc_14><loc_30><loc_43><loc_32></location>It turns out that the following matrix</text>
<formula><location><page_9><loc_40><loc_25><loc_84><loc_29></location>Ω p δ k i + ∂ ( ˜ D k j ˜ n j ) ∂ ˜ n i = 0 (2.46)</formula>
<text><location><page_9><loc_14><loc_20><loc_84><loc_24></location>cannot be solved for Ω p and hence we have to demand the existence of following secondary constraints [37, 38, 41, 42]</text>
<formula><location><page_9><loc_36><loc_15><loc_84><loc_20></location>C i ≡ R i -2 m 2 M 2 p √ g √ ˜ x f ij ˜ n j ≈ 0 . (2.47)</formula>
<text><location><page_9><loc_14><loc_9><loc_84><loc_14></location>Finally we have to proceed to the analysis of the time development of the constraint Σ p ≈ 0. However it turns out that it is very difficult to perform this analysis for Σ p due to the presence of the terms that contain the spatial derivatives of φ A . Then the explicit</text>
<text><location><page_10><loc_14><loc_85><loc_84><loc_89></location>calculation gives { Σ p ( x ) , Σ p ( y ) } /negationslash = 0. For that reason we proceed in a different way when we try to simplify the constraint Σ p . Using C i and Σ i we find that</text>
<formula><location><page_10><loc_33><loc_79><loc_84><loc_84></location>A = 4 m 4 M 4 p g ˜ n i f ij ˜ n j √ ˜ x + F i Σ i + G i C i , (2.48)</formula>
<text><location><page_10><loc_14><loc_75><loc_18><loc_76></location>where</text>
<formula><location><page_10><loc_39><loc_73><loc_84><loc_75></location>A = p A ∂ i φ A f ij ∂ j φ B p B , (2.49)</formula>
<text><location><page_10><loc_14><loc_66><loc_84><loc_72></location>and where F i , G i are the phase space functions whose explicit form is not important for us. Then with the help of (2.48) we express ˜ n i f ij ˜ n j as a function of the phase space variables p A , φ A and g ij , π ij</text>
<formula><location><page_10><loc_32><loc_62><loc_84><loc_66></location>˜ n i f ij ˜ n j = A -F i Σ i -G i C i ( A -F i Σ i -G i C i ) + 4 m 4 M 4 p g . (2.50)</formula>
<text><location><page_10><loc_14><loc_60><loc_35><loc_61></location>In the same way we obtain</text>
<formula><location><page_10><loc_30><loc_52><loc_84><loc_58></location>˜ n i R i = A -F i Σ i -G i C i √ ( A -F i Σ i -G i C i ) + 4 m 4 M 4 p g + ˜ n i C i , (2.51)</formula>
<formula><location><page_10><loc_33><loc_45><loc_84><loc_51></location>˜ n i = -∂ j φ A p A f ji √ A +4 m 4 M 4 p g + ˜ F i Σ i + ˜ G i C i , (2.52)</formula>
<text><location><page_10><loc_14><loc_51><loc_16><loc_52></location>and</text>
<text><location><page_10><loc_14><loc_41><loc_81><loc_45></location>where again ˜ F i , ˜ G i are phase space functions whose explicit form is not needed for us. Now using these results we find that the constraint Σ p takes the form</text>
<formula><location><page_10><loc_24><loc_32><loc_84><loc_40></location>Σ p = ( A -F i Σ i -G i C i +4 m 4 M 4 p g )4 m 4 M 4 p g A -F i Σ i -G i C i +4 m 4 M 4 p g + H i Σ i + p A p A = = p A p A +4 m 4 M 4 p g + H i Σ i ≡ 4 m 4 M 4 p g ˜ Σ p + H i Σ i , (2.53)</formula>
<text><location><page_10><loc_14><loc_28><loc_54><loc_30></location>where we introduced new independent constraint ˜ Σ p</text>
<formula><location><page_10><loc_39><loc_23><loc_84><loc_26></location>˜ Σ p = p A p A 4 m 4 M 4 p g +1 = 0 (2.54)</formula>
<text><location><page_10><loc_14><loc_13><loc_84><loc_21></location>that has precisely the same form as in [51]. Note that the constraint ˜ Σ p has the desired property that { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. As a result we see that it is more natural to consider ˜ Σ p instead of Σ p as an independent constraint. Then the total Hamiltonian, where we include all constraints, takes the form</text>
<formula><location><page_10><loc_26><loc_7><loc_84><loc_12></location>H T = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p ˜ Σ p +Ω i ˜ Σ i +Γ i C i ) . (2.55)</formula>
<text><location><page_11><loc_14><loc_83><loc_84><loc_89></location>Now we are ready to analyze the stability of all constraints that appear in (2.55). First of all we find that π N ≈ 0 is automatically preserved while the preservation of the constraint π i ≈ 0 gives</text>
<formula><location><page_11><loc_29><loc_74><loc_84><loc_82></location>∂ t π i = { π i , H T } ≈ ∫ d 3 x Γ j ( x ) { π i , C j ( x ) } = = -2 m 2 Γ j 1 √ ˜ x ( f ij -f ik ˜ n k f jl ˜ n l ) ≡ -/triangle π i , C j Γ j . (2.56)</formula>
<text><location><page_11><loc_14><loc_70><loc_24><loc_71></location>By definition</text>
<text><location><page_11><loc_59><loc_66><loc_59><loc_68></location>/negationslash</text>
<formula><location><page_11><loc_34><loc_64><loc_84><loc_68></location>det( f ij -f ik ˜ n k f il ˜ n l ) = ˜ x det f ij = 0 (2.57)</formula>
<text><location><page_11><loc_14><loc_59><loc_84><loc_62></location>and hence the matrix /triangle π i , C j is non-singular. Then the only solution of the equation (2.56) is Γ i = 0.</text>
<text><location><page_11><loc_14><loc_55><loc_84><loc_59></location>As the next step we proceed to the analysis of the stability of the constraint ˜ Σ p . As is clear from (2.54) we have</text>
<formula><location><page_11><loc_40><loc_50><loc_84><loc_55></location>{ ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0 . (2.58)</formula>
<text><location><page_11><loc_14><loc_50><loc_59><loc_51></location>Then the time evolution of given constraint takes the form</text>
<formula><location><page_11><loc_30><loc_43><loc_84><loc_48></location>∂ t ˜ Σ p = { ˜ Σ p , H T } ≈ ∫ d 3 x N ( x ) { Σ p , C 0 ( x ) } (2.59)</formula>
<text><location><page_11><loc_14><loc_37><loc_84><loc_41></location>using the fact that ˜ Σ p does not depend on ˜ n i together with Γ i = 0 and also the fact that ˜ Σ p is manifestly diffeomorphism invariant.</text>
<text><location><page_11><loc_17><loc_35><loc_69><loc_37></location>In order to explicitly determine (2.59) we need following expression</text>
<formula><location><page_11><loc_30><loc_28><loc_84><loc_34></location>δ ( √ ˜ x ˜ D k k ) δf ij = √ ˜ x 2 ˜ D j p f pi -1 √ ˜ x ˜ n l f lm δ ( ˜ D m p ˜ n p ) δf ij . (2.60)</formula>
<text><location><page_11><loc_14><loc_25><loc_44><loc_26></location>Then after some calculations we obtain</text>
<formula><location><page_11><loc_19><loc_9><loc_84><loc_23></location>{ ˜ Σ p , ∫ d 3 x N C 0 } = 2 ∂ i [ N ˜ D i j ]˜ n j Σ p + + 1 M 4 p m 4 g p A ∂ i [ N δ ( ˜ D k l ˜ n l ) δf ij C k ∂ j φ A ] -1 M 4 p m 4 g p A ∂ j φ A ∂ i [ N δ ( ˜ D k l ˜ n l ) δf ij ] Σ k + + N ( -˜ D i j ∂ i [˜ n j p A ] p A 2 m 4 p m 4 g + 2 m 2 M 2 p M 4 p m 4 g ˜ D i k p A ∂ i [ √ g √ ˜ xf kj ∂ j φ A ] ) ≈ N Σ II p . (2.61)</formula>
<text><location><page_12><loc_14><loc_86><loc_83><loc_89></location>In order to simplify Σ II p further we use (2.50), (2.51) together with (2.52) to make Σ II p independent on ˜ n i . In fact, using (2.39)(2.40) we obtain</text>
<formula><location><page_12><loc_23><loc_75><loc_84><loc_85></location>Q m p = 1 A +4 M 4 p m 4 g (4 m 4 M 4 p gδ m p + ∂ j φ A p A f jm ∂ p φ B p B ) , ( Q -1 ) m p = A +4 M 4 p m 4 g 4 m 4 M 4 p g ( δ m p -1 A +4 m 4 M 4 p g ∂ j φ A p A f jm ∂ p φ B p B ) (2.62)</formula>
<text><location><page_12><loc_14><loc_71><loc_84><loc_74></location>up to terms proportional to the constraints C i , Σ i . With the help of these results we obtain</text>
<formula><location><page_12><loc_19><loc_63><loc_84><loc_71></location>Σ II p = -˜ D i j ∂ i [˜ n j p A ] p A 2 m 4 p m 4 g + 2 m 2 M 2 p M 4 p m 4 g ˜ D i p p A ∂ i [ √ g √ ˜ xf pj ∂ j φ A ] + F ' i Σ i + G ' i C i ≡ ≡ ˜ Σ II p + F ' i Σ i + G ' i C i , (2.63)</formula>
<text><location><page_12><loc_17><loc_52><loc_17><loc_54></location>/negationslash</text>
<text><location><page_12><loc_14><loc_49><loc_84><loc_62></location>where ˜ x and ˜ D i j are functions of p A , ∂ j φ A and g through the relations (2.50),(2.51) and (2.52). Now we see from (2.59) that the time evolution of the constraint ˜ Σ p ≈ 0 is obeyed on condition when either N = 0 or when ˜ Σ II p = 0. Note that we should interpreted N as the Lagrange multiplier so that it is possible to demand that N = 0 on condition when ˜ Σ II p = 0 on the whole phase space. Of course such a condition is too strong so that it is more natural to demand that ˜ Σ II p ≈ 0 and N = 0. In other words ˜ Σ II p ≈ 0 is the new secondary constraint.</text>
<text><location><page_12><loc_51><loc_50><loc_51><loc_52></location>/negationslash</text>
<text><location><page_12><loc_14><loc_43><loc_84><loc_48></location>In summary we have following collection of constraints: π N ≈ 0 , π i ≈ 0 , C 0 ≈ 0 , C i ≈ 0 , ˜ Σ i ≈ 0 , ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The dynamics of these constraints is governed by the total Hamiltonian</text>
<formula><location><page_12><loc_20><loc_38><loc_84><loc_43></location>H T = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p ˜ Σ p +Ω II p ˜ Σ II p +Ω i ˜ Σ i +Γ i C i ) . (2.64)</formula>
<text><location><page_12><loc_14><loc_34><loc_84><loc_38></location>As the final step we have to analyze the preservation of all constraints. The case of π N ≈ 0 is trivial. For π i ≈ 0 we obtain</text>
<formula><location><page_12><loc_16><loc_27><loc_84><loc_34></location>∂ i π i ( x ) = { π i ( x ) , H T } = ∫ d 3 y (Γ j ( y ) { π i ( x ) , C j ( y ) } +Ω II p ( y ) { π i ( x ) , ˜ Σ II p ( y ) } ) = = Γ j /triangle π i , C j ( x ) = 0 (2.65)</formula>
<text><location><page_12><loc_14><loc_19><loc_84><loc_25></location>due to the crucial fact that ˜ Σ II p does not depend on ˜ n i . This is the main reason why we introduced ˜ Σ II p instead of Σ II p . Then as we argued above the only solution of the equation is Γ i = 0. Now the time development of C i is given by the equation</text>
<formula><location><page_12><loc_24><loc_9><loc_84><loc_18></location>∂ t C i ( x ) = {C i ( x ) , H T } ≈ ≈ ∫ d 3 x ( N ( y ) {C i ( x ) , C 0 ( y ) } + v j ( y ) {C i ( x ) , π j ( y ) } + + Ω p ( y ) { C i ( x ) , ˜ Σ p ( y ) } +Ω II p ( y ) { C i ( x ) , ˜ Σ II p ( y ) }) (2.66)</formula>
<text><location><page_13><loc_14><loc_88><loc_72><loc_89></location>and the time development of the constraint ˜ Σ p is governed by the equation</text>
<formula><location><page_13><loc_25><loc_82><loc_84><loc_87></location>∂ t ˜ Σ p ( x ) = { ˜ Σ p ( x ) , H T } ≈ ∫ d 3 x Ω II p ( y ) { ˜ Σ p ( x ) , ˜ Σ II p ( y ) } . (2.67)</formula>
<text><location><page_13><loc_14><loc_73><loc_84><loc_81></location>As follows from the explicit form of the constraint ˜ Σ II p we see that { ˜ Σ II p ( x ) , ˜ Σ p ( y ) } is non-zero and proportional also to the higher order derivatives of the delta functions. As a consequence we find that the only solution of the equation above is Ω II p = 0. Further we analyze the time evolution of the constraint ˜ Σ II p</text>
<formula><location><page_13><loc_20><loc_64><loc_84><loc_72></location>∂ t ˜ Σ II p ( x ) = { ˜ Σ II p ( x ) , H T } = = ∫ d 3 x ( N ( y ) { ˜ Σ II p ( x ) , C 0 ( y ) } +Ω p ( y ) { ˜ Σ II p ( x ) , ˜ Σ p ( y ) }) = 0 . (2.68)</formula>
<text><location><page_13><loc_14><loc_52><loc_84><loc_62></location>Now from the last equation we obtain Ω p as a function of the phase space variables and N , at least in principle. Then inserting this result into the equation for the preservation of C i (2.66) we determine v j as functions of the phase space variables. Finally note also that the constraint C 0 is automatically preserved due to the fact that Γ i = Ω II p = 0 and also the fact that {C 0 ( x ) , C 0 ( y ) } ≈ 0 as was shown in [40].</text>
<text><location><page_13><loc_14><loc_38><loc_84><loc_53></location>In summary we obtain following picture. We have five the first class constraints π N ≈ 0 , C 0 ≈ 0 , ˜ Σ i ≈ 0 together with eight the second class constraints π i ≈ 0 , C i ≈ 0 and ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The constraints π i ≈ 0 together with C i ≈ 0 can be solved for π i and ˜ n i . Then the constraint ˜ Σ p can be solved for one of the four momenta p A while the constraint ˜ Σ II p can be solved for one of the four φ A . As a result we have 12 gravitational degrees of freedom g ij , π ij ,6 scalars degrees of freedom together with 4 first class constraints C 0 ≈ 0 , ˜ Σ i ≈ 0. Then we find that the number of physical degrees of freedom is 10 which is the correct number of physical degrees of freedom of the massive gravity.</text>
<section_header_level_1><location><page_13><loc_14><loc_35><loc_59><loc_36></location>3. General Non-Linear Massive Gravity Action</section_header_level_1>
<text><location><page_13><loc_14><loc_30><loc_84><loc_33></location>Let us try to apply the procedure performed in previous section to the case of the general non-linear massive gravity whose action takes the form</text>
<formula><location><page_13><loc_18><loc_25><loc_84><loc_29></location>S = M 2 p ∫ d 3 x dt [ N √ g ˜ K ij G ijkl ˜ K kl + √ gNR +2 m 2 √ gMU +2 m 2 N √ gV ] , (3.1)</formula>
<text><location><page_13><loc_14><loc_23><loc_18><loc_25></location>where</text>
<formula><location><page_13><loc_26><loc_21><loc_35><loc_23></location>√ √</formula>
<formula><location><page_13><loc_20><loc_10><loc_77><loc_22></location>U = β 1 ˜ x + β 2 [( ˜ x ) 2 ˜ D i i + ˜ n i f ij ˜ D j k ˜ n k ] + + β 3 [ √ ˜ x ( ˜ D l l ˜ n i f ij ˜ D j k ˜ n k -˜ D i k ˜ n k f ij ˜ D j l ˜ n l ) + 1 2 √ ˜ x 3 ( ˜ D i i ˜ D j j -˜ D i j ˜ D j i )] , V = β 0 + β 1 √ ˜ x ˜ D i i + 1 2 √ ˜ x 2 [ ˜ D i i ˜ D j j + ˜ D i j ˜ D j i ] + + 1 6 β 3 √ ˜ x 3 [ ˜ D i i ˜ D j j ˜ D k k -3 ˜ D i i ˜ D j k ˜ D k j +2 ˜ D i j ˜ D j k ˜ D k i ]</formula>
<formula><location><page_13><loc_80><loc_9><loc_84><loc_10></location>(3.2)</formula>
<text><location><page_14><loc_14><loc_86><loc_84><loc_89></location>Following the analysis performed in the previous section we find the extended Hamiltonian in the form</text>
<text><location><page_14><loc_14><loc_80><loc_18><loc_82></location>where</text>
<formula><location><page_14><loc_29><loc_82><loc_84><loc_86></location>H E = ∫ d 3 x ( N C 0 + v N π N + v i π i +Ω p Σ p +Ω i ˜ Σ i ) , (3.3)</formula>
<formula><location><page_14><loc_26><loc_77><loc_84><loc_81></location>C 0 = 1 √ gM 2 p π ij G ijkl π kl -M 2 p √ gR -2 m 2 M 2 p V + R i ˜ D i j ˜ n j , (3.4)</formula>
<text><location><page_14><loc_14><loc_75><loc_54><loc_76></location>and where the primary constraint Σ p takes the form</text>
<formula><location><page_14><loc_30><loc_71><loc_84><loc_74></location>Σ p : (˜ n i R i +2 m 2 M 2 p √ gU ) 2 + p A M AB p B ≈ 0 . (3.5)</formula>
<text><location><page_14><loc_14><loc_62><loc_84><loc_69></location>As the next step we should analyze the stability of all primary constraints. As in previous section we find that the stability of the constraint π N ≈ 0 implies the secondary constraint C 0 ≈ 0 while the stability of the constraints π i implies set of the secondary constraints C i [37, 38, 41, 42]</text>
<formula><location><page_14><loc_24><loc_52><loc_84><loc_61></location>C i = R i -2 m 2 √ g ˜ n l f lj √ ˜ x [ β 1 δ j i + β 2 √ ˜ x ( δ j i ˜ D m m -˜ D j i )+ + β 3 √ ˜ x 2 ( 1 2 δ j i ( ˜ D m m ˜ D n n -˜ D m n ˜ D n m ) + ˜ D j m ˜ D m i -˜ D j i ˜ D m m )] . (3.6)</formula>
<text><location><page_14><loc_14><loc_21><loc_84><loc_50></location>Now we come to the key point of the analysis which is the requirement of the preservation of the constraint Σ p ≈ 0 during the time evolution of the system. This is very complicated expression which depends on the all phase space variables. Remember that in the minimal case we expressed ˜ n i as functions of g ij , p A and φ A . As a result we found that Σ p can be expressed as a linear combination of C i , Σ i and ˜ Σ p where ˜ Σ p obeys an important property { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0.It would be certainly nice to repeat the same procedure in the case of the constraint Σ p given in (3.5). Unfortunately we are not able to solve the constraint C i in order to express ˜ n i as a function of R i . Consequently we are not able to express the constraint Σ p as a linear combination of the constraints C i and possibly ˜ Σ i , C 0 and the new constraint ˜ Σ p where { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. In other words, despite of the fact we are able to find the primary constraint Σ p ≈ 0 we are not able to determine the additional secondary constraint which is necessary for the elimination of two non-physical phase space modes. As a result the main goal of this paper which was the proof of the absence of the ghosts in the general non-linear massive gravity action in Stuckelberg formalism cannot be completed.</text>
<text><location><page_14><loc_14><loc_17><loc_84><loc_21></location>Acknowledgement: This work was supported by the Grant agency of the Czech republic under the grant P201/12/G028.</text>
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</document>
[
{
"title": "J. Klusoˇn",
"content": "Department of Theoretical Physics and Astrophysics Faculty of Science, Masaryk University Kotl´aˇrsk´a 2, 611 37, Brno Czech Republic E-mail: klu@physics.muni.cz Abstract: In this note we try to prove the absence of the ghosts in case of the general non-linear massive gravity action in Stuckelberg formalism. We argue that in order to find the explicit form of the Hamiltonian it is natural to start with the general non-linear massive gravity action found in arXiv:1106.3344 [hep-th]. We perform the complete Hamiltonian analysis of the Stuckelberg form of the minimal the non-linear gravity action in this formulation and show that the constraint structure is so rich that it is possible to eliminate non-physical modes. Then we extend this analysis to the case of the general non-linear massive gravity action. We find the corresponding Hamiltonian and collection of the primary constraints. Unfortunately we are not able to finish the complete analysis of the stability of all constraints due to the complex form of one primary constraint so that we are not able to determine the conditions under which given constraint is preserved during the time evolution of the system. Keywords: Massive Gravity.",
"pages": [
1
]
},
{
"title": "1. Introduction and Summary",
"content": "The first formulation of the massive gravity was performed by [1] at least in its linearized level as the propagation of the massive graviton above the flat background 1 . Even if the theory seems to be well defined at the linearized level there is a Boulware-Deser ghost [3] in the naive non-linear extension of the Fierz-Pauli formulation. On the other hand recently there has been much progress in the non-linear formulation of the massive gravity without the Boulware-Deser ghost [4, 5] and also [37, 38] 2 . The Hamiltonian treatment of non-linear massive gravity theory was performed in many papers with emphasis on the general proof of the absence of the ghosts in given theory. The first attempt for the analysis of the constraint structure of the non-linear massive gravity was performed in [47]. However it turned out that this analysis was not complete and the wrong conclusion was reached as was then shown in the fundamental paper [40] where the complete Hamiltonian analysis of the gauge fixed form of the general non-linear massive gravity was performed. The fundamental result of given paper is the proof of the existence of two additional constraints in the theory which are crucial for the elimination of non physical modes and hence for the consistency of the non-linear massive gravity at least at the classical level. Then the Hamiltonian analysis of the non-linear massive gravity in the Stuckelberg formulation was performed in [48, 49]. Unfortunately the wrong conclusion was again reached in the first versions of given papers as was then shown in [50] where the absence of the ghosts in the minimal version of non-linear massive gravity was proven for the first time. Then an independent proof of the absence of the ghosts in the minimal version of non-linear massive gravity in the Stuckelberg formulation was presented in [51]. However the proof of the absence of the ghosts in the general form of the non-linear massive gravity in the Stuckelberg formulation is still lacking. The difficulty with the possible Hamiltonian formulation of given theory is that the action depends on the kinetic terms of the Stuckelberg fields in a highly non-linear way so that it seems to be impossible to find an explicit relation between canonical conjugate momenta and the time derivatives of the Stuckelberg fields. On the other hand there exists the formulation of the non-linear massive gravity action with the linear dependence on the kinetic term for the Stuckelberg fields. This is the form of the non-linear massive gravity action that arises from the original one when the redefinition of the shift functions is performed [38, 37, 41, 42]. The goal of this paper is to perform the Hamiltonian analysis of given action and try to identify all constraints. As we argued above the main advantage of the formulation of the non-linear massive gravity with redefined shift function is that the kinetic term for the Stuckelberg fields appears linearly and hence it is easy to find the corresponding Hamiltonian even for the general form of the non-linear massive gravity action. Further it is possible to identify four primary constraints of the theory where the three ones are parts of the generators of the spatial diffeomorphism. Note that the presence of the diffeomorphism constraints is the reflection of the fact that we have manifestly diffeomorphism invariant theory. On the other hand the fourth primary constraint could be responsible for the elimination of the additional non physical mode. However this claim is only true when the requirement of the preservation of given constraint during the time development of the system generates another additional constraint. Unfortunately we find that the original primary constraint cannot provide such additional constraint due to the fact that the Poisson bracket between the primary constraints defined at different space points is non zero. For that reason we should find another constraint that obeys the property that Poisson bracket between these constraints defined at different space points is zero. We find such a constraint in the case of the minimal non-linear massive gravity action and we show that this constraint has the same form as the primary constraint found in [51]. Then we will be able to show that the requirement of the preservation of given constraint during the time evolution of the system implies the additional constraint and these two constraints together allow to eliminate two non-physical modes. This result agrees with previous two independent analysis performed in [50] and in [51]. Unfortunately we are not able to reach the main goal of this paper which is the proof of the absence of the ghosts for the general non-linear massive gravity theory in Stuckelberg formalism. The reason is that we are not able to find the primary constraint that has vanishing Poisson bracket between these constraints defined at different space points and that has the Poisson brackets with another constraints that vanish on the constraint surface. This is crucial condition for the existence of the additional constraint. It is rather worrying that we are not able to finish the Hamiltonian analysis for the general non-linear massive gravity action especially in the light of the very nice proof of the absence of the ghosts in case of the gauge fixed non-linear massive gravity action [40]. However there is a possibility that the proof of the absence of the ghosts for general non-linear massive gravity action in Stuckelberg formalism could be found in the very elegant formulation of the massive and multi metric theories of gravity presented in [23]. We hope to return to this problem in future. The structure of this paper is as follows. In the next section (2) we introduce the non-linear massive gravity action and perform the field redefinition of the shift function. We also we find the Hamiltonian formulation of the minimal form of the non-lineal massive gravity in Stuckelberg formulation with redefined shift functions and we find that given theory is free from the ghosts. Then we extend this approach to the case of the general non-linear massive gravity theory in the section ( ?? ). We identify all primary constraints and discuss the difficulties that prevent us to finish the complete Hamiltonian analysis.",
"pages": [
2,
3,
4
]
},
{
"title": "2. Non-linear Massive Gravity with Redefined Shift Functions",
"content": "As we stressed in the introduction section the goal of this paper is to perform the Hamiltonian analysis of the general non-linear massive gravity with presence of the Stuckelberg fields. It turns out that it is useful to consider this action with redefined shift functions [38, 37, 41, 42]. More explicitly, let us begin with following general form of the non-linear massive gravity action where e k ( A ) are elementary symmetric polynomials of the eigenvalues of A . For generic 4 × 4 matrix they are given by where A µ ν is 4 × 4 matrix and where Of the four β n two combinations are related to the mass and the cosmological constant while the remaining two combinations are free parameters. If we consider the case when the cosmological constant is zero and the parameter m is mass, the four β n are parameterized in terms of the α 3 and α 4 as [5] The minimal action corresponds to β 2 = β 3 = 0 that implies α 3 = α 4 = 1 and consequently β 0 = 3 , β 1 = -1. We consider the massive gravity with that is manifestly diffeomorphism invariant. This can be ensured with the help of 4 scalar fields φ A , A = 0 , 1 , 2 , 3 so that Then we have where we also used 3 + 1 decomposition of the four dimensional metric ˆ g µν [45, 46] Let us now perform the redefinition of the shift function N i that was introduced in [37, 38, 41, 42] where and where we defined f ij as the inverse to f ij in the sense 3 Finally note that the matrix ˜ D i j obeys the equation and also following important identity Let us now concentrate on the minimal form of the non-linear massive gravity action. Using the redefinition (2.8) we find that it takes the form where and where we used the 3 + 1 decomposition of the four dimensional scalar curvature where R is three dimensional scalar curvature and where with inverse Note that in (2.15) we ignored the terms containing total derivatives. Finally note that ˜ K ij is defined as where N i depends on ˜ n i and g through the relation (2.8). At this point we should stress the reason why we consider the non-linear massive gravity action in the form (2.13). The reason is that we want to perform the Hamiltonian analysis for the general non-linear massive gravity action written in the Stuckelberg formalism. It turns out that the action (2.13) has formally the same form as in case of the general nonlinear massive gravity action when we replace U with more general form whose explicit form was determined in [37, 38, 41, 42]. On the other hand the main advantage of the action (2.13) is that it depends on the time derivatives of φ A through the term M and that this term appears linearly in the action (2.13). We should compare this fact with the original form of the non-linear massive gravity action where the dependence on the time derivatives of φ A is highly non-linear and hence it is very difficult to find corresponding Hamiltonian. Explicitly, from (2.13) we find the momenta conjugate to N, ˜ n i and g ij and the momentum conjugate to φ A (2.20) where It turns out that it is useful to write M 2 in the form (2.22) where by definition the matrix M AB obeys following relations together with With the help of these results we find and consequently which however implies following primary constraint Note that using (2.24) we obtain another set of the primary constraints (2.28) Observe that using (2.28) we can write so that we can rewrite Σ p into the form where H i are functions of the phase space variables. As a result we see that it is natural to consider following independent constraint Σ p We return to the analysis of the constraint Σ p below. Now we are ready to write the extended Hamiltonian which includes all the primary constraints where and where we introduced the constraints ˜ Σ i defined as Note that ˜ Σ i is defined as linear combination of the constraints Σ i ≈ 0 together with the constraints π i ≈ 0. To proceed further we have to check the stability of all constraints. To do this we have to calculate the Poisson brackets between all constraints and the Hamiltonian H E . Note that we have following set of the canonical variables g ij , π ij , φ A , p A , ˜ n i , π i and N,π N with non-zero Poisson brackets Now we show that the smeared form of the constraint ˜ Σ i ≈ 0 is the generator of the spatial diffeomorphism. First of all using (2.35) we find which is the correct transformation rule for ˜ n i . Then using (2.35) we find that are the correct transformation rules of the canonical variables under spatial diffeomorphism. To proceed further we need the Poisson bracket between T S ( N i ) and ˜ D i j . It turns out that it is convenient to know the explicit form of the matrix ˜ D i j [37, 38, 41, 42] where Using this expression we can easily determine the Poisson brackets between T S ( N i ) and ˜ D k j . In fact, by definition we have Using (2.39) and the results derived above we find Collecting all these results and after some calculations we find Note also that it is easy to show that following Poisson bracket holds Now we are ready to analyze the stability of all primary constraints. As usual the requirement of the preservation of the constraint π N ≈ 0 implies an existence of the secondary constraint C 0 ≈ 0. However the fact that C 0 is the constraint immediately implies that the constraint ˜ Σ i ≈ 0 is preserved during the time evolution of the system, using (2.43) and (2.44). Now we analyze the requirement of the preservation of the constraints π i ≈ 0 during the time evolution of the system It turns out that the following matrix cannot be solved for Ω p and hence we have to demand the existence of following secondary constraints [37, 38, 41, 42] Finally we have to proceed to the analysis of the time development of the constraint Σ p ≈ 0. However it turns out that it is very difficult to perform this analysis for Σ p due to the presence of the terms that contain the spatial derivatives of φ A . Then the explicit calculation gives { Σ p ( x ) , Σ p ( y ) } /negationslash = 0. For that reason we proceed in a different way when we try to simplify the constraint Σ p . Using C i and Σ i we find that where and where F i , G i are the phase space functions whose explicit form is not important for us. Then with the help of (2.48) we express ˜ n i f ij ˜ n j as a function of the phase space variables p A , φ A and g ij , π ij In the same way we obtain and where again ˜ F i , ˜ G i are phase space functions whose explicit form is not needed for us. Now using these results we find that the constraint Σ p takes the form where we introduced new independent constraint ˜ Σ p that has precisely the same form as in [51]. Note that the constraint ˜ Σ p has the desired property that { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. As a result we see that it is more natural to consider ˜ Σ p instead of Σ p as an independent constraint. Then the total Hamiltonian, where we include all constraints, takes the form Now we are ready to analyze the stability of all constraints that appear in (2.55). First of all we find that π N ≈ 0 is automatically preserved while the preservation of the constraint π i ≈ 0 gives By definition /negationslash and hence the matrix /triangle π i , C j is non-singular. Then the only solution of the equation (2.56) is Γ i = 0. As the next step we proceed to the analysis of the stability of the constraint ˜ Σ p . As is clear from (2.54) we have Then the time evolution of given constraint takes the form using the fact that ˜ Σ p does not depend on ˜ n i together with Γ i = 0 and also the fact that ˜ Σ p is manifestly diffeomorphism invariant. In order to explicitly determine (2.59) we need following expression Then after some calculations we obtain In order to simplify Σ II p further we use (2.50), (2.51) together with (2.52) to make Σ II p independent on ˜ n i . In fact, using (2.39)(2.40) we obtain up to terms proportional to the constraints C i , Σ i . With the help of these results we obtain /negationslash where ˜ x and ˜ D i j are functions of p A , ∂ j φ A and g through the relations (2.50),(2.51) and (2.52). Now we see from (2.59) that the time evolution of the constraint ˜ Σ p ≈ 0 is obeyed on condition when either N = 0 or when ˜ Σ II p = 0. Note that we should interpreted N as the Lagrange multiplier so that it is possible to demand that N = 0 on condition when ˜ Σ II p = 0 on the whole phase space. Of course such a condition is too strong so that it is more natural to demand that ˜ Σ II p ≈ 0 and N = 0. In other words ˜ Σ II p ≈ 0 is the new secondary constraint. /negationslash In summary we have following collection of constraints: π N ≈ 0 , π i ≈ 0 , C 0 ≈ 0 , C i ≈ 0 , ˜ Σ i ≈ 0 , ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The dynamics of these constraints is governed by the total Hamiltonian As the final step we have to analyze the preservation of all constraints. The case of π N ≈ 0 is trivial. For π i ≈ 0 we obtain due to the crucial fact that ˜ Σ II p does not depend on ˜ n i . This is the main reason why we introduced ˜ Σ II p instead of Σ II p . Then as we argued above the only solution of the equation is Γ i = 0. Now the time development of C i is given by the equation and the time development of the constraint ˜ Σ p is governed by the equation As follows from the explicit form of the constraint ˜ Σ II p we see that { ˜ Σ II p ( x ) , ˜ Σ p ( y ) } is non-zero and proportional also to the higher order derivatives of the delta functions. As a consequence we find that the only solution of the equation above is Ω II p = 0. Further we analyze the time evolution of the constraint ˜ Σ II p Now from the last equation we obtain Ω p as a function of the phase space variables and N , at least in principle. Then inserting this result into the equation for the preservation of C i (2.66) we determine v j as functions of the phase space variables. Finally note also that the constraint C 0 is automatically preserved due to the fact that Γ i = Ω II p = 0 and also the fact that {C 0 ( x ) , C 0 ( y ) } ≈ 0 as was shown in [40]. In summary we obtain following picture. We have five the first class constraints π N ≈ 0 , C 0 ≈ 0 , ˜ Σ i ≈ 0 together with eight the second class constraints π i ≈ 0 , C i ≈ 0 and ˜ Σ p ≈ 0 , ˜ Σ II p ≈ 0. The constraints π i ≈ 0 together with C i ≈ 0 can be solved for π i and ˜ n i . Then the constraint ˜ Σ p can be solved for one of the four momenta p A while the constraint ˜ Σ II p can be solved for one of the four φ A . As a result we have 12 gravitational degrees of freedom g ij , π ij ,6 scalars degrees of freedom together with 4 first class constraints C 0 ≈ 0 , ˜ Σ i ≈ 0. Then we find that the number of physical degrees of freedom is 10 which is the correct number of physical degrees of freedom of the massive gravity.",
"pages": [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13
]
},
{
"title": "3. General Non-Linear Massive Gravity Action",
"content": "Let us try to apply the procedure performed in previous section to the case of the general non-linear massive gravity whose action takes the form where Following the analysis performed in the previous section we find the extended Hamiltonian in the form where and where the primary constraint Σ p takes the form As the next step we should analyze the stability of all primary constraints. As in previous section we find that the stability of the constraint π N ≈ 0 implies the secondary constraint C 0 ≈ 0 while the stability of the constraints π i implies set of the secondary constraints C i [37, 38, 41, 42] Now we come to the key point of the analysis which is the requirement of the preservation of the constraint Σ p ≈ 0 during the time evolution of the system. This is very complicated expression which depends on the all phase space variables. Remember that in the minimal case we expressed ˜ n i as functions of g ij , p A and φ A . As a result we found that Σ p can be expressed as a linear combination of C i , Σ i and ˜ Σ p where ˜ Σ p obeys an important property { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0.It would be certainly nice to repeat the same procedure in the case of the constraint Σ p given in (3.5). Unfortunately we are not able to solve the constraint C i in order to express ˜ n i as a function of R i . Consequently we are not able to express the constraint Σ p as a linear combination of the constraints C i and possibly ˜ Σ i , C 0 and the new constraint ˜ Σ p where { ˜ Σ p ( x ) , ˜ Σ p ( y ) } = 0. In other words, despite of the fact we are able to find the primary constraint Σ p ≈ 0 we are not able to determine the additional secondary constraint which is necessary for the elimination of two non-physical phase space modes. As a result the main goal of this paper which was the proof of the absence of the ghosts in the general non-linear massive gravity action in Stuckelberg formalism cannot be completed. Acknowledgement: This work was supported by the Grant agency of the Czech republic under the grant P201/12/G028.",
"pages": [
13,
14
]
}
]
2013IJMPB..2750119B
https://arxiv.org/pdf/1212.0368.pdf
<document>
<section_header_level_1><location><page_1><loc_19><loc_73><loc_78><loc_76></location>MEASUREMENT OF THE THERMAL EXPANSION COEFFICIENT OF AN Al-Mg ALLOY AT ULTRA-LOW TEMPERATURES</section_header_level_1>
<text><location><page_1><loc_22><loc_66><loc_75><loc_70></location>M. BASSAN b,c , B. BUONOMO a , G. CAVALLARI e , E. COCCIA b,c , S. D'ANTONIO b , V. FAFONE b,c , L.G. FOGGETTA a , C. LIGI a ∗ , A. MARINI a , G. MAZZITELLI a , G. MODESTINO a , G. PIZZELLA c,a , L. QUINTIERI a † , F. RONGA a , P. VALENTE d</text>
<text><location><page_1><loc_26><loc_53><loc_71><loc_66></location>a Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Via E. Fermi, 40 - 00044 Frascati, Italy b Istituto Nazionale di Fisica Nucleare - Sezione Roma2, Via della Ricerca Scientifica - 00133 Rome, Italy c Dipartimento di Fisica, Universit'a di Tor Vergata, Via della Ricerca Scientifica - 00133 Rome, Italy d Istituto Nazionale di Fisica Nucleare - Sezione Roma1, Piazzale Aldo Moro 2 - 00185 Rome, Italy e CERN, CH1211, Gen'eve, Switzerland carlo.ligi@lnf.infn.it</text>
<text><location><page_1><loc_22><loc_37><loc_75><loc_50></location>We describe a result coming from an experiment based on an Al-Mg alloy ( ∼ 5% Mg) suspended bar hit by an electron beam and operated above and below the termperature of transition from superconducting to normal state of the material. The amplitude of the bar first longitudinal mode of oscillation, excited by the beam interacting with the bulk, and the energy deposited by the beam in the bar are the quantities measured by the experiment. These quantities, inserted in the equations describing the mechanism of the mode excitation and complemented by an independent measurement of the specific heat, allow us to determine the linear expansion coefficient α of the material. We obtain α = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 for the normal state of conduction in the temperature interval 0 . 9 < T < 2 K and α = [( -2 . 45 ± 0 . 60)+( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 for the superconducting state in the interval 0 . 3 < T < 0 . 8 K.</text>
<text><location><page_1><loc_22><loc_35><loc_62><loc_36></location>Keywords : Thermal expansion; Low temperature; Aluminum alloy.</text>
<section_header_level_1><location><page_1><loc_19><loc_31><loc_32><loc_32></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_19><loc_22><loc_78><loc_30></location>Wrought aluminum-magnesium alloys (International Alloy Designation System: 5XXX, Mg content: 5.6% maximum) are commonly used in applications where good workability, very good resistance to corrosion, high fatigue strength are desidered. Example of applications are: oil, fuel lines and tanks, pressure cryogenic vessels, marine structures and fittings, automotive and architectural components.</text>
<text><location><page_1><loc_19><loc_15><loc_78><loc_21></location>The alloy Al 5056 (5.2% Mg, 0.1% Mn, 0.1% Cr) is the material of the test mass of gravitational wave (GW) resonant antennas AURIGA 1 , EXPLORER 2 , NAUTILUS 3 in Italy and ALLEGRO 4 in the United States, operated at liquid Helium temperature and below. During the last decade, all these detectors took</text>
<text><location><page_2><loc_19><loc_78><loc_78><loc_80></location>part, as a network, in coordinated searches for impulsive GW excitations (no GW events were detected). 5 , 6</text>
<text><location><page_2><loc_19><loc_66><loc_78><loc_77></location>The test mass of the GW resonant antennas is a suspended cylinder ( ∼ 2300 kg mass), acting as a mechanical resonator whose resonances are excited by the incoming GW. The elastic vibrations are converted to electrical signals by a transducer system. The minimum detectable energy of a GW resonant antenna inversely depends on Q , the acoustic quality factor of the material also known as the inverse of the internal friction. Al 5056 was chosen as the material for the GW antennas due to the high values of Q , 7 exhibited at very low temperatures.</text>
<text><location><page_2><loc_19><loc_55><loc_78><loc_66></location>B.L. Baron and R. Hofstadter first measured 8 mechanical oscillations in piezoelectric disks penetrated by high energy electron beams and they outlined the possibility that cosmic ray (CR) events could excite mechanical vibrations in a metallic cylinder at its resonant frequency and that these events could mimic GW events detected by resonant antennas. NAUTILUS has been equipped with a CR detector 9 to study the interactions caused by CRs and to provide a veto against the CR induced events in the antenna.</text>
<text><location><page_2><loc_19><loc_45><loc_78><loc_54></location>Correlating NAUTILUS data with CR observations, the following results were obtained: 1) the rate of high energy signals due to CR showers was larger than expectations 10 , 11 when the antenna was operated at a temperature T = 0 . 14 K, i.e. well below the transition temperature from normal-conducting ( n ) to superconducting ( s ) states of Al 5056; 12 2) this feature was not observed when the antenna was operated at T = 1 . 5 K, i.e. above the transition temperature. 13</text>
<text><location><page_2><loc_19><loc_13><loc_78><loc_44></location>Since these findings had not a straightforward interpretation, we performed an experiment (RAP) aimed at measuring the longitudinal oscillations of a small suspended Al 5056 cylinder hit by an electron beam of known energy. The RAP experiment, performed at the Beam Test Facility (BTF) 14 of the DAFNE Φ-factory complex in the INFN Frascati Laboratory, has shown that 1) the amplitude of the longitudinal oscillations of the cylinder hit by ionizing particles depends on the state of conduction of the material 15 , 16 and that 2) the observed amplitudes are consistent with the amplitudes measured by NAUTILUS in CR events both in ( n ) and ( s ) states. 17 Moreover, the amplitude can be evaluated in the framework of the Thermo-Acoustic model: in the ( n ) state it depends on the ratio of the thermal expansion coefficient to the specific heat, this ratio being part of the definition of the Gruneisen parameter. In the ( s ) state the amplitude also depends on the fractional volume change between the ( s ) and ( n ) states. In principle, the relevant thermophysical properties of the Al 5056 at low temperature are not known with sufficient precision to allow the comparison between experimental and theoretical values of the amplitude predicted by the model. However, our confidence in the model is also based on the small discrepancy, of the order of 15%, between the amplitudes measured in the alloy and those computed by using the data for pure aluminum in the region 4.5 ≤ T ≤ 264 K. 18</text>
<text><location><page_2><loc_21><loc_12><loc_78><loc_13></location>Direct measurements of the Gruneisen parameter of elastic materials have been</text>
<text><location><page_3><loc_19><loc_71><loc_78><loc_81></location>reported: 19 the authors used pulsed electron beams hitting the front surface of a thin slab, and measured the induced motion on the rear face surface. The equation of motion describing thermoelastic effects depends on the Gruneisen parameter and on the energy deposited by the beam in the slab. The measurement of the displacement, the knowledge of the deposited energy and the use of the equation of motion allowed them to directly determine the Gruneisen parameter.</text>
<text><location><page_3><loc_19><loc_58><loc_78><loc_70></location>In a similar way this paper presents experimental results on the linear expansion coefficient of the aluminum alloy above and below the temperature of transition between the ( s ) and ( n ) state. In order to determine the coefficient we use: a) the oscillation amplitude measurements, b) the equations describing the amplitude in the framework of the Thermo-Acoustic model, c) the measurements of the deposited energy by the electron beam, obtained by the product of the measured beam multiplicity and the calculated energy loss in the material by each particle, and d) an independent measurement of the specific heat.</text>
<section_header_level_1><location><page_3><loc_19><loc_54><loc_45><loc_55></location>2. The Thermo-Acoustic model</section_header_level_1>
<section_header_level_1><location><page_3><loc_19><loc_51><loc_30><loc_53></location>2.1. ( n ) state</section_header_level_1>
<text><location><page_3><loc_19><loc_36><loc_78><loc_50></location>We consider a cylinder (radius R , length L , mass M ) that is suspended in correspondence of the middle section, with its axis horizontal. A ionizing particle, after hitting the lateral surface of the cylinder and interacting with the material, generates a pressure pulse in the bulk. This sonic pulse is the result of the local thermal expansion of the material caused by the warming up due to the energy lost by the particle in the material. The sonic pulse determines the excitation of the vibrational oscillation modes of the suspended bar. By introducing a vector field u ( x , t ) to describe the local displacements from equilibrium, we express the amplitude of the k -th longitudinal mode of oscillation as proportional to the quantity: 20 , 21 , 22</text>
<formula><location><page_3><loc_39><loc_27><loc_78><loc_35></location>g k,therm = ∆ P therm ρ A ' I k = γ ρ ∣ ∣ ∣ dW dl ∣ ∣ ∣ I k , (1)</formula>
<text><location><page_3><loc_19><loc_18><loc_78><loc_29></location>∣ ∣ where ∆ P therm is the pressure pulse due to the sonic source described above, ρ is the mass density, dW/dl is the energy loss per unit length of the interacting particle, A ' is the cross section of the tubular zone centered on the particle path in which the effects are generated and I k = ∫ dl ( ∇· u k ( x )) is a line integral over the particle path involving the normal mode of oscillation u k ( x ). In the previous relation γ is the adimensional Gruneisen parameter:</text>
<formula><location><page_3><loc_44><loc_14><loc_78><loc_17></location>γ = βK T ρc V , (2)</formula>
<text><location><page_3><loc_19><loc_10><loc_77><loc_13></location>with β the volume thermal expansion coefficient ( β = 3 α for cubic elements), c V the isochoric specific heat and K T the isothermal bulk module.</text>
<section_header_level_1><location><page_4><loc_19><loc_83><loc_29><loc_84></location>4 C. Ligi et al.</section_header_level_1>
<text><location><page_4><loc_19><loc_74><loc_78><loc_80></location>A thin bar ( R/L /lessmuch 1) hit by a ionizing particle at the center of the lateral surface is a particular case of the more general one that leads to relation (1). In this case, the fundamental longitudinal mode of oscillation is excited to a maximum amplitude given by: 23</text>
<formula><location><page_4><loc_43><loc_68><loc_78><loc_71></location>X = 2 αLW πc V M , (3)</formula>
<text><location><page_4><loc_19><loc_49><loc_78><loc_67></location>and then exponentially decays due to internal friction. Here W is the total energy released in the bar by the particle beam. The experimental conditions of RAP are close to this particular case and the amplitude maximum of the fundamental longitudinal mode of oscillation is the observable measured by the experiment. However, in order to model the amplitude in the most realistic way, we have performed a Monte Carlo (MC) simulation, 18 which takes into account the corrections O [( R/L ) 2 ] for the modes of oscillation of a cylinder, the transverse dimension of the electron beam at the impact point and the trajectories of the secondary particles generated by the electron interactions in the bar. All these effects are summarized in a corrective parameter /epsilon1 and the amplitude maximum of the fundamental longitudinal mode of oscillation is modeled by:</text>
<formula><location><page_4><loc_42><loc_44><loc_78><loc_46></location>X ( n ) = X (1 + /epsilon1 ) (4)</formula>
<text><location><page_4><loc_19><loc_40><loc_78><loc_43></location>The value of /epsilon1 for the aluminum alloy bar used in the experiment is estimated by MC to be -0.04.</text>
<section_header_level_1><location><page_4><loc_19><loc_35><loc_30><loc_36></location>2.2. ( s ) state</section_header_level_1>
<text><location><page_4><loc_19><loc_26><loc_78><loc_34></location>The energy released by the ionizing particles to the material determines the suppression of the superconductivity in a region (Hot Spot) centered around the particle path. The maximum value of the Hot Spot radius, r HS , is obtained by equating the energy lost per unit length by the particle, dW/dl , to the enthalpy variation in the volume undergoing the ( s ) → ( n ) transition at temperature T : 24 , 25 , 26</text>
<formula><location><page_4><loc_38><loc_21><loc_78><loc_25></location>r HS = √ | dW/dl | π ∆ h = √ A '' π , (5)</formula>
<text><location><page_4><loc_19><loc_17><loc_78><loc_20></location>where ∆ h is the enthalpy change per unit volume and A '' is the cross section of the zone switched to the ( n ) state.</text>
<text><location><page_4><loc_19><loc_10><loc_78><loc_17></location>The amplitude of the longitudinal oscillations is the sum of two terms, one related to thermodynamic effects in the ( s ) → ( n ) transition and the other to the thermal effects in the region switched to the ( n ) state, which have been already described in the Section 2.1. The contribution to the amplitude of the cylinder k -th oscillation</text>
<text><location><page_5><loc_19><loc_78><loc_63><loc_81></location>mode due to the ( s ) → ( n ) transition is proportional to: 20 , 21</text>
<formula><location><page_5><loc_34><loc_71><loc_63><loc_78></location>g k,trans = ∆ P trans ρ A '' I k = 1 ρ [ K T ∆ V V + γT ∆ S V ] A '' I k ,</formula>
<text><location><page_5><loc_19><loc_66><loc_78><loc_71></location>where V = V ( s ) and ∆ V , ∆ S are the differences of the volume and the entropy in the two states of conduction. According to the thermodynamics of volume and pressure effects for Type-II superconductors, these differences can be written as a : 27</text>
<formula><location><page_5><loc_33><loc_62><loc_78><loc_65></location>∆ V = V ( n ) -V ( s ) = V H c 4 π ∂H c ∂P + { H 2 c 8 π ∂V ∂P } (6)</formula>
<text><location><page_5><loc_19><loc_60><loc_22><loc_61></location>and</text>
<formula><location><page_5><loc_32><loc_54><loc_78><loc_58></location>∆ S = S ( n ) -S ( s ) = -V H c 4 π ∂H c ∂T -{ H 2 c 8 π ∂V ∂T } (7)</formula>
<text><location><page_5><loc_19><loc_47><loc_78><loc_53></location>Here H c is the superconducting critical magnetic field that is supposed to have the parabolic behavior H c ( t ) = H c (0)(1 -t 2 ), where t = T/T c and T c is the transition temperature. Moreover, the terms in { } brackets are expected to be smaller than the preceding terms 28 and are usually ignored for practical purposes.</text>
<text><location><page_5><loc_19><loc_44><loc_78><loc_47></location>Finally, the knowledge of the specific heat of the material for the ( s ) state, c ( s ) , allows us to compute the variation of the entalpy per unit volume in the form: 29</text>
<formula><location><page_5><loc_40><loc_39><loc_78><loc_42></location>∆ h = C int ( T ) + T ∆ S V (8)</formula>
<text><location><page_5><loc_19><loc_37><loc_23><loc_38></location>with:</text>
<formula><location><page_5><loc_39><loc_33><loc_58><loc_36></location>C int ( T ) = ∫ T c T c ( s ) ( T ' ) dT '</formula>
<text><location><page_5><loc_19><loc_24><loc_78><loc_32></location>The amplitude maximum of the fundamental longitudinal mode of oscillation in the ( s ) state, as already indicated, is the sum of two terms, one related to the transition and the other to the ( n ) contribution from the switched region: X ( s ) = X trans + X ( n ) . By observing that X trans /X ( n ) = g k =0 ,trans /g k =0 ,therm we can describe the amplitude maximum according to the following relation:</text>
<formula><location><page_5><loc_34><loc_15><loc_78><loc_22></location>X ( s ) = X ( n ) { 1 + X trans X ( n ) } = X ( n ) { 1 + [ Π ∆ V V + T ∆ S V ] [∆ h ] -1 } , (9)</formula>
<section_header_level_1><location><page_6><loc_19><loc_83><loc_29><loc_84></location>6 C. Ligi et al.</section_header_level_1>
<text><location><page_6><loc_19><loc_79><loc_67><loc_80></location>where the relation (4) and the definition (2) of γ are used to define:</text>
<formula><location><page_6><loc_41><loc_75><loc_55><loc_78></location>Π = 2 ρL (1 + /epsilon1 ) 3 πM ( X ( n ) /W )</formula>
<text><location><page_6><loc_19><loc_69><loc_78><loc_74></location>We note that the term contained in { } brackets in the relation (9) is independent from W , the energy released by the particle to the bar, and that X ( s ) linearly depends on W through X ( n ) .</text>
<section_header_level_1><location><page_6><loc_19><loc_65><loc_50><loc_66></location>3. Experimental setup and procedures</section_header_level_1>
<text><location><page_6><loc_19><loc_24><loc_78><loc_64></location>The experiment mechanical layout, its cryogenic setup and operations, the electron beam characteristics, the instrumentation and the procedures for calibrations, data taking and analysis have been fully described in Refs. 17-18. Briefly, the dimensions and the mass of the Al 5056 cylindrical bar are R = 0 . 091 m, L = 0 . 500 m, M = 34 . 1 kg, respectively. The bar hangs from the cryostat top by means of a multistage suspension system providing a 150 dB attenuation of the external mechanical noise in the 1700-6500 Hz frequency window. The frequency of the fundamental longitudinal mode of oscillation of the bar is f 0 = 5413 . 6 Hz below T = 4 K. The cryostat is equipped with a dilution refrigerator. The temperatures are measured inside the cryostat by 11 thermometers controlled by two multi-channel resistance bridges and, among them, a calibrated RuO 2 resistor measures the temperature of one of the bar end faces with an accuracy of 0.01 K below T = 4 K. Two piezoelectric ceramics (Pz), electrically connected in parallel, are inserted in a slot cut in the position opposite to the bar suspension point and are squeezed when the bar shrinks. In this Pz arrangement the strain measured at the bar center is proportional to the displacement of the bar end faces. The Pz output is first amplified and then sampled at 100 kHz by an ADC embedded in a VME system dedicated to the data acquisition. A Pz calibration procedure, performed before each run of data taking, provides the factor converting the ADC samples into the displacements of the bar end faces. A software filtering algorithm, known as 'digital lock-in', extracts the Fourier component at the frequency f 0 from the time sequence formed by the ADC samples before and after the beam hit, determining the amplitude of the induced fundamental oscillation. The sign of the amplitude is taken positive or negative according to the sign of the first sample raising above the threshold after the beam hit.</text>
<text><location><page_6><loc_19><loc_11><loc_78><loc_23></location>The BTF beam line delivers to the bar single pulses of ∼ 10 ns duration, containing N e electrons of 510 ± 2 MeV energy. N e ranges from about 5 × 10 7 to 10 9 and is measured with an accuracy of ∼ 3% (for N e > 5 × 10 8 ) by an integrating current transformer placed close to the beam line exit point. The MC simulation, discussed in Section 2, estimates an average energy lost 〈 ∆ E 〉 ± σ ∆ E = 195 . 2 ± 70 . 6 MeV for a 512 MeV electron interacting in the bar and, consequently, the total energy loss per beam pulse is given by W = N e 〈 ∆ E 〉 , σ W = √ N e σ ∆ E . Two sources of error affect the vibration maximum amplitude: the first is an overall systematic error of</text>
<text><location><page_7><loc_19><loc_75><loc_78><loc_80></location>the order of ± 6%, that accounts for the slightly different set-up and calibration procedures implemented in the runs over two years and the second is related to the noise in the measurement of the oscillation amplitude ( ± 1 . 3 × 10 -13 m).</text>
<section_header_level_1><location><page_7><loc_19><loc_73><loc_56><loc_74></location>4. Linear expansion coefficient measurements</section_header_level_1>
<text><location><page_7><loc_19><loc_56><loc_78><loc_72></location>The coefficient α is determined by inserting the measured values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) , X ( s ) ) and the corresponding measurements of the deposited energy ( W ) into the relations modeling the amplitudes (equations (4),(9)). The measurement 30 at very low temperatures of the specific heat of an Al 5056 sample, of the same production batch of the RAP bar, is a fundamental ingredient for the coefficient determination both in ( n ) and ( s ) states. Moreover, the superconducting characterization of the material has shown that T c = 0 . 845 ± 0 . 002 K with a total transition width of about 0.1 K. 30 Thus we determine α for the two states of conduction ( α ( n ) , α ( s ) ) according to this temperature value.</text>
<section_header_level_1><location><page_7><loc_19><loc_52><loc_30><loc_54></location>4.1. ( n ) state</section_header_level_1>
<text><location><page_7><loc_19><loc_39><loc_78><loc_51></location>Values of the amplitude maximum of the fundamental longitudinal mode of oscillation and the related values of the deposited energies by the beam have been measured and analyzed in the temperature interval 0 . 9 ≤ T ≤ 2 K. 17 The values of X ( n ) show a strict linear correlation with the values of W (Fig. 1) and they have been fitted, including the errors, by a line constrained to the origin ( X ( n ) = p 0 W ). The slope of the fit, which has a χ 2 per degree of freedom equal to 0 . 45 / 15, is p 0 = (2 . 24 ± 0 . 05) × 10 -10 m J -1 . From the relation (4) it follows that:</text>
<formula><location><page_7><loc_43><loc_35><loc_78><loc_38></location>α c V = πMp 0 2 L (1 + /epsilon1 ) (10)</formula>
<text><location><page_7><loc_19><loc_27><loc_78><loc_35></location>In the same temperature interval the parametrization c V = CT + DT 3 , where C is the electronic specific heat coefficient per unit volume and D is the lattice contribution, is determined by 30 C = (4 . 382 ± 0 . 117) × 10 -2 J kg -1 K -2 and D = (5 . 20 ± 0 . 37) × 10 -3 J kg -1 K -4 . Inserting this c V parametrization in relation (10) directly gives:</text>
<formula><location><page_7><loc_32><loc_20><loc_78><loc_24></location>α ( n ) = α ( n ) ,e + α ( n ) ,/lscript = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 (11)</formula>
<text><location><page_7><loc_19><loc_16><loc_78><loc_20></location>for the linear expansion coefficient (Fig. 2) expressed in terms of the electronic and lattice components in the temperature interval 0 . 9 ≤ T ≤ 2 K.</text>
<text><location><page_7><loc_19><loc_11><loc_78><loc_17></location>We note that the value of α ( n ) for pure aluminum is reported 31 to be 2 . 7 × 10 -9 K -1 at T = 2 K. This value, which is the one associated to the lowest temperature in the previously cited work, can be compared with α ( n ) = (3 . 2 ± 0 . 2) × 10 -9 K -1 obtained in the present work for Al 5056 at the same temperature.</text>
<figure>
<location><page_8><loc_19><loc_39><loc_77><loc_80></location>
<caption>Fig. 1. Normal state. Values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) ) vs. the deposited energies (W) measured in the temperature interval 0 . 9 ≤ T ≤ 2 K. Data have been fitted by a straight line, constrained to zero, with slope (2 . 24 ± 0 . 05) × 10 -10 m J -1 .</caption>
</figure>
<section_header_level_1><location><page_8><loc_19><loc_29><loc_30><loc_30></location>4.2. ( s ) state</section_header_level_1>
<text><location><page_8><loc_19><loc_26><loc_78><loc_28></location>From eq. (9) we can derive the volume variation among the two states of conduction:</text>
<formula><location><page_8><loc_39><loc_21><loc_78><loc_24></location>∆ V V = 1 Π [ R ∆ h p 0 -T ∆ S V ] (12)</formula>
<text><location><page_8><loc_19><loc_10><loc_78><loc_20></location>where R = X ( s ) /W -p 0 , with p 0 introduced in Section 4.1. The volume variation depends on measured quantities ( X ( s ) , W, p 0 , c V , T, T c ) and calculated ones. The latter are related to the critical field H c ( T = 0) and to its parabolic dependance on the temperature, which also allows us to compute the derivative ∂H c /∂T . If the superconducting properties of Al 5056 can be described by the BCS theory, then H c ( T = 0) ≈ 2 . 42 √ CT c ≈ 70 Oe, C being the electronic specific heat coefficient</text>
<section_header_level_1><location><page_9><loc_52><loc_83><loc_78><loc_84></location>Measurement of the thermal expansion 9</section_header_level_1>
<figure>
<location><page_9><loc_20><loc_39><loc_77><loc_80></location>
<caption>Fig. 2. Normal state. Measured values of the linear expansion coefficient ( α ( n ) ) vs. temperature (T). The thinner lines enclose the region compatible with the uncertainties.</caption>
</figure>
<text><location><page_9><loc_19><loc_10><loc_78><loc_32></location>per unit volume (Section 4.1), here expressed in suitable cgs units (erg cm -3 K -2 ). As already mentioned, the model predicts that the amplitude maximum of the fundamental longitudinal mode of oscillation for both states of conduction ( X ( n ) , X ( s ) ) linearly depends on the energy deposited by the e -beam in the bar. On the contrary, the X ( s ) values for Al 5056 15 , 17 show an increasing deviation from linearity with the increase of the deposited energy W , due to saturation effects discussed in Ref. 17. However, if we restrict our analysis to the data gathered at the lowest deposited energies, these non linearities can safely be neglected, and we can use the relation (12) to compute the volume variation. Fig. 3 shows the values of ∆ V/V versus the temperature T for the data with released energy: 1 . 5 × 10 -3 < W < 9 × 10 -3 J. ∆ V/V data have been fitted by a 2nd order polynomial q 0 + q 1 T + q 2 T 2 obtaining q 0 = ( -1 . 737 ± 0 . 054) × 10 -8 , q 1 = (7 . 334 ± 1 . 785) × 10 -9 K -1 , q 2 = (1 . 766 ± 0 . 187) × 10 -8 K -2 with a χ 2 = 1 . 38</text>
<text><location><page_10><loc_19><loc_79><loc_47><loc_81></location>normalized over 94 degrees of freedom.</text>
<figure>
<location><page_10><loc_19><loc_33><loc_77><loc_75></location>
<caption>Fig. 3. Superconducting state. Measured values of ∆ V/V vs. temperature for beam hits deposing in the bar an energy W < 9 mJ. The thick dotted line is a fit to the data with a 2nd order polynomial. The thinner lines enclose a region compatible with the uncertainties in the fit parameters.</caption>
</figure>
<text><location><page_10><loc_19><loc_11><loc_78><loc_23></location>The use of q 0 and of the relation (6) gives ∂H c /∂P = ( -3 . 12 ± 0 . 10) × 10 -12 T Pa -1 at T = 0 K, a value that can be compared with ( -2 . 67 ± 0 . 06) × 10 -12 T Pa -1 obtained by Harris and Mapother in their experimental study of the critical field of pure aluminum as a function of pressure and temperature. 32 The difference of the thermal expansion coefficients in the normal and superconducting phases is obtained by taking the derivative with respect to T of the difference of volumes: 27 ∆ α = α ( n ) -α ( s ) = 1 3 d dT ∆ V V .</text>
<text><location><page_10><loc_19><loc_10><loc_78><loc_12></location>Under the hypothesis that α ( n ) is still represented by the relation (11) in the inter-</text>
<text><location><page_11><loc_19><loc_78><loc_78><loc_80></location>al 0 . 3 K < T < T c and by using the polynomial fitting for ∆ V/V , the following expression is obtained for that temperature interval:</text>
<formula><location><page_11><loc_35><loc_70><loc_78><loc_76></location>α ( s ) = α ( n ) -∆ α = [( -2 . 45 ± 0 . 60) + ( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 (13)</formula>
<figure>
<location><page_11><loc_20><loc_27><loc_77><loc_68></location>
<caption>Fig. 4. Superconducting state. Values of α ( s ) vs. reduced temperature ( t = T/T c ) for Al 5056 (solid line). For reference, we also show the result for pure Al (dotted line) derived from data of Ref. 32. The thinner lines enclose the regions compatible with the uncertainties in the α ( s ) values.</caption>
</figure>
<text><location><page_11><loc_19><loc_10><loc_78><loc_18></location>This result can be compared with the expansion coefficient of pure aluminum in the ( s ) state, using for pure aluminum the relation α ( s ) = α ( s ) ,e + α /lscript , in which it is assumed that the lattice component does not depend on the conduction state and that its values are given in Ref. 33. The definition (2) gives α ( s ) ,e = ρc V,e ( s ) γ ( s ) ,e / (3 K T ), where the electronic component of c V in the ( s ) state ( c V,e ( s ) ) is taken from the</text>
<text><location><page_12><loc_19><loc_76><loc_78><loc_81></location>work of Phillips, 34 the electronic component of the Gruneisen parameter in the ( s ) state ( γ ( s ) ,e ) from Ref. 35 and K ( s ) T ∼ K ( n ) T = 79 . 4 × 10 9 Pa at low temperatures. 36 The comparison is shown in Fig. 4 as a function of the reduced temperature t .</text>
<section_header_level_1><location><page_12><loc_19><loc_71><loc_31><loc_73></location>5. Conclusions</section_header_level_1>
<text><location><page_12><loc_19><loc_51><loc_78><loc_70></location>In this paper we present an evaluation of the linear expansion coefficient of an AlMg alloy obtained by measuring both the amplitude of the fundamental mode of the longitudinal oscillation excited by electrons interacting in a suspended bar and the energy released in the bar by the electron pulse. The use of this method to determine α is new, although similarities exist with the direct measurement of the the Gruneisen parameter performed by hitting thin slabs with particle beams to generate thermoelastic pulses. No expansion data for aluminum alloys were previously available in the literature for the temperature range 0 . 3 < T < 2 K explored by this experiment. The expansion coefficient is negative in the superconducting state and its absolute value just below T c is larger by an order of magnitude than the value above this temperature. This feature is expected also for pure aluminum, according to an analysis 35 of the measured values of H c ( P, T ) available in Ref. 32.</text>
<text><location><page_12><loc_19><loc_46><loc_78><loc_49></location>This work was partially supported by the EU FP-6 Project ILIAS (RII3-CT-2004506222).</text>
<section_header_level_1><location><page_12><loc_19><loc_40><loc_28><loc_41></location>References</section_header_level_1>
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</document>
[
{
"title": "MEASUREMENT OF THE THERMAL EXPANSION COEFFICIENT OF AN Al-Mg ALLOY AT ULTRA-LOW TEMPERATURES",
"content": "M. BASSAN b,c , B. BUONOMO a , G. CAVALLARI e , E. COCCIA b,c , S. D'ANTONIO b , V. FAFONE b,c , L.G. FOGGETTA a , C. LIGI a ∗ , A. MARINI a , G. MAZZITELLI a , G. MODESTINO a , G. PIZZELLA c,a , L. QUINTIERI a † , F. RONGA a , P. VALENTE d a Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Via E. Fermi, 40 - 00044 Frascati, Italy b Istituto Nazionale di Fisica Nucleare - Sezione Roma2, Via della Ricerca Scientifica - 00133 Rome, Italy c Dipartimento di Fisica, Universit'a di Tor Vergata, Via della Ricerca Scientifica - 00133 Rome, Italy d Istituto Nazionale di Fisica Nucleare - Sezione Roma1, Piazzale Aldo Moro 2 - 00185 Rome, Italy e CERN, CH1211, Gen'eve, Switzerland carlo.ligi@lnf.infn.it We describe a result coming from an experiment based on an Al-Mg alloy ( ∼ 5% Mg) suspended bar hit by an electron beam and operated above and below the termperature of transition from superconducting to normal state of the material. The amplitude of the bar first longitudinal mode of oscillation, excited by the beam interacting with the bulk, and the energy deposited by the beam in the bar are the quantities measured by the experiment. These quantities, inserted in the equations describing the mechanism of the mode excitation and complemented by an independent measurement of the specific heat, allow us to determine the linear expansion coefficient α of the material. We obtain α = [(10 . 9 ± 0 . 4) T +(1 . 3 ± 0 . 1) T 3 ] × 10 -10 K -1 for the normal state of conduction in the temperature interval 0 . 9 < T < 2 K and α = [( -2 . 45 ± 0 . 60)+( -10 . 68 ± 1 . 24) T +(0 . 13 ± 0 . 01) T 3 ] × 10 -9 K -1 for the superconducting state in the interval 0 . 3 < T < 0 . 8 K. Keywords : Thermal expansion; Low temperature; Aluminum alloy.",
"pages": [
1
]
},
{
"title": "1. Introduction",
"content": "Wrought aluminum-magnesium alloys (International Alloy Designation System: 5XXX, Mg content: 5.6% maximum) are commonly used in applications where good workability, very good resistance to corrosion, high fatigue strength are desidered. Example of applications are: oil, fuel lines and tanks, pressure cryogenic vessels, marine structures and fittings, automotive and architectural components. The alloy Al 5056 (5.2% Mg, 0.1% Mn, 0.1% Cr) is the material of the test mass of gravitational wave (GW) resonant antennas AURIGA 1 , EXPLORER 2 , NAUTILUS 3 in Italy and ALLEGRO 4 in the United States, operated at liquid Helium temperature and below. During the last decade, all these detectors took part, as a network, in coordinated searches for impulsive GW excitations (no GW events were detected). 5 , 6 The test mass of the GW resonant antennas is a suspended cylinder ( ∼ 2300 kg mass), acting as a mechanical resonator whose resonances are excited by the incoming GW. The elastic vibrations are converted to electrical signals by a transducer system. The minimum detectable energy of a GW resonant antenna inversely depends on Q , the acoustic quality factor of the material also known as the inverse of the internal friction. Al 5056 was chosen as the material for the GW antennas due to the high values of Q , 7 exhibited at very low temperatures. B.L. Baron and R. Hofstadter first measured 8 mechanical oscillations in piezoelectric disks penetrated by high energy electron beams and they outlined the possibility that cosmic ray (CR) events could excite mechanical vibrations in a metallic cylinder at its resonant frequency and that these events could mimic GW events detected by resonant antennas. NAUTILUS has been equipped with a CR detector 9 to study the interactions caused by CRs and to provide a veto against the CR induced events in the antenna. Correlating NAUTILUS data with CR observations, the following results were obtained: 1) the rate of high energy signals due to CR showers was larger than expectations 10 , 11 when the antenna was operated at a temperature T = 0 . 14 K, i.e. well below the transition temperature from normal-conducting ( n ) to superconducting ( s ) states of Al 5056; 12 2) this feature was not observed when the antenna was operated at T = 1 . 5 K, i.e. above the transition temperature. 13 Since these findings had not a straightforward interpretation, we performed an experiment (RAP) aimed at measuring the longitudinal oscillations of a small suspended Al 5056 cylinder hit by an electron beam of known energy. The RAP experiment, performed at the Beam Test Facility (BTF) 14 of the DAFNE Φ-factory complex in the INFN Frascati Laboratory, has shown that 1) the amplitude of the longitudinal oscillations of the cylinder hit by ionizing particles depends on the state of conduction of the material 15 , 16 and that 2) the observed amplitudes are consistent with the amplitudes measured by NAUTILUS in CR events both in ( n ) and ( s ) states. 17 Moreover, the amplitude can be evaluated in the framework of the Thermo-Acoustic model: in the ( n ) state it depends on the ratio of the thermal expansion coefficient to the specific heat, this ratio being part of the definition of the Gruneisen parameter. In the ( s ) state the amplitude also depends on the fractional volume change between the ( s ) and ( n ) states. In principle, the relevant thermophysical properties of the Al 5056 at low temperature are not known with sufficient precision to allow the comparison between experimental and theoretical values of the amplitude predicted by the model. However, our confidence in the model is also based on the small discrepancy, of the order of 15%, between the amplitudes measured in the alloy and those computed by using the data for pure aluminum in the region 4.5 ≤ T ≤ 264 K. 18 Direct measurements of the Gruneisen parameter of elastic materials have been reported: 19 the authors used pulsed electron beams hitting the front surface of a thin slab, and measured the induced motion on the rear face surface. The equation of motion describing thermoelastic effects depends on the Gruneisen parameter and on the energy deposited by the beam in the slab. The measurement of the displacement, the knowledge of the deposited energy and the use of the equation of motion allowed them to directly determine the Gruneisen parameter. In a similar way this paper presents experimental results on the linear expansion coefficient of the aluminum alloy above and below the temperature of transition between the ( s ) and ( n ) state. In order to determine the coefficient we use: a) the oscillation amplitude measurements, b) the equations describing the amplitude in the framework of the Thermo-Acoustic model, c) the measurements of the deposited energy by the electron beam, obtained by the product of the measured beam multiplicity and the calculated energy loss in the material by each particle, and d) an independent measurement of the specific heat.",
"pages": [
1,
2,
3
]
},
{
"title": "2.1. ( n ) state",
"content": "We consider a cylinder (radius R , length L , mass M ) that is suspended in correspondence of the middle section, with its axis horizontal. A ionizing particle, after hitting the lateral surface of the cylinder and interacting with the material, generates a pressure pulse in the bulk. This sonic pulse is the result of the local thermal expansion of the material caused by the warming up due to the energy lost by the particle in the material. The sonic pulse determines the excitation of the vibrational oscillation modes of the suspended bar. By introducing a vector field u ( x , t ) to describe the local displacements from equilibrium, we express the amplitude of the k -th longitudinal mode of oscillation as proportional to the quantity: 20 , 21 , 22 ∣ ∣ where ∆ P therm is the pressure pulse due to the sonic source described above, ρ is the mass density, dW/dl is the energy loss per unit length of the interacting particle, A ' is the cross section of the tubular zone centered on the particle path in which the effects are generated and I k = ∫ dl ( ∇· u k ( x )) is a line integral over the particle path involving the normal mode of oscillation u k ( x ). In the previous relation γ is the adimensional Gruneisen parameter: with β the volume thermal expansion coefficient ( β = 3 α for cubic elements), c V the isochoric specific heat and K T the isothermal bulk module.",
"pages": [
3
]
},
{
"title": "4 C. Ligi et al.",
"content": "A thin bar ( R/L /lessmuch 1) hit by a ionizing particle at the center of the lateral surface is a particular case of the more general one that leads to relation (1). In this case, the fundamental longitudinal mode of oscillation is excited to a maximum amplitude given by: 23 and then exponentially decays due to internal friction. Here W is the total energy released in the bar by the particle beam. The experimental conditions of RAP are close to this particular case and the amplitude maximum of the fundamental longitudinal mode of oscillation is the observable measured by the experiment. However, in order to model the amplitude in the most realistic way, we have performed a Monte Carlo (MC) simulation, 18 which takes into account the corrections O [( R/L ) 2 ] for the modes of oscillation of a cylinder, the transverse dimension of the electron beam at the impact point and the trajectories of the secondary particles generated by the electron interactions in the bar. All these effects are summarized in a corrective parameter /epsilon1 and the amplitude maximum of the fundamental longitudinal mode of oscillation is modeled by: The value of /epsilon1 for the aluminum alloy bar used in the experiment is estimated by MC to be -0.04.",
"pages": [
4
]
},
{
"title": "2.2. ( s ) state",
"content": "The energy released by the ionizing particles to the material determines the suppression of the superconductivity in a region (Hot Spot) centered around the particle path. The maximum value of the Hot Spot radius, r HS , is obtained by equating the energy lost per unit length by the particle, dW/dl , to the enthalpy variation in the volume undergoing the ( s ) → ( n ) transition at temperature T : 24 , 25 , 26 where ∆ h is the enthalpy change per unit volume and A '' is the cross section of the zone switched to the ( n ) state. The amplitude of the longitudinal oscillations is the sum of two terms, one related to thermodynamic effects in the ( s ) → ( n ) transition and the other to the thermal effects in the region switched to the ( n ) state, which have been already described in the Section 2.1. The contribution to the amplitude of the cylinder k -th oscillation mode due to the ( s ) → ( n ) transition is proportional to: 20 , 21 where V = V ( s ) and ∆ V , ∆ S are the differences of the volume and the entropy in the two states of conduction. According to the thermodynamics of volume and pressure effects for Type-II superconductors, these differences can be written as a : 27 and Here H c is the superconducting critical magnetic field that is supposed to have the parabolic behavior H c ( t ) = H c (0)(1 -t 2 ), where t = T/T c and T c is the transition temperature. Moreover, the terms in { } brackets are expected to be smaller than the preceding terms 28 and are usually ignored for practical purposes. Finally, the knowledge of the specific heat of the material for the ( s ) state, c ( s ) , allows us to compute the variation of the entalpy per unit volume in the form: 29 with: The amplitude maximum of the fundamental longitudinal mode of oscillation in the ( s ) state, as already indicated, is the sum of two terms, one related to the transition and the other to the ( n ) contribution from the switched region: X ( s ) = X trans + X ( n ) . By observing that X trans /X ( n ) = g k =0 ,trans /g k =0 ,therm we can describe the amplitude maximum according to the following relation:",
"pages": [
4,
5
]
},
{
"title": "6 C. Ligi et al.",
"content": "where the relation (4) and the definition (2) of γ are used to define: We note that the term contained in { } brackets in the relation (9) is independent from W , the energy released by the particle to the bar, and that X ( s ) linearly depends on W through X ( n ) .",
"pages": [
6
]
},
{
"title": "3. Experimental setup and procedures",
"content": "The experiment mechanical layout, its cryogenic setup and operations, the electron beam characteristics, the instrumentation and the procedures for calibrations, data taking and analysis have been fully described in Refs. 17-18. Briefly, the dimensions and the mass of the Al 5056 cylindrical bar are R = 0 . 091 m, L = 0 . 500 m, M = 34 . 1 kg, respectively. The bar hangs from the cryostat top by means of a multistage suspension system providing a 150 dB attenuation of the external mechanical noise in the 1700-6500 Hz frequency window. The frequency of the fundamental longitudinal mode of oscillation of the bar is f 0 = 5413 . 6 Hz below T = 4 K. The cryostat is equipped with a dilution refrigerator. The temperatures are measured inside the cryostat by 11 thermometers controlled by two multi-channel resistance bridges and, among them, a calibrated RuO 2 resistor measures the temperature of one of the bar end faces with an accuracy of 0.01 K below T = 4 K. Two piezoelectric ceramics (Pz), electrically connected in parallel, are inserted in a slot cut in the position opposite to the bar suspension point and are squeezed when the bar shrinks. In this Pz arrangement the strain measured at the bar center is proportional to the displacement of the bar end faces. The Pz output is first amplified and then sampled at 100 kHz by an ADC embedded in a VME system dedicated to the data acquisition. A Pz calibration procedure, performed before each run of data taking, provides the factor converting the ADC samples into the displacements of the bar end faces. A software filtering algorithm, known as 'digital lock-in', extracts the Fourier component at the frequency f 0 from the time sequence formed by the ADC samples before and after the beam hit, determining the amplitude of the induced fundamental oscillation. The sign of the amplitude is taken positive or negative according to the sign of the first sample raising above the threshold after the beam hit. The BTF beam line delivers to the bar single pulses of ∼ 10 ns duration, containing N e electrons of 510 ± 2 MeV energy. N e ranges from about 5 × 10 7 to 10 9 and is measured with an accuracy of ∼ 3% (for N e > 5 × 10 8 ) by an integrating current transformer placed close to the beam line exit point. The MC simulation, discussed in Section 2, estimates an average energy lost 〈 ∆ E 〉 ± σ ∆ E = 195 . 2 ± 70 . 6 MeV for a 512 MeV electron interacting in the bar and, consequently, the total energy loss per beam pulse is given by W = N e 〈 ∆ E 〉 , σ W = √ N e σ ∆ E . Two sources of error affect the vibration maximum amplitude: the first is an overall systematic error of the order of ± 6%, that accounts for the slightly different set-up and calibration procedures implemented in the runs over two years and the second is related to the noise in the measurement of the oscillation amplitude ( ± 1 . 3 × 10 -13 m).",
"pages": [
6,
7
]
},
{
"title": "4. Linear expansion coefficient measurements",
"content": "The coefficient α is determined by inserting the measured values of the amplitude maximum of the fundamental longitudinal mode of oscillation ( X ( n ) , X ( s ) ) and the corresponding measurements of the deposited energy ( W ) into the relations modeling the amplitudes (equations (4),(9)). The measurement 30 at very low temperatures of the specific heat of an Al 5056 sample, of the same production batch of the RAP bar, is a fundamental ingredient for the coefficient determination both in ( n ) and ( s ) states. Moreover, the superconducting characterization of the material has shown that T c = 0 . 845 ± 0 . 002 K with a total transition width of about 0.1 K. 30 Thus we determine α for the two states of conduction ( α ( n ) , α ( s ) ) according to this temperature value.",
"pages": [
7
]
},
{
"title": "4.1. ( n ) state",
"content": "Values of the amplitude maximum of the fundamental longitudinal mode of oscillation and the related values of the deposited energies by the beam have been measured and analyzed in the temperature interval 0 . 9 ≤ T ≤ 2 K. 17 The values of X ( n ) show a strict linear correlation with the values of W (Fig. 1) and they have been fitted, including the errors, by a line constrained to the origin ( X ( n ) = p 0 W ). The slope of the fit, which has a χ 2 per degree of freedom equal to 0 . 45 / 15, is p 0 = (2 . 24 ± 0 . 05) × 10 -10 m J -1 . From the relation (4) it follows that: In the same temperature interval the parametrization c V = CT + DT 3 , where C is the electronic specific heat coefficient per unit volume and D is the lattice contribution, is determined by 30 C = (4 . 382 ± 0 . 117) × 10 -2 J kg -1 K -2 and D = (5 . 20 ± 0 . 37) × 10 -3 J kg -1 K -4 . Inserting this c V parametrization in relation (10) directly gives: for the linear expansion coefficient (Fig. 2) expressed in terms of the electronic and lattice components in the temperature interval 0 . 9 ≤ T ≤ 2 K. We note that the value of α ( n ) for pure aluminum is reported 31 to be 2 . 7 × 10 -9 K -1 at T = 2 K. This value, which is the one associated to the lowest temperature in the previously cited work, can be compared with α ( n ) = (3 . 2 ± 0 . 2) × 10 -9 K -1 obtained in the present work for Al 5056 at the same temperature.",
"pages": [
7
]
},
{
"title": "4.2. ( s ) state",
"content": "From eq. (9) we can derive the volume variation among the two states of conduction: where R = X ( s ) /W -p 0 , with p 0 introduced in Section 4.1. The volume variation depends on measured quantities ( X ( s ) , W, p 0 , c V , T, T c ) and calculated ones. The latter are related to the critical field H c ( T = 0) and to its parabolic dependance on the temperature, which also allows us to compute the derivative ∂H c /∂T . If the superconducting properties of Al 5056 can be described by the BCS theory, then H c ( T = 0) ≈ 2 . 42 √ CT c ≈ 70 Oe, C being the electronic specific heat coefficient",
"pages": [
8
]
},
{
"title": "Measurement of the thermal expansion 9",
"content": "per unit volume (Section 4.1), here expressed in suitable cgs units (erg cm -3 K -2 ). As already mentioned, the model predicts that the amplitude maximum of the fundamental longitudinal mode of oscillation for both states of conduction ( X ( n ) , X ( s ) ) linearly depends on the energy deposited by the e -beam in the bar. On the contrary, the X ( s ) values for Al 5056 15 , 17 show an increasing deviation from linearity with the increase of the deposited energy W , due to saturation effects discussed in Ref. 17. However, if we restrict our analysis to the data gathered at the lowest deposited energies, these non linearities can safely be neglected, and we can use the relation (12) to compute the volume variation. Fig. 3 shows the values of ∆ V/V versus the temperature T for the data with released energy: 1 . 5 × 10 -3 < W < 9 × 10 -3 J. ∆ V/V data have been fitted by a 2nd order polynomial q 0 + q 1 T + q 2 T 2 obtaining q 0 = ( -1 . 737 ± 0 . 054) × 10 -8 , q 1 = (7 . 334 ± 1 . 785) × 10 -9 K -1 , q 2 = (1 . 766 ± 0 . 187) × 10 -8 K -2 with a χ 2 = 1 . 38 normalized over 94 degrees of freedom. The use of q 0 and of the relation (6) gives ∂H c /∂P = ( -3 . 12 ± 0 . 10) × 10 -12 T Pa -1 at T = 0 K, a value that can be compared with ( -2 . 67 ± 0 . 06) × 10 -12 T Pa -1 obtained by Harris and Mapother in their experimental study of the critical field of pure aluminum as a function of pressure and temperature. 32 The difference of the thermal expansion coefficients in the normal and superconducting phases is obtained by taking the derivative with respect to T of the difference of volumes: 27 ∆ α = α ( n ) -α ( s ) = 1 3 d dT ∆ V V . Under the hypothesis that α ( n ) is still represented by the relation (11) in the inter- al 0 . 3 K < T < T c and by using the polynomial fitting for ∆ V/V , the following expression is obtained for that temperature interval: This result can be compared with the expansion coefficient of pure aluminum in the ( s ) state, using for pure aluminum the relation α ( s ) = α ( s ) ,e + α /lscript , in which it is assumed that the lattice component does not depend on the conduction state and that its values are given in Ref. 33. The definition (2) gives α ( s ) ,e = ρc V,e ( s ) γ ( s ) ,e / (3 K T ), where the electronic component of c V in the ( s ) state ( c V,e ( s ) ) is taken from the work of Phillips, 34 the electronic component of the Gruneisen parameter in the ( s ) state ( γ ( s ) ,e ) from Ref. 35 and K ( s ) T ∼ K ( n ) T = 79 . 4 × 10 9 Pa at low temperatures. 36 The comparison is shown in Fig. 4 as a function of the reduced temperature t .",
"pages": [
9,
10,
11,
12
]
},
{
"title": "5. Conclusions",
"content": "In this paper we present an evaluation of the linear expansion coefficient of an AlMg alloy obtained by measuring both the amplitude of the fundamental mode of the longitudinal oscillation excited by electrons interacting in a suspended bar and the energy released in the bar by the electron pulse. The use of this method to determine α is new, although similarities exist with the direct measurement of the the Gruneisen parameter performed by hitting thin slabs with particle beams to generate thermoelastic pulses. No expansion data for aluminum alloys were previously available in the literature for the temperature range 0 . 3 < T < 2 K explored by this experiment. The expansion coefficient is negative in the superconducting state and its absolute value just below T c is larger by an order of magnitude than the value above this temperature. This feature is expected also for pure aluminum, according to an analysis 35 of the measured values of H c ( P, T ) available in Ref. 32. This work was partially supported by the EU FP-6 Project ILIAS (RII3-CT-2004506222).",
"pages": [
12
]
}
]
2013IJMPD..2230030L
https://arxiv.org/pdf/1310.8459.pdf
<document>
<section_header_level_1><location><page_1><loc_15><loc_86><loc_85><loc_91></location>Neutrino coupling to cosmological background: A review on gravitational Baryo/Leptogenesis</section_header_level_1>
<text><location><page_1><loc_22><loc_82><loc_77><loc_83></location>Gaetano Lambiase a,b , Subhendra Mohanty c , and A.R. Prasanna c</text>
<text><location><page_1><loc_33><loc_77><loc_34><loc_81></location>a b</text>
<list_item><location><page_1><loc_34><loc_79><loc_67><loc_80></location>University of Salerno, Baronisi, Italy.</list_item>
<text><location><page_1><loc_34><loc_76><loc_66><loc_78></location>INFN , Sezione di Napoli, Italy. and</text>
<text><location><page_1><loc_25><loc_73><loc_75><loc_75></location>c Physical Research Laboratory, Ahmedabad 380009, India.</text>
<section_header_level_1><location><page_1><loc_45><loc_70><loc_54><loc_72></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_54><loc_88><loc_69></location>In this work we review the theories of origin of matter-antimatter asymmetry in the Universe. The general conditions for achieving baryogenesis and leptogenesis in a CPT conserving field theory have been laid down by Sakharov. In this review we discuss scenarios where a background scalar or gravitational field spontaneously breaks the CPT symmetry and splits the energy levels between particles and anti-particles. Baryon or Lepton number violating processes in proceeding at thermal equilibrium in such backgrounds gives rise to Baryon or Lepton number asymmetry.</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_8><loc_88><loc_86></location>It is largely accepted that General Relativity is the best (self-consistent) theory of gravity. It dynamically describes the space-time evolution and matter content in the Universe and is able to explain several gravitational phenomena ranging from laboratory and solar system scales (where it has been mainly tested) to astrophysical and cosmological scales. On cosmological scales, the cornerstone of General Relativity is represented by Hubble expansion, the Big Bang Nucleosynthesis, i.e. the formation of light elements in the early Universe, and cosmic microwave background (CMB) radiation. Despite these crucial predictions, Einstein's theory of gravity is in disagreement with the increasingly high number of observational data, such as those coming for example from SNIA-type, large scale structure ranging from galaxies up to galaxy super-clusters, provided by the advent of the Precision Cosmology and the achievement of high sensitivity of experiments. The experimental evidences that the observable Universe is at the moment expanding in an accelerating phase [1, 2] represents without any doubts the most exciting discovery of the modern Cosmology. As a consequence of this discovery, there has been in the last years more and more interest to understand the evolution not only of the early Universe, but also of the present Universe, and for this formidable task new ideas and theories beyond the standard Cosmology and particle physics have been proposed. Attempts to explain the recent observational data and at the same time try to preserve the conceptual structure of General Relativity, lead cosmologists to introduce two new fundamental concepts: Dark Matter (DM) and Dark Energy (DE). Observational data indicate that a huge amounts of DM and DE are indeed needed to explain the observed cosmic acceleration of the Universe in expansion (as well as all new observational data), and at the moment there are no experimental and theoretical evidence that definitively shed some light on such mysterious components (see [3-16] for DE reviews and [17-26] for DM reviews, and references therein). Moreover, recent data suggest that also in the very early epoch, the Universe was in an accelerated phase. This era is called Inflation, and is able to to solve the problems that affect the standard cosmological model (the Cosmology based on General Relativity): 1) The flatness of the Universe, that is why Ω = ρ/ρ c /similarequal O (1). Here ρ is the average cosmological energy density, and ρ c = 3 H 2 m 2 P / 8 π the critical density. 2) The problems of homogeneity, isotropy, and horizon (which created headache in the frameworks of the standard Friedman-Robertson-Walker (FRW cosmology).</text>
<text><location><page_3><loc_12><loc_84><loc_88><loc_91></location>Inflation provides a natural mechanism of generation of small density perturbations with almost flat spectrum[203]. This agrees with observations. In order to solve all problems of standard FRW cosmology, it is required that the duration of Inflation is</text>
<formula><location><page_3><loc_41><loc_79><loc_88><loc_81></location>N ≡ Ht ∼ 70 -100 . (1)</formula>
<text><location><page_3><loc_12><loc_57><loc_88><loc_77></location>At the end of the Inflationary epoch, according to the standard cosmological model, the Universe is in a cold, low entropy state, and appears baryon symmetric, that is the same amount of matter and antimatter. On the contrary, the present Universe looks baryon asymmetric. The issue that one has to solve is about the physical mechanism occurred during the Universe evolution for which it ends up being matter dominated. Equivalently, what about anti-matter? Theories that try to explain how the asymmetry between baryon and antibaryon was generated in the early phases of the Universe evolution are called baryogenesis . They represent a perfect interplay between particle physics and cosmology.</text>
<text><location><page_3><loc_14><loc_55><loc_69><loc_56></location>The parameter characterizing the baryon asymmetry is defined as</text>
<formula><location><page_3><loc_44><loc_49><loc_56><loc_53></location>η ' ≡ n B -n ¯ B n γ ,</formula>
<text><location><page_3><loc_12><loc_40><loc_88><loc_48></location>where n B ( n ¯ B ) is the number of baryons (antibaryon) per unit volume, and n γ = 2 ξ (3) T 3 π 2 is the photon number density at temperature T . A different definition of the parameter η that refers to the entropy density</text>
<formula><location><page_3><loc_39><loc_36><loc_88><loc_38></location>s = 2 π 2 g eff T 3 = 7 . 04 n γ , (2)</formula>
<formula><location><page_3><loc_44><loc_35><loc_45><loc_36></location>45</formula>
<formula><location><page_3><loc_40><loc_27><loc_60><loc_31></location>η ≡ n B -n ¯ B s = 1 7 . 04 η ' .</formula>
<text><location><page_3><loc_12><loc_22><loc_88><loc_27></location>Finally, the baryon asymmetry can be also expressed in term of the baryonic fraction Ω B = ρ B /ρ c , i.e.</text>
<formula><location><page_3><loc_41><loc_19><loc_59><loc_21></location>η = 2 . 74 × 10 -8 Ω B h 2 ,</formula>
<text><location><page_3><loc_12><loc_15><loc_60><loc_18></location>where h = 0 . 701 ± 0 . 013 is the present Hubble parameter.</text>
<text><location><page_3><loc_12><loc_8><loc_88><loc_15></location>The physics of the CMB temperature anisotropies, which are related to the acoustic oscillations of baryon-photon fluid around the decoupling of photons, provides a strong probe of the baryon asymmetry. In fact, the observation of the acoustic peaks in CMB measured</text>
<text><location><page_3><loc_12><loc_31><loc_36><loc_33></location>g eff = g γ + T 3 ν T 3 γ g ν , is given by</text>
<text><location><page_4><loc_12><loc_87><loc_88><loc_91></location>by WMAP satellite [27], when combined with measurements of large scale structures, leads to following estimation of the parameter η</text>
<formula><location><page_4><loc_24><loc_81><loc_88><loc_84></location>η ( CMB ) ∼ (6 . 3 ± 0 . 3) × 10 -10 0 . 0215 ≤ Ω B h 2 ≤ 0 . 0239 (3)</formula>
<text><location><page_4><loc_12><loc_76><loc_88><loc_80></location>An independent measurement of η can be carried out in the framework of the BBN [28], that gives</text>
<formula><location><page_4><loc_25><loc_72><loc_88><loc_75></location>η ( BBN ) ∼ (3 . 4 -6 . 9) × 10 -10 0 . 017 ≤ Ω B h 2 ≤ 0 . 024 . (4)</formula>
<text><location><page_4><loc_12><loc_62><loc_88><loc_71></location>It is remarkable that two completely different probes of the baryon content of the Universe (the synthesis of light elements occurred during the first 3 minutes of the Universe evolution, and the the photons decoupling occurred when the Universe was 400 thousand years old) give compatible results. This represents one of the great success of modern Cosmology.</text>
<text><location><page_4><loc_12><loc_35><loc_88><loc_61></location>Although many mechanisms have been proposed, the explanation of the asymmetry between matter and antimatter is still an open problem of the modern Cosmology and Particle Physics. In this review we discuss some general topics related to the baryogenesis. The work is divided in two parts. In the first part we recall some models of baryogenesis/leptogenesis, which are mainly based on particle physics (essentially GUT and SUSY). In the second part we discuss different approaches to the baryon asymmetry which rely on the coupling of baryon/lepton currents with the gravitational background. Particular attention will be devoted to the mechanism based on the spin-gravity coupling of neutrinos with the gravitational waves of the cosmological background, which are generated by quantum fluctuations of the inflation field during the inflationary era.</text>
<section_header_level_1><location><page_4><loc_12><loc_30><loc_72><loc_31></location>2. SOME TOPICS OF BARYOGENESIS AND LEPTOGENESIS</section_header_level_1>
<text><location><page_4><loc_12><loc_23><loc_88><loc_27></location>In this Section, we recall some general topics of Baryogenesis and Leptogenesis. More details can be found in [29-34].</text>
<text><location><page_4><loc_12><loc_11><loc_88><loc_21></location>Standard cosmological model is unable to explain the so small magic value of the baryon asymmetry (Eqs. (3) and (4)) and why the Universe starting from an initial baryon symmetry ( n B = n ¯ B ) evolves in a final state such that matter dominates over antimatter ( n B /greatermuch n ¯ B ).</text>
<section_header_level_1><location><page_5><loc_14><loc_89><loc_42><loc_91></location>2.1. The Sakharov conditions</section_header_level_1>
<text><location><page_5><loc_12><loc_82><loc_88><loc_86></location>As pointed out by Sakharov [36], in a CPT conserving theory a baryon asymmetry B may be dynamically generated in the early Universe provided that:</text>
<text><location><page_5><loc_55><loc_72><loc_55><loc_74></location>/negationslash</text>
<unordered_list>
<list_item><location><page_5><loc_12><loc_62><loc_88><loc_80></location>1) There exist interactions that violate the baryon number B . Baryon number violating interactions are required because one starts from a baryon symmetric Universe ( B = 0) to end in a baryon asymmetric Universe ( B = 0). Direct experimental proofs that baryons are not conserved are still missing. From a theoretical point of view, both GUT and the standard electroweak theory (via sphaleron processes) give non conservation of baryon number (notice that another possibility to break all global charges and in particular the baryon charge is related to gravity, as discussed in Ref. [37]).</list_item>
<list_item><location><page_5><loc_12><loc_45><loc_88><loc_60></location>2) The discrete symmetries C and CP must be violated. This condition is necessary in order that matter and antimatter can be differentiated, as otherwise B non-conserving interactions would produce baryons and antibaryons at the same rate thus maintaining the net baryon number to be zero. In contrast to non-conservation of baryons, the breaking of CP -symmetry was discovered in direct experiment ( CP violation has been indeed observed in the kaon system).</list_item>
</unordered_list>
<text><location><page_5><loc_17><loc_32><loc_88><loc_44></location>The C and CP violation imply that B/L violating reactions in the forward and reverse channels do not cancel ( L stands for Lepton number). To see this, consider the process: X → Y + B , where X is the initial state with B = 0, Y the final state with B = 0, and B the excess baryon produced. Suppose that C is a symmetry. Then the C -conjugate process is characterized by the fact that</text>
<formula><location><page_5><loc_38><loc_27><loc_67><loc_30></location>Γ( X → Y + B ) = Γ( ¯ X → ¯ Y + ¯ B ) .</formula>
<text><location><page_5><loc_17><loc_24><loc_61><loc_25></location>The net rate of baryon production evolves in time as</text>
<formula><location><page_5><loc_25><loc_19><loc_80><loc_22></location>dB dt ∝ Γ( ¯ X → ¯ Y + ¯ B ) -Γ( X → Y + B ) = 0 if C is a symmetry</formula>
<text><location><page_5><loc_17><loc_13><loc_88><loc_17></location>Similar arguments hold for CP symmetry. Therefore both C and CP discrete symmetry violation are required to generate a net baryon asymmetry.</text>
<unordered_list>
<list_item><location><page_5><loc_12><loc_7><loc_88><loc_11></location>3) Departure from thermal equilibrium: This condition is required because the statistical distribution of particles and anti-particles is the same if the Hamiltonian commutes</list_item>
</unordered_list>
<text><location><page_6><loc_17><loc_83><loc_88><loc_91></location>with CPT , i.e. [ H,CPT ] = 0, which implies n B = n ¯ B . Hence only a departure from thermal equilibrium, which means that the form of n B, ¯ B has to be modified, can allow for a finite baryon excess (so that n B -n ¯ B = 0).</text>
<text><location><page_6><loc_53><loc_83><loc_53><loc_86></location>/negationslash</text>
<text><location><page_6><loc_17><loc_78><loc_88><loc_82></location>More specifically, consider again the process X → Y + B . If the process is in thermal equilibrium , then by definition</text>
<formula><location><page_6><loc_38><loc_73><loc_67><loc_75></location>Γ( X → Y + B ) = Γ( Y + B → X ) ,</formula>
<text><location><page_6><loc_17><loc_67><loc_88><loc_71></location>so that no net baryon asymmetry can be produced since the inverse process destroy B as fast as the forward process creates it (see also Appendix A).</text>
<text><location><page_6><loc_70><loc_41><loc_70><loc_43></location>/negationslash</text>
<text><location><page_6><loc_12><loc_38><loc_88><loc_64></location>However, as discussed in the seminal paper by Cohen and Kaplan [38], it is possible to generate lepton/baryon asymmetry at thermal equilibrium (without requiring CP violation). The reason is due to the general result that in an expanding Universe at finite temperature, CPT is not a good symmetry, i.e. CPT can be (spontaneously) violated. The third Sakharov criterion is therefore violated. As before pointed out, in fact, CPT invariance requires that the baryon number must be generated out of thermal equilibrium, but CPT invariance requires that the thermal distribution of baryon and anti-baryon will be identical. This condition fails if there is a spontaneous CPT violation in the theory which modifies the baryon-antibaryon spectrum. As a consequence one obtains E particle = E antiparticle which implies n B -n ¯ B = 0.</text>
<text><location><page_6><loc_26><loc_38><loc_26><loc_41></location>/negationslash</text>
<text><location><page_6><loc_12><loc_15><loc_88><loc_38></location>The CPT violation allows for the generation of the baryon asymmetry during an era when baryon(lepton) violating interaction are still in thermal equilibrium. The asymmetry gets frozen at the decoupling temperature T d when the baryon(lepton) violation goes out of equilibrium. The decoupling temperature is calculated by equating the interaction rate of processes Γ and the expansion rate of the Universe represented by the Hubble constant H , Γ( T d ) /similarequal H ( T d ). The scenario underlying these processes in an expanding Universe can be schematized as follows: in the regime Γ /greatermuch H , or T > T d B-asymmetry is generated by B-violating processes at thermal equilibrium; at T = T d , i.e. Γ /similarequal H , the decoupling occurs, and finally when Γ < H , or T < T d the B-asymmetry gets frozen.</text>
<section_header_level_1><location><page_7><loc_14><loc_89><loc_39><loc_91></location>2.2. Leptogenesis scenario</section_header_level_1>
<text><location><page_7><loc_12><loc_69><loc_88><loc_86></location>Leptogenesis is a mechanism, proposed by Fukugita and Yanagida, that allows to convert the lepton asymmetry to baryon asymmetry via electroweak (EW) effects. Even if the baryon number is conserved at high scales, it is possible to generate the baryon asymmetry in the present Universe if lepton asymmetry is generated at either GUT or intermediate scales. This idea attracted much attention in view of discovery of a possible lepton number violation in the neutrino sector [39, 40]. For a recent review see [41, 42] (the role of neutrinos in cosmology has been recently treated in [43]).</text>
<text><location><page_7><loc_12><loc_53><loc_88><loc_68></location>The Leptogenesis scenario is the simplest extension of the standard model able to realize the Sakharov conditions for explaining the matter antimatter asymmetry in the Universe. In this model, the standard model is modified by adding right handed neutrinos which permit the implementation of the see-saw mechanism and provide the explanation of light mass of the standard model neutrinos. At the same time, the augmented model is able to spontaneously generate leptons from the decays of right handed neutrinos.</text>
<text><location><page_7><loc_12><loc_48><loc_88><loc_52></location>For later convenience, we shall discuss in a nutshell the Leptogenesis scenario. The leptonic Lagrangian density is given by (here we follow [44])</text>
<formula><location><page_7><loc_27><loc_43><loc_88><loc_46></location>L = h ∗ β ( ¯ L β φ c ∗ ) E β -λ ∗ αk ( ¯ L α φ ∗ ) N k -1 2 ¯ N j M j N c j + h.c. (1)</formula>
<text><location><page_7><loc_12><loc_10><loc_88><loc_41></location>In this expression L is the Standard Model left-handed doublet, E is the right-handed singlet, N j are the singlet fermions (Majorana neutrinos), α, β are the flavor indices of the Standard Model, i.e. α, β = e, µτ , M is mass matrix and λ Yakawa matrices. Equation (1 is written in a basis where the coupling h and the mass matrix M are diagonal and real, whereas λ is complex. This Lagrangian leads, once the heavy fermions N i are integrated out, to the effective light neutrino masses (see-saw mechanism): m ν αβ = λ αk M -1 k λ βk . The Lagrangian (1) satisfies the Sakharov conditions. In fact it violates the leptonic number L due to λ - and M -terms; CP is violated through the complex Yukawa coupling λ αk ; Since the interactions are only determined by Yukawa's interaction terms, the smallness of these couplings may provide the right conditions for which the interaction rates are smaller than the expansion rate of the Universe, establishing in such a way the condition for the out of equilibrium (i.e. the heavy Majorana fermions can decay out of equilibrium).</text>
<text><location><page_8><loc_14><loc_89><loc_87><loc_91></location>For simplicity consider the lightest Majorana singlet N 1 . It may decay in two channels</text>
<formula><location><page_8><loc_38><loc_84><loc_88><loc_87></location>N 1 → L α φ, N 1 → ¯ L α φ † . (2)</formula>
<text><location><page_8><loc_12><loc_79><loc_88><loc_83></location>As a consequence of the N 1 decay, the net baryon asymmetry can be generated. The parameter η turns out to be</text>
<formula><location><page_8><loc_40><loc_75><loc_88><loc_79></location>η /similarequal 135 ζ (3) 4 π 4 g ∗ C sph η eff /epsilon1 . (3)</formula>
<text><location><page_8><loc_12><loc_64><loc_88><loc_74></location>Here η eff is the efficiency factor which assumes the value in the range 0 < η eff < 1 (owing to inverse decays, washout processes and inefficiency in N 1 production). Below to the freeout temperature T F , the temperature for which Γ( φL → N 1 ) < H , where Γ( φL → N 1 ) /similarequal 1 2 Γ D e -M 1 /T , with</text>
<formula><location><page_8><loc_42><loc_61><loc_88><loc_65></location>Γ D = ( λ † λ ) 11 M 1 8 π , (4)</formula>
<text><location><page_8><loc_12><loc_59><loc_79><loc_60></location>and H is the expansion rate of the Universe during the radiation dominated era,</text>
<formula><location><page_8><loc_42><loc_53><loc_88><loc_57></location>H = 1 . 66 g 1 / 2 ∗ T 2 M 2 P , (5)</formula>
<text><location><page_8><loc_12><loc_48><loc_88><loc_52></location>the density of the fermion N 1 is Boltzmann suppressed ( N 1 ∼ e -M 1 /T ). Therefore, below T F the decay of N 1 contribute to the lepton asymmetry, and the efficiency factor is</text>
<formula><location><page_8><loc_28><loc_42><loc_88><loc_46></location>η eff /similarequal n N 1 ( T F ) n N 1 ( T /greatermuch M 1 ) /similarequal e -M 1 /T F /similarequal m ∗ ˜ m , ˜ m<m ∗ , (6)</formula>
<text><location><page_8><loc_12><loc_40><loc_17><loc_41></location>where</text>
<formula><location><page_8><loc_20><loc_35><loc_88><loc_39></location>˜ m ≡ 8 πv 2 M 2 1 Γ D = ( λ † λ ) 11 v 2 M 1 , m ∗ ≡ 8 πv 2 M 2 1 H ( T = M 1 ) /similarequal 1 . 1 × 10 -3 eV . (7)</formula>
<text><location><page_8><loc_12><loc_29><loc_88><loc_34></location>C sph is a factor that takes into account the dilution of the asymmetry due to fast processes. Finally /epsilon1 is the CP parameter related to the asymmetry in the N 1 decays and defined as</text>
<formula><location><page_8><loc_36><loc_23><loc_88><loc_28></location>/epsilon1 = Γ( N 1 → φL ) -Γ( N 1 → φ † ¯ L ) Γ( N 1 → φL ) + Γ( N 1 → φ † ¯ L ) . (8)</formula>
<text><location><page_8><loc_12><loc_19><loc_88><loc_23></location>Its non vanishing value arises from the interference of three level and one loop amplitudes (complex Yakawa couplings). One gets</text>
<formula><location><page_8><loc_33><loc_12><loc_88><loc_18></location>/epsilon1 = 1 8 π 1 ( λ † λ ) 11 ∑ j I/arrowdblbothv { [ ( λ † λ ) 1 j ] 2 } g ( x j ) , (9)</formula>
<text><location><page_8><loc_12><loc_10><loc_17><loc_12></location>where</text>
<formula><location><page_8><loc_27><loc_6><loc_72><loc_11></location>g ( x ) = √ x [ 2 -x 1 -x -(1 + x ) ln 1 + x x ] , x j ≡ M 2 j M 2 1 .</formula>
<text><location><page_9><loc_12><loc_89><loc_71><loc_91></location>Taking into account Eqs. (6), (9) and (7), the net lepton asymmetry is</text>
<formula><location><page_9><loc_42><loc_84><loc_88><loc_88></location>η /similarequal 10 -3 10 -3 eV ˜ m /epsilon1 . (10)</formula>
<text><location><page_9><loc_12><loc_57><loc_88><loc_82></location>Leptogenesis is then related to Baryogenesis by a phenomenon that happens in the currently accepted Standard Model. Indeed, certain non-perturbative configurations of gauge fields, the sphalerons, can convert leptons into baryons and vice versa. These processes that violate B + L and conserve B -L occur at the electroweak scale. Under normal conditions sphalerons processes are unobservably rare due to the fact that the transition rates are extremely small Γ ∼ e -16 π 2 /g 2 ∼ O (10 -165 ), hence are completely negligible in the Standard Model (at T = 0). However as emphasized by Kuzmin, Rubakov, and Shaposhnikov [40], in the thermal bah provided by the expanding Universe, thermal fluctuations becomes important and B + L violating processes can occur at a significant rate and these processes can be in equilibrium in the expanding Universe (see B).</text>
<text><location><page_9><loc_12><loc_52><loc_88><loc_56></location>Finally from (10) it follows that by requiring η ∼ 10 -10 the lower bound of the mass of the Majorana neutrinos N 1 is</text>
<formula><location><page_9><loc_43><loc_49><loc_88><loc_51></location>M 1 /greaterorsimilar 10 11 GeV , (11)</formula>
<text><location><page_9><loc_12><loc_45><loc_79><loc_47></location>where the light value of the neutrino mass has been used: ˜ m /similarequal (10 -3 -10 -1 )eV.</text>
<section_header_level_1><location><page_9><loc_14><loc_40><loc_40><loc_42></location>2.3. Models of Baryogenesis</section_header_level_1>
<text><location><page_9><loc_12><loc_23><loc_88><loc_37></location>Many models aimed to explain the generation of the baryon asymmetry have been proposed in literature. These are GUT Baryogenesis, Affleck-Dine Baryogenesis and AffleckDine Leptogenesis, Leptogenesis from heavy Majorana neutrinos, Leptogensis from ν R oscillation, Thermal baryogenesis, Electroweak baryogenesis, Spontaneous baryogenesis, Baryogenesis through evaporation of primordial black holes. Details of such mechanisms can be found in [29-33, 45] and references therein. Here we list some of them in Table I.</text>
<unordered_list>
<list_item><location><page_9><loc_15><loc_17><loc_56><loc_20></location>· GUT-Baryogenesis or decay of heavy particles:</list_item>
</unordered_list>
<text><location><page_9><loc_38><loc_11><loc_38><loc_14></location>/negationslash</text>
<text><location><page_9><loc_17><loc_7><loc_88><loc_17></location>Consider the X -boson decays in two channels X → 2 q and X → 2¯ q , with the probabilities given by P X → 2 q = P X → 2¯ q due to CP violation. This implies the excess of baryons over anti-baryons. In the original scenario of GUT baryogenesis one uses the heavy gauge bosons X and Y (leptoquarks), which decay while they decouple from</text>
<text><location><page_10><loc_17><loc_73><loc_88><loc_91></location>equilibrium. This is called delayed decay scenario . It was soon realized that this boson gauge decay does not produce the required baryon asymmetry because that the X and Y boson masses predicted are too low to satisfy the out-of-equilibrium condition (in non-SUSY GUT). The alternative scenario was to use decays of coloured Higgs particles. If more than two Higgs particles exist, sufficiently large baryon asymmetry can be generated (provided that the Kuzmin, Rubakov, Shaposhnikov effect is switched off).</text>
<unordered_list>
<list_item><location><page_10><loc_15><loc_62><loc_88><loc_71></location>· SUSY: Supersymmetry actually opens a number of options. Since Supersymmetry extends the particle content of the theory near the EW scale, the possibility of a strong EW phase transition cannot yet be completely excluded. This revives the hope of explaining baryon asymmetry entirely within the MSSM.</list_item>
</unordered_list>
<text><location><page_10><loc_17><loc_24><loc_88><loc_60></location>Affleck-Dine scenario (1985): This scenario is based on the observation that in SUSY theories ordinary quarks and leptons are accompanied by supersymmetric partners - s-quarks and s-leptons - which are scalars. The corresponding scalar fields carry baryon and lepton number, which can in principle be very large in the case of a scalar condensate (classical scalar field). An important feature of SUSY theories is the existence of flat directions in the superpotential, along which the relevant components of the complex scalar fields ϕ can be considered as massless. The condensate is frozen until supersymmetry breaking takes place. Supersymmetry breaking lifts the flat directions and the scalar fields acquire mass. When the Hubble constant becomes of the order of this mass, the scalar fields starts to oscillate and decays. At this time, B , L , and CP violating terms (for example, quartic couplings λ 1 ϕ 3 ϕ ∗ + c.c. and λ 2 ϕ 4 + c.c. , with complex λ 1 , 2 ) becomes important and a substantial baryon asymmetry can be produced. The scalar particles decay into ordinary quarks and leptons transferring to them the generated baryon asymmetry.</text>
<text><location><page_10><loc_17><loc_13><loc_88><loc_23></location>The Affleck-Dine mechanism can be implemented at nearly any energy scale, even below 200 GeV. By suitable choice of the parameters one can explain almost any amount of baryon asymmetry and this lack of a falsifiable prediction is an unattractive feature of the Affleck-Dine mechanism.</text>
<unordered_list>
<list_item><location><page_10><loc_15><loc_6><loc_88><loc_11></location>· Electroweak baryogenesis ( ∼ TeV ): The asymmetry is generated by phase transitions involving SU (2) × U (1) breaking. The EWBG is assumed to occur during the radiation</list_item>
</unordered_list>
<text><location><page_11><loc_17><loc_31><loc_88><loc_91></location>dominated era of the early Universe, a period in which the SU (2) L × U (1) Y electroweak symmetry is manifest. As the temperature falls down the EW scale ( T EW ∼ 100GeV), the Higgs field acquires an expectation value and the electroweak symmetry is spontaneously broken to the subgroup U (1). The EWBG occurs during this phase transition, and in order that it could be an available mechanism, it is required that the transition is of the first order. Remarkably the EWBG satisfies all the three Sakharov's conditions: 1) The rapid sphaleron transitions in the symmetric phase provide the required violation of the baryon number; 2) The scattering of plasma with bubble walls generates the C and CP asymmetry of the number of particles if the underlying theory does contain terms that violate these discrete symmetries (these processes bias the sphalerons to create more baryon than anti-baryons); 3) The rapid expansion of bubble walls through the plasma induces the departure from the thermal equilibrium. All these conditions are fulfilled by the Standard Model. However, EWBG is unable to explain the observed baryon asymmetry of the Universe if it is only based on the Standard Model. The reason is due to the fact that EW phase transition in the Standard Model is of the first order if the Higgs mass is constrained by m H /lessorsimilar 70GeV, in disagreement with experimental lower bound obtained from LEP II experiment, i.e. m H /greaterorsimilar 114GeV, as well as, from recent LHC results that give a value of the Higgs mass near to 125GeV. Recent studies however, open the possibilities to reconsider EWBG as an available candidate for the generation of baryon asymmetry[46]. Moreover, the EWBG mechanism is also affected by the problem related to the CP violation because the latter generated by the Cabibbo-Kobayashi-Maskawa phase is unable to generate large enough chiral asymmetry. For a recent review on EWBG see[47].</text>
<unordered_list>
<list_item><location><page_11><loc_15><loc_25><loc_88><loc_29></location>· An interesting idea to mention is the Baryogenesis generated through evaporation of primordial BHs[48, 49].</list_item>
</unordered_list>
<section_header_level_1><location><page_11><loc_12><loc_15><loc_88><loc_19></location>3. BARYOGENESIS GENERATED BY COUPLING OF BARYON CURRENTS AND GRAVITATIONAL BACKGROUND</section_header_level_1>
<text><location><page_11><loc_12><loc_8><loc_88><loc_12></location>As we have seen in the previous Section, the Baryo/Leptogenesis is generated in the framework of particle interactions (essentially GUT and SUSY). The gravitational field</text>
<text><location><page_12><loc_30><loc_75><loc_30><loc_78></location>/negationslash</text>
<table>
<location><page_12><loc_12><loc_10><loc_87><loc_88></location>
<caption>TABLE I: Models of baryogenesis</caption>
</table>
<text><location><page_13><loc_12><loc_79><loc_88><loc_91></location>enters marginally in these mechanisms. In the last years, however, many mechanisms have been proposed in which gravity plays a fundamental role in generating the baryo/leptogenesis (see Table II). In these models matter or hadron/lepton currents are coupled with some physical quantity characterizing the gravitational background, such as Ricci curvature or its derivative, Riemann tensor, gravitational waves (GW),</text>
<formula><location><page_13><loc_21><loc_74><loc_79><loc_77></location>L int ∼ J · F , J → ¯ ψγ µ ψ, ¯ ψγ µ γ 5 ψ, φ∂ µ φ ∗ , .... , F → R , ∂R , ∂φ , ... .</formula>
<text><location><page_13><loc_12><loc_69><loc_88><loc_73></location>Typically, the background is the FRW geometry, but there are also models in which the gravitational background is described by black holes physics.</text>
<text><location><page_13><loc_12><loc_55><loc_88><loc_67></location>Gravitational baryogenesis share some basic features of the spontaneous spontaneous (or quintessential) baryogenesis[38]. In this mechanism scalar fields (or their derivatives) couple to matter or hadron/lepton current. To illustrate in some detail the spontaneous baryogenesis, consider a neutral scalar field φ . The interaction between a baryon current J µ B and ∂ µ φ is</text>
<formula><location><page_13><loc_43><loc_52><loc_88><loc_55></location>L = 1 M s J µ B ∂ µ φ, (1)</formula>
<text><location><page_13><loc_54><loc_38><loc_54><loc_40></location>/negationslash</text>
<text><location><page_13><loc_12><loc_26><loc_88><loc_51></location>where M s characterize is a cut-off scale. In a isotropic and homogenous Universe, like FRW Universe, φ does only depend on cosmic time, In such a case only the zero component of the baryon current ( J 0 B = n B with n B the number density of baryons) contribute in (1), L = µn B , where µ ≡ ˙ φ/M s for baryons and µ ≡ -˙ φ/M s for antibaryons. Here is assumed that the current J B is not conserved and that, of course, ˙ φ = 0. The coupling (1) therefore gives rise to an effective chemical potential with opposite sign for B and ¯ B leading to a generation of a net baryon asymmetry even at thermal equilibrium. The latter point bypass the third Sakharov condition because CPT violation occurs owing the Universe expansion. The scalar field could also play the role of DE or DM. Models based on spontaneous (quintessence) baryogenesis are studied in[50-57].</text>
<section_header_level_1><location><page_13><loc_14><loc_20><loc_44><loc_21></location>3.1. Gravitational Baryogenesis</section_header_level_1>
<text><location><page_13><loc_12><loc_10><loc_88><loc_17></location>The key ingredient for the gravitational baryo/leptogenesis is a CP-violating interaction between the derivative of the Ricci scalar curvature R and the B(aryon)/L(epton) current J µ [58, 59]</text>
<formula><location><page_13><loc_40><loc_6><loc_88><loc_10></location>L int = 1 M 2 ∗ √ -gJ µ ∂ µ R (2)</formula>
<text><location><page_14><loc_30><loc_76><loc_30><loc_78></location>/negationslash</text>
<table>
<location><page_14><loc_15><loc_30><loc_85><loc_88></location>
<caption>TABLE II: Models of gravitational baryogenesis</caption>
</table>
<text><location><page_14><loc_12><loc_10><loc_88><loc_27></location>where M ∗ is the cutoff scale of the effective theory. L int is expected in a low energy effective field theory of quantum gravity or Super gravity theories (more specifically it can be obtained in supergravity theories from a higher dimensional operator in the Kahler potential). Moreover, it dynamically breaks the CPT in an expanding Universe. In the standard cosmological model ˙ R vanishes during the radiation era (see below). However, (tiny) deviations from General Relativity prevent the Ricci curvature to vanish, as well as its first time derivative, so that a net lepton asymmetry can be generated.</text>
<text><location><page_14><loc_14><loc_7><loc_88><loc_8></location>To generate a B -asymmetry, it is required that there exist B/L -violating processes in</text>
<text><location><page_15><loc_12><loc_81><loc_88><loc_91></location>thermal equilibrium. In this mechanism, the interaction J µ ∂ µ R gives a contribution to the energy of particles and antiparticles with opposite sign, and thereby dynamically violates CPT . This coupling term modifies thermal equilibrium distribution and the chemical potential</text>
<formula><location><page_15><loc_37><loc_77><loc_88><loc_82></location>µ particle = ˙ R M 2 ∗ = -µ anti -particle (3)</formula>
<text><location><page_15><loc_12><loc_70><loc_88><loc_77></location>driving the Universe towards nonzero equilibrium B/L -asymmetry via the B/L -violating interactions. Once the temperature drops below the decoupling temperature T d the asymmetry can no longer change and is frozen. The net asymmetry is</text>
<formula><location><page_15><loc_45><loc_64><loc_88><loc_69></location>η ≈ ˙ R M 2 ∗ T D . (4)</formula>
<text><location><page_15><loc_12><loc_59><loc_88><loc_63></location>In the cosmological standard model it is assumed that the energy-momentum tensor of classical fields is described by a perfect fluid</text>
<formula><location><page_15><loc_39><loc_54><loc_61><loc_56></location>T µν = diag( ρ, -p, -p, -p ) ,</formula>
<text><location><page_15><loc_12><loc_43><loc_88><loc_52></location>where ρ is the energy density and p the pressure. They are related by the relation p = wρ , w being the adiabatic index. During the radiation dominated era, the equation of the state is p = ρ/ 3, i.e. w = 1 / 3, and the scale factor evolves as a ( t ) = ( a 0 t ) 1 / 2 . The energy density of the (classical) radiation is given by</text>
<formula><location><page_15><loc_42><loc_38><loc_88><loc_42></location>ρ r = T 00 = π 2 g ∗ 30 T 4 , (5)</formula>
<text><location><page_15><loc_12><loc_35><loc_75><loc_36></location>whereas the cosmic time is related to the temperature T of the Universe as</text>
<formula><location><page_15><loc_43><loc_29><loc_88><loc_33></location>1 t 2 = 32 π 3 g ∗ 90 T 4 M 2 P . (6)</formula>
<text><location><page_15><loc_12><loc_27><loc_81><loc_28></location>Moreover, Eq. (5 implies that the expansion rate of the Universe can be written as</text>
<formula><location><page_15><loc_39><loc_21><loc_88><loc_25></location>H = 1 . 6 g 1 / 2 ∗ T 2 M P = -˙ R 4 R . (7)</formula>
<text><location><page_15><loc_12><loc_16><loc_88><loc_20></location>In what follows we shall consider a flat Friedman-Robertson-Walker (FRW) Universe whose element line is</text>
<formula><location><page_15><loc_35><loc_12><loc_88><loc_15></location>ds 2 = dt 2 -a 2 ( t )[ dx 2 + dy 2 + dz 2 ] . (8)</formula>
<text><location><page_15><loc_12><loc_6><loc_88><loc_11></location>From the above considerations it follows that the trace of the energy-momentum tensor of (classical) relativistic fields vanishes, T = ρ -3 p = 0. As a consequence one has R =</text>
<text><location><page_16><loc_12><loc_84><loc_88><loc_91></location>-8 πGT µ µ = 0, and no net baryon asymmetry may be generated. However, a possibility to generate the baryon asymmetry is given by the interaction among massless particles that lead to running coupling constants and hence the trace anomaly [62]</text>
<formula><location><page_16><loc_35><loc_79><loc_65><loc_82></location>T µ µ ∝ β ( g ) F 2 = 0 F 2 = F µν F µν .</formula>
<text><location><page_16><loc_46><loc_79><loc_46><loc_81></location>/negationslash</text>
<text><location><page_16><loc_12><loc_73><loc_88><loc_77></location>In a SU ( N c ) gauge theory with coupling g and N f flavors, the effective equation of state is given by</text>
<formula><location><page_16><loc_25><loc_68><loc_75><loc_73></location>1 -3 w = 5 g 4 96 π 6 [ N c +(5 / 4) N f ] [(11 / 3) N c -(2 / 3) N f ] 2 + (7 / 2) [ N c N f / ( N 2 c -1)] + O ( g 5 )</formula>
<text><location><page_16><loc_12><loc_59><loc_88><loc_68></location>The numerical value of 1 -3 w depends the gauge group and the fermions, and lies in the range 1 -3 w ∼ 10 -2 -10 -1 . The baryon asymmetry turns out to be η = (1 -3 w ) T 5 D M 2 ∗ m 3 P . Gravitational baryogenesis is conceptually similar to spontaneous baryogenesis [38], see Eq. (1). However there some basic differences between the two paradigms:</text>
<unordered_list>
<list_item><location><page_16><loc_15><loc_52><loc_88><loc_57></location>· The scalar field φ has to be added by hand, whereas the term in Eq. (2) is expected to be present in an effective theory of gravity.</list_item>
<list_item><location><page_16><loc_15><loc_38><loc_88><loc_50></location>· The scalar field φ must satisfies specific initial conditions, that is to generate a net asymmetry φ has to evolve homogeneously in one direction versus the other and must be spatially uniform. In the gravitational baryogenesis, instead, the time-evolution of R naturally occurs in a cosmological background and it is highly spatially uniform owing to high homogeneousity of the Universe.</list_item>
<list_item><location><page_16><loc_15><loc_28><loc_88><loc_36></location>· In the regime in which φ oscillates around its minimum ˙ φ is zero, so that the asymmetry is canceled[63], whereas the mean value of ˙ R does not vanish because is proportional to ∼ H 3 .</list_item>
</unordered_list>
<section_header_level_1><location><page_16><loc_14><loc_23><loc_55><loc_25></location>3.2. Genaralised Gravitational baryogenesis</section_header_level_1>
<text><location><page_16><loc_12><loc_16><loc_88><loc_20></location>An interesting model related to the gravitational baryogenesis has been provided by Li, Li and Zhang [51], who consider a generalized coupling of the form</text>
<formula><location><page_16><loc_42><loc_11><loc_88><loc_14></location>L int ∼ J µ ∂ µ f ( R ) , (9)</formula>
<text><location><page_17><loc_12><loc_86><loc_88><loc_91></location>where f ( R ) is a generic function of the scalar curvature. This function has been chosen as f ∼ ln R so that the effective interaction Lagrangian density reads</text>
<formula><location><page_17><loc_42><loc_82><loc_88><loc_86></location>L int ∼ -c ∂ µ R R J µ , (10)</formula>
<text><location><page_17><loc_12><loc_77><loc_88><loc_81></location>where c is a constant fixed to in order to reproduce the observed baryon asymmetry. Following the same reasoning leading to (3) one gets</text>
<formula><location><page_17><loc_36><loc_72><loc_63><loc_76></location>µ particle = -c ˙ R R = -µ anti -particle .</formula>
<text><location><page_17><loc_12><loc_66><loc_88><loc_71></location>During the radiation dominated era one obtains that a net baryon asymmetry can be generated and is given by</text>
<formula><location><page_17><loc_30><loc_59><loc_88><loc_66></location>η = -15 g b 4 π 2 g ∗ c ˙ R RT ∣ ∣ ∣ T D = 15 π 2 cg b H ( T D ) g ∗ T D /similarequal 0 . 1 c T D M P , (11)</formula>
<text><location><page_17><loc_12><loc_16><loc_88><loc_61></location>where Eq. (5) has been used and T D is the decoupling temperature. Moreover, one can also determine an order of magnitude of the absolute neutrino mass compatible with the current cosmological data, i.e. m ν /similarequal O (1)eV. The idea goes along the line traced in Section 4.1. In the Standard Model, B -L symmetry is exactly conserved ( ∂ µ J µ B -L = 0). In [51] the B -L violation is parameterized by higher dimensional operators, i.e. by the dimension 5 operator L ∼ C ¯ llφ † φ (see Eq. (26)). C is a scale of new physics beyond the Standard Model which generates the B -L violations, l and φ are the left-handed lepton and Higgs doublets, respectively. When the Higgs field gets a vacuum expectation value 〈 φ 〉 = v , the lefthanded neutrino becomes massive m ν /similarequal Cv . Comparing the lepton number violating rate induced by the interaction L , Γ ∼ T 3 (Eq. (30)), with the expansion rate of the Universe, H ∼ T 2 (Eq. (5)), one gets the decoupling temperature below which the lepton asymmetry is freeze-out, i.e. T D /similarequal 10 10 GeV. The observed baryon asymmetry η ∼ 10 -10 follows for c ∼ O (1). Then, assuming an approximate degenerate masses, i.e. m ν 1 ∼ m ν 2 ∼ m ν 3 , one gets m ν /lessorsimilar ∞ eV. The current cosmological limit comes from WMAP Collaboration [64] and SDSS Collaboration [65]. The analysis of Ref.[64] gives ∑ i m ν i < 0 . 69 eV. The analysis from SDSS gives[65] ∑ i m ν i < 1 . 7 eV.</text>
<section_header_level_1><location><page_17><loc_14><loc_14><loc_61><loc_15></location>3.3. Baryogenesis in Randall-Sundrum braneworld</section_header_level_1>
<text><location><page_17><loc_12><loc_7><loc_88><loc_11></location>The asymmetry baryon-antibaryon can arise in the Randall-Sundrum brane world model[66] with bulk fields owing to the effects of higher dimensionality. These studies have</text>
<text><location><page_18><loc_12><loc_89><loc_85><loc_91></location>been performed in[67-69]. The total action contains the bulk and brane actions[68, 70]</text>
<formula><location><page_18><loc_19><loc_81><loc_88><loc_88></location>S = S bulk ( (5) R, Λ , Φ) + S brane ( σ ) = (12) = ∫ d 5 x √ G [ M 5 2 (5) R ( G ) -Λ -|∇ x M Φ | 2 ] + ∫ d 4 x √ -g [ σ + L matter ] ,</formula>
<text><location><page_18><loc_12><loc_70><loc_88><loc_81></location>where G MN is the 5-dim bulk metric and G its determinant, g µν the brane induced metric and g the determinant, Λ the bulk cosmological constant, σ the brane tension, and Φ the bulk complex scalar field (localized on the brane as the graviton). Λ and σ are related by Λ = -σ 2 6 M 5 .</text>
<text><location><page_18><loc_12><loc_66><loc_88><loc_70></location>It is worth to write down the the effective theory on the brane. It is derived by making use of the braneworld holography[68, 71]. This method gives</text>
<formula><location><page_18><loc_25><loc_52><loc_88><loc_66></location>S eff /similarequal ∫ d 4 x √ -g [ M 4 2 R ( g ) + L matter -|∇ x ϕ | 2 -(13) -log /epsilon1 4 M 4 4 M 6 5 ( -4 R µν ∇ µ ϕ ∇ ν ϕ + 4 3 R |∇ ϕ | 2 + R µν R µν --R 2 3 + 2 3 |∇ ϕ | 4 +2 | ( ∇ ϕ ) 2 | 2 ) +2 |∇ 2 ϕ | 2 ] +Γ CFT ,</formula>
<text><location><page_18><loc_12><loc_35><loc_88><loc_52></location>Here Γ CFT is the effective action for the holographic CFT on the brane, R ( g ) is the Ricci scalar on the brane, M 2 4 = lM 3 5 = M 2 P plays the role of Planck mass, with l the the curvature radius of the AdS spacetime, and L matter is the Lagrangian density matter localized on the brane. The parameter /epsilon1 determines the renormalization scale of CFT, whereas the field φ corresponds to the zero mode of the bulk complex scalar field Φ localized on the brane ( ϕ could represent squarks or sleptons on the brane carrying baryon/lepton number). Notice that (13) is written as and Hilbert-Einstein action</text>
<formula><location><page_18><loc_39><loc_29><loc_88><loc_35></location>S EH = 1 2 κ 2 ∫ d 4 x √ -g R . (14)</formula>
<text><location><page_18><loc_12><loc_25><loc_88><loc_29></location>plus scalar field (in the so called Jordan frame). In this respect it is similar to scalar tensor theories.</text>
<text><location><page_18><loc_14><loc_22><loc_83><loc_25></location>The current associated to φ , defined as J µ = -iφ ←→ ∇ µ φ ∗ , satisfies the relation[204]</text>
<text><location><page_18><loc_12><loc_10><loc_88><loc_18></location>Assuming a coupling of the form (1), with φ a scalar field on the brane and J µ B replaced by J µ , one obtains (after an integration by parts) that the effective Lagrangian density on interaction is</text>
<formula><location><page_18><loc_29><loc_17><loc_88><loc_23></location>∇ µ J µ /similarequal M 4 4 M 6 5 [ 2 3 J µ ∇ µ R +4 R µν ∇ µ J ν ] + O (log /epsilon1 ) . (15)</formula>
<formula><location><page_18><loc_35><loc_6><loc_88><loc_11></location>L /similarequal M 4 4 M 6 5 φ [ 2 3 J µ ∇ µ R +4 R µν ∇ µ J ν ] . (16)</formula>
<text><location><page_19><loc_12><loc_84><loc_88><loc_91></location>This interaction leads to the baryon asymmetry given in (4). A comparison with (2) suggest M ∗ = fM 3 5 /M 2 4 . To determine the baryon asymmetry one needs to evaluate ˙ R . In the Randall-Sundrum model, the geometrical projection method yields the field equation</text>
<formula><location><page_19><loc_33><loc_79><loc_88><loc_83></location>R µν -1 2 g µν R = 1 M 2 4 T µν + 1 M 6 5 π µν -E µν , (17)</formula>
<text><location><page_19><loc_12><loc_76><loc_83><loc_78></location>where T µν is the energy-momentum tensor on the brane, E µν is the Weyl tensor, and</text>
<formula><location><page_19><loc_25><loc_71><loc_74><loc_75></location>π µν = -1 4 T µα T α ν + 1 12 T µ µ T µν + 1 8 g µν T αβ T αβ -1 24 g µν ( T µ µ ) 2 .</formula>
<text><location><page_19><loc_12><loc_63><loc_88><loc_70></location>In deriving (17) it is assumed the contribution to gravity is dominated by matter field L matter . Notice that the energy-momentum tensor satisfies the continuity equation ∇ µ T µν = 0. The trace of (17) in a FRW Universe reads</text>
<formula><location><page_19><loc_29><loc_58><loc_88><loc_62></location>R = -T µ µ M 2 4 -π ν µ M 6 5 = (1 -3 w ) ρ M 2 4 -(1 + 3 w ) ρ 2 6 M 6 5 , (18)</formula>
<text><location><page_19><loc_12><loc_55><loc_29><loc_56></location>from which it follows</text>
<formula><location><page_19><loc_32><loc_49><loc_88><loc_55></location>˙ R = -3(1 + w ) Hρ [ 1 -3 w M 2 4 -(1 + 3 w ) ρ 3 M 6 5 ] (19)</formula>
<formula><location><page_19><loc_34><loc_46><loc_88><loc_50></location>/similarequal 8 3 Hρ 2 M 6 5 ∼ T 10 M 4 M 6 5 , (20)</formula>
<text><location><page_19><loc_12><loc_37><loc_88><loc_44></location>where (20) follows in a Universe radiation dominated ( w = 1 / 3). One can compute the decoupling temperature[68] T D ∼ M 3 / 2 5 M 1 / 2 4 , so that the bet baryon asymmetry assumes the form</text>
<formula><location><page_19><loc_29><loc_32><loc_88><loc_38></location>η /similarequal 10 -10 ( 10 -3 f ) 2 ( 10 8 GeV M 5 ) 12 ( T D 10 2 . 5 GeV ) 9 , (21)</formula>
<text><location><page_19><loc_12><loc_28><loc_88><loc_32></location>which has been written to emphasize the estimations that the parameters characterizing the theory must assume in order that the observed baryon asymmetry is obtained.</text>
<text><location><page_19><loc_12><loc_22><loc_88><loc_26></location>Other models based on gravitational baryogenesis can be found in[58, 68, 72-78] [61, 7985].</text>
<section_header_level_1><location><page_19><loc_12><loc_17><loc_85><loc_18></location>4. LEPTOGENESIS BY CURVATURE COUPLING OF HEAVY NEUTRINOS</section_header_level_1>
<text><location><page_19><loc_12><loc_7><loc_88><loc_14></location>In this Section, we study the generalization in the matter Lagrangian by including higher order terms in R consistent with general covariance, Lorentz-invariance in a locally inertial frame. The effect of spin-gravity coupling will be neglected (they will be extensively discussed</text>
<text><location><page_20><loc_12><loc_87><loc_88><loc_91></location>in Section 6.1). Therefore we work in the approximation for which the characteristic time of spinor fields variation is smaller than the age of the Universe.</text>
<text><location><page_20><loc_12><loc_81><loc_88><loc_85></location>Consider the action for a four component Dirac fermion ψ which couples to background gravity[86]</text>
<formula><location><page_20><loc_21><loc_75><loc_88><loc_81></location>S m [ g µν , ψ ] = ∫ d 4 x [ i ¯ ψγ µ ( -→ ∂ µ -←-∂ µ ) ψ -h 1 ( R ) ¯ ψψ -ih 2 ( R ) ¯ ψγ 5 ψ ] , (1)</formula>
<text><location><page_20><loc_12><loc_73><loc_69><loc_75></location>where h 1 ( R ) and h 2 ( R ) real valued scalar functions of the curvature,</text>
<formula><location><page_20><loc_30><loc_69><loc_88><loc_71></location>h 1 ( R ) = M + g 1 ( R ) , h 2 ( R ) = M ' + g 2 ( R ) . (2)</formula>
<text><location><page_20><loc_12><loc_63><loc_88><loc_68></location>Here h 1 is a generalization of the neutrino mass term. Note that since ¯ ψγ 5 ψ transforms as a pseudo-scalar, the h 2 term is odd under CP . We write the four-component fermion</text>
<formula><location><page_20><loc_44><loc_54><loc_88><loc_63></location>ψ = ψ L ψ R . (3)</formula>
<text><location><page_20><loc_12><loc_53><loc_73><loc_55></location>The lagrangian in terms of the two-component fields ψ R and ψ L becomes</text>
<formula><location><page_20><loc_28><loc_45><loc_88><loc_52></location>L = iψ † R ¯ σ µ ( -→ ∂ µ -←-∂ µ ) ψ R + iψ † L σ µ ( -→ ∂ µ -←-∂ µ ) ψ L --h 1 ( ψ † R ψ L + ψ † L ψ R ) -i h 2 ( ψ † R ψ L -ψ † L ψ R ) , (4)</formula>
<text><location><page_20><loc_12><loc_40><loc_88><loc_44></location>where σ µ = ( I, σ i ) and ¯ σ µ = ( I, -σ i ) in terms of the Pauli matrices. The h 2 term can be rotated away by a chiral transformation</text>
<formula><location><page_20><loc_37><loc_35><loc_88><loc_38></location>ψ L → e -iα/ 2 ψ L ψ R → e iα/ 2 ψ R . (5)</formula>
<text><location><page_20><loc_12><loc_32><loc_88><loc_33></location>Keeping terms to the linear order in α , we see that the lagrangian (4) changes by the amount</text>
<formula><location><page_20><loc_25><loc_23><loc_88><loc_30></location>δ L = -ψ † R ¯ σ µ ψ R ∂ µ α + ψ † L σ µ ψ L ∂ µ α --h 1 ( iα ) ( ψ † L ψ R -ψ † R ψ L ) -ih 2 ( iα ) ( ψ † L ψ R + ψ † R ψ L ) . (6)</formula>
<text><location><page_20><loc_12><loc_18><loc_88><loc_23></location>Now we choose α = -h 2 /h 1 to eliminate the chiral mass term and obtain for the total Lagrangian</text>
<formula><location><page_20><loc_28><loc_5><loc_88><loc_17></location>L = iψ † R ¯ σ µ ( -→ ∂ µ -←-∂ µ ) ψ R + iψ † L σ µ ( -→ ∂ µ -←-∂ µ ) ψ L --ψ † L σ µ ψ L ∂ µ ( h 2 h 1 ) + ψ † R ¯ σ µ ψ R ∂ µ ( h 2 h 1 ) --1 h 1 ( h 2 1 + h 2 2 ) ( ψ † L ψ R + ψ † R ψ L ) . (7)</formula>
<text><location><page_21><loc_12><loc_81><loc_88><loc_91></location>If h 1 and h 2 are constants then, one can always rotate the axial-mass term away. We will assume that the neutrino mass M /greatermuch g 1 therefore h 1 /similarequal M and since a constant M ' can be rotated away h 2 = g 2 . Further we will assume that the background curvature is only dependent on time. The lagrangian (7) then reduces to the form</text>
<formula><location><page_21><loc_28><loc_68><loc_88><loc_80></location>L = iψ † R ¯ σ µ ( -→ ∂ µ -←-∂ µ ) ψ R + iψ † L σ µ ( -→ ∂ µ -←-∂ µ ) ψ L --ψ † L ψ L ( ˙ g 2 M ) + ψ † R ψ R ( ˙ g 2 M ) --M ( ψ † L ψ R + ψ † R ψ L ) . (8)</formula>
<text><location><page_21><loc_12><loc_67><loc_85><loc_69></location>The equation of motion for the left and the right helicity fermions derived from (8) are</text>
<formula><location><page_21><loc_34><loc_57><loc_88><loc_67></location>i ¯ σ µ ∂ µ ψ R + ( ˙ g 2 M ) ψ R -Mψ L = 0 , iσ µ ∂ µ ψ L -( ˙ g 2 M ) ψ L -Mψ R = 0 . (9)</formula>
<text><location><page_21><loc_12><loc_55><loc_88><loc_57></location>Written in momentum space ψ ( x ) = ψ ( p ) e i ( Et -/vector p · /vectorx ) the equation of motion of ψ R and ψ L are</text>
<formula><location><page_21><loc_33><loc_45><loc_88><loc_55></location>( E R -˙ g 2 M ) ψ R -/vectorσ · /vector pψ R -Mψ L = 0 , ( E L + ˙ g 2 M ) ψ L + /vectorσ · /vector pψ L -Mψ R = 0 . (10)</formula>
<text><location><page_21><loc_12><loc_40><loc_88><loc_45></location>In the limit p /greatermuch M,g 2 the dispersion relations are[87] E R,L /similarequal E p ± 2 ˙ g 2 M , where E p = p + M 2 2 p . The canonical momenta of the ψ L and ψ R fields are as usual</text>
<formula><location><page_21><loc_34><loc_35><loc_88><loc_39></location>π L = ∂ L ∂ ˙ ψ L = iψ † L , π R = ∂ L ∂ ˙ ψ R = iψ † R , (11)</formula>
<text><location><page_21><loc_12><loc_32><loc_49><loc_34></location>so that the canonical Hamiltonian density is</text>
<formula><location><page_21><loc_19><loc_20><loc_88><loc_30></location>H ≡ π L ˙ ψ L + π R ˙ ψ R -L = iψ † L ˙ ψ L + iψ † L σ · ∇ ψ L + iψ † R ˙ ψ R -iψ † R σ · ∇ ψ R + M ( ψ † L ψ R + ψ † R ψ L ) + n L ( ˙ g 2 M ) -n R ( ˙ g 2 M ) , (12)</formula>
<text><location><page_21><loc_12><loc_18><loc_88><loc_20></location>where we have introduced the number density operators of the left and right chirality modes,</text>
<formula><location><page_21><loc_39><loc_13><loc_88><loc_16></location>n L ≡ ψ † L ψ L , n R ≡ ψ † R ψ R . (13)</formula>
<text><location><page_21><loc_12><loc_11><loc_64><loc_12></location>The partition function in terms of this effective Hamiltonian is</text>
<formula><location><page_21><loc_35><loc_6><loc_88><loc_9></location>Z = Tr e -β H ≡ Tr e -β ( H 0 -µ L n L -n R µ R ) , (14)</formula>
<text><location><page_22><loc_12><loc_79><loc_88><loc_91></location>where β = 1 /T and H 0 is the free particle Hamiltonian. We see that when ˙ g 2 is non-zero then the effective chemical potential for the left chirality neutrinos is µ L = -˙ g 2 /M and for the right-chirality neutrinos is µ R = ˙ g 2 /M . In the presence of interactions which change ψ L ↔ ψ R at thermal equilibrium there will be a net difference between the left and the right chirality particles,</text>
<formula><location><page_22><loc_27><loc_69><loc_88><loc_78></location>n R -n L = 1 π 2 ∫ d 3 p [ 1 1 + e β ( E p -µ R ) -1 1 + e β ( E p -µ L ) ] (15) = T 2 3 ˙ g 2 M .</formula>
<text><location><page_22><loc_12><loc_64><loc_88><loc_68></location>Here we consider the simplest case in which h 2 and g 2 are linear function of the curvature R ,</text>
<formula><location><page_22><loc_40><loc_60><loc_59><loc_63></location>h 2 ( R ) = g 2 ( R ) = R M P .</formula>
<text><location><page_22><loc_12><loc_55><loc_88><loc_59></location>The axial term in (1) is a CP violating interaction between fermions and the Ricci curvature described by the dimension-five operator [61, 84]</text>
<formula><location><page_22><loc_39><loc_50><loc_88><loc_53></location>L /upslope CP = √ -g 1 m P R ¯ ψiγ 5 ψ. (16)</formula>
<text><location><page_22><loc_12><loc_33><loc_88><loc_48></location>This operator is invariant under Local Lorentz transformation and is even under C and odd under P and conserves CPT . In a non-zero background R , there is an effective CPT violation for the fermions. Take ψ = ( N R , N c R ) T , where N R is a heavy right handed neutrino and N c R a left handed heavy neutrino, which decay into the light neutrinos. Majorana neutrino interactions with the light neutrinos and Higgs relevant for leptogenesis, are described by the lagrangian</text>
<formula><location><page_22><loc_32><loc_27><loc_88><loc_33></location>L = -h αβ ( ˜ φ † N Rα l Lβ ) -1 2 N c R ˜ MN R + h.c. , (17)</formula>
<text><location><page_22><loc_12><loc_14><loc_88><loc_29></location>where ˜ M is the right handed neutrino mass-matrix, l Lα = ( ν α , e -α ) T L is the left-handed lepton doublet ( α denotes the generation), φ = ( φ + , φ 0 ) T is the Higgs doublet. In the scenario of leptogenesis introduced by Fukugita and Yanagida, lepton number violation is achieved by the decays N R → φ + l L and also N R c → φ † + l L c . The difference in the production rate of l L compared to l c L , which is necessary for leptogenesis, is achieved via the CP violation. In the standard scenario, n ( N R ) = n ( N c R ) as demanded by CPT , but</text>
<formula><location><page_22><loc_35><loc_9><loc_65><loc_12></location>Γ( N R → l L + φ ) = Γ( N c R → l c L + φ † )</formula>
<text><location><page_22><loc_49><loc_9><loc_49><loc_11></location>/negationslash</text>
<text><location><page_23><loc_12><loc_84><loc_88><loc_91></location>due to the complex phases of the Yukawa coupling matrix h αβ , and a net lepton number arises from the interference terms of the tree-level and one loop diagrams (see Section 2 and Ref.[88, 89]).</text>
<text><location><page_23><loc_14><loc_81><loc_87><loc_83></location>In this leptogenesis scenario we have that the decay rates of N R and N c R are the same,</text>
<formula><location><page_23><loc_34><loc_77><loc_65><loc_79></location>Γ( N R → l L + φ ) = Γ( N c R → l c L + φ † ) ,</formula>
<text><location><page_23><loc_12><loc_71><loc_88><loc_75></location>but there is a difference between the heavy light and left chirality neutrinos at thermal equilibrium due to the CP violating gravitational interaction (16),</text>
<formula><location><page_23><loc_38><loc_66><loc_88><loc_70></location>n ( N R ) -n ( N c R ) = T 2 3 ˙ R M P M . (18)</formula>
<text><location><page_23><loc_12><loc_56><loc_88><loc_65></location>The N R ↔ N c R interaction can be achieved by the scattering with a Higgs field. A recent example of leptogenesis due heavy neutrino decay with CP violation in a SO(10) model is described in[90]. In standard SO(10) unification, all Standard Model fermions of a given generation together with a right-handed neutrino are in a 16 representation of SO(10),</text>
<formula><location><page_23><loc_36><loc_49><loc_88><loc_54></location>16 f = ( 1 f + ¯ 5 f + ¯ 10 f ) SU (5) = ( N R +( L, d c ) + ( Q,u c , e c )) (19)</formula>
<text><location><page_23><loc_12><loc_40><loc_88><loc_47></location>The charged fermion and Dirac neutrino mass matrices receive contributions from Yukawa couplings of the form 16 f 16 f H (where H = 10 H , 126 H and/or 120 H ). Majorana masses for the right-handed neutrinos are generated either from</text>
<formula><location><page_23><loc_38><loc_35><loc_88><loc_38></location>16 f 16 f 126 H ⊃ y S ' N c R N R (20)</formula>
<text><location><page_23><loc_12><loc_32><loc_75><loc_34></location>or from the non-renormalizable operators suppressed by some mass scale Λ</text>
<formula><location><page_23><loc_35><loc_28><loc_88><loc_31></location>f Λ 16 f 16 f 16 H 16 H ⊃ f Λ S 2 N c R N R . (21)</formula>
<text><location><page_23><loc_12><loc_12><loc_88><loc_27></location>When the GUT Higgs fields S ' or S acquire a vev , a large Majorana mass M is generated for N R which breaks lepton number spontaneously. This following the see-saw mechanism leads to small neutrino masses at low energies. At temperatures larger than the heavy neutrinos and the GUT Higgs masses one there will be helicity flip scattering interactions like S + N R ↔ S + N c R which change the lepton number (as T > M the helicity and the chirality of N R are same). The interaction rate is</text>
<formula><location><page_23><loc_32><loc_6><loc_88><loc_11></location>Γ( SN R ↔ SN c R ) = 〈 n s σ 〉 = 0 . 12 π ( f Λ ) 2 T 3 . (22)</formula>
<text><location><page_24><loc_12><loc_89><loc_85><loc_91></location>The interactions decouple at a temperature T D . The latter is computed via the equality</text>
<formula><location><page_24><loc_43><loc_85><loc_88><loc_87></location>Γ( T D ) = H ( T d ) , (23)</formula>
<text><location><page_24><loc_12><loc_81><loc_68><loc_82></location>where H = ˙ a/a . From (23) one derives the decoupling temperature</text>
<formula><location><page_24><loc_27><loc_75><loc_88><loc_80></location>T D = 13 . 7 π √ g ∗ ( Λ f ) 2 1 M P = 13 . 7 π √ g ∗ ( 〈 S 〉 2 M ) 2 1 M P , (24)</formula>
<text><location><page_24><loc_12><loc_71><loc_40><loc_74></location>where we have used M = f 〈 S 〉 / Λ.</text>
<text><location><page_24><loc_12><loc_67><loc_88><loc_71></location>From the lepton asymmetry (18) and (2) one obtains the value of frozen in lepton asymmetry as</text>
<formula><location><page_24><loc_33><loc_63><loc_88><loc_67></location>η = n ( N R ) -n ( N c R ) s = 15 2 π 2 g ∗ ˙ R ( T D ) T D MM P , (25)</formula>
<text><location><page_24><loc_12><loc_61><loc_34><loc_62></location>This result agrees with[58].</text>
<text><location><page_24><loc_12><loc_32><loc_88><loc_60></location>Some comments are in order. 1) In the case in which the fermion is, for example, an electron one also gets a splitting of energy levels E ( e R ) -E ( e L ), but this does not lead to lepton generation of lepton asymmetry as both e L and e R carry the same lepton number. 2) In principle, one should also take into account primordial perturbations of the gravitational background (characterized mainly by scalar and tensor perturbations) and of the energy density and pressure, characterized by δρ = δT 0 0 and δpδ j i = δT j i (see for example [91]). These perturbations are related as δp = c 2 s δρ , where c 2 s = w + ρdw/dρ is the adiabatic sound speed squared. For relativistic particles w = 1 / 3 and therefore c 2 s = 1 / 3. As a consequence, the trace of the perturbed energy-momentum tensor vanishes (this is not true in presence of anisotropic shear perturbations), so that according to the gravitational leptogenesis mechanism, no net baryon asymmetry can be generated.</text>
<section_header_level_1><location><page_24><loc_14><loc_26><loc_47><loc_27></location>4.1. Avoiding subsequent wash-out</section_header_level_1>
<text><location><page_24><loc_12><loc_11><loc_88><loc_23></location>The light neutrino asymmetry can be erased by the interactions ν L + φ 0 → ν c L + φ † 0 with the standard model Higgs. To prevent the erasure of the lepton asymmetry by Higgs scattering, we must demand that the lightest heavy neutrino mass be lower than the decoupling temperature of the light-neutrino Higgs interaction, which is calculated as follows. The light neutrino masses arise from an effective dimension five operator (26) which is obtained from</text>
<text><location><page_25><loc_12><loc_89><loc_42><loc_91></location>(17) by heavy neutrino exchange[92]</text>
<formula><location><page_25><loc_22><loc_82><loc_88><loc_88></location>L = C αβ ( l Lα c ˜ φ ∗ )( ˜ φ † l Lβ ) + h.c. (26) = C αβ 2 M ( l Lαa c /epsilon1 am φ m )( l Lβb /epsilon1 bn φ n ) + C ∗ αβ 2 M ( l Lαa /epsilon1 am φ ∗ m )( l Lβb c /epsilon1 bn φ ∗ n ) .</formula>
<text><location><page_25><loc_12><loc_76><loc_88><loc_81></location>Here ˜ φ ≡ iσ 2 φ ∗ = ( -φ 0 ∗ , φ -) T , /epsilon1 ab is the antisymmetric tensor, and a, b.. denote the gauge SU (2) L indices.</text>
<text><location><page_25><loc_14><loc_74><loc_67><loc_75></location>The ∆ L = 2 interactions that result from the operator (26) are</text>
<formula><location><page_25><loc_39><loc_66><loc_88><loc_72></location>ν L + φ 0 ←→ ν R + φ 0 , ν R + φ 0 ∗ ←→ ν L + φ 0 ∗ . (27)</formula>
<text><location><page_25><loc_12><loc_62><loc_62><loc_65></location>The cross section for the interaction ν Lα + φ 0 ↔ ν Rβ + φ 0 is</text>
<formula><location><page_25><loc_44><loc_59><loc_88><loc_63></location>σ = | C αβ | 2 2 M 2 1 π , (28)</formula>
<text><location><page_25><loc_12><loc_54><loc_88><loc_58></location>In the electroweak era, when the Higgs field in (26) acquires a vev , 〈 φ 0 〉 = v = 174 GeV , this operator gives rise to a Majorana neutrino mass matrix</text>
<formula><location><page_25><loc_43><loc_49><loc_57><loc_52></location>m αβ = 4 v 2 C αβ M ,</formula>
<text><location><page_25><loc_12><loc_46><loc_78><loc_48></location>and the cross section (28) can be expressed in terms of light neutrino masses as</text>
<formula><location><page_25><loc_45><loc_42><loc_88><loc_45></location>σ = | m αβ | 2 32 πv 4 . (29)</formula>
<text><location><page_25><loc_12><loc_38><loc_88><loc_41></location>The interaction rate of the lepton number violating scattering ν L + φ 0 ↔ ν R + φ † 0 is given by</text>
<formula><location><page_25><loc_34><loc_34><loc_66><loc_38></location>Γ( ν L + φ 0 ↔ ν R + φ † 0 ) = 0 . 122 16 π m 2 ν T 3 v 4 .</formula>
<text><location><page_25><loc_12><loc_26><loc_88><loc_33></location>The decoupling temperature T l when the interaction rate Γ( T l ) falls below the expansion rate of the Universe (5. The decoupling temperature T l is obtained from equation Γ( T l ) = H ( T l ), where</text>
<formula><location><page_25><loc_33><loc_23><loc_88><loc_27></location>Γ( T l ) = 0 . 122 π | C αβ | 2 T 3 l M 2 = 1 . 7 √ g ∗ T 2 l M P . (30)</formula>
<formula><location><page_25><loc_37><loc_16><loc_88><loc_22></location>T l = 2 × 10 14 ( 0 . 05 eV m ν ) 2 GeV , (31)</formula>
<text><location><page_25><loc_12><loc_21><loc_26><loc_22></location>It turns out that</text>
<text><location><page_25><loc_12><loc_7><loc_88><loc_16></location>The heavy neutrino decays occur at T /similarequal M /similarequal 10 12 GeV, below the temperature T l /similarequal 2 × 10 14 GeV. At temperatures T ∼ T l the light-neutrino lepton number violating interactions are effective. As a consequence, the lepton number asymmetry from the decay of asymmetric number of heavy neutrino decays is not washed out by Higgs scattering with light neutrinos.</text>
<section_header_level_1><location><page_26><loc_12><loc_87><loc_88><loc_91></location>5. MODELS AND TIME VARYING RICCI CURVATURE IN DIFFERENT COSMOLOGICAL SCENARIOS</section_header_level_1>
<text><location><page_26><loc_12><loc_79><loc_88><loc_84></location>We now discuss some cosmological scenarios in which the gravitational leptogenesis mechanism can be realized.</text>
<section_header_level_1><location><page_26><loc_14><loc_74><loc_67><loc_75></location>5.1. Gravitational Leptogenesis in f ( R ) theories of gravity</section_header_level_1>
<text><location><page_26><loc_12><loc_51><loc_88><loc_71></location>As discussed in the Introduction, the observation that the present phase of the expanding Universe is accelerated has motivated in the last years the developments of many models of gravity which go beyond the general relativity, and therefore the standard cosmological model. Among the different approaches, the f ( R )-theories of gravity have received a great attention. The reason relies on the fact that they allow to explain, via a gravitational dynamics, the observed accelerating phase of the Universe, without invoking exotic matter as sources of dark energy. Moreover, they also provide an alternative approach to expliain Dark Matter problem.</text>
<text><location><page_26><loc_12><loc_40><loc_88><loc_50></location>The Lagrangian density of these models does depend on higher-order curvature invariants(see[7, 8, 13-15, 93, 94] and references therein), such as, for example, R 2 , R µν R µν , R /square R , and so on. Here we focalize our attention to f ( R ) models which are a generic function of the Ricci scalar curvature R</text>
<formula><location><page_26><loc_34><loc_35><loc_88><loc_40></location>S = 1 2 κ 2 ∫ d 4 x √ -g f ( R ) + S m [ g µν , ψ ] . (1)</formula>
<text><location><page_26><loc_12><loc_28><loc_88><loc_35></location>In (1), S m is the action of matter and κ 2 = 8 πG = 8 πM -2 P ( M P /similarequal 10 19 GeV is the Planck mass). Cosmological and astrophysics consequences of (1) have been largely studied in literature [10, 11, 23, 24, 95-113].</text>
<text><location><page_26><loc_12><loc_7><loc_88><loc_27></location>f ( R ) gravity provide scenarios that make these models very attractive. In fact [10]: 1) They allow to unify the early-time Inflation and the later-time acceleration of the Universe owing to the different role of the gravitational terms relevant at small and large scales; 2) DM and DE issues can be treated in a unique and unified setting; 3) They provide a framework for the explanation of hierarchy problem and unification of GUT with gravity. However, solar system tests strongly constraint or rule out many f ( R ) models of gravity. Therefore the form of the generic function f ( R ) must be properly constructed. In this respect, available models are:</text>
<unordered_list>
<list_item><location><page_27><loc_15><loc_88><loc_43><loc_91></location>· The Hu and Sawicki model[111]</list_item>
</unordered_list>
<formula><location><page_27><loc_40><loc_84><loc_88><loc_88></location>f ( R ) = -m 2 c 1 ( R 2 /m 2 ) 2 n 1 + c 2 ( R/m 2 ) 2 n , (2)</formula>
<unordered_list>
<list_item><location><page_27><loc_15><loc_79><loc_40><loc_82></location>· The Starobinsky model[114]</list_item>
</unordered_list>
<formula><location><page_27><loc_33><loc_74><loc_88><loc_80></location>f ( R ) = R + λR st [ ( 1 + R 2 R 2 st ) -d -1 ] -αR 2 , (3)</formula>
<unordered_list>
<list_item><location><page_27><loc_15><loc_70><loc_47><loc_72></location>· The Nojiri and Odintsov model[115]</list_item>
</unordered_list>
<formula><location><page_27><loc_42><loc_65><loc_88><loc_69></location>f ( R ) = R + αR l -βR m 1 + γR n . (4)</formula>
<text><location><page_27><loc_12><loc_57><loc_88><loc_63></location>The parameters c 1 , c 2 , d , m , n , l , λ , R st , α , β and γ entering the above equations are free. Their combinations allow to get a description of cosmic acceleration (early and present) of the Universe .</text>
<text><location><page_27><loc_12><loc_51><loc_88><loc_55></location>A characteristic of the models (2)-(4) is that the R -terms can be expanded in the appropriate regimes, reproducing simplest form of f ( R ). A particular subclass is of the form</text>
<formula><location><page_27><loc_42><loc_47><loc_88><loc_49></location>f ( R ) = R + αR n , (5)</formula>
<text><location><page_27><loc_12><loc_40><loc_88><loc_45></location>where α > 0 has the dimensions [energy] -2( n -1) and n > 0. Particularly interesting is the case n = 2 (referred in literature as Starobinsky's model [116])</text>
<formula><location><page_27><loc_42><loc_36><loc_88><loc_38></location>f ( R ) = R + αR 2 . (6)</formula>
<text><location><page_27><loc_12><loc_26><loc_88><loc_34></location>This model (6) has been studied in the framework of astrophysics and cosmology. For instance, gravitational radiation emitted by isolated system constraints the free parameter to | α | /lessorsimilar (10 17 -10 18 )m 2 [117, 118]. Eot-Wash experiments lead instead to the constraints</text>
<formula><location><page_27><loc_42><loc_22><loc_88><loc_25></location>| α | /lessorsimilar 2 × 10 -9 m 2 . (7)</formula>
<text><location><page_27><loc_12><loc_11><loc_88><loc_20></location>More stringent constraints are provided by the Cosmic Microwave Background (CMB) physics. The amplitude of the curvature perturbation corresponding to (6) is P R /similarequal N 2 k 18 π 1 αm 2 P , with N k ∼ 55. Using the WMAP 5-years data[119] ( P R ∼ 2 . 445 × 10 -9 ), it follows that α is constrained as [8]</text>
<formula><location><page_27><loc_44><loc_6><loc_88><loc_9></location>| α | < 10 -39 m 2 . (8)</formula>
<text><location><page_28><loc_12><loc_87><loc_88><loc_91></location>The bound (8) is obtained in the regime R /greatermuch α -1 (in this regime the model describes the inflationary epoch).</text>
<text><location><page_28><loc_12><loc_71><loc_88><loc_85></location>In these models of f ( R ) gravity is implicitly assumed that the chameleon effect[120] holds, which means that the Compton length λ associated to the characteristic scales, coming out from adding (pertubative) higher order terms to the Hilbert-Eisntein action, are smaller or larger in regions with higher or lower matter density. Typically one assumes that λ is constant, so that the theory is viewed as a local effective theory which is valid for a certain range of parameters.</text>
<section_header_level_1><location><page_28><loc_14><loc_65><loc_45><loc_67></location>5.1.1. Field equations in f ( R ) gravity</section_header_level_1>
<text><location><page_28><loc_12><loc_58><loc_88><loc_62></location>The field equations obtained by the variation of the action (1) with respect to the metric are</text>
<formula><location><page_28><loc_31><loc_54><loc_88><loc_58></location>f ' R µν -f 2 g µν -∇ µ ∇ ν f ' + g µν /square f ' = κ 2 T µν , (9)</formula>
<text><location><page_28><loc_12><loc_52><loc_76><loc_54></location>where the prime stands for the derivative with respect to R . The trace reads</text>
<formula><location><page_28><loc_39><loc_47><loc_88><loc_50></location>3 /square f ' + f ' R -2 f = κ 2 T µ µ , (10)</formula>
<text><location><page_28><loc_12><loc_44><loc_71><loc_45></location>In the spatially flat FRW Universe, Eq. (8), Eqs. (9) and (10) become</text>
<formula><location><page_28><loc_39><loc_38><loc_88><loc_42></location>3 f ' H 2 -Rf ' -f 2 +3 Hf '' ˙ R = κ 2 ρ , (11)</formula>
<formula><location><page_28><loc_29><loc_35><loc_88><loc_38></location>-2 f ' H 2 -f ''' ˙ R 2 + f '' ( Hf '' ˙ R -R ) = κ 2 ( ρ + p ) , (12)</formula>
<formula><location><page_28><loc_31><loc_32><loc_88><loc_35></location>3 f ''' ˙ R 2 +3 f '' R +9 Hf '' ˙ R + f ' R -2 f = κ 2 T , (13)</formula>
<text><location><page_28><loc_12><loc_29><loc_87><loc_31></location>Moreover, the Bianchi identities give a further condition on the conservation of the energy</text>
<formula><location><page_28><loc_42><loc_24><loc_88><loc_28></location>˙ ρ +3 ˙ a a ( ρ + p ) = 0 . (14)</formula>
<text><location><page_28><loc_12><loc_18><loc_88><loc_23></location>In what follows, we shall look for those solutions of field equations such that the scale factor evolves as</text>
<formula><location><page_28><loc_39><loc_15><loc_88><loc_18></location>a ( t ) = a 0 t β , H = β t . (15)</formula>
<text><location><page_28><loc_12><loc_12><loc_42><loc_14></location>The scalar curvature turns out to be</text>
<formula><location><page_28><loc_36><loc_7><loc_88><loc_11></location>R = 6(2 H 2 + ˙ H ) = 6 β (2 β -1) t 2 . (16)</formula>
<text><location><page_29><loc_12><loc_84><loc_88><loc_91></location>The f ( R ) model we concern here is that one of Eq. (5). By using Eqs. (11) and (12) and the usual expression relating the energy density and the pressure, p = wρ , where w is the adiabatic index, one gets</text>
<formula><location><page_29><loc_30><loc_78><loc_88><loc_83></location>w = 1 3 + ς ( t ) , ς ≡ 2 3 β ( β + n A β + A -2 β ) /lessmuch 1 , (17)</formula>
<text><location><page_29><loc_12><loc_76><loc_15><loc_77></location>with</text>
<formula><location><page_29><loc_33><loc_72><loc_67><loc_75></location>A ≡ αR n -1 [ β (2 -n ) -( n -1)(2 n -1)] .</formula>
<text><location><page_29><loc_12><loc_70><loc_45><loc_71></location>The energy density ρ assumes the form</text>
<formula><location><page_29><loc_41><loc_63><loc_88><loc_69></location>κ 2 ρ = 3 β 2 t 2 ( 1 + A β ) . (18)</formula>
<text><location><page_29><loc_12><loc_53><loc_88><loc_63></location>Notice that during the radiation dominated era ( β = 1 / 2), to which we are mainly interested, the quantity A vanishes because R = 0, as well as the perturbation ς , and the adiabatic index reduces to the standard value w = 1 / 3. Moreover, our concern is for the regime αR n -1 /lessorsimilar 1.</text>
<section_header_level_1><location><page_29><loc_14><loc_48><loc_38><loc_49></location>5.1.2. Constraints from BBN</section_header_level_1>
<text><location><page_29><loc_12><loc_37><loc_88><loc_45></location>In BBN one has to consider the weak interaction rate of particles ( p, n, e ± and ν ) in thermal equilibrium. For T /greatermuch Q ( Q = m n -m p , where m n,p are the neutron and proton masses), one gets[121-124] Λ( T ) /similarequal qT 5 , where q = 9 . 6 × 10 -46 eV -4 .</text>
<formula><location><page_29><loc_42><loc_6><loc_88><loc_14></location>∣ ∣ ∣ ∣ δT f T f ∣ ∣ ∣ ∣ < 4 . 7 × 10 -4 . (19)</formula>
<text><location><page_29><loc_12><loc_13><loc_88><loc_37></location>The primordial mass fraction of 4 He is estimated by defining Y p ≡ λ 2 x ( t f ) 1+ x ( t f ) , where λ = e -( t n -t f ) /τ . t f and t n are the time of the freeze-out of the weak interactions and of the nucleosynthesis, respectively, τ /similarequal 887sec is the neutron mean life, and x ( t f ) = e -Q /T ( t f ) is the neutron to proton equilibrium ratio. The function λ ( t f ) represents the fraction of neutrons that decay into protons in the time t ∈ [ t f , t n ]. Deviations from Y p (generated by the variation of the freezing temperature T f ) are given by[125-127] δY p = Y p [( 1 -Y p 2 λ ) ln ( 2 λ Y p -1 ) -2 t f τ ] δT f T f . In the above equation we have set δT ( t n ) = 0 because T n is fixed by the deuterium binding energy. The current estimation on[128] Y p , Y p = 0 . 2476 ± δY p , with | δY p | < 10 -4 , leads to</text>
<text><location><page_30><loc_12><loc_85><loc_88><loc_91></location>The freeze-out temperature T is determined by Λ = H . One gets T = T f (1 + δT f T f ), where T f ∼ 0 . 6 MeV and</text>
<formula><location><page_30><loc_31><loc_81><loc_88><loc_86></location>δT f T f = ς 4 π 15 √ πg ∗ 5 1 qm P T 3 f /similarequal 1 . 0024 ( β -1 2 ) . (20)</formula>
<text><location><page_30><loc_12><loc_79><loc_59><loc_80></location>Equations (20) and (19) implies (see also Ref.[129, 130])</text>
<formula><location><page_30><loc_41><loc_74><loc_88><loc_77></location>2 β -1 /lessorsimilar 9 . 4 × 10 -4 . (21)</formula>
<section_header_level_1><location><page_30><loc_14><loc_69><loc_59><loc_71></location>5.1.3. Gravitational leptogenesis induced by f ( R ) gravity</section_header_level_1>
<text><location><page_30><loc_14><loc_65><loc_64><loc_67></location>Using the definition of Ricci scalar curvature (16), it follows</text>
<formula><location><page_30><loc_41><loc_60><loc_88><loc_64></location>˙ R = -12 β (2 β -1) t 3 . (22)</formula>
<text><location><page_30><loc_12><loc_57><loc_59><loc_59></location>Equation (6) then implies (to leading order in (2 β -1))</text>
<formula><location><page_30><loc_35><loc_53><loc_88><loc_57></location>η = 128 π 2 3 √ 5 β (2 β -1) √ πg ∗ T 5 D M 4 P M /similarequal (23)</formula>
<formula><location><page_30><loc_30><loc_47><loc_69><loc_53></location>/similarequal (2 β -1)3 . 4 × 10 -10 10 12 GeV M ( T D 10 15 GeV ) 5 .</formula>
<text><location><page_30><loc_12><loc_34><loc_88><loc_47></location>An inspection of (23) immediately revels that the observed baryon asymmetry can be obtained, for example, for T D ∼ 10 16 GeV, M ∼ 10 12 GeV (see (11) and for example[131]), provided that 2 β -1 /similarequal 2 × 10 -6 . The value of the heavy neutrino mass M ∼ 10 12 GeV is consistent with the atmospheric neutrino scale m ν = 0 . 05 eV, obtained from the see-saw relation m ν = m 2 D /M with the Dirac mass scale m D ∼ O (10) GeV.</text>
<text><location><page_30><loc_14><loc_21><loc_14><loc_24></location>/negationslash</text>
<text><location><page_30><loc_12><loc_19><loc_88><loc_34></location>The lepton asymmetry generated via (23) is passed on to the light neutrino sector when the heavy neutrino decays at temperature T ∼ M ∼ 10 12 GeV. The effects of washed out are avoided by considering the effective (five dimensional) operator violating the lepton number ∆ L = 2, as before discussed. Notice that the baryon asymmetry is generated both for[86] n = 2 and n = 2 The case n < 0 is excluded because these f ( R ) models of gravity are affected by instability problems[7-9].</text>
<section_header_level_1><location><page_30><loc_14><loc_14><loc_70><loc_15></location>5.2. Time varying Ricci curvature from quantum fluctuations</section_header_level_1>
<text><location><page_30><loc_12><loc_7><loc_88><loc_11></location>In this Section we discuss another interesting cosmological scenario in which a non-zero Ricci curvature is generated in the radiation era by back-reaction of quantum fields. Quan-</text>
<text><location><page_31><loc_12><loc_79><loc_88><loc_91></location>tum effects cannot be ignored because they may modify the dynamics of the Universe evolution. In order to incorporate these back-reaction effects in the cosmic evolution of the Universe, General Relativity requires some modification. Again without a complete theory of quantum gravity, one works assuming a semiclassical theory of gravity[132]. In this context, the Einstein field equations are rewritten as[132, 133]</text>
<formula><location><page_31><loc_34><loc_72><loc_88><loc_77></location>R µν -1 2 g µν R = 8 π M 2 P ( T ( cl ) µν + 〈 T ( QM ) µν 〉 ) (24)</formula>
<text><location><page_31><loc_74><loc_56><loc_74><loc_59></location>/negationslash</text>
<text><location><page_31><loc_12><loc_52><loc_88><loc_72></location>where T ( cl ) µν is the stress energy-momentum tensor for the classical field, T ( QM ) µν represents the energy momentum tensor operator generated by quantum fields, and finally 〈 T ( QM ) µν 〉 = 〈 0 | T ( QM ) µν | 0 〉 represents the regularized expectation value of T ( QM ) µν . During the radiation dominated era, although the trace of the classical energy momentum tensor vanishes, T ( cl ) = 0, the presence of the quantum corrections 〈 0 | T ( QM ) µν | 0 〉 implies that the trace is nonvanishing, and therefore a net baryon asymmetry could be generated by having ˙ R = 0. This trace anomaly comes from the infinite counterterms that must be add to the gravitational action to make the trace finite.</text>
<text><location><page_31><loc_12><loc_44><loc_88><loc_51></location>The dynamical evolution of the gravitational background is assumed to be described by the FRW Universe, Eq. (8). The regularized components of the energy-momentum tensor have the form [132, 134]</text>
<formula><location><page_31><loc_36><loc_40><loc_88><loc_43></location>〈 T ( QM ) µν 〉 = k 1 (1) H µν + k 3 (3) H µν , (25)</formula>
<text><location><page_31><loc_12><loc_38><loc_17><loc_39></location>where</text>
<formula><location><page_31><loc_28><loc_29><loc_88><loc_37></location>(1) H µν = 2 R ; µ ; ν -2 g µν /square R +2 RR µν -R 2 2 g µν , (26) (3) H µν = R α µ R να -2 3 RR µν -1 2 R αβ R αβ g µν + R 2 4 g µν ,</formula>
<text><location><page_31><loc_12><loc_10><loc_88><loc_28></location>/square = ∇ µ ∇ µ , and ; stands for covariant derivative. The coefficients k 1 , 3 are constants and come from the regularization process. Their values strictly depend not only on number and types of fields present in the Universe, but also on the method of regularization. Because the methods of regularization affect the the values of k 1 , 3 and more important because of the uncertainty of what fields were present in the very early Universe, they can be considered as free parameters[133, 134]. The tensor (1) H µν satisfies ∇ µ (1) H µ ν = 0. It is obtained by varying the local action</text>
<formula><location><page_31><loc_36><loc_6><loc_64><loc_11></location>(1) H µν = 2 √ -g δ δg µν ∫ d 4 √ -gR 2 .</formula>
<text><location><page_32><loc_12><loc_76><loc_88><loc_91></location>The infinities in 〈 T ( QM ) 〉 are canceled by adding infinite counterterms in the Lagrangian density that describes the gravitational fields. One of these counterterms if of the form √ -gCR 2 , and due to (the logarithmically divergent) constant C , the coefficients k 1 is arbitrary(actually it can be fixed experimentally[205]). As regards (3) H µν , it is covariantly conserved only for conformal flat spacetimes, and cannot be derived by means of the variation of a local action, as for (1) H µν . The coefficient k 3 is given by</text>
<text><location><page_32><loc_12><loc_66><loc_88><loc_71></location>For a SU (5) model, for example, the number of quantum fields take the values N 0 = 34, N 1 / 2 = 45, and N 1 = 24, so that[134] k 3 /similarequal 0 . 07.</text>
<formula><location><page_32><loc_34><loc_71><loc_66><loc_76></location>k 3 = 1 1440 π 2 ( N 0 + 11 2 N 1 / 2 +31 N 1 ) .</formula>
<text><location><page_32><loc_14><loc_64><loc_70><loc_66></location>The explicit expression of the components of (1) H µν and (3) H µν are</text>
<formula><location><page_32><loc_29><loc_61><loc_88><loc_63></location>(1) H = 18(2 HH + ˙ H 2 +10 ˙ HH 2 ) , (27)</formula>
<formula><location><page_32><loc_29><loc_54><loc_71><loc_62></location>00 (1) H ij = 6 ( 2 d 3 H dt 3 +12 HH +14 ˙ HH 2 +7 ˙ H 2 ) g ij , (3) H 00 = 3 H 4 , (3) H 00 = H 2 (4 ˙ H +3 H 2 ) g ij .</formula>
<text><location><page_32><loc_12><loc_51><loc_72><loc_53></location>Applying the regularization procedure one infers the trace anomaly[206]</text>
<text><location><page_32><loc_12><loc_45><loc_50><loc_46></location>that for the FRW Universe assumes the form</text>
<formula><location><page_32><loc_32><loc_46><loc_88><loc_52></location>〈 T ( QM ) µ µ 〉 = k 3 ( R 2 3 -R αβ R αβ ) -6 k 1 /square R, (28)</formula>
<formula><location><page_32><loc_28><loc_35><loc_88><loc_45></location>〈 T ( QM ) 〉 = 36 k 1 ( d 3 H dt 3 +7 HH +4 ˙ H 2 +12 ˙ HH 2 ) + +12 k 3 H 2 ( ˙ H + H 2 ) . (29)</formula>
<text><location><page_32><loc_12><loc_21><loc_88><loc_36></location>During the radiation dominated era, as we have seen, the trace of the energy-momentum tensor of classical fields vanishes T ( cl ) = ρ -3 p = 0. The trace anomaly instead gives a non vanishing contribution. To evaluate it, we point out that both the scale factor a ( t ) = ( a 0 t ) 1 / 2 and the relation between the cosmic time and the temperature T , Eq. (6), should be modified by the back-reaction effects induced by quantum fields. As we shall see below the evolution of the Universe can be described by standard cosmology. In a FRW Universe, the modified</text>
<text><location><page_32><loc_12><loc_18><loc_46><loc_20></location>Einstein field equations assume the form</text>
<formula><location><page_32><loc_27><loc_13><loc_88><loc_18></location>3 H 2 = 8 π M 2 P [ ρ +18 k 1 (2 HH + ˙ H 2 +10 ˙ HH 4 ) + 3 k 3 H 4 ] , (30)</formula>
<formula><location><page_32><loc_34><loc_5><loc_88><loc_10></location>+ k 3 H 2 (4 ˙ H +3 H 2 )] , (32)</formula>
<formula><location><page_32><loc_22><loc_9><loc_88><loc_14></location>3 H 2 +2 ˙ H = 8 π M 2 P [ -p +6 k 1 ( 2 d 3 H dt 3 +12 HH +14 ˙ HH 2 +7 ˙ H 2 ) + (31)</formula>
<text><location><page_33><loc_12><loc_89><loc_29><loc_91></location>from which it follows</text>
<formula><location><page_33><loc_28><loc_79><loc_88><loc_89></location>2 H 2 + ˙ H = 8 π M 2 P [ 6 k 1 ( d 3 H dt 3 +7 HH +4 ˙ H 2 +12 ˙ HH 2 ) +2 k 3 H 2 ( ˙ H + H 2 ) ] . (33)</formula>
<text><location><page_33><loc_12><loc_79><loc_45><loc_80></location>We are looking for solutions of the form</text>
<formula><location><page_33><loc_41><loc_75><loc_88><loc_77></location>H ( t ) = H 0 ( t ) + δ ( t ) , (34)</formula>
<text><location><page_33><loc_85><loc_65><loc_85><loc_68></location>/negationslash</text>
<text><location><page_33><loc_12><loc_61><loc_88><loc_73></location>where δ ( t ) /lessmuch 1 is a perturbation, and H 0 = 1 / 2 t is the Hubble parameter for a radiation dominated universe. The Ricci curvature vanishes, R = 0, as well as its covariant and (cosmic) time derivatives. It then follows that (1) H µν ( H = H 0 ) = 0 whereas (3) H µν = 0 when H = H 0 (see[84] for details). Inserting H given in (34) into Eq. (33), one obtains (to leading order) the solution for δ :</text>
<formula><location><page_33><loc_40><loc_57><loc_88><loc_60></location>δ ( t ) /similarequal k 3 M 2 P 1 t 3 -C 4 M 2 P 1 t 4 . (35)</formula>
<text><location><page_33><loc_12><loc_49><loc_88><loc_56></location>C is a constant of integration. As it can be seen, the M -2 P suppresses considerably the effects of δ on the dynamics of the Universe evolution, and these terms wash-out for large t . The trace anomaly (29) reads</text>
<formula><location><page_33><loc_43><loc_45><loc_57><loc_49></location>〈 T ( QM ) 〉 = -3 k 3 4 t 4 .</formula>
<text><location><page_33><loc_12><loc_40><loc_88><loc_45></location>From R = -8 π M 2 P 〈 T ( QM ) 〉 we find that the parameter characterizing the heavy neutrino asymmetry (25) assumes the form</text>
<formula><location><page_33><loc_30><loc_35><loc_88><loc_40></location>η = k 3 180 πg ∗ √ 32 5 π 15 g 5 ∗ 90 5 T 10 MM 9 P /similarequal 2 . 5 k 3 10 7 T 9 M 1 M 8 P . (36)</formula>
<text><location><page_33><loc_12><loc_19><loc_88><loc_34></location>According to the leptogenesis scenario, the heavy neutrino asymmetry freezes at the decoupling temperature T D when the lepton-number violating interactions ( N R ↔ N c R ) go out of equilibrium. The subsequent decays of these heavy neutrinos into the light standard model particles and the conversion of lepton asymmetry into baryon asymmetry can explain the observed baryon asymmetry of the Universe. In fact, for T D ∼ (10 16 -2 × 10 16 )GeV and M ∼ (10 9 -10 12 ), respectively, one gets η ∼ 10 -11 -10 -10 .</text>
<text><location><page_33><loc_12><loc_14><loc_88><loc_18></location>Let us finally compute the energy density of back-reaction of quantum fields and compare it with the energy density of radiation. From Eqs. (25) it follows</text>
<formula><location><page_33><loc_25><loc_6><loc_88><loc_14></location>〈 ρ 〉 = 〈 T ( QM ) 00 〉 = 18 k 1 ( 2 HH + ˙ H 2 +10 ˙ HH 2 ) +3 k 3 H 4 = (37) = 3 k 3 8 t 4 .</formula>
<text><location><page_34><loc_12><loc_89><loc_70><loc_91></location>The k 1 -term vanishes identically. The total energy density is given by</text>
<formula><location><page_34><loc_31><loc_84><loc_88><loc_88></location>ρ = ρ r + 〈 ρ 〉 = ρ 0 a 4 + A qf a 8 , A qf ≡ 3 k 3 8 a 4 0 . (38)</formula>
<text><location><page_34><loc_12><loc_78><loc_88><loc_83></location>where ρ r is the classical radiation defined in (5). The ratio between the energy densities 〈 ρ 〉 and ρ r reads</text>
<text><location><page_34><loc_12><loc_72><loc_36><loc_74></location>where we have definite T ∗ as</text>
<formula><location><page_34><loc_30><loc_65><loc_69><loc_71></location>T ∗ ≡ [ 80 k 3 π 4 g ∗ ( 15 16 ) 2 ] 1 / 4 M P /similarequal 1 k 1 / 4 3 10 18 GeV .</formula>
<text><location><page_34><loc_12><loc_54><loc_88><loc_64></location>For temperatures T < T ∗ we have that r < 1, i.e. the energy density of quantum fields is subdominant with respect to the energy density of the radiation. In particular, since the decoupling temperature of heavy neutrinos occurs at GUT scales, T D ∼ 10 16 GeV, we infer r ∼ 10 -9 /lessmuch 1 and the back-reaction is indeed subdominant over the radiation density.</text>
<section_header_level_1><location><page_34><loc_14><loc_49><loc_60><loc_50></location>5.3. Gravitational Leptogenesis in Warm Inflation</section_header_level_1>
<text><location><page_34><loc_12><loc_37><loc_88><loc_46></location>A further application of the gravitational leptogenesis scenario is the warm inflation [137] models as there is a large non-zero Ricci curvature from the inflaton potential during inflation and a large temperature where the lepton number violating interaction can be at equilibrium.</text>
<text><location><page_34><loc_12><loc_13><loc_88><loc_36></location>Let us recall the central point underlying the warm inflation idea [138]. In the Inflationary dynamics the scalar field carries most of the energy of the Universe. The inflaton however also interact with other fields, but these interactions plays no role except to give rise to modifications to the effective scalar field through quantum corrections. In the warm inflation scenario, instead, the effect of these interactions is not only to modify the scalar field potential, but also to generate dissipation and fluctuation effects. In order that warm inflation works, it is required that the time scale of quantum mechanical processes leading to the dissipation is much slower than the expansion rate of the Universe (in such a way the whole system, inflaton and fields, would not equally distribute the available energy).</text>
<text><location><page_34><loc_12><loc_7><loc_88><loc_12></location>The Ricci scalar is related to the Hubble expansion rate during inflation as R = -12 H 2 and its time derivative is related to the slow roll parameter /epsilon1 = -˙ H/H 2 as ˙ R = 24 /epsilon1H 3 . The</text>
<formula><location><page_34><loc_42><loc_74><loc_88><loc_79></location>r ≡ 〈 ρ 〉 ρ r = ( T T ∗ ) 4 , (39)</formula>
<text><location><page_35><loc_12><loc_89><loc_51><loc_91></location>lepton asymmetry (25) in warm inflation reads</text>
<formula><location><page_35><loc_31><loc_84><loc_88><loc_88></location>η wI /similarequal A wI /epsilon1H 3 M P T l M , A wI ≡ 180 g ∗ /similarequal O (1) . (40)</formula>
<text><location><page_35><loc_12><loc_81><loc_56><loc_83></location>T l is the light neutrino decoupling temperature (31).</text>
<text><location><page_35><loc_12><loc_76><loc_88><loc_80></location>The power spectrum of curvature perturbation in thermal inflation and the spectral index of scalar perturbations are expressed in terms of H and /epsilon1 [139]</text>
<formula><location><page_35><loc_29><loc_69><loc_88><loc_75></location>P R = ( π 16 ) 1 / 2 H 1 / 2 Γ 1 / 2 T M 2 P /epsilon1 , n s -1 = -27 4 H Γ /epsilon1 , (41)</formula>
<text><location><page_35><loc_12><loc_59><loc_88><loc_69></location>where Γ is the damping parameter in the inflaton equation of motion and represents the coupling between the inflaton and the thermal bath. The WMAP observations [140] provides the amplitude of the curvature power spectrum and of the spectral index, which are given by P R = (2 . 3 ± 0 . 3) × 10 -9 and the spectral index n s = 0 . 951 ± 0 . 017, respectively.</text>
<text><location><page_35><loc_14><loc_57><loc_76><loc_59></location>Combining (40) and (41) one can write /epsilon1 , Γ and M in terms of H and T :</text>
<formula><location><page_35><loc_35><loc_47><loc_88><loc_57></location>Γ = ( 27 √ π 16 ) 2 H 3 T 2 M 4 P 1 P 2 R (1 -n s ) 2 , /epsilon1 = 27 √ π 16 H 2 T 2 M 4 P 1 P 2 R (1 -n s ) , (42)</formula>
<formula><location><page_35><loc_34><loc_43><loc_88><loc_48></location>M = A wI 27 √ π 16 H 5 T M 5 P η wI 1 P 2 R (1 -n s ) , (43)</formula>
<text><location><page_35><loc_12><loc_27><loc_88><loc_42></location>Choosing H /similarequal 8 × 10 12 GeV and T /similarequal 8 × 10 12 GeV, and using, consistently with WMAP data, P R ∼ 2 × 10 -9 and n s = 0 . 968, the net baryon asymmetry η wI /similarequal 10 -10 in the warm inflation scenario follows provided Γ = 7 . 1 × 10 9 GeV, /epsilon1 = 4 . 2 × 10 -6 , and M = 2 . 7 × 10 11 GeV. Finally, from (31) one gets that T l ∼ 10 13 GeV corresponds to the neutrino mass of m ν 3 = 0 . 15eV. Other models on baryo/leptogenesis in warm Inflation scenario have been proposed in [141, 142].</text>
<section_header_level_1><location><page_35><loc_12><loc_19><loc_88><loc_23></location>6. LEPTOGENESIS INDUCED BY SPIN-GRAVITY COUPLING OF NEUTRINOS WITH THE PRIMORDIAL GRAVITATIONAL WAVES</section_header_level_1>
<text><location><page_35><loc_12><loc_7><loc_88><loc_16></location>The behavior of (relativistic) quantum systems in gravitational fields, as well as in inertial fields, play a crucial role for investigating the structure of spacetime at the quantum level [143]. Quantum objects are in fact finer and more appropriate probes of structures that appear classically as results of limiting procedures. On fundamental ground, it is expected</text>
<text><location><page_36><loc_12><loc_71><loc_88><loc_91></location>that only a quantum theory of gravity will be able to provide a definitive answer to questions regarding the fundamental structure of spacetime. However, the extrapolation of General Relativity from ordinary terrestrial scales to Planck's scales is not free from subtle questions and new data coming from modern cosmology, as discussed in the Introduction and previous Sections. Therefore, one considers the gravitational background described by General Relativity, hence considered as a classical filed, and matter as quantized fields propagating in a classical background [132]. In this respect, one is there considering considering the interaction of classical inertial and gravitational fields with (relativistic) quantum objects.</text>
<text><location><page_36><loc_12><loc_44><loc_88><loc_70></location>Observations performed in [144] do confirm that both gravity and inertia interact with quantum systems in ways that are compatible with General Relativity. This, per se, does not represent a test of General Relativity, but shows that the effects of inertia and gravity on wave functions are consistent with covariant generalization of wave equations dictated by General Relativity paradigms. Clearly, to test fundamental theories a central role is played, at level of terrestrial experiments (Earth-bound and near-space experiment), by inertial effects. Their identifications is therefore required with great accuracy and represents a big challenge for future experiments. Inertial effects, on the other hands, provide a guide in the study of relativity because, in all physical situations in which non-locality is not an issue, the equivalence principle ensures the existence of a gravitational effect for each inertial effect.</text>
<text><location><page_36><loc_12><loc_18><loc_88><loc_43></location>Certainly the study of spin-gravity and spin-inertia coupling (as well as spin precession) represents a very active and relevant topics of physics. Experiments of high energy physics indeed typically involve spin-1/2 particles and take place or in a gravitational environment or in non-inertial frames. Thanks to the progress of technology, for example, atomic interferometry and the physics of polarized systems, the effects of the interactions of relativistic quantum particle with gravitation field, i.e. the spin-gravity coupling effects, could be provide new insights of QFT in curved spacetime. Spin-inertia and spin-gravity interactions and their effects in different physical situations are the subject of numerous theoretical (see for example [143, 145-169] and references therein) and experimental efforts[170-174]. Spin precession in inertial and gravitational fields have been studies in[175-180].</text>
<text><location><page_36><loc_12><loc_7><loc_88><loc_17></location>In this Section we shall discuss the effect of spin gravity coupling in a cosmological context. In particular, we are going to discuss the mechanism of Leptgenesis induced by spin-gravity coupling of neutrinos with the cosmological background [83, 181]. The approach is based on QFT in curved spacetime, and in particular we write down the Dirac equation</text>
<text><location><page_37><loc_12><loc_76><loc_88><loc_91></location>in an expanding Universe. To this aim, we use the vierbein formalism. We assume that the early phase of the Universe is described by Inflation which generates the gravitational waves (tensor modes). The latter split in the energy levels of (Majoarana) neutrinos and antineutrinos, which ultimately results in the creation of a lepton asymmetry in the presence of lepton number violating interactions. This mechanism gives rise to the generation of a net leptogenesis.</text>
<section_header_level_1><location><page_37><loc_14><loc_71><loc_84><loc_72></location>6.1. Dirac equation in curved space-time and the fermion dispersion relation</section_header_level_1>
<text><location><page_37><loc_12><loc_56><loc_88><loc_68></location>In passing from flat to curved space time we use the standard prescription ∂ →∇ , and η µν → g µν . The procedure to replace flat space tensors with 'curved space' tensor cannot be extended to the case of spinors. This procedure works with tensors because the tensor representations of GL (4 , R ), i.e. the group of 4 × 4 real matrices, behave like tensors under the subgroup SO (3 , 1). Thus, considering the vector representation, as an example, one gets</text>
<formula><location><page_37><loc_27><loc_51><loc_72><loc_55></location>V ' µ ( x ' ) = ∂x ' µ ∂x ν V ν ( x ) x ' =Λ µ ν x ν ←→ V ' µ ( x ' ) = Λ µ ν V ν ( x )</formula>
<text><location><page_37><loc_12><loc_41><loc_88><loc_50></location>But there are no representation of GL (4 , R ) which behave like spinors under SO (3 , 1), i.e. there not does exists a function of x and x ' which reduces to the usual spinor representation of the Lorentz group ( D (Λ)) for x ' = Λ µ ν x ν . Therefore, to write down the general covariant coupling of spin-1/2 particles to gravity, we have to use the vierbein formalism.</text>
<text><location><page_37><loc_14><loc_38><loc_48><loc_39></location>A vierbein fields (or tretad) is defined as</text>
<formula><location><page_37><loc_41><loc_33><loc_58><loc_37></location>e a µ ( X ) = ∂ξ a ( x ) ∂x µ | x = X</formula>
<text><location><page_37><loc_12><loc_28><loc_88><loc_32></location>where ξ a are the local inertial coordinate, x µ the generic coordinate, ξ a ( x ) → ξ ' a ( x ) = Λ a b ( x ) ξ b ( x ), and Λ T η Λ = η .</text>
<text><location><page_37><loc_12><loc_19><loc_88><loc_27></location>The quantities e a µ ( x ) constitute a set of four coordinate vectors which form a basis for the (flat) tangent space to the curved space at the point x = X . Under the coordinate transformation x → x ' = x ( x ), the vierbeins e a µ ( x ) transform as (see Table III)</text>
<formula><location><page_37><loc_32><loc_16><loc_68><loc_19></location>e ' a µ ( x ' ) = ∂x ν ∂x ' µ e a ν ( x ) , ( ξ ' a ( x ' ) = ξ a ( x )) .</formula>
<text><location><page_37><loc_12><loc_10><loc_88><loc_15></location>The metric g µν ( x ) is related to the vierbein fields by the relation g µν ( x ) = η ab e a µ ( x ) e b ν ( x ), where η ab is the Minkowsky metric (in the local inertial frame). It follows</text>
<formula><location><page_37><loc_15><loc_6><loc_85><loc_9></location>δ µ ν = e µ a ( x ) e a ν ( x ) i.e. e a ν ( x ) is the inverse of e ν a ( x ) → η ab = g µν ( x ) e a µ ( x ) e b ν ( x ) .</formula>
<text><location><page_38><loc_12><loc_72><loc_88><loc_91></location>Spinor fields are coordinate scalars which transforms under local Lorentz transformations as ψ α ( x ) → ψ ' α ( x ) = D αβ [Λ( x )] ψ β ( x ), where D αβ [Λ( x )] is the spinor representation of the Lorentz group and ψ α is the component of the spinor ψ (not be confused with general coordinate indices). Since Λ does depend on x , ∂ µ ψ α does not transform like ψ α under local Lorentz transformations. To obtain a Lagrangian invariant under generic coordinate transformation one has to define the covariant derivative D µ ψ α ≡ ∂ µ ψ α -[Ω µ ] αβ ψ β , where [Ω µ ] αβ is the connection matrix. Therefore one requires D µ ψ α → D αβ [Λ( x )] D µ ψ β ( x ), provided</text>
<formula><location><page_38><loc_32><loc_69><loc_68><loc_72></location>Ω ' µ = D (Λ)Ω µ D -1 (Λ) -( ∂ µ D (Λ)) -1 D -1 (Λ)</formula>
<text><location><page_38><loc_12><loc_67><loc_54><loc_68></location>The connection matrix [Ω µ ] αβ ( x ) can be written as</text>
<formula><location><page_38><loc_38><loc_63><loc_61><loc_66></location>[Ω µ ] αβ ( x ) = i 2 [ S ab ] αβ ω ab µ ( x )</formula>
<text><location><page_38><loc_12><loc_54><loc_88><loc_62></location>where S ab = σ ab 2 = i [ γ a ,γ b ] 2 are the generators of the the Lorentz group in the spinor representation and ω a µ b the spin connections. The spinor representation of the Lorentz group can be written as D [Λ( x )] = exp[ -( i/ 2) S ab θ ab ]. The covariant derivative acts on vierbeins as</text>
<formula><location><page_38><loc_37><loc_51><loc_63><loc_54></location>D µ e a ν = ∂ µ e a ν -Γ λ µν e a λ -ω a µ b e b ν</formula>
<text><location><page_38><loc_12><loc_48><loc_70><loc_50></location>and the condition D µ e a ν = 0 allows to determine the spin connections</text>
<formula><location><page_38><loc_38><loc_42><loc_62><loc_48></location>ω bca = e bλ ( ∂ a e λ c +Γ λ γµ e γ c e µ a ) .</formula>
<text><location><page_38><loc_12><loc_39><loc_88><loc_43></location>Therefore, the general covariant coupling of spin 1 / 2 particles to gravity is given by the Lagrangian</text>
<formula><location><page_38><loc_38><loc_36><loc_88><loc_39></location>L = √ -g ( ¯ ψγ a D a ψ -m ¯ ψψ ) (1)</formula>
<text><location><page_38><loc_12><loc_28><loc_88><loc_35></location>where D a = ∂ a -i 4 ω bca σ bc is the covariant derivative before introduced. The Lagrangian is invariant under the local Lorentz transformation of the vierbein and the spinor fields. By using the Dirac matrices properties</text>
<formula><location><page_38><loc_34><loc_24><loc_65><loc_27></location>γ a [ γ b , γ c ] = η ab γ c + η ac γ c -iε dabc γ g γ 5</formula>
<text><location><page_38><loc_12><loc_22><loc_57><loc_23></location>the Lagrangian density (1) can be written in the form</text>
<text><location><page_38><loc_12><loc_15><loc_17><loc_17></location>where</text>
<formula><location><page_38><loc_32><loc_16><loc_88><loc_21></location>L = det( e ) ¯ ψ ( iγ a ∂ a -m -γ 5 γ d B d ) ψ, (2)</formula>
<formula><location><page_38><loc_37><loc_12><loc_88><loc_14></location>B d = /epsilon1 abcd e bλ ( ∂ a e λ c +Γ λ αµ e α c e µ a ) . (3)</formula>
<text><location><page_38><loc_12><loc_6><loc_88><loc_11></location>In a local inertial frame of the fermion, the effect of a gravitational field appears as a axialvector interaction term shown in L .</text>
<section_header_level_1><location><page_39><loc_14><loc_89><loc_69><loc_91></location>6.2. Neutrinos effective Lagrangian in a local inertial frame</section_header_level_1>
<text><location><page_39><loc_12><loc_74><loc_88><loc_86></location>To determine the dispersion relation of neutrinos propagating in a perturbed FRW Universe, we have to compute B d . Perturbations are generated by quantum fluctuations of the inflaton. Notice that for a FRW Universe the B a -term vanishes due to symmetry of the metric. The general form of perturbations on a flat FRW expanding universe can be written as</text>
<formula><location><page_39><loc_24><loc_71><loc_76><loc_73></location>ds 2 = a ( τ ) 2 [(1 + 2 φ ) dτ 2 -ω i dx i dτ -((1 + 2 ψ ) δ ij + h ij ) dx i dx j ]</formula>
<text><location><page_39><loc_12><loc_58><loc_88><loc_70></location>where φ, ψ are scalar fluctuations, ω i the vector fluctuations and h ij the tensor fluctuations of the metric. Of the ten degrees of freedom in the metric perturbations only six are independent and the remaining four can be set to zero by suitable gauge choice. We work in the TT gauge: h i i = 0, ∂ i h ij = 0. In the TT gauge the perturbed FRW metric can be expressed as</text>
<formula><location><page_39><loc_19><loc_54><loc_81><loc_55></location>ds 2 = a ( τ ) 2 [(1 + 2 φ ) dτ 2 ω dx i dτ (1 + 2 ψ h ) dx 2 (1 + 2 ψ + h ) dx 2</formula>
<formula><location><page_39><loc_38><loc_49><loc_81><loc_55></location>-i --+ 1 -+ 2 -2 h × dx 1 dx 2 -(1 + 2 ψ ) dx 2 3 ] .</formula>
<text><location><page_39><loc_12><loc_46><loc_62><loc_48></location>An orthogonal set of vierbiens e a µ for this metric is given by</text>
<formula><location><page_39><loc_21><loc_33><loc_77><loc_46></location>e a µ = a ( τ ) 1 + φ -ω 1 -ω 2 -ω 3 0 -(1 + ψ ) + h + / 2 h × 0 0 0 -(1 + ψ ) -h + / 2 0 0 0 0 -(1 + ψ ) .</formula>
<text><location><page_39><loc_12><loc_29><loc_88><loc_33></location>For our application we need only the tensor perturbations. Explicit calculations give B a = ( ∂ τ h × , 0 , 0 , ∂ τ h × ). Using ψ = ( ν L , ν R ) T into (1), one gets</text>
<formula><location><page_39><loc_26><loc_20><loc_74><loc_26></location>L = det( e )[( i ¯ ν L γ a ∂ a ν L + i ¯ ν R γ a ∂ a ν R ) + m ¯ ν L ν R + m † ¯ ν R ν L + + B a (¯ ν R γ a ν R -¯ ν L γ a ν L )] .</formula>
<text><location><page_39><loc_12><loc_7><loc_88><loc_19></location>The fermion bilinear term ¯ ψγ 5 γ a ψ is odd under CPT transformation. When one treats B a as a background field then the interaction term in L explicitly violates CPT . When the primordial metric fluctuations become classical, i.e there is no back-reaction of the microphysics involving the fermions on the metric and B a is considered as a fixed external field, then CPT is violated spontaneously. Moreover, we consider only Standard Model fermions,</text>
<text><location><page_40><loc_12><loc_83><loc_88><loc_91></location>so that L ν = +1 for neutrinos and L ¯ ν = -1 for antineutrinos ( L is lepton number), and we consider Majorana spinors ν R = ( ν L ) c , i.e. ν R is the charge-conjugate of ν L . With this choice, the mass term in L is of the Majorana type (the generation index is suppressed).</text>
<section_header_level_1><location><page_40><loc_14><loc_79><loc_87><loc_80></location>6.3. The Leptogenesis mechanism from spin-gravity coupling following Inflation</section_header_level_1>
<text><location><page_40><loc_12><loc_61><loc_88><loc_75></location>According to the general setting, one has to assume that there are GUT processes that violate lepton number above some decoupling temperature T D . Results of Section 4 imply that in the presence of non-zero metric fluctuations, there is a split in energy levels of ν L,R (a different effective chemical potential), so that the dispersion relation of ν L,R fields reads (see also[87]) E L,R ( p ) = E p ∓ ( B 0 -p · B p ) , where as usual E p = p + m 2 / 2 p and p = | p | .</text>
<formula><location><page_40><loc_42><loc_54><loc_88><loc_59></location>∆ n = gT 3 6 ( B 0 T ) . (4)</formula>
<text><location><page_40><loc_12><loc_58><loc_88><loc_62></location>The equilibrium value of lepton asymmetry generated for all T > T D turns out to be (Eq. (15))</text>
<text><location><page_40><loc_12><loc_49><loc_88><loc_54></location>where ∆ n ≡ n ( ν L ) -n ( ν R ) and p /greatermuch m ν and B 0 /lessmuch T has been used. The dependence on B drops out after angular integration</text>
<text><location><page_40><loc_12><loc_44><loc_88><loc_49></location>To evaluate B 0 , we need to compute the spectrum of gravitational waves h ( x , τ ) during inflation. To this aim, we express h × in terms of the creation- annihilation operator</text>
<formula><location><page_40><loc_26><loc_38><loc_88><loc_44></location>h ( x , τ ) = √ 16 π aM p ∫ d 3 k (2 π ) 3 / 2 ( a k f k ( τ ) + a † -k f ∗ k ( τ ) ) e i k · x , (5)</formula>
<text><location><page_40><loc_12><loc_31><loc_88><loc_38></location>where k is the comoving wavenumber, k = | k | , and m P = 1 . 22 10 19 GeV is the Planck mass. Remembering that B 0 = ∂ τ h one gets that the two point correlation function for B 0 is (see C for details)</text>
<formula><location><page_40><loc_21><loc_24><loc_78><loc_30></location>〈 B 0 ( x , τ ) , B 0 ( x , τ ) 〉 = ∫ dk k ( k a ) 2 ( h rad k ) 2 = 4 π ( H I m 2 P T 2 1 . 67 √ g ∗ ) 2 N</formula>
<text><location><page_40><loc_12><loc_22><loc_87><loc_24></location>Therefore, the r.m.s value of spin connection that determines the lepton asymmetry ∆ n is</text>
<formula><location><page_40><loc_29><loc_16><loc_71><loc_22></location>( B 0 ) rms ≡ √ 〈 B 2 0 〉 = 2 √ π ( H I m 2 P T 2 1 . 67 √ g ∗ ) √ N .</formula>
<text><location><page_40><loc_12><loc_12><loc_88><loc_16></location>The lepton asymmetry ∆ n as a function of temperature can therefore be expressed as (considering three neutrino flavors)</text>
<formula><location><page_40><loc_27><loc_6><loc_73><loc_11></location>∆ n ( T ) = gT 3 6 ( B 0 ) rms T = 1 √ π (1 . 67 √ g ∗ ) √ N ( T 4 H I m 2 P ) .</formula>
<text><location><page_41><loc_12><loc_89><loc_68><loc_91></location>The lepton number to entropy density ( s = 0 . 44 g ∗ T 3 ) is therefore</text>
<formula><location><page_41><loc_38><loc_84><loc_62><loc_89></location>η ≡ ∆ n ( T ) s ( T ) /similarequal 2 . 14 T H I √ N m 2 P √ g ∗</formula>
<text><location><page_41><loc_41><loc_48><loc_41><loc_51></location>/negationslash</text>
<text><location><page_41><loc_12><loc_47><loc_88><loc_82></location>The lepton asymmetry is evaluated at the decoupling temperature T D . According to general setting, lepton number asymmetry will be generated as long as the lepton number violating interactions are in thermal equilibrium. Once these reactions decouple at some decoupling temperature T D , which we shall determine, the ∆ n ( T ) /s ( T ) ratio remains fixed for all T < T D . To calculate the decoupling temperature of the lepton number violating processes we turn to a specific effective dimension five operator which gives rise to Majorana masses for the neutrinos (26) The ∆ L = 2 interactions that result from the dimension five operator (26) are given in (27). In absence of GWs, it follows that the forward reactions are equal to the backward reactions, and therefore no net lepton number is generated. On the contrary, in presence of GWs, the forward reactions are different by backward reactions, and the energy levels of the left and right helicity neutrinos are no longer degenerate, E L = E R , which implies a difference in the number density of left and right handed neutrinos (at thermal equilibrium) ∆ n = n ( ν L ) -n ( ν R ) = 0. This process continues till the ∆ L = 2 interactions decouple.</text>
<text><location><page_41><loc_12><loc_40><loc_88><loc_45></location>Next step now is to calculate the decoupling temperature T D by using (23). The interaction rate for the interaction ν Lα + φ 0 ↔ ν Rβ + φ 0 is (see (30))</text>
<formula><location><page_41><loc_38><loc_36><loc_62><loc_40></location>Γ = 〈 n φ σ 〉 = 0 . 122 π | C αβ | 2 T 3 M 2</formula>
<text><location><page_41><loc_12><loc_22><loc_88><loc_35></location>In the electroweak era, when the Higgs field in L W acquires a vev , 〈 φ 〉 = (0 , v ) T (where v = 174 GeV ), the five dimensional Weinberg operator gives rise to a neutrino mass matrix m αβ = v 2 C αβ M . This implies that in our calculations, we can substitute the couplings C αβ M in terms of the light left handed Majorana neutrino mass, i.e. C αβ M → m αβ v 2 . The decoupling temperature T D turns out to be</text>
<formula><location><page_41><loc_39><loc_17><loc_61><loc_21></location>T D = 13 . 68 π √ g ∗ v 4 m 2 ν m P ,</formula>
<text><location><page_41><loc_12><loc_12><loc_88><loc_16></location>where m ν is the mass of the (heaviest) neutrino. Substituting T D into the expression for η , we finally obtain the formula for lepton number</text>
<formula><location><page_41><loc_39><loc_6><loc_88><loc_11></location>η = 92 . 0 ( v 4 H I m 2 ν m 3 P ) √ N . (6)</formula>
<text><location><page_41><loc_77><loc_53><loc_77><loc_56></location>/negationslash</text>
<text><location><page_42><loc_12><loc_73><loc_88><loc_91></location>The input parameters we used for our estimations are: a ) The amplitude of the h × ∼ 10 -6 or equivalently the curvature during inflation H I ∼ 10 14 GeV or the scale of inflation is the GUT scale, V 1 / 4 ∼ 10 16 GeV which is allowed by CMB [182]. b ) Neutrino Majorana mass in the atmospheric neutrino scale[183, 184] m 2 ν ∼ 10 -3 eV 2 . c ) Duration of inflation H I t = N ∼ 100 (needed to solve the horizon and entropy problems in the standard inflation paradigm, Eq. (1)). The parameters entering (6) are well within experimentally acceptable limits. The magnitude of baryogenesis is therefore</text>
<formula><location><page_42><loc_25><loc_68><loc_88><loc_73></location>η = 7 . 4 10 -11 H I 4 10 14 GeV 2 . 5 10 -3 eV 2 m 2 ν √ N 10 ∼ 10 -11 -10 -10 , (7)</formula>
<text><location><page_42><loc_12><loc_60><loc_88><loc_66></location>which is compatible with the previous values (3) and (4). According to [39], a lepton asymmetry generated at an earlier epoch gets converted to baryon asymmetry of the same magnitude by the electroweak sphalerons.</text>
<text><location><page_42><loc_12><loc_33><loc_88><loc_59></location>The mechanism here discussed makes use of the standard QFT in curved space-time, which gives rise to the conventional spin gravity coupling of neutrinos with the gravitational background. This leads to the coupling of axial-vector current with the four-vector B a which accounts for the curved background. As we have seen, left-handed and right-handed fields couple differently to gravity and therefore have different dispersion relations (the equivalence principle is hence violated). Moreover, the model uses the Majorana neutrinos because one needs of a violation of lepton number which generates the lepton asymmetry. In order that spin-gravity coupling of neutrinos with the gravitational background be a viable model, one has to assume that the early Universe is described by Inflation. Remarkably, no free parameters are present in the final expression for the lepton asymmetry given by (6).</text>
<section_header_level_1><location><page_42><loc_14><loc_28><loc_71><loc_29></location>6.4. CPT Violation in an Expanding Universe - String Theory</section_header_level_1>
<text><location><page_42><loc_12><loc_18><loc_88><loc_25></location>Recently Ellis, Mavromatos and Sarkar[185, 186] proposed a model in which they explore the possibility to violate CPT in an expanding Universe in the framework of String Theory, and generate a net baryon/lepton asymmetry via Majorana neutrinos.</text>
<text><location><page_42><loc_12><loc_7><loc_88><loc_17></location>The model goes along the following line. According to String Theory, besides to the spin-2 graviton (described by the usual symmetric tensor g µν ), the theory contains also the spin-0 dilaton field (described by the scalar field Φ) and the anti-symmetric tensor KalbRamond field (described by B µν ). The latter enters into the effective action via the field</text>
<text><location><page_43><loc_12><loc_87><loc_88><loc_91></location>H µνρ = ∂ µ B νρ + p.c. , where p.c. stands for the cyclic permutation of the indices { µ, ν, ρ } ( H µνρ plays the role of torsion[187]). The effective action reads (in the Einstein frame)</text>
<formula><location><page_43><loc_21><loc_80><loc_88><loc_86></location>S = M 2 s V c 16 π ∫ d 4 x √ -g ( R ( g ) -2 ∂ µ Φ ∂ µ Φ -e -4Φ 12 H µνρ H µνρ + O ( α ' ) ) . (8)</formula>
<text><location><page_43><loc_12><loc_73><loc_88><loc_81></location>Here M s = 1 / √ α ' represents the string scale mass, V c the compactification volume which is expressed, together with the compact radii, in terms of the √ α ' units. In this model ne gets that the connections are generalized as</text>
<formula><location><page_43><loc_43><loc_68><loc_88><loc_70></location>¯ Γ λ µν = Γ λ µν + T λ µν , (9)</formula>
<text><location><page_43><loc_12><loc_64><loc_53><loc_66></location>where Γ λ µν are the usual Christoffel symbols, and</text>
<formula><location><page_43><loc_40><loc_59><loc_88><loc_62></location>T λ µν ≡ e -2Φ H λ µν = -T λ νµ . (10)</formula>
<text><location><page_43><loc_12><loc_56><loc_47><loc_58></location>The four-vector B d given in (3) now reads</text>
<formula><location><page_43><loc_37><loc_52><loc_88><loc_54></location>B d = /epsilon1 abcd e bλ ( ∂ a e λ c + ¯ Γ λ αµ e α c e µ a ) . (11)</formula>
<text><location><page_43><loc_12><loc_48><loc_88><loc_49></location>The anti-symmetric tensor can be written in terms of the pseudo-scalar axion-like field b ( x )</text>
<formula><location><page_43><loc_41><loc_43><loc_88><loc_45></location>H µνρ = e 2Φ /epsilon1 µνρσ ∂ σ b , (12)</formula>
<text><location><page_43><loc_12><loc_38><loc_78><loc_42></location>where /epsilon1 0123 = √ -g . Field equations of String theory provides the solution [188]</text>
<formula><location><page_43><loc_39><loc_33><loc_88><loc_38></location>b ( x ) = √ 2 e -Φ 0 √ Q 2 M s √ n t , (13)</formula>
<text><location><page_43><loc_12><loc_23><loc_88><loc_33></location>where Φ 0 is a constant appearing in the time evolution of the dilaton field Φ( t ) = -ln t +Φ 0 , Q 2 > 0 is the central charge deficit and n is an integer associated to the Kac-Moody algebra of the underlying world sheet conformal field theory. According to (9)-(13) one finds that the non vanishing component of B d is</text>
<formula><location><page_43><loc_34><loc_16><loc_88><loc_22></location>B 0 = /epsilon1 ijk T ijk = 6 √ 2 Q 2 e -Φ 0 M s √ n GeV . (14)</formula>
<text><location><page_43><loc_12><loc_15><loc_76><loc_16></location>The net baryon asymmetry between the (Majorana) neutrino-antineutrino is</text>
<formula><location><page_43><loc_40><loc_9><loc_88><loc_14></location>η ∼ B 0 T d ∼ e -Φ 0 M s √ Q 2 √ nT d . (15)</formula>
<section_header_level_1><location><page_44><loc_14><loc_89><loc_81><loc_91></location>6.5. Leptogenesis induced by Einstein-Cartan-Sciama-Kibble torsion field</section_header_level_1>
<text><location><page_44><loc_12><loc_72><loc_88><loc_86></location>Another interesting model of matter-antimatter asymmetry has been proposed by Poplawski [189]. It is based on Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity [190] in which the usual Hilbert-Einstein action (14) incorporates the torsion field. The latter therefore extend General Relativity to include matter with intrinsic spin-1/2, which produce torsion, and provides a more general theory of local gauge with respect to the Poincare group [191] (interesting applications of ECSK can be find in [192, 193] and references therein).</text>
<text><location><page_44><loc_14><loc_69><loc_84><loc_70></location>Spinors coupled to the torsion fields evolves according to Helh-Datta equation [194]</text>
<formula><location><page_44><loc_32><loc_64><loc_88><loc_67></location>ie µ a γ a ∇ a ψ -mψ = -3 8 M 2 P ( ¯ ψγ 5 γ a ψ ) γ 5 γ a ψ, (16)</formula>
<text><location><page_44><loc_12><loc_58><loc_88><loc_62></location>where ∇ a represents the covariant derivative with respect to the affine connection (Christoffel connections). The corresponding equation for the charge conjugate (C) spinor ψ C is</text>
<formula><location><page_44><loc_29><loc_53><loc_88><loc_56></location>ie µ a γ a ∇ a ψ C -mψ C = + 3 8 M 2 P ( ¯ ψ C γ 5 γ a ψ c ) γ 5 γ a ψ C , (17)</formula>
<text><location><page_44><loc_12><loc_42><loc_88><loc_51></location>The equations (16) and (17) are therefore different, leading to a different shift of energy spectrum, generating in such a way an asymmetry. In fact, the energy levels for a free fermion ( X ) and antifermion ( ¯ X ) resulting from ECSK theory are (in the ultrarelativistic limit)</text>
<formula><location><page_44><loc_34><loc_38><loc_88><loc_41></location>E X = p + ακN , E ¯ X = p -ακN , (18)</formula>
<text><location><page_44><loc_12><loc_11><loc_88><loc_37></location>where α is a numerical factor of the order of unity, and N is the inverse normalization of Dirac spinor (it is of the order of N ∼ E 3 ∼ T 3 ). Here X and ¯ X refer to heavy fermion carrying baryon and antibaryon number, respectively, and are dubbed archeons and antiarcheons. They candidate for a possible component of DM. Eqs. (2) and (15) then imply that the net baryon asymmetry is η ∼ T 2 D /M 2 P , where T D is the decoupling temperature that must assume the value T D ∼ 10 13 GEV in order to generate the observed baryon asymmetry. One can estimate the mass m X of the (anti)archeon by equating the decay rate Γ ∼ G 2 F m 5 X 192 π 3 with the expansion rate of the Universe during the radiation era, Eq. (5): m X = m ¯ X ∼ 10TeV. This value is of the same order of magnitude of the maximum energy of a proton beam at LHC ( ∼ 7TeV).</text>
<section_header_level_1><location><page_45><loc_12><loc_89><loc_29><loc_91></location>7. CONCLUSION</section_header_level_1>
<text><location><page_45><loc_12><loc_72><loc_88><loc_86></location>Understanding how the baryon asymmetry of the universe originated is one of the fundamental goals of modern cosmology. As we have seen, particle physics, as well as cosmology, have provided with a number of possibilities. They involve very fascinating physics, but with varying degrees of testability. In fact, all the baryogenesis models are indeed able to derive the correct estimation of η ∼ 10 -10 , but it is rather difficult, if not impossible, to exclude or confirm one or other scenario.</text>
<text><location><page_45><loc_12><loc_37><loc_88><loc_70></location>A possibility to discriminate among the plethora of baryogenesis models is to investigate its predictibility or compatibility with certain form of DM (besides the models discussed in this review, see also[195]). In other words, one should expect that a realistic model of baryogenesis is able to determine both the right value of η , and to explain the magnitude of the ratio ρ B /ρ DM fixed by cosmological observations. An example is provided by the AffleckDine Baryogenesis: The model explains the observed baryon asymmetry of the Universe, and supersymmetric particles are favored candidates of DM in the Universe. In this respect, as discussed in this review, gravitational baryogenesis represents also an interesting framework. In fact because if from one side, in these models one is able to recover the correct estimation of the parameter 1 eta , from the other side they are compatible with the present cosmological data of an epanding Universe, and hence the necessity to invoke new form of energy or matter, hence DM and DE. This is the case of f ( R ) gravity, but also variant of these models and more generally, scalar tensor theories are available candidates.</text>
<text><location><page_45><loc_12><loc_8><loc_88><loc_36></location>A question that arises is whether the universe is baryon-symmetric on cosmological scales, and eventually separated into domains which are either dominated by baryons or antibaryons. One then would expect to detect, due to annihilations, an excess of gamma rays. There are no evidences of the existence of such a cosmic anti-matter. In fact, the analysis of p ¯ p annihilation in gamma rays ( p ¯ p → π 0 → 2 γ ), with E γ ∼ 100MeV, allows to conclude that the nearest rich antimatter region (anti-galaxy) should be away from at distance[196, 197] D ≥ (10 -15)Mpc. This results indicate hence that the patches of matterantimatter should be as large as the presently observable Universe. However, no mechanism is known which is able to explain how to separate out these domains of matter-antimmater. There is anyway an existing as well as many planned experimental activity in searching for cosmic antimatter[198-201].</text>
<text><location><page_46><loc_12><loc_71><loc_88><loc_91></location>It is still unclear to cosmologists and particle physicists what scenario was realized in nature to generate the observed baryon asymmetry in the Universe. What has arisen in the last years is that no model of Baryogenesis is complete without incorporating the idea s underlying the Leptogenesis. This is also supported by recent results of neutrino physics. LHC experiment and the new generation of linear colliders will certainly allow a deeper understanding and a considerable progress of this fundamental problem. In any case, the baryon asymmetry in the Universe furnishes a clear evidence that a new physics, beyond the Standard Model of particle physics and the Standard Cosmological model, is called for.</text>
<section_header_level_1><location><page_46><loc_14><loc_65><loc_30><loc_67></location>Acknowledgments</section_header_level_1>
<text><location><page_46><loc_12><loc_55><loc_88><loc_62></location>The authors thank Prof. Dh. Ahluwalia and Prof. E.H. Chion for invitation to write this review. G. Lambiase thanks the University of Salerno and ASI (Agenzia Spaziale Italiana) for partial support through the contract ASI number I/034/12/0.</text>
<section_header_level_1><location><page_46><loc_14><loc_50><loc_50><loc_51></location>Appendix A: Out equilibrium condition</section_header_level_1>
<text><location><page_46><loc_12><loc_43><loc_88><loc_47></location>For completeness, we discuss on a different setting, the departure from thermal equilibrium.</text>
<text><location><page_46><loc_12><loc_32><loc_88><loc_42></location>If all particles in the Universe were in thermal equilibrium then there would be no preferred direction for the time T and also if B asymmetry could be generated, it would be prevented by CPT invariance. Therefore, also the violation of the CP symmetry would be irrelevant.</text>
<text><location><page_46><loc_12><loc_27><loc_88><loc_31></location>Consider a species of massive particle X in thermal equilibrium at temperatures T /lessmuch m X . Let m X its mass. The number density of these particles is</text>
<formula><location><page_46><loc_38><loc_22><loc_62><loc_25></location>n X ≈ g X ( m X T ) 3 / 2 e -m X T + µ X T ,</formula>
<text><location><page_46><loc_12><loc_8><loc_88><loc_20></location>where, as usual, µ X indicate the chemical potential. The species X are in chemical equilibrium if the rate Γ inel of inelastic scatterings (responsible of the variation of the number of X particles in the plasma, according to the processes X + A → B + C ) is larger than the expansion rate of the Universe Γ inel /greatermuch H . This allows to write a relation among the different chemical potentials of the particles involved in the process µ X + µ A = µ B + µ C . The</text>
<text><location><page_47><loc_12><loc_84><loc_88><loc_91></location>number density of the antiparticle ¯ X which have the same mass of particles X , m ¯ X = m X , but opposite chemical potential µ ¯ X = -µ X due to the process X + ¯ X → γ + γ with µ γ = 0, is</text>
<formula><location><page_47><loc_38><loc_80><loc_61><loc_83></location>n ¯ X ≈ g X ( m X T ) 3 / 2 e -m X T -µ X T</formula>
<text><location><page_47><loc_12><loc_78><loc_76><loc_79></location>If the X particle carries baryon number, then B will get a contribution from</text>
<formula><location><page_47><loc_31><loc_73><loc_69><loc_76></location>B ∝ n X -n ¯ X = 2 g X ( m X T ) 3 / 2 e -m X T sinh µ X T .</formula>
<text><location><page_47><loc_12><loc_59><loc_88><loc_71></location>If there exist B -violating reactions (first Sakharov's condition) for the species X and ¯ X , such as X + X → ¯ X + ¯ X , then the chemical potential is zero, µ X = 0. As a consequence, also the relative contribution of the X particles to the net baryon number vanishes. Therefore, only a departure from thermal equilibrium can allow for a finite baryon excess, that means that the form of n X, ¯ X has to be modified,</text>
<text><location><page_47><loc_12><loc_42><loc_88><loc_58></location>The typical example of the out-of equilibrium decay can be represented by the following steps: Let X be a heavy particle such that m X > T at the decay time, and let X → Y + B the decay process. When the energy of the final state is given by E Y + B ∼ O ( T ), then there is no phase space for the inverse decay to occur. The final state Y + B does not have enough energy to create a heavy particle X (the rate for Y + B → X is Boltzmann suppressed, i.e. Γ( Y + B → X ) ∼ e -m X /T ).</text>
<section_header_level_1><location><page_47><loc_14><loc_38><loc_50><loc_39></location>Appendix B: The physics of Sphalerons</section_header_level_1>
<text><location><page_47><loc_12><loc_25><loc_88><loc_35></location>In the EW theory, the most general Lagrangian invariant under the SM gauge group and containing only color singlet Higgs fields is automatically invariant under global abelian symmetries. The latter are associated to the baryonic and leptonic symmetries. It is hence not possible to violate B and L at tree level, as well as in any order of perturbation theory.</text>
<text><location><page_47><loc_12><loc_9><loc_88><loc_24></location>The perturbative expansion, however, does not describe all the dynamics of the theory. 't Hooft provided a scenario in which nonperturbative effects (instantons) may give rise to processes which the combination B + L is violated, whereas the (orthogonal) combination B -L does not. In some circumstances, such as the early Universe at very high temperature, the processes that violate the baryon and lepton number may be fast enough. These processes may be significant role for baryogenesis mechanisms.</text>
<text><location><page_48><loc_12><loc_87><loc_88><loc_91></location>At the quantum level, the baryon and lepton symmetries are anomalous (triangle anomaly)</text>
<text><location><page_48><loc_12><loc_80><loc_82><loc_82></location>where g, g ' are the gauge coupling of SU (2) L and U (1) Y , n f the number of families,</text>
<formula><location><page_48><loc_28><loc_82><loc_71><loc_88></location>∂J µ B = ∂ µ J µ L = n f ( g 2 32 π 2 W a µν ˜ W aµν -g ' 2 32 π 2 F µν ˜ F µν )</formula>
<formula><location><page_48><loc_42><loc_75><loc_58><loc_79></location>˜ W aµν = 1 2 /epsilon1 µναβ W a αβ</formula>
<text><location><page_48><loc_12><loc_72><loc_46><loc_74></location>the dual of SU (2) L field strength tensor,</text>
<formula><location><page_48><loc_43><loc_67><loc_57><loc_71></location>˜ F µν = 1 2 /epsilon1 µναβ F αβ</formula>
<text><location><page_48><loc_12><loc_61><loc_88><loc_65></location>the dual of U (1) Y field strength tensor. The change of baryon number, which is closely related to the vacuum structure of the theory, is given by</text>
<formula><location><page_48><loc_22><loc_55><loc_77><loc_61></location>∆ B = B ( t f ) -B ( t i ) = ∫ t f t i ∫ d 3 x∂ µ J B µ = n f [ N CS ( t f ) -N CS ( t 0 )] ,</formula>
<text><location><page_48><loc_12><loc_53><loc_15><loc_54></location>with</text>
<formula><location><page_48><loc_27><loc_48><loc_72><loc_54></location>N CS ( t ) = g 2 32 π 2 ∫ d 3 x/epsilon1 ijk Tr ( A i ∂ j A k + 2 3 igA i A j A k ) .</formula>
<text><location><page_48><loc_12><loc_23><loc_88><loc_48></location>N CS is the Chern-Simons number. For vacuum to vacuum transition, the field A represent a (pure) gauge configuration, whereas the Chern-Simons numbers N CS ( t f ) and N CS ( t i ) assume integer values. In a non-abelian gauge theory, there are infinitely many degenerate ground states (labeled by the Chern-Simons number ∆ N CS = ± 1 , ± 2 , ± 3 , . . . ). In field space, the corresponding point are separated by a potential barrier. The height of his barrier gives the sphaleron energy E sp . Because the anomaly, jumps in the Chern-Simons numbers are associated with changes of baryon and lepton number ∆ B = ∆ L = n f ∆ N CS . The smallest jump in the Standard Model is characterized by ∆ B = ∆ L = ± 3. In semiclassical approximation, the probability of tunneling between neighboring vacua is determined by instanton configurations.</text>
<text><location><page_48><loc_14><loc_20><loc_83><loc_22></location>In the Standard Model, SU(2) instanton lead to an effective 12-fermion interaction</text>
<formula><location><page_48><loc_38><loc_16><loc_61><loc_18></location>O B + L = Π i =1 , 2 , 3 q Li q Li q Li l Li ,</formula>
<text><location><page_48><loc_12><loc_12><loc_57><loc_14></location>which describes processes with ∆ B = ∆ L = 3, such as</text>
<formula><location><page_48><loc_31><loc_7><loc_69><loc_10></location>u c + d c + c c → d +2 s +2 b + t + ν e + ν µ + ν τ .</formula>
<text><location><page_49><loc_12><loc_71><loc_88><loc_91></location>The transition rate is given by Γ ∼ e -S inst ∼ O (10 -165 ), where S inst is instanton action. Because the rate is extremely small, B + L violating interactions appear completely negligible in the Standard Model. However this is not true in a thermal bath, and hence in the primordial Universe. As emphasized by Kuzmin, Rubakov, and Shaposhnikov, transition between the gauge vacua occurs not by tunneling but through thermal fluctuations over the barrier. For temperature T > E sp , the suppression in the rate provided by Boltzmann factor disappears and therefore processes that violate B + L can occur at a significant rate. In the expanding Universe these processes can be in equilibrium .</text>
<section_header_level_1><location><page_49><loc_14><loc_65><loc_70><loc_67></location>Appendix C: The power spectrum of the Gravitational Waves</section_header_level_1>
<text><location><page_49><loc_12><loc_58><loc_88><loc_62></location>The function h ( x , τ ) ≡ h appearing in (5) satisfies the equation (from Einstein field equation)</text>
<formula><location><page_49><loc_40><loc_54><loc_60><loc_58></location>∂ 2 τ h +2 ˙ a a ∂ τ h + k 2 h = 0 .</formula>
<text><location><page_49><loc_12><loc_52><loc_77><loc_54></location>The mode functions f k ( τ ) obey the minimally coupled Klein-Gordon equation</text>
<formula><location><page_49><loc_40><loc_46><loc_60><loc_52></location>f '' k + ( k 2 -a '' a ) f k = 0 ,</formula>
<text><location><page_49><loc_12><loc_41><loc_88><loc_46></location>where ' = ∂ τ . During de Sitter era, the scale factor a ( τ ) = -1 / ( H I τ ) where H I is the Hubble parameter, so that a solution is</text>
<formula><location><page_49><loc_39><loc_36><loc_61><loc_41></location>f k ( τ ) = e -ikτ √ 2 k ( 1 -i kτ ) ,</formula>
<text><location><page_49><loc_12><loc_25><loc_88><loc_36></location>which matches the positive frequency 'flat space' solutions e -ikτ / √ 2 k in the limit of kτ /greatermuch 1. Substituting this solution in h ( x , τ ) (5), and using the canonical commutation relation [ a k , a † k ' ] = δ kk ' , we get the standard expression for two point correlation of gravitational waves generated by inflation</text>
<formula><location><page_49><loc_34><loc_19><loc_65><loc_25></location>〈 h ( x , τ ) h ( x , τ ) 〉 inf ≡ ∫ dk k ( | h k | 2 ) inf ,</formula>
<text><location><page_49><loc_12><loc_12><loc_88><loc_19></location>with the spectrum of gravitational waves given by the scale invariant form ( | h k | 2 ) inf = 4 π H 2 I m 2 p . Consider now the GWs modes that re-enter the horizon at the radiation era a ( τ ) ∼ τ . One finds that the GWs have the two point correlation fluctuation</text>
<formula><location><page_49><loc_34><loc_6><loc_65><loc_11></location>〈 h ( x , τ ) h ( x , τ ) 〉 rad ≡ ∫ dk k ( | h k | 2 ) rad ,</formula>
<text><location><page_50><loc_12><loc_89><loc_17><loc_91></location>where</text>
<formula><location><page_50><loc_37><loc_85><loc_62><loc_90></location>h rad k = h inf k a ( T ) k T 2 1 . 67 √ g ∗ m P ,</formula>
<text><location><page_50><loc_12><loc_83><loc_88><loc_85></location>and g = 106 . 7 is the number of relativistic degrees of freedom which for the Standard Model</text>
<text><location><page_50><loc_12><loc_57><loc_88><loc_84></location>∗ There is a stringent constraint H I /M p < 10 -5 from CMB data [202]. This constraint limits the parameter space of interactions that can be used for generating the requisite lepton-asymmetry. In the radiation era, when these modes re-enter the horizon, the amplitude redshifts by a -1 from the time of re-entry. The reason is that in the radiation era a ( τ ) ∼ τ and the equation for f k gives plane wave solutions f k = (1 / √ 2 k ) exp ( -ikτ ). Therefore in the radiation era the amplitudes of h redshifts as a -1 . The gravitational waves inside the horizon in the radiation era will be h rad k = h inf k a k a ( τ ) = h inf k T T k , where h inf k are the gravitational waves generated by inflation, a k and T k are the scale factor and the temperature when the modes of wavenumber k entered the horizon in the radiation era. The horizon entry of mode k occurs when</text>
<formula><location><page_50><loc_40><loc_52><loc_60><loc_56></location>a k H k k = a ( T ) T H k T k k = 1 ,</formula>
<text><location><page_50><loc_12><loc_49><loc_17><loc_51></location>where</text>
<formula><location><page_50><loc_41><loc_47><loc_58><loc_49></location>H k = 1 . 67 √ g ∗ T 2 k /M p</formula>
<text><location><page_50><loc_12><loc_41><loc_88><loc_45></location>is the Hubble parameter at the time of horizon crossing of the k the mode. Solving equation for T k we get</text>
<formula><location><page_50><loc_41><loc_37><loc_59><loc_41></location>T k = 1 1 . 67 √ g ∗ k M p a ( τ ) T .</formula>
<text><location><page_50><loc_12><loc_32><loc_88><loc_36></location>The amplitude of the gravitational waves of mode k inside the radiation horizon is (using the equation for T k and the previous expression for</text>
<formula><location><page_50><loc_36><loc_27><loc_64><loc_32></location>h rad h rad k = h inf k a ( T ) k T 2 1 . 67 √ g ∗ M p .</formula>
<text><location><page_50><loc_12><loc_22><loc_88><loc_26></location>Note that the gravitational wave spectrum inside the radiation era horizon is no longer scale invariant. The gravitational waves in position space have the correlation function</text>
<formula><location><page_50><loc_35><loc_16><loc_64><loc_22></location>〈 h ( x , τ ) h ( x , τ ) 〉 rad = ∫ dk k ( h rad k ) 2 ,</formula>
<text><location><page_50><loc_12><loc_12><loc_88><loc_16></location>and hence for the spin connection B 0 generated by the inflationary gravitational waves in the radiation era, we get</text>
<formula><location><page_50><loc_17><loc_6><loc_82><loc_12></location>〈 B 0 ( x , τ ) B 0 ( x , τ ) 〉 = ∫ dk k ( k a ) 2 ( h rad k ) 2 = 4 π ( H I M 2 p T 2 1 . 67 √ g ∗ ) 2 ∫ k max k min dk k .</formula>
<text><location><page_51><loc_12><loc_79><loc_88><loc_91></location>The spectrum of spin-connection is scale invariant inside the radiation horizon. This is significant in that the lepton asymmetry generated by this mechanism depends upon the infrared and ultraviolet scales only logarithmically. The scales outside the horizon are bluetilted which means that there will be a scale dependent anisotropy in the lepton number correlation at two different space-time points</text>
<formula><location><page_51><loc_37><loc_73><loc_63><loc_76></location>〈 ∆ L ( r )∆ L ( r ' ) 〉 ∼ Ak n , n > 0 ,</formula>
<text><location><page_51><loc_12><loc_70><loc_17><loc_72></location>where</text>
<formula><location><page_51><loc_42><loc_67><loc_58><loc_70></location>∆ L ( r ) ≡ L ( r ) -¯ L,</formula>
<text><location><page_51><loc_12><loc_40><loc_88><loc_66></location>is the anisotropic deviation from the mean value. Unlike in the case of CMB, this anisotropy in the lepton number is unlikely to be accessible to experiments. Nucleosynthesis calculations only give us an average value at the time of nucleosynthesis (when T ∼ 1 MeV ). The maximum value of k are for those modes which leave the de Sitter horizon at the end of inflation. If inflation is followed by radiation domination era starting with the re-heat temperature T RH then the maximum value of k in the radiation era (at temperature T ) is given by k max /a ( T ) = H I ( T T RH ) . The lower limit of k is k min = e -N k max which are the modes which left the de-Sitter horizon in the beginning of inflation ( N is the total e-folding of the scale factor during inflation, N /similarequal 55 -70). The integration over k then yields just the factor ln( k max /k min ) = N .</text>
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<list_item><location><page_60><loc_12><loc_7><loc_88><loc_17></location>[203] Inflation is responsible for the inhomogeneities in the matter distribution (their evolution in fact give rise to the formation of structures, stars, planets) and the inhomogeneities of the CMB. These perturbations are generated by quantum fluctuations of the inflaton, the scalar field that drives the Inflation, and they can be scalar, vectorial or tensorial. The tensor</list_item>
</unordered_list>
<text><location><page_61><loc_16><loc_89><loc_62><loc_91></location>perturbations allow for the primordial gravitational waves.</text>
<text><location><page_61><loc_12><loc_87><loc_73><loc_88></location>[204] Notice that due to U (1) symmetry there is a conserved current given by</text>
<formula><location><page_61><loc_19><loc_80><loc_85><loc_86></location>¯ J µ = J µ -M 4 4 M 6 5 [ 4 R µ ν J n u -R 3 J µ -4 3 |∇ ϕ | 2 J µ + i 4 ( ϕ ∗ ∇ µ ϕ ( ∇ ϕ ∗ ) 2 -ϕ ∇ µ ϕ ∗ ( ∇ ϕ ) 2 ) ] .</formula>
<formula><location><page_61><loc_38><loc_72><loc_67><loc_77></location>k 1 = 1 1440 π 2 ( 1 2 N 0 +3 N 1 / 2 +6 N 1 ) .</formula>
<text><location><page_61><loc_12><loc_76><loc_88><loc_80></location>[205] It is worth nothing that by making use of the dimensional regularization [135], for example, one infers</text>
<text><location><page_61><loc_12><loc_67><loc_88><loc_71></location>[206] The anomaly trace is typically expressed in term of curvature tensors and their covariant derivatives, as well as mass-terms[132, 136], i.e.</text>
<formula><location><page_61><loc_16><loc_61><loc_89><loc_66></location>〈 T µ µ 〉 = α ( N 0 , N 1 / 2 , N 1 ) ( R µν R µν -R 2 3 ) + c 1 ( m ) R + c 2 ( m )+ c 3 /square R -N 0 ∑ i =1 m 2 i 〈 φ 2 i 〉-N 1 / 2 ∑ i =1 m i 〈 ¯ ψ i ψ i 〉 ,</formula>
<text><location><page_61><loc_16><loc_45><loc_88><loc_60></location>where the coefficients c 1 , 2 ( m ) are combinations of (power) mass fields, N 0 , 1 / 2 , 1 is the number of the quantum matter of boson, fermion and vector fields. The coefficients c i are subject to a finite renormalization, becoming free parameters of the theory. For our purpose we can neglect the mass-terms since the fields are relativistic, keeping in mind however, that the procedure of renormalization gives rise to purely geometric terms which appear in the final expression of the trace anomaly.</text>
<section_header_level_1><location><page_62><loc_35><loc_88><loc_65><loc_90></location>TABLE III: Vierbeins transformations</section_header_level_1>
<text><location><page_62><loc_22><loc_83><loc_77><loc_87></location>Under coordinate transformations Under local Lorentz transformations the vierbeins e a µ ( x ) transform as the vierbeins e a µ ( x ) transform as</text>
<formula><location><page_62><loc_27><loc_77><loc_43><loc_79></location>e ' a µ ( x ' ) = ∂x ν ' µ e a ν ( x )</formula>
<formula><location><page_62><loc_35><loc_77><loc_72><loc_79></location>∂x e ' a µ ( x ' ) = Λ a b ( x ) e b ν ( x )</formula>
<text><location><page_62><loc_29><loc_72><loc_41><loc_74></location>ξ ' a ( x ' ) = ξ a ( x )</text>
<text><location><page_62><loc_51><loc_72><loc_75><loc_74></location>whereas x µ does not transform</text>
</document>
[
{
"title": "Neutrino coupling to cosmological background: A review on gravitational Baryo/Leptogenesis",
"content": "Gaetano Lambiase a,b , Subhendra Mohanty c , and A.R. Prasanna c a b INFN , Sezione di Napoli, Italy. and c Physical Research Laboratory, Ahmedabad 380009, India.",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "In this work we review the theories of origin of matter-antimatter asymmetry in the Universe. The general conditions for achieving baryogenesis and leptogenesis in a CPT conserving field theory have been laid down by Sakharov. In this review we discuss scenarios where a background scalar or gravitational field spontaneously breaks the CPT symmetry and splits the energy levels between particles and anti-particles. Baryon or Lepton number violating processes in proceeding at thermal equilibrium in such backgrounds gives rise to Baryon or Lepton number asymmetry.",
"pages": [
1
]
},
{
"title": "1. INTRODUCTION",
"content": "It is largely accepted that General Relativity is the best (self-consistent) theory of gravity. It dynamically describes the space-time evolution and matter content in the Universe and is able to explain several gravitational phenomena ranging from laboratory and solar system scales (where it has been mainly tested) to astrophysical and cosmological scales. On cosmological scales, the cornerstone of General Relativity is represented by Hubble expansion, the Big Bang Nucleosynthesis, i.e. the formation of light elements in the early Universe, and cosmic microwave background (CMB) radiation. Despite these crucial predictions, Einstein's theory of gravity is in disagreement with the increasingly high number of observational data, such as those coming for example from SNIA-type, large scale structure ranging from galaxies up to galaxy super-clusters, provided by the advent of the Precision Cosmology and the achievement of high sensitivity of experiments. The experimental evidences that the observable Universe is at the moment expanding in an accelerating phase [1, 2] represents without any doubts the most exciting discovery of the modern Cosmology. As a consequence of this discovery, there has been in the last years more and more interest to understand the evolution not only of the early Universe, but also of the present Universe, and for this formidable task new ideas and theories beyond the standard Cosmology and particle physics have been proposed. Attempts to explain the recent observational data and at the same time try to preserve the conceptual structure of General Relativity, lead cosmologists to introduce two new fundamental concepts: Dark Matter (DM) and Dark Energy (DE). Observational data indicate that a huge amounts of DM and DE are indeed needed to explain the observed cosmic acceleration of the Universe in expansion (as well as all new observational data), and at the moment there are no experimental and theoretical evidence that definitively shed some light on such mysterious components (see [3-16] for DE reviews and [17-26] for DM reviews, and references therein). Moreover, recent data suggest that also in the very early epoch, the Universe was in an accelerated phase. This era is called Inflation, and is able to to solve the problems that affect the standard cosmological model (the Cosmology based on General Relativity): 1) The flatness of the Universe, that is why Ω = ρ/ρ c /similarequal O (1). Here ρ is the average cosmological energy density, and ρ c = 3 H 2 m 2 P / 8 π the critical density. 2) The problems of homogeneity, isotropy, and horizon (which created headache in the frameworks of the standard Friedman-Robertson-Walker (FRW cosmology). Inflation provides a natural mechanism of generation of small density perturbations with almost flat spectrum[203]. This agrees with observations. In order to solve all problems of standard FRW cosmology, it is required that the duration of Inflation is At the end of the Inflationary epoch, according to the standard cosmological model, the Universe is in a cold, low entropy state, and appears baryon symmetric, that is the same amount of matter and antimatter. On the contrary, the present Universe looks baryon asymmetric. The issue that one has to solve is about the physical mechanism occurred during the Universe evolution for which it ends up being matter dominated. Equivalently, what about anti-matter? Theories that try to explain how the asymmetry between baryon and antibaryon was generated in the early phases of the Universe evolution are called baryogenesis . They represent a perfect interplay between particle physics and cosmology. The parameter characterizing the baryon asymmetry is defined as where n B ( n ¯ B ) is the number of baryons (antibaryon) per unit volume, and n γ = 2 ξ (3) T 3 π 2 is the photon number density at temperature T . A different definition of the parameter η that refers to the entropy density Finally, the baryon asymmetry can be also expressed in term of the baryonic fraction Ω B = ρ B /ρ c , i.e. where h = 0 . 701 ± 0 . 013 is the present Hubble parameter. The physics of the CMB temperature anisotropies, which are related to the acoustic oscillations of baryon-photon fluid around the decoupling of photons, provides a strong probe of the baryon asymmetry. In fact, the observation of the acoustic peaks in CMB measured g eff = g γ + T 3 ν T 3 γ g ν , is given by by WMAP satellite [27], when combined with measurements of large scale structures, leads to following estimation of the parameter η An independent measurement of η can be carried out in the framework of the BBN [28], that gives It is remarkable that two completely different probes of the baryon content of the Universe (the synthesis of light elements occurred during the first 3 minutes of the Universe evolution, and the the photons decoupling occurred when the Universe was 400 thousand years old) give compatible results. This represents one of the great success of modern Cosmology. Although many mechanisms have been proposed, the explanation of the asymmetry between matter and antimatter is still an open problem of the modern Cosmology and Particle Physics. In this review we discuss some general topics related to the baryogenesis. The work is divided in two parts. In the first part we recall some models of baryogenesis/leptogenesis, which are mainly based on particle physics (essentially GUT and SUSY). In the second part we discuss different approaches to the baryon asymmetry which rely on the coupling of baryon/lepton currents with the gravitational background. Particular attention will be devoted to the mechanism based on the spin-gravity coupling of neutrinos with the gravitational waves of the cosmological background, which are generated by quantum fluctuations of the inflation field during the inflationary era.",
"pages": [
2,
3,
4
]
},
{
"title": "2. SOME TOPICS OF BARYOGENESIS AND LEPTOGENESIS",
"content": "In this Section, we recall some general topics of Baryogenesis and Leptogenesis. More details can be found in [29-34]. Standard cosmological model is unable to explain the so small magic value of the baryon asymmetry (Eqs. (3) and (4)) and why the Universe starting from an initial baryon symmetry ( n B = n ¯ B ) evolves in a final state such that matter dominates over antimatter ( n B /greatermuch n ¯ B ).",
"pages": [
4
]
},
{
"title": "2.1. The Sakharov conditions",
"content": "As pointed out by Sakharov [36], in a CPT conserving theory a baryon asymmetry B may be dynamically generated in the early Universe provided that: /negationslash The C and CP violation imply that B/L violating reactions in the forward and reverse channels do not cancel ( L stands for Lepton number). To see this, consider the process: X → Y + B , where X is the initial state with B = 0, Y the final state with B = 0, and B the excess baryon produced. Suppose that C is a symmetry. Then the C -conjugate process is characterized by the fact that The net rate of baryon production evolves in time as Similar arguments hold for CP symmetry. Therefore both C and CP discrete symmetry violation are required to generate a net baryon asymmetry. with CPT , i.e. [ H,CPT ] = 0, which implies n B = n ¯ B . Hence only a departure from thermal equilibrium, which means that the form of n B, ¯ B has to be modified, can allow for a finite baryon excess (so that n B -n ¯ B = 0). /negationslash More specifically, consider again the process X → Y + B . If the process is in thermal equilibrium , then by definition so that no net baryon asymmetry can be produced since the inverse process destroy B as fast as the forward process creates it (see also Appendix A). /negationslash However, as discussed in the seminal paper by Cohen and Kaplan [38], it is possible to generate lepton/baryon asymmetry at thermal equilibrium (without requiring CP violation). The reason is due to the general result that in an expanding Universe at finite temperature, CPT is not a good symmetry, i.e. CPT can be (spontaneously) violated. The third Sakharov criterion is therefore violated. As before pointed out, in fact, CPT invariance requires that the baryon number must be generated out of thermal equilibrium, but CPT invariance requires that the thermal distribution of baryon and anti-baryon will be identical. This condition fails if there is a spontaneous CPT violation in the theory which modifies the baryon-antibaryon spectrum. As a consequence one obtains E particle = E antiparticle which implies n B -n ¯ B = 0. /negationslash The CPT violation allows for the generation of the baryon asymmetry during an era when baryon(lepton) violating interaction are still in thermal equilibrium. The asymmetry gets frozen at the decoupling temperature T d when the baryon(lepton) violation goes out of equilibrium. The decoupling temperature is calculated by equating the interaction rate of processes Γ and the expansion rate of the Universe represented by the Hubble constant H , Γ( T d ) /similarequal H ( T d ). The scenario underlying these processes in an expanding Universe can be schematized as follows: in the regime Γ /greatermuch H , or T > T d B-asymmetry is generated by B-violating processes at thermal equilibrium; at T = T d , i.e. Γ /similarequal H , the decoupling occurs, and finally when Γ < H , or T < T d the B-asymmetry gets frozen.",
"pages": [
5,
6
]
},
{
"title": "2.2. Leptogenesis scenario",
"content": "Leptogenesis is a mechanism, proposed by Fukugita and Yanagida, that allows to convert the lepton asymmetry to baryon asymmetry via electroweak (EW) effects. Even if the baryon number is conserved at high scales, it is possible to generate the baryon asymmetry in the present Universe if lepton asymmetry is generated at either GUT or intermediate scales. This idea attracted much attention in view of discovery of a possible lepton number violation in the neutrino sector [39, 40]. For a recent review see [41, 42] (the role of neutrinos in cosmology has been recently treated in [43]). The Leptogenesis scenario is the simplest extension of the standard model able to realize the Sakharov conditions for explaining the matter antimatter asymmetry in the Universe. In this model, the standard model is modified by adding right handed neutrinos which permit the implementation of the see-saw mechanism and provide the explanation of light mass of the standard model neutrinos. At the same time, the augmented model is able to spontaneously generate leptons from the decays of right handed neutrinos. For later convenience, we shall discuss in a nutshell the Leptogenesis scenario. The leptonic Lagrangian density is given by (here we follow [44]) In this expression L is the Standard Model left-handed doublet, E is the right-handed singlet, N j are the singlet fermions (Majorana neutrinos), α, β are the flavor indices of the Standard Model, i.e. α, β = e, µτ , M is mass matrix and λ Yakawa matrices. Equation (1 is written in a basis where the coupling h and the mass matrix M are diagonal and real, whereas λ is complex. This Lagrangian leads, once the heavy fermions N i are integrated out, to the effective light neutrino masses (see-saw mechanism): m ν αβ = λ αk M -1 k λ βk . The Lagrangian (1) satisfies the Sakharov conditions. In fact it violates the leptonic number L due to λ - and M -terms; CP is violated through the complex Yukawa coupling λ αk ; Since the interactions are only determined by Yukawa's interaction terms, the smallness of these couplings may provide the right conditions for which the interaction rates are smaller than the expansion rate of the Universe, establishing in such a way the condition for the out of equilibrium (i.e. the heavy Majorana fermions can decay out of equilibrium). For simplicity consider the lightest Majorana singlet N 1 . It may decay in two channels As a consequence of the N 1 decay, the net baryon asymmetry can be generated. The parameter η turns out to be Here η eff is the efficiency factor which assumes the value in the range 0 < η eff < 1 (owing to inverse decays, washout processes and inefficiency in N 1 production). Below to the freeout temperature T F , the temperature for which Γ( φL → N 1 ) < H , where Γ( φL → N 1 ) /similarequal 1 2 Γ D e -M 1 /T , with and H is the expansion rate of the Universe during the radiation dominated era, the density of the fermion N 1 is Boltzmann suppressed ( N 1 ∼ e -M 1 /T ). Therefore, below T F the decay of N 1 contribute to the lepton asymmetry, and the efficiency factor is where C sph is a factor that takes into account the dilution of the asymmetry due to fast processes. Finally /epsilon1 is the CP parameter related to the asymmetry in the N 1 decays and defined as Its non vanishing value arises from the interference of three level and one loop amplitudes (complex Yakawa couplings). One gets where Taking into account Eqs. (6), (9) and (7), the net lepton asymmetry is Leptogenesis is then related to Baryogenesis by a phenomenon that happens in the currently accepted Standard Model. Indeed, certain non-perturbative configurations of gauge fields, the sphalerons, can convert leptons into baryons and vice versa. These processes that violate B + L and conserve B -L occur at the electroweak scale. Under normal conditions sphalerons processes are unobservably rare due to the fact that the transition rates are extremely small Γ ∼ e -16 π 2 /g 2 ∼ O (10 -165 ), hence are completely negligible in the Standard Model (at T = 0). However as emphasized by Kuzmin, Rubakov, and Shaposhnikov [40], in the thermal bah provided by the expanding Universe, thermal fluctuations becomes important and B + L violating processes can occur at a significant rate and these processes can be in equilibrium in the expanding Universe (see B). Finally from (10) it follows that by requiring η ∼ 10 -10 the lower bound of the mass of the Majorana neutrinos N 1 is where the light value of the neutrino mass has been used: ˜ m /similarequal (10 -3 -10 -1 )eV.",
"pages": [
7,
8,
9
]
},
{
"title": "2.3. Models of Baryogenesis",
"content": "Many models aimed to explain the generation of the baryon asymmetry have been proposed in literature. These are GUT Baryogenesis, Affleck-Dine Baryogenesis and AffleckDine Leptogenesis, Leptogenesis from heavy Majorana neutrinos, Leptogensis from ν R oscillation, Thermal baryogenesis, Electroweak baryogenesis, Spontaneous baryogenesis, Baryogenesis through evaporation of primordial black holes. Details of such mechanisms can be found in [29-33, 45] and references therein. Here we list some of them in Table I. /negationslash Consider the X -boson decays in two channels X → 2 q and X → 2¯ q , with the probabilities given by P X → 2 q = P X → 2¯ q due to CP violation. This implies the excess of baryons over anti-baryons. In the original scenario of GUT baryogenesis one uses the heavy gauge bosons X and Y (leptoquarks), which decay while they decouple from equilibrium. This is called delayed decay scenario . It was soon realized that this boson gauge decay does not produce the required baryon asymmetry because that the X and Y boson masses predicted are too low to satisfy the out-of-equilibrium condition (in non-SUSY GUT). The alternative scenario was to use decays of coloured Higgs particles. If more than two Higgs particles exist, sufficiently large baryon asymmetry can be generated (provided that the Kuzmin, Rubakov, Shaposhnikov effect is switched off). Affleck-Dine scenario (1985): This scenario is based on the observation that in SUSY theories ordinary quarks and leptons are accompanied by supersymmetric partners - s-quarks and s-leptons - which are scalars. The corresponding scalar fields carry baryon and lepton number, which can in principle be very large in the case of a scalar condensate (classical scalar field). An important feature of SUSY theories is the existence of flat directions in the superpotential, along which the relevant components of the complex scalar fields ϕ can be considered as massless. The condensate is frozen until supersymmetry breaking takes place. Supersymmetry breaking lifts the flat directions and the scalar fields acquire mass. When the Hubble constant becomes of the order of this mass, the scalar fields starts to oscillate and decays. At this time, B , L , and CP violating terms (for example, quartic couplings λ 1 ϕ 3 ϕ ∗ + c.c. and λ 2 ϕ 4 + c.c. , with complex λ 1 , 2 ) becomes important and a substantial baryon asymmetry can be produced. The scalar particles decay into ordinary quarks and leptons transferring to them the generated baryon asymmetry. The Affleck-Dine mechanism can be implemented at nearly any energy scale, even below 200 GeV. By suitable choice of the parameters one can explain almost any amount of baryon asymmetry and this lack of a falsifiable prediction is an unattractive feature of the Affleck-Dine mechanism. dominated era of the early Universe, a period in which the SU (2) L × U (1) Y electroweak symmetry is manifest. As the temperature falls down the EW scale ( T EW ∼ 100GeV), the Higgs field acquires an expectation value and the electroweak symmetry is spontaneously broken to the subgroup U (1). The EWBG occurs during this phase transition, and in order that it could be an available mechanism, it is required that the transition is of the first order. Remarkably the EWBG satisfies all the three Sakharov's conditions: 1) The rapid sphaleron transitions in the symmetric phase provide the required violation of the baryon number; 2) The scattering of plasma with bubble walls generates the C and CP asymmetry of the number of particles if the underlying theory does contain terms that violate these discrete symmetries (these processes bias the sphalerons to create more baryon than anti-baryons); 3) The rapid expansion of bubble walls through the plasma induces the departure from the thermal equilibrium. All these conditions are fulfilled by the Standard Model. However, EWBG is unable to explain the observed baryon asymmetry of the Universe if it is only based on the Standard Model. The reason is due to the fact that EW phase transition in the Standard Model is of the first order if the Higgs mass is constrained by m H /lessorsimilar 70GeV, in disagreement with experimental lower bound obtained from LEP II experiment, i.e. m H /greaterorsimilar 114GeV, as well as, from recent LHC results that give a value of the Higgs mass near to 125GeV. Recent studies however, open the possibilities to reconsider EWBG as an available candidate for the generation of baryon asymmetry[46]. Moreover, the EWBG mechanism is also affected by the problem related to the CP violation because the latter generated by the Cabibbo-Kobayashi-Maskawa phase is unable to generate large enough chiral asymmetry. For a recent review on EWBG see[47].",
"pages": [
9,
10,
11
]
},
{
"title": "3. BARYOGENESIS GENERATED BY COUPLING OF BARYON CURRENTS AND GRAVITATIONAL BACKGROUND",
"content": "As we have seen in the previous Section, the Baryo/Leptogenesis is generated in the framework of particle interactions (essentially GUT and SUSY). The gravitational field /negationslash enters marginally in these mechanisms. In the last years, however, many mechanisms have been proposed in which gravity plays a fundamental role in generating the baryo/leptogenesis (see Table II). In these models matter or hadron/lepton currents are coupled with some physical quantity characterizing the gravitational background, such as Ricci curvature or its derivative, Riemann tensor, gravitational waves (GW), Typically, the background is the FRW geometry, but there are also models in which the gravitational background is described by black holes physics. Gravitational baryogenesis share some basic features of the spontaneous spontaneous (or quintessential) baryogenesis[38]. In this mechanism scalar fields (or their derivatives) couple to matter or hadron/lepton current. To illustrate in some detail the spontaneous baryogenesis, consider a neutral scalar field φ . The interaction between a baryon current J µ B and ∂ µ φ is /negationslash where M s characterize is a cut-off scale. In a isotropic and homogenous Universe, like FRW Universe, φ does only depend on cosmic time, In such a case only the zero component of the baryon current ( J 0 B = n B with n B the number density of baryons) contribute in (1), L = µn B , where µ ≡ ˙ φ/M s for baryons and µ ≡ -˙ φ/M s for antibaryons. Here is assumed that the current J B is not conserved and that, of course, ˙ φ = 0. The coupling (1) therefore gives rise to an effective chemical potential with opposite sign for B and ¯ B leading to a generation of a net baryon asymmetry even at thermal equilibrium. The latter point bypass the third Sakharov condition because CPT violation occurs owing the Universe expansion. The scalar field could also play the role of DE or DM. Models based on spontaneous (quintessence) baryogenesis are studied in[50-57].",
"pages": [
11,
12,
13
]
},
{
"title": "3.1. Gravitational Baryogenesis",
"content": "The key ingredient for the gravitational baryo/leptogenesis is a CP-violating interaction between the derivative of the Ricci scalar curvature R and the B(aryon)/L(epton) current J µ [58, 59] /negationslash where M ∗ is the cutoff scale of the effective theory. L int is expected in a low energy effective field theory of quantum gravity or Super gravity theories (more specifically it can be obtained in supergravity theories from a higher dimensional operator in the Kahler potential). Moreover, it dynamically breaks the CPT in an expanding Universe. In the standard cosmological model ˙ R vanishes during the radiation era (see below). However, (tiny) deviations from General Relativity prevent the Ricci curvature to vanish, as well as its first time derivative, so that a net lepton asymmetry can be generated. To generate a B -asymmetry, it is required that there exist B/L -violating processes in thermal equilibrium. In this mechanism, the interaction J µ ∂ µ R gives a contribution to the energy of particles and antiparticles with opposite sign, and thereby dynamically violates CPT . This coupling term modifies thermal equilibrium distribution and the chemical potential driving the Universe towards nonzero equilibrium B/L -asymmetry via the B/L -violating interactions. Once the temperature drops below the decoupling temperature T d the asymmetry can no longer change and is frozen. The net asymmetry is In the cosmological standard model it is assumed that the energy-momentum tensor of classical fields is described by a perfect fluid where ρ is the energy density and p the pressure. They are related by the relation p = wρ , w being the adiabatic index. During the radiation dominated era, the equation of the state is p = ρ/ 3, i.e. w = 1 / 3, and the scale factor evolves as a ( t ) = ( a 0 t ) 1 / 2 . The energy density of the (classical) radiation is given by whereas the cosmic time is related to the temperature T of the Universe as Moreover, Eq. (5 implies that the expansion rate of the Universe can be written as In what follows we shall consider a flat Friedman-Robertson-Walker (FRW) Universe whose element line is From the above considerations it follows that the trace of the energy-momentum tensor of (classical) relativistic fields vanishes, T = ρ -3 p = 0. As a consequence one has R = -8 πGT µ µ = 0, and no net baryon asymmetry may be generated. However, a possibility to generate the baryon asymmetry is given by the interaction among massless particles that lead to running coupling constants and hence the trace anomaly [62] /negationslash In a SU ( N c ) gauge theory with coupling g and N f flavors, the effective equation of state is given by The numerical value of 1 -3 w depends the gauge group and the fermions, and lies in the range 1 -3 w ∼ 10 -2 -10 -1 . The baryon asymmetry turns out to be η = (1 -3 w ) T 5 D M 2 ∗ m 3 P . Gravitational baryogenesis is conceptually similar to spontaneous baryogenesis [38], see Eq. (1). However there some basic differences between the two paradigms:",
"pages": [
13,
14,
15,
16
]
},
{
"title": "3.2. Genaralised Gravitational baryogenesis",
"content": "An interesting model related to the gravitational baryogenesis has been provided by Li, Li and Zhang [51], who consider a generalized coupling of the form where f ( R ) is a generic function of the scalar curvature. This function has been chosen as f ∼ ln R so that the effective interaction Lagrangian density reads where c is a constant fixed to in order to reproduce the observed baryon asymmetry. Following the same reasoning leading to (3) one gets During the radiation dominated era one obtains that a net baryon asymmetry can be generated and is given by where Eq. (5) has been used and T D is the decoupling temperature. Moreover, one can also determine an order of magnitude of the absolute neutrino mass compatible with the current cosmological data, i.e. m ν /similarequal O (1)eV. The idea goes along the line traced in Section 4.1. In the Standard Model, B -L symmetry is exactly conserved ( ∂ µ J µ B -L = 0). In [51] the B -L violation is parameterized by higher dimensional operators, i.e. by the dimension 5 operator L ∼ C ¯ llφ † φ (see Eq. (26)). C is a scale of new physics beyond the Standard Model which generates the B -L violations, l and φ are the left-handed lepton and Higgs doublets, respectively. When the Higgs field gets a vacuum expectation value 〈 φ 〉 = v , the lefthanded neutrino becomes massive m ν /similarequal Cv . Comparing the lepton number violating rate induced by the interaction L , Γ ∼ T 3 (Eq. (30)), with the expansion rate of the Universe, H ∼ T 2 (Eq. (5)), one gets the decoupling temperature below which the lepton asymmetry is freeze-out, i.e. T D /similarequal 10 10 GeV. The observed baryon asymmetry η ∼ 10 -10 follows for c ∼ O (1). Then, assuming an approximate degenerate masses, i.e. m ν 1 ∼ m ν 2 ∼ m ν 3 , one gets m ν /lessorsimilar ∞ eV. The current cosmological limit comes from WMAP Collaboration [64] and SDSS Collaboration [65]. The analysis of Ref.[64] gives ∑ i m ν i < 0 . 69 eV. The analysis from SDSS gives[65] ∑ i m ν i < 1 . 7 eV.",
"pages": [
16,
17
]
},
{
"title": "3.3. Baryogenesis in Randall-Sundrum braneworld",
"content": "The asymmetry baryon-antibaryon can arise in the Randall-Sundrum brane world model[66] with bulk fields owing to the effects of higher dimensionality. These studies have been performed in[67-69]. The total action contains the bulk and brane actions[68, 70] where G MN is the 5-dim bulk metric and G its determinant, g µν the brane induced metric and g the determinant, Λ the bulk cosmological constant, σ the brane tension, and Φ the bulk complex scalar field (localized on the brane as the graviton). Λ and σ are related by Λ = -σ 2 6 M 5 . It is worth to write down the the effective theory on the brane. It is derived by making use of the braneworld holography[68, 71]. This method gives Here Γ CFT is the effective action for the holographic CFT on the brane, R ( g ) is the Ricci scalar on the brane, M 2 4 = lM 3 5 = M 2 P plays the role of Planck mass, with l the the curvature radius of the AdS spacetime, and L matter is the Lagrangian density matter localized on the brane. The parameter /epsilon1 determines the renormalization scale of CFT, whereas the field φ corresponds to the zero mode of the bulk complex scalar field Φ localized on the brane ( ϕ could represent squarks or sleptons on the brane carrying baryon/lepton number). Notice that (13) is written as and Hilbert-Einstein action plus scalar field (in the so called Jordan frame). In this respect it is similar to scalar tensor theories. The current associated to φ , defined as J µ = -iφ ←→ ∇ µ φ ∗ , satisfies the relation[204] Assuming a coupling of the form (1), with φ a scalar field on the brane and J µ B replaced by J µ , one obtains (after an integration by parts) that the effective Lagrangian density on interaction is This interaction leads to the baryon asymmetry given in (4). A comparison with (2) suggest M ∗ = fM 3 5 /M 2 4 . To determine the baryon asymmetry one needs to evaluate ˙ R . In the Randall-Sundrum model, the geometrical projection method yields the field equation where T µν is the energy-momentum tensor on the brane, E µν is the Weyl tensor, and In deriving (17) it is assumed the contribution to gravity is dominated by matter field L matter . Notice that the energy-momentum tensor satisfies the continuity equation ∇ µ T µν = 0. The trace of (17) in a FRW Universe reads from which it follows where (20) follows in a Universe radiation dominated ( w = 1 / 3). One can compute the decoupling temperature[68] T D ∼ M 3 / 2 5 M 1 / 2 4 , so that the bet baryon asymmetry assumes the form which has been written to emphasize the estimations that the parameters characterizing the theory must assume in order that the observed baryon asymmetry is obtained. Other models based on gravitational baryogenesis can be found in[58, 68, 72-78] [61, 7985].",
"pages": [
17,
18,
19
]
},
{
"title": "4. LEPTOGENESIS BY CURVATURE COUPLING OF HEAVY NEUTRINOS",
"content": "In this Section, we study the generalization in the matter Lagrangian by including higher order terms in R consistent with general covariance, Lorentz-invariance in a locally inertial frame. The effect of spin-gravity coupling will be neglected (they will be extensively discussed in Section 6.1). Therefore we work in the approximation for which the characteristic time of spinor fields variation is smaller than the age of the Universe. Consider the action for a four component Dirac fermion ψ which couples to background gravity[86] where h 1 ( R ) and h 2 ( R ) real valued scalar functions of the curvature, Here h 1 is a generalization of the neutrino mass term. Note that since ¯ ψγ 5 ψ transforms as a pseudo-scalar, the h 2 term is odd under CP . We write the four-component fermion The lagrangian in terms of the two-component fields ψ R and ψ L becomes where σ µ = ( I, σ i ) and ¯ σ µ = ( I, -σ i ) in terms of the Pauli matrices. The h 2 term can be rotated away by a chiral transformation Keeping terms to the linear order in α , we see that the lagrangian (4) changes by the amount Now we choose α = -h 2 /h 1 to eliminate the chiral mass term and obtain for the total Lagrangian If h 1 and h 2 are constants then, one can always rotate the axial-mass term away. We will assume that the neutrino mass M /greatermuch g 1 therefore h 1 /similarequal M and since a constant M ' can be rotated away h 2 = g 2 . Further we will assume that the background curvature is only dependent on time. The lagrangian (7) then reduces to the form The equation of motion for the left and the right helicity fermions derived from (8) are Written in momentum space ψ ( x ) = ψ ( p ) e i ( Et -/vector p · /vectorx ) the equation of motion of ψ R and ψ L are In the limit p /greatermuch M,g 2 the dispersion relations are[87] E R,L /similarequal E p ± 2 ˙ g 2 M , where E p = p + M 2 2 p . The canonical momenta of the ψ L and ψ R fields are as usual so that the canonical Hamiltonian density is where we have introduced the number density operators of the left and right chirality modes, The partition function in terms of this effective Hamiltonian is where β = 1 /T and H 0 is the free particle Hamiltonian. We see that when ˙ g 2 is non-zero then the effective chemical potential for the left chirality neutrinos is µ L = -˙ g 2 /M and for the right-chirality neutrinos is µ R = ˙ g 2 /M . In the presence of interactions which change ψ L ↔ ψ R at thermal equilibrium there will be a net difference between the left and the right chirality particles, Here we consider the simplest case in which h 2 and g 2 are linear function of the curvature R , The axial term in (1) is a CP violating interaction between fermions and the Ricci curvature described by the dimension-five operator [61, 84] This operator is invariant under Local Lorentz transformation and is even under C and odd under P and conserves CPT . In a non-zero background R , there is an effective CPT violation for the fermions. Take ψ = ( N R , N c R ) T , where N R is a heavy right handed neutrino and N c R a left handed heavy neutrino, which decay into the light neutrinos. Majorana neutrino interactions with the light neutrinos and Higgs relevant for leptogenesis, are described by the lagrangian where ˜ M is the right handed neutrino mass-matrix, l Lα = ( ν α , e -α ) T L is the left-handed lepton doublet ( α denotes the generation), φ = ( φ + , φ 0 ) T is the Higgs doublet. In the scenario of leptogenesis introduced by Fukugita and Yanagida, lepton number violation is achieved by the decays N R → φ + l L and also N R c → φ † + l L c . The difference in the production rate of l L compared to l c L , which is necessary for leptogenesis, is achieved via the CP violation. In the standard scenario, n ( N R ) = n ( N c R ) as demanded by CPT , but /negationslash due to the complex phases of the Yukawa coupling matrix h αβ , and a net lepton number arises from the interference terms of the tree-level and one loop diagrams (see Section 2 and Ref.[88, 89]). In this leptogenesis scenario we have that the decay rates of N R and N c R are the same, but there is a difference between the heavy light and left chirality neutrinos at thermal equilibrium due to the CP violating gravitational interaction (16), The N R ↔ N c R interaction can be achieved by the scattering with a Higgs field. A recent example of leptogenesis due heavy neutrino decay with CP violation in a SO(10) model is described in[90]. In standard SO(10) unification, all Standard Model fermions of a given generation together with a right-handed neutrino are in a 16 representation of SO(10), The charged fermion and Dirac neutrino mass matrices receive contributions from Yukawa couplings of the form 16 f 16 f H (where H = 10 H , 126 H and/or 120 H ). Majorana masses for the right-handed neutrinos are generated either from or from the non-renormalizable operators suppressed by some mass scale Λ When the GUT Higgs fields S ' or S acquire a vev , a large Majorana mass M is generated for N R which breaks lepton number spontaneously. This following the see-saw mechanism leads to small neutrino masses at low energies. At temperatures larger than the heavy neutrinos and the GUT Higgs masses one there will be helicity flip scattering interactions like S + N R ↔ S + N c R which change the lepton number (as T > M the helicity and the chirality of N R are same). The interaction rate is The interactions decouple at a temperature T D . The latter is computed via the equality where H = ˙ a/a . From (23) one derives the decoupling temperature where we have used M = f 〈 S 〉 / Λ. From the lepton asymmetry (18) and (2) one obtains the value of frozen in lepton asymmetry as This result agrees with[58]. Some comments are in order. 1) In the case in which the fermion is, for example, an electron one also gets a splitting of energy levels E ( e R ) -E ( e L ), but this does not lead to lepton generation of lepton asymmetry as both e L and e R carry the same lepton number. 2) In principle, one should also take into account primordial perturbations of the gravitational background (characterized mainly by scalar and tensor perturbations) and of the energy density and pressure, characterized by δρ = δT 0 0 and δpδ j i = δT j i (see for example [91]). These perturbations are related as δp = c 2 s δρ , where c 2 s = w + ρdw/dρ is the adiabatic sound speed squared. For relativistic particles w = 1 / 3 and therefore c 2 s = 1 / 3. As a consequence, the trace of the perturbed energy-momentum tensor vanishes (this is not true in presence of anisotropic shear perturbations), so that according to the gravitational leptogenesis mechanism, no net baryon asymmetry can be generated.",
"pages": [
19,
20,
21,
22,
23,
24
]
},
{
"title": "4.1. Avoiding subsequent wash-out",
"content": "The light neutrino asymmetry can be erased by the interactions ν L + φ 0 → ν c L + φ † 0 with the standard model Higgs. To prevent the erasure of the lepton asymmetry by Higgs scattering, we must demand that the lightest heavy neutrino mass be lower than the decoupling temperature of the light-neutrino Higgs interaction, which is calculated as follows. The light neutrino masses arise from an effective dimension five operator (26) which is obtained from (17) by heavy neutrino exchange[92] Here ˜ φ ≡ iσ 2 φ ∗ = ( -φ 0 ∗ , φ -) T , /epsilon1 ab is the antisymmetric tensor, and a, b.. denote the gauge SU (2) L indices. The ∆ L = 2 interactions that result from the operator (26) are The cross section for the interaction ν Lα + φ 0 ↔ ν Rβ + φ 0 is In the electroweak era, when the Higgs field in (26) acquires a vev , 〈 φ 0 〉 = v = 174 GeV , this operator gives rise to a Majorana neutrino mass matrix and the cross section (28) can be expressed in terms of light neutrino masses as The interaction rate of the lepton number violating scattering ν L + φ 0 ↔ ν R + φ † 0 is given by The decoupling temperature T l when the interaction rate Γ( T l ) falls below the expansion rate of the Universe (5. The decoupling temperature T l is obtained from equation Γ( T l ) = H ( T l ), where It turns out that The heavy neutrino decays occur at T /similarequal M /similarequal 10 12 GeV, below the temperature T l /similarequal 2 × 10 14 GeV. At temperatures T ∼ T l the light-neutrino lepton number violating interactions are effective. As a consequence, the lepton number asymmetry from the decay of asymmetric number of heavy neutrino decays is not washed out by Higgs scattering with light neutrinos.",
"pages": [
24,
25
]
},
{
"title": "5. MODELS AND TIME VARYING RICCI CURVATURE IN DIFFERENT COSMOLOGICAL SCENARIOS",
"content": "We now discuss some cosmological scenarios in which the gravitational leptogenesis mechanism can be realized.",
"pages": [
26
]
},
{
"title": "5.1. Gravitational Leptogenesis in f ( R ) theories of gravity",
"content": "As discussed in the Introduction, the observation that the present phase of the expanding Universe is accelerated has motivated in the last years the developments of many models of gravity which go beyond the general relativity, and therefore the standard cosmological model. Among the different approaches, the f ( R )-theories of gravity have received a great attention. The reason relies on the fact that they allow to explain, via a gravitational dynamics, the observed accelerating phase of the Universe, without invoking exotic matter as sources of dark energy. Moreover, they also provide an alternative approach to expliain Dark Matter problem. The Lagrangian density of these models does depend on higher-order curvature invariants(see[7, 8, 13-15, 93, 94] and references therein), such as, for example, R 2 , R µν R µν , R /square R , and so on. Here we focalize our attention to f ( R ) models which are a generic function of the Ricci scalar curvature R In (1), S m is the action of matter and κ 2 = 8 πG = 8 πM -2 P ( M P /similarequal 10 19 GeV is the Planck mass). Cosmological and astrophysics consequences of (1) have been largely studied in literature [10, 11, 23, 24, 95-113]. f ( R ) gravity provide scenarios that make these models very attractive. In fact [10]: 1) They allow to unify the early-time Inflation and the later-time acceleration of the Universe owing to the different role of the gravitational terms relevant at small and large scales; 2) DM and DE issues can be treated in a unique and unified setting; 3) They provide a framework for the explanation of hierarchy problem and unification of GUT with gravity. However, solar system tests strongly constraint or rule out many f ( R ) models of gravity. Therefore the form of the generic function f ( R ) must be properly constructed. In this respect, available models are: The parameters c 1 , c 2 , d , m , n , l , λ , R st , α , β and γ entering the above equations are free. Their combinations allow to get a description of cosmic acceleration (early and present) of the Universe . A characteristic of the models (2)-(4) is that the R -terms can be expanded in the appropriate regimes, reproducing simplest form of f ( R ). A particular subclass is of the form where α > 0 has the dimensions [energy] -2( n -1) and n > 0. Particularly interesting is the case n = 2 (referred in literature as Starobinsky's model [116]) This model (6) has been studied in the framework of astrophysics and cosmology. For instance, gravitational radiation emitted by isolated system constraints the free parameter to | α | /lessorsimilar (10 17 -10 18 )m 2 [117, 118]. Eot-Wash experiments lead instead to the constraints More stringent constraints are provided by the Cosmic Microwave Background (CMB) physics. The amplitude of the curvature perturbation corresponding to (6) is P R /similarequal N 2 k 18 π 1 αm 2 P , with N k ∼ 55. Using the WMAP 5-years data[119] ( P R ∼ 2 . 445 × 10 -9 ), it follows that α is constrained as [8] The bound (8) is obtained in the regime R /greatermuch α -1 (in this regime the model describes the inflationary epoch). In these models of f ( R ) gravity is implicitly assumed that the chameleon effect[120] holds, which means that the Compton length λ associated to the characteristic scales, coming out from adding (pertubative) higher order terms to the Hilbert-Eisntein action, are smaller or larger in regions with higher or lower matter density. Typically one assumes that λ is constant, so that the theory is viewed as a local effective theory which is valid for a certain range of parameters.",
"pages": [
26,
27,
28
]
},
{
"title": "5.1.1. Field equations in f ( R ) gravity",
"content": "The field equations obtained by the variation of the action (1) with respect to the metric are where the prime stands for the derivative with respect to R . The trace reads In the spatially flat FRW Universe, Eq. (8), Eqs. (9) and (10) become Moreover, the Bianchi identities give a further condition on the conservation of the energy In what follows, we shall look for those solutions of field equations such that the scale factor evolves as The scalar curvature turns out to be The f ( R ) model we concern here is that one of Eq. (5). By using Eqs. (11) and (12) and the usual expression relating the energy density and the pressure, p = wρ , where w is the adiabatic index, one gets with The energy density ρ assumes the form Notice that during the radiation dominated era ( β = 1 / 2), to which we are mainly interested, the quantity A vanishes because R = 0, as well as the perturbation ς , and the adiabatic index reduces to the standard value w = 1 / 3. Moreover, our concern is for the regime αR n -1 /lessorsimilar 1.",
"pages": [
28,
29
]
},
{
"title": "5.1.2. Constraints from BBN",
"content": "In BBN one has to consider the weak interaction rate of particles ( p, n, e ± and ν ) in thermal equilibrium. For T /greatermuch Q ( Q = m n -m p , where m n,p are the neutron and proton masses), one gets[121-124] Λ( T ) /similarequal qT 5 , where q = 9 . 6 × 10 -46 eV -4 . The primordial mass fraction of 4 He is estimated by defining Y p ≡ λ 2 x ( t f ) 1+ x ( t f ) , where λ = e -( t n -t f ) /τ . t f and t n are the time of the freeze-out of the weak interactions and of the nucleosynthesis, respectively, τ /similarequal 887sec is the neutron mean life, and x ( t f ) = e -Q /T ( t f ) is the neutron to proton equilibrium ratio. The function λ ( t f ) represents the fraction of neutrons that decay into protons in the time t ∈ [ t f , t n ]. Deviations from Y p (generated by the variation of the freezing temperature T f ) are given by[125-127] δY p = Y p [( 1 -Y p 2 λ ) ln ( 2 λ Y p -1 ) -2 t f τ ] δT f T f . In the above equation we have set δT ( t n ) = 0 because T n is fixed by the deuterium binding energy. The current estimation on[128] Y p , Y p = 0 . 2476 ± δY p , with | δY p | < 10 -4 , leads to The freeze-out temperature T is determined by Λ = H . One gets T = T f (1 + δT f T f ), where T f ∼ 0 . 6 MeV and Equations (20) and (19) implies (see also Ref.[129, 130])",
"pages": [
29,
30
]
},
{
"title": "5.1.3. Gravitational leptogenesis induced by f ( R ) gravity",
"content": "Using the definition of Ricci scalar curvature (16), it follows Equation (6) then implies (to leading order in (2 β -1)) An inspection of (23) immediately revels that the observed baryon asymmetry can be obtained, for example, for T D ∼ 10 16 GeV, M ∼ 10 12 GeV (see (11) and for example[131]), provided that 2 β -1 /similarequal 2 × 10 -6 . The value of the heavy neutrino mass M ∼ 10 12 GeV is consistent with the atmospheric neutrino scale m ν = 0 . 05 eV, obtained from the see-saw relation m ν = m 2 D /M with the Dirac mass scale m D ∼ O (10) GeV. /negationslash The lepton asymmetry generated via (23) is passed on to the light neutrino sector when the heavy neutrino decays at temperature T ∼ M ∼ 10 12 GeV. The effects of washed out are avoided by considering the effective (five dimensional) operator violating the lepton number ∆ L = 2, as before discussed. Notice that the baryon asymmetry is generated both for[86] n = 2 and n = 2 The case n < 0 is excluded because these f ( R ) models of gravity are affected by instability problems[7-9].",
"pages": [
30
]
},
{
"title": "5.2. Time varying Ricci curvature from quantum fluctuations",
"content": "In this Section we discuss another interesting cosmological scenario in which a non-zero Ricci curvature is generated in the radiation era by back-reaction of quantum fields. Quan- tum effects cannot be ignored because they may modify the dynamics of the Universe evolution. In order to incorporate these back-reaction effects in the cosmic evolution of the Universe, General Relativity requires some modification. Again without a complete theory of quantum gravity, one works assuming a semiclassical theory of gravity[132]. In this context, the Einstein field equations are rewritten as[132, 133] /negationslash where T ( cl ) µν is the stress energy-momentum tensor for the classical field, T ( QM ) µν represents the energy momentum tensor operator generated by quantum fields, and finally 〈 T ( QM ) µν 〉 = 〈 0 | T ( QM ) µν | 0 〉 represents the regularized expectation value of T ( QM ) µν . During the radiation dominated era, although the trace of the classical energy momentum tensor vanishes, T ( cl ) = 0, the presence of the quantum corrections 〈 0 | T ( QM ) µν | 0 〉 implies that the trace is nonvanishing, and therefore a net baryon asymmetry could be generated by having ˙ R = 0. This trace anomaly comes from the infinite counterterms that must be add to the gravitational action to make the trace finite. The dynamical evolution of the gravitational background is assumed to be described by the FRW Universe, Eq. (8). The regularized components of the energy-momentum tensor have the form [132, 134] where /square = ∇ µ ∇ µ , and ; stands for covariant derivative. The coefficients k 1 , 3 are constants and come from the regularization process. Their values strictly depend not only on number and types of fields present in the Universe, but also on the method of regularization. Because the methods of regularization affect the the values of k 1 , 3 and more important because of the uncertainty of what fields were present in the very early Universe, they can be considered as free parameters[133, 134]. The tensor (1) H µν satisfies ∇ µ (1) H µ ν = 0. It is obtained by varying the local action The infinities in 〈 T ( QM ) 〉 are canceled by adding infinite counterterms in the Lagrangian density that describes the gravitational fields. One of these counterterms if of the form √ -gCR 2 , and due to (the logarithmically divergent) constant C , the coefficients k 1 is arbitrary(actually it can be fixed experimentally[205]). As regards (3) H µν , it is covariantly conserved only for conformal flat spacetimes, and cannot be derived by means of the variation of a local action, as for (1) H µν . The coefficient k 3 is given by For a SU (5) model, for example, the number of quantum fields take the values N 0 = 34, N 1 / 2 = 45, and N 1 = 24, so that[134] k 3 /similarequal 0 . 07. The explicit expression of the components of (1) H µν and (3) H µν are Applying the regularization procedure one infers the trace anomaly[206] that for the FRW Universe assumes the form During the radiation dominated era, as we have seen, the trace of the energy-momentum tensor of classical fields vanishes T ( cl ) = ρ -3 p = 0. The trace anomaly instead gives a non vanishing contribution. To evaluate it, we point out that both the scale factor a ( t ) = ( a 0 t ) 1 / 2 and the relation between the cosmic time and the temperature T , Eq. (6), should be modified by the back-reaction effects induced by quantum fields. As we shall see below the evolution of the Universe can be described by standard cosmology. In a FRW Universe, the modified Einstein field equations assume the form from which it follows We are looking for solutions of the form /negationslash where δ ( t ) /lessmuch 1 is a perturbation, and H 0 = 1 / 2 t is the Hubble parameter for a radiation dominated universe. The Ricci curvature vanishes, R = 0, as well as its covariant and (cosmic) time derivatives. It then follows that (1) H µν ( H = H 0 ) = 0 whereas (3) H µν = 0 when H = H 0 (see[84] for details). Inserting H given in (34) into Eq. (33), one obtains (to leading order) the solution for δ : C is a constant of integration. As it can be seen, the M -2 P suppresses considerably the effects of δ on the dynamics of the Universe evolution, and these terms wash-out for large t . The trace anomaly (29) reads From R = -8 π M 2 P 〈 T ( QM ) 〉 we find that the parameter characterizing the heavy neutrino asymmetry (25) assumes the form According to the leptogenesis scenario, the heavy neutrino asymmetry freezes at the decoupling temperature T D when the lepton-number violating interactions ( N R ↔ N c R ) go out of equilibrium. The subsequent decays of these heavy neutrinos into the light standard model particles and the conversion of lepton asymmetry into baryon asymmetry can explain the observed baryon asymmetry of the Universe. In fact, for T D ∼ (10 16 -2 × 10 16 )GeV and M ∼ (10 9 -10 12 ), respectively, one gets η ∼ 10 -11 -10 -10 . Let us finally compute the energy density of back-reaction of quantum fields and compare it with the energy density of radiation. From Eqs. (25) it follows The k 1 -term vanishes identically. The total energy density is given by where ρ r is the classical radiation defined in (5). The ratio between the energy densities 〈 ρ 〉 and ρ r reads where we have definite T ∗ as For temperatures T < T ∗ we have that r < 1, i.e. the energy density of quantum fields is subdominant with respect to the energy density of the radiation. In particular, since the decoupling temperature of heavy neutrinos occurs at GUT scales, T D ∼ 10 16 GeV, we infer r ∼ 10 -9 /lessmuch 1 and the back-reaction is indeed subdominant over the radiation density.",
"pages": [
30,
31,
32,
33,
34
]
},
{
"title": "5.3. Gravitational Leptogenesis in Warm Inflation",
"content": "A further application of the gravitational leptogenesis scenario is the warm inflation [137] models as there is a large non-zero Ricci curvature from the inflaton potential during inflation and a large temperature where the lepton number violating interaction can be at equilibrium. Let us recall the central point underlying the warm inflation idea [138]. In the Inflationary dynamics the scalar field carries most of the energy of the Universe. The inflaton however also interact with other fields, but these interactions plays no role except to give rise to modifications to the effective scalar field through quantum corrections. In the warm inflation scenario, instead, the effect of these interactions is not only to modify the scalar field potential, but also to generate dissipation and fluctuation effects. In order that warm inflation works, it is required that the time scale of quantum mechanical processes leading to the dissipation is much slower than the expansion rate of the Universe (in such a way the whole system, inflaton and fields, would not equally distribute the available energy). The Ricci scalar is related to the Hubble expansion rate during inflation as R = -12 H 2 and its time derivative is related to the slow roll parameter /epsilon1 = -˙ H/H 2 as ˙ R = 24 /epsilon1H 3 . The lepton asymmetry (25) in warm inflation reads T l is the light neutrino decoupling temperature (31). The power spectrum of curvature perturbation in thermal inflation and the spectral index of scalar perturbations are expressed in terms of H and /epsilon1 [139] where Γ is the damping parameter in the inflaton equation of motion and represents the coupling between the inflaton and the thermal bath. The WMAP observations [140] provides the amplitude of the curvature power spectrum and of the spectral index, which are given by P R = (2 . 3 ± 0 . 3) × 10 -9 and the spectral index n s = 0 . 951 ± 0 . 017, respectively. Combining (40) and (41) one can write /epsilon1 , Γ and M in terms of H and T : Choosing H /similarequal 8 × 10 12 GeV and T /similarequal 8 × 10 12 GeV, and using, consistently with WMAP data, P R ∼ 2 × 10 -9 and n s = 0 . 968, the net baryon asymmetry η wI /similarequal 10 -10 in the warm inflation scenario follows provided Γ = 7 . 1 × 10 9 GeV, /epsilon1 = 4 . 2 × 10 -6 , and M = 2 . 7 × 10 11 GeV. Finally, from (31) one gets that T l ∼ 10 13 GeV corresponds to the neutrino mass of m ν 3 = 0 . 15eV. Other models on baryo/leptogenesis in warm Inflation scenario have been proposed in [141, 142].",
"pages": [
34,
35
]
},
{
"title": "6. LEPTOGENESIS INDUCED BY SPIN-GRAVITY COUPLING OF NEUTRINOS WITH THE PRIMORDIAL GRAVITATIONAL WAVES",
"content": "The behavior of (relativistic) quantum systems in gravitational fields, as well as in inertial fields, play a crucial role for investigating the structure of spacetime at the quantum level [143]. Quantum objects are in fact finer and more appropriate probes of structures that appear classically as results of limiting procedures. On fundamental ground, it is expected that only a quantum theory of gravity will be able to provide a definitive answer to questions regarding the fundamental structure of spacetime. However, the extrapolation of General Relativity from ordinary terrestrial scales to Planck's scales is not free from subtle questions and new data coming from modern cosmology, as discussed in the Introduction and previous Sections. Therefore, one considers the gravitational background described by General Relativity, hence considered as a classical filed, and matter as quantized fields propagating in a classical background [132]. In this respect, one is there considering considering the interaction of classical inertial and gravitational fields with (relativistic) quantum objects. Observations performed in [144] do confirm that both gravity and inertia interact with quantum systems in ways that are compatible with General Relativity. This, per se, does not represent a test of General Relativity, but shows that the effects of inertia and gravity on wave functions are consistent with covariant generalization of wave equations dictated by General Relativity paradigms. Clearly, to test fundamental theories a central role is played, at level of terrestrial experiments (Earth-bound and near-space experiment), by inertial effects. Their identifications is therefore required with great accuracy and represents a big challenge for future experiments. Inertial effects, on the other hands, provide a guide in the study of relativity because, in all physical situations in which non-locality is not an issue, the equivalence principle ensures the existence of a gravitational effect for each inertial effect. Certainly the study of spin-gravity and spin-inertia coupling (as well as spin precession) represents a very active and relevant topics of physics. Experiments of high energy physics indeed typically involve spin-1/2 particles and take place or in a gravitational environment or in non-inertial frames. Thanks to the progress of technology, for example, atomic interferometry and the physics of polarized systems, the effects of the interactions of relativistic quantum particle with gravitation field, i.e. the spin-gravity coupling effects, could be provide new insights of QFT in curved spacetime. Spin-inertia and spin-gravity interactions and their effects in different physical situations are the subject of numerous theoretical (see for example [143, 145-169] and references therein) and experimental efforts[170-174]. Spin precession in inertial and gravitational fields have been studies in[175-180]. In this Section we shall discuss the effect of spin gravity coupling in a cosmological context. In particular, we are going to discuss the mechanism of Leptgenesis induced by spin-gravity coupling of neutrinos with the cosmological background [83, 181]. The approach is based on QFT in curved spacetime, and in particular we write down the Dirac equation in an expanding Universe. To this aim, we use the vierbein formalism. We assume that the early phase of the Universe is described by Inflation which generates the gravitational waves (tensor modes). The latter split in the energy levels of (Majoarana) neutrinos and antineutrinos, which ultimately results in the creation of a lepton asymmetry in the presence of lepton number violating interactions. This mechanism gives rise to the generation of a net leptogenesis.",
"pages": [
35,
36,
37
]
},
{
"title": "6.1. Dirac equation in curved space-time and the fermion dispersion relation",
"content": "In passing from flat to curved space time we use the standard prescription ∂ →∇ , and η µν → g µν . The procedure to replace flat space tensors with 'curved space' tensor cannot be extended to the case of spinors. This procedure works with tensors because the tensor representations of GL (4 , R ), i.e. the group of 4 × 4 real matrices, behave like tensors under the subgroup SO (3 , 1). Thus, considering the vector representation, as an example, one gets But there are no representation of GL (4 , R ) which behave like spinors under SO (3 , 1), i.e. there not does exists a function of x and x ' which reduces to the usual spinor representation of the Lorentz group ( D (Λ)) for x ' = Λ µ ν x ν . Therefore, to write down the general covariant coupling of spin-1/2 particles to gravity, we have to use the vierbein formalism. A vierbein fields (or tretad) is defined as where ξ a are the local inertial coordinate, x µ the generic coordinate, ξ a ( x ) → ξ ' a ( x ) = Λ a b ( x ) ξ b ( x ), and Λ T η Λ = η . The quantities e a µ ( x ) constitute a set of four coordinate vectors which form a basis for the (flat) tangent space to the curved space at the point x = X . Under the coordinate transformation x → x ' = x ( x ), the vierbeins e a µ ( x ) transform as (see Table III) The metric g µν ( x ) is related to the vierbein fields by the relation g µν ( x ) = η ab e a µ ( x ) e b ν ( x ), where η ab is the Minkowsky metric (in the local inertial frame). It follows Spinor fields are coordinate scalars which transforms under local Lorentz transformations as ψ α ( x ) → ψ ' α ( x ) = D αβ [Λ( x )] ψ β ( x ), where D αβ [Λ( x )] is the spinor representation of the Lorentz group and ψ α is the component of the spinor ψ (not be confused with general coordinate indices). Since Λ does depend on x , ∂ µ ψ α does not transform like ψ α under local Lorentz transformations. To obtain a Lagrangian invariant under generic coordinate transformation one has to define the covariant derivative D µ ψ α ≡ ∂ µ ψ α -[Ω µ ] αβ ψ β , where [Ω µ ] αβ is the connection matrix. Therefore one requires D µ ψ α → D αβ [Λ( x )] D µ ψ β ( x ), provided The connection matrix [Ω µ ] αβ ( x ) can be written as where S ab = σ ab 2 = i [ γ a ,γ b ] 2 are the generators of the the Lorentz group in the spinor representation and ω a µ b the spin connections. The spinor representation of the Lorentz group can be written as D [Λ( x )] = exp[ -( i/ 2) S ab θ ab ]. The covariant derivative acts on vierbeins as and the condition D µ e a ν = 0 allows to determine the spin connections Therefore, the general covariant coupling of spin 1 / 2 particles to gravity is given by the Lagrangian where D a = ∂ a -i 4 ω bca σ bc is the covariant derivative before introduced. The Lagrangian is invariant under the local Lorentz transformation of the vierbein and the spinor fields. By using the Dirac matrices properties the Lagrangian density (1) can be written in the form where In a local inertial frame of the fermion, the effect of a gravitational field appears as a axialvector interaction term shown in L .",
"pages": [
37,
38
]
},
{
"title": "6.2. Neutrinos effective Lagrangian in a local inertial frame",
"content": "To determine the dispersion relation of neutrinos propagating in a perturbed FRW Universe, we have to compute B d . Perturbations are generated by quantum fluctuations of the inflaton. Notice that for a FRW Universe the B a -term vanishes due to symmetry of the metric. The general form of perturbations on a flat FRW expanding universe can be written as where φ, ψ are scalar fluctuations, ω i the vector fluctuations and h ij the tensor fluctuations of the metric. Of the ten degrees of freedom in the metric perturbations only six are independent and the remaining four can be set to zero by suitable gauge choice. We work in the TT gauge: h i i = 0, ∂ i h ij = 0. In the TT gauge the perturbed FRW metric can be expressed as An orthogonal set of vierbiens e a µ for this metric is given by For our application we need only the tensor perturbations. Explicit calculations give B a = ( ∂ τ h × , 0 , 0 , ∂ τ h × ). Using ψ = ( ν L , ν R ) T into (1), one gets The fermion bilinear term ¯ ψγ 5 γ a ψ is odd under CPT transformation. When one treats B a as a background field then the interaction term in L explicitly violates CPT . When the primordial metric fluctuations become classical, i.e there is no back-reaction of the microphysics involving the fermions on the metric and B a is considered as a fixed external field, then CPT is violated spontaneously. Moreover, we consider only Standard Model fermions, so that L ν = +1 for neutrinos and L ¯ ν = -1 for antineutrinos ( L is lepton number), and we consider Majorana spinors ν R = ( ν L ) c , i.e. ν R is the charge-conjugate of ν L . With this choice, the mass term in L is of the Majorana type (the generation index is suppressed).",
"pages": [
39,
40
]
},
{
"title": "6.3. The Leptogenesis mechanism from spin-gravity coupling following Inflation",
"content": "According to the general setting, one has to assume that there are GUT processes that violate lepton number above some decoupling temperature T D . Results of Section 4 imply that in the presence of non-zero metric fluctuations, there is a split in energy levels of ν L,R (a different effective chemical potential), so that the dispersion relation of ν L,R fields reads (see also[87]) E L,R ( p ) = E p ∓ ( B 0 -p · B p ) , where as usual E p = p + m 2 / 2 p and p = | p | . The equilibrium value of lepton asymmetry generated for all T > T D turns out to be (Eq. (15)) where ∆ n ≡ n ( ν L ) -n ( ν R ) and p /greatermuch m ν and B 0 /lessmuch T has been used. The dependence on B drops out after angular integration To evaluate B 0 , we need to compute the spectrum of gravitational waves h ( x , τ ) during inflation. To this aim, we express h × in terms of the creation- annihilation operator where k is the comoving wavenumber, k = | k | , and m P = 1 . 22 10 19 GeV is the Planck mass. Remembering that B 0 = ∂ τ h one gets that the two point correlation function for B 0 is (see C for details) Therefore, the r.m.s value of spin connection that determines the lepton asymmetry ∆ n is The lepton asymmetry ∆ n as a function of temperature can therefore be expressed as (considering three neutrino flavors) The lepton number to entropy density ( s = 0 . 44 g ∗ T 3 ) is therefore /negationslash The lepton asymmetry is evaluated at the decoupling temperature T D . According to general setting, lepton number asymmetry will be generated as long as the lepton number violating interactions are in thermal equilibrium. Once these reactions decouple at some decoupling temperature T D , which we shall determine, the ∆ n ( T ) /s ( T ) ratio remains fixed for all T < T D . To calculate the decoupling temperature of the lepton number violating processes we turn to a specific effective dimension five operator which gives rise to Majorana masses for the neutrinos (26) The ∆ L = 2 interactions that result from the dimension five operator (26) are given in (27). In absence of GWs, it follows that the forward reactions are equal to the backward reactions, and therefore no net lepton number is generated. On the contrary, in presence of GWs, the forward reactions are different by backward reactions, and the energy levels of the left and right helicity neutrinos are no longer degenerate, E L = E R , which implies a difference in the number density of left and right handed neutrinos (at thermal equilibrium) ∆ n = n ( ν L ) -n ( ν R ) = 0. This process continues till the ∆ L = 2 interactions decouple. Next step now is to calculate the decoupling temperature T D by using (23). The interaction rate for the interaction ν Lα + φ 0 ↔ ν Rβ + φ 0 is (see (30)) In the electroweak era, when the Higgs field in L W acquires a vev , 〈 φ 〉 = (0 , v ) T (where v = 174 GeV ), the five dimensional Weinberg operator gives rise to a neutrino mass matrix m αβ = v 2 C αβ M . This implies that in our calculations, we can substitute the couplings C αβ M in terms of the light left handed Majorana neutrino mass, i.e. C αβ M → m αβ v 2 . The decoupling temperature T D turns out to be where m ν is the mass of the (heaviest) neutrino. Substituting T D into the expression for η , we finally obtain the formula for lepton number /negationslash The input parameters we used for our estimations are: a ) The amplitude of the h × ∼ 10 -6 or equivalently the curvature during inflation H I ∼ 10 14 GeV or the scale of inflation is the GUT scale, V 1 / 4 ∼ 10 16 GeV which is allowed by CMB [182]. b ) Neutrino Majorana mass in the atmospheric neutrino scale[183, 184] m 2 ν ∼ 10 -3 eV 2 . c ) Duration of inflation H I t = N ∼ 100 (needed to solve the horizon and entropy problems in the standard inflation paradigm, Eq. (1)). The parameters entering (6) are well within experimentally acceptable limits. The magnitude of baryogenesis is therefore which is compatible with the previous values (3) and (4). According to [39], a lepton asymmetry generated at an earlier epoch gets converted to baryon asymmetry of the same magnitude by the electroweak sphalerons. The mechanism here discussed makes use of the standard QFT in curved space-time, which gives rise to the conventional spin gravity coupling of neutrinos with the gravitational background. This leads to the coupling of axial-vector current with the four-vector B a which accounts for the curved background. As we have seen, left-handed and right-handed fields couple differently to gravity and therefore have different dispersion relations (the equivalence principle is hence violated). Moreover, the model uses the Majorana neutrinos because one needs of a violation of lepton number which generates the lepton asymmetry. In order that spin-gravity coupling of neutrinos with the gravitational background be a viable model, one has to assume that the early Universe is described by Inflation. Remarkably, no free parameters are present in the final expression for the lepton asymmetry given by (6).",
"pages": [
40,
41,
42
]
},
{
"title": "6.4. CPT Violation in an Expanding Universe - String Theory",
"content": "Recently Ellis, Mavromatos and Sarkar[185, 186] proposed a model in which they explore the possibility to violate CPT in an expanding Universe in the framework of String Theory, and generate a net baryon/lepton asymmetry via Majorana neutrinos. The model goes along the following line. According to String Theory, besides to the spin-2 graviton (described by the usual symmetric tensor g µν ), the theory contains also the spin-0 dilaton field (described by the scalar field Φ) and the anti-symmetric tensor KalbRamond field (described by B µν ). The latter enters into the effective action via the field H µνρ = ∂ µ B νρ + p.c. , where p.c. stands for the cyclic permutation of the indices { µ, ν, ρ } ( H µνρ plays the role of torsion[187]). The effective action reads (in the Einstein frame) Here M s = 1 / √ α ' represents the string scale mass, V c the compactification volume which is expressed, together with the compact radii, in terms of the √ α ' units. In this model ne gets that the connections are generalized as where Γ λ µν are the usual Christoffel symbols, and The four-vector B d given in (3) now reads The anti-symmetric tensor can be written in terms of the pseudo-scalar axion-like field b ( x ) where /epsilon1 0123 = √ -g . Field equations of String theory provides the solution [188] where Φ 0 is a constant appearing in the time evolution of the dilaton field Φ( t ) = -ln t +Φ 0 , Q 2 > 0 is the central charge deficit and n is an integer associated to the Kac-Moody algebra of the underlying world sheet conformal field theory. According to (9)-(13) one finds that the non vanishing component of B d is The net baryon asymmetry between the (Majorana) neutrino-antineutrino is",
"pages": [
42,
43
]
},
{
"title": "6.5. Leptogenesis induced by Einstein-Cartan-Sciama-Kibble torsion field",
"content": "Another interesting model of matter-antimatter asymmetry has been proposed by Poplawski [189]. It is based on Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity [190] in which the usual Hilbert-Einstein action (14) incorporates the torsion field. The latter therefore extend General Relativity to include matter with intrinsic spin-1/2, which produce torsion, and provides a more general theory of local gauge with respect to the Poincare group [191] (interesting applications of ECSK can be find in [192, 193] and references therein). Spinors coupled to the torsion fields evolves according to Helh-Datta equation [194] where ∇ a represents the covariant derivative with respect to the affine connection (Christoffel connections). The corresponding equation for the charge conjugate (C) spinor ψ C is The equations (16) and (17) are therefore different, leading to a different shift of energy spectrum, generating in such a way an asymmetry. In fact, the energy levels for a free fermion ( X ) and antifermion ( ¯ X ) resulting from ECSK theory are (in the ultrarelativistic limit) where α is a numerical factor of the order of unity, and N is the inverse normalization of Dirac spinor (it is of the order of N ∼ E 3 ∼ T 3 ). Here X and ¯ X refer to heavy fermion carrying baryon and antibaryon number, respectively, and are dubbed archeons and antiarcheons. They candidate for a possible component of DM. Eqs. (2) and (15) then imply that the net baryon asymmetry is η ∼ T 2 D /M 2 P , where T D is the decoupling temperature that must assume the value T D ∼ 10 13 GEV in order to generate the observed baryon asymmetry. One can estimate the mass m X of the (anti)archeon by equating the decay rate Γ ∼ G 2 F m 5 X 192 π 3 with the expansion rate of the Universe during the radiation era, Eq. (5): m X = m ¯ X ∼ 10TeV. This value is of the same order of magnitude of the maximum energy of a proton beam at LHC ( ∼ 7TeV).",
"pages": [
44
]
},
{
"title": "7. CONCLUSION",
"content": "Understanding how the baryon asymmetry of the universe originated is one of the fundamental goals of modern cosmology. As we have seen, particle physics, as well as cosmology, have provided with a number of possibilities. They involve very fascinating physics, but with varying degrees of testability. In fact, all the baryogenesis models are indeed able to derive the correct estimation of η ∼ 10 -10 , but it is rather difficult, if not impossible, to exclude or confirm one or other scenario. A possibility to discriminate among the plethora of baryogenesis models is to investigate its predictibility or compatibility with certain form of DM (besides the models discussed in this review, see also[195]). In other words, one should expect that a realistic model of baryogenesis is able to determine both the right value of η , and to explain the magnitude of the ratio ρ B /ρ DM fixed by cosmological observations. An example is provided by the AffleckDine Baryogenesis: The model explains the observed baryon asymmetry of the Universe, and supersymmetric particles are favored candidates of DM in the Universe. In this respect, as discussed in this review, gravitational baryogenesis represents also an interesting framework. In fact because if from one side, in these models one is able to recover the correct estimation of the parameter 1 eta , from the other side they are compatible with the present cosmological data of an epanding Universe, and hence the necessity to invoke new form of energy or matter, hence DM and DE. This is the case of f ( R ) gravity, but also variant of these models and more generally, scalar tensor theories are available candidates. A question that arises is whether the universe is baryon-symmetric on cosmological scales, and eventually separated into domains which are either dominated by baryons or antibaryons. One then would expect to detect, due to annihilations, an excess of gamma rays. There are no evidences of the existence of such a cosmic anti-matter. In fact, the analysis of p ¯ p annihilation in gamma rays ( p ¯ p → π 0 → 2 γ ), with E γ ∼ 100MeV, allows to conclude that the nearest rich antimatter region (anti-galaxy) should be away from at distance[196, 197] D ≥ (10 -15)Mpc. This results indicate hence that the patches of matterantimatter should be as large as the presently observable Universe. However, no mechanism is known which is able to explain how to separate out these domains of matter-antimmater. There is anyway an existing as well as many planned experimental activity in searching for cosmic antimatter[198-201]. It is still unclear to cosmologists and particle physicists what scenario was realized in nature to generate the observed baryon asymmetry in the Universe. What has arisen in the last years is that no model of Baryogenesis is complete without incorporating the idea s underlying the Leptogenesis. This is also supported by recent results of neutrino physics. LHC experiment and the new generation of linear colliders will certainly allow a deeper understanding and a considerable progress of this fundamental problem. In any case, the baryon asymmetry in the Universe furnishes a clear evidence that a new physics, beyond the Standard Model of particle physics and the Standard Cosmological model, is called for.",
"pages": [
45,
46
]
},
{
"title": "Acknowledgments",
"content": "The authors thank Prof. Dh. Ahluwalia and Prof. E.H. Chion for invitation to write this review. G. Lambiase thanks the University of Salerno and ASI (Agenzia Spaziale Italiana) for partial support through the contract ASI number I/034/12/0.",
"pages": [
46
]
},
{
"title": "Appendix A: Out equilibrium condition",
"content": "For completeness, we discuss on a different setting, the departure from thermal equilibrium. If all particles in the Universe were in thermal equilibrium then there would be no preferred direction for the time T and also if B asymmetry could be generated, it would be prevented by CPT invariance. Therefore, also the violation of the CP symmetry would be irrelevant. Consider a species of massive particle X in thermal equilibrium at temperatures T /lessmuch m X . Let m X its mass. The number density of these particles is where, as usual, µ X indicate the chemical potential. The species X are in chemical equilibrium if the rate Γ inel of inelastic scatterings (responsible of the variation of the number of X particles in the plasma, according to the processes X + A → B + C ) is larger than the expansion rate of the Universe Γ inel /greatermuch H . This allows to write a relation among the different chemical potentials of the particles involved in the process µ X + µ A = µ B + µ C . The number density of the antiparticle ¯ X which have the same mass of particles X , m ¯ X = m X , but opposite chemical potential µ ¯ X = -µ X due to the process X + ¯ X → γ + γ with µ γ = 0, is If the X particle carries baryon number, then B will get a contribution from If there exist B -violating reactions (first Sakharov's condition) for the species X and ¯ X , such as X + X → ¯ X + ¯ X , then the chemical potential is zero, µ X = 0. As a consequence, also the relative contribution of the X particles to the net baryon number vanishes. Therefore, only a departure from thermal equilibrium can allow for a finite baryon excess, that means that the form of n X, ¯ X has to be modified, The typical example of the out-of equilibrium decay can be represented by the following steps: Let X be a heavy particle such that m X > T at the decay time, and let X → Y + B the decay process. When the energy of the final state is given by E Y + B ∼ O ( T ), then there is no phase space for the inverse decay to occur. The final state Y + B does not have enough energy to create a heavy particle X (the rate for Y + B → X is Boltzmann suppressed, i.e. Γ( Y + B → X ) ∼ e -m X /T ).",
"pages": [
46,
47
]
},
{
"title": "Appendix B: The physics of Sphalerons",
"content": "In the EW theory, the most general Lagrangian invariant under the SM gauge group and containing only color singlet Higgs fields is automatically invariant under global abelian symmetries. The latter are associated to the baryonic and leptonic symmetries. It is hence not possible to violate B and L at tree level, as well as in any order of perturbation theory. The perturbative expansion, however, does not describe all the dynamics of the theory. 't Hooft provided a scenario in which nonperturbative effects (instantons) may give rise to processes which the combination B + L is violated, whereas the (orthogonal) combination B -L does not. In some circumstances, such as the early Universe at very high temperature, the processes that violate the baryon and lepton number may be fast enough. These processes may be significant role for baryogenesis mechanisms. At the quantum level, the baryon and lepton symmetries are anomalous (triangle anomaly) where g, g ' are the gauge coupling of SU (2) L and U (1) Y , n f the number of families, the dual of SU (2) L field strength tensor, the dual of U (1) Y field strength tensor. The change of baryon number, which is closely related to the vacuum structure of the theory, is given by with N CS is the Chern-Simons number. For vacuum to vacuum transition, the field A represent a (pure) gauge configuration, whereas the Chern-Simons numbers N CS ( t f ) and N CS ( t i ) assume integer values. In a non-abelian gauge theory, there are infinitely many degenerate ground states (labeled by the Chern-Simons number ∆ N CS = ± 1 , ± 2 , ± 3 , . . . ). In field space, the corresponding point are separated by a potential barrier. The height of his barrier gives the sphaleron energy E sp . Because the anomaly, jumps in the Chern-Simons numbers are associated with changes of baryon and lepton number ∆ B = ∆ L = n f ∆ N CS . The smallest jump in the Standard Model is characterized by ∆ B = ∆ L = ± 3. In semiclassical approximation, the probability of tunneling between neighboring vacua is determined by instanton configurations. In the Standard Model, SU(2) instanton lead to an effective 12-fermion interaction which describes processes with ∆ B = ∆ L = 3, such as The transition rate is given by Γ ∼ e -S inst ∼ O (10 -165 ), where S inst is instanton action. Because the rate is extremely small, B + L violating interactions appear completely negligible in the Standard Model. However this is not true in a thermal bath, and hence in the primordial Universe. As emphasized by Kuzmin, Rubakov, and Shaposhnikov, transition between the gauge vacua occurs not by tunneling but through thermal fluctuations over the barrier. For temperature T > E sp , the suppression in the rate provided by Boltzmann factor disappears and therefore processes that violate B + L can occur at a significant rate. In the expanding Universe these processes can be in equilibrium .",
"pages": [
47,
48,
49
]
},
{
"title": "Appendix C: The power spectrum of the Gravitational Waves",
"content": "The function h ( x , τ ) ≡ h appearing in (5) satisfies the equation (from Einstein field equation) The mode functions f k ( τ ) obey the minimally coupled Klein-Gordon equation where ' = ∂ τ . During de Sitter era, the scale factor a ( τ ) = -1 / ( H I τ ) where H I is the Hubble parameter, so that a solution is which matches the positive frequency 'flat space' solutions e -ikτ / √ 2 k in the limit of kτ /greatermuch 1. Substituting this solution in h ( x , τ ) (5), and using the canonical commutation relation [ a k , a † k ' ] = δ kk ' , we get the standard expression for two point correlation of gravitational waves generated by inflation with the spectrum of gravitational waves given by the scale invariant form ( | h k | 2 ) inf = 4 π H 2 I m 2 p . Consider now the GWs modes that re-enter the horizon at the radiation era a ( τ ) ∼ τ . One finds that the GWs have the two point correlation fluctuation where and g = 106 . 7 is the number of relativistic degrees of freedom which for the Standard Model ∗ There is a stringent constraint H I /M p < 10 -5 from CMB data [202]. This constraint limits the parameter space of interactions that can be used for generating the requisite lepton-asymmetry. In the radiation era, when these modes re-enter the horizon, the amplitude redshifts by a -1 from the time of re-entry. The reason is that in the radiation era a ( τ ) ∼ τ and the equation for f k gives plane wave solutions f k = (1 / √ 2 k ) exp ( -ikτ ). Therefore in the radiation era the amplitudes of h redshifts as a -1 . The gravitational waves inside the horizon in the radiation era will be h rad k = h inf k a k a ( τ ) = h inf k T T k , where h inf k are the gravitational waves generated by inflation, a k and T k are the scale factor and the temperature when the modes of wavenumber k entered the horizon in the radiation era. The horizon entry of mode k occurs when where is the Hubble parameter at the time of horizon crossing of the k the mode. Solving equation for T k we get The amplitude of the gravitational waves of mode k inside the radiation horizon is (using the equation for T k and the previous expression for Note that the gravitational wave spectrum inside the radiation era horizon is no longer scale invariant. The gravitational waves in position space have the correlation function and hence for the spin connection B 0 generated by the inflationary gravitational waves in the radiation era, we get The spectrum of spin-connection is scale invariant inside the radiation horizon. This is significant in that the lepton asymmetry generated by this mechanism depends upon the infrared and ultraviolet scales only logarithmically. The scales outside the horizon are bluetilted which means that there will be a scale dependent anisotropy in the lepton number correlation at two different space-time points where is the anisotropic deviation from the mean value. Unlike in the case of CMB, this anisotropy in the lepton number is unlikely to be accessible to experiments. Nucleosynthesis calculations only give us an average value at the time of nucleosynthesis (when T ∼ 1 MeV ). The maximum value of k are for those modes which leave the de Sitter horizon at the end of inflation. If inflation is followed by radiation domination era starting with the re-heat temperature T RH then the maximum value of k in the radiation era (at temperature T ) is given by k max /a ( T ) = H I ( T T RH ) . The lower limit of k is k min = e -N k max which are the modes which left the de-Sitter horizon in the beginning of inflation ( N is the total e-folding of the scale factor during inflation, N /similarequal 55 -70). The integration over k then yields just the factor ln( k max /k min ) = N . perturbations allow for the primordial gravitational waves. [204] Notice that due to U (1) symmetry there is a conserved current given by [205] It is worth nothing that by making use of the dimensional regularization [135], for example, one infers [206] The anomaly trace is typically expressed in term of curvature tensors and their covariant derivatives, as well as mass-terms[132, 136], i.e. where the coefficients c 1 , 2 ( m ) are combinations of (power) mass fields, N 0 , 1 / 2 , 1 is the number of the quantum matter of boson, fermion and vector fields. The coefficients c i are subject to a finite renormalization, becoming free parameters of the theory. For our purpose we can neglect the mass-terms since the fields are relativistic, keeping in mind however, that the procedure of renormalization gives rise to purely geometric terms which appear in the final expression of the trace anomaly.",
"pages": [
49,
50,
51,
61
]
},
{
"title": "TABLE III: Vierbeins transformations",
"content": "Under coordinate transformations Under local Lorentz transformations the vierbeins e a µ ( x ) transform as the vierbeins e a µ ( x ) transform as ξ ' a ( x ' ) = ξ a ( x ) whereas x µ does not transform",
"pages": [
62
]
}
]
<document>
<section_header_level_1><location><page_1><loc_17><loc_80><loc_80><loc_85></location>Dark energy with rigid voids versus relativistic voids alone</section_header_level_1>
<text><location><page_1><loc_41><loc_75><loc_67><loc_77></location>Boudewijn F. Roukema 1 , 2</text>
<text><location><page_1><loc_30><loc_60><loc_78><loc_75></location>1 Toru´n Centre for Astronomy Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University ul. Gagarina 11, 87-100 Toru´n, Poland 2 Universit´e de Lyon, Observatoire de Lyon Centre de Recherche Astrophysique de Lyon ´</text>
<text><location><page_1><loc_15><loc_57><loc_93><loc_61></location>CNRS UMR 5574: Universit´e Lyon 1 and Ecole Normale Sup´erieure de Lyon 9 avenue Charles Andr´e, F-69230 Saint-Genis-Laval, France ∗</text>
<text><location><page_1><loc_40><loc_55><loc_68><loc_56></location>boud@astro.uni.torun.pl</text>
<section_header_level_1><location><page_1><loc_47><loc_48><loc_62><loc_50></location>30 March 2013</section_header_level_1>
<text><location><page_1><loc_21><loc_43><loc_77><loc_46></location>Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation</text>
<section_header_level_1><location><page_1><loc_45><loc_36><loc_53><loc_37></location>Abstract</section_header_level_1>
<text><location><page_1><loc_19><loc_21><loc_79><loc_34></location>The standard model of cosmology is dominated-at the present epochby dark energy. Its voids are rigid and Newtonian within a relativistic background. The model prevents them from becoming hyperbolic. Observations of rapid velocity flows out of voids are normally interpreted within the standard model that is rigid in comoving coordinates, instead of allowing the voids' density parameter to drop below critical and their curvature to become negative. Isn't it time to advance beyond nineteenth century physics and relegate dark energy back to the 'no significant evidence' box?</text>
<figure>
<location><page_2><loc_26><loc_64><loc_68><loc_89></location>
<caption>Figure 1: Virialisation fraction f vir ( z ) in Virgo Consortium 256 3 -particle simulations [1, 2] with 240 h -1 Mpc and 85 h -1 Mpc box sizes, shown as continuous thick curves for the Einstein-de Sitter model, and dark energy parameter Ω Λ ( z ) evolution for Ω Λ0 = 0 . 72. Thin curve: 240 h -1 Mpc ΛCDM simulation.</caption>
</figure>
<section_header_level_1><location><page_2><loc_19><loc_52><loc_74><loc_54></location>1 'Dark energy' traces inhomogeneity</section_header_level_1>
<text><location><page_2><loc_19><loc_32><loc_79><loc_51></location>The standard model of cosmology is generally accepted to be a spacetime with a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric [3, 4, 5, 6, 7], solving the Einstein equation rather simply thanks to the assumption of homogeneous density on any spatial slice, and measured to have the Concordance Model [8] values of the matter density and dark energy parameters, Ω m0 ≈ 0 . 32 (e.g. [9]) and Ω Λ0 := 1 -Ω m0 , respectively (hereafter, ΛCDM). However, we live in the inhomogeneous epoch: galaxies and voids certainly exist today. As found by [10], in an Einstein-de Sitter (EdS) FLRW model (i.e. with Ω m0 = 1 , Ω Λ0 = 0), the fraction of matter in a large region that is virialised, f vir , evolves in a very similar way to that of the dark energy parameter in a flat FLRW model with negligible radiation density,</text>
<formula><location><page_2><loc_38><loc_28><loc_79><loc_31></location>Ω Λ ( z ) = 1 -Ω m0 a 3 Ω Λ0 +Ω m0 . (1)</formula>
<text><location><page_2><loc_19><loc_25><loc_52><loc_26></location>This is shown in Fig. 1, where Ω Λ0 = 0 . 68.</text>
<text><location><page_2><loc_19><loc_13><loc_79><loc_25></location>This seems like an extraordinary coincidence. Over the same redshift range during which one expects that the Universe is inhomogeneous, the degree of inhomogeneity, as expressed by f vir ( z ) in an EdS model, approximately follows the proportion of the critical density represented by dark energy, if the dark energy is inferred from forcing a homogeneous model on the observational data. To first order, f vir is not sensitive to the choice of FLRW model (see Fig. 1), so the coincidence also exists for inhomogeneity in a ΛCDM model.</text>
<text><location><page_3><loc_19><loc_87><loc_79><loc_90></location>The simplest inference is that a homogeneous-model-inferred non-zero dark energy parameter is really just a measurement of inhomogeneity .</text>
<text><location><page_3><loc_19><loc_84><loc_79><loc_87></location>What physical link could there be between this inhomogeneity parameter and homogeneous-model-inferred 'dark energy'?</text>
<section_header_level_1><location><page_3><loc_19><loc_76><loc_79><loc_81></location>2 Void dominance: low matter density, high critical density</section_header_level_1>
<text><location><page_3><loc_19><loc_61><loc_79><loc_75></location>The most obvious physical link between inhomogeneity and homogeneousinferred dark energy is the volume dominance of voids compared to virialised regions at recent epochs [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. This is because gravitational collapse implies an increase in density, i.e., a reduction in volume, by a factor of about δ vir ∼ 100-200 (e.g., 8-18 π 2 [21]), so that, to first order, the collapsed matter occupies a negligible fraction of the spatial volume. Thus, recent-epoch spatial volume is overwhelmingly dominated by low-density regions.</text>
<text><location><page_3><loc_19><loc_39><loc_79><loc_61></location>Moreover, there are velocity flows out of the voids, since otherwise, the voids couldn't be nearly empty. Thus, the critical density defining spatial flatness in the voids is higher than it would be in a homogeneous calculation. The standard, FLRW approach insists that space expands uniformly with spatially constant curvature, i.e. space is rigid in comoving coordinates-it is forbidden from bending under the influence of gravity. N -body simulations are typically used to study the Newtonian formation of overdense structures and voids within this rigid, comoving background model. However, both the low density of the voids and the velocity flow out of them imply that the matter density parameter in the voids is sub-critical. Thus, the voids are hyperbolic. Geometrically, this hyperbolicity should also be taken into account, implying an even lower matter density parameter (conservatively, let us ignore this: the effect is small).</text>
<section_header_level_1><location><page_3><loc_19><loc_34><loc_70><loc_36></location>3 Volume-weighted averaged metric</section_header_level_1>
<text><location><page_3><loc_19><loc_19><loc_79><loc_33></location>These arguments can be formalised using the volume-weighted averaging approach to modelling inhomogeneous spatial slices [22, 23, 11, 24, 16], in which the Friedmann equation is generalised to (12) of [13]. For simplicity, let us (i) set the dark energy term to zero, (ii) neglect the kinematic backreaction as much smaller than the curvature backreaction (see [16] for numerical justification), and (iii) combine the curvature parameters into a single full curvature parameter as suggested in [13]. Writing 'k' instead of ' R ', this gives the domain-averaged, effective Friedmann equation</text>
<formula><location><page_3><loc_41><loc_16><loc_79><loc_18></location>Ω eff k ( z ) = 1 -Ω eff m ( z ) . (2)</formula>
<text><location><page_3><loc_19><loc_12><loc_79><loc_15></location>Along a typical, large-scale, random, spacelike or null geodesic over recent epochs, what proportions of the geodesic lie in the emptied and virialised</text>
<text><location><page_4><loc_19><loc_87><loc_79><loc_90></location>regions? The proportions at a given z are, on average, (1 -f vir /δ vir ) : f vir /δ vir , respectively. Given that δ vir ∼ 100-200 and</text>
<formula><location><page_4><loc_44><loc_84><loc_79><loc_86></location>0 ≤ f vir ≤ 1 , (3)</formula>
<text><location><page_4><loc_19><loc_74><loc_79><loc_83></location>less than about 1% of the geodesic falls within the virialised regions. Thus, by starting with a large-scale, high-redshift, 'background' FLRW model-in this case, an EdS model-an effective metric can be written by assuming that the virialised matter contributes negligibly. As in [10], the effective expansion rate is</text>
<formula><location><page_4><loc_38><loc_73><loc_79><loc_74></location>H eff ( z ) = H ( z ) + H pec ( z ) , (4)</formula>
<text><location><page_4><loc_19><loc_63><loc_79><loc_71></location>combining the FLRW expansion H ( z ) with the peculiar velocity gradient (physical, not comoving) across voids, H pec ( z ), estimated numerically from an N -body simulation ([1, 2], a 240 h -1 Mpc box-size EdS simulation; see [10]) The background Hubble constant H bg is set to make the present value consistent with low redshift estimates [25, 26], i.e.</text>
<formula><location><page_4><loc_30><loc_60><loc_79><loc_62></location>H bg := 74km / s / Mpc -H pec (0) ∼ 50km / s / Mpc . (5)</formula>
<text><location><page_4><loc_19><loc_54><loc_79><loc_58></location>Thus, the loss of matter from the voids and the higher critical density in voids both decrease the EdS background value from Ω m , bg = 1 to an effective value of</text>
<formula><location><page_4><loc_32><loc_43><loc_79><loc_53></location>Ω eff m ( z ) ≈ (1 -f vir ) ( H H eff ) 2 Ω m = (1 -f vir ) ( H bg H eff ) 2 Ω m , bg a -3 . (6)</formula>
<text><location><page_4><loc_19><loc_40><loc_46><loc_41></location>The effective radius of curvature is</text>
<formula><location><page_4><loc_38><loc_34><loc_79><loc_39></location>R eff C ( z ) = c aH eff ( z ) √ Ω eff k ( z ) . (7)</formula>
<text><location><page_4><loc_19><loc_32><loc_43><loc_33></location>The effective metric (cf [17]) is</text>
<formula><location><page_4><loc_21><loc_25><loc_79><loc_30></location>d s 2 = -d t 2 + a 2 ( t ) [ d χ eff 2 + R eff C 2 ( sinh 2 χ eff R eff C ) (d θ 2 +cos 2 θ d φ 2 ) ] , (8)</formula>
<text><location><page_4><loc_19><loc_22><loc_50><loc_23></location>where the radial comoving component is</text>
<formula><location><page_4><loc_39><loc_18><loc_79><loc_21></location>d χ eff ( z ) := c a 2 H eff ( z ) d a. (9)</formula>
<figure>
<location><page_5><loc_25><loc_64><loc_68><loc_89></location>
<caption>Figure 2: Distance modulus normalised to the Milne model (Ω m = 0 , Ω Λ = 0 , ∀ z ) for the homogeneous ΛCDM model (top, black), the uncorrected, homogeneous EdS model (bottom, red), and the void-corrected EdS model (middle, thick, green; 'VA' = virialisation approximation).</caption>
</figure>
<section_header_level_1><location><page_5><loc_19><loc_51><loc_79><loc_56></location>4 Matter density parameter and luminosity distances</section_header_level_1>
<text><location><page_5><loc_19><loc_39><loc_79><loc_49></location>Without any attempt to fit this approximation to observational data , apart from (5) above, the correction of the EdS model as presented above gives an effective matter density parameter (6) that drops slowly from its background value of unity at high redshift down to Ω eff m = 0 . 27 at the present epoch z = 0, remarkably close to the last two decades' local estimates of the matter density parameter.</text>
<text><location><page_5><loc_19><loc_36><loc_79><loc_39></location>The effective luminosity distance follows directly from the radial comoving distance and hyperbolicity,</text>
<formula><location><page_5><loc_38><loc_31><loc_79><loc_35></location>d eff L = (1 + z ) R eff C sinh χ eff R eff C . (10)</formula>
<text><location><page_5><loc_19><loc_23><loc_79><loc_30></location>Figure 2 shows that despite the rough nature of the virialisation approximation, it shifts the homogeneous EdS magnitude-redshift relation by a substantial fraction towards the homogeneous ΛCDM relation, and thus, towards the observational supernovae type Ia relation.</text>
<section_header_level_1><location><page_5><loc_19><loc_18><loc_39><loc_20></location>5 Conclusion</section_header_level_1>
<text><location><page_5><loc_19><loc_12><loc_79><loc_17></location>A handful of simple formulae, lying at the heart of homogeneous, spatially rigid cosmology, remain approximately valid when generalised to inhomogeneous, spatially flexible cosmology [13] and applied to what observationally</text>
<text><location><page_6><loc_19><loc_63><loc_79><loc_90></location>and theoretically dominate the present-day spatial volume-the voids. The result is a correction to a large-scale, high-redshift, background Einstein-de Sitter cosmological model. The correction approximately gives the observed lowredshift matter density parameter and nearly matches the type Ia supernovae luminosity distance relation. The amplitude of the correction is unlikely to be much smaller than estimated here. At the present epoch, direct observations [27], N -body simulations [1, 2], and the existence of the cosmic web itself establish inhomogeneous peculiar velocity gradients of ∼ 20-30 km/s/Mpc, forcing, at least, a factor of ∼ (75 / 50) 2 reduction of the Einstein-de Sitter matter density to Ω eff m < 0 . 5, via Eqs (4), (5), and (6). A virialisation fraction of the order of ∼ 50% reduces this to ∼ 0 . 25. Even when forcing the homogeneous FLRW models onto the data, the initial analysis of the Planck Surveyor cosmic microwave background data finds H bg = 67 . 3 ± 1 . 2 km/s/Mpc at z ≈ 1100 [9]-not as low as the velocity gradients imply, but still significantly lower than the low redshift estimates of H eff (0) = 74 . 0 ± 1 . 6 km/s/Mpc ([26, 25], standard error in the mean).</text>
<text><location><page_6><loc_19><loc_45><loc_79><loc_63></location>How could such a simple, back-of-the-envelope calculation have been missed for so long? While the volume-averaging approach to cosmology has been developed over many years (see e.g. [13] for a review), possibly the answer lies, ironically, in confusion between the spacelike, unobserved , comoving, present time slice and the past light cone. A gigaparsec-scale void in the former should have a very weak ( δ ∼ 10 -5 ) underdensity, and our would-be location at its centre would be uncomfortably anti-Copernican. But these are both moot points! On the past light cone, an average , gigaparsec-scale, sub-critical (0 < Ω eff m < 0 . 8) void is perfectly natural, since it is defined by the onset of the virialisation epoch at z < ∼ 3. Moreover, we are naturally located at this pseudo-void's centre, by the nature of the past light cone.</text>
<text><location><page_6><loc_19><loc_38><loc_79><loc_44></location>What is simpler: relativistic, hyperbolic voids, observed by an observer at the tip of the past light cone, with no dark energy parameter? Or rigid, Newtonian voids together with a dark energy parameter that traces the virialisation fraction?</text>
<text><location><page_6><loc_19><loc_28><loc_79><loc_36></location>Acknowledgments: This essay is in part based on work conducted within the 'Lyon Institute of Origins' under grant ANR-10-LABX-66, and on Program Oblicze'n Wielkich Wyzwa'n nauki i techniki (POWIEW) computational resources (grant 87) at the Pozna'n Supercomputing and Networking Center (PCSS).</text>
<section_header_level_1><location><page_6><loc_19><loc_23><loc_34><loc_25></location>References</section_header_level_1>
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[
{
"title": "Dark energy with rigid voids versus relativistic voids alone",
"content": "Boudewijn F. Roukema 1 , 2 1 Toru´n Centre for Astronomy Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University ul. Gagarina 11, 87-100 Toru´n, Poland 2 Universit´e de Lyon, Observatoire de Lyon Centre de Recherche Astrophysique de Lyon ´ CNRS UMR 5574: Universit´e Lyon 1 and Ecole Normale Sup´erieure de Lyon 9 avenue Charles Andr´e, F-69230 Saint-Genis-Laval, France ∗ boud@astro.uni.torun.pl",
"pages": [
1
]
},
{
"title": "30 March 2013",
"content": "Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "The standard model of cosmology is dominated-at the present epochby dark energy. Its voids are rigid and Newtonian within a relativistic background. The model prevents them from becoming hyperbolic. Observations of rapid velocity flows out of voids are normally interpreted within the standard model that is rigid in comoving coordinates, instead of allowing the voids' density parameter to drop below critical and their curvature to become negative. Isn't it time to advance beyond nineteenth century physics and relegate dark energy back to the 'no significant evidence' box?",
"pages": [
1
]
},
{
"title": "1 'Dark energy' traces inhomogeneity",
"content": "The standard model of cosmology is generally accepted to be a spacetime with a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric [3, 4, 5, 6, 7], solving the Einstein equation rather simply thanks to the assumption of homogeneous density on any spatial slice, and measured to have the Concordance Model [8] values of the matter density and dark energy parameters, Ω m0 ≈ 0 . 32 (e.g. [9]) and Ω Λ0 := 1 -Ω m0 , respectively (hereafter, ΛCDM). However, we live in the inhomogeneous epoch: galaxies and voids certainly exist today. As found by [10], in an Einstein-de Sitter (EdS) FLRW model (i.e. with Ω m0 = 1 , Ω Λ0 = 0), the fraction of matter in a large region that is virialised, f vir , evolves in a very similar way to that of the dark energy parameter in a flat FLRW model with negligible radiation density, This is shown in Fig. 1, where Ω Λ0 = 0 . 68. This seems like an extraordinary coincidence. Over the same redshift range during which one expects that the Universe is inhomogeneous, the degree of inhomogeneity, as expressed by f vir ( z ) in an EdS model, approximately follows the proportion of the critical density represented by dark energy, if the dark energy is inferred from forcing a homogeneous model on the observational data. To first order, f vir is not sensitive to the choice of FLRW model (see Fig. 1), so the coincidence also exists for inhomogeneity in a ΛCDM model. The simplest inference is that a homogeneous-model-inferred non-zero dark energy parameter is really just a measurement of inhomogeneity . What physical link could there be between this inhomogeneity parameter and homogeneous-model-inferred 'dark energy'?",
"pages": [
2,
3
]
},
{
"title": "2 Void dominance: low matter density, high critical density",
"content": "The most obvious physical link between inhomogeneity and homogeneousinferred dark energy is the volume dominance of voids compared to virialised regions at recent epochs [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. This is because gravitational collapse implies an increase in density, i.e., a reduction in volume, by a factor of about δ vir ∼ 100-200 (e.g., 8-18 π 2 [21]), so that, to first order, the collapsed matter occupies a negligible fraction of the spatial volume. Thus, recent-epoch spatial volume is overwhelmingly dominated by low-density regions. Moreover, there are velocity flows out of the voids, since otherwise, the voids couldn't be nearly empty. Thus, the critical density defining spatial flatness in the voids is higher than it would be in a homogeneous calculation. The standard, FLRW approach insists that space expands uniformly with spatially constant curvature, i.e. space is rigid in comoving coordinates-it is forbidden from bending under the influence of gravity. N -body simulations are typically used to study the Newtonian formation of overdense structures and voids within this rigid, comoving background model. However, both the low density of the voids and the velocity flow out of them imply that the matter density parameter in the voids is sub-critical. Thus, the voids are hyperbolic. Geometrically, this hyperbolicity should also be taken into account, implying an even lower matter density parameter (conservatively, let us ignore this: the effect is small).",
"pages": [
3
]
},
{
"title": "3 Volume-weighted averaged metric",
"content": "These arguments can be formalised using the volume-weighted averaging approach to modelling inhomogeneous spatial slices [22, 23, 11, 24, 16], in which the Friedmann equation is generalised to (12) of [13]. For simplicity, let us (i) set the dark energy term to zero, (ii) neglect the kinematic backreaction as much smaller than the curvature backreaction (see [16] for numerical justification), and (iii) combine the curvature parameters into a single full curvature parameter as suggested in [13]. Writing 'k' instead of ' R ', this gives the domain-averaged, effective Friedmann equation Along a typical, large-scale, random, spacelike or null geodesic over recent epochs, what proportions of the geodesic lie in the emptied and virialised regions? The proportions at a given z are, on average, (1 -f vir /δ vir ) : f vir /δ vir , respectively. Given that δ vir ∼ 100-200 and less than about 1% of the geodesic falls within the virialised regions. Thus, by starting with a large-scale, high-redshift, 'background' FLRW model-in this case, an EdS model-an effective metric can be written by assuming that the virialised matter contributes negligibly. As in [10], the effective expansion rate is combining the FLRW expansion H ( z ) with the peculiar velocity gradient (physical, not comoving) across voids, H pec ( z ), estimated numerically from an N -body simulation ([1, 2], a 240 h -1 Mpc box-size EdS simulation; see [10]) The background Hubble constant H bg is set to make the present value consistent with low redshift estimates [25, 26], i.e. Thus, the loss of matter from the voids and the higher critical density in voids both decrease the EdS background value from Ω m , bg = 1 to an effective value of The effective radius of curvature is The effective metric (cf [17]) is where the radial comoving component is",
"pages": [
3,
4
]
},
{
"title": "4 Matter density parameter and luminosity distances",
"content": "Without any attempt to fit this approximation to observational data , apart from (5) above, the correction of the EdS model as presented above gives an effective matter density parameter (6) that drops slowly from its background value of unity at high redshift down to Ω eff m = 0 . 27 at the present epoch z = 0, remarkably close to the last two decades' local estimates of the matter density parameter. The effective luminosity distance follows directly from the radial comoving distance and hyperbolicity, Figure 2 shows that despite the rough nature of the virialisation approximation, it shifts the homogeneous EdS magnitude-redshift relation by a substantial fraction towards the homogeneous ΛCDM relation, and thus, towards the observational supernovae type Ia relation.",
"pages": [
5
]
},
{
"title": "5 Conclusion",
"content": "A handful of simple formulae, lying at the heart of homogeneous, spatially rigid cosmology, remain approximately valid when generalised to inhomogeneous, spatially flexible cosmology [13] and applied to what observationally and theoretically dominate the present-day spatial volume-the voids. The result is a correction to a large-scale, high-redshift, background Einstein-de Sitter cosmological model. The correction approximately gives the observed lowredshift matter density parameter and nearly matches the type Ia supernovae luminosity distance relation. The amplitude of the correction is unlikely to be much smaller than estimated here. At the present epoch, direct observations [27], N -body simulations [1, 2], and the existence of the cosmic web itself establish inhomogeneous peculiar velocity gradients of ∼ 20-30 km/s/Mpc, forcing, at least, a factor of ∼ (75 / 50) 2 reduction of the Einstein-de Sitter matter density to Ω eff m < 0 . 5, via Eqs (4), (5), and (6). A virialisation fraction of the order of ∼ 50% reduces this to ∼ 0 . 25. Even when forcing the homogeneous FLRW models onto the data, the initial analysis of the Planck Surveyor cosmic microwave background data finds H bg = 67 . 3 ± 1 . 2 km/s/Mpc at z ≈ 1100 [9]-not as low as the velocity gradients imply, but still significantly lower than the low redshift estimates of H eff (0) = 74 . 0 ± 1 . 6 km/s/Mpc ([26, 25], standard error in the mean). How could such a simple, back-of-the-envelope calculation have been missed for so long? While the volume-averaging approach to cosmology has been developed over many years (see e.g. [13] for a review), possibly the answer lies, ironically, in confusion between the spacelike, unobserved , comoving, present time slice and the past light cone. A gigaparsec-scale void in the former should have a very weak ( δ ∼ 10 -5 ) underdensity, and our would-be location at its centre would be uncomfortably anti-Copernican. But these are both moot points! On the past light cone, an average , gigaparsec-scale, sub-critical (0 < Ω eff m < 0 . 8) void is perfectly natural, since it is defined by the onset of the virialisation epoch at z < ∼ 3. Moreover, we are naturally located at this pseudo-void's centre, by the nature of the past light cone. What is simpler: relativistic, hyperbolic voids, observed by an observer at the tip of the past light cone, with no dark energy parameter? Or rigid, Newtonian voids together with a dark energy parameter that traces the virialisation fraction? Acknowledgments: This essay is in part based on work conducted within the 'Lyon Institute of Origins' under grant ANR-10-LABX-66, and on Program Oblicze'n Wielkich Wyzwa'n nauki i techniki (POWIEW) computational resources (grant 87) at the Pozna'n Supercomputing and Networking Center (PCSS).",
"pages": [
5,
6
]
}
]
2013IJMPD..2242024K
https://arxiv.org/pdf/1312.3462.pdf
<document>
<section_header_level_1><location><page_1><loc_34><loc_89><loc_66><loc_91></location>Matters on a moving brane</section_header_level_1>
<text><location><page_1><loc_25><loc_85><loc_75><loc_87></location>Tomi Sebastian Koivisto 1, ∗ and Danielle Elizabeth Wills 2, †</text>
<text><location><page_1><loc_25><loc_83><loc_26><loc_84></location>1</text>
<text><location><page_1><loc_21><loc_74><loc_79><loc_83></location>Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway 2 Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, UK</text>
<text><location><page_1><loc_39><loc_71><loc_61><loc_73></location>(Dated: November 13, 2018)</text>
<section_header_level_1><location><page_1><loc_45><loc_68><loc_54><loc_70></location>Abstract</section_header_level_1>
<text><location><page_1><loc_14><loc_65><loc_45><loc_67></location>'An idle brane is the devil's workshop.'</text>
<text><location><page_1><loc_12><loc_46><loc_88><loc_61></location>A novel generalisation of the Dirac-Born-Infeld string scenario is described. It is shown that matter residing on the moving brane is dark and has the so-called disformal coupling to gravity. This gives rise to cosmologies where dark matter stems from the oscillations of the open strings along the brane and the transverse oscillations result in dark energy. Furthermore, due to a new screening mechanism that conceals the fifth force from local experiments, one may even entertain the possibility that the visible sector is also moving along the extra dimensions.</text>
<text><location><page_1><loc_12><loc_33><loc_88><loc_36></location>Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation, Submitted 31.3.2013</text>
<text><location><page_2><loc_12><loc_79><loc_88><loc_91></location>The relation between physical and gravitational geometry is a basic issue of fundamental importance to both classical and quantum theory. Gravity is the geometry of space-time described by the metric g µν , which in General Relativity is prescribed dynamics by the Einstein-Hilbert action. The geometry governing the movement of matter fields in the space-time is called the physical geometry ¯ g µν ,</text>
<formula><location><page_2><loc_33><loc_74><loc_88><loc_77></location>S = ∫ d 4 x √ -g R 16 πG + S (matter , ¯ g µν ) . (1)</formula>
<text><location><page_2><loc_12><loc_63><loc_88><loc_72></location>Deviations from the postulate of minimal coupling, ¯ g µν = g µν , have to be constrained experimentally [1]. For simplicity, one may consider such deviations to be given by a single scalar field φ . It can then be argued that the most general physically meaningful relation between the two metrics has the form [2]</text>
<formula><location><page_2><loc_35><loc_59><loc_88><loc_60></location>¯ g µν = C ( φ, X ) g µν + D ( φ, X ) φ ,µ φ ,ν , (2)</formula>
<text><location><page_2><loc_12><loc_50><loc_88><loc_57></location>where X = ( ∂φ ) 2 is the kinetic term of the field. The function C gives the very well known conformal transformation. In this essay we discuss the role of the function D that gives the so-called disformal transformation.</text>
<text><location><page_2><loc_12><loc_26><loc_88><loc_49></location>It is easy to argue that though largely neglected in the past literature, this relation is generic. The Brans-Dicke class of scalar-tensor theories is described by a nontrivial function C ; a very extensively studied example is the f ( R ) type of gravity theory. However, when one considers any more general scalar-tensor theory, or writes a covariant action for the metric involving any other invariant besides R , the function D must appear in the Einstein frame formulation of the theory 1 . Here our starting point, instead of an ad hoc modification of gravity, is a higher dimensional theory with pure Einstein gravity minimally coupled to matter: we will see that the resulting four-dimensional description generically involves a nontrivial function D .</text>
<text><location><page_2><loc_12><loc_13><loc_88><loc_25></location>The primary novel contribution of this essay is to present a generalisation of the DiracBorn-Infeld (DBI) string scenario, that predicts matter couplings of the form (2), with both functions C and D robustly derivable from first principles. It is shown that any matter residing upon the moving brane will have this coupling to our gravitational geometry, when the scalar field φ is then identified with the radial coordinate of the brane in a warped</text>
<figure>
<location><page_3><loc_29><loc_63><loc_70><loc_90></location>
<caption>FIG. 1: Dark matter emerges from oscillations of the open strings along the brane, visualised here as ripples on the surface sourced by the open string endpoints. Dark energy emerges from the oscillations of the open strings transverse to the brane, visualised here as the movement of the brane as a single wave-front or wave-pulse in the radial direction.</caption>
</figure>
<text><location><page_3><loc_62><loc_62><loc_64><loc_66></location>x</text>
<text><location><page_3><loc_12><loc_35><loc_88><loc_44></location>background, the DBI radion. If the brane doesn't intersect the Standard Model stack of branes 2 , the disformally coupled matter would interact only gravitationally with baryonic matter. In particular we propose to associate the U (1) gauge field upon the moving brane with this invisible matter.</text>
<text><location><page_3><loc_12><loc_14><loc_88><loc_34></location>An appealing unification of the cosmological dark sector has then emerged: dark matter stems from the oscillations of the open strings in the directions along the brane, which propagate as a vector field on the world-volume, and the dynamical dark energy field is the scalar DBI radion, which encodes the oscillations of open strings transverse to the brane, see Fig. 1. Gravity emerges from the oscillations of the closed strings in the bulk spacetime. In generic compactifications, one obtains both massive and massless vector fields. In cosmology, the former manifest as dark matter and the latter as 'dark radiation', both of which are non-minimally coupled to the dark energy field via the disformal relation.</text>
<text><location><page_4><loc_12><loc_87><loc_88><loc_91></location>In a warped flux compactification of Type IIB string theory, the ten-dimensional metric takes the form</text>
<formula><location><page_4><loc_29><loc_84><loc_88><loc_86></location>G MN dx M dx N = h -1 / 2 g µν dx µ dx ν + h 1 / 2 g ab dx a dx b , (3)</formula>
<text><location><page_4><loc_12><loc_65><loc_88><loc_82></location>where h = h ( x a ) is the warp factor which depends only on the compact coordinates indexed by a, b = 4 , ..., 9. These are the wrapped dimensions of the Calabi-Yau space. The capital indices M,N = 0 , ..., 9 run over all space-time dimensions, the greek µ, ν = 0 , ..., 3 over the large 4 dimensions. For simplicity we consider a probe D3-brane embedded in this background, as the set-up can readily be generalised to branes of lower codimension. In the Einstein frame the Dirac-Born-Infeld (DBI) action describing the dynamics of a D3-brane is given by</text>
<formula><location><page_4><loc_29><loc_62><loc_88><loc_64></location>S DBI = -(2 π ) -3 glyph[lscript] -4 s ∫ d 4 ξ √ -det( γ µν + e -ϕ 2 F µν ) , (4)</formula>
<text><location><page_4><loc_12><loc_53><loc_88><loc_60></location>where the integration is over the brane coordinates on the world-volume, ξ µ , and the string scale glyph[lscript] 2 s gives the tension of the brane. The dilaton ϕ we assume to be stabilized as usual. The induced metric on the brane is denoted by γ µν . Finally, there appears</text>
<formula><location><page_4><loc_40><loc_49><loc_88><loc_51></location>F µν = B µν +2 πglyph[lscript] 2 s F µν , (5)</formula>
<text><location><page_4><loc_12><loc_37><loc_88><loc_47></location>the gauge invariant combination of the pullback of the NSNS 2-form field B 2 and the field strength F µν of the world-volume U (1) gauge field. Below we will describe how the induced metric γ µν gives rise to the disformal relation in gravity, and the field F µν to disformally coupled matter.</text>
<text><location><page_4><loc_12><loc_32><loc_88><loc_36></location>Let us first focus on the geometry, see Fig. 2. The pullback of the ten dimensional metric onto the brane world-volume is given by</text>
<formula><location><page_4><loc_40><loc_28><loc_88><loc_29></location>γ µν = G MN ∂ µ x M ∂ ν x N . (6)</formula>
<text><location><page_4><loc_12><loc_13><loc_88><loc_25></location>Adopting the static gauge, the D3-brane fills the large dimensions such that ξ µ = x µ . We allow the compact spacetime coordinates which are transverse to the brane to be functions of the four-dimensional world-volume coordinates, x a = x a ( ξ µ ). This entails that the brane may move in the internal space. With these choices, the four dimensional components of the induced metric on the brane world-volume take the form</text>
<formula><location><page_4><loc_39><loc_8><loc_88><loc_11></location>γ µν = G µν + ∂x a ∂ξ µ ∂x b ∂ξ v G ab . (7)</formula>
<text><location><page_5><loc_12><loc_79><loc_88><loc_91></location>It can be justified to consider purely one-dimensional brane trajectories in the radial direction of a warped throat [5]. Returning to Eq. (2), we may identify the scalar field φ with the canonically normalised radial coordinate of the brane, one of the x a 's in Eq. (7). Thus we have obtained the relation (2), where γ µν → ¯ g µν and the functions C ∼ h 1 2 ( φ ) and D ∼ h -1 2 ( φ ) are given by the warp factor.</text>
<figure>
<location><page_5><loc_22><loc_52><loc_76><loc_74></location>
<caption>FIG. 2: A D3-brane moving in the radial direction of a warped throat in a compact Calabi-Yau threefold.</caption>
</figure>
<text><location><page_5><loc_12><loc_24><loc_88><loc_39></location>It is straightforward to compute the determinant in the DBI action explicitly to separate the geometric part into the contributions from the gravitational metric and the scalar field. To complete the picture, we include the D3-brane charge C 4 which is given by the warp factor as ∼ 1 /h , as well as the potential V ( φ ) that may emerge from the coupling of the brane to other sectors and is in principle computable given any explicit set-up. We arrive at the following description of our four-dimensional gravity:</text>
<formula><location><page_5><loc_18><loc_18><loc_88><loc_22></location>S = ∫ d 4 x √ -g [ R 16 πG -1 h ( φ ) ( √ 1 + h ( φ ) X -1 ) -V ( φ ) ] + S ( F µν , ¯ g µν ) , (8)</formula>
<text><location><page_5><loc_12><loc_8><loc_88><loc_17></location>where all the F -dependent terms are collected into an effective matter action. Here we have written γ µν = ¯ g µν , to emphasize that what we have is precisely a nontrivial realization of our starting point (1), supplemented with the Lagrangian for the scalar field that governs the relation (2).</text>
<text><location><page_6><loc_12><loc_76><loc_88><loc_91></location>Let us then consider the matter sources. The vector fields are associated with the open string endpoints on the brane, and may acquire masses via Stuckelberg couplings to bulk two-forms. The masses will depend on the geometry and are inherently very large, such that for the late-time universe these fields will have long decayed into lighter particles, unless they are stable. In the end, for either case, we do expect massive particles on the brane, and by construction, they will be both dark and disformally coupled to gravity.</text>
<text><location><page_6><loc_12><loc_50><loc_88><loc_75></location>Thus we have arrived at a novel generalisation of the coupled quintessence cosmology. In such models, the fine-tuning problem of dark energy could be alleviated by the fact that the energy density is promoted to a dynamical field, and the coincidence problem could be addressed by the direct coupling between the dark matter and dark energy components. A crucial feature is that the coupling will affect the formation of structure in dark matter and thus will modify the predictions for the observed large-scale structure. In the equations governing the evolution of cosmological perturbations, the disformal coupling has been shown to have a considerably richer structure than the purely conformal interaction term [4], and we expect that the precise model presented here can be very efficiently constrained by studying the matter power spectrum at both linear and nonlinear regimes.</text>
<text><location><page_6><loc_12><loc_34><loc_88><loc_48></location>An even more radical possibility emerges due to the recently discovered disformal screening mechanism [6], which can conceal the coupling from local experiments even while it has drastic consequences at cosmological scales. In the DBI string scenario presented above this suggests that our Standard Model could be residing on a moving stack of branes 3 without introducing a cosmological moduli problem. This could potentially bring new classes of string cosmologies with novel late-time phenomenology into serious consideration.</text>
<text><location><page_6><loc_12><loc_13><loc_88><loc_33></location>The rich phenomenology of disformal couplings has very recently attracted growing interest. Whereas the function C in (2) is a local scale transformation that leaves the causal structure untouched, the function D affects angles and thus distorts the light cones. Therefore coupling the electromagnetic field disformally results in a varying-speed-of light theory. Constraints on such couplings have been derived from both high-precision laboratory experiments of low-energy photons [7] and from the cosmological evolution of the cosmic microwave background black-body radiation [8]. Bounds can also be derived by considering the coupling of baryonic fluids in the radiation dominated epoch of the universe evolution [9]. As</text>
<text><location><page_7><loc_12><loc_87><loc_88><loc_91></location>we have shown here, such phenomenology can in fact be considered as a probe of extra dimensional brane movement in a robust string theoretical setting.</text>
<text><location><page_7><loc_12><loc_79><loc_88><loc_85></location>A note added: The theoretical underpinnings and cosmological implications of the scenario outlined here are explored much further in Ref. [10] where we refer the reader for more details. See also [11-13] for related recent studies of disformal couplings.</text>
<section_header_level_1><location><page_7><loc_14><loc_73><loc_30><loc_74></location>Acknowledgments</section_header_level_1>
<text><location><page_7><loc_12><loc_63><loc_88><loc_70></location>TK is supported by the Research Council of Norway and DW by an STFC studentship. We would like to thank Ruth Gregory, David Mota, Ivonne Zavala and Miguel Zumalac'arregui for useful discussions.</text>
<unordered_list>
<list_item><location><page_7><loc_13><loc_52><loc_88><loc_56></location>[1] C. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, 1993), ISBN 9780521439732.</list_item>
<list_item><location><page_7><loc_13><loc_50><loc_52><loc_51></location>[2] J. D. Bekenstein, Phys.Rev. D48 , 3641 (1993).</list_item>
<list_item><location><page_7><loc_13><loc_47><loc_83><loc_48></location>[3] L. Amendola, K. Enqvist, and T. Koivisto, Phys.Rev. D83 , 044016 (2011), 1010.4776.</list_item>
<list_item><location><page_7><loc_13><loc_44><loc_88><loc_45></location>[4] M. Zumalacarregui, T. S. Koivisto, and D. F. Mota, Phys.Rev. D87 , 083010 (2013), 1210.8016.</list_item>
<list_item><location><page_7><loc_13><loc_39><loc_88><loc_43></location>[5] D. A. Easson, R. Gregory, D. F. Mota, G. Tasinato, and I. Zavala, JCAP 0802 , 010 (2008), 0709.2666.</list_item>
<list_item><location><page_7><loc_13><loc_33><loc_88><loc_37></location>[6] T. S. Koivisto, D. F. Mota, and M. Zumalacarregui, Phys.Rev.Lett. 109 , 241102 (2012), 1205.3167.</list_item>
<list_item><location><page_7><loc_13><loc_30><loc_74><loc_32></location>[7] P. Brax, C. Burrage, and A.-C. Davis, JCAP 1210 , 016 (2012), 1206.1809.</list_item>
<list_item><location><page_7><loc_13><loc_28><loc_61><loc_29></location>[8] C. van de Bruck, J. Morrice, and S. Vu (2013), 1303.1773.</list_item>
<list_item><location><page_7><loc_13><loc_25><loc_77><loc_26></location>[9] C. van de Bruck and G. Sculthorpe, Phys.Rev. D87 , 044004 (2013), 1210.2168.</list_item>
<list_item><location><page_7><loc_12><loc_22><loc_58><loc_23></location>[10] T. Koivisto, D. Wills, and I. Zavala (2013), 1312.2597.</list_item>
<list_item><location><page_7><loc_12><loc_20><loc_70><loc_21></location>[11] D. Bettoni and S. Liberati, Phys.Rev. D88 , 084020 (2013), 1306.6724.</list_item>
<list_item><location><page_7><loc_12><loc_17><loc_61><loc_18></location>[12] M. Zumalacrregui and J. Garca-Bellido (2013), 1308.4685.</list_item>
<list_item><location><page_7><loc_12><loc_14><loc_84><loc_15></location>[13] P. Brax, C. Burrage, A.-C. Davis, and G. Gubitosi, JCAP 1311 , 001 (2013), 1306.4168.</list_item>
</unordered_list>
</document>
[
{
"title": "Matters on a moving brane",
"content": "Tomi Sebastian Koivisto 1, ∗ and Danielle Elizabeth Wills 2, † 1 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway 2 Centre for Particle Theory, Department of Mathematical Sciences, Durham University, South Road, Durham, DH1 3LE, UK (Dated: November 13, 2018)",
"pages": [
1
]
},
{
"title": "Abstract",
"content": "'An idle brane is the devil's workshop.' A novel generalisation of the Dirac-Born-Infeld string scenario is described. It is shown that matter residing on the moving brane is dark and has the so-called disformal coupling to gravity. This gives rise to cosmologies where dark matter stems from the oscillations of the open strings along the brane and the transverse oscillations result in dark energy. Furthermore, due to a new screening mechanism that conceals the fifth force from local experiments, one may even entertain the possibility that the visible sector is also moving along the extra dimensions. Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation, Submitted 31.3.2013 The relation between physical and gravitational geometry is a basic issue of fundamental importance to both classical and quantum theory. Gravity is the geometry of space-time described by the metric g µν , which in General Relativity is prescribed dynamics by the Einstein-Hilbert action. The geometry governing the movement of matter fields in the space-time is called the physical geometry ¯ g µν , Deviations from the postulate of minimal coupling, ¯ g µν = g µν , have to be constrained experimentally [1]. For simplicity, one may consider such deviations to be given by a single scalar field φ . It can then be argued that the most general physically meaningful relation between the two metrics has the form [2] where X = ( ∂φ ) 2 is the kinetic term of the field. The function C gives the very well known conformal transformation. In this essay we discuss the role of the function D that gives the so-called disformal transformation. It is easy to argue that though largely neglected in the past literature, this relation is generic. The Brans-Dicke class of scalar-tensor theories is described by a nontrivial function C ; a very extensively studied example is the f ( R ) type of gravity theory. However, when one considers any more general scalar-tensor theory, or writes a covariant action for the metric involving any other invariant besides R , the function D must appear in the Einstein frame formulation of the theory 1 . Here our starting point, instead of an ad hoc modification of gravity, is a higher dimensional theory with pure Einstein gravity minimally coupled to matter: we will see that the resulting four-dimensional description generically involves a nontrivial function D . The primary novel contribution of this essay is to present a generalisation of the DiracBorn-Infeld (DBI) string scenario, that predicts matter couplings of the form (2), with both functions C and D robustly derivable from first principles. It is shown that any matter residing upon the moving brane will have this coupling to our gravitational geometry, when the scalar field φ is then identified with the radial coordinate of the brane in a warped x background, the DBI radion. If the brane doesn't intersect the Standard Model stack of branes 2 , the disformally coupled matter would interact only gravitationally with baryonic matter. In particular we propose to associate the U (1) gauge field upon the moving brane with this invisible matter. An appealing unification of the cosmological dark sector has then emerged: dark matter stems from the oscillations of the open strings in the directions along the brane, which propagate as a vector field on the world-volume, and the dynamical dark energy field is the scalar DBI radion, which encodes the oscillations of open strings transverse to the brane, see Fig. 1. Gravity emerges from the oscillations of the closed strings in the bulk spacetime. In generic compactifications, one obtains both massive and massless vector fields. In cosmology, the former manifest as dark matter and the latter as 'dark radiation', both of which are non-minimally coupled to the dark energy field via the disformal relation. In a warped flux compactification of Type IIB string theory, the ten-dimensional metric takes the form where h = h ( x a ) is the warp factor which depends only on the compact coordinates indexed by a, b = 4 , ..., 9. These are the wrapped dimensions of the Calabi-Yau space. The capital indices M,N = 0 , ..., 9 run over all space-time dimensions, the greek µ, ν = 0 , ..., 3 over the large 4 dimensions. For simplicity we consider a probe D3-brane embedded in this background, as the set-up can readily be generalised to branes of lower codimension. In the Einstein frame the Dirac-Born-Infeld (DBI) action describing the dynamics of a D3-brane is given by where the integration is over the brane coordinates on the world-volume, ξ µ , and the string scale glyph[lscript] 2 s gives the tension of the brane. The dilaton ϕ we assume to be stabilized as usual. The induced metric on the brane is denoted by γ µν . Finally, there appears the gauge invariant combination of the pullback of the NSNS 2-form field B 2 and the field strength F µν of the world-volume U (1) gauge field. Below we will describe how the induced metric γ µν gives rise to the disformal relation in gravity, and the field F µν to disformally coupled matter. Let us first focus on the geometry, see Fig. 2. The pullback of the ten dimensional metric onto the brane world-volume is given by Adopting the static gauge, the D3-brane fills the large dimensions such that ξ µ = x µ . We allow the compact spacetime coordinates which are transverse to the brane to be functions of the four-dimensional world-volume coordinates, x a = x a ( ξ µ ). This entails that the brane may move in the internal space. With these choices, the four dimensional components of the induced metric on the brane world-volume take the form It can be justified to consider purely one-dimensional brane trajectories in the radial direction of a warped throat [5]. Returning to Eq. (2), we may identify the scalar field φ with the canonically normalised radial coordinate of the brane, one of the x a 's in Eq. (7). Thus we have obtained the relation (2), where γ µν → ¯ g µν and the functions C ∼ h 1 2 ( φ ) and D ∼ h -1 2 ( φ ) are given by the warp factor. It is straightforward to compute the determinant in the DBI action explicitly to separate the geometric part into the contributions from the gravitational metric and the scalar field. To complete the picture, we include the D3-brane charge C 4 which is given by the warp factor as ∼ 1 /h , as well as the potential V ( φ ) that may emerge from the coupling of the brane to other sectors and is in principle computable given any explicit set-up. We arrive at the following description of our four-dimensional gravity: where all the F -dependent terms are collected into an effective matter action. Here we have written γ µν = ¯ g µν , to emphasize that what we have is precisely a nontrivial realization of our starting point (1), supplemented with the Lagrangian for the scalar field that governs the relation (2). Let us then consider the matter sources. The vector fields are associated with the open string endpoints on the brane, and may acquire masses via Stuckelberg couplings to bulk two-forms. The masses will depend on the geometry and are inherently very large, such that for the late-time universe these fields will have long decayed into lighter particles, unless they are stable. In the end, for either case, we do expect massive particles on the brane, and by construction, they will be both dark and disformally coupled to gravity. Thus we have arrived at a novel generalisation of the coupled quintessence cosmology. In such models, the fine-tuning problem of dark energy could be alleviated by the fact that the energy density is promoted to a dynamical field, and the coincidence problem could be addressed by the direct coupling between the dark matter and dark energy components. A crucial feature is that the coupling will affect the formation of structure in dark matter and thus will modify the predictions for the observed large-scale structure. In the equations governing the evolution of cosmological perturbations, the disformal coupling has been shown to have a considerably richer structure than the purely conformal interaction term [4], and we expect that the precise model presented here can be very efficiently constrained by studying the matter power spectrum at both linear and nonlinear regimes. An even more radical possibility emerges due to the recently discovered disformal screening mechanism [6], which can conceal the coupling from local experiments even while it has drastic consequences at cosmological scales. In the DBI string scenario presented above this suggests that our Standard Model could be residing on a moving stack of branes 3 without introducing a cosmological moduli problem. This could potentially bring new classes of string cosmologies with novel late-time phenomenology into serious consideration. The rich phenomenology of disformal couplings has very recently attracted growing interest. Whereas the function C in (2) is a local scale transformation that leaves the causal structure untouched, the function D affects angles and thus distorts the light cones. Therefore coupling the electromagnetic field disformally results in a varying-speed-of light theory. Constraints on such couplings have been derived from both high-precision laboratory experiments of low-energy photons [7] and from the cosmological evolution of the cosmic microwave background black-body radiation [8]. Bounds can also be derived by considering the coupling of baryonic fluids in the radiation dominated epoch of the universe evolution [9]. As we have shown here, such phenomenology can in fact be considered as a probe of extra dimensional brane movement in a robust string theoretical setting. A note added: The theoretical underpinnings and cosmological implications of the scenario outlined here are explored much further in Ref. [10] where we refer the reader for more details. See also [11-13] for related recent studies of disformal couplings.",
"pages": [
1,
2,
3,
4,
5,
6,
7
]
},
{
"title": "Acknowledgments",
"content": "TK is supported by the Research Council of Norway and DW by an STFC studentship. We would like to thank Ruth Gregory, David Mota, Ivonne Zavala and Miguel Zumalac'arregui for useful discussions.",
"pages": [
7
]
}
]
